text stringlengths 4 2.78M |
|---|
---
abstract: 'The MuCool R&D program is described. The aim of MuCool is to develop all key pieces of hardware required for ionization cooling of a muon beam. This effort will lead to a more detailed understanding of the construction and operating costs of such hardware, as well as to optimized designs that can be used to build a Neutrino Factory or Muon Collider. This work is being undertaken by a broad collaboration including physicists and engineers from many national laboratories and universities in the U.S. and abroad. The intended schedule of work will lead to ionization cooling being well enough established that a construction decision for a Neutrino Factory could be taken before the end of this decade based on a solid technical foundation.'
address: |
for the MuCool Collaboration\
Physics Division, Illinois Institute of Technology, 3101 S. Dearborn Street, Chicago, Illinois, 60616 USA
author:
- 'Daniel M Kaplan[ ]{}'
title: 'Progress in Muon Cooling Research and Development[$^*$]{}'
---
Introduction
============
The MuCool Collaboration is pursuing a research and development program on hardware that can be used for cooling of muon beams [@MuCool]. Our recent work [@Alsharoa] focuses on muon ionization cooling [@cooling] for a possible Neutrino Factory [@Geer], in which an intense, pure, and collimated neutrino beam is produced via decay of a muon beam circulating in a storage ring. The goal for such a facility is $\sim10^{21}$ neutrinos/year (requiring a similar number of muons/year stored in the ring), to exploit the recently established discovery of neutrino oscillations [@nu-osc].
Muon beams at the required intensity can only be produced into a large phase space, but affordable existing acceleration technologies require a small input beam. This mismatch could be alleviated by developing new, large-aperture, acceleration techniques [@Japan-study], by “cooling" the muon beam to reduce its size, or both. Given the 2.2-$\mu$s muon lifetime, only one cooling technique is fast enough: ionization cooling, in which muons repeatedly traverse an energy-absorbing medium, alternating with accelerating devices, within a strongly focusing magnetic lattice [@cooling; @Kaplan-cooling].
Principle of ionization cooling
===============================
In an ionization-cooling channel, ionization of the energy-absorbing medium decreases all three muon-momentum components without affecting the beam size. By the definition of normalized transverse beam emittance,[^1] $\epsilon_n\approx\sqrt{\sigma_x\sigma_y\sigma_{p_x}\sigma_{p_y}}/mc$ (where $\sigma_x$, $\sigma_{p_x}$, etc. denote the r.m.s. deviations of the beam in position and momentum coordinates, $m$ is the particle mass, and $c$ the speed of light), this constitutes cooling, [*i.e.*]{}, reduction of normalized emittance. This is so since the reduction of each particle’s momentum results in a reduced transverse-momentum spread of the beam as a whole.
At the same time, multiple Coulomb scattering of the muons increases the beam divergence, heating the beam. The equilibrium between these heating and cooling effects is expressed in the following approximate equation for the dependence of $\epsilon_n$ on the distance $s$ traveled through an absorber [@Neuffer2; @Fernow]: $$\frac{{\rm d}\epsilon_n}{{\rm d}s}\ \approx\
-\frac{1}{\beta^2} \left\langle\frac{{\rm d}E_{\mu}}{{\rm d}s}\right\rangle\frac{\epsilon_n}{E_{\mu}}\ +
\ \frac{1}{\beta^3} \frac{\beta_\perp (0.014)^2}{2E_{\mu}m_{\mu}L_R}\,.
\label{eq:cool}$$ Here, angle brackets denote mean value, $\beta$ is the muon velocity in units of $c$, muon energy $E_\mu$ is in GeV, $\beta_\perp$ is the lattice beta function evaluated at the location of the absorber, $m_\mu$ is the muon mass in GeV/$c^2$, and $L_R$ is the radiation length of the absorber medium. (Eq. \[eq:cool\] is derived for the cylindrically-symmetric case of solenoidal focusing, where $\beta_x=\beta_y\equiv\beta_\perp$, by differentiating the expression for $\epsilon_n$ given above.) The first term in Eq. \[eq:cool\] is the cooling term and the second is the heating term. To minimize the heating term, which is proportional to the beta function and inversely proportional to radiation length, it has been proposed [@Status-Report] to use liquid hydrogen (LH$_2$) as the energy-absorbing medium, giving $\langle {\rm d}E_{\mu}/{\rm d}s\rangle\approx 30\,$MeV/m and $L_R=8.7\,$m [@PDG], with superconducting-solenoid focusing to give small $\beta_\perp\sim10\,$cm. (Other possible absorber materials are discussed below.)
Between absorbers, high-gradient acceleration of the muons must be provided to replace the lost longitudinal momentum, so that the ionization-cooling process can be repeated many times. Ideally, the acceleration should exceed the minimum required for momentum replacement, allowing “off-crest" operation. This gives continual rebunching, so that even a beam with large momentum spread remains captured in the rf bucket. Even though it is the absorbers that actually cool the beam, for typical accelerating gradients ($\sim$10MeV/m), the rf cavities dominate the length of the cooling channel (see [*e.g.*]{} Fig. \[fig:SFOFO\]). The achievable rf gradient thus determines how much cooling is practical before an appreciable fraction of the muons have decayed or drifted out of the bucket.
We see from Eq. \[eq:cool\] that the percentage decrease in normalized emittance is proportional to the percentage energy loss, hence cooling in one transverse dimension by a factor $1/e$ requires $\sim$100% momentum loss and replacement. Low beam momentum is thus favored, because it requires less accelerating voltage and because of the increase of d$E$/d$s$ for momenta below the ionization minimum [@PDG]. Most Neutrino Factory and Muon Collider beam-cooling designs and simulations to date have therefore used momenta in the range $150-400$MeV/$c$. (This is also the momentum range in which the pion-production cross section off of thick targets tends to peak and is thus optimal for muon production as well as cooling.) The cooling channel of Fig. \[fig:SFOFO\] is optimized for a mean muon momentum of 200MeV$/c$.
As a muon beam passes through a transverse ionization-cooling lattice, its longitudinal emittance tends to grow, due to such effects as energy-loss straggling. The six-dimensional emittance (approximately the square of the transverse emittance times the longitudinal emittance) typically is reduced despite this longitudinal heating. However, if not controlled, the longitudinal heating leads to beam losses and thus limits the degree of transverse cooling that is practical to achieve. Cooling lattices with longitudinal–transverse emittance exchange (which can cool in all six dimensions simultaneously) have been receiving increasing attention and are discussed in detail elsewhere in these Proceedings [@Palmer-ring]. They have the potential to increase Neutrino Factory performance and decrease cost, and are essential to achieving sufficient cooling for a Muon Collider.
Muon-cooling technology development
===================================
An effective ionization-cooling channel must include low-$Z$ absorbers with (if an intense muon beam is to be cooled) high power-handling capability. To achieve low beta at the absorbers requires either high solenoidal magnetic field or high field gradient [@quad]. To pack as much cooling as possible into the shortest distance requires the highest practical accelerating gradient. The MuCool Collaboration has embarked on R&D on all three of these technologies.
High-gradient normal-conducting rf cavities
-------------------------------------------
An ionization-cooling channel requires insertion of high-gradient rf cavities into a lattice employing strong solenoidal magnetic fields. This precludes the use of superconducting cavities. The lattice of Fig. \[fig:SFOFO\] employs normal-conducting 201-MHz cavities, but R&D is more readily carried out with smaller, higher-frequency devices. Having already embarked on the development of 805-MHz cavities (aimed at a cooling channel for a Muon Collider [@Status-Report]), we have continued to pursue their development, while working out the details of the 201-MHz design in parallel [@Li].
Radio-frequency cavities normally contain a minimum of material in the path of the beam. However, the penetrating character of the muon allows the use of closed-cell (“pillbox") cavities, provided that the cell closures are constructed of thin material of long radiation length. Eq. \[eq:cool\] implies that this material will have little effect on cooling performance as long as its thickness $L$ per cooling cell (at the $\beta_\perp$ of its location in the lattice) has $\beta_\perp L/L_R$ small compared to that of an absorber. Closing the rf cells approximately doubles the on-axis accelerating gradient for a given maximum surface electric field, allowing operation with less rf power and suppressing field emission. Two alternatives have been considered for the design of the cell closures: thin beryllium foils and grids of gas-cooled, thin-walled aluminum tubing. As a fall-back, an open-cell cavity design was also pursued.
So far we have prototyped and tested a 6-cell open-cell cavity, designed at Fermilab, and a single-cell closed-cell cavity, designed at LBNL, both at 805MHz. The tests are carried out in Fermilab’s Laboratory G, where we have installed a high-power 805-MHz klystron transmitter (12-MW peak pulsed power with pulse length of 50$\mu$s and repetition rate of 15Hz), an x-ray-shielded cave, remote-readout test probes, safety-interlock systems, and a control room and workshop area for setup of experiments. The cave also contains a high-vacuum pumping system and water cooling for the cavity. To allow tests of the cooling-channel rf cavities and absorbers in a high magnetic field or high field gradient, a superconducting 5-T solenoid with a room-temperature bore of 44cm was constructed by LBNL and installed in Lab G, with two separate coils that can be run in “solenoid" mode (currents flowing in the same direction) or “gradient" mode (currents in opposite directions).
The open-cell cavity (Fig. \[fig:open-cell\]) was conditioned up to a surface electric field of 54MV/m (on-axis accelerating gradient up to 25MV/m). Electron dark currents and x-ray backgrounds were found to be large and to scale as a high power of the surface field, $\approx E^{10}$ [@Norem]. With a 2.5-T solenoidal field applied, at 54-MV/m surface field, axially focused dark currents ultimately burned a hole in the cavity’s titanium vacuum window. This level of background emission would preclude cavity operation in the required solenoidal field. However, for the same accelerating gradient, the pillbox cavity should operate at approximately half the surface field, corresponding to lower background emission by a factor of order $10^{3}$. Furthermore, an analysis of the observed emission rate in terms of the Fowler-Nordheim formalism [@Fowler-Nordheim] implies an enhancement of the emission probability by a factor of order $10^{3}$ compared to that of a smooth, clean surface [@Norem]. This suggests that an R&D program focused on improving the surface preparation and treatment might reap large improvements.
Tests of the closed-cell prototype have begun. Initial tests up to the design gradient of $\approx$30MV/m were carried out successfully with no applied magnetic field. Upon disassembly, no damage to the windows was observed. The thickness of the cavity’s vacuum windows precluded measurement of low-energy backgrounds. As of July 2002, a thin window has been installed and reconditioning of the cavity for high-gradient operation has started. So far, the gradients are low and x rays and dark currents have not been seen. Our planned program includes tests of the pillbox cavity with solenoidal field, followed by tests of a variety of surface coatings and cleaning and polishing techniques to identify an optimal approach to building high-gradient normal-conducting cavities for operation in high magnetic fields. In parallel, design studies and prototype tests of beryllium foils and aluminum-tube grids will continue.
High-power liquid-hydrogen absorbers
------------------------------------
The development of high-power liquid-hydrogen (LH$_2$) absorbers with thin windows has been a key goal of the MuCool R&D program [@Kaplan-NuFACT01; @Kaplan-windows]. Simulations as well as theory show that scattering in absorber windows degrades muon-cooling performance. To keep this effect to a minimum, the Neutrino Factory Feasibility Study II design [@FS2] calls for aluminum absorber windows of thicknesses given in Table \[tab:lh2\].
--------------------- -------- -------- -------- --------------- --------------------
Length Radius Number Power Al window
\[0pt\][Absorber]{} (cm) (cm) needed (kW) thickness ($\mu$m)
SFOFO 1 35 18 16 æ$\approx$0.3 360
SFOFO 2 21 11 36 $\approx$0.1 220
--------------------- -------- -------- -------- --------------- --------------------
The power dissipated per absorber as given in Table \[tab:lh2\] is within the bounds of high-power liquid-hydrogen targets developed for, and operated in, a variety of experiments [@targets]. However, the highly turbulent fluid dynamics involved in the heat-exchange process necessarily requires R&D for each new configuration. We have identified two possible approaches: a “conventional" flow-through design with external heat exchanger, similar to that used for high-power LH$_2$ targets, and a convection-cooled design, with internal heat exchanger built into the absorber vessel. The convection design has desirable mechanical simplicity and minimizes the total hydrogen volume in the cooling channel (a significant safety concern), but is expected to be limited to lower power dissipation than the flow-through design.
Various scenarios have been discussed involving substantially higher absorber power dissipation: 1) a Neutrino Factory with a more ambitious Proton Driver (4MW proton-beam power on the pion-production target instead of the 1MW assumed in Study-II) is a relatively straightforward and cost-effective design upgrade [@Alsharoa], 2) the “bunched-beam phase rotation" scenario of Neuffer [@Neuffer-bunch] captures $\mu^+$ and $\mu^-$ simultaneously, doubling the absorber power dissipation, and 3) a ring cooler [@Palmer-ring] would entail multiple traversals of each absorber by each muon, potentially increasing absorber power dissipation by an order of magnitude. If all three of these design upgrades are implemented, power dissipations of tens of kilowatts per absorber will result. The large heat capacity of hydrogen means that such levels of instantaneous power dissipation are in principle supportable, but much higher average heat transfer would be needed, possibly requiring higher pressure and thicker windows. More work is needed to assess the muon-cooling performance implications.
The large transverse dimensions of the muon beam require large apertures and correspondingly wide absorbers, while the large energy spread of the beam demands frequent rebunching via rf cavities, favoring thin absorbers. These two requirements lead to the oblate shapes of the SFOFO-cooling-channel absorbers indicated in Table \[tab:lh2\] and shown in Fig. \[fig:SFOFO\]. Since these shapes are wider than they are long, hemispherical windows (which would be thinnest at a given pressure) are ruled out, and we are led to the “torispherical" window shape. Aluminum alloy is a suitable window material, combining long radiation length with good machinability, weldability, and thermal properties. For an ASME-standard torispherical window [@ASME], the required minimum thickness is (essentially) $t = 0.885 P D / ES$, with $P$ the maximum pressure differential, $D$ the window diameter, $E$ the weld efficiency, and $S$ the maximum allowable stress, in this case 25% of the 289-MPa ultimate strength of the 6061-T6 aluminum alloy [@ASME] (the standard alloy for cryogenic and vacuum applications). Taking into account Fermilab’s requirement of safe operation at 25-psi (0.17-MPa) differential pressure [@FNAL-safety], the minimum torispherical window thickness is 760$\mu$m of 6061-T6 for the SFOFO 1 absorber (460$\mu$m for SFOFO 2), far exceeding the thicknesses called for by the Study-II simulation. To meet the Study-II specifications, we devised a new approach to the design and fabrication of thin windows [@Kaplan-windows], in which windows of a custom shape and tapered thickness profile are machined out of a solid disk of material using a numerically-controlled lathe, with an integral flange so that no welds are required and $E=1$. We also devised means to test these nonstandard windows and demonstrate that they meet their specifications and satisfy the applicable safety requirements [@MACC].
Over the past year, as work has continued towards a realistic absorber design, it has become clear that the Fermilab safety code will require external containment of each absorber, to guard against such possibilities as spark damage to a window due to occasional rf-cavity discharges. This doubles the number of windows per absorber, though the containment windows need not be as strong as the absorber windows themselves. We have now developed designs for yet thinner[^2] windows that will allow the Study-II specification to be met even with the additional set of windows per absorber [@Wing]. The old and new window shapes are compared in Fig. \[fig:windows\]. We are also exploring the use of new (lithium–aluminum) alloys, such as the 2195 alloy used in the Space Shuttle (80% stronger than 6061-T6); the resulting thinness of the window may challenge our fabrication techniques, and we will need to certify the new alloy for machinability and high-radiation application.
Other absorber materials
------------------------
Other candidate absorber materials include helium, lithium, lithium hydride, methane, and beryllium. All other things being equal, in principle these would all give worse cooling performance than hydrogen. For fixed $\beta_\perp$, a possible figure of merit is $(L_R\,\langle {\rm d}E/{\rm d}s\rangle_{\rm min})^2$ (proportional to the four-dimensional transverse-cooling rate), normalized to that of liquid hydrogen. Table \[tab:matl\] shows that hydrogen is best by a factor $\approx2$, although its advantage could be vitiated if thick windows are necessary. Furthermore, for sufficiently high focusing-current density, lithium lenses could provide substantially lower $\beta_\perp$ than is practical with solenoids, perhaps sufficient to overcome lithium’s disadvantageous merit factor. Liquids provide high power-handling capability, since the warmed liquid can be moved to a heat exchanger. On the other hand, the higher densities of solids allow the absorber to be located more precisely at the low-beta point of the lattice. Lithium hydride may be usable with no windows, but means would have to be devised to prevent combustion due to contact with moisture, as well as to avoid melting at high power levels. More work will be required to assess these issues in detail.
--------------------- ------------------------------------- -------------- ------------------
$\langle$d$E$/d$s\rangle_{\rm min}$ $L_R$
\[0pt\][Material]{} (MeVg$^{-1}$cm$^{2}$) (gcm$^{-2}$) \[0pt\][Merit]{}
GH$_2$ 4.103 61.28 1.03
LH$_2$ 4.034 61.28 1
He 1.937 94.32 0.55
LiH 1.94 86.9 0.47
Li 1.639 82.76 0.30
CH$_4$ 2.417 46.22 0.20
Be 1.594 65.19 0.18
--------------------- ------------------------------------- -------------- ------------------
It has been pointed out [@Kaplan-NuFACT01; @MCNote195] that gaseous hydrogen (GH$_2$) at high pressure could provide the energy absorption needed for ionization cooling, with significantly different technical challenges than those of a liquid or solid absorber. Table \[tab:matl\] shows that GH$_2$ is actually a slightly better ionization-cooling medium than LH$_2$. In addition, if the hydrogen is allowed to fill the rf cavities, the number of windows in the cooling channel can be substantially reduced, and the length of the channel significantly shortened. Moreover, filling the cavities with a dense gas can suppress breakdown and field emission, via the Paschen effect [@Paschen]. A small business [@MuonsInc] has been formed to pursue this idea, with funding from the U.S. Dept. of Energy’s Small Business Technology Transfer program [@STTR]. Phase I, which includes tests of breakdown in gaseous helium at 805MHz, 80K temperature, and pressures from 1 to 100atm, has been approved. If approved, a follow-on Phase II will explore operation with hydrogen at lower frequency. Successful completion of this program could lead to construction of a prototype gaseous-absorber cooling cell, to be tested at the MuCool Test Area (described next) and perhaps in a future phase of the Muon Ionization Cooling Experiment (MICE) [@MICE].
Test facilities
---------------
To augment the Lab G facility described above, we are building a MuCool Test Area at the end of the Fermilab Linac. This location combines availability of multi-megawatt rf power at both 805 and 201MHz and 400-MeV proton beam at high intensity. Cryogenic facilities will be provided for liquid-hydrogen-absorber and superconducting-magnet operation. The underground enclosure under construction will provide the radiation shielding needed for beam tests of absorber power handling and for high-gradient cavity testing, with the added capability of exploring possible effects on cavity breakdown due to beam irradiation of the cavity walls in a solenoidal magnetic field.
The MuCool program includes engineering tests of ionization-cooling components and systems, but not an actual experimental demonstration of ionization cooling with a muon beam. Such a cooling demonstration (MICE) has been proposed and is discussed elsewhere in these Proceedings [@MICE].
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the U.S. Dept. of Energy, the National Science Foundation, the Illinois Board of Higher Education, and the Illinois Dept. of Commerce and Community Affairs.
References {#references .unnumbered}
==========
[99]{}
See http://www.fnal.gov/projects/muon\_collider/.
M. M. Alsharo’a [*et al.*]{}, “Status of Neutrino Factory and Muon Collider Research and Development and Future Plans," FNAL-PUB-02/149-E (July 19, 2002), submitted to Phys. Rev. ST Accel. Beams, arXiv:hep-ex/0207031.
A. N. Skrinsky and V. V. Parkhomchuk, Sov. J. Part. Nucl. [**12**]{}, 223 (1981); D. Neuffer, Part. Acc. [**14**]{}, 75 (1983); E. A. Perevedentsev and A. N. Skrinsky, in [*Proc. 12th Int. Conf. on High Energy Accelerators*]{}, F. T. Cole and R. Donaldson, eds. (Fermilab, 1984), p 485; R. Palmer [*et al.*]{}, Nucl. Phys. Proc. Suppl. [**51A**]{}, 61 (1996).
S. Geer, Phys. Rev. D [**57**]{}, 6989 (1998); earlier versions of a Neutrino Factory, considered by [*e.g.*]{} D. G. Koshkarev, report CERN/ISR-DI/74-62 (1974), S. Wojicki (unpublished, 1974), D. Cline and D. Neuffer, AIP Conf. Proc. [**68**]{}, 846 (1981), and D. Neuffer, IEEE Trans. Nucl. Sci. [**28**]{}, 2034 (1981), were based on pion injection into a storage ring and had substantially less sensitivity.
Q. R. Ahmad [*et al.*]{} (SNO Collaboration), Phys. Rev. Lett. [**89**]{}, 011301 (2002), Phys. Rev. Lett. [**89**]{}, 011302 (2002), and Phys. Rev. Lett. [**87**]{}, 071301 (2001); Y. Fukuda [*et al.*]{} (Super-Kamiokande Collaboration), Phys. Rev. Lett. [**81**]{}, 1562 (1998) and Phys. Rev. Lett. [**86**]{}, 5651 (2001); B. T. Cleveland [*et al.*]{} (Homestake Collaboration), Astrophys. J. [**496**]{}, 505 (1998); R. Davis, D. S. Harmer, and K. C. Hoffman, Phys. Rev. Lett. [**20**]{}, 1205 (1968).
“A Feasibility Study of A Neutrino Factory in Japan," Y. Kuno, ed., available from http://www-prism.kek.jp/nufactj/index.html; Y. Mori, “Review of Japanese Neutrino Factory R&D," A. Sato, “Beam dynamics studies of FFAG," D. Neuffer, “Recent FFAG studies," S. Machida, “Muon Acceleration with FFAGs," and C. Johnstone, “FFAG with high frequency RF for rapid acceleration," all presented at this Workshop.
An introductory discussion of muon ionization cooling may be found in D. M. Kaplan, “Introduction to muon cooling,” in [*Proc. APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001)*]{}, N. Graf, ed., arXiv:physics/0109061 (2002). More detailed treatments may be found in D. Neuffer, “$\mu^+\mu^-$ Colliders," CERN yellow report CERN-99-12 (1999), K. J. Kim and C. X. Wang, Phys. Rev. Lett. [**85**]{}, 760 (2000), and C. X. Wang and K. J. Kim, “Linear theory of 6-D ionization cooling,” in [*Proc. Snowmass 2001*]{}, [*op. cit.*]{}, SNOWMASS-2001-T502 (2001).
D. Neuffer, in [**Advanced Accelerator Concepts**]{}, F. E. Mills, ed., AIP Conf. Proc. [**156**]{} (American Institute of Physics, New York, 1987), p 201.
R. C. Fernow and J. C. Gallardo, Phys. Rev. E [**52**]{}, 1039 (1995).
C. Ankenbrandt [*et al.*]{}, Phys. Rev. ST Accel. Beams [**2**]{}, 081001 (1999).
K. Hagiwara [*et al.*]{} (Particle Data Group), Phys. Rev. D [**66**]{}, 010001 (2002).
“Feasibility Study-II of a Muon-Based Neutrino Source," S. Ozaki, R. Palmer, M. Zisman, and J. Gallardo, eds., BNL-52623, June 2001, available at http://www.cap.bnl.gov/mumu/studyii/FS2-report.html.
R. Palmer, “Ring Coolers: status and prospects" and “Ring cooler studies," S. Kahn, “Simulation of Balbekov ring with realistic fields," D. Cline, “Progress in the development of a quadrupole ring cooler and possible use for neutrino factories and muon colliders," all presented at this Workshop.
An effort to design quadrupole-focused cooling channels is in progress, but their applicability appears to be limited to the early part of the cooling channel, where relatively large beta functions are appropriate (C. Johnstone, “Quadrupole channel for muon cooling," presented at this Workshop).
D. Li, “201 and 805 MHz cavity developments in MUCOOL," presented at this Workshop.
J. Norem [*et al.*]{}, “Dark Current Measurements of a Multicell, 805 MHz Cavity," submitted to Phys. Rev. ST Accel. Beams (2002); J. Norem, “RF induced backgrounds at MICE," presented at this Workshop.
R. H. Fowler and L. W. Nordheim, Proc. Roy. Soc.(London) [**A119**]{}, 173 (1928).
D. M. Kaplan [*et al.*]{}, “Progress in Absorber R&D for Muon Cooling," to appear in [*Proc. 3rd International Workshop on Neutrino Factory based on Muon Storage Rings (NuFACT’01)*]{}, Tsukuba, Japan, 24–30 May 2001, arXiv:physics/0108027.
D. M. Kaplan [*et al.*]{}, “Progress in Absorber R&D 2: Windows," in [*Proc. 2001 Particle Accelerator Conference*]{}, P. Lucas and S. Webber, eds. (IEEE, Piscataway, NJ, 2001), p 3888 (arXiv:physics/0108028).
R. W. Carr [*et al.*]{}, SLAC-Proposal-E-158, July 1997, and E-158 Liquid Hydrogen Target Milestone Report, http://www.slac.stanford.edu/exp/e158/documents/target.ps.gz (April 21, 1999); E. J. Beise [*et al.*]{}, Nucl. Instrum. Meth. A [**378**]{}, 383 (1996); D. J. Margaziotis, in [*Proc. CEBAF Summer 1992 Workshop*]{}, F. Gross and R. Holt, eds., AIP Conf. Proc. [**269**]{} (American Institute of Physics, New York, 1993), p 531; J. W. Mark, SLAC-PUB-3169 (1984) and references therein.
D. Neuffer, “High frequency buncher and phase rotation," presented at this Workshop.
“ASME Boiler and Pressure Vessel Code," ANSI/ASME BPV-VIII-1 (American Society of Mechanical Engineers, New York, 1980), part UG-32.
“Guidelines for the Design, Fabrication, Testing, Installation and Operation of Liquid Hydrogen Targets," Fermilab, Rev. May 20, 1997.
M. A. Cummings, “Absorber R&D in MUCOOL," presented at this Workshop.
W. Lau, “Hydrogen Absorber Window Design," presented at this Workshop.
R. Johnson and D. M. Kaplan, MuCool Note 195, March 2001 (see http://www-mucool.fnal.gov/notes/notes.html).
J. M. Meek and J. D. Craggs, [**Electrical Breakdown in Gases**]{} (John Wiley & Sons, 1978), p 557.
Muons, Inc., R. Johnson, Principal Investigator, Batavia, Illinois.
See http://sbir.er.doe.gov/SBIR/.
R. Edgecock, “International Muon Ionisation Cooling Experiment: Status and plans," presented at this Workshop; see also http://hep04.phys.iit.edu/cooldemo/.
[^1]: In this expression, for expositional clarity, the effects of possible correlations among the components have been neglected; more rigorously, the normalized transverse emittance is proportional to the square-root of the 4-dimensional covariance matrix of the coordinates $(x,y,p_x,p_y)$ for all particles in the beam.
[^2]: That is, thinner at the window center, where the beam is concentrated. Simulation studies have shown that towards the edges, where the new window design is considerably thicker than the old, the muon rate is sufficiently low that the additional window thickness does not degrade cooling performance appreciably.
|
---
author:
- |
[^1]\
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,\
Chinese Academy of Sciences, Beijing 100190, China\
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China\
E-mail:
title: |
Traps in hadron spectroscopy:\
Thresholds, triangle singularities, …
---
Introduction
============
Color confinement forces us to understand hadron spectroscopy in order to understand the strong interaction at low energies. The empirical knowledge of hadron spectroscopy is provided by experimental observations of hadronic resonances and measuring their properties. Most of the resonances were observed as peaking structures in certain invariant mass distributions, such as the new hadronic(-like) structures observed since 2003 in high energy experiments BaBar, Belle, BESIII, LHCb etc. Many of these new structures do not fit in the expectations of quark models treating mesons and baryons as quark-antiquark and three-quark bound states, respectively. Thus, they are regarded as prominent candidates of exotic hadrons which are expected to exist in the spectrum of quantum chromodynamics (QCD) as well and have not received unambiguous experimental confirmation. Most of these discoveries were made in the heavy-flavor, in particular the heavy quarkonium (the so-called $XYZ$ states), sector. For recent reviews, we refer to Refs. [@Chen:2016qju; @Chen:2016spr; @Lebed:2016hpi; @Esposito:2016noz; @Guo:2017jvc; @Ali:2017jda; @Olsen:2017bmm].
However, it is well-known that resonances do not always appear as peaks. Depending on the presence of coupled channels and/or the interference with background contributions, a resonance may even show up as a dip, see, [*e.g.*]{}, Ref. [@Taylor]. Similarly, not all peaks are due to resonances. Here, by resonances we refer to poles of the $S$-matrix. They are of dynamical origin in the sense that they are generated as poles in the scattering amplitudes by the interactions among quarks and gluons (or among hadrons). This is necessarily a nonperturbative phenomenon. In addition to the dynamical poles, the $S$-matrix also has kinematic singularities. The simplest is the two-body branch points (and the associated cuts) at normal thresholds. A more complicated type is the so-called triangle singularity originating from three on-shell intermediate particles. They emerge in the physical amplitudes and can have observable effects when the kinematics of a process satisfies special conditions.[^2] Sometimes, such kinematic singularities may produce peaks mimicking the behavior of a resonance. They lay traps in hadron spectroscopy. In order to establish an unambiguous hadron spectroscopy, it is thus important to distinguish kinematic singularities from genuine resonances.
Two-body threshold cusps
========================
Denoting the amplitude for producing a pair of particles with masses $m_1$ and $m_2$ in a process as $F(s)$, the two-body unitarity requires $$\text{Im}\, F(s) = T^*(s) \rho(s) F(s) \theta(s-(m_1+m_2)^2),$$ where $T(s)$ is the scattering amplitude, and $\rho(s) = \sqrt{\lambda(s,m_1^2,m_2^2)}/(16\pi s)$, with $\lambda(x,y,z)=x^2+y^2+z^2-2xy-2yz-2zx$ the Källén function, is the two-body phase space factor. One sees that at the threshold, there is a square-root branch point, which leads to a cusp at an $S$-wave threshold.[^3] Since the location and involved hadron masses are fixed, the shape of the cusp measures the interaction strength. A well-known example is provided by the precise measurement of the $\pi\pi$ $S$-wave scattering length from the cusp at the $\pi^+\pi^-$ threshold (discussed first in Ref. [@Meissner:1997fa]) in the $\pi^0\pi^0$ invariant mass distribution of the $K^\pm\to \pi^\pm\pi^0\pi^0$ data by the NA48/2 Collaboration [@Batley:2000zz] (see, [*e.g.*]{}, Refs. [@Cabibbo:2004gq; @Gasser:2011ju] for theoretical discussions). The cusp in this process is moderate because the $\pi\pi$ low-energy interaction is rather weak due to the chiral symmetry breaking of QCD.
Some of the new $XYZ$ states are located close to certain $S$-wave thresholds. For instance, the $X(3872)$ [@Choi:2003ue] and $Z_c(3900)$ [@Ablikim:2013mio; @Liu:2013dau] are very close to the $D\bar D^*$ threshold, the $Z_c(4020)$ [@Ablikim:2013xfr] is close to the $D^*\bar D^*$ threshold, the charged bottomonium-like $Z_b(10610)$ and $Z_b(10650)$ [@Belle:2011aa] are nearby the $B\bar B^*$ and $B^*\bar B^*$ thresholds, respectively, and their quantum numbers are the same as the corresponding $S$-wave meson pairs. Furthermore, all these structures have a narrow width. These facts stimulated models speculating them as threshold cusps [@Bugg:2004rk; @Bugg:2011jr; @Chen:2011pv; @Chen:2013coa; @Swanson:2014tra]. A common feature of these calculations is that they considered the processes with the $Z_{c(b)}$ structures in the inelastic channels, [*i.e.*]{}, in the modes with one pion and one charmonium (bottomonium) (here the channel with the relevant threshold is denoted as “elastic”), so that the final states were produced through the $D^{(*)}\bar D^* (B^{(*)}\bar B^*)$ rescattering at the one-loop level. It seems that experimental data could be reproduced rather well by adjusting the cutoff parameter in the form factor which was introduced to regularize the ultraviolet divergent loop integral. But does this imply that the data suggest these structures to be simply due to coupled-channel threshold cusps, which would mean that there is no nearby pole in the $S$-matrix? To answer this question, one has to analyze elastic processes, as will be discussed below.
The intrinsic assumption of the approaches outlined in the cusp models [@Bugg:2011jr; @Chen:2013coa; @Swanson:2014tra] is that the interactions are perturbative so that the amplitude can be approximated by the one loop result, which does not possess a pole by definition. Let us consider a two-channel problem, say $J/\psi\pi$ and $D\bar D^*+c.c.$[^4] We denote the production vertex for these two modes from some process as $g_\text{in}$ and $g_\text{el}$, respectively, and approximate the tree-level $S$-wave amplitudes for $J/\psi\pi\to D\bar D^*$ and $D\bar D^*\to D\bar D^*$ as constants $C_X$ and $C_D$, respectively. The direct $J/\psi\pi\to J/\psi\pi$ amplitude may be neglected due to the Okubo–Zweig–Iizuka rule. Thus, the cusp models for the production of $J/\psi\pi$ and $D\bar D^*$ may be expressed as the following one-loop amplitudes $$\begin{aligned}
g_\text{in} + g_\text{el}\, G_{\Lambda}(E)\, C_X, \qquad\text{and}\qquad
g_\text{el} \left[1 + G_{\Lambda}(E)\, C_D \right],
\label{eq:1loop}\end{aligned}$$ respectively, where $G_\Lambda(E)$ is the two-point loop function with $D\bar D^*$ as the intermediate states and $\Lambda$ denotes that the loop integral needs to be regularized. The two terms in the second amplitude are represented as (a) and (b) in the left panel of Fig. \[fig:DDst\].
![Left: Tree-level, one-loop and two-loop diagrams for the decay $Y(4260)\to \pi D\bar D^*$. Right: Results for the $D\bar D^*$ invariant mass distribution of the decay $Y(4260)\to \pi D \bar D^*$. The data are from Ref. [@Ablikim:2013xfr] and the results from the tree level, one-loop and two-loop calculations are shown by the dotted (green), solid (red) and dashed (magenta) curves, respectively. The parameters are determined from fitting to data using the one-loop amplitude. The dot-dashed (black) curve shows the one-loop result with the rescattering strength requested to be small to justify a perturbative treatment.[]{data-label="fig:DDst"}](diagrams "fig:"){height="5.cm"}![Left: Tree-level, one-loop and two-loop diagrams for the decay $Y(4260)\to \pi D\bar D^*$. Right: Results for the $D\bar D^*$ invariant mass distribution of the decay $Y(4260)\to \pi D \bar D^*$. The data are from Ref. [@Ablikim:2013xfr] and the results from the tree level, one-loop and two-loop calculations are shown by the dotted (green), solid (red) and dashed (magenta) curves, respectively. The parameters are determined from fitting to data using the one-loop amplitude. The dot-dashed (black) curve shows the one-loop result with the rescattering strength requested to be small to justify a perturbative treatment.[]{data-label="fig:DDst"}](ddstarpi "fig:"){height="5.cm"}
One notices that the $G_{\Lambda}(E)\, C_D$ can be fixed from fitting to the $D\bar D^*$ invariant mass distribution in the near-threshold region (so that the approximation of the contact term as a constant $C_D$ is valid) because $g_\text{el}$ only serves as an overall normalization and does not affect the shape. On the contrary, $G_{\Lambda}(E)\, C_X$ cannot be fixed due to its interference with $g_\text{in}$. Once $G_{\Lambda}(E)\, C_D$ is fixed from fitting to the data using the one-loop amplitude, it is easy to check whether the implicit perturbative assumption is proper by comparing the two-loop, $$g_\text{el} \left[1 + G_{\Lambda}(E)\, C_D + G_{\Lambda}(E)\, C_D\, G_{\Lambda}(E)\, C_D \right],
\label{eq:2loop}$$ with the one-loop result. If the difference is small, the perturbative treatment is valid; otherwise, it would mean that such a model is not self-consistent.
We regularize the loop integral $G_{\Lambda}(E)$ using a Gaussian form factor. Using the one-loop amplitudes given in Eq. , we find that indeed the BESIII data for both the $(D\bar D^*)^-$ invariant mass distribution of the $e^+e^- \to \pi^+ (D\bar D^*)^-$ [@Ablikim:2013xfr] and the $J/\psi\pi^-$ invariant mass distribution of the $e^+e^- \to \pi^+ \pi^-J/\psi$ [@Ablikim:2013mio] can be well described, both measured at the $e^+e^-$ center-of-mass energy $E_\text{cm}=4.26$ GeV. The best fit to the data for the former process is shown as the solid curve in the right panel of Fig. \[fig:DDst\], in comparison with the data. Using the same parameters, the tree-level, which is simply the phase space, and two-loop, Eq. , results are shown as the dotted (green) and dashed (magenta) curves, respectively. It is clear that the two-loop result largely deviates from the one-loop one, which indicates that the interaction determined in this way is nonperturbative or $ G_\Lambda(E)\,C_D$ is of order 1 in the near-threshold region. In fact, if one resums the two-point bubbles up to infinite orders, the resulting amplitude $g_\text{el} /\left[1 -G_{\Lambda}(E)\, C_D \right]$ has a pole close to the $D\bar D^*$ threshold. It is the narrowness of the near-threshold peak that requires the $D\bar D^*$ interaction to be nonperturbative. If we demand the interaction to be perturbative by hand, say by requiring $| G_\Lambda(E)\,C_D| = 1/2$ at the $D\bar D^*$ threshold, we are not able to produce any narrow structure in the $D\bar D^*$ channel and the corresponding result is shown as the dot-dashed (black) curve in the right panel of Fig. \[fig:DDst\]. On the contrary, the data in the inelastic channel, $e^+e^- \to \pi^+ \pi^-J/\psi$ for the $Z_c(3900)$, is not enough to determine the rescattering strength because it cannot be disentangled from the direct production represented by $g_\text{in}$ in the first amplitude in Eq. .[^5]
Therefore, we conclude that a narrow pronounced near-threshold peak cannot be produced solely by a threshold cusp, and it necessarily indicates the existence of a nearby pole which might be even a virtual state pole located in the unphysical Riemann sheet with respect to the elastic channel. In fact, it was suggested in Ref. [@Guo:2013sya] and the $Z_c(3900)$ and $Z_c(4020)$ correspond to virtual state poles, which may be located a few tens of MeV below the corresponding thresholds, and a multi-channel fit using a formalism with the unitarity built in [@Hanhart:2015cua; @Guo:2016bjq] to the Belle data suggests the $Z_b(10610)$ to be a virtual state and the $Z_b(10650)$ to be a resonance. Here we want to briefly comment on the lattice results by the HALQCD [@Ikeda:2016zwx; @Ikeda:2017mee] which suggest the $Z_c(3900)$ is a threshold cusp. In the HALQCD calculation, they derived the $\pi J/\psi$, $\rho\eta_c$ and $D\bar D^*$ coupled-channel potential from lattice with the pion mass between 410 and 700 MeV. From the Lippmann–Schwinger equation, a virtual state pole far from the physical region was found. We will not discuss their method, but only point out that the obtained $D\bar D^*$ invariant mass is too broad to account for the BESIII double $D$-tagged data with little background at $E_\text{cm}=4.26$ GeV [@Ablikim:2015swa].
It is worthwhile to notice that in the above discussion, we have assumed that the production vertex (the $Y\to \pi D\bar D^*$ vertex $g_\text{el}$ in the considered example) does not produce a nontrivial structure. The presence of nearby triangle singularities [@Wang:2013cya; @Wang:2013hga] makes the problem more complicated. However, as will be discussed below, the conclusion that a narrow pronounced near-threshold peak in the elastic channel requires the presence of a nearby pole remains unchanged.
Triangle singularities
======================
The location of a threshold cusp is fixed, but the location of a triangle singularity, which is the leading Landau singularity [@Landau:1959fi] of a triangle diagram, depends crucially on the kinematics, [*i.e.*]{}, on the masses and momenta of the involved particles. Moreover, whether it appears close to the physical region also depends on the kinematics. Coleman and Norton showed that the triangle singularity is on the physical boundary if the process could happen classically, [*i.e.*]{}, all of the three intermediate particles could go on shell and all of the interaction vertices satisfied the energy-momentum conservation [@Coleman:1965xm]. Triangle singularity is a logarithmic branch point, which would produce an infinite reaction rate if it really appears in the physical region. This does never happen because at least one of the three particles must be unstable as a consequence of the on-shell condition. The finite width moves the singularity into the complex energy plane, and the differential reaction rate can have a finite peak due to the proximity of the singularity. Although there have been lots of discussions since the 1960’s, no unambiguous observation of a triangle singularity was achieved in the old days. In recent years, experimental data have been collected in many more processes, and there appeared several candidates which might be explained by or contain a large contribution from triangle singularities. A prominent example is provided by the $\eta(1405)\to\pi\pi\pi$ [@BESIII:2012aa]. The $G$-parity of the pions and the $\eta(1405)$ are negative and positive, respectively, so that this decay breaks isospin symmetry. In Refs. [@Wu:2011yx; @Wu:2012pg], it is proposed that this process can happen by coupling the initial state to the $\bar K K^*$, the $K^*$ decaying into $K\pi$ and the $K\bar K$ rescattering into $\pi\pi$. The rescattering contains both the $f_0(980)$ and $a_0(980)$. Isospin symmetry breaking is derived from the mass differences between the charged and neutral intermediate strange mesons. The kinematics of the $\eta(1405/1475)\to\pi f_0(980)$ allows a triangle singularity close to the physical region, and as a consequence the isospin breaking is tremendously enhanced.[^6] In recent years, triangle singularities were considered in the discussion on light mesons, the $a_1(1420)$ [@Liu:2015taa; @Ketzer:2015tqa; @Aceti:2016yeb], the $f_1(1420)$ [@Debastiani:2016xgg; @Liu:2015taa], and the $f_2(1810)$ [@Xie:2016lvs], on exotic hadron candidates, the $Z_c(3900)$ [@Wang:2013cya; @Wang:2013hga; @Szczepaniak:2015eza; @Szczepaniak:2015hya; @Pilloni:2016obd; @Gong:2016jzb], the $P_c(4450)$ [@Guo:2015umn; @Liu:2015fea; @Guo:2016bkl; @Bayar:2016ftu] and its hidden-strangeness analogue [@Xie:2017mbe], and the $Z_b$ [@Wang:2013hga; @Szczepaniak:2015eza; @Bondar:2016pox], and in the baryon sector, see, [*e.g.*]{}, [@Wang:2016dtb; @Debastiani:2017dlz]. Suggestions of searching for new triangle singularities in $B$ or $B_c$ decays can be found in Refs. [@Liu:2017vsf; @Pavao:2017kcr]. In particular, the recent BESIII observation of the fast variation of the $\psi'\pi$ distribution shapes for the $e^+e^-\to \psi'\pi^+\pi^-$ measured at different collision energies [@Ablikim:2017oaf] could be a hint to the importance of triangle singularities discussed in Ref. [@Liu:2014spa; @Cao:2018].
![A triangle diagram with the internal lines labeled by the masses of the corresponding particles. The two vertical dashed lines refer to the two cuts discussed in the text. []{data-label="fig:triangle"}](trianglediag){width="0.4\linewidth"}
To be more explicit, let us take Fig. \[fig:triangle\] and explain the kinematical region where the triangle singularity occurs on the physical boundary. The diagram can be interpreted as $A$ decays into particles $m_1$ and $m_2$, following by the sequential decay of $m_1$ into $B$ and $m_3$, and $m_2$ and $m_3$ react to generate the external $C$. Notice that $A,B$ and $C$ do not need to be single particles. We consider the rest frame of $A$. All of the intermediate particles are on shell so that the magnitudes of their momenta are fixed in terms of the $A,B,C$ invariant masses. The reactions at all vertices can happen classically means that all particles must move parallel or anti-parallel, and particle $m_3$ emitted from the decay of $m_1$ must move fast enough to catch up with particle $m_2$ in order to react to form the external $C$. Expressing the above conditions mathematically, we get [@Bayar:2016ftu] $$\begin{aligned}
q^{}_{\rm on+} = q^{}_{a-}, \quad \text{with}\quad
q^{}_{{\rm on}+} = \frac1{2 m^{}_A} \sqrt{\lambda(m_A^2,m_1^2,m_2^2)},~~ q^{}_{a-} = \gamma \left( \beta \, E_2^* - p_2^* \right),
\label{eq:ts}\end{aligned}$$ where $E_2^*$ and $p_2^*$ are the energy and the size of the 3-momentum of particle $m_2$ in the $B$ rest frame, $\beta$ is the magnitude of the velocity of $B$ in the rest frame of $A$, and $\gamma= 1/{\sqrt{1-\beta^2}}$ is the Lorentz boost factor. The two momenta given above correspond to the two cuts depicted in Fig. \[fig:triangle\]. One sees that the triangle singularity is on the physical boundary only for very special kinematics. For given masses $m_2$, $m_3$ and invariant masses for external particles, one can work out the special range of $m_1$, as well as the corresponding range of the triangle singularity in, [*e.g.*]{}, the $C$ invariant mass. The ranges can be obtained by requiring $q_{\rm on}$ and $q_{a-}$ to take values in the physical regions. Using the above equation, we find that when $$m_1^2 \in \left[ \frac{m_A^2 m^{}_3 + m_{B}^2 m^{}_2}{m^{}_2+m^{}_3} - m^{}_2 m^{}_3\,,~
\left(m^{}_A-m^{}_2 \right)^2 \right],
\label{Eq:m1range}$$ there is a triangle singularity on the physical boundary, and in terms of the $B$ invariant mass it is within the range $$m_{C}^2 \in \left[ (m_2+m_3)^2,~ \frac{m^{}_A m_3^2 - m_{B}^2
m^{}_2}{m^{}_A-m^{}_2} + m^{}_A m^{}_2 \right].
\label{Eq:m23range}$$ For discussions of these ranges, see, [*e.g.*]{}, Refs. [@Aitchison:1964zz; @Szczepaniak:2015eza; @Liu:2015taa; @Guo:2015umn; @Guo:2016bkl].
![Left: The red cure labeled as “TS arc” represents the trajectory in the $E_\text{cm}$–$M_{J/\psi\pi}$ plane along which the triangle singularity is on the physical boundary. Right: Dependence of the absolute value of the $D_1\bar D D^*$ triangle loop on the incoming energy. []{data-label="fig:ytozpi"}](ytozpi_TSregion "fig:"){height="5.cm"} ![Left: The red cure labeled as “TS arc” represents the trajectory in the $E_\text{cm}$–$M_{J/\psi\pi}$ plane along which the triangle singularity is on the physical boundary. Right: Dependence of the absolute value of the $D_1\bar D D^*$ triangle loop on the incoming energy. []{data-label="fig:ytozpi"}](powercounting_check_AbsJ "fig:"){height="4.9cm"}
In order to make the sensitivity on the kinematics clear, let us take the process $e^+e^-\to J/\psi\pi\pi$, the discovery process of the $Z_c(3900)$, as an example. As first pointed out in Ref. [@Wang:2013cya], the triangle singularity of the $D_1(2420)\bar D D^*$ triangle loop (substituting $m_1,m_2,m_3, B$ and $C$ in Fig. \[fig:triangle\] by $D_1,\bar D, D^*, J/\psi\pi$ and $\pi$, respectively) may play an important role. If we fix the intermediate particles as $D_1\bar D D^*$ with their widths neglected and vary the collision energy $E_\text{cm}$ and the $J/\psi\pi$ invariant mass, Eq. implies that the triangle singularity appears as an arc in the $E_\text{cm}$–$M_{J/\psi\pi}$ plane as shown in the left panel of Fig. \[fig:ytozpi\]. The kinematics for the $Y(4260)\to Z_c(3900)\pi$ is not on the arc, but is only a few tens of MeV away and thus leaves an influence. Taking a 30 MeV constant width for the $D_1$, the absolute values of the $D_1\bar D D^*$ scalar 3-point loop integral are shown in the right panel of Fig. \[fig:ytozpi\]. When $E_\text{cm}=4.29$ GeV, the singularity is away from the physical region only due to the small $D_1$ width, and the loop function has a sharp peak. Decreasing $E_\text{cm}$, the peak becomes less pronounced since the triangle singularity is moving further away from the physical region. Nevertheless, there is always a cusp at the $\bar D D^*$ threshold because they couple in an $S$-wave, as discussed in the last section, and the threshold cusp is a subleading singularity of the triangle diagram. The finite width of the $D_1$ does not smear out this cusp. The sensitivity of the line shape on the incoming energy is one of the keys to reveal the role of kinematic singularities.
The above discussion also implies that the $D_1\bar D D^*$ triangle diagrams have to be included in a realistic analysis of the $Z_c(3900)$. Such an analysis of both the $Y(4260)\to J/\psi\pi\pi$ [@Ablikim:2013mio] and $Y(4260)\to D\bar D^*\pi$ [@Ablikim:2015swa] data was done in Ref. [@Albaladejo:2015lob]. It was found that despite the inclusion of the $D_1\bar D D^*$ loops, fits to the data still led to the presence of a pole corresponding to the $Z_c(3900)$ near the $D\bar D^*$ threshold. Depending on whether the $J/\psi\pi$–$D\bar D^*$ coupled-channel interaction model allows for an energy-dependent term in the potential, the pole can be either a virtual state below the $D\bar D^*$ threshold, which could be a few tens of MeV away, or a resonance above the threshold. However, a later analysis by the JPAC Collaboration using a constant $D_1 D^*\pi$ coupling concluded that the data [@Ablikim:2013mio; @Ablikim:2015tbp; @Ablikim:2015swa; @Ablikim:2015gda] did not allow for distinguishing models with a $Z_c(3900)$ state or not. It is interesting to see whether the conclusion remains if the updated BESIII data on the $e^+e^-\to J/\psi\pi^+\pi^-$ at $E_\text{cm}=4.23$ and 4.26 GeV [@Collaboration:2017njt] are used and the $D_1 D^*\pi$ $D$-wave coupling is properly taken into account. In particular, one sees from Fig. \[fig:ytozpi\] that the triangle singularity should not be important for $E_\text{cm}=4.23$ GeV.
Another interesting occurrence of triangle singularity [@Guo:2015umn; @Liu:2015fea] is related to the narrow structure $P_c(4450)$, which was observed by the LHCb Collaboration [@Aaij:2015tga] in the $J/\psi p$ invariant mass distribution of the decay $\Lambda_b^0\to J/\psi p K^-$. The $P_c(4450)$ is regarded as a candidate of hidden-charm pentaquark states first predicted in Ref. [@Wu:2010jy]. However, it was pointed out in Ref [@Guo:2015umn] that the $P_c(4450)$ mass coincides with the $\chi_{c1}\,p$ threshold and, more importantly, the location of the triangle singularity of the $\Lambda(1890)\chi_{c1}p$ loop diagram. The $\Lambda(1890)$ is a well-established four-star hyperon with $J^P=3/2^+$ and a width of about 100 MeV decaying with a large branching fraction into $N\bar K$ [@Patrignani:2016xqp], and $J/\psi p$ in the final state are produced through the $\chi_{c1}p$ rescattering. The shape produced by the $\Lambda_b^0\to J/\psi p K^-$ scalar 3-point loop integral well reproduces the LHCb peak structure around 4.45 GeV. However, because the $\chi_{c1}p\to J/\psi p$ rescattering strength and the $\Lambda_b\to \Lambda(1890)\chi_{c1}$ decay width are unknown and the presence of many $\Lambda^*$ resonances coupled to $p K^-$, we are not able to predict how large the triangle singularity contribution is.[^7] In view of this, we need to have other methods revealing whether the $P_c(4450)$ is really a pentaquark or not, which is an utmost important question to be answered for the $P_c$’s being the first candidates of quasi-explicitly exotic pentaquark states.[^8] Possible methods include:
- To measure the $\chi_{c1}p$ invariant mass distribution of the decay $\Lambda_b^0\to \chi_{c1}p K^-$. For this process, the $\chi_{c1}p$ pair are in the final state as well as in the intermediate state. Therefore, in addition to the $\Lambda(1890)\chi_{c1}p$ loop diagram, the $\chi_{c1}p K^-$ can also be produced at tree-level by exchanging the $\Lambda(1890)$. The subtle interference between the tree-level and triangle diagrams around the singularity region results in an amplitude which is simply the tree-level one multiplied by a complex phase factor, and there would be no pronounced peak in the projected Dalitz distribution [@Schmid:1967].[^9] Thus, were the $P_c(4450)$ due to a triangle singularity, there would be no narrow near-threshold peak in the $\chi_{c1}p$ invariant mass distribution. Following this suggestion, the LHCb Collaboration measured the branching fraction of $\Lambda_b^0\to \chi_{c1}p K^-$ [@Aaij:2017awb], and the amplitude analysis is on going.
- To determine the quantum numbers of the $J/\psi p$ pair in the $P_c(4450)$ peak structure. For the discussed triangle singularity to produce a narrow peak, the $\chi_{c1}$ and proton need to be in an $S$-wave, and thus the quantum numbers of the rescattered $J/\psi p$ should be $J^P=1/2^+$ or $3/2^+$. So far the quantum numbers have not been unambiguously determined with the latter being one of the preferred [@Jurik:2016bdm].
- To search for the $P_c(4450)$ in reactions with different kinematics to avoid the discussed triangle singularity. Such reactions could be, [*e.g.*]{}, the photoproduction processes [@Wang:2015jsa; @Kubarovsky:2015aaa; @Karliner:2015voa; @Huang:2016tcr], pion induced reactions [@Lu:2015fva; @Liu:2016dli] and heavy ion collisions [@Wang:2016vxa; @Schmidt:2016cmd].
Conclusion
==========
In order to understand the QCD spectrum, we need to search for more candidates of exotic hadrons. We are aware of the existence of possible traps along the way, such as resonance-like structures induced by kinematic-singularities discussed above and due to some other reasons, like the $X(5568)$ reported by the D0 Collaboration [@D0:2016mwd], which finds no reason to exist theoretically [@Burns:2016gvy; @Guo:2016nhb], has no signal in lattice QCD calculations [@Lang:2016jpk], and was not confirmed in other experiments [@Aaij:2016iev; @Sirunyan:2017ofq; @CDF:2017]. In order to escape from these traps, cooperative efforts from experiments, phenomenology and lattice calculations are necessary.
[99]{}
H.-X. Chen, W. Chen, X. Liu and S.-L. Zhu, Phys. Rept. [**639**]{}, 1 (2016).
H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu and S.-L. Zhu, Rept. Prog. Phys. [**80**]{}, 076201 (2017).
R. F. Lebed, R. E. Mitchell and E. S. Swanson, Prog. Part. Nucl. Phys. [**93**]{}, 143 (2017).
A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept. [**668**]{}, 1 (2016).
F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao and B.-S. Zou, Rev. Mod. Phys. [**90**]{}, in print (2018) \[arXiv:1705.00141 \[hep-ph\]\].
A. Ali, J. S. Lange and S. Stone, Prog. Part. Nucl. Phys. [**97**]{}, 123 (2017).
S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. [**90**]{}, in print (2018) \[arXiv:1708.04012 \[hep-ph\]\].
J. R. Taylor, [*Scattering Theory: The Quantum Theory of Nonrelativistic Collisions*]{}, John Wiley & Sons, New York, 1972.
R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, [*The Analytic $S$-Matrix*]{}, Cambridge University Press, Cambridge, 1966.
T.-S. Chang, [*Introduction to Dispersion Relation*]{} (2 volumes, in Chinese, written in 1965), Science Press, Beijing, 1980.
V. Gribov, [*Strong Interactions of Hadrons at High Energies*]{}, Cambridge University Press, Cambridge, 2009.
A. V. Anisovich, V. V. Anisovich, M. A. Matveev, V. A. Nikonov, J. Nyiri and Sarantsev. A.V., [*Three-particle physics and dispersion relation theory,*]{} World Scientific, Singapore, 2013.
I. J. R. Aitchison, arXiv:1507.02697 \[hep-ph\]. U.-G. Mei[ß]{}ner, G. Müller and S. Steininger, Phys. Lett. B [**406**]{}, 154 (1997) Erratum: \[Phys. Lett. B [**407**]{}, 454 (1997)\]. J. R. Batley [*et al.*]{}, Eur. Phys. J. C [**64**]{}, 589 (2009).
N. Cabibbo, Phys. Rev. Lett. [**93**]{}, 121801 (2004).
J. Gasser, B. Kubis and A. Rusetsky, Nucl. Phys. B [**850**]{}, 96 (2011).
S. K. Choi [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**91**]{}, 262001 (2003).
M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**110**]{}, 252001 (2013).
Z. Q. Liu [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**110**]{}, 252002 (2013).
M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**112**]{}, 022001 (2014).
A. Bondar [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**108**]{}, 122001 (2012).
D. V. Bugg, Phys. Lett. B [**598**]{}, 8 (2004).
D. V. Bugg, EPL [**96**]{}, 11002 (2011).
D.-Y. Chen and X. Liu, Phys. Rev. D [**84**]{}, 094003 (2011).
D.-Y. Chen, X. Liu and T. Matsuki, Phys. Rev. D [**88**]{}, 036008 (2013).
E. S. Swanson, Phys. Rev. D [**91**]{}, 034009 (2015).
F.-K. Guo, C. Hidalgo-Duque, J. Nieves and M. P. Valderrama, Phys. Rev. D [**88**]{}, 054007 (2013).
C. Hanhart, Y. S. Kalashnikova, P. Matuschek, R. V. Mizuk, A. V. Nefediev and Q. Wang, Phys. Rev. Lett. [**115**]{}, 202001 (2015).
F.-K. Guo, C. Hanhart, Y. S. Kalashnikova, P. Matuschek, R. V. Mizuk, A. V. Nefediev, Q. Wang and J.-L. Wynen, Phys. Rev. D [**93**]{}, 074031 (2016).
Y. Ikeda [*et al.*]{} \[HAL QCD Collaboration\], Phys. Rev. Lett. [**117**]{}, 242001 (2016).
Y. Ikeda \[for HAL QCD Collaboration\], arXiv:1706.07300 \[hep-lat\].
M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. D [**92**]{}, 092006 (2015).
Q. Wang, C. Hanhart and Q. Zhao, Phys. Rev. Lett. [**111**]{}, 132003 (2013).
Q. Wang, C. Hanhart and Q. Zhao, Phys. Lett. B [**725**]{}, 106 (2013).
L. D. Landau, Nucl. Phys. [**13**]{}, 181 (1959).
S. Coleman and R. E. Norton, Nuovo Cim. [**38**]{}, 438 (1965).
M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**108**]{}, 182001 (2012).
J.-J. Wu, X.-H. Liu, Q. Zhao and B.-S. Zou, Phys. Rev. Lett. [**108**]{}, 081803 (2012).
X.-G. Wu, J.-J. Wu, Q. Zhao and B.-S. Zou, Phys. Rev. D [**87**]{}, 014023 (2013).
X.-H. Liu, M. Oka and Q. Zhao, Phys. Lett. B [**753**]{}, 297 (2016).
M. Mikhasenko, B. Ketzer and A. Sarantsev, Phys. Rev. D [**91**]{}, 094015 (2015).
F. Aceti, L.-R. Dai and E. Oset, Phys. Rev. D [**94**]{}, 096015 (2016).
V. R. Debastiani, F. Aceti, W.-H. Liang and E. Oset, Phys. Rev. D [**95**]{}, 034015 (2017).
J.-J. Xie, L.-S. Geng and E. Oset, Phys. Rev. D [**95**]{}, 034004 (2017). A. P. Szczepaniak, Phys. Lett. B [**747**]{}, 410 (2015). A. P. Szczepaniak, Phys. Lett. B [**757**]{}, 61 (2016).
A. Pilloni [*et al.*]{} \[JPAC Collaboration\], Phys. Lett. B [**772**]{}, 200 (2017).
Q.-R. Gong, J.-L. Pang, Y.-F. Wang and H.-Q. Zheng, arXiv:1612.08159 \[hep-ph\].
F.-K. Guo, U.-G. Meißner, W. Wang and Z. Yang, Phys. Rev. D [**92**]{}, 071502 (2015).
X.-H. Liu, Q. Wang and Q. Zhao, Phys. Lett. B [**757**]{}, 231 (2016).
F.-K. Guo, U.-G. Meißner, J. Nieves and Z. Yang, Eur. Phys. J. A [**52**]{}, 318 (2016).
M. Bayar, F. Aceti, F.-K. Guo and E. Oset, Phys. Rev. D [**94**]{}, 074039 (2016).
J.-J. Xie and F.-K. Guo, Phys. Lett. B [**774**]{}, 108 (2017).
A. E. Bondar and M. B. Voloshin, Phys. Rev. D [**93**]{}, 094008 (2016).
E. Wang, J.-J. Xie, W.-H. Liang, F.-K. Guo and E. Oset, Phys. Rev. C [**95**]{}, 015205 (2017).
V. R. Debastiani, S. Sakai and E. Oset, Phys. Rev. C [**96**]{}, 025201 (2017).
X.-H. Liu and U.-G. Meißner, Eur. Phys. J. C [**77**]{}, 816 (2017).
R. Pavao, S. Sakai and E. Oset, Eur. Phys. J. C [**77**]{}, 599 (2017).
M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. D [**96**]{}, 032004 (2017).
X.-H. Liu, Phys. Rev. D [**90**]{}, 074004 (2014).
Z. Cao [*et al.*]{}, in preparation.
I. J. R. Aitchison, Phys. Rev. [**133**]{}, B1257 (1964).
M. Albaladejo, F.-K. Guo, C. Hidalgo-Duque and J. Nieves, Phys. Lett. B [**755**]{}, 337 (2016). M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**115**]{}, 112003 (2015). M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**115**]{}, 222002 (2015). M. Ablikim [*et al.*]{} \[BESIII Collaboration\], Phys. Rev. Lett. [**119**]{}, 072001 (2017). R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**115**]{}, 072001 (2015). J.-J. Wu, R. Molina, E. Oset and B.-S. Zou, Phys. Rev. Lett. [**105**]{}, 232001 (2010). C. Patrignani [*et al.*]{} \[Particle Data Group\], Chin. Phys. C [**40**]{}, 100001 (2016) and 2017 update. C. Schmid, Phys. Rev. [**154**]{}, 1363 (1967).
A. V. Anisovich and V. V. Anisovich, Phys. Lett. B [**345**]{}, 321 (1995).
R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**119**]{}, 062001 (2017). N. P. Jurik, [*Observation of $J/\psi p$ resonances consistent with pentaquark states in $\Lambda_ b^0\to J/\psi K^-p$ decays*]{}, CERN-THESIS-2016-086.
Q. Wang, X.-H. Liu and Q. Zhao, Phys. Rev. D [**92**]{}, 034022 (2015).
V. Kubarovsky and M. B. Voloshin, Phys. Rev. D [**92**]{}, 031502 (2015).
M. Karliner and J. L. Rosner, Phys. Lett. B [**752**]{}, 329 (2016).
Y. Huang, J.-J. Xie, J. He, X. Chen and H.-F. Zhang, Chin. Phys. C [**40**]{}, 124104 (2016).
Q.-F. Lü, X.-Y. Wang, J.-J. Xie, X.-R. Chen and Y.-B. Dong, Phys. Rev. D [**93**]{}, 034009 (2016).
X.-H. Liu and M. Oka, Nucl. Phys. A [**954**]{}, 352 (2016).
R.-Q. Wang, J. Song, K.-J. Sun, L.-W. Chen, G. Li and F.-L. Shao, Phys. Rev. C [**94**]{}, 044913 (2016).
I. Schmidt and M. Siddikov, Phys. Rev. D [**93**]{}, 094005 (2016). V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. Lett. [**117**]{}, 022003 (2016). T. J. Burns and E. S. Swanson, Phys. Lett. B [**760**]{}, 627 (2016). F.-K. Guo, U.-G. Meißner and B.-S. Zou, Commun. Theor. Phys. [**65**]{}, 593 (2016). C. B. Lang, D. Mohler and S. Prelovsek, Phys. Rev. D [**94**]{}, 074509 (2016). R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. Lett. [**117**]{}, 152003 (2016) Addendum: \[Phys. Rev. Lett. [**118**]{}, 109904 (2017)\]. A. M. Sirunyan [*et al.*]{} \[CMS Collaboration\], arXiv:1712.06144 \[hep-ex\].
T. Aaltonen [*et al.*]{} \[CDF Collaboration\], arXiv:1712.09620 \[hep-ex\].
[^1]: I would like to thank all my collaborators for sharing their insights into the topics discussed here. This work is supported in part by NSFC and DFG through funds provided to the Sino–German Collaborative Research Center “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Grant No. TRR110), by NSFC (Grant No. 11647601), by the Thousand Talents Plan for Young Professionals, and by the CAS Key Research Program of Frontier Sciences (Grant No. QYZDB-SSW-SYS013).
[^2]: For detailed discussions about the triangle singularity and other Landau singularities, we refer to the monographs [@Eden:book; @Chang:book; @Gribov:book; @Anisovich:2013gha] and recent lecture notes [@Aitchison:2015jxa].
[^3]: For higher partial waves, the cusp is smeared by positive powers of momentum in $T(s)$.
[^4]: The required charge conjugation will be kept implicit in the following.
[^5]: This is different from the case of $K^\pm\to \pi^\pm\pi^0\pi^0$ where the two channels $\pi^0\pi^0$ and $\pi^+\pi^-$ are related to each other.
[^6]: Because of isospin braking, the neutral and charged strange meson loops cancel out below the $K^+K^-$ threshold and above the $K^0\bar K^0$ threshold so that the $f_0(980)$ peak is as narrow as about 10 MeV$\sim 2(M_{K^0}-M_{K^+})$.
[^7]: For such a calculation, the three-body unitarity needs to be considered properly, which presents another difficulty.
[^8]: The quantum numbers of the $P_c$ structures can be formed by three light quarks. However, since their masses are above 4 GeV, if they are light baryons they would decay into light hadrons very quickly due to the vast amount of phase space, and the widths would be much larger than those reported by the LHCb Collaboration. Therefore, it is more natural to assume that there are a pair of charm and anticharm quarks inside whose annihilation into light hadrons is suppressed.
[^9]: Corrections to this observation were discussed in Refs. [@Anisovich:1995ab; @Szczepaniak:2015hya].
|
---
abstract: 'In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\bigwedge_{i=1}^n {{\ensuremath{\mathbf{G}}}}{{\ensuremath{\mathbf{F}}}}\varphi_i \vee {{\ensuremath{\mathbf{F}}}}{{\ensuremath{\mathbf{G}}}}\psi_i $, where $\varphi_i$ and $\psi_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.'
author:
- Salomon Sickert
- Javier Esparza
bibliography:
- 'bibliography.bib'
subtitle: Extended Version
title: An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata
---
<ccs2012> <concept> <concept\_id>10003752.10003790.10003793</concept\_id> <concept\_desc>Theory of computation Modal and temporal logics</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003752.10003766.10003770</concept\_id> <concept\_desc>Theory of computation Automata over infinite objects</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
The authors want to thank Orna Kupferman for the suggestion to examine the expressive power of weak alternating automata and the anonymous reviewers for their helpful comments and remarks.
This work is partly funded by the project “Verified Model Checkers” () and partly funded by the under the European Union’s Horizon 2020 research and innovation programme under grant agreement PaVeS (No ).
|
---
abstract: |
We consider a variational approximation scheme for the 3D elastodynamics problem. Our approach uses a new class of admissible mappings that are closed with respect to the space of mappings with finite distortion.\
*Key words and phrases:* elastodynamics, mapping with finite distortion, polyconvexity, variational approximation scheme.
address:
- 'Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, Novosibirsk 630090, Russia'
- 'Peoples’ Friendship University, 6 Miklukho-Maklaya str., Moscow 117198, Russia'
author:
- Anastasia Molchanova
bibliography:
- 'biblio\_ed.bib'
title: A variational approximation scheme for elastodynamic problems using a new class of admissible mappings
---
The motion of a deformable solid body can be written: $$\frac{\partial^2 y}{\partial t^2} = \nabla \cdot S(\nabla y),$$ where $y \colon \Omega \times [t_0,t_1] \to \mathbb{R}^3$ is the displacement and $S$ is the first Piola–Kirchgoff stress tensor. This equation can also be written as a system of conservation laws for the deformation gradient $F = D_x y$ and the velocity $v = {\partial_t} y$ $$\label{eq:conserv_law}
\begin{aligned}
& \frac{\partial}{\partial t} F_{i \alpha} = \frac{\partial}{\partial x_{\alpha}} v_{i},\\
& \frac{\partial}{\partial t} v_{i} = \sum\limits_{\alpha = 1}\limits^{3} \frac{\partial}{\partial x_{\alpha}} S_{i \alpha} (F),
\end{aligned}
\quad i, \alpha = 1, 2, 3.$$ In the case of hyperelastic materials, the tensor $S$ can be expressed as the gradient of a scalar function $W(F)$. This function is called the stored energy function, $W\colon \mathbb{M}^{3\times 3} \to [0, \infty)$, where $\mathbb{M}^{3\times 3}$ stands for $({3 \times 3})$-matrices, i.e.$S(F) = \Big[\dfrac{\partial W}{\partial F_{ij}} (F)\Big]$. In order to prove the existence of solutions to the system of equations in \[eq:conserv\_law\], it would normally be necessary that W be convex. However, this is incompatible with the known physics of elastic materials such as the requirement of frame-indifference, i.e. the principle that certain properties of the system are invariant under arbitrary coordinate transformations [@ColNol1959].
This suggests the replacement of the condition of convexity with a weaker condition such as [*polyconvexity*]{} (for further details see, e.g. [@Ball1977; @Ball2002; @Ciar1988]). More precisely, we may assume that $$W(F) = G(F, { \operatorname {cof} }F, \det F)$$ holds for some convex function $G(F,Z,w)$, where ${ \operatorname {cof} }F$ and $\det F$ are the cofactor matrix (i.e. transposed adjunctive matrix ${ \operatorname {cof} }F = {\rm adj}\, F^T$) and determinant of the matrix $F$ respectively.
At present, the question of the existence of a solution to the elastostatic problem has been thoroughly studied. A review of basic works and open problems can be found, for example, in [@Ball2002]. Furthermore, the reader is referred to [@Daf2010] for local existence of the classical solution of the elastodynamical system with rank-one convex and polyconvex stored energy functions. The existence of global weak solutions, excluding some particular cases [@DiPer1983], is still an open problem. Nevertheless, the existence of a global measure-valued solution was proven in [@DemStuTza2001] using a variational approximation scheme.
The variational approximation method or, in the nomenclature of E. De Giorgi [@DeGio1993], the minimizing movements method, is a method by which the limit of a minimizing sequence of iterations to the variational problem for an appropriate functional is found. The technique developed in [@DemStuTza2001] uses the variational approximation scheme to establish a link between elastostatics and elastodynamics. The method is based on the observation that the solution of the system meets the [*additional conservation laws*]{} (the idea was suggested independently by P.G. Le Floch and T. Qin [@Qin1998]) $$\label{eq:ad_conservation_laws}
\begin{aligned}
\frac{\partial}{\partial t} \det F &
= \sum\limits_{i, \alpha=1}\limits^{3}\frac{\partial}{\partial x_{\alpha}}(({ \operatorname {cof} }F)_{i \alpha} v_i),
\\
\frac{\partial}{\partial t}({ \operatorname {cof} }F)_{k \gamma} &
= \sum\limits_{i, j, \alpha, \beta = 1}\limits^{3} \frac{\partial}{\partial x_{\alpha}}
(\epsilon_{ijk} \epsilon_{\alpha \beta \gamma} F_{j \beta} v_i)
\end{aligned}$$ where $\epsilon_{ijk}$ is the permutation symbol. Then, introducing new variables $Z = { \operatorname {cof} }F$, $w = \det F$ and setting $\Xi = (F, Z, w)$, we can use together with to derive the enlarged system $$\label{enlar_syst}
\begin{aligned}
\partial_t v_i & = \sum\limits_{\alpha,A} \partial_{\alpha}
\left(\frac{\partial G}{\partial \Xi^A}(\Xi) \frac{\partial \Phi^A}{\partial F_{i\alpha}(F)} \right)
= \sum\limits_{\alpha} \partial_{\alpha} (g_{i\alpha} (\Xi; F)), \\
\partial_t \Xi^A & = \sum\limits_{i,\alpha} \partial_{\alpha}
\left(\frac{\partial \Phi^A}{\partial F_{i \alpha}}(F)v_i \right),
\end{aligned}$$ where $$g_{i \alpha} (F, Z, w; F^0) = \frac{\partial G}{\partial F_{i \alpha}}
+ \sum\limits_{j, k, \beta, \gamma}
\frac{\partial G}{\partial Z_{k \gamma}} \epsilon_{i j k} \epsilon_{\alpha \beta \gamma} F^0_{j \beta}
+ \frac{\partial G}{\partial w} ({ \operatorname {cof} }F^0)_{i \alpha}. \\$$ and $\Phi (F) = (F, { \operatorname {cof} }F, \det F)$. Further, time discretization gives the variational problem $$\min\limits_{\mathcal C} \int\limits_{\Omega} \frac{1}{2}(v - v^0)^2 + G(F,Z,w) \, dx,$$ with initial data $v^0(x)$, $F^0(x)$, $Z^0(x)$, $w^0(x)$, and the admissible set involves the “additional” constraints $$\begin{aligned}
\mathcal{C}=
\Bigl\{ (v, F, Z, w)
\in
& L_2 (\Omega) \times L_p (\Omega) \times L_q (\Omega) \times L_r(\Omega),
\: p>4, \: q,\: r \geq 2 :
\\
&\frac{1}{h} (F_{i \alpha} - F^0_{i \alpha}) = \partial_\alpha v_i ,
\\
&\frac{1}{h} (Z_{k \gamma} - Z^0_{k \gamma})
= \sum\limits_{i, j, \alpha, \beta} \epsilon_{i j k}\partial_\alpha
\big( \epsilon_{\alpha \beta \gamma} F^0_{j \beta} v_i \big),
\\
&\frac{1}{h} (w - w^0)
= \sum\limits_{i, \alpha} \partial_\alpha \big(({ \operatorname {cof} }F^0)_{i \alpha} v_i \big)
\Bigr\}.
\end{aligned}$$
In this article, we consider the variational approximation scheme in a new class of admissible mappings, in function classes stemming from quasiconformal analysis, and derive the Euler–Lagrange equations in the cases of smaller regularity ($p \geq 3$), finite distortion ($|F|^3 \leq M w$) and nonnegative Jacobian ($w \geq 0$) requirements. Recall that a mapping $f\colon \Omega \to \mathbb{R}^n$ is called the [*mapping with finite distortion*]{}, $f \in FD(\Omega)$, if $f \in W^1_{1, { {\rm loc} }} (\Omega)$, $J(x, f) \geq 0$ almost everywhere (henceforth abbreviated as a.e.) in $\Omega$ and $$|Df (x)|^n \le K(x) J(x,f) \quad \text{a.e. in } \Omega,$$ where $1 \leq K(x) < \infty$ a.e. in $\Omega$ (see for example [@IwaSve1993]). We also note that the problem of the approximation preserving the constraint $\det F > 0$ is still open, except for the very special case of radial elastodynamics [@MirTza2012]. This condition on the deformation gradient is necessary to ensure that the mappings representing motion are orientation-preserving i.e. that the deformations are physical.
For the sake of simplicity we will work with periodic boundary conditions, i.e. the domain $\Omega$ is taken to be a three dimensional torus. Consider the stored energy function $W \colon \Omega \times \mathbb{M}^3 \to \mathbb{R}$ with the following properties:\
[**(H1)**]{} [**Polyconvexity:**]{} \[cond:polyconvexity\] there exists a convex $C^2$-function $G \colon \mathbb{M}^3\times \mathbb{M}^{3} \times \mathbb{R}_{+} \to \mathbb{R}$ such that for all $F\in \mathbb{M}^3$, $\det F \geq 0$, the equality $$G(F, { \operatorname {cof} }F, \det F) = W(F)$$ holds.\
[**(H2)**]{} [**Coercivity:**]{} \[cond:coer\] there are constants $C_1 > 0$, $C_2 \in \mathbb{R}$, $p \geq 3$, $q$, $r \geq 2$ such that $$\label{neq:coer}
G(F, Z, w) \geq C_1 \bigg(|F|^p + |Z|^{q} + w^r
\bigg) + C_2.$$\
[**(H3)**]{} \[cond:finI\] There is a constant $c > 0$ such that $$G(F, Z, w) \leq c (|F|^p + |Z|^{q}
+ w^r + 1).$$\
[**(H4)**]{} \[cond:limI’\] There is a constant $C > 0$ such that the inequality $$|\partial_{F} G|^{p'} + |\partial_{Z} G|^{\frac{p p'}{p-p'}}
+ |\partial_{w} G|^{\frac{p p'}{p - 2 p'}}
\leq C (|F|^p + |Z|^{q} + w^r + 1)$$ holds for $p' = \frac{p}{p - 1}$ if $p > 3$ and $p' < \frac{3}{2}$ if $p = 3$.
Then the iteration scheme is constructed by solving $$\begin{aligned}
\frac{v_i^J - v_i^{J-1}}{h} & = & \sum\limits_{\alpha, A} \partial_{\alpha}
\left(\frac{\partial G}{\partial \Xi^{A}}(\Xi^{J-1})
\frac{\partial \Phi^A}{\partial F_{i\alpha}(F^{J-1})} \right), \\
\frac{(\Xi^J - \Xi^{J-1})^A}{h} & = & \sum\limits_{i,\alpha} \partial_{\alpha}
\left(\frac{\partial \Phi^A}{\partial F_{i \alpha}}(F^{J-1})v_i^J \right).\end{aligned}$$
The $J$-th iterates are given by $$(v^J, \Xi^J) = (v^J, F^J, Z^J,w^J) = (S_h)^J (v^0, F^0, Z^0, w^0),$$ where a solution operator $S_h$ is defined by
\[enlar\_syst\_h\] $$\begin{aligned}
\frac{1}{h} (v_i - v^0_i) & = &\sum\limits_{\alpha}
\partial_\alpha g_{i \alpha}(F, Z, w; F^0), \label{enlar_syst_h:1} \\
\frac{1}{h} (F_{i \alpha} - F^0_{i \alpha}) & = & \partial_\alpha v_i, \\
\frac{1}{h} (Z_{k \gamma} - Z^0_{k \gamma}) & = & \sum\limits_{i, j, \alpha, \beta}
\partial_\alpha (\epsilon_{i j k} \epsilon_{\alpha \beta \gamma} F^0_{j \beta} v_i), \\
\frac{1}{h} (w - w^0) & = & \sum\limits_{i, \alpha} \partial_\alpha
(({ \operatorname {cof} }F^0)_{i \alpha} v_i).
\end{aligned}$$
Given $M \in L_s (\Omega)$, $s >2$, consider the space $X = L_2 (\Omega) \times L_p (\Omega)
\times L_{q} (\Omega) \times L_r(\Omega)$ and the set of admissible mappings $$\label{enlar_syst_h_weak}
\begin{aligned}
{ \mathcal{A} }=
& \Bigl\{ (v, F, Z, w)
\in X,
\: I(v, F, Z, w) < \infty,
\: |F(x)|^3 \leq M(x) w(x) \text{ a.e.\ in } \Omega,
\\
&
\: w(x) \geq 0 \text{ a.e.\ in } \Omega,
\text{ and for every } \theta \in C_0^\infty (\Omega, \mathbb{R}^3)
\\
&\int\limits_\Omega \theta \frac{1}{h} (F_{i \alpha} - F^0_{i \alpha}) \, dx
= -\int\limits_\Omega v_i \partial_\alpha \theta \, dx,
\\
&\int\limits_\Omega \theta \frac{1}{h} (Z_{k \gamma} - Z^0_{k \gamma}) \, dx
= -\int\limits_\Omega \sum\limits_{i, j, \alpha, \beta} \epsilon_{i j k} \epsilon_{\alpha \beta \gamma} F^0_{j \beta} v_i
\partial_\alpha \theta \, dx,
\\
&\int\limits_\Omega \theta \frac{1}{h} (w - w^0) \, dx
= -\int\limits_\Omega \sum\limits_{i, \alpha} ({ \operatorname {cof} }F^0)_{i \alpha}
v_i \partial_\alpha \theta \, dx
\Bigr\},
\end{aligned}$$
Let the initial data satisfy $y^0 = y(0) \in W^1_p(\Omega) \cap FD(\Omega)$, $v^0 = \partial_t y(0) \in L_2 (\Omega)$, $F^0 = D y^0 \in L_p (\Omega)$, $Z^0 = { \operatorname {cof} }D y^0 \in L_{q} (\Omega)$, $w^0 = \det D y^0 \in L_r(\Omega)$, $|F^0 (x)|^3 \leq M(x) w^0 (x)$, $w^0 (x) \geq 0$ a.e. in $\Omega$ and $$\int\limits_\Omega \frac{1}{2} (v^0)^2
+ G(F^0, Z^0, w^0) \, dx < \infty.$$
It is easy to see that the following assertions hold.
The admissible set ${ \mathcal{A} }$ is nonempty.
The admissible set ${ \mathcal{A} }$ is invariant with respect to the relations $$\begin{aligned}
& \sum\limits_{\alpha} \partial_\alpha Z_{i \alpha} = 0,\\
& \partial_\beta F_{i \alpha} - \partial_\alpha F_{i \beta} = 0.
\end{aligned}$$ In particular, if $F^0$ is a differential then so is $F$, and, thus, there exists the mapping $y \in W^1_p (\Omega)$ such that $\partial_\alpha y_i = F_{i \alpha}$.
Consider the minimization problem for the functional $$\label{eq:minI}
I(v, F, Z, w) = \int\limits_{\Omega} \frac{1}{2} |v - v^0|^2 + G(F, Z, w) \, dx.$$
\[thm:exist&uniq\] There exists $(v, F, Z, w) \in { \mathcal{A} }$ satisfying $$I (v, F, Z, w)
= \inf \limits_{{ \mathcal{A} }} I (v', F', Z', w').$$ Furthermore, if $G$ is a strictly convex function then the minimizer $(v, F, Z, w) \in { \mathcal{A} }$ is unique.
The proof of Theorem \[thm:exist&uniq\] is based on the next theorem.
\[thm:l.c.s.\] Let $\{(v_n, F_n, Z_n, w_n)\}_{n \in \mathbb{N} }\subset { \mathcal{A} }$ and $
S = \sup\limits_{n\in \mathbb{N}} I (v_n, F_n, Z_n, w_n) < \infty
$. Then there exist $(v, F, Z, w)\in X$ and a subsequence $(v_\mu, F_\mu, Z_\mu, w_\mu)$ such that $$\label{condition:convergence}
\begin{cases}
v_\mu \rightharpoonup v & \text{ in } L_2 (\Omega),
\\
F_\mu \rightharpoonup F & \text{ in } L_p(\Omega),
\\
Z_\mu \rightharpoonup Z & \text{ in } L_{q}(\Omega),
\\
w_\mu \rightharpoonup w & \text{ in } L_r(\Omega),
\end{cases}$$ Moreover, $(v, F, Z, w)\in { \mathcal{A} }$ and $$\label{neq:l.c.s.}
I(v, F, Z, w)
\leq \liminf\limits_{n \rightarrow \infty} I (v_n, F_n, Z_n, w_n) = s < \infty.$$
This statement can be proven by applying the techniques and methods of papers [@DemStuTza2001; @VodMol2015; @MolVod2017].
We will now show that the minimizer of (\[eq:minI\]) over the admissible set ${ \mathcal{A} }$ satisfies the weak form of the system of equations . To derive the Euler–Lagrange equations, we assume that the minimizer $( v, F, Z, w) $ meets $w(x) \geq \gamma > 0$ a.e. in $\Omega$.
We fix “direction” ${\theta = (\theta_1, \theta_2, \theta_3) \in C^\infty_0 (\Omega, \mathbb{R}^3)}$ such that $$|F + h D \theta|^n \leq M \big(w + \sum\limits_{i, \alpha} h { \operatorname {cof} }F^0_{i \alpha} \, \partial_\alpha \theta_i \big)
\quad \text{and} \quad
\sum\limits_{i, \alpha} { \operatorname {cof} }F^0_{i \alpha} \, \partial_\alpha \theta_i \in L_{\infty} (\Omega).$$ For $$|\varepsilon| \leq \varepsilon^0 = \frac{\gamma}{\|\sum\limits_{i, \alpha} h
{ \operatorname {cof} }F^0_{i \alpha} \partial_\alpha \theta_i\|_{L_\infty} + 1}$$ we set $$\varepsilon(\delta v_i, \delta F_{i \alpha}, \delta Z_{k \gamma}, \delta w) \\
= \varepsilon \bigg(\theta_i, h \partial_\alpha \theta_i,
\sum\limits_{i, j, \alpha, \beta} h \epsilon_{i j k} \epsilon_{\alpha \beta \gamma}
F^0_{j \beta} \partial_\alpha \theta_i,
\sum\limits_{i, \alpha} h \partial_\alpha { \operatorname {cof} }F^0_{i \alpha} \partial_\alpha \theta_i\bigg)$$ and a variation $$\Xi^\varepsilon = (v^\varepsilon, F^\varepsilon, Z^\varepsilon, w^\varepsilon )
= ( v, F, Z, w) +
\varepsilon(\delta v_i, \delta F_{i \alpha}, \delta Z_{k \gamma}, \delta w).$$
One can readily see that such variations fulfill the conditions since columns of the matrix ${ \operatorname {cof} }F^0$ are divergence-free vector fields. Additional requirements on $\theta$ allow us to conclude that $\Xi^\varepsilon = (v^\varepsilon, F^\varepsilon, Z^\varepsilon, w^\varepsilon )$ belongs to the admissible set ${ \mathcal{A} }$.
Furthermore, using the mean value theorem and Lebesgue’s dominated convergence theorem we find that $$\begin{gathered}
\lim \limits_{\varepsilon \to 0}
\frac{1}{\varepsilon} (I(v_i + \varepsilon\delta v_i, F_{i \alpha} + \varepsilon\delta F_{i \alpha},
Z_{k \gamma} + \varepsilon\delta Z_{k \gamma}, w + \varepsilon\delta w)
- I(v_i, F_{i \alpha}, Z_{k \gamma}, w) )\\
= \int\limits_{\Omega} \sum\limits_{i} \theta_i (v_i - v_i^0)
+ \sum\limits_{i, \alpha} h \partial_\alpha \theta_i \, g_{i \alpha} (F, Z, w; F^0) \, dx \end{gathered}$$ for $\varepsilon^* \in [0, \varepsilon]$. The last equality is the weak form of . To apply the dominated convergence theorem we use the integrability properties of $g_{i \alpha}$ derived from hypothesis **(H4)** and Young’s inequality [@DemStuTza2001].
In the case $p>4$, following to [@DemStuTza2001], we find the mapping $y\colon (0, \infty) \times \Omega \to \mathbb{R}^3$, $y \in W^{1}_{\infty} ([0,\infty];L_2) \cap L_{\infty} ([0,\infty];W^1_p)$ such that the conditions $\partial_t y = v$, $D_x y = F$, ${ \operatorname {cof} }D_x y = Z$, $\det D_x y = w$ are fulfilled. Moreover, they satisfy the weak form of the additional conservation laws and $y$ is the measure-valued solution of the system (\[enlar\_syst\]).
**Acknowledgment.** The author is grateful to professor Sergey Vodop$'$yanov for suggesting the problem and useful discussions.
|
---
abstract: 'We pursue a low-wavenumber, second-order homogenized solution of the time-harmonic wave equation in periodic media with a source term whose frequency resides inside a band gap. Considering the wave motion in an unbounded medium $\mathbb{R}^d$ ($d\geqslant1$), we first use the Bloch transform to formulate an equivalent variational problem in a bounded domain. By investigating the source term’s projection onto certain periodic functions, the second-order model can then be derived via asymptotic expansion of the Bloch eigenfunction and the germane dispersion relationship. We establish the convergence of the second-order homogenized solution, and we include numerical examples to illustrate the convergence result.'
author:
- 'Shixu Meng, Othman Oudghiri-Idrissi, Bojan B. Guzina'
bibliography:
- 'MIG2020.bib'
title: 'A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term'
---
waves in periodic media, dynamic homogenization, finite frequency, band gap, Bloch transform, variational formulation
Introduction {#Introduction}
============
[Let ${\boldsymbol{I}}:=\{\bz: \bz \in \mathbb{Z}^d \}$]{}, and consider the time-harmonic wave equation $$\label{PDE fast}
-\nabla\!\cdot\!\big(G(\br/\eps)\nabla U\big) - \Omega^2\rho(\br/\eps)\hh U \:=\: f(\br) \qquad\text{in~~}\mathbb{R}^d, ~ d\geqslant 1$$ [with highly oscillating coefficients $(\eps\!\to\!0)$]{} at frequency $\Omega$, where $G$ and $\rho$ are $\boldsymbol{I}$-periodic i.e. $$\begin{aligned}
\label{Grho}
G(\br+ \bz) = G(\br), \quad \rho(\br+ \bz) = \rho(\br), \qquad \forall~\br \in \mathbb{R}^d, ~ \bz \in \mathbb{Z}^d.\end{aligned}$$ Here $f$ denotes the source term, while $G$ and $\rho$ are assumed to be real-valued, sufficiently smooth functions bounded away from zero. The regularity on $\rho$ and $G$ can be relaxed and we comment on this later. When $d\!=\!2$, one may interpret (\[PDE fast\]) in the context of anti-plane shear (elastic) waves, in which case $u,G,\rho$ and $f$ take respectively the roles of transverse displacement, shear modulus, mass density, and body force.
We seek a low-wavenumber, second-order asymptotic solution $U_2$ of , formulated in terms of the perturbation parameter $\eps$, such that $$\label{aux1}
\|U-U_2\|_{L^2(\mathbb{R}^d)} = O(\eps^3) \quad \text{as } \eps\to 0.$$ More precisely, we are interested in the respective field equations that $U_2$ and its “mean” i.e. effective counterpart (obtained by projecting $U_2$ onto certain eigenfunction) satisfy. [For completeness, we consider both low and high frequency cases. As will be seen shortly, in the low (resp. high) frequency case $\Omega$ is $O(1)$ (resp. $O(\eps^{-2})$).]{}
Background and motivation
-------------------------
Wave motion in periodic and otherwise microstructured media [@lions1978asymptotic; @JJWM; @kuchment2016overview] has keen applications in science, engineering, and technology owing to the emergence of metamaterials facilitating the phenomena such as cloaking, sub-wavelength imaging, and vibration control [@Capo2009; @Bava2013]. To simulate the underpinning physical processes effectively, of particular interest is the development of asymptotic models. In this vein, homogenized (i.e. effective) models of waves in periodic media that transcend the quasi-static limit have attracted much attention. In particular, effective models using the concept of Bloch waves were studied in [@santosa1991dispersive; @conca2002bloch; @dohnal2014bloch], while [@chen2001dispersive; @wautier2015second] investigated the dispersive wave motion via multi-scale homogenization. A formal link between the homogenization using Bloch waves and two-scale homogenization, [restricted to the periodic index of refraction (no periodicity in the principal part)]{}, was established in [@allaire2016comparison], including an account for the *source term*; the study suggests that the source term must undergo asymptotic correction in order for the equivalence between the two methods to hold. On the other hand, Willis’ approach to dynamic homogenization has brought much attention in the engineering community as a means to deal with periodic and random composites, see for instance [@milton2007modifications; @willis2011effective; @norris2012analytical; @nassar2015willis]. Recently, [@meng2018dynamic] investigated a second-order asymptotic expansion of the Willis’ model, which demonstrates that the source term must be homogenized when considering the two-scale, second-order homogenization of the wave equation. We refer the reader to [@cakoni2016homogenization; @cakoni2019scattering; @lin2018leading] for related works on the scattering by bounded periodic structures.
However, the foregoing studies primarily focused on the low-frequency wave motion, either in the frequency- or time-domain. Recent advances in metamaterials have motivated the studies on high- (or finite-) frequency homogenization [@craster2010high; @harutyunyan2016high]. Within the framework of [Bloch-wave homogenization]{}, [@guzina2019rational] pursued a (second-order) [finite-frequency, finite-wavelength]{} effective description of the wave equation with a source term. To our knowledge, however, there are no convergence results on the [high-frequency homogenization of wave motion in periodic media]{} when the *source term* is present. This motives our work, which considers the situations when the (high or low) frequency resides inside a band gap. For completeness, we note that this subject is also relevant to the asymptotic behavior of the Green’s function near internal edges of the spectra of periodic elliptic operators [@kuchment2012green; @kha2015green]. [In our investigation, we apply various mapping properties of the Floquet-Bloch transform [@lechleiter2017floquet] to formulate the wave motion in $\mathbb{R}^d$ as a variational problem in the Wigner-Seitz cell. This particular approach may play a key role in our ability to extend the current work to asymptotic models of wave motion across periodic surfaces and interfaces of periodic media.]{}
The paper is organized as follows. In Section \[Prelim\], we provide the necessary background on the Floquet-Bloch transform and wave dispersion in periodic media. Section \[Bloch expansion\] formulates the [wave motion in $\mathbb{R}^d$ with highly oscillating periodic coefficients (due to a source term) as a variational problem in a bounded domain]{}, and gives both variational and integral representations of the germane wavefield. Here we also discuss the class of source terms under consideration, and we establish the key properties of this class when projected onto periodic functions. We then show in Section \[Significant contri\] that the principal contribution to the second-order homogenization arises [from the nearest ($p$th) branch of the dispersion relationship]{}. We further give the asymptotics of latter and the corresponding eigenfunction in Section \[Asymptotic eigenfunction dispersion relation\]. These results, together with the asymptotic expansion of the source term, are used to obtain the sought convergence result on the second-order, high-frequency homogenization in Section \[Higher order U\]. The analytical results are illustrated in Section \[Numerical example\] by way of numerical simulations, performed for an example periodic structure.
Bloch transform and dispersion relationship {#Prelim}
===========================================
In this section, we provide preliminary background on the (Floquet-) Bloch transform and the dispersion relationship characterizing the periodic medium featured in .
Bloch transform in $\mathbb{R}^d$
---------------------------------
The key elements of Bloch transform introduced in the sequel are adapted from [@lechleiter2017floquet] (see also [@kuchment2016overview]). To begin with, we introduce the Wigner-Seitz cell $$\begin{aligned}
\label{wgcell}
W_{\!\boldsymbol{I}}:=\{ \tilde{\bx}: \tilde{\bx} \in \mathbb{R}^d, \, -1/2 \le \tilde{\bx}_{1,\cdots,d} \le 1/2 \} \subset \mathbb{R}^d.\end{aligned}$$ The gives the dual or reciprocal lattice ${\boldsymbol{I}}^*\!:=\{ 2\pi \bj: \bj \in \mathbb{Z}^d \}$ and the Brillouin zone $$\begin{aligned}
\label{Brill}
W_{\!{\boldsymbol{I}}^*}:=\{ 2\pi \tilde{\bk}: \tilde{\bk} \in \mathbb{R}^d, \, -1/2 \le \tilde{\bk}_{1,\cdots,d} \le 1/2 \} \subset \mathbb{R}^d.\end{aligned}$$ With reference to the periodicity matrix $\boldsymbol{I}$ and arbitrary wavenumber $\bk\!\in\!\mathbb{R}^d$, any function $\phi(\bx)$ that satisfies $$\begin{aligned}
\phi(\bx+ \bj) \,=\, e^{i \bk \cdot \bj} \es \phi(\bx), \quad \forall\bx \in \mathbb{R}^d, ~ \bj\in\mathbb{Z}^d.\end{aligned}$$ is called $\bk$-quasiperiodic. In passing, we note that any $\boldsymbol{I}$-periodic function becomes $\bk$-quasiperiodic upon multiplication by $e^{i\bk\cdot \bx}$.
With the above definitions, the Bloch transform $\mathcal{J}_{\mathbb{R}^d}$ of a function $\psi\in C_0^\infty(\mathbb{R}^d)$ is given by $$\begin{aligned}
\label{Def Bloch transform}
\mathcal{J}_{\mathbb{R}^d} \psi (\bk;\bx) = \frac{1}{(2\pi)^{d/2}} \sum_{\bj \in \mathbb{Z}^d} \psi(\bx+ \bj) e^{-i \bk \cdot \bj}, \quad \bk,\bx \in \mathbb{R}^d.\end{aligned}$$ It is readily seen that $\mathcal{J}_{\mathbb{R}^d}$ commutes with $\boldsymbol{I}$-periodic functions; namely $q$ is $\boldsymbol{I}$-periodic, then $$\label{Periodic q commute}
\mathcal{J}_{\mathbb{R}^d} (q\psi) (\bk;\bx) = \frac{1}{(2\pi)^{d/2}} \sum_{\bj \in \mathbb{Z}^d}q(\bx+ \bj) \psi(\bx+ \bj) e^{-i \bk \cdot \bj} = q(\bx)\mathcal{J}_{\mathbb{R}^d} \psi (\bk;\bx).$$
To facilitate the ensuing analysis, we introduce several function spaces. We first recall the Fourier transform $$\begin{aligned}
(\mathcal{F} \psi) (\bz) := \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} e^{-i \bz \cdot \bx} \psi(\bx) \ind \bx\end{aligned}$$ for $\psi \in C_0^\infty(\mathbb{R}^d)$ and $\bz \in \mathbb{R}^d$. This transform extends to an isometry on $L^2(\mathbb{R}^d)$ and defines Bessel potential spaces by $$\begin{aligned}
\mathcal{H}^s(\mathbb{R}^d): = \Big\{ \psi \in \mathcal{D}'(\mathbb{R}^d): \,\int_{\mathbb{R}^d} (1+|\bz|^2)^s \es |(\mathcal{F} \psi) (\bz) |^2 \ind \bz < \infty \Big\}, \quad s \in \mathbb{R}, \end{aligned}$$ [where $\mathcal{D}'(\mathbb{R}^d)$ denotes the space of distributions in $\mathbb{R}^d$.]{}
We next introduce periodic Sobolev spaces. Specifically, we denote by $\mathcal{H}^s_{\bk}(W_{\!\boldsymbol{I}})$ the Hilbert space containing all $\bk$-quasiperiodic distributions (which contains the products of all periodic distributions with $e^{i \bk \cdot \bx}$ [@saranen2013periodic]) with finite norm $$\begin{aligned}
\label{Hsk Fourier norm}
\| \tilde{\psi} \|_{\mathcal{H}^s_{\bk}(W_{\!\boldsymbol{I}})}:=\bigg( \sum_{\bj \in \mathbb{Z}^d} (1+|\bj|^2)^s |c_{\bj}|^2 \bigg)^{1/2} < \infty, \quad s \in \mathbb{R},\end{aligned}$$ where $
c_{\bj} := \int_{W_{\!\boldsymbol{I}}} \tilde{\psi}(\tilde{\bx}) e^{-i \bk \cdot \tilde{\bx}} \es \overline{e^{i 2\pi \bj \cdot \tilde{\bx}}} ~\ind \tilde{\bx}$.
Now we are ready to introduce adapted function spaces in $(\bk;\tilde{\bx})$. We denote by $L^2(W_{\!{\boldsymbol{I}}^*};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))$ the Sobolev space containing distributions in $\mathcal{D}'(\mathbb{R}^d \!\times\! \mathbb{R}^d)$ that are (i) $2\pi\boldsymbol{I}$-periodic in the first variable; (ii) first-variable-quasiperiodic with respect to $\boldsymbol{I}$ in the second variable, and (iii) have a finite norm $$\begin{aligned}
\label{Section Bloch transform eqn 10}
\| \tilde{\psi} \|_{L^2(W_{\!{\boldsymbol{I}}^*};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))}&:=& \bigg(\sum_{\bj \in \mathbb{Z}^d} (1+|\bj|^2)^s \int_{W_{\!{\boldsymbol{I}}^*}} |\hat{\psi}_{\boldsymbol{I}}(\bk,\bj)|^2 \ind {\bk} \bigg)^{1/2} \label{L2Hsk Fourier norm}\\
&=& \bigg( \int_{W_{\!{\boldsymbol{I}}^*}} \|\tilde{\psi}(\bk;\cdot)\|^2_{\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}})} \ind {\bk} \bigg)^{1/2} < \infty, \nonumber \end{aligned}$$ where $\hat{\psi}_{\boldsymbol{I}}(\bk,\bj) = \int_{W_{\!{\boldsymbol{I}}}} \tilde{\psi}(\bk;\tilde{\bx}) e^{-i \bk \cdot \tilde{\bx}} \es \overline{e^{i 2\pi \bj \cdot \tilde{\bx}}} \ind \tilde{\bx}$.
With the above definitions in place, we can state the following lemma [@lechleiter2017floquet Theorem 4, Lemma 7].
\[Bloch isomorphism\]
- The Bloch transform $\mathcal{J}_{\mathbb{R}^d}$ given by extends from $C_0^\infty(\mathbb{R}^d)$ to an isometric isomorphism between $L^2(\mathbb{R}^d)$ and $L^2(W_{\!{\boldsymbol{I}}^*};L^2(W_{\!\boldsymbol{I}}))$ with inverse $$\label{Def inverse Bloch transform}
\big(\mathcal{J}^{-1}_{\mathbb{R}^d} \tilde{\psi} \big) (\bx): = \frac{1}{(2\pi)^{d/2}} \int_{W_{\!{\boldsymbol{I}}^*}} \tilde{\psi}(\bk; \bx-\bj) e^{i \bk \cdot \bj} ~\ind \bk,$$ where $\bj \in \mathbb{Z}^d$ is the translation such that $\bx- \bj \in W_{{\boldsymbol{I}}}$. Further for $s \in \mathbb{R}$, the Bloch transform $\mathcal{J}_{\mathbb{R}^d}$ extends from $C_0^\infty(\mathbb{R}^d)$ to an isomorphism between $\mathcal{H}^s(\mathbb{R}^d)$ and $L^2(W_{\!{\boldsymbol{I}}^*}; \mathcal{H}^s_{\bk}(W_{\!\boldsymbol{I}}))$.
- For $m \in \mathbb{N}$ and $u \in \mathcal{H}^m(\mathbb{R}^d)$, the Bloch transform $\mathcal{J}_{\mathbb{R}^d} u(\bk; \tilde{\bx})$ processes weak partial derivatives with respect to $\tilde{\bx} \in W_{\!\boldsymbol{I}}$ in $L^2(W_{\!\boldsymbol{I}})$ up to order $m \in \mathbb{N}$. If $\balpha \in \mathbb{Z}^d$, $\balpha\ge \boldsymbol{0}$ with $|\balpha|\le m$, then $$\begin{aligned}
\label{Bloch inside derivative}
\partial^\balpha_{\tilde{\bx}} (\mathcal{J}_{\mathbb{R}^d} u) (\bk; \tilde{\bx}) = \mathcal{J}_{\mathbb{R}^d} [ \partial^\balpha_{\bx} u ] (\bk; \tilde{\bx}).\end{aligned}$$
Orthonormal basis and dispersion relationship
---------------------------------------------
For any $\bk \in W_{\!\boldsymbol{I}^*}$, let us introduce the second-order elliptic operator on $W_{\!\boldsymbol{I}}$ $$\begin{aligned}
\mathcal{A}(\bk) =-\frac{1}{\rho}\big(\nabla+i \bk ) \!\cdot\!\big[G (\nabla + i\bk) \big].\end{aligned}$$ From the theory of compact self-adjoint operators [@lions1978asymptotic]\[pp 614–619\] and [@wilcox1978theory], there exists a discrete set of eigenvalues $\{\omega^2_m(\bk)\}_{m=0}^\infty$ and eigenfunctions $\{\phi_m(\bk;\cdot)\}_{m=0}^\infty$ such that $$\begin{aligned}
\label{eigenfunction}
\mathcal{A}(\bk) \phi_m(\bk;\cdot) = \omega^2_m(\bk) \phi_m(\bk;\cdot),\end{aligned}$$ where $\phi_m(\bk;\cdot) \in \mathcal{H}^1(W_{\!\boldsymbol{I}})$; $$\begin{aligned}
\langle \rho \phi_m(\bk;\cdot),\phi_n(\bk;\cdot) \rangle := \int_{ W_{\!\boldsymbol{I}}} \rho(\tilde{\bx}) \phi_m(\bk; \tilde{\bx}) \es \overline{\phi_n(\bk; \tilde{\bx})} \es \ind \tilde{\bx} \:=\: \delta_{mn},\end{aligned}$$ and $\{\phi_m(\bk;\tilde{\bx})\}_{m=0}^\infty$ is a complete orthonormal basis in $L^2(W_{\!\boldsymbol{I}})$. Hereon, we use the notation $\langle \cdot \rangle$ to denote the average of a (possibly tensor-valued) function over $W_{\!\boldsymbol{I}}$.
By analogy to we find that for all $\bk \!\in\! W_{\!\boldsymbol{I}^*}$, an equivalent norm for $\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}})$ can be defined as $$\begin{aligned}
\|\nes| \tilde{\psi} |\nes\|_{\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}})}&:=& \bigg( \sum_{m=0}^\infty (1+\omega_m^2)^s \es |\langle \rho \tilde{\psi} , e^{i\bk \cdot } \phi_m \rangle|^2 \bigg)^{1/2} \\
&=& \Big( \| \tilde{\psi} e^{-i\bk \cdot } \|^2_{L^2(W_{\!\boldsymbol{I}}; \rho)}+ \delta_{s1}\| (\nabla+i\bk) (\tilde{\psi} e^{-i\bk \cdot}) \|^2_{L^2(W_{\!\boldsymbol{I}}; G)} \Big)^{1/2} < \infty, \quad s=0,1, \nonumber\end{aligned}$$ where $L^2(W_{\!\boldsymbol{I}};\rho)$ denotes the $\rho$-weighted $L^2$-space in $W_{\!\boldsymbol{I}}$ with norm $\big(\int_{W_{\!\boldsymbol{I}}} \rho |\cdot|^2 \big)^{\frac{1}{2}}$, and the same notation holds for $L^2(W_{\!\boldsymbol{I}};G)$. As a result, $L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))$ (with $s=0,1$) has equivalent norm given by $$\begin{aligned}
\|\nes| \tilde{\psi} |\nes\|_{L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))}&:=& \bigg( \sum_{m=0}^\infty (1+\omega_m^2)^s \int_{W_{\!{\boldsymbol{I}^*}}} |\langle \rho \tilde{\psi}(\bk;\cdot), e^{i\bk \cdot} \phi_m(\bk;\cdot) \rangle|^2 \ind \bk \bigg)^{1/2} ~~~ \label{Convergence Bloch modes}\\
&=& \bigg( \int_{W_{\!{\boldsymbol{I}^*}}} \|\nes|\tilde{\psi}(\bk;\cdot)|\nes\|^2_{\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}})} \ind {\bk} \bigg)^{1/2} < \infty, \quad s=0,1. \nonumber\end{aligned}$$ The above $\|\nes|\cdot |\nes\|$-norm is equivalent to that given by and allows us to write down the series expansion of $\psi \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^0_{\bk} (W_{\!\boldsymbol{I}}))$ as $$\begin{aligned}
\label{Expansion Bloch modes}
\tilde{\psi}(\bk; \tilde{\bx}) = \sum_{m=0}^\infty \langle \rho \tilde{\psi}(\bk;\cdot), e^{i\bk \cdot} \phi_m(\bk;\cdot) \rangle e^{i \bk \cdot \tilde{\bx}} \phi_m(\bk;\tilde{\bx}),\end{aligned}$$ where the convergence is in the $\|\nes|\cdot |\nes\|$-norm.
For future reference we introduce the real Bloch variety, i.e. the dispersion relationship [@kuchment2016overview], as the set $$\begin{aligned}
\cup_{m=0}^\infty\{ (\bk, \omega_m(\bk)); \, \bk \in W_{\!{\boldsymbol{I}^*}} \},\end{aligned}$$ whose $p$th branch is given by $$\begin{aligned}
\{ (\bk, \omega_p(\bk)); \, \bk \in W_{\!{\boldsymbol{I}^*}} \}, \quad p=0,1,2,\ldots.\end{aligned}$$ In this setting, [the union of all band gaps can be conveniently written as]{} $$\begin{aligned}
\label{bg1}
{\{\omega^2 \!\in \mathbb{R}\!:\, \omega^2 \neq \omega_m^2(\bk),~ \bk \in W_{\!\boldsymbol{I}^*}, ~m\in\mathbb{Z}^+\}.}\end{aligned}$$
Finally we state the Bloch expansion theorem [@lions1978asymptotic pp 616–617] (see also [@wilcox1978theory]), which is a direct consequence of the representation and Lemma \[Bloch isomorphism\].
Let $\psi \in \mathcal{H}^s(\mathbb{R}^d)$ with $s=0,1$. Then $$\begin{aligned}
\psi(\bx) &=& \frac{1}{(2\pi)^{d/2}} \int_{W_{\!{\boldsymbol{I}^*}}} \sum_{m=0}^\infty
\langle \rho [\mathcal{J}_{\mathbb{R}^d} \psi](\bk;\cdot), e^{i\bk \cdot \cdot} \phi_m(\bk;\cdot) \rangle
e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk, \label{Lemma Bloch expansion}\end{aligned}$$ and the Parseval’s identity holds $$\begin{aligned}
\|\psi\|_{\mathcal{H}^s(\mathbb{R}^d)}^2 = \| \mathcal{J}_{\mathbb{R}^d} \psi \|^2_{L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))} \approx \|\nes| \mathcal{J}_{\mathbb{R}^d} \psi |\nes\|^2_{L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))}, \label{Parseval's identity}\end{aligned}$$ where $``\approx"$ means that there exists positive constants $c_1$ and $c_2$ such that $$\begin{aligned}
c_1 \|\nes| \mathcal{J}_{\mathbb{R}^d} \psi |\nes\|^2_{L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))} \leqslant \|\psi\|_{\mathcal{H}^s(\mathbb{R}^d)}^2 \leqslant c_2 \|\nes| \mathcal{J}_{\mathbb{R}^d} \psi |\nes\|^2_{L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))}.\end{aligned}$$
For future reference we note that under the assumption of smooth $G(\bx)$ and $\rho(\bx)$, $\phi_p({\bk};\bx)$ and $\omega_p^2({\bk})$ (where $\omega_p^2({\boldsymbol{0}})$ is simple) have Taylor series expansion in $\bk$ for sufficiently small $\|\bk\|$, see for instance [@wilcox1978theory]. The featured restriction on $G$ and $\rho$ could in principle be relaxed, see for instance [@conca2002bloch] the the of case non-smooth coefficients when $p=0$ (corresponding to the first branch). We also illustrate by numerical examples that our convergence results appear to apply for piecewise-constant coefficients.
Formulation of the problem and Bloch-wave solution {#Bloch expansion}
==================================================
[As examined earlier we pursue a low-wavenumber, second-order homogenized solution $U_2$ of that, when $\Omega$ resides inside a band gap, converges to $U$ according to . Since specifies the union of all band gaps for an $\bI$-periodic medium, from the scaling argument we find that for an $\eps\bI$-periodic domain featured in , we have $$\begin{aligned}
\label{bg2}
\Omega^2 \neq \eps^{-2}\omega_m^2(\bk), \quad \bk\in W_{\!\boldsymbol{I}^*}, ~ m\in\mathbb{Z}^+\end{aligned}$$ as an explicit condition that $\Omega$ resides inside a band gap. In the context of the low-wavenumber assumption, we further restrict the analysis by letting $$\begin{aligned}
\label{bg3}
\Omega^2 = \eps^{-2}\omega^2_p(\boldsymbol{0}) + \sigma\hat{\Omega}^2, \quad \sigma=\pm 1\end{aligned}$$ where (i) $\omega^2_p(\boldsymbol{0})$ is a simple eigenvalue; (ii) $\hat{\Omega}=O(1)$, and (iii) $p$ and $\sigma$ are chosen so that is satisfied. In this setting, we can clearly distinguish between the low frequency case ($p=0$, $\omega_0(\boldsymbol{0})=0$, $\Omega=O(1)$) and its high-frequency counterpart ($p\geqslant 1$, $\Omega=O(\eps^{-2})$).]{}
Using the change of variables $$\label{aux2}
\bx=\eps^{-1}\br, \quad \omega=\eps \es \Omega, \quad u(\bx)=U(\br)$$ we can conveniently rewrite as $$\label{PDE}
-\nabla\!\cdot\!\big(G(\bx)\nabla u \big) - \omega^2\rho(\bx)\hh u ~=~ \eps^2 f(\eps \bx) \qquad\text{in~~}\mathbb{R}^d,$$ where $$\label{drf1}
\omega^2 = \omega_p^2(\boldsymbol{0}) + \eps^2 \sigma\es \hat\Omega^2, \quad \hat{\Omega}=O(1)$$ due to . In what follows, we first use the variational formulation of to obtain $u(\bx)$, and then recover $U(\br)$ via .
Variational formulation
-----------------------
We begin with solutions $u\in C_0^\infty(\mathbb{R}^d)$. This allows us to apply the Bloch transform to , resulting in $$\mathcal{J}_{\mathbb{R}^d} \big[ -\nabla\!\cdot\!\big(G\nabla u\big) - \omega^2\rho\hh u \big] (\bk; \wtx)~=~ \eps^2 \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \qquad\text{in~~} W_{\!\boldsymbol{I}}.$$ With the aid of and , we obtain $$\label{WLambda distributional PDE}
-\nabla\!\cdot\!\big(G(\wtx)\nabla [\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx)]\big) - \omega^2\rho(\wtx)\hh [\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx)]~=~ \eps^2 \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \quad \text{in } W_{\!\boldsymbol{I}}.$$ Let $\psi(\bk;\wtx) \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^1_{\bk} (W_{\!\boldsymbol{I}}))$; then multiplying the above equation by $\overline{\psi}$, integrating with respect to $\wtx$ over $W_{\!\boldsymbol{I}}$, and integrating by parts we obtain $$\begin{gathered}
\label{variational Bloch 1}
\int_{W_{\!\boldsymbol{I}}}G(\wtx)\nabla_{\wtx} [\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx)] \overline{\nabla_{\wtx} \psi(\bk;\wtx)} \ind \wtx - \omega^2 \int_{W_{\!\boldsymbol{I}}}\rho(\wtx)\hh [\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx)] \overline{ \psi(\bk;\wtx)} \ind \wtx \\
~=~ \eps^2 \int_{W_{\!\boldsymbol{I}}} \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \overline{ \psi(\bk;\wtx)} \ind \wtx.\end{gathered}$$ On denoting $\wtu = \mathcal{J}_{\mathbb{R}^d} u$, integration of the above equation w.r.t. $\bk$ over $W_{\!{\boldsymbol{I}^*}}$ yields $$\begin{gathered}
\label{variational Bloch}
\int_{W_{\!{\boldsymbol{I}^*}}}\int_{W_{\!\boldsymbol{I}}}G(\wtx)\nabla_{\wtx} \wtu (\bk; \wtx) \overline{\nabla_{\wtx} \psi(\bk;\wtx)} \ind \wtx \ind \bk - \omega^2 \int_{W_{\!{\boldsymbol{I}^*}}}\int_{W_{\!\boldsymbol{I}}}\rho(\wtx)\hh \wtu (\bk; \wtx) \overline{ \psi(\bk;\wtx)} \ind \wtx \ind \bk \\
~=~ \eps^2 \int_{W_{\!{\boldsymbol{I}^*}}} \int_{W_{\!\boldsymbol{I}}} \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \overline{ \psi(\bk;\wtx)} \ind \wtx \ind \bk. \end{gathered}$$
\[variational Bloch Thm\] If $u \in \mathcal{H}^1(\mathbb{R}^d)$ is a solution to , then $\wtu(\bk; \wtx)=\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx) \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^1_{\bk} (W_{\!\boldsymbol{I}}))$ solves . Conversely, if $\wtu(\bk;\wtx) \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^1_{\bk} (W_{\!\boldsymbol{I}}))$ is the solution to , then $u(\bx)=\big(\mathcal{J}^{-1}_{\mathbb{R}^d} \wtu \big) (\bx) \in \mathcal{H}^1(\mathbb{R}^d)$ solves .
From the above arguments we have shown that, for any $u\in C_0^\infty(\mathbb{R}^d)$ [solving ]{} and $\psi(\bk;\wtx) \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^s_{\bk} (W_{\!\boldsymbol{I}}))$, equation holds. Since $C_0^\infty(\mathbb{R}^d)$ is dense in $\mathcal{H}^1(\mathbb{R}^d)$, and the Bloch transform is an isomorphism between $\mathcal{H}^1(\mathbb{R}^d)$ and $L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^1_{\bk} (W_{\!\boldsymbol{I}}))$, therefore the variational formulation holds.
Conversely, if $\wtu(\bk;\wtx) \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^1_{\bk} (W_{\!\boldsymbol{I}}))$ is the solution to , then $$\begin{aligned}
\int_{W_{\!{\boldsymbol{I}^*}}}\int_{W_{\!\boldsymbol{I}}} \Big( \nabla_{\wtx}\!\cdot\! \big(G\nabla_{\wtx} \wtu(\bk; \wtx)\big) + \omega^2\rho\hh \wtu (\bk; \wtx) + \eps^2 \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \Big) \overline{ \psi(\bk;\wtx)} \ind \wtx \ind \bk=0,\end{aligned}$$ which yields $$\begin{aligned}
\nabla_{\wtx}\!\cdot\! \big(G\nabla_{\wtx} \wtu(\bk; \wtx)\big) + \omega^2\rho\hh \wtu (\bk; \wtx) + \eps^2 \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx)=0\end{aligned}$$ in $L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^{-1}_{\bk} (W_{\!\boldsymbol{I}}))$. Since $f \in L^2(\mathbb{R}^d)$, we have that $\mathcal{J}_{\mathbb{R}^d} f \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^{0}_{\bk} (W_{\!\boldsymbol{I}}))$. Since $G$ is smooth, then by standard regularity, $\wtu \in L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^{2}_{\bk} (W_{\!\boldsymbol{I}}))$. Let $u(\bx)=\big(\mathcal{J}^{-1}_{\mathbb{R}^d} \wtu \big) (\bx)$. From Lemma \[Bloch isomorphism\], $u \in \mathcal{H}^2(\mathbb{R}^d)$. On applying the inverse Bloch transform $\mathcal{J}^{-1}_{\mathbb{R}^d}$ to the above equation and making use of , we find that $$\begin{aligned}
- \nabla\!\cdot\! \big(G\nabla u\big) (\bx)- \omega^2\rho\hh u (\bx) =\eps^2 \big[f(\eps \cdot) \big] (\bx) \end{aligned}$$ in $L^2(\mathbb{R}^d)$, whereby $u(\bx)=\big(\mathcal{J}^{-1}_{\mathbb{R}^d} \wtu \big) (\bx) \in \mathcal{H}^1(\mathbb{R}^d)$ solves .
Since $\omega$ belongs to a band gap, then there exists at most one solution to . Further for each $\bk$, the variational problem in $W_{\!\boldsymbol{I}}$ is uniquely solvable (see for instance [@lions1978asymptotic]). As a result, we obtain the following claim.
Assume that $\omega$ belongs to a band gap. Then for any $f \in L^2(\mathbb{R}^d)$, there exists a unique solution $u \in \mathcal{H}^1(\mathbb{R}^d)$ to .
Bloch expansion of $u$
----------------------
Starting from the variational formulation with test function $\psi(\bk;\wtx) = e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)$, we find $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}}G(\wtx)\nabla_{\wtx} [\mathcal{J}_{\mathbb{R}^d} u(\bk; \wtx)] \overline{\nabla_{\wtx} e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)} \ind\wtx \\
- \omega^2 \int_{W_{\!\boldsymbol{I}}}\rho(\wtx)\hh [\mathcal{J}_{\mathbb{R}^d} u(\bk; \wtx)] \overline{e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)} \ind \wtx
= \eps^2 \int_{W_{\!\boldsymbol{I}}} \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \overline{e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)} \ind \wtx. \end{gathered}$$ Integrating by parts the first term in the above equation and using , one obtains $$\begin{gathered}
\omega_m^2 \!\int_{W_{\!\boldsymbol{I}}} \rho(\wtx)[\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx)] \overline{ e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)} \ind \wtx \,-\, \omega^2 \!\int_{W_{\!\boldsymbol{I}}}\rho(\wtx)\hh [\mathcal{J}_{\mathbb{R}^d} u (\bk; \wtx)] \overline{ e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)} \ind \wtx \nonumber \\
= \eps^2 \int_{W_{\!\boldsymbol{I}}} \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \wtx) \overline{ e^{i\bk \cdot \wtx} \phi_m(\bk;\wtx)} \ind \wtx, \label{Section Bloch expansion 1}\end{gathered}$$ i.e. $$\begin{aligned}
&& (\omega_m^2(\bk)- \omega^2) \langle \rho \mathcal{J}_{\mathbb{R}^d}u (\bk; \cdot), e^{i \bk \cdot}\phi_m(\bk;\cdot) \rangle = \eps^2 \langle \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] (\bk; \cdot), e^{i \bk \cdot}\phi_m(\bk;\cdot) \rangle.\end{aligned}$$ Since $\omega$ is fixed and $\omega^2_m(\bk) -\omega^2\not=0$, we thus obtain $$\begin{aligned}
\mathcal{J}_{\mathbb{R}^d}u (\bk; \wtx) &=& \sum_{m=0}^\infty \langle \rho \mathcal{J}_{\mathbb{R}^d}u(\bk; \cdot), e^{i \bk \cdot}\phi_m(\bk;\cdot) \rangle e^{i \bk \cdot \wtx} \phi_m(\bk;\wtx), \nonumber \\
&=& \sum_{m=0}^\infty \frac{\eps^2 \langle \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] ({\bk}; \cdot), e^{i {\bk} \cdot}\phi_m({\bk};\cdot) \rangle }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \wtx} \phi_m(\bk; \wtx), \quad \wtx \in W_{\!\boldsymbol{I}}, \quad \label{Section Bloch expansion 2}\end{aligned}$$ where the convergence is with respect to the $L^2(W_{\!{\boldsymbol{I}^*}};\mathcal{H}^1_{\bk} (W_{\!\boldsymbol{I}}))$-norm. Here, one may directly compute $$\begin{aligned}
&& \langle \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] ( {\bk}; \cdot), e^{i {\bk} \cdot}\phi_m( ;\cdot) \rangle \\
&=& \frac{1}{(2\pi)^{d/2}} \int_{W_{\!\boldsymbol{I}}} \sum_{\bj \in \mathbb{Z}^d} f(\eps \wtx+\eps \bj) e^{-i {\bk} \cdot \bj} \overline{e^{i {\bk} \cdot \wtx}\phi_m( {\bk};\wtx)}\ind \wtx \\
&=& \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by.\end{aligned}$$ With the above equality, we obtain the unique solution $u \in \mathcal{H}^1(\mathbb{R}^d)$ by applying the inverse Bloch transform to , namely $$\begin{aligned}
\label{u bloch 0}
u(\bx) &=& \frac{1}{(2\pi)^{d/2}} \int_{W_{\!{\boldsymbol{I}}^*}} \mathcal{J}_{\mathbb{R}^d}u(\bk; \bx-\bj) e^{i \bk \cdot \bj} ~\ind \bk \nonumber \\
&=& \frac{1}{(2\pi)^{d/2}} \int_{W_{\!{\boldsymbol{I}^*}}} \sum_{m=0}^\infty \frac{\eps^2 \langle \mathcal{J}_{\mathbb{R}^d} \big[f(\eps \cdot) \big] ( {\bk}; \cdot), e^{i {\bk} \cdot \cdot}\phi_m( {\bk};\cdot) \rangle }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk \nonumber \\
&=&\frac{1}{(2\pi)^{d}} \int_{W_{\!{\boldsymbol{I}^*}}} \sum_{m=0}^\infty \frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk,\end{aligned}$$ where $\bj \in \mathbb{Z}^d$ is a translation such that $\bx- \bj \in W_{{\boldsymbol{I}}}$.
Source term
-----------
The source term $f$ plays an important role in the second-order homogenization of wave motion in periodic media and requires an in-depth treatment. To facilitate the analysis, we assume that $f$ is the Fourier transform of a compactly-supported function (multiplied by the $p$th eigenfunction in the high frequency case). To bring specificity to the discussion, we assume the following.
\[assumption f\] Assuming that the driving frequency $\omega$ is given by , the source term $f$ in has the following form $$\begin{aligned}
\label{source1}
f(\bx) = \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\bk) e^{i\bk\cdot \bx} \ind \bk\Big) \rho(\bx/\eps)\phi_p(\boldsymbol{0}; \bx/\eps),\end{aligned}$$ where $\mathcal{F}^{-1}[F] \in L^2(\mathbb{R}^d) \cap L^1(\mathbb{R}^d)$, $F \in L^2(\mathbb{R}^d) \cap L^1(\mathbb{R}^d)$, and $F$ is compactly supported in some open bounded region $Y \subset \mathbb{R}^d$ with $\boldsymbol{0} \in Y$.
When $p=0$, $\phi_p(\boldsymbol{0}; \bx/\eps)$ is a constant. This implies that $f(\bx)/\rho({\bx/\eps})$ is proportional to the inverse Fourier transform of $F$. We also remark that the compact-support requirement on $F$ could be relaxed, for instance by allowing for sufficiently fast decaying functions. We illustrate this claim by a numerical example in Section 7.
\[almost orthogonality\] Let $\phi$ be a bounded $\boldsymbol{I}$-periodic function. Assume that the Fourier series of $\rho(\bx)\overline{\phi_p(\boldsymbol{0}; \bx)} \phi(\bx)$ converges pointwise almost everywhere, i.e. $$\begin{aligned}
\label{Fourier of phiphi}
\rho(\bx)\overline{\phi_p(\boldsymbol{0}; \bx)} \phi(\bx) \sim \sum_{\bn \in \mathbb{Z}^d} a_{\bn} e^{i 2\pi \bn \cdot \bx}.\end{aligned}$$ Then $$\begin{aligned}
\label{lemma 1 limit}
\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx =\eps^{-d}(2\pi)^{d/2} \sum_{\bn \in \mathbb{Z}^d} \overline{a}_{\bn} F\big(\big(\bk+2\pi \bn\big)/\eps\big).\end{aligned}$$ Further for sufficiently small $\eps$, the following claims hold.\
1. If $\bk \in W_{\!{\boldsymbol{I}^*}} \backslash \overline{\eps Y}$, then $$\begin{aligned}
\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx =0.\end{aligned}$$
2. If $\bk \in \eps Y$, then $$\begin{aligned}
\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx =\eps^{-d} (2\pi)^{d/2} \overline{a}_{\boldsymbol{0}} F(\bk/\eps).\end{aligned}$$
Since $f \in L^2(\mathbb{R}^d) \cap L^1(\mathbb{R}^d)$ and $\phi$ is bounded, we find that $$\begin{aligned}
\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx\end{aligned}$$ is convergent as an improper integral. By , we also have $$\begin{aligned}
&&\hspace{-0.5cm}\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx
=\int_{\mathbb{R}^d} \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\boldeta) e^{i\eps \boldeta\cdot \bx} d\boldeta\Big) \rho(\bx)\phi_p(\boldsymbol{0}; \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx \\
&=& \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^{d}} \int_{\mathbb{R}^d} F(\boldeta) e^{i\eps \boldeta\cdot \bx} \rho(\bx) \phi_p(\boldsymbol{0}; \bx)\overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\boldeta \ind\bx. \end{aligned}$$ Now from the pointwise convergence of the Fourier series expansion of $ \rho(\bx)\overline{\phi_p(\boldsymbol{0}; \bx)}\phi(\bx)$ and the dominated convergence theorem, we conclude that $$\begin{aligned}
&&\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx
= \sum_{n \in \mathbb{Z}^d} \frac{1}{(2\pi)^{d/2}} \overline{a}_{\bn} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} F(\boldeta) e^{i\eps \boldeta\cdot \bx} \overline{e^{i\bk \cdot \bx}} e^{-i 2\pi \bn \cdot \bx} \ind\boldeta \ind\bx \\
&=& \sum_{n \in \mathbb{Z}^d} \overline{a}_{\bn} \int_{\mathbb{R}^d}\mathcal{F}[F](-\eps \bx) e^{-i ( \bk + 2 \pi \bn)\cdot \bx} \ind\bx = \eps^{-d} \sum_{n \in \mathbb{Z}^d} \overline{a}_{\bn} \int_{\mathbb{R}^d}\mathcal{F}[F](\by) e^{i ( \bk + 2 \pi \bn)\cdot \by/\eps} \ind \by \\
&=& \eps^{-d}(2\pi)^{d/2} \sum_{\bn \in \mathbb{Z}^d} \overline{a}_{\bn} F(\frac{\bk+2\pi \bn}{\eps}),\end{aligned}$$ where we have applied a change of variable $\by=-\eps \bx$ in the last two steps. This establishes claim .
Let us now prove part (a). Since $\bk \in W_{\!{\boldsymbol{I}^*}}$, we have $-\pi \leqslant {\bk}_{1,\cdots,d} \leqslant \pi $. We assume that $\bk \not \in \eps Y$ with $\boldsymbol{0} \in Y$; then for sufficiently small $\eps$, ${\bk}\not=\boldsymbol{0}$ and $\bk+2\pi \bn \not=\boldsymbol{0}$ for any $\bn \in \mathbb{Z}^d$. Therefore $\eps^{-1}(\bk+2 \pi \bn) \not \in Y$ for sufficiently small $\eps$. By Assumption \[assumption f\], $F$ is compactly supported in $Y \subset \mathbb{R}^d$, whereby $F(\eps^{-1}(\bk+2 \pi \bn))=0$ for any $\bn \in \mathbb{Z}^d$. By virtue of , we thus obtain $$\begin{aligned}
\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx =0.\end{aligned}$$ Lastly we establish part (b). Since $\bk \in \eps Y$, then for sufficiently small $\eps$, assumption $\eps^{-1}(\bk+ 2 \pi \bn) \in Y$ requires that $\bn=\boldsymbol{0}$. This yields $$\begin{aligned}
\int_{\mathbb{R}^d} f(\eps \bx) \overline{e^{i\bk \cdot \bx} \phi(\bx)} \ind\bx =\eps^{-d}(2\pi)^{d/2} \overline{a}_{\boldsymbol{0}} F(\bk/\eps), \end{aligned}$$ which completes the proof.
In Lemma \[almost orthogonality\], we assumed that the Fourier series of $ \rho(\bx)\overline{\phi_p(\boldsymbol{0};\bx)} \phi(\bx)$ converges pointwise almost everywhere, namely $$\begin{aligned}
\rho(\bx)\overline{\phi_p(\boldsymbol{0};\bx)} \phi(\bx) = \sum_{\bn \in \mathbb{Z}^d} a_{\bn} e^{i \bn \cdot \bx}.\end{aligned}$$ This premise holds for instance if: (i) $\rho(\cdot)\overline{\phi_p(\boldsymbol{0};\cdot)}\phi(\cdot) \!\in H^2_{\boldsymbol{0}}(W_{\!\boldsymbol{I}})$, [or]{} (ii) $\rho(\cdot)\overline{\phi_p(\boldsymbol{0};\cdot)}\phi(\cdot)$ is piecewise smooth in one dimension [@folland2009fourier pp 35]. In the former case, a standard regularity result demonstrates that the Fourier coefficients in satisfy $$\begin{aligned}
\sum_{\bn \in \mathbb{Z}^d} |a_{\bn}|^2 |\bn|^4 < \infty\end{aligned}$$ which, together with Holder inequality, yields $$\begin{aligned}
\sum_{\bn \in \mathbb{Z}^d} |a_{\bn}| \leqslant \big(\sum_{\bn \in \mathbb{Z}^d} |a_{\bn}|^2|\bn|^4\big)^{1/2} \big(\sum_{\bn \in \mathbb{Z}^d} \frac{1}{|\bn|^4}\big)^{1/2} < \infty.\end{aligned}$$ This demonstrates that $\sum_{\bn \in \mathbb{Z}^d} a_{\bn} e^{i \bn \cdot \bx}$ [converges uniformly and pointwise]{}. [When]{} $\rho(\bx)\overline{\phi_p(\boldsymbol{0};\bx)}\phi(\bx) \in H^2_{\boldsymbol{0}}(W_{\!\boldsymbol{I}})$, then by the Sobolev embedding theorem [@adams2003sobolev pp 85–86], $ \rho(\bx)\overline{\phi_p(\boldsymbol{0};\bx)}\phi(\bx)$ is continuous in $W_{\!\boldsymbol{I}}$. [In this case]{} we conclude that $\sum_{n \in \mathbb{Z}^d} a_{\bn} e^{i \bn \cdot \bx}$ converges uniformly and pointwise to the continuous function $\rho(\bx)\overline{\phi_p(\boldsymbol{0};\bx)}\phi(\bx)$.
Reduced Bloch expansion of $u$
------------------------------
On recalling the representation of $u$, we can rewrite the latter as $$\begin{gathered}
u(\bx) = \frac{1}{(2\pi)^{d}} \int_{\eps Y} \sum_{m=0}^\infty \frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk \\
+\frac{1}{(2\pi)^{d}} \int_{W_{\!{\boldsymbol{I}^*}}\backslash \overline{\eps Y}} \sum_{m=0}^\infty \frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk.\end{gathered}$$ Since $G$ and $\rho$ are smooth by premise, a standard regularity result yields $\rho\phi_m(\bk;\cdot) \in H^2_{\boldsymbol{0}}(W_{\!\boldsymbol{I}})$. Then by Lemma \[almost orthogonality\], the second term in the above equation vanishes i.e. $$\label{Section solution u 1}
u(\bx)=\frac{1}{(2\pi)^{d}} \int_{\eps Y} \sum_{m=0}^\infty \frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk.$$ [With reference to ]{}, we next deploy the change of variables $\bk=\eps \hat{\bk}$ and $\by = \bz/\eps$, which yields the solution for $U(\br) = u(\bx)$ as $$\label{Section solution U 1}
U(\br) = \frac{1}{(2\pi)^{d}} \int_{Y} \sum_{m=0}^\infty \frac{\eps^2 \int_{\mathbb{R}^d} f(\bz) e^{-i \hat{\bk} \cdot \bz} \overline{ \phi_m( \eps\hat{\bk};\bz/\eps)}\ind \bz }{\omega^2_m(\eps\hat{\bk}) -\omega^2} e^{i \hat{\bk}\cdot \br} \phi_m(\eps\hat{\bk};\br/\eps) \ind \hat{\bk}.$$
Contributions to the second-order approximation {#Significant contri}
===============================================
[Recalling we are interested in approximating, up to the second order, the wave motion at low wavenumbers and driving frequency (inside a band gap) that is $O(\epsilon^2)$ removed from the nearest $p$th branch. In this section, we demonstrate that such second-order model of , or equivalently , originates completely from the $m=p$ term, while the contribution from all other branches is of higher order, more specifically $O(\eps^3)$.]{}
[Contribution from branches $m\neq p$]{}
----------------------------------------
We first establish the following claim.
\[Thm uothers no contri\] [ Let $\up(\bx)$ and $\Up(\br)$ denote the respective contributions of the (nearest) $p$th branch to $u(\bx)$ given by , and $U(\br)$ given by .]{} For sufficiently small $\eps$, one has $$\label{Thm u - uothers eqn 1}
\|u - \up\|_{L^2(\mathbb{R}^d)} \leqslant {C}\, \eps^{3-d/2} \|F\|_{L^2(\mathbb{R}^d)}$$ where [$C$ is a constant independent of $\eps$]{}, and $$\label{u(p)}
\up(\bx)=
\frac{1}{(2\pi)^{d}} \int_{\eps Y} \frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_p( {\bk};\by)}\ind \by }{\omega^2_p(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_p(\bk;\bx) \ind \bk.$$ Equivalently, $$\begin{aligned}
\|U - \Up(\br)\|_{L^2(\mathbb{R}^d)} \leqslant {C}\, \eps^{3} \|F\|_{L^2(\mathbb{R}^d)} \label{Thm u - uothers eqn 2},\end{aligned}$$ where $\Up(\br) = \up(\br/\eps)$.
Let $$\label{Section contri u others}
u^{\mbox{\tiny ($m\!\neq\!p$)}}(\bx) = \frac{1}{(2\pi)^{d}} \int_{\eps Y} \sum_{m\not=p} \frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by }{\omega^2_m(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_m(\bk;\bx) \ind \bk.$$ In then follows that $u - \up = u^{\mbox{\tiny ($m\!\neq\!p$)}}$.
First, we find from assumption that $$\begin{aligned}
&& \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by \nonumber \\
&=&\int_{\mathbb{R}^d} \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\bk) e^{i\bk\cdot \eps \by} \ind \bk\Big) \rho(\by)\phi_p(\boldsymbol{0}; \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by. \label{Section contri others 0-1}\end{aligned}$$ We now seek to estimate this quantity. Our idea is to replace $\phi_p({\bk};\cdot)$ by $\phi_p(\boldsymbol{0};\cdot)$, and then estimate the difference. Since $\phi_p({\bk};\cdot)$ has a convergent Taylor series in $\bk$, for sufficiently small $\eps$ and $\bk \in \eps Y$ we have $$\begin{aligned}
\phi_p({\bk};\cdot) \:=\: \phi_p(\boldsymbol{0};\cdot) + {\bk} \cdot \boldsymbol{\psi}_p({\bk};\cdot) \end{aligned}$$ for some $\boldsymbol{\psi}_p({\bk};\cdot)$ that satisfies $\|\boldsymbol{\psi}_p({\bk};\cdot)\|_{(L^2(W_{\!\boldsymbol{I}}))^d} \leqslant c$, where $c$ is a constant independent of ${\bk}$. We then obtain $$\begin{aligned}
&& \int_{\mathbb{R}^d} \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\bk) e^{i\bk\cdot \eps \by} \ind \bk\Big) \rho(\by)\phi_p(\boldsymbol{0}; \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by \nonumber \\
&=& \int_{\mathbb{R}^d} \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\bk) e^{i\bk\cdot \eps \by} \ind \bk\Big) \rho(\by)\Big(\phi_p({\bk};\by) - {\bk} \cdot \boldsymbol{\psi}_p({\bk};\by) \Big) \cdot e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by. \nonumber \\ \label{Section contri others 1}\end{aligned}$$ We next apply Lemma \[almost orthogonality\] with $\phi_p(\boldsymbol{0}; \by)$ replaced by $\phi_p({\bk};\by)$ and $\psi_p({\bk};\by)$, respectively. [Recalling the notation $\bk = \eps\hat{\bk}$]{}, one finds this applicable since both $\phi_p(\eps \hat{\bk};\by)$ and $\bk \cdot \boldsymbol{\psi}_p(\eps \hat{\bk};\by) = \phi_p(\eps \hat{\bk};\by) - \phi_p(\boldsymbol{0}; \by)$ belong to $\mathcal{H}^2_{\boldsymbol{0}}(W_{\!\boldsymbol{I}})$. This gives $$\begin{gathered}
\int_{\mathbb{R}^d} \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\bk) e^{i\bk\cdot \eps \by} \ind \bk\Big) \rho(\by) \phi_p({\bk};\by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by \\
= \eps^{-d} (2\pi)^{d/2} \langle {\rho \phi_p( {\bk}; \cdot)} \overline{\phi_m( {\bk};{\cdot})} \rangle F(\bk/\eps) =0 \mbox{ (orthogonality)}, \label{Section contri others 2}\end{gathered}$$ and $$\begin{gathered}
\int_{\mathbb{R}^d} \frac{1}{(2\pi)^{d/2}} \Big(\int_{\mathbb{R}^d} F(\bk) e^{i\bk\cdot \eps \by} \ind \bk\Big) \rho(\by) \big(\bk \cdot \boldsymbol{\psi}_p({\bk};\by) \big) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by \\
= \eps^{-d} (2\pi)^{d/2} \langle {\rho \bk \cdot \boldsymbol{\psi}_p( {\bk}; \cdot)} \overline{\phi_m( {\bk};{\cdot})} \rangle F(\bk/\eps).\label{Section contri others 3}\end{gathered}$$ In this setting, can be estimated via , and ; in particular, we find $$\begin{aligned}
&& \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by \:=\, -\bk \eps^{-d} (2\pi)^{d/2} \langle {\rho \bk\cdot \boldsymbol{\psi}_p( {\bk}; \cdot)} \overline{\phi_m( {\bk};{\cdot})} \rangle F(\bk/\eps). \label{Section contri others 4}\end{aligned}$$
To estimate $\|u^{\mbox{\tiny ($m\!\neq\!p$)}}\|_{L^2(\mathbb{R}^d)}$ according to , we write $$\begin{aligned}
\|u^{\mbox{\tiny ($m\!\neq\!p$)}}\|^2_{L^2(\mathbb{R}^d)} &=& \sum_{m\not=p} \int_{\eps Y} \Big| \frac{1}{(2\pi)^d}\frac{\eps^2 \int_{\mathbb{R}^d} f(\eps \by) e^{-i {\bk} \cdot \by} \overline{ \phi_m( {\bk};\by)}\ind \by }{\omega^2_m(\bk) -\omega^2} \Big|^2 \ind \bk,\end{aligned}$$ and make use of to obtain $$\begin{aligned}
\|u^{\mbox{\tiny ($m\!\neq\!p$)}}\|^2_{L^2(\mathbb{R}^d)} &\leq& \frac{1}{(2\pi)^d}\sum_{m\not=p} \int_{\eps Y} \|\bk\|^2 \Big|\frac{\eps^{2-d} \langle {\rho \boldsymbol{\psi}_p( {\bk}; \cdot)} \overline{\phi_m( {\bk};{\cdot})} \rangle F(\bk/\eps) }{\omega^2_m(\bk) -\omega^2} \Big|^2 \ind \bk \\
&=& \frac{\eps^{4-2d} }{(2\pi)^d} \int_{\eps Y} \|\bk\|^2 \Big| F(\bk/\eps) \Big|^2 \sum_{m\not=p} \Big| \frac{\langle {\rho \boldsymbol{\psi}_p( {\bk}; \cdot)} \overline{\phi_m( {\bk};{\cdot})}\rangle}{\omega^2_m(\bk) -\omega^2} \Big|^2\ind \bk. \\\end{aligned}$$ Since $\omega^2_m(\bk) -\omega^2=O(1)$ for any $m\not=p$, from the last equality we find $$\begin{aligned}
\|u^{\mbox{\tiny ($m\!\neq\!p$)}}\|^2_{L^2(\mathbb{R}^d)} &\leqslant& c' \es \frac{\eps^{4-2d} }{(2\pi)^d} \int_{\eps Y} \|\bk\|^2 \Big| F(\bk/\eps) \Big|^2 \sum_{m\not=p} \Big| \langle {\rho \boldsymbol{\psi}_p( {\bk}; \cdot)} \overline{\phi_m( {\bk};{\cdot})}\rangle \Big|^2\ind \bk \\
&\leqslant& c' \es \frac{\eps^{4-2d} }{(2\pi)^d} \int_{\eps Y} \|\bk\|^2 \Big| F(\bk/\eps) \Big|^2 \|\boldsymbol{\psi}_p( {\bk}; \cdot) \|^2_{(L^2(W_{\!\boldsymbol{I}}))^d} \ind \bk, \\\end{aligned}$$ where $c'$ is a constant independent of $\eps$. On recalling that $\|\boldsymbol{\psi}_p({\bk};\cdot)\|_{(L^2(W_{\!\boldsymbol{I}}))^d} \leqslant c$ for some $c$ independent of ${\bk}$, from the above estimate we obtain $$\begin{aligned}
\|u^{\mbox{\tiny ($m\!\neq\!p$)}}\|{^2}_{L^2(\mathbb{R}^d)}
&\leqslant& c\es c' \, \frac{\eps^{4-2d} }{(2\pi)^d} \int_{\eps Y} \|\bk\|^2 \Big| F(\bk/\eps) \Big|^2 \ind \bk \:=\: c\es c' \frac{\eps^{6-d} }{(2\pi)^d} \int_{Y} |\hat{\bk}|^2 \Big| F(\hat{\bk}) \Big|^2 \ind \hat{\bk} \\
&\leqslant& {C^2} \, \eps^{6-d} \|F\|^2_{L^2(\mathbb{R}^d)},\end{aligned}$$ where $C$ is a constant independent of $\eps$. This establishes claim . Equation then follows immediately from via the change of variable.
Contribution from the $p$th branch
----------------------------------
We begin by recalling the expression for $\up$ given by , and Theorem \[Thm uothers no contri\] which states that $u - \up$ does not contribute to the second-order approximation of $u$. To facilitate the analysis, we next seek to simplify the formula for $\up$.
Thanks to Lemma \[almost orthogonality\], one finds that $$\begin{aligned}
\int_{\mathbb{R}^d} f(\eps \by ) e^{-i {\bk} \cdot \by} \overline{ \phi_p( {\bk};\by)}\ind \by =\eps^{-d} (2\pi)^{d/2} \overline{a}_{\boldsymbol{0}} F(\bk/\eps),\end{aligned}$$ where $\overline{a}_{\boldsymbol{0}} = \langle {\rho \phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( {\bk};{\cdot})} \rangle$. As a result, we obtain $$\begin{aligned}
\label{Section u drive eqn 1}
\up(\bx) &=& \frac{1}{(2\pi)^{d/2}} \int_{\eps Y} \frac{\eps^{2-d} \langle \rho {\phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( {\bk};{\cdot})} \rangle F(\bk/\eps) }{\omega^2_p(\bk) -\omega^2} e^{i\bk\cdot \bx} \phi_p(\bk;\bx) \ind \bk.\end{aligned}$$ Making use of the change of variable $\bk=\eps \hat{\bk}$, one further finds that $$\begin{aligned}
\label{Section u drive eqn 2}
\|u^{\mbox{\tiny ($p$)}}\|^2_{L^2(\mathbb{R}^d)}
\:=\: \Big\|\frac{\eps^{2} \langle \rho {\phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( \eps \hat{\bk};{\cdot})} \rangle F(\hat{\bk}) }{\omega^2_p(\eps \hat{\bk}) -\omega^2}\Big\|^2_{L^2( Y)}. ~~~~~~~\end{aligned}$$ To obtain the second-order asymptotic model of $\up$, one accordingly needs the respective second-order approximations of $\phi_p(\eps \hat{\bk};\cdot)$ and $\omega^2_p(\eps \hat{\bk}) $. This motivates our study in the next section.
Asymptotic expansion of $\phi_p({\bk};\cdot)$ and $\omega^2_p({\bk})$ {#Asymptotic eigenfunction dispersion relation}
=====================================================================
As before, we let $\bk=\eps \hat{\bk}$ where $\eps$ is sufficiently small and $\hat{\bk}=O(1)$. Since $\omega_p({\boldsymbol{0}})$ is simple by premise, one easily verifies that $\omega_p(\eps \hat{\bk}) = \omega_p(-\eps \hat{\bk})$ for sufficiently small $\eps$; then $\frac{\partial^q}{\partial {\eps^q}}\omega_0^2(\eps \hat{\bk})=0$ for any odd $q$. As a result, we can assume the following Taylor series expansions to hold $$\begin{aligned}
\phi_p(\eps \hat{\bk};\bx) ~&=&~ \tilde{w}_0(\bx) \;+\; \eps \hh \tilde{ w}_1(\bx) \;+\; \eps^2\hh \tilde{w}_2(\bx) \;+\; \eps^3\hh \tilde{ w}_3(\bx) \;+\; \cdots, \label{wexp 1}\\
\omega_p^2(\eps \hat{\bk}) ~&=&~ \hat{\omega}_0^2 + \eps^2 \hat{\omega}_2^2 + \eps^4 \hat{\omega}_4^2 + \cdots, \label{wexp 2}\end{aligned}$$ where the convergence of is in $L^2(W_{\!\boldsymbol{I}})$.
For clarity of discussion, we will use a short-hand notation where
- $:$ denotes the contraction between two tensors.
- $\{\boldsymbol{\cdot}\}$ denotes tensor averaging over all index permutations; in particular for an $n$th-order tensor $\boldsymbol{\tau}$, one has $$\label{symtot}
\{\boldsymbol{\tau}\}_{j_1,j_2,\ldots j_n} ~=~ \frac{1}{n!}\sum_{(l_1,l_2,\ldots l_n)\in P} \boldsymbol{\tau}_{l_1,l_2,\ldots l_n}, \qquad j_1,j_2,\ldots j_n \in\overline{1,d}$$ where $P$ denotes the set of all permutations of $(j_1,j_2,\ldots j_n)$.
- $\{\boldsymbol{\cdot}\}'$ denotes partial tensor symmetrization according to $$\label{sympart}
\{\boldsymbol{\tau}\}'_{j_1,j_2,\ldots j_n} ~=~ \frac{1}{(n\!-\!1)!}\sum_{(l_2,\ldots l_n)\in Q} \boldsymbol{\tau}_{j_1,l_2,\ldots l_n}, \qquad j_1,j_2,\ldots j_n \in\overline{1,d}$$ where $Q$ denotes the set of all permutations of $(j_2,j_3,\ldots j_n)$.
Leading-order approximation {#Asymptotic eigenfunction dispersion relation subsection 1}
---------------------------
From , it is clear that $\hat{\omega}_0=\omega_p(\boldsymbol{0})$. From and , on the other hand, we have $$\begin{aligned}
\label{eq:A1}
&&\int_{W_{\!\boldsymbol{I}}} \big[G (\nabla + i \eps \hat{\bk}) \big( \tilde{w}_0 + \eps \hh \tilde{ w}_1 + \eps^2\hh \tilde{w}_2 + \eps^3\hh \tilde{ w}_3 +\cdots \big) \big] \cdot\overline{\big(\nabla+ i \eps \hat{\bk} ) \phi} \ind \bx~~~~~~ \\
&&= \big(\hat{\omega}_0^2 + \eps^2 \hat{\omega}_2^2 + \eps^4 \hat{\omega}_4^2 +\cdots \big) \int_{W_{\!\boldsymbol{I}}} \rho \big( \tilde{w}_0 + \eps \hh \tilde{ w}_1 + \eps^2\hh \tilde{w}_2 + \eps^3\hh \tilde{ w}_3 +\cdots \big) \overline{\phi} \ind \bx, \nonumber \end{aligned}$$ for any $\phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}})$.
The $O(1)$ contribution stemming from reads $$\begin{aligned}
\label{eq:A2}
&&\int_{W_{\!\boldsymbol{I}}} \big[G \nabla \tilde{w}_0\big] \cdot \overline{\nabla\phi} \ind \bx \:=\: \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho \tilde{w}_0 \overline{\phi} \ind \bx, \quad \forall \phi \in {\mathcal{H}^1_{\boldsymbol{0}}}(W_{\!\boldsymbol{I}}), \quad\end{aligned}$$ where ${\mathcal{H}^1_{\boldsymbol{0}}}(W_{\!\boldsymbol{I}})$ is given by , noting that the subscript $\bk=\boldsymbol{0}$ indicates periodicity. As a result, the above equation admits a unique solution $\tilde{w}_0(\bx) = \ww_0 \phi_p(\boldsymbol{0}; \bx)$ where $\ww_0$ is a multiplication constant. Since $\phi_p({\bk}; \bx)=\tilde{w}_0(\bx) + O(\|\bk\|)$ for sufficiently small $\|\bk\|$, it is clear that $\ww_0=1$.
Next, the $O(\eps)$ statement reads $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_1 + i \hat{\bk} \tilde{w}_0 \big) \big] \cdot \overline{\nabla\phi} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \nabla \tilde{w}_0 \cdot i \hat{\bk}\big] \overline{ \phi} \ind \bx \\
= \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho \tilde{w}_1 \overline{\phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$ This equation can be solved by $\tilde{ w}_1(\bx) = \bchi^{\mbox{\tiny{(1)}}} (\bx) \cdot i \hat{\bk}\hh w_0 + \bw_1 \cdot i\hat{\bk}\phi_p(\boldsymbol{0}; \bx)$, where $\bchi^{\mbox{\tiny{(1)}}}\in (H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}))^d$ is a *zero-mean* ($\int_{W_{\!\boldsymbol{I}}} \rho\bchi^{\mbox{\tiny{(1)}}} \overline{\phi_p(\boldsymbol{0}; \cdot)} \ind\bx=\boldsymbol{0}$), vector-valued function solving $$\begin{gathered}
\label{ComparisonEqnchi1}
\int_{W_{\!\boldsymbol{I}}} \big( G (\nabla \bchi^{\mbox{\tiny{(1)}}} \!+\boldsymbol{I}\phi_p(\boldsymbol{0}; \cdot)\big) : \overline{\nabla\phi} \ind \bx - \int_{W_{\!\boldsymbol{I}}} \big( G\boldsymbol{I} \nabla \phi_p(\boldsymbol{0}; \cdot) \big) \overline{\phi} \ind \bx \\
= \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho\bchi^{\mbox{\tiny{(1)}}} \overline{ \phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$ We remark that the above equation for $\bchi^{\mbox{\tiny{(1)}}} $ is solvable since $\phi_p(\boldsymbol{0}; \cdot)$ has a constant phase thanks to the assumption that $\omega_p^2(\boldsymbol{0})$ is a simple eigenvalue, see for instance [@guzina2019rational].
The $O(\eps^{2})$ contribution stemming from reads $$\begin{gathered}
\label{w2}
\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_2 + i \hat{\bk} \tilde{w}_1\big) \big] \cdot \overline{\nabla\phi} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_1 + i\hat{\bk} \tilde{w}_0\big) \cdot i \hat{\bk}\big] \overline{ \phi} \ind \bx \\
= \int_{W_{\!\boldsymbol{I}}} \rho \big[ \hat{\omega}_0^2 \tilde{w}_2+ \hat{\omega}_2^2 \tilde{w}_0 \big] \overline{\phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$ Taking $\phi(\bx)=\phi_p(\boldsymbol{0}; \bx)$ and deploying integration by parts, we find $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big[G i \hat{\bk} \tilde{w}_1 \big] \cdot \overline{\nabla\phi_p(\boldsymbol{0}; \cdot)} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_1 + i\hat{\bk} \tilde{w}_0 \big) \cdot i \hat{\bk}\big] \overline{ \phi_p(\boldsymbol{0}; \cdot)} \ind \bx ~~~~ \nonumber\\
=\; \hat{\omega}_2^2 \int_{W_{\!\boldsymbol{I}}} \rho \tilde{w}_0(\cdot) \overline{\phi_p(\boldsymbol{0}; \cdot)} \ind \bx.\end{gathered}$$ On substituting – into the above equation, all the terms involving $\bw_1$ vanish due to the fact that $\phi_p(\boldsymbol{0}; \cdot)$ has a constant phase. Accordingly, we obtain $$\begin{aligned}
\label{ComparisonSecondOrderHomogenizationZerothOrderPDE}
\rho^{\mbox{\tiny{(0)}}} \hat{\omega}^2_2 = -{\bmu}^{\mbox{\tiny{(0)}}} \!: (i\hat{\bk})^2,\end{aligned}$$ where $$\begin{aligned}
\rho^{\mbox{\tiny{(0)}}} &=& {\alpha_p}\es \langle \rho \phi_p(\boldsymbol{0}; \cdot) \overline{\phi_p(\boldsymbol{0}; \cdot) } \rangle, \label{SecondOrderHomogenizationZerothOrderCoefficients-0}\\
{\bmu}^{\mbox{\tiny{(0)}}} &=& {\alpha_p}\es \big\langle\{G( \nabla \bchi^{\mbox{\tiny{(1)}}} + \boldsymbol{I} \phi_p(\boldsymbol{0}; \cdot) ) \overline{ \phi_p(\boldsymbol{0}; \cdot) }\} -
\{G \bchi^{\mbox{\tiny{(1)}}} \otimes \overline{\nabla \phi_p(\boldsymbol{0}; \cdot)}\} \big\rangle, \label{SecondOrderHomogenizationZerothOrderCoefficients}\end{aligned}$$ [and $$\alpha_p = \langle\phi_p(\boldsymbol{0}; \cdot) \overline{\phi_p(\boldsymbol{0};\cdot)}\rangle^{-1}.$$]{}
First-order corrector {#Asymptotic eigenfunction dispersion relation subsection 2}
---------------------
Let $\bchi^{\mbox{\tiny{(2)}}}\in (H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}))^{d\times d}$ denote a *zero-mean* (i.e. $\int_{W_{\!\boldsymbol{I}}} \rho \bchi^{\mbox{\tiny{(2)}}} \overline{\phi_p(\boldsymbol{0}; \cdot)} \ind\bx=\boldsymbol{0}$), tensor-valued function satisfying
$$\begin{gathered}
\label{Comparisonchi2}
\int_{W_{\!\boldsymbol{I}}} \big( G (\nabla \bchi^{\mbox{\tiny{(2)}}} \!+ \{\boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(1)}}}\}' \big) : \overline{\nabla\phi} \ind \bx - \int_{W_{\!\boldsymbol{I}}} G(\{\nabla \bchi^{\mbox{\tiny{(1)}}}\} + \boldsymbol{I} \phi_p(\boldsymbol{0};\cdot)) \overline{\phi} \ind \bx \\
+ \int_{W_{\!\boldsymbol{I}}} \frac{\rho}{\rho^{\mbox{\tiny{(0)}}}} \bmu^{\mbox{\tiny{(0)}}} \phi_p(\boldsymbol{0};\cdot) \overline{\phi} \ind \bx \:=\: \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho\bchi^{\mbox{\tiny{(2)}}} \overline{ \phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$
It is directly verified that the above equation is uniquely solvable thanks to . With such definitions, equation is solved by $\,\tilde{w}_2 (\bx) \,=\, \bchi^{\mbox{\tiny{(2)}}} (\bx) : (i \hat{\bk})^{2}\hh w_0 \,+\, \big(\bchi^{\mbox{\tiny{(1)}}} (\bx) \cdot i\hat{\bk}\big) \big(\bw_1 \cdot i \hat{\bk} \big) \, +\, \bw_2 : (i \hat{\bk})^2 \phi_p(\boldsymbol{0};\bx)$.
Proceeding with the asymptotic analysis, the $O(\eps^3)$ contribution stemming from is found as $$\begin{gathered}
\label{w3}
\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_3 + i \hat{\bk} \tilde{w}_2\big) \big] \cdot \overline{\nabla\phi} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_2 + i\hat{\bk} \tilde{w}_1\big) \cdot i \hat{\bk}\big] \overline{ \phi} \ind \bx \\
= \int_{W_{\!\boldsymbol{I}}} \rho \big[ \hat{\omega}_0^2 \tilde{w}_3+ \hat{\omega}_2^2 \tilde{w}_1 \big] \overline{\phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$ In a manner similar to earlier treatment, taking $\phi(\bx)=\phi_p(\boldsymbol{0}; \bx)$ we find $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big[G i \hat{\bk} \tilde{w}_2 \big] \cdot \overline{\nabla\phi_p(\boldsymbol{0}; \cdot)} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_2 + i\hat{\bk} \tilde{w}_1 \big) \cdot i \hat{\bk}\big] \overline{ \phi_p(\boldsymbol{0}; \cdot)} \ind \bx ~~~~ \nonumber\\
=\; \hat{\omega}_2^2 \int_{W_{\!\boldsymbol{I}}} \rho \tilde{w}_1(\cdot) \overline{\phi_p(\boldsymbol{0}; \cdot)} \ind \bx.\end{gathered}$$ On substituting – into the above equation, all the terms involving $\bw_2$ vanish due to the fact that $\phi_p(\boldsymbol{0}; \cdot)$ has a constant phase. In this way, we obtain $$\begin{aligned}
\label{ComparisonSecondOrderHomogenizationFirstOrderPDE-1}
\big( \bmu^{\mbox{\tiny{(1)}}}\!: (i \hat{\bk})^{3} + {\brho}^{\mbox{\tiny{(1)}}} \!\cdot\! i\hat{\bk} ~\hat{\omega}^2_2 \big)w_0 \,+\, \big( {\bmu}^{\mbox{\tiny{(0)}}}\!: (i \hat{\bk})^{2}+ \rho^{\mbox{\tiny{(0)}}} \hat{\omega}^2_2 \big) \bw_1\cdot i\hat{\bk} ~=\: 0,\end{aligned}$$ where $$\begin{aligned}
\brho^{\mbox{\tiny{(1)}}} &=& {\alpha_p}\es \langle \rho \bchi^{\mbox{\tiny{(1)}}} \overline{\phi_p(\boldsymbol{0}; \cdot) } \rangle =\boldsymbol{0}, \label{SecondOrderHomogenizationFirstOrderCoefficients}\\
{\bmu}^{\mbox{\tiny{(1)}}} &=& {\alpha_p}\es \big \langle \{G( \nabla \bchi^{\mbox{\tiny{(2)}}} + \boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(1)}}} ) \overline{ \phi_p(\boldsymbol{0}; \cdot) } \} -
\{G \bchi^{\mbox{\tiny{(2)}}}\otimes \overline{\nabla \phi_p(\boldsymbol{0}; \cdot) }\} \big \rangle. \end{aligned}$$
\[rem3\] It can be shown that $ {\bmu}^{\mbox{\tiny{(1)}}} = \boldsymbol{0}$. Indeed, with $\phi$ taken as the $k\ell$th component of $\overline{\bchi^{\mbox{\tiny{(2)}}}}$ yields $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big( G (\nabla \bchi^{\mbox{\tiny{(1)}}} \!+\boldsymbol{I}\phi_p(\boldsymbol{0}; \cdot)\big) : {\nabla\bchi^{\mbox{\tiny{(2)}}}_{k\ell} } \ind \bx - \int_{W_{\!\boldsymbol{I}}} \big( G\boldsymbol{I} \nabla \phi_p(\boldsymbol{0}; \cdot) \big) \bchi^{\mbox{\tiny{(2)}}}_{k\ell} \ind \bx \\
=\; \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho\bchi^{\mbox{\tiny{(1)}}} \bchi^{\mbox{\tiny{(2)}}}_{k\ell} \ind \bx, \label{mu1 vanish remark eqn 1}\end{gathered}$$ while with $\phi$ being the $j$th component of $\overline{\bchi^{\mbox{\tiny{(1)}}}_j}$ gives $$\begin{aligned}
&&\int_{W_{\!\boldsymbol{I}}} \big( G (\nabla \bchi^{\mbox{\tiny{(2)}}} \!+ \{\boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(1)}}}\}' \big) : {\nabla \bchi^{\mbox{\tiny{(1)}}}_j} \ind \bx - \int_{W_{\!\boldsymbol{I}}} G(\{\nabla \bchi^{\mbox{\tiny{(1)}}}\} + \boldsymbol{I} \phi_p(\boldsymbol{0};\cdot)) {\bchi^{\mbox{\tiny{(1)}}}_j} \ind \bx \nonumber \\
&& + \int_{W_{\!\boldsymbol{I}}} \frac{\rho}{\rho^{\mbox{\tiny{(0)}}}} \bmu^{\mbox{\tiny{(0)}}} \phi_p(\boldsymbol{0};\cdot) {\bchi^{\mbox{\tiny{(1)}}}_j} \ind \bx= \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho\bchi^{\mbox{\tiny{(2)}}} { \bchi^{\mbox{\tiny{(1)}}}_j}\ind \bx. \label{mu1 vanish remark eqn 2}\end{aligned}$$ Taking the difference of and , we obtain $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big[ \{G( \nabla \bchi^{\mbox{\tiny{(2)}}} + \boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(1)}}} ) { \phi_p(\boldsymbol{0}; \cdot) } \}-
\{G \bchi^{\mbox{\tiny{(2)}}}\otimes {\nabla \phi_p(\boldsymbol{0}; \cdot) }\} \big] \ind \bx \nonumber \\
=\: \frac{1}{\rho^{\mbox{\tiny{(0)}}}} \Big\{ \big(\int_{W_{\!\boldsymbol{I}}} \rho \bchi^{\mbox{\tiny{(1)}}} {\phi_p(\boldsymbol{0}; \cdot) } \ind \bx \big)\otimes {\bmu^{\mbox{\tiny{(0)}}}} \Big\}. \end{gathered}$$ Since $ {\phi_p(\boldsymbol{0}; \cdot) }$ has a constant phase, the right-hand side of the above equation [is proportional to]{} $\brho^{\mbox{\tiny{(1)}}}=\boldsymbol{0}$ and thus vanishes. By the same (constant-phase) argument, the left-hand side is a constant multiplication of ${\bmu}^{\mbox{\tiny{(1)}}} $ and hence $$\label{w1zero}
{\bmu}^{\mbox{\tiny{(1)}}} =\boldsymbol{0}.$$
From Remark \[rem3\] and , we obtain the balance statement $$\begin{aligned}
\label{ComparisonSecondOrderHomogenizationFirstOrderPDE}
\big( {\bmu}^{\mbox{\tiny{(0)}}}\!: (i \hat{\bk})^{2}+ \rho^{\mbox{\tiny{(0)}}} \hat{\omega}^2_2 \big) \bw_1\cdot i\hat{\bk} ~=\: 0,\end{aligned}$$ that is satisfied by constant vector $\bw_1$ thanks to .
Second-order corrector {#Asymptotic eigenfunction dispersion relation subsection 3}
----------------------
Let $\bchi^{\mbox{\tiny{(3)}}}\in (H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}))^{d\times d \times d}$ be the *zero-mean* (i.e. $\int_{W_{\!\boldsymbol{I}}} \rho \bchi^{\mbox{\tiny{(3)}}} \overline{\phi_p(\boldsymbol{0}; \cdot)} \ind\bx=\boldsymbol{0}$) tensor-valued function satisfying $$\begin{gathered}
\label{Comparisonchi3}
\int_{W_{\!\boldsymbol{I}}} \big( G (\nabla \bchi^{\mbox{\tiny{(3)}}} \!+ \{\boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(2)}}}\}' \big) : \overline{\nabla\phi} \ind \bx - \int_{W_{\!\boldsymbol{I}}} G\big(\{\nabla \bchi^{\mbox{\tiny{(2)}}}\} +\{\boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(1)}}}\} \big)\overline{\phi} \ind \bx \\
+ \int_{W_{\!\boldsymbol{I}}} \frac{1}{\rho^{\mbox{\tiny{(0)}}}}\{\rho \bchi^{\mbox{\tiny{(1)}}} \otimes \bmu^{\mbox{\tiny{(0)}}}\}\overline{\phi} \ind \bx \:=\: \hat{\omega}_0^2 \int_{W_{\!\boldsymbol{I}}} \rho\bchi^{\mbox{\tiny{(3)}}} \overline{ \phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$ One easily verifies that the above equation is uniquely solvable due to and . With such definitions, is solved by $\,\tilde{w}_3 (\bx) \,=\, \bchi^{\mbox{\tiny{(3)}}} (\bx) : (i \hat{\bk})^{3}\hh w_0 \,+\, \big(\bchi^{\mbox{\tiny{(2)}}} (\bx) : (i \hat{\bk})^{2} \big)\hh \bw_1\cdot i \hat{\bk} \,+\, \big(\bchi^{\mbox{\tiny{(1)}}} (\bx) \cdot i\hat{\bk}\big) \big(\bw_2 \cdot i \hat{\bk}^2 \big) \, +\, \bw_3 : (i \hat{\bk})^3 \phi_p(\boldsymbol{0};\cdot)$.
To complete the second-order analysis, one must consider the $O(\eps^4)$ contribution to which reads $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_4 + i \hat{\bk} \tilde{w}_3\big) \big] \cdot \overline{\nabla\phi} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_3 + i\hat{\bk} \tilde{w}_2\big) \cdot i \hat{\bk}\big] \overline{ \phi} \ind \bx ~~~~ \nonumber\\
=\, \int_{W_{\!\boldsymbol{I}}} \rho \big[ \hat{\omega}_0^2 \tilde{w}_4+ \hat{\omega}_2^2 \tilde{w}_2+ \hat{\omega}_4^2 \tilde{w}_0 \big] \overline{\phi} \ind \bx, \quad \forall \phi \in H^1_{\boldsymbol{0}}(W_{\!\boldsymbol{I}}).\end{gathered}$$ Taking $\phi(\bx)=\phi_p(\boldsymbol{0}; \bx)$ in the above equation yields $$\begin{gathered}
\int_{W_{\!\boldsymbol{I}}} \big[G i \hat{\bk} \tilde{w}_3 \big] \cdot \overline{\nabla\phi_p(\boldsymbol{0}; \cdot)} \ind \bx -\int_{W_{\!\boldsymbol{I}}} \big[G \big(\nabla \tilde{w}_3 + i\hat{\bk} \tilde{w}_2 \big) \cdot i \hat{\bk}\big] \overline{ \phi_p(\boldsymbol{0}; \cdot)} \ind \bx ~~~~ \nonumber\\
=\, \int_{W_{\!\boldsymbol{I}}} \rho \big[ \hat{\omega}_2^2 \tilde{w}_2+ \hat{\omega}_4^2 \tilde{w}_0 \big]\overline{\phi_p(\boldsymbol{0}; \cdot)} \ind \bx.\end{gathered}$$ On inserting – into the above equation, we find that all the terms involving $\bw_1$ must vanish because $\phi_p(\boldsymbol{0}; \cdot)$ has a constant phase, while all the terms containing $\bw_3$ are necessarily trivial due to and . As a result, we obtain $$\label{ComparisonSecondOrderHomogenizationSecondOrderPDE}
\big( {\bmu}^{\mbox{\tiny{(2)}}} \!: (i\hat{\bk})^4 + {\brho}^{\mbox{\tiny{(2)}}} \!: (i\hat{\bk})^2 ~ \hat{\omega}^2_2 + \rho^{\mbox{\tiny{(0)}}} \hat{\omega}_4^2 \big) w_0 +
\big( {\bmu}^{\mbox{\tiny{(0)}}} \!: (i\hat{\bk})^2 + \rho^{\mbox{\tiny{(0)}}} \hat{\omega}_2^2 \big) \bw_2 : (i\hat{\bk})^2 =0,~~~~$$ where $$\begin{aligned}
\brho^{\mbox{\tiny{(2)}}} &=& {\alpha_p}\es \langle \rho \bchi^{\mbox{\tiny{(2)}}} \overline{\phi_p(\boldsymbol{0}; \cdot) } \rangle=\boldsymbol{0}, \label{SecondOrderHomogenizationSecondOrderCoefficients-0}\\
{\bmu}^{\mbox{\tiny{(2)}}} &=& {\alpha_p}\es \big\langle \{G( \nabla \bchi^{\mbox{\tiny{(3)}}} + \boldsymbol{I} \otimes \bchi^{\mbox{\tiny{(2)}}} ) \overline{ \phi_p(\boldsymbol{0}; \cdot) } \} -
\{G \bchi^{\mbox{\tiny{(3)}}}\otimes \overline{\nabla \phi_p(\boldsymbol{0}; \cdot) }\} \big\rangle. \label{SecondOrderHomogenizationSecondOrderCoefficients}\end{aligned}$$ From and , we also find that $$\begin{aligned}
{\bmu}^{\mbox{\tiny{(2)}}} \!: (i\hat{\bk})^4 + \rho^{\mbox{\tiny{(0)}}} \hat{\omega}_4^2 =0. \end{aligned}$$
With the following lemma, we synthesize the results of asymptotic expansion.
\[asymptotic main phi\_0\] The truncated Taylor series expansions and are given respectively by $$\begin{aligned}
\tilde{w}_0(\bx) &=& \ww_0 \phi_p(\boldsymbol{0};\bx), \label{wexp eigenfunction w0}\\
\tilde{ w}_1(\bx) &=& \bchi^{\mbox{\tiny{(1)}}} (\bx) \cdot i \hat{\bk}\hh \ww_0 + \bw_1 \cdot i\hat{\bk} \phi_p(\boldsymbol{0};\bx), \label{wexp eigenfunction w1}\\
\tilde{ w}_2 (\bx) \,&=&\, \bchi^{\mbox{\tiny{(2)}}} (\bx) : (i \hat{\bk})^{2}\hh \ww_0 \,+\, \bchi^{\mbox{\tiny{(1)}}} (\bx) \otimes \bw_1 : (i \hat{\bk})^2 \, +\, \bw_2 : (i \hat{\bk})^2 \phi_p(\boldsymbol{0};\bx), \label{wexp eigenfunction w2}\\
\tilde{ w}_3 (\bx) &=& \bchi^{\mbox{\tiny{(3)}}} (\bx): (i \hat{\bk})^3 w_0 + (\bchi^{\mbox{\tiny{(2)}}} (\bx) : (i \hat{\bk})^2) \bw_1\cdot i \hat{\bk} \nonumber \\
&& +\bchi^{\mbox{\tiny{(1)}}}(\bx) \cdot i\hat{\bk}\hh (\bw_2 : (i \hat{\bk})^2) + \bw_3: (i \hat{\bk})^3\phi_p(\boldsymbol{0};\bx),\label{wexp eigenfunction w3}\end{aligned}$$ and $$\begin{aligned}
\label{wexp dispersion}
\hat{\omega}_0^2=\omega_p^2(\boldsymbol{0}),\hspace{0.5cm}\hat{\omega}_2^2 = -\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2,\hspace{0.5cm} \hat{\omega}_4^2 = \frac{1}{\rho^{\mbox{\tiny{(0)}}}}\big(\{\frac{{\brho}^{\mbox{\tiny{(2)}}}}{\rho^{\mbox{\tiny{(0)}}}} \otimes {\bmu}^{\mbox{\tiny{(0)}}}\} - {\bmu}^{\mbox{\tiny{(2)}}}\big) \!: (i\hat{\bk})^4, \qquad \end{aligned}$$ where constants $\ww_0$, $\bw_1$ solving , and $\bw_2$ solving satisfy the necessary conditions $$\begin{aligned}
\ww_0 &=& 1, \label{necessary w0}\\
i \hat{\bk} \cdot ( \overline{\bw}_1- \bw_1)&=&0, \label{necessary w1}\\
(i\hat{\bk})^2:\Big[ (\bw_2 + \overline{\bw}_2 ) - (\bw_1 \otimes \overline{\bw}_1) - \langle \rho \bchi^{(1)} \otimes \overline{\bchi^{(1)}} \rangle \Big]&=&0; \quad \label{necessary w2}\end{aligned}$$ $\bchi^{\mbox{\tiny{(1)}}} $ is given by ; $\rho^{\mbox{\tiny{(0)}}}$ and ${\bmu}^{\mbox{\tiny{(0)}}}$ are given by ; $ \bchi^{\mbox{\tiny{(2)}}} $ is given by ; $\brho_2$ and ${\bmu}^{\mbox{\tiny{(2)}}}$ are given by , and $\bchi^{\mbox{\tiny{(3)}}}$ is given by .
We will show shortly that the necessary conditions – are sufficient to derive the second-order asymptotic model.
\[Asymptotic eigenfunction dispersion relation subsection 4\] From the analysis in Sections \[Asymptotic eigenfunction dispersion relation subsection 1\]–\[Asymptotic eigenfunction dispersion relation subsection 3\], [we see that – apply with $w_0=1$, and]{} $$\begin{aligned}
\hat{\omega}_0^2=\omega_p^2(\boldsymbol{0}),\hspace{0.5cm}\hat{\omega}_2^2 &=& -\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2, \hspace{0.5cm}\hat{\omega}_4^2 = -\frac{{\bmu}^{\mbox{\tiny{(2)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^4. \end{aligned}$$
Now we [demonstrate and ]{} by the normalization of eigenfunctions. Specifically, the normalization of $\phi_p(\eps\hat{\bk};\bx)$ yields $$\begin{aligned}
\int_{\mathbb{T}^d} \rho(\bx) \phi_p(\eps\hat{\bk};\bx) \overline{\phi_p(\eps\hat{\bk};\bx)} \ind \bx \:=\: 1.\end{aligned}$$ From the asymptotic expansion of $\phi_p$ according to , we find that the $O(\eps)$ term gives $$\begin{aligned}
i \hat{\bk} \cdot (w_0 \overline{\bw}_1-\overline{w}_0 \bw_1) \rho^{\mbox{\tiny{(0)}}}=i \hat{\bk} |w_0|^2 \cdot \big( {\brho}^{\mbox{\tiny{(1)}}} - \overline{ {\brho}^{\mbox{\tiny{(1)}}} } \big) = 0,\end{aligned}$$ and hence $$\begin{aligned}
i \hat{\bk} \cdot ( \overline{\bw}_1- \bw_1)=0.\end{aligned}$$ Similarly, the $O(\eps^2)$ term gives $$\begin{aligned}
(i\hat{\bk})^2:\big[ (\bw_2 + \overline{\bw}_2 ) - (\bw_1 \otimes \overline{\bw}_1) - \langle \rho \bchi^{(1)} \otimes \overline{\bchi^{(1)}} \rangle \big]=0, \end{aligned}$$ which completes the proof.
Main result of the second-order homogenization {#Higher order U}
==============================================
[Let us first recall the key results established so far. As shown in Theorem \[Thm uothers no contri\], $u - \up$ does not contribute to the second-order approximation of $u$. Consequently, the second-order approximation of $u$ is given by that of $\up$ specified in ]{}.
To expand , we substitute the asymptotic expansion of $\phi_p(\eps \hat{\bk};\cdot)$ given by and Theorem \[asymptotic main phi\_0\] in order to simplify the norm . We first calculate $$\begin{gathered}
\langle \rho {\phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( \eps \hat{\bk};{\cdot})} \rangle \phi_p(\eps \hat{\bk};\cdot) \\
=\Big[\langle \rho {\phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( \boldsymbol{0};{\cdot})} \rangle \big( \overline{w}_0 + \eps \overline{\bw_1 \cdot i \hat{\bk}} + \eps^2 \overline{\bw_2 \cdot (i \hat{\bk})^2 } \big) \Big] \cdot \Big[ w_0\phi_p(\boldsymbol{0}; \cdot) \\
+ \eps \bchi^{\mbox{\tiny{(1)}}} \cdot i \hat{\bk}\hh w_0 + \eps \bw_1 \cdot i\hat{\bk}\phi_p(\boldsymbol{0}; \cdot)
+ \eps^2 \bchi^{\mbox{\tiny{(2)}}} \!:\! (i \hat{\bk})^{2}\hh w_0 \,+\, \eps^2\bchi^{\mbox{\tiny{(1)}}} \otimes \bw_1 \!:\! (i \hat{\bk})^2 \, +\, \eps^2\bw_2 \!:\! (i \hat{\bk})^2 \phi_p(\boldsymbol{0}; \cdot) \Big] \\
= |w_0|^2 \phi_p(\boldsymbol{0}; \cdot) + \eps \Big( |w_0|^2\bchi^{\mbox{\tiny{(1)}}} \cdot i \hat{\bk} + (-w_0 \overline{\bw}_1 +\overline{w}_0 \bw_1)\cdot i \hat{\bk} \phi_p(\boldsymbol{0}; \cdot) \Big) \\
+ \eps^2 (i \hat{\bk})^2 \!:\! \Big( (\bw_2 \overline{w}_0 + \overline{\bw}_2 w_0 )\phi_p(\boldsymbol{0}; \cdot)-\bw_1 \otimes \overline{\bw}_1 \phi_p(\boldsymbol{0}; \cdot)+ (-w_0 \overline{\bw}_1 +\overline{w}_0 \bw_1)\otimes \bchi^{\mbox{\tiny{(1)}}} + |w_0|^2 \bchi^{\mbox{\tiny{(2)}}} \Big).\end{gathered}$$ Using the necessary conditions – , we can simplify the above equation as $$\begin{gathered}
\langle \rho {\phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( \eps \hat{\bk};{\cdot})} \rangle \phi_p(\eps \hat{\bk};\cdot) \\
=\: \phi_p(\boldsymbol{0}; \cdot) + \eps i \hat{\bk} \cdot \bchi^{\mbox{\tiny{(1)}}} + \eps^2 (i \hat{\bk})^2 : \Big( \langle \rho \bchi^{(1)} \otimes \overline{\bchi^{(1)}} \rangle \phi_p(\boldsymbol{0}; \cdot)+ \bchi^{\mbox{\tiny{(2)}}} \Big). \label{Section Higher order eqn 1}\end{gathered}$$ Now we are ready to derive the asymptotic model for $\up$. The leading-order approximation of $\up$ according to is easily obtained by the leading-order asymptotic in (with $\hat{\omega}_2^2$ given by ) and by setting $\epsilon=0$ in , namely $$\begin{aligned}
\up(\bx) &\:=\:& \frac{1}{(2\pi)^{d/2}} \int_{Y} \frac{ \langle \rho {\phi_p(\boldsymbol{0}; \cdot)} \overline{\phi_p( \eps \hat{\bk};{\cdot})} \rangle F(\hat{\bk}) e^{i\eps \hat{\bk} \cdot \bx} \phi_p(\eps \hat{\bk};\bx)}{-\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2 -{\hat{\Omega}}^2} \ind \hat{\bk} \nonumber \\
&\:\approx\:& \frac{1}{(2\pi)^{d/2}} \int_{Y} \frac{ \phi_p(\boldsymbol{0}; \cdot) F(\hat{\bk}) e^{i\eps \hat{\bk} \cdot \bx} }{-\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2 -{\hat{\Omega}}^2} \ind \hat{\bk} \;:=\: \up_0(\bx), \label{Main leading-order u0}\end{aligned}$$ As a result, the leading order approximation of $\Up$ is given by $$\begin{aligned}
\label{Main leading-order U0}
\Up_0(\br) &:=& \up_0(\br/\eps)
= \phi_p(\boldsymbol{0}; \br/\eps) \, W_0({\br}), \end{aligned}$$ where the envelope function $W_0$ reads $$\begin{aligned}
{W_0(\br):=\frac{1}{(2\pi)^{d/2}} \int_{Y} \frac{ F(\hat{\bk}) e^{i \hat{\bk} \cdot \br} }{-\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2 -{\hat{\Omega}}^2} \ind \hat{\bk}} \end{aligned}$$ and satisfies the field equation $$\begin{aligned}
\label{Main envelop W0}
{\bmu}^{\mbox{\tiny{(0)}}} \!:\! \nabla^2 W_0(\bx) + \rho^{\mbox{\tiny{(0)}}} {\hat{\Omega}}^2 W_0(\bx) \:=\, -\rho^{\mbox{\tiny{(0)}}} \mathcal{F}^{-1}[F] (\bx).\end{aligned}$$
Analogously, [from and the first-order approximation of $\up$ can be computed as]{} $$\begin{aligned}
\up_1(\bx)&\::=\:& \frac{1}{(2\pi)^{d/2}} \int_{Y} \frac{ F(\hat{\bk}) e^{i\eps \hat{\bk} \cdot \bx} }{-\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2 -{\hat{\Omega}}^2} \big(\phi_p(\boldsymbol{0}; \cdot) + \eps i \hat{\bk} \cdot \bchi^{\mbox{\tiny{(1)}}} \big)\ind \hat{\bk} \nonumber \\
&\:=\:& \phi_p(\boldsymbol{0}; \bx) \, W_0(\eps\bx) + \bchi^{\mbox{\tiny{(1)}}}(\bx) \cdot \nabla_{\bx}W_0(\eps\bx). \label{Main first-order u1}\end{aligned}$$ This gives the first order approximation of $U$ by $$\begin{aligned}
\label{Main first-order U1}
U_1(\br) &:=& u(\br/\eps)= \phi_p(\boldsymbol{0}; \br/\eps) W_0( \br) + \eps \bchi^{\mbox{\tiny{(1)}}}(\br/\eps) \cdot \nabla_{\br}W_0(\br). \end{aligned}$$
From and , we similarly find the second-order approximation of $\up$ as $$\begin{aligned}
\up_2(\bx) &:=& \frac{1}{(2\pi)^{d/2}} \int_{Y} \Big(\frac{\phi_p(\boldsymbol{0}; \bx) + \eps i \hat{\bk} \cdot \bchi^{\mbox{\tiny{(1)}}} + \eps^2 (i \hat{\bk})^2 \!:\! \big( \langle \rho \bchi^{(1)} \!\otimes \overline{\bchi^{(1)}} \rangle \phi_p(\boldsymbol{0}; \bx)+ \bchi^{\mbox{\tiny{(2)}}} \big) }{-\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!\!:\! (i\hat{\bk})^2 + \eps^2 \frac{1}{\rho^{\mbox{\tiny{(0)}}}} \big(\{\frac{{\brho}^{\mbox{\tiny{(2)}}}}{\rho^{\mbox{\tiny{(0)}}}} \otimes {\bmu}^{\mbox{\tiny{(0)}}}\} - {\bmu}^{\mbox{\tiny{(2)}}}\big) \!\!:\! (i\hat{\bk})^4 -{\hat{\Omega}}^2} \cdot F(\hat{\bk}) e^{i \eps \hat{\bk} \cdot \bx} \Big)\ind \hat{\bk} \nonumber\\
&=& \phi_p(\boldsymbol{0}; \bx ) W_2(\eps \bx) + \bchi^{\mbox{\tiny{(1)}}}(\bx) \cdot \nabla_{\bx}W_2(\eps\bx)
+ \Big( \langle \rho \bchi^{(1)} \otimes \overline{\bchi^{(1)}} \rangle \phi_p(\boldsymbol{0}; \bx)+ \bchi^{\mbox{\tiny{(2)}}}(\bx) \Big) \!:\! \nabla^2_{\bx}W_2(\eps\bx), \notag \\ \label{Main second-order u2}\end{aligned}$$ where the envelope function $W_2$ is given by $$\begin{aligned}
W_2(\br) &:=& \frac{1}{(2\pi)^{d/2}} \int_{Y} \frac{ F(\hat{\bk}) e^{i \hat{\bk} \cdot \br} }{-\frac{{\bmu}^{\mbox{\tiny{(0)}}}}{\rho^{\mbox{\tiny{(0)}}}}\!: (i\hat{\bk})^2 + \eps^2 \frac{1}{\rho^{\mbox{\tiny{(0)}}}} \big(\{\frac{{\brho}^{\mbox{\tiny{(2)}}}}{\rho^{\mbox{\tiny{(0)}}}} \otimes {\bmu}^{\mbox{\tiny{(0)}}}\} - {\bmu}^{\mbox{\tiny{(2)}}}\big) \!: (i\hat{\bk})^4 -{\hat{\Omega}}^2} \ind \hat{\bk} ~~~~~~~~~~~\end{aligned}$$ and satisfies the field equation $$\begin{gathered}
\label{Main envelop W2}
{\bmu}^{\mbox{\tiny{(0)}}} : \nabla_{\br}^2 W_2(\br) + \eps^2 \big( {\bmu}^{\mbox{\tiny{(2)}}} - \{ \frac{{\brho}^{\mbox{\tiny{(2)}}}}{\rho^{\mbox{\tiny{(0)}}}} \otimes {\bmu}^{\mbox{\tiny{(0)}}}\} \big) : \nabla_{\br}^4 W_2(\br) + \rho^{\mbox{\tiny{(0)}}} {\hat{\Omega}}^2 W_2(\br) \\
=\: -\rho^{\mbox{\tiny{(0)}}} \mathcal{F}^{-1}[F] (\br).\end{gathered}$$ We then obtain the second order approximation of $\Up$ as $$\begin{aligned}
\label{Main second-order U2}
\Up_2(\br) &:=& u(\br/\eps) = \phi_p(\boldsymbol{0}; \br/\eps) W_2(\br) + \eps \bchi^{\mbox{\tiny{(1)}}}(\br/\eps) \cdot \nabla_{\br}W_2(\br) \nonumber \\
&& + \eps^2 \Big( \langle \rho \bchi^{(1)} \otimes \overline{\bchi^{(1)}} \rangle \phi_p(\boldsymbol{0}; \br/\eps)+ \bchi^{\mbox{\tiny{(2)}}}(\br/\eps) \Big) : \nabla^2_{\br}W_2(\br).\end{aligned}$$
[Now let us summarize. We recall that the driving frequency $\Omega$, specified by – and featuring parameter $\hat{\Omega}=O(1)$, belongs to a band gap]{}; $(\bk,\omega_p(\bk))$ is the $p$th branch of the dispersion relationship, and $\phi_p(\boldsymbol{\bk}; \cdot)$ is the corresponding eigenfunction given by ; the effective coefficients $\rho^{\mbox{\tiny{(0)}}}$ and ${\bmu}^{\mbox{\tiny{(0)}}}$ are given by and ; $\brho_2$ and ${\bmu}^{\mbox{\tiny{(2)}}}$ are given by and, and the tensor-valued functions $ \bchi^{\mbox{\tiny{(1)}}} $,$ \bchi^{\mbox{\tiny{(2)}}} $, and $ \bchi^{\mbox{\tiny{(3)}}} $ are given respectively by , , and .
\[finalthm\] [Let the source term $f$ in be given by Assumption \[assumption f\] (designed to generate the long-wavelength motion), and let the driving frequency $\Omega$ reside within a band gap according to –. The leading-order ($U_0=\Up_0$), first-order ($U_1=\Up_1$), and second-order ($U_2=\Up_2$) asymptotic approximations of the solution $U$ to are given respectively by –, –, and –. In particular, we have]{} $$\begin{aligned}
\|U-U_0\|_{L^2(\mathbb{R}^d)} = O(\eps),~~\|U-U_1\|_{L^2(\mathbb{R}^d)} = O(\eps^2),~~\|U-U_2\|_{L^2(\mathbb{R}^d)} = O(\eps^3).\end{aligned}$$
By projecting $U_0(\br)$, $U_1(\br)$, and $U_2(\br)$ onto $\phi_p(\boldsymbol{0}; \br/\eps)$ [(with weight $\rho(\br/\eps)$)]{}, one obtains the corresponding effective field equations that describe the leading-order, first-order, and second-order effective i.e. “mean” wave motions, respectively. It can be shown that those effective field equations agree with those obtained via formal two-scale homogenization, both in low- and high-frequency cases.
Numerical examples {#Numerical example}
==================
Consider the wave motion with $d=2$ in a periodic medium depicted in Fig. \[fig1\](a), whose unit (Wigner-Seitz) cell contains centric circular inclusion of radius $a=0.3$, see Fig. \[fig1\](b). The coefficients inside the unit cell are given by $$(G(\bx),\rho(\bx)) \:=\:
\left\{ \begin{array}{ll}
(G_1=1,\rho_1=1), & \|\bx\|>a, \\
(G_2=6,\rho_2=20), & \|\bx\|<a. \end{array} \right.$$ Fig. \[fig1\](c) and Fig. \[fig1\](d) plot respectively the first Brillouin zone and the bend diagram for the problem ($\eps=1$, first 14 branches) that features three complete band gaps. [Without loss of generality, we take the excitation “frequency” as $\omega = i\eps$, i.e. $\omega^2 = -\eps^2$, that (i) formally resides inside a band gap , and (ii) admits asymptotic representation with $p=1$ and $\sigma=-1$.]{} In the context of Assumption \[assumption f\], we consider the source term with $p=1$ and $$\begin{aligned}
\label{Gforce}
F(\hat{\bk}) = \frac{1}{2\sqrt{\pi}} e^{-\frac{\| \hat{\bk }\|^2}{4}}.\end{aligned}$$ In essence, the body force $f(\eps\bx)$ stemming from and corresponds to a Gaussian distribution modulated by the product of the eigenfunction $\phi_1(\boldsymbol{0},\bx)$ and the mass density $\rho(\bx)$, whose standard deviation is $O(\eps^{-1})$ and whose amplitude decays exponentially in time as $e^{-\eps t}$. In the wavenumber-frequency space (described by the bend diagram), the body force “triggers” the acoustic branch $(\bk,\omega_1(\bk))$ near the origin $\bk=\boldsymbol{0}$.
In what follows, we compare an “exact” (numerically computed) response of the medium due to with its leading-, first-, and second-order asymptotic approximations for different values of $\eps$. To this end an “exact” solution is obtained, for each $\eps$, via the finite element platform NGSolve [@NGSolve], where $\mathbb{R}^2$ is approximated by a square domain $$\mathcal{D}_{\tiny{\mbox{N}}}=[-(N+0.5),(N+0.5)]\times[-(N+0.5),(N+0.5)]$$ containing $2N\times 2N$ unit cells. This domain is discretized using finite elements of order 3 and maximum size $h_1= 0.01125\eps$, assuming the homogenous Dirichlet boundary conditions along its boundary $\partial \mathcal{D}_{\tiny{\mbox{N}}}$. The leading-, first- and second-order approximations of the solution are evaluated on a $64N\times 64N$ grid of points $\mathcal{G}_{\tiny{\mbox{N}}}$, by integrating numerically the expressions , and (with $p=1$), [after a change of variable $\hat{\bk}\to \bk=\epsilon\hat{\bk}$]{}, in MATLAB over the first Brillouin zone, [used to approximate]{} [$\epsilon Y= \mathbb{R}^2 $]{} [thanks to the strong exponential decay of]{}[ $F(\eps^{-1}\hat{\bk})$]{}. The eigenfunction $\phi_1$, cell functions $\bchi^{\mbox{\tiny{(1)}}}, \bchi^{\mbox{\tiny{(2)}}}, \bchi^{\mbox{\tiny{(3)}}}$ and effective coefficients $\rho^{\mbox{\tiny{(0)}}},\bmu^{\mbox{\tiny{(0)}}}$ and $\bmu^{\mbox{\tiny{(2)}}}$ featured in the approximations’ expressions are evaluated using NGSolve by discretizing the unit cell with elements of order 5 and maximum length $h_2=0.0075$. The first eigenfunction $\phi_1(\boldsymbol{0},\bx)$ as well as the components of the affiliated cell functions $\bchi^{\mbox{\tiny{(1)}}}$ and $\bchi^{\mbox{\tiny{(2)}}}$ are shown in Fig. \[fig6\].\
Here it should be noted that the perturbation parameter $\eps$ physically signifies the amount of “foray” into the band gap, whereby larger values of $\eps$ inherently yield faster (exponential) decay of the solution away from the source of disturbance, $f(\eps\bx)$. In terms of numerical simulations, this is the key alleviating factor that allows us to approximate the wave motion in $\mathbb{R}^2$ with that in $\mathcal{D}_{\tiny{\mbox{N}}}$ with homogeneous Dirichlet boundary conditions. In this vein, the response of the medium is computed over $\mathcal{D}_{\tiny{\mbox{25}}}$ for $\eps=0.25$, and over $\mathcal{D}_{\tiny{\mbox{20}}}$ for $\eps=0.375$ and $\eps=0.5$. We also remark that the computation numerical responses for $\eps\!<\!0.25$ was not feasible owing to excessive computational effort, in terms of larger values of $N$, required to achieve sufficiently accurate numerical approximation of the wave motion in $\mathbb{R}^2$.
Fig. \[fig4\] plots the distribution over $\mathcal{D}_{\tiny{\mbox{N}}}$ of $f(\eps\bx)$, $u(\bx)$ and $u_m(\bx)$ $(m\!=\!0,1,2)$ for all three values of the perturbation parameter, namely $\eps\in\{0.25,0.375,0.5\}$. An overall agreement between the solutions is evident from the display, as is the increase in fidelity of asymptotic approximation with the order of expansion. A detailed comparison is provided in Fig. \[fig2\] and Fig. \[fig3\], which plot the variations of $u$ and $u_m$ versus $\bx=(x,y_0=const.)$ for $\eps=0.25$ and $\eps=0.5$, respectively. From the displays, we observe that all three approximations provide the same rate of decay as the exact solution; however, the fine solution detail in the vicinity of the peak load ($x\approx 0)$ is accurately approximated only by higher-order models.
For a better insight into the convergence of the asymptotic models $u_m(\bx)$, we next consider the relative approximation errors $e^{\tiny{\mbox{$(m)$}}}_{\tiny{\mbox{M}}}$ ($m=0,1,2$) given by $$\begin{aligned}
\label{relerr}
e^{\tiny{\mbox{$(m)$}}}_{\tiny{\mbox{M}}} (\eps)= \frac{\|u_m-u\|_{L^2(\mathcal{D}_{\tiny{\mbox{M-0.5}}})}} {\|u\|_{L^2(\mathcal{D}_{\tiny{\mbox{M-0.5}}})}},\end{aligned}$$ where $M\leqslant N$. The domain integrals featured in are evaluated numerically by sampling the reference “exact” solution $u$ and its approximations $u_m$ over grid $\mathcal{G}_{\tiny{\mbox{M}}}$. The values of $e^{\tiny{\mbox{$(m)$}}}_{\tiny{\mbox{M}}}(\eps)$ inherently depend on $M$, and were found to numerically converge at $M\!=\!15$ for all three values of the perturbation parameter, $\eps\in\{0.25,0.375,0.5\}$. Fig. \[fig5\] illustrates the observed scaling of $e^{\tiny{\mbox{$(m)$}}}_{\tiny{\mbox{M}}}(\eps)$, on the log-log scale, over this limited range of the perturbation parameter. Specifically, we find that the apparent linear trends (indicated by dashed lines) have slopes $e^{\tiny{\mbox{(0)}}}_{\tiny{\mbox{15}}}=O(\eps^{1.11})$, $e^{\tiny{\mbox{(1)}}}_{\tiny{\mbox{15}}}=O(\eps^{2.00})$ and $e^{\tiny{\mbox{(2)}}}_{\tiny{\mbox{15}}}=O(\eps^{3.06})$, which is in good agreement with the expected result $\lim_{\tiny{\mbox{M}}\to\infty}e^{\tiny{\mbox{(m)}}}_{\tiny{\mbox{M}}}=O(\eps^{m+1})$ due to Theorem \[finalthm\].
Summary {#Summary}
=======
[In this investigation we establish a convergent, second-order asymptotic model of the high-frequency, low-wavenumber wave motion in a highly-oscillating periodic medium driven by a source term. Within this framework, the driving frequency is further assumed to reside inside a band gap, while the source term is restricted to a class of functions which generate the long-wavelength motion. We first use the Bloch transform to formulate an equivalent variational problem in the unit (Wigner-Seitz) cell. By investigating the source term’s projection onto certain periodic functions, we establish a convergent, second-order homogenized model via asymptotic expansionexpansion of (i) the nearest dispersion branch, (ii) germane (low-wavenumber) eigenfunction, and (iii) the source term. A set of numerical results is included to illustrate the utility of the homogenized model in representing, with high fidelity, both macroscopic and microscopic features of the exact wave motion, and to demonstrate the obtained convergence result.]{}
|
---
abstract: 'With the detection of Gravitational waves just about an year ago Einstein‘s general theory of relativity- a space-time theory of gravity, got established on a firmer footing than any other theory in physics. Gravitational waves are just propagating disturbances in the gravitational field of extremely strong sources caused by some catastrophic event associated with cosmic bodies, like binary black hole coalescence, or neutron star mergers. As these events happen very far away in cosmos, hundreds of millions of light years away and the signal strength would be extremely weak, it requires extraordinary detection and analysis technology to observe an event. Luckily the joint collaboration LIGO-VIRGO, have so far detected two events in September and December of 2015 during their analysis of observations made with the laser interferometers over the last few observing sessions. The talk will give a brief theoretical sketch of the analysis required for describing the waves resulting from mass motion in the realm of general relativity, and point out, the serious and sincere efforts of the past fifty years that went into the final success. An attempt will be made to point out the enormous scope that is available for the new generation of students and researchers in pursuing the topic as a new window for possibly viewing the Universe with the implications for the studies of Dark matter, Dark energy and Cosmology as a whole.'
author:
- 'A.R.Prasanna[^1]'
title: 'Gravitational waves-a new window to Cosmos'
---
With Galileo’s telescope, the view of the Universe expanded rapidly from naked eye observations to cover the celestial beauty of Jupiter’s moons, Saturn’s rings and outer planets to the expanding Universe of Hubble that included, Galaxies, Clusters and myriads of extra galactic objects. In 1930, came the next revolution of the enigmatic Radio Universe, lead by Karl Jansky and G Reber, which led one to very high energy cosmic sources like Quasars, Active Galactic Nucleii, Pulsars and more importantly, the most profound all encompassing ‘Cosmic microwave Background’ that indicated the beginning of our Universe. Once the realisation came about emissions from cosmic sources in two different frequencies, the optical and the radio, it was a simple task to look for emissions in other frequencies, IR, UV, X-ray sources. Along with came the bonus of emissions of $ \gamma $-rays , which completed the electromagnetic spectrum of Universe being visible in the entire spectrum from Radio waves to $ \gamma $-rays. While it was known that the emission of radiation from most of these sources were all due to Electromagnetic processes, it was not clear upto 1960s, the source of energy for emissions from objects like Quasars and AGN s, till Hoyle and Fowler put forward the idea of Gravitational collapse of massive stars, within the framework of Einstein’s theory of General Relativity which explained Gravitation as the curvature of space-time.\
Of all the discoveries of the human mind, Einstein’s theory of general relativity is considered to be the most beautiful creation. In fact, it is often said that the special relativity, which forms a strong basis of modern physics, along with quantum mechanics, was ripe to be discovered at the turn of the nineteenth century, and if not Einstein, Poincare or Lorentz would have developed the theory. On the other hand the general theory of relativity, which is the epitome of the world of symmetry, assigning freedom from the confines of coordinate systems (observers) to understand the most important of all the fundamental interactions-Gravity, is completely the work of one individual, arising out of thought experiments instead of laboratory experiments or observations, that preceded all other discoveries in physics.\
The most important features that lead Einstein from special to general relativity are-\
1.The equivalence of inertial and gravitational mass of any body $ M_i = M_g $ known as the principle of equivalence (demonstrated by Eotvos in 1889), also guided by his thought experiment of a freely falling lift with an observer inside.\
Consider the observer in the elevator dropping two coins side by side as shown in figure \[ffe1\].The observer inside will see them hung in air($ a = g $) as both the coins and the elevator are falling with the same acceleration. However, if the observer has a very accurate measuring rod, he will find that with time the two coins appear coming closer towards each other. The explanation to this is quite simple. As the earth’s gravitational field is radially symmetric the paths of the freely falling coins are along lines converging at the center of the earth. It is thus that they appear to be moving towards each other. This means, though the observer cannot measure the gravitational field inside the elevator, he can measure the variation in the field from point to point, or the *relative acceleration* between the particles.
![Freely falling elevator [@arp08][]{data-label="ffe1"}](fig1.eps){width="10cm"}
2\. effect of gravity on light (figure \[ffe2\]) which implies that light has a curved path in a gravitational field, as also there appears a frequency shift depending upon the gravitational potential difference.
![light beam in a freely falling elevator [@arp08][]{data-label="ffe2"}](grfig2.eps){width="6cm"}
$$\label{freqrel}
(\nu_o - \nu_e)/ (\nu_e) = v/c = -gH/c^2,$$
With this Einstein realized that the arena he wanted for general relativity, was the non Euclidean geometry, (non flat),which requirement was satisfied by the Riemannian Geometry, an extension of Gaussian curved geometry, as suggested by his mathematician friend Marcel Grossmann, represented by the metric $ ds^2 = g_{ij} dx^i dx^j $. As one saw in the case of the elevator, two freely falling particles in a gravitational field have a relative acceleration which in Newtonian terms can be seen as follows. Consider two freely falling particles in a gravitational field with their trajectories labeled $ \lambda_1 $ and $\lambda_2 $. At time $ t $ let their positions be P and Q respectively, identified in the associated frame by P $(x^a)$ and Q $(x^a + \eta^a)$, all coordinates being functions of $ t $ \[figure \[2ffpars\]\].
![two freely falling particles [@arp08][]{data-label="2ffpars"}](grfig5.eps){width="8cm"}
\
$ \eta^a $ is a small connecting vector between $ \lambda_1 $ and $\lambda_2 $, $ t $ being the path parameter. One can write, their equations of motion to be $$\label{relacn1}
\ddot x^a = - (\partial^a \phi)_P = -\delta^{ab}\partial_b (\phi)_P\\$$ $$\label{relacn2}
\ddot x^a + \ddot \eta^a = - (\partial^a \phi)_Q = -\delta^{ab}\partial_b (\phi)_Q$$ where $ \phi $ is the gravitational potential in which the particles are falling.\
As $\eta^a $ is very small, one can use Taylor expansion and simplify to get $$\ddot \eta^a = - \eta^k \partial_k \partial^a \phi,$$ which may be written as $$\label{relacn3}
\ddot\eta + K^a_{\phantom {a} b}\eta^b = 0 ;\quad K^a_{\phantom {a} b} = \partial^a \partial_b \phi.$$ $\ts K{^a_b} $ thus represents the relative acceleration between the two particles as they fall freely in the gravitational field of $ \phi $.\
Moving on to general relativistic formulation, one can consider a two surface made of a congruence of geodesics and some connecting vectors $ \eta^i $ as shown figure \[2sfce\].
![congruence of world lines[]{data-label="2sfce"}](grfig6.eps){width="8cm"}
Using the equation for absolute derivatives one can get the geodesic deviation equation $$\label{relaccn3}
\frac{\delta^2 \eta^i}{\delta s^2} + R^i_{\phantom{i} l k j}U^l \eta^j U^k = 0.$$ and then referring it to a local Lorentz frame, one can find the acceleration equation to be $$\label{relaccn4}
\frac{d^2 X^{(a)}}{ds^2} + K^{(a)} _{(c)} X^{(c)} = 0, \qquad K^{(a)} _ {(c)} = \ts R{^{(a)} _{(b)(c)(d)}} u^{(b)} u^{(d)}.$$\
The field equations of general relativity are \[ $ \ts G{_{ij}}\equiv \ts R{_{ij}}-\frac{1}{2} R \ts g{_{ ij}} = \kappa \ts T{_{ij}} $\] along with the Bianchi identities , and consequently the conservation laws \[ $ \ts T{^{ij}_{; j}} = 0 $\].\
If the mass distribution is static, then for some specific symmetrical distributions, a few exact solutions have been obtained as given by the Schwarzschild (spherically symmetric) and the Kerr solutions (axisymmetric) apart from a few cylindrically symmetric solutions, for the cases of uncharged and charged matter that are asymptotically flat. On the other hand, when the matter distribution is nonstatic, then one expects the field surrounding the distribution to vary slowly and the change could propagate all through the space–time as small perturbations of the background field. In such a situation, one could write the general metric solution as, $ g_{ij} = g^b _{ij} + h_{ij}. $ As Einstein had pointed out, things are made much simpler when one assumes, the background metric to be flat, $ g^b _{ij}= \eta_{ij}.$\
Considering the field outside the matter distribution, and substituting for the components of the Ricci tensor, the metric $\eta _{ij} + h_{ij} $ and its derivatives in the equations, $ R_{ij} = 0,$ one finds for the perturbations $ h_{ij} $ the set of equations $$\label{lineqns1}
\square h_{ij} + \ts h{^k _k _{, i j}} - \ts h{^k _i _{, j k}} -\ts h{^k _j _{, i k}} = 0,$$ where $ \square $ represents the usual flat space D’Alembertian, with the second and higher powers of $ h_{ij} $ being ignored.\
It is understood that solutions to this equation cannot be unique, as one can have a general coordinate transformation, and in order to remove this ambiguity, one can choose a particular gauge, and one often chooses the so–called, harmonic or Lorentz (also called deDonder) gauge, as given by $ g^{ij} \ts \Gamma{^k _{ij}} = 0, $ which in terms of $ h $ yield the relations,$$\label{gaugecon1}
\bar h^j _{i,j} = 0, \quad \bar h^j _i = h^j _i - \frac{1}{2}\delta^j _i h^k _k.$$ With this choice of gauge, the equations reduce to the simple flat space wave equation for the tensor potential, $ h_{ij} $, $$\label{wveqn1}
\square h_{ij} = 0.$$ for which one can write the general solution as a superposition of plane monochromatic waves,$$h_{ij} = A_{ij} e^{i k_l x^l} + A^*_{ij} e^{- i k_l x^l},$$ with $ A $ and $ A^* $ representing the complex amplitudes and $k^l $ the wave covector, satisfying the orthogonality relation, $ \eta_{ij} k^i k^j = 0. $\
The gauge condition yields four constraints on the ten complex amplitudes, given by the relation $$\label{gaugecon2}
A_{ij} k^j = \frac{1}{2} A^j _j k_i.$$ However, as the coordinate freedom is still left within the gauge as specified by $$\begin{aligned}
\square \xi_i = 0, \quad A' _{ij} = A_{ij} + k_i \xi \hat \xi_j + k_j \xi \hat \xi_i,\\
\xi^k (x) = i[\hat\xi^k e^{ (i k_l x^l)} + \hat \xi^{*k} e^{ (i k_l x^l)}], \end{aligned}$$ where $ \hat \xi $ are constants by choosing them appropriately, one can make four of the $ A_{ij} $s zero. In order to remove this freedom, one needs four additional constraints, which is achieved by choosing a globally defined time like vector field $ u^i $ such that, $ A^i _j u^j = 0, \, A ^i _i = 0 $. Thus there are eight constraints on the ten complex amplitudes as given by $$A_{i j} u^j = 0, \quad A_{i j} k^j = 0, \quad A ^i _i = 0,$$ indicating that the $ A_{ij}$s are transverse and traceless. Such a choice of gauge is known as T–T gauge or transverse traceless gauge.\
In terms of the metric potentials, the choice of T–T gauge yields, $$h_{i0} = 0, \quad \ts h{_a^j_{, j}} = 0, \quad h^i _i = 0.$$ As there are only two degrees of freedom associated with the waves, it implies physically that there are only two degrees of polarisation associated with these waves.Thus, for a plane gravitational wave propagating along the z-direction, in a Cartesian system, the solution may be written explicitly as $$\begin{aligned}
\label{pgwamp}
\ts h{^ {TT} _{XX}} = -\ts h{^ {TT} _{YY}} = \sR \{a_+ \,e^{[-i\omega (t-z)]}\} \nonumber\\
\ts h{^ {TT} _{XY}} = \ts h{^ {TT} _{YX}} = \sR \{a_\times e^{[-i\omega (t-z)]}\}
\end{aligned}$$ with $ a_+ = A_{11} = - A_{22} $ and $ a_\times = A_{12} = A_{21}, $ denoting the two independent states of polarisation.\
In the case of a monochromatic plane gravitational wave, propagating along the z-direction, the space-time metric is given by $$\label{relaccn2}
ds^2 = dt^2 - (1- h_{XX}) dx^2 - (1 - h_{YY}) dy^2 + 2 h_{XY} dx dy - dz^2$$ and the only components of the curvature tensor that are nonzero are $$\begin{aligned}
\label{rhijk}
(i)\quad &\,\ts R{^x _{0 x 0}} = -\frac{1}{2} \ts h{^{TT} _{XX,00}}, \\
(ii)\quad &\,\ts R{^y _{0 y 0}} = \frac{1}{2} \ts h{^{TT} _{YY,00}}, \\
(iii)\quad &\, \ts R{^x _{0 y 0}} = \,\ts R{^y _{0 x 0}} = -\frac{1}{2}\ts h{^{TT} _{XY,00}}.
\end{aligned}$$ Choosing a comoving frame $ u^i = (1, 0, 0, 0) $ and the deviation vector $ \eta^i = (0, \varepsilon, 0, 0),$ equation yields$$\begin{aligned}
\label{eometa1}
(i)\quad & \frac{\partial^2\eta^x}{\partial t^2} = \frac{1}{2}\ts h{^{TT} _{XX,00}}\;\varepsilon,\\
(ii)\quad & \frac{\partial^2\eta^y}{\partial t^2} = \frac{1}{2}\ts h{^{TT} _{XY,00}}\;\varepsilon.
\end{aligned}$$ On the other hand if the deviation vector $ \eta^i = (0, 0, \varepsilon, 0) $ then the equations are, $$\begin{aligned}
\label{eometa2}
(iii)\quad & \frac{\partial^2\eta^x}{\partial t^2} = \frac{1}{2} \ts h{^{TT} _{XY,00}}\;\varepsilon,\\
(iv)\quad & \frac{\partial^2\eta^y}{\partial t^2} = -\frac{1}{2} \ts h{^{TT} _{XX,00}}\;\varepsilon.
\end{aligned}$$ It is clear from these four equations that the passing wave induces oscillations of the particles in the ring depending upon the nonzero components of the tensor $ h_{ij}. $\
If $ h_{XY} = 0, $ and $ h_{XX} = - h_{YY} \neq 0, $ then the ring of particles oscillates as shown in figure \[gwvring\](b), along the X and Y directions, with $ h_{XX} $ changing sign. On the other hand if the wave is such that, $ h_{XX} , h_{YY} $ are zero but $ h_{XY} \neq 0, $ then the particles oscillates as shown in figure \[gwvring\](c).\
![gravitational wave passing thrugh a ring of particles. (a) Before, (b)wave with + polarisation, and (c) wave with x polarisation[]{data-label="gwvring"}](gravwvring.eps){width="10cm"}
\
**Do these waves carry energy and angular momentum?**\
As has been pointed out in references [@MTW], [@weinberg72], [@ll75], the gravitational field energy cannot be localised and thus it is difficult to separate the source energy and the field energy from the total energy momentum tensor $ T^{ij} $ that appears in the field equations. However, in the case of waves as described here, one has an advantage that in the linearised theory, one can still construct a pseudotensor that characterises the energy momentum for gravitational waves.\
Writing the general energy momentum conservation law coming from the field equations, $ \ts T{_i ^j _{;j}} = 0, $ as [@ll75] $$\label{6conslaw}
\frac{1}{\sqrt{-g}} [\frac{\partial (T_i ^j \sqrt{-g}}{\partial x^j}] - \frac{1}{2} \frac{\partial g_{jk}}{\partial x^i} T^{jk} = 0,$$ one can see that it gives the simple conservation law for the source when the potentials $ g_{ij} $s are constants. Rewriting it as $$\frac{\partial}{\partial x^j}[(-g) \, ( T_i ^j + t_i ^j)] = 0,$$ one can see that the total energy momentum has been separated into a part representing the source energy and the remaining the field energy $ t_i ^j $ called the Landau–Lifshitz pseudotensor, obtained from a super potential, $ \Psi^{i k l} $, defined through the equation $$\label{llpsudotens}
(-g) [( R^{ik} - \frac{1}{2} R g^{ik}) + t^{ik}] = \ts \Psi{^{i k l} _{, l}}.$$ The L–L super potential, when expressed in terms of the metric and its derivatives, is given by [@anderson72] $$\label{psidef}
\ts \Psi{^{i k l}} = \sqrt{-g} \,\delta^i _p \,\{g (g^{kp} g^{lm} - g^{km} g^{lp})\}_ {, m}.$$ With this definition, one then calls the total energy momentum, $ (-g) \,(T^{ik} + t^{ik})$ the ‘effective energy momentum’, of the space-time governed by the chosen metric, that satisfies the usual divergence–free relation, $ (T^{ij} + t^{ij})_{,j} = 0, $ such that one can use the volume integral and recover the effective energy.\
In the case of linearised gravity, with the perturbations defined over a flat background metric ( $ g_{ij} = \eta_{ij} + h_{ij} $), as shown in [@MTW], for the short wave approximation, defined by ($ \lambda / \sR \ll 1, a \ll 1 $), the effective stress tensor averaged over several wavelengths is given by $$t_{ij} = \frac{1}{8\pi}\{< R_{ij}(h^2)>- \frac{1}{2} \ts g{^B_{ij}}<R (h^2)> \},$$ which, for the flat background, yields, in the TT gauge the expression, $$<t_{ij}> = \frac{1}{32 \pi} < \ts h{^{kl} _ {,i}} \, \ts h {_{kl , j}}>.$$ This is also commonly referred to as Issacson stress–energy tensor for gravitational waves, [@schutz], [@isac68], when the averages are taken over one period of oscillation in time and spatial regions of the size of a wavelength of distance in all directions.\
Going back to the field equations one can see that on using the harmonic gauge $ \bar h^j _{i , j} = 0 $, the equations reduce to $$\label{6fldeqns2}
\square \,\bar h_{ij} = -2 \kappa \tau_{ij},$$ whose integrability requires $ (\tau_i ^j)_,j = 0. $\
One can write the solution of in terms of a retarded Green’s function, [@demansk85]which, after integration with respect to t’, yields,$$\label{6soln2}
\bar h_{ij} = 4 \int\{[\frac{\tau_{ij} (x',\,t -\vert x - x' \vert )]}{\vert x - x'\vert}\} \, d^3 x'.$$ As $ \tau_{ij} $ satisfies the conservation law, $ \tau^{ij} _{, j} = 0, $ one can write this as $$\begin{aligned}
\label{6conslaw}
(i)\quad \ts \tau{^{ab} _{,b}} + \ts \tau{^{a0} _{,0}} = 0 \nonumber \\
(ii)\quad \ts \tau{^{0b} _{,b}} + \ts \tau{^{00} _{,0}} = 0.\end{aligned}$$ indices a,b taking values 1,2,3. Taking the appropriate moments of these equations and simplifying one gets finally$$\label{2mom}
\int \tau^{ab} dV = \frac{1}{2} \frac{\partial^2}{\partial t^2}\int \rho(r',t) x^a x^b dV = \frac{1}{2} \ddot I^{ab},$$ where $ I^{ab} $ is the second moment of the mass distribution at the source related to the moment of inertia tensor [@MTW], $$\label{mominr}
\sI^{ab} = \int \rho(r^2 \delta^{ab}- x^a x^b) dV = (\delta^{ab}I^c _c - I ^{ab}),$$ and to the quadrupole moment $ Q^{ab} $ [@ll75] $$Q^{ab} = \int \rho(x,t) (3 x^a x^b - r^2 \delta^{ab}) dV = (3 I^{ab} - \delta^{ab} I^c _c).$$ With this one can finally write down the approximate solution for , $$\label{quadform}
\bar h_{ij} = \frac{-2\Omega^2}{r} I_{ij} e^{[i\Omega (r-t)]},$$ $\Omega $ being the frequency. Equation is the well known ‘quadrupole formula’ for gravitational radiation.\
The individual components of the metric tensor $ h_{ij} $ for a plane gravitational wave in T–T gauge, moving along the z-direction, are now given by $$\begin{aligned}
h_{Zi} = 0,(i= 0,1,2,3);\quad h_{XY} = -\frac{2\Omega^2}{3r} Q_{XY} e^{[i\Omega(r-t)]}\nonumber\\ h_{XX} = - h_{YY} = -\frac{\Omega^2}{3r} (Q_{XX} - Q_{YY}) e^{[i\Omega(r-t)]},\end{aligned}$$ and the energy flux carried along the direction of propagation is $$t^{z0} = (\frac{G}{36 \pi r^2 c^5}) [(\frac{\dddot Q_{XX} - \dddot Q_{YY}}{2})^2 + (\dddot Q_{XY})^2].$$ In order to express the energy and angular momentum carried by the waves, in an invariant form, one can use the 3-dim. symmetric, unit polarisation tensor $ e_{ab} $ [@ll75], which determines the nonzero components of the metric tensor $ h_{ab} $ in the appropriate gauge ($ h_{0a} = h_{a0} = h = 0) $ and satisfies the relations $$\label{poltens1}
e_{0a} = 0, \quad e_{ab} n^b = 0, \quad e_{ab} e^{ab} = 1,$$ $ n^a $ being the unit vector along the direction of wave propagation.\
The intensity of radiation of a given polarisation into a given solid angle $ d\Sigma $ is then $$\label{intensity1}
dI = \frac{1}{72 \pi} (\dddot Q_{ab} e^{ab})^2 d\Sigma,$$ which depends implicitly on the direction $n$, because of the condition of transversality. Summing over all polarisations then gives the total angular distributions,$$\label{intensity2}
dI = (\frac{G}{36 \pi c^5})[ \frac{1}{4}(\dddot Q_{ab} n^a n^b)^2 +\frac{1}{2}\dddot Q_{ab} ^2 - \dddot Q_{ab} \dddot Q^a _c n^b n^c ].$$ The energy loss of the system per unit time can be found by averaging $\frac{dI}{d\Sigma}$ over all directions and multiplying by $ 4\pi $ as given by $$\begin{aligned}
\label{eneangmm}
&\frac{dE}{dt} = -(\frac{G}{45 c^5}) <\dddot Q_{ab} \dddot Q^{ab}> \\
&\frac{dJ_k}{dt} = -(\frac{2G}{45 c^2} )\varepsilon_{klm}\, <\dddot Q^{la} \dddot Q_a ^m> .\end{aligned}$$ In the quadrupole approximation, which also happens to be the lowest order post–Newtonian approximation, one can express the amplitude, frequency, and luminosity of the emitted radiation [@sslr12], which depend only on the density $ \rho $ and velocity fields of the Newtonian system as given by\
(a) the amplitude in the Lorentz gauge,$$h_{ab} = \frac{2}{r}\frac{d^2 Q_{ab}}{dt^2}, \quad Q^{ab} = \int \rho x^a x^b \,d^3x,$$ (b) the frequency, $$\label{gwlumin}
f_0 = \omega_0 / 2 \pi = \sqrt{G \bar \rho /4\pi},$$ where $ \bar\rho $ is the mean density of mass–energy in the source.\
(c) the luminosity expressed in terms of the local stress–energy in the T–T gauge is given by $$L_{gw} =\frac{1}{5}(\Sigma_{j,k} (\dddot Q_{jk})^2 - \frac{1}{3} (\dddot Q)^2,$$ where Q is the trace of $ Q_{jk},$ an equation which may also be used to estimate the back reaction on a system emitting gravitational radiation[@sslr12].\
**Detection of gravitational waves**\
Late last year (2015), the world celebrated the one hundredth anniversary of general relativity and Einstein’s prediction of gravitational waves by detecting through LIGO (Laser Interferometric Gravitational wave Observatory) the first signal of the waves arriving on earth, produced far away in the cosmos, by coalescence of two medium–sized black holes .[@abbott16].\
As one can see from the expression for the energy carried by the wave , its strength is of order $ c^{-5} $, and thus would require an extremely sensitive set of apparatus and very sophisticated methods of data analysis to detect signals of such low strength and to separate them from all other forms of noise.\
The experimental search for gravitational waves from cosmic sources started with J. Weber’s pioneering idea of using a resonant bar detector [@weber60], which was essentially a suspended homogeneous metal bar, on which an impinging gravitational wave would excite mechanical vibrations that could be transferred to electromagnetic signals by piezoelectric transducers which can be amplified and recorded. The excitation is mainly due to the relative acceleration between the particles of the bar caused by the passing wave. When two such antennas separated by a large distance (in the case of Weber, the bars set up were in Maryland, Virginia and Argonne National lab in Chicago) record similar signals coincidentally, it was assumed that the disturbance was caused by a cosmic source far away from the earth and attributed to gravitational wave.
![Joe Weber and his antenna[]{data-label="jweber"}](joeweber1.eps){width="10cm"}
(Although Weber announced the recording of such signals in 1969 claiming the detection of gravitational waves, it was very soon found to be not correct as no other experimental group, even with increased sensitivity systems could find any coincident signal.)\
Though there have been continuous efforts to improve the sensitivities of the bar mode detectors, the attention of the experimental community turned towards the beam mode detectors, where one uses laser interferometry, consisting of four masses hung from vibration–free support systems with their separation being monitored by a highly sophisticated optical system.
![LIGO interferometer[]{data-label="ligo"}](ligo1.eps){width="10cm"}
\
The four masses (mirrors) are placed at the ends of two orthogonal arms such that two are closer to each other with the other two at the far ends of the arms, and the arm’s lengths being almost equal ($ L_1 \simeq L_2 = L, $ such that) the change ($ \bigtriangleup L(t), $ is directly proportional to the output of the interferometer (photodiode). When a gravitational wave passes through such a system, having frequency higher than the pendulum’s natural frequency of $ \sim 1 Hz $, the acceleration induced by the wave pushes the masses (as though they are freely falling) which causes the arm length difference $ \bigtriangleup L = L_1 -L_2 $ to change. Depending upon the polarisation of the impinging wave, ($ h_+ or \,h_\times $) the interferometer’s output would be a linear combination of the two wave fields [@thorne97] $$\label{deltal}
\frac{\bigtriangleup L (t)}{L} = F_+ h_+ (t) + F_\times h_\times (t)\equiv h(t)$$ where $ F_+, F_\times $ are of order unity having quadrupolar dependence upon the direction and orientation to the source [@thorne87]. The $ h(t)$ in is called the strain of the gravitational wave and the time evolution of $ h(t), h_+(t), h_\times (t) $ as waveforms.\
![Typical waveforms from The inspiral of a compact binary computed using Newtonian gravity for the orbital evolution and the qudrupole moment approximation for the wave generation [@abramwvfm][]{data-label="6wvform"}](typwaveforms.eps){width="10cm"}
\
A typical waveform arising out of inspiraling compact binary system appears as shown in figure \[6wvform\], which has been computed using Newtonian gravity for the orbital evolution and the quadrupole-moment approximation for wave generation [@abramwvfm]. As the inspiralling binaries get closer, one finds increasing amplitude and the upward sweeping frequency (often referrred to as chirp) of the waveform, with the amplitude ratio for the two polarisations going as $$\frac{amp\, h_+}{amp\, h_\times} = \frac{2 \cos i}{1 + \cos^2 i},$$ $ i $ being the inclination of the orbit to observer’s line of sight, and the orbital eccentricity determining the waves’ harmonic content. For simplicity, if the orbit is considered circular, then the rate at which the frequency sweeps or ‘chirps’, $ df/dt $ (also referred to as the number of cycles spent near a given frequency $ n = f^2 (df/dt)^{-1}, $ is determined solely by the binary’s chirp mass in terms of the masses of the binary components, $ M_c \equiv \frac{(M_1 M_2)^{3/5}} {(M_1 + M_2)^{1/5}} $. Thus, the amplitudes of the two waveforms ($ h_+, h_\times) $ are determined by the chirp mass, distance to the source, and the orbital inclination. With the preliminary information coming from the qudrupolar (near Newtonian) formula, the general relativistic effects add further information, through the waveform modulation coming from the rate of frequency sweep, depending upon the binary’s dimensionless ratio, $ \eta = \mu /M,$ with $ \mu = M_1 M_2 / (M_1 + M_2) $ the reduced mass and $ M = (M_1 + M_2) $, the total mass, as well as on the spins of the two bodies. Two of the important effects worth noting are (i) the back scattering of waves due to the curvature of the binary space–time [@vish70], producing tails that act back on the binary modifying the inspiral rate that can be measured and (ii) the Lense–Thirring drag arising from the inclinations of the spin axes of the components with respect to the binary’s orbital plane, causing the orbit to precess, which, in turn can modulate the wave forms. In order to incorporate these relativistic modulations of the basic wave forms, while detecting, one uses a technique called the matched filter, where the incoming signals are matched to already prepared theoretical templates with several different combinations of parameters, and the best matched template will give the details of the wave form [@thorne97].\
As Blanchet points out [@blanchlr14], the basic problem that one faces in relating the amplitude $h_{ab}$ seen in the wave zone with the source material stress energy, $ T_{ij} $, is due to the approximation methods in general relativity. While the post–Newtonian methods may appear satisfactory in the weak field limit (valid only in the near zone), its inadequacy appears while trying to include the boundary conditions at infinity, which affects the proper determination of the radiation reaction force. While the post-Minkowskian approximation appears valid all over the space–time as long as the source is weakly gravitating, it faces hurdles while treating the multipole approximation outside the source with respect to the far zone expansion.\
In the early 1970s, while several groups were still trying to check Weber’s claim of the detection of gravitational waves, an altogether different set of observations confirmed the existence of gravitational waves indirectly. Hulse and Taylor, during a routine search for pulsars, from the Arceibo Observatory had recorded several new pulsars, amongst which was the discovery of the first binary pulsar PSR 1913 + 16, which was identified as a set of two neutron stars with almost equal masses, ($ M_p = 1.39\pm 0.15 M_\odot,\, M_c = 1.44\pm 0.15 M_\odot), $ moving on a fairly eccentric orbit ($ e = 0.617155\pm 0.000007 $), quite close to each other having the projected semi–major axis $ a \sin i \sim 7\cdot 10^{10} cm$ [@ht75]. Continuous monitoring of the binary pulsar over the next few years, yielded a much better evaluation of the orbital parameters [@taylor79], which clearly revealed the binary pulsar system to be the best laboratory for testing general relativity. As summarised by Weisberg and Taylor (2005) the measured orbital parameters over the period, 1981– 2003, are as listed in the table below [@wt05]\
[ c c ]{} fitted parameter & value\
\
$ a_P \sin i $(s) & 2.3417725 (8)\
\
$ \omega_0 $ & 292.54487 (8)\
\
$ e $ & 0.6171338\
\
$ \langle \dot\omega\rangle $(deg/yr) & 4.226595 (5)\
\
$ T_0$ & 52144.90097844(5)\
\
$ \gamma $(s) & 0.0042919 (8)\
\
$ P_b $ & 0.322997448930(4)\
\
$ \dot P_b(10^{-12}s/s)$ & -2.4184(9)\
\
\[table 1\]
While the first five parameters of the table are derivable purely from non–relativistic analysis, the next three, the mean rate of advance of periastron $\langle \dot \omega^i\rangle $, gravitational redshift and time-dilation parameter $ \gamma $ and the orbital period derivative $ \dot P_b $ come only from general relativistic corrections.
One of the most important results pointed out was the fact that the orbital period of the system was changing as given by $ \dot P_b $, which can happen only with the loss of the orbital energy bringing the two components closer. Taylor et al. found the secular decrease of the orbital period to be consistent with loss of energy through emission of gravitational radiation as predicted by general relativity [@tw82], which is calculated on the basis of suggestion from Wagoner[@wago75], and Esposito & Harrison[@esphar75], using the analysis of Peters and Mathews [@pm63], as given by $$\begin{aligned}
\label{pbdot}
\dot P_b = -\frac{192 \pi G^{5/3}}{5 c^5}({\frac{P_b}{2\pi}})^{-5/3} ({1-e^2})^{-7/2}\cdot (1+\frac{73}{24}e^2 + \frac{37}{96})\nonumber\\ e^4 [m_P m_c / (m_P + m_c)^{-1/3}].\end{aligned}$$ As the relativistic variables $ \langle \dot \omega\rangle $ and $ \gamma $, both measurable quantities, depend upon the masses of the binary components as given by $$\langle \dot \omega\rangle = 3 G^{2/3} c^{-2} (P_b /2\pi)^{-5/3} (1-e^2)^{-1} (m_p + m_c)^{2/3}$$ and $$\gamma = G^{2/3} c^{-2} e (P_b /2\pi)^{1/3} m_c (m_P + e m_c) (m_p + m_c)^{-4/3},$$ inserting the measured values and solving for the masses, one finds $$m_P = 1.4408 \pm 0.0003 M_\odot, ; \quad m_c = 1.3873 \pm 0.0003 M_\odot.$$ Using these in the above , one can get the orbital period decay rate to be, $ (\dot P_b)_{GR} = (-2.40247 \pm 0.00002)\times 10 ^{-12} s/s. $\
![(a)Orbital decay of PSR 1913+16 during 1975 to 2003,producing the change in period decay (b) orbit changes leading to coalescence schematic [@wt05].\][]{data-label="orbdecay"}](htbinary.eps){width="10cm"}
.
As Damour and Taylor [@dt91] argue, there would be some effect on the periods, both theoretical and observational, as a result of galactic acceleration of the system and the motion of the sun, which in fact has several components that add up to $ (\dot P_b / P_b)^{obs} = -86.79 \pm 0.19 (gal) \pm 0.65(obs) 10^{-18}/sec $, and the corresponding theoretical estimate yields $ (\dot P_b /P_b)^{GR} = -86.0923 \pm 0.0025 (gal) 10^{-18}/sec ,$ yielding the ratio of the observed to the theoretical values of the periods to be\
$$(\dot P_b)^{obs}/(\dot P_b)^{GR} = 1.0081 \pm 0.0022(gal)\pm 0.0076 (obs),$$ which is an excellent agreement. This orbital decay in period due to gravitational radiation damping should cause a shift in the epoch of periastron as shown in the figure \[orbdecay\], where the theoretical curve (solid line) and the observed data points are plotted [@wt05], which shows the remarkable agreement of the data collected over almost thirty years.\
According to Blanchet [@blanchlr14], to observe the final stages of the inspiralling binary coalescence, by the ground based detectors, one requires very high accuracy templates as predicted by general relativity, and this is achieved by using a higher order post–Newtonian wave generation formalism. This has indeed been achieved to a good degree of applicability, and a host of investigations seem to have demonstrated that the post–Newtonian precision required to successfully implement an optimal filtering technique for the existing detectors (LIGO and VIRGO) to correspond upto 3PN order ($ c^{-6} $) for neutron star binaries, beyond the quadrupole moment. ([@cutler93], [@fin93], [@tagoshi94], [@poisson97],[@krolak95],[@damour98]). Whereas these techniques of calculations would suffice to discuss wave emission from binary neutron stars and white dwarfs, they would be found wanting when it comes to the discussion of binary black holes, particularly when one of the components is massive. Modeling the merger of two black holes requires numerical relativity [@sslr12], as calculating the wave forms (templates)requires full solutions of Einstein’s equations.\
As reviewed by Centrella et al [@centbaker10] mergers of comparable-mass black-hole binaries are expected to be among the strongest sources of gravitational waves, wherein the final death spiral of a black-hole binary encompasses three stages called inspiral, merger, and ringdown phases. During the inspiral phase, the orbits of the binaries get circularised due to the emission of gravitational waves and further the black holes spiral together in quasi–circular orbits, as the orbitaltime scale would be much shorter than the timescales on which the orbital parameters change. Due to the large separation between the components one can treat them as point particles and thus apply the orbital dynamics as was done for the case of neutron star binaries [@peters64]. The wave forms can be calculated using the post–Newtonian equations in terms of $ v^2/c^2 \sim GM/Rc^2 $, R being the binary separation [@blanchet06], and one finds that the wave form would have the characteristic of a ‘chirp’, as defined earlier. As the black holes get closer, the weak field limit will not be valid in the merger phase, as the strong field dynamical region of general relativity requires the numerical treatment of Einstein’s equations (a three–dimensional simulation of solutions). At this stage the black holes get close enough to merge and form a single, bigger black hole which could be highly distorted. Finally, this distorted remnant black hole could settle down as a Kerr black hole, after shedding all the nonaxisymmetric modes in the form of gravitational radiation known as ‘ringdown’ phase.\
The order of magnitude estimates for the amplitudes of the waves emitted at different phases is given by [@sslr12].\
(a)the inspiral phase; $ h_b \sim 2 M^2/ r R \simeq ( 2/r) M^{5/3} \Omega^{2/3} $, (M the mass, R is the orbital radius, r distance to the source, $ \Omega $ the orbital angular frequency), with the luminosity, $ L_b \sim (4/5 G)(M c/ R)^5 $. As the orbital radius shrinks, the emitted frequency increases towards a chirp, with chirp time for equal mass binary to be $ t_{chirp} = Mv^2 /2 L_b \sim (5M/8)(M/R)^{-4}$.\
(b) As the merger stage approaches, with the distance between the components closer to the last stable orbit $(R \sim 6M) $, the frequency reaches the value, $ f_{lso} \sim 220 (20 M_\odot /M)Hz.$ In the case of unequal mass binaries the coalescing time as measured from the rate of period change, $ \dot P_b = -\frac{192 \pi}{5}(\frac{2\pi\sM}{P_b})^{5/3},$ is $ t_{chirp} =(\frac{5 M}{96 \nu} ( \frac{M}{R})^{-4}$, where M is the total mass of the two components, and $ \sM = \nu^{3/5} M $, the chirp mass, with $ \nu = \mu/M $. one can see from these numbers that, while the binaries with large mass ratios can spend a long time in highly relativistic orbits, those with equal mass are expected to merge after being in this regime for only a few orbits.\
It may be pointed out that the famous binary, Hulse–Taylor pulsar is expected to merge in just about 300 million years as the orbit is shrinking at the rate of $ \sim 3.1 mm /orbit. $\
In the case of massive black hole binaries, as they will be perturbed as they get closer, it is necessary to understand the evolution of black hole perturbations. Vishveswara [@vish70] was the first to discuss the consequence of black hole perturbation by the back scattering of the gravitational waves, following an approach initiated by Regge and Wheeler [@regwhee57], for the case of Schwarzschild blackhole, which was followed up with detailed discussions by Zerrilli, [@zer70], and later for the perturbations of the Kerr metric by Teukolsky [@teuk72]. However, the most detailed discussion of the perturbations of blackhole spacetimes was done by Chandrasekhar et al, which can be studied from [@chandra].\
These perturbed black holes were found to exhibit ‘quasi–normal modes’ of vibration that emit gravitational radiation whose amplitude, frequency, and damping time are characteristic of the black hole‘s mass and angular momentum, the only two features of a Kerr black hole. The effective amplitude of the waves is of the form $ h_{eff} \sim \frac{4\alpha \nu M}{\pi r}$, which, for a pair of 10$ M_\odot $ black holes, at a distance of about 200 Mpc, turns out to be $ (\frac{\nu}{0.25})(\frac{M}{20 M_\odot})( \frac{200 Mpc}{r}) 10^{-21}$, and for super massive black holes at cosmological distance is $ 3 \times 10^{-17} (\frac{\nu}{0.25})(\frac{M}{2\times 10^6 M_\odot})( \frac{6.5 Gpc}{r}) $[@sslr12]. As the equations of general relativity are a set of coupled nonlinear, second order partial differential equations, the details of the dynamics of the merger of black holes are not accessible for analytic treatment and one resorts to numerical approach.\
As pointed out by several reviewers, Hahn and Lindquist [@hanlin64] seem to be the first in 1964, to have tried the simulation of the dynamics of, head-on collision of two equal mass black holes, using a two dimensional axisymmetric approach, which they found was not being accurate after 50 time steps. Almost after a decade, Smarr et al. reconsidered the problem, using the ADM formalism (canonical 3+1 formalism [@adm62]) with improved coordinate conditions, which led them in spite of the difficulties of instabilities, and large number errors, to some information about the spectrum and total energy of the gravitational waves emitted in the zero frequency limit [@smarr77].\
However, the necessity to use numerical methods and computer simulation gained importance with the attempts to detect gravitational waves in the beam detector–like LIGO, in the nineties, as they are sensitive only at the frequencies emitted by black hole mergers. As the signal–to–noise ratios of ground based detectors are fairly modest, constructing templates to pattern the wave forms for this device was very important for data analysis which required numerical simulations. This activated several groups of numerical relativists trying to develop three dimensional codes for relativistic hydrodynamics using super computers which became important [@centbaker10],[@cghs15]. The successful application of numerical methods and simulations during the period 1990 to 2006, with the revolutionary idea of Pretorius [@pretorius05], advanced the developments in numerical relativity as applied to the detection of gravitational waves immensely, resulting in the final detection of gravitational waves by LIGO/VIRGO collaboration in 2016.\
![The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, right column panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series are filtered with a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and band-reject filters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra[@abbott16][]{data-label="grwvdet"}](grawvdetect.eps){width="10cm"}
It is well known that the electromagnetic waves, passing through the intervening matter between the source and the observer, do undergo some changes like frequency modulation, Landau damping etc.,which in a way gives one the information about the medium through which the waves are passing. Can there be any similar effect on gravitational waves passing through matter? Further the analysis above is restricted to purely linearised solution of Einstein’s equations. As the intervening space is not empty it is important to consider the space-time perturbations for non-flat metric. This question was considered by Ehlers et al, a short discussion of which follows.\
**Propagation of gravitational waves through matter**\
Starting with the perturbations on a non flat background $ g_{ij}= (g_{ij})_B + \hat g_{ij} $ one can set up the equations for perturbations, for the case of perfect fluid distributionas given by[@ep96] $$\begin{aligned}
\label{equnspertbs}
\ts P{^{ij} _ {ab}} \hat g_{ij} \equiv &[(2 h^i _{(a} h^c _{b)}\nabla^j \nabla^c - h^i _a h^j _b \nabla^2 - g^{ij} h^c _a h^d _b \nabla_c \nabla_d) +\alpha h_{ab}(g^{ij} \nabla_u ^2 +\nonumber\\& 2 \nabla^{(i} u^{j)}\nabla_u + 4(\nabla_d u^i \nabla^{[d} u^{j)})) - (\rho - p+ 2\Lambda) h^i _a h^j _b ] (\hat g_{ij}) = 0. \end{aligned}$$ The characteristic equation, given by the zeroth order equation of the above hirarchy,gives the dispersion relation which is expressed as $$(g^{ab}l_a l_b)^2 [(u^a u^b - c_s ^2 h^{ab})l_a l_b ] (u^a l_a)^6 = 0.$$ shows that there exist three modes\
(i) the *gravitational wave* mode, given by the Hamiltonian $ H = \frac{1}{2} g^{ab} l_a l_b$ propagating along the null geodesics having the tangent vector $ T^a = l^a $,\
(ii) the *sound wave* mode given by $ H = \frac{1}{2} [(c_s ^2 h^{ab}-u^a u^b)] $, propagating along the sound rays, with tangent $ T^a = \omega (\frac{c_s k^a}{k} + u^a)$ , and\
(iii)the *matter* mode given by $ H = u^a l_a$ , moving along matter rays, $ T^a = u^a $, From the general formalism,that can be referred to in [@ep96],one sees that while the zeroth order gives the dispersion relation,the first order gives the transport equation from which one can set for the primary amplitudes the set of ordinary differential equations, $$\label{trangrwv}
(\nabla_l + \frac{\theta}{2})\begin{pmatrix}
a^{(0)} _+ \\
a^{(0)} _\times
\end{pmatrix} =0.$$ This implies that the change of the complex vector $ (a_+, a_\times) $ along a ray consists of a rescaling by a positive factor proportional to the square root of the cross–sectional area of a small bundle of rays, just as in the case of gravitational waves in *vacua*. The transport preserves linear, circular, elliptic polarisation, helicity and ellipticity. Further it also implies that the Issacson stress tensor (defined in vacuum)$$\hat T ^{ab} = \frac{1}{4\pi}(\vert a_+\vert ^2 + \vert a_\times\vert^2) l^a l^b,$$ which represents the effective energy momentum tensor of the wave, is conserved, $ \nabla_a \hat T^{ab} = 0. $\
The transport equation for the first order primary amplitudes is then given by,$$\begin{aligned}
\label{transp1}
(\nabla_l + \theta /2) (a_+ ^{(1)}) = &\frac{1}{2}(\rho - p + 2\Lambda)((a_+ ^{(0)}) + \nonumber\\ &+ \frac{1}{2} e^{ij}\{[2 \nabla^ c\nabla_i +\nabla^i \nabla_c)\delta^d _j - \delta^c _i \delta^d _j\nabla^2 - h^{cd}\nabla_j \nabla_i] (v_{0 cd})\}\end{aligned}$$ and a similar one for $ a_\times $.\
Using the definitions of the curvature tensor, $ \ts R{^h _{ijk}} $, and the Weyl tensor, $ \ts C{^h _{ijk}} $, along with the field equations,one finds that in a conformally flat $ (\ts C{^h _{ijk}}) = 0 $ background space time the above equation reduces to $$\begin{aligned}
\label{transp3}
(\nabla_l + \theta /2) (a_+ ^{(1)})+ \frac{R}{3}((a_+ ^{(0)}) = \frac{1}{2} e_+ ^{ij}[4 \nabla_i \nabla^c \delta_j ^d - \delta^c _i \delta^d _j\nabla^2 - h^{cd}\nabla_j \nabla_i)](v_{0 cd}).\end{aligned}$$ which exhibits the possibility of the background curvature R and the nonlinear derivatives of the primary amplitudes $ v_0 $ possibly influencing the transport of $ v_1 $, the correction to the primary amplitude. Instead of a perfect fluid if one has a dissipative fluid with the energy momentum tensor $$T_{ij} = (\rho + p)u_i u_j + p g_{ij} - 2\eta \sigma_{ij} -\zeta\theta h_{ij},$$ where, apart from the usual definitions of $ p, \rho, u^i, $ one has $ \eta, \zeta, \sigma_{ij},\theta $ representing the shear viscosity, the bulk viscosity, the shear tensor, and the scalar of expansion, respectively then the perturbed field equations[@arp99] $$\hat R_{ij} = \kappa [\hat T_{ij} - \frac{1}{2}( g_{ij} \hat T + \hat g_{ij} T)]$$ give, after using the gauge condition $ \hat g_{ab}u^b = 0, $ along with the fact that the unperturbed streamlines are geodesics, the set of equations $$\ts H{^{ij} _{ab}}\hat R_{ij} = (\kappa/2)[\hat g_{ab} (\rho -p + \zeta \theta) + h_{ab}(4\zeta \hat \theta/(1+3c_s ^2))-4 \eta \hat \sigma_{ab}],$$ with $$\begin{aligned}
&\ts H{^{ij} _{ab}} = h^i _a h^j _b - \alpha h_{ab} u^i u^j,\\
&\hat \theta = \hat u^k _{, k}+\frac{g^{ka}}{2} (g_{ka,b} \hat u^b + \hat g_{ka,b}u^b) + \frac{\hat g^{ka}}{2} g_{ka,b} u^b, \\
&\nabla_j \hat u_i = \hat u_{i,j} - \frac{u^b}{2} (\hat g_{ib,j} + \hat g_{jb,i}- \hat g_{ij,b}) + \{ ij,b \} (u_k \hat g^{kb} + \hat u^b). \end{aligned}$$ Applying the high frequency approximation and associated ansatz and equating the coefficients of $ \epsilon $ terms, the leading order $ \epsilon^{-2} $ yields the dispersion relation as earlier, with its determinant being referred to the tetrad $ (u^a, k^a, e^a _1, e^a _2) $ $$l^4 \omega^6 [\omega^2 - c_s ^2 k^2] = 0,$$ which gives, as in the case of perfect fluid distribution, the three modes $ l^2 = 0,$ corresponding to the gravitational waves,moving along the null rays with $ T^a = l^a, $ the sound wave mode $ H = \frac{1}{2} [c_s^2 h^{ab} - u^a u^b)l_a l_b $, with rays having the tangent $T^a = \omega(c_s k^a/k + u^a) $ and the matter mode $ H = u^a l_a $, with tangent vector $ u^a. $\
If one now considers the quasi-parallel transport of $ e^a _1 $ and $ e^a _2 $ as defined in [@epb87], and simplifies the transport equation for the primary amplitudes, one gets the simple relation,[@arp99]$$[l^i\nabla_i + \frac{1}{2} \nabla_i l^i +\kappa\eta\omega] e_+ ^{ab} f_{ab} = 0 \Rightarrow [\nabla_l + \frac{1}{2} \nabla_i l^i + \kappa\eta\omega)\begin{pmatrix}
a^{(0)} _+ \\
a^{(0)} _\times \end{pmatrix} = 0,$$ an equation similar to the one with perfect fluid but with an extra term proportional to the viscosity $ \eta.$\
Writing in terms of the total amplitude $ A^2 = 2(\vert a^+ \vert^2 + \vert a^\times \vert^2 ), $ the equation for the amplitude transport in the dissipative fluid for the gravitational waves, comes out to be \[Prasanna 99\]$$( D + \nabla^i l_i ) A^2 = - 2 \kappa \eta \omega A^2,$$ clearly showing the presence of a damping term due to shear viscosity, which seems to indicate that in the presence of viscosity, the propagation of gravitational waves could be influenced by the medium, a result that requires further investigation.Thus *the detection of gravitational waves could be opening anew window to look at the unknown Cosmos more effectively*.\
Finally it is worthwhile to consider how the LIGO project was commissioned and carried out to finally achieve the findings. “LIGO research is carried out by the LIGO scientific collaboration (LSC), a group of more than 1000 scientists of the collaboration.from universities around the U.S. and in 14 other countries. More than 90 universities and research institutes in the LSC develop detector technology and analyze data; approximately 250 students are strong contributing members of the collaboration.The LSC detector network includes the LIGO interferometers and the GEO600 detector.The GEO team includes scientists at the MPI for Gravitation physics (Albert Einstein Institute AEI), Leibnitz Universitat, Hannover alongwith partners at the university of Glasgow, Cardiff university, the university of Burmingham and few other universities in the UK,and the university of Balearic islands in Spain". The Indian efforts in the successful detection of gravitational waves, has given stimulus to the project LIGO-India, also known as INDIGO which is a planned advanced gravitational wave observatory to be located in India as part of the world wide network. The project received in principle approval from the government of India in March 2016. LIGO-India is planned as a collaborative project betwen a consortium of Indian research institutes and the LIGO laboratory in the USA, along with its international partners in Australia, Germany and the UK. Thus the Indian scientific community from research institutes and universities,faculty and students (of Physics, Mathematics and Engineering) have a very ambitious goal to look forward to in observing analyzing and creating new science of the Cosmos stimulating both the academics and the intellect.
[99]{}
J. L. Anderson,*Principles of Relativity Physics*, Academic Press, (1972) R. Arnowitt, S. Deser, and C. W. Misner, ‘The dynamics of General Relativity’, in ,*Gravitation: An Introduction to Current Research,*,L. Witten, ed. pp. 227–265, (Wiley, New York; London, 1962). S. Chandrasekhar,*The Mathematical Theory of Blackholes*, Clarendon Press, Oxford, (1992) M. Demainski, *Relativistic Astrophysics*, Pergamonn Press, (1985) G.Esposito-Fares,*Binary-pulsar tests of gravity theories* workshop in “Pulsars, Theories and observations”, 2006. L. S. Finn, and D. F. Chernoff, *Phys.Rev.* D, 47, p 2198, (1993).
Ray. d’Inverno,*Introducing Einstein’s Relativity*, Clarendon Press, Oxford,(1992) L. D. Landau and E. M. Lifshitz, *Classical Theory of Fields*, Pergamon Press, (1951) C. W. Misner, K. S. Thorne, and J. A. Wheeler, *Gravitation*, Freeman and Co (1973) A. R. Prasanna, *Space and Time to space–time*, Universities Press, (2008) T. Regge and J. A. Wheeler, *Phys. Rev.*, 108, p 1063, 1957. M. Schawrzschild,*Structure and Evolution of the Stars*, Dover Publications, (1958) B. F. Schutz, *A First Course in General Relativity*, (Cambridge University Press, Cambridge;New York, (2009), 2nd edition.
S. Weinberg, *Gravitation and Cosmology,Principles and Applications of General Relativity*, John Wiley, (1972) B. P. Abbott et al., LIGO collaboration and Virgo collaboration, *Phys.Rev.Letts*, 116, 061102, 2016.
A. Abramovici et al., *Science*, 256, p 325, 1992. L. Blanchet, ‘Gravitational Radiation from post–Newtonian Sources and Inspiralling Compact Binaries’, in *Living Rev. Relativity,* 17, 2, 2014. DOI 10.12942/lrr-2014-2. L. Blanchet, A. Buonanno, and G. Faye, *Phys. Rev.* D, 74, 104034, 2006. J. Centrella, J. Baker, B. J. Kelly and J. R. vanMeter, *Blackhole binaries,gravitational waves and Numerical Relativity*, arXive:1010.5260v2,\[gr-qc\],2010. V. Cardoso, L. Gulatieri, C. Herdiero, and U. Sperhake, ‘Exploring New Physics Frontiers Through Numerical Relativity’, Living Rev. Relativity, 18, 2015. 1.DOI 10.1007/lrr-2015-1. C. Cutler, L. S. Finn, E. Poisson, and G. J. Sussman,*Phys. Rev.* D, 47, p.1511, (1993). C. Cutler and E. E. Flanagan,*Phys. Rev.* D, 49, p 2658, 1994. C. Cutler, et. al., *Phys. Rev. Lett.*, 70, p 2984, (1993). T. Damour, *Phys. Rev. Letts*, 51 (12), p 1019, 1983. T. Damour, B. R. Iyer, and B. S. Sathyaprakash,*Phys. Rev.* D, 57, p 885, 1998. T. Damour and J. H. Taylor,*Ap.J.*, 366, p 501, 1991. J. Ehlers, A. R. Prasanna,and R.A.Breuer, *Class. Quantum. Grav.*, 4, p 253, 1987. J. Ehlers, and A. R. Prasanna, *Class.Quantum.Grav.*,13, p 2231,1996. L. W. Esposito and E.R.Harrison,*Ap. J. Letters*, 196, L1, 1975. S. G. Hahn, and R. W. Lindquist, *Ann. Phys.* 29, p 304, 1964. R. A. Hulse and J. H. Taylor,*Astrophys.J.*, 195, L51, 1975. R. A. Issacson,*Phys.Rev*, 166, p 1263, 1968. A. Kr‘olak, K. D. Kokkotas, and G. Schaefer, *Phys. Rev.* D, 52, p 2089, 1995. P. C. Peters and J. Mathews,*Phys.Rev.*, 131, p 435, 1963. P. C. Peters,*Phys. Rev.*, B 136, p 1224 1964. E. Poisson,*Phys. Rev.* D, 47, p 1497, 1993.; E. Poisson, *Phys. Rev.* D, 52, p 5719, 1995; Erratum: *Phys. Rev.* D, 55, 7980, 1997; E.Poisson, *Phys. Rev.*D, 57, p 5287, 1997. A. R. Prasanna, *Phys.Letts.* A 257, p 120, 1999. F. Pretorius,*Phys. Rev. Lett.* 95, 121101. 2005.; F. Pretorius,*Class. Quantum Grav.* 22, p 425, 2005.;F. Pretorius, *Class. Quantum. Grav.* 23, S 529. 2006.; F. Pretorius, *‘Binary Black Hole Coalescence’,* in *Physics of Relativistic Objects in Compact Binaries: from Birth to Coalescence,* edited by M. Colpi, P. Casella, V. Gorini, U. Moschella, and A. Possenti (Springer, Heidelberg, Germany), p 305–369, 2009 ; eprint arXiv:0710.1338 \[gr-qc\]; F. Pretorius, and D. Khurana,*Class. Quantum Grav.* 24, S83, 2007. T. Regge and J. A. Wheeler, *Phys. Rev.*, 108, p 1063, 1957. B. S. Satyaprakash and B. F. Schutz, *Living Rev.Relativity*, 12, 2009. R. K. Sachs, *Proc. Roy. Soc., London* A 270, p 103, 1962. L. L. Smarr,*Ann. N. Y. Acad. Sciences* 302, p 569, eighth Texas Symposium on Relativistic Astrophysics.(1977.; L. L .Smarr, A. Cadez, B. S. DeWitt, and K. Eppley,*Phys. Rev.* D 14(10), p 2443, 1977.; L. L. Smarr, and J. W. York Jr., 1978,*Phys.Rev.* D 17, p 2529, 1978. H. Tagoshi, and M. Sasaki,*Prog. Theor. Phys.*, 92, p 745,(1994),; H. Tagoshi, S. Mano, and E. Takasugi,*Prog. Theor. Phys.*, 98, p 829, 1997. J. H. Taylor, L. A. Fowler, and P. M. McCulloch, *Nature*, 277, p 437, 1979. J. H. Taylor and J. M. Weisberg, *Ap.J.*, 253. p 908, 1982. S. A. Teukolsky,*Phys.Rev.Letts.*, 29, p 1114, 1972,; *Astrophys.J.*, 185, p 635, 1973. K. S. Thorne, ‘Gravitational Radiation’,in *Three hundred years of Gravitation*, Hawking and Israel,eds, Cambridge.University.Press, 1987. K. S. Thorne,*Reviews in Modern Astronomy*, 10, . R.E. Schielicke,ed (Astronomische Gesellschaft), pp. 1-28. 1997. C. V. Vishveswara, ‘Blackholes for Bedtime’, in *Gravitation,Quanta and the Universe* A. R. Prasanna, J. V. Narlikar an C. V. Vishveswara.eds, Wiley Eastern, 1980. C. V. Vishveswara, *Nature*, 227, p 936, 1970.;C. V. Vishveswara,*Phys.Rev*D 1, p 2870, 1970. R. Wagoner, *Ap. J. Letters*, 196, L 63, 1975. J. M. Weisberg and J. H. Taylor, ‘The Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis’, in *Binary Radio Pulsars,Proceedings of a meeting held at the Aspen Center for Physics*,F.A.Rasio, and I.H.Stairs, eds, USA, 12 – 16 January 2004, ASP Conference Series, vol. 328, pp. 25–32, (Astronomical Society of the Pacific, SanFrancisco, 2005). J. Weber, *Phys.Rev.*,117, p 306, 1960. C. M. Will, ‘The Confrontation between General Relativity and Experiment’, Living Rev. Relativity, 17, 2014.\[cited 2015\]: F. J. Zerilli, *Phys. Rev.*D,2, p 2141, 1970..
[^1]: **Vaidya - Raichaudhuri award lecture**, delivered on 22.September 2016, Department of Physics, Lucknow University
|
---
abstract: 'A magnetostatic spin wave analog of integer quantum Hall (IQH) state is proposed in realistic patterned ferromagnetic thin films. Due to magnetic shape anisotropy, magnetic moments in a thin film lie within the plane, while all spin-wave excitations are fully gapped. Under an out-of-plane magnetic field, the film acquires a finite magnetization, where some of the gapped magnons become significantly softened near a saturation field. It is shown that, owing to a spin-orbit locking nature of the magnetic dipolar interaction, these soft spin-wave volume-mode bands become chiral volume-mode bands with finite topological Chern integers. A bulk-edge correspondence in IQH physics suggests that such volume-mode bands are accompanied by a chiral magnetostatic spin-wave edge mode. The existence of the edge mode is justified both by micromagnetic simulations and by band calculations based on a linearized Landau-Lifshitz equation. Employing intuitive physical arguments, we introduce proper tight-binding models for these soft volume-mode bands. Based on the tight-binding models, we further discuss possible applications to other systems such as magnetic ultrathin films with perpendicular magnetic anisotropy (PMA).'
author:
- Ryuichi Shindou
- 'Jun-ichiro Ohe'
title: Magnetostatic wave analog of integer quantum Hall state in patterned magnetic films
---
u ł[l]{}
introduction
============
Spin-wave propagations in magnetic insulators realize spin transports with less dissipation, [@YIG; @KDG] fostering much prospect for realizations of future spintronic devices. For the purpose of device applications, spin-wave transport in two-dimensional systems such as thin films is expected to have many advantages. In ferromagnet thin film, moments lie within the plane to minimize the magnetostatic energy. A thin film with the in-plane magnetization has a surface spin-wave mode called Damon-Eshbach (DE) mode, [@DE1] where spin wave propagates in a chiral direction transverse to the in-plane moment. The mode realizes a unidirectional spin transport in the two-dimensional (2-$d$) top surface of the film and the counter-propagating transport in the bottom surface. The mode enables a number of spin-wave spintronic devices. [@Kostylev; @Lee; @Schneider; @Sato]
Recently, the present author proposes chiral spin-wave edge mode in a 2-$d$ [*periodically-structured*]{} dipolar magnetic thin film with [*out-plane*]{} ferromagnetic moment. [@SO1; @SO2] The mode has a resonance frequency within a band gap of volume modes, where the gap and multiple-band structure of volume-mode bands come from the 2-$d$ periodic structuring. The chiral direction is transverse to the out-of-plane ferromagnetic moment; the mode realizes a unidirectional spin-wave propagation along the one-dimensional boundary of the plane, instead of along the top (or bottom) surface. Such chiral edge modes could possibly connect various elements in 2-$d$ spin-current circuits in more flexible way than the DE surface mode. Moreover, the chiral direction (whether clockwise or counterclockwise) and number of the edge modes (can be more than one) are determined by a sum of the topological number (Chern integer) defined for the volume-mode bands below the gap. [@TKKN; @Hal; @Hat; @SO1; @SO2] This enables us to control the direction and number of the edge modes in terms of a band gap manipulation, bringing up further prospect for spin-current circuits with richer structures. [@SO1] To make such spin-wave circuits experimentally, it is much more important for theory to propose a number of structured thin films which have these topological modes.
In this paper, we introduce an efficient method of constructing the topological chiral edge modes in realistic dipolar magnetic thin films. We considered that magnetic clusters, either thin rings or circular disks, form a 2-$d$ periodic square lattice. To study their spin-wave excitations, we derive several tight-binding models, using intuitive physical arguments. Based on these models, we show that soft volume-mode bands near the saturation field acquire finite topological Chern integer, resulting in chiral spin wave edge modes within band gaps of volume mode bands.
The organization of the paper is as follows. In sec. II, we consider a ring model; circular magnetic thin rings forming a square lattice (Fig. \[fig:cluster0\](a)). We first introduce an ‘atomic-orbital’ like wavefunction for spin-wave excitations within each ring. Using these atomic orbitals, we construct tight-binding models for soft magnons. The models naturally lead to chiral volume-mode bands and edge modes (Fig. \[fig:1.2-1\] and Fig. \[fig:1.5-2\]). In sec. III, we further extend the argument to a disk model, circular magnetic disks forming a square lattice (Fig. \[fig:cluster0\](b)). The same type of chiral spin-wave edge modes are shown to appear in low-frequency regions near the saturation field (Fig. \[fig:cir-band\]). To justify the existence of the chiral edge modes by a standard method in the field, we also carried out in sec. IV micromagnetic simulations in the proposed magnetic superlattices (Fig. \[fig:mc3\]). In sec. V, we further discuss possible application of the present theory to other systems such as ferromagnetic ultrathin film systems with the perpendicular magnetic anisotropy. Two appendices describe some details useful for understanding the main text. In the appendix A, we describe how wavelength-frequency dispersion relations for spin-wave volume-mode bands and edge mode bands (such as Fig. \[fig:1.2-1\], Fig. \[fig:1.5-2\] and Fig. \[fig:cir-band\]) are calculated from Landau-Lifshitz equations. In appendix B, we construct, in a more expliclit way, an effective tight-binding model for soft spin-wave excitations above the saturation field, which is helpful for understanding sec. II, and sec.IV in detail.
All the results presented in this paper are essentially scalable, since the models do not have any short-range exchange interactions; the saturation field, $H_c$, and spin-wave resonance frequency are scaled only by the saturation magnetization (per volume) $M_s$ (appendix A). We took $M_s$ to be typically on the order of unit in Fig. \[fig:ring-atlv\], Fig. \[fig:1.2-1\], Fig. \[fig:1.5-2\], Fig. \[fig:cir-atlv\] and Fig. \[fig:cir-band\], while it is on the order of GHz (see e.g. Sec.IV).
![ (Color online) Schematic top-view of two-dimensional patterned magnetic thin films (blue region represents magnetic media, while the other stands for the vacuum). An external magnetic field is applied perpendicular to the plane with out-of-plane moments. Either circular rings (a) or disks (b) form a square lattice. We assume that the film is sufficiently thin, so that there is no texture along the direction perpendicular to the plane.[]{data-label="fig:cluster0"}](fig0.eps){width="70mm"}
ring model
==========
To begin with, consider spin-wave excitations in a magnetic circular ring. When a linear dimension of a cross section of the ring is comparable to short-ranged exchange length $l_{\rm ex}$ of a constituent magnetic material, the ring may be treated as a one-dimensional chain of $M$ spins, which are coupled with one another via long-range dipole-dipole interaction. $M$ is the number of the spins along the ring and is on the order of $2\pi r/l_{\rm ex}$ ($r$ is the radius of the ring). Without the field, the magnetostatic energy is minimized by a vortex spin configuration: spins are aligned along the tangential direction of the ring. Under the out-of-plane magnetic field $H$, the vortex spin configuration acquires an out-of-plane moment which becomes fully polarized above the saturation field, $H>H_c$.
Suppose that the amplitude of each spin moment is fixed to be $M_s$. Excitations in each spin comprise two real-valued fields (transverse moments), so that the ring has $M$ numbers of complex-valued spin-wave modes, $\psi(\theta_j)$ with $\theta_j\equiv 2\pi j /M$ ($j=1,\cdots,M$). Under a proper gauge choice (see appendix A), they have total angular momentum $q_J$ as their quantum number, $$\begin{aligned}
\psi_{q_J}(\theta_j+\theta_m)=e^{iq_Jm}\psi_{q_J}(\theta_j), \label{eq2}\end{aligned}$$ which comes from the circular rotational symmetry of a ring. Here $\theta_j\equiv 2\pi j /M$ and $q_J\equiv 2\pi n_J/M$ $(n_J=-M/2,-M/2+1,\cdots,M/2)$. The resonance frequency for these ‘atomic orbitals’ is given as a function of the angular momentum, which forms a frequency band for larger $M$.
At the zero field, all the spins in the circular vortex is along the angular momentum axis (along the tangential direction of the ring), so that the frequency band at $H=0$ becomes essentially same as the ‘backward’ volume modes in an in-plane magnetized thin film [@DV2; @Kalinikos] or cylindrically magnetized nanowire. [@AM] Namely, the band has its resonance frequency minimum at $q_J=\pi$ and its frequency maximum at $q_J=0$. When increasing the out-of-plane field, the maximum and minimum are inverted at some ‘critical’ field below the saturation field, $H=H_d\simeq 0.8H_c$ (Fig. \[fig:ring-atlv\] (a,b)). The resonance frequency mode at $q_J=0$ becomes eventually gapless at $H=H_c$, being consistent with the classical spin configuration which starts to acquire finite in-plane components forming a circular vortex for $H<H_c$. For $H>H_c$, these excitations become gapped again with the minimum being at $q_J=0$. For $H\gg H_c$, a ‘band center’ of these resonance frequency levels converges to usual ferromagnetic resonance (FMR) mode. Note also that two time-reversal-pair modes, $q_J$ and $-q_J$, are degenerate at the zero field, while they are not under a finite field. For the out-of-plane field along $+z$ direction, $\varepsilon_{q_J}<\varepsilon_{-q_J}$ for $q_J\gtrsim 0$ (see Fig. \[fig:ring-atlv\](b,c,d)).
When a circular ring embedded into the square lattice, the quantum number for the atomic orbital reduces to either one of the following four, $q_J=0,\pm\frac{2\pi}{M},\frac{4\pi}{M}$. Namely, each ring feels an anisotropic demagnetization field from its surrounding rings, which respects four-fold rotational symmetry. This mixes any two states whose $q_J$ differs by $\frac{8\pi}{M}$ (Fig. \[fig:ring-atlv\] (c,d)). Under the four-fold rotation, these four atomic orbital wave functions acquire $+1$, $\pm i$ and $-1$, which suggests that they are essentially $s$-wave, $p_{\pm}=p_x\pm ip_y$-wave and $d_{x^2-y^2}$-wave function respectively; $$\begin{aligned}
\psi^{(n)}_{s}\big(\theta_j+\frac{\pi}{2}\big) &=
\psi^{(n)}_{s}\big(\theta_j\big), \label{s-w} \\
\psi^{(n)}_{p_{\pm}}\big(\theta_j+\frac{\pi}{2}\big) &= \pm i \!\
\psi^{(n)}_{p_{\pm}}\big(\theta_j\big), \label{p-w} \\
\psi^{(n)}_{d_{x^2-y^2}}\big(\theta_j+\frac{\pi}{2}\big) &=
- \psi^{(n)}_{d_{x^2-y^2}}\big(\theta_j\big). \label{d-w}\end{aligned}$$ Every fourth levels from below are grouped together in the frequency space, forming a branch specified by the superscript index $n$ (Fig. \[fig:ring-atlv\](d)); every branch includes the four types of wave functions, $s$, $p_{\pm}$, $d_{x^2-y^2}$-wave functions. The corresponding atomic orbital levels are arranged in the frequency space as $$\begin{aligned}
&\varepsilon^{(1)}_s < \varepsilon^{(1)}_{p_+}
< \varepsilon^{(1)}_{p_-}
< \varepsilon^{(1)}_{d_{x^2-y^2}} < \nn \\
&\ \ \varepsilon^{(2)}_{d_{x^2-y^2}} <
\varepsilon^{(2)}_{p_{-}}
< \varepsilon^{(2)}_{p_+} < \varepsilon^{(2)}_{s} <
\varepsilon^{(3)}_{s}
< \varepsilon^{(3)}_{p_+} < \cdots. \label{fre}\end{aligned}$$
![ (Color online) Resonance frequency levels in a magnetic ring (a $M$-spins chain with $M=60$) as a function of the angular momentum $q_J \equiv \frac{2\pi n_J}{M}$ with $n_J=-M/2,-M/2+1,\cdots,M/2$. (a) $H=0.7H_c$ and (b) $H=0.9H_c$. (c) When demagnetization fields from the surrounding magnetic rings are included, the angular momentum $n_J$ is defined mod $4$. (d) Wave functions with $n_J \equiv$ $-1$,$0$,$+1$, $+2$ (mod $4$) are referred to as $P_-$,$S$,$P_+$ $D_{x^2-y^2}$-wave respectively, since they acquire $-i$, $+1$, $+i$ and $-1$ phase under the $\frac{\pi}{2}$ spatial rotation (eqs. (\[s-w\],\[p-w\],\[d-w\])).[]{data-label="fig:ring-atlv"}](fig1.eps){width="85mm"}
When inter-ring ‘exchange’ processes via magnetic dipole-dipole interaction are included, these atomic orbitals constitute extended volume-mode bands. When neighboring branches are sufficiently separated from each other by the anisotropic demagnetization field, the volume-mode bands can be constructed out of each branch separately; $$\begin{aligned}
&\big\{\psi^{(2m+1)}_{s},\psi^{(2m+1)}_{p_+},\psi^{(2m+1)}_{p_-},
\psi^{(2m+1)}_{d_{x^2-y^2}}\big\}, \label{s1} \end{aligned}$$ or $$\begin{aligned}
&\big\{\psi^{(2m+2)}_{d_{x^2-y^2}},\psi^{(2m+2)}_{p_-},
\psi^{(2m+2)}_{p_{+}}, \psi^{(2m+2)}_{s}\big\}. \label{s2}\end{aligned}$$ Each branch provides four volume-mode bands. A qualitative feature of the four volume-mode bands can be roughly captured by a two-orbital model made out of the lower two atomic orbital wave functions within each branch; $$\begin{aligned}
\big\{ \psi^{(2m+1)}_s, \psi^{(2m+1)}_{p_+} \big\}, \ \
{\rm or} \ \ \big\{ \psi^{(2m+2)}_{d_{x^2-y^2}},
\psi^{(2m+2)}_{p_-} \big\}. \label{s3}\end{aligned}$$ This is because lower two atomic orbitals within each branch have less nodes than the other two along the ring. The inter-ring transfer integrals among such two are expected to be larger than those otherwise.
From the symmetry point of view, a nearest neighbor tight-binding model composed of the lower two orbitals is given by; $$\begin{aligned}
\hat{H}_{01} &= \sum_{\bm b} \big(\varepsilon_0 \gamma^{\dagger}_{0,{\bm b}}
\gamma_{0,{\bm b}} + \varepsilon_1\gamma^{\dagger}_{1,{\bm b}}
\gamma_{1,{\bm b}} \big) \nn \\
&\hspace{-0.4cm} - \sum_{\bm b}
\sum_{\mu=x,y} \sum_{\sigma=\pm} \big(a_{00}
\gamma^{\dagger}_{0,{\bm b}}
\gamma_{0,{\bm b}+\sigma{\bm e}_{\mu}}
- a_{11} \gamma^{\dagger}_{1,{\bm b}}
\gamma_{1,{\bm b}+\sigma{\bm e}_{\mu}} \big) \nn \\
& \hspace{0.5cm} - \sum_{\bm b}
\sum_{\sigma=\pm} \big(- \sigma b_{01} \gamma^{\dagger}_{0,{\bm b}}
\gamma_{1,{\bm b}+\sigma{\bm e}_x} + {\rm H.c.} \big) \nn \\
& \hspace{0.5cm} - \sum_{\bm b}
\sum_{\sigma=\pm} \big(-i\sigma b_{01} \gamma^{\dagger}_{0,{\bm b}}
\gamma_{1,{\bm b}+\sigma{\bm e}_y} + {\rm H.c.} \big). \label{sp-model}\end{aligned}$$ Here $\gamma^{\dagger}_{0,{\bm b}}$ ($\gamma_{0,{\bm b}}$) and $\gamma^{\dagger}_{1,{\bm b}}$ ($\gamma_{1,{\bm b}}$) stand for creation (annihilation) operators for parity-even and parity-odd atomic orbitals respectively. The subscript ${\bm b}$ denotes a coordinate of a center of a ring which the orbitals belong to. ${\bm e}_{\mu}$ is the primitive translation vector of the square lattice ($\mu=x,y$). The parity-even atomic orbital refers to $s$-wave or $d_{x^2-y^2}$-wave, while the parity-odd atomic orbital refers to $p_{\pm}$-wave: $$\begin{aligned}
\big\{\varepsilon_0,\varepsilon_1\big\}
=\big\{\varepsilon^{(2m+1)}_{s}, \varepsilon^{(2m+1)}_{p_+}\big\}, \ \
{\rm or} \
\big\{\varepsilon^{(2m+2)}_{d_{x^2-y^2}},
\varepsilon^{(2m+2)}_{p_{-}}\big\}, \nn\end{aligned}$$ so that $\varepsilon_0<\varepsilon_1$. A general observation of orbital shapes suggests that $a_{00}$, $a_{11}$ and $b_{01}$ are all positive real values under a proper gauge choice.
The tight binding Hamiltonian in the momentum space is expanded in term of the Pauli matrices as, $H({\bm k}) = c({\bm k}) \!\ {\bm \sigma}_0
+ \sum^{3}_{j=1} {\bm h}_{j}({\bm k}) \!\ {\bm \sigma}_j$ with $h_3({\bm k}) \equiv \epsilon_0-\epsilon_1 -
2(a_{00}+a_{11})(\cos k_x+\cos k_y)$, $h_1({\bm k})\equiv
2 b_{01} \sin k_y$ and $h_{2}({\bm k}) \equiv 2b_{01} \sin k_x$. In terms of a vector field ${\bm h}({\bm k})$, the topological Chern integer for the two volume-mode bands obtained from this Hamiltonian can be defined as a wrapping number of a normalized vector $\overline{\bm h}({\bm k})\equiv
{\bm h}({\bm k})/|{\bm h}({\bm k})|$. [@Volovik; @Yakovenko; @QWZ] The integer counts how many times the normalized vector wraps the unit sphere, when the momentum ${\bm k}$ wraps around the two-dimensional Brillouin zone with the torus geometry; [@Volovik; @Yakovenko; @QWZ] $$\begin{aligned}
c_{+} = -c_{-} = \int_{[-\pi,\pi]^2} \frac{d^2{\bm k}}{4\pi} \!\ \!\
\overline{\bm h}({\bm k}) \cdot \big(\partial_{k_x}
\overline{\bm h}({\bm k})
\times \partial_{k_y} \overline{\bm h}({\bm k})\big). \nn\end{aligned}$$ Within a two-band model, the integer for the upper band ($c_+$) always has an opposite sign to that for the lower band ($c_-$). When two nearest neighboring rings are spatially proximate to each other, larger exchange integrals realize $\varepsilon_1-\varepsilon_0 < 4 (a_{00}+a_{11})$, which makes the wrapping number to be unit. Namely, the unit vector points at the south pole/north pole ($\bar{\bm h}=(0,0,-1)$/$(0,0,+1)$) at ${\bm k}=(0,0)$/$(\pi,\pi)$, while the vector winds once around the south pole/north pole when ${\bm k}$ rotates once around the ${\bm k}=(0,0)$/$(\pi,\pi)$. This observation suggests that the Chern integers for two bands obtained from eq. (\[sp-model\]) become $\{c_-,c_+\}=\{-1,+1\}$. When the out-of-field direction is reversed, $p_{+}$ and $p_{-}$ are exchanged in
![ (Color online) Wavelength-frequency dispersions for spin-wave excitations for $H_d < H< H_c$ $(H=0.94H_c)$. (a) A side-view of lowest 8 volume-mode bands with the Chern integer. The red bands have $-1$ Chern integer, while blue bands have $+1$. The dispersions are calculated with periodic boundary conditions for both $x$ and $y$-directions. Since the 7th and 8th lowest band have frequency degeneracies around $M$-points, only the sum of their integers is quantized to $+1$. (b) Spin-wave excitations calculated with an open/periodic boundary condition along the $y$/$x$-direction respectively. The resonance frequencies are given as a function of the wave vector along the $x$-direction. The system along the $y$-direction includes 18 square-lattice unit cells ($L=18$). More than $75\%$ of amplitudes of eigen wave functions with red points are localized within $y=1$ and $y=2$, while those with blue points are localized within $y=L-1$ and $y=L$ (edge modes). Compared with Fig. (a), the calculated spectra have additional spin-wave modes which are localized along the edges. (c) With the out-of-plane field up-headed, the chiral edge modes rotate in the counterclockwise way.[]{data-label="fig:1.2-1"}](fig2.eps){width="85mm"}
Fig. \[fig:ring-atlv\](d) and eqs. (\[s1\],\[s2\]), which changes the sign of the last term in eq. (\[sp-model\]) and that of $c_{\pm}$. Note also that, to have the non-zero wrapping number, it is essential that ‘wave function character’ for the lower/higher band at ${\bm k}=(\pi,\pi)$ is parity odd/even atomic orbital, while that at the ${\bm k}=(0,0)$ is parity-even/odd one (‘band inversion’). [@BTZ; @FK] When $\varepsilon_1-\epsilon_0 > 4 (a_{00}+a_{11})$, wave function character of the lower (higher) band at ${\bm k}=(\pi,\pi)$ and that of ${\bm k}=(0,0)$ have same parity, so that the unit vector always stays within the southern hemisphere, irrespective of the momentum ${\bm k}$; the wrapping number always reduces to zero.
The argument so far suggests that, in the presence of larger inter-ring transfer integrals, the distribution of the Chern integers for soft volume-mode bands at $H_d<H<H_c$ can be non-trivial and is composed of a sequence of $\{-1,+1,0,0\}$ from below; $$\begin{aligned}
&\big\{c_1,c_2,c_3,c_4\!\ \big|,c_5,c_6,c_7,c_8\!\ \big|,\cdots\big\} \nn \\
& \ \ = \big\{-1,+1,0,0\!\ \big|,-1,+1,0,0\!\ \big|,\cdots\big\} \label{6}\end{aligned}$$ where $c_n$ denotes the integer for the $n$-th lowest band (see also appendix A for general definition of the topological Chern integer for volume-mode spin-wave bands). An explicit calculation of the Chern integers for volume-mode bands within $H_d < H< H_c$ based on a linearized Landau-Lifshitz equation confirms this feature with a minor modification. In the actual calculation, we also observed that, within each branch, another band inversion is often induced by relatively stronger exchange integrals between higher two atomic orbitals and the 2nd lowest atomic orbital, which transfer the non-zero integer of the 2nd lowest band into the 3rd or 4th lowest bands in each branch, $\{-1,+1,0,0\} \rightarrow
\{-1,0,+1,0\}$ or $\{-1,0,0,+1\}$. Which comes true among these three, i.e. $\{-1,+1,0,0\}$, $\{-1,0,+1,0\}$ and $\{-1,0,0,+1\}$, depends on specific branch and other details, while the integer for the lowest band ($-1$) remains intact in every branch ,e.g. $$\begin{aligned}
&\big\{c_1,c_2,c_3,c_4\!\ \big|,c_5,c_6,c_7,c_8\!\ \big|,\cdots\big\} \nn \\
& \ \ = \big\{-1,0,0,+1\!\ \big|,-1,0,\alpha,1-\alpha \!\ \big|,\cdots\big\},
\label{7}\end{aligned}$$ (Fig. \[fig:1.2-1\](a)).
General arguments [@SO1; @SO2] based on a bulk-edge correspondence in IQH physics [@TKKN; @Hal; @Hat] dictate that the Chern integers for the volume-mode bands shown in eq. (\[7\]) lead to counterclockwise rotating spin-wave edge modes, whose chiral dispersion connects in the frequency space a volume-mode band with $-1$ Chern integer and that with $+1$ Chern integer. In fact, the existence of such chiral edge modes are confirmed by quantitative band calculations based on a linearized Landau-Lifshitz equation with open boundary condition (Fig. \[fig:1.2-1\](b)). Again, reversing the out-of-field direction ($+z \rightarrow - z$) results in the sign change of $c_n$, which changes the chiral direction of the edge modes from counterclockwise to clockwise (Fig. \[fig:1.2-1\](c)).
When the out-of-plane field is less than the ‘critical’ field, $H<H_d$, the lower spin-wave volume mode bands are from those atomic orbitals having higher total angular momentum $q_J = \pi$. Compared to those around $q_J=0$, such orbitals have many nodes along the ring; their wave functions change sign under the translation only by one spin, e.g. $$\begin{aligned}
\psi_{q_J=\pi}(\theta_{j+1})=-\psi_{q_J=\pi}(\theta_j). \nn\end{aligned}$$ Due to this many-node structure, transfer integrals between the higher angular momentum orbitals ($q_J\simeq \pi$) become much smaller than those between orbitals with lower angular momentum ($q_J \simeq 0$). As a result, low-frequency volume-mode bands for $H<H_{d}$ have tiny dispersions, which can hardly fulfill the band inversion condition, $|\epsilon_{1}-\epsilon_{0}| < 4 (a_{00}+a_{11})$; we thus cannot expect the chiral spin-wave edge modes.
![ (Color online) (a) Four corners in a ring (red regions; $\theta=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}$) feel larger demagnetization field than other regions. ${\bm b}$ denotes a coordinate of the center of the ring, while $r$ is a radius of the ring. (b) two-orbital tight-binding model with nearest-neighbor (‘NN’ in the figure) inter-cluster transfer integral (${\bm H}_0$), next-nearest-neighbor $\sigma$-$\sigma$ coupling (‘NNN,$\sigma\sigma$’) inter-cluster transfer integral (${\bm H}_{1}$ with $c$) and next-nearest-neighbor $\pi$-$\pi$ coupling (‘NNN,$\pi\pi$’) inter-cluster transfer integral (${\bm H}_{1}$ with $c'$). The in-phase orbital (red peanut-shape item) at the $x$-link is extended along the $x$-direction, while that at the $y$-link is along the $y$-direction. ${\bm e}_x$ and ${\bm e}_y$ denote the primitive translation vectors of the square lattice. []{data-label="fig:2-orbital-TB"}](fig3.eps){width="80mm"}
Above the saturation field ($H>H_c$), the four-fold rotational anisotropy in the demagnetization field becomes stronger. When the classical spin configuration becomes fully polarized along the out-of-plane field, spins in a ring which are proximate to its four nearest neighboring rings especially feel stronger demagnetization fields than those spins in the ring which are not. In terms of the angle variable $\theta$ defined as ${\bm r} \equiv {\bm b} + r(\cos\theta,\sin\theta)$ (${\bm b}$ denotes a coordinate of a center of the ring at which a spin at ${\bm r}$ is included and $r$ is the radius of the ring; see fig. \[fig:2-orbital-TB\](a)), these spins are at the four corners of a ring, $\theta=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}$ respectively. As a result of this strongly anisotropic demagnetization field, soft spin-wave excitations for $H>H_c$ are highly localized around these four corners.
From this point of view, we made another tight binding model for soft spin-wave bands, which is valid only above the saturation field (appendix B). Thereby, we first took into account proximate ‘exchange process’ which transfers a spin in a corner of a ring into its closest corner of the nearest neighboring ring. The inclusion of such exchange process leads to in-phase and out-of-phase orbital wave functions formed by these two spins. These ‘atomic-orbital’ wave functions are on a center of a link connecting two nearest neighboring rings (red peanut-shape items in Fig. \[fig:2-orbital-TB\](b)). It turns out that, when the field is not too close to the saturation field, the in-phase atomic orbital level becomes lower than the out-of-phase orbital level (see Appendix B for the argument).
The square lattice has two inequivalent links within its unit cell, the link along the $x$-axis (‘$x$-link’) and that along the $y$-axis (‘$y$-link’). Each link provides in-phase and out-of-phase orbital wave functions. Since the out-of-phase wave function has a node at the center, while the in-phase one does not, inter-link transfer integrals between out-of-phase orbitals becomes smaller than those between in-phase orbitals. Being interested in spin-wave bands with larger band width, we focus only on the in-phase orbital wave functions.
A transfer integral between $x$-link and its nearest neighbor $y$-link becomes complex-valued; $$\begin{aligned}
H_{0} &= \sum_{{\bm b}} \Big\{ (i a + b) \!\
\beta^{\dagger}_{{\bm b}+\frac{{\bm e}_y}{2}}
\beta_{{\bm b}+\frac{{\bm e}_x}{2}}
\nn \\
&\ \ \ - (i a+b) \!\
\beta^{\dagger}_{{\bm b}+{\bm e}_y+\frac{{\bm e}_x}{2}}
\beta_{{\bm b}+\frac{{\bm e}_y}{2}} \nn \\
& \ \ \ \ + (i a + b) \!\
\beta^{\dagger}_{{\bm b}+{\bm e}_x + \frac{{\bm e}_y}{2}}
\beta_{{\bm b}+{\bm e}_y+\frac{{\bm e}_x}{2}} \nn \\
& \ \ \ \ \ \ - (i a+ b) \!\
\beta^{\dagger}_{{\bm b}+\frac{{\bm e}_x}{2}}
\beta_{{\bm b}+{\bm e}_x+\frac{{\bm e}_y}{2}} + {\rm h.c.} \Big\}
\label{h0}\end{aligned}$$ with real-valued $a$ and $b$. $\beta_{{\bm b}+\frac{{\bm e}_{x}}{2}}$ and $\beta_{{\bm b}+\frac{{\bm e}_{y}}{2}}$ represent annihilation operators for the in-phase orbital on the $x$-link (whose center is at ${\bm b}+\frac{{\bm e}_{x}}{2}$) and that on $y$-link (at ${\bm b}+\frac{{\bm e}_{y}}{2}$) respectively. ‘$ia\equiv a e^{i\theta}$’ with $\theta=\frac{\pi}{2}$ in eq. (\[h0\]) comes from 90$^{\circ}$ degree angle subtended by the two nearest neighbor orbitals at ${\bm b}+\frac{{\bm e}_x}{2}$ and at ${\bm b}+\frac{{\bm e}_y}{2}$ and a center of the ring at ${\bm b}$. ‘$b$’ in eq. (\[h0\]) results from a finite particle-hole mixing (see appendix B for the derivation of eq. (\[h0\])). A band structure obtained from $H_0$ has two frequency bands which form gapless Dirac cone spectra at ${\bm k}=(\pi,0)$ and $(0,\pi)$.
![ (Color online) Wavelength-frequency dispersions for lowest four volume-mode bands and chiral edge modes in $H>H_c$ (a) $H =1.09 H_c$ (b) $H=1.17 H_c$. The dispersion are obtained with an open/periodic boundary condition along the $y$/$x$-direction, where the resonance frequencies for spin wave excitations are given as a function of the wave vector along the $x$-direction. We used the same system size along the $y$-direction as in Fig. \[fig:1.2-1\] and the same definition of red and blue points as in Fig. \[fig:1.2-1\]. In both (a) and (b), the lowest two volume modes (black points) consist of the in-phase atomic orbitals on $x$-link and $y$-link, while the upper two volume modes mainly consist of out-of-phase orbitals on these two links. The spectra clearly contain a chiral edge mode connecting the lowest two volume-mode bands. Compared to the lowest two bands, the 3rd and 4th lowest bands have smaller band width and no band gap in between. This is because, contrary to the in-phase atomic orbital, the out-of-phase atomic orbital has a node at the center of each link, which results in smaller transfer integrals. Compared (a) and (b), note also that a frequency spacing between the in-phase atomic orbital level and the out-of-phase atomic orbital level increases on increasing the field (see appendix B for the reasoning). []{data-label="fig:1.5-2"}](fig4.eps){width="85mm"}
A finite transfer between the nearest $x$-links and that between the nearest $y$-links endows the gapless Dirac cone spectra with a finite mass. The transfer takes a form of, $$\begin{aligned}
H_1 &= \sum_{{\bm b}} \Big\{c \!\
\beta^{\dagger}_{{\bm b}+\frac{{\bm e}_x}{2}}
\beta_{{\bm b} - \frac{{\bm e}_x}{2}}
+ c \!\ \beta^{\dagger}_{{\bm b}+\frac{{\bm e}_y}{2}}
\beta_{{\bm b} - \frac{{\bm e}_y}{2}} \nn \\
& \ \ \ + c^{\prime} \!\
\beta^{\dagger}_{{\bm b}+{\bm e}_y +\frac{{\bm e}_x}{2}}
\beta_{{\bm b} + \frac{{\bm e}_x}{2}}
+ c^{\prime} \!\
\beta^{\dagger}_{{\bm b}+{\bm e}_x
+\frac{{\bm e}_y}{2}}
\beta_{{\bm b} + \frac{{\bm e}_y}{2}} + {\rm h.c.}
\Big\}, \label{h1} \end{aligned}$$ with real-valued $c$ and $c'$. Now that orbital wave function at the $\mu$-link is extended along the $\mu$-axis, ‘$c$’ stands for the ($\sigma,\sigma$)-coupling next nearest neighbor (NNN) transfer integral, while ‘$c'$’ stands for the $(\pi,\pi)$-coupling NNN transfer integral (Fig. \[fig:2-orbital-TB\](b)). Amplitudes of transfer integrals are inversely proportional to the cubic in distance, so that $|c|>|c'|$. A finite $|c-c'|$ induces a gap in the gapless Dirac cone spectra.
The Chern integers for these two spin-wave bands can be evaluated from the wrapping number of the normalized vector $\overline{\bm h}({\bm k})\equiv {\bm h}({\bm k})/|{\bm h}({\bm k})|$. For eqs. (\[h0\],\[h1\]), $h_1({\bm k})=4b \sin\frac{k_x}{2} \sin \frac{k_y}{2}$, $h_2 ({\bm k})= 4a \cos \frac{k_x}{2} \cos \frac{k_y}{2}$ and $h_3({\bm k})=2(c-c^{\prime})(\cos k_x -\cos k_y)$. When the momentum rotates around ${\bm k}=(\pi,0)$ / $(0,\pi)$, $\overline{\bm h}({\bm k})$ rotates around the south pole/ north pole once for $c>c'$; the winding numbers are $\pm 1$. More generally, the integers for these two bands are $\{c_{-},c_{+}\}=\{+1,-1\}$ from below for $(c-c^{\prime})\cdot a \cdot b >0$, while $\{-1,+1\}$ for $(c-c^{\prime})\cdot a \cdot b < 0$. In either case, there appears a chiral edge mode within the band gap, whose sense of rotation along the boundary is clockwise for the former case, while counterclockwise for the latter case. A primitive evaluation suggests that $a >0$, $b >0$ and $c<c^{\prime}<0$ (appendix B), so that a counterclockwise chiral edge mode is expected. In fact, the counterclockwise chiral edge mode is observed within a band gap between the lowest and the 2nd lowest volume-mode band for a wide field range of $H>H_c$ (Fig. \[fig:1.5-2\]). Contrary to the effective $s$-$p_{\pm}$ model for $H_d <H<H_c$, the band gap and the chiral edge mode in the present two-orbital model persist for a wider range of $H>H_c$. This is because any symmetries in the model requires neither $c = c^{\prime}$ nor $a=0$, while $b$ vanishes only in the large $H$ limit (appendix B). This feature is indeed justified by the micromagnetic simulation in sec. IV.
disk model
==========
Let us next consider spin-wave excitations in circular disk model. We simulate the magnetic disk by a cluster of many spins, each of which has a same volume element. The spins are distributed as homogeneously in space as possible (see the caption of Fig. \[fig:cir-atlv\]). Physically, a linear dimension of the volume element should be on the order of short-ranged exchange interaction length $l_{\rm ex}$. The spins are coupled with one another via magnetic dipole-dipole interaction.
![ (Color online) (a,b) Distribution of resonance frequency levels of magnon modes in a single circular magnetic disk as a function of the out-of-plane field. The four-fold-rotational demagnetization field from other disks are also included in the calculation. The red curve plots the lowest resonance frequency level as a function of the out-of-plane field, while the blue curve represents the highest resonance frequency level. We simulate the circular magnetic disk by a cluster of spins, respecting the circular symmetry as much as possible. For a given radius $R$, we discretize $R$ into $n$ pieces. For the radial coordinate ranging from $[\frac{Rj}{n},\frac{R(j+1)}{n}]$ ($j=0,1,\cdots,n-1$), we discretize the azimuth coordinate into $4(2j+1)$ pieces, so that area of each element is same, $\frac{\pi R^2}{4 n^2}$. We put a spin at the center of each element specified by $(x,y)=r_{j} (\cos \theta_{j,m},\sin\theta_{j,m})$ with $r_j=\frac{R(2j+1)}{2n}$ and $\theta_{j,m}=\frac{\pi (2m+1)}{4(2j+1)}$ ($m=0,\cdots,4(2j-1)$). In the calculation, we take $n=8$, so that a cluster has $256$ spins.[]{data-label="fig:cir-atlv"}](fig5.eps){width="85mm"}
A circular vortex structure minimizes the magnetostatic energy of the disk at the zero field, while the field induces a finite out-of-plane magnetization. Suppose that spins are nearly polarized along the field, while any of them are not yet fully polarized. Being surrounded by many others, spins around the center of a disk feel the strongest demagnetization field, while the demagnetization field around the boundary is smallest. Thus, spins at the boundary become fully polarized first by a relatively lower field, $H_{c,1}$, while spins around the center become fully polarized at last by a relatively higher field, $H_{c,n}(>H_{c,1})$. In the present discrete spin model, these two critical fields encompasses a couple of other critical fields ($H_{c,1}<H_{c,2}<H_{c,3}<\cdots<H_{c,n}$), at which interior spins get fully polarized successively from the outer to the inner on increasing the field.
) (a) wavelength-frequency dispersion for volume-mode bands with the Chern integer ($H=0.94H_{c,1})$. The dispersions are calculated with periodic boundary conditions for both $x$ and $y$-directions. (b) wavelength-frequency dispersion for volume-mode bands and edge-mode bands ($H=0.94H_{c,1}$) calculated with an open/periodic boundary condition along the $y$/$x$-direction. Resonance frequencies are given as a function of the wave vector along the $x$-direction. The system along the $y$-direction includes 9 unit cells ($L=9$). More than $50\%$ of amplitudes of eigen wave functions with red points are localized only within $y=1$, while those for blue points are localized from $y=L$ (edge modes). Compared with Fig. (a), the spectra have additional spin-wave modes which are localized along the edges, whose chiral dispersion connect spin-wave volume modes bands with opposite Chern integers []{data-label="fig:cir-band"}](fig6.eps){width="85mm"}
Correspondingly, spin wave excitations, which are fully gapped at $H=0$, become gapless or significantly softened at each of these critical fields, $H=H_{c,1},H_{c,2},\cdots$ (Fig. \[fig:cir-atlv\]). Especially, the soft magnons around $H=H_{c,1}$ are localized around the boundary of the disk, while those around $H=H_{c,n}$ are localized at the center. In a single magnetic disk, spin-wave excitations have the total angular momentum $q_J$ as a good quantum number. All the soft magnons around these critical fields come from $q_J=0$, so as to be consistent with the classical spin configuration. In the presence of the four-fold rotational demagnetization field, these soft magnons take a form of either $s$-wave ($n_J=0$), $p_{\pm}$-wave ($n_J=\pm1$) or $d_{x^2-y^2}$-wave ($n_J=2$) atomic orbital. As in the ring model, an inter-disk exchange process via the dipolar interaction makes these atomic orbitals to form extended volume-mode bands.
Since the soft magnons around $H = H_{c,1}$ are localized around the boundary of the disk, the inter-disk transfer integrals between these magnons become larger and soft volume-mode bands around $H = H_{c,1}$ become similar to what we observed in the ring model at $H \simeq H_{c}$; the distribution of Chern integers for a set of these four bands becomes either $\{-1,+1,0,0\}$, $\{-1,0,+1,0\}$, or $\{-1,0,0,+1\}$ from below (Fig. \[fig:cir-band\](a)). Again, this leads to a counterclockwise chiral edge mode between these two (Fig. \[fig:cir-band\](b)).
On the other hand, the soft magnons in $H \gtrsim H_{c,n}$ are localized around the center of the disk, so that the inter-disk transfer integrals between these atomic orbitals are very small. As a result, soft volume-mode bands in $H \gtrsim H_{c,n}$ have tiny dispersions, where we cannot expect any band inversion mechanism.
micromagnetic simulation
========================
In order to uphold the existence of the chiral edge mode in the proposed magnetic superlattices, we perform a micromagnetic simulation by solving the Landau-Lifshitz-Gilbert equation in terms of the 4th order Runge-Kutta method with a unit time step 1ps. Fig. \[fig:mc3\] shows an entire magnetic superlattice, which contains 14 $\times$ 14 unit cells with open boundaries. Each unit cell contains 12 ferromagnetic grains, forming a square-shape ring. Each grain is 5-nanometer cube. Note also that, not including any short-range exchange interaction (see below), the following result is scalable; provided that each ferromagnetic grain behaves as single spin, the size of the grain can be much larger than 5-nanometer and the scale of resonance frequency and saturation field still remain unaltered. The saturation magnetization and Gilbert damping coefficient of the ferromagnetic grain are set to 135300 A/m and 1.0$\times$10$^{-5}$ respectively. We regard each nanograin as a uniform magnet, assigning single spin degree of freedom to each grain. Different ferromagnetic nanograins are coupled with one another through the magnetic dipole-dipole interaction. Under a static out-of-plane field (along the $z$ direction) greater than 620 Oe, a stable spin configuration becomes fully polarized along the field, while the configuration acquires finite in-plane components below 620 Oe. We studied spin-wave excitations above the saturation field ($\gtrsim$ 620 Oe).
![ (Color online) magnetic superlattice with 14 $\times$ 14 unit cell. (right) a unit cell contains 12 ferromagnetic grain forming a square ring. Each grain is cubic-shape with its linear dimension $5$ nm. (left) To excite volume-mode/edge-mode excitations, we apply a pulse field at the center/boundary of the superlattice (blue/red crossed point) respectively.[]{data-label="fig:mc3"}](fig7.eps){width="87mm"}
![ (Color online) (a) Contour plot of the integrated power specctrum $A(\omega)$ as a function of the static out-of-plane field $H$, where the initial pulse field is applied at the center of the magnetic superlattice. The out-of-plane field is greater than the saturation field ($\simeq$ 620 Oe). (b) Contour plot of the density of state for volume-mode bands obtained from spin-wave calculations on the same magnetic superlattice. The horizontal axis is the static out-of-plane field, where the unit is taken to be the saturation field. In both figures, darker regions have higher intensities.[]{data-label="fig:bulk"}](fig8.eps){width="87mm"}
To study spin wave modes in a broad frequency range at once, we apply a pulse magnetic field in a transverse ($x$) direction (pulse time $1$ ps and amplitude $1$ Oe). We then calculate a time evolution of magnetization dynamics afterward, and take a Fourier transformation of the transverse moments with respect to time; $$\begin{aligned}
s_{+}(X,Y,\omega) \equiv \sum_{j=0}^{n-1} m_{+}(X,Y,j \Delta T)
\exp\left( 2\pi {\rm i} \omega j \Delta T \right) \label{ps-local} \end{aligned}$$ with $m_{+}(X,Y,t)\equiv m_x(X,Y,t)+
{\rm i}m_y(X,Y,t)$, $\Delta T=100$ ps and $n=1024$. An amplitude of the frequency power spectrum, $|s_{+}(X,Y,\omega)|$, represents a sort of local density of state of spin-wave modes at the resonance frequency $\omega$. When integrated over the two-dimensional space coordinates, ($X,Y$), the power spectrum represents the total density of states at $\omega$; $$\begin{aligned}
A(\omega) \equiv \sum_{X,Y} \big|s_{+}(X,Y,\omega)\big|, \label{ps}\end{aligned}$$
![ (Color online) (a) integrated power spectra calculated with the pulse field at the center (blue) and at the boundary (red). The static out-of-plane field is set to 800 Oe. (b-d) spatial-resolved power spectra $|s_{+}(X,Y,\omega)|$ calculated with the pulse field at the center (blue crossed point); (b) $\omega=$ 6.25 GHz, (c) 6.69 GHz, (d) 6.54 GHz. (e) spatial-resolved power spectrum calculated with the pulse field at the boundary (red crossed point) with $\omega=6.54$ GHz.[]{data-label="fig:lw2"}](fig9.eps){width="95mm"}
(see Fig. \[fig:bulk\] for a comparison between the integrated power spectra and the total density of state obtained from spin-wave calculations). For the purpose of studying volume modes and edge modes selectively, we did two micromagnetic simulations; one with the initial pulse field applied at the center of the system, exciting volume modes, and the other with the pulse field applied near the boundary of the system, exciting edge modes. The power spectra obtained from these separate simulations are regarded as the density of states of volume/edge-mode bands respectively.
![ (Color online) snap shots of a transverse magnetic moment after the a.c. field is applied at $t=0$; (a): $t=100$ns, (b): 200ns, (c) 300ns, (d) t=350, 352, 354ns. The frequency of the a.c. field and the static out-of-plane field is set to 654GHz and 800 Oe respectively. Color specify the sign of the transverse moment (red is for positive and blue is for negative). In (a-c), the spin density propagates in the counterclockwise direction, while, in (d), the node of the transverse moment (indicated by black arrows) moves in the clockwise direction; the phase velocity is opposite to the group velocity.[]{data-label="fig:snap"}](fig10.eps){width="85mm"}
Fig. \[fig:bulk\](a) shows a contour plot of the integrated power spectrum $A(\omega)$ as a function of the static out-of-plane field ($\ge$ 620 Oe). The initial pulse field is applied at the center of the superlattice. On the whole, the spectrum composes of three major responance frequency regimes; for H=$800$Oe, these three are ranged over 6$\sim$7GHz, 8.5$\sim$9GHz, and 9.5$\sim$11GHz respectively. Fig. \[fig:bulk\](b) shows a contour plot of the density of states of volume-mode bands obtained from a spin-wave calculation on the same magnetic superlattice. Since the superlattice has 12 spins within each unit cell, it has 12 volume-mode bands. A comparison reveals that the first and second lowest resonance frequency regimes found in $A(\omega)$ includes two volume-mode bands respectively, while the third resonance frequency regime includes remaining 8 bands. A comparison with the spin-wave analyses also shows that the lowest two volume-mode bands can be well reproduced by the two-orbital tight-binding model introduced in eqs. (\[h0\],\[h1\]); the lowest two bands are mainly composed of the in-phase orbital wavefunction localized at the nearest neighbor $x$-link and that of the $y$-link. Thereby, they are essentially same as the lowest two bands found in the sec. II ($H>H_c$), and thus we expect that the chiral edge mode goes across a band gap between these two (see Fig. \[fig:1.5-2\]).
Fig. \[fig:lw2\](a) shows the integrated power spectra within the lowest resonance frequency regime. The spectrum for volume-mode bands (spectrum obtained with the initial pulse field applied at the center of the system) comprises of two major humps; one ranges from 6.0GHz to 6.4GHz and the other from 6.6GHz to 6.8GHz (black line in Fig. \[fig:lw2\](a)). They correspond to the lowest two volume-mode bands. In fact, the spatial-resolved power spectra within these two frequency regimes are extended over the system (Fig. \[fig:lw2\](b,c)), while the system remains intact against those pulse fields within a band gap regime 6.4GHz $\sim$ 6.6GHz. (Fig. \[fig:lw2\](d)). When the pulse field is applied at the boundary of the system, however, the integrated spectrum has a significant weight within the band gap regime (red line in Fig. \[fig:lw2\](a)). The spatial-resolved spectrum reveals that these weight mainly come from the boundary of the system (Fig. \[fig:lw2\](e)), indicating the existence of edge modes within the band gap regime.
A key feature of the chiral edge mode is a unidirectional propagation of spin wave densities. To confirm this feature, we perform another micromagnetic simulation, applying a.c. transverse field locally at the boundary of the system (red crossed point in Fig. \[fig:mc3\]). We set an external frequency of the a.c. field within the band gap regime; $\omega=6.54$GHz. Fig. \[fig:snap\] shows several snap shots of the transverse magnetization ($m_x(X,Y,t)$) taken after the a.c. field is applied from $t=0$. The snap shots clearly demonstrate a unidirectional propagation of spin densities in the counterclockwise direction. The direction of the propagation is consistent with the sign of the group velocity of the chiral edge modes proposed in the preceding sections. From the snap shots, the group velocity can be estimated to be one unit cell ($a$; linear dimension of the unit cell) per 10 ns, which is on the same order of the band gap divided by $2\pi/a$ (the gap $\sim$ 0.2GHz). The phase velocity of the edge mode is 10 times faster than the group velocity and its sign sometimes becomes opposite to that of the group velocity (Fig. \[fig:snap\](d)). This observation is also consistent with the chiral spin edge mode proposed in the ring model; the chiral dispersion goes across the first Brillouin zone once (Fig. \[fig:1.5-2\](a,b)), so that the sign of the phase velocity can be either same or opposite to the group velocity.
Summary and Discussion
======================
summary of our findings
-----------------------
In this paper, we theoretically explored a realization of topological chiral edge mode for magnetostatic spin wave in patterned magnetic thin films, where magnetic clusters (either rings or disks) form a two-dimensional square lattice. Without external magnetic field, the ground-state spin configuration takes a form of circular vortices within each ring or disk, respecting the square-lattice translational symmetry. Due to the magnetic shape anisotropy, spin-wave excitations are fully gapped at the zero field. When an out-of-plane magnetic field is increased up to a saturation field, forward spin-wave modes within each ring or disk become significantly softened. With the four-fold rotational symmetry of the square lattice, these modes can be regarded as either $s$-wave, $p_x\pm ip_y$-wave or $d_{x^2-y^2}$-wave-like ‘atomic orbitals’. When inter-cluster transfer integrals among these orbital wave functions are larger than frequency spacings among their atomic orbital levels, the band-inversion between the parity-even atomic orbital level ($s$-wave or $d$-wave) and parity-odd orbital level ($p_{\pm}$-wave) leads to a chiral volume-mode bands with finite Chern integers. This results in a chiral (counterclockwise) edge mode within a band gap for the volume-mode bands.
When the system is fully polarized by the out-of-plane field, a strong four-fold rotational anisotropy of the demagnetization coefficient leads to another effective two-bands model. The model is composed of soft magnons localized on the nearest neighbor $x$-link and that on the $y$-link. Since atomic orbital levels for these two are same due to the square-lattice symmetry, transfer integrals between neighboring soft magnons immediately lead to a band inversion mechanism. The two-orbital model has massive Dirac cone like spectra at two inequivalent $X$-points, inside which a chiral (counterclockwise) edge mode appears. The massive Dirac spectra and the edge mode persist for a wide range above the saturation field. This feature is also justified by micromagnetic simulations.
applications to other systems
-----------------------------
In reality, the square-lattice models studied in this paper could be placed on some magnetic substrates. Also, it is experimentally much easier to engrave only a surface of a plane thin film with some periodic structuring. [@Adeyeye; @Gulyaev] The arguments employed in this paper can be also applicable to such systems. For example, consider that a surface of a magnetic film has a number of gutters/cambers forming a square lattice, Fig. \[fig:recession\](a)/(b) respectively. Due to the magnetostatic energy, moments in thinner film regions have stronger easy-plane anisotropy than those in thicker film regions. Therefore, on applying and increasing an out-of-plane field, the moments in thinner regions are expected to become fully polarized along the field at the highest saturation field, while those in the thicker regions do so at the lowest saturation field. This means that, in a system shown in Fig. \[fig:recession\](a), magnons at the gutter region becomes softened around the highest saturation field, forming atomic orbital wave functions. In the other system shown in Fig. \[fig:recession\](b), soft modes near the lowest saturation field are from the camber region. In the presence of the four-fold-rotational symmetry, these orbital wave functions play the role of either parity-even ($d_{x^2-y^2}$ or $s$-waves) orbitals and parity-odd ($p_{\pm}$-waves) orbitals, or the in-phase orbitals localized on the nearest neighbor $\mu$-link ($\mu=x,y$). Thus, provided that neighboring gutters/cambers are proximate to each other, the band inversion mechanisms described in this paper are expected to be valid, leading to a band gap of soft volume-mode bands with a chiral (counterclockwise) edge mode.
![ (Color online) Patterned magnetic films with periodically aligned gutters (a) or cambers (b). Without magnetic crystalline anisotropy (MCA), spins at the thinner regions feel stronger easy-plane anisotropy than those at that thicker regions.[]{data-label="fig:recession"}](fig11.eps){width="75mm"}
The argument is also applicable to thin film ferromagnetic materials with perpendicular magnetic anisotropy (PMA), where relative strength between magnetic shape anisotropy and magnetic crystalline anisotropy (MCA) is controlled by the film thickness. [@Hubert] In an ultrathin film limit (several atomic monolayer), the MCA with easy-axis (out-of-plane) anisotropy dominates over the magnetostatic energy with easy-plane anisotropy, so that magnetic moments are polarized vertically to the plane. It has been experimentally known that increasing film thickness leads to spin-reorientation transition from out-of-plane magnetization to in-plane magnetization, which indicates that magnetic shape anisotropy overcomes the MCA in thicker region. [@Qiu] Around the critical thickness, gapped spin wave modes are expected to become significantly softened.
Regarding the film thickness as alternative to the magnetic field, one could also realize topological chiral spin-wave edge modes [*without any external magnetic field*]{}. For example, consider that a surface of a thin-film PMA material is engraved with cambers with a lattice periodicity as in Fig. \[fig:recession\](b). Suppose that a film-thickness of the camber region is chosen near the critical thickness of the material, so that magnons around camber regions are sufficiently softened, forming orbital wave functions such as $s$, $p_{\pm}$, $d$-waves or in-phase orbitals. When neighboring cambers are put in close contact with one another as in Fig. \[fig:recession\](b), exchange processes due to magnetic dipole interaction give rise to considerable transfer integrals among these orbital wave functions. Although their atomic orbital levels within each camber could be also modified by the MCA energy, we can still expect that larger transfer integrals induce the similar type of the band inversion as discussed in this paper.
a possible experimental method for detecting the chiral spin-wave edge mode
---------------------------------------------------------------------------
The proposed chiral edge modes can be experimentally detected in terms of two coils put along the boundary of the 2-$d$ magnetic superlattice; one is for an input and the other for an output (see Fig. \[fig:1.2-1\](c)). They are spatially separated by tens of the unit cell (for example, 30 unit cells; 0.3mm for a unit cell of 10$\mu$m size). An a.c. electric current in the input coil induces an a.c. magnetic field near the coil, exciting spin waves (electric input). When a frequency of the a.c. current is chosen within the band gap regime, the chiral edge mode will be selectively excited. The excited spin wave propagates along the chiral edge mode and reaches the output coil after a certain time delay (e.g. 0.3$\mu$s according to the simulation in sec. IV). When the spin wave reaches around the output coil, an a.c. electric current with the same frequency will be induced in the output coil (electric detection). When the two coils are exchanged, the spin wave never reaches the output coil, unless it could propagate all the way around the boundary without being dissipated.
When the thickness of the 2-$d$ magnetic superlattice is much larger than short-range exchange interaction length, the input a.c. current can excite not only the proposed topological chiral edge modes but also the conventional chiral surface mode; Damon-Eshbach (DE) surface mode. In such a case, the output a.c. current comprises of two contributions; one from the topological chiral edge mode and the other from the DE mode. In general, these two modes have a number of quantitatively different features. First of all, these modes have quite different group velocities; the group velocity of the topological edge mode linearly depends on the superlattice unit cell size, while that of the DE mode doesn’t depend on the unit cell size. The topological edge mode has a resonance frequency within a band gap regimes for volume mode bands, which is determined by the magnetic superlattice. On the one hand, a resonance frequency regime of the DE mode is determined only by the out-of-plane magnetization $M$ and field $H$, $H<\omega<\sqrt{H(H+4\pi M)}$. When the external frequency is changed within the band gap regime, the phase velocity of the topological mode often changes its sign (see Fig. \[fig:1.5-2\](a,b)), while that of DE mode doesn’t. In actual experiments, we can exploit these distinct features, so as to distinguish the contribution of the topological edge mode from that of the conventional DE surface mode. For example, we can easily differentiate these two contributions [*in time*]{}, by changing the distance between the input and output coils. We can further reduce one or the other, by changing an external frequency of the input a.c. current. Also, by changing the external frequency within the band gap regime, we can see the phase velocity of the topological mode change its sign.
The author acknowledges S. Murakami, E. Saitoh, G. Tatara, Y. Otani, Y. Fukuma, S. Kasai, Y. Suzuki, S. Miwa, Z. Q. Qiu, J. Shi for discussions and informations. This work was partly supported by Grant-in-Aids from the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grants No. 21000004, No. 24740225).
Holstein-Primakoff approximation and topological Chern integer for magnetostatic spin waves
===========================================================================================
In this paper, we considered that magnetic clusters, either thin rings or circular disks, form a 2-$d$ periodic lattice; magnetic superlattice. To study their magnetostatics and dynamics, we used discrete spin models; each cluster is discretized into many spins with small volume element, where the spins are coupled with one another only via magnetic dipole-dipole interaction. We first minimize the magnetostatic energy of the discrete spin models, $$\begin{aligned}
E &\equiv -\frac12 \big(\Delta V\big)^2
\sum^{i\ne j}_{i,j} \sum_{a,b=x,y,z}
M_a({\bm r}_i) f_{ab}({\bm r}_i-{\bm r}_j)
M_b({\bm r}_j) \nn \\
&\hspace{0.5cm}
- H\Delta V \sum_i M_z({\bm r}_i), \label{magsta}\end{aligned}$$ to determine a classical spin configuration ${\bm M}_{0}({\bm r})$. ${\bm r}_i$ specifies a spatial location of a ferromagnetic spin with fixed size of moment $|{\bm M}({\bm r}_i)|=M_s$. $f_{ab}({\bm r}_i-{\bm r}_j)$ is the magnetic dipole-dipole interaction between spin at ${\bm r}_i$ and spin at ${\bm r}_j$; $$\begin{aligned}
f_{ab}({\bm r}) \equiv -\frac{1}{4\pi} \Big(\frac{\delta_{a,b}}{|{\bm r}|^3}
-\frac{3r_a r_b}{|{\bm r}|^5}\Big). \nn \end{aligned}$$ $\Delta V$ denotes a volume element for each spin, whose linear dimension is of the same order of short-ranged exchange length $l_{\rm ex}$; For YIG and Iron, $l_{\rm ex}=18.4$ nm and $2.9$ nm respectively.
Without the field, the energetically stable spin configuration is an array of circular magnetic vortices, [@Hubert; @Cowburn; @Shinjo] respecting the periodicity of the square lattice. Under the out-of-plane field, the configurations acquire finite out-of-plane moments, which will be fully polarized above a saturation field. To obtain spin-wave modes, we linearize the corresponding Landau-Lifshitz equation in favor of fluctuation fields around the classical spin configuration.
In the discrete spin models, the Landau-Lifshitz equation take a form of, $$\begin{aligned}
\partial_t M_a({\bm r}_i)
&= \epsilon_{abc} \Big[ - H \delta_{b,z} \nn \\
& \hspace{1cm}
- \Delta V \sum_{j\ne i} f_{bd} ({\bm r}_i-{\bm r}_j)
M_{d}({\bm r}_j) \Big] M_{c}({\bm r}_i). \nn \end{aligned}$$ Note that the right hand side suggests that the saturation field and characteristic spin-wave resonance frequency are scaled as $M_s \Delta V/l^3$. Here $l$ denotes a distance between the nearest neighbor spins in the discrete spin models and $1/l^3$ comes from the dipole-dipole interaction between them. The small volume element for each spin should be spatially isotropic, such that the discrete spin models can approximately describe the Maxwell equation for magnetic continuum media. This requires $\Delta V\simeq l^3$. As a result, characteristic spin-wave resonance frequencies and saturation field are scaled only by the saturation magnetization of a constituent material.
The equation of motion is linearized with respect to a small transverse field ${\bm m}_{\perp}({\bm r})$ with ${\bm m}_{\perp}({\bm r}) \equiv {\bm M}({\bm r}) - {\bm M}_{0}({\bm r})$ and ${\bm m}_{\perp}({\bm r}) \perp {\bm M}_{0}({\bm r})$. With a local spin frame in which the classical configuration ${\bm M}_0({\bm r})$ becomes fully polarized along the $z$-direction, i.e. ${\bm R}({\bm r}){\bm M}_{0}({\bm r})=M_s {\bm e}_z$ and ${\bm R}({\bm r}){\bm m}_{\perp}({\bm r})
= {\bm m}({\bm r})$, the two transverse moments in the rotated frame ${\bm m}({\bm r})=(m_x({\bm r}),m_{y}({\bm r}))$ comprise creation/annihilation operator for spin wave (magnon); $$\begin{aligned}
m_{\mp}({\bm r}) \equiv m_x({\bm r}) \pm i m_y({\bm r}). \nn \end{aligned}$$ With this magnon field, the linearized equation reduces to a generalized Hermitian eigenvalue problem, $$\begin{aligned}
i\partial_t
\left(\begin{array}{c}
m_{-}({\bm r}_i) \\
m_{+}({\bm r}_i) \\
\end{array}\right)
= \sum_{j} {\bm \sigma}_3 \!\
\big({\bm H}\big)_{{\bm r}_i,{\bm r}_j}
\left(\begin{array}{c}
m_{-}({\bm r}_j) \\
m_{+}({\bm r}_j) \\
\end{array}\right), \label{hami0}\end{aligned}$$ ${\bm \sigma}_3$ is a diagonal matrix which takes $+1$ in the particle space ($m_{+}$) and $-1$ in the hole space ($m_{-}$), reflecting the fact that the magnon obeys the bose statistics. In this particle-hole space, the Hermite matrix is given by the following $2$ by $2$ matrix, $$\begin{aligned}
\big({\bm H}\big)_{{\bm r}_i,{\bm r}_j}
& \equiv - M_s \alpha({\bm r}_i) \delta_{{\bm r}_i,{\bm r}_j}
\left(\begin{array}{cc}
1 & \\
& 1 \\
\end{array}\right) \nn \\
&\hspace{-1.2cm}
- M_s \Delta V (1-\delta_{{\bm r}_i,{\bm r}_j})
\left(\begin{array}{cc}
f_{++}({\bm r}_i,{\bm r}_j) & f_{+-}({\bm r}_i,{\bm r}_j) \\
f_{-+}({\bm r}_i,{\bm r}_j) & f_{--}({\bm r}_i,{\bm r}_j) \\
\end{array}\right). \label{hami} \end{aligned}$$ $\alpha({\bm r}_i)$ denotes the demagnetization coefficient including the static out-of-plane field component; $$\begin{aligned}
\alpha({\bm r}_i) {\bm M}_{0}({\bm r}_i)
= - \Delta V \sum_{j\ne i} {\bm f}({\bm r}_i
-{\bm r}_j) {\bm M}_0({\bm r}_j) - H {\bm e}_z, \nn\end{aligned}$$ where the equality holds true provided that the classical spin configuration gives a local minimum of the magnetostatic energy, eq. (\[magsta\]). $f_{\mu\nu}({\bm r}_i,{\bm r}_j)$ ($\mu=\pm$) in eq. (\[hami\]) represents ‘exchange’ process between ${\bm r}_i$ and ${\bm r}_j$, which gives rise to propagation of magnon excitation under a background of the classical spin configuration. The $2$ by $2$ matrix is defined as $$\begin{aligned}
&\left(\begin{array}{cc}
f_{++}({\bm r},{\bm r}') & f_{+-}({\bm r},{\bm r}') \\
f_{-+}({\bm r},{\bm r}') & f_{--}({\bm r},{\bm r}') \\
\end{array}\right) \nn \\
& \ = \frac12 \left(\begin{array}{cc}
1 & i \\
1 & -i \\
\end{array}\right) \left(\begin{array}{cc}
f_{xx}({\bm r},{\bm r}') & f_{xy}({\bm r},{\bm r}') \\
f_{yx}({\bm r},{\bm r}') & f_{yy}({\bm r},{\bm r}') \\
\end{array}\right)
\left(\begin{array}{cc}
1 & 1 \\
-i & i \\
\end{array}\right). \label{++}\end{aligned}$$ $f_{\alpha\beta}({\bm r},{\bm r}')$ ($\alpha,\beta=x,y,z$) in the right hand side denotes the dipolar interaction in the rotated frame, $$\begin{aligned}
{\bm f}({\bm r},{\bm r}')
\equiv {\bm R}({\bm r}){\bm f}({\bm r}-{\bm r}')
{\bm R}^{t}({\bm r}'). \label{rotated-frame}\end{aligned}$$
To begin with, consider spin-wave exctitations in a circular ring. We treat the ring as a one-dimensional chain of many spins which are equally spaced from respective neighborings and spins along the ring is parameterized by an angle i.e. ${\bm r}_j=r(\cos\theta_j,\sin\theta_j,0)$ with $\theta_j=\frac{2\pi j}{M}$ $(j=1,2,\cdots,M)$. ‘$r$’ denotes the radius of the ring. The classical spin configuration minimizing the magnetostatic energy eq. (\[magsta\]) respects the circular symmetry, $$\begin{aligned}
& {\bm M}_{0}({\bm r}_j) =
M_s (-\sin\varphi\sin\theta_j,\sin\varphi\cos\theta_j,\cos\varphi).
\nn\end{aligned}$$ $\varphi$ denotes a relative angle between each spin and the external magnetic field, which is independent from $j$ due to the circular symmetry. To introduce a magnon and its Hamiltonian in a ring, we take a following local spin frame in eqs. (\[hami\]-\[rotated-frame\]), $$\begin{aligned}
{\bm R}({\bm r}_j) = \left(\begin{array}{ccc}
1 & & \\
& \cos\varphi & -\sin\varphi \\
& \sin\varphi & \cos\varphi \\
\end{array}\right)
\left(\begin{array}{ccc}
\cos\theta_j & \sin\theta_j & \\
-\sin\theta_j & \cos\theta_j & \\
& & 1 \\
\end{array}\right). \nn\end{aligned}$$ Under this gauge, the right hand side of eq. (\[rotated-frame\]) depends only on a relative angle between two magnetic elements along the ring;
$$\begin{aligned}
{\bm f}({\bm r}_i,{\bm r}_j)
&={\bm f}(\theta_i-\theta_j)
= - \frac{1}{32\pi a^3 |\sin\frac{\theta_i-\theta_j}{2}|^3}
\Bigg\{\left(\begin{array}{ccc}
c_{\theta_i-\theta_j} & s_{\theta_i-\theta_j} c_{\varphi} &
s_{\theta_i-\theta_j} s_{\varphi} \\
- s_{\theta_i-\theta_j} c_{\varphi} & c_{\theta_i-\theta_j}
c^2_{\varphi} + s^2_{\varphi} & (c_{\theta_i-\theta_j} -1)c_{\varphi}s_{\varphi} \\
- s_{\theta_i-\theta_j}s_{\varphi} & (c_{\theta_i-\theta_j}-1) c_{\varphi}s_{\varphi}
& s^2_{\varphi} c_{\theta_i-\theta_j} + c^2_{\varphi} \\
\end{array}\right) \nn \\
& \hspace{3cm}
- \frac{3}{2}
\left(\begin{array}{ccc}
-(1-c_{\theta_i-\theta_j}) &
s_{\theta_i-\theta_j} c_{\varphi} & s_{\theta_i-\theta_j} s_{\varphi} \\
- s_{\theta_i-\theta_j} c_{\varphi} & (c_{\theta_i-\theta_j} -1)
c^2_{\varphi} & (c_{\theta_i-\theta_j}-1) c_{\varphi}s_{\varphi} \\
- s_{\theta_i-\theta_j}s_{\varphi} & c_{\varphi}s_{\varphi}
(c_{\theta_i-\theta_j}-1) &
s^2_{\varphi} (c_{\theta_i-\theta_j}-1) \\
\end{array}\right) \Bigg\}
% \end{aligned}$$
with $c_{\theta_i-\theta_j}\equiv \cos(\theta_i-\theta_j)$, $s_{\theta_i-\theta_j} \equiv \sin(\theta_i-\theta_j)$, $c_{\varphi} \equiv \cos\varphi$, and $s_{\varphi}
\equiv \sin\varphi$. The demagnetization coefficient in an isolated ring also respects the circular symmetry; $\alpha({\bm r}_j)=\alpha$. Thus, the magnon Hamiltonian for a circular ring depends only on the relative angle; $$\begin{aligned}
i\partial_t
\left(\begin{array}{c}
m_{-}(\theta_i) \\
m_{+}(\theta_i) \\
\end{array}\right)
= \sum^{M}_{j=1} {\bm \sigma}_3 \!\
\big({\bm H}\big)_{\theta_i-\theta_j}
\left(\begin{array}{c}
m_{-}(\theta_j) \\
m_{+}(\theta_j) \\
\end{array}\right), \label{hami0}\end{aligned}$$ Correspondingly, the spin-wave excitations in a ring are characterized by the angular momentum variable $q_J=\frac{2\pi n_J}{M}$ $(n_J=-\frac{M}{2},-\frac{M}{2}+1
,\cdots,\frac{M}{2})$ associated with the circular symmetry; $$\begin{aligned}
\left(\begin{array}{c}
m_{-}(\theta_i) \\
m_{+}(\theta_i) \\
\end{array}\right) =
\sum_{n_J} e^{in_J \theta_i} \left(\begin{array}{c}
m_{-}(n_J) \\
m_{+}(-n_J) \\
\end{array}\right). \end{aligned}$$ The linearized equation is given by $$\begin{aligned}
i\partial_t
\left(\begin{array}{c}
m_{-}(n_J) \\
m_{+}(-n_J) \\
\end{array}\right)
= {\bm \sigma}_3 \!\ \big({\bm H}\big)_{n_J}
\left(\begin{array}{c}
m_{-}(n_J) \\
m_{+}(-n_J) \\
\end{array}\right),\end{aligned}$$ with $$\begin{aligned}
({\bm H})_{n_J} \equiv \sum_{j} e^{in_J \theta_j}
({\bm H})_{\theta_j}. \nn\end{aligned}$$ The 2 by 2 Hermite matrix $({\bm H})_{n_J}$ is diagonalized for each angular momentum in terms of canonical transformation (2 by 2 paraunitary matrix); $$\begin{aligned}
&({\bm H})_{n_J}
{\bm t}_{n_J} = {\bm \sigma}_3 \!\
{\bm t}_{n_J} E_{n_J}. \end{aligned}$$ with a proper normalization ${\bm t}^{\dagger}_{n_J}{\bm \sigma}_3{\bm t}_{n_J}=1$. Positive definite $E_{n_J}$ stands for a resonance frequency for the spin-wave excitations in a circular ring. Respective spin-wave mode is represented by the two-component vector in the particle-hole space ${\bm t}_{n_J}$; the linearized equation of motion eq. (\[hami0\]) is satisfied by $$\begin{aligned}
\psi_{q_J}(\theta_j) = {\bm t}_{n_J} \!\ e^{in_J \theta_j-iE_{n_J} t}
\label{psi-def}\end{aligned}$$ with $q_J\equiv \frac{2\pi n_J}{M}$. Eq. (\[psi-def\]) with $n_J=-\frac{M}{2},\cdots,\frac{M}{2}-1,\frac{M}{2}$ comprise ‘atomic orbital’ wavefunctions within a circular ring, which are classified by the total angular momentum $q_J$; $$\begin{aligned}
\psi_{q_J}(\theta_j+\theta_m) = e^{iq_J m} \psi_{q_J}(\theta_j). \nn\end{aligned}$$ These wavefunctions gives us bases for tight-binding descriptions of spin-wave excitations in the magnetic superlattice.
To obtain spin-wave dispersion relations for volume modes and edge modes in the magnetic superlattices, we diagonalize eq. (\[hami\]) with a periodic boundary condition along the $x$-direction and an open boundary condition along the $y$-direction. A system typically contains $9$-$18$ square-lattice unit cell along the $y$-direction. We minimize the magnetostatic energy, respecting the periodicity of the square lattice, ${\bm M}_0({\bm r}+{\bm e}_x)={\bm M}_0({\bm r})$. So do $\alpha({\bm r})$, ${\bm R}({\bm r})$ and $({\bm H})_{{\bm r}_i,{\bm r}_j}$; $({\bm H})_{{\bm r}_i+{\bm e}_{x},{\bm r}_j}=
({\bm H})_{{\bm r}_i,{\bm r}_j-{\bm e}_{x}}$. Correspondingly, we diagonalize the following fourier-transformed Hamiltonian, $$\begin{aligned}
({\bm H}_{k})_{{\bm r}_i,{\bm r}_j}
&= - M_s \alpha({\bm r}_i) \delta_{{\bm r}_i,{\bm r}_j} \nn \\
&\ \ \ - M_s \Delta V \left(\begin{array}{cc}
f_{k,++}({\bm r}_i,{\bm r}_j) & f_{k,+-}({\bm r}_i,{\bm r}_j) \\
f_{k,-+}({\bm r}_i,{\bm r}_j) & f_{k,--}({\bm r}_i,{\bm r}_j) \\
\end{array}\right), \label{hermitian} \end{aligned}$$ with $$\begin{aligned}
f_{k,\sigma\sigma'}({\bm r},{\bm r}')
&= e^{-i k({\bm r}-{\bm r}')_x} \times \nn \\
&\ \ \ \sum_{\bm b} (1-\delta_{{\bm r},{\bm r}'-{\bm b}})
f_{\sigma\sigma'}({\bm r},{\bm r}'-{\bm b}) e^{-i k{\bm b}_x}. \nn\end{aligned}$$ with $\sigma,\sigma'=\pm$. The summation over the lattice translational vector ${\bm b}$ is taken only along the $x$-direction and is over a finite range ${\bm b}\equiv n{\bm e}_x$ with $-10\le n\le 10$. Provided that the classical spin configuration ${\bm M}_0({\bm r}_i)$ gives a local minimum for the magnetostatic energy, the linearized Hamiltonian is paraunitarily equivalent to a positive definite diagonal matrix ${\bm E}_{k}$: ${\bm T}^{\dagger}_{k}
{\bm H}_{k} {\bm T}_{k} =
{\bm E}_{k}$ with ${\bm T}^{\dagger}_{k}{\bm \sigma}_3 {\bm T}_{k}
= {\bm \sigma}_3$. Each diagonal element in ${\bm E}_k$ and corresponding column vector in ${\bm T}_{k}$ gives a resonance frequency and wave function for a volume mode and edge mode as a function of the wave vector $k$ along the $x$-direction. With the normalization condition of ${\bm T}_k$ in mind, an amplitude of the wave function for the $n$-th eigen mode at ${\bm r}_i$ is defined as $\sum_{\sigma=\pm} \sigma|({\bm T}_k)_{({\bm r}_j,\sigma|n)}|^2$. We regard the mode as an edge mode, when more than $70\%$ of the amplitude is localized along the boundaries of the system (see also the captions of Figs. \[fig:1.2-1\], \[fig:cir-band\]). Otherwise, we observed that wave functions are usually extended over the system, and thus can be regarded as volume modes.
Dispersion relations for the volume modes are also obtained from calculations with periodic boundary conditions imposed on both $x$ and $y$-direction. The classical ground-state spin configuration respects the periodicity of the square lattice, ${\bm M}_0({\bm r}+{\bm e}_x)={\bm M}_0({\bm r}+{\bm e}_y)
= {\bm M}_0({\bm r})$. We diagonalize eq. (\[hermitian\]) with $k$ being replaced by ${\bm k}=(k_x,k_y)$, in terms of a paraunitary transformation ${\bm T}_{\bm k}$; $$\begin{aligned}
f_{{\bm k},\sigma\sigma'}({\bm r},{\bm r}')
&= e^{-i {\bm k}({\bm r}-{\bm r}')} \times \nn \\
&\ \ \ \sum_{\bm b} (1-\delta_{{\bm r},{\bm r}'-{\bm b}})
f_{\sigma\sigma'}({\bm r},{\bm r}'-{\bm b}) e^{-i {\bm k}{\bm b}}, \nn\end{aligned}$$ with ${\bm b}=n{\bm e}_x+m{\bm e}_y$ and $-10 \le n,m \le 10$. The topological Chern integer for the $j$-th volume mode band is defined by the $j$-the column vector of the paraunitary matrix ${\bm T}_{\bm k}$ as $$\begin{aligned}
c_j \equiv \frac{i \epsilon_{\mu\nu}}{2\pi} \int_{\rm BZ} d^2{\bm k}
\!\ {\rm Tr} \Big[{\bm \Gamma}_j {\bm \sigma}_3
\big(\partial_{k_\mu}{\bm T}^{\dagger}_{\bm k}\big)
{\bm \sigma}_3
\big(\partial_{k_\nu}{\bm T}_{\bm k}\big) \Big]. \nn \end{aligned}$$ Here ${\bm \Gamma}_j$ takes $+1$ in the $(j,j)$ component while $0$ otherwise. $c_j$ takes an integer and describes a topological structure of a wave function for the $j$-th volume mode band in the two-dimensional Brillouin zone (BZ). [@SO1; @TKKN; @Avron]
two-orbital model valid above the saturation field
==================================================
When the classical spin configuration is fully polarized along the out-of-plane field, demagnetization field at the four corner of a ring, ${\bm r}_j={\bm b}_j+r(\cos\theta_j,\sin\theta_j)$ with $\theta_j=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}$, is much stronger than those in the others. Here ${\bm b}_j$ denote a coordinate of a center of the ring at which a spin at ${\bm r}_j$ is included. $r$ is a radius of the ring (Fig. \[fig:cluster\]). As a result, soft volume mode bands are mainly composed of spins localized at $\theta_j=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}$. In such a case, exchange process between nearest neighbor rings becomes even larger than that within a same ring. We thus take into account the former exchange process first, to introduce [*atomic orbital wave functions defined on a link connecting two nearest neighboring rings*]{}.
![ (Color online) (a) Ring is decomposed into four quadrants (grey shadow regions), which are ranged as $-\frac{\pi}{4}\le \theta_j \le \frac{\pi}{4}$ ($n=0$), $\frac{\pi}{4}\le \theta_j \le \frac{3\pi}{4}$ ($n=1$), $\frac{3\pi}{4}\le \theta_j \le \frac{5\pi}{4}$ ($n=2$), $\frac{5\pi}{4}\le \theta_j \le \frac{7\pi}{4}$ ($n=3$) respectively with ${\bm r}_j={\bm b}_j+r(\cos\theta_j,\sin\theta_j)$. Here ${\bm b}_j$ denotes a center of the ring and $r$ is a radius of the ring. (b) One quadrant in a ring and its closest quadrant in the nearest neighbor ring are combined together, to form a cluster (grey shadow region encompassed by a black dotted line). The cluster thus defined is centered at ${\bm b}_j+\frac{{\bm e}_x}{2}$ (a mid-point of the nearest neighbor $x$-link) or ${\bm b}_j+\frac{{\bm e}_y}{2}$ (a mid-point of the $y$-link), where ${\bm e}_x$ and ${\bm e}_y$ are the basic translational vectors.[]{data-label="fig:cluster"}](fig12.eps){width="70mm"}
Specifically, we first decompose every ring, ${\bm r}_j={\bm b}_j+r(\cos\theta_j,\sin\theta_j)$, into four quadrants, which are ranged as $-\frac{\pi}{4}\le \theta_j
\le \frac{\pi}{4}$, $\frac{\pi}{4} \le \theta_j \le \frac{3\pi}{4}$, $\frac{3\pi}{4} \le \theta_j \le \frac{5\pi}{4}$, $\frac{5\pi}{4} \le \theta_j \le \frac{7\pi}{4}$ respectively (Fig. \[fig:cluster\](a)). We then combine one quadrant in a ring ($-\frac{\pi}{4}\le \theta_j
\le \frac{\pi}{4}$ with ${\bm b}_j$ or $\frac{\pi}{4}\le \theta_j
\le \frac{3\pi}{4}$ with ${\bm b}_j$) and its closest quadrant of the nearest neighboring ring ($\frac{3\pi}{4}\le \theta_j
\le \frac{5\pi}{4}$ with ${\bm b}_j+{\bm e}_x$ or $\frac{5\pi}{4}\le \theta_j
\le \frac{7\pi}{4}$ with ${\bm b}_j+{\bm e}_y$ respectively), to make a ‘cluster’ (Fig. \[fig:cluster\](b)). The cluster thus defined is centered at a middle point of the nearest neighbor $x$-link or that of the $y$-link (${\bm b}_j+\frac{{\bm e}_x}{2}$ or ${\bm b}_j+\frac{{\bm e}_y}{2}$ respectively). Correspondingly, we decompose the BdG Hamiltonian in eq. (\[hami\]) into two parts, one is diagonal with respect to a cluster index and the other is off-diagonal with respect to the cluster index; $$\begin{aligned}
\big({\bm H}\big)_{{\bm r}_i,{\bm r}_j}
= \big({\bm H}_0\big)_{{\bm r}_i,{\bm r}_j} +
\big({\bm H}_1\big)_{{\bm r}_i,{\bm r}_j}, \end{aligned}$$ with $$\begin{aligned}
\big({\bm H}_0\big)_{{\bm r}_i,{\bm r}_j} &= - M_s
\alpha({\bm r}_i) \delta_{{\bm r}_i,{\bm r}_j} \left(\begin{array}{cc}
1 & \\
& 1 \\
\end{array}\right) - M_s \Delta V \times \nn \\
&\hspace{1cm} \delta_{[{\bm r}_i],[{\bm r}_j]} \!\
\eta_{{\bm r}_i,{\bm r}_j} \left(\begin{array}{cc}
f_{++}({\bm r}_i,{\bm r}_j) & f_{+-}({\bm r}_i,{\bm r}_j) \\
f_{-+}({\bm r}_i,{\bm r}_j)& f_{--} ({\bm r}_i,{\bm r}_j) \\
\end{array}\right), \nn \\
%
\big({\bm H}_1\big)_{{\bm r}_i,{\bm r}_j} &= - M_s \Delta V
\!\ \eta_{[{\bm r}_i],[{\bm r}_j]} \left(\begin{array}{cc}
f_{++}({\bm r}_i,{\bm r}_j) & f_{+-}({\bm r}_i,{\bm r}_j) \\
f_{-+}({\bm r}_i,{\bm r}_j)& f_{--} ({\bm r}_i,{\bm r}_j) \\
\end{array}\right), \nn \\\end{aligned}$$ where $\eta_{{\bm r}_i,{\bm r}_j}\equiv 1- \delta_{{\bm r}_i,{\bm r}_j}$, $\eta_{[{\bm r}_i],[{\bm r}_j]}\equiv 1- \delta_{[{\bm r}_i],[{\bm r}_j]}$ and $[{\bm r}_i]$ species a cluster in which a spin site ${\bm r}_i$ is included. Now that the spin configuration is fully polarized, we take the following frame in eq. (\[rotated-frame\]), $$\begin{aligned}
{\bm R}({\bm r}_i) \equiv \left(\begin{array}{ccc}
\cos\theta_i & \sin\theta_i & \\
-\sin\theta_i & \cos\theta_i & \\
& & 1 \\
\end{array}\right). \label{rotation2}\end{aligned}$$ with ${\bm r}_i\equiv {\bm b}_i+r(\cos\theta_i,\sin\theta_i)$. With this rotated spin frame, the $2$ by $2$ transfer integrals is given as
$$\begin{aligned}
&\left(\begin{array}{cc}
f_{++}({\bm r}_i,{\bm r}_j) & f_{+-}({\bm r}_i,{\bm r}_j) \\
f_{-+}({\bm r}_i,{\bm r}_j) & f_{--}({\bm r}_i,{\bm r}_j) \\
\end{array}\right) = \frac{1}{4\pi R^3} \!\ \Bigg\{ -
\left(\begin{array}{cc}
e^{-i(\theta_i-\theta_j)} & \\
& e^{i(\theta_i-\theta_j)} \\
\end{array}\right)
+ \frac{3}{2} \left(\begin{array}{cc}
e^{-i(\theta_i-\theta_j)} & e^{-i(\theta_i+\theta_j)+2i\varphi_{ij}} \\
e^{i(\theta_i+\theta_j)-2i\varphi_{ij}} & e^{i(\theta_i-\theta_j)} \\
\end{array}\right) \Bigg\} \label{rotation}\end{aligned}$$
with ${\bm r}_j \equiv {\bm b}_j + r(\cos\theta_j,\sin\theta_j)$, $R\equiv |{\bm r}_i-{\bm r}_j|$ and ${\bm r}_i-{\bm r}_j
\equiv R (\cos\varphi_{ij},\sin\varphi_{ij})$. In the following, we first diagonalize ${\bm H}_0$ to introduce ‘orbital wave functions’ within each cluster. In terms of this orbital basis, we next include ${\bm H}_1$ as a inter-cluster transfer integrals.
To carry out this procedure systematically, we further decompose the diagonal part into two parts, ${\bm H}_0 \equiv {\bm H}^{\prime}_0
+ {\bm H}^{\prime\prime}_0$, where $({\bm H}^{\prime}_{0})_{{\bm r}_i,{\bm r}_j}$ is no-zero if and only if both ${\bm r}_i$ and ${\bm r}_j$ are within the same quadrant, while $({\bm H}^{\prime\prime}_0)_{{\bm r}_i,{\bm r}_j}$ is non-zero if ${\bm r}_i$ is in one quadrant and ${\bm r}_j$ is in the other; ${\bm H}^{\prime\prime}_0$ plays the part of exchange process between the nearest neighbor quadrants. Consider first the lowest eigen basis which diagonalizes ${\bm H}^{\prime}_0$; $$\begin{aligned}
&{\bm H}^{\prime}_{0} |u_{\pm,n,{\bm b}}\rangle
= {\bm \sigma}_3 |u_{\pm,n,{\bm b}}\rangle (\pm E),
\label{basis} \end{aligned}$$ where $\langle u_{\nu,n,{\bm b}}|{\bm \sigma}_3
|u_{\mu,n^{\prime},{\bm b}^{\prime}}\rangle
= \nu \delta_{\nu,\mu}
\delta_{n,n^{\prime}} \delta_{{\bm b},{\bm b}^{\prime}}$ with $\nu,\mu=\pm$. ${\bm b}$ and ${\bm b}^{\prime}$ denote spatial coordinate of (a center of) the ring to which the basis belongs, while the subscripts $n,n^{\prime} (=0,1,2,3)$ specify the quadrant to which the basis belongs. For example, $\langle {\bm r}_j,\tau|u_{\pm,0,{\bm b}}\rangle$ is non-zero only when ${\bm b}={\bm b}_j$ and $-\frac{\pi}{4}\le \theta_j \le \frac{\pi}{4}$ with ${\bm r}_j={\bm b}_j + r(\cos\theta_j,\sin\theta_j)$, while $\langle {\bm r}_j,\tau|u_{\pm,1,{\bm b}}\rangle$ is non-zero only when ${\bm b}={\bm b}_j$ and $\frac{\pi}{4}\le \theta_j \le \frac{3\pi}{4}$ and so on (see also fig. \[fig:cluster\](a) for $n=2,3$). $|u_{+,n,{\bm b}}\rangle$ and $|u_{-,n,{\bm b}}\rangle$ are particle-hole pair to each other, $$\begin{aligned}
\langle {\bm r}_i,\tau|u_{-,n,{\bm b}}\rangle
= ({\bm \sigma}_1)_{\tau\tau^{\prime}}
\langle u_{+,n,{\bm b}}|{\bm r}_i,{\tau}^{\prime}\rangle, \label{p-h} \end{aligned}$$ with the particle-hole index $\tau=1,2$. Due to the four-fold rotational and square-lattice translational symmetries, the lowest eigen frequency in eq. (\[basis\]), $E$, does not depend on $n$ and ${\bm b}$.
Now that $-\alpha({\bm r}_j)$ in ${\bm H}^{\prime}_0$ has deep minima at $\theta_j=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}$ with ${\bm r}_j={\bm b}_j+r(\cos\theta_j,\sin\theta_j)$, the lowest eigen basis is expected to be localized around these valley bottoms, $$\begin{aligned}
&\big\langle {\bm r}_{j}, \tau\big|
u_{+,n,{\bm b}} \big\rangle \simeq \delta_{{\bm b},{\bm b}_j}
\Big(\delta_{n,0}\delta_{\theta_j,0} +
\delta_{n,1}\delta_{\theta_j,\frac{\pi}{2}} \nn \\
&\hspace{2cm} + \delta_{n,2}\delta_{\theta_j,\pi} +
\delta_{n,3}\delta_{\theta_j,\frac{3\pi}{2}} \Big)
\left(\begin{array}{cc}
u \\
v \\
\end{array}\right)_{\tau}. \label{w1}\end{aligned}$$ $(u,v)$ represents a two-component vector in the particle-hole space. Near (but above) the saturation field, the vector is equally-weighted in the particle-hole space, $$\begin{aligned}
\left(\begin{array}{c}
u \\
v \\
\end{array}\right) \simeq \left(\begin{array}{c}
i \\
- i \\
\end{array}\right) \ \ \ {\rm for} \ \ H\gtrsim H_c. \label{eq1}\end{aligned}$$ The relative phase between the particle-component ($u$; $\tau=1$) and the hole component ($v$; $\tau=2$) was taken $-1$, because a condensation of the soft magnon with eq. (\[eq1\]) results in an in-plane component which is [*tangential*]{} to the ring; the in-plane component of the classical spin configuration at $H<H_c$ takes the circular vortex structure within each ring. Note also that, in eq. (\[w1\]), the relative phase among different quadrants was chosen to be $+1$, because of the rotated spin frame, eq. (\[rotation2\]). In the high field limit, the vector is polarized in the particle space, $$\begin{aligned}
\left(\begin{array}{c}
u \\
v \\
\end{array}\right) \rightarrow
\left(\begin{array}{c}
1 \\
0 \\
\end{array}\right) \ \ \ {\rm for} \ \ H\rightarrow \infty. \label{full}\end{aligned}$$
In terms of the lowest eigen basis of ${\bm H}^{\prime}_0$, ${\bm H}_0$ takes a form;
$$\begin{aligned}
{\bm H}_0 =& \sum_{{\bm b}} \Big\{
\left(\begin{array}{cccc}
\gamma^{\dagger}_{0,{\bm b}} & \gamma^{\dagger}_{2,{\bm b}+{\bm e}_x}
& \gamma_{0,{\bm b}} & \gamma_{2,{\bm b}+{\bm e}_x}
\end{array}\right)\left(\begin{array}{cccc}
E & t & 0 & s \\
t & E & s & 0 \\
0 & s & E & t \\
s & 0 & t & E \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{0,{\bm b}} \\
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\end{array}\right) \nn \\
& \hspace{1cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{1,{\bm b}} & \gamma^{\dagger}_{3,{\bm b}+{\bm e}_y}
& \gamma_{1,{\bm b}} & \gamma_{3,{\bm b}+{\bm e}_y}
\end{array}\right)\left(\begin{array}{cccc}
E & t & 0 & s \\
t & E & s & 0 \\
0 & s & E & t \\
s & 0 & t & E \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{1,{\bm b}} \\
\gamma_{3,{\bm b}+{\bm e}_y} \\
\gamma^{\dagger}_{1,{\bm b}} \\
\gamma^{\dagger}_{3,{\bm b}+{\bm e}_y} \\
\end{array}\right)
\Big\} \label{4by4}\end{aligned}$$
where $\gamma^{\dagger}_{n,{\bm b}}$ / $\gamma_{n,{\bm b}}$ denotes a creation / annihilation operator which excites $|u_{+,n,{\bm b}}\rangle$ / $|u_{-,n,{\bm b}}\rangle$ respectively. $t$ and $s$ are real-valued and represent hopping terms between two nearest neighboring quadrants in the particle-particle channel and particle-hole channel respectively, $$\begin{aligned}
t & \equiv \langle u_{+,0,{\bm b}} |
{\bm H}^{\prime\prime}_{0} | u_{+,2,{\bm b}+{\bm e}_x} \rangle
= \langle u_{+,2,{\bm b}+{\bm e}_x} |
{\bm H}^{\prime\prime}_{0} | u_{+,0,{\bm b}} \rangle \nn \\
&= \langle u_{-,0,{\bm b}} | {\bm H}^{\prime\prime}_{0}
| u_{-,2,{\bm b}+{\bm e}_x} \rangle
= \langle u_{-,2,{\bm b}+{\bm e}_x} |
{\bm H}^{\prime\prime}_{0} | u_{-,0{\bm b}} \rangle , \label{t-ha} \end{aligned}$$ $$\begin{aligned}
s & \equiv \langle u_{+,0,{\bm b}} |
{\bm H}^{\prime\prime}_{0} | u_{-,2,{\bm b}+{\bm e}_x} \rangle
= \langle u_{+,2,{\bm b}+{\bm e}_x} |
{\bm H}^{\prime\prime}_{0} | u_{-,0,{\bm b}} \rangle \nn \\
&= \langle u_{-,0,{\bm b}} |
{\bm H}^{\prime\prime}_{0} |
u_{+,2,{\bm b}+{\bm e}_x} \rangle
= \langle u_{-,2,{\bm b}+{\bm e}_x} |
{\bm H}^{\prime\prime}_{0} | u_{+,0,{\bm b}} \rangle . \label{s-ha} \end{aligned}$$ The equalities in eqs. (\[t-ha\],\[s-ha\]) come from the particle-hole symmetry, $\pi$-rotational symmetry and a mirror symmetry combined with the time-reversal. Diagonalization of eq. (\[4by4\]) introduces orbital wave functions on the nearest neighbor $x$-link as, $$\begin{aligned}
&\left(\begin{array}{c}
\beta^{\dagger}_{-,{\bm b}+\frac{{\bm e}_x}{2}} \\
\beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_x}{2}} \\
\beta_{-,{\bm b}+\frac{{\bm e}_x}{2}} \\
\beta_{+,{\bm b}+\frac{{\bm e}_x}{2}} \\
\end{array}\right) = \frac{1}{\sqrt{2}} \times \nn \\
&\hspace{0.2cm}
\left(\begin{array}{cccc}
{\rm ch}_{\frac{\theta}{2}} & {\rm ch}_{\frac{\theta}{2}}
& {\rm sh}_{\frac{\theta}{2}} & {\rm sh}_{\frac{\theta}{2}} \\
- {\rm ch}_{\frac{\theta^{\prime}}{2}} & {\rm ch}_{\frac{\theta^{\prime}}{2}}
& - {\rm sh}_{\frac{\theta^{\prime}}{2}} & {\rm sh}_{\frac{\theta^{\prime}}{2}} \\
%
{\rm sh}_{\frac{\theta}{2}} & {\rm sh}_{\frac{\theta}{2}}
& {\rm ch}_{\frac{\theta}{2}} & {\rm ch}_{\frac{\theta}{2}} \\
-{\rm sh}_{\frac{\theta^{\prime}}{2}} & {\rm sh}_{\frac{\theta^{\prime}}{2}}
& -{\rm ch}_{\frac{\theta^{\prime}}{2}} &
{\rm ch}_{\frac{\theta^{\prime}}{2}} \\
\end{array}\right) \left(\begin{array}{c}
\gamma^{\dagger}_{0,{\bm b}} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\gamma_{0,{\bm b}} \\
\gamma_{2,{\bm b}+{\bm e}_x} \\
\end{array}\right) \label{bog} \end{aligned}$$ with $({\rm ch}_{\frac{\theta}{2}},{\rm sh}_{\frac{\theta}{2}})
\equiv (\cosh \frac{\theta}{2},\sinh \frac{\theta}{2})$ and $$\begin{aligned}
&\cosh\theta = \frac{E+t}{\sqrt{(E+t)^2-s^2}}, \!\
\sinh\theta=\frac{s}{\sqrt{(E+t)^2-s^2}}, \nn \\
&\cosh\theta^{\prime} = \frac{E-t}{\sqrt{(E-t)^2-s^2}}, \!\
\sinh\theta^{\prime}=\frac{-s}{\sqrt{(E-t)^2-s^2}}. \nn\end{aligned}$$ $\beta_{+,{\bm b}+\frac{{\bm e}_x}{2}}$/$\beta_{-,{\bm b}+\frac{{\bm e}_x}{2}}$ is for a ‘in-phase’/‘out-of-phase’ orbital formed by $\gamma_{0,{\bm b}}$ and $\gamma_{2,{\bm b}+{\bm e}_x}$, whose eigen frequency is $\sqrt{(E-t)^2-s^2}$/$\sqrt{(E+t)^2-s^2}$ respectively. Under the rotated spin frame, eq. (\[rotation2\]), these two in fact stand for an ‘in-phase’/‘out-of-phase’ mode formed by a spin at ${\bm r}={\bm b}+(r,0)$ and that at ${\bm r}={\bm r}+{\bm e}_x-(r,0)$ respectively. Similarly, the in-phase/out-of-phase orbitals between $\gamma_{1,{\bm b}}$ and $\gamma_{3,{\bm b}+{\bm e}_y}$ are introduced on the nearest neighboring $y$-link, $\beta_{\pm,{\bm b}+\frac{{\bm e}_y}{2}}$; $$\begin{aligned}
{\bm H}_0 &= 2 \sum_{{\bm b}} \sum_{\mu=x,y}
\Big\{ \sqrt{(E-t)^2-s^2} \!\ \!\ \!\
\beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_{\mu}}{2}}
\beta_{+,{\bm b}+\frac{{\bm e}_{\mu}}{2}} \nn \\
&\hspace{1cm} + \sqrt{(E+t)^2-s^2} \!\ \!\ \!\
\beta^{\dagger}_{-,{\bm b}+\frac{{\bm e}_{\mu}}{2}}
\beta_{-,{\bm b}+\frac{{\bm e}_{\mu}}{2}} \Big\}. \label{h0-app}\end{aligned}$$
An evaluation based on eqs. (\[rotation\],\[t-ha\],\[w1\],\[eq1\],\[full\]) suggests that $t<0$ near (but above) the saturation field while $t>0$ in the high-field limit. The sign change is because the two-component vector $(u,v)$ is equally weighted in the particle-hole space near the saturation field (eq. (\[eq1\])), while it is fully polarized in the particle space in the high-field limit (eq. (\[full\])). In the present model, $t$ changes the sign around $H=1.05 H_c$, where the in-phase orbital level goes below the out-of-phase one in frequency. Thus, in most of the fully polarized regime, we regard that the in-phase orbital at the $x$-link and that at the $y$-link comprises the lowest two.
In terms of the in-phase orbitals on the $x$-link and $y$-link, we next include ${\bm H}_1$ as inter-cluster transfer (hopping) integrals. To this end, we first describe ${\bm H}_1$, using the eigen basis of ${\bm H}^{\prime}_0$, $|u_{\mu,n,{\bm b}}\rangle$ ($n=0,1,2,3$ and $\mu=\pm$). The most dominant inter-cluster transfer integral is mainly from exchange processes between neighboring quadrants within the same ring. In terms of $\gamma^{\dagger}_{n,{\bm b}}$ and $\gamma_{n,{\bm b}}$, they are given by
$$\begin{aligned}
{\bm H}^{NN}_1 =& \sum_{{\bm b}} \bigg\{
\left(\begin{array}{cccc}
\gamma^{\dagger}_{1,{\bm b}} & \gamma^{\dagger}_{0,{\bm b}}
& \gamma_{1,{\bm b}} & \gamma_{0,{\bm b}}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{A}_1 & 0 & \overline{B}_1 \\
\overline{A}^{*}_1 & 0 & \overline{B}_1 & 0 \\
0 & \overline{B}^{*}_1 & 0 & \overline{A}^{*}_1 \\
\overline{B}^{*}_1 & 0 & \overline{A}_1 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{1,{\bm b}} \\
\gamma_{0,{\bm b}} \\
\gamma^{\dagger}_{1,{\bm b}} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\end{array}\right) \nn \\
%
& \hspace{1.0cm}
+ \left(\begin{array}{cccc}
\gamma^{\dagger}_{3,{\bm b}+{\bm e}_y} & \gamma^{\dagger}_{0,{\bm b}}
& \gamma_{3,{\bm b}+{\bm e}_y} & \gamma_{0,{\bm b}}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{A}_2 & 0 & \overline{B}_2 \\
\overline{A}^{*}_2 & 0 & \overline{B}_2 & 0 \\
0 & \overline{B}^{*}_2 & 0 & \overline{A}^{*}_2 \\
\overline{B}^{*}_2 & 0 & \overline{A}_2 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{3,{\bm b}+{\bm e}_y} \\
\gamma_{0,{\bm b}} \\
\gamma^{\dagger}_{3,{\bm b}+{\bm e}_y} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\end{array}\right) \nn \\
%
& \hspace{1.5cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} & \gamma^{\dagger}_{1,{\bm b}}
& \gamma_{2,{\bm b}+{\bm e}_x} & \gamma_{1,{\bm b}}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{A}^{*}_2 & 0 & \overline{B}^{*}_2 \\
\overline{A}_2 & 0 & \overline{B}^{*}_2 & 0 \\
0 & \overline{B}_2 & 0 & \overline{A}_2 \\
\overline{B}_2 & 0 & \overline{A}^{*}_2 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma_{1,{\bm b}} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{1,{\bm b}} \\
\end{array}\right) \nn \\
%
& \hspace{2.0cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{3,{\bm b}+{\bm e}_y} &
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x}
& \gamma_{3,{\bm b}+{\bm e}_y} & \gamma_{2,{\bm b}+{\bm e}_x}
\end{array}\right) \left(\begin{array}{cccc}
0 & \overline{A}_3 & 0 & \overline{B}_3 \\
\overline{A}^{*}_3 & 0 & \overline{B}_3 & 0 \\
0 & \overline{B}^{*}_3 & 0 & \overline{A}^{*}_3 \\
\overline{B}^{*}_3 & 0 & \overline{A}_3 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{3,{\bm b}+{\bm e}_y} \\
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{3,{\bm b}+{\bm e}_y} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\end{array}\right) \nn \\
%
& \hspace{2.0cm}
+ \Big( \big\{{\bm e}_x,{\bm e}_y\big\},
\gamma^{(\dagger)}_{n,\cdots}
\rightarrow \big\{{\bm e}_{y}, -{\bm e}_x\big\},
\gamma^{(\dagger)}_{n+1,\cdots}
\Big) + \nn \\
%
&\hspace{1.0cm}
+ \Big( \big\{{\bm e}_x,{\bm e}_y\big\},
\gamma^{(\dagger)}_{n,\cdots}
\rightarrow \big\{-{\bm e}_{x}, -{\bm e}_y\big\},
\gamma^{(\dagger)}_{n+2,\cdots}
\Big)
+ \Big( \big\{{\bm e}_x,{\bm e}_y\big\},
\gamma^{(\dagger)}_{n,\cdots}
\rightarrow \big\{-{\bm e}_{y}, {\bm e}_x\big\},
\gamma^{(\dagger)}_{n+3,\cdots} \Big)
\bigg\}, \label{nn1}\end{aligned}$$
with $$\begin{aligned}
\overline{A}_1 & \equiv \langle u_{+,1,{\bm b}} |
{\bm H}_{1} | u_{+,0,{\bm b}} \rangle, \label{a1-ha} \\
\overline{B}_1 &\equiv \langle u_{+,1,{\bm b}} |
{\bm H}_{1}| u_{-,0,{\bm b}} \rangle, \label{b1-ha}\end{aligned}$$ $$\begin{aligned}
\overline{A}_2 & \equiv \langle
u_{+,3,{\bm b}+{\bm e}_y} |
{\bm H}_{1} | u_{+,0,{\bm b}} \rangle, \label{a2-ha} \\
\overline{B}_2 &\equiv \langle u_{+,3,{\bm b}+{\bm e}_y}
| {\bm H}_{1} | u_{-,0,{\bm b}} \rangle, \label{b2-ha} \end{aligned}$$ $$\begin{aligned}
\overline{A}_3 &\equiv \langle
u_{+,3,{\bm b}+{\bm e}_y} |
{\bm H}_{1} | u_{+,2,{\bm b}+{\bm e}_x} \rangle, \label{a3-ha} \\
\overline{B}_3 &\equiv \langle u_{+,3,{\bm b}+{\bm e}_y} |
{\bm H}_{1} | u_{-,2,{\bm b}+{\bm e}_x} \rangle. \label{b3-ha} \end{aligned}$$ The 2nd line and 3rd line in the r.h.s. of eq. (\[nn1\]) are related to each other by a combined symmetry between the time-reversal and a in-plane mirror which interchanges $x$-axis and $y$-axis.
The next dominant inter-cluster transfer integrals are between the next-nearest-neighbor (NNN) clusters and they hvae two kinds; one is $(\sigma,\sigma)$-coupling type, which is between two in-phase orbitals on $x$-links connected by ${\bm e}_x$ or those on $y$-links connected by ${\bm e}_y$ (Fig. \[fig:2-orbital-TB\]). The other is $(\pi,\pi)$-coupling type, which is between two in-phase orbitals on $x$-links connected by ${\bm e}_y$ or those on the $y$-links connected by ${\bm e}_x$ (Fig. \[fig:2-orbital-TB\]). They are given by
$$\begin{aligned}
{\bm H}^{NNN,\sigma\sigma}_1 = & \sum_{{\bm b}} \bigg\{
\left(\begin{array}{cccc}
\gamma^{\dagger}_{2,{\bm b}} & \gamma^{\dagger}_{0,{\bm b}}
& \gamma_{2,{\bm b}} & \gamma_{0,{\bm b}}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{C}_1 & 0 & \overline{D}_1 \\
\overline{C}^{*}_1 & 0 & \overline{D}_1 & 0 \\
0 & \overline{D}^{*}_1 & 0 & \overline{C}^{*}_1 \\
\overline{D}^{*}_1 & 0 & \overline{C}_1 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{2,{\bm b}} \\
\gamma_{0,{\bm b}} \\
\gamma^{\dagger}_{2,{\bm b}} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\end{array}\right) \nn \\
%
& \hspace{1cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{2,{\bm b}} & \gamma^{\dagger}_{2,{\bm b}+{\bm e}_x}
& \gamma_{2,{\bm b}} & \gamma_{2,{\bm b}+{\bm e}_x}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{C}_2 & 0 & \overline{D}_2 \\
\overline{C}^{*}_2 & 0 & \overline{D}_2 & 0 \\
0 & \overline{D}^{*}_2 & 0 & \overline{C}^{*}_2 \\
\overline{D}^{*}_2 & 0 & \overline{C}_2 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{2,{\bm b}} \\
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{2,{\bm b}} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\end{array}\right) \nn \\
%
& \hspace{1.5cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{0,{\bm b}} & \gamma^{\dagger}_{0,{\bm b}-{\bm e}_x}
& \gamma_{0,{\bm b}} & \gamma_{0,{\bm b}-{\bm e}_x}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{C}_2 & 0 & \overline{D}_2 \\
\overline{C}^{*}_2 & 0 & \overline{D}_2 & 0 \\
0 & \overline{D}^{*}_2 & 0 & \overline{C}^{*}_2 \\
\overline{D}^{*}_2 & 0 & \overline{C}_2 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{0,{\bm b}} \\
\gamma_{0,{\bm b}-{\bm e}_x} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\gamma^{\dagger}_{0,{\bm b}-{\bm e}_x} \\
\end{array}\right) \nn \\
%
& \hspace{2.0cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x}
& \gamma^{\dagger}_{0,{\bm b}-{\bm e}_x}
& \gamma_{2,{\bm b}+{\bm e}_x} & \gamma_{0,{\bm b}-{\bm e}_x}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{C}_3 & 0 & \overline{D}_3 \\
\overline{C}^{*}_3 & 0 & \overline{D}_3 & 0 \\
0 & \overline{D}^{*}_3 & 0 & \overline{C}^{*}_3 \\
\overline{D}^{*}_3 & 0 & \overline{C}_3 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma_{0,{\bm b}-{\bm e}_x} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{0,{\bm b}-{\bm e}_x} \\
\end{array}\right) \nn \\
%
& \hspace{2.5cm} + \Big({\bm e}_x,
\gamma^{(\dagger)}_{0,\cdots},
\gamma^{(\dagger)}_{2,\cdots}
\rightarrow {\bm e}_y,
\gamma^{(\dagger)}_{1,\cdots},
\gamma^{(\dagger)}_{3,\cdots} \Big)
\bigg\}, \label{nnns}\end{aligned}$$
and $$\begin{aligned}
{\bm H}^{NNN,\pi\pi}_1 = & \sum_{{\bm b}} \bigg\{
\left(\begin{array}{cccc}
\gamma^{\dagger}_{0,{\bm b}+{\bm e}_y}
& \gamma^{\dagger}_{0,{\bm b}}
& \gamma_{0,{\bm b}+{\bm e}_y} & \gamma_{0,{\bm b}}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{E}_1 & 0 & \overline{F}_1 \\
\overline{E}^{*}_1 & 0 & \overline{F}_1 & 0 \\
0 & \overline{F}^{*}_1 & 0 & \overline{E}^{*}_1 \\
\overline{F}^{*}_1 & 0 & \overline{E}_1 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{0,{\bm b}+{\bm e}_y} \\
\gamma_{0,{\bm b}} \\
\gamma^{\dagger}_{0,{\bm b}+{\bm e}_y} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\end{array}\right) \nn \\
%
& \hspace{1cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x+{\bm e}_y}
& \gamma^{\dagger}_{2,{\bm b}+{\bm e}_x}
& \gamma_{2,{\bm b}+{\bm e}_x+{\bm e}_y}
& \gamma_{2,{\bm b}+{\bm e}_x}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{E}^{*}_1 & 0 & \overline{F}_1 \\
\overline{E}_1 & 0 & \overline{F}_1 & 0 \\
0 & \overline{F}^{*}_1 & 0 & \overline{E}_1 \\
\overline{F}^{*}_1 & 0 & \overline{E}^{*}_1 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{2,{\bm b}+{\bm e}_x+{\bm e}_y} \\
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x+{\bm e}_y} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\end{array}\right) \nn \\
%
& \hspace{1.5cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x+{\bm e}_y}
& \gamma^{\dagger}_{0,{\bm b}}
& \gamma_{2,{\bm b}+{\bm e}_x+{\bm e}_y}
& \gamma_{0,{\bm b}}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{E}_2 & 0 & \overline{F}_2 \\
\overline{E}_2 & 0 & \overline{F}_2 & 0 \\
0 & \overline{F}^{*}_2 & 0 & \overline{E}_2 \\
\overline{F}^{*}_2 & 0 & \overline{E}_2 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{2,{\bm b}+{\bm e}_x+{\bm e}_y} \\
\gamma_{0,{\bm b}} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x+{\bm e}_y} \\
\gamma^{\dagger}_{0,{\bm b}} \\
\end{array}\right) \nn \\
%
& \hspace{2.0cm} + \left(\begin{array}{cccc}
\gamma^{\dagger}_{0,{\bm b}+{\bm e}_y}
& \gamma^{\dagger}_{2,{\bm b}+{\bm e}_x}
& \gamma_{0,{\bm b}+{\bm e}_y}
& \gamma_{2,{\bm b}+{\bm e}_x}
\end{array}\right)\left(\begin{array}{cccc}
0 & \overline{E}_2 & 0 & \overline{F}^{*}_2 \\
\overline{E}_2 & 0 & \overline{F}^{*}_2 & 0 \\
0 & \overline{F}_2 & 0 & \overline{E}_2 \\
\overline{F}_2 & 0 & \overline{E}_2 & 0 \\
\end{array}\right) \left(\begin{array}{c}
\gamma_{0,{\bm b}+{\bm e}_y} \\
\gamma_{2,{\bm b}+{\bm e}_x} \\
\gamma^{\dagger}_{0,{\bm b}+{\bm e}_y} \\
\gamma^{\dagger}_{2,{\bm b}+{\bm e}_x} \\
\end{array}\right) \nn \\
%
& \hspace{2.5cm} + \Big({\bm e}_x, {\bm e}_y,
\gamma_{0,\cdots},\gamma^{\dagger}_{0,\cdots},
\gamma_{2,\cdots},\gamma^{\dagger}_{2,\cdots},
\rightarrow {\bm e}_y, {\bm e}_x,
\gamma^{\dagger}_{1,\cdots},\gamma_{1,\cdots},
\gamma^{\dagger}_{3,\cdots},\gamma_{3,\cdots} \Big)
\bigg\}, \label{nnnp}\end{aligned}$$
with $$\begin{aligned}
\overline{C}_1 & \equiv \langle u_{+,2,{\bm b}} |
{\bm H}_{1} | u_{+,0,{\bm b}} \rangle, \label{c1-ha} \\
\overline{D}_1 &= \langle u_{+,2,{\bm b}} |
{\bm H}_{1} | u_{-,0,{\bm b}} \rangle, \label{d1-ha}\end{aligned}$$ $$\begin{aligned}
\overline{C}_2 &
\equiv \langle u_{+,2,{\bm b}} |
{\bm H}_{1} | u_{+,2,{\bm b}+{\bm e}_x} \rangle, \label{c2-ha} \\
\overline{D}_2 &= \langle u_{+,2,{\bm b}} |
{\bm H}_{1} | u_{-,2,{\bm b}+{\bm e}_x} \rangle, \label{d2-ha} \end{aligned}$$ $$\begin{aligned}
\overline{C}_3 & \equiv \langle u_{+,2,{\bm b}+{\bm e}_x} |
{\bm H}_{1} | u_{+,0,{\bm b}-{\bm e}_x} \rangle, \label{c3-ha} \\
\overline{D}_3 &= \langle u_{+,2,{\bm b}+{\bm e}_x} |
{\bm H}_{1} | u_{-,0,{\bm b}-{\bm e}_x} \rangle, \label{d3-ha} \end{aligned}$$ $$\begin{aligned}
\overline{E}_1 & \equiv \langle
u_{+,0,{\bm b}+{\bm e}_y} |
{\bm H}_{1} | u_{+,0,{\bm b}} \rangle, \label{e1-ha} \\
%
\overline{F}_1 &= \langle u_{+,0,{\bm b}+{\bm e}_y} |
{\bm H}_{1} | u_{-,0,{\bm b}} \rangle, \label{f1-ha}. \end{aligned}$$ $$\begin{aligned}
\overline{E}_2 & \equiv \langle
u_{+,2,{\bm b}+{\bm e}_x+{\bm e}_y} |
{\bm H}_{1} | u_{+,0,{\bm b}} \rangle, \label{e2-ha} \\
\overline{F}_2 &= \langle
u_{+,2,{\bm b}+{\bm e}_x+{\bm e}_y} |
{\bm H}_{1} | u_{-,0,{\bm b}} \rangle. \label{f2-ha} \end{aligned}$$ Evaluations based on eqs. (\[rotation\],\[p-h\],\[w1\],\[full\],\[a1-ha\]-\[f2-ha\]) suggests that $\overline{A}_1=ia_1$, $\overline{A}_2= -ia_2$, $\overline{A}_3=ia_3$, $\overline{B}_1
=b_1$, $\overline{B}_2=-b_2$, $\overline{B}_3=b_3$, $\overline{C}_1=c_1$, $\overline{C}_2=-c_2$, $\overline{C}_3=c_3$, $\overline{D}_1=d_1$, $\overline{D}_2=-d_2$, $\overline{D}_3=d_3$, $\overline{E}_1=-e_1$, $\overline{E}_2=e_2$, $\overline{F}_1= f_1$, $\overline{F}_2 = - f_2$ with real and positive $a_1$, $a_2$, $a_3$ ($a_1\gtrsim a_2 \gtrsim a_3>0$), $b_1$, $b_2$, $b_3$ ($b_1\gtrsim b_2 \gtrsim b_3>0$), $c_1$, $c_2$, $c_3$ ($c_1\gtrsim c_2 \gtrsim c_3>0$), $d_1$, $d_2$, $d_3$ ($d_1\gtrsim d_2 \gtrsim d_3>0$), $e_1$, $e_2$, $f_1$, $f_2$, ($e_1 \gtrsim e_2$) and ($f_1 \gtrsim f_2$).
Using eqs. (\[bog\]), we rewrite Eqs. (\[nn1\],\[nnns\],\[nnnp\]) in the basis of the in-phase ($\beta_{+,{\bm b}+\frac{{\bm e}_x}{2}}$, $\beta_{+,{\bm b}+\frac{{\bm e}_y}{2}}$) and out-of-phase ($\beta_{-,{\bm b}+\frac{{\bm e}_x}{2}}$, $\beta_{-,{\bm b}+\frac{{\bm e}_y}{2}}$) orbital wave functions. In most of the fully polarized regime, the in-phase orbital level goes below the out-of-phase orbital level. Focusing on the lowest two volume-mode bands, we thus ignore those transfer integrals which are involved with out-of-phase orbitals. This leads to, $$\begin{aligned}
\overline{\bm H}=\sum_{\bm b}\big\{
\overline{\bm H}_0
+ \overline{\bm H}^{NN}_1 +
\overline{\bm H}^{NNN}_1\big\}, \label{app-total}\end{aligned}$$ $$\begin{aligned}
\overline{\bm H}_0 &= \Delta \!\
\beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_x}{2}}
\beta_{+,{\bm b}+\frac{{\bm e}_x}{2}}
+ \Delta \!\ \beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_y}{2}}
\beta_{+,{\bm b}+\frac{{\bm e}_y}{2}} \label{app-h0}\end{aligned}$$ $$\begin{aligned}
\overline{\bm H}^{NN}_{1} &= (i a + b) \!\
\beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_y}{2}}
\beta_{+,{\bm b}+\frac{{\bm e}_x}{2}}
\nn \\
&\ \ \ - (i a+b) \!\
\beta^{\dagger}_{+,{\bm b}+{\bm e}_y+\frac{{\bm e}_x}{2}}
\beta_{+,{\bm b}+\frac{{\bm e}_y}{2}} \nn \\
& \ \ \ \ + (i a + b) \!\
\beta^{\dagger}_{+,{\bm b}+{\bm e}_x + \frac{{\bm e}_y}{2}}
\beta_{+,{\bm b}+{\bm e}_y+\frac{{\bm e}_x}{2}} \nn \\
& \hspace{0.3cm} - (i a+ b) \!\
\beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_x}{2}}
\beta_{+,{\bm b}+{\bm e}_x+\frac{{\bm e}_y}{2}} + {\rm h.c.}
\label{app-h1}\end{aligned}$$ $$\begin{aligned}
\overline{\bm H}^{NNN}_1 &= c \!\
\beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_x}{2}}
\beta_{+,{\bm b} - \frac{{\bm e}_x}{2}}
+ c \!\ \beta^{\dagger}_{+,{\bm b}+\frac{{\bm e}_y}{2}}
\beta_{+,{\bm b} - \frac{{\bm e}_y}{2}} + \nn \\
& \hspace{-1.2cm} c^{\prime} \!\
\beta^{\dagger}_{+,{\bm b}+{\bm e}_y +\frac{{\bm e}_x}{2}}
\beta_{+,{\bm b} + \frac{{\bm e}_x}{2}}
+ c^{\prime} \!\
\beta^{\dagger}_{+,{\bm b}+{\bm e}_x
+\frac{{\bm e}_y}{2}}
\beta_{+,{\bm b} + \frac{{\bm e}_y}{2}} + {\rm h.c.}, \label{app-h2} \end{aligned}$$ with $$\begin{aligned}
\Delta & = \sqrt{(E-t)^2-s^2} \\
a &= a_1+2a_2+a_3, \label{a-ha} \\
b &=
\frac{(b_1+2b_2+b_3)\cdot s}{\sqrt{(E-t)^2-s^2}}, \label{b-ha} \\
c & = - \frac{(d_1+2d_2+d_3)\cdot s
+ (c_1+2c_2+c_3)\cdot (E-t) }{\sqrt{(E-t)^2-s^2}}, \label{c-ha} \\
c^{\prime} & = \frac{2(f_1+f_2)\cdot s
- 2(e_1+e_2)\cdot (E-t)}{\sqrt{(E-t)^2-s^2}}. \label{cd-ha} \end{aligned}$$ Since the particle space and the hole space is separated by a large frequency spacing, $2\Delta$, we have also omitted hopping terms in particle-particle channel, such as $\beta^{\dagger} \beta^{\dagger}$ and $\beta \beta$. $a \!\ (>0)$ and $b \!\ (>0)$ quantify an imaginary part and real part of the nearest-neighbor inter-cluster transfer integral, while $c \!\ (<0)$ and $c^{\prime} \!\ (<0)$ stand for the $(\sigma,\sigma)$-coupling and the $(\pi,\pi)$-coupling next-nearest-neighbor transfer integrals respectively. An amplitude of transfer integral is inversely proportional to the cubic in distance (eq. (\[rotation\])), so that the $(\sigma,\sigma)$-coupling type is expected to be larger than the $(\pi,\pi)$-coupling type, $|c|>|c^{\prime}|$ (or $c_1+2c_2+c_3>2e_1+2e_2$). Note also that $b\rightarrow 0$ in the limit of $H\rightarrow +\infty$, where $t/E,s/E\rightarrow 0$. By replacing $\beta_{+,{\bm b}+\frac{{\bm e}_x}{2}}$ and $\beta_{+,{\bm b}+\frac{{\bm e}_y}{2}}$ by $\beta_{{\bm b}+\frac{{\bm e}_x}{2}}$ and $\beta_{{\bm b}+\frac{{\bm e}_y}{2}}$ respectively, we have eqs. (\[h0\],\[h1\]).
[99]{} A. A. Serga, A. V. Chumak and B. Hillebrands, J. Phys. D: Appl. Phys. **43**, 264002 (2010). V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D, **43**, 264001 (2010). R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids, **19**, 308 (1961). M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. **87**, 153501 (2005). K. S. Lee and S. K. Kim, J. Appl. Phys. **104**, 053909 (2008). T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. **92**, 022505 (2008). N. Sato, K. Sekiguchi, Y. Nozaki, Appl. Phys. Express. **6**, 063001 (2013). R. Shindou, R. Matsumoto, S. Murakami, and J-i Ohe, Phys. Rev. B, **87**, 174427 (2013). R. Shindou, J-i Ohe, R. Matsumoto, S. Murakami, and E. Saitoh, Phys. Rev. B, **87**, 174402 (2013). D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. **49**, 405 (1982). B. I. Halperin, Phys. Rev. B **25**, 2185 (1982). Y. Hatsugai, Phys. Rev. Lett. **71**, 3697 (1993). R. W. Damon and H. Van De Varrt, J. Appl. Phys. **36**, 3453 (1965). B. A. Kalinikos, and A. N. Slavin, J. Phys. C: Solid State Phys **19**, 7013 (1986). R. Arias, and D. L. Mills, Phys. Rev. B, **63**, 134439 (2001). G. E. Volovik, Sov. Phys. JETP, **67**, 1804 (1988). V. M. Yakovenko, Phys. Rev. Letters, **65**, 251 (1990). X. L. Qi, Y. S. Wu, and S. C. Zhang, Phys. Rev. B **74**, 085308 (2006). B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science **314**, 1757 (2006). L. Fu and C. L. Kane, Phys. Rev. B **76**, 045302 (2007). A. O. Adeyeye and N. Singh, J. Phys. D: Appl. Phys. **41**, 153001 (2008). Y. V. Gulyaev, JETP Lett. **77**, 567 (2003). A. Hubert, and R. Schafer, [*Magnetic Domains*]{} (Springer, Berlin, Germany, 2000). Z. Q. Qiu, J. Pearson, S. D. Bader, Phys. Rev. Letters, **70**, 1006 (1993). R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Letters, **83**, 1042 (1999). T. Shinjo, T. Okuno, R. Hassdorf, and K. Shigeto, and T. Ono, Science, **289**, 930 (2000). J. E. Avron, R. Seiler and B. Simon, Phys. Rev. Letters, **51**, 51 (1983).
|
---
abstract: |
We use deep $J$ ($1.25 \ \mu$m) and $K_s$ ($2.15 \ \mu$m) images of the Antennae (NGC 4038/9) obtained with the Wide-field InfraRed Camera on the Palomar 200-inch telescope, together with the [*Chandra*]{} X-ray source list of @zez02a, to search for infrared counterparts to X-ray point sources. We establish an X-ray/IR astrometric frame tie with $\sim0\farcs5$ rms residuals over a $\sim
4\farcm3$ field. We find 13 “strong” IR counterparts brighter than $K_s = 17.8$ mag and $<1\farcs0$ from X-ray sources, and an additional 6 “possible” IR counterparts between $1\farcs0$ and $1\farcs5$ from X-ray sources. Based on a detailed study of the surface density of IR sources near the X-ray sources, we expect only $\sim 2$ of the “strong” counterparts and $\sim 3$ of the “possible” counterparts to be chance superpositions of unrelated objects.
Comparing both strong and possible IR counterparts to our photometric study of $\sim 220$ IR clusters in the Antennae, we find with a $>$ 99.9% confidence level that IR counterparts to X-ray sources are $\Delta M_{K_s} \sim 1.2$ mag more luminous than average non-X-ray clusters. We also note that the X-ray/IR matches are concentrated in the spiral arms and “overlap” regions of the Antennae. This implies that these X-ray sources lie in the most “super” of the Antennae’s Super Star Clusters, and thus trace the recent massive star formation history here. Based on the $N_H$ inferred from the X-ray sources without IR counterparts, we determine that the absence of most of the “missing” IR counterparts../../../ASTRO-PH/ is not due to extinction, but that these sources are intrinsically less luminous in the IR, implying that they trace a different (possibly older) stellar population. We find no clear correlation between X-ray luminosity classes and IR properties of the sources, though small-number statistics hamper this analysis.
author:
- 'D.M. Clark , S.S. Eikenberry , B.R. Brandl , J.C. Wilson , J.C. Carson , C.P. Henderson , T.L. Hayward , D.J. Barry , A.F. Ptak , and E.J.M. Colbert'
title: 'Infrared Counterparts to [*Chandra*]{} X-Ray Sources in the Antennae'
---
Introduction
============
Recently, high resolution X-ray images using [*Chandra*]{} have revealed 49 point sources in the Antennae [@zez02a]. We will assume a distance to the Antennae of 19.3 Mpc (for $H_{0}$=75 km s$^{-1}$ Mpc$^{-1}$), which implies 10 sources have X-ray luminosities greater than $10^{39}$ ergs s$^{-1}$. Considering new observations of red giant stars in the Antennae indicate a distance of 13.8 Mpc [@sav04], we point out this ultraluminous X-ray source population could decrease by roughly a half. Typically, masses of black holes produced from standard stellar evolution are less than $\sim20$ $M_{\odot}$ [e.g., @fry01]. The Eddington luminosity limit implies that X-ray luminosities $>10^{39}$ ergs s$^{-1}$ correspond to higher-mass objects not formed from a typical star. Several authors [e.g., @fab89; @zez99; @rob00; @mak00] suggest these massive ($10
\sbond 1000$ $M_{\sun}$) compact sources outside galactic nuclei are intermediate mass black holes (IMBHs), a new class of BHs. While IMBHs could potentially explain the observed high luminosities, other theories exist as well, including beamed radiation from a stellar mass BH [@kin01]; super-Eddington accretion onto lower-mass objects [e.g., @moo03; @beg02]; or supernovae exploding in dense environments [@ple95; @fab96].
Compact objects tend to be associated with massive star formation, which is strongly suspected to be concentrated in young stellar clusters [@lad03]. Massive stars usually end their lives in supernovae, producing a compact remnant. This remnant can be kicked out of the cluster due to dynamical interactions, stay behind after the cluster evaporates, or remain embedded in its central regions. This last case is of particular interest to us as the compact object is still [*in situ*]{}, allowing us to investigate its origins via the ambient cluster population. The potential for finding such associations is large in the Antennae due to large numbers of both X-ray point sources and super star clusters; a further incentive for studying these galaxies.
In @bra05 [henceforth Paper I] we presented $J$ and $K_s$ photometry of $\sim220$ clusters in the Antennae. Analysis of ($J -
K_s$) colors indicated that many clusters in the overlap region suffer from 9–10 mag of extinction in the $V$-band. This result contrasts with previous work by @whi02 who associated optical sources with radio counterparts in the Antennae [@nef00] and argued that extinction is not large in this system. Here, we continue our analysis of these Antennae IR images by making a frame-tie between the IR and [*Chandra*]{} X-ray images from @fab01. Utilizing the similar dust-penetrating properties of these wavelengths, we demonstrate the power of this approach to finding counterparts to X-ray sources. By comparing the photometric properties of clusters with and without X-ray counterparts, we seek to understand the cluster environments of these X-ray sources. In §2 we discuss the IR observations of the Antennae. §3 explains our matching technique and the photometric properties of the IR counterparts. We conclude with a summary of our results in §4.
Observations and Data Analysis
==============================
Infrared Imaging
----------------
We obtained near-infrared images of NGC 4038/9 on 2002 March 22 using the Wide-field InfraRed Camera (WIRC) on the Palomar 5-m telescope. At the time of these observations, WIRC had been commissioned with an under-sized HAWAII-1 array (prior to installation of the full-sized HAWAII-2 array in September 2002), providing a $\sim 4.7 \times
4.7$-arcminute field of view with $\sim0\farcs25$ pixels (“WIRC-1K” – see @wil03 for details). Conditions were non-photometric due to patches of cloud passing through. Typical seeing-limited images had stellar full-width at half-maximum of $1\farcs0$ in $K_s$ and $1\farcs3$ in $J$. We obtained images in both the $J$- ($1.25
\mu$m) and $K_s$-band ($2.15 \mu$m) independently. The details of the processing used to obtain the final images are given in Paper 1.
Astrometric Frame Ties
----------------------
The relative astrometry between the X-ray sources in NGC 4038/9 and images at other wavelengths is crucial for successful identification of multi-wavelength counterparts. Previous attempts at this have suffered from the crowded nature of the field and confusion between potential counterparts @zez02b. However, the infrared waveband offers much better hopes for resolving this issue, due to the similar dust-penetrating properties of photons in the [*Chandra*]{} and $K_s$ bands. (See also @bra05 for a comparison of IR extinction to the previous optical/radio extinction work of @whi02.) We thus proceeded using the infrared images to establish an astrometric frame-tie, i.e. matching [*Chandra*]{} coordinates to IR pixel positions.
As demonstrated by @bau00, we must take care when searching for X-ray source counterparts in crowded regions such as the Antennae. Therefore, our astrometric frame-tie used a unique approach based on solving a two-dimensional linear mapping function relating right ascension and declination coordinates in one image with x and y pixel positions in a second image. The solution is of the form:
$$\begin{aligned}
r_1 = ax_1+by_1+c, \\
d_1 = dx_1+dy_1+f\end{aligned}$$
Here $r_1$ and $d_1$ are the right ascension and declination, respectively, for a single source in one frame corresponding to the $x_1$ and $y_1$ pixel positions in another frame. This function considers both the offset and rotation between each frame. Since we are interested in solving for the coefficients $a$–$f$, elementary linear algebra indicates we need six equations or three separate matches. Therefore, we need at least three matches to fully describe the rms positional uncertainty of the frame-tie.
We first used the above method to derive an approximate astrometric solution for the WIRC $K_s$ image utilizing the presence of six relatively bright, compact IR sources which are also present in images from the 2-Micron All-Sky Survey (2MASS). We calculated pixel centroids of these objects in both the 2MASS and WIRC images, and used the 2MASS astrometric header information to convert the 2MASS pixel centroids into RA and Dec. These sources are listed in Table 1. Applying these six matches to our fitting function we found a small rms positional uncertainty of $0\farcs2$, which demonstrates an accurate frame-tie between the 2MASS and WIRC images.
Using the 2MASS astrometric solution as a baseline, we identified seven clear matches between [*Chandra*]{} and WIRC sources, which had bright compact IR counterparts with no potentially confusing sources nearby (listed in Table 2). We then applied the procedure described above, using the [*Chandra*]{} coordinates listed in Table 1 of @zez02a (see that reference for details on the [*Chandra*]{} astrometry) and the WIRC pixel centroids, and derived the astrometric solution for the IR images in the X-ray coordinate frame. For the 7 matches, we find an rms residual positional uncertainty of $\sim
0\farcs5$ which we adopt as our $1 \sigma$ position uncertainty. We note that the positional uncertainty is an entirely [*empirical*]{} quantity. It shows the achieved uncertainty in mapping a target from one image reference frame to the reference frame in another band, and automatically incorporates all contributing sources of uncertainty in it. These include, but are not limited to, systematic uncertainties (i.e. field distortion, PSF variation, etc. in both [*Chandra*]{} and WIRC) and random uncertainties (i.e. centroid shifts induced by photon noise, flatfield noise, etc. in both [*Chandra*]{} and WIRC). Thus, given the empirical nature of this uncertainty, we expect it to provide a robust measure of the actual mapping error – an expectation which seems to be borne out by the counterpart identification in the following section.
To further test the accuracy of our astrometric solution we explored the range in rms positional uncertainties for several different frame-ties. Specifically we picked ten IR/X-ray matches separated by $<$1$\arcsec$, which are listed in Table 3 (see §3.1). Of these ten we chose 24 different combinations of seven matches resulting in 24 unique frame-ties. Computing the rms positional uncertainty for each we found a mean of 0$\farcs$4 with a 1$\sigma$ uncertainty of 0$\farcs$1. Considering the rms positional uncertainty for the frame-tie used in our analysis falls within 1$\sigma$ of this mean rms, this indicates we made an accurate astrometric match between the IR and X-ray.
Infrared Photometry
-------------------
We performed aperture photometry on 222 clusters in the $J$-band and 221 clusters in the $K_s$-band (see also Paper 1). We found that the full width at half maximum (FWHM) was 3.5 pixels ($0\farcs9$) in the $K_s$ image and 4.6 pixels ($1\farcs2$) in the $J$ band. We used a photometric aperture of 5-pixel radius in $K_s$ band, and 6-pixel radius in $J$ band, corresponding to $\sim 3\sigma$ of the Gaussian PSF.
Background subtraction is both very important and very difficult in an environment such as the Antennae due to the brightness and complex structure of the underlying galaxies and the plethora of nearby clusters. In order to address the uncertainties in background subtraction, we measured the background in two separate annuli around each source: one from 9 to 12 pixels and another from 12 to 15 pixels. Due to the high concentration of clusters, crowding became an issue. To circumvent this problem, we employed the use of sky background arcs instead of annuli for some sources. These were defined by a position angle and opening angle with respect to the source center. All radii were kept constant to ensure consistency. In addition, nearby bright sources could shift the computed central peak position by as much as a pixel or two. If the centroid position determined for a given source differed significantly ($>1$ pixel) from the apparent brightness peak due to such contamination, we forced the center of all photometric apertures to be at the apparent brightness peak. For both annular regions, we calculated the mean and median backgrounds per pixel.
Multiplying these by the area of the central aperture, these values were subtracted from the flux measurement of the central aperture to yield 4 flux values for the source in terms of DN. Averaging these four values provided us with a flux value for each cluster. We computed errors by considering both variations in sky background, $\sigma_{sky}$, and Poisson noise, $\sigma_{adu}$. We computed $\sigma_{sky}$ by taking the standard deviation of the four measured flux values. We then calculated the expected Poisson noise by scaling DN to $e^-$ using the known gain of WIRC [2 $e^{-1}$ DN$^{-1}$, @wil03] and taking the square root of this value. We added both terms in quadrature to find the total estimated error in photometry.
We then calibrated our photometry using the bright 2MASS star in Table 1.
Results and Discussion
======================
Identification of IR Counterparts to [*Chandra*]{} Sources
----------------------------------------------------------
We used the astrometric frame-tie described above to identify IR counterpart candidates to [*Chandra*]{} X-ray sources in the WIRC $K_s$ image. We restricted our analysis to sources brighter than $K_s\sim19.4$ mag. This is our $K_s$ sensitivity limit which we define in our photometric analysis below (see §3.2.1). Using the $0\farcs5$ rms positional uncertainty of our frame-tie, we defined circles with 2$\sigma$ and 3$\sigma$ radii around each [*Chandra*]{} X-ray source where we searched for IR counterparts (Figure 1). If an IR source lay within a $1\farcs0$ radius ($2\sigma$) of an X-ray source, we labeled these counterparts as “strong”. Those IR sources between $1\farcs0$ and $1\farcs5$ ($2-3 \sigma$) from an X-ray source we labeled as “possible” counterparts. We found a total of 13 strong and 6 possible counterparts to X-ray sources in the Antennae. These sources are listed in Table 3 and shown in Figure 3. Of the 19 X-ray sources with counterparts, two are the nuclei [@zez02a], one is a background quasar [@cla05], and two share the same IR counterpart. Therefore, in our analysis of cluster properties, we only consider the 15 IR counterparts that are clusters. (While X-42 has two IR counterparts, we chose the closer, fainter cluster for our analysis.)
We then attempted to estimate the level of “contamination” of these samples due to chance superposition of unrelated X-ray sources and IR clusters. This estimation can be significantly complicated by the complex structure and non-uniform distribution of both X-ray sources and IR clusters in the Antennae, so we developed a simple, practical approach. Given the $<0\farcs5$ rms residuals in our relative astrometry for sources in Table 2, we assume that any IR clusters lying in a background annulus with radial size of $2\farcs0$–$3\farcs0$ ($4-6\sigma$) centered on all X-ray source positions are chance alignments, with no real physical connection (see Figure 1). Dividing the total number of IR sources within the background annuli of the 49 X-ray source positions by the total area of these annuli, we find a background IR source surface density of 0.02 arcsecond$^{-2}$ near [*Chandra*]{} X-ray sources. Multiplying this surface density by the total area of all “strong” regions ($1\farcs0$ radius circles) and “possible” regions ($1\farcs0$ – $1\farcs5$ annuli) around the 49 X-ray source positions, we estimated the level of source contamination contributing to our “strong” and “possible” IR counterpart candidates. We expect two with a $1\sigma$ uncertainty of +0.2/-0.1[^1] of the 13 “strong” counterparts to be due to chance superpositions, and three with a $1\sigma$ uncertainty of +0.5/-0.3of the six “possible” counterparts to be chance superpositions.
This result has several important implications. First of all, it is clear that we have a significant excess of IR counterparts within $1\farcs0$ of the X-ray sources – 13, where we expect only two in the null hypothesis of no physical counterparts. Even including the “possible” counterparts out to $1\farcs5$, we have a total of 19 counterparts, where we expect only five from chance superposition. Secondly, this implies that for any given “strong” IR counterpart, we have a probability of $\sim$ 85% ($11/13$ with a $1\sigma$ uncertainty of 0.3) that the association with an X-ray source is real. Even for the “possible” counterparts, the probability of true association is $\sim$50%. These levels of certainty are a tremendous improvement over the X-ray/optical associations provided by @zez02b, and are strong motivators for follow-up multi-wavelength studies of the IR counterparts. Finally, we can also conclude from strong concentration of IR counterparts within $\sim 1 \arcsec$ of X-ray sources that the frame tie uncertainty estimates described above are reasonable.
Figure 2 is a $4\farcm3\times4\farcm3$ $K_s$ image of the Antennae with X-ray source positions overlaid. We designate those X-ray sources with counterparts using red circles. Notice that those sources with counterparts lie in the spiral arms and bridge region of the Antennae. Since these regions are abundant in star formation, this seems to indicate many of the X-ray sources in the Antennae are tied to star formation in these galaxies.
Photometric Properties of the IR Counterparts
---------------------------------------------
### Color Magnitude Diagrams
Using the 219 clusters that had both $J$ and $K_s$ photometry, we made $(J-K_s)$ versus $K_s$ color magnitude diagrams (Figure 4). We estimated our sensitivity limit by first finding all clusters with signal-to-noise $\sim5\sigma$. The mean $J$ and $K_s$ magnitudes for these clusters were computed separately and defined as cutoff values for statistical analyzes. This yielded 19.0 mag in $J$ and 19.4 mag in $K_s$. We note that the X-ray clusters are generally bright in the IR compared to the general population of clusters. While the IR counterpart for one X-ray source (X-32) falls below our $J$-band sensitivity limit, its $K_s$ magnitude is still above our $K_s$ cutoff. Therefore, we retained this source in our analysis.
We then broke down the X-ray sources into three luminosity classes (Figure 4). We took the absorption-corrected X-ray luminosities, $L_X$, as listed in Table 1 of @zez02a for all sources of interest. These luminosities assumed a distance to the Antennae of 29 Mpc. We used 19.3 Mpc (for $H_{0}$=75 km s$^{-1}$ Mpc$^{-1}$) instead and so divided these values by 2.25 as suggested in @zez02a. We defined the three X-ray luminosities as follows: Low Luminosity X-ray sources (LLX’s) had $L_{X}$ $<$ 3$\times$$10^{38}$ ergs s$^{-1}$, High Luminosity X-ray sources (HLX’s) were between $L_{X}$ of 3$\times10^{38}$ergs s$^{-1}$ and 1$\times10^{39}$ ergs s$^{-1}$, while $L_{X}$ $>$ 1$\times10^{39}$ ergs s$^{-1}$ were Ultra-Luminous X-ray Sources (ULX’s). In Figure 4 we designate each IR counterpart according to the luminosity class of its corresponding X-ray source. There does not appear to be a noticeable trend in the IR cluster counterparts between these different groupings.
### Absolute K Magnitudes
To further study the properties of these IR counterparts, we calculated $M_{K_s}$ for all IR clusters. We calculated reddening using the observed colors, $(J-K_s)_{obs}$, (hence forth the “color method”). Assuming all clusters are dominated by O and B stars, their intrinsic $(J-K_s)$ colors are $\sim$0.2 mag. Approximating this value as 0 mag, this allowed us to estimate $A_{K_s}$ as $\simeq$ $(J-K_s)_{obs}$/1.33 using the extinction law defined in @car89 [hereafter CCM]. Since these derived reddenings are biased towards young clusters, they will lead to an overestimate of $M_{K_s}$ for older clusters.
For IR counterparts to X-ray sources, we also computed X-ray-estimated $A_{K_s}$ using the column densities, $N_{H}$, listed in Table 5 of @zez02a. Here, $N_{H}$ is derived by fitting both a Power Law (PL) and Raymond-Smith [RS @ray77] model to the X-ray spectra. Using the CCM law, $A_{K_s}$ is defined as 0.12$A_{V}$. Taking $A_{V}$ = $5\times10^{-22}$ mag cm$^2$ $N_{H}$, we could then derive $A_{K_s}$.
Then we compared $A_{K_s}$ calculated using the “color method” to $A_{K_s}$ found using the above two $N_{H}$ models. We found the “color method” matched closest to $N_{H}(PL)$ for all except one (the cluster associated with [*Chandra*]{} source 32 as designated in @zez02b).
In Figure 5, we plot histograms of the distribution of $K_s$-band luminosity, $M_{K_s}$. Figure 5 displays all clusters as well as over plotting only those with X-ray counterparts. Notice that the clusters with associated X-ray sources look more luminous. To study whether this apparent trend in luminosity is real, we compared these two distributions using a two-sided Kolmogorov-Smirnov (KS) test. In our analysis, we only included clusters below $M_{K_s}$ $<$ -13.2 mag. Restricting our study to sources with “good” photometry, we first defined a limit in $K_s$, 18.2 mag, using the limiting $J$ magnitude, 19.0 mag as stated above, and, since the limit in $K_s$ is a function of cluster color, the median $(J-K_s)$ of 0.8 mag. Subtracting the distance modulus to the Antennae, 31.4 mag, from this $K_s$ limit, we computed our cutoff in $M_{K_s}$. Since all clusters with X-ray sources fall below this cutoff, our subsample is sufficient to perform a statistical comparison.
The KS test yielded a D-statistic of 0.37 with a probability of $3.2\times10^{-2}$ that the two distributions of clusters with and without associated X-ray sources are related. Considering the separate cluster populations as two probability distributions, each can be expressed as a cumulative distribution. The D-statistic is then the absolute value of the maximum difference between each cumulative distribution. This test indicates that those clusters with X-ray counterparts are more luminous than most clusters in the Antennae.
### Cluster Mass Estimates
@whi99 found 70% of the bright clusters observed with the [*Hubble Space Telescope*]{} have ages $<$20 Myr. Therefore, in this study we will assume all clusters are typically the same age, $\sim$20 Myr. This allows us to make the simplifying assumption that cluster mass is proportional to luminosity and ask: Does the cluster mass affect the propensity for a given progenitor star to produce an X-ray binary? We estimated cluster mass using $K_s$ luminosity ($M_{K_s}$). Since cluster mass increases linearly with flux (for an assumed constant age of all clusters), we converted $M_{K_s}$ to flux. Using the data as binned in the $M_{K_s}$ histogram (Figure 5), we calculated an average flux per bin. By computing the fraction of number of clusters per average flux, we are in essence asking what is the probability of finding a cluster with a specific mass. Since those clusters with X-ray sources are more luminous, we expect a higher probability of finding an X-ray source in a more massive cluster. As seen in Figure 6, this trend does seem to be true. Applying a KS-test between the distributions for all clusters and those associated with X-ray sources for clusters below the $M_{K_s}$ completeness limit defined in the previous section, we find a D-statistic of 0.66 and a probability of $7.2\times10^{-3}$ that they are the same. Hence, the two distributions are distinct, indicating it is statistically significant that more massive clusters tend to contain X-ray sources.
While we assume all clusters are $\sim$20 Myr above, we note that the actual range in ages is $\sim$1–100 Myr [@whi99]. Bruzual-Charlot spectral photometric models [@bru03] indicate that clusters in this age range could vary by a factor of roughly 100 in mass for a given $K_s$ luminosity. Thus, we emphasize that the analysis above should be taken as suggestive rather than conclusive evidence, and note that in a future paper (Clark, et al. 2006, in preparation) we explore this line of investigation and the impacts of age variations on the result in depth.
### Non-detections of IR Counterparts to X-ray Sources
To assess whether our counterpart detections were dependent of reddening or their intrinsic brightness, we found limiting values for $M_{K_s}$ for those X-ray sources without detected IR counterparts. We achieved this by setting all clusters $K_s$ magnitudes equal to our completeness limit defined for the CMDs (19.4 mag; see §3.2.1) and then finding $M_{K_s(lim)}$ for each using $A_{K_s}$ calculated for that cluster. Since $M_{K_s(lim)}$ is theoretical and only depends on reddening, we could now find this limit for all X-ray sources using an $A_{K_s}$ estimated from the observed $N_{H}$ values. Thus we considered all IR counterparts (detections) and those X-ray sources without a counterpart (nondetections). If nondetections are due to reddening there should not exist a difference in $M_{K_s(lim)}$ between detections and nondetections. In contrast, if nondetections are intrinsically fainter, we expect a higher $M_{K_s(lim)}$ for these sources. In the case of detections, we considered reddening from both the “color method” and the $N_{H}(PL)$ separately. We could only derive nondetections using $N_{H}(PL)$ reddening. Figure 7 shows $M_{K_s(lim)}$ appears higher for all nondetections. To test if this observation is significant, we applied a KS-test to investigate whether detections and nondetections are separate distributions. We find a D-statistic of 0.82 and probability of $8.8\times10^{-6}$ that these two distributions are the same using the “color method” for detections. Considering the $N_{H}(PL)$ reddening method for detections instead, the D-statistic drops to 0.48 and the probability increases to $3.9\times10^{-2}$. Since both tests indicate these distributions are distinct, the observed high $M_{K_s(lim)}$ for nondetections seems to be real. This leads to the conclusion that these sources were undetected because they are intrinsically IR-faint, and that reddening does not play the dominant role in nondetections.
We summarize these statistics in Table 4. Here we calculated the mean $K_s$, $(J-K_s)$, and $M_{K_s}$ for three different categories: 1) all clusters, 2) clusters only connected with X-ray sources, and 3) these clusters broken down by luminosity class. We also include uncertainties in each quantity. Notice that the IR counterparts appear brighter in $K_s$ and intrinsically more luminous than most clusters in the Antennae, although there is no significant trend in color. We also summarize the above KS-test results in Table 5.
Conclusions
===========
We have demonstrated a successful method for finding counterparts to X-ray sources in the Antennae using IR wavelengths. We mapped [*Chandra*]{} X-ray coordinates to WIRC pixel positions with a positional uncertainty of $\sim 0\farcs5$. Using this precise frame-tie we found 13 “strong” matches ($< 1\farcs0$ separation) and 5 “possible” matches ($1 - 1\farcs5$ separation) between X-ray sources and IR counterparts. After performing a spatial and photometric analysis of these counterparts, we reached the following conclusions:
1\. We expect only 2 of the 13 “strong” IR counterparts to be chance superpositions. Including all 19 IR counterparts, we estimated 5 are unrelated associations. Clearly, a large majority of the X-ray/IR associations are real.
2\. The IR counterparts tend to reside in the spiral arms and bridge region between these interacting galaxies. Since these regions contain the heaviest amounts of star formation, it seems evident that many of the X-ray sources are closely tied to star formation in this pair of galaxies.
3\. A $K_s$ vs. $(J - K_s)$ CMD reveal those clusters associated with X-ray sources are brighter in $K_s$ but there does not seem to be a trend in color. Separating clusters by the X-ray luminosity classes of their X-ray counterpart does not reveal any significant trends.
4\. Using reddening derived $(J - K_s)$ colors as well as from X-ray-derived $N_H$, we found $K_s$-band luminosity for all clusters. A comparison reveals those clusters associated with X-ray sources are more luminous than most clusters in the Antennae. A KS-test indicates a significant difference between X-ray counterpart clusters and the general population of clusters.
5\. By relating flux to cluster luminosity, simplistically assuming a constant age for all clusters, we estimated cluster mass. Computing the fraction of number of clusters per average flux, we estimated the probability of finding a cluster with a specific mass. We find more massive clusters are more likely to contain X-ray sources, even after we normalize by mass.
6\. We computed a theoretical, limiting $M_{K_s}$ for all counterparts to X-ray sources in the Antennae using X-ray-derived reddenings. Comparing detections to non-detections, we found those clusters with X-ray source are intrinsically more luminous in the IR.
In a future paper exploring the effects of cluster mass on XRB formation rate (Clark, et al. 2006a, in preparation), we will investigate the effects of age on cluster luminosity and hence our cluster mass estimates. Another paper will extend our study of the Antennae to optical wavelengths (Clark, et al. 2006b, in preparation). Through an in depth, multi-wavelength investigation we hope to achieve a more complete picture of counterparts to several X-ray sources in these colliding galaxies.
The authors thank the staff of Palomar Observatory for their excellent assistance in commissioning WIRC and obtaining these data. WIRC was made possible by support from the NSF (NSF-AST0328522), the Norris Foundation, and Cornell University. S.S.E. and D.M.C. are supported in part by an NSF CAREER award (NSF-9983830). We also thank J.R. Houck for his support of the WIRC instrument project.
Begelman, M. C. 2002, , 568, L97
Brandl, B. R. et al. 2005, , 635, 280
Bruzual, G. & Charlot, S. 2003, , 344, 1000
Bauer, F. E., Condon, J. J., Thuan, T. X., & Broderick, J. J. 2000, , 129, 547
Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, , 345, 245
Clark, D. M. et al. 2005, , 631, L109
Fabbiano, G. 1989, , 27, 87
Fabbiano, G., Zezas, A., & Murray, S. S. 2001, , 554, 1035
Fabian, A. C. & Terlevich, R. 1996, , 280, L5
Fryer, C. L., & Kalogera, V. 2001, , 554, 548
Gehrels, N. 1986, , 303, 336
King, A. R, Davies, M. B., Ward, M. J., Fabbiano, G., & Elvis, M. 2001, , 552, L109
Makishima, K. et al. 2000, , 535, 632
Moon, D.-S., Eikenberry, S. S., & Wasserman, I. M. 2003, , 586, 1280
Neff, S. G., & Ulvestad, J. S. 2000, , 120, 670
Lada, C. J. & Lada, E. A. 2003, , 41, 57
Plewa, T. 1995, , 275, 143
Raymond, J. C. & Smith, B. W. 1977, ApJS, 35, 419
Roberts, T. P. & Warwick, R. S. 2000, , 315, 98
Saviane, I., Hibbard, J. E., & Rich, R. M. 2004, IAUS, 217, 546
Whitmore, B. C., Zhang, Q., Leitherer, C., Fall, S. M., Schweizer, F., & Miller, B. W. 1999, AJ, 118, 1551
Whitmore, B. C. & Zhang, Q. 2002, , 124, 1418
Wilson, J. C. et al. 2003, Proc. SPIE, 4841, 451
Zezas, A. L., Georgantopoulos, I., & Ward, M. J. 1999, , 308, 302
Zezas, A., Fabbiano, G., Rots, A. H., & Murray, S. S. 2002a, , 142, 239
Zezas, A., Fabbiano, G., Rots, A. H., & Murray, S. S. 2002b, , 577, 710
[lcccc]{} Description & RA (2MASS) & Dec (2MASS) & $J$ & $K_s$\
Bright Star & 12:01:47.90 & -18:51:15.8 & 13.07(0.01) & 12.77(0.01)\
Southern Nucleus & 12:01:53.50 & -18:53:10.0 & 13.45(0.05) & 12.50(0.03)\
Cluster 1 & 12:01:51.66 & -18:51:34.7 & 14.84(0.01) & 14.24(0.01)\
Cluster 2 & 12:01:50.40 & -18:52:12.2 & 14.98(0.02) & 14.06(0.01)\
Cluster 3 & 12:01:54.56 & -18:53:04.0 & 15.05(0.03) & 14.27(0.02)\
Cluster 4 & 12:01:54.95 & -18:53:05.8 & 16.52(0.01) & 14.66(0.01)\
[lccc]{} Source ID & RA ([*Chandra*]{}) & Dec ([*Chandra*]{}) & $K_s$\
6 & 12:01:50.51 & $-$18:52:04.80 & 15.66(0.02)\
10 & 12:01:51.27 & $-$18:51:46.60 & 14.66(0.01)\
24 & 12:01:52.99 & $-$18:52:03.20 & 13.55(0.11)\
29 & 12:01:53.49 & $-$18:53:11.10 & 12.50(0.03)\
34 & 12:01:54.55 & $-$18:53:03.20 & 14.27(0.02)\
36 & 12:01:54.81 & $-$18:52:14.00 & 15.92(0.01)\
37 & 12:01:54.98 & $-$18:53:15.10 & 16.16(0.01)\
[lcccccc]{} & & & $\Delta\alpha$ & $\Delta \delta$ & &\
[*Chandra*]{} Src ID & RA & Dec & (arcsec) & (arcsec) & $J$ & $K_s$\
“Strong” Counterparts\
6 & 12:01:50.51 & -18:52:04.77 & 0.29 & 0.04 & 16.21(0.01) & 15.66(0.02)\
10 & 12:01:51.27 & -18:51:46.58 & 0.24 & 0.31 & 15.57(0.01) & 14.66(0.01)\
11 & 12:01:51.32 & -18:52:25.46 & 0.34 & 0.03 & 18.27(0.01) & 17.37(0.08)\
20 & 12:01:52.74 & -18:51:30.06 & 0.11 & 0.38 & 18.48(0.04) & 17.78(0.02)\
24 & 12:01:52.99 & -18:52:03.18 & 0.07 & 0.82 & 14.37(0.11) & 13.55(0.11)\
26 & 12:01:53.13 & -18:52:05.53 & 0.27 & 0.87 & 15.95(0.01) & 14.71(0.15)\
29 & 12:01:53.49 & -18:53:11.08 & 0.20 & 0.25 & 13.45(0.05) & 12.50(0.03)\
33 & 12:01:54.50 & -18:53:06.82 & 0.11 & 0.99 & 16.71(0.08) & 16.45(0.07)\
34 & 12:01:54.55 & -18:53:03.23 & 0.02 & 0.39 & 15.05(0.03) & 14.27(0.02)\
36 & 12:01:54.81 & -18:52:13.99 & 0.11 & 0.50 & 16.60(0.03) & 15.92(0.01)\
37 & 12:01:54.98 & -18:53:15.07 & 0.13 & 0.10 & 17.55(0.02) & 16.16(0.01)\
39 & 12:01:55.18 & -18:52:47.50 & 0.18 & 0.03 & 17.10(0.07) & 15.71(0.04)\
42 & 12:01:55.65 & -18:52:15.06 & 0.73 & 0.40 & 17.13(0.03) & 16.27(0.06)\
“Possible” Counterparts\
15 & 12:01:51.98 & -18:52:26.47 & 1.09 & 0.84 & 16.63(0.04) & 15.95(0.02)\
22 & 12:01:52.89 & -18:52:10.03 & 0.70 & 1.20 & 15.77(0.01) & 15.13(0.06)\
25 & 12:01:53.00 & -18:52:09.59 & 0.87 & 0.76 & 15.77(0.01) & 15.13(0.06)\
32 & 12:01:54.35 & -18:52:10.31 & 0.92 & 1.39 & 20.30(0.45) & 16.84(0.03)\
35 & 12:01:54.77 & -18:52:52.43 & 0.42 & 0.92 & 16.76(0.02) & 14.88(0.04)\
40 & 12:01:55.38 & -18:52:50.53 & 0.61 & 1.24 & 16.21(0.02) & 15.27(0.04)\
[lcccccc]{} Category & $\overline{K}$ & $\sigma_{\overline{K}}$ & $\overline{(J-K)}$ & $\sigma_{\overline{(J-K)}}$ & $\overline{M_{K_s}}$ & $\sigma_{\overline{M_K}}$\
all clusters & 16.72 & 0.08 & 0.82 & 0.03 & -15.33 & 0.09\
X-ray sources & 15.72 & 0.27 & 0.95 & 0.11 & -16.30 & 0.35\
LLX & 15.85 & 0.36 & 0.84 & 0.11 & -16.16& 0.39\
HLX & 15.09 & 0.37 & 1.13 & 0.28 & -17.14 & 0.54\
ULX & 16.82 & 0.55 & 0.88 & 0.02 & -14.67 & 0.51\
[lcc]{} & Probability & D\
$M_{K_s}$ & $3.2\times10^{-2}$ & 0.37\
Cluster Mass & $9.6\times10^{-2}$ & 0.40\
$M_{K_s(lim)}$ (CM) & $8.8\times10^{-6}$ & 0.82\
$M_{K_s(lim)}$ (NH) & $3.9\times10^{-2}$ & 0.48\
[^1]: Found using confidence levels for small number statistics listed in Tables 1 and 2 of @geh86.
|
---
abstract: |
Young super star clusters and young compact massive star forming regions can provide useful information on their burst properties (age, burst duration, SFR), the upper end of the IMF and yield new constraints on the evolution of massive stars. Through the study of their stellar populations we can in particular extend our knowledge on massive stars to extreme metallicities unavailable for such stars in the Local Group.
Here we summarise the main results from recent studies on two metallicity extremes: Wolf-Rayet and O star populations in very metal-poor BCD, and metal-rich compact nuclear SF regions.
author:
- Daniel Schaerer
- 'Natalia G. Guseva and Yuri I. Izotov'
title: Super star clusters as probes of massive star evolution and the IMF at extreme metallicities
---
Introduction
============
Two “modes” of star formation are observed in (optically or UV selected) starburst galaxies (e.g. Meurer 1995): a young unresolved population responsible for emission of diffuse UV light (Meurer 1995, also Calzetti these proceedings), and compact stellar clusters, losely termed super star clusters (SSCs) hereafter. SSCs have been the focus of numerous recent studies related in particular to the possibility that these clusters may represent the progenitors of globular clusters (cf. Fritze von Alvensleben, Miller, these proceedings). A different aspect is emphasized in the present work. We use spectroscopic observations of young star forming (SF) regions to determine their massive star content with the aim of providing constraints on stellar evolution models and the upper end of the IMF.
SSCs and similar compact young SF regions have the following properties: [*a)*]{} Numerous such objects are known. [*b)*]{} They represent clusters rich enough ($\sim$ 10$^{2-4}$ O stars) such that the IMF can be well populated and stochastical effects (cf. Lançon these proceedings) are negligible. [*c)*]{} A priori the clusters cover a wide range of metallicities, and [*d)*]{} consist likely of a fairly coeval population. Given these properties, SSCs resemble “normal” Galactic of Local Group clusters which represent fundamental test-cases for stellar evolution. The only disadvantage is that their stellar content can only be studied through observations of their integrated light. On the other hand b) and c) represent important advantages for studies focussed on massive stars over using “local” clusters. [*This shows that young SSCs provide ideal samples for studies of massive star evolution in different environments, such as e.g. extreme metallicities largely inaccessible in Local Group objects.*]{}
After a brief introduction on the type of objects used here (Wolf-Rayet rich SF region) we will summarise recent work along these lines.
Wolf-Rayet galaxies and clusters
================================
We will concentrate on the so-called Wolf-Rayet (WR) galaxies (cf.Schaerer 1999b for the latest catalogue), which are objects where broad stellar emission lines (called “WR bumps”, mostly at and ) in the integrated spectrum testify to the presence of WR stars. For the study of massive star populations these objects are ideal since WR stars are the descendents of the most massive stars in a short-lived phase ($M_{\rm ini} \ga 25$ , $t_{\rm WR} \sim 10^{5-6}$ yr). Their detection is also a good age indicator for young systems ($t \la 10$ Myr), and allows good measure of the burst duration and the best direct probe of the upper end of the IMF. An overview of studies on WR populations in starburst regions can be found in the reviews of Schaerer (1998, 1999).
In the context of the present workshop it is important to note that the objects broadly referred to as WR “galaxies” are found among a large variety of objects including BCD, massive spirals, IRAS galaxies, Seyfert 2, and LINERs (see Schaerer 1999b). The “WR rich” regions contained in the spectroscopic observations will thus in general cover quite a large scale of sizes, different morphologies etc. In the case of blue compact dwarfs (BCD), one is however mostly dealing with one or few individual compact regions or SSC dominating the observed light. Although this statement cannot, with few exceptions, be quantified so far for the objects studied below (but see e.g. Conti & Vacca 1994) we will mostly assume that the spectroscopic observations correspond closely to light from one young compact SF region or SSC.
Studies of Wolf-Rayet populations in metal-poor environments
============================================================
The spectroscopic sample of dwarf galaxies from Izotov, Thuan and collaborators, obtained for the main purpose of determining the primordial He abundance and other abundance studies, has proven to be very useful for analysis of massive star populations especially at very low metallicities. Indeed, $\sim$ 20 WR rich regions are found in this sample at metallicities below the SMC ($12 + \log$ O/H $\la$ 8.1) extending to I Zw 18 with $\sim$ 1/50 solar metallicity. No [*bona fide*]{} massive star of such low a metallicity is known in the Local Group!
The analysis of the WR and O star content in these objects has been presented by Guseva et al. (1999, hereafter GIT99). Some of their main results are summarised in Fig. 1, which shows (left panel) the derived WR/(WR+O) number ratio as a function of metallicity from their objects and observations of Kunth & Joubert (1985), Vacca & Conti (1992), and Schaerer et al. (1999a, hereafter SCK99). The left Fig. considerably extends the previous samples (cf. Vacca & Conti 1992, Meynet 1995). The trend of increasing WR/O with metallicity is well understood (Arnault et al. 1989) The comparison with appropriate evolutionary synthesis models (Schaerer & Vacca 1998, SV98; shown as solid lines) calculated for a “standard” Salpeter IMF with $M_{\rm up}=120$ and using the high mass loss Geneva tracks shows a good agreement. This and more direct comparisons of the observed WR features (see Schaerer 1996, de Mello et al. 1998, SCK99, GIT99) indicate that the bulk of the observations are compatible with short (“instantaneous”) bursts with a Salpeter IMF extending to large masses. The short burst durations[^1] derived by SCK99 for the metal-poor objects are also in agreement with the study of Mas-Hesse & Kunth (1999).
Of particular interest for evolutionary models is the relative number of WR stars of the WN (exhibiting H-burning products on their surface) and WC subtypes (He-burning products). The relative lifetimes vary strongly with initial mass and metallicity and are sensitive to various mass loss prescriptions and mixing scenarios currently not well known (see Maeder & Meynet 1994, Meynet these proceedings). The recent high S/N spectra of SCK99 and GIT99 have now allowed to establish number ratios of WC/WN stars in a fair number of WR rich regions. The determinations of GIT99 are shown in Fig. 1 (right panel); similar values are derived by SCK99. The comparison with synthesis models shows a reasonable agreement. To reproduce sufficiently large WC/WN ratios the use of the stellar tracks based on the high mass loss prescription are, however, required as shown by de Mello et al. (1998) and SCK99.
It is understood that part of the “requirement” for the high mass loss (cf. Schaerer 1998) may be compensated by additional mixing processes leading to a similar prolongation of the WR phase (cf. Meynet, these proceedings). In any case in addition to the well known stellar census in the Local Group (cf. Maeder & Meynet 1994 and references therein) the present new data from integrated populations place important constraints on the evolutionary models which have to be matched by successful stellar models. Especially the new studies extend the range of available metallicities to very low $Z$, well beyond the SMC.
Massive stars and the IMF in metal-rich starbursts
==================================================
A small sample of metal-rich (O/H $\ga$ solar) starbursts (4 objects from GIT99 and Mrk 309) have recently been analysed in detail by Schaerer et al. (2000). In this case the observations (high S/N, intermediate resolution optical spectroscopy) correspond to compact nuclear SF regions. Despite this complication we use these objects as a first step to probe the upper end of the IMF at high-metallicity. Subsequent studies of more isolated and simple, cluster-like objects will be undertaken in the future.
For our comparison with evolutionary synthesis models (cf. below) we use following main observational constraints: H$\beta$ and H$\alpha$ equivalent widths serving as age indicator, H$\alpha$/H$\beta$ determining the gaseous extinction, intensities and equivalent widths of the main WR features (4650 bump, ), TiO bands at $\sim$ 6250 and 7200 Å indicating the presence of red supergiants from a population with ages $\ga$ 7–10 Myr, and the overall SED provide an important constraint on the population responsible for the continuum.
Model calculations have been done using the SV98 synthesis code. The basic model parameters we consider are: stellar metallicity, IMF slope and upper mass cut-off, star formation history (instantaneous or extended bursts, constant SF), fraction of Lyc photons absorbed by the gas, stellar extinction (which may differ from gaseous).
Results
-------
The comparison of the observed and predicted WR features is shown in Fig. 2. The observations are well reproduced by Z=0.02 models with a “standard IMF” (Salpeter, M$_{\rm up}=$120 ) assuming extended burst durations of $\Delta t \sim$ 4–10 Myr. The longer burst durations found here are physically plausible and likely explained by the different nature of the metal-rich objects analysed here (larger nuclear regions vs. compact clusters in BCD, cf. SCK99). The corresponding ages of our objects, as indicated by W(H$\beta$), are between $\sim$ 7 and 15 Myr, also in agreement with the presence of the TiO bands. A very good fit is also obtained to the overall SED. This requires, however, an extinction of the stellar continuum which is less than that derived from the gas. The differences are of similar amount that found by other methods in other studies (e.g. Calzetti 1997, Mas-Hesse & Kunth 1999).
In short, we conclude that all the given observational constraints can be well reproduced by models with a Salpeter IMF extending to high masses for a burst scenario with star formation extending over $\sim$ 4–10 Myr. This solution is not unique. Therefore a variety of alternate models have been considered (cf. Schaerer et al. 2000). Regarding the IMF we find that steeper (with slope $\sim$ Miller-Scalo) IMFs are very unlikely.
In view of several studies indicating a possible lack of massive stars in metal-rich environments (e.g. Bresolin et al. 1999, Goldader et al. 1997) we have used the present set of metal-rich WR galaxies to put a [*lower limit*]{} on the value of M$_{\rm up}$ from the strength of the WR features. As mentioned above our data is compatible with a large upper mass cut-off. The real range of values our data is sensitive to is, however, limited; intrinsically younger regions need to be sampled to probe the most massive stars. Adopting a conservative approach (cf. Fig. 3) we obtain M$_{\rm up} \ga$ 30 (for a Salpeter slope). We also find similar values (M$_{\rm up} \sim$ 35–50 ) from comparisons of H$\beta$ equivalent width measurements in metal-rich regions (see Schaerer et al. 2000).
In contrast with previous studies of metal-rich starbursts and related objects based on properties of the ionized gas our work provides first constraints on the upper end of the IMF measured directly from stellar signatures. Future work on larger samples and using detailed coupled stellar population and photoionisation models should be of great interest.
We thank the organisers, especially Ariane Lançon, for this very stimulating and pleasant workshop. We acknowledge support to this collaboration trough INTAS grant 97-0033.
Arnault, P., Kunth, D., Schild, H. 1989, , 224, 73 Calzetti, D., 1997, in “The Ultraviolet Universe at Low and High Redshift: Probing the Progress of Galaxy Evolution”, Eds. W.H. Waller et al., AIP Conference Proceedings, v.408., 403 Conti, P.S., Vacca, W.D., 1994, , 423, L97 de Mello, D.F., Schaerer, D., Heldman, J., Leitherer, C., 1998, , 507, 199 Goldader, J., et al., 1997, , 474, 104 Guseva, N.G., Izotov, Y.I., Thuan, T.X., 1999, , in press (GIT99) Kunth, D., & Joubert, M. 1985, , 142, 411 Maeder, A., Meynet, G., 1994, , 287, 803 Mas-Hesse, J.M., Kunth, D., 1999, , in press Meurer, G., et al., 1995, , 110, 2665 Meynet, G. 1995, , 298, 767 Schaerer, D., 1996, , 467, L17 Schaerer, D., 1998, in “Wolf-Rayet Phenomena in Massive Stars and Starburst Galaxies”, IAU Symp. 193, 539, Schaerer, D., Contini, T., Kunth D., 1999a, , 341, 399 (SCK99) Schaerer, D., Contini, T., Pindao M., 1999b, , 136, 35 Schaerer, D., Guseva, N., Green, R., Izotov, Y.I., Thuan, T.X., 2000, , in preparation Schaerer, D., Vacca, W. D. 1998, , 497, 618 (SV98) Vacca, W.D., Conti, P.S., 1992, ,401, 543
[**Q. (Manfred Pakull):**]{} Some years ago it was thought that WRs do not form in very low Z environments because the massive stellar winds were too weak. Is that still the state of the art in stellar evolution models ? Or are the broad emission features due to a population of H-rich, very young objects like in 30 Dor and NGC 3603 ?
[**A. (Daniel Schaerer:**]{} Indeed the metallicity dependence of mass loss causes an important decrease of the WR population toward low Z. Despite this evolutionary models applying recent mass-loss prescriptions predict [*some*]{} WR stars at the lowest metallicities (1/50 solar) corresponding to I Zw 18 (see de Mello et al. 1998), quite in agreement with the observations. The role of other formation channels (massive close binaries, rotation) at these low Z remains unexplored
In Schaerer et al. (1999, SCK99) we have explored the effect of such R136-like WR stars during H-burning in addition to other WR and Of stars with broad emission lines. Given their high initial mass their integrated contribution to the WR bump should in most cases not be very important compared to “normal” WR stars. An exception could be very young ($\la 2-3$ Myr) and strongly coveal populations. Few such observational cases seem, however, known to date (see SCK99).
[^1]: See Meurer (these proceedings) for a discussion of SF durations.
|
---
abstract: |
We present measurements of the column densities of interstellar [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, and [H$_2$]{} made with the [*Far Ultraviolet Spectroscopic Explorer*]{} ([[*FUSE*]{}]{}), and of [H$\;$[I]{}]{} made with the [*International Ultraviolet Explorer*]{} ([[*IUE*]{}]{}) toward the sdO star LSE 44 \[[($l$,$b$) = (31337, +1349)]{}; d = $554\pm66$ pc; z = $+129\pm15$ pc\]. This target is among the seven most distant Galactic sight lines for which these abundance ratios have been measured. The [H$\;$[I]{}]{} column density was estimated by fitting the damping wings of interstellar [Ly$\alpha$]{}. The column densities of the remaining species were determined with profile fitting analyses, and supplemented with curve of growth analyses for [O$\;$[I]{}]{} and [H$_2$]{}. We find log $N$([D$\;$[I]{}]{}) = $15.87\pm0.08$, log $N$([O$\;$[I]{}]{}) = $17.57{\ensuremath{^{+ 0.21}_{- 0.15}}}$, log $N$([N$\;$[I]{}]{}) = $16.43\pm0.14$, and log $N$([H$\;$[I]{}]{}) = $20.52{\ensuremath{^{+ 0.20}_{- 0.36}}}$ (all errors $2\,\sigma$). This implies D/H = $(2.24{\ensuremath{^{+ 1.39}_{- 1.32}}}) \times
10^{-5}$, D/O = $(1.99{\ensuremath{^{+ 1.30}_{- 0.67}}}) \times 10^{-2}$, D/N = $(2.75{\ensuremath{^{+ 1.19}_{- 0.89}}}) \times 10^{-1}$, and O/H = $(1.13{\ensuremath{^{+ 0.96}_{- 0.71}}})
\times 10^{-3}$. Of the most distant Galactic sight lines for which the deuterium abundance has been measured LSE 44 is one of the few with D/H higher than the Local Bubble value, but D/O toward all these targets is below the Local Bubble value and more uniform than the D/H distribution.
author:
- 'S.D. Friedman, G. Hébrard, T.M. Tripp, P. Chayer, K. R. Sembach'
title: 'The Deuterium, Oxygen, and Nitrogen Abundance Toward LSE 44'
---
Introduction
============
Precise measurements of primordial abundances of the light elements deuterium (D), $^3$He, $^4$He, and $^7$Li relative to hydrogen have been a goal of astronomers for many years. In the standard Big Bang nucleosynthesis model, these quantities are related in a straightforward way to the baryon-to-photon ratio in the early universe, from which $\Omega_{\rm B}$, the fraction of the critical density contributed by baryons, may be determined (Boesgard & Steigman 1985). Since deuterium is easily destroyed in stellar interiors (astration), and no significant production mechanisms have been identified (Reeves et al. 1973; Epstein, Lattimer, & Schramm 1976), the D/H ratio is expected to monotonically decrease with time.
Precise measurements of D/H in the local interstellar medium (ISM) were first made using [[*Copernicus*]{}]{} (Rogerson & York 1973), [[*HST*]{}]{} (Linsky et al. 1995), and [[*IMAPS*]{}]{} (Jenkins et al. 1999; Sonneborn et al. 2000). In the last several years many additional measurements of D/H, D/O, D/N, and related species, have been made using [[*FUSE*]{}]{} (Moos et al. 2002; Hébrard & Moos 2003; Wood et al. 2004). These local ISM measurements represent a lower limit to the primordial value. When corrected for the effects of astration (Tosi et al. 1998), the results should be comparable to those obtained in low-metallicity, high redshift [Ly$\alpha$]{} clouds (Crighton et al. 2004; Kirkman et al. 2003), which should be very nearly the primordial value itself.
As data on more sight lines has accumulated the situation is not as simple as originally envisioned. At high redshift, measurements of D/H vary from $(1.6-
4) \times 10^{-5}$ (Crighton et al. 2004; Pettini & Bowen 2001), despite the fact that the metallicity in all these environments is low enough to suggest that D/H should match the primordial value. More locally, it is now accepted that for sight lines within the Local Bubble, a region of hot, low-density gas (Sfeir et al. 1999) within a distance of $100-150$ pc and log [$N({\rm H \; \mbox{\small\rm I}})$]{}$\la 19.2,$ in which the Sun lies, D/H is nearly constant, but the value is uncertain. Based on measurements toward 16 targets, Wood et al. (2004) state that (D/H)$_{LB}=1.56\pm{0.04} \times 10^{-5}$. Alternatively, (D/H)$_{LB}$ may be inferred from measurements of D/O and O/H. Hébrard & Moos (2003) discuss the advantages of this approach, including potential reduction of systematic errors associated with measuring [$N({\rm D \; \mbox{\small\rm I}})$]{}and [$N({\rm H \; \mbox{\small\rm I}})$]{} directly, which differ by five orders of magnitude. Their approach relies on the observed high spatial uniformity of O/H (Meyer, Jura, & Cardelli 1998; Meyer 2001; André et al. 2003). The result is (D/H)$_{LB}=1.32\pm{0.08} \times 10^{-5}$, about $2\sigma$ different than the Wood et al. value.
At distances greater than $\sim500$ pc, or more precisely, at log [$N({\rm H \; \mbox{\small\rm I}})$]{}$\ga
20.5,$ the few sight lines measured have lower D/H. Based on a suggestion originally made by Jura (1982) and extended by Draine (2004a, 2004b) that under the right thermodynamic conditions deuterium can be depleted onto dust grains, Wood et al. (2004) have interpreted D/H variations as being due to various levels of depletion and, over the longer sight lines, the effects of averaging the D/H concentration over multiple clouds along the sight line. In this scenario, the low value at large distances represents the gas phase abundance in the local disk, (D/H)$_{LDg} = (0.85\pm0.09) \times 10^{-5},$ and one must include the depleted deuterium to determine the total deuterium abundance. An alternate explanation is offered by Hébrard & Moos (2003). Based on D/O, D/N, and D/H measurements, they suggest that the low gas-phase value observed at high column densities and distances is the true Galactic disk value, and there is not a significant reservoir of deuterium trapped in dust grains. D/H has an unusually high value in the Local Bubble if this interpretation is correct. The reason for this high value is not understood. One possibility is the infall of primordial, deuterium-rich material, and another is non-primordial deuterium production, such as in stellar flares. This is discussed further in §5.
To test this hypothesis it is necessary to make additional measurements of [D$\;$[I]{}]{}, [H$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, [H$_2$]{}, and additional species that might yield information about depletion and physical conditions, toward targets with log [$N({\rm H \; \mbox{\small\rm I}})$]{}$\ga 20.5$ In this paper we present the results of an analysis of the sight line toward LSE 44, an sdO star located at a distance of $554\pm66$ pc and well within this column density regime. We derive [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, and [H$_2$]{} column densities with data obtained from the [*Far Ultraviolet Spectroscopic Explorer*]{} ([[*FUSE*]{}]{}) and estimate the [H$\;$[I]{}]{} column density with data from the [*International Ultraviolet Explorer*]{} ([[*IUE*]{}]{}).
In §2 we present a summary of the observations and a description of the data reduction processes. In §3 the stellar properties of LSE 44 are discussed. The details of the analysis of the [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, and [H$_2$]{} column densities are given in §4, and the [H$\;$[I]{}]{} analysis is presented in §5. In §6 we discuss the results of this study.
Observations and Data Processing
================================
The [[*FUSE*]{}]{} instrument consists of four co-aligned spectrograph channels designated LiF1, LiF2, SiC1, and SiC2, named for the coatings on their optics, which were selected to optimize throughput. Each channel illuminates a pair of microchannel plate segments, labeled A and B. The LiF channels have a bandpass of approximately 1000-1187 Å and the SiC channels approximately 910-1090 Å. Thus, each observed spectral line may appear in the spectra from multiple channels. The [[*FUSE*]{}]{} resolution is approximately 20 [km s$^{-1}$]{} (FWHM), but it varies slightly between channels, and within a channel, as a function of wavelength. A detailed description of the [[*FUSE*]{}]{} instrumentation and performance is given by Sahnow et al. (2000).
LSE 44 was observed with [[*FUSE*]{}]{} for a total of 86 ksec under programs P2051602 - P2051609. A total of 119 exposures are in the MAST archive, obtained with various levels of alignment success for the separate channels, and some were obtained when one or both detectors were not at full voltage. The observation log is shown in Table 1, and the total exposure time by channel is shown in Table 2. All data were obtained in time-tag mode with focal-plane offsets (FP-SPLITS) in the MDRS ($20 \arcsec \times 4 \arcsec$) aperture.
The data were reduced using CALFUSE pipeline version 3.0.6. We used the default pixel size of 0.013Å. The signal-to-noise ratio per pixel in the continuum region around 928Å for both SiC1B and SiC2A spectra is typically about 3.5 in individual, well-aligned exposures of $\sim900$s, the maximum duration of any exposure. The individual 1-dimensional spectra were co-added to form the final spectrum for each channel and detector segment separately after removing the relative shifts between individual spectra on the detector caused by image and grating motion (Sahnow et al. 2000). The shifts were determined by cross-correlating the individual spectra over a limited wavelength range which contained prominent spectral features but no airglow lines. Typical shifts were $\lesssim 4$ pixels, or $\lesssim 0.050$Å. The S/N ratios in the summed spectra are approximately 26 and 27, respectively, for the SiC1B and SiC2A channels. Narrow interstellar absorption features in the summed spectra do not exhibit unexpected broadening, indicating that the summing process did not smear the data. Although [[*FUSE*]{}]{} observations of [H$\;$[I]{}]{} and [O$\;$[I]{}]{} lines are sometimes contaminated by airglow emission, data obtained during spacecraft orbital night showed that this was unimportant in our analysis, so the combined day and night data were used.
Figure \[fig\_s1bspec\] shows the co-added SiC1B spectrum of LSE 44, which covers the wavelength range $915 \sim 990$Å. [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, and [H$_2$]{} lines arising in the ISM are identified.
The [H$\;$[I]{}]{} column density along this line of sight was estimated using the Ly$\alpha$ profile from the single high resolution spectrum of LSE 44 taken with [[*IUE*]{}]{}. The 16.5 ksec observation was made on 29 August 1986 (see Table 1), and reduced using IUESIPS. The analysis of this data is discussed in more detail in §4.4.
Stellar Properties of LSE 44
============================
In his survey of subluminous O stars, Drilling (1983) obtained photometric and spectroscopic data for LSE 44. His $UBV$ photometry gives $V = 12.45$, $B-V =
-0.24$, and $U-B = -1.18$ (Table 3), and indicates that the star is relatively hot. He describes the optical spectrum of LSE 44 by saying that the only lines present are the $\lambda$4686 line and the Balmer series. He adds that the appearance of these lines is very similar to those in the spectrum of the sdOB star Feige 110, but the Balmer lines are stronger and the $\lambda$4686 line is weaker in the spectrum of LSE 44 (see, e.g., Heber 1984; Friedman et al. 2002). In order to measure the atmospheric parameters of LSE 44, D. Kilkenny (2004, private communication) obtained an optical spectrum of LSE 44 at the South African Astronomical Observatory with the 1.9-m telescope and CCD spectrograph. The optical spectrum covers the wavelength range 3400 – 5400 Å with a resolution of about $\sim 3$ Å. The normalized spectrum is plotted in Figure \[fig:optical\_spectrum\]. As reported by Drilling (1983), our optical spectrum shows broad Balmer lines starting from H$\beta$ to H$\theta$ and the $\lambda$4686 line, which is the only helium line detected in this spectrum. The K $\lambda$3935 line is also detected in the optical spectrum.
The atmospheric parameters of LSE 44 were obtained by fitting the optical spectrum illustrated in Figure \[fig:optical\_spectrum\] with a grid of synthetic spectra and by using the Marquardt method (see, Bevington & Robinson 1992). This method optimizes the effective temperature, gravity, and helium abundance in order to obtain a model that best matches the observed spectrum. We computed a grid of NLTE H/He atmosphere models with 20,000 K $\leq T_{\rm{eff}} \leq 80$,000 K in steps of 2,000 K, for different values of the surface gravity $4.8
\leq \log g \leq 6.4$ in steps of 0.2 dex, and for different helium abundances $-4.0 \leq \log N({\rm{He}})/N({\rm{H}}) \leq 0.0$ in steps of 0.5 dex. The models were computed with the stellar atmosphere codes TLUSTY and SYNSPEC (see, e.g., Hubeny & Lanz 1995). Figure \[fig:optical\_spectrum\] shows our best fit for the optical spectrum of LSE 44. We obtain $T_{\rm{eff}} = 38$,700$\pm1$,000 K, $\log g = 5.5\pm0.1$, and $\log
N({\rm{He}})/N({\rm{H}}) = -2.8\pm0.1$. We used these atmospheric parameters to compute the stellar Ly$\alpha$ line profile in order to measure the column density toward LSE 44. (see §4.4).
The evolutionary status and physical properties of LSE 44 can be estimated by comparing its position in a $T_{\rm{eff}}$-$g$ diagram with evolutionary sequences. Figure \[fig:teff\_g\_plane\] shows the position of LSE 44 in such a diagram with the positions of a sample of hot subdwarf B stars (sdB). It also shows post-extreme horizontal branch (post-EHB) evolutionary sequences that were computed by B. Dorman (1999, private communication). The position of LSE 44 in the $T_{\rm{eff}}$-$g$ diagram suggests that it is an sdB star that has evolved from the EHB. Even though LSE 44 is classified as an sdO star based on its optical spectrum, it relates to the sdB stars. sdB stars have effective temperatures and gravities in the ranges 24,000 K $\lesssim T_{\rm{eff}}
\lesssim 40$,000 K and $5.0 \lesssim \log g \lesssim 6.2$, and helium abundances that are typically a factor of ten smaller than solar (see, e.g., Saffer et al. 1994). The evolutionary sequences in Figure \[fig:teff\_g\_plane\] represent the evolution of typical sdB stars that have helium-burning cores of $\sim
0.477$ M$_\odot$ and small hydrogen envelopes $M_{\rm{env}} \lesssim
0.0045$ M$_\odot$. Each evolutionary track corresponds to models with about the same helium core mass but with different hydrogen envelope masses. In these models the hydrogen envelopes are too small to ignite the hydrogen burning shell, and therefore the stars appear subluminous, hot, and compact. In Figure \[fig:teff\_g\_plane\] the hatched region corresponds to the zero-age EHB (ZAEHB). In this way, an EHB star evolves from the ZAEHB and moves first to the right and then to the left toward the white dwarf region without reaching the asymptotic giant branch (AGB). A star spends $\sim 10^8$ yr on the EHB and $\sim 10^7$ yr off the EHB en route to the white dwarf region.
The spectral type sdO is a spectroscopic class that includes stars showing strong Balmer lines and lines. Because of its high effective temperature and low helium abundance, LSE 44 appears as an sdO star. The evolutionary tracks surrounding the position of LSE 44 show that stars in this portion of the $T_{\rm{eff}}$-$g$ diagram have left the helium core burning phase and undergo a helium shell burning phase. One can assume that the mass of LSE 44 is $\sim 0.4775$ M$_\odot$ and estimate its distance by using the expression
$$D = \sqrt{\frac{GM}{g}\frac{4\pi\,H_{V}}{F_{0,V}}}\; 10^{0.2(V-A_V)},$$
where $G$ is the gravitational constant, $M$ is the mass of the star, $g$ is the gravitational acceleration, $H_\nu$ is the Eddington flux weighted by the Johnson passband $V$ of Bessel (1990), $F_{0,V}$ is the average absolute flux of Vega at $V$ as specified in Heber et al. (1984), $V$ is the apparent visual magnitude, and $A_V$ is the interstellar absorption. By using the best model fit parameters and the passbands of Bessel (1990) we computed the intrinsic color index $(B-V)_0$ and obtained the color excess $E(B-V)
= 0.05\pm0.03$, which gives $A_V = 0.155\pm0.093$ when using $R_V =
3.1$. The distance to LSE 44 is then $554\pm66$ pc.
The [*FUSE*]{} spectrum of LSE 44 shows many photospheric and interstellar absorption lines. The strongest photospheric lines are the Lyman series that start at Ly$\beta$ and go up to lines close to the series limit. Because of the high gravity the photospheric Lyman lines merge together before reaching the series limit, and the remaining Lyman lines close to this limit are mainly due to interstellar . The other strong photospheric lines are $\lambda$1084, $\lambda\lambda$1175, $\lambda\lambda$1184, $\lambda\lambda$923 and $\lambda$955, $\lambda\lambda$1126 and $\lambda$1066, $\lambda\lambda$1123, $\lambda\lambda$1070, and $\lambda\lambda$937. There are also many fainter unidentified stellar lines that depress the continuum. For instance, Figure \[fig:lse44\_feige110\] shows a comparison between the [*FUSE*]{} spectra of LSE 44 and the sdOB star Feige 110 ($T_{\rm{eff}} = 42$,300 K and $\log g = 5.95$; Friedman et al. 2002). The figure illustrates that both stars have many lines in common, but because of the lack of atomic data, we cannot identify these absorption lines. The lines at 1122.49 Åand 1128.33 Å and the line at 1128.008 Å are practically the only stellar lines that can be identified in these portions of the [*FUSE*]{} spectrum. This example illustrates clearly that blends of unidentified stellar lines with interstellar lines could lead to systematic errors when measuring interstellar column densities.
Table 3 summarizes the important properties of LSE 44 and the sight line to this star.
Interstellar Column Density Analyses
====================================
Two techniques have been used to estimate the column densities in this analysis. The first involves directly measuring the equivalent widths of absorption lines, which are fit to a single-component Gaussian curve-of-growth (COG) (Spitzer 1978). Although it appears from the line widths that the line of sight may traverse more than a single velocity component, no high resolution optical or UV spectrum exists which would allow us to model a more complicated velocity structure. We therefore made the simplifying assumption that a single interstellar cloud with a Maxwellian velocity distribution is responsible for the absorption. The estimates of the column density ($N$) and Doppler parameter ($b$) derived from the COG are likely to include systematic errors due to this assumption. For example, the derived $b-$value will not be a simple quadrature combination of thermal and turbulent velocity components, as it would be for a true single cloud. Instead, it will also reflect the spread in velocities of the multiple clouds that are likely to lie along a sight line of this length. Jenkins (1986) has discussed the systematic errors associated with the COG technique when there are multiple clouds of various strengths along a sight line, and has shown that if many non-saturated lines are used, and if the distribution of $N$ and $b-$values in the clouds is not strongly bimodal, then the column density is likely to be underestimated by $\la$ 15%.
Despite these shortcomings, the COG technique has the virtue of being unaffected in principle by convolution with the instrumental line spread function (LSF). In practice, it must be used with caution because of the problem of blending with neighboring lines and, especially for a target like LSE 44, because it is difficult to establish the proper continuum in some spectral regions, presumably due to the presence of many unidentified stellar absorption features.
The second technique is profile fitting (PF). For all species except [H$\;$[I]{}]{} we used the code [Owens.f]{}, which has been applied to [[*FUSE*]{}]{} spectra for many previous investigations. This code models the observed absorption lines with Voigt profiles using a $\chi^2$ minimization procedure with many free parameters, including the line spread function, flux zero point, gas temperature, and turbulent velocity within the cloud. We split each spectrum into a series of small sub-spectra centered on absorption lines, and fit them all simultaneously. Each fit typically includes about 60 spectral windows, and approximately 100 transitions of [H$\;$[I]{}]{}, [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, and H$_2$ ($J=0$ to $5$). The wavelengths and oscillator strengths of the lines used in the profile fitting analysis are given in Table 4. Note that there are fewer than 100 entries in the table because some lines appear in multiple detector segments. [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, and [N$\;$[I]{}]{} were assumed to be in one component, [H$_2$]{} in a second, and [H$\;$[I]{}]{} in a third. Velocities between windows were allowed to shift to accommodate inaccuracies in the [[*FUSE*]{}]{} wavelength calibration. The line spread function (LSF) is fixed within a spectral window, but is allowed to change in a given detector segment from one window to another, and between segments. This is consistent with the performance of the [[*FUSE*]{}]{} spectrographs (Sahnow et al. 2000). No systematic relationship between the LSF and the derived column density or $b-$value of any species was observed.
Only one interstellar component for a given species was assumed along the sight line although this is unlikely to be true for such a distant target. Hébrard et al. (2002) discuss tests they performed on an extensive list of potential systematic errors using profile fitting, including the single component assumption. In these tests they found that fits obtained with up to five interstellar components gave the same total column density within the 1 $\sigma$ error bars, as long as saturated lines are excluded from the analysis, which we have been careful to do. Thus, we report total integrated column densities along the sight line.
It is possible that the temperature and turbulent velocities differ from one cloud to another. To test this we did profile fits with [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, and [N$\;$[I]{}]{} in three independent components. The column density estimates differ by less than 1 $\sigma$ from those obtained by assuming a single component. The $b-$values also agree with that obtained by assuming a single component, although they are not well-constrained because only unsaturated lines are used. These species could also have different ionization states if they reside in clouds with different temperatures. However, the ionization potentials of [D$\;$[I]{}]{} and [O$\;$[I]{}]{}and, to a slightly lesser extent, [N$\;$[I]{}]{} are so similar that this is unlikely to be a problem.
The [Fe$\;$[II]{}]{} absorption lines detected in the spectrum are abnormally broad for interstellar lines, about 10 [km s$^{-1}$]{}, with a temperature of $\sim2\times10^5$ K, compared to $\sim 6$ [km s$^{-1}$]{} for [N$\;$[I]{}]{} and [O$\;$[I]{}]{} (see §4.2). The cause of this broadening is not known, but [Fe$\;$[II]{}]{} is not present in the stellar atmosphere because the temperature is too high. In the fits we did not include [Fe$\;$[II]{}]{} in the component containing [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, and [N$\;$[I]{}]{}, but this has no effect on the determination of the interstellar column densities of these species.
All laboratory wavelengths and oscillator strengths used in this work are from Morton (2003) and Abgrall et al. (1993a, 1993b). Neither the COG nor profile fitting techniques include oscillator strength uncertainties in the error estimates. The application of [Owens.f]{} to [[*FUSE*]{}]{} data is described in greater detail by Hébrard et al. (2002).
The [D$\;$[I]{}]{} Analysis
---------------------------
We determined [$N({\rm D \; \mbox{\small\rm I}})$]{} using only the profile fitting technique. We attempted to construct a COG, but we were unable to determine unambiguously the proper placement of the continuum in the presence of the strong [H$\;$[I]{}]{} lines separated from the [D$\;$[I]{}]{} lines by +81 [km s$^{-1}$]{}. The situation is particularly difficult for this sight line due to the unusual shape of the neighboring continuum around the $\lambda 916.180$ line, and the unaccounted for absorption on the red wing of the [H$\;$[I]{}]{} $\lambda 920.712$ line. Our tests indicated that failure to recognize such problems caused COG estimates of [$N({\rm D \; \mbox{\small\rm I}})$]{} to be approximately 0.3 dex too low. Once the continuum is defined in the COG analysis, it is fixed. By contrast, although continuum placement errors will also lead to column density errors with profile fitting, this technique allows the shape of the continuum to change as part of the fitting process. The continuum was modeled with polynomials of order $0-4$ to allow for a large range of continuum fits. We required acceptable fits simultaneously to all five [D$\;$[I]{}]{} lines used in the analysis (see below). In addition, removing any of the lines from the analysis did not change the final column density estimate significantly. This was not true for the COG fit, which was highly sensitive to the continuum placement at the $\lambda
916.180$ line. Removal of this line changed the COG result by more than 0.2 dex. Therefore, by virtue of its simultaneous and self-consistent fit over multiple lines, and its sensitivity to the exact way in which [D$\;$[I]{}]{} blends with the blue wing of the neighboring [H$\;$[I]{}]{} line, profile fitting is the best method for measuring the deuterium column density.
Five [D$\;$[I]{}]{} absorption profiles over three separate spectral lines were used in the profile fits: $\lambda916.180$ in the SiC1B channel, and $\lambda919.100$ and $\lambda920.712$ in both the SiC1B and SiC2A channels. Longer wavelength [D$\;$[I]{}]{} lines are too saturated to provide additional constraints on the column density estimate, and the convergence of the [H$\;$[I]{}]{} Lyman series precludes using shorter wavelength lines. The [D$\;$[I]{}]{} $\lambda916.931$ and $\lambda917.880$ lines are too strongly blended with [O$\;$[I]{}]{} and [H$_2$]{} lines to be used. Although the same transition observed in separate [[*FUSE*]{}]{} channels potentially are subject to systematic errors arising from similar effects, such as continuum placement and blending, they do represent independent measurements with distinct spectrographs. Thus, we include all five profiles in our analysis.
To properly measure the interstellar [D$\;$[I]{}]{}, the effects of the stellar [H$\;$[I]{}]{} Lyman series and absorption must be removed. This was done in two ways. First, the stellar model described in §3 was shifted in velocity space and scaled in flux until the stellar [H$\;$[I]{}]{} damping wings matched the observed spectrum away from the cores of the interstellar [H$\;$[I]{}]{} lines. The resulting average velocity difference was $\Delta V \equiv V_* -V_{\rm ISM} = -39\pm5$ [km s$^{-1}$]{} and $-32\pm13$ [km s$^{-1}$]{} for the SiC1B and SiC2A spectra, respectively. This compares favorably to $\Delta V = -33\pm6$ [km s$^{-1}$]{} based on the measured velocities in the high-dispersion [[*IUE*]{}]{} spectrum of photospheric lines (, , , , , , and ) and interstellar lines (, , and ). The spectral resolution of [[*IUE*]{}]{} in this mode is approximately 25 [km s$^{-1}$]{} (FWHM).
The stellar model is only an approximation of the true absorption from the atmosphere of LSE 44. Since not all species present in the atmosphere are properly accounted for in the model, and since some of the atomic constants included in the model are not well-known, using the model may in fact introduce systematic errors. To check for this we also applied the profile fitting without the model, using instead up to a fourth-order polynomial to fit the stellar spectrum and continuum. This method has been used many times before in similar analyses; see, for example, Hébrard & Moos (2003). As described below, the two methods give column densities which agree within the errors.
Several parameters are free to vary through the fits, including the column densities, the radial velocities of the interstellar clouds, and the shapes of the stellar continua. Some instrumental parameters are also free to vary, including the widths of the Gaussian line spread function (LSF), which is convolved with the Voigt profiles within each spectral window. The averaged width of the LSF that we found is 5.8 pixels, with a 0.8-pixel $1\,\sigma$ dispersion (full widths at half maximum). These [*FUSE*]{} pixels are associated with CALFUSE version 3, and are 0.013Å in size. Note that CALFUSE versions 1 and 2, which was used in most previously published [[*FUSE*]{}]{} studies, produced pixels approximately half this size. This explains why the LSF reported previously was approximately about 11 pixels wide (see, e.g., Friedman et al. 2002; Hébrard et al. 2002). Some examples of fits are plotted in Figure \[fig\_owens\].
Using the stellar model we find log $N$([D$\;$[I]{}]{})$_{sm} = 15.84\pm{0.08}$. Using a polynomial fit to the stellar plus interstellar continuum, we find $N$([D$\;$[I]{}]{})$_{poly} = 15.89\pm{0.08}$. These errors include our estimates of the systematic errors associated with the stellar normalization. Our final estimate is the mean of these two values[^1], log [$N({\rm D \; \mbox{\small\rm I}})$]{}= $15.87\pm{0.08},$ where the error of the mean has not been reduced below the individual values since $N$([D$\;$[I]{}]{})$_{sm}$ and $N$([D$\;$[I]{}]{})$_{poly}$ are not independent estimates of the column density.
The [O$\;$[I]{}]{} Analysis
---------------------------
The continuum placement problem is much less severe for the [O$\;$[I]{}]{} lines because the spectral regions adjacent to the lines used in the analysis are generally smooth and well-behaved, without the presence of very strong adjacent lines. Thus, we use both the profile fitting and curve-of-growth techniques to estimate [$N({\rm O \; \mbox{\small\rm I}})$]{}. The profile fitting analysis used only the [O$\;$[I]{}]{} $\lambda 974.070$ line, because this line is almost completely unsaturated. We did verify that the fit to other [O$\;$[I]{}]{} lines in the spectrum was acceptable, but they provide almost no additional constraint on the column density because they are saturated. One might be concerned that an erroneous $f-$value for this transition would cause an inaccurate estimate of [$N({\rm O \; \mbox{\small\rm I}})$]{}. However, Hébrard et al. (2005) discuss this issue, and show that it is consistent with column densities derived from stronger [O$\;$[I]{}]{} lines in the case of BD+28$\arcdeg4211,$ and derived from the weak $\lambda1356$ intersystem line in the case of HD 195965. The [O$\;$[I]{}]{}$\lambda974.070$ line is blended with two [H$_2$]{} lines, Lyman 11-0R(2) $\lambda974.156$ and Werner 2-0Q(5) $\lambda974.287,$ and absorption from these species was accounted for when calculating [$N({\rm O \; \mbox{\small\rm I}})$]{}, as described below. There is also an [H$_2$]{} $J=6$ line at nearly the same wavelength, but the column density in this rotational level is too small to be of concern. The result of our analysis is $N$([O$\;$[I]{}]{})$_{pf} = 17.53{\ensuremath{^{+ 0.25}_{- 0.15}}}.$
To construct the curve-of-growth the equivalent widths of the [O$\;$[I]{}]{} lines were measured in the following way. First, low-order Legendre polynomials were fitted to the local continua around each [O$\;$[I]{}]{} line. The lines were integrated over velocity limits chosen to exclude absorption from adjacent lines. Only isolated lines were selected so that no deblending was required. Scattered light remaining after the CALFUSE 3 reduction is negligibly small, and no additional correction was required. The measured equivalent widths are given in Table 5. The estimated errors from this method are described in detail by Sembach & Savage (1992). They include contributions from both statistical and fixed-pattern noise in the local continuum. Continuum placement is particularly difficult and subject to error due to the presense of many unidentified stellar lines. To estimate the magnitude of such errors, we did trials in which we drew the continuum at its maximum and minimum plausible locations, and compared the calculated equivalent widths with the best estimated placement. We added 2 mÅ to the error of each measured [O$\;$[I]{}]{} line equivalent width to account for these placement errors. This is consistent with the magnitude of this systematic error determined in previous analyses of similar sight lines (e.g., Friedman et al., 2002).
Due to the blending with the [H$_2$]{} lines we initially did not include the [O$\;$[I]{}]{}$\lambda974.070$ line. However, this line is extremely important because it is the only [O$\;$[I]{}]{} line that is not substantially saturated, and therefore provides the greatest constraint on [$N({\rm O \; \mbox{\small\rm I}})$]{}. We constructed curves of growth for the $J=2$ and $J=5$ rotational levels of [H$_2$]{}, calculated column densities and $b-$values, determined the appropriate Voigt profiles, and used them to remove the [H$_2$]{} signature from the blended line. Then the equivalent width of the [O$\;$[I]{}]{} line was measured in the usual way. The [H$_2$]{} analysis is described in Section 4.5. The [O$\;$[I]{}]{} curve of growth is shown as the solid line in Figure \[fig\_OI\_cog\]. The best fit column density is log $N$([O$\;$[I]{}]{})$_{COG} =
17.59{\ensuremath{^{+ 0.13}_{- 0.14}}}$. Combining this with our profile fitting result, we arrive at our best estimate of the [O$\;$[I]{}]{} column density, log [$N({\rm O \; \mbox{\small\rm I}})$]{}= $17.57{\ensuremath{^{+ 0.21}_{- 0.15}}}.$
We note that the [O$\;$[I]{}]{} $\lambda919.917$ line has been excluded from the analysis. We were unable to get a good fit of this line with profile fitting, and it fell well below the curve-of-growth. This is also true for Feige 110 (Hébrard et al. 2005), whose spectrum is similar in many respects to that of LSE 44, as discussed in §3. It is unlikely that this line has a significantly incorrect $f-$value since the absorption from this line is consistent with absorption from other [O$\;$[I]{}]{} lines in, for example, the spectrum of BD+$28\arcdeg4211$ (Hébrard & Moos 2003). It is possible that there is unidentified absorption adjacent to this line, which depresses the local continuum, causing an underestimate of the equivalent width of this line. Alternatively, the location of this line on the COG may be an indication that there are multiple components along the sight line, with different column densities. This would cause a “kink” in the COG, which might be more obvious if additional [O$\;$[I]{}]{} lines were available to include on the plot. However, with no high resolution spectra available, we are unable to confirm this hypothesis. If we include the $\lambda919.917$ line, the single-component COG gives log [$N({\rm O \; \mbox{\small\rm I}})$]{}= $17.52{\ensuremath{^{+ 0.22}_{- 0.23}}}$ and $b =
6.33{\ensuremath{^{+ 0.49}_{- 0.44}}},$ which is shown as the dashed line in Figure \[fig\_OI\_cog\]. This differs from our best estimate column density by less than $1\sigma.$
It is worth emphasizing the importance of non-saturated lines in the determination of [$N({\rm O \; \mbox{\small\rm I}})$]{} even at the expense of the potential introduction of systematic errors associated with using strongly blended lines. Computing a COG excluding the [O$\;$[I]{}]{} $\lambda974.070$ (and $\lambda919.917$) lines gives $N$([O$\;$[I]{}]{}) = $17.22{\ensuremath{^{+ 0.55}_{- 0.29}}}$, about $2\sigma$ less than the result obtained when including the weak line. This is shown as the dotted line in Figure \[fig\_OI\_cog\]. The fit to the remaining spectral lines is quite acceptable, but is clearly inconsistent with the weak line. Indeed, this line was not used in the initial analysis of Feige 110 (Friedman et al. 2002). We have recently included $\lambda974.070$ in a re-analysis of [$N({\rm O \; \mbox{\small\rm I}})$]{} along this sight line. The COG and profile fitting methods are now in good agreement, and the column density estimate has increased by 0.33 dex compared to the original estimate. This is discussed more fully in Hébrard et al. (2005).
The [N$\;$[I]{}]{} Analysis
---------------------------
As was the case with [D$\;$[I]{}]{}, only profile fitting was used to determine [$N({\rm N \; \mbox{\small\rm I}})$]{}. The COG technique was not used because of the uncertainties in establishing continuum levels on each side of the [N$\;$[I]{}]{} lines.
Three lines were used in the profile fitting analysis: $\lambda 951.079,
\lambda 951.295,$ and $\lambda 955.882$. The method is the same as was used for the deuterium analysis, and is described in §4.1. The result is log $N$([N$\;$[I]{}]{}) = $16.43\pm0.14.$
The [H$\;$[I]{}]{} Analysis
---------------------------
In the direction of LSE 44, the [Ly$\alpha$]{} line provides the best constraint on the total [H$\;$[I]{}]{} column density because its radiation damping wings are very strong. However, [Ly$\alpha$]{} is not accessible to [[*FUSE*]{}]{}. [Ly$\beta$]{} displays modest damping wings, and we attempted to model the absorption profile in order to estimate [$N({\rm H \; \mbox{\small\rm I}})$]{}. We were unable to do this primarily because of the presense of two [H$_2$]{} lines, J=0 $\lambda1024.230$ and J=2 $\lambda1026.526$, on the blue and red wings of the [Ly$\beta$]{} absorption, respectively. As discussed in §4.5, all lines of [H$_2$]{}in the J=0 rotational level lie on the flat portion of the COG, so we are unable to make even a rough estimate of the column density. The situation for J=2 is only slightly better, with the weakest lines still considerably saturated. Since these lines are located almost exactly at the regions of the [Ly$\beta$]{} profile which are most sensitive to constraining a model profile, it is impossible to obtain a good estimate of [$N({\rm H \; \mbox{\small\rm I}})$]{}. The higher order Lyman lines in the [[*FUSE*]{}]{}spectrum are even weaker relative to the stellar absorption than is [Ly$\alpha$]{} or [Ly$\beta$]{}. Correcting these lines for stellar absorption would therefore require more precise knowledge of the proper flux scaling of the model, and of the velocity difference between the stellar and interstellar components. We have examined the range of $b-$value/column density combinations allowed by the higher Lyman series lines in the [[*FUSE*]{}]{} spectrum, along with the uncertainties in correcting for the stellar absorption, and we find that these lines do not constrain $N$([H$\;$[I]{}]{}) with sufficient precision to provide an interesting D/H measurement. No suitable [*HST*]{} spectra of the LSE 44 [Ly$\alpha$]{} line have been obtained. Consequently, we have used the only high-dispersion echelle observation of this star obtained with the short wavelength prime camera on [[*IUE*]{}]{} (exposure ID SWP29086) to estimate $N$([H$\;$[I]{}]{}) toward LSE 44. This [[*IUE*]{}]{}mode provides a resolution of $\sim$25 [km s$^{-1}$]{} FWHM and covers the 1150-1950 Årange.
The [[*IUE*]{}]{} observation was obtained on 1986 August 29 with an exposure time of 16.5 ksec. We retrieved the [[*IUE*]{}]{} spectrum from the STScI MAST archive with both IUESIPS and NEWSIPS processing. Since there are several known problems with NEWSIPS processing of high-dispersion observations (Massa et al. 1998) that can adversely affect the $N$([H$\;$[I]{}]{}) measurement, particularly the background subtraction in the vicinity of [Ly$\alpha$]{} where the orders are closely spaced on the detector (Smith 1999), we generally prefer IUESIPS spectra. Nevertheless, we compared the IUESIPS and NEWSIPS versions of the spectrum at the outset. In addition to the standard processing, we applied the Bianchi & Bohlin (1984) correction method to the IUESIPS spectrum to compensate for scattered light from adjacent orders in the background regions near the interstellar [Ly$\alpha$]{} line.
### Analysis Method
To measure the total interstellar [H$\;$[I]{}]{} column density, we used the method of Jenkins (1971); see also Jenkins et al. (1999) and Sonneborn et al. (2000). In brief, we constrained the [H$\;$[I]{}]{} column density using the Lorentzian wings of the [Ly$\alpha$]{} profile, which have optical depth $\tau$ at wavelength $\lambda$ given by $\tau (\lambda ) = N$([H$\;$[I]{}]{})$\sigma (\lambda ) = 4.26 \times 10^{-20}
N$([H$\;$[I]{}]{})$(\lambda - \lambda _{0})^{-2}$. Here $\lambda _{0}$ is the centroid of the interstellar [H$\;$[I]{}]{} absorption in this direction, which was determined from the triplet at 1200 Å. We estimated the value of $N$([H$\;$[I]{}]{}) that provides the best fit to the [Ly$\alpha$]{} profile by minimizing $\chi ^{2}$ using Powell’s method with five free parameters: (1) $N$([H$\;$[I]{}]{}), (2)-(4) three coefficients that fit a second-order polynomial to the continuum (with a model stellar [Ly$\alpha$]{} line superimposed, see below), and (5) a correction for the flux zero level. We then increased (or decreased) $N$([H$\;$[I]{}]{}) while allowing the other free parameters to vary in order to set upper and lower confidence limits based on the ensuing changes in $\chi ^{2}$.
In addition to the [Ly$\alpha$]{} absorption profile due to interstellar [H$\;$[I]{}]{} , a subdwarf O star like LSE 44 will have a substantial stellar [Ly$\alpha$]{} absorption line with broad, Lorentzian wings as well. Neglect of this stellar line when setting the continuum for fitting the interstellar [Ly$\alpha$]{} will lead to a substantial systematic overestimate of $N$([H$\;$[I]{}]{}). We have used the stellar atmosphere models discussed in §3 to account for the stellar [Ly$\alpha$]{} line. The offset of the stellar line centroid with respect to the interstellar lines was determined by comparing the velocities of several stellar and interstellar lines in the [[*FUSE*]{}]{} spectrum, and the uncertainty in the determined offset has a negligible impact on the derived $N$([H$\;$[I]{}]{}). We use the most shallow and deepest stellar profiles derived from a grid of models, as described in §3, to set upper and lower confidence limits on $N$([H$\;$[I]{}]{}) in the following section.
### $N$([H$\;$[I]{}]{}) Results
Figure \[fig\_hifit\] shows the [[*IUE*]{}]{} spectrum of LSE 44; both panels plot the same data but the upper panel shows a broader wavelength range to enable the reader to inspect the fit to the continuum away from the strong [Ly$\alpha$]{}absorption. Overplotted on the spectrum with dotted lines are the fits that provide $2\,\sigma$ upper and lower limits on the interstellar $N$([H$\;$[I]{}]{}). The deepest model stellar [Ly$\alpha$]{} line is assumed for the lower limit on $N$([H$\;$[I]{}]{}), and the most shallow stellar [Ly$\alpha$]{} model is assumed for the upper limit. From these fits we derive log $N$([H$\;$[I]{}]{}) = 20.52$^{+0.20}_{-0.36}$ ($2\,\sigma$).
There are, of course, many potential sources of systematic error in the $N$([H$\;$[I]{}]{}) measurement. We have already discussed the stellar [Ly$\alpha$]{} line. Strong, but nevertheless unrecognized, narrow stellar lines within the [Ly$\alpha$]{} profile are another potential source of systematic error. We have included stellar lines in the continuum regions used to estimate $\sigma$ for the $\chi ^{2}$ calculation, but nevertheless future higher resolution and signal-to-noise observations that resolve strong stellar lines might yield a substantially different result. However, it seems likely that the greatest source of systematic errors in this case is location of the zero-point flux. In a saturated line the core should be broad and completely black. However, examination of Figure \[fig\_hifit\] shows that the core has significant undulations. We have decided that the flattening of the spectrum between approximately 1216.5Å and 1217.8Å gives the best indication of the zero point. The spectral regions used in the fit to determine [$N({\rm H \; \mbox{\small\rm I}})$]{} are indicated in Figure \[fig\_hifit\] by the dark horizontal lines below the spectrum.
We recognize that there are other plausible zero-point placements. For example, Friedman et al. (2005) initially used the spectral region indicated by the gray horizontal line in Figure \[fig\_hifit\] to determine the zero point, completely excluding the flat part in the red side of the airglow emission. The result of this fit was log $N$([H$\;$[I]{}]{}) = 20.41$^{+0.19}_{-0.33}.$ However, we ultimately decided that the flattening to the red of the airglow line better represents the expected shape of the line core, and the undulations on the blue side make this region an unreliable indicator of the zero point. The undulations could have many sources, including improper joining of spectral orders in the [[*IUE*]{}]{} echellogram. We note that if the region on the blue side is used to set the zero point, we obtain a [*lower*]{} value for [$N({\rm H \; \mbox{\small\rm I}})$]{}, which in turn makes D/H even higher than the value we obtain in the next section.
The [H$_2$]{} Analysis
----------------------
Accurate measurements of the column densities of molecular hydrogen in various rotational levels are important so that the [H$_2$]{} lines may be deblended in order to measure the column densities of [D$\;$[I]{}]{} and metals. For example, as noted in §4.2, the weak [O$\;$[I]{}]{} $\lambda974.070$ line is blended with $J=2$ and $J=5$ lines. We attempted to measure $N$([H$_2$]{}) for $J=0-5$ so that we could estimate the total [H$_2$]{} column density and determine the molecular fraction of the gas along the sight line. However, the $J=0$ and $J=1$ rotational levels could not be accurately determined. In each case there are no lines strong enough to show well-developed damping wings, and no lines weak enough to be on the linear portion of the curve-of-growth. The situation is slightly better for $J=2,$ which has some weaker lines but none that are in the true linear regime. We constructed a COG, but the column density is not well-constrained. For $J=3-5,$ weaker lines are available and the errors are lower. The measured equivalent widths are given in Table 6, and the column densities are given in Table 7. The [H$_2$]{} COGs are shown in Figure \[fig\_h2cog\].
Results and Discussion
======================
We have measured the column densities of [D$\;$[I]{}]{}, [O$\;$[I]{}]{}, [N$\;$[I]{}]{}, [H$\;$[I]{}]{}, and [H$_2$]{} along the sight line to LSE 44. This target is of particular interest because it is one of only five for which such values are known for Galactic targets at distances greater than about 500 pc and hydrogen column densities log [$N({\rm H \; \mbox{\small\rm I}})$]{}$>
20.5.$ Table 8 gives a summary of the results of this study, and Figure \[fig\_dhplot\] shows our D/H value compared to previous measurements. Our principle results are D/H = $(2.24{\ensuremath{^{+ 1.39}_{- 1.32}}}) \times 10^{-5},$ D/O = $(1.99{\ensuremath{^{+ 1.30}_{- 0.67}}}) \times 10^{-2}$, and O/H = $(1.13{\ensuremath{^{+ 0.96}_{- 0.71}}})
\times 10^{-3}$. Of the targets with the [H$\;$[I]{}]{} column densities only LSE 44 and PG0038+199 (Williger et al. 2005) have high D/H, about $2.2 \times 10^{-5}.$ D/H for the remaining targets are all $\leq 1.0 \times 10^{-5}.$
The spatial variation of D/H has drawn increased attention in recent years. The large number of measurements made using [[*FUSE*]{}]{}, together with earlier results from [[*IMAPS*]{}]{}, [[*HST*]{}]{}, and [[*Copernicus*]{}]{}, led to initial speculation that the observed variations in D/H and O/H were due to differing levels of astration (Moos et al. 2002). Models of galactic chemical evolution (de Avillez & Mac Low 2002; Chiappini et al. 2002) attempt to account for the spatial variability in terms of mixing scale lengths and times. One of the greatest difficulties for such models is to properly account for the astration factor required to reduce the primordial deuterium abundance to that observed along some sight lines in the Galaxy. The primordial value, estimated from gas-phase measurements of low-metallicity material at high redshift toward 5 quasars, is (D/H)$_{prim} =
2.78{\ensuremath{^{+ 0.44}_{- 0.38}}} \times 10^{-5}$ (Kirkman et al. 2003). This is consistent with the value inferred from the amplitude of the acoustic peaks in cosmic background radiation (CBR) measurements made with [[*WMAP*]{}]{}, (D/H)$_{CBR} =
2.37{\ensuremath{^{+ 0.19}_{- 0.21}}} \times 10^{-5}$ (Spergel et al. 2003). Thus, astration factors of approximately 1.6 or 1.9 are required to account for Local Bubble D/H abundances determined by Wood et al. \[(D/H)$_{LB} = (1.56\pm{0.04}) \times
10^{-5}$\], and Hébrard & Moos \[(D/H)$_{LB} = (1.32\pm{0.08}) \times
10^{-5}$\], respectively. An astration factor of approximately 3 is required to account for the mean D/H value of the four highest column density sight lines in Figure \[fig\_dhplot\](b), and this is pushing the limits of most chemical evolution models (Tosi et al. 1998). These models will also have to account for the deuterium abundance measured in the Complex C, a low-metallicity, high-velocity cloud falling in to the Milky Way. Sembach et al. (2004) found (D/H)$_{complex\,C} = (2.2 \pm 0.7) \times 10^{-5}$ and (D/O)$_{complex\,C} =
0.28 \pm 0.12\, (1\sigma).$
Hébrard & Moos (2003) suggest that the low D/H value at large distances truly reflects the present-epoch D/H abundance. This conclusion is primarily based on D/O and O/H measurements. If this is correct, it presents a challenge to models of galactic chemical evolution, which not only have to account for high astration values, but also the observed spatial variation of these quantities.
Jura (1982) first proposed that deuterium might be depleted onto dust grains in the interstellar medium. Draine (2004a, 2004b) extended this by suggesting that D/H in carbonaceous dust grains can exceed the overall D/H ratio by a factor greater than $10^4$, if the material is in thermodynamic equilibrium. Thus, it is possible that a significant reservoir of deuterium is depleted onto these grains, as long as the metallicity is approximately solar or greater. Prochaska, Tripp, & Howk (2005) have tested this idea by examining the correlation between D/H and the highly depleted ion [Ti$\;$[II]{}]{}. They find these are correlated at greater than 95% confidence level. They emphasize, however, that since not all data points support this relationship, measurements of distant targets, such as the one in the current study, are most needed to further constrain this relationship. At present, we are unable to comment directly on this hypothesis because the [Ti$\;$[II]{}]{} abundance along this sight line is not known.
Wood et al. (2004) and Linsky et al. (2005, in preparation) have applied Draine’s suggestion to the observational data, and suggest the presence of three D/H regimes, as shown in Figure \[fig\_dhplot\]. In the Local Bubble, where log [$N({\rm H \; \mbox{\small\rm I}})$]{}$\la
19.2,$ most of the deuterium is in the gas phase, presumably because of the recent passage of strong shocks from supernovae or stellar winds. Thus, (D/H)$_{LBg} = (1.56 \pm 0.04) \times 10^{-5}$ is relatively high, spatially uniform, and approximates the true local Galactic disk value. At great distances, where log [$N({\rm H \; \mbox{\small\rm I}})$]{}$\ga 20.5$, sight lines traverse many regions, some recently shocked and others not. D/H is likely to be indicative of an average of these regions. However, the highest column density sight lines, which tend to be the lengthiest ones, will be weighted toward the densest, coldest regions, and presumably the most heavily deuterium-depleted clouds. This may explain the low D/H values toward the majority of the most distant targets. In the intermediate column density regime, 19.2 $\la$ log([$N({\rm H \; \mbox{\small\rm I}})$]{}) $\la$ 20.5, sight lines traverse fewer regions with different shock event histories and therefore exhibit more variability.
Hébrard & Moos (2003) show that D/O tends to be uniform within the Local Bubble, with a mean of $(3.84\pm 0.16) \times 10^{-2} (1\sigma),$ and variable beyond it. D/N is similar, but exhibits somewhat more variability due to ionization effects. At the highest column densities D/O and D/N converge to mean values that are lower than the Local Bubble values. Indeed, for all seven sight lines for which log [$N({\rm H \; \mbox{\small\rm I}})$]{} $> 20.1$ and log [$N({\rm D \; \mbox{\small\rm I}})$]{} $>15.4$ (Hoopes, et al. 2003; Wood et al. 2004; Williger et al.), including LSE 44, D/O lies in the relatively narrow range $(1.8 - 2.6) \times
10^{-2}.$ The fact that D/O is much more uniform than D/H is a surprise, and suggests to some investigators that measurements of [$N({\rm H \; \mbox{\small\rm I}})$]{} may suffer from additional unidentified errors (Hébrard et al. 2005).
The $\sim100$ pc length scale in the Local Bubble over which D/H is roughly constant (Figure \[fig\_dhplot\]) may be a natural consequence of supernova-driven mixing within the ISM. Moos et al. (2002) discuss a simple model which has two time scales: the mixing time and the time between supernovae (SNe). In a fully ionized ISM at $\sim10^6$K, the adiabatic sound velocity is $c_s \approx 150$ [km s$^{-1}$]{}. For a spherical parcel of gas with radius 100 pc, the crossing time is $t_s \approx 7 \times 10^5$ years. Other mechanisms, such as the Alfvén crossing time for a cloud with a magnetic field, are generally slower.
A supernova rate of 0.02 yr$^{-1}$ (Cappellaro et al. 1993) spread uniformly over the galactic disk of radius 8.5 kpc, yields approximately $8.8 \times
10^{-11}$ SNe pc$^{-2}$ yr$^{-1}.$ In the 100 pc radius parcel of gas considered above, there would be one supernova every $4 \times 10^5$ yr. Therefore, in this simple model, $t_{SN} \sim t_s.$ Thus, one might expect that within the Local Bubble, the material is well-mixed, and D/H, O/H, and N/H will be approximately constant, with relatively small target-to-target variations. Toward targets at greater distances, different supernovae events may have given rise to interstellar regions with different enrichment levels, and these do not have time to mix thoroughly with each other. See Moos et al. (2002) for additional discussion of this model. More sophisticated treatments of mixing, including the effects of supernovae and infall of material into the Galactic disk, are given by de Avillez (2000), de Avillez & Mac Low (2002), and Chiappini et al. (2002).
If this model is correct, it is hard to explain the uniformity of D/O for the high column density sight lines, unless a different mechanism forces this to be the true Galactic value. In addition, for these targets D/H and O/H vary by factors of 2.9 and 4.6, respectively, but there is no convincing anticorrelation between them, as might be expected if O is produced as D is destroyed in stellar interiors. Steigman (2003) reported a tentative detection of such an anticorrelation based on [[*IMAPS*]{}]{} and the initial set of [[*FUSE*]{}]{} results, all of which had column densities lower than those considered here.
Neither the depletion hypothesis nor the low Galactic D/H hypothesis appears to be consistent with all of the data currently available. Hébrard et al. (2005) show that the dispersion of D/H is greater than that of D/O in the sense that there are no high D/O values at large distances or large column densities. They argue against the depletion hypothesis since for sight lines beyond the Local Bubble, the uniformity of D/O and O/H (Meyer, Jura, & Cardelli 1998; Meyer 2001; André et al. 2003) strongly imply that D/H must be uniform as well, unless D/O and O/H are correlated in exactly the same way. They emphasize that measurements of D/H may be subject to fewer systematic errors then measurments of D/O and O/H. On the other hand, nobody has identified an error that could account for the D/H variability shown in Fig. \[fig\_dhplot\]. Two observational tests will help clarify these interpretations. First, additional measurements of the depletion of refractory species, such as Fe, Si, or Ti (Prochaska, Tripp, & Howk 2005) along these sight lines would reveal whether a correlation exists between depletion and the gas phase abundance of D/H. Such a correlation would provide strong evidence for the depletion hypothesis. Second, Lecour et al. (2005) measured N(HD) and N([H$_2$]{}) along highly reddened sight lines, and inferred (D/H) = $(2.0\pm{1.1}) \times 10^{-6}$ an extremely low value. Linsky et al. (2005, in preparation) suggest that this is due to high levels of deuterium depletion, and they discuss methods to investigate this which would test the depletion hypothesis.
It is possible that the systematic errors in these measurements are larger than the investigators have estimated, leading to erroneous conclusions about the variability of D, O, and N. This may also be the cause of the apparent dispersion in D/H in high redshift environments (Crighton et al. 2004), where very little variability is expected since the gas has experienced little astration and has low metallicity. A potential alternative explanation of the variability would be a mechanism of non-primordial deuterium production (Epstein, Lattimer, & Schramm 1976). Many mechanisms have been examined carefully including, for example, D production in stellar flares (Mullan & Linsky 1999), but none has been shown to produce enough to materially change the composition of the ISM (Prodanović & Fields 2003). See Lemoine et al. (1999) for a review of a variety of potential production mechanisms.
Combining results from Hébrard & Moos (2003), Wood et al. (2004), and Hoopes et al. (2003), it is seen that D/H, O/H, and N/H are factors of $1.5-3$ higher toward LSE 44 than toward most other stars with log [$N({\rm H \; \mbox{\small\rm I}})$]{} $> 20.5.$ This at least suggests the possibility that our estimate of [$N({\rm H \; \mbox{\small\rm I}})$]{} may be underestimated by approximately this amount. We discussed in §4.4 the problems of determining the proper flux zero point in the core of the [H$\;$[I]{}]{} line. If our zero-point determination is correct, then our error in [$N({\rm H \; \mbox{\small\rm I}})$]{} is just consistent with these abundances being a factor of 1.5 greater than our best estimate, but not with a factor of 3. Unfortunately, without improved measurements of the Ly$\alpha$ profile, better estimates of [$N({\rm H \; \mbox{\small\rm I}})$]{} are not likely to be obtained.
Studies by the [[*FUSE*]{}]{} deuterium team have been concentrating on distant targets in order to probe more completely both the transition and high column density regions displayed in Figure \[fig\_dhplot\]. LSE 44 is one of several new targets in this regime. Additional observations of targets at these large distances will be required to determine whether depletion has caused the observed spatial variations in the gas phase D/H ratio, or if the Galactic disk value is really below $10^{-5}$ and different histories of astration are of primary importance, or if some unknown non-primordial source of deuterium is responsible for the variability.
We thank Dave Kilkenny for kindly providing the optical spectrum of LSE 44, Gerry Williger for reducing the optical data, and Brian Wood for providing the data in Figure \[fig\_dhplot\] in numerical form. This work is based on data obtained for the Guaranteed Time Team by the NASA-CNES-CSA [[*FUSE*]{}]{} mission operated by The Johns Hopkins University. This work used the profile fitting procedure [Owens.f]{} developed by M. Lemoine and the French [[*FUSE*]{}]{} team. Financial support has been provided by NASA contract NAS5-32985. SDF is supported by NASA grant NNG04GH19G and TMT by NASA grant NNG04GG73G. GH is also supported by CNES.
Abgrall, H., Roueff, E., Launay, F., Roncin, J.Y., & Subtil, J.L., 1993a, , 101, 273
Abgrall, H., Roueff, E., Launay, F., Roncin, J.Y., & Subtil, J.L., 1993b, , 101, 323 André, M.K. et al. 2003, , 591, 1000
Bessell, M. S. 1990, , 102, 1181
Bevington, P. R., & Robinson, D. K. 1992, Data Reduction and Error Analysis for the Physical Sciences, (2d ed.; New York: McGraw-Hill)
Bianchi, L., & Bohlin, R. C. 1984, A&A, 134, 31
Boesgard, A. & Steigman, G. 1985, , 23, 319
Cappellaro, E. Turatto, M., Benetti, S., Tsvetkov, D.Y., Bartunov, O.S., & Makarova, L.N. 1993, A&A, 273, 383
Chiappini, C., Renda, A., & Matteucci, F. 2002, , 395, 789
Crighton, N.H.M., Webb, J.K., Ortiz-Gil,A., & Fernández-Soto, A. 2004, , 355, 1042
de Avillez, M.A. 2000, 315, 479
de Avillez, M.A. & Mac Low, M.M. 2002, , 581, 1047
Draine, B.T. 2004a, in Origin and Evolution of the Elements, ed. A. McWilliam & M. Rauch (Pasadena,: Carnegie Obs.), 320
Draine, B.T. 2004b, preprint (astro-ph/0410310)
Drilling, J.S. 1983, , 270, L13
Epstein, R. I., Lattimer, J. M., & Schramm, D. N., 1976, , 263, 198.
Friedman, S.D. et al. 2002, , 140, 37
Friedman, S.D. et al. 2005, in “Astrophysics in the Far Ultraviolet”, ASP Conference. Series., Sonneborn, Moos, & Andersson (eds).
Heber, U. 1984, , 136, 331
Heber, U., Hunger, K., Jonas, G., & Kudritzki, R. P. 1984, , 130, 119
Hébrard, G. & Moos, H.W. 2003, , 599, 297
Hébrard, G. et al. 2002, , 140, 103
Hébrard, G. et al. 2005, ApJ, in press. astro-ph/0508611
Hébrard, G. 2005, in “Astrophysics in the Far Ultraviolet”, ASP Conference. Series., Sonneborn, Moos, & Andersson (eds)
Hoopes, C.G, Sembach, K.R., Hébrard, G. Moos, H.W., & Knauth, D.C. 2003, , 586, 1094
Hubeny, I., & Lanz, T. 1995, , 439, 875
Jenkins, E. B. 1971, , 169, 25
Jenkins, E. B. 1986, , 304, 739.
Jenkins, E. B., Tripp, T. M., Woźniak, P. R., Sofia, U. J., & Sonneborn, G. 1999, ApJ, 520, 182
Jura, M. 1982, in Advances in Ultraviolet Astronomy, ed. Y. Kondo (NASA CP-2238; Washington: NASA), 54
Kirkman, D., Tytler, D., Suzuki, N., O’Meara, J.M., & Lubin, D. 2003, , 149, 1
Lacour, S. et al. 2005, A&A, 430, 967
Linsky, J. L. et al. 1995, , 451, 335
Massa, D., Van Steenberg, M. E., Oliversen, N., & Lawton, P., 1998, in Ultraviolet Astrophysics Beyond the [*IUE*]{} Final Archive, ed. W. Wamsteker & R. Gonzalez Riestra (ESA SP-413; Noordwijk: ESA), 723
Meyer, D.M. 2001, in Proc. 17th IAP Colloq., GaseousMatter in Galaxies and Interstellar Space, ed. R. Ferlet, M. Lemoine, J.-M. Desert, &B. Raban (Paris: Editions Frontiéres), 135
Meyer, D.M., Jura, M., & Cardelli, J.A. 1998, , 493,222
Moos, H. W. et al. 2002, , 140, 3
Morton, D.C. 2003, , 149, 205
Mullan, D.J. & Linsky, J.L. 1999, , 511, 502
Pettini, M. and Bowen, D.V. 2001, , 560, 41
Prochaska, J.X., Tripp, T.M., & Howk, J.C. 2005, , 620, L39
Prodanović, T. & Fields B.D. 2003, , 597, 48
Reeves, H., Audouze, J., Fowler, W., & Schramm, D., N. 1973 , 179, 909
Rogerson, J. B. & York, D. G. 1973, , 186, L95
Saffer, R. A., Bergeron, P., Koester, D., & Liebert, J. 1994, , 432, 351
Sahnow, D. J. et al. 2000, , 538, L7
Sembach, K.R. et al. 2004, , 150, 387
Sembach, K. R. & Savage, B. D. 1992, , 83, 147
Sfeir, D.M., Lallement, R., Crifo, F., & Welsh, B.Y. 1999, , 346, 785
Smith, M. A. 1999, PASP, 111, 722
Sonneborn, G., Tripp, T. M., Ferlet, R., Jenkins, E. B., Sofia, U. J., Vidal-Madjar, A., & Woźniak, P. R. 2000, ApJ, 545, 277 Spergel, D. N. et al. 2003, , 148, 175
Spitzer, L. 1978, Physical Processes in the Interstellar Medium (New York: John Wiley)
Steigman, G. 2003, , 586, 1120
Tosi, M. et al. 1998, , 498, 226
Williger et al. 2005, , 625, 210
Wood, B.E. et al. 2004, , 609, 838
[ccccc]{}
[[*FUSE*]{}]{}& 24 April 2002 & P2051602 & 14 & 13.9\
[[*FUSE*]{}]{}& 09 August 2003 & P2051603 & 9 & 6.2\
[[*FUSE*]{}]{}& 10 August 2003 & P2051604 & 14 & 5.6\
[[*FUSE*]{}]{}& 11 August 2003 & P2051605 & 25 & 18.2\
[[*FUSE*]{}]{}& 12 August 2003 & P2051606 & 12 & 8.5\
[[*FUSE*]{}]{}& 03 February 2004 & P2051607 & 20 & 15.7\
[[*FUSE*]{}]{}& 04 February 2004 & P2051608 & 14 & 11.1\
[[*FUSE*]{}]{}& 05 February 2004 & P2051609 & 11 & 6.3\
[[*IUE*]{}]{}& 29 August 1986 & & SWP 29086 & 16.5\
[lc]{} LiF1A & 64.5\
LiF1B & 64.8\
LiF2A & 50.6\
LiF2B & 51.3\
SiC1A & 61.2\
SiC1B & 61.2\
SiC2A & 61.9\
SiC2B & 60.4\
[lcc]{} Spectral Type & sdO & 1\
$(l,b)$ & $(313\fdg37,+13\fdg49)$ & 1\
$d$ (pc)& $554\pm66$ & 2\
$z$ (pc) & $129 \pm 15$ & 2\
$V$ & 12.45 & 1\
$U-B$ & $-1.18$ & 1\
$B-V$ & $-0.24$ & 1\
$E_{B-V}$ & $0.05 \pm 0.03$ & 2\
$T_{\rm eff}$ (K)& $38700 \pm 1000$ & 2\
$\log g$ (cm s$^{-2}$) & $5.5 \pm 0.1$& 2\
log \[$N$(He)/$N$(H)\] & $-2.8\pm0.1$ & 2\
[lcclcc]{} [H$\;$[I]{}]{}& 919.351 & 1.20E-03 & [H$_2$]{} J=3 & 1028.985 & 1.74E-02\
[H$\;$[I]{}]{}& 920.963 & 1.61E-03 & & 1043.502 & 1.08E-02\
[D$\;$[I]{}]{}& 916.18 & 5.78E-04 & & 1056.472 & 9.56E-03\
[D$\;$[I]{}]{}& 919.101 & 1.20E-03 & & 928.436 & 3.30E-03\
[D$\;$[I]{}]{}& 920.712 & 1.61E-03 & & 933.578 & 1.95E-02\
[N$\;$[I]{}]{}& 951.079 & 1.69E-04 & & 934.789 & 7.09E-03\
[N$\;$[I]{}]{}& 951.295 & 2.32E-05 & & 935.573 & 6.57E-03\
[N$\;$[I]{}]{}& 955.882 & 5.88E-05 & & 967.673 & 2.28E-02\
[O$\;$[I]{}]{}& 974.07 & 1.56E-05 & & 970.56 & 9.75E-03\
[Fe$\;$[II]{}]{}& 1055.262 & 6.15E-03 & & 985.962 & 8.28E-03\
[Fe$\;$[II]{}]{}& 1062.152 & 2.91E-03 & [H$_2$]{} J=4 & 1023.434 & 1.05E-02\
[Fe$\;$[II]{}]{}& 1125.448 & 1.56E-02 & & 1032.349 & 1.71E-02\
[Fe$\;$[II]{}]{}& 926.897 & 5.66E-03 & & 1044.542 & 1.55E-02\
[Fe$\;$[II]{}]{}& 935.518 & 2.56E-02 & & 1057.38 & 1.29E-02\
[H$_2$]{} J=0 & 938.465 & 9.51E-03 & & 919.048 & 9.30E-03\
[H$_2$]{} J=1 & 955.708 & 4.23E-03 & & 920.834 & 1.31E-02\
[H$_2$]{} J=2 & 1026.526 & 1.80E-02 & & 933.788 & 1.08E-02\
& 1053.284 & 9.02E-03 & & 935.958 & 1.95E-02\
& 920.241 & 1.68E-03 & & 938.726 & 6.38E-03\
& 927.017 & 2.33E-03 & & 940.384 & 1.99E-03\
& 932.604 & 4.78E-03 & & 955.851 & 1.69E-03\
& 940.623 & 6.03E-03 & & 962.151 & 8.75E-03\
& 941.596 & 3.37E-03 & & 968.664 & 1.26E-02\
& 974.156 & 1.32E-02 & & 970.835 & 1.60E-02\
& 975.344 & 6.64E-03 & & 971.387 & 3.51E-02\
& 984.862 & 8.17E-03 & & 994.227 & 1.38E-02\
& & & [H$_2$]{} J=5 & 1052.496 & 1.11E-02\
& & & & 1061.697 & 1.26E-02\
& & & & 1065.596 & 9.90E-03\
& & & & 916.101 & 2.38E-03\
& & & & 928.76 & 3.91E-03\
& & & & 955.681 & 2.76E-02\
& & & & 974.286 & 3.52E-02\
[lccccc]{} 925.446 & -0.484 & $74.4\pm6.0$ & $71.0\pm4.5$ & &\
971.738 & 1.128 & $112.3\pm6.3$ & $105.2\pm6.4$ & &\
974.070 & -1.817 & $33.3\pm4.8$ & $34.3\pm4.5$ & &\
1039.230 & 0.974 & & & $122.0\pm3.1$ & $120.7\pm3.8$\
[cccccc]{} J=2 & 920.241 & 0.190 & $57.7\pm3.9$ & $53.6\pm5.4$\
& 927.017 & 0.334 & $60.7\pm4.0$ & $61.4\pm2.2$\
& 932.604 & 0.649 & $69.4\pm2.1$ & $74.3\pm2.3$\
& 940.623 & 0.754 & $74.0\pm2.6$ & $77.9\pm5.5$\
& 975.344 & 0.812 & $75.4\pm2.5$ & $77.0\pm1.7$\
J=3 & 933.578 & 1.260 & $82.8\pm2.3$ & $77.5\pm2.1$\
& 934.789 & 0.821 & $65.9\pm2.2$ & $61.0\pm2.4$\
& 936.854 & 0.283 & $36.8\pm2.4$ & $38.9\pm3.9$\
& 951.672 & 1.081 & $77.2\pm2.3$ & $77.5\pm1.9$\
& 960.449 & 0.676 & $56.1\pm2.3$ & $56.1\pm2.3$\
J=4 & 933.788 & 1.002 & $11.3\pm2.7$ & $11.0\pm2.1$\
& 968.664 & 1.086 & $13.4\pm2.1$ & $14.7\pm2.6$\
& 970.835 & 1.192 & $19.2\pm2.2$ & $19.2\pm2.4$\
J=5 & 938.909 & 1.265 & $14.7\pm4.6$ & $16.7\pm5.7$\
& 940.882 & 0.927 & $12.8\pm1.7$ & $11.5\pm1.7$\
& 942.685 & 0.769 & $ 7.1\pm1.8$ & $ 6.9\pm2.5$\
& 958.009 & 0.964 & $14.2\pm1.9$ & ...\
& 974.884 & 1.142 & $11.4\pm2.8$ & $11.9\pm2.7$\
& 983.897 & 1.094 & $14.8\pm1.6$ & ...\
[cc]{} $J = 2$ & $\sim16.20$\
$J = 3$ & $15.52\pm0.13$\
$J = 4$ & $14.27\pm0.08$\
$J = 5$ & $14.19\pm0.11$\
[lc]{} log $N$([D$\;$[I]{}]{}) & $15.87\pm0.08$\
log $N$([O$\;$[I]{}]{}) & $17.57{\ensuremath{^{+ 0.21}_{- 0.15}}}$\
log $N$([N$\;$[I]{}]{}) & $16.43\pm0.14$\
log $N$([H$\;$[I]{}]{}) & $20.52{\ensuremath{^{+ 0.20}_{- 0.36}}}$\
D/H & $(2.24{\ensuremath{^{+ 1.39}_{- 1.32}}}) \times 10^{-5}$\
O/H & $(1.13{\ensuremath{^{+ 0.96}_{- 0.71}}}) \times 10^{-3}$\
N/H & $(8.13{\ensuremath{^{+ 3.09}_{- 2.24}}}) \times 10^{-5}$\
D/O & $(1.99{\ensuremath{^{+ 1.30}_{- 0.67}}}) \times 10^{-2}$\
D/N & $(2.75{\ensuremath{^{+ 1.19}_{- 0.89}}}) \times 10^{-1}$\
[^1]: Unless otherwise noted, errors on all quantities in this paper are $2\,\sigma.$
|
---
abstract: 'Ladder operators for the simplest version of a rationally extended quantum harmonic oscillator (REQHO) are constructed by applying a Darboux transformation to the quantum harmonic oscillator system. It is shown that the physical spectrum of the REQHO carries a direct sum of a trivial and an infinite-dimensional irreducible representation of the polynomially deformed bosonized $\mathfrak{osp}(1|2)$ superalgebra. In correspondence with this the ground state of the system is isolated from other physical states but can be reached by ladder operators via non-physical energy eigenstates, which belong to either an infinite chain of similar eigenstates or to the chains with generalized Jordan states. We show that the discrete chains of the states generated by ladder operators and associated with physical energy levels include six basic generalized Jordan states, in comparison with the two basic Jordan states entering in analogous discrete chains for the quantum harmonic oscillator.'
author:
- |
[**José F. Cariñena${}^a$ and Mikhail S. Plyushchay${}^b$**]{}\
\[8pt\] [ *${}^a$Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain*]{}\
[ *${}^b$Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile* ]{}\
\[4pt\] **
title: |
[ **Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator\
**]{}
---
.5cm
Introduction
============
Darboux transformations, introduced originally as a method to solve linear differential equations and generalized subsequently for Darboux-Crum(-Krein-Adler) transformations [@Darb; @Crum; @Krein; @Adler; @Matveev], find many important applications in physics. For a long period of time they were used in quantum mechanics in factorization method for solving Schrödinger equation [@Schr; @InfHull; @CR00; @MelRos]. Nowadays they are exploited intensively in the context of supersymmetry. These transformations lie in the heart of supersymmetric quantum mechanics [@Witten1; @Witten2; @CoKhSu; @CR01; @CR08]. They are particularly employed for the construction of new solvable and quasi-exactly solvable quantum mechanical systems. The Darboux transformations play a fundamental role in investigation of nonlinear equations in partial derivatives and partial difference equations, where they allow to relate different integrable systems and provide an effective method for the construction of nontrivial solutions for them [@Matveev; @ArPl]. Periodic Darboux chains generate finite-gap systems [@VesSha] in an alternative way to the original algebro-geometric approach [@NMPZ; @GesHol]. Such chains generate also the quantum harmonic oscillator (QHO) system and Painlevé equations [@VesSha; @AdlPai; @Hiet], which are intimately related with isomonodromic deformations of linear systems and integrability properties of nonlinear systems in partial derivatives. Recently, the isomonodromic deformations [@Iso1; @Iso2] and Darboux transformations played a key role in the discovery and investigation of the properties of the new class of exceptional orthogonal polynomials [@Adler; @Dub; @CPRS; @FellSmi; @Exc1; @OdSas; @SasTsZh; @Ques; @Grand; @Sesma; @GGM; @Pupas]. One of such a family corresponds to exceptional Hermite polynomials, which can be obtained by applying Darboux and Darboux-Crum transformations to the QHO system. The quantum mechanical systems appearing in such a way are described by certain rational extensions of the harmonic potential.
A simplest rationally extended quantum harmonic oscillator (REQHO) [@Dub; @CPRS] can be obtained from the QHO system by applying to it Darboux transformation generated by the “Wick-rotated" second excitation of the ground-state. The resulting system is characterized by an infinite tower of equidistant bound states which are separated from the ground-state by a triple energy gap. As a consequence, the general solution of the evolution problem for REQHO, like for the quantum harmonic and isotonic oscillators, is periodic in time with a constant (not depending on energy) period [@Dub; @AsoCar; @CPR07]. For the discussion of different aspects of this quantum mechanical system see Refs. [@Adler; @Dub; @CPRS; @FellSmi; @Sesma; @GGM; @Pupas].
It is known that the spectrum of the QHO carries an infinite-dimensional irreducible representation of the bosonized $\mathfrak{osp}(1\vert 2)$ superalgebra which can be generated by means of the creation and annihilation operators identified as fermionic generators [@CromRit; @defOSP] [^1]. In this context there appears a rather natural question: what are the ladder operators in rationally extended quantum harmonic oscillator systems and what spectrum generating algebras do they produce? In this paper we answer these questions for the simplest case of the REQHO system by employing the properties of the Darboux transformations. A special role in the construction we obtain belongs to generalized Jordan states.
The paper is organized as follows. In the next section we review general properties of the Darboux transformations and related Jordan states. In Section 3 we discuss discrete flows generated by the ladder operators in the QHO system and recall the bosonized superconformal $\mathfrak{osp}(1\vert 2)$ structure appearing in it in the form of the spectrum generating superalgebra. In Section 4 we generate the simplest REQHO by applying Darboux transformation to the QHO. Then we construct ladder operators for REQHO by a Darboux-dressing of the creation and annihilation operators of the QHO system, consider discrete flows and discuss a polynomially deformed bosonized $\mathfrak{osp}(1\vert 2)$ structure in the REQHO that reflects in a coherent way peculiarities of its spectrum. Section 5 is devoted to the conclusion and outlook. In Appendix we describe the action of the ladder operators on non-physical eigenstates of the quantum harmonic oscillator which are closely related to its physical spectrum, and present the construction of the net of associated Jordan and generalized Jordan states.
Darboux transformations and Jordan states
=========================================
Let $\psi_*(x)$ be a solution of the stationary Schrödinger equation $H\psi_*=E_*\psi_*$. For the moment we consider this equation formally as an abstract second order differential equation in which $H=-\frac{d^2}{dx^2}+V(x)$ is a Hamiltonian operator with a real nonsingular on $\R$ potential $V(x)$. A real constant $E_*$ is treated here as an eigenvalue without preoccupying the questions of boundary conditions and normalizability for $\psi_*(x)$. Consequently, we do not distinguish functions $\psi(x)$ and $C\psi(x)$, where $C\in\C$, $C\neq 0$, and assume that modulo such a multiplicative factor $\psi(x)$ is chosen to be a real-valued function. A linearly independent solution for the same eigenvalue $E_*$ can be taken in the form \[secsol\] (x)=\_\*(x)\^x. Due to integration with an indefinite lower limit, function $\widetilde{\psi_*}(x)$ is supposed to be defined up to an additive term proportional to ${\psi_*}(x)$. Assume now that $E_*$ is chosen so that function $\psi_*(x)$ is nodeless, $\psi_*(x)\neq 0$, and introduce the first-order differential operators $$\label{Adef}
A_{\psi_*}=\psi_*\frac{d}{dx}\frac{1}{\psi_*}=
\frac{d}{dx}-\mathcal{W}(x)\,,\qquad
\mathcal{W}(x)= \frac{\psi'_*}{\psi_*} \,,$$ and $A_{\psi_*}^\dagger=-\frac{d}{dx}-\mathcal{W}(x)$, where prime denotes derivative in $x$. Note that as $A_{\psi_*}$ and $A_{\psi_*}^\dagger$ are first-order differential operators, their kernels are one-dimensional, $$\ker\, A_{\psi_*}= {\psi_*},
\qquad \ker\, A^\dag_{\psi_*}=\frac 1{\psi_*}\,.
\label{kernels}$$ These operators provide a factorization of the shifted for the constant $E_*$ Hamiltonian, $H-E_*=A_{\psi_*}^\dagger A_{\psi_*}$. Potential $V(x)$ and superpotential $\mathcal{W}(x)$ are connected by a relation $V(x)-E_*=\mathcal{W}^2+\mathcal{W}'$. The product with the permuted first-order operators, $A_{\psi_*} A_{\psi_*}^\dagger=\breve{H}-E_*$, defines the associated partner system described by the Hamiltonian $\breve{H}=-\frac{d^2}{dx^2}+\breve{V}(x)$ with $\breve{V}(x)-E_*=\mathcal{W}^2-\mathcal{W}'$. From the alternative factorization relations it follows immediately that the first-order operators $A_{\psi_*}$ and $A_{\psi_*}^\dagger $ intertwine quantum Hamiltonians $H$ and $\breve{H}$, \[intertrel\] A\_[\_\*]{} H=A\_[\_\*]{},A\_[\_\*]{}\^=HA\_[\_\*]{}\^. If $\psi_{{}_{E}}$ is a physical (normalizable) or non-physical (non-normalizable) solution of the Schrödinger equation $H\psi_{{}_{E}}=E\psi_{{}_{E}}$ for some eigenvalue $E\neq E_*$, then as a consequence of (\[intertrel\]), $A_{\psi_*}\psi_{{}_{E}}$ will be an eigenstate of $\breve{H}$ of the same, physical or non-physical, nature for the same eigenvalue $E$: $\breve{H}(A_{\psi_*}\psi_{{}_{E}})=E(A_{\psi_*}\psi_{{}_{E}})$. Particularly, for the linear independent solution $\widetilde{\psi_{{}_{E}}}$ constructed from $\psi_{{}_{E}}$ according to the rule (\[secsol\]), $\widetilde{\psi_{{}_{E}}}=\psi_{{}_{E}}(x)\int^x d\xi/\psi^2_{{}_{E}}(\xi)$, we have $\breve{H}(A_{\psi_*}\widetilde{\psi_{{}_{E}}})=E(A_{\psi_*}\widetilde{\psi_{{}_{E}}})$. On the other hand, for $E=E_*$ and $\psi_{{}_{E}}=\widetilde{\psi_*}$ we find that \[Apsi\*\] A\_[\_\*]{}=. The function $\frac{1}{\psi_*}$ is the kernel of the operator $A_{\psi_*}^\dagger$, and therefore is an eigenstate of $\breve{H}$, $(\breve{H}-E_*)\frac{1}{\psi_*}=0$. Analogously, if $\breve{\psi}_{{}_{E}}$ is an eigenfunction of $\breve{H}$ of eigenvalue $E\neq E_*$, then $A_{\psi_*}^\dagger\breve{\psi}_{{}_{E}}$ is an eigenstate of $H$ of the same eigenvalue, $H(A_{\psi_*}^\dagger\breve{\psi}_{{}_{E}})=E(A_{\psi_*}^\dagger \breve{\psi}_{{}_{E}})$. For $E=E_*$ the application of $A_{\psi_*}^\dagger$ to a linearly independent from ${\frac{1}{\psi_*}}$ eigenfunction $\widetilde{\left(\frac{1}{\psi_*}\right)}$ of $\breve{H}$ maps it into the kernel of $A_{\psi_*}$, \[A+psi\*\] A\_[\_\*]{}\^=\_\*, that is the eigenstate of $H$.
The described structure of the Darboux transformations reveals an essential difference between the cases $E\neq E_*$ and $E= E_*$. The action of the Darboux transformation generators on the eigenstates with $E\neq E_*$ is of the two-cyclic nature in the following sense: if $\psi$ is such that $H\psi=E\psi$, then $A_{\psi_*}$ maps this state into an eigenstate of $\breve{H}$, $A_{\psi_*}\psi=\breve{\psi}$, $\breve{H}\breve{\psi}=E\breve{\psi}$, while application of $A_{\psi_*}^\dagger$ to $\breve{\psi}$ reproduces (up to a multiplicative factor) the initial state $\psi$. At the same time, for $E=E_*$ we have $A_{\psi_*}\widetilde{\psi_*}=\frac{1}{\psi_*}$, $A_{\psi_*}^\dagger \frac{1}{\psi_*}=0$ and $A_{\psi_*}^\dagger \widetilde{ \left(\frac{1}{\psi_*}\right)}=\psi_*$, $A_{\psi_*}\psi_*=0$, and no analogous cyclic structure does appear. In this case the functions \[Omega\] \_2(x)=[\_\*(x)]{}\^x()d, \_2(x)=\^x\_\*()()d are the pre-images of $ \widetilde{ \left(\frac{1}{\psi_*}\right)}$ and $\widetilde{\psi_*}$, \[AA+\] A\_[\_\*]{}\_2=\_1, A\_[\_\*]{}\^\_2=\_1. Similarly to the eigenstates of the form (\[secsol\]), functions $\Omega_2(x)$ and $\breve{\Omega}_2(x)$ are defined modulo additive terms ${\psi_*(x)}$ and $\frac{1}{\psi_*(x)}$, respectively. Wave functions (\[Omega\]) are not, however, formal eigenfunctions of $H$ and $\breve{H}$, but as a consequence of (\[AA+\]) they obey the relations $A_{\psi_*} A_{\psi_*}^\dagger A_{\psi_*}\Omega_2=0$ and $A_{\psi_*}^\dagger A_{\psi_*}A_{\psi_*}^\dagger\breve{\Omega}_2=0$. Therefore, \[Jordan\] ([H]{}-E\_\*)\^2\_2=0,(-E\_\*)\^2 \_2=0, and we conclude that $\Omega_2$ and $\breve{\Omega}_2$ are generalized eigenstates of $H$ and $\breve{H}$ of rank $2$ corresponding to the same eigenvalue $E=E_*$. Having in mind a generalization of relations of the form (\[Jordan\]) which appear in the following, see particularly Eq. (\[chi-Jordan\]) below, we refer to $\Omega_2$ and $\breve{\Omega}_2$ as Jordan states of order $2$ [@Jord1; @Jord2]. The states (\[Omega\]) can be generalized further by defining \[Omegan\] \_n(x)=[\_\*(x)]{}\^x\_[n-1]{}()d, \_n(x)=\^x\_\*()\_[n-1]{}()d, where $n=2,3,\ldots$. These states obey the relations $A_{\psi_*}\Omega_n=\breve{\Omega}_{n-1}$, $A_{\psi_*}^\dagger\breve{\Omega}_n=\Omega_{n-1}$. Consequently we find that $\Omega_n$ and $\breve{\Omega}_n$ are annihilated by differential operators of order $n+1$ constructed in terms of $A_{\psi_*}$ and $A_{\psi_*}^\dagger$. Namely, for even $n=2k$ we have $A_{\psi_*}(A_{\psi_*}^\dagger A_{\psi_*})^k\Omega_{2k}=0$, $A_{\psi_*}^\dagger(A_{\psi_*} A_{\psi_*}^\dagger)^k\breve{\Omega}_{2k}=0$, while for odd $n=2k-1$ we obtain $(A_{\psi_*}^\dagger A_{\psi_*})^k\Omega_{2k-1}=0$ and $(A_{\psi_*} A_{\psi_*}^\dagger)^k\breve{\Omega}_{2k-1}=0$, $k=1,\ldots$. In both cases of the even and odd values of $n$ we have $(H-E_*)^{n+1}\Omega_{2n}=0$, $(\breve{H}-E_*)^{n+1}\breve{\Omega}_{2n}=0$, $n=1,2,\ldots$, and $(H-E_*)^{n+1}\Omega_{2n+1}=0$, $(\breve{H}-E_*)^{n+1}\breve{\Omega}_{2n+1}=0$, $n=0,1,\ldots$. Thus, $\Omega_k$ and $\breve{\Omega}_k$ with $k=2n,\, 2n+1$ are Jordan states of order $n+1$. The described properties are illustrated by Figure \[Fig1\].
![ Action of the Darboux transformation generators. []{data-label="Fig1"}](figure1.eps)
0.3cm
Discrete flows in harmonic oscillator system and the $\mathfrak{osp}(1\vert 2)$
================================================================================
Let us consider now the QHO system with $V(x)=x^2$. Its physical bound eigenstates with eigenvalues $E_n=2n+1$ are described by normalizable wave functions $\psi_n(x)=H_n(x)e^{-x^2/2}$, $n=0,1,\ldots$, where $H_n(x)$ are the Hermite polynomials. The change $x\rightarrow ix$ transforms Hamiltonian of the QHO $$\label{HamQHO}
H=-\frac{d^2}{dx^2}+x^2$$ into $-H$, from where it follows that the wave functions $\psi^-_n(x)=\mathcal{H}_n(x)e^{x^2/2}$ with $\mathcal{H}_n(x)\equiv H_n(ix)$ correspond to non-physical (i.e. non-normalizable) eigenstates of $H$ with eigenvalues $E^-_n=-(2n+1)$. The corresponding functions $\widetilde{\psi_n}$ and $\widetilde{\psi^-_n}$ are non-physical (non-normalizable) eigenfunctions of $H$ of eigenvalues $E_n=2n+1$ and $E_n=-(2n+1)$, $n=0,1,\dots$, respectively. They will play important role in the structure and properties of the REQHO system [@Adler; @Dub; @CPRS; @FellSmi; @Sesma; @GGM; @Pupas].
The well known peculiarity of the QHO system in the context of the Darboux transformations is that the choice of $E_*=1$, $\psi_*=\psi_0=e^{-x^2/2}$ gives $\mathcal{W}=-x$, and as the factorizing operators $A_{\psi_*}$ and $A_{\psi_*}^\dagger$ we obtain the usual, up to a multiplicative factor $\sqrt{2}$, creation and annihilation operators, $$\label{a+a-}
a^-=
\frac{d}{dx}+x\,,
\qquad
a^+=(a^-)^\dagger=-\frac{d}{dx}+x\,, \qquad
[a^-,a^+]=2\,.$$ In this case if $N=a^+\,a^-$ denotes the number operator for the QHO (with the spectrum $2n$, $n=0,1,\ldots$, corresponding to a normalization chosen in (\[a+a-\])) we have $$[N,a^\pm]=\pm 2\, a^\pm\,,
\qquad H=N+1\,.
\label{hoNa}$$ As a result, the Darboux-partner system for the QHO turns out to be $\breve{H}=H+2$, which is the same quantum harmonic oscillator but just with the spectrum of physical states shifted in $+2$. Since $\psi^-_0=1/\psi_0$, another choice $E_*=-1$, $\psi_*=\psi^-_0$ changes the role of the Darboux-generating operators, $A_{\psi_*}=a^+$, $A_{\psi_*}^\dagger=a^-$, and the partner system $\breve{H}=H-2$ in this second case is again the quantum harmonic oscillator but with the physical spectrum shifted in $-2$. The action of the ladder operators on physical and associated non-physical eigenstates of the QHO and on the related Jordan and generalized Jordan states is described in Appendix. The corresponding discrete flows generated by $a^-$ and $a^+$ are depicted in Figure \[Fig2\].
![Discrete flows of the ladder operators of the QHO. Operator $a^-$ acts left and up, and $a^+$ acts right and up. []{data-label="Fig2"}](figure2.eps)
In general, because of the two-cyclic structure associated with the Darboux transformations, there appears a supersymmetry in the extended system composed from $H$ and $\breve{H}$. Since for the QHO with its equidistant spectrum the partner generated by the Darboux transformation based on the eigenfunction $\psi_*=\psi_0$ (or on $\psi_*=\psi^-_0=1/\psi_0$) is the same system but just with the spectrum shifted exactly in one energy step $\Delta E=+2$ (or, $\Delta E=-2$), the Darboux transformation generators responsible for supersymmetric structure transmute into the ladder operators for the single harmonic oscillator system. Coherently with this, instead of a usual quantum mechanical supersymmetry of the composed system, single quantum harmonic oscillator itself is characterized by the *bosonized* superconformal $\mathfrak{osp}(1\vert 2)$ structure. The $\mathfrak{osp}(1\vert 2)$ Lie superalgebra is generated here by the set of operators $$\mathcal{L}_\pm=\frac{1}{4}a^\pm\,,\qquad
J_0=\frac{1}{8}\{a^+,a^-\}=\frac{1}{4}H\,, \qquad
J_\pm=\frac{1}{4}(a^\pm)^2\,,
\label{ospGen}$$ with nontrivial (anti)commutation relations \[osp+\] {\_+,\_-}=J\_0,{\_+,\_+}=J\_+,{\_-,\_-}=J\_-, \[osp-\] \[J\_0,\_\]=\_,=-\_+,=\_-, \[so(2,1)\] \[J\_0,J\_\]=J\_,,=-2J\_0. For this superalgebra a reflection operator $\mathcal{R}=(-1)^{N/2}=e^{i\pi N/2}$ plays a role of a $\Z_2$-grading operator, i.e. $$\mathcal{R}^2=1, \qquad \{\mathcal{R},\mathcal{L}_\pm\}=0,\qquad
[\mathcal{R},J_0]=[\mathcal{R},J_\pm]=0\,.\label{grading}$$ The operator $$\label{ospCas}
\mathcal{C}_{\mathfrak{osp}(1\vert 2)}=-J_0^2+\frac{1}{2}(J_+J_-+J_-J_+)+2(\mathcal{L}_+\mathcal{L}_--
\mathcal{L}_-\mathcal{L}_+)$$ is the quadratic Casimir of the $\mathfrak{osp}(1\vert 2)$.
This corresponds to the well known spectrum-generating superalgebra of the QHO [@CromRit; @defOSP], on the physical eigenstates $\psi_n(x)$ of which the infinite-dimensional irreducible representation of the $\mathfrak{osp}(1\vert 2)$ with $\mathcal{C}_{\mathfrak{osp}(1\vert 2)}=-\frac{1}{16}$ is realized. The generators of the $\mathfrak{so}(2,1)$ Lie subalgebra (\[so(2,1)\]) act irreducibly on the eigensubspaces of $\mathcal{R}$ spanned by the states $\psi_n(x)$ with even, $n=2n_+$, and odd, $n=2n_-+1$, $n_\pm=0,1,\dots$, values of $n$, where the operator $J_0$ takes eigenvalues $n_++\frac{1}{4}$ and $n_-+\frac{3}{4}$, respectively. On both these subspaces the $\mathfrak{so}(2,1)$ Casimir operator $\mathcal{C}_{{}{\mathfrak{so}(2,1)}}=-J_0^2+\frac{1}{2}(J_+J_-+J_-J_+)$ takes the same value $\mathcal{C}_{{}{\mathfrak{so}(2,1)}}=\frac{3}{16}$.
In conclusion of this section we note that a structure with a hidden bosonized supersymmetry [@bosSUSY; @PVZ] also appears in periodic finite-gap and reflectionless quantum mechanical systems [@BosHid]. There, however, hidden supersymmetry has a different origin associated with a presence of a nontrivial Lax-Novikov integral in the quantum mechanical systems related to finite-gap and soliton solutions of the Korteweg-de Vries equation.
Discrete flows in the REQHO and deformed superconformal $\mathfrak{osp}(1\vert 2)$ structure
============================================================================================
Take now a non-physical eigenstate $\psi^-_2(x)=(2x^2+1)e^{x^2/2}$ of the harmonic oscillator as the function $\psi_*$ to generate Darboux transformation. This is a nodeless function, and the associated Darboux-transformed system will be given by a non-singular on $\R$ potential. For the sake of simplicity we denote by $A^-$ the corresponding first order operator (\[Adef\]), in which the superpotential $$\mathcal{W}(x)=\frac{d}{dx}\left(\ln \psi^-_{2}\right)=x+\frac{4x}{2x^2+1}=
x+\frac{1}{x+\frac{i}{\sqrt{2}}}+\frac{1}{x-\frac{i}{\sqrt{2}}}\,$$ has simple poles at $\infty$ and $\pm \frac{i}{\sqrt{2}}$. By construction, $A^-\psi^-_{2}=0$, and $A^+\left(\frac{1}{\psi^-_{{}_2}}\right)=0$, where $A^+=(A^-)^\dagger$. A simple computation gives \[Hoscill\] A\^+ A\^-=-+x\^2+5=N+6H\_[\_O]{}, \[HdefO\] A\^- A\^+ =-+x\^2+3+8\_[\_[O]{}]{}, where $N$ is the number operator for the QHO, $N=a^+a^-$. Here $H_{{}_O}$ represents the QHO Hamiltonian shifted by an additive constant $5$. Hamiltonian operator $\breve{H}_{{}_{O}}$ describes the REQHO system with the physical bound states $\Psi_0=\frac{1}{\psi^-_2}=A^-\widetilde{\psi^-_2}$ and $\Psi_{n+1}=A^-\psi_n$ of energies $E_0=0$ and $E_{n+1}=6+2n$, $n=0,1,\ldots$, constructed from the corresponding QHO states.
Let us introduce the third order differential operators \[dressedA\] \^-=A\^-a\^-A\^+,\^+=(\^-)\^=A\^-a\^+A\^+. These are the Darboux-dressed ladder operators of the QHO. The operator $A^+$ maps a physical or non-physical eigenstate of $\breve{H}_{{}_O}$ into an eigenstate (of the same nature) of the QHO, to which $a^-$ or $a^+$ is then applied, and the obtained in this way eigenstate of $H_{{}_O}$ is mapped by $A^-$ into another eigenstate of $\breve{H}_{{}_O}$. Operators (\[dressedA\]) satisfy the following commutation relations with the REQHO Hamiltonian, \[ospdef0\] \[,\^\]=2 \^, for which from now on we use a simplified notation $\breve{H}$. To find (\[ospdef0\]) we used the intertwining relations $A^+\breve{H}=H_{{}_O}A^+$, $A^-H_{{}_O}=\breve{H} A^-$ as well as Eq. (\[hoNa\]). Relation (\[ospdef0\]) is generalized further for \[fHApm\] \[,\^[n]{}\]=2n\^[n]{}, n=1,2,…, and $
f(\breve{H})\mathcal{A}^\pm=\mathcal{A}^\pm
f(\breve{H}\pm 2)
$ for an arbitrary polynomial function $f(\breve{H})$.
The operators $\mathcal{A}^+$ and $\mathcal{A}^-$ transform eigenstates of $\breve{H}$, which are not from their kernels, into eigenstates of $\breve{H}$ with the increased and decreased in two energy values. In this aspect they act analogously to the ladder operators $a^+$ and $a^-$ in the QHO system. There are, however, essential differences. These third order differential operators satisfy relations \[A+A-H\] \^+\^-=(-2)(-6) () ,\^-\^+=(+2)(-4)= (+2), which follow from (\[Hoscill\]), (\[HdefO\]), (\[hoNa\]) and intertwining properties of $A^\pm$, and include the degree three polynomial $\Phi(\lambda)=\lambda (\lambda -2)(\lambda-6)$. From (\[A+A-H\]) and (\[fHApm\]) we also obtain the relations which will be used in what follows: \[AAadd1\] \^[+2]{}\^-=(-2)\^+,\^-\^[+2]{}=(+2)\^+, \[AAadd2\] \^[-2]{}\^+=(+4)\^-,\^+\^[-2]{}=()\^-, \[AAadd3\] \^[+2]{}\^[-2]{}=()(-2),\^[-2]{}\^[+2]{}=(+2)(+4). Both third order polynomials $\Phi(\breve{H})$ and $\Phi(\breve{H}+2)$ in (\[A+A-H\]) include a factor $\breve{H}$. This reflects the essential peculiarity of the REQHO system: its ground-state $\Psi_0$ of zero energy is annihilated by both operators $\mathcal{A}^-$ and $\mathcal{A}^+$, \^-\_0=\^+\_0=0, because $\Psi_0=\frac{1}{\psi^-_2}$ is the kernel of $A^+$.
Consider now other properties of the lowering ladder operator $\mathcal{A}^-$. It also annihilates the first excited physical state $\Psi_1(x)=A^-\psi_0(x)$, $
\mathcal{A}^-\Psi_1=0
$ due to sequential action of the operators $A^+$ and then $a^-$. Moreover, it annihilates a non-physical eigenstate $A^-\psi^-_1$ of $\breve{H}$ by means of transforming it by the second order operator $a^-A^+$ into the kernel of $A^-$. As the kernel of the third-order differential operator $\mathcal{A}^-$ is three-dimensional, it is spanned by the three eigenstates of $\breve{H}$, \[ker-1\] \^-=[span]{} {\_0, A\^-\^-\_1, \_1}, whose eigenvalues $E=0, 2, 6$ correspond to zeros of the third degree polynomial $\Phi(\breve{H})$ in the first equality in (\[A+A-H\]). The operator $\mathcal{A}^-$ acts as a lowering ladder operator, and it is also interesting to look for kernels of powers $(\mathcal{A}^-)^n$ with $n=2,3,\ldots$. *A priori* it is clear that due to the presence of another physical state in the kernel of $\mathcal{A}^-$, which is the first exited state $\Psi_1$ in the spectrum of $\breve{H}$, and of the non-physical state $A^-\psi^-_1$ with eigenvalue $E=2$ located between the energies $E=0$ and $E=6$ of the physical zero modes of $\mathcal{A}^-$, some additional peculiarities have to appear in comparison with the case of the QHO. Note first that (\[fHApm\]) implies that $\ker\, (\mathcal{A}^-)^2$ must be invariant under the action of $\breve{H}$. On the other hand, we remark that $\ker\, \mathcal{A}^-\subset \ker\,(\mathcal{A}^-)^2$. Moreover, $\psi\in \ker\,(\mathcal{A}^-)^2$ if and only if $\mathcal{A}^-(\psi)\in \ker\,\mathcal{A}^-$, and therefore $\ker\, (\mathcal{A}^-)^2$ is generated by $\ker\, \mathcal{A}^-$ and the pre-images under $ \mathcal{A}^-$ of $\Psi_0$, $A^-\psi^-_1$ and $\Psi_1$, i.e. one finds that (\^-)\^2=[span]{} { [ker]{} \^-, A\^-, A\^-\^-\_0, \_2}. Here $\Psi_2=A^-\psi_{1}$ is a physical eigenstate at the next energy level $E=8$, and two other states $A^-\widetilde{\psi_1^-}$ and $A^-\psi^-_0$ are non-physical eigenstates of $\breve{H}$ of energies $E=2$ and $E=4$. Under the action of $\mathcal{A}^-$ the states $A^-\widetilde{\psi_1^-}$, $A^-\psi^-_0$ and $\Psi_2$ are transformed into the states $\Psi_0$, $A^-\psi^-_1$ and $\Psi_1$ from the kernel (\[ker-1\]). One can proceed in this way and identify the action of the decreasing operator on all the physical eigenstates of the system and on the associated non-physical eigenstates of the special form $\widetilde{\Psi_0}$, $A^-\widetilde{\psi_n}$, $n=0,1,\ldots$, and $A^-\psi^-_n$, $A^-\widetilde{\psi^-_n}$, $n=0,1,3,4,5,\ldots$. This action is depicted on Figure \[Fig3\]. The figure also shows that the pre-images of all the indicated physical and non-physical eigenstates of $\breve{H}$ with eigenvalues $E_n=2n$, $n\in \Z$, are contained in the same set of eigenstates with the exception of the three non-physical states $A^-\widetilde{\psi^-_0}$, $\widetilde{\Psi_0}=\widetilde{\left(\frac{1}{\psi^-_2}\right)}$, and $A^-\widetilde{\psi^-_3}$ of the eigenvalues $E=4$, $E=0$ and $E=-2$, respectively. These eigenvalues coincide with the set of zeros of the polynomial $\Phi(\breve{H}+2)$ that appears in the second relation in (\[A+A-H\]).
![Discrete flows of the ladder operators of the REQHO. Operator $\mathcal{A}^-$ acts left and up, and $\mathcal{A}^+$ acts right and up. []{data-label="Fig3"}](figure3.eps)
The preimages of the indicated states are the states $\chi^-_a$, $a=\alpha,\beta,\gamma$, having the structure \[chia-\] \^-\_a(x)=\^x\_0()\^-\_2()(\^\^-\_0 ()\^-\_a()d)d. Here $\rho^-_\alpha=\widetilde{\psi^-_0}$, $\rho^-_\beta=\Omega_{\psi^-_2}$, $\rho^-_\gamma=\widetilde{\psi^-_3}$, and $\mathcal{A}^-\chi^-_\alpha=A^-\widetilde{\psi^-_0}$, $\mathcal{A}^-\chi^-_\beta=\widetilde{\Psi_0}$, $\mathcal{A}^-\chi^-_\gamma=A^-\widetilde{\psi^-_3}$. The states $\chi^-_a$ are not eigenstates of $\breve{H}$ but satisfy relations (-6)\^-\_=A\^-\_0, (-2)(-6)\^-\_=A\^-\^-\_1, (-6)\^-\_=\_0. This implies that the following polynomials in the Hamiltonian $\breve{H}$ annihilate the states $\chi_a^-$: \[chi-Jordan\] (-6)\^2\^-\_=0,(-2)\^2(-6)\^-\_=0,\^2(-6)\^-\_=0. In correspondence with (\[chi-Jordan\]), that generalizes relations (\[Jordan\]), we call the states $\chi^-_a$ the generalized Jordan states of the REQHO since they are destroyed by the polynomials in $\breve{H}$ with different roots. Note also that \[A-4ker\] (\^-)\^3\^-\_=\_0 \^-\_(\^-)\^4, whereas $\chi^-_\beta$ and $\chi^-_\gamma$ are not annihilated by any degree of the ladder operator $\mathcal{A}^-$ and in this aspect they are similar to Jordan state $\chi^-_2$ in the QHO system, see Eq. (\[lam-\]).
The kernel of the raising ladder operator is \^+=[span]{} { A\^-[\^-\_3]{}, \_0, A\^-\^-\_0}. The action of $\mathcal{A}^+$ is illustrated by the same Figure \[Fig3\]. The corresponding shown there generalized Jordan states $\chi^+_a$, $a=\alpha,\beta,\gamma$, are given by relations similar to (\[chia-\]), \[chia+\] \^+\_a(x)=\^x\^-\_0()\^-\_2()(\^\_0 ()\^+\_a()d)d, where $\rho^+_\alpha=\Omega_{\psi^-_2}$, $\rho^+_\beta=\widetilde{\psi^-_1}$, $\rho^+_\gamma=\widetilde{\psi_0}$, and $\mathcal{A}^+\chi^+_\alpha=\widetilde{\Psi_0}$, $\mathcal{A}^+\chi^+_\beta=A^-\widetilde{\psi^-_1}$, $\mathcal{A}^+\chi^+_\gamma=A^-\widetilde{\psi_0}$. The non-physical eigenstates $\widetilde{\Psi_0}$, $A^-\widetilde{\psi^-_1}$ and $A^-\widetilde{\psi_0}$ of $\breve{H}$ appearing here have eigenvalues $E=0$, $E=2$, $E=6$, respectively, which correspond to zeros of the polynomial $\Phi(\breve{H})$ in (\[A+A-H\]). The states $\chi^+_a$ satisfy relations (+2)(-4)\^+\_=A\^-\^-\_3, (-4)\^+\_=\_0, (-4)\^+\_=A\^-\^-\_0. As a consequence, (+2)\^2(-4)\^+\_=0, \^2(-4)\^+\_=0, (-4)\^2\^+\_=0. Note that we have here \[A+4ker\] (\^+)\^3\^+\_=A\^-\^-\_0 \^+\_(\^+)\^4, cf. (\[A-4ker\]). The notations for the generalized Jordan states $\chi^\pm_a$ are chosen so that the ordering in the lower index $a=\alpha, \beta, \gamma$ in wave functions $\chi^+_a$ corresponds to the ordering in energies of the associated non-physical eigenstates $\widetilde{\Psi_0}$, $A^-\widetilde{\psi^-_1}$ and $A^-\widetilde{\psi_0}$. Generalized Jordan state $\chi^-_\alpha$ is characterized by the property (\[A-4ker\]) to be similar to the property (\[A+4ker\]) for $\chi^+_\alpha$. Under subsequent application of the ladder operator $\mathcal{A}^+$ to the state $\chi^+_\beta$ and of the operator $\mathcal{A}^-$ to $\chi^-_\beta$, these states are lifted up to the highest horizontal level shown in Figure \[Fig3\] to which physical eigenstates do belong, while the generalized Jordan states $\chi^+_\gamma$ and $\chi^-_\gamma$ are lifted up by analogous action of the corresponding ladder operator to the lower horizontal level where only non-physical eigenstates of $\breve{H}$ do appear. As in the case of the QHO system, one can proceed and construct iteratively the net of the related generalized Jordan states by finding the pre-images and images of the six basic generalized Jordan states $\chi^\pm_a$, and of the generated in such a way new states under the sequential action of the ladder operators $\mathcal{A}^\pm$.
A complete isolation of the ground-state $\Psi_0$ from other normalizable eigenstates $\Psi_n$ with $n=1,2,\ldots$, reflects here the fact that two irreducible representations of the polynomially deformed superconformal algebra $\mathfrak{osp}(1\vert 2)$ are realized on the physical bound states of the REQHO system. The operators $\breve{\mathcal{L}}_\pm=\frac{1}{4}\mathcal{A}^\pm$ can be identified as the odd generators of the superalgebra, $\{\mathcal{R}, \breve{\mathcal{L}}_\pm\}=0$, while $\breve{J}_0=\frac{1}{4}\breve{H}$ and $\breve{J}_\pm=\frac{1}{4}\mathcal{A}^{\pm 2}$ are its even generators, $[\mathcal{R},\breve{J}_0]=[\mathcal{R},\breve{J}_\pm]=0$. Here, as in the case of the QHO, the operator $\mathcal{R}=(-1)^{N/2}$ with $N=a^+ a^-$ is the $\Z_2$-grading operator, $\mathcal{R}^2=1$. The nontrivial commutation and anti-commutation relations of the deformed $\mathfrak{osp}(1\vert 2)$ superalgebra of the REQHO can be found with the help of relations (\[ospdef0\])–(\[AAadd3\]). They can be presented in the form \[osp+!\] {\_+,\_-}= \_[\_]{}(\_0) \_0,{\_+,\_+}=\_+,{\_-,\_-}=\_-,\[osp-!!\] \[\_0,\_\]=\_,=- \_[\_[J]{}]{}(\_0) \_+,=\_-\_[\_[J]{}]{}(\_0), \[so(2,1)!!\] \[\_0,\_\]=\_,,=-2 \_[\_[JJ]{}]{}(\_0) \_0. The operator-valued coefficients $$\begin{aligned}
&
\mathcal{C}_{{}_{\mathcal{L}\mathcal{L}}}(\breve{J}_0)=2\left(8 \breve{J}_0^2-10 \breve{J}_0 +1\right)\,,
\qquad
\mathcal{C}_{{}_{J\mathcal{L}}}(\breve{J}_0)=16
\left(\breve{J}_0-1\right)\left(3\breve{J}_0-1\right)\,,&\nonumber\\
&\mathcal{C}_{{}_{JJ}}(\breve{J}_0)=
16 \left(2\breve{J}_0-1\right)\left(\breve{J}_0-1\right)\left(24\breve{J}_0^2-14\breve{J}_0+7\right)
&\nonumber\end{aligned}$$ appear here instead of the unit coefficients in the $\mathfrak{osp}(1\vert 2)$ superalgebra (\[osp+\]), (\[osp-\]) and (\[so(2,1)\]) of the QHO. The ground-state $\Psi_0$ is annihilated by all the generators of the superalgebra and carries its trivial one-dimensional representation. On the higher bound states $\Psi_n$, $n=1,2,\ldots$, infinite-dimensional irreducible representation of the superalgebra is realized. The structure with two irreducible representations is reflected coherently in the discrete flows of the ladder operators depicted on Figure \[Fig3\].
In conclusion of this section we note that the case of the deformed $\mathfrak{osp}(1\vert 2)$ superalgebra of the REQHO as well as the $\mathfrak{osp}(1\vert 2)$ Lie superalgebra of the QHO system can be considered as particular cases of the algebra generated by three elements $h$, $\alpha^+$ and $\alpha^-$ subject to the relations \[h,\^\]=2\^, {\^+,\^-}=F(h)+F(h+2), where $F(h)$ is some polynomial [@defOSP]. Such an algebra is characterized by the central element =\^[+2]{}\^[-2]{}+\^+\^-(F(h)-F(h-2))-(F(h))\^2. In the case of the QHO we have a correspondence $\alpha^\pm=a^\pm$, $h=N+1$ and $F(h)=N$. The quadratic Casimir (\[ospCas\]) of the Lie superalgebra $\mathfrak{osp}(1\vert 2)$ generated by the rescaled operators $\alpha^\pm$, $\alpha^{\pm 2}$ and $h$ is nothing else as the rescaled and shifted for additive constant central element $\Xi$, $\mathcal{C}_{\mathfrak{osp}(1\vert 2)}=\frac{1}{16}(\Xi-1)$. For the REQHO system operators $\alpha^\pm$ correspond to the ladder operators $\mathcal{A}^\pm$, and we have $h=\breve{H}$, $F(h)=\Phi(\breve{H})$ with $\Phi(\breve{H})$ defined in (\[A+A-H\]). The superalgebra (\[osp+!\])–(\[so(2,1)!!\]) in this case can be considered as a polynomial deformation of the $\mathfrak{osp}(1\vert 2)$ superalgebra. Using relations (\[A+A-H\]) and (\[AAadd3\]), one can easily check that the central element $\Xi$ reduces here identically to zero.
Conclusion and outlook
======================
To conclude, we list some problems to be interesting for further investigation.
We have constructed ladder operators for the simplest version of the REQHO system by the Darboux-dressing of creation and annihilation operators of the QHO. This was done by means of the first order differential operators $A^-$ and $A^+$ which intertwine the REQHO and QHO Hamiltonians and factorize both of them. The applied procedure here is analogous to the procedure by which nontrivial Lax-Novikov integrals for reflectionless quantum systems are constructed by the Darboux-dressing of the free particle’s momentum operator [@LaxNov]. But the same REQHO system can also be constructed by means of the Darboux-Crum-Krein-Adler procedure based on the usage of several eigenstates of the QHO. In such a case the intertwiners will be higher order differential operators. One can expect that the existence of different Darboux and Darboux-Crum-Krein-Adler transformations should reveal some new interesting aspects in the construction of the ladder operators for the REQHO and related dynamical symmetries (spectrum generating algebras).
There exist other rational extensions of the QHO system. First, the analogs of the REQHO considered here can be generated by taking non-physical nodeless eigenstate $\psi_{2n}^-$ with $n>1$ as a function $\psi_*$ to generate Darboux transformations. The ladder operators for such systems can be constructed in a similar way, by the Darboux-dressing of the ladder operators of the QHO. We can generate also then a corresponding polynomially deformed bosonized $\mathfrak{osp}(1\vert 2)$ superalgebra, whose trivial and infinite-dimensional representations will be realized on physical states of the corresponding rationally extended quantum harmonic oscillator. It is interesting if there will be any essential difference in the structure of the discrete flows generated by the ladder operators in such systems in comparison with the REQHO system considered here. The construction of the ladder operators by taking into account the existence of different Darboux-Crum-Krein-Adler transformations to generate such systems should also reveal a dependence on the order of the polynomial that presents in the structure of the generating function $\psi^-_{2n}(x)$ and on a size of the gap between the isolated ground-state and the infinite tower of equidistant bound states.
A more complicated and a more rich picture from the point of view of the ladder operators and related symmetries can be expected in rationally extended quantum harmonic oscillator systems with number $l>1$ of isolated bound states in the spectrum. There, a priori two essentially different cases should be distinguished. One case is when $l>1$ bound states will be separated from the infinite tower of equidistant bound states without any additional gaps between those $l$ states. Another, more general case is when isolated states include some additional gaps between themselves.
It is known that Jordan states appear in confluent Darboux-Crum transformations [@Jord2]. They, particularly, were employed recently for the design of the PT-symmetric optical systems with invisible periodicity defects as well as completely invisible reflectionless PT-symmetric systems [@Jord1]. It would be interesting to look for possible physical applications of the generalized Jordan states considered here.
The considered REQHO system as well as its generalizations seem also to be interesting from the point of view of possible physical applications since unlike other known deformations of the QHO, e.g. related to the minimal length uncertainty relation [@Kempf; @Rossi], they provide a very specific change of the spectrum. Namely, they add effectively a finite number of bound states in the lower part of the QHO spectrum, separated by an additional (*adjustable*) gap, without disturbing the equidistant character of the rest of the infinite tower of the discrete levels. In this aspect they are very similar, as it has been noted above in another but related context, to the quantum reflectionless systems which add a finite number of discrete bound states into the spectrum of the free particle. Such reflectionless systems are directly related to the soliton solutions to the Korteweg-de Vries and modified Kortweg-de Vries equations, and find a lot of interesting applications in very diverse areas of physics including QCD, cosmology, solid states physics, the physics of polymers, plasma physics, and quantum optics, just to mention a few [^2]. Further results related to the ladder operators in rationally extended harmonic oscillator systems, which exploite the indicated similarity, will be presented elsewhere [@CarPlyprep].
0.4cm
[ ]{} 0.4cm
MSP thanks Ya. Ispolatov for discussions. JFC and MSP acknowledge support from research projects FONDECYT 1130017 (Chile), Proyecto Basal USA1555 (Chile), MTM2015-64166-C2-1 (MINECO, Madrid) and DGA E24/1 (DGA, Zaragoza). MSP is grateful for the warm hospitality at Zaragoza University. JFC thanks for the kind hospitality at Universidad de Santiago de Chile.
Appendix: Discrete chains of the states of the QHO
==================================================
We describe here the action of the ladder operators $a^+$ and $a^-$ on non-physical eigenstates $\widetilde{\psi_n}$ and $\widetilde{\psi_n^-}$ of the QHO and the construction of the associated Jordan and generalized Jordan states.
Making use of the identities $H'_n=2nH_{n-1}$ and $H_n=2xH_{n-1}-H'_{n-1}$ for Hermite polynomials, we obtain the relations \[App1\] \^x d=-\^xd()= -+\^x d, from where we find that \[a-tilde\] a\^-=,n=1,2,…. Application to both sides of this equality of the operator $a^+$ gives \[a+tilde\] a\^+=,n=0,1,…. Changing $x\rightarrow ix$ in (\[a-tilde\]) and (\[a+tilde\]), and taking into account that $a^-\rightarrow ia^+$, we also obtain \[a+tilde-\] a\^+=,n=1,2,…,a\^-=,n=0,1,…. In correspondence with (\[Apsi\*\]) and (\[A+psi\*\]), a\^-==\^-\_0,a\^+=[\_0]{}. We also have \[lam-\] \^-\_2(x)=\_0(x)\^x\^-\_0()()d,a\^-\^-\_2(x)=\^+\_1. This is a Jordan state which obeys the relations $
(H-1)\chi^-_2=\psi_0,
$ $
(H-1)^2\chi^-_2=0,
$ where $H=a^+a^-+1$. Analogously, \[lam+\] \^+\_2(x)=\^x \_0()() d, a\^+\^+\_2=\^-\_1, and $(H+1)\chi^+_2=\psi^-_0$, $
(H+1)^2\chi^+_2=0.
$
Proceeding from the states $\chi^+_2$ and $\chi^-_2$, one can construct an infinite net of related to them Jordan and generalized Jordan states. First, as analogs of $\Omega_n$ and $\breve{\Omega}_n$ defined in (\[Omegan\]) we have the states $\chi^-_n$ and $\chi^+_n$, \[chin-Def\] \^-\_n(x)=\_0(x)\^x\^-\_0()\^+\_[n-1]{}()d,\^+\_n(x)=\_0\^-(x)\^x\_0()\^-\_[n-1]{}()d, where the case $n=1$ is also included by assuming $\chi^-_0\equiv\psi_0$ and $\chi^+_0\equiv \psi^-_0$. These are the higher order Jordan states (\[Omegan\]) generated on the basis of $\psi_*=\psi_0$. They satisfy relations $a^-\chi^-_n=\chi^+_{n-1}$, $a^+\chi^+_n=\chi^-_{n-1}$, and, consequently, $a^-(a^+a^-)^n\chi^-_{2n}=0$, $a^+(a^-a^+)^n\chi^+_{2n}=0$. Therefore, $(H-1)^{n+1}\chi^-_{k}=0$ and $(H+1)^{n+1}\chi^+_{k}=0$ for $k=2n$, $2n+1$.
One can define $\sigma^-_n=a^+\chi^-_n$, $\sigma^+_n=a^-\chi^+_n$, $n=2,\ldots$. These are Jordan states obeying the relations $(H-3)^{n}\sigma^-_{2n}=\psi_1$, $(H-3)^{n}\sigma^-_{2n+1}=\widetilde{\psi_1}$, $(H+3)^n\sigma^+_{2n}=\psi^-_1$, $(H+3)^n\sigma^+_{2n+1}=\widetilde{\psi^-_1}$, and, therefore, $(H-3)^{n+1}\sigma^-_k=0$, $(H+3)^{n+1}\sigma^+_k=0$ for $k=2n$, $2n+1$.
In the same vein the family of Jordan states $\tau^-_n=a^+\sigma^-_n$ and $\tau^+_n=a^-\sigma^+_n$, $n=2,\ldots$, can be defined. They satisfy the relations $(H-5)^{n}\tau^-_{2n}=\psi_2$, $(H-5)^{n}\tau^-_{2n+1}=\widetilde{\psi_2}$, $(H+5)^n\tau^+_{2n}=\psi^-_2$, $(H+5)^n\tau^+_{2n+1}=\widetilde{\psi^-_2}$, and $(H-5)^{n+1}\tau^-_k=0$, $(H+5)^{n+1}\tau^+_k=0$ for $k=2n$, $2n+1$. These discrete flows can be further continued ‘horizontally’.
On the other hand, the states defined via $\gamma^-_n=a^+\sigma^-_n$, $\gamma^+_n=a^-\sigma^+_n$, $n=2,\ldots$, are reduced to linear combinations of the already introduced Jordan states. Namely, $\gamma^-_{n}$ is a linear combination of $\chi^-_n$ and $\chi^-_{n-2}$, and $\gamma^+_{n}$ is a linear combination of $\chi^+_n$ and $\chi^+_{n-2}$.
Consider the states $\lambda^\pm_n$ given by means of relations $a^-\lambda^-_n=\chi^-_{n-1}$, $a^+\lambda^+_n=\chi^+_{n-1}$, $n=3,\ldots$. The states $\lambda^-_n$ can be presented in the form similar to that for $\chi^-_n$ in (\[chin-Def\]) but with $\chi^+_{n-1}$ in the integrand changed for $\chi^-_{n-1}$. Analogously, $\lambda^+_n$ are presented similarly to $\chi^+_n$ in (\[chin-Def\]) with $\chi^-_{n-1}$ in the integrand changed for $\chi^+_{n-1}$. For these states we have relations $(a^+a^-)^na^-\lambda^-_{2n}=0$, $(a^-a^+)^na^+\lambda^+_{2n}=0$, $a^-(a^+a^-)^na^-\lambda^-_{2n+1}=0$, $a^+(a^-a^+)^na^+\lambda^+_{2n+1}=0$. As a consequence, $(H-1)\lambda^-_n=\chi^+_{n-2}$, $(H+1)\lambda^+_n=\chi^-_{n-2}$. Therefore, these are generalized Jordan states which obey the relations $(H-1)(H-3)^n\lambda^-_k=0$, $(H+1)(H+3)^n\lambda^+_k=0$ with $k=2n$, $2n-1$.
Similarly, generalized Jordan states $\mu^\pm_n(x)$ can be defined proceeding from the states $\lambda^\pm_n$ via the relations $a^+\mu^-_n=\lambda^-_{n+1}$, $a^-\mu^+_n=\lambda^+_{n+1}$, $n=2,\ldots$. Then \^-\_n(x)=\^-\_0(x)\^x\_0()\^-\_[n+1]{}()d,\^+\_n(x)=\_0(x)\^x\^-\_0()\^+\_[n+1]{}()d. For these states we have $\chi^-_n=(H+1)\mu^-_n$, $\chi^+_n=(H-1)\mu^+_n$. They are generalized Jordan states obeying the relations $(H+1)(H-1)^{n+1}\mu^-_k=0$, $(H-1)(H+1)^{n+1}\mu^+_k=0$ with $k=2n$, $2n+1$.
The described procedure of the construction of the Jordan and generalized Jordan states can be continued further in the obvious way. The discrete flows corresponding to the action of the ladder operators on the physical and associated non-physical eigenstates of the QHO Hamiltonian and associated Jordan and generalized Jordan states are illustrated by Figure \[Fig2\].
[99]{}
G. Darboux, *“Sur une proposition relative aux équations linéaires,"* C. R. Acad. Sci Paris [**94**]{} (1882) 1456.
M. M. Crum, *“Associated Sturm-Liouville systems,"* [ Quart. J. Math. Oxford [**6**]{} (1955) 121](http://qjmath.oxfordjournals.org/content/6/1/121.full.pdf+html).
M. G. Krein, *“On a continuous analogue of a Christoffel formula from the theory of orthogonal polynomials,"* Dokl. Akad. Nauk SSSR [**113**]{} (1957) 970.
V. E. Adler, *“A modification of Crum’s method,"* [ Theor. Math. Phys. [**101**]{} (1994) 1381](http://link.springer.com/article/10.1007%2FBF01035458).
V. B. Matveev and M. A. Salle, *Darboux Transformations and Solitons* (Springer, Berlin, 1991).
E. Schrödinger, *“A method of determining quantum-mechanical eigenvalues and eigenfunctions,”* Proc. Roy. Irish Acad. (Sect. A) [**46**]{} (1940) 9. L. Infeld and T. E. Hull, *“The factorization method,”* [ Rev. Mod. Phys. [**23**]{} (1951) 21](http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.23.21). J. F. Cariñena and A. Ramos, *“Riccati equation, Factorization Method and Shape Invariance,"* [ Rev. Math. Phys. [**12**]{} (2000) 1279](http://www.worldscientific.com/doi/abs/10.1142/S0129055X00000502) [](http://arxiv.org/abs/math-ph/9910020).
B. Mielnik and O. Rosas-Ortiz, *“Factorization: little or great algorithm?,”* [ J. Phys. A [**37**]{} (2004) 10007](http://iopscience.iop.org/article/10.1088/0305-4470/37/43/001/meta). E. Witten, *“Dynamical Breaking of Supersymmetry,”* [Nucl. Phys. B [**188**]{} (1981) 513](http://www.sciencedirect.com/science/article/pii/0550321381900067). E. Witten, *“Supersymmetry and Morse theory,”* [ J. Diff. Geom. [**17**]{} (1982) 661](http://projecteuclid.org/euclid.jdg/1214437492). F. Cooper, A. Khare and U. Sukhatme, *“Supersymmetry and quantum mechanics,”* [ Phys. Rept. [**251**]{} (1995) 267](http://www.sciencedirect.com/science/article/pii/037015739400080M) [](http://arxiv.org/abs/hep-th/9405029).
J. F. Cariñena, D. J. Fernández and A. Ramos, *“Group theoretical approach to the intertwined Hamiltonians,”* [ Ann. Phys. [**292**]{} (2001) 42](http://www.sciencedirect.com/science/article/pii/S0003491601961792) [](http://arxiv.org/abs/math-ph/0311029). J. F. Cariñena and A. Ramos, *“Generalized Bäcklund-Darboux transformations in one-dimensional quantum mechanics,”* [ Int. J. Geom. Methods Mod. Phys. [**05**]{} (2008) 605](http://www.worldscientific.com/doi/abs/10.1142/S0219887808002989).
A. Arancibia and M. S. Plyushchay, *“Chiral asymmetry in propagation of soliton defects in crystalline backgrounds,”* [ Phys. Rev. D [**92**]{} (2015) 05009](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.92.105009) [](http://arxiv.org/abs/1507.07060). A. P. Veselov and A. B. Shabat, [*“Dressing chains and the spectral theory of the Schrödinger operator,"*]{} [ Funct. Anal. Appl. 27 (1993) 81](http://link.springer.com/article/10.1007%2FBF01085979).
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, *Theory of Solitons* (Plenum, New York, 1984).
F. Gesztesy and H. Holden, *Soliton Equations and their Algebro-Geometric Solutions*, (Cambridge Univ. Press, 2003).
V. E. Adler, [*“Nonlinear chains and Painlevé equations,"*]{} [ Physica D 73 (1994) 335](http://www.sciencedirect.com/science/article/pii/016727899490104X).
R. Willox and J. Hietarinta, [*“Painlevé equations from Darboux chains: I. PIII - PV,"*]{} [ J. Phys. A [**36**]{} (2003) 10615](http://iopscience.iop.org/article/10.1088/0305-4470/36/42/014).
J. J. Duistermaat and F. A. Grünbaum, [*“Differential equations in the spectral parameter,"*]{} [ Comm. Math. Phys. [**103**]{} (1986) 177](http://link.springer.com/article/10.1007/BF01206937).
A. Oblomkov, [*“Monodromy-free Schrödinger operators with quadratically increasing potentials,"*]{} [ Theor. Math. Phys. [**121**]{} (1999) 1574](http://link.springer.com/article/10.1007%2FBF02557204).
S. Yu. Dubov, V. M. Eleonskii and N. E. Kulagin, *“Equidistant spectra of anharmonic oscillators,"* [ Zh. Eksp. Teor. Fiz. 102 (1992) 814 \[Sov. Phys. JETP 75, 446 (1992)\]](http://www.jetp.ac.ru/cgi-bin/e/index/r/102/3/p814?a=list); [ Chaos, [**4**]{} (1994) 47](http://scitation.aip.org/content/aip/journal/chaos/4/1/10.1063/1.166056).
J. F. Cariñena, A. M. Perelomov, M. F. Rañada and M. Santander, *“A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator,"* [ J. Phys. A: Math. Theor. [**41**]{} (2008) 085301](http://iopscience.iop.org/article/10.1088/1751-8113/41/8/085301/meta) [](http://arxiv.org/abs/0711.4899).
J. M. Fellows and R. A. Smith, *“Factorization solution of a family of quantum nonlinear oscillators,"* [ J. Phys. A [**42**]{} (2009) 335303](http://iopscience.iop.org/article/10.1088/1751-8113/42/33/335303/meta).
D. Gómez-Ullate, N. Kamran, and R. Milson, *“An extended class of orthogonal polynomials defined by a Sturm-Liouville problem,"* [ J. Math. Anal. Appl. [**359**]{} (2009) 352](http://www.sciencedirect.com/science/article/pii/S0022247X09004569) [](https://arxiv.org/abs/0807.3939); *“An extension of Bochner’s problem: exceptional invariant subspaces,"* [ J. Approx. Theory [**162**]{} (2010) 987](http://www.sciencedirect.com/science/article/pii/S0021904509001853) [](https://arxiv.org/abs/0805.3376).
S. Odake and R. Sasaki, *“Infinitely many shape invariant potentials and new orthogonal polynomials,* [ Phys. Lett. B [**679**]{} (2009) 414](http://www.sciencedirect.com/science/article/pii/S0370269309009186) [](https://arxiv.org/abs/0906.0142); *“Another set of infinitely many exceptional (X) Laguerre polynomials,* [ Phys. Lett. B [**684**]{} (2010) 173](http://www.sciencedirect.com/science/article/pii/S0370269310000158) [](https://arxiv.org/abs/0911.3442). R. Sasaki, S. Tsujimoto and A. Zhedanov, *“Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations,"* [ J. Phys. A [**43**]{} (2010) 315204](http://iopscience.iop.org/article/10.1088/1751-8113/43/31/315204/meta) [](https://arxiv.org/abs/1004.4711).
C. Quesne, *“Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry,"* [ J. Phys. A [**41**]{} (2008) 392001](http://iopscience.iop.org/article/10.1088/1751-8113/41/39/392001/meta) [](http://arxiv.org/abs/0807.4087); *“Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics,"* [ SIGMA [**5**]{} (2009) 084](http://www.emis.de/journals/SIGMA/2009/084/) [](https://arxiv.org/abs/0906.2331).
Y. Grandati, *“Solvable rational extensions of the isotonic oscillator,"* [ Ann. Phys. [**326**]{} (2011) 2074](http://www.sciencedirect.com/science/article/pii/S000349161100039X) [](https://arxiv.org/abs/1101.0055).
J. Sesma, *“The generalized quantum isotonic oscillator,"* [ J. Phys. A [**43**]{} (2010) 185303](http://iopscience.iop.org/article/10.1088/1751-8113/43/18/185303/meta) [](http://arxiv.org/abs/1005.1227).
D. Gómez-Ullate, Y. Grandati, and R. Milson, *“Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials,"* [ J. Phys. A [**47**]{} (2014) 015203](http://iopscience.iop.org/article/10.1088/1751-8113/47/1/015203/meta) [](http://arxiv.org/abs/1306.5143).
A. M. Pupasov-Maksimov, *“Propagators of isochronous an-harmonic oscillators and Mehler formula for the exceptional Hermite polynomials,"* [ Annals of Physics [**363**]{} (2015) 122](http://www.sciencedirect.com.ezproxy.usach.cl/science/article/pii/S0003491615003565) [](https://arxiv.org/abs/1502.01778).
M. Asorey, J. F. Carinena, G. Marmo and A. Perelomov, *“Isoperiodic classical systems and their quantum counterparts,”* [ Annals Phys. [**322**]{} (2007) 1444](http://www.sciencedirect.com/science/article/pii/S0003491606001461) [](https://arxiv.org/abs/0707.4465). J. F. Cariñena, A. M. Perelomov and M. F. Rañada, *“Isochronous classical systems and quantum systems with equally spaced spectra,”*, [In:]{} [*Particles and Fields: Classical and Quantum*]{}, [Journal of Physics: Conference Series [**87**]{} (2007) 012007](http://iopscience.iop.org/article/10.1088/1742-6596/87/1/012007).
M. de Crombrugghe and V. Rittenberg, *“Supersymmetric quantum mechanics,* [Annals of Physics [**151**]{} (1983) 99](http://www.sciencedirect.com/science/article/pii/0003491683903160).
J. Van der Jeugt and R. Jagannathan, *“Polynomial deformations of osp(1/2) and generalized parabosons,”* [ J. Math. Phys. [**36**]{} (1995) 4507](http://scitation.aip.org/content/aip/journal/jmp/36/8/10.1063/1.530904) [](http://arxiv.org/abs/hep-th/9410145).
S. Krivonos and O. Lechtenfeld, *“SU(2) reduction in $\mathcal{N}=4$ supersymmetric mechanics,"* [ Phys. Rev. D [**80**]{} (2009) 045019](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.80.045019) [](https://arxiv.org/abs/0906.2469).
S. Fedoruk and J. Lukierski, *“Algebraic structure of Galilean superconformal symmetries,"* [Phys. Rev. D [**84**]{} (2011) 065002](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.84.065002) [](http://arxiv.org/abs/1105.3444).
M. S. Plyushchay and A. Wipf, *“Particle in a self-dual dyon background: hidden free nature, and exotic superconformal symmetry,”* [ Phys. Rev. D [**89**]{} (2014) 045017](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.89.045017) [](http://arxiv.org/abs/1311.2195). G. F. de Téramond, H. G. Dosch and S. J. Brodsky, *“Baryon spectrum from superconformal quantum mechanics and its light-front holographic embedding,"* [Phys. Rev. D [**91**]{} (2015) 045040](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.91.045040) [](http://arxiv.org/abs/1411.5243). E. Ivanov, S. Sidorov and F. Toppan, *“Superconformal mechanics in $SU(2|1)$ superspace,"* [Phys. Rev. D [**91**]{} (2015) 085032 ](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.91.085032) [](https://arxiv.org/abs/1501.05622).
F. Correa, V. Jakubsky and M. S. Plyushchay, *“$PT$-symmetric invisible defects and confluent Darboux-Crum transformations,”* [ Phys. Rev. A [**92**]{} (2015) 023839](http://journals.aps.org/pra/abstract/10.1103/PhysRevA.92.023839) [](http://arxiv.org/abs/1506.00991). A. Schulze-Halberg, *“Wronskian representation for confluent supersymmetric transformation chains of arbitrary order,”* [ Eur. Phys. J. Plus [**128**]{} (2013) 68](http://link.springer.com/article/10.1140/epjp/i2013-13068-2). M. S. Plyushchay, *“Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions,”* [ Annals Phys. [**245**]{} (1996) 339](http://www.sciencedirect.com/science/article/pii/S0003491696900123) [](http://arxiv.org/abs/hep-th/9601116); *“Hidden nonlinear supersymmetries in pure parabosonic systems,”* [ Int. J. Mod. Phys. A [**15**]{} (2000) 3679](http://www.worldscientific.com/doi/abs/10.1142/S0217751X00001981) [](http://arxiv.org/abs/hep-th/9903130); V. Jakubsky, L. M. Nieto and M. S. Plyushchay, *“The origin of the hidden supersymmetry,”* [ Phys. Lett. B [**692**]{} (2010) 51](http://www.sciencedirect.com/science/article/pii/S0370269310008270) [](http://arxiv.org/abs/1004.5489). S. Post, L. Vinet and A. Zhedanov, *“Supersymmetric Quantum Mechanics with Reflections,”* [J. Phys. A [**44**]{} (2011) 435301](http://iopscience.iop.org/article/10.1088/1751-8113/44/43/435301/meta) [](http://arxiv.org/abs/1107.5844); V. X. Genest, L. Vinet, Guo-Fu Yu, and A. Zhedanov, *“Supersymmetry of the quantum rotor,"* [](http://arxiv.org/abs/1607.06967).
F. Correa and M. S. Plyushchay, *“Hidden supersymmetry in quantum bosonic systems,”* [ Annals Phys. [**322**]{} (2007) 2493](http://www.sciencedirect.com/science/article/pii/S0003491606002831) [](https://arxiv.org/abs/hep-th/0605104); F. Correa, L. M. Nieto and M. S. Plyushchay, *“Hidden nonlinear supersymmetry of finite-gap Lamé equation,”* [ Phys. Lett. B [**644**]{} (2007) 94](http://www.sciencedirect.com/science/article/pii/S0370269306014274) [](https://arxiv.org/abs/hep-th/0608096); F. Correa, V. Jakubsky, L. M. Nieto and M. S. Plyushchay, *“Self-isospectrality, special supersymmetry, and their effect on the band structure,”* [Phys. Rev. Lett. [**101**]{} (2008) 030403](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.030403) [](http://arxiv.org/abs/0801.1671).
A. Arancibia, J. Mateos Guilarte and M. S. Plyushchay, *“Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems,”* [ Phys. Rev. D [**87**]{} (2013) 045009](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.87.045009) [](http://arxiv.org/abs/1210.3666). A. Kempf, *“Uncertainty relation in quantum mechanics with quantum group symmetry,”* [J. Math. Phys. [**35**]{} (1994) 4483](http://scitation.aip.org/content/aip/journal/jmp/35/9/10.1063/1.530798) [](http://arxiv.org/abs/hep-th/9311147); A. Kempf, G. Mangano and R. B. Mann, *“Hilbert space representation of the minimal length uncertainty relation,"* [ Phys. Rev. D [**52**]{} (1995) 1108](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.52.1108) [](http://arxiv.org/abs/hep-th/9412167).
M. A. C. Rossi, T. Giani and M. G. A. Paris, *“Probing deformed quantum commutators,”* [Phys. Rev. D [**94**]{} (2016) 024014](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.024014) [](https://arxiv.org/abs/1606.03420). A. Arancibia and M. S. Plyushchay, *“Transmutations of supersymmetry through soliton scattering and self-consistent condensates,”* [Phys. Rev. D [**90**]{} (2014) 025008](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.025008) [](https://arxiv.org/abs/1401.6709). J. F. Cariñena and M. S. Plyushchay, in preparation.
[^1]: For some recent investigations on superconformal quantum mechanical symmetry and its applications see [@KriLech; @FedLuk; @PlyWi; @TDB; @IST].
[^2]: See, e.g., [@ArPl; @LaxNov; @APtrans] and references therein.
|
---
abstract: 'The hydrogen transfer reaction catalysed by soybean lipoxygenase (SLO) has been the focus of intense study following observations of a high kinetic isotope effect (KIE). Today high KIEs are generally thought to indicate departure from classical rate theory and are seen as a strong signature of tunnelling of the transferring particle, hydrogen or one of its isotopes, through the reaction energy barrier. In this paper we build a qualitative quantum rate model with few free parameters that describes the dynamics of the transferring particle when it is exposed to energetic potentials exerted by the donor and the acceptor. The enzyme’s impact on the dynamics is modelled by an additional energetic term, an oscillatory contribution known as “gating”. By varying two key parameters, the gating frequency and the mean donor-acceptor separation, the model is able to reproduce well the KIE data for SLO wild-type and a variety of SLO mutants over the experimentally accessible temperature range. While SLO-specific constants have been considered here, it is possible to adapt these for other enzymes.'
author:
- 'S. Jevtic'
- 'J. Anders'
bibliography:
- 'references.bib'
title: A qualitative quantum rate model for hydrogen transfer in soybean lipoxygenase
---
\[sec\_Intro\]Introduction
==========================
Enzymes play a central role in biological functions and are indispensable in many industrial processes [@SBT02]. As such, there is a pressing need to develop theoretical models that fully specify the method by which enzymes catalyse reactions. An enzyme creates an alternative path for a reaction to occur and can greatly speed up the reaction compared to the uncatalysed case; speed-up factors of up to $10^{26}$ have been observed [@ELW12]. As well as being able to simulate the data of known enzymes, it is crucial to find a model that can predict the action of a potential catalyst. This will enable the engineering of new enzymes for reactions that are currently too slow [@HB03].
The standard method for modelling enzyme reaction rates is based on transition state theory (TST) [@ES39]. In TST, the reactants begin in a local minimum of a potential $V(x)$, proceed along a reaction coordinate $x$, and at $x=x_b$ they encounter an energetic barrier of height $V(x_b)$. Thermal excitations from the environment enable the formation of the transition state at the top of the barrier, and crossing the barrier leads to the products being created, see Fig. \[E\_barrier\]. In this picture, the catalyst lowers the energy barrier increasing the likelihood for the transition state to be formed and the transferring particle to hop over the barrier. An alternative transfer mechanism is also possible: the transferring particle may tunnel [@ETunnel] through the barrier instead of hopping over it. This has been discussed in a number of enzymatic systems that catalyse hydrogen transfer and have high kinetic isotope effects (KIEs), such as soybean lipoxygenase (SLO) [@SHS_04; @SHS_07; @SHH_S_2005; @ESS_H_2010; @SH_S_2015; @SLO_DM; @SH_S_2016; @KRK02; @Warshel; @Cha_Klinman_1989; @debate; @debate2; @PB04; @Scrutton_book; @BGM10; @Scrutton_book; @Layfield_SHS; @SS_2008].
In this manuscript, we present a rate model that aims to capture qualitatively the mechanism of hydrogen transfer in enzyme-catalysed reactions. The purpose of the model is to be able to predict the temperature dependence of the KIEs for various enzyme mutants, which are parameterised by a few key parameters. To achieve this, we build a rate model that treats the dynamics of the transferring particle quantum mechanically and also allow the enzyme to sample a range of donor-acceptor configurations through a classical vibrating motion, known as *gating* [@BB92], which arises due to thermal excitations from the environment at temperature $T$.
For concreteness, here we focus on hydrogen and deuterium transfer catalysed by the enzyme SLO, and its mutants. We choose two independent quantities to parametrise the rate of transfer, the average donor-acceptor separation, $R_e$, and the gating frequency, $\Omega$. Our calculated KIEs show good agreement with the experimental KIE curves for wild-type (WT) SLO and four different SLO mutants reported in Refs \[, \] over the measured temperature range of $5-50^\circ$C. The parameter choices provide insight into the physical features that affect the reaction rates and KIEs, such as the donor-acceptor configurations of SLO mutants in comparison to WT. The model also allows us to discuss the magnitude of the tunnelling contribution.
The paper is organised as follows. In section \[sec\_SLOdata\], we introduce the enzyme SLO, summarise pertinent experimental results, and briefly discuss a selection of existing rate models. We present our new qualitative quantum rate model in section \[sec3\] and discuss the key results in section \[sec4\]. In section \[sec5\] we comment on whether tunnelling plays a significant role in SLO enzyme catalysis. Finally, in section \[sec7\], we discuss insights arising from the comparison between the proposed model and the experimental data, and suggest future directions.
![Sketch of the potential energy profile $V(x)$ experienced by a particle transferring from a donor (reactants) to an acceptor (products). The particle, initially bonded to the donor, sits in a potential minimum at position $x_e$. The reaction proceeds along a reaction coordinate $x$ and at position $x_b$ a barrier of height $V(x_b)$ must be overcome if the particle is to break free of the donor and form a new bond with the acceptor. The particle may be thermally excited and hop over the barrier if the transfer is viewed classically, or it may tunnel through the barrier if it is governed by quantum dynamics. Enzymes are believed to catalyse the reaction by lowering the barrier height. []{data-label="E_barrier"}](energybarrier4.pdf){width="3.2in"}
\[sec\_SLOdata\]The enzyme SLO
==============================
Soybean lipoxygenase (SLO) is studied because of its similarities to the mammalian lipoxygenases. These are key components in the production of fatty acids which are required for the functioning of cells [@PA96; @PA97]. Abnormal lipoxygenase activity has been linked with cancer formation, hence these enzymes play an important role in human health and are of particular interest to the pharmaceutical industry [@YF04; @RC98; @MM98]. SLO catalyses the production of fatty acid hydroperoxides and the substrate is linocleic acid [@KRK02]. The reaction consists of a sequence of rapid steps, however, the rate-limiting step is the hydrogen transfer from a carbon atom on linocleic acid to an oxygen molecule. This is the step that is modelled in the quantum dynamical rate model developed in section \[sec3\].
Kinetic isotope effect (KIE)
----------------------------
The first clear deviation from standard enzyme kinetics was reported in the kinetic isotope effect (KIE) of soybean lipoxygenase more than 20 years ago [@GWK94]. The KIE is an experimental tool for testing the mass-dependence of a reaction rate. It is the ratio between two rates: the rate of hydrogen transfer, $k_H$, and the rate of transfer of one of its isotopes, e.g. deuterium, $k_D$. (When the transferring particle is substituted by one of its isotopes this is called the primary KIE, and this is the situation we consider here. Secondary KIEs refer to rate changes that occur when isotopically substituting a non-transferring particle in the reactant.) The isotope substitution does not affect the electrostatic potentials, however, the mass change affects the zero point energy. At 30$^{\circ}$C this can reduce the deuterium rate by a factor of 1.4 - 3 per normal mode (e.g. squeezing or bending modes) leading to increased KIEs. Experimental SLO rates, shown in Fig. \[figKRK02\], exhibit a huge KIE = $k_H / k_D = 81$ at 30$^{\circ}$C.
Mutations
---------
Aside from deuterating the transferring particle, it is possible to *mutate* the enzyme by substituting large clusters of atoms (“residues") with smaller ones [@KRK02; @MTK08; @SLO_DM]. This is carefully done so that the enzyme catalyses the same reaction but the rate is altered. In \[\], several “bulky” residues (leucine (Leu) 546 and 754, and isoleucine (Ile) 553) near the active site of SLO are replaced by the smaller amino acid alanine (Ala). Such mutations modify the active site and so hydrogen will be exposed to a different potential energy barrier. For the mutations Leu$^{546} \rightarrow$ Ala (mutation M1) or Leu$^{754} \rightarrow$ Ala (mutation M2), which are both close to the active site, both rates $k_H$ and $k_D$ significantly drop (about 3 orders of magnitude) in comparison to WT SLO, see Fig. \[figKRK02\]. These findings indicate that wild-type SLO is configured optimally to catalyse this reaction. The KIEs of mutants M1 and M2 are larger than WT, 109 and 112 respectively at 30$^\circ$C, and show stronger variation with temperature. In contrast, the more distant mutation Ile$^{553}\rightarrow$ Ala (mutation M3) barely changes the rate $k_H$, see Fig. \[figKRK02\], but the M3 KIE is more temperature dependent than the KIEs of WT and mutations M1 and M2. These observations were confirmed once more in \[\]. Recently, kinetic data for the SLO *double mutant* (DM) have been obtained [@SLO_DM]. The double mutation makes both replacements M1 and M2 at the same time in SLO. Using two independent experimental methods, hugely inflated KIEs were observed: a KIE of $537\pm 55$ at $35^\circ$C was measured using single-turnover kinetics and a KIE of $729\pm 26$ at $30^\circ$C was measured using steady-state measurements.
Advanced models of enzyme catalysis
-----------------------------------
It is widely accepted that enzymes with high KIEs, such as SLO WT and its mutants, require quantum corrections, such as the inclusion of thermally activated tunnelling [@CON77; @Bell_Tunnel; @Scrutton_book]. Other quantum corrections include making the Wentzel-Kramers-Brillouin (WKB) approximation that treats tunnelling semiclassically [@BB92]. However, for transfer distances of $1~\AA$ and activation energies of $10^{-20}$J in SLO, this approximation is not fully justified [^1]. Nevertheless, semiclassical rate theories have been successful in simulating a variety of non-classical enzymes [@BGM10], including SLO [@Pollak].
An array of quantum rate models has been developed that account for the increased complexity of the rate-determining step in SLO [@KU99; @KRK02; @Warshel; @Mincer_Schwartz; @SS04; @IS2008; @IS2010] for the WT and mutants M1, M2 and M3. A successful framework employing Fermi’s golden rule is presented by Hammes-Schiffer and co-workers [@SHS_AS_2000; @SHS_04; @SHH_S_2005; @SHS_07; @SS_2008; @ESS_H_2010; @AS_SHS_2014; @SLO_DM; @SH_S_2015; @SH_S_2016]. This rate model provides a good fit to the wild-type SLO KIE data[@KRK02] as well as predicting the KIE magnitude and temperature dependence of the mutants M1, M2, M3 (and its variants [@MTK08]) and the double mutant [@SLO_DM]. This approach incorporates hydrogen transfer into Marcus theory through “proton-coupled electron transfer” and combines this with *gating*. Gating is the sampling of different configurations of the active site including close confinement where quantum tunnelling is possible [@BB92]. This sampling is caused by the enzyme’s thermal vibrating motion which reorganises the active site and modulates the barrier.
An intensive computational study was carried out in \[\] using ensemble averaged variational TST with multidimensional tunnelling [@PGT06] to calculate the SLO rate and KIE. The authors reported that hydrogen tunnelling accounted for over 99% of the transfer mechanism in the wild type setting. The KIEs they obtain ($\sim10$) are far lower than the observed value of 81. It is believed that this is due to an underestimation of the barrier height which comes from their computed hydrogen potential energy surface. Manually increasing the barrier height (and width) rapidly leads to an increased KIE.
The above quantum rate models have been shown to match the observed KIE data. However, many of them are rather complex and require the fixing of numerous parameters. Rates calculated for different parameter choices are checked for consistency with the data, but the complexity of how the parameters affect the rates could limit the models’ ability to make predictions for new experiments. We note that apart from the quantum models mentioned above, a semiclassical model, which leads to a Langevin equation including friction, has also shown agreement with the experimental data[@Pollak]. While the individual rates are not specified, this model requires only a single parameter for each mutant, the friction coefficient, to obtain the corresponding KIE curves.
Here we aim to develop a quantum rate model with limited complexity (two parameters) that qualitatively produces the observed KIEs and temperature variation for various mutants. To benchmark the proposed model we will compare its predictions with the conclusions drawn from another two-parameter model that has previously been discussed [@ESS_H_2010; @SH_S_2016].
\[sec3\] A qualitative quantum model with classical gating
==========================================================
Building on previous rate models we propose here a new qualitative model for enzyme-catalysed hydrogen transfer that treats the dynamics of the transferring particle fully quantum mechanically. The model does not make semiclassical approximations, such as WKB. Instead the model calculates coherent quantum dynamics contributions to the rate. These contributions are then averaged over the active site configurations which are sampled by the enzyme’s vibration (gating).
Overview of the rate model
--------------------------
The model assumes that hydrogen, or one of its isotopes, is initially in thermal equilibrium in a potential $V^C$ created by the donor atom (carbon, C). When the acceptor atom (oxygen, O) is brought close by the enzyme, the hydrogen experiences a different potential, $V^{CO}_R$, which is parametrised by the donor-acceptor separation $R$. The hydrogen atom is in a non-stationary state with respect to the new potential $V^{CO}_R$ and this will result in quantum dynamics with the state of hydrogen evolving according to the Schrödinger equation. We obtain a quantum rate $\tau_R$ that quantifies the rate of the hydrogen transferring from the donor to the acceptor for each value of $R$. To obtain a prediction of the experimentally measured rate these quantum rates are then weighted with a classical gating probability $p(R)$ that determines the likelihood of the donor-acceptor distance $R$ being realised in a thermal environment [@BB92]. The variation of this distance over a range $R_i \leq R \leq R_f$ is realised by the enzyme vibration, i.e. “gating”. Averaging over the range of $R$ then gives the overall transfer rate, $$\begin{aligned}
k= \frac{N}{|R_f - R_i|}\int^{R_f}_{R_i} p(R)\, \tau_R \, {\textrm d}R.
\label{Our_avg_k}\end{aligned}$$ Here $N$ is a dimensionless prefactor that accounts for factors that will influence the experimentally measured rate, but do not directly relate to the particle transfer assisted by the enzyme in the rate-limiting step. These factors include the probability of the reactants coming together in the active site in the first place, as well as any other relevant effects due to the environment, for instance the concentration of the solvent. Thus $N$ will be temperature dependent although we expect it to have a significantly weaker temperature dependence than the temperature dependence of the other factor in the rate expression, $\frac{1}{|R_f - R_i|}\int^{R_f}_{R_i} p(R)\, \tau_R \, {\textrm d}R$. We also assume that $N$ is independent of the mass of the transferring particle, i.e. it is the same for all isotopes. This is justified because the charge of the transferring particle, which may cause long-range interactions with the environment, is constant for all isotopes.
This rate expression allows one to predict the mass and temperature dependence of ratios of rates, i.e. KIEs, where environmental effects captured in the prefactor $N$ cancel out. The thermal vibrations of the enzyme influence the reaction by realising a configuration where the donor and acceptor are at a distance $R$ with probability $p(R)$. This probability will be a function of temperature and is determined by two parameters: the mean donor-acceptor distance $R_e$ and the gating frequency $\Omega$ of donor-acceptor oscillations about their equilibrium separation realised by the enzyme. The quantum rate $\tau_R$ derives from the Schrödinger equation of the transferring particle and is thus mass-dependent. In principle, $\tau_R$ is weakly temperature dependent, too, because of thermal occupation of the C-H bond energies before catalysis. However, as we will see this dependence is negligible at biological temperatures.
Choosing a specific hydrogen isotope, and thus the mass, fixes $\tau_R$. The only variables left to obtain the KIEs for each SLO variant are then $R_e$ and $\Omega$, which determine the gating probability $p(R)$. The following subsections will discuss the factors $\tau_R$ and $p(R)$ and their parameter dependence in more detail.
The quantum mechanical transfer rate $\tau_R$
---------------------------------------------
Prior to the transfer event, hydrogen (H) or one of its isotopes, is bonded to the donor carbon (C) atom and in thermal equilibrium with its environment at temperature $T = (k_B \beta)^{-1}$. The initial state of hydrogen is thus the stationary, thermal state $\rho = \sum_{n=0}^{\infty} {|E^C_n\rangle}{\langle E^C_n|} \, e^{- \beta E^C_n}/Z^C_{\beta}$ of the C-H interaction potential $V^C$, where $E^C_n$ are the eigenenergies and ${|E^C_n\rangle}$ the energy eigenstates of $V^C$, and $Z^C_{\beta}$ is the partition function. However, in the biological temperature range $T \in [5^\circ\mbox{C}, 50^\circ\mbox{C}]$ the parameters that determine $V^C$, discussed in subsection \[secParams\], result in excited energy levels of $V^C$ that are too high to be significantly populated. Thus the probability of hydrogen occupying the ground state ${|E_0^{\mathrm{C}}\rangle}$ of the potential $V^\mathrm{C}$ is over $99\%$. Therefore the rate $\tau_R$ will be determined solely by the ground state evolution and so is temperature-independent.
When the enzyme brings the donor and acceptor atoms into close confinement in its active site, with donor-acceptor distance $R$, the hydrogen becomes exposed to an asymmetric double well potential $V_R^{CO}$ due to its interaction with the nearby acceptor oxygen (O) atom. Since the potential is suddenly changed, the initial state (ground state ${|E^C_0\rangle}$ of $V^C$) is no longer a stationary state for the new potential $V_R^{CO}$ and so the probability of finding hydrogen near the acceptor changes over time.
Assuming $V_R^{CO}$ is constant during the small transfer time window $t_{\max}$, then the state of the transferring hydrogen atom at intermediate times $t \in [0, t_{\max}]$ is $${|\psi_R(t)\rangle} =\exp \left( -i \H_R^{CO}t / \hbar \right) {|E^C_0\rangle},$$ where $\H_R^{CO} =\frac{p^{2}}{2m}+V_R^{CO}$ is the Hamiltonian that generates the evolution from the initial ground state ${|E^C_0\rangle}$ and $m$ is the mass of the transferring particle. The potential $V_R^{CO}$ will be a double well potential for larger values of $R$, see Fig. \[fig\_cho\_config\], with further details for $V_R^{CO}$ described below. The probability of hydrogen transfer is the probability of observing it on the acceptor site at time $t$, $\varphi_R(t)= \int_{x_b}^{\infty} |{\langle x|\psi_R(t)\rangle}|^2 \, {\textrm d}x$, i.e. the hydrogen is anywhere to the right of the barrier peak position $x_b$, see Fig. \[E\_barrier\]. If there is no barrier, which can occur when C and O are very close, then $V_R^{CO}$ has a single well and $x_b$ is defined as the position of the minimum of $V_R^{CO}$. Physically this means that, at large donor-acceptor distances, the transferring particle is strongly localised either at the donor or the acceptor, while at smaller distances it is shared between the two.
To obtain a rate constant we average the time-derivative of $\varphi(t)$ over a time window $t_{\max}$. We choose this time window as the smallest timescale on which thermal relaxation will affect the Schrödinger evolution of the system. This damping time is given by $t_{\max} = \frac{\hbar}{\Delta E}$ where $\Delta E$ is the energy gap[@BP_OpenQS] that the transferring particle sees at the donor, i.e. the energy gap between the ground and first excited states of the C-H bond (or C-D bond for deuterium). The Schrödinger evolution of the hydrogen atom then gives rise to the rate $$\begin{aligned}
\label{eqn_tau}
\tau_R &=&\frac{1}{t_{\max}}\int_{0}^{t_{\max}} {{\textrm d}{\varphi}_R(t) \over {\textrm d}t} \, {\textrm d}t.\end{aligned}$$
To calculate the rate $\tau_R$ we now detail the potential $V_R^{CO}$. It is composed of the two Morse potentials seen by the transferring particle due to the presence of the donor and acceptor. The Morse potential has the form $V^Y(x) = D^Y \left(1-e^{- g \, a^Y(x-x^Y_e)}\right)^2$ for each Y-H bond where Y is either C (donor) or O (acceptor). Here $D^Y$ is the well depth, $x^Y_e$ is the equilibrium separation and $a^Y = \omega^Y\sqrt{\mu^Y/2D^Y}$ is the well “curvature”. $\mu^Y$ is the reduced mass between hydrogen and Y, and $\omega^Y$ is the bond frequency. These constants can be found in the literature for the case when only two particles, either C-H or O-H, are bonded. To account for the fact that the transferring particle before (after) the transfer sees the electrostatic potential not just of a single carbon (oxygen) atom, but of these atoms when part of donor (acceptor) molecule, the squeezing parameter $g$ has been introduced in $V^Y(x)$.
Assuming that the donor, hydrogen, and acceptor atoms are collinear in a single reaction coordinate $x$, see Fig. \[fig\_cho\_config\], the combined potential seen by hydrogen at C-O separation $R$ is $V_R^{CO}(x) = V^C(x) + V_R^O(x) - D^O$, obtained by summing $V^C(x) = D^C \left((1-e^{-ga^C(x-x^C_e)}\right)^2$ and $V_R^O(x) = D^O \left(1-e^{-ga^O(-x+(R-x_e^O))}\right)^2$ together with an offset $-D^O$. This offset guarantees that, if the acceptor (O) were moved infinitely far away from the donor (C), the hydrogen would feel no force due to the acceptor. The C-O separation is $R = d + x_e^O + x_e^C$, where $x_e^O$ and $x_e^C$ are fixed, and $d$ can vary, see Fig. \[fig\_cho\_config\]. Note that $d$ defines the separation between the $V^C$ and $V^O_R$ well minima which can deviate from the well-separation of the two wells in the resulting $V^{CO}_R$.
While isotopes of hydrogen will have a different mass from hydrogen, this mass has no effect on the geometry of the system and on the electrostatic forces involved. Consequently, the potentials $V^C, V_R^O, V_R^{CO}$ remain the same for all isotopes. However, the eigenenergies and eigenstates of the corresponding Hamiltonians $\H^C, \H_R^O, \H_R^{CO}$ are mass-dependent, because mass enters the Schrödinger equation through the kinetic term $\frac{p^2}{2m}$. This is what makes the quantum contribution $\tau_R$, and therefore the overall rate $k$, dependent on the mass of the transferring particle. Denoting the hydrogen rate by $k_H$ and the deuterium rate by $k_D$, the KIE is the ratio $k_H/k_D$.
\[sec\_Gating\]Gating and the classical probability $p(R)$
----------------------------------------------------------
Thermal energy from the environment causes the enzyme to vibrate. “Gating” [@BB92] assumes that this enzyme motion is coupled to the active site configuration by making the donor (C) and acceptor (O) oscillate and sample a range of C-O separations $R_i \leq R \leq R_f$. The likelihood of a separation $R$ occurring is governed by the gating probability distribution $p(R)$. It is a Boltzmann distribution $p(R) = e^{-\beta U^{CO}(R)}/Z^{CO}_{\beta}$ for a potential $U^{CO}(R)$ that describes the sampling of donor-acceptor distances around a mean equilibrium position, $R_e$, at inverse temperature $\beta = 1/(k_B T)$, normalised by a partition function $Z^{CO}_{\beta}$. Assuming a quadratic potential, $U^{CO}(R) = \frac{\mu\Omega^2}{2}(R-R_e)^2$, results in a Gaussian gating probability $p(R)$. Here $\mu$ is the C-O reduced mass and $\Omega$ is the gating frequency. The standard deviation of distances sampled about the peak position $R_e$ is $\sigma = \sqrt{k_B T / (\mu\Omega^2)}$.
\[secParams\]Constants and parameters
-------------------------------------
Constants for the C-H and O-H Morse potentials are available from standard chemistry data books [@Chem_Data]: the experimental dissociation energies 413 kJ/mol and 493 kJ/mol (measured from the zero point energies) are used to derive $D^C$ and $D^O$ (measured from the bottom of the well); the equilibrium distances are $x_e^C=1.09~\AA$ and $x_e^O=0.94~\AA$; and the bond frequencies $\omega^C$ and $\omega^O$ are both $3000 \mathrm{cm}^{-1}$ (in units of wavenumbers). The diatomic well curvatures $a^C$ and $a^O$ are calculated using these values. The squeezing parameter $g$ scales the width of the local electrostatic potentials seen by the transferring particle at the donor (and acceptor) in comparison to the widths of diatomic bonds. If the potentials $V^C$ and $V^O$ in the substrate are narrower in comparison to the isolated C-H and O-H bonds respectively, then the bond frequency increases and mathematically this is reflected in a value of $g>1$. Here we choose $g=2.3$ throughout. The quantum transfer rate $\tau_R$ accounts for contributions from transitions between the ground state of the initial potential and various energetic states of the new double well potential. While theoretically all transfers will have a non-zero probability, environmental noise will limit the number of energetic levels that can be reached by the transferring particle. Here we choose to include transfers to the lowest 15 energetic eigenstates of the double well potential.
The donor-acceptor range $R_i \leq R \leq R_f$ governs which rate contributions are included in the rate Eq. . Recall that $R = d + x_e^O + x_e^C$ is determined by the distance $d$ between the $V^C$ and $V^O_R$ well minima. We assume that $d$ cannot be negative, i.e. C and O can be no closer than $R_i = x_e^C+x_e^O = 2.03~\AA$. We choose the maximal value of $d$ to be $3~\AA$ and so $R_f = 5.03~\AA$. While this is the integration range we allow, in the end the probability $p(R)$ “controls” the window of C-O separations that are most relevant in the overall transfer rate $k$, see Eq. .
Physically it is reasonable for the isotope-independent rate prefactor $N$ to depend on the temperature as it reflects the environment’s impact on the rate dynamics. To obtain a specific functional form will require a more detailed model of the environment, following for example Refs \[\] and \[\]. Here we choose the rate prefactor $N$ to be independent of the isotope (H or D) for each SLO mutant, see Table \[table1\], with its value set by fitting the calculated hydrogen rates to the experimental data. This choice fixes the scale for the hydrogen and deuterium rate plots which are shown in Fig. \[fig:ks+KIEs\][**a)**]{} and [**b)**]{}. While these plots could be modified by a temperature-dependent $N$, the relative behaviour of rates captured by KIEs, plotted in Fig. \[fig:ks+KIEs\][**c)**]{} and [**d)**]{}, are unaffected by $N$ even if it is temperature-dependent.
In addition to varying temperature $T$ and isotope mass $m$, we will investigate how mutations of SLO from its wild-type affect the rates and the KIEs predicted with Eq. . Since mutants catalyse the same reaction, the functional form of the Morse potentials $V^C, V^O_R$ and $V_R^{CO}$ experienced by the transferring particle remain the same. However, the enzyme mutations will affect the equilibrium donor-acceptor distance, $R_e$, and the gating frequency, $\Omega$, of the gating distribution $p(R)$, and this determines the likelihood that a potential $V_R^{CO}$ will be seen by the transferring particle. The parameter values of $R_e$ and $\Omega$ for the various SLO mutants are discussed in the next section.
{width="98.00000%"}
\[sec4\] Results
================
The hydrogen and deuterium transfer rates, $k_H$ and $k_D$, for SLO WT and each SLO mutant are calculated with Eq. using the mutant-specific values for the mean C-O separation $R_e$ and the gating frequency $\Omega$, as listed in Table \[table1\]. We first discuss the top four SLO variants, WT, M1, M2 and M3. The values of $R_e$ and $\Omega$ for these are chosen to provide the best fit to the experimental KIE data [@KRK02] with gating frequencies in the physically reasonable range $\Omega \leq 400~\mathrm{cm}^{-1}$. Figs. \[fig:ks+KIEs\] [**a)**]{} and [**b)**]{} show the calculated H and D rates for the four SLO variants WT and M3 ([**a**]{}), and M1 and M2 ([**b**]{}), over the experimental temperature range $5^\circ\mbox{C} \leq T \leq 50^\circ\mbox{C}$ together with the observed rates[@KRK02], c.f. Fig. \[figKRK02\]. The corresponding KIEs are shown in Figs. \[fig:ks+KIEs\] [**c)**]{} and [**d)**]{} together with the ratio of the experimentally measured rates as a function of inverse temperature.
Comparing the experimental and theoretical KIEs we find excellent agreement for the KIE gradient for SLO WT and its mutants M1 and M2. The agreement for the distant mutant M3 is less good than what has been achieved with other models[@ESS_H_2010; @SH_S_2016; @Pollak]. Our calculated M3 deuterium rate does not vary as strongly with temperature as the experimentally observed one. Nevertheless our parameters produce a M3 KIE that is the most temperature dependent of the four SLO variants, WT, M1, M2 and M3, in agreement with experiment.
The SLO WT parameters are $R_e=2.9~\AA $ and $\Omega=400~\mathrm{cm}^{-1}$. Any smaller value of $R_e$ would require an even higher $\Omega$, i.e. result in unrealistically high WT donor-acceptor vibrations. For M3 our equilibrium separation is $R_e = 3.05~\AA$, i.e. $0.15~\AA$ higher than that of WT. The M3 gating frequency is reduced to $\Omega = 325~\mathrm{cm}^{-1}$ and implies an increase in the standard deviation $\sigma$ of the gating in comparison to WT. The SLO WT and M3 gating distributions $p(R)$ at 30$^\circ$C are thus determined by peak and standard deviations $R_e \pm \sigma = 2.9 \pm 0.08~\AA$ and $R_e \pm \sigma = 3.05 \pm 0.10~\AA$, respectively. The values of $R_e$ and $\Omega$ for WT and M3 end up close to the ones reported in Ref. \[\] where a proton-coupled electron transfer model was used to derive the rates. There a choice of $R_e = 2.88~\AA$ and $\Omega = 368.2~\mathrm{cm}^{-1}$ for the WT, and $R_e = 3.08~\AA$ and $\Omega = 295.1~\mathrm{cm}^{-1}$ for M3 (see Table 1 of \[\]), provided a very good fit to the experimental data when an effective mass of $M=10$ amu was chosen for the proton donor-acceptor vibrational mode[@SH_S_2016]. Despite the different approaches, the similarity of the parameter values presented here and in \[\] is particularly noteworthy.
The rates calculated with Eq. for SLO mutants M1 and M2 are displayed in Fig. \[fig:ks+KIEs\][**b)**]{} and show good agreement with the experimental data, which are also shown. To match the experimental KIEs required an increase of $R_e$ by $0.05~\AA$ in comparison to WT and reduction of the gating frequency $\Omega$ by ca. 5%. The parameter values for both M1 and M2 that give the best KIE fit are then $R_e = 2.95~\AA$ and $\Omega = 380~\mathrm{cm}^{-1}$. Thus the sampling range increases only very slightly and peak and standard deviation of $p(R)$ at 30$^\circ$C are $R_e \pm \sigma = 2.95 \pm 0.08~\AA$. The calculated KIEs are displayed in Fig. \[fig:ks+KIEs\][**d)**]{} together with the experimental data points.
[ | m[0.9cm]{} || m[0.9cm]{} | m[1.3cm]{} | m[1.9cm]{} | m[0.8cm]{} |m[1.3cm]{} | ]{} SLO variant & $R_e$ (Å) & $\Omega$ (cm$^{-1}$) & $N$ & KIE & Exp. KIE\
WT & 2.9 & 400 & $4\times 10^{-6}$ & 79 & 81\
M1 & 2.95 & 380 & $6\times 10^{-8}$ & 101 & 109\
M2 & 2.95 & 380 & $1.04\times 10^{-8}$ & 101 & 112\
M3 & 3.05 & 325 & $1\times 10^{-4}$ & 94 & 93\
DM & 3.3 & 495.7 & Fig. \[fig\_KIEs\_DM\] - grey & 563 & 729\
DM & 3.8 & 185.3 & Fig. \[fig\_KIEs\_DM\] - orange & 696 & 729\
Finally, we compare KIE predictions of the developed model against the KIEs reported in a recent SLO double mutant (DM) experiment[@SLO_DM]. Here both the M1 and M2 mutations (Leu$^{546} \rightarrow$ Ala and Leu$^{754} \rightarrow$ Ala) were implemented on a single SLO enzyme. Huge KIEs were observed using two independent methods for the same hydrogen transfer reaction: $537\pm 55$ at $35^\circ$C using single-turnover kinetics and $729\pm 26$ at $30^\circ$C using steady-state measurements. Since these are completely different measurements resulting in systematic errors, it is not possible to use these two data points to conclusively infer the temperature dependence of the DM KIE. We thus calculate a set of possible KIE curves with Eq. , where each curve has a different parameter pair $(R_e, \Omega)$, see Fig. \[fig\_KIEs\_DM\]. All the curves are pinned at $537$ at $35^\circ$C, which is the KIE from Ref. \[\] measured using the more reliable steady-state method.
We find that obtaining a high KIE of 537 at $35^\circ$C requires quite large equilibrium separations, $R_e = 3.3~\AA$ or more, when the gating frequency $\Omega$ is assumed not to exceed $500 \mathrm{cm}^{-1}$. Fig. \[fig\_KIEs\_DM\] shows the calculated KIE curves in the temperature range $5^{\circ}C \leq T \leq 50^{\circ}C$ for parameters in the ranges $3.3~\AA \leq R_e \leq 3.8~\AA$ and $180~\mathrm{cm}^{-1} \leq \Omega \leq 500~\mathrm{cm}^{-1}$. The KIE gradients vary strongly as the parameters are changed. At a high value of $\Omega \approx 500~\mathrm{cm}^{-1}$, which corresponds to a very rigid active site, the donor-acceptor separation is $3.3~\AA$ and these values result in a fairly small variation of the KIE with temperature. Allowing $R_e$ to increase to $3.8~\AA$ implies a frequency of $\Omega \approx 190~\mathrm{cm}^{-1}$ and results in a steep KIE increase of ca. 160 between $35^\circ$C and $30^\circ$C. Calculations for the DM KIE with a different model [@SH_S_2016], have previously suggested an equilibrium separation of $R_e\approx 3.3~\AA$ while the gating frequency was given as $\Omega \approx 280~\mathrm{cm}^{-1}$. This is significantly smaller than the $\Omega \approx 500~\mathrm{cm}^{-1}$ obtained here at the same equilibrium distance, see the grey curve in Fig. \[fig\_KIEs\_DM\]. While this manuscriprt was under review, new experimental data were published in Ref. \[\] that provide further evidence supporting the hypothesis of a SLO DM KIE with a very small temperature dependence.
\[sec5\]Tunnelling contribution
===============================
The presented model allows us to determine whether hydrogen tunnelling contributes to the observed rates. Hydrogen starts with a very high probability of over 99% in the energetic ground state $E_0^C$ of the donor potential $V^{C}$. It is then exposed to the combined donor-acceptor potential $V_{R}^{CO}$. At a given $R$, hydrogen tunnels if its initial energy $E_0^C$ is less than the height of the barrier of the potential $V_{R}^{CO}$. To quantify whether hydrogen tunnelling contributes significantly to the rate $k$ in Eq. we identify the distance $\bar{R}$ that contributes most to it, i.e. $\bar{R}$ is the value of $R$ at which the product $p(R) \tau_R$ is maximised. For example, for WT SLO at 30$^\circ$C the distance contributing most to the rate is found to be $\bar{R} = 2.67~\AA$. At this distance the energy difference between barrier height and initial energy is $E_{\rm diff} = V_{\bar{R}}^{CO} (x_b) - E_0^C \sim 0.47 $ eV $\approx 11$ kcal/mol. We note that this energy barrier is an order of magnitude larger than activation energies obtained from an Arrhenius plot of the experimental data [@KRK02]. The chance that hydrogen is thermally excited from the ground state to the top of the barrier is exponentially suppressed and hence it is highly unlikely that hydrogen hops over the barrier. Thus the qualitative model developed here suggests that the dominant transfer mechanism is tunnelling.
\[sec7\] Discussion and further work
====================================
In this paper we investigated the hydrogen transfer reaction catalysed by soybean lipoxygenase. We developed a qualitative model that gives the temperature dependence of the primary KIE through the rate $k$ given in equation . The model treats the dynamics of the transferring particle quantum mechanically resulting in $\tau_R$, a quantum contribution to the rate $k$. In addition it accounts for the enzyme’s role in the transfer via a classical “gating” rate, $p(R)$, which arises because of the coupling of the enzyme’s vibrations to the donor-acceptor separation.
The quantum rate $\tau_R$ is fixed by the type of chemical reaction - here the hydrogen (deuterium) transfer catalysed by SLO - and depends principally on the mass of the transferring isotope which will be crucial for the KIE. $\tau_R$ is determined by the double-well potential $V^{CO}_R$. One of the parameters that fixes its shape is the squeezing parameter $g$. If $V^{CO}_R$ were the sum of isolated diatomic C-H and O-H potentials, then $g$ would be 1. A larger $g$ makes the individual wells narrower and in our calculation we find that choosing $g=2.3$ gives rate and KIE predictions that are in good agreement with experimental data for SLO WT and all mutants analysed here. This suggests that, when hydrogen (or deuterium) is bonded to the donor or acceptor in the presence of the substrate and the enzyme, then hydrogen experiences a stronger attraction to either C and O during the rate-limiting step. This slightly higher $g$ value of $2.3$ produces a higher barrier in $V^{CO}_R$ and leads to enlarged KIEs. A similar finding was reported in the intensive computational study of SLO, where the barrier had to be increased manually to obtain KIEs that were as high as the experimental ones [@TVLLY06].
With the quantum part of the rate fixed, one is left with the classical gating rate $p(R)$ which is determined by just two free parameters. The donor-acceptor equilibrium separation $R_e$ and the gating frequency $\Omega$ set the average position and spread of the classical gating distribution $p(R)$. It is through changing these two parameters only that we obtain the various rates and KIEs of all the SLO mutants. These two parameters are also the ones that parameterise the KIE curve of each mutant in the non-adiabatic proton-coupled electron transfer reaction model of Ref \[\] with which we compared our results. A conceptually different semiclassical rate model, based on a Caldeira-Leggett type Hamiltonian that results in a Langevin equation, parameterises the KIE curve of each mutant with only a single parameter, the friction coefficient [@Pollak]. The calculated KIE curves are markedly different from the curves reported here, while also showing agreement with the experimental data within the experimental uncertainty.
We found that our model predictions show good agreement with the experimental data for physically reasonable choices of $R_e$ and $\Omega$, see Table \[table1\] and Fig. \[fig:ks+KIEs\]. The general picture that emerges from this model is that in WT the active site is compressed and very rigidly held: its low $R_e$ keeps the donor and acceptor very close on average and its high $\Omega$ indicates little movement around the most likely separation $R_e$, with a spread $\sigma$ of less than $0.1~\AA$. As we saw in Fig. \[fig:ks+KIEs\][**c)**]{} the WT parameter values, which are expected to be close to the “optimal” configuration for SLO, lead to KIEs in the range of 65-100 for biological temperatures with moderate temperature dependence. This suggests that $R_e$ and $\Omega$ are very finely tuned in the SLO WT.
In the mutants, a larger $R_e$ means that the carbon and oxygen are held further apart on average, and this opening of the active site effectively makes the barrier larger. Specifically, while the barrier height most likely seen by the transferring particle in WT is $35.2$ kcal/mol at $R_e = 2.9~\AA$, the most likely barrier height in M1 and M2 is $48.8$ kcal/mol at $R_e = 2.95~\AA$. The higher barrier in M1 and M2 makes the transfer even more difficult and very significantly lowers the rates of these mutants in comparison to WT. As well as an increase in $R_e$, a mutated SLO also has a decreased $\Omega$, implying that the donor and acceptor oscillate from their equilibrium separation over a larger range. The standard deviation $\sigma$ of the sampling distribution $p(R)$ depends on temperature and gating frequency as $\sigma \propto \sqrt{T}/\Omega$. The interplay of larger equilibrium separation and larger gating ranges results in large KIEs that have a much more pronounced temperature dependence than those of WT.
We note that the quantum rate $\tau_R$ is large for small $R$ and decays rapidly with $R$, thus smaller separations result in much more efficient hydrogen transfer. This means that the separations which contribute significantly to the overall rate $k$ in Eq. can be many $\sigma$ smaller than $R_e$. This was observed in section \[sec5\], where we found that the separation $\bar{R}$ that contributes most to the rate $k$ can be much shorter than the most likely equilibrium separation $R_e$. This functional dependence makes the KIEs and their temperature dependence very sensitive to the values of $R_e$ and $\Omega$, particularly as these parameters take on higher values. Generally we find that the KIE increases when (i) $R_e$ is fixed while $\Omega$ is increased and (ii) when $\Omega$ is fixed but $R_e$ increases. These tendencies found here are in agreement with the conclusions of previous works that have investigated the variation of the KIE with $R_e$ and $\Omega$ in SLO [@ESS_H_2010; @SH_S_2016].
The presented model is qualitative and does not include several physical properties that have been considered elsewhere. The true reaction takes place within a three-dimensional potential landscape[@Pollak], whereas the model presented here considers only a one-dimensional double-well potential in which the hydrogen can move, thus stretching the C-H bond only. Omitting rate contributions from other normal modes, such as bending modes[@Pollak], can lead to an overestimate of the tunnelling contribution. The model presented here is also adiabatic, including only a single electronic state. This contrasts with other rate models [@SH_S_2016] where rate contributions arise from multiple non-adiabatic transfers. Future extensions of the model could address multidimensional potentials and non-adiabatic transfers.
We found that while the model presented here is quite simple in its structure and dependence on the particle mass and temperature, it qualitatively produces the rate/KIE behaviour observed in the experiments we compared with for various SLO mutants. The model predictions for the temperature dependence of the KIE for the DM open the possibility to further test the validity of the approach and model.
The values of the physical constants used here, such as the binding energies of hydrogen to donor and acceptor, are specific to SLO and taken from the literature. But it would be straightforward to replace them with relevant constants for other enzyme-catalysed reactions. Future research could thus address the modelling of rates and KIEs of other enzymatic systems that exhibit significant signatures of tunnelling, for instance the Old Yellow Enzyme family of flavoproteins [@HPS09] where particular attention is paid to the role of promoting vibrations. The observed KIEs of these enzymes are not as high as SLO, and tend to be more temperature-dependent than SLO. This could be accounted for by our model by, for instance, choosing a higher $R_e$ and lower $\Omega$ than the SLO WT. Applying the quantum model to these enzymes thus provides a fruitful avenue for testing the importance of the gating frequency and transfer distance in enzymatic systems whose KIEs suggest a large tunnelling contribution.
It is a pleasure to thank Judith Klinman and her group for their hospitality and many insightful discussions. We are also very grateful to Nigel Scrutton and Sam Hay for their instructive comments, and to Tobias Osborne and his group for their hospitality. SJ is funded by an Imperial College London Junior Research Fellowship. SJ also acknowledges the following grants: ERC grants QFTCMPS; SIQS by the cluster of excellence EXC 201 Quantum Engineering and Space-Time Research; and EPSRC grant EP/K022512/1. JA acknowledges support by the Royal Society and EPSRC (EP/M009165/1).
[^1]: The WKB approximation provides a good estimate of the tunnelling probability through a barrier when $\hbar/S \ll 1$. In this case the tunnelling probability is $\propto \exp(-2S/\hbar)$. The action $S = \int_a^b \sqrt{2m[V(x)-E]}dx$ is for a one-dimensional potential $V(x)$ that has a barrier in the range $a \leq x \leq b$, and the incident particle has energy $E$ and mass $m$. For the purposes of an order estimation, let us assume the barrier is rectangular, i.e. $V(x) = V$ for $a \leq x \leq b$ and zero otherwise. Then $S = (b-a)\sqrt{2m[V-E]}$. Typical parameters for the transfer distance $b-a \sim 1~\AA$ and for the activation energy $V-E \sim 1$ kcal/mol $\sim 1\times 10^{-20} $ J (from SLO data in \[\]). Hence $\hbar/S \sim 0.1$ indicating that the WKB regime is not a very good approximation.
|
---
abstract: |
In this paper we present an advanced numerical method to simulate a real life challenging industrial problem that consists of the non-destructive testing in steam generators. We develop a finite element technique that handles the big data numerical set of systems arising when a discretization of the eddy-current equation in three dimensional space is made. Using a high performance technique, our method becomes fully efficient. We provide numerical simulations using the software Freefem++ which has a powerful tool to handle finite element method and parallel computing. We show that our technique speeds up the simulation with a good efficiency factor. ***Keywords***: Maxwell’s Equation, Non-destructive testing, Eddy-Current approximation, Numerical Methods, Parallel Algorithm, High performance computing, Industrial problem.\
***PACS***: 89.75.-k,, 89.75.Da, 05.45.Xt, 87.18.-h\
***MSC***: 34C15, 35Q70
author:
- 'Mohamed Kamel RIAHI$^{\star}$'
bibliography:
- 'biblio.bib'
title: '**A Fast Eddy-current Non Destructive Testing Finite Element Solver in Steam Generator**'
---
[^1]
Introduction and industrial model {#Mission of the Journal}
=================================
Many industrial applications use Eddy-Current Testing (ECT) to evaluate cracks, defaults and other types of anomalies in the material engines. In general, ECT is a non-invasive technique where electric impedance measurement plays an important role in the evaluation of the quality of the tested material. Eddy currents are created through a process called electromagnetic induction. When alternating current is applied near the conductor (such as copper wire), a magnetic field develops in and around this conductor. The measurement of the electrical impedance enables a detection of anomalies in the tested pieces. The computer simulation of the eddy-current problem has an enormous role in the automatization of the ECT, especially when inverse type problems are conducted, such as shape identification of deposits and cracks detection [@1386227; @RPQNE; @bendjoudi; @HuangTakagiFukutomi; @HuangTakagi; @girarclogging; @haddar2015axisymmetric] (we may also refer to [@ndtdatabase] for further lecture). To do so, the direct problem needs to be robust and effective.
In this work, we are concerned with the numerical simulation on parallel computers of the direct problem arising from the ECT of a steam generator (SG). The numerical problem involves the finite element approximation of the eddy-current equations in a time harmonic low frequency regime. Because of the vectorial aspect of the three dimensional unknown, the discretization of the problem leads to a huge and ill-conditioned system. Its numerical resolution is time and memory consuming. This has motivated us to construct an algorithm based on high performance computing tools.
SGs are critical components in nuclear power plants. Heat produced in a nuclear reactor core is transferred as pressurized water at high temperature via the primary coolant loop into an SG, consisting of tubes in U-shape, and boils coolant water in the secondary circuit on the shell side of the tubes into steam. The steam is then delivered to the turbine generating electrical power. The SG tubes are held by the broached quatrefoil tube support plates (TSP) with flow paths between tubes and plates for the coolant circuit.
Without disassembling the SG, the lower part of the tubes – which is very long – is inaccessible for normal inspections. Therefore, an ECT procedure is widely practiced in industry to detect the presence of defects, such as cracks, flaws, inclusions and deposits. In this paper, we go over a specific technique and tools using high performance computing. We provide an efficient and scalable finite element algorithm to solve the ECT problem in the SG. We consider a potential formulation supplemented by a Coulomb gauge condition, where the electric field is represented in a suitable space by a magnetic vector potential and scalar electric potential (see for instance [@MR2680968] and reference therein). This involves a coupled system between the new variables, which we solve using a sophisticated technique.
In the SG case, the ECT procedure consists in withdrawing a probe constituted by two coils that move together all the way through the tube. These coils are called generator and receiver coils, where the generator coil produces the source term current excitation that produces a magnetic field, which in turn penetrate materials nearby and produce an eddy-current by induction. The receiver coil, has a detective role, where it can measure the variation of electric impedance due to the induced eddy-current. The motivation behind our parallelism technique is that the ECT procedure is a complicated task that needs a careful attention starting from the mesh generation regarding the motions of the coils along the scan zone, to the resolution of the coupled system that governs the electric field solution of the eddy-current problem. Indeed, we consider the vectorial ${\textbf{curl\hspace{.01in}}}{\textbf{curl\hspace{.01in}}}$ version of the eddy-current equations in three dimension. Thus the finite element discretization of the problem at hand may significantly lead to a huge linear coupled system, especially when mesh refinement is needed in order to have a good approximation of the exponential decay of the wave when it penetrates materials.
Our work is presented as follows. In section 2 we present the eddy-current equations. We reformulate the problem using potential formulations, then we give the system that has to be solved. Our presentation uses bi linear formulation obtained after setup of variational formulation of the eddy-current problem. In section 3 we In section 3 we describe the numerical method that we use to deal with the particularity of the ECT in SG, where a set of operations has to be done and that involves numerical solution of eddy-current problem. We provide sample of Freefem++ scripting that show how to implements mathematical formulas using the aforementioned software. We give numerical results in section4, that shows the efficiency of our approach in the speedup of the ECT.
Time harmonic Eddy-current equations
====================================
The following description of the eddy-current problem follows [@MR2680968].
The time harmonic Eddy-current approximation of Maxwell’s equations for the electric field ${\mathbf{E}}$ and the magnetic field ${\mathbf{H}}$ and a source excitation term ${\mathbf{J}_e}$ reads: $$\label{MAMF}
\begin{cases}
{\textbf{curl\hspace{.01in}}}{\mathbf{H}}-\sigma {\mathbf{E}}={\mathbf{J}_e}&\text{ on } \Omega,\\
{\textbf{curl\hspace{.01in}}}{\mathbf{E}}- i\omega\mu {\mathbf{H}}= 0.&\text{ on } \Omega.
\end{cases}$$ where $\sigma$ stands for the electric conductivity, $\mu$ stands for the magnetic permeability and $\omega$ stands for the frequency. The computational domain ${\Omega}\subset\mathbb{R}^3$ is a bounded with Lipchitz boundary $\partial{\Omega}$. The above system represents Maxwell-Ampère and Maxwell-Faraday equations, where the current displacement term has been dropped to arrive at the eddy-current model [@Ammari:2000:JEC:354423.354457; @bossavit2004electromagnetisme]. In addition, in order to insure uniqueness of the solution; Eq. is thus supplemented with a boundary condition. In our case we consider a perfect magnetic boundary condition i.e. ${\mathbf{H}}\times{\boldsymbol{\vec{n}}}=0$ on $\partial{\Omega}$ with ${\boldsymbol{\vec{n}}}$ stands for the unit normal vector pointing outward at the domain ${\Omega}$. Our computational domain is decomposed, with respect to the conductivity $\sigma$, to a conductor material ${\Omega}_C$ characterized with $\sigma\neq0$ and insulator ${\Omega}_I$. Thus ${\Omega}={\Omega}_I\cup{\Omega}_C$, where the common interface is denoted as $\Gamma={\Omega}_I\cap\overline{{\Omega}_C}$. Remark that $\Gamma\cap\partial{\Omega}\neq\{0\}$.
From Eq. $_1$ one can see that ${\textbf{curl\hspace{.01in}}}{\mathbf{H}}= {\mathbf{J}_e}$ in the insulating region ${\Omega}_I$. This imposes a condition on the excitation term ${\mathbf{J}_e}$ where it is required to be a divergence free vector field.
In order to cope with the geometry configuration of the computational domain, it is common to use potential formulation to better handle the eddy current problem numerically. In the sequel we consider the magnetic vector potential formulation by introducing the magnetic vector potential ${\mathbf{A}}$ and the scalar electric potential ${\textbf{V}_c}$ (uniquely defined on the conductor ${\Omega}_c$) [@biro2007coulomb] where they satisfy $$\label{defA}
\begin{cases}
{\mathbf{E}}= i\omega{\mathbf{A}}+ {\nabla\textbf{V}_c}&\text{ on } {\Omega},\\
\mu{\mathbf{H}}= {\textbf{curl\hspace{.01in}}}{\mathbf{A}}&\text{ on } {\Omega}.
\end{cases}$$ In order to avoid singularity in this system, regarding the change of variables Eq., and ensure well-posedness; in the sense of the magnetic potential vector ${\mathbf{A}}$ is unique, it is classical and necessary to impose additional gauge conditions. In this work we consider Coulomb gauge conditions which read $$\label{CGCdiv}{\textbf{div}}{\mathbf{A}}=0\text{ in } {\Omega},$$ We also need to impose the boundary condition ${\mathbf{A}}{\cdot}{\boldsymbol{\vec{n}}}= 0$ on $\partial{\Omega}$ in order to close the problem. A classical technique [@MR2298698] incorporates the constraint Eq. using a penalization term $-\frac 1{\tilde\mu}{\nabla}{\textbf{div}}{\mathbf{A}}$, in the Ampère equation, where $\tilde\mu$ is a suitable average of $\mu$ in ${\Omega}$. Therefore a strong formulation (${\mathbf{A}},{\nabla\textbf{V}_c}$) of our eddy-current problem writes $$\label{strongPbAV} \begin{cases}
{\textbf{curl\hspace{.01in}}}\big( \dfrac 1 \mu{\textbf{curl\hspace{.01in}}}{\mathbf{A}}\big) -\frac 1 {\tilde\mu}{\nabla}{\textbf{div}}{\mathbf{A}}-\sigma (i\omega{\mathbf{A}}- {\nabla\textbf{V}_c}) = {\mathbf{J}_e}\!\!\!\!&\text{in }{\Omega},\\
{\textbf{div}}\big( i\omega\sigma{\mathbf{A}}+\sigma{\nabla\textbf{V}_c}\big) = 0&\text{in }{\Omega}_c,\\
\big( \sigma i\omega{\mathbf{A}}+\sigma{\nabla\textbf{V}_c}\big){\cdot}{\boldsymbol{\vec{n}}}= 0&\text{on }\Gamma,\\
{\mathbf{A}}{\cdot}{\boldsymbol{\vec{n}}}= 0 &\text{on }\partial{\Omega},\\
\big(\dfrac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}\big)\times{\boldsymbol{\vec{n}}}= 0 &\text{on }\partial{\Omega},
\end{cases}$$ where ${\textbf{V}_c}$ is determined up to an additive constant.
Consider the space $H(\textbf{curl},\Omega)\cap H_0(\textbf{div},\Omega)$. where $H(\textbf{curl};\Omega):=\{{\bf u}\in (L^2(\Omega)\big)^3 \,|\, {\textbf{curl\hspace{.01in}}}{\bf u}\in(L^2(\Omega)\big)^3 \},$ and $H(\textbf{div};\Omega):=\{ {\bf u}\in(L^2(\Omega)\big)^3\,|\, \nabla{\cdot}{\bf u}\in L^2(\Omega) \},$ also we have $ H_0(\textbf{div};\Omega):=\{{\bf u}\in H(\textbf{div};\Omega)\, |\, {\bf u}{\cdot}{\boldsymbol{\vec{n}}}_{|\partial\Omega}=0\}.$
Let us take test functions $\Phi\in H(\textbf{curl};\Omega)\cap H_0(\textbf{div};\Omega)$ and $\varphi\in H^1(\Omega_c)$ for the Eq. $_1$ and the Eq. $_2$ respectively. After integrating by part we obtain the following weak formulations: $$\begin{aligned}
\int_{\Omega}\dfrac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}{\cdot}{\textbf{curl\hspace{.01in}}}\overline{\Phi} {\,\delta v}+\dfrac{1}{\tilde\mu}\int_{\Omega}{\textbf{div}}{\mathbf{A}}{\textbf{div}}\overline{\Phi} {\,\delta v}\notag\\
\hspace{-2cm}- \int_{\Omega_c}\sigma(i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}){\cdot}\overline{\Phi} {\,\delta v}= \int_{\Omega}{\mathbf{J}_e}{\cdot}\overline{\Phi} {\,\delta v}\label{varfcurl} \\
\hspace{-2cm}\int_{\Omega_c}\sigma\big(i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}\big){\cdot}\overline{\nabla\varphi} {\,\delta v}= 0\label{varflaplace} .\end{aligned}$$
We multiply Eq. by $\dfrac{-1}{i\omega}$ to obtain : $$\displaystyle\dfrac{-1}{i\omega}\int_{\Omega_c}\sigma\big(i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}\big){\cdot}\overline{\nabla\varphi} {\,\delta v}= 0.$$ and couple this with Eq. in a single mixed weak variational formulation, which is written: $$\begin{aligned}
&\int_{\Omega}\dfrac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}{\cdot}\overline{{\textbf{curl\hspace{.01in}}}\Phi} {\,\delta v}+\dfrac{1}{\tilde\mu}\int_{\Omega}{\textbf{div}}{\mathbf{A}}\overline{{\textbf{div}}\Phi} {\,\delta v}\\
&- \dfrac{1}{i\omega}\int_{\Omega_c}\sigma(i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}){\cdot}(i\omega\overline\Phi+\overline{{\nabla}\varphi}) {\,\delta v}\\
&=\int_{\Omega}\!\!\!{\mathbf{J}_e}{\cdot}\overline{\Phi} {\,\delta v}.\end{aligned}$$ For sake of simplicity, we define the sesquilinear form $\mathcal{L}({\mathbf{A}},{\textbf{V}_c},\Phi,\varphi)$ as the right-hand side of the above, which is written: $$\begin{aligned}
\label{sesquil}\displaystyle
\mathcal{L}\big({\mathbf{A}},{\textbf{V}_c},\Phi,\varphi\big):=\int_{\Omega}\dfrac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}{\cdot}\overline{{\textbf{curl\hspace{.01in}}}\Phi} {\,\delta v}\\+\dfrac{1}{\tilde\mu}\int_{\Omega}{\textbf{div}}{\mathbf{A}}\overline{{\textbf{div}}\Phi} {\,\delta v}\notag\\
- \dfrac{1}{i\omega}\int_{\Omega_c}\sigma(i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}){\cdot}(i\omega\overline{\Phi}+\overline{{\nabla}\varphi}) {\,\delta v}\notag.\end{aligned}$$
and which represents a coupled system between the two potential variables. We quote hereafter the part that constitutes the coupled system given as variational forms
$$\begin{aligned}
\mathcal{L}_{11}\big({\mathbf{A}},\Phi\big)=\int_{\Omega}\dfrac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}{\cdot}\overline{{\textbf{curl\hspace{.01in}}}\Phi} {\,\delta v},\\+\dfrac{1}{\tilde\mu}\int_{\Omega}{\textbf{div}}{\mathbf{A}}\overline{{\textbf{div}}\Phi} {\,\delta v}\notag\\
\mathcal{L}_{12}\big({\textbf{V}_c},\Phi\big)=-\int_{\Omega_c}\sigma {\nabla\textbf{V}_c}{\cdot}\overline{\Phi} {\,\delta v}.\\
\mathcal{L}_{21}\big({\mathbf{A}},\varphi\big)=-\int_{\Omega_c}\sigma{\mathbf{A}}{\cdot}\overline{{\nabla}\varphi} {\,\delta v},\\
\mathcal{L}_{22}\big({\textbf{V}_c},\varphi\big)=- \dfrac{1}{i\omega}\int_{\Omega_c}\sigma{\nabla\textbf{V}_c}{\cdot}\overline{{\nabla}\varphi} {\,\delta v}.\end{aligned}$$
We describe briefly in the sequel how one can dismiss the volume part of the TSP by using the appropriate boundary condition since it has high conductivity as compared with the tube, its corresponding skin depth is then very small.
![Triangulation of the conductive materials in the steam generator. Tube and Tube-support-plate (left), and their respective vertical-slice (right).[]{data-label="condpart"}](fig1 "fig:"){height="8cm" width="4cm"} ![Triangulation of the conductive materials in the steam generator. Tube and Tube-support-plate (left), and their respective vertical-slice (right).[]{data-label="condpart"}](fig2 "fig:"){height="8cm" width="4cm"}
Taking into account the effect of the TSP using the 3D model, described above, which requires a very thin mesh size (proportional to the skin depth) inside the TSP and leads to a huge size of the discrete 3D problem. We hereafter explain how one can avoid integration over the volume of the TSP by imposing an appropriate impedance boundary condition (IBC) on its boundary $\Gamma_p$. More precisely it is shown in [@durufle2006higher] that electromagnetic field satisfies $$\label{IBC}
{\boldsymbol{\vec{n}}}\times\frac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}= -\dfrac{1}{\mathcal{Z}_{\Gamma_{p}}}(i\omega{\mathbf{A}}_\tau+{\bf {\nabla}V}_{c,\tau}), \quad\text{ on } \Gamma_{p}.$$ (up to $O(\delta^2)$) on $\Gamma_{p}$. In the above equation; $\mathcal{Z}_{\Gamma_{p}}:= (1-i)\slash(\delta\sigma)$ with the skin depth $\delta:=\sqrt{ 2\slash(\omega\mu\sigma)}$ and the tangential component of the vector field ${\mathbf{A}}_T={\boldsymbol{\vec{n}}}\times\big( {\mathbf{A}}\times{\boldsymbol{\vec{n}}}\big)$ (same apply for ${\textbf{V}_c}$). Therefore, if $\delta$ is sufficiently small i.e. $\omega\sigma_p\mu$ is sufficiently large and Eq. is a very good approximation.
The impedance surface term of the weak formulation of Eq. at the interface $\Gamma_p$ of the TSP is written: $$\label{ibcROTA}\footnotesize
\int_{\Gamma_{p}}\!\!\!\!\!\!\! \big({\boldsymbol{\vec{n}}}\times(\frac{1}{\mu_p}{\textbf{curl\hspace{.01in}}}{\mathbf{A}})\big){\cdot}\overline{{\Phi_{\tau}}} {\,\delta s}=\dfrac{-1}{\mathcal{Z}_{\Gamma_{p}}}\!\!\! \int_{\Gamma_{p}}\!\!\!\! (i\omega{\mathbf{A}_{\tau}}+{\nabla_{\tau}\textbf{V}_c}){\cdot}\overline{{\Phi_{\tau}}} {\,\delta s}$$ $$\label{ibcNORM}\footnotesize
\int_{\Gamma_{p}}\!\!\!\!\!\!\! \sigma_p(i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}){\cdot}{\boldsymbol{\vec{n}}}\,\overline{\varphi} {\,\delta s}=\\\dfrac{-1}{\mathcal{Z}_{\Gamma_{p}}}\int_{\Gamma_{p}}\!\!\!\!(i\omega{\mathbf{A}_{\tau}}+{\nabla_{\tau}\textbf{V}_c}){\cdot}\overline{{\nabla_{\tau}}\varphi} {\,\delta s},$$ We refers for instance to [@haddar:hal-01044648] for details related to the update on the variational formulation with respect to the impedance boundary condition. Here, we shall consider the general case in the numerical experiments, which consist of the volume representation of the TSP in the variational formulation.
The mathematical formulation for the evaluation of the electric impedance ${\mathbf{Z}}$ see [@auld1999review]. It is common to use the following types of signals $$\label{impedmod}
\begin{cases}
{\bf Z}_{FA} = \frac{i}{2}\big({\Delta\mathbf{Z}}_{11}+{\Delta\mathbf{Z}}_{12}\big),\\
{\bf Z}_{F3} =\frac{i}{2} ({\Delta\mathbf{Z}}_{11}-{\Delta\mathbf{Z}}_{22}\big).
\end{cases}$$ which are respectively the absolute signal mode and the differential signal mode. The term ${\mathbf{Z}}_{kl}$ with $k,l$ in $\{1,2\}$ represents the volume impedance measured with the coil $k$ in the electromagnetic field induced by the coil $l$. It is written: $$\begin{aligned}
\label{impedkl}
\displaystyle
{\Delta\mathbf{Z}}_{kl}:=\frac{1}{i\omega}\frac{\mu_\epsilon-\mu_d}{\mu_d\mu_\epsilon}\int_{{\Omega}_{d}}\big( {\textbf{curl\hspace{.01in}}}{\mathbf{A}}_k{\cdot}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}_l^{\epsilon}\big){\,\delta v}\notag\\
+ (\sigma-\sigma_{\epsilon})\!\!\!\int_{{\Omega}_d}\!\!\!\!\!(i\omega{\mathbf{A}}_k + {\nabla\textbf{V}_c}){\cdot}(i\omega{\mathbf{A}}_{\tau,l}^{\epsilon}+{\bf {\nabla}V}^{\epsilon}_{c,l,\tau}) {\,\delta v},\end{aligned}$$ where $i\omega{\mathbf{A}}_{\tau,l}^{\epsilon}+{\bf {\nabla}V}^{\epsilon}_{c,l,\tau}:=:{\mathbf{E}}_{l,\tau}^0$ refers to the tangential component of the electric field propagating in “vacuum”. Rigorously one has to take care of the presence of loop field [@MR3090156] in the vacuum. Their presence is related to the geometry of the computational domain (see [@MR2680968; @bossavit2004electromagnetisme] and reference therein).
As eddy-current approximation of the Maxwell equation can be seen as the limit when the permittivity $\varepsilon$ tends to zero in Maxwell equations. We may use the vacuum as a medium with very low permittivity. In our case we consider a very low conductivity $\sigma_\epsilon\approx \epsilon<<1$ to describe the vacuum instead.
Numerical method
================
We describe in this section our numerical method which consists of the resolution of the coupled systems that solve the eddy-current problems necessary for the evaluation of the electric impedance ${\mathbf{Z}}$ defined at Eq. . We describe the major steps for the non-destructive test procedure and give details related to the resolution where numerical decisions have been made in order to better handle big ill-conditioned systems.
We suppose from now on, that the computational domain $\Omega$ is a Lipschitz polyhedra in $\mathbb{R}^3$, practically $\Omega$ represent the cylinder that envelops the Tube and the TSP. Once the computational domain is fixed, we then introduce a family of triangulation $\mathcal{T}_{\natural}$ of $\Omega$, the subscript $\natural$ stands for the largest length of the edges of the triangles that constitute $\mathcal T_{\natural}$. The Tetrahedrons of $\mathcal T_{\natural}$ match on the interface between the conductive part, i.e. tube and TSP ($\sigma\neq 0$) and the insulator part ($\sigma=0$). The triangulations of the conductive part are is in Figure \[condpart\].
The electric field ${\mathbf{E}}\equiv({\mathbf{A}},{\textbf{V}_c})$ solution of the eddy current equation belongs to $H({\textbf{curl\hspace{.01in}}},\Omega) \cap H({\textbf{div}},\Omega)\times H^{1}(\Omega_c)$. The fact that our computational domain (Cylinder as mentioned before) is a convex polyhedron is very important. It turns out that the space of smooth tangential vector fields $H_\tau^1(\Omega):=\big(H^{1}(\Omega)\big)^3\cap H^{}({\textbf{div}},\Omega)$ coincides with the proper close subspace of $H({\textbf{curl\hspace{.01in}}},\Omega) \cap H({\textbf{div}},\Omega)$. Thanks to this nice property, a finite element numerical approximation based on a nodal finite element will be considered for the electric vector potential ${\textbf{V}_c}$ as well as for the magnetic vector potential ${\mathbf{A}}$ [@biro2007coulomb]. The infinite dimensional space $H_\tau^1(\Omega)\times H^{1}(\Omega)$ is therefore approximated with the finite-dimensional space $\mathcal W_\natural:=\mathcal V_\natural\times \mathcal C_\natural$, where $\mathcal V_\natural$ and $\mathcal C_\natural$ represent the discrete spaces of Lagrange nodal elements defined respectively as $$\begin{aligned}
\mathcal{V}_{\natural}:=\{ {\bf v}_\natural\in (C^0(\Omega))^3 | {\bf v}_{\natural_{|K}}\in (\mathbb P_1)^3, \forall K\in\mathcal T_\natural,\\ {\bf v}_\natural . {\boldsymbol{\vec{n}}}=0 \text{ on } \partial \Omega\}\end{aligned}$$ and $\mathcal{C}_{\natural}:=\{ c_\natural\in C^0(\Omega_c)| v_{\natural_{|K}}\in \mathbb P_1, \forall K\in\mathcal T_{c,\natural}\}$.
In the above $\mathbb P_1$ stands for the space of polynomials of degree less than or equal to $1$. In addition, the boundary conditions (${\mathbf{A}}{\cdot}{\boldsymbol{\vec{n}}}=0$ on $\partial\Omega$) are taken into account via penalization of the vortices that belong on the boundaries.Again thanks to geometric property of our computational domain, imposing this kind of boundary condition becomes trivial.
As stated earlier, the scan procedure consists in moving the probe according to the z-direction all the way through the tube. This leads to a set of scan positions where at each position we need to solve an ill-conditioned huge system. In addition to that, for any position one has to rebuild the triangulation and reassemble the matrices. To avoid this handicap we are led to consider a unique triangulation that includes all probe positions and then consider a factorization using a sophisticated software.
In order to proceed with the parallelization, let us consider $P$ partitions of the computational domain ${\Omega}$, such that ${\Omega}=\cup_{p=1}^{P}{\Omega}^p$. We notice here that the partition is made in terms of the triangulation see Figure \[partitionTriangle\], where after the construction of the mesh, we partition its triangulations using a graph partitioner (e.g. Scotch [@Pellegrini01scotchand] or Metis [@Karypis95metis]).
Both graph partitioners Scotch and Metis are interfaced with the software Freefem++ [@MR3043640], and their utilization is quite easy.
We give a sample Freefem++ script in Table \[partition\], where the graph partition scotch is called at line 6 to build a piecewise function “balance” which takes different values on each subdomain. Freefem++ is therefore able to construct local triangulation according to the values of the function “balance” (see lines 11-12 of Table \[partition\]). We notice that the construction of the function “balance” is performed only on the master processor with rank zero. The function “balance” is thus broadcasted to all processors through MPI (see line 9 of Table \[partition\]). We use this technique to ensure that all processors receive the same function balance and construct a uniform triangulation.
The numerical simulation of the ECT procedure reads as follows
1. Read/build a triangulation of the computational domain ${\Omega}$
2. Partition the triangulation with respect to the arbitrary choice (see command line in Tabular \[partition\])
3. Assemble Morse type sparse-sub-matrices in parallel
4. All\_reduce sparse-sub-matrices (This operation is achieved with MPI\_SUM operation)
5. LU factorization of the global matrix.
6. Solve the set of problems $P_i$ by uniquely changing the right-hand side source term (accordingly to the position of the probe position)
7. Print out the impedance value results in an append writing file (to be sorted).
Let $\Phi_\natural\in \mathcal{V}_{\natural}$ and $\varphi_\natural\in \mathcal{C}_{\natural}$ be test functions for our coupled eddy-current problem, we denote by ${\mathbf{A}}_{\natural}$ and ${\textbf{V}_{c,\natural}}$ our unknowns. Local matrices need the following variational formulation in order to be assembled $$\begin{aligned}
\mathcal{L}^{p}_{11}\big({\mathbf{A}}_{\natural},\Phi_{\natural}\big)\!\!\!\!\!\!&=&\!\!\!\!\!\!\sum_{K\in\mathcal{T}_h\subset{\Omega}^{p}}\int_{K}\dfrac{1}{\mu}{\textbf{curl\hspace{.01in}}}{\mathbf{A}}_{\natural}{\cdot}\overline{{\textbf{curl\hspace{.01in}}}\Phi_{\natural}} {\,\delta v},\notag\\
&\qquad&+\dfrac{1}{\tilde\mu}\int_{K}{\textbf{div}}{\mathbf{A}}_{\natural}\overline{{\textbf{div}}\Phi_{\natural}} {\,\delta v}\\
\mathcal{L}^{p}_{12}\big({\textbf{V}_{c,\natural}},\Phi_{\natural}\big)\!\!\!\!\!\!&=&\!\!\!\!\!\!-\!\!\!\!\sum_{K\in\mathcal{T}_h\subset{\Omega}_c^{p}}\int_{K}\sigma {\textbf{V}_{c,\natural}}{\cdot}\overline{\Phi_{\natural}} {\,\delta v}.\\
\mathcal{L}^{p}_{21}\big({\mathbf{A}}_{\natural},\varphi_{\natural}\big)\!\!\!\!&=&\!\!\!\!-\!\!\!\!\sum_{K\in\mathcal{T}_h\subset{\Omega}_c^{p}}\int_{K}\sigma{\mathbf{A}}_{\natural}{\cdot}\overline{{\nabla}\varphi_{\natural}} {\,\delta v},\\
\mathcal{L}^{p}_{22}\big({\textbf{V}_{c,\natural}},\varphi_{\natural}\big)\!\!\!\!\!\!&=&\!\!\!\!\!\!-\dfrac{1}{i\omega}\!\!\!\!\!\!\sum_{K\in\mathcal{T}_h\subset{\Omega}_c^{p}}\int_{K}\sigma{\textbf{V}_{c,\natural}}{\cdot}\overline{{\nabla}\varphi_{\natural}} {\,\delta v}.\end{aligned}$$ We give in Table \[varf\] a sample-freefem script that shows how to implement such bilinear forms.
$$\begin{array}{lr}
{\bf M}_{11} = \sum_{p=1}^{P}M^{p}_{11}, &\text{ with } { M}^{p}_{11}= \mathcal{L}^{p}_{11}(\Phi_{\natural},\Phi_{\natural}),\\
{\bf M}_{12} = \sum_{p=1}^{P}M^{p}_{12}, &\text{ with } { M}^{p}_{12}=\mathcal{L}^{p}_{12}(\varphi_{\natural},\Phi_{\natural}),\\
{\bf M}_{21} = \sum_{p=1}^{P}M^{p}_{21}, &\text{ with } { M}^{p}_{21}=\mathcal{L}^{p}_{21}(\Phi_{\natural},\varphi_{\natural}),\\
{\bf M}_{22} = \sum_{p=1}^{P}M^{p}_{22}, &\text{ with } { M}^{p}_{22}=\mathcal{L}^{p}_{22}(\varphi_{\natural},\varphi_{\natural}).
\end{array}$$
The sparse matrice summations above are performed through the MPI Reduce operation. Thanks to the Morse sparse format of the matrices, the summations is not a term-by-term addition, but it increases the dimension at each sum, because of the assembly in disjoint subdomains; also all assembly uses the same mesh numbering.
The full discretized system thus reads $$\left(\begin{array}{cc}
{\bf M}^{}_{11} & {\bf M}^{}_{12}\\
{\bf M}{}_{21} & {\bf M}^{}_{22}
\end{array}\right)
\left(\begin{array}{l}{\mathbf{A}}_{\natural}\\{\textbf{V}_{c,\natural}}\end{array}\right) =\left(\begin{array}{l} {{\mathbf{J}_e}}_{,\natural}\\ {\bf 0}\end{array}\right)$$
The right hand side vector ${{\mathbf{J}_e}}_{,\natural}$ stands for the source term, which is basically given by the multiplication of the mass matrix by the interpolation of the analytic source term ${\mathbf{J}_e}$.
int[int] nupart(Th.nt);
fespace p0h(Th,P03d);
p0h balance;
int npart=mpisize;
if(mpirank==0){
scotch(nupart, Th, npart);
for(int i=0;i<Th.nt;i++)
balance[][i] = nupart[i];}
broadcast(processor(0,com),balance[]);
Th3[int] Thpart(npart);
for(int i=0;i<npart;i++) {
Thpart[i]=trunc(Th,balance==i);
Thpart[i]=change(Thpart[i],fregion=i);}
Th=Thpart[0];
for(int i=1;i<npart;i++) Th=Th+Thpart[i];
varf L11([Ax,Ay,Az],[Bx,By,Bz]) =
int3d(Th,mpirank)(1/mu*curl(Bx,By,Bz)*curl(Ax,Ay,Az) )
- int3d(Th,mpirank)( iomega*mu*sigma* [Bx,By,Bz]*[Ax,Ay,Az] )
+ int3d(Th,mpirank)((div(Bx,By,Bz))*(div(Ax,Ay,Az)))
+ on(labelup,labeldown,Az=0.)
+ on(labelmid,Ax=0., Ay=0.);
varf L22(V,qhc) =
int3d(ThC,mpirank)( (-1/iomega) * (
mu*sigmaEpsilon* [dx(qhc),dy(qhc),dz(qhc)]*[dx(V),dy(V),dz(V)] )
+ delta*mu*sigma*V*qhc ) );
varf L12([V],[Bx,By,Bz]) =
- int3d(ThC,mpirank)( mu*sigma* ( dx(V)*Bx+dy(V)*By+dz(V)*Bz) );
varf L21([Ax,Ay,Az],[qhc]) =
- int3d(ThC,mpirank)( mu*sigma* [dx(qhc),dy(qhc),dz(qhc)]*[Ax,Ay,Az] );
matrix <complex> M11,Mp11= L11(VPh,VPh);
matrix <complex> M22,Mp22= L22(PhC,PhC);
matrix <complex> M12,Mp12= L12(PhCS,VPh);
matrix <complex> M21,Mp21= L21(VPh,PhCS);
mpiAllReduce(Mp11,M11,mpiCommWorld,mpiSUM);
mpiAllReduce(Mp12,M12,mpiCommWorld,mpiSUM);
mpiAllReduce(Mp21,M21,mpiCommWorld,mpiSUM);
mpiAllReduce(Mp22,M22,mpiCommWorld,mpiSUM);
matrix<complex> M = [[M11,M12],
[M21,M22]];
set(M,solver=sparsesolver,eps=1.e-16);
![Partition of the Triangulation of the computational domain. Volume tetrahedrons surfaces are presented in top left and bottom right, the other plot correspond to the interfaces as required by the geometry of the SG and also interfaces generated by the partition.[]{data-label="partitionTriangle"}](Th000.pdf "fig:"){height="5cm" width="4cm"} ![Partition of the Triangulation of the computational domain. Volume tetrahedrons surfaces are presented in top left and bottom right, the other plot correspond to the interfaces as required by the geometry of the SG and also interfaces generated by the partition.[]{data-label="partitionTriangle"}](Th001.pdf "fig:"){height="5cm" width="4cm"} ![Partition of the Triangulation of the computational domain. Volume tetrahedrons surfaces are presented in top left and bottom right, the other plot correspond to the interfaces as required by the geometry of the SG and also interfaces generated by the partition.[]{data-label="partitionTriangle"}](Th002.pdf "fig:"){height="5cm" width="4cm"} ![Partition of the Triangulation of the computational domain. Volume tetrahedrons surfaces are presented in top left and bottom right, the other plot correspond to the interfaces as required by the geometry of the SG and also interfaces generated by the partition.[]{data-label="partitionTriangle"}](Th003.pdf "fig:"){height="5cm" width="4cm"}
The boundary condition ${\mathbf{A}}\cdot{\boldsymbol{\vec{n}}}=0$ on the exterior boundary $\partial{\Omega}$ are taken into account using the penalization term, where the linear system is forced to take into account the given values at the boundary mesh points.
Numerical Experiments
=====================
In our application we consider that the excitation source term ${\mathbf{J}_e}$ is uniformly distributed on a support included in ${\Omega}_I$ (principally it models the solenoid source coil of the probing problem). So, let ${\mathbf{J}_e}\in (L^2({\Omega}))^3$ with ${\textbf{div}}{\mathbf{J}_e}=0$ in ${\Omega}_I$ as ${\mathbf{J}_e}:=:{[-y_{|_{{\Omega}_I}},x_{|_{{\Omega}_I}},0]}\slash{\sqrt{x^2+y^2}}$.
As the electric scalar potential ${\textbf{V}_c}$ is determined up to an additive constant, we may numerically impose a supplement condition such that $\int_{{\Omega}_{c_i}} \!\!\!\!\!\!{\nabla\textbf{V}_c}{\,\delta v}= 0$ (${\Omega}_{c_i}$ is any connex component subset of ${\Omega}_c.$). This also could be incorporated under the global problem by penalization $\delta \sigma{\nabla\textbf{V}_c}, \text{ in } {\Omega}_c, \text{ with a small } \delta<<1.$ Therefore we augment the variational formulation $\mathcal{L}^{p}_{22}$ by the former penalization integration.
P in MPI $\text{Assemble}\left(M^0_i\right)$ $\text{Reduce}\left(M^0\right)$ $LU|^0$
---------- ------------------------------------- --------------------------------- --------- --
1 3144.68 – 585.28
2 1202.98 585.74 440.36
4 758.73 575.86 582.76
8 312.08 512.76 501.25
16 163.87 591.56 596.33
32 88.74 542.30 577.28
64 19.02 516.50 562.61
: Parallel Performance in wall-clock time in seconds for the problem without TSP as default. This means the numerical simulation takes the region of the TSP as a “vacuum”.[]{data-label="TableM0"}
P in MPI $\text{Assemble}\left(M^1_i\right)$ $\text{Reduce}\left(M^1\right)$ $LU|^1$
---------- ------------------------------------- --------------------------------- --------- --
1 3237.47 - 341.22
2 1284.52 392.18 332.61
4 764.47 587.49 332.44
8 311.28 687.63 497.75
16 177.14 647.32 368.43
32 42.37 583.32 328.61
64 19.55 598.87 350.10
: Parallel Performance in wall-clock time in seconds for the problem with TSP.[]{data-label="TableM1"}
We give in Table \[TableM0\] and Table \[TableM1\] the performance in term of the execution (wall-clock) time in seconds of our numerical method. In Table \[TableM0\] (respectively Table \[TableM1\]) we report results related to the system without TSP as default (respectively with TSP region as default), where the computed matrix has size $n\times m=991.246\times 991.246$ (respectively $n\times m=1.069.595\times1.069.595$) with $46.664.492$ (respectively $54.350.718$) non-zero coefficients. It is worth recalling that these calculations are necessary for the evaluation of the impedance signals Eq. . As the results indicate, the most memory and time consuming task (the main matrix for the coupled system) is mitigated through high performance computing using parallel resources, where scalability of the operation is clearly exhibited. MPI communications are thus necessary to collect local Morse sparse matrices. These operations, thanks to the optimized communication technique, enable the maintenance of a reasonable average of performance despite the increase of the communicators number. Actually, this fact is balanced with the size of the message that has to be transferred. Indeed, as the communicators numbers increases, the size of the morse sparse matrices decreases. Once the global matrix is assembled and copied (via MPI) to all processors, we are then able to perform a fast factorization. In Figure \[PlotE\] we plot the intensity of the computed electric field when the probe is near to the TSP. We have used Super\_LU [@li05] parallel direct solver for the factorization. This procedure has also a good performance while using several processors. As the scan procedure is parallel by nature (each probe position is totally independent of other positions). Therefore, parallel resources share the effort of solving independently coupled systems according to different probe positions. We report in Figure \[speedup\] the robustness of our method in term of speedup while increasing the number of parallel processors $P$. We evaluate the speedup formula given by $S^{p}=\text{t}_\text{tot}^{p}\slash t^\text{serial}$ with $\text{t}_\text{tot}^{p} = \text{t}_\text{Assemble}^{p} + \text{t}_\text{Reduce}^{p} + \text{t}_\text{Factorize}^{p} + \text{t}_\text{Solve}^{p}$. We have considered $128$ probe positions for the scan procedure where the time to solve the problem is in average $40$ minutes. The results exhibit the fact that our method is fully efficient and scalable in term of parallelism.
![The intensity of the computed Electric field ${\mathbf{E}}=i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}$: YZ-plan slice projection (top left) and XY-plan slice projections (top right). Volume rescale plot (bottom left) and Wireframe projection rescaled plot (bottom right).[]{data-label="PlotE"}](x_sol_magnetude "fig:"){height="4cm" width="4cm"} ![The intensity of the computed Electric field ${\mathbf{E}}=i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}$: YZ-plan slice projection (top left) and XY-plan slice projections (top right). Volume rescale plot (bottom left) and Wireframe projection rescaled plot (bottom right).[]{data-label="PlotE"}](z_sol_magnetude "fig:"){height="4cm" width="4cm"} ![The intensity of the computed Electric field ${\mathbf{E}}=i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}$: YZ-plan slice projection (top left) and XY-plan slice projections (top right). Volume rescale plot (bottom left) and Wireframe projection rescaled plot (bottom right).[]{data-label="PlotE"}](sol_surface "fig:"){height="4cm" width="4cm"} ![The intensity of the computed Electric field ${\mathbf{E}}=i\omega{\mathbf{A}}+{\nabla\textbf{V}_c}$: YZ-plan slice projection (top left) and XY-plan slice projections (top right). Volume rescale plot (bottom left) and Wireframe projection rescaled plot (bottom right).[]{data-label="PlotE"}](sol_wireframe "fig:"){height="4cm" width="4cm"}
![Speedup of the numerical method.[]{data-label="speedup"}](Speedup.pdf){height="6cm" width="6cm"}
Conclusions {#Conclusions}
===========
We have presented in this paper a finite element technique that enables a rapid numerical simulation of the ECT problem in SG. Our approach has been validated through the use of high performance computing in parallel machines. We have shown that our approach is full efficient and leads to a robust and rapid ECT in real life industrial problem.
Acknowledgments {#Acknowledgements .unnumbered}
===============
This work was made possible by the facilities of the New Jersey Institute of Technology Shared Hierarchical Academic Research Computing Network. The author gratefully thanks Professor Frederic Hecht for the earliest discussion on Freefem++-mpi and also for the helpful introductory examples uploaded on the Freefem++ website.
[^1]: Corresponding author. Email: riahi@njit.edu, Tel.: +1 (973)-5966-084.
|
---
abstract: 'We show that atoms or molecules subject to fields that couple their internal and translational (momentum) states may undergo a crossover from randomization (diffusion) to strong localization (sharpening) of their momentum distribution. The predicted crossover should be manifest by a drastic change of the interference pattern as a function of the coupling fields.'
author:
- 'Nir Bar-Gill and Gershon Kurizki'
title: 'Signatures of Strong Momentum Localization via Translational-Internal Entanglement'
---
It is well known that potential energy disorder added to a spatial lattice of energy degenerate sites may cause localization of an otherwise delocalized (hopping) quantum mechanical particle [@Anderson]. This localization is crucially dependent not only on the amount of disorder, but also on the lattice dimensionality [@Anderson; @Akkermans; @Akulin]. Momentum localization may be caused by time-dependent perturbations [@fishman]. Here, we consider the hitherto unexplored analogs of these phenomena in the [*momentum space of a diffracted particle*]{}, due to [*correlations of internal and translational states*]{} within the particle. We show that such translational-internal entanglement (TIE) may incur disorder that causes a crossover from diffusion to strong localization of the momentum distribution and thereby a drastic change in the diffraction pattern, even for particles with few internal levels. The first point we must address is: how to create the intra-particle momentum-space analog of a perfectly ordered lattice, in which all sites have equal energy? The free particle energy-momentum relation implies that each momentum state carries a different energy. An ordered lattice can however be achieved, by correlating (entangling) a selected set of [*discrete*]{} momentum states with eigenstates of different degrees of freedom. This means that the corresponding states $| \vec{k}_n, n \rangle$, where $\hbar \vec{k}_n$ stands for the momentum of state $n$, and $\varepsilon_n$ for the energy eigenvalues of its additional degrees of freedom, satisfy in an ordered, momentum-space “lattice” $$\frac{\hbar^2 k_n^2}{2M} + \varepsilon_n = \frac{\hbar^2 k_{n'}^2}{2M} + \varepsilon_{n'} \label{en_eq}$$ for all $n,n'$. It will be shown later that such TIE may also yield momentum lattices which deviate from condition (\[en\_eq\]) and have [*controllable dimensionality and disorder*]{}. Alternatively to TIE, one may entangle orthogonal momentum components of the particle. However, this has the limitation of complicating the resulting diffraction pattern (see below).
In order to create such a TIE particle, consider a cold atom or molecule (assuming a non-interacting ensemble thereof) initially prepared in a single internal state $| 0 \rangle$ and in a narrow wavepacket of momentum states centered around $\hbar \vec{k}_0$. The state $|\vec{k}_0,0 \rangle$ can next be coupled to a band of $N$ long-lived, non-degenerate states $| \vec{k}_n,n \rangle$ via Raman near-resonant pairs of laser beams (see Fig. \[setup\](a)), analogously to [@chu]. In turn, these $N$ states may be quasi-adiabatically Raman-coupled among themselves. Examples of appropriate systems are atomic ground-state hyperfine/Zeeman sublevels, multiplets of circular Rydberg states, or rovibrational bands of a molecular ground state. Following the outlined quasi-adiabatic Raman sequence, the system occupies a “lattice” of entangled momentum-internal states, whose dimensionality and disorder are controllable by the Raman couplings, as detailed below. The system Hamiltonian $H_S$ and the dipolar system-field interaction Hamiltonian $H_I$ in the Rotating Wave Approximation (RWA) are [@Scully] $$\begin{aligned}
H_S &=& \hbar \sum_{n=1}^N \omega_n^{(g)} |\vec{k}_n,n \rangle_g \langle \vec{k}_n,n |_g \nonumber \\
H_I &=& \hbar \sum_{n=1}^N \left[ \Omega_{ne}^{(+)} e^{-i (\nu_{+,n} t + \vec{q}_{+,n} \cdot \vec{r} )} | \left \{ \vec{k}_e,e \right \} \rangle \langle \vec{k}_n,n |_g \right. \\
&+& \left. \sum_{n'=1}^N \Omega_{en'}^{(-)} e^{i (\nu_{-,n'} t + \vec{q}_{-,n'} \cdot \vec{r})} | \vec{k}_{n'},n' \rangle_g \langle \left \{ \vec{k}_e,e \right \} | \right] + h.c. \nonumber\end{aligned}$$ Here, $|\vec{k}_n,n \rangle_g$ are the states of the stable (long-lived) band with momenta $(\hbar \vec{k}_n)_g$ and energies $\hbar \omega_n^{(g)}$, whereas $| \left \{ \vec{k}_e,e \right \} \rangle$ refer to states of a far-detuned, [*intermediate*]{} unstable (electronically-excited) band. The $+$ and $-$ steps of the Raman coupling consist respectively of virtual (off-resonant) transitions $|\vec{k}_n,n \rangle_g \leftrightarrow |\left \{ \vec{k}_e,e \right \} \rangle \leftrightarrow |\vec{k}_n',n' \rangle_g$ via pairs of laser beams with frequencies $\nu_{+,n}$ and $\nu_{-,n'}$ and respective wavevectors $\vec{q}_{+,n}, \vec{q}_{-,n'}$. The corresponding Rabi frequencies are $\Omega_{ne}^{(+)}$ and $\Omega_{en'}^{(-)}$. The frequencies are chosen such that $\nu_{+,n} - \nu_{-,n'} \simeq \omega_{n'}^{(g)} - \omega_n^{(g)}$. Hence the two-step Raman process $|\vec{k}_n,n \rangle_g \rightarrow |\vec{k}_{n'},n' \rangle_g$ is near-resonant only for a chosen pair of laser beams, causing energy and momentum transfer of $\hbar (\nu_{+,n} - \nu_{-,n'})$ and $\hbar (\vec{q}_{+,n} - \vec{q}_{-,n'})$, respectively. In the frame transforming away the laser frequencies and wavevectors and upon [*adiabatically eliminating*]{} [@Scully] the intermediate unstable states $| \left \{ \vec{k}_e,e \right \} \rangle$ from the Schroedinger equation yields the following pairwise coupling equations for the eigenstate amplitudes $c_n$ and $c_{n'}$ $$\begin{aligned}
\dot{c}_{n'} & \simeq & \frac{i}{\hbar} (E_{n'} c_{n'} + J_{nn'} c_n), \nonumber \\
E_{n'} &=& \hbar (\nu_{+,n} - \nu_{-,n'} + \omega_n^{(g)} - \omega_{n'}^{(g)}), \nonumber \\
J_{nn'} & \simeq & \hbar (\Omega_{ne}^{(+)} \Omega_{en'}^{(-)})/\left[ \nu_{+,n} - (\omega^{(e)} - \omega_n^{(g)}) \right].
\label{H3}\end{aligned}$$ Here we have assumed sufficiently weak fields to neglect power broadening corrections (AC Stark shifts), as well as the off-resonant linewidths of the far-detuned fields $\gamma_{eg} \left[ \Omega_{n,e}^{(\pm)}/(\nu_{\pm,n} - \omega^{(e)} - \omega_{n}^{(g)}) \right]^2 << \gamma_{eg}$.
Even with the near-resonant Raman selectivity imposed on (\[H3\]), it allows for rich, complex dynamics. Here we wish to map (\[H3\]) onto known models of strong localization [@Anderson]. To this end, we require: $$\nu_{+,n} - \nu_{-,n'} + \omega_n^{(g)} - \omega_{n'}^{(g)} = const. ; J_{nn'} = J = const.
\label{H4}$$ These requirements amount to adjusting the frequencies $\nu_{\pm,n}$ and the two-step Rabi-frequency product to be independent of $n,n'$. The momenta $\hbar \vec{k}_{n'} = \hbar (\vec{k}_n + \vec{q}_{+,n} - \vec{q}_{-,n'})$ are separately controlled to give equal diagonal energies (ordered “lattice”) or random energies $E_n$ (disorderd “lattice”). The disorder is measured in terms of the width $\Delta$ of the flat ([*uncorrelated*]{}) distribution of on-diagonal energies, such that the random $E_n \in [-\Delta/2,\Delta/2]$.
\
Our setup allows the creation of momentum-space configurations with [*any effective dimensionality*]{}. This dimensionality is not related to the spatial dimensionality of the atomic or molecular ensemble, but rather to the off-diagonal coupling terms. A 1D momentum-space lattice is created by resonantly coupling states $| \vec{k}_n,n \rangle$ with the states $| \vec{k}_{n+1},n+1 \rangle$ and vice-versa. In order to fulfill periodic boundary conditions, in a system of finite size $N$, we couple $| \vec{k}_N,N \rangle \leftrightarrow | \vec{k}_1,1 \rangle$. The single-particle Hamiltonian describing the system in the entangled basis $|\vec{k}_i,i \rangle$ is represented by the matrix $$\begin{aligned}
H = \left (
\begin{array}{cccccc}
E_1 & -J & 0 & 0 & \cdots & -J \\
-J & E_2 & -J & 0 & 0 & \cdots \\
0 & -J & E_3 & -J & 0 & \cdots \\
\vdots & & & & & \\
-J & 0 & \cdots & 0 & -J & E_N
\end{array}
\right). \label{H1d}\end{aligned}$$ The ability to satisfy Eqs. (\[H3\]), (\[H4\]) so that (\[H1d\]) is realized is numerically demonstrated in Fig. 1(c): laser beam pairs selectively couple level pairs with equal $J$ but random $E_n$.
By adding Raman resonant laser-beam pairs, such that each state (“lattice site”) is coupled to an increasing number of other “sites”, the Hamiltonian emulates a system of higher dimensionality. Thus, 1D, 2D and 3D “lattices” are realized when each “site” has 2, 4 and 6 neighbors, respectively (see Fig. \[setup\](b)). Specifically, we assume the number of sites $N$ equals the system size $L$ in 1D, $L^2$ in 2D and $L^3$ in 3D. Therefore, the neighbors of site $i$ in 1D are $i \pm 1$, in 2D the neighbors $i \pm L$ are added, and in 3D - $i \pm L^2$ are added. This dimensionality argument holds as long as all $N$ “sites” have the same connectivity [@Anderson; @Akkermans; @Akulin], as we assume in the following. In the $N \rightarrow \infty$ (thermodynamic) limit, this system is equivalent to either a perfect or a disordered infinite lattice, depending on $\Delta$.
![Momentum distribution for random systems with $(\Delta/J)=1$ (red-dashed) and $(\Delta/J)=10$ (blue-solid), in a 1D system with $N=10$ states, compared to a periodic (regular) system with $(\Delta/J)=10$ (green-dotted). The random momentum distribution is delocalized for $(\Delta/J)=1$, while strong localization appears for $(\Delta/J)=10$. The periodic distribution is not localized for $(\Delta/J)=10$. A localization “crossover” is found even though a random 1D system should not exhibit a transition, due to finite size effects.[]{data-label="mom_dist_1d"}](mom_dist_1d2){width="0.8\linewidth" height="0.4\linewidth"}
As is well known, in 1D any amount of random disorder causes localization [@Anderson; @Akkermans; @Akulin]. In Fig. \[mom\_dist\_1d\] we present the results of our 1D lattice calculations for the adiabatic ramping-up of the two-photon coupling $J$, and different values of the diagonal disorder $\Delta$. We plot the ground-state 1D momentum distribution for $N=10$ different states $|\vec{k}_n,n \rangle$, a setup not too difficult to realize. It can be seen that finite size effects, resulting from the finite number of states N, are manifest as an [*artificial*]{} “localization crossover” in this 1D configuration (this occurs also in 2D): the localization length increases with a reduction in the strength of the disorder, until, for small enough disorder, the localization length exceeds the length of the system, causing the momentum distribution to appear delocalized.
In order to quantify the localization crossover for each effective dimensionality, one can use the ground-state momentum distribution, and apply such standard measures as the entropy or the Inverse Participation Ratio (IPR) [@Scalettar]. We suggest a different localization measure, appropriate for atomic or molecular interferometry [@Berman; @Arndt]: (a) Consider an internally structured particle that is incident upon an interferometer in a state with a narrow momentum distribution $f (\vec{k} - \vec{k_0})$ around $\vec{k_0}$ and an arbitrary superposition of internal-energy eigenstates $| n \rangle$: $$\begin{aligned}
| \psi_{in} \rangle &=& \int d \vec{k} f (\vec{k}-\vec{k_0}) | \vec{k} \rangle \sum_n c_n | n \rangle; \nonumber \\
\langle \vec{r} | \psi_{in} \rangle &\simeq& e^{i \vec{k_0} \cdot \vec{r}} \sum_n c_n | n \rangle. \label{in}\end{aligned}$$ In a Mach-Zehnder Interferometer (MZI), it is split between two alternative, interfering paths, $\vec{r}_1$ and $\vec{r}_2$. Because of the internal-translational [*factorization*]{}, such a state can exhibit a high-visibility interference pattern [@Arndt], since its propagation in the MZI results in $$\langle \vec{r} | \hat{U_0} |\psi_{in} \rangle = e^{i \vec{k_0} \cdot (\vec{r_1} - \vec{r_2})} \sum_n c_n |n\rangle. \label{standard}$$ (b) By contrast, the output state resulting from TIE propagation in the MZI is $$\langle \vec{r} | \psi_{out} \rangle = \langle \vec{r} | \hat{U}_{TIE} |\psi_{in}\rangle = \sum_n c_n e^{i \vec{k_n} \cdot ( \vec{r_1} - \vec{r_2} ) } | n \rangle, \label{tie_out}$$ where $c_n$ and $\vec{k}_n$ are subject to Eqs. (\[H3\])-(\[H4\]) above. The averaging of the detection probability of state (\[tie\_out\]) over $|n\rangle$ tends to wash out the interference fringes: $$\begin{aligned}
Tr_n \{ \langle \vec{r} | \psi_{out} \rangle \langle \psi_{out} | \vec{r} \rangle \} = \sum_n |c_n |^2 cos^2 [\vec{k}_n \cdot (\vec{r}_1 - \vec{r}_2)]. \label{vis}\end{aligned}$$ Thus, the width of the momentum distribution of such a TIE particle is directly related to the visibility of the interference fringes measured by passing this particle through a MZI. Specifically, for a flat distribution of $|c_n|^2$, the interference pattern (\[vis\]) is $\frac{1}{2} \left[ 1+ sinc(k_0 L) \right]$, assuming $k_n = k_0 n$ and system size $L$. Thus, the visibility scales as $1/L$, approaching zero for $L \rightarrow \infty$. By contrast, for a localized distribution $|c_n|^2 \sim e^{-\gamma n}$, the interference pattern becomes $\frac{1}{2} + \frac{2 \gamma}{4 \gamma^2 + 4 k_0^2} \left[ \gamma cos \left(k_0 L \right) + k_0 sin \left( k_0 L \right) \right]$. The visibility is then $\frac{4 \gamma (\gamma + k_0)}{4 \gamma^2 + 4 k_0^2}$, i.e. it does not depend on the system size $L$ and it approaches $1$ in the localized limit $\gamma >> k_0$. In Fig. \[flow\_1d\](a) we plot the visibility of the interference pattern as a function of the disorder $\Delta/J$ for a 3D system, showing a crossover from a delocalized state (low visibility) to a localized state (high visibility). It is possible to distinguish between an artificial transition caused by finite size effects and a true transition, by checking the [*scaling*]{} of the “crossover” point $(\Delta/J)_C$ as a function of system size. The “crossover” point is found upon adiabatically ramping up the coupling for different values of $\Delta/J$, and calculating the visibility for the resulting momentum distribution. A true transition should occur at the discontinuity point of the derivative of the visibility, i.e. it should jump from zero to its maximal value. Due to finite-size effects, the smoothed “crossover” is at the point of the [*maximal derivative*]{} of the visibility (as a function of $\Delta/J$), which remains continuous. In Fig. \[flow\_1d\](b) it can be seen that for 1D the “crossover” point flows toward $(\Delta/J)_C=0$ (this occurs for 2D as well), indicating that there is no thermodynamical transition and the system is always localized.
A 3D momentum-space lattice, which is more difficult to realize experimentally due to the large number of Raman pairs needed ($N=L^3$, where $L$ is the system size in each dimension), exhibits a true localization transition as $N \rightarrow \infty$. In Fig. \[flow\_1d\](c) we plot the transition point as a function of $N$, displaying the finite size scaling. In this case the transition point flows toward a [*finite, non-zero value*]{}.
Can we emulate an infinite-dimensional (but finite-size) momentum-space lattice? Instead of the matrix (\[H1d\]), infinite dimensionality corresponds to non-zero coupling terms in all off-diagonal elements. Such a setup would obviously require a prohibitively large number of Raman pairs. However, by coupling every level to the initially populated level, for which the number of Raman pairs needed is equivalent to that of the 1D case, the system is governed by the Hamiltonian $$\begin{aligned}
H = \left (
\begin{array}{cccccc}
E_1 & -J & -J & -J & \cdots & -J \\
-J & E_2 & 0 & 0 & 0 & \cdots \\
-J & 0 & E_3 & 0 & 0 & \cdots \\
\vdots & & & & & \\
-J & 0 & \cdots & 0 & 0 & E_N
\end{array}
\right). \label{Hnd}\end{aligned}$$ which is similar to an infinite dimensional system of states $n=2...N$ (in the sense that all eigenstates are thermodynamically delocalized). As in the 1D case, it is known that the Anderson model of infinite dimensionality does not have a phase transition [@Ulmke]. However, here the system will remain delocalized, with the “crossover” point flowing to infinity (or, more precisely, scaling as $N$). This is shown in Fig. \[flow\_1d\](d), where we plot the transition point as a function of the system size. An experimental demonstration may involve ultracold Li atoms, which allow significant momentum to be imparted by laser beams. The atoms can be outcoupled from a Bose-Einstein condensate, and prepared in an initial state $|F=1,m_F=-1,k_0 \rangle$, with a velocity of $v_0 \simeq 10 \frac{cm}{sec}$ and an energy of $E_0 \simeq 74 kHz$. This state can then be Raman-coupled to the Zeeman-split $m$ states of levels $F=1$ and $F=2$, providing a total of $8$ accessible levels. Such a [*small number of levels*]{} cannot reproduce 3D effects, but [*can provide measurable scaling results*]{} for the 1D Hamiltonian (\[H1d\]) and the effective $\infty$D Hamiltonian (\[Hnd\]). These levels are accessible using pairs of laser beams (far detuned by hundreds of GHz from the single-photon resonance), and detuned from each other with an accuracy of $\sim 1kHz$ by means of acousto-optic modulators (AOMs). These laser beam pairs have been numerically shown to create momentum states with energies in the range of $0 - 300 kHz$, whose separation allows the neglect of off-resonant couplings (unaccounted for by Eq. (\[H3\]) - see Fig. 1c). Random energies are realized by randomly setting the angles between the beams. The desired coupling strengths can be achieved with beam power of a few mW and beam waists of $\sim 100 \mu m$. The AOMs allow quasi-adiabatic control over the coupling Raman beams, as required in order to probe ground state properties. A Mach-Zehnder interferometer can be realized by two perpendicular pairs of standing-wave laser beams, forming the two beam splitters of the interferometer (as in [@rempe]). The interference fringes are then recorded by counting the number of atoms in the two scattered clouds in a time-of-flight image. Molecular or Rydberg atom experiments with larger $N$ may be feasible [@Arndt], but merit separate discussion of conditions (\[H3\]) and (\[H4\]).
To conclude, we have studied an intriguing fundamental effect: strong localization of the momentum distribution of particles subject to TIE and inter-state mixing. It may be revealed by interferometry of such particles. Remarkably, even few-level diffracted particles allow for measurable scaling effects that bear the signature of strong localization. We acknowledge the support of the EC (QUACS RTN and SCALA NOE) and ISF.
[99]{}
P. W. Anderson, Phys. Rev. **109**, 5, 1492 (1958).
G. Montambaux and E. Akkermans, [*Physique Mesoscopique des Electrons et des Photons*]{}, EDP Sciences (2004).
V. M. Akulin, [*Coherent Dynamics of Complex Quantum Systems*]{}, Springer (2006).
P. J. Bardroff et. al, Phys. Rev. Lett. **74**, 3959 (1995); F. L. Moore, J. C. Robinson, C. Bharucha, P. E. Williams and M. G. Raizen, Phys. Rev. Lett. **73**, 2974 (1994); S. Fishman, D. R. Grempel and R. E. Prange, Phys. Rev. Lett. **49**, 509 (1982).
M. Kasevich and S. Chu, Phys. Rev. Lett. **67**, 181 (1991).
M. O. Scully and M. S. Zubairy, [*Quantum Optics*]{} (Cambridge University Press, 1997); C. Cohen-Tannoudji, [*Atomic Motion in Laser Light*]{}, (Les Houches, 1990).
R. T. Scalettar, G. G. Batrouni and G. T. Zimanyi, Phys. Rev. Lett **66**, 24, 3144 (1991).
V. Buzek, M. Hillery and L. Mlodinov, Phys. Rev. A **71**, 062104 (2005); M. Arndt et al., Nature **401**, 680 (1999).
P. Berman Ed., Atom Interferometry, Academic Press, NY (1997).
M. Ulmke, V. Janis and D. Vollhardt, Phys. Rev. B **51**, 16, 10411 (1995).
S. Dürr, T. Nonn, and G. Rempe, Nature **395**, 33 (1998).
|
---
author:
- 'E. Kun, Z. Keresztes, L. Á. Gergely'
title: 'Slowly rotating Bose–Einstein Condensate confronted with the rotation curves of 12 dwarf galaxies'
---
[We assemble a database of $12$ dwarf galaxies, for which optical ($R$-band) and near-infrared ($3.6\mu m$) surface brightness density together with spectroscopic rotation curve data are available, in order to test the slowly rotating Bose–Einstein Condensate (BEC) dark matter model.]{} [We aim to establish the angular velocity range compatible with observations, bounded from above by the requirement of finite size halos, to check the modelfits with the dataset, and the universality of the BEC halo parameter $\mathcal{R}$.]{} [We construct the spatial luminosity density of the stellar component of the dwarf galaxies based on their $3.6\mu m$ and R-band surface brightness profiles, assuming an axisymmetric baryonic mass distribution with arbitrary axis ratio. We build up the gaseous component of the mass by employing a truncated disk model. We fit a baryonic plus dark matter combined model, parametrized by the $M/L$ ratios of the baryonic components and parameters of the slowly rotating BEC (the central density $\rho_\mathrm{c}$, size of the BEC halo $\mathcal{R}$ in the static limit, angular velocity $\omega$) to the rotation curve data.]{} [The $3.6\mu m$ surface brightness of six galaxies indicates the presence of a bulge and a disk component. The shape of the $3.6\mu m$ and $R$-band spatial mass density profiles being similar is consistent with the stellar mass of the galaxies emerging wavelength-independent. The slowly rotating BEC model fits the rotation curve of $11$ galaxies out of 12 within $1\sigma$ significance level, with the average of $\mathcal{R}$ as $7.51$ kpc and standard deviation of $2.96$ kpc. This represents an improvement over the static BEC model fits, also discussed. For the well-fitting $11$ galaxies the angular velocities allowing for a finite size slowly rotating BEC halo are less then $2.2\times 10^{-16}$ $s^{-1}$. For a scattering length of the BEC particle of $a\approx 10^6$ fm, as allowed by terrestrial laboratory experiments, the mass of the BEC particle is slightly better constrained than in the static case as $m\in[1.26\times10^{-17}\div3.08\times10^{-17}]$(eV/c$^{2}$).]{}
Introduction
============
The pioneering work by Vera Rubin and her collaborators on optical (H$\alpha$) galaxy rotation curves proved the presence of an unknown form of matter [@Rubin1978; @Rubin1985]. It was followed up by the radio (HI) observations, first systematically conducted by Albert Bosma [e.g. @Bosma1977; @Bosma1981]. Fritz Zwicky also concluded from the dynamic analysis of galaxy clusters the existence of some invisible material [@Zwicky1937], referred as dark matter (DM).
Since then other evidence appeared for matter interacting only gravitationally, such as gravitational lensing [e.g. @Wegg2016; @Chudaykin2016], or measurements on the cosmic microwave background radiation [@Planck2015]. Recent observations with the Planck satellite indicate that the DM makes up about one quarter of the energy of the Universe [@Planck2015; @Planck2018].
Galactic astronomy cannot explain the observed rotation curves through luminous matter alone. Several DM-type mass density profiles were proposed to relax the problem of the missing mass. The Navarro-Frenk-White (NFW) DM model [@NFW1996] emerged from cold DM structure-formation simulations. The pseudo-isothermal halo model [@Gunn1972] has a core-like constant density profile avoiding the density singularity of the NFW model emerging at the center of the galaxies.
Supplementing other viable proposals, @Bohmer2007 considered the possibility that DM could be in the form of a Bose–Einstein Condensate (BEC). They described DM as a non-relativistic, Newtonian gravitational BEC gas, obeying the Gross–Pitaevskii equation with density and pressure related through a barotropic equation of state. They fitted the Newtonian tangential velocity of the model with a sample of rotation curves of low surface brightness and dwarf galaxies, finding good agreement.
@Dwornik2015 tested the BEC DM model against rotation curve data of high and low surface brightness galaxies. Fits were of similar quality for the BEC and NFW DM models, except for the rotation curves exhibiting long flat regions, slightly better favouring the NFW profiles.
@Kun2018mgrbec confronted a non-relativistic BEC model of light bosons interacting gravitationally either through a Newtonian or a Yukawa potential with the observed rotational curves of 12 dwarf galaxies. The rotational curves of 5 galaxies were reproduced with high confidence level by the BEC model. Allowing for a small mass the gravitons resulted in similar performances of the fit. The upper mass limit for the graviton in this approach resulted in $10^{-26}$ eV [c]{}$^{-2}$.
@Zhang2018 derived the tangential velocity of a test particle moving in a slowly rotating Bose–Einstein Condensate (srBEC)-type DM halo. In this paper we confront their model with the rotation curve of 12 dwarf galaxies. The rotational velocity is parametrized by the central density of the srBEC halo ($\rho_\mathrm{c}$), the radius of the static BEC halo ($\mathcal{R}$), and the angular velocity ($\omega$) of the srBEC halo. The value of $\mathcal{R}$ is determined by the scattering length $a$ and the mass $m$ of the DM particle. Therefore $\mathcal{R}$ is expected to be a universal constant and the different size of the srBEC halos should emerge due to the differences in their angular velocity.
In Section \[baryoniccomponent\] we give the contribution of the baryonic component to the galaxy rotation curves. We present the model of the stellar component, we argue for a more sophisticated stellar model producing better results than the widely accepted exponential disk model, and we build up the $3.6 \mu m$ and $R$-band spatial luminosity density models to compare them to each other. At the end of this Section we present the model of the gaseous component. In Section \[section:darkmatter\] we introduce the srBEC model. We address the maximum rotation of the srBEC halos, a novel concept advanced in relation with this model. In Section \[sec:besftmodels\] we present and discuss the best-fit rotation curve models of 12 dwarf galaxies. In Section \[sec:discussion\] we summarize our results and give final remarks.
Baryonic model {#baryoniccomponent}
==============
Stellar component
-----------------
The stellar contribution to rotational curves is derived based on the distribution of the luminous matter, deduced from the surface brightness of the galaxies. We follow [@Tempel2006] to derive the surface brightness density model, assuming the spatial luminosity density distribution of each visible component given by $$l(a)=l(0)\exp\left[ -\left( \frac{a}{ka_\mathrm{0}}\right)^{{1/N}} \right].
\label{eq:spatlumdistr}$$ Here $l(0)=hL {(4\pi q a_\mathrm{0}^3)}^{-1}$ is the central density, where $a_\mathrm{0}$ characterizes the harmonic mean radius of the respective component, and $k$ and $h$ are scaling parameters. Furthermore, $a=\sqrt{r^2+z^2 q^{-2}}$, where $q$ is the axis ratio, and $r$ and $z$ are cylindrical coordinates. From the measurements the projection of $l(a)$ onto the plane of the sky perpendicular to the line of sight, the surface luminosity is derived cf. @Kun2017: $$S(R)=2 \sum_i^n q_i \int_R^\infty \frac{l_\mathrm{i}(a) a}{\sqrt{a^2-R^2}}da.
\label{eq:sr}$$ Here $S(R)$ arises as a sum for $n$ visible components, and we assumed constant axis ratios $q_\mathrm{i}$. Equation (\[eq:sr\]) was fitted to the observed surface luminosity profiles, assuming a constant axis ratio $q$. In the two-component stellar model the spatial mass density is $$\rho(a)= \Upsilon_\mathrm{b} l_\mathrm{b}(a)+\Upsilon_\mathrm{d} l_d(a),
\label{eq:rhoa}$$ where $l_\mathrm{b}(a)$ and $l_\mathrm{d}(a)$ are the spatial luminosity densities of the bulge and disk components, and $\Upsilon_\mathrm{b}$ and $\Upsilon_\mathrm{d}$ are the respective mass-to-light ($M/L$) ratios (given in solar units).
It follows from the Poisson equation that for spheroidal shape matter, the rotational velocity squared in the galactic plane ($z=0$) induced by each stellar component is given by [@Tamm2005]: $$v_{i,*}^2(R)=4 \pi q_\mathrm{i} G \int_0^R \frac{\rho_\mathrm{i}(r) r^2}{(R^2-e_\mathrm{i}^2 r^2)^{1/2}} dr,$$ where $i={b,d}$, $G$ is the gravitational constant, $e_i=(1-q_\mathrm{i}^2)^{1/2}$ is the eccentricity of the $i$th stellar component, and $\rho_\mathrm{i}(r)$ is its mass density.
Exponential disk and Tempel–Tenjes models
-----------------------------------------
----------------------- -------------------------------------- ------------ -------- -------- -------------------------------------- ------------ ---------------- ---------------- ------------------ ------------------ -- --
ID $l(0)_b$ $ka_{0,b}$ $N_b$ $q_b$ $l(0)_\mathrm{d}$ $ka_{0,d}$ $N_\mathrm{d}$ $q_\mathrm{d}$ $L_\mathrm{b}$ $L_\mathrm{d}$
(UGC) $\left(\frac{L_\odot}{kpc^3}\right)$ ($kpc$) $\left(\frac{L_\odot}{kpc^3}\right)$ ($kpc$) ($10^9 L_\odot$) ($10^9 L_\odot$)
1281$^{\mathrm{NIR}}$ - - - - $6.328\cdot10^8$ 0.872 1.112 0.156 - 2.548
1281$^{\mathrm{R}}$ - - - - $3.056\cdot10^8$ 0.89 1.075 0.166 - 1.200
4325$^{\mathrm{NIR}}$ $6.733\cdot10^8$ 0.046 2.236 0.769 $1.884\cdot10^8$ 2.930 0.5 0.110 0.594 2.902
4325$^{\mathrm{R}}$ $3.53\cdot10^7$ 0.612 0.986 0.808 $9.531\cdot10^7$ 2.290 0.758 0.095 0.156 1.346
4499$^{\mathrm{NIR}}$ - - - - $1.079\cdot10^{10}$ 0.085 2.023 0.095 - 2.161
4499$^{\mathrm{R}}$ - - - - $3.693\cdot10^9$ 0.114 1.835 0.099 - 0.659
5721$^{\mathrm{NIR}}$ $6.650\cdot10^9$ 0.095 1.31 0.856 $6.23\cdot10^8$ 1.309 0.401 0.06 0.442 0.387
5721$^{\mathrm{R}}$ $3.020\cdot10^9$ 0.07 1.602 0.800 $3.203\cdot10^8$ 0.732 0.701 0.100 0.300 0.116
5986$^{\mathrm{NIR}}$ $5.170\cdot10^9$ 0.065 2.00 0.81 $1.409\cdot10^9$ 3.011 0.551 0.100 3.468 23.984
5986$^{\mathrm{R}}$ $6.793\cdot10^8$ 0.249 1.168 0.792 $8.064\cdot10^8$ 1.183 1.178 0.089 0.407 6.070
6446$^{\mathrm{NIR}}$ - - - - $1.286\cdot10^{10}$ 0.053 2.217 0.071 - 1.432
6446$^{\mathrm{R}}$ - - - - $3.837\cdot10^9$ 0.122 1.975 0.109 - 1.991
7125$^{\mathrm{NIR}}$ $9.499\cdot10^7$ 1.833 0.7908 0.6174 $5.041\cdot10^7$ 2.716 1.388 0.063 4.379 8.212
7125$^{\mathrm{R}}$ $7.125\cdot10^7 $ 0.331 1.762 0.700 $9.92\cdot10^7$ 2.35 1.134 0.109 1.492 5.974
7151$^{\mathrm{NIR}}$ - - - - $1.314\cdot10^{10}$ 0.374 1.433 0.078 - 8.538
7151$^{\mathrm{R}}$ - - - - $2.341 \cdot 10^9$ 0.359 1.504 0.076 - 1.840
7399$^{\mathrm{NIR}}$ $1.674\cdot10^9$ 0.324 1.096 0.671 $3.763\cdot10^7$ 5.088 0.144 0.072 1.395 1.324
7399$^{\mathrm{R}}$ $1.505\cdot10^9$ 0.071 1.714 0.836 $1.053\cdot 10^8$ 1.511 0.712 0.106 0.289 0.367
7603$^{\mathrm{NIR}}$ - - - - $9.362\cdot10^9$ 0.161 1.479 0.130 - 1.006
7603$^{\mathrm{R}}$ - - - - $1.592\cdot10^9$ 0.539 0.891 0.112 - 0.472
8286$^{\mathrm{NIR}}$ $2.789\cdot10^9$ 0.048 2.063 0.604 $1.286\cdot10^9$ 1.604 0.801 0.107 0.802 7.114
8286$^{\mathrm{R}}$ $6.601\cdot10^8$ 0.058 2.321 0.689 $6.953\cdot10^8$ 1.117 0.845 0.147 1.739 2.061
8490$^{\mathrm{NIR}}$ - - - - $3.080\cdot10^{10}$ 0.07 1.88 0.10 - 1.640
8490$^{\mathrm{R}}$ - - - - $2.582\cdot10^9$ 0.157 1.569 0.246 - 0.755
----------------------- -------------------------------------- ------------ -------- -------- -------------------------------------- ------------ ---------------- ---------------- ------------------ ------------------ -- --
1281 4325 4499 5721 5986 6446 7125 7151 7309 7603 8286 8490
---------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$\sigma$ - 3.808 - 1.473 8.521 - 2.935 - 4.827 - 0.461 -
$\tau$ 2.123 2.156 3.279 3.336 3.951 0.719 1.375 4.640 3.607 2.131 3.452 2.172
The SPARC database [@Lelli2016] offers robust mass models of a sample of 175 disk galaxies with Spitzer $3.6 \mu m$ photometry together with accurate rotation curves, well-suited to test rotation curve models. The largest number of dwarf galaxies were assumed to be bulgeless, and their photometry was fitted by an exponential disk model. The disk model is a widely explored in automatized modelling. We select 12 galaxies from this database, with the longest near-infrared (NIR) surface photometry profiles and accurate rotation curves, for which $R$-band counterparts are also available (this last criterion being motivated in the next subsection).
Unfortunately the exponential disk model in SPARC sometimes underestimates the luminosity of the inner region, in other cases under- or overestimates the outer region (see Fig. \[fig:expttcaomp\_dwarfs\]). Therefore we explored a more sophisticated Tempel–Tenjes model, moreover for half of the galaxies we allowed for both bulge and disk as indicated by their photometric data. We binned the NIR surface brightness profile of the galaxies on a logarithmic scale to smooth out possible small-scale inhomogenities in them. The best-fit baryonic parameters are presented in Table \[table:gx\_photometry\] (the respective galaxy names carry the superscript NIR). In Fig. \[fig:expttcaomp\_dwarfs\] we show the best-fit Tempel–Tenjes models for the chosen galaxies, along with their exponential disk fit from the SPARC. It is clear that the Tempel–Tenjes model has a better fit to the surface brightness data.
Near-infrared and R-band spatial luminosity models {#irrband}
--------------------------------------------------
We took the $R$-band (effective central wavelength $634.9$ nm, FWHM $106.56$ nm) surface brightness data of the same 12 late-type dwarf galaxies from the Westerbork HI survey of spiral and irregular galaxies, to build up their $R$-band photometric models [@Swaters1999; @Swaters2002; @Swaters2009]. These measurements were made with the 2.54 m Isaac Newton Telescope on La Palma in the Canary Islands. We again fitted the data with the Tempel–Tenjes model. Galaxies described by a two-component surface brightness model (bulge+disk) at $3.6\mu m$ are described by a two-component model in the $R$-band too. For the absolute $R$-magnitude of the Sun $\mathcal{M}_{\odot,R}=4.42^m$ [@Binney1998] was adopted. The best-fit parameters describing the $R$-band spatial luminosity density of these 12 dwarf galaxies are given in Table \[table:gx\_photometry\] (the galaxy names carrying the superscript $R$). Compared to the $3.6\mu m$ data, the $R$-band data result in lower luminosities for almost all of the galaxies (one exception is the galaxy UGC6446).
Earlier studies indicate that the near-infrared (NIR) $M/L$ ratio depends weakly on the color, several models predicting its constancy in the NIR over a broad range of galaxy masses and morphologies, for both the bulge and the disk [e.g. @McGaugh2014 and references therein]. The NIR surface photometry provides the most sensitive proxy to the stellar mass, as shown by e.g. @Verheijen2001. We build the spatial mass density distribution of the disk (and bulge) employing the NIR and $R$-band surface brightness models, and Eq. (\[eq:rhoa\]).
The total mass of the stellar component should not depend on the wavelength at which the galaxies are observed. The masses of the bulge and the disk should be the same for the $3.6 \mu m$ and $R$-band measurements, i.e.: $$\begin{aligned}
\Upsilon_\mathrm{NIR, b} L_\mathrm{NIR, b} (=M_\mathrm{NIR, b})=\Upsilon_\mathrm{R, b} L_\mathrm{R, b} (=M_\mathrm{R, b}),\\
\Upsilon_\mathrm{NIR, d} L_\mathrm{NIR, d} (=M_\mathrm{NIR, d})=\Upsilon_\mathrm{R, d} L_\mathrm{R, d} (=M_\mathrm{R, d}),\end{aligned}$$ where $\Upsilon$ is the $M/L$ ratio, $L$ is the total luminosity, $M$ is the total mass. We give the total luminosities in Table \[table:gx\_photometry\] based on the best-fit surface brightness models of the galaxies. Then the $R$-band $M/L$ ratios are: $$\begin{aligned}
\Upsilon_\mathrm{R, b}=\frac{L_\mathrm{NIR, b}}{L_\mathrm{R, b}}\Upsilon_\mathrm{NIR, b}=\sigma \Upsilon_\mathrm{NIR, b},\\
\Upsilon_\mathrm{R, d}=\frac{L_\mathrm{NIR, d}}{L_\mathrm{R, d}}\Upsilon_\mathrm{NIR, d}=\tau \Upsilon_\mathrm{NIR, d},
\label{eq:mperlbulgediskr}\end{aligned}$$ for the bulge and disk, respectively. The values for $\sigma$ and $\tau$ are given in Table \[table:sigmatau\], calculated based on the $R$ to $NIR$ ratio of the total luminosities of the $12$ galaxies of the sample.
In Fig \[fig:rhos\] we plot the mass densities of the 12 dwarf galaxies employing the best-fit surface brightness density models (from Table \[table:gx\_photometry\]) Stellar population models and earlier studies on the conversion between the NIR flux and stellar mass suggest that the typical value of the stellar $M/L$ in NIR should be at about $0.5 M_\odot/L_\odot$ [e.g. @Eskew2012; @McGaugh2014 and references therein]. We assume $\Upsilon_\mathrm{NIR, d}\equiv0.5$ to derive the mass density from the luminosity density, and for those galaxies with bulge $\Upsilon_\mathrm{NIR, b}\equiv0.5$, in order to calculate through Eqs. (\[eq:mperlbulgediskr\]) how much larger is the $M/L$ of the disk (and the bulge where it applies) than that of the NIR $M/L$s. In case of the galaxies of the present sample $\sigma$ (where it applies) and $\tau$ are given in Table \[table:sigmatau\]. The predicted shape of the spatial mass densities is similar for both the $R$- and NIR-bands.
The SPARC $3.6\mu m$ photometry samples the surface brightness of the galaxies from a region five-ten times closer to the centre of the galaxies, out of the same region where the $R$-band observations end. Due to their good resolution, and the fact that they are the closest proxy to the stellar mass distribution, we employ the SPARC $3.6 \mu m$ data to model the stellar component of the baryonic mass of the galaxies in the next section, to test the slowly rotating BEC model.
Gaseous component
-----------------
Observations of galaxies show that for a large fraction of dwarf galaxies the rotation velocity of the gas (measured by emission lines) is close to the rotation velocity of the stellar component (measured by absorption lines) [e.g. @Rhee2004; @Lelli2016]. Therefore for these galaxies it is necessary to involve a gaseous contribution to the baryonic component of their rotation curves. For this purpose we include an additional velocity square of an exponential disk [@BT1987]: $$v_\mathrm{gas}^2(R)= 4 \pi G \Sigma_{0} R_\mathrm{d} y^2 \left[I_0(y) K_0(y)-I_1(y) K_1(y)\right],$$ where $\Sigma_{0}$ is the central surface mass density, $R_\mathrm{d}$ is the scale length of the disk, $y \equiv R/2R_\mathrm{d}$, and $I$ and $K$ are the modified Bessel functions. The mass of the disk within radius $R$ is $$M_\mathrm{d}(R)=2\pi \Sigma_0 R^2_\mathrm{d} \left[ 1 - \exp \left( -\frac{R}{R_\mathrm{d}} \right) \left( 1+\frac{R}{R_\mathrm{d}}\right)\right],$$ while its total mass is: $$M_\mathrm{tot,d}=2 \pi \Sigma_0 R^2_\mathrm{d}.$$ We employ these equation in $R-R_\mathrm{t}$ by a introducing a truncation radius $R_\mathrm{t}$, where $R_\mathrm{t}$ denotes that radius outside of which the gaseous component is not negligible. To build up the contribution of the gaseous component to the baryonic rotation curves we fitted this truncated exponential disk model to the discrete values of the gas velocity given in the SPARC database. The best-fit parameters are given in Table \[table:vrot\_bestfit\].
Dark matter model {#section:darkmatter}
=================
The slowly rotating BEC-type dark matter component
--------------------------------------------------
The equatorial radius of the srBEC DM halo is given by @Zhang2018 $$R_0\left(\frac{\pi}{2}\right)= \frac{\pi}{k}\left(1+\frac{9}{4}\Omega ^2\right),
\label{eq:rnull}$$ where $$\begin{aligned}
\Omega ^2&=&\frac{\omega^2}{2\pi G\rho_\mathrm{c}}=0.02386\times \nonumber\\
&&\times \left(\frac{\omega }{10^{-16}\;{\rm s}^{-1}}\right)^2\times \left(\frac{\rho_\mathrm{c}}{10^{-24}\;{\rm g/cm^3}}\right)^{-1},\end{aligned}$$ $\rho_\mathrm{c}$ is the central density and $\omega$ the angular velocity of the srBEC halo (assumed to be in rigid rotation). In the non-rotating case $\Omega =0$ and $R_0\left(\pi/2\right)=\mathcal{R}=\pi/k$, the radius of the static BEC DM halo $\mathcal{R}$, being determined by the mass $m$ and scattering length $a$ of the DM particle through $$k=\sqrt{\frac{G m^3}{a \hbar ^2}},$$ where $\hbar$ is the reduced Planck-constant. The tangential velocity squared $v_\mathrm{srBEC}^2$ of massive test particles rotating in the BEC galactic DM halo is given in the first order of approximation as in the equatorial plane of the galaxies by $$\begin{aligned}
&&v_\mathrm{srBEC}^2(R)=\frac{4G\rho_\mathrm{c} \mathcal{R}^2}{\pi}\times \nonumber\\
&&\Bigg[\left(1-\Omega ^2\right)\frac{\sin \left(\pi R/\mathcal{R}\right)}{\pi R/\mathcal{R}}-\left(1-\Omega ^2\right)\cos \frac{\pi R}{\mathcal{R}} +\frac{\Omega ^2}{3}\left(\frac{\pi R}{\mathcal{R}}\right)^2\Bigg], \nonumber\\\end{aligned}$$ or equivalently[^1], $$\begin{aligned}
\hspace{-1.5cm}&&v_\mathrm{srBEC}^2\left( \;{\rm km^2/s^2}\right)= 80.861\times \left(\frac{\rho_\mathrm{c}}{10^{-24}\;{\rm g/cm^3}}\right)\times \left(\frac{\mathcal{R}}{{\rm kpc}}\right)^2\times \nonumber\\
\hspace{-1.5cm}&&\Bigg[\left(1-\Omega ^2\right)\left[\frac{\sin \left(\pi R/\mathcal{R}\right)}{\pi R/\mathcal{R}}-\cos \frac{\pi R}{\mathcal{R}}\right] +\frac{\Omega ^2}{3}\left(\frac{\pi R}{\mathcal{R}}\right)^2\Bigg]. \nonumber\\
\label{eq:becv2}\end{aligned}$$
  
 
On the maximal rotation of the slowly rotating BEC halo {#maxspeed}
-------------------------------------------------------
When formulating the srBEC model, [@Zhang2018] applied first order corrections to the density and radius of the DM halo. On Fig. \[fig:multirho1\] we present the density profile of the srBEC halo as a function of the distance from the center of the galaxy measured in its equatorial plane and of the angular velocity of the DM halo, for three values of the central density $\rho_\mathrm{c}$.
For fast rotation the halo density although oscillating, is positive at all radii, meaning that the halo size is infinite. For slower rotation however, at some finite radius the density reaches zero, where the model should have a cut-off (otherwise it is continued through negative densities). The two regimes are separated by a limiting omega value, the fastest angular velocity allowing for a finite srBEC halo (having zero density at a given radius). This limiting $\omega$ increases together with the value of the central density $\rho_\mathrm{c}$.
On Fig. \[fig:multirho2\] we present again the density profile of the srBEC halo, varying this time the size $\mathcal{R}$ of the static BEC halo. The highest $\omega$ giving a finite size halo does not seem to depend on the size of the static BEC halo $\mathcal{R}$, only on the central density $\rho_\mathrm{c}$. While for small $\omega$ the trigonometric term in Eq. (\[eq:becv2\]) dominates, for larger $\omega$ the monotonic $r^2$ term is dominant. In this paper we consider only finite-size srBEC models, thus those possessing an upper limit for $\omega$.
Rotation curve model of 12 dwarf galaxies {#sec:besftmodels}
=========================================
ID $\Sigma_{0}$ $R_\mathrm{d}$ $R_\mathrm{t}$ $M_\mathrm{tot,g}$ $\Upsilon_\mathrm{b}$ $\Upsilon_\mathrm{d}$ $M_\mathrm{tot,s}$ $\rho_\mathrm{c}$ $\mathcal{R}$ $\omega$ $M_\mathrm{srBEC}$ $\chi^2$ $1\sigma$
-------------- ------------------------------------------- -------------------- -------------------- -------------------- ----------------------------------------- ----------------------------------------- -------------------- ---------------------------------------- -------------------- ------------------------------------- -------------------- ---------- -----------
(UGC) $\left(10^7 \frac{M_\odot}{kpc^2}\right)$ $\left(kpc\right)$ $\left(kpc\right)$ $10^9 M_\odot$ $\left(\frac{M_\odot}{L _\odot}\right)$ $\left(\frac{M_\odot}{L _\odot}\right)$ $10^9 M_\odot$ $\left(10^{-24} \frac{g}{cm^3}\right)$ $\left(kpc\right)$ $\left(10^{-16} \frac{1}{s}\right)$ $10^9 M_\odot$
1281 2.34 1.54 1.35 0.35 - 0.10 0.25 1.216 4.992 1.232 3.02 1.29 24.58
4325 2.41 5.35 0.60 4.35 4.12 1.34 6.34 0.77 5.0 1.65 2.15 3.55 4.71
4499 2.78 3.41 1.46 2.03 - 0.52 1.13 0.731 7.409 1.560 6.57 5.39 7.03
5721 2.22 2.43 0 0.82 0.59 2.63 1.28 1.213 6.532 0.676 6.46 7.73 21.35
5986 1.12 9.18 0 5.93 0.09 0.22 5.59 1.447 7.884 2.193 15.67 1.25 12.64
6446 1.83 9.17 1.06 9.67 - 2.01 2.88 0.655 8.413 1.476 8.62 2.69 15.93
7125 1.07 22.29 0 33.4 0.38 0.34 4.49 0.105 14.381 0.702 7.33 1.33 10.42
7151 3.21 2.34 1.57 1.10 - 0.26 2.23 0.745 6.128 2.046 4.19 6.46 9.30
7399 1.92 3.20 0.44 1.24 1.34 0.1 2.00 2.68 6.0 1.89 11.62 6.69 7.03
7603 1.26 2.53 0 0.51 - 0.37 0.38 2.692 3.793 0.978 2.80 8.54 10.42
8286 1.42 5.48 1.00 2.68 0.56 0.47 3.76 0.410 11.540 0.063 11.79 7.26 14.81
8490$^\star$ 2.15 2.59 0.91 0.29 - 1.25 2.06 0.80 8.071 1.64 9.31 39.84 29.93
In the previous sections we gave the contribution of the baryonic sector (Section \[baryoniccomponent\].) and the slowly rotating BEC-type DM halo (Section \[section:darkmatter\].) to the combined rotation curve models. Then the model rotation curve in the equatorial pane of the galaxy is given as [@Rodrigues2018] $$v_\mathrm{rot}=\sqrt{v_\mathrm{gas}|v_\mathrm{gas}|+\Upsilon_\mathrm{b} v_\mathrm{b}|v_\mathrm{b}|+ \Upsilon_\mathrm{d} v_\mathrm{d}|v_\mathrm{d}|+v^2_\mathrm{srBEC}},
\label{eq:vrot}$$ where $v_\mathrm{gas}$, $v_\mathrm{b}$, $v_\mathrm{d}$ and $v_\mathrm{srBEC}$ are the contributions of the gaseous component, the bulge (where it applies), the disk, and the DM halo to the rotation curves.
When fitting Eq. (\[eq:vrot\]) to the observed rotation curves, we apply a non-linear least-squares method to perform the fit with error$^{-2}$ weights, minimizing the residual sum of squares ($\chi^2$) between the data and the model. We are interested in such models, where the mass density of the halo drops to zero for a given radius, therefore we set an upper limit for $\omega$, such that we allow only fits which results in finite size halos (see Section \[maxspeed\]). This limit is dynamically changing during the fit with $\rho_\mathrm{c}$. The fitted parameters are $\Upsilon_\mathrm{b}$ and $\Upsilon_\mathrm{d}$ for the stellar component, $\rho_\mathrm{c}$, $\mathcal{R}$ and $\omega$ for the srBEC component. Fitting the $M/L$ ratios we are able to reveal the maximal performance of the srBEC model.
The parameters of the best-fit galactic rotation curves, composed by a baryonic and a srBEC-type DM component are presented in Table \[table:vrot\_bestfit\], and the best-fit rotation curves are shown in Fig. \[fig:vrot\_dwarfs\] along the observed ones. The combined model fits the dataset within the $1\sigma$ confidence level in case of 11 dwarf galaxies out of 12.
The size of the static BEC halo $\mathcal{R}$ is expected to be uniform for all of the galaxies, as it only depends on the mass and scattering length of the particle forming the BEC halo. From our fitting-procedure the average value of $\mathcal{R}$ emerged as 7.51 kpc, with standard deviation 2.96 kpc (see Table \[table:vrot\_bestfit\]).
In Section \[irrband\] we derived the stellar mass density from the NIR luminosity density of the galaxies assuming $M/L$ ratios equal to $0.5$ in order to calculate how much larger are the R-band $M/L$ ratios compared to the NIR ones, and to plot the NIR and R-band mass density curves (the total mass of the stellar component should not depend on the observational band). By fitting the $M/L$ ratios (together with the srBEC parameters) to the rotational curve data we got different $M/L$s. Hence these galaxies may hold diverse stellar populations resulting in different luminosity characteristics [e.g. @Bell2001; @Bell2003].
Summary and final remarks {#sec:discussion}
=========================
In this paper we assembled photometric data and rotation curves of 12 late-type dwarf galaxies in order to test the srBEC DM–model from the SPARC database ($3.6 \mu m$ photometry) and the Westerbork HI survey of spiral and irregular galaxies ($R$-band photometry). Our particular interests were in 1) establishing the limiting angular velocity below which the model leads to finite size halos, 2) how well the model fits the dataset and 3) whether one of its parameters, the size of the BEC halo $\mathcal{R}$ in the static limit is really universal.
ID $\Upsilon_\mathrm{b}'$ $\Upsilon_\mathrm{d}'$ $\rho_\mathrm{c}'$ $\mathcal{R'}$ $\chi^2$ $1\sigma$
-------------- ----------------------------------------- ----------------------------------------- ---------------------------------------- -------------------- ---------- -----------
(UGC) $\left(\frac{M_\odot}{L _\odot}\right)$ $\left(\frac{M_\odot}{L _\odot}\right)$ $\left(10^{-24} \frac{g}{cm^3}\right)$ $\left(kpc\right)$
1281 - 0.11 1.201 5.011 1.50 25.66
4325$^\star$ 4.87 1.38 0.476 5.253 6.42 5.89
4499 - 0.57 0.679 8.094 6.52 8.18
5721 0.59 2.63 1.214 6.574 7.86 22.64
5986 0.09 0.23 1.359 8.579 1.52 13.74
6446 - 2.03 0.631 9.105 3.31 17.03
7125 0.32 0.98 0.059 15.36 1.56 11.54
7151 - 0.27 0.714 6.806 7.62 10.42
7399 1.34 0.03 2.623 6.221 7.27 8.18
7603 - 0.36 2.763 3.794 10.10 11.54
8286 0.56 0.47 0.411 11.54 7.26 15.94
8490$^\star$ - 1.28 0.753 8.843 43.28 31.00
: Best-fit parameters of the rotational curve models of 12 dwarf galaxies. The rotational curves are composed of baryonic matter and a non-rotating BEC component. The fitted parameters are the M/L of the bulge ($\Upsilon_\mathrm{b}'$, where it applies) and the disk ($\Upsilon_\mathrm{d}'$), the central density of the non-rotating BEC halo ($\rho_\mathrm{c}'$) and size of the static BEC halo $\mathcal{R'}$. The two galaxies that cannot be fitted within $1\sigma$ are marked by $^\star$.[]{data-label="table:vrot_bestfit_nonrotbec"}
We investigated whether the widely employed exponential disk model accurately describes the surface brightness of the galaxies, and found necessary to employ a more complicated model than the exponential one to correctly estimate the luminosity of the inner region of these galaxies. We built up the $3.6\mu m$ and $R$-band spatial luminosity densities of the galaxies fitting the Tempel–Tenjes model to their surface brightness densities. For six galaxies a two-component model (bulge+disk) described their surface brightness density more accurately then the disk model. We found the near infrared luminosity of almost all galaxies larger compared to the $R$-band one, leading to higher $M/L$ ratios in $R$-band in order to generate the same stellar mass. We added a gaseous component by fitting a truncated exponential disk to the gas velocity given in the SPARC database.
The stellar component+gas+slowly rotating BEC combined rotation curve model fits the dataset within the $1\sigma$ confidence level in case of 11 dwarf galaxies out of 12. The size of the static BEC halo $\mathcal{R}$, related to the BEC particle characteristics, hence expected to be the same for all of galaxies, has an average value of $\bar{\mathcal{R}}=7.51$ kpc, with standard deviation as $2.96$ kpc (see Table \[table:vrot\_bestfit\]). The best-fit limiting angular velocity which allows for a finite size slowly rotating BEC halo is $<2.2\times 10^{-16}$ $s^{-1}$ for the well-fitting 11 galaxies. Its average value is $1.32\times 10^{-16}$ $s^{-1}$, with standard deviation $0.66\times 10^{-16}$ $s^{-1}$. Based on the total masses (Table \[table:vrot\_bestfit\]) the slowly rotating BEC-type DM dominates the rotation curves of 9 galaxies out of 12 (exceptions are UGC 4325, UGC 6446, UGC 7125).
The mass $m$ of the BEC particle depends on its scattering length $a$ and the size of the static BEC halo $\mathcal{R}$ [@Bohmer2007]: $$m=6.73\times 10^{-2} [a\mathrm{(fm)}]^{1/3} [\mathcal{R}\mathrm{(kpc)}]^{-2/3} \mathrm{eV}.$$ Terrestrial laboratory experiments render the value of $a$ to be $\approx 10^6$ fm [e.g @Bohmer2007]. Hence the mass of the BEC particle falls into the range $m\in[1.26\times10^{-17}\div3.08\times10^{-17}]$(eV/c$^2$) based on the best-fits of the srBEC model to the rotation curves of the present galaxy sample. The lower limit is given by galaxy UGC7125 having the largest static BEC halo ($\mathcal{R}=14.381$ kpc), and the upper limit from UGC7603 having the smallest one ($\mathcal{R}=3.793$ kpc). It is worth to note, that UGC7125 also has the smallest ($\rho_\mathrm{c}=0.105\times 10^{-24}$ g/cm$^3$), while UGC7603 the largest central density ($\rho_\mathrm{c}=2.692\times 10^{-24}$ g/cm$^3$) among these galaxies. A slightly different lower limit on $m$ emerges when assuming a static BEC model, the mass of the BEC particle falling into the range $m\in[1.21\times10^{-17}\div3.08\times10^{-17}]$(eV/c$^2$). Again, the two limits are constrained by the galaxies UGC7125 (from below, $\mathcal{R'}=15.36$ kpc) and UGC7603 (from above, $\mathcal{R'}=3.794$ kpc). We also note that UGC7125 possesses the longest, while UGC7603 the shortest observed rotation curve in the sample, hence the size of the static BEC halo seems to correlate with the length of the rotation curves.
Finally we discuss whether the slow rotation improves over the fits. By setting $\omega=0$ we fit a static BEC model to the rotational curve data, obtaining best-fit parameters given in Table \[table:vrot\_bestfit\_nonrotbec\]. Comparison shows that the finite size srBEC model gave slightly better fits, which are below $1\sigma$ in 11 cases as compared to only 10 cases for the static BEC halo fits. In the static case the average value of $\mathcal{R'}$ emerged as $8.42$ kpc, with a standard deviation of $3.35$ kpc, as compared to the rotating BEC case with average of $7.51$ kpc and standard deviation of $2.96$ kpc. In a srBEC dark matter halo, the tangential velocity of a test particle is larger than in the static case at the same position [@Zhang2018], also the plateau of the rotation curves is slightly lifted. With $\omega=0$ the fitting process favours larger $\mathcal{R'}$s to lift the plateau to give the same performance. This is why the average value of $\mathcal{R'}$ is larger than that of $\mathcal{R}$..
According to our rotation curve analysis, the srBEC halo with suitable constrained angular velocity values proves to be a viable DM model. However the steep decrease of either the static or the slowly rotating BEC rotation curves raises doubts on whether such a halo could be well fitted with galaxy lensing data.
We thank Tiberiu Harko and Maria Crăciun for suggesting to add the gas component to the baryonic sector. The authors acknowledge the support of the Hungarian National Research, Development and Innovation Office (NKFIH) in the form of the grant 123996. The work of Z. K. was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the UNKP-18-4 New National Excellence Program of the Ministry of Human Capacities.
[31]{} natexlab\#1[\#1]{}
, E. F. & [de Jong]{}, R. S. 2001, ApJ, 550, 212
, E. F., [McIntosh]{}, D. H., [Katz]{}, N., & [Weinberg]{}, M. D. 2003, ApJS, 149, 289
, J. & [Merrifield]{}, M. 1998, [Galactic Astronomy]{}
, J. & [Tremaine]{}, S. 1987, [Galactic dynamics]{}
, C. G. & [Harko]{}, T. 2007, JCAP, 6, 025
, A. 1981, AJ, 86, 1791
, A., [van der Hulst]{}, J. M., & [Sullivan]{}, III, W. T. 1977, A&A, 57, 373
, A., [Gorbunov]{}, D., & [Tkachev]{}, I. 2016, Phys. Rev. D., 94, 023528
, M., [Keresztes]{}, Z., & [Gergely]{}, L. [Á]{}. 2015, in Thirteenth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories, ed. K. [Rosquist]{}, 1279–1281
, M., [Zaritsky]{}, D., & [Meidt]{}, S. 2012, AJ, 143, 139
, J. E. & [Gott]{}, III, J. R. 1972, ApJ, 176, 1
Kun, E., Keresztes, Z., Das, S., & Gergely, L. 2018, Symmetry, 10, 520
, E., [Keresztes]{}, Z., [Simk[ó]{}]{}, A., [Sz[ű]{}cs]{}, G., & [Gergely]{}, L. [Á]{}. 2017, A&A, 608, A42
, F., [McGaugh]{}, S. S., & [Schombert]{}, J. M. 2016, AJ, 152, 157
, S. S. & [Schombert]{}, J. M. 2014, AJ, 148, 77
, J. F., [Frenk]{}, C. S., & [White]{}, S. D. M. 1996, AJ, 462, 563
, [Ade]{}, P. A. R., [Aghanim]{}, N., [et al.]{} 2016, A&A, 594, A13
, [Aghanim]{}, N., [Akrami]{}, Y., [et al.]{} 2018, arXiv e-prints
, G., [Valenzuela]{}, O., [Klypin]{}, A., [Holtzman]{}, J., & [Moorthy]{}, B. 2004, ApJ, 617, 1059
, D. C., [Marra]{}, V., [del Popolo]{}, A., & [Davari]{}, Z. 2018, Nature Astronomy, 2, 668
, V. C., [Burstein]{}, D., [Ford]{}, Jr., W. K., & [Thonnard]{}, N. 1985, , 289, 81
, V. C., [Thonnard]{}, N., & [Ford]{}, Jr., W. K. 1978, ApJL, 225, L107
, R. A. 1999, PhD thesis, , Rijksuniversiteit Groningen, (1999)
, R. A. & [Balcells]{}, M. 2002, A&A, 390, 863
, R. A., [Sancisi]{}, R., [van Albada]{}, T. S., & [van der Hulst]{}, J. M. 2009, A&A, 493, 871
, A. & [Tenjes]{}, P. 2005, A&A, 433, 31
, E. & [Tenjes]{}, P. 2006, MNRAS, 371, 1269
, M. A. W. 2001, ApJ, 563, 694
, C., [Gerhard]{}, O., & [Portail]{}, M. 2016, MNRAS, 463, 557
, X., [Chan]{}, M. H., [Harko]{}, T., [Liang]{}, S.-D., & [Leung]{}, C. S. 2018, ArXiv e-prints
, F. 1937, ApJ, 86, 217
[^1]: This equation corrects Eq. (103) of [@Zhang2018]. When they substituted the definition of $\Omega ^2$ from their Eq. (52) to Eq. (103) missed the term $\left(\rho_\mathrm{c}/10^{-24}\;{\rm g/cm^3}\right)^{-1}$ from the right-side of their Eq. (52) due to a misprint. We thank T. Harko for pointing this out to us.
|
---
abstract: 'Pr $4f$ electronic states in Pr-based filled skutterudites ${\rm Pr}T_4X_{12}$ ($T$=Fe and Ru; $X$=P and Sb) have been studied by high-resolution bulk-sensitive Pr $3d\to4f$ resonance photoemission. A very strong spectral intensity is observed just below the Fermi level in the heavy-fermion system PrFe$_4$P$_{12}$. The increase of its intensity at lower temperatures is observed. We speculate that this is the Kondo resonance of Pr, the origin of which is attributed to the strong hybridization between the Pr $4f$ and the conduction electrons.'
author:
- 'A. Yamasaki,$^a$ S. Imada,$^a$ T. Nanba,$^b$ A. Sekiyama,$^a$ H. Sugawara,$^c$ H. Sato,$^c$ C. Sekine,$^d$ I. Shirotani,$^d$ H. Harima,$^e$ and S. Suga$^a$'
title: 'Possible Kondo resonance in PrFe$_4$P$_{12}$ studied by bulk-sensitive photoemission'
---
Heavy-fermion properties observed in many Ce and U compounds and compounds of some other rare-earth elements emerge when the hybridization between the conduction band in the vicinity of the Fermi level and the $f$ state ($c-f$ hybridization) is moderate. The $4f$ electrons in Pr are more localized and less hybridized with conduction electrons than in Ce. No heavy-fermion Pr compound was known until the discovery of ${\rm PrInAg_2}$ with a large Sommerfeld coefficient reaching $\sim$6.5 J/mol K$^2$. [@Yatskar_PrInAg2] Recently, the heavy electron mass has been found in ${\rm PrFe_4P_{12}}$ under high magnetic field. [@Sugawara_dHvA] In both PrInAg$_2$ and ${\rm PrFe_4P_{12}}$, the crystal-electric field ground state is suggested to be a non-Kramers doublet, [@Yatskar_PrInAg2; @Nakanishi_PFP; @Aoki_PFP] which is nonmagnetic but has an electric quadrupolar degree of freedom. Therefore, the heavy-fermion behaviors in these Pr compounds may result from the quadrupolar Kondo effect, [@Cox_Kondo; @Kelley_PrInAg2] which was first applied to U compounds and is in contrast to the usual spin Kondo effect applied to Ce and Yb compounds.
PrFe$_4$P$_{12}$ is one of the Pr-based filled skutterudites ${\rm Pr}T_4X_{12}$. Among them are ${\rm PrRu_4P_{12}}$ known to show the metal-insulator transition at $\sim 64$ K, [@Sekine_MI] ${\rm PrRu_4Sb_{12}}$ and ${\rm PrOs_4Sb_{12}}$ known as a conventional [@Takeda_Sb] and heavy-fermion [@Bauer_POS] superconductor, respectively. PrFe$_4$P$_{12}$ is particularly interesting due to the phase transition at around 6.5 K [@Torikachvili_PFP] and the Kondo-like behaviors. Recent studies suggest that the phase transition is associated with the ordering of quadrupolar moments. [@Keller; @Iwasa_PFP] In the high-temperature phase, Kondo anomalies are found in the transport properties. [@Sato_PFP] When the low-temperature ordered phase is destroyed by high magnetic field, enormously enhanced cyclotron effective mass ($m_{\rm c}^* \simeq 81m_0$) is observed in the de Haas-van Alphen measurement. [@Sugawara_dHvA] A large electronic specific heat coefficient of $C_{\rm el}/T\sim 1.2{\rm J/K^2 mol}$ is found under 6 T, [@Sugawara_dHvA] which suggests the Kondo temperature $T_{\rm K}$ of the order of 10 K. These facts suggest the following scenario; quadrupolar degree of freedom of the Pr $4f$ state due to the non-Kramers twofold degeneracy leads to the quadrupolar Kondo effect, and the phase transition at 6.5K resulting in the antiquadrupolar ordering is driven by the lifting of the quadrupolar degeneracy. In order for the quadrupolar Kondo effect to take place, $c$–$f$ hybridization must be appreciably strong.
It has recently been demonstrated that high-resolution photoemission (PE) with use of the soft x-ray can reveal bulk electronic states. [@Sekiyama_nature] The bulk sensitivity owes to the long mean free paths of the high-energy photoelectrons. Bulk-sensitive measurement must be crucial in the study of Pr $4f$ states since the $c$–$f$ hybridization in Ce, Sm, and Yb compounds is known to be much weaker at the surface than in the bulk. [@Sekiyama_nature; @Iwasaki_CeMX] In addition, as in the case of Ce systems, one needs to enhance the Pr $4f$ contribution in the PE spectrum by means of resonance photoemission (RPE), otherwise the Pr $4f$ state cannot be accurately distinguished from other states. [@Parks_PES; @Suga_PES; @Kucherenko_PES]
In this paper, we report the results of the bulk-sensitive Pr 3$d\to4f$ RPE measurments for ${\rm PrFe_4P_{12}}$, ${\rm PrRu_4P_{12}}$, and ${\rm
PrRu_4Sb_{12}}$. It is shown that the Pr $3d\to4f$ RPE spectrum of ${\rm PrFe_4P_{12}}$ has much larger spectral weight just below the Fermi level ($E_{\rm F}$) than other systems. We speculate that this spectral weight, which increases at lower temperatures, comes from the Kondo resonance (KR) due to the $c-f$ hybridization. If so, to our knowledge, this is the first observation of the KR in PE of Pr systems.
Single crystals of ${\rm PrFe_4P_{12}}$ and ${\rm PrRu_4Sb_{12}}$, and polycrystals of ${\rm PrRu_4P_{12}}$ were fractured [*in situ*]{} for the soft x-ray absorption (XA) and PE measurements at the BL25SU of SPring-8. [@Saitoh_BL25SU] The total energy resolution of the PE measurement was set to $\sim$ 80 meV in the high-resolution mode and $\sim$ 130 meV, otherwise. The samples were cooled and kept at 20K except for the temperature dependence measurement.
![ (a) On- and off-RPE spectra normalized by the photon flux. Inset: Pr $3d\to4f$ XA spectrum for ${\rm PrFe_4P_{12}}$. Arrows show the energies at which spectra in the main panel were taken. (b) Off-RPE spectra taken at 825 eV in an enlarged intensity scale. []{data-label="Fig_exp"}](1.eps){width="8cm"}
The Pr $3d\to4f$ XA spectrum for ${\rm PrFe_4P_{12}}$ is shown in the inset of Fig.\[Fig\_exp\](a). This spectrum reflects the predominant ${\rm Pr^{3+}}$ ($4f^2$) character in the initial state. [@Thole_XAS] Spectra of ${\rm PrRu_4P_{12}}$ and ${\rm PrRu_4Sb_{12}}$ were also quite similar to this spectrum. Valence-band PE spectra were measured at three photon energies. On-RPE spectra were taken at 929.4 eV, around XA maximum. Off-RPE spectra were taken at 921 and 825 eV, which were quite similar in shape. The on- and off- (921.0 eV) RPE spectra are compared in the main panel of Fig.\[Fig\_exp\](a). We consider that mainly Pr $4f$ contribution is enhanced in the on-RPE spectra, [@PrOn-Off] and therefore that the difference between the on- and off-RPE spectra mainly reflects the Pr $4f$ spectrum.
The off-RPE spectra taken at 825 eV with better statistics are shown in Fig.\[Fig\_exp\](b) in a magnified intensity scale. The valence band between $E_{\rm F}$ and binding energy ($E_{\rm B}$) of $\sim 7$ eV is expected to be composed of Pr $5d$ and $4f$, $T$ $d$, and $X$ $p$ orbitals. Among these, main contribution to the off-RPE spectrum (more than 60 %) comes from the $T$ $d$ states according to the photoionization cross-section. [@Lindau] The off-RPE spectral features are reproduced in the theoretical off-RPE spectra based on the FLAPW and LDA+U band structure calculation (see Fig.\[Fig\_cal\](b)), [@Harima_MI] where the parameter for the on-site Coulomb interaction $U$ of Pr $4f$ electron is set as 0.4 Ry (5.4 eV).
![ Calculated PE spectra based on band structure calculation. Density of states is multiplied by the Fermi-Dirac function for 20 K and is broadened by the Gaussian with the full width at half maximum of 80 meV. (a) Calculated Pr (solid line) and La (dashed line: magnified ten times) $f$ spectra of Pr$T_4X_{12}$ and La$T_4X_{12}$. (b) Calculated off-RPE spectra (solid lines), where partial density of states except for Pr $f$ are multiplied by the cross sections [@Lindau] and summed up. Dashed lines show the contribution of the $T$ $d$ state. []{data-label="Fig_cal"}](2.eps){width="8cm"}
The on-RPE spectra shown in Fig.\[Fig\_exp\](a) are characterized by two features. First, the on-RPE spectra have various multiple peak structures, where peaks (or structures) are indicated by arrows, in contrast to the calculated Pr $f$ PDOS (see Fig.\[Fig\_cal\](a)) that has a strong peak and small structures near $E_{\rm F}$ for all the three compounds. This feature will be interpreted in the next paragraph taking into account the hybridization between the valence band and the Pr $4f$ states ($v-f$ hybridization) in the [*final*]{} states of PE. Second, the intensity near $E_{\rm F}$, [*i.e.*]{}, between $E_{\rm F}$ and $E_{\rm B}\sim 0.3$ eV, is much stronger in PrFe$_4$P$_{12}$ than in other two systems. Such strong intensity at $E_{\rm F}$ is neither found in reported Pr $4f$ spectra. Later in this paper, this feature will be attributed the strong $c-f$ hybridization in the [*initial*]{} state of PrFe$_4$P$_{12}$.
Multiple peak structures observed for various Pr compounds have been interpreted in terms of the $v-f$ hybridization. [@Parks_PES; @Suga_PES; @Kucherenko_PES] We adopt the cluster model, [@Fujimori] [*i.e.*]{}, the simplified version of the single impurity Anderson model (SIAM). [@GS] The part of the valence band that hybridizes strongly with the $4f$ state is expected to be similar between Pr$T_4X_{12}$ and La$T_4X_{12}$. La $f$ PDOS of La$T_4X_{12}$ at a certain energy correspond roughly to the $v-f$ hybridization strength at that energy since La $f$ states below $E_{\rm F}$ comes only from the hybridization with the valence band. As a first approximation, we replace the La $f$ PDOS with two levels, $v_1$ and $v_2$, the energies of which, $E_{\rm B}(v_k)$, are shown by the arrows in Fig.\[Fig\_cal\](a). We now assume that the initial Pr $4f$ state is $|f^2\rangle$. Although it turns out later that deviation from this state is appreciable in PrFe$_4$P$_{12}$, this is a good approximation when discussing the overall spectral features. Then the final states of Pr $4f$ PE are linear combinations of $|f^1\rangle$, $|(f^2)^*\underline{v_1}\rangle$, and $|(f^2)^*\underline{v_2}\rangle$, where $\underline{v_k}$ denotes a hole at $v_k$. Since the resulting $f^2$ state includes all the excited states, it is denoted as $(f^2)^*$ so as to distinguish it from the initial ground state $f^2$. The average excitation energy $E((f^2)^*)-E(f^2)$ is $\sim 1.4$ eV according to the atomic multiplet calculation. [@Thole_XAS] The main origin of this excitation energy is found to be the exchange interaction. The energies of the bare $|(f^2)^*\underline{v_k}\rangle$ with respect to the initial state $|f^2\rangle$ are hence $E_{\rm B}(v_k)+[E((f^2)^*)-E(f^2)]$ and are shown by the thin open and filled bars in the upper pannels of Figs.\[Fig\_clus\] (a)-(c). We take the remaining three parameters, $E_{\rm B}$ of the bare $|f^1\rangle$ ($E_0$), the hybridization between $|f^1\rangle$ and $|(f^2)^*\underline{v_k}\rangle$ ($V_k$), to be free parameters, and numerically solve the $3 \times 3$ Hamiltonian matrix. When the parameters are set as in the upper pannels of Figs.\[Fig\_clus\] (a)-(c), the three final states are obtained as shown in the lower pannels. At each of the three eigen-energies of the final states is placed a set of vertical bars the lengths of which are proportional to the weights of $|f^1\rangle$ (thick filled bar), $|(f^2)^*\underline{v_1}\rangle$ (thin open bar), and $|(f^2)^*\underline{v_2}\rangle$ (thin filled bar). Since we assume that the initial state is $|f^2\rangle$, the Pr $4f$ excitation intensity is proportional to the weight of the $|f^1\rangle$ in each final state. Therefore, the thick filled bars show the obtained line spectrum. The line spectra qualitatively well reproduce the experimentally observed system dependence in the energy positions and intensity ratios of the three-peak structures of the on-RPE spectra (see Fig.\[Fig\_exp\](a)). The present analysis revealed the character of each final states. For example, the final state with the smallest $E_{\rm B}$ is the bonding state between $|f^1\rangle$ and $|(f^2)^*\underline{v_1}\rangle$. The trend in the E$_B$ of bare $|f^1\rangle$ corresponds to some extent with the trend in the peak position of Pr $f$ PDOS in Fig.\[Fig\_cal\](a). The origin of these trends could be that E$_B$ of the Pr $4f$ electron becomes smaller because the negative $X$ ion comes closer to Pr atom in the direction of PrRu$_4$Sb$_{12}$, PrRu$_4$P$_{12}$, PrFe$_4$P$_{12}$.
![ Pr $4f$ spectrum reproduced by the cluster model for (a) ${\rm PrFe_4P_{12}}$, (b) ${\rm PrRu_4P_{12}}$, (c) ${\rm PrRu_4Sb_{12}}$, and (d) surface of ${\rm PrFe_4P_{12}}$. Upper panels: Binding energies of bare $|f^1\rangle$ and $|(f^2)^*\underline{v_k}\rangle$ final states are shown by the bars and the effective hybridization between $|f^1\rangle$ and $|(f^2)^*\underline{v_k}\rangle$ are written. Lower panels: The resultant final states are shown. The thick filled bars show the weights of the $|f^1\rangle$ state in the final states, and correspond to the Pr $4f$ PE line spectrum for the assumed $|f^2\rangle$ initial state. The thin open and filled bars show the weights of $|(f^2)^*\underline{v_1}\rangle$ and $|(f^2)^*\underline{v_2}\rangle$ states, respectively. []{data-label="Fig_clus"}](3.eps){width="8cm"}
The present Pr $4f$ spectrum of ${\rm PrFe_4P_{12}}$ obtained from the bulk-sensitive $3d\to4f$ RPE is qualitatively different from that obtained from the surface-sensitive $4d\to4f$ RPE. [@Ishii] The surface-sensitive spectrum also has a three peak structure but the peak at $E_{\rm B}\sim$ 4.5 eV is the strongest and the intensity at $E_{\rm F}$ is negligible. The origin of the difference is the increase of the localization of $4f$ electrons at the surface, in other words, the increase of the $4f$ binding energy and the decrease of the hybridization. By making such changes in $E_0$ and $V_k$, the surface-sensitive spectrum is reproduced (see Fig.\[Fig\_clus\](d)).
![ (a) High-resolution Pr $3d\to4f$ on- (dots) and off- (solid lines) RPE spectra near $E_{\rm F}$ at 20 K. The vertical lines show the energy positions of the atomic $4f^2$ multiplets with the ground state set at $E_{\rm F}$. (b) Temperature dependence of the on-RPE spectrum of ${\rm PrFe_4P_{12}}$. (c) Calculated partial density of states. []{data-label="Fig_EF"}](4.eps){width="8cm"}
We measured the Pr $3d\to4f$ RPE spectra near $E_{\rm F}$ with high resolution as shown in Fig.\[Fig\_EF\](a). The most prominent feature is the strong peak of ${\rm PrFe_4P_{12}}$ at $E_{\rm B}\simeq100$ meV. The Pr $4f$ spectra of ${\rm PrRu_4P_{12}}$ and ${\rm PrRu_4Sb_{12}}$, on the other hand, decrease continuously with some humps as approaching $E_{\rm F}$. Spectral features similar to ${\rm PrRu_4P_{12}}$ and ${\rm PrRu_4Sb_{12}}$ has been found for very localized Ce systems such as CePdAs, in which Ce $4f$ takes nearly pure $4f^1$ state. [@Iwasaki_CeMX] This indicates that pure $4f^2$ state is realized in ${\rm PrRu_4P_{12}}$ and ${\rm PrRu_4Sb_{12}}$. On the other hand, similarity between the ${\rm PrFe_4P_{12}}$’s and Kondo Ce compound’s spectra [@Sekiyama_nature] suggests that the Pr $4f^2$-dominant Kondo state, with the finite contribution of $4f^1$ or $4f^3$ state, is formed in ${\rm PrFe_4P_{12}}$.
The present energy resolution of $\sim 80$ meV exceeds the characteristic energy $k_{\rm B}T_{\rm K} \sim 1$ meV for ${\rm PrFe_4P_{12}}$. KR has been observed even in such cases, for example, for ${\rm CeRu_2Si_2}$ ($T_{\rm K}\sim 20$ K) [@Sekiyama_nature] and YbInCu$_4$ ($T_{\rm K} \sim 25$ K for $T > 42$ K) [@Sato_Yb] with energy resolution of $\sim 100$ meV.
In the Kondo Ce (Yb) system, the KR is accompanied by the spin-orbit partner, the $E_{\rm B}$ of which corresponds to the spin-orbit excitation, $J=5/2\to 7/2$ ($J=7/2\to 5/2$), of the $4f^1$ ($4f^{13}$)-dominant state. [@GS] A KR in Pr would then be accompanied by satellites corresponding to the excitation from the ground state ($^3H_4$) to excited states ($^3H_5$, $^3H_6$, $^3F_2$, and so on) of the $4f^2$ states. Fig.\[Fig\_EF\](a) shows that the on-RPE spectrum of ${\rm PrFe_4P_{12}}$ have strucures at $\sim 0.3$ and $\sim$0.6 eV which correspond to the lowest few excitation energies.
KR is expected to depend upon temperature reflecting the temperature dependence of the $4f$ occupation number. In fact, a temperature dependence was found as the temperature approaches the suggested $T_{\rm K}\sim 10$ K as shown in Fig.\[Fig\_EF\](b). The temperature dependence was reproducible in both heat-up and cool-down processes. The temperature dependence is characterized not only by the narrowing of the $\sim$ 0.1 eV structure but also by the increase of the weight of all the structures at $\sim$ 0.1, $\sim$ 0.3, and $\sim$ 0.6 eV. Although the former can be attributed at least partly to the thermal broadening, the latter should be attributed to intrisic temperature dependence of the excitation spectrum. Therefore it is quite possible that the $\sim$ 0.1 eV structure is the KR and the $\sim$ 0.3 and $\sim$ 0.6 structures are its satellite structures.
The temperature dependence can be a vital clue to check whether the observed structure is the Kondo peak itself or the tail of the Kondo peak centered above $E_{\rm F}$. These cases correspond respectively to the $c_2|f^2\rangle + c_3|f^3\rangle$ or $d_1|f^1\rangle + d_2|f^2\rangle$ initial states, where the hole or electron in the valence or conduction band is not denoted explicitly. The non-crossing approximation (NCA) calculation based on the SIAM for the Ce system [@Kasai_NCA] shows that, as temperature is lowered, the Kondo tail is sharpened [@Reinert_CeCu2Si2] but the [*weights*]{} of both the Kondo tail and its spin-orbit partner [*decrease*]{} when the spectra are normalized in a similar way as in Fig.\[Fig\_EF\](b). This contradicts with the present temperature dependence for ${\rm PrFe_4P_{12}}$. On the other hand, for Yb systems, it is well known that the intensities of both the Kondo peak itself and its spin-orbit partner increase with decreasing temperature. [@Tjeng_Yb] Since this is consistent with the ${\rm PrFe_4P_{12}}$’s temperature dependence, we tend to believe that the observed structure is the Kondo peak itself, and therefore that the initial state is dominated by $c_2|f^2\rangle + c_3|f^3\rangle$. We consider that the Kondo peak at around $k_{\rm B}T_{\rm K}\sim 1$ meV is broadened due to the energy resolution of $\sim 80 $ meV resulting in the observed structure at $\sim 100$ meV.
Microscopic origin of the $c-f$ hybridization is considered to be the P $3p-$Pr $4f$ mixing since the nearest neighbors of the Pr atom are the twelve P atoms. The large coordination number definitely enhances the effective $p-f$ mixing. It has been pointed out that the calculated P $p$ PDOS of $R$Fe$_4$P$_{12}$ shows a sharp peak in the vicinity of $E_{\rm F}$. [@Sugawara_FS; @Harima_MI] This is also the case for PrFe$_4$P$_{12}$ as shown in Fig.\[Fig\_EF\](c). Therefore, the large P $3p$ PDOS at $E_{\rm F}$ together with the large effective P $3p-$Pr $4f$ mixing is interpreted to be the origin of the Kondo state in ${\rm PrFe_4P_{12}}$.
In conclusion, our analysis of the data suggests that there may be a Kondo resonance (KR) in the $4f$ photoemission spectrum of PrFe$_4$P$_{12}$, whereas no KR was seen in PrRu$_4$P$_{12}$ and PrRu$_4$Sb$_{12}$. The origin of the KR in PrFe$_4$P$_{12}$ is considered to be the Kondo effect caused by the strong hybridization between the Pr $4f$ and P $3p$ states in the vicinity of $E_{\rm F}$.
We would like to thank Profs. O. Sakai and K. Miyake for fruitful discussions. The research was performed at SPring-8 (Proposal Nos. 2001A0158-NS-np and 2002A0433-NS1-np) under the support of a Giant-in-Aid for COE Research (10CE2004) and Scientific Research Priority Area “Skutterudite” (No.15072206) of the Ministry of Education, Culture, Sports, Science, and Technology, Japan.
A. Yatskar, W. P. Beyermann, R. Movshovich, and P. C. Canfield, Phys. Rev. Lett. [**77**]{}, 3637 (1996).
H. Sugawara, T. D. Matsuda, K. Abe, Y. Aoki, H. Sato, S. Nojiri, Y. Inada, R. Settai, and Y. Ōnuki, Phys. Rev. B [**66**]{}, 134411 (2002).
Y. Nakanishi, T. Simizu, M. Yoshizawa, T. D. Matsuda, H. Sugawara, and H. Sato, Phys. Rev. B [**63**]{}, 184429 (2001).
Y. Aoki, T. Namiki, T. D. Matsuda, K. Abe, H. Sugawara, and H. Sato, Phys. Rev. B [**65**]{}, 064446 (2002).
D. L. Cox, Phys. Rev. Lett. [**59**]{}, 1240 (1987).
T. M. Kelley, W. P. Beyermann, R. A. Robinson, F. Trouw, P. C. Canfield, and H. Nakotte, Phys. Rev. B [**61**]{}, 1831 (2000).
C. Sekine, T. Uchiumi, I. Shirotani, and T. Yagi, Phys. Rev. Lett. [**79**]{}, 3218 (1997).
N. Takeda and M. Ishikawa, J. Phys. Soc. Jpn. [**69**]{}, 868 (2000). E. D. Bauer, N. A. Frederick, P.-C. Ho, V. S. Zapf, and M. B. Maple, Phys. Rev. B [**65**]{}, 100506R (2002).
M. S. Torikachvili, J. W. Chen, Y. Dalichaouch, R. P. Guertin, M. W. McElfresh, C. Rossel, M. B. Maple, and G. P. Meisner, Phys. Rev. B [**36**]{}, 8660 (1987).
K. Iwasa, Y. Watanabe, K. Kuwahara, M. Kohgi, H. Sugawara, T. D. Matsuda, Y. Aoki, and H. Sato, Physica B [**312-313**]{}, 834 (2002).
L. Keller, P. Fischer, T. Herrmannsdorfer, A. Donni, H. Sugawara, T. D. Matsuda, K. Abe, Y. Aoki, and H. Sato, J. Alloys Compd. [**323-324**]{}, 516 (2001).
H. Sato, Y. Abe, H. Okada, T. D. Matsuda, K. Abe, H. Sugawara, and Y. Aoki, Phys. Rev. B [**62**]{}, 15125 (2000).
A. Sekiyama, T. Iwasaki, K. Matsuda, Y. Saitoh, Y. Ōnuki, and S. Suga, Nature (London) [**403**]{}, 396 (2000).
T. Iwasaki, A. Sekiyama, A. Yamasaki, M. Okazaki, K. Kadono, H. Utsunomiya, S. Imada, Y. Saitoh, T. Muro, T. Matsushita, H. Harima, S. Yoshii, M. Kasuya, A. Ochiai, T. Oguchi, K. Katoh, Y. Niide, K. Takegahara, and S. Suga, Phys. Rev. B [**65**]{}, 195109 (2002).
R. D. Parks, S. Raaen, M. L. denBoer, Y.-S. Chang, and G. P. Williams, Phys. Rev. Lett. [**52**]{}, 2176 (1984).
S. Suga, S. Imada, H. Yamada, Y. Saitoh, T. Nanba, and S. Kunii, Phys. Rev. B [**52**]{}, 1584 (1995).
Yu. Kucherenko, M. Finken, S. L. Molodtsov, M. Heber, J. Boysen, C. Laubschat, and G. Behr, Phys. Rev. B [**65**]{}, 165119 (2002).
Y. Saitoh, H. Kimura, Y. Suzuki, T. Nakatani, T. Matsushita, T. Muro, T. Miyahara, M. Fujisawa, S. Ueda, H. Harada, M. Kotsugi, A. Sekiyama, and S. Suga, Rev. Sci. Instrum. [**71**]{}, 3254 (2000).
B. T. Thole, G. van der Laan, J. C. Fuggle, G. A. Sawatzky, R. C. Karnatak, and J.-M. Esteva, Phys. Rev. B [**32**]{}, 5107 (1985).
In the on-RPE spectra, not only Pr $4f$ compounent but also other components can be enhanced. [@Olson_La; @Molodtsov_La] The latter may include Pr $5d$, $T$ $d$, and $X$ $p$ states. Contribution from the Pr $5d$ state can be estimated from the La $3d\to 4f$ RPE of La$T_4X_{12}$ (not shown) to be at most of the order of the off-RPE intensity of PrFe$_4$P$_{12}$ (see Fig.\[Fig\_exp\](a)). Also the cross sections of the non-Pr $4f$ components change between on- and off-RPE conditions but these changes are at most 5 % of the off-RPE intensity. There might be further enhancement of the non-Pr $4f$ state due to the interplay of the $c-f$ mixing and the cross term between the Pr $4f$ and other excitation processes. However, since the matrix element of Pr $4f$ is far larger than those of other states judging from the large enhancement, the cross term should be much smaller than the pure Pr $4f$ term.
C. G. Olson, P. J. Benning, M. Schmidt, D. W. Lynch, P. Canfield, and D. M. Wieliczka, Phys. Rev. Lett. [**76**]{}, 4265 (1996).
S. L. Molodtsov, M. Richter, S. Danzenbächer, S. Wieling, L. Steinbeck, and C. Laubschat, Phys. Rev. Lett. [**78**]{}, 142 (1997).
J. J. Yeh and I. Lindau, Atomic Data and Nuclear Data Tables [**32**]{}, 1 (1985).
H. Harima and K. Takegahara, Physica B [**312-313**]{}, 843 (2002).
A. Fujimori, Phys. Rev. B [**27**]{}, 3992 (1983).
O. Gunnarsson and K. Schönhammer, Phys. Rev. B [**28**]{}, 4315 (1983).
H. Ishii, K. Obu, M. Shinoda, C. Lee, and Y. Takayama, J. Phys. Soc. Jpn. [**71**]{}, 156 (2002).
H. Sato, K. Hiraoka, M. Taniguchi, Y. Takeda, M. Arita, K. Shimada, H. Namatame, A. Kimura, K. Kojima, T. Muro, Y. Saitoh, A. Sekiyama, and S. Suga, J. Synchrotron Rad. [**9**]{}, 229 (2002).
S. Kasai, S. Imada, A. Sekiyama, and S. Suga, unpublished.
F. Reinert, D. Ehm, S. Schmidt, G. Nicolay, S. Hüfner, J. Kroha, O. Trovarelli, and C. Geibel, Phys. Rev. Lett. [**87**]{}, 106401 (2001).
L. H. Tjeng, S.-J. Oh, E.-J. Cho, H.-J. Lin, C. T. Chen, G.-H. Gweon, J.-H. Park, J. W. Allen, and T. Suzuki, Phys. Rev. Lett. [**71**]{}, 1419 (1993).
H. Sugawara, Y. Abe, Y. Aoki, H. Sato, M. Hedo, R. Settai, Y. Ōnuki, and H. Harima, J. Phys. Soc. Jpn. [**69**]{}, 2938 (2000).
|
[**Conservation of pseudo-norm\
in ${\cal PT}$ symmetric quantum mechanics** ]{}
Miloslav Znojil
Ústav jaderné fyziky AV ČR, 250 68 Řež, Czech Republic\
e-mail: znojil@ujf.cas.cz
Abstract {#abstract .unnumbered}
========
We show that the evolution of the wave functions in the ${\cal
PT}$ symmetric quantum mechanics is pseudo-unitary. Their pseudo-norm $\langle \psi| {\cal P} | \psi \rangle$ remains time-independent. This persists even if the ${\cal PT}$ symmetry itself becomes spontaneously broken.
PACS 03.65.Bz, 03.65.Ca 03.65.Fd
[, spon.tex file]{}
Introduction
============
Evolution in quantum mechanics [@Landau] |(t)= e\^[-iHt]{} |(0) conserves the norm of a state. The assumption $H = H^\dagger$ of the Hermiticity of the Hamiltonian leads to the time-independence of the probability density, (t)| (t) = (0)| e\^[iH\^t]{}e\^[-iHt]{} | (0) = (0)| (0) . \[dve\] , in the light of the Stone’s theorem, the unitarity of the evolution implies the Hermiticity of the Hamiltonian.
Apparently, there is no space left for the non-Hermitian Hamiltonians which were recently studied by Bender et al [@BBjmp]-[@spont]. The latter formalism may prove inspiring in field theory [@BMa] but, as an extended quantum mechanics, it contradicts the Stone’s theorem. In what follows, we intend to clarify this point.
Section 2 reviews a few basic features of the extended formalism which replaces the Hermiticity $H = H^\dagger$ by its weakening $H
= H^\ddagger$. The latter property (called ${\cal PT}$ symmetry) is explained and several parallels between the symmetries $H=H^\dagger$ and $H=H^\ddagger$ are mentioned. Explicit ${\cal
PT}$ symmetric square well solutions [@SQW] are recalled as an illustration of the whole idea.
The introductory part of section 3 recollects the regularized spiked harmonic oscillator bound states of ref. [@hopt] as a solvable example of a ${\cal PT}$ symmetric system which is defined on the whole real line. With its orthogonality and completeness properties kept in mind, we arrive at the first climax of our paper and formulate the appropriate modification of the conservation law (\[dve\]) in the case of a general non-Hermitian system with the unbroken ${\cal PT}$ symmetry.
At the beginning of section 4 the construction of the harmonic bound states is extended to the domain of couplings where the ${\cal PT}$ symmetry of the wave functions becomes spontaneously broken. We show how all the Hilbert-space-like concepts of orthogonality and completeness of the states may be generalized accordingly. In particular, the ${\cal PT}$ symmetric norm (or rather pseudo-norm) proves so robust that the new conservation law of the type (\[dve\]) remains valid even in the spontaneously broken regime where the energies cease to be real.
${\cal PT}$ symmetric quantum mechanics
=======================================
The concept of the extended, non-Hermitian quantum mechanics with the requirement of ${\cal PT}$ symmetry of its Hamiltonians grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [@Caliceti] have discovered that a low-lying part of the spectrum of the manifestly non-Hermitian cubic anharmonic oscillator $H= p^2 + x^2 + g\,x^3$ is [*real*]{} for the [*purely imaginary*]{} couplings $g$. This establishes many formal analogies with Hermitian oscillators [@Alvarez; @ixna].
A different direction of analysis has been accepted by Buslaev and Grecchi [@BG] who emphasized and employed some parallels between the Hermiticity and ${\cal PT}$ symmetry during their solution of an old puzzle of perturbative equivalence between apparently non-equivalent quartic anharmonic oscillators [@Seiler].
A mathematical background of the non-unique choice of the phenomenological Hamiltonians with real spectra has been pointed out, in non-Hermitian setting, by several authors [@BT]. Bender and Milton [@QED] emphasized the relevance of the unique analytic continuation of boundary conditions for the clarification and consequent explanation of the famous Dyson’s paradox in QED [@Dyson].
In the cubic anharmonic models $H= p^2 + x^2 +i\,g\,x^3$ the reality of the energies at the sufficiently small $g$ [@Caliceti; @Pham] resembles the quartic case. Bender and Boettcher [@BB] attributed this connection to the commutativity of the Hamiltonian with the product of the complex conjugation ${\cal T}$ (which mimics the time reversal) and the parity ${\cal P}$, $H = {\cal PT} H {\cal PT}= H^\ddagger$. An acceptability of this conjecture has been supported by a few partially [@QES] as well as completely [@exact] exactly solvable models.
In the physics community, a steady growth of acceptance of the ${\cal PT}$ symmetric models can be attributed to their possible phenomenological relevance. The cubic $H= p^2 + i\,x^3$ has been found relevant in statistical physics [@Bessis] and its non-linear perturbations $H= p^2 + (i\,x^3)^{1+\delta}$ were studied in field theory [@BMb]. In all these models, a key argument that they can prove useful in some applications has been based on the reality of their spectrum. This argument is slightly misleading as we shall see in what follows.
Solvable illustration: Square well
----------------------------------
The possibility of a spontaneous breakdown of the reality of the energies has been recently studied via a ${\cal PT}-$symmetric quartic oscillator $H= p^2 + i\,g\,x+x^4$ [@Simsek]. In this model, one spots the sequence of “critical" couplings $g_n$ such that, step by step, the lowest real pair of the bound state energies $E_{2n}$ and $E_{2n+1}$ becomes converted into a complex conjugate doublet beyond $g = g_n$. Such a pattern may prove characteristic for a fairly broad class of non-Hermitian interactions. For the sake of simplicity of the whole discussion, let us pick up the Schrödinger bound state problem on a finite interval, \_n(x) = E\_n\_n(x), \_n(-1) = \_n(1)=0, \[SQWL\] equipped with one of the most elementary ${\cal PT}$ symmetric forces $V(x) =i\,T^2\,{\rm sign}\, x$. Then, the ansatz \_n(x) = { (x+1), x < 0,\
C(x-1), x < 0 . with $E = k^2$ defines the solutions via the matching condition at $x=0$. Using an abbreviation $\lambda = p-i\,q$ and rules \^2=k\^2-iT\^2, \^2=k\^2+iT\^2= \^\* we get the elementary matching condition = [purely imaginary]{} which is equivalent to the elementary rule q 2q= -p2p . Its numerical solution has been discussed elsewhere [@SQW]. Still, without any detailed numerical computations one can fairly easily see that in the two-dimensional $p - q$ plane, the left-hand-side function forms a valley with zero minimum along the line $q=0$. The right-hand-side periodically oscillates with an increasing amplitude. One can conclude that the positive solutions $p(q)>0$ of the latter equation form an infinite family of ovals which are symmetric with respect to the $p-$axis. The $n-th$ oval is confined between the zeros of the sine function, i.e., between the two lines $p=(2n+1)\pi/2$ and $p=(2n+2)\pi/2$. With the growth of $n = 0, 1, \ldots$, the ovals are longer as their ends move farther and farther from the $p-$axis. In the $n
\gg 1$ asymptotic region, we get the estimate $p_{end} \approx
(n+3/4)\pi$ and $q_{end} \approx {\rm ln}\, n$.
As long as the definition of $p$ and $q$ implies that $p =
T^2/(2q)$ we get the second curve which is a plain hyperbola in $p - q$ plane. The final solutions (i.e, intersections of this hyperbola with all the ovals) move close to the standard square well solutions in the quasi-Hermitian limit where $n \gg T^2$.
At the opposite extreme, the two lowest real energies determined by the lowest oval cease to exist for the sufficiently strong imaginary part of the force, i.e., for $ T^2
> T^2_{crit} \approx 2\,p_{end}\,q_{end}$. In the light of the previous estimates, the values of these critical points will grow with the number $n$ of the oval in question. At $n=0$ one has $T^2_{crit} \approx 4.48$ [@SQW].
Models with unbroken ${\cal PT}$ symmetry
=========================================
We can summarize that the elementary and exactly solvable ${\cal
PT}-$symmetric square well model has a spectrum $E_n$ which remains real in a certain non-empty interval of couplings $T \in
(0,T_0)$. At the boundary $T = T_0$ with certain exceptional features [@Herbst], the lowest energy doublet $E_{0}$ and $E_{1}$ merges into a single state. The most immediate fructification of this experience lies in the possibility of its transfer to the ${\cal PT}$ symmetric potentials on a “more realistic" infinite interval of coordinates.
${\cal PT}$ symmetric harmonic oscillators \[HOlab\]
----------------------------------------------------
A nontrivial example which is solvable on the full real line is the ${\cal PT}$ symmetric harmonic oscillator described by the differential Schrödinger equation of ref. [@hopt], \_n(x) = E\_n\_n(x), \_n(x) L\_2(-,). \[HO\] This equation with $G > -1/4$ can be interpreted as a confluent hypergeometric equation where an elementary change of the coordinate $x=r+i\,\delta$ eliminates all the unusual imaginary terms. At the same time, due to the analyticity of such a transformation, one can simply keep $r$ on the real line with a small complex half-circle circumventing the singularity in the origin ($r=0$). Without any loss of generality we can then work with the complex general solution of our equation. It is available in closed form, (x) = C\_[(+)]{}r\^[1/2-]{}e\^[r\^2/2]{} \_1F\_1 + + C\_[(-)]{}r\^[1/2+]{}e\^[r\^2/2]{} \_1F\_1 . This facilitates the use of the asymptotic boundary conditions. In the standard way described in any textbook [@Fluegge] the idea works without alterations since the general solutions grow exponentially unless one of the confluent hypergeometric series terminates to a Laguerre polynomial. This gives the compact wave functions \_N(r)=[N]{}r\^[1/2-Q]{}e\^[-r\^2/2]{} L\_n\^[(-Q)]{}(r\^2) with the quasi-parity $Q = \pm 1$, main quantum number $n = 0, 1,
\ldots$ and subscripted index $N = 2n+(1-Q)/2$. The energies E\_N=4n+2-2Q, = , G > - \[spectrum\] remain real for the positive centrifugal-like parameters $\alpha>0$. These energies decrease/grow with $G$ for the quasi-even/quasi-odd quasi-parity $Q$ of the state, respectively.
If necessary, we can return to the Hermitian case and reproduce the usual radial harmonic oscillator solutions, provided only that we cross out the “unphysical" quasi-even states. These states violate the textbook boundary conditions [@Landau]. The only exception concerns the regular limit (one-dimensional case) with $G=0$ (i.e., $\alpha=1/2$) and two parities $Q=\pm 1$. In this sense, the full spectrum (\[spectrum\]) of our complexified oscillator represents a comfortable formal link between the seemingly different one- and three-dimensional Hermitian cases [@creation].
Scalar product and orthogonality
--------------------------------
Let us return to a general Hamiltonian $H = H^\ddagger$ and assume that its spatial parity ${\cal P}$ becomes manifestly broken, $ {\cal P} H {\cal P} = {\cal T} H {\cal T} \neq H$. We may define the quasi-parity as a constant integer $Q_n= (-1)^n$ in the $n-$th state. This generalizes the above square well and harmonic oscillator constructions and applies also to the quartic oscillator of ref. [@Simsek] in a constrained domain of the couplings $g \in (0, g_0)$.
In the first step we introduce the indeterminate scalar product defined by the prescription $\langle \phi | {\cal P} | \psi
\rangle$ of ref. [@whatis]. It does not possess the property of definiteness and defines merely a pseudo-norm. The disappearance of the self-overlap $\langle \psi | {\cal P} | \psi
\rangle =0$ does not imply that the vector $ | \psi \rangle$ must vanish by itself.
The main merit of such a definition of the scalar product lies in the observation that it leads to the usual orthonormality of the left and right eigenstates of the ${\cal PT}$ symmetric Hamiltonians and Schrödinger equations [@ixna], \_n | [P]{}| \_m = Q\_n \_[mn]{}, m , n = 0, 1, …. The completeness relations also acquire the form mentioned in ref. [@whatis], \_[n=0]{}\^| \_n Q\_n \_n | [P]{} = I. The related innovated spectral representation of our non-Hermitian ${\cal PT}$ symmetric Hamiltonians can be written in a bit unusual but fully transparent manner, H= \_[n=0]{}\^| \_n E\_nQ\_n \_n | [P]{}. This enables us to infer that the time evolution of the corresponding system is pseudo-unitary, |(t)= e\^[-iHt]{} |(0)= \_[n=0]{}\^| \_n e\^[-i E\_nQ\_nt]{} \_n | [P]{} |(0), and preserves the value of the scalar product in time, (t)| [P]{}| (t) = (0)| [P]{}| (0) . We have to stress that the Stone’s theorem (which relates the unitary evolution law to the Hermitian underlying Hamiltonians) finds the first interesting extension here.
Spontaneously broken ${\cal PT}$ symmetry
=========================================
Harmonic oscillator inspiration
-------------------------------
Clear parallels between the Hermitian and non-Hermitian ${\cal
PT}$ symmetric Hamiltonians remain marred by the possibility that in the latter case the reality of the spectrum could break down at certain couplings. We have seen that “step-by-step", at an increasing sequence of the couplings, the levels can cease to be real even for the most transparent square well and quartic examples. Here, we are going to explain that from a formal point of view, the break-down of the ${\cal PT}$ symmetry can still remain quite innocent in its physical consequences.
When we return to the harmonic oscillator example (\[HO\]), we can mimic the break-down of the ${\cal PT}$ symmetry when we remove the constraint $G > -1/4$ which was inherited from the standard Hermitian quantum mechanics where its violation would cause the unavoidable fall of the particles into the attractive strong singularity in the origin [@Landau].
In the present non-Hermitian context, one cannot find any persuasive excuse why the smaller couplings $G < -1/4$ could not be admitted as legitimate. Of course, they give the purely imaginary parameters $\alpha = i\,\gamma = i\,\sqrt{1/4-G}$ but the rest of the construction in subsection \[HOlab\] would remain perfectly valid. In particular, the termination of the hypergeometric series will definitely determine the normalizable solutions existing at the complex energies. This is a puzzle which is to be resolved here.
Our illustrative harmonic oscillator energies form the two complex families, E\_N=4n+2-2iQ, = >0 , G < -. \[brspectr\] These energies (as well as their Laguerre-polynomial wave functions) are numbered, as above, by the quasi-parity $Q = \pm 1$ and by the integers $n = 0, 1, \ldots$ in the index $N =
2n+(1-Q)/2$.
We are going to demonstrate now that the similar families of the complex energies can still be interpreted as admissible solutions. We shall see that, rather counterintuitively, there exists in fact no acceptable reason why the complex spectrum (\[brspectr\]) should be forgotten [@Mezincescu].
We have to return to a model-independent argumentation. In the case of the broken symmetry, we shall only assume that the two solutions with $E\neq E^*$ have to be sought simultaneously.
The case of the complex conjugate pairs of the energies
-------------------------------------------------------
In a way inspired by ref. [@spont] we may assume that the ${\cal PT}$ symmetry of the Hamiltonian becomes broken by a pair of the wave functions. One gets the two respective Schrödinger equations H |\_+= E|\_+ , H |\_[-]{}= E\^\*|\_[-]{} . As long as we have $H \neq H^\dagger={\cal P} \,H \,{\cal P}$, we may also re-write our equations in the form of action of the Hamiltonian to the left, \_+ | [P]{} H = E\^\* \_+ |[P]{}, \_[-]{} | [P]{} H = E \_[-]{} |[P]{}. Out of all the possible resulting overlaps, let us compare the following two, \_[+]{} |[P]{} H |\_[+]{}= E\^\*\_[+]{} |[P]{} |\_[+]{} , \_[+]{} | [P]{} H|\_[+]{} = E \_[+]{} |[P]{}|\_[+]{} and, in parallel, \_[-]{} |[P]{} H |\_[-]{}= E\^\*\_[-]{} |[P]{} |\_[-]{} , \_[-]{} | [P]{} H|\_[-]{} = E \_[-]{} |[P]{}|\_[-]{} . These alternatives imply that for $E\neq E^*$ the self-overlaps must vanish, \_+ |[P]{} |\_+= 0, \_[-]{} |[P]{} |\_[-]{}= 0. This leads to several interesting consequences. Firstly, we are free to employ the following less common normalization \_+ |[P]{} |\_[-]{}= \^\*= c, and, wherever needed, re-normalize $c \to \pm 1$. This convention is less common but can be still interpreted as a generalized orthonormality condition in any two-dimensional subspace of the linear pseudo-normalized space of the ${\cal PT}$ symmetry breaking states.
Evolution under the broken ${\cal PT}$ symmetry
-----------------------------------------------
For the sake of simplicity, let us assume that the ${\cal PT}$ symmetry is broken just at the two lowest states. The necessary modification of the completeness relations adds then just the two new terms to the sum over the unbroken $\psi_n$’s, I = | \_+ \_[-]{} | [P]{} + | \_[-]{} \_+ | [P]{} + \_[n=2]{}\^| \_n Q\_n \_n | [P]{} . The forthcoming modification of the spectral decomposition of the Hamiltonian adds the similar two new terms to the sum over the unbroken energies, I = | \_+ \_[-]{} | [P]{} + | \_[-]{} \_+ | [P]{} + \_[n=2]{}\^| \_n E\_n Q\_n \_n | [P]{} . Finally, the pseudo-unitary time development acquires the compact form as well, |(t)= e\^[-iHt]{} |(0)= | \_+ e\^[-iEt]{} \_[-]{} | [P]{} + | \_[-]{} e\^[-iE\^\*t]{} \_+ | [P]{} + \_[n=2]{}\^| \_n e\^[-i E\_nQ\_nt]{} \_n | [P]{} |(0) . It is quite amusing to discover that the value of the scalar product is conserved, (t)| [P]{}| (t) = (0)| [P]{}| (0) . A full parallel to the conventional quantum mechanics is established.
Summary
=======
We may summarize that far beyond the boundaries of the ordinary quantum mechanics, the above-mentioned difficulties with the implications of the Stone’s theorem in ${\cal PT}$ symmetric context were shown related to the “hidden" use of a pseudo-norm. [*Vice versa,*]{} after one admits that the vanishing (pseudo-)norm need not imply the vanishing of the state, a modified version of the Stone’s theorem is recovered. We have shown that the appropriately defined self-overlaps of the states remain unchanged in time not only in the systems characterized by the preserved ${\cal PT}$ symmetry, but also in the domains of couplings where this symmetry is spontaneously broken.
A consistent and complete interpretation of the extended quantum formalism is not at our disposal yet. Still, several new features of it have been revealed here. For example, it seems to mimic some properties of the indefinite metric which is already of quite a common use, say, in relativistic physics.
Acknowledgement {#acknowledgement .unnumbered}
===============
An e-mail-mediated discussion with I. Herbst and G. A. Mezincescu proved inspiring. Partially supported by the GA AS grant Nr. A 104 8004.
[99]{}
L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, London, 1960).
C. M. Bender, S. Boettcher and P. N. Meisinger, J. Math. Phys. 40 (1999) 2201;
C. M. Bender, S. Boettcher and M. Van Savage, J. Math. Phys. 41 (2000) 6381.
C. M. Bender and K. A. Milton, Phys. Rev. D 57 (1998) 3595.
M. Znojil, “PT symmetric square well" (arXiv: quant-ph/0101131), submitted.
M. Znojil, Phys. Lett. [ A 259]{} (1999) 220.
E. Caliceti, S. Graffi and M. Maioli, Commun. Math. Phys. 75 (1980) 51.
G. Alvarez, J. Phys. A: Math. Gen. 27 (1995) 4589;
M. Znojil, J. Phys. A: Math. Gen. 32 (1999) 7419;
B. Bagchi and R. Roychoudhury, J. Phys. A: Math. Gen. 33 (2000) L1;
M. Znojil and G. Lévai, Phys. Lett. [ A 271]{} (2000) 327;
E. Delabaere and D. T. Trinh, J. Phys. A: Math. Gen. 33 (2000) 8771.
F. Fernández, R. Guardiola, J. Ros and M. Znojil, J. Phys. A: Math. Gen. 31 (1998) 10105.
V. Buslaev and V. Grecchi, J. Phys. A: Math. Gen. 26 (1993) 5541.
C. M. Bender and A. Turbiner, Phys. Lett. A 173 (1993) 442;
A. A. Andrianov, F. Cannata, J. P. Dedonder and M. V. Ioffe, Int. J. Mod. Phys. A 14 (1999) 2675;
M. Znojil, F. Cannata, B. Bagchi and R. Roychoudhury, Phys. Lett. B 483 (2000) 284;
B. Bagchi, S. Mallik and C. Quesne, Int. J. Mod. Phys. A, to appear (arXiv: quant-ph/0102093).
C. M. Bender and K. A. Milton, J. Phys. A: Math. Gen. 32 (1999) L87.
E. Delabaere and F. Pham, Phys. Letters A 250 (1998) 25 and 29;
P. Dorey, C. Dunning and R. Tateo, “Spectral equivalences from Bethe Ansatz equations" (arXiv: hep-th/0103051), submitted.
C. M. Bender and S. Boettcher, Phys. Rev. Lett. [ 24]{} (1998) 5243.
C. M. Bender and S. Boettcher, J. Phys. A: Math. Gen. 31 (1998) L273;
M. Znojil, J. Phys. A: Math. Gen. 32 (1999) 4563;
B. Bagchi, F. Cannata and C. Quesne, Phys. Lett. A 269 (2000) 79;
M. Znojil, J. Phys. A: Math. Gen. 33 (2000) 4203 and 6825;
F. Cannata, M. Ioffe, R. Roychoudhury and P. Roy, “A new class of PT-symmertric Hamiltonians" (arXiv: quant-ph/0011089), submitted.
F. Cannata, G. Junker and J. Trost, Phys. Lett. [ A 246]{} (1998) 219;
M. Znojil, Phys. Lett. [ A 264]{} (1999) 108.
B. Bagchi and C. Quesne, Phys. Lett. A 273 (2000) 7165;
M. Znojil, J. Phys. A: Math. Gen. 33 (2000) L61 and 4561;
G. Lévai and M. Znojil, J. Phys. A: Math. Gen. 33 (2000) 7165.
Daniel Bessis, private communication;
M. Znojil and M. Tater, J. Phys. A: Math. Gen. 34 (2001) 1793.
C. M. Bender and K. A. Milton, Phys. Rev. D 55 (1997) R3255.
Bender C M, Berry M, Meisinger P N, Savage V M and Simsek M 2001 J. Phys. A: Math. Gen. 34 L31.
M. Znojil, “Annihilation and creation operators anew" (arXiv: hep-th/0012002), submitted.
M. Znojil, “What is PT symmetry?" (arXiv: quant-ph/0103054), submitted.
G. A. Mezincescu, J. Phys. A: Math. Gen. 33 (2000) 4911 and private communication.
A. Khare and B. P. Mandel, Phys. Lett. A 272 (2000) 53.
R. Seznec and J. Zinn-Justin, J. Math. Phys. 20 (1979) 1398;
J. Avron and R. Seiler, Phys. Rev. D 23 (1981) 1316;
A. Andrianov, Ann. Phys. 140 (1982) 82.
F. J. Dyson, Phys. Rev. 85 (1952) 631.
I. Herbst, private communication.
S. Flügge, Practical Quantum Mechanics I (Springer, Berlin, 1971), p. 166.
|
---
abstract: |
The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie group $G$ are associated to a $G$-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The $G$-strand itself is the map $g(t,s): \mathbb{R}\times \mathbb{R}\to G$, where $t$ and $s$ are the independent variables of the $G$-strand equations. The Euler-Poincaré reduction of the variational principle leads to a formulation where the dependent variables of the $G$-strand equations take values in the corresponding Lie algebra $\mathfrak{g}$ and its co-algebra, $\mathfrak{g}^*$ with respect to the pairing provided by the variational derivatives of the Lagrangian.
We review examples of different $G$-strand constructions, including matrix Lie groups and diffeomorphism group. In some cases the $G$-strand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.
address: |
$^a$ Department of Mathematics, Imperial College, London SW7 2AZ, UK\
$^b$Department of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
author:
- 'Darryl D. Holm$^a$, Rossen I. Ivanov$^b$'
title: ' Euler-Poincaré equations for $G$-Strands'
---
Introduction
============
We give a brief account of the $G$-strand construction, which gives rise to equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie group $G$ associated to a $G$-invariant Lagrangian. Our presentation reviews our previous works [@Ho-Iv-Pe; @Ho-Iv1; @Ho-Iv2; @FDT; @HoLu2013] and is aimed to illustrate the $G$-strand construction with several simple but instructive examples. The following examples are reviewed here:
\(i) $SO(3)$-strand equations for the so-called continuous spin chain. The equations reduce to the integrable chiral model in their simplest (bi-invariant) case.
\(ii) $SO(3)$ - anisotropic chiral model, which is also integrable,
\(iii) ${\rm Diff}(\mathbb{R})$-strand equations. These equations are in general non-integrable; however they admit solutions in $2+1$ space-time with singular support (e.g., peakons). Peakon-antipeakon collisions governed by the ${\rm Diff}(\mathbb{R})$-strand equations can be solved *analytically*, and potentially they can be applied in the theory of image registration.
Ingredients of Euler–Poincaré theory for Left $G$-Invariant Lagrangians
=======================================================================
Let $G$ be a Lie group. A map $g(t,s): \mathbb{R}\times
\mathbb{R}\to G$ has two types of tangent vectors, $\dot{g} := g_t
\in T G$ and $g' :=g_s \in T G$. Assume that the Lagrangian density function $ L(g,\dot{g},g') $ is left $G$-invariant. The left $G$–invariance of $L$ permits us to define $l:
\mathfrak{g}\times \mathfrak{g} \rightarrow \mathbb{R}$ by $$L(g,\dot{g},g')=L(g^{-1}g,g^{-1}\dot{g},g^{-1}g')\equiv l(g^{-1}
\dot{g} , g^{-1} g' ).$$ Conversely, this relation defines for any reduced lagrangian $l=l({\sf u},{\sf v}) : \mathfrak{g}\times \mathfrak{g}
\rightarrow \mathbb{R} $ a left $G$-invariant function $ L : T
G\times TG \rightarrow \mathbb{R} $ and a map $g(t,s):
\mathbb{R}\times \mathbb{R}\to G$ such that $${\sf u} (t,s) := g^{ -1} g_t (t,s) =g^{ -1}\dot{g}(t,s)
\quad\hbox{and}\quad {\sf v} (t,s) := g^{ -1} g_s (t,s)= g^{ -1}
g' (t,s) .$$
The left-invariant tangent vectors ${\sf u} (t,s)$ and ${\sf v}
(t,s)$ at the identity of $G$ satisfy $${\sf v}_t - {\sf u}_s = -\,{\rm ad}_{\sf u}{\sf v} \,.
\label{zero-curv1}$$
The proof is standard and follows from equality of cross derivatives $g_{ts}=g_{st}$.
Equation (\[zero-curv1\]) is usually called a [[******]{}zero-curvature relation]{}.
\[ Euler-Poincaré theorem for left-invariant Lagrangians\]\[lall\]$\,$
With the preceding notation, the following two statements are equivalent:
1. Variational principle on $T G\times TG$ $\,\,$ $
\delta \int _{t_1} ^{t_2} L(g(t,s), \dot{g} (t,s), g'(t,s) )
\,ds\,dt = 0 $ holds, for variations $\delta g(t,s)$ of $ g (t,s)
$ vanishing at the endpoints in $t$ and $s$. The function $g(t,s)$ satisfies Euler–Lagrange equations for $L$ on $G$, given by $$\label{EL-eqns}
\frac{\partial L}{\partial g} - \frac{\partial}{\partial
t}\frac{\partial L}{\partial g_t} - \frac{\partial}{\partial
s}\frac{\partial L}{\partial g_s} = 0.$$
2. The constrained variational principle[^1] $$\label{variationalprinciple}
\delta \int _{t_1} ^{t_2} l({\sf u}(t,s), {\sf v}(t,s)) \,ds\,dt
= 0$$ holds on $\mathfrak{g}\times\mathfrak{g}$, using variations of $
{\sf u} := g^{ -1} g_t (t,s)$ and ${\sf v}:= g^{ -1} g_s(t,s) $ of the forms $$\label{epvariations}
\delta {\sf u} = \dot{{\sf w} } + {\rm ad}_{\sf u}{\sf w}
\quad\hbox{and}\quad \delta {\sf v} = {\sf w}\,' + {\rm ad}_{\sf
v} {\sf w} \,,$$ where ${\sf w}(t,s) :=g^{ -1}\delta g \in \mathfrak{g}$ vanishes at the endpoints. The [[******]{}Euler–Poincaré]{} equations hold on $\mathfrak{g}^*\times\mathfrak{g}^*$ ([[******]{}$G$-strand equations]{})
$$\begin{aligned}
\frac{d}{dt} \frac{\delta l}{\delta {\sf u}} -
\operatorname{ad}_{{\sf u}}^{\ast} \frac{ \delta l }{ \delta {\sf u}}
+ \frac{d}{ds} \frac{\delta l}{\delta {\sf v}} -
\operatorname{ad}_{{\sf v}}^{\ast} \frac{ \delta l }{ \delta {\sf v}}
=
0 \quad\hbox{ \& }\quad
\partial_{s}{\sf u} - \partial_t{\sf v} = [\,{\sf u},\,{\sf v}\,] = {\rm ad}_{\sf u}{\sf v}
\label{GSeqns}\end{aligned}$$
where $({\rm ad}^*:
\mathfrak{g}\times\mathfrak{g}^*\to\mathfrak{g}^*)$ is defined via $({\rm ad}:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g})$ in the dual pairing $\langle \,\cdot\,,\,\cdot\,\rangle:
\mathfrak{g}^*\times\mathfrak{g}\to\mathbb{R}$ by, $$\begin{aligned}
\left\langle {\rm ad}^*_{\sf u}\frac{\delta \ell}{\delta{\sf u}}
\,,\, {\sf v} \right\rangle_\mathfrak{g} = \left\langle
\frac{\delta \ell}{\delta{\sf u}} \,,\, {\rm ad}_{\sf u}{\sf v}
\right\rangle_\mathfrak{g}.\end{aligned}$$
In 1901 Poincaré in his famous work proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equations are equivalent to a new system of differential equations defined on the product of the configuration space with the Lie algebra. These equations are called now in his honor Euler-Poincaré equations. In modern language the contents of the Poincaré’s article [@Poincare] is presented for example in [@Ho2011GM2; @Marle]. English translation of the article [@Poincare] can be found as Appendix D in [@Ho2011GM2].
$G$-strand equations on matrix Lie algebras
===========================================
Denoting ${\sf m}:=\delta \ell/\delta{\sf u}$ and ${\sf n}:=\delta
\ell/\delta{\sf v}$ in $\mathfrak{g}^*$, the $G$-strand equations become $${\sf m}_t + {\sf n}_{s} - {\rm ad}^*_{\sf u}{\sf m}
- {\rm ad}^*_{\sf v}{\sf n}
=0 \quad\hbox{and}\quad
\partial_t{\sf v} -\partial_{s}{\sf u} + {\rm ad}_{\sf u}{\sf v} =
0.$$ For $G$ a semisimple *matrix Lie group* and $\mathfrak{g}$ its *matrix Lie algebra* these equations become $$\label{MatrAlgEq} \begin{split} {\sf m}^{T}_t + {\sf n}^{T}_{s} +
{\rm ad}_{\sf u}{\sf m}^{T}
+ {\rm ad}_{\sf v}{\sf n}^{T}
=&0, \\
\partial_t{\sf v} -\partial_{s}{\sf u} + {\rm ad}_{\sf u}{\sf v}=& 0
\end{split}$$ where the ad-invariant pairing for semisimple matrix Lie algebras is given by
$$\Big\langle{{{\sf m}}}\,,\,{{{\sf n}}}\Big\rangle=\frac{1}{2}\tr({\sf m}^T{\sf n}),$$the transpose gives the map between the algebra and its dual $(\,\cdot\,)^{T}:
\mathfrak{g}\to\mathfrak{g}^*$. For semisimple matrix Lie groups, the adjoint operator is the matrix commutator. Examples are studied in [@Ho-Iv-Pe; @Ho-Iv2; @FDT].
Lie-Poisson Hamiltonian formulation
===================================
Legendre transformation of the Lagrangian $\ell({{{\sf u}},{{\sf
v}}}):\, \mathfrak{g}\times \mathfrak{g}\to\mathbb{R}$ yields the Hamiltonian $h({{{\sf m}},{{\sf v}}}):\, \mathfrak{g}^*\times
\mathfrak{g}\to\mathbb{R}$ $$h({{{\sf m}}},{{{\sf v}}}) = \Big\langle{{{\sf m}}}\,,\,{{{\sf
u}}}\Big\rangle - \ell({{{\sf u}},{{\sf v}}}) \,.
\label{leglagham} \vspace{-3mm}$$
Its partial derivatives imply $$\begin{aligned}
\frac{\delta l}{\delta {{\sf u}}} = {{\sf m}} \,,\quad
\frac{\delta h}{\delta {{\sf m}}} = {{\sf u}} \quad\hbox{and}\quad
\frac{\delta h}{\delta {{\sf v}}} = -\,\frac{\delta \ell}{\delta
{{\sf v}}} = {\sf v} .\end{aligned}$$
These derivatives allow one to rewrite the Euler-Poincaré equation solely in terms of momentum ${{\sf m}}$ as $$\label{hameqns-so3}
\begin{split}
{\partial_t} {{\sf m}} &= {\rm ad}^*_{\delta h/\delta {{\sf m}}}\,
{{\sf m}} + \partial_{s} \frac{\delta h}{\delta {{\sf v}}} - {\rm
ad}^*_{{\sf v}}\,\frac{\delta h}{\delta {{\sf v}}}
\, ,\\
\partial_t {{\sf v}}
&= \partial_{s}\frac{\delta h}{\delta {{\sf m}}} - {\rm
ad}_{\delta h/\delta {{\sf m}}}\,{{\sf v}} \,.
\end{split}$$ Assembling these equations into Lie-Poisson Hamiltonian form gives, $$\label{LP-Ham-struct-symbol1}
\frac{\partial}{\partial t}
\begin{bmatrix}
{{\sf m}}
\\
{{\sf v}}
\end{bmatrix}
=
\begin{bmatrix}
{\rm ad}^\ast(\,\cdot\,) {{\sf m}}
&\hspace{5mm}
\partial_s - {\rm ad}^*_{{\sf v}}
\\
\partial_s + {\rm ad}_{{\sf v}}
&\hspace{5mm} 0
\end{bmatrix}
\begin{bmatrix}
\delta h/\delta{{\sf m}} \\
\delta h/\delta{{\sf v}}
\end{bmatrix}$$
The Hamiltonian matrix in equation (\[LP-Ham-struct-symbol1\]) also appears in the Lie-Poisson brackets for Yang-Mills plasmas, for spin glasses and for perfect complex fluids, such as liquid crystals.
Example: The Euler-Poincaré PDEs for the $SO(3)$-strand and the chiral model. The $2$-time spatial and body angular velocities on $\mathfrak{so}(3)$
====================================================================================================================================================
Let us make the following explicit identification: $${\sf u}=\left(\begin{array}{ccc}
0 & -u_3 & u_2 \\
u_3 & 0 & -u_1 \\
-u_2 & u_1 & 0 \\
\end{array}\right)\in \mathfrak{g}\quad \leftrightarrow \quad \boldsymbol{{{\sf u}}}\equiv\left(\begin{array}{c}
u_1 \\
u_2 \\
u_3 \\
\end{array}\right)\in \mathbb{R}^3 \label{eqomegaL}$$
and similarly for $\boldsymbol{{{\sf v}}}$. In terms of the corresponding group element $g(s,t)$, describing rotation, $ {\sf
u}(t,{s})=g^{-1}\partial_t g(t,{s}) $ and $ {\sf
v}(t,{s})=g^{-1}\partial_{s} g(t,{s}) $ resemble $2$ body angular velocities. For $G=SO(3)$ and Lagrangian $ \ell (\boldsymbol{{{\sf
u}},\,{{\sf v}}} ): \mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}, $ in $1+1$ space-time the Euler-Poincaré equation becomes
$$\frac{\partial}{\partial t} \frac{\delta\ell}{\delta \boldsymbol{{{\sf u}}}}
+ \boldsymbol{{{\sf u}}}\times\frac{\delta\ell}{\delta \boldsymbol{{{\sf u}}} }
=
- \left(\frac{\partial}{\partial {s}} \frac{\delta\ell}{\delta \boldsymbol{{{\sf v}}} }
+
{\boldsymbol{{{\sf v}}}}\times
\frac{\delta\ell}{\delta \boldsymbol{{{\sf v}}} }\right)
\,,
\label{2timeEP-SO3}$$
and its auxiliary equation becomes $$\frac{\partial}{\partial t} \boldsymbol{{{\sf v}}}
=
\frac{\partial}{\partial {s}}{\boldsymbol{{{\sf u}}}}
+
{\boldsymbol{{{\sf v}}}}\times{\boldsymbol{{{\sf u}}}}
\,.
\label{aux-eqn-2timeX}$$
The Hamiltonian form of these equations on $\mathfrak{so}(3)^*$ are obtained from the Legendre transform relations $$\begin{aligned}
\frac{\delta \ell}{\delta \boldsymbol{{{\sf u}}} }
= \boldsymbol{{{\sf m}}}
\,,\quad
\frac{\delta h}{\delta \boldsymbol{{{\sf m}}}} = \boldsymbol{{{\sf u}}}
\quad\hbox{and}\quad
\frac{\delta h}{\delta \boldsymbol{{{\sf v}}} }
= -\,\frac{\delta \ell}{\delta \boldsymbol{{{\sf v}}} }
\,.\end{aligned}$$
Hence, the Euler-Poincaré equation implies the Lie-Poisson Hamiltonian structure in vector form
$$\label{LP-Ham-struct-so3}
\partial_t
\begin{bmatrix}
\boldsymbol{{{\sf m}}}
\\
\boldsymbol{{{\sf v}}}
\end{bmatrix}
=
\begin{bmatrix}
\boldsymbol{{{\sf m}}}\times
& \partial_s
+ \boldsymbol{{{\sf v}}}\times
\\
\partial_s
+ \boldsymbol{{{\sf v}}}\times
& 0
\end{bmatrix}
\begin{bmatrix}
\delta h/\delta\boldsymbol{{{\sf m}}} \\
\delta h/\delta\boldsymbol{{{\sf v}}}
\end{bmatrix}.$$
This Poisson structure appears in various other theories, such as complex fluids and filament dynamics.
When $$\label{spinchainLagr}\ell=\frac12 \int
(\boldsymbol{{{\sf u}}}\cdot A \boldsymbol{{{\sf u}}}
+\boldsymbol{{{\sf v}}}\cdot B \boldsymbol{{{\sf
v}}})\,ds$$ this is the $SO(3)$ *spin-chain model*, which is in general non-integrable- eq. (\[2timeEP-SO3\]) and (\[aux-eqn-2timeX\]) give: $$\frac{\partial}{\partial t} A \boldsymbol{{{\sf
u}}} + \boldsymbol{{{\sf u}}}\times A \boldsymbol{{{\sf u}}}+
\frac{\partial}{\partial {s}} B \boldsymbol{{{\sf v}}} +
{\boldsymbol{{{\sf v}}}}\times B \boldsymbol{{{\sf v}}} =0\,,
\label{SO3 spin chain1}$$ $$\frac{\partial}{\partial t} \boldsymbol{{{\sf v}}} =
\frac{\partial}{\partial {s}}{\boldsymbol{{{\sf u}}}} +
{\boldsymbol{{{\sf v}}}}\times{\boldsymbol{{{\sf u}}}}
\,.\label{SO3 spin chain2}$$
When $A=-B=1$, this is the $SO(3)$ *chiral model*, which is an integrable Hamiltonian system.
$$\boldsymbol{{{\sf u}}}_t - \boldsymbol{{{\sf v}}}_s =0\,,
\label{SO3 chiral1}$$
$$\boldsymbol{{{\sf v}}}_t - {\boldsymbol{{{\sf u}}}}_s +
{\boldsymbol{{{\sf u}}}}\times{\boldsymbol{{{\sf v}}}}=0
\,.\label{SO3 chiral2}$$
Integrability
=============
Some of the $G$-strands models are well known integrable models. They have a [*zero-curvature representation*]{} for two operators $L$ and $M$ of the form $$\begin{aligned}
L_t - M_s + [L,M] = 0, \label{ZC-comrel}\end{aligned}$$which is the compatibility condition for a pair of linear equations $$\psi_s = L\psi, \quad\hbox{and}\quad \psi_t = M\psi.$$
For the SO(3) chiral model for example these operators are $$\begin{split} L&=\frac{1}{4}\left[(1+\lambda)({\sf u}-{\sf v})-
\left(1+\frac{1}{\lambda}\right)({\sf u}+{\sf v}) \right],\\
M&=-\frac{1}{4}\left[(1+\lambda)({\sf u}-{\sf v})+
\left(1+\frac{1}{\lambda}\right)({\sf u}+{\sf v}) \right].
\end{split}$$
Another integrable matrix example: $SO(3)$ anisotropic Chiral model [@Ch1981] $$\label{XY-eqn-1a}
\begin{split}
\partial_t{\boldsymbol{\mathsf{v}}}(t,{s}) - \partial_{s}{\boldsymbol{\mathsf{u}}}(t,{s})
+{\boldsymbol{\mathsf{u}}}\times
P{\boldsymbol{\mathsf{v}}}-{\boldsymbol{\mathsf{v}}}\times
P{\boldsymbol{\mathsf{u}}}=0 \,,
\\
\partial_{s}{\boldsymbol{\mathsf{v}}}(t,{s}) - \partial_t{\boldsymbol{\mathsf{u}}}(t,{s})
- {\boldsymbol{\mathsf{v}}}\times P
{\boldsymbol{\mathsf{v}}}+{\boldsymbol{\mathsf{u}}}\times
P{\boldsymbol{\mathsf{u}}}=0 \,.
\end{split}$$
$P=\text{diag}(P_1,P_2,P_3)$ is a constant diagonal matrix. Under the linear change of variables $$\boldsymbol{\mathsf{X}} = \boldsymbol{\mathsf{u}} -
\boldsymbol{\mathsf{v}} \quad\hbox{and}\quad
\boldsymbol{\mathsf{Y}} = -\,\boldsymbol{\mathsf{u}} -
\boldsymbol{\mathsf{v}} \label{changeXY-uv}$$ equations (\[XY-eqn-1a\]) acquire the form of the following $SO(3)$ anisotropic chiral model,
$$\label{uv-eqn-1}
\begin{split}
\partial_t{\boldsymbol{\mathsf{X}}}(t,{s}) +\partial_{s}{\boldsymbol{\mathsf{X}}}(t,{s})
+{\boldsymbol{\mathsf{X}}}\times P{\boldsymbol{\mathsf{Y}}}&=0
\,,\\
\partial_t{\boldsymbol{\mathsf{Y}}}(t,{s}) - \partial_{s}{\boldsymbol{\mathsf{Y}}}(t,{s})
+{\boldsymbol{\mathsf{Y}}}\times P {\boldsymbol{\mathsf{X}}}&=0
\,.
\end{split}$$
The system (\[uv-eqn-1\]) represents two *cross-coupled* equations for ${\boldsymbol{\mathsf{X}}}$ and ${\boldsymbol{\mathsf{Y}}}$. These equations preserve the magnitudes $|{\boldsymbol{\mathsf{X}}}|^2$ and $|{\boldsymbol{\mathsf{Y}}}|^2$, so they allow the further assumption that the vector fields $(\boldsymbol{\mathsf{X}},\boldsymbol{\mathsf{Y}})$ take values on the product of unit spheres $\mathbb{S}^2 \times \mathbb{S}^2
\subset \mathbb{R}^3 \times\mathbb{R}^3$. The anisotropic chiral model is an integrable system and its Lax pair in terms of $(\boldsymbol{\mathsf{u}},\boldsymbol{\mathsf{v}})$ utilizes the following isomorphism between $\mathfrak{so}(3) \oplus
\mathfrak{so}(3)$ and $\mathfrak{so}(4)$: $$A({\boldsymbol{\mathsf{u}}},{\boldsymbol{\mathsf{v}}})=\left(
\begin{matrix}
0 & u_3&-u_2 & v_1\\
-u_3 & 0&u_1 & v_2\\
u_2 & -u_1&0 & v_3 \\
-v_1 & -v_2&-v_3 & 0
\end{matrix}
\right) . \label{Mso4-def}$$
The system (\[XY-eqn-1a\]) can be recovered as a compatibility condition of the operators $$\begin{aligned}
L&=&\partial_{s}-A({\boldsymbol{\mathsf{v}}},{\boldsymbol{\mathsf{u}}})(\lambda\,{\rm Id}+J),\\
M&=&\partial_t-A({\boldsymbol{\mathsf{u}}},{\boldsymbol{\mathsf{v}}})(\lambda\,{\rm
Id}+J), \label{L-Mpair}\end{aligned}$$ where the diagonal matrix $J$ is defined by $$J = -\frac{1}{2}\text{diag}(P_1,P_2,P_3,P_1+P_2+P_3).
\label{J-def}$$
This Lax pair is due to Bordag and Yanovski [@BoYa1995]. The $O(3)$ anisotropic chiral model can be derived as an Euler-Poincaré equation from a Lagrangian with quadratic kinetic and potential energy. The details are presented in [@Ho-Iv-Pe].
If ${\sf P}={\rm Id}$, equations (\[XY-eqn-1a\]) recover the $SO(3)$ chiral model.
The ${\rm Diff}(\mathbb{R})$-strand system
===========================================
The constructions described briefly in the previous sections can be easily generalized in cases where the Lie group is the group of the Diffeomorphisms. Consider Hamiltonian which is a right-invariant bilinear form given by the $H^1$ Sobolev inner product $$H(u,v)\equiv \frac{1}{2}\int
_{\mathcal{M}}(uv+u_xv_x) dx.$$ The manifold $\mathcal{M}$ is $\mathbb{S}^1$ or in the case when the class of smooth functions vanishing rapidly at $\pm \infty$ is considered, we will allow $\mathcal{M} \equiv \mathbb{R}$.
Let us introduce the notation $u(g(x))\equiv u\circ g$. If $g(x)\in G$, where $G\equiv \text{Diff}(\mathcal{M})$, then $$H(u,v)=H(u\circ g, v\circ g)$$ is a right-invariant $H^1$ metric.
Let us further consider an one-parametric family of diffeomprphisms, $g(x,t)$ by defining the $t$ - evolution as $$\dot{g}=u(g(x,t),t), \qquad g(x,0)=x, \qquad \text{i.e.} \qquad
\dot{g}=u\circ g \in T_g G;$$ $u=\dot{g}\circ g^{-1}\in
\mathfrak{g}$, where $\mathfrak{g}$, the corresponding Lie-algebra is the algebra of vector fields, $\text{Vect}(\mathcal{M})$. Now we recall the following result:
(A. Kirillov, 1980, [@K81; @K93]) The dual space of $\mathfrak{g}$ is a space of distributions but the subspace of local functionals, called the regular dual $\mathfrak{g}^*$ is naturally identified with the space of quadratic differentials $m(x)dx^2$ on $\mathcal{M}$. The pairing is given for any vector field $u\partial_x\in
\text{Vect}(\mathcal{M})$ by
$$\langle mdx^2, u\partial_x\rangle=\int_{\mathcal{M}}m(x)u(x)dx$$
The coadjoint action coincides with the action of a diffeomorphism on the quadratic differential:
$$\text{Ad}_g^*:\quad mdx^2\mapsto m(g)g_x^2dx^2$$ and $$\text{ad}_{u}^*=2u_x+u\partial_x$$ Indeed, a simple computation shows that $$\begin{aligned}
\langle\text{ad}_{u\partial_x}^*
mdx^2,v\partial_x\rangle&=&\langle
mdx^2,[u\partial_x,v\partial_x]\rangle=\int_{\mathcal{M}}m(u_xv-v_xu)dx=
\nonumber
\\ \int_{\mathcal{M}}v(2mu_x+um_x)dx&=&\langle(2mu_x+um_x)dx^2,v\partial_x\rangle, \nonumber \end{aligned}$$ i.e. $\text{ad}_{u}^*m=2u_xm+um_x$.
The ${\rm Diff}(\mathbb{R})$-strand system arises when we choose $G={\rm Diff}(\mathbb{R})$. For a two-parametric group we have two tangent vectors $$\partial_t{g}= u \circ g
\quad\hbox{and}\quad
\partial_s{g}= v \circ g
\,,$$ where the symbol $\circ$ denotes composition of functions.
In this right-invariant case, the $G$-strand PDE system with reduced Lagrangian $\ell(u,v)$ takes the form,
$$\begin{aligned}
\begin{split}
\frac{\partial}{\partial t} \frac{\delta\ell}{\delta u} +
\frac{\partial}{\partial s} \frac{\delta\ell}{\delta v } &= -\,
{\rm ad}^*_{u}\frac{\delta\ell}{\delta u} - {\rm ad}^*_{v
}\frac{\delta\ell}{\delta v } \,,
\\
\frac{\partial v}{\partial t} - \frac{\partial u}{\partial s} &=
{\rm ad}_u v \,.
\end{split}
\label{Gstrand-eqn1R}\end{aligned}$$
Of course, the distinction between the maps $({u},{v}):
\mathbb{R}\times \mathbb{R}\to \mathfrak{g}\times\mathfrak{g}$ and their pointwise values $({u}(t,s),{v}(t,s))\in
\mathfrak{g}\times\mathfrak{g}$ is clear. Likewise, for the variational derivatives ${\delta\ell}/{\delta {{u}}}$ and ${\delta\ell}/{\delta v}$.
The ${\rm Diff}(\mathbb{R})$-strand Hamiltonian structure
=========================================================
Upon setting $m={\delta\ell}/{\delta u }$ and $n={\delta\ell}/{\delta v }$, the right-invariant ${\rm
Diff}(\mathbb{R})$-strand equations in (\[Gstrand-eqn1R\]) for maps $\mathbb{R}\times\mathbb{R}\to G={\rm Diff}(\mathbb{R})$ in one spatial dimension may be expressed as a system of two 1+2 PDEs in $(t,s,x)$, $$\begin{aligned}
\begin{split}
m_t + n_s &= -\, {\rm ad}^*_{u }m - {\rm ad}^*_{v }n = - (u m)_x -
m u_x - (v n)_x - nv_x \,,
\\ \bigskip
v_t - u _s &= -\,{\rm ad}_v u = -u v_x + v u_x \,.
\end{split}
\label{Gstrand-eqn2R}\end{aligned}$$ The Hamiltonian structure for these ${\rm
Diff}(\mathbb{R})$-strand equations is obtained by Legendre transforming to $$h(m,v)=\langle m,\, u\rangle - \ell(u,\,v) \,.$$
One may then write the equations (\[Gstrand-eqn2R\]) in Lie-Poisson Hamiltonian form as
$$\frac{d}{dt}
\begin{bmatrix}
m \\ v
\end{bmatrix}
=
\begin{bmatrix}
-\,{\rm ad}^*(\,\cdot\,) m &\quad \partial_s + {\rm ad}^*_v
\\
\partial_s - {\rm ad}_v &\quad 0
\end{bmatrix}
\begin{bmatrix}
{\delta h}/{\delta m} = u
\\
{\delta h}/{\delta v} = -\, n
\end{bmatrix}.
\label{1stHamForm}$$
Peakon solutions of the ${\rm Diff}(\mathbb{R})$-strand equations
=================================================================
With the following choice of Lagrangian, $$\ell (u ,v ) = \frac12 \|u \|^2_{H^1} - \frac12 \|v \|^2_{H^1} \,,
\label{Gstrand-pkn-Lag}$$ the corresponding Hamiltonian is positive-definite and the ${\rm
Diff}(\mathbb{R})$-strand equations (\[Gstrand-eqn2R\]) admit peakon solutions in *both* momenta $$m=u-u_{xx} \quad \text{ and} \quad n=-(v-v_{xx}),$$ with continuous velocities $u$ and $v$. This is a two-component generalization of the CH equation.
The ${\rm Diff}(\mathbb{R})$-strand equations (\[Gstrand-eqn2R\]) admit singular solutions expressible as linear superpositions summed over $a\in\mathbb{Z}$ $$\begin{aligned}
\begin{split}
m(s,t,x) &= \sum_a M_a(s,t)\delta(x-Q^a(s,t)) \,,\\
n(s,t,x) &= \sum_a N_a(s,t)\delta(x-Q^a(s,t)) \,,
\\
u(s,t,x) & =K*m=\sum_a M_a(s,t) K(x,Q^a) \,,\\
v(s,t,x) & = -K*n=-\sum_a N_a(s,t) K(x,Q^a) \,,
\end{split}
\label{Gstrand-singsolns1}\end{aligned}$$ that are *peakons* in the case that $K(x,y)= \frac12
e^{-|x-y|}$ is the Green function the inverse Helmholtz operator $(1-\partial_x^2)$: $$(1-\partial_x^2)K(x,0)=\delta(x)$$
The solution parameters $\{Q^a(s,t), M_a(s,t), N_a(s,t)\}$ with $a\in\mathbb{Z}$ that specify the singular solutions (\[Gstrand-singsolns1\]) are determined by the following set of evolutionary PDEs in $s$ and $t$, in which we denote $
K^{ab}:=K(Q^a,Q^b) $ with integer summation indices $a,b,c,e\in\mathbb{Z}$: $$\begin{aligned}
\begin{split}
\partial_t Q^a(s,t) &= u(Q^a,s,t) = \sum_b M_b(s,t) K^{ab}
\,,\\
\partial_s Q^a(s,t) &= v(Q^a,s,t) =-\sum_b N_b(s,t) K^{ab}
\,,\\
\partial_t M_a(s,t) &= -\, \partial_s N_a
-\sum_c (M_aM_c-N_aN_c) \frac{\partial K^{ac}}{\partial Q^a}
\quad\hbox{(no sum on $a$),}
\\
\partial_t N_a(s,t) &=-\partial_s M_a
+ \sum_{b,c,e} (N_bM_c - M_bN_c) \frac{\partial K^{ec}}{\partial Q^e} (K^{eb}-K^{cb})(K^{-1})_{ae}
\,.
\end{split}
\label{Gstrand-eqns}\end{aligned}$$
The last pair of equations in (\[Gstrand-eqns\]) may be solved as a system for the momenta, i.e., Lagrange multipliers $(M_a,N_a)$, then used in the previous pair to update the support set of positions $Q^a(t,s)$.
Single-peakon solution of the of the ${\rm Diff}(\mathbb{R})$-strand system
===========================================================================
The single-peakon solution of the ${\rm Diff}(\mathbb{R})$-strand equations (\[Gstrand-eqn2R\]) is straightforward to obtain from (\[Gstrand-eqns\]). Combining the equations in (\[Gstrand-eqns\]) for a single peakon shows that $Q^1(s,t)$ satisfies the Laplace equation, $$(\partial_s^2 - \partial _t^2)Q^1(s,t)=0\,.
\label{Q-1-peak}$$ Thus, any function $h(s,t)$ that solves the wave equation provides a solution $Q^1=h(s,t)$. From the first two equations in (\[Gstrand-eqns\]) $$M_1(s,t)=\frac{1}{K_0}h_t (s,t)
\qquad
N_1(s,t)=\frac{1}{K_0}h_s (s,t),
\label{MN-1-peak}$$ where $K_0=K(0,0)$.
The solutions for the single-peakon parameters $Q^1, M_1$ and $N_1$ depend only on one function $h(s,t)$, which in turn depends on the $(s,t)$ boundary conditions. The shape of the Green’s function comes into the corresponding solutions for the peakon profiles $$u(s,t,x) = M_1(s,t) K(x,Q^1(s,t)) \,,\qquad v(s,t,x) =- N_1(s,t)
K(x,Q^1(s,t)) \,.$$
Peakon-Antipeakon collisions on a ${\rm Diff}(\mathbb{R})$-strand
=================================================================
Denote the relative spacing $X(s,t)=Q^1-Q^2$ for the peakons at positions $Q^1(t,s)$ and $Q^2(t,s)$ on the real line and the Green’s function $K=K(X)$. Then the first two equations in (\[Gstrand-eqns\]) imply $$\begin{aligned}
\begin{split}
\partial_t X &= (M_1-M_2)(K_0-K(X))
\,,\\
\partial_s X &= - (N_1-N_2)(K_0-K(X))
\,,\end{split} \label{Qdiff-eqns}\end{aligned}$$ where $K_0=K(0)$.
The second pair of equations in (\[Gstrand-eqns\]) may then be written as $$\begin{aligned}
\begin{split}
\partial_t M_1 &= - \partial_s N_1 - (M_1M_2- N_1N_2)K'(X)
\,,\\
\partial_t M_2 &= - \partial_s N_2 + (M_1M_2- N_1N_2)K'(X)
\,,\\
\partial_t N_1 &= - \partial_s M_1 + (N_1M_2-M_1N_2)
\frac{K_0-K}{K_0+K}K'(X)
\,,\\
\partial_t N_2 &= - \partial_s M_2 + (N_1M_2-M_1N_2)
\frac{K_0-K}{K_0+K} K'(X) \,.
\end{split}
\label{Gstrand-pp}\end{aligned}$$ Asymptotically, when the peakons are far apart, the system (\[Gstrand-pp\]) simplifies, since $\frac{K_0-K}{K_0+K}\to1$ and $K'(X)\to0$ as $|X|\to\infty$.
The system (\[Gstrand-pp\]) has two immediate conservation laws obtained from their sums and differences, $$\begin{aligned}
\begin{split}
\partial_t (M_1+M_2) &= -\, \partial_s (N_1+N_2)
\,,\\
\partial_t (N_1-N_2) &=-\partial_s (M_1-M_2)
\,.\end{split} \label{pp-CLs}\end{aligned}$$ These may be resolved by setting $$\begin{aligned}
\begin{split}
M_1-M_2 &= \frac{\partial_t X}{K_0-K} \,,\qquad N_1-N_2 = -
\frac{\partial_s X}{K_0-K}
\,,\\
M_1+M_2 &= \partial_s\phi \,,\qquad N_1+N_2 = -\,\partial_t\phi
\,,
\end{split}
\label{pp-Xpotentials}\end{aligned}$$ and introducing two potential functions, $X$ and $\phi$, for which equality of cross derivatives will now produce the system of equations (\[Qdiff-eqns\]) and (\[Gstrand-pp\]).
A simplification.
=================
A simplification arises if $\phi=0$, in which case the collision is perfectly antisymmetric, as seen from equation (\[pp-Xpotentials\]). This is the peakon-antipeakon collision, for which the equation for $X$ reduces to $$\begin{aligned}
(\partial_t^2 - \partial _s^2) X &+ \frac{K'}{2(K_0-K)} (X_t^2-
X_s^2) = 0 \,.\end{aligned}$$ This equation can be easily rearranged to produce a linear equation: $$\begin{aligned}
(\partial_t^2 - \partial _s^2) F(X) = 0 \,,\quad\hbox{where}\quad
F(X) = \int_{X_0}^X (K_0-K(Y))^{-1/2}\,dY \,.\end{aligned}$$ When $K(Y)=\frac{1}{2}e^{-|Y|}$, we have $$\begin{aligned}
F(X) = \sqrt{2}\int_{X_0}^X \frac{1}{\sqrt{1-e^{-|Y|}}}\,dY
.\label{F-express}\end{aligned}$$
We can take for simplicity $X_0=0$, this would change $F(X)$ only by a constant. The computation gives $$F(X)=
2\sqrt{2}\,\text{sign}(X)\cosh^{-1}\left(e^{|X|/2}\right)$$. Hence the solution $X(t,s)$ can be expressed in terms of any solution $h(t,s)$ of the linear wave equation $(\partial_t^2 -
\partial _s^2)h(t,s)=0$ as $$\begin{aligned}
X(t,s) = \pm \ln \left({\,\rm cosh}^2(h(t,s))\right) \,.
\label{X-express}\end{aligned}$$ $h(t,s)$ is any solution of the wave equation. $$M_1=-M_2 =
\frac{\partial_t X}{2(K_0-K(X))} \,,\qquad N_1=-N_2 = -
\frac{\partial_s X}{2(K_0-K(X))}.$$
Complex ${\rm Diff}(\mathbb{R})$-strand equations {#complex-rm-diffmathbbr-strand-equations .unnumbered}
==================================================
The ${\rm Diff}(\mathbb{R})$-strands may also be *complexified*. Upon complexifying $(s,t)\in
\mathbb{R}^2\to(z,\bar{z})\in \mathbb{C}$ where $\bar{z}$ denotes the complex conjugate of $z$ and setting $\partial_z{g}= u \circ
g$ and $\partial_{\bar{z}}{g}= \bar{u} \circ g$ the Euler-Poincaré $G$-strand equations in (\[Gstrand-eqn2R\]) become $$\begin{aligned}
\begin{split}
\frac{\partial}{\partial z} \frac{\delta\ell}{\delta u} +
\frac{\partial}{\partial \bar{z} } \frac{\delta\ell}{\delta {
\bar{u}} } &= -\, {\rm ad}^*_{u}\frac{\delta\ell}{\delta u} - {\rm
ad}^*_{{\bar{u}} }\frac{\delta\ell}{\delta { \bar{u}} } \,,
\\
\frac{\partial {\bar{u}}}{\partial z} - \frac{\partial
u}{\partial \bar{z}} &= {\rm ad}_u { \bar{u}} \,.
\end{split}
\label{ComplexGstrand-eqns1}\end{aligned}$$ Here the Lagrangian $\ell$ is taken to be real: $$\begin{aligned}
\ell
(u, {\bar{u}}) = \frac12\|\nu\|_{H^1}^2 = \frac12\int u\,
(1-\partial_x^2) \, {{ \bar{u}}} \,dx . \, \label{ComplexNorm-ell}\end{aligned}$$
Upon setting $m={\delta\ell}/{\delta u}$, $\bar{m}={\delta\ell}/{\delta {\bar{u}} }$, for the real Lagrangian $\ell$, equations (\[ComplexGstrand-eqns1\]) may be rewritten as
$$\begin{aligned}
\begin{split}
m_{z} + \bar{m}_{\bar{z}} &= -\, {\rm ad}^*_{u }m - {\rm
ad}^*_{{\bar{u}} }\,\bar{m} = - (u m)_x - mu_x - ({\bar{u}}
\,\bar{m})_x - \bar{m}\,{\bar{u}}_x \,,
\\ \\
{\bar{u}}_z- u _{\bar{z}} &= -\,{\rm ad}_{{\bar{u}}} \,u = -u \,{
\bar{u}}_x + { \bar{u}}\,u_x \,,
\end{split}
\label{ComplexGstrand-eqns2}\end{aligned}$$
where the independent coordinate $x\in\mathbb{R}$ is on the real line, although coordinates $ (z,\bar{z})\in\mathbb{C}$ are complex, as are solutions $u$, and $m=u-u_{xx}$. This is a possible comlexification of the Camassa-Holm equation. These equations are invariant under two involutions, $P$ and $C$, where $$P: (x,m)\to(-x,-m)
\quad \hbox{and} \quad
C: \hbox{Complex conjugation.}$$ They admit singular solutions just as before, modulo $\mathbb{R}\times\mathbb{R}\to\mathbb{C}$. For real variables $m=\bar{m}$, $u={\bar{u}}$ and real evolution parameter $z=\bar{z}=:t$, they reduce to the CH equation. Their travelling wave solutions and other possible CH complexifications are studied in [@Ho-Iv1].
Conclusions {#conclusion-sec .unnumbered}
===========
The $G$-strand equations comprise a system of PDEs obtained from the Euler-Poincaré (EP) variational equations for a $G$-invariant Lagrangian, coupled to an auxiliary *zero-curvature* equation. Once the $G$-invariant Lagrangian has been specified, the system of $G$-strand equations in (\[MatrAlgEq\]) follows automatically in the EP framework. For matrix Lie groups, some of the the $G$-strand systems are integrable. The single-peakon and the peakon-antipeakon solution of the ${\rm Diff}(\mathbb{R})$-strand equations (\[Gstrand-eqn2R\]) depends on a single function of $s,t$. The *complex* ${\rm Diff}(\mathbb{R})$-strand equations and their peakon collision solutions have also been solved by elementary means. The stability of the single-peakon solution under perturbations into the full solution space of equations (\[Gstrand-eqn2R\]) would be an interesting problem for future work.
Acknowledgments
===============
We are grateful for enlightening discussions of this material with F. Gay-Balmaz, T. S. Ratiu and C. Tronci. Work by RII was partially supported by the Science Foundation Ireland (SFI), under Grant No. 09/RFP/MTH2144. Work by DDH was partially supported by Advanced Grant 267382 FCCA from the European Research Council.
References {#references .unnumbered}
==========
[9]{}
Bordag, L. A. and Yanovski, A. B. \[1995\] Polynomial Lax pairs for the chiral $O(3)$ field equations and the Landau-Lifshitz equation. [*J. Phys. A: Math. Gen.*]{} [**28**]{}, 4007–4013.
Cherednik, I. \[1981\] On the integrability of the 2-dimensional asymmetric chiral $O(3)$ field equations and their quantum analogue, [*J. Nuc. Phys*]{}, [**33**]{}, 278–282, in Russian.
Gay-Balmaz, F., Holm, D. D. and Ratiu, T. S. \[2013\] Integrable G-Strands on semisimple Lie groups, arXiv:1308.3800 \[math-ph\] <http://arxiv.org/pdf/1308.3800v1.pdf>
Holm, D. D. \[2011\] [*Geometric Mechanics II: Rotating, Translating and Rolling*]{}, World Scientific: Imperial College Press, Singapore, 2nd edition (2011).
Holm, D. D. and Ivanov, R. I. \[2013\] $G$-Strands and peakon collisions on Diff$(\mathbb{R})$ [*SIGMA*]{} [**9**]{} 027, 14 pages.\
<http://arxiv.org/pdf/1211.6931v1.pdf>
Holm, D. D. and Ivanov, R. I. \[2013\] Matrix $G$-Strands, arxiv 1305.4010,\
<http://arxiv.org/pdf/1305.4010v1.pdf>
Holm, D. D., Ivanov, R. I. and Percival, J. R. \[2012\] $G$-Strands, *Journal of Nonlinear Science* [**22**]{}, (4) 517–551.\
<http://arxiv.org/pdf/1109.4421.pdf>
Holm, D. D. and Lucas, A. M. \[2013\] Toda lattice G-Strands, arXiv:1306.2984 \[nlin.SI\]\
<http://arxiv.org/pdf/1306.2984v1.pdf>
Kirillov, A. \[1981\] The orbits of the group of diffeomorphisms of the circle, and local Lie superalgebras, [*Funct. Anal. Appl.*]{} [**15**]{}, 135-136 (English); Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 75-76 (Russian).
Kirillov, A. \[1993\] The orbit method. II. Infinite-dimensional Lie groups and Lie algebras. Representation theory of groups and algebras, [*Contemp. Math.*]{} [**145**]{}, Amer. Math. Soc., Providence, RI, pp. 33-63.
Marle, C.-M. \[2013\] On Henry Poincaré’s note “Sur une forme nouvelle des équations de la Mécanique", *J. Geom. Symm. Phys.*, [**29**]{} 1-38.
Poincaré, H. \[1901\] Sur une forme nouvelle des équations de la Mécanique, *C. R. Acad. Sci. Paris*, [**CXXXII**]{} 369-371.
[^1]: As with the basic Euler–Poincaré equations, this is not strictly a variational principle in the same sense as the standard Hamilton’s principle. It is more like the Lagrange d’Alembert principle, because we impose the stated constraints on the variations allowed.
|
---
abstract: 'The system formed in ultrarelativistic heavy-ion collisions behaves as a nearly-perfect fluid. This collective behavior is probed experimentally by two-particle azimuthal correlations, which are typically averaged over the properties of one particle in each pair. In this Letter, we argue that much additional information is contained in the detailed structure of the correlation. In particular, the correlation matrix exhibits an approximate factorization in transverse momentum, which is taken as a strong evidence for the hydrodynamic picture, while deviations from the factorized form are taken as a signal of intrinsic, “nonflow” correlations. We show that hydrodynamics in fact predicts factorization breaking as a natural consequence of initial state fluctuations and averaging over events. We derive the general inequality relations that hold if flow dominates, and which are saturated if the matrix factorizes. For transverse momenta up to 5 GeV, these inequalities are satisfied in data, but not saturated. We find factorization breaking in event-by-event ideal hydrodynamic calculations that is at least as large as in data, and argue that this phenomenon opens a new window on the study of initial fluctuations.'
author:
- 'Fernando G. Gardim'
- Frédérique Grassi
- Matthew Luzum
- 'Jean-Yves Ollitrault'
title: 'Breaking of factorization of two-particle correlations in hydrodynamics'
---
Introduction
============
In relativistic heavy-ion collision experiments a large second Fourier harmonic is observed in two-particle correlations as a function of relative azimuthal angle [@Ackermann:2000tr; @Adler:2003kt; @Back:2004mh; @Aamodt:2010pa]. This has long been considered a sign of significant collective behavior [@Ollitrault:1992bk], or “elliptic flow”, indicating the existence of a strongly-interacting, low-viscosity fluid [@Romatschke:2007mq]. However, only recently has it been realized that *all* such correlations observed between particles separated by a large relative pseudorapidity could be explained by this collective behavior [@Mishra:2007tw; @Sorensen:2008dm; @Takahashi:2009na; @Hama:2009vu; @Andrade:2010xy; @Sorensen:2010zq; @Alver:2010gr; @Staig:2010pn; @Luzum:2010sp], at least for the bulk of the system.
One significant piece of evidence for this view was the recent observation of the factorization [@Aamodt:2011by; @Alver:2010rt; @Chatrchyan:2012wg; @ATLAS:2012at] of two-particle correlations into a product of a function of properties of only one of the particles times a function of the properties of the second. Specifically, for pairs of particles in various bins of transverse momentum $p_T$, factorization of each Fourier harmonic was tested as [@Aamodt:2011by]: $$\label{test}
V_{n\Delta} (p_T^a, p_T^b) \equiv \left\langle \cos n(\phi^a - \phi^b) \right\rangle \overset{?}{=} v_n (p_T^a) \times v_n (p_T^b) ,$$ where the brackets indicate an average over pairs of particles ($a$ and $b$) coming from the same event as well as an average over a set of collision events, and $\phi^a(\phi^b)$ is the azimuthal angle of particle $a(b)$. The left-hand side is a (symmetric) function of two variables, $p_T^a$ and $p_T^b$, and in general may not factorize into a product of a function $v_n$ of each variable individually. The fact that this factorization holds at least approximately, then, is a non-trivial observation about the structure of the correlation.
While most known sources of non-flow correlations do not factorize at low $p_T$ [@Kikola:2011tu], a type of factorization comes naturally in a pure hydrodynamic picture where particles are emitted independently. They thus have no intrinsic correlations with other particles, carrying only information about their orientation with respect to the system as a whole. This causes the two-particle probability distribution in a single collision event to factorize [@Dinh:1999mn] into a product of one-particle distributions, $$\label{hydro}
\frac {dN_{pairs}}{d^3p^a d^3p^b} \overset{\rm{(flow)}}{=} \frac {dN}{d^3p^a}\times \frac {dN}{d^3p^b}.$$
Inspired by this fact, it has often been stated [@ATLAS:2012at; @Adare:2011hd; @collaboration:2011hfa] that the factorization test in Eq. should work perfectly in hydrodynamics. The observed approximate factorization was hailed as a success for the flow interpretation of correlations, while small deviations from the factorized form was interpreted as a gradual breakdown of the hydrodynamic description with increasing transverse momentum, and of increasing contribution from other sources of correlations.
In this work, we show that factorization as in Eq. is not necessarily present even in an ideal hydrodynamic system governed by Eq. , because of event-by-event fluctuations [@Miller:2003kd; @Alver:2006wh; @Alver:2010gr]. These stem from quantum fluctuations: the collision takes place over a very short timescale, and takes a snapshot of the wavefunction of incoming nuclei. In the presence of fluctuations, we show that the correlation matrix satisfies general inequalities, which are saturated by Eq. . We test these inequalities on ALICE data and point out where breaking of factorization occurs. We then illustrate with a full event-by-event hydrodynamic calculation that the same deviation seen in experiment is also present in ideal hydrodynamics.
Hydrodynamics and two-particle correlations
===========================================
We begin by recalling the discussion originally found in Ref. [@Luzum:2011mm]. In a pure hydrodynamic picture, particles are emitted independently from the fluid at the end of the system evolution according to some underlying one-particle probability distribution. One can write any such distribution as a Fourier series in the azimuthal angle $\phi$ of the particles $$\label{complexfourier}
\frac {2\pi} {N} \frac {dN}{d\phi} = \sum_{n=-\infty}^{\infty} V_n(p_T,\eta) e^{-i n \phi},$$ where $V_n = \{e^{in\phi}\}$ is the $n$th complex Fourier flow coefficient, and curly brackets indicate an average over the probability density in a single event. Writing $V_n=v_ne^{in\Psi_n}$, where $v_n$ is the (real) anisotropic flow coefficient and $\Psi_n$ the corresponding phase, and using $V_{-n}=V_n^*$ (where $V_n^*$ is the complex conjugate of $V_n$), this can be rewritten as $$\label{realfourier}
\frac {2\pi} {N} \frac {dN}{d\phi} = 1 + 2\sum_{n=1}^\infty
v_n(p_T,\eta) \cos n\left(\phi - \Psi_n(p_T,\eta)\right).$$ Note that, for this form to describe an arbitrary distribution, both $v_n$ and $\Psi_n$ may depend on transverse momentum $p_T$ and pseudorapidity $\eta$.
In this picture, the relation in Eq. holds, and a complex Fourier harmonic of the two-particle correlation factorizes in each event as: $$\begin{aligned}
\label{complex}
\left\{ e^{in(\phi^a - \phi^b)} \right\} &=&
\left\{e^{in\phi^a}\right\} \left\{ e^{-in\phi^b} \right\}\cr
&=& V_n^{a}V_n^{b*}=v_n^a v_n^b e^{in (\Psi_n^a - \Psi_n^b)}.\end{aligned}$$ This factorization only holds in a single hydro event. Both the magnitudes and the phases of anisotropic flow fluctuate event to event [@Miller:2003kd; @Alver:2006wh; @Alver:2010gr]. The experimental quantity, Eq. , is then obtained by averaging over events: $$\label{complexav}
V_{n\Delta} (p_T^a, p_T^b) =\left\langle V_n^{a}V_n^{b*}\right\rangle=
\left\langle v_n^a v_n^b e^{in (\Psi_n^a -
\Psi_n^b)}\right\rangle$$ Due to parity symmetry, only the real part remains after this average, hence the cosine in Eq. .
From this relation alone, one can make the following general statements about the event-averaged correlation matrix: the diagonal elements must be positive, and the off-diagonal elements must satisfy a Cauchy-Schwarz inequality, $$\begin{aligned}
V_{n\Delta}(p_T^a,p_T^a) &\geq 0 ,\\
\label{CS}
V_{n\Delta}(p_T^a,p_T^b)^2 &\leq V_{n\Delta}(p_T^a,p_T^a)V_{n\Delta}(p_T^b,p_T^b) .\end{aligned}$$ Factorization, Eq. , implies that the second inequality is saturated, i.e., equality is achieved. Thus, while flow does not necessarily imply factorization, any violation of these inequalities is an unambiguous indication of the presence of non-flow correlations.
An inspection of published data from the ALICE Collaboration [@Aamodt:2011by] shows that these inequalities are indeed violated in certain regimes [@Ollitrault:2012cm]. For $n=3$, diagonal elements $V_{3\Delta}(p_T^a,p_T^a)$ are negative above 5 GeV for 0–10% centrality, and above 4 GeV for 40–50% centrality. This is a clear indication that there are nonflow correlations at high $p_T$. For instance, the correlation between back-to-back jets typically yields a relative angle $\Delta\phi\sim\pi$, thus producing a negative $V_{3\Delta}$ at high $p_T$. For $n=1$, diagonal elements are negative not only at high $p_T$ (with a slightly higher threshold than for $n=3$), but also for $p_T$ between 1 and 1.5 GeV. This is believed to be caused by the correlation from global momentum conservation [@Retinskaya:2012ky; @ATLAS:2012at], but it is interesting to note that its effect can be noticed by a simple inspection of elements.
{width="\linewidth"}
In order to check the validity of the second inequality , we introduce the ratio
$$\label{correlation}
r_n\equiv\frac{V_{n\Delta}(p_T^a,p_T^b)}{\sqrt{V_{n\Delta}(p_T^a,p_T^a)V_{n\Delta}(p_T^b,p_T^b)}},$$
which is defined when diagonal elements $V_{n\Delta}(p_T^a,p_T^a)$ and $V_{n\Delta}(p_T^b,p_T^b)$ are both positive, and lies between $-1$ and $+1$ if Eq. holds. Factorization corresponds to the limit $r_n=\pm 1$. Figure \[fig:cauchy\] displays $r_2$ and $r_3$ as a function of $p_T^a$ and $p_T^b$ for Pb-Pb collisions at 2.76 TeV, 0–10% centrality. ALICE results for $r_2$ satisfy the inequalities at all $p_T$. When both particles are below $1.5$ GeV, the inequality is saturated, $r_2=1$, within errors. As soon as one of the particles is above $1.5$ GeV, however, $r_2$ is smaller than unity, and the difference with unity increases with the difference $p_T^a-p_T^b$. Results for $r_3$ are qualitatively similar below 5 GeV, with larger error bars. However, $r_3$ is closer to 1 than $r_2$ between 2 and 3 GeV. The values of $r_n$ for mid-central collisions (40-50% centrality, not shown) are comparable to the values for central collisions, although $r_2$ is slightly closer to 1.
The ALICE collaboration concluded from their analysis that factorization holds approximately for $n>1$ and $p_T$ below $4$ GeV. However, their results actually show evidence for a slight breaking of factorization for $n=2$, as soon as one of the particles has $p_T>1.5$ GeV. Even though factorization is broken, the general inequalities implied by flow are satisfied for $n=2$ and $n=3$ below 5 GeV for central collisions. It is therefore worth investigating in more detail to what extent the breaking of factorization which is seen experimentally can be understood within hydrodynamics.
First, we recall under which conditions factorization holds in hydrodynamics. It implies that the Cauchy-Schwarz inequality is saturated. By inspection of Eq. , this in turn implies that the complex flow vectors $V_n^a$ and $V_n^b$ are linearly dependent. This is true only under the following assumptions:
1. By parity symmetry, $\Psi_n^a-\Psi_n^b = 0$ in each event. I.e., $\Psi_n$ does not depend on $p_T$, which removes the exponential from the right-hand side of Eq. .
2. $v_n(p_T)$ changes from event to event by only a global factor, with no $p_T$-dependent fluctuations. $v_n(p_T)$ in the right-hand side of Eq. then represents the rms value over events.
In general, fluctuations ensure that these conditions are not met exactly, and the factorization of Eq. will not be perfect. Within hydrodynamics, the ratio $r_n$ in Eq. has a simple interpretation. Inserting Eq. into Eq. , one obtains
$$\label{rhydro}
r_n=\frac{\langle V_n^{a*}V_n^{b}\rangle}{\sqrt{\langle |V_n^a|^2\rangle
\langle |V_n^b|^2\rangle}},$$
The ratio $r_n$ thus represents the linear correlation between the complex flow vectors at momenta $p_T^a$ and $p_T^b$. Since in each event, $V_n^a$ is a smooth function of $p_T^a$, one expects that the correlation is stronger when $p_T^a\simeq p_T^b$, and decreases as the difference between $p_T^a$ and $p_T^b$ increases, as a result of the decoherence induced by initial fluctuations. ALICE data confirm this qualitative expectation.
Note that even in a single hydrodynamic event, factorization holds in the complex form Eq. , but is broken if one takes the real part before averaging over particle pairs, as in Eq. . The ratio $r_n$ in Eq. is then $\cos n(\Psi_n^a-\Psi_n^b)$, which is smaller than unity as soon as the flow angle $\Psi_n$ depends on $p_T$.
The question then becomes: how large are factorization-breaking effects in hydrodynamics, and do they have the same properties as seen in data? If purely hydrodynamic calculations give the same result as experiment, then the observed breaking of factorization may not indicate the presence of non-flow correlations.
Ideal hydrodynamic calculations
===============================
To illustrate these concepts we perform calculations using the NeXSpheRIO model [@Hama:2004rr]. This model solves the equations of relativistic ideal hydrodynamics with fluctuating initial conditions given by the NeXuS event generator [@Drescher:2000ha]. It has proven succesful in reproducing RHIC results, in particular the structure of two-particle angular correlations in Au-Au collisions at the top RHIC energy [@Takahashi:2009na]. It has recently been shown to reproduce the whole set of measured anisotropic flow data [@Gardim:2012yp; @Gardim:2011qn; @DerradideSouza:2011rp]. Our calculations are therefore performed for Au-Au collisions at the top RHIC energy, not for Pb-Pb collisions at LHC energy, as would be appropriate for a direct quantitative comparison with ALICE data. Our results are merely meant as a proof of concept, and as a prediction for measurements at RHIC. Note that the main source of fluctuations (namely, the finite number of nucleons within the nucleus) is identical in both cases.
We run $30000$ NeXuS events, which are then sorted into $10\%$ centrality bins defined by the number of participant nucleons, and then evolved hydrodynamically. Anisotropic flow is calculated accurately in every event [@Gardim:2011xv]. The ratio $r_n$ is displayed in Fig. \[fig:cauchy\] for $n=2$ and $n=3$. Deviations from the factorization limit $r=1$ are already seen at low momentum but become larger as the difference between $p_T^a$ and $p_T^b$ increases, as expected from the general arguments above. Surprisingly, the breaking of factorization appears [ *larger*]{} in hydrodynamics than in experiment.
{width="\linewidth"}
The ALICE collaboration has studied factorization by performing a global fit of the measured correlation $V_{n\Delta}(p_T^a,p_T^b)$ by the right-hand side of Eq. , where $v_n(p_T)$ is a fit parameter [@Aamodt:2011by]. The ratio of the measured correlation to the best fit differs from unity if factorization is broken. We can apply the same procedure to our hydrodynamic results. The result is shown in Fig. \[fig:globalfit\]. Again, hydrodynamic calculations and experimental data show similar trends, with the noticeable difference that the breaking of factorization is significantly [*stronger*]{} in ideal hydrodynamics than in data.
Several effects can explain this discrepancy. First, the average $p_T$ is significantly larger at LHC than at RHIC [@Floris:2011ru], so that it might be more natural to compare, e.g., 4 GeV at RHIC to 5 GeV at LHC, rather than doing the comparison at the same $p_T$. The second effect is viscosity, which is neglected in our calculation. Shear viscosity, in particular, tends to damp the effect of initial fluctuations [@Schenke:2010rr]. It is therefore natural that it will also decrease the breaking of factorization induced by initial fluctuations. A similar observation is that the linear correlation between the initial eccentricity and the final anisotropic flow is stronger in viscous hydrodynamics [@Niemi:2012aj] than in ideal hydrodynamics [@Gardim:2011xv].
Conclusions
===========
We have demonstrated that the detailed structure of two-particle angular correlations contains much more information than traditional analyses of anisotropic flow, where the correlation is averaged over one of the particles [@Luzum:2012da]. Even though such two-dimensional analyses are much more demanding in terms of statistics than traditional analyses, they bring new, independent insight into the underlying physics of flow fluctuations.
In particular, we have shown that quantum fluctuations in the wavefunction of incoming nuclei result in a decoherence in the angular correlations produced by collective flow, which becomes increasingly important as the difference between particle momenta increases. Due to this effect, factorization of angular correlations is broken even if collective flow is the only source of correlations. Our numerical calculations show that factorization breaking can be as strong in hydrodynamics as in experimental data, thereby suggesting that all correlations below $p_T\sim 5$ GeV (for central Pb-Pb collisions at LHC near midrapidity) may actually be dominated by flow. The sensitivity of this decoherence phenomenon to viscosity has not yet been investigated, but we anticipate that factorization should be restored as viscosity increases, thus potentially offering a new means of constraining the viscosity from data. On the other hand, thermal fluctuations should be considered along with viscosity [@Kapusta:2011gt], and may also contribute to factorization breaking.
Decoherence also provides a natural explanation for the important observation that event-by-event fluctuations reduce elliptic flow at high $p_T$ [@Andrade:2008xh], thus improving agreement between hydrodynamics and experimental data. Indeed, $v_2$ at high $p_T$ is inferred from azimuthal correlations between a high $p_T$ particle and all other particles — mostly low $p_T$ particles, and these azimuthal correlations are reduced due to the decoherence phenomenon. Note that the other main explanation for the reduction of $v_2$ at high $p_T$, viscosity, typically relies on the assumption of a quadratic momentum dependence of the viscous correction to the distribution function at freeze-out $\delta f$, which may not be correct [@Dusling:2009df].
In this paper, we have focused on the transverse momentum dependence of the correlations. The rapidity dependence of the correlation is also worth investigating. In particular, it was recently observed that azimuthal correlations decrease as a function of the relative pseudorapidity [@Pandit:2012rm], at variance with common lore that correlations due to flow are essentially independent of rapidity. While standard models of initial conditions do predict a mild rapidity dependence of azimuthal correlations [@Bozek:2010vz; @Dusling:2009ni], longitudinal fluctuations [@Pang:2012he] could also produce a decoherence effect similar to the one studied here. The detailed structure of two-particle correlations as a function of both particle momenta thus opens a new window on the study of flow fluctuations.
We thank Yogiro Hama for useful discussion. This work is funded by FAPESP under projects 09/50180-0 and 09/16860-3, by the FAPESP/CNRS grant 2011/51854-0, and by CNPq under project 301141/2010-0. ML is supported by the European Research Council under the Advanced Investigator Grant ERC-AD-267258.
[99]{}
K. H. Ackermann [*et al.*]{} \[STAR Collaboration\], Phys. Rev. Lett. [**86**]{}, 402 (2001) \[nucl-ex/0009011\]. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**91**]{}, 182301 (2003) \[nucl-ex/0305013\]. B. B. Back [*et al.*]{} \[PHOBOS Collaboration\], Phys. Rev. C [**72**]{}, 051901 (2005) \[nucl-ex/0407012\]. KAamodt [*et al.*]{} \[ALICE Collaboration\], Phys. Rev. Lett. [**105**]{}, 252302 (2010) \[arXiv:1011.3914 \[nucl-ex\]\]. J. -Y. Ollitrault, Phys. Rev. D [**46**]{}, 229 (1992). P. Romatschke and U. Romatschke, Phys. Rev. Lett. [**99**]{}, 172301 (2007) \[arXiv:0706.1522 \[nucl-th\]\]. A. P. Mishra, R. K. Mohapatra, P. S. Saumia and A. M. Srivastava, Phys. Rev. C [**77**]{}, 064902 (2008) \[arXiv:0711.1323 \[hep-ph\]\]. P. Sorensen, arXiv:0808.0503 \[nucl-ex\]. J. Takahashi, B. M. Tavares, W. L. Qian, R. Andrade, F. Grassi, Y. Hama, T. Kodama and N. Xu, Phys. Rev. Lett. [**103**]{}, 242301 (2009) \[arXiv:0902.4870 \[nucl-th\]\]. Y. Hama, R. P. G. Andrade, F. Grassi and W. -L. Qian, Nonlin. Phenom. Complex Syst. [**12**]{}, 466 (2009) \[arXiv:0911.0811 \[hep-ph\]\]. R. P. G. Andrade, F. Grassi, Y. Hama and W. -L. Qian, Phys. Lett. B [**712**]{}, 226 (2012) \[arXiv:1008.4612 \[nucl-th\]\]. P. Sorensen, J. Phys. G [**37**]{}, 094011 (2010) \[arXiv:1002.4878 \[nucl-ex\]\]. B. Alver and G. Roland, Phys. Rev. C [**81**]{}, 054905 (2010) \[Erratum-ibid. C [**82**]{}, 039903 (2010)\] \[arXiv:1003.0194 \[nucl-th\]\]. P. Staig and E. Shuryak, Phys. Rev. C [**84**]{}, 034908 (2011) \[arXiv:1008.3139 \[nucl-th\]\]; Phys. Rev. C [**84**]{}, 044912 (2011) \[arXiv:1105.0676 \[nucl-th\]\]. M. Luzum, Phys. Lett. B [**696**]{}, 499 (2011) \[arXiv:1011.5773 \[nucl-th\]\]. K. Aamodt [*et al.*]{} \[ALICE Collaboration\], Phys. Lett. B [**708**]{}, 249 (2012) \[arXiv:1109.2501 \[nucl-ex\]\]. B. Alver [*et al.*]{} \[PHOBOS Collaboration\], Phys. Rev. C [**81**]{}, 034915 (2010) \[arXiv:1002.0534 \[nucl-ex\]\]. S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Eur. Phys. J. C [**72**]{}, 2012 (2012) \[arXiv:1201.3158 \[nucl-ex\]\]. G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Rev. C [**86**]{}, 014907 (2012) \[arXiv:1203.3087 \[hep-ex\]\]. D. Kikola, L. Yi, S. I. Esumi, F. Wang and W. Xie, Phys. Rev. C [**86**]{}, 014901 (2012) \[arXiv:1110.4809 \[nucl-ex\]\]. P. M. Dinh, N. Borghini and J. -Y. Ollitrault, Phys. Lett. B [**477**]{}, 51 (2000) \[nucl-th/9912013\]. A. Adare, J. Phys. G [**38**]{}, 124091 (2011) \[arXiv:1107.0285 \[nucl-ex\]\]. J. Jia, J. Phys. G [**38**]{}, 124012 (2011) \[arXiv:1107.1468 \[nucl-ex\]\]. M. Miller and R. Snellings, nucl-ex/0312008. B. Alver [*et al.*]{} \[PHOBOS Collaboration\], Phys. Rev. Lett. [**98**]{}, 242302 (2007) M. Luzum, J. Phys. G [**38**]{}, 124026 (2011) \[arXiv:1107.0592 \[nucl-th\]\]. J. -Y. Ollitrault and F. G. Gardim, arXiv:1210.8345 \[nucl-th\]. E. Retinskaya, M. Luzum and J. -Y. Ollitrault, Phys. Rev. Lett. [**108**]{}, 252302 (2012) \[arXiv:1203.0931 \[nucl-th\]\]. Y. Hama, T. Kodama and O. Socolowski, Jr., Braz. J. Phys. [**35**]{}, 24 (2005) \[hep-ph/0407264\].
H. J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog and K. Werner, Phys. Rept. [**350**]{}, 93 (2001) F. G. Gardim, F. Grassi, M. Luzum and J. -Y. Ollitrault, Phys. Rev. Lett. [**109**]{}, 202302 (2012) \[arXiv:1203.2882 \[nucl-th\]\]. F. G. Gardim, F. Grassi, Y. Hama, M. Luzum and J. -Y. Ollitrault, Phys. Rev. C [**83**]{}, 064901 (2011) \[arXiv:1103.4605 \[nucl-th\]\]. R. D. de Souza, J. Takahashi, T. Kodama and P. Sorensen, Phys. Rev. C [**85**]{}, 054909 (2012) \[arXiv:1110.5698 \[hep-ph\]\]. F. G. Gardim, F. Grassi, M. Luzum and J. -Y. Ollitrault, Phys. Rev. C [**85**]{}, 024908 (2012) \[arXiv:1111.6538 \[nucl-th\]\]. M. Floris, J. Phys. G [**38**]{}, 124025 (2011) \[arXiv:1108.3257 \[hep-ex\]\]. B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. [**106**]{}, 042301 (2011) \[arXiv:1009.3244 \[hep-ph\]\]. H. Niemi, G. S. Denicol, H. Holopainen and P. Huovinen, arXiv:1212.1008 \[nucl-th\]. M. Luzum and J. -Y. Ollitrault, arXiv:1209.2323 \[nucl-ex\]. J. I. Kapusta, B. Muller and M. Stephanov, Phys. Rev. C [**85**]{}, 054906 (2012) \[arXiv:1112.6405 \[nucl-th\]\]. R. P. G. Andrade, F. Grassi, Y. Hama, T. Kodama and W. L. Qian, Phys. Rev. Lett. [**101**]{}, 112301 (2008) \[arXiv:0805.0018 \[hep-ph\]\]. K. Dusling, G. D. Moore and D. Teaney, Phys. Rev. C [**81**]{}, 034907 (2010) \[arXiv:0909.0754 \[nucl-th\]\]. Y. Pandit \[STAR Collaboration\], arXiv:1209.0244 \[nucl-ex\]. P. Bozek, W. Broniowski and J. Moreira, Phys. Rev. C [**83**]{}, 034911 (2011) \[arXiv:1011.3354 \[nucl-th\]\]. K. Dusling, F. Gelis, T. Lappi and R. Venugopalan, Nucl. Phys. A [**836**]{}, 159 (2010) \[arXiv:0911.2720 \[hep-ph\]\]. L. Pang, Q. Wang and X. -N. Wang, Phys. Rev. C [**86**]{}, 024911 (2012) \[arXiv:1205.5019 \[nucl-th\]\].
|
---
abstract: 'Quantum coherence and quantum correlations are of fundamental and practical significance for the development of quantum mechanics. They are also cornerstones of quantum computation and quantum communication theory. Searching physically meaningful and mathematically rigorous quantifiers of them are long-standing concerns of the community of quantum information science, and various faithful measures have been introduced so far. We review in this paper the measures of discordlike quantum correlations for bipartite and multipartite systems, the measures of quantum coherence for any single quantum system, and their relationship in different settings. Our aim is to provide a full review about the resource theory of quantum coherence, including its application in many-body systems, and the discordlike quantum correlations which were defined based on the various distance measures of states. We discuss the interrelations between quantum coherence and quantum correlations established in an operational way, and the fundamental characteristics of quantum coherence such as their complementarity under different basis sets, their duality with path information of an interference experiment, their distillation and dilution under different operations, and some new viewpoints of the superiority of the quantum algorithms from the perspective of quantum coherence. Additionally, we review properties of geometric quantum correlations and quantum coherence under noisy quantum channels. Finally, the main progresses for the study of quantum correlations and quantum coherence in the relativistic settings are reviewed. All these results provide an overview for the conceptual implications and basic connections of quantum coherence, quantum correlations, and their potential applications in various related subjects of physics.'
author:
- 'Ming-Liang Hu'
- Xueyuan Hu
- Jieci Wang
- Yi Peng
- 'Yu-Ran Zhang'
- Heng Fan
nocite: '[@*]'
title: Quantum coherence and geometric quantum discord
---
Introduction {#sec:1}
============
Quantum correlations and quantum coherence are two fundamental concepts in quantum theory [@Nielsen]. While quantum correlations characterize the quantum features of a bipartite or multipartite system, quantum coherence was defined for the integral system. Despite this obvious difference and as the different embodiments of the unique characteristics of a quantum system, they are also intimately related to each other which can be scrutinized from different perspectives. Moreover, from a practical point of view, quantum correlations and quantum coherence are also invaluable physical resources for quantum information and computation tasks [@Nielsen], hence remain research focuses since the early days of quantum mechanics.
The quantum correlation in a system can be characterized and quantified from different perspectives. Historically, there are various categories of quantum correlation measures being proposed, prominent examples include the widely-studied Bell-type nonlocal correlation [@rev-Bell] and quantum entanglement [@RMPE]. At the beginning of this century, a new framework for quantifying quantum correlations was formulated by @qd01, as well as by @qd02 in the study of quantum discord (QD). Within this framework, an abundance of discordlike quantum correlation measures were proposed and studied from different aspects in the past years [@RMP; @Adesso-jpa].
Quantum coherence, another embodiment of the superposition principle of quantum states, is essential for many novel and intriguing characteristics of quantum systems [@Ficek]. It is also of equal importance as quantum correlations in the studies of both bipartite and multipartite systems [@RMP]. Constructing a mathematically rigorous and physically meaningful framework for its characterization and quantification was a main pursue of researchers in quantum community, as this is not only essential for quantum foundations, but can also provides the basis for its potential applications in a wide variety of promising subjects, such as quantum thermodynamics [@ther1; @ther2; @ther3; @ther4], reference frames [@frame], and quantum biology [@biolo].
Coherence is a main concern of quantum optics historically, and it was usually studied from the perspective of phase space distributions or multipoint correlations. Instead of reviewing these approaches, we focus on the recent developments about quantitative characterization of coherence [@coher], the essence of which is the adoption of the viewpoint of treating quantum coherence as a physical resource, just like the resource theory of entanglement. A big advantage of this approach is that the related resource theory of entanglement can be adapted for introducing similar measures of quantum coherence [@RMPE].
The geometric characterization of quantum correlations in a bipartite system have been studied extensively in the past five years, with the corresponding measures being defined based on various (pseudo) distance measures of two states. The central idea for this kind of definition is that the distance between the considered state and the state without the desired property is a measure of that property [@reqd]. The similar idea can also be used to characterize quantum coherence of a system, and along this line, many quantum coherence measures have been proposed. We will review these coherence measures and show their interrelations with quantum correlations. Moreover, we will also discuss the other major progresses achieved in studying fundamental problems of geometric quantum correlations and quantum coherence, their dynamics under noisy quantum channels, and their applications in the study of many-body systems.
During the preparation of this review, we became aware of another nice review work by @Adesso. Though seemingly discussing a similar topic, our concerns are different. @Adesso reviewed the resource theory of coherence, while our concerns are quantum coherence, quantum correlations, and their intrinsic connections. In our work, besides presenting a comprehensive view of the main developments of quantum coherence and quantum correlations, we try to summarize and reformulate some calculations scattering in a large number of literature. In particular, we have also reviewed in detail the main progresses for the study of quantum correlations and quantum coherence in the relativistic settings. We hope that this review may be useful for both beginners and seniors in quantum information science.
This review is organized as follows. In Section \[sec:2\], we first briefly recall the framework for defining QD and the seminal measure of QD defined via the discrepancy between quantum mutual information (QMI) and the classical aspect of correlations. Then, we present in detail the recently introduced geometric quantum discords (GQDs) and measurement-induced nonlocality (MIN) defined by virtue of various distance measures of quantum states, e.g., the Hilbert-Schmidt (HS) norm, the trace norm, the Bures distance, the Hellinger distance, the relative entropy, and the Wigner-Yanase (WY) skew information. We will also present a review concerning the fundamental connections among these correlation measures and the recent developments about their potential role in typical tasks of quantum information processing.
In Section \[sec:3\], we first review the basic aspects related to the resource theory of quantum coherence including structure of the corresponding free states and free operations. We then present a detailed review on those recent developments of the quantification of quantum coherence. They can be classified roughly into the following categories: the distance-based measure of coherence, the entanglement-based measure of coherence, the convex roof measure of coherence, the robustness of coherence, the Tsallis relative entropy measure of coherence, and the skew-information-based measure of coherence. For every category, there are also different definitions of coherence measures. Moreover, some of their extensions valid for infinite-dimensional systems and some other coherence measures defined within slightly different frameworks, for example, the different construction of free operations, will also be reviewed in this section.
In Section \[sec:4\], we review recent developments on interpretations of the aforementioned quantum coherence measures. First, we show from an operational perspective that quantum coherence is intimately related to quantum correlations in bipartite and multipartite systems, highlighting the fundamental position of coherence in quantum theory. Then, we review the complementarity relations of quantum coherence under both mutually unbiased bases (MUBs) and incompatible bases, as well as the complementarity relation between coherence and mixedness and between coherence and path distinguishability. Finally, the distillation and dilution of quantum coherence with different forms of restricted operations and the average coherence of randomly sampled states are discussed.
In Section \[sec:5\], we consider the role of quantum coherence in some quantum information processing tasks, including quantum state merging, Deutsch-Jozsa algorithm, Grover search algorithm, and deterministic quantum computation with one qubit (DQC1). The role of coherence in the quantum metrology tasks such as phase discrimination and subchannel discrimination is also reviewed. Of course, as quantum coherence underlies different forms of quantum correlations which are essential for quantum information, we focus here only on those of the closely related topics.
In Section \[sec:6\], we discuss dynamics of quantum coherence and QD, mainly concentrating on their singular behaviors in open quantum systems. In this section, we first review frozen phenomena of quantum coherence and QD which are preferable for information processing tasks. Then, we discuss potential ways for protecting and enhancing quantum coherence and QD. Two closely related problems, i.e., the resource creating and breaking powers of quantum channel and the factorization relation for the evolution equation of QD and quantum coherence are described in detail.
In Section \[sec:7\], we consider quantum coherence in explicit physical systems. We employ the various spin-chain model to show that quantum coherence can be used to study the long-range order, valence-bond-solid states, localized and thermalized states, and quantum phase transitions of the many-body systems. This shows that the resource theory of quantum coherence is not only of fundamental but is also of practical significance.
In Section \[sec:8\], we present a review on recent progresses of quantum coherence and QD in relativistic settings, including their behaviors for the free field modes, for curved spacetime and expanding universe, and for noninertial cavity modes. We provide a summary of the Unruh temperature, Hawking temperature, expansion rate of the universe, accelerated motion of cavities and detectors, and boundary conditions of the field on quantum correlations and quantum coherence. Quantum correlations for particle detectors and the dynamical Casimir effects on these correlations are also provided here.
Finally, in Section \[sec:9\], we present a concluding remark on the main results of this review. We hope the review be helpful for further exploration in these fields. Several open questions are also raised for possible future research.
Geometric quantum correlation measures {#sec:2}
======================================
The widely used discordlike quantum correlation measures proposed in the past ten years can be categorized roughly into two different families, namely, those based on the entropy theory, and those based on various distance measures of quantum states. A detailed overview of the first category has already been given by @RMP, and there are no new measures being proposed along this line in recent years, so we will recall only the original definition of QD and several of the related entropic measures for self consistency of this review. Our main concern will be the second category of discordlike quantum correlation measures. Most of them are proposed after the year 2012, and have been proven to be well defined. The related notions and approaches used in their definitions have also been proven useful for introducing coherence measures. In particular, this allows us to put the discordlike correlations and quantum coherence on an equal footing, which facilitates one’s investigation of the interrelation between these two different quantifiers of quantumness. Moreover, as the definitions of GQD and quantum coherence have something in common, the well developed methods for calculating GQD are also enlightening for deriving analytical expressions of the related coherence measures.
To begin with, we recall the concept of QD that was framed from the viewpoints of information theory. Within the seminal framework formulated by @qd01 as well as @qd02, it was defined based on the partition of the total correlation in state $\rho_{AB}$ into two different parts, that is, the classical part and the quantum part. The QMI $I(\rho_{AB})$ was used as a measure of total correlation, and it reads $$\label{new1-1}
I(\rho_{AB})=S(\rho_A)+S(\rho_B)-S(\rho_{AB}),$$ with $S(\rho_X)=-{\mathrm{tr}}(\rho_X\log_2 \rho_X)$ ($X=A$, $B$, or $AB$) being the von Neumann entropy.
Similar to most correlation measures whose quantification implies measurement, the classical correlation was also defined from a measurement perspective. @qd01 proposed four defining conditions for a classical measure $J(\rho_{AB})$: (+1) $J(\rho_{AB}) = 0$ for any $\rho_{AB}=\rho_A\otimes \rho_B$, (+2) it should be locally unitary invariant, (+3) It is nonincreasing under local operations, and (+4) $J(\rho_{AB})=S(\rho_A)=S(\rho_B)$ for pure states. Based on these conditions, they defined the classical correlation as the maximum information about one party (say $B$) of a bipartite system that can be extracted by performing the positive operator valued measure (POVM) on the other party (say $A$). If the POVM $\{E_k^A\}$ with elements $E_k^A=M_k^{A\dagger} M_k^A$ is performed on party $A$, one can obtain the postmeasurement state of the system and the conditional state of party $B$ as $$\label{new1-2}
\rho'_{AB}=\sum_k M_k^A\rho_{AB}M_k^{A\dagger},~~
\rho_{B|E_k^A}=\frac{\mathrm{tr}_{A}(E_k^A\rho_{AB})}{p_k},$$ where $p_k=\mathrm{tr}(E_k^A\rho_{AB})$ is the probability for obtaining the outcome $k$. The classical correlation is given by $$\label{new1-3}
J_A(\rho_{AB})=S(\rho_B)-\min_{\{E_k^A\}}S(B|\{E_k^A\}),$$ where $$\label{new1-4}
S(B|\{E_k^A\})=\sum_{k}p_{k}S(\rho_{B|E_k^A}),$$ is the averaged conditional entropy of the postmeasurement state $\rho'_{AB}$.
The QD is then defined by the discrepancy between $I(\rho_{AB})$ and $J_A(\rho_{AB})$ as $$\label{new1-5}
D_A(\rho_{AB})=I(\rho_{AB})-J_A(\rho_{AB})=\min_{\{E_k^A\}}S(B|\{E_k^A\})-S(B|A),$$ where $S(B|A)=S(\rho_{AB})-S(\rho_A)$ is the conditional entropy. Of course, one can also define $D_B(\rho_{AB})$ by performing the measurements on party $B$. In general, $D_A(\rho_{AB}) \neq
D_B(\rho_{AB})$, that is, the QD is an asymmetric quantity.
The QD defined above is based on POVM, but the corresponding maximization is generally a notoriously challenging task. So sometimes one can also consider the set of rank-one projectors $\{\Pi_k^A\}$, for which the postmeasurement state turns out to $$\label{new1-6}
\rho'_{AB}=\sum_k p_k \Pi_k^A \otimes \rho_{B|\Pi_k^A}.$$ This yields $I(\rho'_{AB})= S(\rho_B)-\sum_k p_k
S(\rho_{B|\Pi_k^A})$, and the QD becomes $$\label{new1-7}
D_A(\rho_{AB})=I(\rho_{AB})-\max_{\{\Pi_k^A\}}I(\rho'_{AB}).$$ Hence the intuitive meaning of QD can be interpreted as the minimal loss of correlations due to the local projective measurements $\{\Pi_k^A\}$. Indeed, it suffices to consider only the projective measurements $\{\Pi_k^A\}$ for two-qubit states in most cases [@QD-ana1; @QD-ana2]. Moreover, the optimal measurement strategy (over four-element POVM) for obtaining classical correlation $J_A(\rho_{AB})$ in many-body system has been studied by @Amico.
Similarly, a symmetric version of QD was also proposed. It reads [@qdsym1; @qdsym2] $$\label{new1-8}
D_s(\rho_{AB})=I(\rho_{AB})-\max_{\{\Pi_k^A\otimes \Pi_l^B\}}I(\rho''_{AB}),$$ where $\rho''_{AB}=\sum_k p_{kl} \Pi_k^A \otimes \otimes \Pi_l^B$.
Apart from the above QD measure, @thermalqd presented a slightly different discordlike correlation measure which was called thermal QD, and is defined as $$\label{new1-9}
\tilde{D}_A(\rho_{AB})=\min_{\{\Pi_k^A\}}[S(\rho'_A)+S(B|\{\Pi_k^A\})]-S(\rho_{AB}),$$ where $\rho'_A$ is the reduced state of $\rho'_{AB}$ in Eq. , and $S(\rho'_A)=H(\{p_k\})$, with $H(\{p_k\})$ being the Shannon entropy function and the probability $p_k={\mathrm{tr}}(\Pi_k^A
\rho_{AB}\Pi_k^A)$.
Apart from the above entropic measure and the other related entropic measures summarized in detail by @RMP, QD can also be measured from a geometric aspect. The motivation for this approach is very similar to the geometric measure of entanglement first introduced by @GME1 and further extended by @GME2. For pure state $|\psi\rangle$, they proposed to adopt the minimal squared distance between $|\psi\rangle$ and the set of separable pure states $|\phi\rangle$ to characterize its entanglement, that is, by minimizing $\min_{|\phi\rangle}\||\psi\rangle-
|\phi\rangle\|^2$. Based on this, one can derive the following geometric entanglement measure $$\label{new1-10}
E_g(\psi)= \min_{|\phi\rangle} (1-|\langle\psi|\phi\rangle|^2)
= 1-\max_{|\phi\rangle} |\langle\psi|\phi\rangle|^2,$$ where $\psi=|\psi\rangle\langle\psi|$. If $\psi$ is a bipartite state, $E_g(\psi)=1- \lambda_{\max}^{1/2}$ [@GME1], where $\lambda_{\max}^{1/2}$ is the maximal Schmidt coefficient corresponding to the Schmidt decomposition of $|\psi\rangle$ of the following form $$\label{schmidt}
|\psi\rangle=\sum_i \sqrt{\lambda_i}
|\varphi_i^A\rangle\otimes|\varphi_i^B\rangle.$$ For a mixed state described by density operator $\rho$, the geometric entanglement measure can be defined in terms of the convex roof construction $$\label{eq2b-5}
E_g(\rho)= \min_{\{p_i,\psi_i \}}\sum_i p_i E_g(\psi_i),$$ where $\psi_i=|\psi_i\rangle\langle\psi_i|$, and the minimization is with respect to the possible decompositions of $$\label{decomp}
\rho= \sum_i p_i \psi_i.$$ Though the calculation of $E_g(\rho)$ for general mixed states is a daunting task, for any two-qubit state $\rho$, it can be evaluated analytically as $$\label{GME-qubit}
E_g(\rho)= \frac{1-\sqrt{1-C^2(\rho)}}{2},$$ where $C(\rho)$ is the concurrence of $\rho$ [@concur2]. We refer to the work of @GME3 for a comparison of different geometric entanglement measures.
In the following, we review in detail the geometric measure of discordlike correlations. The motivation for such kind of measures may be fourfold. First, the definition of geometric correlations are based on the idea that *a distance from a given state to the closest state without the desired property is a measure of that property* [@reqd], thereby one can quantify amount of correlations by the distance of the considered state to the set of states without the desired property. This endows the resulting geometric measure a clear geometric interpretation. Second, the geometric measures are preferable due to their analytically computable for a wide regime of states. In particular, the theory for geometric entanglement measure is historically well developed, while the features of various distance measures of states are also intensively investigated. The corresponding results can be borrowed for studying geometric discordlike correlations. Thirdly, it is hard to generalize the concept of the entropic discord to multipartite scenario as it is based on QMI which is not defined for multipartite systems. But the geometric approach enables one to define discordlike correlations which are completely applicable for multipartite states. Finally, the geometric discordlike correlations have also been shown to be related to some quantum information processing tasks, thereby endows them with an actual meaning.
Once a distance measure of quantum states is chosen, the corresponding GQD measure will be determined by the set of classical states. In general, the definition of classical states is not unique and different types are studied in different contexts, see, e.g., the work of @QD-ana2 and references therein. We refer the following two slightly different types of them which are within the theory of discordlike correlations: partial classical states and total classical states. In the case where there are only two subsystems, they are usually called one-sided (classical-quantum or quantum-classical) and two-sided (classical-classical) classical states. The set contains mixtures of locally distinguishable states and include the set of product states as its subset. They are defined to be classical as the total correlation (measured by QMI) contained in them is the same as the classical correlation [@qd01; @qd02]. In fact, for any partial (total) classical state, there exist local (tensor product of local) POVM such that the postmeasurement state is the same as the premeasurement one. Contrary, if there dose not exist such a POVM, the considered state is said to be discordant.
Apart from classical states, we will also consider the sets of locally invariant states and locally invariant projective measurements which are utilized in defining MIN. Here, by saying a projective measurement to be locally invariant we mean that it does not disturb the reduced state (say $\rho_A$) of a bipartite system $AB$. It constitutes a subset of the full set of local projective measurements. Moreover, the set of locally invariant states are also different from that of the above-mentioned classical (i.e., zero-discord) states, as some of the classical states may have nonvanishing MIN [@min].
In the following discussion of discordlike correlations other than that measured by relative entropy, we focus our attention mainly on bipartite states. But most of them can be generalized directly to multipartite scenario due to the definite structure of total classical states [@reqd].
Geometric quantum discord {#sec:2A}
-------------------------
The starting point for the definition of GQD is the identification of the set $\mathcal{CQ}$ of classical-quantum (i.e., zero-discord with respect to subsystem $A$) states. For a bipartite state in the Hilbert space $\mathcal {H}_{AB}$, the classical-quantum states can be written as $$\label{eq1-1}
\chi=\sum_i p_k \Pi_k^A\otimes \rho_k^B,$$ which is a convex combination of the tensor products of the orthogonal projector $\Pi_k^A$ in $\mathcal{H}_A$ and an arbitrary density operator $\rho_k^B$ in $\mathcal{H}_B$, with $\{p_k\}$ being any probability distribution. Intuitively, $\chi$ of Eq. is said to be classical-quantum as there exists at least one measurement on subsystem $A$ for which $B$ is not affected. or in other words, by measuring $A$ one extracts no information about $B$ as the entropy $S(\rho_B)$ and the residual entropy $\sum_k p_kS(\rho_k^B)$ for the conditional ensemble $\{p_k,\rho_k^B\}$ after an optimal local POVM is performed on $A$ are the same. Indeed, within the framework of @qd01 and @qd02, one can also check directly that the classical correlation contained in $\chi$ is zero.
With $\mathcal{CQ}$ in hand, the category of GQDs for a state $\rho$ can be characterized by its closest (pseudo) distance to the zero-discord state in set $\mathcal{CQ}$ (see Fig. \[fig:hu1\]). More specifically, it can be formalized in the general form $$\label{eq1-2}
D_\mathcal{D}(\rho)= \min_{\chi\in\mathcal{CQ}}\mathcal{D}(\rho,\chi),$$ where $\mathcal{D}(\rho,\chi)$ is a suitable distance measure of quantum states which should satisfy certain natural restrictions in order for the GQD to be well defined, for example, it should be nonnegative, and should be nonincreasing under the action of completely positive and trace preserving (CPTP) map. In certain specific situations, some equivalent forms of $\mathcal{D}(\rho,
\chi)$ may be used as well. As the distance between two quantum states can be measured from different aspects, the GQDs can be defined accordingly, provided that they satisfy the conditions for a faithful measure of quantum correlation [@qd01]. Moreover, while the GQD defined in Eq. can increase under local operations on party $A$ by its definition, it should not be increased by local operations on the unmeasured party $B$ [@Piani].
Likewise, one could write directly the set $\mathcal{QC}$ of quantum-classical states and define the GQD with respect to subsystem $B$, or the set $\mathcal{CC}$ of classical-classical states and define the GQD with respect to total system $AB$. The classical-classical states can be written as $\chi'=\sum_i p_{kl}
\Pi_k^A\otimes \Pi_l^B$, and there exists at least one local measurements for which it is not affected. In what follows we consider the GQD defined with respect to $A$. Its definition with respect to $B$ or $AB$ is similarly.
This definition of GQD is somewhat different from the initially proposed entropic measure of QD [@qd01]. But it should also satisfy the similar necessary conditions in order for it to be a bona fide measure of quantum correlation, e.g., it is non-negative, vanishes only for zero-discord states, keeps invariant under local unitary transformations, and is nonincreasing under local operations.
### Hilbert-Schmidt norm of discord
By using the HS norm as a measure of the distance between two states, @GQDdefined the GQD of $\rho$ as $$\label{gqd-1}
D_G(\rho)=\min_{\chi\in \mathcal{CQ}}\|\rho-\chi\|_2^2,$$ with $\|X\|_2$ denoting the HS norm which is defined as $\|X\|_2=\sqrt{{\mathrm{tr}}(X^\dag X)}$.
@bound1 further proved that the above definition of GQD is completely equivalent to $$\label{gqd-v2}
D_G(\rho)=\min_{\Pi^A}\|\rho-\Pi^A(\rho)\|_2^2,$$ where $\Pi^A=\{\Pi_k^A\}$ is the local von Neumann measurements on party $A$ which sum to the identity (i.e., $\sum_k \Pi_k^A=
{\openone}_A$), and $$\label{poststate}
\Pi^A(\rho)= \sum_k (\Pi_k^A\otimes{\openone}_B)\rho (\Pi_k^A\otimes{\openone}_B).$$ As the set $\{\Pi^A(\rho)\}$ of postmeasurement states is generally a subset of the full set $\mathcal{CQ}$ of classical states, the equivalence between the above two definitions implies that one only need to take the minimization over $\{\Pi^A(\rho)\}$, and this greatly simplifies the estimation of $D_G(\rho)$. Moreover, Eq. also reveals that $D_G(\rho)$ basically measures how much a measurement on party $A$ does disturbs other parts of the state.
@qdiscord put forward another discord measure which was termed as $q$-discord. It reads $$\label{eq-qdiscord}
D_q(\rho)=\min_{\Pi^A} S_q(\Pi^A[\rho])- S_q(\rho),$$ where $S_q(\rho)$ is the Tsallis $q$-entropy defined as [@qentropy] $$\label{eq-qentropy}
S_q(\rho)=\frac{1-{\mathrm{tr}}\rho^q}{q-1}.$$ It reduces to $-{\mathrm{tr}}(\rho\ln\rho)$ when $q\rightarrow 1$. Moreover, one can obtain immediately from Eq. that $D_2(\rho)={\mathrm{tr}}\rho^2-{\mathrm{tr}}(\Pi^A[\rho])^2$, thus $D_G(\rho)$ can also be retrieved from the $q$-discord by setting $q = 2$.
This GQD measure is favored for its ease of computation. In particular, by noting that any two-qubit state $\rho$ can be represented as $$\label{gqd-ad1}
\rho=\frac{1}{4} \Bigg({\openone}_4 +\vec{x}\cdot\vec{\sigma}\otimes{\openone}_2
+{\openone}_2\otimes\vec{y}\cdot\vec{\sigma}
+\sum_{i,j=1}^3r_{ij}\sigma_i\otimes\sigma_j\Bigg),$$ @GQD derived the explicit formula for $D_G(\rho)$, which is given by $$\label{gqd-2}
D_G(\rho)=\frac{1}{4}\left(\|x\|_2^2+\|R\|_2^2-k_{\max} \right),$$ where $\|\vec{x}\|_2^2=\sum_{i=1}^3 x_i^2$, $\|R\|_2^2={\mathrm{tr}}(R^T R)$, and $k_\mathrm{max}$ is the largest eigenvalue of the matrix $K=\vec{x}\vec{x}^T+RR^T$, where $R=(r_{ij})$ is a $3\times 3$ real matrix, and the superscript $T$ denotes transpose of vectors or matrices.
Using the same method, the GQD for any qubit-qutrit state $\rho$ was obtained as [@gqd-ana] $$\label{gqd-3}
D_G(\rho)=\frac{1}{6}\|\vec{x}\|_2^2+\frac{1}{4}\|R\|_2^2-k_{\max},$$ where $\|\vec{x}\|_2^2$, $\|R\|_2^2$, and $k_\mathrm{max}$ are similar to those for the two-qubit case, with however $x_i={\mathrm{tr}}\rho(\sigma_i\otimes {\openone}_3)$, $R$ becomes a $3\times 8$ matrix with the elements $r_{ij}={\mathrm{tr}}\rho(\sigma_i\otimes \lambda_j)$, $\lambda_j$ ($j=1,2,\ldots,8$) are the Gell-Mann matrices, and $K=\vec{x}\vec{x}^T/6+RR^T/4$.
Moreover, for the $(d\times d)$-dimensional Werner state $\rho_W$ and isotropic state $\rho_I$ of the following form $$\label{mina-4}
\begin{split}
&\rho_W=\frac{d-x}{d^3-d}{\openone}_{d^2}+\frac{dx-1}{d^3-d}\sum_{ij}|ij\rangle\langle ji|, ~~x\in[-1,1],\\
&\rho_I=\frac{1-x}{d^2-1}{\openone}_{d^2}+\frac{d^2 x-1}{d^3-d}\sum_{ij} |ii\rangle\langle jj|, ~~x\in[0,1],
\end{split}$$ the HS norm of GQD can be obtained analytically as [@bound1] $$\label{mina-v2}
\begin{split}
&D_G(\rho_W)=\frac{(dx-1)^2}{d(d-1)(d+1)^2},\\
&D_G(\rho_I)=\frac{(d^2 x-1)^2}{d(d-1)(d+1)^2}.
\end{split}$$ For any $(m\times n)$-dimensional bipartite state, it can always be decomposed as $$\begin{aligned}
\label{mina-1}
\rho=\sum_{ij}r_{ij}X_i\otimes Y_j,\end{aligned}$$ where $\{X_i\!:i=0,1, \ldots,m^2-1\}$ ($X_0={\openone}_m/ \sqrt{m}$) is the orthonormal operator basis for subsystem $A$ that satisfy ${\rm
tr} (X_i^\dag X_{i'})=\delta_{ii'}$ (likewise for $Y_j$), the HS norm of GQD is showed to be lower bounded by [@bound1] $$\label{mina-v3}
D_G(\rho)\geq {\mathrm{tr}}(CC^T) -\sum_{j=1}^m \lambda_j=\sum_{j=m+1}^{m^2} \lambda_j,$$ with $\lambda_j$ representing eigenvalues of the matrix $CC^T$ arranged in nonincreasing order (counting multiplicity), and $C=(r_{ij})$ is a $m^2\times n^2$ matrix.
@bound2 also obtained a different tight lower bound of $D_G(\rho)$, which is given by $$\label{mina-v4}
D_G(\rho)\geq \|x\|^2+ \|R\|^2-\sum_{j=1}^{m-1}\eta_j,$$ where $\eta_j$ are eigenvalues of the matrix $m^2 n(xx^T+RR^T)/2$ arranged in nonincreasing order (counting multiplicity). Here, we have denoted by $x=(r_{10},r_{20},\cdots,r_{m^2-1,0})^T$, and $R=(r_{kl})$ with $k=1,2,\cdots, m^2-1$, and $l=1,2,\cdots, n^2-1$. Moreover, note that our decomposed form of $\rho$ in Eq. is slightly different from that given by @bound2, thus induces the seemingly different but essentially the same expressions of the lower bound of $D_G(\rho)$.
Different from the GQD of Eq. , @Guo2015ijtp proposed another measure of quantumness by using the average distance between the reduced state $\rho_B= {\mathrm{tr}}_A \rho$ and the $i$th output reduced state of subsystem $B$ after the local von Neumann measurements were performed on $A$. Let $\mathcal{H}_A
\otimes \mathcal{H}_B$ with $\dim \mathcal{H}_A=m$ and $\dim
\mathcal{H}_B =n\geq m$ be the state space of a bipartite system. The measure is then defined by $$\begin{aligned}
\label{avhs}
D_G^{av}(\rho)= \sup_{\Pi^A} \sum_k p_k \|\rho_B-\rho^B_k\|_2^2,\end{aligned}$$ where the supremum is taken over the full set of local von Neumann measurements $\Pi^A=\{\Pi_k^A\}$, and $\rho_k^B={\mathrm{tr}}_A(\Pi_k^A\otimes
{\openone}_B)\rho(\Pi_k^A\otimes {\openone}_B)$. It was showed that only the product states do not contain this kind of quantumness, that is, $D_G^{av}(\rho)=0$ only for $\rho=\rho_A\otimes\rho_B$. So it captures quantumness of a state which is different from that captured by the QD defined within the framework of @qd02.
While the GQD given in Eq. is analytical computable for any two-qubit state, it is noncontractive, i.e., its value may be changed even by local reversible operations on the unmeasured party $B$, so it was thought to be not well defined [@Piani]. But it does play a role in some quantum information tasks, see Sec. \[sec:2C\]. Due to this reason, it is desirable to find ways of characterizing and quantifying GQD using other distance measures of states.
### Trace norm of discord
In stead of using the HS norm, @TDD considered the possibility of using the general Schatten $p$-norm to measure quantum correlations. The Schatten $p$-norm for a matrix $M$ is defined as $$\label{Schatten}
\|M\|_p=\{{\mathrm{tr}}[M^\dag M]^{p/2}\}^{1/p},$$ which reduces to the HS norm if $p=2$, and the trace norm if $p=1$. By using multiplicative property of the Schatten $p$-norm under tensor products, @TDD showed that the corresponding GQD is well defined only for $p=1$. Based on this fact, they introduced the trace norm of discord as $$\label{eq1a-1}
D_T(\rho)=\min_{\chi\in \mathcal{CQ}}\|\rho-\chi\|_1,$$ and for $2\times n$ dimensional state $\rho$ (i.e., $A$ is a qubit), the optimal $\chi$ can also be obtained from the subset $\Pi^A(\rho)$ [@TDD1], with $\Pi^A=\{\Pi_k^A\}$ being the set of local projective measurements, i.e., $$\label{eq-tdd}
D_T(\rho)=\min_{\Pi^A} \|\rho-\Pi^A(\rho)\|_1.$$ The calculation of $D_T(\rho)$ is a hard task, and there is no analytical solution for it in general cases. For the two-qubit Bell-diagonal states $$\label{eq-bell}
\rho^{\rm Bell}=\frac{1}{4}\left({\openone}_4+\sum_{i=1}^3 c_i
\sigma_i\otimes\sigma_i\right),$$ it can be derived as $$\label{eq1a-bell2}
D_T(\rho^{\rm Bell})= {\rm int} \{|c_1|,|c_2|, |c_3|\},$$ with ${\rm int}\{\cdot\}$ denoting the intermediate value. The closest $\chi_\rho$ is still a Bell-diagonal state with the only nonzero parameter $c_k$ corresponding to $|c_k|=\max\{|c_1|,|c_2|,|c_3|\}$.
Moreover, for two-qubit *X* state $\rho^X$ which contains nonzero elements only along the main diagonal and anti-diagonal in the computational basis $\{|00\rangle,|01\rangle,
|10\rangle,|11\rangle\}$, the trace norm of discord is given by [@TDD-ana] $$\begin{aligned}
\label{eq1a-2}
D_{T}(\rho^X)=\sqrt{\frac{\xi_1^2\xi_{\rm max}-\xi_2^2\xi_{\rm min}}
{\xi_{\rm max}-\xi_{\rm min}+\xi_1^2-\xi_2^2}},\end{aligned}$$ where $$\begin{aligned}
\label{eq1a-v2}
\begin{aligned}
& \xi_{1,2}=2(|\rho_{23}|\pm |\rho_{14}|),~
\xi_3=1-2(\rho_{22}+\rho_{33}),\\
& \xi_{\rm max}=\max\{\xi_3^2,\xi_2^2+x_{A3}^2\},~
x_{A3}=2(\rho_{11}+\rho_{22})-1.
\end{aligned}\end{aligned}$$ For higher-dimensional states, @TDD-ana1 considered a simplified version of $D_T(\rho)$ defined also by Eq. , and obtained its analytical solution for certain very special kinds of qutrit-qutrit states, e.g., the maximally entangled states and the Werner states.
The trace norm of discord could also be connected to quantum correlations such as entanglement witness. We refer to the work of @witness for a comprehensive review of entanglement witnesses. In general, an entanglement witness $W$ is an Hermitian operator for which ${\mathrm{tr}}(W\rho)$ takes negative value for at least one entangled state and non-negative values for all separable states. By minimizing over the compact subset $\mathcal{M}$ of the set of entanglement witnesses $\mathcal {W}$, one can obtain the optimal entanglement witness, and define the quantifier $$\begin{aligned}
\label{eq1a-3}
E_w(\rho)=\max\{0,-\min_{W\in\mathcal {M}}{\mathrm{tr}}(W\rho)\},\end{aligned}$$ as an entanglement measure [@OEW].
@TDD-en proved that $D_T(\rho)$ is lower bounded by $$\begin{aligned}
\label{eq1a-v3}
D_T(\rho)\geq \max\{0, -\min_{\{W\in\mathcal{W}| -{\openone}\leq W\leq{\openone}\}}{\mathrm{tr}}(W\rho)\}.\end{aligned}$$ As $E_w(\rho)$ is in fact the negativity $\mathcal{N}(\rho)$ [@negat] for $\mathcal{M}= \{W^{T_A}\in\mathcal{W}| 0\leq
W^{T_A}\leq {\openone}\}$ [@OEW], and the robustness of entanglement $R_r(\rho)/d$ for $\mathcal{M}= \{W\in\mathcal{W}|{\mathrm{tr}}W=1\}$ [@roe; @TDD-en1], both of which are obviously equal to or smaller than the optimal entanglement witness showed on the right-hand side of Eq. , we also have $$\begin{aligned}
\label{eq1a-vv4}
D_T(\rho)\geq \mathcal {N}(\rho),~~ D_T(\rho)\geq R_r(\rho)/d.\end{aligned}$$
While Eq. gives a proper quantum correlation measure, @togd further defined the corresponding geometric classical and total correlations using the trace norm. By fixing $\chi\in
\Pi^A(\rho)$ and denoting $\tilde{\Pi}^A$ the corresponding optimal measurement operator for obtaining $D_T(\rho)$ (the minimization over different $\Pi^A(\rho)$ is equivalent to the minimization over $\mathcal{CQ}$ for qubit states), they defined the geometric classical correlation $C_T(\rho)$ and total correlation $T_T(\rho)$ as (see Fig. \[fig:hu2\]) $$\begin{aligned}
\label{eq1a-4}
\begin{aligned}
& C_T(\rho)=\|\tilde{\Pi}^A (\rho)-\tilde{\Pi}^A(\pi_{\rm rd})\|_1, \\
& T_T(\rho)=\|\rho-\pi_{\rm rd}\|_1,
\end{aligned}\end{aligned}$$ with $\pi_{\rm rd}=\rho_A\otimes\rho_B$ being product of the reduced density matrices of $\rho$.
For the Bell-diagonal states $\rho^{\rm Bell}$ of Eq. , @togd further obtained $$\begin{aligned}
\label{eq1a-v4}
\begin{aligned}
& C_T(\rho^{\rm Bell})=c_+, \\
& T_T(\rho^{\rm Bell})=\frac{1}{2}[c_{+} +\max\{c_+, c_0+c_-\}],
\end{aligned}\end{aligned}$$ with $c_+$, $c_-$, and $c_0$ being the maximum, minimum, and intermediate values of $\{|c_1|,|c_2|, |c_3|\}$, respectively. This yields the superadditivity relation: $T_T\leq C_T+D_T$.
In fact, $\tilde{\Pi}^A(\pi_{\rm rd})$ in Eq. may be not the closest state to $\tilde{\Pi}^A (\rho)$, and $\pi_{\rm rd}$ composed of the reduced density matrices may also be not the closest product state to $\rho$. This stimulates more general definitions of geometric classical correlation and total correlation. Without loss of generality, one can denote by $\chi_\rho$ for the state with closest trace distance to $\rho$ \[note that $\tilde{\Pi}^A (\rho)$ is optimal only for $A$ being a qubit\], and $\mathcal {P}$ the set of local product states of the subsystems. Based on these, @traceqd2 defined (see Fig. \[fig:hu2\]) $$\begin{aligned}
\label{eq1a-5}
\begin{aligned}
& \tilde{C}_T(\rho)=\min_{\pi\in \mathcal{P}}\|\chi_\rho-\pi\|_1, \\
& \tilde{T}_T(\rho)=\min_{\pi\in \mathcal{P}}\|\rho-\pi\|_1,
\end{aligned}\end{aligned}$$ and derived analytically $$\label{eq1a-v5}
\tilde{C}_T(\rho^{\rm Bell})=\sqrt{1+c_+}-1,$$ where the closest state $\pi_{\chi_\rho}$ to $\chi_\rho$ is given by $\pi_{\chi_\rho}=\tilde{\rho}_A\otimes\tilde{\rho}_B$ with $\tilde{\rho}_A=({\openone}_2+a_k\sigma_k)/2$ and $\tilde{\rho}_B=
({\openone}_2+b_k\sigma_k)/2$. The index $k$ corresponds to the maximum of $|c_k|=c_+$, and $a_k=b_k |c_k|/c_k$. Clearly, $\pi_{\chi_\rho}$ is not the product of its marginals, and is not even a Bell-diagonal state in general.
For the family of classical-quantum or quantum-classical bipartite states, @noncommutativity introduced another quantum correlation measure based on the non-commutativity of quantum observables, where the trace norm of the commutators of the ensemble state of one subsystem is used. To be explicit, for classical-quantum state $\chi$ of Eq. described by the ensemble $\{X_i\}$ with $X_i=p_i\rho_i$, they found the quantity $$\begin{aligned}
\label{eq-nc}
D(\chi)=\sum_{i>j} \|[X_i,X_j]\|_1,\end{aligned}$$ satisfy the following properties of correlations: $D(\chi)\geq 0$, and the equality holds when subsystem $B$ is also classical. Moreover, it is local unitary invariant, and is nonincreasing when an ancillary system is introduced.
The above result was further extended by @Guo2016srep, who proposed to define GQD for any $\rho$ in a similar manner. Let $\{|i^A\rangle\}$ be an orthonormal basis of $\mathcal{H}_A$. Then any state $\rho$ acting on $\mathcal{H}_A \otimes \mathcal{H}_B$ can be represented by $$\begin{aligned}
\label{eq-guo}
\rho=\sum_{i,j}|i^A\rangle\langle j^A| \otimes B_{ij}.\end{aligned}$$ with $B_{ij}$ being operators in $\mathcal{H}_B$. The non-commutativity measure of GQD for $\rho$ is defined by $$\begin{aligned}
\label{eq-guo2}
D_N(\rho)\coloneqq \frac{1}{2}\sum_{(ij)\neq(kl)}\|[B_{ij},B_{kl}]\|_1,\end{aligned}$$ under the trace norm, and $$\begin{aligned}
\label{eq-guo3}
D'_N(\rho)\coloneqq \frac{1}{2}\sum\limits_{(ij)\neq(kl)}\|[B_{ij},B_{kl}]\|_2,\end{aligned}$$ under the HS norm, where the commutator $[X,Y]=XY-YX$, and the summation is over all different pairs of $\{B_{ij}\}$.
These two measures of GQD can be calculated easily for $\rho$ of arbitrary dimension. In particular, for pure state $\psi=
|\psi\rangle\langle\psi|$ with Schmidt decomposition of Eq. , analytical solutions of them are given by $$\begin{aligned}
\label{eq-guo4}
\begin{aligned}
& D_N(\psi)=2\sum_{i,j}\lambda_i\lambda_j\left(\sum_{(k,l)\in \Omega} \lambda_k\lambda_l\right), \\
& D'_N(\psi)=2\sum_{i,j}\lambda_i\lambda_j\left(\sum_{(k,l)\in \Omega'} \lambda_k\lambda_l\right)+\sqrt{2},
\end{aligned}\end{aligned}$$ where $\Omega=\{(k,l)\}$ with $i<k\leq j\leq l$ or $k=i$, $l=j$ if $i<j$, and $i\leq k<l$ if $i=j$, while $\Omega'=\{(k,l)\}$ with $i<k\leq j\leq l$ if $i<j$, and $i\leq k<l$ if $i=j$.
It was showed via several examples that they can reflect the amount of the original QD. In particular, these two measures disappear if and only if the corresponding state is zero discordant. Here, we would like to further point out that this is in fact a direct consequence of the result of @chenlin, in which a necessary and sufficient condition for vanishing QD has been proven. It says that $\rho$ has zero QD if and only if all the operators $\rho_{B|ij}$ commute with each other for any orthonormal basis $\{|i^A\rangle\}$ in $\mathcal {H}_A$, where $$\label{chenlin}
\rho_{B|ij}\coloneqq \langle i^A|\rho|j^A\rangle.$$ It is obvious that $B_{ij}$ in Eq. is the same as $\rho_{B|ij}$ of the above equation.
### Bures distance of discord
The distance between two states $\rho$ and $\sigma$ can also be quantified by the Bures distance $$\begin{aligned}
\label{Bures}
\mathcal{D}_{B}(\rho,\sigma)= 2[1- \sqrt{F(\rho,\sigma)}],\end{aligned}$$ where $$\begin{aligned}
\label{fidelity}
F(\rho,\sigma)=\big[{\mathrm{tr}}(\sqrt{\rho} \sigma\sqrt{\rho})^{1/2}\big]^2,\end{aligned}$$ is the Uhlmann fidelity [@Nielsen]. The Bures distance satisfy the preferable properties of joint convexity, i.e., $$\label{fidelity2}
\begin{aligned}
\mathcal{D}_B(p_1\rho_1+p_2\rho_2,p_1\sigma_1+p_2\sigma_2)
\leq & p_1 \mathcal{D}_B(\rho_1,\sigma_1) \\
&+ p_2 \mathcal{D}_B(\rho_2,\sigma_2),
\end{aligned}$$ and it is also monotonous under CPTP maps. It has been used to quantify entanglement [@bures-e1; @bures-e2], and there are some equivalent definitions of Bures distance discord. @BDD proposed to define it as $$\begin{aligned}
\label{eq1b-1}
D_B (\rho)=(2+\sqrt{2})\big[1-\sqrt{F_{\rm max}(\rho)}\big],\end{aligned}$$ where $F_{\rm max}(\rho)=\max_{\chi\in\mathcal{CQ}}F(\rho,\chi)$ represents the maximum of the Uhlmann fidelity, and the constant $2+
\sqrt{2}$ is introduced for the normalization of it for two-qubit maximally discordant states. Moreover, the square root of $D_B(\rho)$ in Eq. equals to that defined by @bures2.
There are several cases that the evaluation of $F_{\rm max}(\rho)$, and thus $D_B (\rho)$ can be simplified:
\(1) For arbitrary pure state $|\psi\rangle$, the maximum Uhlmann fidelity can be obtained as $F_{\rm max} (|\psi\rangle
\langle\psi|)=\mu_{\rm max}$, with $\mu_{\rm max}$ being the largest Schmidt coefficient of $|\psi\rangle$ [@BDD].
\(2) For any Bell-diagonal state $\rho^{\rm Bell}$ of Eq. , we have [@bures2; @bures3] $$\begin{aligned}
\label{eq1b-2}
F_{\rm max}(\rho^{\rm Bell})&=&\frac{1}{2}+\frac{1}{4}\max_{\langle ijk\rangle}
\bigg[\sqrt{(1+c_i)^2-(c_j-c_k)^2} \nonumber\\
&&+\sqrt{(1-c_i)^2-(c_j+c_k)^2}\bigg],\end{aligned}$$ where the maximum is taken over all the cyclic permutations of $\{1,2,3\}$.
\(3) For general $(2\times n)$-dimensional state, although there is no analytical solution, the maximum of the Uhlmann fidelity can be calculated as [@bures3] $$\begin{aligned}
\label{eq1b-3}
F_{\rm max}(\rho)=\frac{1}{2}\max_{||\vec{u}=1||}
\left(1-{\mathrm{tr}}\Lambda(\vec{u})
+2\sum_{k=1}^{n_B}\lambda_k(\vec{u})\right),\end{aligned}$$ where $\lambda_k(\vec{u})$ are eigenvalues of $$\begin{aligned}
\label{eq1b-4}
\Lambda(\vec{u})= \sqrt{\rho}(\sigma_{\vec{u}}\otimes {\openone}_B)\sqrt{\rho}\end{aligned}$$ arranged in nonincreasing order, and $\sigma_{\vec{u}}=\vec{u}\cdot
\vec{\sigma}$, with $\vec{u}=(\sin\theta\cos\phi,\sin\theta\sin\phi,
\cos\theta)$ being a unit vector in $\mathbb{R}^3$, and $n_B$ the dimension of $\mathcal {H}_B$.
### Relative entropy of discord
The relative entropy of a state $\rho$ to another state $\sigma$ is defined as $$\begin{aligned}
\label{new2-1}
S(\rho\|\sigma)={\mathrm{tr}}(\rho\log_2\rho)- {\mathrm{tr}}(\rho\log_2\sigma),\end{aligned}$$ which is non-negative, and can sometimes be infinite. Though technically the relative entropy does not has a geometric interpretation as $S(\rho\|\sigma)\neq S(\sigma\|\rho)$ in general, it can be recognized as a (pseudo) distance measure of quantum states.
The relative entropy has been used to define quantum entanglement, $$\begin{aligned}
\label{new2-2}
E_R=\min_{\sigma\in \mathcal{S}} S(\rho\|\sigma),\end{aligned}$$ which is indeed the minimal relative entropy of $\rho$ to the set $\mathcal{S}$ of separable states [@re-en1; @bures-e1]. In the same spirit, one can use it to define the discordlike correlation measures. @reqd made the first attempt in this direction by introducing the relative entropy of discord $D_R$ and the relative entropy of dissonance $Q_R$, They are defined, respectively, to be the minimal relative entropy of $\rho$ and $\sigma$ (the closest separable state to $\rho$) to the set of classical states $\mathcal{C}$ (here, by saying a state to be classical, we mean that it is classical with respect to all of its subsystems, which is similar to the set of $\mathcal{CC}$ states for bipartite systems), and can be written explicitly as $$\begin{aligned}
\label{new2-3}
D_R =\min_{\chi\in \mathcal{C}} S(\rho\|\chi), ~~
Q_R =\min_{\chi\in \mathcal{C}} S(\sigma\|\chi),\end{aligned}$$ which are applicable for the general bipartite and multipartite states. In particular, $Q_R(\sigma)$ reveals a kind of quantum correlation excluding quantum entanglement.
@reqd also showed that $D_R$ and $Q_R$ are equivalent to $$\begin{aligned}
\label{new2-4}
D_R=S(\chi_\rho)-S(\rho), ~~
Q_R=S(\chi_\sigma)-S(\rho),\end{aligned}$$ where $S(\chi_\rho)=\min_{|\vec{k}\rangle}S(\sum_{\vec{k}}
|\vec{k}\rangle\langle \vec{k}|\rho|\vec{k}\rangle\langle \vec{k}|)$ with $\{|\vec{k}\rangle\}$ forming the eigenbasis of $\chi_\rho$, and likewise for $S(\chi_\sigma)$. So the optimization in Eq. is reduced to the optimization of the von Neumann entropy $S(\chi_\rho)$ and $S(\chi_\sigma)$.
In a similar manner to Eqs. and , @reqd defined the total correlation and classical correlation as $$\begin{aligned}
\label{new2-5}
\begin{aligned}
& T_\rho=S(\rho\|\pi_\rho)=S(\pi_\rho)-S(\rho), \\
& T_\sigma=S(\sigma\|\pi_\sigma)=S(\pi_\sigma)-S(\sigma), \\
& C_\rho=S(\chi_\rho\|\pi_{\chi_\rho})=S(\pi_{\chi_\rho})-S(\chi_\rho), \\
& C_\sigma=S(\chi_\sigma\|\pi_{\chi_\sigma})=S(\pi_{\chi_\sigma})-S(\chi_\sigma),
\end{aligned}\end{aligned}$$ where $\pi_\rho=\pi_1\otimes \ldots\otimes\pi_N$ ($\pi_k$ is the reduced density operator of the $k$th subsystem of $\rho$), and likewise for $\pi_\sigma$, $\pi_{\chi_\rho}$, and $\pi_{\chi_\sigma}$. From these definitions, one can obtain the following additivity relations $$\begin{aligned}
\label{new2-6}
T_\rho+L_\rho=D_R+C_\rho,~~
T_\sigma+L_\sigma=Q_R+C_\sigma,\end{aligned}$$ where $L_\rho= S(\pi_{\chi_\rho})-S(\pi_\rho)$ and $L_\sigma=
S(\pi_{\chi_\sigma})-S(\pi_\sigma)$.
For two-qubit Bell-diagonal states of Eq. , if we rewrite it as $\rho^{\mathrm{Bell}}=\sum_i \lambda_i
|\Psi_i\rangle\langle\Psi_i|$, where $\lambda_i$ are arranged in nonincreasing order and $|\Psi_i\rangle$ ($i=1,2,3,4$) are the four Bell states, then the closest separable state to it is given by [@bures-e1] $$\begin{aligned}
\label{new2-7}
\sigma= \sum_{i=1}^4 p_i |\Psi_i\rangle\langle\Psi_i,\end{aligned}$$ where $p_1=1/2$ and $p_i=\lambda_i/[2(1-\lambda_1)]$ for $i\neq 1$. Similarly, the closest classical state to $\rho^{\mathrm{Bell}}$ is given by [@reqd] $$\begin{aligned}
\label{new2-8}
\begin{aligned}
\chi_\rho =&\frac{q_\rho}{2}[|\Psi_1\rangle\langle\Psi_1|+|\Psi_2\rangle\langle\Psi_2|] \\
&+\frac{1-q_\rho}{2}[|\Psi_3\rangle\langle\Psi_3|+|\Psi_4\rangle\langle\Psi_4|],
\end{aligned}\end{aligned}$$ with $q_\rho=\lambda_1+\lambda_2$, and the closest classical state to $\sigma$ can be obtained directly by substituting $q_\rho$ with $q_\sigma=p_1+p_2$.
Moreover, it is worthwhile to note that for any bipartite state $\rho_{AB}$, the relative entropy of discord $D_R(\rho_{AB})$ equals to the zero-way quantum deficit $$\begin{aligned}
\label{new2-9}
\Delta^{\O}(\rho_{AB}) = \min_{\Pi^A \otimes \Pi^B}
S(\rho_{AB}\|\Pi^A \otimes \Pi^B [\rho_{AB}]),\end{aligned}$$ which is also a discordlike quantum correlation measure and was defined originally from the perspective of work extraction from the quantum system coupled to a heat bath [@deficit]. The above equation thereby endows $\Delta^{\O}(\rho_{AB})$ a geometric interpretation, that is, it corresponds to the minimal relative entropy between $\rho_{AB}$ and the full set of postmeasurement states $\Pi^A \otimes \Pi^B [\rho_{AB}]$.
The one-way quantum deficit can also be expressed by using the quantum relative entropy as [@deficit] $$\begin{aligned}
\label{new2-10}
\Delta^{\rightarrow}(\rho_{AB}) = \min_{\Pi^A}S(\rho_{AB}\|\Pi^A[\rho_{AB}]),\end{aligned}$$ and it also equals to the minimal relative entropy between $\rho_{AB}$ and the set $\mathcal{CQ}$ of classical-quantum states. Furthermore, $\Delta^{\rightarrow}(\rho_{AB})$ also equivalents to the thermal QD $\tilde{D}_A(\rho_{AB})$ introduced by @thermalqd.
### Hellinger distance of discord
Although in most cases the quantum correlation measure is defined as a direct function of the density operator $\rho$ itself, its other forms may also be very useful. For example, with roots in the well-know notion of WY skew information [@WYD], the square root $\sqrt{\rho}$ has been used to study the local quantum uncertainty (LQU) of a single system [@LQU].
By using the square root form of a density operator, @HDD introduced a new quantifier of the GQD, for which we call it the Hellinger distance discord. It can be recognized as a modified version of the GQD proposed by @GQD, and reads $$\label{eq1c-1}
D_H(\rho)=2\min_{\Pi^A}\parallel\sqrt{\rho}-\Pi^A(\sqrt{\rho})\parallel_2^2,$$ where the minimum is taken over $\Pi^A= \{\Pi_k^A\}$, with $$\label{eq1c-2}
\Pi^A(\sqrt{\rho})= \sum_k (\Pi_k^A\otimes {\openone}_B)\sqrt{\rho}(\Pi_k^A\otimes {\openone}_B),$$ and ${\openone}_B$ is the identity operator in $\mathcal {H}_B$.
The Hellinger distance discord is well defined. It is locally unitary invariant, and vanishes if and only if $\rho$ is a classical-quantum state. It also keeps invariant when adding a local ancilla to the unmeasured party, i.e., $D_H(\rho_{A:BC})=
D_H(\rho_{A:B})$ for $\rho_{A:BC}=\rho_{AB} \otimes\rho_C$. This property averts the fault encountered when measuring GQD via the HS norm [@Piani]. Moreover, it is similar to the squared form of the Hellinger distance defined as $$\label{eq1c-v2}
d_H^2(\rho,\chi)=\frac{1}{2}{\mathrm{tr}}\{(\sqrt{\rho}-\sqrt{\chi})^2\},$$ and this is the reason for it to be called the Hellinger distance discord.
For pure state $\psi=|\psi\rangle\langle\psi|$ with the Schmidt decomposition of Eq. , the Hellinger distance discord can be obtained as $D_H(\psi)=1-\sum_i \lambda_i^2$, which is the same as that of the GQD based on the HS norm [@GQD]. Moreover, for the Bell-diagonal state $\rho^{\rm Bell}$, it is given by $$\begin{aligned}
\label{eq1c-3}
D_H(\rho^{\rm Bell})=1-\frac{1}{4}(h^2+\max_i\{d_i^2\}),\end{aligned}$$ where $h=\sum_i \sqrt{\lambda_i}$, $d_i=h-2\sqrt{\lambda_4}
-2\sqrt{\lambda_i}$ ($i=1,2,3$), and $\lambda_i$ are eigenvalues of $\rho^{\rm Bell}$ given by $$\begin{aligned}
\label{eq1c-v3}
\begin{aligned}
& \lambda_1=\frac{1}{4}(1-c_1+c_2+c_3),~ \lambda_2=\frac{1}{4}(1+c_1-c_2+c_3), \\
& \lambda_3=\frac{1}{4}(1+c_1+c_2-c_3),~ \lambda_4=\frac{1}{4}(1-c_1-c_2-c_3).
\end{aligned}\end{aligned}$$ For $(2\times n)$-dimensional state $\rho$ with the decomposed form of $$\begin{aligned}
\label{eq-sqrt}
\sqrt{\rho}=\sum_{ij} \gamma_{ij}X_i\otimes Y_j,\end{aligned}$$ where $\{X_i\!:i=0,1,2,3\}$ and $\{Y_j\!:j=0,1,\ldots,n^2-1\}$ constitutes the orthonormal operator bases for the Hilbert spaces $\mathcal {H}_A$ and $\mathcal {H}_B$, the Hellinger distance discord can be calculated as [@HDD] $$\begin{aligned}
\label{eq1c-4}
D_H(\rho)=2(1-||\bold{r}||_2^2-\mu_{\max}),\end{aligned}$$ where $||\bold{r}||_2^2=\sum_j \gamma_{0j}^2$, and $\mu_{\max}$ represents the largest eigenvalue of the matrix $\Gamma\Gamma^\dag$, with $\Gamma=(\gamma_{ij})_{ i=1,2,3;j=0,1,\cdots,n^2-1}$.
### Local quantum uncertainty
The WY skew information was defined as follows [@WYD] $$\begin{aligned}
\label{eq2f-1}
\mathsf{I}^p(\rho,K)= -\frac{1}{2}{\mathrm{tr}}\{[\rho^p,K][\rho^{1-p},K]\},\end{aligned}$$ with $p\in(0,1)$, and when $p=1/2$ (we omit the superscript in $\mathsf{I}^p(\rho,K)$ for brevity), $$\begin{aligned}
\label{eq2f-2}
\mathsf{I}(\rho,K)= -\frac{1}{2}{\mathrm{tr}}\{[\sqrt{\rho},K]^2\}=\frac{1}{2}\|[K,\sqrt{\rho}]\|_2^2,\end{aligned}$$ was also termed the WY skew information, where $K$ denotes the observable to be measured (a self-adjoint operator). $\mathsf{I}(\rho,K)$ measures the information content embodied in a state that is skewed to the chosen observable $K$, and is bounded above by the variance of $K$, i.e., $$\begin{aligned}
\label{eq2f-v2}
\mathsf{I}(\rho,K)\leq \langle K^2\rangle_\rho-\langle K\rangle^2_\rho,\end{aligned}$$ where the equality holds for pure states. This equation shows that $\mathsf{I}(\rho,K)$ is indeed a lower bound of the weighted statistical uncertainty about $K$ (measured by the variance of $K$) for any possible state preparation. For $\rho=\sum_i \lambda_i
|\psi_i\rangle\langle\psi_i|$, one can further obtain $$\label{eq1d-2}
\mathsf{I}(\rho,K)=\frac{1}{2}\sum_{ij}(\sqrt{\lambda_i}-\sqrt{\lambda_j})^2 K_{ij}^2,$$ where the overlap $K_{ij}=|\langle\psi_i|K|\psi_j\rangle|$.
Compared to the variance of $K$, the skew information has many advantages. In particular, it possesses preferable properties which are useful for defining quantum correlations, e.g., it is nonnegative and vanishes if and only if $[\rho,K]=0$, it is convex, that is, $\mathsf{I}(\sum_i p_i \rho_i,K)\leq \sum_i p_i
\mathsf{I}(\rho_i,K)$. Indeed, the skew information can used to characterize uncertainty relation [@luoprl], while a correlation measure based on it has also been introduced [@skew].
@LQU proposed to use the WY skew information to quantify LQU of a bipartite state $\rho$. They chose the local observable $K^\Lambda=K_A^\Lambda\otimes {\openone}_B$ ($\Lambda$ denotes spectrum of $K_A^\Lambda$ that are nondegenerate as this corresponds to maximally informative observables on $A$) and defined the LQU as $$\label{eq1d-3}
\mathcal {U}_A^\Lambda(\rho)=\min_{K^\Lambda} \mathsf{I}(\rho,K^\Lambda),$$ which is not only a measure of uncertainty, but also a well-defined quantum correlation measure. In particular, for $(2\times
n)$-dimensional state $\rho$, by dropping the superscript $\Lambda$ for brevity and choosing the nondegenerate observables as $K_A=\vec{n}\cdot\vec{\sigma}^A$, with $\vec{\sigma}=(\sigma_1,
\sigma_2,\sigma_3)$ being the vector of Pauli operators, the LQU can be derived as [@LQU] $$\begin{aligned}
\label{eq1d-4}
\mathcal{U}_A(\rho)=1-\lambda_{\max}(W_{AB}),\end{aligned}$$ where $\lambda_{\max}(W_{AB})$ denotes the maximum eigenvalue of the matrix $W_{AB}$ whose elements are given by $$\begin{aligned}
\label{eq1d-5}
(W_{AB})_{ij}={\mathrm{tr}}\{\sqrt{\rho}(\sigma_i^A\otimes {\openone}_B)
\sqrt{\rho}(\sigma_j^B\otimes {\openone}_B)\},\end{aligned}$$ from which one can obtain that for pure state $|\psi\rangle$, the LQU reduces to the linear entropy of entanglement $$\begin{aligned}
\label{eq1d-v5}
\mathcal{U}_A(|\psi\rangle\langle\psi|)= 2[1-{\mathrm{tr}}(\rho_A)^2],\end{aligned}$$ where $\rho_A={\mathrm{tr}}_B (|\psi\rangle\langle\psi|)$.
In fact, for arbitrary $(2\times n)$-dimensional state $\rho$, as $K_A=\vec{n}\cdot\vec{\sigma}^A$ is a root-of-unity local unitary operation, which implies $K_Af(\rho)K_A=f(K_A\rho K_A)$ for arbitrary function $f(\cdot)$, thus $$\label{eq1d-6}
\begin{aligned}
\mathcal{U}_A(\rho)=& 1-{\mathrm{tr}}\{\sqrt{\rho}K_A\sqrt{\rho}K_A\},\\
=& 1-{\mathrm{tr}}\{\sqrt{\rho}\sqrt{K_A\rho K_A}\},\\
=& d_H^2(\rho,K_A\rho K_A),
\end{aligned}$$ while $\{\Pi_k^A\}$ of Eq. can be written as $\Pi_{1,2}^A= ({\openone}_2\pm K_A)/2$, which gives $$\label{eq1d-7}
\begin{aligned}
\big[\sqrt{\rho}-\Pi^A(\sqrt{\rho})\big]^2
= & \frac{1}{4}(\rho+K_A\rho K_A-\sqrt{\rho}K_A\sqrt{\rho}K_A \\
& -K_A\sqrt{\rho}K_A\sqrt{\rho}),
\end{aligned}$$ then by combining this with Eqs. , , and , one can obtain $$\label{eq1d-v7}
\mathcal{U}_A(\rho)= 2D_H(\rho).$$ This equation establishes a direct connection between LQU and the Hellinger distance discord, thereby gives LQU a geometric interpretation, although it applies only for $(2\times
n)$-dimensional states.
By restricting $K_A$ to rank-one projectors, @yucslqu further defined a measure of quantum correlation for arbitrary bipartite state as follows $$\label{eq1d-8}
Q_A(\rho)= \min_{K_A}\sum_{i=1}^{m}
\mathsf{I}(\rho,K_A^i\otimes {\openone}_B),$$ where the minimization is taken over the set of $K_A=
\{|i^A\rangle\langle i^A|\}$, and $m$ is the dimension of $\mathcal{H}_A$. The measure $Q_A(\rho)$ vanishes if and only if $\rho$ is classical-quantum correlated (i.e., $\rho\in
\mathcal{CQ}$). Moreover, it is locally unitary invariant, and is contractive under CPTP map on the unmeasured party $B$.
@yucslqu also gave a numerical method for calculating $Q_A(\rho)$ which uses the technique of approximate joint diagonalization. Moreover, for any pure state $|\psi\rangle$ and $(2\times n)$-dimensional state $\rho$, analytical solutions of $Q_A(\rho)$ can be obtained, which equal half of $\mathcal{U}_A(\rho)$. For the $(d\times d)$-dimensional Werner state $\rho_W$ and isotropic state $\rho_I$ of the form of Eq. , one has $$\label{eq1d-9}
\begin{aligned}
& Q_A(\rho_W)=\frac{d-x-\sqrt{(d^2-1)(1-x^2)}}{2(d+1)},\\
& Q_A(\rho_I)=\frac{1-2\sqrt{(d^2-1)(1-x)x}+(d^2-2)x}{d(d+1))}.
\end{aligned}$$
### Negativity of quantumness
The quantumness in a bipartite or multipartite state can also be quantified by virtue of the amount of entanglement created between the considered system and the measurement apparatus in a local measurement. @entqd made such an attempt along this line. For a bipartite state $\rho_{AB}$ and a measurement apparatus $M$ prepared in an initial state $|0^M\rangle$, they proved that the created minimum distillable entanglement between $M$ and $AB$ equals to the one-way deficit $\Delta^{\rightarrow}(\rho^{AB})$ \[see Eq. \], that is, $$\label{eq1h-0}
\Delta^{\rightarrow}(\rho^{AB})= \min_{U_{MAB}}E_D^{M:AB}(U_{MAB}\rho_{MAB}U_{MAB}^\dagger),$$ where $\rho_{MAB}=|0^M\rangle\langle 0^M|\otimes \rho_{AB}$, $U_{MAB}=U_{MA}\otimes{\openone}_B$ and $U_{MA}$ denotes only those unitary operators which give $\sum_k \Pi_k^A\rho_{AB}\Pi_k^A={\mathrm{tr}}_M
(U\rho_{MAB}U^\dagger)$. Therefore, the above equation establishes a quantitative connection between discordlike quantum correlation and entanglement.
@TDD1 further proposed several discordlike measures of quantumness by using this approach. First, they introduced the measurement interaction $V_{A\mapsto AA'}$ described by a linear isometry from $A$ to a bipartite system $AA'$, i.e., $$\label{eq1h-1}
V_{A\mapsto AA'}|a_i\rangle= |a_i\rangle |i'\rangle, ~ \forall i,$$ with $\{|a_i\rangle\}$ being the basis of system $A$, and $\{|i'\rangle\}$ is the computational basis of system $A'$. Then, for any $N$-partite system described by density operator $\rho_{\bm{A}}$ (we denote $\bm{A}={A_1 A_2 \ldots A_N}$ for short) and the chosen subsystems $\Sigma\subseteq\{A_1 A_2 \ldots A_N\}$ for which the measurements are performed, the corresponding premeasurement state reads $$\label{eq1h-2}
\tilde{\rho}_{\Xi}=\Bigg(\bigotimes_{i\in\Sigma}V_{i\mapsto ii'}\Bigg)
\rho_{\bm{A}} \Bigg(\bigotimes_{i\in\Sigma}V_{i\mapsto
ii'}\Bigg)^\dagger,$$ where $\Xi=\bm{A} \cup \Sigma'$.
By using negativity as a measure of entanglement [@negat], @TDD1 defined negativity of quantumness as the minimum entanglement created between the system and the apparatus, that is, $$\label{eq1h-3}
Q_\mathcal {N}^\Sigma(\rho_{\bm{A}})\coloneqq \min_{\tilde{\rho}^{\Xi}}
\mathcal{N}_{\bm{A}:\Sigma'}(\tilde{\rho}_{\Xi}),$$ where the minimization is taken over all possible $\tilde{\rho}_{\Xi}$ obtained with different choice of basis for the system $\bm{A}$. This measure is showed to be nonnegative for any $\rho_{\bm{A}}$, and vanishes if and only if $\rho_{\bm{A}}$ is classical on the subsystems $\Sigma$ to be measured. When $\Sigma=
\{k_1,k_2,\ldots, k_n\}$ ($n<N$), $Q_\mathcal {N}^\Sigma
(\rho_{\bm{A}})$ is said to be the partial negativity of quantumness and is equivalent to [@TDD1] $$\label{eq1h-4}
Q_\mathcal {N}^\Sigma(\rho_{\bm{A}})=\min_{\bigotimes_{k\in\Sigma}\mathcal{B}_k}
\frac{1}{2} \left( \sum_{i_{k_1},i_{k_2},\ldots,i_{k_n}}
\| \rho_{i_{k_1}, i_{k_2},\ldots, i_{k_n}}\|_1 -1 \right),$$ where $\mathcal{B}_k= \{|a^{(k)}_{i_k}\rangle\}$ denotes basis of the $k$th subsystem, and $$\label{eq1h-5}
\rho_{i_{k_1}, i_{k_2},\ldots, i_{k_n}}=\langle a_{i_{k_1}}^{(k_1)}
a_{i_{k_2}}^{(k_2)}\dots a_{i_{k_n}}^{(k_n)}| \rho_{\bm{A}} |
a_{i_{k_1}}^{(k_1)} a_{i_{k_2}}^{(k_2)}\dots
a_{i_{k_n}}^{(k_n)}\rangle.$$ When $\Sigma=\bm{A}$, one obtains the total negativity of quantumness, and it is equivalent to [@TDD1] $$\label{eq1h-6}
Q_\mathcal {N}^{\bm{A}}(\rho^{\bm{A}})= \min_{\bigotimes_{k=1}^N \mathcal{B}_k}
\frac{1}{2} \Big( \| \rho^{\bm{A}}\|_{l_1} -1 \Big),$$ where the minimization is taken over different choices of factorized basis $\bigotimes_{k=1}^N \mathcal{B}_k$ and the $l_1$ norm is also calculated in the same basis.
For the case of bipartite state $\rho_{AB}$ with $\dim A=2$ (i.e., $A$ is a qubit), @TDD1 further showed that $$\label{eq1h-7}
\begin{aligned}
Q_\mathcal {N}^A(\rho_{AB})& = \frac{1}{2}\min_{\Pi^A}\| \rho_{AB}-\Pi^A(\rho_{AB})\|_1, \\
& = \frac{1}{2}\min_{\sigma\in \mathcal {C}\mathcal {Q}}\|
\rho_{AB}-\sigma\|_1,
\end{aligned}$$ where $\Pi^A=\{\Pi_i^A\}$ with $\Pi_i^A= |a_i\rangle\langle a_i|$ being the local projective measurements on $A$. It implies that the minimization over the full set of classical-quantum states can be simplified to the minimization only over the full set of postmeasurement states. Similarly, the total negativity of quantumness for the above-mentioned $\rho_{AB}$ is given by $$\label{eq1h-8}
\begin{aligned}
Q_\mathcal{N}^{AB}(\rho_{AB})& = \frac{1}{2}\min_{\Pi^A\otimes\Pi^B}\|\rho_{AB}-\Pi^A\otimes\Pi^B(\rho_{AB})\|_{l_1}, \\
& = \frac{1}{2}\min_{\sigma\in \mathcal {CC}}\| \rho_{AB}-\sigma\|_{l_1}
\end{aligned}$$ where the local projective measurements $\Pi^A\otimes \Pi^B$ are defined with respect to the factorized basis $\mathcal{B}_A\otimes
\mathcal{B}_B$, and the $l_1$ norm in the first line is also calculated with the same basis, while that in the second line is calculated with respect to the eigenbasis of $\sigma$ (if the eigenbasis are degenerate then it is chosen optimally to minimize the distance by default).
For certain special $\rho_{AB}$, analytical solutions of $Q_\mathcal
{N}^A(\rho_{AB})$ and $Q_\mathcal {N}^{AB}(\rho_{AB})$ can be obtained. For example, for the two-qubit $\rho_{AB}$ with $\rho_A={\openone}_2/2$, one has $Q_\mathcal {N}^A(\rho_{AB})=
\mathrm{int} \{s_1, s_2, s_3\}/2$, where $s_i$ is the singular value of $R=(r_{ij})$ with $r_{ij}={\mathrm{tr}}(\rho_{AB}\sigma_i \otimes
\sigma_j)$. If $\rho_{AB}$ belongs to the Bell-diagonal states, one can further obtain $Q_\mathcal {N}^{AB}(\rho_{AB})=\mathrm{int}
\{s_1, s_2, s_3\}/2$. Moreover, for Werner states and isotropic states given in Eq. , one has [@TDD1] $$\label{eq1h-9}
\begin{aligned}
& Q_\mathcal {N}^A(\rho_W)= Q_\mathcal {N}^{AB}(\rho_W) = \frac{|dx-1|}{2(d+1)}, \\
& Q_\mathcal {N}^A(\rho_I)= Q_\mathcal {N}^{AB}(\rho_I) = \frac{|d^2 x-1|}{d+1}.
\end{aligned}$$
Measurement-induced nonlocality {#sec:2B}
-------------------------------
Apart from the various Bell-type nonlocality widely studied in the literature [@rev-Bell], the nonlocality of a system can also be studied from other aspects. One typical research direction in recent years is initialized by @min, who proposed the notion of MIN. In this subsection, we will review in detail various geometric measures of them. They were all defined from the measurement perspective, and were motivated by those of the discordlike correlation measures [@RMP]. We shall focus mainly on the bipartite systems described by the density operator $\rho$ in the Hilbert space $\mathcal {H}_A\otimes \mathcal {H}_B$. But the related concepts and ideas can in fact be generalized to multipartite systems straightforwardly.
Different from the definitions of GQDs in the above section, and motivated by the consideration that the state of a bipartite system may be disturbed by a measurement on one party (say $A$) of the considered system, one can define the MIN as the maximal distance that a state $\rho$ to the set $\mathcal {L}$ of locally invariant quantum states, namely $$\label{min}
N(\rho)=\max_{\delta\in\mathcal{L}}\mathcal{D}(\rho,\delta),$$ where the locally invariant of $\delta$ means that $\delta=\sum_k
\Pi_k^A\rho\Pi_k^A$ for all $\Pi^A=\{\Pi_k^A\}$ satisfying $\sum_k
\Pi_k^A \rho_A \Pi_k^A=\rho_A$.
By adopting different distance measures, one can define different measures of MIN which possess distinct novel characteristics. Moreover, for bipartite state $\rho$ with nondegenerate reduced state $\rho_A$, the MIN measures can readily be obtained as the optimal measurements $\tilde{\Pi}^A=\{\tilde{\Pi}^A_i\}$ are induced by the spectral resolutions of $\rho_A=\sum_i p_i^A
\tilde{\Pi}_i^A$. But when $\rho_A$ is degenerate, an optimization procedure should be performed. In fact, seeking the optimal measurements in order to extract various measurement-based correlations (including MIN) is an important task for characterizing quantumness of a state [@QD-ana2; @Amico].
### Hilbert-Schmidt norm of MIN
The notion of MIN was introduced by @min. They used the HS norm as a measure of distance, and defined the MIN as $$\label{smin}
N_G(\rho)=\max_{\Pi^A} \|\rho-\Pi^A(\rho)\|_2^2,$$ with $\Pi^A$ being the locally invariant projective measurements. $N_G(\rho)$ characterizes the maximal global disturbance caused by the locally invariant measurements, in the sense that it corresponds to the maximal square HS distance between the postmeasurement state $\Pi^A(\rho)$ and the premeasurement state $\rho$. The way for revealing nonlocal feature of a state $\rho$ by doing local measurements on one of its subsystem is somewhat similar to the notion of localizable entanglement which was also defined based on local measurements on fixed subsystems of $\rho$ [@localent1; @localent2].
This MIN measure is showed to have the basic properties: (+1) $N_G(\rho)\geqslant 0$, and the inequality holds for any product state. (+2) it is locally unitary invariant, namely, $N_G(U_{AB}\rho U_{AB}^\dag)=N_G(\rho)$, $\forall$ $U_{AB}=U_A\otimes U_B$. (+3) For the case of nondegenerate reduced state $\rho_A=\sum_k \lambda_k
|k\rangle\langle k|$, the optimal $\tilde{\Pi}^A$ is given by $\tilde{\Pi}^A(\rho)=\sum_k |k\rangle\langle k|\rho |k\rangle\langle
k|$.
For $(m\times n)$-dimensional states of Eq. , $N_G(\rho)$ is upper bounded by $\sum_{i=1}^{m^2-m}\lambda_i$, where $\lambda_i$ ($i=1,2,\ldots, m^2-1$) denote the eigenvalues of $RR^T$ in nonincreasing order, $R=(r_{ij})$ with $i,j\geq1$ is a real matrix.
This MIN measure can be derived analytically for a wide range of quantum states, which include the pure states, the bipartite states $\rho_{AB}$ with $A$ being a qubit, certain higher dimensional states with symmetry, as well as certain bound entangled states [@minbd] and other special states with degenerate $\rho_A$ [@minar]. Some of the results are summarized as follows:
\(1) For pure state $|\psi\rangle$ with the Schmidt decomposition of Eq. , one has $$\label{mina-2}
N_G(|\psi\rangle\langle\psi|)= 1-\sum_k \lambda_k^2.$$ (2) For bipartite state $\rho$ of Eq. with $\mathrm{dim} \mathcal{H}_A=2$, one has $$\label{mina-3}
N_G(\rho)=\left\{
\begin{aligned}
&||R||_2^2-\frac{1}{||\vec{x}||_2^2}\vec{x}^T R R^T \vec{x} &\text{if}~\vec{x}\neq 0,\\
&||R||_2^2-\lambda_{\text{min}}(RR^T) &\text{if}~\vec{x}=0.
\end{aligned} \right.$$ where $\lambda_{\text{min}}(RR^T)$ denotes the smallest eigenvalue of $R R^T$, and $\vec{x}=(r_{10}, r_{20}, r_{30})^T$.
\(3) For $\rho_W$ and $\rho_I$ of Eq. , one has [@min2] $$\label{mina-5}
\begin{aligned}
& N_G(\rho_W)=\frac{(dx-1)^2}{d(d+1)(d^2-1)},\\
& N_G(\rho_I)=\frac{(d^2 x-1)^2}{d(d+1)(d^2-1)}.
\end{aligned}$$ @Guo2013jpa-min given a necessary and sufficient condition for nullity of the HS norm of MIN. Let $\rho$ be a bipartite state acting on $\mathcal{H}_A\otimes \mathcal{H}_B$, and write $\rho=\sum_{i,j}A_{ij}\otimes |i^B\rangle\langle j^B|$ \[similar to Eq. \], they showed that $N_G(\rho)=0$ if and only if the $A_{ij}$s are mutually commuting normal operators, and each eigenspace of $\rho_A=\sum_i A_{ii}$ is contained in some eigenspace of $A_{ij}$, $\forall~ i,j$.
Furthermore, it is showed that for a zero-MIN state $\rho$ with $\dim \mathcal{H}_A \geqslant 3$, any local channel acting on party $A$ cannot create MIN if and only if either it is a completely contractive channel or it is a nontrivial isotropic channel [@Guo2013jpa-min]. For the qubit case this property is an additional characteristic of the completely contractive channel or the commutativity-preserving unital channel. That is, MIN can also be created under local operations and classical communication (LOCC).
### Trace norm of MIN
Similar to the GQD measured with the HS norm, the flaw of $N_G(\rho)$ is that it is also noncontractive under CPTP maps. Explicitly, it can be increased or decreased by trivial local reversible operations on the unmeasured party $B$. For example, a map $\mathcal{E}_B(\rho)= \rho\otimes \rho_C$ leads to $N_G(\rho_{A:BC})= N_G(\rho) {\mathrm{tr}}(\rho_C)^2$. As the purity ${\mathrm{tr}}\rho_C^2\leq 1$, this equality means that the MIN is decreased by simply introducing an uncorrelated local ancillary system. As a matter of fact, the flaw of the HS norm of MIN, being noncontractive under CPTP maps, is the same flaw for every HS norm measure of correlation. Despite this flaw, the MIN defined originally using the HS norm, in particular its motivation, inspires one to introduce most of the subsequent MIN measures.
Motivated by using the trace norm to measure GQD [@TDD], @trmin proposed that this norm can also be used to measure MIN, with the explicit expression $$\label{minb-1}
N_T(\rho)=\max_{\Pi^A}\|\rho-\Pi^A(\rho)\|_1.$$ This definition, although amends slightly the definition of Eq. , avoids successfully its non-contractivity problem. One can show that $N_T(\rho)$ is nonincreasing under any CPTP map $\mathcal {E}_B$ [@trmin], i.e., $N_T(\rho)\geq N_T(\mathcal
{E}_B[\rho])$. The proof is as follows: Let $\bar{\Pi}^A$ be the optimal measurement for obtaining $N_T(\rho)$, and $\tilde{\Pi}^A$ be the optimal measurement for obtaining $N_T(\mathcal
{E}_B[\rho])$, then as $\mathcal{E}_B$ and $\tilde{\Pi}^A$ commute, we obtain $\tilde{\Pi}^A (\mathcal{E}_B [\rho])=\mathcal{E}_B
(\tilde{\Pi}^A [\rho])$, and therefore $$\label{minb-2}
\begin{aligned}
N_T(\rho)&= \|\rho-\bar{\Pi}^A (\rho)\|_1 \\
&\geq \|\rho-\tilde{\Pi}^A (\rho)\|_1 \\
&\geq \|\mathcal{E}_B(\rho)-\mathcal{E}_B (\tilde{\Pi}^A [\rho])\|_1 \\
&= N_T(\mathcal {E}_B[\rho]),
\end{aligned}$$ where the first inequality comes from the fact that $\tilde{\Pi}^A
\neq \bar{\Pi}^A$ in general, and the second inequality is due to the contractivity of the trace norm under CPTP map. Therefore, $N_T(\rho)$ circumvents successfully the problem incurred for $N_G(\rho)$.
In purse of the analytical solutions of $N_T(\rho)$, some main results are as follows:
\(1) For $(2\times n)$-dimensional state $|\psi\rangle$ with the Schmidt decomposition of Eq. , one has $N_T(|\psi\rangle\langle\psi|)=2\sqrt{\lambda_1 \lambda_2}$.
\(2) For two-qubit state $\rho$ decomposed as Eq. , with the addition $r_{ij}=0$ for $i\neq j$, we have $$\label{minb-4}
N_T(\rho)=\left\{
\begin{aligned}
&\frac{\sqrt{\chi_{+}}+\sqrt{\chi_{-}}}{||\vec{x}||_1}
&\text{if}~\vec{x}\neq 0,\\
&2\max\{|r_{11}|,|r_{22}|,|r_{33}|\} &\text{if}~\vec{x}=0,
\end{aligned} \right.$$ where the corresponding parameter are $$\label{minb-v4}
\begin{aligned}
&\chi_\pm=\alpha\pm 4\sqrt{\beta} |\vec{x}|,~
\alpha=|\vec{r}|^2 |\vec{x}|^2-|\vec{r}\cdot \vec{x}|^2,\\
&\vec{r}=(r_{11},r_{22},r_{33}),~
\beta=\sum_{\langle ijk\rangle}x_i^2 r_{jj}^2 r_{kk}^2,
\end{aligned}$$ and the summation in the second line of the above equation runs over all the cyclic permutations of $\{1,2,3\}$.
\(3) For $\rho_W$ and $\rho_I$ of Eq. , solutions of the the trace norm MIN are given, respectively, by $$\begin{aligned}
\label{minb-5}
N_T(\rho_{W})=\frac{|dx-1|}{d+1}, ~~N_T(\rho_I)=\frac{2|d^2 x-1|}{d(d+1)},\end{aligned}$$ and by comparing them with Eq. , one can see that for the present cases, the two MIN measures $N_G$ and $N_T$ give qualitatively the same descriptions of nonlocality.
### Bures distance of MIN
By changing the maximization of Eq. , one can define the Bures distance of MIN as follows [@trmin] $$\begin{aligned}
\label{minc-1}
N_B(\rho)= \max_{\Pi^A}\{1 -\sqrt{F(\rho,\Pi^A(\rho)}\},\end{aligned}$$ where $\Pi^A$ is still the locally invariant measurements on party $A$, and $F(\rho,\sigma)$ is the Uhlmann fidelity defined in Eq. .
Compared with the former two measures of MIN, the calculation of the present MIN is more complicated. But when $A$ is a qubit, the minimum Uhlmann fidelity $F_{\min}(\rho,\Pi^A[\rho])= \min_{\Pi^A}
F(\rho,\Pi^A[\rho])$ can be calculated via Eq. , with however, the maximization being replaced by the minimization.
For Bell-diagonal state $\rho^{\rm Bell}$ of Eq. , its square root can be derived explicitly, from which $F_{\min}(\rho^{\rm Bell}, \Pi^A[\rho^{\rm Bell}])$ can be calculated as $$\label{minc-3}
F_{\min} =\frac{1}{2} \left(1+\min_{\{\theta,\phi\}}
\sqrt{b_3^2+(b_{13}^2+b_{21}^2 \sin^2\phi)\sin^2\theta}\right),$$ where $b_{ij}^2=b_i^2-b_j^2$, $b_i=8(t_0^2+t_i^2)-1$ ($i=1,2,3$), and by writing $c_{\rm sum}=c_1+c_2+c_3$, we have $$\begin{aligned}
\label{minc-4}
\begin{aligned}
t_0=&\frac{1}{8}\sqrt{1-c_{\rm sum}}+\frac{1}{8}\sum_{k=1}^3 \sqrt{1+c_{\rm sum}-2c_k},\\
t_i=&-\frac{1}{8}\sqrt{1-c_{\rm sum}}+\frac{1}{8}\sum_{k=1}^3\sqrt{1+c_{\rm sum}-2c_k}\\
&-\frac{1}{4}\sqrt{1+c_{\rm sum}-2c_i}.
\end{aligned}\end{aligned}$$ From Eq. one can see that $F_{\min}$ equals to $(1+|b_1|)/2$ if $|b_1|\leqslant\min\{|b_2|,|b_3|\}$, $(1+|b_2|)/2$ if $|b_2|\leqslant \min\{|b_1|, |b_3|\}$, and $(1+|b_3|)/2$ otherwise.
### Relative entropy of MIN
The relative entropy can also be recognized as a (pseudo) distance measure of quantum states, though technically it does not has a geometric interpretation as it is not symmetric, i.e., $S(\rho\|\sigma) \neq S(\sigma\|\rho)$ in general. It has been used to define the relative entropy of discord and quantum dissonance [@reqd].
@remin introduced the relative entropy of MIN as $$\label{mind-1}
N_R(\rho)=\max_{\Pi^A}S(\rho||\Pi^A[\rho]),$$ where $\Pi^A(\rho)=\sum_i \Pi_i^A \rho \Pi_i^A$, and $\{\Pi_i^A\}$ is the set of locally invariant projective measurements.
This MIN measure has been showed to be well defined. It possesses the same basic properties (+1), (+2), and (+3) as that of the HS norm of MIN. Furthermore, $N_R(\mathcal {E}_B[\rho])\leq N_R(\rho)$ for any CPTP map $\mathcal
{E}_B$ on the unmeasured party $B$ [@enmin]. It is also intimately related to the HS norm of MIN, $N_{E}(\rho)\geq
N_G^2(\rho)/(2\ln 2)$ [@remin].
It vanishes for the classical-quantum state $\chi$ with nondegenerate reduced density operator $\chi_A={\mathrm{tr}}_B\chi$, or for $\chi$ with degenerate $\chi_A$ and $\rho^B_k=\rho^B_l$ ($\forall$ $k,l$), see Eq. . Moreover, it is lower bounded by $-S(A|B)$ and upper bounded by $\min\{I(\rho), S(\rho_A)\}$, with $S(A|B)=S(\rho)-S(\rho_B)$ the conditional entropy [@enmin]. For $\rho^{\rm Bell}$ of Eq. , analytical solution of it is given by $$\label{mind-2}
\begin{split}
N_R(\rho^{\rm Bell})=& 1+H\left(\frac{1+c_{-}}{2}\right)+\frac{1-c_{\rm sum}}{4}\log_2\frac{1-c_{\rm sum}}{4} \\
&+\sum_{k=1}^3\frac{1+c_{\rm sum}-2c_k}{4}\log_2 \frac{1+c_{\rm sum}-2c_k}{4},
\end{split}$$ with $H(\cdot)$ being the binary Shannon entropy function.
The measure $N_R(\rho)$ is equivalent to that of the entropic MIN defined as the maximal discrepancy between QMI of the pre- and post-measurement states as [@enmin] $$\label{mind-3}
N_{E}(\rho)=I(\rho)-\min_{\Pi^A}I[\Pi^A(\rho)],$$ where $I(\rho)$ is the QMI given by Eq. .
This MIN quantifies in fact, the maximal loss of total correlations under locally non-disturbing measurements $\Pi^A$. Moreover, as $\rho$ and $\Pi^A(\rho)$ have the same reduced states, $N_E(\rho)$ defined above is equivalent to $$\label{mind-4}
N_E(\rho)=\max_{\Pi^A} S(\Pi^A[\rho])-S(\rho).$$ Thus, this measure of MIN quantifies also the maximal increment of von Neumann entropy induced by $\Pi^A$. Moreover, as the entropy of a state measures how much uncertainty there is in it, $N_E(\rho)$ can also be interpreted as the maximal increment of our uncertainty about the considered system induced by the locally invariant measurements.
### Skew information measure of MIN
Apart from measuring uncertainty in a state, the WY skew information has also been proposed to measure MIN. Its definition is as follows [@simin] $$\label{mine-1}
N_{SI}(\rho)=\max_{\tilde{K}^A}\sum_{i=1}^m
\mathsf{I}(\rho,\tilde{K}_i^A\otimes {\openone}_B),$$ which is in some sense dual to the correlation measure given in Eq. , with however the rank-one projectors $\tilde{K}^A=\{\tilde{K}_i^A\}$ are restricted to those which do not disturb $\rho_A={\mathrm{tr}}_B \rho$.
This MIN measure is invariant under locally unitary operations, contractive under CPTP map $\mathcal {E}_B$ on party $B$, and vanishes for all the product states and the classical-quantum states with nondegenerate reduced state $\rho_A$. For general state the calculation of $N_{SI}(\rho)$ is difficult. But if we decompose $\sqrt{\rho}$ as Eq. , an upper bound can be obtained as follows [@simin] $$\label{mine-2}
N_{SI}(\rho)\leq1- \sum_{i=1}^{m-1}\mu_i,$$ with $\mu_i$ ($i=1,2,\ldots,m^2$) being the eigenvalues of $\Gamma\Gamma^T$ listed in decreasing order (counting multiplicity), and $\Gamma=(\gamma_{ij})$ is the $(m^2\times n^2)$-dimensional correlation matrix.
For the pure states $\psi=|\psi\rangle\langle\psi|$, $N_{SI}(\psi)=N_G(\psi)$, while for the bipartite states $\rho$ with $A$ being a qubit, one has $N_{SI}(\rho)=1-\mu_1$ if $\vec{u}= 0$, and $$\label{mine-3}
N_{SI}(\rho)=1-\frac{1}{2}{\rm tr}\left(\left(\begin{array}{cc}
1 & \vec{u}_0 \\
1 & -\vec{u}_0
\end{array}\right)
\Gamma\Gamma^T\left(\begin{array}{cc}
1 & \vec{u}_0 \\
1 & -\vec{u}_0
\end{array}\right)^T\right),$$ if $~\vec{u}\neq 0$. Here, $\vec{u}=(u_1,u_2,u_3)$ with $u_i={\mathrm{tr}}(\rho_A\sigma_i) /\sqrt{2}$, and $\vec{u}_0=\vec{u}
/|\vec{u}|$. Moreover, for $\rho_W$ and $\rho_I$ of Eq. , one has $$\begin{aligned}
\label{mine-4}
\begin{split}
& N_{SI}(\rho_{W})=\frac{1}{2}\left(\frac{d-x}{d+1}-\sqrt{\frac{d-1}{d+1}(1-x^2)}\right),\\
& N_{SI}(\rho_I)=\frac{1}{d}\left(\sqrt{(d-1)x}-\sqrt{\frac{1-x}{d+1}}\right)^2.
\end{split}\end{aligned}$$ Similar to the above measure, @uin introduced another MIN-like nonlocality measure which was termed as uncertainty-induced nonlocality. It takes the form $$\begin{aligned}
\label{mine-5}
\mathcal {U}_{SI}(\rho)=\max_{K^A} \mathsf{I}(\rho,K^A\otimes {\openone}_B),\end{aligned}$$ where $K^A$ is a Hermitian observable with nondegenerate spectrum, and $[K^A,\rho_A]=0$. This measure is locally unitary invariant, nonincreasing under any CPTP map on the unmeasured party $B$. Moreover, it can also be interpreted by the Hellinger distance via the equality $$\begin{aligned}
\label{mine-6}
\mathcal {U}_{SI}(\rho)= \max_{K^A}d_H^2(\rho,K^A\rho K^A).\end{aligned}$$ For $(2\times n)$-dimensional state of Eq. , the uncertainty-induced nonlocality can be obtained explicitly as $$\label{mine-7}
\mathcal {U}_{SI}(\rho)=\left\{
\begin{aligned}
& 1-\lambda_{\min}(W_{AB}) &\text{if}~\vec{x}= 0, \\
& 1-\frac{1}{|\vec{x}|^2}\vec{x}^T W_{AB} \vec{x} &\text{if}~\vec{x}\neq
0,
\end{aligned} \right.$$ where $\vec{x}=(r_{10},r_{20},r_{30})^T$, and $\lambda_{\min} (W)$ is the smallest eigenvalue of the $3\times 3$ matrix $W_{AB}$, the elements of which is given by Eq. .
### Generalization of the MIN measures
The MIN measures we reviewed in the above sections reveal in fact only partial information about nonlocal features of a state, as they are defined based on the one-sided locally invariant measurements, thus those measures are all asymmetric. But a local state with respect to one party may be nonlocal with respect to another party. From this respect of view, it is significant to extend their definitions to more general case of two-sided locally invariant measurements. This gives the symmetric measure of MIN which can be written as $$\begin{aligned}
\label{minf-1}
\tilde{N}(\rho)= \max_{\tilde{\delta}\in\mathcal{L}}D(\rho,\tilde{\delta}),\end{aligned}$$ with $\tilde{\delta}$ being the two-sided locally invariant states in the sense that $\Pi^{AB}\tilde{\delta} \Pi^{AB}= \tilde{\delta}$ (with $\Pi^{AB}=\Pi^A\otimes\Pi^B$) should be satisfied, and $\sum_k
\Pi_k^A \rho_A \Pi_k^A=\rho_A$ and $\sum_k \Pi_k^B \rho_B
\Pi_k^B=\rho_B$ for any bipartite state $\rho$.
As an explicit example, we list the symmetric MIN measure defined based on the HS norm, i.e., $\tilde{N}_G(\rho)= \max_{\Pi^{AB}}
\|\rho-\Pi^{AB}(\rho)\|_2^2$. This measure is locally unitary invariant, and vanishes for the product states. For the pure state $\psi=|\psi\rangle \langle\psi|$, we have $\tilde{N}_G(\psi)=
N_G(\psi)$ [@tsmin]. In fact, $\tilde{N}_G(\rho)$ can also be extended to $N$-partite quantum states. The definition can be written in the same form of Eq. , with however the locally invariant measurements $\Pi^{A_1} \otimes\Pi^{A_2}\otimes
\ldots\otimes \Pi^{A_N}$, with $\sum_k \Pi_k^{A_i} \rho_{A_i}
\Pi_k^{A_i}= \rho_{A_i}$ for $i=\{1,2,\ldots,N\}$, and $\rho_{A_i}$ the reduced state of the subsystem $A_i$. But now the evaluation of their analytical expression becomes a hard work.
Applications of geometric quantum discord {#sec:2C}
-----------------------------------------
Up to now, we have presented an overview of the formal definitions and related formulae of the discordlike correlations defined via different distances. In general, these measures are conceptually different, and it is natural to wonder in what context one is more or less useful than the other. As a matter of fact, these measures capture different characteristic features of a state, and may have different physical implications and potential applications, e.g., the LQU (equivalent to the Hellinger distance of discord for any two-qubit state) guarantees a minimum precision of phase estimation [@LQU], the GQD defined via the relative entropy enables a direct comparison of it with the relative entropy of entanglement, and the negativity of quantumness can be connected to the negativity of entanglement. The above correlations defined with different distances may play role in different quantum information protocols, e.g., the HS norm of discord bounds from above fidelity of quantum teleportation and remote state preparation. Moreover, these discordlike correlations may reveal different aspects of the physical properties of a many-body system, and this will be discussed in Sec. \[sec:7\] of this review.
### Quantum teleportation
To teleport a state from one party to another spatially separated party, the sender Alice and the receiver Bob should share a quantum channel $\rho$, and one can achieve a perfect teleportation if $\rho$ is maximally entangled [@qtp0]. However, entanglement of $\rho$ is the prerequisite but not the only key elements for accomplishing the teleportation protocol. This is because for the non-maximally entangled channel, the fidelity of teleportation is not proportional to the amount of entanglement in $\rho$, e.g., it has been showed that the purity of $\rho$ is also a crucial element in determining the quality of the teleportation protocol [@Hupla].
When the channel is composed of a general two-qubit state $\rho$ as given by Eq. , the average teleportation fidelity, based on the assumption that Bob can perform all kinds of recovery operations to his qubit, can be derived as $\bar{F}=1/2+{\mathrm{tr}}\sqrt{R^\dagger R}/6$ [@fidelity1]. By considering a normalized version of the HS norm of discord $$\label{new4-1}
\tilde{D}_{G}(\rho)=\frac{d}{d-1}D_{G}(\rho),$$ @int-gqd identified a connection between an upper bound of $\tilde{D}_{G}(\rho)$ and $\bar{F}(\rho)$. The bound of $\tilde{D}_{G}(\rho)$ was derived by using the Weyl’s theorem, and is given by $$\label{new4-2}
\tilde{D}_{G}^{\max}(\rho)=\frac{1}{3}\left[\|R^2\|-k_{\max}(RR^T)\right],$$ where $k_{\max}(RR^T)$ represents the largest eigenvalue of $RR^T$. As ${\mathrm{tr}}\sqrt{R^\dagger R}\geqslant \|R^2\|$, one can show that $$\label{new4-3}
\bar{F}(\rho)\geqslant \frac{1+\tilde{D}_{G}^{\max}(\rho)}{2}.$$ On the other hand, by using the relations $3\bar{F}(\rho)-2\leq
\mathcal {N}(\rho)$ [@nega1] and $\mathcal{N}^2(\rho) \leq
\tilde{D}_G(\rho)$ [@nega2] for all two-qubit states, with $\mathcal{N}(\rho)$ being an entanglement measure called negativity [@negat], one can obtain $$\label{new4-4}
\bar{F}(\rho)\leqslant \frac{2+\sqrt{\tilde{D}_G(\rho)}}{3}.$$ These two equations show that the GQD bounds the average teleportation fidelity. But a direct quantitative connection between the various discordlike quantum correlation measures and $\bar{F}$ does not exist.
### Remote state preparation
Remote state preparation (RSP) is a quantum protocol for remotely preparing a quantum state by LOCC [@rsp0]. To accomplish this task, the two participants, Alice and Bob, also need to share a correlated channel. But different from the protocol of quantum teleportation [@qtp0], Alice knows what state to be transmitted in advance, so the amount of required classical information can be reduced.
If the shared state is maximally entangled, one can accomplish a perfect state preparation. Otherwise, the fidelity of the protocol may be reduced. @rsp considered such a problem. They considered the channel to be a general two-qubit state of the form of Eq. , and Alice wants to prepare a qubit state $\rho(\vec{s})=(I+\vec{s}\cdot \sigma)/2$ with the Bloch vector $\vec{s}$ in the plane orthogonal to the direction $\vec{\beta}$. To this purpose, she performs the local measurements $\Pi_\alpha^A=[I+\alpha\vec{\alpha}\cdot \vec{\sigma}]/2$ along the direction $\vec{\alpha}$ and informs Bob of her outcome $\alpha=\pm
1$. The Bloch vector of Bob’s state can then be obtained as $$\label{new4-5}
\vec{y}_\alpha= \frac{\vec{y}+\alpha R^T\vec{\alpha}}
{1+\alpha\vec{x}\cdot \vec{\alpha}}.$$ If Alice’s outcome is $\alpha=-1$, Bob applies a $\pi$ rotation about $\vec{\beta}$ to his system, whereas no operation is required for $\alpha=1$. After these conditional operations, the Bloch vector of Bob’s resulting state becomes the following mixture $$\label{new4-6}
\vec{r}=p_{+} \vec{y}_{+} + p_{-}R(\pi)\vec{y}_{-}.$$ where $p_\alpha=(1+\alpha\vec{\alpha}\cdot \vec{x})/2$ is the probability for Alice’s measurement outcome $\alpha$.
To evaluate the efficiency of the RSP protocol, @rsp defined the payoff-function $\mathcal{P}=(\vec{r}\cdot \vec{s})^2$ which is proportional to the fidelity $F={\mathrm{tr}}[\rho(\vec{r})\rho(\vec{s})]=(1+
\vec{r} \cdot \vec{s})/2$. For the present case, $\mathcal{P}$ can be derived explicitly as $$\label{new4-7}
\mathcal {P}= (\vec{\alpha}^T R \vec{s})^2
= \sum_{j=1}^3[\alpha_j (r_{j1}s_1+r_{j2}s_2+r_{j3}s_3)]^2,$$ and by optimizing over Alice’s choice of $\vec{\alpha}$, one can obtain $$\label{new4-8}
\mathcal {P}_{opt}=\sum_{j=1}^3 (r_{j1}s_1+r_{j2}s_2)^2.$$ Finally, the expected payoff is averaged over the distribution $\vec{s}$ and minimized over all possible choices of $\vec{\beta}$. The corresponding RSP-fidelity is given by $$\label{new4-9}
\mathcal{F}=\frac{1}{2}(E_2+E_3),$$ where $E_1\geqslant E_2\geqslant E_3$ are the eigenvalues of $R^T R$ arranged in nonincreasing order. Clearly, $\mathcal{F}$ vanishes if and only if $E_2=E_3=0$, which corresponds to a zero-discord state.
Moreover, if the local Bloch vector $\vec{x}$ of $\rho_{AB}$ is parallel to the eigenvector corresponding to largest eigenvalue of $R^T R$, the HS norm of discord is given by $D_G=\mathcal{F}/2$ [@rsp], which endows the GQD an operational interpretation.
Note that the nonvanishing GQD in the channel state is a necessary but not sufficient condition for RSP, as it has been found that there are discordant states which yields zero RSP-fidelity, e.g., the family of two-qubit states described by the real density matrix with $\rho_{11}-\rho_{22} = \rho_{44}- \rho_{33}$, $\rho_{14}=
\rho_{23}=0$, and $\rho_{12}=\rho_{13}=\rho_{24}=\rho_{34}$ [@rsp1].
### Phase estimation
@LQU considered a phase estimation task in which a bipartite state $\rho$ is utilized as a probe. In this task, a local unitary operation $U_\phi$ is performed on subsystem $A$ of this system, therefore an unknown phase $\phi$ is encoded to it and $\rho$ is transformed to $\rho_\phi=(U_\phi\otimes {\openone}) \rho (U_\phi^\dagger
\otimes{\openone})$. One’s goal is to estimate as precisely as possible the parameter $\phi$. For a given probe state $\rho$, one can optimize the measurements performed on $\rho_\phi$ to achieve the Cramér-Rao bound [@PD] $$\label{pd01}
\mathrm{Var}(\tilde{\phi}_{\mathrm{best}})= \frac{1}{N\mathcal{F} (\rho_\phi)},$$ where $\mathrm{Var} (\tilde{\phi}_{\mathrm{best}})$ is the variance of the best unbiased estimator $\tilde{\phi}_{\mathrm{best}}$, $N$ is the times of independent measurements, and $$\label{QFI}
\mathcal{F}(\rho_\phi)={\mathrm{tr}}(\rho_\phi L_\phi^2),$$ is the quantum fisher information, with $L_\phi$ being the symmetric logarithmic derivative determined by $$\label{SLD}
\frac{\partial{\rho_\phi}}{\partial{\phi}}= \frac{1}{2}(L_\phi\rho_\phi+\rho_\phi
L_\phi).$$ For the above phase estimation task, @LQU proved that $$\label{pd02}
\mathrm{Var} (\tilde{\phi}_{\mathrm{best}}) \leq \frac{1}{4N\mathcal{U}_A^\Lambda(\rho)},$$ hence the inverse of the LQU limits the achievable precision of the estimated phase parameter $\phi$. This gives an operational interpretation of the LQU.
Quantum coherence measures {#sec:3}
==========================
Different from quantum correlations which are defined in the framework of bipartite and multipartite scenarios, quantum coherence is related to the characteristics of the whole system. In general, the starting point for the resource theoretic characterization of a quantum character, e.g., quantum entanglement [@RMPE] and QD [@RMP], is the identification of free states which can be created at no cost and free operations which transform any free state into free state.
In a manner similar to the resource framework of entanglement where the free states are identified as those of the separable one and the free operations are specified by the LOCC, the set $\mathcal {I}$ of free states for quantum coherence encompasses those of the incoherent states which are diagonal in the prefixed reference basis $\{|i\rangle\}_{i=1}^d$, and take the form [@coher] $$\label{eq2-1}
\delta=\sum_{i=1}^d \delta_i |i\rangle\langle i|,$$ for a $d$-dimensional Hilbert space.
Within the framework of @coher, the set of free operations are those of the incoherent operations (IO) which can be specified by the Kraus operators $\{K_i\}$ satisfying $\sum_i K_i^\dag
K_i={\openone}$. Based on the measurements with and without subselection, @coher further identified two different classes of IO:
\(A) The incoherent completely positive and trace preserving (ICPTP) operations which act as $\Lambda(\rho)= \sum_i K_i \rho K_i^\dag$. Here, all $K_i$ are of the same dimension, and should obey the property $K_i \delta K_i^\dag/p_i\in\mathcal{I}$ for arbitrary $\delta\in\mathcal{I}$, with $p_i={\mathrm{tr}}(K_i \rho K_i^\dag)$ being the probability for obtaining the result $i$.
\(B) The incoherent operations with subselection for which the output measurement results are retained. They also require $K_i \delta
K_i^\dag/p_i\in \mathcal{I}$ to be satisfied for any $\delta\in\mathcal{I}$. But the dimension of $K_i$ may be different, that is, different $K_i$ may corresponds to different output spaces.
In general, a Kraus operator for an IO can be represented as $K_i=\sum_i c_i |f(i)\rangle\langle i|$, with the coefficient $c_i\in [0,1]$ and $f(i)$ a function on the index set [@qcd1].
As showed through explicit examples by @meas0 and proved strictly by @Sun, the Kraus operators related to incoherent operations $\Lambda$ are very limited. There is at most one nonzero entry in every column of $K_i$, and the number of possible structure of $K_i$ (a legal structure stands for a possible arrangement of nonzero entries in $K_i$) is $m^n$ for $K_i$ being the $m\times n$ matrices. @structure further discussed the problem relevant to the number of Kraus operators in a general quantum operation. For a system of dimension $d$, it has been found that any IO admits a decomposition with at most $d^4+1$ Kraus operators. For $d=2$ and 3, this number can be improved to 5 and 39, respectively.
Equipped with the sets of incoherent states and IO, @coher presented the defining conditions for a faithful coherence measure $C(\rho)$ which is a function that maps state $\rho$ to a nonnegative real value:
(C1) Nonnegativity, i.e., $C(\rho)\geq 0$, and $C(\delta)=0$ iff $\delta\in\mathcal {I}$.
(C2a) Monotonicity under ICPTP map, $C(\rho)\geq C(\Lambda[\rho])$.
(C2b) Monotonicity under selective IO on average, that is, $C(\rho)\geq \sum_i p_i C(\rho_i)$.
(C3) Convexity under mixing of states, i.e., $\sum_i p_i C(\rho_i)
\geq C(\sum_i p_i \rho_i)$, with $\{p_i\}$ being the probability distribution.
Note that condition (C2b) is stronger than (C2a), as its combination with (C3) automatically imply (C2a). In general, a real-valued function $C(\rho)$ is called a coherence measure if it satisfies the above four conditions. If only the conditions (C1), (C2a), and (C2b) are satisfied, $C(\rho)$ is usually called a coherence monotone.
A dual notion to incoherent states is the maximally coherent state, which can serve as a unit for defining coherence measure [@coher]. It takes the form $$\label{eq2-2}
|\Psi_d\rangle=\frac{1}{\sqrt{d}}\sum_{i=1}^d |i\rangle,$$ for which any other $\rho$ in the same Hilbert space can be generated with certainty by merely IO on it.
@cotr considered problem of general pure states transformation under IO by using the majorization theory [@major]. For states $|\psi\rangle= \sum_{i=1}^d \psi_i
|i\rangle$ and $|\phi\rangle=\sum_{i=1}^d \phi_i |i\rangle$, with the parameters $\{|\psi_i|\}$ and $\{|\phi_i|\}$ being arranged in nonincreasing order, they proved that $|\psi\rangle$ can be transformed to $|\phi\rangle$ via IO if and only if $(|\psi_1|^2,
|\psi_2|^2, \cdots, |\psi_d|^2)^T$ is majorized by $(|\phi_1|^2,|\phi_2|^2, \cdots, |\phi_d|^2)^T$, i.e., $$\label{majorization}
(|\psi_1|^2,|\psi_2|^2,\cdots,|\psi_d|^2)^T \prec (|\phi_1|^2,|\phi_2|^2,\cdots,|\phi_d|^2)^T.$$ Moreover, by applying the general unitary incoherent operations $$\label{UIO}
U_I=\sum_j e^{i\theta_j}|\alpha_j\rangle\langle j|$$ on $|\Psi_d\rangle$, with $\{\alpha_j\}$ being a relabeling of $\{j\}$, @mcvs found that the complete set $\mathcal {M}$ of maximally coherent states is composed of $\rho^{\rm mcs}
=|\Psi_d^{\rm mcs}\rangle\langle\Psi_d^{\rm mcs}|$, with $$\label{Peng}
|\Psi_d^{\rm mcs}\rangle= \frac{1}{\sqrt{d}} \sum_je^{i\theta_j} |j\rangle.$$ Building upon this, they proposed that $U_I$ are the unique quantum operations that preserve the coherence of a state, and suggested an additional condition for a valid coherence measure, i.e.,
(C4) $C(\rho)$ should assign a maximal value only to $\rho^{\rm
mcs}$.
@Tong proposed an alternative framework for defining coherence, in which their first two conditions are the same as (C1) and (C2a), while (C2b) and (C3) are replaced by one condition, that is, the additivity requirement of coherence for subspace-independent states. To be precise, for $\rho_1$ and $\rho_2$ in two different subspaces, the amount of coherence contained in $\rho=p_1\rho_1
\oplus p_2\rho_2$ (with $p_1$ and $p_2$ being probabilities) should be neither more nor less than the average coherence of $\rho_1$ and $\rho_2$ due to the block-diagonal structure of $\rho$. Hence, a reasonable measure of coherence should satisfy the condition $$\label{eq2-3}
C(p_1\rho_1 \oplus p_2\rho_2)=p_1 C(\rho_1)+p_2 C(\rho_2),$$ the above condition, together with (C1) and (C2a), have been showed to be equivalent to the four conditions introduced by @coher.
While the the set of free or incoherent states is widely accepted, there is no general consensus on the set of free operations in the resource theory of coherence. Apart from the above mentioned IO, there are other forms of free operations being introduced based on different physical or mathematical motivations up to date. Three typical ones are as follows:
\(1) Maximally incoherent operations (MIO). It refers to the set of physically realizable quantum operations $\Phi$ which maps incoherent states into incoherent states, i.e., $\Phi(\mathcal{I})
\in \mathcal {I}$ [@cof]. Obviously, this is the most general class of operations which do not create coherence from incoherent states.
\(2) Dephasing-covariant incoherent operations (DIO). The relevant set of it is a subset of MIO with the additional property $[\Delta,\Phi]=0$ [@DIO1; @DIO2; @Marvianpra]. That is, it admits $\Lambda[\Delta(\rho)]=\Delta[\Lambda(\rho)]$.
\(3) Strictly incoherent operations (SIO). This type of operations also admit an incoherent Kraus decomposition $\{K_i\}$ for which not only $K_i$ but also $K_i^\dagger$ ($\forall~ i$) is incoherent . That is, $\Delta(K_i\rho K_i^\dagger)= K_i
\Delta(\rho) K_i^\dagger$, where $$\label{eq2d-v2}
\Delta(\rho)=\sum_i \langle i|\rho|i\rangle |i\rangle\langle i|,$$ denotes full dephasing of $\rho$ in the basis $\{|i\rangle
\}_{i=1}^d$. It is the most general class of operations which do not use coherence, and admits a decomposition with at most $\min\{d^4+1,
\sum_{k=1}^d d!/(k-1)!\}$ Kraus operators [@structure] .
The inclusion relation of the above free operations are given by $\mathrm{SIO}\subset \mathrm{IO} \subset \mathrm{MIO}$ and $\mathrm{SIO}\subset \mathrm{DIO} \subset \mathrm{MIO}$.
The definitions of incoherent state $\delta$ and maximally coherent state $|\Psi_d\rangle$ imply that the related coherence measure will be a basis dependent quantity. This is because any density operator can be diagonalized in the reference basis spanned by its eigenvectors, hence casting a doubt on the rationality of this framework. But the recent progresses, particularly those studied from an operational perspective, still yields physically meaningful results. Moreover, in practice the reference basis are usually chosen according to the physical problem under consideration. All these indicate that the study of coherence measure has its own irreplaceable role.
Up to now, there are a number of quantum coherence measures being proposed in the literature (see Fig. \[fig:coh\]), where some of them satisfy the defining conditions, while the others satisfy only partial of these conditions. We review them in detail in the following.
Distance-based measures of coherence {#sec:3A}
------------------------------------
With the advent of quantum information science, geometric approaches are used to treat a huge class of problems such as the characterization and quantification of various quantum features. Analogously to the resource theory of entanglement for which the free operations are described by LOCC, the free states correspond to the separable states, and the entanglement can be defined by a distance between the considered state and the set of separable states, it is natural to quantify coherence of a state by utilizing a distance measure because coherence is also placed in a resource theoretic framework. To be explicit, one can quantify the amount of coherence contained in a state $\rho$ by using the minimal distance between $\rho$ and the set $\mathcal {I}$ of incoherent states, i.e., $$\label{eq2a-1}
C_\mathcal{D}(\rho)= \min_{\delta\in \mathcal{I}}\mathcal{D}(\rho,\delta),$$ where $\mathcal{D}(\rho,\delta)$ denotes certain distance measures of quantum states.
By its definition of Eq. , the condition (C1) is fulfilled for the distance measure which gives $\mathcal{D}(\rho,\delta) =0$ if and only if $\rho=\delta$, while (C2a) can be fulfilled when $\mathcal{D}$ is monotonous under the action of CPTP maps, i.e., $\mathcal{D} (\rho,\delta)\geq
\mathcal{D}(\Lambda[\rho],\Lambda[\delta])$. Moreover, (C3) is also fulfilled if $\mathcal{D}$ is jointly convex, i.e., $\mathcal{D}
(\sum_i p_i \rho_i,\sum_i p_i\sigma_i)\leq \sum_i p_i
\mathcal{D}(\rho_i,\sigma_i)$.
### Relative entropy of coherence
The relative entropy has been adopted to quantify entanglement, QD, and MIN. @coher showed that it can also serve as a valid tool for quantifying coherence. To be explicit, they defined $$\label{eq2a-2}
C_r(\rho)= \min_{\delta\in \mathcal{I}}S(\rho\|\delta)
= S(\rho_{\rm diag})-S(\rho),$$ where $\rho_{\rm diag}$ denotes the diagonal part of $\rho$.
This is an entropic measure of coherence which has a clear physical interpretation, as $C_r(\rho)$ equals to the optimal rate of the distilled maximally coherent states by IO in the asymptotic limit of many copies of $\rho$ [@qcd1].
For the one-qubit state $\rho= ({\openone}_2+\vec{r}\cdot
\vec{\sigma})/2$, with $\vec{r}\in \mathbb{R}^3$ and $|\vec{r}|\leq
1$, and $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ is the vector of Pauli matrices, if one chooses the reference basis as the eigenbasis of $\hat{n}\cdot \vec{\sigma}$ ($\hat{n}$ is a unit vector in $
\mathbb{R}^3$), then $$\label{eq2a-v2}
C_r(\rho)= H\left(\frac{1+\vec{r}\cdot\hat{n}}{2}\right)
-H\left(\frac{1+\vec{r}}{2}\right),$$ and $H(\cdot)$ is the binary Shannon entropy function.
For the two-qubit Bell-diagonal states of Eq. , @froz1 found that the closest incoherent state $\delta$ with respect to the bona fide distance measure of coherence (e.g., the relative entropy of coherence) is still a Bell-diagonal state with vanishing local Bloch vectors along $x$ and $y$ directions, and for the particular case of $c_2=-c_1 c_3$, $\delta$ reduces to the diagonal part of $\rho^{\rm Bell}$.
@RQC and @maxco studied maximal coherence of a state $\rho$ under generic reference basis. When the dimension of $\rho$ is $d$, they showed that the maximal coherence is given by $$\label{eq-maxrc}
C_r^{\max}(\rho)= \log_2 d -S(\rho).$$ @maxco also derived the corresponding unitary operator which transforms the computational basis to the optimal basis such that the maximal $C_r^{\max}(\rho)$ is obtained. It is given by $U=VH^\dag$, with the column vectors of $V$ being the eigenvectors of $\rho$, and $H$ is the rescaled complex Hadamard matrices. In fact, the Fourier matrix (a subset of the complex Hadamard matrices) $\bold{F}_d$ with elements $[\bold{F}_d]_{\mu\nu}= e^{i2\pi
\mu\nu/d}/\sqrt{d}$ is suffice for this purpose.
### $l_1$ norm of coherence
Intuitively, the superposition corresponds to the nonvanishing off-diagonal elements of the density operator description of a quantum state with respect to the selected reference basis. Starting from this consideration, @coher showed that the $l_1$ norm can also serve as a bona fide measure of coherence. To be explicit, they defined it as $$\label{eq2a-3}
C_{l_1}(\rho)= \min_{\delta\in \mathcal {I}}\|\rho-\delta\|_{l_1}
= \sum_{i\neq j}|\langle i|\rho|j\rangle|,$$ which equals to sum of the absolute values of the off-diagonal elements of $\rho$, and is favored for its ease of evaluation. Apart from convexity, it also satisfy the inequality $C_{l_1}(p_1
\rho_1+p_2\rho_2) \geq |p_1 C_{l_1}(\rho_1)-p_2 C_{l_1}(\rho_2)|$ [@triangle]. Moreover, for any bipartite pure state $|\psi\rangle_{AB}$, the $l_1$ norm of coherence equals to twice of its negativity which is a measure of quantum entanglement [@negat].
For pure state $\psi=|\psi\rangle\langle\psi|$, a relation between $C_{l_1}(\psi)$ and $C_r(\psi)$ has also been established [@meas1], which is given by $$\label{eq2a-4}
C_{l_1}(\psi)\geq \max\{C_r(\psi),2^{C_r(\psi)}-1\},$$ where $C_{l_1}(\psi)$ equals to $C_r(\psi)$ if and only if the diagonal elements of $\psi$ are (up to permutation) either $\{1,0,\ldots,0\}$ or $\{1/2,1/2,0, \ldots ,0\}$. It has also been proven that [@Rana] $$\label{eq2a-n1}
C_{l_1}^2(\psi)\leq \frac{d(d-1)C_r(\psi)}{\sqrt{2}},$$ where $d=\mathrm{rank}(\psi)$ is the rank of $\psi$. If $d>2$, one can further obtain $C_{l_1}(\psi)-C_r(\psi)\leq d-1-\log_2 d$. @Rana also proved a sharpest bound of $C_r(\psi)$ in terms of $C_{l_1}(\psi)$ as follows: $$\label{eq2a-n2}
\begin{aligned}
H(\alpha)+(1-\alpha)\log_2 (d-1) & \leq C_r(\psi) \\
& \leq H(\beta)+(1-\beta)\log_2 (n-1),
\end{aligned}$$ where $n$ equals $C_{l_1}+1$ if $C_{l_1}$ is an integer, and $[C_{l_1}]+2$ ($[C_{l_1}]$ is the integer part of $C_{l_1}$) otherwise. By denoting $x=C_{l_1}+1$, the other two parameters are given by $$\label{eq2a-n3}
\begin{aligned}
& \alpha=\frac{2+(d-2)(d-C_{l_1})+2\sqrt{(d-1)(d-x)x}}{d^2},\\
& \beta=\frac{2+(n-2)(n-C_{l_1}) -2\sqrt{(n-1)(n-x)x}}{n^2}.
\end{aligned}$$ For general states $\rho$, one has $$\label{eq2a-5}
C_{l_1}(\rho)\geq C_r(\rho)/ \log_2 d.$$ @meas1 have also conjectured $C_{l_1}(\rho)\geq C_r(\rho)$, but it was proved only for pure states, single-qubit states, and pseudopure states $\rho=p\psi+(1-p)\delta$ ($\forall \delta\in
\mathcal {I}$ and $p\in[0,1]$), while for general case, one can only prove $C_{l_1}(\rho)\geq 2^{C_r(\rho)}-1$ [@Rana], which is somewhat similar to the case for pure states showed in Eq. .
Moreover, for any single-qubit state $\rho$, by using convexity of $C_r$ and the inequality $2\min\{x,1-x\}\leq H(x)\leq
2\sqrt{x(1-x)}$, $\forall x\in[0,1]$, @Rana further proved the following relation $$\label{eq2a-n4}
\begin{aligned}
1-H\left(\frac{1-C_{l_1}(\rho)}{2} \right)& \leq C_r(\rho)\leq H\left(\frac{1-\sqrt{1-C_{l_1}^2(\rho)}}{2} \right) \\
& \leq C_{l_1}(\rho),
\end{aligned}$$ where the equality holds when $\rho$ is either incoherent or maximally coherent.
In fact, any state $\rho$ can be decomposed as $$\label{eq2d-4}
\rho = \frac{1}{d}{\openone}_d+\frac{1}{2}\sum_{i=1}^{d^2 -1} x_i X_i,$$ where $x_i={\mathrm{tr}}(\rho X_i)$, and $\{X_i/\sqrt{2}\}$ ($X_0={\openone}_d
/\sqrt{d}$) is the orthonormal operator bases for $\mathcal {H}$ (e.g., $X_i$ is the Pauli matrices when $d=2$, and the Gell-Mann matrices when $d=3$). Then if one arranges elements of $\vec{X}$ as $$\label{eq2a-n5}
\vec{X}=\{u_{12},v_{12}, \ldots, u_{d-1,d},v_{d-1,d}, w_1,\ldots, w_{d-1}\},$$ with the elements $$\label{eq2d-5}
\begin{split}
& u_{jk}= |j\rangle\langle k| + |k\rangle\langle j|, ~
v_{jk}= -i(|j\rangle\langle k| - |k\rangle\langle j|), \\
& w_{l}= \sqrt{\frac{2}{l(l+1)}}\sum_{j=1}^l (|j\rangle\langle j|
- l |l+1\rangle\langle l+1|),
\end{split}$$ where $j,k\in \{1,2,\ldots, d\}$ with $j<k$, $l\in \{1,2,\ldots,
d-1\}$ (the symbol $i$ in $v_{jk}$ is the imaginary unit), one has [@comp2] $$\label{eqa-01}
C_{l_1}(\rho)=\sum_{r=1}^{(d^2-d)/2}\sqrt{x_{2r-1}^2+ x_{2r}^2}.$$ By using Eq. , @RQC showed that the maximal $C_{l_1}(\rho)$ under generic basis is upper bounded by $$\label{eq2a-n6}
C_{l_1}^{\max}(\rho)\leq \sqrt{\frac{d^2-d}{2}}|\vec{x}|,$$ where $|\vec{x}|$ is length of the vector $(x_1,x_2,\cdots,
x_{d^2-1})$.
### Trace norm of coherence
Apart from the $l_1$ norm, one may wonder whether the general $l_p$ and Schatten-$p$ matrix norm can be adopted for defining coherence measures. In general, the answer to this question is negative. For example, @coher have considered the HS norm (i.e., $p=2$, for which it is also known as the Frobenius norm) measure of coherence defined as $$\label{eq2a-nn3}
C_{l_2}(\rho)=\min_{\delta\in \mathcal {I}}\|\rho-\delta\|^2_{2}
=\sum_{i\neq j}|\langle i|\rho|j\rangle|^2,$$ and showed through an counterexample that it does not satisfy condition (C2b). @meas1 further showed that any coherence measure defined via the $l_p$ norm or the Schatten-$p$ norm with $p\geq 2$ violates (C2b).
For the case of $p=1$ which corresponds to the trace norm (i.e., the Schatten-$1$ norm), if one defines $$\label{eq2a-6}
C_{tr}(\rho)=\min_{\delta\in \mathcal {I}}\|\rho-\delta\|_1,$$ with $\|M\|_1={\mathrm{tr}}\sqrt{M^\dag M}$ denoting the trace norm of the matrix $M$, the conditions (C1), (C2a), and (C3) for $C_{tr}(\rho)$ to be a proper coherence measure have been proven, but the work of @Tong showed that the condition of Eq. may be violated, thus proved it not to be a proper coherence measure.
For certain special classes of states, e.g., $\rho$ of one qubit or having possible nonzero elements along only the main diagonal and anti-diagonal (i.e., the *X* states), $C_{tr}(\rho)$ has already been proven to be a coherence monotone, and the corresponding optimal incoherent state is given by $\rho_{\rm diag}$ [@froz1]. Moreover, for the state $\rho$ with all of its non-diagonal elements equal to each other, i.e., $\rho_{ij}=a$ ($\forall i\neq j$), the trace norm of coherence can be derived analytically as $$\label{eq2a-v6}
C_{tr}(\rho)= 2(d-1)|a|,$$ where $d=\dim\rho$, and the closest incoherent state is $\delta^\star= \rho_\mathrm{diag}$ [@traceana]. For a restricted family of SIO [@qcd1], i.e., those of the SIO whose Kraus operators are $(2\times d)$-dimensional, the trace norm of coherence for this family of $\rho$ was also showed to satisfy the four conditions for a reliable quantum coherence measure [@traceana]. But its monotonicity under general IO may does not hold.
When restricted to pure states $|\psi\rangle$, it is possible to identify structure of the optimal incoherent state under the trace norm of coherence. As for any pure state $|\psi\rangle$, one can find a diagonal unitary matrix $U$ and a permutation matrix $P$ which gives $PU|\psi\rangle=|x\rangle$, with entries $x_1\geq
\cdots\geq x_d\geq 0$, the calculation can be performed to $|x\rangle$ only. By using the approximation theory, @pure found that $|\Psi_d\rangle$ of Eq. is the unique state that maximizing the trace norm of coherence, for which $C_{tr}^{\max} =2(1-1/d)$. The optimal incoherent state to $|x\rangle$ and the corresponding trace norm of coherence are given, respectively, by $$\label{eq2a-7}
\begin{aligned}
& \delta_{\rm opt}={\rm diag} \{\alpha_1,\cdots,\alpha_k,0,\cdots,0\},\\
& C_{tr}(|x\rangle\langle x|)=2(q_k s_k+m_k),
\end{aligned}$$ where $$\label{eq2a-8}
\alpha_i=\frac{x_i-q_k}{s_k-k q_k},$$ and $k$ is the maximum integer satisfying $$\label{eq2a-9}
x_k>q_k\coloneqq \frac{1}{2ks_k}\left( p_k+\sqrt{p_k^2+4km_k s_k^2}\right),$$ with the parameters $$\label{eq2a-v9}
s_l=\sum_{i=1}^l x_i,~m_l=\sum_{i=l+1}^d x_i^2,
~ p_l=s_l^2-l m_l-1$$ for all $l\in\{1,2,\cdots,d\}$.
To avoid the perplexity for $C_{tr}(\rho)$ of Eq. , i.e., the non-monotonicity of the trace norm of coherence under general incoherent operations, @Tong further proposed a modified version of trace norm of coherence by introducing a control parameter $\lambda$, and defined it as $$\label{eq-mtc}
C'_{tr}(\rho)= \min_{\lambda\geq 0, \delta\in \mathcal {I}}
\|\rho-\lambda\delta\|_1,$$ and proved that it satisfies the conditions (C1), (C2a) and Eq. , that is to say, it satisfies all the conditions for a reliable measure of quantum coherence. As its relation with other coherence measures, we have $$\label{eq-mtc2}
C'_{tr}(\rho)\leq C_{tr}(\rho)\leq C_{l_1}(\rho),$$ where the first inequality is obvious from their definitions in Eq. and , and the second one is due to $\|\cdot\|_1\leq\|\cdot\|_{l_1}$ for any Hermitian operator.
For one-qubit state, $ C'_{tr}(\rho)= C_{tr}(\rho)=C_{l_1}(\rho)$, and the optimal parameter $\lambda^\star=1$ and the optimal $\delta^\star = \rho_\mathrm{diag}$ [@froz1; @feism]. For general state $\rho$, determination of the analytical solution of $C'_{tr}(\rho)$ is possible only for certain special family of states. For example, the class of maximally coherent mixed states (MCMS) with respect to the $l_1$ norm of coherence, up to incoherent unitaries, is given by [@comp2] $$\label{eq4b-4}
\rho_{\rm mcms}=\frac{1-p}{d}{\openone}+p|\Psi_d\rangle\langle\Psi_d|,$$ for which the modified trace norm of coherence can be obtained analytically as $ C'_{tr}(\rho_{\rm mcms})=p$, with the optimal $\lambda^\star=1-p$ and $\delta^\star={\openone}_d/d$.
Entanglement-based measure of coherence {#sec:3B}
---------------------------------------
In a way analogous to the entanglement activation via local von Neumann measurements , one can also introduce the operational coherence measure with the help of IO.
Given a system $S$ in the state $\rho_S$ and an ancilla $A$ initialized in the pure state $|0^A\rangle$, @meas2 considered incoherent operations $\Lambda^{SA}$ on the combined system $SA$. By denoting $E_\mathcal{D}= \min_{\chi\in\mathcal {S}}
\mathcal{D}(\rho,\chi)$ a distance-based entanglement monotone and $C_\mathcal{D}$ the corresponding coherence monotone as given in Eq. , with $\mathcal{D}$ any contractive distance measure of quantum states and $\mathcal{S}$ the set of separable states, they found that the generated entanglement $E_\mathcal{D}^{S:A}$ is bounded from above by $$\label{eq2b-1}
E_\mathcal{D}^{S:A}(\Lambda^{SA}[\rho^S\otimes |0^A\rangle\langle 0^A|])\leq C_\mathcal{D}(\rho^S),$$ which means that when $\rho^S$ is incoherent, the IO cannot generate entanglement between $S$ and $A$.
Particularly, when $\mathcal{D}$ is the relative entropy and $d_A\geq d_S$ with $d_{A,S}=\dim\mathcal {H}_{A,S}$, then there always exists an incoherent operation (i.e., the generalized <span style="font-variant:small-caps;">cnot</span> operation) $$\label{eq-cnot}
U_{\textsc{cnot}}= \sum_{i=0}^{d_S-1}\sum_{j=0}^{d_S-1}|i,i\oplus j\rangle^{SA}\langle ij|
+ \sum_{i=0}^{d_S-1}\sum_{j=d_S}^{d_A-1}|ij \rangle^{SA}\langle ij|,$$ where $\oplus$ represents addition modulo $d_S$. This unitary operation maps the state $\rho^S\otimes |0^A\rangle\langle 0^A|$ to $$\label{eq2b-2}
\Lambda^{SA}[\rho^S\otimes |0^A\rangle\langle 0^A|] = \sum_{ij}\rho^S_{ij}
|ij\rangle^{SA}\langle ij|,$$ and henceforth Eq. is saturated: $$\label{eq2b-3}
E_{r}^{S:A}(\Lambda^{SA}[\rho^S\otimes |0^A\rangle\langle 0^A|]) = C_{r}(\rho^S),$$ which can be proved immediately by Eq. and the lower bound $-S(A|S)$ of $ E_{r}^{S:A}(\rho^{SA})$, where $S(A|S)$ is the conditional entropy. Then, @meas2 proposed to define coherence of $\rho^S$ as the maximal entanglement of $SA$ generated by IO, that is, $$\label{eq2b-4}
C_E(\rho^S)=\lim_{d_A\rightarrow \infty}\{\sup_{\Lambda^{SA}}E^{S:A}
(\Lambda^{SA}[\rho^S\otimes|0^A\rangle\langle 0^A|]) \},$$ with $E$ being an arbitrary entanglement measure, and $C_E$ will satisfy the four conditions of @coher if $E$ is convex as well.
For the geometric measure of entanglement $E_g(\rho)=
1-\max_{\sigma\in \mathcal{S}}F(\rho,\sigma)$ \[see also Eq. \] [@fident], the associated coherence measure can be evaluated as $$\label{eq2b-6}
C_g(\rho)=1-\max_{\delta\in\mathcal{I}}F(\rho,\delta),$$ and for $\rho$ of single-qubit state, $$\label{eq2b-7}
C_g(\rho)=\frac{1}{2}\left(1-\sqrt{1-4|\rho_{12}|^2}\right),$$ which is an increasing function of $C_{l_1}(\rho)= 2|\rho_{12}|$.
Moreover, for pure state $\psi=|\psi\rangle\langle\psi|$, as $F(\psi,\sigma)= |\langle\psi|\sigma|\psi\rangle|$, we have $$\label{eq-cg}
C_g(\psi)=1-\max_i\{\psi_{ii}\},$$ with $\psi_{ii}$ being the diagonal elements of $\psi$. For general mixed states, the calculation of $C_g(\rho)$ is formidably difficult, hence derives some bounds of it is significant. By using the relations among fidelity $F(\rho,\sigma)$, sub-fidelity $E(\rho,\sigma)$, and super-fidelity $G(\rho,\sigma)$, @longgl obtained lower and upper bounds of $C_g(\rho)$. The sub-fidelity and super-fidelity were defined as [@subsuper] $$\label{eq-ss}
\begin{aligned}
& E(\rho,\sigma)={\mathrm{tr}}(\rho\sigma)+\sqrt{2[({\mathrm{tr}}(\rho\sigma))^2-{\mathrm{tr}}(\rho\sigma\rho\sigma)]}, \\
& G(\rho,\sigma)={\mathrm{tr}}(\rho\sigma)+\sqrt{(1-{\mathrm{tr}}\rho^2)(1-{\mathrm{tr}}\sigma^2)},
\end{aligned}$$ and based on the relation $E(\rho,\sigma)\leq F(\rho,\sigma) \leq
G(\rho,\sigma)$ (the equality holds for one-qubit state or at least one of $\rho$ and $\sigma$ is pure), they found $$\label{eq-longgl}
\begin{aligned}
& 1-\frac{1}{d}-\frac{d-1}{d}\sqrt{1-\frac{d}{d-1}\left({\mathrm{tr}}\rho^2-\sum\nolimits_i \rho_{ii}^2\right)} \\
& \leq C_g(\rho) \leq \min\left\{1-\max_i\{\rho_{ii}\},1-\sum\nolimits_i b_{ii}^2\right\},
\end{aligned}$$ where $b_{ii}$ is related to the square root of $\sqrt{\rho}=
\sum_{ij}b_{ij}|i\rangle\langle j|$.
Convex roof measure of coherence {#sec:3C}
--------------------------------
### Intrinsic randomness of coherence
In the framework of quantum theory, measurement of quantum states induce intrinsically random outputs in general, and this randomness indicates genuine quantumness of a system. Based on this consideration, @meas4 proposed a convex roof measure for coherence, which has been proved to satisfy the four conditions of @coher. We call it intrinsic randomness of coherence.
For pure states $\psi=|\psi\rangle\langle\psi|$, the intrinsic randomness can be quantified by Shannon entropy of the probability distribution $\{p_i\}$ of the measurement outcomes which reads $$\label{eq2c-1}
R_I(\psi)=H(\{p_i\})=-\sum_i p_i\log_2 p_i,$$ with $H(\{p_i\})$ denotes the Shannon entropy of the probability distribution $\{p_i\}$, with $p_i={\mathrm{tr}}(E_i \psi)$ and $\{E_i\}$ is the set of measurement operators. $R_I$ characterizes also the degree of uncertainty related to the measurement outcomes, namely, the outcomes that cannot be predicted by blindly guessing. When restricted to projective measurements for which $E_i=|i\rangle\langle i|$ with $\{|i\rangle\}$ the reference basis, the right-hand side of Eq. is $S(\psi_{\rm diag})$, henceforth, the intrinsic randomness $R_I(\psi)$ equals the relative entropy of coherence $C_r(\psi)$.
For general case of mixed states $\rho$, @meas4 defined the intrinsic randomness $R_I(\rho)$ by utilizing convex roof construction, that is, $$\label{eq2c-2}
R_I(\rho)=\min_{\{p_i,\psi_i\}}\sum_i p_i R_I(\psi_i),$$ where $\sum_i p_i=1$, $\psi_i=|\psi_i\rangle\langle\psi_i|$, and the minimum is taken over all possible pure state decompositions of $\rho$ given in Eq. .
Eq. establishes a convex roof definition of quantum coherence, which bears some resemblance with the convex roof measures of entanglement such as entanglement of formation $E_f(\rho)=\min_{\{p_i, \psi_i\rangle}\sum_i p_i S({\mathrm{tr}}_B \psi_i)$ and the geometric entanglement $E_g(\rho)=\min_{\{p_i,
\psi_i\}}\sum_i p_i E_g (\psi_i)$. It was also termed as superposition of formation by @cof, and coherence of formation by @qcd1.
Apart from pure states, $R_I(\rho)$ is analytically computable for one-qubit states, i.e., $$\label{eq-ri}
R_I(\rho)=H\left(\frac{1+\sqrt{1-C_z^2(\rho)}}{2} \right),$$ where $C_z(\rho)=2|\rho_{12}|$ is termed as coherence concurrence, as it is given by $$\label{eq-rand}
C_z(\rho)= |\lambda_1 -\lambda_2|,$$ with $\lambda_i$ being square roots of the eigenvalues of the product matrix $$\label{eq-cc}
R=\rho\sigma_x\rho^*\sigma_x,$$ where $\rho^*$ is the conjugation of $\rho$, and $\sigma_x$ is the first Pauli matrix.
### Coherence concurrence
Using the fact that any $\rho$ can be decomposed as Eq. , @gaoyan found that the $l_1$ norm of coherence for $\rho$ is equivalent to $$\label{gaoyan}
C_{l_1}(\rho)=\sum_{j<k}\bigg |\sqrt{\eta_1^{jk}}-\sqrt{\eta_2^{jk}}\bigg |,$$ where $\eta_1^{jk}$ and $\eta_2^{jk}$ are nonzero eigenvalues of the matrix $$\label{gaoyan2}
R^{jk}=\rho u_{jk}\rho^* u_{jk},$$ and the generalized Gell-Mann matrices $\{u_{jk}\}$ as given in Eq. . When $d=2$, $R^{jk}$ is just that of $R$ given in Eq. . For pure state $\psi=|\psi\rangle\langle\psi|$, the above equation further gives $$\label{gaoyan3}
C_{l_1}(\psi)=\sum_{j<k} |\langle\psi|u_{jk}|\psi^*\rangle |,$$ which is direct consequence of $R^{jk}=|\psi\rangle\langle\psi|
u_{jk}|\psi^*\rangle\langle\psi^*|u_{jk}$. Motivated by this fact, @gaoyan proposed a convex roof measure of coherence $$\label{eq-gaoyan}
C_{\rm con}(\rho)=\min_{\{p_i,\psi_i\}}\sum_i p_i C_{l_1}(\psi_i),$$ where the minimization is with respect to the possible pure state decompositions of $\rho$ given in Eq. . $C_{\rm
con}(\rho)$ is termed as coherence concurrence, as it is very similar to the entanglement concurrence given by $$\label{gaoyan4}
C_E(\rho)=\min_{\{p_i,\psi_i\}}\sum_i p_i C_E(\psi_i),$$ with $C_E(\psi_i)=[2(1-{\mathrm{tr}}\rho_A^2)]^{1/2}$, and $\rho_A={\mathrm{tr}}_B
\psi_i$ is the reduced density matrix of $\psi_i$. For two-qubit state, $C_E(\rho)$ is analytically solved as $C_E(\rho)=\max\{0,
2\lambda_{\max}-\sum_{j=1}^4 \lambda_j\}$ (with $\lambda_j$s being eigenvalues of $\rho(\sigma_y\otimes\sigma_y)\rho^*(\sigma_y\otimes
\sigma_y)$), and it is linked to the entanglement of formation as $E_f=H([1+(1-C_E^2)^{1/2}]/2)$, with $H(\cdot)$ being the binary Shannon entropy function [@concur1; @concur2].
It can be proved that this measure satisfy all the conditions (C1), (C2a), (C2b), and (C3) for a reliable measure of quantum coherence. Moreover, it is bounded from below by the $l_1$ norm of coherence, i.e., $C_{\rm con}(\rho) \geq C_{l_1}(\rho)$, which can be derived directly by combining the definition and the convexity of it.
### Fidelity-based measure of coherence
@liuqip proposed a convex roof measure of coherence based on fidelity and showed that it fulfills the four conditions introduced by @coher. For any pure state $\psi=|\psi\rangle
\langle\psi|$, it is defined as $$\label{eq-cf1}
C_F(\psi)=\min_{\delta\in \mathcal {I}} \sqrt{1-F(\psi,\delta)},$$ which is very similar to $C_g(\rho)$ of Eq. , and can be derived analytically as $C_F(\psi)=C_g^{1/2}(\psi)$ \[see Eq. \]. For general mixed state $\rho$, it is defined via a convex roof construction, that is $$\label{eq-cf2}
C_F(\rho)=\min_{\{p_i,\psi_i\}}\sum_i p_i C_F(\psi_i),$$ and the minimization is taken over all the possible pure state decompositions of $\rho$, see Eq. . Moreover, when $\rho$ is of a single-qubit state, $C_F(\rho)= C_g^{1/2}(\rho)$, see Eq. .
### The rank-measure of coherence
The Schmidt rank for a pure state and the Schmidt number which is an extension of the Schmidt rank to mixed states have been shown to be useful for defining entanglement measures [@snumber]. The Schmidt rank $r(\psi)$ for a $(d\times d')$-dimensional pure state $\psi=|\psi\rangle\langle\psi|$ is the number of nonzero coefficients in its Schmidt decomposition of Eq. , and the Schmidt number for a general $(d\times d')$-dimensional mixed state $\rho_{AB}$ is defined as [@srank] $$\label{rank1}
r(\rho_{AB})=\min_{\{p_i,\psi_i\}}\max_i r(\psi_i),$$ where $\psi_i=|\psi_i\rangle\langle\psi_i|$, and the minimization is taken over all the pure state decompositions of $\rho$, see Eq. .
In analogous to Schmidt rank and Schmidt number, the coherence rank $r_C(\psi)=\mathrm{rank}(\psi)$ is the number of nonzero coefficients $\alpha_i$ for a pure state $|\psi\rangle=\sum_i
\alpha_i |i\rangle$. It can serve as a coherence measure of $\psi$. For a general mixed state $\rho$, @cnumber introduced a convex roof measure of coherence termed as coherence number. It reads $$\label{eq-cnumber}
r_C(\rho)=\min_{\{p_i,\psi_i\}}\max_i r_C(\psi_i),$$ where the minimization is taken over the pure state decompositions of $\rho$ showed in Eq. . This is a coherence monotone under IO, as it satisfies the conditions (C1), (C2a), and (C2b) of @coher.
It is obvious that $r_C(|\Psi_d\rangle)=d$ for the maximally incoherent state $|\Psi_d\rangle$, and $r_C(\delta)=1$ for any incoherent state $\delta$. Of course, one can also take logarithm to $r_C(\rho)$ and define the coherence monotone as [@oneshot] $$\label{eq-oneshot}
C_0(\rho)= \log_2 r_C(\rho),$$ and now $C_0(\delta)$ equals zero when $\delta$ is incoherent.
Robustness of coherence {#sec:3D}
-----------------------
Given an arbitrary state $\rho$ on the Hilbert space $\mathcal {H}$, its convex mixture with another state $\tau$ on the same space may be coherent or incoherent. In another word, a proper choice of $\tau$ and weight factor $s$ of mixing may destroy the coherence in $\rho$. Based on this fact and stimulated by similar definitions for various quantum correlation monotones, @meas6 introduced a new coherence measure which was called robustness of coherence (RoC). It is defined as $$\label{eq2d-1}
C_R(\rho)= \min_{\tau\in\mathcal{D}(\mathbb{C}^d)}\left\{ s\geq 0 \bigg|
\frac{\rho+s\tau}{1+s}\coloneqq \delta\in\mathcal{I}\right\},$$ where $\mathcal{D}(\mathbb{C}^d)$ is the convex set of density operators on $\mathcal{H}$.
Alternatively, the RoC can also be defined as [@asymm] $$\label{eq2d-new1}
C_R(\rho)=\min_{\delta\in \mathcal {I}}\{s\geq 0| \rho\leq
(1+s)\delta\}.$$ The RoC is proved to be a full coherence monotone [@meas6]. That is to say, it satisfies the conditions required by the framework for a resource theory of quantum coherence [@coher]. It is also analytically computable for $\rho$ being arbitrary one-qubit and pure states, as well as for those $\rho$ with possible nonzero elements along only the main diagonal and anti-diagonal (i.e., the so-called $X$ states). For those $\rho$, $C_R(\rho)= \sum_{i\neq
j}|\rho_{ij}|$ in the reference basis $\{|i\rangle\}$, hence equals to the related $l_1$ norm of coherence $C_{l_1}(\rho)$.
For general $\rho$, @meas6 constructed a semidefinite program for calculating $C_R(\rho)$ numerically, and obtained tight lower bounds of RoC of the following form [@asymm] $$\label{eq2d-2}
C_R(\rho)\geq \frac{\|\rho-\Delta(\rho)\|_2^2}{\|\Delta(\rho)\|_\infty}
\geq \frac{\|\rho-\Delta(\rho)\|_2^2}{\|\Delta(\rho)\|_2}
\geq \|\rho-\Delta(\rho)\|_2^2,$$ where the operator norm $\|M\|_\infty=\lambda_{\max}$, with $\lambda_{\max}$ being the largest singular value of $M$, and $\|M\|_2$ is the HS norm.
@meas6 also obtained bounds of $C_R(\rho)$ via the $l_1$ norm of coherence which is analytically computable, i.e., $$\label{eq2d-3}
(d-1)^{-1}C_{l_1}(\rho)\leq C_R(\rho)\leq C_{l_1}(\rho),$$ where the upper bound is tight obviously, and the lower bound is saturated by the family of $\rho=(1+p){\openone}/d-p|\Psi_d\rangle\langle\Psi_d|$, with $0\leq p\leq
1/(d-1)$. It has also been proven that [@Rana] $$\label{eq2d-v3}
C_r(\rho)\leq \log_2 [1+C_R(\rho)].$$ Moreover, the RoC is also showed to be upper bounded by [@asymm] $$\label{roc2}
C_R(\rho)\leq d\|\rho\|_\infty -1,$$ where $\|\rho\|_\infty$ denotes the largest singular value of $\rho$ (also known as the operator norm). This inequality implies that $C_R(\rho)$ takes its maximum value $d-1$ only if $\rho$ is a maximally coherent pure state.
From Eq. one can verify directly that the optimization in Eq. can be restricted to the subset of $\tau\in\mathcal{D}(\mathbb{C}^d)$ given by $$\label{eq2d-6}
\tau_{\rm sub} = \frac{1}{d}{\openone}_d-\frac{1}{2}\sum_{i=1}^{d_0} x_i X_i
+\frac{1}{2}\sum_{i=d_0+1}^{d^2-1} y_i X_i,$$ where $d_0=(d^2-d)/2$. That is to say, the optimization can be performed over all possible $\{y_i\}$ such that $\tau_{\rm sub}$ is a physically allowed state.
Experimentally, @roc-exp have explored the RoC for one-qubit states. They developed two different methods to measure directly the quantum coherence, i.e., the interference-fringe method and the witness-observable method. For the former one, they showed experimentally that the sweeping on ancilla state is necessary only along the equatorial pure states, while for the latter one, the optimal witness operator is $$\label{eq2d-v6}
W^*=\cos\varphi_\rho \sigma_1+ \sin\varphi_\rho\sigma_2.$$ They have also compared the experimental results with those calculated via state tomography and found a high coincidence of them.
Stimulated by the resource weight-based quantification of quantum features such as the best separable approximation of entangled states [@weight1], the steering weight [@weight2], and the measurement incompatibility weight [@weight2], @coh-weight proposed a similar measure of coherence which they termed as coherence weight. It is defined as $$\label{eq2d-cw}
C_w(\rho)= \min_{\delta\in \mathcal {I}, \tau\in\mathcal{D}(\mathbb{C}^d)}
\{s\geq 0| \rho= (1-s)\delta+s\tau\},$$ which can be interpreted operationally as the minimal number of coherent states (combination with the maximal number of incoherent states) needed to prepare the considered state $\rho$ on average. $C_w(\rho)$ was showed to satisfy all the four conditions proposed by @coher, thus constitutes a bona fide measure of coherence. For certain special states, the coherence weight can be obtained analytically, e.g., for the pure states we always have $C_w(\rho)=1$, while for Werner state $\rho_W$ of Eq. , we have $$\label{eq2d-cw2}
C_w(\rho_W)=C_R(\rho_W)=C_{l_1}(\rho_W)=\frac{1-dx}{d+1}.$$ Similar to the definition of RoC in Eq. , the coherence weight can also be defined as $$\label{eq2d-cw3}
C_w(\rho)=\min_{\delta\in \mathcal {I}}\{s\geq 0| \rho\geq
(1-s)\delta\}.$$ @coh-weight also obtained lower bounds of $C_w(\rho)$ as $$\label{eq2d-cw4}
C_w(\rho)\geq \frac{\|\rho-\Delta(\rho)\|_2^2}{\|\rho\|_\infty}
\geq \|\rho-\Delta(\rho)\|_2^2,$$ and proved its relation with the other quantum coherence measures, i.e., $$\label{eq2d-cw5}
\begin{aligned}
& C_w(\rho)\geq \frac{1}{d-1}C_{l_1}(\rho)\geq \frac{1}{d-1}C_R(\rho),\\
& C_w(\rho)\geq \frac{1}{\ln d}C_r(\rho).
\end{aligned}$$ Moreover, the coherence weight $C_w$ is showed to satisfy the following relation $$\label{eq2d-cw6}
C_w(\rho_1\otimes \rho_2)\leq C_w(\rho_1)+C_w(\rho_2)-C_w(\rho_1)C_w(\rho_2),$$ for any quantum states $\rho_1$ and $\rho_2$, while for RoC, we have $$\label{eq2d-cw7}
C_R(\rho_1\otimes \rho_2)\leq C_R(\rho_1)+C_R(\rho_2)+C_R(\rho_1)C_R(\rho_2).$$
Tsallis relative entropy measure of coherence {#sec:3E}
---------------------------------------------
@meas5 studied the problem for quantifying coherence via the Tsallis $\alpha$ relative entropies, which are functionals of powers of density matrices, and formulated family of coherence measures defined by quantum divergences of the Tsallis type.
As an extension of the standard quantum relative entropy, the Tsallis $\alpha$ divergence is given by $$\label{eq2e-1}
\begin{small}
D_{\alpha}(\rho\|\sigma)=\left\{ \begin{aligned}
&\frac{{\mathrm{tr}}(\rho^\alpha\sigma^{1-\alpha})-1}{\alpha-1},
&&{\rm if~} {\rm ran}(\rho)\subseteq {\rm ran}(\sigma), \\
& +\infty, &&{\rm otherwise},
\end{aligned} \right.
\end{small}$$ with $\alpha$ being a positive number. The trace is taken over ${\rm
ran}(\sigma)$ for $\alpha>1$, while for $\alpha\in(0,1)$ this condition is not necessary. Here, ${\rm ran}(\rho)$ denotes the range of $\rho$, and likewise for ${\rm ran}(\sigma)$.
Then, motivated by Eq. , @meas5 proposed to quantify coherence of $\rho$ as $$\label{eq2e-v1}
C_\alpha(\rho)= \min_{\delta\in \mathcal{I}}
D_\alpha(\rho\|\delta),$$ and further proved that it can be evaluated analytically as $$\label{eq2e-2}
C_\alpha(\rho)= \frac{1}{\alpha-1}\left\{ \left(\sum_i \langle i|
\rho^\alpha|i\rangle^\alpha\right)^{1/\alpha}-1\right\},$$ and for $\alpha=1$, it reduces to the relative entropy of coherence in Eq. , while for the specific case of $\alpha=2$, we have $$\label{eq2e-3}
C_2(\rho)=\left( \sum_j \sqrt{\sum_i |\langle i|\rho|j\rangle|^2} \right)^2-1,$$ The coherence measure based on the quantum $\alpha$ divergence is bounded above by $$\label{eq2e-4}
C_{\alpha}(\rho) \leq\left\{ \begin{aligned}
& -\ln_\alpha \frac{1}{d{\mathrm{tr}}\rho^2}, &&{\rm if~} \alpha\in(0,2], \\
& \frac{d{\mathrm{tr}}\rho^2 \varsigma^{\alpha-2}-1}{\alpha-1},
&&{\rm if~} \alpha\in(2,+\infty),
\end{aligned} \right.$$ where $\varsigma=\{(d-1)(d{\mathrm{tr}}\rho^2-1)\}^{1/2}+1$, which is intimately related to the mixedness of $\rho$, hence is experimentally accessible.
To be a reliable quantifier of coherence, the proposed functional should obey the four conditions derived via the resource theoretic framework of coherence [@coher]. For $C_\alpha(\rho)$, it vanishes if and only if $\rho\in \mathcal{I}$, namely, it satisfies the condition (C1). Furthermore, (C2a) is satisfied for $\alpha\in(0,2]$, which is a direct result of $D_\alpha
(\Lambda[\rho],\Lambda[\sigma])\leq D_\alpha(\rho,\sigma)$ for the Tsallis $\alpha$ divergence. Thirdly, by denoting $\delta^*$ the closest incoherent state to $\rho$, and $q_i={\mathrm{tr}}(K_i \delta^*
K_i^\dag)$, $p_i={\mathrm{tr}}(K_i \rho K_i^\dag)$, and $\rho_i=E_i\rho
E_i^\dag /p_i$, @meas5 derived a generalized form of the monotonicity formulation applicable for $C_{\alpha}(\rho)$. It is given by $$\label{eq2e-5}
C_{\alpha}(\rho)\geq \sum_i p_i^\alpha q_i^{1-\alpha}C_\alpha
(\rho_i),$$ which reduces to the usual monotonicity formula for standard relative entropy when $\alpha=1$. Note that the coherence measure of Eq. may violate the monotonicity condition (C2b) for $\alpha\neq 1$. Finally, the Tsallis $\alpha$ relative entropy measure of coherence is also convex for $\alpha\in (0,2]$, this is because the Tsallis divergence $D_\alpha$ is a convex function of density matrices in the same region of $\alpha$.
If one takes logarithm (with base 2) to the first term of Eq. , then in the limit of $\alpha\rightarrow \infty$, we can recover the maximum relative entropy defined as $$\label{eq2e-6}
D_{\max}(\rho\|\sigma)= \min\{\lambda| \rho\leq 2^\lambda \sigma\},$$ where $\lambda\geq 0$. It is also an important concept in quantum information science [@MRE1; @MRE2; @MRE3; @MRE4]. In a recent work, @Buprl and @oneshot proposed to use it as a basis for defining a coherence measure which was termed as the maximum relative entropy of coherence. It reads $$\label{eq-mre1}
C_\mathrm{max}(\rho)\coloneqq \min_{\delta\in \mathcal{I}}D_{\max}(\rho\|\delta),$$ and has been proven to obey the conditions (C1), (C2a), and C(2b) introduced by @coher, so it is a coherence monotone under MIO [@oneshot]. Nevertheless, it does not satisfy the convexity condition (C3), and it is only quasiconvex, that is, $C_\mathrm{max}
(\sum_i p_i \rho_i)\leq \max_i C_\mathrm{max}(\rho_i)$. Moreover, by comparing the above equation with Eq. , one can also found that $C_\mathrm{max}(\rho)$ can be linked quantitatively to the RoC as $$\label{eq-mre2}
C_\mathrm{max}(\rho)=\log_2 [1+C_R(\rho)],$$ thus similar to RoC, if $\exists U$ such that $(U\rho
U^\dagger)_{kl}=|\rho_{kl}|$, then a closed formula for $C_\mathrm{max}(\rho)$ can also be obtained. A class of $\rho$ where such a requirement is satisfied consists all the one-qubit states, the pure states, and the $X$ states.
Besides $C_\mathrm{max}(\rho)$, @Buprl further proposed the $\varepsilon$-smoothed maximum relative entropy of coherence, which was defined as $$\label{eq-mre3}
C_\mathrm{max}^\varepsilon(\rho)\coloneqq \min_{\rho'\in B_\varepsilon
(\rho)}C_\mathrm{max}(\rho'),$$ where $B_\varepsilon (\rho) \coloneqq \{\rho'\geq 0 \!: \|\rho'-
\rho\|_1 \leq \varepsilon, {\mathrm{tr}}\rho'\leq {\mathrm{tr}}\rho\}$. It has been shown that $\lim_{\varepsilon\rightarrow 0, n\rightarrow\infty}
C_\mathrm{max}^\varepsilon(\rho^{\otimes n})=n C_r(\rho)$. That is, $C_\mathrm{max}^\varepsilon(\rho)$ is equivalent to the relative entropy of coherence in the asymptotic limit.
@DIO2 put forward another similar coherence measure as $$\label{eq-mre4}
C_{\Delta,\mathrm{max}}(\rho)\coloneqq \min\{\lambda| \rho\leq 2^\lambda \Delta(\rho)\},$$ which has also been proven to be a coherence monotone under DIO [@oneshot; @DIO2] and it is also quasiconvex.
Skew-information-based measure of coherence {#sec:3F}
-------------------------------------------
Soon after the work of @coher, @meas8 proposed a new method to quantify the amount of coherence in a state. It is defined based on the WY skew information, and has later been proven to violate one of the reliability criteria for a bona fide coherence monotone [@meas9]. But due to the experimentally accessibility, it may still be of interest to the quantum community.
### Definition and properties
For a system to be measured, the uncertainty of the outputs comes from both ignorance of the mixture of the state and the truly quantum part related to state collapse induced by measurements. @meas8 proposed that the latter feature (i.e., the truly quantum uncertainty of a measurement) is an embodiment of quantum coherence, and can be reliably quantified by the WY skew information.
The skew information $\mathsf{I}(\rho,K)$ is nonnegative and vanishes if and only if $[\rho,K]=0$, namely, if and only if $\rho$ is diagonal in the basis defined by $K$, hence it fulfills condition (C1). Furthermore, $\mathsf{I}(\rho,K)$ is a convex function of density matrices, thus (C3) is also fulfilled. But $\mathsf{I}
(\rho,K)$ does not fulfill the other axiomatic postulates for a faithful coherence measure [@coher], e.g., @meas9 have constructed a series of phase sensitive IO $\Lambda$ for which $\mathsf{I}(\rho,K)\leq \mathsf{I}(\Lambda[\rho],K)$.
But the above fact does not affect the status of $\mathsf{I}(\rho,K)$ as a well-accepted measure of asymmetry, which is defined with respect to a given symmetry group $\mathsf{G}$, and includes as a special case the group $\mathsf{U}(1)$ used for defining quantum coherence. In fact, previously @Marvian have proposed such an asymmetry measure, where they used the more general $\mathsf{I}^p(\rho,L)$ of Eq. with $p\in(0,1)\cup(1,2]$, i.e., $$\label{wysk1}
S_{L,p}(\rho)={\mathrm{tr}}(\rho L^2)-{\mathrm{tr}}(\rho^p L \rho^{1-p}L),$$ and $L$ represents an arbitrary generator of the group. If $\rho$ is symmetric relative to group $\mathsf{G}$ \[$U_g(\rho)=\rho$ for all group elements $g\in\mathsf{G}$\], then $S_{L,p}(\rho)=0$, see @Marvian and @Marvian-phd. In the same work, the authors also proposed several other measures of asymmetry, for example, those based on the Holevo quantity, the trace norm, and the relative Rényi entropy \[see Eq. \]. Moreover, @asymm also proposed a measure of asymmetry which they called robustness of asymmetry. It is defined in a manner very similar to RoC in Eq. , with only $\delta$ being replaced by the symmetric state relative to a group $\mathsf{G}$.
In a more general sense, all the coherence measures defined in the framework of @coher constitute a proper subset of measures of asymmetry. Asymmetry measures the extent to which the symmetry relative to a group of translations such as time translations or phase shifts is broken, wherein the translationally invariant operations $\Lambda_{\rm TI}$ play a central role. To be explicit, an asymmetry measure $f$ from states to reals should satisfy the inequality [@Marvianpra] $$\label{wysk2}
f(\Lambda_\mathrm{TI}[\rho])\leq f(\rho).$$ This answers why the WY skew information which measures asymmetry relative to the group of translations generated by an observable $H$ cannot serve as a measure of coherence, as there are IO which are not translationally invariant. For more detailed explanations about the relations between coherence and asymmetry, and a comparison of different notions of coherence, see the recent works of @Marvianpra and @Marvianpra2.
### Tight lower bounds
By adopting the inequality $$\label{wysk3}
{\mathrm{tr}}\{[\rho,K]^2\}\geq 2{\mathrm{tr}}\{[\rho^{1/2},K]^2\},$$ @meas8 derived the following lower bound of $\mathsf{I}(\rho,K)$, $$\label{eq2f-3}
\mathsf{I}(\rho,K)\geq \mathsf{I}^L (\rho,K)=-\frac{1}{4}{\mathrm{tr}}\{[\rho,K]^2\},$$ and demonstrated that it can be experimentally evaluated efficiently without tomographic state reconstruction of the full density matrix.
@bound also established a lower bound for the skew information. By discussing a dynamical process with the evolved state $\rho_\phi=U_\phi \rho_0 U_\phi^\dag$ and the observable $K_\phi$ generating its evolution, they derived $$\label{eq2f-4}
\mathsf{I}(\rho_\phi,K_\phi)\geq \frac{\hbar^2}{2}\bigg|\frac{\partial}
{\partial\phi}\cos[\mathcal{L}(\rho_0,\rho_\phi)]\bigg|^2,$$ where $\mathcal{L}(\rho_0,\rho_\phi)=\arccos[{\mathrm{tr}}(\rho_0^{1/2}
\rho_\phi^{1/2})]$ is the Hellinger angle, and $K_\phi$ is connected to the unitary operator $U_\phi$ via $$\label{eq2f-5}
K_\phi=-i\hbar U_\phi\frac{\partial U_\phi^\dag}{\partial\phi},$$ with $\phi$ being an arbitrary parameter encoded in $U_\phi$. Eq. indicates that the lower bound of the evolved WY skew information $\mathsf{I}(\rho_\phi,K_\phi)$ is determined by change rate of the distinguishability between the initial and the evolved states.
### Modified version of coherence measure
To avoid the problem occurred for $\mathsf{I}(\rho,K)$, @Yucsskw further proposed a similar definition of coherence measure still by using the WY skew information, which is very similar to the definition of quantum correlation measure $Q_A(\rho)$ given in Eq. . To be explicit, by denoting $\{|k\rangle\}$ the reference basis, the new coherence measure is defined as $$\label{eq2f-6}
C_{sk}(\rho)=\sum_k \mathsf{I}(\rho,|k\rangle\langle k|).$$ which has been proven to satisfy all the conditions for a bona fide measure of quantum coherence [@coher]. It can be linked to the task of quantum phase estimation. For the special case of single-qubit state, $C_{sk}(\rho)$ is also qualitatively equivalent to the asymmetry measure $\mathsf{I}(\rho,K)$ given by @meas8.
To calculate $C_{sk}(\rho)$ for a given state $\rho$, one can also use its equivalent form $$\label{yucs1}
C_{sk}(\rho)=1-\sum_k \langle k|\sqrt{\rho} |k\rangle^2,$$ which can be further written in a distance-based form $C_{sk}(\rho)=1- [\max_{\delta\in\mathcal {I}}f(\rho,\delta)]^2$, where $f(\rho,\delta)= {\mathrm{tr}}(\sqrt{\rho}\sqrt{\delta})$. The optimal $\delta^\star$ can be derived as $$\label{yucs2}
\delta^\star=\sum_k \frac{\langle k|\sqrt{\rho} |k\rangle^2}
{\sum_{k'}\langle k'|\sqrt{\rho} |k'\rangle^2}.$$ Moreover, by using Eq. and the inequality $\langle
k|\sqrt{\rho}|k\rangle \geq \langle k|\rho |k\rangle$, a connection between $C_{sk}(\rho)$ and the HS norm of coherence measure $C_{l_2}(\rho)$ given in Eq. can be established as follows $$\label{eq2f-7}
\frac{1}{2}C_{l_2}(\rho)\leq C_{sk}(\rho) \leq 1-{\mathrm{tr}}\rho^2 + C_{l_2}(\rho).$$ Since $C_{l_2}(\rho)$ is experimentally measurable, the above relation provides a way for estimating bounds of $C_{sk}(\rho)$.
Coherence of Gaussian states {#sec:3G}
----------------------------
In real experiments, there exists very relevant physical situations for which the systems under scrutiny are of infinite-dimensional (e.g., quantum optics states of light and Gaussian states). Hence, the characterization and quantification of coherence in these systems are also required.
### Coherence in the Fock space
A typical class of infinite-dimensional system is the bosonic system in the Fock space, which is describable using the Fock basis $\{|n\rangle\}_{n=0}^\infty$. Here, $|n\rangle$ is the eigenstate of the number operator $\hat{a}^\dag \hat{a}$, and $\hat{a}^\dag$ and $\hat{a}$ are the bosonic creation and annihilation operators.
By generalizing the set $\mathcal{I}$ of incoherent states as those with $\delta=\sum_{n=0}^\infty \delta_n |n\rangle\langle n|$, and incoherent operations described by the Kraus operators $\{K_n\}$ satisfying $\sum_{n=0}^\infty K_n^\dag K_n={\openone}$ and $K_n\mathcal{I}K_n^\dag \subset \mathcal{I}$, @gaus1 studied the problem of quantification of coherence in this system. For this purpose, they first proposed a new criterion that $C(\rho)$ should satisfy in order to circumvent the divergence problem of $C(\rho)$, which may be termed as the mean energy constraints,
(C5) If the first-order moment, the average particle number $\bar{n}=\langle \hat{a}^\dag \hat{a} \rangle$ is finite, it should fulfill $C(\rho)<\infty$.
Based on this new criterion, @gaus1 proved that the relative entropy of coherence in Eq. is also a proper coherence measure for the infinite-dimensional systems. But the $l_1$ norm of coherence of Eq. , despite its simple structure and intuitive meaning, does not satisfy the condition (C5), hence fails to serve as a proper measure of coherence for the infinite-dimensional systems.
Referring to the coherence measure in infinite-dimensional systems, one may also concern about the counterparts of Eqs. and , i.e., the incoherent state and the maximally coherent state. For the $d$-mode Fock space $\mathcal {H}=\otimes_
{i=1}^d \mathcal {H}_i$ with the basis $|\boldsymbol{n} \rangle=
\otimes_{i=1}^d |n_i\rangle$ and probability distributions $\{p_{\boldsymbol{n}}\}$, the incoherent state is given by [@gaus2] $$\label{eq2g-1}
\delta_{\rm th}(\boldsymbol{n})=\otimes_{i=1}^d \rho_{\rm
th}(\bar{n}_i),$$ where $\bar{n}_i=\langle \hat{a}_i^\dag\hat{a}_i\rangle$, and $\rho_{\rm th}(\bar{n}_i)$ is just the thermal state for the $i$th-mode Fock space, $$\label{eq2g-2}
\rho_{\rm th}(\bar{n}_i)=\sum_{n=0}^\infty \frac{\bar{n}_i^n}
{(\bar{n}_i+1)^{n+1}}|n\rangle\langle n|.$$ Moreover, the maximally coherent state is given by [@gaus1] $$\label{eq2g-3}
|\Psi_m^d\rangle=\sum_{\boldsymbol{n}} \frac{\bar{n}_t^{|\boldsymbol{n}|_1/2}}
{[(\bar{n}_t+1)^{|\boldsymbol{n}|_1+1}\mathbb{C}_{|\boldsymbol{n}|_1
+d-1}^{d-1}]^{1/2}}|\boldsymbol{n}\rangle,$$ where $|\boldsymbol{n}|_1=\sum_{i=1}^d n_i$, and $\bar{n}_t=\sum_
{\boldsymbol{n}}p_{\boldsymbol{n}}|\boldsymbol{n}|_1$ denotes the average total particle number which is finite. The corresponding maximal coherence is given by $$\label{eq2g-4}
C_r^{\max}=C_{r,d=1}^{\max}+\sum_{n=0}^{\infty}\frac{\bar{n}_t^n}
{(\bar{n}_t+1)^{n+1}}\log_2\mathbb{C}_{n+d-1}^{d-1},$$ with $$\label{eq2g-v4}
C_{r,d=1}^{\max}=(\bar{n}+1)\log_2(\bar{n}+1)-\bar{n}\log_2\bar{n},$$ being the maximal coherence for the single-mode case ($\bar{n}_t=\bar{n}$).
### Analytic formulas
A state is said to be Gaussian if its characteristic function $\chi(\rho,\lambda)={\mathrm{tr}}[\rho D(\lambda)]$ is Gaussian, where $D(\lambda)$ is the displacement operator. A Gaussian state is fully describable using the covariance matrix $\gamma$ (with entries $\gamma_{kl}$) and displacement vector $\vec{\upsilon}=(\upsilon_1,
\upsilon_2)^T$, $\rho=\rho(\gamma,\vec{\upsilon})$. The incoherent thermal state $\rho_{\rm th}(\bar{n})$ corresponds to $\gamma=
(2\bar{n}+1){\openone}$ and $\vec{\upsilon}=(0,0)^T$, where the superscript $T$ denotes transpose.
For the case of $d$-mode Gaussian states $\rho(\gamma,
\vec{\upsilon})$, by denoting $x_i=[\det \gamma^{(i)}]^{1/2}$ square of the determinant of the covariance matrix $\gamma^{(i)}$ for the $i$th mode, @gaus2 obtained analytical formula for the relative entropy of coherence, which is given by $$\label{eq2g-5}
\begin{aligned}
C_r(\rho)=&\sum_{i=1}^d \left(\frac{x_i-1}{2}\log_2\frac{x_i-1}{2}-\frac{x_i+1}{2}\log_2\frac{x_i+1}{2}\right)\\
&+\sum_{i=1}^d[(\bar{n}_i+1)\log_2(\bar{n}_i+1)-\bar{n}_i\log_2\bar{n}_i],
\end{aligned}$$ where $\bar{n}_i$ can be written in terms of the covariance matrix $\gamma^{(i)}$ and displacement vector $\vec{\upsilon}$ as $$\label{eq2g-6}
\bar{n}_i=\frac{1}{4}\{\gamma_{11}^{(i)}+\gamma_{22}^{(i)}+[\upsilon^{(i)}_1]^2+[\upsilon^{(i)}_2]^2-2\},$$ from which one can also see that the maximally coherent state should be pure, i.e., $x_i=1$, $\forall~i\in\{1,\cdots,d\}$.
### Coherence of coherent states
For the set $\{|\alpha\rangle\}$ of coherent states which spans an infinite-dimensional Hilbert space, a direct application of the resource theory of quantum coherence formulated by @coher is not applicable. This is because states of the considered set are not only overcomplete but also may be not linearly independent. To circumvent this perplexity, @continuous developed an orthogonalization procedure which allows for defining coherence measure for arbitrary superposition of coherent states, whether they are orthogonal or not.
For a given density operator $\rho_A$, let $\rho_{AB}^{(0)}=\rho_A
\otimes |0^B\rangle\langle 0^B|$ and $\rho_{AB}^{(i)}=
U_{\alpha^{(i)}} \rho_{AB}^{(i-1)} U_{\alpha^{(i)}}^\dagger$ ($i=1,
\ldots,N$). Then if one denotes $|\alpha^{(i)}\rangle$ for the coherent state admitting ${\mathrm{tr}}(|\alpha^{(i)}\rangle \langle
\alpha^{(i)}|\otimes |0^B\rangle\langle 0^B|\rho_{AB}^{(i-1)})
=\max_{\{|\alpha\rangle\}} {\mathrm{tr}}(|\alpha \rangle \langle \alpha
|\otimes |0^B\rangle\langle 0^B|\rho_{AB}^{(i-1)})$, the $N$th Gram-Schmidt unitary $U_{\mathrm{GS}}^{(N)}=U_{\alpha^{(N)}} \ldots
U_{\alpha^{(1)}}$, where the <span style="font-variant:small-caps;">cnot</span> type unitary is given by $$\label{eq2g-7}
\begin{aligned}
U_{\alpha^{(i)}}=& \mathbb{I}\otimes \mathbb{I}+|\alpha^{(i)}\rangle\langle \alpha^{(i)}|\otimes
(|\beta^{(i)}\rangle\langle 0^B|+|0^B\rangle\langle \beta^{(i)}| \\
&-|0^B\rangle\langle 0^B|-|\beta^{(i)}\rangle\langle \beta^{(i)}|).
\end{aligned}$$ Using the Gram-Schmidt unitary, @continuous defined the $N$ coherence for a general state $\rho_A$ as follows $$\label{eq2g-8}
C_{\alpha}(\rho_A, N)=\inf_{\rho_{AE}\in\mathcal {E}\atop U_{\mathrm{GS}}\in \mathcal{S}^{(N)}}
C(\Phi_{\mathrm{GS}}^{(N)}[\rho_{AE}]),$$ where $C$ denotes any faithful coherence measure, $\mathcal{S}^{(N)}$ denotes the full set of $N$th Gram-Schmidt unitaries, $\mathcal {E}=\{\rho_{AE}|{\mathrm{tr}}_E \rho_{AE}=\rho_A\}$ is the set of extensions of $\rho_A$, and $$\label{eq2g-9}
\Phi_{\mathrm{GS}}^{(N)}[\rho_A]=\frac{\Pi_{\mathrm{GS}}^{(N)}
[U_{\mathrm{GS}}^{(N)}(\rho_A\otimes |0^B\rangle\langle 0^B|)
U_{\mathrm{GS}}^{(N)\dagger}]\Pi_{\mathrm{GS}}^{(N)}}{{\mathrm{tr}}\{\Pi_{\mathrm{GS}}^{(N)}
[U_{\mathrm{GS}}^{(N)}(\rho_A\otimes |0^B\rangle\langle 0^B|)
U_{\mathrm{GS}}^{(N)\dagger}]\Pi_{\mathrm{GS}}^{(N)}\}},$$ with the projector $$\label{eq2g-10}
\Pi_{\mathrm{GS}}^{(N)}= \sum_{i=1}^N |\alpha^{(i)}\rangle\langle\alpha^{(i)}|
\otimes|\beta^{(i)}\rangle\langle\beta^{(i)}|.$$ Then, an $\varepsilon$-smoothed version of $N$ coherence can be written as $$\label{eq2g-11}
C_{\alpha}^\varepsilon(\rho_A, N,)=\inf_{\rho'_A\in \mathcal{B}_\varepsilon(\rho_A)}
C_{\alpha}(\rho'_A, N),$$ where $\mathcal{B}_\varepsilon(\rho_A)=\{\rho'_A|\frac{1}{2}
\|\rho'_A-\rho_A\|_1\leq\varepsilon \}$. Finally, the $\alpha$ coherence is defined as the smoothed $N$ coherence in the asymptotic limit, that is, $$\label{eq2g-12}
C_{\alpha}(\rho_A)= \lim_{\varepsilon\rightarrow 0}\lim_{N\rightarrow\infty}
C_{\alpha}^\varepsilon(\rho_A, N,).$$ @continuous proved that $C_{\alpha}(\rho_A)=0$ if and only if $\rho_A$ is a classical state, and it is also a nonclassicality measure.
Generalized coherence measures {#sec:3H}
------------------------------
When the constraints imposed by the axiomatic-like postulates of @coher are somewhat released, one may introduce other measures of quantum coherence that are physically relevant. These measures may also have potential applications under specific contexts.
### Basis-independent coherence measure
For a state $\rho$ in the $d$-dimensional Hilbert space, @frobe formulated the following basis-independent measure of quantum coherence $$\label{eq2h-1}
C_{BI}(\rho)=\sqrt{\frac{d}{d-1}}\|\rho-\rho_{\rm mm}\|_2,$$ which is proportional to the HS distance between $\rho$ and the maximally mixed state $\rho_{\rm mm}={\openone}_d/d$, and the parameter before $\|\cdot\|_2$ is introduced for normalizing $C_{\rm
free}(\rho)$.
The $C_{BI}(\rho)$ can be calculated analytically as $$\label{eq2h-2}
C_{BI}(\rho)= \sqrt{\frac{dP-1}{d-1}}
= \sqrt{\frac{d\mathcal{I}_{BZ}}{d-1}},$$ where $P={\mathrm{tr}}\rho^2$ is the purity of $\rho$, and $P-1/d$ equals to the Brukner-Zeilinger invariant information $$\label{eq2h-3}
\mathcal{I}_{BZ}(\rho)=\sum_{i=1}^m \sum_{j=1}^d \left[
{\mathrm{tr}}(\Pi_{ij}\rho)-\frac{1}{d}\right]^2,$$ thereby endows $C_{BI}(\rho)$ a clear physical meaning. Here, $\{\Pi_{ij}\}$ denote eigenvectors of the mutually complementary observables, for example, for the case of $d=2$, they are those of the Pauli operators $\sigma_x$, $\sigma_y$, and $\sigma_z$.
The coherence measure $C_{BI}(\rho)$ is unitary invariant (a trait that distinguishes it from other coherence measures), takes the maximum 1 for all the pure states, and is nonincreasing under the action of any unital channel $\Lambda_u$, i.e., $C_{BI}(\rho) \geq
C_{BI}(\Lambda_u[\rho])$. Moreover, it also provides loose lower bounds for the $l_1$ norm of coherence and trace norm of coherence, i.e., $$\label{eq2h-4}
\begin{split}
& C_{l_1}(\rho)\leq \sqrt{d(d-1)}C_{BI}(\rho),\\
& C_{tr}(\rho)\leq \sqrt{d-1}C_{BI}(\rho).
\end{split}$$ As the measures of quantum coherence is basis dependent, it is of particular interest to consider the maximum amount of coherence attainable by varying the reference basis, and define $$\label{maxcoh}
C^{\max}(\rho)=\max_{U} C(U\rho U^\dag),$$ with $C$ being any valid coherence measure and $\{U\}$ the set of unitary operations.
@Yucs investigated problems of such kind. For a $d$-dimensional state $\rho$, they proved $$\label{maxcoh2}
C_{r}^{\max}(\rho)=\log_2 d-S(\rho),~~
C_{l_2}^{\max}(\rho)={\mathrm{tr}}\rho^2-\frac{1}{d},$$ where $C_r(\rho)$ and $C_{l_2}(\rho)$ denote, respectively, the relative entropy and the HS norm measure of coherence. In particular, $C_{r}^{\max}(\rho)$ equals to the relative entropy between $\rho$ and the corresponding maximally mixed state $\rho_{\rm mm}$, and $C_{2}^{\max} (\rho)$ equals to the squared HS norm between $\rho$ and $\rho_{\rm mm}$. Both $C_{r}^{\max}(\rho)$ and $C_{2}^{\max}(\rho)$ take the maximum value for pure states and zero for incoherent states. They also possess preferable features of coherence measures such as: (+1) invariant under unitary operations; (+2) convexity under mixing of states; (+3) monotonicity under the unitary operations $\{U_i| \sum p_i U_i^\dag U_i={\openone}, U_i^\dag U_i={\openone}\}$ and SIO.
For some other widely used coherence measures including the RoC, the coherence weight, and the modified skew information measure of coherence, $C^{\max}(\rho)$ defined in Eq. can also be obtained analytically, which reads [@Husf] $$\label{maxcoh3}
\begin{aligned}
& C_R^{\max}(\rho)=d\lambda_{\max}-1,~~
C_w^{\max}(\rho)=1-d\lambda_{\min}, \\
& C_{sk}^{\max}(\rho)= 1-\frac{1}{d}\left( \sum_i \sqrt{\lambda_i}\right)^2.
\end{aligned}$$ where $\{\lambda_i\}$ denote the eigenvalues of $\rho$, with $\lambda_{\max}=\max\{\lambda_i\}$ and $\lambda_{\min}= \min
\{\lambda_i\}$.
@coh-pur also considered the maximal coherence achievable by performing unitary operations on a state $\rho$, with however a different set of free operations from that of @coher was used when defining $C(\rho)$. To be explicit, they used the set of MIO $\Lambda_{\mathrm{MIO}} [\delta]\in \mathcal{I}$ ($\forall \delta\in
\mathcal{I}$) instead of the traditional IO as the free operations, which was first suggested by @cof. Clearly, the set of IO is a subset of MIO, and thus any coherence monotone with respect to MIO is also a coherence monotone with respect to IO, but the inverse may not always be true, e.g., the $l_1$ norm of coherence and the coherence of formation is an IO monotone but not a MIO monotone [@ncgc].
Based on the above setting, @coh-pur showed that the following state $$\label{maxcohmio}
\rho_{\max}=\sum_{n=1}^d p_n |n_+\rangle\langle n_+|,$$ is a MCMS with respect to any MIO monotone, where $\{p_n\}$ is the probability distribution, and $\{|n_+\rangle\}$ denotes a MUB with respect to the incoherent basis $\{|i\rangle\}$, i.e., $|\langle
i|n_+\rangle|=1/d$, $\forall i, n_+$ [@mubs1; @mubs2]. The Eq. can be proved straightforwardly by noting that $\Lambda_{\mathrm{MIO}}[\rho_{\max}]=U\rho_{\max}U^\dag$ if one chooses Kraus operators of $\Lambda_{\mathrm{MIO}}$ as $K_n=U|n_+\rangle\langle n_+|$. This is because $C(U\rho_{\max}U^\dag)= C(\Lambda_{\mathrm{MIO}}[\rho_{\max}]) \leq
C(\rho_{\max})$, where the inequality is due to the monotonicity of a coherence measure under MIO. By the way, one can also show that $\Lambda_{\mathrm{MIO}}$ of this type yields $\Lambda_{\mathrm{MIO}}
[\delta]={\openone}/d$ and $\Lambda_{\mathrm{MIO}}[\rho]\in \rho_{\max}$.
Furthermore, when the coherence is measured by the shortest distance between $\rho$ and the set $\mathcal{I}$ of incoherent states \[see Eq. , with only the IO being replaced by MIO\], we have $$\label{maxc1}
C^{\max}(\rho)=C(\tilde{U}\rho \tilde{U}^\dag)\leq
\mathcal{D}(\tilde{U}\rho\tilde{U}^\dag,{\openone}/d)=
\mathcal{D}(\rho,{\openone}/d),$$ where $\tilde{U}$ is the optimal unitary for obtaining $C^{\max}(\rho)$, and the inequality comes from the fact that ${\openone}/d$ is not necessary the closest incoherent state to $\tilde{U}\rho \tilde{U}^\dag$, while the last equality is due to the unitary invariance of $\mathcal{D}$. Moreover, by denoting $\Delta_+$ the full dephasing of $\rho$ in the maximally coherent basis $\{n_+\rangle\}$ \[see the definition in Eq. \], one can obtain $\Delta_+[\rho_{\max}]=\rho_{\max}$ and $\Delta_+[\delta] ={\openone}/d$. Thus by denoting $\tilde{\delta}$ the closest incoherent state to $\rho_{\max}$, we have $$\label{maxc2}
\begin{split}
C^{\max}(\rho)&\geq C(\rho_{\max})=\mathcal{D}(\rho_{\max},\tilde{\delta})\\
&\geq \mathcal{D}(\Delta_+[\rho_{\max}],\Delta_+[\tilde{\delta}])\\
&= \mathcal{D}(\rho_{\max},{\openone}/d)= \mathcal{D}(\rho,{\openone}/d),
\end{split}$$ then by combining the above two equations, one can obtain $$\label{maxcohana}
C^{\max}(\rho)=C(\rho_{\max})=\mathcal{D}(\rho,{\openone}/d),$$ for any contractive distance measure $\mathcal{D}$ of two states.
When $\rho$ is incoherent, i.e., $\rho=\delta$, Eq. corresponds to the maximum amount of coherence (quantified by any faithful measures) generated from a given incoherent state $\delta$. By focusing on the two-qubit states only, and arranging $\delta$’s diagonal elements as $\delta_1\leq \delta_2\leq \delta_3\leq
\delta_4$ and taking the relative entropy as a measure of coherence, @Sun obtained solutions of Eq. for specified types of $U$. They are $$\label{eq5b-2}
\begin{aligned}
& C^{\rm max}_{r,1}=1+H(\delta_1+\delta_3)-\sum_i \delta_i\log_2\delta_i,\\
& C^{\rm max}_{r,2}=2-\sum_i \delta_i\log_2\delta_i,
\end{aligned}$$ where $C^{\rm max}_{r,1}$ ($C^{\rm max}_{r,2}$) is the optimal coherence created under local unitaries $U_A\otimes {\openone}$ ($U_A\otimes U_B$), with the corresponding $$\begin{aligned}
\label{eq5b-v2}
U_A=U_B=|+\rangle\langle 1|+|-\rangle\langle 0|,\end{aligned}$$ where $|\pm\rangle=(|0\rangle\pm |1\rangle)/\sqrt{2}$, and $H(\cdot)$ is the binary Shannon entropy. For the kernel $U_d$ of nonlocal $U$ in the Cartan decomposed form, @Sun found that they cannot outperform the local unitaries on creating quantum coherence, but it remains open if this is also true for more general nonlocal $U$.
### Genuine quantum coherence
The core for the resource theory of quantum coherence is incoherent states and incoherent operations. @gqc introduced a slightly different types of incoherent operations which he called genuinely incoherent operations (GIO). These operations preserve all incoherent states, i.e., $$\label{GIO}
\Lambda_{\rm gi}(\delta)=\delta,$$ for all $\delta\in\mathcal{I}$. This definition of GIO ensures their independency on the explicit forms of the Kraus decompositions.
It has been proved that the Kraus operators of $\Lambda_{\rm gi}$ are diagonal in the prefixed reference basis, and $$\label{GIO2}
\Lambda_{\rm gi}(\cdot)=\sum_k p_k U_k (\cdot) U_k^\dag,$$ for the single-qubit case, with $\{p_k\}$ being the probability distribution and $U_k=\sum_l e^{i\phi_{lk}}|k\rangle\langle k|$ the unitary. Moreover, the GIO includes the nondegenerate thermal operations $\Lambda_{\rm th}$ as a special case, while itself belongs to the set of translationally invariant operations $\Lambda_{\rm ti}$ [@gqc]. Given a system state $\rho^S$ with the Hamiltonian $\hat{H}_S$, and the thermal state $\rho_{\rm th}^E=
e^{-\beta\hat{H}_E}/{\mathrm{tr}}e^{-\beta\hat{H}_E}$ with $\hat{H}_E$ the environmental Hamiltonian and $\beta=1/T$ the inverse temperature, $\Lambda_{\rm th}$ and $\Lambda_{\rm ti}$ are defined by $$\label{eq2h-5}
\begin{aligned}
& \Lambda_{\rm th}(\rho^S)={\mathrm{tr}}_E (U\rho^S\otimes \rho_{\rm th}^E U^\dag),\\
& \Lambda_{\rm ti}(e^{-i\hat{H}_S t}\rho e^{i\hat{H}_S t})=e^{-i\hat{H}_S t}
\Lambda_{\rm ti}(\rho)e^{i\hat{H}_S t},
\end{aligned}$$ where $U$ is the unitary which preserves the total energy of the considered system plus its environment.
Under the set of GIO, @gqc proposed the prerequisites for a function to be a genuine coherence measure. They are analogous to those labeled as (C1), (C2a), (C2b), and (C3), with only the IO being replaced by the GIO. @gqc called a measure respecting the first three prerequisites a genuine coherence monotone. As the GIO is a strict subset of the general IO, the $l_1$ norm, relative entropy, and intrinsic randomness measures of coherence, as well as the RoC are all genuine coherence monotones.
Apart from the above measures, the WY skew information $\mathsf{I}(\rho,K)$ obeys (C1), (C2a), and (C3). The distance-based measure of genuine coherence $$\label{eq2h-6}
G_p(\rho)= \min_{\delta\in\mathcal{I}} \|\rho- \delta\|_p,$$ also obeys these three conditions, and for the special case of $p=2$, $G_2(\rho)=\|\rho- \Delta(\rho)\|_2$, where the closest incoherent state is $\Delta(\rho)=\sum_i \langle i|\rho|i\rangle
|i\rangle\langle i|$. For other cases, $\Delta(\rho)$ is not the closest state for obtaining $G_p(\rho)$, but @gqc showed that $$\label{eq2h-7}
\tilde{G}_p(\rho)= \|\rho- \Delta(\rho)\|_p,$$ is also a valid genuine coherence measure, as it obeys the conditions (C1), (C2a), and (C3).
### Quantification of superposition
As a generalization of the resource theories of coherence, @superposition introduced a similar framework for quantifying superposition. In their framework, the set $\mathcal{F}$ of free states is comprised of the states that can be represented as statistical mixtures of linear independent (not necessarily orthogonal) basis states $\{|c_i\rangle\}_{i=1}^d$. To be explicit, these superposition-free states are given by $$\label{eq2h-8}
\varsigma=\sum_{i=1}^d \varsigma_i |c_i\rangle\langle c_i|,$$ where $\varsigma_i\geq 0$ and $\sum_i \varsigma_i=1$. Those states that are not free are called superposition states. Similarly, the quantum operation $\Phi(\rho)=\sum_i K_i \rho K_i^\dagger$ is said to be superposition-free if the Kraus operator gives the map $K_i\varsigma K_i^\dagger/{\mathrm{tr}}(K_i\varsigma K_i^\dagger) \in \mathcal
{F}$ ($\forall K_i$), that is, every $K_i$ (hence $\Phi$) maps the superposition-free state to another superposition state. @superposition showed that such a $K_i$ is of the following general form: $$\label{eq2h-9}
K_i=\sum_k c_i(k)|c_{f_i(k)}\rangle\langle c_k^\bot|,$$ where $|c_k^\bot\rangle$ are called reciprocal states which satisfy $\langle c_k^\bot|c_l\rangle=\delta_{kl}$, $c_i(k)$ are coefficients, and $f_i(k)$ are index functions.
@superposition presented the defining conditions for a faithful superposition measure $M(\rho)$. These conditions are very similar to those for a faithful coherence measure [@coher]. The difference is that $\Lambda(\rho)$ and $\delta$ in (C1), (C2a), (C2b), and C(3) were replaced by $\Phi(\rho)$ and $\varsigma$, respectively. Then, in a similar manner to the definition of $C_{\mathcal{D}} (\rho)$ in Eq. , one can define $$\label{eq2h-10}
M_{\mathcal{D}}(\rho)= \min_{\varsigma\in\mathcal{F}}D(\rho,\varsigma).$$ For explicit distance measures, @superposition proved the superposition measures including the relative entropy of superposition, the $l_1$ norm of superposition, and the robustness of superposition. They are similar to the coherence measures $C_r(\rho)$, $C_{l_1}(\rho)$, and $C_R(\rho)$ defined respectively, in Eqs. , , and . Apart from these, @superposition also proved the rank-measure of superposition $$\label{eq2h-11}
M_\mathrm{rank}(\rho)=\min_{\{p_i,\psi_i\}}\sum_i p_i \log[r_S(\psi_i)],$$ where the superposition rank $r_S(\psi_i)$ is the number of nonzero $\alpha_n^{(i)}$ for $|\psi_i\rangle=\sum_n \alpha_n^{(i)}
|c_n\rangle$, and the minimization is taken over all pure state decompositions of $\rho$ showed in Eq. .
Before ending this section, we remark here that while various coherence measures have been introduced, @tongdm proposed a proposal for estimating their values with limited experimental data available. Their approach is based on the optimization of a Lagrangian function and the limited expectation value of certain Hermitian operators, and can be applied to any coherence measure $C(\rho)$ that is continuous and convex.
Interpretation of quantum coherence {#sec:4}
===================================
Coherence is not only a basic feature signifying quantumness in an integral system, but also a common prerequisite for different forms of quantum correlations when composite systems are considered. Apart from its characterization and quantification, it has also been shown to be intimately related to many other quantities manifesting quantumness of states, and fundamental problems of quantum mechanics such as complementarity and uncertainty relations. All these have triggered the community’s interest in investigating it from different perspectives, which endows quantum coherence clear operational interpretations and physical meanings.
Coherence and quantum correlations {#sec:4A}
----------------------------------
In the seminal work of @coher and the subsequent stream of works, the quantum coherence measures are defined for single systems. Contrary to it, the traditional manifestation of quantumness for a system, e.g., quantum correlations, are defined in a scenario which involves at least two parties. In fact, both quantum coherence and quantum correlations arise from the superposition principle of quantum mechanics, hence it is essential to study the interrelation between them. The main progresses up to now are summarized in Fig. \[fig:coqd\], and we review them in detail in the following.
### Coherence and entanglement
@meas2 made a first step toward the above problem. By considering the setting where a coherent state $\rho^S$ is coupled to an incoherent ancilla initially in the vacuum state $|0^A\rangle$, they showed that the generated entanglement between $S$ and $A$ is upper bounded by the coherence of $S$. The bound can be saturated for certain contractive distance measures, hence yields a natural family of entanglement-based coherence measures, see Sec. \[sec:2B\] for more detail.
@gaoyan considered a very similar problem to that of @meas2. They used the coherence concurrence and entanglement concurrence, and found that the generated entanglement concurrence from the initial state $\rho^S\otimes |0^A\rangle\langle 0^A|$ is upper bounded by $$\label{qi01}
C_E(\Lambda^{SA}[\rho^S\otimes|0^A\rangle\langle 0^A|])\leq
C_{con}(\rho^S),$$ and when $\rho^S$ is a two-qubit state while the ancilla $A$ is also a qubit, the above equality is saturated. Moreover, by applying the generalized <span style="font-variant:small-caps;">cnot</span> gate of Eq. , they also found a lower bound of the created entanglement $$\label{qi02}
C_E(\Lambda^{SA}[\rho^S\otimes|0^A\rangle\langle 0^A|])\geq \sqrt{\frac{2}{d(d-1)}}
C_{\rm con}(\rho^S).$$ Apart from the link to entanglement, @coqd introduced the concept of correlated coherence, and argued that it can be connected to QD and entanglement. Their key idea is by distinguishing the coherence in $\rho^{AB}$ as local and nonlocal, i.e., by dividing the total coherence into two different portions: those stored locally in the subsystems, and those stored only in the correlated states. Based on this starting point, they defined the correlated coherence as $$\label{eq3a-1}
C_{cc}(\rho^{AB})=C(\rho^{AB})-C(\rho^A)-C(\rho^B),$$ which is a nonnegative quantity.
By choosing tensor products of the eigenvectors $\{|i_A\rangle\}$ (for $\rho^A$) and $\{|j_B\rangle\}$ (for $\rho^B$) as reference basis (for degenerate case, they will chosen to be those minimize $C_{cc}$), $C_{cc}(\rho^{AB})=0$ if and only if $\rho^{AB}\in
\mathcal{CC}$. Similarly, $C_{cc}(\rho^{AB})= C_{cc}
(\Delta^A[\rho^{AB}])$ if and only if $\rho^{AB}\in\mathcal{CQ}$. Based on these observations, @coqd defined $$\label{eq3a-2}
E_{cc}(\rho^{AB})\coloneqq \min C_{cc}(\rho^{AA'BB'}),$$ and showed that it possesses the preferable properties for an entanglement monotone. Here, the minimization is taken over the full set of unitarily symmetric extensions of $\rho^{AB}$ satisfying $$\label{eq3a-3}
U_{AA'}\otimes U_{BB'} (U_{\rm SWAP}\rho^{AA'BB'}U_{\rm SWAP}^\dag)
U_{AA'}^\dagger\otimes U_{BB'}^\dagger= \rho^{AA'BB'},$$ where $\rho^{AB}={\mathrm{tr}}_{A'B'} \rho^{AA'BB'}$ for all the local unitaries $U_{AA'}$ and $U_{BB'}$, and $U_{\rm SWAP}$ is the swap operator.
@NAQC examined the steered coherence from another perspective. In their framework, Alice and Bob hold respectively, qubit $A$ and $B$ of $\rho^{AB}$, and agree on the observables $\{\sigma_1, \sigma_2,\sigma_3\}$ in advance. Alice then measures $\sigma_i$ on her qubit and informs Bob of her choice $\sigma_i$ and outcome $a\in\{0,1\}$. Bob computes the coherence of his conditional states $\{p(a|\sigma_i), \rho_{B|\sigma_i^a}\}$ in the eigenbasis of either $\sigma_j$ or $\sigma_k$ ($j,k\neq i$) randomly, which can be written as $\sum_a p(a|\sigma_i) C^{\sigma_j}(\rho_{B|\sigma_i^a})$. Here, $p(a|\sigma_i)$ is the probability for Alice’s outcome $a$ when she measures $\sigma_i$, and $\rho_{B|\sigma_i^a}$ is the corresponding postmeasurement state of $B$. By averaging over Alice’s possible measurements and Bob’s allowable eigenbasis sets, one can obtain $$\label{eq-hu3}
\begin{aligned}
& C_{l_1}^{na}(\rho^{AB})= \frac{1}{2}\sum_{i,j,a\atop i\neq j} p(a|\Pi_i) C^{\sigma_j}_{l_1}(\rho_{B|\sigma_i^a}),\\
& C_{r}^{na}(\rho^{AB})= \frac{1}{2}\sum_{i,j,a\atop i\neq j} p(a|\Pi_i) C^{\sigma_j}_{re}(\rho_{B|\sigma_i^a}).
\end{aligned}$$ As for any single-partite state $\rho$, we have $$\label{eq-hu4}
\begin{aligned}
& \sum_{j=1}^3 C^{\sigma_j}_{l_1}(\rho)\leq \sqrt{6},\\
& \sum_{j=1}^3 C^{\sigma_j}_{r}(\rho)\leq C_2^m
=3H(1/2+\sqrt{3}/6),
\end{aligned}$$ it is said that a nonlocal advantage of quantum coherence is achieved on $B$ when $C_{l_1}^{na}(\rho^{AB})>\sqrt{6}$ or $C_r^{na}(\rho^{AB}) >C_2^m$. @NAQC showed that any two-qubit state that can achieve a nonlocal advantage of quantum coherence is quantum entangled. Moreover, the interplay between nonlocal advantage of quantum coherence and Bell nonlocality was also established for two-qubit states [@NAQC1].
The above framework was extended to $(d\times d)$-dimensional state $\rho^{AB}$, in which the Pauli observables are replaced by the set of mutually unbiased observables $\{A_i\}$ [@NAQC2]. Now, the average coherence for Bob’s conditional states is $$\label{eq-hu5}
C^{na}(\rho_{AB})= \frac{1}{d}\sum_{i,j,a \atop i\neq j} p(a|A_i)
C^{A_j}(\rho_{B|A_i^a}),$$ and $C^{na}(\rho_{AB})>C^m$ captures the existence of nonlocal advantage of quantum coherence in state $\rho_{AB}$. When one adopts the $l_1$ norm of coherence and relative entropy of coherence, the state-independent bound $C^m$ is given by $(d-1)\sqrt{d(d+1)}$ and $(d+1)\log_2{d}- {(d-1)^2 \log_2(d-1)}/ {d(d-2)}$, respectively [@NAQC2].
Similarly, one can also formulate other framework for capturing the nonlocal advantage of quantum coherence in a state, e.g., after Alice executing one round of measurements and announced her choice $A_i$ and outcomes $a$, Bob can measure the coherence of his conditional states only in the preagreed basis spanned by the eigenvectors of $A_{\alpha_i}$, with $\{\alpha_i\}$ being any one of the possible permutations of the elements of $\{i\}$. This can give some new insights on the interrelation between coherence and quantum correlations [@NAQC2].
### Coherence and quantum discord
Since the coherence measures reviewed in Sec. \[sec:3\] are basis dependent, they can be changed by unitary operations. Based on this consideration, @Sun introduced a basis-free coherence measure of the following form $$\label{eq3b-1}
C^{\rm free}(\rho)=\min_{\vec{U}} C(\bm{U}\rho\bm{U}),$$ where $\bm{U}=U_1\otimes U_2\otimes\cdots \otimes U_N$ for a $N$-partite state $\rho$. It is in fact the minimum coherence created by local unitary operations.
By putting the measures of coherence and QD on an equal footing, that is, to quantify the both via relative entropy, @Sun found that $C^{\rm free}(\rho)$ defined above equals to the QD $D_r(\rho)= \min_{\chi\in\mathcal{C}} S(\rho\|\chi)$ ($\mathcal {C}$ is the set of classical states) introduced by @reqd, i.e., $$\label{eq3b-v1}
C^{\rm free}(\rho) =D_r(\rho),$$ thereby establishes a direct connection between coherence of a $N$-partite state in the product basis $\{|\bm{i} \rangle \coloneqq
\otimes_{k=1}^N |i_k\rangle\}$ (with $\{|i_k\rangle\}_{i=1}
^{d_k}\}$ and $d_k=\dim\mathcal{H}_k$) and QD of this state with the same multipartite divisions.
For bipartite states $\rho$ and local von Neumann measurements $\{\Pi_k^A\}$, @ulrc1 found that the difference between relative entropy of coherence for $\rho$ and the postmeasurement state $\rho'=\sum_k (\Pi_k^A \otimes {\openone}_B)\rho(\Pi_k^A\otimes
{\openone}_B)$ equals to the one-way quantum deficit [@def1; @def2], i.e., $$\label{eq3b-v2}
C_r(\rho)- C_r(\rho')=\Delta^{\rightarrow}(\rho).$$ For a system with $N$-partite division $S=\{S_1,S_2, \cdots,S_N\}$ and product reference basis $\{|\bm{i}\rangle\}$, one can define the sets of $S_k$-incoherent states $\mathcal{I}_{S_k}$ with elements $\delta^{S_k}=\sum_i p_i |i_k\rangle\langle i_k|\otimes
\rho_{\tilde{S}|k}$ ($\tilde{S}=S-S_k$), incoherent operations $\{\Lambda^{S_k}\}$ which map $\mathcal{I}_{S_k}$ to itself, and coherence measure $C_{\mathcal{D}}^{S_k}(\rho)= \min_{\delta\in
\mathcal{I}_{S_k}} \mathcal {D}(\rho,\delta)$, all with respect to the local basis $\{|i_k\rangle\}$. When $\mathcal{D}$ is the relative entropy, we have $$\label{eq3b-2}
C_r^{S_k}(\rho)= S(\rho\|\Delta^{S_k}[\rho]),$$ where $\Delta^{S_k}[\rho]$ is the full dephasing of $\rho=\rho^{S_1
S_2\cdots S_N}$ in the basis $\{|i_k\rangle\}$ of party $S_k$. The GQD $D(\rho)\coloneqq \min_{\chi\in \mathcal{C}} \mathcal{D}
(\rho,\chi)$, and for $\mathcal{D}$ to be the relative entropy, $D(\rho)= \min_{\Delta} S(\rho\|\Delta[\rho])$, with $\Delta=\otimes_{k=1}^N \Delta^{S_k}$ [@RMP]. Moreover, the global discord $$\label{eq3b-n2}
\bar{D}(\rho)=\min_{\Delta} [S(\rho\|\Delta[\rho])- \sum_k
S(\rho^{S_k}\|\Delta^{S_k}[\rho^{S_k}]),$$ and the usual asymmetric discord $$\label{eq3b-n3}
D_{\tilde{S}|S_k}(\rho)=\min_{\Delta^{S_k}}[S(\rho\|
\Delta^{S_k}[\rho])-S(\rho^{S_k}\|\Delta^{S_k}[\rho^{S_k}])].$$ By considering an analogous setting to that constructed by @meas2, i.e., the system $S$ and an incoherent ancilla is prepared initially in the product state $\rho^S\otimes
|0^A\rangle\langle 0^A|$, @Mile studied, from the perspective of coherence consumption and discord generation, the interplay between quantum coherence and QD. First, they found that $$\label{eq3b-3}
D(\Lambda^{SA}[\rho^S\otimes\rho^A])\leq C_{\mathcal {D}}(\rho^S),$$ for any contractive measure of $\mathcal{D}$, i.e., the generated discord is upper bounded by the initial coherence in $\rho^S$. In particular, if $d_A\geq d_S$, the equality holds for $\mathcal{D}$ to be the relative entropy or the Bures distance. Second, for $\rho^{S_1 S_2\cdots S_N}=\otimes_{k=1}^N \rho^{S_k}$, the sum of coherence consumed for all subsystems bounding the amount of global discord that can be generated by IO, i.e., $$\label{eq3b-4}
\bar{D}(\Lambda[\rho^{S_1 S_2\cdots S_N}])\leq \sum_k \delta C_r(\rho^{S_k}),$$ where $\delta C_r =C_r^i-C_r^t$, with $C_r^i$ ($C_r^t$) being coherence of the state prior to the measurement (after the measurement). Similarly, for the asymmetric discord, $$\label{eq3b-as4}
D_{S_2|S_1} (\Lambda^{S_1}[\rho^{S_1 S_2}])\leq \delta
C(\rho^{S_1}),$$ for $\rho^{S_1S_2}=\rho^{S_1}\otimes\rho^{S_2}$.
In a recent paper, @RQC introduced the concept of relative quantum coherence (RQC), which is the coherence of one state in the reference basis spanned by the eigenvectors of another one. To be explicit, for $\rho$ and $\sigma$ in the same Hilbert space $\mathcal {H}$, and the eigenvectors of $\sigma$ being given by $\{|\psi_i \rangle\}$, with the corresponding eigenvalues $\{\epsilon_i\}$, the RQC is given by $$\begin{aligned}
\label{eq-hu1}
C(\rho,\sigma)= C^{\Xi}(\rho),\end{aligned}$$ where $C^{\Xi}(\rho)$ denotes any *bona fide* measure of quantum coherence defined in the reference basis $\Xi$.
When the quantum coherence is measured by the $l_1$ norm, they showed that the QD $D_A(\rho_{AB})$ is bounded from above by the discrepancy between the RQC for the total system and that for the subsystem to be measured in the definition of QD, that is $$\label{eq-hu2}
D_A(\rho_{AB}) \leqslant C_{\rm re}(\rho_{AB},\tilde{\rho}_{\mathbb{PQ}})
-C_{\rm re}(\rho_A,\tilde{\rho}_\mathbb{P}),$$ where $\tilde{\rho}_{\mathbb{PQ}}$ denotes the optimal postmeasurement state for obtaining the QD, and $\tilde{\rho}_\mathbb{P}$ is the reduced state of $\tilde{\rho}_{\mathbb{PQ}}$. This upper bound can also be saturated when the state $\rho_{AB}$ is quantum-classical correlated.
Similarly, for the symmetric QD $D_{s}(\rho_{AB})= I(\rho_{AB})-
I(\tilde{\rho}_\mathbb{PQ})$ defined via two-sided optimal measurements $\{\tilde{\Pi}_k^A \otimes \tilde{\Pi}_l^B\}$ [@qdsym1; @qdsym2], it was further showed that [@RQC] $$\label{eq3-9}
\begin{split}
D_{s}(\rho_{AB}) = C_{\rm re}(\rho_{AB},\tilde{\rho}_\mathbb{PQ})
-C_{\rm re}(\rho_A,\tilde{\rho}_\mathbb{P})
-C_{\rm re}(\rho_B,\tilde{\rho}_\mathbb{Q}),
\end{split}$$ which implies that $D_{s}(\rho_{AB})$ is nonzero if and only if there exists RQC not localized in the subsystems. This establishes a direct connection between the RQC discrepancy and the symmetric discord.
For a bipartite system $AB$ described by the density operator $\rho^{AB}$, @steer considered the maximum amount of coherence created at party $B$ by the procedure of steering on $A$, and defined the steered coherence $$\label{eq5b-4}
C_{\rm str}(\rho^{AB})\coloneqq \inf_{e^B}[\max_{E^A} C_{l_1}(e^B,\rho^B_i)],$$ where $E^A=\{E_k^A\}$ represents the set of POVM operators, and $\rho^B_i={\mathrm{tr}}_A (E_i^A\otimes{\openone}_B\rho^{AB})/p_i$, $p_i={\mathrm{tr}}(E_i^A\otimes{\openone}_B\rho^{AB})$. The infimum over the eigenbasis $e^B=\{e_k^B\}$ of $\rho^B$ is necessary only when it is degenerate. The motivation for the definition of $C_{\rm str}(\rho^{AB})$ is very similar to the concept of localizable entanglement which is indeed the maximum entanglement that can be localized, on average, between two parties of a multipartite system, by performing local measurements on the other parties [@localent1; @localent2].
The steered coherence is showed to have several preferable properties of $C_{\rm str}(\rho^{AB})$, e.g., it vanishes when $\rho^{AB}$ is quantum-classical correlated, takes the maximum for all pure entangled states with full Schmidt rank $d_B$, and is locally unitary invariant. Moreover, $C_{\rm str}(\rho^{AB})$ may be increased by the local operations $\Lambda_B$ on $B$ prior to the steering on $A$. For two-qubit states, this is achievable if and only if $\Lambda_B$ is neither unital nor semi-classical. All these properties are very similar to those of the various discordlike quantum correlation measures [@RMP].
For two-qubit states $\rho^{AB}$, the maximum steered coherence is given by [@steer] $$\label{eq5b-5}
C_{\rm str}(\rho^{AB})=\inf_{\vec{n}_B\in \mathbb{R}^3,|\vec{n}_B|=1}
\left\{\max_{\vec{m}\in \mathbb{R}^3,|\vec{m}|=1}
\bigg|\frac{R^T\vec{m}\times\vec{n}_B}{1+\vec{a}\cdot \vec{m}}\bigg|\right\},$$ where $R$ is a $3\times 3$ matrix with elements $R_{ij}={\mathrm{tr}}\rho^{AB}(\sigma_i\otimes\sigma_j)$, $\vec{m}$ is a vector related to the POVM $E^A=({\openone}+\vec{m}\cdot \vec{\sigma})/2$, $\vec{a}={\mathrm{tr}}(\rho^A\vec{\sigma})$ is the local Bloch vector of $\rho^A$ (similarly for $\vec{b}$), and $\vec{n}_B=\vec{b}/
|\vec{b}|$.
### Coherence and measurement-induced disturbance
Based on quantum steering, @Huxy introduced the steering-induced coherence, and explored its connection with a quantum correlation measure known as measurement-induced disturbance [@MID]. For a bipartite system described by density operator $\rho^{AB}$ and shared by two players Alice and Bob, if Alice performs a local quantum measurement $\Xi^A=\{|\xi^A_i\rangle\langle
\xi_i^A|\}$ on her subsystem, she will obtain the outcome $i$ with probability $p_i={\mathrm{tr}}(\Xi_i^A\otimes{\openone}_B\rho)$, and Bob’s subsystem is steered to $\rho_i^B={\mathrm{tr}}_A(\Xi_i^A\otimes{\openone}_B
\rho)/p_i$. After multi-rounds of measurements, Bob will have the ensemble $\{p_i,\rho_i^B\}$. see Fig. \[fig:coqd\](e) for an illustration of the scheme
From the above scheme, @Huxy defined the steering-induced coherence as $$\label{eq3c-1}
\bar{C}(\rho^{AB})=\inf_{e^B}[\max_{\Xi^A}\sum_i p_i C(e^B,\rho^B_i)],$$ where Bob’s reference basis is chosen to be eigenbasis $e^B=\{|e_k^B\rangle\}$ of $\rho^B$, and the infimum in Eq. is necessary only when $\rho^B$ is degenerate. $\bar{C}(\rho)$ characterizes Alice’s ability to steer coherence on Bob’s side, and has also been proven satisfying the necessary requirements of a faithful coherence measure.
Based on the observation that the symmetric measurement-induced disturbance equals to coherence of $\rho^{AB}$ in the tensor-product eigenbasis of $\rho^A$ and $\rho^B$, @Huxy further considered the asymmetric measurement-induced disturbance $$\label{eq3c-a1}
Q_B(\rho^{AB})=\inf_{E^B}\mathcal{D}(\rho^{AB}, E^B[\rho^{AB}]),$$ with $E^B=\{|e_k^B\rangle\langle e_k^B|\}$ the locally invariant projective measurements on $B$. Their results show that when being quantified by the same distance measure, $\bar{C}(\rho^{AB})$ is bound from above by $Q_B(\rho^{AB})$, i.e., $$\label{eq3c-2}
\bar{C}(\rho^{AB})\leq Q_B(\rho^{AB}),$$ and the equality holds for the maximally correlated state $$\label{eq-mcs}
\rho_{\rm mc}= \sum_{i,j}\rho_{ij}|ii\rangle\langle jj|,$$ when the relative entropy quantifiers of them are adopted, for which they both equal to $S(\rho^B_{\rm mc})-S(\rho^{AB}_{\rm mc})$. Moreover, for the two-qubit state and the $l_1$ norm quantifiers, the upper bound is also saturated.
@MIAC considered a very similar coherence steering scheme to that of @Huxy. The difference is that they used the computation basis $\{|i\rangle\}_{i=1}^d$, and discussed the amount of coherence gain with classical correlation of the premeasurement state $\rho^{AB}$. They defined the measurement-induced average coherence $\bar{C}_r^P$ and measurement-induced average total coherence $\bar{C}^T_r$ as $$\label{eq-mia1}
\begin{aligned}
& \bar{C}^P_r(\rho^B)=\sum_i p_i C_r(\{|i\rangle\},\rho^B_i),\\
& \bar{C}^T_r(\rho^B)=\sum_i p_i C^{\max}_r(\rho^B_i),
\end{aligned}$$ where $C^{\max}_r(\rho^B_i)$ is the maximal attainable relative entropy of coherence under generic basis, see Eq. . The corresponding coherence gain are given by $$\label{eq-mia2}
\begin{aligned}
& \Delta C_r^P= \bar{C}^P_r(\rho^B)-C_r(\rho^B),\\
& \Delta C_r^T=\bar{C}^T_r(\rho^B)-C_r^T(\rho^B).
\end{aligned}$$ Based on these definitions, they found that $$\label{eq-yucs}
\Delta C_r^P\leq \Delta C_r^T,$$ which can be seen from $\Delta C_r^P-\Delta C_r^T=\sum_i p_i
S((\rho_i^B)_\mathrm{diag})- S(\rho^B_{\mathrm{diag}})\geq 0$. Moreover, the two coherence gains are proved to be upper bounded by the classical correlation (with respect to subsystem $A$) present in the premeasurement state $\rho^{AB}$, i.e., $$\label{eq-yucs2}
\max\{\Delta C_r^P,\Delta C_r^T\}\leq J_A(\rho^{AB}),$$ where $J_A(\rho^{AB})\coloneqq S(\rho^B)-\min_{\{E_k^A\}}\sum_k q_k
S(\rho_{B|E_k^A})$, and $\{E_k^A\}$ represents local positive operator valued measurements for defining classical correlation and QD [@qd02]. Here, we provide a slightly different proof of the above equation from @MIAC. First, $\Delta C_r^T \leq
J_A(\rho^{AB})$ can in fact be obtained directly from $\Delta
C_r^T=S(\rho^B)- \sum_i p_i S(\rho_i^B)$ and the ensemble $\{p_i,S(\rho_i^B)\}$ obtained with the measurement operators $\Xi^A$ may not be optimal for attaining $J_A(\rho^{AB})$. Second, $\Delta C_r^P \leq J_A(\rho^{AB})$ is due to Eq. .
Moreover, when a maximization process is performed over all possible $\Xi^A$, just like that of Eq. , the statements in the above two equations still holds. The optimized coherence gain $\Delta C_r^P$ equals zero if and only if $\rho^{AB}=\sum_i
A_{ii}\otimes|i^A\rangle \langle i^A|$ \[cf. Eq. for the meaning of $A_{ij}$\] or a product state, while $\Delta C_r^T$ equals zero if and only if $\rho^{AB}=\rho^A\otimes \rho^B$.
### Distribution of quantum coherence
The distribution of quantum coherence among the subsystems of a multipartite system is also an interesting research direction. In fact, for quantum correlation measures such as entanglement and QD, similar problems have been studied via various monogamy inequalities, see, e.g., the works of @mono1 [@mono2; @mono3]. For different coherence measures, one can also derive monogamy inequalities that impose limits on their shareability among multipartite systems. For example, for the $l_1$ norm of coherence, it is direct to show that $$\label{monoga}
C_{l_1}(\rho^{A_1 A_2 \cdots A_N})\geq \sum_{i} C_{l_1} (\rho^{A_i}).$$ for any multipartite system described by the density operator $\rho^{A_1 A_2 \cdots A_N}$, with $\rho^{A_i}$ ($i=1,2,\ldots,N$) being the reduced density operators.
For bipartite state $\rho^{AB}$ with the reduced states $\rho^A=
{\mathrm{tr}}_B\rho^{AB}$ and $\rho^B={\mathrm{tr}}_A\rho^{AB}$, @ulrc1 proved that the relative entropy of coherence respects the monogamy relation $$\label{eq3d-1}
C_r(\rho^{AB})\geq C_r(\rho^A)+C_r(\rho^B).$$ In fact, for the multipartite system described by the density operator $\rho^{A_1 A_2 \cdots A_N}$, an application of Eq. immediately yields $$\label{eq3d-v1}
C_r(\rho^{A_1 A_2 \cdots A_N})\geq \sum_{i} C_r (\rho^{A_i}).$$ But a similar monogamy relation does not hold for the general bipartite division of tripartite states. Even for the pure state $\rho^{ABC}=|\psi\rangle^{ABC} \langle\psi|$ with $\rho^{AB}=
{\mathrm{tr}}_C\rho^{ABC}$ and $\rho^{AC}={\mathrm{tr}}_B\rho^{ABC}$, the relation $$\label{eq3d-2}
C_r(\rho^{ABC})\geq C_r(\rho^{AB})+C_r(\rho^{AC}),$$ holds with a very strong constraint, that is, there should exist some real-valued parameters $\lambda\in[0,1]$ such that [@ulrc2] $$\label{eq3d-v2}
\lambda S(\rho^{AB}_{\rm diag})\leq S(\rho^{AB}),~~
(1-\lambda)S(\rho^{AC}_{\rm diag})\leq S(\rho^{AC}).$$ @dist also explored the distribution of quantum coherence among constituents of a $N$-partite system. By introducing a quantum version of the Jensen-Shannon divergence (QJSD): $$\label{eq3d-3}
\mathcal {J}(\rho,\sigma)=\frac{1}{2}[S(\rho\| (\rho+\sigma)/2)+
S(\sigma\|(\rho+\sigma)/2)],$$ or equivalently, $$\label{eq3d-v3}
\mathcal{J}
(\rho,\sigma)=S\left(\frac{\rho+\sigma}{2}\right)-\frac{1}{2}S(\rho)-\frac{1}{2}S(\sigma),$$ and using its square root as the distance measure of two states, i.e., $\mathcal {D}=\mathcal {J}^{1/2}$, they defined $C(\rho)$ of Eq. as the total coherence, and $$\label{eq3d-v4}
C_I (\rho)=\min_{\delta_S \in\mathcal{I}_S} \mathcal{D}(\rho,\delta_S),$$ as the intrinsic coherence which excludes the contribution of the subsystems, and $$\label{eq3d-v5}
C_L (\rho)= \mathcal{D} (\delta_S^\star, \delta^\star),$$ as the local coherence. Here, $\mathcal {I}_S$ is comprised of the states $\delta_S=\sum_k p_k \tau_{k,1}^b \otimes \cdots \otimes
\tau_{k,N}^b$ obtained by choosing all the possible basis set $\{|b_{l,n}\rangle \langle b_{l,n}|\}$ (with $\tau_{l,n}^b=\sum_l
p_{l,n} |b_{l,n}\rangle \langle b_{l,n}|$), while $\delta^\star$ and $\delta_S^\star$ are the closest states for obtaining $C(\rho)$ and $C_I(\rho)$, respectively.
Building upon the above definitions, one has $$\label{eq3d-v6}
C\leq C_L+C_I\leq \sum_n C_{L,n}+C_I,$$ where the first inequality is a direct result of the metric properties of $\mathcal{D}$ (i.e., the triangle inequality), and the second one is due to the subadditivity of $C_L$ for product $\delta_S^\star$. In particular, for the $N$-partite system, one can divide it into different subsystems, and calculate the corresponding coherence. For example, we denote by $C_{1:2}$ and $C_{1:23}$ the intrinsic coherence between subsystems 1 and 2, and between subsystem 1 and the combined subsystem 23, and similarly for other cases. In this way, @dist defined the multipartite monogamy of coherence as $$\label{eq3d-v7}
M=\sum_{n=2}^N C_{1:n}- C_{1:2\cdots N},$$ which is monogamous for $M\leq 0$ and polygamous otherwise. They have calculated $M$ for the three-qubit *W* and GHZ states, and showed the validity of it on analyzing coherence distribution in spin systems of the Heisenberg model.
Considering the fact that the coherence of a state $\rho$ cannot be larger than the average coherence of its ensemble states $\{p_i,\rho_i\}$ [@coher], @acc-coh studied the distribution of quantum coherence from another perspective. For $\rho=\sum_ip_i\rho_i$, they introduced a quantity which they called it accessible coherence, $$\label{eq3d-v8}
C^{\mathrm{acc}}(\rho)= \sum_i p_i C(\rho_i)-C(\rho),$$ which characterizes the coherence one gains when knowing the information of the ensemble $\{p_i,\rho_i\}$. Moreover, if a maximization is taken over all state decompositions of $\rho=\sum_i p_i\rho_i$, one can obtain the maximal accessible coherence. In fact, the maximization is only necessary to be taken over the pure state decompositions of $\rho$ due to compact convexity of the density matrix set.
For a bipartite state $\rho^{AB}$, they further defined the remaining coherence as $$\label{eq3d-v9}
\begin{aligned}
C^{\mathrm{rem}}(\rho^{AB})=&C(\rho^{AB})-C(\rho^A)-C(\rho^B)\\
&-C^{\mathrm{acc}}(\rho^A)-C^{\mathrm{acc}}(\rho^B).
\end{aligned}$$ That is to say, the coherence in $\rho^{AB}$ are divided into the local coherence and the local accessible coherence in its subsystems plus the remaining coherence. Through explicit examples, they showed that there are states for which the local coherence and local accessible coherence vanishes, while the remaining coherence survives. Moreover, the remaining coherence can also be qualitatively different when being measured by the relative entropy and the $l_1$ norm, e.g., there are cases for which $C^{\mathrm{rem}}_{l_1}(\rho^{AB})=0$ and $C^{\mathrm{rem}}_r
(\rho^{AB})>0$.
For the skew-information-based coherence measure of Eq. , @Yucsskw showed that for bipartite pure state $|\psi\rangle_{AB}$ with the reduced states $\rho^A$ and $\rho^B$, the following polygamy relation holds $$\label{eq3d-v10}
[1-C_{sk}(\rho^A)][1-C_{sk}(\rho^B)]\geq 1-C_{sk}(|\psi\rangle_{AB}),$$ while for the mixed state $\rho^{AB}$, one has [@Yucsskw] $$\label{eq3d-5}
[1-C_{sk}(\rho^A)][1-C_{sk}(\rho^B)]\geq \sum_{kk'}\langle kk'|\rho^{AB}|kk'\rangle^2,$$ and $$\label{eq3d-new5}
[1-C_{sk}(\rho^A)][1-C_{sk}(\rho^B)]\geq \frac{1}{c_s}[1-C_{sk}(\rho^{AB})]^2,$$ where the right-hand side of Eq. equals to ${\mathrm{tr}}(\rho^{AB})^2 -C_{l_2}(\rho^{AB})$, and $c_s=[r-\sum_i
C_{sk}(\rho^{Ai})][r-\sum_i C_{sk}(\rho^{Bi})]$, with $r=\mathrm{rank}(\rho^{AB})$, and $\rho^{Ai}$ and $\rho^{Bi}$ are the reduced states of the $i$th eigenstate of $\rho^{AB}$.
### State ordering under different coherence measures
As various measures of quantum coherence have been put forward up to now, one may wonder whether they impose the same state ordering or not, just as the similar problem encountered when comparing various entanglement [@ordere] and discordlike correlation measures [@orderd1; @orderd2].
When considering two measures of quantumness of a state denoted by $Q_1$ and $Q_2$, if $$\label{eq3e-1}
Q_1(\rho_1)\leq Q_1(\rho_2) \Longleftrightarrow
Q_2(\rho_1)\leq Q_2(\rho_2),$$ for arbitrary two states $\rho_1$ and $\rho_2$, then they are said to give the same state ordering. Otherwise, they give inconsistent descriptions of quantumness.
By concentrating on the coherence measures, @orderc examined state ordering problem imposed by the $l_1$ norm of coherence, relative entropy of coherence, and coherence of formation. Through explicit examples, they found that these measures also impose different orderings of states. In particular, for all measures of coherence that are equivalent for pure states, they must impose different orderings for general mixed states.
Complementarity of quantum coherence {#sec:4B}
------------------------------------
As the measures of quantum coherence are basis dependent, a natural question that arises is how they behave when different bases are involved?
### Mutually unbiased bases
For the $l_1$ norm and relative entropy measures of coherence, @comp1 studied tradeoffs between coherence of the MUBs. Here, two observables and the resulting basis sets are said to be mutually unbiased if the measurement outcomes of either one with respect to any eigenstate of the other is uniformly distributed, i.e., the probability distribution is $\{1/d,\cdots, 1/d\}$ for a $d$-dimensional Hilbert space $\mathcal {H}$. For example, for qubits the three Pauli observables $\sigma_x$, $\sigma_y$, and $\sigma_z$ are mutually unbiased. In fact, when $d=\dim\mathcal{H}$ is a prime power, there always exists a complete set of $d +1$ MUBs [@mubs1; @mubs2].
If one uses the $l_1$ norm of coherence $C_{l_1}(A_j,\rho)$ as a quantifier, with $\{A_j\}_{j=1}^{d+1}$ being the MUBs and $A_j=
\{|a_i^{(j)} \rangle\}_{i=1}^d$, @comp1 obtained $$\label{eq4a-1}
C_{l_1}(A_j,\rho)\leq \sqrt{d(d-1)[P(\rho)-P(A_j|\rho)]},$$ where $P(\rho)={\mathrm{tr}}\rho^2$ and $P(A_j|\rho)=\sum_i \langle
a_i^{(j)}|\rho|a_i^{(j)}\rangle^2$ are called the quantum and classical purities, respectively. On the other hand, from the equality $\rho=\sum_j \rho(A_j)- {\openone}$ [@mubs3], with $\rho(A_j)=\Delta(A_j,\rho)$ denoting full dephasing of $\rho$ in the basis $A_j$ \[see Eq. \], one can prove $\sum_j
P(A_j|\rho)=1+P(\rho)$, hence $$\label{eq4a-2}
\sum_{j=1}^{d+1} C_{l_1}^2 (A_j,\rho)\leq d(d-1)[dP(\rho)-1].$$ This is the complementarity relation for $l_1$ norm of coherence under the complete set of MUBs. It establishes connection between coherence and purity of a state, and bounds from above distribution of coherence as well. This bound is tight as it is saturated by the following states $$\label{eq4a-3}
\rho_\epsilon=\frac{\epsilon}{d-1}{\openone}+\frac{d(1-\epsilon)-1}{d-1}
|a_i^{(j)}\rangle\langle a_i^{(j)}|,$$ with $0\leq\epsilon\leq 1$.
Similarly, @comp1 derived a complementarity relation for the relative entropy of coherence $$\label{eq4a-4}
\begin{aligned}
\sum_{j=1}^{d+1}C_r(A_j,\rho)\leq & (d+1)[\log_2 d-S(\rho)]\\
& -\frac{(d-1)[dP(\rho)-1]}{d(d-2)}\log_2(d-1),
\end{aligned}$$ this bound is saturated for the maximally coherent state $|\Psi_d\rangle$, and the second term on the right-hand side reduces to $[P(\rho)-0.5]\log_2 e$ for $d=2$.
@comp1 also defined the mean coherence $\bar{C}(\rho)$ and the root mean square coherence ${\rm rms}[C(\rho)]$ as $$\label{eq4a-5}
\begin{aligned}
& \bar{C}(\rho)=\int dU C(U\rho U^\dag,\rho),\\
& {\rm rms}[C(\rho)]=\left[\int dU C^2(U\rho U^\dag,\rho)\right]^{1/2},
\end{aligned}$$ where $\{U\}$ denote the unitaries which transform one basis set to another one, and $dU$ is the normalized invariant Haar measure over $\{U\}$. Based on this, one can obtain $$\label{eq4a-6}
\begin{aligned}
& \bar{C}_{l_1}(\rho)\leq {\rm rms}[C_{l_1}(\rho)]\leq \sqrt{\frac{d(d-1)
[dP(\rho)-1]}{d+1}},\\
& \bar{C}_r(\rho)=\sum_{n=2}^d \frac{1}{n}\log_2 e-[S(\rho)-Q(\rho)],
\end{aligned}$$ where $\{\lambda_i\}_{i=1}^d$ are eigenvalues of $\rho$, and $$\label{eq4a-7}
Q(\rho)=-\sum_{i=1}^d \frac{\lambda_i^d\log_2\lambda_i}
{\prod_{j\neq i} (\lambda_i-\lambda_j)},$$ is the quantum subentropy [@sube1].
### Incompatible bases
Following the established notions for various entropic uncertainty relations (EURs) \[see the review paper of @eur0\], @incbs further discussed tradeoff relations between quantum coherence of the MUBs.
First, for single-partite quantum state $\rho$, it follows immediately from the EUR $H(P)+H(Q)\geq \log_2 (1/c)+S(\rho)$ that $$\label{eq4a-new6}
C_r(Q,\rho)+C_r(R,\rho)\geq \log_2(1/c)-S(\rho),$$ where $$\label{EUR1}
c=\max_{k,l}|\langle\psi_k^Q|\psi_l^R\rangle|^2,$$ with $\{|\psi_k ^Q \rangle\}$ and $\{|\psi_l^R\rangle\}$ denoting respectively, the eigenstates of the two incompatible observables $Q$ and $R$. Similarly, by using the EUR derived by @ur1 and @ur2, one can obtain two new lower bounds for the sum of coherence, which are as follows $$\label{EUR2}
\begin{aligned}
& C_r(Q,\rho)+C_r(R,\rho)\geq \log_2(1/c)[1-S(\rho)], \\
& C_r(Q,\rho)+C_r(R,\rho)\geq H\left(\frac{1+\sqrt{2c-1}}{2}\right)-2S(\rho),
\end{aligned}$$ and they may be more or less optimal than the bound of Eq. due to the different values of $c$ and the form of $\rho$.
For single-qubit state $\rho$, @Maxf obtained new lower bounds for the sum of coherence measures under two incompatible bases. By denoting $P'=2{\mathrm{tr}}\rho^2-1$, these bounds can be written explicitly as $$\label{EUR3}
\begin{aligned}
& C_r(Q,\rho)+C_r(R,\rho)\geq H\left(1+\frac{\sqrt{P'}(2\sqrt{c}-1)}{2} \right)-S(\rho),\\
& C_{l_1}(Q,\rho)+C_{l_1}(R,\rho)\geq 2\sqrt{P'c(1-c)}, \\
& R_I(Q,\rho)+R_I(R,\rho)\geq H\left( \frac{1+\sqrt{1-4P'(\sqrt{c}-c)}}{2}\right).
\end{aligned}$$ Second, for the bipartite state $\rho^{AB}$, by using the quantum-memory-assisted EUR [@eur1] $$\label{EUR4}
S(Q|B)+S(R|B)\geq \log_2(1/c)+S(A|B),$$ and taking the eigenstates $\Xi=\{|\psi_k^X\rangle
|\varphi_j^B\rangle\}$ ($X=\{Q,R\}$, and $|\varphi_j^B\rangle$ is the eigenstate of $\rho^B={\mathrm{tr}}_A \rho^{AB}$) as the basis, we have $$\label{eq4a-8}
C_r(\Xi,\rho^{AB})+C_r(\Xi,\rho^{AB})\geq \log_2(1/c)-S(A|B),$$ where the bound on the right-hand side can be further tightened by using the concept of QD [@eur2].
Similarly, if one considers the mutually incompatible observables $\{Q_1,Q_2,\cdots,Q_n\}$, with the corresponding eigenstate bases $\{|\psi_{k_1}^{Q_1}\rangle |\psi_{k_2}^{Q_2}\rangle\cdots
|\psi_{k_n}^{Q_n}\rangle\}$ and $\Xi_i=\{|\psi_{k_i}^{Q_i}\rangle
|\varphi_j^B\rangle\}$, then by using the formulas established by @eur3, one can obtain $$\label{eq4a-9}
\begin{aligned}
& \sum_{i=1}^n C_r(Q_i,\rho)\geq \log_2 (1/b)-S(\rho),\\
& \sum_{i=1}^n C_r(\Xi_i,\rho^{AB})\geq \log_2 (1/b)-S(A|B),
\end{aligned}$$ and this can be considered as an extension of the results for two mutually unbiased observables.
### Complementarity between coherence and mixedness
For the class of states with fixed mixedness, the amount of quantum coherence contained in them may be different. Using the linear entropy measure of mixedness $$\label{eq4b-1}
M_l(\rho)=\frac{d}{d-1}(1-{\mathrm{tr}}\rho^2),$$ and the $l_1$ norm of coherence given in Eq. , @comp2 derived a tradeoff relation between the two quantities, $$\label{eq4b-2}
\frac{C_{l_1}^2(\rho)}{(d-1)^2}+M_l(\rho)\leq 1,$$ where the first term on the left-hand side can be seen as the square of the normalized coherence, $\tilde{C}_{l_1} (\rho) \coloneqq
C_{l_1} (\rho)/(d-1)$. It sets a fundamental limit to the amount of coherence that can be extracted from the class of states with equal mixedness, and vice versa.
Moreover, in the same vein with the definition of maximally entangled mixed states [@mems1; @mems2; @mems3], @comp2 considered the class of MCMS given in Eq. , and found that the upper bound in Eq. is saturated, as it gives $$\label{eq4b-3}
C_{l_1}(\rho_{\rm mcms})= p(d-1),~
M_l(\rho_{\rm mcms})= 1-p^2,$$ for $1\leq p\leq 1$. In fact, $\rho_{\rm mcms}$ constitutes also the class of states with maximal mixedness for fixed coherence.
Although @comp2 pointed out that similar tradeoffs apply to the relative entropy of coherence \[i.e., $C_r(\rho)+S(\rho) \leq 1$, which is incorrect as $C_r(\rho)+ S(\rho)= S(\rho_{\rm diag})\leq
\log_2 d$\], and the fidelity-based measure of coherence for single qubit state \[see Eq. \], it remains as a challenge to generalize the complementarity relation to other coherence measures which are on equal footing with the $l_1$ norm of coherence. Some other progress have been made, e.g., @longgl have showed that if the mixedness of $\rho$ is defined via the fidelity as $$\label{eq4b-v3}
M_g(\rho)=F(\rho,{\openone}/d)=\frac{1}{d}({\mathrm{tr}}\sqrt{\rho})^2,$$ then by combing this with Eq. and further using the mean inequality $\sum_i x_i\leq (d\sum\nolimits_i x_i^2)^{1/2}$ ($\forall x_i\in\mathbb{R}$), it is easy to see that $$\label{eq4b-v4}
C_g(\rho)+M_g(\rho)\leq 1-\sum_i b_{ii}^2+\frac{1}{d}\left(\sum_i b_{ii}\right)^2\leq 1,$$ with equality holding for $\rho_\mathrm{mcms}$ of Eq. .
Stimulated by the work of @purity and @ther4, a resource theory of purity was established by @coh-pur. In this framework, the only free state is the maximally mixed state $\rho_{\mathrm{mm}}={\openone}/d$, and the set of free operations is the unital operations $\Lambda_{\mathrm{U}}$. A functional $P(\rho)$ is said to be a purity monotone if it is nonnegative and $P(\Lambda_{\mathrm{U}}[\rho]) \leq P(\rho)$. $P(\rho)$ is a purity measure if it further satisfies the additivity property $P(\rho\otimes\sigma)=P(\rho)+ P(\sigma)$ and normalization condition $P(|\psi\rangle)=\log_2 d$. Moreover, it is convex if $\sum_i p_i P(\rho_i)\geq P(\sum_i p_i\rho_i)$.
Based on the above framework, @coh-pur introduced a coherence-based purity monotone $$\label{purity}
P_C(\rho)\coloneqq \max{\Lambda_{\mathrm{U}}} C(\Lambda_{\mathrm{U}}[\rho])
=C(\rho_{\max}),$$ with $C$ being any MIO monotone. When $C$ is defined by the contractive distance $\mathcal{D}$, the combination of the above equation with Eq. further gives $P_C(\rho)=\mathcal{D}(\rho,{\openone}/d)$. This shows another connection between purity of a state and the maximum amount of quantum coherence achievable by suitable unitary operation.
Moreover, @coh-pur also introduced a Rényi $\alpha$-entropy purity measure $$\label{purity2}
P_\alpha(\rho)= \log_2 d -\frac{1}{1-\alpha}\log_2\{{\mathrm{tr}}(\rho^\alpha)\},$$ for $\alpha\geq 0$, which is also convex when $\alpha\in[0,1]$. In particular, the Rényi $2$-entropy purity measure $P_2(\rho)=
\log_2\{d{\mathrm{tr}}(\rho^2)\}$ is quantitatively related to the linear entropy of purity ${\mathrm{tr}}\rho^2$, and when $\alpha\rightarrow 1$, we have the traditional relative entropy of purity $$\label{purity3}
P_r(\rho)=\log_2 d-S(\rho).$$
Duality of coherence and path distinguishability {#sec:4C}
------------------------------------------------
With roots in quantum optics, quantum coherence lies at the heart of interference phenomenon. The presence of coherence in a quantum system can be seen as a manifestation of the wave nature of it [@wapa1; @wapa2], while the path distinguishability or which-path information signifies its complementarity aspect, i.e., the particle nature of it. The quantitative connections between them can be investigated in the context of unambiguous quantum state discrimination (UQSD) or ambiguous quantum state discrimination (AQSD), which are implementable in interference experiments.
### Unambiguous quantum state discrimination
@wapa1 proposed to use quantum coherence to signify the wave nature of a particle, and the upper bound of the success probability of UQSD to signify its particle aspect. Let $\{|\xi_i\rangle\}$ be a collection of states which may be nonorthogonal, then the task of UQSD is to find with certainty which of them is the given one, see @uqsd1 [@uqsd2] and references therein.
In the $N$-path interference experiment, we denote by $\{|\psi_i\rangle\}$ the orthogonal basis state of the path. Then if the initial state of the particle entering the interferometer is $$\label{eq4c-0}
|\psi\rangle_s=\sum_{i=1}^N c_i |\psi_i\rangle,$$ with $\sum_i |c_i|^2=1$, and the related detector state is $|0_i\rangle$. Their combined state after the interaction operation is $$\label{par-dec}
|\psi\rangle_{sd}=\sum_{i=1}^N c_i |\psi_i\rangle\otimes |\xi_i\rangle.$$ To discriminate the which-path information, the experimenter can perform measurements on the detector states. The probability for successfully discriminating them is proved to be bounded from above by [@uqsd1; @uqsd2] $$\label{eq4c-1}
p_{\rm uqsd}\leq \mathcal{D}_Q \coloneqq 1-\frac{1}{N-1}\sum_{i\neq j}|c_i c_j|
|\langle\xi_i|\xi_j\rangle|,$$ where $\mathcal{D}_Q$ sets a limit to the ability of the experimenter to distinguish between the states $\{|\xi_i\rangle\}$ (and hence $\{|\psi_i\rangle\}$), although it may not be achievable in real experiments.
On the other hand, the postmeasurement state of the particle is $$\label{eq4c-2}
\rho'_s= \sum_{i,j}c_i c_j^* \langle \xi_j|\xi_i\rangle |\psi_i
\rangle\langle\psi_j|,$$ hence $$\label{eq4c-3}
C_{l_1}(\rho'_s)=\sum_{i\neq j}|c_i c_j| |\langle\xi_j|\xi_i\rangle|,$$ in the path basis $\{|\psi_i\rangle\}$. Based on these, @wapa1 derived the following relation $$\label{eq4c-4}
\frac{C_{l_1}(\rho'_s)}{N-1}+\mathcal{D}_Q=1,$$ It characterizes in a quantitative way the wave-particle duality. In particular, for two- and three-path situations with uniform $|c_i|$, the normalized coherence $\tilde{C}_{l_1}= C_{l_1}/(N-1)$ is also quantitatively related to the interference fringe visibility $\mathcal{V}$, namely, $\tilde{C}_{l_1}=\mathcal {V}$ and $\tilde{C}_{l_1}=2\mathcal {V}/(3-\mathcal {V})$, respectively [@wapa1].
Moreover, for the case of initially mixed particle state $\rho_s=\sum_i \rho_{ij}|\psi_i\rangle\langle\psi_j|$ and pure detector state, or the more general case of both initially mixed particle and detector states, @wapa1 showed that the equality of Eq. becomes inequality $$\label{eq4c-5}
\frac{C_{l_1}(\rho'_s)}{N-1}+\mathcal{D}_Q\leq 1.$$ By using a slightly different path distinguishability $\mathcal{D}=[\mathcal{D}_Q(2-\mathcal{D}_Q)]^{1/2}$, @waveparticle further gave an equivalent form of Eq. , i.e., $$\label{eq4c-v5}
\frac{C_{l_1}^2(\rho'_s)}{(N-1)^2} + \mathcal{D}^2 \leq 1,$$ which is similar to the complementarity relation $\mathcal
{D}^2+\mathcal {V}^2\leq 1$ given by @mierd.
In we put a screen behind the slits, the interference pattern of the particle is described by the probability density of particle hitting the screen at particular position. @multipath considered one such kind of problem. For the particle-detector state $
|\psi\rangle_{sd}$ of Eq. , the expression for the pattern on the screen is given by $|\langle x|\psi\rangle_{sd}|^2$. For an $N$-slit experiment with the width of the slits being $\epsilon$, and the distance between any two neighboring slits (between the slits and the screen) is $\ell$ ($D$), then if we assume that the state that emerges from the $j$th slit is a Gaussian along the $x$ axis and centered ad $x_j=j\ell$, the state of the particle hitting the screen at a position $x$ will be [@multipath] $$\label{eq4c-n5}
\langle x|\psi(t)\rangle_{sd}= A_t \sum_{j=1}^N c_j \exp{\left[-\frac{(x-j\ell)^2}
{\epsilon^2+i\lambda D/\pi}\right]}|\xi_j\rangle,$$ where $A_t=\{2/[\pi(\epsilon+i\lambda D/\pi\epsilon)]\}^{1/4}$ ($i$ is the imaginary unit), and $|\psi(t)\rangle_{sd}$ is the evolved state with $|\psi(0)\rangle_{sd}$ being given by Eq. .
Then by using the facts that $\epsilon^2 \ll (\lambda D/\pi)^2$ and the distance between the primary maxima $\lambda D/\ell\gg j\ell$, one can obtain that the intensity of the fringe $I(x)=|\langle
x|\psi(t)\rangle_{sd}|^2$ at position $x$ is given by @multipath $$\label{intensity}
\begin{split}
I(x) =& |A_t|^2 \exp{\left[-\frac{2\epsilon^2 x^2}{(\lambda D/\pi)^2}\right]}
\left\{1+\sum_{j\neq k} |c_j c_k|\right. \\
& \left.\times|\langle\xi_j|\xi_k\rangle|\cos\left[\frac{2\pi x\ell(k-j)}
{\lambda D}+\theta_k-\theta_j\right]\right\},
\end{split}$$ where we have defined $c_k|\xi_k\rangle=|c_k| |\xi'_k\rangle
e^{i\theta_k}$, with $ |\xi'_k\rangle$ being real.
By choosing $\theta_k=\theta_j$ ($\forall k, j$), one can obtain from the above equation that at positions $x_m=m\lambda D/\ell$ $(m\in\mathbb{Z})$ of the primary maxima, the intensity of the fringe is given by $$\label{inten1}
I_{\max} = |A_t|^2 \exp{\left[-\frac{2\epsilon^2 x_m^2}{(\lambda D/\pi)^2}\right]}
\left(1+\sum_{j\neq k} |c_j c_k| |\langle\xi_j|\xi_k\rangle|\right).$$ Moreover, when a phase randomizer is applied to the setup such that the phases of the incoming state at different slits are randomized (i.e., the incoming state becomes incoherent), the cosine term of Eq. will disappear, thus we have $$\label{inten2}
I_{\mathrm{inc}} = |A_t|^2 \exp{\left[-\frac{2\epsilon^2 x_m^2}{(\lambda
D /\pi)^2}\right]}.$$ Finally, by combining the above two results with Eq. , one can obtain directly that $$\label{inten3}
\frac{I_{\max}-I_{\mathrm{inc}}}{I_{\mathrm{inc}}}=C_{l_1}(\rho'_s),$$ In fact, from Eq. one can see directly that when the which-path information is completely indistinguishable (i.e., $\langle\xi_j| \xi_k\rangle=1$, $\forall k,j$), the intensity of the interference fringe at the primary maximum $x_m=m\lambda D/\ell$ is given by $$\label{inten4}
I_{\max}^\parallel = |A_t|^2 \exp{\left[-\frac{2\epsilon^2 x_m^2}{(\lambda D/\pi)^2}\right]}
\left(1+\sum_{j\neq k} |c_j c_k|\right).$$ Similarly, when the which-path information is completely distinguishable (i.e., $\langle\xi_j| \xi_k\rangle=0$, $\forall
k,j$), the intensity of the interference fringe at the primary maximum $x_m=m\lambda D/\ell$ turns out to be $$\label{inten5}
I_{\max}^\perp = |A_t|^2 \exp{\left[-\frac{2\epsilon^2 x_m^2}{(\lambda
D/\pi)^2}\right]}.$$ Then it is obvious that $$\label{inten6}
\frac{I_{\max}^\parallel-I_{\max}^\perp}{I_{\max}^\perp}=C_{l_1}(\rho'_s).$$ The implementation of the above scheme for measuring quantum coherence depends essentially on whether there exits such a path detector which is (at least) capable of making the which-path information completely indistinguishable and distinguishable.
### Ambiguous quantum state discrimination
Different from UQSD, one always has a result in the AQSD experiments, but it may be right or wrong, and the task of the experimenter is to minimize the probability of being wrong to its theoretical limit [@mierd], hence it is also known as minimum-error state discrimination .
By using the $l_1$ norm of coherence to characterize the wave nature, and an upper bound on the average probability $p_{\rm aqsd}$ of successfully discriminating the path information to characterize the particle nature, @wapa2 derived the following duality relation between them $$\label{eq4c-6}
\left(\frac{C_{l_1}(\rho'_s)}{N-1}\right)^2+
\left(\frac{Np_{\rm aqsd}-1}{N-1}\right)^2 \leq 1,$$ where $\rho'_s$ is the same as Eq. . To discriminate the which-path information, the experimenter can perform POVM on the reduced detector state $\rho_{d}=\sum_i |c_i|^2 \rho^\xi_i$ (with $\rho^\xi_i= |\xi_i\rangle \langle \xi_i|$), and the average probability $p_{\rm aqsd}=\sum_i |c_i|^2 {\mathrm{tr}}(\Pi_i|\xi_i \rangle
\langle\xi_i|)$ is showed to be upper bounded by $$\label{eq4c-8}
p_{\rm aqsd}\leq \frac{1}{N}+\frac{1}{2N}\sum_{ij}\|\Lambda_{ij}\|_1,$$ where $\{\Pi_i\}$ is the set of POVM, while $\Lambda_{ij}=|c_i|^2
\rho^\xi_i-|c_j|^2 \rho^\xi_j$ is the Helstron matrix of the state pair $(\rho^\xi_i, \rho^\xi_j)$, and $$\label{eq4c-v8}
\|\Lambda_{ij}\|_1=2\sqrt{\left(\frac{|c_i|^2+|c_j|^2}{2}\right)^2-|c_i
c_j|^2|\langle\xi_i|\xi_j\rangle|^2}.$$ @wapa2 also derived a duality relation between the relative entropy of coherence and path distinguishability, i.e., $$\label{eq4c-9}
C_r (\rho'_s)+H(M\!:\!D)\leq H(\{p_i\}),$$ where $H(M\!:\!D)=H(\{p_j\})+H(\{q_i\})-H(\{p_{ij}\})$ is the mutual information of $D=\{p_j\}$ and $M=\{q_i\}$, and $H(\cdot)$ is the Shannon entropy, with the probabilities $p_{ij}= {\mathrm{tr}}(\Pi_i
\rho^\xi_j)p_j$, $p_j=\sum_i p_{ij}=|c_j|^2$, and $q_i=\sum_j
p_{ij}={\mathrm{tr}}(\Pi_i\rho_d)$.
Eq. holds as well even if its second term on the left-hand side is replaced by the accessible information $I_{acc}$, which is defined as the maximum value of $H(M\!:\!D)$ over all possible POVMs, and characterizes how well an experimenter can do at inferring the detector states.
Recently, it has been pointed out by @waveparticle that the upper bound of Eq. may not be saturated for general pure states $|\psi\rangle_{sd}$ if $N\geq 3$. It is also doubted as the two terms on its left-hand side (which characterize the wave and particle nature of a quanton, respectively) can increase or decrease simultaneously.
Distillation and dilution of quantum coherence {#sec:4D}
----------------------------------------------
### Standard coherence distillation and dilution
In the same spirit as entanglement distillation and entanglement formation, one can also consider the tasks of coherence distillation and coherence formation by incoherent operations $\Lambda$, see Fig. \[fig:hu4\](a). The former corresponds to the transformation of a general state $\rho$ to the maximally coherent one, i.e., the distillation of $\rho$ to $\Psi_2= |\Psi_2\rangle \langle\Psi_2|$, while the latter is the formation of $\rho$ from $\Psi_2$. The corresponding optimal rate can be recognized as an operational measure of coherence.
In the asymptotic setting (i.e., infinitely many copies of $\rho$), @qcd1 defined the distillable coherence as the maximal rate at which $\Psi_2$ can be obtained from $\rho$, i.e., $$\label{eq4d-1}
C_d(\rho)=\sup\{R:\lim_{n\rightarrow \infty\atop\varepsilon\rightarrow 0}
(\inf_{\Lambda}\|\Lambda(\rho^{\otimes n})
-\Psi_2^{\otimes \lfloor nR\rfloor }\|_1)\leq \varepsilon\},$$ and the coherence cost which is the minimal rate at which $\Psi_2$ has to be consumed for formatting $\rho$ is dually to Eq. , with only the supremum being replaced by the infimum, and $\Lambda$ acting on $\Psi_2^{\otimes \lceil nR\rceil
}$. The central results are that $C_d(\rho)$ equals to the relative entropy of coherence $C_r(\rho)$, while $C_c(\rho)$ equals to the coherence of formation $$\label{cof}
C_f(\rho)\coloneqq \min_{\{p_i,\psi_i\}}\sum_i p_i S(\Delta [\psi_i]),$$ which involves a minimization over all pure state decompositions of $\rho$ showed in Eq. [@cof], and the additivity of $C_f$ and $C_r$ implies that both $C_d$ and $C_c$ are additive as well.
Moreover, different from the possible bound entanglement in a state, @qcd1 found that there is no bound coherence, that is, there does not exist quantum state for which its creation consumes coherence while no coherence could be distilled from it. Thus, $$\label{cof2}
C_d(\rho)=0\Rightarrow C_c(\rho)=0,$$ which reveals that quantum coherence in any state is always distillable.
@Buprl considered an one-shot version of coherence distillation. They defined the relevant coherence cost for formatting a quantum state $\rho$ under MIO as $$\label{eq4d-15}
C_{c,\mathrm{MIO}}^{(1),\varepsilon}(\rho)=\inf_{\{\Lambda_{\mathrm{MIO}}\}\atop M\in \mathbb{Z}}
\{\log_2 M|F[\rho,\Lambda_{\mathrm{MIO}}(\Psi_+^M)]\geq 1-\varepsilon\},$$ where $\Psi_+^M=|\Psi_+^M\rangle\langle\Psi_+^M|$ with $|\Psi_{+}^M\rangle=\sum_{i=1}^M |i\rangle/\sqrt{M}$, and $F(\cdot)$ is the Uhlmann fidelity given in Eq. \[note that it equals to the square of that adopted by @Buprl\]. Then, they showed that, for any $\varepsilon>0$, one has $$\label{eq4d-16}
C_{\max}^{2\sqrt{\varepsilon}}(\rho)\leq C_{c,\mathrm{MIO}}^{(1),\varepsilon}(\rho),$$ so the smooth maximum relative entropy of coherence bounds from below the one-shot coherence cost.
Similar to the one-shot coherence distillation, one can also consider the one-shot version of coherence dilution, in which the corresponding coherence cost reads $$\label{eq4d-17}
C_\mathcal{O}^\varepsilon (\rho)=\inf_{\Lambda\in \mathcal {O}}
\{\log_2 M|F[\rho,\Lambda(\Psi_+^M)]\geq 1-\varepsilon\},$$ where $\mathcal{O}$ is one of the free operations $\{\mathrm{MIO,DIO,IO, SIO}\}$. To establish an operational interpretation for the coherence measure, @oneshot further introduced an $\varepsilon$-smoothed coherence measure $C^\varepsilon (\rho)=\min_{\rho'\in B_\varepsilon(\rho)}C(\rho')$, where $ B_\varepsilon(\rho)=\{\rho'| F(\rho,\rho'\geq
1-\varepsilon)\}$. Based on these preliminaries, they proved that for any $\varepsilon>0$, we have $$\label{eq4d-18}
\begin{aligned}
& C_{\max}^\varepsilon (\rho)\leq C_{\mathrm{MIO}}^\varepsilon (\rho)\leq C_{\max}^\varepsilon (\rho)+1,\\
& C_{\Delta,\max}^\varepsilon (\rho)\leq C_{\mathrm{DIO}}^\varepsilon (\rho)\leq C_{\Delta,\max}^\varepsilon (\rho)+1, \\
& C_{\mathrm{MIO}}^\varepsilon (\rho)= C_{\mathrm{SIO}}^\varepsilon (\rho)= C_0^\varepsilon(\rho),
\end{aligned}$$ and in the asymptotic limit, the $\varepsilon$-smoothed coherence equivalents either to the relative entropy of coherence [@coher] or to the coherence of formation [@qcd1], i.e., $$\label{eq4d-19}
\begin{aligned}
& C_{\mathrm{MIO}}^\infty (\rho)= C_{\mathrm{DIO}}^\infty (\rho)=C_r(\rho),\\
& C_{\mathrm{IO}}^\infty(\rho)=C_{\mathrm{SIO}}^\infty(\rho)=C_f(\rho).
\end{aligned}$$ where $C_{\mathcal{O}}^\infty (\rho)= \lim_{\varepsilon
\rightarrow0,n\rightarrow \infty} C_{\mathcal {O}}^\varepsilon
(\rho^{\otimes n})/n$. This result, together with that of @qcd1, implies that the role of MIO and IO in the asymptotic scenario of coherence dilution is the same as we also have $C_{\mathrm{IO}}^\infty (\rho)=C_r(\rho)$.
### Assisted coherence distillation
In analogy to assisted entanglement distillation, @qcd2 investigated the task of assisted coherence distillation in the setting of local quantum-incoherent operations and classical communication (LQICC), see Fig. \[fig:hu4\](b). In this task, two players, Alice and Bob, share $n$ copies of $\rho^{AB}$, and Alice’s objective is to help Bob to distill as much quantum coherence as possible. Different from the allowable LOCC in assisted entanglement distillation, LQICC represents quantum operations $\Lambda_{QI}$ that are general on Alice’s side and incoherent on Bob’s side. In this setting, the set $\mathcal {QI}$ of free states called the quantum-incoherent (QI) states are given by $\chi^{AB}=\sum_i p_i
\rho^A_i \otimes |i^B\rangle\langle i^B|$, with $p_i$ the probabilities, $\rho^A_i$ arbitrary states for subsystem $A$, and $\{|i^B\rangle\}$ the incoherent basis for subsystem $B$.
Formally, the generated maximum coherence is called “coherence of collaboration" (CoC) for two-way communication, and “coherence of assistance" (CoA) in the one-way situation, for which Alice holds a purifying state and only she is allowed to announce the measurement results. @qcd2 defined the optimal rate of distillable CoC as $$\label{eq4d-2}
C_d^{A|B}(\rho)=\sup\{R:\lim_{n\rightarrow \infty}
(\inf_{\Lambda_{QI}}\|\Lambda_{QI}(\rho^{\otimes n})
-\Psi_2^{\otimes \lfloor nR\rfloor }\|_1)=0\},$$ where $C_d^{A|B}$ is upper bounded by the QI relative entropy $C_r^{A|B}$, i.e., $$\label{eq4d-v2}
C_d^{A|B}(\rho^{AB})\leq C_r^{A|B}(\rho^{AB}),$$ with equality holding for any pure state, and $$\label{eq4d-3}
C_r^{A|B}(\rho^{AB})=\min_{\chi^{AB}\in \mathcal {QI}}S(\rho^{AB}\|\chi^{AB})
=S(\Delta^B[\rho^{AB}])-S(\rho^{AB}),$$ where $\Delta^B[\rho^{AB}]$ denotes dephasing of $\rho^{AB}$ in the incoherent basis of $B$.
Moreover, when extended to the situation with $N\geq 2$ assistants, the global operations across all auxiliary systems do not necessarily outperform the local operations on generating coherence, e.g., for the initial state $|\Psi\rangle^{A_1\cdots A_N B}$ with $B$ being a qubit, local operations on $A_1,\cdots, A_N$ together with classical communication are enough to localize maximum coherence on $B$.
@qcd2 also proposed quantitative definitions of CoA and the regularized CoA, which are given respectively, by $$\label{eq4d-4}
C_a(\rho)=\max_{\{p_i,\psi_i\}}\sum_i p_i C_r(\psi_i),~
C_a^\infty(\rho)=\lim_{n\rightarrow\infty}\frac{1}{n}C_a(\rho^{\otimes n}),$$ and the maximization is taken over the pure state decompositions $\rho$ showed in Eq. . Moreover, for qubit states $\rho$, the CoA is showed to be additive, i.e., $C_a(\rho^{\otimes
n})= n C_a (\rho)$.
Notably, there exists some resemblance between CoA of the state $\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$ and entanglement of assistance (EoA) of the related maximally correlated state $\rho_{\rm mc}$ of Eq. , that is, $$\label{EoA}
C_a(\rho)= E_a(\rho_{\rm mc}),~ C_a^\infty(\rho)= E_a^\infty
(\rho_{\rm mc})=S(\Delta[\rho]).$$ Moreover, for pure state $|\psi\rangle^{AB}$, the CoC equals to the regularized CoA, i.e., $$\label{eq4d-7}
C_d^{A|B}(|\psi\rangle^{AB})=C_a^\infty (\rho^B)=S(\Delta[\rho^B]),$$ which immediately yields that the maximum extra coherence that Bob can gain \[compared with the standard distillation protocol [@qcd1]\] with Alice’s assistance equals to the von Neumann entropy of $\rho^B$.
By replacing LQICC with the local incoherent operations and classical communication (LIOCC), @qcd3 further studied the coherence-entanglement tradeoffs in a task similar to @qcd2, but now both the two parties’ operations are restricted to be local incoherent, see Fig. \[fig:hu4\](c). In this new setting, if we denote by $R^A$ ($R^B$) the rate of coherence formation for Alice (Bob), and $E^{co}$ that of entanglement formation between Alice and Bob, then the triple $(R^A, R^B,E^{co})$ is achievable if for every $\varepsilon>0$ there exists a LIOCC $\Lambda_{II}$ and integer $n$ such that $$\label{eq4d-8}
\|\Lambda_{II}(\Psi_2^{\otimes\lceil n(R^A+\varepsilon)\rceil}
\otimes \Psi_2^{\otimes\lceil n(R^B+\varepsilon)\rceil}
\otimes\Phi_2^{\otimes\lceil n(E^{co}+\varepsilon)\rceil})
-\rho^{\otimes n}\|_1 \leq \varepsilon,$$ where the two $\Psi_2$ belong to Alice and Bob, respectively, while $\Phi_2=|\Phi_2\rangle \langle\Phi_2|$ with $|\Phi_2\rangle=
(|00\rangle+|11\rangle) /\sqrt{2}$ is shared between them. Similarly, $(R^A, A^B,E^{co})$ is the achievable coherence-entanglement distillation triple if $$\label{eq4d-9}
\|\Lambda_{II}(\rho^{\otimes n})-
\Psi_2^{\otimes\lfloor n(R^A-\varepsilon)\rfloor}
\otimes \Psi_2^{\otimes\lfloor n(R^B-\varepsilon)\rfloor}
\otimes\Phi_2^{\otimes\lfloor n(E^{co}-\varepsilon)\rfloor} \|_1
\leq \varepsilon.$$ For pure states $\Psi=|\Psi\rangle^{AB}\langle\Psi|$, @qcd3 obtained the possible optimal triples of resource formation $$\label{eq4d-10}
\begin{aligned}
& (R^A, A^B,E^{co})=(0,S(B|A)_{\Delta(\Psi)}, S(A)_{\Delta(\Psi)}),\\
& (R^A, A^B,E^{co})=(S(A)_{\Delta(\Psi)},S(B|A)_{\Delta(\Psi)},E(\Psi)),\\
& (R^A, A^B,E^{co})=(0, 0, S(AB)_{\Delta(\Psi)}),
\end{aligned}$$ with $S(X)_{\Delta(\Psi)}$ \[$S(X|Y)_{\Delta(\Psi)}$\] the von Neumann entropy (conditional entropy) of $\Delta(\Psi)$, and $E(\Psi)=
S(A)_{\Psi}$ the entanglement of $\Psi$. Using monotonicity of a LIOCC monotone $$\label{eq4d-v10}
C_L(\rho^{AB})=\min_{\{p_i,\psi_i\}} \sum_i p_i C_L(\psi_i),$$ with $\rho^{AB}=\sum_i p_i \psi_i$, $\psi_i = |\psi_i\rangle\langle
\psi_i|$, and $$\label{eq4d-n10}
C_L(\psi_i)= S(A)_{\Delta(\psi_i)}+ S(B)_{\Delta(\psi_i)}-E(\psi_i),$$ @qcd3 also derived the optimal resource distillation tripes $$\label{eq4d-11}
\begin{aligned}
& (R^A, A^B,E^{co})=(S(A)_{\Delta(\Psi)}-E(\Psi), S(B)_{\Delta(\Psi)},0),\\
& (R^A, A^B,E^{co})=(0,S(B|A)_{\Delta(\Psi)},I(A\!:\!B)_{\Delta(\Psi)}),
\end{aligned}$$ where $I(A\!:\!B)_{\Delta(\Psi)}$ is the mutual information of ${\Delta(\Psi)}$.
It is evident that the distillable coherence rate sum $C_D^{\rm
LIOCC}=R^A+R^B$ that can be distilled simultaneously at Alice and Bob’s side is constrained by their shared entanglement. By further defining two similar quantities $C_D^{\rm Global}$ and $C_D^{\rm
LIO}$, the former with global incoherent operations, and the latter with local incoherent operations without classical communication, @qcd3 found that for $\Psi$, the differences $$\label{eq4d-v11}
\begin{aligned}
& \delta(\Psi)= C_D^{\rm Global}(\Psi)-C_D^{\rm LIOCC}(\Psi),\\
& \delta_c(\Psi)=C_D^{\rm LIOCC}(\Psi)-C_D^{\rm LIO}(\Psi),
\end{aligned}$$ are given by $$\label{eq4d-12}
\delta(\Psi)=E(\Psi)-I(A\!:\!B)_{\Delta(\Psi)},~~
\delta_c(\Psi)=E(\Psi).$$ They describe, respectively, the extra coherence rates that can only be distilled by nonlocal incoherent operations and by using the data communicated via a classical channel.
@RCC considered a similar scenario of collaborative creation of coherence, see Fig. \[fig:hu4\](d). Here, two parties share a state $\rho^{AB}$, and their aim is to create coherence on $A$ with the help of quantum operation solely on $B$ and one-way classical communication from $B$ to $A$. They called this as remote creation of coherence (RCC), and obtained relations between the created coherence and entanglement of $\rho^{AB}$. By using the operator-sum representation $\mathcal {E}(\cdot)= \sum_i E_i(\cdot)E_i^\dag$, and denoting $\tilde{\rho}^A={\mathrm{tr}}_B ({\openone}_A\otimes\mathcal {E})
\rho^{AB}/p'$ (with $p'={\mathrm{tr}}({\openone}_A \otimes\mathcal {E})\rho^{AB}$ the probability of getting $\tilde{\rho}^A$), they proved that the RCC $C(\tilde{\rho}^A)=0$ if and only if $\rho^{AB}=\sum_i p_i
\sum_k q^i_k |k\rangle\langle k| \otimes \rho^B_i$, namely, it is an incoherent-quantum state.
For the initial pure state $|\psi^{AB}\rangle$ with vanishing coherence on $A$, the RCC is nonzero if and only if there exists a basis $\{|i\rangle\}$ which gives $[N, (\langle i|\otimes {\openone})
|\psi^{AB}\rangle\langle\psi^{AB}|(|i\rangle\otimes {\openone})] \neq 0$, with $N=\sum_i E_i^\dag E_i\leq {\openone}$. The amount of RCC measured by the $l_1$ norm is bounded above by $$\label{eq4d-13}
C_{l_1}(\tilde{\rho}^A)\leq \frac{E(|\psi^{AB}\rangle)}{p'}\sqrt{\sum_{j<i}|N_{ji}|^2},$$ where $E(|\psi^{AB}\rangle)$ denotes the concurrence of $|\psi^{AB}\rangle$, and $N_{ji}$s are matrix elements of $N$ under the Schmidt decomposition basis of $\rho^B={\mathrm{tr}}_A(|\psi^{AB}\rangle
\langle\psi^{AB}|)$. Furthermore, if the channel $\mathcal{E}$ is trace preserving, the average RCC $$\label{eq4d-v13}
\bar{C}_{l_1}(|\psi^{AB} \rangle)\coloneqq \sum_i p_i C_{l_1}(\tilde{\rho}^A_i),$$ with $\tilde{\rho}^A_i={\mathrm{tr}}_B [({\openone}_A\otimes E_i) |\psi^{AB}\rangle
\langle\psi^{AB}|({\openone}_A\otimes E_i)]/p_i$ and $p_i={\mathrm{tr}}[({\openone}_A\otimes E_i) |\psi^{AB}\rangle
\langle\psi^{AB}|({\openone}_A\otimes E_i)]$, has the following bound $$\label{eq4d-14}
\bar{C}_{l_1}^{A|B}(|\psi^{AB}\rangle) \leq \frac{d}{2}
E(|\psi^{AB}\rangle)\bar{C}_{l_1}^{A|B}(|\Phi^{AB}\rangle),$$ with $|\Phi^{AB}\rangle$ being the maximally entangled state in the Schmidt decomposition basis of $|\psi^{AB}\rangle$, and for $d=2$ case, the equality in the above equation holds. This also establish an operational connection between created coherence of a subsystem and entanglement of the composite system, although applies only for the initial pure states.
Average coherence of randomly sampled states {#sec:4E}
--------------------------------------------
As is known, some measures of quantumness manifest concentration effect, e.g., the random bipartite pure states sampled from the uniform Haar measure are typically maximally entangled [@avee]. Along the same line, @avec studied the coherence properties of pure states chosen randomly from the uniform Haar measure, and found that most of them possess almost the same amount of coherence which are not typically maximally coherent.
For the Haar distributed random pure states $\psi=|\psi\rangle
\langle \psi|$ with dimension $d\geq 3$, they considered the average coherence of the form $$\label{Haar1}
\bar{C}(\psi)\coloneqq \int d(\psi) C(\psi)=\int
d\mu(U)C(U|1\rangle\langle 1| U^\dag),$$ with $U$ being sampled from the uniform Haar distribution, and $C(\psi)$ can be any faithful measure of coherence.
First, for the relative entropy of coherence, its average over all $\psi$ was found to be $$\label{Haar2}
\bar{C}_r(\psi)= H_d-1,$$ with $H_d=\sum_{k=1}^d (1/k)$ the $d$th harmonic number, and the logarithm in Eq. is with respect to natural base here. The probability for $|C_r(\psi)-(H_d-1)|> \epsilon$ is upper bounded by $2e^{-{d \epsilon^2}/{36\pi^3\ln 2 \ln ^2 d}}$, hence the randomly chosen $\psi$ with $C_r(\psi)$ not close to $H_d-1$ is exponentially small. This is the concentration phenomenon for relative entropy of coherence. It reveals that most Haar distributed random $\psi$ have $H_d-1$ amount of coherence, which is solely determined by the parameter $d$.
Second, for $l_1$ norm of coherence, they found that the mean classical purity $\bar{P}(\Delta[\psi])$ averaged over the Haar distributed $\psi$ is given by $2/(d+1)$. The probability for $|P(\Delta[\psi])-2/(d+1)|>\epsilon$ is $2e^{-{d \epsilon^2}/
{18\pi^3\ln 2}}$, which is also exponentially small for $\epsilon\rightarrow 0$. Thus by using the upper bound of $C_{l_1}(\psi)$ given in Eq. , one has $$\label{Haar3}
\bar{C}_{l_1}(\psi)\leq \sqrt{d(d-1)^2/(d+1)}.$$ Finally, to show most of the Haar distributed pure states are not typically maximally coherent, @avec calculated the average trace distance between $\rho^\psi_{\rm diag}$ and the maximally mixed state $\rho_{\rm mm}$ (which is the optimal $\delta$ for $|\Psi_d\rangle$). The result shows that $$\label{Haar4}
\bar{\mathcal{D}} (\rho^\psi_{\rm diag},\rho_{\rm mm})=2(1-1/d)^d,$$ which approaches $2/e$ in the limit of $d\rightarrow \infty$. The probability for a divergence of the amount $\epsilon$ is $2e^{-{d
\epsilon^2}/ {18\pi^3\ln 2}}$, which is arbitrary small for $\epsilon\rightarrow 0$. This shows that the optimal $\delta$ for the majority of $C_r(\psi)$ are not $\rho_{\rm mm}$, hence the random Haar distributed $\psi$ are not maximally coherent.
In fact, for the uniformly distributed pure states, the average $l_1$ norm of coherence can be obtained. The corresponding analytical result is derived by @onenorm, which is given by $$\label{Haar5}
\bar{C}_{l_1}(\psi)=\frac{(d-1)\pi}{4},$$ and there is no concentration phenomenon for it, this is because the probability for $C_{l_1}(\psi)$ not close to $(d-1)\pi/4$ is given by $2e^{-{4\epsilon^2}/ {9d\pi^3 \ln 2}}$, which is finite when $d\rightarrow \infty$. But the scaled $l_1$ norm of coherence $C_{l_1}(\rho)/(d-1)$ concentrates around $\pi/4$ for very large values of $d$, and the probability for a divergence of the amount $\epsilon$ is given by $2e^{-{4(d-1)^2\epsilon^2}/ {9d\pi^3 \ln
2}}$.
Similarly, for $d$-dimensional randomly mixed states sampled from various induced measures, @avec-mix considered the average relative entropy of coherence $$\label{Haar6}
\begin{split}
\bar{C}_r(\alpha,\gamma) &\coloneqq \int d\mu_{\alpha,\gamma}(\rho)C_r(\rho)\\
&= \int d\mu_{\alpha,\gamma}(U\Lambda U^\dag)C_r(U\Lambda U^\dag),
\end{split}$$ with $\Lambda=\mathrm{diag}\{\lambda_1,\cdots,\lambda_d\}$, and $U\Lambda U^\dag$ is the isospectral full-ranked density matrices (i.e., the spectra of $\Lambda$ is nondegenerate), and $\mu_{\alpha,\gamma}$ is the normalized probability measure on the set of density matrix $\mathcal{D}(\mathbb{C}^d)$.
For the special case of mixed $\rho$ sampled from induced measures obtained via partial tracing of the Haar distributed $dd'$-dimensional ($d'\geq d$) pure states $\psi$, the average coherence can be further obtained analytically as $\bar{C}_r(\alpha,\gamma) =(d-1)/2d'$ for $(\alpha,\gamma)=
(d'-d+1,1)$. If $d$ is further restricted to $d\geq 3$, the probability for $|C_r(\alpha,\gamma)-(d-1)/2d'|> \epsilon$ is bounded from above by $2e^{-{dd' \epsilon^2}/{144\pi^3\ln 2 \ln ^2
dd'}}$. Hence, nearly all $\rho$ obtained via partial tracing over the uniformly Haar distributed random pure bipartite states $\psi$ in the Hilbert space $\mathcal{H}_{d'}$ have coherence approximately equal to the average relative entropy of coherence. These results were further extended by the same author in a recent work [@avec-mix2].
Quantum coherence in quantum information {#sec:5}
========================================
Quantum state merging {#sec:5A}
---------------------
For a quantum protocol with two or more parties, e.g., the simplest case of two players, Alice and Bob, one may wonder how much coherence is localized (or consumed) at Bob’s side, and simultaneously, how much entanglement is established (or consumed) for Alice and Bob, after finishing the pre-designed computation procedure?
@qsm explored such a problem. They discussed the protocol of quantum state merging under IO, which they called incoherent quantum state merging, and is indeed an analog of the standard state merging with general quantum operations [@qsm1]. In this task, Alice, Bob, and a referee share the state $\rho^{RAB}$. Alice and Bob also have access to $\Phi_2$ at rate $E$, and Bob has access to $\Psi_2$ at rate $C$. The goal is for them to merge the state of $AB$ on Bob’s side by LQICC, i.e., Alice performs general quantum operations, while Bob is restricted to IO only.
By denoting $E=E_i-E_t$ and $C=C_i-C_t$, with $E_i$ and $E_t$ ($C_i$ and $C_t$) being the entanglement rate of $AB$ (local coherence of $B$) before and after the state merging protocol, @qsm showed that the entanglement-coherence pair $(E,C)$ is achievable if there exists $E_i$, $E_t$, $C_i$, $C_t$, and sufficiently large integers $n$ such that $$\label{eq4e-1}
\begin{aligned}
& \|\Lambda_{QI}[\rho_i^{\otimes n}\otimes\Phi_2^{\lfloor(E_i+\delta)n\rfloor}
\otimes\Psi_2^{\otimes \lfloor (C_i+\delta n)\rfloor}]\\
& - \rho_t^{\otimes n}\otimes\Phi_2^{\otimes\lceil E_t n\rceil}
\otimes\Psi_2^{\otimes\lceil C_t n\rceil} \|_1 \leq \varepsilon,
\end{aligned}$$ is satisfied for every $\varepsilon>0$ and $\delta>0$. Moreover, $\rho_i=\rho^{RAB}\otimes |0^{\tilde{B}}\rangle\langle
0^{\tilde{B}}|$, $\rho_t= \rho^{R\tilde{B}B}\otimes
|0^A\rangle\langle 0^A|$, and $|0^{\tilde{B}}\rangle$ is the initial state of the auxiliary system $\tilde{B}$ (with the same dimension as $A$) belong to Bob. $E>0$ ($C>0$) corresponds to entanglement (coherence) consumption in the task of state merging, while $E<0$ ($C<0$) corresponds to the reverse situation, i.e., the merging protocol is achievable for free, with the additional gain of entanglement (coherence) at rate $|E|$ ($|C|$).
On the above basis, @qsm found that the sum of $E$ and $C$ is upper bounded by a nonnegative quantity, i.e., $$\label{eq4e-2}
E+C \leq S(\Delta^{AB} [\rho^{RAB}])
-S(\Delta^B[\rho^{RAB}]),$$ where $\Delta^{AB}$ and $\Delta^{B}$ are the same as that in Eq. . The equality holds for any pure state $\rho^{RAB}$, for which $(E,C)$ reduces to $(E_0,0)$, with $E_0=S(\bar{\rho}^{AB})
-S(\bar{\rho}^B)$ and $\bar{\rho}^{AB}=\Delta(\rho^{AB})$. It implies that whenever $E<0$, we must have $C\geq 0$, and vice versa. Therefore, there is no state merging procedure for which entanglement and coherence can be gained simultaneously. This can be recognized as an operational complementarity relation between entanglement of a bipartite state and quantum coherence of its reduction.
Deutsch-Jozsa algorithm {#sec:5B}
-----------------------
The Deutsch-Jozsa algorithm is one of the first quantum algorithms in quantum information science. It uses quantum coherence as a resource, and this enables its speedup compared with that of the classical counterpart [@Deutsch].
By considering a discrete quantum walk version, @DJ-coherence studied the Deutsch-Jozsa algorithm. It is performed in three steps: (+1) let the particle sitting at the edge between 0 and $A$ (with the state denoting as $|0,A\rangle$) traverses the vertex $A$, which transforms it to $U_A|0,A\rangle=\sum_{j=0}^N |A,j\rangle
/\sqrt{(N+1)}$; (+2) It goes further from $A$ to $B$ between which there are $N$ paths, and there will be a phase $e^{i\phi_j}$ ($\phi_j=0$ or $\pi$) being added after traversing the vertex of the $j$th path, thus the state becomes $(|0,-1\rangle+
\sum_{j=1}^{N} e^{i\phi_j} |j,B\rangle)/ \sqrt{(N+1)}$; (+3) Finally, the particle traversing the vertex $B$ and it is transformed into $$\label{eq-dj1}
\begin{split}
& \frac{1}{\sqrt{N+1}}|-1,-2\rangle+\frac{1}{N+1}\sum_{j,k=1}^{N}
e^{i\phi_j}e^{2\pi ijk/(N+1)}|B,k\rangle\\
& +\frac{1}{N+1}\sum_{j=1}^{N} e^{i\phi_j}|B,N+1\rangle.
\end{split}$$ To discuss in a quantitative way how quantum coherence affects performance of the algorithm, @DJ-coherence further introduced a qubit (with the initial state $|0\rangle$) to every path of the graph. By supposing the qubit state $|\mu_j\rangle=
\alpha_j|0\rangle_j +\beta_j |1\rangle_j$ after traversing the $j$th vertex, and defining $|\eta\rangle_j =|\mu_j\rangle\prod_{k\neq j}^N
|0\rangle_k$ and $|\eta\rangle_0 = \prod_{k=0}^N |0\rangle_k$, their state after passing through the $N$ paths will be $$\label{eq-dj0}
|\Psi\rangle_{in}=(|0,-1\rangle|\eta_0\rangle+\sum_{j=1}^{N} e^{i\phi_j}
|j,B\rangle|\eta_j\rangle)/\sqrt{(N+1)},$$ for which the $l_1$ norm of coherence is given by $C_{l_1}(|\Psi\rangle_{in})=\sum_{j\neq k}^N
|\langle\eta_k|\eta_j\rangle|/(N+1)$, and the output state after the vertex $B$ is $$\label{eq-dj2}
\begin{split}
|\Psi\rangle_{out}=& \frac{1}{\sqrt{N+1}}|-1,-2\rangle|\eta_0\rangle\\
& +\frac{1}{N+1}\sum_{j,k=1}^{N} e^{i\phi_j}e^{2\pi ijk/(N+1)}|B,k\rangle|\eta_j\rangle\\
& +\frac{1}{N+1}\sum_{j=1}^{N}|\eta_j\rangle e^{i\phi_j}|B,N+1\rangle,
\end{split}$$ then the the probability of finding the particle on the edge between $B$ and $N + 1$ is $$\label{eq-dj3}
\begin{split}
p &= |\langle B,N+1|\Psi\rangle_{out}|^2 \\
&= \frac{1}{(N+1)^2}\sum_{j,k=1}^{N} e^{i(\phi_j-\phi_k)}\langle\eta_k|\eta_j\rangle\\
&\leq \frac{N}{(N+1)^2}+\frac{C_{l_1}(|\Psi\rangle_{in})}{N+1}.
\end{split}$$ Clearly, the amount of coherence in the system limits our ability to distinguish between the constant case (i.e., all $\phi_j$ are the same and thus $p$ takes the maximum value) and the balanced case (half of $\phi_j$ are zero and half of $\phi_j$ are $\pi$, thus $p$ takes the minimum value).
When one have not detect the particle in the edge between $B$ and $N+1$, one can guess we have the balanced case. @DJ-coherence calculated the error probability for the classical and quantum Deutsch-Jozsa algorithm after $m$ trials of the discrete quantum walk experiments, and found they are given respectively, by $$\label{eq-dj4}
p_{error}^{\mathrm{class}} = \frac{1}{2^m},~
p_{error}^{\mathrm{quant}} = \frac{1}{2}(1-v)^m,$$ with $v=\langle\eta_k|\eta_j\rangle$ is supposed to be positive for all $j\neq k$. Thus if $v$ is larger than a critical value, the quantum algorithm always outperforms its classical counterparts.
Grover search algorithm {#sec:5C}
-----------------------
The Grover search algorithm is another important algorithm in the developments of quantum information science [@Grover]. The pursue of the reason for the speedup of this algorithm attract researchers’ interest for many years.
For an $N$-qubit database initialized as $$\label{Grover1}
|\psi_0\rangle=\sqrt{\frac{j}{N}}|X\rangle+ \sqrt{\frac{N-j}{N}}|X^\perp\rangle,$$ where $|X\rangle=\sum_{x_s}|x_s\rangle/\sqrt{j}$, $|X^\perp\rangle=\sum_{x_n}|x_n\rangle/\sqrt{N-j}$, and $j$ represents the number of solutions. To optimize the success probability, one can perform the Grover operation $$\label{Grover2}
G=OD,$$ with $O={\openone}- 2|X\rangle\langle X|$ and $D=2|\psi_0\rangle
\langle\psi_0|-{\openone}$. After $r$ iterations of the Grover operation $G$, the initial state $|\psi_0\rangle$ turns to be $$\label{Grover3}
|\psi_r\rangle\equiv G^r|\psi_0\rangle=\sin\alpha_r|X\rangle+ \cos\alpha_r|X^\perp\rangle,$$ with $\alpha_r=(2r+1)\arctan \sqrt{j/(N-j)}$. The success probability for finding the correct result is $p(r)=\sin^2
\alpha_r$, and the optimal times of search is given by $r_{\mathrm{opt}}=CI[(\pi-\alpha)/2\alpha]$, with $CI[x]$ denoting the closest integer to $x$.
@Liusy calculated the relative entropy and the $l_1$ norm of coherence for $|\psi_r\rangle$, and found that the success probability $p(r)$ depends on the amount of quantum coherence remaining in $|\psi_r\rangle$. To be explicit, $$\label{Grover4}
\begin{aligned}
& C_r(|\psi_r\rangle)=H(p)+\log_2 (N-j)+p\log_2\frac{j}{N-j},\\
& C_{l_1}(|\psi_r\rangle)=\left[\sqrt{jp}+\sqrt{(N-j)(1-p)}\right]^2-1,
\end{aligned}$$ both of which decrease with the increasing value of $p$. Therefore, the larger the quantum coherence depletion (or equivalently, the less the remaining quantum coherence in $|\psi_r\rangle$), the bigger the success probability one can obtain.
@Liusy also calculated the quantum coherence depletion for the generalized Grover search algorithm [@Grover2], and found that the required optimal search time may increase with the increasing quantum coherence depletion. Moreover, quantum correlations such as quantum entanglement and QD cannot be directly related to the success probability or the optimal search time.
Deterministic quantum computation with one qubit {#sec:5D}
------------------------------------------------
The DQC1 algorithm is the first algorithm that shows quantum computation can outperform those of the classical computation even without entanglement [@DQC1-2; @DQC1-3]. The standard DQC1 algorithm starts with an initial product state $|0\rangle\langle
0|\otimes ({\openone}/2^n)$, and then it was transformed into $$\label{eq-dqc1}
\tilde{\rho}^{AR}= \frac{1}{2}\left({\openone}_2\otimes
\frac{{\openone}_{2^n}}{2^n}+|0\rangle\langle 1|\otimes
\frac{U^\dag}{2^n}+|1\rangle\langle 0|\otimes \frac{U}{2^n}\right),$$ after performing a Hadamard operation on the first qubit, who then served as the control qubit when a controlled unitary operation $U$ is performed on the target qubits in the maximally mixed state ${\openone}_{2^n}/2^n$ [@DQC1]. The goal of this algorithm is to estimate the normalized trace of $U$.
As the reduced states of the control qubit after the series operations is given by $$\label{eq-dqc2}
\tilde{\rho}^A=\frac{1}{2}
\left(\begin{array}{cc}
1 & \frac{{\mathrm{tr}}U^\dag}{2^n} \\
\frac{{\mathrm{tr}}U}{2^n} & 1
\end{array}\right),$$ the estimation can be finalized by measurements of the ancilla in an appropriate basis, i.e., $\langle \sigma_x+ i\sigma_y
\rangle_{\tilde{\rho}_A}={\mathrm{tr}}U/ 2^n$.
For the above fashion of DQC1, several works have been undertaken to understand the origin of its superiority over the classical algorithm [@DQC1-mid; @DQC1-qd; @GQD]. Recently, @Mile provided a further viewpoint of its superiority from the perspective of quantum coherence. By choosing the computational basis for the ancilla qubit and the the eigenbasis of $U$ for the register as the reference basis, they studied how the interplay between coherence consumption and creation of QD works in DQC1, and showed that $$\label{eq-dqc3}
\bar{D}(\tilde{\rho}^{AR})\leq \delta C(\rho^A), ~
\bar{D}_{R|A}(\tilde{\rho}^{AR})\leq \delta C(\rho^A),$$ which are direct consequences of Eqs. and . When being measured by the relative entropy, the coherence consumption can be obtained (note that $\rho^A$ is maximally coherent) from Eq. as $$\label{eq-dqc4}
\delta C(\rho^A)=C_r(\rho^A)-C_r(\tilde{\rho}^A)=H_2\left(\frac{1-|{\mathrm{tr}}U|/2^n}{2}\right),$$ where $H_2(\cdot)$ is the binary Shannon entropy function. It shows that the speedup of this algorithm always corresponds to the consumption of quantum coherence in the ancilla. When there is no coherence to be consumed, we must have $|{\mathrm{tr}}U|=2^n$, and thus $U=e^{i\phi}{\openone}$ for some $\phi$.
By considering a duplication of the DQC1 protocol termed as nonlocal deterministic quantum computation with two qubits (NDQC2), i.e., the collaborative task of estimating the product of normalized traces of two unitaries without obtaining the individual trace value of each unitary, @NDQC2 found that its computational advantage can be achieved with quantum states that have no quantum entanglement and QD. To interpret this phenomenon, they introduced an operational definition of nonclassical correlations, that is, a state $\rho^{AB}$ is said to be nonclassical if it enables a collaborative task only using correlated inputs and measurement results of correlations more efficiently than any classical algorithm. Based on this framework, they defined $$\label{eq-dqc5}
C^{\mathrm{net}}(\rho^{AB})=C(\rho^{AB})-C(\rho^A)-C(\rho^B),$$ and suggested that this quantity can be used for interpreting the efficiency of the NDQC2 protocol, as its quantum advantage is achieved only when $C^{\mathrm{net}}(\rho^{AB})>0$.
For the relative entropy of coherence, from Eq. it is clearly that $C_r^{\mathrm{net}}(\rho^{AB})\geq 0$, and it takes the maximum for the maximally coherent states of the form of Eq. . Moreover, $\rho^{AB}$ is a classical-classical state if and only if $C^{\mathrm{net}} (\rho^{AB})=0$ for certain reference bases. Operationally, when there are no local coherence, i.e., when $C(\rho^A)=C(\rho^B)=0$, two spatially separated parties (Alice and Bob) cannot distil quantum coherence on neither sides using LICC if and only if $C_r^{\mathrm{net}}(\rho^{AB})=0$. This shows another physical implication of the net global coherence as a primitive property of quantum systems which is distinct from those captured by entanglement or QD.
Based on the aforementioned facts, @NDQC2 also gave a basis dependent characterization of nonclassical states, that is, a state $\rho^{AB}$ is said to be nonclassical if and only if $$\label{eq-dqc6}
C_{r,\max}^{\mathrm{net}}(\rho^{AB})= \max_{\{|i\rangle_A|j\rangle_B\}}
C_r^{\mathrm{net}}(\rho^{AB})
>0,$$ and it vanishes only for the product states $\rho^{AB}=\rho^A\otimes
\rho^B$. On the contrary, as $C_r^{\mathrm{net}}(\rho^{AB})=
I(\rho^{AB})- I(\Delta[\rho^{AB}])$, then when it is minimized over the reference bases, we obtain the symmetric discord [@RQC], see Eq. .
Quantum metrology {#sec:5E}
-----------------
Considering an explicit metrology task, i.e., the phase discrimination (PD) game. In this game, a particle in the state $\rho\in \mathcal{D} (\mathbb{C}^d)$ passes through a black box, after which an unknown phase was encoded to it as $U_\phi\rho
U_\phi^\dag$, with $U_\phi=\sum_{j=0}^{d-1} e^{ij\phi}|j\rangle
\langle j|$, $\phi\in \mathbb{R}$, and $\{|j\rangle\}$ being the reference basis. For a collection of pairs $\Theta=
\{p_k,\phi_k\}_{k=0}^{m-1}$, the goal of the PD game is to predict the phases $\{\phi_k\}$ with success probability as high as possible. In general, the optimal probability can be obtained by optimizing over all measurements $\{M_k\}$, and it is given by $$\label{eq-metrology}
p_\Theta^{\mathrm{succ}}(\rho)=\max_{\{M_k\}}\sum_k p_k {\mathrm{tr}}[U_{\phi_k}\rho
U_{\phi_k}^\dag \rho],$$ @meas6 showed that for the above metrology task, the optimal probability can be linked to RoC of the state $\rho$, i.e., $$\label{metro1}
\max_{\Theta}\frac{p_\Theta^{\mathrm{succ}}(\rho)}{p_\Theta^{\mathrm{succ}}(\mathcal{I})}
=1+C_R(\rho),$$ where the maximum is achieved for $\Theta^\star= \{1/d,2\pi
k/d\}_{k=0}^{d-1}$, and $p_\Theta^{\mathrm{succ}}(\mathcal{I})$ is the corresponding classical probability obtained only by guessing. Therefore, $C_R(\rho)$ quantifies the quantum advantage of the PD task, thus suggests a prominent role of RoC in quantum information processing.
@Yucsskw also investigated a similar metrology task. The difference is that they linked the skew-information-based coherence measure $C_{sk}(\rho)$ in Eq. to uncertainty of the estimated phase. By using the quantum Cramér-Rao bound $$\label{metro2}
(\delta\phi_k)^2 \geq \frac{1}{N \mathcal{F}(\rho_\phi)},$$ with $\mathcal{F}(\rho_\phi)$ being the quantum Fisher information given in Eq. , they showed that $$\label{metro3}
\frac{1}{4N C_{sk}(\rho)}\leq \sum_k (\delta\phi_k^\star)^2 \leq \frac{1}{8N C_{sk}(\rho)},$$ where $\delta\phi_k^\star$ denotes the optimal variance. It shows that the measurement precision can be increased by increasing RoC of the state $\rho$.
By considering the subchannel discrimination task which is a generalization of the PD task, @Buprl provided an operation interpretation for the maximum relative entropy of coherence defined in Eq. . Here, the subchannel refers to a map that is linear completely positive and trace nonincreasing, and an instrument $\mathfrak{I}$ for a channel $\mathcal{E}= \sum_a
\mathcal{E}_a$ is a collection of subchannels $\{\mathcal{E}_a\}$. The optimal probability of successfully discriminating the subchannels in $\mathfrak{I}$ reads $$\label{metro4}
P_\mathfrak{I}^{\mathrm{succ}}(\rho)= \max_{\{M_k\}}\sum_a {\mathrm{tr}}[\mathcal{E}_a(\rho)M_k],$$ where the optimization is taken over the POVM $\{M_k\}$. If we are restricted to the set of incoherent states, the resulting optimal probability turns out to be $P_\mathfrak{I}^{\mathrm{succ}}(\mathcal
{I})= \max_{\sigma\in \mathcal {I}}P_\mathfrak{I}^{\mathrm{succ}}
(\sigma)$. @Buprl showed that the optimal maximum advantage achievable in subchannel discrimination can be characterized by the maximum relative entropy of coherence. To be precise, we have $$\label{metro5}
2^{C_{\max}(\rho)}=\max_{\mathfrak{I}} \frac{P_{\mathfrak{I}}^{\mathrm{succ}}(\rho)}
{P_{\mathfrak{I}}^{\mathrm{succ}}(\mathcal{I})},$$ which is very similar to Eq. as $C_\mathrm{max}
(\rho)$ is connected to $C_R(\rho)$ via $C_\mathrm{max}(\rho)=\log_2
[1+C_R(\rho)]$, see Eq. .
Quantum correlations and coherence under quantum channels {#sec:6}
=========================================================
As a precious physical resource for implementing quantum computation and communication tasks that are otherwise impossible classically, and due to the obvious fact that nearly all quantum systems are inevitably interact with their surroundings which may cause decoherence and other negative effects, the study of GQD and quantum coherence, in particular, the control and maintenance of them in noisy environments, is of equal importance to the study of other similar problems such as quantum correlation measures [@RMP; @Licf].
Frozen phenomenon of QD and quantum coherence {#sec:6A}
---------------------------------------------
### Freezing of quantum discord
@frozen-epl investigated the family of local quantum channels under the action of which the QD is preserved for all bipartite states. By using a result of @Petz which says that $$\label{free1}
S(\rho\|\sigma)= S(T[\rho]\|T[\sigma]),$$ if and only if the map $\rho\mapsto T[\rho]$ and $\sigma\mapsto
T[\sigma]$ are invertible, they showed that the mutual-information-based QD is frozen for all states if and only if the channels are invertible. Explicitly, by denoting $\Lambda_A$ ($\Lambda_B$) the quantum channel acting on party $A$ ($B$), then $$\label{free2}
D_A(\rho^{AB})=D_A(\Lambda_A\otimes\Lambda_B [\rho^{AB}]),$$ if and only if there exists $\Lambda_A^*$ and $\Lambda_B^*$ such that $$\label{fro-01}
\begin{aligned}
&(\Lambda_A^*\otimes\Lambda_B^*)(\Lambda_A\otimes\Lambda_B)[\rho^{AB}]=\rho^{AB},\\
&(\Lambda_A^*\otimes\Lambda_B^*)(\Lambda_A\otimes\Lambda_B) [\rho^A\otimes\rho^B]=
\rho^A\otimes\rho^B.
\end{aligned}$$ Moreover, for a distance measure of two states that is monotonic under the action of quantum channel, the corresponding GQD defined based on it is frozen if and only if the local quantum channels $\Lambda_A$ and $\Lambda_B$ are invertible. The related distance measures include those based on the trace norm and Uhlmann fidelity.
For certain quantum channels, the QD may be frozen for a restricted family of states. @frozen-pra studied such a problem. They considered the phase damping channel whose action on a state can be described by $\Lambda_{pd}(\rho)=\sum_i E_i\rho E_i^{\dag}$, with $$\label{eq-pd}
E_0={\rm diag}\{1,p(t)\},~
E_1={\rm diag}\{0,\sqrt{1-p^2(t)}\},$$ being the Kraus operators, and $p(t)$ a time-dependent parameter containing the information of the channel. For the initial Bell-diagonal states $\rho^{\rm Bell}$ of Eq. with one subsystem subjecting to the channel $\Lambda_{pd}$, they obtained necessary and sufficient conditions for freezing QD, which are given in terms of the triple $(c_1,c_2,c_3)$. Explicitly, the QD in $\rho^{\rm Bell}$ is frozen if and only if $$\label{fro-02}
\begin{aligned}
&c_2=-c_1c_3,~ |c_1| > |c_3| \\
\rm {or}~ & c_1=-c_2c_3, ~ |c_2| >|c_3|.
\end{aligned}$$ The above condition is also of special importance for studying the universal freezing of geometric quantum correlations [@frozen-universal]. Besides these, they also generalized their results to an extended family of two-qubit states $$\label{free3}
\rho=\rho^{\rm Bell}+\frac{1}{4}(c_{12}\sigma_1\otimes\sigma_2+ c_{21} \sigma_2\otimes
\sigma_1),$$ and obtained a similar necessary and sufficient conditions.
@frozen-ohmic also studied the frozen phenomenon of QD in dephasing reservoir. The difference is that they considered the explicit Ohmic-type spectrum given by $$\label{free4}
J(\omega)=\omega^s \omega_c^{1-s}e^{-\omega/\omega_c},$$ with $\omega_c$ being cutoff frequency of the reservoir, and it is said to be sub-Ohmic if $0<s < 1$, Ohmic if $s = 1$, and super-Ohmic if $s > 1$. For a subset of the initial Bell-diagonal state $$\label{free5}
\rho^{\rm Bell}_{\rm sub}=\frac{1+c}{2}|\Psi^{\pm}\rangle
\langle\Psi^{\pm}|+\frac{1-c}{2}|\Phi^{\pm}\rangle
\langle\Phi^{\pm}|,$$ with $|\Psi^{\pm}\rangle=(|00\rangle\pm |11\rangle)/\sqrt{2}$ and $|\Phi^{\pm}\rangle=(|01\rangle\pm |10\rangle)/\sqrt{2}$, they obtained the expression of the evolved QD, and found that if $e^{-\Lambda(\bar{t})}=c$ \[$\Lambda(t)$ is the dephasing factor\], then there will be a transition from classical decoherence to quantum decoherence. But if there is no solution for $e^{-\Lambda(\bar{t})}
=c$, the QD will be frozen forever, with the time-invariant value $$\label{free6}
D_A(\rho^{\rm Bell}_{\rm sub})=\frac{1+c}{2}\log_2(1+c) +\frac{1-c}{2} \log_2 (1-c).$$ The $(s,c)$ region for which the frozen condition is satisfied is determined by temperature of the reservoir. For the zero-temperature case, they obtained numerically the corresponding $(s,c)$ region, which shrinks with the increase of $c$ and vanishes when $c\gtrsim
0.16$.
For two qubits prepared initially in the Bell-diagonal states described by the triple $(c_1,c_2,c_3)$, there may be universal freezing of GQD defined based on the distance measures $\mathcal{D}$ of quantum states that satisfy the following conditions: (+1) contractivity under CPTP maps, (+2) invariant under transposition, and (+3) convexity under mixing of states. Distance of this type include the relative entropy, the squared Bures distance, the squared Hellinger distance, and the trace distance.
From the above conditions, @frozen-universal considered the initial Bell-diagonal state $\rho^{\rm Bell}$ of Eq. with the triple $(c_1,-c_1c_3,c_3)$, and proved that its distance to $(c_1,0,0)$ \[$(0,0,c_3)$\] is independent of $c_1$ ($c_3$). Moreover, one of the closest classical state to $\rho^{\rm Bell}$ is still a Bell-diagonal state $(s_1,s_2,s_3)$ with however only one of $s_k$ is nonzero, and for the special case $c_2=-c_1c_3$, the closest classical state further reduces to $(c_1,0,0)$ if $|c_1|\geqslant
|c_3|$, and $(0,0,c_3)$ otherwise. This extends the results of Eq. . As it shows Eq. holds for general distance measure of states satisfying the above three conditions.
Based on these formulas, @frozen-universal found that when the two qubits are subject to independent phase flip (similar for bit flip and bit-phase flip) channels, the GQDs satisfying the above three conditions will be frozen in the time interval $t<t^*=-(1/2\gamma) \ln(|c_3(0)|/ |c_1(0)|)$. As this conclusion depends only on the proposed properties of distance measures of states, it shows the universal freezing of geometric quantum correlations.
@traceqd1 studied the trace norm of discord for a two-qubit system (initially prepared in the Bell-diagonal state) passes through the local bit flip, phase flip, bit-phase flip, and generalized amplitude damping channels. Through detailed analysis with different initial state parameters $(c_1,c_2,c_3)$, they found that the trace norm of discord exhibits the phenomenon of freezing behavior during its evolution process. @traceqd2 discussed the trace norm of discord with the locally applied phase-flip channels and random external fields and observed the freezing phenomenon. They also compared dynamics of the total and classical classical correlations defined via the trace norm, see Eq. . Moreover, the trace norm, Bures distance, and Hellinger distance measure of GQD for two non-interacting qubits subject to two-sided and one-sided thermal reservoirs have also been investigated [@husun]. In fact, the frozen phenomenon of various GQDs were proved to be universal by @bures2 and @frozen-universal.
### Freezing of quantum coherence
For a $N$-qubit quantum system subject to local independent and identical decohering environments, @froz1 studied decay dynamics of coherence and provided important insights between them and the discordlike correlation measures. The extension of two-qubit Bell-diagonal states, i.e., $$\label{cfree1}
\rho^{\rm Bell}_N= \frac{1}{2^N}\left({\openone}_2^{\otimes N}+\sum_{i=1}^3 c_i
\sigma_i^{\otimes N}\right),$$ are also described by the triple $(c_1,c_2,c_3)$. For a system with even number of qubits, @froz1 found that if $$\label{cfree2}
c_2= (-1)^{N/2}c_1 c_3,$$ then all bona fide distance-based coherence measures will be permanently frozen for local bit flip channel (similar result can be obtained for local bit-phase flip channel by exchanging $c_1$ and $c_2$). These include the relative entropy of coherence for general even $N$, and the trace norm of coherence for $N=2$.
Moreover, for general one-qubit state (i.e., $\rho^{\rm Bell}_N$ with $N=1$) subject to bit flip channel, the $l_1$ norm of coherence is frozen forever if $c_2=0$, while for the two-qubit state of Eq. with the elements of $T$ vanishing for all non-diagonal elements, it is frozen when the parameters $$\label{cfree3}
x_2=y_2=0,~T_{22}=uT_{11},~~(-1\leq u\leq 1).$$ Experimentally, the freezing phenomenon for relative entropy of coherence [@coher], fidelity-based measure of coherence [@meas2], and trace norm of coherence [@froz1] for two and four qubits exposing to the phase damping channel were observed in an nuclear magnetic resonance system [@expfreeze].
If an incoherent operation satisfy not only $K_i\mathcal{I}
K_i^\dag\subset \mathcal{I}$, but also the additional constraint $K_i^\dag \mathcal{I} K_i\subset \mathcal{I}$ for all $K_i$, then it is said to be strictly incoherent [@qcd1]. Their Kraus operators contain at most one nonzero entry in each row and each column, and incoherent channels of such type cover the paradigmatic source of noises in quantum information science, e.g., the bit flip, phase flip, bit-phase flip, depolarizing, amplitude damping, and phase damping channels.
By restricting to strictly incoherent channels, @froz2 established a measure-independent freezing condition of coherence, which states that for any initial state of a system, all measures of its coherence are frozen if and only if its relative entropy of coherence is frozen. The proof for this claim comprises two essential steps. First, if $\delta^\star$ is the closest incoherent state to $\rho$ in the definition of $C_r(\rho)$, then $\Lambda[\delta^\star]$ is the closest state to $\Lambda[\rho]$ for $C_r(\Lambda[\rho])$. Second, if the channel maps $\rho(0)$ to $\rho(t)$, i.e., $\Lambda[\rho(0)]=\rho(t)$, then one can always construct an incoherent operation which gives the map $\mathcal
{R}[\rho(t)]=\rho(0)$.
For a system of $N$ qubits interacting independently with $N$ bit flip (not necessary to be identical) channels, @froz2 further identified two families of states for which all measures of quantum coherence are frozen, they are given respectively by:
\(1) $|\varphi^{\pm}_l\rangle=(|l\rangle\pm |\bar{l}\rangle)
/\sqrt{2}$, with the sequences $l=l_1 l_2 \cdots l_N$, $\bar{l}=\bar{l}_1 \bar{l}_2 \cdots \bar{l}_N$, $l_1=0$, $l_{i\neq
1}=\{0,1\}$, $\bar{l}_i=1-l_i$.
\(2) $\rho=\sum_l p_l [p|\varphi^{+}_l\rangle
\langle\varphi_l^{+}|+(1-p)|\varphi^{-}_l \rangle\langle
\varphi_l^{-}|]$, with $0\leq p\leq 1$, and $p_l$ being any probability distribution.
By decomposing state $\rho$ as Eq. , and using a transformation matrix $T$ to describe the action of $\mathcal {E}$, i.e., $$\label{eq-master}
\mathcal{E}^\dag (X_i)=\sum_j T_{ij}X_j,$$ @fac3 also derived a condition for freezing the $l_1$ norm of coherence. They found that when $T_{k0}=0$ for $k\in
\{1,2,\ldots,d^2-d\}$, and $T^S$ (the submatrix of $T$ consisting $T_{ij}$ with $i$ ranging from 1 to $d^2-d$ and $j$ from 1 to $d^2-1$) is a rectangular block diagonal matrix, with the main diagonal blocks $$\label{eq5a-1}
T^S_r =\left(\begin{array}{cc}
T_{2r-1,2r-1} & T_{2r-1,2r} \\
T_{2r,2r-1} & T_{2r,2r}
\end{array}\right)
~(r\in \{1,\dots,d_0\}),$$ being orthogonal matrices, i.e., $(T^S_r)^T T^S_r = {\openone}_2$, the $l_1$ norm of coherence for $\rho^{\hat{n}}$ will be frozen during the entire evolution. Here, $\rho^{\hat{n}}$ represents states with the characteristic vectors $\vec{x}$ \[see Eq. \] along the same or completely opposite directions but possessing different lengths.
Enhancing the quantum resources via quantum operations {#sec:6B}
------------------------------------------------------
Since QD and quantum coherence are both quantum properties of quantum states, the ability of a quantum channel to create and/or enhance strength of QD or quantum coherence is related to the quantumness of the channel. It is then of interest to study whether a channel has the ability to create quantum resources, and how many quantum resources the channel can create or enhance.
### Creation of quantum discord from classical states
When a bipartite system is coupled to a common bath, it was proved that a Markovian dissipative quantum channel can generate QD from some bipartite product states if and only if it cannot be reduced to individual decoherence channels independently acting on each qudit [@create-collective]. Further, if the subsystems initially share classical correlations, even local operations can create QD.
The local creation of quantum correlations was first studied by @create-prl. They investigated the completely decohering (or semiclassical) channel $\Lambda_{sc}$ described by $$\label{semic}
\Lambda_{sc}(\rho)=\sum_k p_k(\rho)|k\rangle\langle k|,$$ and the unital channel $\Lambda_{u}$ which keeps the maximally mixed state invariant. For a single qubit of a multiqubit system subject to a channel, they proved strictly that $\Lambda_{sc}$ and $\Lambda_u$ are the only two types of channels that cannot create quantum correlations. Equivalently, for qubit systems, a necessary and sufficient condition for a local channel to create quantum correlation is that the channel is neither completely decohering nor unital. The geometric quantum correlations defined based on contractive distance measures of states are further showed to be nonincreasing under local semiclassical channels and local unital channels. However, this result does not hold for states with higher dimensions. For multi-qudit states with dimension $d\geq 3$, even unital channels can create quantum correlation.
@create-huxy defined commutativity-preserving channels $\Lambda_{cp}$ as those preserve the commutativity of any two input states, that is, $$\label{eq-cp}
[\rho, \sigma]=0\Rightarrow [\Lambda_{cp}(\rho),\Lambda_{cp}(\sigma)]=0.$$ For any finite-dimensional multipartite systems, they proved that a channel $\Lambda$ acting locally on $B$ of a quantum-classical state can create quantum correlation if and only if $\Lambda\notin
\Lambda_{cp}$. For qubit case, a commutativity-preserving channel is either a completely decohering channel or a unital channel. For qutrit case, a commutativity-preserving channel is either a completely decohering channel or an isotropic channel, which is defined as $$\label{cre-01}
\Lambda_{iso}(\rho)=p\Gamma(\rho)+(1-p) \frac{{\openone}}{d},$$ with $p$ being the parameter for ensuring CPTP of $\Lambda_{iso}$, and $\Gamma$ is either a unitary operation \[$-1/(d-1)\leqslant p
\leqslant 1$\] or is unitarily equivalent to transpose \[$-1/(d-1)
\leqslant p \leqslant 1/(d+1)$\].
@create-huxy also conjectured that for systems with dimension higher than 3, a commutativity-preserving channel is also either isotropic or completely decohering. @create-jpa further gave an affirmative answer to this conjecture. They proved that $\Lambda$ acting on party $B$ of a system cannot create QD (i.e., $D_B(\rho_{AB})=0 \Rightarrow D_B({\openone}\otimes
\Lambda[\rho_{AB}])=0$) if and only if it is either a completely decohering channel or a nontrivial isotropic channel ($\Lambda_{iso}$ with $p\neq 0$). Channels of these types are also showed to preserve commutativity and normality of quantum states. Furthermore, $\Lambda_B$ which yields $D_B(
\rho_{AB})=0\Leftrightarrow D_B(\Lambda_B[\rho_{AB}])=0$ are restricted only to the nontrivial isotropic channels.
@create-pra considered the local creation of QD by a Markovian amplitude-damping channel described by $\Lambda_{ad}(\rho)=\sum_i E_i\rho E_i^{\dag}$, with $$\label{cre-02}
E_0=|0\rangle\langle 0|+ \sqrt{1-p(t)} |1\rangle\langle 1|,~
E_1=\sqrt{p(t)}|0\rangle \langle 1|.$$ For the initial state $$\label{cre-03}
\rho =\frac{1}{2}(|0\rangle \langle 0| \otimes \tau_{0}+|1\rangle
\langle 1|\otimes\tau_{1}),$$ with the length of local Bloch vectors for $\tau_{0}$ and $\tau_{1}$ equal to each other (i.e., $|\vec{s}_0|=|\vec{s}_1|=s$), they obtained $$\label{cre-04}
\begin{aligned}
D_B(\rho)= & h\left(\frac{1+s|\cos(\varphi/2)|}{2}\right)+
h\left(\frac{1+s|\sin(\varphi/2)|}{2}\right)\\
& -h\left(\frac{1+s}{2}\right)-1,
\end{aligned}$$ where $\varphi$ is the angle between $\vec{s}_0$ and $\vec{s}_1$. Using this formula, they showed rigorously that $\Lambda_{ad}$ acting on $B$ can create QD from $\rho$. In fact, it is easy to check that $\Lambda_{ad}$ is neither a completely decohering nor a unital channel, hence the local creation of QD in the present case is easy to understand from the result of @create-huxy.
Now we have reviewed the conditions on the quantum channels which has the ability to increase quantum correlations. An equally important problem is to characterize the quantum states whose quantum correlations can be increased locally. @increase-huxy studied this problem by employing the tool of quantum steering ellipsoids [@qse]. They considered the amplitude damping channel acting on qubit $B$ of a Bell-diagonal state of Eq. . For such a state, both $\mathcal E_A$ and $\mathcal E_B$ are unit spheres shrunk by $c_1,\ c_2$ and $c_3$ in the $x,\ y$ and $z$ direction, respectively. It is observed that, the local increase of discord occurs when $|c_1|\gg |c_2|,|c_3|$. An interesting consequence is that, the local quantum operation can increase the QD of an entangled state.
### Enhancing the coherence via quantum operations
Quantum coherence measures should be monotonically decreasing under IO, which are a strict subset of non-coherence-generating (NC) channels. The behavior of different measures of coherence under the action of NC channels was studied by @ncgc. While the relative entropy of coherence was proved monotone under all NC channels, the coherence of formation $C_f$ can be increased by some NC channels. An example was presented that $C_f$ of a two-qubit state is increased when a NC qubit channel is acting on one of the two qubits. Here, the NC channel is chosen as $\Lambda(\cdot)=
E_1(\cdot)E_1^\dagger+E_2(\cdot)E_2^\dagger$, with $$\label{ncchannel}
E_1=\frac{1}{2}\left(\begin{array}{cc}
1 & 0\\
-1 & \sqrt2
\end{array}\right),~
E_2=\frac{1}{2}\left(\begin{array}{cc}
1 & \sqrt2\\
1 & 0
\end{array}\right),$$ and the two-qubit input state is $\Psi^+\equiv |\Psi^+\rangle
\langle\Psi^+|$ with $|\Psi^+\rangle=(|00\rangle+|11\rangle)/
\sqrt{2}$. It was checked that the $C_f$ of the output state is strictly larger than the input state, i.e., $C_f({\openone}\otimes
\Lambda(\Psi^+))>1=C_f(\Psi^+)$. Interestingly, the channel $\Lambda(\cdot)$ can never increase $C_f$ of a single-qubit state, so the ability of this channel to increase coherence is enhanced when extending to composed Hilbert space. The reason for this enhancement is that, the local NC operation turn the quantum correlation into the local coherence, and meanwhile increase the quantum coherence of the total state.
Although the amount of coherence for a state cannot be enhanced under IO by definition, this does not prevent us from obtaining probabilistically a postmeasurement state with enhanced coherence when selective measurements are allowed, e.g., by retaining those $\rho_n=K_n\rho K_n^\dagger/p_n$ \[$p_n={\mathrm{tr}}(K_n\rho K_n^\dagger)$\] that satisfy $C(\rho_n)> C(\rho)$ and discarding the other $\rho_n$, one can obtain a mixed state $\sum_{n|C(\rho_n)> C(\rho)}p_n\rho_n$ with enhanced coherence. @SSIO considered one such problem. By taking the $l_1$ norm of coherence as a measure and considering the stochastic strictly incoherent operation $\Lambda_s$ whose Kraus operators $\{K_n\}_{n=1}^L$ (a subset of SIO) fulfilling $\sum_{n=1}^L K_n^\dagger K_n\leq {\openone}$, they obtained the maximum attainable coherence for the postmeasurement state $\Lambda_s[\rho]$, which reads $$\label{eq-ssio01}
\max_{\Lambda_s} C_{l_1}(\Lambda_s[\rho])=\lambda_{\max}
\left(\rho_{\mathrm{diag}}^{-1/2}|\rho|\rho_{\mathrm{diag}}^{-1/2} \right)-1,$$ where $\lambda_{\max}(\cdot)$ is the largest eigenvalue of $\rho_{\mathrm{diag}}^{-1/2}|\rho|\rho_{\mathrm{diag}}^{-1/2}$, and $|\rho|$ is a matrix obtained from $\rho$ by taking absolute values to all its elements. @SSIO also constructed the Kraus operator and the corresponding optimal probability for obtaining Eq. . If $\rho$ is irreducible, by denoting $|\varphi_{\max} \rangle=(\varphi_1, \varphi_2, \ldots,
\varphi_d)^T$ ($d= \dim\rho$) the eigenvector corresponding to the largest eigenvalue of $\rho_{\mathrm{diag}}^{-1/2}|\rho|
\rho_{\mathrm{diag}}^{-1/2}$ and $U_{\mathrm{in}}$ an arbitrary incoherent unitary matrix, one has $$\label{eq-ssio02}
\begin{aligned}
& K'= \min_i \frac{\sqrt{\rho_{ii}}}{\varphi_i} U_{\mathrm{in}}\mathrm{diag}
\left( \frac{\varphi_1}{\sqrt{\rho_{11}}}, \frac{\varphi_2}{\sqrt{\rho_{22}}},
\ldots,\frac{\varphi_d}{\sqrt{\rho_{dd}}}\right), \\
& p_{\max}(\rho)= \min_i \frac{\rho_{ii}}{\varphi_i^2},
\end{aligned}$$ and if $\rho$ is reducible, i.e., it can be transformed by a permutation matrix into $p_1\rho_1 \oplus p_2\rho_2\oplus \ldots
\oplus p_n\rho_n \oplus \bold{0}$, one has $$\label{eq-ssio03}
\begin{aligned}
& K'= U_{\mathrm{in}}(K'_1\oplus K'_2 \oplus\ldots\oplus K'_n \oplus\bold{0}), \\
& p_{\max}(\rho)= \sum_{\lambda_{\max}^\alpha
= \lambda_{\max}}p_\alpha p_{\max}(\rho_\alpha),
\end{aligned}$$ with $$\label{eq-ssio04}
K'_\alpha=\min_i \frac{\sqrt{\rho_{ii}^\alpha}}{\varphi_i^\alpha}
\mathrm{diag} \left( \frac{\varphi_1^\alpha}{\sqrt{\rho_{11}^\alpha}},
\frac{\varphi_2^\alpha}{\sqrt{\rho_{22}^\alpha}},\ldots,\frac{\varphi_d^\alpha}
{\sqrt{\rho_{dd}^\alpha}}\right).$$
### Energy cost of creating quantum coherence
As it has been shown in the above sections, quantum operations acting on an incoherent state can map it to be incoherent. The amount of quantum coherence in a non-maximally coherent state can also be enhanced via some quantum operations. When the quantum operations are restricted to be unitary, it has been shown by @RQC, @maxco, and @Yucs that the maximal achieved relative entropy of coherence is $\log_2 d-S(\rho)$ for any initial state $\rho$, see Eq. . When $\rho$ is incoherent, this is also the maximal coherence created by unitary operation.
In a similar manner, @energy also considered the maximal creation of quantum coherence. They considered the initial state to be the thermal state of a system, and adopted eigenbasis $\{|j\rangle\}$ of the system Hamiltonian $\hat{H}$ as the reference basis. The initial thermal state $\rho^T=e^{-\hat{H}/T}$ ($T$ is the temperature) before acting the unitary operation is incoherent. The maximum amount of relative entropy of coherence created by using unitary operations thus has the same form as Eq. , i.e., $C_r^{\rm max}(\rho^f)=\log_2 d-S(\rho^T)$, where $\rho^f=U\rho^T U^\dag$ denotes the output state after performing the unitary operation.
To construct the corresponding optimal unitary operations, @energy used the maximally coherent basis $\{|\phi_j\rangle\}$ which are very similar to that of the maximally entanglement basis (e.g., for the two-qubit case they are the four Bell states). Here, all $\{|\phi_j\rangle\}$ have maximal value of coherence, and they are orthogonal to each other. To be explicit, they can be written as $$\label{eq-mcb1}
|\phi_j\rangle=\frac{1}{\sqrt{d}}\sum_{m=0}^{d-1} e^{i\frac{2\pi jm}{d}}|m\rangle,$$ which is in fact a map of $\mathbb{Z}= \sum_m e^{2\pi i m/d}
|m\rangle\langle m|$ on the maximally coherent state $|\Psi_d\rangle$, and $i$ in the superscript is the imaginary unit. Then they gave an unitary operation $$\label{eq-mcb2}
U=\sum_{j=0}^{d-1} |\phi_j\rangle\langle j|,$$ for which the output state after the action of it is given by $$\label{eq-mcb3}
\rho^f=\frac{1}{Z}\sum_{j=0}^{d-1} e^{-E_j/T}|\phi_j\rangle\langle\phi_j|,$$ thus $S(\rho^f_\mathrm{diag})=\log_2 d$.
On the other hand, during these processes of coherence creation and coherence enhancement, a supply of external energy is needed. Hence, it is natural to inquire if there are quantitative connections between the created quantum coherence and the amount of energy cost? In general, the energy cost is given by $\Delta E={\mathrm{tr}}(U\rho U^\dag
\hat{H})- {\mathrm{tr}}(\rho\hat{H})$.
For the case of initial thermal state $\rho^T$, by using the fact that ${\mathrm{tr}}(\rho^f \hat{H})={\mathrm{tr}}(\rho^f_\mathrm{diag} \hat{H})$ as $\hat{H}$ is diagonal, and the maximum entropy principle which says that the thermal state has maximum entropy among all states with a fixed average energy [@mep1; @mep2], one can show that with limited energy cost $\Delta E$, the maximum created relative entropy of coherence is bounded from above by $$\label{eq-bound}
C_r^{\max}(\Delta E)\leq S(\rho^{T'})-S(\rho^T),$$ where $\rho^{T'}$ is the thermal state at the higher temperature $T'$ such that $$\label{eq-bound2}
\Delta E={\mathrm{tr}}(\rho^{T'}\hat{H})-{\mathrm{tr}}(\rho^T\hat{T}).$$ To obtain the maximal coherence with limited energy $\Delta E$, i.e., to saturate the upper bound of Eq. , one should find an optimal $U$ such that the diagonal part of $\rho^f$ equals to $\rho^{T'}$. @energy proved strictly that there always exists such an (real) unitary. The derivation of such an unitary for single-qubit state is easy, but for higher dimensional case it turns out to be very complicated.
For multipartite system, @energy also compared the amounts of quantum coherence and quantum total correlations (measured by the quantum mutual information) by using the same unitary operations. For the noninteracting system described by the Hamiltonian $\hat{H}=\sum_{k=1}^N \hat{H}_k$ ($\hat{H}_k$ is the Hamiltonian for subsystem $k$), and starting from the initial product thermal states $\prod_k \otimes \rho_k^{T}$ of each subsystems, the maximum created quantum mutual information with limited energy $\Delta E$ is given by $$\label{eq-energy}
I^{\max}(\Delta E)=\sum_k [S(\rho_k^{T'})-S(\rho_k^T)],$$ where the optimal unitary transforms the initial state to a final state $\rho^f$ whose all marginals are thermal states at a higher temperature $T'$, i.e., $\rho^{T'}_k=e^{-\hat{H}_k/T'}/{\mathrm{tr}}e^{-\hat{H}_k/T'}$.
As $\rho^f_\mathrm{diag}$ and the products of the marginals $\prod_k
\otimes\rho^{T'}_k$ have the same average energy, the maximum entropy principle implies that $S(\rho^f_\mathrm{diag})\leq
S(\prod_k \otimes \rho_k^{T'})$. Hence, when the maximum correlation is created among the multipartite system, the corresponding coherence is upper bounded by it. Contrary, if maximum coherence is created in the multipartite system, then the diagonal part of the output $\rho^f$ will be a thermal state at temperature $T'$, and it has the same average energy with the products of the marginals $\prod_k\otimes\rho_k^f$ (with $\rho^f_k={\mathrm{tr}}_{l\neq k}\rho^f$) due to the product structure of $\hat{H}$, the maximum entropy principle implies $S(\rho^f_\mathrm{diag})\geq \sum_k S(\rho^f_k)$. Hence, in this case the created correlation turns to be bounded from above by the maximum created coherence. As for the problem of whether the maximum quantum coherence and correlation can be created simultaneously, the study of the two-qubit case shows that the answer to this may be negative [@energy].
Resource creating and breaking power {#sec:6C}
------------------------------------
When a quantum channel has the ability to create or enhance quantum resources, it is of interest to quantify this ability. This quantification can be regarded an intrinsic property of the channel. As a dual problem, the power of a channel to decrease or destroy the quantum resources, also attracts some research interest.
### Quantum correlating power
The quantum correlating power (QCP) is defined as the maximum amount of quantum correlations that can be created when the channel acts locally on one party of a multipartite system [@PhysRevA.87.032340], i.e., $$\label{qcp1}
\mathcal{Q}(\Lambda)\coloneqq \max_{\rho\in \mathcal{CQ}} Q(\Lambda\otimes {\openone}(\rho)),$$ where $Q$ is a bona fide measure of quantum correlation, and $\mathcal {CQ}$ denotes the set of classical-quantum states.
The QCP is an intrinsic attribute of a channel, which quantifies the channels’s ability to create quantum correlations. In its definition, the maximization is taken over the set of quantum-classical states. The input states that correspond to the maximization are called the optimal input states, which are proved to be in the set of classical-classical (CC) states $$\label{qcp2}
{\mathcal{CC}}=\left\{\rho|\rho=\sum_i p_i|\psi_i\rangle_A
\langle\psi_i|\otimes|\phi_i\rangle_B\langle\phi_i|\right\}.$$ where $\{|\psi_i\rangle_A\}$ and $\{|\phi_i\rangle_B\}$ are orthogonal basis of $\mathcal H_A$ and $\mathcal H_B$, respectively. The proof can be easily sketched. For any output state $\rho'$ that corresponds to a general QC input state, one can find a CC state whose corresponding output state $\rho$ can be transformed to $\rho'$ by a local channel on $B$, i.e., $\rho'={\openone}\otimes
\lambda_B (\rho)$. From the contractivity of the measure $Q$ under CPTP map, we have $Q(\rho)\geq Q(\rho')$. Hence the definition of QCP can be optimized to $$\label{qcp3}
\mathcal Q(\Lambda)\coloneqq \max_{\rho\in{\mathcal{CC}}}
Q(\Lambda\otimes{\openone}(\rho)).$$ A channel with larger amount of QCP is more quantum, in the sense of the ability to create quantum correlations. Hence it is of interest to find out the channels with the most QCP. It can be proved that, the local single-qubit channel which maximum QCP can be found in the set of the following channels $$\label{qcp4}
\mathcal{MP}=\left\{\Lambda|\Lambda(\cdot)=\sum_{i=0}^1
|\phi_i\rangle\langle\alpha_i|(\cdot)|\alpha_i\rangle\langle\phi_i|\right\},$$ where $|\phi_0\rangle$ and $|\phi_1\rangle$ are two nonorthogonal pure states.
@PhysRevA.88.012315 also studied the superadditivity of QCP. Its says that two zero-QCP channels can constitute a positive-QCP channel. The phase damping channel $\Lambda_{pd}$ was used as an example to show how this property works. The corresponding Kraus operators are given in Eq. . $\Lambda_{pd}$ is unital and thus $\mathcal {Q}(\Lambda_{pd})=0$. For a four-qubit initial state shared between Alice ($AA'$) and Bob ($BB'$) $$\label{qcp5}
\rho_{AA'BB'}=\frac14\sum_{i,j}|ij\rangle_{AA'}\langle
ij|\otimes|\psi_{ij}\rangle_{BB'}\langle \psi_{ij}|,$$ where $$\label{qcp6}
\begin{aligned}
& |\psi_{00}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle),~
|\psi_{11}\rangle= \frac{1}{\sqrt{2}}(|0+\rangle+|1-\rangle),\\
& |\psi_{01}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle),~
|\psi_{10}\rangle=\frac{1}{\sqrt{2}}(|0-\rangle-|1+\rangle).
\end{aligned}$$ Since $|\psi_{ij}\rangle$ are orthogonal to each other, the quantum correlation on Bob is zero. After the action of $\Lambda_{pd}^B
\otimes \Lambda_{pd}^{B'}$ on $B$ and $B'$, the output state becomes $\rho'_{AA'BB'} ={\openone}_{AA'}\otimes\Lambda_{pd}^B \otimes
\Lambda_{pd}^{B'}(\rho_{AA'BB'})$. Because $[\Lambda_{pd}\otimes
\Lambda_{pd} (\psi_{00}), \Lambda_{pd}\otimes\Lambda_{pd}
(\psi_{11})] =\frac{1}{8}\tilde{i} p\sqrt{1-p} (\sigma^y\otimes
\sigma^z +\sigma_z\otimes\sigma^y)\neq0$, the output state $\rho'_{AA'BB'}$ is not a QC state. Therefore, the quantum correlation on Bob’s qubits $BB'$ is created by the channel $\Lambda_{pd}^B \otimes\Lambda_{pd}^{B'}$.
### Cohering and decohering power
For a system traversing a quantum channel $\mathcal{E}$, the amount of coherence contained in it may be increased or decreased. Building upon this fact and in the same spirit as defining entangling power and discording power \[see @dpower and references therein\], one can also consider the ability of $\mathcal{E}$ on producing or destroying coherence, and introduce the concepts of cohering and decohering power of $\mathcal{E}$.
@cpower defined the cohering power of $\mathcal{E}$ as the maximum coherence (measured in some way) that it can produce from the full set $\mathcal{I}$ of incoherent states, and the decohering power as the maximum amount of coherence lost after all the maximal-coherence-value states $\rho^{\rm mcs} \in\mathcal{M}$ [@mcvs] passing through this channel. To be precise, $$\label{eq5b-3}
\begin{aligned}
& {\rm CP}(\mathcal{E})=\max_{\delta\in\mathcal{I}}C(\mathcal{E}[\delta]),\\
& {\rm DP}(\mathcal{E})=C(\rho^{\rm mcs})-\min_{\rho^{\rm mcv}
\in \mathcal{M}}C(\mathcal{E}[\rho^{\rm mcs}]),
\end{aligned}$$ and the optimization can in fact be restricted to pure states, in particular, for ${\rm CP}(\mathcal{E})$ the maximization can be taken with only the basis states $\{|k\rangle\}$.
Focusing on single-qubit states and coherence measured by the $l_1$ norm and skew information, they calculated ${\rm CP}(\mathcal{E})$ and ${\rm DP}(\mathcal{E})$ for the depolarizing and bit flip (similarly for bit-phase and phase flip) channels, and showed that the unitary channel has equal cohering and decohering power in any basis, i.e., ${\rm CP}(U)={\rm DP}(U)$. Moreover, for $N$ qubits subjecting to $N$ independent unitary channels, the cohering power $$\label{eq5b-n3}
{\rm CP}(\otimes_{i=1}^N U_i)= \prod_{i=1}^N [{\rm CP}(U_i)+1]-1,$$ while the decohering power is bounded from below by $$\label{eq5b-n4}
{\rm DP}(\otimes_{i=1}^N U_i)\geq 2^N-\prod_{i=1}^N [2-{\rm DP}(U_i)],$$ and apart from the very special case of ${\rm DP}(U_i)=0$, $\forall
U_i$, ${\rm DP}(\otimes_{i=1}^N U_i)$ approaches $2^N-1$ when $N\rightarrow\infty$, hence the coherence in this state will be completely deteriorated for infinitely large $N$.
@Bukf obtained the analytical solutions of the cohering power. First, when the coherence is measured by the $l_1$ norm, they showed that $$\label{eq5b-n5}
{\rm CP}_{l_1}(U)= \|U\|_{1\rightarrow1}^2-1,$$ where $\|U\|_{1\rightarrow1}=\max_{1\leq j\leq d}\{\sum_{i=1}^d
|U_{ij}|\}$ is the matrix norm, and for $N$-qubit system, the Hadamard gate $H^{\otimes N}$ \[with $H=(\sigma_x+\sigma_z)/
\sqrt{2}$\] was showed to have maximum cohering power. When we adopt the relative entropy of coherence, it is given by $$\label{eq5b-n6}
{\rm CP}_r (U)= \max_{1\leq j\leq d}H(|U_{1j}|^2,\ldots,|U_{dj}|^2),$$ where $H(p_1,\ldots,p_d)$ denotes the Shannon entropy.
For general quantum channel $\mathcal{E}$, although there is no analytical solutions for the cohering power, they were showed to satisfy the additivity relation $$\label{eq5b-n7}
\begin{aligned}
& {\rm CP}_{l_1}(\mathcal{E}_1 \otimes \mathcal{E}_2)+1=({\rm
CP}_{l_1} (\mathcal{E}_1)+1)({\rm CP}_{l_1}(\mathcal{E}_2)+1),\\
& {\rm CP}_{r} (\otimes_{i=1}^N \mathcal {E}_i)=
\sum_{i=1}^N {\rm CP}_{r}(\mathcal {E}_i).
\end{aligned}$$ Moreover, one may consider a slightly different definition of cohering power $$\label{eq5b-n8}
{\rm CP}^{(\rho)}(\mathcal{E})= \max_{\rho\in \mathcal {D}_\mathcal
{H}} \{C(\mathcal{E}[\rho])-C(\rho)\},$$ which characterizes the maximum enhancement of quantum coherence after the action of the channel $\mathcal{E}$, it was found that for the 2-dimensional system, the two different cohering powers measured by the $l_1$ norm are always the same for any unitary channel, but when $d \geq 3$ or when the coherence is measured by the relative entropy, they can be different.
On the other hand, we know that the action of $\mathcal{E}$ on $\rho$ can be implemented by an IO $\Lambda$ on the product state of $\rho$ and an ancillary state $\sigma$, i.e., $\Lambda(\rho\otimes
\sigma)=\mathcal{E}(\rho)\otimes\sigma'$ \[see, e.g., the work of @coher\]. Start from this point of view, @Bukf further gave an interpretation of the cohering power. To be explicit, they showed that the minimal amount of coherence of $\sigma$ is just the cohering power of $\mathcal{E}$, i.e., $C(\sigma)\geq {\rm CP}
(\mathcal {E})$.
@cpower2 also explored the cohering and decohering powers of various typical channels, with however the coherence being measured by the relative entropy. These include the amplitude damping, phase damping, depolarizing, as well as the bit flip, bit-phase flip and phase flip channels. They also found that the cohering power can be enhanced by applying weak measurement and reversal operation to the qubit.
For the HS norm measure of quantum coherence, @cgp discussed the cohering power of the unitary and unital channels. As the HS norm is not an monotonic quantity under general CP maps, they restricted in their work only to those of the unital incoherent CP maps, under the action of which the HS norm of coherence is monotonically decreasing. By denoting $\Delta(\rho)$ the full dephasing of $\rho$ in a given basis \[see Eq. \] and $\tilde{\Delta}={\openone}-\Delta$ the the complementary projection of $\Delta$, they defined the cohering power of the quantum channel as the average coherence generated from an uniformly distributed incoherent states, i.e., $$\label{eq5b-n9}
C_{av}(\mathcal{E})=\langle C_{l_2}(\mathcal{E}_{\rm off}[\psi])
\rangle_\psi,$$ where $\mathcal{E}_{\rm off}=\tilde{\Delta}\mathcal{E}\Delta$, and the average is taken over the ensemble of pure states $\psi=|\psi\rangle\langle\psi|$ sampled randomly from the uniform Haar measure. That is to say, the uniform ensemble of incoherent states are generated by dephasing $\{\psi\}$.
When the channel is unitary, they found that the cohering power can be obtained analytically. For a $d$-dimensional system, it is given by $$\label{eq5b-n10}
C_{av}(U)=\frac{1}{d+1}\left(1-\frac{1}{d}\sum_{i,j}|\langle
i|U|j\rangle|^4\right),$$ where the upper bound $C_{av}^{\max}(U)=(d-1)/d(d+1)$ is achieved when the basis $\{|i\rangle\}$ and $\{U|i\rangle\}$ are mutually unbiased, and the lower bound $C_{av}(U)=0$ is achieved when $U$ is an incoherent operation, i.e., $[U,\Delta]=0$.
Similarly, when the channel $\mathcal{E}$ is unital, i.e., $\mathcal{E}({\openone}/d) ={\openone}/d$, with $\{A_k\}$ being the corresponding Kraus operators, the cohering power is given by $$\label{eq5b-n11}
C_{av}(\mathcal{E})=\frac{1}{d(d+1)}\sum_{i,l\neq m}\Big|\sum_k
(A_k)_{li}(A_k)^*_{mi}\Big|^2,$$ but now $C_{av}(\mathcal{E})=0$ does not always implies $[\mathcal
{E},\Delta]=0$, that is to say, the cohering power $C_{av}(\mathcal{E})$ for unital channel is not faithful.
In fact, as the unitary operation $U$ is a subset of the unital operation, $C_{av}(\mathcal{E})$ also covers the result of $C_{av}(U)$. Moreover, the above equation is equivalent to $$\label{eq5b-n12}
C_{av}(\mathcal {E})= \frac{1}{d+1}\{{\mathrm{tr}}[S\tilde{\omega}(\mathcal
{E})]-{\mathrm{tr}}[S\omega(\mathcal {E})]\},$$ where $\tilde{\omega}(\mathcal{E})=\mathcal{E}^{\otimes
2}(\rho_{\mathrm{mm}})$, $\omega(\mathcal{E})=(\Delta
\mathcal{E})^{\otimes 2}[\rho_{\mathrm{mm}}]$, $\rho_{\mathrm{mm}}=
{\openone}/d$, and $S=\sum_{ij}|ij\rangle\langle ji|$ is the <span style="font-variant:small-caps;">swap</span> operator.
@cgp1 also examined power of the dephasing channel $\Delta_{B'}$ described by the projector $B'=\{|i'\rangle\langle
i'|\}$ (i.e., $\Delta_{B'}[\rho]=\sum_{i'} \langle i'|\rho
|i'\rangle |i'\rangle\langle i'|$) on generating quantum coherence defined with respect to the basis $B=\{|i\rangle\langle i|\}$. To be explicitly, they defined the cohering power as $$\label{eq5b-n13}
C^B(\Delta_{B'})=\int d\mu_{\mathrm{unif}} (\delta)
C^B\left(\Delta_{B'}[\delta]\right),$$ where $d\mu_{\mathrm{unif}} (\delta)$ denotes the uniform measure in the $(d-1)$-dimensional simplex. $C^B(\Delta_{B'})$ is in fact the average coherence generated from uniformly distributed incoherent states $\delta\in \mathcal {I}$. When using the relative entropy of coherence as a quantifier, they proved that $$\label{eq5b-n14}
C_r^B(\Delta_{B'})= \tilde{Q}(X_U X_U^T)-\tilde{Q}(X_U),$$ and $C_r^B(\Delta_{B'})=0$ if and only if the dephasing operators $\Delta_{B'}$ and $\Delta_{B}$ commute. Here, $X_U$ is bistochastic with elements $(X_U)_{ij}=|\langle i|U|j\rangle|^2$, $U$ is the unitary operator ensures $|i'\rangle= U|i\rangle$, $\forall i$, and $\tilde{Q}(X)=\sum_j Q(\bm{p}_j)/d$, with $Q(\bm{p}_j)$ being the subentropy of the column vector $\bm{p}_j$ with elements $(\bm{p}_j)_i=(X_U)_{ij}$.
When one uses the HS norm of coherence, the power turns out to be [@cgp1] $$\label{eq5b-n15}
C_{l_2}^B(\Delta_{B'})= \frac{1}{d(d+1)}{\mathrm{tr}}[X_U X_U^T(1-X_U X_U^T)],$$ and it is bounded from above by $(d-1)/4d(d+1)$, which is just one-quarter of the maximum $C_{av}^{\max}(U)$.
### Coherence-breaking channels
@breaking investigated coherence breaking channels (CBC) which were defined as those of the incoherent channels who destroy completely the coherence of any input state. They also discussed the selective CBC for which the Kraus operators $\{K_n\}$ give $K_n\rho
K_n^\dag \in \mathcal{I}$, and found that they are equivalent to CBC, i.e., the two sets $\mathcal {S}_{\rm cbc}=\mathcal{S}_{\rm
scbc}$. The CBC are subsets of the entanglement-breaking channels [@breaking-e] and quantum-classical channels.
When a channel $\Phi\in \mathcal {S}_{\rm cbc}$, then $\Phi(|i\rangle\langle j|)$ is diagonal for any two incoherent basis states $|i\rangle$ and $|j\rangle$, and the action of $\Phi$ on $\rho$ can be written as $$\label{CBC01}
\Phi(\rho)=\sum_i |i\rangle\langle i| {\mathrm{tr}}(\rho F_i),$$ with $\{F_i\}$ being the set of positive semidefinite operators satisfying $\sum_i F_i={\openone}$. For the special case of single-qubit state $\rho=({\openone}+\vec{r}\cdot\vec{\sigma})/2$, the channel $$\label{CBC02}
\Phi(\rho)=\frac{1}{2}[{\openone}+(M\vec{r}+\vec{n}) \cdot\vec{\sigma}],$$ belongs to CBC if the nonzero elements of $M$ and $\vec{n}$ lie only in the third row of them.
@breaking further introduced a notion which they termed as coherence-breaking index. It concerns the iterative actions of an incoherent quantum channel $\Phi$ on a given state, and can be defined explicitly as $$\begin{aligned}
\label{eq5b-add}
n(\Phi)=\min\{n\geqslant 1\!: \Phi^n\in\mathcal {S}_{\rm cbc}\},\end{aligned}$$ that is, $n(\Phi)$ characterizes the minimum number of iterations of $\Phi$ such that $\Phi^n(\rho)\in \mathcal{I}$ for any $\rho$. Clearly, if $\Phi$ is already a CBC, then $n(\Phi)=1$, while for the case of $n(\Phi)= \infty$, $\Phi^n$ is not a CBC.
For the single-qubit case, they also investigated the sudden death phenomenon for the $l_1$ norm of coherence by using the result of @fac3. Explicitly, the occur of coherence sudden death is only determined by forms of the incoherent channel and is independent of the initial state. But this does not apply to high dimensional states.
Evolution equation of quantum correlation and coherence {#sec:6D}
-------------------------------------------------------
Refereing to various measures of quantumness in open system, the search of certain dynamical law governing their evolution is of practical significance, as this can simplify the assessment of their robustness against decoherence. For entanglement measured by concurrence, its evolution was found to obey a factorization law for the initial two-qubit states and arbitrary quantum channels [@fac1; @fac2], we review here the similar problems for geometric quantum correlations and quantum coherence.
### Evolution equation of geometric quantum correlation
Similar to the evolution equation of entanglement measured by concurrence [@fac1], @fac4 found that when a bipartite system traverses the local quantum channel, the evolution of GQD may demonstrate a factorization decay behavior.
For a bipartite state $\rho$ decomposed as $$\begin{aligned}
\label{eq1}
\rho = -\frac{1}{d_A d_B} {\openone}_{AB}
+\rho_A\otimes \frac{1}{d_B}{\openone}_B+\frac{1}{d_A}
{\openone}_A\otimes\rho_B+\rho_{co},\end{aligned}$$ where the reduced states $\rho_A = {\mathrm{tr}}_B \rho$, $\rho_B={\mathrm{tr}}_A \rho$, and the traceless ‘correlation operator’ $\rho_{co}$ are give by $$\label{eq2}
\begin{split}
& \rho_A = \frac{1}{d_A}{\openone}_A+\vec{x}\cdot\vec{X},~
\rho_B = \frac{1}{d_B}{\openone}_B+\vec{y}\cdot\vec{Y},\\
& \rho_{co} = \sum_{i=1}^{d_A^2-1}\sum_{j=1}^{d_B^2-1}t_{ij} X_i \otimes Y_j,
\end{split}$$ they proved that when the channel $\mathcal {E}$ gives $\mathcal
{E}(\varrho)= q(t)\varrho$, with $\varrho = \vec{x} \cdot \vec{X}
\otimes {\openone}_B/d_B+\rho_{co}$, the evolution of $D_p(\mathcal
{E}[\rho])$ fulfills the factorization decay behavior $$\begin{aligned}
\label{eq6}
D_p(\mathcal{E}[\rho]) = |q(t)|^p D_p(\rho),\end{aligned}$$ which is solely determined by the product of the initial $D_p(\rho)$ and a channel-dependent factor $|q(t)|$, and $$\begin{aligned}
\label{eq3}
D_p(\rho) = \mathop{\rm opt}_{\Pi^A\in \mathcal {M}}
\parallel\rho-\Pi^A(\rho)\parallel_p^p,\end{aligned}$$ with opt representing the optimization over some class $\mathcal
{M}$ of the local measurements $\Pi^A=\{\Pi_k^A\}$ acting on party $A$.
By turning to the Heisenberg picture to describe the action of $\mathcal {E}$ (with the Kraus operators $\{E_j\}$), i.e., $\mathcal
{E}^\dag(X_i)=\sum_j E_j^\dag X_i E_j$ (see Fig. \[fig:hu3\]), they identified the family of states for which the factorization relation holds. Explicitly, if $\mathcal {E}_1^\dag (X_i)=q_A X_i$ for all $\{X_i\}$, and $\mathcal {E}_2^\dag (Y_j) = q_B Y_j$ for all $\{Y_j\}$, then Eq. holds for the families of $\rho$ with $$\label{eq-c1}
\begin{aligned}
(1)&~ \mbox{arbitrary}~ \rho_A,~ \rho_B,~ \rho_{co} ~~
(\mbox{for \,} \mathcal {E}_1 \otimes {\openone}_B), \\
(2)&~ \rho_A = \frac{1}{d_A}{\openone}_A, \mbox{or~} \rho_{co}=0 ~~
(\mbox{for \,} \mathcal {E}_1 \otimes \mathcal {E}_2~ \mbox{with~} \mathcal {E}_2\neq
{\openone}_B),
\end{aligned}$$ while $\mathcal {E}_1^\dag (X_k) = q_A X_k$ only for $\{X_k\}$ with $k = \{k_1,\ldots,k_\alpha\}$ ($\alpha < d_A^2-1$), and $\mathcal
{E}_2^\dag (Y_l) = q_B Y_l$ only for $\{Y_l\}$ with $l=\{l_1,\ldots,l_\beta\}$ ($\beta < d_B^2-1$), Eq. holds for the families of $\rho$ with $$\label{eq-c2}
\begin{aligned}
(1)&~ \rho_A = \rho_A^{(1)},~ \rho_{co} = \rho_{co}^{(1)} ~~
(\mbox{for \,} \mathcal{E}_1 \otimes {\openone}_B),\\
(2)&~ \rho_A = \frac{1}{d_A}{\openone}_A,~ \rho_{co} = \rho_{co}^{(2)},~
\mbox{or~} \rho_{co} = 0~
(\mbox{for \,} {\openone}_A \otimes \mathcal {E}_2),\\
(3)&~ \rho_A = \frac{1}{d_A}{\openone}_A,~ \rho_{co} = \rho_{co}^{(3)},~
\mbox{or~} \rho_A = \rho_A^{(1)},~\rho_{co} = 0 \\
&~ (\mbox{for \,} \mathcal {E}_1 \otimes \mathcal {E}_2),
\end{aligned}$$ where $$\label{eq-cc12}
\begin{split}
& \rho_A^{(1)} = \frac{1}{d_A}{\openone}_A +
\sum_{k=k_1}^{k_\alpha} x_k X_k,~
\rho_{co}^{(1)} = \sum_{k=k_1}^{k_\alpha}\sum_{j=1}^{d_B^2-1}
t_{k j} X_k \otimes Y_j,\\
& \rho_{co}^{(2)} = \sum_{i=1}^{d_A^2-1}\sum_{l=l_1}^{l_\beta}
t_{i l} X_i \otimes Y_l,~
\rho_{co}^{(3)} = \sum_{k=k_1}^{k_\alpha}\sum_{l=l_1}^{l_\beta}
t_{k l} X_k \otimes Y_l.
\end{split}$$ Besides the above statements, they also discussed the case of symmetric GQD, the families of states for which Eq. holds are similar to those of Eqs. and , and we do not list them here again.
### Evolution equation of quantum coherence
For quantum coherence measured by $l_1$ norm, @fac3 explored its evolution for a $d$-dimensional system traversing the quantum channel $\mathcal {E}$. As for any master equation which is local in time, whether Markovian, non-Markovian, of Lindblad form or not, one can always construct a linear map which gives $\rho(t)= \mathcal {E}
(\rho(0))$ (the opposite case may not always be true), and the linear map can be expressed in the Kraus-type representations [@me]. If the map $\mathcal{E}$ is CPTP, then one can explicitly construct the Kraus operators $\{E_\mu\}$ such that $\mathcal
{E}(\rho)= \sum_\mu E_\mu \rho E_\mu^\dag$.
For $\rho$ of Eq. , one can turn to the Heisenberg picture to describe $\mathcal{E}$ via the map $$\begin{aligned}
\label{eq-ee1}
\mathcal{E}^\dag (X_i)= \sum_\mu E_\mu^\dag X_i E_\mu,\end{aligned}$$ which gives $ x'_i= {\mathrm{tr}}(\rho\mathcal{E}^\dag [X_i])$. As any Hermitian operator $\mathcal{O}$ on $\mathbb{C}^{d \times d}$ can always be decomposed as $\mathcal{O}=\sum_{i=0}^{d^2-1} r_i X_i$ ($r_i \in \mathbb{R}$), $\mathcal{E}^\dag (X_i)$ can be further characterized by the transformation matrix $T$ defined in Eq. , namely, $\mathcal {E}^\dag (X_i)=
\sum_{j=0}^{d^2-1} T_{ij}X_j$, where $T_{ij}={\mathrm{tr}}(\mathcal{E}^\dag
[X_i]X_j)/2$, and $X_0=\sqrt{2/d}{\openone}_d$. Clearly, $T_{00}=1$, and $T_{0j}=0$ for $j \geq 1$. This further gives $$\begin{aligned}
\label{eq-ee2}
x'_i=\sum_{j=0}^{d^2-1} T_{ij} x_j.\end{aligned}$$ By classifying state $\rho$ of Eq. into different families: $\rho= \{\rho^ {\hat{n}}\}$, with $\rho^{\hat{n}}=
{\openone}_d/d+ \chi \hat{n}\cdot\vec{X}/2$ ($\hat{n}$ is a unit vector in $\mathbb{R}^{d^2-1}$, and $\chi$ is smaller than $\sqrt{2(d-1)/d}$ as ${\mathrm{tr}}(\rho^{\hat{n}})^2=|\chi|^2/2+ 1/d$), one can found that if $T_{k0}=0$ for $k \in \{1,2,\ldots,d^2-d\}$, then the evolution of $C_{l_1}(\mathcal{E}[\rho^{\hat{n}}])$ obeys the factorization relation $$\begin{aligned}
\label{eq5c-1}
C_{l_1}(\mathcal {E}[\rho^{\hat{n}}])= C_{l_1}(\rho^{\hat{n}})
C_{l_1}(\mathcal {E}[\rho_p^{\hat{n}}]),\end{aligned}$$ with $$\begin{aligned}
\label{eq-ee3}
\rho_p^{\hat{n}}= \frac{1}{d}{\openone}+\frac{1}{2}\chi_p\hat{n}
\cdot\vec{X},\end{aligned}$$ being the probe state, and $\chi_p = 1/\sum_{r=1}^{d_0} (n_{2r-1}^2+
n_{2r}^2)^{1/2}$.
As a corollary of the above equation, one can also show that if the operator $A=\sum_\mu E_\mu E_\mu^\dag$ is diagonal, then the evolution of $C_{l_1}(\mathcal{E}[\rho^{\hat{n}}])$ is governed by Eq. . Moreover, if more restrictions are imposed on the quantum channel, e.g., if a channel $\mathcal{E}$ yields $\mathcal{E} ^\dag (X_k)= q(t) X_k$ for $\{X_k\}_{k= k_1, \ldots,
k_\beta}$ ($\beta \leq d^2-d$), with $q(t)$ containing information on $\mathcal{E}$’s structure, then $$\label{eq5c-2}
C_{l_1}(\mathcal{E}[\rho])=|q(t)| C_{l_1}(\rho),$$ holds for all the $$\label{eq5c-3}
\rho = \frac{1}{d}{\openone}_d +\frac{1}{2}\sum_{k=k_1}^{k_\beta}
x_k X_k+\frac{1}{2}\sum_{l=d^2-d+1}^{d^2-1} x_l X_l.$$ The channel $\mathcal{E}$ satisfying the requirements includes the Pauli channel (covers bit flip, phase flip, bit-phase flip, phase damping, and depolarizing channels) and Gell-Mann channel given by @fac3, and the generalized amplitude damping channel. They also constructed a quantum channel $\mathcal{E}_{\rm G}$ for which $C_{l_1}(\mathcal{E}_{\rm G}[\rho])$ obeys Eq. for arbitrary initial state.
Preservation of GQD and quantum coherence {#sec:6E}
-----------------------------------------
Along with the similar line for exploring quantum entanglement and entropic discord dynamics in open quantum systems, some works have also been devoted to the study of dynamical behaviors of various GQDs and coherence monotones. Apart from those focused on identifying freezing conditions discussed above, the others are aimed at seeking flexible methods to control their evolution. We summarize the key results in this section, mainly for the qubit states and typical noisy sources in quantum information processing.
@orderd1 discussed robustness of the HS norm of discord for two qubits coupled to a multimode vacuum electromagnetic field, and found that the robustness can be enhanced if appropriate local unitary operations were performed on the initial state of the system. @hutian discussed trace norm of discord and Bures norm of discord for a two-qubit system subject to independent and common zero-temperature bosonic structured reservoirs. The results showed that the two GQDs can be preserved well or even be improved and generated by the noisy process of the common reservoir. If one can detuning the transition frequency of the qubits to large enough values, the long-time preservation of these two GQDs in independent reservoirs can also be achieved. Moreover, it was found that the decay rates of GQD can be retarded apparently by properly choosing the Heisenberg type interaction of two qubits when they are embedded in two independent Bosonic structured reservoirs [@Lizhao].
If the noisy channel (or reservoir) coupled to the central system is non-Markovian, the backflow of information from the reservoir to the system can induce damped oscillation behaviors of the GQD. For two qubits subject to bosonic structured reservoirs with Lorentzian and Ohmic-like spectra, the relation between behaviors of GQDs and the extent of non-Markovianity of the reservoir have been studied [@hulian]. By analyzing their dependence on a factor whose derivative signifies the (non-)Markovianity of the dynamics, it was demonstrated that the non-Markovianity induced by the backflow of information from the reservoirs to the system enhances the GQDs in most of the parameter regions.
For a single qubit subjecting to pure dephasing channel with the Ohmic-like spectral densities, @trap1 compared the coherence evolution behaviors (measured by the $l_1$ norm and relative entropy) with different system and bath parameters. They found that the initial system-bath correlations are preferable for realizing long-lived coherence in the super-Ohmic baths, and the region of coherence trapping is enlarged with increasing the correlation parameter. For the given initial state with equal amplitudes, they obtained numerically the optimal Ohmicity parameter $\mu\doteq 1.46$ for the most efficient coherence trapping, which is independent of the coupling constant and the correlation parameter.
The atomic system is also an important candidate for various quantum information processing tasks. For the static polarizable two-level atoms interacting with a fluctuating vacuum electromagnetic field, @LiuTian2016a explored the coherence dynamics measured by the $l_1$ norm and relative entropy, both for an initial product state of the atoms and the field. The results show that for the initial one-qubit pure states and two-qubit Bell-diagonal states, the coherence cannot be protected for non-boundary electromagnetic field. Contrarily, when there is a reflecting boundary, the coherence will be trapped if the atom is close to the boundary and transversely polarized. The coherence can also be protected to some extent for other specific polarization directions. All these show that the coherence behavior is position and polarization dependent.
Quantum coherence and GQD in many-body systems {#sec:7}
==============================================
Quantum coherence can be regarded as a fundamental property in quantum realm. The many-body systems of condensed matter physics possess various quantum characteristics which may have no classical analogue. In this sense, the exploring of quantum coherence in many-body systems may lead some intriguing connections and may also result in developments in both research areas. We next start from a interesting concept in condensed matter physics.
Off-diagonal long-range order and $l_1$ norm of coherence {#sec:7A}
---------------------------------------------------------
In theory of superconductivity, one of the well-known properties is the off-diagonal long-range order (ODLRO). Apparently, this property is related with $l_1$ norm quantum coherence measure, which uses the summation of the off-diagonal elements norm of a (reduced) density matrix quantifying coherence.
Let us consider a $\eta$-pairing state as an example. We start from the Hamiltonian of the Hubbard model, $$\begin{aligned}
H =& -\sum _{\sigma ,\langle i,j\rangle }\left( c_{j,\sigma}^{\dagger }
c_{k,\sigma }+c_{k,\sigma }^{\dagger }c_{j,\sigma}\right)\\
& +U\sum _{j=1}^L\left( n_{j\uparrow }-\frac {1}{2}\right)\left(
n_{j\downarrow }-\frac {1}{2}\right) ,
\end{aligned}$$ where $\sigma =\uparrow ,\downarrow $, and $\langle j,k\rangle $ is considered as a pair of the nearest-neighboring sites, $c_{j\sigma
}^{\dagger }$ and $c_{j\sigma }$ are the creation and annihilation operators of fermions. The $\eta $-pairing operators at lattice site $j$ are defined as $$\eta _j=c_{j\uparrow}c_{j\downarrow}, ~
\eta^{\dagger}_j=c_{j\downarrow}^{\dagger}c_{j\uparrow}^{\dagger}, ~
\eta_j^z=-\frac {1}{2}n_j+\frac {1}{2},$$ and they constitutes a SU(2) algebra. The $\eta $ operators are defined as $\eta =\sum \eta_j$ and $\eta^{\dagger}= \sum
\eta_j^{\dagger}$. The $\eta $-pairing state is defined as [@CNYang1989; @EsslerKorepin; @FanLloyd] $$\begin{aligned}
|\Psi \rangle =(\eta^{\dagger})^N|\mathrm{vac}\rangle\end{aligned}$$ where $|{\rm vac}\rangle $ is the vacuum state, and $|\Psi \rangle$ is an eigenstate of the Hubbard model. We can find that the $\eta
$-pairing state is actually the completely symmetric state with $N$ sites filled while the other $L-N$ sites unfilled up to a normalization factor. The ODLRO of this $\eta$-pairing state is shown as $$\begin{aligned}
C_{odlro}=\frac {\langle \Psi |\eta _k^{\dagger }\eta_l|\Psi \rangle}
{\langle \Psi |\Psi \rangle }=\frac {N(L-N)}{L(L-1)},
~~k\not=l.\end{aligned}$$ The off-diagonal element $C_{odlro}$ is a constant which does not depend on the distance $|k-l|$, in particular when $|k-l|\rightarrow
\infty $.
We may find that the density matrix $\rho =|\Psi\rangle\langle\Psi
|$ of the $\eta$-pairing state is a $L$-qubit state, the quantity $C_{odlro}$ corresponds to one off-diagonal element of $\rho $. The $l_1$ norm of $\eta$-pairing state can be calculated as $$\begin{aligned}
C_{l_1}(\rho)= L(L-1)C_{odlro}.\end{aligned}$$ In this sense, the ODLRO is directly related with $l_1$ norm of coherence. As we mentioned that the coherence measure depends on a specified basis, here the definition of ODLRO naturally provides the basis by which the quantum coherence can be quantified.
By using this example, we try to show that further perspective study can be expected concerning about the quantum coherence in many-body systems.
Quantum coherence of valence-bond-solid state {#sec:7B}
---------------------------------------------
Haldane conjectured that antiferromagnetic spin chains will be gapless for half-odd-integer spins and gapped for integer spins [@Haldaneconj1; @Haldaneconj2]. The Affleck-Kennedy-Lieb-Tasaki (AKLT) model [@AKLT1; @AKLT2] is a spin-1 chain in bulk and spin-$1/2$ at the two ends, which agrees with the Haldane conjecture. The ground state of AKLT model is known as the valence-bond-solid (VBS) state. The Hamiltonian of the AKLT model is written as $$H=\sum _{j=1}^{N-1}\left(\vec {S}_j\cdot \vec {S}_{j+1}
+\frac{1}{3}(\vec {S}_j\cdot \vec {S}_{j+1})^2\right)
+\pi _{0,1}+\pi_{N,N+1},$$ where $\vec {S}$ is the spin-1 operator in bulk, $\pi$ describes the interaction of spin-1 in bulk and spin-[1/2]{} at one end.
The ground state, VBS state, is written as $$\begin{aligned}
|G\rangle =(\otimes_{j=1}^NP_{j \bar{j}})|\Psi^-\rangle _{\bar
{0}1}|\Psi^-\rangle_{\bar {1}2} \cdots |\Psi^-\rangle_{\bar{N}N+1},\end{aligned}$$ where $|\Psi ^-\rangle =(|\uparrow \downarrow \rangle -|\downarrow
\uparrow\rangle)/\sqrt{2}$ is a singlet state which corresponds to the operator form, $(a_{\bar {i}}^{\dagger }b_i^{\dagger }-b_{\bar
{i}}^{\dagger }a_i^{\dagger })|{\rm vac}\rangle$, where $a^{\dagger}$ and $b^{\dagger }$ are bosonic creation operators. The projector $P$ maps a two-qubit state, which is a four-dimensional Hilbert space, on a symmetric subspace which is three-dimension for spin-1 operator $\vec{S}$. So the VBS state is constructed by a chain of singlet states under the projection of $P$ at the bulk sites and leaves the spin-$1/2$ at two ends (see Fig. \[fig:fan\]).
By using the teleportation technique sequentially [@FKR], the VBS state takes a form like the following $$\begin{aligned}
|G\rangle &=& \frac {1}{3^{N/2}}\sum _{\alpha _j=1}^3|\alpha_1\rangle
\cdots |\alpha _N\rangle \nonumber \\
&& \times\left[{\openone}\otimes(\sigma_{\alpha_N}\cdots\sigma_{\alpha_1})\right]
|\Psi^-\rangle_{\bar{0},N+1},\end{aligned}$$ where ${\openone}$ is the identity operator, $\sigma_1$, $\sigma_2$, $\sigma_3$ are Pauli matrices. It is proved that the reduced density matrix of continuously $L$ bulk spins is invariant which does not depend on its position in the spin chain, it can be written as $$\begin{aligned}
\rho_L=\frac {1}{3}\sum _{\alpha,\alpha '}f_{\alpha \alpha '}
|\alpha_1\rangle \langle \alpha_1'|\cdots|\alpha_L\rangle \langle
\alpha'_L|,\end{aligned}$$ where parameters $f_{\alpha \alpha'}$ can be determined as $$f_{\alpha\alpha'}={\mathrm{tr}}({\openone}\otimes V_{\alpha })|\Phi^-\rangle
\langle\Psi^-|(I\otimes V_{\alpha'}^{\dagger}),$$ with $V_{\alpha }= \sigma_{\alpha_L}\cdots \sigma_{\alpha_1}$, and $V_{\alpha'}= \sigma_{\alpha'_L}\cdots \sigma_{\alpha'_1}$.
Here we consider the measure of relative entropy of coherence. Due to the form of $f_{\alpha \alpha'}$, one may find that the diagonal matrix $\rho ^{diag}_L$ of $\rho _L$ is a completely mixed state with tensor product of $L$ identities and a normalization factor, so we have $$\begin{aligned}
\label{volumepart}
S(\rho_L^{\rm diag})= L\log_2 3.\end{aligned}$$ This quantity corresponds to the volume quantity of the bulk $L$ spins. On the other hand, we know [@FKR] that the von Neumann entropy of $\rho _L$ equals to the von Neumann entropy of the state of two ends which is a Werner state $$\label{areapart}
\begin{aligned}
& S(\rho_L)=S(\tilde\rho_L), \\
& \tilde{\rho}_L= \frac{1}{4}(1-p_L){\openone}+ p_L|\Psi^-\rangle \langle \Psi^-|,
\end{aligned}$$ where $p_L=(-1/3)^L$. We know that the entropy of $L$ bulk spins, $S(\rho_L)$, reaches to a constant $2$ exponentially fast in terms of the number of bulk spins $L$.
With combination of those analysis, we can find that the relative entropy of coherence of bulk $L$ spins, $$\begin{aligned}
C_r(\rho_L)= S(\rho_L^{\rm diag})-S(\rho_L).\end{aligned}$$ By substituting the results of Eqs. (\[volumepart\]-\[areapart\]) into the definition, the coherence of the VBS state can be obtained straightforwardly. We would like to remark that this result is independent of the basis chosen because of the complete mixed form of $\rho _L^{\rm diag}$ is invariant for different bases.
For gapped one-dimensional system, the above results can be interpreted as, $$\begin{aligned}
C_r(\rho_L)={\rm volume}-{\rm constant}.\end{aligned}$$ Let us point out that the “constant” term corresponds to the area law [@HammaArea; @PlenioRev].
Perspectively, it is worth exploring whether the relative entropy of coherence can be generally written as the difference between volume of the studied subsystem and the boundary term of area law, $$\begin{aligned}
C_r= {\rm volume}-{\rm area~law~(boundary)}.\end{aligned}$$ Further evidences of this expectation are necessary. Recently, it is shown that the volume effect can be found for the *XY* model, where the factor for the volume term is also important which can be used to distinguish different quantum phases [@WangZhengAn].
In the seminal work of @VLRK, two different behaviors of entanglement entropy for one-dimensional gapped and gapless models were proposed. For gapless models like critical spin chains, the entanglement of the ground state for a bulk of spins grows logarithmically in number of particles in the bulk. The prefactor of the logarithm term is related with the central charge of the conformal field theory. The same scaling behavior holds also for mean entanglement at criticality for a class of strongly random quantum spin chains [@RefaelMoore]. For a gapped model, the entanglement approaches a constant bound. Additionally, the topological entanglement entropy and entanglement spectrum are studied based on the theory of entanglement [@HammaArea; @KP; @LW; @LH]. The entanglement entropy for a pure state is defined as the von Neumann entropy of the reduced density matrix of its subsystem. Rényi entropies parametrized by a parameter $\alpha$ are the generalizations of the von Neumann entropy. So topological entanglement entropy can also be generalized as topological entanglement Rényi entropies [@Flammia]. However, it is found that all topological Rényi entropies are the same, which is due to the fact that Rényi entropies are additive and the studied density matrix takes a product form. In correspondence, those results can be further explored from point of view of quantum coherence. There are some works about the characteristics of quantum phase transitions by quantum coherence, as presented next.
Quantum coherence and correlations of localized and thermalized states {#sec:7C}
----------------------------------------------------------------------
Quantum dynamics of isolated quantum systems far from equilibrium has recently been extensively studied [@Eisert2015]. By principles of statistical mechanics, it is known that the non-equilibrium state will evolve to a thermalized state which is ergodic [@Thermalization-Deutsch; @Thermalization-Srednicki; @Thermalization-Rigol], and no quantum correlation is expected to exist. For an isolated quantum system initially in a pure state, the time evolution is unitary transformation which keeps the system in a pure state. The thermalization means that the reduced density matrix of a subsystem, which is relatively small compared with the whole system, takes the form of a thermal state, $\rho^S= e^{-\beta
H^{S}}/Z$, where $\beta$ is the inverse of temperature, $H^{S}$ is the Hamiltonian of the studied subsystem *S*, and $Z= {\mathrm{tr}}e^{-\beta H^{S}}$ is the partition function.
On the other hand, it is pointed out that the disorder may prevent the system from thermalizing, resulting in localized state. In general, there are two different types of localization, the single-particle localization in name of Anderson localization [@Anderson58] and the many-body localization [@MBL-Altshuler; @MBL-Polyakov]. The many-body localization is induced by competition between interactions and disorder, in contrast, Anderson localization is only due to disorder but without interaction. Besides, thermalization cannot happen for integrable models because of the constraints imposed by infinite number of conserved quantities. There are various signatures in characterizing the thermalized states and localized states. Here we will review the properties of quantum coherence and quantum correlations for those states.
### Entanglement entropy
The mechanism of thermalization is based on the eigenstate thermalization hypothesis, and the thermal state is ergodic [@Thermalization-Rigol; @AltmanVosk; @NandkishoreHuse]. Those facts lead to that the thermalized state $\rho^{S}$ takes a diagonal form, so state $\rho^{S}$ possesses neither quantum coherence, nor quantum correlations. On the other hand, the von Neumann entropy of the thermal state, $S(\rho ^{S})=-{\mathrm{tr}}(\rho^{S}\log_2\rho^{S})$ satisfies the volume law, implying that it is proportional to the number of the particles $L$ of the subsystem *S*. Considering that the isolated system is always in pure state, the von Neumann entropy of the reduced density matrix is the entanglement entropy itself. Thus in thermal phase, the entanglement entropy satisfies the volume law. If the initial state of the system is a highly excited state such as Néel state, the thermalized state approaches to the completely mixed state corresponding to infinite temperature, so entanglement entropy approaches $L$, corresponding to particle number of the subsystem for spin-1/2 particles. Note that $L$ is the upper bound of the entanglement entropy.
The quantum dynamics of localizations, both Anderson localization and many-body localization, and thermalization can be well characterized by behaviors of entanglement entropy. Suppose the initial state is a product state like Néel state, which is also a highly excited state, the initial entropy will be zero for the subsystem *S*. For thermalization, the entropy will increase quickly and approaches its upper bound. For Anderson localization, similarly, the entropy will quickly saturate its bound but the bound is much smaller than that of the thermalized state. In contrast, many-body localization is a consequence of the competition between particle interactions and disorder. The localized state breaks the ergodicity and eigenstate thermalization hypothesis. The entanglement entropy does not obey the volume law. Instead, the entropy for the stationary state demonstrates a long time slow increase characterized as logarithmic increasing in time [@MBL-Log1; @MBL-Log2; @MBL-Log3; @MBL-Log4; @MBL-entropyincrease1; @MBL-entropyincrease2], or algebraic with power-law interactions [@MBL-entropyincrease3]. Reminding that the coherence can be quantified as the diagonal entropy subtracting the von Neumann entropy, the coherence of the subsystem will demonstrate decrease for many-body localization.
Both Anderson localization and many-body localization have been realized experimentally. In system of trapped ions with long-range interactions, the growth of entanglement is shown by measuring the quantum Fisher information [@MBL-experiment]. Recently, the entanglement entropy logarithmic increase in time of many-body localization is successfully demonstrated in a 10-qubit superconducting quantum simulation based on single-shot state tomography measurement [@XuKai18]. The many-body localization and thermalization can also be distinguished by energy spectrum of the system. In thermal phase, the energy levels of the system tend to repel one another and their statistics are Wigner-Dyson distribution, while for many-body localized state, the energy levels show a Poisson statistics [@energy-statistics1; @energy-statistics2; @energy-statistics3]. These phenomena are also demonstrated experimentally [@energy-statistics4].
### Entanglement spectrum
The entanglement entropy of a ground state is just one quantity based on entanglement spectrum [@LH] due to Schmidt decomposition for a pure state. The entanglement spectrum possesses more information which may be invisible for entanglement entropy. For a ground state $|G\rangle$, the reduced density matrix for a bulk of $L$ sites is written as $\rho_L$, which can be rewritten as $$\begin{aligned}
\label{entanglementHamiltonian}
\rho_L= e^{-H_\mathrm{E}}.\end{aligned}$$ The entanglement spectrum is the energy spectrum of the so-defined entanglement Hamiltonian $H_\mathrm{E}$. It is shown that there is one-to-one correspondence between low-energy edge states of the system with open boundary condition and the low-lying eigenstates of the entanglement Hamiltonian [@Fidkowski; @QiXiaoLiang].
The entanglement spectrum of the ground state for a topological Chern insulator with disorder exhibits level repulsion, which is consistent with Wigner-Dyson distribution. This result in addition with energy spectrum and Chern number can be used to describe transition of Chern insulator to Anderson-insulator [@Prodan]. The many-body localization and thermalization and the periodically driven systems, which are known as Floquet systems, can be characterized by entanglement spectrum [@GNRegnault]. The level statistics of the entanglement spectrum in the thermalizing phase is governed by an appropriate random matrix ensemble. The Floquet entanglement spectrum has similar results showing a result beyond eigenstate thermalization hypothesis. In many-body localized phase, the entanglement spectrum shows level repulsion and obeys a semi-Poisson distribution. Also, the dynamical many-body localization is observed in an integrable system with periodically driven [@Keser16]. Perspectively, the study of quantum benchmarks such as quantum coherence, entanglement may be performed for those systems and phases like Floquet topological insulators induced by disorder [@Titum15].
Quantum coherence and quantum phase transitions {#sec:7D}
-----------------------------------------------
Quantum phase transition (QPT) describes an abrupt change for properties of the ground state of a many-body system driven by its quantum fluctuations. It is a purely quantum process and is caused by variation of the system parameters of the Hamiltonian, such as the spin coupling and external magnetic field [@QPTbook]. With the development of quantum entanglement theory, it is natural to study QPT of a many-body system from the point of view of entanglement. Indeed, it has been found that the singularity and extreme point of entanglement or its derivative can be used for detecting QPTs. An overview for the related progress can be found in the work of @RMP2 and @ZengBei.
As quantum coherence measures defined within the framework of @coher are also quantitative characterizations of the quantum feature of a system, they are hoped to play a role in studying quantum phase transitions (QPTs) of the many-body systems. We review briefly in this section some main progress for such studies.
@qptc1 demonstrated role of the coherence susceptibility on studying QPTs at both absolute zero and finite temperatures. Here, the coherence susceptibility is defined as the first derivative of the relative entropy of coherence $C_r(\rho)$, that is, $$\begin{aligned}
\chi^{\rm co}= \frac{\partial C_r(\rho)}{\partial \lambda},\end{aligned}$$ with $\lambda$ being a characteristic parameter of the system Hamiltonian. For the transverse Ising model with the Hamiltonian $H_I$, the spin-1/2 Heisenberg *XX* model with $\hat{H}_{XX}$, and the Kitaev honeycomb model with $\hat{H}_K$, described by $$\begin{aligned}
\label{Ising}
\begin{aligned}
& \hat{H}_I= -\sum_{i=1}^N \sigma_i^z \sigma_{i+1}^z-\lambda\sum_{i=1}^N \sigma_x, \\
& \hat{H}_{XX}=-\frac{1}{2}\sum_{i=1}^N(\sigma_i^x \sigma_{i+1}^x +
\sigma_i^y \sigma_{i+1}^y)- \lambda\sum_{i=1}^N \sigma_i^x,\\
& \hat{H}_K=- \sum_{\alpha=\{x,y,z\}}J_\alpha \sum_{i,j\in \alpha-{\rm links}}
\sigma_i^\alpha \sigma_j^\alpha,
\end{aligned}\end{aligned}$$ where $\lambda$ is the strength of the external magnetic field in units of the interaction energy, they showed that apart from the figure of merit that this method requires no prior knowledge of order parameter (the same as those based on entanglement and discord), the coherence susceptibility pinpoints not only the exact QPT points via its singularity with respect to $\lambda$, but also the temperature frame of quantum criticality. In particular, the latter has been considered to be a superiority of the coherence susceptibility method.
@qptc2 showed validity of the Skew-information-based coherence measure $\mathsf{I} (\rho,K)$ (will be called the $K$ coherence for brevity) and its lower bound $\mathsf{I}^L (\rho,K)$ on studying QPTs [@meas8]. They considered the spin-1/2 Heisenberg *XY* model described by the Hamiltonian $$\begin{aligned}
\begin{aligned}
\hat{H}=& -\frac{\lambda}{2} \sum_i [(1+\gamma) \sigma_i^x \sigma_{i+1}^x
+(1-\gamma)\sigma_i^y\sigma_{i+1}^y]\\
& - \sum_i\sigma_i^z,
\end{aligned}\end{aligned}$$ with $0\leq \gamma\leq 1$ being the anisotropy parameter, and $\lambda$ strength of the inverse magnetic field. They calculated the single-spin coherence $\mathsf{I}(\rho, \sigma^\beta)$, two-spin local coherence $\mathsf{I}(\rho, \sigma^\beta\otimes{\openone}_2)$ ($\beta=x,y,z$), and their lower bounds. The numerical results show that the divergence of the first derivatives of $\mathsf{I}
(\rho,\sigma^x)$ and $\mathsf{I} (\rho,\sigma^{x,y,z} \otimes
{\openone}_2)$ (including their lower bounds) with respect to $\gamma$ pinpoint exactly the transition point $\gamma_c=1$ \[$\mathsf{I}
(\rho,\sigma^y\otimes {\openone}_2)$ fails for the special case $\gamma=0.5$\], while the derivatives of $\mathsf{I} (\rho,
\sigma^{x,y,z}\otimes {\openone}_2)$ also detect the factorization point $\lambda_f\sim 1.1547$. Moreover, the performance of $\mathsf{I}^L(\rho,\sigma^x)$ in detecting QPTs at relatively high temperatures outperforms that of $\mathsf{I}(\rho,\sigma^x)$ for the considered model. A review of these results in addition with some quantum correlations are presented in [@Entropy].
@qptc3 studied quantum coherence measured by the WY skew information on diagnosing critical points of the spin-1/2 transverse field *XY* model with the *XZY$-$YZX* type of three-spin interactions. The Hamiltonian is given by $$\label{eq6-1}
\begin{aligned}
\hat{H}=&-\sum_{i=1}^N \bigg[\frac{1+\gamma}{2}\sigma_i^x\sigma_{i+1}^x
+\frac{1-\gamma}{2}\sigma_i^y\sigma_{i+1}^y+h\sigma_l^z \\
&+\frac{\alpha}{4} (\sigma_{i-1}^x\sigma_i^z\sigma_{i+1}^y
-\sigma_{i-1}^y\sigma_i^z\sigma_{i+1}^x)\bigg].
\end{aligned}$$ By examining the single-spin $\sigma^{x,y,z}$ coherence \[i.e., $K=\sigma^x$, $\sigma^y$, or $\sigma^z$ in Eq. \], the two-spin local $\sigma^{x,y,z}$ coherence \[i.e., $K=\sigma^x\otimes
{\openone}_2$, $\sigma^y\otimes {\openone}_2$, or $\sigma^z\otimes {\openone}_2$\], and their lower bounds $\mathsf{I}^L(\rho,K)$ [@meas8], @qptc3 found that if the three-spin interaction $\alpha=0$ and the external magnetic field $h<1$, the single-spin $\sigma^{x,y,z}$, two-spin local $\sigma^z$ coherence, and their lower bounds are extremal at the critical point $\gamma_c=0$ of anisotropy transition. But the two-spin local $\sigma^x$ ($\sigma^y$) coherence and its lower bound decrease (increase) with the increasing $\gamma$. Their first derivative with respect to $\gamma$ are minimal (maximum) at the critical point $\gamma_c=0$, and show scaling behaviors with respect to $\log N$, i.e., $d
\mathsf{Q} /d\gamma=a_1+a_2\log_2 N$, where $\mathsf{Q}=\mathsf{I}
(\rho,K)$ or $\mathsf{I}^L(\rho,K)$, and $a_1$ and $a_2$ are the system-dependent parameters.
When the three-spin interaction is introduced, there will be a gapless phase in the range $h\in[h_{c1},h_{c2}]$ for $\gamma<\alpha$. The system undergoes two QPTs (second order transitions) with increasing $h$, the first from gapped phase to gapless phase when $h$ increases from $h<h_{c1}$ to $h>h_{c1}$, and the second from the gapless phase to gapped phase when $h$ increases from $h<h_{c2}$ to $h>h_{c2}$. For this case, it was found that both the single-spin and two-spin local $\sigma^{x,y,z}$ coherence and their lower bounds are affected by the existence of $\alpha$ only in the gapless phase, and the two critical points $h_{c1}$ and $h_{c2}$ of the gapless phase can be pinpointed by the extremal points of their first derivatives. But different from that of $\alpha=0$, there are no size effect of the corresponding derivatives of coherence around the critical points, that is, the derivatives for different $N$ are almost the same.
@qptc4 also showed effectiveness of the two-spin local $\sigma^\beta$ ($\beta=x,y,z$) coherence $\mathsf{I}^L
(\rho,\sigma^\beta)$ in detecting QPTs of different physical systems. For the *XY* model with transverse magnetic fields and *XZX+YZY* type of three-spin interactions \[the Hamiltonian is similar as that of Eq. , with only the terms in the second line being replaced by $\alpha(\sigma_{i-1}^x \sigma_i^z
\sigma_{i+1}^x+ \sigma_{i-1}^y\sigma_i^z\sigma_{i+1}^y)$\]. Contrary to the case of $h=0.5$ studied in @qptc3, when $h=\alpha=0$, it was found that while the extremal of $\mathsf{I}^L (\rho,
\sigma^z\otimes {\openone}_2)$ can pinpoint the critical points of QPT at $\gamma_c=0$, $\mathsf{I}^L (\rho,\sigma^x\otimes {\openone}_2)$ \[$\mathsf{I}^L (\rho,\sigma^y\otimes {\openone}_2)$\] increases (decreases) with $\gamma$, and its first derivative with respect to $\gamma$ is maximal (minimal) at the first-order QPT point $\gamma_c=0$. For the second-order QPT at $h_c=1$, the derivative of $\mathsf{I}^L (\rho,\sigma^x\otimes {\openone}_2)$ with respect to $h$ at $h_c=1$ show a size-dependent scaling behavior, which implies that it will be divergent in the thermodynamic limit. Even at finite temperature, $\mathsf{I}^L (\rho,\sigma^x\otimes {\openone}_2)$ and its first derivative can also detect the second order QPT at $\alpha=0.5$ for $\gamma= 0.5$ and $h=0$. Moreover, they also showed that the two-spin local coherence can detect QPTs for the one-dimensional half-filled Hubbard model with both on-site and nearest-neighboring interactions and topological phase transition for the Su-Schrieffer-Heeger model.
The amount of WY skew information $\mathsf{I}(\rho,K)$ is determined by the observable one chooses. @luoprl introduced a quantity $$Q(\rho)=\sum_i \mathsf{I}(\rho,X_i)$$ where $\{X_i\}$ is the set of observables which constitute an orthonormal basis, and proved it to be independent of the choice of $\{X_i\}$ [@luopra]. Based on this, @qptc5 explored its role in detecting critical points of QPTs and the factorization transition of the spin model. For the density matrix $\rho^{AB}$ of two neighboring spins, they calculated $$F(\rho^{AB})=Q_A(\rho^{AB})- Q_A(\rho^A\otimes\rho^B),$$ with $Q_A(\rho^{AB})=\sum_i \mathsf{I} (\rho^{AB},X_i\otimes
{\openone}_B)$, and likewise for $Q_A(\rho^A\otimes\rho^B)$. For the *XY* model with transverse magnetic fields, their results show that the second-order QPT from the ferromagnetic to the paramagnetic phase (Ising transition) can be detected by the minimum of the first derivative of $F(\rho^{AB})$ with respect to $h$. At the vicinity of the transition point $h_c=1$, $\partial F(\rho^{AB})/\partial h$ shows a size-dependent scaling behavior, and is logarithmic divergent in the thermodynamic limit. On the other hand, the first-order transition from a ferromagnet with magnetization in the $x$ direction to one with magnetization in the $y$ direction (anisotropy transition) at $\gamma_c=0$ can be detected directly by the minimum of $F(\rho^{AB})$, but its first derivative with respect to $\gamma$ is continuous and size-independent. Moreover, it was found that both $\partial F(\rho^{AB})/\partial h$ and $\partial
F(\rho^{AB}) /\partial \gamma$ are discontinuous along the curve $h^2+\gamma^2 =1$. The emergence of the discontinuity pinpoints the factorization transition for ground states of the considered system, and has its roots in the elements of $\rho^{1/2}$.
Compared with the spin-1/2 models, various high-spin systems show richer phase diagrams. @qptc6 considered the spin-1 *XXZ* model and bilinear biquadratic model, with the Hamiltonian $$\label{eq6-2}
\begin{aligned}
& \hat{H}_{XXZ}=\sum_i(S_i^x S_{i+1}^x+S_i^y S_{i+1}^y+\Delta S_i^z S_{i+1}^z), \\
& \hat{H}_{BB}=\sum_i [\cos\theta (\bold{S}_i \cdot\bold{S}_{i+1})
+\sin\theta(\bold{S}_i\cdot \bold{S}_{i+1})^2]
\end{aligned}$$ where $\bold{S}_i= (S_i^x,S_i^y,S_i^z)$ are spin-1 operators. For the spin-1 *XXZ* model, the relative entropy and $l_1$ norm of coherence for two neighboring spins, and the local two-spin $S^x$ and $S^y$ coherence cannot detect the Kosterlitz-Thouless QPT at $\Delta_{c2} \approx 0$, while their inflection points detect the Ising type second-order QPT at $\Delta_{c2}\approx 1.185$. Moreover, the extremum of single-spin $S^x$ coherence pinpoints the SU(2) symmetry point $\Delta= 1$. For the spin-1 bilinear biquadratic model, the single spin density matrix is diagonal in $S_z$ basis for all values of the anisotropy parameter, so all coherence measures are zero. On the other hand, the transition point can be identified by mutual information and discord, which coincidences to both the infinite order Kosterlitz-Thouless transition and the SU(3) symmetry point $\theta= 0.25\pi$.
GQD and quantum phase transition {#sec:7E}
--------------------------------
The singularity or extreme point of QD can be used for detecting QPTs. @qptd studied one such problem. They considered a general Heisenberg *XXZ* model with the Hamiltonian $$\label{qpt-01}
\hat{H}= J \sum_{i=1}^N (\sigma_i^x \sigma_{i+1}^x+\sigma_i^y \sigma_{i+1}^y
+\Delta\sigma_i^z \sigma_{i+1}^z),$$ and by setting $J=1$, they calculated QD of Eq. as well as its first and second order derivatives for thermal states of the neighboring spins, and showed that it can efficiently detect the QPT points $\Delta=\pm 1$ for this model even at finite temperature, while the entanglement measured by entanglement of formation does not. This shows potential role of QD in investigating QPT. In particular, it is very important for experimental characterization of QPTs as in principle one is unable to reach a zero temperature experiment.
@qpt-qd studied QD for ground states of the transverse Ising and Heisenberg *XXZ* model, and found that the amount of QD increases close to the QPT points. Indeed, there are many other related works discussing role of the QD defined in Eq. in detecting QPTs, and we refer to the work of @RMP for a detailed overview in this respect. In what follows, we focus on role of GQDs on studying QPTs in various many-body systems.
@togd examined ground state properties of the Heisenberg *XXZ* model Eq. by setting $J=-1$. By employing the trace norm of discord as a quantifier of correlation, they found that $D_T(\rho)$ defined in Eq. as well as $C_T(\rho)$ and $T_T(\rho)$ defined in Eq. detects successfully the first-order phase transition at $\Delta=1$. On the other hand, the infinite-order QPT at $\Delta=-1$ can only be detected by the classical correlation $C_T(\rho)$, while $D_T(\rho)$ and $T_T(\rho)$ failed. This seems to casting a doubt on the usefulness of GQD, but for other many-body systems it may work effectively for detecting phase transition points.
By using the quantum renormalization group method, @qpt08 studied HS norm of discord for ground states of the Heisenberg *XXZ* model with Dzyaloshinskii-Moriya (DM) interaction. The Hamiltonian reads $$\label{qpt-02}
\begin{aligned}
\hat{H}= & \frac{J}{4}\sum_{i=1}^N [\sigma_i^x \sigma_{i+1}^x
+\sigma_i^y \sigma_{i+1}^y+\Delta\sigma_i^z \sigma_{i+1}^z \\
&+ D(\sigma_i^x\sigma_{i+1}^y-\sigma_i^y\sigma_{i+1}^x)].
\end{aligned}$$ Their calculation shows that the HS norm of discord can effectively characterize the QPT point $\Delta_c= \sqrt{1+D^2}$ separating the spin-fluid phase and the Néel phase.
@qpt01 also studied quantum correlations for the ground state properties of several three different spin models, but they used the trace norm of discord. First, for the *XXZ* model given in Eq. , they found that the trace norm of discord detects successfully the QPT point $\Delta_c$. Second, for the Ising model with DM interaction, $$\label{qpt-03}
\hat{H}= \frac{J}{4}\sum_{i=1}^N [\sigma_i^z \sigma_{i+1}^z
+ D(\sigma_i^x\sigma_{i+1}^y-\sigma_i^y\sigma_{i+1}^x)],$$ it was showed that the trace norm of discord can also be used to detect the critical point $D=1$ which separates the antiferromagnet phase and chirality phase. @qpt01 also considered the Heisenberg *XXZ* model with staggered DM interaction, with the Hamiltonian being given by $$\label{qpt-04}
\begin{aligned}
\hat{H}= & \frac{J}{4}\sum_{i=1}^N [(1+\Delta)\sigma_i^x \sigma_{i+1}^x
-(1-\Delta)\sigma_i^y \sigma_{i+1}^y \\
&+ D(\sigma_i^x\sigma_{i+1}^y+\sigma_i^y\sigma_{i+1}^x)],
\end{aligned}$$ and their result showed that the trace norm of discord also detects successfully the region $|\Delta|\leq \sqrt{1+D^2}$ in which the system is in the Néel phase. Concerning entanglement properties of this model, we refer to the work of @Mafw
For the *XX* model with transverse magnetic field as showed in Eq. , @qpt03 found that the trace norm of discord can also effectively characterize the second-order QPT occurs at $\lambda_c=1$ which separates the ferromagnetic and paramagnetic phases. @qpt04 studied QPT in an Ising-*XXZ* diamond model. By analyzing scaling behavior of the trace norm of discord for the thermal state, they found that around the critical lines, its first-order derivative exhibits a maximal at finite temperature and diverges when $T\rightarrow 0$.
Moreover, @qpt05 studied the problem of many-body localization (MBL) in a spin-1/2 Heisenberg model with random on-site disorder of strength $h$. The Hamiltonian is $$\label{qpt-05}
\hat{H}= \frac{1}{2}\sum_{i=1}^N [J(\sigma_i^x \sigma_{i+1}^x
+\sigma_i^y \sigma_{i+1}^y+\sigma_i^z \sigma_{i+1}^z)
+h_i \sigma_i^z],$$ where $h_i$ are uniformly distributed random numbers in the interval $[-h,h]$. They founded that the derivatives of the trace norm of discord of Eq. and the geometric classical and total correlations of Eq. give the range $h_c/J \in [3,4]$ for the MBL critical point. This estimate is in accordance with the result $h_c/J\sim 3.8$ given in the literature [@qpt06; @qpt07].
Quantum correlations and coherence in relativistic settings {#sec:8}
===========================================================
Since the early 20th century, much efforts have been put forward to bridge the gap between quantum mechanics and relativity theory, which are two fundamentals of modern physics. The reconciliation between them gives birth for quantum field theory (QFT), and several predictions have been made based on this theory. A fundamental prediction in QFT is that the particle content of a quantum field is observer dependent, such a phenomenon is named Unruh effect [@unruh1976; @unruhreview]. Again, the phenomenon of a quantum field is in vacuum state as observed by a freely falling observer of an eternal black hole, while it is a thermal state for a observer who hovers outside the event horizon the black hole. Such a phenomenon is named Hawking effect. The study of quantum correlation in a relativistic framework is not only helpful to understand some of the key questions in quantum information theory, but also plays an important role in the black hole entropy and black hole information paradox [@Hawking76; @Terashima]. Following the pioneering work of @SRQIT1, many authors have studied quantum correlations in relativistic setting from different aspects.
Quantum correlations for free field modes {#sec:8A}
-----------------------------------------
For a free mode scalar field, the dynamics of the field obeys the Klein-Gordon (KG) equation [@Birrelldavies] $$\begin{aligned}
\label{K-G Equation}
\frac{1}{\sqrt{-g}}\frac{{\partial}}{\partial x^{\mu}}
\left(\sqrt{-g}g^{\mu\nu}\frac{\partial\phi}{\partial x^{\nu}}\right)=0,\end{aligned}$$ where $g$ is the determinant of the metric $g_{\mu\nu}$ [@wald94]. Similarly, the motion equation of a Dirac field $\Psi$ in a background reads $$\begin{aligned}
\label{DiracEquation}
i\gamma^{\mu}(x)\left(\frac{\partial}{\partial x^{\mu}}- \Gamma_{\mu}\right)\Psi=m\Psi,\end{aligned}$$ where the background-dependent Dirac matrices $\gamma^{\mu}(x)$ relate to the matrices in flat space through $\gamma^{\mu}(x)=
e^{\mu}_{a}(x)\bar{\gamma}^{a}$, and $\bar{\gamma}^{a}$ are the flat-space Dirac matrices. Here, $$\Gamma_\mu = \frac{1}{8} [\gamma^\alpha,\gamma^\beta]e_\alpha^\nu
e_{\beta\nu;\mu},$$ are the spin connection coefficients. Throughout this section we set $G=c=\hbar=\kappa_{B}=1$.
The field (either scalar field or Dirac field) can be quantized in terms of a complete set of modes $u_{k} (x, \eta)$, which is an orthonormal basis of solutions of the scalar field (or Dirac field). That is, $$\Phi(\Psi)= \int d^3 k (a_{k} u_{k} + a^{\dagger}_{k} u^*_{k}),$$ where $\Phi$ denotes the scalar field and $\Psi$ denotes the Dirac field, $k$ is the wave vector labeling the modes and for massless fields $\omega=|k|$.
For a scalar field, the positive and negative frequency modes satisfy the canonical commutation relations $[a_{k},
a^{\dagger}_{k'}]= \delta^3({k}-{k}')$, while for the Dirac fields the anti-commutation relations $\{ a_{k}, a^{\dag}_{k'}\}=
\delta^3(k-k')$ should be satisfied. The annihilation operators $a_{k}$ define the vacuum state $\vert0\rangle$ through $a_{k}
\vert0\rangle=0,\,\,\forall\, k$. A different inequivalent choice of modes $\{ \tilde{u}_k \}$ might exist which satisfies the same equation of motion in different spacetime background. For example, the appropriate coordinates to describe the accelerated observer’s motion is the Rindler coordinates $(\eta,\xi)$, which is given by the transformation $$\label{Rindlerc}
t= a^{-1}e^{a\xi}\sinh(a\eta),~
x=a^{-1}e^{a\xi}\cosh(a\eta).$$ Solving the KG equation or Dirac equation in the Rindler coordinates, we obtain some sets of positive frequency modes propagating in the regions I and II of the Rindler spacetime, respectively. For free scalar fields, the positive frequency modes can be used to expand the field as [@Schuller-Mann] $$\label{Firstexpand}
\Phi= \int d\omega[\hat{a}^{I}_{\omega}\Phi^{+}_{{\omega},\text{I}}
+\hat{b}^{I\dag}_\omega \Phi^{-}_{{\omega},\text{I}}
+\hat{a}^{II}_{\omega}\Phi^{+}_{{\omega},\text{II}}
+\hat{b}^{II\dag}_{\omega}\Phi^{-}_{{\omega},\text{II}}],$$ where $\hat{a}^{I}_{\omega}$ and $\hat{b}^{I\dag}_{\omega}$ are the bosonic annihilation and anti-boson creation operators in the Rindler region $I$, and $\hat{a}^{II}_{\omega}$ and $\hat{b}^{II\dag}_{\omega}$ are the bosonic annihilation and creation operators in the region $II$.
The quantum field theory for Dirac fields is constructed by expanding the field in terms of the positive and negative frequency modes [@Alsing2006] $$\label{Firstexpandd}
\Psi= \int d\mathbf{k}[\hat{c}^{I}_{\mathbf{k}}\Psi^{I+}_{\mathbf{k}}
+\hat{d}^{I\dag}_{\mathbf{k}}\Psi^{I-}_{\mathbf{k}}
+\hat{c}^{II}_{\mathbf{k}}\Psi^{II+}_{\mathbf{k}}
+\hat{d}^{II\dag}_{\mathbf{k}}\Psi^{II-}_{\mathbf{k}}],$$ where $\hat{c}^{I}_{\mathbf{k}}$ and $\hat{d}^{I\dag}_{\mathbf{k}}$ are the fermionic annihilation and creation operators acting on the state in region $I$, and $\hat{c}^{II}_{\mathbf{k}}$ and $\hat{d}^{II\dag}_{\mathbf{k}}$ are the fermionic operators in the region $II$. The above positive and negative frequency modes are defined in terms of the future-directed timelike Killing vector in different regions, in Rindler region $I$ the Killing vector is $\partial_\eta$ and in the region $II$ the Killing vector is $\partial_{-\eta}$.
After some calculations, the Minkowski vacuum is found to be an entangled two-mode squeezed state for a free scalar field $$\label{vacuums}
|0_\omega\rangle_\text{M}=\frac{1}{\cosh r_\omega^{2}}
\sum_{n=0}^{\infty}\tanh r_\omega^{n}|nn\rangle_{\omega},$$ where $\cosh r=(1-e^{-2\pi\omega/a})^{-1/2}$ and $a$ is Rob’s acceleration. For a free Dirac field, the Minkowski vacuum has the following form $$\label{Dirac-vacuum1}
|0_{\mathbf{k}}\rangle_{M}= \cos r|0_{\mathbf{k}}\rangle_{I}|0_{-\mathbf{k}}\rangle_{II}+\sin r
|1_{\mathbf{k}}\rangle_{I}|1_{-\mathbf{k}}\rangle _{II},$$ where $\cos r=(e^{-2\pi\omega/a}+1)^{-1/2}$.
### Quantum entanglement
@Schuller-Mann studied quantum entanglement between two free bosonic modes as observed by two relatively accelerated observers. They found that the quantum entanglement is an observer-dependent quantity in noninertial frames. A maximally entangled initial state in an inertial frame becomes less entangled under the influence of the Unruh effect. In the infinite acceleration limit, the distillable entanglement for the final state of the scalar field vanishes. @Alsing2006 studied the entanglement between two free modes of a Dirac field in noninertial frames. They found that entanglement between the Dirac modes is destroyed by the Unruh effect. Differently, the entanglement of the fermionic modes asymptotically reaches a nonzero minimum value in the infinite acceleration limit.
@LingYi studied entanglement of the electromagnetic field in a noninertial reference frame. They employed the photon helicity entangled state and found that the logarithmic negativity of the final state remains the same as those in the inertial reference frame, which is completely different from that of the particle number entangled state. @Panqiyuan investigated the entanglement between two modes of free scalar and Dirac fields. They proved that the different behavior of the field modes is owing to the in equivalence of the quantization of the free field modes in the Minkowski and the Rindler coordinates. In the infinite-acceleration limit, the mutual information equals to the half mutual information of the initial state, which is independent of the initial state and the type of field.
@Adesso2007 studied the distribution of entanglement between modes of a free scalar field from the perspective of observers in uniform acceleration. We consider a two-mode squeezed state of the field from an inertial perspective, and analytically study the degradation of entanglement due to the Unruh effect, in the cases of either one or both observers undergoing uniform acceleration. The effect of Unruh effect on a quantum radiation can be described by a two-mode squeezing operator acting on the input state of the quantum system. In the phase space the symplectic phase-space representation, $S_{B, \bar B}(r)$ for the two-mode squeezing transformation is [@Adesso2007] $$\begin{aligned}
\label{cmtwomode}
S_{B,\bar B}(r)=\left(
\begin{array}{cc}
\cosh r {\openone}_2 & \sinh r Z_2 \\
\sinh r Z_2 & \cosh r {\openone}_2 \\
\end{array}
\right),\end{aligned}$$ where ${\openone}_2$ is a $2\times 2$ identity matrix and $Z_2=
\mathrm{diag} \{1,-1\}$. After the transformation, the final state of the entire three-mode system is given by the covariance matrix [@Adesso2007] $$\begin{aligned}
\label{in34}
\nonumber\sigma^{\rm }_{AB \bar B}(s,r) &=&
\big[{\openone}_A \oplus S_{B,\bar B}(r)\big]
\big[\sigma^{\rm (M)}_{AB}(s) \oplus {\openone}_{\bar B}\big]
\big[{\openone}_A \oplus S_{B,\bar B}(r)\big]\\
&=& \left(
\begin{array}{ccc}
\mathcal{\sigma}_{A} & \mathcal{E}_{AB} & \mathcal{E}_{A\bar B} \\
\mathcal{E}^{\sf T}_{AB} & \mathcal{\sigma}_{B} & \mathcal{E}_{B\bar B} \\
\mathcal{E}^{\sf T}_{A\bar B} & \mathcal{E}^{\sf T}_{B\bar B} & \mathcal{\sigma}_{\bar B} \\
\end{array}
\right),\end{aligned}$$ where the diagonal elements are $$\begin{aligned}
& \mathcal{\sigma}_{A}= \cosh(2s){\openone}_2, \\
& \mathcal{\sigma}_{B}=[\cosh(2s) \cosh^2(r) + \sinh^2(r)]{\openone}_2, \\
& \mathcal{\sigma}_{\bar B}=[\cosh^2(r) + \cosh(2s) \sinh^2(r)]{\openone}_2,
\end{aligned}$$ and the non-diagonal elements have the following forms: $$\begin{aligned}
& \mathcal{E}_{AB}=[\cosh(r) \sinh(2s)]Z_2, \\
& \mathcal{E}_{B\bar B}=[\cosh^2(s) \sinh(2r)]Z_2, \\
& \mathcal{E}_{A\bar B}=[\sinh(2s) \sinh(r)]Z_2.
\end{aligned}$$ It was found that for two observers undergoing the finite acceleration, the entanglement vanishes between the lowest-frequency modes. The loss of entanglement is precisely explained as a redistribution of the inertial entanglement into the multipartite quantum correlations among accessible and inaccessible modes from a noninertial perspective. The classical correlations are also lost from the perspective of two accelerated observers but conserved if one of the observers remains inertial.
@Leon2009 investigated the effect of Unruh effect on spin and occupation number entanglement of Dirac fields in the noninertial frame. They analyzed spin Bell states and occupation number entangled state in a relativistic setting, obtained their entanglement dependence on the acceleration. They showed that the acceleration produces a qubit$\times$four-level quantum system state for the spin case, while there is always qubit$\times$qubit for the spinless case despite their apparent similitude. The entanglement degradation in the spin case is greater than in the spinless case. They as well introduced a procedure to consistently erase the spin information from the system and preserving occupation numbers at the same time. @MannVillalba09 studied the speeding-up degradation of entanglement as a function of acceleration for the free scalar field in an accelerated frame.
@Moradi studied the distillability of entanglement of bipartite mixed states of two modes of a free Dirac field in accelerated frames. It was showed that there are some certain value of accelerations which will change the state from a distillable one into separable one. @Doukas09 studied the loss of spin entanglement for accelerated electrons in electric and magnetic fields by using an open quantum system. They found that the proper time for the extinguishment of entanglement is proportional to the inverse of the acceleration cubed at high Rindler temperature. @Ostapchuk09 studied the generation of entangled fermions by accelerated measurements on the vacuum. [@Aspachs:2010] find that the Unruh-Hawking effect acts on a quantum system as a bosonic amplification channel.
@wangjing2010 studied the dynamics of quantum entanglement for Dirac field when the field interacts with noise environment in noninertial frames. They found that the decoherence induced by the noise environment and loss of the entanglement generated by the Unruh effect will influence each other remarkably. In the case of the total system interact with noise environment, the sudden death of entanglement may appear for any acceleration. However, sudden death may only occur when the acceleration parameter is greater than a critical point when only Rob’s qubit under decoherence.
@Hwang2011 examined the entanglement of a tripartite of scalar field when one of the three parties is moving with uniform acceleration. The tripartite entanglement exhibits a decreasing behavior but does not completely vanish in the infinite acceleration limit, which is different from the behavior of bipartite entanglement. This fact indicates that the quantum information processing tasks using tripartite entanglement may be possible even if one of the parties approaches to the horizon of the Rindler spacetime.
@wang2011 investigated tripartite entanglement of a fermionic system when one or two subsystems accelerated. They found that all the one-tangles decrease with increasing acceleration but never reduce to zero for any acceleration, which is different from the scalar case of scalar field. It was shown that the system has only tripartite entanglement when one or two subsystems with accelerated motion, which means that the acceleration does not effect the entanglement structure of the quantum states. The tripartite entanglement of the case of two observers accelerated decreases much quicker than the one-observer-accelerated case.
@Olson11 studied quantum entanglement between the future region and the past region in the quantum vacuum of the Rindler spacetime. The massless free scalar fields within the future and past light cone was quantized as independent systems. The initial vacuum between the future and past regions became an entangled state of these systems, which exactly mirrors the prepared entanglement between the space-like separated Rindler wedges. This lead to the notion of time-like entanglement. They described an detector which would exhibit thermal response to the vacuum and discussed the feasibility of detecting the Unruh effect.
@wangjing2012 discussed the system-environment dynamics of Dirac fields for amplitude damping and phase damping channels in noninertial systems. They found that the thermal fields generated by the Unruh thermal bath promotes the sudden death of entanglement between the subsystems while postpone the sudden birth of entanglement between the environments. However, no entanglement was generated between the system and environment when the system coupled with the phase damping environment.
@Montero13 argued that in the infinite acceleration limit, the entanglement in a bipartite system of the fermionic field must be independent of the choice of Unruh modes. Therefore, to compute field entanglement in relativistic quantum information, the tensor product structures should be modified to give rise to physical results.
@Khan14 studied the dynamics of tripartite entanglement for Dirac fields through linear contraction criterion in the noninertial frames. It is found that the entanglement measurement is not invariant with respect to the partial realignment of different subsystems if one observer is accelerated case, while it is invariant in the two observers accelerated case. It is shown that entanglement are not generated by the acceleration of the frame for any bipartite subsystems. Unlike the bipartite states, the genuine tripartite entanglement does not completely vanish in both one observer accelerated and two observers accelerated cases even in the limit of infinite acceleration.
@Dai15JHEP discussed the entanglement of two accelerated Unruh-Wald detectors which couple with real scalar fields. The found that the bipartite entanglement of the two qubits suddenly dies when the acceleration of one or more qubits are large enough, which is a result of Unruh thermal bath. @Dai16PRD studied the entanglement of three accelerated qubits, each of them is locally coupled with the real scalar field, without causal influence among the qubits or among the fields. The obtained how the entanglement depends on the accelerations of the three qubits and found that all kinds of entanglement would sudden death if at least two of three qubits have large enough accelerations.
@Metwally studied the possibility of recovering the entanglement of accelerated qubit and qutrit systems by using weak-reverse measurements. It is found that the accelerated coded local information in the qutrit system is more robust than that encoded in the qubit system. In addition, the non-accelerated information in the qubit system is not affected by the local operation compared with that depicted on qutrit system.
### Discordlike correlations
@Datta09 discussed the QD between two free modes of a scalar field which are observed by two relatively accelerated observers. It was showed that finite amount of QD exists in the regime where there is no distillable entanglement due to the Unruh effect. In addition, they provided evidence for a nonzero amount of QD in the limit of infinite acceleration. @Martin2010c studied the behavior of classical and quantum correlations in a spacetime with an event horizon, comparing fermionic with bosonic fields. They showed the emergence of conservation laws for classical correlations and quantum entanglement, pointing out the crucial role that statistics plays in the entanglement tradeoff across the horizon.
@wang2010c investigated the distribution of classical correlations and QD of Dirac modes among different regions in a noninertial frames. They found that for the Dirac field, the classical correlation decreases with increasing acceleration, which is different from the scalar field case where the classical correlation is independent of the acceleration.
@Ramzan studied the dynamics of GQD and MIN for noninertial observers at finite temperature. It was found that the GQD can be used to distinguish the Bell, Werner, and general type initial quantum states. In addition, sudden transition in the behavior of GQD and MIN depends on the mean photon number of the local environment. In the case of environmental noise is introduced in the system, this effect becomes more prominent. In the case of depolarizing channel, the environmental noise is found to have stronger affect on the dynamics of GQD and MIN as compared to the Unruh effect. @qiang15 investigated the distribution of GQD among all possible bipartite divisions of a tripartite system for the free Dirac field modes in noninertial frames. As a comparison, they also discussed the geometric measure of entanglement for the same quantum state.
### Quantum coherence
@chencoherence investigated the behavior of quantum coherence for free scalar and Dirac modes as detected by accelerated observers. They showed that the relative entropy of coherence is destroyed as increasing acceleration of the detectors. In addition, the shared coherence between the accelerated observers vanishing in the infinite acceleration limit for the scalar field, but tends to a non-vanishing value for the Dirac field.
@huangcoh studied the freezing condition of coherence for accelerated free modes in a relativistic setting beyond the single-mode approximation. They also discussed the behavior of cohering power and decohering power under the Unruh channel. It was found that the quantum coherence can be distributed between different modes, but the coherence lost in the particle mode sector is not transferred entirely to the antiparticle mode sector. They also demonstrated that the robustness of quantum coherence are better than entanglement under the influence of Unruh effect.
@LiuTian2016a [@LiuTian2016b] investigated the dynamics of quantum coherence of two-level atoms interacting with the electromagnetic field in the absence and presence of boundaries. They found that for the two-level systems, the quantum coherence cannot be protected from noise without boundaries. However, in the presence of a boundary, the insusceptible of the quantum coherence can be fulfilled when the atoms is close to the boundary and is transversely polarizable. In addition, in the presence of two parallel reflecting boundaries, for some special distances the quantum coherence of atoms can be shielded from the influence of external environment when the atoms have a parallel dipole polarization at arbitrary location between these two boundaries.
Free field modes beyond the single-mode approximation {#sec:8B}
-----------------------------------------------------
@SMA2009 introduced an arbitrary number of accessible modes when analyzing the Unruh effect on bipartite entanglement degradation. @Bruschi2010 performed that an inertial observer has the freedom to create excitations in any accessible modes $\Omega_j, \forall j$ rather than a typical mode. Therefore, one cannot maps a single-frequency Minkowski mode into a set of single frequency Rindler modes in an accelerated setting [@Bruschi2010]. That is, the single-mode approximation should be relaxed in a general setting. To overcome the shortage of the single-mode approximation, one should employ the Unruh basis which provides an intermediate step between the Minkowski modes and Rindler modes. The relations between the Unruh and the Rindler operators are $$\label{Unruhop}
\begin{aligned}
& C_{\omega,\text{\text{R}}}=\left(\cosh r_{\omega}\, \hat{a}_{{\omega},\text{I}}-\sinh r_{\omega}\, \hat{b}^\dagger_{{\omega},\text{II}}\right),\\
& C_{\omega,\text{\text{L}}}=\left(\cosh r_{\omega}\, \hat{a}_{{\omega},\text{II}}-\sinh r_{\omega}\, \hat{b}^\dagger_{{\omega},\text{I}}\right),\\
& D^\dagger_{\omega,\text{\text{R}}}=\left(-\sinh r_{\omega}\, \hat{a}_{{\omega},\text{I}}+\cosh r_{\omega}\, \hat{b}^\dagger_{{\omega},\text{II}}\right),\\
& D^\dagger_{\omega,\text{\text{L}}}=\left(-\sinh r_{\omega}\, \hat{a}_{{\omega},\text{II}}+\cosh r_{\omega}\, \hat{b}^\dagger_{{\omega},\text{I}}\right),
\end{aligned}$$ where $\sinh r_{\omega}=(e^{2\pi\omega/a}-1)^{-1/2}$. The Unruh modes are positive-frequency combinations of plane waves in the Minkowski spacetime, but enjoy an important property: they are mapped into single frequency Rindler modes.
For the free scalar fields, the generic Rindler Fock state $|nm,pq\rangle_{\omega}$ describing both boson and antiboson is
$$\begin{aligned}
\label{shortnot}
|nm,pq\rangle_{\omega}\coloneqq \frac{\hat{a}_{{\omega},\text{out}}^{\dagger n}}{\sqrt{n!}}
\frac{\hat{b}^{\dagger m}_{\omega,\text{in}}}{\sqrt{m!}}\frac{\hat{b}^{\dagger p}_{\omega,\text{out}}}{\sqrt{p!}}
\frac{\hat{a}^{\dagger q}_{\omega,\text{in}}}{\sqrt{q!}}|0\rangle_S,\end{aligned}$$
where the $\pm$ sign is the notation for boson and antiboson, respectively. This allows us to rewrite the Unruh vacuum as [@Fabbri; @Bruschi2] $$\label{vacuumba}
|0_\omega\rangle_\text{U}=\frac{1}{\cosh r_\omega^{2}}\sum_{n,m=0}^{\infty}\tanh r_\omega^{n+m}|nn,mm\rangle_{\omega},$$ where $|0_\omega\rangle_\text{U}$ is a shortcut notation used to underline that each Unruh mode $\omega$ is mapped into a single frequency Rindler mode $\omega$.
One particle Unruh states are defined as $|1_{j}\rangle^+_{\text{U}}
=c_{\omega,\text{U}}^\dagger|0\rangle_\text{H}$, $|1_{j}
\rangle^-_{\text{U}}=d_{\omega,\text{U}}^\dagger|0\rangle_\text{H}$, where $|0\rangle_\text{H}$ denotes the Hartle-Hawking vacuum. The particle and antiparticle creation operators for Unruh modes are defined as $$\label{creat}
c_{\omega,\text{U}}^\dagger= q_{\text{R}}C^\dagger_{\omega,\text{R}}+q_{\text{L}}C^\dagger_{\omega,\text{L}},~
d_{\omega,\text{U}}^\dagger= q_{\text{R}}D^\dagger_{\omega,\text{R}}+q_{\text{L}}D^\dagger_{\omega,\text{L}},$$ where $q_\text{\text{R}}$ and $q_\text{\text{L}}$ satisfy $|q_\text{\text{R}}|^2+|q_\text{\text{L}}|^2=1$. The operator $c_{\omega,\text{U}}^\dagger$ in Eq. means the creation of a pair of particles [@Bruschi2; @jieci1], i.e., a boson with mode $\omega$ in the Rindler region [*I*]{} and an antiboson in the Rindler region [*II*]{}. Similarly, the creation operator $d_{\omega,\text{U}}^\dagger$ denotes that an antiboson and a boson are created in Rindler region [*I*]{} and [*II*]{}, respectively.
@SMA2009 introduced an arbitrary number of accessible modes for the Dirac field. They proved that under the single-mode approximation a fermion only has a few accessible levels due to Pauli exclusion principle, which is different from the bosonic fields which has infinite number of excitable levels. This was argued to justify entanglement survival in the fermionic case at the infinite acceleration limit under the single-mode approximation. By relaxing the single-mode approximation, entanglement loss for the Dirac field mode is limited, which comes from fermionic statistics through the characteristic structure. In addition, the surviving entanglement in the infinite acceleration limit is found to be independent of the the type of fermionic field and the number of considered accessible modes.
@Bruschi2010 addressed the validity of the single-mode approximation and discussed the behavior of Unruh effect beyond the single-mode approximation. They argued that the single-mode approximation is not valid for arbitrary states in a relativistic setting. In addition, some corrections to previous studies on relativistic quantum information beyond the single-mode approximation are performed both for the bosonic and fermionic cases. They also exhibited a sequence of wave packets where such approximation is justified subject to the peaking constraints which set by some appropriate Fourier transforms.
@Bruschi2 analyzed the tradeoff of quantum entanglement between particle and anti-particle modes of a charged bosonic field in a noninertial frame beyond the single-mode approximation. They found that the redistribution of entanglement between bosonic and antibosonic modes does not prevent the entanglement from vanishing in the limit of infinite acceleration. That is, they included antiparticles in the study of bosonic entanglement by analyzing the charged bosonic case and find that mode entanglement always vanishes in this limit. This supports the conjecture that the main differences in the behavior of entanglement in the bosonic field mode and fermionic field mode case are due to the difference between the Bose-Einstein statistics and the Fermi-Dirac statistics.
@Brown2012 demonstrated that quantum correlations measured by the GQD decays to zero in the limit of infinite acceleration, which is in contrast with previous research showing that the degradation of QD vanish in this limit. They argued that the usable quantum correlations measured by GQD in the large acceleration regime appear severely limited for any protocols. In addition, vanishing of the GQD implies a significant limitation on the usable quantum correlations for large accelerations.
@Tian13 studied the MIN for both Dirac and Bosonic fields in non-inertial frames beyond the single-mode approximation. They found that two different behaviors exist between the Dirac and scalar fields are: (i) the MIN for Dirac fields persists for any acceleration, while for Bosonic fields this quantity does decay to zero in the limit of infinite acceleration; (ii) the dynamic behaviors of the MIN for scalar fields is quite different from the Dirac fields case in the accelerated frame. In addition, the MIN is found to be more general than the quantum nonlocality related to violation of Bell inequalities.
@Richter studied the entanglement of Unruh modes shared by two accelerated observers and find some differences in the robustness of entanglement for these states under the effect of Unruh thermal bath. For the initial state prepared in Bell states of free bosonic and a fermionic modes, they found that the states $\Psi_\pm$ are entangled for any finite accelerations. However, the states $\Phi_\pm$ exists entanglement sudden death for some finite accelerations due to the effect of Unruh radiation. They also considered the differences in the behavior of entanglement for fermionic modes and discussed the role that is played by particle statistics. These results suggest that the degradation of entanglement in noninertial frames strongly depends on the occupation patterns of the constituent states.
Curved spacetime and expanding universe {#sec:8C}
---------------------------------------
### In the background of a black hole
As discussed in Refs. [@Schuller-Mann; @Panqiyuan2008], the role of a Rindler observer in the accelerated frame corresponds to a Schwarzschild observer in the background of a black hole [@Fabbri]. In addition, it was found that the effect of Hawking radiation of the black hole on a quantum system can be described by a bosonic amplification channel [@Aspachs:2010]. In this case, we assume Alice stays stationary at an asymptotically flat region of an external black hole, and Bob is a Schwarzschild observer who hovers near the event horizon of a black hole. The spacetime background near a static and asymptotically flat Schwarzschild black hole, is described by $$\label{matric}
\begin{split}
ds^2 =&-(1-\frac{2M}{r}) dt^2+(1-\frac{2M}{r})^{-1} dr^2 \\
&+r^2(d\theta^2 +sin^2\theta d\varphi^2),
\end{split}$$ where $M$ represents the mass of the black hole.
Solving the KG equation or Dirac equation near the event horizon of the black hole, one can obtain a set of positive frequency modes propagating in the exterior and interior regions of the event horizon. Here, we introduce the quantization of Dirac fields, in this case the positive (fermions) frequency solutions are found to be $$\label{inside and outside mode}
\Psi^{+}_{I,\mathbf{k}}=\mathcal {G}e^{-i\omega \mathcal {U}},~
\Psi^{+}_{II,\mathbf{k}}=\mathcal {G}e^{i\omega \mathcal {U}},$$ where $\mathcal {U}=t-r_{*}$ and $\mathcal{G}$ is a 4-component Dirac spinor, $\mathbf{k}$ is the wave vector used to label the modes and for massless Dirac field we have $\omega=|\mathbf{k}|$.
In terms of these basis, the Dirac field $\Psi$ can be expanded as $$\label{Firstexpand02}
\Psi =\int d\mathbf{k}[\hat{a}^{out}_{\mathbf{k}}\Psi^{+}_{out,\mathbf{k}}
+\hat{b}^{out\dag}_{\mathbf{-k}}\Psi^{-}_{out,\mathbf{k}}
+ \hat{a}^{in}_{\mathbf{k}}\Psi^{+}_{in,\mathbf{k}}
+\hat{b}^{in\dag}_{\mathbf{-k}}\Psi^{-}_{in,\mathbf{k}}],$$ where $\hat{a}^{out}_{\mathbf{k}}$ and $\hat{b}^{out\dag}_{\mathbf{k}}$ are the fermionic annihilation and antifermion creation operators acting on the state of the exterior region of the black hole, and $\hat{a}^{in}_{\mathbf{k}}$ and $\hat{b}^{in\dag}_{\mathbf{k}}$ are the operators acting on the state in the interior region of the black hole. These operators $\hat{a}^{out}_{\mathbf{k}}$ satisfy the canonical anticommutation relations $$\begin{split}
& \{\hat{a}^{out}_{\mathbf{k}}, \hat{a}^{out}_
{\mathbf{k'}}\}=\delta_{\mathbf{k}\mathbf{k'}},\\
& \{\hat{a}^{out}_{\mathbf{k}},\hat{a}^{out\dagger}_{\mathbf{k'}}\}
=\{\hat{a}^{out\dagger}_{\mathbf{k}},\hat{a}^{out\dagger}_{\mathbf{k'}}\}=0
\end{split}$$ where $\{.,.\}$ denotes the anticommutator.
Making analytic continuation for Eq. according to the suggestion of Damour-Ruffini [@D-R], a set of Kruskal modes is obtained. The Kruskal modes can be used to define the Hartle-Hawking vacuum, corresponding to Minkowski vacuum in an inertial frame [@Fabbri]. These two sets of operators are related to each other by the Bogoliubov transformation $$\begin{gathered}
\label{Bogoliubov transformation}
\tilde{a}=\int_{k'} dk'\left[\alpha_{k k'} a_{k'}+\beta^*_{k k'} a_{k'}^{\dagger}\right],\end{gathered}$$ where $\alpha_{k k'}$ and $\beta_{k k'}$ are the Bogoliubov coefficients, which encode information about the spacetime. To quantize the Dirac filed beyond the single-mode approximation [@SMA2009; @Bruschi2010], we construct a different set of operators in the inside and outside regions of the black hole, which are $$\label{Dirac-ex}
\begin{aligned}
& \tilde{c}^\dagger_{\mathbf{k},R}= \cos r \hat{a}^{out\dagger}_{\mathbf{k}}- \sin r \hat{b}^{in\dagger}_{\mathbf{-k}} \\
& \tilde{c}^\dagger_{\mathbf{k},L}= \cos r \hat{a}^{in\dagger}_{\mathbf{k}}- \sin r \hat{b}^{out\dagger}_{\mathbf{-k}},
\end{aligned}$$ where $$\cos r = (e^{-8\pi\omega M}+1)^{-1/2},~ \sin r = (e^{8\pi\omega
M}+1)^{-1/2}.$$ A relevant set of annihilation operators can be constructed in a analogous way. These modes with subscripts L and R are left and right Unruh modes. After some calculations, the Hartle-Hawking vacuum is found to be $|0\rangle_H = \bigotimes_{\mathbf{k}}
|0_{\mathbf{k}}\rangle_{K}$, where $$\label{Dirac-vacuum}
\begin{aligned}
|0_{\mathbf{k}}\rangle_{K}=&\cos^2 r|0000\rangle -\sin r |0011\rangle \\
&+\sin r\cos r|1100\rangle-\sin^2 r|1111\rangle.
\end{aligned}$$ In the last-written equation $$|mnm'n'\rangle=|m_{\mathbf{k}} \rangle^{+}_{out}
|n_{-\mathbf{k}}\rangle^{-}_{in} |m'_{-\mathbf{k}}\rangle^{-}_{out}
|n'_{\mathbf{k}} \rangle^{+}_{in},$$ with $\{|n_{-\mathbf{k}}\rangle^{-}_{in}\}$ and $\{|n_{\mathbf{k}}
\rangle^{+}_{out}\}$ being the orthonormal bases of the inside and outside the event horizon of the black hole, and the $\{+,-\}$ is used to indicate the fermion and antifermion vacuum states.
@Panqiyuan2008 discussed the effect of the Hawking temperature of a static and asymptotically flat black hole on the entanglement and teleportation for the free scalar modes. It was demonstrated that the fidelity of teleportation decreases as the Hawking temperature of the black increases, which indicates the thermal bath induced by the Hawking radiation destroys the quantum channel. The final state are absent of any distillable entanglement in the infinite Hawking temperature limit, which corresponds to the case of the black hole evaporating completely.
@gesang studied the dynamics of entanglement and the fidelity of teleportation in the background of a rotating black hole with extra dimensions. The metric of a $d$-dimensional black hole is given by $$\label{schw}
\begin{aligned}
ds^2=&-\left[1-\Bigl(\frac{r_{h}}{r} \Bigr)^{d-3}\right]dt^2
+\left[1- \Bigl(\frac{r_{h}}{r}
\Bigr)^{d-3}\right]^{-1}dr^2\\
&+r^2d\Omega^2_{d-2},
\end{aligned}$$ where $r_{h}$ is the event horizon of the black hole with area $A_{d}=r^{d-2}_{h}\Omega_{d-2}$, and $\Omega_{d-2}$ is the volume of a unit $(d-2)$-sphere. From Eq. , one can obtain the mass of the $d$-dimensional black hole, which is $$\label{mbh} M=\frac{(d-2)r^{d-3}_{h}\Omega_{d-2}}{16\pi G_{d}},$$ for the $d$-dimensional Newton’s constant $G_{d}$. They discussed how the extra dimensions, the black hole’s mass and angular momentum parameter, and mode frequency would influence the behavior of quantum entanglement and fidelity in the curved spacetime. They showed that a maximally entangled initial state which is prepared in an inertial frame becomes less entangled in the curved space due to the Hawking radiation. In addition, the degree of entanglement and fidelity of quantum teleportation were found to be degraded with increasing extra dimension parameter and surface gravity of the black hole.
@wang2009plb studied quantum entanglement of the coupled massive scalar field in the spacetime of a Garfinkle-Horowitz-Strominger dilation black hole. The metric for a Garfinkle-Horowitz-Strominger dilation black hole spacetime is [@Horowitz] $$\label{gem1}
\begin{aligned}
ds^2=&- \left(\frac{r-2M}{r-2\alpha}\right)dt^2+\left(\frac{r-2M}{r-2\alpha}\right)^{-1} dr^2 \\
&+ r(r-2\alpha)d \Omega^2,
\end{aligned}$$ where $M$ is the mass of the black hole and $\alpha$ is the dilation charge. It was found that entanglement does not depend on the coupling between the scalar field and the gravitational field and the mass of the field. As the dilation parameter $\alpha$ increases, entanglement is destroyed by the Hawking effect. It is interesting to note that in the limit of $\alpha=M$, corresponding to the case of an extreme black hole, the system has no entanglement for any initial state, which its mutual information equals to a nonvanishing minimum value.
@wangpan2010 studied the quantum projective measurements and generation of entangled Dirac particles in the background of a Schwarzschild Black under the single mode approximation. They found that the measurements performed by Bob who locates near the event horizon of the Schwarzschild black hole creates entangled particles. The particles can be detected by Alice who stays stationary at the asymptotically flat region. In addition, the degree of entanglement increases when the Hawking temperature increases. @dengwang2010 studied how the Hawking effect of a black hole influence the entanglement distillability of Dirac fields in the Schwarzschild spacetime. It was found that entanglement distillability of the states are influenced both by the Hawking temperature of the black hole and energy of the fields. Although the parameter of the generic entangled states affects the entanglement, it would not change the range in which the states are entangled for the case of generic entangled states.
@martin2010a analyzed the entanglement degradation provoked by the Hawking effect in a bipartite system near the event horizon of a Schwarzschild black hole beyond the single mode approximation. They determined the degree of entanglement as a function of the frequency of the field modes, the distance of the accelerated observer to the event horizon, and the mass of the black hole. They found that, in the case of Rob is far off the black hole, all the interesting phenomena occur in the vicinity and the presence of event horizons do not effectively degrade the entanglement. They also discussed the localization of Alice and Rob states in the curved spacetime.
@martin2010b studied the generation of quantum entanglement in the formation of a black hole. It was found that a field in a dynamical gravitational collapse the vacuum of a field can evolve to an entangled state. They quantified and discussed the origin of this entanglement and found that for micro-black hole formation and the final stages of evaporating black holes, it could even reach the maximal entanglement limit. In addition, fermions are found to be more sensitive than bosons to the quantum entanglement generation, which is helpful in finding experimental evidence of quantum Hawking effect in analog gravity models.
@wang2010plb studied how the Hawking radiation influence the redistribution of the entanglement and mutual information in the Schwarzschild spacetime. It was shown that the physically accessible correlations degrade while the unaccessible correlations increase under the Hawking thermal bath. This is partly because the initial correlations prepared in an inertial frame are redistributed between all the bipartite subsystems. In the limit case that the temperature tends to infinity, the accessible mutual information equals to just half of its initial value. They also studied the influence of Hawking radiation on the redistribution of the entanglement and mutual information of free Dirac field modes in the Schwarzschild spacetime [@wang2010plb]. The results showed that the physically accessible correlations degrade while the unaccessible correlations increase with increasing Hawking temperature. That is, the initial quantum entanglement prepared in inertial frame are redistributed between all the bipartite modes due to the influence of Hawking effect. In the limit of infinite Hawking temperature, the physically accessible mutual information equals to just half of its initial value. In addition, the unaccessible mutual information between mode $A$ and $II$ equals to the mutual information between mode $A$ and $I$.
@Hosler2012 discussed quantum communication between an observer who free falls into the black hole and an observer hovering over the horizon of a Schwarzschild black hole. It was found that the communication channels degrades due to the effect of the Unruh-Hawking noise. It was showed that for bosonic quantum communication using single-rail and dual-rail encoding, the classical channel capacity reduces to a finite value and the quantum coherent tends to zero by ignoring time dilation which affects all channels equally. That is, quantum coherence is fully removed at infinite acceleration, whereas classical correlation still exist in this limit.
@jieci1 studied the dynamics of the discord-type quantum correlation, the measurement-induced disturbance, and classical correlation of Dirac fields in the background of the Garfinkle-Horowitz-Strominger dilation black hole. They showed that all the above mentioned correlations are destroyed as the increase of black hole’s dilation charge. Comparing to the inertial systems, the quantum correlation measured by QD is always not symmetric with respect to the measured subsystems, while the measurement-induced disturbance is always symmetric. In addition, the symmetry of QD is found to be influenced by the spacetime curvature produced by the dilation of the black hole.
@hejuan16 discussed the MIN for Dirac particles in the Garfinkle-Horowitz-Strominger dilation spacetime. They found that as the dilation parameter increases, the physical accessible MIN decreases monotonically. The physical accessible correlation is found to be nonzero when the Hawking temperature is infinite. This is different from the case of scalar fields and owns to the statistical differences between the Fermi-Dirac fields and the Bose-Einstein fields. They also derived the boundary of the MIN related to Bell-violation and found that the former is more general than the Bell nonlocality.
The behavior of monogamy deficit and monogamy asymmetry of quantum steering under the influence of the Hawking effect is studied in [@wangjing2018]. In the curved spacetime, the monogamy of quantum steering shows an extreme scenario: the first part of a tripartite system cannot individually steer two other parties, while it can steer the collectivity of them. In addition, the monogamy deficit of Gaussian steering are generated due to the influence of the Hawking thermal bath.
### In an expanding universe
The spacetime of a homogeneous and isotropic expanding universe is described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is $$ds^2=dt^2-[a(t)]^2 (dx^2),$$ for a two-dimensional geometry. By defining the conformal time coordinate $\eta$, the FLRW metric equation is rewritten as $$\label{MconFLRW}
ds^2=[a(\eta)]^2(d\eta^2- dx^2),$$ where $[a(\eta)]^2=C(\eta)$ is the conformal scale factor. To solve the KG equation in the asymptotic past $\eta\rightarrow-\infty$ and the asymptotic future $\eta\rightarrow+\infty$ region, we choose the following conformal scale factor $$C(\eta)=1+\epsilon \tanh(\rho\eta)\,,$$ where $\epsilon$ and $\rho$ are parameters controlling the total volume and rapidity of the expansion, respectively. In the asymptotic past and future, the FLRW universe is asymptotically flat. The asymptotic solutions of the KG equation in the past and asymptotic future are $$\begin{aligned}
& u^{\text{in}}_k(x,\eta)&\underset{\eta\rightarrow-\infty}{\longrightarrow}
\frac{1}{2\sqrt{\pi \omega_{\text{in}}}}e^{i (kx-\omega_{\text{in}}\eta)},\\
& u^{\text{out}}_k(x,\eta)&\underset{\eta\rightarrow\infty}{\longrightarrow}
\frac{1}{2\sqrt{\pi \omega_{\text{out}}}}e^{i (kx-\omega_{\text{out}}\eta)},
\end{aligned}$$ where $\omega_{\text{out/in}}=\sqrt{k^2+m^2(1\pm\epsilon)}$. Considering the properties of hypergeometric functions, the Bogoliubov coefficient matrix of the scalar field in the FLRW spacetime is calculated in diagonal form. After some calculations, the Bogoliubov transform between operators is found to be $$a_{k, in}=\alpha_k^* a_{k,out}-\beta_k^* a^\dagger_{-k,out},$$ and $$a^\dagger_{k,in}=\alpha_k a^\dagger_{k,out}-\beta_k a_{-k,out}$$ where $$\begin{aligned}
&\alpha_k= \sqrt{\frac{\omega_{\text{out}}}{\omega_{\text{in}}}}
\frac{\Gamma([1-(i\omega_{\text{in}}/\rho)])
\Gamma(-i\omega_{\text{out}}/\rho)}{\Gamma([1-(i\omega_{+}/\rho)])
\Gamma(-i\omega_{+}/\rho)}, \\
&\label{betak}\beta_k= \sqrt{\frac{\omega_{\text{out}}}{\omega_{\text{in}}}}
\frac{\Gamma([1-(i\omega_{\text{in}}/\rho)])
\Gamma(i\omega_{\text{out}}/\rho)}{\Gamma([1+(i\omega_{-}/\rho)])
\Gamma(i\omega_{-}/\rho)}.
\end{aligned}$$ To quantize Dirac fields in the FLRW spacetime [@Birrelldavies; @dun1; @Bergs], an appropriate choice for the conformal factor $a(\eta)$ is [@dun1; @Birrelldavies] $$a(\eta)=1+\epsilon(1+\tanh\rho\eta).$$ Similarly, one may obtain the solution of the Dirac fields that behaving as in the asymptotic past and future region of the FLRW spacetime. Then we calculate the relation between the operators in the asymptotic future and past region and quantize the field. For the Dirac field, the Bogoliubov transformations [@Birrelldavies] between $in$ and $out$ modes are $$\psi_{in}^{(\pm)}(k)= {\cal A}_{k}^{\pm}\psi^{(\pm)}_{out}(k)
+{\cal B}_{k}^{\pm}\psi^{(\mp)*}_{out}(k),$$ where the Bogoliubov coefficients ${\cal A}^{\pm}_k$ and ${\cal
B}^{\pm}_k$ that take the form $$\begin{aligned}
& {\cal A}^{\pm}_k=
\sqrt{\frac{\omega_{out}}{\omega_{in}}}\frac{\Gamma(1-\frac{i\omega_{in}}{\rho})
\Gamma(-\frac{i\omega_{out}}{\rho})}{\Gamma(1-i\zeta_{(+)}^{\mp}/\rho)
\Gamma(-i\zeta_{(+)}^{\pm}/\rho)},\\
&{\cal B}^{\pm}_k=\sqrt{\frac{\omega_{out}}{\omega_{in}}}
\frac{\Gamma(1-\frac{i\omega_{in}}{\rho})
\Gamma(\frac{i\omega_{out}}{\rho})}{\Gamma(1+i\zeta_{(-)}^{\pm}/\rho)
\Gamma(i\zeta_{(-)}^{\mp}/\rho)}.
\end{aligned}$$ @Ball2006 studied entanglement for scalar field modes in the two-dimensional asymptotically flat Robertson-Walker expanding spacetime. They showed that the expanding universe generates entanglement between modes of the scalar field, which conversely encodes information of the underlying spacetime structure. They calculated the entanglement in the far future, for the scalar field residing in the vacuum state in the distant past. They pointed out how the cosmological parameters of the toy Robertson-Walker spacetime can be extracted from quantum correlations between the field modes.
@Ahn2007 considered the entanglement of two-mode squeezed states for scalar fields in the Riemannian spacetime. The system is prepared as a two mode squeezed state for continuous variables from an inertial point of view. The initial system is prepared in Unruh mode $A$ and mode $B$ in an inertial frame with the covariance matrix $$\begin{aligned}
\label{inAR}
\sigma^{\rm (M)}_{AB}(s)=\left(
\begin{array}{cc}
\mathcal{A}_i(s) & \mathcal{E}_i(s) \\
\mathcal{E}^T_i(s) & \mathcal{B}_i(s) \\
\end{array}
\right),\end{aligned}$$ where $\mathcal{A}_i(s)=\mathcal{B}_i(s)=\cosh(2s){\openone}_2$, and $\mathcal{E}_i(s)=\sinh (2s)Z_2$. This setting allows the use of entanglement measure for continuous variables, which can be applied to discuss free and bound entanglement from the point of view from noninertial observer.
@Fuentes2010 found that entanglement was generated between modes of Dirac fields in a two-dimensional Robertson-Walker universe. The entanglement generated by the expansion of the universe is lower than for the bosonic case for some fixed conditions [@Ball2006]. It was also found that the entanglement for Dirac fields codifies more information about the underlying spacetime structure than the bosonic case, thereby allowing us to reconstruct more information about the history of the expanding universe. This highlights the importance of the difference between the bosonic and the fermionic statistics to account for relativistic effects on the entanglement of field modes.
@Fengj2013 investigated quantum teleportation between the conformal observer Alice and the inertial observer Bob in de Sitter space with both free scalar modes and cavity modes. The fidelity of the teleportation is found to be degraded in both cases, which is due to the Gibbons-Hawking effect associated with the cosmological horizon of the de Sitter space. In both schemes, the cutoff at Planck-scale causes extra modifications to the fidelity of the teleportation comparing with the standard Bunch-Davies choice.
@Moradi14 studied the spin-particles entanglement between two modes of Dirac field in the expanding Robertson-Walke spacetime. They calculated the Bogoliubov transformations for spin-particles between the asymptotic flat remote past and far future regions. It was showed that the particles-antiparticles entanglement creation when passing from remote past to far future due to the articles creation, while particles entanglement in the remote past degrades into the far future. They derived analytical expressions of logarithmic negativity both for spin particles and for spin-less ones as function of the density of the created particles. In addition, they highlighted the role of spin of particles for the dynamics of entanglement in the Robertson-Walke spacetime.
@Fengj2014 studied the quantum correlations and quantum channel of both free scalar and Dirac modes in de Sitter space. They found that the entanglement between the free field modes is degraded due to the radiation associated with the cosmological horizon. They constructed proper Unruh modes admitting general $\alpha$-vacua beyond the single-mode approximation and found a convergent feature of both the bosonic and fermionic cases. In particular, the convergent points of fermionic entanglement are found to dependent on the choice of $\alpha$. Moreover, an one-to-one correspondence between the convergent points of entanglement and zero capacity of quantum channels in the de Sitter space was proved.
@wangtian2015 studied the parameter estimation for excitations of Dirac fields in the expanding Robertson-Walker universe. The optimal precision of the estimation was found to depend on the dimensionless mass $\tilde{m}$ and dimensionless momentum $\tilde{k}$ of the Dirac particles. The precision of the estimation was obtained by choosing the probe state as an eigenvector of the hamiltonian. This is because the largest quantum fisher information can be obtained by performing projective measurements implemented by the projectors onto the eigenvectors of specific probe states.
@Pierini investigated the effects of spin on entanglement arising in Dirac field in the Robertson-Walker universe. They present an approach to treat the case which only requires charge conservation, and the case which also requires angular momentum conservation. It was found that in both situations entanglement originated from the vacuum have the same behaviors and does not qualitatively deviates from the spinless case. Differences only arise for the case in which particles or antiparticles are present in the input state.
@liunana studied the thermodynamical properties of scalar fields in the Robertson-Walker spacetime. They treated scalar fields in the curved spacetime as a quantum system undergoing a non-equilibrium transformation. The out-of-equilibrium features were studied via a formalism which was developed to derive emergent irreversible features and fluctuation relations beyond the linear response regime. They applied these ideas to the expanding universe scenario, therefore the assumptions on the relation between entropy and quantum matter is not required. They provided a fluctuation theorem to understand particle production due to the universe expansion.
Noninertial cavity modes {#sec:8D}
------------------------
@Downes2011 proposed a scheme for storing quantum correlations in the field modes of moving cavities in a flat spacetime. In contrast to previous work where quantum correlations degradation due to the Unruh-Hawking effect, they found that entanglement in such systems is protected. They further discussed the establishment of entanglement and found that the generation of maximally entangled states between the cavities is in principle possible. Like free field modes, the dynamics of the scalar field inside the cavity is also given by the KG equation given in Eq. . Under the Dirichlet boundary conditions, solutions of the KG equation are given by the plane waves $$u_{n}(t,x)= \frac{1}{\sqrt{n\pi}}\sin\left(\frac{n\pi}{L}[x-x_1]\right)
e^{-\frac{in\pi}{L}t},$$ and the scalar field contained within the cavity walls is $$\hat{\phi}_A(t,x)=\sum_n(u_{n}(t,x)\hat{a}_n+u_{n}^*(t,x)\hat{a}_n^{\dagger}),$$ where $\hat{a}_n^{\dagger}$ and $\hat{a}_n$ are the creation and annihilation operators, with $[\hat{a}_n, \hat{a}^{\dagger}_
{n^{\prime}}]=\delta_{n n^{\prime}}$. The Dirichlet boundary conditions describe the perfectly reflecting mirrors of the scalar field which is set to vanish on the boundary. Here, Alice’s cavity is inertial and Rob’s cavity is described by a uniformly accelerating boundary condition.
The world line of Rob’s cavity is described by the Rindler coordinates $(\eta,\xi)$ given in Eq. . We assume that Rob is stationary at spatial location $\xi=\xi_1$ for all $\eta$, his trajectory in the Minkowski coordinates has the form $x_1(t)=(t^2+X_1^2)^{1/2}$, where $X_1=a^{-1}e^{a\xi_1}$, and Rob’s proper acceleration is given by $\alpha=X_1^{-1}$. Rob’s cavity consists of two mirrors, one at $\xi_1$ and the other at $\xi_2$ and stationary with respect to him.
Then, one let Alice and Rob to meet at $t=0$ with their mirrors aligned, which fixes the position of Alice’s cavity as $x_1=X_1$ and the length of Rob’s cavity at $t=0$ to be $X_2-X_1=L$. Therefore, the length of Rob’s cavity in Rindler coordinates is $L^{\prime}=
\frac{1}{a} \ln\left(1+aL\right)$ for all $t$ for fixed $a$. The boundary conditions $\phi[\eta,\xi_1]=\phi[\eta,\xi_2]=0$ in this case are time-independent since the length $L^{\prime}$ is a constant. The solutions of the KG equation takes the form $$v_{n}(\eta,\xi)= \frac{1}{\sqrt{n\pi}}\sin\left(\frac{n\pi}{L^{\prime}}\xi\right)
e^{-\frac{in\pi}{L^{\prime}}\eta},$$ where $n\in \{1,2,\ldots\}$. Therefore, the scalar field inside the cavity is $$\hat{\phi}_R(\eta,\xi)= \sum_n(v_n(\eta,\xi)
\hat{b}_n+v_n^*(\eta,\xi)\hat{b}_n^{\dagger}),$$ from Rob’s perspective, where $\hat{b}_n^{\dagger}$ and $\hat{b}_n$ are creation and annihilation operators with $[\hat{b}_n,
\hat{b}^{\dagger}_{n^{\prime}}]=\delta_{n n^{\prime}}$. The vacuum state is defined by $\hat{b}_n|0\rangle_R=0$, $\forall n$, where the subscript $R$ indicates Rindler cavity. Assuming the cavity’s mirrors is perfectly reflecting. Then one can obtain that if Rob prepares the cavity in a given Rindler state, it will remain in the same state for all times [@Avagyan2002].
@Bruschi2012 studied whether the nonuniform motion degrades entanglement of a relativistic quantum field that is localized both in space and in time. The field modes in each cavity are discrete and have the frequencies $\omega_n \coloneqq \sqrt{M^2 + \pi^2 n^2}
/ \delta$, where $M \coloneqq \mu\delta$ and the quantum number are $n\in \{1,2,\ldots\}$. We then assume Rob undergoes accelerated motion. The trajectories of the cavities is given in Fig. \[PIchi\]. They denote $U_n$ as Rob’s field modes with positive frequency $\omega_n$ before the acceleration and denote $\bar{U}_n$ as Rob’s field modes after the acceleration. The two sets of modes are related by the Bogoliubov transformation $$\bar{U}_m= \sum_n \, \bigl( \alpha_{mn} U_n + \beta_{mn} U^*_n \bigr),$$ where the Bogoliubov coefficient matrices $\alpha$ and $\beta$ are determined by the motion of the cavity during the acceleration [@Birrelldavies]. Here, the proper acceleration at the center of the cavity is $h/\delta$, where the parameter $h$ should satisfy $h<2$ to ensure the acceleration at the left end of the cavity is finite. In the region II, the scalar field positive frequency modes with respect to $\xi$ are a discrete set $V_n$ with $n\in
\{1,2,\ldots\}$, and their frequencies at the center of the cavity are $\tilde\Omega_n = (\pi h n)/[2\delta \mathrm{atanh}(h/2)]$ with respect to the proper time $\tau$.
![The figure shows the trajectories of the cavities [@Bruschi2012]: Alice’s cavity keeps inertial, Rob’s cavity is inertial in region I and is again inertial in region III, Rob’s cavity is accelerated in region II. Here, $\bar\eta$ is the duration of the acceleration.[]{data-label="PIchi"}](figure8.eps)
The Bogoliubov transformation between the two sets of modes can be computed at the junction $t=0$ [@Birrelldavies]. The coefficient matrices $\alpha$ and $\beta$, have small $h$ expansions and have the form $$\label{eq:mrcoeffs}
\begin{aligned}
& \alpha_{nn}^0 = 1-{\tfrac {1}{240}}\,{\pi }^{2}{n}^{2}{h}^{2} + O(h^4), \\[1ex]
& \alpha_{mn}^0 = \sqrt {mn} \, {\frac { \bigl( -1+ {( -1 )}^{m-n}\bigr) }{{\pi }^{2} {( m-n )}^{3}}}h + O(h^2), \\[-1ex]
& \beta_{mn}^0 = \sqrt {mn} \, {\frac { \bigl(1 - {( -1 )}^{m-n} \bigr)}{{\pi}^{2} {( m+n )}^{3}}}h + O(h^2),
\end{aligned}$$ where $m\ne n$. Then one can calculate the state of the cavity modes after the acceleration.
@Friis2012 analyzed quantum entanglement and nonlocality of a massless Dirac field confined to a cavity. The world tube of the cavity consists of inertial and uniformly accelerated segments, and the accelerations are assumed to be small but the travel time is arbitrarily long. The quantum correlations between the field modes in the inertial cavity and the accelerated cavity modes are periodic in the durations of the individual trajectory segments. They found that the loss of quantum correlations can be entirely avoided by tuning the relative durations of the segments. Compared with bosonic correlations, it is easier to calculate the quantum correlations in the fermionic Fock space because the relevant density matrices act in lower dimensional Hilbert space due to the fermionic statistics. Therefore, it is possible to quantify the quantum correlations not only in terms of the entanglement negativity but also in terms of the CHSH inequality.
@Brenna studied the effects of different boundary conditions and coupling forms on the response of a accelerated particle detector in optical cavities. Specifically, they considered cavity fields with periodic, Dirichlet, and Neumann boundary conditions. They demonstrated that the Unruh effect does indeed occur in a cavity, which is independent of the boundary conditions. They found the thermalization properties of the accelerated detector: an accelerated detector evolves to a thermal state whose temperature increases linearly with its acceleration. In a non-perturbatively way, it was proven that if the switching process is smooth enough, the detector is thermalized to the Unruh temperature, which is independent of the type of coupling and the boundary conditions.
@Bruschi2013 discussed how the accelerated motion of a quantum system can be used to generate quantum gates. They present a class of sample travel scenarios in which the nonuniform relativistic motion of a cavity is used to generate two-mode quantum gates in a quantum system with the continuous variables. They found that the degree of entanglement between the cavity modes are produced through resonance of the cavity which appears by repeating periodically trajectory. In addition, they obtained analytical expression of the generated entanglement in terms of the magnitude and direction of the acceleration. The cavity modes are assumed to be initially at rest and the cavity trajectories are constructed through the Bogoliubov transformations. In the covariance matrix formalism, the Bogoliubov transformations are represented by the symplectic matrix $\mathcal{S}$, which has the form $$\begin{aligned}
s_{kk'}\label{nico}&=& \left(
\begin{array}{cccc}
\Re({A_{kk'}- B_{kk'}})&\Im({A_{kk'}+B_{kk'}}) \\
-\Im({A_{kk'}- B_{kk'}})&\Re({A_{kk'}+B_{kk'}})
\end{array}
\right),\end{aligned}$$ where $A_{kk'}$ and $B_{kk'}$ are the Bogoliubov coefficients associated with the trajectory. By assuming $h=aL\ll1$, the Bogoliubov coefficients can be expanded to the first order in $h$ as $$A_{kk'}=G_k\delta_{kk'}+A_{kk'}^{(1)},~
B_{kk'}=B_{kk'}^{(1)},$$ where $G_{k}=e^{i\omega_{k}T}$ are the phases of the state during segments of free evolution, $T$ denotes the total proper time of the segment, and the superscript $(1)$ denotes the first order in $h$. If the cavity is prepared initially in the vacuum state, the reduced state of the modes after an $N$-segment trajectory is found to be $$\sigma_N=(S_{kk'}^N)^{Tp} S_{kk'}^N,$$ where $$\begin{aligned}
S_{kk'}&=& \left( \begin{array}{cc}
s_{kk}&s_{kk'}\\
s_{k'k}&s_{k'k'}
\end{array}
\right),\end{aligned}$$ and the transformation $S^N_{kk'}$ corresponds to two mode squeezer that is a two mode entangling gate.
@Bruschi2013b studied the mode-mixing quantum gates and entanglement in nonuniform accelerated cavities. It was showed that the periodic accelerated motion of the cavity can produce entangling quantum gates between different frequency modes. The resonant condition in the cavities which associates with particle creation is an important feature of the dynamical Casimir effect. It was found that a second resonance, which has attracted less attention because it produces negligible particles, generates a beam splitting quantum gate. This quantum gate leads to a resonant enhancement of quantum entanglement, which can be regarded as the most important evidence of acceleration effects in mechanical oscillators.
@Friis2013 analyzed relativistic quantum information for quantized scalar, spinor, and photon fields in an accelerated mechanically rigid cavity in the perturbative small acceleration formalism. The scalar field was analyzed with Neumann and Dirichlet boundary conditions, and the photon field was discussed under conductor boundary conditions. The massive Dirac spinor is analyzed with dimensions transverse to the acceleration. It was found that for smooth accelerations, the unitarity of time evolution holds, while for discontinuous accelerations, it fails in 4-dimensions and higher spacetime. The experimental scenario proposed in [@Bruschi2013b] for the scalar field can also apply to the photon field.
@Pozas analyzed the harvesting of classical and quantum correlations from vacuum for particle detectors. They demonstrated how the spacetime dimensionality, the detectors’ physical size, and their internal energy structure would impact the detectors’ harvesting ability. They revealed several dependence on these parameters that can optimize the harvesting of quantum entanglement and classical correlations. Furthermore, they found that to harvest vacuum entanglement, smooth switching is more efficient than sudden switching, especially in the case of the detectors are spacelike separated.
@Regula2016 investigated entanglement generated between the modes of two uniformly accelerated bosonic cavities when interacting with a two-level system. It was found that the inertial and the accelerated cavity become entangled by letting an atom emitting an excitation when it passes through the cavities, but the generated entanglement is degraded against the effects of acceleration. The generated entanglement is affected not only by the accelerated motions of the cavities but also by its transverse dimension which plays the role of an effective mass. In addition, they found that the extra spatial dimensions contribute to the mass of the field. Therefore, if the massless bosonic field is used, the degradation effect of entanglement should not occur.
Unruh-DeWitt detectors {#sec:8E}
----------------------
To model the response of an accelerated detector in a quantum field, the Unruh-DeWitt detector model was performed [@UW84]. This model consists of a two-level non-interacting atom, which couples to the external scalar field along its world line in a point-like manner [@wald94]. The response of the Unruh-DeWitt detector depends on its trajectory and the state of the field. For definiteness and without loss of generality, we consider an uniformly accelerated detector, whose world line is given by Eq. .
Here we study two detectors, one named Alice keeps static and the other one named Rob moves with uniform acceleration $a$ for a time duration $\Delta$. Alice’s detector is assumed always switched off and Rob’s detector is switched on during the time duration $\Delta$. The total Hamiltonian of the system is $$\label{totalh}
H_{A\, R\, \phi} = H_A + H_R + H_{KG} + H^{R\phi}_{\rm int},$$ where $H_{A}=\Omega A^{\dagger}A$, $H_{R}=\Omega R^{\dagger}R$, and $\Omega$ is the energy gap of the detectors. The interaction Hamiltonian $H^{R\phi}_{\rm int}(t)$ between Rob’s detector and the external scalar field is $$\label{int}
H^{R\phi}_{\rm int}(t)= \epsilon(t)\int_{\Sigma_t} d^3 {\bf x} \sqrt{-g} \phi(x)
[\chi({\bf x})R +\overline{\chi}({\bf x})R^{\dagger}],$$ where $g\equiv {\rm det} (g_{ab})$, and $g_{ab}$ is the Minkowski metric. Moreover, $$\chi(\mathbf{x})=(\kappa\sqrt{2\pi})^{-3}
\exp(-\mathbf{x}^{2}/2\kappa^{2}),$$ is a Gaussian coupling function which vanishes outside a small volume around the detector. This model describes a point-like detector which only interacts with its neighbor fields. The total initial state of detectors-field system has the form $$\label{IS}
|\Psi_{t_0}^{AR\phi}\rangle=|\Psi_{AR}\rangle\otimes|0_{M}\rangle,$$ where $$|\Psi_{AR}\rangle=\sin\theta |0_{A}\rangle|1_{R}\rangle+
\cos\theta|1_{A}\rangle |0_{R}\rangle,$$ is the initial state of the detectors, and $|0_{M}\rangle$ represents that the external scalar field is in vacuum state from an inertial perspective.
In weak coupling case, the final state $|\Psi^{R \phi}_{t =
t_0+\Delta} \rangle$ at time $t_0+\Delta$ can be calculated by employing the first order of perturbation over the coupling constant $\epsilon$ [@UW84]. After some calculations, one can find that the final state $|\Psi^{R \phi}_{t}\rangle$ at time $t=t_0+\Delta$ is $$\label{primeira_ordem}
|\Psi^{R \phi}_{t} \rangle = [I - i(\phi(f)R + \phi(f)^{\dagger}
R^{\dagger})] |\Psi^{R \phi}_{t_0} \rangle,$$ where $$\label{phi(f1)}
\begin{aligned}
\phi(f) &\equiv \int d^4 x \sqrt{-g}\chi(x)f \\
&= i [a_{RI}(\overline{u E\overline{f}})-a_{RI}^{\dagger}(u Ef)],
\end{aligned}$$ is the operator of the external scalar field [@Landulfo; @wald94], and $f \equiv \epsilon(t) e^{-i\Omega t}\chi ({\bf x})$ is a compact support complex function. In addition, $u$ is the positive frequency part from a solution of the KG equation in the Rindler metric [@Landulfo; @wald94], and $E$ is the difference between the advanced Green function and the retarded Green functions.
@Landulfo investigated how the teleportation of a quantum channel is affected by the Unruh effect when one of the entangled detector is accelerated for a finite amount of proper time. They performed a detailed analysis of how the acceleration of the detector and the Unruh effect on the entangled qubit system. The mutual information and concurrence between the two detectors are calculated and showed that the latter has a sudden death at some fixed finite acceleration. Similarly, the teleportation fidelity exhibits sudden death behavior via the Unruh effect. The values of quantum entanglement and mutual information depend on the time interval along which one of the detectors is accelerated.
@Landulfo1 analyzed the dynamics of QD and classical correlation for a pair of Unruh-DeWitt detectors when one of them is uniformly accelerated, and showed that the discord-type quantum correlation is completely destroyed under the influence of Unruh thermal bath when one detector is in the limit of infinite acceleration, while the classical correlation is nonzero for any acceleration. In particular, unlike the quantum entanglement, the discord-type quantum correlations exhibits sudden-change behavior at some certain acceleration parameter. They also discussed how their results can be interpreted when one of the detector hovers near the event horizon of a Schwarzschild black hole.
@Ostapchuk12 discussed the dynamics of quantum entanglement between a pair of Unruh-DeWitt detectors, one keeps inertial in the flat spacetime, and the other non-uniformly accelerated in some specified way. Each of the detectors coupled to the external scalar quantum field in an indirectly way. The primary problem involving nonuniformly accelerated detectors in an event horizon is absent and the Unruh temperature cannot be well defined. By numerical calculation, they demonstrated that the quantum entanglement in the weak-coupling limit like those of an oscillator in a bath of time-depending “temperature" proportional to the proper acceleration of the detector, with oscillatory modifications due to non-adiabatic effects.
Different from the Unruh-DeWitt detector model, a localized solution to the problem of entanglement degradation in relativistic settings were performed by @Doukas13 [@Dragan13]. They prepared a two mode squeezed state between two observers, the inertial Alice and the accelerated Rob. The initial state is $$\hat{S}_{\text{AB}} |0\rangle_\text{M} =
\exp[s(\hat{a}^\dagger\hat{b}^\dagger -\hat{a}
\hat{b})]|0\rangle_\text{M},$$ where the annihilation operators $\hat{a}$ and $\hat{b}$ are associated with two localized and spatially separated scalar modes $\phi_\text{A}(x,t)$ and $\phi_\text{B}(x,t)$, respectively. From the perspective of the accelerated observer, the covariance matrix of the state has the form [@Dragan13] \[phi(ef)\]
$$\label{covariance}
\begin{aligned}
\sigma=& \openone+2\langle\hat{n}\rangle_U
\begin{pmatrix}
0&0&0&0\\
0&0&0&0\\
0&0&1&0 \\
0&0&0&1
\end{pmatrix}
+2\sinh^2 s
\begin{pmatrix}
|\alpha|^2&0&0&0\\
0&|\alpha|^2&0&0\\
0&0&|\beta+\beta'^\star|^2&2\,\text{Im}(\beta\beta') \\
0&0&2\,\text{Im}(\beta\beta')&|\beta-\beta'^\star|^2
\end{pmatrix} \\
&+ \sinh 2s
\begin{pmatrix}
0&0&-\text{Re}[\alpha(\beta+\beta'^\star)]&-\text{Im}[\alpha(\beta-\beta'^\star)]\\
0&0&-\text{Im}[\alpha(\beta+\beta'^\star)]&\text{Re}[\alpha(\beta-\beta'^\star)]\\
-\text{Re}[\alpha(\beta+\beta'^\star)]&-\text{Im}[\alpha(\beta+\beta'^\star)]&0&0 \\
-\text{Im}[\alpha(\beta-\beta'^\star)]&\text{Re}[\alpha(\beta-\beta'^\star)]&0&0
\end{pmatrix},
\end{aligned}$$
where $$\label{noise}
\langle\hat{n}\rangle_U = \sum_{k}\frac{|(\psi_\text{B},w_{Ik})|^2}{e^{\frac{2\pi |k| c^2}{a}}-1},~
w_{Ik}=\frac{1}{\sqrt{4\pi |k| c}}e^{i(k\xi-|k|c\tau)},$$ is the average number of Unruh particles seen by an accelerated detector in the vacuum [@Dragan2012].
@Dragan13 studied the amount of entanglement by using the localized projective detection model and found that the quantum correlations are able to extract from the initial state. It was found that the Unruh thermal noise plays only a minor role in the degradation process of entanglement. The dominant source of degradation is the mismatch between the mode Rob observed in the squeezed state and the mode which is detectable from the accelerated frame. In addition, leakage of initial mode through the Rindler horizon places a limit on Rob’s ability to fully measure the state, which leads to an inevitable loss of entanglement that even cannot be corrected by changing the hardware design of the detectors.
@Doukas13 investigated the quantum entanglement and discord extractable from a two mode squeezed state as considered by two detectors, one inertial and the other accelerated. They found that for large accelerations, the quantum system using localized modes produces qualitatively different properties than that of Unruh modes. Specifically, the quantum entanglement of the given quantum state undergoes a sudden death for some finite acceleration while the discord asymptotes to zero in the infinite acceleration limit.
@tian2014 studied the dynamics of freely falling and static two-level detectors interacting with quantized scalar field in de Sitter spacetime. The atomic transition rates is found to depend on both the parameter of de Sitter spacetime and the motion of atoms. They found that the steady states for both cases are always purely thermal states, regardless of the initial states of the detectors. In addition, it was found that the thermal baths will generate entanglement between the freely falling atom and its auxiliary partner. They also calculated the proper time for extinguishment of the entanglement between the detectors.
@Lin2015 studied quantum teleportation modeled by Unruh-DeWitt detectors which initially coupled to a common field. An unknown coherent state of the inertia detector is teleported to the agent Rob with relativistic motion, using a detector pair initially entangled and shared by these two agents. The results showed that average fidelity of the teleportation always drops below the best fidelity value from classical teleportation before the detector pair becomes disentangled. The distortion of the detectors’ state can suppress the fidelity significantly even if the detectors are still strongly entangled around the light cone. They pointed out that the dynamics of entanglement are not directly related to the fidelity of quantum teleportation between the detectors observed in Minkowski frame or in quasi-Rindler frame.
@Menezes investigated the radiative processes of entangled and accelerated atoms interacting with an electromagnetic field prepared in the Minkowski vacuum. They discussed the structure of the variation rate of the atomic energy for two atoms moving in different world lines. The contributions of vacuum fluctuations and radiation reaction were identified to the generation of entanglement to the decay of entangled states. The situation where two static atoms are coupled independently to two spatially separated cavities at different temperatures is resembled by the results. In addition, it was found that one of antisymmetric Bell state is a decoherence-free state for equal accelerations.
@wangtian2016 studied how the Unruh thermal noise influences the quantum coherence and compared its behavior with entanglement of the same system. They discussed the frozen condition of coherence and find that the decoherence of detectors’ quantum state is irreversible under the effect of Unruh thermal bath without any boundary. Comparing with entanglement which reduces to zero for a finite acceleration, the coherence-type quantum correlation approaches zero only in the limit of an infinite acceleration. They found that the evolution of the detectors’ state after the interaction described by the Hamiltonian can also be represented by $$\begin{aligned}
\rho_{t}^{AR} =\sum_{\mu \nu} M^{A}_\mu \otimes M^{R}_\nu |\Psi_{AR}\rangle\langle\Psi_{AR}|
(M^{A}_\mu \otimes M^{R}_\nu)^{\dag},\end{aligned}$$ where $M_{\mu}^{A}$ and $M_{\mu}^{R}$ are the Kraus operators. The Kraus operators act on Rob’s state are $$\begin{aligned}
\begin{split}
& M_1^{R}=\left(\begin{array}{cc}
\sqrt{1-q}&0\\
0&\sqrt{1-q}
\end{array}\right), M_2^{R}=\left(\begin{array}{cc}
0&0 \\
v\sqrt{q}&0
\end{array}\right), \\
&M^{R}_3 =\left(\begin{array}{cc}
0&v\\
0&0
\end{array}\right).
\end{split}\end{aligned}$$ Unlike $M_{\mu}^{R}$, $M_{\mu}^{A}$ is an identity matrix because Alice’s detector keeps static.
The dynamics of steering between two correlated Unruh-Dewitt detectors when one detector interacts with external scalar field was studied in [@liuwang2018]. The quantum steering is found to be very fragile under the influence of Unruh thermal noise. In addition, the quantum steering experience “sudden death" for some accelerations, which are quite different from other quantum correlations in the same system.
Quantum correlations and the dynamical Casimir effect {#sec:8F}
-----------------------------------------------------
Like the Unruh effect, dynamical Casimir effect is an important prediction of QFT in relativistic setting [@moore]. The dynamical Casimir effect predicts that relativistic motion of boundary conditions would generate pairs of photons from vacuo. Such prediction has been experimentally observed in a superconducting circuit architecture [@wilson]. The modulation of boundary condition, theoretically created by a mirror at relativistic speeds, was achieved by high-frequency modulation of the external magnetic flux threading a superconducting quantum interferometric device [@wilson]. The experimental demonstration of the dynamical Casimir effect has triggered a renewed interest in it and has paved the way for the analysis of the role of Casimir radiation as a resource for quantum information tasks [@discordsabin].
To understand the creation of photons from vacuum fluctuations when the boundaries of the electromagnetic field are modulated, one should quantize the field. In the 2011 experimental observation of this phenomenon, the relativistic moving mirror was simulated by a superconducting quantum interferometric device interrupting a superconducting transmission line [@wilson]. The electromagnetic field confined in the transmission line can be described by a flux operator $\Phi(x,t)$, which obeys the KG equation. The solution of the KG equation in the plane-waves basis is [@johan; @johan2]: $$\label{flujo}
\Phi(x,t)= \sqrt{\frac{\hbar Z_0}{4\pi}} \int\frac{d\omega}{\sqrt{\omega}}(\hat{a}
(\omega)e^{i(-\omega t+kx)}+\hat{b}(\omega)e^{i(-\omega t-kx)}),$$ where $k=\omega/v$, and $v$ is the speed of light in the transmission line, $Z_0=\sqrt{L_0/C_0}$ is the characteristic impedance. In Eq. , $a(\omega)$ is the annihilation operator of photons that moves into the mirror and $b(\omega)$ denotes the annihilation operator of the photons moving away from the mirror. For sufficiently large superconducting quantum interferometric device plasma frequency [@johan], the charging energy is small compared with the Josephson energy $E_J(t)=
E_J[\Phi(x,t)]$. Therefore, the superconducting quantum interferometric device can provide a boundary condition to the flux field which is analogous to the boundary condition produced by a relativistic moving mirror. That is, $$\label{boundary}
\frac{(2\pi)^2}{\phi^2_0}\Phi(0,t)+\frac{1}{L_0}\left.\frac{\partial\Phi(x,t)}{\partial x}\right|_{x=0}=0,$$ where $L_0$ is the characteristic inductance per unit length, and the additional term associated with the capacitance is neglected, $\Phi_0=h/{2e}$ is the magnetic flux quantum. Inserting Eq. into the boundary condition Eq. , one obtain $$\begin{aligned}
\int_0^\omega \frac{d\omega}{\sqrt{\omega}}ik\frac{L_{J}(t)}{L_0}(\hat{b}(\omega)-\hat{a}(\omega))e^{-i\omega t}& \\
= \int_0^\omega \frac{d\omega}{\sqrt{\omega}}(\hat{a}(\omega) + \hat{b}(\omega))e^{-i\omega t},&
\end{aligned}$$ where $L_J(t)=\frac{\phi^2_0/(2\pi)^2}{E_J(t)}$ is the tunable Josephson inductance. Then the pair creation of photons in dynamical Casimir effect can be calculated using scattering theory which describes how the time-dependent boundary condition mixes the input and output modes. By employing the methods discussed in [@johan; @johan2], we obtain the Bogoliubov transformation between the incoming and outgoing modes, which relates the input and output vacuum state.
@Felicetti discussed how the ultrafast modulation of the qubit-field coupling strength between a superconducting qubit and a single mode of a superconducting resonator mimics the motion of the qubit at relativistic speeds. When the qubit follows an effective oscillatory motion, they find two different regimes. The system is found to experience unbounded photon generation or resemble the anti-JC dynamics, which depends on the oscillation frequency. Moreover, by combining the performed technique with the dynamical Casimir physics, the toolbox for studying relativistic phenomena with superconducting circuits can be enhanced.
@Friis13 analyzed the effect of relativistic motion on the fidelity of continuous variable protocol for quantum teleportation and proposed a state-of-the-art technology experiment to test their results. They computed the bounds for the fidelity of teleportation when one of the observers moves with nonuniform acceleration for a finite time, which is degraded due to the observer’s motion. The effects of time evolution can be removed by applying time dependent local operations and the effects of acceleration on the fidelity can be isolated in this way. In addition, the origin of the fidelity loss of the quantum teleportation has the same physical regime for particle generation due to motion-underlying the Unruh (or Hawking) radiation or the dynamical Casimir effect.
@Alhambra studied the Casimir-Polder forces experienced by atoms or molecules in optical cavities. They model the quantum systems as qubits, and the electromagnetic field components are modeled as scalar fields with Dirichlet or Neumann boundary conditions. The light-matter interaction model is used to compute the Casimir and Casimir-Polder effects. They found that the diamagnetic term can qualitatively change the Casimir-type forces, or in other words, it can turn a repulsive force into an attractive force and vice versa. To be specific, when this term is present, the atoms are attracted to plates with Dirichlet boundary conditions, while the plate-atom forces are repulsive without this term. They also considered the Neumann boundary condition for the atom with or without diamagnetic coupling term in a cavity, where the forces are found to have opposite sign to that of the Dirichlet cavity. In addition, the microscopic-macroscopic transition was studied in this system, and the results showed that the atoms start to affect the Casimir force similarly to a dielectric medium for increasing number of atoms in the cavity.
@Marino studied the thermal and nonthermal signatures of the Unruh effect in Casimir-Polder forces. They found that the Casimir-Polder forces between two uniformly accelerated atoms exhibit a transition from the short distance thermal-like behavior to a long distance nonthermal behavior. The former is predicted by the Unruh effect and the latter is associated with the breakdown of local inertial descriptions of the system. This effect extends the Unruh thermal response detected by an accelerated observer to the spatially extended system of two particles. They identified the characteristic length scale with the acceleration of the two atoms for this crossover. Their results were derived separating at fourth order in perturbation theory and radiation reaction field to the Casimir-Polder interaction between a pair of atoms separated by a constant distance and linearly coupled to a scalar field.
@steersabin investigated the generation of the Einstein-Podolsky-Rosen steering and Gaussian interferometric power under the influence of the dynamical Casimir effect. They computed the quantum steering and the interferometric power generated in the superconducting waveguide interrupted by the superconducting quantum interferometric device. It has been shown that, similar with entanglement and QD [@discordsabin], the value of the experimental driving amplitude and velocity should be higher than a critical value to overcome the initial thermal noise and to create quantum steering. Conversely, the interferometric power is nonzero for any experimental value of the amplitude and velocity and increases with the increasing average number of thermal photons. In other words, any nonzero squeezing produces interferometric power, while a certain value of squeezing is required to generate quantum steering.
@coherencesabin studied how the dynamical Casimir effect influences the behavior of quantum coherence for Gaussian states in continuous variables. They found that quantum coherence is significantly different from zero for any value of the external pump amplitude for the realistic experimental parameters. This means that the Casimir radiation creates quantum coherence for any value of the pump amplitude. In addition, quantum coherence is always greater than QD and entanglement and exhibits a remarkable robustness again thermal noise. They believe that quantum coherence is a more suitable figure of merit of the quantum character for the dynamical Casimir effect since the experimental requirements for obtaining a dynamical Casimir effect state with finite coherence are less than that of entanglement or QD.
Conclusions {#sec:9}
===========
Quantum coherence and quantum correlations are fundamental notions of quantum theory. Although the former was defined with respect to a single system, while the latter were for bi- and multipartite systems, they are intimately related to, and can be transformed into each other through an operational way. When considering their characterization and quantification, there are many different methods apply to different situations. They can also be quantified in a similar manner, e.g., by using the (pseudo) geometric distance of two states. This unified framework provides the possibility for understanding the intrinsic connections between these two basic notions.
We concentrated in this work the recent progresses on the above two notions, mainly those discussed from the resource theoretic perspective. After a short introduction, we reviewed in Secs. \[sec:2\] and \[sec:3\] the various quantifiers of geometric quantum correlations and quantum coherence, which include their definitions and calculations, and some quantitative relations between these measures. As these measures are defined by an optimal (minimum or maximum) distance between the considered state and the set of states without the quantum property one want to characterizes, they can be categorized as the geometric characterization of quantumness.
Building upon the above basic notions, we reviewed in Sec. \[sec:4\] the interpretations of the resource theory of quantum coherence. These include the inter-conversions between coherence and quantum correlations such as entanglement and QD established in an operational way, their role on signifying the wave nature of quantum particle, and the various complementarity relations. This section also covers a review of the distillation and formation of quantum coherence for which different free operations and communication schemes are used.
In Sec. \[sec:5\], we summarized the recent investigations of typical quantum algorithms from the perspective of quantum coherence, which include the protocol of quantum state merging, the Deutsch-Jozsa algorithm, the Grover search algorithm, the DQC1 algorithm, and quantum metrology such as the PD task. All these show that the various quantitative measure of coherence can provide new viewpoints on the origin of superiority of quantum information processing tasks.
In Sec. \[sec:6\], we reviewed the recent progresses on control of quantum correlations and quantum coherence in noisy channels. We first showed that the various quantum correlation and quantum coherence may be frozen for special forms of initial states, and there is universal freezing phenomenon for the distance-based measure of them. We also showed local and nonlocal creation of quantum correlation and quantum coherence, the cohering power of quantum channels, as well as the evolution equation and preservation of quantumness.
In Sec. \[sec:7\], we showed some applications of quantum coherence in the related subjects of condensed matter physics. As explicit examples, we summarized role of quantum coherence on studying the long-range order, VBS state, and quantum phase transitions of the many-body systems. These reveal from one side the potential of characterizing quantum coherence from a quantification perspective.
Finally, we showed in Sec. \[sec:8\] some progresses for the study of quantum correlations and quantum coherence in relativistic settings. These involve their behaviors for the free field modes with and beyond single-mode approximation, for curved spacetime and expanding universe, for noninertial cavity modes, as well as quantum correlations for particle detectors and the dynamical Casimir effects on these correlations. The dynamics of quantum correlations and quantum coherence under the influence of Unruh temperature, Hawking temperature, expansion rate of the universe, accelerated motion of cavities and detectors, and boundary conditions of the field, have been reviewed. The advantage and disadvantage of free modes and local modes for the implement of quantum information processing tasks in noninertial frames and curved spacetime are also discussed.
Despite the main progresses summarized above which are of broad interest, there are still many challenging problems need to be solved in the future. We think, some of the valuable research directions may be of the following.
The characterization and quantification of quantum coherence under extended family of free operations. Up to now, most of the proposed coherence measures were based on the axiomatic postulates of @coher, where some of them may be extended in different circumstances. There may exist other coherence measures which are both mathematically rigorous and physically significant, e.g., those under incoherence preserving operations, translationally invariant operations, SIO, and GIO. Moreover, if the postulates (C1-C3) of @coher are somewhat released, other measures of quantum coherence that are physically relevant may exist. The physical meanings of those coherence measures are worth exploring. We believe that the searching process for various coherence quantifications will deepen our understanding of quantum theory, and new findings can also be expected.
The intrinsic connections between quantum coherence and quantum correlations may be another topic needs to be further considered. Although for relative entropy of coherence, there are some progresses being achieved in the past two years along this line, for most of the other measures the interconversion between coherence and quantum correlations still remain to be exploited. In particular, while the role of quantum correlations (entanglement, discord, *etc.*) in explicit quantum communication and computation tasks have been proved, the role of quantum coherence seems not to be so convinced. The investigation of their relations and their interconversion can thus provide interpretations of quantum coherence from a practical perspective. Moreover, the quantum coherence measures are questioned as they are basis dependent, the establishments of their connections with the basis-independent quantum correlation measures are therefore also significant from a theoretic point of view.
When considering a real physical system, the detrimental effects of environments are unavoidable. Though the related decoherence process have been analyzed extensively via decay of various quantum correlation measures, the quantum coherence measures defined in a rigorous framework provide in a real sense the tool for a quantitative analysis of the decoherence process. Moreover, the robustness of quantum correlations and quantum coherence against the detrimental effects of environment are also different. In general, the former are more fragile under environment coupling than the latter. But if the two can be converted into each other efficiently, one can store the quantum correlation of a bi- or multipartite system by converting it to coherence of a single system, and then convert it back into quantum correlations when being used.
As a resource theory of quantum coherence, what is really the resource aspect of them are in fact seldom considered. As far as we know, the coherence measures have already been related to quantum protocols such as state merging and quantum state discrimination, but we think these are by no means the only two roles they will be played in explicit quantum tasks. Further research, some of the ideas can be borrowed from the study of quantum correlations, may help to reveal their potential role as a physical resource. Moreover, the applications of these coherence measures in other subjects of physics, e.g., whether they can serve as useful order parameters for exploiting novel properties of many-body systems, may be nontrivial topics of future research.
Just as those exciting findings of this field in the past few years, the solve of the (no limited to) above problems, in our opinion, will continue to impact the development of basic fundamental quantum theory and the implementation of various new quantum technologies which strongly depend on quantum correlations and quantum coherence.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by NSFC (Grants No. 11675129, 91536108, 11675052, and 11774406), National Key R & D Program of China (Grant Nos. 2016YFA0302104, 2016YFA0300600), New Star Project of Science and Technology of Shaanxi Province (Grant No. 2016KJXX-27), CAS (Grant No. XDB), Hunan Provincial Natural Science Foundation of China (2018JJ1016), and New Star Team of XUPT, .
[42]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
Aaronson, B., R. L. Franco, G. Compagno, G. Adesso, 2013a, [New J. Phys. ]{}[**15**]{}, 093022. Aaronson, B., R. L. Franco, G. Adesso, 2013b, [Phys. Rev. A ]{}[**88**]{}, 012120. Aberg, J., 2006, arXiv:quant-ph/0612146. Adesso, G., I. Fuentes-Schuller, M. Ericsson, 2007, [Phys. Rev. A ]{}[**76**]{}, 062112. Adesso, G., T. R. Bromley, M. Cianciaruso, 2016, [J. Phys. A ]{}[**49**]{}, 473001. Affleck, A., T. Kennedy, E. H. Lieb, H. Tasaki, 1987, [Phys. Rev. Lett. ]{}[**59**]{}, 799. Affleck, A., T. Kennedy, E. H. Lieb, H. Tasaki, 1988, [Commun. Math. Phys. ]{}[**115**]{}, 477. Ahn, D., M. S. Kim, 2007, [Phys. Lett. A ]{}[**366**]{}, 202. Alhambra, A. M., A. Kempf, E. Martin-Martinez, 2014, [Phys. Rev. A ]{}[**89**]{}, 033835. Alsing, P. M., I. Fuentes-Schuller, R. B. Mann, T. E. Tessier, 2006, [Phys. Rev. A ]{}[**74**]{}, 032326. Altman, E., R. Vosk, 2015, Ann. Rev. Condens. Matter Phys. [**6**]{}, 383. Amico, L., R. Fazio, A. Osterloh, V. Vedral, 2008, [Rev. Mod. Phys. ]{}[**80**]{}, 517. Amico, L., D. Rossini, A. Hamma, V. E. Korepin, 2012, [Phys. Rev. Lett. ]{}[**108**]{}, 240503. Anderson, P. W., 1958, Phys. Rev. [**109**]{}, 1492. Andersson, E., J. D. Cresser, M. J. W. Hall, 2007, [J. Mod. Opt. ]{}[**54**]{}, 1695. Aspachs, M., G. Adesso, I. Fuentes, 2010, [Phys. Rev. Lett. ]{}[**105**]{}, 151301. Atas, Y. Y., E. Bogomolny, O. Giraud, G. Roux, 2013, [Phys. Rev. Lett. ]{}[**110**]{}, 084101. Avagyan, R. M., A. A. Saharian, A. H. Yeranyan, 2002, [Phys. Rev. D ]{}[**66**]{}, 085023.
Bagan, E., J. A. Bergou, S. S. Cottrell, M. Hillery, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 160406. Bai, Y. K., Y. F. Xu, Z. D. Wang, 2014, [Phys. Rev. Lett. ]{}[**113**]{}, 100503. Ball, J. L., I. Fuentes-Schuller, F. P. Schuller, 2006, [Phys. Lett. A ]{}[**359**]{}, 550. Bardarson, J. H., F. Pollmann, J. E. Moore, 2012, [Phys. Rev. Lett. ]{}[**109**]{}, 017202. Bartlett, S. D., T. Rudolph, R. W. Spekkens, 2007, [Rev. Mod. Phys. ]{}[**79**]{}, 555. Basko, D. M., I. L. Aleiner, B. L. Altshuler, 2006, [Ann. Phys. ]{}[**321**]{}, 1126. Baumgratz, T., M. Cramer, M. B. Plenio, 2014, [Phys. Rev. Lett. ]{}[**113**]{}, 140401. Bennett, C. H., G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, 1993, [Phys. Rev. Lett. ]{}[**70**]{}, 1895. Bennett, C. H., D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, 1996, [Phys. Rev. A ]{}[**54**]{}, 3824. Bennett, C. H., D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, W. K. Wootters, 2001, [Phys. Rev. Lett. ]{}[**87**]{}, 077902. Bera, M. N., T. Qureshi, M. A. Siddiqui, A. K. Pati, 2015, [Phys. Rev. A ]{}[**92**]{}, 012118. Bergstrom, L., A. Goobar, 2006, Cosmology and Particle Astrophysics, second ed., Spring. Berta, M., M. Christandl, R. Colbeck, J. M. Renes, R. Renner, 2010, [Nat. Phys. ]{}[**6**]{}, 659. Bhatia, R., 1997, Matrix Analysis, Springer-Verlag, New York. Biham, E., O. Biham, D. Biron, M. Grassl, D. A. Lidar, 1999, [Phys. Rev. A ]{}[**60**]{}, 2742. Birrell, N. D., P. C. W. Davies, 1982, Quantum Fields in Curved Space, Cambridge University Press, Cambridge. Bohigas, O., M. J. Giannoni, C. Schmit, 1984, [Phys. Rev. Lett. ]{}[**52**]{}, 1. Brandão, F. G. S. L., 2005, [Phys. Rev. A ]{}[**72**]{}, 022310. Brandão, F. G. S. L., R. O. Vianna, 2006, [Int. J. Quantum Inf. ]{}[**4**]{}, 331. Brandão, F. G. S. L., N. Datta, 2011, IEEE Trans. Inf. Theor. [**57**]{}, 1754. Brenna, W. G., E. G. Brown, R. B. Mann, E. Martin-Martinez, 2013, [Phys. Rev. D ]{}[**88**]{}, 064031. Bromley, T. R., M. Cianciaruso, G. Adesso, 2015, [Phys. Rev. Lett. ]{}[**114**]{}, 210401. Brown, E. G., K. Cormier, E. Martin-Martinez, R. B. Mann, 2012, [Phys. Rev. A ]{}[**86**]{}, 032108. Bruschi, D. E., J. Louko, E. Martin-Martinez, A. Dragan, I. Fuentes, 2010, [Phys. Rev. A ]{}[**82**]{}, 042332. Bruschi, D. E., I. Fuentes, J. Louko, 2012a, [Phys. Rev. D ]{}[**85**]{}, 061701. Bruschi, D. E., A. Dragan, I. Fuentes, J. Louko, 2012b, [Phys. Rev. A ]{}[**86**]{}, 025026. Bruschi, D. E., A. Dragan, A. R. Lee, I. Fuentes, J. Louko, 2013a, [Phys. Rev. Lett. ]{}[**111**]{}, 090504. Bruschi, D. E., A. Dragan, J. Louko, D. Faccio, I. Fuentes, 2013b, [New J. Phys. ]{}[**15**]{}, 073052. Bu, K., Swati, U. Singh, J. Wu, 2016a, [Phys. Rev. A ]{}[**94**]{}, 052335. Bu, K., U. Singh, L. Zhang, J. Wu, 2016b, arXiv:1603.06715. Bu, K., A. Kumar, L. Zhang, J. Wu, 2017a, [Phys. Lett. A ]{}[**381**]{}, 1670. Bu, K., U. Singh, S. M. Fei, A. K. Pati, J. Wu, 2017b, [Phys. Rev. Lett. ]{}[**119**]{}, 150405. Bu, K., N. Anand, U. Singh, 2018, [Phys. Rev. A ]{}[**97**]{}, 032342. Buscemi, F., N. Datta, 2010, IEEE Trans. Inf. Theor. [**56**]{}, 1447.
Cheng, C. C., Y. Wang, J. L. Guo, 2016, [Ann. Phys. ]{}[**374**]{}, 237. Cheng, W. W., X. Y. Wang, Y. B. Sheng, L. Y. Gong, S. M. Zhao, J. M. Liu, 2017, [Sci. Rep. ]{}[**7**]{}, 42360. Cakmak, B., G. Karpat, F. F. Fanchini, 2015, Entropy [**17**]{}, 790. Céleri, L. C., A. G. S. Landulfo, R. M. André, G. E. A. Matsas, 2010, [Phys. Rev. A ]{}[**81**]{}, 062130. Chang, L., S. Luo, 2013, [Phys. Rev. A ]{}[**87**]{}, 062303. Chen, B., S. M. Fei, 2018, [Quantum Inf. Process. ]{}[**17**]{}, 107. Chen, L., E. Chitambar, K. Modi, G. Vacanti, 2011, [Phys. Rev. A ]{}[**83**]{}, 020101. Chen, L., M. Aulbach, M. Hajdušek, 2014, [Phys. Rev. A ]{}[**89**]{}, 042305. Chen, J., S. Grogan, N. Johnston, C. K. Li, S. Plosker, 2016a, [Phys. Rev. A ]{}[**94**]{}, 042313. Chen, J. J., J. Cui, Y. R. Zhang, H. Fan, 2016b, [Phys. Rev. A ]{}[**94**]{}, 022112. Chen, X., C. F. Wu, H. Y. Su, C. L. Ren, J. L. Chen, 2016c, arXiv: 1601.02741. Cheng, W. W., J. X. Li, C. J. Shan, L. Y. Gong, S. M. Zhao, 2015, [Quantum Inf. Process. ]{}[**14**]{} 2535. Cheng, S., M. J. W. Hall, 2016, [Phys. Rev. A ]{}[**92**]{}, 042101. Chin, S., 2017, [Phys. Rev. A ]{}[**96**]{}, 042336. Chitambar, E., A. Streltsov, S. Rana, M. N. Bera, G. Adesso, M. Lewenstein, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 070402. Chitambar, E., G. Gour, 2016a, [Phys. Rev. Lett. ]{}[**117**]{}, 030401. Chitambar, E., G. Gour, 2016b, [Phys. Rev. A ]{}[**94**]{}, 052336. Chitambar, E., M.-H. Hsieh, 2016, [Phys. Rev. Lett. ]{}[**117**]{}, 020402. Cianciaruso, M., T. R. Bromley, W. Roga, R. L. Franco, G. Adesso, 2015, [Sci. Rep. ]{}[**5**]{}, 10177. Ciccarello, F., V. Giovannetti, 2012, [Phys. Rev. A ]{}[**85**]{}, 010102. Ciccarello, F., T. Tufarelli, V. Giovannetti, 2014, [New J. Phys. ]{}[**16**]{}, 013038. Coles, P. J., M. Berta, M. Tomamichel, S. Wehner, 2015, [Rev. Mod. Phys. ]{}[**89**]{}, 015002. Costa, A. C. S., R. M. Angelo, 2013, [Phys. Rev. A ]{}[**87**]{} 032109. Crispino, L. C. B., A. Higuchi, G. E. A. Matsas, 2008, [Rev. Mod. Phys. ]{}[**80**]{}, 787.
Dai, Y., Z. Shen, Y. Shi, 2015, JHEP [**09**]{}, 071. Dai, Y., Z. Shen, Y. Shi, 2016, [Phys. Rev. D ]{}[**94**]{}, 025012. Dai, Y., W. You, Y. Dong, C. Zhang, 2017, [Phys. Rev. A ]{}[**96**]{}, 062308. Dakić, B., V. Vedral, Č. Brukner, 2010, [Phys. Rev. Lett. ]{}[**105**]{}, 190502. Dakić, B., Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, C. Brukner, P. Walther, 2012, [Nat. Phys. ]{}[**8**]{}, 666. Damour, T., R. Ruffini, 1976, [Phys. Rev. D ]{}[**14**]{}, 332. Datta, A., A. Shaji, C. M. Caves, 2008, [Phys. Rev. Lett. ]{}[**100**]{}, 050502. Datta, A., S. Gharibian, 2009, [Phys. Rev. A ]{}[**79**]{}, 042325. Datta, A., 2009, [Phys. Rev. A ]{}[**80**]{}, 052304. Datta, N., 2009b, IEEE Trans. Inf. Theory [**55**]{}, 2816. Datta, N., 2009c, [Int. J. Quantum Inf. ]{}[**7**]{}, 475. Datta, N., T. Dorlas, R. Jozsa, F. Benatti, 2014, J. Math. Phys. [**55**]{}, 062203. Debarba, T., T. O. Maciel, R. O. Vianna, 2012, [Phys. Rev. A ]{}[**86**]{}, 024302. Deutsch, J. M., 1991, [Phys. Rev. A ]{}[**43**]{}, 2046. Deutsch, D., R. Jozsa, 1992, Proc. R. Soc. London A, [**439**]{}, 553. Deng, J., J. Wang, J. Jing, 2010, [Phys. Lett. B ]{}[**695**]{}, 495. de Vicente, J. I., A. Streltsov, 2017, [J. Phys. A ]{}[**50**]{}, 045301. Dillenschneider, R., 2008, [Phys. Rev. B ]{}[**78**]{}, 224413. Doukas, J., L. Hollenberg, 2009, [Phys. Rev. A ]{}[**79**]{}, 052109. Downes, T. G., I. Fuentes, T. C. Ralph, 2011, [Phys. Rev. Lett. ]{}[**106**]{}, 210502. Doukas, J., E. G. Brown, A. Dragan, R. B. Mann, 2013, [Phys. Rev. A ]{}[**87**]{}, 012306. Dragan, A., J. Doukas, E. Martin-Martinez, D. E. Bruschi, 2013a, Class. Quantum Grav. [**30**]{}, 235006. Dragan, A., J. Doukas, E. Martin-Martinez, 2013b, [Phys. Rev. A ]{}[**87**]{}, 052326. Du, S., Z. Bai, 2015, [Ann. Phys. ]{}[**359**]{}, 136. Du, S., Z. Bai, Y. Guo, 2015, [Phys. Rev. A ]{}[**91**]{}, 052120. Duncan, A., 1978, [Phys. Rev. D ]{}[**17**]{}, 964.
Eisert, J., M. Cramer, M. B. Plenio, 2010, [Rev. Mod. Phys. ]{}[**82**]{}, 277. Eisert, J., M. Friesdorf, C. Gogolin, 2015, [Nat. Phys. ]{}[**11**]{}, 124. Englert, B.-G., 1996, [Phys. Rev. Lett. ]{}[**77**]{}, 2154. Essler, F. H. L., V. E. Korepin, K. Schoutens, 1992, [Phys. Rev. Lett. ]{}[**68**]{}, 2960.
Fabbri, A., J. Navarro-Salas, 2005, Modeling Black Hole Evaporation, Imperial College Press, London. Fan, H., V. Korepin, V. Roychowdhury, 2004, [Phys. Rev. Lett. ]{}[**93**]{}, 227203. Fan, H., S. Lloyd, 2005, [J. Phys. A ]{}[**38**]{}, 5285. Farías, O. J., C. L. Latune, S. P. Walborn, L. Davidovich, P. H. S. Ribeiro, 2009, Science [**324**]{}, 1414. Felicetti, S., C. Sabin, I. Fuentes, L. Lamata, G. Romero, E. Solano, 2015, [Phys. Rev. B ]{}[**92**]{}, 064501. Feng, J., W. L. Yang, Y. Z. Zhang, H. Fan, 2013, [Phys. Lett. B ]{}[**719**]{}, 430. Feng, J., Y. Z. Zhang, M. D. Gould, H. Fan, C. Y. Sun, W. L. Yang, 2014, [Ann. Phys. ]{}[**351**]{}, 872. Ficek, Z., S. Swain, 2005, Quantum Interference and Coherence: Theory and Experiments, Springer series in optical sciences. Fidkowski, L., 2010, [Phys. Rev. Lett. ]{}[**104**]{}, 130502. Filho, J. L. C. C., A. Saguia, L. F. Santos, M. S. Sarandy, 2017, [Phys. Rev. B ]{}[**96**]{}, 014204. Flammia, S. T., A. Hamma, T. L. Hughes, X. G. Wen, 2009, [Phys. Rev. Lett. ]{}[**103**]{}, 261601. Friis, N., A. R. Lee, D. E. Bruschi, J. Louko, 2012, [Phys. Rev. D ]{}[**85**]{}, 025012. Friis, N., A. R. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson, I. Fuentes, 2013a, [Phys. Rev. Lett. ]{}[**110**]{}, 113602. Friis, N., A. R. Lee, J. Louko, 2013b, [Phys. Rev. D ]{}[**88**]{}, 064028. Fuentes-Schuller, I., R. B. Mann, 2005, [Phys. Rev. Lett. ]{}[**95**]{}, 120404. Fuentes, I., R. B. Mann, E. Martín-Martínez, S. Moradi, 2010, [Phys. Rev. D ]{}[**82**]{}, 045030.
Galve, F., G. L. Giorgi, R. Zambrini, 2011, [Europhys. Lett. ]{}[**96**]{}, 40005. Galve, F., F. Plastina, M. G. A. Paris, R. Zambrini, 2013, [Phys. Rev. Lett. ]{}[**110**]{}, 010501. Garfinkle, D., G. T. Horowitz, A. Strominger, 1991, [Phys. Rev. D ]{}[**43**]{}, 3140. Ge, X., S. P. Kim, 2008, Class. Quant. Grav. [**25**]{}, 075011. Genovese, M., 2005 [Phys. Rep. ]{}[**413**]{}, 319. Geraedts, S. D., R. Nandkishore, N. Regnault, 2016, [Phys. Rev. B ]{}[**93**]{}, 174202. Giorgi, G. L., 2013, [Phys. Rev. A ]{}[**88**]{}, 022315. Giovannetti, V., S. Lloyd, L. Maccone, 2011, Nat. Photonics **5**, 222. Girolami, D., G. Adesso, 2011, [Phys. Rev. A ]{}[**84**]{}, 052110. Girolami, D., 2014, [Phys. Rev. Lett. ]{}[**113**]{}, 170401. Girolami, D, M. Paternostro, G. Adesso, 2011, [J. Phys. A ]{}[**44**]{}, 352002. Girolami, D., T. Tufarelli, G. Adesso, 2013, [Phys. Rev. Lett. ]{}[**110**]{}, 240402. Goold, J., C. Gogolin, S. R. Clark, J. Eisert, A. Scardicchio, A. Silva, 2015, [Phys. Rev. B ]{}[**92**]{}, 180202. Gornyi, I. V., A. D. Mirlin, D. G. Polyakov, 2005, [Phys. Rev. Lett. ]{}[**95**]{}, 206603. Gour, G., 2005, [Phys. Rev. A ]{}[**71**]{}, 012318. Gour, G., M. P. Müller, V. Narasimhachar, R. W. Spekkens, N. Y. Halpern, 2015, [Phys. Rep. ]{}[**583**]{}, 1. Grover, L. K., 1997, [Phys. Rev. Lett. ]{}[**79**]{}, 325. Gühne, G., G. Tóth, 2009, [Phys. Rep. ]{}[**474**]{}, 1. Guo, Y., 2013, [Int. J. Mod. Phys. B ]{}[**27**]{}, 1350067. Guo, Y., 2016, [Sci. Rep. ]{}[**6**]{}, 25241. Guo, Y., J. C. Hou, 2013a, [J. Phys. A ]{}[**46**]{}, 155301. Guo, Y., J. Hou, 2013b, [J. Phys. A ]{}[**46**]{}, 325301. Guo, Y., X. Li, B. Li, H. Fan, 2015, [Int. J. Theor. Phys. ]{}[**54**]{}, 2022.
Haikka, P., T. H. Johnson, S. Maniscalco, 2013, [Phys. Rev. A ]{}[**87**]{}, 010103. Hawking, S. W., 1976, [Phys. Rev. D ]{}[**14**]{}, 2460. Haldane, F. D. M., 1983a, [Phys. Lett. A ]{}[**93**]{}, 464. Haldane, F. D. M., 1983b, [Phys. Rev. Lett. ]{}[**50**]{}, 1153. Hamma, A., R. Ionicioiu, P. Zanardi, 2005, [Phys. Rev. A ]{}[**71**]{}, 022315. Hamieh, S., R. Kobes, H. Zaraket, 2004, [Phys. Rev. A ]{}[**70**]{}, 052325. Hassan, A. S. M., P. S. Joag, 2013, [Europhys. Lett. ]{}[**103**]{}, 10004. Hassan, A. S. M., B. Lari, P. S. Joag, 2012, [Phys. Rev. A ]{}[**85**]{}, 024302. Hayden, P., D. W. Leung, A. Winter, 2006, [Commun. Math. Phys. ]{}[**265**]{}, 95. He, J., S. Xu, L. Ye, 2016, [Phys. Lett. B ]{}[**756**]{}, 278. Henderson, L., V. Vedral, 2001, [J. Phys. A ]{}[**34**]{}, 6899. Hillery, M., 2016, [Phys. Rev. A ]{}[**93**]{}, 012111. Horodecki, R., P. Horodecki, M. Horodecki, 1995, [Phys. Lett. A ]{}[**200**]{}, 340. Horodecki, R., M. Horodecki, P. Horodecki, 1996, [Phys. Lett. A ]{}[**222**]{}, 21. Horodecki, M., P. Horodecki, J. Oppenheim, 2003a, [Phys. Rev. A ]{}[**67**]{}, 062104. Horodecki, M., P. W. Shor, M. B. Ruskai, 2003b, [Rev. Mod. Phys. ]{}[**15**]{}, 629. Horodecki, M., P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De), U. Sen, B. Synak, 2005a, [Phys. Rev. A ]{}[**71**]{}, 062307. Horodecki, M., J. Oppenheim, A. Winter, 2005b, Nature [**436**]{}, 673. Horodecki, M., P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen, U. Sen, B. Synak-Radtke, 2005c, [Phys. Rev. A ]{}[**71**]{}, 062307. Horodecki, R., P. Horodecki, M. Horodecki, K. Horodecki, 2009, [Rev. Mod. Phys. ]{}[**81**]{}, 865. Hosler, D., C. van de Bruck, P. Kok, 2012, [Phys. Rev. A ]{}[**85**]{}, 042312. Huber, M., M. Perarnau-Llobet, K. V. Hovhannisyan, P. Skrzypczyk, C. Klöckl, N. Brunner, A. Acín, 2015, [New J. Phys. ]{}[**17**]{}, 065008. Hu, M. L., 2011, [Phys. Lett. A ]{}[**375**]{}, 2140. Hu, X., 2016, [Phys. Rev. A ]{}[**94**]{}, 012326. Hu, M. L., H. Fan, 2012a, [Ann. Phys. ]{}[**327**]{}, 2343. Hu, M. L., H. Fan, 2012b, [Ann. Phys. ]{}[**327**]{}, 851. Hu, M. L., H. Fan, 2015a, [New J. Phys. ]{}[**17**]{}, 033004. Hu, M. L., H. Fan, 2015b, [Phys. Rev. A ]{}[**91**]{}, 052311. Hu, X., H. Fan, 2015c, [Phys. Rev. A ]{}[**91**]{}, 022301. Hu, M. L., H. Fan, 2016a, [Sci. Rep. ]{}[**6**]{}, 29260. Hu, X., H. Fan, 2016b, [Sci. Rep. ]{}[**6**]{}, 34380. Hu, M. L., H. Fan, 2017, [Phys. Rev. A ]{}[**95**]{}, 052106. Hu, M. L., H. Fan, 2018, arXiv:1804.04517. Hu, M. L., H. L. Lian, 2015, [Ann. Phys. ]{}[**362**]{}, 795. Hu, M. L., J. Sun, 2015, [Ann. Phys. ]{}[**354**]{}, 265. Hu, M. L., D. P. Tian, 2014, [Ann. Phys. ]{}[**343**]{}, 132. Hu, X., Y. Gu, Q. Gong, G. Guo, 2011, [Phys. Rev. A ]{}[**84**]{}, 022113. Hu, X., H. Fan, D. L. Zhou, W. M. Liu, 2012, [Phys. Rev. A ]{}[**85**]{}, 032102. Hu, X., H. Fan, D. L. Zhou, W. M. Liu, 2013a, [Phys. Rev. A ]{}[**87**]{}, 032340. Hu, X., H. Fan, D. L. Zhou, W. M. Liu, 2013b, [Phys. Rev. A ]{}[**88**]{}, 012315. Hu, X., A. Milne, B. Zhang, H. Fan, 2015, [Sci. Rep. ]{}[**6**]{}, 19365. Hu, M. L., S. Q. Shen, H. Fan, 2017, [Phys. Rev. A ]{}[**96**]{}, 052309. Hu, M. L., X. M. Wang, H. Fan, 2018, arXiv:1802.03540. Huang, Z., H. Situ, C. Zhang, 2016, arXiv:1609.02295. Huang, Z., H. Situ, C. Zhang, 2017, [Int. J. Theor. Phys. ]{}[**56**]{}, 2178. Hwang, M.-R., D. Park, E. Jung, 2011, [Phys. Rev. A ]{}[**83**]{}, 012111.
Ishizaka, S., T. Hiroshima, 2000, [Phys. Rev. A ]{}[**62**]{}, 022310. Ivanovic, I., 1981, [J. Phys. A ]{}[**14**]{}, 3241.
Jakóbczyk, L., A. Frydryszak, P. [Ł]{}ugiewicz, 2016, [Phys. Lett. A ]{}[**380**]{}, 1535. Jaynes, E. T., 1957a, Phys. Rev. [**106**]{}, 620. Jaynes, E. T., 1957b, Phys. Rev. [**108**]{}, 171. Jevtic, S., M. Pusey, D. Jennings, T. Rudolph, 2014, [Phys. Rev. Lett. ]{}[**113**]{}, 020402. Jozsa, R., N. Linden, 2003, Proc. R. Soc. A [**459**]{}, 2011. Johansson, J. R., G. Johansson, C. M. Wilson, F. Nori, 2009, [Phys. Rev. Lett. ]{}[**103**]{}, 147003. Johansson, J. R., G. Johansson, C. M. Wilson, P. Delsing, F. Nori, 2013, [Phys. Rev. A ]{}[**87**]{}, 043804.
Karpat, G., Z. Gedik, 2011, [Phys. Lett. A ]{}, [**375**]{}, 4166. Karpat, G., B. Çakmak, F. F. Fanchini, 2014, [Phys. Rev. B ]{}[**90**]{}, 104431. Pozas-Kerstjens, A., E. Martin-Martinez, 2015, [Phys. Rev. D ]{}[**92**]{}, 064042. Keser, A. C., S. Ganeshan, G. Refael, V. Galitski, 2016, [Phys. Rev. B ]{}[**94**]{}, 085120. Khan, S., 2014, [Ann. Phys. ]{}[**348**]{}, 270. Kitaev, A., J. Preskill, 2006, [Phys. Rev. Lett. ]{}[**96**]{}, 110404. Knill, E., R. Laflamme, 1998, [Phys. Rev. Lett. ]{}[**81**]{}, 5672. Konrad, T., F. deMelo, M. Tiersch, C. Kasztelan, A. Aragão, A. Buchleitner, 2008, [Nat. Phys. ]{}[**4**]{}, 99. Korzekwa, K., M. Lostaglio, D. Jennings, T. Rudolph, 2014, [Phys. Rev. A ]{}[**89**]{}, 042122.
Laflamme, R., D. G. Cory, C. Negrevergne, L. Viola, 2002, Quantum Inf. Comput. [**2**]{}, 166. Lambert, N., Y.-N. Chen, Y.-C. Cheng, C.-M. Li, G.-Y. Chen, F. Nori, 2013, [Nat. Phys. ]{}[**9**]{}, 10. Landulfo, A. G. S., G. E. A. Matsas, 2009, [Phys. Rev. A ]{}[**80**]{}, 032315. Lei, S., P. Tong, 2016, [Quantum Inf. Process. ]{}[**15**]{}, 1811. Leon, J., E. Martin-Martinez, 2009, [Phys. Rev. A ]{}[**80**]{}, 012314. Levi, E., M. Heyl, I. Lesanovsky, J. P. Garrahan, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 237203. Levin, M., X. G. Wen, 2006, [Phys. Rev. Lett. ]{}[**96**]{}, 110405. Lewenstein, M., A. Sanpera, 1998, [Phys. Rev. Lett. ]{}[**80**]{}, 2261. Li, H., F. D. Haldane, 2008, [Phys. Rev. Lett. ]{}[**101**]{}, 4962. Li, L., Q. W. Wang, S. Q. Shen, M. Li, 2016, [Europhys. Lett. ]{}[**114**]{}, 10007. Li, Z., X. M. Wang, W. Zhou, M. L. Hu, 2017, [Sci. Rep. ]{}[**7**]{}, 3342. Li, Y. C., H. Q. Lin, 2016, [Sci. Rep. ]{}[**6**]{}, 26365. Liu, S., L. Z. Mu, H. Fan, 2015a, [Phys. Rev. A ]{}[**91**]{}, 042133. Liu, C. L., X. D. Yu, G. F. Xu, D. M. Tong, 2016a, [Quantum Inf. Process. ]{}[**15**]{}, 4189. Liu, F., F. Li, J. Chen, W. Xing, 2016b, [Quantum Inf. Process. ]{}[**15**]{}, 3459. Liu, X. B., Z. H., Tian, J. C., Wang, J. L., Jing, 2016c, [Ann. Phys. ]{}[**366**]{}, 102. Liu, X. B., Z. H., Tian, J. C., Wang, J. L., Jing, 2016d, [Quantum Inf. Process. ]{}[**15**]{}, 3677. Liu, N., J. Goold, I. Fuentes, V. Vedral, K. Modi, D. E. Bruschi, 2016e, Class. Quantum Grav. [**33**]{}, 035003. Liu, C. L., Y. Q. Guo, D. M. Tong, 2017a, [Phys. Rev. A ]{}[**96**]{}, 062325. Liu, C. L., D. J. Zhang, X. D. Yu, Q. M. Ding, L. J. Liu, 2017b, [Quantum Inf. Process. ]{}[**16**]{}, 198. Liu, T. H., J. Wang, J. Jing, 2018, [Ann. Phys. ]{}[**390**]{}, 334. Lin, S. Y., C. H. Chou, B. L. Hu, 2015, [Phys. Rev. D ]{}[**91**]{}, 084063. Ling, Y., S. He, W. Qiu, H. Zhang, 2007, [J. Phys. A ]{}[**40**]{}, 9025. Lostaglio, M., D. Jennings, T. Rudolph, 2015a, [Nat. Commun. ]{}[**6**]{}, 6383. Lostaglio, M., K. Korzekwa, D. Jennings, T. Rudolph, 2015b, [Phys. Rev. X ]{}[**5**]{}, 021001. Luitz, D. J., N. Laflorencie, F. Alet, 2015, [Phys. Rev. B ]{}[**91**]{}, 081103. Luo, S. L., 2003, [Phys. Rev. Lett. ]{}[**91**]{}, 180403. Luo, S. L., 2006, [Phys. Rev. A ]{}[**73**]{}, 022324. Luo S., 2008, [Phys. Rev. A ]{}[**77**]{}, 022301. Luo, S., S. Fu, 2010a, [Phys. Rev. A ]{}[**82**]{}, 034302. Luo, S., S. Fu, 2010b, [Europhys. Lett. ]{}[**92**]{}, 20004. Luo S., S. Fu, 2011, [Phys. Rev. Lett. ]{}[**106**]{}, 120401. Luo S., S. Fu, C. H. Oh, 2012, [Phys. Rev. A ]{}[**85**]{}, 032117.
Ma, F. W., S. X. Liu, X. M. Kong, 2011, [Phys. Rev. A ]{}[**84**]{}, 042302. Ma, T., M. J. Zhao, Y. K. Wang, S. M. Fei, 2014, [Sci. Rep. ]{}[**4**]{}, 6336. Ma, J., B. Yadin, D. Girolami, V. Vedral, M. Gu, 2016a, [Phys. Rev. Lett. ]{}[**116**]{}, 160407. Ma, T., M. J. Zhao, S. M. Fei, G. L. Long, 2016b, [Phys. Rev. A ]{}[**94**]{}, 042312. Ma, T, M. J. Zhao, H. J. Zhang, S. M. Fei, G. L. Long, 2017, [Phys. Rev. A ]{}[**95**]{}, 042328. Modak, R., S. Mukerjee, 2015, [Phys. Rev. Lett. ]{}[**115**]{}, 230401. Malvezzi, A. L., G. Karpat, B. Çakmak, F. F. Fanchini, T. Debarba, R. O. Vianna, 2016, [Phys. Rev. B ]{}[**93**]{}, 184428. Mani, A., V. Karimipour, 2015, [Phys. Rev. A ]{}[**92**]{}, 032331. Mann, R. B., V. M. Villalba, 2009, [Phys. Rev. A ]{}[**80**]{}, 022305. Marvian, I., 2012, Symmetry, Asymmetry and Quantum Information, Ph.D. Thesis, University of Waterloo, https://uwspace.uwaterloo.ca/handle/10012/7088. Marvian, I., R. W. Spekkens, 2014, [Nat. Commun. ]{}[**5**]{}, 3821. Marvian, I., R. W. Spekkens, 2016, [Phys. Rev. A ]{}[**94**]{}, 052324. Marvian, I., R. W. Spekkens, P. Zanardi, 2016, [Phys. Rev. A ]{}[**93**]{}, 052331. Marino, J., A. Noto, R. Passante, 2014, [Phys. Rev. Lett. ]{}[**113**]{}, 020403. Martín-Martínez, E., J. León, 2009, [Phys. Rev. A ]{}[**80**]{}, 042318. Martín-Martínez, E., J. Leon, 2010, [Phys. Rev. A ]{}[**81**]{}, 032320. Martín-Martínez, E., L. J. Garay, J. León, 2010a, [Phys. Rev. D ]{}[**82**]{}, 064006. Martín-Martínez, E., L. J. Garay, J. León, 2010b, [Phys. Rev. D ]{}[**82**]{}, 064028. Menezes, G., N. F. Svaiter, [Phys. Rev. A ]{}[**93**]{}, 052117. Metwally, N., 2016, [Europhys. Lett. ]{}[**116**]{}, 60006. Mirafzali, S. Y., I. Sargolzahi, A. Ahanj, K. Javidan, M. Sarbishaei, 2013, [Quantum Inf. Comput. ]{}[**13**]{}, 479. Misra, A., U. Singh, S. Bhattacharya, A. K. Pati, 2016, [Phys. Rev. A ]{}[**93**]{}, 052335. Miszczak, J.A., Z. Pucha[ł]{}a, P. Horodecki, A. Uhlmann, K. Zyczkowski, 2009, [Quantum Inf. Comput. ]{}[**9**]{}, 103. Modi, K., T. Paterek, W. Son, V. Vedral, M. Williamson, 2010, [Phys. Rev. Lett. ]{}[**104**]{}, 080501. Modi, K., A. Brodutch, H. Cable, T. Paterek, V. Vedral, 2012, [Rev. Mod. Phys. ]{}[**84**]{}, 1655. Mondal, D., T. Pramanik, A. K. Pati, 2017, [Phys. Rev. A ]{}[**95**]{}, 010301. Montero, M., E. Martín-Martínez, 2012, [Phys. Rev. A ]{}[**85**]{}, 024301. Montealegre, J. D., F. M. Paula, A. Saguia, M. S. Sarandy, 2013, [Phys. Rev. A ]{}[**87**]{}, 042115. Moradi, S., R. Pierini, S. Mancini, 2014, [Phys. Rev. D ]{}[**89**]{}, 024022. Moore, G. T., 1970, J. Math. Phys. [**11**]{}, 2679. Moradi, S., 2009, [Phys. Rev. A ]{}[**79**]{}, 064301.
Nakano, T., M. Piani, G. Adesso, 2013, [Phys. Rev. A ]{}[**88**]{}, 012117. Nandkishore, R., D. A. Huse, 2015, Annu. Rev. Condens. Matter Phys. [**6**]{}, 15. Napoli, C., T. R. Bromley, M. Cianciaruso, M. Piani, N. Johnston, G. Adesso, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 150502. Nandkishore, R., D. A. Huse, 2015, Ann. Rev. Condens. Matter Phys. [**6**]{}, 15. Narasimhachar, V., G. Gour, 2015, [Nat. Commun. ]{}[**6**]{}, 7689. Nielsen, M. A., I. L. Chuang, 2000, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge.
Oganesyan, V., D. A. Huse, 2007, [Phys. Rev. B ]{}[**75**]{}, 155111. Okrasa, M., Z. Walczak, 2012, [Europhys. Lett. ]{}[**98**]{}, 40003. Olson, S. J., T. C. Ralph, 2011, [Phys. Rev. Lett. ]{}[**106**]{}, 110404. Ollivier, H., W. H. Zurek, 2001, [Phys. Rev. Lett. ]{}[**88**]{}, 017901. Oppenheim, J., M. Horodecki, P. Horodecki, R. Horodecki, 2002, [Phys. Rev. Lett. ]{}[**89**]{}, 180402. Osborne, T. J., F. Verstraete, 2006, [Phys. Rev. Lett. ]{}[**96**]{}, 220503. Ostapchuk, D. C. M., R. B. Mann, 2009, [Phys. Rev. A ]{}[**79**]{}, 042333. Ostapchuk, D, C. M., S. Y. Lin, R. B. Mann, B. L. Hu, 2012, JHEP [**07**]{}, 072.
Pan, Q., J. L. Jing, 2008a, [Phys. Rev. A ]{}[**77**]{}, 024302. Pan, Q., J. L. Jing, 2008b, [Phys. Rev. D ]{}[**78**]{}, 065015. Pati, A. K., M. M. Wilde, A. R. Usha Devi, A. K. Rajagopal, Sudha, 2012, [Phys. Rev. A ]{}[**86**]{}, 042105. Paul, T., T. Qureshi, 2017, [Phys. Rev. A ]{}[**95**]{}, 042110. Paula, F. M., T. R. de Oliveira, M. S. Sarandy, 2013a, [Phys. Rev. A ]{}[**87**]{}, 064101. Paula, F. M., J. D. Montealegre, A. Saguia, T. R. de Oliveira, M. S. Sarandy, 2013b, [Europhys. Lett. ]{}[**103**]{}, 50008. Peng, Y., Y. Jiang, H. Fan, 2016, [Phys. Rev. A ]{}[**93**]{}, 032326. Peres, A., P. F. Scudo, D. R. Terno, 2002, [Phys. Rev. Lett. ]{}[**88**]{}, 230402. Peters, N. A., J. B. Altepeter, D. Branning, E. R. Jeffrey, T.-C. Wei, P. G. Kwiat, 2004, [Phys. Rev. Lett. ]{}[**92**]{}, 133601. Petz, D., 2003, [Rev. Mod. Phys. ]{}[**15**]{}, 79. Piani, M., 2012, [Phys. Rev. A ]{}[**86**]{}, 034101. Piani, M., M. Cianciaruso, T. R. Bromley, C. Napoli, N. Johnston, G. Adesso, 2016, [Phys. Rev. A ]{}[**93**]{}, 042107. Pierini, R., S. Moradi, S. Mancini, 2016, [Int. J. Theor. Phys. ]{}[**55**]{}, 3059. Pires, D. P., L. C. Céleri, D. O. Soares-Pinto, 2015, [Phys. Rev. A ]{}[**91**]{}, 042330. Pino, M., 2014, [Phys. Rev. B ]{}[**90**]{}, 174204. Ponte, P., Z. Papi[ć]{}, F. Huveneers, D. A. Abanin, 2015, [Phys. Rev. Lett. ]{}[**114**]{}, 140401. Popp, M., F. Verstraete, M. A. Martín-Delgado, J. I. Cirac, 2005, [Phys. Rev. A ]{}[**71**]{}, 042306. Prodan, E., T. L. Hughes, B. A. Bernevig, 2010, [Phys. Rev. Lett. ]{}[**105**]{}, 115501. Pusey, M. F., 2015, J. Opt. Soc. Am. B [**32**]{}, A56.
Qi, X. L., H. Katsura, A. W. W. Ludwig, 2012, [Phys. Rev. Lett. ]{}[**108**]{}, 196402. Qi, X., T. Gao, F. L. Yan, 2017, [J. Phys. A ]{}[**50**]{}, 285301. Qiang, W. C., L. Zhang, 2015, [Phys. Lett. B ]{}[**742**]{}, 383. Qiu, D., 2002, [Phys. Lett. A ]{}[**303**]{}, 140. Qureshi, T, M. A. Siddiqui, 2017, [Ann. Phys. ]{}[**385**]{}, 598.
Radhakrishnan, C., M. Parthasarathy, S. Jambulingam, T. Byrnes, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 150504. Ramzan, M., 2013, [Quantum Inf. Process. ]{}[**12**]{}, 2721. Rana, S., P. Parashar, M. Lewenstein, 2016, [Phys. Rev. A ]{}[**93**]{}, 012110. Rana, S., P. Parashar, A. Winter, M. Lewenstein, 2017, [Phys. Rev. A ]{}[**96**]{}, 052336. Rastegin, A. E., 2016, [Phys. Rev. A ]{}[**93**]{}, 032136. Rana, S., P. Parashar, 2013, [Quantum Inf. Process. ]{}[**12**]{}, 2523. Refael, G., J. E. Moore, 2004, [Phys. Rev. Lett. ]{}[**93**]{}, 260602. Regula, B., A. R. Lee, A. Dragan, I. Fuentes, 2016, [Phys. Rev. D ]{}[**93**]{}, 025034. Richter, B., Y. Omar, 2015, [Phys. Rev. A ]{}[**92**]{}, 022334. Rigol, M., V. Dunjko, M. Olshani, 2008, Nature [**452**]{}, 854. Roushan, P., C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, J. Martinis, 2017, Science [**358**]{}, 1175.
Sabín, C., I. Fuentes, G. Johansson, 2015, [Phys. Rev. A ]{}[**92**]{}, 012314. Sabín, C., G. Adesso, 2015, [Phys. Rev. A ]{}[**92**]{}, 042107. Sachdev, S., 2000, Quantum Phase Transitions, Cambridge University Press, Cambridge. Samos-Sáenz de Buruaga, D. N., C. Sabín, 2017, [Phys. Rev. A ]{}[**95**]{}, 022307. Satyabrata, A., B. Subhashish, 2012, [Phys. Rev. A ]{}[**86**]{}, 062313. Sánchez-Ruiz, J., 1998, [Phys. Lett. A ]{}[**244**]{}, 189. Shahandeh, F., A. P. Lund, T. C. Ralph, 2017, arXiv:1706.00478. Shao, L. H., Z. Xi, H. Fan, Y. Li, 2015, [Phys. Rev. A ]{}[**91**]{}, 042120. Shimony, A., 1995, Ann. N.Y. Acad. Sci. [**755**]{}, 675. Shi, H. L., S. Y. Liu, X. H. Wang, W. L. Yang, Z. Y. Yang, H. Fan, [Phys. Rev. A ]{}[**95**]{}, 032307. Silva, I. A., A. M. Souza, T. R. Bromley, M. Cianciaruso, R. Marx, R. S. Sarthour, I. S. Oliveira, R. L. Franco, S. J. Glaser, E. R. deAzevedo, D. O. Soares-Pinto, G. Adesso, 2016, [Phys. Rev. Lett. ]{}[**117**]{}, 160402. Singh, U., M. N. Bera, H. S. Dhar, A. K. Pati, 2015, [Phys. Rev. A ]{}[**91**]{}, 052115. Singh, U., A. K. Pati, M. N. Bera, 2016a, Mathematics [**4**]{}, 47. Singh, U., L. Zhang, A. K. Pati, 2016b, [Phys. Rev. A ]{}[**93**]{}, 032125. Situ, H., X. Hu, 2016, [Quantum Inf. Process. ]{}[**15**]{}, 4649. Skrzypczyk, P., M. Navascués, D. Cavalcanti, 2014, [Phys. Rev. Lett. ]{}[**112**]{}, 180404. Smith, J., A. Lee, P. Richerme, B. Neyerhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, C. Monroe, 2016, [Nat. Phys. ]{}[**12**]{}, 907. Spehner, D., M. Orszag, 2013, [New J. Phys. ]{}[**15**]{}, 103001. Spehner, D., M. Orszag, 2014, [J. Phys. A ]{}[**47**]{}, 035302. Song, X. K., T. Wu, S. Xu, J. He, L. Ye, 2014, [Ann. Phys. ]{}[**349**]{}, 220. Srednicki, M., 1994, [Phys. Rev. E ]{}[**50**]{}, 888. Streltsov, A., H. Kampermann, D. Bru[ß]{}, 2010a, [New J. Phys. ]{}[**12**]{}, 123004. Streltsov, A., H. Kampermann, D. Bru[ß]{}, 2010b, [New J. Phys. ]{}[**12**]{}, 123004. Streltsov, A., H. Kampermann, D. Bru[ß]{}, 2011a, [Phys. Rev. Lett. ]{}[**107**]{}, 170502. Streltsov, A., H. Kampermann, D. Bru[ß]{}, 2011b, [Phys. Rev. Lett. ]{}[**106**]{}, 160401. Streltsov, A., G. Adesso, M. Piani, D. Bru[ß]{}, 2012, [Phys. Rev. Lett. ]{}[**109**]{}, 050503. Streltsov, A., U. Singh, H. S. Dhar, M. N. Bera, G. Adesso, 2015, [Phys. Rev. Lett. ]{}[**115**]{}, 020403. Streltsov, A., E. Chitambar, S. Rana, M. N. Bera, A. Winter, M. Lewenstein, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 240405. Streltsov, A., G. Adesso, M. B. Plenio, 2017a, [Rev. Mod. Phys. ]{}[**89**]{}, 041003. Streltsov, A., S. Rana, P. Boes, J. Eisert, 2017b, [Phys. Rev. Lett. ]{}[**119**]{}, 140402. Streltsov, A., H. Kampermann, S. Wölk, M. Gessner, D. Bru[ß]{}, 2018, [New J. Phys. ]{}[**20**]{}, 053058. Styliaris,G., L. C. Venuti, P. Zanardi, 2018, [Phys. Rev. A ]{}[**97**]{}, 032304.
Tan, K. C., H. Kwon, C.-Y. Park, H. Jeong, 2016, [Phys. Rev. A ]{}[**94**]{}, 022329. Tan, K. C., T. Volkoff, H. Kwon, H. Jeong, 2017, [Phys. Rev. Lett. ]{}[**119**]{}, 190405. Terashima, H., 2000, [Phys. Rev. D ]{}[**61**]{}, 104016. Terhal, B. M., P. Horodecki, 2000, [Phys. Rev. A ]{}[**61**]{}, 040301. Theurer, T., N. Killoran, D. Egloff, M. B. Plenio, 2017, [Phys. Rev. Lett. ]{}[**119**]{}, 230401. Tian, Z. H., J. C. Wang, J. L. Jing, 2013, [Ann. Phys. ]{}[**322**]{}, 98. Tian, Z. H., J. Jing, 2013, [Ann. Phys. ]{}[**333**]{}, 76. Tian, Z. H., J. Jing, 2014, [Ann. Phys. ]{}[**350**]{}, 1. Titum, P., N. H. Lindner, M. C. Rechtsman, G. Refael, 2015, [Phys. Rev. Lett. ]{}[**114**]{}, 056801. Tsallis, C., 1988, J. Stat. Phys. [**52**]{}, 479.
Unruh, W. G., 1976, [Phys. Rev. D ]{}[**14**]{}, 870. Unruh, W. G., R. M. Wald, 1984, [Phys. Rev. D ]{}[**29**]{}, 1047.
Vedral, V., M. B. Plenio, M. A. Rippin, P. L. Knight, 1997, [Phys. Rev. Lett. ]{}[**78**]{}, 2275. Vedral, V., M. B. Plenio, 1998, [Phys. Rev. A ]{}[**57**]{}, 1619. Verstraete, F., H. Verschelde, 2002, [Phys. Rev. A ]{}[**66**]{}, 022307. Verstraete, F., M. Popp, J. I. Cirac, 2004, [Phys. Rev. Lett. ]{}[**92**]{}, 027901. Verstraete, F., K. Audenaert, B. D. Moor, 2001, [Phys. Rev. A ]{}[**64**]{}, 012316. Vidal G., R. F. Werner, 2002, [Phys. Rev. A ]{}[**65**]{}, 032314. Vidal, G., B. Tarrach, 1999, [Phys. Rev. A ]{}[**59**]{}, 141. Vidal, G., J. I. Latorre, E. Rico, A. Kitaev, 2003, [Phys. Rev. Lett. ]{}[**90**]{}, 227902. Virmani, S., M. Plenio, 2000, [Phys. Lett. A ]{}[**268**]{}, 31.
Wald, R. M., 1994, Quantum Field Theory in Curved Spacetimes and Black Hole Thermodynamics, The University of Chicago Press, Chicago. Wang, J. C., Q. Y. Pan, S. B. Chen, J. L. Jing, 2009, [Phys. Lett. B ]{}[**692**]{}, 202. Wang, J. C., Q. Y. Pan, J. L. Jing, 2010a, [Ann. Phys. ]{}[**325**]{}, 1190. Wang, J. C., Q. Y. Pan, J. L. Jing, 2010b, [Phys. Lett. B ]{}[**692**]{}, 202. Wang, J. C., J. F. Deng, J. L. Jing, 2010c, [Phys. Rev. A ]{}[**81**]{}, 052120. Wang, J. C., Z. H. Tian, J. L. Jing, H. Fan, 2014a, [Sci. Rep. ]{}[**4**]{}, 7195. Wang, J. C., Jing, J. L., H. Fan, 2014b, [Phys. Rev. D ]{}[**90**]{}, 025032. Wang, J. C., Z. H. Tian, J. L., Jing, H. Fan, 2015, Nucl. Phys. B [**892**]{}, 390. Wang, J. C., Z. H. Tian, J. L. Jing, H. Fan, 2016a, [Phys. Rev. A ]{}[**93**]{}, 062105. Wang, J. C., H. X. Cao, J. L. Jing, H. Fan, 2016b, [Phys. Rev. D ]{}[**93**]{}, 125011. Wang, Z., Y. L. Wang, Z. X. Wang, 2016c, [Quantum Inf. Process. ]{}[**15**]{}, 4641. Wang, Y. T., J. S. Tang, Z. Y. Wei, S. Yu, Z. J. Ke, X. Y. Xu, C. F. Li, G. C. Guo, 2017, [Phys. Rev. Lett. ]{}[**118**]{}, 020403. Wang, J. C., J. L. Jing, H. Fan, 2018a, Annalen Der Physik [**530**]{}, 1700261. Wang, Z. A., Y. Zeng, H. F. Lang, Q. T. Hong, J. Cui, H. Fan, 2018b, arXiv:1802.08475. Wang, J. C., J. L. Jing, 2010, [Phys. Rev. A ]{}[**82**]{}, 032324. Wang, J. C., J. L. Jing, 2011, [Phys. Rev. A ]{}[**83**]{}, 022314. Wang, J. C., J. L. Jing, 2012, [Ann. Phys. ]{}[**327**]{}, 283. Wei, T. C., P. M. Goldbart, 2003, [Phys. Rev. A ]{}[**68**]{}, 042307. Werlang, T., C. Trippe, G. A. P. Ribeiro, G. Rigolin, 2010, [Phys. Rev. Lett. ]{}[**105**]{}, 095702. Wigner, E. P., M. M. Yanase, 1963, Proc. Natl. Acad. Sci. U.S.A. [**49**]{}, 910. Wilson, C. M., G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, P. Delsing, 2011, Nature [**479**]{}, 376. Wootters, W. K., 1998, [Phys. Rev. Lett. ]{}[**80**]{}, 2245. Winter, A., D. Yang, 2016, [Phys. Rev. Lett. ]{}[**116**]{}, 120404. Wootters, W. K., Found. Phys. 1986, [**16**]{}, 391. Wootters, W. K., B. D. Fields, 1989, [Ann. Phys. ]{}[**191**]{}, 363. Wu, S, U. V. Poulsen, K. M[ø]{}lmer, 2009, [Phys. Rev. A ]{}[**80**]{}, 032319. Wu, S., J. Zhang, C. Yu, H. Song, 2014, [Phys. Lett. A ]{}[**378**]{}, 344-347.
Xi, Z., X. Wang, Y. Li, 2012, [Phys. Rev. A ]{}[**85**]{}, 042325. Xi, Z., Y. Li, H. Fan, 2015, [Sci. Rep. ]{}[**5**]{}, 10922. Xu, J., 2016, [Phys. Rev. A ]{}[**93**]{}, 032111. Xu, J. S., C. F. Li, 2013, [Int. J. Mod. Phys. B ]{}[**27**]{}, 1345054. Xu, K., J. J. Chen, Y. Zeng, Y. R. Zhang, C. Song, W. X. Liu, Q. J. Guo, P. F. Zhang, D. Xu, H. Deng, K. Q. Huang, H. Wang, X. B. Zhu, D. N. Zheng, H. Fan, 2018, [Phys. Rev. Lett. ]{}[**120**]{}, 050507.
Yang, C. N., 1989, [Phys. Rev. Lett. ]{}[**63**]{}, 2144. Yao, Y., X. Xiao, L. Ge, C. P. Sun, 2015, [Phys. Rev. A ]{}[**92**]{}, 022112. Yao, Y., G. H. Dong, X. Xiao, C. P. Sun, 2016a, [Sci. Rep. ]{}[**6**]{}, 32010. Yao, Y., G. H. Dong, L. Ge, M. Li, C. P. Sun, 2016b, [Phys. Rev. A ]{}[**94**]{}, 062339. You, B., L. X. Cen, 2012, [Phys. Rev. A ]{}[**86**]{}, 012102. Yu, C. S., S. X. Wu, X. Wang, X. X. Yi, H. S. Song, 2014, [Europhys. Lett. ]{}[**107**]{}, 10007. Yu, X. D., D. J. Zhang, C. L. Liu, D. M. Tong, 2016a, [Phys. Rev. A ]{}[**93**]{}, 060303. Yu, X. D., D. J. Zhang, G. F. Xu, D. M. Tong, 2016b, [Phys. Rev. A ]{}[**94**]{}, 060302. Yu, C. S., S. R. Yang, B. Q. Guo, 2016c, [Quantum Inf. Process. ]{}[**15**]{}, 3773. Yu, C. S., 2017, [Phys. Rev. A ]{}[**95**]{}, 042337. Yuan, X., H. Zhou, Z. Cao, X. Ma, 2015, [Phys. Rev. A ]{}[**92**]{}, 022124. Yuan, X., G. Bai, T. Peng, X. Ma, 2017, [Phys. Rev. A ]{}[**96**]{}, 032313.
Zanardi, P., G. Styliaris, L. C. Venuti, 2017, [Phys. Rev. A ]{}[**95**]{}, 052306. Zeng, B., X. Chen, D. L. Zhou, X. G. Wen, 2015, arXiv:1508.02595. Zhang, L., 2017, [J. Phys. A ]{}[**50**]{}, 155303. Zhang, S., Y. Feng, X. Sun, M. Ying, 2001, [Phys. Rev. A ]{}[**64**]{}, 062103. Zhang, Y. J., W. Han, Y. J. Xia, Y. M. Yu, H. Fan, 2015, [Sci. Rep. ]{}[**5**]{}, 13359. Zhang, Y. R., L. H. Shao, Y. Li, H. Fan, 2016, [Phys. Rev. A ]{}[**93**]{}, 012334. Zhang, H. J., B. Chen, M. Li, S. M. Fei, G. L. Long, 2017a, [Commun. Theor. Phys. ]{}[**67**]{}, 166. Zhang, J., S. R. Yang, Y. Zhang, C. S. Yu, 2017b, [Sci. Rep. ]{}[**7**]{}, 45598. Zhang, L., U. Singh, A. K. Pati, 2017c, [Ann. Phys. ]{}[**377**]{}, 125. Zhang, D. J., C. L. Liu, X. D. Yu, D. M. Tong, 2018, [Phys. Rev. Lett. ]{}[**120**]{}, 170501. Zhao, Q., Y. Liu, X. Yuan, E. Chitambar, X. Ma, 2018, [Phys. Rev. Lett. ]{}[**120**]{}, 070403. Žnidarič, M., T. Prosen, P. Prelovšek, 2008, [Phys. Rev. B ]{}[**77**]{}, 064426. Zurek, W. H., 2003, [Phys. Rev. A ]{}[**67**]{}, 012320.
|
---
abstract: 'We report high angular resolution (3$\arcsec$) Submillimeter Array (SMA) observations of the molecular cloud associated with the Ultra-Compact HII region G5.89-0.39. Imaged dust continuum emission at 870$\mu$m reveals significant linear polarization. The position angles (PAs) of the polarization vary enormously but smoothly in a region of 2$\times$10$^{4}$ AU. Based on the distribution of the PAs and the associated structures, the polarized emission can be separated roughly into two components. The component “x” is associated with a well defined dust ridge at 870 $\mu$m, and is likely tracing a compressed B field. The component “o” is located at the periphery of the dust ridge and is probably from the original B field associated with a pre-existing extended structure. The global B field morphology in G5.89, as inferred from the PAs, is clearly disturbed by the expansion of the HII region and the molecular outflows. Using the Chandrasekhar-Fermi method, we estimate from the smoothness of the field structures that the B field strength in the plane of sky can be no more than 2$-$3 mG. We then compare the energy densities in the radiation, the B field, and the mechanical motions as deduced from the C$^{17}$O 3-2 line emission. We conclude that the B field structures are already overwhelmed and dominated by the radiation, outflows, and turbulence from the newly formed massive stars.'
author:
- 'Ya-Wen Tang'
- 'Paul T. P. Ho'
- Josep Miquel Girart
- Ramprasad Rao
- Patrick Koch
- 'Shih-Ping Lai'
title: 'Evolution of Magnetic Fields in High Mass Star Formation: SMA dust polarization image of the UCHII region G5.89-0.39'
---
Introduction
============
One of the main puzzles in the study of star formation is the low star formation efficiency in molecular clouds. Since molecular clouds are known to be cold, the thermal pressure is small. Hence, if there are no other supporting forces against gravity, the free-fall time scale will be short and the star formation rate will be much higher than what is observed. Magnetic (B) fields have been suggested to play the primary role in providing a supporting force to slow down the collapsing process (see the reviews by Shu et al. (1999) and Mouschovias & Ciolek (1999)). In these models, the B field is strong enough and has an orderly structure in the molecular cloud. The B field lines, which are anchored to the ionized particles, will then be dragged in along the direction of accretion, only when the ambipolar diffusion process allows the neutral component to slip pass the ionized component. In the standard low-mass star formation model (Galli & Shu 1993; Fiedler & Mouschovias 1993), an hourglass-like B field morphology is expected with an accreting disk near the center of the pinched field. Alternatively, turbulence has also been suggested as a viable source of support against contraction (see reviews by Mac Low & Klessen (2004) and Elmegreen & Scalo (2004)). The relative importance of B field and turbulence continues to be a hot topic as the two methods of support will lead to different scenarios for the star formation process.
Compared with the low mass stars, the formation process of high mass stars is really poorly understood. High mass star forming regions, because of their rarity, are usually at larger distances and are always located in dense and massive regions, because they are typically formed in a group. Hence, both poor resolution and complexity have hampered past observational studies. Furthermore, the environments of high mass star forming regions are very different from the low mass case because of higher radiation intensity, higher temperature, and stronger gravitational fields. Will the B fields in massive star forming sites have a similar morphology to the low mass cases?
Polarized emission from dust grains can be used to study the B field in dense regions, because the dust grains are not spherical in shape. They are thought to be aligned with their minor axes parallel to the B field in most of the cases, even if the alignment is not magnetic (Lazarian 2007). Due to the differences in the emitted light perpendicular and parallel to the direction of alignment, the observed thermal dust emission will be polarized, the direction of polarization is then perpendicular to the B field. Although the alignment mechanism of the dust grains has been a difficult topic for decades (see review by Lazarian 2007), the radiation torques seem to be a promising mechanism to align the dust grains with the B field (e.g., Draine & Weingartner 1996; Lazarian & Hoang 2007). However, other processes such as mechanical alignments by outflows can also be important.
Polarized dust emission has been detected successfully at arcsecond scales. The best example might be the low mass star forming region NGC 1333 IRAS 4A (Girart, Rao & Marrone 2006), which reveals the classic predicted hourglass B field morphology. Results on the massive star formation regions, such as W51 e1/e2 cores (Lai et al. 2001), NGC2024 FIR5 (Lai et al. 2002), DR21 (OH) (Lai et al. 2003), G30.79 FIR 10 (Cortes et al. 2006) and G34.4+0.23 MM (Cortes et al. 2008), typically show an organized and smooth B field morphology. However, this could be due to a lack of spatial resolution. Indeed, for the nearby high mass cases such as Orion KL (Rao et al. 1998) and NGC2071IR (Cortes, Crutcher, & Matthews 2006), abrupt changes of the polarization direction on small physical scales have been seen, which may suggest mechanical alignments by outflows as proposed by these authors. Whether high mass star forming regions will all show complicated B field structures on small scales remains to be examined.
In this study, we report on one of the first SMA measurements of dust polarization for a high mass star forming region, G5.89-0.39 (hereafter, G5.89). The linearly polarized thermal dust emission is used to map the B field at $\sim$3$\arcsec$ resolution, and the C$^{17}$O 3-2 line is used to study the structure and kinematics of the dense molecular cloud. The description of the source, the observations and the data analysis, the results, and the discussion are in Sec. 2, 3, 4 and 5, respectively. The conclusions and summary are in Sec. 6.
Source Description
==================
G5.89 is a shell-like ultracompact HII (UCHII) region (Wood et al. 1989) at a distance of 2 kpc (Acord et al. 1998). The UCHII region is 0.01 pc in size, and its dynamical age is 600 years, estimated from the expansion velocity (Acord et al. 1998). Observations of the K$_{s}$ and $L'$ magnitudes and color by Feldt et al. (2003) suggest that G5.89 contains an O5 V star.
Just as in other cases of massive stars, G5.89 contains most likely a cluster of stars. The detections of associated H$_{2}$O masers (Hofner & Churchwell 1996), OH masers (Stark et al. 2007; Fish et al. 2005) and class I CH$_{3}$OH masers (Kurtz et al. 2004), suggest that multiple stars have formed in this region. Furthermore, the morphology of the detected molecular outflows also suggest the presence of multiple driving sources, because different orientations are observed in different tracers. In CO 1-0, the large scale outflow is almost in the east-west direction (Harvey & Forveille 1988; Watson et al. 2007). In C$^{34}$S and the OH masers, the outflow is in the north-south direction (Cesaroni et al. 1991; Zijlstra et al. 1990). In SiO 5$-$4, the outflow is at a position angle (PA) of 28$\degr$ (Sollins et al. 2004). In the CO 3-2 line, the outflows (Hunter et al. 2008) are in the north-south direction and at the PA of 131$\degr$, and the latter one is associated with the Br$\gamma$ outflow (Puga et al. 2006). In addition, the detected 870 $\mu$m emission has also been resolved into multiple peaks (labelled in Fig. 1(a); Hunter et al. 2008). The different masers, the multiple outflows, and the multiple dust peaks, are all consistent with the formation of a cluster of young stars.
G5.89 should be expected to have a substantial impact on its environment. In terms of the total energy in outflows in this region, G5.89 is definitely one of the most powerful groups of outflows ever detected (Churchwell 1997).
Observation and Data Analysis
=============================
The observations were carried out on July 27, 2006 and September 10, 2006 using the Submillimeter Array (Ho, Moran & Lo (2004))[^1] in the compact configuration, with 7 of the 8 antennas available for both tracks. The projected lengths of the baselines ranged from 6.5 to 70 k$\lambda$ ($\lambda\approx$870$\mu$m). Therefore, our observational results are insensitive to structures larger than 39$\arcsec$. The SMA receivers are intrinsically linearly polarized and only one polarization is available at the current time. Thus, quarter-wave plates (see Marrone & Rao 2008) were installed in order to convert the linear polarization (LP) to circular polarization (CP). The quarter-wave plates were rotated by 90$ \degr$ on a 5 minutes cycle using a Walsh function to switch between 16 steps in order to sample all the 4 Stokes parameters. The integration time spent on the source in each step was approximately 15 seconds. The overhead required in switching between the different states was approximately 5 seconds. In each cycle all four cross-correlations (LL, LR, RL, and RR) were each calculated 4 times. The data were then averaged over this complete cycle in order to obtain quasi simultaneous dual polarization visibilities. We assume that the smearing due to the change of the polarization angles on this time scale is negligible.
The local oscillator frequency was tuned to 341.482 GHz. With a 2 GHz bandwidth in each sideband we were able to cover the frequency range from 345.5 to 347.5 GHz and from 335.5 to 337.5 GHz in the upper and lower sideband, respectively. The correlator was set to a uniform frequency resolution of 0.65 MHz ($\sim$ 0.7 km s$^{-1}$) for both sidebands. While our main emphasis was to map the polarized continuum emission from the dust, we were also able to detect a number of molecular lines simultaneously. These results will be published separately.
Generally, the conversions of the LP to CP of the receivers are not perfect. This non-ideal characteristic of the receiver will cause an unpolarized source to appear polarized, which is known as instrumental polarization or leakage. Nevertheless, these leakage terms (see Sault, Hamaker, & Bregman 1996) can be calibrated by observing a strong linearly polarized quasar. In this study, the leakage and bandpass were calibrated by observing 3c279 for the first track and 3c454.3 for the second track. Both sources were observed for 2 hours while they were transiting in order to get the best coverage of parallactic angles. The leakage terms are frequency dependent, $\sim$1% and $\sim$3% for the upper and lower sideband before the calibration, respectively. After calibration, the leakage is less than 0.5$\%$ in both sidebands. Besides the calibration for the polarization leakage, the amplitudes and phases were calibrated by observing the quasars 1626-298 and 1924-292 every 18 minutes. These two gain calibrators in both tracks were used because of the availabilities of the calibrators during the observations. Finally, the absolute flux scale was calibrated using Callisto.
The data were calibrated and analyzed using the MIRIAD package (Sault, Teuben, & Wright 1995). After the standard gain calibration, self-calibration was also performed by selecting the visibilities of G5.89 with uv distances longer than 30 k$\lambda$. As a result, the sidelobes and the noise level of the Stokes $I$ image were reduced by a factor of 2. In order to get the images from the measured visibilities, the task INVERT in the MIRIAD package was used. The Stokes $Q$ and $U$ maps are crucial for the derivation of the polarization segments. We used the dirty maps of $Q$ and $U$ to derive the polarization to avoid a possible bias introduced from the CLEAN process. The Stokes $I$ map shown in this paper is after CLEAN.
The Stokes $I$, $Q$ and $U$ images of the continuum were constructed with natural weighting in order to get a better S/N ratio for the polarization. The final synthesized beam is $3\arcsec.0 \times1\arcsec.9$ with the natural weighting. The C$^{17}$O images are presented with a robust weighting 0.5 in order to get a higher angular resolution, and the synthesized beam is 2.8$\arcsec \times$1.8$\arcsec$ with PA of 13$\degr$. The noise levels of the $I$, $Q$ and $U$ images are $\sim$ 30, 5 and 5 mJy Beam$^{-1}$, respectively. Note that the noise level of the Stokes $I$ image is much larger than the ones in the Stokes $Q$ and $U$ images. The large noise level of the $I$ image is most likely due to the extended structure, which can not be recovered with our limited and incomplete uv sampling. The strength (I$_{p}$) and percentage ($P$) of the linearly polarized emission is calculated from: $I_{p}^{2} = Q^2 +U^2-\sigma_{Q,U}^{2}$ and $P$ = I$_{p}$/I, respectively. The term $\sigma_{Q,U}$ is the noise level of the Stokes $Q$ and $U$ images, and it is the bias correction due to the positive measure of I$_{p}$. The noise of I$_{p}$ ($\sigma_{I_p}$) is thus 5 mJy Beam$^{-1}$. The presented polarization is derived using the task IMPOL in the MIRIAD package, where the bias correction of $\sigma_{I_p}$ is included.
Results
=======
In this section, we present the observational results of the dust continuum and the dust polarization at 870$\mu$m, and the C$^{17}$O 3-2 emission line. No polarization was detected in the CO 3-2 emission line.
Continuum Emission
------------------
The total continuum emission at 870 $\mu$m, shown in Fig. 1(a), is resolved with a total integrated flux density 12.6$\pm$1.3 Jy. In general, the morphology of the continuum emission at 870$\mu$m is similar to the emission at 1.3 mm by Sollins et al. (2004). However, the 870$\mu$m emission peaks at $\sim$ 1$\arcsec$ west of the position of the O5 star, which is offset toward the north-west by $\sim$1$\arcsec$.7 from the peak of the 1.3 mm continuum emission. Because there is still a significant contribution from the free-free emission to the continuum at 870 $\mu$m and at 1.3mm, the differences between the 870 $\mu$m and 1.3 mm maps most likely result from the increasing contribution from the dust emission as compared to the free-free emission at shorter wavelengths. Due to the importance of a correct dust continuum image in the derivation of the polarization, we describe here how the free-free continuum was estimated and removed from the 870$\mu$m total continuum emission.
### Removing the free-free emission
The free-free continuum at 2cm (shown in color scale in Fig.1 (a) and (b)) was imaged from the VLA archival database observed on August 7, 1986. The VLA synthesized beam of the 2cm free-free image is 0$ \arcsec.92\times$0$\arcsec$.45 with natural weighting of the uv data. Since the free-free shell is expanding at a rate of 2.5 mas year$^{-1}$ (Acord et al. 1998), at a distance of 2 kpc, this expansion motion over the intervening 20 years is negligible within the synthesized beam of our SMA observation.
The contribution from the free-free continuum was removed by the following steps. Firstly, we adopted a spectral index $\alpha=-$0.154 calculated in Hunter et al. (2008) for the free-free continuum emission between 2cm to 870$\mu$m. The resulting estimated free-free continuum strength at 870$\mu$m was 4.9 Jy. Secondly, we further assumed that the morphology of the free-free continuum at 870 $\mu$m and at 2cm were identical. We then smoothed the VLA 2cm image to the SMA resolution and scaled the total flux density to 4.9 Jy. Finally, we subtracted this image from the total continuum at 870 $\mu$m. The resultant 870 $\mu$m dust continuum image is shown in Fig. 1(b). The total flux density of the dust continuum is therefore 7.7$\pm$0.8 Jy.
### Dust continuum: mass and morphology
The corresponding gas mass (M$_{gas}$) was calculated from the flux density of the dust continuum at 870 $\mu$m following Lis et al. (1998):
$$\label{mh2}
M_{gas} = \frac{2\lambda^{3} Ra \rho d^{2}}{3hcQ(\lambda)J(\lambda,T_d)}S(\lambda)$$
Here, we assumed a gas-to-dust mass ratio $R$ of 100, a grain radius $a$ of 0.1 $\mu$m, a mean grain mass density $\rho$ of 3 g cm$^{-3}$, a distance to the source $d$ of 2 kpc, a dust temperature $T_{d}$ of 44 K, an observed flux density S$(\lambda)$ of 7.7 Jy, the Planck factor $J(\lambda, T_{d})=[exp(hc/\lambda k
T_{d})-1]^{-1}$. $h$, $c$ and $k$ are the Planck constant, the speed of light and the Boltzmann constant, respectively. The grain emissivity $Q$($\lambda$) was estimated to be $1.5\times 10^{-5}$ after assuming $Q(350\mu m)$ of $7.5\times 10^{-4}$ and $\beta$ of 2 (cold dust component), and using the relation $Q(\lambda)=Q(350\mu m)(350\mu m/\lambda)^{\beta}$ (Hunter et al. 2000). As suggested in the same paper, the dust emission can be modeled by two temperature components, with the emission dominated by the colder component at T$_{d}$ $\sim$ 44 K. We adopted this value for T$_{d}$, and therefore, the mass given here refers only to the cold component and is an underestimate of the total mass. The derived gas mass of the dust core M$_{gas}$ is $\sim$ 300 M$_{\sun}$, with a number density $n_{H_{2}}=$ 5.3$\times$10$^{6}$ cm$^{-3}$ averaged over the emission region. The sizescale along the line of sight is assumed to be 0.13 pc, which is the diameter of the circle with the equivalent emission area.
The dust emission presented in Fig. 1(b) has an extension toward the northeast, east and southwest and has a steep roll off on the northwestern edge of the ridge. In the higher angular resolution (0.8$\arcsec$) observation at the same wavelength by Hunter et al. (2008), the dust core is resolved into 5 peaks, where the two strongest peaks align in the north-south direction to the west of the O5 star. The dust continuum emission associated with SMA-N, SMA-1 and SMA-2 is called *sharp dust ridge* hereafter because of its strong emission and its morphology. There is no peak detected at the position of the O5 star. It is likely that the O5 star is located in a dust-free cavity, as proposed by Feldt et al. (1999) and Hunter et al. (2008).
Dust polarization
-----------------
We first compare the dust polarization derived from the 870 $\mu$m total continuum (Fig. 1(c)) and from the 870 $\mu$m dust continuum (Fig. 1(d)). In both cases the derived polarization is at the same location with the same PAs. The only difference of the polarization in Fig. 1(c) and 1(d) is that the percentage of polarization near the HII region is increased in Fig. 1(d). This is because of the fact that the free-free continuum is not polarized, and the $Q$ and $U$ components are not affected by the free-free continuum subtraction. Therefore, the expected polarization percentage will increase when the free-free continuum is removed from the 870 $\mu$m continuum. The total detected polarized intensity I$_{p}$ is 59 mJy. All the polarization shown in the figures besides Fig. 1(c) is calculated from the derived dust continuum image. The off-set positions, percentages and PAs of the polarization segments are listed in Table 1.
### Morphology of the detected polarization
The polarized emission is not uniformly distributed. Detected polarization at 2$\sigma_{I_p}$ are shown as blue segments and detections above 3$\sigma_{I_p}$ are shown by red segments. Most of the polarized emission is located in the northern half of the dust core close to the HII region and appears as 4 patches, mostly with $\sigma_{I_p} \geq 3$ (Fig. 2(a) in color scale). There is a sharp gap where no polarization is detected extending from the NE to the SW across the O star. The southern half of the dust core is free of polarization, except for a few positions at the edge of the dust core. However, the polarization in the south half of the dust core is at 2 to 3$\sigma_{I_p}$ level only. We will focus our discussions on the more significant detections in the core of the cloud.
We separate the polarized emission into two groups. We are guided principally by the fact that one group is associated with the periphery of the total dust emission, while the other group tracks the strongest parts of the total dust emission. The polarized patches to the east of the O star and to the west of the Br$\gamma$ outflow source have similar PAs of $\sim 50
\degr$ (Fig. 2(b)). These polarization segments are located at the fainter edges of the higher resolution 870 $\mu$m dust continuum image (Fig. 2(c); Hunter et al. 2008) and at the less steep part of the 3$\arcsec$ resolution image (this paper). This may suggest that this polarization originates from a more extended overall structure, rather than from the detected condensations. Therefore, these polarization segments are suggested to be the component “o” (defined in the next section). The rest of the polarization in the northern part is all next to the sharp gap where no polarization is detected. Most of the polarization is on the 870$\mu$m *sharp dust ridge* observed with 0.8$\arcsec$ resolution, except for the ones at the NE and SW ends where the polarization patches stretch toward the extended structure. At these NE and SW ends the polarization is probably the sum of the extended and the condensed structures. These polarization segments are suggested to belong to the component “x”.
The 0.8$\arcsec$ resolution observations show that there is a hole in the southern part of the detected dust continuum. This hole is not resolved with the 3$\arcsec$ synthesized beam of our map. That may explain why polarization is not detected at this position. Here, and also for the dust ridge sharply defined with 0.8$\arcsec$ resolution, the dust polarization is sensitive to the underlying structures and can help to identify unresolved features which are smaller than our resolution.
### Distribution of the polarization segments
The detected PAs vary enormously over the entire map, ranging from $-60\degr$ to 61$\degr$ (Fig. 3(a)). Nevertheless, they vary smoothly along the dust ridge and show organized patches. We have roughly separated the polarized emission into two different components according to their locations (as discussed in Sec. 4.2.1) and their PAs. The “o” component is probably from an extended structure with PAs ranging from 33$\degr$ to 61 $\degr$. The mean PA weighted with the observational uncertainties of component “o” is 49$\pm$3$\degr$, with a standard deviation of 11$\degr$. The “x” component associated with the *sharp dust ridge* has PAs ranging from $-60\degr$ to 4$\degr$. Its weighted mean PA is $-$24$\pm$1$\degr$, with a standard deviation of 18$\degr$. If the polarization were not separated into two components, the weighted mean PA is $-$9$\degr$ with a standard deviation of 39$\degr$.
The relation between the percentage of polarization and the intensity is shown in Fig. 3(b). The percentage of polarization decreases towards the denser regions, which has already been seen for other star formation sites, such as the ones listed in Sec. 1. This is possibly due to a decreasing alignment efficiency in high density regions, because the radiation torques are relatively ineffective (Lazarian & Hoang 2007). It can also be due to the geometrical effects, such as differences in the viewing angles (Gonçalves et al. 2005), or due to the results from averaging over a more complicated underlying field morphology.
C$^{17}$O 3-2 emission line
---------------------------
In order to trace the physical environments and the gas kinematics in G5.89, we choose to use the C$^{17}$O 3-2 emission line because of its relatively simple chemistry. The critical density of C$^{17}$O 3-2 is $\sim$ 10$^{5}$ (cm$^{-3}$), assuming a cross-section of 10$^{-16}$ (cm$^{-2}$) and a velocity of 1 km s$^{-1}$, and therefore, it will trace both the relative lower (n$_{H_2}$ $\sim$10$^{5}$ (cm$^{-3}$)) and higher (n$_{H_2}$$\sim$ 10$^{6}$ (cm$^{-3}$)) density regions. Although its critical density is much smaller than the estimated gas density of 5.3$\times$10$^{6}$ (cm$^{-3}$) from the dust continuum, it is apparently tracing the same regions as the dust continuum because of the similar morphology of the integrated intensity image, shown in the next section. We therefore assume that the kinematics traced by C$^{17}$O represents the bulk majority of the molecular cloud and that it is well correlated with the dust continuum.
### Morphology of C$^{17}$O 3-2 emission
The emission of the C$^{17}$O 3-2 line covers a large velocity range, from $-$7 to 28 km s$^{-1}$, as shown in the channel maps in Fig. 4. The majority of the gas traced by the C$^{17}$O 3-2 line is relatively quiescent and has a morphology similar to the 870$\mu$m dust continuum emission. Besides the components which trace the dust continuum, an arc feature is seen in the south-east corner of the panel covering 10 to 15 km s$^{-1}$. There is no associated 870$\mu$m dust continuum detected at this location, probably due to the low total column density or mass of this feature. Another feature seen in the more quiescent gas is the clump extending towards the south of the dust core (see the panel covering 6 to 10 km s$^{-1}$ in Fig. 4). This clump has a similar morphology as seen in the 870 $\mu$m dust continuum where no polarization has been found. At the higher velocity ends, i.e. from $-$7 to $-$3 km s$^{-1}$ and from 23 to 28 km s$^{-1}$, the emission appears at the 870$\mu$m dust ridge. This suggests that at the *sharp dust ridge*, there are high velocity components besides the majority of quiescent material. Furthermore, the brightest HII features appear correlated with the strongest C$^{17}$O emission, especially at low velocities (v$_{lsr}=$ 6 to 15 km s$^{-1}$), which may point toward an interaction between the molecular gas and the HII region.
The total integrated intensity (0th moment) image (Fig. 5(*upper-panel*)) of the C$^{17}$O 3-2 emission line shows a similar morphology as the 870 $\mu$m dust continuum. The morphology of the C$^{17}$O gas to the west of the O star is similar to the dense dust ridge, i.e. there is an extension from north to south. The steep roll off of the dust continuum in the north-west and an extension from NE to the west of the O star are also seen in C$^{17}$O. Besides these similar features to the dust continuum, a strong C$^{17}$O peak is found at position A, where no dust continuum peak is detected. This feature A likely does not have much mass, and we will not discuss its properties further in this paper.
### Total gas mass from C$^{17}$O 3-2 line
The total gas mass M$_{gas}$ in this region can be derived from the C$^{17}$O 3-2 line. This provides a complementary estimate, which is independent from the mass derived from the dust continuum in Eq. 1. Assuming that the observed C$^{17}$O 3-2 line is optically thin and in local thermal equilibrium (LTE), the mean column density $N_{C^{17}O}$ is calculated following the standard derivation of radiative transfer (see Rohlfs & Wilson 2004): $$\label{1}
N_{C^{17}O} = 1.3 \times 10^{13} \times
\frac{T_{R3-2}\triangle V}{D(n,T_{k})}$$
Here, the T$_{R3-2}\triangle$V term is the mean flux density of the entire emission region in K km s$^{-1}$. The D parameter depends on the number density $n$ and the kinetic temperature T$_{k}$ and is given by: $$\label{1}
D(n,T_{k})=f_{2}[J_{\nu}(T_{ex})-J_{\nu}(T_{bk})][1-exp(-16.597/T_{ex})],$$
where f$_2$ is the population fraction of C$^{17}$O molecules in the J$=$2 state. T$_{ex}$ and T$_{bk}$ are the excitation and background temperatures, respectively. The adopted value of D is 1.5 from the LVG calculation by Choi, Evans II $\&$ Jaffe (1993). In their calculation, this D value is correct within a factor of 2 for 10 $<$ T$_{k}$$<$ 200 K in the LTE condition. The total gas mass M$_{gas}$ is given by:
$$\label{2}
M_{gas} = \mu m_{H_{2}} d^2 \Omega \frac{N_{C^{17}O}}{X_{C^{17}O}}$$
$\mu$ is 1.3, which is a correction factor for elements heavier than hydrogen. m$_{H_2}$ is the mass of a hydrogen molecule. $d$ and $\Omega$ are the distance to the source and the solid angle of the emission, respectively. The C$^{17}$O abundance $X_{C^{17}O}$ is assumed to be [5 $\times$ 10$^{-8}$]{} (Frerking & Langer 1982; Kramer et al. 1999). The derived mean $N_{C^{17}O}$ is [2$\times$10$^{16}$ cm$^{-2}$]{}. The mean gas number density $n_{H_{2}}$ is [1.6$\times$10$^{6}$ $cm^{-3}$]{}, assuming the size of the molecular cloud is 0.13 pc along the line of sight, which is the diameter of the circle with the equivalent emission area. The derived M$_{gas}$ from the C$^{17}$O 3-2 emission is $\sim$100 M$_{\sun}$.
The gas mass calculated using the C$^{17}$O 3-2 line is a factor of 3 smaller than the value derived from the dust continuum (300 M$_{\sun}$). This difference has also been seen in the C$^{17}$O survey towards the UCHII regions by Hofner et al. (2000). Their M$_{gas}$ estimated from the measurement of the C$^{17}$O emission tends to be a factor of 2 smaller than the measurement from the dust continuum. The uncertainty of the estimate here possibly results from the assumptions of the dust emissivity, the gas to dust ratio, the abundance of the C$^{17}$O, and from the possibility that C$^{17}$O might not be entirely optically thin.
Discussion
==========
We discuss the possible reasons of the non-detected polarization in the CO 3-2 line in the next paragraph. In order to interpret our results, we have also analyzed the kinematics of the molecular cloud in G5.89 using the C$^{17}$O 3-2 1st and 2nd moment images, the position velocity (PV) diagrams, and the spectra at various positions. The strength of the B field inferred from the dust polarization is calculated using the Chandrasekhar-Fermi method. A possible scenario of the dust polarization is discussed based on the calculation of the mass to flux ratio and the energy density.
CO 3-2 polarization
-------------------
Under the presence of the B field, the molecular lines can be linearly polarized if the molecules are immersed in an anisotropic radiation field and the rate of radiative transitions is at least comparable with the rate of collisional transitions. This effect is called the Goldreich-Kylafis (G-K) effect (Goldreich & Kylafis (1981); Kylafis (1983)). The G-K effect provides a viable way to probe the B field structure of the molecular cores, because the polarization direction is either parallel or perpendicular to the B field. The degree of the polarization depends on several factors: the degree of anisotropy; the ratio of the collision rate to the radiative rate; the optical depth of the line; and the angle between the line of sight, the B field, and the axis of symmetry of the velocity field. In general, the maximum polarization occurs when the line optical depth is $\sim$ 1 (Deguchi & Watson 1984). Although the predicted polarization can be as high as 10%$-$20%, the G-K effect is only detected in a limited number of star formation sites: the molecular outflow as traced by the CO molecular lines with BIMA in the source NGC 1333 IRAS 4A (Girart & Crutcher 1999), and the outer low-density envelope in G34.4+0.23 MM (Cortes et al. 2008), G30.79 FIR 10 (Cortes & Crutcher 2006) and DR 21(OH) (Lai et al. 2003). High resolution observations are required to separate regions with different physical conditions.
We have checked the polarization in the molecular lines. No detection in the CO 3-2 and other emission lines was found. The molecular outflows as seen in the CO 3-2 and SiO 8-7 emission lines will be shown in Tang et al. (in prep.). We briefly discuss the possible reasons for the lack of polarization in the molecular lines here.
One possible reason is the high optical depth ($\tau$) of the CO 3-2 line. It has been shown by Goldreich and Kylafis (1981) that the percentage of polarization depends on the value of $\tau$, decreasing rapidly as the line becomes optically thick. When corrected for multi-level populations, Deguchi & Watson (1984) suggested that the percentage of polarization decreases further by about a factor of 2. In G5.89, $\tau$ of the CO 3$-$2 line is $\sim$10 at v$_{lsr}$ = 25 km s$^{-1}$ (Choi et al. 1995), which is the channel where the emission is strongest in our SMA observation. Note that this emission does not peak at the systematic velocity ($v_{sys}$) of 9.4 km s$^{-1}$, which is most likely due to the missing extended structure which our observation cannot reconstruct. We then estimate that the expected percentage of polarization will be about 1.5$\%$, or a polarized flux density of 0.5 Jy Beam$^{-1}$ for the CO 3-2 line, which is below our sensitivity.
Besides an optimum $\tau$, the anisotropic physical conditions, such as the velocity gradient and the density of the molecular cloud, are needed to produce a polarized component from the spectral line. The fraction and direction of polarization will also change as a function of $\tau$ if there are external radiation sources nearby (Cortes et al. 2005). Here, we are not able to distinguish between these possible reasons.
The kinematics traced by C$^{17}$O 3-2 emission line
----------------------------------------------------
As shown in Sec. 4.3.1, high velocity components of the molecular gas are traced by the C$^{17}$O 3-2 emission near the HII region. Here, we examine the kinematics in G5.89.
The intensity weighted velocity (1st moment) image provides the information on the line-of-sight motion (mean velocity). The molecular cloud is red-shifted with respect to the v$_{sys}$ of 9.4 (km s$^{-1}$) in the NW (position $B$) and SE (position $D$) of the O5 star (middle panel of Fig. 5). Next to the south of the O5 star, a blue-shifted clump with respect to 9.4 km s$^{-1}$ is detected. The molecular cloud in G5.89 has significant variations in mean velocity within a radius of 5$\arcsec$ around the O5 star.
To further investigate the relative motions, the total velocity dispersion $\delta v_{total}$ (2nd moment) image is also presented (Fig. 5 (*lower-panel*)). $\delta v_{total}$ is related to the spectral linewidth at full-width half maximum (FWHM) for a Gaussian line profile: FWHM $=$ 2.355$\delta v_{total}$. Around the HII region in G5.89, $\delta v_{total}$ has a maximum of $\sim$ 6 km s$^{-1}$ (FWHM $\sim$ 14 km s$^{-1}$) near the O5 star and decreases in the regions away from the O5 star. In terms of mean velocity and velocity dispersion, the molecular gas near the HII region is clearly disturbed. Besides the feature near the HII region, the velocity dispersion along the *sharp dust ridge* is larger and has a correspondent extension (NE$-$SW), which suggests that the molecular cloud along the *sharp dust ridge* is more turbulent (see also Sec. 4.3.1). This enhanced turbulent motion supports our separation of the polarized emission into component “o” and “x” in Sec. 4.2.2. These two polarized components are most likely tracing different physical environments.
The PV plots cut at various PAs at the position of the O5 star and cut along the extension in the NE and SW direction on the 2nd moment image (white segments on the lower panel of Fig. 5) are shown in Fig. 6. The strongest emission is at $v_{sys}$ with an extension of 18$\arcsec$, which suggests that the majority of the gas is quiescent. Besides the quiescent gas, a ring-like structure, indicated as red-dashed ellipses in Fig. 6, can be seen clearly, especially at the PA of 60$\degr$ to 100$\degr$. Both an infalling motion (e.g. Ho & Young 1996) and an expansion can produce a ring-like structure in the PV plots. In an infalling motion, the expected free-fall velocity is $\sim$ 5 km s$^{-1}$ for a central mass of 50 M$_{\sun}$ at a distance of 2$\arcsec$ from the central star. This is smaller than the value measured in the ring-like structure in G5.89. This C$^{17}$O 3-2 ring-like structure in the PV plots is therefore more likely tracing the expansion along with the HII region because of its high velocity ($\pm$10 km s$^{-1}$) and its dimension (2$\arcsec$ in radius). However, the ring structure is not complete. This may be because the material surrounding the HII region is not homogeneously distributed, or the HII region is not completely surrounded by the molecular gas.
Besides the expansion motion along with the HII region, there are higher velocity components extending up to 30 km s$^{-1}$ (red-shifted) and $-$5 km s$^{-1}$ (blue-shifted) (Fig. 6). The high velocity structure extending from the position of 2.5$\arcsec$ to the velocity of $\sim$30 km s$^{-1}$ is clearly seen in the PV cuts at PA of 0$\degr$ to 40$\degr$ (indicated as cyan arcs in Fig. 6). These high velocity components are probably due to the sweeping motion of the molecular outflows in G5.89, because there is no other likely energy source which can move the material to such a high velocity. From the PV plots at the position of the O5 star at various PAs and the PV plot cut along the *sharp dust ridge*, we conclude that the molecular cloud is most likely both expanding along with the HII region and being swept-up by the molecular outflows, all in addition to the bulk of the quiescent gas.
The examination of the spectra at various positions also helps to analyze the kinematics in G5.89. The spectra (Fig. 7) near the HII ring (positions $C$, $D$, $F$, $G$ and $H$) have broad line-widths. Furthermore, the spectra are not Gaussian-like, or with distinct components at high velocities ($\pm$ 10 km s$^{-1}$). At position $F$ and $H$, both spectra show a strong peak at v$_{sys}$. The high velocity wing at the position $F$ is red-shifted, and it is blue-shifted at position H. This is consistent with the NS molecular outflow. The molecular gas near the positions $E$ and $I$ is more quiescent because of its narrow linewidth. The spectrum taken at position $I$ has a FWHM of 4 km s$^{-1}$ and a peak intensity at $\sim$ 7 km s$^{-1}$. Comparing with the spectra at other positions, the cloud around the position $I$ is relatively quiescent and unaffected by the HII region or the outflows. This cloud in the south near the position $I$ may be a more independent component which is further separated along the line of sight. We conclude that the C$^{17}$O 3-2 spectra demonstrate that the kinematics and morphology have been strongly affected by the expansion of the HII region. The nearly circular structure in the PV plots, and in the channel maps near the systemic velocity, as well as the spectra, suggest that a significant part of the mass has been pushed by the HII region. An impact from the molecular outflow can also be seen in the PV plots and spectra.
Estimate of the B field strength
--------------------------------
The B field strength projected in the plane of sky (B$_{\bot}$) can be estimated by means of the Chandrasekhar-Fermi (CF) method (Chandrasekhar & Fermi 1953; Falceta-Gonçalves, Lazarian, & Kowal 2008). In general, the CF method can be applied to both dust and line polarization measurements. We apply the CF method only to the dust continuum polarization, because there is no line polarization detected in this paper. Although the dust grains can also be mechanically aligned, the existing observational evidence in NGC 1333 IRAS4A (Girart et al. 2006) demonstrates that the dust grains can align with the B field in the low mass star formation regions. Here, we assume that the dust grains also align with the B field in G5.89.
The strength of B$_{\bot}$ can be calculated from: $$\label{1}
B_{\bot} = Q \sqrt{4\pi\bar{\rho}}\frac{\delta v_{los, A}}{\delta\phi}
= 63 \sqrt{n_{H_2}}\frac{\delta v_{los, A}}{\delta \phi}$$
Here, B$_{\bot}$ is in the unit of mG. The term Q is a dimensionless parameter smaller than 1. Q is $\sim$0.5 (Ostriker, Stone & Gammie 2001), depending on the inhomogeneities within the cloud, the anisotropies of the velocity perturbations, the observational resolution and the differential averaging along the line of sight. The term $\bar{\rho}$ is the mean density. $\delta \phi$ is the dispersion of the polarization angles in units of degree. $\delta
v_{los, A}$ is the velocity dispersion along the line of sight in units of km s$^{-1}$, which is associated with the Alfvénic motion. $n_{H{2}}$ is the number density of H$_{2}$ molecules in units of 10$^{7}$ cm$^{-3}$. It has been shown numerically that the CF method is a good approximation for $\delta \phi <
25^{\degr}$ (Ostriker, Stone, & Gammie (2001)).
$\delta v_{los, A}$ is estimated from $\delta v_{total}$ in the 2nd moment image (lower panel in Fig. 5). $\delta v_{total}$ contains the information of the dispersions caused by the Alfvénic turbulent motion ($\delta \textit{v}_{los, A}$) and the dispersions caused by the HII expansion and outflow motions ($\delta \textit{v}_{bulk}$). The relation of these three components is:
$$\label{1}
\delta v_{total} \sim \sqrt{\delta v_{los, A}^{2}+\delta v_{bulk}^{2}}$$
Here, we neglect the minor contributions from the thermal Doppler motions. The measured $\delta v_{total}$ at the positions of detected polarization are listed in Table 1. $\delta v_{total}$ is in the range of 1 to 6 km s$^{-1}$. However, the molecular gas near the HII region is clearly disturbed by both the HII expansion and the molecular outflows (see Sec. 4.3.1 and 5.2). Therefore, the detected $\delta v_{total}$ at these positions is dominated by the bulk motion. Since $\delta v_{total}$ in the relatively quiescent regions is more likely tracing the Alfvénic motion only, we adopt the minimum value $\delta v_{total}$ of 1 km s$^{-1}$ at these positions for $\delta v_{los, A}$ in order to derive B$_{\bot}$.
The term $n_{H_{2}}$ is $\sim$3$\times$10$^{6}$ (cm$^{-3}$), estimated from the averaged $n_{H_{2}}$ from the 870$\mu$m dust continuum and the C$^{17}$O 3-2 line emission (Sec. 4.1.2 and Sec. 4.3.2). $\delta\phi$ in Eq. 4 can be extracted from the observed standard deviation of the PAs $\delta \phi_{obs}$. $\delta
\phi_{obs}$ contains both the observational uncertainty $\sigma_{\phi,obs}$ and $\delta \phi$. The relation is: $\delta\phi_{obs}^{2}$ = $\delta\phi^{2} + \sigma_{\phi,obs}^{2}$. Since the polarization in G5.89 results probably from two different systems (discussed in Sec. 4.2.1 and 4.2.2), it is more reasonable to separate these two groups when deriving $\delta\phi$. The derived $\sigma_{\phi,obs}$, $\delta\phi_{obs}$ and $\delta\phi$ are 3$\degr$ and 11$\degr$ and $\sim$11$\degr$ for component “o”, and 2$\degr$, 18$\degr$ and $\sim$18$\degr$ for component “x”, respectively. By using Eq. (4), the derived B$_{\bot}$ is 3mG and 2mG for component “o” and “x”, respectively.
The estimated B$_{\bot}$ is highly uncertain. Due to the bulk motions, it is difficult to extract the $\delta v_{los, A}$ component from the observed $\delta v_{total}$. The uncertainty introduced from $\delta v_{los, A}$ is within a factor of 6. Of course, the grouping of “o” and “x” components of the polarization, as motivated in Sec. 4.2.1, 4.2.2 and 5.2, is not a unique interpretation. If $\delta \phi$ is calculated without grouping, a more complex model of the larger scale B field morphology is needed to calculate the deviation due to the Alfvénic motion. More observations with sufficient uv coverage are required to establish such a model. Based on the standard deviation $\delta \phi$ of 39$\degr$ from the detected polarization without subtracting the larger scale B field and without grouping, the calculated lower limit of B$_{\bot}$ is $\sim$1mG. Therefore, the estimated B$_{\bot}$ from the grouping of component “o” and “x” seems reasonable. The value is comparable to the ones estimated via the CF method in other massive star formation regions with an angular resolution of a few arcseconds: $\sim$ 1mG in DR 21(OH) (Lai et al. 2003) and $\sim$1.7 mG in G30.79 FIR 10 (Cortes & Crutcher 2006). Moreover, B$_{\bot}$ is similar to B$_{\parallel}$ measured from the Zeeman pairs of the OH masers by Stark et al. (2007), ranging from $-$2 to 2 mG. Although B$_{\parallel}$ measured from OH masers is most likely tracing special physical conditions, such as shocks or dense regions, it is the only direct measurement of B$_{\parallel}$ in G5.89, and hence, is of interest to compare. Assuming B$_{\bot}$ and B$_{\parallel}$ have the same strengths of 2mG, the total B field strength in G5.89 is $\sim$ 3 mG.
Collapsing cloud or not?
------------------------
The mass to flux ratio $\lambda$, a crucial parameter for the magnetic support/ambipolar diffusion model, can be calculated from: $\lambda$ = 7.6 $\times$ 10$^{-21}$ $\frac{N_{H_{2}}}{B}$ (Mouschovias & Spitzer 1976; Nakano & Nakamura 1978). $N_{H_2}$ is in cm$^{-2}$. $B$ is in $\mu$G. In the case of $\lambda$ $<$ 1, the cloud is in a subcritical stage and magnetically supported. In the case of $\lambda$ $>$ 1, the cloud is in a collapsing stage.
Since there is no observation of the B field strength as a function of position in the entire cloud, we assume that the B field is uniform with the strength of 3 mG in the entire cloud when $\lambda$ is calculated. For consistency, when comparing with the kinetic pressure in the next section, $N_{H_{2}}$ is derived from the C$^{17}$O 3-2 emission (section 4.3.2). The derived $\lambda$ in G5.89 is $>$1 in most parts of the molecular cloud, as shown in the upper panel of Fig. 8. If the statistical geometrical correction factor of $\frac{1}{3}$ is considered (Crutcher 2004), the $\lambda_{corr}$ in the *sharp dust ridge* is still close to 1, whereas at the positions of the component “o” and the outer part, it is much smaller than 1. This suggests that G5.89 is probably in a supercritical phase near the HII region and in a subcritical phase in the outer part of the dust core.
This conclusion is based on the assumption that the B field in the entire cloud is uniform with a strength of 3 mG. This assumption seems to be crucial at first glance. However, the derived $\lambda$ increases from 0.1 to 2.5 toward the UCHII region, which is due to the high contrast of the column density across the cloud. Unless the actual B field strength differs by a factor of 25 across the region and compensates for the contrast in the column density, such a variation of $\lambda$ in G5.89 is indeed possible.
Compressed field?
-----------------
The coincident location of the detected polarization of component “x” and the *sharp dust ridge* is quite interesting. One possible scenario is that the B field lines are compressed by the shock front, i.e. HII expansion. In a magnetized large molecular cloud with a B field traced by a component “o”, and with a shock sent out from the east of the narrow dust ridge, we expect a rapid change of the polarization PA. This is similar to the results in magnetohydrodynamic simulations by Krumholz et al. (2007). Because of our limited angular resolution, polarization with a large dispersion in the PAs over a small physical scale will be averaged out within the synthesized beam. In our result, in fact, there is a gap where polarized emission is not detected right next to the *sharp dust ridge*, and a series of OH masers are detected in this gap. Note that the OH masers are most likely from the shock front. From the discussion in Sec. 5.2, evidence for the molecular cloud expanding with the HII region is found in the molecular gas traced by the C$^{17}$O 3-2 emission. The 870$\mu$m *sharp dust ridge* can be explained by the swept-up material along with the molecular gas from the HII expansion. In this scenario, the component “o” is tracing the B field in the pre-shock region, while the “x” component is tracing the compressed field.
However, the swept-up flux density (summation of the flux density of SMA-N, SMA-1 and SMA-2 reported in Hunter et al. 2008) is $\sim$ 20% of the total detected flux density in this paper. This requires a huge amount of energy to sweep up the material with this mass. Is the radiation pressure (P$_{rad}$) from the central star large enough to overcome the kinetic pressure (P$_{kin}$) and the B field pressure (P$_{B}$)? Here we compare these pressure terms.
P$_{rad}$ can be calculated from the luminosity in G5.89 following the equation: P$_{rad}$ $=$ $\frac{L}{cA}$, where $L$, $c$ and A are the luminosity, speed of light and the area, respectively. Since the G5.89 region is dense, most of the radiation is absorbed and redistributed into the surrounding material. The total far infrared luminosity of G5.89 is 3$\times$10$^{5}$L$_{\sun}$ (Emerson, Jennings, & Moorwood 1973) and the radius of the HII region at 2cm is $\sim$ 2$\arcsec$ (4000 AU). The energy density and hence, the radiation pressure (P$_{rad}$) in the sphere with a radius of 2$\arcsec$ is 8.5$\times$10$^{-7}$ (dyne cm$^{-2}$).
P$_{kin}$ is calculated by using the 0th moment ($MOM0$) and 2nd moment ($MOM2$) images of the C$^{17}$O 3-2 line: $$\label{1}
P_{kin} = \frac{1}{2} \rho \delta v_{total}^2 = 3.4 \times10^{-9} \times (MOM0) \times
(MOM2)^2$$
where P$_{kin}$ is in dyne cm$^{-2}$, $\rho$ is the gas density in g cm$^{-3}$ and $\delta v_{total}$ is the velocity dispersion in cm s$^{-1}$, $MOM0$ is in units of K km s$^{-1}$, and $MOM2$ is in units of km s$^{-1}$. $\rho$ is calculated following Sec. 4.3.2, with the size of 0.13 pc for the molecular cloud along the line of sight. The derived P$_{kin}$ image is shown in the middle and lower panels of Fig. 8. P$_{kin}$ is in the range of $\sim$ 1$\times$10$^{-9}$ to 1.4$\times$10$^{-6}$ dyne cm$^{-2}$. P$_{kin}$ is calculated under the assumption that the length along the line of sight is uniform in G5.89, which is the main bias in the calculation. The estimated total B field strength is 3 mG, thus the B-field pressure (P$_B$) is 3.6$\times$10$^{-7}$ (dyne cm$^{-2}$). Although the upper limit of P$_{kin}$ is 1.6 times larger than P$_{rad}$ at a radius of 2$\arcsec$, any variation of the structure in the direction along the line of sight in G5.89 - which is most likely the case - will affect the estimated P$_{kin}$. Nevertheless, P$_{rad}$ is at the same order as P$_{kin}$ and P$_{B}$ at the radius of 2$\arcsec$ around the O5 star. Therefore, in terms of pressure, the radiation from the central star is likely sufficient to sweep up the material and compress the B field lines along the narrow dust ridge. $\lambda_{corr}$ close to 1 near the UCHII region suggests that the B fields play a minor role as compared with the gravity.
The B field direction traced by component “x” is parallel to the major axis of this sharp dust ridge, which is also seen in some other star formation sites such as Cepheus A (Curran $\&$ Chrysostomou 2007) and DR 21(OH) (Lai et al. 2003). However, in most of the cases, the detected B field direction is parallel to the minor axis of the dust ridge, e.g. W51 e1/e2 cores (Lai et al. 2001) and G34.4+0.23 MM (Cortes et al. 2008), which agrees with the ambipolar diffusion model. A possible explanation is that the polarization is from the swept-up material, which interacts with the original dense filament. Thus, the polarization here may represent the swept-up field lines. This scenario is supported by the energy density and also the morphology of the field lines in the case of G5.89.
Comparison with Other Star Formation Sites
------------------------------------------
The detected B field structure in the G5.89 region is more complicated than the B fields in other massive star formation sites detected so far with interferometers. Both the compressed field structure and the more organized larger scale B field are detected in G5.89.
The B field lines vary smoothly in the cores at earlier star formation stages, such as the W51 e1/e2 cores (Lai et al. 2001), G34.4+0.23 MM (Cortes et al. 2008) and NGC 2024 FIR 5 (Lai et al. 2002). These cores are still in a collapsing stage (Ho et al. 1996 ; Ramesh et al. 1997; Mezger et al. 2001). Among these observations, the B fields inferred from the dust polarization show an organized structure over the scale of 10$^{5}$ AU (15$\arcsec$) at a distance of 7 kpc in the W51 e1/e2 cores and also on the scale of 10$^{5}$ AU (35$\arcsec$) at a distance of 3.9 kpc in G34.4+0.23 MM. In contrast, the B fields in NGC 2024 FIR 5, which is closer at a distance of 415 pc, show an hourglass morphology on a scale of 4$\times$10$^{3}$ AU (9$\arcsec$). Such small scale structures would not be resolved in the current data of W51 e1/e2 core and G34.4+0.23 MM due to the resolution effect. Compared to these sources, G5.89 is more complicated with polarization structures on both small (4$\times$10$^{3}$ AU) and large (2$\times$10$^{4}$ AU) scales. However, higher angular resolution polarization images of the cores in the earlier stages are necessary in order to compare the B field morphology with the later stages in the massive star formation process. At this moment, we cannot conclude at which stage the B field structures become more complex.
Currently, the best observational evidence supporting the theoretical accretion model is the polarization observation of the source NGC 1333 IRAS 4A (Girart, Rao & Marrone 2006) carried out with the SMA. The NGC 1333 IRAS 4A is a low mass star formation site, at a distance of $\sim$300 pc. The detected pinched B-field structure is at a scale of 2400 AU (8$\arcsec$). If NGC 1333 IRAS 4A were at a distance of 2kpc, we could barely resolve it at our resolution of 3$\arcsec$. Higher angular resolution polarization measurements are required to resolve the underlying structure in G5.89.
Conclusions and Summary
=======================
High angular resolution (3$\arcsec$) studies at 870$\mu$m have been made of the magnetic (B) field structures, the dust continuum structures, and the kinematics of the molecular cloud around the Ultra-Compact HII region G5.89-0.39. The goal is to analyze the role of the B field in the massive star forming process. Here is the summary of our results:
1. The gas mass (M$_{gas}$) is estimated from the dust continuum and from the C$^{17}$O 3$-$2 emission line. The continuum emission at 870 $\mu$m is detected with its total flux density of 12.6$\pm$1.3 Jy. After removing the free-free emission from the detected continuum, the flux density of the 870 $\mu$m dust continuum is 7.7 Jy, which corresponds to M$_{gas}$ $\sim$300 M$_{\sun}$. M$_{gas}$ derived from the detected C$^{17}$O 3-2 emission line is $\sim$100 M$_{\sun}$, which is 3 times smaller than the value derived from the dust continuum. The discrepancy of M$_{gas}$ derived from the dust continuum and the C$^{17}$O emission line is also seen in other UCHII regions, e.g. Hofner et al. (2000). The lower values measured from C$^{17}$O could be due to optical depth effects or abundance problems.
2. The linearly polarized 870 $\mu$m dust continuum emission is detected and resolved. The dust polarization is not uniformly distributed in the entire dust core. Most of the polarized emission is located around the HII ring, and there is no polarization detected in the southern half of the dust core except at the very southern edges. The position angles (PAs) of the polarization vary enormously but smoothly in a region of 2$\times$10$^{4}$ AU (10$\arcsec$), ranging from $-$60$\degr$ to 61$\degr$. Furthermore, the polarized emission is from organized patches, and the distribution of the PAs can be separated into two groups. We suggest that the polarization in G5.89 traces two different components. The polarization group “x”, with its PAs ranging from $-$60$\degr$ to $-$4$\degr$, is located at the 870$\mu$m *sharp dust ridge*. In contrast, the group “o”, with its PAs ranging from 33$\degr$ to 61$\degr$, is at the periphery of the *sharp dust ridge*. The inferred B field direction from group “x” is parallel to the major axis of the 870$\mu$m dust ridge. One possible interpretation of the polarization in group “x” is that it may represent swept-up B field lines, while the group “o” traces more extended structures. In the G5.89 region, both the large scale B field (group “o”) and the compressed B field (group “x”) are detected.
3. By using the Chandrasekhar-Fermi method, the estimated strength of B$_{\bot}$ from component “o” and from component “x” is in between 2 to 3 mG, which is comparable to the Zeeman splitting measurements of B$_{\parallel}$ from the OH masers, ranging from $-$2 to 2 mG by Stark et al. (2007). The derived lower limit of B$_{\bot}$ from the detected polarization without grouping and without modeling larger scale B field is $\sim$1mG. Assuming that B$_{\bot}$ and B$_{\parallel}$ have the same strengths of 2 mG in the entire cloud, the derived $\lambda$ increases from 0.1 to 2.5 toward the UCHII region, which is due to the high contrast of the column density across the cloud. Unless the actual B field strength differs by a factor of 25 across the region and compensates for the contrast in the column density, such a variation of $\lambda$ in G5.89 is suggested. The corrected mass to flux ratio ($\lambda_{corr}$) is closer to 1 near the HII region and is much smaller than 1 in the outer parts of the dust core. G5.89 is therefore most likely in a supercritical phase near the HII region.
4. The kinematics of the molecular gas is analyzed using the C$^{17}$O 3-2 emission line. From the analysis of the channel maps, the position velocity plots and spectra, the molecular gas in the G5.89 region is expanding along with the HII region, and it is also possibly swept-up by the molecular outflows. Assuming the size along the line of sight is uniform in G5.89, P$_{kin}$ is in the range of $\sim$ 1$\times$10$^{-9}$ to 1.4$\times$10$^{-6}$ dyne cm$^{-2}$. The calculated radiation pressure (P$_{rad}$) at a radius of 2$\arcsec$ and the B field pressure (P$_B$) with a field strength of 3mG are 8.5$\times$10$^{-7}$ and 3.6$\times$10$^{-7}$ dyne cm$^{-2}$, respectively. Although the upper limit of P$_{kin}$ is 1.6 times larger than P$_{rad}$ at a radius of 2$\arcsec$, any variation of the structure in the direction along the line of sight in G5.89 - which is most likely the case - will affect the estimated P$_{kin}$. Nevertheless, P$_{rad}$ is on the same order as P$_{kin}$ and P$_{B}$ at the radius of 2$\arcsec$ around the O5 star. The scenario that the matter and B field in the 870$\mu$m *sharp dust ridge* have been swept-up is supported in terms of the available pressure.
G5.89 is in a more evolved stage as compared with the corresponding structures of other sources in the collapsing phase. The morphologies of the B field in the earlier stages of the evolution show systematic or smoothly varying structures, e.g. on the scale of 10$^{5}$ AU for W51 e1/e2 and G34.4+0.23 MM, and on the scale of 4$\times$10$^{3}$ AU for NGC 2024 FIR 5. With the high resolution and high sensitivity SMA data, we find that the B field morphology in G5.89 is more complicated, being clearly disturbed by the expansion of the HII region and the molecular outflows. The large scale B field structure on the scale of 2$\times$10$^{4}$ AU in G5.89 can still be traced with dust polarization. From the analysis of the C$^{17}$O 3-2 kinematics and the comparison of the available energy density (pressure), we propose that the B fields have been swept up and compressed. Hence, the role of the B field evolves with the formation of the massive star. The ensuing luminosity, pressure and outflows overwhelm the existing B field structure.
Acord, J. M., Churchwell, E., & Wood, D. O. S. 1998, ApJ, 495, 107 Cesaroni, R., Walmsley, C. M., Koempe, C., & Churchwell, E. 1991, A&A, 252, 278
Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113 Choi, M., Evans II, N., & Jaffe, D. T. 1993, ApJ, 417, 624
Churchwell 1997, ApJ, 479, L59
Cortes, P. C., Crutcher, R. M., & Watson, W. D. 2005, 628, 780 Cortes, P., & Crutcher, R. M. 2006, ApJ, 639, 965
Cortes, P., Crutcher, R. M., & Matthews, B. 2006, ApJ, 650, 246
Cortes, P., Crutcher, R. M., Shepherd, D. S., & Bronfman, L. 2008, ApJ, 676, 464
Crutcher, R. M. 2004, ApSS, 292, 225
Curran, R. L. & Chrysostomou, A. 2007, MNRAS, 382, 699
Deguchi, S. & Watson, W. 1984, ApJ, 285, 126 Draine, & Weingartner 1996, ApJ, 470, 551
Elmegreen, B. G., & Scalo, J. 2004, ARA&A, 42, 211
Emerson, J. P., Jennings, R. E., & Moorwood, A. F. M. 1973, ApJ, 184, 401
Falceta-Gonçalves, D., Lazarian, A., & Kowal, G. 2008, ApJ, 679, 537
Feldt, M., Stecklum, B., Henning, Th., Launhardt, R., & Hayward, T. L. 1999, A&A, 346, 243
Feldt, M., Puga, E., Lenzen, R., Henning, Th., Brandner, W., Stecklum, B., Lagrange, A.-M., Gendron, E., & Rousset, G. 2003, ApJ, 599, L91
Fiedler, R. A., & Mouschovias, T. Ch. 1993, ApJ, 415, 680
Fish, V. L, Reid, M. J., Argon, A. L., & Zheng, X.-W. 2005, ApJS, 160, 220
Frerking, M. A., & Langer, D. L., & Wilson, W. W. 1982, ApJ, 262, 590
Galli, D., & Shu, F. H. 1993, ApJ, 417, 243
Girart, J. M., Crutcher, R. M. & Rao, R. 1999, ApJ, 525, L109
Girart, J. M., Rao, R., & Marrone, D. P. 2006, Sci, 313, 812 Goldreich, P., & Kylafis, N. D. 1981, ApJ, 243, 75 Gonçalves, J., Galli, D., & Walmsley, M. 2005, A&A, 430, 979
Harvey, P. M., & Forveille, T. 1988, A&A, 197, L19 Ho, P. T. P., Moran, J. M., & Lo, K. Y. 2004, ApJ, 616, 1
Ho, P. T. P., & Young, L. M. 1996, ApJ, 472, 742
Hofner, P., & Churchwell 1996, A&AS, 120, 283
Hofner, P., Wyrowski, F., Walmsley, C. M., & Churchwell, E. 2000, ApJ, 536, 393
Hunter, T. R., Churchwell, E., Watson, C., Cox, P., Benford, D. J., & Roelfsema, P. R. 2000, AJ, 119, 2711 Hunter, T. R., Brogan, C. L., Indebetouw, R., & Cyganowski, C. J. 2008, 680, 127
Kramer, C., Alves, J., Lada, C., Lada, E., Sievers, A., Ungerechts, H., & Walmsley, M. 1999, A&A, 342, 257
Krumholz, M., Stone, J. M., & Gardiner, T. A. 2007, ApJ, 671, 518 Kurtz, S., Hofner, P., & Alvarez, C. V. 2004, ApJS, 155, 149
Kylafis, N. D. 1983, ApJ, 267, 137
Lai, S.-P., Crutcher, R. M., Girart, J. M., & Rao, R. 2001, ApJ, 561, 864
Lai, S.-P., Crutcher, R. M., Girart, J. M., & Rao, R. 2002, ApJ, 566, 925 Lai, S.-P., Girart, J. M., & Crutcher, R. M. 2003, ApJ, 598, 392 Lazarian, A. 2007, Journal of Quantitative Spectroscopy & Radiative Transfer, 106, 255
Lazarian, A. & Hoang, T. 2007, MNRAS, 378, 910
Lis, D. C., Serabyn, E., Keene, Jocelyn, Dowell, C. D., Benford, D. J., Phillips, T. G., Hunter, T. R., Wang, N. 1998, 509, 299 Mac Low
Mac Low, M.-M., & Klessen, R. S. 2004, Rev. Mod. Phys., 76, 125
Marrone, D. & Rao, R. 2008, arXiv:0807.2255
Mezger, P. G., Sievers, A. W., Haslam, C. G. T., Kreysa, E., Lemke, R., Mauersberger, R., & Wilson, T. L. 1992, A&A, 256, 631
Mouschovias, T. Ch. 1976, ApJ, 207, 141
Mouschovias, T. Ch., & Spitzer, L. 1976, ApJ, 210, 326
Mouschovias, T. Ch. & Ciolek, G. E. 1999, in The Origin of Stars and Planetary Systems, ed. C. J. Lada & N. D. Kylafis (Kluwer: Dordrecht), p. 305
Nakano, T., & Nakamura, T. 1978, PASJ, 30, 681
Ostriker, E. C., Stone, J. M., & Gammie, C. F. 2001, ApJ, 546, 980
Puga, E., Feldt, M., Alvarez, C., Henning, Th., Apai, D., Coarer, E. Le, Chalabaev, A., & Stecklum, B. 2006, ApJ, 641, 373
Ramesh, B., Bronfman, L., & Deguchi, S. 1997, PASJ, 49, 307
Rao, R., Crutcher, R. M., Plambeck, R. L., & Wright, M. C. H. 1998, ApJ, 502, L75 Rohlfs, K., & Wilson, T. L. 2004, Tools of Raido Astronomy (4th ed; Berlin: Springer) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, in ASP Conf. Ser. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes (San Francisco: ASP), 433
Sault, R. J., Hamaker, J. P., & Bregman, J. D. 1996, A&AS, 117, 149
Shu, F., Allen, A., Shang, H., Ostriker, E. C., & Li, Z.-Y. 1999, in The Origin of Stars and Planetary Systems, ed. Charles J. Lada & Nikolaos D. Kylafis, (Kluwer: Dordrecht), p. 193
Sollins, P. K., Hunter, T. R., Battat, J., Beuther, H., Ho, P. T. P., Lim, J., Liu, S. Y., Ohashi, N., Sridharan, T. K., Su, Y. N., Zhao, J.-H., & Zhang, Q. 2004, ApJ, 616, 35
Stark, D. P., Goss, W. M., Churchwell, E. Fish, V. L., & Hoffman, I. M. 2007, ApJ, 656, 943
Watson, C., Churchwell, E., Zweibel, E. G., & Crutcher, R. M. 2007, ApJ, 657, 318
Wood, D. O. S., & Churchwell, E. 1989, ApJS, 69, 831
Zijlstra, A. A., Pottasch, S. R., Engels, D., Roelfsema, P. R., Hekkert, P. T., & Umana, G. 1990, MNRAS, 246, 217
![*(a)* The SMA 870 $\mu$m total continuum image (contours) overlayed on the VLA 2cm continuum (free-free continuum; color scale). The white contours represent the continuum emission strength at 3, 5, 10, 15, 20, 25 ... 60 and 65 $\sigma$ levels, and the black contours in the center represent 70, 80, 90, 100 and 110 $\sigma$ levels, where [$1\sigma$ is 30 mJy Beam$^{-1}$]{}. The star marks the O star detected by Feldt et al. (2003). The asterisk marks the origin of the Br$\gamma$ outflow detected by Puga et al. (2006). The SMA and VLA synthesized beams are shown as white and black ellipses at the lower-left corner, respectively. The white “+” mark the positions of the sub-mm peaks identified in Hunter et al. (2008). *(b)* The same as in (a), with the white contours representing the SMA 870 $\mu$m dust continuum (after the subtraction of the free-free continuum). The contours start from and step in $3\sigma$, where [$1\sigma$ is 30 mJy Beam$^{-1}$]{}. The color wedge on the upper-right edge represents the strength of the 2cm free-free continuum in the units of Jy Beam $^{-1}$. The red and blue arrows indicate the axes of the molecular outflows. The outflows in the N-S and NW-SW direction in the west of O5 star are identified in Hunter et al. (2008). The 3rd outflow in the east of O5 star is identified in Tang et al. (in prep.). *(c)* The polarization (red segments) derived by using image (a). The length of the red segment represents the percentage of the polarized intensity. The 870 $\mu$m continuum is shown both in white contours with the steps as in Fig. (a) and in the color scale. *(d)* The polarization (red segments) derived by using image (b). The 870 $\mu$m dust continuum is again shown both in white contours with the steps as in Fig. (b) and in the color scale. The color wedge at the lower-right edge shows the strength of the dust continuum in the units of Jy Beam$^{-1}$. In *(c)* and *(d)*, the polarization plotted is above 3$\sigma_{I_p}$. []{data-label=""}](f1.eps)
![(a) The 870$\mu$m dust continuum (white and grey contours) overlaid on the polarized intensity (I$_{P}$) image (color scale). The contours plotted are the same as in Fig. 1(b). The color wedge shows the strength of polarization intensity in units of mJy Beam$^{-1}$. The smallest white open circles and plus signs mark the positions of the Zeeman pairs of the OH maser (Stark et al. 2007) with different polarimetries. The other symbols are the same as in Fig. 1. The larger solid white circles mark component “o”, defined in Sec. 4.2.2. (b) The polarization (red and blue segments) overlaid on the 870 $\mu$m dust continuum (black contours) and the 2cm free-free continuum emission (color scale). In red and blue segments are polarization segments above 3$\sigma_{I_p}$ and between 2 to 3$\sigma_{I_p}$, respectively. The contours, star, asterisk, circles and plus signs are all the same as in (a). The color wedge shows the strength of the 2cm free-free emission in units of Jy Beam$^{-1}$. The ellipses in the lower-left corner are the synthesized beams of this paper, shown in black, and of the 2cm free-free continuum image, shown in white. (c) The inferred B field (red segments) overlaid on the 870 $\mu$m dust continuum (blue contours) in this paper and in Hunter et al. (2008) (grey scale). The ellipses in the lower-left corner are the synthesized beams of this work, shown in black, and of Hunter et al. (2008), shown in white. The white crosses mark the sub-mm sources detected by Hunter et al. (2008). The triangles mark the positions of the H$_{2}$ knots identified in Puga et al. (2006).[]{data-label="pol"}](f2_a "fig:") ![(a) The 870$\mu$m dust continuum (white and grey contours) overlaid on the polarized intensity (I$_{P}$) image (color scale). The contours plotted are the same as in Fig. 1(b). The color wedge shows the strength of polarization intensity in units of mJy Beam$^{-1}$. The smallest white open circles and plus signs mark the positions of the Zeeman pairs of the OH maser (Stark et al. 2007) with different polarimetries. The other symbols are the same as in Fig. 1. The larger solid white circles mark component “o”, defined in Sec. 4.2.2. (b) The polarization (red and blue segments) overlaid on the 870 $\mu$m dust continuum (black contours) and the 2cm free-free continuum emission (color scale). In red and blue segments are polarization segments above 3$\sigma_{I_p}$ and between 2 to 3$\sigma_{I_p}$, respectively. The contours, star, asterisk, circles and plus signs are all the same as in (a). The color wedge shows the strength of the 2cm free-free emission in units of Jy Beam$^{-1}$. The ellipses in the lower-left corner are the synthesized beams of this paper, shown in black, and of the 2cm free-free continuum image, shown in white. (c) The inferred B field (red segments) overlaid on the 870 $\mu$m dust continuum (blue contours) in this paper and in Hunter et al. (2008) (grey scale). The ellipses in the lower-left corner are the synthesized beams of this work, shown in black, and of Hunter et al. (2008), shown in white. The white crosses mark the sub-mm sources detected by Hunter et al. (2008). The triangles mark the positions of the H$_{2}$ knots identified in Puga et al. (2006).[]{data-label="pol"}](f2_b "fig:")
![(a) Distribution of the PAs (defined in the range $-$90$\degr$ to 90$\degr$) of the polarization in G5.89. (b) Total intensity (I) versus percentage (%) of the polarized flux density. In both (a) and (b) panels, the statistics are from the detected polarization segments above 3 $\sigma_{I_p}$ confidence level. The “cross” represents the component “x”, which is associated with the sharp dust ridge. The “circle” represents the component “o”, which is associated with the extended structure.[]{data-label="vec_dist"}](f3.ps)
[^1]: The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.
|
---
abstract: '[*Phenotyping electronic health records (EHR)*]{} focuses on defining meaningful patient groups (e.g., heart failure group and diabetes group) and identifying the temporal evolution of patients in those groups. Tensor factorization has been an effective tool for phenotyping. Most of the existing works assume either a static patient representation with aggregate data or only model temporal data. However, real EHR data contain both temporal (e.g., longitudinal clinical visits) and static information (e.g., patient demographics), which are difficult to model simultaneously. In this paper, we propose [*T*]{}emporal [*A*]{}nd [*S*]{}tatic [*TE*]{}nsor factorization ([`TASTE`]{}) that jointly models both static and temporal information to extract phenotypes. [`TASTE`]{}combines the PARAFAC2 model with non-negative matrix factorization to model a temporal and a static tensor. To fit the proposed model, we transform the original problem into simpler ones which are optimally solved in an alternating fashion. For each of the sub-problems, our proposed mathematical re-formulations lead to efficient sub-problem solvers. Comprehensive experiments on large EHR data from a heart failure (HF) study confirmed that [`TASTE`]{}is up to $14 \times$ faster than several baselines and the resulting phenotypes were confirmed to be clinically meaningful by a cardiologist. Using 80 phenotypes extracted by [`TASTE`]{}, a simple logistic regression can achieve the same level of area under the curve (AUC) for HF prediction compared to a deep learning model using recurrent neural networks (RNN) with 345 features.'
author:
- |
Ardavan Afshar$^1$, Ioakeim Perros$^2$, Haesun Park$^1$\
Christopher deFilippi$^3$, Xiaowei Yan$^4$, Walter Stewart, Joyce Ho$^5$, Jimeng Sun$^1$\
$^1$Georgia Institute of Technology, ,$^2$Health at Scale $^3$INOVA,\
$^4$Sutter Health, $^5$Emory University\
bibliography:
- 'main.bib'
title: '[`TASTE`]{}: Temporal and Static Tensor Factorization for Phenotyping Electronic Health Records'
---
|
---
abstract: |
We show that the sheaves of algebras of generalized functions $\Omega\to \mathcal{G}(\Omega)$ and $\Omega\to
\mathcal{G}^{\infty}(\Omega)$, $\Omega$ are open sets in a manifold $X$, are supple, contrary to the non-suppleness of the sheaf of distributions.
address:
- |
Department of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Serbia\
Tel.: +381-21-4852860, Fax: +381-21-6350458
- |
Department of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Serbia\
Tel.: +381-21-4852790, Fax: +381-21-6350458
author:
- Stevan Pilipović
- Milica Žigić
title: Suppleness of the sheaf of algebras of generalized functions on manifolds
---
Introduction and definitions {#sec:intro}
============================
The aim of this paper is to give a complete answer to the question concerning suppleness of sheaves of certain generalized function algebras. This question is discussed in [@msd] and here it is completely solved. Note that Bros and Iagolnitzer [@bros] conjectured that the analytic singular support (analytic wavefront set) for distributions is decomposable. Bengel and Schapira [@ben] have studied this decomposition by considering Cousin’s problem with bounds in a tuboid. In [@eida] authors have studied microlocal decomposition for ultradistributions and ultradifferentiable functions. They used the Laubin decomposition of delta distribution [@laub] for the proof in this setting. We consider in this paper the algebra $\mathcal{G}$ of generalized functions containing the Schwartz distributions space $\mathcal{D}'$ as a subspace so that all the linear operations on $\mathcal{D}'$ are preserved within $\mathcal{G}$. We refer to [@bia], [@col], [@co1], [@gfks], [@gkos] and [@ober; @001] for the theory of generalized function algebras and applications to non-linear and linear problems with non-smooth coefficients. Such algebras are also called Colombeau algebras, since he was the first one who introduced and analyzed such algebras. The geometric theory of algebras of generalized functions [@gkos] is further developed in papers [@kosv], [@ks2], [@ksv], [@ksv2], [@ksv3], [@sv]. In these papers applications to general relativity show the strong impact of the new approach developed by the authors through the analysis of PDE on manifolds with singular metrics and, in particular, in Lie group analysis of differential equations (see [@gfks], [@gkos], [@gksv], [@ko]). A version of this theory, which is the object of the present article, is initiated in [@dd], [@ks]. The sheaf properties of generalized function algebras are investigated in [@dp], [@msd].
In this paper we are interested in an important sheaf property, the suppleness. It is known that the sheaves of Schwartz distributions $\Omega\to \mathcal{D}'(\Omega)$ and of smooth functions $\Omega\to
C^\infty(\Omega)$, where $\Omega$ varies through all open sets of a manifold $X$, are not supple. The extensions of these sheaves $\Omega\to \mathcal{G}(\Omega)$ and $\Omega\to
\mathcal{G}^\infty(\Omega)$, $\Omega$ are open sets of a manifold $X$, which are actually sheaves of algebras of generalized functions, are supple. The proof of this assertion is the subject of this paper.
Generalized functions on $\mathbb{R}^d$ {#0.1}
---------------------------------------
We recall the main definitions. Let $\Omega$ be an open set in $\mathbb{R}^{d}$ and $\mathcal{E}(\Omega)$ be the space of nets of smooth functions. Then the set of moderate nets $\mathcal{E}_{M}(\Omega)$, respectively of negligible nets $\mathcal{N}(\Omega),$ consists of nets $(f_{\varepsilon })_{\varepsilon\in(0,1)}\in
\mathcal{E}(\Omega)$ with the properties $$(\forall K\subset\subset \Omega)\;(\forall n\in\mathbb{N})\;(\exists
a\in\mathbb{R})\;(\sup\limits_{x\in K}|f^{(n)}_{\varepsilon
}(x)|=O(\varepsilon^{a})),$$$$\mbox{ respectively, } \;(\forall K\subset\subset \Omega)\;(\forall
n\in\mathbb{N})\;(\forall b\in\mathbb{R})\;(\sup\limits_{x\in
K}|f^{(n)}_{\varepsilon}(x)|=O(\varepsilon^{b}))$$ ($O$ is the Landau symbol “big O” and $K\subset\subset\Omega$ means that $K$ is compact in $\Omega$ or that $\bar{K}$ is compact in $\Omega$.) Both spaces are algebras and the latter is an ideal of the former.
The algebra of generalized functions $\mathcal{G}(\Omega)$ is defined as the quotient $\mathcal{G}(\Omega)=\mathcal{E}_M(\Omega)/\mathcal{N}(\Omega).$ This is also a differential algebra. If the nets $(f_\varepsilon)_\varepsilon$ consist of constant functions on $\Omega $ (i.e. supremums over the compact set $K$ reduce to the absolute value), then one obtains the corresponding spaces $\mathcal{E}_M$ and $\mathcal{N}_0.$ They are algebras, $\mathcal{N}_0$ is an ideal in $\mathcal{E}_M$ and, as a quotient, one obtains the algebra of generalized complex numbers $\bar{\mathbb{C}}=\mathcal{E}_M/\mathcal{N}_0$ (or $\bar{\mathbb{R}}$). It is a ring, not a field.
The embedding of the Schwartz distributions in $\mathcal{E}^{\prime}(\Omega)$ is realized through the sheaf homomorphism $ \mathcal{E}^{\prime}(\Omega)\ni f\mapsto
[(f\ast\phi_{\varepsilon}|_{\Omega})_\varepsilon]\in\mathcal{G}(\Omega),
$ where the fixed net of mollifiers $(\phi_{\varepsilon})_{\varepsilon}$ is defined by $\phi_{\varepsilon}=\varepsilon ^{-d}\phi(\cdot/\varepsilon),\;
\varepsilon<1,$ where $\phi\in\mathcal{S}(\mathbb{R}^{d})$ satisfies $$\int
\phi(t)dt=1,\;\int t^{m}\phi(t)dt=0,m\in\mathbb{N}_{0}^{n},|m|>0.$$ ($t^{m}=t_{1}^{m_{1}}...t_{n}^{m_{n}}$ and $|m|=m_{1}+...+m_{n}.$) In fact $ \mathcal{E}^{\prime}(\Omega)$ is embedded into the space $\mathcal{G}_c(\Omega)$ of compactly supported generalized functions. This sheaf homomorphism, extended onto $\mathcal{D}^{\prime}$, gives the embedding of $\mathcal{D}^{\prime
}(\Omega)$ into $\mathcal{G}(\Omega).$
The algebra of generalized functions $\mathcal{G}^\infty(\Omega)$ is defined in [@ober; @001] as the quotient of $\mathcal{E}^\infty_{M}(\Omega)$ and $\mathcal{N}(\Omega),$ where $\mathcal{E}^\infty_{M}(\Omega)$ consists of nets $(f_{\varepsilon
})_{\varepsilon\in(0,1)}\in \mathcal{E}(\Omega)^{(0,1)}$ with the properties $$(\forall K\subset\subset \Omega) (\exists a\in\mathbb{R}) (\forall
n\in\mathbb{N}) (\sup_{x\in
K}|f^{(n)}_{\varepsilon}(x)|=O(\varepsilon^{a})),$$
Note that $\mathcal{G}^\infty$ is a subsheaf of $\mathcal{G}.$
Generalized functions on a manifold {#0.2}
-----------------------------------
We will recall the main definitions and assertions following [@gkos]. Let $X$ be a smooth Hausdorff paracompact manifold. We denote by $\mathcal{U}=\{(V_{\alpha},\psi_{\alpha}):\alpha\in
\Lambda\}$ an atlas on $X$, $\Lambda$ is the index set.
We use $\mathcal{P}(X,E)$ to denote the space of linear differential operators $\Gamma(X,E)\rightarrow\Gamma(X,E)$, where $E$ is a vector bundle on $X$ and $\Gamma(X,E)$ is the space of smooth sections of the vector bundle $E$ over $X$. Particularly, if $E=X\times
\mathbb{R}$ we write $\mathcal{P}(X)$ instead of $\mathcal{P}(X,E)$. We denote by $\frak X(X)$ the space of smooth vector fields on $X$.
Let $\mathcal{E}(X):=(C^{\infty}(X))^{(0,1)}$ and $(u_{\varepsilon})_{\varepsilon}\in \mathcal{E}(X)$. Then the following statements are equivalent:
1. $(\forall K\subset\subset X)\;(\forall P\in
\mathcal{P}(X))\;(\exists N\in \mathbb{N})\;(\sup\limits_{p\in
K}|Pu_{\varepsilon}(p)|=O(\varepsilon^{-N}))$;
2. $(\forall K\subset\subset X)\;(\forall k\in \mathbb{N}_{0})\;(\exists N\in
\mathbb{N})\;(\forall\xi_{1},...,\xi_{k}\in
\frak{X}(X))$\
$(\sup\limits_{p\in
K}|L_{\xi_{1}}...L_{\xi_{k}}u_{\varepsilon}(p)|=O(\varepsilon^{-N})),$ ($L_{\xi_{i}}$ is the Lie derivative);
3. For any chart $(V,\psi)$: $(u_{\varepsilon}\circ
\psi^{-1})_{\varepsilon}\in \mathcal{E}_{M}(\psi(V))$.
Denote by $\mathcal{E}_{M}(X)$ the subset of $\mathcal{E}(X)$ defined by any of the conditions 1, 2 or 3. We call it the space of moderate nets on the manifold $X$. The space of negligible nets is defined as:
$$\mathcal{N}(X):=\{(u_{\varepsilon})_{\varepsilon}\in \mathcal{E}_{M}(X):\forall K\subset\subset X\;\forall m\in \mathbb{N}:
\;\sup\limits_{p\in K}|u_{\varepsilon}(p)|=O(\varepsilon^{m})\}.$$
An algebra of generalized functions on the manifold $X$ is defined as the quotient space $\mathcal{G}(X):=\mathcal{E}_{M}(X)/\mathcal{N}(X).$ Elements of $\mathcal{G}(X)$ are written as $u=[(u_{\varepsilon})_{\varepsilon}]=(u_{\varepsilon})_{\varepsilon}+\mathcal{N}(X)$. As one can expect, $\mathcal{E}_{M}(X)$ is a differential algebra (with respect to Lie derivatives) and $\mathcal{N}(X)$ is a differential ideal in it. Moreover, $\mathcal{E}_{M}(X)$ and $\mathcal{N}(X)$ are invariant with respect to any $P\in
\mathcal{P}(X)$. Thus $Pu:=[(Pu_{\varepsilon})_{\varepsilon}]$ is a well-defined element of $\mathcal{G}(X)$.
Let $u\in \mathcal{G}(X)$ and let $X'$ be an open set on a manifold $X$. The restriction of a generalized function $u$, denoted by $u|_{X'}\in \mathcal{G}(X')$, is represented by $(u_{\varepsilon}|_{X'})_{\varepsilon}+\mathcal{N}(X')$. The support of a generalized function $u$, denoted by ${\rm supp}\;u$, is defined as the complement of the union of open sets $X'\subseteq X$ such that $u|_{X'}=0$.
The algebra $\mathcal{G}^{\infty}(X)$ is defined as a subalgebra of $\mathcal{G}(X)$ satisfying $u\in \mathcal{G}^{\infty}(X)$ if there exists a representative $(u_\varepsilon)_\varepsilon$ of $u$ so that for any chart $(U,\varphi)$, $(u_\varepsilon\circ
\varphi^{-1})_\varepsilon\in \mathcal{G}^{\infty}(\varphi(U))$.
Now we recall the sheaf properties of the space $\mathcal{G}(X)$ (see [@gkos]) and $\mathcal{G}^{\infty}(X)$.
A generalized function $u$ on $X$ allows the following local description via the correspondence: $\mathcal{G}(X)\ni u\mapsto
(u_{\alpha})_{\alpha \in A}$, where $u_{\alpha}:=u\circ
\psi_{\alpha}^{-1}\in \mathcal{G}(\psi_{\alpha}(V_{\alpha}))$. We call $u_{\alpha}$ the local expression of $u$ with respect to the chart $(V_{\alpha},\psi_{\alpha})$. Then $\mathcal{G}(X)$ can be identified with the set of all families $(u_{\alpha})_{\alpha}$ of generalized functions $u_{\alpha}\in
\mathcal{G}(\psi_{\alpha}(V_{\alpha}))$ satisfying the transformation law $$u_{\alpha}|_{\psi_{\alpha}(V_{\alpha}\cap V_{\beta})}=
u_{\beta}\circ\psi_{\beta}\circ\psi_{\alpha}^{-1}|_{\psi_{\alpha}(V_{\alpha}\cap
V_{\beta})}$$ for all $\alpha,\beta\in A$ with $V_{\alpha}\cap
V_{\beta}\neq\emptyset$.
It is well known that $\Omega \to \mathcal{G}(\Omega)$, $\Omega$ are open sets in $X$, is a fine and soft sheaf of $\mathbb{K}$-algebras on $X$. Thus, $\mathcal{G}$ is defined directly as a quotient sheaf of the sheaves of moderate modulo negligible sections. Similarly, $\Omega\to \mathcal{G}^{\infty}(\Omega)$, $\Omega$ open in $X$, is a fine and soft sheaf.
Supple sheaves {#sec:supple}
==============
Recall [@war], if $\mathcal{F}$ is a sheaf over the differential manifold $X$ and $U\subset X$ open than a continuous map $f:U\to
\mathcal{F}$ such that $\pi\circ f=id$ is called a section of $\mathcal{F}$ over $U$. The set of sections of $\mathcal{F}$ over $U$ is denoted by $\Gamma(U,\mathcal{F})$.
\[savitljiv\] Let $\mathcal{F}$ be a sheaf over the topological space $X$. Then, $\mathcal{F}$ is a supple sheaf if for all $f\in \Gamma(U,\mathcal{F})$, $U$ open in $X$, the following is true: If ${\rm supp}\;f=Z=Z_{1}\cup Z_{2}$, where $Z_{1}$ and $Z_{2}$ are arbitrary closed sets of $X$, then there exist $f_{1},f_{2}\in \Gamma(U,\mathcal{F})$ such that $${\rm
supp}\;f_{1}\subseteq Z_{1},\; {\rm supp}\;f_{2}\subseteq
Z_{2}\;\mbox{ and}\; f=f_{1}+f_{2}.$$
It will be shown that the sheaf of algebras $\Omega\to
\mathcal{G}(\Omega)$, $\Omega$ varies over all open sets of the manifold $X$, is supple, but it is not flabby. It is well known that $\mathcal{D}'$ is not supple. We give an example which shows this.
\[dist koja nije supple\] Consider $$f(x)=\sum\limits_{n=1}^{\infty}(\delta(x+\frac{1}{n^2})-\delta(x-\frac{1}{n^2}))+\delta(x),$$ where $\delta$ is the delta distribution. The support of this distribution is $$Z=\{\frac{1}{n^2}:\;n\in \mathbb{N}\}\cup\{-\frac{1}{n^2}:\;n\in
\mathbb{N}\}\cup \{0\}.$$ One can see that the closed set $Z$ is a union of two closed sets $$\displaystyle
Z_{1}=\{\frac{1}{n^2}:\;n\in \mathbb{N}\}\cup\{0\}\;\mbox{ and}\;
\displaystyle Z_{2}=\{-\frac{1}{n^2}:\;n\in \mathbb{N}\}\cup\{0\}.$$ Then distributions $f_{1}$ and $f_{2}$ (in order to satisfy Definition \[savitljiv\]) should be of the form $$f_{1}=\sum\limits_{n=1}^{\infty}\delta(x-\frac{1}{n^2})+C_{1}\delta(x)\;\mbox
{and} \;
f_{1}=\sum\limits_{n=1}^{\infty}\delta(x+\frac{1}{n^2})+C_{2}\delta(x),$$ since ${\rm supp\;}f_{1}\subseteq Z_{1}$, ${\rm
supp\;}f_{2}\subseteq Z_{2}$ and $f=f_{1}+f_{2}$. However, it is known that $f_{1}$ and $f_{2}$ are not distributions since they are infinite sums of shifted delta distributions so that their supports have zero as the accumulation point.
We will prove the next theorem:
\[GX je savitljiv snop\]$\Omega\to\mathcal{G}(\Omega)$, $\Omega$ open in $X$, is a supple sheaf.
**Proof.** For the set $A$, we will denote by $A^\varepsilon$ the set $A^{\varepsilon}=\{x\in {\mathbb{R}^d}:\;d(x,A)<\varepsilon\},\;\varepsilon<1$, where $d$ is a distance on $X$. The notation $L(x,r)$ stands for the open ball of radius $r>0$ centered in $x\in X$, i.e. $L(x,r)=\{y\in X:\;d(x,y)<r\}$.
We divide the proof of this theorem into two parts (I) and (II) and use the following two simple assertions:
1. Let $A$ be a measurable set in ${\mathbb{R}^d}$. Then there exists a generalized function $\eta=[(\eta_\varepsilon)_\varepsilon]\in
\mathcal{G}({\mathbb{R}^d})$ such that $$\eta_{\varepsilon}(x):=\left\{\begin{array}{ll}
1, & x\in A\\
0, & x\in {\mathbb{R}^d}\setminus A^{\varepsilon}
\end{array}\right.$$ and $0\leq|\eta_\varepsilon(x)|\leq 1,\;x\in {\mathbb{R}^d}$. More precisely, $\eta_\varepsilon$ is defined to be $1_A\ast \phi_\varepsilon$, where $1_A$ is the characteristic function of $A$, $\phi$ is a compactly supported smooth function so that $\int_{{\mathbb{R}^d}}\phi(t)dt=1$ and $\phi_\varepsilon(x)=1/\varepsilon^d\phi(x/\varepsilon)$.
2. Let $\delta>0$ and $Z_{1}$ and $Z_{2}$ arbitrary closed sets of ${\mathbb{R}^d}$. Then there exists a closed set $\widetilde{Z_{1}^{\delta}}\supset Z_{1}$ such that $Z_{1}\cap
Z_{2}=\widetilde{Z_{1}^{\delta}}\cap Z_{2}$ and $d(x,Z_{1})\leq
\delta$, $x\in \widetilde{Z_{1}^{\delta}}$.
We define $$\widetilde{Z_{1}^{\delta}}=\{x\in {\mathbb{R}^d}:\;d(x,Z_{1})\leq \delta\;\wedge d(x,Z_{1})\leq d(x,Z_{2})\}.$$
\(I) Now we show that $\Omega\to\mathcal{G}(\Omega)$, $\Omega$ open in ${\mathbb{R}^d}$, is a supple sheaf.
It is enough to prove the assertion for $U={\mathbb{R}^d}$ and $f\in \mathcal{G}({\mathbb{R}^d})=\Gamma({\mathbb{R}^d},\mathcal{G})$ with the property ${\rm
supp}\;f=Z$ and let $Z=Z_{1}\cup Z_{2}$, where $Z_{1}$ and $Z_{2}$ are arbitrary closed sets. Let $\delta>0$ and define $\widetilde{Z_{1}^{\delta}}$ as in Assertion 2. Next by Assertion 1, let $\eta_{\varepsilon}\in C^{\infty}({\mathbb{R}^d}),\;\varepsilon\in(0,1)$, such that $$\eta_{\varepsilon}(x)=\left\{\begin{array}{ll}
1, & x\in \widetilde{Z_{1}^{\delta}}\\
0, & x\in X\setminus
(\widetilde{Z_{1}^{\delta}})^{\varepsilon},\;\varepsilon\in (0,1),
\end{array}\right.$$ with $(\eta_\varepsilon)_\varepsilon\in \mathcal{E}_M({\mathbb{R}^d})$.
Let $f_{1}=[(f_{\varepsilon}\eta_{\varepsilon})_{\varepsilon}]$ and $f_{2}=[(f_{\varepsilon}(1-\eta_{\varepsilon}))_{\varepsilon}]$. Then $f_{1},f_{2}\in \mathcal{G}({\mathbb{R}^d})$ and $f=[(f_{\varepsilon}\eta_{\varepsilon})_{\varepsilon}]+
[(f_{\varepsilon}(1-\eta_{\varepsilon}))_{\varepsilon}]$. So, we have to show that ${\rm supp}\;(f_{1})\subseteq Z_{1}$ and ${\rm
supp}\;(f_{2})\subseteq Z_{2}$.
We show the inclusion ${\rm supp}\;(f_{1})\subseteq Z_{1}$ by showing that for any point $x\notin Z_{1}$ there exists a neighborhood $X'$ of $x$ such that $(f_{1\varepsilon}|_{X'})_{\varepsilon}\in \mathcal{N}(X')$ (according to the definition of the support this means $x\notin {\rm
supp}\;f_{1}$). Let $x\in {\mathbb{R}^d}\setminus Z_{1}$. Then we have $A=d(x,Z_{1})>0$ and there are two possibilities: $x\in
\widetilde{Z_{1}^{\delta}}$ and $x\notin
\widetilde{Z_{1}^{\delta}}$. If $x\in \widetilde{Z_{1}^{\delta}}$ then the ball $X'=L(x,\frac{A}{2})$ has no intersection with $Z_{1}$ and $Z_{2}$ because in the set $\widetilde{Z_{1}^{\delta}}$ we have $d(x,Z_{1})\leq d(x,Z_{2})$. So $X'$ does not intersect the set $Z=Z_{1}\cup Z_{2}$. From $$|f_{1\varepsilon}(y)|=|f_{\varepsilon}(y)\eta_{\varepsilon}(y)|\leq
|f_{\varepsilon}(y)| \;\mbox{for all}\; y\in X'$$ and $(f_{\varepsilon}|_{X'})_{\varepsilon}\in \mathcal{N}(X')$, we have $(f_{1\varepsilon}|_{X'})_{\varepsilon}\in \mathcal{N}(X')$. Finally, if $x\notin\widetilde{Z_{1}^{\delta}}$ then $B=d(x,\widetilde{Z_{1}^{\delta}})>0$ (since $\widetilde{Z_{1}^{\delta}}$ is a closed set). Let $X'=L(x,\frac{B}{2})$. Then $$f_{1\varepsilon}(y)=f_{\varepsilon}(y)\eta_{\varepsilon}(y)=f_{\varepsilon}(y)\cdot
0=0,\;y\in X',$$ where $\varepsilon<\frac{B}{2}$. Again, we have $(f_{1\varepsilon}|_{X'})_{\varepsilon}\in \mathcal{N}(X')$.
Similarly, we show the second inclusion ${\rm
supp}\;(f_{2})\subseteq Z_{2}$. Let $x\notin Z_{2}$. Our aim is to show that $x\notin {\rm supp}\;f_{2}$. There are two possibilities: $x\notin Z_{1}$ and $x\in Z_{1}$. If $x\notin Z_{1}$, we also have $x\notin Z_{2}$ and so $x\notin Z$. Then there exists a neighborhood $W$ of $x$ such that $(f_{\varepsilon}|_{W})_{\varepsilon}\in \mathcal{N}(W)$, since ${\rm supp}\;f\subseteq Z$. We take $X'=W$. Then, clearly, $$|f_{2\varepsilon}(y)|=|f_{\varepsilon}(y)(1-\eta_{\varepsilon}(y))|\leq
|f_{\varepsilon}(y)|,\;y\in X',\;\varepsilon<1.$$ Since $(f_{\varepsilon}|_{X'})_{\varepsilon}\in \mathcal{N}(X')$ we also have that $(f_{2\varepsilon}|_{X'})_{\varepsilon}\in
\mathcal{N}(X')$. The second possibility is $x\in Z_{1}$. Let $H=d(x,Z_{2})>0$ and $H'=\min \{H,\delta\}$. Note that $d(x,Z_{1})=0$ since $x\in Z_{1}$. Then for $X'=L(x,\frac{H'}{2})$ we have $X'\subseteq
\widetilde{Z_{1}^{\delta}}$ (since $H'\leq \delta$ and for all $y\in
X'$ holds $d(y,Z_{1})<\frac{H'}{2}\leq d(y,Z_{2})$). So $X'$ has no intersection with $Z_{2}$ (since $H'\leq H$). We have $$f_{2\varepsilon}(y)=f_{\varepsilon}(y)(1-\eta_{\varepsilon}(y))=f_{\varepsilon}(y)\cdot
0=0,\;y\in X',\;\varepsilon<1.$$ Again, $(f_{2\varepsilon}|_{X'})_{\varepsilon}\in \mathcal{N}(X')$. This finishes the proof of the suppleness of $\Omega
\to\mathcal{G}(\Omega)$, $\Omega$ open in ${\mathbb{R}^d}$.
\(II) Let $Z\subseteq X$ be closed. Let $Z=Z_1\cup Z_2$ where $Z_1$ and $Z_2$ are closed and let $f\in
\mathcal{G}(X)=\Gamma(X,\mathcal{G})$ such that ${\rm
supp}\;(f)\subseteq Z$. Cover $X$ by a family of chart neighborhoods $\mathcal{U}$ and let $\{\tilde{\chi}_{\alpha}:\;\alpha\in
\Lambda\}$ be a partition of unity subordinated to $\mathcal{U}$. Set $$\chi_\alpha=\frac{\tilde{\chi}_\alpha}{(\sum_{\alpha\in \Lambda}\tilde{\chi}^{2}_{\alpha})^{1/2}}.$$ So, we obtain the family of functions $\{\chi_\alpha:\;\alpha\in
\Lambda\}$ such that $\{{\rm supp}\;(\chi_\alpha):\;\alpha\in
\Lambda\}$ is locally finite and $\sum_{\alpha\in
\Lambda}\chi^{2}_{\alpha}=1$. Hence, one can write $$\label{rastavljane f} f=\sum_{\alpha\in \Lambda}
\chi^{2}_{\alpha} f=\sum_{\alpha\in \Lambda}\chi_\alpha(\chi_\alpha
f).$$ For the functions $\chi_\alpha f\;\alpha\in
\Lambda$ we have ${\rm supp}(\chi_\alpha f)$ is closed in some $U_\alpha\in \mathcal{U}$, where the results hold (due to part (I) of this proof). Precisely, one can see ${\rm supp}(\chi_\alpha
f)\subseteq Z\cap U_\alpha$ as $${\rm supp}(\chi_\alpha f)=(Z_1\cap ({\rm supp}(\chi_\alpha
f)))\cup (Z_2\cap ({\rm supp}(\chi_\alpha f)))\subseteq(Z_1\cap
U_\alpha)\cup(Z_2\cap U_\alpha),$$ where the sets $Z_1\cap ({\rm
supp}(\chi_\alpha f))$ and $Z_2\cap ({\rm supp}(\chi_\alpha f))$ are closed.
Applying part (I) of this proof to $\chi_\alpha f,\;\alpha\in
\Lambda$ we obtain $f^{\alpha}_{1},f^{\alpha}_{2}\in
\mathcal{G}(U_\alpha)$ such that $$\chi_\alpha f=f^{\alpha}_{1}+f^{\alpha}_{2},\;{\rm
supp}(f^{\alpha}_{1})\subseteq Z_1\cap U_\alpha\subseteq Z_1,\;{\rm
supp}(f^{\alpha}_{2})\subseteq Z_2\cap U_\alpha\subseteq Z_2.$$ According to (\[rastavljane f\])
$f=\sum_{\alpha\in
\Lambda}\chi_\alpha(f^{\alpha}_{1}+f^{\alpha}_{2})=\sum_{\alpha\in
\Lambda}\chi_\alpha f^{\alpha}_{1}+\sum_{\alpha\in \Lambda}
\chi_\alpha f^{\alpha}_{2}$.
Set $f_1=\sum_{\alpha\in \Lambda}\chi_\alpha f^{\alpha}_{1}$ and $f_2=\sum_{\alpha\in \Lambda} \chi_\alpha f^{\alpha}_{2}$. Then $f_1,f_2\in \mathcal{G}(X)$ and ${\rm supp}f_1\subseteq Z_1,\;{\rm
supp}f_2\subseteq Z_2$.
$\hfill\Box$
\[GXinf je savitljiv snop\]$\Omega\to\mathcal{G}^{\infty}(\Omega)$, $\Omega$ open in $X$, is a supple sheaf.
**Proof.** Suppleness of the sheaf $\Omega\to
\mathcal{G}^{\infty}(\Omega)$ can be proved using the same ideas as in the proof of Theorem \[GX je savitljiv snop\]. Let $Z\subseteq
X$ be closed. Let $Z=Z_1\cup Z_2$ where $Z_1$ and $Z_2$ are closed and let $f\in \mathcal{G}^\infty(X)$ such that ${\rm
supp}\;(f)\subseteq Z$. Now we have to construct generalized functions $f_1, f_2\in \mathcal{G}^\infty(X)$ such that ${\rm
supp}f_1\subseteq Z_1,\;{\rm supp}f_2\subseteq Z_2$. In order to obtain $f_1, f_2\in \mathcal{G}^\infty(X)$ we will take $\hat{\eta}$ to be a generalized function from $ \mathcal{G}^{\infty}(X)$ (see Assertion 1). We will replace $\varepsilon$ by $|\ln\varepsilon|^{-1}$ and then the generalized function $\hat{\eta}=[(\hat{\eta}_\varepsilon)_\varepsilon]$ will be $$\hat{\eta}_{\varepsilon}(x)=\left\{\begin{array}{ll}
1, & x\in A\\
0, & x\in {\mathbb{R}^d}\setminus A_{|\ln\varepsilon|^{-1}}
\end{array}\right..$$
This finishes the proof, since for $f=[(f_\varepsilon)_\varepsilon]\in\mathcal{G}^\infty(X)$ generalized functions $\hat{f}_{1}=[(f_{\varepsilon}\hat{\eta}_{\varepsilon})_{\varepsilon}]$ and $\hat{f}_{2}=[(f_{\varepsilon}(1-\hat{\eta}_{\varepsilon}))_{\varepsilon}]$ are in $\mathcal{G}^\infty(X)$ and following the proof of Theorem \[GX je savitljiv snop\] (replacing $\varepsilon$ by $|\ln\varepsilon|^{-1}$ ) we obtain ${\rm
supp}\;\hat{f}_{1}\subseteq Z_{1}$ and ${\rm
supp}\;\hat{f}_{2}\subseteq Z_{2}$.
$\hfill\Box$
Let us remark at the end that $\mathcal{G}(X)$ and $\mathcal{G}^{\infty}(X)$ are not flabby sheaves (see [@msd], Remark on page 95). If we take $X=\mathbb{R}$ and $X'=(0,\infty)$ then one can not extend the generalized function $[(\varepsilon^{-1/x})_{\varepsilon}]$, defined on $(0,\infty)$, to the whole space $\mathbb{R}$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The research is supported by Ministry of Science and Technological Development, Republic of Serbia, project 144016.
[00]{}
G. Bengel, P. Schapira, Décomposition microlocale analytique des distributions, Ann. Inst. Fourier, Grenoble 29, 101-124 (1979)
H. A. Biagioni, A Nonlinear Theory of Generalized Functions, Springer-Verlag, Berlin-Hedelberg-New York (1990)
J. Bros, D. Iagolnitzer, Support essentiel et structure analytique des distribution, Séminaire Goulaouic-Lions-Schwartz, Exposé 18 (1975-1976)
J. F. Colombeau, New Generalized Functions and Multiplications of Distributions, North Holland, Amsterdam (1984)
J. F. Colombeau, Elementary Introduction in New Generalized Functions, North Holland, Amsterdam (1985)
J.W. De Roever, M. Damsma, Colombeau algebras on a $C^\infty$-manifold, Indag. Math. N.S. 2, 341-358 (1991)
N. Djapić, S. Pilipović, Microlocal analysis of Colombeau’s generalized functions on a manifold, Indag. Math. N.S. 7, 293–309 (1996)
A. Eida, S. Pilipović, On the microlocal decomposition of same classes of hyperfunctions, Math. Proc. Camb. Phil. Soc. 125, 455-461 (1999)
M. Grosser, E. Farkas, M. Kunzinger, R. Steinbauer, On the foundations of nonlinear generalized functions I, II, Mem. Amer. Math. Soc. 153 (2001)
M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Kluwer Academic Publishers, Dordrecht (2001)
M. Grosser, M. Kunzinger, R. Steinbauer, J. Vickers, A global theory of algebras of generalized functions, Adv. Math. 166, 50-72 (2002)
M. Kunzinger, M. Oberguggenberger, Group analysis of differential equations and generalized functions, SIAM J. Math. Anal. 31, 1192-1213 (2000)
M. Kunzinger, M. Oberguggenberger, R. Steinbauer, J. Vickers, Generalized flows and singular ODEs on differentiable manifolds, Acta Appl. Math. 80, 221-241 (2004)
M. Kunzinger, R. Steinbauer, Foundations of a nonlinear distributional geometry, Acta Appl. Math. 71, 179-206 (2002)
M. Kunzinger, R. Steinbauer, Generalized pseudo-Riemannian geometry, Trans. Amer. Math. Soc. 354, 4179-4199 (2002)
M. Kunzinger, R. Steinbauer, J. Vickers, Sheaves of nonlinear generalized functions and manifold-valued distributions, Trans. Amer. Math. Soc. 361, 5177-5192 (2009)
M. Kunzinger, R. Steinbauer, J. Vickers, Generalised connections and curvature, Math. Proc. Cambridge Philos. Soc. 139, 497–521 (2005)
M. Kunzinger, R. Steinbauer, J. Vickers, Intrinsic characterization of manifold-valued generalized functions, Proc. London Math. Soc. 87, 451-470 (2003)
P. Laubin, Front d’onde analytique et décomposition microlocale des distributions, Ann. Inst. Fourier, Grenoble 33, 179-199 (1983)
M. Oberguggenberger, Multiplication of distributions and application to partial differential equations, Pitman Res. Notes Math. Ser. 259, Longman, Harlow (1992)
M. Oberguggenberger, S. Pilipović, D. Scarpalezos, Local properties of Colombeau generalized functions, Math. Nachr. 256, 88-99 (2003)
R. Steinbauer, J. Vickers, The use of generalized functions and distributions in general relativity, Classical Quantum Gravity 23, 91-114 (2006)
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, Glenview, Illinois, London (1971)
|
---
abstract: 'The total cross sections for the elastic electroproduction of and mesons for $>$ 8 and $\langle W \rangle \simeq 90$ are measured at HERA with the H1 detector. The measurements are for an integrated electron$-$proton luminosity of $\simeq$ 3 pb$^{-1}$. The dependences of the total virtual photon$-$proton () cross sections on $Q^2$, $W$ and the momentum transfer squared to the proton ($t$), and, for the $\rho$, the dependence on the polar decay angle ($\cos \theta^*$), are presented. The : cross section ratio is determined. The results are discussed in the light of theoretical models and of the interplay of hard and soft physics processes.'
---
23.0cm 16.0cm -1.0cm 0.0cm
\#1\#2\#3 [[*Ann. Rev. Nucl. Part. Sci.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Erratum*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*ibid.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Int. J. Mod. Phys.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*JETP Lett.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Mod. Phys. Lett.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Nucl. Instr. Meth.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Nuovo Cim.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Nucl. Phys.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Phys. Lett.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Phys. Rep.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Phys. Rev.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Phys. Rev. Lett.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Prog. Th. Phys.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Rev. Mod. Phys.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Rep. Prog. Phys.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Sov. J. Nucl. Phys.*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Sov. Phys. JEPT*]{} [**\#1**]{} (\#2) \#3]{} \#1\#2\#3 [[*Zeit. Phys.*]{} [**\#1**]{} (\#2) \#3]{}
[DESY 96-023 ISSN 0418-nnnn]{}\
[February 1996]{}\
**[ Elastic Electroproduction of and Mesons\
at large at HERA\
]{}**
H1 Collaboration\
S. Aid$^{14}$, V. Andreev$^{26}$, B. Andrieu$^{29}$, R.-D. Appuhn$^{12}$, M. Arpagaus$^{37}$, A. Babaev$^{25}$, J. Bähr$^{36}$, J. Bán$^{18}$, Y. Ban$^{28}$, P. Baranov$^{26}$, E. Barrelet$^{30}$, R. Barschke$^{12}$, W. Bartel$^{12}$, M. Barth$^{5}$, U. Bassler$^{30}$, H.P. Beck$^{38}$, H.-J. Behrend$^{12}$, A. Belousov$^{26}$, Ch. Berger$^{1}$, G. Bernardi$^{30}$, R. Bernet$^{37}$, G. Bertrand-Coremans$^{5}$, M. Besançon$^{10}$, R. Beyer$^{12}$, P. Biddulph$^{23}$, P. Bispham$^{23}$, J.C. Bizot$^{28}$, V. Blobel$^{14}$, K. Borras$^{9}$, F. Botterweck$^{5}$, V. Boudry$^{29}$, A. Braemer$^{15}$, W. Braunschweig$^{1}$, V. Brisson$^{28}$, D. Bruncko$^{18}$, C. Brune$^{16}$, R. Buchholz$^{12}$, L. Büngener$^{14}$, J. Bürger$^{12}$, F.W. Büsser$^{14}$, A. Buniatian$^{12,39}$, S. Burke$^{19}$, M.J. Burton$^{23}$, G. Buschhorn$^{27}$, A.J. Campbell$^{12}$, T. Carli$^{27}$, F. Charles$^{12}$, M. Charlet$^{12}$, D. Clarke$^{6}$, A.B. Clegg$^{19}$, B. Clerbaux$^{5}$, S. Cocks$^{20}$, J.G. Contreras$^{9}$, C. Cormack$^{20}$, J.A. Coughlan$^{6}$, A. Courau$^{28}$, M.-C. Cousinou$^{24}$, G. Cozzika$^{10}$, L. Criegee$^{12}$, D.G. Cussans$^{6}$, J. Cvach$^{31}$, S. Dagoret$^{30}$, J.B. Dainton$^{20}$, W.D. Dau$^{17}$, K. Daum$^{35}$, M. David$^{10}$, C.L. Davis$^{19}$, B. Delcourt$^{28}$, A. De Roeck$^{12}$, E.A. De Wolf$^{5}$, M. Dirkmann$^{9}$, P. Dixon$^{19}$, P. Di Nezza$^{33}$, W. Dlugosz$^{8}$, C. Dollfus$^{38}$, J.D. Dowell$^{4}$, H.B. Dreis$^{2}$, A. Droutskoi$^{25}$, D. Düllmann$^{14}$, O. Dünger$^{14}$, H. Duhm$^{13}$, J. Ebert$^{35}$, T.R. Ebert$^{20}$, G. Eckerlin$^{12}$, V. Efremenko$^{25}$, S. Egli$^{38}$, R. Eichler$^{37}$, F. Eisele$^{15}$, E. Eisenhandler$^{21}$, R.J. Ellison$^{23}$, E. Elsen$^{12}$, M. Erdmann$^{15}$, W. Erdmann$^{37}$, E. Evrard$^{5}$, A.B. Fahr$^{14}$, L. Favart$^{5}$, A. Fedotov$^{25}$, D. Feeken$^{14}$, R. Felst$^{12}$, J. Feltesse$^{10}$, J. Ferencei$^{18}$, F. Ferrarotto$^{33}$, K. Flamm$^{12}$, M. Fleischer$^{9}$, M. Flieser$^{27}$, G. Flügge$^{2}$, A. Fomenko$^{26}$, B. Fominykh$^{25}$, J. Formánek$^{32}$, J.M. Foster$^{23}$, G. Franke$^{12}$, E. Fretwurst$^{13}$, E. Gabathuler$^{20}$, K. Gabathuler$^{34}$, F. Gaede$^{27}$, J. Garvey$^{4}$, J. Gayler$^{12}$, M. Gebauer$^{36}$, A. Gellrich$^{12}$, H. Genzel$^{1}$, R. Gerhards$^{12}$, A. Glazov$^{36}$, U. Goerlach$^{12}$, L. Goerlich$^{7}$, N. Gogitidze$^{26}$, M. Goldberg$^{30}$, D. Goldner$^{9}$, K. Golec-Biernat$^{7}$, B. Gonzalez-Pineiro$^{30}$, I. Gorelov$^{25}$, C. Grab$^{37}$, H. Grässler$^{2}$, R. Grässler$^{2}$, T. Greenshaw$^{20}$, R. Griffiths$^{21}$, G. Grindhammer$^{27}$, A. Gruber$^{27}$, C. Gruber$^{17}$, J. Haack$^{36}$, D. Haidt$^{12}$, L. Hajduk$^{7}$, M. Hampel$^{1}$, W.J. Haynes$^{6}$, G. Heinzelmann$^{14}$, R.C.W. Henderson$^{19}$, H. Henschel$^{36}$, I. Herynek$^{31}$, M.F. Hess$^{27}$, W. Hildesheim$^{12}$, K.H. Hiller$^{36}$, C.D. Hilton$^{23}$, J. Hladký$^{31}$, K.C. Hoeger$^{23}$, M. Höppner$^{9}$, D. Hoffmann$^{12}$, T. Holtom$^{20}$, R. Horisberger$^{34}$, V.L. Hudgson$^{4}$, M. Hütte$^{9}$, H. Hufnagel$^{15}$, M. Ibbotson$^{23}$, H. Itterbeck$^{1}$, A. Jacholkowska$^{28}$, C. Jacobsson$^{22}$, M. Jaffre$^{28}$, J. Janoth$^{16}$, T. Jansen$^{12}$, L. Jönsson$^{22}$, K. Johannsen$^{14}$, D.P. Johnson$^{5}$, L. Johnson$^{19}$, H. Jung$^{10}$, P.I.P. Kalmus$^{21}$, M. Kander$^{12}$, D. Kant$^{21}$, R. Kaschowitz$^{2}$, U. Kathage$^{17}$, J. Katzy$^{15}$, H.H. Kaufmann$^{36}$, O. Kaufmann$^{15}$, S. Kazarian$^{12}$, I.R. Kenyon$^{4}$, S. Kermiche$^{24}$, C. Keuker$^{1}$, C. Kiesling$^{27}$, M. Klein$^{36}$, C. Kleinwort$^{12}$, G. Knies$^{12}$, T. Köhler$^{1}$, J.H. Köhne$^{27}$, H. Kolanoski$^{3}$, F. Kole$^{8}$, S.D. Kolya$^{23}$, V. Korbel$^{12}$, M. Korn$^{9}$, P. Kostka$^{36}$, S.K. Kotelnikov$^{26}$, T. Krämerkämper$^{9}$, M.W. Krasny$^{7,30}$, H. Krehbiel$^{12}$, D. Krücker$^{2}$, U. Krüger$^{12}$, U. Krüner-Marquis$^{12}$, H. Küster$^{22}$, M. Kuhlen$^{27}$, T. Kurča$^{36}$, J. Kurzhöfer$^{9}$, D. Lacour$^{30}$, B. Laforge$^{10}$, R. Lander$^{8}$, M.P.J. Landon$^{21}$, W. Lange$^{36}$, U. Langenegger$^{37}$, J.-F. Laporte$^{10}$, A. Lebedev$^{26}$, F. Lehner$^{12}$, C. Leverenz$^{12}$, S. Levonian$^{26}$, Ch. Ley$^{2}$, G. Lindström$^{13}$, M. Lindstroem$^{22}$, J. Link$^{8}$, F. Linsel$^{12}$, J. Lipinski$^{14}$, B. List$^{12}$, G. Lobo$^{28}$, H. Lohmander$^{22}$, J.W. Lomas$^{23}$, G.C. Lopez$^{13}$, V. Lubimov$^{25}$, D. Lüke$^{9,12}$, N. Magnussen$^{35}$, E. Malinovski$^{26}$, S. Mani$^{8}$, R. Maraček$^{18}$, P. Marage$^{5}$, J. Marks$^{24}$, R. Marshall$^{23}$, J. Martens$^{35}$, G. Martin$^{14}$, R. Martin$^{20}$, H.-U. Martyn$^{1}$, J. Martyniak$^{7}$, T. Mavroidis$^{21}$, S.J. Maxfield$^{20}$, S.J. McMahon$^{20}$, A. Mehta$^{6}$, K. Meier$^{16}$, T. Merz$^{36}$, A. Meyer$^{14}$, A. Meyer$^{12}$, H. Meyer$^{35}$, J. Meyer$^{12}$, P.-O. Meyer$^{2}$, A. Migliori$^{29}$, S. Mikocki$^{7}$, D. Milstead$^{20}$, J. Moeck$^{27}$, F. Moreau$^{29}$, J.V. Morris$^{6}$, E. Mroczko$^{7}$, D. Müller$^{38}$, G. Müller$^{12}$, K. Müller$^{12}$, P. Murín$^{18}$, V. Nagovizin$^{25}$, R. Nahnhauer$^{36}$, B. Naroska$^{14}$, Th. Naumann$^{36}$, P.R. Newman$^{4}$, D. Newton$^{19}$, D. Neyret$^{30}$, H.K. Nguyen$^{30}$, T.C. Nicholls$^{4}$, F. Niebergall$^{14}$, C. Niebuhr$^{12}$, Ch. Niedzballa$^{1}$, H. Niggli$^{37}$, R. Nisius$^{1}$, G. Nowak$^{7}$, G.W. Noyes$^{6}$, M. Nyberg-Werther$^{22}$, M. Oakden$^{20}$, H. Oberlack$^{27}$, U. Obrock$^{9}$, J.E. Olsson$^{12}$, D. Ozerov$^{25}$, P. Palmen$^{2}$, E. Panaro$^{12}$, A. Panitch$^{5}$, C. Pascaud$^{28}$, G.D. Patel$^{20}$, H. Pawletta$^{2}$, E. Peppel$^{36}$, E. Perez$^{10}$, J.P. Phillips$^{20}$, A. Pieuchot$^{24}$, D. Pitzl$^{37}$, G. Pope$^{8}$, S. Prell$^{12}$, R. Prosi$^{12}$, K. Rabbertz$^{1}$, G. Rädel$^{12}$, F. Raupach$^{1}$, P. Reimer$^{31}$, S. Reinshagen$^{12}$, H. Rick$^{9}$, V. Riech$^{13}$, J. Riedlberger$^{37}$, F. Riepenhausen$^{2}$, S. Riess$^{14}$, E. Rizvi$^{21}$, S.M. Robertson$^{4}$, P. Robmann$^{38}$, H.E. Roloff$^{36}$, R. Roosen$^{5}$, K. Rosenbauer$^{1}$, A. Rostovtsev$^{25}$, F. Rouse$^{8}$, C. Royon$^{10}$, K. Rüter$^{27}$, S. Rusakov$^{26}$, K. Rybicki$^{7}$, N. Sahlmann$^{2}$, D.P.C. Sankey$^{6}$, P. Schacht$^{27}$, S. Schiek$^{14}$, S. Schleif$^{16}$, P. Schleper$^{15}$, W. von Schlippe$^{21}$, D. Schmidt$^{35}$, G. Schmidt$^{14}$, A. Schöning$^{12}$, V. Schröder$^{12}$, E. Schuhmann$^{27}$, B. Schwab$^{15}$, F. Sefkow$^{12}$, M. Seidel$^{13}$, R. Sell$^{12}$, A. Semenov$^{25}$, V. Shekelyan$^{12}$, I. Sheviakov$^{26}$, L.N. Shtarkov$^{26}$, G. Siegmon$^{17}$, U. Siewert$^{17}$, Y. Sirois$^{29}$, I.O. Skillicorn$^{11}$, P. Smirnov$^{26}$, J.R. Smith$^{8}$, V. Solochenko$^{25}$, Y. Soloviev$^{26}$, A. Specka$^{29}$, J. Spiekermann$^{9}$, S. Spielman$^{29}$, H. Spitzer$^{14}$, F. Squinabol$^{28}$, R. Starosta$^{1}$, M. Steenbock$^{14}$, P. Steffen$^{12}$, R. Steinberg$^{2}$, H. Steiner$^{12,40}$, B. Stella$^{33}$, A. Stellberger$^{16}$, J. Stier$^{12}$, J. Stiewe$^{16}$, U. Stö[ß]{}lein$^{36}$, K. Stolze$^{36}$, U. Straumann$^{38}$, W. Struczinski$^{2}$, J.P. Sutton$^{4}$, S. Tapprogge$^{16}$, M. Taševský$^{32}$, V. Tchernyshov$^{25}$, S. Tchetchelnitski$^{25}$, J. Theissen$^{2}$, C. Thiebaux$^{29}$, G. Thompson$^{21}$, P. Truöl$^{38}$, J. Turnau$^{7}$, J. Tutas$^{15}$, P. Uelkes$^{2}$, A. Usik$^{26}$, S. Valkár$^{32}$, A. Valkárová$^{32}$, C. Vallée$^{24}$, D. Vandenplas$^{29}$, P. Van Esch$^{5}$, P. Van Mechelen$^{5}$, Y. Vazdik$^{26}$, P. Verrecchia$^{10}$, G. Villet$^{10}$, K. Wacker$^{9}$, A. Wagener$^{2}$, M. Wagener$^{34}$, A. Walther$^{9}$, B. Waugh$^{23}$, G. Weber$^{14}$, M. Weber$^{12}$, D. Wegener$^{9}$, A. Wegner$^{27}$, T. Wengler$^{15}$, M. Werner$^{15}$, L.R. West$^{4}$, T. Wilksen$^{12}$, S. Willard$^{8}$, M. Winde$^{36}$, G.-G. Winter$^{12}$, C. Wittek$^{14}$, E. Wünsch$^{12}$, J. Žáček$^{32}$, D. Zarbock$^{13}$, Z. Zhang$^{28}$, A. Zhokin$^{25}$, M. Zimmer$^{12}$, F. Zomer$^{28}$, J. Zsembery$^{10}$, K. Zuber$^{16}$, and M. zurNedden$^{38}$ $\:^1$ I. Physikalisches Institut der RWTH, Aachen, Germany$^ a$\
$\:^2$ III. Physikalisches Institut der RWTH, Aachen, Germany$^ a$\
$\:^3$ Institut für Physik, Humboldt-Universität, Berlin, Germany$^ a$\
$\:^4$ School of Physics and Space Research, University of Birmingham, Birmingham, UK$^ b$\
$\:^5$ Inter-University Institute for High Energies ULB-VUB, Brussels; Universitaire Instelling Antwerpen, Wilrijk; Belgium$^ c$\
$\:^6$ Rutherford Appleton Laboratory, Chilton, Didcot, UK$^ b$\
$\:^7$ Institute for Nuclear Physics, Cracow, Poland$^ d$\
$\:^8$ Physics Department and IIRPA, University of California, Davis, California, USA$^ e$\
$\:^9$ Institut für Physik, Universität Dortmund, Dortmund, Germany$^ a$\
$ ^{10}$ CEA, DSM/DAPNIA, CE-Saclay, Gif-sur-Yvette, France\
$ ^{11}$ Department of Physics and Astronomy, University of Glasgow, Glasgow, UK$^ b$\
$ ^{12}$ DESY, Hamburg, Germany$^a$\
$ ^{13}$ I. Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany$^ a$\
$ ^{14}$ II. Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany$^ a$\
$ ^{15}$ Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany$^ a$\
$ ^{16}$ Institut für Hochenergiephysik, Universität Heidelberg, Heidelberg, Germany$^ a$\
$ ^{17}$ Institut für Reine und Angewandte Kernphysik, Universität Kiel, Kiel, Germany$^ a$\
$ ^{18}$ Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic$^ f$\
$ ^{19}$ School of Physics and Chemistry, University of Lancaster, Lancaster, UK$^ b$\
$ ^{20}$ Department of Physics, University of Liverpool, Liverpool, UK$^ b$\
$ ^{21}$ Queen Mary and Westfield College, London, UK$^ b$\
$ ^{22}$ Physics Department, University of Lund, Lund, Sweden$^ g$\
$ ^{23}$ Physics Department, University of Manchester, Manchester, UK$^ b$\
$ ^{24}$ CPPM, Université d’Aix-Marseille II, IN2P3-CNRS, Marseille, France\
$ ^{25}$ Institute for Theoretical and Experimental Physics, Moscow, Russia\
$ ^{26}$ Lebedev Physical Institute, Moscow, Russia$^ f$\
$ ^{27}$ Max-Planck-Institut für Physik, München, Germany$^ a$\
$ ^{28}$ LAL, Université de Paris-Sud, IN2P3-CNRS, Orsay, France\
$ ^{29}$ LPNHE, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France\
$ ^{30}$ LPNHE, Universités Paris VI and VII, IN2P3-CNRS, Paris, France\
$ ^{31}$ Institute of Physics, Czech Academy of Sciences, Praha, Czech Republic$^{ f,h}$\
$ ^{32}$ Nuclear Center, Charles University, Praha, Czech Republic$^{ f,h}$\
$ ^{33}$ INFN Roma and Dipartimento di Fisica, Universita “La Sapienza”, Roma, Italy\
$ ^{34}$ Paul Scherrer Institut, Villigen, Switzerland\
$ ^{35}$ Fachbereich Physik, Bergische Universität Gesamthochschule Wuppertal, Wuppertal, Germany$^ a$\
$ ^{36}$ DESY, Institut für Hochenergiephysik, Zeuthen, Germany$^ a$\
$ ^{37}$ Institut für Teilchenphysik, ETH, Zürich, Switzerland$^ i$\
$ ^{38}$ Physik-Institut der Universität Zürich, Zürich, Switzerland$^ i$\
$ ^{39}$ Visitor from Yerevan Phys. Inst., Armenia\
$ ^{40}$ On leave from LBL, Berkeley, USA\
$ ^a$ Supported by the Bundesministerium für Forschung und Technologie, FRG, under contract numbers 6AC17P, 6AC47P, 6DO57I, 6HH17P, 6HH27I, 6HD17I, 6HD27I, 6KI17P, 6MP17I, and 6WT87P\
$ ^b$ Supported by the UK Particle Physics and Astronomy Research Council, and formerly by the UK Science and Engineering Research Council\
$ ^c$ Supported by FNRS-NFWO, IISN-IIKW\
$ ^d$ Supported by the Polish State Committee for Scientific Research, grant nos. 115/E-743/SPUB/P03/109/95 and 2 P03B 244 08p01, and Stiftung für Deutsch-Polnische Zusammenarbeit, project no.506/92\
$ ^e$ Supported in part by USDOE grant DE F603 91ER40674\
$ ^f$ Supported by the Deutsche Forschungsgemeinschaft\
$ ^g$ Supported by the Swedish Natural Science Research Council\
$ ^h$ Supported by GA ČR, grant no. 202/93/2423, GA AV ČR, grant no. 19095 and GA UK, grant no. 342\
$ ^i$ Supported by the Swiss National Science Foundation\
Introduction {#sect:intro}
============
The study of elastic production of vector mesons in photo- and leptoproduction (Fig. \[fig:diag\]a) in fixed target experiments has provided information on the hadronic component of the photon and on the nature of diffraction.
With the advent of the electron-proton collider HERA, there is renewed interest in vector meson production, in particular at large ( is minus the square of the exchanged photon four-momentum). HERA experiments have observed in deep-inelastic scattering that the proton structure function $F_2$ increases rapidly [@F2] with increasing , the invariant mass, which is in striking contrast with the slow rise of the total cross section at ${\mbox{$Q^2$}}\simeq 0$ [@totalgammaxsection]. In addition, the rise of $F_2$ already at relatively low values (${\mbox{$Q^2$}}\ {\raisebox{-0.5mm}{$\stackrel{<}{\scriptstyle{\sim}}$}}\ 2$ ) indicates that the transition between these two behaviours is rapid [@F2_low_Q2]. The study of the elastic production of vector mesons is expected to provide useful information about these different regimes, and in particular about the transition between them.
The subject of this paper [@Bxl_papers] is the study of elastic and meson electroproduction at large (${\mbox{$Q^2$}}> 8 $ ) and high ( $\simeq 90$ ), in the reactions $$e \ p \rightarrow e \ {\mbox{$\varrho$}}\ p \\ ;\
{\mbox{$\varrho$}}\rightarrow \pi^+ \ \pi^-, \label{eq:rh}$$ $$e \ p \rightarrow e \ {\mbox{$J/\psi$}}\ p \\ ;\
{\mbox{$J/\psi$}}\rightarrow l^+ \ l^- \\ ;\
l = e \ {\rm or} \ \mu. \label{eq:jpsi}$$
After the presentation of the event selection and of the mass distributions, the total cross sections and the differential distributions for reactions (\[eq:rh\]) and (\[eq:jpsi\]) are studied. Features relevant for the study of the transition regime are then discussed, in particular the energy dependence of the cross section, the distribution ( is the square of the four-momentum transferred from the photon to the target), the vector meson polarisation, and the evolution of the : cross section ratio. The results are discussed in the light of the interplay of hard and soft physics processes. Results on production in a similar kinematic range have been presented by the ZEUS Collaboration [@ZEUS].
Models and phenomenology {#sect:models}
========================
The elastic production of light vector mesons, in particular of mesons, by real or quasi-real photons (${\mbox{$Q^2$}}\simeq 0$), a process called hereafter photoproduction, exhibits numerous features typical of soft, hadron-like interactions. These are a small angle peak in the distribution of the vector meson scattering angle with respect to the incident photon beam direction (“elastic peak”), a steepening of this distribution with increasing energy (“shrinkage”), for $W\ {\raisebox{-0.5mm}{$\stackrel{>}{\scriptstyle{\sim}}$}}\ 10$ a slow increase with energy of the cross section, and s-channel helicity conservation ($SCHC$).
These observations support the vector meson dominance model (VDM) [@VDM; @Bauer], according to which a photon with energy greater than a few behaves predominantly as the superposition of the lightest (, , ) $J^{PC}=1^{--}$ mesons. In this framework, the cross section for elastic vector meson production is related to the total meson$-$proton cross section through the optical theorem, and the energy dependence is related to the “universal” energy dependence of the total hadron$-$proton cross section. Neglecting a contribution (“reggeon exchange”) which decreases with energy approximately as $s^{-0.5}$, the latter is parameterised at high energy as $\sigma_{tot} \propto
{\mbox{$s$}}^{\delta}$ with $\delta \simeq 0.0808$, $\sqrt{{\mbox{$s$}}}$ being the hadron$-$proton centre of mass energy [@DL_hadronxsection]. This is expressed in Regge theory as due to soft pomeron exchange. The elastic production of mesons has been studied extensively by fixed target experiments in photoproduction [@Bauer; @gp_rho], for intermediate [@moder_qsq_rho] and for ${\raisebox{-0.5mm}{$\stackrel{>}{\scriptstyle{\sim}}$}}\ 6$ [@EMC_NMC_rho]. Also at HERA, the photoproduction of mesons exhibits the characteristic features of soft interactions [@H1_Z_g_rho].
In contrast, the VDM approach does not give a satisfactory description of photoproduction data [@gp_jpsi; @moder_qsq_jpsi; @EMC_jpsi]: the cross section is smaller than the VDM prediction (see [@DL_1995]) and it increases significantly faster with energy than expected from soft pomeron exchange. This is also observed by the HERA experiments [@H1_Z_g_jpsi].
Using the hard scale provided by the mass of the charm quark, the fast increase of the photoproduction cross section was predicted in the framework of QCD by Ryskin [@Ryskin]. In his model, the interaction between the proton and the $c \bar c$ pair in the photon is mediated by a gluon ladder, and non-perturbative effects are included in the nucleon parton distribution.
At high energy and high , meson production is modelled using two different approaches based on QCD, which both refer to the pomeron as basically a two gluon system (see [@Low_Nussinov]). The meson production is thus related to the gluon distribution in the proton, but the two approaches emphasize respectively soft or hard behaviour.
In the soft model initially proposed by Donnachie and Landshoff [@DL_rho; @Cudell; @DL_1995], the gluons are described non-perturbatively and the elastic cross section is expected to increase slowly with energy: $d\sigma / dt\ (t=0) \propto W^{4 \delta}$, with $\delta \simeq 0.0808$. The vector mesons are predicted to be mostly longitudinally polarised, the cross section to fall as $Q^{-6}$ and the cross section to be comparable to that of the .
In the second approach, perturbative QCD calculations similar to the work of Ryskin have been performed by several groups [@Kope; @Brodsky; @Ginzburg], a hard scale being provided by the photon virtuality. A major prediction of this model is a rapid increase of the cross section with , as a consequence of the rise of the gluon distribution in the proton. It is stressed, however, that non-perturbative processes are also expected to be present in an intermediate energy region [@Kope; @F_K_S]. The scattering amplitude is obtained in these calculations from the convolution of the hadronic wave function of the photon, of the scattering amplitude of this hadronic component, and of the final state vector meson wave function. This is because the photon is viewed at sufficient energy as the coherent superposition of hadronic states formed well before the target (essentially $q \bar q$ pairs), whereas the final state meson is formed beyond it, the corresponding time scales being much longer than the interaction time. As a consequence, the spatial dimensions of the hadronic wave function are an essential parameter in the interplay of perturbative and non-perturbative effects. It is predicted that the hard effects should show up earlier for small size objects than for large ones, and for longitudinally than for transversely polarised photons. The dependence of the cross section is predicted, as in the non-perturbative approach, to fall as $Q^{-6}$ but when the evolution of the parton distribution and quark Fermi motion are taken into account, it was pointed out that the distribution is expected to be harder [@F_K_S]. In view of the more compact wave function, the : cross section ratio has been predicted [@F_K_S] to exceed, at very high energy, the value 8 : 9 obtained from SU(4) and the quark counting rule [@quark-counting_rule].
In addition to these two “microscopic” approaches inspired by QCD, calculations are also performed for elastic vector meson production on the basis of multiple pomeron exchange, with the effective pomeron intercept depending on the photon virtuality [@Kaidalov]. A stronger increase of the cross section with energy is again predicted than for soft pomeron exchange with $\delta \ \simeq 0.0808$.
Detector and event selection {#sect:detector}
============================
The data presented here correspond to an integrated luminosity of 2.8 for the and 3.1 for the mesons. They were collected in 1994 using the H1 detector. HERA was operated with 27.5 positrons and 820 protons[^1]. The detector is described in detail in ref. [@detector].
The event final state corresponding to reactions (\[eq:rh\]) and (\[eq:jpsi\]) consists of the scattered positron and two particles of opposite charges, originating from a vertex situated in the nominal $e^+p$ interaction region. In most cases, the scattered proton remains inside the beam pipe because of the small momentum transfer to the target in elastic interactions.
In the range studied here, the positron is identified as an electromagnetic cluster with an energy larger than 12 GeV, reconstructed in the backward electromagnetic calorimeter (BEMC) and associated with a hit in the proportional chamber (BPC) placed in front of it at 141 cm from the nominal interaction vertex[^2]. The BEMC covers the polar angles $151^\circ <\rm \theta < 176^\circ$. Its electromagnetic energy resolution is $\sigma_{E}/E \simeq 6 - 7$% in the energy range of the positrons selected for the present studies. The BPC angular acceptance is $155.5^\circ < \theta < 174.5^\circ$. The scattered positron polar angle $\rm{\theta_e}$ is determined from the positions of the BPC hit and of the interaction vertex. The trigger used for the present analyses requires the presence of a total energy larger than 10 deposited in the BEMC, outside a square of $32 \times 32\ {\rm cm^2}$ around the beam pipe. Additional cuts, similar to those used for the structure function analysis \[1a\], are applied to the hit and cluster position and shape in order to provide high trigger efficiency and good quality positron measurement. These cuts are complemented by the selection of events with $>$ 8 .
The decay pions ( events) or leptons ( events) are detected in the central tracking detector, consisting mainly of two coaxial cylindrical drift chambers, 2.2 m long and respectively of 0.5 and 1 m outer radius. The charged particle momentum component transverse to the beam direction is measured in these chambers by the track curvature in the 1.15 T magnetic field generated by the superconducting solenoid which surrounds the inner detector, with the field lines directed along the beam axis. Two polygonal drift chambers with wires perpendicular to the beam direction, located respectively at the inner radius of the two coaxial chambers, are used for a precise measurement of the particle polar angle. For the present analyses, two tracks with transverse momenta $p_{t}$ larger than 0.1 GeV/c are required to be reconstructed in the central region of the tracker. For production, the polar angles must lie in the range $25^\circ < \theta < 155^\circ$, corresponding to particles completely crossing the inner cylindrical drift chamber for interactions at the nominal vertex position. For production, the accepted range is extended to $20^\circ < \theta < 160^\circ$ in order to increase the statistics as much as possible while keeping good detection efficiency. The vertex position is reconstructed using these tracks. No other track linked to the interaction vertex is allowed in the tracking detector, except possibly the positron track. To suppress beam-gas interactions, the vertex must be reconstructed within 30 cm of the nominal interaction point in $z$, which corresponds to 3 times the width of the vertex distribution. In addition, the accepted events are restricted to the range $40 < {\mbox{$W$}}< 140$ for mesons and $30 < {\mbox{$W$}}< 150$ for mesons.
The tracking detector is surrounded by a liquid argon calorimeter situated inside the solenoid and covering the polar angular range $4^\circ < \rm \theta < 153^\circ$ with full azimuthal coverage. In the case of elastic interactions, the calorimeters should register only activity associated with the decay particles or the positron. However, due to noise in the calorimeters and to the small pile-up from different events, elastic and production can be accompanied by the presence of additional energy clusters. These are considered in terms of the variable , defined as the energy of the most energetic cluster which is not associated with a track. The distribution shows a peak at small values, attributed mostly to elastic and diffractive interactions, and a broad maximum at higher energies. The cut $<$ 1 is applied to enhance exclusive production. This is discussed in section \[sect:mass\], together with the effect of the cut $<$ 0.5 .
The sample of events containing a positron and a vector meson candidate includes two main contributions: elastic production (see Fig. \[fig:diag\]a), defined by reactions (\[eq:rh\]) and (\[eq:jpsi\]), and events where the proton is diffractively excited into a system $X$ of mass $M_X$, which subsequently dissociates (Fig. \[fig:diag\]b). Non-resonant background is also present. It is possible to identify most of the “proton dissociation” events with the components of the H1 detector placed in the forward region [@diffrpaper], namely the forward part of the liquid argon calorimeter $4^\circ \leq \theta \leq 10^\circ$, the forward muon detectors (arrays of muon chambers placed around the beam pipe in the proton direction, $3^\circ \leq \theta \leq 17^\circ$) and the proton remnant tagger (an array of scintillators placed 24 m downstream of the interaction point, $0.06^\circ \leq \theta \leq 0.17^\circ$). When particles from the diffractively excited system interact in the beam pipe and the collimators, the interaction products can be detected in these forward detectors. The events are tagged as due to proton dissociation by the presence of a cluster with energy $E^{LAr}_{fw}$ larger than 1 (0.75 for the candidates) at an angle $\theta^{LAr}_{fw} < 10 ^\circ$ in the liquid argon calorimeter, or by at least 2 pairs of hits in the forward muon detectors (one hit pair is compatible with noise), or by at least one hit in the proton remnant tagger. This last criterion is not used for the events, since they have a flatter distribution than the events and would be partially vetoed by the proton tagger.
In order to minimise the effects of QED radiation in the initial state, the difference between the total energy and the total longitudinal momentum of the positron and the two particles emitted in the central part of the detector is required to be larger than 45 GeV. If no particle, in particular a radiated photon, has escaped detection in the backward direction, should be twice the incident positron energy, i.e. 55 .
The selection criteria for the two samples, supplemented by the mass selections discussed in section \[sect:mass\], are summarised in Table \[tab:sel\].
-- -- --
-- -- --
: Selection criteria for and events.[]{data-label="tab:sel"}
Kinematics and cross section definitions {#sect:kin}
========================================
The kinematics of reactions (\[eq:rh\]) and (\[eq:jpsi\]) are described with the variables commonly used for deep-inelastic interactions. In addition to (the square of the $e^+p$ centre of mass energy), and , it is useful to define the two Bjorken variables $ y= p \cdot q / p \cdot k $ (in the proton rest frame, the energy fraction transferred from the positron to the hadrons) and $ x= {\mbox{$Q^2$}}/ 2p \cdot q $, where $k$, $p$, $q$ are, respectively, the four-momenta of the incident positron, of the incident proton and of the virtual photon. These variables[^3] obey the relations ${\mbox{$Q^2$}}= x y s $ and $W^2 = {\mbox{$Q^2$}}\ (\frac{1}{{\mbox{$x$}}} -1)$.
The kinematical variables can be reconstructed from four measured quantities: the energies and the polar angles of the scattered positron and of the vector meson. With the “double angle” method [@Koijman_Workshop] used for the present analyses, and are computed using the polar angles $\theta$ and $\gamma$ of the positron and of the vector meson, which are well measured: $${\mbox{$Q^2$}}= 4 E_0^2 \ \frac {\sin\gamma \ (1 + \cos \theta)}
{\sin\gamma + \sin\theta - \sin(\gamma + \theta)} ,
\label{eq:qsq}$$ $$y = \frac {\sin\theta \ (1 - \cos \gamma)}
{\sin\gamma + \sin\theta - \sin(\gamma + \theta)} ,
\label{eq:y}$$ where $E_0$ is the energy of the incident positron.
The meson momentum components are obtained from the measured decay products. The momentum of the scattered positron is computed from and , which provides better precision than the direct measurement. The energy transfer to the proton being negligible, the absolute value of is given by: $${\mbox{$|t|$}}\simeq (\vec{p}_{tp})^{2} = (\vec{p}_{te} + \vec{p}_{tv})^2, \label{eq:t}$$ where $\vec{p}_{tp}$, $\vec{p}_{te}$ and $\vec{p}_{tv}$ are, respectively, the momentum components transverse to the beam direction of the final state proton, positron and vector meson[^4].
The fourth quantity which is directly measured, the positron energy, is used to compute the variable : $$E-p_z = (E_e + E_v) - (p_{ze} + p_{zv}), \label{eq:eminpz}$$ $E_e$ and $E_v$ being the energies of the scattered positron and of the vector meson, and $p_{ze}$ and $p_{zv}$ their momentum components parallel to the beam direction.
The cross section for elastic electroproduction of a vector meson $V$ can be converted into a cross section using the relation $$\frac{d^2 \sigma_{tot} (ep \rightarrow e V p)}{dy \ dQ^2} =
\Gamma \ \sigma_{tot} (\gamma ^*p \rightarrow V p) = \
\Gamma \ \sigma_{T} (\gamma ^*p \rightarrow V p) \ (1 + \varepsilon \ {\mbox{$R$}}),
\label{eq:sigma}$$ where $\sigma_{tot}$, $\sigma_{T}$ and $\sigma_{L}$ are the total, transverse and longitudinal cross sections, $$R\ = \sigma_{L} / \sigma_{T},
\label{eq:Rdef}$$ and $\Gamma$ is the flux of transverse virtual photons given by $$\Gamma = \frac {\alpha_{em} \ (1-{\mbox{$y$}}+{\mbox{$y$}}^2/2)} {\pi\ {\mbox{$y$}}\ {\mbox{$Q^2$}}};
\label{eq:flux}$$ $\varepsilon$ is the polarisation parameter $$\varepsilon = \frac{1 - {\mbox{$y$}}}{1-{\mbox{$y$}}+{\mbox{$y$}}^2/2}.
\label{eq:epsil}$$
Information on the vector meson production process can be obtained from the angular distributions of the decay particles. In particular, the probability for the meson to be longitudinally polarised can be determined from the distribution of , where $\theta^*$ is the angle, in the rest frame, between the direction of the positively charged decay pion and the direction in the centre of mass system (helicity frame) [@Bauer; @Schilling_Wolf]: $$\frac {{\rm d}N} {{\rm d}{\mbox{$\cos\theta^*$}}} \propto 1 - {\mbox{$r_{00}^{04}$}}+ (3 \ {\mbox{$r_{00}^{04}$}}-1) \ \cos^2\theta^* .
\label{eq:costhst}$$ With the assumption of s-channel helicity conservation ($SCHC$), is related to : $${\mbox{$R$}}= \frac{1}{\varepsilon} \ \frac{{\mbox{$r_{00}^{04}$}}}{1-{\mbox{$r_{00}^{04}$}}} .
\label{eq:R}$$
Mass distributions and final samples {#sect:mass}
====================================
Fig. \[fig:mass\]a shows for the selected events (Table \[tab:sel\]) the distribution of the invariant mass in the range $<$ 2 , obtained by assigning the pion mass to the particles detected in the central tracker. The mass distribution without the and cuts (insert in Fig. \[fig:mass\]a) is seen to peak at small values. The cut strongly reduces the background of events containing neutral particles, and enhances the peak. The cut is very effective in rejecting non-resonant events containing, in addition to the candidate and the positron, particle(s) with a significant transverse momentum, which is not used to compute $\vec{p}_{tp}$ in eq. (\[eq:t\]). This cut also enhances the elastic production signal compared to the background of proton dissociation events, which are known to have a flatter distribution. In total, 180 events are found in the peak region with $0.6 < {\mbox{$m_{\pi^+\pi^-}$}}< 1.0$ .
(160,80) (1,0) (39.63,36) (80,0)
In Fig. \[fig:mass\]a, events compatible with the mass, when the charged particles detected in the central tracker are considered as kaons, have been removed ($m_{K^+K^-} < 1.04$ ). Assuming that vector mesons are produced according to the quark counting rule with the SU(3) ratios : : = 9 : 1 : 2, the Monte Carlo simulation described below indicates that the remaining and reflections contribute, in the range (0.4 $-$ 0.6) , 2.0% of the signal in the peak, and 0.7% in the range (0.6 $-$ 1.0) . These contributions were subtracted statistically.
The distribution is described by a relativistic Breit-Wigner function, over a non-resonant background attributed to incompletely reconstructed diffractive photon dissociation. The Breit-Wigner function has the form [@Jackson] $$\frac {dN({\mbox{$m_{\pi\pi}$}})} {d{\mbox{$m_{\pi\pi}$}}} = \frac {{\mbox{$m_{\pi\pi}$}}\ {\mbox{$m_{\rho}$}}\ {\mbox{$\Gamma ({\mbox{$m_{\pi\pi}$}})$}}}
{({\mbox{$m_{\rho}^2$}}- {\mbox{$m_{\pi\pi}^2$}})^2 + {\mbox{$m_{\rho}^2$}}\ {\mbox{$\Gamma^2({\mbox{$m_{\pi\pi}$}})$}}},
\label{eq:b_w}$$ with the mass dependent width $${\mbox{$\Gamma ({\mbox{$m_{\pi\pi}$}})$}}= {\mbox{$\Gamma_{\rho}$}}\ (\frac {q^*} {q_0^*})^3 \
\frac {2} {1 + (q^* / q^*_0)^2}.
\label{eq:GJ}$$ Here is the resonance mass and the width; $q^*$ is the pion momentum in the ($\pi^+\pi^-$) rest frame, and $q^*_0$ this momentum when ${\mbox{$m_{\pi^+\pi^-}$}}= {\mbox{$m_{\rho}$}}$.
The background has been parameterised using the distribution $$\frac {dN({\mbox{$m_{\pi\pi}$}})} {d{\mbox{$m_{\pi\pi}$}}} = {\alpha_1 \ ({\mbox{$m_{\pi\pi}$}}- 2 {\mbox{$m_{\pi}$}})^{\alpha_2} \ e^{-\alpha_3 {\mbox{$m_{\pi\pi}$}}}},
\label{eq:ps_bg}$$ where is the pion mass and $\alpha_1$, $\alpha_2$ and $\alpha_3$ are free parameters. This form, which includes a two pion threshold and an exponential fall off, is in qualitative agreement with the background shape in the insert of Fig. \[fig:mass\]a.
With these parameterisations, the resonance mass is 763 $\pm$ 10 and the width is $176 \pm 23$ , in agreement with the Particle Data Group (PDG) values of 770 and 151 [@PDG]. No skewing to low values of is needed to describe the shape (for the Ross-Stodolsky parameterisation [@RS], the skewing exponent is found to be $n = 0.3 \pm 0.5$).
Two alternative forms have been used for the resonance width: $${\mbox{$\Gamma ({\mbox{$m_{\pi\pi}$}})$}}= {\mbox{$\Gamma_{\rho}$}}\ (\frac {q^*} {q_0^*})^3 \ \frac {{\mbox{$m_{\rho}$}}} {{\mbox{$m_{\pi\pi}$}}}
\label{eq:Gmm}$$
and $${\mbox{$\Gamma ({\mbox{$m_{\pi\pi}$}})$}}= {\mbox{$\Gamma_{\rho}$}}\ (\frac {q^*} {q_0^*})^3.
\label{eq:G1}$$ They also give a meson mass and width compatible with the PDG data. The non-resonant background under the peak is estimated to be 11 $\pm$ 6%. The error includes the uncertainty on the resonance parameterisation and on the background shape, estimated by using for the latter an alternative linearly decreasing form.
The distribution of the invariant mass for the selected events in the region is presented in Fig. \[fig:mass\]b. The mass is $3.13 \pm 0.03$ . The peak width is slightly larger than, but compatible with, the expectation obtained from the detector simulation. No selection is applied since the distribution is significantly flatter than for the mesons (see Fig. \[fig:modt\_1\]). No cut is required in view of the small background in this high mass region (compare Fig. \[fig:mass\]b and insert in Fig. \[fig:mass\]a): all candidate events have $<$ 1.35 , of which 4 have larger than 1 .
A sample of 31 candidate events is thus selected with the cut $| {\mbox{$m_{l^+l^-}$}}-m_{\psi} |$ $< 300\ {\rm MeV/c^2}$, where $m_{\psi}$ is the meson mass. The non-resonant background is estimated by fitting the sidebands using an exponential distribution and amounts to roughly 20% (6.8 events). No lepton identification is required, but 10 of the candidate events contain two identified electrons and 7 contain two identified muons.
One event with $>$ 1 is a $\psi^\prime \rightarrow {\mbox{$J/\psi$}}\pi^0 \pi^0$ candidate, with one identified muon and neutral clusters detected in the electromagnetic part of the liquid argon calorimeter, attributed to the interaction of photons from $\pi^0$ meson decay. The measured mass is $3.16 \pm 0.04$ . The invariant mass computed using the two charged tracks and the neutral clusters is $3.67 \pm 0.09$ , in excellent agreement with the Particle Data Group value (3.69 ) [@PDG].
Kinematical characteristics of the selected events are summarised in Table \[tab:events\].
-------- ---------------- ----------------
\[\] $13.4 \pm 0.4$ $17.7 \pm 1.5$
\[\] $81 \pm 2$ $92 \pm 6$
\[\] $25.5 \pm 0.1$ $24.9 \pm 0.3$
\[\] $3.4 \pm 0.1$ $3.9 \pm 0.2$
\[\] $4.2 \pm 0.1$ $6.0 \pm 0.3$
\[\] $1.7 \pm 0.1$ $2.3 \pm 0.2$
: Averages of kinematical variables characterising the selected events.[]{data-label="tab:events"}
Corrections and simulations {#sect:sim}
===========================
Table \[tab:corr\] summarises the correction factors applied to the selected samples to take account of detector acceptance and efficiencies, smearing effects, losses due to the selection criteria and remaining backgrounds.
Most corrections are estimated using a Monte Carlo simulation based on the vector meson dominance model, which permits variation of the , and dependences, as well as of the value of [@Benno]. The H1 detector response is simulated in detail, and the events are subjected to the same reconstruction and analysis chain as the data.
The accuracies of the () variable measurements are for , 3.3 (4.3) , for , 0.4 (0.4) , for , 0.06 (0.10) . The values of the slopes are little affected by the detector resolution.
The scattered positron selection criteria induce dependent losses for $\leq$ 12 . The error quoted in Table \[tab:corr\] corresponds to a systematic uncertainty on the positron direction of 2 mrad. The charged track selection criteria induce dependent losses for small and high , depending on the accepted range in the two selections. A ${\mbox{$Q^2$}}-{\mbox{$W$}}$ correlation of the losses is observed, and taken into account in the corrections. The correction for the cut in the sample is computed using the measured slope (see section \[sect:t\]).
--------------------------------- ----------------- -----------------
trigger
positron acceptance ( dep.) $1.16 \pm 0.03$ $1.15 \pm 0.03$
BPC hit $-$ cluster link
tracker acceptance ( dep.) $1.07 \pm 0.01$ $1.29 \pm 0.03$
track reconstr. (per track) $1.05 \pm 0.03$ $1.03 \pm 0.03$
track $p_{tmin}$ (per track) $1.02 \pm 0.01$ $1.00 \pm 0.01$
cut $1.03 \pm 0.02$
cut $1.02 \pm 0.01$ $1.01 \pm 0.01$
cut $1.03 \pm 0.03$
forward det. cuts ( dep.) $1.04 \pm 0.02$ $1.03 \pm 0.03$
mass selection $1.22 \pm 0.01$ $1.00 \pm 0.02$
non-resonant background $0.89 \pm 0.06$ $0.78 \pm 0.14$
proton dissoc. background $0.91 \pm 0.08$ $0.75 \pm 0.11$
$\phi$ and $\omega$ background $0.99 \pm 0.01$
photon flux / bin integration $1.00 \pm 0.04$ $1.00 \pm 0.07$
radiative corrections $0.96 \pm 0.03$ $1.00 \pm 0.04$
luminosity
: Correction factors and systematic errors, averaged over the data samples.[]{data-label="tab:corr"}
The choice of the cut for the sample is a compromise between the loss of elastic events to which a cluster is accidentally associated in the calorimeter, and the presence of non-resonant background in the final sample. The loss is estimated using a Monte Carlo simulation which includes random noise in the calorimeters superimposed on elastic events. A simulation indicates that 2% of the elastic events with $<$ 0.5 are lost because the proton has acquired sufficient $\vec{p}_t$ to interact in the beam pipe walls, giving interaction products which are registered in the proton tagger; this loss is thus dependent. For the sample, 1.5% of the events are lost because of interaction products registered in the forward muon detectors. Another 2% loss for both samples is due to spurious hits in the latter.
In view of the uncertainty on the high mass shape of the resonance, the cross section is quoted in this paper for $<$ 1.5 , i.e. $\simeq m_{\rho} + 5 \ \Gamma_{\rho}$. The uncertainty is larger in the present case than for low energy data, for which a natural cut off is imposed by the limited available energy. With this definition of resonance production line shape, the correction for the mass selection 0.6 $<$ $<$ 1.0 is respectively $21\%$ and $23\%$ for parameterisations (\[eq:GJ\]) and (\[eq:Gmm\]).
A simulation was performed in order to estimate the contribution to the final samples of proton dissociation events which are not tagged by the forward detectors. The distribution of the target mass $M_X$ is parameterised as $1/M_X^2$. High mass states are assumed to decay according to the Lund string model [@jetset] or, alternatively, to a final state with particle multiplicity following the KNO scaling law and isotropic phase space distribution. In the resonance domain, the mass distribution follows measurements from $p$ dissociation on deuterium [@Goulianos] and resonance decays are described according to their known branching ratios. The decay particles are followed through the beam pipe walls and the forward detectors.
For the sample, the correction factor for the contamination of undetected proton dissociation events in the selected sample is $0.91 \pm 0.08$. This number is obtained from the number of measured events tagged and not tagged by the forward detectors, and from the detection probabilities provided by the Monte Carlo simulation. No assumption needs to be made for the ratio of proton dissociation to elastic events. The error is a conservative estimate taking into account the uncertainties on the efficiencies of the forward detectors for tagging proton dissociation events and on the dissociation model.
The correction factor for unobserved proton dissociation background in the selected sample, for which the proton tagger is not used, is $0.75 \pm 0.11$.
The cross section measurements are given in the QED Born approximation for electron interactions. The effects of higher order processes are estimated using the HERACLES 4.4 generator [@Heracles].
Radiative corrections for the sample are of the order of 4% after the cut $>$ 45 , and are weakly dependent on and . A systematic error of 3% is obtained by varying the effective dependence of the cross section from $Q^{-4}$ to $Q^{-6}$ and by modifying the dependence from a constant to a linearly increasing form[^5]. The small value of the correction is due to the high cut resulting from the good BEMC resolution; for the chosen value of the cut, small smearing effects are observed.
For the sample, the radiative corrections determined using the measured and dependences of the cross section vary from $+2$% to $-2$%.
Results {#sect:results}
=======
Electroproduction cross sections {#sect:xsect}
--------------------------------
The and data are grouped in several (, ) bins. Table \[tab:xsect\] gives, for each bin, the number of events, the integrated $ep$ cross section and the cross section obtained using relation (\[eq:sigma\]) for a given ($Q_0^2,W_0$) value, taking into account the observed dependence across the bin. All known smearing, acceptance and background effects are corrected for. For the sample, each event is weighted using the differential flux factor given by eq. (\[eq:flux\]). A 4% systematic error accounts for the uncertainty in the and dependences of the cross section used for the bin size integration and the bin centre correction. For the sample, in view of the small statistics, the photon flux is integrated over each (,) bin. Since the data span a large range in and , this leads to a systematic error on the cross section of the order of 7%.
The integrated cross section for meson electroproduction with $<$ 1.5 is $$\sigma(e\ p \rightarrow e\ {\mbox{$\varrho$}}\ p) = 96 \pm 7\ (stat.) \pm 13\ (syst.)\ {\rm pb},
\label{eq:xrho}$$ for $>$ 8 and $40 < {\mbox{$W$}}< 140$ .
The cross section for meson electroproduction, taking into account the ${\mbox{$J/\psi$}}\rightarrow$ 2 leptons branching fraction 0.12 [@PDG], is $$\sigma(e\ p \rightarrow e\ {\mbox{$J/\psi$}}\ p) = 100 \pm 20\ (stat.) \pm 20\ (syst.)\ {\rm pb},
\label{eq:xjpsi}$$ for $>$ 8 and $30 < {\mbox{$W$}}< 150$ .
The ratio : is $0.64 \pm 0.13$ for = 10 and $1.3 \pm 0.5$ for = 20 .
[|l|l|l|]{}\
&\
& $40 < W < 80$ & $80 < W < 140$\
number of events & 57 & 47\
total correction factor & $1.53 \pm 0.22 $ & $1.76 \pm 0.25 $\
integrated $ep$ cross section \[pb\] & $31.5 \pm 4.2 \pm 4.5 $ & $29.8 \pm 4.3 \pm 4.2 $\
$Q_0^2$ \[\], $W_0$ \[\] & 10, 65 & 10, 115\
cross section \[nb\] & $25.8 \pm 3.4 \pm 3.7 $ & $29.4 \pm 4.3 \pm 4.2 $\
&\
& $40 < W < 80$ & $80 < W < 140$\
number of events & 39 & 37\
total correction factor & $1.32 \pm 0.19 $ & $1.23 \pm 0.18 $\
integrated $ep$ cross section \[pb\] & $18.5 \pm 3.0 \pm 2.7 $ & $16.3 \pm 2.7 \pm 2.4 $\
$Q_0^2$ \[\], $W_0$ \[\] & 20, 65 & 20, 115\
cross section \[nb\] & $5.0 \pm 0.8 \pm 0.7 $ & $5.0 \pm 0.8 \pm 0.7 $\
\
&\
& $30 < W < 90$ & $90 < W < 150$\
number of events & 15 & 16\
total correction factor & $1.50 \pm 0.27 $ & $0.92 \pm 0.17 $\
integrated $ep$ cross section \[pb\] & $61 \pm 18 \pm 12 $ & $40 \pm 12 \pm 8 $\
$Q_0^2$ \[\], $W_0$ \[\] & 16, 65 & 16, 115\
cross section \[nb\] & $7.8 \pm 2.2 \pm 1.6 $ & $12.2 \pm 3.4 \pm 2.5 $\
&\
& $8 < {\mbox{$Q^2$}}< 12$ & $12 < {\mbox{$Q^2$}}< 40$\
number of events & 10 & 21\
total correction factor & $1.82 \pm 0.33 $ & $0.90 \pm 0.16 $\
integrated $ep$ cross section \[pb\] & $49 \pm 18 \pm 10 $ & $51 \pm 13 \pm 10 $\
$Q_0^2$ \[\], $W_0$ \[\] & 10, 88 & 20, 88\
cross section \[nb\] & $17.6 \pm 6.3 \pm 3.7 $ & $6.6 \pm 1.6 \pm 1.4 $\
Momentum transfer distributions {#sect:t}
-------------------------------
The distributions of the selected and events (Fig. \[fig:modt\_1\]) show the forward exponential peaking $\propto e^{bt}$ characteristic of elastic interactions.
For the sample, the slope of the distribution is computed taking into account the contributions of the non-resonant and proton dissociation backgrounds estimated in sections \[sect:mass\] and \[sect:sim\]. Their exponential slopes were taken to be respectively $0.15 \pm 0.10$ and $2.5 \pm 1.0$ , which is consistent with the dependence of event samples which approximate these contributions. The results quoted below are rather insensitive to the choice of these slopes. The distribution is corrected for detector effects, including the loss of elastic events tagged by the forward detectors.
A fit for $<$ 0.6 gives for the elastic slope the value = $7.0 \pm 0.8 \pm 0.4 \pm 0.5$ ($\chi^2 = 9.6 \ / 10$ d.o.f.) for $>$ 8 and $40 < W < 140$ . The first error corresponds to the statistical precision of the fit. The second describes the spread of the fits according to the choice of the range (0.4 to 0.6 ) and of the cut value. The third comes from the uncertainties in the sizes and shapes of the backgrounds; it is dominated by the error on the total contribution of the non-resonant background. The slope measured by the ZEUS Collaboration is = $5.1~ _{-0.9} ^{+1.2} \pm 1.0$ [@ZEUS]. The values of the slopes for two and two domains are given in Table \[tab:slope\].
[|c|c|]{}\
$8 < {\mbox{$Q^2$}}< 12$ & $12 < {\mbox{$Q^2$}}< 50$\
= $7.8 \pm 1.0 \pm 0.7$ & = $5.7 \pm 1.3 \pm 0.7$\
\
$40 < W < 80$ & $80 < W < 140$\
= $6.2 \pm 1.0 \pm 0.7$ & = $8.0 \pm 1.3 \pm 0.7$\
For the sample, the slope value for $<$ 1.0 is = $3.8 \pm 1.2 \ _{-1.6}^{+2.0}$ , after subtraction of the proton dissociation and non-resonant backgrounds with slopes = 2 . The systematic error is estimated by varying the background contributions by one standard deviation and their slopes between 0 and 3 . The combined value of the HERA experiments \[17a-b\] for the slope in photoproduction is = $4.0 \pm 1.0 $ . Three events have $>$ 1.1 (see Fig. \[fig:modt\_1\]b), of which one has $>$ 1 . These 3 events contribute 15% to the cross section quoted in this paper. The $\psi^\prime$ candidate event (see section \[sect:mass\]) has $= 0.16$ , being computed including the neutral clusters attributed to $\pi ^0$ mesons ( = 26.3 , = 72.6 ).
Fig. \[fig:modt\_2\]a shows that for the domain of the HERA experiments the decrease with rising of the slope for elastic production is similar to that observed at lower .
The comparison of the NMC and H1 results for $\simeq$ 10 (Fig. \[fig:modt\_2\]b) shows an increase of the slope with energy. This shrinkage of the elastic peak with (or $\sqrt {{\mbox{$s$}}}$) is observed in diffractive hadron interactions [@Goulianos] and in photoproduction (see the comparison of fixed target and HERA results in Fig. \[fig:modt\_2\]b). In the framework of Regge theory, for pomeron exchange and in terms of the exponential parameterisation, the shrinkage of the elastic peak can be written $$b(W^2) = b(W^2=W_0^2) + 2 \ \alpha^\prime \ \ln(W^2 / W_0^2),
\label{eq:b}$$ where $\alpha^\prime$ is the slope of the effective pomeron Regge trajectory: $$\alpha_{{{\rm l \! P }}}(t) = \alpha_{{{\rm l \! P }}}(0)+ \alpha^\prime \ {\mbox{$t$}}. \label{eq:pom}$$ Applying relation (\[eq:b\]) to the = 10 results (with statistical and systematic errors combined quadratically) gives for $\alpha^\prime$ the value 0.41 $\pm$ 0.18 , in agreement with a value of 0.25 deduced from hadronic interactions [@D_L_alphaprim]. For the H1 data alone, there is also an indication for an increase of the slope with (see Table \[tab:slope\]).
dependence of the cross sections {#sect:q2}
---------------------------------
The dependence of the total cross section ($\sigma_{tot} = \sigma_T + \varepsilon \ \sigma_L$) for the elastic meson production by virtual photons (Fig. \[fig:q2\]a) can be described by $Q^{-2n}$ with $n = 2.5 \pm 0.5 \pm 0.2$. In extracting the dependence on of the cross section, correction has been made for the presence of non-resonant background, for which $n = 1.5 \pm 0.2$ as obtained from the events with $>$ 0.5 or $>$ 1 . The second error on $n$ reflects the uncertainty on the background size and shape and the spread of the results according to the details of the fitting procedure. The cross section dependence for the present data is close to that obtained by NMC ($n = 2.02 \pm 0.07$) and by ZEUS ($n = 2.1 \pm 0.4\ ^{+0.7}_{-0.3}$ for $0.0014 < x < 0.004$). It should be noted, however, that the NMC data span a large range in the polarisation parameter $\varepsilon$ from $\varepsilon = 0.50$ at = 2.5 to $\varepsilon = 0.80$ for $>$ 10 , whereas the HERA data are for $\varepsilon = 0.99$. Although the dependence is probably sensitive to this kinematical effect, it was not taken into account because the evolution of with in the NMC data (see eq. \[eq:sigma\]) is not published. The differences in the absolute normalisations are discussed in section \[sect:w\].
The dependence of the production cross section at HERA is shown in Fig. \[fig:q2\]b. The errors on the high data points include the uncertainty in the dependence of the background. The evolution from photoproduction to high is well described by $1 \ / \ ({\mbox{$Q^2$}}+ m_\psi^2)^n$ with $n = 1.9 \pm 0.3\ ({\it stat.})$. This is similar to the dependence of the low energy EMC results [@EMC_jpsi], for which a fit of the data shown in Fig. \[fig:q2\]b gives $n = 1.7 \pm 0.1$.
dependence of the cross sections {#sect:w}
---------------------------------
The dependence of the (for $<$ 1.5 ) and production cross sections is shown on Fig. \[fig:w\] for $=$ 10 and 20 . These values are chosen in order to minimise the bin centre corrections for the analysis.
The ZEUS results in Fig. \[fig:w\]a have been scaled to the values of the H1 measurements using the dependence of the latter (see section \[sect:q2\]). These results can be directly compared to the H1 results although they include no explicit cut-off on the mass. Indeed, the cross sections quoted in [@ZEUS] are determined assuming a non-relativistic Breit-Wigner mass distribution with mass independent width, integrated over the full kinematical range. It turns out that this procedure leads to a cross section closely similar to that obtained using the relativistic form of eq. (\[eq:b\_w\]-\[eq:GJ\]) for $<$ 1.5 , as is done in the present experiment. The ZEUS results are higher than those of H1, but the discrepancy is not very significant when the overall 31% systematic error on the ZEUS results [@ZEUS], which is not included in the plot, is taken into account. Other differences between the results of the two experiments are observed: both the ZEUS (section \[sect:t\]) and $\cos\theta^*$ distributions (section \[sect:cos\]) are flatter than for H1. It is worth emphasizing in this context that the event by event selection in the H1 analysis, which uses the forward detectors, provides a very clean elastic sample.
The NMC results shown in Fig. \[fig:w\]a are obtained from the published dependence of the cross section for interactions on deuterium (Fig. 4 of \[11d\]), which has small nuclear corrections [@Sandacz; @Nuov_cim]. They are corrected for the different values of the polarisation parameter $\varepsilon$ in the two experiments (the NMC measurement of is used). The published NMC cross section was computed using for the resonance the Breit-Wigner parameterisation given by eq. (\[eq:b\_w\]) and (\[eq:Gmm\]), for $<$ 1.5 [@Sandacz]. The results of the two experiments can thus be directly compared. An overall 20% systematic uncertainty \[11d\] is not included in the plot.
A significant increase with energy of the elastic cross section is observed from the NMC to the HERA domains. Following section \[sect:models\], it is parameterised as $d\sigma / dt \ (t=0)\ \propto {\mbox{$W$}}^{4 \delta}$. The fitted values of $\delta$ are for elastic production: $${\mbox{$Q^2$}}= 10\ {\mbox{${\rm GeV}^2$}}\ :\ \ \delta = 0.14 \pm 0.05,
\label{eq:del1}$$ $${\mbox{$Q^2$}}= 20\ {\mbox{${\rm GeV}^2$}}\ : \ \ \delta = 0.10 \pm 0.06.
\label{eq:del2}$$ The errors result from the combination of statistical and systematic errors of both experiments, including the 20% normalisation uncertainty for NMC.
The measurements (\[eq:del1\]) and (\[eq:del2\]) take into account the following effects:
$-$ the $d\sigma / dt \ (t=0)$ cross sections are obtained by multiplying the total cross sections by the corresponding slopes. The H1 slopes given in Table \[tab:slope\] were used and the corresponding NMC slopes were computed[^6] according to the shrinkage description given by eq. (\[eq:b\]) with $\alpha^\prime = 0.25$ .
$-$ the cross section definition[^7] contains a kinematical factor due to phase space integration, involving the centre of mass energy , the mass squared of the particles and . This factor is not part of the study of the interaction dynamics contained in the evolution of the matrix element. As the values considered here are rather large compared to $W^2$ for the NMC experiment and small for H1, there is a rising contribution to the dependence amounting to 12% (26%) between NMC and H1 energies for ${\mbox{$Q^2$}}= 10$ (20) . This corresponds to a decrease of $\delta$ by 0.02 (0.03), which is included in the measurements (\[eq:del1\]) and (\[eq:del2\]). Additional effects may have to be taken into account:
$-$ model predictions are often computed for the longitudinal cross section $\sigma_L$ and not for the total cross section $\sigma_{T} + \varepsilon \ \sigma_L$. Taking into account the difference in the values for the two experiments (see section \[sect:cos\]), the $\delta$ values for $\sigma_L$ alone would be increased by 0.02 with respect to the values (\[eq:del1\]) and (\[eq:del2\]) (a possible dependence of is not considered).
$-$ at the NMC energies, reggeon exchange could contribute significantly to elastic production. Following the parameterisation obtained by Donnachie and Landshoff for the forward amplitude (eq. (9) of [@DL_1995]) and assuming it holds for high , the contributions to the $d\sigma / dt \ (t=0)$ cross sections of the purely reggeon exchange and of the reggeon$-$pomeron interference term are, respectively, 4% and 28% of the pomeron exchange contribution (0% and 4% at HERA). To extract the forward differential cross sections from the measured total cross sections, assumptions have to be made concerning the relevant slopes. For the purely reggeon exchange term, the value 0.83 is used for the slope of the Regge trajectory [@Cudell_Kang], with the same parameter $b(W_0^2) = 2.5$ as for the pomeron ($W_0 = 1$ ). The slope for the interference term is chosen as the average of the pomeron and the reggeon slopes. When these contributions are subtracted, the value of $\delta$ is increased by 0.02.
Not including an error for theoretical uncertainties, the values of $\delta$ for $\sigma_L$ and pomeron exchange only are $ 0.18 \pm 0.05\ {\rm for} \ {\mbox{$Q^2$}}= 10\ {\mbox{${\rm GeV}^2$}}$ and $ 0.14 \pm 0.06\ {\rm for} \ {\mbox{$Q^2$}}= 20\ {\mbox{${\rm GeV}^2$}}.$
The dependence of production is presented in Fig. \[fig:w\]b[^8] for $\simeq$ 0, 10 and 20 . A steep increase of the photoproduction cross section is observed from low energy to the HERA experiments. For higher values, a similar increase is observed between the EMC and the H1 measurements. However, quantitative comparisons should be taken with caution in view of normalisation uncertainties and the possible presence of inelastic background in the fixed target data.
decay angular distribution {#sect:cos}
---------------------------
The acceptance corrected distribution for the selected sample is shown in Fig. \[fig:costhst\]a. After subtraction of the non-resonant background, which is consistent with being flat in , and correction for detector effects, the fit of eq. (\[eq:costhst\]) to this distribution gives = $0.73 \pm 0.05 \pm 0.02$. The first error is statistical, the second reflects the uncertainty on the background subtraction. Assuming $SCHC$, relation (\[eq:R\]) gives $R = \sigma_L \ / \sigma_T = 2.7\ _{-0.5\ -0.2}^{+0.7\ +0.3}$, with ${\mbox{$ \langle \varepsilon \rangle $}}$ = 0.99.
The value of is shown in Fig. \[fig:costhst\]b together with fixed target measurements[^9] and given in Table \[tab:R\] for two values of and of . Compared with results at low , a clear increase of with is observed.
[|c|c|]{}\
$8 < {\mbox{$Q^2$}}< 12$ & $12 < {\mbox{$Q^2$}}< 50$\
${\mbox{$R$}}= 2.2\ _{-0.5}^{+0.8}$ & ${\mbox{$R$}}= 4.0\ _{-1.2}^{+2.4}$\
\
$40 < W < 80$ & $80 < W < 140$\
${\mbox{$R$}}= 2.2\ _{-0.5}^{+0.8}$ & ${\mbox{$R$}}= 3.7\ _{-1.1}^{+1.9}$\
An attempt was made to test the hypothesis [@Ginzburg] that the meson should be completely longitudinally polarised for $\gg$ $\Lambda_{QCD}^2$, by dividing the data with $<$ 0.5 into two samples with respectively smaller and larger than 0.15 . The predicted effect was not observed, but the imposed cut is probably too low to provide a sensitive test of the prediction.
Discussion and conclusions
==========================
The production of elastic and mesons by virtual photons has been measured at HERA with the H1 detector. Samples of 180 and 31 events, respectively, have been collected with $>$ 8 and $40 < {\mbox{$W$}}< 140$ (30 $-$ 150 for the ), for an integrated luminosity of 2.8 (3.1) pb$^{-1}$. Most of the proton dissociation background is removed using the forward components of the H1 detector and the small residual backgrounds are corrected for.
A major interest of the study of elastic vector meson production is that predictions have been proposed for the differential cross sections both in the framework of a soft, non-perturbative approach, and based on perturbative QCD calculations for hard processes.
The main difference between the predictions of the soft and of the hard approaches concerns the rise of the cross section with energy, expected respectively to be slow or fast. For production at high , the dependence of the cross section attributed to pomeron exchange is parameterised as $d\sigma / dt \ (t=0)\ \propto {\mbox{$W$}}^{4 \delta}$. Using the NMC and the present H1 results, the measured values of $\delta$ for the total cross section $\sigma_T + \varepsilon \ \sigma_L$ are $ \delta = 0.14 \pm 0.05\ {\rm for} \ {\mbox{$Q^2$}}= 10\ {\mbox{${\rm GeV}^2$}}$ and $ \delta = 0.10 \pm 0.06\ {\rm for} \ {\mbox{$Q^2$}}= 20\ {\mbox{${\rm GeV}^2$}}.$ For the longitudinal cross section $\sigma_L$ alone, and taking into account possible reggeon exchange, these values would be $ \delta = 0.18 \pm 0.05$ and $ \delta = 0.14 \pm 0.06$, respectively.
For the soft pomeron model of Donnachie-Landshoff, the expected value is 0.08. For the hard approach, it is presumably in the range $0.20 - 0.25$ \[3b\]. The present measurements thus lie between the values expected for these two types of models.
The suggestion [@Brodsky] that the cross section measurement may provide information on the gluon distribution in the proton is applicable only when the hard regime is reached. As this condition does not seem to be fulfilled for the present and ranges, an attempt to extract the gluon distribution from these data seems premature.
In contrast, the production cross section for $>$ 8 increases strongly from the fixed target to the HERA region. This increase is of the same order as in photoproduction. This indicates that a hard regime is reached for production already at low , which could be related to the smaller spatial extent of the wave function and the large scale provided by the charm quark mass.
A major result of the present measurement is the similarity of the cross sections for and elastic production. Whereas photoproduction, which is suppressed by factors of 100 to 1000 with respect to , is not well described by the “quark counting rule”, quark flavour symmetry appears to be approximately restored for of 10 to 20 . Such a behaviour is expected both in the soft and the hard models. However, this evolution is observed to be faster than for some hard models (a ratio 1/2 has been proposed for $\simeq \ 100 $ [@F_K_S]).
The dependence of the production differential cross sections for and mesons are found to be well described at low values by exponential dependences $e^{bt}$. For the sample, is $7.0 \pm 0.8\ (stat.) \pm 0.4\ (syst.) \pm 0.5\ (bg.)$ . This value is smaller than for photoproduction at HERA, showing that the decrease of the slope with rising observed in fixed target experiments extends to the HERA regime. This behaviour can be attributed to the decrease of the $q \bar q$ transverse separation in the photon with rising .
The evolution of the slope with is sensitive to the interplay of soft and hard effects in production. There is an indication that the shrinkage of the elastic peak observed in hadron interactions and photoproduction also occurs in the present electroproduction data, at large . This is as expected from Regge predictions based on soft pomeron exchange, in contrast with the little shrinkage predicted in perturbative calculations for hard processes [@F_K_S].
For production, the distribution is well described with ${\mbox{$b$}}\ = 3.8 \pm 1.2 \ _{-1.6}^{+2.0}$ , which is smaller than for the . This difference can be qualitatively explained by the fact that the wave function is more compact than the wave function.
The dependence of the total cross section can be described by a power law $Q^{-2n}$, with $n = 2.5 \pm 0.5 \pm 0.2$. This distribution is slightly harder than initially expected both for non-perturbative two gluon exchange and for hard QCD calculations ($\propto Q^{-6}$). It is compatible with predictions taking into account the evolution of parton densities and the transverse motion of the quarks in the photon [@F_K_S]. It should be noted that the calculations are performed for $\sigma_{L}$ whereas the present measurement is of $\sigma_{tot}$, which includes a transverse contribution at the level of 20$-$25%, with presumably a steeper dependence [@Kope; @Brodsky].
The dependence of the production cross section is parameterised as $1 / (Q^2 + m_\psi^2)^n$, with $n = 1.9 \pm 0.3\ (stat.)$, which is similar to fixed target results.
The polar angle distribution of the decay pions indicates that mesons are mostly longitudinally polarised: the spin density matrix element is $0.73 \pm 0.05 \pm 0.02$. Assuming [*SCHC*]{}, $R = \sigma_L \ / \sigma_T$ is $2.7\ _{-0.5\ -0.2}^{+0.7\ +0.3}$. The increase of with , which is predicted both by the non-perturbative model and the QCD calculations, is observed. The indication in the present data of an increase of with suggests, in the framework of hard physics, that perturbative features at high and moderate are indeed more important for longitudinal than for transverse photons.
In conclusion, the study of and meson production offers a contrasting picture.
For mesons, hard physics effects are probably at work already for very small . The present high data support this interpretation, albeit with limited statistical precision.
For mesons, the , , and dependences of the cross section allow a more detailed study, from which a mixed picture emerges. It can be speculated that the present data correspond to a transition regime, with interplay of hard processes, amenable to perturbative description, and soft processes, requiring a non-perturbative approach. The dependence of the cross section does not provide conclusive evidence in favour of a purely soft or a purely hard model. The indication of shrinkage of the elastic peak with is important because it shows a continuation with increasing of the photoproduction behaviour and is at variance with expectations for a purely hard behaviour. The observation that SU(4) flavour symmetry is restored in the : cross section ratio, which is a striking feature of the present measurements, is expected in both models. The study of the and distributions also does not discriminate between the soft and hard models, since their predictions are similar.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to the HERA machine group whose outstanding efforts made this experiment possible. We appreciate the immense effort of the engineers and technicians who constructed and maintained the detector. We thank the funding agencies for their financial support of the experiment. We wish to thank the DESY directorate for the support and hospitality extended to the non-DESY members of the collaboration. We thank further J.-R. Cudell, B. Kopeliovich, A. Sandacz and H. Spiesberger for useful discussions.
[19]{}
a\. T. Ahmed et al., H1 Coll., ;\
b. M. Derrick et al., ZEUS Coll., . M. Derrick et al., ZEUS Coll., ;\
T. Ahmed et al., H1 Coll., ;\
M. Derrick et al., ZEUS Coll., ;\
S. Aid et al., H1 Coll., . a. M. Derrick et al., ZEUS Coll., [*Measurement of the Proton Structure Function $F_2$ at low x and low at HERA,*]{} preprint DESY-95-193 (1995);\
b. H1 Coll., [*A measurement and QCD Analysis of the Proton Structure Function $F_2(x,{\mbox{$Q^2$}})$ at HERA,*]{} to be publ. Preliminary H1 results were presented at the Int. Europhys. Conf. on HEP, Brussels, 1995:\
H1 Coll., [*Exclusive Production in Deep Inelastic Scattering Events at HERA*]{}, EPS-0480;\
H1 Coll., [* Meson Production in Deep Inelastic Scattering at HERA*]{}, EPS-0469. M. Derrick et al., ZEUS Coll., . J.J. Sakurai, ;\
J.J. Sakurai and D. Schildknecht, . T.H. Bauer et al., , and references therein. A. Donnachie and P.V. Landshoff, . a. R.M. Egloff et al., ;\
b. D. Aston et al., . a. P. Joos et al., ;\
b. C. del Papa et al., ;\
c. D.G. Cassel et al., ;\
d. W.D. Shambroom et al., CHIO Coll., ;\
e. B.Z. Kopeliovich and P. Marage, . a. J.J. Aubert et al., EMC Coll., ;\
b. J. Ashman et al., EMC Coll., ;\
c. P. Amaudruz et al., NMC Coll., ;\
d. M. Arneodo et al., NMC Coll., . a. M. Derrick et al., ZEUS Coll., 39;\
b. S. Aid et al., H1 Coll., [*Elastic Photoproduction of Mesons at HERA,*]{} preprint DESY-95-251 (1995), to be publ. in . a. U. Camerini et al.,
[^1]: For the studies only positron data are used, whereas the small amount of data taken with $e^-p$ scattering is included in the sample; in this report, [*positrons*]{} refers both to positrons and electrons.
[^2]: The forward ($+z$) direction, with respect to which polar angles are measured, is defined as that of the incident proton beam, the backward direction is that of the positron beam.
[^3]: In this paper, the positron and proton masses are neglected.
[^4]: The lowest value kinematically allowed, $t_{min} \simeq ({\mbox{$Q^2$}}+ m^2_{V})^2 \ m^2_p\ / y^2 s^2$, is negligible in this experiment.
[^5]: In practice, the input to the program is an effective “$F_2$ structure function” parameterisation, with the chosen and dependences of the $ep$ cross section for vector meson production.
[^6]: For ${\mbox{$Q^2$}}= 10$ , the use of the measured NMC slope instead would lead to an additional increase of $\delta$ by 0.03. No measurement of the NMC slope is published for ${\mbox{$Q^2$}}= 20$ .
[^7]: See eq. (23.32) and (23.36), p. 1292 of [@PDG].
[^8]: All results presented in Fig. \[fig:w\]b have been rescaled to take into account the latest measurements of the branching fractions: $B({\mbox{$J/\psi$}}\rightarrow e^+e^-) = 5.99 \pm 0.25 \%,
B({\mbox{$J/\psi$}}\rightarrow \mu^+\mu^-) = 5.97 \pm 0.25 \%$ [@PDG].
[^9]: EMC measurements \[11a\] with largest errors have been omitted.
|
---
abstract: 'The TOTEM experiment at LHC has chosen the triple Gas Electron Multiplier (GEM) technology for its T2 telescope which will provide charged track reconstruction in the rapidity range 5.3$<$$|\eta|$$<$6.5 and a fully inclusive trigger for diffractive events. GEMs are gas-filled detectors that have the advantageous decoupling of the charge amplification structure from the charge collection and readout structure. Furthermore, they combine good spatial resolution with very high rate capability and a good resistance to radiation. Results from a detailed T2 GEM simulation and from laboratory tests on a final design detector performed at CERN are presented.'
address:
- 'Pisa INFN, Largo B. Pontecorvo, 3 - 56127 Pisa, Italy'
- 'Physics Department, Siena University, Via Roma, 56 - 53100 Siena, Italy'
- 'CERN, EP Division, 1211 Geneva 23, Switzerland'
author:
- 'S. Lami, G. Latino [^1], E. Oliveri, L. Ropelewski, N. Turini'
title: 'A triple-GEM telescope for the TOTEM experiment'
---
INTRODUCTION
============
The TOTEM [@Totem] experiment at the LHC collider will measure the total $pp$ cross section with a precision of about 1$\div$2$\%$, the elastic $pp$ cross section over a wide range in -t, up to $10$GeV$^2$, and will study diffractive dissociation processes. Relying on the “luminosity independent method” the evaluation of the total cross section with such a small error will require simultaneous measurements of the $pp$ elastic scattering cross section $d\sigma /dt$ down to $-t \sim 10^{-3}$GeV$^2$ (to be extrapolated to $t$ = 0) as well as of the $pp$ inelastic interaction rate with a good rapidity coverage up to the very forward region. Roman Pots (RP) stations at 147m and at 220m on both sides from the Interaction Point (IP), equipped with “edgeless planar silicon” detectors, will provide the former measurement. The latter will be achieved by two inelastic telescopes, T1 and T2, placed in the forward region of the CMS experiment on both sides of the IP. T1, using “Cathode Strip Chambers”, will cover the rapidity range 3.1$<$$|\eta|$$<$4.7 while T2, based on “Triple-GEM” technology, will extend charged track reconstruction to the rapidity range 5.3$<$$|\eta|$$<$6.5. These detectors will also allow common CMS/TOTEM diffractive studies with an unprecedented coverage in rapidity. The T2 telescope will be placed 13.56m away from IP and the GEMs employed will have an almost semicircular shape, with an inner radius matching the beam pipe. Each arm of T2 will have a set of 20 triple-GEM detectors combined into 10 aligned semi-planes mounted on each side of the vacuum pipe (Figure \[fig:T2\_telescope\]).
-0.3in 0.2in ![One arm of TOTEM T2 Telescope.[]{data-label="fig:T2_telescope"}](T2_Gem_Telescope.eps "fig:") -0.3in
-0.3in
GEM TECHNOLOGY
==============
The CERN developed GEM technology [@GEM_Sauli] has already been successfully adopted in other experiments such as COMPASS and LHCb and has been considered for the design of TOTEM very forward T2 telescopes thanks to its characteristics: large active areas, good position and timing resolution, excellent rate capability and radiation hardness. Furthermore, GEM detectors are also characterized by the advantageous decoupling of the charge amplification structure from the charge collection and readout structure which allows an easy implementation of the design for a given apparatus. The T2 GEMs use the same baseline design as the one adopted in COMPASS [@GEM_Compass]: each GEM foil consists of thin copper clad polymer foil of 50$\mu$m, with copper layers of 5$\mu$m on both sides, chemically perforated with a large number of holes of 70$\mu$m in diameter. A potential difference around 500V applied between the two copper electrodes generates an electric field of about 100kV/cm in the holes which therefore can act as multiplication channels (gains of order $10 \div 10^2$) for electrons created in a gas (Ar/CO$_2$ (70/30 $\%$) for T2) by an ionizing particle. The triple-GEM structure, realized by separating three foils by thin (2$\div$3mm) insulator spacers, is adopted in order to reduce sparking probabilities while reaching high total gas gains, of order $10^4 \div 10^5$, in safe conditions. The read-out boards will have two separate layers with different patterns: one with 256x2 concentric circular strips, 80$\mu$m wide and with a pitch of 400$\mu$m, allowing track radial reconstruction with ${\sigma}_R$ down to 70$\mu$m, and the other with a matrix of 24x65 pads of 2x2 to 7x7mm$^2$ in size from inner to outer circle, providing level-1 trigger information as well as track azimuthal reconstruction.
T2 TRIPLE-GEM SIMULATION
========================
A detailed simulation of T2 triple-GEM detector has been developed starting from the existing implementation for the GEMs used at LHCb [@GEM_Sim]. The general framework is relying on several packages allowing a complete and detailed “step by step” simulation, for a given gas mixture and detector geometry, for the several underlying processes: starting from the primary ionization up to the spatial and timing properties of the collected signals. The main framework is implemented in [*Garfield*]{}; the electric field mapping is simulated with [*Maxwell*]{}; the electron/ion drift velocity and diffusion coefficients are evaluated with [*Magboltz*]{}; Townsend and attachment coefficients are simulated by [*Imonte*]{}; the energy loss by a given ionizing particle in gas and the cluster production process are evaluated by [*Heed*]{}. As an example, Figure \[fig:Pad\_Weighting\_Field\] reports the simulation for the “weighting field” ${\vec{E}}_K^W(x)$ (defined by putting at 1V the given readout electrode while keeping all the others at 0V) for a pad electrode. Signal induction is then derived via the Ramo theorem: $I_k = -q\vec{v}(x)\times {\vec{E}}_K^W(x)$.
-0.3in ![Simulation of the weighting field for a T2 GEM pad electrode.[]{data-label="fig:Pad_Weighting_Field"}](RO_board_simII.eps "fig:"){width="1.0\linewidth"} -0.2in
-0.4in
From the reconstruction of the full process chain leading to signal collection, with proper modeling of lateral electron cluster diffusion through each GEM foil, the expected signal for a MIP particle has been derived for both strips and pads for typical values of the electric field in the drift and induction zones between GEM foils (E$_{d/t}$ $\sim$ 3kV/cm). Timing properties, such as a typical signal time delay(duration) of $\sim$60(50) ns, have been found consistent with preliminary test beam studies on prototypes. Furthermore, the study of signals as a function of distance from electron cluster centroid, when combined with expected signal processing by the readout electronics, has shown a typical strip cluster size of 2$\div$3 channels (1$\div$2 for pads), which is consistent with COMPASS test beam results [@GEM_Compass]. Ongoing test beam activities, performed with final production detectors read by final design electronics (digital readout via VFAT chip), are expected to allow an improved test and tuning of current simulation.
TEST ACTIVITIES AT CERN
=======================
Two final full size detectors, whose components were provided by CERN, have been assembled by an italian private company [@GeA], and then tested at CERN Gas Detector Development Laboratory with a Cu X-Ray source ($K_{\alpha /\beta}$ = 8/8.9 KeV). These activities involved studies on: general functionality, absolute gain, strip/pad charge sharing, energy resolution, time stability and response uniformity. In particular, the analysis of signals simultaneously collected from 8 strip/pad electrodes allowed to check the most important detector parameters. Figure \[fig:GainCalib\] shows the total effective gain $G_T$ [^2] for both strip and pad readout channels as a function of the applied HV: an expected gain of 8$\div$10$\times$$10^3$ for a typical HV value of -4kV is observed.
-0.3in ![Strip/Pad gain as a function of the applied high voltage.[]{data-label="fig:GainCalib"}](Pad_str_GAINCalibration.eps "fig:"){width="1.\linewidth"} -0.4in
-0.35in
The study of strip/pad cluster charge sharing showed the expected correlation between the two clusters (Figure \[fig:ChShar\]). A slightly higher charge collected by strips (about 10$\div$15$\%$), considering the typical higher strip cluster size, is consistent with the design for an optimal setup of the common readout chip.
-0.3in -0.05in -0.3in
-0.3in
The evaluation of energy resolution represents another important detector test as it is related to the quality and uniformity of GEM foils. In fact, a not uniform gain over the irradiated zone will results in an anomalous broadening of the peak in the response spectrum. An energy resolution of $\sim$ 20$\%$, in terms of FWHM for the leading 8 KeV peak, was found to be well in agreement with the expected design performance of the detector.
Furthermore, time stability of signal has been tested with continuous detector irradiation over more than one hour and response uniformity checked by randomly moving the X-Ray source over the detector surface.
In conclusion detector performances well within expectations have been observed. A more extensive test on ten production detectors will be performed at the incoming test beam activities.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are particularly grateful to A. Cardini and D. Pinci of LHCb Collaboration for very precious initial input on triple-GEM simulation.
[9]{} “TOTEM Technical Design Report”, CERN-LHCC-2004-002; addendum CERN-LHCC-2004-020 (2004). F. Sauli, Nucl. Inst. $\&$ Meth [**A**]{}386 (1997) 531. C. Altunbas [*et al.*]{}, Nucl. Inst. $\&$ Meth [**A**]{}490 (2002) 177. W. Bonivento [*et al.*]{}, IEEE Trans.Nucl.Sci.49 (2002) 1638. G$\&$A Engineering, Carsoli (Aquila) - Italy, www.gaengineering.com.
[^1]: Corresponding author. Phone: +39-050-2214439. E-mail address: giuseppe.latino@pi.infn.it
[^2]: This parameter can be derived according to equation $G_T = I_{tot}/(e\cdot n\cdot f)$, from total readout current ($I_{tot}$) and X-Ray interaction rate ($f$) measurements, knowing the average number of electrons produced by an interacting X-Ray ($n$).
|
---
author:
- 'Andr[é]{} G. Moreira[^1] and Roland R. Netz[^2]'
date: 'Received: date / Revised version: date'
title: Virial expansion for charged colloids and electrolytes
---
Introduction {#chapterD:intro}
============
The two-component hard-core plasma (TCPHC) has been used for a long time as an idealized model for electrolyte solutions. In this model, also known as “primitive model,” the ions are spherical particles that interact with each other via the Coulomb potential and a hard-core potential, which avoids the collapse of oppositely charged particles onto each other. In the general asymmetric case, the positive ions have a charge $q_+ \, e$ (where $e$ is the elementary charge) and ionic diameter $d_+$, while the negative ions have a charge $- q_- e$ and diameter $d_-$. The particles are immersed in a structureless solvent whose presence is felt only through the value of the dielectric constant of the medium, and the system is (globally) electroneutral.
The TCPHC in this formulation is a gross simplification of real systems. When two ions are at distances of the order of the size of the solvent molecules, the assumption of a continuum solvent breaks down, giving rise to so-called solvation forces[@israelachvili:book]. Such effects also lead to non-local contributions to the dielectric constant. Since the dielectric constant of ions or colloids typically differs from the surrounding aqueous medium, one also expects dispersion forces to act between these particles. All these effects, which contribute to “ion-specific” effects[@ninham-yaminsky:97], and are possibly more relevant than previously thought[@alfridsson-al:2000], are not accounted for in the TCPHC, which treats the solvent—usually water—as a structureless medium within which the charged particles are embedded, neglecting the molecular arrangement that occurs around the ions.
Nevertheless, even with such simplifications, the TCPHC is far from being amenable to an exact treatment. A better understanding of this model is a necessary step if one wishes to develop more realistic approaches to charged systems. In this article, we turn our attention to the low-density, or virial, expansion of the TCPHC. Since we have exact results for the thermodynamic properties at low concentrations using field-theoretic methods, we can obtain useful information on dilute systems with otherwise arbitrary hard-core radii and charge valences (like colloidal suspensions). In particular, we can test various approximations to treat the TCPHC, like the mapping of a colloidal solution on an effective one-component plasma.
Due to the long-range character of the Coulomb potential, it is not easy to obtain the thermodynamic behavior of the TCPHC through the usual methods of statistical mechanics. For example, it can be shown[@mayer:50; @friedman:book; @mcquarrie:book] that the straightforward application of the cluster expansion to the TCPHC leads to divergent virial coefficients. Mayer[@mayer:50] proposed a solution to this problem through an infinite resummation of the cluster diagrams, carried out such that the divergent contributions to the virial expansion are canceled. With this, he was able to obtain explicitly the first term in the virial (or low-density) expansion that goes beyond the ideal gas, which turns out to be the well-known Debye-H[ü]{}ckel limiting law[@debye-hueckel:23]. Haga[@haga:53] carried the expansion further and went up to order $5/2$ in the ionic density. More or less at the same time, Edwards[@edwards:59] also obtained the virial expansion of the TCPHC by mixing cluster expansion and field theory. In both cases only equally-sized ions were considered.
The aforementioned methods typically depend on drawing, counting and recollecting the cluster diagrams which give finite contributions to the expansion up to the desired order in the density. This can be quite a formidable task, and unfortunately it is easy to “forget” diagrams that are relevant to the series (see for instance the comment on pp. 222–223 of Ref. [@friedman:book]). Also, the generalization to ions with different sizes is rather complicated[@friedman:book]. Besides, the final results are typically not obtained in closed form, i.e., the final expressions depend on infinite sums that usually have to be evaluated numerically, which reflects the infinite diagrammatic resummation.
We generalize here a novel field-theoretic technique[@roland-orland:tcp], introduced for the symmetric TCPHC ($q_+ = q_-$ and $d_+=d_-$). We obtain the exact low-density expansion of the *asymmetric* (both in size and charge) TCPHC. This method does not use the cluster expansion (and resummation) and yields analytic, closed-form results. We go up to order $5/2$ in the volume fraction of a system where the sizes and the charge valences of positive and negative ions are unconstrained, that is, the results we obtain can be applied, without modifications, to both electrolyte solutions (where anions and cations have approximately the same size and valence) and to colloidal suspensions (where the macro- and counterions have sizes and valences that can be different by orders of magnitude). In the colloidal limit, we demonstrate how to map the TCPHC on an effective one-component hard-core plasma (OCPHC) and obtain the corrections due to the exclusion of the background (i.e. the counterions) from the colloidal particles. For electrolytes, we obtain effective ionic sizes in solution, or thermodynamic diameters, from experimental data for the mean activity. The method developed here allows, in principle, to measure the diameters of cations and anions as *independent* variables, given that the range of experimental data extends to low enough densities such that the expansion used here is valid. It should be noted that the diameters obtained in this way are fundamentally different from the hydrodynamic diameters.
This article is organized as follows. In section \[chapterD:method\] we describe in detail the steps that lead to the low-density expansion. Readers which are not interested in this derivation can go directly to section \[chapterD:applications\], where we apply the expressions obtained to two particular problems, viz., (i) colloids, where one of the charged species is much smaller and much less charged than the other one and (ii) the mean activity coefficient (which is related to the exponential of the chemical potential) of electrolyte solutions, which is available from experiments and can be compared to our results. Finally, section \[chapterD:conclusions\] contains some concluding remarks.
The method {#chapterD:method}
==========
We begin our calculation by assuming a system with $N_+$ positively charged particles with charge valence $q_+$ and diameter $d_+$, and $N_-$ negatively charged particles with charge valence $q_-$ and diameter $d_-$. Global electroneutrality of the system will be imposed at a later stage of the calculation. The canonical partition function $\mathcal{Z}$ is given by $$\label{partition1}
\mathcal{Z}=\frac{1}{N_+! N_-!}
\int \prod\limits_{i=1}^{N_+} \frac{{\rm d}{\mathbf{r}}_i^{(+)}}{\lambda^3_{T,+}}
\prod\limits_{j=1}^{N_-} \frac{{\rm d}{\mathbf{r}}_j^{(-)}}{\lambda^3_{T,-}}
\, \exp \Bigl(- \frac{\mathcal{H}}{k_B T} \Bigr)$$ where $\lambda_{T,+}$ and $\lambda_{T,-}$ are the thermal wavelengths, and ${\mathbf{r}}{^{(+)}}_i$ and ${\mathbf{r}}_i^{(-)}$ are the positions of positively and negatively charged particles. The Hamiltonian $\mathcal{H}$ is given by $$\begin{gathered}
\label{hamiltonian}
\frac{\mathcal{H}}{k_B \, T}
= -E_{{\mathrm{self}}} + \frac{1}{2}
\sum_{\alpha, \beta =+,-}
\int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \,
\hat{\rho}_{\alpha}({\mathbf{r}}) \omega_{\alpha \beta}({\mathbf{r}}-{\mathbf{r}}')
\hat{\rho}_{\beta}({\mathbf{r}}') \\
+ \frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \,
[\hat{\rho}_{+}({\mathbf{r}})-\hat{\rho}_{-}({\mathbf{r}})] v_c({\mathbf{r}}-{\mathbf{r}}')
[\hat{\rho}_{+}({\mathbf{r}}')-\hat{\rho}_{-}({\mathbf{r}}')]\end{gathered}$$ where the charge-density operators of the ions are defined as $$\begin{split}
\label{density}
\hat{\rho}_{+}({\mathbf{r}}) &= q_+ \sum\limits_{i=1}^{N_+} \delta({\mathbf{r}}-{\mathbf{r}}{^{(+)}}_i) \\
\hat{\rho}_{-}({\mathbf{r}}) &= q_- \sum\limits_{i=1}^{N_-} \delta({\mathbf{r}}-{\mathbf{r}}{^{(+)}}_i),
\end{split}$$ and $\delta({\mathbf{r}}-{\mathbf{r}}')$ is the Dirac delta function. The indices $\alpha$ and $\beta$ in Eq. (\[hamiltonian\]) stand for $+$ and $-$, and the sum over $\alpha$ and $\beta$ in Eq. (\[hamiltonian\]) runs over all possible permutations (viz. $++$, $--$, $-+$ and $+-$), i.e., we consider a different short-ranged potential $\omega_{\alpha \beta}$ for each combination. The Coulomb potential is given by $v_c({\mathbf{r}}) = \ell_B / r$, where $\ell_B \equiv e^2 / ( 4 \,\pi \, \varepsilon \, k_B \, T )$ is the Bjerrum length, defined as the distance at which the electrostatic energy between two elementary charges equals the thermal energy $k_B \, T$. Finally, $E_{{\mathrm{self}}}$ is the self-energy of the system $$\label{selfE}
E_{{\mathrm{self}}}= \frac{N_+ q_+^2 }{2} [\omega_{++}(0) + v_c(0)] +
\frac{N_- q_-^2}{2} [\omega_{--}(0) + v_c(0)]$$ which cancels the diagonal terms in Eq. (\[hamiltonian\]).
We proceed by applying the Hubbard-Stratonovich transformation, the essence of which is given by $$\begin{gathered}
\label{HS}
{\mathrm{e}^{ -\frac{1}{2}
\int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \, \hat{\rho}({\mathbf{r}}) v({\mathbf{r}}-{\mathbf{r}}')
\hat{\rho}({\mathbf{r}}')}} = \\
\frac{\int {\cal D}\phi \,
{\mathrm{e}^{-\frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \,
\phi({\mathbf{r}}) v^{-1}({\mathbf{r}}-{\mathbf{r}}') \phi({\mathbf{r}}') - \imath \int {\rm d}{\mathbf{r}} \,
\phi({\mathbf{r}}) \hat{\rho}({\mathbf{r}})}}}{\int {\cal D}\phi \,
{\mathrm{e}^{ -\frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \,
\phi({\mathbf{r}}) v^{-1}({\mathbf{r}}-{\mathbf{r}}') \phi({\mathbf{r}}') }}}\end{gathered}$$ where $v({\mathbf{r}})$ is some general potential and $\int {\cal D}\phi$ denotes a path integral over the fluctuating field $\phi$. While this transformation can be used without problems when $v({\mathbf{r}})$ is the Coulomb potential, for a short-ranged potential this can be more problematic: for instance, a hard-core potential does not even have a well-defined inverse function. We will anyway take this formal step for the short-ranged potential, and, as we will see later, the way we handle the resulting expressions leads to finite (and consistent) results, viz., the virial coefficients[@note1].
Applying Eq. (\[HS\]) to Eq. (\[partition1\]) we obtain the partition function in field-theoretic form $$\label{partition2}
\mathcal{Z}= \int \frac{{\cal D}\psi_+ \, {\cal D}\psi_-}{\mathcal{Z}_{\psi}}
\frac{{\cal D}\phi}{\mathcal{Z}_{\phi}}
\, {\mathrm{e}^{-\bar{\mathcal{H}}_0}} \, W_+ \, W_-$$ with the action $$\begin{gathered}
\label{H0}
\bar{\mathcal{H}}_0 = \frac{1}{2} \sum_{\alpha, \beta =+,-}
\int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \, \psi_{\alpha}({\mathbf{r}})
\omega^{-1}_{\alpha \beta}({\mathbf{r}}-{\mathbf{r}}') \psi_{\beta}({\mathbf{r}}') \\
+\frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \, \phi({\mathbf{r}})
v_c^{-1}({\mathbf{r}}-{\mathbf{r}}') \phi({\mathbf{r}}').\end{gathered}$$ The inverse potentials are formally defined as the solution of the equation $$\sum\limits_{\beta=+,-} \int {\rm d}{\mathbf{r}}' \, \omega_{\alpha \beta}({\mathbf{r}}-{\mathbf{r}}')
\omega^{-1}_{\beta \gamma}({\mathbf{r}}'-{\mathbf{r}}'') =
\delta_{\alpha \gamma} \, \delta({\mathbf{r}}- {\mathbf{r}}'')$$ ($\delta_{\alpha \gamma}$ is the Kronecker delta) and $$\int {\rm d}{\mathbf{r}}' \, v_c({\mathbf{r}}-{\mathbf{r}}') v^{-1}_c({\mathbf{r}}'-{\mathbf{r}}'')=
\delta({\mathbf{r}}-{\mathbf{r}}'').$$ For the Coulomb potential, $v_c^{-1}({\mathbf{r}}) = - \nabla^2 \delta({\mathbf{r}}) / 4 \pi \ell_B$. We also define $$\label{W+}
W_{\alpha} \! = \! \frac{1}{N_{\alpha}!} \Biggl[
{\mathrm{e}^{ q_{\alpha}^2 \bigl[\omega_{\alpha \alpha}(0)+v_c(0) \bigr]/2}}
\int \frac{{\rm d}{\mathbf{r}}}{\lambda^3_{T,\alpha}} \,
{\mathrm{e}^{-\imath q_{\alpha} \bigl[\psi_{\alpha}({\mathbf{r}})+
\alpha \, \phi({\mathbf{r}}) \bigr]}} \Biggr]^{N_{\alpha}}$$ and the normalization factors $$\mathcal{Z}_{\psi}= \int {\cal D}\psi_+ \, {\cal D}\psi_-
\, {\mathrm{e}^{-\frac{1}{2} \sum\limits_{\alpha \beta}
\int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \, \psi_{\alpha}({\mathbf{r}})
\omega^{-1}_{\alpha \beta}({\mathbf{r}}-{\mathbf{r}}') \psi_{\beta}({\mathbf{r}}')}}$$ and $$\label{Zphi}
\mathcal{Z}_{\phi}= \int {\cal D}\phi \,
{\mathrm{e}^{-\frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \, \phi({\mathbf{r}})
v_c^{-1}({\mathbf{r}}-{\mathbf{r}}') \phi({\mathbf{r}}')}}.$$
In order to make the calculations simpler we use the grand-canonical ensemble. This is achieved through the transformation $$\label{partition3}
\mathcal{Q} = \sum_{N_+, N_- =0}^\infty \Lambda_+^{N_+} \Lambda_-^{N_-} \mathcal{Z},$$ where $\Lambda_+$ and $\Lambda_-$ are, respectively, the fugacities (exponential of the chemical potential) of the positively and negatively charged particles. We perform the sum over $N_+$ and $N_-$ without constraints, i.e., without imposing the electroneutrality condition . Imposing this condition before going to the grand-canonical ensemble makes the calculations much more difficult. Later, electroneutrality will be imposed order-by-order in the low-density expansion in a consistent way, any infinities arising from the non-neutrality of the system will then be automatically canceled.
In its full form, the grand-canonical partition function $\mathcal{Q}$ reads $$\begin{gathered}
\label{partition4}
\mathcal{Q} = \int \frac{{\cal D}\psi_+ \, {\cal D}\psi_-}{\mathcal{Z}_{\psi}}
\frac{D\phi}{\mathcal{Z}_{\phi}}
\, \exp \Biggl(
\frac{\Lambda_+}{\lambda^3_{T,+}} \int {\rm d}{\mathbf{r}} \, h_+({\mathbf{r}}) {\mathrm{e}^{-i q_+ \phi({\mathbf{r}})}} \\
+ \frac{\Lambda_-}{\lambda^3_{T,-}} \int {\rm d}{\mathbf{r}} \, h_-({\mathbf{r}}) {\mathrm{e}^{i q_- \phi({\mathbf{r}})}}
-\bar{\mathcal{H}}_0 \Biggr),\end{gathered}$$ where $\bar{\mathcal{H}}_0$ is given in Eq. (\[H0\]). We defined the local non-linear operator $$\label{h}
h_{\alpha}({\mathbf{r}}) \equiv \exp \Bigl(\frac{q_{\alpha}^2 }{2} \bigl[
\omega_{\alpha \alpha}(0)+ v_c(0) \bigr] - \imath
q_{\alpha} \psi_{\alpha}({\mathbf{r}}) \Bigr)$$ (as before, $\alpha$ stands for both $+$ and $-$). At this point we rescale the fugacities such that $\lambda_+ = \Lambda_+/\lambda^3_{T,+}$ and $\lambda_- = \Lambda_-/\lambda^3_{T,-}$, i.e., the fugacities have from now on dimensions of inverse volume.
Introducing the Debye-H[ü]{}ckel propagator, $$\label{vDH-1}
v_{{\mathrm{DH}}}^{-1}({\mathbf{r}}-{\mathbf{r}}') = v_c^{-1}({\mathbf{r}}-{\mathbf{r}}') +
I_2 \delta({\mathbf{r}}-{\mathbf{r}}')$$ with the ionic strength $$I_2 = q_+^2 \lambda_+ + q_-^2 \lambda_-$$ and after some algebraic manipulations, we finally obtain the grand-canonical free energy density. It is defined through $g \equiv - \ln \bigl( \mathcal{Q} \bigr) / V$, and reads $$\begin{gathered}
\label{free-energy1}
g = - \lambda_+ - \lambda_- -
\frac{1}{2} I_2 v_c(0) -
\frac{1}{V} \ln \Bigl(
\frac{\mathcal{Z}_{{\mathrm{DH}}}}{\mathcal{Z}_{\phi}} \Bigr) \\
- \frac{1}{V} \ln \Bigl\langle {\mathrm{e}^{\lambda_+ \int {\rm d}{\mathbf{r}} \, Q_+({\mathbf{r}}) +
\lambda_- \int {\rm d}{\mathbf{r}} \, Q_-({\mathbf{r}})}} \Bigr\rangle,\end{gathered}$$ where $V$ is the volume of the system and the brackets $\langle \cdots \rangle$ denote averages over the fluctuating fields $\phi$ and $\psi_\alpha$ with the propagators $\omega^{-1}_{\alpha \beta}$ and $v^{-1}_{{\mathrm{DH}}}$. $\mathcal{Z}_{{\mathrm{DH}}}$ is defined as $$\label{ZDH}
\mathcal{Z}_{{\mathrm{DH}}}= \int {\cal D}\phi \,
{\mathrm{e}^{-\frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \, \phi({\mathbf{r}})
v_{{\mathrm{DH}}}^{-1}({\mathbf{r}}-{\mathbf{r}}') \phi({\mathbf{r}}')}}$$ and the functions $Q_+({\mathbf{r}})$ and $Q_-({\mathbf{r}})$ are defined by $$\label{Q+}
Q_{\alpha}({\mathbf{r}}) = h_{\alpha}({\mathbf{r}}) {\mathrm{e}^{-\imath \alpha \, q_{\alpha} \phi({\mathbf{r}})}} - 1
+ \frac{1}{2} q_{\alpha}^2 \phi^2({\mathbf{r}}) - \frac{1}{2} q_{\alpha}^2 v_c(0).$$
In the preceding steps we obtained the exact expression for the grand-canonical free energy density $g$, still without imposing electroneutrality. In order to obtain the low-density expansion of the free energy, we now (i) expand $g$ in powers of $\lambda_+$ and $\lambda_-$ (up to a order $5/2$), (ii) calculate the concentrations of positive and negative particles and impose electroneutrality *consistently*, order-by-order, and (iii) make a Legendre transformation back to the canonical ensemble.
Expansion in powers of the fugacities
-------------------------------------
We start the expansion of $g$ by noting that the Fourier transform of the Coulomb potential is ${\widetilde{v}}_c({\mathbf{k}}) = 4 \pi \ell_B /k^2$. Using this, one is able to express $v_c(0)$ as $$\label{vc(0)}
v_c(0) = \int \frac{{\rm d} {\mathbf{k}}}{(2 \pi )^3} \frac{4 \pi \ell_B}{k^2}.$$ It is easy to show that $$\label{logZZ}
\frac{1}{V} \ln \Biggl( \frac{\mathcal{Z}_{{\mathrm{DH}}}}{\mathcal{Z}_{\phi}}
\Biggr) =
-\frac{1}{2} \int \frac{{\rm d}{\mathbf{k}}}{(2 \pi )^3} \ln
\Biggl( 1 + \frac{4 \pi \ell_B I_2}{k^2} \Biggr).$$ Using Eqs. (\[vc(0)\]) and (\[logZZ\]), one finds $$\label{g_exp_1}
\frac{1}{2} I_2 v_c(0) +
\frac{1}{V} \ln \Biggl( \frac{\mathcal{Z}_{{\mathrm{DH}}}}{\mathcal{Z}_{\phi}} \Biggr)
= \frac{1}{12 \pi} \Bigl[ 4 \pi \ell_B I_2 \Bigr]^{3/2}.$$ Introducing the dimensionless quantity $${\Delta}v_0 = \sqrt{4 \pi \ell_B^3 I_2}$$ and expanding the last term in the rhs of Eq. (\[free-energy1\]) in cumulants of $Q_+({\mathbf{r}})$ and $Q_-({\mathbf{r}})$, one obtains the low-fugacity expansion $$\begin{gathered}
\label{free-energy2}
g = - \lambda_+ - \lambda_- - \frac{{\Delta}v_0^3}{12 \pi \ell_B^3}
-\lambda_+ Z^{+}_1 - \lambda_- Z^{-}_1 - \frac{\lambda_+^2}{2} Z_2^{++} \\
- \frac{\lambda_-^2}{2} Z_2^{--} - \lambda_+ \lambda_- Z_2^{+-} +
O(\lambda^3),\end{gathered}$$ where the coefficients are given by $$\label{z1+}
Z_1^{+}=\frac{1}{V} \int {\rm d}{\mathbf{r}} \, \bigl\langle Q_+({\mathbf{r}}) \bigr\rangle$$ and an analogous formula for $Z_1^{-}$, and $$\label{z1++}
Z_2^{++}=\frac{1}{V} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \,
\Bigl\{ {\bigl\langle Q_+({\mathbf{r}}) Q_+({\mathbf{r}}') \bigr\rangle} -
{\bigl\langle Q_+({\mathbf{r}}) \bigr\rangle} {\bigl\langle Q_+({\mathbf{r}}') \bigr\rangle} \Bigr\}$$ and similar formulas for $Z_2^{--}$ and $Z_2^{+-}$. The symbol $O(\lambda^3)$ in Eq. (\[free-energy2\]) means that any other contribution to the expansion will be of order 3 or higher, i.e., with terms like $\lambda_+^3$, $\lambda_+^2 \lambda_-$, etc. Clearly, the expectation values in Eqs. (\[z1+\]) and (\[z1++\]) contain additional dependencies on the fugacity $\lambda_\alpha$ via the DH propagator, Eq.(\[vDH-1\]), but, and this stands at the very core of our method, all expectation values can itself be expanded with respect to $\lambda_\alpha$ and have finite values as $\lambda_\alpha \rightarrow 0$
In order to do a full expansion of Eq. (\[free-energy2\]), one needs first to calculate the coefficients $Z_1^+$, etc., in Eq. (\[free-energy2\]), for which the averages given in the appendix are needed, cf. Eq. (\[a\_av\_1\]–\[a\_av\_last\]). Since $v_c(0) - v_{{\mathrm{DH}}}(0) = {\Delta}v_0$, we obtain $$\label{Z1}
Z_1^{\alpha} = {\mathrm{e}^{q^2_{\alpha} {\Delta}v_0 / 2}} - 1 - \frac{1}{2} q_{\alpha}^2 {\Delta}v_0$$ and $$\begin{gathered}
\label{Zlast}
Z_2^{\alpha \beta} = \int {\rm d}{\mathbf{r}} \, \Biggl\{
{\mathrm{e}^{[q_{\alpha}^2 + q_{\beta}^2] {\Delta}v_0 / 2}}
\Bigl[ {\mathrm{e}^{-q_{\alpha} q_{\beta} [\omega_{\alpha \beta}({\mathbf{r}}) + \alpha \beta \,
v_{{\mathrm{DH}}}({\mathbf{r}})]}}
-1 \Bigr] \\ + \frac{1}{2} q_{\alpha}^2 q_{\beta}^2 v_{{\mathrm{DH}}}^2({\mathbf{r}})
\Bigl[ 1 - {\mathrm{e}^{q_{\alpha}^2 {\Delta}v_0 /2}} - {\mathrm{e}^{q_{\beta}^2 {\Delta}v_0 /2}} \Bigr] \Biggr\}.\end{gathered}$$ Note that $$\label{vDH}
v_{{\mathrm{DH}}}({\mathbf{r}}) = \frac{\ell_B}{r} {\mathrm{e}^{-{\Delta}v_0 r / \ell_B}}$$ was defined (through its inverse function) in Eq. (\[vDH-1\]). We now introduce the hard-core through the short-range potentials $$\label{hard-core}
\omega_{\alpha \beta}({\mathbf{r}}) =
\begin{cases}
+ \infty& \text{if } r < \bigl( d_{\alpha} + d_{\beta} \bigr) / 2, \\
0 & \text{otherwise,}
\end{cases}$$ where the indices $\alpha$ and $\beta$ stand again for $+$ and $-$; $d_+$ and $d_-$ are respectively the (effective) ionic diameters of the positive and negative particles.
Since the expressions for $Z_1^+$, etc., do depend on $\lambda_+$ and $\lambda_-$, and in order to have a consistent expansion of $g$ in the fugacities, one has to expand Eqs. (\[Z1\]–\[Zlast\]) in powers of $\lambda_+$ and $\lambda_-$, up to the appropriate order, before inserting them into $g$, Eq. (\[free-energy2\]). By doing this consistently up to order $5/2$ in the fugacities, one obtains the rescaled grand-canonical free energy density $$\begin{gathered}
\label{free-energy3}
{\widetilde{g}} \equiv d_+^3 g = \\
-{\widetilde{\lambda}}_+ - {\widetilde{\lambda}}_-
- m_1 {\widetilde{\lambda}}_+^2 - m_2 {\widetilde{\lambda}}_-^2 -
m_3 {\widetilde{\lambda}}_+ {\widetilde{\lambda}}_-
- \Bigl[ n_1 {\widetilde{\lambda}}_+^2 + \\
n_2 {\widetilde{\lambda}}_-^2 + n_3 {\widetilde{\lambda}}_+
{\widetilde{\lambda}}_- \Bigr] {\Delta}v_0 -
\Bigl[ p_1 {\widetilde{\lambda}}_+^2 +
p_2 {\widetilde{\lambda}}_-^2 + p_3 {\widetilde{\lambda}}_+
{\widetilde{\lambda}}_- \Bigr] \ln {\Delta}v_0 \\
-\Bigl[ r_1 {\widetilde{\lambda}}_+^2 +
r_2 {\widetilde{\lambda}}_-^2 + r_3 {\widetilde{\lambda}}_+ {\widetilde{\lambda}}_- \Bigr]
{\Delta}v_0 \ln {\Delta}v_0
- \Bigl[ s_1 {\widetilde{\lambda}}_+ \\ + s_2 {\widetilde{\lambda}}_- \Bigr] {\Delta}v_0^2 -
\Bigl[ t_0 + t_1 {\widetilde{\lambda}}_+ + t_2 {\widetilde{\lambda}}_- \Bigr] {\Delta}v_0^3
+ \Omega_0 \Bigl[ q_+ {\widetilde{\lambda}}_+ - q_- {\widetilde{\lambda}}_- \Bigr] \\ -
{\Delta}v_0 \Biggl\{ \Omega_1 \Bigl[ q_+ {\widetilde{\lambda}}_+ - q_-
{\widetilde{\lambda}}_- \Bigr]^2 -
\Omega_0 \Bigl[ q_+^4 {\widetilde{\lambda}}_+^2 + q_-^4 {\widetilde{\lambda}}_-^2 \\ -
2 q_+ q_- \Bigl[ \frac{q_+^2+q_-^2}{2} \Bigr] {\widetilde{\lambda}}_+
{\widetilde{\lambda}}_- \Bigr]
\Biggr\}+ O(\tilde{\lambda}^3),\end{gathered}$$ where we used dimensionless fugacities $${\widetilde{\lambda}}_+ \equiv d_+^3 \lambda_+$$ and $${\widetilde{\lambda}}_- \equiv d_+^3 \lambda_-.$$ In this expansion, we utilized that ${\Delta}v_0$ scales like ${\widetilde{\lambda}}^{1/2}$. The coefficients $m_1$, etc. are given explicitly in the appendix, Eqs. (\[a\_coeff\_1\]–\[a\_coeff\_last\]).
The coefficients $\Omega_0$ and $\Omega_1$ are given by the divergent integrals $$\begin{split}
\Omega_0 &= 2 \pi \ell_B \int\limits_0^{\infty} {\rm d}r \, r \\
\Omega_1 &= 2 \pi \int\limits_0^{\infty} {\rm d}r \, r^2.
\end{split}$$ These terms are present in Eq. (\[free-energy3\]) because global charge neutrality has not been yet demanded. By imposing this condition, these divergent terms cancel exactly, as is shown next.
Imposing electroneutrality {#chapterD:neutral}
--------------------------
The electroneutrality condition ensures that the global charge of the system is zero, i.e., $q_+ N_+ = q_- N_-$. In the grand-canonical ensemble, $N_+$ and $N_-$ are no longer fixed numbers but average values. This means that the electroneutrality condition in the grand-canonical ensemble is given by $$\label{neutro}
q_+ \langle N_+ \rangle = q_- \langle N_- \rangle.$$ Defining the rescaled ion density $${\widetilde{c}}_+ = d_+^3 \langle N_+ \rangle / V,$$ it is easy to show that $$\label{concentra}
{\widetilde{c}}_+ = - {\widetilde{\lambda}}_+ \frac{\partial {\widetilde{g}}}
{\partial {\widetilde{\lambda}}_+}$$ with an analogous formula for ${\widetilde{c}}_-$; ${\widetilde{c}}_+$ is the volume fractions of the positive ions[@note_volfrac]. As one imposes Eq. (\[neutro\]), the fugacities will depend on each other in a non-trivial way such that the system is, on average, neutral. If the system were totally symmetric (i.e., $q_+=q_-$ and $d_+=d_-$), this dependence would be given by the relation ${\widetilde{\lambda}}_+ = {\widetilde{\lambda}}_-$[@roland-orland:tcp]. However, this is not the case here: one needs to find the relation between the fugacities order-by-order.
First, assume that ${\widetilde{\lambda}}_-$ can be expanded in terms of ${\widetilde{\lambda}}_+$ such that $$\begin{gathered}
\label{lambda-}
{\widetilde{\lambda}}_- = a_0 {\widetilde{\lambda}}_+
+ a_1 {\widetilde{\lambda}}_+^{3/2} +
a_2 {\widetilde{\lambda}}_+^2 +
a_3 {\widetilde{\lambda}}_+^2 \ln {\widetilde{\lambda}}_+ \\
+ a_4 {\widetilde{\lambda}}_+^{5/2} +
a_5 {\widetilde{\lambda}}_+^{5/2} \ln {\widetilde{\lambda}}_+ +
O({\widetilde{\lambda}}_+^3).\end{gathered}$$ This is inspired by the expanded form of the grand-canonical free energy ${\widetilde{g}}$, Eq.(\[free-energy3\]).
After calculating ${\widetilde{c}}_+$ and ${\widetilde{c}}_-$ from ${\widetilde{g}}$, Eq. (\[free-energy3\]), according to Eq.(\[concentra\]), and insertion into the electroneutrality condition Eq. (\[neutro\]), we substitute the fugacity ${\widetilde{\lambda}}_-$ by its expanded form Eq. (\[lambda-\]). This leads to the expanded form (up to order ${\widetilde{\lambda}}_+^{5/2}
\ln {\widetilde{\lambda}}_+$) of the electroneutrality condition. Solving it consistently, order-by-order, yields the values of the coefficients in Eq.(\[lambda-\]), which ensure electroneutrality order by order. For instance, at linear order in ${\widetilde{\lambda}}_+$, the expanded form of Eq. (\[neutro\]) reads $$q_+ - a_0 q_- = 0,$$ which naturally gives $a_0 = q_+ / q_-$. With the knowledge of $a_0$, one can solve the next-order term (in this case ${\widetilde{\lambda}}_+^{3/2}$) and obtain the value of $a_1$, and so on. The resulting coefficients $a_0$ up to $a_5$ are given in the appendix—cf. Eqs. (\[a\_a0\]–\[a\_a5\]).
As this order-by-order neutrality condition is imposed, one notices that the terms $\Omega_0$ and $\Omega_1$ in Eq. (\[free-energy3\]) are exactly canceled in a natural way, without any further assumptions. The resulting expression for ${\widetilde{g}}$, now expanded only in one of the fugacities (in this case ${\widetilde{\lambda}}_+$), is then a well behaved expansion (we omit this expression here since it is quite lengthy). With this, one can finally obtain the canonical free energy (as a density expansion) through a Legendre transform.
The canonical free energy
-------------------------
In order to obtain the free energy in the canonical ensemble, we use the back-Legendre-transformation $$\label{legendre}
{\widetilde{f}} = {\widetilde{g}} + {\widetilde{c}}_+
\ln \bigl( {\widetilde{\lambda}}_+ \bigr) +
{\widetilde{c}}_- \ln \bigl( {\widetilde{\lambda}}_- \bigr),$$ where the dimensionless canonical free energy density is $${\widetilde{f}} \equiv d_+^3 F / V k_B T$$ (F is the canonical free energy). Note that at this point ${\widetilde{\lambda}}_-$ is a function of ${\widetilde{\lambda}}_+$, according to Eq. (\[lambda-\]).
The first step to obtain ${\widetilde{f}}$ is to invert the expression given in Eq. (\[concentra\]) such that ${\widetilde{\lambda}}_+$ is obtained as an expansion in ${\widetilde{c}}_+$, neglecting any terms of order ${\widetilde{c}}_+^3$ or higher. With this, we obtain ${\widetilde{\lambda}}_+ = {\widetilde{\lambda}}_+({\widetilde{c}}_+)$: plugging this into Eq.(\[legendre\]), we finally obtain ${\widetilde{f}}$, which reads $$\begin{gathered}
\label{f}
{\widetilde{f}}={\widetilde{f}}_{{\mathrm{id}}} + B_{{\mathrm{DH}}} {\widetilde{c}}_+^{3/2} +
B_2 {\widetilde{c}}_+^2 + B_{2{\mathrm{log}}} {\widetilde{c}}_+^2 \ln {\widetilde{c}}_+
\\ + B_{5/2} {\widetilde{c}}_+^{5/2} + B_{5/2{\mathrm{log}}} {\widetilde{c}}^{5/2}_+ \ln {\widetilde{c}}_+
+O\bigl(\tilde{c}_+^3 \bigr).\end{gathered}$$ Defining the valence ratio parameter $$\eta \equiv q_- / q_+,$$ the diameter ratio parameter $$\xi \equiv d_- / d_+,$$ and the coupling parameter $$\epsilon_+ \equiv q_+^2 \ell_B / d_+$$ (which is the ratio between the Coulomb energy at contact between two positive ions and the thermal energy $k_B T$), the coefficients in Eq. (\[f\]) can be explicitly written as $$\label{ideal}
{\widetilde{f}}_{{\mathrm{id}}} = {\widetilde{c}}_+ \ln {\widetilde{c}}_+ + \frac{{\widetilde{c}}_+}{\eta}
\ln \Bigl( \frac{{\widetilde{c}}_+}{\eta} \Bigr) -
\Bigl[ 1 + \frac{1}{\eta} \Bigr] {\widetilde{c}}_+$$ which is the ideal contribution to the free energy, $$\label{BDH}
B_{{\mathrm{DH}}}= - \frac{2}{3}
\sqrt{\pi \epsilon_+^3 [1+\eta]^3}$$ which is the coefficient of the Debye-H[ü]{}ckel limiting law term (order $3/2$ in ${\widetilde{c}}_+$). It is useful to define the function $$\label{Hdefine}
H(x) = \frac{11}{6} - 2 \gamma + \frac{1}{x^3} {\mathrm{e}^{-x}}
\bigl[ 2 - x + x^2 \bigr] - \Gamma(0,x) - \ln x,$$ where $\gamma$ is the Euler’s constant and $\Gamma(a,b)$ is the incomplete Gamma-function[@abramowitz:book]. Using $H(x)$, the higher order coefficients can be explicitly written as $$\begin{gathered}
\label{B2}
B_2 = - \frac{\pi}{3} \epsilon_+^3 \Biggl\{ - H(\epsilon_+) - \ln \epsilon_+
+ 2 \eta^2 \Biggl[ H\Bigl(-\frac{2 \eta \epsilon_+}{1+\xi} \Bigr) \\ +
\ln\Bigl( \frac{2 \eta \epsilon_+}{1+\xi} \Bigr) \Biggr]
- \eta^4 \Biggl[ H \Bigl( \frac{\eta^2 \epsilon_+}{\xi} \Bigr) +
\ln\Bigl( \frac{\eta^2 \epsilon_+}{\xi} \Bigr) \Biggr] \\
- 2 \eta^2 \bigl[ 1 - \eta^2 \bigr] \ln\eta
+ \frac{1}{2} \bigl[ 1-\eta^2 \bigr]^2
\ln \bigl( 36 \pi \epsilon_+^3 [1+\eta] \bigr) \Biggl\}\end{gathered}$$ $$\label{B2log}
B_{2{\mathrm{log}}} = - \frac{\pi}{6} \epsilon_+^3 \bigl[ 1-\eta^2 \bigr]^2,$$ $$\begin{gathered}
B_{5/2} = \frac{2}{3} \bigl[ \pi \epsilon_+^3 \bigr]^{3/2}
\bigl[1 + \eta \bigr]^{1/2} \Biggl\{
\frac{5}{8} + H\bigl(\epsilon_+ \bigr) + \ln \bigl( \epsilon_+ \bigr) \\
+ \eta^6 \Biggl[ \frac{5}{8} +
H\Bigl( \frac{\eta^2 \epsilon_+}{\xi} \Bigr) +
\ln \Bigl( \frac{\eta^2 \epsilon_+}{\xi} \Bigr) \Biggr]+
2 \eta^3 \Biggl[ \frac{5}{8} + H\Bigl(-\frac{2 \eta \epsilon_+}{1+\xi} \Bigr) \\
+ \ln\Bigl(\frac{2 \eta \epsilon_+}{1+\xi} \Bigr) \Biggr]
+ \frac{1}{8} \bigl[ 1 + \eta \bigr]^2 \bigl[ 5 - 12 \eta +
17 \eta^2 - 12 \eta^3 + 5 \eta^4 \bigr] \\
- 2 \eta^3 \bigl[ 1 + \eta^3 \bigr] \ln \eta
- \frac{1}{2} \bigl[ 1 + \eta^3 \bigr]^2
\ln\Bigl( 64 \pi \epsilon_+^3 \bigl[ 1 + \eta \bigr] \Bigr)
\Biggr\}\end{gathered}$$ and $$B_{5/2{\mathrm{log}}} = - \frac{1}{3} \bigl[ \pi \epsilon_+^3 \bigr]^{3/2}
\bigl[1 + \eta \bigr]^{1/2} \bigl[ 1 + \eta^3 \bigr]^2.$$
The free energy Eq. (\[f\]) is the exact low density expansion of the asymmetric TCPHC and constitutes the main result of this paper. The only parameter that is demanded to be small is the ion density ${\widetilde{c}}_+$, that means, this result is non-perturbative in the coupling $\epsilon_+$, charge ratio $\eta$ and size ratio $\xi$.
We chose the positive ions as the “reference species” (i.e., the expansion is done with respect to ${\widetilde{c}}_+$) without any loss of generality, since the relation between ${\widetilde{c}}_+$ and ${\widetilde{c}}_-$ is fixed through the electroneutrality condition. As consistency checks, we notice that our expression for ${\widetilde{f}}$ is symmetric, as expected, with respect to the simultaneous exchange of $d_+$ with $d_-$ and $q_+$ with $q_-$. Also, in the limit $d_+ = d_-$ and $q_+ = q_-$, we recover the same expression as previously calculated in Ref. [@roland-orland:tcp] for totally symmetric systems. Finally, as one turns off the charges in the system (or equivalently, as one takes the limit $\epsilon_+ \rightarrow 0$), the pure hard-core fluid is recovered, i.e., ${\widetilde{f}}$ becomes the usual virial expansion with $B_{3/2}$ and $B_{5/2}$ equal to zero and $B_2$ given by the second virial coefficient of a two-component hard-core gas, $B_2^{\rm HC} = B_2(\epsilon_+ \rightarrow 0)$, which in our units reads $$\label{BHC}
B_2^{{\mathrm{HC}}}=\frac{\pi}{3}
\biggl[ 2 + \frac{2 \xi^3}{\eta^2} + \frac{[1+\xi]^3}{2 \eta} \biggr].$$ This limit can be also understood as the high-temperature regime: as the thermal energy largely exceeds the Coulomb energy at contact, the hard core interaction becomes the only relevant interaction between the particles.
Results {#chapterD:applications}
=======
The virial coefficients
-----------------------
The behavior of the coefficients $B_2$ and $B_{5/2}$ as functions of the coupling parameter $\epsilon_+$ are depicted in Figs. \[fig:fig1\] and \[fig:fig2\], the behavior of $B_{\rm 2log}$ and $B_{\rm 5/2log}$ is rather trivial and not shown graphically. In Fig. \[fig:fig1\] the ionic diameters of positive and negative ions are kept equal, $d_+=d_-$, and the ratio between the charge valences, $\eta$, is varied, while in Fig. \[fig:fig2\] the charge valences are equal and the ratio between the ionic diameters is varied. These figures highlight the fact that both coefficients diverge as $\epsilon_+$ goes to infinity. In this limit, $$\label{B2_asy}
B_2 \approx - \frac{\pi}{4} \frac{[1 + \xi]^4}{\eta^2}
\frac{1}{\epsilon_+} \exp\Bigl(\frac{2 \eta \epsilon_+}{1+\xi} \Bigr)$$ and $$\label{B52_asy}
B_{5/2} \approx \frac{\pi^{3/2}}{2} \frac{[1+\xi]^4 \sqrt{1+\eta}}{\eta}
\sqrt{\epsilon_+} \exp\Bigl(\frac{2 \eta \epsilon_+}{1+\xi} \Bigr).$$ Note that $B_{5/2}$ diverges faster (and with opposite sign) than $B_2$, reflecting that this low-density expansion is badly converging, as we will discuss further below. The exponentially divergent behavior of $B_2$ and $B_{5/2}$ when $\epsilon_+ \rightarrow \infty$ is due to the increasing importance of the interaction between oppositely charged particles (ionic pairing) as the coupling parameter increases[@bjerrum:26; @fisher-levin:93; @yeh-zhou-stell:96], corresponding for instance to lower temperatures. This is confirmed by noting that the argument in the exponential occurring in both asymptotic forms Eqs. (\[B2\_asy\]) and (\[B52\_asy\]), viz. $2 \, \eta \, \epsilon_+ / [1 + \xi]$, can be re-expressed as $2 \, q_+ \, q_- \, \ell_B / [d_+ + d_-]$, which is the coupling between positive and negative ions (in this case, the ratio between the Coulomb contact energy between oppositely charged ions and the thermal energy $k_B T$).
To estimate roughly the ionic density $\tilde{c}_+$ up to which the expansion in Eq. (\[f\]) is expected to be valid, we use the following simple criterion: the terms proportional to $B_{\rm DH}$ and $B_{5/2}$ give the same contribution to the free energy when $|B_{\rm DH} |\tilde{c}_+^{3/2} =| B_{5/2} | \tilde{c}_+^{5/2}$ or, in other words, at a critical density $\tilde{c}_+ =
|B_{\rm DH}/B_{5/2}|$. Note that we set up this criterion separately for the integer and fractional coefficients, since the scaling behavior for the two different classes is very different and no meaningful results can be obtained by mixing them. The analogous criterion for the integer terms leads to a critical density $\tilde{c}_+ = (1+1/\eta) /|B_{2}|$ where for the ideal contribution in Eq. \[ideal\] we replace the logarithmic term by a linear one (which is the ideal contribution to the osmotic pressure). We tacitly assume that the higher-order terms (which we have not calculated) show the same relative behavior, an assumption which seems plausible to us but which we cannot check. In Fig. \[figratio\] we show the estimate for the critical densities obtained from a) the ratio of the integer terms and from b) the ratio of the fractional terms. In a) the critical density settles at a finite value for vanishing coupling parameter $\varepsilon_+ \rightarrow 0$, and decreases to zero for increasing coupling parameter (the divergence in the curves at finite value of $\varepsilon_+$ is not significant since it is caused by a change in sign of $B_2$.). This means that for small coupling constants the integer coefficients are expected to show regular convergence behavior for not too large concentrations. For large values of $\tilde{c}_+$ on the other hand, the range of densities which can be described by the expansion quickly goes to zero. In b) the displayed behavior is more complex. The expansion of the coefficient $B_{5/2}$ for small values of the coupling parameter $\varepsilon_+$ leads to $$\begin{aligned}
B_{5/2}&=&
\pi^{3/2} \sqrt{1+\eta} (\xi-1)^2(\xi+1) \varepsilon_+^{3/2} \\
&&-\pi^{3/2} \sqrt{1+\eta} (2+2\eta^2 \xi^2+\eta(1+\xi)^2) \nonumber
\varepsilon_+^{5/2}\end{aligned}$$ plus terms with scale as higher powers in $\varepsilon_+$. The important point is that except for the singular case $\xi=1$, that is for particles of identical radius, the leading term of $B_{5/2}$ scales as $ \varepsilon_+^{3/2}$ and thus identical to the lower leading term $B_{\rm DH}$. In the case $\xi=1$ the leading term scales as $\varepsilon_+^{5/2}$ and thus shows a different scaling behavior than $B_{\rm DH}$. In Fig. \[figratio\]b) this difference is cleary seen: For $\xi=1$ the curves diverge as $\varepsilon_+ \rightarrow 0$ and the convergence of the fractional density terms in the series is expected to be guaranteed. For $\xi =0.1$ the behavior is similar to the results in a), showing a saturation at a finite value as $\varepsilon_+ \rightarrow 0$ (and an unsignificant divergence at finite $\varepsilon_+$). All curves go to zero as the coupling parameter grows. In conclusion, the reliability of the low-density expansion becomes worse as the coupling parameter increases, but one can always find a window of small densities within which the expansion should work.
Finally, we mention that in a different calculation scheme, based on an integral-equation-procedure within the so-called mean-spherical-model (MSM) approximation, similar results for the free energy and other thermodynamic functions have been obtained[@waisman-lebowitz:72; @Palmer; @Blum]. Those results also reproduce the limiting laws, namely the hard-core behavior as the coupling parameter goes to zero, and the leading Debye-Hückel correction at low density, and give a quite satisfactory description even for much larger values of the density. It is important to note, however, that the next-leading (beyond Debye-Hückel) terms in the low-density expansion are not correctly reproduced by the MSM approximation.
The colloidal limit
-------------------
In colloidal suspensions, flocculation or coagulation (driven by attractive van der Waals interaction between colloidal particles) can be prevented by the presence of repulsive electrostatic forces. These suspensions are typically quite dilute, with volume fractions of colloidal particles usually not higher than a few percent. The macro-particles normally have dimensions[@ottewill:review; @gonnet-al:94; @kurth-al:2000] ranging from 10 to 1000 nm and charges of several thousands $e$ (elementary charge), with much smaller counterions that have a charge of a few $e$. In such systems the charge and size asymmetry between ions and counterions is immense; the TCPHC with unconstrained charge and size asymmetry is a suitable model for dilute colloidal suspensions, and the free energy Eq. (\[f\]) can be used to study the thermodynamic behavior of such systems.
With this in mind, let us take the following limit: assume the parameters describing the positive ions (representing the colloids) $d_+$, $q_+$ and ${\widetilde{c}}_+$ fixed, and make both $\eta \equiv q_-/q_+$ and $\xi \equiv d_-/d_+$ vanishingly small (this has to be done with some care, since the limit $\eta=0$ is not well-defined). One can then rewrite the free energy Eq. (\[f\]) up to second order in ${\widetilde{c}}_+$ as $$\begin{gathered}
\label{f-collo}
{\widetilde{f}}^{{\mathrm{cl}}} = {\widetilde{f}}_{{\mathrm{id}}} +
+ B^{{\mathrm{cl}}}_{{\mathrm{DH}}} {\widetilde{c}}_+^{3/2} +
( B_2^{{\mathrm{HC,cl}}} + B^{{\mathrm{cl}}}_2) {\widetilde{c}}_+^2 \\
+ B^{{\mathrm{cl}}}_{2{\mathrm{log}}} {\widetilde{c}}_+^2 \log {\widetilde{c}}_+ + O({\widetilde{c}}_+^{5/2})\end{gathered}$$ where ${\widetilde{f}}_{{\mathrm{id}}}$ is the ideal term Eq. (\[ideal\]) and $$\label{f-collo-hc}
B_2^{{\mathrm{HC,cl}}} = \frac{\pi}{3} \biggl[ \frac{2 \xi^3}{\eta^2} +
\frac{[1+\xi]^3}{2 \eta} \biggr]$$ is the hard core contribution due to the counterion–counterion and macroion–counterion interactions only, without the macroion–macroion contributions, which explains the difference to the full hard-core virial coefficient in Eq.(\[BHC\]). It is necessary to treat this term separately, as it does not have a well-defined behavior in the double limit $\eta \rightarrow 0$ and $\xi \rightarrow 0$. The other coefficients in Eq.(\[f-collo\]) follow from Eqs.(\[BDH\]), (\[B2\]), (\[B2log\]) by performing the limits $B^{{\mathrm{cl}}}_{{\mathrm{DH}}} = B_{{\mathrm{DH}}} (\eta \rightarrow 0, \xi \rightarrow 0)$, $B^{{\mathrm{cl}}}_{{\mathrm{2}}} = B_{{\mathrm{2}}} (\eta \rightarrow 0, \xi \rightarrow 0)
-B_2^{{\mathrm{HC,cl}}}$, $B^{{\mathrm{cl}}}_{{\mathrm{2log}}} = B_{{\mathrm{2log}}} (\eta \rightarrow 0, \xi \rightarrow 0)$, and are given by $$\label{BDHcollo}
B^{{\mathrm{cl}}}_{{\mathrm{DH}}} = -\frac{2}{3} \sqrt{\pi \epsilon_+^3},$$ $$\label{B2collo}
B^{{\mathrm{cl}}}_2 = \frac{\pi}{3} \epsilon_+^3 \Bigl\{ H\bigl(\epsilon_+ \bigr)
- \frac{1}{2} \ln\bigl(36 \pi \epsilon_+ \bigr) \Bigr\}
+\frac{\pi}{2} \epsilon_+,$$ $$\label{B2logcl}
B^{{\mathrm{cl}}}_{2{\mathrm{log}}} = -\frac{\pi}{6} \epsilon_+^3.$$ It is to be noted that the contributions in ${\widetilde{f}}_{{\mathrm{id}}}$ and $B_2^{\rm HC,cl}$ contain terms that scale with $1/\eta$. For dilute colloidal systems therefore the ideal contribution of the counterions dominates the free energy and cannot be neglected. Effects due to the electrostatic interaction between the ions are corrections to the ideal behavior.
At this point we briefly introduce yet another model that is also widely used to describe charged systems. It is the one-component plasma (OCP), which in its simplest form consists of a collection of $N$ equally charged point-like particles immersed in a neutralizing background that ensures the global charge neutrality of the system (in the TCPHC electroneutrality is ensured by oppositely charged particles). The OCP, or its quantum mechanical counter-part (“jellium”) has been used in different contexts in physics, as for instance to describe degenerate stellar matter (the interior of white dwarfs or the outer layer of neutron stars) and the interior of massive planets like Jupiter. Another example comes from condensed matter physics, where jellium is often used as a reference state when calculating the electronic structure of solids. For reviews see Refs. [@ishimaru:82; @baus-hansen:80; @abe:59].
When the particles have a hard core, the OCP is called one-component hard-core plasma (OCPHC): what we will see next is that the electrostatic contribution to the free energy in dilute colloidal suspensions can be almost described through the OCPHC. If we compare the coefficients Eqs. (\[BDHcollo\]–\[B2log\]) with the ones previously obtained[@roland-orland:tcp] for the low density expansion of the OCPHC $$\label{focpRol}
{\widetilde{f}}_{\rm OCPHC}={\widetilde{c}} \ln {\widetilde{c}} -{\widetilde{c}} + d_{3/2} {\widetilde{c}}^{3/2} +
d_2 {\widetilde{c}}^2 + d_{\ln2} {\widetilde{c}}^2 \ln {\widetilde{c}} + O({\widetilde{c}}^{5/2}),$$ where ${\widetilde{c}}$ is the volume fraction of particles in the OCPHC, we find (cf. Eqs. (51), (56) and (57) of Ref. [@roland-orland:tcp]) $$\label{bcocp1}
B^{\rm cl}_{{\mathrm{DH}}} = d_{3/2},$$ $$\label{bcocp2}
B^{\rm cl}_2 = d_2 + \frac{\pi}{2} \epsilon_+$$ and $$\label{bcocp3}
B^{\rm cl}_{2{\mathrm{log}}} = d_{\ln2},$$ where the notation used in Ref. [@roland-orland:tcp] is kept in Eq. (\[focpRol\]) and on the rhs of Eqs. (\[bcocp1\]–\[bcocp3\]). The comparison between $B^{\rm cl}_2$ and $d_2$ is shown in Fig. \[fig:fig3\], where we rescaled both coefficients by their value at vanishing coupling, $d_2(\epsilon_+ =0)=B_2^{\rm cl} (\epsilon_+ =0)= 2 \pi/3$. This shows that, including the order ${\widetilde{c}}_+^2 \ln {\widetilde{c}}_+$ (i.e., for very dilute colloidal suspension), the OCPHC is almost recovered, except an additional term in Eq.(\[bcocp2\]). This additional term can be quite important at intermediate values of $\epsilon_+$ since it changes the sign of the virial coefficient, as seen in Fig. 3.
This is in fact what one would intuitively expect: each macroion has around it a very large number of small neutralizing counterions which act like a background. Our results show that the background formed by counterions, which is a deformable background since the counterion distribution is not uniform, acts almost like a rigid homogeneous background, as it is assumed in OCP calculations. However, there is a small difference between the TCPHC in this limit and the OCPHC: in the latter, the background penetrates the particles, while in the TCPHC it cannot (for a thorough discussion of this difference see [@Hansen]). In our calculation, this is reflected in the ${\widetilde{c}}_+^2$ term, where $\pi \epsilon_+ / 2$ is the positive extra cost in the free energy that the OCPHC has to pay (at this order) to expel the background from the hard-core particles. The first consequence of the non-penetrating background is that the effective charge of the colloids increases by exactly the amount of background charge that is expelled from the colloidal interior. A simple calculation shows that the increased effective charge $q_+^{\rm eff}$ of colloids for the non-penetrating background turns out to be $$\label{qeff}
q_+^{\rm eff} = \frac{q_+}{1- \pi \tilde{c}_+ /6}.$$ In the low-density expansion of the OCPHC, Eq.(\[focpRol\]), one would have to reexpand all coefficients with respect to the density, using Eq.(\[qeff\]). However, since the leading term which depends on the charge $q_+$ goes like $\tilde{c}_+^{3/2}$, the effect would come in at order $\tilde{c}_+^{5/2}$ and therefore is not responsible for the additional term in Eq.(\[bcocp2\]). The reason for the extra term in Eq.(\[bcocp2\]) has to do with an increase of the self-free-energy of a colloidal particle, which can be understood in the following way: Assume that the OCPHC is very dilute, such that each colloidal particle and its neutralizing background form a neutral entity (in the spirit of the cell model, see for instance Ref. [@alexander-al:84]) that can be regarded independent from the other particles. The free energy difference *per colloidal particle* between a system without penetrating background and with penetrating background (Fig. \[fig:bolas\_ocp\]) is then given by two contributions: one coming from the entropy lost by the background (formed by $N_-$ counterions) since it cannot penetrate the macroions, given by $$\frac{\Delta F_{en}}{N_+ k_B T} =
-\frac{N_-}{N_+} \ln(1- \pi \tilde{c}_+/6) =
\frac{N_-}{N_+} \frac{\pi {\widetilde{c}}_+}{6} + O({\widetilde{c}}^2_+).$$ This is one of the contributions to the second virial coefficient due to the hard-core interaction, and already included in Eq. (\[f-collo-hc\]). This term does not depend on the coupling $\epsilon_+$, and therefore has nothing to do with the extra term in Eq.(\[bcocp2\]). The second contribution to the self-energy is electrostatic in origin. Defining the charge distributions for the situations in Fig. 4 as ${\rho}_a({\mathbf{r}})= q_+ \delta({\mathbf{r}}) - \rho_0$ when the background can enter the colloids and ${\rho}_b({\mathbf{r}})= q_+ \delta({\mathbf{r}}) - \rho_0 \bigl[1-\theta(d_+/2 - r)]$ when the background cannot enter the colloids (where $\rho_0=c_+ q_+$ is the background charge density and $\theta(x)=1$ if $x>0$ and $0$ otherwise), the electrostatic self-energy difference between the two cases reads $$\begin{gathered}
\frac{\Delta F_{el}}{N_+ k_B T} = \frac{1}{2} \int {\rm d}{\mathbf{r}} {\rm d}{\mathbf{r}}' \,
\biggl\{ {\rho}_b({\mathbf{r}}) v_c({\mathbf{r}}-{\mathbf{r}}') {\rho}_b({\mathbf{r}}') \\ -
{\rho}_a({\mathbf{r}}) v_c({\mathbf{r}}-{\mathbf{r}}') {\rho}_a({\mathbf{r}}') \biggr\} \\
= \rho_0 q_+ 4 \pi \ell_B \int_{0}^{d_+/2} {\rm d}r \, r + O(c_+^2) \\
= \frac{\pi}{2} {\widetilde{c}}_+ \epsilon_+ + O(c_+^2).\end{gathered}$$ It follows that the free energy difference per volume is given by $$\Delta \tilde{f} = \frac{ \tilde{c}_+ \Delta F_{el}}{N_+ k_BT}
= \frac{\pi \tilde{c}_+^2 \epsilon_+}{ 2}.$$ Note that this is a positive contribution to the ${\widetilde{c}}^2$ term with the same coefficient as the extra term in Eq. (\[bcocp2\]). Therefore, we conclude that this extra term is due to the electrostatic energy associated with expelling the background from the colloidal volume. This is in accord with more general results on the difference between OCPHC models with penetrating and excluded neutralising backgrounds[@Hansen]. In summary, whenever using the OCPHC to describe highly asymmetric charged systems such as colloids, one has to to take into account the exclusion of the background from the macroions. As we demonstrated, if this is taken into account, then the TCPHC maps exactly (at least up to order ${\widetilde{c}}_+^2$) onto the OCPHC.
Ionic activity and diameters {#chapterD:radii}
----------------------------
In electrochemistry, it is usual to define the mean activity $\lambda_{\pm}$ of an electrolyte as $$\label{ma}
\lambda_{\pm} \equiv \Bigl[ \lambda_+^{q_-} \lambda_-^{q_+}
\Bigr]^{ 1 / [q_+ + q_-]},$$ where $\lambda_+$ and $\lambda_-$ are respectively the fugacities of the positive and the negative ions. The mean activity coefficient ${\mathrm{f}}_{\pm}$ is the ratio between the mean activity of the electrolyte and that of an ideal gas, in general given by $$\label{mac}
{\mathrm{f}}_{\pm} =
\exp \Bigl( \frac{q_-}{q_+ + q_-}
\frac{\partial {\widetilde{f}}_{{\mathrm{ex}}}}{\partial {\widetilde{c}}_+} \Bigr),$$ for a two-component system where the positively charged particles are used as reference species in the same way as in Eq. (\[f\]). ${\widetilde{f}}_{{\mathrm{ex}}}$, the excess free energy, is the difference between the free energy of the interacting systems and the free energy of an ideal gas, i.e., ${\widetilde{f}}_{{\mathrm{ex}}} = {\widetilde{f}} - {\widetilde{f}}_{{\mathrm{id}}}$.
There are different ways of measuring ${\mathrm{f}}_{\pm}$, as for instance, through the change of the freezing point of the solvent (usually water) with the addition of salt[@robinson-stokes:book], by measuring the change of the potential difference between the electrodes of a concentration cell as salt is added[@butler-roy:91; @robinson-stokes:book] (potentiometry), or by direct measurement of the solvent activity through vapor exchange between a solution with known activity and the sample[@rard-platford:91; @robinson-stokes:book] (isopiestic). Although dating from the early nineteen hundreds, these are still the most common techniques used today, especially the potentiometry, which is regarded as the most precise technique of all. The values of ${\mathrm{f}}_{\pm}$ are tabled as function of the salt concentration for many different electrolytes[@conway:book; @parsons:book; @robinson-stokes:book].
From the free energy Eq. (\[f\]) and the definition Eq. (\[mac\]) we can obtain the low density expansion for the mean activity coefficient of a $q_+ \! : \! q_-$ salt. To compare with experimental results, it is useful to note that ${\widetilde{c}}_+ = 6.022 \times 10^{-4} q_- d_+^3 \varrho$, where $d_+$ is in [Å]{}ngstr[ö]{}ms and $\varrho$ is the salt concentration in moles/liter. After the appropriate expansion, ${\mathrm{f}}_{\pm}$ reads $$\begin{gathered}
\label{mac-expd}
{\mathrm{f}}_{\pm} = 1 + \nu_{{\mathrm{DH}}} \varrho^{1/2} +
\nu_1 \varrho +\nu_{1{\mathrm{log}}} \varrho \ln \varrho +
\nu_{3/2} \varrho^{3/2} \\ +
\nu_{3/2{\mathrm{log}}} \varrho^{3/2} \ln \varrho + O(\varrho^2)\end{gathered}$$ (the order $3/2$ in the mean activity coefficient is the one consistent with a free energy given up to $5/2$). The coefficients $\nu_{{\mathrm{DH}}}$, $\nu_1$, etc. are given in the appendix, Eqs. (\[a\_nuI\]–\[a\_nuE\]). Experimentally determined activity coefficients are typically measured at constant pressure, while the theoretical results are obtained for constant volume. Note that in the limit of vanishing density, the distinction between the activities calculated in the Lewis-Randall or in the McMillan-Mayer descriptions (constant pressure versus constant volume ensemble) becomes negligible[@pailthorpe-al:82] in comparison to the first corrections to ideal behavior. The experimental data at low density can be directly compared with the theory without the need to convert between the two ensembles.
At infinite dilution, the mean activity coefficient Eq. (\[mac-expd\]) goes to $1$, which is the prediction for an ideal gas. The first correction to the ideal behavior is the term $\nu_{{\mathrm{DH}}} \varrho^{1/2}$, which is the prediction one obtains from the Debye-H[ü]{}ckel limiting law (DHLL), and is independent of the ionic diameters, see Eq. (\[a\_nuI\]). This means that there is always a range of concentrations where different salts (but with the same $q_+$ and $q_-$) will deviate from ideality, but have the same activity. At higher concentrations, other terms have to be taken into account, and the ionic sizes begin to play an important role. In fact, as one fixes $\ell_B$ ($\simeq 7.1$ [Å]{} in water at 25 $^o$C) and the valences $q_+$ and $q_-$, the only free parameters in the rhs of Eq. (\[mac-expd\]) are the ionic diameters $d_+$ and $d_-$, which can be used in the theoretical predictions to fit the experimental values. This leads to effective equilibrium values of the ionic diameters (when in solution), which we call the “thermodynamic diameter”, in contrast to the bare diameter[@israelachvili:book] (obtained through crystallographic methods) and the hydrodynamic diameter (obtained from mobility measurements[@wennerstroem:book]). By construction, it is this diameter which should be used in equilibrium situations if one wants to describe an electrolyte solution as a TCPHC, as for example computer simulations of electrolytes.
### Fitting assuming one mean diameter
We now show the fitting procedure assuming that $d_+ = d_- = d$, where $d$ is the mean diameter. We also restrict here this procedure to $1:1$ salts, where more experimental data is available at reasonably low densities (below $\sim 0.05$ mole/liter). For asymmetric salts, this method demands precise data at the range below $\sim 0.01$ mole/liter, where Debye-H[ü]{}ckel is often assumed to account for all effects and few experimental points are available.
The assumption of equal ionic sizes has been often used in the past to fit activities to modified Debye-H[ü]{}ckel theories[@robinson-stokes:book] to account for the ionic sizes (as previously mentioned, the Debye-H[ü]{}ckel limiting law is insensitive to it). Since we force the two diameters to be equal, we only need the expansion for $f_{\pm}$ up to linear order since this will determine uniquely the single unknown parameter. From the experimental data ${\mathrm{f}}^{exp}_{\pm}$[@sheldlovsky:50; @conway:book] we subtract the diameter-independent term and obtain the difference $$\label{delta-mac1}
{\Delta}{\mathrm{f}}_{\pm} \equiv {\mathrm{f}}^{exp}_{\pm} - \nu_{{\mathrm{DH}}} \varrho^{1/2}
= \nu^{exp}_1 \varrho + O(\varrho^{(3/2)}).$$ The fit to this function leads to the coefficient of the linear term $\nu^{exp}_1$, and that can be used to determine the diameter by solving the equation $\nu_1 (d) = \nu^{exp}_1$ (with $d$ as unknown). In Fig. \[fig:fit1\_11\] we show the experimental[@sheldlovsky:50; @conway:book] $f_{\pm}$ for HCl, NaCl and KCl and the theoretical results (up to linear order in $\varrho$) using, respectively, the diameters $4.0$ [Å]{}, $3.7$ [Å]{} and $3.6$ [Å]{}. These values come from the fitting described above applied to the experimental data in the range $\varrho = 0.01$ to $0.05$ mole/liter. The diameters obtained are very close to the ones obtained in Ref. [@sheldlovsky:50] with a similar fitting, but using the DH theory with an approximate way to incorporate the ionic sizes.
The fact that $d_{{\mathrm{HCl}}} > d_{{\mathrm{NaCl}}} > d_{{\mathrm{KCl}}}$, which is the opposite to the sequence of bare diameters, is a consequence of the existence of hydration shells around the ions which tend to be larger for smaller bare ion size[@israelachvili:book]. Notice however that the values obtained here lie between the bare diameters and the hydrated values available in the literature. This is not surprising since the effective hard-core size simultaneously reflects both the presence of the hydration shells and the “softness” of the outer water layer as two oppositely charged ions approach each other.
### Fitting assuming two diameters
If we now assume that both $d_+$ and $d_-$ are unconstrained, we have to use one more term in the expansion of the activity coefficient and solve the coupled equations $$\label{system}
\begin{split}
\nu_1(d_+,d_-) &= \nu^{exp}_1 \\
\nu_{3/2}(d_+,d_-) &= \nu^{exp}_{3/2}
\end{split}$$ where the coefficients $\nu^{exp}_1$ and $\nu^{exp}_{3/2}$ are obtained by fitting the function $$\begin{aligned}
{\Delta}{\mathrm{f}}_{\pm} &= &{\mathrm{f}}^{exp}_{\pm} - \nu_{{\mathrm{DH}}} \varrho^{1/2} -
\nu_{1{\mathrm{log}}} \varrho \ln \varrho \nonumber \\
& =& \nu_1^{exp} \varrho + \nu^{exp}_{3/2} \varrho^{3/2} +
O(\varrho^{3/2} \ln \varrho).\end{aligned}$$ For the actual fitting we divide by the density and obtain $$\label{fitdefine}
\frac{\Delta {\mathrm{f}}_{\pm}}{\varrho} = \nu^{exp}_1 + \nu^{exp}_{3/2} \varrho^{1/2}.$$ Note that we subtracted the term proportional to $\varrho \ln \varrho$, since like the DH term it is independent of the ionic diameters. What is interesting about this fitting procedure is that the two ionic diameters are independent parameters, that is, the effective sizes obtained through this fitting do not depend on the size of a “reference” ion, contrary to what happens when using crystallographic methods[@schwister-al:book]. The method we show here can in principle lead to useful results, as long as the experimental data is accurate enough at very low densities, as we now demonstrate.
Fig. \[fig:map\] shows the mapping between the parameter space of ionic diameters and the two coefficients $\nu_1$ and $\nu_{3/2}$ (for a $1:1$ salt in water at room temperature). The mapping is restricted to the range $3$ [Å]{} $ < d_+ < 27$ [Å]{} and $3$ [Å]{} $ < d_- < 27$ [Å]{}. This allows the “inverse mapping” between the ($\nu_{1}$;$\nu_{3/2}$)–space and the ($d_-$;$d_+$)–space: any system with ionic diameters within the values above should have the coefficients $\nu_1$ and $\nu_{3/2}$ inside the region in the ($\nu_{1}$;$\nu_{3/2}$)–space, as shown in Fig. \[fig:map\]. One problem becomes obvious: the coefficient $\nu_{3/2}$ is larger than the coefficient of the linear term, $\nu_1$, meaning that experimental data at very low concentrations are needed. As an example, Fig. \[fig:mac2\] shows $\Delta {\mathrm{f}}_{\pm} / \varrho$ for NaCl[@conway:book] as a function of $\varrho^{1/2}$, which should be asymptotically linear in the limit $\varrho \rightarrow 0$ and therefore give the coefficients $\nu_1$ and $\nu_{3/2}$ by a simple linear fit according to Eq.(\[fitdefine\]). The lines shown are two possible asymptotic linear fits to the experimental points (line $a$ is given by $\Delta f_{\pm} / \varrho = 2.59-9.94 \varrho^{1/2}$, and line $b$ by $ 2.47-6.00 \varrho^{1/2}$). The inset to Fig. \[fig:mac2\] gives the corresponding positions in the ($\nu_{1}$;$\nu_{3/2}$)–space (cf. Fig.\[fig:map\]) for the two lines. Clearly, only one of the fits (line $b$) leads to reasonable values for the diameters (between $3$ and $9$ [Å]{} as seen from the position of point $b$ in the inset, $d_+=3.8$ [Å]{} and $d_-=5.4$ [Å]{} as obtained by solving Eq. (\[system\])), while the other fit (line $a$) leads to unreasonable values for the diameters (outside the range $3$ to $27$ [Å]{}). In other words, the asymptotic extrapolation of the experimental data is very sensitive to small errors, and in order to obtain the diameters with this method one needs more accurate data at very low densities (which to the best of our knowledge is not available in the literature). Although NaCl was used as example, the situation is identical for other salts.
Conclusions {#chapterD:conclusions}
===========
Using field theoretic methods we obtained the exact low density (“virial”) expansion of the TCPHC up to order $5/2$ in density. In its general form, the model can be applied to both electrolyte solutions and dilute colloidal suspensions (when the van der Waals forces are unimportant); the free energy derived here provides a unified way for handling both systems in the limit of low concentration. As the calculations show, the generalization to short-ranged potentials other than hard core is possible.
The behavior of the coefficients $B_2$ and $B_{5/2}$ suggests that the series is badly convergent, meaning that the inclusion of higher order terms does not necessarily extend the validity of the free energy to larger densities. We saw that the divergent behavior of these coefficients is related to the ionic pairing[@bjerrum:26; @fisher-levin:93; @yeh-zhou-stell:96] which is favored as the coupling increases (this is not present e.g. in the OCPHC[@roland-orland:tcp]). This also means that such a low-density expansion is not very useful to study the phase behavior[@three-component] of ionic systems: in the situation of phase-separation between a very dilute phase and a dense phase, typically the average density in the dense phase already falls outside the range of validity of the low density expansion.
In applying our results to colloidal systems we concluded that at low density the counterion entropic contribution dominates over the electrostatic contribution due to the macroions; the latter contribution can be described, in this limit and up to this order, by an OCPHC corrected by effects due to the exclusion of counterions from the colloidal particles.
Finally, we used the theoretical results for the mean activity coefficient to fit experimental data and extract effective ionic sizes. In the simplest fitting, where we assumed the two ionic diameters to be equal, we obtained sizes that are reasonable and close to what one would expect from the results obtained by other methods. For the more interesting case where the two ionic sizes are taken as free parameters and determined independently, we showed that one would need more experimental data for the mean activity coefficient at very low densities (which, to the best of our knowledge, is not available in the literature) to obtain the correct values for the diameters. With the proper experimental data it would then be a simple matter to obtain the effective thermodynamic ionic sizes, which could serve as useful input for computer simulations of two-component-plasmas with hard-cores to model electrolyte solutions.
We thank N. Brilliantov and H. L[ö]{}wen for useful discussions. AGM acknowledges the support from FCT through grant PRAXIS XXI/BD/13347/97 and DFG.
Appendix {#appendix .unnumbered}
========
Averages needed for $Z_1$ and $Z_2$ {#averages-needed-for-z_1-and-z_2 .unnumbered}
-----------------------------------
The following expressions were used to obtain Eqs. (\[Z1\]–\[Zlast\]). In order to have more compact formulas we use the Greek letters $\alpha$ and $\beta$ instead of $+$ or $-$. For instance, $\alpha \, q_{\alpha}$ means both $+ q_+$ and $- q_-$. $$\label{a_av_1}
{\bigl\langle h_{\alpha}({\mathbf{r}}) \bigr\rangle} = {\mathrm{e}^{ q_{\alpha}^2 v_c(0) / 2 }},$$ $${\bigl\langle {\mathrm{e}^{-\imath \alpha q_\alpha \phi({\mathbf{r}})}} \bigr\rangle} =
{\mathrm{e}^{-q_{\alpha}^2 v_{{\mathrm{DH}}}(0) / 2}},$$ $${\bigl\langle h_{\alpha}({\mathbf{r}}) h_{\beta}({\mathbf{r}}') \bigr\rangle} =
{\mathrm{e}^{- \omega_{\alpha \beta}({\mathbf{r}}-{\mathbf{r}}') +
\bigl[ q_{\alpha}^2 v_c(0) + q_{\beta}^2 v_c(0) \bigr] / 2}} ,$$ $$\begin{split}
{\bigl\langle {\mathrm{e}^{-\imath \alpha \, q_{\alpha} \phi({\mathbf{r}}) - \imath \beta \,
q_{\beta} \phi({\mathbf{r}}')}} \bigr\rangle} &=
{\mathrm{e}^{- \bigl[ q_{\alpha}^2 v_{{\mathrm{DH}}}(0) + q_{\beta}^2 v_{{\mathrm{DH}}}(0) \bigr] / 2}} \\ & \times
{\mathrm{e}^{-\alpha \beta \, q_{\alpha} q_{\beta} v_{{\mathrm{DH}}}({\mathbf{r}}-{\mathbf{r}}')}}
\end{split}$$ $$\label{a_av_last}
{\bigl\langle \phi^2({\mathbf{r}}') {\mathrm{e}^{- \imath \alpha \, q_{\alpha} \phi({\mathbf{r}})}} \bigr\rangle} =
{\mathrm{e}^{-q_{\alpha}^2 v_{{\mathrm{DH}}}(0) / 2}} \Bigl[ v_{{\mathrm{DH}}}(0) -
q_{\alpha}^2 v^2_{{\mathrm{DH}}}({\mathbf{r}}-{\mathbf{r}}') \Bigr].$$
The brackets $\langle \cdots \rangle$ denote averages over the fields $\phi$ and $\psi_\alpha$ where $\omega^{-1}$ and $v_{{\mathrm{DH}}}^{-1}$ are the propagators.
The coefficients in the grand-canonical free energy {#the-coefficients-in-the-grand-canonical-free-energy .unnumbered}
---------------------------------------------------
We give here the explicit expressions for the coefficients $\tilde{g}$, Eq. (\[free-energy3\]). Using the function $H(x)$, defined in Eq.(\[Hdefine\]), the coefficients read $$\label{a_coeff_1}
m_1 = \frac{\pi}{3} \frac{q_+^6 \ell_B^3}{d_+^3}
\biggl\{ - H\Bigl( \frac{q_+^2 \ell_B}{d_+} \Bigr) +
\ln \Bigl(\frac{3 d_+}{\ell_B} \Bigr) \biggr\},$$ $$m_2 = \frac{\pi}{3} \frac{q_-^6 \ell_B^3}{d_+^3}
\biggl\{ - H\Bigl( \frac{q_-^2 \ell_B}{d_-} \Bigr)
+ \ln \Bigl(\frac{3 d_-}{\ell_B} \Bigr) \biggr\},$$ $$m_3 = \frac{2 \pi}{3} \frac{q_+^3 q_-^3 \ell_B^3}{d_+^3}
\biggl\{ H\Bigl( \frac{2 q_+ q_- \ell_B}{d_+ + d_-} \Bigr)
- \ln \Bigl(\frac{3 [d_+ + d_-]}{2 \ell_B} \Bigr) \biggr\},$$ $$n_1 = \frac{\pi}{3} \frac{q_+^8 \ell_B^3}{d_+^3}
\biggl\{ - \frac{5}{8} - 2 H\Bigl( \frac{q_+^2 \ell_B}{d_+} \Bigr)
+ \ln \Bigl(\frac{12 d_+^2}{\ell_B^2} \Bigr) \biggr\},$$ $$n_2 = \frac{\pi}{3} \frac{q_-^8 \ell_B^3}{d_+^3}
\biggl\{ - \frac{5}{8} - 2 H\Bigl( \frac{q_-^2 \ell_B}{d_-} \Bigr)
+ \ln \Bigl(\frac{12 d_-^2}{\ell_B^2} \Bigr) \biggr\},$$ $$\begin{split}
n_3 &= \frac{2 \pi}{3} \frac{q_+^3 q_-^3 \ell_B^3}{d_+^3}
\biggl\{ - \frac{5}{8} q_+ q_- + \frac{[q_+ - q_-]^2}{2}
H\Bigl( - \frac{2 q_+ q_- \ell_B}{d_+ + d_-} \Bigr) \\
&+ q_+ q_- \ln \Bigl(\frac{2 [d_+ + d_-]}{\ell_B} \Bigr) -
\frac{q_+^2 + q_-^2}{2} \ln \Bigl(\frac{3 [d_+ + d_-]}{2 \ell_B} \Bigr) \biggr\},
\end{split}$$ $$p_1 = \frac{\pi}{3} \frac{q_+^6 \ell_B^3}{d_+^3},
\qquad
p_2 = \frac{\pi}{3} \frac{q_-^6 \ell_B^3}{d_+^3},
\qquad
p_3 = - \frac{2 \pi}{3} \frac{q_+^3 q_-^3 \ell_B^3}{d_+^3},$$ $$r_1 = \frac{2 \pi}{3} \frac{q_+^8 \ell_B^3}{d_+^3},
\qquad
r_2 = \frac{2 \pi}{3} \frac{q_-^8 \ell_B^3}{d_+^3},$$ $$r_3 = \frac{2 \pi}{3} \frac{q_+^3 q_-^3 \ell_B^3}{d_+^3}
\Bigl[ q_+ q_- - \frac{q_+^2 + q_-^2}{2} \Bigr],$$ $$\label{a_coeff_last}
s_1 = \frac{q_+^4}{8},
\quad
s_2 = \frac{q_-^4}{8},
\quad
t_0 = \frac{d_+^3}{12 \pi \ell_B^3},
\quad
t_1 = \frac{q_+^6}{48},
\quad
t_2 = \frac{q_-^6}{48}.$$
The coefficients in the fugacity {#the-coefficients-in-the-fugacity .unnumbered}
--------------------------------
In Eq. (\[lambda-\]) we define the fugacity of negative ions ${\widetilde{\lambda}}_-$ as an expansion in terms of ${\widetilde{\lambda}}_+$. The coefficients $a_0$, $a_1$, etc. are determined by the condition of global electroneutrality. They are given by $$\label{a_a0}
a_0=\frac{q_+}{q_-},$$ $$a_1=\frac{q_+^2}{q_-} \bigl[ q_+^2 - q_-^2 \bigr]
\sqrt{\frac{\pi \ell_B^3}{d_+^3} \Bigl[1 + \frac{q_-}{q_+} \Bigr]},$$ $$\begin{gathered}
a_2= \frac{q_+}{q_-^2} \bigl[ -2 m_2 q_+ + 2 m_1 q_- + m_3 [q_1-q_2] \bigr] +
\frac{\pi}{6} \frac{q_+^2 \ell_B^3}{q_- d_+^3} \bigl[q_+ + q_- \bigr]^2 \\ \times
\bigl[ 7 q_+^3 - 9 q_+^2 q_- + 2 q_-^3 \bigr] +
\frac{\pi}{3} \frac{q_+^2 \ell_B^3}{q_- d_+^3}
\bigl[ q_+^5 - q_+^3 q_-^2 + q_+^2 q_-^3
-q_-^5 \bigr] \\ \times
\ln \Bigl( \frac{4 \pi \ell_B^3}{d_+^3}
q_+ \bigl[ q_+ + q_- \bigr] \Bigr),\end{gathered}$$ $$a_3 = \frac{\pi}{3} \frac{q_+^2 \ell_B^3}{q_- d_+^3}
\bigl[ q_+^5 - q_+^3 q_-^2 + q_+^2 q_-^3
-q_-^5 \bigr],$$ $$\begin{gathered}
a_4 = \frac{\ell_B^{3/2}}{24 q_-^3} \sqrt{\pi q_+ \bigl(q_+ + q_- \bigr)} \\ \times
\Bigl\{ 24 n_1 \bigl[ 5 q_+ q_-^2 - q_-^3 \bigr]
+24 n_2 \bigl[ q_+^3 - 5 q_+ q_-^2 \bigr]
\\ +72 n_3 \bigl[ q_+^2 q_- - q_+ q_-^2 \bigr]
+24 m_1 \bigl[ q_+^2 q_-^3 - 3 q_+ q_-^4 \bigr]
+24 m_2 \bigl[ -4 q_+^4 q_- \\ - q_+^3 q_-^2 + 7 q_+^2 q_-^3 \bigr]
+12 m_3 \bigl[ 2 q_+^4 q_- - q_+^3 q_-^2 - 6 q_+^2 q_-^3 +5 q_+ q_-^4 \bigr] \\
+\frac{\pi \ell_B^3}{d_+^3} q_+^2 q_-^2 [q_+ + q_-]^2
\bigl[ 26 q_+^5 - 34 q_+^4 q_- - 31 q_+^3 q_-^2 + 67 q_+^2 q_-^3 \\ -
45 q_+ q_-^4 + 17 q_-^5 \bigr] \Bigr\} +
\frac{\pi^{3/2}}{6} \frac{q_+^{5/2} \ell_B^{9/2}}{q_- d_+^{9/2}}
\bigl[q_+ - q_- \bigr] \bigl[q_+ + q_- \bigr]^{5/2} \\ \times
\bigl[ 10 q_+^4 -11 q_+^3 q_- + 13 q_+^2 q_-^2 -4 q_+ q_-^3 +
3 q_-^4 \bigl] \\ \times \ln \Bigl( \frac{4 \pi \ell_B^3}{d_+^3} q_+ \bigl[q_+ +
q_- \bigr] \Bigr),\end{gathered}$$ $$\begin{gathered}
\label{a_a5}
a_5 = \frac{\pi^{3/2}}{6} \frac{q_+^{5/2} \ell_B^{9/2}}{q_- d_+^{9/2}}
\bigl[q_+ - q_- \bigr] \bigl[q_+ + q_- \bigr]^{5/2}
\bigl[ 10 q_+^4 -11 q_+^3 q_- \\ + 13 q_+^2 q_-^2 -4 q_+ q_-^3 +
3 q_-^4 \bigl] \ln \Bigl( \frac{4 \pi \ell_B^3}{d_+^3} q_+ \bigl[q_+ +
q_- \bigr] \Bigr).\end{gathered}$$ Notice that $m_1$, $m_2$, etc. were defined in Eqs. (\[a\_coeff\_1\]–\[a\_coeff\_last\]).
The coefficients in the mean activity coefficient {#the-coefficients-in-the-mean-activity-coefficient .unnumbered}
-------------------------------------------------
In Eq. \[mac-expd\] we obtained the low-density expansion of the mean activity coefficient of ionic solutions where the ions have valences $q_+$ and $q_-$ and effective diameters $d_+$ and $d_-$ (in [Å]{}ngstr[ö]{}ms). Defining $\omega \equiv 6.022 \times 10^{-4} \ell_B^3 q_+^6 q_-$ (with $\eta \equiv q_-/q_+$ and $\xi \equiv d_-/d_+$), the coefficients in Eq. \[mac-expd\] read $$\label{a_nuI}
\nu_{{\mathrm{DH}}} = - \eta \sqrt{\pi \omega [1+\eta]},$$ $$\begin{gathered}
\label{a_nu1}
\nu_1 = \frac{\pi \omega \eta}{6 [1+\eta]}
\Bigl\{ -1 + 3 \eta + 8 \eta^2 + 3 \eta^3 -\eta^4
+ 4 \Bigl[
H\bigl(\epsilon_+ \bigr) + \ln\bigl( \epsilon_+ \bigr)
\Bigr] \\
+4 \eta^4 \Bigl[
H\bigl(\frac{\eta^2 \epsilon_+}{\xi} \bigr) +
\ln\bigl(\frac{\eta^2 \epsilon_+}{\xi} \bigr)
\Bigr]
-8 \eta^2 \Bigl[
H\bigl(- \frac{2 \eta \epsilon_+}{1+\xi} \bigr) +
\ln\bigl(\frac{2 \eta \epsilon_+}{1+\xi} \bigr)
\Bigr] \\
+8 \eta^2 [1-\eta^2] \ln(\eta)
- \bigl[ 2 - 4 \eta^2 +2 \eta^4 \bigr]
\ln \bigl( 36 \pi \omega [1+\eta] \bigr) \Bigl\},\end{gathered}$$ $$\nu_{1{\mathrm{log}}}= - \frac{\pi \omega \eta}{3} [1-\eta]^2 [1+\eta],$$ $$\begin{gathered}
\label{a_nu32}
\nu_{3/2} = \frac{\eta [\pi \omega]^{3/2}}{24 \sqrt{1+\eta}}
\Bigl\{ 42 - 6\eta -14 \eta^2 +68 \eta^3 -14 \eta^4 -6 \eta^5
+42 \eta^6 \\
+ 40 \bigl[ 1- \frac{2 \eta}{5} \bigr] \Bigl[
H\bigl(\epsilon_+ \bigr) + \ln\bigl( \epsilon_+ \bigr)
\Bigr]
+ 40 \eta^6 \bigl[ 1- \frac{2}{5 \eta} \bigr] \Bigl[
H\bigl(\frac{\eta^2 \epsilon_+}{\xi} \bigr) \\ +
\ln\bigl(\frac{\eta^2 \epsilon_+}{\xi} \bigr)
\Bigr]
+ 112 \eta^3 \Bigl[
H\bigl(- \frac{2 \eta \epsilon_+}{1+\xi} \bigr) +
\ln\bigl(\frac{2 \eta \epsilon_+}{1+\xi} \bigr)
\Bigr] \\
- 16 \eta^3 \bigl[7 -2 \eta^2 + 5 \eta^3 \bigr] \ln(\eta)
+ 8 \eta \bigl[1-\eta^2 \bigr]^2 \ln \bigl(36 \pi \omega [1+\eta] \bigr) \\
-20 \bigl[1+\eta^3 \bigr]^2 \ln \bigl(64 \pi \omega [1+\eta] \bigr)
\Bigl\},\end{gathered}$$ $$\begin{gathered}
\label{a_nuE}
\nu_{3/2{\mathrm{log}}} = - \frac{\eta [\pi \omega]^{3/2}}{6} \sqrt{1+\eta} \\ \times
\bigl[ 5 - 7 \eta + 7 \eta^2 + 7 \eta^3 -7 \eta^4 + 5 \eta^5 \bigr].\end{gathered}$$
[10]{}
J. Israelachvili, [*Intermolecular and surface forces*]{}, 2nd ed. (Academic Press, London, 1991).
B. Ninham and V. Yaminsky, Langmuir [**13**]{}, 2097 (1997).
M. Alfridsson, B. Ninham, and S. Wall, Langmuir [**16**]{}, 10087 (2000).
J. E. Mayer, J. Chem. Phys. [**18**]{}, 1426 (1950).
H. L. Friedman, [*Ionic solution theory*]{} (Interscience, New York, 1962).
D. A. McQuarrie, [*Statistical mechanics*]{} (Harper Collins, New York, 1976).
P. Debye and E. H[ü]{}ckel, Physik. Z. [**24**]{}, 185 (1923).
E. Haga, J. Phys. Soc. Jpn. [**8**]{}, 714 (1953).
S. F. Edwards, Philos. Mag. [**4**]{}, 1171 (1959).
R. R. Netz and H. Orland, Eur. Phys. J. E [**1**]{}, 67 (2000).
This can be checked in our calculations by turning-off the charges (or simply be setting $\ell_B =0$) in the final result.
The definition of volume fraction usually found in the literature—normally represented as $\eta$—relates to our definition through $\eta = \pi {\widetilde{c}}
/ 6$.
M. Abramowitz and I. A. Stegun, [*Handbook of mathematical functions*]{} (Dover, New York, 1965).
N. Bjerrum, kgl. Danske Videnskab. Selskab, Math.–fys. Medd. [**7**]{}, 1 (1926); *Selected papers*, pp. 108–119, Einar Munksgaard, Copenhagen (1949).
M. E. Fisher and Y. Levin, Phys. Rev. Lett. [**71**]{}, 3826 (1993).
S. Yeh, Y. Zhou, and G. Stell, J. Phys. Chem. [**100**]{}, 1415 (1996).
E. Waisman and J. L. Lebowitz, J. Chem. Phys. [**56**]{}, 3086 (1972); J.Chem. Phys. [**56**]{}, 3093 (1972).
R.G. Palmer and J.D. Weeks, J. Chem. Phys. [**58**]{}, 4171 (1973).
L. Blum, Mol. Phys. [**30**]{}, 1529 (1975).
R. H. Ottewill, in [*An introduction to polymer colloids*]{}, edited by F. Candau and R. H. Ottewill (Kluwer academic publishers, Dordrecht, 1990).
C. Bonnet-Gonnet, L. Belloni, and B. Cabane, Langmuir [**10**]{}, 4012 (1994).
D. G. Kurth [*et al.*]{}, Chem. Eur. J. [**6**]{}, 385 (2000).
S. Ishimaru, Rev. Mod. Phys. [**54**]{}, 1017 (1982).
M. Baus and J. Hansen, Phys. Rep. [**59**]{}, 1 (1980).
R. Abe, Prog. Theor. Phys. [**22**]{}, 213 (1959).
J.-P. Hansen, J. Phys. C [**14**]{}, L151 (1981).
S. Alexander [*et al.*]{}, J. Chem. Phys. [**80**]{}, 5776 (1984).
R. A. Robinson and R. H. Stokes, [*Electrolyte solutions*]{} (Butterworths, London, 1959).
J. N. Butler and R. N. Roy, in [*Activity coefficients in electrolyte solutions*]{}, 2nd ed., edited by K. S. Pitzer (CRC Press, Boca Raton, 1991).
J. A. Rard and R. F. Platford, in [*Activity coefficients in electrolyte solutions*]{}, 2nd ed., edited by K. S. Pitzer (CRC Press, Boca Raton, 1991).
B. E. Conway, [*Electrochemical data*]{} (Elsevier, Amsterdam, 1952).
R. Parsons, [*Handbook of electrochemical constants*]{} (Butterworths, London, 1959).
B. A. Pailthorpe, D. J. Mitchell, and B. W. Ninham, J. Chem. Soc., Faraday Trans. 2 [**80**]{}, 115 (1984).
D. F. Evans and H. Wennerstr[ö]{}m, [*The colloidal domain*]{} (VCH, New York, 1994).
T. Sheldlovsky, J. Am. Chem. Soc. [**72**]{}, 3680 (1950).
See for instance K. Schwister, *Taschenbuch der Chemie*, 2nd ed. (Fachbuchverlag Leipzig, Leipzig, 1999).
A. G. Moreira and R. R. Netz, Eur. Phys. J. D [**13**]{}, 61 (2001).
[^1]: Present address: Materials Research Laboratory, University of California, Santa Barbara, CA 93106, USA
[^2]: Present address: Sektion Physik, Ludwig-Maximilians-Universität, Theresienstr. 37, 80333 München, Germany
|
---
abstract: 'Squeezing ensemble of spins provides a way to surpass the standard quantum limit (SQL) in quantum metrology and test the fundamental physics as well, and therefore attracts broad interest. Here we propose an experimentally accessible protocol to squeeze a giant ensemble of spins via the geometric phase control. Using the cavity-assisted Raman transitions in a double $\Lambda$-type system, we realize an effective Dicke model. Under the condition of vanishing effective spin transition frequency, we find a particular evolution time where the cavity decouples from the spins and the spin ensemble is squeezed considerably. Our scheme has the potential to improve the sensitivity in quantum metrology with spins by about two orders.'
author:
- 'Keyu Xia (夏可宇)'
title: 'Squeezing giant spin states via geometric phase control in cavity-assisted Raman transitions'
---
[UTF8]{}
Spins, due to the merit of their long decoherence, have been widely used for ultrasensitive sensing of various signals [@NatPhys.3.227; @NatPhys.10.21; @Science.339.561; @PhysRevX.4.021045; @PhysRevLett.110.160802; @PhysRevLett.112.160802; @PhysRevLett.110.130802; @PhysRevX.5.041001; @NatCommun.6.8251; @PhysRevA.92.043409]. However, the precision of the conventional measurement with spins is bounded by the shot noise or the SQL [@Science.306.1330; @Science.344.1486]. Quantum spin squeezing and entanglement can surpass the SQL and therefore boost the sensitivity in quantum measurements to approach the Heisenberg limit [@Science.306.1330; @PhysRep.509.89].
To exploit the power of the spin-squeezed state (SSS), various methods have been proposed using quantum measurement [@AtomicSpinSqu2; @AtomicSpinSqu1; @AtomicSpinSqu4], quantum bath engineering [@AtomicSpinSqu3], converting entanglement to squeezing [@PhysRevLett.109.173603] and cavity feedback [@CavityFeedback1; @CavityFeedback2], typically for atomic ensembles. The state-of-the-art experiment has achieved $20~\deci\bel$ squeezing of half a million ultracold Rb atoms in a natural trap [@AtomicSpinSqu2]. Recently, Bennett et al. show the potential to squeeze $100$ nitrogen-vacancy (NV) spins in diamond via the Tavis-Cummings interaction with a nanomechanical resonator, mediated by strain [@PhysRevLett.110.156402]. Their scheme inevitably and sensitively suffers to the large thermal excitation of mechanical resonator. Zhang’s and our works show that the NV centers can also couple to a mechanical resonator mediated by a giant magnetic gradient and the geometric phase control can be used to squeeze NV centers. Taking the merit of the geometric phase protocol robust again various noises, the squeezing is immune to thermal excitation [@PhysRevA.92.013825; @KXJT]. However, the giant magnetic gradient causes large Zemman splitting in NV centers and is highly localized in nanometer region. As a result, the available number of spins is limited up to $20$ [@PhysRevA.92.013825; @KXJT]. Cavity-assisted Raman transition (CART) has been proposed and then demonstrated for Dicke model quantum phase transitions [@PhysRevA.75.013804; @PhysRevLett.113.020408; @SuperfluidGas]. Here we aim to provide an experimentally feasible scheme to squeeze millions or even trillions spins using CART.
In this letter, we propose a scheme for squeezing in a transient way a large ensemble of spins in an optical cavity via the geometric phase control, avoiding the complex configuration in squeezing spins via quantum measurement, quantum bath engineering or feedback. We couple the ensemble of ultracold alkali atoms or negatively charged silicon-vacancy (SiV$^-$) color centers in diamond or a superfluid gas formed in Bose-Einstein condensate (BEC) to the cavity. Using CART, we create an effective Dicke model for the spin-photon interaction. In a special arrangement, the effective resonance frequency, $\omega_c$, of the cavity is much larger than the effective transition frequency of the spins. At a particular time, $t=2\pi/\omega_c$, the spin and cavity decouples. At the same time, the ensemble of spins accumulates a geometric phase due to the collective interaction with the cavity and are collectively twisted along one axis of the Bloch sphere of spins. As a result, the cavity squeezes the spins considerably. Because the spins can be optically initialized to their ground state and the thermal excitation of the optical cavity is vanishing small even at room temperature, our scheme has an advantage that the thermal noise can be neglected in squeezing.
![(Color online) Level diagram for showing two CARTs. We consider a cavity QED system in which an ensemble of spins (cold atoms or SiV centers or BEC) with double $\Lambda$ configuration is trapped in a good cavity. In combination with the cavity mode, two classical laser fields, $\Omega_r$ and $\Omega_s$, (red and brown) drives the spins to form Raman transitions between states $|e\rangle$ and $|g\rangle$.[]{data-label="fig:Level"}](fig1.jpg "fig:"){width="0.6\linewidth"}\
We start the discussion of our work by describing the system. Our configuration is a cavity electrodynamics (QED) system in which an ensemble of $N_a$ double $\Lambda$-type systems is trapped. The level diagram of the system is depicted in Fig. \[fig:Level\]. Each $\Lambda$-type system has two optical excited states $|r\rangle$ and $|s\rangle$, and two metastable states $|g\rangle$ and $|e\rangle$. The state $|j\rangle$ has energy $\hbar \omega_j$ ($j=r,s,g,e$). We assume that the excited states, $|r\rangle$ ($|s\rangle$) decay to the two ground states, $|g\rangle$ and $|e\rangle$, with the rates of $\gamma_{rg}$ and $\gamma_{re}$ ($\gamma_{sg}$ and $\gamma_{se}$), respectively. The cavity mode, $\hat{c}$, with resonance frequency $\omega_\text{cav}$ and decay rate $\kappa$, drives the transition $|g\rangle \leftrightarrow |r\rangle$ ($|e\rangle \leftrightarrow |s\rangle$) with strength $g_r$ ($g_s$). The classical laser fields drive atomic transitions $|e\rangle \leftrightarrow |r\rangle$ and $|g\rangle \leftrightarrow |s\rangle$ with Rabi frequency $\Omega_r$ and $\Omega_s$, respectively and detuning $\Delta_r=(\omega_r-\omega_e)-\omega_{lr}$ and $\Delta_s=(\omega_s-\omega_g)-\omega_{ls}$, respectively. $\omega_{lr}$ and $\omega_{ls}$ are the carrier frequencies of the laser fields $\Omega_r$ and $\Omega_s$. The paired interaction, $g_r$ and $\Omega_r$, $g_s$ and $\Omega_s$, forms two CARTs. Each CART drives the transition between two ground states. Combining these two CARTs, we obtain the Dicke Hamiltonian [@PhysRevA.75.013804] which is the key of our geometric phase control.
Before we go to the model, lets first briefly discuss three possible implementations using ultracold alkali atoms, SiV$^-$ centers in diamond or a superfluid gas. All three systems for implementations can be effectively treated as an ensemble of spin-$1/2$ systems in the Dicke model. As an example, we consider an ensemble of ultracold $^{87}\text{Rb}$ atoms for the first implementation [@PhysRevLett.113.020408; @AtomData]. We choose $|r\rangle=|5^2P_{3/2},F^\prime=2,m_{F^\prime}=1\rangle$, $|s\rangle=|5^2P_{3/2},F^\prime=2,m_{F^\prime}=2\rangle$, $|g\rangle=|5^2S_{1/2},F=1,m_{F}=1\rangle$ and $|e\rangle=|5^2S_{1/2},F=2,m_{F}=2\rangle$ in the $D_2$ line of $^{87}\text{Rb}$ atom. According to atomic data [@AtomData], the dipole moments are $d_{rg}=d_{re}=-\sqrt{1/8} d$ for the transitions $|r\rangle\leftrightarrow |g\rangle$ and $|r\rangle\leftrightarrow |e\rangle$, $d_{sg}=\sqrt{1/4} d$ for $|s\rangle\leftrightarrow |g\rangle$, and $d_{se}= \sqrt{1/6} d$ for $|s\rangle\leftrightarrow |e\rangle$, with $d=3.584\times 10^{-29} ~\coulomb\cdot \meter$. In such configuration, the cavity mode can be a linear-polarized field and the cavity-atom interaction is strong due to the large dipole-dipole moments. Other hyperfine levels can be effectively decoupled due to the large detuning which can also be adjusted with a constant magnetic field $B_c$ [@PhysRevLett.113.020408]. The each excited state decays at a rate of $\gamma \sim 2\pi \times 6~\mega\hertz$ [@PhysRevLett.113.020408; @AtomData], yielding $\gamma_{rg}=\gamma_{re}=2\pi \times 3~\mega\hertz$, $\gamma_{rg}=2\pi \times 3.6~\mega\hertz$, and $\gamma_{se}=2\pi \times 2.4~\mega\hertz$ for different branches. Interestingly, we can also squeeze an ensemble of solid-state spins, SiV$^-$ centers in diamond trapped in a cavity [@PhysRevLett.112.160802]. The SiV$^-$ centers in diamond cut with $\{111\}$ surface have shown a double $\Lambda$-type configuration [@PhysRevLett.113.263601; @PhysRevLett.113.263602; @PhysRevLett.112.036405]. To use SiV centers for our scheme, we take $|s\rangle=|^2\text{\bf E}_u,e_-^u,\uparrow\rangle$, $|r\rangle=|^2\text{\bf E}_u,e_-^u,\downarrow\rangle$, $|e\rangle=|^2\text{\bf E}_g,e_+^u,\uparrow\rangle$, $|g\rangle=|^2\text{\bf E}_g,e_+^u,\downarrow\rangle$, respectively [@NJP.17.043011]. The relaxation rate, $\Gamma$, of the spin ground state is negligible ($2.4 ~\milli\second$), but the pure dephasing, $\Gamma_\phi$, is about $2\pi \times 3.5~\mega\hertz$ [@PhysRevLett.113.263601; @PhysRevLett.113.263602]. While, the relaxation of the optical excited states, $|r\rangle$ and $|s\rangle$, is negligible at cryogenic temperature [@NJP.17.043011]. We assume $d_{rg}=d_{re}=d_{sg}=d_{se}$. At $T=1~\kelvin$, we can take $\gamma_{rg}=\gamma_{re}=\gamma_{sg}=\gamma_{se}=2\pi \times 3.7~\mega\hertz$. More remarkably, our protocol can squeeze the momentum of a superfluid gas which can also construct the double $\Lambda$-type configuration [@SuperfluidGas], taking $|r\rangle = |\pm \hbar k, 0\rangle^\prime$, $|s\rangle = |0,\pm \hbar k\rangle^\prime$, $|g\rangle=|0,0\rangle$ and $|e\rangle= |\pm \hbar k, \pm \hbar k\rangle$. The Dicke model driving the effective transition between $|0,0\rangle$, the atomic zero-momentum state, and $|\pm \hbar k, \pm \hbar k\rangle$, the symmetric superposition of momentum states can be created via the CART. The effective energy of the cavity and the spin can be controlled via the optical trapping potential, the photon-spin coupling, the detuning $\Delta_c$ and the atom-induced dispersive shift of the cavity resonance $UB$ [@SuperfluidGas]. The energy of the state $|\pm \hbar k, \pm \hbar k\rangle$ is lifted relative to the state $|0,0\rangle$ by twice the recoil energy that $\omega_q=2\pi\times 28.6~\kilo\hertz$ [@SuperfluidGas]. While the effective energy, $\hbar\omega_c=\hbar \Delta_c - UB$ is typically much larger than $\hbar\omega_q$. In the experiment, the single-atom coupling $\eta>2\pi \times 0.9~\kilo\hertz$ is achieved. In the end of our numerical investigation, we will numerically evaluate the squeezing parameter as a function of the number of spins and then estimate the achievable squeezing degree for a large ensemble by fitting the numerical data.
We now go to derive the Dicke Hamiltonian governing the evolution of system. We transform the system into the interaction picture by introducing the unitary transformation $\hat{U}(t)=\exp(-iH_0 t)$ with $H_0=\sum_j \omega_g |g_j\rangle \langle g_j| + \omega_e |e_j\rangle \langle e_j| + (\omega_{lr}+ \omega_e) |r_j\rangle \langle r_j| + (\omega_{ls}+ \omega_g) |s_j\rangle \langle s_j| + \omega_\text{cav}^\prime \hat{c}^\dag \hat{c}$, as in [@PhysRevA.75.013804]. We set $\omega_{ls}-\omega_{lr}=2(\omega_e- \omega_g)$ that $\omega_\text{cav}^\prime=\omega_{lr}+(\omega_e-\omega_g)= \omega_{ls}-(\omega_e-\omega_g)$. Thus we obtain the Hamiltonian in the interaction picture, $$\begin{split}
H = \delta_\text{cav} & \sum_j (\Delta_r |r_j\rangle \langle r_j| + \Delta_s |s_j\rangle \langle s_j|) \; \\
& + \sum_j \left(g_r e^{-ik r_j}\hat{c}^\dag |g_j\rangle \langle r_j| + g_s e^{-ik r_j}\hat{c}^\dag |e_j\rangle \langle s_j| + H.c. \right) \;\\
& + \sum_j \left(\frac{\Omega_r}{2} e^{ik_{lr} r_j} |r_j\rangle \langle e_j| + \frac{\Omega_s}{2} e^{ik_{ls} r_j} |s_j\rangle \langle g_j| + H.c.\right) \; ,
\end{split}$$ where $k=\omega_\text{cav}/C$, $k_{lr}=\omega_{lr}/C$ and $k_{ls}=\omega_{ls}/C$ with $C$ is the light velocity in vacuum are the wave vector of the cavity mode and the classical laser fields, $r_j$ is the position of the $j$th spin. We assume $k\approx k_{lr} \approx k_{ls}$. Taking $|\Delta_{r,s}| \gg \Omega_{r,s}, g_{r,s}, \gamma$, we adiabatically eliminate the optical excited states $|r_j\rangle$ and $|s_j\rangle$, and neglect the constant energy terms to arrive at the Dicke model Hamiltonian for the collective coupling of the ground states $|g_j\rangle$ and $|e_j\rangle$ [@PhysRevA.75.013804; @SupplementaryInf], $$\label{eq:HDicke}
H_\text{Dicke} = \omega_c \hat{c}^\dag \hat{c} + \omega_q J_z + 2\sqrt{N_a}\lambda (\hat{c}^\dag+ \hat{c})\bar{J}_x \;,$$ where $\omega_c= \delta_\text{cav} -\frac{1}{2}N_a \left(\frac{|g_r|^2}{\Delta_r} + \frac{|g_s|^2}{\Delta_s} \right)$, $\omega_q = \frac{|\Omega_s|^2}{4\Delta_s}-\frac{|\Omega_r|^2}{4\Delta_r}$ caused by the ac Stark shifts. Namely, the two-photon detuning in the CARTs is $\delta_\text{cav}$. In Hamiltonian Eq. (\[eq:HDicke\]), we define the collective operators for the spins, $J_z=\sum_j(|e_j\rangle \langle e_j|-|g_j\rangle \langle g_j|)/2$, $J_+=J_-^\dag=\sum_j |e_j\rangle \langle g_j|$ and $\bar{J}_x=(J_+ + J_-)/2\sqrt{N_a}$. Here we, for our purpose of squeezing spins, choose $\frac{|g_r|^2}{\Delta_r}=\frac{|g_s|^2}{\Delta_s}$ and $\lambda=\frac{\Omega_r^* g_r}{2\Delta_s}=\frac{\Omega_s g_s^*}{2\Delta_s}$ by controlling the detuning and the classical driving. Essentially, these conditions requires $\Delta_r/\Delta_s= |d_{rg}|^2/|d_{se}|^2$ and $\Omega_r/\Omega_s=d_{rg}/d_{se}$ when the dipole moments $d_{rg,se}$, $g_{r,s}$ and $\Omega_{r,s}$ are real numbers. As a results, $\omega_q=0$ is obtained. We will also investigate the case of $\omega_q \neq 0$ for a general discussion of squeezing BEC. We can consider the ensemble of spins as a resonator with annihilation operator $\hat{a}$ under the Holstein-Primakoff (HP) transformation that $J_z=(\hat{a}^\dag \hat{a} - \mathscr{N}/2)$, $J_+=\hat{a}^\dag \sqrt{\mathscr{N}-\hat{a}^\dag \hat{a}}$, $J_-= \sqrt{\mathscr{N}-\hat{a}^\dag \hat{a}}\hat{a}$, and $\bar{J}_x=(\hat{a}^\dag \sqrt{\mathbf{I}-\hat{a}^\dag \hat{a}/N_a} + \sqrt{\mathbf{I}-\hat{a}^\dag \hat{a}/N_a}\hat{a})/2$ [@HPTransf; @DickeQPT1], where $\mathscr{N}=N_a \mathbf{I}$. In the ideal case of $\omega_q=0$, we rewrite the Hamiltonian in the interaction picture of $\omega_c \hat{c}^\dag\hat{c}$ as $$\label{eq:V}
V_x=2\sqrt{N_a}\lambda( e^{i\omega_c t}\hat{c}^\dag + e^{-i\omega_c t}\hat{c}) \bar{J}_x \;.$$
Now we go to the geometric phase control of the evolution of the system. By applying the Magnus’s formula [@MagnusFormula], the dynamics for the system is governed exactly, in the absence of decoherence, by the unitary operator $U_x(t)=e^{iN_a\theta(t) \bar{J}_x^2} e^{2\lambda/\omega_c(\alpha(t)\hat{c}^\dag -\alpha^*(t)\hat{c})\bar{J}_x}$, where $\alpha(t)=1-e^{i\omega_c t}$, and $\theta(t)=\left(\frac{2\lambda}{\omega_c}\right)^2 (\omega_c t -\sin\omega_c t)$. $\theta(t)$ is the accumulated geometric phase only dependent on the global geometric features of operators and is robust against random operation errors [@PhysRevLett.90.160402]. Note that the spin-cavity coupling is modulated quickly by the periodic function $\alpha(t)$. At $t_m=2m\pi/\omega_c$ for an integer $m$, $\alpha(t_m)$ vanishes, $\theta(t_m)=2m\pi \left(\frac{2\lambda}{\omega_c}\right)^2$ and the spins decouple from the cavity. As a result, the evolution operator for the spin ensemble takes an explicit form, $$\label{eq:Utm}
U_x(t_m)=e^{iN_a\theta(t_m) \bar{J}_x^2} \;.$$ Given the initial state $|\Psi(0)\rangle$ for the spin ensemble, the generated state after one period, i.e. at $t_1$ is $|\Psi(t_1)\rangle=U_x(t_1)|\Psi(0)\rangle$. It is noticeable that the squeezing degree of the SSS only depends on the accumulated geometric phase $\theta(t_1)$, which can be adjusted with the classical driving and the detuning.
The power of our protocol in squeezing spins is limited by the discrepancy of $\omega_q$ from zero and the decoherence of system. Although we set $\omega_q=0$ for the analysis of ideal geometric phase control, the protocol actually works efficiently when $\omega_c \gg \omega_q$. In comparison with the protocol using a mechanical resonator to enable the geometric phase control [@PhysRevA.92.013825; @KXJT], the crucially detrimental thermal noise is negligible in our scheme because the thermal excitation of the optical cavity is vanishing small and the spins can be optically polarized in the ground state $|g_j\rangle$. The decay of excited states $|r_j\rangle$ and $|s_j\rangle$ can introduce some coherence to the evolution via CARTs but is suppressed by the large detuning [@RamanModel]. Threfore, the decay of the cavity is the main decoherence source. Another decoherence source is the pure dephasing, $\Gamma_\phi$, of the ground state $|e_j\rangle$. To taking into account the influence of the imperfection in $\omega_q$ and the decoherence, we numerically solve the quantum Langevin equation in the HP picture [@RamanModel; @SupplementaryInf], $$\label{eq:MEq}
\begin{split}
\partial \rho/\partial t = & -i [H_\text{Dicke},\rho] + \mathscr{L}(\sqrt{\Gamma_\phi/2}J_z)\rho + \mathscr{L}_c(\sqrt{\kappa}\hat{c})\rho \;,
\end{split}$$ where $\mathscr{L}_c(\hat{A})\rho =\hat{A}\rho \hat{A}^\dag - \frac{1}{2} \hat{A}^\dag \hat{A} \rho -\frac{1}{2} \rho \hat{A}^\dag \hat{A}$.
In our three implementations, the dark states of spins are rarely excited, thanks to the small inhomogeneous broadening of the excited state. Therefore, we focus on the symmetric states with the total spin $J=N_a/2$. The state of spin ensemble can be fully described by set of the Dicke state $|J,m\rangle$ with $m\in \{-J,-J+1,\cdots, J-1,J\}$ in the spin picture, which is equivalent to the Fock state $|J+m\rangle$ in the Bosonic or HP picture. In the later, the squeezing degree of spin states $\{|g\rangle, |e\rangle\}$ of spin ensemble can be evaluated by the squeezing parameter defined by Wineland et al. as $\xi_R^2=\left(\frac{N_a}{2|\langle \vec{J}\rangle|} \right)^2 \xi_s^2$ [@PhysRep.509.89], where $|\langle \vec{J}\rangle|=\sqrt{\langle J_x\rangle^2 + \langle J_y\rangle^2 + \langle J_z\rangle^2}$ and the squeezing parameter $\xi_s^2=1+2\langle \hat{a}^\dag \hat{a}\rangle -2 \frac{\langle (\hat{a}^\dag \hat{a})^2\rangle}{N_a}-2|\langle \bar{J}_x^2\rangle|$ is given by Kitagawa and Ueda [@PhysRep.509.89]. The squeezing is optimal at $\theta_\text{opt}=6^{-1/6}(N/2)^{-2/3}$ [@KXJT]. Correspondingly, the phase uncertainty in quantum metrology with such SSS can be reduced down to $\delta\phi=\xi_R/\sqrt{N}$, improved by a factor of $\xi_R$.
Next we go to evaluate the squeezing parameter by solving the master equation Eq. (\[eq:MEq\]). The cavity decay and the imperfection in $\omega_q$ dominantly limit the attainable squeezing parameter. We first study the squeezing parameter for $N_a=50$ spins at time $t_1=2\pi/\omega_c$ for different ratios $\kappa/\omega_c$ and $\omega_q/\omega_c$, as shown in Fig. \[fig:xiR\](a). The squeezing is maximal around $\theta_\text{opt}$ in the case of small $\omega_q/\omega_c$. When $\kappa=0$ and $\omega_q=0$, we obtain $\xi_R^2=9.6 ~\deci\bel$. The squeezing parameter reduces slightly for $\kappa\leq 0.01 \omega_c$ ($\omega_q=0$). Even when the cavity decay increases to a relative large number, $\kappa= 0.1 \omega_c$, $\xi_R^2=7.7~\deci\bel$ is still achieved. In contrast, the imperfection in $\omega_q$ has stronger effect on the squeezing. The squeezing parameter for $\kappa=0$ and $\omega_q/\omega_c=0.01$ is very close to that for $\kappa/\omega_c=0.01$ and $\omega_q=0$, while it deteriorates considerably when $\omega_q$ increases to $0.1\omega_c$. In this case, the maximal available squeezing parameter decreases to $\xi_R^2=6.1~\deci\bel$ at a reduced optimal geometric phase of $\theta=0.7\theta_\text{opt}$. In experiments, we can adjust the classical driving and the detuning so that $\omega_q<0.01 \omega_c$ to guarantee the optimal squeezing at $\theta\approx \theta_\text{opt}$. The pure dephasing has the strongest influence on squeezing because it destroys the coherence among spins. A small pure dephasing of $\Gamma_\phi/\omega_c=0.01$ causes the maximal squeezing parameter to decrease from $\sim 10 ~\deci\bel$ at $\theta=\theta_\text{opt}$ to $6.1 ~\deci\bel$ at $\theta=0.6 \theta_\text{opt}$. When $\Gamma_\phi/\omega_c=0.05$, the maximal squeezing degree reduces by $50\%$, to $3 ~\deci\bel$.
![(Color online) (a) Squeezing parameter $\xi_R^2$ for $N=50$ spins as a function of the geometric phase, $\theta$, at different cavity decay rate, $\kappa$ (lines without markers, $\omega_q=\Gamma_\phi=0$), spin transition frequency, $\omega_q$ (grouped lines with $*$ markers, $\kappa=\Gamma_\phi=0$), the pure dephasing, $\Gamma_\phi$ (grouped lines with o markers, $\omega_q=\kappa=0$); (b) Squeezing parameter $\xi_R^2$ as a function of the number of spins at different $\kappa$. $\Gamma_\phi=0, \omega_q=0$ in (b).[]{data-label="fig:xiR"}](fig2.jpg "fig:"){width="1\linewidth"}\
It is always desired to provide a prediction for the attainable squeezing parameter for a large ensemble. To provide such prediction, we calculate the squeezing parameter as the number of spins, see Fig. \[fig:xiR\] (b). Considering $\omega_q/\omega_c\ll 1$ available in most cases, we set $\omega_q=0$ for simplicity. The squeezing parameter is well fitted by $\xi_R^2=1.4 N^{-2/3}$ when $\kappa/\omega_c\leq 0.01$. It decreases to $1.4N^{-0.56}$ with increasing the cavity decay to $\kappa/\omega_c= 0.1$. Typically, $\kappa/\omega_c\leq 0.01$ is achievable using current available experimental technology for $N_a\sim 10^6$ ultracold atoms. It means that our geometric phase control protocol can achieve a phase uncertainty $\delta\phi\propto N^{-5/6}$, approaching the Heisenberg limit of $\delta\phi\propto N^{-1}$. In above investigation, we neglect the small decoherence terms of spins. Next, we investigate the available squeezing degree for up to $100$ spins by solving the master equation with the spin decoherence and using experimental available numbers for parameters. In doing so, we can provide a rough estimation of the achievable squeezing parameter for $10^6$ spins by fitting the numerical data. We first find the geometric phase $\theta_\text{max}$ to achieve the maximal squeezing degree for $N_a=50$ spins. It is found that $\theta_\text{max}=\theta_\text{opt}$ for cold Rb atoms, $\theta_\text{max}=0.8\theta_\text{opt}$ for BEC and $\theta_\text{max}=0.5\theta_\text{opt}$ for SiV$^-$ centers. Then we calculate the squeezing parameter as $N_a$ varying but with $\theta=\theta_\text{max}$ fixed. After simulation, we will discuss the realistic parameters for the predicted squeezing degree for each implementation. In all of three implementations, we set $\Omega_r^2/\Delta_r^2=\Omega_s^2/\Delta_s^2<0.001$ for simplicity, which are achievable as the discussion of experimental accessible parameters below. The decoherence of spins for each sample uses experimental data.
![(Color online) Squeezing parameter at a particular geometric phase $\theta_\text{max}$ as a function of $N_a$ using experimental available parameters for three implementations using Rb atoms (blue line), BEC (yellow line), and SiV (Fuchsia). $\omega_c=2\pi\times 5.88 ~\mega\hertz$, $\kappa=2\pi\times 70 ~\kilo\hertz$, $\omega_q=0$, $\theta_\text{max}=\theta_\text{opt}$ for Rb atoms, $\omega_c=2\pi\times 500 ~\kilo\hertz$, $\kappa=2\pi\times 70 ~\kilo\hertz$, $\omega_q=2\pi\times 28.6~\kilo\hertz$, $\theta_\text{max}=0.8\theta_\text{opt}$ for BEC and $\omega_c=2\pi\times 350~\mega\hertz$, $\kappa=2\pi\times 1 ~\mega\hertz$, $\omega_q=0$, $\theta_\text{max}=0.5\theta_\text{opt}$ for SiV centers. The lines are fitted (black dashed lines) with $\xi_R^2= 1.4 N^{-0.64}$ for Rb atoms, $\xi_R^2= 1.4 N^{-0.46}$ for BEC and $\xi_R^2= 0.36 N^{-0.1}$ for SiV$^-$ centers.[]{data-label="fig:ExpParam"}](fig3.jpg "fig:"){width="0.8\linewidth"}\
It can be seen from see Fig. \[fig:ExpParam\] that the largest squeezing of $\xi_R^2= 1.4 N^{-0.64}$ can be expected using an ensemble of cold alkali atoms like Rb atoms, because the total decoherence of ground states of the alkali atoms is small and the effective transition frequency $\omega_q$ can be vanishing small. Due to the large pure dephasing of SiV centers, we can only achieve squeezing of $0.36 N^{-0.1}$. According to [@SuperfluidGas], the decoherence of BEC is negligible but $\omega_q=2\omega_r$ is nonzero. Taking $\omega_q=2\pi\times 28.6~\kilo\hertz$ [@SuperfluidGas], we obtain the squeezing parameter of $\xi_R^2=1.4N^{-0.46}$.
Our spin-squeezing protocol via geometric phase control can be realized in various systems. For example, we can squeeze $N_a=10^6$ cold Rb atoms. Using the experimentally available parameters [@AtomicSpinSqu2; @PhysRevLett.113.020408], we choose $\omega_c=2\pi\times 5.88 ~\mega\hertz$, $\kappa=2\pi\times 70 ~\kilo\hertz$, $\omega_q=0$, $g_r=-\sqrt{3/4}g_s=2\pi \times 1.1 ~\mega\hertz$, $\Delta_s=\frac{4}{3}\Delta_r=2\pi\times 5~\giga\hertz$, $\Omega_s=\frac{\Delta_s}{50}$ and $\Omega_r=-\sqrt{\frac{3}{4}\Omega_s}$ yielding $\lambda/2\pi=-12.7~\kilo\hertz$, and $\frac{|g_s|}{\Delta_s}, \frac{|g_r|}{\Delta_r}<3\times 10^{-4}$, $\frac{\Omega_r}{2\Delta_r}\sim -1.1\times 10^{-2}, \frac{\Omega_s}{2\Delta_s}\sim -8.7\times 10^{-3}$. According to the prediction in Fig. \[fig:ExpParam\], the ensemble of $10^6$ Rb atoms can be squeezed by $\xi_R^2 \approx 37 ~\deci\bel$, and the phase uncertainty in measurement with squeezed spins is $\delta\phi \sim 1/N^{-0.82}$, very close to the Heisenberg limit. If we trap billion [@BillionAtoms1] or trillions [@PhysRevLett.101.073601] cold atoms in the cavity, we are potentially able to obtain a squeeze degree of $\xi_R^2=56~\deci\bel$ or even $\xi_R^2=75~\deci\bel$, respectively. The superfluid gas has the smallest decoherence but $\omega_q=2\pi\times 28.6~\kilo\hertz$ [@SuperfluidGas]. We take, $\kappa=2\pi \times 70~\kilo\hertz$, $\Delta_c/2\pi=-4~\mega\hertz$, $UB/2\pi= -3.5~\mega\hertz$ yielding $\omega_c/2\pi= 500~\kilo\hertz$, and assume $\lambda=2\pi\times 0.88~\kilo\hertz$. Correspondingly, the superfluid gas including $10^6$ ultracold atoms can be squeezed by $\xi_R^2 \approx 26 ~\deci\bel$. It is worth noting that this is the first proposal for quantum squeezing momentum of BEC. Our protocol can only squeeze one-million SiV$^-$ centers by $10.4 ~\deci\bel$ because SiV$^-$ centers has a pure dephasing of $\Gamma_\phi/2\pi =3.5~\mega\hertz$ [@PhysRevLett.113.263601; @PhysRevLett.113.263602]. To achieve it, we take $\kappa=2\pi \times 1~\mega\hertz$,$\omega_c=2\pi \times 350~\mega\hertz$, $\Delta=\Delta_r=\Delta_s=2\pi\times 10~\giga\hertz$, $\Omega_r=\Omega_s=\Delta/30$, and a large single-atom coupling $g_r=g_s=2\pi\times 46~\mega\hertz$, leading to $\frac{|g_s|}{\Delta_s}= \frac{|g_r|}{\Delta_r}=4.6\times 10^{-3}$, $\frac{\Omega_r}{2\Delta_r}=\frac{\Omega_s}{2\Delta_s}=0.017$. Such coupling strength requires a mode volume of cavity $V_c> 3000 ~\micro\meter^3$ if the dipole moment $d>10^{-29}~\coulomb \cdot \meter^3$.
Using the CARTs in spins, we have proposed a geometric phase control scheme to squeeze ensemble of spin. The available squeezing with increasing the number of spins has been numerically studied and can be tens of dB. The protocol is free of the detrimental thermal noise which heavily destroys the squeezing in mechanical resonator-based schemes. Our scheme paves a way to prepare the quantum state of a large ensemble of spins for achieving ultrasensitive quantum sensing.
The work is partly supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQuS), Project No. CE110001013.
[10]{}
D. Budker and M. Romalis. Optical magnetometry. , 3:227–234, 2007.
Fazhan Shi, Xi Kong, Pengfei Wang, Fei Kong, Nan Zhao, Ren-Bao Liu, and Jiangfeng Du. Sensing and atomic-scale structure analysis of single nuclear-spin clusters in diamond. , 10:21–25, 2014.
T. Staudacher, F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. Meriles, F. Reinhard, and J. Wrachtrup. Nuclear magnetic resonance spectroscopy on a $($5$-\text{Nanometer})^3$ sample volume. , 339:561–563, 2013.
R. J. Sewell, M. Napolitano, N. Behbood, G. Colangelo, F. Martin Ciurana, and M. W. Mitchell. Ultrasensitive atomic spin measurements with a nonlinear interferometer. , 4:021045, 2014.
D. Sheng, S. Li, N. Dural, and M. V. Romalis. Subfemtotesla scalar atomic magnetometry using multipass cells. , 110:160802, 2013.
K. Jensen, N. Leefer, A. Jarmola, Y. Dumeige, V. M. Acosta, P. Kehayias, B. Patton, and D. Budker. Cavity-enhanced room-temperature magnetometry using absorption by nitrogen-vacancy centers in diamond. , 112:160802, 2014.
Kejie Fang, Victor M. Acosta, Charles Santori, Zhihong Huang, Kohei M. Itoh, Hideyuki Watanabe, Shinichi Shikata, and Raymond G. Beausoleil. High-sensitivity magnetometry based on quantum beats in diamond nitrogen-vacancy centers. , 110:130802, 2013.
Thomas Wolf, Philipp Neumann, Kazuo Nakamura, Hitoshi Sumiya, Takeshi Ohshima, Junichi Isoya, and Jörg Wrachtrup. Subpicotesla diamond magnetometry. , 5:041001, 2015.
Liang Jin, Matthias Pfender, Nabeel Aslam, Philipp Neumann, Sen Yang, Jörg Wrachtrup, and Ren-Bao Liu. Proposal for a room-temperature diamond maser. , 6:8251, 2015.
Keyu Xia, Nan Zhao, and Jason Twamley. Detection of a weak magnetic field via cavity-enhanced faraday rotation. , 92:043409, 2015.
Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum-enhanced measurements: Beating the standard quantum limit. , 306:1330, 2004.
Sydney Schreppler, Nicolas Spethmann, Nathan Brahms, Thierry Botter, Maryrose Barrios, and Dan M. Stamper-Kurn. Optically measuring force near the standard quantum limit. , 344:1486–1489, 2014.
Jian Ma, Xiaoguang Wang, C. P. Sun, and Franco Nori. Quantum spin squeezing. , 509:89–165, 2011.
Onur Hosten, Nils J. Engelsen, Rajiv Krishnakumar, and Mark A. Kasevich. Measurement noise $100$ times lower than the quantum-projection limit using entangled atoms. , 529:505, 2016.
C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Bookjans, and M. S. Chapman. Spin-nematic squeezed vacuum in a quantum gas. , 8:305, 2012.
T. Fernholz, H. Krauter, K. Jensen, J. F. Sherson, A. S. Sørensen, and E. S. Polzik. Spin squeezing of atomic ensembles via nuclear-electronic spin entanglement. , 101:073601, 2008.
Emanuele G. Dalla Torre, Johannes Otterbach, Eugene Demler, Vladan Vuletic, and Mikhail D. Lukin. Dissipative preparation of spin squeezed atomic ensembles in a steady state. , 110:120402, 2013.
Leigh M. Norris, Collin M. Trail, Poul S. Jessen, and Ivan H. Deutsch. Enhanced squeezing of a collective spin via control of its qudit subsystems. , 109:173603, 2012.
Ian D. Leroux, Monika H. Schleier-Smith, and Vladan Vuleti ć. Orientation-dependent entanglement lifetime in a squeezed atomic clock. , 104:250801, 2010.
Ian D. Leroux, Monika H. Schleier-Smith, and Vladan Vuleti ć. Implementation of cavity squeezing of a collective atomic spin. , 104:073602, 2010.
S. D. Bennett, N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, and M. D. Lukin. Phonon-induced spin-spin interactions in diamond nanostructures: Application to spin squeezing. , 110:156402, 2013.
Yan-Lei Zhang, Chang-Ling Zou, Xu-Bo Zou, Liang Jiang, and Guang-Can Guo. Phonon-induced spin squeezing based on geometric phase. , 92:013825, 2015.
Keyu Xia and Jason Twamley. . 2016.
F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael. Proposed realization of the dicke-model quantum phase transition in an optical cavity qed system. , 75:013804, 2007.
Markus P. Baden, Kyle J. Arnold, Arne L. Grimsmo, Scott Parkins, and Murray D. Barrett. Realization of the dicke model using cavity-assisted raman transitions. , 113:020408, 2014.
Kristain Baumann, Christine Guerlin, Ferdinand Brennecke, and Tilman Esslinger. Dicke quantum phase transition with a superfluid gas in an optical cavity. , 464:1301–1306, 2010.
D. Steck. http://steck.us/alkalidata. revision 2.1.5, January 13, 2015.
Benjamin Pingault, Jonas N. Becker, Carsten H. H. Schulte, Carsten Arend, Christian Hepp, Tillmann Godde, Alexander I. Tartakovskii, Matthew Markham, Christoph Becher, and Mete Atatüre. All-optical formation of coherent dark states of silicon-vacancy spins in diamond. , 113:263601, 2014.
Lachlan J. Rogers, Kay D. Jahnke, Mathias H. Metsch, Alp Sipahigil, Jan M. Binder, Tokuyuki Teraji, Hitoshi Sumiya, Junichi Isoya, Mikhail D. Lukin, Philip Hemmer, and Fedor Jelezko. All-optical initialization, readout, and coherent preparation of single silicon-vacancy spins in diamond. , 113:263602, 2014.
Christian Hepp, Tina Müller, Victor Waselowski, Jonas N. Becker, Benjamin Pingault, Hadwig Sternschulte, Doris Steinmüller-Nethl, Adam Gali, Jeronimo R. Maze, Mete Atatüre, and Christoph Becher. Electronic structure of the silicon vacancy color center in diamond. , 112:036405, 2014.
Kay D Jahnke, Alp Sipahigil, Jan M Binder, Marcus W Doherty, Mathias Metsch, Lachlan J. Rogers, Neil B. Manson, Mikhail D. Lukin, and Fedor Jelezko. Electron-phonon processes of the silicon-vacancy centre in diamond. , 17:043011, 2015.
See the supplementary information.
T. Holstein and H. Primakoff. Field dependence of the intrinsic domain magnetization of a ferromagnet. , 58:1098–1113, 1940.
D. Nagy, G. Kónya, G. Szirmai, and P. Domokos. Dicke-model phase transition in the quantum motion of a bose-einstein condensate in an optical cavity. , 104:130401, 2010.
W. Magnus. On the exponential solution of differential equations for a linear operator. , 7:649, 1954.
A. Carollo, I. Fuentes-Guridi, M. Fran ça Santos, and V. Vedral. Geometric phase in open systems. , 90:160402, 2003.
Florentin Reiter and Anders S. Sørensen. Effective operator formalism for open quantum systems. , 85:032111, 2012.
A. Ridinger, S. Chaudhuri, T. Salez, U. Eismann, D. R. Fernandes, K. Magalhes, D. Wilkowski, C. Salomon, and F. Chevy. Large atom number dual-species magneto-optical trap for fermionic $^6$li and $^40$k atoms. , 65:223, 2011.
T. Fernholz, H. Krauter, K. Jensen, J. F. Sherson, A. S. Sørensen, and E. S. Polzik. Spin squeezing of atomic ensembles via nuclear-electronic spin entanglement. , 101:073601, 2008.
|
---
abstract: 'Distribution grids refer to the part of the power grid that delivers electricity from substations to the loads. Structurally a distribution grid is operated in one of several radial/tree-like topologies that are derived from an original loopy grid graph by opening switches on some lines. Due to limited presence of real-time switch monitoring devices, the operating structure needs to be estimated indirectly. This paper presents a new learning algorithm that uses only nodal voltage measurements to determine the operational radial structure. The algorithm is based on the key result stating that the correct operating structure is the optimal solution of the minimum-weight spanning tree problem over the original loopy graph where weights on all permissible edges/lines (open or closed) is the variance of nodal voltage difference at the edge ends. Compared to existing work, this spanning tree based approach has significantly lower complexity as it does not require information on line parameters. Further, a modified learning algorithm is developed for cases when the input voltage measurements are limited to only a subset of the total grid nodes. Performance of the algorithms (with and without missing data) is demonstrated by experiments on test cases.'
author:
-
bibliography:
- '../../Bib/FIDVR.bib'
- '../../Bib/SmartGrid.bib'
- '../../Bib/voltage.bib'
- '../../Bib/trees.bib'
title: Learning Topology of the Power Distribution Grid with and without Missing Data
---
Power Distribution Networks, Power Flows, Spanning Tree, Graphical Models, Load estimation, Voltage measurements, Missing data, Computational Complexity
Introduction {#sec:intro}
============
Distribution grids constitute the low voltage segment of the power system delivering electricity from substations to end-users. Both structurally and operationally the distribution grids are distinct from the transmission (high voltage) portion of the power system. A typical distribution grid is operated as a collection of disjoint tree graphs, each growing from substations at the root to customers. However, the complete layout of the distribution system is loopy to allow multiple alternatives for the trees to energize operationally. Switching from one layout to another, implemented through switch on/off devices placed on many segments of the distribution grid [@distgridpart1], can take place rather often, in some cases few times an hour. (See Fig. \[fig:city\] for the illustration.) More frequent reconfiguration of the distribution is also promoted by recent in-mass integration of smart meters, PMUs [@phadke1993synchronized] and smart devices, such as deferrable loads and energy storage devices. Mixed operational responsibilities in monitoring and operations, as well as the growing role of the new smart devices and controls, make fast and reliable estimation of the operational configuration of the distribution grid an important practical task, complicated by the lack of real-time, line-based measurements. In such a scenario, to estimate the distribution grid operational topology one ought to rely only on nodal measurements of voltage and end-user consumption. Notice, that brute force (combinatorial) check of topologies for the nodal measurement consistency is prohibitively expensive with the complexity growing exponentially with the number of loops in the grid layout.
In this work we focus on beating the naive exponential complexity of the operational topology learning task by exploring power flow specific correlations between available nodal measurements. In particular, *we develop a spanning tree algorithm that reconstructs the radial operational topology from the original loopy layout by using functions of nodal voltage magnitudes as edge weights.* Computational complexity of this algorithm is order $O(n\log n)$ in the size of the loopy graph’s edge set. Moreover, the algorithm is generalized to the case when some nodes are hidden.
Prior Work
----------
Several approaches in the past have been made to learn the topology of power grids under different operating conditions and available measurements. [@he2011dependency] uses a Markov random field model for bus phase angles to build a dependency graph to identify faults in the grids. [@bolognani2013identification] presents a topology identification algorithm for distribution grids that uses the signs of elements in the inverse covariance matrix of voltage measurements. [@berkeley] compares available time-series observations from smart meters with a database of permissible signatures to identify topology changes. This is similar to envelope comparison schemes used in parameter estimation [@sandia1; @sandia2]. For available line flow measurements, topology estimation using maximum likelihood tests was analyzed in [@ramstanford]. In our own prior work [@distgridpart1; @distgridpart2], we analyzed an iterative greedy structure learning algorithm using trends in second order moments of voltages. [@distgridpart2] also presented the first attempt at topology learning from incomplete voltage data where nodes with missing voltages are separated by greater than two hops. The aforementioned approaches are specific to power grid graphs and typically not linked to research in probabilistic Graphical Models (GM) [@wainwright2008graphical] used to study statistics of images, languages, social networks, and communication schemes. Learning generic (loopy) structures from pair-wise correlations in a GM is a difficult task, normally based on the maximal likelihood [@wainwright2008graphical] with regularization for sparsity [@ravikumar2010high] and greedy schemes utilizing conditional mutual information [@anandkumar2011high; @netrapalli2010greedy]. However, the GM-based learning simplifies dramatically when used, following the famous Chow-Liu approach [@chow1968approximating], to reconstruct the spanning tree maximizing edge-factorized mutual information. [@choi2011learning] generalizes this technique to learn tree structured GMs with latent variables (missing data) using information distances as edge weights.
Contribution of This Work
-------------------------
Following [@distgridpart1; @distgridpart2], we consider linear lossless AC power flow models (also called, following [@89BWa; @89BWb] Lin-Dist-Flow) and assume that fluctuations of consumption at the nodes are uncorrelated. In this setting, our main result states that reconstruction of the operating grid topology is equivalent to solving the minimum weight spanning tree problem defined over the loopy graph of the grid layout where edge weights are given by *variances in voltage magnitude differences across the edges*. We use this result to formulate the operating topology as a spanning tree reconstruction problem that needs only empirical voltage magnitude measurements as input. As spanning trees can be efficiently reconstructed, our learning algorithm has much lower average and worst-case computational complexity compared to existing techniques [@bolognani2013identification; @distgridpart2]. While our algorithm does not require knowing line impedances, these can be used to estimate additionally statics of power consumption. Further, we extend the topology learning algorithm to the case with missing voltage data. The extension works provided nodes with missing data are separated by at least two hops from each other and covariances of nodal power consumption are available. Compared to our prior work [@distgridpart2] on learning with missing data, the spanning tree approach has lower complexity. It also allows extension to cases with lesser restrictions on missing data. Our algorithm shows some commonality with the GM based spanning-tree learning of [@choi2011learning]. However the key difference is that our approach relies principally on the Kirchoff’s laws of physical network flows contrary to the measure of conditional independence utilized in [@chow1968approximating; @choi2011learning]. Thus, voltage magnitude based edge weights used in our work are not restricted to satisfy graph additivity unlike information distances in GM. Further, in the case with missing data, we use power flow relations between nodal voltages and injections that, to the best of our knowledge, do not have an analog in GM learning literature. We highlight the performance of our algorithm through experiments on test distribution grids for both cases, with or without missing data.
The rest of the manuscript is organized as follows. Section \[sec:structure\] introduces notations, nomenclature and power flow relations in the distribution grids. Section \[sec:trends\] describes important features of the nodal voltage magnitudes. This Section also contains the proof of our main – spanning tree learning/reconstruction – theorem. Algorithm reconstructing operational spanning tree in the case of complete visibility (voltage magnitudes are observed at all nodes) is discussed in Section \[sec:algo1\]. Modification of the algorithm which allows for some missing data (at the nodes separated by at least two hopes) is described in Section \[sec:missing\]. This Section also contains a brief discussion of some other extensions/applications of our approach. Simulation results of our learning algorithm on a test radial network are presented in Section \[sec:experiments\]. Finally, Section \[sec:conclusions\] contains conclusions and discussion of future work.
Distribution Grid: Structure and Power Flows {#sec:structure}
============================================
**Radial Structure**: The original distribution grid is denoted by the graph ${\cal G}=({\cal V},{\cal E})$, where ${\cal V}$ is the set of buses/nodes of the graph and ${\cal E}$ is the set of all undirected lines/edges (open or operational). We denote nodes by alphabets ($a$, $b$,...) and the edge connecting nodes $a$ and $b$ by $(ab)$. The operational grid has a ‘radial’ structure as shown in Fig. \[fig:city\]. In general, the operational grid is a collection of $K$ disjoint trees, $\cup_{i=1,\cdots,K}{\cal T}_i$ where each tree’s root node has degree one (connected by one edge) and represents a substation.
In this paper, we will mainly focus on grids where the operational structure consists of only one tree $\cal T$ with nodes ${\cal V}_{\cal T}$ and operational edge set ${\cal E}_{\cal T} \subset {\cal E}$. Generalization to the case with multiple disjoint trees will be discussed along side major results.
**Power Flow (PF) Models**: Let $z_{ab}=r_{ab}+i x_{ab}$ denote the complex impedances of a line $(ab)$ ($i^2=-1$). Here $r_{ab}$ and $x_{ab}$ are line resistance and reactance respectively. Kirchhoff’s laws express the complex valued power injection at a node $a$ in tree ${\cal T}$ as $$\begin{aligned}
P_a =p_a+i q_a= \underset{b:(ab)\in{\cal E}_{\cal T}}{\sum}\frac{v_a^2-v_a v_b\exp(i\theta_a-i\theta_b)}{z_{ab}^*}\label{P-complex1}\end{aligned}$$ where the real valued scalars, $v_a$, $\theta_a$, $p_a$ and $q_a$ denote the voltage magnitude, voltage phase, active and reactive power injection respectively at node $a$. $V_a (= v_a\exp(i\theta_a))$ and $P_a$ denote the nodal complex voltage and injection respectively. One node (substation/root node in our case) is considered as reference and the voltage magnitude and phase at every non-substation node are measured relative to the reference values. As the complex power injection at the reference bus is given by negation of the sum of injections at other buses, without a loss of generality the analysis can be limited to a reduced system, where one ignores reference substation bus voltages and power injections. Under realistic assumption that losses of both active and reactive power in lines of a distribution system are small, Eq. (\[P-complex1\]) can be linearized as follows.
**Linear Coupled (LC) model** [@distgridpart1; @distgridpart2]: In this model, phase difference between neighboring nodes and magnitude deviations ($v_a -1=\varepsilon_a$) from the reference voltage are assumed to be small. The PF Eqs. (\[P-complex1\]) are linearized jointly over both voltage magnitude and phase to give: $$\begin{aligned}
\varepsilon = H^{-1}_{1/r}p + H^{-1}_{1/x}q~~ \theta = H^{-1}_{1/x}p - H^{-1}_{1/r}q \label{PF_LPV_p}\end{aligned}$$ Here, $p,q,\varepsilon$ and $\theta$ are the vectors of real power, reactive power, voltage magnitude deviation and phase angle respectively at the non-substation nodes of the reduced system. $H_{1/r}$ and $H_{1/x}$ denote the reduced weighted Laplacian matrices for $\cal T$ where reciprocal of resistances and reactances are used respectively as edge weights. The reduced Laplacian matrices are of full rank and constructed by removing the row and column corresponding to the reference bus from the true Laplacian matrix. [@distgridpart1] shows that the LC-PF model is equivalent to the LinDistFlow model [@89BWa; @89BWb; @89BWc], if deviations in voltage magnitude are assumed to be small and thus ignored. (Notice, that if line resistances are equated to zero, the LC-PF model reduces to the DC PF model [@abur2004power] used for transmission grids.) We can express means $(\mu_{\theta}, \mu_{\varepsilon})$ and covariance matrices $(\Omega_{\varepsilon}, \Omega_{\theta}, \Omega_{\theta\varepsilon})$ of voltage magnitude deviations and phase angles in terms of corresponding statistics of power injections using Eq. (\[PF\_LPV\_p\]) as shown below. Other quantities can be similarly determined. $$\begin{aligned}
\mu_{\theta} &= H^{-1}_{1/x}\mu_p - H^{-1}_{1/r}\mu_q,~~\mu_\varepsilon = H^{-1}_{1/r}\mu_p + H^{-1}_{1/x}\mu_q\label{means}\\
\Omega_{\varepsilon} &= H^{-1}_{1/r}\Omega_{p}H^{-1}_{1/r} + H^{-1}_{1/x}\Omega_qH^{-1}_{1/x}+H^{-1}_{1/r}\Omega_{pq}H^{-1}_{1/x}\nonumber\\
&~+H^{-1}_{1/x}\Omega_{qp}H^{-1}_{1/r}\label{volcovar1}\end{aligned}$$
In the next Section, we derive key results for functions of nodal voltages in a radial distribution grid that will subsequently be used in the topology learning algorithm.
Properties of Voltage Magnitudes in Radial Grids {#sec:trends}
================================================
Consider grid tree $\cal T$ with operational edge set ${\cal E}_{\cal T}$. Let ${\cal P}^{a}_{\cal T}$ denote the set of edges in the unique path from node $a$ to the root node (reference bus) in tree ${\cal T}$. A node $b$ is termed as a descendant of node $a$ if ${\cal P}^{b}_{\cal T}$ includes some edge $(ac)$ connected to node $a$. We use $D^{a}_{\cal T}$ to denote the set of descendants of $a$. By definition, $a \in D^{a}_{\cal T}$. If $b$ is an immediate descendant of $a$ ($(ab) \in {\cal E}_{\cal T}$), we term $a$ as parent and $b$ as its child. These definitions are illustrated in Fig \[fig:picHinv\].
![The Figure shows distribution grid tree with substation/root node colored in red. Here, nodes $a$ and $c$ are descendants of node $a$. Dotted lines represent the paths from nodes $a$ and $d$ to the root node. The paths’ common edges give $H_{1/r}^{-1}(a,d) = r_{be}+ r_{e0}$. \[fig:picHinv\]](picHinvnew.pdf){width="24.00000%" height=".23\textwidth"}
Due to the radial topology of $\cal T$, the inverse of the reduced weighted graph Laplacian matrix $H_{1/r}$ has the following structure (see Section $4$ in [@distgridpart1] for details). $$\begin{aligned}
H_{1/r}^{-1}(a,b)&= \sum_{(cd) \in {\cal P}^a_{\cal T}\bigcap {\cal P}^b_{\cal T}} r_{cd} \label{Hrxinv}\end{aligned}$$ Thus, the $(a,b)^{th}$ entry in $H^{-1}_{1/r}$ is given by the sum of line resistances of edges that are included in the path to the root from either node as shown in Fig. \[fig:picHinv\]. For nodes $a$ and its parent $b$ in tree ${\cal T}$ (see Fig. \[fig:picHinv\]), it follows from Eq. (\[Hrxinv\]) that $$\begin{aligned}
{\huge H}_{1/r}^{-1}(a,c)-{\huge H}_{1/r}^{-1}(b,c) &&=\begin{cases}r_{ab} & \quad\text{if node $c \in D^a_{\cal T}$}\\
0 & \quad\text{otherwise,} \end{cases} \label{Hdiff}\end{aligned}$$ We use Eqs. (\[Hrxinv\]) and (\[Hdiff\]) to prove our results on voltage magnitude relations. The results hold under the following assumptions.
**Assumption $1$:** Power Injection at different nodes are not correlated, while active and reactive injections at the same node are positively correlated. Mathematically, $\forall a,b$ non-substation nodes $$\begin{aligned}
\Omega_{qp}(a,a) > 0,~\Omega_p(a,b) = \Omega_q(a,b)= \Omega_{qp}(a,b) = 0 \nonumber\end{aligned}$$ Note that this is a valid assumption for many distribution grids due to independence between different nodal load fluctuations and alignment/correlations between same node’s active and reactive power usage.
Under Assumption $1$, we state the following result without proof. (See [@distgridpart2] for details.)
\[Theorem1\_LC\] [@distgridpart2 Theorem 1] If node $a \neq b$ is a descendant of node $b$ on tree ${\cal T}$ then $\Omega_{\varepsilon}(a,a) > \Omega_{\varepsilon}(b,b)$.
Next, we define the term $\phi_{ab} = \mathbb{E}[(\varepsilon_a - \mu_{\varepsilon_a})-(\varepsilon_b-\mu_{\varepsilon_b})]^2 $, which is the variance of the difference in voltage magnitudes between nodes $a$ and $b$. $$\begin{aligned}
\phi_{ab} = \Omega_{\varepsilon}(a,a) - 2\Omega_{\varepsilon}(a,b) + \Omega_{\varepsilon}(b,b) \label{expand}\end{aligned}$$ where $\Omega_{\varepsilon}$ is given by Eq. (\[volcovar1\]). Expressing Eq. (\[expand\]) in terms of the four matrices that constitute $\Omega_{\varepsilon}$ and then using Eq. (\[Hrxinv\]) leads to the following expansion of $\phi_{ab}$ over power injections. $$\begin{aligned}
&\phi_{ab} = \smashoperator[lr]{\sum_{d \in {\cal T}}}(H^{-1}_{1/r}(a,d)- H^{-1}_{1/r}(b,d))^2\Omega_p(d,d)\nonumber\\
&+(H^{-1}_{1/x}(a,d)- H^{-1}_{1/x}(b,d))^2 \Omega_q(d,d)+2\left(H^{-1}_{1/r}(a,d)- H^{-1}_{1/r}(b,d)\right)\nonumber\\
&\left(H^{-1}_{1/x}(a,d)- H^{-1}_{1/x}(b,d)\right)\Omega_{pq}(d,d) \label{usediff_1}\end{aligned}$$
The next result identifies trends in $\phi_{ab}$ along the radial grid. Note that the first two cases in Lemma \[Lemmacases\] are proven in [@distgridpart2]. The additional final case is opposite of the first case and helps develop our new learning scheme presented later in this paper.
\[Lemmacases\] For three nodes $a \neq b \neq c$ in grid tree ${\cal T}$, $\phi_{ab} < \phi_{ac}$ holds for the following cases:
1. Node $a$ is a descendant of node $b$ and node $b$ is a descendant of node $c$ (see Fig. \[fig:item1\]).
2. Nodes $a$ and $c$ are descendants of node $b$ and the path from $a$ to $c$ passes through node $b$ (see Fig. \[fig:item2\]).
3. Nodes $c$ is a descendant of node $b$ and node $b$ is a descendant of node $a$ (see Fig. \[fig:item3\]).
We give the proof for Case $3$ depicted in Fig. \[fig:item3\]. In this case, ${\cal P}^b_{\cal T}- {\cal P}^a_{\cal T} \subseteq {\cal P}^c_{\cal T}-{\cal P}^a_{\cal T}$, where ${\cal P}^a_{\cal T}$ is the set of edges in the unique path from node $a$ to the root node of ${\cal T}$. Further, the sets of descendants of $a,b$ and $c$ satisfy $D^c_{\cal T} \subseteq D^b_{\cal T} \subseteq D^a_{\cal T}$. From Fig. \[fig:item3\], it is clear that any node $d$ belongs to either $D_{\cal T}^c$, $D_{\cal T}^b - D_{\cal T}^c$, $D^a_{\cal T} - D^b_{\cal T}$ or ${\cal V}_{\cal T} - D^a_{\cal T}$. When $d \in D_{\cal T}^c$, using Eq. (\[Hrxinv\]), we have, $$\begin{aligned}
H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d) &= \smashoperator[lr]{\sum_{(ef) \in {\cal P}^b_{\cal T}-{\cal P}^a_{\cal T}}}r_{ef} < \smashoperator[r]{\sum_{(ef) \in {\cal P}^c_{\cal T}-{\cal P}^a_{\cal T}}}r_{ef}\label{f1}\\
\Rightarrow ~H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d) &< H^{-1}_{1/r}(c,d)-H^{-1}_{1/r}(a,d)\label{first}\end{aligned}$$ For node $d \in D^b_{\cal T} - D^c_{\cal T}$, one derives $$\begin{aligned}
&H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d) = \smashoperator[lr]{\sum_{(ef) \in {\cal P}^b_{\cal T}-{\cal P}^a_{\cal T}}} r_{ef} ~~~~< \smashoperator[r]{\sum_{(ef) \in {\cal P}^c_{\cal T}\cap {\cal P}^{d}_{\cal T}-{\cal P}^a_{\cal T}} } r_{ef}\label{f2}\\
\Rightarrow ~&H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d) < H^{-1}_{1/r}(c,d)-H^{-1}_{1/r}(a,d)\label{second}\end{aligned}$$ For $d \in D^a_{\cal T} - D^b_{\cal T}$, one derives $$\begin{aligned}
&H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d)= \smashoperator[lr]{\sum_{(ef) \in {\cal P}^b_{\cal T}\cap {\cal P}^{d}_{\cal T}-{\cal P}^a_{\cal T}}}r_{ef}~~= \smashoperator[r]{\sum_{(ef) \in {\cal P}^c_{\cal T}\cap {\cal P}^{d}_{\cal T}-{\cal P}^a_{\cal T}} }r_{ef}\label{f3}\\
\Rightarrow~& H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d) = H^{-1}_{1/r}(c,d)-H^{-1}_{1/r}(a,d)\label{third}\end{aligned}$$ Finally for $d \in {\cal T} - D^a_{\cal T}, H^{-1}_{1/r}(b,d)-H^{-1}_{1/r}(a,d) = H^{-1}_{1/r}(c,d)-H^{-1}_{1/r}(a,d) = 0$. Such inequalities also hold for $H_{1/x}^{-1}$ matrix. Using the inequalities in Eqs. (\[first\], \[second\],\[third\]) for $H^{-1}_{1/r}$ and $H^{-1}_{1/x}$ with Eq. (\[usediff\_1\]) results in $\phi_{ab} < \phi_{ac}$ for Case $3$. The proofs for the other cases ($1$ and $2$) can be done in a similar way and they are thus skipped.
Further, the following results hold for operational edges in $\cal T$.
\[Lemmacases2\] Let $(ab)$ and $(bc)$ be operational edges in $\cal T$
1. If node $a$ is the parent of node $b$ (see Fig. \[fig:item3\]) then $$\begin{aligned}
\phi_{ab} = \smashoperator[lr]{\sum_{d \in D_{\cal T}^b}}r_{ab}^2\Omega_p(d,d)+x_{ab}^2 \Omega_q(d,d)+2r_{ab}x_{ab}\Omega_{pq}(d,d)\nonumber\end{aligned}$$
2. If node $b$ is the parent of node $c$ and child of node $a$ (see Fig. \[fig:item3\]), then $$\begin{aligned}
\phi_{ac} &= \smashoperator[lr]{\sum_{d \in D_{\cal T}^c}}(r_{ab}+r_{bc})^2\Omega_p(d,d)+(x_{ab}+x_{bc})^2 \Omega_q(d,d)\nonumber\\ &+2(r_{ab}+r_{bc})(x_{ab}+x_{bc})\Omega_{pq}(d,d) \nonumber\\ &+ \smashoperator[lr]{\sum_{d \in D_{\cal T}^b- D_{\cal T}^c}}r_{ab}^2\Omega_p(d,d)+x_{ab}^2 \Omega_q(d,d)+2r_{ab}x_{ab}\Omega_{pq}(d,d)\nonumber\\
&> \phi_{ab} + \phi_{bc} \label{equal1}\end{aligned}$$
3. If node $b$ is the parent of both nodes $a$ and $c$ (see Fig. \[fig:item2\]), then $$\begin{aligned}
\phi_{ac} &= \smashoperator[lr]{\sum_{d \in D_{\cal T}^a}}r_{ab}^2\Omega_p(d,d)\nonumber+x_{ab}^2 \Omega_q(d,d)+2r_{ab}x_{ab}\Omega_{pq}(d,d)\nonumber\\
&+ \smashoperator[lr]{\sum_{d \in D_{\cal T}^c}}r_{bc}^2\Omega_p(d,d)+x_{bc}^2 \Omega_q(d,d)+2r_{bc}x_{bc}\Omega_{pq}(d,d)\nonumber\\
&= \phi_{ab} + \phi_{bc} \label{equal2}\end{aligned}$$
<!-- -->
1. We use Eq. (\[Hdiff\]) in Eq. (\[usediff\_1\]) as $(ab)$ is an edge.
2. We follow the proof in Lemma \[Lemmacases\]. The result holds as the left sides of Eqs. (\[f1\]),(\[f2\]),(\[f3\]) here are given by $(r_{ab}+r_{bc})$, $r_{ab}$ and $0$ respectively. The inequality in (\[equal1\]) is derived by applying Statement $1$ for edges $(ab)$ and $(bc)$ and noting that $(y_1+y_2)*(y_3+y_4)>y_1y_3+y_2y_4$ holds for positive reals $y_1,y_2, y_3,y_4$.
3. We use the same technique as above. Here $D_{\cal T}^c$ and $D_{\cal T}^a$ are disjoint. Using this fact along with Eq. (\[Hdiff\]) for edges $(ab)$ and $(bc)$ results in the equality (\[equal2\]).
It is worth mentioning that all three statements in Lemma \[Lemmacases2\] involve line impedances corresponding to edges $(ab)$ and $(bc)$ only. In the following sections, we use these results to design our topology learning algorithm.
Structure Learning with Full Observation {#sec:algo1}
========================================
Our main result for topology learning using voltage magnitude measurements is formulated using Lemma \[Lemmacases\].
\[main\] Let the weight of each permissible edge $(ab) \in {\cal E}$ of the original loopy graph be $\phi_{ab} = \mathbb{E}[(\varepsilon_a-\mu_{\varepsilon_a}) -(\varepsilon_b-\mu_{\varepsilon_b})]^2$. Then operational edge set ${\cal E}_{\cal T}$ in radial grid $\cal T$ forms the minimum weight spanning tree of the original graph.
From Lemma \[Lemmacases\], it is clear that for each node $a$, the minimum value of $\phi_{ab}$ along any path in $\cal T$ (towards or away from the root node) is attained at its immediate neighbor $b$ on that path, connected by edge $(ab) \in {\cal E}_{\cal T}$. The minimum spanning tree for the original loopy graph with $\phi$’s as edge weights is thus given by the operational edges in the radial tree.
Note that if node $a$ is taken as the substation/root node ($\varepsilon_{a}=0$), the weight of any edge $(ab)$ is given by $\phi_{ab} = \Omega_{\varepsilon}(b,b)$. As mentioned in Section \[sec:structure\], the substation has one child. In the spanning tree construction, the root is thus connected to the node with lowest variance of voltage magnitude. This is in agreement with Theorem \[Theorem1\_LC\].
**Algorithm $1$:** The input consists of voltage magnitude readings for all non-substation buses in the system. An observer computes $\phi_{ab}$ for all permissible edges $(ab) \in {\cal E}$ (including those with the root node) and identifies edges in the minimum spanning tree as the set of operational edges ${\cal E}_{\cal T}$. The root node is restricted to have a single edge. Note that Algorithm $1$ does not need any information on line parameters (resistances and reactances) or on statistics of active and reactive nodal power consumption. If impedances of lines in $\cal E$ and phase angle measurements at all nodes are known, Eqs. (\[PF\_LPV\_p\]), (\[means\]) and (\[volcovar1\]) can be used to estimate means and covariances of each node’s power injection.
**Input:** $m$ voltage magnitude deviations $\varepsilon$ for all nodes, set of all edges $\cal T$.\
**Output:** Operational Edge set ${\cal E}_{\cal T}$.
$\forall (ab) in {\cal E}$, compute $\phi_{ab} = \mathbb{E}[(\varepsilon_a-\mu_{\varepsilon_a}) -(\varepsilon_b-\mu_{\varepsilon_b})]^2$ Find minimum weight spanning tree from $\cal E$ with $\phi_{ab}$ as edge weights. Limit degree of substation to $1$. ${\cal E}_{\cal T} \gets $ [edges in spanning tree]{}
**Algorithm Complexity:** Using Kruskal’s Algorithm [@kruskal1956shortest; @Cormen2001], the minimum spanning tree from $\cal E$ edges can be computed in $O(|{\cal E}|\log|{\cal E}|)$ operations. This is a great improvement over previous iterative or matrix inversion based techniques which scaled as $O(N^3)$, where $N = |{\cal V_{\cal T}}|$ is the number of nodes in the grid. If $\cal E$ is not known or corresponds to the complete graph, Algorithm $1$’s complexity is $O(N^2\log N)$, i.e. it still compares favorably with the prior scheme.
**Extension to Multiple Trees:** If multiple trees exist in the grid, voltage magnitudes at nodes $a$ and $b$ belonging to disjoint trees will be independent. Thus, $\phi_{ab} = \Omega_{\varepsilon}(a,a) + \Omega_{\varepsilon}(b,b)$. This result can be used to separate nodes into disjoint groups before running Algorithm $1$ to generate the operational tree in each group.
In the next Section, we extend our spanning tree based algorithm to consider cases where information is missing at some fraction of nodes.
Structure Learning with Missing Data {#sec:missing}
====================================
In a realistic power grid, communication packet drops or random noise events may erase voltage magnitude measurements for node set ${\cal M}$ in $\cal T$. Following [@distgridpart2], we consider arbitrary placement of unobserved nodes with the following restriction.
**Assumption $2$:** Missing nodes are separated by greater than two hops in the grid tree $\cal T$.
Note that under assumption $1$, an observable node cannot be connected to two or more unobserved nodes. (We plan to analyze extensions beyond Assumption $2$ in future work.) Additionally, we assume that the adversary estimates or has access to historical information for the values of $\Omega_p, ~\Omega_q$ and $\Omega_{pq}$ covariance matrices for all nodes and impedances of all possible lines in $\cal E$.
To reconstruct operational topology in the presence of missing data, we first construct the minimum weight observable spanning tree ${\cal T}_{\cal M}$ using $\phi_{ab} = \mathbb{E}[(\varepsilon_a-\mu_{\varepsilon_a}) -(\varepsilon_b-\mu_{\varepsilon_b})]^2$ as edge weights between observable nodes. We then analyze edges in tree ${\cal T}_{\cal M }$ and detect unobserved node locations. Consider the situation shown in Fig. \[fig:missing1\] where information from the leaf node $l$ is missing. By Assumption $2$, information from its parent ($q$) and grandparent ($w$) are observed in ${\cal T}_{\cal M}$. Note that $\phi_{qw}$ satisfies Statement $1$ in Lemma \[Lemmacases2\]. If all other descendants of $q$ are known, statement $1$ of the Lemma can be used to identify the existence of unobserved node $l$. We now discuss the identification of a non-leaf node with missing information. Assume that information is missing at the node $b$ in Fig. \[fig:missing1\]. $b$’s parent $a$ and children node set ${\cal C} = \{c_1, c_2, c_3, c_4\}$ comprise its one-hop neighborhood, and are observable under Assumption $2$. Using Cases $1$ and $3$ in Lemma \[Lemmacases\], $\operatorname*{arg\,min}_{d \in D^b_{\cal T} - \{b\}} \phi_{ad} \in {\cal C}$ and $\operatorname*{arg\,min}_{d \in {{\cal V}_{\cal T} - D^b_{\cal T}}} \phi_{c_id} =a \forall c_i \in {\cal C}$. Thus, descendants of $b$ are connected to the rest of ${\cal T}_{\cal M}$ through edges between its one-hop neighbors (set $C$ and $a$). The following theorem gives the edge configurations possible in ${\cal T}_{\cal M}$ for $a$ and nodes in $\cal C$.
\[permissiblecases\] Let $\operatorname*{arg\,min}_{c_i \in {\cal C}} \phi_{bc_i} = c^*$. No edge $(c_ic_j)$ between children nodes $c_i, c_j \neq c^*$ exists in ${\cal T}_{\cal M}$. All nodes in set ${\cal C}^1= \{c_i\in {\cal C}, \phi_{ac_i} < \phi_{c^*c_i}\}$ are connected to node $a$, while all nodes in ${\cal C}^2 = {\cal C}-{\cal C}^1$ are connected to $c^*$.
Consider any node pair $c_i,c_j\neq c^*$ in $C$. Using Eq. (\[equal2\]) in Lemma \[Lemmacases2\] and definition of $c^*$, $\phi_{c_ic_j} = \phi_{bc_i}+ \phi_{bc_j} <\phi_{bc_i}+ \phi_{bc^*} = \phi_{c_ic^*}$. Thus, any possible edge between children nodes must include node $c^*$. The edges for each node in sets ${\cal C}^1$ and ${\cal C}^2$ follow immediately by comparing weights with $c^*$ and $a$.
Theorem \[permissiblecases\] does not specify if edge $(ac^*)$ exists in ${\cal T}_{\cal M}$. In fact node $c^*$ will be connected to a node $c\dag \in {\cal C}^1$ instead of $a$ if $\phi_{ac\dag} < \phi_{c^*c\dag} <\phi_{ac^*}$ holds. There are thus two permissible configurations $A$ and $B$ (see Figs. \[fig:missing2\], \[fig:missing3\]) in ${\cal T}_{\cal M}$ for connections between one hop neighbors of non-leaf unobservable node $b$. Note that one of sets ${\cal C}^1$ or ${\cal C}^2$ may be empty as well.
Any two nodes in $\cal C$ are children of node $b$ and thus satisfy Statement $3$ in Lemma \[Lemmacases2\]. Observe that for both configurations $A$ and $B$, this result holds for $c^*$ and any of its children in ${\cal T}_{\cal M}$ that belong to $\cal C$. The result also holds for $c^*$ and its parent in configuration $B$. On the other hand, any node in $\cal C$ and $a$ are actually separated by node $b$ and thus it satisfies Statement $2$ in Lemma \[Lemmacases2\]. This result thus holds for node $a$ and any of its children from $\cal C$. Statements $2$ and $3$ in Lemma \[Lemmacases2\] can hence be used to identify unobservable node $b$ in Algorithm $2$. **Algorithm $2$:** Assume that information is missing at the set ${\cal M}$, thus leaving only ${\cal V}_{\cal T} - {\cal M}$ observable. Covariance matrices for power injection at all nodes of the observed set are assumed known to the observer along with impedances of all lines in $\cal E$. Algorithm $2$, first, constructs spanning tree ${\cal T}_{\cal M}$ for observed nodes using edge weights for all node combinations given by $\phi$. Observed nodes in ${\cal T}_{\cal M}$ are then arranged in reverse topological order (decreasing depth from root node). This is done as unobserved node locations are iteratively searched from leaf sites inward towards the root (see Step \[step1\]). For each leaf $b$ with parent $a$, Steps \[step2\] to \[step3\] checks if edge $(ab) \in {\cal E}_{\cal T}$ with or without some unobserved leaf node $h$ connected to $b$. For undecided nodes in $C$, the Algorithm first checks for configuration $A$ or $B$ described in the preceding discussion. Step \[step4\] determines if nodes in $C$ and $a$ are separated by a unobserved node $h$ using Statement $2$ in Lemma \[Lemmacases2\]. If such a node doesn’t exist, Step \[step5\] search for a unobserved node that is parent of both nodes in $C$ and node $a$ using using Statement $3$ in Lemma \[Lemmacases2\]. Nodes $a$ and set $C$ are removed from the observed tree ${\cal T}_{\cal M}$ in each iteration and discovered edges are added to ${\cal E}_{\cal T}$. Further, injection covariances at the recently identified descendants are added for use in later checks involving results from Lemma \[Lemmacases2\]. Note that only in the final case (Step \[step5\]), the unobserved node $h$ is not removed from set $M$ as its parent node has not been determined yet. This process is iterated by picking a new node $a$ with all children as leaf nodes until no nodes with missing information remain to be discovered.
**Complexity:** Computing the spanning tree for observed nodes has complexity $O((N-|{\cal M}|)^2\log(N-|{\cal M}|))$. Sorting observed nodes in topological order is done in linear time ($O(N-|{\cal M}|)$) [@Cormen2001]. Finally, checking (Steps \[step1\], \[step2\], \[step3\], \[step4\]) for all iterations has complexity $O((N- |{\cal M}|)|{\cal M}|)$ as total observed nodes and edges number $O((N- |{\cal M}|))$ and searching over unobserved nodes takes at most $|{\cal M}|$ steps. The overall complexity of Algorithm $2$ is thus $O((N-|{\cal M}|)^2\log(N-|{\cal M}|)+(N- |{\cal M}|)|{\cal M}|)$ which is $O(N^2\log N)$ in the worst case. Note that this is also the worst-case complexity of Algorithm $1$.
**Relation to Learning Probabilistic Graphical Model:** It is worth noting that in the tree-structured GM learning [@choi2011learning], edge $(ac^*)$ always exists due to the graph-additivity of edge weights and configuration $B$ in Fig. \[fig:missing3\] is not realized. The inequality in Eq. (\[equal1\]) of Lemma \[Lemmacases2\] shows that $\phi$ may be strictly increasing with the number of graph hops and thus it does not satisfy graph additivity in general. Non-additivity of edge weights makes our topology learning approach a generalization of the additive model in [@choi2011learning] .
**Extensions:** We briefly mention two extensions of Algorithm $2$, planning to analyze these in details in the future. First, Algorithm $2$ can be used for structure learning *when injection covariances at unobserved nodes are not known*. Here each unobserved node must have at least two children for unique identification. Second, Algorithm $2$ will be extended to operate *when unobserved nodes are separated by $2$ hops*. In this case, permissible configurations in addition to $A$ and $B$ (see Fig. \[fig:missing\]) need to be checked. A modification of Statement $2$ in Lemma \[Lemmacases2\] will be used to detect unobserved nodes. In the following Section, we discuss the performance of our designed algorithms through experiments on test networks.
**Input:** Injection covariances $\Omega_p, \Omega_q, \Omega_{pq}$ of all nodes, Missing nodes Set ${\cal M}$, $m$ voltage deviation observations $\varepsilon$ for nodes in ${\cal V}_{\cal T} -{\cal M}$, set of all edges $\cal T$ with line impedances.\
**Output:** Operational Edge set ${\cal E}_{\cal T}$.
$\forall$ observable nodes $a,b$, compute $\phi_{ab} = \mathbb{E}[(\varepsilon_a-\mu_{\varepsilon_a}) -(\varepsilon_b-\mu_{\varepsilon_b})]^2$ Find minimum weight spanning tree ${\cal T}_{\cal M}$ with $\phi_{ab}$ as edge weights. Limit degree of substation to $1$. Sort nodes in ${\cal T}_{\cal M}$ in reserve topological order. Select node $a$ whose children set ${\cal C}$ in ${\cal T}_{\cal M}$ consists only of leaf nodes \[step1\] \[step2\] ${\cal E}_{\cal T} \gets {\cal E}_{\cal T} \cup \{(ab)\}$, ${\cal C} \gets {\cal C}-\{b\}$, Add injection covariance of $b$ to $a$. Remove node $b$ from ${\cal T}_{\cal M}$. ${\cal E}_{\cal T} \gets {\cal E}_{\cal T} \cup \{(ab), (bh)\}$, ${\cal M} \gets {\cal M}-\{h\}$, ${\cal C} \gets {\cal C}-\{b\}$, Add injection covariance of $b$ and $h$ to $a$. Remove node $b$ from ${\cal T}_{\cal M}$. \[step3\] \[step4\] ${\cal E}_{\cal T} \gets {\cal E}_{\cal T} \cup \{(ah)\} \cup \{(ch) \forall c \in {\cal C}\}$, ${\cal M} \gets {\cal M}-\{h\}$, ${\cal C} \gets \emptyset$, Add injection covariances $\forall c \in {\cal C}$ and $h$ to $a$. Remove nodes in ${\cal C}$ from ${\cal T}_{\cal M}$. Pick $b \in {\cal C}$. Find $h \in {\cal M}$ s..t. $\phi_{ab}$ satisfy Statement $3$ in Lemma \[Lemmacases2\] with $h$ as parent and $D^b_{\cal T} = \{b\}$ , $D^a_{\cal T} = \{a\}$. \[step5\] ${\cal E}_{\cal T} \gets {\cal E}_{\cal T} \cup \{(ah)\} \cup \{(ch) \forall c \in {\cal C}\}$, ${\cal C} \gets \emptyset$, Add injection covariances of $a$ and $\forall c \in {\cal C}$ to $h$. Remove $a$ and nodes in ${\cal C}$ from ${\cal T}_{\cal M}$.
Experiments {#sec:experiments}
===========
Here we demonstrate performance of Algorithm $1$ in determining the operational edge set ${\cal E}_{\cal T}$ of the radial grid ${\cal T}$. We consider a radial network [@testcase2; @radialsource] with $29$ load nodes and one substation as shown in Fig. \[fig:case\]. In each of our simulation runs, we first collect complex power injection samples at the non-substation nodes from a multivariate Gaussian distribution that is uncorrelated between different nodes as per Assumption $1$. We use LC-PF model to generate nodal voltage magnitude measurements. Finally, we introduce $30$ additional edges (at random) forming the loopy edge set ${\cal E}$. The additional edges are given random impedances comparable to those of operational lines. We, first, test performance of the Algorithm $1$ for the case where locations of edges in the set $\cal E$ and voltage magnitude measurements at all non-substation nodes are available. We show results for topology learning for this case in Fig. \[fig:plotadjerrors\]. Note that the estimation is extremely accurate and average errors expressed relative to the size of the operational edge set) decay to zero at the sample sizes less than $50$. We also estimate covariance matrices of complex nodal power injections using the just reconstructed radial operating topology and plot results in Fig. \[fig:plotcoverrors\]. For covariance estimation, line impedances of the set $\cal E$ and samples of phase angle measurements are used along with voltage magnitude samples as input. The relative errors in this case decay exponentially with increase in the number of the measurement samples.
Next, we present simulations for Algorithm $2$ where the operational grid structure is reconstructed in the presence of unobserved nodes. We consider three cases with information at the nodes $4$, $6$ and $8$ missing. The location of the unobserved nodes are selected at random in accordance with Assumption $2$. Voltage magnitudes at the unobserved nodes are removed from the input data. Covariance of power injections at all the load nodes and impedances of all the lines within the loopy edge set $\cal E$ are provided as input to the observer. The average number of errors shown in Fig. \[fig:plotmissing\] decreases steadily with increase in the number of samples. This tendency is seen clearly for all the cases of the unobserved node sets. Further, the average errors increase with increase in the number of unobserved nodes for a fixed number of measurement samples. The average errors produced by Algorithm $2$ are significantly lower in comparison with the respective algorithm from [@distgridpart2], however (and as expected) the Algorithm is significantly less accurately than Algorithm $1$ where all nodes are observed.
Conclusions {#sec:conclusions}
===========
Identifying the operational edges in the distribution grids is critical for real-time control and reliable management of different grid operations. In this paper, we study the problem of learning the radial operating structure from a dense loopy grid graph. Under an LC (linear coupled) power flow model, we show that if edge weights between load nodes are defined as the variance of the difference of their voltage magnitudes, the minimum weight spanning tree optimization over the loopy physical layout outputs operational radial structure. Using this spanning tree property, we design a fast structure learning algorithm that uses only nodal voltage magnitude measurements for the input. We then extend the spanning tree based framework to learn the operational structure when available voltage measurements are limited to a subset of the grid nodes. For unobserved nodes separated by greater than three hops, the learning algorithm is able to identify locations of the missing measurements by verifying properties of our voltage magnitude based edge weights. In this case, statistics of nodal injections and line impedances are used as a part of the input. We demonstrate good performance of the learning algorithm through experiments on distribution grid test cases. Finally, we discuss how voltage magnitude based edge weights in our algorithm generalizes edge metrics used in learning schemes of probabilistic GMs. In future we plan to generalize our approach reducing restrictions, e.g. allowing unobserved nodes to be separated by less than two hops and utilizing less information about nodal consumption.
|
---
abstract: '[*The research on aerial manipulation systems has been increased rapidly in recent years. These systems are very attractive for a wide range of applications due to their unique features. However, dynamics, control and manipulation tasks of such systems are quite challenging because they are naturally unstable, have very fast dynamics, have strong nonlinearities, are very susceptible to parameters variations due to carrying a payload besides the external disturbances, and have complex inverse kinematics. In addition, the manipulation tasks require estimating (applying) a certain force of (at) the end-effector as well as the accurate positioning of it. Thus, in this article, a robust force estimation and impedance control scheme is proposed to address these issues. The robustness is achieved based on the Disturbance Observer (DOb) technique. Then, a tracking and performance low computational linear controller is used. For teleoperation purpose, the contact force needs to be identified. However, the current developed techniques for force estimation have limitations because they are based on ignoring some dynamics and/or requiring of an indicator of the environment contact. Unlike these techniques, we propose a technique based on linearization capabilities of DOb and a Fast Tracking Recursive Least Squares (FTRLS) algorithm. The complex inverse kinematics problem of such a system is solved by a Jacobin based algorithm. The stability analysis of the proposed scheme is presented. The algorithm is tested to achieve tracking of task space reference trajectories besides the impedance control. The efficiency of the proposed technique is enlightened via numerical simulation.* ]{}'
author:
- Ahmed Khalifa
- 'Mohamed Fanni\'
bibliography:
- 'My\_Ref.bib'
title: 'Quadrotor Manipulation System: Development of a Robust Contact Force Estimation and Impedance Control Scheme Based on DOb and FTRLS'
---
Introduction
============
Recently, Unmanned Aerial Vehicles (UAVs) especially multi-rotors type, receive great attention due to their higher degree of mobility, speed and capability to access to regions that are inaccessible to ground vehicles. However, UAV as a standalone vehicle has a limited functionality to the search and surveillance applications.
Due to their superior mobility, much interest is given to utilize them for aerial manipulation and thus the application of UAV manipulation systems have been expanded dramatically. Applications of such systems include inspection, maintenance, structure assembly,firefighting, rescue operation, surveillance, or transportation in locations that are inaccessible, very dangerous or costly to be accessed from the ground.
Research on quadrotor-based aerial manipulation can be divided into different approaches based on the tool attached to the UAV including gripper based [@mellinger2011design], cables based [@goodarzi2015geometric; @guerrero2017swing], multi-DoF robotic manipulator based[@kim2013aerial; @fanni2017new], multi-DoF dual-arms manipulator based [@korpela2014towards], compliant manipulator -based [@bartelds2016compliant], Hybrid rigid/elastic-joint manipulator [@yuksel2016aerial].
In the gripper/ tool-based approach, the attitude of the payload/tool is restricted to that of the quadrotor, and hence, the resulting aerial system has independent 4 DOFs; three translational DOFs and one rotational DOF (Yaw), i.e., the gripper/tool cannot posses pitch or roll rotation without moving horizontally. The second approach is to suspend a payload with cables but this approach has a drawback that the movement of the payload cannot be always regulated directly. To cope up with these limitations, another approach is developed in which a quadrotor is equipped with a robotic manipulator that can actively interact with the environment. Very few reports exist in the literature that investigate the combination of aerial vehicle with robotic manipulator. Kinematic and dynamic models of the quadrotor combined with arbitrary multi-DOF robot arm are derived using the Euler-Lagrangian formalism in [@lippiello2012cartesian]. In [@orsag2013modeling], a quadrotor with light-weight manipulators, three 2-DOF arms, are tested. In [@kim2013aerial], an aerial manipulation using a quadrotor with a 2-DOF robotic arm is presented but with certain topology that disable the system from making arbitrary position and orientation of the end-effector. In this system, the axes of the manipulator joints are parallel to each other and parallel to one in-plane axis of the quadrotor. Thus, the system cannot achieve orientation around the second in-plane axis of the quadrotor without moving horizontally.
From the above discussion, the current introduced systems in the literature that use a gripper suffers from the limited allowable DOFs of the end-effector. The other systems have a manipulator with either two DOFs but in certain topology that disables the end-effector to track arbitrary 6-DOF trajectory, or more than two DOFs which decreases greatly the possible payload carried by the system.
In [@new_quad_manp; @khalifa2013adaptive; @fanni2017new], the authors propose a new aerial manipulation system that consists of 2-link manipulator, with two revolute joints whose axes are perpendicular to each other and the axis of the first joint is parallel to one in-plane axis of the quadrotor. Thus, the end-effector is able to reach arbitrary position and orientation without moving horizontally with minimum possible actuators.
In order to achieve position holding during manipulation, uncertainties and disturbances in the system such as wind, contact forces, measurement noise have to be compensated by using a robust control scheme. Disturbance Observer (DOb)-based controller is used to achieve a robust motion control [@li2014disturbance; @chen2012disturbance]. The DOb estimates the nonlinear terms and uncertainties then compensates them such that the robotic system acts like a multi-SISO linear systems. Therefore, it is possible to rely on a standard linear controller to design the controller of the outer loop such that the system performance can be adjusted to achieve desired tracking accuracy and speed. In [@sariyildiz2015nonlinear; @choi2014simplified; @dong2014high], DOb-based motion control technique is applied to robotic-based systems and gives efficient results.
In the motion control of the aerial manipulator, achievement of the compliance control is very important because the compliance motion makes possible to perform flexible motion of the manipulator according to desired impedance [@barbalata2018position]. This is very critical demand in applications such as demining and maintenance. In the compliance control, end-effector position and generated force of the manipulator are controlled according to the reaction force detected by the force sensor. In this method, the desired impedance is selected arbitrary in the controller. However, the force sensor is essential to detect the reaction force as presented in [@seraji1997force; @love1995environment; @singh1995analysis]. On line identified environment impedance has also been used for transparency in teleoperation systems [@misra2006environment]. These problems are more severe when environment displays sudden changes in its dynamic parameters which cannot be tracked by the identification process. In [@hashtrudi1996adaptive], it is found that in order to faithfully convey to the operator the sense of high frequency chattering of contact between the slave and hard objects, faster identification and structurally modified methods were required. However, these methods need the measurement of force.
Several techniques are proposed to estimate the contact force and the environment dynamics. In [@sariyildiz2014guide], the DOb and Recursive Least Squares (RLS) are used to estimate the environment dynamics. However, in this method, two DObs are used besides the RLS, and the estimation of contact force is activated only during the instance of contacting, thus there is a need to detect the instant at which the contact occurs. However, this is not practical approach especially if we target autonomous system. In [@murakami1993force; @eom1998disturbance; @van2011estimating; @phong2012external; @colome2013external; @alcocera2004force], several techniques are proposed to achieve force control without measuring the force. However, these techniques are based on ignoring some dynamics and external disturbances which will produce inaccurate force estimation. In [@forte2012impedance; @lippiello2012exploiting], an impedance control is designed for aerial manipulator without the need to measure/estimate the contact force. However, in such work, the authors neglect some dynamics as well as external disturbances, in addition to, the proposed algorithm is model-based and it does not have a robustness capability. In [@ruggiero2014impedance], a scheme is proposed which allows a quadrotor to perform tracking tasks without a precise knowledge of its dynamics and under the effect of external disturbances and unmodeled aerodynamics. In addition, this scheme can estimate the external generalized forces. However, as the authors claim, this estimator can work perfectly with constant external disturbances. In addition, the estimated forces contain many different types of forces such as wind, payload, environment impacts, and unmodeled dynamics. Thus, it can not isolate the end-effector force only from the others. The authors in [@tomic2014unified] present a model-based method to estimate the external wrench of a flying robot. However, this method assumes that there are no modeling errors and no external disturbance. Moreover, it estimates the external force as one unite and it can not distinguish between external disturbance and the end-effector force which we need to calculate for teleoperation purposes. In addition, it uses a model based control which needs a full knowledge of the model.
In this article, a new scheme is proposed to cope up with these limitations of the currently developed techniques to solve the issues of this complicated multibody robotic system. Firstly, a DOb inner loop is used to estimate both the system nonlinearities and all external forces to compensate for them, as a result, the system acts like a linear decoupled MIMO system. Secondly, a fast tracking RLS algorithm is utilized with the linearization capabilities of DOb to estimate the contact force, in addition to, it enables the user to sense the contact force at the end-effector that it is not available in the current developed schemes. Thirdly, a model-free robust impedance control of the quadrotor manipulation system is implemented. The DOb is designed in the quadrotor/joint space while the impedance control is designed in the task space such that the end-effector can track the desired task space trajectories besides applying a specified environment impedance. Thus, Fourthly, a Jacobian based algorithm is proposed to transform the control signal from the task space to the quadrotor/joint space coordinates. The rigorous stability analysis of the proposed scheme is presented. Finally, the system model is simulated in MATLAB taking in to considerations all the non-idealities and based on real parameters to emulate a real system.
System Modeling {#se:model}
===============
Fig. \[3D-CAD-MODEL\] presents a 3D CAD model of the proposed quadrotor-based aerial manipulator. The system is composed of a manipulator mounted on the bottom center of a quadrotor.
![3D CAD model of the proposed quadrotor-based aerial manipulator[]{data-label="3D-CAD-MODEL"}](3D-CAD-MODEL.jpg){width="0.5\columnwidth"}
System geometrical frames, which are assumed to satisfy the Denavit-Hartenberg (DH) convention, are illustrated in Fig. \[frames\]. The manipulator has two revolute joints. The axis of the first revolute joint, $z_0$, is parallel to the quadrotor $x$-axis. The axis of the second joint, $z_1$, is normal to that of the first joint and hence it is parallel to the quadrotor $y$-axis at the extended configuration. Therefor, the pitching and rolling rotation of the end-effector is allowable independently from the horizontal motion of the quadrotor. Hence, with this proposed aerial manipulator, it is possible to manipulate objects with arbitrary location and orientation. Consequently, the end-effector can make motion in 6-DOF with minimum possible number of actuators/links that is critical factor in flight.
![Quadrotor-based aerial manipulator with relevant frames[]{data-label="frames"}](frames.jpg){width="0.7\columnwidth"}
The quadrotor components are designed to achieve a payload capacity of $500$ g. Asctec pelican quadrotor [@asctec] is utilized as a quadrotor platform. The maximum thrust force for each rotor is $6$N. The arm is designed so that the total weight of the arm is $200$ g, it has a maximum reach in the range of $22$, and it can carry a payload of $200$ g. It has three DC motors, (HS-422 (Max torque = $0.4$ N.m) for gripper, HS-5485HB (Max torque = $0.7$ N.m) for joint $1$, and HS-422 (Max torque = $0.4$ N.m) for joint $2$).
The angular velocity of each rotor $j$ is $\Omega_j$ and it generates thrust force $F_j$ and drag moment $M_j$ that are given by $$F_j = K_{f_j} \Omega_j^2,
\label{thrust}$$ $$M_j = K_{m_j} \Omega_j^2,
\label{dragmoment}$$ where $K_{f_j}$ and $K_{m_j}$ are the thrust and drag coefficients.
Kinematics
----------
Let $\Sigma_b$, $O_{b}$- $x_b$ $y_b$ $z_b$, represents the quadrotor body-fixed reference frame with origin at the quadrotor center of mass, see Fig. \[frames\]. Its position with respect to the world-fixed inertial reference frame, $\Sigma$, $O$- $x$ $y$ $z$, is given by the $(3 \times 1)$ vector $p_b=[x, y, z]^T$, while its orientation is given by the rotation matrix $R_b$ which is given by $$R_b= \begin{bmatrix}
C_{\psi} C_{\theta} & S_{\phi} S_{\theta} C_{\psi}-S_{\psi} C_{\phi} & S_{\psi} S_{\phi}+C_{\psi} S_{\theta} C_{\phi} \\
S_{\psi} C_{\theta} & C_{\psi} C_{\phi}+ S_{\psi} S_{\theta} S_{\phi} & S_{\psi} S_{\theta} C_{\phi}-C_{\psi} S_{\phi} \\
-S_{\theta} & C_{\theta} S_{\phi} & C_{\theta} C_{\phi} \\
\end{bmatrix},
\label{eq:Rb}$$ where $\Phi_b$=$[\psi,\theta,\phi]^T$ is the triple $ZYX$ yaw-pitch-roll angles. Note that $C(.)$ and $S(.)$ are short notations for $cos(.)$ and $sin(.)$. Let us consider the frame $\Sigma_e$, $O_{2}$- $x_2$ $y_2$ $z_2$, attached to the end-effector of the manipulator, see Fig. \[frames\]. Thus, the position of $\Sigma_e$ with respect to $\Sigma$ is given by $$p_e = p_b + R_b p^b_{eb},
\label{eq:pe}$$ where the vector $p^b_{eb}$ describes the position of $\Sigma_e$ with respect to $\Sigma_b$ expressed in $\Sigma_b$. The orientation of $\Sigma_e$ can be defined by the rotation matrix $$R_e = R_b R^b_e,
\label{eq:Re_cpct}$$ where $R^b_e$ describes the orientation of $\Sigma_e$ w.r.t $\Sigma_b$. The linear velocity $\dot{p}_e$ of $\Sigma_e$ in the world-fixed frame is obtained by the differentiation of (\[eq:pe\]) as $$\dot{p}_e = \dot{p}_b - Skew(R_b p^b_{eb}) \omega_b + R_b \dot{p}^b_{eb},
\label{eq:pde}$$ where $Skew(.)$ is the $(3 \times 3)$ skew-symmetric matrix operator [@spong2006robot], while $\omega_b$ is the angular velocity of the quadrotor expressed in $\Sigma$. The angular velocity $\omega_e$ of $\Sigma_e$ is expressed as $$\omega_e = \omega_b + R_b \omega^b_{eb},
\label{eq:we}$$ where $\omega^b_{eb}$ is the angular velocity of the end-effector relative to $\Sigma_b$ and is expressed in $\Sigma_b$.
Let $\Theta = [\theta_1, \theta_2]^T$ be the $(2 \times 1)$ vector of joint angles of the manipulator. The $(6 \times 1)$ vector of the generalized velocity of the end-effector with respect to $\Sigma_b$, $v^b_{eb} = [\dot{p}^{bT}_{eb},\omega^{bT}_{eb}]^T$, can be expressed in terms of the joint velocities $\dot{\Theta}$ via the manipulator Jacobian $J^b_{eb}$ [@Tsai], such that $$v^b_{eb} = J^b_{eb} \dot{\Theta}.
\label{eq:vbeb}$$
From (\[eq:pde\]) and (\[eq:we\]), the generalized end-effector velocity, $v_e = [\dot{p}^T_e, \omega^T_e]^T$, can be expressed as
$$v_e = J_b v_b + J_{eb} \dot{\Theta},
\label{eq:ve}$$
where $v_b = [\dot{p}^T_b,\omega^T_b]^T$, $J_b= \begin{bmatrix}
I_3 & -Skew(R_b p^b_{eb})\\
O_3 & I_3
\end{bmatrix},$ $J_{eb}= \begin{bmatrix}
R_b & O_3\\
O_3 & R_b
\end{bmatrix} J^b_{eb}$,\
where $I_m$ and $O_m$ denote $(m \times m)$ identity and $(m \times m)$ null matrices, respectively. If the attitude of the vehicle is expressed in terms of yaw-pitch-roll angles, then (\[eq:ve\]) will be $$v_e = J_b Q_b \chi_b + J_{eb} \dot{\Theta},
\label{eq:veph}$$ with $\chi_b= \begin{bmatrix}
p_b \\
\Phi_b
\end{bmatrix},$ $Q_b= \begin{bmatrix}
I_3 & O_3\\
O_3 & T_b
\end{bmatrix},$ where $T_b$ describes the transformation matrix between the angular velocity $\omega_b$ and the time derivative of Euler angles $\dot{\Phi}_b$, and it is given as $$T_b(\Phi_b)= \begin{bmatrix}
0 & -S(\psi) & C(\psi) C(\theta) \\
0 & C(\psi) & S(\psi) C(\theta) \\
1 & 0 & -S(\theta) \\
\end{bmatrix}.
\label{eq:Tb}$$
Since the vehicle is an under-actuated system, i.e., only $4$ independent control inputs are available for the 6-DOF system, the position and the yaw angle are usually the controlled variables. The pitch and roll angles are used as intermediate control inputs to control the horizontal position. Hence, it is worth rewriting the vector $\chi_b$ as follows $
\chi_b= \begin{bmatrix}
\eta_b \\
\sigma_b
\end{bmatrix},
$ $
\eta_b= \begin{bmatrix}
p_b \\
\psi
\end{bmatrix},
$ $
\sigma_b= \begin{bmatrix}
\theta \\
\phi
\end{bmatrix}.
$
Thus, the differential kinematics (\[eq:veph\]) will be $$\begin{aligned}
v_e &= J_{\eta} \dot{\eta}_b + J_{\sigma} \dot{\sigma}_b + J_{eb} \dot{\Theta}\\
&=J_{\zeta} \dot{\zeta} + J_{\sigma} \dot{\sigma}_b,
\end{aligned}
\label{eq:vediv}$$ where $\zeta = [\eta_b^T,\Theta^T]^T$ is the vector of the controlled variables, $J_{\eta}$ is composed by the first 4 columns of $J_b Q_b$, $J_{\sigma}$ is composed by the last 2 columns of $J_b Q_b$, and $J_{\zeta} = [J_{\eta}, J_{eb}]$.
If the end-effector orientation is expressed via a triple of Euler angles, $ZYX$, $\Phi_e$, the differential kinematics (\[eq:vediv\]) can be rewritten in terms of the vector $\dot{\chi}_e = [\dot{p}^T_e, \dot{\Phi}_e^T]^T$ as follows $$\begin{aligned}
\dot{\chi}_e &= Q_e^{-1}(\Phi_e) v_e\\
&=Q_e^{-1}(\Phi_e) [J_{\zeta} \dot{\zeta} + J_{\sigma} \dot{\sigma}_b],
\end{aligned}
\label{eq:xedot}$$ where $Q_e$ is the same as $Q_b$ but it is a function of $\Phi_e$ instead of $\Phi_b$.
Dynamics
--------
The equations of motion of the proposed robot have been derived in details in [@new_quad_manp]. The dynamical model of the quadrotor-manipulator system can be reformulated in a matrix form as $$M(q) \ddot{q} + C(q,\dot{q}) \dot{q} + G(q) + \tau_w + \tau_l=\tau, \qquad \tau = B u,
\label{eq:dyn_gen}$$ where $q=[x, y, z, \psi, \theta, \phi, \theta_1, \theta_2]^T$ $ \in R^{8}$ represents the vector of the generalized coordinates, $M$ $ \in R^{8 \times 8}$ denotes the symmetric and positive definite inertia matrix of the system, $C$ $ \in R^{8 \times 8}$ represents the Coriolis and centrifugal terms, $G$ $ \in R^{8}$ represents the gravity term, $\tau_w $ $ \in R^{8}$ is vector of the external disturbances, $\tau_l $ $ \in R^{8}$ is vector of the contact force effect, $\tau$ $ \in R^{8}$ is the generalized input torques/forces, $u = [F_1, F_2, F_3, F_4, \tau_{m_1}, \tau_{m_2}]^T$ is vector of the actuator inputs, and $B= H N$ is the input matrix which is used to produced the body forces and moments from the actuator inputs. The control matrix, $N$, is given as $$N= \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 \\
\gamma_1 & -\gamma_2 & \gamma_3 & -\gamma_4 & 0 & 0 \\
-d & 0 & d & 0 & 0 & 0 \\
0 & -d & 0 & d & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix},
\label{eq:N}$$
where $\gamma_j=K_{m_j}/K_{f_j}$, and $H$ $ \in R^{8 \times 8}$ is matrix that transforms body input forces to be expressed in $\Sigma$ and is given by $$H= \begin{bmatrix}
R_b & O_3 & O_2 \\
O_3 & T_b^T R_b & O_2 \\
O_{2x3} & O_{2x3} & I_2
\end{bmatrix}.
\label{eq:H}$$ The environment dynamics, contact force, $\tau_l$, can be modeled as following: $$\begin{split}
\tau_l = J^T F_e,\\
F_e= S_c \chi_e + D_c \dot{\chi}_e,
\end{split}
\label{eq:Tl}$$ where $S_c = diag\{S_{c_1}, S_{c_2}, S_{c_3}, S_{c_4}, S_{c_5}, S_{c_6}\}$ and $D_c = diag\{D_{c_1}, D_{c_2}, D_{c_3}, D_{c_4}, D_{c_5}, D_{c_6}\}$ represent the environment stiffness and the environment damping, receptively.
The wind dynamics, $\tau_w$, can be modeled as following [@windmodel; @hsu2011verifying; @andrews2012modeling]:
The average wind velocity is determined by $$V_{wz} = V_{w_{z_0}} \frac{z}{z_0},
\label{eq:Vwz}$$ where $V_{wz}$ is the wind velocity at altitude $z$, $V_{w_{z_0}}$ is the specified (measured) wind velocity at altitude $z_0$. To simulate wind disturbances, it is worth calculating the wind force, $F_{w}$, which influences the platform than the wind velocity. This force can be determined by $$F_{w} = 0.61 * A_e V_{wz}^2,
\label{eq:Fw}$$ where $0.61$ is used to convert wind velocity to pressure, and $A_e$ is the influence effective area which depends on the quadrotor structure and its orientation.
This force can be projected on the axes of frame $\Sigma$ as $$\begin{split}
F_{wx} = f_{wx_1} z^2 sin(\theta) + f_{wx_2} z^2 cos(\theta), \\
F_{wy} = f_{wy_1} z^2 sin(\phi) + f_{wy_2} z^2 cos(\phi),
\end{split}
\label{eq:Fwxy}$$ where $f_{wx_1} = 0.61 * A_{e_1} (\frac{V_{w_{z_0}}}{z_0})^2 cos(\psi_w)$, $f_{wx_2} = 0.61 * A_{e_2} (\frac{V_{w_{z_0}}}{z_0})^2 cos(\psi_w)$, $f_{wy_1} = 0.61 * A_{e_1} (\frac{V_{w_{z_0}}}{z_0})^2 sin(\psi_w)$, $f_{wy_2} = 0.61 * A_{e_2} (\frac{V_{w_{z_0}}}{z_0})^2 sin(\psi_w)$, $\psi_w$ represents the angle of wind direction, and both $A_{e_1}$ and $A_{e_2}$ depend on the quadrotor design parameters.
Controller Design {#se:control}
=================
Control Objectives {#sse:cntrl_obj}
------------------
Our goal is to design of estimation and control system to achieve the following objectives:
1. Robust Stability: The robotic system in Fig. \[fig:mpc\_dob\_fncblg\] is robust and stable against the external disturbances, parameters uncertainties, and noises.
2. Force Estimation: The end-effector contact force has to be estimated with fast response and the estimation error tends to zero as the time tends to $\infty$.
3. $6$-DOF Impedance Control: In the presence of the applied force/desired impedance at the end-effector, the end-effector tracking error tends to zero as time tends to $\infty$.
To this end, we propose a control scheme as shown in Fig. \[fig:mpc\_dob\_fncblg\]. In this control strategy, the system nonlinearities, external disturbances (wind), $\tau_{w}$, and contact force, $\tau_{l}$, are treated as disturbances, $\tau^{dis}$, that will be estimated, $\hat{\tau}^{dis}$, and compensated by the DOb in the inner loop. The system can be now tackled as linear SISO plants. The output of DOb with system measurements of both joint and task spaces variables are used as the inputs to the FTRLS to obtain the end-effector contact force $\hat{F}_e$. The task space impedance control is used in the external loop of DOb and its output is transformed to the joint space through a transformation algorithm.
![Functional block diagram of the proposed control scheme[]{data-label="fig:mpc_dob_fncblg"}](force_dob_fncblg.jpg){width="0.9\columnwidth"}
Disturbance Observer Loop {#sse:dob}
-------------------------
A block diagram of the DOb inner loop is shown in Fig. \[fig:interlp\_DOb\]. In this figure, $M_n$ $\in R^{8 \times 8}$ is the system nominal inertia matrix, $\tau$ and $\tau^{des}$ are the robot and desired inputs, respectively, $P=diag([g_1,...,g_i,...,g_8])$ with $g_i$ is the bandwidth of the $i^{th}$ variable of $q$, $Q(s)=diag([\frac{g_1}{s+g_1} ,...,\frac{g_i}{s+g_i},...,\frac{g_8}{s+g_8}])$ $\in R^{8 \times 8}$ is the matrix of the low pass filter of DOb. The DOb requires velocity measurement. Practically, the velocity have to be fed through a low pass filter, $Q_v(s)=diag([\frac{g_{v_1}}{s+g_{v_1}},...,\frac{g_{v_i}}{s+g_{v_i}},...,\frac{g_{v_8}}{s+g_{v_8}}])$ $\in R^{8 \times 8}$, and with cut-off frequency of $P_v=diag([g_{v_1},...,g_{v_i},...,g_{v_8}])$. $\tau^{dis}$ represents the system disturbances, and $\hat{\tau}^{dis}$ is the estimated disturbances.
If we apply the concept of disturbance observer to the proposed system, the independent coordinate control is possible without considering coupling effect of other coordinates. The coupling terms such as centripetal and Coriolis and gravity terms are considered as disturbance and compensated by feed forward the estimated disturbance torque.
The disturbance $\tau^{dis}$ can be assumed as $$\begin{split}
\tau^{dis} = (M(q) - M_n) \ddot{q} + \tau^d,\\
\tau^d=C(q,\dot{q}) \dot{q} + G(q) + d_{ex}.
\label{eq:sys_dis}
\end{split}$$ Substituting from (\[eq:sys\_dis\]), then (\[eq:dyn\_gen\]) can be rewritten as $$M_n \ddot{q} + \tau^{dis} = \tau.
\label{eq:sys_eq_new}$$ The control input, $\tau$, see Fig. \[fig:interlp\_DOb\], is given as $$\begin{split}
\tau = \frac{1}{(1-Q(s))}[M_n \ddot{q}^{des} - Q(s) M_n \ddot{q}], \\ = M_n \ddot{q}^{des} + M_n P e_v, \quad e_v = \dot{q}^{des} - \dot{q}.
\label{eq:tau}
\end{split}$$ Applying this control law results in $$\begin{split}
M(q) \dot{e}_v + C(q,\dot{q}) e_v + K_v e_v = \delta,
\\ K_v = P M_n,
\label{eq:dyn_gen_dobeD}
\end{split}$$ where $$\begin{split}
\delta = \Delta M(q) \ddot{q}^{des} + C(q,\dot{q}) \dot{q}^{des} + G(q) + d_{ex}, \\ \Delta M(q) = M(q) - M_n.
\label{eq:delta}
\end{split}$$
Stability of this inner loop can be proved as following:
To simplify the analysis, let us ignore the effect of the velocity filter which will be considered later.
Let us use a Lyapunov function as $$V=\frac{1}{2} e_v^T M(q) e_v.
\label{eq:lyp_fun}$$ The time derivative of this function is $$\dot{V}= e_v^T M(q) \dot{e}_v + \frac{1}{2} e_v^T \dot{M}(q) e_v.
\label{eq:lyp_der1}$$ Substituting from (\[eq:dyn\_gen\_dobeD\]), then (\[eq:lyp\_der1\]) becomes $$\dot{V}= e_v^T \delta - e_v^T K_v e_v + \frac{1}{2} e_v^T (\dot{M}(q) - 2 C(q,\dot{q}))e_v.
\label{eq:lyp_der2}$$
To complete this proof, the properties of the dynamic equation of motion (\[eq:dyn\_gen\]) will be utilized. Theses properties are [@from2014vehicle; @spong2006robot]:
$$\lambda_{min} \norm{\nu}^2 \leq \nu^T M(q) \nu \leq \lambda_{max} \norm{\nu}^2,
\label{eq:M_pro}$$
$$\nu^T(\dot{M}(q) - 2 C(q,\dot{q}))\nu=0,
\label{eq:d/dt_pro}$$
where $\nu \in R^8$ represents a $8$-dimensional vector, and $\lambda_{min}$ and $\lambda_{max}$ are positive real constants.
Substituting from (\[eq:d/dt\_pro\]), then (\[eq:lyp\_der2\]) will be $$\dot{V}= e_v^T \delta - e_v^T K_v e_v.
\label{eq:lyp_der3}$$ From property (\[eq:M\_pro\]), one can get $$\dot{V} \leq -\gamma V + \sqrt{\frac{2 V}{\lambda_{min}}} |\delta|, \quad
\gamma = \frac{2K_v}{\lambda_{max}}.
\label{eq:lyp_der4}$$ From the analysis presented in [@sadegh1990stability], (\[eq:lyp\_der4\]) can be reformulated as $$\norm{e_v}_p \leq \frac{1}{\gamma} + \sqrt{\frac{2}{\lambda_{min}}} (\frac{2}{p\gamma})^{\frac{1}{p}} \sqrt{V(0,e_v(0))} \norm{\delta}_p.
\label{eq:ev_final}$$ Thus, the error dynamics is $L_p$ input/output stable with respect to the pair ($\delta$,$e_v$) for all $p \in [1,\infty]$ with the assumption that the system states, $q$ and $\dot{q}$, are bounded.
If one considers the effect of using the velocity filter, then the characteristic equation of the inner loop is $$P_{c_i} = s^2 + g_{v_i} s + \alpha_i g_i g_{v_i},
\label{eq:ch_eq}$$ where $\alpha_i = \frac{M_{n_{ii}}}{M_{ii}}$.
To improve the robustness, the damping coefficient of this equation, which is $ 0.5 \sqrt{\frac{g_i g_{v_i}}{\alpha g_i}}$, should larger than or equal $0.707$ . Therefore, the following inequality $$\alpha g_i \leq \frac{g_{v_i}}{2},
\label{eq:gi_gv}$$ should be hold. Recasting (\[eq:gi\_gv\]) with respect to $K_v$ gives to $$\frac{K_{v_i} }{M_{ii}}\leq \frac{g_{v_i}}{2}.
\label{eq:gi_gv_Kv}$$
Summarizing, (\[eq:lyp\_der4\]) shows that the stability and robustness of the control system is enhanced by increasing $K_v$, i.e., by increasing $M_n$ and $P$, but without violating the robustness constraint given in (\[eq:gi\_gv\_Kv\]).
If the DOb performs well, that is $\hat{\tau}^{dis}$ = $\tau^{dis}$, the dynamics from the DOb loop input $\tau^{des}$ to the output of the system is given as $$M_n \ddot{q}=\tau^{des}.
\label{eq:sys_eq_reduced}$$ Since $M_n$ is assumed to be a diagonal matrix, the system can be considered as a decoupled linear multi SISO systems as $$M_{n_{ii}} \ddot{q}_i=\tau_i^{des},
\label{eq:sys_eq_decoupled}$$ or in the acceleration space as: $$\ddot{q}_i=\ddot{q}_i^{des}.
\label{eq:sys_eq_decoupled_acc}$$
The next step is to design an Impedance tracking based controller in the outer loop for the system of (\[eq:sys\_eq\_decoupled\_acc\]).
![Block diagram of DOb internal loop[]{data-label="fig:interlp_DOb"}](interlp_DOb.jpg){width="0.7\columnwidth"}
Fast Tracking Recursive Least Squares {#sse:sprls}
-------------------------------------
In this part, we develop a technique which utilizes a Fast Tracking Recursive Least Squares (FTRLS) to estimate the contact force with the aid of the DOb linearization capabilities. The FTRLS algorithm is one of the fast online least squares-based identification methods used for the identification of environments with varying dynamic parameters [@hu2014least; @wang2015recursive]. To apply FTRLS, the dynamic equations (\[eq:dyn\_gen\]- \[eq:Fwxy\]) have to be parametrized (i.e., to be product of measurement data regressor and dynamic parameters) as follows:
The system dynamic part, $\tau_{int}= M(q) \ddot{q} + C(q,\dot{q}) \dot{q} + G(q)$, can be rewritten as the product of data regressor, $Y_i(q,\dot{q},\ddot{q})$, and platform parameters, $h_i$. The environment dynamics, $\tau_l$, can be reformulated as $Y_l(q,\dot{q},\ddot{q},\chi_e,\dot{\chi}_e)*h_l$, where, $Y_l=J^T Y_e$, $Y_e$ is a function of the end effector states, ($\chi_e$,$\dot{\chi}_e$), and $h_l$ is the environment parameters $S_c$ and $D_c$. Finally, the wind effect is formulated as $Y_w(z,\theta,\phi) * h_w$, where $h_w$ is the wind parameters. Thus, the total dynamics can be reformulated as $$\begin{split}
\tau = Y * h, \\
Y=[Y_i, Y_l, Y_w], \\
h=[h_i, h_l, h_w]^T,
\label{eq:reg_tot}
\end{split}$$ where $Y$ $\in R^{8 \times 40}$ and $h$ $\in R^{40}$ are the data regressor and parameters vector of (\[eq:dyn\_gen\]), respectively.
The parameter estimation error is $$\tilde{h}(t) = h - \hat{h}(t),
\label{eq:SPRLS_eh}$$ while the estimation error is $$\tilde{\tau}(t) = \tau(t) - Y(t) \hat{h}(t) = Y(t) \tilde{h}(t).
\label{eq:SPRLS_ey}$$ By minimizing a cost function with respect to the parameter estimation error, one can find the time derivative of the estimated parameters vector, $\hat{h}$, as following $$\frac{d}{dt}\hat{h}(t)= R(t) Y^{T}(t) \tilde{\tau}(t),
\label{eq:SPRLS_hd}$$ where $R(t)$ is the parameters’ covariance matrix, and it can be calculated from $$\frac{d}{dt}R^{-1}(t) = -\eta_h(t) R^{-1}(t) + Y^{T}(t) Y(t),
\label{eq:SPRLS_R}$$ where $\eta_h$ is the forgetting factor, and it is given as $$\eta_h(t) = \eta_h^{min} + (1-\eta_h^{min}) 2^{(-NINT(\gamma_g \norm{\tilde{\tau}(t)}^2))},
\label{eq:SPRLS_Rg}$$ where $\eta_h^{min}$ is a constant representing the minimum forgetting factor, $NINT(.)$ is the round-off operator, and $\gamma_g$ is a design constant. This adaptive formulation of the forgetting factor enables the RLS to track the non-stationary parameters to be estimated.
The convergence/stability ($\tilde{h}(t) \longrightarrow 0$) proof of this algorithm can be implemented as following:
Let us assume the Lyapunov function as $$V(t) = \tilde{h}^T(t) R^{-1}(t) \tilde{h}(t).
\label{eq:SPRLS_lyap}$$ If $R^{-1}(t)$ is chosen to be positive definite, then $V(t)$ will be positive definite. To prove the positive definiteness of $R^{-1}(t)$, let us use the solution of the differential equation (\[eq:SPRLS\_R\]) which is $$\begin{gathered}
R^{-1}(t) = \Phi_h(t,t_0) R^{-1}(t_0) \Phi_h^T(t,t_0) + \\ \int_{t_0}^{t} \Phi_h(t,\varrho) Y^T(\varrho) Y(\varrho) \Phi_h^T(t,\varrho) d\varrho,
\label{eq:SPRLS_R_sol}\end{gathered}$$ where $\Phi_h^T(t,t_0)$ is the state transition matrix of a system described by $\dot{\upsilon}(t) = - \frac{1}{2}\eta_h \upsilon(t)$. Thus, by choosing $R^{-1}(t_0) > 0$, then the first term in (\[eq:SPRLS\_R\_sol\]) will be positive definite. The second term is also positive definite. As a result, the proposed covariance matrix update formula is positive definite, and thus, the chosen Lyapunov function (\[eq:SPRLS\_lyap\]) is positive definite.
The time derivative of Lyapunov function is $$\dot{V}(t) = 2 \tilde{h}^T R^{-1} \dot{\tilde{h}} + \tilde{h}^T \dot{R^{-1}} \tilde{h}.
\label{eq:SPRLS_lyapd}$$ However, by differentiating both sides of (\[eq:SPRLS\_eh\]) with respect to time, one can find that $\dot{\tilde{h}} = - \dot{\hat{h}}$, by substituting from the proposed formula of $\dot{\hat{h}}$ (\[eq:SPRLS\_hd\]) and (\[eq:SPRLS\_ey\]), then $$\dot{\tilde{h}} = - R Y^T Y \tilde{h}.
\label{eq:SPRLS_lyapd2}$$ Substituting from (\[eq:SPRLS\_lyapd2\]) in (\[eq:SPRLS\_lyapd\]), then $\dot{V}(t)$ will be $$\dot{V}(t) = -\tilde{h}^T [2 Y^T Y - \dot{R^{-1}}] \tilde{h}.
\label{eq:SPRLS_lyapd3}$$ Substituting from the proposed formula (\[eq:SPRLS\_R\]) for $\dot{R^{-1}}$ into (\[eq:SPRLS\_lyapd3\]), then $$\dot{V}(t) = -\tilde{h}^T [Y^T Y +\eta_h(t) R^{-1}(t)] \tilde{h}.
\label{eq:SPRLS_lyapd4}$$ Thus, the time derivative of $V(t)$ is negative definite which ensures the asymptotic stability of the estimation error ($\tilde{h}(t) \longrightarrow 0$ as $t \longrightarrow \infty$)
Finally, for both teleoperation impedance control purposes, the user can calculate the estimated environment impedance, contact force, from $$\begin{split}
\hat{\tau}_l = Y_l \hat{h}_l,\\
\hat{F}_e = Y_e \hat{h}_l.
\end{split}
\label{eq:SPRLS_thl}$$ Therefore, unlike the current the developed schemes, with this technique, one can isolate and estimate the end-effector contact force apart from the whole estimated forces in the systems.
Impedance Control {#sse:impd_ctrl}
-----------------
The objective of the impedance control is to regulate the end-effector interaction force, which may vary due to the uncertainty in the location of the interaction point and/or the structural properties of the environment, besides achieving task space trajectory tracking. The linear impedance control is designed in the task space. This is based on the linearization effect of the designed DOb in the joint space. The desired acceleration in the task space, $\ddot{\chi}^{des}_e$, can be calculated from $$\ddot{\chi}^{des}_e = \ddot{\chi}_{e,r} + S_{c,d} (\chi_{e,r} - \chi_{e}) + D_{c,d} (\dot{\chi}_{e,r} - \dot{\chi}_{e}) - \hat{F}_e,
\label{eq:xe_des}$$ where $S_{c,d}$ and $D_{c,d}$ are the desired values of $S_{c}$ and $D_{c}$ respectively, which determine the desired impedance that the end-effector will apply to the environment.
A complete and detailed block diagram of the proposed control scheme is illustrated in Fig. \[fig:detail\_impd\_dob\]. Quadrotor position and yaw rotation are the controlled variables, while pitch and roll angles are used as intermediate control inputs to achieve the desired $x$ and $y$. Therefore, the proposed scheme has two DOb-based controllers include one for $\zeta=[x, y, z, \psi, \theta_1, \theta_2]^T$ (with $M_{n_\zeta}$, $P_{\zeta}$, $Q_{\zeta}$) and the other for $\sigma_b = [\theta, \phi]^T$ (with $M_{n_\sigma}$, $P_{\sigma}$, $Q_{\sigma}$). The desired 6-DOF trajectories for the end-effector’s ($\chi_{e,r}$), their actual values calculated by the forward kinematics, and the estimated end-effector force, are applied to the impedance control algorithm, $K_e$ that is given in (\[eq:xe\_des\]). Then, a transformation from task space to joint space is done by using (\[eq:qdd\_des\]) to get $\ddot{\zeta}^{des}$. The desired acceleration in the joint space, $\ddot{\zeta}^{des}$, can be calculated by differentiating (\[eq:xedot\]) with respect to time as $$\ddot{\zeta}^{des} = J_{\zeta}^{-1} (Q_e \ddot{\chi}^{des}_e + \dot{Q}_e \dot{\chi}_{e,r} - \dot{J}_{\zeta} \dot{\zeta} - J_{\sigma} \ddot{\sigma}_b - \dot{J}_{\sigma} \dot{\sigma}_b),
\label{eq:qdd_des}$$
The desired acceleration in quadrotor/joint space, $\ddot{\zeta}^{des}$, is then applied to the DOb of the independent coordinates, $\zeta$, to produce $\tau_{\zeta}$. The desired values for the intermediate DOb controller, $\sigma_{b,r}$, are obtained from the output of position controller, $\tau_{\zeta}$, through the following simplified nonholonomic constraints relation $$\sigma_{b,r} = \frac{1}{\tau_{\zeta}(3)}
\begin{bmatrix}
C(\psi) & S(\psi) \\
S(\psi) & -C(\psi)
\end{bmatrix}
\begin{bmatrix}
\tau_{\zeta}(1) \\
\tau_{\zeta}(2)
\end{bmatrix}.
\label{eq:sigmad}$$ The external controller of second DOb controller, /$\tau_{\sigma}$, is used as a PD controller with velocity feedback, $K_{\sigma}$, as following: $$\ddot{\sigma}^{des} = K_{p_{\sigma}} (\sigma_{b,r} - \sigma_{b}) - K_{d_{\sigma}} \dot{\sigma}_{b}.
\label{eq:K_sig}$$ After that, $\ddot{\sigma}^{des}$ is applied to the second DOb to generate $\tau_{\sigma}$.
It is konwn that the response of the $\sigma$ controller must be much faster than that of the position controller. This can be achieved by the tuning parameters of both DOb and PD of $\sigma$-controller.
The output of two controllers are converted to the required forces/torques applied to quadrotor/manipulator by $$u= B_6^{-1} \begin{bmatrix}
\tau_{\zeta}(3,4)\\
\tau_{\sigma}\\
\tau_{\zeta}(5,6)
\end{bmatrix},
\label{eq:ufinal}$$ where $B_6$ $\in R^{6 \times 6}$ is part of $B$ matrix and it is given by $B_6 = B(3:8,1:6)$.
{width="90.00000%"}
Simulation Results {#se:sim}
==================
In this section, the presented aerial manipulation robot model with the proposed control technique is implemented in MATLAB/SIMULINK.
Simulation Environment
----------------------
For a more realistic simulation studies, the following setup have been made:
- Linear and angular position and orientation of the quadrotor are available at rate of $1$ KHz. In [@achtelik2011onboard], a scheme is proposed to measure and estimate the vehicle (Asctec Pelican Quadrotor) states based on IMU and Onboard camera in both indoors and outdoors.
- The joints angles are measured at rate of $1$ KHz and angular velocities are estimated by a low pass filter.
- The measured signal are affected by a normally distributed measurement noise with mean of $10^{-3}$ and standard deviation of $5 \times 10^{-3}$.
- $1$ KHz Control loop.
- To test the robustness against model uncertainties, a step disturbance is applied at $15$ s to both the inertia matrix, $M(q)$, and the control matrix, $N$, (Actuators’ losses) with $10 \% $ error.
Par. Value Unit Par. Value Unit
----------- ----------------------- --------------------- ----------- ----------------------- ---------------------
$m$ $1$ $kg$ $L_2$ $85\times10^{-3}$ $m$
$d$ $223 \times 10^{-3}$ $m$ $m_0$ $30\times10^{-3}$ $kg$
$I_x$ $13.2 \times 10^{-3}$ $N.m.s^2$ $m_1$ $55\times10^{-3}$ $kg$
$I_y$ $12.5 \times 10^{-3}$ $N.m.s^2$ $m_2$ $112\times10^{-3}$ $kg$
$I_z$ $23.5 \times 10^{-3}$ $N.m.s^2$ $I_r$ $33.2 \times 10^{-6}$ $N.m .s^2$
$L_0$ $30\times10^{-3}$ $m$ $L_1$ $70\times10^{-3}$ $m$
$K_{F_1}$ $1.6\times10^{-5}$ $kg.m.rad^{-2}$ $K_{F_2}$ $1.2\times10^{-5}$ $kg.m.rad^{-2}$
$K_{F_3}$ $1.7\times10^{-5}$ $kg.m.rad^{-2}$ $K_{F_4}$ $1.5\times10^{-5}$ $kg.m.rad^{-2}$
$K_{M_1}$ $3.9\times10^{-7}$ $kg.m^{2}.rad^{-2}$ $K_{M_2}$ $2.8\times10^{-7}$ $kg.m^{2}.rad^{-2}$
$K_{M_3}$ $4.4\times10^{-7}$ $kg.m^{2}.rad^{-2}$ $K_{M_4}$ $3.1\times10^{-7}$ $kg.m^{2}.rad^{-2}$
: System Parameters[]{data-label="sys_par"}
The desired trajectories of the end-effector are generated to follow a circular helix, while its orientation follows quintic polynomial trajectories [@spong2006robot]. Parameters of the proposed algorithm are presented in Table \[tab:DOb\_par\]. The controller is tested to achieve task space trajectory tracking under the effect of the contact force, wind disturbances, and measurement noise.
$Parameter$ $Value$ $Par.$ $Val.$
--------------------------------- ------------------------------------------- --------------- ------------
$M_{n_\zeta}$ $diag\{0.02, 0.02, 2, 0.05, 0.01, 0.01\}$ $A_{e_1}$ $0.16$
$S_{c,d}$ $diag\{20, 20, 30, 50, 100, 500\}$ $D_{c}$ $0.01 I_6$
$D_{c,d}$ $diag\{15, 15, 25, 100, 100, 100\}$ $g_{v_i}$ $100$
$M_{n_\sigma}$ $diag\{0.05, 0.05\}$ $A_{e_2}$ $0.032$
$\eta^{min}_{h}$ $0.8$ $\gamma_{g}$ $5$
$S_{c}$ $0.1 I_6$ $z_0$ $1$
$K_{p_\sigma}$ / $K_{d_\sigma}$ $20 I_2$ $V_{w_{z_0}}$ $3$
: Controller parameters[]{data-label="tab:DOb_par"}
Estimation of the End-effector Contact Force
--------------------------------------------
Fig. \[fig:Fe\] shows the response of the proposed algorithm to estimate the environment effect/end-effector contact force. From this figure, it is possible to recognize that the norm of the end-effector generalized force has maximum value of $1$ N/N.m at the beginning of operation due to the time taken by the DOb to estimate the system dynamics and external forces. This initial time is about $3$ s. After that period the error norm decreases gradually. The norm of estimation error in both $x$ and $y$ directions have maximum value of $0.03$ N with sinusoidal shape which is due to the sinusoidal motion in theses axes. While the norm value in $z$ direction have maximum value of $0.005$ N. In both $\phi$ and $\theta$ directions, the norms reach value of $0.007$ N.m. The maximum value of the norm in the $\psi$ direction is about $0.0015$ N.m. Thus, it is possible to appreciate the estimation performance of the end-effector generalized forces. Consequently, one can contend that the second control objective is achieved.
![Error norm of estimation of the environment dynamics/contact force[]{data-label="fig:Fe"}](Fnorm_TS_Inv_Kin.pdf){width="0.7\columnwidth"}
Impedance Control {#impedance-control}
-----------------
Fig. \[fig:inv\_kin\_pp\_results\] shows the response of the system in the task space (the actual end-effector position and orientation can be found from the forward kinematics). From this figure, it is possible to recognize that the controller has good tracking of the desired trajectories of the end-effector (i.e., the tracking error tends to zero as with time). Moreover, it is clear that the capability of the proposed technique to recover the trajectory tracking in the presence of parameters uncertainties which are applied at instant $15$ s. As we see, in the $x$, $y$, $z$, and $\psi$ directions, there is no effect on the tracking. However, in the $\theta$ and $\phi$ directions, the uncertainty effect appears and the controller can recover the tracking quickly.
[c]{}\
Fig. \[fig:DOb\_3d\] shows the motion of the end-effector in the 3D dimension (The markers represent the orientation ). These results show that the proposed impedance motion control scheme provides a robust performance to track the desired end-effector trajectories as well as achieve the desired compliance/impedance effect on the environment taken into consideration the external disturbances and noises. As a result, one can claim that the three control objectives are achieved.
![3D trajectory of the end-effector pose (The marker represents the end-effector orientation; Green, Blue, and Red for x-,y, and z-axis, respectively)[]{data-label="fig:DOb_3d"}](3D_TS_Inv_Kin.pdf){width="0.5\columnwidth"}
Conclusion {#se:concl}
==========
The problem of the contact force estimation and impedance control of an aerial manipulation robot is presented with a new solution. A brief presented of the system modeling is given. DOb-based system linearization is implemented in the quadrotor/joint space. A DOb is used in the inner loop is to achieve robust linear input/output behavior of the system by compensating disturbances, measurement noise, and uncertainties. Contact force/environment impedance is estimated based on FTRLS and DOb which appear efficient estimation results and stability guarantee. Then, a linear impedance control is designed and implemented in the task space. The inverse kinematics problem is solved by utilization of the system Jacobian. The controller is tested to achieve trajectory tracking under the effect of external wind disturbances, parameters uncertainty, and measurement noise. Numerical results enlighten the efficiency of the proposed control scheme.
|
---
author:
- |
Alexander Shashkin, Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia\
Sergey Kravchenko, Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A.
title: 'Quantum phase transitions in two-dimensional electron systems'
---
Strongly and Weakly Interacting 2D Electron Systems
===================================================
Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, $r_s^*=E_{ee}/E_F$. Assuming that the effective electron mass is equal to the band mass, the interaction parameter $r_s^*$ in the single-valley case reduces to the Wigner-Seitz radius, $r_s=1/(\pi
n_s)^{1/2}a_B$, and therefore increases as the electron density, $n_s$, decreases (here $a_B$ is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [@SK:wigner34], a ferromagnetic Fermi liquid with spontaneous spin ordering [@SK:stoner46], a paramagnetic Fermi liquid [@SK:landau57], etc. In the strongly-interacting limit ($r_s\gg1$), no analytical theory has been developed to date. According to numerical simulations [@SK:tanatar89], Wigner crystallization is expected in a very dilute regime, when $r_s$ reaches approximately 35. Refined numerical simulations [@SK:attaccalite02] have predicted that prior to the crystallization, in the range of the interaction parameter $25\leq
r_s\leq35$, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, $r_s\sim1$, the electron liquid is expected to be paramagnetic, with the effective mass, $m$, and Landé $g$ factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.
In real 2D electron systems, the inherent disorder leads to a drastic change of the above picture, which significantly complicates the problem. According to the scaling theory of localization [@SK:abrahams79], all electrons in a disordered infinite noninteracting 2D system become localized at zero temperature and zero magnetic field. At finite temperatures, regimes of strong and weak localizations are distinguished: (i) if the conductivity of the 2D electron layer is activated, the resistivity diverges exponentially as $T\rightarrow0$; and (ii) in the opposite limit of weak localization the resistivity increases logarithmically with decreasing temperature, an effect originating from the increased probability of electron backscattering from impurities to the starting point. Interestingly, the incorporation of weak interactions ($r_s<1$) between the electrons promotes the localization [@SK:altshuler80]. However, for weak disorder and $r_s\geq1$ a possible metallic ground state was predicted [@SK:finkelstein83].
In view of the competition between the interactions and disorder, high- and low-disorder limits can be considered. In highly-disordered electron systems, the range of low densities is not accessible as the strong (Anderson) localization sets in. This corresponds to the weakly-interacting limit in which an insulating ground state is expected. The case of low-disordered electron systems is much more interesting because low electron densities corresponding to the strongly-interacting limit become accessible. According to the renormalization group analysis for multi-valley 2D systems [@SK:punnoose05], strong electron-electron interactions can stabilize the metallic ground state, leading to the existence of a metal-insulator transition in zero magnetic field.
In quantizing magnetic fields, the interaction strength is characterized by the ratio between the Coulomb energy and the cyclotron splitting. In the ultra-quantum limit, it is similar to the interaction parameter $r_s^*$. Within the concept of single-parameter scaling for noninteracting 2D electrons [@SK:pruisken88], there is only one extended state in the Landau level, and the localization length diverges at the center of the Landau level [@SK:iordansky82]. For consistency with the scaling theory of localization in zero magnetic field, it was predicted that extended states in the Landau levels cannot disappear discontinuously with decreasing magnetic field but must “float up” (move up in energy) indefinitely in the limit [@SK:khmelnitskii84] of $B\rightarrow0$. The corresponding phase diagram plotted in disorder versus inverse filling factor ($1/\nu=eB/hcn_s$) plane is known as the global phase diagram for the quantum Hall effect (QHE) [@SK:kivelson92]. As long as no merging of the extended states was considered to occur, their piercing of the Fermi level was predicted to cause quantization of the Hall conductivity in weak magnetic fields [@SK:khmelnitskii92]. The case of strongly interacting 2D electrons in the quantum Hall regime has not been considered theoretically. In the very dilute regime, there are theoretical predictions that Wigner crystallization is promoted in the presence of a magnetic field (see, e.g., Ref. [@SK:lozovik75]).
In this chapter, attention is focused on experimental results obtained in low-disordered strongly interacting 2D electron systems, in particular, (100)-silicon metal-oxide-semiconductor field-effect transistors (MOSFETs). Due to the relatively large effective mass, relatively small dielectric constant, and the presence of two valleys in the spectrum, the interaction parameter in silicon MOSFETs is an order of magnitude bigger at the same electron density than in the 2D electron system in GaAs/AlGaAs heterostructures. Except at very low electron densities, the latter electron system can be considered weakly interacting. It is worth noting that the observed effects of strong electron-electron interactions are more pronounced in silicon MOSFETs compared to GaAs/AlGaAs heterostructures, although the fractional QHE, which is usually attributed to electron-electron interactions, has not been reliably established in silicon MOSFETs.
Proof of the Existence of Extended States in the Landau Levels {#SK:proof}
==============================================================
In a magnetically quantized 2D electron system, the Landau levels bend up at the sample edges due to the confining potential, and edge channels are formed where these intersect the Fermi energy (see, e.g., Ref. [@SK:halperin82]). There arises a natural question as to whether the current in the quantum Hall state[^1] flows in the bulk or at the edges of the sample. Although the Hall conductivity $\sigma_{xy}$ was not directly measured in early experiments on the QHE, it seemed obvious that this value corresponds to the Hall resistivity $\rho_{xy}$, in agreement with the concept of currents that flow in the bulk [@SK:QHE87]; it stands to reason that finite $\sigma_{xy}$ would give evidence for the existence of extended states in the Landau levels [@SK:halperin82; @SK:levine83]. This concept was challenged by the edge current model [@SK:buttiker88]. In the latter approach extended states in the bulk are not crucial and the problem of current distributions in the QHE is reduced to a one-dimensional task in terms of transmission and reflection coefficients as defined by the backscattering current at the Fermi level between the edges. Importantly, if the edge current contributes significantly to the net current, conductivity/resistivity tensor inversion is not justified, because the conductivities $\sigma_{xx}$ and $\sigma_{xy}$ are related to the bulk of the 2D electron system. That is to say, a possible shunting effect of the edge currents in the Hall bar (rectangular) geometry makes it impossible to extract the value $\sigma_{xy}$ from the magnetotransport data for $\rho_{xx}$ and $\rho_{xy}$.
To verify whether or not the Hall conductivity is quantized, direct measurements of $\sigma_{xy}$ are necessary, excluding the shunting effect of the edge currents. Being equivalent to Laughlin’s *gedanken* experiment [@SK:laughlin81], such measurements were realized using the Corbino (ring) geometry which allows separation of the bulk contribution to the net current (see, e.g., Ref. [@SK:dolgopolov92d]). A Hall charge transfer below the Fermi level between the borders of a Corbino sample is induced by a magnetic field sweep through the generated azimuthal electric field. If the dissipative conductivity $\sigma_{xx}\rightarrow0$, no discharge occurs, allowing determination of the transferred charge, $Q=\sigma_{xy}\pi
r_{\rm{eff}}^2c^{-1}\delta B$, where $r_{\rm{eff}}$ is the effective radius. The induced voltage, $V=Q/C$, which is restricted due to a large shunting capacitance, $C$, changes linearly with magnetic field with a slope determined by $\sigma_{xy}$ in the quantum Hall states until the dissipationless quantum Hall state breaks down (Fig. \[SK:charge\]). The fact that the quantization accuracy of $\sigma_{xy}$ (about 1%) is worse compared to that of $\rho_{xy}$ may be attributed to non-constancy of the effective area in not very homogeneous samples. Thus, the Hall current in the QHE flows not only at the edges but also in the bulk of the 2D electron system through the extended states in the filled Landau levels.
The finite Hall conductivity measured in the Corbino geometry in the arrangement of Laughlin’s *gedanken* experiment establishes the existence of extended states in the Landau levels for both strongly and weakly interacting 2D electron systems. Note that the insignificance of edge-channel effects in transport experiments is verified in the usual way by coincidence of the results obtained in Hall bar and Corbino geometries.
Metal-insulator Transitions in Perpendicular Magnetic Fields
============================================================
Metal-insulator transitions were studied for the quantum Hall phases and the insulating phase at low electron densities. The insulating phase was attributed to possible formation of a pinned Wigner crystal [@SK:andrei88; @SK:pudalov90; @SK:santos92a]. However, floating-up of the extended states relative to the Landau level centers and a close similarity of all insulating phases have been found experimentally [@SK:shashkin93; @SK:shashkin94a; @SK:shashkin94b]. Thus, the experimental results excluded the formation of a pinned Wigner crystal in available samples, but supported the existence of a metallic state in zero field. It was also found that the bandwidth of the extended states in the Landau levels is finite, which is in contradiction to scaling arguments. Strangely, the latter experimental result has not attracted much of theorists’ attention.
Floating-up of Extended States {#SK:floating-up}
------------------------------
The first experimental results on the metal-insulator phase diagram at low temperatures in low-disordered silicon MOSFETs [@SK:shashkin93] already revealed discrepancies with the theory (Fig. \[SK:floating\](a)). In that paper, a somewhat arbitrary criterion for the longitudinal conductivity, $\sigma_{xx}=e^2/20h$, was used to map out the phase boundary that corresponds to the Anderson transition to the regime of strong localization. However, first, the phase boundary was shown to be insensitive to the choice of the cutoff value (see, e.g., Ref. [@SK:dolgopolov92b]). Second, that particular cutoff value is consistent with the results obtained for quantum Hall states by a vanishing activation energy combined with a vanishing nonlinearity of current-voltage characteristics when extrapolated from the insulating phase [@SK:shashkin94a].[^2] The metallic phase surrounds each insulating phase as characterized by the dimensionless Hall conductivity, $\sigma_{xy}h/e^2$, that counts the number of quantum levels below the Fermi level.[^3] This indicates that the extended states indeed do not disappear discontinuously. Instead, with decreasing magnetic field they float up in energy relative to the Landau level centers and merge forming a metallic state in the limit of $B=0$ (for more on this, see Sec. \[SK:zero\]). This contradicts the theoretical scenario that in the limit of zero magnetic field the extended states should float up indefinitely in energy [@SK:khmelnitskii84] leading to an insulating ground state. Besides, the experimental phase boundary at low electron densities oscillates as a function of $B$ with minima corresponding to integer filling factors. The phase boundary oscillations manifest themselves in that the magnetoresistance at electron densities near the $B=0$ metal-insulator transition oscillates with an amplitude that diverges as $T\rightarrow0$ [@SK:pudalov90]. The regions in which the magnetoresistance diverges are referred to as the reentrant insulating phase.
The topology of the observed metal-insulator phase diagram[^4] is robust, being insensitive to the method for spotting the phase boundary [@SK:shashkin94a; @SK:kravchenko95b] and to the choice of 2D carrier system [@SK:hilke00]. This robustness was verified using a criterion of vanishing activation energy and vanishing nonlinearity of current-voltage characteristics as extrapolated from the insulating phase, allowing more accurate determination of the Anderson transition [@SK:shashkin94a]. A method that had been suggested in Ref. [@SK:glozman95] was also applied for similar silicon MOSFETs [@SK:kravchenko95b]. The extended states were studied by tracing maxima in the longitudinal conductivity in the ($B,n_s$) plane (Fig. \[SK:floating\](b)) and good agreement with the aforementioned results was found. A similar merging of at least the two lowest extended states was observed in a more strongly disordered 2D hole system in a Ge/SiGe quantum well [@SK:hilke00] (Fig. \[SK:dultz\](a)). The extended states were associated either with maxima in $\rho_{xx}$ and/or $d\rho_{xy}/dB$, or with crossing points of $\rho_{xx}$ at different temperatures. It is noteworthy that a bad combination of the criterion for determining the phase boundary and the 2D carrier system under study may lead to a failure in mapping out the phase diagram down to relatively weak magnetic fields. In Ref. [@SK:glozman95], extended states were studied by measuring maxima in the longitudinal conductivity in the ($B,n_s$) plane for the strongly-disordered 2D electron system in GaAs/AlGaAs heterostructures (Fig. \[SK:dultz\](b)). Because of strong damping of the Shubnikov-de Haas oscillations in low magnetic fields, the desired region on the phase diagram below 2 T was not accessible in that experiment. This invalidates the claim of Glozman *et al*. [@SK:glozman95] that the extended states do not merge. The behavior of the lowest extended state in Fig. \[SK:dultz\](b), which Glozman *et al*. [@SK:glozman95] claim to float up above the Fermi level as $B\rightarrow0$, simply reflects the occurrence of a phase boundary oscillation minimum at filling factor $\nu=2$, similar to both the minimum at $\nu=1$ in Fig. \[SK:dultz\](a) and to the case of silicon MOSFETs (Fig. \[SK:floating\]). Such a minimum manifests itself in that there exists a minimum in $\rho_{xx}$ at integer $\nu\ge1$ that is straddled by the insulating phase.
To this end, all available data for the metal-insulator phase diagrams agree well with each other, except those in the vicinity of $B=0$. In weak magnetic fields, experimental results obtained in 2D electron systems with high disorder are not method-independent. Glozman *et al*. [@SK:glozman95] found that the cutoff criterion yields basically a flat phase boundary towards $B=0$, which is in agreement with the data for silicon MOSFETs (Fig. \[SK:floating\](a)). On the contrary, Hilke et al. [@SK:hilke00] employed the method based on temperature dependencies of $\rho_{xx}$ and obtained a turn up on the phase boundary in Fig. \[SK:dultz\](a). Note that the validity of the data for the lowest extended state at magnetic fields $\leq1.5$ T in Fig. \[SK:dultz\](a) is questionable because the weak temperature dependencies of $\rho_{xx}$ as analyzed by Hilke *et al*. [@SK:hilke00] cannot be related to either an insulator or a metal.
As a matter of fact, the weak-field problem, whether or not there is an indefinite rise of the phase boundary as $B\rightarrow0$, is a problem of the existence of a metal-insulator transition at $B=0$ and $T=0$. In dilute 2D electron systems with low enough disorder, the resistivity, $\rho$, strongly drops with decreasing temperature [@SK:sarachik99; @SK:abrahams01], providing an independent way of facing the issue. Given strong temperature dependencies of $\rho$, those with $d\rho/dT>0$ ($d\rho/dT<0$) can be associated with a metallic (insulating) phase [@SK:sarachik99; @SK:abrahams01]. If extrapolation of the temperature dependencies of $\rho$ to $T=0$ is valid, the curve with $d\rho/dT=0$ should correspond to the metal-insulator transition (see Sec. \[SK:zero\]). As long as in more-disordered 2D carrier systems the metallic ($d\rho/dT>0$) behavior is suppressed (see, e.g., Refs. [@SK:hanein98; @SK:pudalov01]) or disappears entirely, it is definitely incorrect to extrapolate those weak temperature dependencies of $\rho$ to $T=0$ with the aim to distinguish between insulator and metal.
Another point at which one can compare experiment and theory is the oscillating behavior of the phase boundary that restricts the insulating phase with $\sigma_{xy}=0$ (see, e.g., Fig. \[SK:floating\]). Note that the oscillations persist down to the magnetic fields corresponding to the fillings of more than one Landau level. The oscillation period includes the following stages. With decreasing magnetic field the lowest extended states follow the Landau level, float up in energy relative to its center, and merge with extended states in the next quantum level. No merging was present in the original theoretical considerations [@SK:khmelnitskii84; @SK:kivelson92; @SK:khmelnitskii92], leading to discrepancies between experiment and theory. Recently, theoretical efforts have been concentrated on modifications of the global phase diagram for the QHE to reach topological compatibility with the observed metal-insulator phase diagram. Although floating and/or merging of the extended states can be obtained in the calculations, the oscillations of the phase boundary at low electron densities have not yet been described theoretically.
Similarity of the Insulating Phase and Quantum Hall Phases {#SK:similarity}
----------------------------------------------------------
The insulating phase at low electron densities was considered to be a possible candidate for a pinned Wigner crystal. It was argued that its aforementioned reentrant behavior is a consequence of the competition between the QHE and the pinned Wigner crystal [@SK:pudalov90]. Another supporting argument was strongly nonlinear current-voltage characteristics in the insulating phase which were attributed to depinning of the Wigner crystal. Similar features of the insulating phase in a 2D electron (near $\nu=1/5$) [@SK:andrei88] and 2D hole (near $\nu=1/3$) [@SK:santos92a] systems in GaAs/AlGaAs heterostructures with relatively low disorder were also attributed to a pinned Wigner crystal which is interrupted by the fractional quantum Hall state. An alternative scenario was discussed in terms of percolation metal-insulator transition [@SK:dolgopolov92b; @SK:dolgopolov92a; @SK:dolgopolov92c]. To distinguish between the two scenarios, the behavior of activation energy and current-voltage characteristics in the insulating phase was studied and compared to that in quantum Hall phases [@SK:shashkin94a; @SK:shashkin94b].
In contrast to the low-density insulating phase, the way of determining the current-voltage characteristics of the quantum Hall phases is different for Corbino and Hall bar geometries. In the former the dissipationless Hall current does not contribute to the dissipative current that is proportional to $\sigma_{xx}$, allowing straightforward measurements of current-voltage curves for all insulating phases. In the latter the two current channels are connected through edge channels (see Sec. \[SK:proof\]), and current-voltage characteristics correspond to quantum-Hall-effect breakdown curves. The dissipative backscattering current, $I$, that flows between opposite edge channels is balanced by the Hall current in the filled Landau levels associated with the longitudinal voltage, $V_{xx}$. As long as $\sigma_{xx}\ll\sigma_{xy}$, the quantized value of $\sigma_{xy}$ is a factor that allows determination of $I=\sigma_{xy}V_{xx}$ and the Hall voltage, $V=I_{sd}/\sigma_{xy}$, from the experimental breakdown dependence of $V_{xx}$ on source-drain current, $I_{sd}$. The dependence $V(I)$ is a current-voltage characteristic, which is equivalent to the case of Corbino geometry [@SK:shashkin94a] (Fig. \[SK:iv\]). Not only are the current-voltage curves similar for all insulating phases, but they also behave identically near the metal-insulator phase boundaries (Fig. \[SK:Vc\](a)). The dependence of the critical voltage, $V_c$, on the distance from the phase boundary is close to a parabolic law [@SK:dolgopolov92b]. The phase boundary position determined by a vanishing $V_c$ is practically coincident with that determined by a vanishing activation energy, $E_a$, of electrons from the Fermi level $E_F$ to the mobility edge, $E_c$ (Fig. \[SK:Vc\](b)). The value $E_a$ is determined from the temperature dependence of the conduction in the linear interval of current-voltage curves, which is activated at not too low temperatures [@SK:adkins76]; note that it transforms into variable range hopping as $T\rightarrow0$ (see below). The activation energy changes linearly with the distance from the phase boundary, reflecting constancy of the thermodynamic density of states near the transition point (see also Sec. \[SK:zero\]). The threshold behavior of the current-voltage characteristics is caused by the breakdown in the insulating phases. The breakdown occurs when the localized electrons at the Fermi level gain enough energy to reach the mobility edge in an electric field, $V_c/d$, over a distance given by the localization length, $L$ [@SK:shashkin94a; @SK:polyakov93]: $$eV_cL/d=|E_c-E_F|,\label{SK:break}$$ where $d$ is the corresponding sample dimension. The values $E_a$ and $V_c$ are related through the localization length which is temperature independent and diverges near the transition as $L(E_F)\propto |E_c-E_F|^{-s}$ with exponent $s$ close to unity, in agreement with the theoretical value $s=4/3$ in the classical percolation problem [@SK:shklovskii84]. The value of the localization length is practically the same near all metal-insulator phase boundaries, which indicates that even quantitatively, all insulating phases are very similar. Note that since the localization length in Eq. (\[SK:break\]) is small compared to the sample sizes, the phase boundary position determined by the diverging localization length refers to an infinite 2D system. As inferred from the vanishing of both $E_a$ and $V_c$ at the same point (see Fig. \[SK:Vc\](b)), possible shifts of the mobility threshold due to finite sample dimensions are small, which justifies extrapolations to the limit of $L\rightarrow\infty$.
The consequences of the method include the following. (i) As long as no dramatic changes occur in transport properties, this excludes the pinned Wigner solid as the origin for the insulating phase at low electron densities in available samples of low-disordered silicon MOSFETs. (ii) The metal-insulator phase diagram of Fig. \[SK:floating\](a) is verified and substantiated. (iii) The existence of a metal-insulator transition in zero magnetic field is supported (see Sec. \[SK:zero\]). (iv) The bandwidth of the extended states in the Landau levels is finite. All of these are also valid for relatively low-disordered 2D carrier systems in GaAs/AlGaAs heterostructures with the distinction that fractional quantum Hall phases are involved. Yet, the topology of the phase diagram remains unchanged, including the oscillating behavior of the phase boundary that restricts the low-density insulating phase. Additional confirmation of the percolation transition to the low-density insulating phase in GaAs/AlGaAs heterostructures was obtained by studies of the high-frequency conductivity [@SK:li94] and time-resolved photoluminescence of 2D electrons [@SK:kukushkin93], as discussed in Ref. [@SK:shashkin94b].
The insulating phase at low electron densities is special in what follows. Deep in the insulating state and at low temperatures the variable-range-hopping regime occurs in which the conductivity $\sigma_{xx}$ is small compared to its peak value [@SK:shklovskii84]. In this regime it was predicted that the deviation, $\Delta\sigma_{xy}$, of $\sigma_{xy}$ from its quantized value in strong magnetic fields is much smaller than $\sigma_{xx}\propto\exp(-(T_0/T)^{1/2})$ [@SK:wysokinski83]: $\Delta\sigma_{xy}\propto\sigma_{xx}^\gamma$ with exponent $\gamma\approx 1.5$. A finite $\rho_{xy}$ contrasted by diverging $\rho_{xx}$ was found in calculations of the $T=0$ magnetotransport coefficients in the insulating phase with vanishing $\sigma_{xx}$ and $\sigma_{xy}$ [@SK:viehweger90]. Such a behavior of $\rho_{xx}$ and $\rho_{xy}$ indicates a special quadratic relation between conductivities: $\sigma_{xy}\propto\sigma_{xx}^2$. Moreover, it was shown that $\rho_{xy}$ is close to the classical value ($B/n_sec$) [@SK:zhang92], providing arguments for the existence of a Hall insulator phase [@SK:kivelson92]. Indeed, values $\rho_{xy}$ close to $B/n_sec$ were experimentally found in the low-density insulating phase. Thus, the distinction of the Hall insulator phase from the quantum Hall phases, i.e., the absence of extended states below the Fermi level, becomes evident when expressed in terms of $\rho_{xx}$ and $\rho_{xy}$.
Scaling and Thermal Broadening {#SK:scaling}
------------------------------
It was predicted that the localization length diverges as a power law at a single energy, $E^*$, which is the center of the Landau level [@SK:iordansky82]: $L(E)\propto |E-E^*|^{-s}$. An idea to check this prediction based on low-temperature measurements of $\sigma_{xx}$ [@SK:aoki85] was quickly developed to a concept of single-parameter scaling [@SK:pruisken88]. It was suggested that the magnetoresistance tensor components are functions of a single variable that is determined by the ratio of the dephasing length, $L_d(T)\propto T^{-p/2}$ (where $p$ is the inelastic-scattering-time exponent), and the localization length. The concept was claimed to be confirmed by measurements of temperature dependencies of the peak width, $\Delta B$, in $\rho_{xx}$ (or $\sigma_{xx}$) and the maximum of $d\rho_{xy}/dB$ in a highly-disordered 2D electron system in InGaAs/InP heterostructures, yielding $\Delta B\propto
T^\kappa$, where $\kappa=p/2s\approx 0.4$ [@SK:wei88]. Later, both deviations in the power law and different exponents in the range between $\kappa=0.15$ and $\kappa=1$ were observed for other 2D carrier systems, different Landau levels, and different disorder strengths (see, e.g., Refs. [@SK:dolgopolov91a; @SK:koch91a; @SK:wakabayashi89; @SK:wei92]). Importantly, the scaling analysis of experimental data in question is based on two unverified assumptions: (i) zero bandwidth of the extended states in the Landau levels; and (ii) constancy of the thermodynamic density of states in the scaling range. If either assumption is not valid, this may lead, at least, to underestimating the experimental value of exponent $\kappa$.
The method of vanishing activation energy and vanishing nonlinearity of current-voltage characteristics as extrapolated from the insulating phase shows that the former assumption is not justified. Also, measurements of the peak width in $\rho_{xx}$ as a function of temperature in low-disordered silicon MOSFETs yield a linear dependence which extrapolates to a finite peak width [@SK:shashkin94a] as $T\rightarrow0$ (Fig. \[SK:width\](a)). Very similar temperature and frequency dependencies were observed in highly-disordered 2D carrier systems in GaAs/AlGaAs heterostructures [@SK:balaban98] and Ge/SiGe heterostructures [@SK:hilke97]. It is noteworthy that a similar behavior is revealed if the data from the publications, which claim the observation of scaling, is plotted on a linear rather than logarithmic scale (see, e.g., Fig. \[SK:width\](b)); finite values of the peak width as $T\rightarrow0$ are even more conspicuous for the data of Refs. [@SK:dolgopolov91a; @SK:koch91a; @SK:li09]. The reason for the ambiguity is quite simple: within experimental uncertainty, it is difficult, especially on a logarithmic scale, to distinguish between sublinear/superlinear fits to the data and linear fits which do not have to run through the origin. Note that attempts were made to relate the finite peak width as $T\rightarrow0$ to the dephasing length reaching the sample size [@SK:koch91a; @SK:li09]. However, the suggested finite-size effect is not supported by experimental data, because in different samples with different sizes, the disorder is also different. It is the disorder, rather than the sample size, that may be responsible for the behavior of the values measured in different samples.
Although lack of data in most of the above experimental papers does not allow one to verify the validity of both assumptions, it is very likely that there is no qualitative difference between all of the discussed results. As a matter of fact, they can be described by a linear, or weakly sub-linear temperature dependence with a finite offset at $T=0$. This is concurrent with the results obtained by vanishing activation energy and vanishing nonlinearity of current-voltage characteristics as extrapolated from the insulating phase. So, the single-parameter scaling is not confirmed by the experimental data which establish the finite bandwidth of the extended states in the Landau levels.
There is an alternative and simple explanation of the temperature dependence of the peak width in $\rho_{xx}$ in terms of thermal broadening. Within a percolation picture, if the activation energy $E_a\sim k_BT$, the conduction is of the order of the maximum $\sigma_{xx}$ so that the value of $\sim k_BT$ gives a thermal shift of the effective mobility edge corresponding to the $\sigma_{xx}$ peak width [@SK:shashkin94a]. Although the concept of thermal broadening has been basically ignored in the literature in the search for less trivial data interpretations, it looks as if no experimental results go beyond this, favoring the concept of single-parameter scaling. Once the behavior of the localization length is not reflected by the temperature-dependent peak width in $\rho_{xx}$, no experimental support is provided for numerical calculations of the localization length which give a somewhat larger exponent $s\approx
2$ compared to $s=4/3$ in classical percolation problem (see, e.g., Ref. [@SK:huckestein95]).
Zero-field Metal-insulator Transition {#SK:zero}
=====================================
In contrast to the case of quantizing magnetic fields, no extended states are expected in zero magnetic field, at least for weakly-interacting 2D electron systems. The criterion of vanishing activation energy and vanishing nonlinearity of current-voltage characteristics as extrapolated from the insulating phase, however, results in an opposite conclusion. To sort out this inconsistency, further support by independent experimental verifications is needed.
Another criterion is based on the analysis of the temperature dependencies of the resistivity at $B=0$. Provided these are strong, those with positive (negative) derivative $d\rho/dT$ are indicative of a metal (insulator) [@SK:sarachik99; @SK:abrahams01]; note that in the vicinity of the transition, $\rho(T)$ dependencies obey the scaling law with exponent $\kappa\approx1$, which is consistent with the concept of thermal broadening/shift by the value $\sim k_BT$ of the effective mobility edge in the insulating phase (see Sec. \[SK:scaling\]). If extrapolation of $\rho(T)$ to $T=0$ is valid, the critical point for the metal-insulator transition is given by $d\rho/dT=0$. In a low-disordered 2D electron system in silicon MOSFETs, the resistivity at a certain electron density shows virtually no temperature dependence over a wide range of temperatures [@SK:sarachik99; @SK:kravchenko00b] (Fig. \[SK:flat\](a)). This curve separates those with positive and negative $d\rho/dT$ nearly symmetrically at temperatures above 0.2 K [@SK:abrahams01]. Assuming that it remains flat down to $T=0$, one obtains the critical point which corresponds to a resistivity $\rho\approx 3h/e^2$.
Recently, these two criteria have been applied simultaneously to the 2D metal-insulator transition in low-disordered silicon MOSFETs [@SK:shashkin01b; @SK:jaroszynski02]. In zero magnetic field, both methods yield the same critical density $n_c$ (Figs. \[SK:flat\](b) and \[SK:nc2\](a)). Since one of the methods is temperature independent, this equivalence strongly supports the existence of a metal-insulator transition at $T=0$ in $B=0$. This also adds confidence that the curve with zero derivative $d\rho/dT$ will remain flat (or at least will retain finite resistivity value) down to zero temperature. Additional confirmation in favor of zero-temperature zero-field metal-insulator transition is provided by magnetic measurements [@SK:shashkin05], as described in the next section. It is argued that the metal-insulator transition in silicon samples with very low disorder potential is driven by interactions. This is qualitatively different from a localization-driven transition in more-disordered samples that occurs at appreciably higher densities.
For 2D electron systems both with high disorder in zero magnetic field (see Sec. \[SK:floating-up\]) and in parallel magnetic fields, the metallic ($d\rho/dT>0$) behavior is suppressed [@SK:hanein98; @SK:pudalov01; @SK:shashkin01b; @SK:shashkin06a] or disappears entirely, and extrapolation of the weak $\rho(T)$ dependence to $T=0$ is not justified, invalidating the derivative criterion for the critical point for the metal-insulator transition (Figs. \[SK:flat\](c) and \[SK:nc2\](b)). Once one of the two methods fails, it remains to be seen how to verify the conclusion as inferred from the other method. This makes uncertain the existence of a zero-temperature metal-insulator transition in 2D electron systems both with high disorder in zero magnetic field and in parallel magnetic fields.
Owing to its simplicity, the derivative method is widely used for describing metallic ($d\rho/dT>0$) and insulating ($d\rho/dT<0$) temperature dependencies of resistance in a restricted temperature range. However, to avoid confusion with metallic and insulating phases, one should employ alternative methods for determining the metal-insulator transition point. Such methods, including a vanishing activation energy and noise measurements, have been applied to highly-disordered 2D carrier systems [@SK:jaroszynski02; @SK:bogdanovich02]. Being similar, they yield lower critical densities $n_c$ for the metal-insulator transition compared to those obtained using formally the derivative criterion. This simply reflects the fact that the metallic ($d\rho/dT>0$) behavior is suppressed. The critical density $n_c$, at which the exponential divergence of the resistivity as $T\rightarrow0$ ends, increases naturally with disorder strength. It also increases somewhat with parallel magnetic field, saturating above a certain field, as was found in dilute silicon MOSFETs [@SK:dolgopolov92c; @SK:shashkin01b].
Possible Ferromagnetic Transition
=================================
After a strongly enhanced ratio $gm$ of the spin and the cyclotron splittings was found at low electron densities in silicon MOSFETs [@SK:kravchenko00a], it became clear that the system behavior was well beyond the weakly interacting Fermi liquid. It was reported that the parallel magnetic field required to produce complete spin polarization, $B_c\propto n_s/gm$, tends to vanish at a finite electron density $n_\chi\approx 8\times 10^{10}$ cm$^{-2}$, which is close to the critical density $n_c$ for the metal-insulator transition in this electron system [@SK:shashkin01a; @SK:vitkalov01; @SK:kravchenko02] (Fig. \[SK:Bc\]). These findings point to a sharp increase of the spin susceptibility, $\chi\propto gm$, and possible ferromagnetic instability in dilute silicon MOSFETs. The fact that $n_\chi$ is close to the critical density $n_c$ indicates that the metal-insulator transition in silicon samples with very low disorder potential is a property of a clean 2D system and is driven by interactions [@SK:shashkin01a]. A similar although less pronounced behavior was observed in other 2D carrier systems [@SK:gao02]. The experimental results indicated that in silicon MOSFETs it is the effective mass, rather than the $g$ factor, that sharply increases at low electron densities [@SK:shashkin02b] (Fig. \[SK:sigma\](a)). They also indicated that the anomalous rise of the resistivity with temperature is related to the increased mass. The magnitude of the mass does not depend on the degree of spin polarization, which points to a spin-independent origin of the effective mass enhancement [@SK:shashkin03a]. It was found that the relative mass enhancement is system- and disorder-independent and is determined by electron-electron interactions only [@SK:shashkin07].
In addition to transport measurements, thermodynamic measurements of the magnetocapacitance and magnetization of a 2D electron system in low-disordered silicon MOSFETs were performed, and very similar results for the spin susceptibility, effective mass, and $g$ factor were obtained [@SK:khrapai03a; @SK:shashkin06b; @SK:anissimova06] (Fig. \[SK:sigma\](b)). The Pauli spin susceptibility behaves critically close to the critical density $n_c$ for the $B=0$ metal-insulator transition: $\chi\propto n_s/(n_s-n_\chi)$. This is in favor of the occurrence of a spontaneous spin polarization (either Wigner crystal or ferromagnetic liquid) at low $n_s$, although in currently available samples, the residual disorder conceals the origin of the low-density phase. The effective mass increases sharply with decreasing density while the enhancement of the $g$ factor is weak and practically independent of $n_s$. Unlike in the Stoner scenario, it is the effective mass that is responsible for the dramatically enhanced spin susceptibility at low electron densities.
Thus, the experimental results obtained in low-disordered silicon MOSFETs indicate that on the metallic side the metal-insulator transition is driven by interactions, while on the insulating side this is still a classical percolation transition with no dramatic effects from interactions. One can consider the metal-insulator transition in the cleanest of currently available samples as a quantum phase transition, even though the problem of the competition between metal-insulator and ferromagnetic transitions is not yet resolved. It is not yet clear whether or not electron crystallization expected in the low-density limit is preceded by an intermediate phase like ferromagnetic liquid.
Outlook
=======
Critical analysis of the available experimental data for 2D electron systems both in zero and in quantizing magnetic fields shows that consequences of the scaling theory of localization for noninteracting 2D electrons are not confirmed. The main points to be addressed by theory are the problem of finite bandwidth of the extended states in the Landau levels and that of a quantum phase transition in low-disordered 2D electron systems in zero magnetic field, including the competition between metal-insulator and ferromagnetic transitions. Recently, some progress has been made in describing the behavior of low-disordered strongly interacting 2D electron systems in zero magnetic field: it has been shown that the metallic ground state can be stabilized by electron-electron interactions [@SK:punnoose05]. It is possible that it may also be necessary to take into account electron-electron interactions to describe the quantum phase transitions that are characterized by the finite bandwidth of the extended states in the Landau levels.
The finding that in dilute 2D electron systems the spin susceptibility tends to diverge due to strong increase in the effective mass remains basically unexplained, and the particular mechanism leading to the effect remains to be seen. It is worth discussing the latest theoretical developments which are claimed to be valid for the strongly-interacting limit. According to the renormalization group analysis for multi-valley 2D systems, the effective mass dramatically increases at disorder-dependent density for the metal-insulator transition while the $g$ factor remains nearly intact [@SK:punnoose05]. However, the prediction of disorder-dependent effective mass is in contradiction to the experiment. Besides, the results of Ref. [@SK:punnoose05] are valid only in the near vicinity of the metal-insulator transition, while the tendency of the spin susceptibility to diverge can be traced up to the densities exceeding $n_c$ by a factor of a few. In the Fermi-liquid-based model of Ref. [@SK:khodel05], a flattening at the Fermi energy in the spectrum has been predicted that leads to a diverging effective mass. Still, the expected dependence of the effective mass on temperature is not confirmed by the experimental data. The strong increase of the effective mass has been obtained, in the absence of the disorder, by solving an extended Hubbard model using dynamical mean-field theory [@SK:pankov08]. This is consistent with the experiment, especially taking into account that the relative mass enhancement has been experimentally found to be independent of the level of the disorder. The dominant increase of $m$ near the onset of Wigner crystallization follows also from an alternative description of the strongly-interacting electron system beyond the Fermi liquid approach (see, *e.g.*, Ref. [@SK:spivak04]).
On the experimental side, progress in the fabrication of increasingly high mobility Si, Si/SiGe, and GaAs-based devices will open up the possibility of probing the intrinsic properties of clean 2D electron systems at still lower densities, where electron-electron interactions are yet stronger and, presumably, the previously observed behaviors will be yet more pronounced. Moreover, as high-mobility devices made with other semiconductors become available, further tests of the universality of the observed phenomena will add to our knowledge of 2D quantum phase transitions.
[apssamp]{} E. Wigner, Phys. Rev. [**46**]{}, 1002 (1934). E. C. Stoner, Rep. Prog. Phys. [**11**]{}, 43 (1946). L. D. Landau, Sov. Phys. JETP [**3**]{}, 920 (1957). B. Tanatar and D. M. Ceperley, Phys. Rev. B [**39**]{}, 5005 (1989). C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Phys. Rev. Lett. [**88**]{}, 256601 (2002). E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. [**42**]{}, 673 (1979). B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev. Lett. [**44**]{}, 1288 (1980). A. M. Finkelstein, Sov. Phys. JETP [**57**]{}, 97 (1983); Z. Phys. B [**56**]{}, 189 (1984); C. Castellani, C. Di Castro, P. A. Lee, and M. Ma, Phys. Rev. B [**30**]{}, 527 (1984). A. Punnoose and A. M. Finkelstein, Science [**310**]{}, 289 (2005). A. M. M. Pruisken, Phys. Rev. Lett. [**61**]{}, 1297 (1988). S. V. Iordansky, Solid State Commun. [**43**]{}, 1 (1982); T. Ando, J. Phys. Soc. Jpn. [**52**]{}, 1740 (1983); H. Aoki, J. Phys. C [**16**]{}, 1893 (1983). D. E. Khmelnitskii, Phys. Lett. A [**106**]{}, 182 (1984); R. B. Laughlin, Phys. Rev. Lett. [**52**]{}, 2304 (1984). S. A. Kivelson, D. H. Lee, and S. C. Zhang, Phys. Rev. B [**46**]{}, 2223 (1992). D. E. Khmelnitskii, Helv. Phys. Acta [**65**]{}, 164 (1992). Y. E. Lozovik and V. I. Yudson, JETP Lett. [**22**]{}, 11 (1975). B. I. Halperin, Phys. Rev. B [**25**]{}, 2185 (1982). K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. [**45**]{}, 494 (1980). *The Quantum Hall Effect*, ed. by R. E. Prange and S. M. Girvin (Springer-Verlag, 1987). H. Levine, S. B. Libby, and A. M. M. Pruisken, Phys. Rev. Lett. [**51**]{}, 1915 (1983). M. Büttiker, Phys. Rev. B [**38**]{}, 9375 (1988). R. B. Laughlin, Phys. Rev. B [**23**]{}, 5632 (1981); A. Widom and T. D. Clark, J. Phys. D [**15**]{}, L181 (1982). V. T. Dolgopolov, A. A. Shashkin, N. B. Zhitenev, S. I. Dorozhkin, and K. von Klitzing, Phys. Rev. B [**46**]{}, 12560 (1992). E. Y. Andrei, G. Deville, D. C. Glattli, F. I. B. Williams, E. Paris, and B. Etienne, Phys. Rev. Lett. [**60**]{}, 2765 (1988). M. D’Iorio, V. M. Pudalov, and S. G. Semenchinsky, Phys. Lett. A [**150**]{}, 422 (1990). M. B. Santos, Y. W. Suen, M. Shayegan, Y. P. Li, L. W. Engel, and D. C. Tsui, Phys. Rev. Lett. [**68**]{}, 1188 (1992). A. A. Shashkin, G. V. Kravchenko, and V. T. Dolgopolov, JETP Lett. [**58**]{}, 220 (1993). A. A. Shashkin, V. T. Dolgopolov, and G. V. Kravchenko, Phys. Rev. B [**49**]{}, 14486 (1994). A. A. Shashkin, V. T. Dolgopolov, G. V. Kravchenko, M. Wendel, R. Schuster, J. P. Kotthaus, R. J. Haug, K. von Klitzing, K. Ploog, H. Nickel, and W. Schlapp, Phys. Rev. Lett. [**73**]{}, 3141 (1994). V. T. Dolgopolov, G. V. Kravchenko, A. A. Shashkin, and S. V. Kravchenko, Phys. Rev. B [**46**]{}, 13303 (1992). S. V. Kravchenko, W. Mason, J. E. Furneaux, and V. M. Pudalov, Phys. Rev. Lett. [**75**]{}, 910 (1995). M. Hilke, D. Shahar, S. H. Song, D. C. Tsui, and Y. H. Xie, Phys. Rev. B [**62**]{}, 6940 (2000). I. Glozman, C. E. Johnson, and H. W. Jiang, Phys. Rev. Lett. [**74**]{}, 594 (1995). M. P. Sarachik and S. V. Kravchenko, Proc. Natl. Acad. Sci. USA [**96**]{}, 5900 (1999). E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. [**73**]{}, 251 (2001); S. V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys. [**67**]{}, 1 (2004). Y. Hanein, U. Meirav, D. Shahar, C. C. Li, D. C. Tsui, and H. Shtrikman, Phys. Rev. Lett. [**80**]{}, 1288 (1998); M. Y. Simmons, A. R. Hamilton, M. Pepper, E. H. Linfield, P. D. Rose, D. A. Ritchie, A. K. Savchenko, and T. G. Griffiths, Phys. Rev. Lett. [**80**]{}, 1292 (1998). V. M. Pudalov, G. Brunthaler, A. Prinz, and G. Bauer, cond-mat/0103087. V. T. Dolgopolov, G. V. Kravchenko, and A. A. Shashkin, JETP Lett. [**55**]{}, 140 (1992). V. T. Dolgopolov, G. V. Kravchenko, A. A. Shashkin, and S. V. Kravchenko, JETP Lett. [**55**]{}, 733 (1992). C. J. Adkins, S. Pollitt, and M. Pepper, J. Phys. C [**37**]{}, 343 (1976). D. G. Polyakov and B. I. Shklovskii, Phys. Rev. B [**48**]{}, 11167 (1993). B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer, New York, 1984). Y. P. Li, PhD thesis, Princeton University, 1994. I. V. Kukushkin and V. B. Timofeev, Sov. Phys. Usp. [**36**]{}, 549 (1993). K. I. Wysokinski and W. Brenig, Z. Phys. B [**54**]{}, 11 (1983). O. Viehweger and K. B. Efetov, J. Phys. Condens. Matter [**2**]{}, 7049 (1990). S. C. Zhang, S. Kivelson, and D. H. Lee, Phys. Rev. Lett. [**69**]{}, 1252 (1992). H. Aoki and T. Ando, Phys. Rev. Lett. [**54**]{}, 831 (1985). H. P. Wei, D. C. Tsui, M. A. Paalanen, and A. M. M. Pruisken, Phys. Rev. Lett. [**61**]{}, 1294 (1988). V. T. Dolgopolov, A. A. Shashkin, B. K. Medvedev, and V. G. Mokerov, Sov. Phys. JETP [**72**]{}, 113 (1991). S. Koch, R. J. Haug, K. von Klitzing, and K. Ploog, Phys. Rev. Lett. [**67**]{}, 883 (1991). J. Wakabayashi, M. Yamane, and S. Kawaji, J. Phys. Soc. Jpn. [**58**]{}, 1903 (1989); T. Wang, K. P. Clark, G. F. Spencer, A. M. Mack, and W. P. Kirk, Phys. Rev. Lett. [**72**]{}, 709 (1994). H. P. Wei, S. Y. Lin, D. C. Tsui, and A. M. M. Pruisken, Phys. Rev. B [**45**]{}, 3926 (1992). N. Q. Balaban, U. Meirav, and I. Bar-Joseph, Phys. Rev. Lett. [**81**]{}, 4967 (1998); D. Shahar, M. Hilke, C. C. Li, D. C. Tsui, S. L. Sondhi, J. E. Cunningham, and M. Razeghi, Solid State Commun. [**107**]{}, 19 (1998). M. Hilke, D. Shahar, S. H. Song, D. C. Tsui, Y. H. Xie, and D. Monroe, Phys. Rev. B [**56**]{}, R15545 (1997). W. Li, C. L. Vicente, J. S. Xia, W. Pan, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**102**]{}, 216801 (2009). B. Huckestein, Rev. Mod. Phys. [**67**]{}, 357 (1995). S. V. Kravchenko and T. M. Klapwijk, Phys. Rev. Lett. [**84**]{}, 2909 (2000). A. A. Shashkin, S. V. Kravchenko, and T. M. Klapwijk, Phys. Rev. Lett. [**87**]{}, 266402 (2001). J. Jaroszyński, D. Popović, and T. M. Klapwijk, Phys. Rev. Lett. [**89**]{}, 276401 (2002). A. A. Shashkin, Phys. Usp. [**48**]{}, 129 (2005). A. A. Shashkin, E. V. Deviatov, V. T. Dolgopolov, A. A. Kapustin, S. Anissimova, A. Venkatesan, S. V. Kravchenko, and T. M. Klapwijk, Phys. Rev. B [**73**]{}, 115420 (2006). S. Bogdanovich and D. Popović, Phys. Rev. Lett. [**88**]{}, 236401 (2002); R. Leturcq, D. L’Hote, R. Tourbot, C. J. Mellor, and M. Henini, Phys. Rev. Lett. [**90**]{}, 076402 (2003). S. V. Kravchenko, A. A. Shashkin, D. A. Bloore, and T. M. Klapwijk, Solid State Commun. [**116**]{}, 495 (2000). A. A. Shashkin, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev. Lett. [**87**]{}, 086801 (2001). S. A. Vitkalov, H. Zheng, K. M. Mertes, M. P. Sarachik, and T. M. Klapwijk, Phys. Rev. Lett. [**87**]{}, 086401 (2001). S. V. Kravchenko, A. A. Shashkin, and V. T. Dolgopolov, Phys. Rev. Lett. [**89**]{}, 219701 (2002). V. T. Dolgopolov and A. Gold, JETP Lett. [**71**]{}, 27 (2000). X. P. A. Gao, A. P. Mills Jr., A. P. Ramirez, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. [**89**]{}, 016801 (2002); J. Zhu, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. [**90**]{}, 056805 (2003); T. Okamoto, M. Ooya, K. Hosoya, and S. Kawaji, Phys. Rev. B [**69**]{}, 041202 (2004). A. A. Shashkin, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev. B [**66**]{}, 073303 (2002). G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B [**64**]{}, 214204 (2001). A. A. Shashkin, M. Rahimi, S. Anissimova, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev. Lett. [**91**]{}, 046403 (2003). A. A. Shashkin, A. A. Kapustin, E. V. Deviatov, V. T. Dolgopolov, and Z. D. Kvon, Phys. Rev. B [**76**]{}, 241302(R) (2007). V. S. Khrapai, A. A. Shashkin, and V. T. Dolgopolov, Phys. Rev. Lett. [**91**]{}, 126404 (2003). A. A. Shashkin, S. Anissimova, M. R. Sakr, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev. Lett. [**96**]{}, 036403 (2006). S. Anissimova, A. Venkatesan, A. A. Shashkin, M. R. Sakr, S. V. Kravchenko, and T. M. Klapwijk, Phys. Rev. Lett. [**96**]{}, 046409 (2006). V. A. Khodel, J. W. Clark, and M. V. Zverev, Europhys. Lett. [**72**]{}, 256 (2005). S. Pankov and V. Dobrosavljević, Phys. Rev. B [**77**]{}, 085104 (2008). B. Spivak and S. A. Kivelson, Phys. Rev. B [**70**]{}, 155114 (2004).
[^1]: In this state the Hall resistivity, $\rho_{xy}=h/\nu e^2$, is quantized at integer filling factor $\nu$, accompanied by vanishing longitudinal resistivity, $\rho_{xx}$ [@SK:klitzing80].
[^2]: Note that for the lowest-density phase boundary, a lower value $\sigma_{xx}^{-1}\approx 100$ kOhm at a temperature $\approx 25$ mK follows from the latter method.
[^3]: In bivalley (100)-silicon MOSFETs, spin and valley degeneracies of the Landau level should be taken into account.
[^4]: We refer here to merging of the extended states and, hence, the presence of direct transitions between the insulating phase with $\sigma_{xy}=0$ and quantum Hall phases with $\sigma_{xy}h/e^2>1$.
|
---
abstract: 'We study the phonon modes in single-walled MoS$_{2}$ nanotubes via the lattice dynamics calculation and molecular dynamics simulation. The phonon spectra for tubes of arbitrary chiralities are calculated from the dynamical matrix constructed by the combination of an empirical potential with the conserved helical quantum numbers $(\kappa, n)$. In particular, we show that the frequency ($\omega$) of the radial breathing mode is inversely proportional to the tube diameter ($d$) as $\omega=665.3/d$ [cm$^{-1}$]{}. The eigen vectors of the first twenty lowest-frequency phonon modes are illustrated. Based on these eigen vectors, we demonstrate that the radial breathing oscillation is disturbed by phonon modes of three-fold symmetry initially, and the tube is squashed by the modes of two-fold symmetry eventually. Our study provides fundamental knowledge for further investigations of the thermal and mechanical properties of the MoS$_{2}$ nanotubes.'
author:
- 'Jin-Wu Jiang'
- 'Bing-Shen Wang'
- Timon Rabczuk
title: 'Phonon Modes in Single-Walled Molybdenum Disulphide (MoS$_{2}$) Nanotubes: Lattice Dynamics Calculation and Molecular Dynamics Simulation'
---
Introduction
============
Molybdenum Disulphide (MoS$_{2}$) is a semiconductor with a bulk bandgap above 1.2 [eV]{},[@KamKK] which can be further manipulated by changing its thickness,[@MakKF] or through application of mechanical strain.[@FengJ2012npho; @LuP2012pccp] This finite bandgap is a key reason for the excitement surrounding MoS$_{2}$ as compared to graphene as graphene has a zero bandgap.[@NovoselovKS2005nat] Because of its direct bandgap and also its well-known properties as a lubricant, MoS$_{2}$ has attracted considerable attention in recent years.[@WangQH2012nn; @ChhowallaM] For example, Radisavljevic et al.[@RadisavljevicB2011nn] demonstrated the application of single-layered MoS$_{2}$ (SLMoS$_{2}$) as a good transistor. The strain and the electronic noise effects were found to be important for the SLMoS$_{2}$ transistor.[@ConleyHJ; @SangwanVK; @Ghorbani-AslM; @CheiwchanchamnangijT]
Besides electronic properties, there has been increasing interest in the thermal and mechanical properties of MoS$_{2}$. Several recent works have addressed the thermal transport properties of SLMoS$_{2}$ in both ballistic and diffusive transport regimes.[@HuangW; @JiangJW2013mos2; @VarshneyV; @JiangJW2013sw] The mechanical behavior of SLMoS$_{2}$ has been investigated experimentally.[@BertolazziS; @CooperRC2013prb1; @CooperRC2013prb2] For the theoretical part, we have examined the size and edge effects on the Young’s modulus of the SLMoS$_{2}$ based on the Stillinger-Weber (SW) potential.[@JiangJW2013sw] Quite recently, we derived an analytic formula for the elastic bending modulus of the SLMoS$_{2}$, where the importance of the finite thickness effect was revealed.[@JiangJW2013bend]
A fundamental property related to the thermal and mechanical behaviors is the lattice dynamics property of MoS$_{2}$, i.e the phonon spectrum or phonon modes. The thermal conductivity in the semiconductor MoS$_{2}$ is dominated by the lattice thermal transport, which is contributed by the phonon mode. Different mechanical processes are governed by the corresponding phonon modes. For instance, the nanomechanical resonant oscillation takes advantage of the bending mode (flexure mode). Recently, there have been increasing experimental and theoretical efforts on the lattice dynamics properties of the single-layer or few-layer MoS$_{2}$ nanosheets.[@WakabayashiN; @JimenezSS; @LeeC; @SanchezAM; @ZengHL; @ZhaoYY; @RiceC; @ZhangXprb2013; @LanzilloN2013] However, up to date, the lattice dynamics of the single-walled MoS$_{2}$ nanotube (SWMoS$_{2}$NT) was investigated only by few works, which show different lattice properties in the tube structure from the planar sheet structure, because of the special line group in the nanotube.[@DobardzicE; @DamnjanovicM2008mmp]
In this paper, we perform the lattice dynamics study and the molecular dynamics simulation for the lattice properties of the SWMoS$_{2}$NT with arbitrary chiralities. The phonon spectrum is calculated from the dynamics matrix with the force constant matrix obtained from the SW potential. The helical quantum numbers are used to denote the phonon mode. The eigen vector of the first twenty lowest-frequency phonon modes are presented. Furthermore, we show that the radial breathing mechanical oscillation is initially disturbed by the three-fold modes and eventually destroyed by the two-fold modes.
\[0.9\][![(Color online) The configuration of MoS$_{2}$. Mo atoms are represented by big balls (gray online). S atoms are presented by small balls (yellow online). Above panel: top view of the two-dimensional MoS$_{2}$ honeycomb lattice. Three lattice directions are depicted by arrows, $\vec{R}_{\rm arm}=n_{1}\vec{a}_{1}+n_{1}\vec{a}_{2}$, $\vec{R}_{\rm zig}=n_{1}\vec{a}_{1}$, and $\vec{R}_{\rm chiral}=2n_{2}\vec{a}_{1}+n_{2}\vec{a}_{2}$. Bottom panels: three SWMoS$_{2}$NTs are obtained by rolling up the above MoS$_{2}$ sheet onto a cylindrical surface with the three corresponding lattice vectors as the circumference.[]{data-label="fig_cfg"}](cfg.eps "fig:"){width="8cm"}]{}
\[0.8\][{width="\textwidth"}]{}
\[0.8\][{width="\textwidth"}]{}
\[0.8\][{width="\textwidth"}]{}
\[0.8\][{width="\textwidth"}]{}
Structure and Calculation Details
=================================
$$\begin{aligned}
\vec{r}_{m,l}=m\vec{H}+l\frac{\vec{R}}{N}, \hspace{0.25cm}
\mbox{or} \hspace{0.25cm}
\vec{r}_{q_{1},q_{2}}=q_{1}\vec{a}_{1}+q_{2}\vec{a}_{2},
\label{eq_rvec}\end{aligned}$$ in which $(q_{1}, q_{2})$ and $(m l)$ are related to each other as: $$\begin{aligned}
m=(n_{1}q_{2}-n_{2}q_{1})/N, \hspace{0.25cm}
l=q_{1}p_{2}-q_{2}p_{1}\;.
\label{eq_ml}\end{aligned}$$ $\vec{b}_{H}$ and $\vec{b}_{R}$ are reciprocal unit vectors corresponding to $(\vec{H}, \frac{\vec{R}}{N})$. Any wave vector in the reciprocal space can be written as $$\begin{aligned}
\vec{K}=\frac{\kappa}{2\pi}\vec{b}_{H}+\frac{n}{N}\vec{b}_{R}\;.\end{aligned}$$ $(\kappa n)$ are the two helical quantum numbers. In the first Brillouin zone, $\kappa\in(-\frac{1}{2}, \frac{1}{2}]$ and $n$ is an integer in $(-\frac{N}{2}, \frac{N}{2}]$.
Phonon Dispersion of SWMoS$_{2}$NT
==================================
All screw and rotational symmetry operations in the SWMoS$_{2}$NT form the space group of this tubal system, which is named line group.[@VujicicM; @BozovicIB1978jpamg; @BozovicIB1981jpamg; @MilosevicI] According to the irreducible representation of the line group, we have the generalized Bloch theory for the vibration displacement of each atom $(mls)$ in the phonon mode $(\kappa n \tau)$,[@MiloevicI1993prb; @PopovVN1999prb; @DobardzicE2003prb; @JiangJW2006; @DakicB2009jpamt] $$\begin{aligned}
\vec{u}(mls) & = & \frac{1}{\sqrt{MN}}\sum_{\kappa n \tau}e^{i\left(\kappa m+\frac{2\pi}{N}nl\right)}R\left(ml\right)\vec{\xi}^{(\tau)}(\kappa n|00s)\hat{Q}_{\kappa n}^{\tau}
\label{eq_bloch}\end{aligned}$$ where $\tau$ is the branch index. $M$ is the total number of the screw symmetry operation. $N$ is the greatest common divisor of $n_{1}$ and $n_{2}$. It is the number of pure rotational symmetry operations. $M\times N$ gives the total number of unit cells in the tube. $s=1,2,3$ corresponds to the three atoms S, Mo, and S in the unit cell. $R\left(ml\right)$ is a rotation matrix for rotation around the axial direction for angle $\phi_{ml} = m\alpha+l\frac{2\pi}{N}$. $\vec{\xi}^{(\tau)}(\kappa n|00s)$ is the vibration displacement for atom $(00s)$ in the phonon mode denoted by $\left(\kappa n \tau\right)$. $\hat{Q}_{\kappa n}^{\tau}$ is the helical quantum state of the phonon mode $\left(\kappa n \tau\right)$.
Applying the generalized Bloch theory into the phonon dynamical equations, we can obtain the dynamics matrix of the SWMoS$_{2}$NT as, $$\begin{aligned}
D_{ss'}\left(\kappa n\right) & = & \frac{1}{\sqrt{m_{s}m_{s'}}}\sum_{(mls')}\Phi(00s|mls')R\left(ml\right)e^{i\left(\kappa m+\frac{2\pi}{N}nl\right)},
\label{eq_D}\end{aligned}$$ where $m_{s}$ is the mass of atom $s$. For MoS2, the dynamical matrix is a $9\times9$ matrix, since there are three atoms in each unit cell. The diagonalization of the dynamical matrix results in nine phonon modes denoted by the helical index $\left(\kappa n \tau\right)$, with $\tau=1,2,3,...,9$.
$\Phi(00s|mls')$ is the force constant matrix. In our calculation, this force constant matrix is obtained using finite differential method based on a newly developed SW potential. The SW potential is developed for SLMoS$_{2}$, including two-body and three-body interactions. We apply this potential to describe the atomic interaction within the SWMoS$_{2}$NT. All SW potential parameters can be found in our previous work.[@JiangJW2013sw]
Fig. \[fig\_dispersion\] shows the phonon spectrum in four SWMoS$_{2}$NTs with almost the same diameter of 18.0 [Å]{}. The spectrum is symmetric about $\kappa=0$ in all tubes, so only curves with $\kappa\in[0, 0.5]$ are displayed. Panel (a) is the phonon spectrum for armchair tube (15, 15). There are nine curves for each helical quantum number $n$, corresponding to the nine degrees of freedom in each unit cell. There are four acoustic branches. The longitudinal acoustic (LA) and twisting (TW) phonon branches have the same helical quantum number $n=0$. The other two acoustic branches are the transverse acoustic (TA) phonons with $n=\pm 1$. One of the TA mode locates at $\kappa=\alpha$ and $n=1$, which is depicted by a solid arrow on the $\kappa$ axis. The other TA mode locates at $\kappa=-\alpha$ and $n=-1$. The two TA modes are also called flexure modes, and they have a parabolic spectrum due to the quasi-one-dimensional nature of the nanotubes. Panel (b) shows the phonon spectrum of the zigzag tube (26, 0). Panels (c) and (d) show the phonon spectrum for chiral tubes. In tube (16, 13), the quantum number $n$ has only one value, i.e $n=0$, so there is only nine curves in this figure.
One significant feature of Fig. \[fig\_dispersion\] is the combination of an empirical many-body potential (SW) with the helical quantum numbers $(\kappa n)$. In previous studies, the helical quantum numbers are used only in dynamical matrix constructed from force field models. The empirical potential can give more comprehensive information for the phonon spectrum. In particular, it fulfills the rigid rotational symmetry, leading to the zero frequency of the TW mode. Furthermore, the helical quantum numbers correspond to the actual symmetric operations (screw and pure rotational symmetric operations) in the SWMoS$_{2}$NT. Hence they are the good quantum numbers for the phonon modes in the system.[@DobardzicE2003prb] The phase ‘good’ means that these quantum numbers are conserved during the phonon-assisted physical process. For instance, it has been shown that these helical quantum numbers give a natural (the simplest) selection rule for the phonon-phonon scattering process in the lattice thermal transport of the single-walled boron nitride nanotube.[@JiangJW2011bntube] Here, we have successfully combined the helical quantum numbers with empirical potential; thus take advantage of both parts. Hence, Fig. \[fig\_dispersion\] provides a ‘good’ recipe for further investigations of the phonon-assisted physical processes in SWMoS$_{2}$NTs.
We now pay attention to the low-frequency modes in SWMoS$_{2}$NTs, because these modes are easier to be excited in practice. Fig. \[fig\_u\_arm\] shows the first twenty lowest-frequency modes in the armchair tube (15, 15). Three big translational cells are used in this calculation. The periodic boundary condition is applied in the axial direction. These modes locate in the $\Gamma$ point in the Brillouin zone; i.e, the linear quantum number is zero. The linear quantum number is the wave vector corresponding to the big translational operation. Numbers in the figure are the frequency of each mode in the unit of cm$^{-1}$. The figure is plotted with XCRYSDEN.[@xcrysden] The arrow attached to each atom represents the component of this atom in the eigen vector of the phonon mode.
\[1.0\][![(Color online) The diameter dependence for the frequency of the RBM in armchair, zigzag, and chiral tubes. The solid line (blue online) is the fitting function, $y=665.3/x$, for all data.[]{data-label="fig_frequency"}](frequency.eps "fig:"){width="8cm"}]{}
\[0.9\][![(Color online) The total kinetic energy time history during the radial breathing oscillation in SWMoS$_{2}$NTs of different chiralities. Tubes oscillate like a breather. (a) Armchair (15, 15). (b) Zigzag (26, 0). (c) Chiral (20, 10). Arrows depict the life time, at which the radial breathing oscillation starts to be disturbed. The inset in each panel shows the life time versus actuation parameter $\alpha$. The solid line is the exponential fitting to the calculated data.[]{data-label="fig_energy"}](energy.eps "fig:"){width="8cm"}]{}
\[0.8\][{width="\textwidth"}]{}
The first four modes are the acoustic phonon modes with zero frequency. Their helical quantum numbers are $(\kappa n) = (0, 0)$, $(\alpha, 1)$, $(-\alpha, -1)$, and $(0, 0)$. In particular, the fourth mode is the TW mode, resulting from the particular hollow cylindrical structure of the SWMoS$_{2}$NT. These four acoustic modes have important contribution to the thermal transport of SWMoS$_{2}$NTs. The fifth and sixth modes have the two-fold symmetry, which are denoted by $(\kappa n)=(0,2)$. From their peculiar vibration morphology and their pretty low frequencies, it can be easily imagined that the vibration of these modes is likely to squash the tube configuration, leading to possible instability of the SWMoS$_{2}$NT. The seventh and eighth modes have a three-fold symmetry, which are denoted by $(\kappa n)=(0,3)$. The thirteenth mode is the radial breathing mode (RBM), which is denoted by $(\kappa n)=(0,0)$. The frequency of the RBM is sensitive to the tube diameter, due to its vibration morphology and this mode is Raman active, so its frequency is usually used to estimate the tube diameter in the experiment. Figs. \[fig\_u\_zig\] and \[fig\_u\_chiral\] show the first twenty lowest-frequency modes in tubes (26, 0) and (20, 10), respectively. In particular, the RBM has close frequency among these three tubes of quite different chiralities, since the chirality only takes effect on the order of[@JiangJW2006] $1/r^{3}$.
RBM mode
========
In the above, we have studied the full phonon spectrum for SWMoS$_{2}$NTs. The rest of this paper is devoted to the discussion of the RBM, since this mode plays an important role in the tubal structure. We will study the diameter dependence and nonlinear properties of the RBM. Fig. \[fig\_frequency\] shows the frequency of the RBM in armchair, zigzag, and chiral tubes, i.e $(n_{1},n_{1})$, $(n_{1},0)$, and $(2n_{2},n_{2})$.
The frequency of all tubes can be fitted to function $\omega=665.3/d$ [cm$^{-1}$]{}, where $d$ is the diameter. The frequency of the RBM in SWMoS$_{2}$NTs is much lower than that in the single-walled carbon nanotube with the same diameter. The coefficient here 665.3 is about one third of the value of 2295.6 in the single-walled carbon nanotube.[@PopovVN1999prb] It is because the frequency of the RBM is related to the tensile mechanical properties,[@RaravikarNR] and the mechanical strength is weaker in the SWMoS$_{2}$NT.[@JiangJW2013sw] Furthermore, Mo and S atoms are heavier than the C atom in carbon nanotubes. Fig. \[fig\_frequency\] can be useful in the estimation of the diameter for SWMoS$_{2}$NTs in the experiment.
We further investigate the nonlinear effect on the RBM. The molecular dynamics is performed to simulate the radial breathing oscillation of SWMoS$_{2}$NTs (15, 15), (26, 0), and (20, 10). The periodic boundary condition is applied in the axial direction. The tube is optimized to the energy minimum configuration. At the optimized configuration, the initial velocity distributions are set according to the vibration morphology of the RBM. The total kinetic energy corresponding to this initial velocity distribution is $\Delta E =\alpha NE_{0}$, where $\alpha$ is the energy actuation parameter.[@JiangJW2012jap] $N$ is the total atom number. $E_{0}=0.54$ [meV]{} is a small piece of energy quantum for a single atom. The tube is allowed to oscillate within the NVE ensemble with the initial velocity distribution. The total kinetic energy and the potential energy exchange between each other during this radial breathing oscillation.
In the initial oscillation stage, there is only a single oscillation mode, i.e the radial breathing oscillation. If the oscillation is in the linear regime, i.e it has a small amplitude, this radial breathing oscillation can be preserved for a very long time and other vibration modes are seldom excited. However, if the oscillation is in the nonlinear regime, i.e with larger actuation parameter $\alpha$, then the other oscillation modes will be excited quickly. As a result, the radial breathing oscillation decays quickly, owning to the nonlinear induced phonon-phonon scattering between the RBM and the other excited modes.
Fig. \[fig\_energy\] shows the kinetic energy time history during the radial breathing oscillation of the three tubes. To study the nonlinear effect, we have shown only simulations with large actuation parameters $\alpha=10$, 30, and 50 in the figure. The radial breathing oscillation will not decay if very small actuation parameter is used, eg. $\alpha=1.0$, which will be similar as the graphene nanomechanical resonator.[@JiangJW2012nanotechnology] Fig. \[fig\_energy\] shows that in the initial stage the total kinetic energy oscillates between zero and a maximum value, which reflects the good resonant oscillation of the SWMoS$_{2}$NT. After some time, the radial breathing oscillation starts to be disturbed by some other vibration modes, which are excited by the nonlinear effect relating to the large actuation energy. Arrows in the figure depict the life time, at which the radial breathing oscillation starts to be disturbed. The inset in each panel shows the life time versus actuation parameter $\alpha$. The inset shows that the life time decays exponentially with increasing actuation parameter $\alpha$. The solid line is the exponential fitting to the calculated data. This set of simulations display that it is important to use small actuation parameter for the radial breathing oscillation, if this oscillation is used as the nanomechanical resonator.[@RaravikarNR]
In all of the three tubes, for very large actuation parameters such as $\alpha=30$ and 50, the kinetic energy increases sharply after the radial breathing oscillation is disturbed. It indicates that these tubes undergo a structure transition, which releases lots of potential energy. This potential energy is converted into the kinetic energy in the system. Fig. \[fig\_cfg\_md\] illustrates the structure evolution during the radial breathing oscillation in the SWMoS$_{2}$NT (20, 10) with actuation parameter $\alpha=50.0$. These snap-shots are produced by OVITO.[@ovito] Similar phenomena have been observed in the molecular dynamics simulation of tubes (15, 15) and (26, 0). The arrow attached to each atom represents the velocity of the atom at that moment. At $t=0$ [ps]{}, the radial breathing oscillation is actuated by adding an initial velocity distribution, which follows the eigen vector of the RBM (13rd mode) in Fig. \[fig\_u\_chiral\]. Configurations at $t=14.7$ and 16.0 [ps]{} show that the radial breathing oscillation is disturbed gradually by the seventh and eighth modes (with three-fold symmetry) in Fig. \[fig\_u\_chiral\] due to the nonlinear effect relating to large actuation energy. The radial breathing oscillation is completely replaced by these three-fold modes after $t=18.6$ [ps]{}. These three-fold oscillation is clearly demonstrated by the two configurations at $t=21.0$ and 22.3 [ps]{}, which exactly follow the eigen vector of the two three-fold modes. This shows that the three-fold mode is also a type of stable oscillation, i.e this oscillation does not destroy the tubal structure. However, after $t=94.7$ [ps]{}, the fifth and sixth modes (with two-fold symmetry) in Fig. \[fig\_u\_chiral\] start to be excited. These two-fold modes are unstable due to their special vibration morphology and their low frequencies. A low frequency indicates that these modes are able to deform the tubal structure quite a lot with small amount of energy. Indeed, the tube starts to be squashed by these two-fold modes after $t=105.1$ [ps]{}.
Conclusion
==========
In conclusion, we study the lattice dynamics properties of the SWMoS$_{2}$NT with arbitrary chiralities. In the construction of the $9\times 9$ dynamical matrix, the force constant matrix is calculated from the SW potential and the phonon modes are denoted by helical quantum numbers, which correspond to the line group in the system. The frequency and eigen vector of the low-frequency phonon modes are analyzed. The frequency of the RBM is found to be inversely proportional to the tube diameter as $\omega=665.3/d$ [cm$^{-1}$]{}, where $d$ is the tube diameter. We perform the molecular dynamics simulation to investigate the radial breathing mechanical oscillation, and find that the life time of the radial breathing oscillation decays exponentially with increasing oscillation energy. More specifically, in the initial stage, this radial breathing oscillation is disturbed by the three-fold modes. The SWMoS$_{2}$NTs are squashed by the two-fold modes in the final stage, if the oscillation is actuated into nonlinear regime.
**Acknowledgements** The work is supported by the Recruitment Program of Global Youth Experts of China (JWJ) and the German Research Foundation (DFG).
[48]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{}
|
---
abstract: |
We study the combinatorial geometry of a random closed multicurve on a surface of large genus $g$ and of a random square-tiled surface of large genus $g$. We prove that primitive components $\gamma_1, \dots,\gamma_k$ of a random multicurve $m_1\gamma_1+\dots +m_k\gamma_k$ represent linearly independent homology cycles with asymptotic probability $1$ and that all its weights $m_i$ are equal to $1$ with asymptotic probability $\sqrt{2}/2$. We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability $1$.
We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus $g$ are both very well-approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of $3g-3$ elements. In particular, we prove that the expected value of these quantities is asymptotically equivalent to $(\log(6g-6) + \gamma)/2 +
\log 2$.
These results are based on our formula for the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform asymptotic formula for intersection numbers of $\psi$-classes on $\overline{{\mathcal{M}}}_{g,n}$ for large $g$ proved by A. Aggarwal in 2020.
address:
- ' LaBRI, Domaine universitaire, 351 cours de la Libération, 33405 Talence, FRANCE '
- ' Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, FRANCE '
- ' St. Petersburg Department, Steklov Math. Institute, Fontanka 27, St. Petersburg 191023, and Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29B, St.Petersburg 199178 Russia'
- ' Center for Advanced Studies, Skoltech; Institut de Mathématiques de Jussieu – Paris Rive Gauche, Case 7012, 8 Place Aurélie Nemours, 75205 PARIS Cedex 13, France'
author:
- Vincent Delecroix
- Élise Goujard
- Peter Zograf
- Anton Zorich
date: 'July 8, 2020'
title: 'Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves '
---
Introduction and statements of main results {#s:intro}
===========================================
We aim to study random multicurves on surfaces of large genus $g$. Before proceeding to the statements of our main results we consider the classical setting of random integers and of random permutations which allow to set up the concept of a random compound object. For more information on the probabilistic analysis of decomposition of combinatorial objects into elementary components we recommend the monograph of R. Arratia, , S. Tavaré [@Arratia:Barbour:Tavare]. An enlightening introduction can be found in the blog post of T. Tao [@Tao].
**Prime decomposition of a random integer.** The Prime Number Theorem states that an integer number $n$ taken randomly in a large interval $[1,N]$ is prime with asymptotic probability $\frac{\log N}{N}$. Denote by $\omega(n)$ the number of prime divisors of an integer $n$ counted without multiplicities. In other words, if $n$ has prime decomposition $n=p_1^{m_1}\dots p_k^{m_k}$, let $\omega(n)=k$. By the Erdős–Kac theorem [@Erdos:Kac], the centered and rescaled distribution prescribed by the counting function $\omega(n)$ tends to the normal distribution: $$\label{eq:CLT1}
\lim_{N\to+\infty}\frac{1}{N}
{\operatorname{card}}\left\{n\le N\,\Big|\,
\frac{\omega(n)-\log\log N}{\sqrt{\log\log N}}
\le x\right\}=\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^x e^{-\tfrac{t^2}{2}} dt\,.$$ The subsequent papers of A. Rényi and P. Turán [@Renyi:Turan], and of A. Selberg [@Selberg] describe the rate of convergence.
**Cycle decomposition of a uniform random permutation.** Denote by ${{\mathrm K}}_n(\sigma)$ the number of disjoint cycles in the cycle decomposition of a permutation $\sigma$ in the symmetric group $S_n$. Consider the uniform probability measure on $S_n$. A random permutation $\sigma$ of $n$ elements has exactly $k$ cycles in its cyclic decomposition with probability ${{\mathbb P}}\big({{\mathrm K}}_n(\sigma) = k\big) =
\frac{s(n,k)}{n!}$, where $s(n,k)$ is the unsigned Stirling number of the first kind. It is immediate to see that ${{\mathbb P}}\big({{\mathrm K}}_n(\sigma) = 1\big) = \tfrac{1}{n}$. V. L. Goncharov proved in [@Goncharov] the following expansions for the expected value and for the variance of ${{\mathrm K}}_n$ as $n\to+\infty$: $$\label{eq:mean:variance:perm}
{{\mathbb E}}({{\mathrm K}}_n) = \log n + \gamma + o(1)\,,
\qquad
{{\mathbb V}}({{\mathrm K}}_n) = \log n + \gamma - \zeta(2) + o(1)\,,$$ as well as the following central limit theorem $$\label{eq:CLTperm}
\lim_{n\to+\infty}\frac{1}{n!}
{\operatorname{card}}\left\{\sigma\in S_n\,\Big|\,
\frac{{{\mathrm K}}_n(\sigma)-\log n}{\sqrt{\log n}}
\le x\right\}=\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^x e^{-\tfrac{t^2}{2}} dt\,.$$
As can be seen in and in , the number of cycles in the cycle decomposition of a random permutation is of the order of $\log n$, so cycles are “rare events”. In such situation one expects the distribution to be close to a Poisson distribution. Recall that the *Poisson distribution with parameter $\lambda$* is $$\label{eq:Poisson:def}
{\mathrm{Poi}}_\lambda(k) = e^{-\lambda}\ \frac{\lambda^k}{k!}\,,
\qquad\text{where }k=0,1,2,\dots$$ H. Hwang proved in [@Hwang:PhD] that the distribution of the random variable ${{\mathrm K}}_n$ is approximated by the Poisson distribution ${\mathrm{Poi}}_{\log n}$ in a very strong sense, which can be formalized as “*mod-Poisson converges with parameter $\log n$ and limiting function $1/\Gamma(t)$*”, using the terminology of E. Kowalski and A. Nikeghbali [@KowalskiNikeghbali]. We discuss the notion of mod-Poisson convergence in Section \[ss:non:uniform:permutations\]. We emphasize that such approximation is much stronger than the central limit theorem.
The result of H. Hwang [@Hwang:stirling] (representing a particular case of results in [@Hwang:PhD Chapter 5]) implies, that for large $n$ and for any positive $x$, the distribution of the number of cycles is uniformly well-approximated in a neighborhood of $x\log n$ by the Poisson distribution with parameter $\log n+{\operatorname{a}}(x)$, where the explicit correctional constant term ${\operatorname{a}}(x)$ is completely determined by the limiting function and does not depend on $n$. Namely, for any $x > 0$ we have uniformly in $0 \leq k \leq x \log n$ $$\label{eq:LDperm}
{{\mathbb P}}\big({{\mathrm K}}_n = k + 1\big)
= \frac{(\log n)^{k}}{n \cdot k!}
\left( \frac{1}{\Gamma(1 + \frac{k}{\log n})} + O\left( \frac{k}{(\log n)^2} \right) \right).$$
**Shape of a random multicurve on a surface of fixed genus.** Consider a smooth oriented closed surface $S$ of genus $g$. A *multicurve* $\gamma=\sum m_i\gamma_i$ (as the one in the picture by D. Calegari from [@Calegari] presented in Figure \[fig:multicurve\]) is a formal weighted sum of curves $\gamma_i$ with strictly positive integer weights $m_i$ where $\gamma_1,\dots,\gamma_k$ is a collection of non-contractible simple curves on $S$ that are pairwise non-isotopic. Following the usual convention, we do not distinguish between the free homotopy class of a multicurve and the multicurve itself.
Every multicurve $\gamma=\sum m_i\gamma_i$ defines a *reduced* multicurve $\gamma_{\mathit{reduced}}=\sum \gamma_i$. Note that the number of reduced multicurves on a surface of a fixed genus $g$ considered up to the action of the mapping class group ${\operatorname{Mod}}_g$ is finite. We say that two multicurves have the same topological type if they belong to the same orbit of ${\operatorname{Mod}}_g$. For example, a simple closed curve has one of the following topological types: either it is non-separating, or it separates the surface into subsurfaces of genera $g'$ and $g - g'$ for some $1 \leq g' \leq g/2$.
Multicurves on a closed surface of genus $g$ (considered up to free homotopy) are parameterized by integer points ${\mathcal{ML}}_g({{\mathbb Z}})$ in the space of measured laminations ${\mathcal{ML}}_g$ introduced by W. Thurston [@Thurston]. Any hyperbolic metric on $S$ provides a length function $\ell$ that associates to a closed curve $\gamma$ the length $\ell(\gamma)$ of its unique geodesic representative. The length function $\ell$ extends to multicurves as $\ell(\gamma) = m_1 \ell(\gamma_1) + \ldots + m_k
\ell(\gamma_k)$. Fixing some upper bound $L$ for the length of a multicurve, one can consider the finite set of multicurves of length at most $L$ on $S$ with respect to the length function $\ell$. See also the paper of M. Mirzakhani [@Mirzakhani:Thesis] and works of V. Erlandsson, H. Parlier, K. Rafi and J. Souto [@Rafi:Souto], [@Erlandsson:Parlier:Souto] and [@Erlandsson:Souto] for alternative ways to measure the length of a multicurve.
Choosing the uniform measure on all integral multicurves of length at most $L$ and letting $L$ tend to infinity we define a “random multicurve” on a surface of fixed genus $g$ in the same manner as we considered “random integers”, see Section \[ss:Frequencies:of:simple:closed:curves\] for details. We emphasize that studying asymptotic statistical geometry of multicurves as the bound $L$ tends to infinity we always keep the genus $g$, considered as a parameter, fixed. One can ask, for example, what is the probability that a random simple closed curve separates the surface of genus $g$ in two components? Or, more generally, what is the probability that the reduced multicurve $\gamma_{\mathit{reduced}}$ associated to a random multicurve $\gamma$ separates the surface of genus $g$ into several components? With what probability a random multicurve $m_1\gamma_1+m_2\gamma_2+\dots+m_k\gamma_k$ has $k=1,2,\dots,3g-3$ primitive connected components $\gamma_1,\dots,\gamma_k$? What are the typical weights $m_1,\dots,m_k$?
A beautiful answer to all these questions was found by M. Mirzakhani in [@Mirzakhani:grouth:of:simple:geodesics]. She expressed the frequency of multicurves of any fixed topological type in terms of the intersection numbers $\int_{\overline{{\mathcal{M}}}_{g',n'}}\psi_1^{d_1}\dots\psi_{n'}^{d_{n'}}$, where $2g'+n'\le 2g$. (These intersection numbers are also called *correlators of Witten’s two dimensional topological gravity*). For small genera $g$ the formula of M. Mirzakhani provides explicit rational values for the quantities discussed above. For example, the reduced multicurve associated to a random multicurve on a surface of genus $2$ without cusps as in Figure \[fig:multicurve\] separates the surface with probability $\tfrac{67}{315}$ and has $1$, $2$ or $3$ components with probabilities $\tfrac{7}{27},\tfrac{5}{9},\tfrac{5}{27}$ respectively.
The formulae of Mirzakhani are applicable to surfaces of any genera. The exact values of the intersection numbers can be computed through Witten–Kontsevich theory [@Witten], [@Kontsevich]. However, despite the fact that these intersection numbers were extensively studied, there were no uniform estimates for Witten correlators for large $g$ till the recent results of A. Aggarwal [@Aggarwal:intersection:numbers]. This is one of the reasons why the following question remained open.
\[question:multicurves\] What shape has a random multicurve on a surface of large genus?
The current paper aims to answer this question to some extent. Denote by $K_g(\gamma)$ the number of components $k$ of the multicurve $\gamma=\sum_{i=1}^k m_i\gamma_i$ counted without multiplicities.
\[th:multicurves:a:b:c\] Consider a random multicurve $\gamma=\sum_{i=1}^k
m_i\gamma_i$ on a surface $S$ of genus $g$. Let $\gamma_{\mathit{reduced}}=\gamma_1+\dots+\gamma_k$ be the underlying reduced multicurve. The following asymptotic properties hold as $g\to+\infty$.
- The multicurve $\gamma_{\mathit{reduced}}$ does not separate the surface (i.e. $S-\sqcup\gamma_i$ is connected) with probability which tends to 1.
- The probability that a random multicurve $\gamma=\sum_{i=1}^k m_i\gamma_i$ is primitive (i.e. that $m_1=m_2=\dots=1$) tends to $\frac{\sqrt{2}}{2}$.
- For any sequence of positive integers $k_g$ with $k_g =
o(\log g)$ the probability that a random multicurve $\gamma=\sum_{i=1}^{k_g} m_i\gamma_i$ is primitive (i.e. that $m_1=\dots=m_{k_g}=1$) tends to $1$ as $g\to +\infty$.
There is no contradiction between parts (b) and (c) of the above Theorem since in (c) we consider only those random multicurves for which the underlying primitive multicurve has an imposed number $k_g$ of components, while in (b) we consider all multicurves. In other words, in part (c) we consider the conditional probability. Part (b) of the above Theorem admits the following generalization.
\[th:multicurves:bounded:weights\] For any positive integer $m$, the probability that all weights $m_i$ of a random multicurve $\gamma=m_1\gamma_1+m_2\gamma_2+\dots$ on a surface of genus $g$ are bounded by a positive integer $m$ (i.e. that $m_1\le m, m_2\le m,\dots$) tends to $\sqrt{\frac{m}{m+1}}$ as $g\to+\infty$.
We describe the probability distribution of the random variable $K_g(\gamma)$ later in this section. However, to follow comparison with prime decomposition of random integers and with cycle decomposition of random permutations we present here the central limit theorem stated for random multicurves.
\[thm:CLT:multicurve\] Choose a non-separating simple closed curve $\rho_g$ on a surface of genus $g$. Denote by $\iota(\rho_g,\gamma)$ the geometric intersection number of $\rho_g$ and $\gamma$. The centered and rescaled distribution defined by the counting function $K_g(\gamma)$ tends to the normal distribution: $$\begin{gathered}
\lim_{g\to+\infty}
\sqrt{\frac{3\pi g}{2}}\cdot 12g
\cdot (4g-4)!
\cdot\left(\frac{9}{8}\right)^{2g-2}
\\
\lim_{N\to+\infty}
\frac{1}{N^{6g-6}}
{\operatorname{card}}\Bigg(\bigg\{\gamma\in{\mathcal{ML}}_g({{\mathbb Z}})\,\bigg|\,
\iota(\rho_g,\gamma)\le N\quad \text{and}
\\
\frac{K_g(\gamma)-\frac{\log g}{2}}{\sqrt{\frac{\log g}{2}}}
\le x\bigg\}
/{\operatorname{Stab}}(\rho_g)\Bigg)
=\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^x e^{-\tfrac{t^2}{2}} dt\,.\end{gathered}$$
Here ${\operatorname{Stab}}(\rho_g)\subset{\operatorname{Mod}}_g$ is the stabilizer of the simple closed curve $\rho_g$ in the mapping class group ${\operatorname{Mod}}_g$.
In plain words, the above theorems say that the components $\gamma_1,\dots,\gamma_k$ of a random multicurve $\gamma=\sum_{i=1}^k m_i\gamma_i$ on a surface of large genus $g$ have all chances to go around $k$ independent handles, where $k$ is close to $\frac{1}{2}\log g$, and that with a high probability all the weights $m_i$ of a random multicurve are uniformly small. In particular, with probability greater than $0.7$ a random multicurve is primitive, i.e. all the weights $m_i$ are equal to $1$.
Our description of the asymptotic geometry of random multicurves on surfaces of large genus and of random square-tiled surfaces of large genera relies on fundamental recent results [@Aggarwal:intersection:numbers] of A. Aggarwal, who proved, in particular, the large genus asymptotic formulae for the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ and for the intersection numbers of $\psi$-classes on $\overline{{\mathcal{M}}}_{g,n}$, conjectured by the authors in [@DGZZ:volume].
**Random square-tiled surfaces of large genus.** A *square-tiled surface* is a closed oriented quadrangulated surface (i.e. a surface built by gluing identical squares along their edges), such that the quadrangulation satisfies the following properties. Consider the flat metric on the surface induced by the flat metric on the squares. We assume that edges of the squares are identified by isometries, which implies that the induced flat metric is non-singular on the complement of the vertices of the squares. We require that the parallel transport of a vector $\vec v$ tangent to the surface along any closed path avoiding conical singularities brings the vector $\vec v$ either to itself or to $-\vec v$. In other words, we require that the holonomy group of the metric is ${{\mathbb Z}}/2{{\mathbb Z}}$ (compared to ${{\mathbb Z}}/4{{\mathbb Z}}$ for a general quadrangulation). This holonomy assumption implies that defining some edge to be “horizontal” or “vertical” we uniquely determine for each of the remaining edges, whether it is “horizontal” or “vertical”. Speaking of square-tiled surfaced we always assume that the choice of horizontal and vertical edges is done.
Our holonomy assumption implies that the number of squares adjacent to any vertex is even. In this article we restrict ourselves to consideration of square-tiled surfaces with no conical singularities of angle $\pi$. In other words, vertices adjacent to exactly two squares are not allowed. Square-tiled surfaces satisfying the above restrictions can be seen as integer points in the total space ${\mathcal{Q}}_g$ of the vector bundle of holomorphic quadratic differentials over the moduli space of complex curves ${\mathcal{M}}_g$.
A stronger restriction on the quadrangulation imposing trivial linear holonomy to the induced flat metric defines *Abelian* square-tiled surfaces; they correspond to integer points in the total space ${\mathcal{H}}_g$ of the vector bundle of holomorphic Abelian differentials over the moduli space of complex curves ${\mathcal{M}}_g$. The subset of square-tiled surfaces having prescribed linear holonomy and prescribed cone angle at each conical singularity corresponds to the set of integer points in the associated *stratum* in the moduli space of quadratic or Abelian differentials respectively.
A square-tiled surface admits a natural decomposition into maximal horizontal cylinders. For example, the square-tiled surface in the left picture of Figure \[fig:square:tiled:surface:and:associated:multicurve\] (which, for simplicity of illustration, contains conical points with cone angles $\pi$) has four maximal horizontal cylinders highlighted by different shades of grey. Two of these cylinders are composed of two horizontal bands of squares. Each of the remaining two cylinders is composed of a single horizontal band of squares.
For any positive integer $N$, the set ${\mathcal{ST}{\hspace*{-3pt}}_{g}}(N)$ of square-tiled surfaces of genus $g$ having no singularities of angle $\pi$ and having at most $N$ squares in the tiling is finite. Choosing the uniform measure on the set ${\mathcal{ST}{\hspace*{-3pt}}_{g}}(N)$ and letting the bound $N$ for the number of squares tend to infinity, we define a “random square-tiled surface” of fixed genus $g$ in the same manner as we considered “random multicurves” on a fixed surface, see Section \[ss:Frequencies:of:square:tiled:surfaces\] for details. We emphasize that studying asymptotic statistical geometry of square-tiled surfaces as the bound $N$ tends to infinity we always keep the genus $g$, considered as a parameter, fixed. One can study the decomposition of a random square-tiled surface into maximal horizontal cylinders in the same sense as we considered prime decomposition of random integers or cyclic decomposition of random permutations.
For each stratum in the moduli space of Abelian differentials, we computed in [@DGZZ:one:cylinder:Yoccoz:volume] the probability that a random square-tiled surface in this stratum has a single cylinder in its horizontal cylinder decomposition. This result can be seen as an analog of the Prime Number Theorem for square-tiled surfaces. In particular, using results [@Aggarwal:Volumes] and [@Chen:Moeller:Sauvaget:Zagier] we proved that for strata of Abelian differentials corresponding to large genera, this probability is asymptotically $\frac{1}{d}$, where $d$ is the dimension of the stratum. However, more detailed description of statistics of square-tiled surfaces in individual strata of Abelian differentials is currently out of reach with the exception of several low-dimensional strata. Conjecturally, for any stratum of Abelian differentials of dimension $d$, the statistics of the number of maximal horizontal cylinders of a random square-tiled surface in the stratum becomes very well-approximated by the statistics of the number ${{\mathrm K}}_n(\sigma)$ of disjoint cycles in a random permutation of $d$ elements as $d\to+\infty$; see Section \[s:speculations\] for details.
In the current paper we address more general question.
\[question:square:tiled\] What shape has a random square-tiled surface of large genus assuming that it does not have conical points of angle $\pi$?
Denote by $K_g(S)$ the number of maximal horizontal cylinders in the cylinder decomposition of a square-tiled surface $S$ of genus $g$.
\[th:square:tiled:a:b:c\] A random square-tiled surface $S$ of genus $g$ with no conical singularities of angle $\pi$ has the following asymptotic properties as $g\to+\infty$.
- All conical singularities of $S$ are located at the same leaf of the horizontal foliation and at the same leaf of the vertical foliation with probability which tends to 1.
- The probability that each maximal horizontal cylinder of $S$ is composed of a single band of squares tends to $\frac{\sqrt{2}}{2}$.
- For any sequence of positive integers $k_g$ with $k_g =
o(\log g)$ the probability that each maximal horizontal cylinder of a random $k_g$-cylinder square-tiled surface of genus $g$ is composed of a single band of squares tends to $1$ as the genus $g$ tends to $+\infty$.
Similarly to the case of multicurves, part (b) of the above Theorem admits the following generalization.
\[th:square:tiled:bounded:weights\] For any $m\in{{\mathbb N}}$, the probability that all maximal horizontal cylinders of a random square-tiled surface of genus $g$ have at most $m$ bands of squares tends to $\sqrt{\frac{m}{m+1}}$ as $g\to+\infty$.
We state now the central limit theorem for square-tiled surfaces.
\[thm:CLT:square:tiled\] The centered and rescaled distribution defined by the counting function $K_g(S)$ tends to the normal distribution as $g \to +\infty$: $$\begin{gathered}
\lim_{g\to+\infty}
3\pi g\cdot\left(\frac{9}{8}\right)^{2g-2}
\\
\lim_{N\to+\infty}
\frac{1}{N^{6g-6}}
{\operatorname{card}}\left\{S\in{\mathcal{ST}{\hspace*{-3pt}}_{g}}(N)\,\bigg|\,
\frac{k(S)-\tfrac{\log g}{2}}{\sqrt{\tfrac{\log g}{2}}}
\le x\right\}
=\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^x e^{-\frac{t^2}{2}} dt\,.\end{gathered}$$
**Approach to the study of random multicurves and of random square-tiled surfaces of large genera: from $p_g(k)$ to $q_g(k)$.** It is time to admit that the parallelism between Theorems \[th:multicurves:a:b:c\]–\[thm:CLT:multicurve\] and respectively Theorems \[th:square:tiled:a:b:c\]–\[thm:CLT:square:tiled\] is not accidental.
Recall that we denote by $K_g(\gamma)$ the number of components $k$ of the multicurve $\gamma=\sum_{i=1}^k
m_i\gamma_i$ on a surface of genus $g$ counted without multiplicities and by $K_g(S)$ the number of maximal horizontal cylinders in the cylinder decomposition of a square-tiled surface $S$ of genus $g$. The following theorem is a direct corollary of Theorem 1.21 from Section 1.8 in [@DGZZ:volume]. (For the sake of completeness we reproduce the original Theorem in Section \[ss:Frequencies:of:square:tiled:surfaces\] below.)
\[th:same:distribution\] For any genus $g\ge 2$ and for any $k\in{{\mathbb N}}$, the probability ${p}_g(k)$ that a random multicurve $\gamma$ on a surface of genus $g$ has exactly $k$ components counted without multiplicities coincides with the probability that a random square-tiled surface $S$ of genus $g$ has exactly $k$ maximal horizontal cylinders: $$\label{eq:same:p:g}
{p}_g(k)
={{\mathbb P}}\big(K_g(\gamma)=k\big)
={{\mathbb P}}\big(K_g(S)=k\big)\,.$$ In other words, $K_g(\gamma)$ and $K_g(S)$, considered as random variables, determine the same probability distribution ${p}_g(k)$, where $k=1,2,\dots,3g-3$.
The Theorem above shows that Questions 1 and 2 are, basically, equivalent. The description of the large genus asymptotic properties of the resulting probability distribution ${p}_g(k)$ can be seen as the main unified goal of the current paper.
The starting point of our approach to the study of the probability distribution ${p}_g(k)$ is the the formula for the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ of the moduli space of holomorphic quadratic differentials derived in our recent paper [@DGZZ:volume]. This formula represents ${\operatorname{Vol}}{\mathcal{Q}}_g$ as a finite sum of contributions of square-tiled surfaces of all possible topological types (Section \[ss:intro:Masur:Veech:volumes\] describes this in detail). However, the number of such topological types grows exponentially as genus grows. Moreover, the contribution of square-tiled surfaces of a fixed topological type to ${\operatorname{Vol}}{\mathcal{Q}}_g$ is expressed in terms of the intersection numbers of $\psi$-classes (Witten correlators) which are difficult to evaluate explicitly in large genera.
We conjectured in [@DGZZ:volume] that in large genera, the dominant part of the contribution to ${\operatorname{Vol}}{\mathcal{Q}}_g$ comes from square-tiled surfaces having all conical singularities at the same horizontal level. The topological type (see Section \[ss:MV:volume\] for the rigorous definition of the “topological type”) of such square-tiled surfaces is completely determined by the number $k$ of maximal horizontal cylinders which varies from $1$ to $g$. This conjecture suggested a strategy for overcoming the first difficulty, reducing the study of all immense variety of topological types of square-tiled surfaces to the study of $g$ explicit topological types. We also conjectured in [@DGZZ:volume] that under certain assumptions on $g$ and $n$, the intersection numbers $\int_{\overline{{\mathcal{M}}}_{g,n}}\psi_1^{d_1}\dots\psi_{n}^{d_{n}}$ are uniformly well-approximated by an explicit closed expression in the variables $d_1, \dots, d_n$, and that the error term becomes uniformly small with respect to all possible partitions $d_1+\dots+d_n=3g-3+n$ for large values of $g$. This conjecture suggested a plan for overcoming the second difficulty reducing analysis of volume contributions of square-tiled surfaces of $g$ distinguished topological types to analysis of closed expressions in multivariate harmonic sums. Such analysis led us, in particular, to the conjectural large genus asymptotics of the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$.
In terms of the probability distributions, we replace the original distribution ${p}_g(k)$ with an auxiliary probability distribution ${q}_g(k)$ in this approach. The distribution ${q}_g(k)$ describes the contributions of square-tiled surfaces of $g$ distinguished topological types (corresponding to the situation when all conical singularities are located at same horizontal layer and the surface has $k=1,\dots,g$ maximal horizontal cylinders), where, moreover, we replace the Witten correlators with the corresponding asymptotic expressions. The precise definition of ${q}_g(k)$ is given in Equation in Section \[ss:Volume:contribution:single:vertex\]. Informally, our conditional asymptotic result in [@DGZZ:volume] stated that for large genera $g$ the auxiliary distribution ${q}_g(k)$ well-approximates the original probability distribution ${p}_g(k)$ modulo the conjectures mentioned above.
Deep analysis of volume contributions of square-tiled surfaces of different topological types was performed by A. Aggarwal in [@Aggarwal:intersection:numbers]. Moreover, in the same paper A. Aggarwal established uniform asymptotic bounds for Witten correlators using elegant approach through biased random walk. In particular, he proved all conjectures from [@DGZZ:volume] (in a stronger form) transforming conditional results from [@DGZZ:volume] into unconditional statements.
In the current paper we follow the original approach, approximating the probability distribution ${p}_g(k)$ with a slight modification of the probability distribution ${q}_g(k)$ as described above. However, the fine asymptotic analysis of A. Aggarwal allows to state that ${q}_g(k)$ “well-approximates” ${p}_g(k)$ in much stronger sense than it was claimed in the original preprint [@DGZZ:volume]. Moreover, we realized that our “slight modification of the probability distribution ${q}_g(k)$” has combinatorial interpretation of independent interest and admits a detailed description based on technique developed by H. Hwang in [@Hwang:PhD].
Having explained the scheme of our approach we can state now the main results concerning the probability distribution ${p}_g(k)$. We start with a formal definition of the “slight modification of the probability distribution ${q}_g(k)$” through random permutations. It plays an important role in the current paper.
**Random non-uniform permutations and distribution $\boldsymbol{q_{n,\infty,1/2}}$.** Let $\theta$ be a sequence $\{\theta_k\}_{k \geq 1}$ of non-negative real numbers. Given a permutation $\sigma \in
S_n$ with cycle type $(1^{\mu_1} 2^{\mu_2} \ldots
n^{\mu_n})$, where $1\cdot\mu_1+2\cdot\mu_2+\dots+n\cdot\mu_n=n$, we define its *weight* $w_\theta(\sigma)$ by the following formula: $$w_\theta(\sigma) = \theta_1^{\mu_1} \theta_2^{\mu_2} \cdots \theta_n^{\mu_n}.$$ To every collection of positive numbers $\theta = \{\theta_k\}_{k \geq 1}$, we associate a probability measure on the symmetric group $S_n$ by means of the weight function defined above: $$\label{eq:proba:sym:group}
{{\mathbb P}}_{\theta,n}(\sigma) := \frac{w_{\theta}(\sigma)}{n!\cdot W_{\theta,n}}\,,
\quad \text{where} \qquad
W_{\theta,n} := \frac{1}{n!} \sum_{\sigma \in S_n} w_\theta(\sigma)
\quad \text{and} \quad k\in{{\mathbb N}}\,.$$ Denote by ${{\mathbb P}}_{n,\infty,1/2}$ the non-uniform probability measure on the symmetric group $S_n$ associated to the collection of strictly positive numbers $\theta_k=\zeta(2k)/2$, where $k=1,2,\dots$ and $\zeta$ is the Riemann zeta function. Consider the random variable ${{\mathrm K}}_n(\sigma)$ on the symmetric group $S_n$, where the random permutation $\sigma$ corresponds to the law ${{\mathbb P}}_{n,\infty,1/2}$ and ${{\mathrm K}}_n(\sigma)$ is the number of disjoint cycles in the cycle decomposition of such random permutation $\sigma$. The random variable ${{\mathrm K}}_n(\sigma)$ takes integer values in the range $[1,n]$. We introduce the following notation: $$\label{eq:q:3g:minus:3:infty:1:2:as:proba}
q_{n,\infty,1/2}(k)
= {{\mathbb P}}_{n, \infty, 1/2}\big({{\mathrm K}}_{3g-3}(\sigma) = k\big)$$ for the law of the random variable ${{\mathrm K}}_{n}(\sigma)$ with respect to the probability measure ${{\mathbb P}}_{n,\infty,1/2}$. We prove in Section \[s:sum:over:single:vertex:graphs\] series of results which informally can be summarized by the following claim: *the probability distribution $q_{3g-3, \infty,
1/2}$ well-approximates the probability distribution ${q}_g$.* We admit that the approximating distribution ${q}_g$ will be formally defined only later, namely, in Equation in Section \[ss:Volume:contribution:single:vertex\] and, strictly speaking, would not be used explicitly. The above claim explains, however, our interest for the probability distribution $q_{3g-3, \infty, 1/2}$ which would be actually used for approximation. An important step of comparison of distributions ${p}_g$ and $q_{3g-3,
\infty, 1/2}$ is established in Lemma \[lem:cycle:distribution:vs:harmonic:sum\] stated and proved in Section \[ss:non:uniform:permutations\]. Theorems \[thm:permutation:mod:poisson:introduction\] and \[thm:permutation:asymptotics\] below carry comprehensive information on the probability distribution $q_{3g-3, \infty, 1/2}(k)
={{\mathbb P}}_{3g-3,\infty,1/2}\big({{\mathrm K}}_{3g-3}(\sigma) = k\big)$.
\[thm:permutation:mod:poisson:introduction\] Let ${{\mathbb P}}_{n,\infty,1/2}$ be the probability distribution on $S_n$ associated to the collection $\theta_k =
\zeta(2k)/2$. Then for all $t \in {{\mathbb C}}$ we have as $n \to
+\infty$ $$\label{eq:mod:Poisson:for:permutations}
{{\mathbb E}}_{n,\infty,1/2}\left(t^{{{\mathrm K}}_n}\right) =
(2n)^{\tfrac{t-1}{2}}
\cdot \frac{t\cdot\Gamma(\tfrac{3}{2})}{\Gamma(1+\tfrac{t}{2})}\
\left(1 + O \left( \frac{1}{n} \right) \right)\,,$$ where the error term is uniform for $t$ in any compact subset of ${{\mathbb C}}$.
For any $\lambda > 0$, let $u_{\lambda, 1/2}(k)$ for $k\in{{\mathbb N}}$ be the coefficients of the following Taylor expansion $$\label{eq:poisson:gamma:1:2}
e^{\lambda (t-1)}\cdot
\displaystyle
\frac{t \cdot \Gamma\left(\tfrac{3}{2}\right)}{\Gamma\left(1 + \tfrac{t}{2}\right)}
=
\sum_{k \geq 1} u_{\lambda, 1/2}(k) \cdot t^k$$ Recall that $\Gamma\left(\tfrac{3}{2}\right)=\frac{\Gamma\left(\tfrac{1}{2}\right)}{2}=\frac{\sqrt{\pi}}{2}$. We have $$u_{\lambda,1/2}(k)
=\sqrt{\pi}\cdot e^{-\lambda}\cdot
\frac{1}{k!}\cdot
\sum_{i=1}^k \binom{k}{i} \cdot \phi_i \cdot
\left(\frac{1}{2}\right)^i \cdot \lambda^{k-i}\,.$$ where the $\phi_j$ are defined through the Taylor expansion $$\label{eq:Taylor:for:1:Gamma}
\frac{t}{\Gamma(1+t)}
= \frac{1}{\Gamma(t)}
= \sum_{j=1}^{+\infty} \phi_j\cdot\frac{t^j}{j!}\,.$$ The first values are given by $$\phi_1=1;\quad
\phi_2=2\gamma;\quad
\phi_3=3\big(\gamma^2-\zeta(2)\big).$$ Theorem \[thm:permutation:mod:poisson:introduction\] has the following consequence.
\[cor:approximation:q:u:introduction\] Uniformly in $k \geq 1$ we have as $n \to \infty$ $$q_{n, \infty, 1/2}(k) = u_{\lambda_{n}, 1/2}(k) + O \left(\frac{1}{n}\right)$$ where $\lambda_{n} = \frac{\log(2n)}{2}$.
Theorem \[thm:permutation:mod:poisson:introduction\] and its Corollary \[cor:approximation:q:u:introduction\] are particular cases of Theorem \[cor:approximation:q:u\] and Corollary \[cor:permutation:mod:poisson\] stated and proved in Section \[s:mod:poisson\]. We also illustrate the numerical aspects of this approximation in Section \[s:numerics\].
\[thm:permutation:asymptotics\] Let $\lambda_n = \log(2n)/2$. Then, for any $x > 0$, we have uniformly in $0 \leq k \leq x \lambda_n$ the following asymptotic behavior as $n\to+\infty$ $$\begin{gathered}
\label{eq:th:permutations:probabiliy:k}
q_{n,\infty,1/2}(k+1)=
{{\mathbb P}}_{n,\infty,1/2}({{\mathrm K}}_n(\sigma) = k + 1)
=\\=
e^{-\lambda_n}
\cdot \frac{(\lambda_n)^k}{k!}
\cdot \left(
\frac{\sqrt{\pi}}{2 \Gamma\left(1 + \frac{k}{2 \lambda_n}\right)}
+ O\left(\frac{k}{(\log n)^2}\right)
\right)\,.\end{gathered}$$ For any $x > 1$ such that $x \lambda_n$ is an integer we have $$\begin{gathered}
\label{eq:th:permutations:tail}
\sum_{k=x\lambda_n+1}^n q_{n,\infty,1/2}(k+1)
={{\mathbb P}}_{n,\infty,1/2}\big({{\mathrm K}}_n(\sigma)
> x\lambda_n + 1\big)
=\\=
\frac{(2n)^{-\tfrac{x \log x - x +1}{2}}}
{\sqrt{2\pi\lambda_n x} }
\cdot
\frac{x}{x - 1}
\cdot \left(
\frac{\sqrt{\pi}}{2 \Gamma\left(1 + \frac{x}{2}\right)}
+ O\left( \frac{1}{\log n} \right)\right)\,,\end{gathered}$$ where the error term is uniform over $x$ in compact subsets of $(1, +\infty)$. Similarly, for any $0 < x < 1$ such that $x \lambda_n$ is an integer we have $$\begin{gathered}
\label{eq:th:permutations:head}
\sum_{k=0}^{x\lambda_n} q_{n,\infty,1/2}(k+1)
={{\mathbb P}}_{n,\infty,1/2}\big({{\mathrm K}}_n(\sigma) \leq x\lambda_n + 1\big)
=\\=
\frac{(2n)^{-\tfrac{x \log x - x +1}{2}}}
{\sqrt{2\pi\lambda_n x} }
\cdot
\frac{x}{1 - x}
\cdot \left(
\frac{\sqrt{\pi}}{2 \Gamma\left(1 + \frac{x}{2}\right)}
+ O\left( \frac{1}{\log n} \right)\right)\,,\end{gathered}$$ where the error term is uniform over $x$ in compact subsets of $(0, 1)$.
Theorem \[thm:permutation:asymptotics\] is a particular case of Corollary \[cor:multi:harmonic:asymptotic:all:k\] stated and proved in Section \[ss:LD:and:CLT\]. Note that for $x \not= 1$, we have $x \log x - x + 1 > 0$. Hence, Equations and provide explicit polynomial bounds in $n$ for the tails of the distribution.
\[rm:lambda:plus:c\] Let $\displaystyle G(x) = \frac{\sqrt{\pi}}{2 \Gamma(1 + \tfrac{x}{2})}$ and define $$\label{eq:correctional:term}
{\operatorname{a}}(x)= \frac{ \log G(x) }{x - 1}.$$ Since $\log G(1) = 0$, the function ${\operatorname{a}}(x)$ admits a continuous extension at $x=1$ $$\lim_{x \to 1} {\operatorname{a}}(x) = G'(1) = \frac{\gamma}{2} + \log(2) - 1\,,$$ where $\gamma$ is the Euler–Mascheroni constant. Now for any $x > 0$, uniformly for $0 \leq k \leq x \lambda_n$ we have $$(\lambda_n)^k \cdot G \left( \frac{k}{\lambda_n} \right)
=
e^{-{\operatorname{a}}\left(\frac{k}{\lambda_n}\right)} \left(\lambda_n + {\operatorname{a}}\left( \tfrac{k}{\lambda_n} \right)\right)^k
\cdot
\left(1 + O\left( \frac{k}{\lambda_n^2} \right) \right) \,.$$ We can hence rewrite the right-hand side of : for any $x > 0$ we have uniformly in $0 \leq k \leq x \lambda_n$ the following asymptotic behavior as $n\to+\infty$ $$q_{n,\infty,1/2}(k+1)=
e^{-\left(\lambda_n + {\operatorname{a}}\left(\frac{k}{\lambda_n}\right)\right)}
\cdot \frac{\left(\lambda_n + {\operatorname{a}}\left( \tfrac{k}{\lambda_n} \right)\right)^k}{k!}
\cdot \left( 1 + O\left(\frac{k}{(\log n)^2}\right) \right)\,.$$ In the latter expression, the right-hand side reads as the value of a Poisson random variable with parameter $\lambda_n + {\operatorname{a}}\left(\frac{k}{\lambda_n}\right)$.
The extended version of the above results as well as the closely related notion of mod-Poisson convergence are discussed in Section \[ss:non:uniform:permutations\]. The above theorems follow from singularity analysis at the boundary of the domain of definition of holomorphic functions representing the relevant generating series performed by H. Hwang in [@Hwang:PhD].
**Properties of the probability distribution $\boldsymbol{{p}_g(k)}$.** The key theorems below strongly rely on asymptotic analysis of the Masur–Veech volume of the moduli space of quadratic differentials performed by A. Aggarwal in [@Aggarwal:intersection:numbers] and on uniform asymptotic bounds for Witten correlators obtained in [@Aggarwal:intersection:numbers].
\[thm:mod:poisson:pg\] Let $K_g$ be the random variable satisfying the probability law . For all $t\in{{\mathbb C}}$ such that $|t|<\frac{8}{7}$ the following asymptotic relation is valid as $g\to+\infty$: $$\label{eq:mod:poisson:pg:introduction}
{{\mathbb E}}\left(t^{K_g}\right) =
(6g-6)^{\tfrac{t-1}{2}}
\cdot \frac{t \Gamma(\tfrac{3}{2})}{\Gamma(\tfrac{t}{2})}\
\left(1 + o(1)\right)\,.$$ Moreover, for any compact set $U$ in the open disk $|t|<\frac{8}{7}$ there exists $\delta(U)>0$, such that for all $t\in U$ the error term has the form $O(g^{-\delta(U)})$.
Note that the right-hand side of expression is very close to the right-hand side of the analogous expression from Theorem \[thm:permutation:mod:poisson:introduction\] evaluated at $n=3g-3$.
We expect that the mod-Poisson convergence holds in a large domain than the disk $|t|<\frac{8}{7}$. If our guess is correct, the asymptotics below for the distribution $p_g$ should hold for larger interval of $x$ than described below. We also expect that the mod-Poisson convergence analogous to holds for all non-hyperelliptic components of all strata of holomorphic quadratic differentials; see Conjecture \[conj:quadratic:strong:form\] in Section \[s:speculations\] for more details.
\[thm:pg:asymptotics\]
Let $\lambda_{3g-3} = \log(6g-6)/2$. For any $x\in
\left[0, \frac{1}{\log\frac{9}{4}}\right)$ we have uniformly in $0 \leq k \leq x \lambda_{3g-3}$ $$\begin{gathered}
\label{eq:pg:equivalent}
{p}_g(k + 1) =
{{\mathbb P}}\big(K_g(\gamma) = k + 1\big)
=\\=
e^{-\lambda_{3g-3}}
\cdot \frac{\lambda_{3g-3}^{k}}{k!}
\cdot \left(
\frac{\sqrt{\pi}}{2 \Gamma\left(1 + \frac{k}{2 \lambda_{3g-3} }\right)}
+ O\left( \frac{k}{(\log g)^2} \right)\right)\,.\end{gathered}$$ For any $x \in (1, 1.236\,]$ such that $x \lambda_{3g-3}$ is an integer we have $$\begin{gathered}
\label{eq:tail:p:1:236}
\sum_{k = x \lambda_{3g-3} + 1}^{3g-3} {p}_g(k + 1)
= {{\mathbb P}}\big(K_g(\gamma) > x \lambda_{3g-3} + 1\big)
=\\=
\frac{(6g-6)^{-\tfrac{x \log x - x + 1}{2}}}{ \sqrt{2 \pi \lambda_{3g-3} x} }
\cdot
\frac{x}{x - 1}
\cdot \left(
\frac{\sqrt{\pi}}{2 \Gamma\left(1 + \frac{x}{2} \right)}
+ O\left( \frac{1}{\log g} \right)\right)\,,\end{gathered}$$ where the error term is uniform over $x$ in compact subsets of $(1, 1.236\,]$. Similarly for any $x \in (0, 1)$ such that $x \lambda_{3g-3}$ is an integer we have $$\begin{gathered}
\label{eq:head:p:0:1}
\sum_{k = 0}^{x \lambda_{3g-3}} {p}_g(k + 1)
= {{\mathbb P}}\big(K_g(\gamma) \leq x \lambda_{3g-3} + 1\big)
=\\=
\frac{(6g-6)^{-\tfrac{x \log x - x + 1}{2}}}{ \sqrt{2 \pi \lambda_{3g-3} x} }
\cdot
\frac{x}{1 - x}
\cdot \left(
\frac{\sqrt{\pi}}{2 \Gamma\left(1 + \frac{x}{2} \right)}
+ O\left( \frac{1}{\log g} \right)\right)\,,\end{gathered}$$ where the error term is uniform over $x$ in compact subsets of $(0, 1)$. Finally, $$\begin{gathered}
\label{eq:tail:1:4}
\sum_{k=\lfloor 0.09 \log g \rfloor}^{\lceil 0.62\log g\rceil}
{p}_g(k)
=\\
= {{\mathbb P}}\Big(0.09 \log g < K_g(\gamma) < 0.62 \log g\Big)
= 1 - O\left((\log g)^{24} g^{-1/4}\right)\,.\end{gathered}$$
Similarly to Remark \[rm:lambda:plus:c\], Equation tells, in particular, that any $x$ in the interval $[0, 1.236]$ (which carries, essentially, all but $O(g^{-1/4})$ part of the total mass of the distribution) and for large $g$, the values ${p}_g(k+1)$ for $k$ in a neighborhood of $x\frac{\log g}{2}$ of size $o(\log g)$ are uniformly well-approximated by the Poisson distribution ${\mathrm{Poi}}_\lambda(k)$ with parameter $\lambda=\frac{\log(6g-6)}{2}+{\operatorname{a}}(x)$, where ${\operatorname{a}}(x)$ is defined in .
The approximation results given in Theorem \[thm:permutation:mod:poisson:introduction\] for $q_{n,\infty,1/2}$ and in Theorem \[thm:mod:poisson:pg\] for $p_g$ imply an asymptotic expansion of the moments that we present now. Recall that the *Stirling number of the second kind*, denoted $S(i,j)$, is the number of ways to partition a set of $i$ objects into $j$ non-empty subsets.
\[th:pg:cumulants\] For any fixed $k\in{{\mathbb N}}$ the difference between the $i$-th moments of random variables with the probability distributions ${p}_g$ and $q_{3g-3,\infty,1/2}$ tends to zero as $g\to+\infty$.
Furthermore, the $i$-th cumulant $\kappa_i(K_g(\sigma))$ of the random variable $K_g$ associated to the probability distribution ${p}_g$ admits the following asymptotic expansion: $$\begin{gathered}
\label{eq:k:cumulant}
\kappa_i(K_g)
=\frac{\log(6g-6)}{2} + \frac{\gamma}{2} + \log 2 -
\\
- \sum_{j=2}^i S(i,j)
\cdot
(-1)^{j} \cdot \zeta(j) \cdot (j-1)! \cdot \left(2^{j} - 1 \right)
\cdot \left(\frac{1}{2}\right)^j
+ O\left(\frac{1}{g}\right)
\ \text{as }g\to+\infty\,,\end{gathered}$$ where $S(i,j)$ are the Stirling numbers of the second kind. In particular, the mean value ${{\mathbb E}}(K_g)$ and the variance ${{\mathbb V}}(K_g)$ satisfy: $$\begin{aligned}
{{\mathbb E}}(K_g) & = \kappa_1(K_g) = \frac{\log(6g-6)}{2} + \frac{\gamma}{2} + \log 2 + o(1)\,,\\
{{\mathbb V}}(K_g) & = \kappa_2(K_g) = \frac{\log(6g-6)}{2} + \frac{\gamma}{2} + \log 2
- \frac{3}{4} \zeta(2) + o(1)\,,\end{aligned}$$ where $\gamma = 0.5572\ldots$ denotes the Euler–Mascheroni constant. The third and the fourth cumulants $\kappa_3(K_g)$ and $\kappa_4(K_g)$ admit the following asymptotic expansions: $$\begin{aligned}
\kappa_3(K_g) & = \frac{\log(6g-6)}{2} + \frac{\gamma}{2} + \log 2
- \frac{9}{4} \zeta(2) + \frac{7}{4} \zeta(3) + o(1)\,,\\
\kappa_4(K_g) & = \frac{\log(6g-6)}{2} + \frac{\gamma}{2} + \log 2
- \frac{21}{4} \zeta(2) + \frac{21}{2} \zeta(3)
- \frac{45}{8} \zeta(4) + o(1)\,.\end{aligned}$$
**Other approaches to random multicurves.** One more interesting aspect of geometry of random multicurves is the lengths statistics of simple closed hyperbolic geodesics associated to components of multicurves of fixed topological type. M. Mirzakhani studied in [@Mirzakhani:statistics] random pants decompositions of a hyperbolic surface of genus $g$. She considered the orbit ${\operatorname{Mod}}_g\gamma$ of a multicurve $\gamma=\gamma_1+\dots+\gamma_{3g-3}$ corresponding to a fixed pants decomposition. Choosing multicurves in this orbit of hyperbolic length at most $L$ she got a finite collection of multicurves. Letting $L\to+\infty$ she defined a *random pants decomposition*. M. Mirzakhani proved in Theorem 1.2 of [@Mirzakhani:statistics] that under the normalization $x_i=\frac{\ell(g\cdot\gamma_i)}{L}$ for $i=1,\dots,3g-3$, the lengths statistics of components of a random pair of pants has the limiting density function $const\cdot x_1\dots x_{3g-3}$ with respect to the Lebesgue measure on the unit simplex. F. Arana-Herrera and M. Liu independently proved in [@Arana:Herrera:Equidistribution:of:horospheres], [@Arana:Herrera:Counting:multi:geodesics] and in [@Mingkun] a generalization of this result to arbitrary multicurves. In terms of square-tiled surfaces the resulting hyperbolic lengths statistics coincides with statistics of flat lengths of the waist curves of maximal horizontal cylinders of the square-tiled surface (see Section 1.9 in [@DGZZ:volume]). It would be interesting to study implications of these results to the large genus limit.
In the regime where one considers simple closed curves of lengths at most $L$ for any fixed $L>0$ and lets the genus tend to $+\infty$, a very precise description of the distribution of lengths was provided by M. Mirzakhani and B. Petri in [@Mirzakhani:Petri].
It would be interesting to establish relations between random multicurves and a general framework of random partitions introduced by A. M. Vershik in [@Vershik].
**Random quadrangulations versus random square-tiled surfaces.** In this article we are concerned with random square-tiled surfaces, which are a particular case of random quadrangulations, which are themselves a particular case of random combinatorial maps (surfaces obtained from gluing polygons). The two latter families have a much longer mathematical history. The two important parameters are the number of polygons $N$ and the genus $g$.
Surfaces obtained by random gluing of polygons have been studied for a long time. Their enumeration can be traced back to the works of W. T. Tutte [@Tutte] for $g=0$ and of T. R. S. Walsh and A. B. Lehman [@Walsh:Lehman] for arbitrary $g$. In particular, their results allow to compute the probability of getting a closed surface of genus $g$ as a result of a random pairwise gluing of the sides of a $2n$-gon. Somewhat later J. Harer and D. Zagier [@Harer:Zagier] were able to enumerate genus $g$ gluings of a $2n$-gon in a more explicit and effective way. This was a crucial ingredient in their computation of the orbifold Euler characteristic of the moduli space $\mathcal{M}_g$ of complex algebraic curves.
Surfaces obtained from randomly glued polygons have been studied since a long time in physics in relation to string theory and quantum gravity as in the paper of V. Kazakov, I. Kostov, A. Migdal [@Kazakov:Kostov:Migdal]. In this approach one often works with surfaces of genus zero and with several perturbative terms corresponding to surfaces of low genera.
In the case $g=0$ and $N \to+\infty$, the Brownian map has been shown to be the scaling limit of various models of combinatorial maps, see the surveys of G. Miermont [@Miermont] and of J.-F. Le Gall [@LeGall] and the references therein. Combinatorial maps also admit local limits, as proved, in particular, in the papers of O. Angel and O. Schramm [@Angel:Schramm], of M. Krikun [@Krikun], of P. Chassaing and B. Durhuus [@Chassaing:Durhuus], of L. Ménard [@Menard]. In higher but fixed genus, the scaling limits giving rise to higher genera Brownian maps have been investigated by J. Bettinelli in [@Bettinelli10; @Bettinelli12].
Surfaces obtained by gluing polygons without restriction on the genus have been studied by R. Brooks and E. Makover in [@Brooks:Makover], by in [@Gamburd], by S. Chmutov and B. Pittel in [@Chmutov:Pittel], by A. Alexeev and P. Zograf in [@Alexeev:Zograf], and by T. Budzinski, N. Curien and B. Petri in [@Budzinski:Curien:Petri:a; @Budzinski:Curien:Petri:b]. In this approach the genus $g$ of the resulting surface is a random variable whose expectation is proportional to the number of polygons $N$. See also the recent paper of S. Shresta [@Shrestha] studying square-tiled surfaces in a similar context.
Finally, in the regime $g = \theta N$ with $\theta \in
[0,\tfrac{1}{2})$ a local limit has been conjectured by N. Curien in [@Curien] and recently proved by T. Budzinski and B. Louf in [@Budzinski:Louf].
Note that our approach is different from all approaches mentioned above. We fix the genus of the surface, and consider square-tiled surfaces tiled with at most $N$ squares (or geodesic multicurves of length bounded by some large number $L$). We define asymptotic frequencies of square-tiled surfaces or of geodesic multicurves of a fixed combinatorial type by passing to the limit when $N$ (respectively $L$) tends to infinity. Only when the resulting limiting frequencies (probabilities) are already defined in each individual genus we study their behavior in the regime when the genus becomes very large. This approach is natural in the context of dynamics of polygonal billiards, dynamics of interval exchange transformations and of translation surfaces, and in the context of geometry and dynamics on the moduli space of quadratic differentials.
Note also that all but negligible part of our square-tiled surfaces of genus $g$ have $4g-4$ vertices of valence $6$, while all other vertices have valence $4$, and the number of such vertices is incomparably larger than $g$. This is one more substantial difference between our random surface model and the random quadrangulations considered in the probability theory literature where, usually, there is no such degree constraint imposed and vertices, typically, have arbitrary degrees even if the resulting surface has genus $0$. As a result, our square-tiled surfaces locally look like a tiling of ${{\mathbb R}}^2$ by squares except around $4g-4$ conical singularities with cone angle $3\pi$. This is not the case for a random planar quadrangulation.
A regime similar to ours was used by H. Masur, K. Rafi and A. Randecker who studied in [@Masur:Rafi:Randecker] the covering radius of random translation surfaces (corresponding to Abelian differentials) and by M. Mirzakhani, who studied in [@Mirzakhani:random] random hyperbolic surfaces of fixed large genus $g$.
**Structure of the paper. ** To make the current paper self-contained, we reproduce in Section \[s:Background:material\] all necessary background material. We start by recalling in Section \[ss:MV:volume\] the definition of the Masur–Veech volume of the moduli space of quadratic differentials ${\mathcal{Q}}_g$. We sketch in Section \[ss:Square:tiled:surfaces:and:associated:multicurves\] how Masur–Veech volumes are related to count of square-tiled surfaces. In the same section we associate to every square-tiled surface a multicurve and we recall the notion of a stable graph, particularly important in the framework of the current paper. We present in Section \[ss:intro:Masur:Veech:volumes\] the formula for the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ and a theorem of A. Aggarwal on the asymptotic value of this volume for large genera $g$. The reader interested in more ample information is addressed to the original papers [@DGZZ:volume] and [@Aggarwal:intersection:numbers] respectively. In Section \[ss:Frequencies:of:simple:closed:curves\] we recall Mirzakhani’s count [@Mirzakhani:grouth:of:simple:geodesics] of frequencies of multicurves. In Section \[ss:Frequencies:of:square:tiled:surfaces\] we explain why Questions \[question:multicurves\] and \[question:square:tiled\] are equivalent and demystify Theorem \[th:same:distribution\]. In Section \[ss:conjecture:on:correlators\] we recall the recent breakthrough results of A. Aggarwal [@Aggarwal:intersection:numbers] on large genus asymptotics of Witten correlators.
In Section \[s:sum:over:single:vertex:graphs\] we recall general background from the works of H. K. Hwang [@Hwang:PhD], and of E. Kowalski, P.-L. Méliot, A. Nikeghbali, D. Zeindler [@KowalskiNikeghbali], [@NikeghbaliZeindler], [@FerayMeliotNikeghbali] on random permutations and on mod-Poisson convergence and apply this general technique to the probability distribution $q_{3g-3,\infty,1/2}$. In particular, we prove Theorems \[thm:permutation:mod:poisson:introduction\] and \[thm:permutation:asymptotics\].
We then introduce a probability distribution ${p^{(1)}}_g(k)$ of the random variable $K_g(\gamma)=K_g(S)$ restricted to non-separating random multicurves $\gamma$ on a surface of genus $g$ (equivalently restricted to random square-tiled surfaces of genus $g$ having single horizontal critical level). Using the results of A. Aggarwal [@Aggarwal:intersection:numbers] on asymptotics of Witten correlators we prove that the distribution $q_{3g-3,\infty,1/2}$ very well-approximates the distribution ${p^{(1)}}_g$ (namely, that they share the same mod-Poisson convergence but ${p^{(1)}}_g$ has smaller radius of convergence). This allows us to extend all the results obtained for random permutations to these special random multicurves (special random square-tiled surfaces).
It remains, however, to pass from the special multi-curves (and square-tiled surfaces) to general ones. The necessary estimates are prepared in Section \[s:disconnecting:multicurves\]. In a sense, this step was already performed by A. Aggarwal in [@Aggarwal:intersection:numbers], who proved a generalization of our conjecture from [@DGZZ:volume] claiming that random multicurves (random square-tiled surfaces) which do not contribute to the distribution ${p^{(1)}}_g$ become rare in large genera. This justifies the fact that the distribution ${p^{(1)}}_g$ well-approximates the distribution ${p}_g$. However, to prove this statement in a much stronger form stated in the current paper we have to adjust certain estimates from Sections 9 and 10 from the original paper [@Aggarwal:intersection:numbers] to our current needs.
We recommend to readers interested in all details of Section \[s:disconnecting:multicurves\] to read it in parallel with Sections 9 and 10 of the original paper [@Aggarwal:intersection:numbers]. (Actually, we recommend reading the entire paper [@Aggarwal:intersection:numbers] of A. Aggarwal. We have no doubt that the reader looking for a deep understanding of the subject would appreciate beauty, strength and originality of the proofs and ideas in [@Aggarwal:intersection:numbers] as we do.)
Having obtained all necessary estimates in Section \[s:disconnecting:multicurves\] we prove in Section \[s:proofs\] that the distribution ${p^{(1)}}_g$ well-approximates the distribution ${p}_g$. By transitivity this implies that the distribution $q_{3g-3,\infty,1/2}$ well-approximates the distribution ${p}_g$. We show in Section \[s:proofs\] how the properties of $q_{3g-3,\infty,1/2}$ derived in Section \[s:sum:over:single:vertex:graphs\] imply all our main results.
In Section \[s:numerics\] we compare our theoretical results with experimental and numerical data. We complete by suggesting in Section \[s:speculations\] a conjectural description of the combinatorial geometry of random Abelian square-tiled surfaces of large genus and of random square-tiled surfaces restricted to any non-hyperelliptic component of any stratum in the moduli space of Abelian or quadratic differentials of large genus.
This article is born from Appendices D–F of the original preprint [@DGZZ:volume]. The latter contained several conjectures and derived from them all other results as “conditional theorems”. All these conjectures were proved by A. Aggarwal; see Theorems \[conj:Vol:Qg\], \[th:correlators:upper:bound\], \[th:asymptotics:of:correlators:lower\], \[th:asymptotics:of:correlators:upper\], and Corollary \[conj:one:vertex:dominates:for:fixed:k\] in the current paper or Theorems 1.7 and Propositions 1.2, 4.1, 4.2, 10.7 respectively in the original paper [@Aggarwal:intersection:numbers]. Moreover, most of the results are proved in [@Aggarwal:intersection:numbers] in a much stronger form than we initially conjectured. Combining our initial approach with these recent results of A. Aggarwal and elaborating close ties with random permutations allowed us to radically strengthen the initial assertions from [@DGZZ:volume].
**Acknowledgements.** We are very much indebted to A. Aggarwal for transforming our dreams into reality by proving all our conjectures from [@DGZZ:volume]. We also very much appreciate his advices, including the indication on how to compute multi-variate harmonic sums, which was crucial for making correct predictions in [@DGZZ:volume]. His numerous precious comments on the preliminary versions of this paper allowed us to correct a technical mistake and numerous typos and improve the presentation.
Results of this paper were directly or indirectly influenced by beautiful and deep ideas of Maryam Mirzakhani.
We thank S. Schleimer, who was the first person to notice that our experimental data on statistics of cylinder decompositions of random Abelian square-tiled surfaces seems to have resemblance with statistics of cycle decomposition of random permutations.
We thank F. Petrov for the reference to the paper [@Goncharov] in the context of cycle decomposition of random permutations.
We thank M. Bertola, A. Borodin, G. Borot, D. Chen, A. Eskin, V. Feray, M. Kazarian, S. Lando, M. Liu, H. Masur, M. Möller, B. Petri, K. Rafi, A. Sauvaget, J. Souto, D. Zagier and D. Zvonkine for useful discussions.
We thank B. Green for the talk at the conference “CMI at 20” and T. Tao for his blog both of which were very inspiring for us.
We thank D. Calegari for kind permission to use a picture from his book [@Calegari] in Figure \[fig:multicurve\].
We are grateful to MPIM in Bonn, where part of this work was performed, to Chebyshev Laboratory in St. Petersburg State University, to MSRI in Berkeley and to MFO in Oberwolfach for providing us with friendly and stimulating environment.
Background material {#s:Background:material}
===================
Masur–Veech volume of the moduli space of quadratic differentials {#ss:MV:volume}
-----------------------------------------------------------------
Consider the moduli space ${\mathcal{M}}_{g,n}$ of complex curves of genus $g$ with $n$ distinct labeled marked points. The total space ${\mathcal{Q}}_{g,n}$ of the cotangent bundle over ${\mathcal{M}}_{g,n}$ can be identified with the moduli space of pairs $(C,q)$, where $C\in{\mathcal{M}}_{g,n}$ is a smooth complex curve with $n$ (labeled) marked points and $q$ is a meromorphic quadratic differential on $C$ with at most simple poles at the marked points and no other poles. In the case $n=0$ the quadratic differential $q$ is holomorphic. Thus, the *moduli space of quadratic differentials* ${\mathcal{Q}}_{g,n}$ is endowed with the canonical symplectic structure. The induced volume element ${d\!{\operatorname{Vol}}}$ on ${\mathcal{Q}}_{g,n}$ is called the *Masur–Veech volume element*. (In the next Section we provide alternative more common definition of the Masur–Veech volume element.)
A non-zero quadratic differential $q$ in ${\mathcal{Q}}_{g,n}$ defines a flat metric $|q|$ on the complex curve $C$. The resulting metric has conical singularities at zeroes and simple poles of $q$. The total area of $(C,q)$ $${\operatorname{Area}}(C,q)=\int_C |q|$$ is positive and finite. For any real $a > 0$, consider the following subset in ${\mathcal{Q}}_{g,n}$: $${\mathcal{Q}}^{{\operatorname{Area}}\le a}_{g,n} := \left\{(C,q)\in{\mathcal{Q}}_{g,n}\,|\, {\operatorname{Area}}(C,q) \le a\right\}\,.$$ Since ${\operatorname{Area}}(C,q)$ is a norm in each fiber of the bundle ${\mathcal{Q}}_{g,n} \to {\mathcal{M}}_{g,n}$, the set ${\mathcal{Q}}^{{\operatorname{Area}}\le a}_{g,n}$ is a ball bundle over ${\mathcal{M}}_{g,n}$. In particular, it is non-compact. However, by the independent results of H. Masur [@Masur:82] and W. Veech [@Veech:Gauss:measures], the total mass of ${\mathcal{Q}}^{{\operatorname{Area}}\le
a}_{g,n}$ with respect to the Masur–Veech volume element is finite. Following a common convention we define the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ as $$\label{eq:def:Vol:Q:g:n}
{\operatorname{Vol}}{\mathcal{Q}}_{g,n}
=(12g-12+4n)\cdot{\operatorname{Vol}}{\mathcal{Q}}^{{\operatorname{Area}}\le\frac{1}{2}}_{g,n}\,.$$
Square-tiled surfaces, simple closed multicurves and stable graphs {#ss:Square:tiled:surfaces:and:associated:multicurves}
------------------------------------------------------------------
We have already mentioned that a non-zero meromorphic quadratic differential $q$ on a complex curve $C$ defines a flat metric with conical singularities. One can construct a discrete collection of quadratic differentials of this kind by assembling together identical flat squares in the following way. Take a finite set of copies of the oriented $1/2 \times 1/2$-square for which two opposite sides are chosen to be horizontal and the remaining two sides are declared to be vertical. Identify pairs of sides of the squares by isometries in such way that horizontal sides are glued to horizontal sides and vertical sides to vertical. We get a topological surface $S$ without boundary. We consider only those surfaces obtained in this way which are connected and oriented. The form $dz^2$ on each square is compatible with the gluing and endows $S$ with a complex structure and with a non-zero quadratic differential $q$ with at most simple poles. The total area ${\operatorname{Area}}(S,q)$ is $\frac{1}{4}$ times the number of squares. We call such surface a *square-tiled surface*.
(0,0)(175,10) (2,-13)[$2\gamma_1$]{} (113,-23)[$\gamma_2$]{} (1.5,-46)[$\phantom{2}\gamma_3$]{} (23,-77)[$2\gamma_4$]{}
(0,0)(-26,-6)(4,-20)[$2\gamma_1$]{} (53.5,-36.5)[$\gamma_2$]{} (15,-54)[$\gamma_3$]{} (20,-82)[$2\gamma_4$]{}
Suppose that the resulting closed *square-tiled surface* has genus $g$ and $n$ conical singularities with angle $\pi$, i.e. $n$ vertices adjacent to only two squares. For example, the square-tiled surfaces in Figure \[fig:square:tiled:surface:and:associated:multicurve\] has genus $g=0$ and $n=7$ conical singularities with angle $\pi$. Consider the complex coordinate $z$ in each square and a quadratic differential $(dz)^2$. It is easy to check that the resulting square-tiled surface inherits the complex structure and globally defined meromorphic quadratic differential $q$ having simple poles at $n$ conical singularities with angle $\pi$ and no other poles. Thus, any square-tiled surface of genus $g$ having $n$ conical singularities with angle $\pi$ canonically defines a point $(C,q)\in{\mathcal{Q}}_{g,n}$. Fixing the size of the square once and forever and considering all resulting square-tiled surfaces in ${\mathcal{Q}}_{g,n}$ we get a discrete subset ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}$ in ${\mathcal{Q}}_{g,n}$.
Define ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(N)\subset{\mathcal{ST}{\hspace*{-3pt}}_{g,n}}$ to be the subset of square-tiled surfaces in ${\mathcal{Q}}_{g,n}$ tiled with at most $N$ identical squares. Square-tiled surfaces form a lattice in period coordinates of ${\mathcal{Q}}_{g,n}$, which justifies the following alternative definition of the Masur–Veech volume: $$\label{eq:Vol:sq:tiled}
{\operatorname{Vol}}{\mathcal{Q}}_{g,n}
= 2(6g-6+2n)\cdot
\lim_{N\to+\infty}
\frac{{\operatorname{card}}({\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(2N))}{N^{d}}\,,$$ where $d=6g-6+2n=\dim_{{{\mathbb C}}}{\mathcal{Q}}_{g,n}$. In this formula we assume that *all* conical singularities of square-tiled surfaces are labeled (i.e., counting square-tiled surfaces we label not only $n$ simple poles but also all zeroes).
**Multicurve associated to a cylinder decomposition.** Any square-tiled surface admits a decomposition into maximal horizontal cylinders filled with isometric closed regular flat geodesics. Every such maximal horizontal cylinder has at least one conical singularity on each of the two boundary components. The square-tiled surface in Figure \[fig:square:tiled:surface:and:associated:multicurve\] has four maximal horizontal cylinders which are represented in the picture by different shades. For every maximal horizontal cylinder choose the corresponding waist curve $\gamma_i$.
By construction each resulting simple closed curve $\gamma_i$ is non-periferal (i.e. it does not bound a topological disk without punctures or with a single puncture) and different $\gamma_i, \gamma_j$ are not freely homotopic on the underlying $n$-punctured topological surface. In other words, pinching simultaneously all waist curves $\gamma_i$ we get a legal stable curve in the Deligne–Mumford compactification $\overline{{\mathcal{M}}}_{g,n}$.
We encode the number of circular horizontal bands of squares contained in the corresponding maximal horizontal cylinder by the integer weight $H_i$ associated to the curve $\gamma_i$. The above observation implies that the resulting formal linear combination $\gamma=\sum
H_i\gamma_i$ is a simple closed integral multicurve in the space ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$ of measured laminations. For example, the simple closed multicurve associated to the square-tiled surface as in Figure \[fig:square:tiled:surface:and:associated:multicurve\] has the form $2\gamma_1+\gamma_2+\gamma_3+2\gamma_4$.
Given a simple closed integral multicurve $\gamma$ in ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$ consider the subset ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\gamma)\subset{\mathcal{ST}{\hspace*{-3pt}}_{g,n}}$ of those square-tiled surfaces, for which the associated horizontal multicurve is in the same ${\operatorname{Mod}}_{g,n}$-orbit as $\gamma$ (i.e. it is homeomorphic to $\gamma$ by a homeomorphism sending $n$ marked points to $n$ marked points and preserving their labeling). Denote by ${\operatorname{Vol}}(\gamma)$ the contribution to ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ of square-tiled surfaces from the subset ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\gamma)\subset{\mathcal{ST}{\hspace*{-3pt}}_{g,n}}$: $${\operatorname{Vol}}(\gamma)
= 2(6g-6+2n)\cdot
\lim_{N\to+\infty}
\frac{{\operatorname{card}}({\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(2N)\cap{\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\gamma))}{N^{d}}\,.$$ The results in [@DGZZ:meanders:and:equidistribution] imply that for any $\gamma$ in ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$ the above limit exists, is strictly positive, and that $$\label{eq:Vol:Q:as:sum:of:Vol:gamma}
{\operatorname{Vol}}{\mathcal{Q}}_{g,n}
=\sum_{[\gamma]\in\mathcal{O}} {\operatorname{Vol}}(\gamma)\,,$$ where the sum is taken over representatives $[\gamma]$ of all orbits $\mathcal{O}$ of the mapping class group ${\operatorname{Mod}}_{g,n}$ in ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$.
\[def:asymptotic:probability\] Formula allows to interpret the ratio ${\operatorname{Vol}}(\gamma)/{\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ as the *asymptotic probability* to get a square-tiled surface in ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\gamma)$ taking a random square-tiled surface in ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(N)$ as $N\to+\infty$. We will also call the same quantity by the *frequency* of square-tiled surfaces of type ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\gamma)$ among all square-tiled surfaces.
**Stable graph associated to a multicurve.** Following M. Kontsevich [@Kontsevich] we assign to any multicurve $\gamma$ a *stable graph* $\Gamma(\gamma)=\Gamma(\gamma_{\mathit{reduced}})$. The stable graph $\Gamma(\gamma)$ is a decorated graph dual to $\gamma_{reduced}$. It consists of vertices, edges, and “half-edges” also called “legs”. Vertices of $\Gamma(\gamma)$ represent the connected components of the complement $S_{g,n}\setminus\gamma_{\mathit{reduced}}$. Each vertex is decorated with the integer number recording the genus of the corresponding connected component of $S_{g,n}\setminus\gamma_{reduced}$. By convention, when this number is not explicitly indicated, it equals to zero. Edges of $\Gamma(\gamma)$ are in the natural bijective correspondence with curves $\gamma_i$; an edge joins a vertex to itself when on both sides of the corresponding simple closed curve $\gamma_i$ we have the same connected component of $S_{g,n}\setminus\gamma_{reduced}$. Finally, the $n$ punctures are encoded by $n$ *legs*. The right picture in Figure \[fig:square:tiled:surface:and:associated:multicurve\] provides an example of the stable graph associated to the multicurve $\gamma$.
Pinching a complex curve of genus $g$ with $n$ marked points by all components of a reduced multicurve $\gamma_{reduced}$ we get a stable complex curve representing a point in the Deligne–Mumford compactification $\overline{\mathcal{M}}_{g,n}$. In this way stable graphs encode the boundary cycles of $\overline{\mathcal{M}}_{g,n}$. In particular, the set ${\mathcal{G}}_{g,n}$ of all stable graphs is finite. It is in the natural bijective correspondence with boundary cycles of $\overline{\mathcal{M}}_{g,n}$ or, equivalently, with ${\operatorname{Mod}}_{g,n}$-orbits of reduced multicurves in ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$.
Formula for the Masur–Veech volumes {#ss:intro:Masur:Veech:volumes}
-----------------------------------
In this section we introduce polynomials $N_{g,n}(b_1, \ldots, b_n)$ that appear in different contexts, in particular, in the formula for the Masur–Veech volume.
Let $g$ be a non-negative integer and $n$ a positive integer. Let the pair $(g,n)$ be different from $(0,1)$ and $(0,2)$. Let $d_1,\dots,d_n$ be an ordered partition of $3g - 3 + n$ into a sum of non-negative integers, $|d|=d_1+\dots+d_n=3g-3+n$, let $\boldsymbol{d}$ be a multiindex $(d_1,\dots,d_n)$ and let $b^{2\boldsymbol{d}}$ denote $b_1^{2d_1}\cdot\cdots\cdot b_n^{2d_n}$.
Define the following homogeneous polynomial $N_{g,n}(b_1,\dots,b_n)$ of degree $6g-6 + 2n$ in variables $b_1,\dots,b_n$ in the following way. $$\label{eq:N:g:n}
N_{g,n}(b_1,\dots,b_n)=
\sum_{|d|=3g-3+n}c_{\boldsymbol{d}} b^{2\boldsymbol{d}}\,,$$ where $$\label{eq:c:subscript:d}
c_{\boldsymbol{d}}=\frac{1}{2^{5g-6+2n}\, \boldsymbol{d}!}\,
\langle \tau_{d_1} \dots \tau_{d_n}\rangle_{g,n}$$ $$\label{eq:correlator}
\langle \tau_{d_1} \dots \tau_{d_n}\rangle_{g,n}
=\int_{\overline{{\mathcal{M}}}_{g,n}} \psi_1^{d_1}\dots\psi_n^{d_n}\,,$$ and $\boldsymbol{d}!=d_1!\cdots d_n!$. Note that $N_{g,n}(b_1,\dots,b_n)$ contains only even powers of $b_i$, where $i=1,\dots,n$.
Following [@AEZ:genus:0] we consider the following linear operators ${\mathcal{Y}}(\boldsymbol{H})$ and ${\mathcal{Z}}$ on the spaces of polynomials in variables $b_1,b_2,\dots$, where $H_1, H_2, \dots$ are positive integers. The operator ${\mathcal{Y}}(\boldsymbol{H})$ is defined on monomials as $$\label{eq:cV}
{\mathcal{Y}}(\boldsymbol{H})\ :\quad
\prod_{i=1}^{k} b_i^{m_i} \longmapsto
\prod_{i=1}^{k} \frac{m_i!}{H_i^{m_i+1}}\,,$$ and extended to arbitrary polynomials by linearity. The operator ${\mathcal{Z}}$ is defined on monomials as $$\label{eq:cZ}
{\mathcal{Z}}\ :\quad
\prod_{i=1}^{k} b_i^{m_i} \longmapsto
\prod_{i=1}^{k} \big(m_i!\cdot \zeta(m_i+1)\big)\,,$$ and extended to arbitrary polynomials by linearity.
Given a stable graph ${\Gamma}$ denote by $V(\Gamma)$ the set of its vertices and by $E(\Gamma)$ the set of its edges. To each stable graph $\Gamma\in{\mathcal{G}}_{g,n}$ we associate the following homogeneous polynomial $P_\Gamma$ of degree $6g-6+2n$. To every edge $e\in E(\Gamma)$ we assign a formal variable $b_e$. Given a vertex $v\in V(\Gamma)$ denote by $g_v$ the integer number decorating $v$ and denote by $n_v$ the valency of $v$, where the legs adjacent to $v$ are counted towards the valency of $v$. Take a small neighborhood of $v$ in $\Gamma$. We associate to each half-edge (“germ” of edge) $e$ adjacent to $v$ the monomial $b_e$; we associate $0$ to each leg. We denote by $\boldsymbol{b}_v$ the resulting collection of size $n_v$. If some edge $e$ is a loop joining $v$ to itself, $b_e$ would be present in $\boldsymbol{b}_v$ twice; if an edge $e$ joins $v$ to a distinct vertex, $b_e$ would be present in $\boldsymbol{b}_v$ once; all the other entries of $\boldsymbol{b}_v$ correspond to legs; they are represented by zeroes. To each vertex $v\in E(\Gamma)$ we associate the polynomial $N_{g_v,n_v}(\boldsymbol{b}_v)$, where $N_{g,v}$ is defined in . We associate to the stable graph $\Gamma$ the polynomial obtained as the product $\prod b_e$ over all edges $e\in E({\Gamma})$ multiplied by the product $\prod N_{g_v,n_v}(\boldsymbol{b}_v)$ over all $v\in V({\Gamma})$. We define $P_\Gamma$ as follows: $$\begin{gathered}
\label{eq:P:Gamma}
P_\Gamma(\boldsymbol{b})
=
\frac{2^{6g-5+2n} \cdot (4g-4+n)!}{(6g-7+2n)!}\cdot
\\
\frac{1}{2^{|V({\Gamma})|-1}} \cdot
\frac{1}{|\operatorname{Aut}({\Gamma})|}
\cdot
\prod_{e\in E({\Gamma})}b_e\cdot
\prod_{v\in V({\Gamma})}
N_{g_v,n_v}(\boldsymbol{b}_v)
\,.\end{gathered}$$
The Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ of the stratum of quadratic differentials with $4g-4+n$ simple zeros and $n$ simple poles has the following value: $$\label{eq:square:tiled:volume}
{\operatorname{Vol}}{\mathcal{Q}}_{g,n}
= \sum_{{\Gamma}\in {\mathcal{G}}_{g,n}} {\operatorname{Vol}}(\Gamma)\,,$$ where the contribution of an individual stable graph $\Gamma$ has the form $$\label{eq:volume:contribution:of:stable:graph}
{\operatorname{Vol}}(\Gamma)={\mathcal{Z}}(P_\Gamma)\,.$$
\[rk:volume:contribution\] The contribution of any individual stable graph has the following natural interpretation. We have seen that stable graphs $\Gamma$ in ${\mathcal{G}}_{g,n}$ are in natural bijective correspondence with ${\operatorname{Mod}}_{g,n}$-orbits of *reduced* multicurves $\gamma_{\mathit{reduced}}=\gamma_1+\gamma_2+\dots$, where simple closed curves $\gamma_i$ and $\gamma_j$ are not isotopic for any $i\neq j$. Let $\Gamma\in{\mathcal{G}}_{g,n}$, let $k=|V(\Gamma)|$, let $\gamma_{\mathit{reduced}}=\gamma_1+\dots+\gamma_k$ be the reduced multicurve associated to $\Gamma$. Let $\gamma_{\boldsymbol{H}}
=\gamma(\Gamma,\boldsymbol{H})=
H_1\gamma_1+\dots H_k\gamma_k$, where $\boldsymbol{H}=(H_1,\dots,H_k)\in{{\mathbb N}}^k$. We have $$\label{eq:Vol:Gamma}
{\operatorname{Vol}}(\Gamma)=
\sum_{H\in{{\mathbb N}}^k}
{\operatorname{Vol}}\big(\Gamma,\boldsymbol{H}\big)$$ where the contribution ${\operatorname{Vol}}\big(\Gamma,\boldsymbol{H})$ of square-tiled surfaces with the horizontal cylinder decomposition of type $(\Gamma,\boldsymbol{H})$ to ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ is given by the formula: $$\label{eq:contribution:of:gamma:to:volume}
{\operatorname{Vol}}\big(\Gamma,\boldsymbol{H}\big)
={\mathcal{Y}}(\boldsymbol{H})(P_\Gamma)\,.$$
In other words, we can rearrange the sum in as $$\label{eq:Vol:Q:g:n:as:sum}
{\operatorname{Vol}}{\mathcal{Q}}_{g,n}
=\sum_{[\gamma]\in\mathcal{O}} {\operatorname{Vol}}(\gamma)
=\sum_{\Gamma\in{\mathcal{G}}_{g,n}}
\sum_{\{[\gamma]\,| \Gamma(\gamma)=\Gamma\}} {\operatorname{Vol}}(\gamma)
\,,$$ where $$\sum_{\{[\gamma]\,|\, \Gamma(\gamma)=\Gamma\}} {\operatorname{Vol}}(\gamma) = {\operatorname{Vol}}(\Gamma)\,.$$ In this way we can extend Definition \[def:asymptotic:probability\] and speak of *asymptotic probability* of getting a square-tiled surface in ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\Gamma)=\cup_{\{[\gamma]\,|\,
\Gamma(\gamma)=\Gamma\}}{\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(\gamma)$ taking a random square-tiled surface in ${\mathcal{ST}{\hspace*{-3pt}}_{g,n}}(N)$ as $N\to+\infty$. In the same way we define *frequency* of square-tiled surfaces having exactly $k$ maximal horizontal cylinders among all square-tiled surfaces of genus $g$.
In particular, we define the quantity ${{\mathbb P}}\big(K_g(S)=k\big)$ from Equation as $$\label{eq:Proba:K:g:S}
{{\mathbb P}}\big(K_g(S)=k\big)=
\frac{1}{{\operatorname{Vol}}{\mathcal{Q}}_g}\cdot
\sum_{\substack{\Gamma \in {\mathcal{G}}_g\\|V(\Gamma)| = k}}
{\operatorname{Vol}}(\Gamma)\,.$$
We complete this section with the theorem which is one of the two keystone results on which rely all further asymptotic results of the current paper. Morally, it serves to establish explicit normalization allowing to pass from a finite measure with unspecified total mass to a specific probability measure. This statement was conjectured in [@DGZZ:volume] and proved in [@Aggarwal:intersection:numbers Theorem 1.7].
\[conj:Vol:Qg\] The Masur–Veech volume of the moduli space of holomorphic quadratic differentials has the following large genus asymptotics: $$\label{eq:Vol:Qg}
{\operatorname{Vol}}{\mathcal{Q}}_g
=
\frac{4}{\pi}
\cdot\left(\frac{8}{3}\right)^{4g-4}\cdot\big(1+o(1)\big)
\quad\text{as }g\to+\infty\,.$$
\[rm:expansion:of:error:term\] The exact values of ${\operatorname{Vol}}{\mathcal{Q}}_g$ for $g\le 250$ (and more) can be obtained by combining results of D. Chen, M. Möller, A. Sauvaget [@Chen:Moeller:Sauvaget] with the results of M.Kazarian [@Kazarian] or with the results of D. Yang, D. Zagier and Y. Zhang [@Yang:Zagier:Zhang]. Supported by serious data analysis, the authots of [@Yang:Zagier:Zhang] conjecture that the error term in admits an asymptotic expansion in $g^{-1}$ with the leading term $-\frac{\pi^2}{144}\cdot\frac{1}{g}$ and with explicit coefficients for the terms $g^{-2}$ and $g^{-3}$. In Theorem \[thm:generating:series:vol\], using a refinement of the estimates from [@Aggarwal:intersection:numbers] we prove that the error term $o(1)$ in \[eq:Vol:Qg\] can be improved to a finer estimate $O(g^{-1/4})$.
Conjectural generalization of formula to all strata of meromorphic quadratic differentials and numerical evidence beyond this conjecture are presented in [@ADGZZ:conjecture]. Actually, [@Aggarwal:intersection:numbers Theorem 1.7] proves the volume asymptotics in the more general setting for ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ under assumption that the number $n$ of simple poles satisfies the relation $20n<\log g$.
Frequencies of multicurves (after M. Mirzakhani) {#ss:Frequencies:of:simple:closed:curves}
------------------------------------------------
Recall that two integral multicurves on the same smooth surface of genus $g$ with $n$ punctures *have the same topological type* if they belong to the same orbit of the mapping class group ${\operatorname{Mod}}_{g,n}$.
We change now flat setting to hyperbolic setting. Following M. Mirzakhani, given an integral multicurve $\gamma$ in ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$ and a hyperbolic surface $X\in{\mathcal{T}}_{g,n}$ consider the function $s_X(L,\gamma)$ counting the number of simple closed geodesic multicurves on $X$ of length at most $L$ of the same topological type as $\gamma$. M. Mirzakhani proves in [@Mirzakhani:grouth:of:simple:geodesics] the following Theorem.
For any rational multi-curve $\gamma$ and any hyperbolic surface $X\in{\mathcal{T}}_{g,n}$, $$\label{eq:frequency:c}
s_X(L,\gamma)\sim B(X)\cdot\frac{c(\gamma)}{b_{g,n}}\cdot L^{6g-6+2n}\,,$$ as $L\to+\infty$.
The factor $B(X)$ in the above formula has the following geometric meaning. Consider the unit ball $B_X=\{\gamma\in{\mathcal{ML}}_{g,n}\,|\,\ell_X(\gamma)\le 1\}$ defined by means of the length function $\ell_X$. The factor $B(X)$ is the Thurston’s measure of $B_X$: $$B(X)=\mu_{\mathrm{Th}}(B_X)\,.$$
The factor $b_{g,n}$ is defined as the average of $B(X)$ over ${\mathcal{M}}_{g,n}$ viewed as the moduli space of hyperbolic metrics, where the average is taken with respect to the Weil–Petersson volume form on ${\mathcal{M}}_{g,n}$: $$\label{eq:b:g:n}
b_{g,n}=\int_{{\mathcal{M}}_{g,n}} B(X)\,dX\,.$$
Mirzakhani showed that $$\label{eq:b:g:n:as:sum:of:c:gamma}
b_{g,n}=\sum_{[\gamma]\in\mathcal{O}(g,n)} c(\gamma)\,,$$ where the sum of $c(\gamma)$ taken with respect to representatives $[\gamma]$ of all orbits $\mathcal{O}(g,n)$ of the mapping class group ${\operatorname{Mod}}_{g,n}$ in ${\mathcal{ML}}_{g,n}({{\mathbb Z}})$. This allows to interpret the ratio $\tfrac{c(\gamma)}{b_{g,n}}$ as the probability to get a multicurve of type $\gamma$ taking a “large random” multicurve (in the same sense as the probability that coordinates of a “random” point in ${{\mathbb Z}}^2$ are coprime equals $\tfrac{6}{\pi^2}$).
In particular, we define the quantity ${{\mathbb P}}\big(K_g(\gamma)=k\big)$ from Equation as $$\label{eq:Proba:K:g:gamma}
{{\mathbb P}}\big(K_g(\gamma)=k\big)=
\frac{1}{b_g}\cdot
\sum_{[\gamma]\in\mathcal{O}_k(g)}
c(\gamma)\,,$$ where, $b_g=b_{g,0}$ and $\mathcal{O}_k(g)\subset\mathcal{O}(g)=\mathcal{O}(g,0)$ is the subcollection of orbits of those multicurves $\gamma$, for which $\gamma_{\mathit{reduced}}$ has exactly $k$ connected components.
M. Mirzakhani found an explicit expression for the coefficient $c(\gamma)$ and for the global normalization constant $b_{g,n}$ in terms of the intersection numbers of $\psi$-classes.
Frequencies of square-tiled surfaces of fixed combinatorial type {#ss:Frequencies:of:square:tiled:surfaces}
----------------------------------------------------------------
The following Theorem bridges flat and hyperbolic count.
For any integral multicurve $\gamma\in{\mathcal{ML}}_{g,n}({{\mathbb Z}})$, the volume contribution ${\operatorname{Vol}}(\gamma)$ to the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ coincides with the Mirzakhani’s asymptotic frequency $c(\gamma)$ of simple closed geodesic multicurves of topological type $\gamma$ up to the explicit factor $const_{g,n}$ depending only on $g$ and $n$: $$\label{eq:Vol:gamma:c:gamma}
{\operatorname{Vol}}(\gamma)
=const_{g,n}\cdot c(\gamma)\,,$$ where $$\label{eq:const:g:n}
const_{g,n}
=2\cdot(6g-6+2n)\cdot
(4g-4+n)!\cdot 2^{4g-3+n}\cdot$$
Definitions and and Formulae and combined with relation imply that ${{\mathbb P}}\big(K_g(\gamma)=k\big)
={{\mathbb P}}\big(K_g(S)=k\big)$.
For any admissible pair of non-negative integers $(g,n)$ different from $(1,1)$ and $(2,0)$, the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_{g,n}$ and the average Thurston measure of a unit ball $b_{g,n}$ are related as follows:
$$\label{eq:Vol:g:n:b:g:n}
{\operatorname{Vol}}{\mathcal{Q}}_{g,n}
=2\cdot(6g-6+2n)\cdot
(4g-4+n)!\cdot 2^{4g-3+n}\cdot
b_{g,n}\,.$$
In Theorem 1.4 in [@Mirzakhani:earthquake] M. Mirzakhani established the relation $${\operatorname{Vol}}{\mathcal{Q}}_{g}
=
const_{g}\cdot
b_{g}\,,$$ where $b_{g}$ is computed in Theorem 5.3 in [@Mirzakhani:grouth:of:simple:geodesics]. However, Mirzakhani does not give any formula for the value of the normalization constant $const_g$ presented in . This constant was recently computed by F. Arana–Herrera [@Arana:Herrera] and by L. Monin and I. Telpukhovskiy [@Monin:Telpukhovskiy] simultaneously and independently of us by different methods. The same value of $const_{g,n}$ is obtained by V. Erlandsson and J. Souto in [@Erlandsson:Souto] through an approach different from all the ones mentioned above.
Uniform large genus asymptotics of correlators (after A. Aggarwal) {#ss:conjecture:on:correlators}
------------------------------------------------------------------
We denote by $\Pi(m,n)$ the set of nonnegative compositions of an integer $m$ as sum of $n$ non-negative integers. For any nonnegative composition $\boldsymbol{d}\in\Pi(3g-3+n,n)$ define ${\varepsilon}(\boldsymbol{d})$ through the following equation: $$\label{eq:ansatz}
\langle \tau_{d_1} \dots \tau_{d_n}\rangle_{g,n}
=
\frac{(6g-5+2n)!!}{(2d_1+1)!!\cdots(2d_n+1)!!}
\cdot\frac{1}{g!\cdot 24^g}
\cdot\big(1+{\varepsilon}(\boldsymbol{d})\big)\,.$$ By construction, the intersection numbers are nonnegative rational numbers, so $\varepsilon\big(\boldsymbol{d}\big)\ge -1$ for any $\boldsymbol{d}\in\Pi(3g-3+n,n)$. We conjectured in [@DGZZ:volume] that ${\varepsilon}(\boldsymbol{d})$ tends to zero uniformly for all nonnegative compositions $\boldsymbol{d}\in
\Pi(3g-3+n,n)$ as soon as $n\le 2\log g$ and $g\to+\infty$. This conjecture was proved in much stronger form in the recent paper of A. Aggarwal [@Aggarwal:intersection:numbers].
The following Theorem corresponds to [@Aggarwal:intersection:numbers Proposition 1.2].
\[th:correlators:upper:bound\] Let $n \in \mathbb{Z}_{\ge 1}$ and $\boldsymbol{d}
\in \mathbb{Z}_{\ge 0}^n$ satisfy $|\boldsymbol{d}|
= 3g + n - 3$, for some $g \in \mathbb{Z}_{\ge 0}$. Then, $$\label{eq:Aggarwal:Prop:1:2}
1+{\varepsilon}(\boldsymbol{d})
\le \left( \frac{3}{2} \right)^{n-1}\,.$$
The next Theorem corresponds to [@Aggarwal:intersection:numbers Proposition 4.1].
\[th:asymptotics:of:correlators:lower\] Let $g > 2^{15}$ and $n \ge 1$ be integers such that $g >
30 n$, and let $\boldsymbol{d}\in \Pi(3g-3+n,n)$. Then we have $$\label{eq:asymptotics:of:correlators:lower}
{\varepsilon}(\boldsymbol{d})
\ge -20\cdot\frac{(n + 4\log g)}{g}\,.$$
Finally, the following Theorem corresponds to [@Aggarwal:intersection:numbers Proposition 4.2].
\[th:asymptotics:of:correlators:upper\] Let $g > 2^{30}$ and $n \ge 1$ be integers such that $g >
800 n^2$, and let $\boldsymbol{d}\in \Pi(3g-3+n,n)$. Then we have $$\label{eq:asymptotics:of:correlators:upper}
1+{\varepsilon}(\boldsymbol{d})
\le \exp\left(625\cdot\frac{(n + 2\log g)^2}{g}\right)\,.$$
We proved in [@DGZZ:volume] explicit sharp upper and lower bounds for $2$-correlators.
Random non-separating multicurves and non-uniform random permutations {#s:sum:over:single:vertex:graphs}
=====================================================================
Consider the stable graph ${\Gamma}_k(g)$ having a single vertex, decorated with genus $g-k$, and having $k$ loops, see the left picture in Figure \[fig:two:non:separating\]. This stable graph corresponds to multicurves on a closed surface of genus $g$, for which the components $\gamma_1,\dots,\gamma_k$ of the underlying reduced multicurve $\gamma_{\mathit{reduced}}=\gamma_1+\dots+\gamma_k$ represent $k$ linearly independent homology cycles. The square-tiled surfaces associated to this stable graph have single horizontal singular layer and $k$ maximal horizontal cylinders.
(0,0)(135,10) (44,-14)[$g-k$]{} (0,0)[$\overbrace{\rule{70pt}{0pt}}^{k\text{ loops}}$]{} (110,7)[$\overbrace{\rule{55pt}{0pt}}^k$]{} (100,-26)[$\underbrace{\rule{175pt}{0pt}}_g$]{}
Recall from Section \[ss:intro:Masur:Veech:volumes\] that by ${\operatorname{Vol}}({\Gamma}_k(g))$ we denote the volume contribution from all square-tiled surfaces corresponding to the stable graph ${\Gamma}_k(g)$. By ${\operatorname{Vol}}\big({\Gamma}_k(g),(m_1, \ldots, m_k))$ we denote the volume contribution from those square-tiled surfaces corresponding to the stable graph ${\Gamma}_k(g)$ for which one maximal horizontal cylinder is filled with $m_1$ bands of squares, another cylinder is filled with $m_2$ bands of squares, and so on up to the $k$th maximal horizontal cylinder, which is filled with $m_k$ bands of squares. The corresponding multicurve has the form $m_1\gamma_1+\dots+m_k\gamma_k$, where $\gamma_1,\dots,\gamma_k$ are as described above. By we have $${\operatorname{Vol}}({\Gamma}_k(g)) =
\sum_{\substack{m_1, \ldots, m_k\\1\le m_i \le m\ \text{for }i=1,\dots,k}}
{\operatorname{Vol}}\big({\Gamma}_k(g),(m_1, \ldots, m_k)).$$
In this section we prove the following result, which relies on the uniform asymptotics of Witten correlators proved by A. Aggarwal (see Theorems \[th:correlators:upper:bound\]–\[th:asymptotics:of:correlators:upper\] in the current paper or Propositions 1.2, 4.1, 4.2 respectively in the original paper [@Aggarwal:intersection:numbers] of A. Aggarwal).
\[thm:generating:series:vol:1\] Let $m \in {{\mathbb N}}\cup \{+\infty\}$. For any complex number $t$, in the disk $|t|<2$ we have as $g \to +\infty$ $$\begin{gathered}
\label{eq:generating:series:vol:1}
\sum_{k=1}^{g}\quad
\sum_{\substack{m_1, \ldots, m_k\\1\le m_i \le m\ \text{for }i=1,\dots,k}}
{\operatorname{Vol}}(\Gamma_k(g), (m_1, \ldots, m_k)) \cdot t^k
=\\=
\frac{2\sqrt{2} \left(\frac{2m}{m+1}\right)^{t/2}}{\sqrt{\pi} \cdot \Gamma(\frac{t}{2})}
\cdot (3g-3)^{\frac{t-1}{2}} \cdot \left( \frac{8}{3} \right)^{4g-4}
\left(1 + O\left( \frac{(\log g)^2}{g}\right) \right)\,,\end{gathered}$$ where for every compact subset $U$ of the complex disk $|t|<2$ the error term is uniform over $m \in {{\mathbb N}}\cup \{+\infty\}$ and $t\in U$. In particular, for $m=+\infty$ and $t=1$ we obtain $$\label{eq:sum:Vol:Gamma:k}
\sum_{k=1}^g {\operatorname{Vol}}(\Gamma_k(g))
=
\frac{ 4 }{ \pi} \left( \frac{8}{3} \right)^{4g-4}
\left(1 + O\left( \frac{(\log g)^2}{g}\right) \right)\,.$$
We prove Theorem \[thm:generating:series:vol:1\] in Section \[ss:from:q:to:p1\]. We note that asymptotics was first obtained by A. Aggarwal in [@Aggarwal:intersection:numbers Proposition 8.3]. Our refinement consists in the bound $O\left( \frac{(\log
g)^2}{g}\right)$ for the error term. Conjecturally, the bound can be further improved to $O\left(
\frac{1}{g}\right)$; see Remark \[rm:expansion:of:error:term\].
Volume contribution of stable graphs with a single vertex {#ss:Volume:contribution:single:vertex}
---------------------------------------------------------
In this section, we show how to express an approximate value of the contribution ${\operatorname{Vol}}({\Gamma}_k(g))$ of square-tiled surfaces corresponding to the stable graph ${\Gamma}_k(g)$ to the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ in terms of the following normalized weighted multi-variate harmonic sum.
\[def:hkzk\] Let $m \in {{\mathbb N}}\cup \{+\infty\}$ and let $\alpha$ be a positive real number. For integers $k, n$ such that $1 \leq k \leq n$, define $$\label{eq:multiple:harmonic:sum:def}
{\widetilde{H}}_{n,m,\alpha}(k) = \frac{\alpha^k}{k!} \sum_{j_1+\dots+j_k=n}
\frac{\zeta_m(2 j_1) \cdot \zeta_m(2 j_2) \cdots \zeta_m(2j_k)}{j_1 \cdot j_2 \cdots j_k}\,,$$ where the sum is taken over all $k$-tuples $(j_1, j_2, \ldots , j_k) \in {{\mathbb N}}^k$ of positive integers summing up to $n$ and $$\zeta_m(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \ldots + \frac{1}{m^s}$$ is the partial zeta function.
The particular cases of the above numbers, namely, $$\begin{aligned}
{H}_k(n)=\sum_{j_1+\dots+j_k=n}&
\frac{1}{j_1\cdot j_2\cdots j_k}
= k! \cdot {\widetilde{H}}_{n,1,1}(k)
\\\
{Z}_k(n)=\sum_{j_1+\dots+j_k=m}&
\frac{\zeta(2j_1)\cdots\zeta(2j_k)}
{j_1\cdot j_2\cdots j_k}
= k! \cdot {\widetilde{H}}_{n,\infty,1}(k)\end{aligned}$$ appeared in the preprint [@DGZZ:volume]; the asymptotic expansions for these quantities were obtained by A. Aggarwal in Sections 6 and 7 of [@Aggarwal:intersection:numbers]. The framework which we develop here allows to treat all normalized weighted multi-variate harmonic sums ${\widetilde{H}}_{n,m,\alpha}(k)$ in a unified way.
\[th:bounds:for:Vol:Gamma:k:g\] There exists a constant $C_1$ such that for sufficiently large $g\in{{\mathbb N}}$ the following property holds. For any couple $m,k$, such that $m \in {{\mathbb N}}\cup
\{+\infty\}$, $k\in{{\mathbb N}}$, $800 k^2\le g$, we have $$\begin{gathered}
\label{eq:bounds:in:terms:of:H:k:gminus3}
\sum_{\substack{m_1, \ldots, m_k\\1\le m_i \le m\ \text{for }i=1,\dots,k}}
{\operatorname{Vol}}\big({\Gamma}_k(g), (m_1, \ldots, m_k)\big)
=\\=
\frac{2 \sqrt{2}}{\sqrt{\pi}} \cdot \sqrt{3g-3}
\cdot \left(\frac{8}{3}\right)^{4g-4}
\cdot{\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)
\cdot\big(1 + \varepsilon_1(g,k) \big)
\,,\end{gathered}$$ where $|\varepsilon_1(g,k)|\le
C_1\cdot\cfrac{(k+2\log g)^2}{g}$.
There exists a constant $C_2$ such that for all triples $(g,k,m)$, where $g\in{{\mathbb N}}$, $g\ge 2$; $k\in{{\mathbb N}}$, $k\le g$; $m \in {{\mathbb N}}\cup \{+\infty\}$, we have $$\begin{gathered}
\label{eq:Vol:Gamma:k:upper:bound}
\sum_{\substack{m_1, \ldots, m_k\\1\le m_i \le m\ \text{for }i=1,\dots,k}}
{\operatorname{Vol}}\big({\Gamma}_k(g), (m_1, \ldots, m_k)\big)\le
\\
\le C_2\cdot
\sqrt{g}\cdot
\left(\frac{8}{3}\right)^{4g-4}\cdot
{\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)\cdot
\left( \frac{9}{4} \right)^k
\,,\end{gathered}$$ where ${\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)$ is the normalized weighted multi-variate harmonic sum defined in .
In order to prove Theorem \[th:bounds:for:Vol:Gamma:k:g\] we first state and prove Lemma \[lm:cgk:asymptotics\] below.
Let $\textbf{D}=(D_1,\dots,D_k)\in\Pi(3g-3+2k,k)$. Define $c_{g,k}(\boldsymbol{D})$ as $$\begin{gathered}
\label{eq:sum:of:normalized:correlators}
c_{g,k}(\boldsymbol{D}):=
\frac{g!\cdot(3g-3+2k)!}{(6g+4k-5)!}
\cdot\frac{3^g}{2^{3g-6+5k}}\cdot
\\
\cdot
\sum_{d_{1,1}+d_{1,2}=D_1}\dots\sum_{d_{k,1}+d_{k,2}=D_k}
\int_{\overline{{\mathcal{M}}}_{g,2k}}
\psi_1^{d_{1,1}}\psi_2^{d_{1,2}}
\dots \psi_{2k-1}^{d_{k,1}}\psi_{2k}^{d_{k,2}}
\cdot\prod_{j=1}^k \cfrac{(2D_j+2)!}{d_{j,1}!\cdot d_{j,2}!}\,.\end{gathered}$$
The following result is a corollary of the uniform asymptotics of Witten correlators proved by A. Aggarwal (see Theorems \[th:correlators:upper:bound\]–\[th:asymptotics:of:correlators:upper\] in the current paper or Propositions 1.2, 4.1, 4.2 respectively in the original paper [@Aggarwal:intersection:numbers] of ).
\[lm:cgk:asymptotics\] There exists a constant $C_3$ such that for sufficiently large $g\in{{\mathbb N}}$ and for $k\in{{\mathbb N}}$ satisfying $800 k^2\le g$ we have $$\label{eq:cgk:at:most:logg}
|c_{g,k}(\boldsymbol{D}) - 1| \le
C_3 \cdot \frac{(k+2\log g)^2}{g}\,.$$ For any positive integers $g,k\in{{\mathbb N}}$ satisfying $1\le k\le g$ and $g\ge 2$, we have $$\label{eq:9:4:power:k}
c_{g,k}(\boldsymbol{D}) \le \left( \frac{9}{4} \right)^{k}.$$
Passing to double factorials and applying definition of ${\varepsilon}(\boldsymbol{d})$ we get $$\begin{gathered}
c_{g,k}(\boldsymbol{D})
=\frac{g!}{2^{3g-3+2k}\cdot(6g+4k-5)!!}
\cdot\frac{3^g}{2^{3g-6+5k}}\cdot
\\
\cdot
\sum_{d_{1,1}+d_{1,2}=D_1}\dots\sum_{d_{k,1}+d_{k,2}=D_k}
\langle\tau_{d_{1,1}}\tau_{d_{1,2}}
\dots \tau_{d_{k,1}}\tau_{d_{k,2}}\rangle_{g,2k}
\\
\prod_{j=1}^k\left(
\frac{(2d_{j,1}+1)!}{d_{j,1}!}\cdot\frac{(2d_{j,2}+1)!}{d_{j,2}!}
\cdot\binom{2D_j+2}{2d_{j,1}+1}\right)
=\\=
\frac{1}{2^{6g-6+5k}}
\sum_{d_{1,1}+d_{1,2}=D_1}\dots\sum_{d_{k,1}+d_{k,2}=D_k}
\left(\big(1+{\varepsilon}(\boldsymbol{d})\big)
\cdot\prod_{j=1}^k
\binom{2D_j+2}{2d_{j,1}+1}\right)\,.\end{gathered}$$ Applying the combinatorial identity $$\sum_{m=0}^{n-1}\binom{2n}{2m+1}=2^{2n-1}$$ we get $$\begin{gathered}
\sum_{d_{1,1}+d_{1,2}=D_1}\dots\sum_{d_{k,1}+d_{k,2}=D_k}
\prod_{j=1}^k
\binom{2D_j+2}{2d_{j,1}+1}
=\\=
\left(\prod_{j=1}^k \sum_{d_{j,1}=0}^{D_j}\binom{2D_j+2}{2d_{j,1}+1}\right)
=\left(\prod_{j=1}^k 2^{2D_j+1}\right)
=
2^{6g-6+5k}\,.\end{gathered}$$ The claim that bound is valid for sufficiently large $g$ now follows from combination of bounds and from Theorems \[th:asymptotics:of:correlators:lower\] and \[th:asymptotics:of:correlators:upper\] of A. Aggarwal (see Propositions 4.1, 4.2 respectively in the original paper [@Aggarwal:intersection:numbers]).
For $k\ge 2$ the universal bound follows from the universal bound from Theorem \[th:correlators:upper:bound\] of A. Aggarwal (see Proposition 1.2 in the original paper [@Aggarwal:intersection:numbers]), using the fact that for $k \ge 2$ we have $1 + (3/2)^{2k-1}
\le (3/2)^{2k}$.
We prove in Proposition 4.1 in [@DGZZ:volume] that ${\varepsilon}(\boldsymbol{d})\le 0$ for any $\boldsymbol{d}\in\Pi(3g-1,2)$. This implies bound for $k=1$, which completes the proof of the Lemma.
Let us denote $$\label{eq:V:m:k}
V_{m,k}(g) =\!\!\!\!
\sum_{\substack{m_1, \ldots, m_k\\1\le m_i \le m\\ \text{for }i=1,\dots,k}}
\!\!\!{\operatorname{Vol}}({\Gamma}_k(g), (m_1, \ldots, m_k))
\quad\text{and}\quad
V_{m}(g)=\sum_{k=1}^g V_{m,k}(g)\,.$$ The automorphism group $\operatorname{Aut}({\Gamma}_k(g))$ consists of all possible permutations of loops composed with all possible flips of individual loops, so $$|\operatorname{Aut}(\Gamma)|=2^k\cdot k!\,.$$ The graph ${\Gamma}_k(g)$ has a single vertex, so $|V({\Gamma}_k(g)|=1$. Thus, applying to ${\Gamma}_k(g)$ we get $$\begin{gathered}
\label{eq:Vol:Gamma:k:g:init}
V_{m,k}(g)
=\frac{2^{6g-5} \cdot (4g-4)!}{(6g-7)!}
\cdot\\
1\cdot\frac{1}{2^k\cdot k!}
\cdot
\sum_{\substack{\boldsymbol{H} = (m_1, \ldots, m_k)\\m_1, \ldots, m_k \leq m}}
{\mathcal{Y}}\big(\boldsymbol{H},
b_1 b_2\dots b_k \cdot
N_{g-k,2k}(b_1,b_1,b_2,b_2,\dots,b_k,b_k)
\big)
=\\=
\frac{(4g-4)!}{(6g-7)!}
\cdot \frac{2^{6g-5}}{2^k\cdot k!}
\cdot\frac{1}{2^{5(g-k)-6+4k}}
\cdot\\
\sum_{\boldsymbol{d}\in\Pi(3g-3-k,2k)}
\frac{\langle\tau_{\mathbf{d}}\rangle_{g-k,2k}}{\mathbf{d}!}\,\cdot\,
\prod_{i=1}^k
\Big((2d_{2i-1}+2d_{2i}+1)!\,\cdot\,\zeta_m(2d_{2i-1}+2d_{2i}+2)\Big)\,.\end{gathered}$$
Rewrite the latter sum using notations $\textbf{D}=(D_1,\dots,D_k)\in\Pi(3g-3-k,k)$ and $c_{g-k,k}(\boldsymbol{D})$ defined by . Adjusting expression given for genus $g$ to genus $g-k$ we get $$\begin{gathered}
\sum_{\boldsymbol{d}\in\Pi(3g-3-k,2k)}
\frac{\langle\tau_{\mathbf{d}}\rangle_{g-k,2k}}{\mathbf{d}!}\,\cdot\,
\prod_{i=1}^k
\Big((2d_{2i-1}+2d_{2i}+1)!\,\cdot\,\zeta_m(2d_{2i-1}+2d_{2i}+2)\Big)
=\\=
\sum_{\boldsymbol{D}\in\Pi(3g-3-k,k)}
c_{g-k,k}(\boldsymbol{D})\cdot
\frac{(6(g-k)+4k-5)!}{(g-k)!\cdot(3(g-k)-3+2k)!}
\cdot\frac{2^{3(g-k)-6+5k}}{3^{(g-k)}}
\cdot\\
\cdot\prod_{j=1}^k \frac{\zeta_m(2D_j+2)}{2D_j+2}\,,\end{gathered}$$ which allows to rewrite as $$\begin{gathered}
\label{eq:V:m:k:g}
V_{m,k}(g)
=\\=
\left(\frac{(4g-4)!}{(6g-7)!}
\cdot 2^{g+1}
\cdot\frac{1}{k!}\right)
\cdot
\left(\frac{(6g-2k-5)!}{(g-k)!\cdot(3g-3-k)!}
\cdot\frac{2^{3g-6+2k}}{3^{g-k}}\right)\cdot
\\
\sum_{\boldsymbol{D}\in\Pi(3g-3-k,2k)}
\frac{c_{g-k,k}(\boldsymbol{D})}{2^k}
\cdot\prod_{j=1}^k \frac{\zeta_m(2D_j+2)}{D_j+1}\,.\end{gathered}$$
Let us define $$c^{min}_{g-k,k} := \min_{\boldsymbol{D}} c_{g-k,k}(\boldsymbol{D})
\quad \text{and} \quad
c^{max}_{g-k,k} := \max_{\boldsymbol{D}} c_{g-k,k}(\boldsymbol{D})\,.$$
Rearranging factors with factorials, collecting powers of $2$ and $3$, and passing to notation ${\widetilde{H}}_{m,3g-3,\frac{1}{2}}(k)$ for the multivariate harmonic sum we get the following bounds: $$\begin{gathered}
\label{eq:Vol:Gamma:k:g:intermed}
c^{min}_{g-k,k}\le V_{m,k}(g)\cdot
\\
\left(
\frac{(6g-2k-5)!}{(6g-7)!}
\cdot\frac{(4g-4)!}{(g-k)!\cdot(3g-3-k)!}
\cdot \frac{2^{4g-5+2k}}{3^{g-k}}
\cdot {\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)
\right)^{-1}
\\
\le c^{max}_{g-k,k}\,.\end{gathered}$$
We start by proving the first assertion of the theorem represented by relation . We rewrite the product of factorials in as $$\begin{gathered}
\label{eq:factorials}
\frac{(6g-2k-5)! \cdot (4g-4)!}{(6g-7)! \cdot (g-k)! \cdot (3g-3-k)!}
=
(6g-6) \cdot
\binom{4g-4}{g-1} \cdot
\frac{\frac{(3g-3)!}{(3g-3-k)!} \cdot \frac{(g-1)!}{(g-k)!}}{\frac{(6g-6)!}{(6g-2k-5)!}} \\
= (6g-6) \cdot \binom{4g-4}{g-1} \cdot
\frac{(3g-3)^k \cdot (g-1)^{k-1}}{(6g-6)^{2k-1}}
\big(1 + \varepsilon_3(g,k)\big)= \\
= \sqrt{3g-3} \cdot \sqrt{\frac{2}{\pi}} \cdot \frac{2^{8g-6-2k}}{3^{3g-4+k}}
\big(1 + \varepsilon_4(g,k) \big)\,.\end{gathered}$$ Note that there exist constants $C'_3$ and $a_0$ such that for any integer $a$ satisfying $a\ge a_0$ and for any $b\in{{\mathbb N}}$, satisfying $b\le \sqrt a$, we have $$a^b \left( 1 - C'_3 \cdot \frac{b^2}{a} \right)
\le
\frac{a!}{(a-b)!}
\le
a^b \left( 1 + C'_3\cdot \frac{b^2}{a} \right)\,.$$ This implies that there exists $g_0$ such that for any $g\in{{\mathbb N}}$ satisfying $g\ge g_0$ and for any $k\in{{\mathbb N}}$ satisfying $800 k^2\le g$ we have the bound $$|{\varepsilon}_3(g,k)|\le 1+C_3\cdot\frac{k^2}{g}$$ for the error term in the second line of . Let $$\binom{4g-4}{g-1}
=\sqrt{\frac{2}{\pi (3g-3)}} \cdot \frac{2^{8g-8}}{3^{3g-3}}
\cdot\big(1+\varepsilon_5(g)\big)\,.$$ There exist constants $C_5$ and $g_1$ such that for any $g\in{{\mathbb N}}$ satisfying $g\ge g_1$ we have $$|\varepsilon_5(g)|\le C_5\cdot
\frac{1}{g}\,.$$ The latter two bounds imply that there exist a constant $C_4$ and a constant $g_2$ such that for any $g\in{{\mathbb N}}$ satisfying $g\ge g_2$ we have the bound $$|\varepsilon_4(g)|\le C_4\cdot
\frac{k^2}{g}\,,$$ for the error term on the right-hand side of the third line of . Using the latter bound and collecting powers of $2$, of $3$ and of $g$, we can rewrite in the following way: $$\begin{gathered}
\label{eq:to:serve:for:c:min:max:equals:1}
c^{min}_{g-k,k}
\left(1 - C_4\cdot \frac{k^2}{g}\right)
\leq \\
\le \cfrac{V_{m,k}(g)}
{\frac{2 \sqrt{2}}{\sqrt{\pi}}
\cdot\left(\frac{8}{3}\right)^{4g-4}
\cdot \sqrt{3g-3}
\cdot {\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)}
\le
c^{max}_{g-k,k} \left(1 + C_4\cdot \frac{k^2}{g}\right)\,.\end{gathered}$$ Now, using the bound from the first part of Lemma \[lm:cgk:asymptotics\] we get .
The proof of the upper bound is similar. For the product of factorials we use the bound $$\begin{gathered}
\label{eq:bound:for:factorials}
\frac{\frac{(3g-3)!}{(3g-3-k)!} \cdot \frac{(g-1)!}{(g-k)!}}{\frac{(6g-6)!}{(6g-2k-5)!}}
=
\prod_{i=0}^{k-1} \frac{3g-3-i}{6g-6-2i}
\prod_{i=1}^{k-1} \frac{g-i}{6g-5-2i}
=\\
=\frac{1}{12^k} \prod_{i=1}^{k-1} \frac{6g-6i}{6g-5-2i}
=\frac{1}{12^k} \left(1+\frac{1}{6g-7}\right) \prod_{i=2}^{k-1} \frac{6g-6i}{6g-5-2i}
\le \frac{6}{5}\cdot \frac{1}{12^k}\,.\end{gathered}$$ valid for any couple $(g,k)$ of positive integers satisfying $g\ge 2$ and $k\le g$. Here we used the inequality $6g-6i < 6g-5-2i$ valid for any integer $g,i$ such that $g\ge 2$ and $i \geq 2$. We also used the inequality $1/(6g-7)\le 1/5$ valid for any integer $g\ge 2$. The upper bound for $c^{max}_{g-k,k}$ was established in Equation in the second part of Lemma \[lm:cgk:asymptotics\]. Plugging this bound for $c^{max}_{g-k,k}$ in and the bound for the product of factorials in and proceeding as before we obtain the result.
Define $$\begin{gathered}
\label{eq:V:k:g:tilde}
\tilde V_{m,k}(g)
=\left(\frac{(4g-4)!}{(6g-7)!}
\cdot 2^{g+1}
\cdot\frac{1}{k!}\right)
\cdot
\left(\frac{(6g-2k-5)!}{(g-k)!\cdot(3g-3-k)!}
\cdot\frac{2^{3g-6+2k}}{3^{g-k}}\right)\cdot
\\
\sum_{\boldsymbol{D}\in\Pi(3g-3-k,2k)}
\frac{1}{2^k}
\cdot\prod_{j=1}^k \frac{\zeta(2D_j+2)}{D_j+1}\,.\end{gathered}$$ This expression is obtained by replacing $c_{g-k,k}(\boldsymbol{D})$ with $1$ in . We have seen that this is equivalent to replacing the Kontsevich–Witten correlators in the right-hand side of formula for $V_{m,k}(g)$ by the asymptotic expression from Section \[ss:conjecture:on:correlators\]. We are now ready to give the formal definition of the the approximating distribution $q_g(k)$ informally described in Section \[s:intro\].
Define the probability distribution ${q}_g(k)$ as $$\label{eq:def:q_g}
{q}_g(k) := \frac{\tilde V_{\infty,k}(g)}{\tilde V_{\infty}(g)}\,,
\quad \text{where} \quad
\tilde V_{\infty}(g) := \sum_{k=1}^{g} \tilde V_{\infty,k}(g)\,.$$
It follows from the proof of of Theorem \[th:bounds:for:Vol:Gamma:k:g\] that for sufficiently large $g\in{{\mathbb N}}$ and for $k\in{{\mathbb N}}$ satisfying $k^2\le g$ we have the bounds for $\tilde V_{m,k}(g)$ analogous to , where $c^{max}_{g-k,k}$ and $c^{max}_{g-k,k}$ are replaced with $1$. This implies that $\tilde V_{m,k}(g)$ satisfies the lower bound $$\label{eq:V:tilde:lower}
\tilde V_{m,k}(g)\ge
\frac{2 \sqrt{2}}{\sqrt{\pi}}
\cdot\left(\frac{8}{3}\right)^{4g-4}
\cdot \sqrt{3g-3}
\cdot {\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)
\cdot \left(1 + O\left(\frac{k^2}{g}\right)\right)\,,$$ where the constant in the error term is uniform for $k\in{{\mathbb N}}$ satisfying $k^2\le g$.
The upper bound for the expression in factorials on the left-hand side of can be expressed for large $g$ as $\frac{1}{12^k}\cdot\left(1+O\left(g^{-1}\right)\right)$. Thus, analog of for $\tilde V_{m,k}(g)$, where $c^{max}_{g-k,k}$ and $c^{max}_{g-k,k}$ are replaced with $1$ implies that that for sufficiently large $g\in{{\mathbb N}}$ and for any $k\in{{\mathbb N}}$ we have the upper bound $$\label{eq:V:tilde:upper}
\tilde V_{m,k}(g)\le
\frac{2 \sqrt{2}}{\sqrt{\pi}}
\cdot\left(\frac{8}{3}\right)^{4g-4}
\cdot \sqrt{3g-3}
\cdot {\widetilde{H}}_{3g-3,m,\frac{1}{2}}(k)
\cdot \left(1 + O\left(\frac{1}{g}\right)\right)\,.$$
Multi-variate harmonic sums and random non-uniform permutations {#ss:non:uniform:permutations}
---------------------------------------------------------------
In this section we analyze the normalized weighted multi-variate harmonic sum from Definition \[def:hkzk\] and Theorem \[th:bounds:for:Vol:Gamma:k:g\]. We show how these kind of sums naturally appear in the study of random permutations in the symmetric group.
Let us recall the settings from Section \[s:intro\]. Let $\theta = \{\theta_k\}_{k \geq 1}$ be non-negative real numbers. From now on we assume for simplicity that $\theta_1 > 0$. Recall that given a permutation $\sigma \in
S_n$ with cycle type $(1^{\mu_1} 2^{\mu_2} \ldots
n^{\mu_n})$ we define its *weight* as $$w_\theta(\sigma) := \theta_1^{\mu_1} \theta_2^{\mu_2} \cdots \theta_n^{\mu_n}.$$ To every sequence $\theta = \{\theta_k\}_{k \geq 1}$ we associate a probability measure on the symmetric group $S_n$ as in by setting $${{\mathbb P}}_{\theta,n}(\sigma) := \frac{w_{\theta}(\sigma)}{n!\cdot W_{\theta,n}}
\qquad \text{where} \qquad
W_{\theta,n} := \frac{1}{n!} \sum_{\sigma \in S_n} w_\theta(\sigma).$$ Constant weights $\theta_i = 1$ correspond to the uniform measure on $S_n$. More generally, the probability measures on $S_n$ obtained from constant weights $\theta_i = \alpha$ are called *Ewens measure*. The following lemma identifies our normalized weighted multi-variate harmonic sums from Definition \[def:hkzk\] as total contribution of permutations having exactly $k$ cycles to the sum $W_{\theta,n}$.
\[lem:cycle:distribution:vs:harmonic:sum\] Let $\theta = \{\theta_k\}_{k \geq 1}$ be non-negative real numbers and consider the associated probability measure ${{\mathbb P}}_{\theta,n}$ on the symmetric group $S_n$ for some $n$. Then $$\label{eq:def:weight:of:permutation}
\frac{1}{n!} \cdot
\sum_{\substack{\sigma \in S_n\\ {{\mathrm K}}_n(\sigma) = k}}
w_\theta(\sigma)
=
\frac{1}{k!} \cdot \sum_{i_1 + \cdots + i_k = n}
\frac{\theta_{i_1} \theta_{i_2} \cdots \theta_{i_k}}{i_1 \cdots i_k}\,,$$ where ${{\mathrm K}}_n(\sigma)$ is the number of cycles in the cycle decomposition of $\sigma$ and the sum in the right hand-side is taken over positive integers $i_1, \ldots, i_k$. In other words, we have the identity in the ring ${{\mathbb Q}}[[t,z]]$ of formal power series in $t$ and $z$ $$\label{eq:generating:series:multi:harmonic}
\sum_{n \geq 1} \sum_{\sigma \in S_n} w_\theta(\sigma) t^{{{\mathrm K}}_n(\sigma)} \frac{z^n}{n!}
=
\exp \left(t \sum_{k \geq 1} \theta_k \frac{z^k}{k} \right).$$
The first several terms of the expansion of in $z$ have the following form: $$\begin{aligned}
\exp \left(t \sum_{k \geq 1} \theta_k \frac{z^k}{k} \right)
=
&1 + \\
&\left(\theta_1 t\right) z + \\
&\left(\theta_2 t + \theta_1^2 t^2\right) \frac{z^2}{2!} + \\
&\left(2 \theta_3 t + 3 \theta_1 \theta_2 t^2 + \theta_1^3 t^3\right) \frac{z^3}{3!} + \\
&\ldots\end{aligned}$$
From each permutation $\sigma$ in $S_n$ and a composition $(i_1,
\ldots, i_k)$ of $n$ we build the following permutation $\widetilde{\sigma}$ with $k$ cycles (in cycle notation) $$\big(\sigma(1), \sigma(2), \ldots, \sigma(i_1)\big)
\big(\sigma(i_1+1), \ldots, \sigma(i_1+i_2)\big) \cdots
\big(\sigma(i_1 + \cdots + i_{k-1}+1), \ldots, \sigma(n)\big).$$ Here the cycles of $\widetilde{\sigma}$ are ordered from $1$ to $k$ so that the first cycle has length $i_1$, the second has length $i_2$, etc. Since each cycle is defined up to cyclic ordering, for each fixed $(i_1, \ldots, i_k)$ we obtain the same permutation (with ordered cycles) $i_1 \cdot i_2 \cdots i_k$ times. Hence the number $$n! \frac{\theta_{i_1} \theta_{i_2} \cdots \theta_{i_k}}{i_1 i_2 \cdots i_k}.$$ is the weighted count of permutations with $k$ labelled cycles of lengths $i_1$, …, $i_k$. Now summing over all possible compositions $(i_1, \ldots, i_k)$ of $n$ and dividing by $k!$ gives the weighted sum of permutations having exactly $k$ cycles.
We see that the normalized weighted multi-variate harmonic sums ${\widetilde{H}}_{n,m,\alpha}(k)$ defined in represent the total weight of permutations having exactly $k$ disjoint cycles in their cycle decomposition, where the weights $w_\theta(\sigma)$ correspond to the sequence $\theta_k =
\alpha \zeta_m(2k)$, $k\in{{\mathbb N}}$. Thus, Lemma \[lem:cycle:distribution:vs:harmonic:sum\] implies the following relation for the generalization of the quantities $q_{n,\infty,\alpha}$ defined in for arbitrary $m\in{{\mathbb N}}\cup\{+\infty\}$: $$\label{eq:q:n:m:alpha}
q_{n,m,\alpha}(k)
={{\mathbb P}}_{n, m, \alpha}\big({{\mathrm K}}_n(\sigma) = k\big)
=\frac{{\widetilde{H}}_{n,m,\alpha}(k)}{W_{n,m,\alpha}}\,,$$ where $$\label{eq:W:n:m:alpha}
W_{n,m,\alpha}
=\sum_{k=1}^n {\widetilde{H}}_{n,m,\alpha}(k)
=\sum_{k=1}^n
\frac{1}{n!} \cdot
\sum_{\substack{\sigma \in S_n\\ {{\mathrm K}}_n(\sigma) = k}}
w_\theta(\sigma)
\,.$$
Theorem \[th:bounds:for:Vol:Gamma:k:g\] relates the contributions ${\operatorname{Vol}}\big({\Gamma}_k(g)\big)$ of stable graphs ${\Gamma}_k(g)$ to the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ to the total weight of permutations having exactly $k$ disjoint cycles in their cycle decomposition, with the weights $w_\theta(\sigma)$ corresponding to the sequence $\theta_k =
\tfrac{1}{2}\zeta(2k)$, $k\in{{\mathbb N}}$, that is to the normalized weighted multi-variate harmonic sums with $m=+\infty$ and $\alpha=\tfrac{1}{2}$.
The unsigned Stirling numbers of the first kind $s(n,k)$ corresponding to the uniform distribution on $S_n$ satisfy $s(n,k) = n! \cdot {\widetilde{H}}_{n,1,1}(k)$.
\[thm:multi:harmonic:sum:total:weight\] Let $t$ be a complex number and $m \in {{\mathbb N}}\cup
\{+\infty\}$. Then $$\label{eq:multi:harmonic:sum:total:weight}
\sum_{k=1}^n {\widetilde{H}}_{n,m,\alpha}(k) t^k =
\frac{\left( \frac{2m}{m+1} \right)^{\alpha t} n^{\alpha t - 1}}{\Gamma(\alpha t)}
\left(1 + O\left(\frac{1}{n}\right)\right)\,,$$ where the error term is uniform in $t$ over compact subsets of complex numbers.
Here we use the convention $$\left.\frac{2m}{m+1}\right\vert_{m=+\infty}=2\,.$$
A version of Theorem \[thm:multi:harmonic:sum:total:weight\] stated for the values $m=1$ and $m=+\infty$; $\alpha=\frac{1}{2}$; $t=1$ of the parameters, which are particularly important in the context of the current paper, was stated as a conjecture in the preprint [@DGZZ:volume] and was first proved by A. Aggarwal in [@Aggarwal:intersection:numbers Proposition 7.2]. We suggest here a proof of Theorem \[thm:multi:harmonic:sum:total:weight\] based on technique of H. Hwang [@Hwang:PhD] applied to the generating function in the right-hand side of Equation . We discovered this approach for ourselves after the paper [@Aggarwal:intersection:numbers] was available.
We will use the following elementary facts in the proof Theorem \[thm:multi:harmonic:sum:total:weight\].
\[lem:g:m\] Let $m \in {{\mathbb N}}\cup \{+\infty\}$. The series $$g_m(z) = \sum_{k \geq 1} \zeta_m(2k) \frac{z^k}{k}$$ converges in the unit disk $|z|<1$. Considered as as a holomorphic function, it extends to ${{\mathbb C}}\setminus [1,
+\infty)$. Moreover, as $z \to 1$ inside ${{\mathbb C}}\setminus [1,
+\infty)$ we have $$\label{eq:g:m:z}
g_m(z) = - \log(1 - z) + \log\left( \frac{2m}{m+1} \right) + O(1 - z).$$
Expanding the definition of the partial zeta function $\zeta_m$ and changing the order of summation we find the alternative formula $$g_m(z) = - \sum_{n = 1}^m \log\left(1 - \frac{z}{n^2}\right)\,,$$ which proves the first assertion of the Lemma.
Now, we have $$g_m(z) = - \log(1 - z) -
\sum_{n=2}^m \log\left(1 - \frac{1}{n^2}\right)
+O(z-1).$$ For finite $m$, we can rewrite the constant term as $$-\sum_{n = 2}^m \log (1 - \frac{1}{n^2})
= \sum_{n = 2}^m (2 \log(n) - \log(n-1) - \log(n+1))
= \log \left( \frac{2 m}{m + 1} \right).$$ The case $m=+\infty$ is obtained by passing to the limit.
Theorem \[thm:multi:harmonic:sum:total:weight\] can be derived as a corollary of Theorem 12 of [@Hwang:PhD] (see also Lemma 2.13 in [@NikeghbaliZeindler]). To make the proof tractable we provide here a complete argument based on the asymptotic analysis performed in the classical book by P. Flajolet and R. Sedgewick [@Flajolet:Sedgewick].
By Lemma \[lem:cycle:distribution:vs:harmonic:sum\] we have $$\label{eq:g:alpha:m}
\sum_{k=1}^n {\widetilde{H}}_{n,m,\alpha}(k) t^k = [z^n] \exp(t \alpha g_{m}(z))\,,$$ where $g_m(z)$ is the function defined in Lemma \[lem:g:m\]. Plugging the asymptotic expansion into we obtain the following expansion as $z \to 1$ inside ${{\mathbb C}}\setminus [1, +\infty)$: $$\label{eq:proof:of:Th;3:7}
\exp(\alpha t g_m(z))
= \left(\frac{1}{1-z}\right)^{\alpha t}
\cdot
\left(\frac{2 m}{m+1}\right)^{\alpha t}
\cdot
(1 + O(z - 1))\,,$$ where the error term is uniform in $t$ over compact subsets of complex numbers. Now by [@Flajolet:Sedgewick Theorem VI.I] we have $$[z^n] \left(\frac{1}{1-z}\right)^{\alpha t}
=
\frac{n^{\alpha t - 1}}{\Gamma(\alpha t)}
\left( 1 + O\left( \frac{1}{n} \right)\right)\,,$$ where the error term is uniform in $t$ over compact subsets of complex numbers. The term $\left(\frac{2 m}{m+1}\right)^{\alpha t}$ in does not depend on $z$. In order to bound the contribution of the error term $\left(\frac{1}{1-z}\right)^{\alpha t}
\cdot(1 + O(z - 1))$ in we use the following estimate [@Flajolet:Sedgewick Theorem VI.3]: $$[z^n] O\left(\left(\frac{1}{1-z}\right)^{s - 1}\right)
=
O\left(n^{s - 2}\right)\,.$$ Hence $$[z^n] \exp(t g_{\alpha,m}(z)) = \frac{n^{\alpha t - 1}}{\Gamma(\alpha t)}
\cdot
\left(\frac{2 m}{m+1}\right)^{\alpha t}
\cdot
\left( 1 + O\left( \frac{1}{n} \right)\right)$$ and the theorem is proved.
Mod-Poisson convergence {#s:mod:poisson}
-----------------------
In this section we recall some facts about mod-Poisson convergence of probability distributions. As a direct corollary of Theorem \[thm:multi:harmonic:sum:total:weight\] we derive mod-Poisson convergence of the probability distribution $q_{n, m, \alpha}=
{{\mathbb P}}_{n, m, \alpha}\big({{\mathrm K}}_n(\sigma) = k\big)$ of the number of cycles associated by Lemma \[lem:cycle:distribution:vs:harmonic:sum\] to the normalized weighted multi-variate harmonic sums ${\widetilde{H}}_{n,m,\alpha}(k)$. For details we refer to the monograph of V. Féray, P.-L. Méliot and A. Nikeghbali [@FerayMeliotNikeghbali] and, for the particular case of uniformly distributed random permutations, to the original article of A. Nikehgbali and D. Zeindler [@NikeghbaliZeindler].
Given a probability distribution $p(k)$ of a random variable $X$ taking values in non-negative integers, we associate to it the *generating series* $$\label{eq:generating:series:F}
F_p(t) = \sum_{k =1}^{+\infty} p(k) t^k\,.$$ The generating series of the Poisson distribution defined in is $e^{\lambda (t-1)}$.
Recall that given two independent discrete random variables with non-negative integer values $X$ and $Y$ with distributions $p_X(k)$ and $p_Y(k)$ respectively, the distribution of their sum $X + Y$ is the convolution $$p_{X+Y}(k) = \sum_{i+j=k} p_X(i)\cdot p_Y(j).$$ The generating series of $p_{X+Y}$ is the product of the generating series of $p_{X}$ and $p_{Y}$: $$\label{eq:convolution:generating:series}
F_{X+Y} = F_X F_Y.$$ We are particularly interested in the situations when we have a sequence of distributions that are close to the convolution of the Poisson distribution with a varying parameter $\lambda_n$ which tends to $+\infty$ as $n\to+\infty$ and an additional fixed distribution.
\[def:mod:poisson:convergence\] Let $p_n$ be a sequence of probability distributions on the non-negative integers; let $\lambda_n$ be a sequence of positive real numbers tending to $+\infty$ as $n\to+\infty$; let $R \in (1, +\infty]$; let $G(t)$ be a function on the disk $|t|\le R$ in ${{\mathbb C}}$ and let ${\varepsilon}_n$ be a sequence of positive real numbers converging to zero. We say that $p_n$ *converges mod-Poisson with parameters $\lambda_n$, limiting function $G$, radius $R$ and speed ${\varepsilon}_n$* if for all $t\in{{\mathbb C}}$ such that $|t|<R$ we have $$\label{eq:def:mod:poisson}
F_{p_n}(t) = e^{\lambda_n (t - 1)} \cdot G(t)
\cdot (1 + O({\varepsilon}_n))\,,$$ where the error term $O({\varepsilon}_n)$ is uniform over $t$ varying in compact subsets of the complex disk $|t|<R$.
We say that a sequence $X_n$ of random variables taking values in non-negative integers *converges mod-Poisson* if the sequence of the associated probability distributions $p_n$ converges mod-Poisson, where $p_n(k)={{\mathbb P}}(X_n=k)$ for $k=0,1,\dots$.
The term $e^{\lambda_n (t - 1)} \cdot G(t)$ in the right hand side of is the product of the generating series of ${\mathrm{Poi}}_{\lambda_n}$ with $G(t)$. In other words, it looks like . Note, however, we emphasize that $G(t)$ is not necessarily the generating series of a probability distribution.
Note that Equation implies that for any $n$ we have $F_{p_n}(1)=1$. Thus, condition from the definition of mod-Poisson convergence implies that $$\label{eq:G:1:equals:1}
G(1)=1\,.$$
\[rk:mod:poisson:diff\] Let us emphasize that our definition of mod-Poisson convergence differs from [@FerayMeliotNikeghbali] in that we take generating series ${{\mathbb E}}(t^X)$ of random variables instead of the moment generating function ${{\mathbb E}}(e^{z X})$. One can pass from one to the other by setting $t = e^z$. In particular, our assumption that $G$ is analytic at $t=0$ is not a requirement in the definition of [@FerayMeliotNikeghbali]. This extra assumption allows us to control the asymptotics of $p_n(k)$ when $k$ is in the range $k \ll \log n$.
Let ${{\mathbb P}}_{n,m,\alpha}$ be the discrete probability measure on the symmetric group $S_n$ corresponding to the weights $w_\theta(\sigma)$ associated to the sequence $\theta_i=\alpha\cdot \zeta_m(2i)$ for $i=1,2\dots$ as defined in . Recall that Lemma \[lem:cycle:distribution:vs:harmonic:sum\] and, more specifically, Equation expresses the probability distribution $q_{n,m,\alpha}(k)
={{\mathbb P}}_{n, m, \alpha}\big({{\mathrm K}}_n(\sigma) = k\big)$ through multivariate harmonic sums ${\widetilde{H}}_{n,m,\alpha}(k)$ defined in . The corollary below is a more general version of Theorem \[thm:permutation:mod:poisson:introduction\] from the introduction.
\[cor:permutation:mod:poisson\] Let ${{\mathrm K}}_n(\sigma)$ be the number of cycles of a permutation $\sigma$ in the symmetric group $S_n$. Let ${{\mathbb E}}_{n, m, \alpha}$ be the expectation with respect to the probability measure ${{\mathbb P}}_{n, m, \alpha}$ on $S_n$ as in .
For all $t \in {{\mathbb C}}$ we have as $n \to +\infty$ $$\label{eq:permutation:mod:poisson}
{{\mathbb E}}_{n, m, \alpha}\left(t^{{{\mathrm K}}_n}\right)
= \left(\frac{2m}{m+1} n\right)^{\alpha (t - 1)}
\cdot \frac{\Gamma(\alpha)}{\Gamma(\alpha t)}
\left(1 + O \left( \frac{1}{n} \right) \right)\,,$$ Moreover, the convergence in is uniform for $t$ in any compact subset of ${{\mathbb C}}$.
In other words the sequence of random variables ${{\mathrm K}}_n$ with respect to the probability measures ${{\mathbb P}}_{n,m,\alpha}$ converges mod-Poisson with parameter $\lambda_n=\alpha \log\left(\frac{2m}{m+1} n\right)$, limiting function $\Gamma(\alpha) / \Gamma(\alpha t)$, radius $R=+\infty$ and speed $1/n$.
Let us define $$G_n(t) := \sum_{k=1}^n {\widetilde{H}}_{n,m,\alpha}(k) t^k.$$ Formula for an abstract generating function combined with Formula for ${{\mathbb P}}_{n, m,
\alpha}\big({{\mathrm K}}_n(\sigma) = k\big)$ give the following expression for the generating series in the left-hand side of : $${{\mathbb E}}_{n,m,\alpha}\left(t^{{{\mathrm K}}_n(\sigma)}\right)
=
\frac{G_n(t)}{G_n(1)}$$ and the corollary now directly follows from Formula from Theorem \[thm:multi:harmonic:sum:total:weight\].
Generalizing $u_{\lambda, 1/2}$ defined in let us define $$\label{eq:poisson:gamma}
e^{\lambda (t-1)}\cdot
\frac{t \cdot \Gamma(1 + \alpha)}{\Gamma(1 + t \alpha)}
=
\sum_{k \geq 1} u_{\lambda, \alpha}(k) \cdot t^k.$$
\[cor:approximation:q:u\] Let $m\in{{\mathbb N}}\cup\{\infty\}$ and let $\alpha$ be a positive real. Uniformly in $k \geq 1$ we have as $n \to +\infty$ $$q_{n, m, \alpha}(k)
= u_{\lambda_n, \alpha}(k) + O \left(\frac{1}{n}\right)\,,$$ where $\lambda_n=\alpha \log\left(\frac{2m}{m+1} n\right)$.
Note that $$\frac{\Gamma(\alpha)}{\Gamma(\alpha t)}
=\frac{\alpha t\cdot\Gamma(\alpha)}{\alpha t\cdot\Gamma(\alpha t)}
=\frac{t \cdot \Gamma(1 + \alpha)}{\Gamma(1 + t \alpha)}\,.$$ Let $$F_1(t) := \sum_{k \geq 1} q_{n,m,\alpha}(k) t^k
\quad
F_2(t) :=
e^{\lambda (t-1)} \cdot \frac{t \cdot \Gamma(1 + \alpha)}{\Gamma(1 + t \alpha)}
=
\sum_{k \geq 1} u_{\lambda_n, \alpha}(k) t^k.$$ Both $F_1(t)$ and $F_2(t)$ are holomorphic in ${{\mathbb C}}$. Since $F_2(t)$ does not vanish we have uniformly in $t \in D(0, 1+{\varepsilon})$ $$F_1(t) - F_2(t) = O\left(\frac{1}{n}\right).$$ Using the saddle-point bound (18) from [@Flajolet:Sedgewick Proposition IV.1] with radius $R=1$ we obtain uniformly in $k \geq 1$ $$q_{n,m,\alpha}(k) - u_{\lambda_n, \alpha}(k)
=
O\left(\frac{1}{n}\right).$$
Large deviations and central limit theorem {#ss:LD:and:CLT}
------------------------------------------
Having proved the mod-Poisson convergence in Corollary \[cor:permutation:mod:poisson\], we could derive most of the following large deviation results by referring to Theorem 3.2.2 from the monograph of V. Feray, , A. Nikeghbali [@FerayMeliotNikeghbali] (see also Example 3.2.6 of the same monograph providing more details in the case of uniform random permutations). However, as we mentioned in Remark \[rk:mod:poisson:diff\], the monograph [@FerayMeliotNikeghbali] uses slightly weaker definition of mod-Poisson convergence which does not allow to study the probability distribution in the range of values of the random variable of the order $o(\lambda_n)$. To overcome this diffiulty we rely on Theorem 14 in [@Hwang:PhD] and on Theorem 2 in [@Hwang:distances] due to H. Hwang.
\[thm:large:deviations\] Let $\{X_n\}_n$ be a sequence of random variables taking values in non-negative integers that converges mod-Poisson with parameters $\lambda_n$, limiting function $G(t)$, radius $R$ and speed at least $\lambda_n^{-1}$. Assume furthermore that $G(0) \not= 0$.
For any $x \in (0, R)$, uniformly in $0 \leq k \leq x \lambda_n$ we have as $n \to +\infty$ $$\label{eq:LD:fixed:k}
{{\mathbb P}}(X_n = k)
=
e^{-\lambda_n} \frac{\lambda_n^k}{k!}
\cdot (G(k / \lambda_n) + O((k+1) / (\lambda_n)^2))\,.$$ For all $x \in (1,R)$ such that $x \lambda_n$ is an integer $$\label{eq:LD:k:bigger:than}
{{\mathbb P}}(X_n > x \lambda_n)
=
\frac{e^{-\lambda_n (x \log x - x + 1)}}{\sqrt{2 \pi \lambda_n x}}
\cdot
\frac{x}{x-1}
\cdot
(G(x) + O(\lambda_n^{-1}))$$ where the error term is uniform over $x$ in compact subsets of $(1,R)$. Similarly, for all $x \in (0, 1)$ such that $x \lambda_n$ is an integer $$\label{eq:LD:k:smaller:than}
{{\mathbb P}}(X_n \leq x \lambda_n)
=
\frac{e^{-\lambda_n (x \log x - x + 1)}}{\sqrt{2 \pi \lambda_n x}}
\cdot
\frac{x}{1-x}
\cdot
(G(x) + O(\lambda_n^{-1}))$$ where the error term is uniform over $x$ in compact subsets of $(0,1)$.
Note that by Stirling formula, for $x = \frac{k}{\lambda_n}$ we have $$\frac{e^{- \lambda_n (x \log x - x + 1)}}{\sqrt{2\pi x\lambda_n}}
=
e^{-\lambda_n} \frac{(\lambda_n)^k}{k!}
(1 + O((\log n)^{-1}))\,.$$
Note also that $x \log(x) - x + 1$ is convex and attains it minimum at $x=1$ for which it has value zero. Hence both quantities in the right-hand sides of and are exponentially decreasing in $n$.
\[Remark:shift\] If the limiting function $G$ vanishes at $0$ we can apply the following trick. Let $a\in{{\mathbb N}}$ be the order of the zero. Then the sequence of shifted variables $X_n - a$ converges mod-Poisson with the same parameters and radius but with the limiting function $t^{-a} G(t)$ which does not vanish anymore at zero. We can then apply Theorem \[thm:large:deviations\] to $X_n - a$.
Since [@FerayMeliotNikeghbali Theorem 3.2.2] is stated for the more general mod-$\phi$ convergence let us explain how their notations translate in our context. Because we use Poisson variables we have $\eta(t) = e^t - 1$ whose Legendre-Fenchel transform is $F(x) = x \log x - x - 1$. Because of this $h = \log x$. The limiting function is $\phi(e^z) = G(z)$. This difference of notation for the limiting functions is due to the fact that we used generating series ${{\mathbb E}}(t^X)$ instead of moment generating functions ${{\mathbb E}}(e^{z X})$.
The statement below is a generalization of Theorem \[thm:permutation:asymptotics\] from Section \[s:intro\] to arbitrary probability measure ${{\mathbb P}}_{n,m,\alpha}$.
\[cor:multi:harmonic:asymptotic:all:k\] Let $\alpha > 0$, $m \in{{\mathbb N}}\cup\{+\infty\}$ and let ${{\mathbb P}}_{n,m,\alpha}$ be the probability measure as in . Let $\lambda_n := \alpha \log\left(\frac{2m}{m+1} n\right)$. Let $x > 0$. Then, uniformly in $0 \leq k \leq x \lambda_n$, we have $$\begin{gathered}
\label{eq:LD1:q}
q_{n,m,\alpha}(k+1)=
{{\mathbb P}}_{n,m,\alpha}({{\mathrm K}}_n = k+1)
=\\
=e^{-\lambda_n} \frac{(\lambda_n)^{k}}{k!}
\left( \frac{\Gamma(1+\alpha)}{\Gamma\left(1 + \alpha \frac{k}{\lambda_n}\right)}
+ O\left(\frac{k+1}{(\log n)^2} \right) \right)\,.\end{gathered}$$ For $x \in (1, +\infty)$ such that $x \lambda_n$ is an integer we have $$\begin{gathered}
\label{eq:LD2:q}
\sum_{k=x\lambda_n+1}^n q_{n,m,\alpha}(k+1)=
{{\mathbb P}}_{n,m,\alpha}\big({{\mathrm K}}_n > x \lambda_n +1\big)
=\\
=\frac{e^{-\lambda_n (x \log x - x + 1)}}{\sqrt{2 \pi x \lambda_n}} \cdot
\frac{x}{x - 1}
\left( \frac{\Gamma(1+\alpha)}{\Gamma(1 + \alpha x)}
+ O\left(\frac{1}{\log n} \right) \right)\,.\end{gathered}$$ where the error term is uniform over $x$ in compact subsets of $(1, +\infty)$ and for $x \in (0, 1)$ such that $x \lambda_n$ is an integer we have $$\begin{gathered}
\label{eq:LD3:q}
\sum_{k=0}^{x\lambda_n} q_{n,m,\alpha}(k+1)=
{{\mathbb P}}_{n,m,\alpha}\big({{\mathrm K}}_n \leq x \lambda_n +1\big)
=\\
=\frac{e^{-\lambda_n (x \log x - x + 1)}}{\sqrt{2 \pi x \lambda_n}} \cdot
\frac{x}{1 - x}
\left( \frac{\Gamma(1+\alpha)}{\Gamma(1 + \alpha x)}
+ O\left(\frac{1}{\log n} \right) \right)\,.\end{gathered}$$ where the error term is uniform over $x$ in compact subsets of $(0, 1)$
By Corollary \[cor:permutation:mod:poisson\], the sequence of random variables ${{\mathrm K}}_n(\sigma)$ with respect to the probability measures ${{\mathbb P}}_{n,m,\alpha}$ on the symmetric group $S_n$ converges mod-Poisson with parameters $\lambda_n = \alpha \log\left(\frac{2m}{m+1}
n\right)$, limiting function $\Gamma(\alpha) /
\Gamma(\alpha t)$, radius $R=+\infty$ and speed $1/n$. The limiting function $\Gamma(\alpha) / \Gamma(\alpha t)$ has zero of the first order at $t=0$, so we have to apply the trick described in Remark \[Remark:shift\]. The sequence of random variables ${{\mathrm K}}_n-1$ converges mod-Poisson with the same radius $R=+\infty$ and speed $1/n$ and has the limiting function $\Gamma(\alpha) / (t\cdot\Gamma(\alpha t))$. Applying the identity $\Gamma(z+1)=z\Gamma(z)$ we conclude that the new limiting function $$G(t)
=\frac{\Gamma(\alpha)}{t\cdot\Gamma(\alpha t)}
=\frac{\alpha\Gamma(\alpha)}{\Gamma(1+\alpha t)}
=\frac{\Gamma(1+\alpha)}{\Gamma(1+\alpha t)}$$ does not vanish at $t=0$ and Theorem \[thm:large:deviations\] becomes applicable to the sequence of random variables $X_n-1$.
\[cor:multi:harmonic:asymptotic:all:k:bis\] Let $\alpha$ be a positive real number and let $m\in{{\mathbb N}}\cup\{+\infty\}$. Let ${\widetilde{H}}_{n,m,\alpha}$ be the normalized weighted multi-variate harmonic sum .
Let $\{k_n\}_n$ be a sequence of integers such that $k_n = O(\log n)$. Then as $n \to
+\infty$ we have $$\begin{gathered}
\label{eq:H:n:m:alpha}
{\widetilde{H}}_{n,m,\alpha}(k_n)=
\\
=\frac{\alpha^{k_n}}{n}
\frac{\left(\log n+\log\left(\tfrac{2m}{m+1}\right)\right)^{k_n-1}}
{(k_n-1)!}
\cdot\left(
\frac{1}
{\Gamma\left(1 + \alpha \frac{k_n-1}{\lambda_n}\right)}
+ O\left(\frac{k_n-1}{(\log n)^2} \right) \right)\,.\end{gathered}$$
If, moreover, $k_n = o(\log n)$, then as $n \to
+\infty$ we have $$\begin{gathered}
\label{eq:H:n:m:alpha:o}
{\widetilde{H}}_{n,m,\alpha}(k_n)
=\\
=\frac{\alpha^{k_n}}{n}
\frac{\left(\log n+\log\left(\tfrac{2m}{m+1}\right)\right)^{k_n-1}}
{(k_n-1)!}
\left(1+\frac{\gamma\cdot (k_n-1)}{\log n}
+O\left(\left(\frac{k_n}{\log n}\right)^2 \right)\right)\,.\end{gathered}$$
Applying with $\lambda_n = \alpha \log\left(\frac{2m}{m+1} n\right)$ we get $$\begin{gathered}
q_{n,m,\alpha}(k_n)
=e^{-\lambda_n}
\frac{(\lambda_n)^{k_n-1}}{(k_n-1)!}
\left(
\frac{\Gamma(1+\alpha)}
{\Gamma\left(1 + \alpha \frac{k_n-1}{\lambda_n}\right)}
+ O\left(\frac{k_n-1}{(\log n)^2} \right) \right)
=\\
=\left(\frac{2m}{m+1}\right)^{-\alpha} n^{-\alpha}
\frac{\left(\alpha\log\left(\tfrac{2m}{m+1}n\right)\right)^{k_n-1}}
{(k_n-1)!}
\left(
\frac{\Gamma(1+\alpha)}
{\Gamma\left(1 + \alpha \frac{k_n-1}{\lambda_n}\right)}
+ O\left(\frac{k_n-1}{(\log n)^2} \right) \right)\,.\end{gathered}$$ Applying Equation with the value $t=1$, we get $$W_{n,m,\alpha}=
\sum_{k=1}^n {\widetilde{H}}_{n,m,\alpha}(k) =
\frac{\alpha\left( \frac{2m}{m+1} \right)^{\alpha} n^{\alpha - 1}}
{\Gamma(1+\alpha)}
\left(1 + O\left(\frac{1}{n}\right)\right)\,,$$ where we used the identity $\alpha\Gamma(\alpha)=\Gamma(1+\alpha)$. By definition of $q_{n,m,\alpha}(k)$ we have $${\widetilde{H}}_{n,m,\alpha}(k)
=q_{n,m,\alpha}(k)\cdot W_{n,m,\alpha}$$ Multiplying the two expressions computed above we get .
To prove we use the asymptotic expansion $$\frac{1}{\Gamma(1+t)}=1+\gamma t + O(t^2)\quad
\text{as }\ t\to 0\,,$$ where $\gamma = 0.5572\ldots$ denotes the Euler–Mascheroni constant.
Note that for the values of parameters $m=\alpha=1$ and for the constant sequence $k_n=2$ for $n=1,2,\dots$, the expansion gives $${\widetilde{H}}_{n,1,1}(k_n)
=\frac{1}{n}\log n
\left(1+\frac{\gamma}{\log n}
+O\left(\frac{1}{(\log n)^2}\right)\right)$$ corresponding to the classical formula $$\frac{1}{2}\sum_{i=1}^n\frac{1}{j\cdot (n-j)}
=\frac{1}{n}\left(\log n+\gamma+O\left(\frac{1}{n}\right)\right).$$
The following strong form of the central limit theorem corresponds to Theorem 3.3.1 of [@FerayMeliotNikeghbali].
\[thm:CLT\] Let $\{X_n\}_n$ be a sequence of random variables on the non-negative integers that converges mod-Poisson with parameters $\lambda_n$. Let $x_n$ be a sequence of real numbers with $x_n = o( (\lambda_n)^{1/6})$. Then as $n \to +\infty$ $${{\mathbb P}}\left( \frac{X_n - \lambda_n}{\sqrt{\lambda_n}} \le x_n\right)
=
\left(\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{x_n} e^{-\tfrac{t^2}{2}} dt\right) (1 + o(1))$$
Note that, contrarily to the large deviations, the radius $R$, the limiting function $G$ and the speed ${\varepsilon}_n$ of the mod-Poisson convergence are irrelevant in the above Theorem.
Let $\alpha > 0$, $m\in{{\mathbb N}}\cup\{+\infty\}$ and let ${{\mathbb P}}_{n,m,\alpha}$ be the probability distribution on the symmetric group defined in . Let $\lambda_n := \alpha \log\left(\frac{2m}{m+1} n\right)$ and $x_n$ be a sequence of real numbers with $x_n = o( (\lambda_n)^{1/6})$. Then as $n \to +\infty$ $${{\mathbb P}}_{n,m,\alpha}
\left( \frac{{{\mathrm K}}_n - \lambda_n}{\sqrt{\lambda_n}}
\le x_n\right)
=
\left(\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{x_n} e^{-\tfrac{t^2}{2}} dt\right) (1 + o(1)).$$
By Corollary \[cor:permutation:mod:poisson\], the sequence of random variables ${{\mathrm K}}_n$ converges mod-Poisson, so Theorem \[thm:CLT\] is applicable to this sequence.
Moments of the Poisson distribution {#ss:Moments:of:Poisson:distribution}
-----------------------------------
Recall that given a non-negative integer $n$ and a positive real number $\lambda$, the $n$-th moment $P_n(\lambda)$ of a random variable corresponding to the Poisson distribution ${\mathrm{Poi}}_\lambda$ with parameter $\lambda$ is defined as $$\label{eq:def:poisson:moments}
P_n(\lambda) := e^{-\lambda} \cdot
\sum_{k = 0}^{+\infty} k^n \frac{\lambda^k}{k!}.$$
Recall that given two integers $n,k$ satisfying $1\le
k\le n$, the *Stirling number of the second kind*, denoted $S(n,k)$, is the number of ways to partition a set of $n$ objects into $k$ non-empty subsets. It is well-known that the Stirling number of the second kind satisfy the following recurrence relation: $$\label{eq:Stirling:2:reccurrence}
S(n+1,k)=k\cdot S(n,k)+S(n,k-1)\,,$$ and are uniquely determined by the initial conditions, where we set by convention: $S(0,0)=1$ and $S(n,-1)=S(n,0)=S(0,n)=S(n,n+1)=0$ for $n\in{{\mathbb N}}$.
Though the following statement is well-known, see, for example, [@Riordan], its proof is so short that we present it for the sake of completeness.
\[lem:poisson:moments\] For any $n\in{{\mathbb N}}$, the expression $P_n(\lambda)$ defined in coincides with the following monic polynomial in $\lambda$ of degree $n$: $$\label{eq:P:n:lambda:Stirling}
P_n(\lambda) = \sum_{k=0}^n S(n,k) \lambda^k\,,$$ where $S(n,k)$ are the Stirling numbers of the second kind.
The polynomials $P_n(\lambda)$ are sometimes called *Touchard polynomials*, *exponential polynomials* or *Bell polynomials*. For $n\le 4$ the polynomials $P_n(\lambda)$ have the following explicit form: $$\label{eq:first:four:P:lambda}
\begin{aligned}
P_0(\lambda) &= 1\,, \\
P_1(\lambda) &= \lambda\,, \\
P_2(\lambda) &= \lambda^2 + \lambda\,, \\
P_3(\lambda) &= \lambda^3 + 3\lambda^2 + \lambda\,, \\
P_4(\lambda) &= \lambda^4 + 6\lambda^3 + 7\lambda^2 + \lambda\,.
\end{aligned}$$
Let $X$ be a random variable with distribution ${\mathrm{Poi}}_\lambda$ and let $$\phi(z) = {{\mathbb E}}(e^{z X}) = \sum_{n=0}^{+\infty} {{\mathbb E}}(X^n)
\frac{z^n}{n!}$$ be its moment generating series. Then $$\phi(z) = \sum_{k \geq 0} e^{-\lambda} \frac{\lambda^k}{k!} e^{z k}
= e^{-\lambda} \sum_{k \geq 0} \frac{(\lambda e^z)^k}{k!}
= e^{\lambda (e^z - 1)}.$$ By definition, $P_n(\lambda) = \frac{d^n}{dz^n} \phi(z)
|_{z=0}$. We claim that for any $n=0,1,\dots$ the following identity holds: $$\label{eq:poisson:stirling}
\frac{d^n}{dz^n} \phi(z)
= \sum_{k=0}^n S(n,k)\cdot (\lambda e^z)^k\cdot \phi(z)\,.$$ Indeed $\frac{d}{dz} \phi(z) = \lambda e^z \phi(z)$ and, hence, the identity holds for $n=0$ and $n=1$. Taking the derivative of the expression in the right hand side of we obtain $$\begin{aligned}
\frac{d}{dz}
\sum_{k=0}^n S(n,k)\cdot \big(\lambda e^z\big)^k\cdot \phi(z)
&=
\sum_{k=0}^n S(n,k)\cdot
\Big(k\cdot\big(\lambda e^z\big)^k + \big(\lambda e^z\big)^{k+1}\Big)\cdot \phi(z) \\
&=\sum_{k=0}^{n+1}
\Big(S(n,k-1) + k\cdot S(n,k)\Big)\cdot (\lambda e^z)^k\cdot
\phi(z)\,.\end{aligned}$$ We recognize the recurrence relations for Stirling numbers of the second kind, which proves identity . Taking $z=0$ in we obtain .
Moment expansion {#ss:moment:expansion}
----------------
In this section we analyze the asymptotic expansions of cumulants of probability distributions that satisfies mod-Poisson convergence. We then apply it to the probability distribution $q_{n,m,\alpha}(k)=
\frac{{\widetilde{H}}_{n,m,\alpha}(k)}{W_{n,m,\alpha}}$ (see Definition \[def:hkzk\] and ).
The *cumulants* $\kappa_i(X)$ of a random variable $X$ are the coefficients of the expansion $$\log {{\mathbb E}}(e^{t X}) = \sum_{i \geq 1} \kappa_i(x) \frac{t^i}{i!}.$$ The first cumulant $\kappa_1(X) = {{\mathbb E}}(X)$ is the mean and the second cumulant $\kappa_2(X) = {{\mathbb V}}(X) = {{\mathbb E}}(X^2) - {{\mathbb E}}(X)^2$ is the variance. The cumulants are combinations of moments, but contrarily to moments, cumulants are additive: if $X$ and $Y$ are independent then $\kappa_i(X + Y) = \kappa_i(X) + \kappa_i(Y)$.
If $X$ is a Poisson random variable with parameter $\lambda$ then $$\log {{\mathbb E}}(e^{t X}) = \lambda (e^t - 1).$$ This implies that all cumulants of a Poisson random variable are equal to $\lambda$. The Theorem below proves that when a sequence of random variables converges mod-Poisson, the main contribution to the cumulants comes from the Poisson part while an explicit correction comes from the logarithmic derivative of the limiting function.
\[thm:cumulants\] Let $X_n$ be a sequence of probability distributions that converges mod-Poisson with parameters $\lambda_n$ limiting function $G$ and speed ${\varepsilon}_n$ as $n\to+\infty$. Then for all $i \geq
1$, as $n \to +\infty$ we have the following asymptotic equivalence of the $i$-th cumulant $$\label{eq:cumulants:abstract}
\kappa_i(X_n) = \lambda_n + \sum_{k=1}^i S(i,k)
\cdot \delta_k + O({\varepsilon}_n)\,,$$ where $S(i,k)$ are the Stirling numbers of the second kind and $\delta_k = \frac{d^k}{dt^k} \log G(t)|_{t=1}$ are the values of the logarithmic derivatives of the limiting function $G$ at $t=1$.
We warn the reader that the error term $O({\varepsilon}_n)$ in is uniform in $n$ but not in $i$.
\[rm:cumulants:do:not:depend:on:R\] Note that the Theorem above is valid for any radius of convergence $R$ as soon as $R>1$. The latter inequality makes part of Definition \[def:mod:poisson:convergence\] of mod-Poisson convergence.
By Definition \[def:mod:poisson:convergence\] of the mod-Poisson convergence we have $${{\mathbb E}}(e^{z X_n}) = e^{\lambda_n (e^z - 1)} G(e^z) (1 + O({\varepsilon}_n))\,,$$ see . We have seen in that our definition of mod-Poisson convergence implies that $G(1) = 1$. Hence, there exists a radius $R'$ such that for $|z| <R'$ we have $G(e^z) \not\in [-\infty,0)$. On the disk $|z| <R'$ we can take the principal determination of the logarithm to obtain $$\log {{\mathbb E}}(e^{z X_n}) = \lambda_n (e^z - 1) + \log G(e^z) + O({\varepsilon}_n).$$ which can be rewritten as $$\label{eq:moment}
\sum_{i \geq 1} (\kappa_i({{\mathrm K}}_n) - \lambda_n - \Delta_i) \cdot \frac{z^i}{i!} = O({\varepsilon}_n)$$ where $\Delta_i := \frac{d^i}{dz^i} \log G(e^z)|_{z=0}$. Let $g^{(i)}(t) = \frac{d^i}{dt^i} \log G(t)$. The rest of the proof is similar to the proof of of Lemma \[lem:poisson:moments\]. Namely, we claim that for $i \geq 1$ we have $$\label{eq:cumulant:stirling2}
\frac{d^i}{dz^i} \log G(e^z) = \sum_{k=1}^i S(i,k) g^{(k)}(e^z) e^{kz}.$$ It is indeed the case for $i=1$ and when differentiating once the formula in the right hand side we obtain $$\begin{aligned}
\frac{d}{dz} \sum_{k=1}^i S(i,k) g^{(k)}(e^z) e^{kz} &=
\sum_{k=1}^i (S(i,k) g^{(k+1)}(e^z) e^{(k+1)z} + k S(i,k) g^{(k)}(e^z) e^{kz}) \\
&= \sum_{k=1}^{i+1} (S(i,k-1) + k S(i,k)) g^{(k)}(e^z) e^{k z}.\end{aligned}$$ We recognize the recurrence relation for the unsigned Stirling numbers of the second kind. This proves the claim. Now let $\delta_i = g^{(i)}(1)$. Specializing at $z=0$ we obtain $\Delta_i = \sum_{k=1}^i S(i,k) \delta_k$.
Now, since the radius of convergence in is positive, we obtain $$\kappa_i({{\mathrm K}}_n) - \lambda_n - \delta_i(\alpha) = O({\varepsilon}_n)$$ (where the error term depends on $i$). This concludes the proof.
Recall that for $m \geq 0$, the $m$-th polygama function is defined as $$\label{eq:polygamma}
\psi^{(m)}(z) = \frac{d^{m+1}}{dz^{m+1}} \log \Gamma(z).$$
\[cor:cumulants:q\] Let $m\in{{\mathbb N}}\cup\{+\infty\}$, $\alpha \geq 0$ and let ${{\mathrm K}}_n$ be the random variable corresponding to the probability law $q_{n,m,\alpha}$ defined in . Then for any $i\in{{\mathbb N}}$ we have the following asymptotic equivalence for the $i$-th cumulant of ${{\mathrm K}}_n$ as $n \to +\infty$: $$\label{eq:cumulant:q:n:m:alpha}
\kappa_i({{\mathrm K}}_n) = \alpha \log\left(\frac{2m}{m+1} \cdot n\right)
- \sum_{k=1}^i S(i,k) \cdot \psi^{(k-1)}(\alpha) \cdot \alpha^k
+ O\left(\frac{1}{n}\right)\,,$$ where $S(i,k)$ is the Stirling number of the second kind and $\psi^{(j)}$ is the polygamma function.
By Corollary \[cor:permutation:mod:poisson\], the random variables ${{\mathrm K}}_n$ with respect to ${{\mathbb P}}_{n,m,\alpha}$ converges mod-Poisson with parameters $\lambda_n = \alpha \log\left(\frac{2m}{m+1} n\right)$, limiting function $G(t) = \Gamma(\alpha) / \Gamma(\alpha
t)$ and rate $O(1/n)$. The logarithmic derivatives of the limiting function are expressed in terms of the polygamma function by the following relation: $$\frac{d^k}{dt^k} \log \frac{\Gamma(\alpha)}{\Gamma(\alpha t)}
=-\frac{d^k}{dt^k}\log\Gamma(\alpha t)
=- \Gamma(\alpha)
\cdot\alpha^k
\cdot\psi^{(k-1)}(\alpha t)\,.$$ Applying Equation from Theorem \[thm:cumulants\] we obtain the desired relation .
The derivatives of the polygamma functions at rational points have explicit expressions in terms of $\zeta$-values. The following lemma provides the values of these derivatives at $1$ and at $1/2$ relevant for the purposes of the current paper. These formulae reproduce Formulae 6.4.2 and 6.4.4, at page 260 of [@Abramowitz:Stegun]. The proofs can be found, for example, in the paper [@Choi:Cvijovic] of J. Choi and D. Cvijović.
We have $$\psi^{(0)}(1) = -\gamma
\qquad
\psi^{(0)}(1/2) = -\gamma - 2 \log 2$$ and for $m \geq 1$ $$\begin{aligned}
\psi^{(m)}(1) &= (-1)^{m+1} \, \zeta(m+1) \, m! \\
\psi^{(m)}(1/2) &= (-1)^{m+1} \, \zeta(m+1) \, m! \, \left(2^{m+1} - 1 \right)\,.\end{aligned}$$
In the special case $m=1$ and $\alpha=1$ which corresponds to the uniform distribution on $S_n$ we obtain $$\begin{aligned}
\kappa_1(q_{n,1,1}) &= \log n + \gamma + O(1/n) \\
\kappa_2(q_{n,1,1}) &= \log n + \gamma - \zeta(2) + O(1/n) \\
\kappa_3(q_{n,1,1}) &= \log n + \gamma - 3 \zeta(2) + 2 \zeta(3) + O(1/n) \\
\kappa_4(q_{n,1,1}) &= \log n + \gamma - 7 \zeta(2) + 12 \zeta(3) - 4 \zeta(4) + O(1/n).\end{aligned}$$ We recover the expression obtained by V. L. Goncharov [@Goncharov] for the expectation and variance of the cycle count of random uniform permutations in $S_n$.
From $q_{3g-3,\infty,1/2}$ to ${p^{(1)}}_g$ {#ss:from:q:to:p1}
-------------------------------------------
Recall that ${\operatorname{Vol}}\big({\Gamma}_k(g),(m_1, \ldots, m_k))$ denotes the volume contribution from those square-tiled surfaces corresponding to the stable graph ${\Gamma}_k(g)$ for which the first maximal horizontal cylinder is filled with $m_1$ bands of squares, the second cylinder is filled with $m_2$ bands of squares, and so on up to the $k$-th maximal horizontal cylinder, which is filled with $m_k$ bands of squares. Recall also that for any $m\in{{\mathbb N}}\cup\{\infty\}$ we defined in the quantities $$V_{m,k}(g) =\!\!\!\!
\sum_{\substack{m_1, \ldots, m_k\\1\le m_i \le m\ \text{for }i=1,\dots,k}}
\!\!\!{\operatorname{Vol}}({\Gamma}_k(g), (m_1, \ldots, m_k))
\quad\text{and}\quad
V_{m}(g)=\sum_{k=1}^g V_{m,k}(g)\,.$$ We define the probability distribution ${p^{(1)}}_{g,m}(k)$ for $k=1,\dots,g$ as $$\label{eq:ProbaCylsOne}
{p^{(1)}}_{g,m}(k):=\frac{V_{m,k}(g)}{V_m(g)}\,.$$ We will sometimes denote ${p^{(1)}}_{g,\infty}$ just by ${p^{(1)}}_g$. In this section we use estimates and obtained in Theorem \[th:bounds:for:Vol:Gamma:k:g\] for $V_{m,k}(g)$ and our study of the normalized weighted harmonic sums ${\widetilde{H}}_{n,m,\alpha}$ performed in the previous sections to deduce properties of the probability distribution ${p^{(1)}}_{g,m}$. We now state and prove a lemma that we will use in the proof of Theorem \[thm:generating:series:vol:1\].
\[prop:1:2:versus:9:8\] Let $m$ be in ${{\mathbb N}}\cup\{+\infty\}$. For any $t\in{{\mathbb C}}$ satisfying $|t|<2$ we have the following estimates as $n \to +\infty$ $$\begin{aligned}
\label{eq:tail:parameter:change}
\sum_{k = \lceil 22 \cdot \log(n) \rceil+1}^{n}
{\widetilde{H}}_{n,m,9/8}(k) |t|^k
&=
\left|
\sum_{k=1}^n
{\widetilde{H}}_{n,m,1/2}(k) t^k\right|
\cdot o\left(n^{-1}\right)\!\!\,,
\\
\label{eq:tail:with:t}
\sum_{k = \lceil 22 \cdot \log(n) \rceil+1}^{n}
{\widetilde{H}}_{n,m,1/2}(k) |t|^k
&=
\left|
\sum_{k=1}^n
{\widetilde{H}}_{n,m,1/2}(k) t^k\right|
\cdot o\left(n^{-1}\right)\!\!\,,\end{aligned}$$ where the error term is uniform over $t$ varying in compact subsets of the complex disk $|t|<2$.
It follows from definition of the weighted multi-variate harmonic sum ${\widetilde{H}}_{n,m,\alpha}(k)$ that for any $n,m,k$ we have ${\widetilde{H}}_{n,m,1/2}(k)
\le {\widetilde{H}}_{n,m,9/8}(k)$. Thus, estimate implies estimate and it is sufficient to prove .
We consider separately the cases $|t| \leq 1/e$ and $1/e\le |t|<2$. We start with the case $|t| \leq 1/e$. Using the fact that we have a generating series of a probability distribution, and that for $|t|\in[0,1/e]$ and positive $k$ the power $|t|^k$ is bounded from above by $e^{-k}$, we get the following estimate valid for any $|t|\in[0,1/e]$ and any $m\in{{\mathbb N}}\cup\{+\infty\}$: $$\begin{gathered}
\label{eq:estimate:t:small}
\frac{1}{W_{n,m,9/8}}
\sum_{k = \lceil 22 \cdot \log(n) \rceil+1}^{n}
{\widetilde{H}}_{n,m,9/8}(k)\ |t|^k
\le\\
\le\frac{1}{W_{n,m,9/8}}
\sum_{k = \lceil 22 \log n\rceil+1}^n
{\widetilde{H}}_{n,m,9/8}(k)\ |t|^{1+22\log n}
\le\\
\le \left(\frac{1}{W_{n,m,9/8}}
\sum_{k=1}^n {\widetilde{H}}_{n,m,9/8}(k)\right)\cdot
|t|\cdot e^{-22\log n}
= |t|\cdot n^{-22}\,,\end{gathered}$$ where $W_{n,m,9/8}$ is the sum over $k$ of ${\widetilde{H}}_{n,m,9/8}(k)$ as defined in . On the other hand, using the identity $z\Gamma(z)=\Gamma(1+z)$ and applying Equation from Theorem \[thm:multi:harmonic:sum:total:weight\] for respectively $\alpha = 1/2$ and $\alpha=9/8$ we have as $n
\to +\infty$ $$\begin{aligned}
\label{eq:H:1:2:t}
\sum_{k=1}^n {\widetilde{H}}_{n,m,1/2}(k)\ t^k &=
t\cdot n^{-t/2}\cdot
\frac{\left( \frac{2m}{m+1} \right)^{t/2}}{2\Gamma(1+t/2)}
\left(1 + O\left(\frac{1}{n}\right)\right)\,,
\\
\notag
\sum_{k=1}^n {\widetilde{H}}_{n,m,9/8}(k)\ t^k &=
t\cdot n^{t/8}\cdot
\frac{9\left(\frac{2m}{m+1} \right)^{9t/8}}
{8\Gamma(1+9t/8)}
\left(1 + O\left(\frac{1}{n}\right)\right)\,,\end{aligned}$$ where the error terms are uniform over the compact complex disk $|t|\le 2$. In particular, for $\alpha=9/8$ and $t=1$ we get $$W_{n,m,9/8} \sim \frac{\left(\frac{2m}{m+1}\right)^{9/8} n^{1/8}}{\Gamma(9/8)}.$$ The latter asymptotic equivalence combined with imply that uniformly for $t\in[0,1/e]$ we have: $$\label{eq:o:for:t:le:1:e}
\sum_{k = \lceil 22 \cdot \log(n) \rceil+1}^{n}
{\widetilde{H}}_{n,m,9/8}(k)\ |t|^k
=O(n^{1/8})\cdot |t|\cdot n^{-22}\,.$$ Recall that $1/\Gamma(1+z)$ is an entire function having zeroes at negative integers and at no other points. Thus, for all $m\in{{\mathbb N}}\cup\{\infty\}$ and for all $t$ satisfying $|t| \le 1/e$ we have $$\min_{|t|\le\frac{1}{e}}
\left|\frac{\left( \frac{2m}{m+1} \right)^{t/2}}{2\Gamma(1+t/2)}\right|
=C>0\,.$$ Together with the latter bound implies that for $|t|\le 1/e$ and for sufficiently large $n$ we have $$\left|\sum_{k=1}^n {\widetilde{H}}_{n,m,1/2}(k)\ t^k\right|
\ge 2C\cdot |t|\cdot n^{|t|/2}
\ge 2C\cdot |t|\cdot n^{1/(2e)}\,.$$ The latter inequality combined with asymptotic estimate implies the desired relation for $|t|\le 1/e$.
We now prove for $|t|\ge
1/e$. In this regime the proof is based on Corollary \[cor:multi:harmonic:asymptotic:all:k\].
Choose any real parameter $R$ satisfying $1/e<R<2$. From now on we assume that the complex variable $t$ belongs to the closed annulus $1/e\le |t| \le R$. All the estimates below are uniform for $t\in[1/e,R]$, but the constants might depend on the choice of $R$.
We start with several preparatory remarks. We consider the function $$f(y)=\frac{1}{2}(y \log y - y + 1)\,,$$ where $y>0$. We note that the function $f$ is strictly convex with a minimum at $y=1$, where $f(1)=0$. We will need the following inequalities for $f(44/9)$: $$\label{eq:f:of:10}
f(44/9) > 1\,,$$ and $$\label{eq:f:for:1:e}
\max_{1/e\le|t|\le 2}
\left(\frac{13}{8}|t|
-\frac{9}{4} f(44/9)\cdot |t| \right)
=
\left(\frac{13}{8}|t|
-\frac{9}{4} f(44/9)\cdot |t| \right)\!\Bigg\vert_{|t|=1/e}
< -1\,.$$ We denote $
\lambda_{m,n}
= \frac{\log\left(\frac{2m}{m+1} \cdot n \right)}{2}
$. For $n\ge 3$ and any $m\in{{\mathbb N}}\cup\{\infty\}$ we have $$\label{eq:log:2:lambda:log}
\frac{\log(n)}{2}
\le \frac{\log\left(\frac{2m}{m+1} \cdot n \right)}{2}
< \log n\,,$$ which implies, in particular, that for real positive $y$ we have $$\label{eq:o:n:power:minus:1}
e^{- \lambda_{m,n} (y \log y - y + 1)}
\le e^{- \tfrac{\log n}{2} (y \log y - y + 1)}
=n^{-f(y)}\,.$$ The next remark is particularly important for the proof. It follows directly from definition of the weighted multi-variate harmonic sum ${\widetilde{H}}_{n,m,\alpha}(k)$ that $$\label{eq:multi:variate:harmonic:rescale}
{\widetilde{H}}_{n,m,\alpha}(k)\, t^k
= {\widetilde{H}}_{n,m,\alpha t}(k)\,.$$ We also get $$W_{n,m,\alpha t}
=\sum_{k=1}^n{\widetilde{H}}_{n,m,\alpha t}(k)
=\sum_{k=1}^n{\widetilde{H}}_{n,m,\alpha}(k)\, t^k\,.$$ Using this remark we can rewrite the asymptotic estimate (which we aim to prove for $|t|\in[1/e,R]$) in the following equivalent form: $$\label{eq:H:9:8:t:W:1:2:t}
\frac{1}{|W_{n,m,t/2}|}
\sum_{k = \lceil 22 \log(n) \rceil+1}^{n}
{\widetilde{H}}_{n,m,|9t/8|}(k) \stackrel{?}{=} o(n^{-1})\,.$$
Now everything is ready for the proof of Proposition \[prop:1:2:versus:9:8\] for $|t|\in[1/e,R]$. By Theorem \[thm:multi:harmonic:sum:total:weight\] for $\alpha=1/2$ and $\alpha=9/8$ we have uniformly in $t$ in the annulus $1/e\le|t|\le R$ $$\label{eq:W:with:t}
W_{n,m,\alpha t}
=\frac{\alpha t\left( \frac{2m}{m+1} \right)^{\alpha t}
n^{\alpha t - 1}}{\Gamma(1+\alpha t)}
\left(1 + O\left(\frac{1}{n}\right)\right)
=O\left(n^{\alpha t-1}\right)
\quad\text{as}\ n \to +\infty\,.$$ Recall definition of $q_{n,m,\alpha}(k)$ and apply estimate from Corollary , where we let $\alpha=9|t|/8$. Under such choice of $\alpha$, the variable $\lambda_n$, present in , takes the following value: $\lambda_n = \frac{9|t|}{8} \log\left(\frac{2m}{m+1} n\right)=
(9|t|/4)\lambda_{m,n}$. For any $y>1$ we have $$\begin{gathered}
\label{eq:tail:q:9:8:t}
\frac{1}{W_{n,m,9|t|/8}}
\sum_{k=\lceil y\lambda_{n}\rceil+1}^{n}
{\widetilde{H}}_{n,m,9|t|/8}(k)
=\sum_{k=\lceil y\cdot (9|t|/4)\cdot\lambda_{m,n}\rceil+1}^{n}
q_{n,m,9|t|/8}(k)
=\\
=\frac{e^{-(9|t|/4)\cdot\lambda_{m,n} (y \log y - y + 1)}}
{\sqrt{2\pi\cdot y\cdot (9|t|/4)\lambda_{m,n}}}
\frac{y}{(y - 1)}
\cdot O(1)
= o\left(n^{-\tfrac{9|t|}{4}\cdot f(y)} \right)\,,\end{gathered}$$ where we used for the rightmost equality.
Note that $|t|\le R< 2$. This implies that $$\min_{1/e\le|t|\le R}
\left|\frac{t\left( \frac{2m}{m+1} \right)^{\alpha t}}
{2\Gamma(1+t/2)}\right|=C'(R)>0\,.$$ This observation combined with imply that $$\frac{W_{n,m,9|t|/8}}{|W_{n,m,t/2}|}
=O\left(n^{\tfrac{9|t|}{8}+\tfrac{|t|}{2}}\right)
=O\left(n^{\tfrac{13|t|}{8}}\right)\,.$$ uniformly in $1/e\le|t|\le R$. Combining the latter estimate with we obtain $$\begin{gathered}
\frac{1}{|W_{n,m,t/2}|}
\sum_{k=\lceil y\lambda_{n}\rceil+1}^{n}
{\widetilde{H}}_{n,m,9|t|/8}(k)
=\\
=\frac{W_{n,m,9|t|/8}}{|W_{n,m,t/2}|}
\cdot
\frac{1}{W_{n,m,9|t|/8}}
\sum_{k=\lceil y\lambda_{n}\rceil+1}^{n}
{\widetilde{H}}_{n,m,9|t|/8}(k)
=O\!\left(n^{\tfrac{13|t|}{8}}\right)
o\!\left(n^{-\tfrac{9|t|}{4}\cdot f(y)}\right).\end{gathered}$$ We now choose $y=44/9$. Applying , we conclude that for $1/e\le t\le R$ we have uniformly in $t$ $$\label{eq:larger:sum:9:8}
\frac{1}{|W_{n,m,t/2}|}
\sum_{k=\left\lceil\tfrac{44}{9}\lambda_{n}\right\rceil+1}^{n}
{\widetilde{H}}_{n,m,9|t|/8}(k)
=o(n^{-1})\,.$$ It remains to note that for $|t|\le R<2$ and for $n\ge 3$ we have $$\frac{44}{9}\lambda_n
=\frac{44}{9}\cdot\frac{9}{4}\cdot|t| \cdot\frac{\log\left(\frac{2m}{m+1} \cdot n \right)}{2}
< 11\cdot |t|\cdot\log n
< 22\cdot\log n\,.$$ This implies that the sum on the left-hand side of is contained in the sum on the left-hand side of . Thus, implies and, hence, it implies the equivalent estimate in the regime $1/e\le |t|\le R<2$.
Now everything is ready to prove Theorem \[thm:generating:series:vol:1\].
The main ingredients on the proof are the asymptotic equivalence and the upper bound from Theorem \[th:bounds:for:Vol:Gamma:k:g\] combined with Proposition \[prop:1:2:versus:9:8\]. We use abbreviation . We split the sum into two parts $\sum_{k=1}^g V_{m,k}(g) \cdot t^k=\Sigma_1+\Sigma_2$, where $$\Sigma_1=\sum_{k=1}^{\lceil 22 \cdot \log(3g-3) \rceil}
V_{m,k}(g) \cdot t^k\,,
\qquad\qquad
\Sigma_2=\sum_{k=\lceil 22 \cdot \log(3g-3) \rceil+1}^g
V_{m,k}(g) \cdot t^k$$ and evaluate the two sums separately. Using from Theorem \[th:bounds:for:Vol:Gamma:k:g\] we get $$\begin{gathered}
\label{eq:head:contribution}
\Sigma_1=\frac{2\sqrt{2}}{\sqrt{\pi}} \cdot \sqrt{3g-3}
\left(\frac{8}{3}\right)^{4g-4}
\cdot\\
\cdot\left(
\sum_{k=1}^{\lceil 22 \cdot \log(3g-3) \rceil}
{\widetilde{H}}_{3g-3,m,1/2}(k) t^k \right)
\left(1 + O\left( \frac{(\log g)^2}{g}\right) \right)\,.\end{gathered}$$ Applying from Proposition \[prop:1:2:versus:9:8\] combined with from Theorem \[thm:multi:harmonic:sum:total:weight\], where we let $\alpha=1/2$ and $n=3g-3$, we get $$\begin{gathered}
\label{eq:tmp:tmp}
\sum_{k=1}^{\lceil 22 \cdot \log(3g-3) \rceil}
{\widetilde{H}}_{3g-3,m,1/2}(k) t^k
=\\
=\sum_{k=1}^{3g-3}
{\widetilde{H}}_{3g-3,m,1/2}(k) t^k
-
\sum_{k=\lceil 22 \cdot \log(3g-3) \rceil+1}^{3g-3}
{\widetilde{H}}_{3g-3,m,1/2}(k) t^k
=\\
=\left(
\sum_{k=1}^{3g-3}
{\widetilde{H}}_{3g-3,m,1/2}(k) t^k \right)
\left(1 - O\left(g^{-1}\right) \right)
=\\
=\frac{\left( \frac{2m}{m+1} \right)^{t/2}
(3g-3)^{t/2 - 1}}{\Gamma(t/2)}
\cdot\left(1 + O\left(g^{-1}\right) \right)
\,.\end{gathered}$$ Plugging the latter expression in we get $$\begin{gathered}
\label{eq:nose}
\Sigma_1=\sum_{k=1}^{\lceil 22 \cdot \log(3g-3) \rceil}
V_{m,k}(g) \cdot t^k
=\\=
\frac{2\sqrt{2} \left(\frac{2m}{m+1}\right)^{t/2}}{\sqrt{\pi} \cdot \Gamma(\frac{t}{2})}
\cdot (3g-3)^{\frac{t-1}{2}} \cdot \left( \frac{8}{3} \right)^{4g-4}
\left(1 + O\left( \frac{(\log g)^2}{g}\right) \right)\,,\end{gathered}$$ where for every compact subset $U$ of the complex disk $|t|<2$ the error term is uniform over $m \in {{\mathbb N}}\cup \{+\infty\}$ and $t\in U$. Note that the expression on the right-hand side of the latter equation coincides with the right-hand side of from Theorem \[thm:generating:series:vol:1\].
Using , from Theorem \[th:bounds:for:Vol:Gamma:k:g\], we get the following bound for the second sum: $$\label{eq:tail:contribution}
|\Sigma_2|
\le C_2\cdot \sqrt{g}
\cdot\left(\frac{8}{3}\right)^{4g-4}
\sum_{k=\lceil 22 \cdot \log(3g-3) \rceil+1}^g
{\widetilde{H}}_{3g-3,m,9/8}(k)
\cdot |t|^k\,.$$ Combining from Proposition \[prop:1:2:versus:9:8\] with and comparing the resulting expressions for $\Sigma_1$ from and for $|\Sigma_2|$ from we conclude that $\Sigma_2=\Sigma_1\cdot o\left(g^{-1}\right)$ uniformly over $m\in{{\mathbb N}}\cup\{\infty\}$ and over $t$ in any compact subset $U$ of the complex disk $|t|<2$. This completes the proof of Theorem \[thm:generating:series:vol:1\].
We show now that Theorem \[thm:generating:series:vol:1\] implies the following result.
\[cor:mod:poisson:for:vol:1\] For any $m\in{{\mathbb N}}\cup\{+\infty\}$ the family of probability distributions $\{{p^{(1)}}_{g,m}\}_{g \geq 2}$ defined in converges mod-Poisson with radius $R=2$, parameters $\lambda_{3g-3}
= \tfrac{\log\left( \tfrac{2m}{m+1} \cdot (3g-3)\right)}{2}$, limiting function $\frac{\Gamma\left(\frac{1}{2}\right)}{\Gamma \left( \frac{x}{2}\right)}$ and speed $O \left( \frac{(\log g)^2}{g}
\right)$.
Let $$\Phi_g(t) = \sum_{k=1}^{g} V_{m,k}(g) t^k\,.$$ The above sum coincides with expression from Theorem . By definition , the generating series $F(t)$ of the probability distribution ${p^{(1)}}_{g,m}$ is $\Phi_g(t) / \Phi_g(1)$. Applying Equation we get $$F(t)=\frac{\Phi_g(t)}{\Phi_g(1)}
=
\left(\frac{2m}{m+1} \cdot (3g-3) \right)^{\tfrac{t-1}{2}}
\frac{\Gamma(1/2)}{\Gamma(t/2)} \cdot
\left(1 + O\left( \frac{(\log g)^2}{g}\right) \right)\,.$$ We conclude that the generating series satisfies all conditions of Definition \[def:mod:poisson:convergence\] of mod-Posson convergence, with parameters $\lambda_{3g-3}=\log(\frac{2m}{m+1} \cdot (3g-3))/2$, limiting function $\frac{\Gamma(1/2)}{\Gamma(t/2)}$, radius $R=2$ and speed $\frac{(\log g)^2}{g}$.
We complete this Section with a proof of the assertion stated in Section \[s:intro\] claiming that *the probability distribution $q_{3g-3, \infty, 1/2}$ well-approximates the probability distribution ${q}_g(k)$*. We admit that we would not use this statement in this particular form, so we provide this justification just for the sake of completeness.
Consider the asymptotic relation from Proposition \[prop:1:2:versus:9:8\] in which we let $t=1$, $n=3g-3$ and $m=+\infty$. We get $$\sum_{k = \lceil 10 \log(n) \rceil+1}^{3g-3}
{\widetilde{H}}_{3g-3,\infty,1/2}(k)
=
\left(
\sum_{k=1}^{3g-3}
{\widetilde{H}}_{3g-3,\infty,1/2}(k)\right)
\cdot o\left(n^{-1}\right)\!\!\,.$$ The latter relation combined with and imply the following asymptotic relations for the probability distribution ${q}_g$ defined in . For $k\in{{\mathbb N}}$, $k^2\le g$, we have $${q}_g(k)=
q_{3g-3,\infty,\frac{1}{2}}(k)
\cdot
\left(1 + O\left( \frac{k^2}{g}\right) \right)\,.$$ The following asymptotic bound is valid as $g\to+\infty$: $$\sum_{k=\lceil 10\log g\rceil+1}^g
{q}_g(k)
=O\left(\frac{(\log g)^3}{g}\right)\,.$$
Analogous considerations imply that for $k\in{{\mathbb N}}$, $800 k^2\le g$, we have $$\label{eq:ProbaCylsOne:k}
{p^{(1)}}_{g,\infty}(k)
=q_{3g-3,\infty,\frac{1}{2}}(k)
\cdot
\left(1 + O\left( \frac{(k+2\log g)^2}{g}\right) \right)\,,$$ and $$\sum_{k=\lceil 10\log g\rceil+1}^g
{p^{(1)}}_{g,\infty}(k)
=O\left(\frac{(\log g)^3}{g}\right)\,.$$
Contribution of separating multicurves {#s:disconnecting:multicurves}
======================================
In Section \[s:sum:over:single:vertex:graphs\] we studied the volume contributions ${\operatorname{Vol}}(\Gamma_E(g))$ of stable graphs $\Gamma_E(g)$ with a single vertex and with $E$ loops. In particular, Theorem \[thm:generating:series:vol:1\] provides precise asymptotics for the generating series $\sum_{E \geq 1}
{\operatorname{Vol}}\Gamma_E(g) t^E$ as $g \to +\infty$. In this section we study the volume contribution of the remaining stable graphs.
In Section \[ss:generic:tail:bound\] we provide some simple estimates of tails of certain series related to Poisson distribution. In Sections \[ss:V:2\] and \[ss:V:ge:3\] we prove necessary minor refinements of estimates from [@Aggarwal:intersection:numbers] to bound respectively the volume contributions of stable graphs with 2 vertices and for the volume contribution coming from stable graphs with at least 3 vertices. We emphasize that the main asymptotic analysis of volume contributions of various stable graphs was already performed by A. Aggarwal in [@Aggarwal:intersection:numbers]. Our presentation in Sections \[ss:V:2\] and \[ss:V:ge:3\] closely follows original presentation in respectively Sections 9 and 10 of [@Aggarwal:intersection:numbers], where we perform more or less straightforward adjustment of the original bounds for the sums to bounds for the associated generating series which we need in the context of the current paper.
Following A. Aggarwal let us introduce the following notations for the contributions of stable graphs with a given number of vertices.
\[def:volume:contributions\] Let $g$ be an integer satisfying $g \geq 2$. Given a stable graph $\Gamma \in {\mathcal{G}}_g$ we denote by $V(\Gamma)$ and $E(\Gamma)$ respectively the set of vertices and the set of edges of $\Gamma$. We denote by $|V(\Gamma)|$ and $|E(\Gamma)|$ the cardinalities of these sets. We define $$\Upsilon_g^{(V)} :=
\sum_{\substack{\Gamma \in {\mathcal{G}}_g\\|V(\Gamma)| = V}} {\operatorname{Vol}}(\Gamma)\,;
\qquad
\Upsilon_g^{(V; E)} :=
\sum_{\substack{\Gamma \in {\mathcal{G}}_g\\|V(\Gamma)|
= V\\|E(\Gamma)| = E}} {\operatorname{Vol}}(\Gamma)\,,$$ where ${\operatorname{Vol}}(\Gamma)$ is the contribution of the stable graph $\Gamma$ to the Masur-Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ as given in .
Note that by we have $${\operatorname{Vol}}{\mathcal{Q}}_g = \sum_{V=1}^{2g-2} \Upsilon_g^{(V)}
\quad \text{and} \quad
\Upsilon_g^{(V)} = \sum_{E = 1}^{3g-3} \Upsilon_g^{(V; E)}\,.$$ We also have $\Upsilon_g^{(1; E)} = {\operatorname{Vol}}\Gamma_E(g)$.
The following propositions are refinements of Propositions 8.4 and 8.5 respectively from the original paper [@Aggarwal:intersection:numbers] of A. Aggarwal.
\[prop:up:bound:V2\] There exists constants $B_2$ and $g_2$ such that for any couple $g, t$, satisfying $g\in{{\mathbb N}}$, $g\ge g_2$, and $0\le t\le \frac{44}{19}$, we have $$\label{eq:up:bound:V2}
\left(\frac{8}{3}\right)^{-4g}
\cdot
\sum_{E = 1}^{3g-3} \Upsilon_g^{(2; E)} t^E
\le B_2 \cdot
t \cdot
(\log g)^{14} \cdot
g^{\frac{2t - 3}{2}}\,.$$
We prove Proposition \[prop:up:bound:V2\] in Section \[ss:V:2\].
\[prop:up:bound:V3\] There exists constants $B_3$ and $g_3$ such that for any triple $V,g, t$, satisfying $V\in{{\mathbb N}}$, $V\ge 3$, $g\in{{\mathbb N}}$, $g\ge g_3$, and $0 \le t\le \frac{44}{19}$, we have $$\label{eq:up:bound:V3}
\left(\frac{8}{3}\right)^{-4g} \cdot
\sum_{V = 3}^{2g-2} \sum_{E=1}^{3g-3}
\Upsilon_g^{(V; E)} t^E
\le B_3\cdot
t \cdot
(\log g)^{24} \cdot
g^{\tfrac{9 t - 10}{4}}\,.$$
We prove Proposition \[prop:up:bound:V3\] in Section \[ss:V:ge:3\].
Bound for tail contribution to moments of Poisson distribution {#ss:generic:tail:bound}
--------------------------------------------------------------
Recall that given a non-negative integer $n$ and a positive real number $\lambda$, the $n$-th moment $P_n(\lambda)$ of a random variable corresponding to the Poisson distribution ${\mathrm{Poi}}_\lambda$ with parameter $\lambda$ is defined as $$P_n(\lambda) := e^{-\lambda} \cdot
\sum_{k = 0}^{+\infty} k^n \frac{\lambda^k}{k!}.$$ In the next two sections we will use several times the following upper bound for the tail of the above expression.
\[lem:exp:tail:bound\] Let $\lambda$ and $x$ be strictly positive real numbers and let $n$ be an integer satisfying $n \geq 0$. Then $$\label{eq:tail:upper:bound}
\sum_{k = \lceil x \lambda \rceil}^{+\infty}
\frac{k^n \cdot \lambda^k}{k!} \leq
P_n(x \lambda) \cdot \exp\big(-\lambda (x \log x - x)\big)\,.$$
We are interested in the case where $x$ is fixed and $\lambda$ tends to infinity. Note that the term $x \log x
- x$ is positive for $x > e = 2.712\ldots$, so for $x>e$ we prove an exponential decay in $\lambda$ of the expression in the left hand side of .
Let $\theta \geq 0$. Then for $k \geq \lceil \lambda x \rceil$ we have $\exp (\theta (k - \lambda x)) \geq 1$. Hence $$\sum_{k = \lceil x \lambda \rceil}^{+\infty} \frac{k^n \cdot \lambda^k}{k!}
\leq
\sum_{k = 0}^{+\infty} \frac{\exp(\theta(k - \lambda x)) \cdot k^n \cdot \lambda^k}{k!}
=
e^{-\theta \lambda x} \sum_{k=0}^{+\infty} \frac{k^n \cdot (e^\theta \cdot \lambda)^k}{k!}\,.$$ We get a sum as in . By Lemma \[lem:poisson:moments\] we have $$\sum_{k \geq 0} \frac{k^n \cdot (e^\theta \lambda)^k}{k!}
= \exp(e^\theta \lambda) \cdot P_n(e^\theta \lambda).$$ The tail bound is obtained by taking $\theta = \log x$.
We note that analogous Lemma 2.4 of A. Aggarwal [@Aggarwal:intersection:numbers] provides a similar upper bound for the case $n=0$ given as $$\label{eq:Lemma:2::Amol}
\sum_{k = \lfloor (1 + 2\delta) R \rfloor}^{+\infty}
\frac{R^k}{k!}
\leq
\exp \left(- R \left( \delta \log(1+\delta) + \frac{\log(\delta)}{R} - 1\right) \right)\,.$$ We will need a slightly stronger estimate. Bound above provides exponential decay as long as $\log(1 + \delta) > 1/\delta$ which corresponds to $\delta > 1.23997\ldots$ In comparison, bound reads as $$\sum_{k = \lceil (1 + 2\delta) R \rceil}^{+\infty}
\frac{R^k}{k!}
\leq
\exp \big(-R ((1+2\delta) \log (1 + 2 \delta) - (1 + 2 \delta) \big)\,,$$ which provides exponential decay as long as $1 + 2 \delta > e$ which corresponds to $\delta > 0.8591409$.
Volume contribution of stable graphs with $2$ vertices {#ss:V:2}
------------------------------------------------------
Following the original approach of A. Aggarwal [@Aggarwal:intersection:numbers], we consider the following refinement of the quantity $\Upsilon_g^{(V;E)}$ introduced in Definition \[def:volume:contributions\].
Given a stable graph $\Gamma \in {\mathcal{G}}_g$ denote by $S(\Gamma)$ the number of self-edges of $\Gamma$ (edges with their two ends on the same vertex). Denote by $T(\Gamma)$ the set of simple edges of $\Gamma$ (edges with their two ends at distinct vertices). The set $E(\Gamma)$ decomposes as the disjoint union $E(\Gamma) =
S(\Gamma) \sqcup T(\Gamma)$. Following [@Aggarwal:intersection:numbers Definition 8.6] let $$\Upsilon_g^{(V; S, T)} :=
\sum_{\substack{\Gamma \in {\mathcal{G}}_g\\
|V(\Gamma)| = V\\
|S(\Gamma)| = S\\|T(\Gamma)| = T}}
{\operatorname{Vol}}(\Gamma)\,.$$ The number $\Upsilon^{(V; S, T)}_g$ is non-zero only when the following three conditions are simultaneously satisfied: $V-1 \leq T$ (connectedness of the graph), $S +
T - V + 1 \leq g$ (bound on the genus) and $V \leq 2g-2$ (“stability” of the graph). In particular, the number $\Upsilon^{(V; S, T)}_g$ is non-zero only when the following bounds are simultaneously satisfied: $0\le S\le g$ and $V-1\le T\le 3g-3$.
Following [@Aggarwal:intersection:numbers Lemma 9.5] we split the stable graphs with $2$ vertices into three groups corresponding to the following ranges of parameters $S$ and $T$. We have $S\ge g-1$ for the first collection of stable graphs. We have $T>13$ and $S\le g-2$ for the second collection. We have $T\le 13$ and $S\le g-2$ for the third collection. Lemmas \[lm:V:2:large:S\], \[lm:V:2:large:T\], \[lm:V:2:small:T\] provide upper bounds for the respective contributions to the sum in Proposition \[prop:up:bound:V2\]. As we already mentioned, our proofs of Lemmas \[lm:V:2:large:S\], \[lm:V:2:large:T\], \[lm:V:2:small:T\] are obtained by elementary adjustment of bounds elaborated by A. Aggarwal in [@Aggarwal:intersection:numbers Section 9].
\[lm:V:2:large:S\] For any non-negative real number $t$ and for any integer $g$ satisfying $g\ge 2$ the following bound is valid $$\label{eq:V:2:large:S}
\left(\frac{8}{3}\right)^{-4g}
\sum_{\substack{T\ge 1\\S\ge g-1 }}
\Upsilon^{(2;S,T)}_g
\cdot t^{S+T}
\le 2^{11}\cdot \big(t^g+t^{g+1}\big)
\cdot g^{3/2}
\cdot\left(\frac{9}{8}\right)^{g}
\frac{(\log g+7)^g}{(g-1)!}
\,.$$
All possible stable graphs with 2 vertices and with $S \ge
g-1$ split into the following three types:
1. $1+[(g-1)/2]$ stable graphs with $S=g-1$, $T=2$ and genera (decorations) $g_1=g_2=0$ at the two vertices;
2. $g-1$ graphs with $S=g-1$, $T=1$ and $g_1=1$, $g_2=0$;
3. $1+[(g-1)/2]$ graphs with $S=g$, $T=1$ and $g_1=g_2=0$.
By Equation (9.14) in [@Aggarwal:intersection:numbers] for any of these graphs $\Gamma$ we have $$\label{eq:Z:of:Gamma}
\left(\frac{8}{3}\right)^{-4g}{\operatorname{Vol}}(\Gamma)\le
2^{10}(S^2+T^2)g^{-3/2}\xi_1\xi_2
\frac{(\log g+7)^{S+T-1}}{2^S S! T!}\,.$$ Here $\xi_i$, $i=1,2$, are defined in Equation (9.1) in [@Aggarwal:intersection:numbers] as $$\xi_i=\max_{\boldsymbol{d}\in\Pi(3g_i+2s_i+T+3,2s_i+T)}
\big(1+{\varepsilon}(\boldsymbol{d})\big)\,,
$$ where $g_i$ and $s_i$ are respectively the genera and the number of self edges at the $i$-th vertex, for $i=1,2$ and ${\varepsilon}(\boldsymbol{d})$ is defined in . The bound from Theorem \[th:correlators:upper:bound\] of A. Aggarwal (see Proposition 1.2 in the original paper [@Aggarwal:intersection:numbers]) implies that for the stable graphs with $V=2$ we have $$\xi_1\xi_2
\le\left(\frac{3}{2}\right)^{2E-2}
\le\left(\frac{3}{2}\right)^{2g}$$ and provides the following bound for any stable graph with $V=2$ and $S\ge g-1$: $$\label{eq:Z:of:Gamma:bis}
\left(\frac{8}{3}\right)^{-4g}{\operatorname{Vol}}(\Gamma)
\le
2^{10}(g^2+1)g^{-3/2}
\cdot\left(\frac{3}{2}\right)^{2g}
\frac{(\log g+7)^g}{2^g (g-1)!}
\,.$$ We have seen that when $V=2$ and $S\ge g-1$ there are $g-1$ stable graphs of type (II) for which $S+T=g$ and there are at most $g+1$ stable graphs of types (I) and (III) counted together for which $S+T=g+1$. Thus we get $$\begin{gathered}
\left(\frac{8}{3}\right)^{-4g}\cdot
\sum_{\substack{T\ge 1\\ S \ge g-1}}
\Upsilon^{(2;S,T)}_g \cdot t^{S + T}
\le\\
\le\Big((g-1)t^g+(g+1)t^{g+1}\Big)\cdot
2^{10}\cdot(g^2+1)\cdot g^{-3/2}
\cdot\left(\frac{3}{2}\right)^{2g}
\frac{(\log g+7)^g}{2^g (g-1)!}
\le\\
\le 2^{11}\cdot\big(t^g+t^{g+1}\big)
\cdot g^{3/2}
\cdot\left(\frac{9}{8}\right)^{g}
\frac{(\log g+7)^g}{(g-1)!}
\,.\end{gathered}$$
\[lm:V:2:large:T\] There exists a constant $C_7$ such that for any non-negative real number $t$ and for any integer $g$ satisfying $g\ge 2$ we have $$\begin{gathered}
\label{eq:V2:2}
\left(\frac{8}{3}\right)^{-4g}
\sum_{\substack{14 \leq T \leq 3g-3\\0\leq S \leq g-2}}
\Upsilon^{(2;S,T)}_g
\cdot t^{S+T}
\le\\
\le C_7\cdot t^{14}\cdot \exp(189t/8)
\cdot(\log g +7)^{13}
\cdot g^{(27t - 56)/8}\,.\end{gathered}$$
By Equation (9.17) from Lemma 9.5 in [@Aggarwal:intersection:numbers], in the case $T>13$ and $S\le g-2$ we have for $g$ large enough $$\left(\frac{8}{3}\right)^{-4g}
\Upsilon^{(2;S,T)}_g
\le 2^{55} g^{-7}\left(\frac{9}{4}\right)^E
\frac{(\log g +7)^{E-1}}{2^S\, S!\, T!}\,.$$ By the binomial expansion we have $$\sum_{\substack{S+T=E\\S,T \geq 0}}
\frac{1}{2^S\, S!\, T!}=
\frac{1}{E!}\left(\frac{1}{2}+1\right)^E=
\frac{1}{E!}\left(\frac{3}{2}\right)^E\,.$$ Note that the set ${\mathcal{G}}_g$ of stable graphs of any fixed genus $g$ is finite. Hence, there exists a constant $C'_7$ such that for any $g\ge 2$ and for any fixed $E$ we have $$\begin{gathered}
\left(\frac{8}{3}\right)^{-4g}
\cdot
\sum_{\substack{14 \le T \le 3g-3\\0 \le S \le g-2\\S+T=E}}
\Upsilon^{(2;S,T)}_g
\le\\
\le C'_7\cdot g^{-7}
\cdot(\log g +7)^{-1}
\cdot\left(\frac{9}{4}\right)^E
\cdot(\log g +7)^{E}
\sum_{\substack{S+T=E\\S,T \geq 0}}
\frac{1}{2^S\, S!\, T!}
=\\=
C'_7\cdot g^{-7}
\cdot(\log g +7)^{-1}
\cdot\left(\frac{27}{8}\right)^E
\cdot(\log g +7)^{E}
\cdot\frac{1}{E!}\,.\end{gathered}$$ Multiplying each term by $t^E=t^{S+T}$ and taking the sum with respect to $E$ we obtain $$\begin{gathered}
\left(\frac{8}{3}\right)^{-4g}
\cdot
\sum_{\substack{14 \le T \le 3g-3\\0 \le S \le g-2}}
\Upsilon^{(2;S,T)}_g
\cdot t^{S+T}
\le\\
\le
C'_7\cdot g^{-7}
\cdot(\log g +7)^{-1}
\sum_{E=14}^{+\infty}
\left(\frac{27}{8}\right)^E
\cdot(\log g +7)^{E}\cdot t^E
\cdot\frac{1}{E!}
\le\\
\le C'_7\cdot g^{-7}
\cdot(\log g +7)^{-1}
\cdot
\left(
\frac{27t}{8}
\cdot
(\log g +7)
\right)^{14}
\cdot\exp \left(
\frac{27t}{8}
\cdot
(\log g +7)
\right)
=\\=
C_7\cdot t^{14}
\cdot \exp(189t/8)
\cdot(\log g +7)^{13}
\cdot g^{(27t - 56)/8}\,,\end{gathered}$$ where we used the inequality $\sum_{k=n}^{+\infty} \frac{x^k}{k!} \le x^n\exp(x)$, which is valid for any non-negative $x$, and where we let $C_7=C'_7\cdot\left(\frac{27}{8}\right)^{14}$.
\[lm:V:2:small:T\] There exists a constant $C_8$ such that for any non-negative real number $t$ and for any integer $g$ satisfying $g\ge 2$ we have $$\begin{gathered}
\label{eq:V2:3}
\left(\frac{8}{3}\right)^{-4g}\cdot
\sum_{\substack{1\le T\le 13\\0\le S\le g-2}}
\Upsilon^{(2;S,T)}_g
\cdot t^{S + T}
\le C_8\cdot t (1+t)^{14}
\cdot \exp(63t/4)
\cdot \\
\cdot (\log g+7)^{14} \cdot \Big(g^{(2t - 3)/2} + g^{(9t-28)/4}\Big)\,.\end{gathered}$$
It follows from Equation (9.18) from Lemma 9.5 from [@Aggarwal:intersection:numbers] that there exists a constant $C_8$ such that in the case $T\le
13$ and $S\le g-2$ the following bound is valid for any integer $g$ satisfying $g\ge 2$: $$\left(\frac{8}{3}\right)^{-4g}
\Upsilon^{(2;S,T)}_g
\le C_8\cdot\frac{(\log g+7)^{S+12}}{S!}
\left(g^{-3/2}(S^2+1)+g^{-7}\left(\frac{9}{4}\right)^S\right)\,.$$ Multiplying by $t^E=t^{S+T}$ and taking the sum over $1 \le T \le 13$ and over $0 \le S \le g-2$ we obtain the following bound: $$\begin{gathered}
\label{eq:bound:T:13}
\left(\frac{8}{3}\right)^{-4g}\cdot
\sum_{\substack{1\le T\le 13\\0\le S\le g-2}}
\Upsilon^{(2;S,T)}_g
\cdot t^{S + T}
\le\\
\le C_8 \cdot\Big(t+t^2+\dots+t^{13}\Big)
\cdot(\log g+7)^{12}
\cdot \Big(g^{-3/2} \Sigma_1 + g^{-7} \Sigma_2\Big)
\le\\
\le C_8\cdot t \cdot(1+t)^{12}
\cdot(\log g+7)^{12}
\cdot \Big(g^{-3/2} \Sigma_1 + g^{-7} \Sigma_2\Big)\,,\end{gathered}$$ where $$\begin{aligned}
\Sigma_1 &=
\sum_{S=0}^{+\infty} (S^2+1) \frac{(t \cdot (\log g+7))^S}{S!}\,,
\\
\Sigma_2 &=
\sum_{S=0}^{+\infty}
\frac{\big(\frac{9t}{4} (\log g+7)\big)^S}{S!}
=\exp(63t/4)\cdot g^{9t/4}\,.\end{aligned}$$ The sum $\Sigma_1$ can be decomposed into two sums of the form , where we take $n=2$ and $n=0$ respectively and where we let $\lambda=t \cdot (\log g+7)$. Applying Lemma \[lem:poisson:moments\] and taking into consideration that $P_2(\lambda)=\lambda^2+\lambda$ by , we get $$\begin{gathered}
\Sigma_1 =
\sum_{S=0}^{+\infty} (S^2+1)\cdot
\frac{(t \cdot (\log g+7))^S}{S!}
=e^\lambda\cdot\big(P_2(\lambda)+1\big)
=\\
=\exp(7t)\cdot g^t\cdot
\Big(1+t\cdot(\log g+7)+t^2\cdot(\log g+7)^2\Big)
\le\\
\le
\exp(7t)\cdot g^t\cdot
(1+t)^2 (\log g + 7)^2\,.\end{gathered}$$ Plugging the resulting bounds for the sums $\Sigma_1$ and $\Sigma_2$ into we obtain the bound $$\begin{gathered}
\left(\frac{8}{3}\right)^{-4g}\cdot
\sum_{\substack{1\le T\le 13\\0\le S\le g-2}}
\Upsilon^{(2;S,T)}_g
\cdot t^{S + T}
\le C_8\cdot t \cdot (1+t)^{14}
\cdot(\log g+7)^{14}
\cdot\\
\Big(g^{-3/2}\cdot
\exp(7t)\cdot g^t
+
g^{-7}\cdot
\exp(63t/4)\cdot g^{9t/4}
\Big)
=\\
=C_8\cdot t (1+t)^{14}
\cdot(\log g+7)^{14}\cdot
\\
\Big(\exp(7t) \cdot g^{(2t - 3)/2}
+\exp(63t/4)\cdot g^{(9t-28)/4}
\Big)\,.\end{gathered}$$ Now it remains to notice that $7t \le 63t/4$ to get the desired bound.
By taking the sum of the bounds , and from respectively Lemmas \[lm:V:2:large:S\], \[lm:V:2:large:T\] and \[lm:V:2:small:T\] covering all possible combinations of $S$ and $T$ we obtain $$\begin{gathered}
\label{eq:up:bound:V2:long}
\left(\frac{8}{3}\right)^{-4g}
\cdot
\sum_{E = 1}^{3g-3} \Upsilon_g^{(2; E)} t^E
\le 2^{11}\cdot\big(t^g+t^{g+1}\big)
\cdot g^{3/2}
\cdot\left(\frac{9}{8}\right)^{g}
\frac{(\log g+7)^g}{(g-1)!}
+\\+
C_7\cdot t^{14}\cdot \exp(189t/8)
\cdot(\log g +7)^{13}
\cdot g^{(27t - 56)/8}
+\\+
C_8\cdot t \cdot (1+t)^{14} \cdot \exp(63t/4)
\cdot(\log g+7)^{14}\cdot \big(g^{(2t - 3)/2} + g^{(9t-28)/4}\big)
\,.\end{gathered}$$ Now note that $$\begin{aligned}
\frac{9t-28}{4} \le \frac{2t - 3}{2}
&\qquad\text{for }t \le \frac{22}{5}=4.4\quad\text{and}
\\
\frac{27t - 56}{8} \le \frac{2t-3}{2}
&\qquad\text{for }t \le \frac{44}{19}\approx 2.3\,.
\end{aligned}$$ Note also that for the particular value $t=\frac{44}{19}$ of $t$ for which the powers of $g$ in the second and in the third term in coincide, the power of $(\log g +7)$ is larger in the third term. Since by construction $C_8>0$, there exists a constant $g_0$ such that for any $g\ge g_0$ and any $t$ in the interval $\left[0,\frac{44}{19}\right]$ the expression $$C_8\cdot t \cdot (1+t)^{14}
\cdot(\log g+7)^{14}
\cdot\exp(63t/4)
\cdot g^{(2t - 3)/2}$$ dominates the sum . This completes the proof of Proposition .
Volume contribution of stable graphs with $3$ and more vertices {#ss:V:ge:3}
---------------------------------------------------------------
In this section we adjust the bound for the sum of contributions of stable graphs with $3$ and more vertices to the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ elaborated by A. Aggarwal in [@Aggarwal:intersection:numbers Section 10] to bounds for the associated generating series in a variable $t$.
The Lemma below is based on Proposition 10.4 in [@Aggarwal:intersection:numbers] and is parallel to [@Aggarwal:intersection:numbers Lemma 10.5].
\[lem:upgraded:Aggarwal:10.5\] For any couple of integers $g$ and $V$ satisfying $g \geq 2$, $V \geq 2$, and for any non-negative real $t$ we have $$\begin{gathered}
\label{eq:upgraded:Aggarwal:10.5}
\left(\frac{8}{3}\right)^{-4g} \sum_{S=0}^{g}
\Upsilon_g^{(V; S, T)} \cdot t^{S+T}
\le T^{2V+Y+1}\cdot\frac{A_{g,t}^T}{T!}\cdot
\\
\cdot 2^{12}\cdot2^{23V}
\cdot\frac{1}{V^{3V}}
\cdot(\log g + 7)^{-1}
\cdot g^{1/2-V}\cdot
\\
\cdot\Big(
1+2^V \big(A_{g,t}+A_{g,t}^V\big)
+A_{g,t}^V (2^V + A_{g,t}) \exp(A_{g,t})
\Big)\,.\end{gathered}$$ where we use the notations $Y = \min(2T, 3V)$ and $\displaystyle A_{g,t} =
\frac{9t}{8}\cdot \left(\log g + 7\right)$.
It follows from Proposition 10.4 in [@Aggarwal:intersection:numbers] that $$\label{eq:sum:over:S}
\left(\frac{8}{3}\right)^{-4g}
\sum_{S=0}^{g} \Upsilon_g^{(V; S, T)} \cdot t^{S+T}
\le
2^{11} \cdot B_{T,V} \cdot
\left( \Sigma_1 + \Sigma_2 \right)\,,$$ where $$B_{T,V} =
2T\cdot g^{1/2-V}\cdot
2^{20V}\cdot \left( \frac{9t}{4} \right)^T
\left( \frac{T}{V} \right)^{2V}
\frac{ (\log g + 7)^{T-1}\cdot (2T-1)!!}{V^V (2T-Y)!}\,,$$ and $$\Sigma_1 = 1+\sum_{S=1}^{V-1} S\cdot A_{g,t}^S \\
\quad \text{and} \quad
\Sigma_2 = \sum_{S=V}^{g} S \frac{A_{g,t}^S}{(S - V)!}\,.$$ Transforming the bound in [@Aggarwal:intersection:numbers Proposition 10.4] to the above form we used the following trivial observations. Since $V\ge 2$ we have $T\ge 1$. The case $S=0$ corresponds to the constant summand “$1$” in $\Sigma_1$. In the remaining cases, that is when $S\ge 1$, we used the bound $S+T\le 2TS$ for the factor $(S+T)$ present in [@Aggarwal:intersection:numbers Proposition 10.4] which is valid for all $S,T\in{{\mathbb N}}$.
Using the trivial inequality $(V-1)^2<2^V$, valid for any $V\in{{\mathbb N}}$, we obtain the following bound on the first sum: $$\Sigma_1 \leq 1+(V-1)^2 \big(A_{g,t}+A_{g,t}^V\big)
\le 1+2^V \big(A_{g,t}+A_{g,t}^V\big)\,.$$ The sum $\Sigma_2$ is a part of an infinite sum of the form taken with parameters $n=1$ and $\lambda=A_{g,t}$. Applying Lemma \[lem:poisson:moments\] and using the fact that $P_1(\lambda)=\lambda$ by , we get the following bound: $$\Sigma_2 \leq \sum_{S=V}^{+\infty} S \frac{ A_{g,t}^S }{(S-V)!}
=
A_{g,t}^V \sum_{S=0}^{+\infty} (V+S) \frac{A_{g,t}^S}{S!}
=A_{g,t}^V (V + A_{g,t}) \exp(A_{g,t})\,.$$ Applying the trivial bound $V\le 2^V$ and taking the sum we obtain $$\label{eq:V3:Sigma:Sigma2:up:bound}
\Sigma_1 + \Sigma_2 \le
1+2^V \big(A_{g,t}+A_{g,t}^V\big)
+A_{g,t}^V (2^V + A_{g,t}) \exp(A_{g,t})\,.$$
Using $(2T)^Y (2T-Y)! \ge (2T)! = 2^T T! (2T-1)!!$ and $Y \le 3V$ we obtain the following bound for $B_{T,V}$, $$\begin{gathered}
\label{eq:BTV:up:bound}
B_{T,V} =
2T\cdot g^{1/2-V}\cdot
2^{20V}\cdot \left( \frac{9t}{4} \right)^T
\frac{T^{2V}}{V^{2V}}
\cdot\frac{ (\log g + 7)^{T-1}}{V^V}
\cdot \frac{(2T-1)!!}{(2T-Y)!}
\le\\
\le 2^{1+20V+Y}
\cdot(\log g + 7)^{-1}
\cdot g^{1/2-V}
\cdot T^{2V+Y+1}
\cdot \frac{A_{g,t}^T}{V^{3V} T!}
\le\\
2\cdot 2^{23V}
\cdot(\log g + 7)^{-1}
\cdot g^{1/2-V}
\cdot\frac{T^{2V+Y+1}}{V^{3V}}
\cdot\frac{A_{g,t}^T}{T!}
\,.\end{gathered}$$ Putting together and into we obtain .
We perform now summation over the variable $T$. The following statement is an adjustment of [@Aggarwal:intersection:numbers Lemma 10.6] to generating series in $t$.
\[lem:upgraded:Aggarwal:10.6\] There exist constants $C_9$ and $C_{10}$ such that for any couple of integers $g$ and $V$ satisfying $g \geq 2$, $V \geq 2$, and for any non-negative real $t$ we have $$\begin{gathered}
\label{eq:upgraded:Aggarwal:10.6}
\left( \frac{8}{3} \right)^{-4g}
\sum_{E} \Upsilon_g^{(V; E)} t^E
\le t\cdot C_9
\cdot \exp(63 t / 4)
\cdot g^{\tfrac{9t+2}{4}}
\cdot\\
\cdot\left(
\left(
\frac{C_{10}\cdot V^{1/2}}{g} \right)^V
+
\left(
\frac{C_{10}
\cdot t^8
\cdot V^{1/2}\cdot (\log g + 7)^8}{g} \right)^V
\right)
\,.\end{gathered}$$
First note that the second and the third lines of expression do not depend on the variable $T$. To bound the sum $$\sum_{T=V-1}^{3g-3}
T^{2V+Y+1}\cdot\frac{A_{g,t}^T}{T!}\,,$$ where $Y = \min(2T, 3V)$, we bound separately the following three partial sums: $$\begin{aligned}
\Sigma_1 &=
\sum_{T=V-1}^{\lfloor 3V/2 \rfloor}
T^{2V+2T+1} \frac{A_{g,t}^T}{T!}
\\
\Sigma_2 &=
\sum_{T=\lceil 3V / 2 \rceil}^{6V+1}
T^{5V+1} \frac{A_{g,t}^T}{T!}
\\
\Sigma_3&=\sum_{T=6V+2}^{+\infty}
T^{5V+1} \frac{A_{g,t}^T}{T!}\,,
\end{aligned}$$ where we use the same notation $A_{g,t} = \frac{9 t}{8}(\log g + 7)$ as in Lemma \[lem:upgraded:Aggarwal:10.5\].
To bound $\Sigma_1$ and $\Sigma_2$ we use twice the inequality $T! \geq e^{-T} T^T$ to obtain $$\begin{gathered}
\Sigma_1 \le
\sum_{T=1}^{\lfloor 3V/2 \rfloor}
(e\cdot A_{g,t})^T
\cdot T^{2V+T+1}
\le\left(\frac{3V}{2} \right)^{2V+1+3V/2}
\cdot\sum_{T=1}^{\lfloor 3V/2 \rfloor}
(e\cdot A_{g,t})^T
\\
\le\left(\frac{3V}{2} \right)^{7V/2+1}
\left(\frac{3V}{2} \right)
\Big(e A_{g,t}+(e A_{g,t})^{3V/2}\Big)
=\\=
\Big(e A_{g,t}+(e A_{g,t})^{3V/2}\Big)
\left(\frac{3V}{2} \right)^{7V/2+2}\,.\end{gathered}$$ Similarly, $$\begin{gathered}
\Sigma_2
\le
\sum_{T=\lceil 3V / 2 \rceil}^{6V+1}
T^{5V+1-T} (e\cdot A_{g,t})^T
\le
(6V+1)^{5V+1-3V/2}
\cdot \sum_{T=\lceil 3V / 2 \rceil}^{6V+1}
(e\cdot A_{g,t})^T
\le\\
\le
(6V+1)^{7V/2+1}
\cdot(6V+1)\cdot
\Big((e\cdot A_{g,t})^{3V/2}+(e\cdot A_{g,t})^{6V+1}\Big)
\le\\ \le
\Big((e\cdot A_{g,t})^{3V/2}+(e\cdot A_{g,t})^{6V+1}\Big)
\cdot(7V)^{7V/2+2}
\,.\end{gathered}$$ To bound the third sum, we use the inequality $$\frac{T^{5V+1}}{T!}\le \frac{6^{5V+1}}{(T - 5V - 1)!}\,,$$ valid for $T > 6V+1$, and the inequality $\sum_{k=n}^{+\infty} \frac{x^k}{k!} \le x^n\exp(x)$, valid for any non-negative $x$: $$\begin{gathered}
\Sigma_3=\sum_{T=6V+2}^{+\infty}
T^{5V+1} \frac{A_{g,t}^T}{T!}
\le 6^{5V+1} \sum_{T=6V+2}^{+\infty}
\frac{A_{g,t}^T}{(T - 5V - 1)!}= \\
= (6 A_{g,t})^{5V+1} \sum_{T=V+1}^{+\infty} \frac{A_{g,t}^T}{T!}
\le (6 A_{g,t})^{5V+1} \sum_{T=V}^{+\infty} \frac{A_{g,t}^T}{T!}
\le\\
\le (6 A_{g,t})^{5V+1} A_{g,t}^{V}\exp(A_{g,t})
=(6 A_{g,t})^{6V+1}
\cdot \exp(A_{g,t})\,.\end{gathered}$$ Collecting the terms and applying the bounds $V^2\le 4^V$ and $A_{g,t}^{3V/2}\le A_{g,t}+A_{g,t}^{7V/2}$ we get $$\begin{gathered}
\Sigma_1+\Sigma_2+\Sigma_3
\le
\Big(e A_{g,t}+(e A_{g,t})^{3V/2}\Big)\cdot
\left(\frac{3V}{2} \right)^{7V/2+2}
+\\+
\Big((e\cdot A_{g,t})^{3V/2}+(e\cdot A_{g,t})^{6V+1}\Big)
\cdot(7V)^{7V/2+2}
+
(6 A_{g,t})^{6V+1}
\cdot \exp(A_{g,t})
\le\\
\le
3e\cdot A_{g,t}\cdot\Big(\big(1+(e\cdot A_{g,t})^{6V}\big)
\cdot(7V)^{7V/2+2}
+
(6 A_{g,t})^{6V}
\cdot \exp(A_{g,t})\Big)
\le\\
\le 10 t\cdot (\log g + 7)
\cdot\Big(49\cdot 4^V\big(1+(e\cdot A_{g,t})^{6V}\big)
\cdot(7V)^{7V/2}
+
(6 A_{g,t})^{6V}\cdot \exp(A_{g,t})\Big)\,.\end{gathered}$$ Combining the resulting bound with we get $$\begin{gathered}
\left( \frac{8}{3} \right)^{-4g}
\sum_{E} \Upsilon_g^{(V; E)} t^E
\le t\cdot g^{1/2-V}\cdot
\\
\cdot 10\cdot 2^{12}
\cdot\left(\frac{2^{23}}{V^3}\right)^V
\cdot\Big(49\cdot 4^V\big(1+(e\cdot A_{g,t})^{6V}\big)
\cdot(7V)^{7V/2}
+
(6 A_{g,t})^{6V}
\exp(A_{g,t})\Big)
\cdot
\\
\cdot\Big(
1+2^V \big(A_{g,t}+A_{g,t}^V\big)
+A_{g,t}^V (2^V + A_{g,t}) \exp(A_{g,t})
\Big)\,.\end{gathered}$$ Expanding the product of the terms located in the second and in the third lines of the expression above we get a sum of $15$ non-negative terms, where every term has the form $$a\cdot b^V\cdot (A_{g,t})^{cV+d}\cdot V^{\alpha V}\cdot\exp(k A_{g,t})\,$$ with constants $a,b,c,d,\alpha,k$ specific for each summand, but satisfying, however, the following common conditions. We always have $a,b>0$, $c\in\{0,1,6,7\}$, $d\in\{0,1\}$, $\alpha\le\frac{1}{2}$, and $k\in\{0,1,2\}$. It remains to note that since $A_{g,t}\ge 0$, we have $\exp(2A_{g,t})\ge \exp(A_{g,t})\ge \exp(0)$. Note also, that since $V\ge 2$, our restrictions on $c$ and $d$ imply that $A_{g,t}^{cV+d}\le\max(1,A_{g,t}^{8V})$. These observations imply that each of the terms can be bounded from above by the expression $$a\cdot b^V\cdot \big(1+A_{g,t}^{8V}\big)\cdot V^{V/2}\cdot\exp(2 A_{g,t})\,.$$ Recall that $A_{g,t} = \frac{9 t}{8}(\log g + 7)$, so $\exp(2A_{g,t})=g^{\tfrac{9}{4}t}\exp(63t/4)$. The above observations imply that letting $C_9=15a'$, where $a'$ is the maximal value of the parameter $a$ over $15$ terms, and letting $C_{10}=\left(\frac{9}{8}\right)^8 b'$, where $b'$ is the maximal value of the parameter $b$ over $15$ terms, we complete the proof of .
By Lemma \[lem:upgraded:Aggarwal:10.6\], we have that for all non-negative real $t$ we have $$\begin{gathered}
\left( \frac{8}{3} \right)^{-4g}
\sum_{V = 3}^{2g-2} \sum_{E \geq 1}
\Upsilon_g^{(V; E)} t^E
\le
t\cdot C_9
\cdot \exp(63 t / 4)
\cdot g^{\tfrac{9t+2}{4}}\cdot
\\
\cdot\left(
\sum_{V = 3}^{2g-2}
\left(
\frac{C_{10}
\cdot V^{1/2}}{g} \right)^V
+
\sum_{V = 3}^{2g-2}
\left(
\frac{C_{10}
\cdot t^8
\cdot V^{1/2}\cdot (\log g + 7)^8}{g} \right)^V
\right)
\,.\end{gathered}$$ Let us denote by $$a_V :=
\left(\frac{C_{10}
\cdot t^8
\cdot V^{1/2}\cdot (\log g + 7)^8}{g}\right)^V$$ the term in the second sum. Then $$\begin{gathered}
\frac{a_{V+1}}{a_V}
=
\frac{C_{10}
\cdot t^8
\cdot (1+1/V)^{V/2} \cdot (V+1)^{1/2} \cdot (\log g + 7)^8}{g}
\leq\\
\le\frac{C_{10}
\cdot t^8
\cdot e^{1/2} \cdot (2g-1)^{1/2} \cdot (\log g + 7)^8}{g}.\end{gathered}$$ In particular, since $t$ is bounded, there exists $g_3$ such that for $g\ge g_3$ we have $\frac{a_{V+1}}{a_V} \leq 1/2$ for all $V$. Hence $$\sum_{V = 3}^{2g-2}
\left(
\frac{C_{10}
\cdot t^8
\cdot V^{1/2}\cdot (\log g + 7)^8}{g} \right)^V\le
2
\left(
\frac{C_{10}
\cdot t^8
\cdot 3^{1/2}\cdot (\log g + 7)^8}{g} \right)^3\,.$$ Applying analogous bound for the first sum and collecting the estimates we get $$\begin{gathered}
\left( \frac{8}{3} \right)^{-4g}
\sum_{V = 3}^{2g-2} \sum_{E \geq 1}
\Upsilon_g^{(V; E)} t^E
\le\\
\\
\le t\cdot C_9
\cdot \exp(63 t / 4)
\cdot g^{\tfrac{9t+2}{4}}
\cdot g^{-3}\cdot
2\cdot
\left(C_{10}
\cdot 3^{1/2}\right)^3
\left(1+t^{24}\cdot(\log g + 7)^{24}\right)\\
\le B_3\cdot t \cdot g^{\tfrac{9t - 10}{4}}
\cdot(\log g + 7)^{24}\,,\end{gathered}$$ where $$B_3
=C_9 \cdot \exp\left(\frac{63}{4}\cdot\frac{44}{19}\right)
\cdot 2
\cdot \left(C_{10}\cdot 3^{1/2}\right)^3
\cdot 2\cdot \left(\frac{44}{19}\right)^{24}\,.$$
Proofs {#s:proofs}
======
We proved in Section \[ss:from:q:to:p1\] mod-Poisson convergence of the distribution ${p^{(1)}}_g$ corresponding to volume contributions of stable graphs with a single vertex. In Section \[ss:from:p1:to:p\] we apply the results collected in Section \[s:disconnecting:multicurves\] on volume contributions of stable graphs with two and more vertices to prove that the distribution ${p^{(1)}}_g$ well-approximates the distribution ${p}_g$. In Section \[ss:Remaining:proofs\] we present the remaining proofs of Theorems stated in Section \[s:intro\].
From ${p^{(1)}}_g$ to ${p}_g$ {#ss:from:p1:to:p}
-----------------------------
For any $k$ denote $$\label{eq:vol:k:cyl}
{\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g
:= \sum_{\substack{{\Gamma}\in {\mathcal{G}}_{g}\\|E(\Gamma)|=k}}
{\operatorname{Vol}}(\Gamma)\,,$$ the contribution to ${\operatorname{Vol}}{\mathcal{Q}}_g$ of stable graphs with exactly $k$ edges. Using this notation, the probability distribution ${p}_g(k)$ defined in Theorem \[th:same:distribution\] can be rewritten as $${p}_g(k) = \frac{{\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g}{{\operatorname{Vol}}{\mathcal{Q}}_g}.$$ Recall that we also have a probability distribution $q_{3g-3, \infty, 1/2}(k)$ defined in and evaluated in that corresponds to the number of cycles in a random permutation of $3g-3$ elements according to the probability distribution ${{\mathbb P}}_{3g-3, \infty, 1/2}$ on $S_{3g-3}$, see Lemma \[lem:cycle:distribution:vs:harmonic:sum\].
We gather the results from Sections \[s:sum:over:single:vertex:graphs\] and \[s:disconnecting:multicurves\] in the following two statements.
\[thm:generating:series:vol\] For $t\in{{\mathbb C}}$ satisfying $|t|\le 4/5$ we have as $g \to
+\infty$ $$\sum_{k=1}^{3g-3} {\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g\ t^k
=\frac{2t}{\sqrt{\pi}\, \Gamma\!\left(1+\frac{t}{2}\right)}
(6g-6)^{\tfrac{t-1}{2}} \left( \frac{8}{3} \right)^{4g-4}
\!\left(1 + O\left(g^{\tfrac{t}{2}-1} (\log g)^{24} \right)\! \right),$$ where the error term is uniform for $t$ in the disk $|t|\le 4/5$.
For $t\in{{\mathbb C}}$ satisfying $4/5\le |t| <8/7$ we have as $g \to
+\infty$ $$\sum_{k=1}^{3g-3} {\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g\ t^k
=\frac{2t}{\sqrt{\pi} \, \Gamma\!\left(1+\frac{t}{2}\right)}
(6g-6)^{\tfrac{t-1}{2}} \left( \frac{8}{3} \right)^{4g-4}
\!\left(1 + O\left(g^{\tfrac{7 t}{4} - 2} (\log g)^{24} \right)\! \right)\,,$$ where the constant in the error term is uniform for $t$ in any compact subset of the annulus $4/5\le |t| <8/7$. In particular, for $t=1$ we get $$\label{eq:Vol:Q:with:error:term}
{\operatorname{Vol}}({\mathcal{Q}}_g) =
\sum_{k=1}^{3g-3} {\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g\
=\frac{4}{\pi}
\cdot \left( \frac{8}{3} \right)^{4g-4}
\cdot \left(1 + O\left(g^{-1/4} \cdot (\log g)^{24}\right)\right)\,.$$
We note that the asymptotic formula for ${\operatorname{Vol}}({\mathcal{Q}}_g)$ without explicit error term as in was conjectured in [@DGZZ:volume] and proved in [@Aggarwal:intersection:numbers Theorem 1.7]. See also Remark \[rm:expansion:of:error:term\] for the discussion of the expected optimal error term.
\[thm:p\_g:individual:estimates\] For any $k$ satisfying $k\le \frac{\log g}{\log\frac{9}{4}}$ we have $${\operatorname{Vol}}_{\textrm{k-cyl}} {\mathcal{Q}}_g
=
{\operatorname{Vol}}\Gamma_k(g) \left(1 +
O\left((\log g)^{25} \cdot
g^{-1+\tfrac{k \log 2}{\log g}}\right)\right)\,.$$ For any $x$ satisfying $x<\frac{2}{\log\frac{9}{2}}$ and for all $k$ satisfying $\frac{\log g}{\log\frac{9}{4}}\le k \le x\log g$ we have $${\operatorname{Vol}}_{\textrm{k-cyl}} {\mathcal{Q}}_g
=
{\operatorname{Vol}}\Gamma_k(g) \left(1 +
O\left((\log g)^{25} \cdot
g^{-2+\tfrac{k \log\frac{9}{2}}{\log g}}\right)\right)\,.$$ For $k\le \frac{3}{4\log 2} \log g$ we have $$\label{eq:p:through:q:minus:1:4}
p_g(k)
=q_{3g-3,\infty,\frac{1}{2}}(k)
\cdot \left(1 + O\left(g^{-1/4} \cdot (\log g)^{24}\right)\right)\,.$$ For $k$ in the range $\frac{3\log g}{4\log 2}\le
k\le \frac{\log g}{\log\frac{9}{4}}$ we have $$\begin{aligned}
p_g(k)
&=q_{3g-3,\infty,\frac{1}{2}}(k)
\cdot \left(1 + O\left(g^{-1}\cdot 2^k\cdot (\log g)^{25}\right)\right)
\\
&=q_{3g-3,\infty,\frac{1}{2}}(k) \left(1 +
O\left((\log g)^{25} \cdot
g^{-1+\tfrac{k \log 2}{\log g}}\right)\right)\,.
\end{aligned}$$ For $k$ in the range $\frac{\log g}{\log\frac{9}{4}}\le k \le x\log g$ we have $$\begin{aligned}
p_g(k)
&=q_{3g-3,\infty,\frac{1}{2}}(k)
\cdot \left(1 + O\left(g^{-1}
\cdot\left(\frac{9}{2}\right)^k
\cdot (\log g)^{25}\right)\right)
\\
&=q_{3g-3,\infty,\frac{1}{2}}(k) \left(1 +
O\left((\log g)^{25} \cdot
g^{-2+\tfrac{k \log\frac{9}{2}}{\log g}}\right)\right)\,,
\end{aligned}$$ where all above estimates are uniform in the corresponding ranges of $k$.
Using the notations from Definition \[def:volume:contributions\] we decompose $${\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g = \Upsilon_g^{(1; k)} + \Upsilon_g^{(2;k)} + \sum_{V \geq 3} \Upsilon_g^{(V; k)}.$$ Here $\Upsilon_g^{(1; k)} = {\operatorname{Vol}}\Gamma_k(g)$. Note that $8/7<2$, so Theorem \[thm:generating:series:vol:1\] gives the uniform asymptotic equivalence for the first term. Applying the identity $\frac{t}{2}\,\Gamma\!\left(\frac{t}{2}\right)=\Gamma\!\left(1+\frac{t}{2}\right)$ we set $m=+\infty$ and rewrite as $$\label{eq:sum:Upsilon:1:t:k}
\sum_{k \geq 1} \Upsilon_g^{(1; k)} t^k
=
\frac{2t}{\sqrt{\pi} \, \Gamma\left(1+\frac{t}{2}\right)}
(6g-6)^{\tfrac{t-1}{2}} \left( \frac{8}{3} \right)^{4g-4}
\left(1 + O\left( \frac{(\log g)^2}{g}\right) \right)\,.$$ The bounds for the contributions of the second and third terms are provided by Propositions \[prop:up:bound:V2\] and \[prop:up:bound:V3\] respectively. We have for $|t|\in[0,44/19]$ and, hence, for $|t|\in[0,8/7]$: $$\begin{aligned}
\left( \frac{8}{3} \right)^{-4g}
\sum_{k \geq 1} \Upsilon_g^{(2; k)} |t|^k
&\le g^{\tfrac{|t|-1}{2}}
\cdot O\left(g^{\tfrac{|t|-2}{2}} (\log g)^{14} \right)
\\
\left( \frac{8}{3} \right)^{-4g}
\sum_{k \geq 1} \sum_{V \geq 3} \Upsilon_g^{(V; k)} |t|^k
&\le g^{\tfrac{|t|-1}{2}}
\cdot O\left(g^{\tfrac{7|t|-8}{4}} (\log g)^{24} \right)\,.\end{aligned}$$ Hence $$\begin{gathered}
\label{eq:two:error:terms}
\sum_{k \geq 1} {\operatorname{Vol}}_{k\textit{-cyl}} {\mathcal{Q}}_g t^k
=\\
= \left(\sum_{k \geq 1} \Upsilon_g^{(1; k)} t^k\right)
\cdot\left(1 + O\left((\log g)^{14} \cdot g^{(|t|-2)/2}\right)
+ O\left((\log g)^{24} \cdot g^{(7|t|-8)/4}\right)\right)\,.\end{gathered}$$ Note that $\frac{|t|-2}{2}\ge -1$, so the error term $O\left((\log g)^{14} \cdot g^{(|t|-2)/2}\right)$ dominates the error term $O\left((\log g)^2\cdot g^{-1}\right)$ coming from . Note also that $$\frac{|t|-2}{2} \ge \frac{7|t|-8}{4}
\quad\text{for}\quad |t|\le \frac{4}{5}
\qquad\text{and}\qquad
\frac{|t|-2}{2} \le \frac{7|t|-8}{4}
\quad\text{for}\quad |t|\ge \frac{4}{5}\,.$$ This shows which of the two error terms in dominates on which interval of the values $|t|$. Plugging into , taking into consideration the observation concerning the domination of the error terms, and taking the maximum of powers $14$ and $24$ of logarithms to cover the case $|t|=\frac{4}{5}$, we complete the proof of Theorem \[thm:generating:series:vol\]. Note that passing to we used that $\Gamma(3/2)=\sqrt{\pi}/2$.
In the proof of Theorem \[thm:p\_g:individual:estimates\] we use the following saddle point bound which corresponds to Equation (18) from [@Flajolet:Sedgewick Proposition IV.1].
Let $f(z)$ be analytic in the disk $|z|<R$ with $0<R\le\infty$. Define $M(f;r)$ for $r\in(0,R)$ by $M(f;r):=\sup_{|z|=r}|f(z)|$. Then, one has for any $r$ in $(0,R)$, the family of saddle-point upper bounds $$\label{eq:FS:saddle:point:bound}
\left[z^n\right]f(z)\le \frac{M(f;r)}{r^n}
\qquad\text{implying}\qquad
\left[z^n\right]f(z)\le \inf_{r\in(0,R)} \frac{M(f;r)}{r^n}\,.$$
Let $\delta=\frac{44}{19}$. From Propositions \[prop:up:bound:V2\] and \[prop:up:bound:V3\] we have for all $t$ in the interval $[0,\delta)$ the bounds $$\begin{aligned}
\sum_{k \geq 1} \Upsilon_g^{(2; k)} t^k
&\leq
B_2 \cdot \left( \frac{8}{3} \right)^{4g}
\cdot t \cdot g^{\tfrac{2t - 3}{2}} \cdot (\log g)^{14}
\\
\sum_{k \geq 1} \sum_{V \geq 3} \Upsilon_g^{(V; k)} t^k
&\leq
B_3 \cdot \left( \frac{8}{3} \right)^{4g}
\cdot t \cdot g^{\tfrac{9 t - 10}{4}} \cdot (\log g)^{24}\,.\end{aligned}$$ Combining these bounds with we obtain for all non-negative integer $k$ $$\begin{aligned}
\Upsilon_g^{(2; k)}
&\leq
B_2 \cdot \left( \frac{8}{3} \right)^{4g} \cdot (\log g)^{14}
\cdot g^{-3/2} \cdot \inf_{t \in (0, \delta)}
\left(t^{1-k} g^t\right)\,,
\\
\sum_{V \geq 3} \Upsilon_g^{(V; k)}
&\leq
B_3 \cdot \left( \frac{8}{3} \right)^{4g} \cdot (\log g)^{24}
\cdot g^{-5/2} \cdot \inf_{t \in (0, \delta)}
\left(t^{1-k} g^\frac{9t}{4}\right)\,.\end{aligned}$$ For the rest of the proof we assume that $t$ is real and is contained in the interval $[0,\delta)$. The minima of $t^{1-k} g^t$ and $t^{1-k} g^{\frac{9t}{4}}$ on $[0,+\infty)$ are reached at $t =
\frac{k-1}{\log g}$ and at $t = \frac{k-1}{\frac{9}{4}
\log g}$ respectively. Hence, for $k-1 \leq \delta\cdot\log g$, we obtain the following bounds: $$\begin{aligned}
\Upsilon_g^{(2;k)}
&\leq
B_2 \cdot \left( \frac{8}{3} \right)^{4g} \cdot (\log g)^{14}
\cdot g^{-3/2} \cdot (\log g)^{k-1} \cdot
\left(\frac{e}{k-1}\right)^{k-1}\,, \\
\sum_{V \geq 3} \Upsilon_g^{(V; k)}
&\leq
B_3 \cdot \left( \frac{8}{3} \right)^{4g} \cdot (\log g)^{24}
\cdot g^{-5/2} \cdot \left( \frac{9}{4} \right)^{k-1}
(\log g)^{k-1} \left( \frac{e}{k-1} \right)^{k-1}\,.\end{aligned}$$ Now, for $g$ large enough we have $\sqrt{2 \pi \delta \log g} \leq \log g$. Hence, by Stirling formula, for $g$ large enough and for all $k-1 \leq \delta\cdot\log g$ we have $$\begin{aligned}
\Upsilon_g^{(2; k)}
&\leq
B_2 \cdot \left( \frac{8}{3} \right)^{4g} \cdot (\log g)^{15}
\cdot g^{-3/2} \cdot \frac{(\log g)^{k-1}}{(k-1)!}\,,
\\
\sum_{V \geq 3} \Upsilon_g^{(V; k)}
&\leq
B_3 \cdot \left( \frac{8}{3} \right)^{4g} \cdot (\log g)^{25}
\cdot g^{-5/2}
\cdot \left( \frac{9}{4}\right)^{k-1}
\cdot \frac{(\log g)^{k-1}}{(k-1)!}\,.\end{aligned}$$
Combining expression (in which we set $m=+\infty$) for $\Upsilon_g^{(1; k)}= {\operatorname{Vol}}\Gamma_k(g)$ from Theorem \[th:bounds:for:Vol:Gamma:k:g\] with expression for ${\widetilde{H}}_{3g-3,\infty,1/2}(k)$ from Corollary \[cor:multi:harmonic:asymptotic:all:k:bis\] we get the following asymptotics for $\Upsilon^{(1; k)}$ as $g \to +\infty$: $$\begin{gathered}
\Upsilon_g^{(1; k)}
= \sqrt{\frac{2}{\pi}}
\frac{1}{\sqrt{3g-3}}
\left( \frac{8}{3} \right)^{4g-4}
\left(\frac{1}{2} \right)^{k-1}
\frac{\big(\log(6g-6)\big)^{k-1}}{(k-1)!}\cdot
\\
\cdot\left(
\frac{1}
{\Gamma\left(1 + \tfrac{k-1}{\log(6g-6)}\right)}
+ O\left(\frac{k-1}{(\log g)^2} \right) \right)\,.\end{gathered}$$ For all $k-1 < \delta \log g$ the rightmost factor in the above expression is greater than or equal to $1/\Gamma(1+44/19)+O((\log g)^{-1})$. We have $1/\Gamma(1+44/19)>1/3$. Hence, for all $k-1 < \delta \log g$ and as $g \to
+\infty$ we have $$\label{eq:conditional}
\frac{\displaystyle
\Upsilon_g^{(2; k)} + \sum_{V \geq 3} \Upsilon_g^{(3; k)}}
{\Upsilon_g^{(1; k)}}
=
O\left((\log g)^{25} \cdot
\max\left(g^{-1} \cdot 2^k,
g^{-2} \cdot
\left(\frac{9}{2}\right)^k
\right)\right).$$ Rewriting $$2^{k} = g^{\tfrac{k}{\log g} \log 2}
\qquad\text{and}\qquad
\left(\frac{9}{2}\right)^k
=g^{\tfrac{k}{\log g} \left(\log 9 -\log 2\right)}$$ we obtain $$\max\left(g^{-1} \cdot 2^k,
g^{-2} \cdot
\left(\frac{9}{2}\right)^k
\right)
=
g^{\max\left(-1+\tfrac{k\log 2}{\log g},
-2+\tfrac{k}{\log g} \left(\log 9 -\log 2\right)
\right)}$$ Solving the linear equation we find that $$\label{eq:winning:error:term}
\begin{aligned}
-2+x\log\tfrac{9}{2} \le -1+x\log 2
&\quad\text{for}\quad
x\le \frac{1}{\log\frac{9}{4}}\approx 1.23315\,,
\\
-1+x\log 2 \le -2+x\log\tfrac{9}{2}<0
&\quad\text{for}\quad
\frac{1}{\log\frac{9}{4}}\le x <
\frac{2}{\log\frac{9}{2}}\approx 1.32972\,.
\end{aligned}$$ Note that $\delta=\frac{44}{19}\approx 2.31$, so $\delta>\frac{2}{\log\frac{9}{2}}$. Note also that $\Upsilon_g^{(1; k)}={\operatorname{Vol}}\Gamma_k(g)$. We conclude that for any $x$ satisfying $x <
\frac{2}{\log\frac{9}{2}}$ we have $$\begin{gathered}
{\operatorname{Vol}}_{\textrm{k-cyl}} {\mathcal{Q}}_g
=
\Upsilon_g^{(1; k)} \left( 1 + \frac{\displaystyle \Upsilon_g^{(2; k)} + \sum_{V \geq 3} \Upsilon_g^{(3; k)}}{\Upsilon_g^{(1; k)}} \right)
=\\
=\begin{cases}
{\operatorname{Vol}}\Gamma_k(g) \left(1 +
O\left((\log g)^{25} \cdot
g^{-1+\tfrac{k \log 2}{\log g}}\right)\right)
&\text{for}\quad
k\le \frac{\log g}{\log\frac{9}{4}}\,;
\\
{\operatorname{Vol}}\Gamma_k(g) \left(1 +
O\left((\log g)^{25} \cdot
g^{-2+\tfrac{k \log\frac{9}{2}}{\log g}}\right)\right)
&\text{for}\quad
\frac{\log g}{\log\frac{9}{4}}\le k \le x\log g\,,
\end{cases}\end{gathered}$$ uniformly in the corresponding range of $k$. This completes the proof of the first assertion of Theorem \[thm:p\_g:individual:estimates\]. By we have $$\frac{\Upsilon_g^{(1; k)}}{\Upsilon_g^{(1)}}=
{p^{(1)}}_{g,\infty}(k)
=q_{3g-3,\infty,\frac{1}{2}}(k)
\cdot
\left(1 + O\left( \frac{(k+2\log g)^2}{g}\right) \right)$$ uniformly for all $k\le \tfrac{2\log g}{\log\frac{9}{2}}$. By Equation from Theorem \[thm:generating:series:vol\], we have $${\operatorname{Vol}}{\mathcal{Q}}_g =
\Upsilon_g^{(1)} \left(1 + O\left((\log g)^{24} g^{-1/4}\right)\right)\,.$$ Note that $$\begin{aligned}
-1+x\log 2 \le -\frac{1}{4}
&\quad\text{for}\quad
x\le \frac{3}{4 \log 2}\approx 1.08202\,,
\\
-\frac{1}{4} \le -1+x\log 2
&\quad\text{for}\quad
x\ge \frac{3}{4 \log 2}\,.
\end{aligned}$$ Combining the latter considerations with we conclude that for any $x<\frac{2}{\log\frac{9}{2}}$ we have $$\begin{gathered}
p_g(k)=
\\=\begin{cases}
q_{3g-3,\infty,\frac{1}{2}}(k)
\left(1 + O\left((\log g)^{25} \cdot
g^{-\tfrac{1}{4}}
\right) \right)
&\text{for}\quad
k\le \frac{3\log g}{4\log 2}\,;
\\
q_{3g-3,\infty,\frac{1}{2}}(k) \left(1 +
O\left((\log g)^{25} \cdot
g^{-1+\tfrac{k \log 2}{\log g}}\right)\right)
&\text{for}\quad
\frac{3\log g}{4\log 2}\le
k\le \frac{\log g}{\log\frac{9}{4}}\,;
\\
q_{3g-3,\infty,\frac{1}{2}}(k) \left(1 +
O\left((\log g)^{25} \cdot
g^{-2+\tfrac{k \log\frac{9}{2}}{\log g}}\right)\right)
&\text{for}\quad
\frac{\log g}{\log\frac{9}{4}}\le k \le x\log g\,,
\end{cases}\end{gathered}$$ uniformly for all $k$ in the corresponding ranges. Theorem \[thm:p\_g:individual:estimates\] is proved.
Remaining proofs {#ss:Remaining:proofs}
----------------
Next we deduce the statements stated in Section \[s:intro\] from Theorems \[thm:generating:series:vol\] and \[thm:p\_g:individual:estimates\].
Let $K_g(\gamma)$ be the number of components of a random multicurve $\gamma$ on a surface of genus $g$. Let $$F_g(t) = \sum_{k=1}^{+\infty} {\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g\ t^k.$$ By definition $F_g(1) = {\operatorname{Vol}}{\mathcal{Q}}_g$ and we have $${{\mathbb E}}_g(t^{K_g(\gamma)}) = \frac{F_g(t)}{F_g(1)}.$$ Applying Theorem \[thm:generating:series:vol\] we obtain the result.
We say that a multicurve $\gamma=m_1\gamma_1+\dots+m_k\gamma_k$ is *non-separating* if primitive components $\gamma_1,
\dots,\gamma_k$ of $\gamma$ represent linearly independent homology cycles. Otherwise we say that a multicurve is *separating*. Clearly, $S\setminus\{\gamma_1\cup\dots\cup\gamma_k\}$ is connected if and only if $\gamma$ is non-separating, so non-separating multicurves correspond to stable graphs with a single vertex, while separating multicurves correspond to stable graphs with two and more vertices.
The Corollary below is a quantitative version of an analogous statement [@Aggarwal:intersection:numbers Proposition 10.7] due to A. Aggarwal. We originally conjectured a weaker form of this assertion in [@DGZZ:volume Conjecture 1.33].
\[cor:separating:conditional\] \[conj:one:vertex:dominates:for:fixed:k\] The following estimate is uniform for $k$ in the interval $\left[1, \frac{\log g}{\log\frac{9}{4}}\right]$ as $g \to +\infty$: $${{\mathbb P}}\big(\text{$\gamma$ is separating}\ |\ K_g(\gamma) = k\big)
= O\left((\log g)^{25}\cdot g^{-1+x\log 2}\right).$$ For any $x$ satisfying $\frac{1}{\log\frac{9}{4}}\le x<\frac{2}{\log\frac{9}{2}}$ the following estimate is uniform for $k$ in the interval $\left[\frac{\log g}{\log\frac{9}{4}}, x \log g\right]$ as $g \to +\infty$: $${{\mathbb P}}\big(\text{$\gamma$ is separating}\ |\ K_g(\gamma) = k\big)
= O\left((\log g)^{25}\cdot g^{-2+x\log\tfrac{9}{2}}\right).$$ Furthermore, for any fixed $k$ we have $${{\mathbb P}}(\text{$\gamma$ is separating}\ |\ K_g(\gamma) = k)
= O\left((\log g)^{25}\cdot g^{-1}\right).$$
By definition, $${{\mathbb P}}(\text{$\gamma$ is separating}\ |\ K_g(\gamma) = k)
= \frac{\Upsilon_g^{(2; k)} + \sum_{V \geq 3} \Upsilon_g^{(3; k)}}{{\operatorname{Vol}}_{k\textit{-cyl}}{\mathcal{Q}}_g}
\leq
\frac{\Upsilon_g^{(2; k)} + \sum_{V \geq 3} \Upsilon_g^{(3; k)}}{\Upsilon_g^{(1; k)}}.$$ Equation in the proof of Theorem \[thm:p\_g:individual:estimates\] and analysis of the error term following it provides an upper bound for the right hand-side of the expression above from which the corollary follows.
We proved in [@DGZZ:volume Theorem 1.27], that for $k=1$ we, actually, have the following exponential decay $${{\mathbb P}}(\text{$\gamma$ is separating}\ |\ K_g(\gamma) = 1) \sim \sqrt{\frac{2}{3 \pi g}} \cdot 4^{-g}\,.$$
It follows from combination of [@Aggarwal:intersection:numbers Propositions 8.3–8.5] proved by A. Aggarwal that asymptotically, as $g\to+\infty$, the relative contribution to the Masur–Veech volume ${\operatorname{Vol}}{\mathcal{Q}}_g$ coming from all stable graphs in ${\mathcal{G}}_g$ which have more than one vertex, tends to zero. Translated to the language of multicurves or to the language of square-tiled surfaces, this statement corresponds to assertion (a) of Theorems \[th:multicurves:a:b:c\] and \[th:square:tiled:a:b:c\].
In terms of the results of the current paper, the same statement can be justified comparing Theorems \[thm:generating:series:vol:1\] and \[thm:generating:series:vol\] and observing that the asymptotics of $\sum_{k \geq 1} {\operatorname{Vol}}({\Gamma}_k(g))$ and of ${\operatorname{Vol}}({\mathcal{Q}}_g)$ are the same up to a factor which tends to $1$ as $g\to+\infty$.
Assertion (b) is a particular case, corresponding to the value $m=1$ of the parameter $m$, of more general Theorems \[th:multicurves:bounded:weights\] and \[th:square:tiled:bounded:weights\]. These Theorems are proved independently below.
Let us prove assertion (c). By Theorem \[thm:p\_g:individual:estimates\], in the range $k = o(\log g)$ the volume contributions ${\operatorname{Vol}}_{\textrm{k-cyl}} {\mathcal{Q}}_g$ and $\Upsilon_g^{(1; k)}$ are asymptotically equivalent. We have $$\Upsilon_g^{(1; k)} = \sum_{m_1, \ldots, m_k \geq 1}
{\operatorname{Vol}}\big(\Gamma_g(k), (m_1, \ldots m_k)\big)\,,$$ where the contribution of primitive multicurves is equal to ${\operatorname{Vol}}\big(\Gamma_g(k), (1, \ldots 1)\big)$. By Theorem \[th:bounds:for:Vol:Gamma:k:g\], the contribution to ${\operatorname{Vol}}{\mathcal{Q}}_g$ coming from all non-separating multicurves and from all primitive non-separating multicurves are respectively proportional to ${\widetilde{H}}_{3g-3, \infty, 1/2}(k)$ and to ${\widetilde{H}}_{3g-3, 1, 1/2}(k)$ with the same coefficient of proportionality $\frac{2 \sqrt{2}}{\sqrt{\pi}} \cdot \sqrt{3g-3} \cdot \left(\frac{8}{3}\right)^{4g-4}$. By Corollary \[cor:multi:harmonic:asymptotic:all:k:bis\], the quantities ${\widetilde{H}}_{3g-3, \infty, 1/2}$ and ${\widetilde{H}}_{3g-3, 1, 1/2}(k)$ are asymptotically equivalent in the range $[0, o(\log g)]$, which completes the proof.
Taking the ratio of expression from Theorem \[thm:generating:series:vol:1\] evaluated at $t=1$ over expression from the same Theorem we get $$\lim_{g\to+\infty}
\frac{\displaystyle\sum_{k=1}^{3g-3}
\sum_{m_1, \ldots, m_k \le m}
{\operatorname{Vol}}(\Gamma_k(g), (m_1, \ldots, m_k))}
{\displaystyle\sum_{k=1}^{3g-3}
{\operatorname{Vol}}\big({\Gamma}_k(g)\big)}
= \sqrt{\frac{m}{m+1}}\,.$$ Since the contribution from stable graphs with $V \geq 2$ vertices is negligible, it is a fortiori negligible when we consider bounded multiplicities $m_i\le m$. Hence, we have as $g \to +\infty$ the asymptotics $$\begin{gathered}
\lim_{g\to+\infty}
\frac{\displaystyle\sum_{k=1}^{+\infty}
\sum_{m_1, \ldots, m_k \le m}
{\operatorname{Vol}}(\Gamma_k(g), (m_1, \ldots, m_k))}
{\displaystyle\sum_{k=1}^{+\infty}
{\operatorname{Vol}}\big({\Gamma}_k(g)\big)}
=\\
=\lim_{g\to+\infty}
\frac{\displaystyle \sum_{\Gamma \in {\mathcal{G}}_g}
\sum_{m_1, \ldots, m_k \le m}
{\operatorname{Vol}}(\Gamma, (m_1, \ldots, m_k))}{{\operatorname{Vol}}({\mathcal{Q}}_g)}\,,\end{gathered}$$ which concludes the proof.
The central limit theorem for $K_g(\gamma)$ follows from the general Theorem \[thm:CLT\] that holds under mod-Poisson convergence. The mod-Poisson convergence was stated in Theorem \[thm:mod:poisson:pg\]. It only remains to justify the normalization used in Theorem \[thm:CLT:multicurve\] and in Theorem \[thm:CLT:square:tiled\].
It follows from that $${\operatorname{card}}({\mathcal{ST}{\hspace*{-3pt}}_{g}}(N))\sim m(g)\cdot N^{6g-6}\,,$$ where $$m(g)=\frac{{\operatorname{Vol}}{\mathcal{Q}}_g}{(12g-12)\cdot 2^{6g-6}}\,.$$ By the central limit theorem (Theorem \[thm:CLT\]) we obtain $$\begin{gathered}
\lim_{g\to+\infty}
\lim_{N\to+\infty}\
\frac{1}{m(g)\cdot N^{6g-6}}
\cdot{\operatorname{card}}\left\{S\in{\mathcal{ST}{\hspace*{-3pt}}_{g}}(N)\,\bigg|\,
\frac{K_g(S)-\lambda_{3g-3}}{\sqrt{\lambda_{3g-3}}}
\le x\right\}
\\
=\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^x e^{-\frac{t^2}{2}} dt\,.\end{gathered}$$ It remains to use from Theorem \[conj:Vol:Qg\] of A. Aggarwal (see Theorem 1.7 in the original paper [@Aggarwal:intersection:numbers]) to compute $$\frac{1}{m(g)}
=\frac{(12g-12)\cdot 2^{6g-6}}{{\operatorname{Vol}}{\mathcal{Q}}_g}
\sim 3\pi g\cdot\left(\frac{9}{8}\right)^{2g-2}\,,$$ which proves Theorem \[thm:CLT:square:tiled\].
The proof of Theorem \[thm:CLT:multicurve\] analogous.
M. Mirzakhani proved in [@Mirzakhani:Thesis] that for any integral multicurve $\lambda\in{\mathcal{ML}}({{\mathbb Z}})$ one has $${\operatorname{card}}\left(\left\{\gamma\in{\mathcal{ML}}_g({{\mathbb Z}})\,\bigg|\,
\iota(\lambda,\gamma)\le N\right\}/{\operatorname{Stab}}(\lambda)\right)
\sim \tilde c(\lambda)\cdot N^{6g-6}\,.$$ Now let $\lambda=\rho_g$, where $\rho_g$ is a simple closed non-separating curve on a surface of genus $g$. Note that the stable graph associated to $\rho_g$ is $\Gamma_1(g)$ and that the associated weight $m_1$ is equal to $1$. By [@Erlandsson:Souto Proposition 8.8] the asymptotic frequency $\tilde c(\rho_g)$ in the expression above is proportional to the asymptotic frequency $c(\rho_g)$ defined in with the following factor: $$c(\rho_g)=2^{2g-3}\cdot \tilde{c}(\rho_g)\,.$$ Combining this relation with and (where we let $n=0$) we get $${\operatorname{Vol}}(\Gamma_1(g),1)=2(6g-6)\cdot(4g-4)!
\cdot 2^{6g-6}\cdot\tilde{c}(\rho_g)\,,$$ Since ${\operatorname{Vol}}(\Gamma_1(g))=\zeta(6g-6)\cdot{\operatorname{Vol}}(\Gamma_1(g),1)$ and since $\zeta(6g-6)\sim 1$ as $g\to+\infty$ we conclude that $$\frac{1}{\tilde c(g)}
\sim
\frac{12g\cdot(4g-4)!\cdot 2^{6g-6}}{{\operatorname{Vol}}(\Gamma_1(g))}
\sim \sqrt{\frac{3\pi g}{2}}\cdot 12g
\cdot (4g-4)!
\cdot\left(\frac{9}{8}\right)^{2g-2}
\,,$$ where we used $${\operatorname{Vol}}{\Gamma}_1(g)
=\sqrt{\frac{2}{3\pi g}}
\cdot\left(\frac{8}{3}\right)^{4g-4}
\cdot (1+o(1))
\quad\text{as }g\to+\infty\,.$$ that is obtained by a combination of Theorem \[th:bounds:for:Vol:Gamma:k:g\] and Corollary \[cor:multi:harmonic:asymptotic:all:k:bis\]. Actually, the latter asymptotic equivalence was originally proved in Equation (4.5) from Theorem 4.2 in [@DGZZ:volume].
At the current stage, we can prove mod-Poisson convergence of $p_g$ only for a relatively small radius $R=8/7\approx
1.14286$. Thus, a straightforward application of Corollary \[cor:multi:harmonic:asymptotic:all:k\] to $p_g$ does not provide sufficiently strong estimates for the distribution $p_g$. This is why we proceed differently.
Relation follows from combination of relations for $q_{3g-3,\infty,1/2}(k)$ with relations expressing $p_g(k)$ through $q_{3g-3,\infty,1/2}(k)$ proved in Theorem \[thm:p\_g:individual:estimates\].
To estimate the left and right tails of the distribution $p_g$ we use relation $$p_g(k)
=q_{3g-3,\infty,1/2}(k)
\cdot \left(1 + O\left(g^{-1/4} \cdot (\log g)^{24}\right)\right)\,,$$ proved in Theorem \[thm:p\_g:individual:estimates\] for $k$ satisfying $k\le \frac{3}{4\log 2} \log g$.
Estimate for the left tails follows directly from Equation of Theorem \[thm:permutation:asymptotics\].
For the right tail, the equivalence between $p_g(k)$ and $q_{3g-3,\infty,1/2}(k)$ is not known beyond $\frac{3}{4 \log 2}$ and we need to pass to estimates on the complementary event. Let $\lambda_{3g-3}=\frac{1}{2}\log(6g-6)$. Relation implies, that for $x$ in the range $0\le x \le \frac{3}{2\log 2}\approx 2.16$ we have $$\sum_{k=1}^{\lceil x\lambda_{3g-3}\rceil} p_g(k)
=\left(
\sum_{k=1}^{\lceil x\lambda_{3g-3}\rceil}
q_{3g-3,\infty,1/2}(k)
\right)
\cdot \left(1 + O\left(g^{-1/4} \cdot (\log g)^{24}\right)\right)\,.$$ In particular, this relation is applicable to $x_1=1.236$ and to $x_2=1.24$. Passing to complementary probabilities, we get for for $0\le x \le \frac{3}{2\log 2}$: $$\label{eq:two:error:terms:for:p}
\sum_{k=\lceil x\lambda_{3g-3}\rceil+1}^{3g-3} p_g(k)
=
\sum_{k=\lceil x\lambda_{3g-3}\rceil+1}^{3g-3}
q_{3g-3,\infty,1/2}(k)
+ O\left(g^{-1/4} \cdot (\log g)^{24}\right)\,.$$ We use now relation from Theorem \[thm:permutation:asymptotics\] for the bound for the tail of distribution $q_{3g-3,\infty,1/2}$. Applying with $n=3g-3$, $\lambda=\frac{1}{2}\log(6g-6)$, we get $$\begin{aligned}
\sum_{k=\lceil x\lambda_{3g-3}\rceil+1}^{3g-3}
q_{3g-3,\infty,1/2}(k)
&=O\Big(\exp\big(-\lambda_{3g-3}
(-x \log x - x + 1)\big)\Big)\,.
\\
&=O\left(g^{\left(-x\log x-x+1\right)/2}\right)\,.
\end{aligned}$$ For $x_1=1.236$ we have $$(-x_1\log x_1-x_1+1)/2>-0.249\,.$$ Since the function $-x\log x-x+1$ takes value $0$ at $x=1$ and is monotonously decreasing on $[1,+\infty]$ we conclude that for $x\in(1,x_1]$ we have $$g^{-1/4} \cdot (\log g)^{24}
=o\left(g^{\left(-x\log x-x+1\right)/2}
\cdot \frac{1}{\log g}\right)\,.$$ This implies that for this range of $x$ the error term in the right-hand side of is negligible with respect to the error term in evaluated with parameters $n=3g-3$, $\lambda=\frac{1}{2}\log(6g-6)$, and follows.
For $x=x_2=1.24$ we have $$(-x_2\log x_2-x_2+1)/2< -0.253\,,$$ so $$O\left(g^{\left(-x\log x-x+1\right)/2}\right)
=o\left(g^{-1/4}\right)\,.$$ Taking into consideration monotonicity of $-x\log x-x+1$ for $x\ge 1$, this implies that for $x\ge x_2$ the first summand in the right-hand side of becomes negligible with respect to the second summand. Note also that $$x_2\lambda_{3g-3}=0.62\log(6g-6)<0.62\log g +0.62\log 6<0.62\log g +1.12\,.$$ Thus, the sum of $q_{3g-3,\infty,1/2}(k)$ starting from $k=\lfloor 0.62 \log g\rfloor+1$ might contain at most three extra terms with respect to the sum starting from $\lceil x_2\lambda_{6g-6}\rceil+1$. Clearly, each of these three terms has order $o\left(g^{-1/4}\right)$. We have proved that $$\sum_{k=\lceil x_0\log g\rceil+1}^{3g-3}
q_{3g-3,\infty,1/2}(k)
=o\left(g^{-1/4}\right)\,.$$ Plugging the above estimates in we obtain our estimate for the right part. To estimate the left part we rely that we already proved. It is sufficient to notice that for $x_0 = 0.18$ we have $$- (x_0 \log x_0 - x_0 + 1) / 2 < -0.255$$ and follows.
The convergence mod-Poisson of $p_g(k)$ proved in Theorem \[thm:mod:poisson:pg\] together with the general asymptotics of cumulants in Theorem \[thm:cumulants\] implies Theorem \[th:pg:cumulants\].
Numerical and experimental data and further conjectures
=======================================================
Numerical and experimental data {#s:numerics}
-------------------------------
In this section we compare the distribution $p_g(k)$ of the number of components of a random multicurve in genus $g$ (see Theorems \[th:multicurves:a:b:c\] and \[thm:pg:asymptotics\] from Section \[s:intro\]) with the approximation given by the mod-Poisson convergence.
Recall that for any $\lambda > 0$, we defined in the real numbers $u_{\lambda, 1/2}(k)$, for $k\in{{\mathbb N}}$, as the coefficients of the Taylor expansion of $$e^{\lambda (t-1)}\cdot
\displaystyle
\frac{t \cdot \Gamma\left(\tfrac{3}{2}\right)}{\Gamma\left(1 + \tfrac{t}{2}\right)}
=
\sum_{k \geq 1} u_{\lambda, 1/2}(k) \cdot t^k$$ We have the formula $$u_{\lambda,1/2}(k)
=\sqrt{\pi}\cdot e^{-\lambda}\cdot
\frac{1}{k!}\cdot
\sum_{i=1}^k \binom{k}{i} \cdot \phi_i \cdot
\left(\frac{1}{2}\right)^i \cdot \lambda^{k-i}\,,$$ where $\phi_k$ are the coefficients of the Taylor series of $1 / \Gamma(t)$. Even though the sequence $\{u_{\lambda, 1/2}(k)\}_{k \geq 1}$ is not a probability distribution we refer to this collection of numbers as the $({\mathrm{Poi}}_\lambda, \Gamma(\frac{1}{2}))$-distribution
Corollary \[cor:approximation:q:u:introduction\] shows that $q_g(k)$ (and hence also $p_g(k)$) is well-approximated by $u_{\lambda_{3g-3}, 1/2}(k)$. Theorem \[thm:pg:asymptotics\] also shows that $u_{\lambda_{3g-3}, 1/2}(k)$ can be approximated by a much simpler formula, namely $$u_{\lambda_{3g-3}, 1/2}(k+1) \sim p_g(k + 1) \sim
e^{-\lambda_{3g-3}} \frac{(\lambda_{3g-3})^{k}}{k!} \frac{\sqrt{\pi}}{2 \cdot \Gamma(1 + k / 2 \lambda_{3g-3})}.$$
In the tables below and in Figures \[fig:poisson:versus:experimental:14\], \[fig:poisson:versus:experimental:26\] and \[fig:poisson:versus:experimental:ratio\] we refer to the approximation given by the function $u_{\lambda_{3g-3},1/2}$ as the $({\mathrm{Poi}}_{\lambda_{3g-3}}, \Gamma(\frac{1}{2}))$-approximation, and to the approximation in the right-hand side of the latter expression as “LLT”-approximation (for “Local Limit Theorem”). We provide numerical data comparing the three quantities in the tables below. For $g=14$ the distribution $p_{14}$ was rigorously computed as sequence of explicit rational numbers. For $g=26$ the distribution $p_{26}$ was computed experimentally, collecting statistics of random integra; generalized interval exchange transformations (linear involutions). The graphic comparison of this data is presented in Figures \[fig:poisson:versus:experimental:14\]–\[fig:poisson:versus:experimental:ratio\].
$$\begin{array}{l|ccc}
k & p_{26}
& ({\mathrm{Poi}}_{\lambda_{3\cdot 26-3}}, \Gamma(\frac{1}{2}))
& \text{LLT}
\\
\hline &&&\\
[-\halfbls]
1 & 0.0713 & 0.0724 & 0.0724 \\
2 & 0.2009 & 0.2022 & 0.1974 \\
3 & 0.2679 & 0.2675 & 0.2559 \\
4 & 0.2260 & 0.2251 & 0.2123 \\
5 & 0.1369 & 0.1361 & 0.1276 \\
6 & 0.0634 & 0.0633 & 0.0596 \\
7 & 0.0237 & 0.0237 & 0.0226 \\
8 & 0.0073 & 0.0073 & 0.0072 \\
9 & 0.0019 & 0.0019 & 0.0020 \\
10 & 0.0005 & 0.0004 & 0.0005 \\
11 & 0.0001 & 0.0001 & 0.0001 \\
\end{array}$$
Further conjectures {#s:speculations}
-------------------
Recall that the square-tiled surfaces which we study in this paper are integer points in the total space of the the bundle of quadratic differentials ${\mathcal{Q}}_g$ over ${\mathcal{M}}_g$. Recall that Abelian square-tiled surfaces correspond to integer points in the total space of the Hodge bundle ${\mathcal{H}}_g$ over ${\mathcal{M}}_g$. In this section, we present two conjectures on asymptotic statistics of cylinder decomposition of random Abelian square-tiled surfaces. We also present a conjecture on asymptotic statistics of cylinder decomposition of random square-tiled surfaces in individual strata of holomorphic quadratic differentials.
We conjecture the following mod-Poisson convergence.
\[conj:abelian:weak:form\] Let $p^{Ab}_g(k)$ be the probability that a random Abelian square-tiled surface in ${\mathcal{H}}_g$ has $k$ cylinders. Then for all $x > 0$, uniformly for $k$ in $\{0, 1, \ldots, \lfloor x \log(g) \rfloor\}$ we have as $g \to \infty$ $$p^{Ab}_g(k + 1)
=
e^{-\mu_g} \cdot \frac{(\mu_g)^k}{k!}
\cdot
\left( \frac{1}{\Gamma(t)}+ o(1)) \right).$$ where $\mu_g = \log(4g - 3)$.
In plain words, Conjecture \[conj:abelian:weak:form\] implies that the statistics ${p}^{Ab}_g (k)$ becomes practically indistinguishable from the statistics of the number of disjoint cycles in the cycle decomposition of a random permutation in $S_{4g-3}$, with respect to the uniform probability measure on the symmetric group of $4g-3$ elements. The latter was denoted by ${{\mathbb P}}_{4g-3,\infty,1}$ in Section \[s:sum:over:single:vertex:graphs\].
Recall that both the space ${\mathcal{Q}}_g$ and ${\mathcal{H}}_g$ of respectively quadratic and Abelian differentials are stratified by the partition of order of the zeros. The parameters $6g-6$ and $4g-3$ that appear in the mod-Poisson convergence of random square-tiled surfaces coincide with the dimensions $\dim_{{\mathbb C}}{\mathcal{Q}}_g = 6g-6$ and $\dim_{{\mathbb C}}{\mathcal{H}}_g =
4g-3$. We conjecture the following strong form of mod-Poisson convergence uniform for all non-hyperelliptic connected components of all strata.
\[conj:quadratic:strong:form\] There exist a constant $R_2>1$ such that the mod-Poisson convergence as in Theorem \[thm:mod:poisson:pg\] but with radius $R_2$ holds uniformly for all non-hyperelliptic components of strata of holomorphic quadratic differentials.
More precisely, let ${\mathcal{C}}$ be a non-hyperelliptic component of a stratum of holomorphic quadratic differentials. Let $p_{\mathcal{C}}(k)$ denote the probability that a random quadratic square-tiled surface in ${\mathcal{C}}$ has $k$ cylinders. Then $$\sum_{k \geq 1} p_{\mathcal{C}}(k) t^k
=
(\dim_{{\mathbb C}}{\mathcal{C}})^\frac{t - 1}{2} \cdot
\frac{\sqrt{\pi}}{\Gamma(t/2)}
\left(1 + O \left( \frac{1}{\dim {\mathcal{C}}} \right) \right)\,,$$ where the error term is uniform over all non-hyperelliptic components of all strata of quadratic differentials and uniform over all $t$ over compact subsets of the complex disk $|t|<R_2$.
\[conj:abelian:strong:form\] Conjecture \[conj:abelian:weak:form\] holds uniformly for all non-hyperelliptic connected components of all strata of Abelian differentials.
More precisely, let $x > 0$. Let ${\mathcal{C}}$ be a non-hyperelliptic connected component of a stratum of Abelian differentials. Let $p_{\mathcal{C}}(k)$ denote the probability that a random Abelian square-tiled surface in ${\mathcal{C}}$ has $k$ cylinders. Then $$p^{Ab}_{\mathcal{C}}(k)
=
e^{-\mu_g} \cdot \frac{(\mu_g)^k}{k!}
\cdot
\left( \frac{1}{\Gamma(t)}+ o(1)) \right).$$ where $\mu_g = \log(4g - 3)$. and the error term is uniform over all non-hyperelliptic components of all strata of Abelian differentials and for $k$ in $\{0, 1, \ldots, \lfloor x \log(g) \rfloor\}$.
Conjecture \[conj:abelian:strong:form\] is based on analyzing huge experimental data. We experimentally collected statistics of the number $K_{{\mathcal{C}}}(S)$ of maximal horizontal cylinders in cylinder decompositions of random square-tiled surfaces in about $30$ connected components ${\mathcal{C}}$ of strata in genera from $40$ to $10\,000$. In particular, the least squares linear approximation for components ${\mathcal{C}}$ of dimension $\dim_{{\mathbb C}}{\mathcal{C}}$ between $400$ and $20\,000$ gives: $$\begin{aligned}
{{\mathbb E}}(K_{\mathcal{C}}) &\sim 0.999 \log \dim_{{\mathbb C}}{\mathcal{C}}+ 0.581\\
{{\mathbb V}}(K_{\mathcal{C}}) &\sim 0.996 \log \dim_{{\mathbb C}}{\mathcal{C}}-1.043\end{aligned}$$ (compare to ). Visually the graphs of distributions $p^{Ab}_{{\mathcal{C}}}(k)$ and $\frac{s(\dim {\mathcal{C}},k)}{(\dim {\mathcal{C}})!}$ are, basically, indistinguishable for large genera.
[CMSZag]{}
M. Abramowitz, I. A. Stegun eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th ed.). New York: Dover. pp. 258–259.
A. Aggarwal, *Large Genus Asymptotics for Volumes of Strata of Abelian Differentials*, to appear in J. Amer. Math. Soc., `arXiv:1804.05431`.
A. Aggarwal, *Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials*, (2020) `2004.05042`.
O. Angel, O. Schramm, *Uniform infinite planar triangulations* Commun. Math. Phys. **241** (2003), No. 2–3, 191–213.
A. Aggarwal, V. Delecroix, E. Goujard, P. Zograf, A. Zorich, *Conjectural large genus asymptotics of Masur–Veech volumes and of area Siegel–Veech constants of strata of quadratic differentials*, to appear in Arnold Math. Journal (2020); `arXiv:1912:11702`.
A. Alexeev and P. Zograf, *Random matrix approach to the distribution of genomic distance*, Journal of Computational Biology **21** (2014), no. 8, 622–631.
R. Arratia, A. D. Barbour, S. Tavaré, Logarithmic combinatorial structures: A probabilistic approach. EMS Monographs in Mathematics. Zürich (2003) 363 p.
F. Arana–Herrera, *Equidistribution of horospheres on moduli spaces of hyperbolic surfaces*, `arXiv:1912.03856`.
F. Arana–Herrera, *Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani’s asymptotics for simple closed hyperbolic geodesics*, to appear in Journal of Modern Dynamics, **16** (2020), `arXiv:1902.05626`.
F. Arana–Herrera, *Counting multi-geodesics on hyperbolic surfaces with respect to the lengths of individual components*, `arXiv:2002.10906`.
J. Athreya, A. Eskin, and A. Zorich, *Right-angled billiards and volumes of moduli spaces of quadratic differentials on ${{\mathbb C}\!\operatorname{P}^1}$*, Ann. Scient. ENS, 4ème série, **49** (2016), 1307–1381.
A. D. Barbour, P. Hall, *On the rate of Poisson convergence* Math. Proc. Camb. Philos. Soc. **95** (1984), 473–480.
N. Batir, *Bounds for the gamma function*, Results Math. **72** (2017), no. 1–2, 865–874.
J. Bettinelli *Scaling Limits for Random Quadrangulations of Positive Genus*, Electronic Journal of Probability **15** (2010), 1594–1644.
J. Bettinelli *The Topology of Scaling Limits of Positive Genus Random Quadrangulations*, The Annals of Probability **40** (2012), 1897–1944.
R. Brooks and E. Makover, *Random construction of Riemann surfaces*, J. of Differential Geometry **68** (2004), no. 1, 121–157.
T. Budzinski, N. Curien, and B. Petri, *Universality for random surfaces in unconstrained genus*, Electron. J. Combin. **26** (2019), no. 4, Paper 4.2, 34 pp.
T. Budzinski, N. Curien, B. Petri, *The diameter of random Belyǐ surfaces* `arXiv:1910.11809`.
T. Budzinski and B. Louf, *Local limits of uniform triangulations in high genus*, `arXiv:1902.00492`.
D. Calegari, Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2007.
P. Chassaing, B. Durhuus, *Local limit of labeled trees and expected volume growth in a random quadrangulation* Annals of Probability **34**, No. 3 (2006), 879–917.
D. Chen, M. Möller, A. Sauvaget with an appendix by G. Borot, A. Giacchetto, D. Lewanski, *Masur–Veech volumes and intersection theory: the principal strata of quadratic differentials*, `arXiv:1912.02267` (2019).
J. Choi, D. Cvijović, *Values of the polygamma functions at rational arguments*, J. Phys. A **40** (2007), no. 50, 15019–15028.
D. Chen, M. Möller, A. Sauvaget, D. Zagier, *Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials*, Inventiones Mathematicae (2020); `arXiv:1901.01785`.
S. Chmutov and B. Pittel, *The genus of a random chord diagram is asymptotically normal*, Journal of Combinatorial Theory, Series A **120** (2013), no. 1, 102–110.
N. Curien, *Planar stochastic hyperbolic triangulations*, Probab. Theory Relat. Fields **165**, No. 3–4 (2016), 509–540.
V. Delecroix, E. Goujard, P. Zograf, A. Zorich, *Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves*, `arXiv:1908.08611` (2019).
V. Delecroix, E. Goujard, P. Zograf, A. Zorich, *Enumeration of meanders and Masur-Veech volumes*, Forum of Mathematics $\Pi$ (2020), **8**, e4, 80p.
V. Delecroix, E. Goujard, P. Zograf, A. Zorich, with appendix of Ph. Engel, *Contribution of one-cylinder square-tiled surfaces to Masur–Veech volumes*, `arXiv:1903.10904`, to appear in Asterisque **415** no. 1 (2020).
P. Erdős, M. Kac, *On the Gaussian law of errors in the theory of additive functions*, Proc. Nat. Acad. Sci. USA **25** (1939), 206–207.
V. Erlandsson, H. Parlier and J. Souto, *Counting curves, and the stable length of currents*, Journal of the EMS **22**, no. 6 (2020), 1675–1702.
V. Erlandsson and J. Souto, Geodesic currents and Mirzakhani’s curve counting, (2020).
V. Féray, P.-L. Méliot, A. Nikeghbali, Mod-$\phi$ convergence. Normality zones and precise deviations, Springer (2016), 152p.
Ph. Flajolet, R. Sedgewick, Analytic Combinatorics. Cambridge University Press, 2009.
A. Gamburd, *Poisson–Dirichlet distribution for random Belyǐ surfaces*, Annals of Probability **34**, No. 5 (2006), 1827–1848.
V. L. Goncharov, *Some facts from combinatorics*, Izvestia Akad. Nauk USSR, Ser. Mat., **8** (1944), 3–48.
J. Harer, D. Zagier, *The Euler characteristic of the moduli space of curves*, Invent. Math. **85:3** (1986), 457–485.
H. K. Hwang, *Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques*, Ph.D. thesis, École Polytechnique, 1994.
H. K. Hwang, *Asymptotics of Poisson approximation to random discrete distrubtions: An analytic approach*, Adv. in Appl. Probab. **31**, No. 2 (1999), 448–491.
H. K. Hwang, *Asymptotic expansions for the Stirling numbers of the first kind*, Journal of Combinatorial Theory Ser. A, **71** (1995), 343–351.
V. A. Kazakov, I. K. Kostov, A. A. Migdal, *Critical properties of randomly triangulated planar random surfaces*, Phys. Lett. B **157** (1985), no. 4, 295–300
M. Kazarian, *Recursion for Masur–Veech volumes of moduli spaces of quadratic differentials*, to appear in Journal of Inst. Math. Jussieu (2020); `arXiv:1912.10422`.
M. Krikun, *Local structure of random quadrangulations*, `arxiv:0512304`
M. Kontsevich, *Intersection theory on the moduli space of curves and the matrix Airy function*, Comm. Math. Phys. **147** (1992), 1–23.
E. Kowalski, A. Nikeghbali, *Mod-Poisson convergence in probability and number theory*, Int. Math. Res. Not. **18** (2010), 3549–3587.
J.-F. Le Gall, *Brownian geometry*, (Takagi lectures), Japanese Jour. of Math. **14** (2019), no. 2, 135–174.
M. Liu, *Length statistics of random multicurves on closed hyperbolic surfaces*, `arXiv:1912:11155`.
H. Masur, *Interval exchange transformations and measured foliations*, Ann. of Math., [**115**]{} (1982), 169–200.
H. Masur, K. Rafi, A. Randecker, *Expected covering radius of a translation surface*, `arXiv:1809.10769`.
L. Ménard, *The two uniform infinite quadrangulations of the plane have the same law*, Ann. Inst. Henri Poincaré, Probab. Stat. **46** (2010), No. 1, 190–208.
G. Miermont, *Aspects of random maps*, Lecture notes of Saint-Flour school 2014, `http://perso.ens-lyon.fr/gregory.miermont/coursSaint-Flour.pdf`.
M. Mirzakhani, *Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves*, Ph.D. Thesis, Harvard University, 2004.
M. Mirzakhani, *Growth of the number of simple closed geodesics on hyperbolic surfaces*, Annals of Math. (2) **168** (2008), no. 1, 97–125.
M. Mirzakhani, *Ergodic theory of the earthquake flow*, IMRN **3** (2008), 1–39.
M. Mirzakhani, *Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus*, Jour. of Differential Geometry **94** (2013), no. 2, 267–300.
M. Mirzakhani, *Counting mapping class group orbits on hyperbolic surfaces*, (2016) `arXiv:1601.03342`.
M. Mirzakhani, B. Petri, *Lengths of closed geodesics on random surfaces of large genus*, Comment. Math. Helv. **94** (2019), no. 4, 869–889.
M. Mirzakhani, P. Zograf, *Towards large genus asymtotics of intersection numbers on moduli spaces of curves*, GAFA **25** (2015), 1258–1289.
L. Monin, V. Telpukhovskiy, *On normalizations of Thurston measure on the space of measured laminations*, Topology Appl.**267** (2019), 106878, 12 pp; `arXiv:1902.04533`.
A. Nikeghbali, D. Zeindler *The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles*, Ann. Inst Henri Poincaré Probab. Stat., **49**, n. 4 (2013), 961–981.
J. Pitman, *Combinatorial stochastic processes. Ecole d’Eté de Probabilités de Saint-Flour XXXII – 2002*, Lecture Notes in Mathematics, Springer, 256 p. (2006).
K. Rafi, J. Souto, *Geodesic currents and counting problems*, GAFA **29** (2019), no. 3, 871–889.
A. Rényi, P. Turán, *On a theorem of Erdős–Kac*, Acta Arith. **4** (1958), 71–84.
J. Riordan, *Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions*, Annals of Mathematical Statistics, **8** (1937)(2), 103–111.
A. Selberg, *Note on a paper by L. G. Sathe*, J. Indian Math. Soc. B. **18** (1954), 83–87.
S. Shrestha, *The topology and geometry of random square-tiled surfaces*, (2020); `arXiv:2005.00099`.
T. Tao, *The Poisson-Dirichlet process, and large prime factors of a random number*, `https://terrytao.wordpress.com/2013/09/21/the-poisson-dirichlet-process-and` `-large-prime-factors-of-a-random-number`
W. P. Thurston, Geometry and topology of three-manifolds, Lecture Notes, Princeton University, 1979.
W. T. Tutte, *A census of of planar maps*, Canad. J. Math. **15** (1963), 249–271.
W. Veech, *Gauss measures for transformations on the space of interval exchange maps*, Annals of Math., **115** (1982), 201–242.
A. M. Vershik, *Statistical mechanics of combinatorial partitions, and their limit shapes*, Functional Analysis and Its Applications **30** (1996), 90–105.
T. R. S. Walsh, A. B. Lehman, *Counting rooted maps by genus. I*, Journal of Combinatorial Theory B, **13** (1972), 192–218.
E. Witten, *Two-dimensional gravity and intersection theory on moduli space*, in Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.
D. Yang, D. Zagier and Y. Zhang, *Asymptotics of the Masur–Veech Volumes*, (2020); `arXiv:2005.02275`.
|
---
abstract: 'This document presents the tool named “Application of Hoare Logic and Dijkstra’s Weakest Proposition Calculus to Biological Regulatory Networks Using Path Programs with Branching First-Order Logic Operators” or [*Hoare-fol*]{} for short. This tool consists in an implementation of the theoretical work developed in [@Bernot2019Genetically] and contains the following features: (1) computation of the weakest precondition of a Hoare triple, (2) simplification of this weakest precondition using De Morgan laws and partial knowledge on the initial state, and (3) translation into Answer Set Programming to allow a solving of all compatible solutions.'
author:
- Maxime Folschette
bibliography:
- 'biblio.bib'
title: 'The [*Hoare-fol*]{} Tool'
---
Introduction
============
Biological Regulatory Networks
------------------------------
Algebraic models [@Kauffman1969Metabolic; @Thomas1973Boolean; @DeJong2002Modeling] are noteworthy in the field of systems biology for their ease of use. Indeed, contrary to other formalisms such as ordinary differential equation-based models, their complexity remains very low as they do not require to compute an analytical solution. Furthermore, they require much less parameters to function, meaning that they are of great help when too many system parameters are unknown, while still yielding results on the modeled system’s behavior.
However, less parameters does not mean no parameters. As a consequence, finding one or several acceptable sets of parameters can still be a challenge, especially if the model is big and that this task cannot be tackled by hand.
The focus of this work is on *Thomas’ formalism*, which is typically used to represent Biological Regulatory Networks (BRNs) consisting in interacting components such as genes, proteins, external influences... Formally, it takes the form of a graph in which nodes model components with discrete expression levels and edges model the interactions between these components. More precisely, a specific extension of this formalism is considered, featuring hyperarcs labeled with logic formulas and called *multiplexes*, that are useful to reduce the number of parameters [@Khalis2009Gene]. An example of such a graph is given in Figure \[fig:toy-thomas\].
![\[fig:toy-thomas\] Toy example of [@Bernot2019Genetically] representing an incoherent feedforward loop.](toy-example.png){width=".5\linewidth"}
Parameters in Thomas’ Formalism
-------------------------------
Yet, mutiplexes are not sufficient to specify some aspects of the dynamics:
- how interactions play together when several of them point to the same node (that is, which logical gate is used),
- in the case of multi-valued models, how “strong” an interaction is (for instance, does it attract the component to level 1 or to level 2),
- in the case where it is not specified in the graph, whether an interaction “pulls” a component up (activation) or down (inhibition).
To represent this information, parameters for Thomas’ formalism were proposed in [@Snoussi1989Qualitative] and are now considered as part of the formalism. They are often denoted with the letter $k$ and are uniquely characterized by a couple $(v, \omega)$ where $v$ is the variable it refers to and $\omega$ is the set of its active predecessors. In other words, when the active predecessors of $v$ (that is, having an effective influence on $v$) are exactly the set $\omega$, then this variable $v$ is attracted towards the expression level $k_{v, \omega}$.
A set of parameters covering the whole model is called a *parametrization*, and allows to compute, in each possible state, the global “focal point” towards which the model is dynamically attracted. A parametrization is equivalent to a complete set of logic gates (and, or, etc.) between interactions towards the same node, and can also be equivalently translated into an activation function that takes the current state as input, and outputs the set of possible next states. The parametrization representation is preferred here because it is central to Thomas’ formalism and has been tailored to this kind of representaiton.
Context of the [*Hoare-fol*]{} tool
-----------------------------------
The present work relies on the work developed in [@Bernot2019Genetically] which aims at providing a way to filter out unwanted parametrizations for a given model, based on known possible dynamical behaviors. This work relies on the classical Hoare logic [@Hoare1969Axiomatic] by adapting it to Thomas’ formalism: imperative programs become dynamical path programs (that represent a possible dynamical behavior of the model), and pre- and postconditions on the program’s variables become conditions on the initial and final states and on the parametrization. It also relies on Dijkstra’s weakest preconditon calculus [@Dijkstra1978Guarded] to compute such information.
The rest of this docuemnt describes an implementation of this work under the form of the tool [*Hoare-fol*]{}. This implementation is written in OCaml and allows an export of the results to Clingo’s Answer Set Programming (ASP) [@Gebser2016Theory; @Baral1994Logic] in order to enumerate all solutions.
This tool is a follow-up to the work started in [@Folschette2011Application], where two unsuccessful approaches were taken:
- using Coq to formally prove the new Hoare logic developed,
- using OCaml with functions to encode formulas (pre- and postconditions).
Both methods were not suited for weakest precondition calculus and manipulation, thus giving impractical or partial results. This document, however, proposes a working proof-of-concept of such an implementation.
Implementation
==============
The general idea of this implementation is to represent all conditions (pre- and postdconditions, and conditions of multiplexes) and dynamical path programs as symbolic trees in OCaml, by defining constructs for each type of node. This representation allows to easily manipulate them in order to perform precondition calculus, simplification, translation to ASP, and so on.
Note however that in the current version of the implementation, all information (model, conditions, processings) must be provided as OCaml definitions in the main program[^1].
Model, Program and Formula Definitions
--------------------------------------
First, the Thomas model takes the form of two lists representing the component nodes (`vars`) and multiplex nodes (`mults`) with all related information (predecessors, conditions, etc.). Each element of these lists are couples where the first element is a string giving the name of the component (variable or multiplex) and the second is the information attached. In the case of `vars`, the second element is also a couple where the first element is the upper bound of the variable (integer) and the second is the list of precedessor multiplex names (list of strings). Regarding `mults`, the second element is the *multiplex formula*, which follows the grammar given in Figure \[fig:grammar-mult\] and which itself contains information about its predecessors (variables or multiplexes). For instance, the model of Figure \[fig:toy-thomas\] taken from [@Bernot2019Genetically] is represented in OCaml as the listing of Figure \[fig:toy-ocaml\].
$\begin{array}{r@{\ }ll}
\varphi :==
& \texttt{MPropConst}(b)
& \text{Constant proposition: $b$ is either True or False} \\
| & \texttt{MPropUn}(n, \varphi)
& \text{Unary proposition: $n$ is the negation} \\
| & \texttt{MPropBin}(o, \varphi, \varphi)
& \text{Binary proposition: $o$ is a connective ($\land$, $\lor$, ...)} \\
| & \texttt{MRel}(c, \psi, \psi)
& \text{Comparison: $c$ is a comparator ($=$, $>$, $\geq$, ...)} \\
| & \texttt{MAtom}(v, i)
& \text{Atom on a variable: means $(v \geq i)$} \\
| & \texttt{MMult}(m)
& \text{Atom on a multiplex: recalls the formula of $m$}
\vspace*{.5em} \\
\psi :==
& \texttt{MExprBin}(o, \psi, \psi)
& \text{Arithmetic operation: $o$ is an operator ($+$ or $-$)} \\
| & \texttt{MExprConst}(i)
& \text{Constant: $i$ is an integer}
\end{array}$
Then, general-purpose *formulas* can be defined with another grammar defined in Figure \[fig:grammar-formula\] which is close to the multiplex grammar. Such formulas will be used to define postconditions in order to perform weakest precondition calculus, but they could also be used to describe other kinds of conditions. As a matter of fact, they are also used to express conditions and invariants in `If` and `While` control flow instructions. For instance, the postcondition $(a = 1 \land b = 0)$ can be expressed with: $$\texttt{\begin{tabular}{r@{}l}
let my\_post = Pr&opBin(And, \\
& Rel(Eq, ExprVar "a", ExprConst 1),\\
& Rel(Eq, ExprVar "b", ExprConst 0)) ;;
\end{tabular}}$$
$\begin{array}{r@{\ }ll}
\Phi :==
& \texttt{PropConst}(b)
& \text{Constant proposition: $b$ is either True or False} \\
| & \texttt{PropUn}(n, \Phi)
& \text{Unary proposition: $n$ is the negation} \\
| & \texttt{PropBin}(c, \Phi, \Phi)
& \text{Binary proposition: $o$ is a connective ($\land$, $\lor$, ...)} \\
| & \texttt{Rel}(r, \Psi, \Psi)
& \text{Comparison: $c$ is a comparator ($=$, $>$, $\geq$, ...)} \\
| & \texttt{FreshState}(\Phi)
& \text{Formula on a fresh set of variables}
\vspace*{.5em} \\
\Psi :==
& \texttt{ExprBin}(o, \Psi, \Psi)
& \text{Arithmetic operation: $o$ is an operator ($+$ or $-$)} \\
| & \texttt{ExprVar}(v)
& \text{Valuation of variable: the value of variable $v$} \\
| & \texttt{ExprParam}(v, \omega)
& \text{Valuation of parameter: the value of parameter $k_{v, \omega}$} \\
| & \texttt{ExprConst}(i)
& \text{Constant: $i$ is an integer}
\end{array}$
Finally, an imperative *path program* can be defined with the grammar given in Figure \[fig:grammar-prog\] which copies classical imperative program instructions (assignments) and control flow (`If`, `While`) but also adds descriptions for possible ($\exists$) and mandatory ($\forall$) dynamical branchings. As an example, the path program $(b+; c+; b-)$ can be expressed with: $$\texttt{let my\_prog = Seq(Seq(Incr "b", Incr "c"), Decr "b") ;;}$$
$\begin{array}{r@{\ }ll}
\Pi :==
& \texttt{Skip}
& \text{Does nothing} \\
| & \texttt{Set}(v, i)
& \text{Assignment: $v \leftarrow i$} \\
| & \texttt{Incr}(v)
& \text{Increment: $v+$, i.e., $v \leftarrow v + 1$} \\
| & \texttt{Decr}(v)
& \text{Decrement: $v-$, i.e., $v \leftarrow v - 1$} \\
| & \texttt{Seq}(\Pi, \Pi)
& \text{Instructions sequence} \\
| & \texttt{If}(\Phi, \Pi, \Pi)
& \text{If-then-else conditional} \\
| & \texttt{While}(\Phi, \Phi, \Pi)
& \text{While loop: requires a condition and a loop invariant} \\
| & \texttt{Forall}(\Pi, \Pi)
& \text{Dynamical branching: both behaviors are possible} \\
| & \texttt{Exists}(\Pi, \Pi)
& \text{Dynamical branching: at least one behavior is possible} \\
| & \texttt{Assert}(\Phi)
& \text{Assertion: the formula is true at this point}
\end{array}$
Note that in the main OCaml file, the model specification should be written just after the multiplex formula grammar definition, while the formulas and path programs to process should be defined at the end of the file.
Useful Functions
----------------
Taking into consideration the program and the post-condition, the weakest precondition can be computed with the `wp` function: $$\texttt{let my\_wp = wp my\_prog my\_post ;;}$$
A simplification can be applied on any formula with the `simplify` function. If an initial state and a parametrization are (partially) known, one can also “refine” (that is, strengthen) the weakest precondition with the same function: $$\texttt{let simpl\_wp = simplify my\_wp known\_vars known\_params ;;}$$ where `my_wp` is the weakest precondition (or any other formula to simplify), and `known_vars` and `known_params` are association lists giving the known values of any number of variables and parameters. If no such infomation is given, empty lists (`[]`) are to be provided. In any case, the `simplify` function replaces all variables and parameters given in these lists by their provided values, and attempts to perform basic simplifications on the formula, following De Morgan’s laws.
At any point, functions are provided to translate a formula (`string_of_formula`), an arithmetic expression (`string_of_expr`) or an imperative path program (`string_of_prog` and `string_of_prog_indent`) into a pretty-printable string.
Finally, one can translate a formula (typically, the simplified and refined weakest precondition) into Answer Set Programming (ASP) that can be read by Clingo by using function `write_example`: $$\texttt{write\_example my\_wp "file.lp" ;;}$$ This translation is made by creating an ASP atom for each node of the OCaml representation of the formula. This atom is used in rules such that it reflect its semantics. For instance, consider the conjunction $f = a \land b$ between some subformulas $a$ and $b$, which ASP representations are atoms `atom0` and `atom1`. This fomrula would be translated into another atom, say `atom2`, and the conjunction would be encoded by the ASP rule: $$\texttt{atom2 :- atom0, atom1.}$$ Arithmetic expressions are also translated into their ASP equivalent, while variables and parameters are each assigned to an ASP variable.
Note that the ASP variables that represend parameters are labeled with integers rather than with the explicit names of the resource set $\omega$. In order to obtain the correspondence between the ASP variable names and the original parameters, one can use the `asp_params` function.
See the “Sandbox” part at the end of the main OCaml file for examples on how to use theses functions to obtain results on the model example.
Contents and Usage
==================
The [*Hoare-fol*]{} tool is freely available at <https://gitlab.cristal.univ-lille.fr/mfolsche/hoare-fol> under the MIT license[^2].
The tool can be run in a Unix compatible terminal. Pease refer to the `README.txt` file for information on the requirements and how to run the main file.
The `main.ml` OCaml file contains the main program with somme example applications on the model of Figure \[fig:toy-ocaml\]. It requires OCaml 4.03 to be executed, but the latest version[^3] is recommended. The command line to run this file is: $$\texttt{ocaml main.ml}$$
If ASP files are produced by the execution, they can be fed to Clingo[^4] with the following command: $$\texttt{clingo 0 file.lp}$$ Note that the command line option `0` means “enumerate all solutions”. It can be replaced by a non-null integer to indicate the maximum number of solutions to enumerate
The provided script `run-all.sh` allows to run all `.lp` files with Clingo and store the results in `.lp.out` files: $$\texttt{bash run-all.sh}$$
Limitations
===========
This implementation comes as a proof of concept, and as such still has a number of limitations.
The biggest theoretical limitations are linked to the `While` loops that are rather difficult to express, and which support is limited in the current version of this tool:
- An explicit loop invariant has to be provided for the `While` loops. However, [@Bernot2019Genetically] propose a method to automatically infer a weakest invariant with the following approach:
- Start with the most general invariant.
- Run the loop and remove values that lead out of boundaries.
- Repeat until reaching a fixpoint.
Since values are finite (variables take bounded discrete values), this is ensured to end.
- The weakest preconditions of `While` loops are expressed as formulas in a special context (`FreshState`, defining a “fresh” set of variables) which is currently not explored nor simplified buy the `simplify` function. The simplifications should also apply to these formulas, probably with the same simplification rules, but by taking care of not performing refining on variables in such a context.
There also are technical limitations regarding the ASP output:
- The output of Clingo can be difficult to read, as variables are all encoded with dummy names. The `asp_params` outputs the correspondence between ASP and model variables, bot does not provide pretty-printing nor replace one with the other in the output. A more explicit encoding could be found to ease direct reading of the solutions.
- The Clingo solving can be really long for some formulas, especially if there are a lot of solutions. This limitation seems hard to fix; working on the formula instead of on the set of solutions seems to be the best alternative in this case.
Finally, there also are obvious limitations on the source code itself:
- Both model and processings have to be hard-coded in the main file, and at specific locations. A parser should be added to load a model from a file, or the main file without examples should be turned into a module.
- Functions related to Hoare logic should be purified (they are currently closures on `vars` and `mults`, which partly causes the previous limitation).
Conclusion
==========
This paper presents an implementation of [@Bernot2019Genetically] which applies Hoare logic to Thomas’ formalism in order to infer constraints on the model’s parameter values. It relies on a symbolic representation of logic formulas and imperative programs in order to compute the weakest precondition of a given couple of program and postcondition. It is written in OCaml and allows an ouput of the formulas in ASP (compatible with Clingo 5) to enumerate solutions. Although there are theoretical and technical limitations, especially when `While` loops are involved, or regarding hard-coded features, this work only aims at being the basis of other works that could require such a framework. This has already been the case with [@Behaegel2017Constraint] which re-uses its main concepts, and applies them to a hybrid extension of Thomas’ formalism.
[^1]: Which is of course neither developer-friendly nor a good practise, but this tool only intends to be a proof-of-concept.
[^2]: <https://opensource.org/licenses/MIT>
[^3]: OCaml version 4.09 at the time of writing
[^4]: Produced scripts are intended for Clingo 5. This feature has not been tested with Clingo 4 although the syntax should be compatible. It is compatible with Clingo 3, but it is advised to comment out the `#hide.` directive in the produced scripts to hide uninteresting atoms.
|
---
abstract: 'We study the behaviour of solutions of linear non-autonomous parabolic equations subject to Dirichlet or Neumann boundary conditions under perturbation of the domain. We prove that Mosco convergence of function spaces for non-autonomous parabolic problems is equivalent to Mosco convergence of function spaces for the corresponding elliptic problems. As a consequence, we obtain convergence of solutions of non-autonomous parabolic equations under domain perturbation by variational methods using the same characterisation of domains as in elliptic case. A similar technique can be applied to obtain convergence of weak solutions of parabolic variational inequalities when the underlying convex set is perturbed.'
address: |
School of Mathematics and Statistics\
University of Sydney, NSW 2006 Australia
author:
- Parinya Sa Ngiamsunthorn
bibliography:
- 'database.bib'
date: 'March 10, 2011'
title: An abstract approach to domain perturbation for parabolic equations and parabolic variational inequalities
---
Introduction {#sec:intro}
============
The primary aim of this paper is to study convergence properties of solutions of linear *non-autonomous* parabolic equations under perturbation of the domain. We consider a sequence of bounded open sets $(\Omega_n)_{n=1}^\infty$ in $\mathbb R^N$ converging to a bounded open set $\Omega$ and investigate the behaviour of solutions of the following parabolic equations $$\label{eq:paraIntro}
\left\{
\begin{aligned}
\frac{\partial u}{\partial t} + \mathcal A_n(t) u &= f_n(x,t) &&\quad \text { in } \Omega_n \times (0,T]\\
\mathcal B_n(t) u &= 0 &&\quad \text{ on } \partial \Omega_n \times (0,T]\\
u(\cdot,0) &= u_{0,n} &&\quad \text{ in } \Omega_n , \\
\end{aligned}
\right .$$ where $\mathcal A_n$ is an elliptic operator of the form $$\mathcal A_n(t) u := - \partial_i[a_{ij}(x,t) \partial_j u + a_i(x,t) u]
+ b_i(x,t) \partial_i u + c_0(x,t) u,$$ and $\mathcal B_n(t)$ is one of the following boundary conditions $$\begin{aligned}
\mathcal B_n(t) u &:= u &&\quad \text{Dirichlet boundary condition} \\
\mathcal B_n(t) u &:= [a_{ij}(x,t) \partial_j u + a_i(x,t)u] \;\nu_i &&\quad
\text{Neumann boundary condition.}
\end{aligned}$$ In abstract form, can be written as $$\label{eq:paraAbsIntro}
\left \{
\begin{aligned}
u'(t) + A_n(t) u &= f(t) \quad \text{ for } t \in (0,T]\\
u(0) &= u_{0,n}
\end{aligned}
\right .$$ in a Banach space $V_n$, where $V_n := H^1_0(\Omega_n)$ for Dirichlet problems or $V_n := H^1(\Omega_n)$ for Neumann problems. We refer to Section \[sec:prelim\] for the precise framework of these parabolic equations. We are particularly interested in singular domain perturbation so that change of variables is not possible on these domains. Typically, the common examples include a sequence of dumbbell shape domains with shrinking handle, and a sequence of domains with cracks. Moreover, we mostly do not assume any smoothness of $\Omega_n$ and $\Omega$. The second aim of this paper is to study a similar convergence properties of solutions of parabolic variational inequalities $$\label{eq:parVarIneqKIntro}
\left\{
\begin{aligned}
\langle u'(t), v-u(t) \rangle + \langle A(t)u(t), v-u(t) \rangle
- \langle f(t) , v - u(t) \rangle &\geq 0, \quad \forall v \in K \\
u(0) &= u_0.
\end{aligned}
\right .$$ on $(0,T)$ when we perturb the underlying convex set $K$ in the problem.
To deal with non-autonomous parabolic equations, it is common to apply variational methods. In this paper we prove that under suitable assumptions on domains, a sequence of solutions $u_n$ of converges to the solution $u$ of a linear non-autonomous parabolic equation on the limit domain $\Omega$ ( with $n$ deleted). This result sometimes refers to *stability* of solutions under domain perturbation or *continuity* of solutions with respect to the domain. The method presented in this work is rather an abstract approach. In particular, it can be applied to obtain stability of solutions under domain perturbation for both Dirichlet problems (Theorem \[th:unifConvSolDir\]) and Neumann problems (Theorem \[th:unifConvSolNeu\]). In general, it is more difficult when handling Neumann boundary condition. We cannot simply consider the trivial extension by zero outside the domain for functions in the sobolev space $H^1(\Omega)$ because the extended function does not belong to $H^1(\mathbb R^N)$. Moreover there is no smooth extension from $H^1(\Omega)$ to $H^1(\mathbb R^N)$ as we do not impose any regularity of the domain. This means the compactness result for a sequence of solutions $u_n$ in [@MR1404388 Lemma 2.1] cannot be applied in the case of Neumann problems. However, our abstract approach can be applied to Neumann problems. We refer to Section \[sec:stabilityParaEq\] for the study on stability of solutions of non-autonomous parabolic equations under domain perturbation.
The key result that enables us to determine a sufficient condition on domains for which solutions converge under domain perturbation is Theorem \[th:equiMosco\]. In particular, Theorem \[th:equiMosco\] shows that continuity of solutions for non-autonomous problems can be deduced from the corresponding elliptic problems via *Mosco convergence*. We refer to Section \[sec:moscoConv\] for the definition and results on Mosco convergence. A similar deduction is well-known for autonomous parabolic equations. This is simply because we can apply semigroup methods together with convergence result of degenerate semigroups due to Arendt [@MR1832168 Theorem 5.2]. In Section 6 of the same paper, stability of solutions of Dirichlet heat equation is given as an example. Further examples on other boundary conditions including Neumann and Robin boundary conditions can be found in [@MR2119988 Section 6]. Indeed, for quasilinear parabolic equations, Simondon [@MR1855977] also obtained continuity of solutions of parabolic equations under Dirichlet boundary condition using a similar equivalence of Mosco convergences between certain Banach spaces. However, Theorem \[th:equiMosco\] can be seen as an abstract generalisation of [@MR1855977]. We show equivalence between Mosco convergences of various *closed and convex subsets* of a Banach space rather than Mosco convergences of a particular choice of closed subspaces of a Banach space. The obvious reason for this generalisation is that Mosco convergence was originally introduced in [@MR0298508] for convex sets and was the main tool to establish convergence properties of solutions of elliptic variational inequalities when the convex set is perturbed. The second advantage of Theorem \[th:equiMosco\] is that we do not only show the equivalence between Mosco convergence of convex subsets of the Bochner-Lebesgue space $L^2((0,T),V)$ and Mosco convergence of convex subsets of the corresponding Banach space $V$ but also show that they are equivalent to Mosco convergence of convex subsets of the Bochner-Sobolev spaces $W((0,T),V,V')$. Hence a similar technique can be applied to obtain stability of solutions of parabolic variational inequalities when the underlying convex set is perturbed (Theorem \[th:strongConvSolParVar\]). We study convergence of solutions of parabolic variational inequalities in Section \[sec:appParVar\].
An important consequence of Theorem \[th:equiMosco\] is that the same conditions for a sequence of domains give stability of solutions under domain perturbation for both parabolic and elliptic equations. We refer to [@MR1822408; @MR1951783; @MR1995490; @MR1955096] for the study of domain perturbation for elliptic equations using Mosco convergence.
Preliminaries on parabolic equations and parabolic variational inequalities {#sec:prelim}
===========================================================================
In this section we state some basic results on variational methods for parabolic equations and a variational formulation for parabolic inequalities.
Suppose $V$ is a real separable and reflexive Banach space and $H$ is a separable Hilbert space such that $V$ is dense in $H$. By identifying $H$ with its dual space $H'$, we consider the following *evolution triple* $$V \overset{d}{\hookrightarrow} H \overset{d}{\hookrightarrow} V'.$$ Throughout this paper, we denote by $(\cdot|\cdot)$, the scalar product in $H$ and $\langle \cdot,\cdot \rangle$, the duality paring between $V'$ and $V$. For an interval $(a,b) \subset \mathbb R$, we denote by $L^2((a,b),V)$ the Bochner-Lebesgue space. We define the Bochner-Sobolev space $$W((a,b),V,V') := \{ u \in L^2((a,b),V) : u'\in L^2((a,b),V') \},$$ where $u'$ is the derivative in the sense of distributions taking values in $V'$. The space $W((a,b),V,V')$ is a Banach space when equipped with the following norm $$\|u\|_W := \Big (\int_a^b \|u(t)\|^2_V \;dt + \int_a^b \|u'(t)\|^2_{V'} \;dt
\Big )^{1/2}.$$ It is well known that $W((a,b),V,V') \hookrightarrow C([a,b],H)$, where the space of $H$-valued continuous functions $C([a,b],H)$ equipped with the uniform norm ([@MR1156075 Theorem I1.3.1]). Moreover, for $u,v \in W((a,b),V,V')$ and $a_0,
b_0 \in [a,b]$ with $a_0 < b_0$ we have the integration by parts formula $$\label{eq:intByParts}
(u(b_0)|v(b_0))-(u(a_0)|v(a_0))
= \int_{a_0}^{b_0} \langle u'(t),v(t) \rangle + \langle v'(t),u(t) \rangle \;dt.$$ Let $I,J$ be two sets, we write $J \subset \subset I$ if $\bar{J} \subset I$. For a subset $X$ of a Banach space $V$, we define the closed convex hull by $$\overline{\text{conv}} (X) := \overline{ \left \{ \sum_{i=1}^k \alpha_i x_i \mid x_i \in X,
\alpha_i \in \mathbb R, \alpha_i \geq 0,
\sum_{i=1}^k \alpha_i =1, k =1,2, \ldots \right \}}.$$
For each $t \in [0,T]$, suppose $a(t;\cdot,\cdot)$ is a continuous bilinear form on $V$ satisfying the following hypothesis:
- for every $u,v \in V$, the map $t \mapsto a(t;u,v)$ is measurable.
- there exists a constant $M >0$ independent of $t \in [0,T]$ such that $$\label{eq:biFormCont}
|a(t;u,v)| \leq M \|u\|_V \|v\|_V ,$$ for all $u,v \in V$.
- there exist $\alpha >0$ and $\lambda \in \mathbb R$ such that $$\label{eq:biFormCoer}
a(t;u,u) + \lambda \|u\|^2_H \geq \alpha \|u\|^2_V ,$$ for all $u \in V$.
It follows that for each $t \in [0,T]$ and $u \in V$ the bilinear form $a(t;\cdot,\cdot)$ induces a continuous linear operator $A(t) \in \mathscr L(V,V')$ with $$\langle A(t)u, v \rangle = a(t;u,v) ,$$ for all $u,v \in V$. We easily see from that $\sup_{t \in [0,T]} \|A(t)\|_{\mathscr L(V,V')} \leq M$.
Parabolic equations {#subsec:varPara}
-------------------
Let us consider the abstract parabolic equation $$\label{eq:paraAbs}
\left \{
\begin{aligned}
u'(t) + A(t) u &= f(t) \quad \text{ for } t \in (0,T] \\
u(0) &= u_0 ,\\
\end{aligned}
\right .$$ where $u_0 \in H$ and $f \in L^2((0,T),V')$. A function $u \in W(0,T,V,V')$ satisfying is called a *variational solution*. It is well known that $u$ is a variational solution of if and only if $u \in L^2((0,T),V)$ and $$\label{eq:solFormula}
\begin{aligned}
- \int_0^T (u(t)|v) \phi'(t) \;dt + \int_0^T a(t;u(t),v) \phi(t) \;dt \\
= (u_0|v) \phi(0) + \int_0^T \langle f(t), v \rangle \phi(t) \;dt,
\end{aligned}$$ for all $v \in V$ and for all $\phi \in \mathscr D([0,T))$. The existence and uniqueness of solution is given in the following theorem (see, for example, [@MR1156075 XVIII §3] and [@MR1033497 §23.7]).
\[th:ExiUniParEq\] Given $f \in L^2((0,T),V')$ and $u_0 \in H$, there exists a unique variational solution of satisfies $$\label{eq:unifBoundSol}
\|u \|_{W(0,T,V,V')} \leq C \Big (\|u_0 \|_H + \|f\|_{L^2((0,T),V')}
\Big ).$$ Moreover, if $\lambda =0$ in the variational solution satisfies $$\label{eq:boundOfSol}
\|u(t)\|^2_H + \alpha \int_0^t \|u(s)\|^2_V \;ds
\leq \|u_0\|^2_H + \alpha^{-1} \int_0^t \|f(s)\|^2_{V'} \;ds,$$ for all $t \in [0,T]$.
Note that $v(t) := e^{-\lambda t}u(t)$ is a variational solution of with $A(t)$ replaced by $A(t) + \lambda$. Hence we can assume without loss of generality that $\lambda = 0$ in .
Let $\Omega$ be an open bounded set in $\mathbb R^N$. Let $D \subset \mathbb R^N$ be a ball such that $\Omega \subset D$. We shall consider a closed subspace $V$ of $H^1(\Omega)$ with $H^1_0(\Omega) \subset V \subset H^1(\Omega)$. We take $H:= L^2(\Omega)$ and consider the evolution triple $ V \overset{d}{\hookrightarrow} H \overset{d}{\hookrightarrow} V'$. In this paper, we study bilinear forms $a(t;\cdot,\cdot)$ for $t \in [0,T]$ given by $$\label{eq:biFormEx}
a(t;u,v) := \int_{\Omega}[ a_{ij}(x,t) \partial_j u+ a_i(x,t)u] \partial_i v
+ b_i(x,t) \partial_i u v + c_0(x,t) uv \;dx,$$ for $u,v \in V$. In the above, we use summation convention with $i,j$ running from $1$ to $N$. Also, we assume $a_{ij}, a_i, b_i, c_0$ are functions in $L^{\infty}(D \times (0,T))$ and there exists a constant $\alpha > 0$ independent of $(x,t) \in \Omega \times (0,T)$ such that $$a_{ij}(x,t) \xi_i \xi_j \geq \alpha |\xi|^2 ,$$ for all $\xi \in \mathbb R^N$. It is clear that the map $t \mapsto a(t;u,v)$ is measurable for all $u,v \in V$. Moreover, it can be verified that the form $a(t;\cdot,\cdot)$ defined above satisfies and (see [@MR1156075]). Let $\mathcal A(t)$ be a differential operator on $V$ defined by $$\label{eq:calAop}
\mathcal A(t) u := - \partial_i[a_{ij}(x,t) \partial_j u + a_i(x,t) u]
+ b_i(x,t) \partial_i u + c_0(x,t) u .$$ Given $u_0 \in L^2(D)$ and $f \in L^2(D \times (0,T))$, we consider the following parabolic boundary value problem $$\label{eq:para}
\left\{
\begin{aligned}
\frac{\partial u}{\partial t} + \mathcal A(t) u &= f(x,t) &&\quad \text { in } \Omega \times (0,T]\\
\mathcal B(t) u &= 0 &&\quad \text{ on } \partial \Omega \times (0,T]\\
u(\cdot,0) &= u_0 &&\quad \text{ in } \Omega , \\
\end{aligned}
\right .$$ where $\mathcal B(t)$ is one of the following boundary conditions $$\begin{aligned}
\mathcal B(t) u &:= u &&\quad \text{Dirichlet boundary condition} \\
\mathcal B(t) u &:= [a_{ij}(x,t) \partial_j u + a_i(x,t)u] \;\nu_i &&\quad
\text{Neumann boundary condition}
\end{aligned}$$ It is well known that we can consider the boundary value problem as an abstract equation by taking $V = H^1_0(\Omega)$ for Dirichlet boundary problem or $V=H^1(\Omega)$ for Neumann boundary problem ([@MR1033497 Corollary 23.24]).
Parabolic variational inequalities
----------------------------------
Suppose that $K$ is a closed and convex subset of $V$. We denote by $$L^2((0,T), K) := \{ u \in L^2((0,T),V) \mid u(t) \in K \text{ a.e.}\}.$$ For each $t \in (0,T)$, suppose $a(t; \cdot, \cdot)$ is a continuous bilinear form on $V$ satisfying and . As before, we denote the induced linear operator by $A(t)$. Given $u_0 \in K$ and $f \in L^2((0,T),V')$, we wish to find $u$ such that for a.e. $t \in (0,T)$, $u(t) \in K$ and $$\label{eq:parVarIneqK}
\left\{
\begin{aligned}
\langle u'(t), v-u(t) \rangle + \langle A(t)u(t), v-u(t) \rangle
- \langle f(t) , v - u(t) \rangle &\geq 0, \quad \forall v \in K \\
u(0) &= u_0.
\end{aligned}
\right .$$ A function $u \in W((0,T),V,V')$ satisfying is called a *strong solution* of parabolic variational inequality . In this paper, we are mainly interested in a weak formulation of the problem. There are various (slightly different) definitions of weak solution of parabolic variational inequalities (see e.g.[@MR0428137],[@MR2210083],[@MR0259693], [@MR0296479]). We shall define a weak notion of solution similar to the one in [@MR2210083] as follows.
\[def:weakSolIneq\] A function $u$ is a *weak solution* of parabolic variational inequality if $u \in L^2((0,T),K)$ and $$\label{eq:weakSolParVar}
\begin{aligned}
\int_0^T \langle v'(t), v(t)-u(t) \rangle
+ \langle A(t)u(t), v(t)-u(t) \rangle
- \langle f(t), v(t)-u(t) \rangle \;dt\\
+ \frac{1}{2} \|v(0)-u_0\|^2_H
\geq 0,
\end{aligned}$$ for all $v \in W((0,T),V,V') \cap L^2((0,T),K)$.
The existence and uniqueness of weak solutions of parabolic variational inequalities have been studied by various authors according to their definitions. In our case, we can state the result in the following theorem.
Given $u_0 \in K$ and $f \in L^2((0,T),V')$. There exists a unique weak solution $u$ of the parabolic variational inequality satisfying $u \in
L^{\infty} ((0,T),H)$.
Note that the existence of our weak solution follows immediately from the existence results in [@MR0259693 Theorem 6.2]. The uniqueness can be proved in the same way as in [@MR0296479 Theorem 2.3].
Mosco convergence {#sec:moscoConv}
=================
We often consider Mosco convergence as introduced in [@MR0298508] when dealing with a sequence of functions belonging to a sequence of function spaces. In this section, we establish the key result which enables us to study domain perturbation for parabolic problems via the corresponding elliptic problems. We prove that Mosco convergence of function spaces for non-autonomous parabolic problems is equivalent to Mosco convergence of function spaces for the corresponding elliptic problems. Throughout this section, we assume that $V$ is a reflexive and separable Banach space, and $K_n, K$ are closed and convex subsets of $V$. We start by giving a definition of Mosco convergence in various spaces including $V$, $L^2((0,T),V)$ and $W((0,T),V,V')$.
\[def:moscoEll\] We say that $K_n$ converges to $K$ in the sense of Mosco if the following conditions hold
- For every $u \in K$ there exists a sequence $u_n \in K_n$ such that $u_n \rightarrow u$ strongly in $V$.
- If $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in K_{n_k}$ for every $k$ and $u_n \rightharpoonup u$ weakly in $V$, then $ u \in K$.
There is an alternative definition of Mosco convergence defined in terms of Kuratowski limits. A general result on Mosco convergence and equivalence of these definitions can be found in [@MR773850 Chapter 3].
As discussed in Section \[sec:prelim\], solutions of parabolic equations and parabolic variational inequalities are functions in $L^2((0,T),V)$. Thus, it is worthwhile to study Mosco convergence in $L^2((0,T),V)$. We denote by $$L^2((0,T), K) := \{ u \in L^2((0,T),V) \mid u(t) \in K \text{ a.e.}\},$$ and $$C([0,T],K) := \{ u \in C([0,T],V) \mid u(t) \in K \quad \forall t \in [0,T] \}.$$ It can be verified that $L^2((0,T),K)$ is a closed and convex subset of $L^2((0,T),V)$. We next state Mosco convergence of function spaces for parabolic problems.
\[def:moscoPara\] We say that $L^2((0,T),K_n)$ converges to $L^2((0,T),K)$ in the sense of Mosco if the following conditions hold
- For every $u \in L^2((0,T),K)$ there exists a sequence $u_n \in L^2((0,T),K_n)$ such that $u_n \rightarrow u$ strongly in $L^2((0,T),V)$.
- If $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in L^2((0,T),K_{n_k})$ for every $k$ and $u_k \rightharpoonup u$ weakly in $L^2((0,T),V)$, then $ u \in L^2((0,T),K)$.
It is also useful to define a similar Mosco convergence in $W((0,T),V,V')$ when studying domain perturbation for parabolic variational inequalities.
\[def:moscoPara2\] We say that $W((0,T),V,V') \cap L^2((0,T),K_n)$ converges to $W((0,T),V,V') \cap L^2((0,T),K)$ in the sense of Mosco if the following conditions hold
- for every $u \in W((0,T),V,V') \cap L^2((0,T),K)$ there exists a sequence $u_n \in W((0,T),V,V') \cap L^2((0,T),K_n)$ such that $u_n$ converges strongly to $u$ in $W((0,T),V,V')$.
- if $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in W((0,T),V,V') \cap L^2((0,T),K_{n_k})$ for every $k$, $u_k \rightharpoonup u$ weakly in $L^2((0,T),V)$ and $u'_k \rightharpoonup w$ weakly in $L^2((0,T),V')$, then $u' = w$ and $u \in W((0,T),V,V') \cap L^2((0,T),K)$.
The following theorem is the key result of this paper.
\[th:equiMosco\] The following assertions are equivalent:
- $K_n$ converges to $K$ in the sense of Mosco.
- $L^2((0,T),K_n)$ converges to $L^2((0,T),K)$ in the sense of Mosco.
- $W((0,T),V,V') \cap L^2((0,T),K_n)$ converges to $W((0,T),V,V') \cap L^2((0,T),K)$ in the sense of Mosco.
Before proving the equivalence of Mosco convergences in Theorem \[th:equiMosco\], we require some technical lemmas.
\[lem:convInt\] For a bounded open interval $(a,b) \subset \mathbb R$, let $u \in L^2((a,b),K)$. If $\phi \in \mathscr D((a,b))$ such that $\int_a^b \phi(t) \;dt
=1$ then $\int_a^b u(t) \phi(t) \;dt \in K$.
Since $K$ is closed and convex, $\int_a^b u(t) \phi(t) \;dt \in \overline{\text{\upshape{conv}}}\{u(t) \mid t \in (a,b)\} \subset K$ for all $u \in L^2((a,b),K)$.
\[lem:mollifierB\] Let $I = (a,b)$ be a bounded open interval in $\mathbb R$. If $u \in L^2(I,V)$ and $\int_{I} u(t) \phi(t) \;dt \in K$ for all $\phi \in \mathscr D(I)$ with $\int_{I} \phi(t) \;dt =1$, then $u \in L^2(J,K)$ for all $J =(c,d) \subset \subset I$.
Let $\eta \in \mathscr D(\mathbb R)$ be the standard mollifier. For $\epsilon >0$, we define $\eta_{\epsilon}(t) = \frac{1}{\epsilon}
\eta(\frac{t}{\epsilon})$ so that $\eta_{\epsilon} \in \mathscr D(\mathbb R)$ with $\int_{\mathbb R} \eta_{\epsilon}(t) \;dt = 1$ and $\operatorname{supp}(\eta_{\epsilon}) \subset
(-\epsilon, \epsilon)$. Consider the mollified function $u_{\epsilon} := \eta_{\epsilon} \ast u$. For a.e. $t \in I$, we have $$\begin{aligned}
\|u_{\epsilon}(t) - u(t)\|_V
&= \left \| \int_{t-\epsilon}^{t+\epsilon} \eta_{\epsilon}(t-s)[u(s) -u(t)] \;ds \right \|_V \\
&\leq \frac{1}{\epsilon} \int_{t-\epsilon}^{t+\epsilon} \eta \Big (\frac{t-s}{\epsilon} \Big ) \|u(s)-u(t)\|_V \;ds \\
&\leq C \frac{1}{\epsilon} \int_{t-\epsilon}^{t+\epsilon} \|u(s)-u(t)\|_V \;ds.\\
\end{aligned}$$ By Lebesgue’s differentiation theorem for vector valued functions (Theorem III.12.8 of [@MR1009162]), $u_{\epsilon}(t) \rightarrow u(t)$ in $V$ a.e. $t \in I$. By the definition of $u_{\epsilon}$, $$u_{\epsilon} (t) = \int_{I} \eta_{\epsilon}(t-s) u(s) \;ds =: \int_{I} u(s) \phi_{\epsilon}(s) \;ds ,$$ where we set $\phi_{\epsilon}(s) := \eta_{\epsilon}(t-s)$. Let $J \subset \subset I$. For $t \in J$, we can choose $\epsilon$ sufficiently small so that $\operatorname{supp}(\phi_{\epsilon}) = (t-\epsilon, t+\epsilon) \subset I$. It follows from the assumption that $u_{\epsilon}(t) \in K$ for all $t \in J$. Since $K$ is a closed subset of $V$, the limit point $u(t) \in K$ a.e. $t \in J$. Hence $u \in L^2(J,K)$ as required.
\[lem:densityOfCK\] The set $C([0,T],K)$ is dense in $L^2((0,T),K)$.
Note first that the lemma is trivial if $K$ is a subspace of $V$ (i.e. $K$ is a Banach space)[@MR1033497 Theorem 23.2 (c)]. Let $u \in L^2((0,T),K)$. We choose a function $\phi \in \mathscr D
((0,T))$ with $\int_0^T \phi(t) \;dt = 1$. It follows from Lemma \[lem:convInt\] that $\xi := \int_0^T u(t) \phi(t) \;dt \in K$. Define the extended function $\tilde u \in L^2((-1,T+1),K)$ by $$\tilde u (t) := \left \{
\begin{aligned}
& \xi &&\quad \text{on } (-1, 0) \cup
(T,T+1) \\
& u(t) &&\quad \text{on } (0,T) .
\end{aligned}
\right .$$ By a mollification argument, the function $u_{\epsilon} := \eta_{\epsilon} \ast \tilde u$ belongs to $C(\mathbb R, V)$. Moreover, $u_{\epsilon}$ converges to $\tilde u$ in $L^2((-1,T+1),V)$. By Choosing $0< \epsilon < 1$, $u_{\epsilon} (t) \in K$ for all $t \in [0,T]$. Therefore, the restriction of $u_{\epsilon}$ on $[0,T]$ belongs to $C([0,T],K)$ and converges to $u$ in $L^2((0,T),V)$ as $\epsilon \rightarrow 0$.
\[lem:densityOfWK\] The set $C^{\infty}([0,T],V) \cap C([0,T],K)$ is dense in $W((0,T),V,V')
\cap L^2((0,T),K)$.
Let $u \in W((0,T),V,V') \cap L^2((0,T),K)$. For $\delta > 0$, we define the stretching map $S_{\delta} :[0,T]
\rightarrow [-\delta, T+\delta]$ by $$\label{eq:stretchMap}
S_{\delta} (t) := \Big (\frac{T+2\delta}{T} \Big ) t - \delta .$$ We define $u_{\delta} \in W((-\delta, T+\delta),V,V')
\cap L^2((-\delta,T+\delta),K)$ by $u \circ S_{\delta}^{-1}$. It can be shown that the restriction of $u_{\delta}$ on $(0,T)$ converges to $u$ in $W((0,T),V,V')$ as $\delta \rightarrow 0$. Let $\eta_{\epsilon}$ be a mollifier. For $t \in [0,T]$ and $\epsilon < \delta$, the translation of $\eta_{\epsilon}$ by $t$ (denoted by $\eta_{\epsilon, t}$) belongs to $\mathscr D((-\delta,T+\delta))$. Hence if $\epsilon < \delta$, $\eta_{\epsilon} \ast u_{\delta}$ belongs to $C^{\infty}([0,T],V) \cap
C([0,T],K)$. Moreover, a mollification argument shows that $\eta_{\epsilon} \ast u_{\delta}$ converges to $u_{\delta}$ in $W((0,T),V,V')$ as $\epsilon \rightarrow 0$. The result then follows.
\[prop:convCombUdelta\] Suppose Mosco condition $(M1)$ is satisfied. For $\delta \geq 0$, let $A_{\delta,n} :=\Big \{ \sum_{i=1}^m \phi_i(t) v_i, m \in \mathbb N \Big \}$, where $$\label{eq:convCombAn}
\left \{
\begin{aligned}
&v_i \in K_n,\phi_i \in C^{\infty}([-\delta,T+\delta])
\quad \text{ for all } i=1, \ldots,m , \\
&0 \leq \phi_i(t) \leq 1 \quad \text{ for all } t \in [-\delta,T+\delta]
\text{ and for all } i=1, \ldots,m, \\
&\sum_{i=1}^m \phi_i(t) = 1 \quad \text{ for all } t \in [-\delta,T+\delta].
\end{aligned}
\right .$$ If $ u_{\delta} \in C([-\delta,T+\delta],K)$, then there exists a sequence of functions $u_{\delta,n} \in A_{\delta,n}$ such that $u_{\delta,n}(t) \rightarrow u_{\delta}(t)$ in $V$ uniformly on $[-\delta, T+\delta]$ as $n \rightarrow \infty$.
Let $u_{\delta} \in C([-\delta,T+\delta],K)$. We extend $u_{\delta}$ to $\tilde u_{\delta}
\in C(\mathbb R, K)$ by $$\tilde u_{\delta}(t) := \left \{
\begin{aligned}
& u_{\delta}(-\delta) &&\quad \text{on } (-\infty, -\delta) \\
& u_{\delta}(t) &&\quad \text{on } [-\delta,T+\delta] \\
& u_{\delta}(T+\delta) &&\quad \text{on } (T+\delta, \infty) .
\end{aligned}
\right .$$ Let $\epsilon > 0$ be arbitrary. We denote by $B(t):= B_V(\tilde u_{\delta}(t), \epsilon/2)$ the open ball in $V$ about $\tilde u_{\delta}(t)$ of radius $\epsilon/2$. Let us construct an open covering $\mathscr
O$ of $(-\delta-1, T+\delta+1)$ by $$\mathscr O = \{\tilde u_{\delta}^{-1}(B(t)) \cap
(-\delta-1, T+\delta+1) \}_{t \in [-\delta,T+\delta]} .$$ Since $\mathscr O$ is also an open covering of the compact set $[-\delta,T+\delta]$, there exists a finite subcovering $$\tilde {\mathscr O} = \{\tilde u_{\delta}^{-1}(B(t_i)) \cap
(-\delta-1, T+\delta+1) \}_{i=1,\ldots,m} ,$$ where $t_i \in [-\delta,T+\delta]$ for all $i=1,\ldots,m$. We can assume that $t_1 < t_2 < \ldots < t_m$ and $t_1 = -\delta$, $t_m = T+\delta$ (add them if required) so that $\tilde {\mathscr O}$ is an open covering of $[-\delta-1/2, T+\delta+1/2]$. For each $i \in \{1,\ldots, m\}$, we have $u_{\delta}(t_i) \in K$. Thus, by Mosco condition (M1), there exists $v_{i,n} \in K_n$ such that $ \|v_{i,n} - u_{\delta}(t_i) \|_V < \epsilon/2$ if $n > N_i$ for some $N_i \in \mathbb N$. Let $N := \max_{i=1,\ldots,m} N_i$. It follows that $\|v_{i,n} - u_{\delta}(t_i) \|_V < \epsilon/2$ if $n > N$ for all $i \in \{1, \ldots, m \}$.
Choose a smooth partition of unity $\{\phi_i\}_{i=1,\ldots m}$ for $[-\delta-1/2, T+\delta+1/2]$ subordinate to $\tilde {\mathscr O}$. Precisely, we choose $\phi_i$ such that $\phi_i \in C^{\infty}_0(\tilde u_{\delta}^{-1}(B(t_i)) \cap
(-\delta-1, T+\delta+1))$ and $\sum_{i=1}^m \phi_i(t) =1$ for all $t \in [-\delta-1/2, T+\delta+1/2]$. Define a function $u_{\delta,n}$ on $(-\delta-1,T+\delta+1)$ by $$u_{\delta,n}(t) := \sum_{i=1}^m \phi_i (t) v_{i,n}.$$ It is clear that the restriction of $u_{\delta,n}$ on $[-\delta,T+\delta]$ belongs to $A_{\delta,n}$ if $n > N$. Moreover, for $t \in [-\delta,T+\delta]$, $$\begin{aligned}
\|u_{\delta,n}(t) - u_{\delta}(t) \|_V
&\leq \sum_{i=1}^m \phi_i(t) \|v_{i,n} - u_{\delta}(t)\|_V \\
&\leq \sum_{i=1}^m \phi_i(t) \|v_{i,n} - u_{\delta}(t_i)\|_V +
\sum_{i=1}^m \phi_i(t) \|u_{\delta}(t_i) - u_{\delta}(t)\|_V \\
&< \epsilon/2 + \epsilon/2 = \epsilon,
\end{aligned}$$ if $n > N$. Note that $m$ and $N$ chosen above depend on $\epsilon$. As the above argument holds for each fixed $\epsilon$, we conclude that for every $\epsilon > 0$, there exists a sequence $u_{\delta,n}^{\epsilon} \in A_{\delta,n}$ and $N(\epsilon) \in \mathbb N$ such that $$\| u_{\delta,n}^{\epsilon}(t) - u_{\delta}(t)\|_V \leq \epsilon,$$ for all $t \in [-\delta,T+\delta]$ if $n > N(\epsilon)$.
In particular, for every $k \in \mathbb N$ we can find a sequence $u_{\delta,n}^{k} \in A_{\delta,n}$ and $N_k \in \mathbb N$ such that $$\label{eq:diagonalRangeUdelta}
\| u_{\delta,n}^{k}(t) - u_{\delta}(t)\|_V \leq \frac{1}{k},$$ for all $t \in [-\delta,T+\delta]$ if $n > N_k$. By choosing inductively we can assume that $N_k < N_{k+1}$ for all $k \in \mathbb N$. We extract a sequence of the form $$u_{\delta,1}^1, u_{\delta,2}^1, \ldots, u_{\delta,(N_1+1)}^1, \ldots, u_{\delta, N_2}^1,
u_{\delta,(N_2+1)}^2, \ldots, u_{\delta,N_3}^2,
u_{\delta,(N_3+1)}^3, \ldots, u_{\delta, N_4}^3,
\ldots$$ so that the $n$-th element of this sequence belongs to $A_{\delta,n}$ for all $n \in \mathbb N$. Moreover, by , we see that this sequence converges to $u_{\delta}$ uniformly with respect to $t \in [-\delta,T+\delta]$ as $n \rightarrow \infty$. This proves the statement of the proposition.
We are now in a position to prove our main result.
The proof is divided into four parts including $(i) \Rightarrow (ii)$, $(ii) \Rightarrow (i)$, $(i) \Rightarrow (iii)$ and $(iii) \Rightarrow (i)$. For $(i) \Rightarrow (ii)$, we actually show that $(M1) \Rightarrow (M1')$ and $(M2) \Rightarrow (M2')$. The other three directions are proved in the same way.
$(i) \Rightarrow(ii)$: Let $u \in L^2((0,T),K)$. By the density of $C([0,T],K)$ in $L^2((0,T),K)$ (Lemma \[lem:densityOfCK\]), we may assume that $u \in C([0,T],K)$. We apply Proposition \[prop:convCombUdelta\] with $\delta = 0$ to obtain a sequence of functions $u_n \in L^2((0,T),K_n)$ such that $u_n(t) \rightarrow u(t)$ in $V$ uniformly on $[0,T]$. The uniform convergence on $[0,T]$ implies that $u_n \rightarrow u$ in $L^2((0,T),V)$, showing $(M1')$. To prove condition $(M2')$, suppose $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in
L^2((0,T),K_{n_k})$ for every $k$ and $u_k \rightharpoonup u$ in $L^2(0,T),V)$. By the definition of weak convergence, $$\label{eq:weakCovMoscoiToii}
\int_0^T \langle w(t), u_k(t) \rangle \;dt \rightarrow
\int_0^T \langle w(t), u(t) \rangle \;dt,$$ for all $w \in L^2((0,T),V')$. By taking $w$ of the form $w = \xi \phi(t)$ where $\xi \in V'$ and $\phi \in \mathscr
D((0,T))$ in and applying a basic property of Bochner-Lebesgue space [@MR1033497 Proposition 23.9(a)], it follows that $$\label{eq:weakConvIntUkPhi}
\int_0^T u_k(t) \phi(t) \;dt \rightharpoonup
\int_0^T u(t) \phi(t) \;dt$$ weakly in $V$ for all $\phi \in \mathscr D((0,T))$. Let $\phi_0 \in \mathscr D((0,T))$ with $\int_0^T \phi_0(t) \;dt = 1$ and define $\zeta_k := \int_0^T u_k(t) \phi_0(t) \;dt$. Lemma \[lem:convInt\] implies that $\zeta_k \in K_{n_k}$ for all $k \in \mathbb N$. Since $\zeta_k \rightharpoonup \zeta := \int_0^T u(t) \phi_0(t) \;dt$ by , Mosco condition $(M2)$ implies that $\zeta \in K$. We now extend $u_k$ to $\tilde u_k \in L^2((-1,T+1),K_{n_k})$ by $$\label{eq:extWeakMosco}
\tilde u_k (t) := \left \{
\begin{aligned}
& \zeta_k &&\quad \text{on } (-1, 0) \cup
(T,T+1) \\
& u_k(t) &&\quad \text{on } (0,T) .
\end{aligned}
\right .$$ It can be easily seen that $\tilde u_k \rightharpoonup \tilde u$ weakly in $L^2((-1,T+1),V)$ , where $\tilde u$ defined as with $k$ deleted. Using the definition of weak convergence in $L^2((-1,T+1),V)$ and a similar argument as above, we obtain $\int_{-1}^{T+1} \tilde u_k(t) \phi(t) \;dt \rightharpoonup
\int_{-1}^{T+1} \tilde u(t) \phi(t) \;dt$ weakly in $V$ for all $\phi \in \mathscr D((-1,T+1))$. In particular, taking $\phi \in
\mathscr D((-1,T+1))$ with $\int_{-1}^{T+1} \phi(t) \;dt =1$, we have $\int_{-1}^{T+1} \tilde u_k(t) \phi(t) \;dt \in K_{n_k}$ converges weakly to $\int_{-1}^{T+1} \tilde u(t) \phi(t) \;dt$ in $V$. Thus, Mosco condition $(M2)$ implies $\int_{-1}^{T+1} \tilde u(t) \phi(t) \;dt \in K$ for all $\phi \in \mathscr D((-1,T+1))$ with $\int_{-1}^{T+1} \phi(t) \;dt
=1$. By Lemma \[lem:mollifierB\], we conclude that $u \in L^2((0,T),K)$ and Mosco condition $(M2')$ follows. $(ii) \Rightarrow (i)$: Let $u \in K$. Define $v \in L^2((0,T),K)$ by the constant function $v(t) :=u$ for $t \in (0,T)$. By condition $(M1')$, there exists $(v_n)_{n \in \mathbb N}$ with $v_n \in L^2((0,T),K_n)$ such that $v_n \rightarrow v$ in $L^2((0,T),V)$. Let $\phi_0 \in \mathscr
D((0,T))$ with $\int_0^T \phi_0(t) \;dt = 1$. We show that the sequence $(u_n)_{n \in \mathbb N}$ defined by $u_n := \int_0^T v_n(t) \phi_0(t) \;dt$ gives Mosco condition $(M1)$. First note that $u_n \in K_n$ for all $n \in \mathbb N$ by Lemma \[lem:convInt\]. Moreover, $$\begin{aligned}
\| u_n - u \|_V &= \Big \| \int_0^T v_n(t) \phi_0(t) \;dt - u \Big \|_V \\
&= \Big \| \int_0^T [v_n(t) \phi_0(t) - v(t) \phi_0(t)] \;dt \Big \|_V \\
&\leq \int_0^T |\phi_0(t)|\|v_n(t) -v(t)\|_V \;dt \\
&\leq \sqrt T \Big ( \int_0^T \|v_n(t) - v(t)\|_V^2 \;dt
\Big )^{\frac{1}{2}} \|\phi_0\|_{\infty} \\
&\rightarrow 0,
\end{aligned}$$ as $n \rightarrow \infty$. To prove condition $(M2)$, suppose $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in K_{n_k}$ for every $k$ and $u_k \rightharpoonup u$ in $V$. Define $v_k \in L^2((0,T),K_{n_k})$ by the constant function $v_k(t) := u_k$ for $t \in (0,T)$. It can be easily verified that $v_k \rightharpoonup v$ in $L^2((0,T),V)$, where $v$ is the constant function $v(t):= u$ for $t \in (0,T)$. It follows from Mosco condition $(M2')$ that $v \in L^2((0,T),K)$. Hence $u \in K$ as required. $(i) \Rightarrow (iii)$: Let $u \in W((0,T),V,V') \cap L^2((0,T),K)$. By Lemma \[lem:densityOfWK\], we may assume that $u \in C^{\infty}([0,T],V) \cap C([0,T],K)$. For $\delta > 0$, we define the stretched function $u_{\delta} \in C^{\infty}([-\delta,T+\delta],V) \cap
C([-\delta,T+\delta],K)$ by $u_{\delta} = u \circ S_{\delta}^{-1}$, where $S_{\delta}$ is the stretching map given by . It can be shown that the restriction of $u_{\delta}$ on $[0,T]$ converges to $u$ in $W((0,T),V,V')$ as $\delta \rightarrow 0$. By Proposition \[prop:convCombUdelta\], there exists a sequence of functions $u_{\delta,n} \in A_{\delta,n}$ such that $u_{\delta,n} (t) \rightarrow
u_{\delta}(t)$ uniformly on $[-\delta, T+\delta]$ as $n \rightarrow
\infty$. Let $\eta_{1/j}$ be a mollifier. For $t \in [0,T]$ and $j > 1/{\delta}$, the translation of $\eta_{1/j}$ by $t$ (denoted by $\eta_{1/j, t}$) belongs to $\mathscr D((-\delta,T+\delta))$. Hence if $j > 1/{\delta}$, we have $\eta_{1/j} \ast u_{\delta,n} \in C^{\infty}([0,T],V) \cap
C([0,T],K_n)$. By continuity of convolution and the well known fact on the $r$-th order derivative that $$\frac{d^r}{dt^r}(\eta_{1/j} \ast u_{\delta,n})
= \frac{d^r}{dt^r} \eta_{1/j} \ast u_{\delta,n}
= \eta_{1/j} \ast \frac{d^r}{dt} u_{\delta,n},$$ we deduce that $\eta_{1/j} \ast u_{\delta,n} \rightarrow \eta_{1/j}
\ast u_{\delta}$ in $C^{\infty}([0,T],V)$ as $n \rightarrow \infty$. Similarly, $\eta_{1/j} \ast u_{\delta} \rightarrow u_{\delta}$ in $C^{\infty}([0,T],V)$ as $j \rightarrow \infty$. The above shows that we can construct a function of the form $\eta_{1/j} \ast u_{\delta,n} \in W((0,T),V,V') \cap L^2((0,T),K_n)$ converging to $u$ in $W((0,T),V,V')$. Hence Mosco condition $(M1")$ follows. To prove condition $(M2")$, suppose $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in
W((0,T),V,V') \cap L^2((0,T),K_{n_k})$ for every $k$, $u_k \rightharpoonup u$ in $L^2((0,T),V)$ and $u'_k \rightharpoonup w$ in $L^2((0,T),V')$. Since $V$ is continuously embedded in $V'$, it follows immediately that $u'=w$ and hence $u \in W((0,T),V,V')$ (see [@MR1033497 Proposition 23.19]). Using $(i) \Rightarrow (ii)$, specifically Mosco condition $(M2')$, we conclude that $u \in W((0,T),V,V') \cap L^2((0,T),K)$.
$(iii) \Rightarrow (i)$: Let $u \in K$. Define $v \in W((0,T),V,V') \cap L^2((0,T),K)$ by the constant function $v(t) :=u$ for $t \in (0,T)$. By condition $(M1")$, there exists $(v_n)_{n \in \mathbb N}$ with $v_n \in W((0,T),V,V') \cap L^2((0,T),K_n)$ such that $v_n \rightarrow v$ in $W((0,T),V,V')$. In particular, $v_n$ converges strongly to $v$ in $L^2((0,T),V)$. By the same argument as in the proof of $(ii) \Rightarrow (i)$, we can show that $u_n := \int_0^T v_n(t) \phi_0(t) \;dt$, for some $\phi_0 \in \mathscr D((0,T))$ with $\int_0^T \phi_0(t) \;dt =1$ establishes Mosco condition $(M1)$. To prove condition $(M2)$, suppose $(n_k)$ is a sequence of indices converging to $\infty$, $(u_k)$ is a sequence such that $u_k \in K_{n_k}$ for every $k$ and $u_k \rightharpoonup u$ in $V$. Define $v_k \in W((0,T),V,V') \cap L^2((0,T),K_{n_k})$ by the constant function $v_k(t) := u_k$ for $t \in (0,T)$. By the same argument as in the proof of $(ii) \Rightarrow (i)$, we have $v_k \rightharpoonup v$ in $L^2((0,T),V)$, where $v(t):= u$ for $t \in (0,T)$. Moreover, it is clear that $v'_k = 0$ for all $k \in \mathbb N$ and hence $v'_k \rightharpoonup v'=0$ in $L^2((0,T),V')$. We apply $(M2")$ to deduce that $v \in W((0,T),V,V') \cap L^2((0,T),K)$. Hence $u \in K$.
Application in domain perturbation for parabolic equations {#sec:stabilityParaEq}
==========================================================
In this section, we study the behaviour of solutions of parabolic equations subject to Dirichlet boundary condition and Neumann boundary condition under domain perturbation. Let $\Omega_n, \Omega$ be bounded open sets in $\mathbb R^N$ and $D \subset \mathbb R^N$ be a ball such that $\Omega_n , \Omega \subset D$ for all $n \in \mathbb N$. Suppose $a_{ij}, a_i, b_i, c_0$ are functions in $L^{\infty}(D \times (0,T))$ and $a_{ij}$ satisfies ellipticity condition. More precisely, there exists $\alpha > 0$ such that $a_{ij}(x,t) \xi_i \xi_j \geq \alpha |\xi|^2$ for all $\xi \in \mathbb R^N$. We consider the evolution triple $V_n \overset{d}{\hookrightarrow} H_n \overset{d}{\hookrightarrow} V_n'$, where we choose
- $V_n = H^1_0(\Omega_n)$ and $H_n = L^2(\Omega_n)$ for Dirichlet problem
- $V_n = H^1(\Omega_n)$ and $H_n = L^2(\Omega_n)$ for Neumann problem.
For $t \in (0,T)$, suppose $a_n(t;\cdot, \cdot)$ is a bilinear form on $V_n$ defined by $$a_n(t;u,v) := \int_{\Omega_n} [a_{ij}(x,t) \partial_j u + a_i(x,t)u] \partial_i v
+ b_i(x,t) \partial_i u v + c_0(x,t) uv \;dx.$$ It follows that for all $n \in \mathbb N$, there exist three constants $M >0, \alpha >0$ and $\lambda \in \mathbb R$ independent of $t \in [0,T]$ such that $$\label{eq:biFormContOnN}
|a_n(t;u,v)| \leq M \|u\|_{V_n} \|v\|_{V_n} ,$$ for all $u,v \in V_n$ and $$\label{eq:biFormCoerOnN}
a_n(t;u,u) + \lambda \|u\|^2_{H_n} \geq \alpha \|u\|^2_{V_n} ,$$ for all $u \in V_n$. Given $u_{0,n} \in L^2(D)$ and $f_n \in L^2(D\times (0,T))$, let us consider the following boundary value problem in $\Omega_n \times (0,T]$. $$\label{eq:paraOmegaN}
\left \{
\begin{aligned}
\frac{\partial u}{\partial t} + \mathcal A_n(t) u &= f_n(x,t) &&\quad \text { in } \Omega_n \times (0,T]\\
\mathcal B_n(t) u &= 0 &&\quad \text{ on } \partial \Omega_n \times (0,T]\\
u(\cdot,0) &= u_{0,n} &&\quad \text{ in } \Omega_n , \\
\end{aligned}
\right .$$ where $\mathcal A_n$ and $\mathcal B_n$ are operators on $V_n$ given by $$\mathcal A_n(t) u := - \partial_i[a_{ij}(x,t) \partial_j u + a_i(x,t)u] + b_i(x,t) \partial_i u
+ c_0(x,t) u ,$$ and $\mathcal B_n$ is one of the following $$\begin{aligned}
\mathcal B_n(t) u &:= u &&\quad \text{Dirichlet boundary condition} \\
\mathcal B_n(t) u &:= [a_{ij}(x,t) \partial_j u + a_i(x,t)u] \;\nu_i
&&\quad \text{Neumann boundary condition.} \\
\end{aligned}$$ We wish to show that a sequence of solutions of the above parabolic equations in $\Omega_n \times (0,T]$ converges to the solution of the following limit problem $$\label{eq:paraOmega}
\left\{
\begin{aligned}
\frac{\partial u}{\partial t} + \mathcal A(t) u &= f(x,t) &&\quad \text { in } \Omega \times (0,T]\\
\mathcal B(t) u &= 0 &&\quad \text{ on } \partial \Omega \times (0,T]\\
u(\cdot,0) &= u_0 &&\quad \text{ in } \Omega. \\
\end{aligned}
\right .$$ However, we will consider the boundary value problems and in the abstract form. As discussed in Section \[sec:prelim\], we can write as $$\label{eq:paraAbsOmegaN}
\left \{
\begin{aligned}
u'(t) + A_n(t) u &= f_n(t) \quad \text{ for } t \in (0,T] \\
u(0) &= u_{0,n},\\
\end{aligned}
\right .$$ where $A_n(t) \in \mathscr L(V_n , V_n')$ is the operator induced by the bilinear form $a_n(t;,\cdot,\cdot)$. Similarly, we write as $$\label{eq:paraAbsOmega}
\left \{
\begin{aligned}
u'(t) + A(t) u &= f(t) \quad \text{ for } t \in (0,T] \\
u(0) &= u_0.\\
\end{aligned}
\right .$$ Throughout this section, we denote the variational solution of by $u_n$ and the variational solution of by $u$. We illustrate an application of Mosco convergence to obtain stability of variational solutions under domain perturbation. The proof is motivated by the techniques presented in [@MR1404388]. However, we replace the notion of convergence of domains ((3.5) and (3.6) in [@MR1404388]) by Mosco convergence. It is not difficult to see that the assumption on domains in [@MR1404388] implies that $H^1_0(\Omega_n)$ converges to $H^1_0(\Omega)$ in the sense of Mosco.
Dirichlet problems {#subsec:dirProb}
------------------
When the domain is perturbed, the variational solutions belong to different function spaces. We often extend functions by zero outside the domain. We embed the spaces $H^1_0(\Omega_n)$ into $H^1(D)$ by $v \mapsto \tilde v$, where $\tilde v = v$ on $\Omega_n$ and $\tilde v = 0$ on $D \backslash \Omega_n$. Similarly, we may consider the embedding $L^2((0,T),H^1_0(\Omega_n))$ into $L^2((0,T),H^1(D))$ by $w(t) \mapsto \tilde w(t)$ for a.e. $t \in (0,T)$. Note that the trivial extension $\tilde v$ also acts on $L^2(\Omega_n)$ into $L^2(D)$.
Let us take $V := H^1(D)$, $K_n := H^1_0(\Omega_n)$ and $K := H^1_0(\Omega)$, and consider Mosco convergence of $K_n$ to $K$. In this case $K_n$ and $K$ are closed and convex subsets of $V$ in the sense of the above embedding. In fact, $K_n$ and $K$ are closed subspace of $V$. The main application of Theorem \[th:equiMosco\] is to show that the variational solution $u_n$ of converges to the variational solution $u$ of by applying various Mosco conditions.
\[th:weakConvSolDir\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n} \rightarrow \tilde u_0$ in $L^2(D)$. If $H^1_0(\Omega_n)$ converges to $H^1_0(\Omega)$ in the sense of Mosco, then $\tilde u_n$ converges weakly to $\tilde u$ in $L^2((0,T),H^1(D))$.
Since $f_n \rightarrow f$ in $L^2((0,T),L^2(D))$ and $u_{0,n} \rightarrow u_0$ in $L^2(D)$, it follows from that $\|u_n \|_{W(0,T,V_n,V_n')}$ is uniformly bounded. Hence $\tilde u_n$ is uniformly bounded in $L^2((0,T),H^1(D))$. We can extract a subsequence (denoted again by $u_n$), such that $\tilde u_n \rightharpoonup w$ in $L^2((0,T),H^1(D))$. Mosco condition $(M2')$ (from Theorem \[th:equiMosco\]) implies that $w \in L^2((0,T),H^1_0(\Omega))$. It remains to show that $w=u$ in $L^2((0,T),H^1_0(\Omega))$.
Let $\xi \in H^1_0(\Omega)$ and $\phi \in \mathscr D([0,T))$. Mosco condition ($M1)$ implies that there exists $\xi_n \in H^1_0(\Omega_n)$ such that $\tilde \xi_n \rightarrow \tilde \xi$ in $H^1(D)$. As $u_n$ is the variational solution of , we get from that $$\begin{aligned}
- \int_0^T (u_n(t)|\xi_n) \phi'(t) \;dt + \int_0^T a_n(t; u_n(t), \xi_n ) \phi(t) \;dt \\
= (u_{0,n}|\xi_n) \phi(0) + \int_0^T \langle f_n(t), \xi_n \rangle \phi(t) \;dt .
\end{aligned}$$ By letting $n \rightarrow \infty$, we get $$\label{eq:limitIsWsolDir}
\begin{aligned}
- \int_0^T (w(t)|\xi) \phi'(t) \;dt + \int_0^T a(t; w(t), \xi ) \phi(t) \;dt \\
= (u_{0}|\xi) \phi(0) + \int_0^T \langle f(t), \xi \rangle \phi(t) \;dt .
\end{aligned}$$ Hence $w$ is a variational solution of . By the uniqueness of solution, we conclude that $w=u$ in $L^2((0,T),H^1_0(\Omega))$ and the whole sequence converges.
\[lem:L2weakLimUnDir\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}\rightarrow \tilde u_0$ in $L^2(D)$. If $H^1_0(\Omega_n)$ converges to $H^1_0(\Omega)$ in the sense of Mosco, then for each $t \in [0,T]$ we have $\tilde u_n(t)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$.
Since $\tilde f_n$ is uniformly bounded in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}$ is uniformly bounded in $L^2(D)$, we have from that $$\max_{t \in [0,T]} \|\tilde u_n(t)\|_{L^2(D)} \leq M,$$ for some $M > 0$. Hence for a subsequence denoted again by $u_n(t)$, there exists $w \in L^2(\Omega)$ such that $\tilde u_n(t)_{|_{\Omega}} \rightharpoonup w$ in $L^2(\Omega)$. Let $\xi \in H^1_0(\Omega)$ and $\phi \in \mathscr D((0,t])$. Mosco condition $(M1)$ implies that there exists $\xi_n \in H^1_0(\Omega_n)$ such that $\tilde \xi_n \rightarrow \tilde \xi$ in $H^1(D)$. As $u_n$ is the variational solution of , we have $$\begin{aligned}
- \int_0^t (u_n(s)|\xi_n) \phi'(s) \;ds + \int_0^t a_n(s;u_n(s),\xi_n) \phi(s) \;ds \\
= -(\tilde u_n(t)| \tilde \xi_n)_{L^2(D)} \phi(t) + \int_0^t \langle f_n(s), \xi_n \rangle \phi(s) \;ds.
\end{aligned}$$ Now $$(\tilde u_{0,n})| \tilde \xi_n)_{L^2(D)} = ( \tilde u_{0,n} | \tilde \xi_n)_{L^2(\Omega)}+
(\tilde u_{0,n}| \tilde \xi_n)_{L^2(D \backslash \Omega)}.$$ Since $\tilde \xi_n \rightarrow \tilde \xi$ in $L^2(D)$, we have $\tilde \xi_n |_{\Omega} \rightarrow \xi$ in $L^2(\Omega)$ and $ \tilde \xi_n |_{(D \backslash \Omega)} \rightarrow 0$ in $L^2(D \backslash \Omega)$. Applying the dominated convergence theorem in the second term above and using the weak convergence of initial condition $u_{0,n}$ in the first term above, we see that $$(u_{0,n}| \xi_n)_{L^2(\Omega_n)} \rightarrow ( u_0 |\xi)_{L^2(\Omega)}.$$ Hence, $$\label{eq:weakL2}
\begin{aligned}
- \int_0^t (u(s)|\xi) \phi'(s) \;ds + \int_0^t a(s;u(s),\xi) \phi(s) \;ds \\
= -(w|\xi)_{L^2(\Omega)} \phi(t) + \int_0^t \langle f(s), \xi \rangle \phi(s) \;ds,
\end{aligned}$$ as $n \rightarrow \infty$. As $u$ is the variational solution of , a similar equation holds with $(w| \xi)_{L^2(\Omega)}$ replaced by $(u(t)|\xi)_{L^2(\Omega)}$. Therefore $(w| \xi)_{L^2(\Omega)}=(u(t)|\xi)_{L^2(\Omega)}$ for all $\xi \in H^1_0(\Omega)$. By the density of $H^1_0(\Omega)$ in $L^2(\Omega)$, $w =u(t)$. Hence, for subsequences $\tilde u_n(t)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$. By the uniqueness, the whole sequence $\tilde u_n(t)_{|_{\Omega}}$ converges weakly to $u(t)$ in $L^2(\Omega)$.
\[rem:dirWeakInitialF\] In fact, we only require that $\tilde f_n |_{\Omega} \rightharpoonup f$ weakly in $L^2((0,T),L^2(\Omega))$ and $\tilde u_{0,n}|_{\Omega} \rightharpoonup u_0$ weakly in $L^2(\Omega)$ to obtain the conclusion of Theorem \[th:weakConvSolDir\] and Lemma \[lem:L2weakLimUnDir\] as done in [@MR1404388].
Next we show the strong convergence of solutions. The assumptions on strong convergence of initial values $u_{0,n}$ and inhomogeneous data $f_n$ are required in the proof below (see also Remark \[rem:NoNeedStrongConvDir\] below).
\[th:strongConvSolDir\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}\rightarrow \tilde u_0$ in $L^2(D)$. If $H^1_0(\Omega_n)$ converges to $H^1_0(\Omega)$ in the sense of Mosco, then $\tilde u_n$ converges strongly to $\tilde u$ in $L^2((0,T),H^1(D))$.
We have $\tilde u_n$ converges weakly to $\tilde u$ in $L^2((0,T),H^1(D))$ from Theorem \[th:weakConvSolDir\]. Mosco condition $(M1')$ (from Theorem \[th:equiMosco\]) implies that there exists $w_n \in L^2((0,T),H^1_0(\Omega_n))$ such that $\tilde w_n \rightarrow \tilde u$ in $L^2((0,T),H^1(D))$. For $t \in [0,T]$, we consider $$\label{eq:dNDir}
\begin{aligned}
d_n(t) &= \frac{1}{2} \|\tilde u_n(t) - \tilde u(t) \|^2_{L^2(D)}
+ \alpha \int_0^t \|\tilde u_n(s) - \tilde w_n(s) \|^2_{H^1(D)} ds.
\end{aligned}$$ By (with $\lambda =0$), we have $$\label{eq:dNleqDir}
\begin{aligned}
d_n(t) &\leq \frac{1}{2} \|\tilde u_n(t)\|^2_{L^2(D)} + \int_0^t a_n(s;u_n(s),u_n(s)) \;ds \\
& \quad + \frac{1}{2} \|\tilde u(t)\|^2_{L^2(D)} + \int_0^t a_n(s; w_n(s), w_n(s)) \;ds \\
& \quad -(\tilde u_n(t)| \tilde u(t))_{L^2(D)} - \int_0^t a_n(s; u_n(s),w_n(s)) \;ds \\
& \quad - \int_0^t a_n(s; w_n(s), u_n(s)) \;ds,\\
\end{aligned}$$ for all $n \in \mathbb N$. It can be easily seen from the weak convergence of $\tilde u_n$ and the strong convergence of $\tilde w_n$ to $\tilde u$ in $L^2((0,T),H^1(D))$ that $$\label{eq:wUnsWnDir}
\begin{aligned}
&\lim_{n \rightarrow \infty}
\Big [\int_0^t a_n(s;u_n(s),w_n(s)) \;ds + \int_0^t a_n(s;w_n(s),u_n(s)) \;ds \Big ] \\
& \quad = 2 \int_0^t a(s;u(s),u(s)) \;ds ,
\end{aligned}$$ and $$\label{eq:sWnWnDir}
\lim_{n \rightarrow \infty} \int_0^t a_n(s;w_n(s),w_n(s)) \;ds
= \int_0^t a(s;u(s),u(s)) \;ds.$$ Also, by lemma \[lem:L2weakLimUnDir\], we have $$\label{eq:limUnNormDir}
\lim_{n \rightarrow \infty} (\tilde u_n(t) | \tilde u(t))_{L^2(D)}
= \lim_{n \rightarrow \infty} (\tilde u_n(t)_{|_{\Omega}} | u(t))_{L^2(\Omega)}
= \| u(t)\|^2_{L^2(\Omega)}.$$ Finally, as $u_n$ is the variational solution of we get from that $$\begin{aligned}
\frac{1}{2} \|u_n(t)\|^2_{L^2(\Omega_n)}+ \int_0^t a_n(s; u_n(s),u_n(s)) \;ds \\
=\frac{1}{2} \|u_n(0)\|^2_{L^2(\Omega_n)} + \int_0^t \langle f_n(s), u_n(s) \rangle \;ds.
\end{aligned}$$ By the assumption that $\tilde u_{0,n} \rightarrow \tilde u_0$ strongly in $L^2(D)$ and $\tilde f_n \rightarrow \tilde f$ strongly in $L^2((0,T),L^2(D))$, we get $$\label{eq:limUnPlusAnDir}
\begin{aligned}
& \lim_{n \rightarrow \infty} \Big [ \frac{1}{2} \|u_n(t)\|^2_{L^2(\Omega_n)}+ \int_0^t a_n(s; u_n(s),u_n(s)) \;ds \Big ]\\
& \quad =\frac{1}{2} \|u(0)\|^2_{L^2(\Omega)} + \int_0^t \langle f(s), u(s) \rangle \;ds \\
& \quad =\frac{1}{2} \|u(t)\|^2_{L^2(\Omega)}+ \int_0^t a(s; u(s),u(s)) \;ds.
\end{aligned}$$ Hence, it follows from – that $d_n(t) \rightarrow 0$ for all $t \in [0,T]$. This shows pointwise convergence of $\tilde u_n(t)$ to $\tilde u(t)$ in $L^2(D)$. Moreover, by taking $t=T$ we get $$\begin{aligned}
&\int_0^T \|\tilde u_n(s)-\tilde u(s)\|^2_{H^1(D)} \;ds \\
& \quad \leq \int_0^T \|\tilde u_n(s)-\tilde w_n(s)\|^2_{H^1(D)} \;ds
+ \int_0^T \|\tilde w_n(s)-\tilde u(s)\|^2_{H^1(D)} \;ds \\
& \quad \rightarrow 0,
\end{aligned}$$ as $n \rightarrow \infty$. This proves the strong convergence $\tilde u_n \rightarrow \tilde u$ in $L^2((0,T),H^1(D))$.
In the next theorem we prove convergence of solutions in a stronger norm. We show that Mosco convergence is sufficient for uniform convergence of solutions in $L^2(D)$ with respect to $t \in [0,T]$. We require the following result on Mosco convergence of $H^1_0(\Omega_n)$ to $H^1_0(\Omega)$ [@MR1955096 Proposition 6.3].
\[lem:M1capH10\] The following statements are equivalent.
1. Mosco condition $(M1)$: for every $w \in H^1_0(\Omega)$, there exists a sequence $w_n \in H^1_0(\Omega_n)$ such that $\tilde w_n \rightarrow \tilde w$ in $H^1(D)$.
2. $\text{\upshape{cap}} (K \cap \Omega_n^c) \rightarrow 0$ as $n \rightarrow \infty$ for all compact set $K \subset \Omega$.
\[th:unifConvSolDir\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}\rightarrow \tilde u_0$ in $L^2(D)$. If $H^1_0(\Omega_n)$ converges to $H^1_0(\Omega)$ in the sense of Mosco, then $\tilde u_n$ converges strongly to $\tilde u$ in $C([0,T],L^2(D))$.
We notice from the proof of Theorem \[th:strongConvSolDir\] that $$\int_0^t \|\tilde u_n(s) - \tilde w_n(s)\|^2_{H^1(D)} \;ds \rightarrow 0$$ uniformly with respect to $t \in [0,T]$. Indeed, by $$\int_0^t \|\tilde u_n(s) - \tilde w_n(s)\|^2_{H^1(D)} \;ds \leq \alpha^{-1} d_n(T)$$ for all $n \in \mathbb N$ and for all $t \in [0,T]$. Moreover, it is clear that , and hold uniformly on $[0,T]$. It remains to show uniform convergence of .
Fix $s \in [0,T]$. For $\epsilon > 0$ arbitrary, we choose a compact set $K \subset \Omega$ such that $\|u(s)\|_{L^2(\Omega \backslash K)} \leq \epsilon/2$. Since $u \in C([0,T],L^2(\Omega))$, there exists $\eta > 0$ only depending on $\epsilon$ such that $\|u(t) - u(s) \|_{L^2(\Omega)} \leq \epsilon/2$ for all $t \in (s-\eta, s+\eta) \cap [0,T]$. It follows that $$\label{eq:normUOutsideK}
\|u(t)\|_{L^2(\Omega \backslash K)}
\leq \|u(t) - u(s) \|_{L^2(\Omega)} + \|u(s)\|_{L^2(\Omega \backslash K)}
\leq \epsilon/2 + \epsilon/2 = \epsilon,$$ for all $t \in (s-\eta, s+\eta) \cap [0,T]$. We next choose a cut-off function $\phi \in C^{\infty}_0(\Omega)$ such that $0 \leq \phi \leq 1$ and $\phi =1$ on $K$. Since $H^1_0(\Omega_n)$ converges to $H^1_0(\Omega)$ in the sense of Mosco, we have from Lemma \[lem:M1capH10\] that $ \text{cap}(\text{supp}(\phi) \cap \Omega_n^c) \rightarrow 0$. By definition of capacity, there exists a sequence $\xi_n \in C^{\infty}_0(\Omega)$ such that $0 \leq \xi_n \leq 1$, $\xi_n = 1$ on a neighborhood of $\text{supp}(\phi) \cap \Omega_n^c$ and $\| \xi_n\|_{H^1(\mathbb R^N)} \leq \text{cap}(\text{supp}(\phi) \cap \Omega_n^c) + 1/n$. Define $\phi_n := (1-\xi_n) \phi$. We have that $\phi_n \in C^{\infty}_0(\Omega_n)$ and $\phi_n \rightarrow \phi$ in $L^2(D)$. Consider $$\label{eq:unifUnUExpandDir}
\begin{aligned}
&\left |(\tilde u_n(t) -\tilde u(t) |\tilde u(t))_{L^2(D)} \right | \\
&\leq \left | (\tilde u_n(t)|\phi_n \tilde u(t))_{L^2(D)} - (\tilde
u(t)| \phi \tilde u(t))_{L^2(D)} \right | \\
& \quad + \left | (\tilde u_n(t)|(1-\phi_n) \tilde u(t))_{L^2(D)} -
(\tilde u(t)| (1-\phi) \tilde u(t))_{L^2(D)} \right | \\
& \leq \left | (\tilde u_n(t)|\phi_n \tilde u(t))_{L^2(D)} - (\tilde
u(t)| \phi \tilde u(t))_{L^2(D)} \right | \\
& \quad + \left | (\tilde u_n(t) -\tilde u(t) |(1-\phi_n) \tilde u(t))_{L^2(D)} \right |
+ \left |(\tilde u(t)| (\phi - \phi_n) \tilde u(t))_{L^2(D)} \right |.\\
\end{aligned}$$ We prove that each term on the right of is uniformly small for $t \in (s-\eta, s+\eta) \cap [0,T]$ if n is sufficiently large. For the first term, applying integration by parts formula and the definition of variational solutions, we obtain $$\label{eq:firstTermUnif}
\begin{aligned}
& (\tilde u_n(t)|\phi_n \tilde u(t))_{L^2(D)} \\
&= (u_{0,n}|\phi_n u_0)_{L^2(D)} + \int_0^t \langle u_n'(s), \phi_n \tilde u(s) \rangle \;ds
+ \int_0^t \langle u'(s), \phi_n \tilde u_n(s) \rangle \;ds \\
&= (u_{0,n}|\phi_n u_0)_{L^2(D)} + \int_0^t \langle f_n(s), \phi_n \tilde u(s) \rangle \;ds
- \int_0^t a_n(s; u_n(s), \phi_n \tilde u(s)) \;ds \\
&\quad + \int_0^t \langle f(s), \phi_n \tilde u_n(s) \rangle \;ds
- \int_0^t a(s; u(s), \phi_n \tilde u_n(s)) \;ds.\\
\end{aligned}$$ It can be easily verified using Dominated Convergence Theorem that $\phi_n \tilde u \rightarrow \phi \tilde u$ in $L^2((0,T),L^2(D))$. Moreover, $$\begin{aligned}
&\int_0^T \|\phi_n \tilde u_n(t) - \phi \tilde u(t)\|^2_{L^2(D)} \;dt \\
&\leq \int_0^T \|\phi_n \tilde u(t) - \phi \tilde u(t)\|^2_{L^2(D)} \;dt
+ \int_0^T \|\phi_n \tilde u_n(t) - \phi_n \tilde u(t)\|^2_{L^2(D)} \;dt \\
&\leq \int_0^T \|\phi_n \tilde u(t) - \phi \tilde u(t)\|^2_{L^2(D)} \;dt
+ \|\phi_n\|^2_{\infty} \int_0^T \| \tilde u_n(t) - \tilde u(t)\|^2_{L^2(D)} \;dt. \\
\end{aligned}$$ It follows also that $\phi_n \tilde u_n \rightarrow \phi \tilde u$ in $L^2((0,T),L^2(D))$. Taking into consideration that $\tilde u_{0,n} \rightarrow \tilde u_0$ and $\tilde f_n \rightarrow \tilde f$, we conclude form that $$\label{eq:firstTUnif2}
(\tilde u_n(t)|\phi_n \tilde u(t))_{L^2(D)} \rightarrow (\tilde u(t)|\phi \tilde u(t))_{L^2(D)}$$ uniformly with respect to $t \in [0,T]$. For the last term on the right of , applying a similar argument as above, we write $$(\tilde u(t)|\phi_n \tilde u(t))_{L^2(D)}
= (u_0|\phi_n u_0)_{L^2(D)} + 2 \int_0^t \langle u'(s), \phi_n \tilde u(s) \rangle \;ds.$$ We conclude that $$\label{eq:lastTUnif}
(\tilde u(t)|\phi_n \tilde u(t))_{L^2(D)} \rightarrow (\tilde u(t)|\phi \tilde u(t))_{L^2(D)}$$ uniformly with respect to $t \in [0,T]$. Finally, for the second term on the right of , we notice that $0 \leq 1- \phi_n \leq 1$ on $\Omega$ and $1- \phi_n = 1-(1-\xi_n)\phi
= \xi_n$ on $K$. Moreover, using and the assumption that $\tilde u_{0,n} \rightarrow \tilde u_0$ and $\tilde f_n \rightarrow \tilde f$, there exists a constant $M_0 > 0$ such that $$\label{eq:constM0}
\|\tilde u_n(t)\|_{L^2(D)}, \|\tilde u(t)\|_{L^2(D)} \leq M_0,$$ for all $t \in [0,T]$. Hence, by Cauchy-Schwarz inequality and , $$\label{eq:secondTUnif1}
\begin{aligned}
\left | (\tilde u_n(t) -\tilde u(t) |(1-\phi_n) \tilde u(t))_{L^2(D)} \right |
&\leq \|\tilde u_n(t) - \tilde u(t)\|_{L^2(D)} \|(1- \phi_n) \tilde u(t)\|_{L^2(D)} \\
&\leq 2M_0 \left ( \|u(t)\|^2_{L^2(\Omega \backslash K)} + \|\xi_n u(t) \|_{L^2(K)} \right ) \\
&\leq 2M_0 \left ( \epsilon + \|\xi_n u(t) \|_{L^2(K)} \right ), \\
\end{aligned}$$ for all $t \in (s-\eta, s+\eta) \cap [0,T]$ and for all $n \in \mathbb N$. Since $\xi_n \rightarrow 0$ in $L^2(D)$, a standard argument using Dominated Convergence Theorem implies that $\xi_n u(s) \rightarrow 0$ in $L^2(\Omega)$. Hence, there exists $N_{s,\epsilon} \in \mathbb N$ such that $\|\xi_n u(s)\|_{L^2(\Omega)} \leq \epsilon/2$ for all $n \geq N_{s,\epsilon}$. Therefore, $$\begin{aligned}
\|\xi_n u(t)\|_{L^2(K)}
&\leq \|\xi_n u(s)\|_{L^2(K)} + \| \xi_n u(t) - \xi_n u(s)\|_{L^2(K)} \\
&\leq \|\xi_n u(s)\|_{L^2(K)} + \| u(t) - u(s)\|_{L^2(K)} \\
&\leq \epsilon/2 + \epsilon/2 = \epsilon,
\end{aligned}$$ for all $t \in (s-\eta, s+\eta) \cap [0,T]$ and for all $n \geq N_{s,\epsilon}$. It follows from that $$\label{eq:secondTUnif2}
\left | (\tilde u_n(t) -\tilde u(t) |(1-\phi_n) \tilde u(t))_{L^2(D)} \right |
\leq 2M_0 ( \epsilon + \epsilon)
= 4M_0 \epsilon,$$ for all $t \in (s-\eta, s+\eta) \cap [0,T]$ and for all $n \geq N_{s,\epsilon}$. Therefore, by , , , and , we conclude that there exist $\tilde N_{s,\epsilon} \in \mathbb N$ and a positive constant $C$ such that $$\left |(\tilde u_n(t) -\tilde u(t)|\tilde u(t))_{L^2(D)} \right | \leq C \epsilon,$$ for all $t \in (s-\eta, s+\eta) \cap [0,T]$ and for all $n \geq \tilde N_{s,\epsilon}$.
Finally, as $[0,T]$ is a compact interval and $\eta$ only depends on $\epsilon$, it follows that $(\tilde u_n(t)|\tilde u(t))_{L^2(D)} \rightarrow (\tilde u(t)|\tilde
u(t))_{L^2(D)}$ uniformly with respect to $t \in [0,T]$.
\[rem:NoNeedStrongConvDir\] In fact, we can conclude that $\tilde u_n \rightarrow \tilde u$ in $L^2((0,T),L^2(D))$ directly from the weak convergence of solutions in Theorem \[th:weakConvSolDir\] and the compactness result in [@MR1404388 Lemma 2.1]. This means we only require that $\tilde f_n |_{\Omega} \rightharpoonup f$ weakly in $L^2((0,T),L^2(\Omega))$ and $\tilde u_{0,n} |_{\Omega} \rightharpoonup u_0$ weakly in $L^2(\Omega)$. Under the same assumptions, we can restate convergence result in Theorem \[th:unifConvSolDir\] as $\tilde u_n \rightarrow \tilde u$ in $C([\delta,T],L^2(D))$ for all $\delta \in (0,T]$, as appeared in [@MR1404388]. The reason we impose stronger assumptions on the initial data and the inhomogeneous terms is to avoid using [@MR1404388 Lemma 2.1], which is not applicable to Neumann problems, and illustrate a technique that can be applied to both boundary conditions.
It is known that stability under domain perturbation of solution of elliptic equations subject to Dirichlet boundary condition can be obtained from Mosco convergence of $H^1_0(\Omega_n)$ to $H^1_0(\Omega)$ [@MR1955096]. Hence we can use the same criterion on $\Omega_n$ and $\Omega$ to conclude the stability of solutions of parabolic equations. In particular, the conditions on domains given in [@MR1955096 Theorem 7.5] implies convergence of solutions of non-autonomous parabolic equations subject to Dirichlet boundary condition under domain perturbation.
Neumann problems {#subsec:neuProb}
----------------
It is more complicated for Neumann problems because the trivial extension by zero outside the domain of a function $u_n$ in $H^1(\Omega_n)$ does not belong to $H^1(D)$. In addition, as we do not assume any smoothness of domains, there is no smooth extension operator from $H^1(\Omega_n)$ to $H^1(D)$. In order to study the limit of $u_n \in H^1(\Omega_n)$ when the domain is perturbed, we embed the space $H^1(\Omega_n)$ into the following space $$H^1(\Omega_n) \hookrightarrow L^2(D) \times L^2(D, \mathbb R^N)$$ by $$v_n \mapsto (\tilde v_n, \tilde \nabla v_n),$$ where $\tilde v_n(x) =v(x)$ if $x \in \Omega_n$ and $\tilde v_n(x) = 0$ if $x \in D \backslash \Omega_n$. Similarly, $\tilde \nabla v_n(x) = \nabla v(x)$ if $x \in \Omega_n$ and $\tilde \nabla v_n(x) = 0$ if $x \in D \backslash \Omega_n$. Note that $\tilde \nabla v_n$ is not the gradient of $\tilde v_n$ in the sense of distribution. By a similar embedding for $H^1(\Omega)$, we can consider Mosco convergence of $$K_n := \{ (\tilde v_n, \tilde \nabla v_n) \in L^2(D) \times
L^2(D,\mathbb R^N) \mid v_n \in H^1(\Omega_n) \}$$ to $$K:= \{ (\tilde v, \tilde \nabla v) \in L^2(D) \times
L^2(D,\mathbb R^N) \mid v \in H^1(\Omega) \}$$ in $V:=L^2(D) \times L^2(D,\mathbb R^N)$. In this case, $K_n$ and $K$ are closed subspace of $V$. For simplicity, we use the term $H^1(\Omega_n)$ converges in the sense of Mosco to $H^1(\Omega)$ for $K_n$ and $K$ above.
When dealing with parabolic equations, we regard the space $L^2((0,T),V)$ as $L^2((0,T),L^2(D)) \times L^2((0,T),L^2(D,\mathbb R^N))$ via the isomorphism between them. Hence $$\begin{aligned}
L^2((0,T),K_n) &\equiv \{ (\tilde w_n, \tilde \nabla w_n)
\mid w_n \in L^2((0,T),H^1(\Omega_n) \} \\
&\subset L^2((0,T),L^2(D)) \times L^2((0,T),L^2(D,\mathbb R^N)),
\end{aligned}$$ and $$\begin{aligned}
L^2((0,T),K) &\equiv \{ (\tilde w, \tilde \nabla w)
\mid w \in L^2((0,T),H^1(\Omega) \} \\
&\subset L^2((0,T),L^2(D)) \times L^2((0,T),L^2(D,\mathbb R^N)).
\end{aligned}$$
As in the case of Dirichlet problem, we apply various Mosco conditions from Theorem \[th:equiMosco\] to prove that the variational solution $u_n$ converges to the variational solution $u$.
\[th:weakConvSolNeu\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n} \rightarrow \tilde u_0$ in $L^2(D)$. If $H^1(\Omega_n)$ converges to $H^1(\Omega)$ in the sense of Mosco, then $\tilde u_n$ converges weakly to $\tilde u$ in $L^2((0,T),L^2(D))$ and $\tilde \nabla u_n$ converges weakly to $\tilde\nabla u$ in $L^2((0,T),L^2(D,\mathbb R^N))$.
By a similar argument as in the proof of Theorem \[th:weakConvSolDir\], we have the uniform boundedness of $(\tilde u_n, \tilde \nabla u_n)$ in $L^2((0,T),L^2(D)) \times L^2((0,T),L^2(D,\mathbb R^N))$. We can extract a subsequence (denoted again by $u_n$), such that $\tilde u_n \rightharpoonup w$ in $L^2((0,T),L^2(D))$ and $\tilde \nabla u_n \rightharpoonup (v_1,\ldots,v_N)$ in $L^2((0,T),L^2(D, \mathbb R^N))$. Mosco condition $(M2')$ (from Theorem \[th:equiMosco\]) implies that $w \in L^2((0,T),H^1(\Omega))$.
To show that $w=u$, we let $\xi \in H^1(\Omega)$ and $\phi \in \mathscr D([0,T))$ and then use Mosco convergence of $H^1(\Omega_n)$ to $H^1(\Omega)$. In the same way as the proof of Theorem \[th:weakConvSolDir\], we get holds for all $\xi \in H^1(\Omega)$ and all $\phi \in \mathscr D([0,T))$. Hence by the uniqueness of solution, $w=u$ in $L^2((0,T),H^1(\Omega))$ and the whole sequence converges.
\[lem:L2weakLimUnNeu\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}\rightarrow \tilde u_0$ in $L^2(D)$. If $H^1(\Omega_n)$ converges to $H^1(\Omega)$ in the sense of Mosco, then for each $t \in [0,T]$ we have $\tilde u_n(t)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$.
We use the same argument as in the proof of Lemma \[lem:L2weakLimUnDir\] with Mosco convergence of $H^1_0(\Omega_n)$ to $H^1_0(\Omega)$ replaced by Mosco convegence of $H^1(\Omega_n)$ to $H^1(\Omega)$ and the fact that $H^1(\Omega)$ is also dense in $L^2(\Omega)$.
\[rem:NeuWeakInitialF\] As remarked in the case of Dirichlet problems, we only require that $\tilde f_n |_{\Omega} \rightharpoonup f$ weakly in $L^2((0,T),L^2(\Omega))$ and $\tilde u_{0,n}|_{\Omega} \rightharpoonup u_0$ weakly in $L^2(\Omega)$ to obtain the conclusion of Theorem \[th:weakConvSolNeu\] and Lemma \[lem:L2weakLimUnNeu\].
We next show the strong convergence.
\[th:strongConvSolNeu\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}\rightarrow \tilde u_0$ in $L^2(D)$. If $H^1(\Omega_n)$ converges to $H^1(\Omega)$ in the sense of Mosco, then $\tilde u_n$ converges strongly to $\tilde u$ in $L^2((0,T),L^2(D))$ and $\tilde \nabla u_n$ converges strongly to $\tilde\nabla u$ in $ L^2((0,T),L^2(D,\mathbb R^N))$.
The proof is similar to the one in Theorem \[th:strongConvSolDir\]. We show some details here for the sake of completeness. By Theorem \[th:weakConvSolNeu\], $(\tilde u_n, \tilde \nabla u_n)$ converges weakly to $(\tilde u, \tilde \nabla u)$ in $L^2((0,T),L^2(D)) \times L^2((0,T),L^2(D,\mathbb R^2))$. Since $u \in L^2((0,T),H^1(\Omega))$, Mosco condition $(M1')$ (from Theorem \[th:equiMosco\]) implies that there exists $w_n \in L^2((0,T),H^1(\Omega_n))$ such that $\tilde w_n \rightarrow \tilde u$ in $L^2((0,T),L^2(D))$ and $\tilde \nabla w_n \rightarrow \tilde\nabla u$ in $L^2((0,T),L^2(D,\mathbb R^N))$. For $t \in [0,T]$, we consider $$\label{eq:dNNeu}
\begin{aligned}
d_n(t) &= \frac{1}{2} \| \tilde u_n(t) - \tilde u(t) \|^2_{L^2(D)}
+ \alpha \int_0^t \| u_n(s) -w_n(s) \|^2_{L^2(\Omega_n)} \;ds \\
&\quad + \alpha \int_0^t \| \nabla u_n(s) - \nabla w_n(s) \|^2_{L^2(\Omega_n, \mathbb R^N)} \;ds \\
&= \frac{1}{2} \| \tilde u_n(t) - \tilde u(t) \|^2_{L^2(D)}
+ \alpha \int_0^t \| u_n(s) -w_n(s) \|^2_{H^1(\Omega_n)} \;ds.
\end{aligned}$$ By (with $\lambda =0$), we can show that $d_n$ satisfies for all $n \in \mathbb N$. It can be easily seen from the weak convergence of $(\tilde u_n, \tilde \nabla u_n)$ and the strong convergence of $(\tilde w_n, \tilde \nabla w_n)$ to $(\tilde u, \tilde \nabla u)$ in $L^2((0,T),L^2(D)) \times L^2((0,T),L^2(D,\mathbb R^N))$ that and also hold. By using Lemma \[lem:L2weakLimUnNeu\] instead of Lemma \[lem:L2weakLimUnDir\], we obtain . Finally, is also valid for Neumann problem. Hence, $d_n(t) \rightarrow 0$ for all $t \in [0,T]$. This shows pointwise convergence of $\tilde u_n(t)$ to $\tilde u(t)$ in $L^2(D)$. Moreover, by taking $t=T$ we get $$\begin{aligned}
&\int_0^T \|\tilde u_n(s)-\tilde u(s)\|^2_{L^2(D)} \;ds \\
& \quad \leq \int_0^T \|\tilde u_n(s)-\tilde w_n(s)\|^2_{L^2(D)} \;ds
+ \int_0^T \|\tilde w_n(s)-\tilde u(s)\|^2_{L^2(D)} \;ds \\
& \quad \rightarrow 0,
\end{aligned}$$ and $$\begin{aligned}
&\int_0^T \|\tilde \nabla u_n(s)-\tilde\nabla u(s)\|^2_{L^2(D,\mathbb R^N)} \;ds \\
& \quad \leq \int_0^T \|\tilde \nabla u_n(s)-\tilde\nabla w_n(s)\|^2_{L^2(D,\mathbb R^N)} \;ds
+ \int_0^T \|\tilde \nabla w_n(s)-\tilde \nabla u(s)\|^2_{L^2(D,\mathbb R^N)} \;ds \\
& \quad \rightarrow 0.
\end{aligned}$$ This proves the strong convergence $\tilde u_n \rightarrow \tilde u$ in $L^2((0,T),L^2(D))$ and $\tilde \nabla u_n \rightarrow \tilde\nabla u$ in $ L^2((0,T),L^2(D, \mathbb R^N))$.
Recall that we embed the space $K = H^1(\Omega)$ in $V = L^2(D) \times L^2(D, \mathbb R^N)$. If $v$ is a function in $W((0,T), H^1(\Omega), H^1(\Omega)')$, then $v' \in L^2((0,T), H^1(\Omega)')$. It is not always true that we can embed $v'(t) \in V' = L^2(D) \times L^2(D, \mathbb R^N)$ a.e. $t \in (0,T)$ and claim that $v \in W((0,T),V,V') \cap L^2((0,T),K)$. However, a similar argument as in the proof of Theorem \[th:equiMosco\] $(i) \Rightarrow (iii)$ for Mosco condition (M1") gives the following result.
\[lem:M1ddreplaceNeu\] Suppose that $H^1(\Omega_n)$ converges to $H^1(\Omega)$ in the sense of Mosco, If $w \in C^{\infty}([0,T],H^1(\Omega))$ then there exists $w_n \in C^{\infty}([0,T],H^1(\Omega_n))$ such that $\tilde w_n$ converges to $\tilde w$ in $C^{\infty}([0,T],L^2(D))$.
We note that Proposition \[prop:convCombUdelta\] gives uniform convergence of the approximation sequence in $V= L^2(D) \times L^2(D, \mathbb R^N)$. The proof follows the same arguments as in the proof of Theorem \[th:equiMosco\] $(i) \Rightarrow (iii)$. The only difference is that we assume here $w \in C^{\infty}([0,T],H^1(\Omega))$. Hence the stretched function $w_{\delta} = w \circ S_{\delta}^{-1}$ belongs to $C^{\infty}([-\delta,T+\delta],H^1(\Omega))$. We point out that, by using uniform continuity of the $k$-th order derivative $w^{(k)}$ on $[0,T]$, the restriction of $w_{\delta}$ on $[0,T]$ converges to $w$ in $C^{\infty}([0,T],H^1(\Omega))$. This gives the required convergence in $C^{\infty}([0,T],L^2(D))$.
Using the above lemma, we show in the next theorem that the solution $u_n$ of indeed converges uniformly with respect to $t \in [0,T]$.
\[th:unifConvSolNeu\] Suppose $\tilde f_n \rightarrow \tilde f$ in $L^2((0,T),L^2(D))$ and $\tilde u_{0,n}\rightarrow \tilde u_0$ in $L^2(D)$. If $H^1(\Omega_n)$ converges to $H^1(\Omega)$ in the sense of Mosco, then $\tilde u_n$ converges strongly to $\tilde u$ in $C([0,T],L^2(D))$.
As in the proof of Theorem \[th:unifConvSolDir\], it requires to show uniform convergence of $(\tilde u_n(t)|\tilde u(t))_{L^2(D)} \rightarrow (\tilde
u(t)|\tilde u(t))_{L^2(D)}$.
Let $\epsilon > 0$ arbitrary. By a similar argument as in the proof Lemma \[lem:densityOfWK\], we have the density of $C^{\infty}([0,T], H^1(\Omega))$ in $W((0,T),H^1(\Omega), H^1(\Omega)')$. Since the solution $u$ is in the space $W((0,T),H^1(\Omega), H^1(\Omega)')$, there exists $w \in C^{\infty}([0,T], H^1(\Omega))$ such that $$\|w - u\|_{W((0,T),H^1(\Omega), H^1(\Omega)')} \leq \varepsilon.$$ As $W((0,T),H^1(\Omega), H^1(\Omega)')$ is continuously embedded in $C([0,T], L^2(\Omega))$, we can indeed choose $w \in C^{\infty}([0,T], H^1(\Omega))$ such that $$\label{eq:wuLessEps}
\|w(t) - u(t)\|_{L^2(\Omega)} \leq \varepsilon,$$ for all $t \in [0,T]$. By Lemma \[lem:M1ddreplaceNeu\], there exists $w_n \in C^{\infty}([0,T], H^1(\Omega_n))$ such that $\tilde w_n \rightarrow \tilde w$ in $C^{\infty}([0,T],L^2(D))$. We can write $$\label{eq:unifNeu3T}
\begin{aligned}
\left | (\tilde u_n(t)- \tilde u(t)|\tilde u(t))_{L^2(D)} \right |
&\leq \left | (\tilde u_n(t)|\tilde w_n(t))_{L^2(D)} - (\tilde u(t)|\tilde w(t))_{L^2(D)} \right | \\
&\quad + \left | (\tilde u_n(t) - \tilde u(t)|\tilde u(t) -\tilde w(t))_{L^2(D)} \right | \\
&\quad + \left | (\tilde u_n(t)|\tilde w(t) - \tilde w_n(t))_{L^2(D)} \right |, \\
\end{aligned}$$ for all $n \in \mathbb N$. Since $u_n$ is a solution of , $$\begin{aligned}
(\tilde u_n(t) |\tilde w_n(t))_{L^2(D)}
&= (\tilde u_{0,n} |\tilde w_n(0))_{L^2(D)}
+ \int_0^t \langle f_n(s), w_n(s) \rangle \;ds \\
&\quad + \int_0^t \langle w_n'(s), u_n(s) \rangle \;ds
- \int_0^t a_n(s; u_n(s),w_n(s)) \rangle \;ds, \\
\end{aligned}$$ for all $n \in \mathbb N$. Taking into consideration that $\tilde u_{0,n} \rightarrow \tilde u_0$ and $f_n \rightarrow f$, we conclude that $$\label{eq:firstTNeu}
(\tilde u_n(t)|\tilde w_n(t))_{L^2(D)} \rightarrow (\tilde u(t)|\tilde w(t))_{L^2(D)}$$ uniformly with respect to $t \in [0,T]$. Moreover, by and the uniform boundedness of solutions as in , $$\label{eq:secondTNeu}
\begin{aligned}
&\left | (\tilde u_n(t) - \tilde u(t)|\tilde u(t) -\tilde w(t))_{L^2(D)} \right | \\
&\leq \| \tilde u_n(t) - \tilde u(t)\|_{L^2(D)} \|\tilde u(t) -\tilde w(t)\|_{L^2(D)} \\
&\leq 2M_0 \epsilon, \\
\end{aligned}$$ for all $t \in [0,T]$ and for all $n \in \mathbb N$. Finally, as $\tilde w_n \rightarrow \tilde w$ in $C^{\infty}([0,T],L^2(D))$, there exists $N_{\epsilon} \in \mathbb N$ such that $$\| \tilde w_n(t) - \tilde w(t) \|_{L^2(D)} \leq \epsilon,$$ for all $t \in [0,T]$ and for all $n \geq N_{\epsilon}$. Hence, $$\label{eq:lastTNeu}
\begin{aligned}
\left | (\tilde u_n(t)|\tilde w_n(t) -\tilde w(t))_{L^2(D)} \right |
&\leq \|\tilde u_n(t)\|_{L^2(D)} \| \| \tilde w_n(t) -\tilde w(t)\|_{L^2(D)} \\
&\leq M_0 \epsilon, \\
\end{aligned}$$ for all $t \in [0,T]$ and for all $n \geq N_{\epsilon}$. Therefore, by – , there exists $\tilde N_{\epsilon} \in \mathbb N$ and a positive constant $C$ such that $$\left |(\tilde u_n(t) -\tilde u(t)|\tilde u(t))_{L^2(D)} \right | \leq C \epsilon,$$ for all $t \in [0,T]$ and for all $n \geq \tilde N_{\epsilon}$. As $\epsilon >0$ was arbitrary, this proves the required uniform convergence of $(\tilde u_n(t)|\tilde u(t))_{L^2(D)} \rightarrow (\tilde u(t)|\tilde
u(t))_{L^2(D)}$ with respect to $t \in [0,T]$.
We can use the same criterion on $\Omega_n$ and $\Omega$ as in Neumann elliptic problems to conclude the stability of solutions of Neumann parabolic equations under domain perturbation. In particular, for domains in two dimensional spaces, the conditions on domains given in [@MR1822408 Theorem 3.1] implies convergence of solutions of non-autonomous parabolic equations subject to Neumann boundary condition.
\[rem:domainCrackNeu\] The assumptions on strong convergence of $\tilde u_{0,n}$ and $\tilde f_n$ can be weaken if we impose some regularity of the domains. We give an example of domains $\Omega_n$ satisfying the *cone condition* (see [@MR0450957 Section 4.3]) uniformly with respect to $n \in \mathbb N$.
Let $N=2$ and let $$\begin{aligned}
\Omega &:= \{ x \in \mathbb R^2 : |x| <1 \} \backslash \{(x_1,0): 0 \leq x_1 <1 \}, \\
\Omega_n &:= \{ x \in \mathbb R^2 : |x| <1 \} \backslash \{(x_1,0): \delta_n \leq x_1 <1 \},
\end{aligned}$$ where $\delta_n \searrow 0$. This example is an exterior perturbation of the domain, that is $\Omega \subset \Omega_{n+1} \subset \Omega_n$ for all $n \in \mathbb N$. It is easy to see that $\Omega$ and $\Omega_n$ satisfy the cone condition uniformly with respect to $n \in \mathbb N$, but $H^1(\Omega)$ and $H^1(\Omega_n)$ do not have the extension property. Moreover, these domains satisfy the conditions in [@MR1822408]. Hence, $H^1(\Omega_n)$ converges to $H^1(\Omega)$ in the sense of Mosco. Note that here we take $D$ to be the open unit disk center at $0$ in $\mathbb R^2$. In this example, we only need that $\tilde f_n |_{\Omega} \rightharpoonup f$ in $L^2((0,T),L^2(\Omega))$ and $\tilde u_{0,n} |_{\Omega} \rightharpoonup u_0$ in $L^2(\Omega)$ to conclude the convergence of solutions $\tilde u_n \rightarrow \tilde u$ in $C([\delta, T], L^2(D))$ for all $\delta \in (0,T]$. In addition, if $\tilde u_{0,n} \rightarrow \tilde u_0$ in $L^2(D)$, then the assertion holds for $\delta = 0$.
To see this, we note from Lemma \[lem:L2weakLimUnNeu\] (taking Remark \[rem:NeuWeakInitialF\] into account) that $\tilde u_n(t)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$ weakly for all $t \in [0,T]$. Since $u \in L^2((0,T),H^1(\Omega))$, we have $u(t) \in H^1(\Omega)$ for almost everywhere $t \in (0,T)$. Fix now such $t \in (0,T)$. By the continuity of the solutions $u_n \in C([0,T],L^2(\Omega_n))$, for each $n \in \mathbb N$ we can choose $\rho_n > 0$ such that $$\|u_n(s) -u_n(t) \|_{L^2(\Omega_n)} \leq \frac{1}{n}$$ for all $s \in (t-\rho_n, t+\rho_n) \cap (0,T)$. As $u_n \in L^2((0,T),H^1(\Omega_n))$ we can choose $t_n \in (t-\rho_n, t+\rho_n) \cap (0,T)$ such that $u_n(t_n) \in H^1(\Omega_n)$ for all $n \in \mathbb N$. For these choices of $t_n$, we have $ \|u_n(t_n) -u_n(t) \|_{L^2(\Omega_n)} \rightarrow 0$ as $n \rightarrow \infty$. It follows that $\tilde u_n(t_n)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$ weakly. Since $\Omega \subset \Omega_n$ for all $n \in \mathbb N$, the restriction $ u_{n}(t_n)_{|_{\Omega}}$ belongs to $H^1(\Omega)$ for all $n \in \mathbb N$. Hence it follows from the weak convergence of $\tilde u_n(t_n)_{|_{\Omega}} = u_n(t_n)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$ that $$\int_{\Omega} \partial_j \big( u_{n}(t_n)_{|_{\Omega}} \big ) \phi dx
= -\int_{\Omega} u_{n}(t_n)_{|_{\Omega}} \partial_j \phi dx
\rightarrow -\int_{\Omega} u(t) \partial_j \phi dx
= \int_{\Omega} \partial_j u (t) \phi dx,$$ for all $\phi \in C^{\infty}_c(\Omega)$ for $j =1,2$. This means $\nabla u_{n}(t_n)_{|_{\Omega}} \rightharpoonup \nabla u(t)$ in $L^2(\Omega, \mathbb R^2)$. Thus, $ u_{n}(t_n)_{|_{\Omega}}$ is bounded in $H^1(\Omega)$. As $\Omega$ is bounded and satisfies the cone condition, we have from the Rellich-Kondrachov theorem that the embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ is compact (see [@MR0450957 Theorem 6.2]). Therefore, $u_{n}(t_n)_{|_{\Omega}}$ has a subsequence which converges strongly in $L^2(\Omega)$. Since we have a prior knowledge of weak convergence $u_n(t_n)_{|_{\Omega}} \rightharpoonup u(t)$ in $L^2(\Omega)$, we conclude that the whole sequence $u_n(t_n)_{|_{\Omega}} \rightarrow u(t)$ in $L^2(\Omega)$ strongly. By the choices of $t_n$, we conclude that $u_n(t)_{|_{\Omega}} \rightarrow u(t)$ in $L^2(\Omega)$ strongly. Since the above argument works for almost everywhere $t \in (0,T)$, we deduce from the dominated convergence theorem that $\tilde u_n |_{\Omega} = u_n |_{\Omega} \rightarrow u$ in $L^2((0,T),L^2(\Omega))$ strongly. As the cutting line is a set of measure zero in $\mathbb R^2$, we have $\tilde u_n \rightarrow \tilde u$ in $L^2((0,T),L^2(D))$. By extracting a subsequence (indexed again by $n$), we can choose $\delta$ arbitrarily closed to zero in $(0,T]$ such that $\tilde u_n(\delta) \rightarrow \tilde u(\delta)$ in $L^2(D)$. The required convergence in $C([\delta,T],L^2(D))$ follows from the argument in the proof of Theorem \[th:strongConvSolNeu\] and Theorem \[th:unifConvSolNeu\] with the integration taken over $[\delta,T]$ instead of $[0,T]$ (see also the proof of [@MR1404388 Therem 3.5]).
Application in Parabolic Variational Inequalities {#sec:appParVar}
=================================================
In the previous section we have seen some applications of Theorem \[th:equiMosco\] when $K_n$ and $K$ are closed subspaces of $V$. In this section, we consider the case when $K_n$ and $K$ are just closed and convex subsets of $V$. We show here a similar convergence properties of solutions of parabolic variational inequalities.
Let $K_n, K$ be closed and convex subsets in $V$. For each $t \in (0,T)$, suppose $a(t; \cdot, \cdot)$ is a continuous bilinear form on $V$ satisfying and . For simplicity, we assume that $\lambda = 0$ in . We denote by $A(t)$ the linear operator induced by $a(t; \cdot, \cdot)$. Let us consider the following parabolic variational inequalities. Given $u_{0,n} \in K_n$ and $f_n \in L^2((0,T),V')$, we want to find $u_n$ such that for a.e. $t \in (0,T)$, $u_n(t) \in K_n$ and $$\label{eq:parVarIneqKn}
\left\{
\begin{aligned}
\langle u'(t), v-u(t) \rangle + \langle A(t)u(t), v-u(t) \rangle
- \langle f_n(t) , v - u(t) \rangle &\geq 0, \quad \forall v \in K_n \\
u(0) &= u_{0,n}.
\end{aligned}
\right .$$ When $K_n, f_n$ and $u_{0,n}$ converge to $K, f$ and $u_{0}$, we wish to obtain convergence results of weak solution of to the following limit inequalities. $$\label{eq:parVarIneqKK}
\left\{
\begin{aligned}
\langle u'(t), v-u(t) \rangle + \langle A(t)u(t), v-u(t) \rangle
- \langle f(t) , v - u(t) \rangle &\geq 0, \quad \forall v \in K \\
u(0) &= u_{0}.
\end{aligned}
\right .$$ Throughout this section, we denote the weak solution of by $u_n$ and the weak solution of by $u$. The notion of our weak solutions is given in Definition \[def:weakSolIneq\].
\[th:boundedParVar\] Suppose $f_n \rightarrow f$ in $L^2((0,T),V')$, $u_{0,n} \rightharpoonup u_0$ in $V$ and $u_{0,n} \rightarrow u_0$ in $H$. Then the sequence of weak solutions $u_n$ is bounded in $L^2((0,T),V)$.
Let $v \in W((0,T),V,V') \cap L^2((0,T),K)$ be the constant function defined by $v(t) := u_0$ for $t \in [0,T]$. Similarly, $v_n \in W((0,T),V,V') \cap L^2((0,T),K_n)$ defined by $v_n(t) := u_{0,n}$ for $t \in [0,T]$. It follows that $v_n
\rightharpoonup v$ in $L^2((0,T),V)$. Since $u_n$ is a weak solution of , $$\begin{aligned}
&\int_0^T \langle A(t)u_n(t), u_n(t)-v_n(t) \rangle \;dt \\
& \quad \leq \int_0^T \langle v_n'(t), v_n(t)-u_n(t) \rangle
- \langle f_n(t), v_n(t)-u_n(t) \rangle \;dt
+ \frac{1}{2} \|v_n(0) -u_{0,n} \|^2_H \\
& \quad = - \int_0^T \langle f_n(t), v_n(t)-u_n(t) \rangle \;dt.
\end{aligned}$$ Thus, $$\begin{aligned}
&\int_0^T \langle A(t)u_n(t) - A(t)v_n(t), u_n(t)-v_n(t) \rangle \;dt \\
&\quad \leq \int_0^T \langle A(t)v_n(t), v_n(t) -u_n(t) \rangle
- \langle f_n(t), v_n(t)-u_n(t) \rangle \;dt \\
&\quad \leq \| A(t)v_n -f_n \|_{L^2((0,T),V')} \|v_n -u_n\|_{L^2((0,T),V)}.
\end{aligned}$$ By the coerciveness of $A(t)$, $$\alpha \|u_n -v_n\|_{L^2((0,T),V)} \leq \|A(t)v_n -f_n \|_{L^2((0,T),V')}.$$ We conclude from the weak convergences of $v_n$ and $f_n$ that $u_n$ is bounded in $L^2((0,T),V)$.
\[th:weakConvSolParVar\] Suppose $f_n \rightarrow f$ in $L^2((0,T),V')$, $u_{0,n} \rightharpoonup u_0$ in $V$ and $u_{0,n} \rightarrow u_0$ in $H$. If $K_n$ converges to $K$ in the sense of Mosco, then the sequence of weak solutions $u_n$ converges weakly to $u$ in $L^2((0,T),V)$.
By Theorem \[th:boundedParVar\], we can extract a subsequence of $u_n$ (denoted again by $u_n$) such that $u_n \rightharpoonup \kappa$ in $L^2((0,T),V)$. Since $u_n \in L^2((0,T),K_n)$, we apply Mosco condition $(M2')$ (from Theorem \[th:equiMosco\]) to deduce that the weak limit $\kappa \in L^2((0,T),K)$. By the uniqueness of weak solution, it suffices to prove that $\kappa$ satisfies (with $u$ replaced by $\kappa$) in the definition of weak solution.
By Mosco condition $(M1')$, there exists $w_n \in L^2((0,T),K_n)$ such that $w_n \rightarrow \kappa$ in $L^2((0,T),V)$. Let $v \in W((0,T),V,V')
\cap L^2((0,T),K)$. We again apply Theorem \[th:equiMosco\] for Mosco condition $(M1")$ to get a sequence of functions $v \in W((0,T),V,V')\cap L^2((0,T),K_n)$ such that $v_n \rightarrow v$ in $W((0,T),V,V')$. For each $n \in \mathbb N$, $$\begin{aligned}
&\langle A(t)w_n(t), v_n(t)-u_n(t) \rangle \\
&\quad= \langle A(t)u_n(t), v_n(t)-u_n(t) \rangle
+ \langle A(t)w_n(t) - A(t)u_n(t), v_n(t) -u_n(t) \rangle \\
&\quad= \langle A(t)u_n(t), v_n(t)-u_n(t) \rangle
+ \langle A(t)w_n(t) - A(t)u_n(t), w_n(t) -u_n(t) \rangle \\
&\quad \quad + \langle A(t)w_n(t) - A(t)u_n(t), v_n(t) -w_n(t) \rangle.
\end{aligned}$$ Hence, by definition of weak solution on $K_n$ and coerciveness of $A(t)$, $$\begin{aligned}
&\int_0^T \langle v_n'(t), v_n(t)-u_n(t) \rangle
+ \langle A(t)w_n(t), v_n(t)-u_n(t) \rangle \;dt \\
&\quad - \int_0^T \langle f_n(t), v_n(t)-u_n(t) \rangle \;dt
+ \frac{1}{2} \|v_n(0) - u_{0,n}\|^2_H \\
&= \int_0^T \langle v_n'(t), v_n(t)-u_n(t) \rangle
+ \langle A(t)u_n(t), v_n(t)-u_n(t) \rangle \;dt \\
&\quad - \int_0^T \langle f_n(t), v_n(t)-u_n(t) \rangle \;dt
+ \frac{1}{2} \|v_n(0) - u_{0,n}\|^2_H \\
&\quad + \int_0^T \langle A(t)w_n(t) - A(t)u_n(t), w_n(t) -u_n(t) \rangle \;dt \\
& \quad + \int_0^T \langle A(t)w_n(t) - A(t)u_n(t), v_n(t) -w_n(t) \rangle \;dt \\
& \geq \int_0^T \langle A(t)w_n(t) - A(t)u_n(t), v_n(t) -w_n(t) \rangle \;dt,
\end{aligned}$$ for all $n \in \mathbb N$. Letting $n \rightarrow \infty$, $$\begin{aligned}
&\int_0^T \langle v'(t), v(t)-\kappa(t) \rangle
+ \langle A(t)\kappa(t), v(t)-\kappa(t) \rangle \;dt \\
&\quad - \int_0^T \langle f(t), v(t)-\kappa(t) \rangle \;dt
+ \frac{1}{2} \|v(0) - u_0\|^2_H
\geq 0.
\end{aligned}$$ This implies $\kappa$ is a weak solution of as required.
We finally prove strong convergence of solutions.
\[th:strongConvSolParVar\] Suppose $f_n \rightarrow f$ in $L^2((0,T),V')$, $u_{0,n} \rightharpoonup u_0$ in $V$ and $u_{0,n} \rightarrow u_0$ in $H$. If $K_n$ converges to $K$ in the sense of Mosco, then the sequence of weak solutions $u_n$ converges strongly to $u$ in $L^2((0,T),V)$.
By the coerciveness of $A(t)$, $$\label{eq:limInfSol}
\liminf_{n \rightarrow \infty}
\int_0^T \langle A(t)u_n(t) - A(t)u(t), u_n(t) -u(t) \rangle \;dt \geq 0.$$ For each $\epsilon > 0$, we define $u_{\epsilon}$ by $$\begin{aligned}
\epsilon u'_{\epsilon} + u_{\epsilon}& = u \\
u_{\epsilon} (0)& = u_0.
\end{aligned}$$ Then $u_{\epsilon} \in W((0,T),V,V') \cap L^2((0,T),K)$ and $u_{\epsilon} \rightarrow u$ in $L^2((0,T),V)$ as $\epsilon \rightarrow
0$ (see in the proof of [@MR0296479 Theorem 2.3]). For each $\epsilon > 0$, Mosco condition $(M1")$ (from Theorem \[th:equiMosco\]) implies that there exists $u_{\epsilon,n} \in W((0,T),V,V') \cap L^2((0,T),K_n)$ such that $u_{\epsilon,n} \rightarrow u_{\epsilon}$ in $W((0,T),V,V')$ as $n \rightarrow \infty$. Since $u_n$ is a weak solution of , $$\begin{aligned}
&\int_0^T \langle A(t)u_n(t), u_n(t)-u(t) \rangle \;dt \\
&\leq \int_0^T \langle v'_n(t),v_n(t)- u_n(t) \rangle \;dt
- \int_0^T \langle f_n(t),v_n(t)- u_n(t) \rangle \;dt \\
&\quad + \frac{1}{2} \|v_n(0) - u_{0,n} \|^2_H
+ \int_0^T \langle A(t)u_n(t),v_n(t)- u(t) \rangle \;dt,
\end{aligned}$$ for all $v \in W((0,T),V,V') \cap L^2((0,T),K)$. In particular, taking $v_n = u_{\epsilon,n}$, $$\begin{aligned}
&\int_0^T \langle A(t)u_n(t), u_n(t)-u(t) \rangle \;dt \\
&\leq \int_0^T \langle u'_{\epsilon,n}(t),u_{\epsilon,n}(t)- u_n(t) \rangle \;dt
- \int_0^T \langle f_n(t),u_{\epsilon,n}(t)- u_n(t) \rangle \;dt \\
&\quad + \frac{1}{2} \|u_{\epsilon,n}(0) - u_{0,n} \|^2_H
+ \int_0^T \langle A(t)u_n(t),u_{\epsilon,n}(t)- u(t) \rangle \;dt.
\end{aligned}$$ Letting $n \rightarrow \infty$, we obtain $$\begin{aligned}
&\limsup_{n \rightarrow \infty}
\int_0^T \langle A(t)u_n(t), u_n(t)-u(t) \rangle \;dt \\
&\leq \int_0^T \langle u'_{\epsilon}(t),u_{\epsilon}(t)- u(t) \rangle \;dt
- \int_0^T \langle f(t),u_{\epsilon}(t)- u(t) \rangle \;dt \\
&\quad + \frac{1}{2} \|u_{\epsilon}(0) - u_0 \|^2_H
+ \int_0^T \langle A(t)u(t),u_{\epsilon}(t)- u(t) \rangle \;dt \\
&= -\epsilon \int_0^T \|u'_{\epsilon}(t)\|^2_H \;dt
- \int_0^T \langle f(t),u_{\epsilon}(t)- u(t) \rangle \;dt \\
&\quad + \int_0^T \langle A(t)u(t),u_{\epsilon}(t)- u(t) \rangle \;dt \\
&\leq - \int_0^T \langle f(t),u_{\epsilon}(t)- u(t) \rangle \;dt
+ \int_0^T \langle A(t)u(t),u_{\epsilon}(t)- u(t) \rangle \;dt.
\end{aligned}$$ This is true for any $\epsilon > 0$. Hence, by letting $\epsilon
\rightarrow 0$, $$\limsup_{n \rightarrow \infty}
\int_0^T \langle A(t)u_n(t), u_n(t)-u(t) \rangle \;dt \leq 0.$$ On the other hand, the weak convergence of $u_n$ in Theorem \[th:weakConvSolParVar\] implies $$\lim_{n \rightarrow 0}
\int_0^T \langle A(t)u(t), u_n(t)-u(t) \rangle \;dt = 0.$$ Thus, $$\label{eq:limSupSol}
\limsup_{n \rightarrow \infty}
\int_0^T \langle A(t)u_n(t) - A(t)u(t), u_n(t) -u(t) \rangle dt \leq 0.$$ It follows from the coerciveness of $A(t)$, and that $$\alpha \|u_n -u \|^2_{L^2((0,T),V)}
\leq \int_0^T \langle A(t)u_n(t) - A(t)u(t), u_n(t) -u(t) \rangle \;dt
\rightarrow 0,$$ as $n \rightarrow \infty$. Therefore, $u_n \rightarrow u$ in $L^2((0,T),V)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank D. Daners for helpful discussions and suggestions.
|
[Bi-Hamiltonian Structure in Serret-Frenet Frame]{}
E Abadoğlu and H. Gūmral
Department of Mathematics, Yeditepe University
Kayişdaği 34750 İstanbul Turkey
eabadoglu@yeditepe.edu.tr, hgumral@yeditepe.edu.tr
Nov 15, 2007
**Abstract** We reduced the problem of constructing bi-Hamiltonian structure in three dimensions to the solution of a Riccati equation in moving coordinates of Serret-Frenet frame. We then show that either the linearly independent solutions of the corresponding second order equation or the normal vectors of the moving frame imply two compatible Poisson structures.
Introduction
============
The discovery of completely integrable nonlinear evolution equations as well as the algebraic and geometric structures associated with them has triggered an intensive search of finite dimensional dynamical systems resembling the similar properties. The bi-Hamiltonian structure as an underlying geometrical framework for complete integrability provoked the revival of Poisson structures of finite dimensional dynamical systems (see [@olver] and the references therein for details and comparison).
Several works [@nambu]-[@hojman] on construction of conserved quantities, on Hamiltonian structures and on integrability of three dimensional systems have led to a systematic investigation using Poisson geometry, Frobenius integrability theorem and unfoldings of foliations [phd]{}, [@hasan]. We presented correspondence between Poisson structures and integrable one forms and utilized this to obtain criteria for local and global existence of Poisson structures. We also obtained the local result that any two Poisson structures can be made into a compatible pair to form a bi-Hamiltonian structure.
The restrictions on Poisson matrices imposed by the Jacobi identity and the compatibility condition for two of them to form a bi-Hamiltonian structure are the most serious conditions requiring deliberate actions against their simple presence as one scalar equation in three dimensions [benitofairen]{}-[@ben]. In [@hasan], using an invariance property, we reduced the Jacobi identity to a nonlinear equation in ratios of components of the Poisson matrix. This scalar equation in one unknown function was shown to contain sufficient information for constructing the Poisson structure completely and was recognized to be the Riccati equation in [haas]{}.
The possibility to determine the Poisson matrix by a single function straightens out the difficulties in general Hamiltonian systems which does not fall into the classes of canonical Hamiltonian or Lie-Poisson (i.e. with linear Poisson structure) equations, arising from the absence of coordinates similar to the canonical Darboux coordinates of symplectic geometry. The Darboux-Weinstein theorem [@weinstein] describes the local structure of a Poisson manifold as a space foliated by symplectic submanifolds. The foliation depends on the rank of the structure which is an invariant of the Poisson matrix. As a result, the Poisson matrix consists of a constant submatrix whose rank is the rank of the Poisson structure and some additional nonlinear part. In three dimensions, one must solve for at least one unknown function to determine the Poisson matrix completely.
The role of the constants of motion, in particular the Casimirs, in linearization and integration of the equations associated with the Jacobi identity were also discussed in [@benitofairen]-[@haas]. Yet, there is no direct relations of the Jacobi identity and the compatibility condition to the theory of linear differential equations which may be one of the elegant ways to avoid some deceptive conclusions from these simple looking differential equations ignoring their nonlinear character. The main source of these confusions is to endeavor to exploit local criteria for global results without questioning any suspicious obstructions such as the one we have found for the Darboux-Halphen system [@hasan].
In [@hasan] we also observed through several examples that some coordinate transformations may cast the dynamical systems into a form where, in spite of much higher degree of nonlinearity, the integration for the conserved quantities and hence the manifestation of the bi-Hamiltonian structures become more efficient. The non-covariance of the general Hamiltonian formulation is a source of the common belief that the existence of Hamiltonian structure and the integrability of dynamical systems rely heavily upon the coordinates in which they are represented as well as the prefered parametrization of the solution curves. Such coordinates were found to be important in numerical integrations as well. It is shown in [capel]{} that, a necessary condition for numerical solution algorithms to preserve the conserved quantities of dynamical systems is their resemblance, as products of a skew symmetric matrix and a gradient vector, to Hamiltonian systems. See also [@bulent] for numerical schemes applied to some concrete examples of systems under consideration. In three dimensions, the Nambu mechanics [@nambu] is the only and generic (up to a conformal factor) framework which enables us to identify such coordinates. Namely, the Nambu structure is a manifestation of the bi-Hamiltonian structure, hence integrability, in a frame with coordinate vectors consisting of the dynamical vector field and gradients of two conserved Hamiltonians [hasan]{}.
The Nambu representation obviously requires the integration of the system for Hamiltonian functions. In our study of the Darboux-Halphen system, we were able to obtain obstraction for the global integration of such quantities [@hasan]. To our knowledge, this is the only example which exhibits, along with a rich geometric structure, differentiation between local criteria and global availability. On the other hand, the local version of the Nambu mechanics has not yet been appeared in the literature.
In this work, we shall show that the coordinates associated with the Serret-Frenet frame is the one we sought for three dimensional systems. In these coordinates the Jacobi identity linearizes through the Riccati equation and the compatibility follows from the Hamilton‘s equations. Obstructions to the global constructions of the bi-Hamiltonian structures are encoded in the helicities and the cross-helicity of the unit vectors spanning the normal plane to the vector field associated with the dynamical system. We shall construct a bi-Hamiltonian moving frame for the local Nambu representation.
In the next section we review the properties of bi-Hamiltonian systems in three dimensions. In section three we introduce the Serret-Frenet frame associated with a vector in three dimension. We express the Jacobi identity in Serret-Frenet frame and show that in moving coordinates it reduces to a Riccati equation. In section four, we shall show that Poisson structures constructed from the solutions of the Riccati equations and/or the normal vectors are all compatible via Hamilton‘s equations of motion. We then conclude the existence of bi-Hamiltonian structures.
Hamiltonian Systems in Three Dimensions
=======================================
Following [@hasan], we shall summarize the necessary ingradients of the bi-Hamiltonian formalism in three dimensions. For $\mathbf{x=}\left\{
x^{i}\right\} =(x,y,z)\in
\mathbb{R}
^{3}$, $t\in
\mathbb{R}
$ and dot denoting the derivative with respect to $t$, we consider the autonomous differential equations$$\overset{\cdot }{\mathbf{x}}=\mathbf{v}\left( \mathbf{x}\right) \label{e1}$$associated with a three-dimensional smooth vector field $\mathbf{v}.$This equation is said to be Hamiltonian if the vector field can be written as $$\mathbf{v}\left( \mathbf{x}\right) =\Omega \left( \mathbf{x}\right) \left(
dH\left( \mathbf{x}\right) \right) \label{e2}$$where $H\left( \mathbf{x}\right) $ is the Hamiltonian function and $\Omega
\left( \mathbf{x}\right) $ is the Poisson bi-vector (skew-symmetric, contravariant two-tensor) subjected to the Jacobi identity $$\left[ \Omega \left( \mathbf{x}\right) ,\Omega \left( \mathbf{x}\right) \right] =0 \label{e3}$$defined by the Schouten bracket. In coordinates, if $\partial _{i}=\partial
/\partial x^{i}$ the Poisson bi-vector is $\Omega \left( \mathbf{x}\right)
=\Omega ^{jk}\left( \mathbf{x}\right) \partial _{j}\wedge \partial _{k}$ with summation over repeated indices. Then the Jacobi identity reads$$\Omega ^{i[j}\partial _{i}\Omega ^{kl]}=0 \label{e3c}$$where $[jkl]$ denotes the antisymmetrization over three indices. It follows that in three dimensions the Jacobi identity is a single scalar equation. One can exploit the vector calculus and the differential forms in three dimensions to have a more transparent understanding of Hamilton‘s equations as well as the Jacobi identity. Using the isomorphism $$J_{i}=\varepsilon _{ijk}\Omega ^{jk}\qquad i,j,k=1,2,3 \label{e4}$$between skew-symmetric matrices and (pseudo)-vectors defined by the completely antisymmetric Levi-Civita tensor $\varepsilon _{ijk}$ we can write the Hamilton’s equations $\left( \ref{e2}\right) $ in vector form $$\mathbf{v}=\mathbf{J}\times \nabla H \label{e5}$$and in this notation the Jacobi identity $\left( \ref{e4}\right) $ becomes $$\mathbf{J}\cdot \left( \nabla \times \mathbf{J}\right) =0 \label{e6}$$In this form, the Jacobi identity is recognized to be equivalent to the Frobenius integrability condition for the vector $\mathbf{J}$, or equivalently, the condition for the one form $J=J_{i}dx^{i}$ to define a foliation of codimension one in three dimensional space [@tondeur], [reinhart]{}, [@hasan].
A distinguished property of Poisson structures in three dimensions is the invariance of the Jacobi identity under the multiplication of the Poisson vector $\mathbf{J}\left( \mathbf{x}\right) $ by an arbitrary but non-zero factor. More precisely, one can easily show that under the transformation $$\mathbf{J}\left( \mathbf{x}\right) \rightarrow f(\mathbf{x})J\left( \mathbf{x}\right) \label{e6a}$$of Poisson vector the Jacobi identity transforms as$$\mathbf{J}\cdot \left( \nabla \times \mathbf{J}\right) \rightarrow \left( f(\mathbf{x})\right) ^{2}\mathbf{J}\cdot \left( \nabla \times \mathbf{J}\right)
\label{e6b}$$which manifests the invariance property. The identities
$$\mathbf{J}\cdot \mathbf{v}=0,\text{ \ \ \ }\nabla H\cdot \mathbf{v}=0
\label{e11}$$
follows directly from the Hamilton’s equations $\left( \ref{e5}\right) $, the second of which is the expression for the conservation of the Hamiltonian function.
A three dimensional vector $\mathbf{v}\left( \mathbf{x}\right) $ is said to be bi-Hamiltonian if there exist two different compatible Hamiltonian structures. In the notation of equation $\left( \ref{e5}\right) $, this implies$$\mathbf{v}=\mathbf{J}_{1}\times \nabla H_{2}=\mathbf{J}_{2}\times \nabla
H_{1} \label{e11a}$$for the dynamical equations. The compatibility condition for $\mathbf{J}_{1}$ and $\mathbf{J}_{2}$ is defined by the Jacobi identity for the Poisson vector $\mathbf{J}_{1}+c\mathbf{J}_{2}$ for arbitrary constant $c.$ Namely, $\mathbf{J}_{1}$ and $\mathbf{J}_{2}$ are compatible Poisson vectors provided they satisfy$$\mathbf{J}_{1}\cdot \left( \nabla \times \mathbf{J}_{2}\right) +\mathbf{J}_{2}\cdot \left( \nabla \times \mathbf{J}_{1}\right) =0. \label{e11b}$$The invariance properties of the Jacobi identity and the Hamiltonian functions enable one to extend the constant $c$ to be a function of the conserved Hamiltonians. More precisely, the Jacobi identity for the combination $\mathbf{J}_{1}+c\mathbf{J}_{2}$ of Poisson vectors gives
$$(\mathbf{J}_{1}\times \mathbf{J}_{2})\cdot \nabla c=(\mathbf{J}_{1}\cdot
\left( \nabla \times \mathbf{J}_{2}\right) +\mathbf{J}_{2}\cdot \left(
\nabla \times \mathbf{J}_{1}\right) )c \label{e36}$$
which reduces to equation $\left( \ref{e11b}\right) $ whenever $c$ is a constant. This linear equation can always be solvable for the function $c$ resulting in considerable relaxation in the compatibility condition. That means, locally every pair of Poisson vectors can be made compatible.
It follows from the bi-Hamiltonian equations $\left( \ref{e11a}\right) $ and the identities $\left( \ref{e11}\right) $ that $\mathbf{J}_{1}\times \nabla
H_{1}=\mathbf{J}_{2}\times \nabla H_{2}=0$. That is, the Hamiltonian of one structure is the Casimir function of the other. Thus, a three dimensional dynamical system can be defined to be integrable if it is a Hamiltonian system with one Casimir. In this case, the flow can be represented by the intersection of surfaces defined by constant values of the integrals of motion (see [@hasan] and references below for further details and examples).
Jacobi Identity in Serret-Frenet Frame
======================================
Let $\left( \mathbf{t},\mathbf{n},\mathbf{b}\right) $ denote the Serret-Frenet frame associated with a differentiable curve $t\rightarrow
\mathbf{x}(t)$ in some domain of the three dimensional space $\mathbb{R}
^{3}$. Throughout, $\nabla =\left( \partial _{x},\partial _{y},\partial
_{z}\right) $ will denote the usual gradient operator in local Cartesian coordinates. Given a vector field $\mathbf{v}$, the unit tangent vector $\mathbf{t},$ the unit normal $\mathbf{n},$ and the unit bi-normal $\mathbf{b}
$ can be constructed immediately as
$$\begin{array}{ccccc}
\mathbf{t}\left( \mathbf{x}\right) =\frac{\mathbf{v}\left( \mathbf{x}\right)
}{\left\Vert \mathbf{v}\left( \mathbf{x}\right) \right\Vert } & \quad &
\mathbf{n}\left( \mathbf{x}\right) =\frac{\mathbf{t}\times \left( \nabla
\times \mathbf{t}\right) }{\left\Vert \mathbf{t}\times \left( \nabla \times
\mathbf{t}\right) \right\Vert } & \quad & \mathbf{b}\left( \mathbf{x}\right)
=\mathbf{t}\left( \mathbf{x}\right) \times \mathbf{n}\left( \mathbf{x}\right)\end{array}
\label{e15}$$
and they form a right-handed orthonormal frame except those vector fields $\mathbf{v}$ satisfying the condition imposed by $$\mathbf{t}\times \left( \nabla \times \mathbf{t}\right) =0. \label{e15a}$$It can be deduced from the vector identity $2\mathbf{t}\times (\nabla \times
\mathbf{t)=\nabla (t\cdot t)+2t\cdot \nabla t=}$ $\mathbf{2t\cdot \nabla t}$ that this condition excludes essentially the flows with constant unit tangent and the points $\mathbf{x}$ at which the unit normal $\mathbf{n}$ (hence the bi-normal $\mathbf{b}$) have zeros. That is, the cases one cannot have a Serret-Frenet frame. To avoid this we may assume that $$\left( \nabla \times \mathbf{t}\right) \neq \lambda \left( \mathbf{x}\right)
\mathbf{t} \label{e15b}$$for arbitrary nonzero function $\lambda \left( \mathbf{x}\right) $. That is, we exclude the dynamical systems whose unit tangent vectors are the eigenvectors of the curl operator [@moses], [@benn].
We introduce the directional derivatives along the triad $\left( \mathbf{t},\mathbf{n},\mathbf{b}\right) $ as $$\begin{array}{ccccc}
\partial _{s}=\mathbf{t}\cdot \nabla & \quad & \partial _{n}=\mathbf{n}\cdot
\nabla & \quad & \partial _{b}=\mathbf{b}\cdot \nabla\end{array}
\label{e16}$$so that the variables $(s,n,b)$ are the coordinates associated with the Serret-Frenet frame. By inverting equations $\left( \ref{e16}\right) $ we get the expression
$$\nabla =\mathbf{t}\partial _{s}+\mathbf{n}\partial _{n}+\mathbf{b}\partial
_{b} \label{e17}$$
for the Cartesian gradient in Serret-Frenet frame. Since $\mathbf{t}\times
\nabla \times \mathbf{t=t\cdot \nabla t=\partial }_{s}\mathbf{t}$ the definition of the normal vector reduces to one of the Serret-Frenet equations justifying the name for the moving frame introduced [@andrade].
It follows from the identity in equation $\left( \ref{e11}\right) $ that the Poisson vector $\mathbf{J}$ has no component along the unit tangent vector $\mathbf{t}.$Hence, we set $$\mathbf{J}=\alpha \mathbf{n}+\beta \mathbf{b} \label{e26}$$for unknown functions $\alpha \left( \mathbf{x}\right) $ and $\beta \left(
\mathbf{x}\right) $ satisfying $\alpha ^{2}+\beta ^{2}\neq 0$. Using derivatives in Cartesian variables we find the expression$$\begin{array}{lll}
\mathbf{J}\cdot \left( \nabla \times \mathbf{J}\right) & = & \left( \beta
\nabla \alpha -\alpha \nabla \beta \right) \cdot \mathbf{t}+\alpha ^{2}\mathbf{n}\cdot \left( \nabla \times \mathbf{n}\right) +\beta ^{2}\mathbf{b}\cdot \left( \nabla \times \mathbf{b}\right) \\
& & \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }+\alpha \beta
\left( \mathbf{n}\cdot \left( \nabla \times \mathbf{b}\right) +\mathbf{b}\cdot \left( \nabla \times \mathbf{n}\right) \right)\end{array}
\label{e28}$$for the Jacobi identity. Assuming $\alpha \neq 0$ and defining the function $\mu =\beta /\alpha $ the Jacobi identity for $\alpha (\mathbf{n}+\mu \mathbf{b)}$ gives $$\mathbf{t\cdot }\nabla \mu =\mathbf{n\cdot }\nabla \times \mathbf{n}+\mu
\left( \mathbf{n\cdot }\nabla \times \mathbf{b}+\mathbf{b\cdot }\nabla
\times \mathbf{n}\right) +\mu ^{2}\mathbf{b\cdot }\nabla \times \mathbf{b}
\label{e30}$$which is an equation involving only the unknown function $\mu $. Obviously, this simplification is a manifestation of the invariance of the Jacobi identity under the multiplication of $\ \mathbf{J}$ by an arbitrary but non-zero function. Similarly, we may assume $\beta \neq 0$ and define $\eta
=-1/\mu =-\alpha /\beta $ for which the Jacobi identity for the combination $\beta (\mathbf{b}-\eta \mathbf{n)}$ becomes$$\mathbf{t\cdot }\nabla \eta =\mathbf{b\cdot }\nabla \times \mathbf{b}-\eta
\left( \mathbf{n\cdot }\nabla \times \mathbf{b}+\mathbf{b\cdot }\nabla
\times \mathbf{n}\right) +\eta ^{2}\mathbf{n\cdot }\nabla \times \mathbf{n.}
\label{e32}$$
To this end, we define the scalar quantities measuring the non-integrability (in the sense of Frobenius) of each unit vector in the Serret-Frenet triad$$\Omega _{\mathbf{t}}=\mathbf{t\cdot }\left( \nabla \times \mathbf{t}\right) ,\text{ \ }\ \Omega _{\mathbf{n}}=\mathbf{n\cdot }\left( \nabla \times
\mathbf{n}\right) ,\text{ \ \ }\Omega _{\mathbf{b}}=\mathbf{b\cdot }\left(
\nabla \times \mathbf{b}\right) \label{e33}$$ the first of which is necessarily not equal to any eigenvalue of the curl operator by the assumption \[e15b\]. The integration of these quantities over the three space gives the so called helicities [@batch], [arnold]{} associated with the triad. We also introduce the sum
$$\Omega _{\mathbf{nb}}=\mathbf{n\cdot }\nabla \times \mathbf{b}+\mathbf{b\cdot }\nabla \times \mathbf{n} \label{e33b}$$
of the cross-helicities for the normal and bi-normal vectors.
Let $\mathbf{J}=\alpha \left( \mathbf{n}+\mu \mathbf{b}\right) $ (or $\mathbf{J}=\beta \left( \mathbf{b}-\eta \mathbf{n}\right) $). Then, the Jacobi identity for $\mathbf{J}$ in moving coordinates is given by the Riccati equation $$\begin{aligned}
\partial _{s}\mu &=&\Omega _{\mathbf{n}}+\mu \Omega _{\mathbf{nb}}+\mu
^{2}\Omega _{\mathbf{b}} \label{e33ii} \\
\text{(or\ \ \ \ \ \ }\partial _{s}\eta &=&\Omega _{\mathbf{b}}-\eta \Omega
_{\mathbf{nb}}+\eta ^{2}\Omega _{\mathbf{n}}\text{ \ \ \ \ with \ \ }\mu
=-1/\eta \text{)}\end{aligned}$$
Thus, in the moving coordinates, the Jacobi identity or, equivalently, the existence of Poisson structure is expressible as a differential equation in arclength coordinates only. It may be interesting to note that the equation named after Jacopo Francesco Riccati originated from his investigations of curves whose radii of curvature depend only on a single variable [@ince]. The disappearence of the moving coordinates $n$ and $b$ from the Jacobi identity will become clear in the last section. In fact, they correspond to local conserved quantities and, as discussed in [@hasan] for the globally integrable cases, may appear arbitrarily in the Poisson vectors.
The Riccati equation $\left( \ref{e33ii}\right) $ is equivalent to the linear second order equation $$\begin{aligned}
\partial _{ss}^{2}u-\left( \frac{\partial _{s}\Omega _{\mathbf{b}}}{\Omega _{\mathbf{b}}}+\Omega _{\mathbf{nb}}\right) \partial _{s}u+\Omega _{\mathbf{n}}\Omega _{\mathbf{b}}u &=&0\text{\ \ \ if }\ \Omega _{\mathbf{b}}\neq 0
\label{e34} \\
\text{(or \ }\partial _{ss}^{2}v-\left( \frac{\partial _{s}\Omega _{\mathbf{n}}}{\Omega _{\mathbf{n}}}-\Omega _{\mathbf{nb}}\right) \partial _{s}v+\Omega
_{\mathbf{n}}\Omega _{\mathbf{b}}v &=&0\text{ }\ \text{\ if }\Omega _{\mathbf{n}}\neq 0\text{\ )}\end{aligned}$$with the solutions being related by$$\mu =-\frac{\partial _{s}\ln u}{\Omega _{\mathbf{b}}}\quad \text{\ if }\
\Omega _{\mathbf{b}}\neq 0\ \ \ \ \ \ \ \ \ \ \text{(or}\ \ \ \ \eta =-\frac{\partial _{s}\ln v}{\Omega _{\mathbf{n}}}\quad \text{\ if }\ \Omega _{\mathbf{n}}\neq 0\text{).} \label{e35}$$For the Poisson vectors of the above proposition at least one of the equations in $\left( \ref{e34}\right) $ possesses two linearly independent solutions.
The emergence of the Riccati equation as the Jacobi identity may be interpreted as a relation between nonlinearity and superposition. The Jacobi identity as well as the compatibility condition for Poisson vectors are nonlinear restrictions on some linear combinations of basis vectors. Both are scalar equations in three dimensions. The Riccati equation, on the other hand, is known to be the only scalar equation admitting a nonlinear superposition principle [@bountis].
Compatibility conditions
========================
To construct the bi-Hamiltonian structure, the Poisson vectors constructed from the linearly independent solutions of the Riccati equation must be compatible. That means, their linear combinations must also satisfy the Jacobi identity. Although, the multiplicative factors are left arbitrary in the construction of Poisson vectors they become important in the compatibility condition. Apart from the general result discussed in section two, we shall restrict ourselves to the case where $c$ is a constant in the combination $\mathbf{J}_{1}+c\mathbf{J}_{2}$. We shall show that the compatibility follows from the Hamilton‘s equations.
First, we have the following result obtained by direct computation from the compatibility condition
Let $\alpha _{i}$ and $\mu _{i}$ be non zero and be different functions for $i=1,2$. For $\Omega _{\mathbf{b}}\neq 0$, the Poisson vectors $\mathbf{J}_{i}=\alpha _{i}(\mathbf{n+}\mu _{i}\mathbf{b)}$ are compatible if$$\partial _{s}\ln \frac{\alpha _{2}}{\alpha _{1}}=(\mu _{1}-\mu _{2})\Omega _{\mathbf{b}}. \label{e37}$$
In obtaining Eq \[e37\] we used the Riccati equations to eliminate derivatives of functions $\mu _{1}$ and $\mu _{2}$. Next result shows that above equation is always satisfied, via Hamilton‘s equations, by Poisson vectors constructed from the solutions of the Riccati equation.
Let $\mathbf{J}=\alpha (\mathbf{n+\mu b)}$ and $H$ define a Poisson structure for the dynamical system associated with $\mathbf{v}$. Then$$\partial _{s}\ln \frac{\parallel \mathbf{v\parallel }}{\alpha }-\mathbf{n}\cdot \nabla \times \mathbf{b}=\mu \Omega _{\mathbf{b}} \label{e38}$$
With the Poisson vector in the assumptions we write the dynamical system as $\mathbf{t}=\parallel \mathbf{v\parallel }^{-1}\mathbf{J}\times \nabla
H=\parallel \mathbf{v\parallel }^{-1}\alpha (\mathbf{n+}\mu \mathbf{b})\times \nabla H$. Taking cross products with $\mathbf{n}$ and $\mathbf{b}$ we get$$\mathbf{b=}\parallel \mathbf{v\parallel }^{-1}(\alpha \nabla H-\mathbf{J(n\cdot \nabla }H\mathbf{))}\text{ \ \ \ \ \ \ \ \ }\mathbf{n=-}\parallel
\mathbf{v\parallel }^{-1}(\alpha \mu \nabla H-\mathbf{J(b\cdot \nabla }H\mathbf{))} \label{e39}$$from which we obtain $\nabla H=\alpha ^{=2}(1+\mu ^{2})^{=1}(\mathbf{J(J\cdot }\nabla H)+\mathbf{J}^{\perp }\parallel \mathbf{v\parallel }).$Here, we define $\mathbf{J}^{\perp }=\alpha (\mathbf{b-}\mu \mathbf{n)}$ for convenience. The integrability condition $\nabla \times \nabla H=0$ for the Hamiltonian function results, after taking dot product with $\mathbf{J,}$ in$$\partial _{s}\ln \frac{\parallel \mathbf{v\parallel }}{\alpha ^{2}(1+\mu
^{2})}=\frac{\mathbf{J}\cdot \nabla \times \mathbf{J}^{\perp }}{\alpha
^{2}(1+\mu ^{2})^{2}} \label{e40}$$where we used $\mathbf{J}\cdot \mathbf{J}^{\perp }=0$. The manipulations leading to the result is now straightforward and requires only the use of Jacobi identity for the derivative of $\mu $.
The proof of compatibility becomes obvious once we write equation \[e38\] for each Poisson vector and subtract them. Similar results may be obtained for the Poisson vectors in the form $\mathbf{J}=\beta \left( \mathbf{b}-\eta
\mathbf{n}\right) $ by assuming $\Omega _{\mathbf{n}}\neq 0$ and using the Riccati equation for the variable $\eta $.
We shall analyse the case $\Omega _{\mathbf{b}}=0$. This is the condition for the unit vector $\mathbf{b}$ to satisfy the Jacobi identity. On the other hand, $\Omega _{\mathbf{b}}=0$ reduces the Riccati equation to a linear first order equation resulting in one linearly independent solution for the construction of a Poisson vector of the form $\mathbf{J}=\alpha (\mathbf{n+}\mu \mathbf{b)}$.
The compatibility condition for Poisson vectors $\mathbf{J}=\alpha (\mathbf{n+}\mu \mathbf{b)}$ and $\mathbf{b}$ is $\ \partial _{s}\alpha +\Omega _{\mathbf{nb}}=0$. This is satisfied by the Hamilton‘s equations.
The equation can easily be obtained from the compatibility condition. Equation \[e38\] with $\Omega _{\mathbf{b}}=0$ holds for the Hamiltonian structure for $\mathbf{J}$. For the Hamiltonian structure with the Poisson vector $\mathbf{b}$, we have $\mathbf{t}=\parallel \mathbf{v\parallel }^{-1}\mathbf{b}\times \nabla H$ for some Hamiltonian function $H$. Taking cross-product with $\mathbf{b}$, solving for $\nabla H$ and taking the dot product of the equation resulting from the integrability condition $\nabla
\times \nabla H=0$, we find $\partial _{s}\mathbf{\ln }\parallel \mathbf{v\parallel =-b}\cdot \nabla \times \mathbf{n}$ which yields the result.
Thus, the Poisson vectors obtained from solutions of Riccati equations are always compatible.
Bi-Hamiltonian structure
========================
We shall combine the results of the previous two sections on construction of Poisson vectors and their compatibility to present the main result. This will include the remaining case where the Poisson vectors are defined by the normal and bi-normal unit vectors of the Serret-Frenet triad. In connection with this particular structure we shall first relate the present work to the existing examples of bi-Hamiltonian dynamical systems in the literature and then present the local form of the Nambu mechanics.
Every three dimensional dynamical system possesses two compatible Poisson vectors.
If both $\Omega _{\mathbf{n}}$ and $\Omega _{\mathbf{b}}$ are non-zero, then any of the two Riccati equations which produce the same result by construction, give two Poisson structures coming from the linearly independent solutions of the corresponding second order equation. If $\Omega
_{\mathbf{b}}=0$ (or $\Omega _{\mathbf{n}}=0$) then the first (the second) Riccati equation becomes linear with one linearly independent solution. Note that the sum $\Omega _{\mathbf{nb}}$ of the cross-helicities of the normal vectors involves as an integrating factor in the integration for this Poisson structure. The other Poisson structure is defined by the bi-normal vector $\mathbf{b}$ (or the normal vector $\mathbf{n}$) since $\Omega _{\mathbf{b}}=0$ (or $\Omega _{\mathbf{n}}=0$) is the Jacobi identity for this. The compatibility conditions for these cases are shown to be satisfied via Hamilton‘s equations in the previous section. If, we have both $\Omega _{\mathbf{n}}\mathbf{=}0$ and $\Omega _{\mathbf{b}}=0$ then $\mathbf{n}$ and $\mathbf{b}$ satisfies Jacobi identity and they become Poisson vectors we sought. The compatibility condition is $\Omega _{\mathbf{nb}}=0$. This implies that the function $\mu $ (or $\eta $) must be a non-zero constant in the vector $\mathbf{J}$ for which the Riccati equation is now trivial. The Hamilton‘s equations with the Poisson vectors $\mathbf{n}$ and $\mathbf{b}$ implies $\partial _{s}\mathbf{\ln }\parallel \mathbf{v\parallel =n}\cdot
\nabla \times \mathbf{b}$ and $\partial _{s}\mathbf{\ln }\parallel \mathbf{v\parallel =-b}\cdot \nabla \times \mathbf{n}$, respectively. Eliminating the vector $\mathbf{v}$, we obtain the compatibility condition $\Omega _{\mathbf{nb}}=0$.
The last case of the theorem gives clues to relate the present work to the existing bi-Hamiltonian dynamical systems in the literature, in particular, to the Nambu mechanics. Recall that the three dimensional dynamical systems admitting bi-Hamiltonian structure with $(\mathbf{J}_{1},H_{2})$ and $(\mathbf{J}_{2},H_{1})$ are of the form of equation \[e11a\] with $\mathbf{J}_{1}$ and $\mathbf{J}_{2}$ satisfying Frobenius integrability conditions as Jacobi identities. In case that these vectors are globally integrable they are related to the conserved Hamiltonians by $\mathbf{J}_{i}=\varphi
_{i}\nabla H_{i}$ $i=1,2$ for some arbitrary non-zero functions $\varphi
_{i} $. In this case, the dynamical system has the form $\mathbf{v=\psi }\nabla H_{1}\mathbf{\times }\nabla H_{2}$ first studied by Nambu in [nambu]{}. All explicitly constructed bi-Hamiltonian systems in three dimension has this form [@nutku], [@nutk], [@hasan], [@aygur]. The flow lines coincide with the intersections of the surfaces defined by contant values of the Hamiltonians. Any constant appearing in this bi-Hamiltonian picture can be taken as arbitrary functions of $H_{1}$ and $H_{2}$.
Our aim is to find the local version of this bi-Hamiltonian representation in three dimensions. This we shall do by first proving the more general result that the local existence of a Hamiltonian function of the proposed form is equivalent to the existence of a Poisson vector (see also [@bben]).
There exists non-zero functions $H$ and $\varphi $ with $\mathbf{J=}\varphi \nabla H$ whenever $\mathbf{J=n+\mu b}$ satisfies the Jacobi identity.
The condition $\nabla \times \nabla H\equiv 0$ together with the form of $\mathbf{J}$ gives $\nabla \times (\mathbf{n+\mu b)=a}\times (\mathbf{n+\mu b)}$ where $\mathbf{a}=\nabla \ln \sqrt{1+\mu ^{2}}/\varphi $. Taking dot products with the unit vectors of the Serret-Frenet triad we obtain three equations one of which is algebraic and the other two expressing the $s$ and $n$ derivatives of the function $\mu $. All three contains terms involving the vector $\mathbf{a}$. Eliminating this term for the $s$ derivative results in the Riccati equation \[e33ii\] for the Poisson vector $\mathbf{J}$. The remaining two equations determine the function $\varphi $ and the dependence of $\mu $ on the variable $n$.
Thus, finding two linearly independent solutions of the Ricatti equation completely determines the local bi-Hamiltonian structure.
In the local picture of the present work the normal coordinates $n$ and $b$ represents the local conserved quantities and they appear arbitrarily in the Poisson structures. The normal vectors $\mathbf{n}$ and $\mathbf{b}$ defining the bi-Hamiltonian structure in the last case thus corresponds, in the globally integrable case, to the gradients of Hamiltonian functions defining Poisson vectors. Therefore, the manifestly bi-Hamiltonian equation $\mathbf{t=n\times b}$ corresponding to the last case of the above proposition is the local version of the Nambu structure.
[99]{} P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, New York, 1993.
Y. Nambu, Generalized Hamiltonian Dynamics, Phys. Rev. D 7 (1973) 2405-2412
M. Kus, Integrals of motion for the Lorenz system, J. Phys. A: Math. Gen. 16 (1983) L689-L691.
J. M. Strelcyn and S. Wojciechowski, A method of finding integrals for three-dimensional dynamical systems, Phys. Lett. A133 (1988) 207-212.
B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J. M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary
differential equations in $\mathbb{R}
^{3}$: The Lotka-Volterra system, Physica A 163 (1990) 683-722.
Y. Nutku, Hamiltonian structure of the Lotka-Volterra equations, Phys. Lett. A, 145 (1990) 27-28.
Y. Nutku, Bi-Hamiltonian structure of the Kermack-McKendrick model for epidemics, J. Phys. A: Math. Gen. 23 (1990) L1145-L1146.
H. J. Giacomini, C. E. Repetto and O. P. Zandron, Integrals of motion for three-dimensional non-Hamiltonian dynamical systems, J. Phys. A: Math. Gen. 24 (1991) 4567-4574.
S. A. Hojman, Quantum algebras in classical mechanics, J. Phys. A: Math. Gen., 24 (1991) L249-L254
H. Gūmral, PhD Thesis, Bilkent University, Ankara, 1991.
H. Gūmral and Y. Nutku, Poisson structure of dynamical systems with three degrees of freedom, J.Math.Phys. 34 (1993) 5691-5723.
B. Hernández-Bermejo and V. Fairén, Separation of variables in the Jacobi identities, Phys. Lett. A 271 (2000) 258–263.
B. Hernández-Bermejo, New solutions of the Jacobi equations for three-dimensional Poisson structures, J. Math. Phys. 42 (2001) 4984-4996.
B. Hernández-Bermejo, Characterization and global analysis of a family of Poisson structures, Phys. Lett. A 355 (2006) 98–103
B. Hernández-Bermejo, New solution family of the Jacobi equations: Characterization, invariants, and global Darboux analysis, J. Math. Phys. 48 (2007) 022903-022914.
F. Haas and J. Goedert, On the Generalized Hamiltonian Structure of 3D Dynamical Systems, Phys. Lett. A 199 (1995) 173-179.
A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983) 523-557.
G. R. W. Quispel and H. W. Capel, Solving ODEs numerically while preserving a first integral, Phys. Lett. A218 (1996) 223-228.
B. Karasōzen, Poisson integrators, Mathematical and Computer Modelling, 40 (2004) 1225-1244.
P. Tondeur, Foliations on Riemannian Manifolds, Springer, Berlin, 1988.
B. L. Reinhart, Differential Geometry of Foliations, Springer, Berlin, 1983.
H. E. Moses, Eigenfunctions of the Curl Operator, Rotationally Invariant Helmholtz Theorem, and Applications to Electromagnetic Theory and Fluid Mechanics, SIAM J. Appl. Math. 21 (1971) 114-144.
I. M. Benn and J. Kress, Force-free fields from Hertz potentials, J. Phys. A: Math. Gen. 29 (1996) 6295-6304.
L.C. Garcia de Andrade, Topology of magnetic helicity in torsioned filaments in Hall plasmas, astro-ph/07104594
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, 1998.
E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.
T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A25 (1982) 1257-1264.
B. Hernández-Bermejo, A constant of motion in 3D implies a local generalized Hamiltonian structure, Phys. Lett. A234 (1997) 35-40.
A. Ay, M. Gũrses and K. Zheltukhin, Hamiltonian Equations in $\mathbb{R}$${{}^3}$, J. Math. Phys. 44 (2003) 5688-5705.
|
-2.5cm -1.5cm =15.5cm
[Radiative corrections to $b\to c\tau\bar\nu_\tau$]{}
1.0cm
[**Andrzej Czarnecki**]{}\
0.3cm [*Institut für Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany*]{}\
1.0cm [**Marek Jeżabek**]{}\
0.3cm [*Institute of Nuclear Physics, Kawiory 26a, PL-30055 Cracow, Poland*]{}\
[*Institut für Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany*]{}\
0.7cm [and]{}\
0.7cm [**Johann H. Kühn**]{}\
0.3cm [*Institut für Theoretische Teilchenphysik, D-76128 Karlsruhe, Germany*]{}
1.5cm
[Abstract]{}
Analytical calculation is presented of the QCD radiative corrections to the rate of the process $b\to c\tau\bar\nu_\tau$ and to the $\tau$ lepton longitudinal polarization in $\tau\bar\nu_\tau$ rest frame. The results are given in the form of one dimensional infrared finite integrals over the invariant mass of the leptons. We argue that this form may be optimal for phenomelogical applications due to a possible breakdown of semilocal hadron-parton duality in decays of heavy flavours.
The semileptonic decay rate of $B$ mesons is one of the key ingredients in the determination of weak mixing angles. Transitions of the $b$ quark to the charmed as well as to the up quark have been analysed in great detail, exploiting either the inclusive decay rate or exclusive channels. The analysis of the inclusive rate is, however, affected by the uncertainty in the $b$ mass and by bound state corrections. This problem is only partly circumvented by fixing $m_b-m_c$ through the difference of bottom and charmed meson masses and by relating the bound state effects to phenomenological constants that can be determined from other observables in the context of the heavy quark effective theory (HQET). The decay rate is, furthermore, affected by perturbative QCD corrections, which have been calculated analytically up to order $\alpha_s$ for arbitrary $b$ and $c$ masses and massless leptons [@CM; @JK; @Nir]. Numerical results for perturbative QCD corrections to the partial decay rate $b\to \tau\bar\nu X$ have been recently obtained in Ref.[@FLNN]. The bound state corrections to this decay chanel up to order ${1/ m_b^2}$ are also known [@FLNN; @Koyrakh; @BKPS] in the context of the HQET. The comparison between decays into light ($\mu$, $e$) and heavy ($\tau$) leptons may furthermore help to test the theoretical approach and allow to fix some of the free parameters. Including $b\to u$ transitions, four kinematically different leptonic decay modes are thus available for the comparison.
In this paper analytical results for the decay rate $b\to c\tau\bar\nu_\tau$ are presented in the form of an one dimensional integral over the invariant mass squared ${\rm w}^2$ of the leptonic system. This formulation allows, at least in principle, the separation of the region of relatively small ${\rm w}^2$, where the inclusive parton model description based on local parton-hadron duality should work, from the region of large ${\rm w}^2$ where only one or few resonances are produced and the duality between the parton and hadron description may be doubtful[^1]. In the region of large ${\rm w}^2$ i.e. close to the Shifman-Voloshin limit[@VS] ${\rm w}^2= {\rm w}^2_{max}$ the theoretical description of lepton spectra based on summation over exclusive channels is particularly simple and reliable. The HQET approach for exclusive decays[@IW; @Neubert] on the other hand becomes quite involved if not impractical in the region of small ${\rm w}^2$ which is dominated by multiparticle final states.
The calculation of the lowest order rate as well as corrections can be related in a straightforward way to the corresponding calculations for the decay into a virtual $W$ boson with the subsequent integration over the mass of the $l\bar\nu$ system. The differential decay rate is proportional to $${\cal H}^{\alpha\beta}\,{\cal L}_{\alpha\beta}\,
{\rm dPS}(b\to c\tau\bar\nu)$$ ${\cal H}_{\alpha\beta}$ depends on quark and gluon fields and $$\begin{aligned}
{\cal L}_{\alpha\beta}(\tau ;\nu) &\sim&
\sum_{s}
\left[\,\bar u_\tau\gamma_\alpha(1
-\gamma_5) v_\nu\,\right]\,
\left[\,\bar u_\tau\gamma_\beta (1-\gamma_5)v_\nu\,\right]^\dagger
\nonumber\\
&\sim& \nu_\alpha \tau_\beta +
\tau_\alpha \nu_\beta - \nu\cdot\tau g_{\alpha\beta}
- {\rm i}\varepsilon_{\alpha\beta\gamma\delta}\nu^\gamma\tau^\delta
\label{eq:Lab}\end{aligned}$$ where $\tau^\alpha$ and $\nu^\alpha$ are the four-momenta of $\tau$ and $\bar\nu_\tau$. The phase space for the decay of $b$ into $c\tau\bar\nu$ is, in the standard way, decomposed into a sequence of two-particle final states $${\rm dPS}(b\to c\tau\bar\nu)\, \sim\,
{\rm d}{\rm w}^2\, {\rm dPS}(b\to c{\rm w})\,
{\rm dPS}({\rm w}\to \tau\bar\nu)$$ where ${\rm w}^\alpha=\tau^\alpha +\nu^\alpha$. Then, it is straightforward to show that $$\begin{aligned}
\lefteqn{
\int {\rm dPS}({\rm w}\to \tau\bar\nu){\cal L}_{\alpha\beta}\, \sim
{\cal A}\left(m_\tau^2/{{\rm w}}^2\right)\,T^{(0)}_{\alpha\beta}\,+
{\cal B}\left(m_\tau^2/{{\rm w}}^2\right)\,T^{(1)}_{\alpha\beta} =
}
\nonumber\\
&& {\textstyle
\left(\, 1 - {m_\tau^2/ {{\rm w}}^2}\, \right)^2
\left[\,\left(\,1 + {2m_\tau^2/ {{\rm w}}^2}\,\right)
{\rm w}_\alpha {\rm w}_\beta
- \left(\,{{\rm w}}^2+ {1\over 2} m_\tau^2\, \right)\,g_{\alpha\beta}
\right]
}
\label{eq:intL}\end{aligned}$$ where $$\begin{aligned}
T^{(0)}_{\alpha\beta} &=&
{\rm w}_\alpha {\rm w}_\beta
\nonumber\\
T^{(1)}_{\alpha\beta} &=&
{\rm w}_\alpha {\rm w}_\beta - {\rm w}^2 g_{\alpha\beta}\end{aligned}$$ It follows that the decay rate can be split accordingly into two incoherent terms which are related to weak decays of a heavy quark $Q$ into another quark $q$ and a real spin one or a spin zero boson. The relative weight of spin one versus spin zero contributions is governed by their respective spectral functions and it can be derived from eq.(\[eq:intL\]). For massless leptons ${\cal A}(0)=0$ and ${\cal B}(0)=1$, and therefore only the spin one (transversal) component ($\sim T^{(1)}_{\alpha\beta}$) contributes. The result is given in [@JK]. For fixed ${\rm w}^2$ this contribution to the rate can be obtained from the formula for $t\to bW$, the top quark decay into $b$ quark and a real $W$ boson [@JK][^2]. Multiplying this formula by ${\cal B}\left(m_\tau^2/{\rm w}^2\right)$ one obtains the contribution of the spin one component for $m_\tau\ne 0$. The other (longitudinal) contribution ($\sim T^{(1)}_{\alpha\beta}$) can be in an analogous way related to the (yet unobserved) decay $t\to bH^+$ where $H^+$ denotes a charged Higgs boson. Let $Q$ and $q$ denote the four-momenta of the quarks in the initial and in the final state. For a two-body decay mode the momentum of the $W$ boson is $W= Q-q$. In Born approximation $$\begin{aligned}
W_\mu \bar u(q)\,\gamma^\mu(1-\gamma_5)\, u(Q) &=&
\bar u(q)\,[\,(\hat Q - \hat q)(1-\gamma_5)\,]\, u(Q)
\nonumber\\
&=& \bar u(q)\,[\,(M-m)+(M+m)\gamma_5\,]\, u(Q)
\label{eq:redu}\end{aligned}$$ where the equations of motion $$\begin{aligned}
(\hat Q -M)\,u(Q)=0
\qquad\quad {\rm and} \qquad\quad
\bar u(q)\,(\hat q-m)=0\end{aligned}$$ have been used. The last line in (\[eq:redu\]) can be interpreted as the amplitude of the decay $Q\to qH$ where $H$ is a spin zero boson whose coupling to the weak quark current is given by $$g = (M-m) + (M-m)\gamma_5$$ Although not applicable for individual Feynman diagrams, the same relation holds true for the longitudinal contribution ($\sim W_\mu$) to the decay amplitude when ${\cal O}(\alpha_s)$ QCD corrections are included [@CD1]. Therefore this contribution to the rate $b\to c\tau\bar\nu_\tau$ can be extracted from a formula describing $t\to bH^+$ which has been given in [@CD]; cf. Model I therein.
Let us define now the following dimensionless quantities $$\textstyle{
\rho= {m_c^2/ m_b^2}\qquad\quad
\eta = {m_\tau^2/ m_b^2}\qquad\quad
{\rm and} \qquad\quad
t = {{\rm w}^2/m_b^2 } }$$ and the kinematic functions $$\begin{aligned}
p_0(t) &=& (1-t+\rho)/2
\nonumber\\
p_3(t) &=& \sqrt{p_0^2 - \rho}
\nonumber\\
p_\pm(t) &=& p_0 \pm p_3 = 1 - w_\mp(t)
\nonumber\\
Y_p(t) &=& {\textstyle{1\over 2}}\ln\left(p_+/ p_-\right)
=\ln\left(p_+/\sqrt{\rho}\right)
\nonumber\\
Y_w(t) &=& {\textstyle{1\over 2}}\ln\left(w_+/w_-\right)
=\ln\left(w_+/\sqrt{t}\right)
\label{eq:rapid}\end{aligned}$$ where ${\rm w}^\alpha=\tau^\alpha +\nu^\alpha$ and ${\rm w}^2$ denotes the effective mass of $\tau\bar\nu_\tau$. The partial rate of the decay $b\to c\tau\bar\nu_\tau$ is given by $$\Gamma(b\to c\tau\bar\nu_\tau)\:=\:
\int^{(1-\sqrt{\rho})^2}_\eta
{{\rm d}\Gamma\over {\rm d}t}\,{\rm d}t
\label{eq:Gamtot}$$ with the differential rate $$\begin{aligned}
\lefteqn{
{{\rm d}\Gamma\over {\rm d}t} = \Gamma_{bc}\,
\left( 1-{\eta \over t}\right)^2 \, \left\{
\left( 1+{\eta \over 2t}\right)
\left[ {\cal F}_0(t) - {2\alpha_s\over 3 \pi} {\cal F}_1(t)\right]
+ {3\eta \over 2t}
\left[ {\cal F}_0^s(t) - {2\alpha_s\over 3 \pi} {\cal F}_1^s(t)\right]
\right\} }
\nonumber\\
\label{main} \\
\lefteqn{
\Gamma_{bc} =
{G_F^2 m_b^5 |V_{cb}|^2\over 192 \pi^3}
}\end{aligned}$$ $$\begin{aligned}
{\cal F}_0(t) &=& 4 p_3\,
\left[ \, (1-\rho)^2 + t(1+\rho) - 2 t^2\, \right]
\\
{\cal F}_0^s(t)
&=& 4p_3\,\left[\, (1-\rho)^2 -t ( 1+\rho)\, \right]
\\
{\cal F}_1(t)&=& {\cal A}_1 \Psi + {\cal A}_2 Y_w
+ {\cal A}_3 Y_p + {\cal A}_4 p_3 \ln\rho + {\cal A}_5 p_3
\\
{\cal F}_1^s(t)&=& {\cal B}_1 \Psi + {\cal B}_2 Y_w
+ {\cal B}_3 Y_p + {\cal B}_4 p_3 \ln\rho + {\cal B}_5 p_3
\label{massive}\end{aligned}$$ $$\begin{aligned}
\Psi &=& 8 \ln (2 p_3) -2\ln t\, +\,
\left[ 2{\rm Li}_2 (w_-) -2{\rm Li}_2 (w_+)
+4 {\rm Li}_2 ({2p_3/ p_+}) \right.
\nonumber \\ && \left. \hskip65pt
-4Y_p \ln({2p_3/ p_+})
-\ln p_- \ln w_+ + \ln p_+ \ln w_- \right]\, 2p_0/p_3\end{aligned}$$ $$\begin{aligned}
{\cal A}_1 &=& {\cal F}_0(t)
\nonumber\\
{\cal A}_2 &=&
- 8 (1-\rho) \left[ 1 +t -4 t^2-
\rho (2-t) +\rho^2 \right]
\nonumber\\
{\cal A}_3 &=&
- 2 \left[ 3 + 6 t -21 t^2 + 12 t^3
-\rho (1+12t+5t^2) + \rho^2(11+2t)- \rho^3 \right]
\nonumber\\
{\cal A}_4 &=&
- 6 \left[ 1 + 3 t - 4 t^2
-\rho (4-t) + 3 \rho^2 \right]
\nonumber\\
{\cal A}_5 &=&
- 2 \left[ 5+9t-6t^2 -\rho( 22 - 9t) + 5 \rho^2 \right]
\\
{\cal B}_1 &=& {\cal F}^s_0(t)
\nonumber\\
{\cal B}_2 &=& -8 (1-\rho)
\left[(1-\rho)^2-t (1+\rho)\right]
\nonumber\\
{\cal B}_3 &=& - 4(1 - \rho)^4/t
-2 (-1 + 3 \rho + 15 \rho^2 - 5 \rho^3)
+ 8 (1 + \rho) t
- 6 (1 + \rho) t^2
\nonumber\\
{\cal B}_4 &=& -4 (1-\rho)^3 /t
-2 (1-\rho) (1-11 \rho)
+6 (1 + 3 \rho) t
\nonumber\\
{\cal B}_5 &=& -6 (1-3 \rho) (3-\rho)
+18 t (1+\rho)\end{aligned}$$ The integral in eq.(\[eq:Gamtot\]) can be easily performed for the Born contribution. It reads:\
$$\begin{aligned}
\lefteqn{
\Gamma_0(b\to c\tau\bar\nu_\tau) =
\Gamma_{bc}\,
\left\{\,
24\,\left[\,\eta^2(1-\rho^2)\,{\cal Y}_w
+ \rho^2(1-\eta^2)\,{\cal Y}_p\,\right]\, +\,
\right. }
\nonumber\\ &&
\left.
2{\cal P}_3\,\left(1-7\eta-7\eta^2+\eta^3-7\rho
+12\eta\rho -7\eta^2\rho
-7\rho^2 -7\eta\rho^2+\rho^3 \right)
\,\right\}
\label{eq:Gam0tot}\end{aligned}$$ where $${\cal P}_3 = p_3(\eta),\qquad\quad
{\cal Y}_p = Y_p(\eta)\qquad\quad {\rm and} \qquad\quad
{\cal Y}_w = Y_w(\eta).$$ In principle the first order QCD correction can be also expressed in terms of polylogarithms. In particular for $\eta=0$ the formula (12) of [@Nir] is obtained. However, the complete result is lenghty. From the practical point of view it is much simpler to evaluate the integral in eq.(\[eq:Gamtot\]) numerically. Moreover, as it has been explained, for $t$ close to the Shifman-Voloshin limit the exclusive description is preferable, thus, for applications, it may be better to perform this integral only over a part of the available phase space. In recent articles [@SV; @LSW] the size of ${\cal O}(\alpha_s^2)$ corrections has been estimated using the scheme of Brodsky, Lepage and Mackenzie [@BLM] for fixing the scale $\mu$ of $\alpha_s$. It has turned out that this scale is rather low. This suggests that next-to-leading QCD corrections are large. The most serious problems arise in the region close to the no-recoil $t=t_{max}$ point. It is well known, cf. [@ACCMM; @JK1], that at the boundaries of the available phase space logarithmic divergences may appear as remnants of infrared divergences. This phenomenon arises when real radiation becomes supressed relative to virtual corrections. This is exactly what happens for $t$ in the neighbour of $t_{max}$ and once again one is led to the conclusion that the inclusive parton-like description may break down there.
In the massless limit $\rho\to 0$ which corresponds to $b\to u\tau\bar\nu_\tau$ transition the functions in eq.(\[massive\]) simplify considerably $$\begin{aligned}
{\cal F}_0(t) &=& 2 (1-t)^2 (1+2t)
\\
{\cal F}_0^s(t)
&=& 2(1-t)^2
\\
{\cal F}_1(t) &=&
{\cal F}_0(t)\,
{\textstyle
\left[\,
{2\over3}\pi^2
+4{\rm Li}_2(t)+2\ln t\ln(1-t)\,\right]
}
- (1-t)(5+9t - 6t^2)
\nonumber\\ &&
+\, 4t(1-t-2t^2)\ln t
+ 2(1-t)^2(5+4t)\ln(1-t)
\label{eq:F1y0}
\\
{\cal F}_1^s(t)&=&
\textstyle{
{\cal F}_0^s(t)\,
\left[{2\over 3}\pi^2
+ 4{\rm Li}_2(t) -{9\over2}
+\ln(1-t)\left(2\ln t-{2\over t}+5\right) \right]
}
+ 4(1-t)t\ln t
\nonumber\\ &&
\label{eq:F1sy0}\end{aligned}$$ After integration over $t$ one derives the following expression for the total partial rate of $b\to u\tau\bar\nu_\tau$ $$\begin{aligned}
\lefteqn{{1\over \Gamma_{bu}}\Gamma(b\to u\tau\bar \nu_\tau)=
1-8\eta+8\eta^3-\eta^4-12\eta^2\ln\eta}
\nonumber\\&&
-{2\alpha_s\over 3\pi}\left\{
(-1 + \eta) (75 - 539 \eta - 476 \eta^2 + 18 \eta^3)/12
\right.
\nonumber\\&& \qquad \quad
+ (3 - 24 \eta - 36 \eta^2 + 16 \eta^3 - 2 \eta^4) \pi^2/3
\nonumber\\&& \qquad \quad
+ 72 \eta^2 \left[\zeta(3)-{\rm Li}_3(\eta)\right]
+ 2 (1 - 8 \eta + 36 \eta^2 + 16 \eta^3 - 2 \eta^4) {\rm Li}_2(\eta)
\nonumber\\&& \qquad \quad
+ (1 - \eta^2) (31 - 320 \eta + 31 \eta^2) \ln(1 - \eta)/6
\nonumber\\&& \qquad \quad
+ \left[2 \eta + 15 \eta^2 - 94 \eta^3/3 + 31 \eta^4/6
- 8 \eta^2 \pi^2 + 24 \eta^2 {\rm Li}_2(\eta)
\right. \qquad \quad
\nonumber\\&& \left.\left. \qquad \qquad
+ 2 (1 - \eta^2) (1 - 8 \eta +\eta^2) \ln(1 - \eta)\right] \ln(\eta)
\right\}\end{aligned}$$ $$\begin{aligned}
\lefteqn{
\Gamma_{bu} =
{G_F^2 m_b^5 |V_{ub}|^2\over 192 \pi^3} }\end{aligned}$$ which agrees with eqs.(2.21) and (3.6) in a recent preprint [@BBBG].
It has been argued [@kalinowski] that the $\tau$ polarization in $b\to c\tau\bar\nu_\tau$ is particularly sensitive to deviations from the Standard Model. The longitudinal component of $\tau$ polarization can either be measured with reference to its direction of flight in the $b$ rest frame or, alternatively, relative to its direction of flight in the $\tau\bar\nu$ rest frame. To evaluate analytically QCD corrections to the longitudinal $\tau$ polarization in the $b$ rest frame is a demanding task. Previous experience from a similar calculation for the decay of polarized top quarks indicates that these corrections are typically quite small through most of the kinematical range. To substantiate this claim, the longitudinal $\tau$ polarization in the $\tau\bar\nu$ rest frame is evaluated including QCD corrections.
For V-A weak current the leptonic tensor $$\begin{aligned}
{\cal L}_{\alpha\beta}(\tau,s;\nu) &\sim&
\left[\,\bar u_\tau\gamma_\alpha(1
-\gamma_5) v_\nu\,\right]\,
\left[\,\bar u_\tau\gamma_\beta (1-\gamma_5)v_\nu\,\right]^\dagger
\nonumber\\
&\sim& \nu_\alpha {\cal T}_\beta +
{\cal T}_\alpha \nu_\beta - \nu\cdot{\cal T} g_{\alpha\beta}
- {\rm i}\varepsilon_{\alpha\beta\gamma\delta}\nu^\gamma
{\cal T}^\delta
\label{eq:Labs}\end{aligned}$$ where $${\cal T}^\alpha = {\textstyle{1\over 2}}
\left(\,\tau^\alpha -m_\tau s^\alpha\,\right)$$ and $s^\alpha$ is the $\tau$ polarization fourvector. The helicity states in the $\tau\bar\nu$ rest frame are obtained for $$m_\tau s^\alpha = \pm \left( \tau^\alpha
- {\textstyle{2\eta\over t-\eta}} \nu^\alpha \right)$$ It is evident that the net polarization can be calculated in the same way as the total rate, decomposing again the spin dependent term into longitudinal and transversal parts. For the positive helicity of $\tau$ one derives the following expression for the differential rate $$\begin{aligned}
\lefteqn{
{1\over \Gamma_{bc}}\,{{\rm d}\Gamma^{(+)}\over {\rm d}t} =
{\eta\over 2t}
\left( 1-{\eta \over t}\right)^2 \, \left\{
{\cal F}_0(t) + 3 {\cal F}_0^s(t)
- {2\alpha_s\over 3 \pi}
\left[\,
{\cal F}_1(t) + 3 {\cal F}_1^s(t)\,\right]
\right\} }
\label{polar}\end{aligned}$$ For the negative helicity one has $$\begin{aligned}
\lefteqn{
{{\rm d}\Gamma^{(-)}\over {\rm d}t} =
{{\rm d}\Gamma\over {\rm d}t} -
{{\rm d}\Gamma^{(+)}\over {\rm d}t}
}
\label{polar1}\end{aligned}$$ and the net helicity in the $\tau\nu_\tau$ rest frame is equal to $${\rm P} = - 1 \, + \,
{ 2
\int^{(1-\sqrt{\rho})^2}_\eta
{{\rm d}\Gamma^{(+)}\over {\rm d}t}\,{\rm d}t
\over
\int^{(1-\sqrt{\rho})^2}_\eta
{{\rm d}\Gamma\over {\rm d}t}\,{\rm d}t
}$$
In the subsequent discussion the mass difference $m_b-m_c$ will be fixed through the HQET relation $$m_b-m_c = [ (3m_{B^*}+m_{B})-(3m_{D^*}+m_{D})]/4 + \ldots$$ The most important corrections to the above relation arise from the kinetic energy of heavy quarks in $B$ and $D$ mesons. On physical grounds one expects this contribution to increase the difference between $m_b$ and $m_c$. We take $m_b- m_c=$ 3.4 GeV in numerical calculations and $m_b$ is varied between 4.5 and 5.0 GeV. In order to estimate effects of QCD corrections on measurable quantities we neglects all the problems with the parton-hadron duality and the scale ambiguity. The following results have been obtained: $$\begin{aligned}
&&\Gamma(b\to c\tau\bar\nu_\tau) =
\Gamma_{0}(b\to c\tau\bar\nu_\tau) \,
\left[\, 1 + ( -0.450\pm 0.016)\,\alpha_s\, \right]
\nonumber\\
&&\Gamma(b\to c e\bar\nu_e) =
\Gamma_{0}(b\to c e\bar\nu_e) \,
\left[\, 1 + ( -0.545\pm 0.025)\,\alpha_s\, \right]
\nonumber\\
&&R=BR(b\to\tau X)/BR(b\to eX) = R_0
\left[\, 1 + ( 0.094\mp 0.009)\,\alpha_s\, \right]
\nonumber\\
&&{\rm P} = -(0.293\mp 0.002)
\left[\, 1 + ( 0.140\mp 0.015)\,\alpha_s\, \right]
\nonumber\\
&&<\, t \,>(b\to\tau X)= (0.34\pm 0.03)
\left[\, 1 + ( 0.016\pm 0.003)\,\alpha_s\, \right]
\nonumber\\
&&<\, t \,>(b\to e X)= (0.20\pm 0.02)
\left[\, 1 + ( 0.035\pm 0.007)\,\alpha_s\, \right]
\nonumber\end{aligned}$$ where $<\, t \,>(\ldots)$ denote the average values of $t$ for the corresponding decay chanels. It is evident that the ${\cal O}(\alpha_s)$ corrections practically cancel in the ratio $R$ of the branching ratios as well as in the result for the polarization $P$. The moments $<\, t \,>(b\to\tau X)$ and $<\, t \,>(b\to e X)$ are also insensitive to $\alpha_s$ corrections. On the other hand all these quantities are sensitive to the quark masses and therefore they may be used for fixing $m_b$ and $m_c$.
Acknowledgements {#acknowledgements .unnumbered}
================
MJ would like to thank Kostya Chetyrkin for helpful discussions. This work was supported in part by KBN grant 2P30225206, by DFG contract 436POL173193S and by Graduiertenkolleg Elementarteilchenphysik at the University of Karlsruhe.
[10]{} N. Cabibbo and L. Maiani, Phys. Lett. B79 (1978) 109. M. Je[ż]{}abek and J.H. K[ü]{}hn, Nucl. Phys. B314 (1989) 1. Y. Nir, Phys. Lett. B221 (1989) 184. A.F. Falk, Z. Ligeti, M. Neubert and Y. Nir, Phys. Lett. B326 (1994) 145. L. Koyrakh, Phys. Rev. D49 (1994) 3379. S. Balk, J.G. Körner, D. Pirjol and K. Schilcher, Z. Phys. C64 (1994) 37. B. Blok, R.D. Dikeman and M. Shifman, preprint TPI-MINN-94/23-T. M.B. Voloshin and M.A. Shifman, Yad. Fiz. 47 (1988) 801. N. Isgur and M.B. Wise, Phys. Lett. B232 (1989) 113; B237 (1990) 527. for a recent theoretical update see: M. Neubert, Phys.Lett. B338 (1994) 84. A. Czarnecki and S. Davidson, in: A. Astbury et al. (eds.), [*Collider Physics*]{}, Proceedings of $8^{th}$ Lake Louise Winter Institute, World Scientific, Singapore, 1993, p.330. A. Czarnecki and S. Davidson, Phys.Rev. D48 (1993) 4183. E. Bagan, P. Ball, V.M. Braun and P. Gosdzinsky, preprint MPI-PhT/94-49. J. Kalinowski, Phys.Lett. B245 (1990) 201. B.H. Smith and M.B. Voloshin, preprint TPI-MINN-94/16-T. M. Luke, M.J. Savage and M.B. Wise, preprints UTPT 94-24 and UTPT 94-27. S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, Phys.Rev. D28 (1983) 228. G. Altarelli et al., Nucl.Phys. B208 (1982) 365. M. Je[ż]{}abek and J.H. K[ü]{}hn, Nucl. Phys. B320 (1989) 20.
[^1]: In a recent preprint [@BDS] the breakdown of the local parton-hadron duality has been invoked as the origin of problems with the semileptonic decay rate of $D$ mesons in the framework of HQET. Let us remark that for large ${\rm w}^2$ the kinetic energy of the hadronic system in $B$ decays can be similar to that in $D$ decays. Thus the semileptonic branching ratios for $b$ decays may be also affected for the effective mass of the hadronic system close to the resonance region.
[^2]: Note that the rates of up and down type quark decays into their respective isospin partners are of course identical, in contrast to the shapes of the lepton spectra.
|
---
abstract: 'Supplemental information for a Letter reporting the rate of coalescences inferred from of coincident Advanced LIGO observations surrounding the transient signal [GW150914]{}. In that work we reported various rate estimates whose 90% fell in the range . Here we give details of our method and computations, including information about our search pipelines, a derivation of our likelihood function for the analysis, a description of the astrophysical search trigger distribution expected from merging , details on our computational methods, a description of the effects and our model for calibration uncertainty, and an analytic method of estimating our detector sensitivity that is calibrated to our measurements.'
bibliography:
- 'LIGO-P1500217\_GW150914\_Rates.bib'
- '../macros/GW150914\_refs.bib'
title: 'Supplement: The Rate of Binary Black Hole Mergers Inferred from Advanced LIGO Observations Surrounding [GW150914]{}'
---
=1
\[BH\][black hole]{} \[BBH\][binary black hole]{} \[CI\][credible interval]{} \[CL\][credible level]{} \[CBC\][compact binary coalescence]{} \[EOB\][effective one body]{} \[GW\][gravitational wave]{}
The first detection of a signal from a merging system is described in @GW150914-DETECTION. @RatesLetter reports on inference of the local merger rate from surrounding Advanced LIGO observations. This Supplement provides supporting material and methodological details for @RatesLetter, hereafter referred to as the Letter.
Search Pipelines {#suppsec:search-description}
================
Both the [`pycbc`]{} and [`gstlal`]{} pipelines are based on matched filtering against a bank of template waveforms. See @GW150914-CBC for a detailed description of the pipelines in operation around the time of [GW150914]{}; here we provide an abbreviated description.
In the [`pycbc`]{} pipeline, the single-detector is re-weighted by a chi-squared factor [@Allen:2004gu] to account for template-data mismatch [@babak:2012zx]; the re-weighted single-detector are combined in quadrature to produce a detection statistic for search triggers.
The [`gstlal`]{} pipeline’s detection statistic, however, is based on a likelihood ratio [@Cannon2013; @Cannon2015] constructed from the single-detector and a signal-consistency statistic. An analytic estimate of the distribution of astrophysical signals in multiple-detector and signal consistency statistic space is compared to a measured distribution of single-detector triggers without a coincident counterpart in the other detector to form a multiple-detector likelihood ratio.
Both pipelines rely on an empirical estimate of the search background, making the assumption that triggers of terrestrial origin occur independently in the two detectors. The background estimate is built from observations of single-detector triggers over a long time ([`gstlal`]{}) or through searching over a data stream with one detector’s output shifted in time relative to the other’s by an interval that is longer than the light travel time between detectors, ensuring that no coincident astrophysical signals remain in the data ([`pycbc`]{}). For both pipelines it is not possible to produce an instantaneous background estimate at a particular time; this drives our choice of likelihood function as described in Section \[suppsec:likelihood\].
The [`gstlal`]{} pipeline natively determines the functions $p_0(x)$ and $p_1(x)$ for its detection statistic $x$. For this analysis a threshold of ${\ensuremath{x_\mathrm{min}}}= 5$ was applied, which is sufficiently low that the trigger density is dominated by terrestrial triggers near threshold. There were $M=15\,848$ triggers observed above this threshold in the 17 days of observation time analyzed by [`gstlal`]{}.
For [`pycbc`]{}, the quantity $x'$ is the re-weighted detection statistic.[^1] We set a threshold ${\ensuremath{x_\mathrm{min}}}' = \newsnrthresh{}$, above which $M' = 270$ triggers remain in the search. We use a histogram of triggers collected from time-shifted data to estimate the terrestrial trigger density, $p_0\left(x'\right)$, and a histogram of the recovered triggers from the injection sets described in Section [2.2]{} of the Letter to estimate the astrophysical trigger density, $p_1\left(x'\right)$. These estimates are shown in Figure \[fig:supp-p0p1\]. The uncertainty in the distribution of triggers from this estimation procedure is much smaller than the uncertainty in overall rate from the finite number statistics (see, for example, Figure \[fig:foreground-background\]). The empirical estimate is necessary to properly account for the interaction of the various single- and double-interferometer thresholds in the [`pycbc`]{} search [@GW150914-CBC]. At high , where these thresholds are irrelevant, the astrophysical triggers follow an approximately flat-space volumetric density (see Section \[suppsec:universal\]) of $$\label{eq:newsnr-foreground}
p_{1}(x') \simeq \frac{3 {\ensuremath{x_\mathrm{min}}}'^3}{x'^4},$$ but they deviate from this at smaller due to threshold effects in the search.
For the [`pycbc`]{} pipeline, a detection statistic $x' \geq \loudnewsnr{}$ corresponds to an estimated search of one per century.
Derivation of Poisson Mixture Model Likelihood {#suppsec:likelihood}
==============================================
In this section we derive the likelihood function in Eq.[(3)]{} of the Letter. Consider first a search of the type described in Section \[suppsec:search-description\] over $N_T$ intervals of time of width $\delta_i$, $\left\{i = 1, \ldots, N_T\right\}$. Triggers above some fixed threshold occur with an instantaneous rate in time and detection statistic $x$ given by the sum of the terrestrial and astrophysical rates: $$\label{eq:time-snr-rate-sum}
{\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t \mathrm{d} x}}}}(t,x) = R_0(t) p_0(x; t) + R_1(t) V(t)
p_1(x; t),$$ where $R_0(t)$ is the instantaneous rate (number per unit time) of terrestrial triggers, $R_1(t)$ is the instantaneous rate density (number per unit time per unit comoving volume) of astrophysical triggers, $p_0$ is the instantaneous density in detection statistic of terrestrial triggers, $p_1$ is the instantaneous density in detection statistic of astrophysical triggers, and $V(t)$ is the instantaneous sensitive comoving redshifted volume [@GW150914-ASTRO see also Eq.[(15)]{} of the Letter] of the detectors to the assumed source population. The astrophysical rate $R_1$ is to any reasonable approximation constant over our observations so we will drop the time dependence of this term from here on.[^2] Note that $R_0$ and $R_1$ have different units in this expression; the former is a rate (per time), while the latter is a *rate density* (per time-volume). The density $p_1$ is independent of source parameters as described in Section \[suppsec:universal\]. Let $$\label{eq:dNdt}
{\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t}}}} \equiv \int \mathrm{d} x \, {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t \mathrm{d} x}}}} =
R_0(t) + R_1 V(t).$$
If the search intervals $\delta_i$ are sufficiently short, they will contain at most one trigger and the time-dependent terms in Eq. will be approximately constant. Then the likelihood for a set of times and detection statistics of triggers, $\left\{ (t_j, x_j) | j = 1, \ldots, M \right\}$, is a product over intervals containing a trigger (indexed by $j$) and intervals that do not contain a trigger (indexed by $k$) of the corresponding Poisson likelihoods $$\begin{gathered}
\label{eq:segmented-likelihood}
\mathcal{L} = \left\{ \prod_{j = 1}^M {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t \mathrm{d} x}}}}\left(
t_j, x_j\right) \exp\left[ -\delta_j {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t}}}}\left( t_j
\right) \right] \right\} \\ \times \left\{ \prod_{k = 1}^{N_T - M}
\exp\left[ - \delta_k {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t}}}}\left( t_k \right)\right]\right\}\end{gathered}$$ (cf. @Farr2015 [Eq. (21)] or @Loredo1995 [Eq.(2.8)]).[^3] Now let the width of the observation intervals $\delta_i$ go to zero uniformly as the number of intervals goes to infinity. Then the products of exponentials in Eq. become an exponential of an integral, and we have $$\label{eq:continuous-likelihood}
\mathcal{L} = \prod_{j=1}^M \left[ {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t \mathrm{d} x}}}}\left( t_j,
x_j\right) \right] \exp\left[ -N \right],$$ where $$\label{eq:dNdt-integrated}
N = \int \mathrm{d} t \, {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{N}}{{\ensuremath{\mathrm{d}}}{t}}}}$$ is the expected number of triggers of both types in the total observation time $T$.
As discussed in Section \[suppsec:search-description\], in our search we observe that $R_0$ remains approximately constant and that $p_0$ retains its shape over the observation time discussed here; this assumption is used in our search background estimation procedure [@GW150914-CBC]. The astrophysical distribution of triggers is universal (Section \[suppsec:universal\]) and also time-independent. Finally, the detector sensitivity is observed to be stable over our of coincident observations, so $V(t) \simeq \mathrm{const}$ [@GW150914-CALIBRATION]. We therefore choose to simply ignore the time dimension in our trigger set. This generates an estimate of the rate that is sub-optimal (i.e. has larger uncertainty) but consistent with using the full data set to the extent that the detector sensitivity varies in time; since this variation is small, the loss of information about the rate will be correspondingly small. We *do* capture any variation in the sensitivity with time in our Monte-Carlo procedure for estimating ${\ensuremath{\left\langle VT \right\rangle}}{}$ that is described in Section [2.2]{} of the Letter.
If we ignore the trigger time, then the appropriate likelihood to use is a marginalization of Eq. over the $t_j$. Let $$\begin{gathered}
\label{eq:time-marg-like}
\bar{\mathcal{L}} \equiv \int \left[\prod_j {\ensuremath{\mathrm{d}}}t_j\right] \, \mathcal{L} \\ =
\prod_j \left[ \Lambda_0 p_0\left( x_j \right) + \Lambda_1 p_1\left(
x_j \right) \right] \exp\left[- \Lambda_0 - \Lambda_1 \right],\end{gathered}$$ where $$\label{eq:L0-marg}
\Lambda_0 p_0(x) = \int {\ensuremath{\mathrm{d}}}t \, R_0(t) p_0\left( x; t \right),$$ and $$\label{eq:L1-marg}
\Lambda_1 p_1(x) = \int {\ensuremath{\mathrm{d}}}t \, R_1 V(t) p_1\left( x; t \right),$$ with $$\label{eq:norm-L0L1}
\int {\ensuremath{\mathrm{d}}}x \, p_0(x) = \int {\ensuremath{\mathrm{d}}}x \, p_1(x) = 1.$$ If we assume that $R_1$ is constant in (comoving) time, and measure $p_1(x)$ by accumulating recovered injections throughout the run as we have done, then this expression reduces to the likelihood in Eq.[(3)]{} of the Letter. A similar argument with an additional population of triggers produces Eq. [(10)]{} of the Letter.
The Expected Number of Background Triggers
------------------------------------------
The procedure for estimating $p_0(x)$ in the [`pycbc`]{} pipeline also provides an estimate of the mean number of background events per experiment $\Lambda_0$ [@GW150914-CBC]. The procedure for estimating $p_0$ used in the [`gstlal`]{} pipeline, however, does not naturally provide an estimate of $\Lambda_0$; instead [`gstlal`]{} estimates $\Lambda_0$ by fitting the observed number of triggers to a Poisson distribution. We have chosen to leave $\Lambda_0$ as a free parameter in our canonical analysis with a broad prior and infer it from the observed data, rather than using the [`pycbc`]{} background estimate to constrain the prior, which would result in a much narrower posterior on $\Lambda_0$. This is equivalent to the [`gstlal`]{} procedure for $\Lambda_0$ estimation in the absence of signals; the presence of a small number of signals in our data here do not substantially change the $\Lambda_0$ estimate due to the overwhelming number of background triggers in the data set.
Using a broad prior on $\Lambda_0$ is *conservative* in the sense that it will broaden the posterior on $\Lambda_1$ from which we infer rates. However, because there are so many more triggers in both searches of terrestrial origin than astrophysical there is little correlation between $\Lambda_0$ and $\Lambda_1$, and so there is little difference between the posterior we obtain on $\Lambda_1$ and the posterior we would have obtained had we implemented the tight prior on $\Lambda_0$. Figure \[fig:Lambda0-Lambda1\] shows the two-dimensional posterior we obtain from Eq. [(5)]{} of the Letter on $\Lambda_0$ and $\Lambda_1$.
We have checked that using a $\delta$-function prior $$\label{eq:pycbc-Lambda0-prior}
p\left( \Lambda_0 \right) = \delta \left( \Lambda_0 - \pycbcFixedBgCounts{} \right)$$ in the [`pycbc`]{} analysis that is the result of the pipeline $\Lambda_0$ estimate from timeslides[^4] [@GW150914-CBC] and using a looser prior that is the result of a [`gstlal`]{} estimate on a single set of time-slid data produces no meaningful change in our results. Figure \[fig:Lambda0-prior-sensitivity\] shows our canonical rate posterior inferred with the [`pycbc`]{} $\Lambda_0$ prior in Eq. and our canonical broad prior.
Universal Astrophysical Trigger Distribution {#suppsec:universal}
============================================
Both the [`pycbc`]{} and [`gstlal`]{} pipelines rely on the as part of their detection statistic, $x$. The of an astrophysical trigger is a function of the detector noise at the time of detection and the parameters of the trigger. @Schutz2011 and @Chen2014 demonstrate that the distribution of the expected $\langle \rho \rangle$ in a simple model of a detection pipeline that simply thresholds on , $\rho \geq \rho_\mathrm{th}$, with sources in the local universe is *universal*, that is, independent of the source properties. It follows $$\label{eq:snr-universal}
p\left( \langle \rho \rangle \right) = \frac{3
\rho_\mathrm{th}^3}{\langle \rho\rangle^4}.$$ This result follows from the fact that the expected value of the in a matched-filter search for signals scales inversely with transverse comoving distance [@Hogg1999]: $$\label{eq:snr-scaling}
\langle \rho \rangle = \frac{A\left( m_1, m_2, \vec{a}_1, \vec{a}_2, S(f), z \right) B \left(
\mathrm{angles} \right)}{D_M},$$ where $A$ is an amplitude factor that depends on the intrinsic properties (source-frame masses and spins) of the source, the detector sensitivity expressed as a noise power spectral density $S(f)$ as a function of observer frequency and redshift $z$, and $B$ is an angular factor depending on the location of the source in the sky and the relative orientations of binary orbit and detector. The redshift enters $A$ only through shifting the source waveform to lower frequency at higher redshift, changing $A$ because the sensitivity varies with observer frequency $f$. For the redshifts to which we are sensitive to in this observation period this effect on $A$ is small.
If we assume that the distribution of source parameters is constant over the range of distances to which we are sensitive, and ignore the small redshift-dependent sensitivity correction mentioned above, then the distribution of will be governed entirely by the distribution of distances of the sources, which, in the local universe is approximately $$\label{eq:approx-dL-dist}
p\left( D_M \right) \propto D_M^2,$$ yielding the distribution of given in Eq..
Both the [`pycbc`]{} and [`gstlal`]{} pipelines use goodness-of-fit statistics in addition to and employ a more complicated system of thresholds than this simple model, but the empirical distribution of detection statistics remains, to an approximation suitable for our purposes, independent of the source parameters. Figure \[fig:universal\] shows the distribution of recovered detection statistics for the various injection campaigns with varying source distribution used to estimate sensitive time-volumes in the [`pycbc`]{} pipeline. In each injection campaign $\mathcal{O}(1000)$ signals were recovered. For loud signals, the detection statistic is proportional to in this pipeline, and the distribution is not sensitive to the complicated thresholding in the pipeline, so we recover Eq. ; for quiet signals the interaction of various single-detector thresolds in the pipeline causes the distribution to deviate from this analytic approximation, but it remains independent of the distribution of sources. Note that the *empirical* distribution of detection statistics, not the analytic one, forms the basis for $p_1$, the foreground distribution used in this rate estimation work.
To quantify the deviations from universality, we have preformed two-sample tests between all six pairings of the sets of detections statistics recovered in the injection campaigns described in Sections 2 and 3 of the Letter and featured in Figure \[fig:universal\]. The most extreme $p$-value occurred with the comparison between the injection set with masses drawn flat in $\log m$ and the one with masses drawn from a power law (both described in Section 3 of the Letter); this test gave a $p$-value of $0.013$. Given that we have performed six identical comparisons we cannot reject the null hypothesis that the empirical distributions used for rate estimation from the [`pycbc`]{} pipeline are identical even at the relatively weak significance $\alpha = 0.05$. Certainly any differences in detection statistic distribution attributable to the population are far too small to matter with the few astrophysical signals in our data set (compared with $\mathcal{O}(1000)$ recovered injections in each campaign).
Because the distribution of detection statistics is, to a very good approximation, *universal*, we cannot learn anything about the source population from the detection statistic alone; we must instead resort to followup [@Veitch2015; @GW150914-PARAMESTIM] of triggers to determine their parameters. The parameters of the waveform template that produced the trigger can be used to guess the parameters of the source that generated that trigger, but the bias and uncertainty in this estimate are very large compared to the estimate. We therefore ignore the parameters of the waveform template that generated the trigger in the assignment of triggers to classes.
Count Posterior {#suppsec:count-posterior}
===============
We impose a prior on the $\Lambda$ parameters of: $$\label{eq:two-pop-prior-supp}
p\left(\Lambda_1,\Lambda_0\right) \propto
\frac{1}{\sqrt{\Lambda_1}} \frac{1}{\sqrt{\Lambda_0}}.$$
The posterior on expected counts is proportional to the product of the likelihood from Eq. [(3)]{} of the Letter and the prior from Eq. : $$\begin{gathered}
\label{eq:two-pop-posterior-supp}
p\left(\Lambda_1, \Lambda_0 | \left\{ x_j | j = 1, \ldots, M\right\}
\right) \\ \propto \left\{ \prod_{j = 1}^{M} \left[ \Lambda_1
p_1\left(x_j\right) + \Lambda_0 p_0\left( x_j \right) \right] \right\}
\\ \times \exp\left[ -\Lambda_1 - \Lambda_0 \right]
\frac{1}{\sqrt{\Lambda_1 \Lambda_0}}.\end{gathered}$$ For estimation of the Poisson rate parameter in a simple Poisson model, the Jeffreys prior is $1/\sqrt{\Lambda}$. With this prior, the posterior mean on $\Lambda$ is $N+1/2$ for $N$ observed counts. With a prior proportional to $1/\Lambda$ the mean is $N$ for $N>0$, but the posterior is improper when $N=0$. For a flat prior, the mean is $N+1$. Though the behaviour of the mean is not identical with our mixture model posterior, it is similar; because we find $\left\langle \Lambda_1 \right\rangle \gg 1/2$, the choice of prior among these three reasonable options has little influence on our results here.
For the [`pycbc`]{} data set we find the posterior median and $90\%$ credible range $\Lambda_1 = \countonetwopop$ above our threshold. For the [`gstlal`]{} set we find the posterior median and 90% credible range $\Lambda_1=\countonetwopopGSTLAL{}$. Though we have only one event ([GW150914]{}) at exceptionally high significance, and one other at marginal significance (), the counting analysis shows these to be consistent with the possible presence of several more events of astrophysical origin at lower detection statistic in both pipelines.
The thresholds applied to the [`pycbc`]{} and [`gstlal`]{} triggers for this analysis are *not* equivalent to each other in terms of either or false alarm rate; instead, both thresholds have been chosen so that the rate of triggers of terrestrial origin ($\Lambda_0 p_0$) dominates near threshold. Since the threshold is set at *different* values for each pipeline, we do not expect the counts to be the same between pipelines.
The estimated astrophysical and terrestrial trigger rate densities (Eq. [(1)]{} of the Letter) for [`pycbc`]{} are plotted in Figure \[fig:foreground-background\]. We select triggers from a subset of the search parameter space (i.e. our bank of template waveforms) that contains [GW150914]{} as well as the mass range considered for possible alternative populations of binaries in Section [3]{} of the Letter. There are $M' = \mcoincs{}$ two-detector coincident triggers in this range in the [`pycbc`]{} search [@GW150914-CBC]. Figure \[fig:foreground-background\] also shows an estimate of the density of triggers that comprise our data set which agrees well with our inference of the trigger rate.
Based on the probability of astrophysical origin inferred for from the two-component mixture model in Eq. and shown in Figure \[fig:pfore\], we introduce a third class of signals and use a three-component mixture model with expected counts $\Lambda_0$ (terrestrial), $\Lambda_1$ ([GW150914]{}-like), and $\Lambda_2$ (-like) to infer rates in Sections [2.1]{} of the Letter and [2.2]{} of the Letter.
We use the Stan and [`emcee`]{} Markov-Chain Monte Carlo samplers [@Foreman-Mackey2013; @stan2015; @pystan2015] to draw samples from the posterior in Eq. [(5)]{} of the Letter for the two pipelines. We have assessed the convergence and mixing of our chains using empirical estimates of the autocorrelation length in each parameter [@Sokal1996], the Gelman-Rubin $R$ convergence statistic [@Gelman1992], and through visual inspection of chain plots. By all measures, the chains appear well-converged to the posterior distribution.
Table \[tab:count-vt-table\] contains the full results on expected counts and associated sensitive time-volumes for both pipelines.
[lcccc]{} [GW150914]{} & & & &\
& & & &\
Both & & & &\
Flat in log mass & & & &\
Power Law (-2.35) & & & &\
Calibration Uncertainty {#suppsec:calibration-uncert}
=======================
The LIGO detectors are subject to uncertainty in their calibration, in both the measured amplitude and phase of the gravitational-wave strain. @GW150914-CALIBRATION discusses the methods used to calibrate the strain output of the detector during the of coincident observations discussed here. @GW150914-CALIBRATION estimates that the reported strain is accurate to within 10% in amplitude and 10 degrees in phase between $20\,\mathrm{Hz}$ and $1\,\mathrm{kHz}$ throughout the observations.
The reported by our searches are quadratically sensitive to calibration errors because they are maximized over arrival time, waveform phase, and a template bank of waveforms [@Allen1996; @Brown2004]. @GW150914-CBC demonstrates that the other search pipeline outputs are also not affected to a significant degree by the calibration uncertainty present during our observing run. Therefore, we ignore effects of calibration on the pipeline detection statistics $x$ and $x'$ we use here to estimate rates from the [`pycbc`]{} and [`gstlal`]{} pipelines.
The amplitude calibration uncertainty in the detector results at leading order in a corresponding uncertainty between the luminosity distances of sources measured from real detector outputs [@GW150914-PARAMESTIM] and the luminosity distances used to produce injected waveforms used to estimate sensitive time-volumes in this work. A 10% uncertainty in $d_L$ at these redshifts corresponds to an approximately 30% uncertainty in volume. We model this uncertainty by treating ${\ensuremath{\left\langle VT \right\rangle}}$ as a parameter in our analysis, and imposing a log-normal prior: $$\label{eq:VT-prior}
p\left( \log {\ensuremath{\left\langle VT \right\rangle}}\right) \propto N\left( \log \mu,
\frac{\sigma}{\mu} \right),$$ where $\mu$ is the Monte-Carlo estimate of sensitive time-volume produced from the injection campaigns described in Section [2.2]{} of the Letter and $$\label{eq:VT-sigma}
\sigma^2 = \sigma_\mathrm{cal}^2 + \sigma_\mathrm{stat}^2,$$ with $\sigma_\mathrm{cal} = 0.3 \mu$ and $\sigma_\mathrm{stat}$ is the estimate of the Monte-Carlo uncertainty from the finite number of recovered injections reported above. In all cases $\sigma_\mathrm{cal} \gg \sigma_\mathrm{stat}$.
Since the likelihood in Eqs. [(3)]{} of the Letter or [(10)]{} of the Letter does not constrain ${\ensuremath{\left\langle VT \right\rangle}}$ independently of $R$, sampling over ${\ensuremath{\left\langle VT \right\rangle}}$ at the same time as $\Lambda$ and $R$ has the effect of convolving the log-normal distribution of ${\ensuremath{\left\langle VT \right\rangle}}$ with the posterior on $\Lambda$ in the inference of $R$. In spite of the 30% relative uncertainty in ${\ensuremath{\left\langle VT \right\rangle}}$ from calibration uncertainty, the counting uncertainty on $R$ from the small number of detected events dominates the width of the posterior on $R$.
Analytic Sensitivity Estimate {#suppsec:analytic-vt}
=============================
As a rough check on our ${\ensuremath{\left\langle VT \right\rangle}}{}$ estimates and the integrand $\mathrm{d} {\ensuremath{\left\langle VT \right\rangle}}/ \mathrm{d}z$, we find that the following approximate, analytic procedure also produces a good approximation to the pycbc Monte-Carlo estimate in Table \[tab:count-vt-table\].
1. Generate inspiral–merger–ringdown waveforms in a single detector at various redshifts from the source distribution $s(\theta)$ with random orientations and sky positions.
2. Using the high-sensitivity early Advanced LIGO noise power spectral density from @Aasi2013ObsScenario, compute the in a single detector.
3. Consider a signal found if the is greater than .
Employed with the source distributions described above, this approximate procedure yields ${\ensuremath{\left\langle VT \right\rangle}}_1 \simeq \approxVTone$ and ${\ensuremath{\left\langle VT \right\rangle}}_2 \simeq \approxVTtwo$ for the sensitivity to the two classes of merging system. Figure \[fig:dVTdz\] shows the sensitive time-volume integrand, $$\label{eq:vt-integrand}
{\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{{\ensuremath{\left\langle VT \right\rangle}}{}}}{{\ensuremath{\mathrm{d}}}{z}}}} \equiv T \frac{1}{1+z} {\ensuremath{\dfrac{{\ensuremath{\mathrm{d}}}{V_c}}{{\ensuremath{\mathrm{d}}}{z}}}} \int {\ensuremath{\mathrm{d}}}\theta \, s(\theta) f(z, \theta)$$ estimated from this procedure for systems with various parameters superimposed on the Monte-Carlo estimates from the injection campaign described above.
[^1]: When quoting pipeline-specific values we distinguish [`pycbc`]{} quantities with a prime.
[^2]: The astrophysical rate can, in principle, also depend on redshift, but in this paper we assume that the coalescence rate is constant in the comoving frame.
[^3]: There is a typo in Eq. (2.8) of @Loredo1995. The second term in the final bracket is missing a factor of $\delta t$.
[^4]: While the statistical uncertainty on the pipeline $\Lambda_0$ estimate is not precisely zero, $\sigma_{\Lambda_0}/ \Lambda_0 \lesssim 10^{-3}$, it is so small that a $\delta$-function prior is appropriate.
|
---
author:
- Tongfeng Weng
- Jie Zhang
- Moein Khajehnejad
- Michael Small
- Rui Zheng
- Pan Hui
title: Navigation by anomalous random walks on complex networks
---
Introduction {#introduction .unnumbered}
============
Complex networks are ubiquitous in the real world ranging from sociology to biology and technology [@Barabasi2012]. Going beyond the interesting topological properties, quantifying the impact of structural organization of networks on transport processes has become one of the most important topics. As a paradigmatic transport process, random walks on complex networks have been intensively studied [@JDNoh2004; @SCondamin2007; @SHwang2012; @Perra2012; @YLin2014]. A variety of measurements including mean first passage time (MFPT) [@JDNoh2004], first passage time [@SHwang2012], and average trapping time [@YLin2014] have been proposed, providing a comprehensive characterization of random walks on networks. Moreover, these studies also facilitate our understanding of diverse dynamical processes on networks including epidemic spreading [@ZMYang2012], synchronization [@PSSkardal2014], and transportation [@GLi2010].
However, for random walks, the walker is confined only to the neighbourhood of a node in each jump, which cannot model some real situations [@FDPatti2015], and also impedes search and transport efficiency on networks [@SHwang2012]. This limitation is circumvented by the model of Lévy walks in natural condition [@GMViswanathan2010; @DRaichlen2014]. Recently, intensive attention has been devoted to anomalous random walks on networks, such as Lévy walks [@APRiascos2012; @APRiascos2014], traditional web surfing [@ANLangville2006], and even electric signals transmitted in brain networks [@FDPatti2015]. One striking feature of anomalous random walks is having the long-range hopping (i.e., the walker can hop to far away nodes not directly connected to its current position). In fact, the occurrence of long-range hopping is frequently encountered in our life. For example, we usually communicate with people socially close to us, but also occasionally with those that are unconnected [@APRiascos2014]. Analogously, when doing web surfing, one usually proceeds by following the hyperlinks but casually may open a new tab to look for the related topic [@FDPatti2015]. Although it is widely agreed that anomalous random walks represent an important branch of search and transport processes on networks, how to characterize anomalous random walks and specifically, to uncover the interplay between their dynamics and the underlying network structure has not been addressed. Traditional measurements like the mean first passage time neglect the difference between the cost associated with the nearest-neighbor jump and the long-range hopping, therefore cannot properly characterize anomalous random walks on networks.
In this paper, we propose the mean first traverse distance that represents the expected traverse distance required by a walker moving from a source node to a target node. Importantly, this allows the cost associated with the hopping to be taken into account in the characterization of anomalous random walks; this therefore overcomes the problems of traditional measurements adopted in general random walks. We obtain analytically the MFTD and the global MFTD on arbitrary networks. Results on Lévy walks demonstrate that these measurements can effectively characterize the relationship between network structure and anomalous random walks. Interestingly, when applied to the PageRank search, we demonstrate that the optimal damping factor occurs at around 0.85 in real web networks which is consistent with our empirical finding. The new metric enables effective characterization of dynamics of anomalous random walks on networks, which promises more efficient search and transport processes on networks.
Results {#results .unnumbered}
=======
**The MFTD of anomalous random walks** We start from an undirected network consisting of $N$ nodes. The connectivity of nodes is fully described by a symmetric adjacency matrix $A$, whose entry $a_{ij}=1$ (0) if nodes $i$ and $j$ are (not) connected. For anomalous random walks, at each time step, the walker jumps from current node $i$ to node $j$ with a nonzero transition probability $p_{ij}$ regardless of the connection profile of node $i$. Take Lévy walks on networks for example, the transition probability is defined as $p_{ij}={d_{ij}^{-\alpha}}/{\sum_{k}d_{ik}^{-\alpha}}$, where $\alpha$ is the tuning exponent lying in the interval $0\leq{\alpha}<\infty$ and $d_{ij}$ is the shortest path length between nodes $i$ and $j$ [@APRiascos2012]. To characterize anomalous random walks, we propose a concept of the MFTD $l_{ij}$, which is the expected distance taken by a walker to first reach node $j$ starting from node $i$. Intuitively, the traverse distance in one-step jump is shorter for a walker when nodes are directly connected, while this distance tends to be larger for indirectly linked nodes. Inspired by the empirical findings that the lengths of links usually obey a power law distribution [@Li2011], we adopt the power function $c_{ij}=d_{ij}^{\beta}$ to describe the effective distance of one-step jump, where $\beta$ named the cost exponent is a nonnegative value. In this situation, if the first step of the walk is to node $j$, the expected traverse distance required is $d_{ij}^{\beta}$; if it is to some other node $k$, the expected traverse distance becomes $l_{kj}$ plus $d_{ik}^{\beta}$ for the previous step already taken. Thus, we have $$\ l_{ij}=p_{ij}d_{ij}^{\beta}+\sum_{k\neq{j}}p_{ik}(l_{kj}+d_{ik}^{\beta}).
\label{101}$$ Using the Markov chains theory [@Grinstead2006; @Kemeny1960], the MFTD $l_{ij}$ of a anomalous random walk (see appendices) becomes $$\ l_{ij}=T_{ij}\sum_{k}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)w_{k}+\sum_{k}(z_{ik}-z_{jk})\left(\sum_{m}p_{km}d_{km}^{\beta}\right),
\label{102}$$ where $w_{k}$ is the $k$th component of the stationary distribution of the anomalous random walk, $T_{ij}$ is the MFPT from node $i$ to node $j$, and $z_{ij}$ is an element of the fundamental matrix $Z=(I-P+W)^{-1}$. Specifically, when $\beta=0$, the effective distances of one-step jump are same (i.e., $c_{ij}=1$). In this situation, it is easy to verify that the MFTD $l_{ij}$ reduces to the MFPT $T_{ij}$, which means that our paradigm can incorporate the commonly used MFPT as a special case. To further evaluate the search efficiency of an anomalous walker, we calculate the global MFTD $\langle{L}\rangle$ by averaging Eq. (\[102\]) over all pairs of source and target nodes, that is, $$\ \langle{L}\rangle=\frac{1}{N(N-1)}\sum_{i}^{N}\sum_{j}^{N}l_{ij}.
\label{32}$$ Plugging Eq. (\[102\]) into Eq. (\[32\]), we have $$\ \langle{L}\rangle=\langle{T}\rangle{\sum_{i}\left(\sum_{j}p_{ij}d_{ij}^{\beta}\right)w_{i}},
\label{33}$$ where $\langle{T}\rangle$ is the average of MFPTs over all pairs of nodes in the networks (see appendices). Here, $\langle{L}\rangle$ quantifies the ability of the anomalous walker to search and transport at the global scale on the network. In this context, smaller $\langle{L}\rangle$ represents a more effective way of achieving mobility. In the following we will demonstrate how these measurements can effectively characterize diverse anomalous random walks on networks.
**The MFTD scheme for characterizing Lévy walks** We first address a specific anomalous random walk — Lévy walks on networks. A Lévy walk exerts a power-law transition probability with the distance given by $p_{ij}={d_{ij}^{-\alpha}}/{\sum_{k}d_{ik}^{-\alpha}}$. Clearly, the tuning exponent $\alpha$ plays an important role in controlling the trade off between short-range and long-range jumping in one step, which in turn fully determines the behaviors of the Lévy walk. Specially, when $\alpha$ is very small, the walker visits all nodes with approximately equivalent probability. In contrast, the walker possibly only hop to the nearest neighbors at an extremely large $\alpha$. In this context, the Lévy walk degenerates to the generic random walk [@JDNoh2004]. Using the balance condition, the stationary distribution of the Lévy walk can be expressed as $$\ w_{i}=\frac{\sum_{k}d_{ik}^{-\alpha}}{\sum_{i}\sum_{j}d_{ij}^{-\alpha}}.
\label{5}$$ Inserting the above equation and the transition probability into Eq. (\[102\]) yields $$\ l_{ij}=\frac{z_{jj}-z_{ij}}{\sum_{m}d_{jm}^{-\alpha}}\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}+\sum_{k}(z_{ik}-z_{jk})\frac{\sum_{m}d_{km}^{\beta-\alpha}}{\sum_{m}d_{km}^{-\alpha}}.
\label{6}$$ Similar calculation applied to Eq. (\[33\]), the global MFTD $\langle{L}\rangle$ of Lévy walks reads $$\ \langle{L}\rangle=\langle{T}\rangle{\frac{\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}}{\sum_{i}\sum_{j}d_{ij}^{-\alpha}}}.
\label{7}$$ To test the validity of Eq. (\[7\]), we report both the numerical and theoretical results of the global MFTD for Lévy walks taking place in planar Sierpiński gasket [@JJKozak2002] and the (1,2)-flower model [@HDRozenfeld2007]. These two networks are typical hierarchical nets having the same number of nodes and edges but exhibiting apparently distinct structure organizations, which can favor us to explore how the network structure influences the behavior of a Lévy walk directly. To achieve the numerical results, we compute the traverse distance required for a walker to travel from a source node to a target node chosen randomly and average over the ensemble of 50,000 independent runs for each test. Fig. \[f1\] shows an excellent agreement between numerics and Eq. (\[7\]) for the different cost exponents $\beta$. In particular, when $\beta=0$, the minimum of $\langle{L}\rangle$ occurs at $\alpha=0$ regardless of the network structures, which reproduces the previous results based on the MFPT [@YLin2013]. However, this result is unreasonable in practice without considering the distinct costs induced by the nearest-neighborhood jumps and the long-range hops. In contrast, we find that when $\beta>0$, the profiles of different network organizations show clearly distinct behaviors. Specially, the profiles of the planar Sierpiński gasket display a clear minimum in the medium range $\alpha$, which minimizes the search distance, (i.e., the global mean first traverse distance). However, such behavior is absent for the (1,2)-flower model for $\beta>0$, where they present a clearly monotonous tendency, see Fig. \[f1\] (b). Such difference can be intuitively explained when referring to their topological properties. Specially, the Sierpiński gasket is a fractal network without the “small-world” property [@JJKozak2002], see its topological structure in Fig. \[f1\] (a). In contrast, the (1,2)-flower network has the “small-world” feature and the “scale-free” characteristics [@HDRozenfeld2007], as shown in Fig. \[f1\] (b). Meanwhile, we also notice that, when $\alpha$ is large, the transition probability of the Lévy walk degenerates to a generic random walk. Thereby, all curves approach a fixed value for $\alpha>9$, see in Fig. \[f1\] (a) and (b), as expected.
![The global MFTD $\langle{L}\rangle$ as a function of $\alpha$, for Lévy walks on (a) the planar Sierpiński gasket and (b) the (1,2) flower model with the same size $N=366$ nodes and $\beta=0, 0.5, 1$, respectively. Symbols represent the values of $\langle{L}\rangle$ found numerically, while solid lines correspond to the theoretical prediction of Eq. (\[7\]). Error bars represent the mean first traverse distance $\langle{L}\rangle$ over 20 tests and each test is averaged over the ensemble of 50,000 independent runs.[]{data-label="f1"}](1.eps){width="100.00000%" height="0.32\textheight"}
To further demonstrate the difference induced by network structure, we observe the size effect on the global MFTD $\langle{L}\rangle$ of the planar Sierpiński gasket and the (1,2)-flower model. We find that the profiles of each network present the same tendency for different network sizes $N$, see Fig. \[f5\] (a) and (b). Interestingly, the result presented in Fig. \[f5\] (a) clearly shows the presence of a minimum $\langle{L}\rangle$ for different network sizes at the same exponent $\alpha=2.8$. The way in which $\langle{L}\rangle$ scales with network size $N$ on the planar Sierpiński gasket seems to follow rather different behaviors depending on the tuning exponent $\alpha$. Specially, when $\alpha{\neq}2.8$, the global MFTD $\langle{L}\rangle$ follows a power law with network size $N$, see in Fig. \[f5\] (c). It is supported by observing the almost invariant values of the successive slopes $\delta_{s}$ obtained from $ln\langle{L}\rangle$ versus $lnN$, as shown in the inset of Fig. \[f5\] (c). Conversely, for $\alpha=2.8$, the successive slopes $\delta_{s}$ present a clearly decreasing tendency. However, for the (1,2)-flower model, the $\langle{L}\rangle$ follows approximately a power law with network size $N$, see in Fig. \[f5\] (d). Note that here we choose the cost exponent $\beta=1$ for convenience. However, such behavior of $\langle{L}\rangle$ versus $N$ is general for an arbitrary cost exponent $\beta$.
![The global MFTD $\langle{L}\rangle$ as a function of $\alpha$ for Lévy walks on (a) the planar Sierpiński gasket and (b) the (1,2) flower model over different network sizes $N$. The behaviors of $\langle{L}\rangle$ versus $N$ for different exponents $\alpha$ on the planar Sierpiński gasket (c) and the (1,2) flower model (d). In the insets, we show the plots of the successive slopes $\delta_{s}$ obtained from $ln\langle{L}\rangle$ versus $lnN$. Note that here we set the cost exponent $\beta=1$.[]{data-label="f5"}](2.eps){width="100.00000%" height="0.65\textheight"}
Clearly, from Eq. (\[7\]), the cost exponent $\beta$ plays an important role in controlling the search efficiency for Lévy walks. In order to explore how the optimal search efficiency of a Lévy walk changes with respect to the cost exponent $\beta$, we investigate the interplay between $\beta$ and $\alpha$ for various networks including three synthetic models (the Barabási-Albert (BA) model [@ABarabasi1999], the planar Sierpiński gasket [@JJKozak2002], and the (u,v)-flower model [@HDRozenfeld2007]) and two real networks (the “Dolphin” network [@DLusseau2003] and an e-mail network [@RGuimera2003]). Here, for a fair comparison, we calculate the measurement $log_{N}\langle{L}\rangle$ in the $(\alpha,\beta)$ plane for eliminating the size effect of networks. Generally, regions with smaller $log_{N}\langle{L}\rangle$ indicate an efficient way of search and transport based on Lévy walks. Fig. \[f2\] shows contour maps of $log_{N}\langle{L}\rangle$ in the $(\alpha,\beta)$ plane computed for these selected networks. Interestingly, we find that distinct network structures lead to different patterns in the corresponding $(\alpha,\beta)$ plane. Specifically, the $(\alpha,\beta)$ planes generated from networks having the “small-world” characteristics, such as the BA model and the (1,2)-flower model, demonstrate an “estuary” pattern, implying that Lévy walks are not the optimal way to search when $\beta>0.4$. In contrast, typical fractal networks without the “small-world” property, for example, the planar Sierpiński gasket and the (4,5)-flower model, result in a striking “flame” in the $(\alpha,\beta)$ planes, suggesting that there exists an optimal tuning exponent $\alpha$, which minimizes the traverse distance for a broad range of cost exponents $\beta$. However, none of these patterns match the ones found in the Dolphin network and the e-mail network, whose $(\alpha,\beta)$ planes show “rippled” features, meaning that the optimal exponent $\alpha$ gradually increases with the cost exponent $\beta$. The $(\alpha,\beta)$ plane uncovers the relationship between network structure and the behavior of Lévy walks, which provides information to help designing more effective search strategies and transport mechanisms in different environments.
![The measurement $log_{N}\langle{L}\rangle$ in the $(\alpha,\beta)$ parameter plane of (a) the BA model, (b) the (1,2)-flower model, (c) planar Sierpiński gasket, (d) the (4,5)-flower model, (e) the “Dolphin” network [@DLusseau2003], and (f) the e-mail network [@RGuimera2003].[]{data-label="f2"}](3.eps){width="95.00000%" height="0.8\textheight"}
Furthermore, we follow the spirit of the MFPT, and extract more statistics from the MFTD. Here, we introduce the average trapping distance (ATD) defined as follows: $$\ L_{j}=\frac{1}{1-w_{j}}\sum_{m=1}^{N}w_{m}l_{mj}.
\label{27}$$ The ATD $L_{j}$ quantifies the mean of MFTD $l_{mj}$ to the trap node $j$, taken over all starting points with the stationary distribution. Submitting the results of Eqs. (\[5\]) and (\[6\]) into Eq. (\[27\]) yields (see appendices) $$\ L_{j}\approx{\frac{z_{jj}}{K_{j}}\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}},
\label{28}$$ where $K_{j}=\sum_{m}d_{jm}^{-\alpha}$ named the long-range degree of node $j$ [@APRiascos2012]. Specifically, when $\alpha$ is small, the diagonal values of $Z$ are almost same. In this context, a clear scaling behavior emerges such that $L_{j}\sim{K_{j}^{-1}}$ regardless of the underlying network structure. This is supported by observing the plots of $lnL_{j}$ vs $lnK_{j}$ shown in Fig. \[f3\] (a) and (b). With an increase of $\alpha$, the slope of $lnL_{j}$ versus $lnK_{j}$ gradually decreases and finally asymptotically approaches to that of random walks as described in Ref. . Results demonstrate the important role of $\alpha$ in shaping the ATD. Meanwhile, from Eq. (\[28\]), it is easy to verify that the relationship between $lnL_{j}$ and $ln K_{j}$ does not depend on the cost exponent $\beta$. So, the profiles present a similar tendency for different cost exponents $\beta$ as illustrated in Fig. \[f3\] (c) and (d). We further find a linear relationship between $lnL_{j}$ and $\beta$, when fixing the tapping position $j$ and the tuning exponent $\alpha$, see the insets in Fig. \[f3\] (c) and (d). The results are consistent with our theoretical prediction of the relationship $lnL_{j}\sim{C\beta}$, where $C$ is a constant value related to the fractal dimension of a given network (see appendices).
![Plots of $lnL_{j}$ versus $lnK_{j}$ are presented for (a) the (1,2)-flower model and (b) the BA model with network size $N=3282$. The same plots with respect to different cost exponents $\beta$ for (c) the (1,2)-flower mode and (d) the BA model under the cost exponent $\alpha=3$. In the inset, we show $lnL_{j}$ versus $\beta$ for different trapping nodes $j$. Note that here the values of $lnL_{j}$ are calculated based on the Eq. (\[27\]).[]{data-label="f3"}](4.eps){width="100.00000%" height="0.65\textheight"}
**The optimal condition of the PageRank search based on the MFTD theory** We finally apply the MFTD theory to characterize the famous PageRank search [@ANLangville2006]. The PageRank search is widely used to compute the relevance of web pages. The transition probability $p_{ij}$ of the PageRank search is $$\ p_{ij}={\mu}\frac{a_{ij}}{k_{i}}+(1-\mu){\frac{1}{N}}
\label{31}$$ where $k_{i}=\sum_{l}a_{il}$ is the degree of node $i$ and $\mu$ is the damping factor lying in the range $[0,1]$. We investigate the global MFTD $\langle{L}\rangle$ for the PageRank search on two real networks (web-Stanford [@JLeskovec2009] and Ego-Facebook [@JLeskovec2012]). The results presented in Fig. [\[f4\]]{} (a) and (b) indicate the existence of a minimum $\langle{L}\rangle$ for different cost exponents $\beta$ at the same value of the damping factor $\mu{\approx}0.85$, where optimal search is achieved. This is further supported by observing the contour maps of the $(\mu,\beta)$ plane, where for $\mu{\approx}0.85$, the global MFTD $\langle{L}\rangle$ is near its minimum value for a very broad range of $\beta$, see in Fig. [\[f4\]]{} (c) and (d). This can explain why the ad hoc damping factor of the PageRank search is suggested to be set around 0.85. Moreover, we notice that the minimum $\langle{L}\rangle$ of the PageRank search is much smaller than that of generic random walks (i.e., $\mu=1$), which in some extent demonstrates the advantage of taking the PageRank search instead of generic random walks.
![The global MFTD $\langle{L}\rangle$ as a function of the damping factor $\mu$, for the PageRank search on (a) web-Stanford [@JLeskovec2009] and (b) Ego-Facebook [@JLeskovec2012]. Symbols correspond to the theoretical prediction of Eq. (\[33\]). The global MFTD $\langle{L}\rangle$ in the $(\beta,\mu)$ parameter plane of (c) web-Stanford and (d) Ego-Facebook. Note that the web-Stanford network used here is a subgraph extracted from the original one for computation convenience with $N=2004$.[]{data-label="f4"}](5.eps){width="100.00000%" height="0.65\textheight"}
Discussion {#discussion .unnumbered}
==========
In summary, we have introduced the concept of the MFTD, a measure that takes into account of the cost of jumps in anomalous random walks, therefore is particularly suited to capture the interplay between the diffusion dynamics of anomalous random walks and underlying network structures. We obtain an exact expression for the MFTD and the global MFTD of anomalous random walks on complex networks. We show that our paradigm provides a unified scheme to characterize diffusion processes on networks, which incorporates the commonly used MFPT as a special case.
We then demonstrate the effectiveness of these measures by applying them to Lévy walks. We find that distinct network structures result in different patterns in the $(\alpha,\beta)$ planes, which explores the effect of the cost exponent $\beta$ on behaviors of Lévy walks with respect to network structure. Moreover, when addressing the trapping problem of Lévy walks, we find that its behavior only depends on the tuning exponent $\alpha$ irrespective of the cost exponent $\beta$. In particularly, when $\alpha$ is smaller, it presents a uniformly scaling feature regardless of network structure. These findings enrich our understanding of interplay between dynamics of Lévy walks and network structure. To implement Lévy walks, we need to compute all shortest paths of a network which involves high computational costs for large networks. In practice, one can use several excellent algorithms such as the preprocessing algorithm [@CSommer2014], which is one of possible solution for this problem. Nonetheless, the results show that for a broad range of the cost exponent $\beta$, the global MFTD $\langle{L}\rangle$ of Lévy walks is much smaller than that of generic random walks, which demonstrates the efficient for search and transport based on Lévy walks.
Finally, application to the famous PageRank search shows that the empirical damping factor is optimal at 0.85 for the cost exponent lying in the range $[0.9,1.3]$, at which individuals can optimize search in traverse distance. It is suggested that the time required for opening a new tab is approximately equivalent to that of following the hyperlinks for several turns for a related topic search. Thereby, in practice, the damping factor of the PageRank search is chosen around 0.85. Overall, our findings offer a new framework to understand the diffusion dynamics of anomalous random walks on complex networks.
Appendices {#appendices .unnumbered}
==========
**The analytic expression of mean first traverse distance** We follow the derivation of MFPT in Ref. to calculate the MFTD on networks. We consider an arbitrary finite network consisting of $N$ nodes. The connectivity is represented by the adjacency matrix $A$, whose entries $a_{ij}=1$ (or 0) if there is (not) a link from nodes $i$ to $j$. Let $D$ denote the distance matrix with elements $d_{ij}$ representing the shortest path length from node $i$ to node $j$. In the process of anomalous random walks, at each step, the walker starting from node $i$ arrives to node $j$ with a non-zero transition probability $p_{ij}$ regardless of the connectivity between nodes $i$ and $j$. If the first step of the walk is to node $j$, the expected traverse distance required is $d_{ij}^{\beta}$; if it is to some other node $k$, the expected traverse distance becomes $l_{kj}$ plus $d_{ik}^{\beta}$ for the previous step already taken. Thus, we obtain $$\ l_{ij}=p_{ij}d_{ij}^{\beta}+\sum_{k\neq{j}}p_{ik}(l_{kj}+d_{ik}^{\beta}),
\label{51}$$ where $l_{ij}$ is the mean first traverse distance from node $i$ to node $j$. Since $l_{jj}=0$, Eq. (\[51\]) can be rewritten as $$\ l_{ij}=\sum_{m}{p_{im}d_{im}^{\beta}}+\sum_{k}p_{ik}l_{kj}.
\label{52}$$ Let $r_{i}$ denote the mean first return distance to node $i$ starting from node $i$. In the same manner, $r_{i}$ can be represented as $$\ r_{i}=\sum_{k}p_{ik}(l_{ki}+d_{ik}^{\beta}).
\label{53}$$ Combining Eq. (\[52\]) and Eq. (\[53\]) together, we obtain the relation $$\ (I-P)L=C-R,
\label{54}$$ where $I$ denotes the identity matrix, and $$L=\left(\begin{array}{cccc}
l_{11}&l_{12}& \cdots &l_{1n}\\
l_{21}&l_{22}&\cdots&l_{2n}\\
\vdots&\vdots&\vdots&\vdots\\
l_{n1}&l_{2n}&\cdots&l_{nn}
\end{array} \right),$$
$$C=\left(\begin{array}{cccc}
\sum_{k}p_{1k}d_{1k}^{\beta}&\sum_{k}p_{1k}d_{1k}^{\beta}
& \cdots&\sum_{k}p_{1k}d_{1k}^{\beta}\\
\sum_{k}p_{2k}d_{2k}^{\beta}& \sum_{k}p_{2k}d_{2k}^{\beta}& \cdots&\sum_{k}p_{2k}d_{2k}^{\beta}\\
\vdots&\vdots&\vdots&\vdots\\
\sum_{k}p_{Nk}d_{Nk}^{\beta}& \sum_{k}p_{Nk}d_{Nk}^{\beta}&\cdots& \sum_{k}p_{Nk}d_{Nk}^{\beta}
\end{array} \right),$$
$$R=\left(\begin{array}{cccc}
r_{1}&0& \cdots &0\\
0&r_{2}&\cdots&0\\
\vdots&\vdots&\vdots&\vdots\\
0&0&\cdots&r_{n}
\end{array} \right).$$
Multiplying both sides of Eq. (\[54\]) by the matrix $W=\left(\begin{array}{cccc}
w_{1}&w_{2}&\cdots&w_{N}\\
w_{1}&w_{2}&\cdots&w_{N}\\
\vdots&\vdots&\vdots&\vdots\\
w_{1}&w_{2}&\cdots&w_{N}
\end{array} \right)$ with the element $w_{i}$ being the $i$th component of the stationary distribution, and using the fact that $$\ W(I-P)=0
\label{55}$$ gives $$\ WC-WR=0.
\label{56}$$ From Eq. (\[56\]), the mean first return distance $r_{i}$ reads $$\ r_{i}=\frac{\sum_{k}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)w_{k}}{w_{i}}.
\label{57}$$ Since the matrix $(I-P+W)$ has an inverse [@Grinstead2006], we denote $Z=(I-P+W)^{-1}$. Multiplying both sides of Equation (\[54\]) by $Z$ and using the fact that $$\ I-W=Z(I-P)
\label{58}$$ gives $$\ L=ZC-ZR+WL.
\label{59}$$ From the above equation, $l_{ij}$ and $l_{jj}$ can be expressed as $$\ l_{ij}= \sum_{k}z_{ik}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)-z_{ij}r_{j}+(wL)_{j}
\label{10}$$ and $$\ l_{jj}= \sum_{k}z_{jk}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)-z_{jj}r_{j}+(wL)_{j}.
\label{11}$$ Since $l_{jj}=0$ and using Eq. (\[57\]), one has $$\ l_{ij}=T_{ij}\sum_{k}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)w_{k}+\sum_{k}(z_{ik}-z_{jk})\left(\sum_{m}p_{km}d_{km}^{\beta}\right),
\label{12}$$ where $T_{ij}=\frac{z_{jj}-z_{ij}}{w_{j}}$ is the mean first passage time.
**The analytic expression of global mean first traverse distance** To further evaluate the search efficiency based on anomalous random walks, we introduce the global mean first traverse distance defined as $$\ \langle{L}\rangle=\frac{1}{N(N-1)}\sum_{i}\sum_{j}{l_{ij}}.
\label{13}$$ Plugging Eq. (\[12\]) into Eq. (\[13\]), we obtain $$\ \langle{L}\rangle=\langle{T}\rangle{\sum_{k}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)w_{k}}+\frac{1}{N(N-1)}\sum_{i}\sum_{j}\sum_{k}(z_{ik}-z_{jk})\left(\sum_{m}p_{km}d_{km}^{\beta}\right),
\label{14}$$ where $\langle{T}\rangle=\frac{1}{N(N-1)}\sum_{i}\sum_{j}{T_{ij}}$ is the global mean first passage time. Since column vectors of the matrix $C$ are the same, the column vectors of the matrix $ZC$ is also the same. Then, the last term of Eq. (\[14\]) will vanish due to that $$\ \sum\sum{(ZC-(ZC)^{T}})=0,
\label{15}$$ where the matrix $(ZC)^{T}$ represents the transpose of matrix $ZC$. So, the expression for $\langle{L}\rangle$ is reduced to $$\ \langle{L}\rangle=\langle{T}\rangle{\sum_{k}\left(\sum_{m}p_{km}d_{km}^{\beta}\right)w_{k}}.
\label{16}$$
**The analytic expression of average trapping distance for Lévy walks** We now study the trapping problem for Lévy walks at an arbitrarily given node. Let $L_{j}$ be the average trapping distance, which is the mean of MFTD $L_{ij}$ to the trap node $j$, taken over the stationary distribution defined as follows: $$\ L_{j}=\frac{1}{1-w_{j}}\sum_{i=1}^{N}w_{i}l_{ij}.
\label{17}$$ Substituting the expression of $l_{ij}$ in Eq. (\[6\]) and $w_{j}$ in Eq. (\[5\]) into Eq. (\[17\]) gives $$\ L_{j}=\frac{1}{1-w_{j}}\sum_{i=1}^{N}w_{i}\left(\frac{z_{jj}-z_{ij}}{w_{j}}\frac{\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}}{\sum_{i}\sum_{j}d_{ij}^{-\alpha}}\right)+\frac{1}{1-w_{j}}\sum_{i=1}^{N}w_{i}\left(\sum_{k}(z_{ik}-z_{jk})\left(\frac{\sum_{m}d_{km}^{\beta-\alpha}}{\sum_{m}d_{km}^{-\alpha}}\right)\right).
\label{18}$$ Using the fact that $wZ=w$ [@Grinstead2006] and with some calculation one obtains
$$\ L_{j}=\frac{1}{1-w_{j}}\frac{z_{jj}}{w_{j}}\frac{\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}}{\sum_{i}\sum_{j}d_{ij}^{-\alpha}}+\frac{1}{1-w_{j}}\sum_{k}z_{jk}\left(\frac{\sum_{m}d_{km}^{\beta-\alpha}}{\sum_{m}d_{km}^{-\alpha}}\right).
\label{19}$$
Empirically we find that the simulation values of the last term is far less than that of the first term and can be neglected in the analysis. In this context, Eq .(\[19\]) reduces to $$\ L_{j}\approx{\frac{z_{jj}}{K_{j}}\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}},
\label{20}$$ where $K_{j}=\sum_{m}d_{jm}^{-\alpha}$ named the long-range degree of node $j$ [@APRiascos2012]. Here, we omit the value $w_{j}$ as it can be approximated as zero when the network size $N$ is very large. Moreover, for the fractal network with the fractal dimension $d_{f}$, the network diameter $M$ can be approximated as $M\sim{N^{\frac{1}{d_{f}}}}$. Approximating $M$ as a continuous variable, the term $\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}$ scales as [@Li2013] $$\
\sum_{i}\sum_{j}d_{ij}^{\beta-\alpha}\sim{N\int_{1}^{M}x^{\beta-\alpha}
x^{d_{f}-1}dx}\sim
\begin{cases}
N\frac{N^{\frac{d_{f}+\beta-\alpha}{d_{f}}}-1}{\beta+d_{f}-\alpha}, & \alpha\neq{d_{f}+\beta} \\
\frac{Nln{N}}{d_{f}}, & \alpha=d_{f}+\beta\\
\end{cases}.
\label{30}$$ Plugging Eq. (\[30\]) into Eq. (\[20\]), we have a linear relationship between $lnL_{j}$ and $\beta$ (i.e., $lnL_{j}\sim{C\beta}$ where $C$ is a constant value determined by the fractal dimension $d_{f}$), when the position of the trapping node $j$ and the tuning exponent $\alpha$ are fixed.
[10]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
. ** ****, ().
& . ** ****, ().
, , , & . ** ****, ().
, & . ** ****, ().
*et al.* . ** ****, ().
& . ** ****, ().
& . ** ****, ().
, & . ** ****, ().
*et al.* . ** ****, ().
, & . ** ****, ().
. ** ****, ().
*et al.* . ** ****, ().
& . ** ****, ().
& . ** ****, ().
& ().
, , & . ** ****, ().
& ().
& ().
& . ** ****, ().
, & . ** ****, ().
& . ** ****, ().
& . ** ****, ().
*et al.* . ** ****, ().
, , , & . ** ****, ().
, , & . ** ****, ().
& . ** ****, ().
*et al.* . ** ****, ().
. ** ****, ().
Acknowledgements {#acknowledgements .unnumbered}
================
J.Z. is supported by National Science Foundation of China NSFC 61104143. We thank Kai Zhang for useful discussions and helps.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
T. F. Weng, J. Zhang, M. Small and P. Hui designed the research, performed the research, and wrote the manuscript. K. Moein and R. Zheng analyzed data and performed research. All authors reviewed the manuscript.
Additional information {#additional-information .unnumbered}
======================
**Competing financial interests:** The authors declare no competing financial interests.
|
---
abstract: 'Tunneling from a two–dimensional contact into quantum–Hall edges is considered theoretically for a case where the barrier is extended, uniform, and parallel to the edge. In contrast to previously realized tunneling geometries, details of the microscopic edge structure are exhibited directly in the voltage and magnetic–field dependence of the differential tunneling conductance. In particular, it is possible to measure the dispersion of the edge–magnetoplasmon mode, and the existence of additional, sometimes counterpropagating, edge–excitation branches could be detected.'
author:
- 'U. Zülicke'
- 'E. Shimshoni'
- 'M. Governale'
title: 'Momentum-Resolved Tunneling into Fractional Quantum Hall Edges'
---
The quantum Hall (QH) effect[@qhe-sg] arises due to incompressibilities developing in two–dimensional electron systems (2DES) at special values of the electronic sheet density $n_0$ and perpendicular magnetic field $B$ for which the [*filling factor*]{} $\nu = 2\pi\hbar c\, n_0/|e B|$ is equal to an integer or certain fractions. The microscopic origin of incompressibilities at fractional $\nu$ is electron–electron interaction. Laughlin’s trial–wave–function approach [@rbl:prl:83] successfully explains the QH effect at $\nu=\nu_{1,p}\equiv 1/(p+1)$ where $p$ is a positive even integer. Our current microscopic understanding of why incompressibilities develop at many other fractional values of the filling factor, e.g., $\nu_{m,p}\equiv m/(m p + 1)$ with nonzero integer $m\ne\pm 1$, is based on hierarchical theories [@fdmh:prl:83; @bih:prl:84; @jain:prl:89].
The underlying microscopic mechanism responsible for creating charge gaps at fractional $\nu$ implies peculiar properties of low–energy excitation in a finite quantum–Hall sample which are localized at the boundary [@ahmintro]. For $\nu=\nu_{m,p}$, $m$ branches of such edge excitations [@ahm:prl:90; @wen:prb:90; @wen:int:92; @wen:adv:95] are predicted to exist which are realizations of strongly correlated chiral one–dimensional electron systems called [*chiral Luttinger liquids*]{} ($\chi$LL). Extensive experimental efforts were undertaken recently to observe $\chi$LL behavior because this would yield an independent confirmation of our basic understanding of the fractional QH effect. In all of these studies[@webb:ssc:96; @amc:prl:96; @amc:prl:98; @amc:prl:01; @matt:prl:01; @hilke:prl:01], current–voltage characteristics yielded a direct measure of the energy dependence of the [*tunneling density of states*]{} for the QH edge. This quantity generally contains information on global dynamic properties as, e.g., excitation gaps and the orthogonality catastrophe, but lacks any momentum resolution. Power–law behavior consistent with predictions from $\chi$LL theory was found[@webb:ssc:96; @amc:prl:96; @matt:prl:01] for the edge of QH systems at the Laughlin series of filling factors, i.e., for $\nu=
\nu_{1,p}$. However, at hierarchical filling factors, i.e., when $\nu=\nu_{m,p}$ with $|m|>1$, predictions of $\chi$LL theory are, at present, not supported by experiment [@amc:prl:98; @amc:prl:01]. This discrepancy inspired theoretical works, too numerous to cite here, from which, however, no generally accepted resolution emerged. Current experiments[@hilke:prl:01] suggest that details of the edge potential may play a crucial rôle. New experiments are needed to test the present microscopic picture of fractional–QH edge excitations.
![Schematic picture of tunneling geometry. Two mutually perpendicular two-dimensional electron systems are realized, e.g., in a semiconductor heterostructure. An external magnetic field is applied such that it is perpendicular to one of them (2DES$_\perp$) but in-plane for the other one (2DES$_\parallel$). When 2DES$_\perp$ is in the quantum-Hall regime, chiral edge channels form along its boundary (indicated by broken lines with arrows). Where they run parallel to 2DES$_\parallel$, electrons tunnel between edge states in 2DES$_\perp$ and plane-wave states in 2DES$_\parallel$ with the [*same*]{} quantum number $p_y$ of momentum component parallel to the barrier. Experimentally, the differential tunneling conductance $dI/dV$ is measured. \[setup\]](setup.eps){width="3.2in"}
Here we consider a tunneling geometry that is particularly well-suited for that purpose, see Fig. \[setup\], and which has been realized recently for studying the integer QH effect in cleaved-edge overgrown semiconductor heterostructures[@matt:physe:02]. In contrast to previous experiments, it provides a [*momentum–resolved*]{} spectral probe of QH edge excitations[^1]. With both the component of canonical momentum parallel to the barrier and energy being conserved in a single tunneling event, strong resonances appear in the differential tunneling conductance $dI/dV$ as a function of the transport voltage and applied magnetic field. Similar resonant behavior for tunneling via extended uniform barriers has been used recently[@jpe:apl:91; @cav:prl:96; @eaves:sci:00; @ophir:sci:02] to study the electronic properties of low–dimensional electron systems. It has also been suggested as a tool to observe spin–charge separation in Luttinger liquids[@hekk:prl:99] and the interaction–induced broadening of electronic spectral functions at single-branch QH edges[@uz:prb-rc:96]. Here we find that the number of resonant features in $dI/dV$ corresponds directly to the number of chiral edge excitations present. Edge–magnetoplasmon dispersion curves can be measured and power laws related to $\chi$LL behavior be observed. Momentum–resolved tunneling spectroscopy in the presently considered geometry thus constitutes a powerful probe to characterize the QH edge microscopically.
![Spectral functions for two–branch hierarchical fractional–QH edges at bulk filling factor $2/3$ \[panel a)\] and $2/5$ \[panel b)\], where the charged (edge–magnetoplasmon) mode is assumed to be left–moving. a) We show $A_{2/3}^{(0)}(q,\varepsilon
)\equiv A_{2/3}^{(1)}(q,\varepsilon)$ for a fixed value of $q$. Note the similarity with the spectral function of a spinless Luttinger liquid[@med:prb:92; @voit:prb:93]. The only difference is that, in our case, velocities of right–moving and left–moving plasmon modes are not equal. b) $A_{2/5}^{(0)}(q,\varepsilon)\equiv
A_{2/5}^{(-1)}(q,\varepsilon)$ at fixed $q$. It is reminiscent of the spectral function for a spinful $\chi$LL exhibiting spin–charge separation[@fab:prl:93; @voit:prb:93] but differs due to the absence of any algebraic divergence at $-v_{\text{n}}q$. \[spectr\]](spectr.eps){width="3.3in"}
To compute the tunneling conductances, we apply the general expression for the current obtained to lowest order in a perturbative treatment of tunneling[@mahan]: $$\begin{aligned}
\label{geniv}
I(V)&=&\frac{e}{\hbar^2}\sum_{\vec k_\parallel, n, X} |t_{\vec
k_\parallel,n,X}|^2 \int\frac{d\varepsilon}{2\pi}\left\{
n_{\text{F}}(\varepsilon)-n_{\text{F}}(\varepsilon+eV)\right\}
\nonumber \\ && \hspace{2cm}\times A_\parallel (\vec k_\parallel,
\varepsilon) \, A_\perp(n, X, \varepsilon+eV) \, .\end{aligned}$$ Here $A_\parallel$ and $A_\perp$ denote single-electron spectral functions for 2DES$_\parallel$ and 2DES$_\perp$, respectively. (See Fig. \[setup\]). We use a representation where electron states in the first are labeled by a two–dimensional wave vector[^2] $k_\parallel=(k_y, k_z)$, while the quantum numbers of electrons in 2DES$_\perp$ are the Landau–level index $n$ and guiding–center coordinate $X$ in $x$ direction. We assume that 2DES$_\parallel$ is located at $x=0$. The simplest form of the tunneling matrix element $t_{\vec k_\parallel,n,X}$ reflecting translational invariance in $y$ direction yields $$t_{\vec k_\parallel,n,X}=t_n(X)\,\,\delta (k_y-k)\quad ,$$ where $k\equiv X/\ell^2$ with the magnetic length $\ell=\sqrt{\hbar
c/|e B|}$. The dependence of $t_n(X)$ on $X$ results form the fact that an electron from 2DES$_\perp$ occupying the state with quantum number $X$ is spatially localized on the scale of $\ell$ around $x=
X$. The overlap of its tail in the barrier with that of states from 2DES$_\parallel$ will drop precipitously as $X/\ell$ gets large. Finally, $n_{\text{F}}(\varepsilon)=[\exp(\varepsilon/k_{\text{B}}
T)+1]^{-1}$ is the Fermi function. In the following, we use the expression $A_\parallel(\vec k_\parallel,\varepsilon)=2\pi\delta(
\varepsilon-E_{\vec k_\parallel})$ which is valid for a clean system of noninteracting electrons[^3]. Here $E_{\vec k_\parallel}$ denotes the electron dispersion in 2DES$_\parallel$.
The spectral function of electrons in 2DES$_\perp$ depends crucially on the type of QH state in this layer. At integer $\nu$, when single–particle properties dominate and disorder broadening is neglected, it has the form $$\label{intspec}
A_\perp(n,X,\varepsilon)\equiv A_n(k,\varepsilon)=2\pi\delta(
\varepsilon-E_{nk})\quad ,$$ where $E_{nk}$ is the Landau–level dispersion. Strong correlations present at fractional $\nu$ alter the spectral properties of edge excitations. In the low–energy limit, it is possible to linearize the lowest–Landau–level dispersion around the Fermi point $k_{\text{F}}$. At the Laughlin series $\nu=1/(p+1)$ and for short-range interactions present at the edge, the spectral function was found[@wen:int:92; @jujo:prl:96] to be $$\label{laughspec}
A_{\frac{1}{p+1}}(q,\varepsilon) = \frac{z}{p!}
\left(\frac{q}{2\pi/L_y}\right)^p \delta\left(\varepsilon-r
\hbar v_{\text{e}}q\right) .$$ Here $q\equiv k-k_{\text{F}}$, $r=\pm$ distinguishes the two chiralities of edge excitations, $L_y$ is the edge perimeter, $v_{\text{e}}$ the edge–magnetoplasmon velocity, and $z$ an unknown normalization constant. The power–law prefactor of the $\delta$–function in Eq. (\[laughspec\]) is a manifestation of $\chi$LL behavior.
The main focus of our work is the sharp QH edge at hierarchical filling factors. Here we provide explicitly the momentum–resolved spectral functions for $\nu=\nu_{\pm 2,p}$[^4]. Microscopic theories[@ahm:prl:90; @wen:int:92] predict the existence of two Fermi points $k_{\text{Fo}}$ and $k_{\text{Fi}}$ which correspond to outer and inner single-branch chiral edges of QH fluids at Laughlin–series filling factors $\nu_{\text{o}}^\pm=1/(p\pm 1)$ and $\nu_{\text{i}}^\pm=\pm 1/[(2p\pm
1)(p\pm 1)]$, respectively. The negative sign of $\nu_{\text{i}^-}$ indicates that the inner edge mode is counterpropagating. We have used standard bosonization methods[@vondelft] applied to fractional–QH edges[@wen:int:92] for the calculation of the spectral functions. As these have not been obtained before, we briefly discuss their main features here.
According to $\chi$LL theory, the existence of two Fermi points gives rise to a discrete infinite set of possible electron tunneling operators at the edge. This is because an arbitrary number $N$ of fractional–QH quasiparticles with charge equal to $e
\nu_{\text{o}}^\pm$ can be transferred to the inner edge after an electron has tunneled into the outer one[@wen:int:92]. Each of these processes gives rise to a separate contribution to the electronic spectral function at the edge which is of the general form
$$\begin{aligned}
A^{(N)}_{\nu_{\pm 2,p}}(q,\varepsilon)&=&\frac{2\pi z}{\Gamma
(\eta^{(N)}_1)\Gamma(\eta^{(N)}_2)}\left(\frac{L_y/2\pi\hbar}{|v_1
\mp v_2|}\right)^{\eta^{(N)}_1+\eta^{(N)}_2-1}\left|\varepsilon-
r\hbar v_1 q \right|^{\eta^{(N)}_2-1}\left|\varepsilon\mp r\hbar
v_2 q \right|^{\eta^{(N)}_1-1}\nonumber \\ && \hspace{2cm}\times
\left\{\Theta\left(r\hbar v_1 q - \varepsilon\right)\Theta\left(\pm
\varepsilon-r\hbar v_2 q\right) + \Theta\left(\varepsilon - r\hbar
v_1 q\right)\Theta\left(r\hbar v_2 q\mp\varepsilon\right)\right\}.\end{aligned}$$
Here $q\equiv k-k_{\text{F}}^{(N)}$, where $k_{\text{F}}^{(N)}=
k_{\text{Fo}}-N\nu_{\text{o}}^\pm(k_{\text{Fo}}-k_{\text{Fi}})$. The velocities $v_1>v_2>0$ of normal–mode edge–density fluctuations and the exponents $\eta_{1,2}^{(N)}$ depend strongly on microscopic details of the edge, e.g., the self-consistent edge potential and inter–edge interactions. We focus here on the experimentally realistic case when inner and outer edges are strongly coupled and the normal modes correspond to the familiar[@cllreview] charged and neutral edge-density waves[^5]. In this limit, we have[@cllreview; @uz:prb:98] $v_1=v_{\text{c}}\sim {\mathcal O}
(\log[L_y/\ell])$, $v_2=v_{\text{n}}\sim {\mathcal O}(1)$ (where c and n denote charged and neutral, respectively), and the exponents assume universal values: $\eta_1^{(N)}=\eta_{\text{c}}\equiv p\pm1/
2$, $\eta_2^{(N)}=\eta_{\text{n}}^{(N)}\equiv(2 N\pm 1)^2/2$. Note that exponents are generally larger than unity except for $N=0,\mp
1$ where $\eta_2^{(N)}=1/2$. In the latter case, an algebraic singularity appears in the spectral function. This is illustrated in Fig. \[spectr\]. Such divergences will be visible as strong features in the differential tunneling conductance; see below. Contributions to the spectral function for all other values of $N$ do not show such divergences and will give rise only to a featureless background in the conductance.
With spectral functions for 2DES$_\perp$ at hand, we are now able to calculate tunneling transport. We focus first on the case when 2DES$_\perp$ is in the QH state at $\nu=1$. For realistic situations, the differential tunneling conductance $dI/dV$ as a function of voltage $V$ and magnetic field $B$ will exhibit two lines of strong maxima whose positions in $V$–$B$ space are given by the equations
\[edgedisp\] $$\begin{aligned}
\label{firstmax}
E_{0 k_V} &=& \varepsilon_{\text{F}\perp} \quad , \\
\label{secdmax}
E_{0 k_{\text{F}\parallel}} &=& \varepsilon_{\text{F}\perp} + e V
\quad .\end{aligned}$$
Here $k_V=\sqrt{2 m (\varepsilon_{\text{F}\parallel}- eV)/\hbar^2}$ and $k_{\text{F}\parallel}$, the Fermi wave vector in 2DES$_\parallel$, are the extremal wave vectors for which momentum–resolved tunneling occurs. Fermi energies in 2DES$_{\perp,
\parallel}$ are denoted by $\varepsilon_{\text{F}\perp,\parallel}$. Eqs. (\[edgedisp\]) can be used to extract the lowest–Landau–level dispersion $E_{0 k}$ from maxima in the experimentally obtained $dI/dV
$, thus enabling microscopic characterization of real QH edges.
When 2DES$_\perp$ is in a QH state at a Laughlin–series filling factor $\nu_{1,p}$, it supports a single branch of edge excitations just like at $\nu=1$, and the calculation of the differential tunneling conductance proceeds the same way. The major difference is, however, the vanishing of spectral weight at the Fermi point of the edge; compare Eqs. (\[intspec\]) and (\[laughspec\]). This results in the suppression of maxima described by Eq. (\[firstmax\]), while those given by Eq. (\[secdmax\]) remain. The intensity of the latter rises along the curve as a power law with exponent $p$.
Finally, we discuss the case of hierarchical filling factors $\nu_{\pm
2, p}$ which are expected to support two branches of edge excitations. To be specific, we consider filling factors $2/3$ and $2/5$. In both cases, there are many contributions to the spectral function and, hence, the differential tunneling conductance. However, only two of these exhibit algebraic singularities. It turns out that these singularities give rise to either a strong maximum or a finite step in the differential tunneling conductance, depending on the sign of voltage. (See Fig. \[twobranch\]). The strong maximum results from a logarithmic divergence that occurs when $eV=\hbar v_{\text{c}}(
k_{\text{F}}^{(N)}-k_{\text{F}\parallel})$. Both the maximum and the step edge follow the dispersion of the charged edge–magnetoplasmon mode and would therefore enable its experimental investigation. Most importantly, however, the two spectral functions with singularities exhibit them slightly shifted in guiding–center, i.e., $k$ direction by an amount $\nu_{\text{o}}^\pm(k_{\text{Fo}}-k_{\text{Fi}})$. Hence, two maxima and a double–step feature should appear in the differential tunneling conductance whose distance in magnetic–field direction will be a measure of the separation of inner and outer edges. Observation of this doubling would yield an irrefutable confirmation of the expected multiplicity of excitation branches at hierarchical QH edges.
![Gray–scale plot of singular contributions to the differential conductance for tunneling into the two–branch QH edge at filling factor $2/3$. A qualitatively similar plot is obtained for filling factor $2/5$. Note the strong maximum rising as a power law for negative bias, which is continued as a step edge for positive bias. Its position in the $eV$–$\delta_N$ plane follows a line whose slope corresponds to the edge–magnetoplasmon velocity $v_{\text{c}}$. To obtain the plot, we have linearized the spectrum in 2DES$_\parallel$ and absorbed the magnetic–field dependence into the parameter $\delta_N=k_{\text{F}}^{(N)} - k_{\text{F}\parallel}$. As there are two such singular contributions to $dI/dV$ with $N=0,1$ which have different $\delta_N$, a doubling of resonant features shown in this plot would be observed experimentally.\[twobranch\]](g23fig.eps){width="3.2in"}
In conclusion, we have calculated the differential conductance for momentum–resolved tunneling from a 2DES into a QH edge. Maxima exhibited at $\nu=1$ follow two curves in $V$–$B$ parameter space whose expression we give in terms of the lowest–Landau–level dispersion. Their explicit form enables edge–dispersion spectroscopy. At Laughlin–series filling factors, $\chi$LL behavior results in the suppression of one of these maxima and characteristic power–law behavior exhibited by the other one. The multiplicity of edge modes at hierarchical filling factors corresponds directly to the multiplicity of maxima in the differential tunneling conductance.
We thank M. Grayson and M. Huber for many useful discussions and comments on the manuscript. This work was supported by DFG Grant No. ZU 116 and the DIP project of BMBF. U.Z. enjoyed the hospitality of Sektion Physik at LMU München when finishing this work.
[30]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, eds., ** (, , ), ed.
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, in **, ed. by (, , ).
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , (); .
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ** (, , ).
, ****, ().
, ****, ().
, in **, ed. by (, , ).
, , , ****, ().
[^1]: Tunneling from a [*three–dimensional*]{} contact into a QH edge, measured in Refs. , cannot resolve momentum even with perfect translational invariance parallel to the edge. The latter is destroyed anyway, in real samples, by dopant–induced disorder in the bulk contact. See also a related tunneling spectroscopy of parallel QH edges by W. Kang [*et al.*]{}, Nature (London) [**403**]{}, 59 (2000).
[^2]: Here we neglect magnetic–field–induced subband mixing in 2DES$_\parallel$. While this can be straightforwardly included, it will typically result in small quantitative changes only.
[^3]: Broadening due to scattering from disorder or interactions can be straightforwardly included and does not change the main results of our study.
[^4]: Generalization to $|n|>2$ is possible but does not add qualitatively new physical insight.
[^5]: Expressions for the general case will be given elsewhere.
|
---
abstract: 'When dealing with continuous single-objective problems, multimodality poses one of the biggest difficulties for global optimization. Local optima are often preventing algorithms from making progress and thus pose a severe threat. In this paper we analyze how single-objective optimization can benefit from multiobjectivization by considering an additional objective. With the use of a sophisticated visualization technique based on the multi-objective gradients, the properties of the arising multi-objective landscapes are illustrated and examined. We will empirically show that the multi-objective optimizer MOGSA is able to exploit these properties to overcome local traps. The performance of MOGSA is assessed on a testbed of several functions provided by the COCO platform. The results are compared to the local optimizer Nelder-Mead.'
author:
- |
Vera Steinhoff\
Statistics and Optimization\
University of M[ü]{}nster\
M[ü]{}nster, Germany\
`v.steinhoff@uni-muenster.de`\
Pascal Kerschke\
Statistics and Optimization\
University of M[ü]{}nster\
M[ü]{}nster, Germany\
`kerschke@uni-muenster.de` Christian Grimme\
Statistics and Optimization\
University of M[ü]{}nster\
M[ü]{}nster, Germany\
`christian.grimme@uni-muenster.de`
bibliography:
- 'somogsa.bib'
title: |
Empirical Study on\
the Benefits of Multiobjectivization for\
Solving Single-Objective Problems
---
Introduction
============
Optimization is essentially everywhere. Whether it is in the field of production, logistics, in medicine or biology; everywhere the global optimal solution or the set of global optimal solutions is sought. However, most real-world problems are of non-linear nature and naturally multimodal which poses severe problems to global optimization. Multimodality, the existence of multiple (local) optima, is regarded as one of the biggest challenges for continuous single-objective problems [@preuss2015]. A lot of algorithms get stuck searching for the global optimum or are requiring many function evaluations to escape local optima. One of the most popular strategies for dealing with multimodal problems are population-based methods like evolutionary algorithms due to their global search abilities [@beyer2001ES].
In this paper we will examine another approach of coping with local traps, namely *multiobjectivization*. By transforming a single-objective into a multi-objective problem, we aim at exploiting the properties of multi-objective landscapes. So far, the characteristics of single-objective optimization problems have often been directly transferred to the multi-objective domain. Due to the existence of multiple objectives to be optimized, visualization seemed to be much more difficult. Thus, multimodality was regarded as an even bigger challenge for multi-objective problems. However, as empirically shown and hypothesized in papers by Kerschke and Grimme [@GrimmeKT2019Multimodality], local optima in continuous multi-objective optimization should not be regarded as challenges but as chances for finding the global efficient set. By visualizing multi-objective problems in gradient-based heatmaps, insights into the landscape and specific characteristics such as basins of attraction and efficient sets could be gained [@MOlandscapes; @kerschke2018search]. Furthermore, local efficient sets were observed to be cut by ridges whereas global efficient sets should be ridge-free. Based on these observations, the Multi-Objective Gradient Sliding Algorithm (MOGSA) was developed to follow the path of the multi-objective gradients and explore the efficient sets to finally find the globally efficient one [@GrimmeKT2019Multimodality]. Instead of getting trapped in a local optima, MOGSA crosses ridges and thereby efficiently steers into more promising basins of attraction. To overcome the problems caused by multimodality in the single-objective case, we build on previous research of Kerschke and Grimme [@GrimmeKT2019Multimodality; @MOlandscapes] and use a multi-objective optimizer to tackle single-objective problems.
[|p[16.2cm]{}|]{}
------------------------------------------------------------------------
*Contribution:* The local optimizer MOGSA will be applied to a set of initially single-objective problems, which were transformed into multi-objective ones by adding an additional objective. We show how locality in the multi-objective landscape can guide the algorithm towards the global optimum of the single-objective problem. Our aim is not to challenge existing algorithms but to show that multi-objective methods can help in overcoming the difficulties of multimodal single-objective optimization. Our experiments, which are based on the well-known black-box optimization benchmark (BBOB) [@Definition_noislessFunctions] – a suite of continuous test functions with various difficulties – reveal that MOGSA is able to solve even highly multimodal single-objective problems.\
------------------------------------------------------------------------
\
For our experiments, we make use of the COCO (COmparing Continuous Optimizers) platform [@hansen2016cocoplat], which provides tools for performance assessment, as well as access to the aforementioned BBOB test suite. The performance of the multi-objective local optimizer MOGSA is compared to the single-objective local optimizer Nelder-Mead [@nelder1965simplex]. To further understand the effect of multiobjectivization, we will visually analyze the corresponding landscapes and study MOGSA’s search behavior on the problems. In Section \[sec:background\] we will formally define single-objective and multi-objective problems, as well as their properties. Furthermore, we will give an overview of related work on multiobjectivization. Afterwards, our concept is explained in detail in Section \[sec:concept\]. Subsection \[sec:SOtoMO\] describes and explains our procedure of transforming single-objective into multi-objective problems. Subsection \[sec:MOlandscape\] underlines this approach by means of a visual investigation of the resulting multi-objective landscape. The local optimizer MOGSA and our adaptations to it are described conceptually in the following Subsection \[sec:mogsa\]. Subsection \[sec:testEnvironment\] introduces the test environment of the COCO platform. Our experiments are described and analyzed in Section \[sec:eval\]. First, the experimental setup is defined in Subsection \[sec:setup\], then the results of the runs in the COCO framework, as well as those of the analysis of the multi-objective landscapes, are illustrated and analyzed in Subsection \[sec\_results\]. At last, our work is concluded in Section \[sec:concl\].
Background {#sec:background}
==========
To introduce the topic, we will provide definitions for single-objective and multi-objective optimization. Furthermore, the basic concept of the multi-objective visualization technique, which is fundamental for our work, is introduced. Afterwards, we will give an overview of related work dealing with multiobjectivization.
In the following, we will study unconstrained single-objective optimization problems of the form: $$\label{eqn1}
\min_{x \in X} f_i(x)$$ where $X\subseteq\mathbb{R}^d$ is the set of feasible solutions and $f_i:X\to\mathbb{R}$ is the objective function mapping the $d$-dimensional continuous decision vector $x$ to one function value. If $f(x)$ is to be maximized, it is equivalent to minimizing $-f(x)$.
For solving multimodal problems, Evolutionary Algorithms (EA) are often used [@back1997handbook; @beyer2001ES]. Inspired by the biological evolution, the algorithms use mechanisms such as mutation, recombination, and natural selection in an evolutionary loop for finding an optimal configuration. With this strategy, a diverse population is maintained where each individual contains information that can be considered for new search points [@schwefel1993evolution; @beyer2001ES]. Further ways of handling multimodal optimization by means of EAs are discussed in depth by Preuss [@preuss2015]. Due to their population-based nature, EAs are powerful general purpose solvers and therefore very popular in domains like multi-objective optimization [@MOEAs].
A multi-objective function $f:X\to\mathbb{R}^p$ with $$f(x)=(f_1(x), ..., f_p(x))^T\in\mathbb{R}^p$$ is the collection of $p$ single-objective functions as in Eq. (\[eqn1\]) with $i=1, ..., p$, which are mapped into a $p$-dimensional continuous objective space. Unlike in single-objective optimization, the aim of multi-objective optimization is to obtain a set of trade-off solutions rather than a single optimal solution. To compare the different solutions, the principle of domination is generally used. To get an understanding of multimodality and the multi-objective version of a local and global optimum in the decision as well as in the objective space, we will provide some definitions as provided in [@kerschke2018search; @MOlandscapes].
Vector $\textbf{a}=(a_1, ..., a_n)\in\mathbb{R}^n$ vector $\textbf{b}=(b_1, ..., b_n)\in\mathbb{R}^n$ (**a $\prec$ b**), iff $a_i \leq b_i$ for all $i\in{1,...,n}$, including at least one element $k \in {1, ..., n}$ in which $a_k<b_k$.
A set $A\subseteq\mathbb{R}^n$ is called if and only if there do not exist two open and disjoint subsets $U_1, U_2\subseteq \mathbb{R}^n$ such that $A \subseteq (U_1 \cup U_2), (U_1 \cap A) \neq \emptyset$, and $(U_2 \cap A) \neq \emptyset$. Further let $B \subseteq \mathbb{R}^n$. A subset $C \subseteq B$ is a of $B$ if and only if $C \neq \emptyset$ is connected, and $\nexists D$ with $D\subseteq B$ such that $C\subset D$.
An observation $\textbf{x}\in X$ is called if there is an open set $U\subseteq \mathbb{R}^d$ with $x \in U$ such that there is no (further) point $\tilde{x} \in (U \cap X)$ that fulfills **$f(\tilde{x}) \prec f(x)$**. The set of all locally efficient points of $X$ is denoted $X_{LE}$, and each connected component of $X_{LE}$ forms a (of **f**).
For local efficient points in the continuous multi-objective case, Fritz John [@FritzJohn] stated necessary conditions which were extended by Kuhn and Tucker[@kuhn1951] to a sufficient condition . Let $x\in X$ be a local efficient point of $X$ and all $p$ single-objective functions of **f** be continuously differentiable in $\mathbb{R}^n$. Then, there is a vector $v\in\mathbb{R}^m$ with $0 \leq v_i \le 1, i=0,...,m$, and $\sum_{i=1}^m v_i=1$, such that
$$\label{eqn_fritzJohn}
\sum \limits_{i=1}^m v_i \nabla f_i(x)=0.$$
In case of local efficient points the gradients cancel each other out given a suitable weighting vector $v$.
An observation $\textbf{x} \in X$ is said to be or , if and only if there exists no further observation $\textbf{$\tilde{x}$} \in X$ that dominates **x**. The set of all global efficient points is denoted $X_E$, and each connected component of $X_E$ is a .
The image (under **f**) of a problem’s local efficient set is called , whereas the respective image of the union of global efficient sets is called .
Since 2016, a Dutch-German research group, whose members are from the Universities of M[ü]{}nster (Germany) and Leiden (The Netherlands), has conducted joint research in the field of multi-objective optimization (e.g., [@kerschke2016towards; @MOlandscapes; @GrimmeKT2019Multimodality; @grimme2019sliding]). They enabled a better understanding of the interactions between the objectives by proposing a way of visualizing the landscapes of continuous multi-objective problems by means of two-dimensional gradient-based heatmaps [@MOlandscapes]. Dividing the search space into discrete points, the normalized (approximated) gradients of every objective are aggregated for each grid point. On the path towards the corresponding local efficient set, these multi-objective gradients are accumulated determining the height of the decision vectors and providing information on the closeness of an efficient set. The closer a point is to an efficient point (or set), the smaller is its accumulated value.
These visualizations reveal basins of attraction of local and global efficient sets and ridges in the decision as well as in the objective space. A basin of attraction is formed by all those points whose multi-objective gradient is directed to the same efficient set. When starting in any point belonging to the same basin, one would thus run to the same efficient set by following the multi-objective gradients. Due to superposition of basins of attraction, some are cut by other basins resulting in visible ridges. As it is assumed that ridges only appear for local efficient sets, they could offer a way to escape locality. Based on these insights, the local optimizer MOGSA was developed exploiting the properties of multimodal multi-objective problems [@GrimmeKT2019Multimodality]. By sliding down the multi-objective gradient hill, exploring the sets and jumping across ridges, the algorithm finally finds the global efficient set (see Section \[sec:mogsa\] for further details).
In an attempt of utilizing this (beneficial) behavior for single-objective problems, this paper investigates the transformation of originally single-objective into multi-objective problems. This process is called multiobjectivization and was introduced by Knowles et al. [@Knowles2001] who differentiated two types of multiobjectivization: the decomposition of the original function into several components, and the consideration of additional objectives to the original function. Knowles et al. were the first demonstrating the positive effect of the reduction of local optima in search space by decomposition. On two different combinatorial optimization problems (the traveling salesman problem and the hierarchical-if-and-only-if function) the hill climber algorithm based on multiobjectivization outperformed the comparable single-objective optimization algorithm [@Knowles2001].
Since then, several authors followed this idea by conducting theoretical and empirical studies on this topic. Jensen [@Jensen2004] empirically showed the benefits of multiobjectivization by considering additional objectives that he called *helper-objectives*. These are conflicting with the primary objective but lead to a diversified population which helped to avoid local optima. His experiments reveal significant improvements compared to the average performance of standard genetic algorithms. For the approach to work, the number of simultaneously used helper-objectives should be low. A high number of helper-objectives in the same run can only be successfully used when changed dynamically. Positive effects of considering one additional objective are reported by Neumann and Wegener [@neumannWegener2008]. Their theoretical results reveal an improved search behavior of evolutionary algorithms.
That multiobjectivization can have both beneficial and detrimental effects is shown for the decomposition [@Handl2008] as well as for the consideration of additional objectives [@Brockhoff2007]. In most research on multiobjectivization, evolutionary algorithms are used for problem solving. The main argument in favor of multiobjectivization is, that with a multi-objective instead of a single-objective problem, more information are available that the algorithm can use for improving its search behavior. Contrary to previous research, we will use a local and deterministic multi-objective optimizer to exploit the properties of multi-objective landscapes. The multi-objective setting is generated by considering one additional function.
Concept {#sec:concept}
=======
Local optima can be deceptive traps for algorithms – especially when they follow a descent. To avoid stagnation and find the global optimum of single-objective problems, we aim at taking advantage of the properties of continuous multimodal multi-objective problems for the single-objective case. To this end, we will use the principle of multiobjectivization, which is described in Subsection \[sec:SOtoMO\], followed by a figurative explanation of our approach. Next, in Subsection \[sec:MOlandscape\], we will use a simplified example to provide a visual understanding of multi-objective problems and their properties. The latter are exploited by the local optimizer MOGSA, whose behavior, including the minor modifications made to it, is explained conceptually in Subsection \[sec:mogsa\]. At last, the test environment used for analyzing our approach is described in Subsection \[sec:testEnvironment\].
General Understanding of Multiobjectivization {#sec:SOtoMO}
---------------------------------------------
Multiobjectivization is the transformation of an originally single-objective into a multi-objective problem. In the following, we will explain our procedure and properties of multi-objective landscapes.
As described above, multiobjectivization can either be achieved by the decomposition of a single-objective problem or by the addition of supplementary objectives. Within this work, we are interested in the multiobjectivization through the addition of one objective. To the initial single-objective optimization problem $f_1$ an additional second single-objective problem $f_2$ will be added. Thereby, a network will emerge, in which all optima of the first objective $f_1$ are connected to all optima of the second problem $f_2$. The arising connections indicate possible areas of efficient sets that can be exploited [@MOlandscapes]. For not making the landscape more confusing but providing a clear way for the algorithm, we selected a unimodal function for $f_2$. This way, all efficient sets should point in the direction of the only optimum of the second function which we define as:
$$\label{eqn_sphere}
f_2(x)=\sum \limits_{i=1}^d (x_i-s_i)^2$$
in the dimension $d$ with a predefined vector $s\in X$ and $X\subseteq\mathbb{R}^d$. The global optimum of that sphere function is located in $s$. For the heatmaps and thus for the multi-objective optimizer MOGSA only the gradient direction and the distance to the next efficient set are important. Since the gradient length as such is not relevant, the form of the added function $f_2$ – meaning whether it is stretched, compressed, moved up or down – does not make any difference.
An important property of continuous multi-objective problems that we will use in the following was stated in Eq. (\[eqn\_fritzJohn\]). For the bi-objective case this means that the gradients of both objectives become anti-parallel for local efficient sets. The angle between the two gradients of local efficient points equals $180^\circ$ as the gradients point in two opposing directions and only differ in length.
For the optimization process the second objective does not result in further costs as it is implemented as a constant. This is possible since the function’s derivative is known as $\nabla f_2(x)=\sum_{i=1}^d 2\cdot x_i - 2\cdot s_i$. The additional second objective $f_2$ only serves to create the multi-objective landscape with all its characteristics that can be exploited. Traps posed by local optima in the single-objective case are replaced by efficient sets that should guide the search to the global efficient set in the multi-objective setting. To ensure visualization we set the dimension $d=2$ for all considered problems.
Multiobjectivization and the Resulting Landscape {#sec:MOlandscape}
------------------------------------------------
In the following, we will provide a figurative description of the process of multiobjectivization and the properties of the resulting multi-objective problems. The principle of our procedure will be illustrated by a rather simple scenario with the initial single-objective function $y=f_1(x)=x_1^4-5\cdot x_1^2+x_1+x_2^2+3$. Figure \[fig:sop2\] depicts this multimodal single-objective problem with its two peaks in a three-dimensional plot. With a function value of $-4.86$ the global minimum is located in $(-1.63, 0)$, the local one with a fitness of $-1.69$ in $(1.53, 0)$. Imagining the behavior of a basic gradient descent algorithm starting in $(2.5, 2)$, it is obvious that the algorithm would converge to the local optimum by following the negated gradients. There, it would finally get stuck as no step in any direction would bring an improvement. The algorithm stops without noticing that there are other optima or rather that the found solution is not the best.
To avoid such a stagnation, the single-objective problem will be transferred into a multi-objective problem by adding an additional function $f_2$. By visualizing the created bi-objective problem in a gradient-based heatmap, the efficient sets become visible as well as their respective basins of attraction. Due to the interaction effects between the objectives, the basins of attraction superpose each other resulting in visible ridges along the borders of the basins. The superposed structure is schematically depicted in Figure \[fig:basinCut\]. The basins of attraction can be regarded as funnels towards their respective efficient set. Picturing a ball following the multi-objective gradients, it would roll down the dotted path towards the efficient set that is displayed by the colored horizontal lines. Ridges, illustrated by the vertical red lines, cut efficient sets due to the superposition of basins of attraction. All efficient sets that are cut by a ridge are assumed to be locally efficient. Consequently, global efficient sets should be ridge-free [@GrimmeKT2019Multimodality]. By following the multi-objective gradients and walking along the efficient sets to the next basin of attraction one should find the global efficient set.
The global optimum of both single-objective problems $f_1$ and $f_2$ is equally globally efficient from a multi-objective point of view. No improvement in the dimension of the respective objective is possible. Therefore, we aim at finding the global efficient set that leads us to the global optimum of the initial single-objective problem $f_1$. For this purpose, we use the local multi-objective optimizer MOGSA that explores the efficient sets, jumps over the ridges into other basins of attraction until the global efficient set without any ridges is found.
The transformation of a single-objective into a multi-objective problem is visually illustrated in Figure \[fig:heats\]. On the left side, the decision space of the single-objective problem from Figure \[fig:sop2\] is depicted in a heatmap. The different colors denote the distance – w.r.t. the gradient descent – to the efficient set of the respective basin of attraction. The darker the red, the further away is the point. The efficient sets – here, the two optima since this is the single-objective case – are colored in blue, usually surrounded by an area of green to yellow.
[0.475]{} ![\[fig:heats\] Depiction of two gradient-based heatmaps. On the left, the decision space of the already introduced bimodal single-objective problem is displayed. Adding a sphere function to this problem making it multi-objective results in the decision space of the right image.](fig_sop_-5-5.png "fig:"){width="\textwidth"} \[fig:heatSOP\]
[0.475]{} ![\[fig:heats\] Depiction of two gradient-based heatmaps. On the left, the decision space of the already introduced bimodal single-objective problem is displayed. Adding a sphere function to this problem making it multi-objective results in the decision space of the right image.](fig_mop.png "fig:"){width="\textwidth"} \[fig:heatMOP\]
Adding the additional function $f_2$ from Eq. (\[eqn\_sphere\]), with its optimum in $s=(-3.5, -2.5)$, to the previously discussed scenario results in the heatmap on the right side. Now, the efficient sets and their basins of attraction become visible. A ridge cuts the local efficient set (the set on the right side of the image) while the global efficient set on the left side is ridge-free. At each ridge-free end of the efficient sets an optimum of one of the two functions is located.
Search Behavior and Implementation of MOGSA {#sec:mogsa}
-------------------------------------------
Ridges help the optimizer MOGSA to escape local efficient sets. As proposed by Grimme and Kerschke [@GrimmeKT2019Multimodality], the algorithm takes advantage of the properties of the multi-objective problems’ landscapes in two repeating phases. Following the problem’s multi-objective gradients in the first phase, MOGSA finds an efficient set that can be explored in the second one. We will make use of the local optimizer with slight adaptions to support our approach. First, we will describe the search behavior of MOGSA in a simplified setting, supported by a schematic illustration. Afterwards, the two phases and our changes made are described and explained in detail.
#### Way of MOGSA:
The interaction of the two repeating phases is schematically depicted in Figure \[fig:wayMOGSA\]. The presented bi-objective problem is comprised of one bimodal function $f_1$ and one unimodal function $f_2$. The optima are indicated by dots (green for the first, yellow for the second objective). The blue lines illustrate the efficient sets which are located in different basins of attraction (area within the red borders). Being cut by a ridge, the efficient set in the right basin of attraction represents the local efficient set. The global efficient set and thereby the global optima of both single-objective functions are located in the left basin of attraction.\
Starting in point x, MOGSA executes the first phase by following the multi-objective gradient to find a point on the local efficient set in the respective basin (1). From there, the single-objective gradient of the first function $f_1$ is followed until the (green) optimum is reached (2). Going back to the first found point on that efficient set, the exploration phase is repeated for the second objective $f_2$ until the end of the set is reached and a ridge found (3). With this, the second phase stops and the first one is started again searching for the efficient set in the new basin of attraction (4). Once the set has been reached, $f_1$ is followed until the (global) optimum is found (5). Eventually, the algorithm would jump over it in a first try but with a local search, the optimum would be found. Next, $f_2$ is followed from the first found point on that set (6). As both sides of the efficient set are ridge-free, MOGSA stops successfully with the global efficient set and thus the global optimum of the first function found.
![\[fig:wayMOGSA\] Schematic view on the decision space of a bi-objective problem with two basins of attraction (encircled in red). The optima of both single-objective functions are indicated by dots (green for the first, yellow for the second objective) being located on two efficient sets (blue lines). The dotted arrows display the chronological search behavior of MOGSA starting in point x.](WegMOGSA.png){width="60.00000%"}
#### First Phase:
The aim of this phase is to find an efficient set. As a first step, the multi-objective gradient is computed for the current position by combining the normalized (approximated) single-objective gradients. The combined gradient contains information about the direction to the attracting efficient set. In the bi-objective case, two single-objective gradients pointing in the same direction result in a large multi-objective gradient, i.e., a length close to two. A small multi-objective gradient or rather a zero-gradient on the other hand means that the single-objective gradients point in opposite directions. In this case, an efficient set is reached and the local search stops. Otherwise, a scaled step in the direction of the multi-objective gradient’s descent is taken. If the step leads to a crossing of the boundaries, the algorithm is placed back on the bounds. When the algorithm would have to be placed at the same position a second time, it is randomly restarted within the feasible area. Gradient-descent steps are performed until an efficient set is reached (or crossed). If the set has been crossed, the last two consecutive points are located on opposite sides of the attracting efficient set. As their multi-objective gradients are both pointing towards the efficient set, they are pointing in nearly opposing directions. Thus, the angle between them will be large. If the angle is bigger than $90^\circ$, an interval bisection procedure weighted by the ratio of the lengths of the two gradients is performed. This way, a point on the efficient set will be found.
#### Second Phase:
Having found an efficient set, it will be explored in the second phase. From the locally efficient point that was found in the first phase, MOGSA starts with following the normalized and scaled gradient of the first objective $f_1$ as long as the step size is not too small. As in the first phase, a possible leaving of the feasible area is checked. This would result in following the second objective $f_2$ from the initially found efficient point of that set. However, when staying within the bounds, MOGSA continues to follow the single-objective gradient of $f_1$ until one of the following three scenarios occurs: the algorithm could have landed directly in the optimum, passed an optimum, or left the efficient set due to a ridge.\
A length of (nearly) zero for the single-objective gradient indicates that the algorithm has found the single-objective optimum of $f_1$. As a next step, the gradient of $f_2$ would be followed to check whether the currently exploring efficient set could be globally efficient. This is likely if the set is not cut by a ridge.\
In the other two scenarios, the efficient set was left. When the last point on the efficient set $x^{(t)}$ and the point after the step taken $x^{(t+1)}$ are still in the same basin of attraction, MOGSA has passed an optimum or the optimum is located close to $x^{(t+1)}$. This is indicated by an angle of bigger than $90^\circ$ between the single-objective gradient of $x^{(t)}$ and the one of $x^{(t+1)}$. Both gradients are pointing to the same optimum but from different sides. To find the local single-objective optimum of $f_1$, the Nelder-Mead algorithm is used, originally published in 1965 by John Nelder and Roger Mead [@nelder1965simplex]. The direct search method does not require any derivative information and uses the concept of a simplex. Instead of a single starting point, $d+1$ vertices are used for $d$-dimensions. In every iteration, one vertex is exchanged by a new point. Since we are only interested in the optima of the first function, the local search was added to the proposed algorithm and is solely performed in the direction of $f_1$. Once the optimum has been found, the second phase starts again from the point that was found first on the set. This time, the second objective’s gradient is followed.\
In the event of having left the basin of attraction, the algorithm has crossed a ridge to an adjacent basin. For a point on the efficient set, the angle between the two single-objective gradients is $180^\circ$. Leaving the basin of attraction leads to a new attracting efficient set. The currently followed single-objective gradient should still point to the same optimum and thus in the same direction as before. For the gradient of the other function on the other hand, the attracting optimum has changed by entering the new basin of attraction. In most cases, the angle between the two single-objective gradients of $x^{(t+1)}$ is smaller than $90^\circ$. The existence of a ridge indicates a local efficient set that we are not looking for. Thus, MOGSA will not start exploring the previous efficient set in the direction of the second objective but stay in the newly found basin. There, the first phase is started again to find a new efficient set.
The two phases are repeated until a set without any ridges and thus a probable global efficient set is found. Each point the algorithm considers a probable optimum is saved and evaluated in the end. For preventing MOGSA from getting trapped in an endless loop between different efficient sets (e.g., from efficient set a into efficient set b, back to a and again to b), every probable optimum is compared to all the already visited optima. In case the algorithm runs into the same optimum again, a random point within the given bounds becomes the starting point for the first phase.
Test Environment {#sec:testEnvironment}
----------------
As already stated within the introduction, our aim is to analyze how the properties of multi-objective landscapes can be exploited for single-objective optimization. We want to explore whether the multi-objective optimizer MOGSA is able to find the global optimum of a single-objective problem that is transferred to a bi-objective problem through multiobjectivization. For testing the search behavior on many single-objective problems with different characteristics and difficulties, we use the benchmark platform COCO [@hansen2016cocoplat]. In this section we will introduce the platform and the performance measure used for our experiments.
The COCO platform is used for comparing continuous optimizers in a black-box setting. For this, an experimental framework as well as post-processing facilities are provided. Different test suites contain benchmark functions for single-objective optimization with and without noise as well as noiseless bi-objective problems. We use the 24 noiseless single-objective test functions of the Black-Box Optimization Benchmark (BBOB) [@Definition_noislessFunctions]. The benchmark functions are categorized in five subgroups according to similar properties such as multimodality, ill-conditioning, and non-separability of the variables. Furthermore, different instances, e.g., shifted, rotated and stretched versions of each problem are available. For each of the 24 problems of the BBOB test suite multiple instances are provided. This way, an algorithm can be exposed to a variety of function characteristics. A problem or problem instance triplet $p^3=(d, f_{\theta}, \theta_i)$ is formally defined by the search space dimension $d$, the objective function $f$ to be minimized and its instance parameters $\theta_i$ of instance $i$. A set of parametrized benchmark functions $f_\theta :\mathbb{R}^d \rightarrow \mathbb{R}^m, \theta \in \Theta$ and the corresponding problems $p^3$ are considered. For performance assessment, it is aggregated over all $\theta_i$-values.
For our experiments, we consider the 24 problems from the BBOB suite as potential functions of interest ($f_1$), which we aim to optimize. The multiobjectivization process is part of the algorithm and thus independent of the first objective. The added second function $f_2$ will not be changed, neither within the optimization process of a problem nor for optimizing other functions. The additional objective can be regarded as a constant. Thus, COCO only considers the first function. Note that the usage of the second objective does not generate any further costs in terms of function evaluations – as $f_2$ as well as its gradient $\nabla f_2$ are known.
Denoted as the runtime, the number of function evaluations conducted on a given problem is the central performance measure of COCO. When the algorithm reaches or surpasses a prescribed target function value $f_{target}$, the problem is considered as being solved regardless of whether the algorithm continues running or not. Function evaluations conducted after having solved the problem are not recorded anymore. The target function value allows small deviations ($\Delta f$) from the best value possible to reach ($f_{opt}$) and is thus defined as $f_{target}=f_{opt}+\Delta f$. The optimal function value is defined for each instance of the benchmark problems individually. The smallest considered precision to reach is $10^{-8}$ [@experimentalSetup]. For the experiments, different target precisions are defined. From each run on the problem instance triple $p^3$ runtime measurements are obtained for each target value reached in this run.
Ways for aggregating the resulting runtimes are the average runtimes (aRT) and the empirical (cumulated) distribution function (ECDF) of runtimes [@hansen2016cocoplat]. As we are primarily interested in the success of the algorithm and the precision of the found optimum, we will focus on the ECDF that displays the proportion of problems solved within a specified budget. The runtime distributions can be displayed for each problem individually but also aggregated over the function groups or over all functions.
Evaluation {#sec:eval}
==========
This section deals with the realization and the analysis of our concept with the aim of reducing local traps for single-objective problems by multiobjectivization. To conduct experiments on several single-objective functions and assess the performance of the two local optimizers Nelder-Mead and MOGSA, the COCO framework is used. Furthermore, a visualization technique based on multi-objective gradients offers insights into the landscape of the multi-objective problems.
First, the setup for experiments conducted by the COCO framework is described. Then, the results of this experiment, the visual inspection of the landscape, and the behavior of MOGSA are described and analyzed.
Experimental Setup {#sec:setup}
------------------
The performance of MOGSA on the initial single-objective problem is assessed by the COCO framework [@hansen2016cocoplat]. To evaluate the transformation of adding an additional function, the local multi-objective optimizer MOGSA is compared to the local single-objective optimizer Nelder-Mead on a testbed with several different optimization problems. In this Section, the characteristics of the machine and the environment used for generating the results of the comparison are provided.
The tests have been run on a Windows 10 Education (2018) computer with AMD Ryzen 5 3500U at 2.10 GHz with 8 GB RAM. Python 2.7.17 (64 bit) was used to run MOGSA and the test environment that is provided by the COCO framework. The simplex algorithm Nelder-Mead was provided by the package skipy.optimize of Python [@scipy; @2020SciPy-NMeth].
A detailed description of the experimental setup and of how the results are displayed with COCO can be found in [@experimentalSetup]. The Nelder-Mead algorithm and MOGSA have been run on each problem of the noiseless BBOB test suite. The suite contains continuous single-objective benchmark functions with different properties. A documentation of the test functions can be found in [@Definition_noislessFunctions]. For each of the 24 problems 15 different function instances (Ntrail) were considered (and optimized). The performance is evaluated over all Ntrial trials. For both algorithms, nothing is known about the system which has to be optimized apart from the search domain $(-5, 5)^d$ and the starting point. In our experiments, we define the search space dimension $d=2$. To be able to evaluate the overall behavior of both algorithms, each started from ten different points within the search domain: $(5,5)$, $(-5,5)$, $(5,-5)$, $(-5,-5)$, $(2,4)$, $(4,2)$, $(-2.5,4)$, $(1,-2)$, $(0,0)$, $(-4.5,0)$. These points were manually spread over the search space and are intended to generate results that are independent of possible advantages in functional structures.\
For each of the ten runs on all function instances of the test suite the same parameter setting, the same initialization procedure, the same budget and the same termination criteria were used. The maximum budget for evaluations done by an optimization algorithm is *dimension$\times$budget-multiplier* with a fixed budget-multiplier of 100 for our experiments.
The number and the initialization of the variables for MOGSA were set as proposed in [@GrimmeKT2019Multimodality]. Only the scaling factors $\sigma_1$ and $\sigma_2$ for the step size for both of the two phases of MOGSA were changed. A step size of one would be too big so that the algorithm would not be able to identify the structure of the problems. Instead, the algorithm would leave basins of attraction without finding the efficient set. Thus, we defined $\sigma_1=\sigma_2=0.1$. As the stopping criterion we implemented the variable *steps*. Running the first phase of finding an efficient set counts as one step. Internally, this phase stops when an efficient set is found or more than $1,\!000$ new x-values were generated while following the multi-objective gradient. For phase two, every movement from one point $x_t$ to another point $x_{t+1}$ for following $f_1$ or $f_2$ counts as one step. When MOGSA has not found an efficient set without any ridge within 1000 steps, the algorithm stops. The transformation of a single-objective benchmark problem into a multi-objective one is internally performed by MOGSA. As in the introductory problem in Figure \[fig:heats\], we defined $s=(-3.5, -2.5)$ for the additional second objective $f_2$ from Eq. (\[eqn\_sphere\]).
For visualizing the generated problems as gradient-based heatmaps, we used R version 4.0 [@R], as well as the R-packages moPLOT [@SchaepermeierGK2020], ggplot2 [@gglot2] and smoof [@smoof].
Experimental Results {#sec_results}
--------------------
From each of the ten starting points, experiments were conducted for the simplex algorithm Nelder-Mead and the multi-objective optimizer MOGSA. Both algorithms were run on 15 instances of each of the 24 functions $(f_1-f_{24})$ of the COCO framework with the previously described setup. For MOGSA, a fixed additional second objective was added to create a multi-objective landscape. The generated results are discussed and compared in the following. Furthermore, individual problems are illustrated in gradient-based heatmaps and visually analyzed.
### Performance Assessment
First, we will look at the runtime distributions aggregated over all 24 functions of the BBOB test suite and explain the plots which resulted from the experiments. Afterwards, we will focus on specific functions. In Figure \[fig:coco\_mogsa\], the empirical cumulated distribution functions (ECDF) of runtimes of Nelder-Mead (left) and MOGSA (right) have been depicted. Both plots show the ECDF of runtimes on a set of problems formed by $24\times 15$ function instances in dimension $d=2$, each with 51 target precisions between $10^2$ and $10^{-8}$ uniform on a log-scale. The precision is displayed on the y-axis. A value of $0.0$ corresponds to a precision of $10^2$, a value of $1.0$ to a precision of $10^{-8}$. On the x-axis, the number of objective function evaluations divided by the dimension $d$ is represented on a log-scale. Each colored line in the plot represents one of the ten runs whose starting points are indicated next to the abbreviated name of the algorithm (NM for Nelder-Mead, MO for MOGSA). The light thick line portrays the artificial best algorithm of BBOB-2009 as reference algorithm. The crosses on the plots illustrate the median of the maximal length of the unsuccessful runs to solve the problems aggregated within the ECDF.
The ECDF can be read in two different ways when regarding one of the axes as the independent variable and the other one as the fixed. In the plot of Nelder Mead, for example, we can find the number of function evaluations (x-axis) necessary for all runs to solve 40% of the problem-target combinations (fixed y-axis). Solving 40% of the problems (on average) corresponds to reaching a target precision of $10^{-2}$. That precision was reached by all of the ten runs within $10^2\cdot d = 200$ function evaluations (value of 2 on the x-axis).
![\[fig:coco\_mogsa\] ECDF of the runtime of ten runs of Nelder-Mead (left) and MOGSA (right) in dimension 2 over 51 targets. Each run was started in a different starting point. The ECDF is aggregated over all functions of the BBOB test suite for each run. ](COCO/nelder_noiselessall.png "fig:"){width="48.50000%"} ![\[fig:coco\_mogsa\] ECDF of the runtime of ten runs of Nelder-Mead (left) and MOGSA (right) in dimension 2 over 51 targets. Each run was started in a different starting point. The ECDF is aggregated over all functions of the BBOB test suite for each run. ](COCO/mog_noiselessall.png "fig:"){width="48.50000%"}
Comparing the two plots, we observe that aggregated over all target-problem combinations, MOGSA obtains better results than Nelder-Mead concerning the final reached precision. All runs of MOGSA solved at least 70% of the target-problem combinations after $10^7\cdot 2$ function evaluations and thus achieved an average precision of at least $10^{-5}$. For Nelder-Mead this is the best achieved value of all the runs. In comparison, the best run of MOGSA even solved 85% of the problems. With a small number of function evaluations, however, Nelder-Mead finds points with better function values. Only from a number of $10^4\cdot 2$ function evaluations on, MOGSA achieves better results.
These plots of the aggregated ECDF over all functions indicate a success for our procedure. However, we need to examine the functions separately as each possesses different characteristics. Besides, in the set of benchmark functions are some unimodal problems. These are not of our interest as we aim to overcome local traps which do not exist for problems with only one optimum. Instead of directly running to that optimum, MOGSA would find the efficient set first to explore it. Although it finds the optimum – with the same precision as Nelder-Mead – it takes more function evaluations. The only unimodal function on which Nelder-Mead performed better than MOGSA concerning the target precision is the Step Ellipsoidal function $f_7$. However, apart from a small area close to the global optimum the gradient is zero almost everywhere. As MOGSA, contrary to Nelder-Mead, uses the approximated gradients for its search, this result is not surprising.
For analyzing whether the multi-objective landscape can be exploited for multimodal single-objective problems, we will focus on the multimodal problems of the COCO framework in the following. These include the two highly multimodal functions $f_3$ and $f_4$ from the separable function group, all multimodal functions with an adequate global structure $(f_{15} - f_{19})$ as well as all multimodal functions with a weak global structure $(f_{20} - f_{24})$. The respective plots of the conducted experiments in the COCO framework are presented in Figures \[fig:results\_3-16\], \[fig:results\_17-20\] and \[fig:results\_21-24\].
![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/nelder_pprldmany_f003_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/mog_pprldmany_f003_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/nelder_pprldmany_f004_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/mog_pprldmany_f004_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/nelder_pprldmany_f015_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/mog_pprldmany_f015_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/nelder_pprldmany_f016_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_3, f_4, f_{15},$ and $f_{16}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_3-16"}](COCO/mog_pprldmany_f016_02D.png "fig:"){width="43.00000%"}
![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/nelder_pprldmany_f017_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/mog_pprldmany_f017_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/nelder_pprldmany_f018_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/mog_pprldmany_f018_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/nelder_pprldmany_f019_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/mog_pprldmany_f019_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/nelder_pprldmany_f020_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{17}, f_{18}, f_{19},$ and $f_{20}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_17-20"}](COCO/mog_pprldmany_f020_02D.png "fig:"){width="43.00000%"}
![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/nelder_pprldmany_f021_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/mog_pprldmany_f021_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/nelder_pprldmany_f022_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/mog_pprldmany_f022_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/nelder_pprldmany_f023_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/mog_pprldmany_f023_02D.png "fig:"){width="43.00000%"}\
![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/nelder_pprldmany_f024_02D.png "fig:"){width="43.00000%"} ![ECDF of runtimes on $f_{21}, f_{22}, f_{23},$ and $f_{24}$ in dimension 2 over 51 targets. On the left the runtime of Nelder-Mead from ten different starting points is displayed. The plots on the right show the same for MOGSA.[]{data-label="fig:results_21-24"}](COCO/mog_pprldmany_f024_02D.png "fig:"){width="43.00000%"}
A success of applying MOGSA on the transformed single-objective problems is already evident in the upper three plots of Figure \[fig:results\_3-16\]. The separable Rastrigin function $f_3$, the Bueche-Rastrigin function $f_4$ and the non-separable Rastrigin function $f_{15}$ are structured and highly multimodal thus posing a lot of local traps. Still, compared to Nelder-Mead (but also in total) MOGSA performed well. The plots for the runs of MOGSA on the right side reveal that the algorithm reached a value very close to the target function value with a precision of around $10^{-6}$ from six, for $f_{15}$ from seven of the ten starting values after $10^7\cdot 2$ function evaluations. Nelder-Mead, on the other hand, reached that precision only for one, for $f_3$ for two starting points. All other runs solved only 10 to 20% of the problems. It is noticeable that MOGSA either solved around 80% or only 10 to 20% of the problems for different starting points. The latter was only the case for four, on $f_{15}$ for three of the runs. This discrepancy between the reached precisions depending on the starting value is also observed for the highly multimodal Composite Griewank-Rosenbrock function $f_{19}$ in Figure \[fig:results\_17-20\]. In every second run, 100% of the problems were solved whereas in the other runs only 40% were solved. The same discrepancy can be identified for Nelder-Mead on this problem although with a lower number of successful runs.
Of all the considered multimodal functions, the Weierstrass function $f_{16}$, whose results are displayed in the bottom plot of Figure \[fig:results\_3-16\], is the one where MOGSA performed best compared to Nelder-Mead. The landscape of the Weierstrass function is highly rugged and moderately repetitive with a global optimum that is not unique. For seven starting values, MOGSA solved the problems to 100%. In the other three runs a precision of $10^{-4}, 10^{-6}$ and $10^{-7}$ was still reached which corresponds to having solved the problems to 60, 80 and 90%. The best run of Nelder-Mead starting in $(-2.5,4)$ solved 90% the problems. In all the other runs, however, only 20 to 30% of the problems could be solved. The results indicate that Nelder-Mead eventually got stuck in local traps while MOGSA did not. While sliding from basin to basin, MOGSA can find the global optimum or at least optima with a function value very close to the best value.
On all the other multimodal functions MOGSA also reached better target precision values than Nelder-Mead. For $f_{17}$ and its counterpart $f_{18}$ at the top of Figure \[fig:results\_17-20\] an overall higher percentage of solved problems can be observed after $10^7\cdot 2$ function evaluations. Even on the Lunacek bi-Rastrigin function in Figure \[fig:results\_21-24\] the values of the reached target precision, for the best run $10^{-1}$, were better for all runs of MOGSA than for those of Nelder-Mead. On $f_{20}$, $f_{21}$ and $f_{23}$ the percentage of problems solved by Nelder-Mead differs between the runs. MOGSA performs slightly better than the best run of Nelder-Mead but for all runs. For $f_{22}$ both algorithms solved 100% of the problems.
In summary, the plots of the ECDF of runtimes reveal that MOGSA is able to solve multimodal single-objective problems after having them transferred into multi-objective ones. Noticeably, on some of the problems the result seemed to depend on the starting value. However, even if the global optimum is not always found, an acceptable target precision is reached for almost all problems. After $10^7\cdot 2$ runs the best value found was always better than the one of Nelder-Mead for all multimodal problems. Though MOGSA can definitely not be compared to the artificial best algorithm of BBOB-2009, the results are satisfactory especially considering that it is a local search and the problems provided possess a highly multimodal landscape as well as other difficulties. To get an understanding of how these landscapes look like in a multi-objective setting, a visual analysis will be conducted in the following.
### Visual Analysis
In the following, we will take a closer look at the heatmaps of individual function instances of the COCO framework after having added the sphere function $f_2$. Furthermore, we will visualize MOGSA’s search trajectory based on its different starting points.
As already stated above, our focus is not on unimodal functions. However, a visualization of an ellipsoidal function $f_1$ with an added sphere function $f_2$ (see Figure \[fig:ellipsoidal\]) reveals that the global optimum of the unimodal function $f_1$ is, other than expected, not located on the efficient set. Figure \[fig:ellipsoidal\] depicts a 3D plot of the single-objective function $f_1$ on the left side. On the right side the decision space of the multi-objective problem is displayed in a gradient-based heatmap. As expected, the heatmap of the multi-objective problem being comprised of two unimodal functions shows only one basin of attraction and one efficient set. The objectives have no local optima that could constitute a trap and thus, no ridges exist. Nevertheless, only the optimum of the sphere function, indicated by the black dot, is located on the straight efficient set. When looking at the 3D plot of $f_1$ one can see that the function has a broad valley. There, the difference in height is rather small so that all gradients around the valley point into that valley instead of towards the best solution. Therefore, the efficient set in the multi-objective setting is always the path between the optimum of $f_2$ and the valley instead of the global optimum. Thus, it is representing the shortest path from the optimum of $f_2$ towards the valley of $f_1$. Since the behavior of MOGSA is based on the structure of the landscape, the algorithm walks as expected to that efficient set first. When reaching the end of the set by following $f_1$ and landing in the valley, the algorithm finds the global optimum of $f_1$ with the help of the local search. The phenomenon of the optimum of $f_1$ not being located on the efficient set is observed for several unimodal functions, all with a rather broad valley. Although our aim is not to solve unimodal problems, this shows that the global optima of both $f_1$ and $f_2$ are not always located on an efficient set, at least for unimodal functions.
[0.5]{}
[0.4]{} ![\[fig:ellipsoidal\] The left image displays an ellipsoidal function $f_1$ in a 3D plot. The heatmap on the right shows a bi-objective problem comprised of $f_1$ and an additional sphere function $f_2$. The global optima of both single-objective problems are depicted by the black triangle for $f_1$ and the dot for $f_2$.](x_-35_-25_BBOB_2_2_1.png "fig:"){width="\textwidth"}
For the multimodal problems of the COCO framework we see that the landscape after the multiobjectivization is sometimes also different than expected. Figure \[fig\_f21\] pictures the heatmap of the decision space of the Gallagher’s Gaussian 101-me Peaks function $f_1$ with the sphere function $f_2$. Due to the multimodality a lot of basins of attraction as well as efficient sets that are cut by a ridge are visible. Only one efficient set is ridge-free. Surprisingly, only the global optimum of $f_2$ is located on that set, illustrated by the black dot. Wherever MOGSA starts, it ends in that set by exploring the other efficient sets, jumping from basin to basin when facing a ridge until the ridge-free set is found. However, the global optimum of the first function (black triangle) is located on the end of another set that is cut by a ridge. Whether MOGSA finds that global optimum or not depends on where it starts. When the respective efficient set is on the way that MOGSA takes, it will explore the set, find and save the global optimum of $f_1$. Although the algorithm does not regard the point found as globally efficient at first due to the existing ridge, it recognizes the globality when comparing all saved points before stopping. Still, it is possible that MOGSA does not run in the basin of that set and thus does not find the global optimum of $f_1$. This would be the case when the starting point is for example in the upper part of the heatmap. When starting somewhere in the lower part, for example in $(-1, -3)$, it finds the global optimum of $f_1$ on the way to the ridge-free efficient set. Based on this heatmap we can state that the global optima of a bi-objective problem can be located on different efficient sets and thus in different basins of attraction. As already assumed during the evaluation of the results generated by the COCO framework, the heatmap also indicates that the starting point has an influence on the success of the optimization process.
![\[fig\_f21\] Heatmap of the decision space of a multi-objective problem which is comprised of the Gallagher’s Gaussian 101-me Peaks function (in the COCO framework $p^3=(2, 21, 1)$) with its optimum illustrated by the black triangle, and the sphere function whose optimum is represented by the black dot.](x_-20_20_BBOB_2_21_1_p.png){width="50.00000%"}
Figure \[fig:rastr4\] confirms this theory. The heatmap depicts a multi-objective problem which is comprised of the Rastrigin function $f_1$ and the sphere function $f_2$. The global optima are represented by the black triangle for $f_1$ and the dot for $f_2$. Since $f_1$ has a lot of local optima, many efficient sets exist that are all oriented towards the global optimum of $f_2$. As in the previous heatmap, a ridge-free global efficient set on which the global optima of both single-objective functions are located does not exist. Contrary to the optimum of $f_2$, the global optimum of $f_1$ is not visible and no difference to the other efficient sets can be seen. The behavior of MOGSA on this multi-objective problem is depicted by the colored dots. The blue land represents the way for starting in $(3,1)$, yellow for $(2,4)$, green for $(2,2)$, black for $(-3,2)$, dark red for $(2.5,4.5)$, and purple for $(2,-2)$.
On the given landscape, MOGSA behaves as expected for all starting points. First, the efficient set of the respective basin of attraction is found which will be explored in the second phase. By following the gradient of $f_1$, an optimum is found and the gradient of $f_2$ is followed. This is done until a ridge cuts the set or the global optimum of $f_2$ is found. For some ways we detect that the path MOGSA takes deviates slightly from the efficient sets, for example for the blue and the dark red path. However, if the set is only left a little it does not have an impact on the search as MOGSA still follows the single-objective gradient, in the cases of the deviation the one of $f_2$. This means that the algorithm is still walking towards the optimum to which the efficient sets are not pointing as direct as the algorithm walks. The next ridge is found regardless of small deviations from the efficient set. In case of no ridge but an optimum, there are no deviations and MOGSA walks straight to that optimum.
Regardless of the starting point, the global optimum of $f_2$ is found in each of the runs as all efficient sets point towards that optimum where the search stops. The global optimum of $f_1$ on the other hand, is only found when the respective efficient set is located on the way towards the optimum of $f_2$. From the six starting points this is only the case for the green and the blue path. MOGSA is not able to walk up the ridges so that once it left a basin it is not able to go back. Thus, the success of the search highly depends on the starting point.
![\[fig:rastr4\] Heatmap of the decision space of a multi-objective problem which is comprised of the Rastrigin function $f_1$ with its optimum illustrated by the black triangle, and the sphere function $f_2$ whose optimum is represented by the black dot. The colored dots on the heatmap show the way of MOGSA starting in six different points.](fig_rastrSix.png){width="75.00000%"}
Both discussed problems are highly multimodal each with further characteristics. Considering the other multimodal problems from the COCO framework we can say that all of them have serious difficulties that MOGSA has to face. Some of these problems are depicted in the heatmaps of Figure \[fig:heatmaps\_coco\] after the multiobjectivization. They underline our observation that the two global optima of bi-objective problems are not always located on the same efficient set. Sometimes one or even both global optima are not even visible like in the heatmap at the bottom left.
![Heatmaps for the decision space of different problem instances of the COCO framework with an added sphere function $f_2$ whose optimum is located in the black dot. The problem instances: $p^3=(2, 16, 1)$ at the top left next to $p^3=(2, 17, 1)$, at the bottom left $p^3=(2, 19, 3)$ and on the bottom right $p^3=(2, 20, 1)$. Their optima are illustrated by the black triangle.[]{data-label="fig:heatmaps_coco"}](x_35_-25_BBOB_2_16_1.png "fig:"){width="44.00000%"} ![Heatmaps for the decision space of different problem instances of the COCO framework with an added sphere function $f_2$ whose optimum is located in the black dot. The problem instances: $p^3=(2, 16, 1)$ at the top left next to $p^3=(2, 17, 1)$, at the bottom left $p^3=(2, 19, 3)$ and on the bottom right $p^3=(2, 20, 1)$. Their optima are illustrated by the black triangle.[]{data-label="fig:heatmaps_coco"}](x_-35_-25_BBOB_2_17_1_p.png "fig:"){width="44.00000%"}\
![Heatmaps for the decision space of different problem instances of the COCO framework with an added sphere function $f_2$ whose optimum is located in the black dot. The problem instances: $p^3=(2, 16, 1)$ at the top left next to $p^3=(2, 17, 1)$, at the bottom left $p^3=(2, 19, 3)$ and on the bottom right $p^3=(2, 20, 1)$. Their optima are illustrated by the black triangle.[]{data-label="fig:heatmaps_coco"}](x_-35_-25_BBOB_2_19_3.png "fig:"){width="44.00000%"} ![Heatmaps for the decision space of different problem instances of the COCO framework with an added sphere function $f_2$ whose optimum is located in the black dot. The problem instances: $p^3=(2, 16, 1)$ at the top left next to $p^3=(2, 17, 1)$, at the bottom left $p^3=(2, 19, 3)$ and on the bottom right $p^3=(2, 20, 1)$. Their optima are illustrated by the black triangle.[]{data-label="fig:heatmaps_coco"}](x_-35_-25_BBOB_2_20_1_p.png "fig:"){width="44.00000%"}
Conclusion {#sec:concl}
==========
Multimodality is a crucial factor defining the hardness of single-objective problems. With our approach of transforming single-objective into multi-objective problems to exploit the properties of their landscape, we show a way to deal with local optima. In the multi-objective setting, local optima do not pose a threat anymore as the optimizer MOGSA is able to overcome the traps and even benefit from local optima, which guide the algorithm to global efficient sets. Although the functions on which we conducted our experiments possess severe difficulties as shown in the visualizations, the algorithm was able to find global optima or reach a function value close to the final target value of $f_1$. On some problems, however, the structure was too complicated. Nevertheless, one should not forget that MOGSA is a deterministic local optimizer and even if the global optimum of $f_1$ was not found, our experiments reveal that it is not because of local traps. With the ability to slide through the landscape of multi-objective problems exploring efficient sets, MOGSA is visiting several optima on its way without getting stuck. In the gradient-based heatmaps we observe that global optima are not always visible, at least not for both single-objective functions. In this case, it often depends on the starting position of the algorithm which always finds the way to the optimum of $f_2$. If the global optimum of $f_1$, the single-objective problem we want to optimize, is found or not depends on whether it is located on the path MOGSA takes.
This opens up opportunities for future work. With the knowledge of being able to find the global optimum from specific starting points, the choice of the *right* starting point could be analyzed and improved. Possibly, the application of an evolutionary algorithm could help to select the best starting point after using the mechanisms of mutation and recombination. Another way of building up on this work is an analysis of the location of the additional second objective $f_2$. Having shown that multiobjectivization can also be beneficial for other than evolutionary algorithms, a dynamic environment could improve the local search as well. The concept mostly applied in evolutionary optimization for maintaining diversity [@branke2012dynamic] could be transferred to our local search. Still aiming at optimizing a constant function $f_1$, the location of the additional objective $f_2$ could be changed during the search process. Without getting stuck in local traps, the diversity could lead to the exploration of a broader area of the multi-objective landscape. Further research can be conducted in the field of optimizing MOGSA and its parameter-setting. Especially the step-size could be focused on for ideally adjusting the size automatically to the landscape. All in all, we have shown that – by adding an additional objective – multiobjectivization can be beneficial for multimodal continuous single-objective optimization and highlighted potential for further investigation.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors acknowledge support by the [*European Research Center for Information Systems (ERCIS)*](https://www.ercis.org).
|
---
abstract: |
We present a computational approach for generating Markov bases for multi-way contingency tables whose cells counts might be constrained by fixed marginals and by lower and upper bounds. Our framework includes tables with structural zeros as a particular case. Instead of computing the entire Markov basis in an initial step, our framework finds sets of local moves that connect each table in the reference set with a set of neighbor tables. We construct a Markov chain on the reference set of tables that requires only a set of local moves at each iteration. The union of these sets of local moves forms a dynamic Markov basis. We illustrate the practicality of our algorithms in the estimation of exact p-values for a three-way table with structural zeros and a sparse eight-way table. Computer code implementing the methods described in the article as well as the two datasets used in the numerical examples are available as supplemental material.\
[**Keywords:**]{} Contingency tables, Exact tests, Markov bases, Markov chain Monte Carlo, Structural zeros.
author:
- 'Adrian Dobra[^1]'
bibliography:
- 'dobra-jcgs-paper.bib'
title: Dynamic Markov Bases
---
Introduction {#sec:intro}
============
Sampling from sets of contingency tables is key for performing exact conditional tests. Such tests arise by eliminating nuisance parameters through conditioning on their minimal sufficient statistics [@agresti-1992]. They are needed when the validity of asymptotic approximations to the null distributions of test statistics of interest is questionable or when no such approximations are available. Kreiner argues against the use of large-sample $\chi^2$ approximations for goodness-of-fit tests for large sparse tables, while Haberman raises similar concerns for tables having expected cell counts that are small and large. The problem is further compounded by the existence of structural zeros or by limits on the values allowed on each cell, e.g. occurrence matrices in ecological studies [@chen-jasa2005].\
One of the earlier algorithms for sampling two-way contingency tables with fixed row and column totals is due to Mehta and Patel . Other key developments include the importance sampling approaches of Booth and Butler , Chen et al. , Chen et al. and Dinwoodie and Chen . Various Markov chain algorithms have been proposed by Besag and Clifford , Guo and Thompson , Forster et al. and Caffo and Booth . A very good review is presented in Caffo and Booth .\
One of the central contributions to the literature was the seminal paper by Diaconis and Sturmfels . They generate tables in a reference set $T$ through a Markov basis. The fundamental concept behind a Markov basis is easily understood by considering all the possible pairwise differences of tables in $T$, i.e. $\mathcal{M} = \{ n^{\prime}-n^{\prime\prime}:n^{\prime},n^{\prime\prime}\in T\}$. The elements of $\mathcal{M}$ are called moves. Any table $n^{\prime}\in T$ can be transformed in another table $n^{\prime\prime}\in T$ by applying the move $n^{\prime\prime}-n^{\prime}\in \mathcal{M}$. Clearly not all the moves in $\mathcal{M}$ are needed to connect any two tables in $T$ through a series of moves. A Markov basis for $T$ is obtained by eliminating some of the moves in $\mathcal{M}$ such that the remaining moves still connect $T$. Generating a Markov basis is in the most general case a computationally difficult task that is solved using computational algebraic techniques. The simplest Markov basis contains only moves with two entries equal to $1$, two entries equal to $-1$ and the remaining entries equal to zero. It connects all the two-way tables with fixed row and columns totals [@diaconis-sturmfels-1998]. These primitive moves extend to decomposable log-linear models as described in Dobra . A divide-and-conquer technique for the determination of Markov bases for reducible log-linear models is given in Dobra and Sullivant . Additional information on Markov bases can be found in Drton et al. .\
In this paper we focus on the general problem of the determination of a Markov basis for sets of multi-way tables defined by fixed marginals and by lower and upper bounds constraints on each cell count. Bounds constraints arise in disclosure limitation from information deemed to be public at a certain time [@willenborg-dewaal-2000]. In ecological inference lower bounds constrains are induced by individual-level information [@wakefield-2004]. Noteworthy theoretical contributions on Markov bases for bounded tables include Aoki and Takemura , Rapallo , Rapallo and Rogantin , Aoki and Takemura , and Rapallo and Yoshida . Unfortunately, it is quite difficult to carry out a principled assessment of the practical value of their algebraic statistics results for tables with more than two dimensions due to the absence of dedicated software that would make these methods accessible to lay users.\
So far the papers dedicated to Markov bases have attempted to generate them in a preliminary step that needs to be completed before the corresponding random walk can be started. In practice this step can be computationally prohibitive to perform because the resulting Markov bases contain a very large number of elements even for three-way tables [@deloera-onn-2005]. The Markov bases repository of Kahle and Rauh (`http://mbdb.mis.mpg.de`) is very useful for understanding the complexity of the moves even for simple, non-decomposable log-linear models. We avoid this major computational hurdle by developing dynamic Markov bases. Such bases do not have to be generated in advance. Instead, at each iteration of our Markov chain algorithm we sample from a set of local moves that connect the table that represents the current state of the chain with a set of neighbor tables. Our computational approach extends the applicability of Markov bases to examples that could not be handled with other approaches presented in the literature.\
The structure of the paper is as follows. In Section \[sec:framework\] we present the notations and the setting of our framework. In Section \[sec:dmb\] we introduce dynamic Markov bases and present two algorithms for sampling multi-way tables. In Section \[sec:art\] we discuss these algorithms in the context of the importance sampling approaches of Booth and Butler and Chen et al. . In Sections \[sec:markovchain\] and \[sec:finding\] we give our Markov chain algorithm based on dynamic Markov bases. In Section \[sec:examples\] we illustrate the applicability of our methodology for a three-way table with structural zeros and a sparse eight-way table. In Section \[sec:conclusions\] we make concluding remarks.
Notations and Framework {#sec:framework}
=======================
Let $X=(X_{1},X_{2},\ldots,X_{k})$ be a vector of discrete random variables. Variable $X_{j}$ takes values $x_{j}\in \mathcal{I}_j=\{1,2,\ldots,I_j\}$, $I_j\ge 2$. Consider a contingency table $n=\{n(i)\}_{i\in \mathcal{I}}$ of observed counts associated with $X$, where $\mathcal{I}=\operatorname{\times}\limits_{j=1}^{k}I_{j}$ are cell indices. The set $\mathcal{I}$ is assumed to be ordered lexicographically, so that $\mathcal{I}=\left\{ i^{1},i^{2},\ldots,i^{m_{c}}\right\}$, where $i^{1}=(1,\ldots,1,1)$, $i^{2}=(1,\ldots,1,2)$, $i^{m_{c}}=(I_{1},I_{2},\ldots,I_{k})$ and $m_{c}=I_{1}\cdot I_{2}\cdot\ldots\cdot I_{k}$ is the total number of cells. With this ordering the $k$-dimensional array $n=\{n(i)\}_{i\in \mathcal{I}}$ is written as a vector $n=\left\{ n(i^{1}),n(i^{2}),\ldots,n(i^{m_{c}})\right\}$. For $C\subset K=\{1,\ldots,k\}$, the $C$-marginal $n_{C}=\{ n_C(i_C)\}_{i_C\in \mathcal{I}_C}$ of $n$ is the cross-classification associated with the sub-vector $X_{C}$ of $X$, where $\mathcal{I}_C=\operatorname{\times}_{j\in C}\mathcal{I}_j$. The grand total of $n$ is $n_{\emptyset}$.\
Consider two other $k$-way tables $n^L$ and $n^U$ that define lower and upper bounds for $n$. The role of these bounds is to specify various constraints that might exist for the cell entries of $n$. For example, a structural zero in cell $i\in \mathcal{I}$ is specified as $n^L(i)=n^U(i)=0$. Zero-one tables are expressed by taking $n^L(i)=0$ and $n^U(i)=1$ for all $i\in \mathcal{I}$. In addition to the bounds constraints, the cell entries of $n$ can be required to satisfy a set of linear constraints induced by a set of fixed marginals $\{ n_{C}:C\in \mathcal{C}\}$, where $\mathcal{C}=\{ C_1,\ldots,C_q\}$, with $C_j\subset K$ for $j=1,\ldots,q$. We let $\mathcal{A}$ be a log-linear model whose minimal sufficient statistics are $\{ n_{C}:C\in \mathcal{C}\}$. We define the set of tables that are consistent with the minimal sufficient statistics of $\mathcal{A}$ and with the bounds $n^L$ and $n^U$: $$\begin{aligned}
\label{eq:T}
T=\left\{ n^{\prime}=\{n^{\prime}(i)\}_{i\in \mathcal{I}}: n^{\prime}_{C_j}=n_{C_j},\mbox{ for }j=1,\ldots,q, n^L(i)\le n'(i)\le n^U(i),\mbox{ for }i\in \mathcal{I}\right\}.\end{aligned}$$ We assume that $n\in T$, that is, the bounds constraints $n^L$ and $n^U$ are not at odds with the observed data. The set $T$ induces bounds constraints $L(i)$ and $U(i)$ on each cell entry $n(i)$, $i\in \mathcal{I}$: $$\begin{aligned}
L(i) = \min\left\{ n^{\prime}(i): n^{\prime}\in T\right\},\quad U(i) = \max\left\{ n^{\prime}(i): n^{\prime}\in T\right\}.\end{aligned}$$ These bounds are possibly tighter than the initial bounds $n^L$ and $n^U$, i.e. $$0\le n^{L}(i)\le L(i)\le n^{\prime}(i)\le U(i)\le n^{U}(i)\le n_{\emptyset},$$ for $i\in \mathcal{I}$ and $n^{\prime}\in T$. They can be determined by integer programming algorithms [@boyd-vandenberghe-2004] or by other methods such as the generalized shuttle algorithm [@dobra-fienberg-2008]. The constraints that define $T$ can lead to the exact determination of some cell counts. More explicitly, we consider $\mathcal{S}\subset \mathcal{I}$ to be the set of cells such that $L(i)<U(i)$. This means that all the tables in $T$ have the same counts for the cells in $\mathcal{I}\setminus \mathcal{S}$. We note that the determination of $\mathcal{S}$ needs to be made based on the bounds $L=\left\{L(i)\right\}_{i\in \mathcal{I}}$ and $U=\left\{U(i)\right\}_{i\in \mathcal{I}}$ and not on $n^{L}$ and $n^{U}$. Thus the set $T$ comprises all the integer arrays $n^{\prime}$ that satisfy the equality constraints $$\begin{aligned}
\label{eq:linearconstraintsT}
n^{\prime}_{C_{j}}\left( i_{C_{j}}\right) & = & n_{C_{j}}\left( i_{C_{j}}\right), \mbox{ for }i_{C_{j}}\in \mathcal{I}_{C_{j}}, j=1,2,\ldots,q,\\
n^{\prime}(i) & = & n(i),\mbox{ for } i\in \mathcal{I}\setminus\mathcal{S},\nonumber\end{aligned}$$ as well as the bounds constraints $$\begin{aligned}
\label{eq:boundsconstraintsT}
& L(i)\le n^{\prime}(i)\le U(i),\; \mbox{ for } i\in \mathcal{S}.& \end{aligned}$$ By ordering the cell indices $\mathcal{I}$ lexicographically the equality constraints (\[eq:linearconstraintsT\]) can be written as a linear system of equations $$\begin{aligned}
\label{eq:linsysaxb}
A n^{\prime} & = & b,\end{aligned}$$ where $A$ is a $m_{r}\operatorname{\times}m_{c}$ matrix with elements equal to $0$ or $1$, $m_{r}=\sum\limits_{j=1}^{q}\mid\mathcal{I}_{C_{j}}|+|\mathcal{I}\setminus\mathcal{S}|$ and $b=An$ is a $m_{r}$-dimensional column vector. Here $|E|$ denotes the number of elements of a set $E$. In order to simplify the notations we subsequently assume that $\mathcal{S}=\mathcal{I}$ with the understanding that the determination of $\mathcal{S}$ is key and needs to be completed before our algorithms are applied.\
Two distributions defined on $T$ play a key role in statistical analyses. They are the uniform and the hypergeometric distributions $$\begin{aligned}
\label{eq:targetdistrib}
P_U(n^{\prime}) = \frac{1}{|T|}, \mbox { and }
P_H(n^{\prime}) = \frac{\left[ \prod\limits_{i\in \mathcal{I}} n^{\prime}(i)!\right]^{-1}}{\sum\limits_{n^{\prime\prime}\in T}\left[ \prod\limits_{i\in \mathcal{I}} n^{\prime\prime}(i)!\right]^{-1}},\end{aligned}$$ for $n'\in T$. In the most general case, the normalizing constants of $P_{H}(\cdot)$ and $P_U(\cdot)$ can be computed only if $T$ can be enumerated. Sundberg developed a formula for the normalizing constant of $P_H(\cdot)$ if $\mathcal{A}$ is decomposable and there are no bounds constraints (i.e. $n^L(i)=0$ and $n^U(i)=n_{\emptyset}$ for all $i\in \mathcal{I}$). Sampling from $P_U(\cdot)$ is relevant for estimating the number of tables in $T$ [@chen-jasa2005; @dobra-jspi2006] or for performing the conditional volume test [@diaconis-efron-1985]. The hypergeometric distribution $P_H(\cdot)$ arises by conditioning on the log-linear model $\mathcal{A}$ and the set of tables $T$ under multinomial sampling. Haberman proved that the the log-linear interaction terms cancel out, which leads to equation (\[eq:targetdistrib\]).\
Sampling from $P_U(\cdot)$ and $P_H(\cdot)$ is straightforward if $T$ can be explicitly determined, but this task is computationally infeasible for most real-world datasets. The goal of this paper is to develop a sampling procedure from $P_H(\cdot)$ and $P_U(\cdot)$ for any set of tables $T$ induced by a set of fixed marginals and lower and upper bounds arrays.
Dynamic Markov Bases {#sec:dmb}
====================
Producing an entire Markov basis up-front is computationally expensive; it also makes random walks impractical for reference sets $T$ involving sparse high-dimensional tables. Such bases contain an extremely large number of moves that are difficult to handle in the rare cases when they can actually be found using an algebra package. However, one does not necessarily need to know the entire Markov basis in order to run a Markov chain on $T$. The Markov bases we introduce in this section are dynamic because they are not generated ahead of time. They consist of sets of moves that connect a given table $n^{*}\in T$ with a set of neighbor tables $\mbox{nbd}_{T}(n^{*})\subseteq T$. The union of the sets of neighbor tables should be symmetric (i.e., $n^{\prime}\in \mbox{nbd}_{T}(n^{\prime\prime})$ if and only if $n^{\prime\prime}\in \mbox{nbd}_{T}(n^{\prime})$), and their union should connect $T$, i.e., $$\begin{aligned}
\label{eq:localbasis}
\bigcup_{n^{\prime}\in T} \{ n^{\prime\prime}-n^{\prime}:n^{\prime\prime}\in \mbox{nbd}_{T}(n^{\prime})\}\end{aligned}$$ is a Markov basis for $T$. The moves given by the difference between a table $n^{\prime}$ and one of its neighbors $n^{\prime\prime}\in \mbox{nbd}_{T}(n^{\prime})$ are called local.\
The sets of neighbors are determined as follows. For two integers $a\le b$, we denote $(a:b)=\{a,a+1,\ldots,b\}$. We define $(a:b)=\emptyset$ if $a>b$. Let $\Delta_{m_{c}}$ denote the set of all permutations of $(1:m_{c})$. For a permutation $\delta\in \Delta_{m_{c}}$, we define the set of tables $T_{\delta}$ that is obtained by reordering the cell counts of tables in $T$ according to $\delta$. The re-ordered version $n^{*}_{\delta}\in T_{\delta}$ of $n^{*}$ is such that $n^{*}_{\delta}(i^{j}) = n^{*}(i^{\delta(j)})$ for $1\le j\le m_{c}$. The difference between $T_{\delta}$ and $T$ relates to the ordering of their cells. We have $T=T_{\delta_{0}}$ where $\delta_{0}\in \Delta_{m_{c}}$, $\delta_{0}(j)=j$ for $1\le j\le m_{c}$.\
For a table $n^{*}\in T$ and an index $s\in (1:m_{c})$, we define the set of tables that have the same counts in cells $\{i^{\delta(1)},\ldots,i^{\delta(s)}\}$ as table $n^{*}$: $$\begin{aligned}
\label{eq:tdelta}
T_{\delta,s}(n^{*}_{\delta}) = \left\{ n^{\prime}_{\delta}\in T_{\delta}:n^{\prime}_{\delta}\left(i^{j}\right)=n^{*}_{\delta}\left(i^{j}\right), \mbox{ for }j=1,\ldots,s\right\}.\end{aligned}$$ We define $T_{\delta,0}(n^{*}_{\delta}) = T_{\delta}$. We have $T_{\delta,m_{c}}(n^{*}_{\delta}) = \{n^{*}_{\delta}\}$, and $n^{*}_{\delta}\in T_{\delta,s}(n^{*}_{\delta})$ for any $s\in (0:m_{c})$. The sets of tables $T_{\delta,s}(n^{*}_{\delta})$ become smaller as the number of common cells increases, i.e. $T_{\delta,s}(n^{*}_{\delta}) \supseteq T_{\delta,s'}(n^{*}_{\delta})$ for $0\le s \le s'\le m_{c}$. We consider the minimum and the maximum values of cell $i^{j}$ in the set of tables $T_{\delta,s}(n^{*}_{\delta})$, i.e. $$\begin{aligned}
L_{\delta,n^{*},s}(i^{j}) = \min\left\{ n^{\prime}_{\delta}(i^{j}):n^{\prime}_{\delta}\in T_{\delta,s}(n^{*}_{\delta}) \right\},\quad U_{\delta,n^{*},s}(i^{j}) = \max\left\{ n^{\prime}_{\delta}(i^{j}):n^{\prime}_{\delta}\in T_{\delta,s}(n^{*}_{\delta}) \right\}.\end{aligned}$$ Remark that $L_{\delta,n^{*},s}(i^{j}) =U_{\delta,n^{*},s}(i^{j})=n^{*}_{\delta}(i^{j})$ for $j\in (1:s)$. Determining the minimum and maximum values for the remaining cells without exhaustively enumerating $T_{\delta,s}(n^{*}_{\delta})$ can be done by computing the integer lower and upper bounds induced on each cell by the constraints that define this set of tables. For $j\in ((s+1):m_{c})$, $L_{\delta,n^{*},s}(i^{j})$ and $U_{\delta,n^{*},s}(i^{j})$ are the solutions of the linear programming problems $$\begin{aligned}
\label{eq:linprogbounds}
\mbox{minimize} & \pm n^{\prime}_{\delta}(i^{j}) &\\
\mbox{subject to} & An^{\prime} = b, &\nonumber \\
& L(i)\le n^{\prime}(i)\le U(i), & \mbox{for } i\in \mathcal{I},\nonumber \\
& n^{\prime}_{\delta}\left(i^{j}\right)=n^{*}_{\delta}\left(i^{j}\right), & \mbox{for } j=1,\ldots,s,\nonumber\\
& n^{\prime}(i) \in {\mathbb{N}}, & \mbox{for } i\in \mathcal{I}.\nonumber \end{aligned}$$ Here ${\mathbb{N}}$ is the set of nonnegative integers. Computationally it is quite demanding to determine the integer bounds $L_{\delta,n^{*},s}(i^{j})$ and $U_{\delta,n^{*},s}(i^{j})$, hence we approximate them with the integer counterparts of the real bounds $L^{R}_{\delta,n^{*},s}(i^{j})$ and $U^{R}_{\delta,n^{*},s}(i^{j})$. These real bounds are calculated by solving the optimization problems (\[eq:linprogbounds\]) without the constraints $n^{\prime}(i) \in {\mathbb{N}}$, for $i\in \mathcal{I}$. In general, we have $$L_{\delta,n^{*},s}(i^{j}) \ge \left\lceil L^{R}_{\delta,n^{*},s}(i^{j})\right\rceil, \quad U_{\delta,n^{*},s}(i^{j}) \le \left\lfloor U^{R}_{\delta,n^{*},s}(i^{j})\right\rfloor.$$ We denote by $\lceil a\rceil$ and $\lfloor a\rfloor$ the smallest integer greater than or equal to $a$ and the largest integer smaller than or equal to $a$, respectively. For the purpose of implementing the procedures described in this paper the approximation given by rounding the real bounds seems to perform well.\
We describe a method for randomly sampling a table in $T_{\delta}$. Algorithm \[alg:randomtable\] generates a feasible table by sequentially sampling the count of each cell given that the counts of the cells preceding it in the reordering of $\mathcal{I}$ defined by $\delta$ have already been fixed. The permutation $\delta$ defines the order in which the cell counts are sampled. The set of possible values of each cell are defined by the lower and upper bounds induced by the constraints that define $T$ and the cell counts already determined. This procedure is employed at each iteration of the sequential importance sampling (SIS) algorithm [@chen-annals2006; @dinwoodie-chen-2010] and has also been suggested, in various forms, in other papers [@chen-jasa2005; @dobra-jspi2006; @chen-jcgs2007]. We note that the determination of multi-way tables through a sequential adjustment of cell bounds appears in earlier writings such as Dobra who proposes a branch-and-bound algorithm for enumerating all the multi-way tables consistent with a set of linear and bounds constraints, as well as Dobra et al. and Dobra and Fienberg who develop the generalized shuttle algorithm.
Consider a table $n^{\prime}_{\delta}$ whose cells are currently unoccupied. Set $s\leftarrow 1$. Calculate the updated bounds for cell $i^{s}$. If $s=1$, set $L^{\prime}_{\delta,n^{\prime},0}(i^{1})=L(i^{\delta(1)})$ and $U^{\prime}_{\delta,n^{\prime},0}(i^{1})=U(i^{\delta(1)})$. Otherwise solve the linear programming problems (\[eq:linprogbounds\]) to determine the real bounds for cell $i^{s}$ and set $L^{\prime}_{\delta,n^{\prime},s-1}(i^{s})=\left \lceil L^{R}_{\delta,n^{\prime},s-1}(i^{s})\right\rceil$ and $U^{\prime}_{\delta,n^{\prime},s-1}(i^{s})=\left\lfloor U^{R}_{\delta,n^{\prime},s-1}(i^{s})\right\rfloor$. STOP. Set $n^{\prime}_{\delta}(i^{s})\leftarrow L^{\prime}_{\delta,n^{\prime},s-1}(i^{s})$. Sample a cell value $n^{\prime}_{\delta}(i^{s})$ from a discrete distribution $f^{\left(L^{\prime}_{\delta,n^{\prime},s-1}(i^{s}):U^{\prime}_{\delta,n^{\prime},s-1}(i^{s})\right)}(\cdot)$ with support $\left( L^{\prime}_{\delta,n^{\prime},s-1}(i^{s}):U^{\prime}_{\delta,n^{\prime},s-1}(i^{s})\right)$. Go to the next cell by setting $s\leftarrow s+1$.
Algorithm \[alg:randomtable\] ends at line 6 without returning a table if the combination of cell values chosen at the previous iterations does not correspond with any table in $T_{\delta}$. Such combinations could arise because there are gaps between the bounds that correspond with integers for which there do not exist any tables in $T$ associated with them. This issue has been properly recognized and discussed in Chen et al. who also propose conditions which they call the sequential interval property that check whether gaps exist for certain tables and configurations of fixed marginals. To the best of the authors’ knowledge, there are no computational tools that implement these conditions. Once such tools become available, Algorithm \[alg:randomtable\] could be improved by replacing lines 5 and 6 with a procedure for identifying which integers in $\left( L^{\prime}_{\delta,n^{\prime},s-1}(i^{s}):U^{\prime}_{\delta,n^{\prime},s-1}(i^{s})\right)$ actually correspond to least one table in $T$. This set of integers becomes the support of the discrete distribution from line 11. With this refinement Algorithm \[alg:randomtable\] will always return a valid table. In the numerical examples from Section \[sec:examples\] we use the reciprocal distribution $f_{r}^{L,U}(v) \propto 1/(1+v)$ to sample a cell value at line 11. Other possible choices include the uniform distribution $f_{u}^{L,U}(v) = 1/(U-L+1)$ or the hypergeometric distribution $f_{h}^{L,U}(v) = {U \choose v}{U \choose L+U-v}/{2U \choose L+U}$.\
Algorithm \[alg:randomtable\] finds any table $n^{*}_{\delta}\in T_{\delta}$ with strictly positive probability $$\begin{aligned}
\label{eq:sampleneighbor}
\pi_{\delta,f^{(L:U)}(\cdot)}(n^{*}_{\delta}) \propto \prod\limits_{s=1}^{m_{c}} f^{(L^{\prime}_{\delta,n^{*},s-1}(i^{s}):U^{\prime}_{\delta,n^{*},s-1}(i^{s}))}(n^{*}_{\delta}(i^{s})).\end{aligned}$$ We define the neighbors of $n^{*}_{\delta}$ as the set of tables returned by Algorithm \[alg:randomtable\], i.e. $\mbox{nbd}_{T_{\delta}}(n^{*}_{\delta})=T_{\delta}$. The corresponding set of local moves (\[eq:localbasis\]) is a Markov basis for $T_{\delta}$. Since $T$ and $T_{\delta}$ are in a one-to-one correspondence, this is also a Markov basis for $T$. This Markov basis is dynamic because its moves are sampled using Algorithm \[alg:randomtable\] from the distribution (\[eq:sampleneighbor\]).\
Algorithm \[alg:randomtable\] returns a table in $T_{\delta}$ only after it has computed lower and upper bounds for each cell in $\mathcal{I}$. Calculating $2m_{c}$ bounds to generate one feasible table could be quite expensive especially for high-dimensional sparse tables. The counts of zero that characterize such tables are likely to make quite a few cells take only one possible value given the current values of the cells that have been already fixed – see lines 8 and 9. Therefore the efficiency of Algorithm 1 can be increased by identifying these fixed-value cells without computing bounds. We consider an array $x=\{ x(i^{1}),\ldots,x(i^{m_{c}})\}$. We transform the linear system of equations (\[eq:linsysaxb\]) defined by the equality constraints (\[eq:linearconstraintsT\]) by reordering the columns $i^{1},\ldots,i^{m_{c}}$ of the matrix $A$ according to $\delta$. The reordered versions of $A$ and $x$ are $A_{\delta}$ and $x_{\delta}$. The column of $A_{\delta}$ that corresponds with $x_{\delta}(i^{j})$ is equal with the column of $A$ that corresponds with $x(i^{\delta(j)})$. An equivalent form of the linear system (\[eq:linsysaxb\]) is $$\begin{aligned}
\label{eq:linsysdelta}
A_{\delta}x_{\delta} = b.\end{aligned}$$ We take the augmented $m_{r}\times (m_{c}+1)$ matrix $[ A_{\delta}\mid b]$ obtained by stacking $A_{\delta}$ and $b$ along side each other. We determine the reduced row echelon form (RREF) $[\widehat{A}_{\delta}\mid\widehat{b}]$ of $[ A_{\delta}\mid b]$ using Gauss-Jordan elimination with partial pivoting – see, for example, Shores . The linear system (\[eq:linsysdelta\]) is equivalent with $$\begin{aligned}
\label{eq:rref}
\widehat{A}_{\delta}x_{\delta} = \widehat{b},\end{aligned}$$ whose number of rows $m_{r}^{\prime}\le \min\{m_{r},m_{c}\}$ is equal with the rank of $A$. Since the linear system (\[eq:rref\]) has fewer equations than the initial linear system (\[eq:linsysaxb\]), it is more efficient to make use of it when defining the linear programming problems (\[eq:linprogbounds\]). A smaller number of constraints translates into reduced computing times in the determination of the bounds in line 4 of Algorithm \[alg:randomtable\]. Furthermore it is possible to re-arrange the columns of $\widehat{A}_{\delta}$ and the coordinates of $x_{\delta}$ such that the system (\[eq:rref\]) is written as $$\begin{aligned}
\label{eq:boundfree}
I_{m_{r}^{\prime}} x_{\delta}^{B} + \widehat{A}_{\delta}^{R} x_{\delta}^{F} = \widehat{b},\end{aligned}$$ where $x_{\delta}^{B}$ represent the $m_{r}^{\prime}$ bound variables of the equivalent linear systems (\[eq:linsysaxb\]) and (\[eq:linsysdelta\]), $x_{\delta}^{F}$ is the $(m_{c}-m_{r}^{\prime})$-dimensional vector of free variables and $I_{l}$ is the $l$-dimensional identity matrix. Once the values of the free cells $x_{\delta}^{F}$ are fixed, the values of the bound cells are immediately determined: $$\begin{aligned}
\label{eq:boundsfromfree}
x_{\delta}^{B} = \widehat{b} - \widehat{A}_{\delta}^{R} x_{\delta}^{F}.\end{aligned}$$
Consider a table $n^{\prime}_{\delta}$ whose cells are currently unoccupied. Find the RREF of the linear system (\[eq:linsysdelta\]). Sample values for the free cells $\{(n^{\prime}_{\delta})^{F}_{j}:1\le j\le (m_{c}-m^{\prime}_{r})\}$ as described in lines 4-13 of Algorithm \[alg:randomtable\]. Determine the values of the bound cells $(n^{\prime}_{\delta})^{B}$ using equation (\[eq:boundsfromfree\]). STOP.
This leads us to a new version Algorithm \[alg:randomtablerref\] of Algorithm \[alg:randomtable\]. The successful determination of a table in $T_{\delta}$ using Algorithm \[alg:randomtablerref\] requires the calculation of $2(m_{c}-m^{\prime}_{r})$ bounds instead of $2m_{c}$ bounds as in Algorithm \[alg:randomtable\]. Furthermore, the calculation of these bounds is faster because the reduced system (\[eq:rref\]) is used. Lines 2 and 4 of Algorithm \[alg:randomtablerref\] can be implemented efficiently using BLAS (Basic Linear Algebra Subprograms) Fortran routines for matrix manipulations, thus overall Algorithm \[alg:randomtablerref\] has a significant computational gain over Algorithm \[alg:randomtable\]. We point out that the determination of the RREF should be done for the system (\[eq:linsysdelta\]) and not for the initial system (\[eq:linsysaxb\]) since each permutation of cell indices could lead to different sets of bound and free cells. Empirically we observed that calculating the RREF is computationally inexpensive and can be efficiently performed at each application of Algorithm \[alg:randomtablerref\]. Line 5 of Algorithm \[alg:randomtablerref\] is needed because certain combinations of values for the free cells might not correspond to any table in $T$, in which case negative integers are found in one or several bound cells. When computing the lower bounds $L^{F}_{\delta,n^{\prime},j}$ and the upper bounds $U^{F}_{\delta,n^{\prime},j}$ for the $j$-th free cell $(n^{\prime}_{\delta})^{F}_{j}$ in line 3 of Algorithm \[alg:randomtablerref\], we add the linear constraints associated with the sampled values of the first $(j-1)$ free cells and make use of the reduced system (\[eq:rref\]) in the corresponding linear programming problems (\[eq:linprogbounds\]). The probability that Algorithm \[alg:randomtablerref\] samples a table $n^{\prime}_{\delta}\in T_{\delta}$ is strictly positive: $$\begin{aligned}
\label{eq:sampleneighborrref}
\pi^{F}_{\delta,f^{(L:U)}(\cdot)}(n^{*}_{\delta}) \propto \prod\limits_{j=1}^{m_{c}-m^{\prime}_{r}} f^{(L^{F}_{\delta,n^{\prime},j}:U^{F}_{\delta,n^{\prime},j})}((n^{\prime}_{\delta})^{F}_{j}).\end{aligned}$$
State of the Art {#sec:art}
================
Algorithm \[alg:randomtable\] is key for the sequential importance sampling (SIS) algorithm [@chen-annals2006; @dinwoodie-chen-2010]. Tables from $T_{\delta}$ are sampled from the discrete distribution given in equation (\[eq:sampleneighbor\]) and are further used to calculate importance sampling estimates of various quantities of interest. For example, when calculating exact p-values, tables sampled from the uniform and hypergeometric distributions $P_{U}(\cdot )$ and $P_{H}(\cdot )$ given in equation (\[eq:targetdistrib\]) are needed, but cannot be obtained through a direct sampling procedure. Instead, tables sampled with Algorithm \[alg:randomtable\] are obtained, but these tables yield reliable estimates of exact p-values only if the discrete distribution (\[eq:sampleneighbor\]) is close to the target distributions $P_{U}(\cdot )$ or $P_{H}(\cdot )$. Various cells orderings $\delta\in \Delta_{m_{c}}$ and discrete distributions $f^{(L:U)}(\cdot)$ lead to discrete distributions (\[eq:sampleneighbor\]) that could be quite far from a desired target distribution on $T$. Unfortunately there is no well defined computational procedure that allows the selection of $\delta\in \Delta_{m_{c}}$ and $f^{(L:U)}(\cdot)$ for any set of tables $T$ and any target distribution on $T$. The SIS algorithm as described by Chen et al. , Chen et al. , Chen performs well for many applications, but completely fails for the two numerical examples we discuss in Section \[sec:examples\] that involve a three-way table with structural zeros and a sparse eight-way binary table. In a recent contribution, Dinwoodie and Chen propose a procedure for sequentially updating the discrete distribution $f^{(L:U)}(\cdot)$ from line 11 of Algorithm \[alg:randomtable\] as a function of the previously sampled cell values. With this improved version of SIS they obtain more promising results for the sparse eight-way binary table example. However, there is no theoretical argument which shows that the examination of other examples will not lead to situations in which SIS does not perform well due to the inability of Algorithm \[alg:randomtable\] to sample tables that receive high probabilities under the target distribution. Replacing Algorithm \[alg:randomtable\] with the more efficient Algorithm \[alg:randomtablerref\] in an importance sampling procedure leads to improved computing times, but does not solve the critical issues related to finding appropriate choices of $\delta$ and $f^{(L:U)}(\cdot)$.\
Booth and Butler proposed another approach for sampling multi-way tables. They start with a log-linear model $\mathcal{A}$ with minimal sufficient statistics $\left\{ n_{C}:C\in \mathcal{C}\right\}$ (see Section \[sec:framework\]) and consider the expected cell values $\hat{\mu}=\left\{ \hat{\mu}(i^{1}),\ldots,\hat{\mu}(i^{m_{c}})\right\}$ under $\mathcal{A}$. Their sampling method is designed for a reference set of tables $T$ specified by the marginals $\left\{ n_{C}:C\in \mathcal{C}\right\}$. Since the ability to compute the expected cell values $\hat{\mu}$ is key, their framework does not extend to sets of tables that are also consistent with some lower and upper bounds $n^{L}$ and $n^{U}$. Therefore the sampling method of Booth and Butler has a more limited domain of applicability than Algorithms \[alg:randomtable\] and \[alg:randomtablerref\]. Booth and Butler consider a permutation of cell indices $\delta \in \Delta_{m_{c}}$ and partition the reordered cells $x_{\delta}$ as bound cells $x_{\delta}^{B}$ and free cells $x_{\delta}^{F}$ as in equation (\[eq:boundfree\]). Furthermore, they assume that the cell counts follow independent normal distributions $x_{\delta}(i^{j})\sim {\mathsf{N}}(\hat{\mu}(i^{j}),\hat{\mu}(i^{j}))$, $1\le j\le m_{c}$, which implies that the joint distribution of the free cells follows a multivariate normal distribution $$\begin{aligned}
\label{eq:mvnfree}
x_{\delta}^{F} & \sim & {\mathsf{N}}_{m_{c}-m^{\prime}_{r}}\left( \hat{\mu}_{\delta},\hat{V}^{\delta}\right),\end{aligned}$$ where $\hat{V}^{\delta}=(\hat{v}^{\delta}_{j_{1}j_{2}})$ is a covariance matrix that depends on $\hat{\mu}$ and the counts in the marginals $\left\{ n_{C}:C\in \mathcal{C}\right\}$. Algorithm \[alg:boothbutler\] outlines the method for sampling tables from $T_{\delta}$ introduced by Booth and Butler .
Consider a table $n^{\prime}_{\delta}$ whose cells are currently unoccupied. Partition the cells as bound $x_{\delta}^{B}$ and free $x_{\delta}^{F}$. Sample $\gamma_{1} \sim {\mathsf{N}}\left((\hat{\mu}_{\delta})_{1},\hat{v}^{\delta}_{11}\right)$, the marginal distribution of $(x_{\delta})^{F}_{1}$ as derived from the joint distribution (\[eq:mvnfree\]). Set $(n^{\prime}_{\delta})^{F}_{1} = [\gamma_{1}]$. Here $[a]$ represents the nearest integer to $a$. Sample from the marginal distribution of $(x_{\delta})^{F}_{j}$ conditional on the current values of the preceding free cells as derived from the joint distribution (\[eq:mvnfree\]): $$\hspace{-1cm}\gamma_{j} \sim {\mathsf{N}}\left( {\mathsf{E}}[(x_{\delta})^{F}_{j}\mid (x_{\delta})^{F}_{(1:(j-1))}=(n_{\delta}^{\prime})^{F}_{(1:(j-1))}],{\mathsf{Var}}[(x_{\delta})^{F}_{j}\mid (x_{\delta})^{F}_{(1:(j-1))}=(n_{\delta}^{\prime})^{F}_{(1:(j-1))}]\right).$$ Set $(n_{\delta}^{\prime})^{F}_{j} = [\gamma_{j}]$. Determine the values of the bound cells $(n^{\prime}_{\delta})^{B}$ using equation (\[eq:boundsfromfree\]). STOP.
It is worthwhile to compare how Algorithms \[alg:randomtablerref\] and \[alg:boothbutler\] differ. A contingency table in $T_{\delta}$ is determined in Algorithm \[alg:randomtablerref\] by sequentially calculating lower and upper bounds associated with the free cell whose value is sampled next, which entails solving $2(m_{c}-m_{r}^{\prime})$ optimization problems. In Algorithm \[alg:boothbutler\] the calculation of bounds is replaced by simulations from multivariate normal distributions whose means and variances are obtained through fast matrix operations [@booth-butler-1999]. Since the determination of bounds comes at a higher computational cost, Algorithm \[alg:boothbutler\] is much faster than Algorithm \[alg:randomtablerref\]. Unfortunately, Algorithm \[alg:boothbutler\] gives no guarantees that it will actually identify any table in $T_{\delta}$. A necessary condition for the successful generation of a table in $T_{\delta}$ is that the values sampled at lines 3 and 6 of Algorithm \[alg:boothbutler\] are actually between their lower and upper bounds calculated at line 3 of Algorithm \[alg:randomtablerref\]. To the best of the authors’ knowledge, there does not exist any proof of this claim. In fact, this necessary condition is not mentioned in Booth and Butler or in the subsequent work of Caffo and Booth . From a theoretical perspective, there is no justification why Algorithm \[alg:boothbutler\] should successfully output a feasible table in $T_{\delta}$. Furthermore, there is no justification why Algorithm \[alg:boothbutler\] should be able to sample any table in $T_{\delta}$ with strictly positive probability. Despite being faster, Algorithm \[alg:boothbutler\] should not be preferred to Algorithm \[alg:randomtablerref\] due to its lack of theoretical underpinning. In addition, Algorithm \[alg:randomtablerref\] can be used to sample from reference sets of tables defined by bounds constraints in addition to linear constraints induced by fixed marginals, while Algorithm \[alg:boothbutler\] cannot be used in such general situations because it relies on the calculations of MLEs associated with a log-linear model.\
Algorithm \[alg:boothbutler\] is employed by Booth and Butler to develop an importance sampling approach for producing Monte Carlo estimates of exact p-values. Caffo and Booth slightly modify Algorithm \[alg:boothbutler\] by fixing a random number of free cells to develop a Markov chain algorithm for conditional inference. Both papers present successful applications of Algorithm \[alg:boothbutler\] in generating feasible tables from a reference set $T$. However, such examples cannot substitute the need to provide rigorous proofs justifying the applicability of Algorithm \[alg:boothbutler\]. Without such proofs one cannot know when to expect Algorithm \[alg:boothbutler\] to succeed or to fail.\
For these reasons the existent literature does not seem to contain a reliable method for calculating exact p-values that works for arbitrary multi-way tables subject to linear and bounds constraints. In the next section we propose a new Markov chain algorithm that makes use of Algorithm \[alg:randomtablerref\] to sample tables from a reference set. There is a significant advantage of using Algorithm \[alg:randomtablerref\] in the context of a Markov chain algorithm as opposed to an importance sampling procedure such as SIS: the discrete distribution (\[eq:sampleneighbor\]) becomes a proposal distribution for generating the candidate for the next state of the chain. The accuracy of the resulting exact p-values estimates is tied significantly less to how close the discrete distribution (\[eq:sampleneighbor\]) is to the hypergeometric or uniform target distributions. Moreover, the instances in which Algorithm \[alg:randomtablerref\] ends without successfully generating a table in the reference set are thrown out in an importance sampling method. On the other hand, a Markov chain procedure makes use of all the output from Algorithm \[alg:randomtablerref\] even if no feasible table was identified.
The Proposed Markov Chain Algorithm {#sec:markovchain}
===================================
We present a Markov chain algorithm that samples from a distribution $P_{*}(\cdot )$ on the reference set of tables $T$ whose key component is the dynamic Markov bases introduced in Section \[sec:dmb\]. Algorithm \[alg:randomtablerref\] generates feasible tables given an ordering of the cell indices $\mathcal{I}$ induced by a permutation $\delta\in \Delta_{m_{c}}$. The partitioning of the cells as bound and free as well as the sequence in which the values of the free cells are sampled are a function of the choice of $\delta$. The linear and bounds constraints that define $T$ and the sequence of free cells associated with permutations $\delta$ translate into various lower and upper bounds for the possible values of a particular cell. Empirically we observed that some tables in $T$ receive very high probabilities (\[eq:sampleneighbor\]) of being sampled under some permutations in $\Delta_{m_{c}}$, but under other permutations the same probabilities could be very low. Characterizing the relationship between the discrete distribution (\[eq:sampleneighbor\]) and a distribution on $T$ as a function of various cell orderings and distributions $f^{(L:U)}(\cdot)$ is a difficult problem that is currently open. The mixing time of a Markov chain that calls Algorithm \[alg:randomtablerref\] to generate candidate tables could vary considerably if the permutation $\delta\in \Delta_{m_{c}}$ remains fixed across iterations. Since there are no theoretical results that would allow one to produce cell orderings that lead to smaller mixing times, we develop a Markov chain with state space $T\operatorname{\times}\Delta_{m_{c}}$ with stationary distribution $$\begin{aligned}
\label{eq:jointstationary}
{\mathsf{Pr}}( n,\delta) & = & {\mathsf{Pr}}(n\mid \delta){\mathsf{Pr}}(\delta).\end{aligned}$$ Conditional on $\delta \in \Delta_{m_{c}}\setminus \{\delta_{0}\}$, a table $n\in T=T_{\delta_{0}}$ is transformed in a table $n_{\delta}\in T_{\delta}$ with the same cell counts but a different ordering of its cells. This implies ${\mathsf{Pr}}(n\mid \delta) = {\mathsf{Pr}}(n_{\delta})$. Sampling from the joint distribution (\[eq:jointstationary\]) is relevant in this context only if $P_{*}(\cdot )$ coincides with the marginal distribution ${\mathsf{Pr}}(n) = \sum_{\delta\in \Delta_{m_{c}}}{\mathsf{Pr}}(n_{\delta}){\mathsf{Pr}}(\delta)$. This condition is satisfied for ${\mathsf{Pr}}(n_{\delta})=P_{*}(n_{\delta})$ if $P_{*}(\cdot )$ is invariant to cell orderings, i.e. $P_{*}(n_{\delta}) = P_{*}(n)$ for all $\delta \in \Delta_{m_{c}}$. The uniform and hypergeometric distributions $P_{U}(\cdot )$ and $P_{H}(\cdot )$ from equation (\[eq:targetdistrib\]) are indeed order invariant. Since there is no reason to favor a cell ordering over another, we assume a uniform distribution ${\mathsf{Uni}}_{\Delta_{m_{c}}}(\cdot)$ on the set of possible permutations of $\mathcal{I}$. Therefore the stationary distribution (\[eq:jointstationary\]) is $$\begin{aligned}
\label{eq:jointstationarygood}
{\mathsf{Pr}}( n,\delta) & = & P_{*}(n_{\delta}){\mathsf{Uni}}_{\Delta_{m_{c}}}(\delta).\end{aligned}$$ We note that the marginal distribution of (\[eq:jointstationarygood\]) associated with $\delta$ is again uniform.\
We start the chain by sampling a permutation $\delta^{(0)} \sim {\mathsf{Uni}}_{\Delta_{m_{c}}}(\cdot)$ and using Algorithm \[alg:randomtablerref\] with cell ordering $\delta^{(0)}$ to sample a table $n^{(0)} \in T$. Given a current state $(n^{(t)},\delta^{(t)})$, we sample a new permutation $\delta^{(t+1)} \sim {\mathsf{Uni}}_{\Delta_{m_{c}}}(\cdot)$. We also sample a candidate table $n^{*}_{\delta^{(t+1)}}\in T_{\delta^{(t+1)}}$ from a proposal distribution $q_{\delta^{(t+1)}}(n^{(t)}_{\delta^{(t+1)}},\cdot)$. If we did not obtain a feasible table (i.e., $n^{*}\notin T$), the next state of the chain is $(n^{(t)},\delta^{(t+1)})$. If $n^{*}\in T$, the next state $(n^{(t+1)},\delta^{(t+1)})$ is $(n^{*},\delta^{(t+1)})$ with the Metropolis-Hastings probability $$\begin{aligned}
\label{eq:acceptanceprob}
\min\left\{ 1, \frac{P_{*}(n^{*}_{\delta^{(t+1)}})q_{\delta^{(t+1)}}(n^{*}_{\delta^{(t+1)}},n^{(t)}_{\delta^{(t+1)}})}{P_{*}(n^{(t)}_{\delta^{(t+1)}})q_{\delta^{(t+1)}}(n^{(t)}_{\delta^{(t+1)}},n^{*}_{\delta^{(t+1)}})}\right\}.\end{aligned}$$ Otherwise we set $n^{(t+1)}=n^{(t)}$. A sufficient condition for the irreducibility of this Markov chain is the positivity of the instrumental distribution, i.e. $$\begin{aligned}
\label{eq:positivity}
q_{\delta}(n^{\prime}_{\delta},n^{\prime\prime}_{\delta})>0, \mbox{ for every }(n^{\prime}_{\delta},n^{\prime\prime}_{\delta})\in T_{\delta}\times T_{\delta} \mbox{ and } \delta\in \Delta_{m_{c}}.\end{aligned}$$ It is possible to employ Algorithm \[alg:randomtablerref\] to generate candidate tables $n^{*}$. In this case the Markov chain stays at its current state if Algorithm \[alg:randomtablerref\] does not generate a feasible table in $T$. If a feasible candidate table $n^{*}$ is identified, the acceptance probability (\[eq:acceptanceprob\]) is calculated based on the proposal distribution $q_{\delta^{(t+1)}}(n^{(t)}_{\delta^{(t+1)}},n^{*}_{\delta^{(t+1)}}) = \pi_{\delta^{(t+1)},f^{(L:U)}(\cdot)}(n^{*}_{\delta^{(t+1)}})$ – see equation (\[eq:sampleneighbor\]). Since Algorithm \[alg:randomtablerref\] can return any table in $T$, the positivity condition (\[eq:positivity\]) is satisfied. Unfortunately the probability of proposing a candidate table at iteration $t$ from the reference set is independent of $n^{(t)}$. This leads to an erratic behavior of the Markov chain with very small acceptance rates for new candidate tables.\
A better option is to sample candidate tables $n^{*}$ that have a number $M$ of cell counts in common with $n^{(t)}$. The maximum value for $M$ is $(m_{c}-m^{\prime}_{r}-1)$. Recall that $(m_{c}-m^{\prime}_{r})$ is the number of free cells associated with $T$. For a permutation $\delta\in \Delta_{m_{c}}$, we construct a proposal distribution $q_{\delta}(n^{(t)}_{\delta},\cdot)$ as follows. We partition $T_{\delta}\setminus \{n_{\delta}^{(t)}\}$ in subsets of tables that have the counts of the first $M$ free cells equal with the counts of the first $M$ free cells of $n^{(t)}_{\delta}$ but the count of the $(M+1)$-th free cell different than $(n^{(t)}_{\delta})^{F}_{M+1}$: $$\begin{aligned}
\label{eq:tdeltadecomp}
T_{\delta}\setminus\{n_{\delta}^{(t)}\} & = &\operatorname*{\cup}\limits_{M=0}^{m_{c}-m^{\prime}_{r}-1} \mbox{nbd}_{\delta,M}(n^{(t)}_{\delta}),
\end{aligned}$$ where $$\begin{aligned}
\mbox{nbd}_{\delta,M}(n^{(t)}_{\delta}) & = & T^{\prime}_{\delta,M}(n^{(t)}_{\delta})\setminus T^{\prime}_{\delta,M+1}(n^{(t)}_{\delta}), \mbox{ and }\\
T^{\prime}_{\delta,M}(n^{(t)}_{\delta}) & = & \left\{ n^{\prime}_{\delta}\in T_{\delta}:(n^{\prime}_{\delta})^{F}_{j}=(n^{(t)}_{\delta})^{F}_{j}, \mbox{ for }j=1,\ldots,M\right\}.\end{aligned}$$ We remark that $T^{\prime}_{\delta,m_{c}-m^{\prime}_{r}}(n^{(t)}_{\delta})=\{n^{(t)}_{\delta}\}$. We refer to the tables in $\mbox{nbd}_{\delta,M}(n^{(t)}_{\delta})$ as the neighbors of $n^{(t)}_{\delta}$ of order $M$. Any table in $T_{\delta}\setminus \{n_{\delta}^{(t)}\}$ is the neighbor of $n^{(t)}_{\delta}$ of a particular order. For certain values of $M\in (0:(m_{c}-m^{\prime}_{r}-1))$, there might not exist a neighbor of $n^{(t)}_{\delta}$ of order $M$.\
We modify Algorithm \[alg:randomtablerref\] into Algorithm \[alg:mcmctable\] such that the feasible tables it generates are neighbors of order $M$ of table $n^{(t)}_{\delta}$. Line 2 of Algorithm \[alg:mcmctable\] guarantees that, if a feasible table is returned, then this table belongs to $T^{\prime}_{\delta,M}(n^{(t)}_{\delta})$. By eliminating $(n^{(t)}_{\delta})^{F}_{M+1}$ from the possible values of the free cell $(n^{\prime}_{\delta})^{F}_{M+1}$ in line 3, we guarantee that Algorithm \[alg:mcmctable\] does not return a table that belongs to $T^{\prime}_{\delta,M+1}(n^{(t)}_{\delta})$. Algorithm \[alg:mcmctable\] samples a table $n^{\prime}_{\delta}\in \mbox{nbd}_{\delta,M}(n^{(t)}_{\delta})$ with strictly positive probability: $$\begin{aligned}
\label{eq:probM}
\pi^{F}_{\delta,f^{(L:U)}(\cdot),M} (n^{\prime}_{\delta}\mid n^{(t)}_{\delta}) \propto f^{\left(L^{F}_{\delta,n^{\prime},M}:U^{F}_{\delta,n^{\prime},M}\right)\setminus \{ (n^{(t)}_{\delta})^{F}_{M+1}\}}((n^{\prime}_{\delta})^{F}_{M+1}) \prod\limits_{j=M+2}^{m_{c}-m^{\prime}_{r}} f^{(L^{F}_{\delta,n^{\prime},j}:U^{F}_{\delta,n^{\prime},j})}((n^{\prime}_{\delta})^{F}_{j}).\end{aligned}$$
Consider a table $n^{\prime}_{\delta}$ whose cells are currently unoccupied. Set the counts of the first $M$ free cells of $n^{\prime}_{\delta}$ to the corresponding counts of $n^{(t)}_{\delta}$, i.e. $$(n^{\prime}_{\delta})^{F}_{j} = (n^{(t)}_{\delta})^{F}_{j}, \mbox{ for } j=1,\ldots,M.$$ Sample the values of the remaining free cells $(n^{\prime}_{\delta})^{F}_{j}$, $j=M+1,\ldots,m_{c}-m_{r}^{\prime}$ using line 3 of Algorithm \[alg:randomtablerref\]. When sampling the value of the $(M+1)$-th free cell, eliminate $(n^{(t)}_{\delta})^{F}_{M+1}$ from the set of possible values of this cell. Attempt to determine a full table $n^{\prime}_{\delta}$ as described in lines 5-12 of Algorithm \[alg:randomtablerref\].
We consider the set of all the local moves associated with $n^{(t)}_{\delta}$ in $T_{\delta}$: $$\begin{aligned}
\label{eq:mdelta}
\mathcal{M}_{\delta}(n^{(t)}_{\delta}) & = &\left\{ n^{\prime}_{\delta}-n^{(t)}_{\delta}:n^{\prime}_{\delta}\in T_{\delta}\setminus\{n_{\delta}^{(t)}\}\right\}.\end{aligned}$$ Their union $\operatorname*{\cup}_{n_{\delta}^{(t)}\in T_{\delta}}\mathcal{M}_{\delta}(n^{(t)}_{\delta})$ is a Markov basis for $T_{\delta}$. The decomposition (\[eq:tdeltadecomp\]) of $T_{\delta}\setminus\{n_{\delta}^{(t)}\}$ as sets of neighbor tables of $n_{\delta}^{(t)}$ of various orders translates into a corresponding decomposition of the set of local moves associated with $n^{(t)}_{\delta}$ in $T_{\delta}$: $$\begin{aligned}
\label{eq:localmovesdec}
\mathcal{M}_{\delta}(n^{(t)}_{\delta}) = \operatorname*{\cup}\limits_{M=0}^{m_{c}-m^{\prime}_{r}-1} \mathcal{M}_{\delta,M}(n^{(t)}_{\delta}),\end{aligned}$$ where $\mathcal{M}_{\delta,M}(n^{(t)}_{\delta})= \left\{ n^{\prime}_{\delta}-n^{(t)}_{\delta}:n^{\prime}_{\delta}\in \mbox{nbd}_{\delta,M}(n^{(t)}_{\delta})\right\}$. We dynamically generate local moves in $\mathcal{M}_{\delta}(n^{(t)}_{\delta})$ as follows. We consider a discrete distribution $g_{T}(\cdot)$ that gives a strictly positive probability to each integer in $(0:(m_{c}-m^{\prime}_{r}-1))$. We draw $M\sim g_{T}(\cdot)$ then employ Algorithm \[alg:mcmctable\] to sample a table $n^{*}_{\delta}\in \mbox{nbd}_{\delta}(n^{(t)}_{\delta})$. This gives us a local move $n^{*}_{\delta}-n^{(t)}_{\delta}\in \mathcal{M}_{\delta,M}(n^{(t)}_{\delta})$ without having to determine the entire set $\mathcal{M}_{\delta,M}(n^{(t)}_{\delta})$. We use this procedure to sample candidate tables for the Markov chain algorithm with stationary distribution (\[eq:jointstationarygood\]). The corresponding instrumental distribution is given by $$\begin{aligned}
\label{eq:proposalmixture}
q_{\delta}(n^{(t)}_{\delta},n^{*}_{\delta}) & = & \sum_{M=0}^{m_{c}-m^{\prime}_{r}-1} g_{T}(M) \pi^{F}_{\delta,f^{(L:U)}(\cdot),M} (n^{*}_{\delta}\mid n^{(t)}_{\delta}) I_{\left\{n^{*}_{\delta}\in \mbox{nbd}_{\delta,M}(n^{(t)}_{\delta})\right\}}.\end{aligned}$$ We remark that $n^{*}_{\delta}\in \mbox{nbd}_{\delta,M}(n^{(t)}_{\delta})$ implies $n^{(t)}_{\delta}\in \mbox{nbd}_{\delta,M}(n^{*}_{\delta})$ and $n^{(t)}_{\delta}\notin \mbox{nbd}_{\delta,M^{\prime}}(n^{*}_{\delta})$ for $M^{\prime}\ne M$. Therefore, in order to calculate the Metropolis-Hastings acceptance ratio (\[eq:acceptanceprob\]), we need to evaluate only one component of the mixture distribution (\[eq:proposalmixture\]). For $n^{*}_{\delta}\in \mbox{nbd}_{\delta,M}(n^{(t)}_{\delta})$, we have $$\begin{aligned}
\frac{q_{\delta}(n^{*}_{\delta},n^{(t)}_{\delta})}{q_{\delta}(n^{(t)}_{\delta},n^{*}_{\delta})} & = & \frac{\pi^{F}_{\delta,f^{(L:U)}(\cdot),M} (n^{(t)}_{\delta}\mid n^{*}_{\delta})}{\pi^{F}_{\delta,f^{(L:U)}(\cdot),M} (n^{*}_{\delta}\mid n^{(t)}_{\delta})}.\end{aligned}$$ This makes the computing effort needed to run the resulting Markov chain quite manageable. The chain is irreducible because the positivity condition (\[eq:positivity\]) is satisfied as a result of $\operatorname*{\cup}_{n_{\delta}^{(t)}\in T_{\delta}}\mathcal{M}_{\delta}(n^{(t)}_{\delta})$ being a Markov basis for $T_{\delta}$.\
The choice of the discrete distribution $g_{T}(\cdot)$ is crucial for a good performance of the chain. The values of $M$ sampled from $g_{T}(\cdot)$ need to maintain a balance between making large jumps in $T$ (hence being more likely to reject the move) and making small jumps in $T$ (hence being less likely to reject the move, but spending many iterations around similar tables). Specifying a reasonable distribution $g_{T}(\cdot)$ could be a daunting task since it needs to be tailored specifically for $T$. In the next section we give a coherent procedure for finding $g_{T}(\cdot)$ based on a flexible algorithm for exploring an arbitrary target set of tables.
The Algorithm for Finding $g_{T}(\cdot)$ {#sec:finding}
========================================
We present a method for producing a discrete distribution $g_{T}(\cdot)$ required in the specification of the proposal distribution (\[eq:proposalmixture\]). Our approach is based on a repeated approximation of the number of free cells whose counts need to be fixed before all the other counts are uniquely determined. An upper bound for this number is the total number of free cells $(m_{c}-m_{r}^{\prime})$ – see Section \[sec:dmb\]. However, due to the particular configurations of small and large counts of the tables in $T$ and to the presence of the bounds constraints (\[eq:boundsconstraintsT\]), this number can actually be anywhere between $1$ and $(m_{c}-m_{r}^{\prime})$.\
We assume that $T$ contains at least two tables. We consider a permutation $\delta\in \Delta_{m_{c}}$ and a table $n^{*}\in T$. We let $\mathcal{F}_{\delta}\subset \mathcal{I}$ be the indices of the $(m_{c}-m_{r}^{\prime})$ free cells associated with $T$ and $\delta$ – see Section \[sec:dmb\]. We take $\delta^{\prime}\in \Delta_{m_{c}}$ such that $\left\{i^{\delta^{\prime}(1)},i^{\delta^{\prime}(2)},\ldots,i^{\delta^{\prime}(m_{c}-m_{r}^{\prime})}\right\}=\mathcal{F}_{\delta}$. Recall from equation (\[eq:tdelta\]) that $T_{\delta^{\prime},s}(n^{*}_{\delta^{\prime}})$ represents the set of tables in $T$ that have the same counts in cells $\{i^{\delta^{\prime}(1)},\ldots,i^{\delta^{\prime}(s)}\}$ as table $n^{*}$. There exists a unique index $s(\delta^{\prime},n^{*})\in (0:(m_{c}-m_{r}^{\prime}-1))$ such that
- (C1) $T_{\delta^{\prime},s(\delta^{\prime},n^{*})}(n^{*}_{\delta^{\prime}})$ contains at least one table in $T$ that is different than $n^{*}$;
- (C2) $T_{\delta^{\prime},s(\delta^{\prime},n^{*})+1}(n^{*}_{\delta^{\prime}})=\{n^{*}_{\delta^{\prime}}\}$.
If $T_{\delta^{\prime},s}(n^{*}_{\delta^{\prime}})$ contains a table different than $n^{*}_{\delta^{\prime}}$, there must exist $j\in ((s+1):(m_{c}-m_{r}^{\prime}))$ such that the lower and upper bounds of the corresponding cell are different: $$\begin{aligned}
\label{eq:boundsdifferent}
\left\lceil L^{R}_{\delta^{\prime},n^{*},s}(i^{\delta^{\prime}(j)})\right\rceil < \left\lfloor U^{R}_{\delta^{\prime},n^{*},s}(i^{\delta^{\prime}(j)})\right\rfloor.\end{aligned}$$ This condition is necessary, but it is not sufficient. That is, equation (\[eq:boundsdifferent\]) might hold while still $T_{\delta^{\prime},s}(n^{*}_{\delta^{\prime}})=\{n^{*}_{\delta^{\prime}}\}$. As such, the computation of real bounds cannot substitute actually checking that $T_{\delta^{\prime},s}(n^{*}_{\delta^{\prime}})\setminus \{n^{*}_{\delta^{\prime}}\}\ne \emptyset$, but such a check is computationally expensive. We reduce this computing effort by first determining an upper bound for $s(\delta^{\prime},n^{*})$.\
Algorithm \[alg:binarysearch\] performs a binary search to determine the maximum index $s^{b}(\delta^{\prime},n^{*})< (m_{c}-m_{r}^{\prime})$ such that $$\begin{aligned}
\label{eq:boundsequal}
\left\lceil L^{R}_{\delta^{\prime},n^{*},s^{b}(\delta^{\prime},n^{*})+1}(i^{\delta(j)})\right\rceil = n^{*}(i^{\delta(j)})= \left\lfloor U^{R}_{\delta^{\prime},n^{*},s^{b}(\delta^{\prime},n^{*})+1}(i^{\delta(j)})\right\rfloor,\end{aligned}$$ for $j\in ((s^{b}(\delta^{\prime},n^{*})+2):(m_{c}-m_{r}^{\prime}))$, but $\left\lceil L^{R}_{\delta^{\prime},n^{*},s^{b}(\delta^{\prime},n^{*})}(i^{\delta(j)})\right\rceil < \left\lfloor U^{R}_{\delta^{\prime},n^{*},s^{b}(\delta^{\prime},n^{*})}(i^{\delta(j)})\right\rfloor$ for at least one index $j\in ((s^{b}(\delta^{\prime},n^{*})+1):(m_{c}-m_{r}^{\prime}))$. Condition (\[eq:boundsdifferent\]) is always satisfied for $s=0$ as long as $T$ contains at least two tables. Moreover, condition (\[eq:boundsdifferent\]) is never satisfied for $s=m_{c}-m_{r}^{\prime}$ since $T_{\delta^{\prime},m_{c}-m_{r}^{\prime}}(n^{*}_{\delta^{\prime}})=\{ n^{*}_{\delta^{\prime}}\}$. At the completion of Algorithm \[alg:binarysearch\], the index $s^{b}(\delta^{\prime},n^{*})$ is returned and we still need to determine $s(\delta^{\prime},n^{*})$. Since $T_{\delta^{\prime},s^{b}(\delta^{\prime},n^{*})+1}(n^{*}_{\delta^{\prime}})=\{n^{*}_{\delta^{\prime}}\}$, condition (C2) is satisfied and hence $s(\delta^{\prime},n^{*})\le s^{b}(\delta^{\prime},n^{*})$. Algorithm \[alg:sdelta\] starts with $s^{b}(\delta^{\prime},n^{*})$ as the initial guess for the value of $s(\delta^{\prime},n^{*})$ and sequentially decreases this guess until condition (C1) is also satisfied.\
Set $s_{1} \leftarrow 0$ and $s_{2} \leftarrow (m_{c}-m_{r}^{\prime})$. Set $s \leftarrow \lfloor(s_{1}+s_{2})/2\rfloor$. Calculate the real lower and upper bounds $L^{R}_{\delta^{\prime},n^{*},s}(i^{\delta(j)})$ and $U^{R}_{\delta^{\prime},n^{*},s}(i^{\delta(j)})$. Set $s_{2}\leftarrow s$. Set $s_{1}\leftarrow s$. Return $s^{b}(\delta^{\prime},n^{*})\leftarrow s_{1}$.
Set free cells $\{ i^{\delta^{\prime}(1)},\ldots,i^{\delta^{\prime}(s)}\}$ to the corresponding values from $n^{*}$. Sample the values of the remaining components of the vector of free cells $x_{\delta}^{F}$ using lines 5-14 of Algorithm \[alg:randomtable\]. Attempt to determine a full table $n^{\prime}\in T$ as described in lines 5-12 of Algorithm \[alg:randomtablerref\].
Algorithm \[alg:sdelta\] returns a value of $s$ that is less or equal than $s(\delta,n^{\prime})$. However, for our purposes, this lower bound is sufficient. Algorithm \[alg:gdot\] estimates the discrete distribution $g_{T}(\cdot)$ by repeatedly calling Algorithms \[alg:randomtablerref\], \[alg:binarysearch\] and \[alg:sdelta\] for a large number of iterations $i_{max}$. The value of $g_{T}(j)$, $j\in (0:(m_{c}-m_{r}^{\prime}-1))$, is proportional with the number of iterations in which keeping $j$ counts of free cells fixed resulted in the successful sampling of a feasible table different than some other randomly generated feasible table. The initialization from line 2 of Algorithm \[alg:gdot\] assures that the distribution $g_{T}(\cdot)$ returned by the procedure satisfies $g_{T}(j) > 0$ for any $j\in (0:(m_{c}-m_{r}^{\prime}-1))$ which is a condition required to generate all the local moves from equation (\[eq:mdelta\]). Algorithm \[alg:gdot\] effectively explores the set of tables $T$ and identifies a distribution $g_{T}(\cdot)$ based on this exploration. We remark that we have not made any assumptions about a parametric form for $g_{T}(\cdot)$. The structure of $T$ dictates the probabilities that define $g_{T}(\cdot)$ which leads to a very flexible choice of $g_{T}(\cdot)$ which is adapted to the structure of $T$.
Consider a vector $G$ with indices $(1:(m_{c}-m_{r}^{\prime}))$. Set $G(j) \leftarrow 1$ for $j\in (1:(m_{c}-m_{r}^{\prime}))$. Call Algorithm \[alg:randomtablerref\] until it generates a random table $n^{*}\in T$. Generate a random permutation $\delta\in \Delta_{m_{c}}$ and find the RREF of the linear system (\[eq:linsysdelta\]). Call Algorithm \[alg:binarysearch\] to determine $s^{b}(\delta^{\prime},n^{*})$. Call Algorithm \[alg:sdelta\] to determine $s\le s(\delta^{\prime},n^{*})$. Set $G(s) \leftarrow G(s)+1$. Set $g_{T}(j) = G(j+1)/i_{max}$ for $j\in (0:(m_{c}-m_{r}^{\prime}-1))$.
Examples {#sec:examples}
========
We illustrate the use of the Markov chain algorithm with dynamic Markov bases in two examples. The first example involves a three-way table with structural zeros, while the second example involves a sparse eight-way table. Both examples have been chosen to show the effectiveness of the Markov chain algorithm described in Sections \[sec:markovchain\] and \[sec:finding\] with respect to competing approaches proposed in the literature. For both examples, we have been unable to generate a Markov basis using the computational algebraic techniques of Diaconis and Sturmfels , which renders their sampling approach inapplicable. The sequential importance sampling (SIS) algorithm of Chen et al. is applicable, but fails to provide any meaningful results by giving estimates equal to $1$ for all the p-values we calculate. The Markov chain algorithm of Caffo and Booth (CB, henceforth) as implemented in the R package `exactLoglinTest` [@caffo-2006] is not applicable for tables with structural zeros, hence it does not produce any estimates for our first example.\
We run $100$ independent Markov chains of length 2500000 with a burn-in time of 25000 iterations. The chains were run with the dynamic Markov bases approach and the CB algorithm. The SIS algorithm was run until it generated an equal number of sampled tables. We sampled from the hypergeometric distribution $P_{H}(\cdot)$ and calculated estimates for the exact p-values associated with the $X^{2}$ and $G^{2}$ statistics for the all two-way interaction model. We run $100$ replicates of Algorithm \[alg:gdot\] for $100000$ iterations to find the distribution $g_{T}(\cdot)$ that defines the Metropolis-Hastings proposal distribution (\[eq:proposalmixture\]).\
We estimate the Monte Carlo error using the non-overlapping batch means method of Geyer . Each of the $100$ independent chains was divided in $10$ batches of size $250000$. The standard error of an exact p-value estimate is the sample standard error of the p-value estimates corresponding with the $1000$ resulting batches. The Monte Carlo errors are given after the “$\pm$” sign following the Monte Carlo estimate of the exact p-value. We report the computing time necessary to generate one batch of $250000$ iterations throughout. We use OpenMPI (`http://www.open-mpi.org/`) to obtain batches by running independent processes on several processors.\
We performed our computations on a Mac Pro computer with 2 x 2.26 GHz quad-core Intel Xeon processors with 16 GB of memory. We report the mean elapsed computing time in seconds with standard errors calculated across the replicates. We wrote our own C++ implementation of the SIS algorithm by following the description from Chen et al. . The tables have been sampled in SIS using Algorithm \[alg:randomtable\] with cell values generated from the hypergeometric distribution $f_{h}(\cdot)$. We implemented the algorithms described in Sections \[sec:dmb\], \[sec:markovchain\] and \[sec:finding\] in C++. The linear programming problems (\[eq:linprogbounds\]) have been solved with IBM ILOG CPLEX Optimizer (`http://www.ibm.com`) routines. All the code and the datasets needed to replicate the numerical results from this section are available as supplemental materials.
NBER data
---------
Table \[table:nber\] is a $4\times 5\times 4$ cross-classification of $4345$ individuals by occupational groups (O1 – “self-employed, business”, O2 – “self-employed, professional”, O3 – “teacher”, O4 – “salary-employed”), aptitude levels (A) and educational levels (E). It was collected in a 1969 survey of the National Bureau of Economic Research (NBER) – see Table 3-6 page 45 from Fienberg . The horizontal lines denote structural zeros. The ten structural zeros under O3 and E1, E2 are associated with teachers being required to have higher education levels. The other two structural zeros under O2 can be motivated in a similar manner.\
The number of degrees of freedom for the all two-way interaction model is calculated by subtracting the number of structural zeros from $36$ – the number of degrees of freedom corresponding with a $4\times 5\times 4$ table without structural zeros. Bishop et al. argue that the number of degrees of freedom must be increased by the number of structural zeros that are present in marginal tables that are among the minimal sufficient statistics of the log-linear model considered. In this case there are two such counts present in the aptitude by educational levels marginal. The resulting number of degrees of freedom is $36-12+2=26$. The observed value of the likelihood-ratio test statistic is $G^2=15.91$ which leads to an asymptotic p-value for the all two-way interactions model of $0.938$. The observed value of the $X^2$ test statistic is $17.1$ which leads to an asymptotic p-value of $0.906$.\
The Markov chains with dynamic Markov bases lead to an estimate of the $G^{2}$ exact p-value of $0.9650\pm 0.0037$ and to an estimate of the $X^{2}$ exact p-value of $0.9134\pm 0.0068$. Figure \[fig:nberconvergence\] shows the convergence of the Markov chain algorithm from Section \[sec:markovchain\] across the $100$ chains. We remark that the large sample size of the NBER data leads to a good agreement between the asymptotic and the exact p-values. The computing time for one batch of $250000$ tables for our Markov chain algorithm is $1882.58\pm 1.33$ seconds. Figure \[fig:nberdiagnostic\] shows the estimated distribution function $g_{T}(\cdot)$ obtained from $10$ million iterations of Algorithm \[alg:gdot\]. The number of free cells is $26$ hence its domain is $\{0,1,\ldots,25\}$. We see that the mode of $g_{T}(\cdot)$ is $g_{T}(24)=0.604$ which represents the estimated probability of obtaining a feasible table different than the current table after fixing the values of $24$ free cells. The running time of Algorithm \[alg:gdot\] is $1266\pm 0.93
$ seconds per $100000$ iterations.
E1 E2 E3 E4 E1 E2 E3 E4
---- ---- ------ ------- ------ ------ -- ---- ---- ------- ------- ------- -------
O1 A1 $42$ $55$ $22$ $3$ O3 A1 – – $1$ $19$
A2 $72$ $82$ $60$ $12$ A2 – – $3$ $60$
A3 $90$ $106$ $85$ $25$ A3 – – $5$ $86$
A4 $27$ $48$ $47$ $8$ A4 – – $2$ $36$
A5 $8$ $18$ $19$ $5$ A5 – – $1$ $14$
O2 A1 $1$ $2$ $8$ $19$ O4 A1 $172$ $151$ $107$ $42$
A2 $1$ $2$ $15$ $33$ A2 $208$ $198$ $206$ $92$
A3 $2$ $5$ $25$ $83$ A3 $279$ $271$ $331$ $191$
A4 $2$ $2$ $10$ $45$ A4 $99$ $126$ $179$ $97$
A5 – – $12$ $19$ A5 $36$ $35$ $99$ $79$
: \[table:nber\] NBER data. The grand total of this table is $4345$.
Rochdale data
-------------
The data in Table \[tab:rochdaledata\] is a cross-classification of eight binary variables relating women’s economic activity and husband’s unemployment from a survey of households in Rochdale – see Whittaker page 279. The variables are as follows: $a$, wife economically active (no,yes); $b$, age of wife $>38$ (no,yes); $c$, husband unemployed (no,yes); $d$, child $\le 4$ (no,yes); $e$, wife’s education, high-school+ (no,yes); $f$, husband’s education, high-school+ (no,yes); $g$, Asian origin (no,yes); $h$, other household member working (no,yes). There are $665$ individuals cross-classified in $256$ cells, which means that the mean number of observations per cell is $2.6$. The table has $165$ counts of zero and $217$ other cells contain at most three observations.\
Whittaker argues that this table is sparse and subsequently that the applicability of any asymptotic results relating to the limiting distributions of goodness-of-fit statistics for log-linear models becomes questionable due to the zeros present in marginals of dimension three or more. The likelihood-ratio test statistic for the all two-way interaction model is $G^{2}=144.59$, while the observed $X^{2}$ test statistic is $258.65$. The all two-way interaction model has 219 degrees of freedom, which leads to asymptotic p-values of 1 for the $G^{2}$ statistic and of $0.034$ for the $X^{2}$ statistic.\
The Markov chain algorithm with dynamic Markov bases and the CB algorithm give similar estimates of the exact p-values. More specifically, the exact $G^{2}$ p-value is estimated to be $0.1668\pm 0.0684$ by our approach and $0.1644\pm 0.0443$ by the CB approach. The exact $X^{2}$ p-value is estimated to be $0.1642\pm 0.0524$ by our approach and $0.1717\pm 0.1101$ by the CB approach. The Monte Carlo standard errors for the $G^{2}$ p-value of both Markov chain algorithms are comparable, but the CB algorithm gives a larger standard error when computing the $X^{2}$ p-value. In a recent paper, Dinwoodie and Chen report two different estimates ($0.223\pm 0.091$ and $0.186\pm 0.041$) of the exact $G^{2}$ p-value obtained with their new version of the SIS algorithm based on two cell orderings. We found estimates equal to $1$ for both the $G^{2}$ and $X^{2}$ exact p-values using our implementation of the SIS algorithm of Chen et al. .\
Figure \[fig:rochdaleconvergence\] illustrates the convergence of the Markov chain algorithm from Section \[sec:markovchain\] across its $100$ replicates. Its running time is $18821.75\pm 304.01$ seconds per $250000$ iterations. Our Markov chain algorithm makes use of an estimate of the discrete distribution $g_{T}(\cdot)$ that is obtained by running Algorithm \[alg:gdot\] for $10$ million iterations. It takes approximately $8813.6\pm 40.62$ seconds per $100000$ sampled tables to obtain the distribition $g_{T}(\cdot)$ from Figure \[fig:grochdale\].
------ ------ ------ ----- ------ ------ ----- ----- ------ ------ ----- ----- ------ ------ ----- -----
$5$ $0$ $2$ $1$ $5$ $1$ $0$ $0$ $4$ $1$ $0$ $0$ $6$ $0$ $2$ $0$
$8$ $0$ $11$ $0$ $13$ $0$ $1$ $0$ $3$ $0$ $1$ $0$ $26$ $0$ $1$ $0$
$5$ $0$ $2$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $0$
$4$ $0$ $8$ $2$ $6$ $0$ $1$ $0$ $1$ $0$ $1$ $0$ $0$ $0$ $1$ $0$
$17$ $10$ $1$ $1$ $16$ $7$ $0$ $0$ $0$ $2$ $0$ $0$ $10$ $6$ $0$ $0$
$1$ $0$ $2$ $0$ $0$ $0$ $0$ $0$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
$4$ $7$ $3$ $1$ $1$ $1$ $2$ $0$ $1$ $0$ $0$ $0$ $1$ $0$ $0$ $0$
$0$ $0$ $3$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
$18$ $3$ $2$ $0$ $23$ $4$ $0$ $0$ $22$ $2$ $0$ $0$ $57$ $3$ $0$ $0$
$5$ $1$ $0$ $0$ $11$ $0$ $1$ $0$ $11$ $0$ $0$ $0$ $29$ $2$ $1$ $1$
$3$ $0$ $0$ $0$ $4$ $0$ $0$ $0$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
$1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
$41$ $25$ $0$ $1$ $37$ $26$ $0$ $0$ $15$ $10$ $0$ $0$ $43$ $22$ $0$ $0$
$0$ $0$ $0$ $0$ $2$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $3$ $0$ $0$ $0$
$2$ $4$ $0$ $0$ $2$ $1$ $0$ $0$ $0$ $1$ $0$ $0$ $2$ $1$ $0$ $0$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
------ ------ ------ ----- ------ ------ ----- ----- ------ ------ ----- ----- ------ ------ ----- -----
: \[tab:rochdaledata\]Rochdale data from Whittaker . The cells counts are written in lexicographical order with $h$ varying fastest and $a$ varying slowest. The grand total of this table is $665$.
Conclusions {#sec:conclusions}
===========
In this paper we introduced dynamic Markov bases and proposed a Markov chain algorithm for sampling tables based on them. Our methods are applicable off-the-shelf to calculate exact p-values for reference sets of tables defined by any type of linear and bounds constraints. The choice of distribution $g_{T}(\cdot)$ that is used in the mixture instrumental distribution (\[eq:proposalmixture\]) is key for a successful application of the Markov chain algorithm described in Section \[sec:markovchain\]. The running time of our sampling approach is a function of the expected number of optimization problems (\[eq:linprogbounds\]), i.e. $$\begin{aligned}
Q(g_{T}) & = & 2(m_{c}-m^{\prime}_{r}-E_{g_{T}}(M)),\end{aligned}$$ that need to be solved to generate one candidate table from (\[eq:proposalmixture\]). In our NBER data example, the number of free cells is $26$ which yields $Q(g_{T})=5.46$ for the distribution $g_{T}(\cdot)$ from Figure \[fig:nberdiagnostic\]. By comparison, if we would work with the uniform distribution $g_{T}(\cdot)={\mathsf{Uni}}_{(0:(m_{c}-m^{\prime}_{r}-1))}(\cdot)$ in the instrumental distribution (\[eq:proposalmixture\]), the expected number of optimization problems increases to $Q({\mathsf{Uni}}_{(0:25)})=27$. For the Rochdale data example we obtain $Q(g_{T})=134.1$ for the distribution $g_{T}(\cdot)$ from Figure \[fig:grochdale\] and $Q({\mathsf{Uni}}_{(0:218)})=220$. As such, Algorithm \[alg:gdot\] is quite effective in determining distributions $g_{T}(\cdot)$ that are lead to Markov chains with dynamic Markov bases with good mixing properties and reasonable running times. Finding suitable distributions $g_{T}(\cdot)$ that are properly adapted to a reference set of tables $T$ in the absence of a well-defined procedure could be detrimental in practice, hence Algorithm \[alg:gdot\] should be seen as integral part of the dynamic Markov bases methodology we proposed.\
We hope that the basic idea of generating only the moves needed to complete one iteration of the random walk will be adopted by other researchers since it is a more practical alternative to the determination of the entire Markov basis in one computationally intensive step as it was originally suggested in Diaconis and Sturmfels . Relevant questions relate to studying the theoretical properties of dynamic Markov bases using algebraic statistics in the spirit of Rapallo , Aoki and Takemura and Rapallo and Yoshida . These research directions should be added to the list of open problems related to Markov bases presented in Yoshida .\
Supplemental Material {#supplemental-material .unnumbered}
=====================
[**Computer Code and Data**]{}: Supplemental materials for this article are contained in a single zip archive and can be obtained in a single download. This archive contains the datasets NBER and Rochdale (in text files) as well as the C++ source code to run the algorithms described in this article (the Markov chain based on the dynamic Markov bases and the sequential importance sampling algorithm). A detailed description of the files contained in this archive is contained in a README.txt file enclosed in the archive.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by a seed grant from the Center of Statistics and the Social Sciences, University of Washington. The author thanks Anna Klimova for her assistance with some of the numerical results presented in the paper. The author thanks three anonymous reviewers and the AE for their helpful comments.
[^1]: Adrian Dobra is Assistant Professor, Departments of Statistics, Biobehavioral Nursing and Health Systems and the Center for Statistics and the Social Sciences, University of Washington, Seattle, WA 98195-4322 (email: adobra@uw.edu).
|
---
author:
- 'A. Bayo'
- 'D. Barrado'
- 'N. Huélamo'
- 'M. Morales-Calderón'
- 'C. Melo'
- 'J. Stauffer'
- 'B. Stelzer'
bibliography:
- './biblio.bib'
subtitle: 'II. On rotation, activity and other properties of spectroscopically confirmed members of Collinder 69. [^1]'
title: 'Spectroscopy of Very Low Mass Stars and Brown Dwarfs in the Lambda Orionis Star Forming Region.'
---
[Most observational studies so far point towards brown dwarfs sharing a similar formation mechanism as the one accepted for low mass stars. However, larger databases and more systematic studies are needed before strong conclusions can be reached. ]{} [In this second paper of a series devoted to the study of the spectroscopic properties of the members of the Lambda Orionis Star Forming Region, we study accretion, activity and rotation for a wide set of spectroscopically confirmed members of the central star cluster Collinder 69 to draw analogies and/or differences between the brown dwarf and stellar populations of this cluster. Moreover, we present comparisons with other star forming regions of similar and different ages to address environmental effects on our conclusions.]{} [We study prominent photospheric lines to derive rotational velocities and emission lines to distinguish between accretion processes and chromospheric activity. In addition, we include information about disk presence and X-ray emission. ]{} [We report very large differences in the disk fractions of low mass stars and brown dwarfs ($\sim$58%) when compared to higher mass stars (26$^{+4}_{-3}$ %) with 0.6 M$_{\odot}$ being the critical mass we find for this dichotomy. As a byproduct, we address the implications of the spatial distribution of disk and diskless members in the formation scenario of the cluster itself. We have used the H$\alpha$ emission to discriminate among accreting and non-accreting sources finding that 38$^{+8}_{-7}$ % of sources harboring disks undergo active accretion and that his percentage stays similar in the substellar regime. For those sources we have estimated accretion rates. Finally, regarding rotational velocities, we find a high dispersion in v$\sin(i)$ which is even larger among the diskless population. ]{}
Introduction
============
This is the second paper of a series devoted to studying, from a spectroscopic point of view, the young population present in several associations belonging to the Lambda Orionis Star Forming Region (LOSFR). In this paper, we systematically analyze several properties of the confirmed members of the central cluster Collinder 69 (C69); including the presence of disks, accretion onto the central star/brown dwarf, rotation and activity.
In [@Bayo11] (from now on Paper I), we presented a very detailed and complete spectroscopic census of Collinder 69; the oldest ($\sim$ 5 – 12 Myr) of the associations belonging to the LOSFR. In short, this star forming region is located at $\sim$400 pc [@Murdin77] representing the head of the Orion giant. Its center is dominated by the O8III multiple star $\lambda$ Orionis and comprises both recently formed stars from 0.2 M$_{\odot}$ to 24 M$_{\odot}$ and dark clouds actively forming stars.
The main goals of this paper are to study in detail the properties of the C69 confirmed members (e.g. rotation, activity, disk fractions and accretion rates), analyze the different populations within the cluster, and compare our results with other low mass star forming regions of similar and different ages. In fact, the evolutionary status of C69 seems to be specially suited to the study of disk evolution and accretion at the low end of the mass spectrum. According to our observations, in almost every bin in mass (for masses lower or equal to 0.6M$_{\odot}$, $\sim$3700K assuming a 5 Myr isochrone from @Baraffe98), we find a diversity of disk harboring sources: from those with optically thick disks undergoing active accretion (onto the central star) to others that have completely lost their primordial circumstellar disks. Finally, and as a byproduct of this study, we try to put our findings in context of the current theory of how the LOSFR as a whole was formed (triggered by a supernova, see @DM01, and more discussion on this scenario on Paper I).
This paper is organized as follows: In Section \[sec:data\] we provide a description of the data analyzed. In Sections \[sec:rotvel\] to \[sec:distrib\] properties such as rotation velocity, activity levels, accretion processes, disks, variability and spatial distribution of the population of C69 are studied. And, finally, our summary conclusions are presented in Section \[sec:conclusions\].
Unlike in Paper I, we have grouped the most interesting/puzzling objects into different categories (following the subsections of Sections \[sec:rotvel\] and \[sec:acacc\]) and discuss their peculiarities in Appendix A.
Observations and data analysis {#sec:data}
==============================
In this work we make use of an extensive spectral database that our group has been gathering during more than seven years and that is described in Paper I. Furthermore, we make use of the measurements and parameters derived for confirmed members published in [@DM99; @DM01; @Barrado07; @Sacco08; @Maxted08; @MoralesPhD; @BayoPhD; @Barrado11] and [@Franciosini11].
Spectroscopic database
----------------------
In total, we analyzed spectra obtained by us corresponding to $\sim$100 confirmed members, with several objects observed more than once, plus data corresponding to $\sim$70 objects that were studied in at least one of the papers cited before and for which we do not have our own observations.
In short, our own database comprises optical spectra of confirmed members of C69 with temperatures between $\sim$2800 and $\sim$4700 K. The spectra have resolutions in the range 600-11250, wavelength coverages from $\sim$5000 Å up to $\sim$10400 Å and have been obtained with different instruments and telescopes at Mauna Kea, Las Campanas, CAHA and VLT.
The data reduction of most of our campaigns was performed with IRAF[^2] following the standard steps. For the analysis of the spectra, motivated by the large amount of data, we developed a tool that provides for any given set of lines in an automatic manner (instrumental response corrected) full width half maximum (FWHM), full width at ten per cent of the flux (FW$_{10\%}$), and equivalent widths (EWs). The description of this routine and a study on the effect of the resolution on parameter determination is given in Appendix A of Paper I.
Photometric and X-ray database
------------------------------
In addition to the spectroscopic database, we have made use of the photometric dataset described in Paper I and analyzed in detail in [@MoralesPhD; @BayoPhD] and Morales-Calderon et al 2012 in prep., and of the X-ray data presented in [@Barrado11] and [@Franciosini11]. The photometric database includes photometry from the optical to the mid-infrared (MIR).
Taking a 5 Myr isochrone as a reference [@Baraffe98; @Chabrier00; @Allard03], the completeness of the optical data is located at $\sim$20 M$_{\rm Jup}$ for the whole cluster and even down to $\sim$3 M$_{\rm Jup}$ for a field to the East of the center of the cluster. In the near infrared (NIR), the data is complete down to $\sim$10 M$_{\rm Jup}$ in almost the whole cluster and down to $\sim$30 M$_{\rm Jup}$ in the outskirts. For the MIR, our database is complete at 3.6 $\mu$m down to $\sim$40 M$_{\rm Jup}$.
Finally the X-ray data is complete down to $\sim$0.3M$_{\sun}$ with detections in [@Barrado11] for confirmed members as low as $\sim$0.1M$_{\sun}$.
Fundamental parameters database
-------------------------------
We have adopted the results from [@MoralesPhD] and [@Barrado07] regarding the presence of circumstellar disks and substellar analogs. Objects with NIR and/or MIR slope above photospheric values are considered to harbor disks. This slope is analyzed as in [@Lada06] based on the $\alpha$ parameter.
The photometric database described before was used to feed VOSA [@Bayo08] and determine, via Spectral Energy Distribution (SED) fit, the effective temperatures and the bolometric luminosities for all confirmed members. These two parameters were used in Paper I, along with theoretical isochrones from the Lyon group [@Baraffe98; @Chabrier00; @Allard03], to determine masses for the confirmed members.
For the mass determination we assumed an age of 5Myr for C69 since that was the best fit to the produced HR diagram. We note, however, that in Paper I we also set an upper-limit for the age of C69 of 20Myr. We will address this point again in Section \[subsec:diskfrac\].
Rotational velocities {#sec:rotvel}
=====================
For the sample of sources observed with the highest resolution (those observed with Magellan/MIKE, R$\sim$11250; 14 sources in total), we estimated their projected rotational velocity. We add to this sample the objects with $v\sin(i)$ determination in [@Sacco08] to study the rotation of $\sim$25 members of C69.
Technique
---------
We based our determination of the projected rotational velocity on the comparison of the observed spectra with Kurucz models [@Castelli97] synthesized for different rotational velocities. Since our estimation of $v\sin(i)$ is based on comparisons with models of a specific T$_{\rm eff}$, and there is a known dispersion between effective temperatures based on spectral types and those derived using models (T$_{\rm eff}$ scales, see Paper I), we followed a three step process:
1. First, we derived T$_{\rm eff}$ and log(g) by comparison with theoretical models. We built a grid of synthetic spectra using different collections: Kurucz [@Castelli97], NextGen [@Hauschildt99] and Dusty [@Allard03] covering effective temperatures in the range 5000-2000 K, assuming solar metallicities, and a range of log(g) between 3.5 and 5.0. We lowered the resolution of both our spectra and the Kurucz models to R$\sim$200, which is the resolution provided in the public servers by the Lyon group, in order to perform a direct fitting process. Table \[tab:paramMIKE\] shows the best fitting set of parameters for each case.
2. Next, we prepared a second grid of high resolution Kurucz models (the ones we could generate with the same resolution as our Magellan/MIKE spectra), for different values of v$\sin(i)$ for direct comparison.
For the comparison we chose the wavelength range from 6090 Å to 6130 Å because the signal-to-noise ratio (SNR) of our spectra was not very high in all cases, and, for the range of temperatures to be analyzed, two of the most prominent photospheric absorption lines (from Ca I) are located within this region. Besides, for the grid of high resolution models, we assume log(g) of 4.0 dex that is a suitable value (according to models) for these young cool confirmed members of C69.
3. Since our high resolution grid of models did not include models from the Lyon group, we only analyzed sources where Kurucz was the best choice in Step 1, or those for which the difference in reduced $\chi^2$ between Kurucz and NextGen was not significant (LOri050, LOri055, LOri057, LOri060 and LOri075).
We performed a model fit to the Ca I lines described in Step 2 (see Fig. \[vsini\]) in order to estimate the rotational velocities of our sources. Prior to this, we included the instrumental response measured on the MIKE lamp arc. The mean FWHM value measured on the lamp spectra was 0.4914 Å, corresponding to a velocity of $\sim$20 km/s (the lower limit we can provide for several cases).
Main results
------------
The estimated rotational velocities from this work are provided in Table \[tab:paramMIKE\], and also in Table \[tab:paramTOTAL\] where we add those determined by [@Sacco08]. In general, the range of values determined in both studies agree very well. In particular, there are three sources for which both @Sacco08 and us have estimated v$\sin(i)$. In two of these cases, LOri055 and LOri057, both estimations are in very good agreement, but in the case of LOri060 we provide a value $\sim$10 km/s higher than the one obtained by @Sacco08. For this source, our determination of the rotational velocity suffers from the fact that the S/N of the spectrum was the lowest of the sample and, for this reason, the wings of the Ca I doublet are not so “clean" as in the other cases. Therefore, we take their upper-limit of $\sim$ 20 km/s as a more robust measurement.
In Fig. \[fig:SpTvsini\], we display the projected rotational velocity as a function of spectral type for those objects where a value of v$\sin(i)$ (either calculated by us or from @Sacco08) is available. Although a significant fraction of the measurements are upper limits, we can see how, as expected given the youth of the sample, the general trend for objects in C69 is to rotate faster than the old disk population of low-mass stars and brown dwarfs from [@Mohanty03] (starred symbols).
Furthermore, although the dispersion of v$\sin(i)$ values among the members of C69 is large; if we consider the NIR and MIR slope as a proxy for disk presence, we see how this dispersion is larger among the diskless population than among disk-harboring sources. Although we are dealing with small number statistics, this could be the result of the disk locking effect [@Bouvier97], that explains the dispersion in rotational velocities as a result of different decoupling timescales between the star and the disk.
\[tab:paramMIKE\]
------------------------ -------------------- ------ --------------- -------- ------------ --------- ----------------------- --------------
Source Mdl$^{\mathrm{1}}$ SpT T$_{\rm eff}$ log(g) v$\sin(i)$ Acc$^2$ $\log(\dot{M}_{acc})$ Binary$^{3}$
(K) (km/s) (M$_{\odot}$/yr)
LOri017$^{\mathrm{*}}$ Kur – 4250 4.0 70 —
LOri031 Kur M3.5 3750 4.0 40 N
LOri042 Kur M4.0 3750 4.0 30 N
LOri050 NG M4.0 3700 4.0 60 Y -10.878$\pm$0.05 M08, S08
LOri055 NG M4.0 3700 4.0 $<$20 N
LOri056 NG M4.5 3700 4.0 — N
LOri057 NG M5.5 3700 4.0 $<$20 N
LOri058 NG M3.5 3500 4.0 — N
LOri059 NG M4.5 3500 4.0 — N
LOri060 NG M4.0 3500 4.0 20-40 N
LOri063 NG M4.0 3700 4.0 — Y
LOri068 NG M4.5 3700 4.0 — N
LOri073 NG M5.0 3700 4.0 — N
LOri075 NG M5.0 3400 4.0 65 N M08, B11
------------------------ -------------------- ------ --------------- -------- ------------ --------- ----------------------- --------------
: T$_{\rm eff}$, log(g) and rotational velocities derived by comparison with models for the sample of sources observed with Magellan/MIKE. We also display other relevant information about the sources such as accretion rate (derived in this work) or binarity.
$^{\mathrm{*}}$There were problems on the extraction of several orders. We have used the T$_{\rm eff}$ corresponding to the SED fit provided in Paper I.\
$^{1}$ Model collection corresponding to the best fitting model: NG = Nextgen; Kur = Kurucz\
$^{2}$ Y indicates that the object fulfills the accretion criterion by @Barrado03 (those marked with N do not).\
$^{3}$ Binary system according to @Maxted08 [M08], @Sacco08 [S08] or this work (B11).
![Example of v$\sin(i)$ estimation for LOri031. In black we have plotted the science spectrum and on top of it with different colors the Kurucz model synthesized for 3750 K, log(g) of 4.0 and five different velocities. The synthetic spectra have been produced with the same resolution as the observed one.[]{data-label="vsini"}](./LOri031.eps){width="9.0cm"}
![Spectral Type vs v$\sin(i)$ for confirmed members of C69 from this work and [@Sacco08] (black dots). For comparison, we include the old disk population brown dwarfs from [@Mohanty03] with five pointed star symbols. From the C69 members we have highlighted those exhibiting peculiarities: large open squares surround spectroscopic binaries, large open circles represent objects with infrared excess (assumed to be a signpost of the presence of a disk) and red dots have been plotted on top of those objects classified as accretors according to the saturation criterion by [@Barrado03] (see text for details). Finally LOri075 is highlighted with a label and is discussed in Appendix \[subsec:rotvel:ps\][]{data-label="fig:SpTvsini"}](./Spt_vsini.eps){width="9.cm"}
Activity and accretion {#sec:acacc}
======================
Activity, accretion, and mass loss processes can be studied through the analysis of emission lines in the spectrum of young stellar and substellar objects. We can group these emission lines into two categories: (i) The forbidden emission lines of \[OI\] ($\lambda$5577, 6300, 6364 Å), \[SII\] ($\lambda$6717, 6731 Å) and \[NII\] ($\lambda$6548, 6581 Å) that have been attributed in the literature to low density regions such as winds, and are therefore a tracer of the mass loss process (see @Shu94 [@Hartmann94] and @Hartmann99). (ii) The permitted emission lines of He I ($\lambda$6678 Å), H$\alpha$ ($\lambda$6563 Å) and the CaII triplet ($\lambda$8498, 8542, 8662 Å) which are characteristic of classical T Tauri stars and their substellar analogs and trace accretion processes, although they are also known to be signposts of chromospheric activity [@Martin01; @Natta01; @Natta02; @Testi02; @Mohanty03b; @Jayawardhana02a; @Jayawardhana02b; @Jayawardhana03a; @Jayawardhana03b; @Barrado02; @Barrado03b; @White03].
Given the evolutionary status of the members of C69 and the environment where they are located, most of the analyzed spectra show a rich variety of emission lines. We summarize the results from our analysis in the following subsections.
Spectroscopic emission lines among the C69 sample {#subsec:el}
-------------------------------------------------
For our sample, the forbidden emission lines (given their narrow nature) were only detected in some of the higher resolution spectra (see tables \[tab:emissionlinesMIKELRIS\] and \[tab:emissionlinesLRISFLAMES\]), but since most of these detections are quite marginal we cannot be certain whether the signal comes from the source itself or from the nebular environment. The Ca II triplet was covered in very few of our runs and only detected in one object (C69-IRAC-005) that is analyzed in detail in Appendix \[subsubsec:Haacc:ps\].
On the other hand, we have measurement/s of H$\alpha$ for 156 sources out of the 172 spectroscopically confirmed members. The vast majority of these sources(140 objects, $\sim$90% of the sample) show the line in emission. The EW of this line is used to distinguish between chromospheric or accretion origin of the emission in Section \[subsec:Haacc\].
Regarding He I; we have detected this line in 24 objects, all in emission, and we provide the EWs in tables \[tab:emissionlinesMIKELRIS\] and \[tab:emissionlinesLRISFLAMES\]. All these sources present also intense H$\alpha$ emission (with a minimum EW of $\sim$5Å) and six of them are classified as accretors in Section \[subsec:Haacc\]. Regarding the mass range where He I is detected; only six of the sources are in the BD domain and the rest have masses above the hydrogen burning limit.
Further comparison of the He I and H$\alpha$ EWs of the 24 sources with He I detection shows a clear correlation among them; the stronger the H$\alpha$ emission, the stronger the He I one; but our sample is not statistically significant to derive any quantitative relationship between them.
The fact that we detect He I mainly in stars and not in brown dwarfs is not surprising since the typical structure of the He I line is much narrower than that of H$\alpha$, for example, and therefore this emission can be hidden if the spectrum does not have high SNR and resolution. In addition to this technical caveat to detect He I, the extended nebular emission of the cluster has to be taken into account when dealing with fiber spectrographs. In particular, in two cases from the VLT/FLAMES spectra, the helium line could be contaminated by nebular emission (see in Fig. \[fig:HalphaFLAMES\] the distribution of the sky fibers and in Section \[subsec:Haacc\] the quantification of this contamination for the H$\alpha$ line).
![Contours corresponding to an H$\alpha$ image of C69 (extracted from the H$\alpha$ six arcmin resolution all sky survey compiled by @Finkbeiner03). We have plotted the location of the C69 members using different symbols (all of them in black, see legend of the figure) according to the instrument with which the individual spectrum was obtained. As can be seen a significant fraction of the sources have been observed more than once by us. The locations of sky fibers in the VLT/FLAMES campaign are displayed as blue filled dots and the position of Lambda Orionis (roughly marking the center of the cluster) is highlighted with a large blue five-pointed star. []{data-label="fig:HalphaFLAMES"}](./HalphaCont_v2.eps){width="9.cm"}
Variability connected to activity {#subsec:var}
---------------------------------
Some hints on the chromospheric activity among the late-type population of C69 have already been studied in Paper I through the analysis of the alkali lines. Another clear signpost of activity in young stellar and substellar objects is variability. This variability has been observed not only in the continuum but also in a variety of lines, and its dependence with the spectral type has also been addressed in the literature (examples can be found in @Soderblom93 [@Stauffer97; @Barrado01]). In Paper I, we reported a large fraction ($\gtrsim$35%) of the members of C69 showing such variability in alkali lines (lithium and sodium).
Most of the sources showing variability in the alkali lines also show variability in the H$\alpha$ emission, and in Paper I we already demonstrated that the differences we see in, for example the EW of the lines, are larger than those we would expect just because we are comparing observations with different spectral resolutions.
On the other hand, we find a couple of objects exhibiting significant variability in H$\alpha$ and no variability in the alkali lines (see Table \[Tab:variability\], the top set of sources). The most extreme example is LOri073; this object not only shows variations in the measured EW of H$\alpha$, but also in the profile of the line. As can be seen in Fig. \[fig:varHa\], while we observed a single peak line profile with VLT/FLAMES, a double peak structure was observed several years before with Magellan/MIKE. This more complex profile is very similar, although narrower, to that presented in [@Fernandez04] for the weak-line T Tauri star V410 Tau: we can see the narrow emission peak slightly red-shifted from the rest velocity, as well as the shallower component and the blue-shifted absorption that could suggest the presence of a wind.
![H$\alpha$ profile of LOri073 observe at high resolution in two different epochs: with VLT/FLAMES (in red, R$\sim$8600) in Jan 2008 and with Magellan/MIKE (in black, R$\sim$11250) in Dec 2002.[]{data-label="fig:varHa"}](./LOri073_profile.eps){width="8.0cm"}
Finally, another three particularly interesting variable objects are discussed in Appendix \[subsec:var:ps\]. They present the peculiarity that according to one of the H$\alpha$ measurements, the emission is too large to have a purely chromospheric activity origin (see Table \[Tab:variability\], second sub-set of objects), but for all the other measurements this is not the case.
Name EW(H$\alpha$) EW(H$\alpha$) S08 Ins. Res Mass(M$_{\odot}$) Class Disk Li I Li I S08 KI NaI
------ ----------------- ------------------- -------------------- ------------------- ------- ------ --------------- ---------- --------------- ---------------
-10.91$\pm$1.04 MIKE R$\sim$11250 0.62$\pm$0.03
-4.54 $\pm$0.19 FLAMES R$\sim$8600 0.66$\pm$0.05
-3.36$\pm$0.11 CAFOS R$\sim$600 0.29$\pm$0.07 2.57$\pm$0.43
-7.72 $\pm$0.19 TWIN R$\sim$1100 0.49$\pm$0.21 2.36$\pm$0.18
-6.82$\pm$0.52 MIKE R$\sim$11250 0.69$\pm$0.09
-14.39$\pm$0.74 LRIS R$\sim$2650 0.53$\pm$0.05
-12.81$\pm$0.86 MIKE R$\sim$11250 0.94$\pm$0.05
-10.20$\pm$0.89 LRIS R$\sim$950 2.82$\pm$0.38
-21.3$\pm$1.00 BC R$\sim$2600 1.66$\pm$0.89
-22.82$\pm$0.62 TWIN R$\sim$1100 1.77$\pm$0.33 2.34$\pm$0.46
-11.93$\pm$1.12 LRIS R$\sim$2650 0.23$\pm$0.07 2.17$\pm$0.46
-23.2$\pm$12.3 TWIN R$\sim$1100 2.05$\pm$0.70
-13.47$\pm$1.11 FLAMES R$\sim$8600 0.76$\pm$0.06
-19.62$\pm$0.52 FLAMES R$\sim$8600 0.57$\pm$0.11
-8.70$\pm$0.45 LRIS R$\sim$2650 0.50$\pm$0.15 2.62$\pm$0.57
$^{\mathrm{*}}$The signal to noise ratio of the TWIN spectrum of LOri091 is very low. This is clear in the very large error on the determination of the EW. Therefore this object is not considered variable.\
$^{\mathrm{**}}$Value from [@Maxted08].\
H$\alpha$ emission as a proxy for accretion {#subsec:Haacc}
-------------------------------------------
We note once again that the H$\alpha$ emission can have different origins, and although it is commonly used as a proxy for active accretion, some considerations have to be taken into account depending on the nature of the source and its surroundings.
First of all, and since cool objects are known to be very active, the H$\alpha$ emission can be chromospheric and not related to accretion processes. Some limits to this chromospheric contribution have been suggested in the literature: [@White03] proposed EWs of 10Å and 20Å for spectral types K3-M2 and later than M2, respectively. On the other hand, @Barrado03 proposed a more spectral dependent relationship mainly focused on late K, M and L dwarfs (and therefore more suitable for our study). This empirical criterion is based on the saturation limit of chromospheric activity ($[L({\rm H}\alpha)/L_{\rm bol}$\] = -3.3) and it is shown in Fig \[fig:HaSpT\] as the boundary to discriminate between accreting and non-accreting stars and brown dwarfs. On the other hand, when trying to extract accurate measurements from the H$\alpha$ emission to disentangle between activity and accretion, we are also forced to take into account the environment where the sources are located. As mentioned before, clusters like C69, possess a non-negligible nebular component (see Fig. \[fig:HalphaFLAMES\]), and therefore one has to make sure that the nebular component is subtracted properly from the spectrum of each science target. This is not an issue, for instance, when dealing with long-slit spectroscopy (like most of our campaigns) but can induce some bias when observing with fiber spectrographs if the “sky fibers" are not placed carefully. In Fig. \[fig:HalphaFLAMES\] we show the contours corresponding to an H$\alpha$ image (from @Finkbeiner03, with low spatial resolution $\sim$6’) of the LOSFR, where we have highlighted with crosses the sources observed with VLT/FLAMES and with blue filled circles the locations of the sky fibers used to correct our measurements. At first sight, some structure can be inferred in the nebular emission surrounding our science targets. To better characterize this effect, we studied the variations of the H$\alpha$ nebular emission with the sky fibers: we measured a mean full width at 10% of $\sim$41 km/s with a standard deviation of $\sim$3 km/s. We computed a median sky fiber and corrected the science spectra with that median; therefore the dispersion measured in those fibers translated into an added $\sim$7% uncertainty in our measurements.
The original accretion criterion provided in @Barrado03 shows the limiting EW(H$\alpha$) as a function of the spectral type. Since for most of the sources from [@DM99; @DM01] and a fraction of those from @Sacco08 we do not have spectral type determination, we have used the temperature scale derived in Paper I to translate the original criterion into a T$_{\rm eff}$ vs. EW(H$\alpha$) relationship.
As mentioned in Section \[subsec:el\], from the total census of 172 spectroscopically confirmed members of C69, there are 16 sources for which we do not have a measurement of the equivalent width of H$\alpha$ and therefore we cannot apply the criterion. Those 16 sources present very dispersed properties such as effective temperature, disk presence, etc; and therefore they should not produce any bias in the statistics derived for the whole cluster.
In Fig \[fig:HaSpT\] we show the accretion criterion applied to the 156 confirmed members with measurements of the EW(H$\alpha$). In order to be consistent; for objects having a spectral type determination, we have also translated them into effective temperatures using the temperature scale from Paper I (as we did for the criterion itself). On the other hand, for objects without spectral type determination, we have assumed the T$_{\rm eff}$ derived from the SED fit performed with VOSA (@Bayo08, 2012, submitted). Applying the saturation criterion, we classified 9$^{+3}_{-2}$% of the members as accretors (14 red dots in the figure; see column “Acc" from Table \[tab:paramTOTAL\]).
To relate this percentage to the disk presence, in Fig \[fig:HaSpT\], we have also included the information regarding the infrared class (as a proxy for the presence of disk, from @Barrado07 and Morales-Calderón et al. 2012, in prep.) with larger circumferences surrounding Class II sources and/or those with a MIR slope compatible with thick, thin or transition disks. If we only consider the 37 members showing signposts of harboring disks and with available measurements of EW(H$\alpha$), we estimate that 38$^{+8}_{-7}$ % show active accretion according to the saturation criterion.
To focus now on the substellar population of C69; if we consider substellar those sources with estimated masses lower than 0.1M$_{\odot}$ to take into account uncertainties in the mass determination, we find eight brown dwarfs harboring disks and three of them to be accreting according to the criterion. This leaves us with a fraction of 37.5$^{+18}_{-13}$ % of accretors among the disk-beating BDs, very similar to that of the stellar population and comparable with the substellar one provided in [@Scholz07] for the similar age cluster Upper Sco (31%, 4 our of 13 objects).
![H$\alpha$ equivalent width versus the effective temperature for confirmed members of C69. We display data from @DM99 [@DM01] with filled squares, and from [@Sacco08] with filled diamonds. Our data are displayed with solid circles. In every sample, red symbols are used for those sources classified as accretors. Overlapping large circles highlight sources exhibiting excess in the Spitzer/IRAC photometry. For some sources we had more than one epoch of data (either ours or from @DM99 [@DM01; @Sacco08]). These different measurements for individual objects appear joint with a solid line. The short-dashed line corresponds to the saturation criterion defined by @Barrado03. A vertical dotted line highlights the substellar frontier for an estimated age of 5 Myr according to the isochrones from the Lyon group. Particular sources discussed in Appendix \[subsubsec:Haacc:ps\] are highlighted with grey labels.[]{data-label="fig:HaSpT"}](./Teff_Ha_acc_v2_3.eps){width="9.cm"}
In Table \[tab:paramTOTAL\], we present the measured EWs of the H$\alpha$ line of the sources in the sample. For those classified as accretors, we have estimated the mass accretion rate using the measured FW$_{10\%}$(H$\alpha$) and the following relationship derived by @Natta04 [^3]:
$$\log(\dot{M}_{acc}) = -12.89(\pm0.3)+9.7(\pm0.7)\times 10^{-3} {\rm FW}_{10\%}(H\alpha)$$
![Mass accretion rates versus mass of the central object for accreting objects. In red and blue we show members of C69 (red dots for measurements from this work and blue squares for those from @Sacco08, the masses of LOri061 and LOri063 have been slightly shifted so that in the figure the comparison between the measurements of two studies is clearer) and in black members of the slightly younger cluster $\sigma$ Ori (measurements from @Rigliaco11). We have included blue and red vertical bars to highlight special cases where more than one measurement is available for a given source (see Appendix \[subsubsec:Haacc:ps\] for details). In all cases, the masses shown in this figure are the average between the one derived from the L$_{\rm bol}$ and the one derived from the T$_{\rm eff}$. The error bars display the differences among these determinations, see Paper I for details. Particular sources discussed in Appendix \[subsubsec:Haacc:ps\] are highlighted with grey labels. []{data-label="fig:M_Macc"}](./Mass_Macc_2.eps){width="9.0cm"}
The accretion rates calculated in this manner are also listed in Table \[tab:paramTOTAL\]. We note that for DM006, classified as accretor, we did not have a measurement of the FW$_{10\%}$ and therefore we could not estimate the accretion rate. In Fig. \[fig:M\_Macc\] we show an accretion rate versus mass diagram where we compare the values estimated in this work and those provided in [@Sacco08] also for C69; with those derived in [@Rigliaco11] for the slightly younger cluster $\sigma$ Orionis ($\sim$3 Myr according to their HR diagram).
[@Rigliaco11] suggested in their work that there are two trends on this diagram with an inflection point at $\sim$0.45M$_{\odot}$. The mass range in C69 for which we have detected active accreting sources does not allow us to check this feature. Besides, the C69 sample is smaller and the dispersion of our measurements is much larger than that derived for $\sigma$ Orionis by [@Rigliaco11].
This larger dispersion can arise at a first stage from the different methodology used to estimate the mass accretion rate. While [@Sacco08] and this work use the H$\alpha$ emission, “contaminated" by chromospheric activity as already discussed, [@Rigliaco11] use U-band photometry; a “cleaner" methodology. On the other hand, C69 is likely older than $\sigma$ Orionis and the accretion disks of the former may be in a different evolutionary state than those of the latter.
Overall, our measurements are consistent with the idea that the accretion rate scales with the mass of the central object for low-mass stars. But, given the dispersion obtained, this is just a very rough trend. Individual sources from Figs. \[fig:HaSpT\] and \[fig:M\_Macc\] are analyzed in Appendix \[subsubsec:Haacc:ps\] Finally, to have a better understanding of the relation of the H$\alpha$ emission with the accretion process, we have tried to correlate that emission with disk properties derived mainly from the mid-infrared photometry.
The theoretical disk models used to interpret the IRAC \[3.6\]-\[4.5\] vs \[5.8\]-\[8.0\] color-color diagram by [@Allen04] suggest that the accretion rates increase from the bottom-left to the top-right of the Class II region, due to the increase of both the disk emission and the wall emission. In Fig. \[fig:IRAC\_ccd\_acc\] we show the mentioned IRAC color-color diagram for the members of C69 (spectroscopically confirmed members compilation from Paper I). We have included information regarding the presence of disks (large red circles), the intensity of the emission of H$\alpha$ (sized blue squares) and the classification as accretors (in red). The general trend agrees with the disk theory since objects with larger H$\alpha$ equivalent widths (up to accreting sources) have redder colors. A similar trend can be observed in the right panel of the same figure where we display the mid infrared SED slope as a function of effective temperature. Objects with optically thick disks seem to exhibit more intense H$\alpha$ emission. On the other hand, as it was already clear in Fig. \[fig:HaSpT\], a large fraction of the sources ($\sim$65%) harboring disks in C69 do not seem to be accreting from their disks. We will analyze this fact in more detail in section \[subsec:holes\].
{width="8.7cm"} {width="8.7cm"}
Disk with low H$\alpha$. Binaries clearing the inner disks? {#subsec:holes}
-----------------------------------------------------------
In the previous section we highlighted the presence of a large population of disk-bearing sources that do not show any signpost of active accretion. In Fig. \[fig:SED\_classII\_not\_ac\] we show examples of SEDs corresponding to this class of sources. We looked for some characteristic that would differentiate these objects from the others in our sample (other than the measured H$\alpha$ equivalent width). Their effective temperatures are mainly colder than $\sim$3750K, but, as explained in the next section, that is characteristic of all the disk population of C69.
One possibility for these “quiet" disks would be that they are dissipating their inner disks, but we see no difference in the near-infrared colors with respect to the actively accreting disks.
Besides, adopting the characterization of the $\alpha$ parameter from [@Lada06], these objects harbor mainly optically thick disks (60% of them), but there are also sources with optically thin disks and the so called transition/cold disks [@Merin10]. There are several mechanisms to explain the evolutionary status of these transition disks; one of the most attractive, in the context of planet formation, is the clearing of the inner disk by a giant planet in its earliest stages of formation (see the first observational candidate for this scenario in @Huelamo11). Another possibility is tidal truncation in close binaries [@Ireland08]. This seems to be the mechanism at work in at least one of the cases of the low H$\alpha$ transition disks of C69; LOri043 that was classified as SB2 by [@Maxted08]. On the other hand, LOri043, is the only documented spectroscopic binary in this sample of quiet disks.
Finally, only two of the sources from this set have been detected in X-rays, suggesting that these are not particularly active objects either. In conclusion, we could not find any parameter (other than the H$\alpha$ emission) that unites these objects or differentiates them from the actively accreting population.
![SEDs of the sources classified as Class II based on their IRAC colors and showing EW(H$\alpha$) low enough not to be classified as accretors according to [@Barrado03].[]{data-label="fig:SED_classII_not_ac"}](./SED_classII_no_ac.eps){width="9cm"}
Disk vs diskless populations {#sec:distrib}
============================
By the end of the previous section, we showed that we cannot trace differences between the actively accreting - quiet disk population of C69. In this section we will analyze the clear distinction between disk and diskless sources, in terms of their X-ray emission and their mass functions.
H$\alpha$ and X-ray {#subsec:HaXrays}
-------------------
As stated in Section 2; [@Barrado11] presented the analysis of the XMM–Newton observations of two fields in C69. Several months later, [@Franciosini11] complemented the study by adding an extra field that covers the vicinity of the massive star $\lambda$ Ori, roughly at the center of the cluster.
These X-ray observations should trace well the weak-line T Tauri (and substellar analogs) population of C69; therefore we have combined the data from the two studies and correlate it with our census of spectroscopically confirmed members. In Fig. \[fig:Halpha\_Xrays\] we illustrate this advantage of the X-ray observations to unveil the weak-line T Tauri population. We show every member of C69 lying in the field covered by XMM-Newton observations. We have highlighted in blue objects above the completeness limit of 0.3M$_{\odot}$. We see how most of the sources classified as weak-line T Tauri, according to the saturation criterion, are detected in X-rays. We also show that the objects that are not detected in X-rays are preferentially those harboring optically thick disks (both, active accretors and non-accreting sources). A total of 9 sources out of 12 with masses above 0.3M$_{\sun}$, harboring disks and within the XMM-Newton fields of view are not detected in X-rays and are labeled in Fig. \[fig:Halpha\_Xrays\] This dichotomy is shown even clearer in Fig. \[fig:Halpha\_Lx\], where we show the X-ray luminosity vs bolometric luminosity ratio as a function of H$\alpha$ equivalent width. Here, we see how most objects with EW(H$\alpha$) between $\sim$5–20 Å and not harboring disks are detected, while those showing infrared excess are not. We have highlighted five of the non-detected sources discussed in the previous paragraph for which [@Barrado11] provide upper-limits of the X-ray luminosity. The remaining four non-detections are located towards the center of the cluster, within the field of view of the observations by [@Franciosini11] and no upper-limit for the X-ray luminosity of the sources is provided in that work.
{width="18cm"}
![EW(H$\alpha$) vs L$_{\rm X}$/L$_{\rm bol}$ diagram for the members from Paper I in the area covered in X-rays by either [@Barrado11] or [@Franciosini11]. As in previous figures we use large open circles to highlight objects showing mid-infrared excess and red dots for those classified as accretors. Five pointed stars indicate that the estimated mass of the object is larger than the critical mass 0.6M$_{\odot}$ explained in section \[subsec:diskfrac\][]{data-label="fig:Halpha_Lx"}](./Ha_Lx_2.eps){width="9cm"}
Disk fractions {#subsec:diskfrac}
--------------
![Cumulative (normalized) mass functions for sources showing signs of harboring disks (infrared excess), plotted with large circles surrounding the points of the histogram, together with the cumulative mass function of sources detected by IRAC showing photospheric MIR slope. The vertical red dashed lines highlights the mass ($\sim$0.6M$_{\odot}$) at which the disk population is almost complete.[]{data-label="fig:mass_cum_func"}](./phot_IRAC_vs_excess.eps){width="9.cm"}
Regarding the formation mechanism of brown dwarfs; if they are just a scaled down version of stars, one would think that the disk life-times above and below the hydrogen burning limit should be the same. On the other hand, there is some evidence for longer life-times for infrared excesses in very low-mass stars and brown dwarfs than in higher-mass stars [@Lada06; @Allers07].
To test these possible differences, instead of estimating a typical star and brown dwarf disk fraction, we have divided our census of confirmed members into two sets: sources showing some infrared excess, and objects detected also in the mid-infrared, but exhibiting purely photospheric colors. The first set contains 43 sources and the second one 115. We propose this approach to avoid choosing a fixed frontier in mass between stars and brown dwarfs so that our findings are easier to compare with other studies. Regarding the sources with excess, we have considered, in the same set, all kinds of MIR slopes and shapes: optically thick, optically thin and transition disks. We have computed the cumulative mass functions for both samples and the result is shown in Fig. \[fig:mass\_cum\_func\]: there is a clear difference at $\sim$0.6M$_{\odot}$ ($\sim$M2 spectral type). While the mass function of the diskless population rises up to $\sim$1.1M$_{\odot}$, the disk population is almost completely composed of sources with masses lower or equal than 0.6M$_{\odot}$. In other words: while we find diskless objects for every bin in mass, sources more massive than $\sim$0.6M$_{\odot}$ seem to have lost their disks already. We tested the dependence of this change of behavior with the 5 Myr age assumed for C69 to estimate the masses. While an older age will significantly affect the mass determination of the lowest mass members of C69; that is not the case for sources with masses above 0.3M$_{\odot}$ according to the isochrones by [@Baraffe98; @Chabrier00; @Allard03], and therefore our result is robust against changes in the age determination of C69. On the other hand, if we use these two cumulative fractions but without normalization to estimate the disk fraction as a function of the mass, we can see that the situation is more complex. In Fig. \[fig:disk\_frac\] and Table \[tab:diskfraction\] we show the ratio of the two cumulative functions; that is, for a given mass $M_i$, we provide $n_i/N_i$ where $n_i$ is the total number of sources with mass $\le M_i$ and infrared excess, and $N_i$ is the total number of sources with mass $\le M_i$ ($n_i$ plus the sources with purely photospheric infrared slopes and masses $\le M_i$). For the error treatment, since our sample is very large, for masses larger or equal than 0.3M$_{\odot}$ we can derive standard Poisson uncertainty limits and for the lower masses we have used the approach described in [@Burgasser03].
Regarding the completeness of the diskless population, even though the X-ray observations are only complete down to 0.3M$_{\odot}$ we are confident that this does not induce any bias in our analysis. The reason is that every source that in the spectroscopic confirmation turned out to be later than M3.5 and was detected in X-rays, had previously already been selected as an optical photometric candidate. Therefore the optical photometric selection was as good as the X-rays one picking up the low to very low-mass members of C69.
The disk fraction function seems to peak (77%) at the brown dwarf boundary, dropping abruptly with increasing mass up to $\sim$0.3M$_{\odot}$. It then stabilizes at $\sim$33% before falling again for masses higher than $\sim$0.6M$_{\odot}$. The total stellar disk fraction is 26$^{+4}_{-3}$ %
We must note that the extremely high disk fraction for substellar objects should be taken as an upper-limit. In this low mass regime, we have some sources that have not been detected in the two reddest channels of Spitzer/IRAC (5.8 and 8.0 $\mu$m). We did not consider those sources since we only used sources with photospheric MIR SED to estimate the disk fractions. If we were certain that those sources do not show excess (and the fact that they are not detected at those red wavelengths is a good indicator of that), the percentage would decrease down to $\sim$58% This fraction is still larger than that derived by [@Barrado07] for the same cluster ($\sim$40%). This is not too surprising since in this study we consider the fraction with any kind of disk and not only Class II sources as in [@Barrado07]. On the other hand, the difference with the value derived by [@Scholz07] for Upper Sco ($\sim$37%), a similar age cluster; although significant, could just be caused by small number statistics or again, use of different criterion to infer whether a source is harboring a disk or not.
![Disk fraction as a function of the mass of the members of C69. For the ratio we use the same sets of objects as in the previous figure.[]{data-label="fig:disk_frac"}](./disk_fractions_mass_function_2.eps){width="9.cm"}
$<$ M(M$_{\odot}$) Disk frac (%) $<$ M(M$_{\odot}$) Disk frac (%)
-------------------- ---------------- -------------------- ----------------
0.1 $\le$ 77 0.75 31$^{+4}_{-4}$
0.15 $\le$ 54 0.8 30$^{+4}_{-3}$
0.2 44$^{+8}_{-7}$ 0.85 30$^{+4}_{-4}$
0.25 41$^{+7}_{-6}$ 0.9 29$^{+4}_{-4}$
0.3 35$^{+6}_{-5}$ 0.95 28$^{+4}_{-3}$
0.35 34$^{+6}_{-5}$ 1.0 27$^{+4}_{-3}$
0.5 34$^{+5}_{-4}$ 1.05 27$^{+4}_{-3}$
0.55 32$^{+5}_{-4}$ 1.1 26$^{+4}_{-3}$
0.7 32$^{+4}_{-4}$ 1.7 26$^{+4}_{-3}$
: Disk fraction as a function of mass.[]{data-label="tab:diskfraction"}
Spatial distribution
--------------------
In Fig. \[fig:dist\_Halpha\] we show the spatial distribution of C69 spectroscopically confirmed members (by @DM99 [@DM01; @Sacco08; @Maxted08] or this work) including information about X-ray emission (from @Barrado11 and/or @Franciosini11), the presence of a disk [@Barrado07] and whether the disk is accreting or not. As can be seen in the figure, the sources with a disk show a higher concentration towards the center of the cluster with respect to the diskless population (contrary to what one would expect according to the supernovae scenario and already suggested in @Barrado07). In fact, if we assume that actively accreting systems are younger than those which do not show any sign of accretion, the youngest population of C69 seems to be clustered either around the central star $\lambda$ Ori or to the South-West.
We have computed the two-sided Kuiper statistic (invariant Kolmogorov-Smirnov), and the associated probability that any of the previously mentioned populations were drawn from the same distribution. The tests reveal that the cumulative distribution function of Class II candidates is very different from that of Class III, with a 99.9% probability that both populations have been drawn from different distributions.
The conclusion from this test is that objects with masses lower than 0.6M$_{\odot}$ have been less efficient, in the life time of C69, in loosing their circumstellar material.
![Spatial distribution of the spectroscopically confirmed members of C69. Active accretors according to the saturation criterion are highlighted in red. Information regarding infrared excess (as a proxy for the presence of disk, from @Barrado07) is provided by surrounding those sources with larger circumferences. The more massive population of C69 ($\lambda$ Ori itself and the B stars) is shown with grey four-pointed stars. Finally, the members with X-ray detections are shown with five-pointed blue stars.[]{data-label="fig:dist_Halpha"}](./distrib_disk_Xrays.eps){width="9.cm"}
Summary and conclusions {#sec:conclusions}
=======================
We have analyzed the spectroscopic properties of the very complete sample of members of C69 compiled in [@Bayo11]. Using different spectroscopic features we have tried to better understand the similarities and differences between the disk and diskless populations and the stellar and substellar ones. Our main results can be summarized as follows:
1. We have estimated the rotational velocities for eight members. We find a high dispersion in the v$\sin(i)$ values, being larger among the diskless population. We interpret this as a result of disk locking in some of the C69 members.
2. We have studied the variability of the H$\alpha$ emission line in 142 members classified as magnetically active (non-accreting sources). We have also identified candidates that might have experienced flares during the epoch of observation.
3. We have tried to disentangle the activity and accretion contributions to the H$\alpha$ emission. In this context, for those sources showing large H$\alpha$ equivalent widths (larger than the one expected to arise from chromospheric activity) we have derived accretion rates using mainly the full width at 10% of the flux of the H$\alpha$ emission line (but also the Ca II IRT) obtaining a very large spread of values. Once individual cases are analyzed, these spread values are still compatible with those previously reported in the literature for objects of similar mass and age [@Rigliaco11]. When considering all the confirmed members with H$\alpha$ measurements, we estimate a 9$^{+3}_{-2}$ % fraction of accretors in C69. If we only consider objects with disks, 38$^{+8}_{-7}$ % show active accretion.
4. We have studied the relation H$\alpha$ – disk properties. While the general trend expected from disk models applies, we also identify a pretty large population of “quiet" disks. Objects showing clear mid-infrared excess (with a variety of SED shapes) but H$\alpha$ levels compatible with arising from pure chromospheric activity and not from the interaction with the disk via accretion.
5. We find a stellar disk fraction of 26$^{+4}_{-3}$ % and we can put some limits to the substellar disk fraction (for the faintest members, due to sensitivity limitations, the lack of detection at 5.8 and/or 8 micron does not imply the presence or absence of a substellar disk) of 58%. These fractions do not compare too badly (taking into account the difference in procedure followed to estimate the fraction) with that previously provided for C69 itself by [@Barrado04] ($\sim$40%) and is significantly higher than that derived by [@Scholz07] for the cluster of similar age Upper Sco ($\sim$37%).
6. Regarding the accretion fraction in the substellar domain, we do not see dramatic changes from the global fraction of C69. This accretion fraction for brown dwarfs varies from 30% to $\sim$43% according to our uncertainties in mass determination, This range is also compatible with the one provided by [@Scholz07] for the substellar population of Upper Sco.
7. We have confirmed that X-ray observations [@Barrado11; @Franciosini11] are extremely efficient recovering the Class III population of C69 in the intermediate to low-mass range.
8. We have compared the mass function of the disk and diskless populations of C69 finding that 0.6M$_{\odot}$ seems to be the critical mass below which a significant fraction of the members preserve their disks. This result implies different disk lifetimes for different stellar masses. In particular, for masses lower than 0.6M$_{\odot}$ we have shown that the disk fraction rises very steeply with the caveat that in the brown dwarf domain the fraction provided should be taken as an upper-limit.
9. We have studied the spatial distribution of the disk-harboring population of C69 and we find that, opposite to what [@DM01] derived from their more massive members sample; the density of disk-sources is larger closer to the center of the cluster, which is inconsistent with the SN scenario invoked to explain the origin of the Lambda Orionis star forming region. In addition, the winds (current or in the recent past) from the massive star $\lambda$ Ori, seem not to have affected the distribution of disk and diskless cluster members.
A. Bayo would like to thank B. Montesinos for the interesting discussion about the rotational velocities determination and H. Bouy and M. Lopez del Fresno for very useful advices in statistics. This publication makes use of VOSA, developed under the Spanish Virtual Observatory project supported from the Spanish MICINN through grant AYA2008-02156. This work was co-funded under the Marie Curie Actions of the European Commission (FP7-COFUND) and Spanish grants AYA2010-21161-C02-02, CDS2006-00070 and PRICIT-S2009/ESP-1496.
Particular sources {#AP_PS}
==================
In this appendix we provide further analysis for sources showing peculiarities in the properties studied in Sections \[sec:rotvel\] and \[sec:acacc\]. To keep consistency throughout the paper we have grouped the interesting sources from Section \[sec:acacc\] following the same subsection scheme.
Rotational velocities {#subsec:rotvel:ps}
---------------------
[**LOri075:**]{} This source has been classified as single-line spectroscopic binary (SB1) by @Maxted08 (but no binarity signpost has been reported in @Sacco08). According to @Maxted08, the spectral lines for this star show rotational broadening; they compared them to those of a narrow-lined star of similar spectral type and estimated a projected rotational velocity of v$\sin(i) \sim$ 65 km/s. They classified the source as SB1, but they also noted that there is an asymmetry in the cross-correlation function (CCF) in the form of a blue-wing, particularly when the measured radial velocity corresponds to a red-shift. Therefore they suggested that the fainter component in this binary was detected but unresolved in their spectra.
We have detected a double peaked structure in H$\alpha$ and Li I in our Magellan/MIKE spectra which made us believe that we had spectroscopically resolved the source. While the origin of double peak in Li I should be related to binarity, the H$\alpha$ one could arise from an accreting companion for example. Further research on the structure of some photospheric lines marginally confirms this idea. In Fig \[fig:LOri075SB\] we show the double peaked structure found in some of the most prominent photospheric lines for this object (given the low temperature of the source, these “most prominent lines” are still very weak). We have measured a mean relative velocity of $\sim$45 km/s ($\sigma \sim$ 9 km/s). We have synthesized a 3500 K (log(g) = 4.0) Kurucz spectra (an effective temperature $\sim$100 K hotter than the one estimated for our source, but the coolest temperature for the Kurucz collection) in the region of the Ba $\lambda$5535Å line with the same resolution and a rotational velocity close to the one derived by @Maxted08 ($\sim$50 km/s). We have checked that the closest line in the synthetic spectra has a relative velocity of $\sim$110 km/s, much higher than those measured by us. We show on the right-hand side panel that the relative velocity derived for the photospheric lines does not agree with the one that would be measured from the H$\alpha$ profile. This fact and the weakness of the lines measured force us to consider the resolution of the binary as tentative. We must note anyway that the environmental H$\alpha$ component (see Fig \[fig:HalphaFLAMES\]) of the region or a possible accreting companion could change the relative velocity of the peaks of this emission line.
![[**Left:**]{} Double peaked structure found in photospheric lines for LOri075. Rest frame velocity and mean relative velocity of the second peak are indicated with dashed lines. The shaded (blue) rectangle shows the $\pm 3 \sigma$ area of the second peak location. [**Right:**]{} H$\alpha$ profile of the same source. Note how the secondary peak dashed line location (calculated from the photospheric lines) does not agree with the position of the peak (see text for details).[]{data-label="fig:LOri075SB"}](./LOri075_SB2_v2.eps){width="9.0cm"}
Activity and accretion {#activity-and-accretion}
----------------------
### Variability connected to activity {#subsec:var:ps}
In our study of the H$\alpha$ variable sources, we find a sub-set of five objects for which the criterion from [@Barrado03] applied to spectra taken at different epochs provides contradictory results. While for some measurement of the object taken in one epoch (EW$(H\alpha)_1$), the source would be classified as accretor; for a different epoch measurement of the same object (EW$(H\alpha)_2$), the H$\alpha$ emission could be explained purely in terms of activity.
![Comparisons of the H$\alpha$ line profiles for different observations of the same two objects: [**Upper panel:**]{} LOri068; one of the objects we suspect experimented a flare during the Keck/LRIS observations (R$\sim$2700). Some asymmetry can be seen in the line even though the resolution of the spectrum is moderate. [**Lower panel:**]{} LOri109; another object suspected to have a flare for which no asymmetry in the line has been found.[]{data-label="fig:lineprofile"}](./LOri068_profile.eps "fig:"){width="9.0cm"} ![Comparisons of the H$\alpha$ line profiles for different observations of the same two objects: [**Upper panel:**]{} LOri068; one of the objects we suspect experimented a flare during the Keck/LRIS observations (R$\sim$2700). Some asymmetry can be seen in the line even though the resolution of the spectrum is moderate. [**Lower panel:**]{} LOri109; another object suspected to have a flare for which no asymmetry in the line has been found.[]{data-label="fig:lineprofile"}](./LOri109_profile.eps "fig:"){width="9.0cm"}
- [**LOri068**]{} and [**LOri109**]{} were observed twice during our campaigns and LOri068 was also observed by [@Sacco08]. Both objects are classified as diskless sources based on their IRAC slopes, and while our Li I measurements agree among themselves (and for the case of LOri068, with the one provided by @Sacco08), in one spectrum for each source, the H$\alpha$ emission is much more intense than in the others. We believe those spectra were taken while the objects were experiencing a flare. In Fig \[fig:lineprofile\] we compare the line profiles of those objects in “steady" and “flared" states; and we show that, while in the case of LOri068, we can see some asymmetry in the line profile for the intense emission (which would indicate mass motion), that is not the case for LOri109 (even though for the latter the change in EW is much stronger).
- [**LOri091**]{} cannot be classified with certainty as a variable source. There is one measurement of H$\alpha$ clearly off from the other two available, but that measurement corresponds to a TWIN spectrum of very poor S/N which translates into a large uncertainty on the continuum, and therefore a very large error-bar in the measurement.
- [**LOri075**]{} is an un-resolved (or marginally resolved, see subsection \[subsec:rotvel:ps\]) double system, and therefore variability in the measured H$\alpha$ is not surprising.
- [**LOri080**]{} is a puzzling case. We observed the object twice; in 2003 (at Las Campanas) and 2005 (at Calar Alto; see Paper I for a description of the instrumentation used in each case), and both measurements agree within the errors. These measurements place LOri080 at the border of being classified as accreting according to the saturation criterion (see next subsection), although the object shows no infrared excess in the IRAC data. No peculiarity has otherwise been found regarding the profiles of the lines in either spectra. On the other hand, [@Sacco08], also observed LOri080 with FLAMES and found an EW for H$\alpha$ significantly lower.
### H$\alpha$ emission as a proxy for accretion {#subsubsec:Haacc:ps}
In the process of determining accretion fractions, ratios and their relation with the disk properties, we encountered several particular cases that we describe below.
[**LOri161:**]{} Is the brown dwarf from Fig \[fig:HaSpT\] that, even though its H$\alpha$ emission places it well above the saturation criterion, has not been classified as harboring a disk according to its SED. The issue with this very faint source is that it was not detected in IRAC channels three and four (5.8 and 8.0 micron, respectively). Since the sensitivity of these channels is lower than that of one and two (3.6 and 4.5 micron), it could be the case that this object indeed has a disk that we are not sensitive to and that is undergoing accretion. In that scenario, the estimated accretion rate according to the ${\rm FW}_{10\%}(H\alpha)$ would be $\sim 1\times 10^{-10}$ M$_{\odot}/yr$, which is much lower than the accretion rate derived for LOri156 ($\sim 9.5\times 10^{-9}$ M$_{\odot}/yr$), also a brown dwarf with the same spectral type and discussed later on in this section, but harboring an optically thick disk.
An example of such a disk would be a transitional disk, where the excess would be only detectable at larger wavelengths. We checked the new release of the WISE catalog (in the preliminary version the source is not detected) and we found a counterpart within 1". Unfortunately, although the photometry at the largest wavelengths ($\sim$11 and 20 micron) shows a clear excess, these measurements have been classified as “U" (upper limit), and therefore we cannot confirm that this source does harbor a disk. [**C69XE-009:**]{} An X-ray candidate from [@Barrado11] confirmed spectroscopically as C69 member in Paper I. This object is right at the limit of the saturation criterion; based on its SED it was classified as a candidate transition disk, but the linear fit to the mid-infrared slope is photospheric. Given that it is clearly an active object (detected in X-rays), and that the disk possibility is based on a very slight excess detected only in one infrared band, we assume that the H$\alpha$ emission has its origin in chromospheric activity and not accretion.
[**C69-IRAC-005:**]{} Is the source from Fig. \[fig:M\_Macc\] exhibiting the largest accretion rate based on the ${\rm FW}_{10\%}(H\alpha)$ ($\log(\dot{M}) = -5.56\pm0.25$) and the Spitzer/IRAC photometry suggests that it harbors an optically thick disk. This particular source was observed with CAFOS in low resolution mode, with a wider wavelength coverage than the other instruments used (see Paper I). Thus, we have been able to obtain a different estimation for the accretion rate based on the equivalent width measurement of the components of the CaII triplet (at 8498 Å, 8542 Å and 8662 Å). This emission could be a sign of chromospheric activity too, as in the case of H$\alpha$; but the obtained equivalent widths for the triplet are too large; placing our measurements in the broad-line component of the unresolved line structures that is generally related to accretion (see @Comeron03 [@Mohanty05]). Furthermore, as in @Comeron03, the CaII triplet line ratios are very close to 1:1:1 (quite different from the 1:9:5 expected ratio for optically thin emission).
We used the following equations to estimate the accretion rate from the CaII triplet (these equations were derived from the accretion line profile study by @Muzerolle98 and are further discussed in @Comeron03): $$\log(\dot{M}_{acc}) = -34.15 + 0.89 \log(F_{\rm CaII(\lambda8542)})$$ $$F_{\rm CaII(\lambda8542)}=4.72 \times 10^{33} EW({\rm CaII(\lambda8542)}) \times 10^{-0.4(m_{\lambda}-0.54A_{V})}$$ where $F_{\rm CaII(\lambda8542)}$ is the flux in the line, $m_{\lambda}$ is the magnitude of the star at $\lambda$8542, and $A_V$ is the visual extinction translated to the wavelength of the line of study using @Fitzpatrick99 relations. Since the bluest photometric point that we had for this object is the 2MASS J magnitude, we used the best fitting model to the SED of the source as a scaling factor to estimate $m_{\lambda}$. On the other hand, according to the intrinsic colors by @Leggett92 and our determination of the spectral type (M3), we find a very low $A_V$ value of 0.03 mag (quite lower than the average value of 0.36 mag derived for the cluster by @Duerr82, but neither of them would significantly affect this estimation).
We obtained an accretion rate value of $\sim$3$\times$10$^{-7}$ M$_{\odot}$/yr; almost an order of magnitude lower than the one obtained based on H$\alpha$, which gives us an idea of the caveats of estimating accretion rates from measurements that can be well contaminated by activity or even by wind contributions. On the other hand, even with the two estimations differing by such a large factor, this object still seems to be experiencing heavy accretion. We have compared its spectrum with that of C69-IRAC-002 (another M3 star, observed with the same setup, harboring a disk but with H$\alpha$ compatible with pure chromospheric activity and no CaII emission) looking for veiling emission, and no blue excess has been found in the source (further than a marginal excess right in the blue edge of the spectra that we think corresponds to an instrumental signature rather than a real excess). This result is not surprising since the wavelength coverage starts at 6200 Å, and veiling in young stars is normally detected at bluer wavelengths. Therefore, we would need further spectroscopic measurements to confirm the presence of veiling in this source. [**LOri050, LOri061 and LOri063**]{} are the other sources with more than one estimation of $\dot{\rm M}_{\rm acc}$.\
[**LOri050**]{} is a spectroscopic binary according to [@Sacco08] and [@Maxted08]; it has been classified as Class II according to its mid-infrared photometry and we obtained two spectra with different instrumentation (see Paper I for details). According to the H$\alpha$ emission, in both cases, the object is above the saturation criterion. The estimated accretion rates for both measurements agree well within the errors ($\log(\dot{M}) =$ -11.09$\pm$0.05, -10.91$\pm$0.05). Therefore, we are observing a very interesting system with a total stellar mass of $\sim$0.3 M$_{\odot}$ and a circumbinary disk actively accreting.\
For [**LOri063**]{}, on the other hand, the two available accretion rate estimations (from @Sacco08 and this work) differ by more than an order of magnitude ($\log(\dot{M}) =$ -10.7$\pm$0.3, -11.87$\pm$0.07). LOri063 harbors an optically thick disk according to its IRAC photometry, and the change in the full width at 10% of the flux in H$\alpha$ is also reflected in the change in EW of the line ($>$9Å).\
Finally, our measurement of [**LOri061**]{} does not agree at all with that from [@Sacco08] (two orders of magnitude difference, $\log(\dot{M}) =$ -10.2$\pm$0.3, -7.65$\pm$0.05). We believe this difference arises from how sensitive the measurement of the ${\rm FW}_{10\%}(H\alpha)$ is to the local continuum determination. In Fig. \[fig:LOri061\] we illustrate the case graphically. While our automatic procedure (see Appendix A of Paper I for details) identifies a local continuum, the thick light green line, other routines fitting global continuum could base their measurements on the teal line. This difference in the determination of the “real base" of the line yields the large discrepancy in the estimated accretion rate. We must note in any case that among our data on accretors, we do not have other sources where the H$\alpha$ profile can provoke this confusion in the continuum determination.
![Detail of the H$\alpha$ emission of LOri061 in the CAHA/TWIN spectrum. Note the dependence on the measurement of the full width at 10% of the flux with the pseudo-continuum choice, in particular for the light green and the teal cases (for a complete description on the process to determine the different continuums see Appendix A of Paper I). []{data-label="fig:LOri061"}](./LOri061.jpg){width="7.5cm"}
[**LOri126, LOri140 and LOri156**]{} are the three brown dwarfs (LOri126 is right at the limit between BD and very low mass star depending on the method used to estimate its mass, see Paper I) from Fig. \[fig:M\_Macc\] exhibiting very large accretion rates ($\log(\dot{M}) = -8.78\pm0.10, -8.88\pm0.14, -8.02\pm0.10$, respectively). According to their mid-infrared slope, the three targets harbor optically thick disks. And according to their very large H$\alpha$ equivalent widths, they are well above the saturation criterion.
With such high accretion rates some veiling (due to excess emission from the accretion shock) could be expected in these sources (as it is the case for LS-RCrA 1, @Barrado04a). In order to study this possibility, for each brown dwarf, we selected a non-accreting Class III source with very similar spectral type (one half subclass) and we compared the strength of several TiO molecular bands in both spectra. As can be seen in Fig \[fig:Accretors\] no significant differences are found in the continuum level of any pair of sources. In fact, in the three cases, $r_{\lambda}$, defined as $F(\lambda)_{\rm excess}/F(\lambda)_{\rm photosphere}$ is negligible. Whilst for LS-RCrA 1, @Barrado04a found that $r_{\lambda}$ varies from $\sim$1 to $\sim$0.25 for the wavelength range 6200–6750 Å we find a horizontal slope in this interval. The only cases where a linear horizontal $r_{\lambda}$ does not work are located on the very edges of the detector, and therefore we can conclude that no veiling is detected in any of the spectra in the studied wavelength range (this does not imply that some veiling cannot be present at bluer wavelengths).
{width="18.0cm"}
[**C69-IRAC-006 and C69-IRAC-007:**]{} Both sources are classified as Class II based on their IRAC photometry, and they show double-peaked structure of the H$\alpha$ emission as can be seen in Fig \[fig:doublepeak\]. Whilst the sky subtraction for C69-IRAC-007 worked very well, some residual could remain in the case of C69-IRAC-006 (although we do not see any structure on other, very narrow, “sky lines”).
This double peak is not present either in the other emission line detected in both spectra (He I) or in the absorption lines, discarding to a certain extent the possibility of these sources being spectroscopic binaries (SB2). We do not have an estimation of the rotational velocities of either given the resolution of the VLT/FLAMES observations (R$\sim$8000), but these almost symmetrical double peak structures in H$\alpha$ have been reproduced with models for higher mass stars with rapid rotation seen almost pole on (see @Muzerolle03 and references therein).
The accretion rate calculated for C69-IRAC-007 is shown in Table \[tab:paramTOTAL\] since this source fulfills the @Barrado03 criterion ($\log(\dot{M}) = -7.05\pm0.02$). On the other hand, although the measured H$\alpha$ equivalent width of C69-IRAC-006 lies well below the saturation criterion of [@Barrado03], the wide FW$_{10\%}$ measured ($\sim$190 km/s) places this object very close to the limit of accretors according to @Natta04). In addition, note the resemblance of the H$\alpha$ profile of C69-IRAC-006 with that of Cha H$\alpha$2, an accreting brown dwarf, modeled in detail (and showing peculiarities attributed to the presence of an outflow) in [@Natta04].
![Double peaked structure in the H$\alpha$ emission of C69-IRAC-007 and C69-IRAC-006.[]{data-label="fig:doublepeak"}](./double-peak.eps){width="9.0cm"}
[^1]: Based on the ESO observing programs 080.C-0592 and 078.C-0124; and observing programs from Calar Alto, Keck, Subaru and Magellan.
[^2]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. under contract to the National Science Foundation.
[^3]: The FW$_{10\%}$(H$\alpha$) of the sources are typically above the threshold of 200 km/s determined by @Natta04 and both quantities, accretion rates and FW$_{10\%}$(H$\alpha$), are provided in Table \[tab:paramTOTAL\]
|
---
abstract: 'A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrapping clusters exist on the torus. We apply this approach to percolation in continuum models, finding overlaps between objects with real-valued positions and orientations. In particular, we find precise values of the percolation transition for disks, squares, rotated squares, and rotated sticks in two dimensions, and confirm that these transitions behave as conformal field theory predicts. The running time and memory use of our algorithm are essentially linear as a function of the number of objects at criticality.'
author:
- Stephan Mertens
- Cristopher Moore
bibliography:
- 'percolation.bib'
- 'mertens.bib'
title: Continuum Percolation Thresholds in Two Dimensions
---
Introduction {#sec:intro}
============
For more than 50 years, percolation theory has been used to model static and dynamic properties of porous media and other disordered physical systems [@gilbert:61; @stauffer:aharony; @sahimi]. Most natural systems correspond to continuum percolation, yet most analytical and numerical work has focused on lattice percolation. This is reasonable since continuum and lattice percolation lie in the same universality class. For properties that are non-universal, however, such as the location of the threshold, one has to study discrete and continuum models individually, and it is also satisfying to confirm universality experimentally by measuring critical exponents and crossing probabilities.
In this contribution we discuss an algorithm to compute the location of the transition in continuum percolation models. The algorithm works in arbitrary dimensions, and for arbitrarily shaped objects; here we focus on two-dimensional percolation with disks, squares that are aligned or randomly rotated, and randomly rotated sticks (see Figure \[fig:examples\]). Our algorithm is an adaption of the union-find algorithm of Newman and Ziff [@newman:ziff:01], the fastest known algorithm for lattice percolation. We show that it can be adapted to continuum percolation with the aid of some simple additional data structures, and we back up our claim by computing numerical values of the transition points that extend the accuracy of previously known values by several orders of magnitude.
![(Color online) Continuum percolation with disks, randomly rotated sticks, and aligned or rotated squares. In each example, the wrapping cluster is marked by color. []{data-label="fig:examples"}](examples){width="\columnwidth"}
In two-dimensional continuum percolation, a number $n$ of penetrable objects are thrown at random in a square of size $L^2$. If the mean density $\rho=n/L^2$ is finite as $n$ and $L$ go to infinity, the spatial distribution of the objects’ centers is a Poisson point process with density $\rho$. The system percolates if there exists a cluster of overlapping objects that spans the square. We follow [@newman:ziff:01] in using periodic boundary conditions, and focusing on clusters that wrap around horizontally, vertically, or both. These wrapping clusters display better finite-size effects than crossing clusters on open boundary conditions.
If each object has area $a$, then the probability that a percolating cluster exists in the limit $L \to \infty$ clearly depends only on the product $\eta = \rho a$. This dimensionless quantity is called the *filling factor*. It also gives the total fraction $\phi$ of the plane covered by the objects, $$\label{eq:2}
\phi = 1 - {\mathrm{e}}^{-\eta} \, .$$ While we expect continuum percolation to be in the same universality class for any fixed shape, the location of the transition, i.e., the critical filling factor $\eta_c$, depends on the shape of the objects. We write $\eta_c^\medcirc$, $\eta_c^\Box$, $\eta_c^\Diamond$, and $\eta_c^\times$ for the percolation of disks, aligned squares, randomly rotated squares, and randomly rotated sticks. In defining $\eta$, we treat sticks of length $\ell$ as if they have area $a=\ell^2$.
$\eta_c^\medcirc$ $\eta_c^\Box$ $\eta_c^\Diamond$ $\eta_c^\times$
---------- ------------------- --------------- ------------------- -----------------
previous 1.128085(2) 1.0982(3) 0.9819(6) 5.63726(2)
our work 1.12808737(6) 1.09884280(9) 0.9822723(1) 5.6372858(6)
$\phi_c^\medcirc$ $\phi_c^\Box$ $\phi_c^\Diamond$ $\phi_c^\times$
---------- ------------------- --------------- ------------------- -----------------
previous 0.6763475(6) 0.6665(1) 0.6254(2) 0.99643738(7)
our work 0.67634831(2) 0.66674349(3) 0.62554075(4) 0.996437475(2)
Table \[tab:etas\] lists the most accurate numerical values for $\eta_c$ from previous work and the work presented here. The best previous results on disk percolation are due to Quintanilla, Torquato, and Ziff [@quintanilla:ziff:07] who varied the density of the Poisson process as a function of position and kept track of the front of the connected cluster. The best previous results on aligned squares are due to Torquato and Jaio [@torquato:jiao:12], who rescale an initial set of particles so that its density is close to rigorous bounds. The best previous results on rotated squares are due to Baker et al. [@baker:etal:02]. The best previous results on sticks are due to Li and Zhang [@li:zhang:09], who used an approach similar to ours but with open boundary conditions.
Our results are consistent with the rigorous bounds $$\begin{aligned}
1.127 &\,\leq\, &\eta_c^\medcirc \,\leq\, 1.12875 \nonumber \\
1.098 &\,\leq\, &\eta_c^\Box \,\leq\, 1.0995 \, , \label{eq:balister}\end{aligned}$$ computed with $99.99\%$ confidence by Balister, Bollobás and Walters [@balister:05] using a Monte Carlo estimate of a high-dimensional integral. On the other hand, it is a little sad to dash the hope—which one might have entertained after reading [@baker:etal:02; @torquato:jiao:12], and which is just barely consistent with —that $\phi_c^\Box$ is exactly $2/3$.
In the following sections, we review the union-find algorithm of [@newman:ziff:01], how it finds wrapping clusters in periodic boundary conditions, and how we extend it to the continuous case. We show that the running time of our algorithm is essentially linear in the number of objects, i.e., linear in $L^2$. In addition to estimating the threshold, we also measure the finite-size exponent $\nu$, giving strong evidence that these continuum models are in the same universality class as lattice percolation. Finally, we find that the probability of a wrapping cluster at criticality is precisely that predicted by conformal field theory.
The Algorithm
=============
We will simulate percolation in the microcanonical ensemble, i.e., where the number $n$ of objects in the square is fixed. In each trial, we add one object at a time, stopping as soon as a percolating cluster appears. Following [@newman:ziff:01], we keep track of the connected components at each step using the union-find data structure. In union-find, each cluster is represented uniquely by one of its members. We have access to two functions: ${\texttt{find}}(i)$, which finds the representative $r(i)$ of the cluster to which object $i$ belongs, and ${\texttt{merge}}(i,j)$, which merges $i$’s cluster and $j$’s cluster together into a single one with the same representative.
Internally, union-find works in a very simple way. Each object $i$ is linked to a unique “parent” $p(i)$ in the same cluster, except for the representative which has no parent. When we call ${\texttt{find}}(i)$, it follows the links from $i$ to its parent $p(i)$, its grandparent $p(p(i))$, and so on, until it reaches $i$’s representative $r(i)$. Similarly, ${\texttt{merge}}(i,j)$ uses ${\texttt{find}}(i)$ and ${\texttt{find}}(j)$ to obtain $r(i)$ and $r(j)$, and declares one of them to be the parent of the other, unless $r(i)=r(j)$ and they are already in the same cluster.
The running time of ${\texttt{find}}(i)$ is proportional to the length of the path from $i$ to $r(i)$. If ${\texttt{merge}}(i,j)$ sensibly links the smaller cluster to the larger one, setting $p(r(i))=r(j)$ whenever $i$’s cluster is smaller than $j$’s, a simple inductive argument shows that these paths never exceed $\log_2 n$ in length. However, we can make these paths even shorter using a trick called *path compression*. Since $r(i)$ is the representative of every object $j$ along the path from $i$ to $r(i)$, we can set $p(j)=r(i)$ for all of them, linking them directly to their representative so that ${\texttt{find}}$ will work in a single step the next time we call it.
As a result, the *amortized cost* of the ${\texttt{find}}$ and ${\texttt{merge}}$ operations—that is, the average cost per operation over the course of many operations—is nearly constant. Specifically, it is proportional to $\alpha(n)$, when $\alpha$ is the inverse of the Ackermann function [@tarjan:75]. The Ackermann function grows faster than any primitive recursive function, i.e., any function that can be computed with a fixed number of for-loops: faster than an exponential, an iterated tower of exponentials, and so on [@noc]. As a consequence, $\alpha(n)$ grows incredibly slowly, and the smallest value of $n$ such that $\alpha(n) > 4$ is so large that it can only be written with exotic notation. Thus the total running time for $n$ objects is essentially $O(n)$.
![When we call ${\texttt{find}}(i)$, we split and shorten the path from $i$ to its representative $r(i)$ by setting the parent of each object along the path to be its grandparent. This turns a path of length $\ell$ into two paths of length $\ell/2$.[]{data-label="fig:path-splitting"}](splitting){width="0.3\columnwidth"}
In our implementation, we employ a form of path compression that is faster and almost as effective: we link each object $j$ on the path to its grandparent, setting $p(j)=p(p(j))$. This is known as *path splitting*, since it turns a path of length $\ell$ into two paths of length $\ell/2$, or $(\ell+1)/2$ and $(\ell-1)/2$ if $\ell$ is odd, as shown in Figure \[fig:path-splitting\]. It has the advantage of requiring only one pass along the path, and it takes just one line of code (e.g. [@newman:ziff:01 Appendix A]). Like path compression, it guarantees an amortized running time of $O(\alpha(n))$ [@tarjan:van-leeuwen:84].
For lattice percolation as in [@newman:ziff:01], each time we add a new occupied site, we can check which of its neighbors are occupied, and ${\texttt{merge}}$ them together with the new site. In the continuous case, we have more work to do: if we add a new disk (say), we have to find which nearby disks it intersects. To do this efficiently, we divide the plane into square bins as shown in Figure \[fig:bins\]. Each disk belongs to whichever bin its center lies in. The width of each bin is the diameter of the disks, so that a disk in a given bin can only intersect with other disks in that bin or the eight bins in its neighborhood.
On average, the number of disks in each bin is a constant proportional to $\rho$, so we can find all the disks intersecting with each new one in constant time. We use the same approach for the other shapes; for rotated squares of width $\ell$, the bins need to have width $\sqrt{2} \ell$. A similar approach for rotated sticks was used in [@li:zhang:09].
![(Color online) We divide the plane into square bins whose width equals the diameter of the disks. Each disk in a given bin (dashed) can only intersect with other disks in the same bin, or in the eight neighboring bins.[]{data-label="fig:bins"}](bins){width="0.9\columnwidth"}
If we wished to detect crossing clusters—those that connect, say, the top and bottom edges of the square—we could add two special objects to the union-find data structure, which are connected by fiat to all the disks in the bins along the top or bottom edge. We would then check, at each step, whether these two objects are in the same cluster. However, as discussed below and in [@newman:ziff:01], the finite-size scaling is much better if we use periodic boundary conditions instead, and look for clusters that wrap around the torus horizontally or vertically.
We detect these wrapping clusters using a technique originally used for detecting crossing clusters in the Potts model [@machta:etal:96]. We associate a vector with each object in the union-find data structure, recording the displacement between it and its parent. In principle this displacement is real-valued, but it suffices to record an integer vector giving the displacement between their respective bins. When we compress and splint a path, we sum these vectors to get the total displacement between each object on the path and its new parent.
Now suppose that ${\texttt{merge}}(i,j)$ finds that two overlapping disks $i$ and $j$ are already in the same cluster. Object $i$ now has two paths to its representative; one that goes through its own parent, and another that consists of hopping to $j$ and then going through $j$’s parents. We sum the displacement vectors along both these paths. If these sums are the same, then the cluster is simply-connected. But if they differ by $\pm L$ in either coordinate, then the cluster has a nontrivial winding number around one or both directions on the torus.
Like the union-find algorithm itself, the time it takes to sum these vectors is proportional to the length of the paths from $i$ and $j$ to their representative. As Figure \[fig:times\] shows, the total running time of our entire algorithm—the time it takes to carry out a trial on a lattice of size $L$, adding objects one at a time until a wrapping cluster appears—is essentially linear in the number $n$ of objects at criticality, or equivalently linear in $L^2$. It slows down somewhat when the computer is forced to switch to parts of its memory with slower access, but this only affects the leading constant.
Analysis and Results
====================
If in each trial we stop at the first $n$ where a wrapping cluster appears, then the estimated probability $P_L(a,n)$ that a wrapping cluster exists in the microcanonical ensemble with $n$ objects of area $a$ is the fraction of trials that stop on or before the $n$th step. To obtain the probability $R_L(\eta)$ of percolation in the grand canonical ensemble with filling fraction $\eta$, we convolve $P_L$ with the Poisson distribution with mean $\lambda = \rho L^2 = \eta L^2 / a$: $$\label{eq:convolution}
R_L(\eta) = {\mathrm{e}}^{-\lambda}
\sum_{n=0}^\infty \frac{\lambda^n}{n!} \,P_L(a,n) \, .$$ To avoid numerical difficulties where the numerator and denominator are both very large, we compute Poisson weights $w_n \propto \lambda^n/n!$ inductively in two sequences $w_{\bar{n}-k}$ and $w_{\bar{n}+k}$ to the left and right of the peak at $\bar{n}=\lfloor \lambda \rfloor$, where we define $w_{\bar{n}}=1$: $$w_{\bar{n}-k} = \begin{cases}
1 & \text{for $k=0$} \\
\frac{\bar{n}-(k-1)}{\lambda}\, w_{\bar{n}-(k-1)} & \text{for $k=1,2,\ldots$}
\end{cases}$$ and $$w_{\bar{n}+k} = \begin{cases}
1 & \text{for $k=0$} \\
\frac {\lambda}{\bar{n}+k} \, w_{\bar{n}+k-1}& \text{for $k=1,2,\ldots$}
\end{cases}$$ The sum only needs to be computed for a finite number of terms. In one direction, we only need to sum down to the smallest $n$ where $P_L(a,n)$ is nonzero, i.e., the smallest value of $n$ where we observed a wrapping cluster in at least one trial. In the other direction, once we pass the largest $n$ where a wrapping cluster first appeared, then $P_L(a,n)=1$. At that point, we sum the remaining terms until they are zero to within the numerical precision of the computer. We then normalize the entire sum by dividing by $\sum w_n$.
Equipped with the data from the microcanonical simulations and this convolution routine, we compute the wrapping probability functions $R_L(\eta)$ for various system sizes $L$ and shapes. Like [@newman:ziff:01], we look for several kinds of wrapping in particular. Specifically:
- $R^e_L(\eta)$ is the probability of any kind of wrapping cluster. This is indicated by a winding number that is nonzero in either coordinate.
- $R^h_L(\eta)$ is the probability of a cluster that wraps horizontally. This is indicated by a winding number that is nonzero in the first coordinate.
- $R^b_L(\eta)$ is the probability of a cluster that wraps both horizontally and vertically. This is indicated by a single winding number that is nonzero in both coordinates, or a pair of winding numbers that are nonzero in the first and second coordinates respectively.
- $R^1_L(\eta)$ is the probability of a cluster that wraps horizontally, but not vertically. This is indicated by a winding number that is nonzero in only the first coordinate.
For any $L$ and any $\eta$, these probabilities obey $$R_L^e = 2 R_L^h - R_L^b = 2 R_L^1 + R_L^b \, .$$ We assume here that the torus is square, so that horizontal and vertical wrapping probabilities are equal.
Note that if the first nonzero winding number observed in a given trial is nonzero in both coordinates, then a cluster of type $1$ (horizontal but not vertical) does not occur at all in that trial. Thus $R^1_L(\eta)$ does not tend to $1$ as $\eta$ increases.
In practice, we focused on $R_L^e$ and $R_L^b$. In each run, we recorded the number of objects $n^h$ at which horizontal wrapping first occurred, and the number $n^v$ at which vertical wrapping first occurred. Then $n^e = \min(n^h,n^v)$ and $n^b = \max(n^h,n^v)$ are our estimates, in that run, of the values of $n$ at which $R^e_L$ and $R^b_L$ jump from $0$ to $1$.
A beautiful fact is that, even though the percolation threshold $\eta_c$ is not known for any of our models, conformal field theory implies exact values for these probabilities at the transition in the limit $L \to \infty$ [@pinson:94; @newman:ziff:01]. Specifically, $$\label{eq:cft-values}
\begin{aligned}
R_\infty^h &= 0.521\,058\,289\,248\,821\,787\,848... \\
R_\infty^e &= 0.690\,473\,724\,570\,168\,677\,230... \\
R_\infty^b &= 0.351\,642\,853\,927\,474\,898\,465... \\
R_\infty^1 &= 0.169\,415\,435\,321\,346\,889\,383...
\end{aligned}$$ For each $L$, and each type of wrapping cluster, we can estimate the critical filling factor $\eta_L$ as the solution of the equation $$\label{eq:estimator}
R_L(\eta_L) = R_\infty \, .$$ For instance, Figure \[fig:span\] shows $R^e_L(\eta)$ for disks for $L$ ranging up to $512$. The filling factors $\eta_L$ where these curves cross $R_\infty^e$ rapidly converge to $\eta_c$.
The rate of convergence is determined by two factors. The first comes from the fact that the width of the transition window from $R_L \approx 0$ to $R_L \approx 1$ scales as $L^{-1/\nu}$ where $\nu=4/3$ is a universal critical exponent for two-dimensional percolation. This scaling holds even for small systems, as can be seen in Figure \[fig:steigung\], where we plot the slope of $R_L$ at the estimated critical filling factor $\eta_L$. The slope scales perfectly like $L^{3/4}$.
The second factor comes from the fact that $R_L(\eta)$ not only becomes steeper but also moves upward in the critical region (see the inset in Figure \[fig:span\]). To measure the contribution from this effect, we computed the difference $R_L(\eta_c) - R^e_\infty$ using the previously best known value for $\eta_c$ from Table \[tab:etas\]. This difference scales like $L^{-2}$, as can be seen from Figure \[fig:disks-fss\]. The exponent $-2$ correponds to the leading irrelevant renormalization exponent $y_i$ in the Kac table [@hu:bloete:deng:13]. Note that the periodic boundary conditions are responsible for this decay. With open boundary conditions, this factor scales as $L^{-1}$ [@hovi:aharony:96], leading to more severe finite-size effects.
These two factors combine to give $$\label{eq:estimator-convergence}
\eta_L-\eta_c \sim L^{-2-1/\nu} = L^{-11/4}\,$$ for the rate of convergence. Hence we expect a straight line if we plot $\eta_L$ vs. $L^{-11/4}$, and this is exactly what we observe in Figure \[fig:etac\]. Extrapolating this line to zero then gives our estimates of $\eta_c$ shown in Table \[tab:etas\].
How do we compute the error bars in our estimates of $\eta_c$? First consider the fluctuations in $R_L(\eta)$. Each of our microcanonical experiments contributes to our estimate of $R_L(\eta)$ for all $\eta$ through the convolution . We can imagine this as choosing $n$ from the Poisson distribution, adding $n$ objects, and returning an estimate of $R_L(\eta) = 1$ or $0$ depending on whether they percolate or not. If we perform $N$ trials, the number of trials that return $1$ is binomially distributed with mean $R_L(\eta) N$, and averaging gives an estimate of $R_L(\eta)$ with standard deviation $$\label{eq:sigma-R}
\sigma_{R_L} = \sqrt{\frac{R_L(\eta)\,\big(1-R_L(\eta)\big)}{N}} \, .$$ Depending on which kind of wrapping cluster we are looking for, this is roughly $0.4 N^{-1/2}$.
When we look for the $\eta_L$ where $R_L(\eta)$ crosses $R_\infty$, the error on $\eta_L$ is given by $$ \sigma_{\eta_L} = \frac{\sigma_{R_L}}{R'_L(\eta_L)} \, .$$ Since the slope $R'_L(\eta_L)$ grows as $0.361 L^{3/4}$ (see Figure \[fig:steigung\]) this gives $$\sigma_{\eta_L} \approx N^{-1/2} \,L^{-3/4} \, .$$ These are the error bars shown in Figure \[fig:etac\].
The extrapolated value for $\eta_c$ is computed from simulations for $D$ different system sizes $L$, which in a weighted linear regression as in Figure \[fig:etac\] yields an error roughly $\sqrt{D}$ times smaller than the error bars of the underlying data points.
Finally, we average our estimates of $\eta_c$ from $R_L^e$ and $R_L^b$. Assuming that these estimates are only weakly correlated reduces the error bars by another factor of $\sqrt{2}$.
The error bars shown in Table \[tab:etas\] are the result of simulating roughly $D=50$ system sizes ranging from $L=8$ to $L=2048$, with sample sizes $N$ ranging from $10^{10}$ for the systems with $L \le 100$, to $10^9$ for $100 < L \le 500$, to $10^6$ for $500 < L \le 2048$.
We ran these simulations in parallel on several computer clusters with greatly varying computational power. In total, our simulations would have taken about 400 years if done only on the laptop on which this paper was written.
Conclusions
===========
We have shown that the union-find approach to estimating percolation thresholds introduced by Newman and Ziff [@newman:ziff:01] can be applied in the continuous case. With the help of an algorithm for estimating $\eta_c$ that runs in essentially linear time as a function of the number of objects at criticality, we have obtained new estimates for $\eta_c$ in a variety of continuum percolation models that are several orders of magnitude more accurate than previous results. In the process, we have confirmed the predictions of conformal field theory for these models, both for the finite-size scaling exponent $\nu$ and the probabilities that various kinds of wrapping clusters exist at $\eta_c$ on periodic boundary conditions.
S.M. thanks the Santa Fe Institute for their hospitality. C.M. is supported by the National Science Foundation through grant CCF-1219117 and by the Air Force Office of Scientific Research and the Defense Advanced Research Projects Agency through grant FA9550-12-1-0432. We are grateful to Robert Ziff and Mark Newman for helpful conversations.
|
---
abstract: 'A widely applicable Bayesian information criterion [@Watanabe2013] is applicable for both regular and singular models in the model selection problem. This criterion tends to overestimate the log marginal likelihood. We identify an overestimating term of a widely applicable Bayesian information criterion. Adjustment of the term gives an asymptotically unbiased estimator of the leading two terms of asymptotic expansion of the log marginal likelihood. In numerical experiments on regular and singular models, the adjustment resulted in smaller bias than the original criterion.'
author:
- |
Toru Imai[^1]\
Kyoto University
bibliography:
- 'WBICa.bib'
title: On the overestimation of widely applicable Bayesian information criterion
---
\#1
[*Keywords:*]{} marginal likelihood, singular fluctuation, singular model, WBIC.
Introduction
============
Evaluation on the log marginal likelihood is an important issue in the model selection problem, and a number of studies have been conducted (see, for example, [@Konishi2007]). [@Schwarz1978] proposed the Bayesian information criterion, BIC, which gives an approximation of the log marginal likelihood. However, BIC requires regularity conditions and therefore covers only regular models.
On the other hand, [@Watanabe2013] proposed the widely applicable Bayesian information criterion, WBIC, which can be applied to both regular and singular models. Unfortunately, WBIC tends to overestimate the log marginal likelihood in numerical experiments [@Friel2017]. In order to prove this overestimation, we need to identify three components: the $O_p(1)$ term of WBIC, the multiplicity of the real log canonical threshold, and the $O_p(1)$ term of the log marginal likelihood. However, it is challenging to identify the second and third components.
The aim of this paper is to identify the explicit constant order term of WBIC that causes an overestimation.
Widely applicable Bayesian information criterion
================================================
Let $X^n=(X_1,...,X_n)$ denote a sample of $n$ independent and identically distributed observations with each $X_i \in R^h$ drawn from a data generating distribution $q$. Let $M$ be a $d$-dimensional model with associated parameters $\theta \in \Omega \subset R^{d}$, where $\Omega$ is a parameter space. Let $p(X^n \!\mid\! \theta, M)$ be the likelihood function and $\varphi(\theta \!\mid\! M)$ a prior distribution. The log marginal likelihood $\log L(M)$ for model $M$ is defined as $$\log L(M) := \log \int_{\Omega} p(X^n \!\mid\! \theta, M) \varphi(\theta \!\mid\! M) d\theta.$$
A statistical model is termed regular if the mapping from a model parameter to a probability distribution is one-to-one and if the Fisher information matrix is positive definite. Otherwise, a statistical model is called singular. In this paper, we assume that $p(X^n \!\mid\! \theta, M)$ is differentiable and that its first derivative function is not a constant.
For any integrable function $f(\theta)$ and a non-negative real variable $t$, let $E_\theta^t \{ f(\theta) \}$ and $V_\theta^t \{ f(\theta) \}$ be defined as $$\begin{aligned}
E_\theta^t \{ f(\theta) \} &=& \left\{ \int_{\Omega} p(X^n \!\mid\! \theta, M)^t \varphi (\theta \!\mid\! M) d\theta \right\}^{-1 }\int_{\Omega} f(\theta) p(X^n \!\mid\! \theta, M)^t \varphi (\theta \!\mid\! M) d\theta, \\
V_\theta^t \{ f(\theta) \} &=& E_\theta^t \{ f(\theta)^2 \} - \Bigl[E_\theta^t \{f(\theta) \} \Bigr]^2,\end{aligned}$$ respectively. Here, $t$ is called an inverse temperature.
Let $F(t)$ be defined as $$F(t) := \log \int_\Omega p(X^n \!\mid\! \theta, M)^t \varphi(\theta \!\mid\! M) d\theta.$$ Then, $F(0)=0, F(1)=\log L(M)$ by definition, and a simple calculation gives $$\begin{aligned}
\frac{d}{dt} F(t) &=& E_\theta^t \{ \log p(X^n \!\mid\! \theta, M) \}, \\
\frac{d^2}{dt^2} F(t) &=& V_\theta^t \{ \log p(X^n \!\mid\! \theta, M) \}.\end{aligned}$$
Thus, we obtain the standard thermodynamic identity: $$\log L(M) = \int_0^1 \frac{d}{dt} F(t) dt = \int_0^1 E_\theta^t \{ \log p(X^n \!\mid\! \theta, M) \} dt.$$
By the Cauchy-Schwarz inequality, we have $d^2 F(t)/dt^2 = V_\theta^t \{ \log p(X^n \!\mid\! \theta, M) \} > 0$. Therefore, $d F(t)/dt = E_\theta^t \{ \log p(X^n \!\mid\! \theta, M) \} $ is an increasing function. Hence, by the mean value theorem, there exists a unique temperature $t^* \in (0,1)$ such that $$\log L(M) = E_\theta^{t^*} \{ \log p(X^n \!\mid\! \theta, M) \}.$$ Based on this fact, WBIC [@Watanabe2013] is defined as $${\rm WBIC} = E_\theta^{t_w} \{ \log p(X^n \!\mid\! \theta, M) \},$$ where $t_w = (\log n)^{-1}.$
The singular learning theory by [@Watanabe2009b] requires the following four assumptions.
\[assumption1\] The set of parameters $\Omega$ is a compact set in $R^d$ and can be defined by analytic functions $\pi_1,..., \pi_k$; $$\Omega = \{ \theta \in R^d : \pi_1(\theta)\ge 0,..., \pi_k(\theta)\ge 0 \}.$$
\[assumption2\] The prior distribution $\varphi(\theta)$ can be decomposed as the product of a non-negative analytic function $\varphi_1$ and a positive differentiable function $\varphi_2$; $$\varphi(\theta) = \varphi_1(\theta) \varphi_2(\theta).$$
\[assumption3\] Let $s \ge 6$ and $$L^s(q) = \left\{ f(x) : \Bigl(\int |f(x)|^s q(x)dx \Bigr)^{1/s} < \infty \right\}$$ be a Banach space. There exists an open set $\Omega' \supset \Omega$ such that for $\theta \in \Omega'$ the map $\theta \mapsto \log q(x)/p(x\mid\theta, M)$ is an $L^s(q)$-valued analytic function.
\[assumption4\] Let $\Omega_\epsilon$ be the set $$\Omega_\epsilon = \{ \theta \in \Omega : K(\theta) \leq \epsilon \},$$ where $K(\theta) = \int q(x) \log q(x)/p(x \!\mid\! \theta, M) dx$. There exists a pair of positive constants $(\epsilon, c)$ such that $$E \{ \log q(X)/p(X \!\mid\! \theta, M) \} \ge c E \left[ \{ \log q(X)/p(X \!\mid\! \theta, M) \}^2 \right], \quad \forall \theta \in \Omega_\epsilon.$$
Under assumptions 1–4, [@Watanabe2009b] showed that $$\log L(M) = \log p(X^n \!\mid\! \theta_0, M) - \lambda \log n + (m-1) \log \log n + O_p(1), \label{asymp_lml}$$ where $ \theta_0$ is the parameter that minimizes the Kullback-Leibler divergence from a data-generating distribution to a statistical model, and $\lambda$ and $m$ are termed the real log canonical threshold and its multiplicity, respectively. The negative real log canonical threshold $(-\lambda)$ is defined as the largest pole of the zeta function $\zeta(z)$: $$\zeta(z) = \int_\Omega K(\theta)^z \varphi(\theta) d\theta,$$ where $K(\theta) = \int q(x) \log q(x)/p(x \!\mid\! \theta, M) dx$ and $z$ is a complex variable. The multiplicity $m$ of the real log canonical threshold is defined as the order of the largest pole of the zeta function $\zeta(z)$. Determining real log canonical thresholds and their multiplicities is generally challenging.
In addition, [@Watanabe2013] showed $$E( {\rm WBIC}) = E\{ \log p(X^n \!\mid\! \theta_0, M) \} - \lambda \log n + O(1).$$ In the next section, we identify the explicit constant order term.
Constant order term of WBIC
============================
The Gibbs training loss $GL(t)$ is defined as $$GL(t) = - E_\theta^t \{ \log p(X^n \!\mid\! \theta, M) \}/n.$$ From the definition, we have $$nGL(t_w) = - {\rm WBIC}. \label{gibbs_wbic}$$ On the other hand, Theorems 6.8 and 6.10 of the book [@Watanabe2009b] lead to $$E\{nGL(t) \} = -E\{ \log p(X^n \!\mid\! \theta_0, M) \} + \frac{\lambda}{t} -\nu(t) + o(1), \label{gibbs_exp}$$ where $\nu(t)$ is called the singular fluctuation and is defined as $$\nu(t) = \lim_{n \to \infty} \frac{t}{2} E \left[ \sum_{i=1}^n V_\theta^{t} \{ \log p({X}_i \!\mid\! {\theta}, M) \} \right].$$ Equations (\[gibbs\_wbic\]) and (\[gibbs\_exp\]) lead to the following proposition:
\[prop1\] Under assumptions 1–4, we have $$E({\rm WBIC}) = E \left\{ \log p(X^n \!\mid\! \theta_0, M) \right\} - \lambda \log n + \nu(t_w) + o(1). \label{asymp_wbic}$$
Since $\nu(t_w)$ is always positive by definition, equations (\[asymp\_lml\]) and (\[asymp\_wbic\]) cause WBIC to overestimate the leading two terms of the asymptotic expansion of the log marginal likelihood.
Let an estimator of the singular fluctuation $\hat{\nu}(t)$ be defined as $$\hat{\nu}(t) = \frac{t}{2} \left[ \sum_{i=1}^n V_\theta^{t} \{ \log p({X}_i \!\mid\! {\theta}, M) \} \right].$$ From Proposition \[prop1\] and the definition of the singular fluctuation and its estimator, we obtain the following corollary.
\[cor1\] Under assumptions 1–4, we have $$E\{ {\rm WBIC} - \hat{\nu}(t_w) \} = E\left\{ \log p(X^n \!\mid\! \theta_0, M) \right\} - \lambda \log n + o(1). \label{asymp_wbic_nu}$$
A simple example
================
To demonstrate Corollary \[cor1\], we compute the explicit form of WBIC and $\hat{\nu}(t_w)$ for a simple model $M_N$ that was considered by [@Friel2008], [@Friel2017], and [@Gelman2013]. Let $x^n=\{x_i \mid i=1,...,n \}$ be independent and identically distributed observations, $x_i \sim N(\theta_0, 1)$, and the prior of $\theta$ is $N(m, v)$. Then, the posterior distribution is $N(m_t, v_t)$, where $m_t=(nt+1/v)^{-1}(nt\overline{x}+m/v), v_t=(nt+1/v)^{-1}$, and $\overline{x}=n^{-1}\sum_{i=1}^n x_i$. Therefore, a simple computation gives $$\begin{aligned}
{\rm WBIC} &=& -\frac{n}{2}\log 2\pi - \frac{1}{2}\left( \sum_{i=1}^n x_i^2 \right) + n \overline{x} m_{t_w} -\frac{n}{2}(v_{t_w} + m_{t_w}^2) \\
&=& -\frac{n}{2}\log 2\pi - \frac{1}{2}\left\{ \sum_{i=1}^n (x_i-\theta_0)^2 \right\} - \frac{1}{2}\log n + \frac{n}{2}(\overline{x} -\theta_0)^2 +o_p(1) \\
&=& \log p(x^n \!\mid\! \theta_0, M_N) - \frac{1}{2}\log n + \frac{n}{2}(\overline{x} -\theta_0)^2 +o_p(1).\end{aligned}$$ In addition, for a large sample size $n$, the central limit theorem leads to $${\rm WBIC} = \log p(x^n \!\mid\! \theta_0, M_N) - \frac{1}{2}\log n + \frac{1}{2} +o_p(1). \label{wbic_simple_eg}$$ On the other hand, a simple calculation leads to $$V_\theta^{t} \{ \log p({x}_i \!\mid\! {\theta}, M_N) \} = v_t(x_i - m_t)^2 + \frac{1}{2}v_t^2,$$ and from the definition of $\hat{\nu}(t)$, $$\hat{\nu}(t) = \frac{ntv - tv}{ntv+1}\frac{s_x^2}{2} + \frac{ntv}{2(ntv+1)^3}(m-\overline{x})^2 + \frac{tnv^2}{4(ntv+1)^2},$$ where $s_x^2=(n-1)^{-1}\sum_{i=1}^n (x_i-\overline{x})^2$. Therefore, $$E\{ \hat{\nu}(t) \} = \frac{1}{2} + o(1), \label{nu_simple_eg}$$ for any $t$. Equations (\[wbic\_simple\_eg\]) and (\[nu\_simple\_eg\]) lead to $$E\{ {\rm WBIC} - \hat{\nu}(t_w) \} = E\{ \log p(x^n \!\mid\! \theta_0, M_N) \} - \frac{1}{2}\log n +o(1).$$
Numerical evaluation
====================
Linear regression model
-----------------------
The radiata pine dataset ($n=42$) was used in the book by [@Williams1959], and $y_i$ denotes the maximum compression strength parallel to the grain, $x_i$ the density, and $z_i$ the resin-adjusted density. [@Friel2012] and [@Friel2017] considered the two non-nested linear regression models: $$\begin{aligned}
M_1: y_i &=& \alpha + \beta(x_i-\overline{x})+\epsilon_i, \quad \epsilon_i \sim N(0, \tau^{-1}), \\
M_2: y_i &=& \gamma + \delta(z_i-\overline{z})+\eta_i, \quad \eta_i \sim N(0, \kappa^{-1}),\end{aligned}$$ where $\overline{x}=n^{-1}\sum_{i=1}^n x_i$ and $\overline{z}=n^{-1}\sum_{i=1}^n z_i$. They supposed the priors of $(\alpha, \beta)$ and $(\gamma, \delta)$ had mean $(3000,185)$ with precision $\tau Q$ and $\kappa Q$ respectively, where $Q$ is the diagonal matrix such that $Q_{(11)}=0.06, Q_{(22)}=6$. A gamma prior with shape $a=6$ and rate $b=600^2$ was chosen for $\tau$ and $\kappa$.
The exact evaluation of the log marginal likelihood was derived by [@Friel2012] and [@Friel2017]: $$\begin{aligned}
\log L(M_1) &=& -\frac{n}{2}\log \pi +\frac{a}{2}\log b + \log \frac{\Gamma\{(n+a)/2 \}}{\Gamma(a/2) } \nonumber \\
&&+ \frac{1}{2} \log \frac{\det (Q)}{\det (M)} - \frac{n+a}{2} \log(y^T R y + b), \end{aligned}$$ where $y=(y_1,...,y_n)^T, M=X^TX + Q$, and $R=I- XM^{-1}X^T$ with $X$ the $n \times 2$ matrix such that $X_{(i1)}=1, X_{(i2)}=x_i$ and $I$ the $2 \times 2$ identity matrix. Obviously, $\log L(M_2)$ has the same expression.
We conducted 1000 independent computations of the Hamiltonian Monte Carlo method, implemented using the R package [*RStan*]{} [@RStan], to obtain the posteriors for computing WBIC and WBIC $-\hat{\nu}(t_w)$.
------------------------ ----------------- ---------- ----------------- ----------
Methods $\log L(M_{1})$ $\log L(M_{2})$
mean s.d. mean s.d.
Exact Evaluation -310.128 - -301.704 -
WBIC -308.091 (0.0263) -299.326 (0.0272)
WBIC $-\hat{\nu}(t_w)$ -310.100 (0.0387) -300.833 (0.0365)
------------------------ ----------------- ---------- ----------------- ----------
: Radiata pine dataset ($n=42$). Comparison of the evaluations of the log marginal likelihood for linear regression models. Figures in parentheses give the standard deviations
\[tab1\]
Table \[tab1\] shows the results of the estimates of WBIC and WBIC $-\hat{\nu}(t_w)$. Comparing WBIC and WBIC $-\hat{\nu}(t_w)$, the adjustment by the estimate of the singular fluctuation reduces the bias for both models.
Normal mixture model
--------------------
In this section, we consider the following mixture model with two normal distributions: $$M_M: \alpha N(\mu_1, 1) + (1-\alpha) N(\mu_2, 1).$$ When the data-generating distribution is $N(0, 1)$, [@Aoyagi2010a] showed that the real log canonical threshold $\lambda$ is 3/4 and its multiplicity $m$ is 1. Therefore, the log marginal likelihood $\log L(M_{M})$ is $$\log L(M_{M}) = \log p(X^n \!\mid\! \theta_0, M_M) - 3/4 \log n + O_p(1).$$
We conducted 1000 simulations to compute WBIC, WBIC $-\nu(t_w)$ and the Monte Carlo evaluation of $\log L(M_{M})$ for each sample size $n=50, 200$. We set the data-generating distribution $N(0, 1)$, and set the priors $\alpha \sim {\rm Unif}(0,1)$, $\mu_1, \mu_2 \sim N(0, 10)$. For Monte Carlo evaluation, we used standard Monte Carlo sampling with $10^7$ draws from the priors based on Neal’s method [@Neal1999]. We used the Hamiltonian Monte Carlo method, implemented with the R package [*RStan*]{} [@RStan], to obtain the posteriors for computing WBIC and $\hat{\nu}(t_w)$.
------------------------ -------- -------- --------- --------
Methods $n=50$ $n=200$
mean s.d. mean s.d.
Monte Carlo Evaluation -74.32 (4.56) -288.84 (9.61)
WBIC -73.36 (4.59) -287.81 (9.59)
WBIC $-\hat{\nu}(t_w)$ -73.90 (4.73) -288.40 (9.68)
------------------------ -------- -------- --------- --------
: Comparison of the approximations of the log marginal likelihood for Gaussian mixture models. Figures in parentheses give the standard deviations
\[tab2\]
Table \[tab2\] shows the results of the estimates of WBIC, WBIC $-\hat{\nu}(t_w)$ and the Monte Carlo evaluation. Comparing WBIC and WBIC $-\hat{\nu}(t_w)$, the values of the adjusted WBIC are closer to those of the Monte Carlo evaluation than those of WBIC. The standard deviations of WBIC $-\hat{\nu}(t_w)$ are slightly larger than those of WBIC.
Discussion
==========
This paper identified the overestimating constant order term of WBIC, which is the singular fluctuation $\nu(t_w)$ with the temperature $t_w=(\log n)^{-1}$. The adjustment of WBIC by the estimator of singular fluctuation gives an asymptotically unbiased estimator for the leading two terms of the asymptotic expansion of the log marginal likelihood. Further work remains to be done regarding the higher asymptotic terms of the log marginal likelihood, including the term $(m-1) \log \log n$ and the $O_p(1)$ term in equation (\[asymp\_lml\]). Another future task is to construct an unbiased estimator of the singular fluctuation, which will reduce the bias of adjusting WBIC by the estimator of singular fluctuation when using a small sample size.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author was funded by the Japan Agency for Medical Research and Development.
[^1]: imai.toru.7w@kyoto-u.ac.jp
|
---
abstract: 'A phenomenological analysis of the scalar glueball and scalar meson spectra is carried out by using the $AdS/QCD$ framework in the bottom-up approach. We make use of the relation between the mode functions in $AdS/QCD$ and the wave functions in Light-Front $QCD$ to discuss the mixing of glueballs and mesons.'
author:
- Matteo Rinaldi
- Vicente Vento
- Risto Orava
title: Looking for pure glueball states
---
Introduction
============
Glueballs have been a matter of theoretical study and experimental search since the formulation of the theory of the strong interaction Quantum Chromodynamics (QCD)[@Fritzsch:1973pi; @Fritzsch:1975wn]. QCD sum rules [@Shifman:1978bx; @Mathieu:2008me], QCD based models [@Mathieu:2008me] and Lattice QCD computations both with sea quarks [@Gregory:2012hu] and in the pure glue theory [@Morningstar:1999rf; @Chen:2005mg; @Lucini:2004my] have been used to determine their spectra and properties. However, due to the lack of phenomenological support much debate has been associated with their properties [@Mathieu:2008me]. Glueballs, if they exist, will mix with meson states of the same quantum numbers, and therefore their direct characterization is difficult to disclose.
Recently we have analyzed the glueball spectrum from the point of view of the $AdS/QCD$ correspondence in the so-called bottom-up approach [@Vento:2017ice; @Rinaldi:2017wdn]. Following these investigations, we analyze here phenomenological consequences of our studies in the scalar sector by comparing theoretical results with data. The experimentally determined scalar “mesons” with spin parities $J^{PC}= 0^{++}$ are known as the $f_0$ mesons [@Patrignani:2016xqp]. Our aim is to describe the glueball lattice spectrum by means of the $AdS/QCD$ correspondence and to compare it with the spectrum of $f_0$’s. Only when the masses of the glueballs and the $f_0$’s are close mixing is to be expected [@Vento:2004xx]. However, if the masses are close, but the dynamics generating the resonance states is different, mixing will not happen [@Vento:2015yja]. Therefore, we are looking for meson and glueball states with similar masses but generated by different dynamics. These scalar particles will appear in the phenomenologial spectrum as mostly glueball or mostly meson.
We use for our analysis the the so-called bottom-up approach of the $AdS/QCD$ correspondence [@Polchinski:2000uf; @Brodsky:2003px; @DaRold:2005mxj]. This approach describes glueball and meson dynamics in a very transparent fashion[@Erlich:2005qh; @Karch:2006pv; @Rinaldi:2017wdn; @BoschiFilho:2002vd; @Colangelo:2007pt]. In section II we describe the scalar glueball spectrum and the meson spectrum in several $AdS/QCD$ appraoches. In section III we will discuss glueball-meson mixing and finally in the conclusions we extract some consequences of our analysis.
Scalar glueball and scalar meson spectrum in a bottom-up approach
=================================================================
The so-called botton-up approach on the AdS/CFT correspondence starts from QCD and attempts to construct its five-dimensional holographic dual. One implements duality in nearly conformal conditions defining QCD on the four dimensional boundary and introducing a bulk space which is a slice of $AdS_5$ whose size is related to $\Lambda_{QCD}$ [@Polchinski:2000uf; @Brodsky:2003px; @Erlich:2005qh; @DaRold:2005mxj].
The metric of this space can be written as
$$ds^2=\frac{R^2}{z^2} (dz^2 + \eta_{\mu \nu} dx^\mu dx^\nu) + R^2 d\Omega_5,
\label{metric5}$$
where $\eta_{\mu \nu}$ is the Minkowski metric and the size of the slice in the holographic coordinate $0< z < z_{max}$ is related to the scale of QCD, $ z_{max} =\frac{1}{\Lambda_{QCD}}$. This is the so called hard wall approximation. Later on, in order to reproduce the Regge trajectories, the so called soft wall approximation was introduced [@Karch:2006pv; @Capossoli:2015ywa]. Within the bottom-up strategy and in both, hard wall and the soft wall approaches, glueballs arising from the correspondence of fields in $AdS_5$ have been studied in refs. [@BoschiFilho:2002vd; @Colangelo:2007pt]. In our recent work we have described the scalar glueball spectrum as a graviton in $AdS_5$ with a modified soft wall metric. In Fig. \[GlueballFit\] we reproduce the results of refs. [@Rinaldi:2017wdn; @BoschiFilho:2002vd; @Colangelo:2007pt] for the glueball spectrum.
The $AdS/QCD$ models provide us with a succession of mass modes of differential equations, which in general, one has to obtain numerically. An exception is the standard soft wall model where the expression turns out to be analytic, $\omega{^2_k} = (4k +8)$, where $k = 0,1,2\ldots$ is the mode number [@Colangelo:2007pt]. In order to reach the experimantal results we have to multiply by an energy scale factor $\varepsilon$, i.e. $m^2_k = \varepsilon^2 \omega^2_k = \varepsilon^2 (4k +8)$, where $\varepsilon$ is its energy scale. To determine $\varepsilon$ we use here the technique, developed in ref.[@Rinaldi:2017wdn] . We display the spectrum obtained by fitting one $AdS$ mode to a physical state, for example in the case of glueballs we fit the lowest mode to the lowest lying glueball and determine an initial value for $\varepsilon$. We then proceed by seeding the rest of the lattice data on the plot and by varying slightly $\varepsilon$ we get a best fit to the whole spectrum. The lattice data used are shown in Table \[masses\] [@Morningstar:1999rf; @Chen:2005mg; @Lucini:2004my] [^1]. We also show the results for the tensor glueball states since theory predicts degeneracy between the scalar an the tensor glueball for the soft wall models.
In Fig. \[GlueballFit\] we show the results for all models analyzed. All models lead to a reasonable fit of the data. The difference between them arises for heavy states. The soft wall dilaton model has a quadratic behavior that softens the slope at high energies. The hard wall and soft wall graviton model have an almost linear behavior in the region studied showing no softening of the slope.
\[htb\]
[|c c c c c c c|]{} $J^{PC}$& $0^{++}$&$2^{++}$&$0^{++}$&$2^{++}$&$0^{++}$&$0^{++}$\
MP & $1730 \pm 94$ & $2400 \pm122$ & $2670 \pm 222 $& & &\
YC & $1719 \pm 94$ & $2390 \pm124$ & & & &\
LTW & $1475 \pm 72$ & $2150 \pm 104$ & $2755 \pm 124$& $2880 \pm 164 $& $3370
\pm 180$& $3990 \pm 277$\
In Fig. \[MesonFit\] we show the soft wall model fit to the PDG meson spectrum [@Patrignani:2016xqp]. Many authors have argued that the $f_0(500)$ is not a conventional meson state but a tetraquark or a hybrid [@Mathieu:2008me; @Tanabashi:2018oca]. The figure shows that once the $f_0(500)$ is taken out of the meson spectrum the soft wall model fit is excellent. From now on we will omit the $f_0(500)$ from our discussion. The hard wall model does not have the correct Regge behavior for mesons [@Erlich:2005qh; @Karch:2006pv] and therefore we will omit the hard wall model from further discussion.
In Fig. \[SpectrumFit\] we show a fit to the experimental scalar meson data [@Patrignani:2016xqp] and to the glueball lattice data. We have used for the meson fit the standard soft wall model [@Colangelo:2008us], which is the same for the soft wall graviton model [@Rinaldi:2017wdn]. The left figure shows the fit to the glueballs in the soft wall dilaton model [@Colangelo:2007pt] and the figure on the right the fit in the soft wall model graviton model [@Rinaldi:2017wdn]. The soft wall model for the mesons follows the equation $m^2_k =(\varepsilon)^2 (4k+6)$ and we get for $\varepsilon = 410$ MeV. For the soft wall dilaton model the spectrum is described by $m^2_k =(\varepsilon')^2 (4k+8)$ with $\varepsilon' = 710$ MeV, and the soft wall graviton model fit has no analytical solution, it is given by a numerical function $m_k= \varepsilon'' f(k)$, and the shown fit is for $\varepsilon''=370$ MeV. Note the similarity of the scales for mesons and glueballs in the soft wall graviton model.
Besides the reasonable quality of the fits, the result we would like to stress is the difference between the slopes of the meson and glueball fits for large mode numbers. Consequently as we go to higher mass states, for a fixed mass, the mode function of the meson will have a much larger mode number then that of the glueball. Thus, the mode function for the meson will oscillate more than that of the glueball. Intuitively one expects that mixing is not very strong between those states. We proceed to analyze the consequences of this observation quantitatively in what follows.
Glueball- Meson mixing
======================
One of the problems in glueball phyics is the fact that glueball candidates always appear strongly mixed with mesons states [@Mathieu:2008me; @Vento:2015yja]. Mixing usually occurs if two states have similar masses and the same quantum numbers. Thus the scalar glueballs might mix with scalar mesons. The study of the $f_0$ spectrum from this perspective has led to the result that if glueballs exist, and there is no reason for its non-existence, either the $f_0(1500)$ or the $f_0(1710)$ might have a large glueball component (see [@Mathieu:2008me; @Vento:2015yja] and references therein). Our aim here is not to contribute to this discussion, but in view of the structure of the spectra, to look for dynamical regions were mixing is not favorable and therefore states with mostly gluonic valence structure might exist. The presence of almost pure glueball states and the study of their decays would help understand many properties of QCD related to the physics of gluons.
In order to proceed with the discussion let us consider the holographic light-front representation of the equation of motion, in $AdS$ space. The latter can be recast in the form of a light-front Hamiltonian [@Brodsky:2003px]
$$H_{LC} |\Psi_k> = M^2 |\Psi_k>.$$
In the $AdS/QCD$ light-front framework the above relation becomes a Schödinger type equation
$$\left(-\frac{d^2}{d z^2} + V(z) \right) \Phi(z) = M^2 \Phi(z)$$
where $z$ and $M^2$ in this equation are adimensional. The holographic light-front wave function are defined by $\Phi_k(z) =\; <z|\Psi_k>$ and are normalized as
$$<\Psi_k|\Psi_k> = \int dz |\Phi_k(z)|^2 = 1.$$
Its eigenmodes determine the mass spectrum.
Within this procedure, mixing occurs when the hamiltonian is not diagonal in a subspace determined by at least two states, e.g. a meson and a glueball states, { $|\Psi_m>, |\Psi_g>$ }. Let us recall the discussion of two state mixing. For notational simplicity we use a linear hamiltonian model. A matrix representation of the hamiltonian in a two dimensional meson-glueball subspace is given by
$$[H]= \left( \begin{array}{cc}
m & \alpha \\
\alpha & m + \Delta m \end{array} \right) ,$$
where $\alpha = <\Psi_m|H_{mg}|\Psi_g>$, $m = <\Psi_m|H_{mm}|\Psi_m>$ and $m + \Delta m = <\Psi_g|H_{gm}|\Psi_g>$. We are assuming for simplicity $\alpha$ real and positive and $\Delta m$ positive. Let us furthermore proceed under the assumption $m > > \Delta m > \alpha$ as guided by the $1/N$ expansion of QCD, where $m \sim O(1)$, $\Delta m \sim O(1/N)$ and $\alpha \sim O((1/N)^{3/2})$ [@Vento:2004xx].
After diagonalization the eigenstates become
$$M= m + \frac{\Delta m}{2} \pm \sqrt{ \alpha^2 + \left(\frac{\Delta m}{2}\right)^2}.$$
Let us discuss the case which leads to small mixing, i.e. $\Delta m >> 2 \alpha$. Then $$\begin{aligned}
M_+ & = & m + \Delta m + \frac{\alpha^2}{\Delta m} \\
M_- & = & m - \frac{\alpha^2}{\Delta m}\end{aligned}$$
The eigenstates to first order in $\alpha/\Delta m$ become for $M_+$, $ (\alpha/\Delta m,1)$, and for $M_-$ , $(1, -\alpha/\Delta m)$. Note that if $\Delta m \sim m >> \alpha$ there is little mixing. Thus states with equal quantum numbers like the scalar glueball and the scalar meson mix if their masses are similar and the mixing term in the hamiltonian is not too small. If $\Delta m = 0$ mixing is maximal independent of the value of $\alpha$ since the hamiltonian can be written as $m I + \alpha \sigma_x$, where I is the unit matrix and $\sigma_x$ the corresponding Pauli matrix. Thus for fixed $\alpha$ mixing varies from maximal to small as $\Delta m$ varies from 0 to $m$ with the assumption $m >> \alpha$. For fixed $\Delta m < m$ mixing depends on $\alpha$. The smaller $\alpha$ the smaller the mixing.
The meson spectrum in the soft wall approach leads to a complete orthonormal set of analytic mode functions whose functional form will be shown shortly, for the moment we label them ${\Psi_k(z), k= 0,1 , 2, \ldots}$. Let $\Psi_n$ be the meson mode of mass $m$. Since $\Delta m$ is small there exists a $\Psi_{n'}$, $n'> n$, such that its mass $m' \sim m + \Delta m$. If $|\Psi_{n'}>$ mixes with $|\Psi_g>$, then
$$\alpha = \sum_k <\Psi_n|H_{mg}|\Psi_k><\Psi_k|\Psi_g> = <\Psi_n|H_{mg}|\Psi_{n'}> <\Psi_{n'}|\Psi_g>,$$
Since the meson states form an orthonormal complete set $H_{mg}|\Psi_{n'} > = h_n |\Psi_n> + ...$, but only the first term in the sum remains in the matrix element. Thus $\alpha$ is proportional to the overlap factor.
Let us study these overlap factors in the two soft wall models for glueballs and mesons.
In the soft wall dilaton model the normalized mode functions for the scalar glueballs are given by [@Colangelo:2007pt]
$$\Psi_{n_g}(z)=\sqrt{(n_g+1)(n_g+2)/2} \; e^{-z^2/2} \; z^{5/2} \; _1F_1(-n_g, 3, z^2)$$
and for the scalar mesons by [@Colangelo:2008us]
$$\Psi_{n}(z)= \sqrt{2(n+1)} \; e^{-z^2/2} \;z^{3/2} \; _1F_1(-n, 2, z^2)$$
where $_1F_1$ is the Kummer confluent hypergeometric function, and $n_g$ the glueball mode number and $n$ the meson mode number. The overlap factor is defined as
$$<\Psi_n|\Psi_{n_g}> = <G|M> = \sqrt{2(n+1)(n_g+1)(n_g+2)} \int_0^\infty dz e^{-z^2} z^4 \; _1F_1(-n_g, 3, z^2) \; _1F_1(-n, 2, z^2),$$
and the overlap probability for no mixing $P_{NM} =1.- |<G|M>|^2$. For the soft wall graviton model the meson mode functions are the same as above and the graviton wave functions have been obtained numerically. In Fig.\[MesonGlueballModes\] we show the mode functions for the $n_g=2$ glueball mode and a possible mixing mode at $n =10$ as seen from Fig. \[SpectrumFit\] . The figure shows the big difference between the glueball modes in both models. The extension of the glueball mode in the soft wall graviton model is determined by the exponential in the potential and therefore all modes die at $z \sim 2$ irrespective of their mode number. In the case of the soft wall dilaton model the extension is governed by the mode number. This difference in structure implies that in the graviton model the overlap factor is oscillating and extends over many modes even reaching large meson mode numbers $n$, while in the dilaton model only a few modes close to $n_g$ are contributing to the overlap as shown in Fig. \[OverlapDilatonGraviton\].
Looking at Figs. \[SpectrumFit\] and \[OverlapDilatonGraviton\], the favorable mixing scenario is mostly excluded in the case of heavy glueballs and mesons, since the mass condition is satisfied for very different mode numbers, e.g. for $n_g=2$ the favorable meson modes occur for $n \sim 10$. This condition reduces the overlap probability for mixing dramatically. In the soft wall dilaton model the overlap probability is extremely small for the required mode number differences, while in the graviton model it oscillates at the level of maximum $10\%$ percent overlap probabilities. The outcome of our analysis is that the $AdS/QCD$ approach predicts the existence of almost pure glueball states in the scalar sector in the mass range above $2$ GeV.
Conclusion
==========
We have performed a phenomenological analysis of the scalar glueball and scalar meson spectrum based on the $AdS/QCD$ correspondence within the soft wall dilaton and graviton approaches. Theoretical outcomes have been compared with lattice $QCD$ data for the scalar and the experimental $f_0$ spectrum of the PDG tables. We have noted that the slope of the glueball spectrum as a function of mode number is bigger that that of the meson spectrum in both approaches and therefore for higher lying almost degenerate glueball and meson states, their mode numbers differ considerably. Assuming a light-front quantum mechanical description of $AdS/QCD$ correspondence we have shown that the overlap probability of glueballs with mesons is small and therefore one expects little mixing in the high mass sector. Therefore, this is the kinematical region to look for almost pure glueball states. At present, large statistics of Central Exclusive Process (CEP) data is being collected by the LHC experiments, and we expect exciting new results to appear concerning the higher mass gluon enriched process.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge Sergio Scopetta, Tatiana Tarutina and Marco Traini for discussions. VV thanks the hospitality extended to him by the University of Perugia and the INFN group in Perugia. MR was a Severo Ochoa postdoctoral fellow at IFIC during the initial stages of this work. This work was supported in part by Mineco under contract FPA2013-47443-C2-1-P, Mineco and UE Feder under contract FPA2016-77177-C2-1-P, GVA- PROMETEOII/2014/066 and SEV-2014-0398.
[99]{}
H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. [**47B**]{} (1973) 365. H. Fritzsch and P. Minkowski, Phys. Lett. [**56B**]{} (1975) 69. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B [**147**]{} (1979) 385. doi:10.1016/0550-3213(79)90022-1
V. Mathieu, N. Kochelev and V. Vento, Int. J. Mod. Phys. E [**18**]{} (2009) 1 \[arXiv:0810.4453 \[hep-ph\]\]. V. Vento, Phys. Rev. D [**73**]{} (2006) 054006 doi:10.1103/PhysRevD.73.054006 \[hep-ph/0401218\]. [@Vento:2004xx]
V. Vento, Eur. Phys. J. A [**52**]{} (2016) no.1, 1 doi:10.1140/epja/i2016-16001-x \[arXiv:1505.05355 \[hep-ph\]\]. E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, C. Richards and E. Rinaldi, JHEP [**1210**]{} (2012) 170 \[arXiv:1208.1858 \[hep-lat\]\].
C. J. Morningstar and M. J. Peardon, Phys. Rev. D [**60**]{} (1999) 034509 \[hep-lat/9901004\]. Y. Chen, A. Alexandru, S. J. Dong, T. Draper, I. Horvath, F. X. Lee, K. F. Liu and N. Mathur [*et al.*]{}, Phys. Rev. D [**73**]{} (2006) 014516 \[hep-lat/0510074\].
B. Lucini, M. Teper and U. Wenger, JHEP [**0406**]{} (2004) 012 doi:10.1088/1126-6708/2004/06/012 \[hep-lat/0404008\].
V. Vento, Eur. Phys. J. A [**53**]{} (2017) no.9, 185 doi:10.1140/epja/i2017-12378-2 \[arXiv:1706.06811 \[hep-ph\]\].
M. Rinaldi and V. Vento, doi:10.1140/epja/i2018-12600-9 arXiv:1710.09225 \[hep-ph\],arXiv:1712.06936 \[hep-ph\]..
C. Patrignani [*et al.*]{} \[Particle Data Group\], Chin. Phys. C [**40**]{} (2016) no.10, 100001. doi:10.1088/1674-1137/40/10/100001 M. Tanabashi [*et al.*]{} \[Particle Data Group\], Phys. Rev. D [**98**]{} (2018) no.3, 030001. doi:10.1103/PhysRevD.98.030001
J. Polchinski and M. J. Strassler, hep-th/0003136.
S. J. Brodsky and G. F. de Teramond, Phys. Lett. B [**582**]{} (2004) 211 \[hep-th/0310227\].
L. Da Rold and A. Pomarol, Nucl. Phys. B [**721**]{} (2005) 79 \[hep-ph/0501218\].
J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. Lett. [**95**]{} (2005) 261602 \[hep-ph/0501128\].
A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D [**74**]{} (2006) 015005 \[hep-ph/0602229\].
H. Boschi-Filho and N. R. F. Braga, JHEP [**0305**]{} (2003) 009 doi:10.1088/1126-6708/2003/05/009 \[hep-th/0212207\].
P. Colangelo, F. De Fazio, F. Jugeau and S. Nicotri, Phys. Lett. B [**652**]{} (2007) 73 \[hep-ph/0703316\].
E. Folco Capossoli and H. Boschi-Filho, Phys. Lett. B [**753**]{}, 419 (2016)
P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri, Phys. Rev. D [**78**]{} (2008) 055009 doi:10.1103/PhysRevD.78.055009 \[arXiv:0807.1054 \[hep-ph\]\].
[^1]: We have not included the lattice results from the unquenched calculation [@Gregory:2012hu] to be consistent, which however, in this range of masses and for these quantum numbers are in agreement with the shown results within errors.
|
---
abstract: 'We consider the setting of component-based design for real-time systems with critical timing constraints. Based on our earlier work, we propose a compositional specification theory for timed automata with I/O distinction, which supports substitutive refinement. Our theory provides the operations of parallel composition for composing components at run-time, logical conjunction/disjunction for independent development, and quotient for incremental synthesis. The key novelty of our timed theory lies in a weakest congruence preserving safety as well as bounded liveness properties. We show that the congruence can be characterised by two linear-time semantics, *timed-traces* and *timed-strategies*, the latter of which is derived from a game-based interpretation of timed interaction.'
author:
- Chris Chilton
- Marta Kwiatkowska
- Xu Wang
bibliography:
- 'timed-bib.bib'
title: |
Revisiting Timed Specification Theories:\
A Linear-Time Perspective
---
Introduction {#intro}
============
Component-based design methodologies can be encapsulated in the form of compositional specification theories, which allow the mixing of specifications and implementations, admit substitutive refinement to facilitate reuse, and provide a rich collection of operators. Previously [@ESOP], we developed a linear-time specification theory for reasoning about untimed components that interact by synchronisation of input and output (I/O) actions, inspired by interface automata [@henzinger-ia]. Models can be specified operationally by means of transition systems augmented by an inconsistency predicate on states, or declaratively using traces. The theory admits non-determinism, a refinement preorder based on traces, and the operations of parallel composition, conjunction and quotient. The refinement is strictly weaker than alternating simulation and is actually the weakest pre-congruence preserving inconsistent states. This implies that our refinement is *substitutive*, meaning component $A$ refines component $B$ iff $A$ can replace $B$ in any environmental context without introducing additional errors.
In this paper we target component-based development for real-time systems with critical timing constraints. We formulate a timed extension of the linear-time specification theory of [@ESOP], by allowing for both operational descriptions of components, as well as declarative specifications based on traces. Our operational models are based on a variant of timed automata with I/O distinction (although we do not insist on input-enabledness, cf [@Kaynar]), augmented by two special states: $\bot$ for safety and bounded-liveness errors, and $\top$ for timestop. Trace-based declarative specifications are shown to be a suitable semantic domain for the operational models. In addition to timed-trace semantics, we present timed-strategy semantics, which coincides with the former but relates our work closer to the timed-game frameworks used by [@larsen-timedio] and [@henzinger-timedia]. The *substitutive refinement* of our framework gives rise to the weakest congruence preserving $\bot$, and is shown to coincide across all our formalisms.
Amongst notable works in the literature, we briefly mention a theory of timed interfaces [@henzinger-timedia] and a theory of timed specifications [@larsen-timedio]. Timed interface theory contributes a framework based on timed games to formalise notions such as interfaces and compatibility, and also provides a parallel composition operator. However, the work cannot be considered a specification theory as it does not deal with the notion of refinement for component substitution or the operations of conjunction, disjunction and quotient. In this respect, [@larsen-timedio] provides a complete theory; however, the refinement is a timed version of the alternating simulation originally defined for interface automata [@henzinger-ia]. Consequently, it is too strong for determining when a component can be safely substituted with another (cf the example in Figure \[fig:strategy-equivalence-strong\]).
#### Outline.
In Section \[frame\] we introduce timed I/O automata, their semantic mapping to timed I/O transition systems, and supply the operational definitions for the operations of parallel composition, conjunction, disjunction and quotient. In Section \[sec:game\] we use the timed-game framework to introduce timed-strategy semantics, which we relate to the operational framework. Similarly in Section \[sec:ts\], we present timed-trace semantics and relate these to the operational definitions. Section \[sec:comp\] discusses related work, and finally Section \[sec:concl\] concludes.
Formal Framework {#frame}
================
In this section we introduce timed I/O automata, timed I/O transition systems and a semantic mapping from the former to the latter. Timed I/O automata are compact representations of timed I/O transition systems. Our theory will be developed using timed I/O transition systems, which are endowed with a richer repertoire of semantic machinery.
Timed I/O Automata {#sec:time}
------------------
#### Clock constraints.
Given a set $X$ of real-valued clock variables, a *clock constraint* over $X$, $cc: CC(X)$, is a boolean combination of atomic constraints of the form $x \bowtie d$ and $x - y \bowtie d$ where $x,y \in X$, $\bowtie \in \{\leq,<, =, >,\geq\}$, and $d \in \mathbb{N}$.
A *clock valuation* over $X$ is a map $t$ that assigns to each clock variable $x$ in $X$ a real value from $\mathbb{R}^{\geq 0}$. We say $t$ satisfies $cc$, written $t \in cc$, if $cc$ evaluates to true under valuation $t$. $t + d$ denotes the valuation derived from $t$ by increasing the assigned value on each clock variable by $d \in \mathbb{R}^{\geq 0}$ time units. $t[rs \mapsto 0]$ denotes the valuation obtained from $t$ by resetting the clock variables in $rs$ to $0$. Sometimes we use $0$ for the clock valuation that maps all clock variables to $0$.
A *timed I/O automaton* (TIOA) is a tuple $(C, I, O, L, l^0, AT,$ $ Inv, coInv)$, where:
- $C \subseteq X$ is a finite set of clock variables
- $A$ ($= I \cup O$) is a finite alphabet, where $I$ and $O$ are disjoint sets of input actions and output actions respectively
- $L$ is a finite set of *locations*
- $l^0 \in L$ is the *initial location*
- $AT \subseteq L \cross CC(C) \cross A \cross 2^{C} \cross L$ is a set of *action transitions*
- $Inv : L \fun CC(C)$ and $coInv : L \fun CC(C)$ assign *invariants* and *co-invariants* to states, each of which is a downward-closed clock constraint.
We use $l,l',l_i$ to range over $L$ and use $l {\xrightarrow{g,a,rs}} l'$ as a shorthand for $(l, g, a, rs, $ $l') \in AT$. $g: CC(C)$ is the enabling guard of the transition, $a \in A$ the action, and $rs$ the subset of clock variables to be reset.
Our TIOAs are similar to existing variants of timed automata with input/output distinction, except for the introduction of co-invariants and non-insistence on input-enabledness. While invariants specify the bounds beyond which time may not progress, co-invariants specify the bounds beyond which the system will *time-out* and enter error states. Our TIOAs can be used to describe both the assumptions made by the component on the inputs, together with the guarantees provided by the component on the outputs. Such assumptions and guarantees can be time constrained: guards on output transitions express safety timing guarantees, while guards on input transitions express safety timing assumptions; invariants (urgency) express liveness timing guarantees on outputs while co-invariants (time-out) express liveness timing assumptions on inputs.
When components interact together, we check whether the guarantees they provide meet the assumptions they make on each other. If not, there are two types of errors:
- An input arrives in a state and at a time when it is not expected (i.e. not satisfying the guards on the input transitions). This is a *safety error*.
- An input does not arrive in a state within a time bound (specified by a co-invariant) as expected. This is a *bounded-liveness error*.
#### Example.
Figure \[fig:automata\] depicts TIOAs representing a job scheduler together with a printer controller. The invariant at location $A$ of the scheduler forces a bounded-liveness guarantee on outputs in that location. As time must be allowed to progress beyond $t=100$, the $start$ action must be fired within the range $0\leq t \leq 100$. After $start$ has been fired, the clock $x$ is reset to $0$ and the scheduler waits (possibly indefinitely) for the job to $finish$. If the job does finish, the scheduler is only willing for this to take place between $5\leq t \leq 8$ after the job started (safety assumption), otherwise an unexpected input error will be thrown.
The controller waits for the job to $start$, after which it will wait exactly $1$ time unit before issuing $print$ (forced by the invariant $y\leq 1$ on state $2$ and the guard $y=1$). The controller now requires the printer to indicate the job is $printed$ within $10$ time units of being sent to the printer, otherwise a time-out error on inputs will occur (co-invariant $y\leq 10$ in state $3$ as liveness assumption). After the job has finished printing, the controller must indicate to the scheduler that the job has $finish$ed within $5$ time units.
{width="\textwidth"}
\[fig:automata\]
Timed Actions and Words
-----------------------
In this section we introduce some notation relating to timed actions and timed words that will be of use to us in later sections.
#### Timed actions.
For a set of input actions $I$ and a set of output actions $O$, define $tA= I \uplus O \uplus \mathbb{R}^{>0}$ to be the set of *timed actions*, $tI= I \uplus \mathbb{R}^{>0}$ to be the set of *timed inputs*, and $tO= O \uplus \mathbb{R}^{>0}$ to be the set of *timed outputs*. We use symbols like $\alpha$, $\beta$, etc. to range over $tA$.
#### Timed words.
A *timed word* (ranged over by $w, w', w_i$ etc.) is a finite mixed sequence of positive real numbers ($\mathbb{R}^{> 0}$) and visible actions such that *no two numbers are adjacent to one another*. For instance, $\langle 0.33, a, 1.41, b, c, 3.1415 \rangle$ is a timed word denoting the observation that action $a$ occurs at $0.33$ time units, then another $1.41$ time units lapse before the simultaneous occurrence of $b$ and $c$, which is followed by $3.1415$ time units of no event occurrence. The empty word is denoted by $\epsilon$.
#### Operations on timed words.
We use $last(w)$ to denote the last element in the sequence $w$, and $l(w)$ to indicate the length, which is obtained as the sum of all the reals in $w$. Concatenation of timed words $w$ and $w'$ is obtained by appending $w'$ onto the end of $w$ and coalescing adjacent reals (summing them). For instance, $\langle a, 1.41 \rangle$ $\cat \langle 0.33, b, 3.1415 \rangle$ = $\langle a, (1.41 + 0.33), b, 3.1415 \rangle$ = $\langle a, 1.74, b, 3.1415 \rangle$. Prefix/extension are defined as usual by concatenation, and we use $\leq$ for the prefix partial order. We write $w \upharpoonright tA_0$ for the projection of $w$ onto timed alphabet $tA_0$, which is defined by removing from $w$ all actions not inside $tA_0$ and coalescing adjacent reals.
Semantics as Timed I/O Transition Systems {#sec:semantics}
-----------------------------------------
The semantics of TIOAs are given as timed I/O transition systems, which are a special class of infinite labelled transition systems.
A *timed I/O transition system* (TIOTS) is a tuple ${\mathcal{P}}= \langle I, O, S,$ $ s^0, \rightarrow \rangle$, where: $I$ and $O$ are the input and output actions respectively, $S$ is a set of states, $s^0$ is the designated initial state, and $\rightarrow \subseteq S \cross I \uplus O \uplus \mathbb{R}^{>0} \cross S$ is the action and time-labelled transition system.
The states of the TIOTS for a TIOA capture the configurations of the automaton, i.e. its location and clock valuation. Therefore, each state of the TIOTS is a pair drawn from $L\times \mathbb{R}^{C}$, which we refer to as the set of *plain states*, denoted $P$. In addition, we introduce two special states $\bot$ and $\top$, which are required for the semantic mapping of disabled inputs/outputs, invariants and co-invariants.
$\bot$ is called the *inconsistent state*, representing safety and bounded-liveness errors. $\top$ is the so-called *timestop state*, representing the *magic moment* from which no error can occur.[^1]
An intuitive way to understand $\top$ and $\bot$ is from an input/output game perspective. The component controls output and delay while the environment controls input. $\bot$ is the losing state for the environment. So a disabled input at a state $p$ is equated to an input transition from $p$ to $\bot$. $\top$ is the losing state for the component. So a disabled output/delay at $p$ is equated to an output/delay transition from $p$ to $\top$. Thus we can have two semantics-preserving transformations on TIOTSs.
The *$\bot$-completion* of a TIOTS ${\mathcal{P}}$, denoted ${\mathcal{P}}^{\bot}$, adds an $a$-labelled transition from $p$ to $\bot$ for every $p \in P_{{\mathcal{P}}}$ and $a \in I$ s.t. $a$ is not enabled at $p$. $\bot$-completion will make a TIOTS *input-receptive*, i.e. input-enabled at all states. The *$\top$-completion* of a TIOTS ${\mathcal{P}}$, denoted ${\mathcal{P}}^{\top}$, adds an $\alpha$-labelled transition from $p$ to $\top$ for every $p \in P_{{\mathcal{P}}}$ and $\alpha \in tO$ s.t. $\alpha$ is not enabled at $p$.
Furthermore, for technical convenience (e.g. ease of defining time additivity), the definition of TIOTSs requires that 1) $\top$ is a *quiescent state*, i.e. a state in which the set of outgoing transitions are all self-loops, one for each $d \in \mathbb{R}^{>0}$, and 2) $\bot$ is a *chaotic state*, i.e. a state in which the set of outgoing transitions are all self-loops, one for each $\alpha \in tA$. The set of all possible states is denoted $S = P \uplus \{\bot,\top\}$. We use $p,p',p_i$ to range over $P$ while $s,s',s_i$ range over $S$.
The transition relation $\rightarrow$ of the TIOTS is derived from the execution semantics of the TIOA.
Let ${\mathcal{P}}$ be a TIOA. The semantic mapping of ${\mathcal{P}}$ is a TIOTS $\langle I, O, S, s^0, \rightarrow \rangle$, where:
- $S = (L \times \mathbb{R}^C) \uplus \{\bot, \top\}$
- $s^0 = \top$ providing $0 \notin Inv(l^0)$, $s^0 = \bot$ providing $0 \in Inv(l^0) \wedge \neg coInv(l^0)$ and $s^0 = (l^0, 0)$ providing $0 \in Inv(l^0) \wedge coInv(l^0)$,
- $\rightarrow$ is the smallest relation satisfying:
1. If $l {\xrightarrow{g,a,rs}} l'$, $t'= t [rs \mapsto 0]$, $t \in Inv(l) \wedge coInv(l) \wedge g$, then:
1. *plain action:* $(l, t) {\xrightarrow{a}} (l', t')$ providing $t' \in Inv(l') \wedge coInv(l')$
2. *error action:* $(l, t) {\xrightarrow{a}} \bot$ providing $t' \in Inv(l') \wedge \neg coInv(l')$
3. *magic action:* $(l, t) {\xrightarrow{a}} \top$ providing $t' \in \neg Inv(l')$.
2. *plain delay:* $(l, t) {\xrightarrow{d}} (l, t+d)$ if $t,t+d \in Inv(l) \wedge coInv(l)$
3. *time-out delay:* $(l, t) {\xrightarrow{d}} \bot$ if $t \in Inv(l) \wedge coInv(l)$, $t+d \notin coInv(l)$ and $\exists 0 < \delta \leq d: t+\delta \in Inv(l) \wedge \neg coInv(l)$.
Note that our semantics tries to minimise the use of transitions leading to $\top/\bot$ states. Thus there are no delay transitions leading to $\top$. This creates implicit timestops, which we capture using the concept of *semi-timestop* (i.e. semi-$\top$). We say a plain state $p$ is a *semi-$\top$* iff 1) all output transitions enabled in $p$ or any of its time-passing successors lead to the $\top$ state, and 2) there exists $d \in \mathbb{R}^{>0}$ s.t. $p {\xrightarrow{d}} \top$ or $d$ is not enabled in $p$. Thus a semi-$\top$ is a state in which it is impossible for the component to avoid the timestop without suitable inputs from the environment.
#### TIOTS terminology.
A TIOTS is *time additive* providing $p {\xrightarrow{d_1+d_2}} s'$ iff $p {\xrightarrow{d_1}} s$ and $s {\xrightarrow{d_2}} s'$ for some $s$. In the sequel of this paper we only consider TIOTSs that are time-additive.
We say a TIOTS is *deterministic* iff there is no ambiguous transition in the TIOTS, i.e. $s {\xrightarrow{\alpha}} s' \wedge s {\xrightarrow{\alpha}} s''$ implies $s'=s''$.
Given a TIOTS ${\mathcal{P}}$, a timed word can be derived from a finite execution of ${\mathcal{P}}$ by extracting the labels in each transition and coalescing adjacent reals. The timed words derived from such executions are called *traces* of ${\mathcal{P}}$. We use $tt, tt', tt_i$ to range over the set of traces and use $s^0 {\xRightarrow{tt}} s$ to denote a finite execution that produces trace $tt$ and leads to $s$.
Operational Specification Theory {#sec:optheory}
--------------------------------
In this section we develop a compositional specification theory for TIOTSs based on the operations of parallel composition $\parallel$, conjunction $\wedge$, disjunction $\vee$ and quotient $\%$. The operators are defined via transition rules that are a variant on synchronised product.
Parallel composition yields a TIOTS that represents the combined effect of its operands interacting with one another. The remaining operations must be explained with respect to a refinement relation, which corresponds to safe-substitutivity in our theory. A TIOTS is a refinement of another if it will work in any environment that the original worked in without introducing safety or bounded-liveness errors. Conjunction yields the coarsest TIOTS that is a refinement of its operands, while disjunction yields the finest TIOTS that is refined by both of its operands. The operators are thus equivalent to the join and meet operations on TIOTSs[^2]. Quotient is the adjoint of parallel composition, meaning that ${\mathcal{P}}_0 \% {\mathcal{P}}_1$ is the coarsest TIOTS such that $({\mathcal{P}}_0 \% {\mathcal{P}}_1) {\|}{\mathcal{P}}_1$ is a refinement of ${\mathcal{P}}_0$.
Let ${\mathcal{P}}_i= \langle I_i, O_i, S_i, s_i^0, \rightarrow_i \rangle$ for $i \in \{0,1\}$ be two TIOTSs that are both $\bot$ and $\top$-completed, satisfying (wlog) $S_0\cap S_1 = \{\bot,\top\}$. The composition of ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ under the operation $\otimes\in \{\parallel, \wedge, \vee, \%\}$, written ${\mathcal{P}}_0 \otimes {\mathcal{P}}_1$, is only defined when certain *composability* restrictions are imposed on the alphabets of the TIOTSs. ${\mathcal{P}}_0\parallel {\mathcal{P}}_1$ is only defined when the output sets of ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ are disjoint, because an output should be controlled by at most one component. Conjunction and disjunction are defined only when the TIOTSs have *identical alphabets* (i.e. $O_0 = O_1$ and $I_0 = I_1$). This restriction can be relaxed at the expense of more cumbersome notation, which is why we focus on the simpler case in this paper. For the quotient, we require that the alphabet of ${\mathcal{P}}_0$ *dominates* that of ${\mathcal{P}}_1$ (i.e. $A_1 \subseteq A_0$ and $O_1 \subseteq O_0$), in addition to ${\mathcal{P}}_1$ being a deterministic TIOTS. As quotient is a synthesis operator, it is difficult to give a definition using just *state-local* transition rules, since quotient needs global information of the transition systems. This is why we insist on ${\mathcal{P}}_1$ being deterministic[^3].
$\parallel$ $\top$ $p_0$ $\bot$
------------- -------- -------------------------- --------
$\top$ $\top$ $\top$ $\top$
$p_1$ $\top$ $p_0 \!\!\cross\!\! p_1$ $\bot$
$\bot$ $\top$ $\bot$ $\bot$
: State representations under composition operators.
$\wedge$ $\top$ $p_0$ $\bot$
---------- -------- -------------------------- --------
$\top$ $\top$ $\top$ $\top$
$p_1$ $\top$ $p_0 \!\!\cross\!\! p_1$ $p_1$
$\bot$ $\top$ $p_0$ $\bot$
: State representations under composition operators.
$\vee$ $\top$ $p_0$ $\bot$
-------- -------- -------------------------- --------
$\top$ $\top$ $p_0$ $\bot$
$p_1$ $p_1$ $p_0 \!\!\cross\!\! p_1$ $\bot$
$\bot$ $\bot$ $\bot$ $\bot$
: State representations under composition operators.
$\%$ $\top$ $p_0$ $\bot$
-------- -------- -------------------------- --------
$\top$ $\bot$ $\bot$ $\bot$
$p_1$ $\top$ $p_0 \!\!\cross\!\! p_1$ $\bot$
$\bot$ $\top$ $\top$ $\bot$
: State representations under composition operators.
\[table:composition\]
Let ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ be TIOTSs composable under $\otimes \in \{\parallel, \wedge, \vee, \%\}$. Then ${\mathcal{P}}_0\otimes {\mathcal{P}}_1 = \langle I, O, S, s^0, \rightarrow\rangle$ is the TIOTS where:
- If $\otimes = \parallel$, then $I = (I_0\cup I_1) \setminus O$ and $O=O_0\cup O_1$
- If $\otimes \in \{ \wedge,\vee\}$, then $I = I_0 = I_1$ and $O=O_0=O_1$
- If $\otimes =\%$, then $I = I_0 \cup O_1$ and $O=O_0\setminus O_1$
- $S = P_0 \times P_1 \uplus P_0 \uplus P_1 \uplus \{\top, \bot\}$
- $s^0 = s^0_0 \otimes s^0_1$
- $\rightarrow$ is the smallest relation containing $\rightarrow_0 \cup \rightarrow_1$, and satisfying the rules:
${p_0 {\xrightarrow{\alpha}}_{0} s_0'} \ \ {p_1 {\xrightarrow{\alpha}}_{1} s_1'} \over {p_0 \otimes p_1 {\xrightarrow{\alpha}} s_0' \otimes s_1'}$ ${p_0 {\xrightarrow{a}}_0 s_0'} \ \ a \notin A_1 \over {p_0 \otimes p_1} {\xrightarrow{a}} {s_0' \otimes p_1}$ ${p_1 {\xrightarrow{a}}_0 s_1'} \ \ a \notin A_0 \over {p_0 \otimes p_1} {\xrightarrow{a}} {p_0 \otimes s_1'}$
We adopt the notation of $s_0 \otimes s_1$ for states, where the associated interpretation is supplied in Table \[table:composition\]. Furthermore, given two plain states $p_i=(l_i, t_i)$ for $i \in \{0,1\}$, we define $p_0 \times p_1 = ((l_0,l_1),t_0 \uplus t_1)$.
Table \[table:composition\] tells us how states should be combined under the composition operators. From the environment’s point of view, $\top$ refines plain states, which in turn refines $\bot$. For parallel, a state is magic if one component state is magic, and a state is error if one component is error while the other is not magic. For conjunction, encountering error in one component implies the component can be discarded and the rest of the composition behaves like the other component. The conjunction table follows the intuition of the join operation on the refinement preorder. Similarly for disjunction. Quotient is the adjoint of parallel composition. If the second component state does not refine the first, the quotient will try to rescue the refinement by producing $\top$ (so that its composition with the second will refine the first). If the second component state does refine the first, the quotient will produce the least refined value so that its composition with the second will not break the refinement.
An *environment* for a TIOTS ${\mathcal{P}}$ is any TIOTS ${\mathcal{Q}}$ such that the alphabet of ${\mathcal{Q}}$ is *complementary* to that of ${\mathcal{P}}$, meaning $I_{\mathcal{P}}=O_{\mathcal{Q}}$ and $O_{\mathcal{P}}=I_{\mathcal{Q}}$. Refinement in our framework corresponds to contextual substitutability, in which the context is an arbitrary environment.
\[defn:op-refine\] Let ${\mathcal{P}}_{imp}$ and ${\mathcal{P}}_{spec}$ be TIOTSs with identical alphabets. ${\mathcal{P}}_{imp}$ *refines* ${\mathcal{P}}_{spec}$, denoted ${\mathcal{P}}_{spec} \sqsubseteq {\mathcal{P}}_{imp}$, iff for all environments ${\mathcal{Q}}$, ${\mathcal{P}}_{spec} \parallel {\mathcal{Q}}$ is $\bot$-free implies ${\mathcal{P}}_{imp} \parallel {\mathcal{Q}}$ is $\bot$-free. We say ${\mathcal{P}}_{imp}$ and ${\mathcal{P}}_{spec}$ are *substitutively equivalent*, i.e. ${\mathcal{P}}_{spec} \simeq {\mathcal{P}}_{imp}$, iff ${\mathcal{P}}_{imp} \sqsubseteq {\mathcal{P}}_{spec}$ and ${\mathcal{P}}_{spec} \sqsubseteq {\mathcal{P}}_{imp}$.
It is obvious that $\simeq$ induces an equivalance on TIOTSs and no equivalence that preserves the $\bot$ state can be weaker than $\simeq$. In the sequel we will give two concrete characterisations of $\simeq$ and show that $\simeq$ is also a congruence w.r.t. the parallel composition, conjunction, disjunction and quotient operators.
The operational definition of quotient requires that ${\mathcal{P}}_1$ is determinised, which can be accomplished by a modified subset construction procedure on $({\mathcal{P}}_1^{\bot})^{\top}$. If the current state subset $S_0$ contains $\bot$, it reduces $S_0$ to $\bot$; if $\bot \notin S_0 \neq \{\top\}$, it reduces $S_0$ by removing any potential $\top$ in $S_0$. As expected, the determinisation of ${\mathcal{P}}$, denoted ${\mathcal{P}}^D$, is substitutively equivalent to ${\mathcal{P}}$.
Any TIOTS is substitutively equivalent to a deterministic TIOTS.
Equipped with determinisation, quotient is a fully defined operator on any pair of TIOTSs. Furthermore, we can give an alternative (although substitutively equivalent) formulation of quotient as the derived operator $({\mathcal{P}}_0^{\neg} \parallel {\mathcal{P}}_1)^{\neg}$, where $\neg$ is a mirroring operation that first determinises its argument, then interchanges the input and output sets, as well as the $\top$ and $\bot$ states.
#### Example.
Figure \[fig:product\] shows the parallel composition of the job scheduler with the printer controller. In the transition from $B4$ to $A1$, the guard combines the effects of the constraints on the clocks $x$ and $y$. As $finish$ is an output of the controller, it can be fired at a time when the scheduler is not expecting it, meaning that a safety error will occur. This is indicated by the transition to $\bot$ when the guard constraint $5\leq x \leq 8$ is not satisfied.
{width="70.00000%"}
\[fig:product\]
Timed I/O Game {#sec:game}
==============
Our specification theory can be understood from a game theoretical point of view. It is an input-output game between a *component* and an *environment* that uses a *coin* to break ties. The specification of a component (in the form of a TIOA or TIOTS) is built to encode the set of strategies possible for the component in the game (just like an NFA encodes a set of words).
- Given two TIOTSs ${\mathcal{P}}$ and ${\mathcal{Q}}$ with identical alphabets, we say ${\mathcal{P}}$ is a partial unfolding [@wang12] of ${\mathcal{Q}}$ if there exists a function $f$ from $S_{{\mathcal{P}}}$ to $S_{{\mathcal{Q}}}$ s.t. 1) $f$ maps $\top$ to $\top$, $\bot$ to $\bot$, and plain states to plain states, 2) $f(s^0_{{\mathcal{P}}})= s^0_{{\mathcal{Q}}}$, and 3) $p {\xrightarrow{\alpha}}_{{\mathcal{P}}} s \implies f(p) {\xrightarrow{\alpha}}_{{\mathcal{Q}}} f(s)$.
- We say an acyclic TIOTS is a *tree* if 1) there does not exist a pair of transitions in the form of $p {\xrightarrow{a}} p''$ and $p' {\xrightarrow{d}} p''$, 2) $p {\xrightarrow{a}} p'' \wedge p' {\xrightarrow{b}} p''$ implies $p=p'$ and $a=b$ and 3) $p {\xrightarrow{d}} p'' \wedge p' {\xrightarrow{d}} p''$ implies $p=p'$.
- We say an acyclic TIOTS is a *simple path* if 1) $p {\xrightarrow{a}} s' \wedge p {\xrightarrow{\alpha}} s''$ implies $s'=s''$ and $a=\alpha$ and 2) $p {\xrightarrow{d}} s' \wedge p {\xrightarrow{d}} s''$ implies $s'=s''$.
- We say a simple path ${\mathcal{L}}$ is a *run* of ${\mathcal{P}}$ if ${\mathcal{L}}$ is a partial unfolding of ${\mathcal{P}}$.
#### Strategies.
A *strategy* ${\mathcal{G}}$ is a deterministic tree TIOTS s.t. each plain state in ${\mathcal{G}}$ is ready to accept all possible inputs by the environment, but allows a single move (delay or output) by the component, i.e. $eb_{{\mathcal{G}}}(p) = I \uplus mv_{{\mathcal{G}}}(p)$ s.t. $mv_{{\mathcal{G}}}(p) = \{a\}$ for some $a \in O$ or $\{\} \subset mv_{{\mathcal{G}}}(p) \subseteq \mathbb{R}^{>0}$, where $eb_{{\mathcal{G}}}(p)$ denotes the set of enabled timed actions in state $p$ of LTS ${\mathcal{G}}$, and $mv_{{\mathcal{G}}}(p)$ denotes the unique component move allowed by ${\mathcal{G}}$ at $p$.
A TIOTS ${\mathcal{P}}$ *contains* a strategy ${\mathcal{G}}$ if ${\mathcal{G}}$ is a partial unfolding of $({\mathcal{P}}^{\bot})^{\top}$. The set of strategies[^4] contained in ${\mathcal{P}}$ is denoted $stg({\mathcal{P}})$. Since it makes little sense to distinguish strategies that are isomorphic, we will freely use strategies to refer to their isomorphism classes and write ${\mathcal{G}}={\mathcal{G}}'$ to mean ${\mathcal{G}}$ and ${\mathcal{G}}'$ are isomorphic.
Let us give some examples in Figure \[fig:strategy-equivalence-strong\]. For the sake of simplicity we use two untimed transition systems ${\mathcal{P}}$ and ${\mathcal{Q}}$, which have identical alphabets $I=\{e,f\}$ and $O=\{a,b,c\}$, to illustrate the idea of strategies. The transition systems use solid lines while strategies use dotted lines. Plain states are unmarked while the $\top$ and $\bot$ states are marked by $\top$ and $\bot$ resp.[^5] We show four strategies of ${\mathcal{P}}$ and two strategies of ${\mathcal{Q}}$ on the right hand side of ${\mathcal{P}}$ and ${\mathcal{Q}}$ resp. in Figure \[fig:strategy-equivalence-strong\]. (They are not the complete sets of strategies for ${\mathcal{P}}$ and ${\mathcal{Q}}$.) Note that the strategies $3$ and $4$ own their existence to the $\top$ completion.
{width="80.00000%"}
\[fig:strategy-equivalence-strong\]
#### Comparing strategies.
When the game is played, the component tries to avoid reaching $\top$ while the environment tries to avoid reaching $\bot$. Different strategies in $stg({\mathcal{P}})$ vary in their effectiveness to achieve the objective. Such effectiveness can be compared if two strategies closely resemble each other: we say ${\mathcal{G}}$ and ${\mathcal{G}}'$ are *affine* if $s^0_{{\mathcal{G}}} {\xRightarrow{tt}} p$ and $s^0_{{\mathcal{G}}'} {\xRightarrow{tt}} p'$ implies $mv_{{\mathcal{G}}}(p) = mv_{{\mathcal{G}}'}(p')$. Intuitively, it means ${\mathcal{G}}$ and ${\mathcal{G}}'$ propose the same move at the ‘same’ states. For instance, the strategies $1$, $3$ and $A$ in Figure \[fig:strategy-equivalence-strong\] are pairwise affine and so are the strategies $2$, $4$ and $B$.
Given two affine strategies ${\mathcal{G}}$ and ${\mathcal{G}}'$, we say ${\mathcal{G}}$ is *more aggressive* than ${\mathcal{G}}'$, denoted ${\mathcal{G}}\preceq {\mathcal{G}}'$, if 1) $s^0_{{\mathcal{G}}'} {\xRightarrow{tt}} \bot$ implies there is a prefix $tt_0$ of $tt$ s.t. $s^0_{{\mathcal{G}}} {\xRightarrow{tt_0}} \bot$ and 2) $s^0_{{\mathcal{G}}} {\xRightarrow{tt}} \top$ implies there is a prefix $tt_0$ of $tt$ s.t. $s^0_{{\mathcal{G}}'} {\xRightarrow{tt_0}} \top$. Intuitively, it means ${\mathcal{G}}$ can reach $\bot$ faster but $\top$ slower than ${\mathcal{G}}'$. $\preceq$ forms a partial order over $stg({\mathcal{P}})$, or more generally, over any set of strategies with identical alphabets. For instance, strategy $A$ is more aggressive than $1$ and $3$, while strategy $B$ is more aggressive than $2$ and $4$.
When the game is played, the component ${\mathcal{P}}$ prefers to use the maximally aggressive strategies in $stg({\mathcal{P}})$[^6]. Thus two components that differ only in non-maximally aggressive strategies should be equated. We define the *strategy semantics* of component ${\mathcal{P}}$ to be $[{\mathcal{P}}]_s = \{{\mathcal{G}}' \, | \, \exists {\mathcal{G}}\in stg({\mathcal{P}}) : {\mathcal{G}}\preceq {\mathcal{G}}' \}$, i.e. the upward-closure of $stg({\mathcal{P}})$ w.r.t. $\preceq$.
#### Game rules.
When a component strategy ${\mathcal{G}}$ is played against an environment strategy ${\mathcal{G}}'$, at each game state (i.e. a product state $p_{{\mathcal{G}}} \times p_{{\mathcal{G}}'}$) ${\mathcal{G}}$ and ${\mathcal{G}}'$ each propose a move (i.e. $mv_{{\mathcal{G}}}(p_{{\mathcal{G}}})$ and $mv_{{\mathcal{G}}'}(p_{{\mathcal{G}}'})$). If one of them is a delay and the other is an action, the action will prevail. If both propose delay moves (i.e. $mv_{{\mathcal{G}}}(p_{{\mathcal{G}}}), mv_{{\mathcal{G}}'}(p_{{\mathcal{G}}'}) \subseteq \mathbb{R}^{\geq 0}$), the smaller one (w.r.t. set containment) will prevail.
Since a delay move proposed at a strategy state is the maximal set of possible delays enabled at that state, the next move proposed at the new state after firing the set must be an action move (due to time additivity). Thus a play cannot have two consecutive delay moves.
If, however, both propose action moves, there will be a tie, which will be resolved by tossing the coin. For uniformity’s sake, the coin can be treated as a special component. A strategy of the coin is a function $h$ from $tA^*$ to $\{0,1\}$. We denote the set of all possible coin strategies as $H$.
A play of the game can be formalised as a composition of three strategies, one each from the component, environment and coin, denoted ${\mathcal{G}}_{{\mathcal{P}}} \parallel_h {\mathcal{G}}_{{\mathcal{Q}}}$. At a current game state $p_{{\mathcal{P}}} \times p_{{\mathcal{Q}}}$, if the prevailing action is $\alpha$ and we have $p_{{\mathcal{P}}} {\xrightarrow{\alpha}} s'_{{\mathcal{P}}}$ and $p_{{\mathcal{Q}}} {\xrightarrow{\alpha}} s'_{{\mathcal{Q}}}$, then the next game state is $s_{{\mathcal{P}}} \parallel s_{{\mathcal{Q}}}$. The play will stop when it reaches either $\top$ or $\bot$. The composition will produce a simple path ${\mathcal{L}}$ that is a run of ${\mathcal{P}}\parallel {\mathcal{Q}}$. Since ${\mathcal{P}}\parallel {\mathcal{Q}}$ gives rise to a *closed system* (i.e. the input alphabet is empty), a run of ${\mathcal{P}}\parallel {\mathcal{Q}}$ is a strategy of ${\mathcal{P}}\parallel {\mathcal{Q}}$.
This is crucial since it reveals that strategy composition of ${\mathcal{P}}$ and ${\mathcal{Q}}$ is closely related to their parallel composition: $stg({\mathcal{P}}\parallel {\mathcal{Q}}) = \{{\mathcal{G}}_{{\mathcal{P}}} \parallel_h {\mathcal{G}}_{{\mathcal{Q}}} \, | \, {\mathcal{G}}_{{\mathcal{P}}} \in stg({\mathcal{P}}), {\mathcal{G}}_{{\mathcal{Q}}} \in stg({\mathcal{Q}}) $ and $h \in H\}$.
#### Parallel composition.
Strategy composition, like component (parallel) composition, can be generalised to any pair of components ${\mathcal{P}}$ and ${\mathcal{Q}}$ with *composable alphabets*. That is, $O_{{\mathcal{P}}} \cap O_{{\mathcal{Q}}} = \{\}$. For such ${\mathcal{P}}$ and ${\mathcal{Q}}$, ${\mathcal{G}}_{{\mathcal{P}}} \parallel_h {\mathcal{G}}_{{\mathcal{Q}}}$ gives rise to a tree rather than simple path TIOTS. That is, at each game state $p_{{\mathcal{P}}} \times p_{{\mathcal{Q}}}$, besides firing the prevailing $\alpha \in tO_{{\mathcal{P}}} \cup tO_{{\mathcal{Q}}}$, we need also to fire 1) all the synchronised inputs, i.e. $e \in I_{{\mathcal{P}}} \cap I_{{\mathcal{Q}}}$, and reach the new game state $s_{{\mathcal{P}}} \parallel s_{{\mathcal{Q}}}$ (assuming $p_{{\mathcal{P}}} {\xrightarrow{e}} s_{{\mathcal{P}}}$ and $p_{{\mathcal{Q}}} {\xrightarrow{e}} s_{{\mathcal{Q}}}$) and 2) all the independent inputs, i.e. $e \in (I_{{\mathcal{P}}} \cup I_{{\mathcal{Q}}}) \setminus (A_{{\mathcal{P}}} \cap A_{{\mathcal{Q}}})$, and reach the new game state $s_{{\mathcal{P}}} \times p_{{\mathcal{Q}}}$ or $p_{{\mathcal{P}}} \times s_{{\mathcal{Q}}}$. It is easy to verify that ${\mathcal{G}}_{{\mathcal{P}}} \parallel_h {\mathcal{G}}_{{\mathcal{Q}}}$ is a strategy of ${\mathcal{P}}\parallel {\mathcal{Q}}$.
#### Conjunction/disjunction.
Besides strategy composition, *strategy conjunction* ($\&$) and *strategy disjunction* ($+$) are also definable. They are binary operators defined only on pairs of affine strategies. We define ${\mathcal{G}}\& {\mathcal{G}}' = {\mathcal{G}}\wedge {\mathcal{G}}'$ and ${\mathcal{G}}+ {\mathcal{G}}' = {\mathcal{G}}\vee {\mathcal{G}}'$. Note that if ${\mathcal{G}}$ and ${\mathcal{G}}'$ are not affine, ${\mathcal{G}}\wedge {\mathcal{G}}'$ and ${\mathcal{G}}\vee {\mathcal{G}}'$ do not necessarily produce a strategy. For instance the disjunction of the strategies $1$ and $2$ in Figure \[fig:strategy-equivalence-strong\] will produce a transition system that stops to output after the $a$ transition.
#### Refinement.
Strategy semantics induce an equivalence on TIOTSs. That is, ${\mathcal{P}}$ and ${\mathcal{Q}}$ are *strategy equivalent* iff $[{\mathcal{P}}]_s = [{\mathcal{Q}}]_s$. However, strategy equivalence is too fine for the purpose of *substitutive refinement* (cf Definition \[defn:op-refine\]). For instance, transition systems ${\mathcal{P}}$ and ${\mathcal{Q}}$ in Figure \[fig:strategy-equivalence-strong\] are substitutively equivalent, but are not strategy equivalent, because $1$, $2$, $3$ and $4$ are strategies of ${\mathcal{Q}}$ (due to upward-closure w.r.t. $\preceq$), but $A$ and $B$ are not strategies of ${\mathcal{P}}$.
However, we demonstrate that *substitutive equivalence is reducible to strategy equivalence* providing we perform *disjunction closure* on strategies.
Given a pair of affine component strategies ${\mathcal{G}}_0$ and ${\mathcal{G}}_1$, ${\mathcal{G}}_0 \parallel_h {\mathcal{G}}$ and ${\mathcal{G}}_1 \parallel_h {\mathcal{G}}$ are $\bot$-free for some environment strategy ${\mathcal{G}}$ and $h \in H$ iff ${\mathcal{G}}_0 + {\mathcal{G}}_1 \parallel_h {\mathcal{G}}$ is $\bot$-free.
We say $\Pi^+$ is a *disjunction closure* of $\Pi$ iff it is the least superset of $\Pi$ s.t. ${\mathcal{G}}+ {\mathcal{G}}' \in \Pi^+$ for all pairs of affine strategies ${\mathcal{G}}, {\mathcal{G}}' \in \Pi^+$. It is easy to see the disjunction closure operation preserves the upward-closedness of strategy sets.
Given TIOTSs ${\mathcal{P}}$ and ${\mathcal{Q}}$, ${\mathcal{P}}\sqsubseteq {\mathcal{Q}}$ iff $[{\mathcal{Q}}]_s^+ \subseteq [{\mathcal{P}}]_s^+$.
For instance, the disjunction of strategies $1$ and $3$ produces $A$, while the disjunction of strategies $2$ and $4$ produces $B$. Thus $[{\mathcal{P}}]_s^+ = [{\mathcal{Q}}]_s^+$,
#### Relating operational composition to strategies.
The operations of parallel composition, conjunction and disjunction defined on the operational models of TIOTSs (Section \[sec:optheory\]) can be characterised by simple operations on strategies in the game-based setting.
For $\parallel$-composable TIOTSs ${\mathcal{P}}$ and ${\mathcal{Q}}$, $[{\mathcal{P}}\parallel {\mathcal{Q}}]_s^+ = \{ {\mathcal{G}}_{{\mathcal{P}}\parallel {\mathcal{Q}}} \, | \, \exists {\mathcal{G}}_{{\mathcal{P}}} \in [{\mathcal{P}}]_s^+, {\mathcal{G}}_{{\mathcal{Q}}} \in [{\mathcal{Q}}]_s^+, h \in H : {\mathcal{G}}_{{\mathcal{P}}} \parallel_h {\mathcal{G}}_{{\mathcal{Q}}} \preceq {\mathcal{G}}_{{\mathcal{P}}\parallel {\mathcal{Q}}} \}$.
For $\vee$-composable TIOTSs ${\mathcal{P}}$ and ${\mathcal{Q}}$, $[{\mathcal{P}}\vee {\mathcal{Q}}]_s^+= ([{\mathcal{P}}]_s^+ \cup [{\mathcal{Q}}]_s^+)^+$.
For $\wedge$-composable TIOTSs ${\mathcal{P}}$ and ${\mathcal{Q}}$, $[{\mathcal{P}}\wedge {\mathcal{Q}}]_s^+= [{\mathcal{P}}]_s^+ \cap [{\mathcal{Q}}]_s^+$.
For $\%$-composable TIOTSs ${\mathcal{P}}$ and ${\mathcal{Q}}$, $[{\mathcal{P}}\% {\mathcal{Q}}]_s^+= \{{\mathcal{G}}_{{\mathcal{P}}\% {\mathcal{Q}}} \, | \, \forall {\mathcal{G}}_{{\mathcal{Q}}} \in [{\mathcal{Q}}]_s^+, h \in H: {\mathcal{G}}_{{\mathcal{P}}\% {\mathcal{Q}}} \parallel_h {\mathcal{G}}_{{\mathcal{Q}}} \in [{\mathcal{P}}]_s^+ \}$.
Thus conjunction and disjunction are the join and meet operations and quotient produces the coarsest TIOTS s.t. $({\mathcal{P}}_0 \% {\mathcal{P}}_1) {\|}{\mathcal{P}}_1$ is a refinement of ${\mathcal{P}}_0$.
For any TIOTS ${\mathcal{P}}$, $[{\mathcal{P}}^{\neg}]_s^+= \{{\mathcal{G}}_{{\mathcal{P}}^{\neg}} \, | \, \forall {\mathcal{G}}_{{\mathcal{P}}} \in [{\mathcal{P}}]_s^+, h \in H: {\mathcal{G}}_{{\mathcal{P}}^{\neg}} \parallel_h {\mathcal{G}}_{{\mathcal{P}}} $ is $\bot$-free$ \}$.
$\simeq$ is a congruence w.r.t. $\parallel$, $\vee$, $\wedge$ and $\%$ subject to composability.
#### Summary.
Strategy semantics has given us a weakest $\bot$-preserving congruence (i.e. $[{\mathcal{P}}]_s^+$) for timed specification theories based on operators for (parallel) composition, conjunction, disjunction and quotient. Strategy semantics captures nicely the game-theoretical nature as well as the operational intuition of the specification theories. However, in a more declarative manner, the equivalence can also be characterised by timed traces, as we see in the next section.
Declarative Specification Theory {#sec:ts}
================================
In this section, we develop a compositional specification theory based on timed traces. We introduce the concept of a timed-trace structure, which is an abstract representation for a timed component. The timed-trace structure contains essential information about the component, for checking whether it can be substituted with another in a safety and liveness preserving manner.
Given any TIOTS ${\mathcal{P}}= \langle I, O, S, s^0, \rightarrow \rangle$, we can extract three sets of traces from $({\mathcal{P}}^{\bot})^{\top}$: $TP$ (plain traces) is a set of timed traces leading to plain states, $TE$ (error traces) a set of timed traces leading to $\bot$ and $TM$ (magic traces) a set of timed traces leading to $\top$. The three sets contain sufficient but not necessary information for our substitutive refinement, which is designed to preserve $\bot$ rather than $\top$. For instance, adding any trace $tt \in TE$ to $TP$ should not change the semantics of the component; similarly it is true for removing any trace $tt \in TP$ from $TM$. Based on a slight abstraction of the three sets we can define a *triple-trace structure* as the semantics of ${\mathcal{P}}$.
${{\mathcal{TT}({\mathcal{P}})}} := (I, O, TT, TR, TE)$, where $TT := TE \cup TP \cup TM$ is the set of all traces and $TR := TE \cup TP$ the set of realisable traces.
Obviously, $TE$ is extension-closed. $TT$ is non-empty and prefix-closed. $TR$ is prefix-closed and *fully branching*[^7] w.r.t. $TT$ (i.e. $tt \cat \langle \alpha \rangle \in TT$ for all $tt \in TR$ and $\alpha \in tA$). $TT \setminus TR$ is time-extension closed (i.e. $tt \in X \implies tt \cat \langle d\rangle \in X$) and any pair of traces from $TT \setminus TR$ that are related by extension are related by time-extension.
From hereon let ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ be two TIOTSs with triple trace structures ${{\mathcal{TT}({\mathcal{P}}_i)}} := (I_i, O_i, TT_i, TR_i, TE_i)$ for $i \in \{0,1\}$. Define $\bar{i}=1-i$.
The substitutive refinement relation $\sqsubseteq$ in Section \[sec:optheory\] can equally be characterised by means of trace containment. Consequently, ${{\mathcal{TT}({\mathcal{P}}_0)}}$ can be regarded as providing an alternative encoding of the set $[{\mathcal{P}}_0]_s^+$ of strategies.
${\mathcal{P}}_0 \sqsubseteq {\mathcal{P}}_1$ iff $TT_1 \subseteq TT_0$, $TR_1 \subseteq TR_0$ and $TE_1 \subseteq TE_0$.
We are now ready to define the timed-trace structure semantics for the operators of our specification theory. Intuitively, the timed-trace semantics mimic the synchronised product of the operational definitions in Section \[sec:optheory\]. An important fact utilised in formulating these operations on traces is that for any trace $tt \in tA^*$ and TIOTS ${\mathcal{P}}$, either $tt$ is a trace of ${\mathcal{P}}$ or there is some prefix $tt_0$ of $tt$ s.t. $tt_0$ is an error or magic trace of ${\mathcal{P}}$.
#### Parallel composition.
The idea behind parallel composition is that the projection of any trace in the composition onto the alphabet of one of the components should be a trace of that component.
If ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ are $\parallel$-composable, then ${{\mathcal{TT}({\mathcal{P}}_0 \parallel {\mathcal{P}}_1)}} = (I, O, TT, $ $TR, TE)$ where $I=(I_0\cup I_1) \setminus O$, $O=O_0\cup O_1$ and the trace sets are given by:
- $TE = \{ tt | tt \upharpoonright tA_i \in TE_i \wedge tt \upharpoonright tA_{\bar{i}} \in TR_{\bar{i}} \} \cdot tA^*$
- $TR = TE \uplus \{ tt | tt \upharpoonright tA_i \in (TR_i \setminus TE_i) \wedge tt \upharpoonright tA_{\bar{i}} \in (TR_{\bar{i}} \setminus TE_{\bar{i}}) \} $
- $TT = TR \uplus \{ tt | tt \upharpoonright tA_i \in (TT_i \setminus TR_i) \wedge tt_0 < tt \upharpoonright tA_{\bar{i}} \implies tt_0 \in (TR_{\bar{i}} \setminus TE_{\bar{i}}) \} \cdot \mathbb{R}^{\geq 0}$.
The above says $tt$ is an error trace if the projection of $tt$ on one component is an error trace while the projection of $tt$ on the other component is not a magic trace. $tt$ is a realisable trace if $tt$ is either an error trace or a plain trace. $tt$ is a plain trace if the projection of $tt$ on both components are plain traces. Finally, $tt$ is a magic trace if its projection on one component is a magic trace, while the projection of all strict prefixes of $tt$ on the other component is a plain trace.
#### Disjunction.
From any composite state in the disjunction of two components, the composition should only be willing to accept inputs that are accepted by both components, but should accept the union of outputs. After witnessing an output enabled by only one of the components, the disjunction should behave like that component. Because of the way that $\bot$ and $\top$ work in Table \[table:composition\], this loosely corresponds to taking the union of the traces from the respective components.
If ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ are $\vee$-composable, then ${{\mathcal{TT}({\mathcal{P}}_0 \vee {\mathcal{P}}_1)}} = (I, O, TR_0 \cup TR_1 \cup TM, TR_0 \cup TR_1, TE_0 \cup TE_1)$, where $I = I_0 = I_1$, $O=O_0 = O_1$ and $TM = \{ tt | tt \in (TT_i \setminus TR_i) \wedge \exists tt_0 \leq tt : tt_0 \in (TT_{\bar{i}} \setminus TR_{\bar{i}}) \} \cdot \mathbb{R}^{\geq 0}$.
Essentially, $tt$ is a magic trace if it is a magic trace on one component while one of its prefixes is a magic trace on the other component. The realisable and error traces are simply the union of the corresponding traces on ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$.
#### Conjunction.
Similarly to disjunction, from any composite state in the conjunction of two components, the composition should only be willing to accept outputs that are accepted by both components, and should accept the union of inputs, until a stage when one of the component’s input assumptions has been violated, after which it should behave like the other component. Because of the way that both $\bot$ and $\top$ work in Table \[table:composition\], this essentially corresponds to taking the intersection of the traces from the respective components.
If ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$ are $\wedge$-composable, then ${{\mathcal{TT}({\mathcal{P}}_0 \wedge {\mathcal{P}}_1)}} = (I, O, (TR_0 \cap TR_1) \cup TM,$ $TR_0 \cap TR_1, TE_0 \cap TE_1)$, where $I = I_0 = I_1$, $O=O_0 = O_1$ and $TM = \{ tt | tt \in (TT_i \setminus TR_i) \wedge tt_0 < tt \implies tt_0 \in TR_{\bar{i}} \} \cdot \mathbb{R}^{\geq 0}$.
A trace $tt$ is a magic trace if it is a magic trace on one of the components, and all strict prefixes of the trace are realisable by the other component. The realisable and error traces are simply the intersection of the corresponding traces on ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$.
#### Quotient.
Quotient ensures its composition with the second component is a refinement of the first. Given the synchronised running of ${\mathcal{P}}_0$ and ${\mathcal{P}}_1$, if ${\mathcal{P}}_0$ is in a more refined state than ${\mathcal{P}}_1$, the quotient will try to rescue the refinement by taking $\top$ as its state (so that its composition with ${\mathcal{P}}_1$’s state will refine ${\mathcal{P}}_0$’s). If ${\mathcal{P}}_0$ is in a less or equally refined state than ${\mathcal{P}}_1$’s, the quotient will take the worst possible state without breaking the refinement. \[sec:ts-quotient\]
If ${\mathcal{P}}_0$ dominates ${\mathcal{P}}_1$, then ${{\mathcal{TT}({)}}{\mathcal{P}}_0 \% {\mathcal{P}}_1} = (I, O, TT, TR, TE)$, where $I= I_0 \cup O_1$, $O=O_0\setminus O_1$, and the trace sets satisfy:
- $TE = \{ tt | (tt \in TE_0 \wedge tt_0 < tt \implies tt_0 \upharpoonright tA_1 \notin TE_1) \vee
(tt \upharpoonright tA_1 \in (TT_1 \setminus TR_1) \wedge tt_0 < tt \implies tt_0 \notin TT_0 \setminus TR_0) \} \cdot tA^*$
- $TR = TE \uplus \{ tt | tt \in (TR_0 \setminus TE_0) \wedge tt \upharpoonright tA_1 \in (TR_1 \setminus TE_1) \} $
- $TT = TR \uplus \{ tt | (tt \in (TT_0 \setminus TR_0) \wedge tt_0 \leq tt \implies tt_0 \upharpoonright tA_1 \in TR_1) \vee
(tt \upharpoonright tA_1 \in TE_1 \wedge tt_0 \leq tt \implies tt_0 \notin TE_0) \}$.
The above says $tt$ is an error trace if either 1) $tt$ is an error trace in ${\mathcal{P}}_0$, but the projection of any strict prefix of $tt$ on ${\mathcal{P}}_1$ is not an error trace, or 2) the projection of $tt$ on ${\mathcal{P}}_1$ is a magic trace, but no strict prefix of $tt$ is a magic trace in ${\mathcal{P}}_0$. $tt$ is a magic trace if either 1) $tt$ is a magic trace in ${\mathcal{P}}_0$, but the projection of any prefix of $tt$ is not a magic trace in ${\mathcal{P}}_1$, or 2) the projection of $tt$ on ${\mathcal{P}}_1$ is an error trace, but no prefix of $tt$ is an error trace in ${\mathcal{P}}_0$.
Mirroring of triple trace structures is straightforward: ${{\mathcal{TT}({\mathcal{P}}_0)}}^\neg = (O_0, I_0, $ $TT_0,$ $ TT_0 \setminus TE_0, TT_0 \setminus TR_0)$. This is because dealing with traces means we have implicit determinism, so we can skip the determinisation step. Consequently, quotient can also be defined as the derived operator $({{\mathcal{TT}({\mathcal{P}}_0)}}^\neg \parallel {{\mathcal{TT}({\mathcal{P}}_1)}})^\neg$.
Comparison with Related Works {#sec:comp}
=============================
Based on linear-time, our timed theory owes much to the pioneering work of trace theories in asynchronous circuit verification, such as Dill’s trace theory [@dill-trace-theory]. Our mirror operator is essentially a timed extension of the mirror operator from asynchronous circuit verification. The definition of quotient based on mirroring (for the untimed case) was first presented by Verhoeff as his Factorisation Theorem [@verhoeff-thesis].
Our work is also deeply influenced by the work of [@henzinger-timedia] on timed games, with some modifications. Firstly, a TIOTS is regarded as a set of component strategies, rather than a timed game graph. We adopt most of the game rules in [@henzinger-timedia], except that, due to our requirement that proposed delay moves are maximal delays allowed by a strategy, a play cannot have consecutive delay moves. This enables us to avoid the complexity of time-blocking strategies and blame assignment, but does not ensure non-Zenoness[^8]. Secondly, we do not use timestop/semi-timestop to model time errors (i.e. bounded-liveness errors). Rather, we introduce the explicit inconsistent state $\bot$ to model both time and immediate (i.e. safety) errors. Timestop is used to model the magic state, which can simplify the definition of parallel, conjunction and quotient and enables us to avoid the complexity of having two transition relations and well-formedness of timed interfaces. Last but not least, our work is related to [@larsen-timedio], as both devise a complete timed specification theory. The major differences lie in the use of timed alternating simulation as refinement in [@larsen-timedio], while ours is linear-time. An advantage of our work is that refinement is the weakest congruence preserving inconsistency, while beneficial in [@larsen-timedio] is the algorithmic efficiency of branching-time simulation checking. Moreover, [@larsen-timedio] has fully implemented the timed-game algorithms.
We briefly mention other related works, which include timed modal transition systems [@Bert09; @Cerans93], the timed I/O model [@Kaynar] and embedded systems [@Thiele06; @Lee07].
Conclusions {#sec:concl}
===========
We have formulated a rich compositional specification theory for components with real-time constraints based on a linear-time notion of substitutive refinement. The operators of hiding and renaming can also be defined, according to our past experiences [@WangKwiat07]. We believe that our theory can be reformulated as a timed extension of Dill’s trace theory [@dill-trace-theory]. Future work will include an investigation of realisability and assume-guarantee reasoning.
#### Acknowledgments.
The authors are supported by EU FP7 project CONNECT and ERC Advanced Grant VERIWARE.
[^1]: For instance, a location with $true$ as co-invariant and $false$ as invariant is mapped to $\top$, while a location with $true$ as invariant and $false$ as co-invariant is mapped to $\bot$. A location with $false$ for both invariant and co-invariant is mapped to $\top$ since invariants have priority over co-invariants according to our semantics; whereas a location with $x \leq 0$ as invariant and $true$ as co-invariant is mapped to a plain state.
[^2]: As we write $A\sqsubseteq B$ to mean $A$ is refined by $B$, our operators $\wedge$ and $\vee$ are reversed in comparison to the standard symbols for meet and join.
[^3]: Technically speaking, the problem lies in that state quotient operator is right-distributive but not left-distributive over state disjunction (cf Table \[table:composition\]).
[^4]: In this paper we use a set of strategies (say $\Pi$) to mean a set of strategies with identical alphabets
[^5]: To simplify drawing, multiple copies of $\top$ and $\bot$ are allowed but the self-loops on them are omitted.
[^6]: This is because our semantics is designed to preserve $\bot$ rather than $\top$.
[^7]: This is due to $\top/\bot$-completion.
[^8]: Zeno behaviours (infinite action moves within finite time) in a play are not regarded as abnormal behaviours in our semantics.
|
---
abstract: 'For a regular chain $R$, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of $R$, that is, the set $\overline{W(R)} \setminus W(R)$. Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of $R$. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in higher dimensions. We provide experimental results illustrating the benefits of our algorithms.'
---
=1
[ **An Algorithm for Computing the Limit Points of the Quasi-component of a Regular Chain** ]{}\
[ Parisa Alvandi, Changbo Chen, Marc Moreno Maza ]{}\
ORCCA, University of Western Ontario (UWO)\
London, Ontario, Canada\
[palvandi@uwo.ca, changbo.chen@gmail.com, moreno@csd.uwo.ca]{}\
Recent advances on the theory of regular chains In this talk, two recent advances on the theory of regular chains will be discussed. We shall start with a theoretical question, summarized below and which is one of the facets of the so-called Ritt problem. A software presentation of the new support for Maple’s solve command with semi-algebraic systems will conclude the talk.
Given a regular chain R, we propose an algorithm which computes the non-trivial limit points of the quasi-component Q of R, that is, the points that belong to the Zariski closure of Q but not to Q. Our procedure relies on Puiseux series expansions and avoids the computation of a system of generators of the saturated ideal of R. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in higher dimensions. This is a joint work with Parisa Alvandi and Marc Moreno Maza.
Introduction
============
The theory of regular chains, since its introduction by J.F. Ritt [@Ritt32], has been applied successfully in many areas including parametric algebraic systems [@ChGa91a], differential systems [@ChGa93; @BLOP95; @Hubert00], difference systems [@GaoHoevenLuoYuan2009], intersection multiplicity [@DBLP:conf/casc/MarcusMV12], unmixed decompositions [@Kalk98] and primary decomposition [@ShiYo96] of polynomial ideals, cylindrical algebraic decomposition [@CMXY09], parametric [@yhx01] and non-parametric [@DBLP:journals/jsc/ChenDMMXX13] semi-algebraic systems. Today, regular chains are at the core of algorithms for triangular decomposition of polynomial systems, which are available in several software packages [@LeMoXi04; @DingkangWang; @EpsilonWang]. Moreover, these algorithms provide back-engines for computer algebra system front-end solvers, such as [Maple]{}’s [solve]{} command.
One of the algorithmic strengths of the theory of regular chains is its [*regularity test*]{} procedure. Given a polynomial $p$ and a regular chain $R$, both in a multivariate polynomial ring ${\K}[X_1, \ldots, X_n]$ over a field , this procedure computes regular chains $R_1, \ldots, R_e$ such that $R_1, \ldots, R_e$ is a decomposition of $R$ in some technical sense [^1] and for each $1 \leq i \leq e$ the polynomial $p$ is either null or regular modulo the saturated ideal of $R_i$. Thanks to the D5 Principle [@D5], this regularity test avoids factorization into irreducible polynomials and involves only polynomial GCD and resultant computations.
One of the technical difficulties of this theory, however, is the fact that regular chains do not fit well in the “usual algebraic-geometric dictionary” (Chapter 4, [@DCJLDOS97]). Indeed, the “good” zero set encoded by a regular chain $R$ is a constructible set $W(R)$, called the [*quasi-component*]{} of $R$, which does not correspond exactly to the “good” ideal encoded by $R$, namely , the [*saturated ideal*]{} of $R$. In fact, the affine variety defined by equals $\overline{W(R)}$, that is, the Zariski closure of $W(R)$.
For this reason, a decomposition algorithm, such as the one of M. Kalkbrener [@Kalk98] (which, for an input polynomial ideal ${\cal I}$ computes regular chains $R_1, \ldots, R_e$ such that $\sqrt{{\cal I}}$ equals the intersection of the radicals of the saturated ideals of $R_1, \ldots, R_e$) can not be seen as a decomposition algorithm for the variety $V({\cal I})$. Indeed, the output of Kalkbrener’s algorithm yields $V({\cal I}) = \overline{W(R_1)} \, \cup \, \cdots \, \cup \, \overline{W(R_e)}$ while a decomposition of the form $V({\cal I}) = {W(R_1)} \, \cup \, \cdots \, \cup \, {W(R_f)}$ would be more explicit.
Kalkbrener’s decompositions, and in fact all decompositions of differential ideals [@ChGa93; @BLOP95; @Hubert00] raise another notorious issue: the [*Ritt problem*]{}, stated as follows. Given two regular chains (algebraic or differential) $R$ and $S$, check whether the inclusion of saturated ideals ${\mbox{{\rm sat}$(R)$}} \subseteq {\mbox{{\rm sat}$(S)$}}$ holds or not. In the algebraic case, this inclusion can be tested by computing a set of generators of ${\mbox{{\rm sat}$(R)$}}$ , using Gröbner bases. In practice, this solution is too expensive for the purpose of removing redundant components in Kalkbrener’s decompositions and only some criteria are applied [@DBLP:journals/jsc/LemaireMPX11]. In the differential case , there has not even an algorithmic solution.
In the algebraic case, both issues would be resolved if one would have a practically efficient procedure with the following specification: for the regular chain $R$ compute regular chains $R_1, \ldots, R_e$ such that we have $\overline{W(R)} = W(R_1) \, \cup \, \cdots \, \cup \, W(R_e)$. If in addition, such procedure does not require a system of generators of ${\mbox{{\rm sat}$(R)$}}$, this might suggest a solution in the differential version of the Ritt problem.
In this paper, we propose a solution to this algorithmic quest, in the algebraic case. To be precise, our procedure computes the [*non-trivial limit points*]{} of the quasi-component $W(R)$, that is, the set ${\mbox{${\lim}(W(R))$}} := \overline{W(R)} \setminus W(R)$. This turns out to be $\overline{W(R)} \, \cap \, V(h_R)$, where $V(h_R)$ is the hypersurface defined by the product of the initials of $R$. We focus on the case where the saturated ideal of $R$ has dimension one. In Section \[sec:discussion\], we sketch a solution in higher dimension.
When the regular chain $R$ consists of a single polynomial $r$, primitive w.r.t. its main variable, one can easily check that ${\mbox{${\lim}(W(R))$}} = V(r, h_R)$ holds. Unfortunately, there is no generalization of this result when $R$ consists of several polynomials, unless $R$ enjoys remarkable properties, such as being a [*primitive regular chain*]{} [@DBLP:journals/jsc/LemaireMPX11]. To overcome this difficulty, it becomes necessary to view $R$ as a “parametric representation” of the quasi-component $W(R)$. In this setting, the points of ${\mbox{${\lim}(W(R))$}}$ can be computed as limits (in the usual sense of the Euclidean topology [^2]) of sequences of points along “branches” (in the sense of the theory of algebraic curves) of $W(R)$ . It turns out that these limits can be obtained as constant terms of convergent Puiseux series defining the “branches” of $W(R)$ in the neighborhood of the points of interest.
Here comes the main technical difficulty of this approach. When computing a particular point of ${\mbox{${\lim}(W(R))$}}$, one needs to follow one branch per defining equation of $R$. Following a branch means computing a truncated Puiseux expansion about a point. Since the equation of $R$ defining a given variable, say $X_j$, depends on the equations of $R$ defining the variables $X_{j-1}, X_{j-2}, \ldots$, the truncated Puiseux expansion for $X_j$ is defined by an equation whose coefficients involve the truncated Puiseux expansions for $X_{j-1}, X_{j-2}, \ldots$.
From Sections \[sec:theory\] to \[sec:principle\], we show that this principle indeed computes the desired limit points. In particular, we introduce the notion of a [*system of Puiseux parametrizations of a regular chain*]{}, see Section \[sec:theory\]. This allows to state in Theorem \[thrm:PuiseuxParamRC\] a concise formula for ${\mbox{${\lim}(W(R))$}}$ in terms of this latter notion. Then, we estimate to which accuracy one needs to effectively compute such a system of Puiseux parametrizations in order to deduce ${\mbox{${\lim}(W(R))$}}$, see Theorem \[Theorem:bound-general\] in Section \[sec:bounds\].
In Section \[section\], we report on a preliminary implementation of the algorithms presented in this paper. We evaluate our code by applying it to the question of removing redundant components in Kalkbrener’s decompositions and observe the benefits of this strategy.
In order to facilitate the presentation of those technical materials, we dedicate Section \[sec:limitpoints\] to the case of regular chains in $3$ variables. Section \[sec:preliminaries\] briefly reviews notions from the theories of regular chains and algebraic curves. We conclude this introduction with a detailed example.
Consider the regular chain $R=\{r_1,r_2\}\subset \K[X_1,X_2,X_3]$ with $r_1=X_1X_2^2+X_2+1,r_2=(X_1+2)X_1X_3^2+(X_2+1)(X_3+1)$. Then, we have $h_R=X_1(X_1+2)$. To determine , we need to compute Puiseux series expansions of $r_1$ about $X_1=0$ and $X_1=-2$. We start with $X_1=0$. The two Puiseux expansions of $r_1$ about $X_1=0$ are: $$\begin{array}{c}[X_1=T,X_2={\frac {-{T}^{2}-T}{T}}+O(T^2)],$$\
$$[X_1=T,X_2={\frac{-1+{T}^{2}+T}{T}}+O(T^2)].\end{array}$$ The second expansion does not result in a new limit point. After, substituting the first expansion into $r_2$, we have: $$\begin{array}{ll}\hspace*{-0.2cm}r_2'&= r_2(X_1=T,X_2={\frac {-{T}^{2}-T}{T}}+O(T^2),X_3)\\&= T\left( \left( T+2 \right) {X_3}^{2}+ \left( O(T^3) {-5\,{T}^{2}-2\,{T}-1}+1 \right) \left( X_3+1 \right) \right). \end{array}$$ Now, we compute Puiseux series expansions of $r_2'$ which are $$\begin{array}{c}[T=T,X_3=1+T+O(T^2)],$$\
$$[T=T, X_3=-1/2-1/4\,T+O(T^2)].\end{array}$$ So the regular chains $\{X_1,X_2+1,X_3-1\}$ and $\{X_1,X_2+1,X_3+1/2\}$ give the limit points of $W(R)$ about $X_1=0$.
Next, we consider $X_1=-2$. We compute Puiseux series expansions of $r_1$ about the point $X_1=-2$. We have: $$\begin{array}{c}[X_1=T-2,X_2=1+1/3\,T+O(T^2)],$$\
$$[X_1=T-2,X_2=-1/2-1/12\,T+O(T^2)].\end{array}$$ After substitution into $r_2$, we obtain: $$\begin{array}{c}
\begin{array}{ll}r_{12}' &= r_2(X_1=T-2,X_2=1+1/3\,T+O(T^2),X_3) \\&= \left( T-2 \right) T{X_3}^{2}+ \left( 2+1/3\,T+O(T^2) \right) \left( X_3+1 \right)\end{array} \\
\begin{array}{ll}r_{22}'&= r_2([X_1=T-2,X_2=-1/2-1/ 12\,T+O(T^2)) \\ &= \left( T-2 \right) T{X_3}^{2}+ \left( 1/2-1/12\,T+O(T^2) \right) \left( X_3+1 \right).\end{array}
\end{array}$$ So those Puiseux expansions of $r_{12}'$ and $r_{22}'$ about $T=0$ which result in a limit point are as follows:
- for $r_{12}'$: $[T=T,X_3= \frac{{T}^{2}-T}{{T}}+O(T^2)]$
- for $r_{22}'$: $ [T=T,X_3= \frac{4
\,{T}^{2}-T }{T}+O(T^2)]$
Thus, the limit points of $R$ about the point $X_1=-2$ can be represented by the regular chains $\{ X_1+2,X_2-1,X_3+1\}$ and $\{X_1+2,X_2+1/2,X_3+1\}$.
One can check that a triangular decomposition of the system $R \, \cup \, \{X_1\}$ is $\{X_2+1,X_1\}$ and, thus, does not yield ${\mbox{${\lim}(W(R))$}} \, \cap \, V(X_1)$, but in fact a superset of it.
Preliminaries {#sec:preliminaries}
=============
This section is a brief review of various notions from the theories of regular chains, algebraic curves and topology. For these latter subjects, our references are the textbooks of R.J. Walker [@RW78], G. Fischer [@GF01] and J. R. Munkres [@Mun00]. The notations and hypotheses introduced in this section are used throughout the sequel of the paper.
[**Multivariate polynomials.**]{} Let ${\K}$ be a field which is algebraically closed. Let $X_1 < \cdots < X_s$ be $s \geq 1$ ordered variables. We denote by ${\K}[X_1, \ldots, X_s]$ the ring of polynomials in the variables $X_1, \ldots, X_s$ and with coefficients in ${\K}$. For a non-constant polynomial $p\in {\K}[X_1, \ldots, X_s]$, the greatest variable in $p$ is called [*main variable*]{} of $p$, denoted by ${\mbox{{\rm mvar}$(p)$}}$, and the leading coefficient of $p$ w.r.t. ${\mbox{{\rm mvar}$(p)$}}$ is called [*initial*]{} of $p$, denoted by ${\mbox{{\rm init}$(p)$}}$.
[**Zariski topology.**]{} We denote by ${\A}^s$ the [*affine $s$-space*]{} over ${\K}$. An [*affine variety*]{} of ${\A}^s$ is the set of common zeroes of a collection $F \subseteq {\K}[X_1, \ldots, X_s]$ of polynomials. The [*Zariski topology*]{} on ${\A}^s$ is the topology whose closed sets are the affine varieties of ${\A}^s$. The [*Zariski closure*]{} of a subset $W \subseteq {\A}^s$ is the intersection of all affine varieties containing $W$. This is also the set of common zeroes of the polynomials in ${\K}[X_1, \ldots, X_s]$ vanishing at any point of $W$.
[**Relation between Zariski topology and the Euclidean topology.**]{} When ${\K} = {\C}$, the affine space ${\A}^s$ is endowed with both Zariski topology and the Euclidean topology. The basic open sets of the Euclidean topology are the balls while the basic open sets of Zariski topology are the complements of hypersurfaces. A Zariski closed (resp. open) set is closed (resp. open) in the Euclidean topology on ${\A}^s$. The following properties emphasize the fact that Zariski topology is coarser than the Euclidean topology: every nonempty Euclidean open set is Zariski dense and every nonempty Zariski open set is dense in the Euclidean topology on ${\A}^s$. However, the closures of a constructible set in Zariski topology and the Euclidean topology are equal. More formally, we have the following (Corollary 1 in I.10 of [@MUM99]) key result. While Zariski topology is coarser than the Euclidean topology, we have the following (Corollary 1 in I.10 of [@MUM99]) key result. Let $V \subseteq {\A}^s$ be an irreducible affine variety and $U \subseteq V$ be open in the Zariski topology induced on $V$. Then the closure of $U$ in Zariski topology and the closure of $U$ in the Euclidean topology are both equal to $V$.
[**Limit points.**]{} Let $(X, {\tau})$ be a topological space. A point $p \in X$ is a [*limit*]{} of a sequence $(x_n, n\in {\N})$ of points of $X$ if, for every neighborhood $U$ of $p$, there exists an $N$ such that, for every $n \geq N$, we have $x_n \in U$; when this holds we write $\lim_{n \rightarrow \infty} \, x_n = p$. If $X$ is a Hausdorff space then limits of sequences are unique, when they exist. Let $S \subseteq X$ be a subset. A point $p \in X$ is a [*limit point*]{} of $S$ if every neighborhood of $p$ contains at least one point of $S$ different from $p$ itself. Equivalently, $p$ is a limit point of $S$ if it is in the closure of $S \setminus \{p\}$. In addition, the closure of $S$ is equal to the union of $S$ and the set of its limit points. If the space $X$ is sequential, and in particular if $X$ is a metric space, the point $p$ is a limit point of $S$ if and only if there exists a sequence $(x_n, n\in {\N})$ of points of $S \setminus \{ p \}$ with $p$ as limit. In practice, the “interesting” limit points of $S$ are those which do not belong to $S$. For this reason, we call such limit points [*non-trivial*]{} and we denote by ${\mbox{${\lim}(S)$}}$ the set of non-trivial limit points of $S$. [**Regular chain.**]{} A set $R$ of non-constant polynomials in ${\K}[X_1, \ldots, X_s]$ is called a [*triangular set*]{}, if for all $p,q \in R$ with $p \neq q$ we have ${\mbox{{\rm mvar}$(p)$}} \neq {\mbox{{\rm mvar}$(q)$}}.$ For a nonempty triangular set $R$, we define the [*saturated ideal*]{} of $R$ to be the ideal ${\langleR\rangle}:h_R^{\infty}$, where $h_R$ is the product of the initials of the polynomials in $R$. The empty set is also regarded as a triangular set, whose saturated ideal is the trivial ideal $\langle 0 \rangle$. From now on, $R$ denotes a triangular set of ${\K}[X_1, \ldots, X_s]$. The ideal has several properties, in particular it is unmixed [@BLM01c]. We denote its height by $e$, thus has dimension $s - e$. Without loss of generality, we assume that ${\K}[X_1, \ldots, X_{s-e}] \cap {\mbox{{\rm sat}$(R)$}}$ is the trivial ideal $\langle 0 \rangle$. For all $1 \leq i \leq e$, we denote by $r_i$ the polynomial of $R$ whose main variable is $X_{i+s-e}$ and by $h_i$ the initial of $r_i$. Thus $h_R$ is the product $h_1 \cdots h_{e}$. We say that $R$ is a [*regular chain*]{} whenever $R$ is empty or $\{ r_1, \ldots, r_{e-1} \}$ is a regular chain and $h_e$ is regular modulo the saturated ideal . The regular chain $R$ is said [*strongly normalized*]{} whenever $h_R \in {\K}[X_1, \ldots, X_{s-e}]$ holds. If $R$ is not strongly normalized, one can compute a regular chain $N$ which is strongly normalized and such that ${\mbox{{\rm sat}$(R)$}} = {\mbox{{\rm sat}$(N)$}}$ and $V(h_N) = V(\widehat{h_R})$ both hold, where $\widehat{h_R}$ is the iterated resultant of $h_R$ w.r.t $R$. See [@DBLP:journals/jsc/ChenM12].
[**Limit points of the quasi-component of a regular chain.**]{} We denote by $W(R) := V(R) \setminus V(h_R)$ the [*quasi-component*]{} of $R$, that is, the common zeros of $R$ that do not cancel $h_R$. The above discussion implies that the closure of $W(R)$ in Zariski topology and the closure of $W(R)$ in the Euclidean topology are both equal to $V({\mbox{{\rm sat}$(R)$}})$, that is, the affine variety of ${\mbox{{\rm sat}$(R)$}}$. We denote by $\overline{W(R)}$ this common closure. We call [*limit points*]{} of $W(R)$ the elements of ${\mbox{${\lim}(W(R))$}}$.
[**Weak primitivity, primitive regular chain.**]{} Let be a commutative ring with unity. Let $p=a_0+\cdots+a_dx^d\in {\B}[X]$ be a univariate polynomial over ${\B}$ with degree $d\geq 1$. We say that $p$ is [*weakly primitive*]{} if for any $\beta\in {\B}$ such that $a_d$ divides $\beta a_i$, for all $0\leq i\leq d-1$, the element $a_d$ divides $\beta$ as well. The notion of weak primitivity, introduced in [@DBLP:journals/jsc/LemaireMPX11] is a generalization of the ordinary notion of primitivity over a unique factorization domain (UFD). The regular chain $R = \{ r_1, \ldots, r_e \}$ is said [*primitive*]{} if for all $1\leq j \leq e$, the polynomial $r_j$ is weakly primitive in ${\B}_j[X_j]$ where ${\B}_j$ is the residue class ring defined by ${\B}_j={\K}[X_1,\ldots,X_{j-1}]/ \langle r_1, \ldots, r_{j-1} \rangle$. Note that this definition implies that $r_1$ is primitive (in the usual sense) as a polynomial over ${\K}[X_1,\ldots,X_{s-e}]$. The notion of primitive regular chains is motivated by the following result. The regular chain $R$ is primitive if and only if $R$ generates its saturated ideal, that is, $\langle R \rangle = {\mbox{{\rm sat}$(R)$}}$ holds.
[**Rings of formal power series.**]{} Recall that ${\K}$ is an algebraically closed field. From now on, we further assume that ${\K}$ is topologically complete. Hence ${\K}$ may be the field ${\C}$ of complex numbers but not the algebraic closure of the field ${\Q}$ of rational numbers. We denote by ${\K}[[X_1, \ldots, X_s]]$ and ${\K} \langle X_1, \ldots, X_s \rangle$ the rings of formal and convergent power series in $X_1, \ldots, X_s$ with coefficients in ${\K}$. Note that the ring ${\K} \langle X_1, \ldots, X_s \rangle$ is a subring of ${\K}[[X_1, \ldots, X_s]]$. When $s=1$, we write $T$ instead of $X_1$. Thus ${\K}[[T]]$ and ${\K} \langle T \rangle$ are the rings of formal and convergent univariate power series in $T$ and coefficients in ${\K}$. For $f \in {\K}[[X_1, \ldots, X_s]]$, its [*order*]{} is defined by $${\rm ord}(f) = \left\{ \begin{array}{lr}
{\rm min} \{ d \mid f_{(d)} \neq 0 \} & {\rm if} \ f \neq 0, \\
\infty & {\rm if} \ f = 0.
\end{array}
\right.$$ where $f_{(d)}$ is the [*homogeneous part*]{} of $f$ in degree $d$. ${\rm min} \{ d \mid f_{(d)} \neq 0 \}$ if $f \neq 0$ and by $ \infty $ otherwise, where $f_{(d)}$ is the [*homogeneous part*]{} of $f$ in degree $d$. Recall that ${\K}[[X_1, \ldots, X_s]]$ is topologically complete for Krull Topology and that ${\K} \langle X_1, \ldots, X_s \rangle$ is a Banach Algebra for the norm defined by ${ \parallel f \parallel }_{\rho} = {\Sigma}_e \, |a_e| {\rho}^e$ where $f = {\Sigma}_e \, a_e X^e \in {\K}[[ X_1, \ldots, X_s ]]$ and ${\rho} = ({\rho}_1, \ldots, {\rho}_s) \in {\R}_{> 0}^s$. Recall that ${\K}[[X_1, \ldots, X_s]]$ is topologically complete for Krull Topology and that ${\K} \langle X_1, \ldots, X_s \rangle$ is a Banach Algebra. We denote by ${\cal M}_s $ the only maximal ideal of ${\K}[[X_1, \ldots, X_s]]$, that is, $${\cal M}_s= \{ f \in {\K}[[X_1, \ldots, X_s]] \mid {\rm ord}(f) \geq 1 \}.$$ ${\cal M}_s= \{ f \in {\K}[[X_1, \ldots, X_s]] \mid {\rm ord}(f) \geq 1 \}.$ Let $f \in {\K}[[ X_1, \ldots, X_s ]]$ with $f \neq 0$. Let $k \in {\N}$. We say that $f$ is (1) [*general*]{} in $X_s$ if $f \neq 0 \mod{ {\cal M}_{s-1}}$, (2) [*general*]{} in $X_s$ of order $k$ if we have ${\rm ord}( f \mod{ {\cal M}_{s-1} }) = k$.
[**Weierstrass polynomial.**]{} Assume furthermore that $f = \sum_{j=0}^{k} \, f_j X_s^j$ holds with $f_j \in {\K} [[ X_1, \ldots, X_{s-1} ]]$ for $j = 0 \cdots k$ and $f_k \neq 0$. Then, we say that $f$ is a [*Weierstrass polynomial*]{} if $f_0, \ldots, f_{k-1} \in {\cal M}_{s-1}$ and $f_k =1$ both hold.
[**Weierstrass preparation theorem.**]{} Let $f \in {\K}[[ X_1, \ldots, X_s ]]$ with $f \not\equiv 0 \mod{ {\cal M}_{s-1}}$. Then, there exists a unit ${\alpha} \in {\K}[[ X_1, \ldots, X_s ]]$, an integer $d \geq 0$ and a Weierstrass polynomial $p$ of degree $d$ such that we have $f = {\alpha} p$. Further, this expression for $f$ is unique. Moreover, if $f \in {\K}\langle X_1, \ldots, X_s \rangle$ holds then ${\alpha} \in {\K}\langle X_1, \ldots, X_s \rangle$ and $p \in {\K}\langle X_1, \ldots, X_{s-1} \rangle [ X_s]$ both hold.
[**Hensel Lemma.**]{} Let $f = X_s^k + a_1 X_s^{k-1} + \cdots + a_k$ with $a_k, \ldots, a_1 \in {\K} \langle X_1, \ldots, X_{s-1} \rangle$. Let $k_1, \ldots, k_r$ be positive integers and $c_{1}, \ldots, c_{r}$ be pairwise distinct elements of ${\K}$ such that we have $
f \equiv (X_s - c_1)^{k_1} \cdots (X_s - c_r)^{k_r}
\ \mod{ {\cal M}_{s-1} }.
$ Then, there exist $f_1, \ldots, f_r \in {\K} \langle X_1, \ldots, X_{s-1} \rangle [X_s]$ all monic in $X_s$ s.t. we have (1) $f = f_1 \cdots f_r$, (2) ${\deg}(f_j, X_s) = k_j$, for all $j = 1, \ldots, r$, (3) $f_j \equiv (X_s - c_j)^{k_j} \ \mod{ {\cal M}_{s-1} }$, for all $j = 1, \ldots, r$.
[**Formal Puiseux series.**]{} We denote by ${\K} [[ T^{*} ]] \ = \
\bigcup_{n=1}^{\infty} \, {\K} [[ T^{\frac{1}{n}} ]]$ the ring of [*formal Puiseux series*]{}. For a fixed ${\varphi} \in {\K} [[ T^{*} ]]$, there is an $n \in {\N}_{>0}$ such that ${\varphi} \in {\K} [[ T^{\frac{1}{n}} ]]$. Hence $
{\varphi} = \sum_{m=0}^{\infty} \, a_m T^{\frac{m}{n}}$, [where]{} $\ a_m \in {\K}$. We call [*order of*]{} ${\varphi}$ the rational number defined by $
{\rm ord}( {\varphi} ) \ = \ {\rm min} \{ \frac{m}{n} \ \mid \ a_m \neq 0 \}
\geq 0.
$ We denote by ${\K} (( T^{*} ))$ the quotient field of ${\K} [[ T^{*} ]]$.
[**Convergent Puiseux series.**]{} Let ${\varphi} \in {\C} [[ T^{*} ]]$ and $n \in {\N}$ such that ${\varphi}=f(T^{\frac{1}{n}})$ with $f\in{\C}[[T]]$ holds. We say that the Puiseux series ${\varphi}$ is [*convergent*]{} if we have ${f} \in {\C} \langle T \rangle$. Convergent Puiseux series form an integral domain denoted by ${\C} \langle T^{*} \rangle$; its quotient field is denoted by ${\C} ( \langle T^{*} \rangle )$. For every ${\varphi} \in {\C}(( T^{*} ))$, there exist $n \in {\Z}$, $r \in {\N}_{>0}$ and a sequence of complex numbers $a_n, a_{n+1}, a_{n+2}, \ldots$ such that we have $${\varphi} \ = \ \sum_{m=n}^{\infty} \, a_m T^{\frac{m}{r}} \ \ {\rm and} \ \
a_n \neq 0.$$ ${\varphi} \ = \ \sum_{m=n}^{\infty} \, a_m T^{\frac{m}{r}}$ and $a_n \neq 0.$ Then, we define ${\rm ord}({\varphi}) = \frac{n}{r}.$
[**Puiseux Theorem.**]{} If has characteristic zero, the field ${\K} (( T^{*} ))$ is the algebraic closure of the field of formal Laurent series over ${\K}$. Moreover, if ${\K} = {\C}$, the field ${\C} ( \langle T^{*} \rangle )$ is algebraically closed as well. From now on, we assume ${\K} = {\C}$.
[**Puiseux expansion.**]{} Let $\B=\C((X^{*}))$ or ${\C} ( \langle X^{*} \rangle )$. Let $f\in\B[Y]$, where $d := \deg(f, Y) >0$. Let $h := {\mbox{{\rm lc}$(f, Y)$}}$. According to Puiseux Theorem, there exists ${\varphi_i}\in \B$, $i=1,\ldots,d$, such that $\frac{f}{h}=(Y-\varphi_1)\cdots(Y-\varphi_d)$. We call $\varphi_1, \ldots, \varphi_d$ the [*Puiseux expansions*]{} of $f$ at the origin.
[**Puiseux parametrization.**]{} Let $f\in\C\langle X \rangle[Y]$. A parametrization of $f$ is a pair $(\psi(T), \varphi(T))$ of elements of $\C\langle T \rangle$ for some new variable $T$, such that (1) $f(\psi(T), \varphi(T))=0$ holds in $\C\langle T \rangle$, (2) we have $0<{\mbox{{\rm ord}$(\psi(T))$}}$, and (3) $\psi(T)$ and $\varphi(T)$ are not both in $\C$. The parametrization $(\psi(T), \varphi(T))$ is [*irreducible*]{} if there is no integer $k>1$ such that both $\psi(T)$ and $\varphi(T)$ are in $\C\langle T^k\rangle$. We call an irreducible parametrization $(\psi(T), \varphi(T))$ of $f$ a [*Puiseux parametrization*]{} of $f$, if there exists a positive integer $\varsigma$ such that $\psi(T)=T^{\varsigma}$. The index $\varsigma$ is called the [*ramification index*]{} of the parametrization $(T^{\varsigma}, \varphi(T))$. It is intrinsic to $f$ and $\varsigma\leq \deg(f, Y)$. Let $z_1,\ldots,z_{\varsigma}$ denote the primitive roots of unity of order $\varsigma$ in $\C$. Then $\varphi(z_iX^{1/\varsigma})$, for $i=1,\ldots,\varsigma$, are $\varsigma$ Puiseux expansions of $f$.
We conclude this section by a few lemmas which are immediate consequences of the above review.
\[Lemma:limitWR1\] We have: ${\mbox{${\lim}(W(R))$}} = \overline{W(R)} \cap V(h_R)$. In particular, ${\mbox{${\lim}(W(R))$}}$ is either empty or an affine variety of dimension $s-e-1$.
To Do.
\[Lemma:limitWR2\] If $R$ is a primitive regular chain, that is, if $R$ is a system of generators of its saturated ideal, then we have ${\mbox{${\lim}(W(R))$}} = V(R) \cap V(h_R)$.
To Do.
\[Lemma:limitWR3\] If $N$ is a strongly normalized regular chain such that ${\mbox{{\rm sat}$(R)$}} = {\mbox{{\rm sat}$(N)$}}$ and $V(h_N) = V(\widehat{h_R})$ both hold, then we have ${\mbox{${\lim}(W(R))$}} \subseteq {\mbox{${\lim}(W(N))$}}$.
\[Lemma:limitWR4\] Let $x \in {\A}^s$ such that $x \not\in W(R)$. Then $x \in {\mbox{${\lim}(W(R))$}}$ holds if and only if there exists a sequence $({\alpha}_n, n \in {\N})$ of points in ${\A}^s$ such that ${\alpha}_n \in W(R)$ for all $n \in {\N}$ and $\lim_{n \rightarrow \infty} \, {\alpha}_n = x$.
\[Lemma:limitWR5\] Recall that $R$ writes $\{ r_1, \ldots, r_{e} \}$. If $e > 1$ holds, writing $R' = \{ r_1, \ldots, r_{e-1} \}$ and $r = r_{e}$, we have $${\mbox{${\lim}(W(R' \cup r))$}} \ \subseteq \
{\mbox{${\lim}(W(R'))$}} \, \cap \,
{\mbox{${\lim}(W(r))$}}.$$
\[Lemma:NegativePowers\] Let ${\varphi} \in {\C} ( \langle T^{*} \rangle )$ and let $p/q \in {\Q}$ be the order of ${\varphi}$. Let $({\alpha}_n, n \in {\N})$ be a sequence of complex numbers converging to zero and let $N$ be a positive integer such that $({\varphi}({\alpha}_n), n \geq N)$ is well defined. Then, if $p/q < 0$ holds, the sequence $({\varphi}({\alpha}_n), n \geq N)$ escapes to infinity while if $p/q \geq 0$, the sequence $({\varphi}({\alpha}_n), n \geq N)$ converges to the complex number ${\varphi}(0)$.
To Do.
Basic techniques {#sec:limitpoints}
================
This section is an overview of the basic techniques of this paper. This presentation is meant to help the non-expert reader understand our objectives and solutions. In particular, the results of this section are stated for regular chains in three variables, while the statements of Sections \[sec:theory\] to \[sec:principle\] do not have this restriction.
Recall that $R \subseteq {\C}[X_1, \ldots, X_s]$ is a regular chain whose saturated ideal has height $1 \leq e \leq s$. As mentioned in the introduction, we mainly focus on the case $e = s-1$, that is, has dimension one. Lemma \[Lemma:limitWR1\] and the assumption $e = s-1$ imply that ${\mbox{${\lim}(W(R))$}}$ consists of finitely many points.
We further assume that $R$ is strongly normalized, thus we have $h_R$ lies in ${\C}[X_1]$. Lemma \[Lemma:limitWR2\] and the assumption $h_R \in {\C}[X_1]$ imply that computing ${\mbox{${\lim}(W(R))$}}$ reduces to check, for each root $\alpha \in {\C}$ of ${h_R}$ whether or not there is a point $x \in {\mbox{${\lim}(W(R))$}}$ whose $X_1$-coordinate is $\alpha$. Without loss of generality, it is enough to develop our results for the case $\alpha = 0$. Indeed, a change of coordinates can be used to reduce to this latter assumption.
We start by considering the case $n=2$. Thus, our regular chain $R$ consists of a single polynomial $r_1 \in {\C}[X_1, X_2]$ whose initial $h_1$ satisfies $h_1(0) = 0$. Lemma \[Lemma:LWRdim1OnePoly\] provides a necessary and sufficient condition for a point of $({\alpha}, {\beta}) \in {\A}^2$, with $\alpha = 0$, to satisfy $({\alpha}, {\beta}) \in {\mbox{${\lim}(W( \{ r_1 \} ))$}}$.
Let $d$ be the degree of $r_1$ in $X_2$. Applying Puiseux Theorem, we consider ${\varphi}_1, \ldots, {\varphi}_d \in {\C}(\langle X_1^{*} \rangle)$ such that the following holds $$\label{eq:facto1}
\frac{r_1}{h_1} = (X_2 - {\varphi}_1) \cdots (X_2 - {\varphi}_d)$$ in ${\C}(\langle X_1^{*} \rangle) [X_2]$. We assume that the series ${\varphi}_1, \ldots, {\varphi}_d $ are numbered in such a way that each of ${\varphi}_1, \ldots, {\varphi}_c $ has a non-negative order while each of ${\varphi}_{c+1}, \ldots, {\varphi}_d $ has a negative order, for some $c$ such that $0 \leq c \leq d$.
\[Lemma:LWRdim1OnePoly\] With $h_1(0)=0$, for all $\beta \in {\C}$, the following two conditions are equivalent
1. $(0, \beta) \in {\mbox{${\lim}(W(r_1))$}}$ holds,
2. there exists $1 \leq j \leq c$ and a sequence $({\alpha}_n, n \in {\N})$ of complex numbers such that the sequence $( {\varphi}_j ({\alpha}_n), n \in {\N})$ is well defined, we have $h_1({\alpha}_n) \neq 0$ for all $n \in {\N}$ and we we have $$\lim_{n \rightarrow \infty} \, {\alpha}_n = 0 \ \ {\rm and} \ \
\lim_{n \rightarrow \infty} \, {\varphi}_j ({\alpha}_n) = {\beta}.$$
We first prove the implication $(ii) \Rightarrow (i)$. Equation (\[eq:facto1\]) together with $(ii)$ implies $({\alpha}_n, {\varphi}_j ({\alpha}_n)) \in V(r_1)$ for all $n \in {\N}$. Since we also have $({\alpha}_n, {\varphi}_j ({\alpha}_n)) \not\in V(h_1)$ for all $n \in {\N}$ and $\lim_{n \rightarrow \infty} \, ({\alpha}_n, {\varphi}_j ({\alpha}_n)) = (0, {\beta})$, we deduce $(i)$, thanks to Lemma \[Lemma:limitWR4\].
We now prove the implication $(i) \Rightarrow (ii)$. By Lemma \[Lemma:limitWR4\], there exists a sequence $(({\alpha}_n, {\beta}_n), n \in {\N})$ in ${\A}^2$ such that for all $n \in {\N}$ we have: (1) $h_1({\alpha}_n) \neq 0$, (2) $r_1 ({\alpha}_n, {\beta}_n) = 0$, and (3) $\lim_{n \rightarrow \infty} \, ({\alpha}_n, {\beta}_n) = (0, {\beta})$. Since $\lim_{n \rightarrow \infty} \, {\alpha}_n = 0$, each series ${\varphi}_1 ({\alpha}_n), \ldots, {\varphi}_d ({\alpha}_n)$ is well defined for $n$ larger than some positive integer $N$. Hypotheses (1) and (2), together with Equation (\[eq:facto1\]), imply that for all $n \geq N$ the product $$({\beta}_n - {\varphi}_1({\alpha}_n)) \cdots
({\beta}_n - {\varphi}_c({\alpha}_n))
({\beta}_n - {\varphi}_{c+1}({\alpha}_n)) \cdots
({\beta}_n - {\varphi}_d({\alpha}_n))$$ is $0$. Since $\lim_{n \rightarrow \infty} \, {\beta}_n = {\beta}$, and by definition of the integer $c$, each of the sequences $({\beta}_n - {\varphi}_1({\alpha}_n)), \ldots,
({\beta}_n - {\varphi}_c({\alpha}_n))$ converges while each of the sequences $({\beta}_n - {\varphi}_{c+1}({\alpha}_n)), \ldots,
({\beta}_n - {\varphi}_d({\alpha}_n))$ escapes to infinity. Thus, for $n$ large enough the product $({\beta}_n - {\varphi}_1({\alpha}_n)) \cdots
({\beta}_n - {\varphi}_c({\alpha}_n))$ is zero. Therefore, one of sequences $({\beta}_n - {\varphi}_1({\alpha}_n)), \ldots,
({\beta}_n - {\varphi}_c({\alpha}_n))$ converges to $0$ and the conclusion follows.
Lemmas \[Lemma:NegativePowers\] and \[Lemma:LWRdim1OnePoly\] immediately imply the following.
\[Propo:LWRdim1OnePoly1\] With $h_1(0) = 0$, for all $\beta \in {\C}$, we have $$(0, \beta) \in {\mbox{${\lim}(W(r_1))$}} \ \ \iff \ \
\beta \in \{ {\varphi}_1(0), \ldots, {\varphi}_c(0) \}.$$
We observe that ${\mbox{${\lim}(W(r_1))$}}$ can be computed in another way. Let us denote by $p_1$ the primitive part of $r_1$ over ${\C}[X_1]$. It is easy to check that ${\mbox{${\lim}(W(r_1))$}} = {\mbox{${\lim}(W(p_1))$}}$ holds. Therefore, with Lemma \[Lemma:limitWR2\], we deduce another characterization of ${\mbox{${\lim}(W(r_1))$}}$, which also derives from Theorem 4.1 in [@RW78].
\[Propo:LWRdim1OnePoly1\] With $h_1(0) = 0$, for all $\beta \in {\C}$, we have $$(0, \beta) \in {\mbox{${\lim}(W(r_1))$}} \ \ \iff \ \
(0, \beta) \in V(r_1).$$
Next, we consider the case $n=3$. Hence, our regular chain $R$ consists of two polynomials $r_1 \in {\C}[X_1, X_2]$ and $r_2 \in {\C}[X_1, X_2, X_3]$ with respective initials $h_1$ and $h_2$. We assume that $0$ is a root of the product $h_1 h_2$ and we are looking for all ${\beta} \in {\C}$ and all ${\gamma} \in {\C}$ such that $(0, {\beta}, {\gamma}) \in {\mbox{${\lim}(W(r_1, r_2))$}}$.
Lemma \[Lemma:limitWR5\] tells us that $(0, {\beta}, {\gamma}) \in {\mbox{${\lim}(W(r_1, r_2))$}}$ implies $(0, {\beta}) \in {\mbox{${\lim}(W(r_1))$}}$. This observation together with Proposition \[Propo:LWRdim1OnePoly1\] yields immediately the following.
\[Propo:LWRdim1OTowPoly1\] With $h_1(0) = 0$ and $h_2(0) \neq 0$, assuming that $r_1$ is primitive over ${\C}[X_1]$, for all ${\beta} \in {\C}$ and all ${\gamma} \in {\C}$, we have $$(0, \beta, \gamma) \in {\mbox{${\lim}(W(r_1, r_2))$}} \ \ \iff \ \
(0, \beta, \gamma) \in V(r_1, r_2).$$
We turn now our attention to the case $h_1(0) = h_2(0) = 0$. Since $(0, {\beta}) \in {\mbox{${\lim}(W(r_1))$}}$ is a necessary condition for $(0, {\beta}, {\gamma}) \in {\mbox{${\lim}(W(r_1, r_2))$}}$ to hold we apply Proposition \[Propo:LWRdim1OnePoly1\] and assume $\beta \in \{ {\varphi}_1(0), \ldots, {\varphi}_c(0) \}$. Without loss of generality, we further assume $\beta = 0$. For each $1 \leq j \leq c$, such that ${\varphi}_j(0) = 0$ holds, we define the univariate polynomial $f^j_2 \in {\C}(\langle X_1^{*} \rangle)[X_3]$ by $$\label{eq:f2}
f^j_2(X_1,X_3) = r_2(X_1, {\varphi}_j(X_1), X_3).$$ Let $b$ be the degree of $f^j_2$. Applying again Puiseux theorem, we consider ${\psi}_1, \ldots, {\psi}_b \in {\C}(\langle X_1^{*} \rangle)$ such that the following holds $$\label{eq:facto2}
\frac{f_2^j}{h_2} = (X_3 - {\psi}_1) \cdots (X_3 - {\psi}_b)$$ in ${\C}(\langle X_1^{*} \rangle) [X_3]$. We assume that the series ${\psi}_1, \ldots, {\psi}_b $ are numbered in such a way that each of ${\psi}_1, \ldots, {\psi}_a $ has a non-negative order while each of ${\psi}_{a+1}, \ldots, {\psi}_b $ has a negative order, for some $a$ such that $0 \leq a \leq b$.
\[Lemma:LWRdim1TwoPolys\] For all $\gamma \in {\C}$, the following two conditions are equivalent.
1. $(0, 0, \gamma) \in {\mbox{${\lim}(W(r_1, r_2))$}}$ holds,
2. there exist integers $j,k$ with $1 \leq j \leq c$ and $1 \leq k \leq a$, and two sequences $({\alpha}_n, n \in {\N})$, $({\beta}_n, n \in {\N})$ of complex numbers such that:
1. the sequences $( {\varphi}_j ({\alpha}_n), n \in {\N})$ and $( {\psi}_k ({\beta}_n), n \in {\N})$ are well defined,
2. $h_1({\alpha}_n) \neq 0$ and $h_2({\alpha}_n) \neq 0$, for all $n \in {\N}$,
3. ${\beta}_n = {\varphi}_j ({\alpha}_n)$, for all $n \in {\N}$,
4. $\lim_{n \rightarrow \infty} \,
({\alpha}_n, {\beta}_n, {\psi}_k ({\beta}_n)) = (0, 0, \gamma)$.
Proving the implication $(ii) \Rightarrow (i)$ is easy. We now prove the implication $(i) \Rightarrow (ii)$. By Lemma \[Lemma:limitWR4\], there exists a sequence $(({\alpha}_n, {\beta}_n, {\gamma}_n), n \in {\N})$ in ${\A}^3$ s.t. for all $n \in {\N}$ we have: (1) $h_1({\alpha}_n) \neq 0$, (2) $h_2({\alpha}_n) \neq 0$, (3) $r_1 ({\alpha}_n, {\beta}_n) = 0$, (4) $r_2 ({\alpha}_n, {\beta}_n, {\gamma}_n) = 0$, (5) $\lim_{n \rightarrow \infty} \, ({\alpha}_n, {\beta}_n, {\gamma}_n) = (0, 0, {\gamma})$. Following the proof of Lemma \[Lemma:LWRdim1OnePoly\], we know that for $n$ large enough the product $({\beta}_n - {\varphi}_1({\alpha}_n)) \cdots
({\beta}_n - {\varphi}_c({\alpha}_n))$ is zero. Therefore, from one of the sequences $({\beta}_n - {\varphi}_1({\alpha}_n)), \ldots,
({\beta}_n - {\varphi}_c({\alpha}_n))$, say the $j$-th, one can extract an (infinite) sub-sequence whose terms are all zero. Thus, without loss of generality, we assume that ${\beta}_n = {\varphi}_j ({\alpha}_n)$ holds, for all $n \in {\N}$. Hence, for all $n \in {\N}$, we have $f^j_2({\alpha}_n,{\gamma}_n) = r_2({\alpha}_n, {\beta}_n, {\gamma}_n) = 0$. Together with Equation (\[eq:facto2\]) and following the proof of Lemma \[Lemma:LWRdim1OnePoly\], we deduce the desired result.
Lemmas \[Lemma:NegativePowers\] and \[Lemma:LWRdim1TwoPolys\] immediately imply the following.
\[Propo:LWRdim1OneTwoPolys\] For all $\gamma \in {\C}$, the following two conditions are equivalent.
1. $(0, 0, \gamma) \in {\mbox{${\lim}(W(r_1, r_2))$}}$ holds,
2. there exist integers $j,k$ with $1 \leq j \leq c$ and $1 \leq k \leq a$, such that ${\varphi}_j (0) = 0$ and ${\psi}_k (0) = \gamma$.
Therefore, applying Puiseux theorem to $r_1$ and $f^j_2$, then checking the constant terms of the series ${\psi}_1, \ldots, {\psi}_b $ provides a way to compute all $\gamma \in {\C}$ such that $(0, 0, \gamma)$ is a limit point of $W(r_1, r_2)$. Theorem \[thrm:PuiseuxParamRC\] in Sections \[sec:theory\] states this principle formally for an arbitrary regular chain $R$ in dimension one.
Finally, one should also consider the case $h_1(0) \neq 0, h_2(0) = 0$. In fact, it is easy to see that this latter case can be handled in a similar manner as the case $h_1(0) = 0, h_2(0) = 0$.
Puiseux expansions of a regular chain {#sec:theory}
=====================================
In this section, we introduce the notion of Puiseux expansions of a regular chain, motivated by the work of [@MaurerJoseph80; @Mo92] on Puiseux expansions of space curves.
\[Lemma:V0\] Let $R=\{r_1,\ldots,r_{s-1}\}\subset\C[X_1<\cdots<X_s]$ be a strongly normalized regular chain whose saturated ideal has dimension one. Recall that $h_R(X_1)$ denotes the product of the initials of polynomials in $R$. Let $\rho>0$ be small enough such that the set $0<{\lvertX_1\rvert}<\rho$ does not contain any zeros of $h_R$. Denote by $U_{\rho} := \{x=(x_1,\ldots,x_s)\in\C^s \mid 0<{\lvertx_1\rvert}<\rho \}$. Denote by $V_{\rho}(R) := V(R)\cap U_{\rho}$. Then we have $W(R)\cap U_{\rho}=V_{\rho}(R)$. Let $R' := \{ {\mbox{{\rm primpart}$(r_1)$}},\ldots,{\mbox{{\rm primpart}$(r_{s-1})$}}\}$. Then $V_{\rho}(R)=V_{\rho}(R')$.
Let $x\in W(R)\cap U_{\rho}$, then $x\in V(R)$ and $x\in U_{\rho}$ hold, which implies that $W(R)\cap U_{\rho}\subseteq V(R)\cap U_{\rho}$. Let $x\in V(R)\cap U_{\rho}$. Since $U_{\rho}\cap V(h_R)=\emptyset$, we have $x\in W(R)$. Thus $V(R)\cap U_{\rho}\subseteq W(R)\cap U_{\rho}$. So $W(R)\cap U_{\rho}=V_{\rho}(R)$. Similarly we have $V_{\rho}(R)=V_{\rho}(R')$.
Let $W \subseteq \C^{s}$. Denote ${\rm lim}_{0}(W):=\{x=(x_1,\ldots,x_s)\in\C^s\mid x\in {\mbox{${\lim}(W)$}}
\ {\rm and} \ x_1=0\}$.
\[Lemma:lim0\] Let $R=\{r_1,\ldots,r_{s-1}\}\subset\C[X_1<\cdots<X_s]$. Then we have ${\rm lim}_{0}(W(R))={\rm lim}_{0}(V_{\rho}(R))$.
By Lemma \[Lemma:V0\], we have $W(R)\cap U_{\rho}(R)=V_{\rho}(R)$. On the other hand ${\rm lim}_{0}(W(R))={\rm lim}_{0}(W(R)\cap U_{\rho}(R))$. Thus ${\rm lim}_{0}(W(R))={\rm lim}_{0}(V_{\rho}(R))$ holds.
\[Lemma:zero-series\] Let $R$ be as in Lemma \[Lemma:V0\]. For $1\leq i\leq s-1$, let $d_i := \deg(r_i, X_{i+1})$. Then $R$ generates a zero-dimensional ideal in $\C(\langle X_1^*\rangle)[X_2,\ldots,X_s]$. Let $V^{*}(R)$ be the zero set of $R$ in $\C(\langle X_1^*\rangle)^{s-1}$. Then $V^{*}(R)$ has exactly $\prod_{i=1}^{s-1} d_i$ points, counting multiplicities.
It follows directly from the definition of regular chain, Bezout bound and the fact that $\C(\langle X_1^*\rangle)^{s-1}$ is an algebraically closed field.
\[Definition:PuiseuxExpansionOfRC\] We use the notions in Lemma \[Lemma:zero-series\]. Each point in $V^{*}(R)$ is called a [*Puiseux expansion*]{} of $R$.
Let $m={\lvertV^*(R)\rvert}$. Write $V^*(R)= \{\Phi_1, \ldots, \Phi_m \}$ with $\Phi_i=(\Phi_i^1(X_1),\ldots,\Phi_i^{s-1}(X_1))$, for $i=1,\ldots,m$. Let $\rho>0$ be small enough such that for $1 \leq i \leq m$, $ 1 \leq j \leq s-1$, each $\Phi_i^j(X_1)$ converges in $0<{\lvertX_1\rvert}<\rho$. We define $
V^*_{\rho}(R):=\cup_{i=1}^m \{x\in\C^s\mid 0<{\lvertx_1\rvert}<\rho, x_{j+1}=\Phi_i^j(x_1), j=1,\ldots,s-1\}.
$
\[Theorem:VStarPho\] We have $V^*_{\rho}(R)=V_{\rho}(R)$.
We prove this by induction on $s$. For $i=1,\ldots,s-1$, recall that $h_i$ is the initial of $r_i$. If $s=2$, we have $$r_1(X_1,X_2)=h_1(X_1)\prod_{i=1}^{d_1}(X_2-\Phi_i^1(X_1)).$$ So $V^*_{\rho}(R)=V_{\rho}(R)$ clearly holds.
Write $R=R'\cup\{r_{s-1}\}$, $X' = X_2, \ldots, X_{s-1}$, $X=(X_1,X',X_{s})$, $x' = x_2, \ldots, x_{s-1}$, $x=(x_1,x',x_{s})$, and $m'={\lvertV^*(R')\rvert}$. For $i=1,\ldots,m$, let $\Phi_i=(\Phi_i', \Phi_i^{s-1})$, where $\Phi_i'$ stands for $\Phi_i^1, \ldots, \Phi_i^{s-2}$. Assume the theorem holds for $R'$, that is $V^*_{\rho}(R')=V_{\rho}(R')$. For any $i=1,\ldots,m'$, there exist $i_k\in\{1,\ldots,m\}$, $k=1,\ldots,d_{s-1}$ such that $$\label{equation:expansion}
r_{s-1}(X_1,X'=\Phi_i', X_{s})=h_1(X_1)\prod_{k=1}^{d_{s-1}}(X_s-\Phi_{i_k}^{s-1}(X_1)).$$ Note that $V^*(R)=\cup_{i=1}^{m'}\cup_{k=1}^{d_{s-1}}\{ (X'=\Phi_i', X_s=\Phi_{i_k}^{s-1})\}$. Therefore, by induction hypothesis and Equation (\[equation:expansion\]), we have $$\begin{array}{rcl}
V^*_{\rho}(R) &=&\cup_{i=1}^{m'}\cup_{k=1}^{d_{s-1}}\{x\mid x\in U_{\rho}, x'=\Phi_i'(x_1), x_s=\Phi_{i_k}^{s-1}(x_1)\}\\
&=&\cup_{k=1}^{d_{s-1}}\{x\mid (x_1,x')\in V^*_{\rho}(R'), x_s=\Phi_{i_k}^{s-1}(x_1)\}\\
&=&\{x\mid (x_1,x')\in V^*_{\rho}(R'), r_{s-1}(x_1,x',x_s)=0\}\\
&=&\{x\mid (x_1,x')\in V_{\rho}(R'), r_{s-1}(x_1,x',x_s)=0\}\\
&=&V_{\rho}(R).
\end{array}$$
\[Theorem:limit\] Let $V^*_{\geq 0}(R):=\{\Phi=(\Phi^1,\ldots,\Phi^{s-1})\in V^*(R)\mid {\mbox{{\rm ord}$(\Phi^j)$}}\geq 0, j=1,\ldots,s-1 \}$. Then we have $${\rm lim}_0(W(R))=\cup_{\Phi\in V^*_{\geq 0}(R)} \{(X_1=0, \Phi(X_1=0))\}.$$
By definition of $V^*_{\geq 0}(R)$, we immediately have $${\rm lim}_0(V^*_{\rho}(R))=\cup_{\Phi\in V^*_{\geq 0}(R)} \{(X_1=0, \Phi(X_1=0))\}.$$ Next, by Theorem \[Theorem:VStarPho\], we have $V^*_{\rho}(R)=V_{\rho}(R)$. Thus, we have ${\rm lim}_0(V^*_{\rho}(R))={\rm lim}_0(V_{\rho}(R))$. Besides, with Lemma \[Lemma:lim0\], we have ${\rm lim}_{0}(W(R))={\rm lim}_{0}(V_{\rho}(R))$. Thus the theorem holds.
\[Definition:PuiseuxParamRC\] Let $V^*_{\geq 0}(R):=\{\Phi=(\Phi^1,\ldots,\Phi^{s-1})\in V^*(R)\mid {\mbox{{\rm ord}$(\Phi^j)$}}\geq 0, j=1,\ldots,s-1 \}$. Let $M={\lvertV^*_{\geq 0}(R)\rvert}$. For each $\Phi_i=(\Phi_i^1,\ldots,\Phi_i^{s-1})\in V^*_{\geq 0}(R)$, $1\leq i\leq M$, we know that $\Phi_i^j\in\C(\langle X_1^*\rangle)$. Moreover, by Equation (\[equation:expansion\]), we know that for $j=1,\ldots,s-1$, $\Phi_i^j$ is a Puiseux expansion of $r_{j}(X_1,X_2=\Phi_i^1,\ldots,X_j=\Phi_i^{j-1} , X_{j+1})$. Let $\varsigma_{i,j}$ be the ramification index of $\Phi_i^j$ and $(T^{\varsigma_{i,j}}, X_{j+1}=\varphi_i^j(T))$, where $\varphi_i^j\in\C\langle T\rangle$, be the corresponding Puiseux parametrization of $\Phi_i^j$. Let $\varsigma_i$ be the least common multiple of $\{\varsigma_{i,1},\ldots,\varsigma_{i,s-1}\}$. Let $g_{i}^{j}=\varphi_i^j(T=T^{\varsigma_i/\varsigma_{i,j}})$. We call the set $\frak{G}_R := \{(X_1=T^{\varsigma_{i}}, X_2=g_i^1(T),\ldots,X_{s}=g_i^{s-1}(T)), i=1,\ldots,M\}$ a [*system of Puiseux parametrizations*]{} of $R$.
\[thrm:PuiseuxParamRC\] We have $${\rm lim}_0(W(R))=\frak{G}_R(T=0).$$
It follows directly from Theorem \[Theorem:limit\] and Definition \[Definition:PuiseuxParamRC\].
Puiseux parametrization in finite accuracy {#sec:newton}
==========================================
In this section, we define the Puiseux parametrizations of a polynomial $f\in\C\langle X\rangle[Y]$ in finite accuracy, see Definition \[Definition:PuiseuxParaFinite\].
This definition is different from the usual sense of Puiseux parametrizations up to some accuracies in the sense that we do not require the singular part of Puiseux parametrizations to be computed.
For $f\in\C\langle X\rangle[Y]$, we define the approximation $\widetilde{f}$ of $f$ for a given finite accuracy, see Definition \[Definition:approximation\]. This approximation $\widetilde{f}$ of $f$ is a polynomial in $\C[X, Y]$. In Section \[sec:bounds\], we prove that in order to compute a Puiseux parametrizations of $f$ of a given accuracy, it suffices to compute a Puiseux parametrization of $\widetilde{f}$ of some finite accuracy.
In this section, we review and adapt the classical Newton-Puiseux algorithm to compute Puiseux parametrizations of a polynomial $f\in\C[X, Y]$ of a given accuracy. Since we do not need to compute the singular part of Puiseux parametrizations, the usual requirement ${\mbox{{\rm discrim}$(f, Y)$}}\neq 0$ is dropped.
\[Definition:approximation\] Let $f=\sum_{i=0}^{\infty} a_i X^i \in \C[[X]]$. For any $\tau\in\N$, let $f^{(\tau)} := \sum_{i=0}^{\tau} a_iX^i$. We call $f^{(\tau)}$ the [*polynomial part*]{} of $f$ [*of accuracy*]{} $\tau+1$. Now, let $f=\sum_{i=0}^{d} a_i(X) Y^i\in\C\langle X\rangle[Y]$. For any $\tau\in\N$, we call $\widetilde{f}^{(\tau)} := \sum_{i=0}^d a_i^{(\tau)} Y^i$ the [*approximation of $f$ of accuracy $\tau+1$*]{}.
\[Definition:PuiseuxParaFinite\] Let $f\in\C\langle X\rangle[Y]$, $\deg(f, Y)>0$. Let $\sigma, \tau\in{\N}_{>0}$ and $g(T)=\sum_{k=0}^{\tau-1}b_k T^k$. The pair $(T^{\sigma}, g(T))$ is called [*a Puiseux parametrization of $f$ of accuracy $\tau$*]{} if there exists an irreducible Puiseux parametrization $(T^{\varsigma}, \varphi(T))$ of $f$ such that
$\sigma$ divides $\varsigma$.
${\mbox{{\rm gcd}$(\sigma, b_0,\ldots,b_{\tau-1})$}}=1$.
$g(T^{\varsigma/\sigma})$ is the polynomial part of $\varphi(T)$ of accuracy $(\varsigma/\sigma)(\tau-1)+1$.
Note that if $\sigma=\varsigma$, then $g(T)$ is simply the polynomial part of $\varphi(T)$ of accuracy $\tau$.
We borrow the following notion from [@Duv89] in order to state an algorithm for computing Puiseux parametrizations.
A $\C$-term[^3] is defined as a triple $t=(q, p,\beta)$, where $q$ and $p$ are coprime integers, $q>0$ and $\beta \in {\C}$ is non-zero. A $\C$-expansion is a sequence $\pi=(t_1,t_2,\ldots)$ of $\C$-terms, where $t_i=(q_i, p_i,\beta_i)$, We say that $\pi$ is [*finite*]{} if there are only finitely many elements in $\pi$.
\[Definition:ExpansionParametrization\] Let $\pi=(t_1,\ldots,t_N)$ be a finite $\C$-expansion. We define a pair $(T^{\sigma}, g(T))$ of polynomials in ${\C}[T]$ in the following manner:
if $N=1$, set $\sigma=1$, $g(T)=0$ and $\delta_N = 0$,
otherwise, let $a := \prod_{i=1}^N q_i$, $c_i := \sum_{j=1}^i\left( p_j\prod_{k=j+1}^Nq_k \right)$ ($1\leq i\leq N$), and $\delta_i := c_i/{\mbox{{\rm gcd}$(a, c_1,\ldots,c_N)$}}$ ($1\leq i\leq N$). Set $\sigma := a/{\mbox{{\rm gcd}$(a, c_1,\ldots,c_N)$}}$ and $g(T) := \sum_{i=1}^N \beta_i T^{\delta_i}$.
We call the pair $(T^{\sigma}, g(T))$ [*the corresponding Puiseux para-metrization of $\pi$ of accuracy $\delta_N+1$*]{}. Denote by [ConstructParametrization]{} an algorithm to compute $(T^{\sigma}, g(T))$ from $\pi$.
Let $f\in\C\langle X\rangle[Y]$ and write $f$ as $f(X, Y) := \sum_{i=0}^d \left(\sum_{j=0}^{\infty} a_{i, j}X^j \right) Y^i$. The [*Newton Polygon*]{} of $f$ is defined as the lower part of the convex hull of the set of points $(i,j)$ in the plane such that $a_{i,j}\neq 0$.
Let $f\in\C\langle X\rangle[Y]$. Next we present an algorithm, called [NewtonPolygon]{} to compute the segments in the Newton Polygon of $f$. This algorithm is from R.J. Walker’s book [@RW78].
${\mbox{{\sf NewtonPolygon}$(f, I)$}}$
[**Input:**]{} A polynomial $f\in\C\langle X\rangle[Y]$; a controlling flag $I$, whose value is $1$ or $2$.
[**Output:**]{} The Newton Polygon of $f$. If $I=1$, only segments with non-positive slopes are computed. If $I=2$, only segments with negative slopes are computed.
[**Description:**]{}
Write $f$ as $f=\sum_{i=0}^d b_{i}(X)Y^i$, where $b_i(X)=\sum_{j=0}^{\infty} a_{i,j}X^j$.
For $0\leq i\leq d$, define $\delta_i := {\mbox{{\rm ord}$(b_i)$}}$.
For $0\leq i\leq d$, we plot the points $P_i$ with coordinates $(i,\delta_i)$; we omit $P_i$ if $\delta_i=\infty$.
We join $P_0$ to $P_d$ with a convex polygonal arc each of whose vertices is a $P_i$ and such that no $P_i$ lies below the arc.
If $I=1$, output all segments with non-positive slopes in the polygon; if $I=2$, output all segments with negative slopes in the polygon.
Let $f\in\C\langle X\rangle[Y]$. We denote by ${\mbox{{\sf NewtonPolygon}$(f, I)$}}$ an algorithm to compute the segments in the Newton Polygon of $f$, where $I$ is flag controlling the algorithm specification as follows. If $I=1$, only segments with non-positive slopes are computed. If $I=2$, only segments with negative slopes are computed. Such an algorithm can be found in [@RW78].
Next we present the specification of several other sub-algorithms which are necessary to present Algorithm \[alg:newton\] for computing Puiseux parametrization of some finite accuracy as defined in Definition \[Definition:PuiseuxParaFinite\].
${\mbox{{\sf NewPolynomial}$(f, t, \ell)$}}$
[**Input:**]{} $f\in\C[X, Y]$; a $\C$-term $t=(q, p, \beta)$; $\ell\in\N$.
[**Output:**]{} A polynomial $X^{-\ell}f(X^q, X^p(\beta + Y))\in\C[X, Y]$.
${\mbox{{\sf SegmentPoly}$(f, \Delta)$}}$
[**Input:**]{} $f\in\C[X, Y]$; $\Delta$ is a segment of the Newton Polygon of $f$.
[**Output:**]{} A quadruple $(q, p, \ell, \phi)$ such that the following holds
$q, p,\ell\in\N$; $\phi\in\C[Z]$; $q$ and $p$ are coprime, $q>0$.
For any $(i,j)\in\Delta$, we have $qj+pi=\ell$.
Let $i_0:={\mbox{{\rm min}$(\{i\mid (i,j)\in\Delta\})$}}$, we have $\phi=\sum_{(i,j)\in\Delta}a_{i,j}Z^{(i-i_0)/q}.$
Algorithm \[alg:newton\] terminates and is correct.
It directly follows from the proof of Newton-Puiseux algorithm in Walker’s book [@RW78], the relation between $\C$-expansion and Puiseux parametrization discussed in Duval’s paper [@Duv89], and Definitions \[Definition:ExpansionParametrization\] and \[Definition:PuiseuxParaFinite\].
\[alg:newton\]
Computing in finite accuracy {#sec:approximation}
============================
Let $f\in\C\langle X\rangle[Y]$. In this section, we consider the following problems.
Is it possible to use an approximation of $f$ of some finite accuracy $m$ in order to compute a Puiseux parametrization of $f$ of some finite accuracy $\tau$?
If yes, how to deduce $m$ from $f$ and $\tau$?
Provide a bound on $m$.
Theorem \[Theorem:finite-bound\] provides the answers to $(a)$ and $(b)$ while Lemma \[Lemma:generic\] answers $(c)$.
The proof of Theorem \[Theorem:finite-bound\] relies on Lemma \[Lemma:one-iteration\]. In Lemma \[Lemma:one-iteration\], we consider one iteration in Newton-Puiseux’s algorithm for computing Puiseux parametrizations. Let $f$ and $f_1$ be respectively the input and output polynomial of the iteration, which corresponds exactly to the input and output of algorithm [NewPolynomial]{} in Section \[sec:newton\]. It provides the estimates on the accuracy required for coefficients of $f\in\C\langle X\rangle[Y]$ in order to compute the approximation of $f_1\in\C\langle X_1\rangle[Y_1]$ of a given accuracy.
\[Lemma:GFsubs\] Let $\underline{X}=X_1,\ldots,X_s$ and $\underline{Y}=Y_1,\ldots,Y_m$. For $g_1, \ldots, g_s \in {\C}[[ \underline{Y} ]]$, with ${\rm
ord}(g_i) \geq 1$, there is a ${\C}$-algebra homomorphism (called the [*substitution homomorphism*]{})
$ {\Phi}_g:
\begin{array}{rcl} {\C}[[ \underline{X} ]] & \longrightarrow & {\C}[[ \underline{Y} ]] \\
f & \longmapsto & f(g_1(\underline{Y}),\ldots,
g_s(\underline{Y})).
\end{array}
$
Moreover, if $g_1, \ldots, g_s $ are convergent power series, then we have ${\Phi}_g({\C} \langle \underline{X} \rangle) \subseteq {\C}
\langle \underline{Y} \rangle$ holds.
Let $f=\sum a_{\mu\nu}X^{\mu}Y^{\nu}\in\C[[X, Y]]$. The [*carrier*]{} of $f$ is defined as $${\rm carr}(f)=\{(\mu,\nu)\in\N^2 \ \mid \ a_{\mu\nu}\neq 0\}.$$
\[Lemma:one-iteration\] Let $f\in \C\langle X \rangle[Y]$. Let $d := \deg(f, Y)>0$. Let $q\in {\N}_{>0}$, $p, \ell\in\N$ and assume that $q$ and $p$ are coprime. Let $\beta\neq 0 \in\C$. Assume that $q, p, \ell$ define a line $ L : qj + pi = \ell$ in $(i, j)$ plane such that
There are at least two points $(j_1, i_1)\in{\mbox{{\rm carr}$(f)$}}$ and $(j_2, i_2)\in {\mbox{{\rm carr}$(f)$}}$ on $L$ with $i_1\neq i_2$.
For any $(j, i)\in{\mbox{{\rm carr}$(f)$}}$, we have $qj + pi \geq \ell$.
Let $f_1 := X_1^{-\ell}f(X_1^q, X_1^p(\beta + Y_1))$. Then, we have the following results
We have $f_1\in\C\langle X_1 \rangle[Y_1]$.
For any given $m_1\in\N$, there exists a finite number $m\in\N$ such that the approximation of $f_1$ of accuracy $m_1$ can be computed from the approximation of $f$ of accuracy $m$.
Moreover, it suffices to take $m=\lfloor \frac{m_1+\ell}{q} \rfloor$.
Since $q >0$ holds, we know that ${\mbox{{\rm ord}$(X_1^q)$}}=q>0$ holds. We also have $f(X_1^q, X_1^p(\beta + Y_1))\in\C\langle X_1 \rangle[Y_1]$. Let $f(X, Y) := \sum_{i=0}^d \left(\sum_{j=0}^{\infty} a_{i, j}X^j \right) Y^i$. Then we have $
f_1(X_1, Y_1) = \sum_{i=0}^d \left(\sum_{j=0}^{\infty} a_{i, j} X_1^{(qj+pi-\ell)} \right) (\beta + Y_1)^i.
$ Since for any $(j, i)\in{\mbox{{\rm carr}$(f)$}}$, we have $qj + pi \geq \ell$, the power of $X_1$ cannot be negative. By Lemma \[Lemma:GFsubs\], we have $f_1\in\C\langle X_1 \rangle[Y_1]$. That is $(i)$ holds.
We prove $(ii)$. We have $$\begin{array}{rcl}
&&f_1(X_1, Y_1)~\mbox{mod}~\langle X_1^{m_1} \rangle \\
&=& \sum_{i=0}^d \left(\sum_{qj+pi-\ell < m_1} a_{i, j} X_1^{(qj+pi-\ell)} \right) (\beta + Y_1)^i.
\end{array}$$ Since $q\in {\N}_{>0}$ and $m_1$, $\ell$ and $i$ are all finite, we know that $j$ has to be finite. In other words, there exists a finite $m$ such that the approximation of $f_1$ of accuracy $m_1$ can be computed from the approximation of $f$ of accuracy $m$. That is, $(ii)$ holds.
Since the first $m_1$ terms of $f_1$ depends on the $j$-th terms of $f$, which satisfies the constraint $qj+pi-\ell < m_1$, we have $
j < \frac{(m_1+\ell)-pi}{q} \leq \frac{(m_1+\ell)}{q}.
$ Let $m'$ be the the maximum of these $j$’s . Now we have $
m'-1 < \frac{(m_1+\ell)}{q}.
$ Since $m'$ is an integer, we have $
m'\leq \lfloor \frac{(m_1+\ell)}{q}\rfloor
$ holds. Let $m=\lfloor \frac{(m_1+\ell)}{q}\rfloor$. Next we show shat $m_1\geq 1$ implies that $m\geq 1$ holds. If there is at least one point $(i,j)\in L$ such that $j\geq 1$, then we have $\ell\geq q$, which implies $m\geq 1$. If the $j$-coordinates of all points on $L$ is $0$, then $q=1$ and $\ell=0$, which implies also $m\geq 1$. Thus $(iii)$ is proved. The proof is routine.
\[Remark:monotone\] We use the same notations as in the previous Lemma. In particular, let $f(X, Y) := \sum_{i=0}^d \left(\sum_{j=0}^{\infty} a_{i, j}X^j \right) Y^i$ and $f_1 := X_1^{-\ell}f(X_1^q, X_1^p(\beta + Y_1))$. For a fixed term $a_{i,j}X^jY^i$ of $f$, it appears in $f_1$ as $$a_{i,j}X_1^{qj+pi-\ell}(\beta+Y_1)^i
= \sum_{k=0}^i \left({i \choose k}\beta^{i-k} a_{i,j}X_1^{qj+pi-\ell} \right) Y_1^{k} .$$ For two fixed terms $a_{i,j_1}X^{j_1}Y^i$ and $a_{i,j_2}X^{j_2}Y^i$ of $f$ with $j_1<j_2$, they respectively appear in $f_1$ as $$a_{i,j_1}X_1^{q{j_1}+pi-\ell}(\beta+Y_1)^i
=\sum_{k=0}^i \left({i \choose k}\beta^{i-k} a_{i,j_1}X_1^{q{j_1}+pi-\ell} \right) Y_1^{k}$$ and $$a_{i,j_2}X_1^{q{j_2}+pi-\ell}(\beta+Y_1)^i
=\sum_{k=0}^i \left({i \choose k}\beta^{i-k} a_{i,j_2}X_1^{q{j_2}+pi-\ell} \right) Y_1^{k}.$$ since $q{j_1}+pi-\ell<q{j_2}+pi-\ell$, we know that for any fixed $k$, $a_{i,j_2}X^{j_2}Y^i$ always contributes strictly higher order of powers of $X_1$ than $a_{i,j_1}X^{j_1}Y^i$ in $f_1$.
\[Remark:polygon\] Let $f(X, Y) := \sum_{i=0}^d \left(\sum_{j=0}^{\infty} a_{i, j}X^j \right) Y^i$. For $0\leq i\leq d$, let $a_{i,j^*}$ be the first nonzero coefficient among $\{a_{i,j}| 0\leq j < \infty\}$. We observe that the Newton polygon of $f$ is completely determined by $a_{i,j^*}$, $0\leq i\leq d$.
\[Theorem:finite-bound\] Let $f\in \C\langle X \rangle[Y]$. Let $\tau\in{\N}_{>0}$. Let $\sigma\in{\N}_{>0}$ and $g(T)=\sum_{k=0}^{\tau-1} b_k T^k$. Assume that $(T^{\sigma}, g(T))$ is a Puiseux parametrization of $f$ of accuracy $\tau$. Then one can compute a finite number $m\in \N$ such that $(T^{\sigma}, g(T))$ is a Puiseux parametrization of accuracy $\tau$ of the approximation of $f$ of accuracy $m$. We denote by ${\sf AccuracyEstimate}$ an algorithm to compute such $m$ from $f$ and $\tau$.
Let $f_0 := f$, $X_0 := X$ and $Y_0 := Y$. For $i=1, 2, \ldots$, Newton-Puiseux’s algorithm computes numbers $q_{i}, p_{i}, {\ell}_{i}, \beta_{i}$ and the transformation $$f_i := X_i^{-\ell_{i}}f_{i-1}(X_i^{q_{i}}, X_i^{p_{i}}(\beta_{i} + Y_i))$$ such that the assumption of Lemma \[Lemma:one-iteration\] is satisfied.
By Lemma \[Lemma:one-iteration\], we know that for any $i$, a given number of terms of the coefficients of $f_i$ in $Y_i$ can be computed from a finite number of terms of the coefficients of $f_{i-1}$ in $Y_{i-1}$. Thus for any $i$, a given number of terms of the coefficients of $f_i$ in $Y_i$ can be computed from a finite number of terms of the coefficients of $f$ in $Y$.
On the other hand, the construction of Newton-Puiseux’s algorithm and Remark \[Remark:polygon\] tell us that there exists a finite $M$, such that $\sigma$ and all the terms of $g(T)$ can be computed from a finite number of terms of the coefficients of $f_i$ in $Y_i$, $i=1,\ldots,M$.
Thus we conclude that there exists a finite number $m\in \N$ such that $(T^{\sigma}, g(T))$ is a Puiseux parametrization of accuracy $\tau$ of the approximation of $f$ of accuracy $m$. By Lemma \[Lemma:one-iteration\] and the construction of Newton-Puiseux’s algorithm, we conclude that there exists a finite number $m\in \N$ such that $(T^{\sigma}, g(T))$ is a Puiseux parametrization of accuracy $\tau$ of the approximation of $f$ of accuracy $m$.
Next we show that there is an algorithm to compute $m$. We initially set $m' := \tau$. Let $f_0 := \sum_{i=0}^d \left(\sum_{j=0}^{m'} a_{i, j}X^j \right) Y^i$. That is, $f_0$ is the approximation of $f$ of accuracy $m'+1$. We run Newton-Puiseux’s algorithm to check whether the terms $a_{k,m'}X^{m'}Y^k$, $0\leq k\leq d$, make any contributions in constructing the Newton Polygons of all $f_i$. If at least one of them make contributions, we increase the value of $m'$ and restart the Newton-Puiseux’s algorithm until none of the terms $a_{k,m'}X^{m'}Y^k$, $0\leq k\leq d$, makes any contributions in constructing Newton Polygons of all $f_i$. By Remark \[Remark:monotone\], we can set $m := m'$. We set $m := m'$.
\[Lemma:prime-chain\] Let $d, \tau\in{\N}_{>0}$. Let $a_{i,j}$, $0\leq i\leq d$, $0\leq j < \tau $, and $b_k$, $0\leq k < \tau$ be symbols. Write $
{\bf a}=(a_{0,0},\ldots,a_{0,\tau-1},\\
\ldots,a_{d,0}, \ldots,a_{d,\tau-1})
$ and ${\bf b}=(b_0,\ldots,b_{\tau-1})$. Let $f({\bf a}, X, Y)= \sum_{i=0}^d \left(\sum_{j=0}^{\tau-1} a_{i, j}X^j \right) Y^i\in\C[{\bf a}][X, Y]$. Let $g({\bf b}, X)=
\sum_{k=0}^{\tau-1} \\
b_kX^k\in\C[{\bf b}][X]$. Let $p := f({\bf a}, X, Y =g({\bf b}, X))$. Let $F_k := {\mbox{{\rm coeff}$(p, X^k)$}}$, $0\leq k < \tau - 1\}$, and $F := \{F_0,\ldots, F_{\tau-1}\}$. Then under the order ${\bf a} < {\bf b}$ and $b_0<b_1<\cdots <b_{\tau-1}$, $F$ forms a zero-dimensional regular chain in $\C({\bf a})[{\bf b}]$ with main variables $(b_0,b_1,\ldots,b_{\tau-1})$ and main degrees $(d, 1,\ldots,1)$. In addition, we have
$F_0=\sum_{i=0}^d a_{i,0}b_0^i$ and
${\mbox{{\rm init}$(F_1)$}}=\cdots={\mbox{{\rm init}$(F_{\tau-1})$}}=\sum_{i=1}^d i\cdot a_{i, 0}b_0^{i-1}$.
Write $p=\sum_{i=0}^d \left(\sum_{j=0}^{\tau-1} a_{i, j}X^j\right) \left( \sum_{k=0}^{\tau-1} b_kX^k \right)^i $ as a univariate polynomial in $X$. Observe that $F_0=\sum_{i=0}^d a_{i,0}b_0^i$. Therefore $F_0$ is irreducible in $\C({\bf a})[{\bf b}]$. Moreover, we have ${\mbox{{\rm mvar}$(F_0)$}}=b_0$ and ${\mbox{{\rm mdeg}$(F_0)$}}=d$.
Since $d>0$, we know that $a_{1, 0} \left( \sum_{k=0}^{\tau-1} b_kX^k \right)$ appears in $p$. Thus, for $0\leq k <\tau$, $b_k$ appears in $F_k$. Moreover, for any $k\geq 1$ and $i<k$, $b_k$ can not appear in $F_i$ since $b_k$ and $X^k$ are always raised to the same power. For the same reason, for any $i>1$, $b_k^i$ cannot appear in $F_k$, for $1\leq k <\tau$. Thus $\{F_0,\ldots,F_{\tau-1}\}$ is a triangular set with main variables $(b_0,b_1,\ldots,b_{\tau-1})$ and main degrees $(d, 1,\ldots,1)$.
Moreover, we have ${\mbox{{\rm init}$(F_1)$}}=\cdots={\mbox{{\rm init}$(F_{\tau-1})$}}=\sum_{i=1}^d i\cdot a_{i, 0}b_0^{i-1}$, which is coprime with $F_0$. Thus $F= \{F_0,\ldots, F_{\tau-1}\}$ is a regular chain.
\[Lemma:generic\] Let $f= \sum_{i=0}^d \left(\sum_{j=0}^{\infty} a_{i, j}X^j \right) Y^i\in\C[[X]][Y]$. Assume that $\deg(f, Y)>0$ and $f$ is general in $Y$. Let $\varphi(X)=\sum_{k=0}^{\infty} b_kX^k\in\C[[X]]$ such that $f(X, \varphi(X))=0$ holds. Let $\tau>0\in\N$. Then “generically”, $b_i$, $0\leq i < \tau$, can be completely determined by $\{a_{i,j}\mid 0\leq i\leq d, 0\leq j<\tau\}$.
By $f(X, Y)=0$, we know that $f(X, Y)=0~\mbox{mod}~\langle X^{\tau}\rangle$. Therefore, we have $$\sum_{i=0}^d \left(\sum_{j<\tau} a_{i, j} X^{j} \right)\left(\sum_{k<\tau} b_kX^k \right)^i=0 ~\mbox{mod}~\langle X^{\tau}\rangle.$$ Let $p=\sum_{i=0}^d \left(\sum_{j<\tau} a_{i, j} X^{j} \right)\left(\sum_{k<\tau} b_kX^k \right)^i$. Let $F_i := \{{\mbox{{\rm coeff}$(p, X^i)$}}$, $0\leq i < \tau\}$, and $F := \{F_0,\ldots, F_{\tau-1}\}$. Since $f$ is general in $Y$ and $f(X, \varphi(X))=0$, there exists $i^*>0$ such that $a_{i^*,0}\neq 0$. By Lemma \[Lemma:prime-chain\], we have $F_0=\sum_{i=0}^d a_{i,0}b_0^i$. Thus $b_0$ can be completely determined by $a_{i,0}$, $0\leq i\leq d$. In order to completely determine $b_1,\ldots,b_{\tau-1}$, it is enough to gurantee ${\mbox{{\rm res}$(F_0, F_i, b_0)$}}\neq 0$ holds. Therefore the values of $b_k$, $0\leq k < \tau$ can be completely determined from almost all the values of $a_{i,j}$, $0\leq i\leq d$, $0\leq j < \tau $. It follows directly from Lemma \[Lemma:prime-chain\].
Accuracy estimates {#sec:bounds}
==================
Let $R := \{r_1(X_1,X_2),\ldots,
r_{s-1}(X_1,\ldots,X_s)\}\subset\C[X_1<\cdots<X_s]$ be a strongly normalized regular chain. In this section, we show that to compute the limit points of $W(R)$, it suffices to compute the Puiseux parametrizations of $R$ of some accuracy. Moreover, we provide accuracy estimates in Theorem \[Theorem:bound-general\].
\[Lemma:subs\] Let $f\in\C[X_1,\ldots,X_s]$. Let $g_1,\ldots,g_s\in \C\langle
T\rangle$. Then $f(g_1,\ldots,g_s)\in\C\langle T \rangle$.
It follows immediately from the fact that $f$ is a polynomial and $\C\langle T\rangle$ is a ring.
\[Lemma:general\] Let $f=a_d(X)Y^d+\cdots+a_0(X)\in\C\langle X\rangle[Y]$, where $d>0$ and $a_d(X)\neq 0$. For $0\leq i\leq d$, let $\delta_i
:= {\mbox{{\rm ord}$(a_i)$}}$. Let $k := {\mbox{{\rm min}$(\delta_0,\ldots,\delta_d)$}}$. Let $\widetilde{f} := f/X^k$. Then we have $\widetilde{f}\in\C\langle
X\rangle[Y]$ and $\widetilde{f}$ is general in $Y$. This process of producing $\widetilde{f}$ from $f$ is called [“ *making $f$ general”*]{} and denote by [MakeGeneral]{} an operation which produces $\widetilde{f}$ from $f$.
Since $k= {\mbox{{\rm min}$(\delta_0,\ldots,\delta_d)$}}$, there exists $i$, $1\leq
i\leq d$, such that $k=\delta_i$. Moreover, for all $1\leq j\leq
d$, we have $\delta_j \geq k$. Thus for every such $i$, we have ${\mbox{{\rm ord}$(a_i(X)/X^k)$}}=0$ and $a_j(X)/X^k\in \C\langle X\rangle$, $0\leq
j\leq d$. This shows that $\widetilde{f}\in\C\langle X\rangle[Y]$ and $\widetilde{f}$ is general in $Y$. The proof is routine.
The following lemma shows that computing limit points reduces to making a polynomial $f$ general.
\[Lemma:Walker\] Let $f\in\C\langle X\rangle[Y]$, where $\deg(f, Y)>0$. Assume that $f$ is general in $Y$. Let $\rho>0$ be small enough such that $f$ converges in ${\lvertX\rvert}<\rho$. Let $V_{\rho}(f) := \{(x,y)\in\C^2\mid 0<{\lvertx\rvert}<\rho, f(x,y)=0\}$. Then we have ${\rm lim}_0(V_{\rho}(f))=\{(0, y)\in\C^2\mid f(0,y)=0\}$.
Let $(X=T^{\varsigma_i}, Y=\varphi_i(T))$, $1\leq i\leq c\leq d$, be the Puiseux parametrizations of $f$. By Lemma \[Lemma:V0\] and Theorem \[thrm:PuiseuxParamRC\], we have ${\rm lim}_0(V_{\rho}(f))=\cup_{i=1}^c\{(0, y)\in\C^2\mid y=\varphi_i(0)\}$. Let $(X=T^{\sigma_i}, g_i(T))$, $i=1,\ldots,c$, be the corresponding Puiseux parametrizations of $f$ of accuracy $1$. By Theorem \[Theorem:finite-bound\], there exists an approximation $\widetilde{f}$ of $f$ of some finite accuracy such that $(X=T^{\sigma_i}, g_i(T))$, $i=1,\ldots,c$, are also Puiseux parametrizations of $\widetilde{f}$ of accuracy $1$. Thus, we have $\varphi_i(0)=g_i(0)$, $i=1,\ldots,c$. Since $\widetilde{f}$ is also general in $Y$, by Theorem 2.3 of Walker [@RW78], we have $\cup_{i=1}^c\{(0, y)\in\C^2\mid y=g_i(0)\}=\{(0, y)\in\C^2\mid \widetilde{f}(0,y)=0\}$. Since $\widetilde{f}(0,y)=f(0,y)$, the Lemma holds.
\[Lemma:finite\] Let $a(X_1,\ldots,X_s)\in\C[X_1,\ldots,X_s]$. Let $g_i=\sum_{j=0}^{\infty}c_{ij}T^j\in \C\langle T \rangle$. We write $a(g_1,\ldots,g_s)$ as $\sum_{k=0}^{\infty}b_kT^k$. To compute a given $b_k$, one only needs the set of coefficients $\{c_{i, j}\mid 1\leq
i\leq s, 0\leq j\leq k\}$.
We observe that any $c_{i, j}$, where $j > k$, does not make any contribution to $b_k$. The proof is routine.
\[Lemma:general-generic\] Let $f=a_d(X)Y^d+\cdots+a_0(X)\in\C\langle X\rangle[Y]$, where $d>0$, and $a_d(X)\neq 0$. Let $\delta := {\mbox{{\rm ord}$(a_d(X))$}}$. Then “generically”, a Puiseux parametrization of $f$ of accuracy $\tau$ can be computed from an approximation of $f$ of accuracy $\tau+\delta$.
Let $\widetilde{f} := {\sf MakeGeneral}(f)$. Observe that $f$ and $\widetilde{f}$ have the same system of Puiseux parametrizations. Then the conclusion follows from Lemma \[Lemma:general\] and Lemma \[Lemma:generic\].
\[Theorem:bound-general\] Let $R := \{r_1(X_1,X_2),\ldots,
r_{s-1}(X_1,\ldots,X_s)\}\subset\C[X_1<\cdots<X_s]$ be a strongly normalized regular chain. For $1\leq i\leq s-1$, let $h_i := {\mbox{{\rm init}$(r_i)$}}$, $d_i := \deg(r_i, X_{i+1})$ and $\delta_i := {\mbox{{\rm ord}$(h_i)$}}$. We define $f_i$, $2\leq i\leq s-1$, and $\varsigma_j$, $T_j$, $\varphi_j(T_j)$, $1\leq j\leq s-2$, as follows
Let $(X_1=T_1^{\varsigma_1}, X_2=\varphi_1(T_1))$ be a Puiseux parametrization of $r_1(X_1,X_2)$.
Let $f_i := r_i(X_1= T_1^{\varsigma_1},X_2=\varphi_1(T_1), \ldots,X_i=\varphi_{i-1}(T_{i-1}), \\
X_{i+1})$.
Let $(T_{i-1}=T_i^{\varsigma_i}, X_{i+1}=\varphi_i(T_i))$ be a Puiseux parametrization of $f_i$.
Then we have the following results:
Let $T_0 := X_1$, for $0\leq i\leq s-2$, define $g_i(T_{s-2}):=T_{s-2}^{\prod_{k=i+1}^{s-2}\varsigma_k}$, then we have $T_i = g_i(T_{s-2})$.
We have $f_{s-1}\in\C\langle T_{s-2}\rangle[X_{s}]$.
There exist numbers $\tau_1,\ldots,\tau_{s-2}\in\N$ such that in order to make $f_{s-1}$ general in $X_{s}$, it suffices to compute the polynomial parts of $\varphi_i$ of accuracy $\tau_i$, $1\leq i\leq s-2$. Moreover, if we write the algorithm ${\sf AccuracyEstimate}$ for short as $\theta$, the accuracies $\tau_i$ can be computed in the following manner
let $\tau_{s-2} := (\prod_{k=1}^{s-2} \varsigma_k)\delta_{s-1}+1$
let $\tau_{i-1} := {\mbox{{\rm max}$(\theta(f_i, \tau_i), (\prod_{k=1}^{i-1} \varsigma_k)\delta_{s-1}+1)$}}$, for $s-2 \geq i \geq 2$.
Generically, for $1\leq i\leq s-3$, we can choose $\tau_i=(\prod_{k=1}^{s-2} \varsigma_k)(\sum_{k=2}^{s-1}\delta_i)+1$.
The indices $\varsigma_k$ can be replaced with $d_k$, $k=1,\ldots,s-2$.
We prove $(i)$ by induction. Clearly $(i)$ holds for $i=s-2$. Suppose it holds for $i$. Then we have $$T_{i-1} = T_i^{\varsigma_i}
= \left( T_{s-2}^{\prod_{k=i+1}^{s-2}\varsigma_k} \right)^{\varsigma_i}
= \left( T_{s-2}^{\prod_{k=i}^{s-2}\varsigma_k} \right)$$ Therefore $(i)$ holds also for $i-1$. So $(i)$ holds for all $0\leq i\leq s-2$.
Note that $$\label{equation:fs-1}
\begin{array}{rcl}
f_{s-1} &=& r_{s-1}(X_1= T_1^{\varsigma_1},X_2=\varphi_1(T_1), \ldots, \\
&& ~\qquad X_{s-2}=\varphi_{s-3} (T_{s-3}), X_{s-1}=\varphi_{s-2}(T_{s-2}), X_{s})\\
&=& r_{s-1}(X_1=g_0(T_{s-2}),X_2=\varphi_1(g_1(T_{s-2})), \ldots, \\
&& ~\qquad X_{s-2}=\varphi_{s-3}(g_{s-3}(T_{s-2})), \\
&& ~\qquad X_{s-1}=\varphi_{s-2}(g_{s-2}(T_{s-2})), X_{s})\\
\end{array}$$ Since ${\mbox{{\rm ord}$(g_i(T_{s-2}))$}}>0$ for all $0\leq i\leq s-2$, by Lemma \[Lemma:GFsubs\], $(ii)$ holds. It is clear that $(ii)$ holds.
Note that $g_0(T_{s-2})= T_{s-2}^{\prod_{k=1}^{s-2}\varsigma_k}$. Since ${\mbox{{\rm ord}$(h_{s-1}(X_1))$}}=\delta_{s-1}$, we have $${\mbox{{\rm ord}$(h_{s-1}(X_1=g_0(T_{s-2})))$}}=\left(\prod_{k=1}^{s-2} \varsigma_k\right)\delta_{s-1}.$$ Let $\tau_{s-2} := (\prod_{k=1}^{s-2} \varsigma_k)\delta_{s-1}+1$. By Lemma \[Lemma:general\], to make $f_{s-1}$ general in $X_s$, it suffices to compute the polynomial parts of the coefficients of $f_{s-1}$ of accuracy $\tau_{s-2}$.
By Lemma \[Lemma:finite\], and Equation (\[equation:fs-1\]), we need to compute the polynomial parts of $\varphi_i(g_i(T_{s-2}))$, $1\leq i\leq s-2$, of accuracy $\tau_{s-2}$. Since ${\mbox{{\rm ord}$(g_i(T_{s-2}))$}}=\prod_{k=i+1}^{s-2}\varsigma_k$, to achieve this accuracy, it’s enough to compute the polynomial parts of $\varphi_i$ of accuracy $(\prod_{k=1}^i \varsigma_k)\delta_{s-1} + 1$, for $1\leq i\leq s-2$.
On the other hand, since $f_i = r_i(X_1= T_1^{\varsigma_1},X_2=\varphi_1(T_1),
\ldots, X_i=\varphi_{i-1}(T_{i-1}), X_{i+1})$ and $(T_{i-1}=T_i^{\varsigma_i}, X_{i+1}=\varphi_i(T_i))$ is a Puiseux parametrization of $f_i$, by Theorem \[Theorem:finite-bound\] and Lemma \[Lemma:finite\], to compute the polynomial part of $\varphi_i$ of accuracy $\tau_i$, we need the polynomial part of $\varphi_{i-1}$ of accuracy $\theta(f_i,\tau_i)$.
Thus, take $\tau_{s-2} := (\prod_{k=1}^{s-2} \varsigma_k)\delta_{s-1}+1$ and $\varphi_{i-1}=
{\mbox{{\rm max}$(\theta(f_i, \tau_i),(\prod_{k=1}^{i-1} \varsigma_k)\delta_{s-1}+1)$}}$ for $2\leq i\leq s-2$ will guarantee $f_{s-1}$ can be made general in $X_{s}$. So $(iii)$ holds. By Lemma \[Lemma:general-generic\], generically we can choose $\theta(f_i, \tau_i)=\tau_i+(\prod_{k=1}^{i-1}\sigma_k)\delta_i$, $2\leq i\leq s-2$. Therefore $(iv)$ holds. Since we have $\varsigma_k\leq d_k$, $1\leq k\leq s-2$, $(iv)$ holds.
Algorithm {#sec:principle}
=========
In this section, we provide a complete algorithm for computing the non-trivial limit points of the quasi-component of a one-dimensional strongly normalized regular chain based on the results of the previous sections.
\[algorithm:limitatzero\]
\[algorithm:limit\]
Note that line $9$ of Algorithm \[algorithm:limitatzero\] computes Puiseux parametrizations of $f_i$ of accuracy $\tau_i$. Thus $(\phi(T_i),\varphi(T_i))$ at line $10$ cannot have negtive orders.
If the D5 principle is applied to Algorithms \[algorithm:limitatzero\] and \[algorithm:limit\], the limit points of $W(R)$ can be represented by a finite family of regular chains.
\[Proposition:newton\] Algorithm \[algorithm:limit\] is correct and terminates.
It follows from Theorem \[thrm:PuiseuxParamRC\], Theorem \[Theorem:finite-bound\], Theorem \[Theorem:bound-general\] and Lemma \[Lemma:Walker\].
Experimentation {#section}
===============
We have implemented Algorithm \[algorithm:limit\] of Section \[sec:principle\], which computes the limit points of the quasi-component of a one-dimensional strongly normalized regular chain. The implementation is based on the library [[[RegularChains]{}]{}]{} and the command [algcurves\[puiseux\]]{} of [[[Maple]{}]{}]{}. The code is available at <http://www.orcca.on.ca/~cchen/ACM13/LimitPoints.mpl>. This preliminary implementation relies on algebraic factorization, whereas, as suggested in [@Duv89], applying the D5 principle, in the spirit of triangular decomposition algorithms, for instance [@CM12], would be sufficient when computations need to split into different cases. This would certainly improve performance greatly and this enhancement is work in progress.
As pointed out in the introduction, the computation of the limit points of the quasi-component of a regular chain can be applied to removing redundant components in a Kalkbrener triangular decomposition. In Table \[table:redundant\], we report on experimental results of this application.
The polynomial systems listed in this table are one-dimensional polynomial systems selected from the literature [@CGLMP07; @CM12]. For each system, we first call the [Triangularize]{} command of the library [[[RegularChains]{}]{}]{}, with the option “[’normalized=’strongly’, ’radical’=’yes’]{}”. For the input system, this process computes a Kalkbrener triangular decomposition ${\cal R}$ where the regular chains are strongly normalized and their saturated ideals are radical. Next, for each one-dimensional regular chain $R$ in the output, we compute the limit points $\lim(W(R))$, thus deducing a set of regular chains $R_1, \ldots, R_e$ such the union of their quasi-components equals the Zariski closure $\overline{W(R)}$. The algorithm [Difference]{} [@CGLMP07] is then called to test whether or not there exists a pair $R, R'$ of regular chains of ${\cal R}$ such that the inclusion $\overline{W(R)} \, \subseteq \, \overline{W(R')}$ holds.
In Table \[table:redundant\], the column T and \#(T) denote respectively the timings spent by [Triangularize]{} and the number of regular chains returned by this command; the column d-1 and d-0 denote respectively the number of $1$-dimensional and $0$-dimensional regular chains, whose sum is exactly \#(T); the column R and \#(R) denote respectively the timings spent on removing redundant components in the output of [Triangularize]{} and the number of regular chains in the output irredundant decomposition. As we can see in the table, most of the decompositions are checked to be irredundant, which we could not do before this work by means of triangular decomposition algorithms. In addition, the three redundant $0$-dimensional components in the Kalkbrener triangular decomposition of system f-744 are successfully removed. Therefore, we have verified experimentally the benefits provided by the algorithms presented in this paper.
Sys T \#(T) d-1 d-0 R \#(R)
--------------- -------- ------- ----- ----- --------- -------
f-744 14.360 4 1 3 432.567 1
Liu-Lorenz 0.412 3 3 0 216.125 3
MontesS3 0.072 2 2 0 0.064 2
Neural 0.296 5 5 0 1.660 5
Solotareff-4a 0.632 7 7 0 32.362 7
Vermeer 1.172 2 2 0 75.332 2
Wang-1991c 3.084 13 13 0 6.280 13
: Removing redundant components.[]{data-label="table:redundant"}
Concluding remarks {#sec:discussion}
==================
We conclude with a few remarks about special cases and a generalization of the algorithms presented in this paper.
[**Reduction to strongly normalized chains.**]{} Using the hypotheses of Lemma \[Lemma:limitWR3\], we observe that one can reduce the computation of to that of . Indeed, under the assumption that has dimension one, both and are finite. Once the set is computed, one can easily check which points in do not belong to $W(R)$ and then deduce . This reduction to strongly normalized regular chains has the advantage that $h_N$ is a univariate polynomial in ${\C}[X_1]$, which simplifies the presentation of the basic ideas of our algorithms, see Section \[sec:limitpoints\]. However, it has two drawbacks. First the coefficients of $N$ are generally much larger than those of $R$. Secondly, may also be much larger than . A detailed presentation of a direct computation of , without reducing to , will be done in a future paper.
[**Shape lemma case.**]{} Here, by reference to the paper [@BeckerMoraMarinariTraverso1994] (which deals with polynomial ideals of dimension zero) we assume that, for $2 \leq i \leq e$, the polynomial $r_i$ involves only the variables $X_1, X_2, X_i$ and that ${\deg}(r_i, X_i) = 1$ holds. In this case, computing Puiseux series expansions is required only for the polynomial of $R$ of lower rank, namely $r_1$. In this case, the algorithms presented in this paper are much simplified. However, for the specific purpose of solving polynomial systems via triangular decompositions, reducing to this Shape lemma case, via a random change of coordinates, has a negative impact on performance and software design, for many problems of practical interest. In contrast, the point of view of the work initiated in this paper is two-fold: first, deliver algorithms that do not require any genericity assumptions; second develop criteria that take advantage of specific properties of the input systems in order to speedup computations. Yet, in our implementation, several tricks are used to avoid unnecessary Puiseux series expansions, such as applying the theorem (see [@GF01] p.113) on the continuity of the roots of a parametric polynomial. [**Handling the case where has dimension greater than 1.**]{} From now on, has dimension $s-e \geq 2$. We use the notations of Lemma \[Lemma:limitWR5\] and recall that each point of is in particular a point of . Since we know how to compute when $R'$ consists of a single polynomial, we assume, by induction that a triangular decomposition of has been computed in the form ${W(R_1)} \, \cup \, \cdots \, \cup \, {W(R_f)}$ for regular chains $R_1, \ldots, R_f$.
We observe that a point $p \in {\mbox{${\lim}(W(R'))$}}$ can be “extended” to a point of in two ways. First, if $p$ does not cancel the initial of $r_e$ (which can be tested algorithmically), then, by applying the theorem (see again [@GF01] p.113) on the continuity of the roots to $r_e$, we extend $p$ with the $X_s$-roots of $r_e$, after specializing $(X_1, \ldots, X_{s-1})$ to $p$. From now on, we assume that $p$ cancels the initial $h_e$ of $r_e$. In this case, we compute a truncated Puiseux parametrization about $p$ using the regular chain $R_i$ such that $p \in W(R_i)$ holds. After substitution into the polynomial $r_e$, we apply Puiseux theorem and compute the limit points of $W(R' \cup r_e)$ extending $p$, in the manner of the algorithms of Section \[sec:principle\].
There are new challenges, however, w.r.t. to the one-dimensional case. First, parametrizations may involve now more than one parameter. When this happens, one should use Jung-Abhyankar theorem [@APGR00] instead of the Puiseux theorem. The second difficulty is that ${\mbox{${\lim}(W(R'))$}} \, \cap \, V(h_e)$ may be infinite. This will not happen, however, if has dimension at most $2$ and $h_e$ is regular w.r.t. . This second assumption can be regarded as a genericity assumption. Thus the algorithms presented can easily be extended to dimension two, under that assumption, which can be tested algorithmically. Overcoming in higher dimensions this cardinality issue with ${\mbox{${\lim}(W(R'))$}} \, \cap \, V(h_e)$, requires to understand which “configurations” are essentially the same. Since , as an algebraic set, can be described by finitely many regular chains, this is, indeed, possible and work in progress.
[10]{}
M. E. Alonso, T. Mora, G. Niesi, and M. Raimondo. An algorithm for computing analytic branches of space curves at singular points. In [*Proc. of the 1992 International Workshop on Mathematics Mechanization*]{}, pages 135–166. International Academic Publishers, 1992.
E. Becker, T. Mora, M. G. Marinari, and C. Traverso. The shape of the shape lemma. In [*[Proc. of ISSAC’94]{}*]{}, pages 129–133. ACM Press, 1994.
F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. . In [*Proc. of ISSAC’95*]{}, pages 158–166, 1995.
F. Boulier, F. Lemaire, and M. [Moreno Maza]{}. . Technical Report LIFL 2001–09, Université Lille I, LIFL, 2001.
C. Chen, J. H. Davenport, J. P. May, M. Moreno Maza, B. Xia, and R. Xiao. Triangular decomposition of semi-algebraic systems. , 49:3–26, 2013.
C. Chen, O. Golubitsky, F. Lemaire, M. [[Moreno Maza]{}]{}, and W. Pan. Comprehensive triangular decomposition. In [*[Proc. of CASC’07]{}*]{}, volume 4770 of [*Lecture Notes in Computer Science*]{}, pages 73–101. Springer Verlag, 2007.
C. Chen and M. Moreno Maza. Algorithms for computing triangular decomposition of polynomial systems. , 47(6):610–642, 2012.
C. Chen and M. [Moreno Maza]{}. Algorithms for computing triangular decomposition of polynomial systems. , 47(6):610 – 642, 2012.
C. Chen, M. [Moreno Maza]{}, B. [Xia]{}, and L. Yang. Computing cylindrical algebraic decomposition via triangular decomposition. In [*[Proc. of ISSAC’09]{}*]{}, pages 95–102, 2009.
S. C. Chou and X. S. Gao. . In [*Proc. ISSAC’91*]{}, pages 122–127, Bonn, Germany, 1991.
S. C. Chou and X. S. Gao. A zero structure theorem for differential parametric systems. , 16(6):585–596, 1993.
D. Cox, J. Little, and D. O’Shea. . Spinger-Verlag, 2nd edition, 1997.
J. [Della Dora]{}, C. Dicrescenzo, and D. Duval. About a new method for computing in algebraic number fields. In [*Proc. of EUROCAL’ 85 Vol. 2*]{}, volume 204 of [*Lect. Notes in Comp. Sci.*]{}, pages 289–290. Springer-Verlag, 1985.
D. Dominique. Rational [Puiseux]{} expansions. , 70(2):119–154, 1989.
G. Fischer. . American Mathematical Society, United States of America, 2001.
X. S. Gao, J. [Van der Hoeven]{}, Y. Luo, and C. Yuan. Characteristic set method for differential-difference polynomial systems. , 44:1137–1163, 2009.
É. Hubert. Factorization free decomposition algorithms in differential algebra. , 29(4-5):641–662, 2000.
M. Kalkbrener. Algorithmic properties of polynomial rings. , 26(5):525–581, 1998.
F. Lemaire, M. [Moreno Maza]{}, and Y. [Xie]{}. The [[[RegularChains]{}]{}]{} library. In [*Maple 10*]{}, Maplesoft, Canada, 2005. Refereed software.
François Lemaire, Marc Moreno Maza, Wei Pan, and Yuzhen Xie. When does $<$t$>$ equal sat(t)? , 46(12):1291–1305, 2011.
S. Marcus, M. Moreno Maza, and P. Vrbik. On [[F]{}]{}ulton’s algorithm for computing intersection multiplicities. In [*Proc. of CASC’12*]{}, pages 198–211, 2012.
J. Maurer. Puiseux expansion for space curves. , 32:91–100, 1980.
D. Mumford. . Springer, 2nd. edition, 1999.
J. R. Munkres. . Prentic Hall, United States of America, 2nd. edition, 2000.
A. Parusinski and G. Rond. The [[A]{}]{}bhayankar-[[J]{}]{}ung theorem. , 365:29–41, 2012.
J. F. Ritt. , volume 14. American Mathematical Society, New York, 1932.
T. Shimoyama and K. Yokoyama. Localization and primary decomposition of polynomial ideals. , 22(3):247–277, 1996.
R. J. Walker. . Springer-Verlag, Berlin-New York, 1978.
D. K. Wang. The [[[Wsolve]{}]{}]{} package. .
D. M. Wang. Epsilon 0.618. wang/epsilon.
L. Yang, X. R. Hou, and B. Xia. A complete algorithm for automated discovering of a class of inequality-type theorems. , 44(6):33–49, 2001.
[^1]: The radical of the saturated ideal of $R$ is equal to the intersection of the radicals of the saturated ideals of $R_1, \ldots, R_e$.
[^2]: This identification of the closures of $W(R)$ in Zariski topology and the Euclidean topology holds when is .
[^3]: It is a simplified version of Duval’s definition.
|
---
author:
- 'Aykut İşleyen, Yasemin Vardar, , and Cagatay Basdogan,'
bibliography:
- 'IEEEabrv.bib'
- 'draft.bib'
title: Tactile Roughness Perception of Virtual Gratings by Electrovibration
---
screens have been used in a wide range of portable devices nowadays, but our interactions with these devices mainly involve visual and auditory sensory channels. While a commercial touch screen today can easily detect finger position and hand gestures, it provides limited tactile feedback. However, tactile feedback can be used as an additional sensory channel to convey information and also reduce the perceptual and cognitive load on the user. Currently, friction modulation is the most promising approach to display tactile feedback through a touch screen. In this regard, there are two promising techniques: ultrasonic and electrostatic actuation. In the case of ultrasonic actuation [@61; @65; @79; @60; @64; @95; @96; @97], the surface is vibrated at an ultrasonic resonance frequency. As a result, a squeezed thin film of air between finger and the surface is formed. This layer breaks the direct contact of finger with the surface and hence leads to a reduction in friction. On the other hand, electrostatic actuation [@2; @1; @4; @5; @12] increases the friction between finger and surface by electroadhesion. When an alternating voltage is applied to the conductive layer of a capacitive touch screen, an attractive electrostatic force is generated in the normal direction between the finger and the surface. By controlling the amplitude, frequency, and waveform of the input voltage, the frictional force between the sliding finger and the touch screen can be modulated. This technology has a great potential especially in mobile applications including online shopping, games, education, data visualization, and development of aids for blind and visually impaired. In this context, one important aim is to render realistic virtual textures on touch screens. Texture information on touch screens would improve the user experience in daily activities. For example, feeling the simulated texture of a jean before purchasing it from Internet would certainly be more motivating for online shoppers. However, our knowledge on tactile perception of virtual textures displayed by friction modulation is quite limited though tactile perception of real textures has been already investigated extensively in the literature. Based on the multi-dimensional-scaling (MDS) studies conducted by Hollins et. al. [@29; @45], there are three independent perceptual dimensions in texture perception: roughness, hardness, and warmness. Among the three dimensions, roughness is arguably the most important dimension in tactile perception of textures.
To investigate the roughness perception of real textures, several types of stimuli have been used; raised dots with controlled height and density [@90; @36], dithered cylindrical raised elements [@18; @26; @19], and metal plates with linear gratings [@20; @23; @30]. These studies have shown that size of the tactile elements (i.e gratings, dots, cones) and the spacing between them are critical parameters in roughness perception. Moreover, Hollins et al. [@19] and Klatzky and Lederman [@52] found that the underlying mechanism behind roughness perception is different for micro-textures (textures having inter-element spacing approximately smaller than 0.2 mm) and macro-textures (textures having inter-element spacing approximately larger than 0.2 mm). At macro-textural scale, Lederman and colleagues [@18; @20; @23; @30] observed that groove width (GW) has a greater effect on perceived roughness than ridge width (RW). This observation has been supported by other studies later [@90; @36; @46] reporting that the perceived roughness increases with the groove width (and hence with the spatial period) until it saturates.
In contrast to the extensive literature on real textures [@29; @45; @90; @36; @18; @26; @19; @52; @20; @23; @30; @46; @98], the number of studies investigating the roughness perception of virtual textures rendered on a touch surface by electrovibration is limited. The studies in this area have mainly focused on the estimation of perceptual thresholds for periodic stimuli so far, but not their roughness perception. Bau et al. measured the sensory thresholds of electrovibration using sinusoidal inputs applied at different frequencies [@2]. They showed that the change in threshold voltage as a function of frequency followed a U-shaped curve similar to the one observed in vibrotactile studies. Later, Wijekoon et al. [@12], followed the work of [@3], and investigated the perceived intensity of friction generated by electrovibration. Their experimental results showed that the perceived intensity was logarithmically proportional to the amplitude of the applied voltage signal.
Additionally, there are also studies that investigate the underlying perceptual mechanism of virtual textures. Vardar et al. [@4; @51] studied the effect of input voltage waveform on our tactile perception of electrovibration. Through psychophysical experiments with 8 subjects, they showed that humans were more sensitive to tactile stimuli generated by square wave voltage than sinusoidal one at frequencies below $60$ Hz. They showed that Pacinian channel was the primary psychophysical channel in the detection of the electrovibration stimuli, which is most effective to tactile stimuli at frequencies around $250$ Hz. Hence, the stronger tactile sensation caused by a low-frequency square wave was due to its high-frequency components stimulating the Pacinian channel.
There are only a few studies in the literature on roughness perception of virtual gratings rendered by electrovibration. Ilkhani et. al [@67] conducted multidimensional scaling analysis (MDS) on data-driven textures taken from Penn Haptic Texture Toolkit [@73] and concluded that roughness is one of the main dimensions in tactile perception of virtual textures. Vardar et al. [@53] investigated the roughness perception of four waveforms; sine, square, triangular and saw-toothed waves with spatial period varying from $0.6$ to $8$ mm. The width of periodic high friction regimes (analogous to ridge width) was taken as $0.5$ mm, while the width of the low friction regimes (analogous to groove width) was varied. The finger velocity was controlled indirectly by displaying a visual cursor moving at $50$ mm/s. The results showed that square waveform was perceived as the roughest, while there was no significant difference between the other three waveforms. Vardar et al. [@89] also investigated the interference of multiple tactile stimuli under electrovibration. This interference is called tactile masking and can cause deficits in perception. They showed that sharpness perception of virtual edges displayed on touch screens depends on the “haptic contrast” between background and foreground tactile stimuli, which varies as a function of masking amplitude and activation levels of frequency-dependent psychophysical channels. This outcome suggests that tactile perception of virtual gratings can be altered by masking since they are constructed by a series of rising and falling virtual edges.
As it is obvious from the above paragraph, the number of studies on roughness perception of virtual textures rendered on a touch surface by electrovibration is only a few and the underlying perceptual mechanisms have not been established yet. In this study, we investigate how the perceived roughness of real and virtual square gratings change as a function of spatial period and normal force applied by finger to the touch screen. Earlier studies on real gratings [@18; @19; @22; @24; @21; @25] mostly investigated the roughness perception but not paid sufficient attention to the contact interactions. However, it was shown in [@20] and [@23] that perceived roughness increases with increasing normal force applied by finger. Also, Taylor and Lederman [@28] showed that skin penetration into the inter-element spacing might predict the perceived roughness as a function of groove width and normal force. Since friction modulation displays cannot explicitly render surface topography in the normal direction, it is expected that perception of virtual gratings do not perfectly match their real counterparts. Our results also suggest that perception of real and virtual textures is mediated by different mechanisms.
Virtual Texture Rendering
=========================
The virtual textures should be rendered as realistically as possible to be able to investigate the perceptual differences between them and their real counterparts systematically. However, the best method for rendering realistic virtual textures on touchscreens has yet to be developed. In this section, we explain our virtual texture rendering method. In order to render virtual square gratings that mimic the real ones, we first investigated the contact interactions between human finger and real square gratings. For this purpose, we recorded the contact forces and analysed them in both time and frequency domain.
We manufactured square gratings from plexiglass using a laser cutter in different groove widths. We fixed the ridge width of the gratings as $1$ mm and varied the groove width from $1.5$ mm to $7.5$ mm (corresponds to varying spatial period from $2.5$ mm to $8.5$ mm) to produce $6$ different gratings, similar to the ones utilized in the earlier texture studies (see Table \[table:stimulus2\]). To analyze the frequency spectrum of contact forces, we selected one of the gratings with a spatial period of $2.5$ mm and recorded the frictional forces acting on the finger of one participant (i.e. the experimenter) while he slides his finger on the grating with a velocity of $50$ mm/s under a constant normal force of $0.75$ N. As shown in Fig. \[fig:realforce\]a, the period of the tangential force signal was $0.05$ sec, corresponding to a temporal frequency of $20$ Hz ($50/2.5$). Hence, the finger spends 0.02 and 0.03 secs on each ridge and groove respectively, leading to a duty cycle of $0.4$ ($0.02/0.05$). The power spectrum of the force signal (Fig. \[fig:realforce\]b) revealed a series of peaks with decreasing magnitude at frequencies that are integer multiples of the temporal frequency. This spectrum resembles to the power spectrum of a pulse train signal. In order to generate virtual gratings having the similar frequency spectrum of the real gratings, we modulated a low frequency pulse train voltage with a high frequency carrier voltage signal as suggested in [@56; @82; @93]. As discussed by Shultz et. al [@82], impedance of the gap between fingerpad and touch screen causes a volatile transition of force dynamics in the frequency range of $20 - 200$ Hz. They argued that amplitude modulation with a high frequency carrier voltage signal avoids this transition regime and hence, the modulated voltage signal results in a tangential force signal with a rectified DC component coming from the envelope signal and an AC component coming from the carrier signal (Fig. \[fig:realforce\]c). If the frequency of carrier signal is selected as higher than the human vibrotactile threshold level of $1$ kHz [@57; @58], then the AC component of the resulting tangential force is not perceived by the user.
![(a) Friction force signal acquired from a real grating (spatial period $=$ $2.5$ mm) under the constant normal force of $0.75$ N and the targeted exploration speed of $50$ mm/s. The period of the signal is approximately $0.05$ sec, which corresponds to the temporal frequency of $20$ Hz ($50/2.5$). (b) Power spectrum of the tangential force signal shown in (a); peaks appear at the integer multiples of the temporal frequency. (c) Tangential force signal acquired from the corresponding virtual grating displays high and low friction regimes. (d) Power spectrum of the tangential force signal shown in (c).[]{data-label="fig:realforce"}](fig_record.png){width="3.5"}
The signal modulation technique discussed above, in fact, creates periodic widths of high and low friction regimes (zones) on the surface of touch screen (Fig. \[fig:periyodik-real-virtual\]). For example, if we design a virtual grating using the envelope frequency of $20$ Hz, duty cycle of $0.4$ (high friction width/spatial period), and carrier frequency of $3$ kHz, then the frequency spectrum of the resulting tangential force signal (Fig. \[fig:realforce\]d) resembles to the one observed for the real grating (Fig. \[fig:realforce\]b). However, we should note that the resulting high and low friction zones in our design depend on the selected exploration speed. If the exploration speed is changed, the temporal signals should be redesigned accordingly.
![Grooves and ridges of a real grating are rendered as high and low friction widths (regimes) in the corresponding virtual grating, respectively. Finger penetrates into the real grating but not the virtual grating since grating height cannot be rendered explicitly by electrovibration. In our approach, the virtual gratings are designed in temporal domain and not in spatial domain, hence the finger cannot be partly on a high friction zone and partly on a low friction zone.[]{data-label="fig:periyodik-real-virtual"}](finger_periodical.png){width="3.5"}
Psychophysical Experiments
==========================
We conducted psychophysical experiments on roughness perception of real and virtual gratings. In particular, we investigated the effect of spatial period and normal force on perceived roughness of real and virtual gratings. We initially aimed to conduct the experiments under $3$ different normal forces ($0.25$, $0.75$, $1.75$ N) and $6$ different spatial periods ($2.5$, $3$, $3.5$, $4.5$, $6.5$, $8.5$ mm), displayed multiple times in random order. However, during our preliminary studies, we observed that the tactile sensitivity of the participants was reduced due to finger wear when the number of trials was high. Hence, we simplified our experimental design and divided the experiments into two sets, executed in multiple sessions in different days to prevent finger wear. These two sets of experiments were performed for both real and virtual gratings for comparison. In the first set (Exp. 1), we conducted roughness estimation experiments for $6$ different spatial periods ($2.5$, $3$, $3.5$, $4.5$, $6.5$, $8.5$ mm) under the normal force of $0.75$ N. In the second set (Exp. 2), we conducted roughness estimation experiments for $3$ different normal forces ($0.25$, $0.75$, $1.75$ N) under $2$ spatial periods of $2.5$ mm and $8.5$ mm.
Experimental Setup
------------------
The experimental setups for investigating the roughness perception of real and virtual gratings were slightly different (Fig. \[fig:exp\_set\]). For both setups, a compact monitor displaying a visual cursor moving with a speed of $50$ mm/s was placed below the real grating surface and the virtual one displayed through the touch screen to adjust the exploration speed of the participants. A force sensor (Nano17, ATI Inc.) was also placed under the grating surface to acquire normal and tangential forces in each trial. The force sensor had a sampling rate of $10$ kHz. In addition, an IR frame with a positional resolution of $1$ mm and a sampling rate of $85$ Hz was placed above the grating surface to monitor the exploration speed of the participants.
To display virtual gratings, a surface capacitive touch screen (SCT3250, 3M Inc.) with dimensions of $20$ x $15$ cm was used (see Fig. \[fig:exp\_set\]b). Voltage signal applied to the conductive layer of touch screen was generated by a DAQ card (USB-6251, National Instruments Inc.) working at a sampling rate of $10$ kHz. The signal was boosted by an amplifier (E-413, PI Inc.) before transmitted to the touch screen. As mentioned earlier, the real gratings were manufactured from plexiglass using a laser cutter. Each real grating surface had a length of $100$ mm and a width of $30$ mm.
![The experimental setup for investigating tactile roughness perception of real (a) and virtual (b) gratings.[]{data-label="fig:exp_set"}](exp_set_2.png){width="3.5"}
Participants
------------
Both experiments (Exp. 1, Exp. 2) were conducted with $2$ different groups of $10$ participants. The average ages of the participants in Exp. 1 and Exp. 2 were $24.9$ $\pm$ $1.3$ and $24.3$ $\pm$ $1.2$, respectively. Both groups were made of $5$ male and $5$ female participants. All participants were senior undergraduate or graduate university students and right-handed. They washed their hands with soap and rinsed with water before each session of the experiment. Moreover, their index finger and the touch screen were cleaned using ethanol before each session. Participants read and signed the consent form approved by Ethical Committee for Human Participants of Koç University.
Experimental Procedure
----------------------
In both experiments, the same experimental protocol was followed. The participants were instructed to sit on a chair and move their index fingers on the grating in tangential direction back and forth only once while synchronizing their finger movements with the movement of a visual cursor displayed by a monitor (see Fig. \[fig:exp\_set\]). The participants were allowed to replay each stimulus only once by pressing the ’$0$’ button on a numpad. To prevent any noise affecting their tactile perception, they were asked to wear noise cancellation headphones. Before the experiment, the participants were given instructions about the experiment and presented with a training session displaying all stimuli of that session once in random order. In both experiments, it took about $30$ minutes for each participant to complete one session including training. If the magnitude of mean normal force applied by the participant was $\pm30\%$ off the desired value, the participants were prompted to repeat the trial. In the case of real gratings, the participants sat on a chair at a table and extended their dominant arm under a curtain that prevented them from seeing the grating surface and the experimenter seated on the opposite site (see Fig. \[fig:exp\_set\]a). The experimenter manually changed the real grating surfaces during the experiments and it took around $3$ seconds to make the change. On the other hand, the virtual gratings were displayed automatically by the computer after each trial with no external intervention by the experimenter (Fig. \[fig:exp\_set\]b). After each trial, the participants entered their ratings of the stimulus using a small numpad. In both cases (real and virtual), participants were allowed to enter any positive number as their magnitude estimation of tactile roughness. They could see their responses on the user interface and could change it until they hit the ’return’ button. After hitting the ’return’ button on numpad, a new grating was displayed to the participants for exploration.
Experiment 1
------------
In Exp. 1, we investigated the effect of spatial period on roughness perception of real and virtual gratings separately for the normal force of $0.75$ N.
### Stimuli {#stimuli .unnumbered}
We selected the spatial periods of the real and virtual gratings in reference to the earlier studies on real gratings (see Table \[table:stimulus2\]). The voltage signal for virtual gratings was generated using the amplitude modulation technique discussed in Section 2. The frequency of the carrier signal was fixed at $3$ kHz, but the frequency of the envelope signal and the duty cycle were set according to the desired spatial period (see Table \[table:stimulus2\]). The number of real and virtual gratings displayed separately to participants was $108$ ($6$ spatial periods x $6$ repetitions x $3$ sessions). Hence, each session of the experiments for real and virtual gratings consisted of $36$ stimuli.
\[table:stimulus2\]
{width="7"}
Experiment 2
------------
In Exp. 2, we investigated the effect of normal force on roughness perception of real and virtual gratings separately for 2 different spatial periods of $2.5$ and $8.5$ mm.
### Stimuli {#stimuli-1 .unnumbered}
The normal forces used in the experiment were $0.25$ N (low), $0.75$ N (medium), and $1.75$ N (high). The number of real and virtual gratings displayed separately to participants was $108$ ($3$ normal forces x $6$ repetitions x $2$ spatial periods x $3$ sessions). Hence, each session of the experiments for real and virtual gratings consisted of $36$ stimuli.
Results
=======
Data Analysis
-------------
We used the same data analysis procedure for both experiments. First, we discarded the outliers of roughness estimates in each session using Peirce’s criterion. Then, we normalized roughness estimates of each participant using the method suggested in [@59]. We first computed the geometric mean of roughness estimates for each session, $GM_S$, and then, the geometric mean of all sessions, $GM_{TOTAL}$. Finally, we calculated the normalized estimates for each session by multiplying each estimate with $GM_{TOTAL}/GM_S$.
We used the position data acquired by IR frame to calculate the average finger speed of participants in each trial. If the actual exploration speed of a participant was $\pm30\%$ off the targeted value of $50$ mm/s, all the related data of that trial was discarded to obtain more consistent results.
During each trial, both normal and tangential forces were recorded using the force sensor. A data segment of $1$ second was chosen symmetrically with respect to the location of force sensor (i.e. the mid-point of travel distance) for each trial. A bandpass filter having the cut-off frequencies of $1.25$ Hz and $1$ kHz was applied to this data segment. The following metrics were calculated for the filtered force data of each trial: average tangential force, ($F_t$), average normal force, and root mean square (rms) of rate of change in tangential force ($dF_t/dt$). Each metric was normalized between $0$ and $1$ for each session and then the average of all sessions was considered as the mean value of the participant. The average of the mean values of all participants were reported in the plots (Fig. \[fig:exp1\] and Fig. \[fig:exp2\]).
Results of Experiment 1
-----------------------
Table \[table:normalforce\_speed1\] shows the average exploration speeds and average normal forces applied by the participants. Normalized roughness estimates (means and standard mean errors) of real and virtual gratings are plotted as a function of spatial period for the normal force of $0.75$ N in Fig. \[fig:exp1\]a.
\[table:normalforce\_speed1\]
The results were analyzed using one-way ANOVA repeated measures. The results showed that spatial period had a significant effect on the perceived roughness of both real and virtual gratings (p $<$ $0.01$). Bonferroni corrected paired t-tests showed that the difference in roughness estimates of real gratings was significant up to the spatial periods of $4.5$ mm, as reported in the earlier studies. On the other hand, the difference in roughness estimates of virtual gratings was significant for spatial periods higher than $3.5$ mm (p $<$ $0.01$).
The average tangential force ($F_t$) and rate of change in tangential force ($dF_t/dt$) are plotted as a function of spatial period for real and virtual gratings in Fig. \[fig:exp1\]b and Fig. \[fig:exp1\]c, respectively. We analyzed these results using one-way ANOVA repeated measures again. The results showed that spatial period had a significant effect on $F_t$ and $dF_t/dt$ for both real and virtual gratings (p $<$ $0.01$). Bonferroni corrected paired t-tests showed that, for real gratings, the difference in $F_t$ was significant for the spatial periods below $8.5$ mm (p $<$ $0.01$) while it was significant for the spatial periods above $3$ mm (p $<$ $0.01$) for virtual gratings. Moreover, the difference in $dF_t/dt$ was statistically significant for spatial periods below $4.5$ mm for real gratings (p $<$ $0.01$) and above $3$ mm for virtual gratings (p $<$ $0.01$).
{width="6.8"}
Results of Experiment 2
-----------------------
Table \[table:normalforces3\] shows the average exploration speeds and the average normal forces applied by participants. The normalized roughness estimates (means and standard mean errors) of real and virtual gratings were plotted as a function of normal forces ($0.25$ N, $0.75$ N, and $1.75$ N) for the spatial periods of $2.5$ mm and $8.5$ mm (see Fig. \[fig:exp2\]a).
\[table:normalforces3\]
The results were analyzed using two-way ANOVA repeated measures. The results showed that both spatial period and normal force had a significant effect on the perceived roughness of both real and virtual gratings (p $<$ $0.01$). Bonferroni corrected paired t-tests showed that, for both real and virtual gratings, the differences in roughness estimates were significant for both spatial periods under all normal forces (p $<$ $0.01$). Average tangential force ($F_t$) and rate of change in tangential force $dF_t/dt$ are plotted as a function of normal forces for both spatial periods in Fig. \[fig:exp2\]b and \[fig:exp2\]c, respectively. We also analyzed these results using two-way ANOVA repeated measures. The results showed that for both real and virtual gratings, spatial period and normal force, had a significant effect on $F_t$ and $dF_t/dt$. Bonferroni corrected paired t-tests showed that, for both real and virtual gratings, the differences in $F_t$ and $dF_t/dt$ were significant for all spatial periods and normal forces (p $<$ $0.01$).
Discussion
==========
To investigate the roughness perception of real and virtual gratings, we conducted $2$ psychophysical experiments. The results showed that there are perceptual differences between real and virtual gratings.
The results obtained for real gratings in the first experiment (Exp. 1) are inline with the results of earlier studies, which reported that perceived tactile roughness of macro gratings increases with increasing spatial period [@26; @20; @30; @27; @86]. However, this increase in perceived roughness saturates around the spatial period of $4.5$ mm [@26; @20; @30; @27; @9; @31; @39], as observed in our study (see Fig. \[fig:lit\]). Our results also show that the tangential force and its rate of change follow similar trends (Fig. \[fig:exp1\]). On the other hand, the results obtained for virtual gratings in our study deviated significantly from those of the real ones. The roughness estimates of participants for virtual gratings, in contrast to the real ones, followed a decreasing trend with increasing spatial period. In fact, this is not surprising since grating height, which is important for activating SA1-afferents, cannot be rendered explicitly by electrovibration, hence, tactile perception of virtual gratings is expected to be different than that of the real ones.
In contrast to the extensive literature on real textures, the number of studies investigating the roughness perception of virtual textures is limited and mostly conducted with force feedback devices. Smith et al. [@94] rendered virtual gratings varying in spatial period from $1.5$ to $8.5$ mm using a force feedback device, which could only display tangential forces resisting to the planar movements, and observed a decrease in roughness perception as spatial period was increased. Unger et al. [@9] investigated the roughness perception of periodic virtual gratings using a force feedback device and the results suggested significant influence of virtual probe diameter. They observed a decrease in roughness perception with increasing spatial periods when virtual textures in macro size were explored with a point-probe (having an infinitely small diameter). On the other hand, there was a monotonic increase in perceived roughness for increasing spatial period from $1$ to $6$ mm for spherical virtual probes having a diameter varying from $0.25$ to $1.5$ mm in [@9].
![Our results are in parallel with the literature, [@26; @9; @30]. Perceived roughness of real gratings scanned by finger and virtual gratings rendered by a force feedback device and scanned by a virtual probe having a finite size tip follow an increasing trend, while perceived roughness of virtual gratings rendered by electrovibration and scanned by finger and rendered by a force feedback device and scanned a virtual probe having an infinitely small tip follow a decreasing trend.[]{data-label="fig:lit"}](lit_new.png){width="3.5"}
In summary, it appears that the trend for perceived roughness of virtual gratings explored by a force feedback device, having a finite-size virtual tip, in the earlier studies resembles to that of real gratings explored by finger. On the other hand, the trend for perceived roughness of virtual gratings explored by a force feedback device, having an infinitely small virtual tip, resembles to that of the virtual gratings displayed by friction modulation explored by finger, as in our study. In electrovibration, fingerpad does not penetrate into virtual gratings since the surface topography cannot be displayed explicitly, only the tangential friction force between finger and surface is modulated periodically. As spatial period increases, the effective tangential force acting on the finger is reduced, leading to a decrease in perceived roughness.
A similar argument applies to a point probe exploring virtual gratings displayed by a force feedback device. Since surface topography can be displayed by a force feedback device and a point probe can fully penetrate into virtual grooves, the effective grating height to be overcome by the probe does not change, but the magnitude of the effective tangential force acting on the participant is reduced with increasing spatial period. As the probe size increases, the effective grating height to overcome is reduced since the probe can no longer penetrate into the gratings completely, leading to a reduction in the magnitude of effective tangential force acting on the finger. On the other hand, if the probe size is kept constant and the spatial period is increased, the probe penetrates more into the virtual grooves and the effective grating height increases until it reaches to a saturation value, as observed in tactile exploration of real gratings with a finger (Fig. \[fig:lit\]).
In the second experiment (Exp. 2), we observed that higher normal force resulted in an increase in perceived roughness of real gratings. This result is also consistent with the results of earlier studies ([@20; @23; @28; @32]) and can be also explained by our hypothesis on fingerpad penetration discussed above. As the normal force applied by the finger increases, the amount of penetration into the grooves, and hence, the effective height of the gratings to be traversed by the finger increases. As a result, the perceived roughness also increases (Table \[fig:discuss\_figure\]).
\[fig:discuss\_figure\]
In contrast to the real gratings, higher normal force caused a decline in perceived roughness of virtual gratings in our study. Although increasing normal force increases the apparent contact area of finger, the tactile effect of electrovibration appears to decrease. It was also interesting to observe that while the tangential force acting on the participants’ finger, ($F_t$), increased with higher normal force (Fig. \[fig:exp2\]b), its rate of change, $dF_t/dt$, also increased for real gratings but decreased for virtual gratings displayed by electrovibration (Fig. \[fig:exp2\]b). This result suggests that $dF_t/dt$ could be a better indicator of the perceived roughness for both real and virtual gratings (see Table \[fig:discuss\_figure\]). A similar conclusion was also achieved by Smith et. al. [@26]. They conducted psychophysical experiments with real gratings and reported that roughness perception is better correlated with the rate of change of tangential force rather than its magnitude.
Conclusion
==========
In this study, we investigated the tactile roughness perception of real gratings made of plexiglass and virtual gratings displayed by electrovibration through a touch screen. We conducted $2$ psychophysical experiments to investigate the effect of spatial period and the normal force applied by the finger on roughness perception of real and virtual gratings. Earlier studies on real macro textures repeatedly showed that, increasing spatial period and normal force result in an increase in perceived roughness. Our results on real gratings were also inline with the earlier literature. On the other hand, the results on virtual gratings displayed by electrovibration showed that tactile roughness perception followed a decreasing trend as a function of spatial period and applied normal force.
We argue that the difference in roughness perception of real and virtual gratings can be explained by the amount of fingerpad penetration into the gratings. This finding was consistent with the tangential force profiles recorded for both real and virtual gratings. In particular, the rate of change in tangential force ($dF_t/dt$) as a function of spatial period and normal force followed trends similar to those obtained for the perceived roughness of real and virtual gratings (Table \[fig:discuss\_figure\]). We suggest that larger spatial period and higher normal force resulted in more penetration of fingerpad into the grooves of real gratings, which in turn, resulted in an increase in the tangential force applied by the participant to overcome a ridge. Hence, the recorded tangential force, $F_t$, and its rate of change, $dF_t/dt$, increased as the spatial period was increased in real gratings. For virtual gratings, on the other hand, in which there was only friction modulation and no fingerpad penetration, the participants’ finger traversed fewer number of high friction zones for larger spatial periods, and hence $F_t$ and $dF_t/dt$ decreased as the spatial period was increased.
Nonetheless, we need to clarify that, although real textures carry both spatial and temporal information, their virtual counterparts in this study were rendered based on the temporal frequency information. Therefore, the variation in exploration speed might have a stronger influence on the roughness perception of virtual textures than the real ones in our study.
In our study, we did not explicitly investigate the temporal effects of changing finger velocity on roughness perception of virtual gratings, which we plan to do so in the near future. We will also expand our study to investigate the individual effects of wavelength and duty cycle on our roughness perception of micro textures.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to express their gratitude to the members of RML and participants who showed great patience throughout the experiments. Moreover, A.İ. would like to acknowledge Merve Adl[i]{} for her unconditional support.
[Aykut İşleyen]{} received the BSc degree in mechanical engineering and physics from Bogazici University, Istanbul, in 2016. He is currently a MSc student in mechanical engineering department of Koc University. He is also a member of Robotics and Mechatronics Laboratory, Koc University. His research interests include haptic interfaces, tactile perception, and psychophysics.
[Yasemin Vardar]{} received the BSc degree in mechatronics engineering from Sabanci University, Istanbul, in 2010, the MSc degree in systems and control from the Eindhoven University of Technology, in 2012, and the PhD degree in mechanical engineering from Koc University, Istanbul, in 2018. She is currently a post-doctoral researcher with the Max Planck Institute for Intelligent Systems. Before starting her PhD study, she conducted research on control of high precision systems in ASML, Philips, and TNO Eindhoven. Her research interests include haptics science and applications. She is a member of the IEEE.
[Cagatay Basdogan]{} received the Ph.D. degree in mechanical engineering from Southern Methodist University in 1994. He is a faculty member in the mechanical engineering and computational sciences and engineering programs of Koc University, Istanbul, Turkey. He is also the director of the Robotics and Mechatronics Laboratory at Koc University. Before joining Koc University, he worked at NASAJPL/Caltech, MIT, and Northwestern University Research Park. His research interests include haptic interfaces, robotics, mechatronics, biomechanics, medical simulation, computer graphics, and multi-modal virtual environments. He is currently the associate editor in chief of IEEE Transactions on Haptics and serves in the editorial boards of IEEE Transactions on Mechatronics, Presence: Teleoperators and Virtual Environments, and Computer Animation and Virtual Worlds journals. He also chaired the IEEE World Haptics conference in 2011.
|
---
abstract: 'A graph is [*$(c_1, c_2, \cdots, c_k)$-colorable*]{} if the vertex set can be partitioned into $k$ sets $V_1,V_2, \ldots, V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar graph without $4$- and $5$-cycles is $(1, 1, 0)$-colorable and $(3,0,0)$-colorable. This is a relaxation of the Steinberg Conjecture that every planar graph without $4$- and $5$-cycles are properly $3$-colorable (i.e., $(0,0,0)$-colorable).'
address: ' Department of Mathematics, College of William and Mary, Williamsburg, VA 23185.'
author:
- Owen Hill
- Gexin Yu
title: 'A relaxation of Steinberg’s Conjecture[^1]'
---
Introduction
============
It is well-known that the problem of deciding whether a planar graph is properly $3$-colorable is NP-complete. Grötzsch in 1959 [@G59] showed the famous theorem that every triangle-free planar graph is $3$-colorable. A lot of research was devoted to find sufficient conditions for a planar graph to be $3$-colorable, by allowing a triangle together with some other conditions. One of such efforts is the following famous conjecture made by Steinberg in 1976.
All planar graphs without $4$-cycles and $5$-cycles are $3$-colorable.
Not much progress in this direction was made until Erdös proposed to find a constant $C$ such that a planar graph without cycles of length from $4$ to $C$ is $3$-colorable. Borodin, Glebov, Raspaud, and Salavatipour [@BGRS05] showed that $C\le 7$. For more results, see the recent nice survey by Borodin [@B12].
Yet another direction of relaxation of the Conjecture is to allow some defects in the color classes. A graph is [*$(c_1, c_2, \cdots, c_k)$-colorable*]{} if the vertex set can be partitioned into $k$ sets $V_1,V_2, \ldots, V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. Thus a $(0,0,0)$-colorable graph is properly $3$-colorable.
Eaton and Hull [@EH99] and independently Škrekovski [@S99] showed that every planar graph is $(2,2,2)$-colorable (actually choosable). Xu [@X08] proved that all planar graphs with no adjacent triangles or $5$-cycles are $(1,1,1)$-colorable. Chang, Havet, Montassier, and Raspaud [@CHMR11] proved that all planar graphs without $4$-cycles or $5$-cycles are $(2,1,0)$-colorable and $(4,0,0)$-colorable. In this paper, we further prove the following relaxation of the Steinberg Conjecture.
\[110-coloring\] All planar graphs without $4$-cycles and $5$-cycles are $(1,1,0)$-colorable.
\[300-coloring\] All planar graphs without $4$-cycles and $5$-cycles are $(3,0,0)$-colorable.
We will use a discharging argument in the proofs. We let the initial charge of vertex $u\in G$ be $\mu(u)=2d(u)-6$, and the initial charge of face $f$ be $\mu(f)=d(f)-6$. Then by Euler’s formula, we have $$\label{euler}
\sum_{v\in V(G)} \mu(u)+\sum_{f\in F(G)} \mu(f)=-12.$$
Our goal is to show that we may re-distribute the charges among vertices and faces so the final charges of the vertices and faces are non-negative, which would be a contradiction. In the process of discharging, we will see that some configurations prevent us from showing some vertices or faces to have non-negative charges. Those configurations will be shown to be reducible configurations, that is, a valid coloring outside of the configurations can be extended to the whole graph. It is worth to note that in the proof of Theorem \[110-coloring\], we prove a somewhat global structure, a special chain of triangles, to be reducible.
The following are some simple observations about the minimal counterexamples to the above theorems.
\[fact\] Among all planar graphs without $4$-cycles and $5$-cycles that are not $(1,1, 0)$-colorable or $(3,0,0)$-colorable, let $G$ be one with minimum number of vertices. Then\
(a) $G$ contains no $2^-$ vertices.\
(b) a $k$-vertex in $G$ can have $\alpha\le \lfloor \frac{k}{2} \rfloor$ incident $3$-faces, and at most $k-2\alpha$ pendant $3$-faces.
We will use the following notations in the proofs. A [*$k$-vertex*]{} ($k^+$-$vertex$, $k^-$-vertex) is a vertex of degree $k$ (at least $k$, at most $k$ resp.). The same notation will apply to faces. An [*$(\ell_1, \ell_2, \ldots, \ell_k)$-face*]{} is a $k$-face with incident vertices of degree $\ell_1, \ell_2, \ldots, \ell_k$. A [*bad $3$-vertex*]{} is a $3$-vertex on a $3$-face. A face $f$ is a [*pendant $3$-face*]{} to vertex $v$ if $v$ is adjacent to some bad $3$-vertex on $f$. The [*pendant neighbor*]{} of a $3$-vertex $v$ on a $3$-face is the neighbor of $v$ not on the $3$-face. A vertex $v$ is [*properly colored*]{} if all neighbors of $v$ have different colors from $v$. A vertex $v$ is [*nicely colored*]{} if it shares colors with at most $\max\{s_i-1, 0\}$ neighbors, thus if a vertex $v$ is nicely colored by a color $c$ which allows deficiency $s_i>0$, then an uncolored neighbor of $v$ can be colored by $c$.
In the next section, we will give a proof to Theorem \[110-coloring\]; and in the last section, we will give a proof to Theorem \[300-coloring\].
$(1,1,0)$-coloring of planar graphs
===================================
We will use a discharging argument in our proof. First we will prove some reducible configurations.
Let $G$ be a minimum counterexample to Theorem \[110-coloring\], that is, $G$ is a planar graph without $4$-cycles and $5$-cycles, and $G$ is not $(1,1,0)$-colorable, but any proper subgraph of $G$ is $(1,1,0)$-colorable.
The following is a very useful tool in the proofs.
\[extending-lemma\] Let $H$ be a proper subgraph of $G$ so that there is a $(1,1,0)$-coloring of $G-H$. If vertex $v\in H$ satisfies either (i) $3$ neighbors of $v$ are colored, with at least two properly colored, or (ii) $4$ neighbors of $v$ are colored, all properly, then the coloring of $G-H$ can be extended to $G-(H-v)$.
\(i) Let $v\in H$ be a vertex with $3$ colored neighbors, two of which are properly colored, such that the coloring of $G-H$ can not be extended to $v$. Since $v$ is not $(1,1, 0)$-colorable, the three neighbors of $v$ must have different colors, and furthermore, two of the colored neighbors cannot be properly colored, a contradiction to the assumption that two of the colored neighbors of $v$ are properly colored.
\(ii) Let $v\in H$ be a vertex of degree $4$ with all neighbors properly colored such that the coloring of $G-H$ can not be extended to $v$. Then due to the coloring deficiencies, $v$ must have at least $2$ neighbors colored by $1$, at least $2$ neighbors colored by $2$, and at least $1$ neighbor colored by $1$. Then $v$ has at least five colored neighbors, a contradiction.
\[334-face\] There is no $(3,3,4^-)$-face in $G$.
Let $uvw$ be a $(3,3,4^-)$-face in $G$ with $d(u)=d(v)=3$ and $d(w)\le 4$. Then $G$\\$\{u,v,w\}$ is $(1,1,0)$-colorable. Color $w$ and $v$ properly, then $u$ is colorable by Lemma \[extending-lemma\], thus $G$ is $(1,1,0)$-colorable, a contradiction.
\[5-vertex\] There is no $5$-vertex that is incident to two $(3,4^-,5)$-faces and adjacent to a $3$-vertex in $G$.
![Figure for Lemma \[5-vertex\][]{data-label="f1"}](5vert.pdf)
Let $v$ be a $5$-vertex with neighbors $u, w, x, y, z$ so that $wx, yz\in E(G)$ and $d(u)=d(x)=d(z)=3$ and $d(w), d(y)\le 4$ (See Figure \[f1\]). By the minimality of $G$, $G$\\$\{u,v,w,x,y,z\}$ is $(1,1,0)$-colorable. Properly color $u$, $w$, and $y$, then properly color $x$ and $z$. For $v$ to not be colorable, $v$ must have two neighbors colored by $1$, two neighbors colored by $2$ and one neighbor colored by $3$. Since the $w,x$ and $y,z$ vertex pairs must be colored differently, one of them must have the colors $1$ and $2$. W.l.o.g. we can assume that $w$ is colored by $1$ and $x$ by $2$. Then since $w$ is properly colored, we can either recolor $x$ by $1$ or $3$, and color $v$ by $2$ obtaining a coloring of $G$, a contradiction.
\[333-vertices\] No $3$-vertex in $G$ can be adjacent to two other $3$-vertices. In particular, the $3$-vertices on a $(3,3, 5^+)$-face must have another neighbor with degree four or higher.
Let $v$ be a $3$-vertex with $x$ and $y$ being two neighbors of degree $3$. By the minimality of $G$, $G$\\$\{v,x,y\}$ is $(1,1,0)$-colorable. Then we can first properly color $x$ and $y$, and then by Lemma \[extending-lemma\] color $v$ to get a coloring of $G$, a contradiction.
\[344-face\] The pendant neighbor of the $3$-vertex on a $(3,4,4)$-face must have degree $4$ or higher.
![Figure for Lemma \[344-face\][]{data-label="fig2"}](344face.pdf "fig:") 0.4in ![Figure for Lemma \[344-face\][]{data-label="fig2"}](344face2.pdf "fig:")
Let $vxy$ be a $(3,4,4)$-face in $G$ such that the pendant neighbor $u$ of the $3$-vertex $v$ has degree $3$ (See Figure \[fig2\]). By the minimality of $G$, $G$\\$\{u,v\}$ is $(1,1,0)$-colorable. We properly color $u$ and then color $v$ differently from both $x$ and $y$. If $u$ and $v$ are not both colored by $3$, then we get a coloring for $G$, a contradiction, so we may assume both $u$ and $v$ are colored by $3$. This means that both $u$ and $v$ have two remaining neighbors colored by $1$ and $2$. Let $x$ and $y$ be colored by $1$ and $2$ respectively. The neighbors of $x$ must be colored by $1$ and $3$ or else we could recolor $v$ by $1$ and $x$ by $3$ if necessary to obtain a coloring of $G$. Likewise, the neighbors of $y$ must be colored by $2$ and $3$. In this case we switch the colors of $x$ and $y$ and color $v$ by $1$ to obtain a coloring of $G$, a contradiction again.
Let a [*$(T_0, T_1, \ldots, T_n)$-chain*]{} be a sequence of triangles, $T_0, T_1, \ldots, T_n$, such that (i) $T_0$ is a $(3,4,4)$-face and $T_n$ is a $(3^+, 4, 4^+)$-face, and all other triangles are $(4, 4,4)$-faces, and (ii) for $0\le i\le n-1$, $T_i$ and $T_{i+1}$ share a $4$-vertex $t_i$. In a $(T_0, T_1, \ldots, T_n)$-chain, let $x_i\in T_i$ for $0\le i\le n$ be a non-connecting $4^+$-vertex.
Let a *special 4-vertex* be a $4$-vertex that is incident to one $3$-face and has two pendant $3$-faces, and let a $3$-face be a *special 3-face* if it has at least one special $4$-vertex. Let a *good 4-vertex* be a $4$-vertex with only one incident $3$-face and at most one pendant $3$-face.
We will prove in the following lemmas that a $(3,4,4)$-face $T_0$ may get help in discharging from a $(3^+, 4^+, 5^+)$-face or special $3$-face $T_n$ through a $(T_0, T_1, \ldots, T_n)$-chain.
There are no special $(3,4,4)$-faces in $G$.
Let $uvw$ be a special $(3,4,4)$-face in $G$ such that $d(v)=d(w)=4$. W.l.o.g. we can assume that $v$ is a special $4$-vertex with pendant neighbors $v_1$ and $v_2$. By the minimality of $G$, $G$\\$\{u,v,v_1,v_2,w\}$ is $(1,1,0)$-colorable. We can properly color $w$ and $u$ in that order then properly color $v_1$ and $v_2$. Then by Lemma \[extending-lemma\], we can color $u$, obtaining a coloring of $G$, a contradiction.
The following is a very useful tool in extending a coloring to a chain.
\[chain-extending-lemma\] Consider a $(T_0, T_1, \cdots,T_n)$-chain with $n\geq 1$ and $T_n$ being a $(4,4^-,k)$-face. If $G$\\$\{T_0, T_1, \cdots, T_{n-1}\}$ has a coloring such that the $k$-vertex of $T_n$ is properly colored, or it shares the same color with the $4^-$-vertex, then the coloring can be extended to $G$.
We assume that the $(4,4^-,k)$-face $T_n$ has $k$-vertex $x_n$ and $4^-$-vertex $t_n$. Also let $G$\\$\{T_0, T_1, \cdots, T_{i-1}\}$ has a coloring such that $x_n$ is properly colored or shares the same color with $t_n$ and $G$ does not have a $(1,1,0)$-coloring. Finally let $u$ be the $3$-vertex of $T_0$ and let $w$ be the pendant neighbor of $u$.
We consider two cases. First let $n=1$. If $x_1$ and $t_1$ have the same color, then we can properly color $x_0$ and $t_0$ in that order, thus by Lemma \[extending-lemma\] we can color $u$ so $G$ has a $(1,1,0)$-coloring, a contradiction. So we know that $x_1$ and $t_1$ must be colored differently, and further $x_1$ is colored properly. We can properly color $x_0$. If $x_0$ and $w$ share the same color then we can color $t_0$ by Lemma \[extending-lemma\] and properly color $u$, a contradiction. So we may assume that $x_0$ and $w$ are colored differently. If any two of $x_0, x_1,$ and $t_1$ are colored the same then we could color $t_0$ properly and color $u$ by Lemma \[extending-lemma\], a contradiction. Since $x_0, x_1,$ and $t_1$ are colored differently, if $x_0$ is not colored by $3$ then we could color $t_0$ by the same color as $x_0$ and properly color $u$, a contradiction. So $x_0$ must be colored by $3$ and w.l.o.g. we can assume that $w$ is colored by $1$. Since $x_1$ is properly colored, it must be colored by $2$, or we could color $t_0$ by $1$ and properly color $u$, a contradiction. It follows that $t_1$ is colored by $1$. If $t_1$ is colored properly, then we could color $t_0$ by $1$ and properly color $u$, a contradiction, so we may assume that $t_1$ is not colored properly. Further, neither $z$ nor $z'$ (the two other neighbors of $t_1$) can be colored by $2$, or we could recolor $t_1$ properly, then color $t_0$ by $1$ and $u$ properly, a contradiction. So we color $t_1$ by $2$ and $t_0$ by $1$, and properly color $u$, a contradiction.
Now we assume that $n\geq 2$. For all $j: 1\leq j\leq n$, properly color $x_{n-j}$ and color $t_{n-j}$ by Lemma \[extending-lemma\], or properly if possible. Then since $x_1$ was properly colored, and $t_1$ was colored after $x_1$, either $x_1$ remains properly colored, or $t_1$ has the same color as $x_1$. Also, we know that $T_1$ must be a $(4,4,4)$-face, so by the previous case, we can extend the coloring to $T_0$ and get a coloring of $G$, a contradiction.
\[chain-to-344\] There is no $(T_0, \ldots, T_n)$-chain so that (i) $n\geq 1$ and $T_n$ is a special $(4,4,4)$-face or (ii) $n\geq 2$ and $T_n$ is a $(3,4,k)$-face or (iii) $n=1$ and $T_n$ is a $(3,4,4^-)$-face.
Let $T_0=ux_0t_0$ be a $(3,4,4)$-face with $d(u)=3$.
\(i) Let $v$ be a special $4$-vertex of $T_n$ and let $y$ and $z$ be the neighbors of $v$ other than $t_n$ and $x_n$. Let $S=\{t_i, x_i: 0\leq i\leq n-1\}$. By the minimality of $G$, $G\setminus (S\cup \{u,v,x_n,y,z\})$ has a $(1,1,0)$-coloring. Properly color $x_n$, $y$ and $z$, then by Lemma \[extending-lemma\] color $v$. Then, either $x_n$ remains properly colored or $v$ shares the same color, so by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1, \cdots, T_{n-1}\}$ to obtain a coloring of $G$.
\(ii) Let $v$ be the $3$-vertex of $T_n$ and let $S=\{t_i, x_i: 0\leq i\leq n-1\}$. By the minimality of $G$, $G\setminus(S\cup \{u,v\})$ has a $(1,1,0)$ coloring. Properly color $v$ and $x_{n-1}$. Then by Lemma \[extending-lemma\], we can color $t_{n-1}$. Either $x_{n-1}$ remains properly colored or $t_{n-1}$ shares the same color, so by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1, \cdots, T_{n-2}\}$ to obtain a coloring of $G$.
\(iii) Assume that $n=1$ and $T_n$ is a $(3,4,4)$-face with $3$-vertex $v$. By the minimality of $G$, $G\setminus\{t_0, u, v, x_0, x_1\}$ has a $(1,1,0)$-coloring. Properly color $x_0$ and $u$ in that order and properly color $x_1$ and $v$ in that order. Then $t_0$ has four neighbors colored, all properly, so by Lemma \[extending-lemma\] we can color $t_0$ to get a coloring for $G$.
**Remark:** By above lemma, a $(T_0, T_1)$-chain with $T_1$ being a $(3,4,5^+)$-face is not necessarily reducible. Let a *bad $(3,4,5^+)$-face* be a $(3,4,5^+)$-face that shares a $4$-vertex with a $(3,4,4)$-face.
\[chain-to-itself\] There is no $(T_0, \ldots, T_n)$-chain with $T_i=T_n$ for some $i\not=n$.
![Figure for Lemma \[chain-to-itself\][]{data-label="fig3"}](cyclepicture.pdf)
Let $(T_0, \ldots,T_n)$-chain be a chain with $T_i=T_n$ for some $i<n$. Let $u$ be the $3$-vertex of $T_0$ and let $S=\{t_j,x_j: 0\leq j\leq n-1\}$. Since $T_i=T_n$, the vertex that would have been labelled $x_i$ is instead labelled $t_{n-1}$ (See Figure \[fig3\]). By the minimality of $G$, $G\setminus (S\cup\{u\})$ is $(1,1,0)$-colorable. Start by properly coloring $x_{i+1}$, $x_{i+2}$, and $t_{i+1}$. Then for all $j: i+2\leq j\leq n-2$, properly color $x_{j+1}$ and color $t_j$ by Lemma \[extending-lemma\]. Next, properly color $t_{n-1}$, and we have two cases:
**Case 1:** $i=0$. We can properly color $u$, then color $t_i$ by Lemma \[extending-lemma\] to get a coloring of $G$, a contradiction.
**Case 2:** $i>0$. We can then color $t_i$ by Lemma \[extending-lemma\] and then either $t_{n-1}$ is properly colored, or $t_{i}$ shares the same color, so by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1, \cdots, T_{i-1}\}$ to obtain a coloring of $G$, a contradiction.
\[existence-chain\] For each $(3,4,4)$-face $T_0$ without good $4$-vertices, there exist two chains, $(T_0, \ldots, T_n)$-chain and $(T_0, \ldots, T_m')$-chain, such that $T_n$ and $T_m'$ are either bad $(3,4,5^+)$-faces, $(4,4^+,5^+)$-faces, or $(4,4,4)$-faces with a good $4$-vertex. Furthermore, $T_n\not=T_m'$.
As $G$ is finite, any chain of triangles in $G$ must be finite. By Lemma \[chain-to-344\] and \[chain-to-itself\], no chain of triangles in $G$ can end with a special $3$-face or a non-bad $(3,4,5^+)$-face, thus it must end with a bad $(3,4,5^+)$-face or a $(4,4^+,4^+)$-face. Since a $(4,4,4)$-face in a chain can not be a special $3$-face, any chain of triangles in $G$ must end with a bad $(3,4,5^+)$-face, a $(4,4^+,5^+)$-face or a $(4,4,4)$-face with a good $4$-vertex.
Now we assume that $T_n = T_m$. Then by Lemma \[chain-extending-lemma\], $T_n$ must be a $(4,4,5^+)$-face, and since $G$ has no 4- and 5-cycles, $n+m\geq 6$. Assume that $n\leq m$. Let $S=\{t_i, x_i: 1\leq i\leq n-1\}$, where $S=\emptyset$ if $n=1$, and $S'=\{t_j', x_j': 0\leq j\leq m-1\}$ and let $u$ be the $3$-vertex of $T_0$. By the minimality of $G$, $G$\\$S\cup S'\cup\{u\}$ has a $(1,1,0)$-coloring. We have two cases:
If $n=1$, properly color $x_{m-1}'$ and $t_{m-1}'$. Then, by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1', \cdots T_{m-2}'\}$ to obtain a coloring of $G$, a contradiction.
If $n\geq 2$, then properly color $x_{n-1}$, $t_{n-1}$ and $x_{m-1}'$ in that order, then by Lemma \[extending-lemma\] we can color $t_{m-1}'$. If $n\geq 3$, for all $i: 2\leq i\leq n-1$, properly color $x_{n-i}$ and by Lemma \[extending-lemma\] we can color $t_{n-1}$. Then since either $x_{m-1}'$ is still properly colored or shares the same color as $t_{m-1}'$, by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1', \cdots, T_{m-2}'\}$ to obtain a coloring of $G$, a contradiction.
We will now prove some lemmas which will ensure that bad $(3,4,5^+)$-faces will have extra charge to help $(3,4,4)$-faces.
\[bad-345s\] A $5$-vertex incident to a bad $(3,4,5)$-face cannot be incident to another bad $(3,4,5)$-face or a $(3,3,5)$-face.
![Figure for Lemma \[bad-345s\][]{data-label="figure4"}](2bad345.pdf)
We only show the case when a $5$-vertex $v$ is incident to two bad $(3,4,5)$-faces, and it is very similar (and easier!) to show the case when it is incident to a bad $(3,4,5)$-face and a $(3,3,5)$-face.
Let $v$ be a $5$-vertex that is incident two bad $(3,4,5)$-faces, $f_1$ and $f_2$, and let $u$ be a $k$-vertex adjacent $v$ (see Figure \[figure4\]). Let $f_3$ be the $(3,4,4)$-face sharing a $4$-vertex with $f_1$ and let $f_4$ be the $(3,4,4)$-face sharing a $4$-vertex with $f_2$. Let $f_3$ and $f_4$ have outer $4$-vertices of $x$ and $x'$ respectively and $3$-vertices of $y$ and $y'$ respectively. Also, let $f_1$ and $f_2$ have $4$-vertices $z$ and $z'$. Then, by the minimality of $G$, $G$\\$\{f_1, f_2, f_3, f_4\}$ has a $(1,1,0)$-coloring.
If $u$ is colored by $1$ or $2$, then we can color $v$ by $3$ and color the $3$-vertices of $f_1$ and $f_2$ properly. Since $v$ is properly colored, by Lemma \[chain-extending-lemma\] we can extend the coloring to $f_1$ and $f_3$. Then, since $v$ is colored by $3$, it would remain properly colored, so again by Lemma \[chain-extending-lemma\] we can extend the coloring to $f_2$ and $f_4$ to get a coloring of $G$.
If $u$ is colored by $3$, then we properly color $x$ and $x'$ then properly color $y$ and $y'$. We then properly color $z$ and $z'$. If either $z$ or $z'$ is colored by $3$, then we can properly color the $3$-vertices of $f_1$ and $f_2$ and color $v$ by either $1$ or $2$ getting a coloring for $G$. So we can assume neither is colored by $3$, and w.l.o.g. we can assume that $z$ is colored by $1$. Then since $z$ and $z'$ are properly colored, we can color the $3$-vertices of $f_1$ and $f_2$ by either $1$ or $3$. Then since $v$ will have at most one neighbor colored by $2$, and that neighbor colored properly, we can color $v$ by $2$ to obtain a coloring for $G$.
\[35k-pendant-neighbor\] A $(3,5,k)$-face in $G$ that is incident a $5$-vertex that is also incident to a bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face will have a pendant neighbor that is a $4^+$-vertex.
![Figure for Lemma \[35k-pendant-neighbor\][]{data-label="figure5"}](bad345pend344and35k.pdf)
Let $f_1$ be a $(3,5,k)$-face in $G$ with a $5$-vertex $v$, a $3$-vertex $u$, and a pendant neighbor $u'$ that is a $3$-vertex. Let the $k$-vertex of $f_1$ be $w$. Let $v$ be incident a bad $(3,4,5)$-face $f_2$ with neighbor $(3,4,4)$-face $f_3$, and let $v$ have a pendant $(3,4,4)$-face $f_4$. Let the $3$-vertex of $f_4$ be $x$ and the $4$-vertices of $f_4$ be $y$ and $z$ (See Figure \[figure5\]). By the minimality of $G$, $G$\\$\{f_2, f_3, u, u', x\}$ has a $(1,1,0)$-coloring. Properly color $x$. If $w$ and $x$ share the same color, then we can properly color $u'$ and $u$, then properly color $v$ and the $3$-vertex of $f_2$. Then the coloring can be extended to $f_3$ by Lemma \[chain-extending-lemma\], obtaining a coloring of $G$. So we can assume that $w$ and $x$ are colored differently. If $x$ is colored by $1$ or $2$ (w.l.o.g. we may assume that $x$ is colored by $1$), then we can color $u'$ properly and color $u$ by $1$. Then we can properly color $v$ and properly color the $3$-vertex of $f_2$. Finally we can apply Lemma \[chain-extending-lemma\] to extend the coloring to $f_3$, obtaining a coloring of $G$. So we can assume that $x$ is colored by $3$.
Since $x$ is colored by $3$, we may assume that $w$ is colored by $1$. Properly color $u'$ and color $u$ by $2$. Since $x$ is properly colored, $y$ and $z$ must be colored by $1$ and $2$. W.l.o.g. let $y$ be colored by $1$. Then to avoid being able to re-color $x$ by $1$, the two other neighbors of $y$ must be colored $1$ and $3$. For similar reasons the other two neighbors of $z$ must be colored $2$ and $3$. Then switch the colors of $y$ and $z$ and color $x$ by $1$ or $2$ and color $v$ by $3$, we can color the $3$-vertex of $f_2$ properly and by Lemma \[chain-extending-lemma\], extend the coloring to $f_3$, obtaining a coloring of $G$.
\[355-two-bad-345\] A $(3,5,5)$-face in $G$ can not have both $5$-vertices also be incident to bad $(3,4,5)$-faces and have pendant $(3,4,4)$-faces.
![Figure for Lemma \[355-two-bad-345\][]{data-label="figure6"}](355twobad345.pdf)
Let $uvw$ be a $(3,5,5)$-face in $G$ where $d(v)=d(w)=5$ and $u$ has pendant neighbor $u'$. Also let $v$ and $w$ both be incident bad $(3,4,5)$-faces, $f_1$ and $f_2$ with neighbor $(3,4,4)$-faces $f_3$ and $f_4$ respectively and let $v$ and $w$ have pendant $(3,4,4)$-faces. Let the pendant $(3,4,4)$-faces to $v$ and $w$ have $3$-vertices $x$ and $x'$ respectively (See Figure \[figure6\]). By the minimality of $G$, $G$\\$\{f_1, f_2, f_3, f_4, u, x, x'\}$ has a $(1,1,0)$-coloring.
Properly color $x$ and $x'$. If either $x$ or $x'$ has a coloring different from $u'$, w.l.o.g. we can assume $x$, then we color $u$ the same as $x$. We can properly color $w$ and $v$ in that order, then properly color the $3$-vertices of $f_1$ and $f_2$. Then by Lemma \[chain-extending-lemma\] we can extend the coloring to $f_3$ and $f_4$ to obtain a coloring of $G$. So we can assume that $x$, $x'$, and $u'$ are colored the same. If $x$ is colored by $3$, since $x$ is properly colored, $y$ and $z$ must be colored by $1$ and $2$. Then to avoid being able to re-color $x$ by $1$, the other two neighbors of $y$ must be colored $1$ and $3$. For similar reasons the other two neighbors of $z$ must be colored $2$ and $3$. Then we can switch the colors of $y$ and $z$ and color $x$ differently from $u'$. Then we follow the above procedure to obtain a coloring for $G$.
So we may assume that w.l.o.g. $x$, $x'$, and $u'$ are all colored by $1$. Then we color $u$ by $2$ and $w$ by $3$. Color the $3$-vertex of $f_2$ properly and by Lemma \[chain-extending-lemma\], extend the coloring to $f_4$. We now have $v$ adjacent to $3$ differently and properly colored vertices. Properly color the outer $4$-vertex and the $3$-vertex of $f_3$ in that order, then properly color the $4$-vertex of $f_1$. If it is colored by $3$, then properly color the $3$-vertex of $f_1$ and color $v$ by either $1$ or $2$ to obtain a coloring of $G$. If it is not colored by $3$, then w.l.o.g. we can assume that it is colored by $1$. Then since it is properly colored, we can color the $3$-vertex of $f_1$ by either $1$ or $3$ and color $v$ by $2$, obtaining a coloring of $G$.
\[bad-345-45k-with-chain\] A $5$-vertex in $G$ that is incident a bad $(3,4,5)$-face and has a pendant $(3,4,4)$-face cannot also be incident a $(4,4^+,5)$-face $T_n$ that is in a $(T_0, \ldots, T_n)$-chain.
![Figure for Lemma \[bad-345-45k-with-chain\][]{data-label="f7"}](bad345pend344and445chain.pdf)
Let $v$ be a $5$-vertex in $G$ that is incident a bad $(3,4,5)$-face $f_1$ with neighbor $(3,4,4)$-face $f_2$. Let $v$ have a pendant $(3,4,4)$-face with $3$-vertex $w$ and $4$-vertices $y$ and $z$. Also let $v$ be incident a $(4,4^+,5)$-face $T_n$ such that there exists a chain of triangles from $T_0$ to $T_n$. Let the $4^+$-vertex of $T_n$ be $w$. Let $S=\{t_i, x_i: 0\leq i\leq n-1\}$ and let $u$ be the $3$-vertex of $T_0$ (See Figure \[f7\]). By the minimality of $G$, $G\setminus(S\cup \{f_1, f_2, u, x\})$ has a $(1,1,0)$-coloring.
Properly color $x$. If $x$ and $w$ are colored the same then we can properly color $x_{n-1}$, $t_{n-1}$, and $v$. If $n=1$, then by Lemma \[extending-lemma\], we can color $u$. If $n\geq 2$, then by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1, \cdots,T_{n-1}\}$. Then we can properly color the $3$-vertex of $f_1$ and by Lemma \[chain-extending-lemma\] we can extend the coloring to $f_2$ obtaining a coloring for $G$. So we can assume that $x$ and $w$ are colored differently.
Let $x$ be colored 1 or 2 and w.l.o.g. we can assume that $x$ is colored by $1$. Then we can properly color $x_{n-1}$ and color $t_{n-1}$ by $1$. Since $w$ and $x$ are colored differently, either $x_{n-1}$ and $t_{n-1}$ are both colored properly or share the same color. If $n=1$, then either we can color $u$ properly or we can color $u$ by Lemma \[extending-lemma\]. If $n\geq 2$, then by Lemma \[chain-extending-lemma\] we can extend the coloring to $\{T_0, T_1, \cdots, T_{n-1}\}$. Then since $t_{n-1}$ and $x$ are colored the same we can properly color $v$ and the $3$-vertex of $f_1$. By Lemma \[chain-extending-lemma\] we can extend the coloring to $f_2$ to obtain a coloring of $G$.
So let $x$ be colored by $3$ (then $w$ is colored $1$ or $2$). Then $y$ and $z$ must be colored by $1$ and $2$, respectively. To avoid being able to re-color $x$ by $1$ or 2, the two other neighbors of $y$ must be colored $1$ and $3$ and the two other neighbors of $z$ must be colored $2$ and $3$. Then we switch the colors of $y$ and $z$ and re-color $x$ to be the same as $w$, and proceed as above to get a coloring for $G$.
\[bad-346\] Every $6$-vertex in $G$ that is incident a bad $(3,4,6)$-face can be incident at most two $(3,4^-,6)$-faces.
![Figure for Lemma \[bad-346\][]{data-label="f8"}](bad346.pdf)
Let $v$ be a $6$-vertex in $G$. Let $vwx$ be a bad $(3,4,6)$-face with $d(w)=3$ and neighbor $(3,4,4)$-face $xyz$ with $3$-vertex $y$. Let $v$ also be incident non-bad $(3,4,6)$-faces $t_1t_2v$ and $u_1u_2v$ where $d(t_1)=d(u_1)=4$ (See Figure \[f8\]). By the minimality of $G$, $G$\\$\{t_1, t_2, u_1, u_2, v, w, x, y, z\}$ has a $(1,1,0)$-coloring. Properly color $t_1$, $t_2$, $u_1$, and $u_2$. If the color set of $\{t_1, t_2, u_1, u_2\}$ is not $\{1,2,3\}$, then we can properly color $v$ and $w$. Then by Lemma \[chain-extending-lemma\], we can extend the coloring to $x, y,$ and $z$, obtaining a coloring of $G$. So we can assume that the color set of $\{t_1, t_2, u_1, u_2\}$ includes $1, 2,$ and $3$.
If two of $\{t_1, t_2, u_1, u_2\}$ are colored by $3$, then we can color $z$, $y$, and $x$ properly. If $x$ is colored by $3$, then we can color $w$ properly and color $v$ by $1$ or $2$ to get a coloring of $G$. If $x$ is colored by $1$ or $2$, then since $x$ is properly colored we can color $w$ by $3$ or the same as $x$. Then we can color $v$ differently from $3$ and $x$ to obtain a coloring of $G$.
So we can assume that exactly one of vertices in the set $\{t_1, t_2, u_1, u_2\}$ is colored by $3$. Then w.l.o.g. we may assume that the color set of $\{t_1,t_2\}$ is $\{1,3\}$ and the color set of $\{u_1, u_2\}$ is $\{1,2\}$. Since $u_1$ and $u_2$ were colored properly, the outside neighbor of $u_2$ must be $3$. Let $u_1$ be colored by $1$, then since it is colored properly we can recolor $u_2$ by $1$. Then we can color $v$ and $w$ properly, and extend to $x$, $y$, and $z$ to obtain a coloring of $G$. So we can assume that $u_1$ is colored by $2$.
Now color $z$, $y$, and $x$ properly in that order. If $x$ is colored by $3$ then color $w$ properly. If $w$ is colored by $1$, then color $v$ by $2$ to get a coloring for $G$. If $w$ is colored by $2$, then since $u_1$ is colored properly recolor $u_2$ by $2$ and color $v$ by $1$ to get a coloring for $G$. So we can assume that $x$ is colored by $1$ or $2$. Since $x$ is properly colored we can color $w$ by $3$ or the same as $x$. Then either $1$ or $2$ but not both is in the color set of $\{x,w\}$. If $1$ is in the color set, then $v$ will have only one neighbor colored by $2$ so we can color $v$ by $2$ and obtain a coloring of $G$. If $2$ is in the color set, then $v$ will have two neighbors colored by $1$, but we can recolor $u_2$ by $2$ and color $v$ by $1$ to obtain a coloring of $G$.
The following lemma says that a $3$-face with $k$ vertices of degree $4$ can have at most $k$ chains of triangles ending at it.
\[at-most-two-344\] If a $(T_0, T_1, \ldots, T_n)$-chain and a $(T_0', T_1', \ldots, T_m')$-chain with $T_m'=T_n$ satisfy $T_{n-1}\cap T_n=\{t_n\}=T_{m-1}'\cap T_m'$, then $T_0=T_0'$.
For otherwise, the two chains have a common $(4,4,4)$-face $T$ so that $T=T_a$ and $T=T_b'$. Then we would have a $(T_0, T_1, T_{a-1},T, T_{b-1}', \ldots, T_1', T_0')$-chain. But by Lemma \[chain-to-344\], this chain cannot exist in $G$.
As we mentioned in the introduction, we set the initial charge of a vertex $v$ to be $\mu(v)=2d(v)-6$ and the initial charge of a face $f$ to be $\mu(f)=d(f)-6$. For the discharging procedure we must introduce the notion of a bank, which serves as a temporary placeholder for charges. We set the bank with initial charge zero and will show it has a non-negative final charge.\
The following are the rules for discharging:
1. Each $4$-vertex gives $\frac{1}{2}$ to each pendant $3$-face and the rest to the incident $3$-faces evenly.
2. Every $6$-vertex gives $\frac{9}{4}$ to incident bad $(3,4,6)$-faces, $2$ to other incident $(3,4^-,6)$-faces and $\frac{3}{2}$ to all other incident $3$-faces; every $7^+$-vertex gives $\frac{9}{4}$ to all incident $3$-faces.
3. Every $6^+$-vertex gives $\frac{1}{2}$ to all pendant $3$-faces.
4. Every $(4^+, 4^+, 5^+)$-face and every $(4,4,4)$-face with a good $4$-vertex give $\frac{1}{2}$ to the bank and every bad $(3,4,5^+)$-face gives $\frac{1}{4}$ to the bank.
5. The bank gives $\frac{1}{2}$ to each $(3, 4, 4)$-face without good $4$-vertices.
6. Every $5$-vertex gives
(a) $2$ to each incident $(3,3,5)$-face and $9/4$ to each incident bad $(3,4,5)$-face.
(b) $7/4$ to incident non-bad $(3,4,5)$-faces when also incident a bad $(3,4,5)$-face, and gives $2$ to incident non-bad $(3,4,5)$-faces otherwise.
(c) $5/4$ to incident $(3,5^+,5^+)$-faces when also incident to a bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face, and gives $3/2$ to incident $(3,5^+,5^+)$-faces otherwise.
(d) $3/2$ to all $(4,4^+,5)$-faces with a chain of triangles to a $(3,4,4)$-face and gives $1$ to $(4,4^+,5)$-faces otherwise.
(e) $1/2$ to each pendant $(3,4^-,4^-)$-face and $(3,3,k)$-face and $1/4$ to all other pendant $3$-faces.
Let $v$ be a $k$-vertex. By Proposition \[fact\], $k\geq 3$.
For $k=3$, the final charge $\mu^*(v)$ of $v$ is $\mu^*(v)=\mu(v)=0$.
For $k=4$, by (R1), the final charge of $v$ is $0$. We note that $v$ gives at least $1$ to each incident $3$-face, and gives at least $3/2$ to $3$-faces when $v$ is a good $4$-vertex.
For $k=5$, if $v$ has at most one incident $3$-face, then by (R6a) and (R6e), $\mu^*(v)\geq \mu(v)-\frac{9}{4}\cdot 1-\frac{1}{2}\cdot 3=1/4>0$. Let $v$ have two incident $3$-faces $f_1$ and $f_2$ and a pendant $3$-face $f_3$.
Let $f_3$ be a $(3,4^-,4^-)$-face. When $f_1$ is a bad $(3,4,5)$-face, by Lemma \[5-vertex\] $f_2$ cannot be a $(3,4^-,5)$-face. By Lemma \[bad-345-45k-with-chain\], if $f_2$ is a $(4,4^+,5)$-face, then there is no chain of triangles from some $(3,4,4)$-face to $f$, so by (R6a), (R6c), (R6d), and (R6e), $\mu^*(v)\geq \mu(v)-\frac{1}{2}\cdot 1-\frac{9}{4}\cdot 1-\frac{5}{4}\cdot 1=0$. When $f_1$ is a non-bad $(3,4,5)$-face, then by Lemma \[5-vertex\], $f_2$ cannot be a $(3,4^-,5)$-face, so by (R6b), (R6c), (R6d), and (R6e), $\mu^*(v)\geq \mu(v)-\frac{1}{2}\cdot 1-2\cdot 1-\frac{3}{2}\cdot 1=0$. When neither $f_1$ nor $f_2$ are $(3,4^-,5)$-faces, by (R6c), (R6d), and (R6e), $\mu^*(v)\geq \mu(v)-\frac{1}{2}\cdot 1-\frac{3}{2}\cdot 2=\frac{1}{2}>0$.
Now let $f_3$ be a $(3,4,5)$-face. When $f_1$ or $f_2$ is $(3,4^-,5)$-face, by Lemma \[5-vertex\], the other one cannot be a $(3,4^-,5)$-face, so by (R6b), (R6c), (R6d), and (R6e), $\mu^*(v)\geq \mu(v)-\frac{1}{4}\cdot 1-\frac{9}{4}\cdot 1-\frac{3}{2}\cdot 1=0$. When neither $f_1$ nor $f_2$ are $(3,4^-,5)$-faces, by rules (R6c), (R6d), and (R6e), $\mu^*(v)\geq \mu(v)-\frac{1}{4}\cdot 1-\frac{3}{2}\cdot 2=\frac{3}{4}>0$.
Finally, let $v$ have two incident $3$-faces $f_1$ and $f_2$, and no pendant $3$-face. If $f_1$ is a bad $(3,4,5)$-face, then by Lemma \[bad-345s\], $f_2$ cannot also be a bad $(3,4,5)$-face or a $(3,3,5)$-face. Then by (R6), $\mu^*(v)\geq \mu(v)-\frac{9}{4}\cdot 1-\frac{7}{4}\cdot 1=0$. If neither $f_1$ nor $f_2$ is a bad $(3,4,5)$-face, then by (R6b), (R6c), and (R6d), $\mu^*(v)\geq \mu(v)-2\cdot 2=0$.
For $k=6$, if $v$ is incident to at most two $3$-faces, then by (R2) and (R3), $\mu^*(v)\geq \mu(v)-\frac{9}{4}\cdot 2-\frac{1}{2}\cdot 2=\frac{1}{2}$. So we can assume that $v$ is incident to three $3$-faces. If $v$ is incident a bad $(3,4,6)$-face then by Lemma \[bad-346\] only one other incident $3$-face can be a $(3,4^-,6)$-face. So by (R2), $\mu^*(v)\geq \mu(v)-\frac{9}{4}\cdot 2-\frac{3}{2}\cdot 1=0$. If $v$ is not incident a bad $(3,4,6)$-face, then by (R2), $\mu^*(v)\geq \mu(v)-2\cdot 3=0$.
For $k\geq 7$, if $k$ is odd, then $\mu^*(v)\geq \mu(v)-\frac{k-1}{2}\cdot \frac{9}{4}-\frac{1}{2}\cdot 1=2k-6-\frac{9k-9}{8}-\frac{4}{8}=\frac{7k-43}{8}\geq \frac{3}{4}$. If $k$ is even, then $\mu^*(v)\geq \mu(v)-\frac{k}{2}\cdot \frac{9}{4}=2k-6-\frac{9k}{8}=\frac{7k-48}{8}\geq 1$.
Now let $f$ be a $k$-face. Since $G$ is a simple graph, $k\geq 3$. By the condition that there is no $4$-cycle and $5$-cycle, $k=3$ or $k\geq 6$. Since no faces above degree $3$ are involved in the discharging procedure, the final charge of $6^+$-face $f$ is $\mu^*(f)=\mu(f)=d(f)-6\geq 0$.
For $k=3$, by Lemma \[334-face\], we have no $(3,3,4^-)$-faces, but we still have a few different cases:
**Case 1:** Face $f$ is a $(3,3,5^+)$-face. By Lemma \[333-vertices\], $f$ will have two pendant neighbors of degree $4$ or higher. So by (R1), (R2), (R4), and (R7), $\mu^*(f)\geq (3-6)+2\cdot 1+\frac{1}{2}\cdot 2=0$.
**Case 2:** Face $f$ is a $(3,4,4)$-face. By Lemma \[344-face\], $f$ will have a pendant neighbor of degree $4$ or higher. If $f$ has a good $4$-vertex, then by (R1), $\mu^*(f)\geq \mu(f)+\frac{3}{2}\cdot 1+1\cdot 1+\frac{1}{2}\cdot 1=0$. If $f$ has no good $4$-vertices, then by (R5), $f$ receives $1/2$ from the bank, so $\mu^*(f)=\mu(f)+1\cdot 2+\frac{1}{2}\cdot 1+\frac{1}{2}=0$.
**Case 3:** Face $f$ is a bad $(3,4,5)$-face. By (R1), (R4) and (R6a), $\mu^*(f)=\mu(f)+1\cdot 1+\frac{9}{4}\cdot 1-\frac{1}{4}\cdot 1=0$.
**Case 4:** Face $f$ is a non-bad $(3,4,5)$-face. If the $5$-vertex of $f$ is not incident a bad $(3,4,5)$-face, then by (R1) and (R6b), $\mu^*(f)=\mu(f)+1\cdot 1+2\cdot 1=0$. If the $5$-vertex of $f$ is incident a bad $(3,4,5)$-face, then by Lemma \[35k-pendant-neighbor\], $f$ has a pendant neighbor of degree $4$ or higher. So by (R1), (R6b), and (R6e), $\mu^*(f)\geq \mu(f)+1\cdot 1+\frac{7}{4}\cdot 1+\frac{1}{4}\cdot 1=0$.
**Case 5:** Face $f$ is a $(3,4,6)$-face. If $f$ is a bad $(3,4,6)$-face, then by (R1), (R2), and (R4), $\mu^*(f)=\mu(f)+1\cdot 1+\frac{9}{4}\cdot 1-\frac{1}{4}\cdot 1=0$. If $f$ is a non-bad $(3,4,6)$-face then by (R1) and (R2), $\mu^*(f)=\mu(f)+1\cdot 1+2\cdot 1=0$.
**Case 6:** Face $f$ is a $(3,4,7^+)$-face. By (R1) and (R2), $\mu^*(f)=\mu(f)+1\cdot 1+\frac{9}{4}\cdot 1=\frac{1}{4}$.
**Case 7:** Face $f$ is a $(3,5,5)$-face. If neither $5$-vertex of $f$ is also incident to a bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face, then by (R6c), $\mu^*(f)=\mu(f)+\frac{3}{2}\cdot 2=0$. If one of the $5$-vertices of $f$ is also incident to a bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face then by Lemma \[35k-pendant-neighbor\], $f$ must have a pendant neighbor of degree $4$ or higher. In addition, by Lemma \[355-two-bad-345\] the other $5$-vertex of $f$ cannot have both an incident bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face. So by (R6c) and (R6e), $\mu^*(f)=\mu(f)+\frac{5}{4}\cdot 1+\frac{1}{4}\cdot 1+\frac{3}{2}\cdot 1=0$.
**Case 8:** Face $f$ is a $(3,5,6^+)$-face. If the $5$-vertex of $f$ is not incident to a bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face then by (R2) and (R6c), $\mu^*(f)\geq \mu(f)+\frac{3}{2}\cdot 2=0$. If the $5$-vertex of $f$ has both an incident bad $(3,4,5)$-face and a pendant $(3,4^-,4^-)$-face, then by Lemma \[35k-pendant-neighbor\] $f$ must have a pendant neighbor of degree $4$ or higher. So by (R2), (R6c), and (R6e), $\mu^*(f)\geq \mu(f)+\frac{5}{4}\cdot 1+\frac{1}{4}\cdot 1+\frac{3}{2}\cdot 1=0$.
**Case 9:** Face $f$ is a $(3,6^+,6^+)$-face. By (R2), $\mu^*(f)\geq \mu(f)+\frac{3}{2}\cdot 2=0$.
**Case 10:** Face $f$ is a $(4,4,4)$-face. If $f$ has no good $4$-vertices then by (R1), $\mu^*(f)=\mu(f)+1\cdot 3=0$. If $f$ has a good $4$-vertex then by (R1) and (R4), $\mu^*(f)\geq \mu(f)+1\cdot 2+\frac{3}{2}\cdot 1-\frac{1}{2}\cdot 1=0$.
**Case 11:** Face $f$ is a $(4^+,4^+,5^+)$-face. If $f$ has no chains of triangles to a $(3,4,4)$-face, then each incident vertex gives at least $1$ to $f$, so $\mu^*(f)\geq \mu(f)+1\cdot 3=0$. If $f$ has a chain of triangles to a $(3,4,4)$-face then by (R6d), at least one vertex must give $\frac{3}{2}$ to $f$, so combined with (R4), $\mu^*(v)\geq \mu(v)+1\cdot 2+\frac{3}{2}\cdot 1-\frac{1}{2}\cdot 1=0$.
Finally, we show that the bank has a non-negative charge. By Lemma \[existence-chain\], for each $(3,4,4)$-face without good $4$-vertices in $G$, there exist at least two chains of triangles from the $(3,4,4)$-face to a bad $(3,4,5^+)$-face, a $(4,4,4)$-face with a good $4$-vertex, or a $(4^+,4^+,5^+)$-face. Then by Lemma \[at-most-two-344\], there exist at most two chains of triangles to $(4^+,4^+,5^+)$-face from $(3,4,4)$-faces and at most one chain of triangles to a $(3,4,5^+)$-face from $(3,4,4)$-faces. So we can see the transfer of charge from triangles with extra charge to the bank and back to $(3,4,4)$-faces is a transfer of $\frac{1}{4}$ charge over each chain of triangles. Each $(4,4,4)$-face with a good $4$-vertex and $(4^+,4^+,5^+)$-face gives $\frac{1}{2}$ to the bank, and the bank will give at most $\frac{1}{4}\cdot 2$ to $(3,4,4)$-faces for each $(4,4,4)$-face with a good $4$-vertex or $(4^+,4^+,5^+)$-face. Also, each bad $(3,4,5^+)$-face gives $\frac{1}{4}$ to the bank, and the bank will give at most $\frac{1}{4}\cdot 1$ to $(3,4,4)$-faces for each bad $(3,4,5^+)$-face. Hence the bank will always have a non-negative charge.
This completes the discharging, showing that the final charges of all faces, vertices, and the bank are non-negative, a contradiction to . This completes the proof of Theorem $1.1$.
$(3,0,0)$-coloring of planar graphs
====================================
In this section, we give a proof for Theorem \[300-coloring\]. Our proof will again use a discharging method. Let $G$ be a minimum counterexample to Theorem \[300-coloring\], that is, $G$ is a planar graph without $4$-cycles and $5$-cycles and is not $(3,0,0)$-colorable, but any proper subgraph of $G$ is properly $(3,0,0)$-colorable. We may assume that vertices colored by $1$ may have up to three neighbors colored by $1$.
The following is a very useful tool to extend a coloring on a subgraph of $G$ to include more vertices.
\[extend-coloring\] Let $H$ be a proper subgraph of $G$. Given a $(3,0,0)$-coloring of $G-H$, if two neighbors of $v\in H$ are colored so that one is a $5^-$-vertex and the other is nicely colored, then the coloring can be extended to $G-(H-v)$ such that $v$ is nicely colored by $1$.
Let $H$ be a subgraph of $G$ such that $G-H$ has a $(3,0,0)$-coloring. Let $v\in H$ have neighbors $u$ and $w$ that are colored. Let $d(u)\leq 5$ and let $w$ be nicely colored. Color $v$ by $1$. Since $w$ is nicely colored, if this coloring is invalid, then $u$ must be colored by $1$. In addition, $u$ must have at least $3$ neighbors colored by $1$. To avoid recoloring $u$ by $2$ or $3$, $u$ must have at least one neighbor of color $2$ and at least one neighbor of color $3$. This implies that $d(u)\geq 6>5$, a contradiction. So $v$ is colorable by $1$. In addition, since the deficiency of color $1$ is $3$ and $v$ only has $2$ neighbors, it follows that $v$ is nicely colored.
\[3-to-6\] Every $3$-vertex in $G$ has a $6^+$-vertex as a neighbor.
Let $v$ be a vertex in $G$ such that each neighbor vertex of $v$ has degree $5$. By the minimality of $G$, $G-v$ is $(3,0,0)$-colorable. If two vertices in $N(v)$ share the same color, then $v$ can be properly colored, so we can assume all the neighbors of $v$ are colored differently. Let $u$ be the neighbor of $v$ that is colored by $1$. Then $u$ must have $3$ neighbors colored by $1$ to forbid $v$ to be colored by $1$. In addition, $u$ must have neighbors colored by $2$ and $3$ to forbid $v$ to be colored by $2$ or $3$. Then, $u$ has at least $6$ neighbors, a contradiction.
Let a $(3,3,3^+)$-face to be *poor* if the pendant neighbors of the two $3$-vertices have degrees at most $5$. A $(3, 3^+, 3^+)$-face is *semi-poor* if exactly one of the pendant neighbors of the $3$-vertices has degree $5$ or less. A $3$-face is *non-poor* if each $3$-vertex on it has the pendant neighbor being a $6^+$-vertex. Finally, a *poor 3-vertex* is a $3$-vertex on a poor or semi-poor $3$-face that has a $5^-$-vertex as its pendant neighbor.
\[336-face\] All (3,3,6$^-$)-faces in G are non-poor.
For all $(3,3,5^-)$-faces in $G$, the proof is trivial by Lemma \[3-to-6\]. Let $uvw$ be a $(3,3,6)$-face in $G$ with $d(u)=d(v)=3$ such that the pendant neighbor $v'$ of $v$ has degree at most $5$. By the minimality of $G$, $G$\\$\{u,v\}$ is $(3,0,0)$-colorable. Properly color $u$ and color $v$ differently than both $w$ and $v'$. Then $u$ and $v$ are both colored by $2$ or $3$, w.l.o.g. assume $2$. This means that $u'$ and $v'$ share the same color (where $u'$ is the pendant neighbor of $u$), different from the color of $w$.
Let $w$ be colored by $1$, then to avoid being able to recolor $u$ or $v$ by $1$, $w$ must have $3$ outer neighbors colored by $1$. Then $w$ can be recolored by $2$ or $3$ depending on the color of its fourth colored neighbor. We recolor $w$ by $2$ or $3$ and recolor $u$ and $v$ by $1$ to get a coloring of $G$, a contradiction.
So we may assume that $w$ is colored by $3$, and that $u'$ and $v'$ are colored by $1$. To avoid recoloring $v$ by $1$, $v'$ must have at least $3$ neighbors colored by $1$. In addition, to avoid recoloring $v'$ by $2$ or $3$ and coloring $v$ by $1$, $v'$ must have neighbors colored by both $2$ and $3$. This contradicts that $v'$ has degree less than $6$.
\[incident-poor\] No vertex $v\in V(G)$ can have $\lfloor \frac{d(v)}{2} \rfloor$ incident poor $3$-faces.
Let $v$ be a $k$-vertex in $G$ with $\lfloor \frac{k}{2} \rfloor$ incident poor $(3,3,k)$-faces. Let $u_1, u_2, \cdots, u_k$ be the neighbors of $v$, and let $u_i'$ be the pendant neighbor if $u_i$ is in a poor $3$-face. Note that $d(u_i')\le 5$ and we know that all except possibly $u_k$ are in poor $3$-faces.
By the minimality of $G$, $G$\\$\{v, u_1, u_2, \cdots, u_{k-1}\}$ is $(3,0,0)$-colorable. If $d(v)$ is odd, then by Lemma \[extend-coloring\], for all $i$ with $1\leq i\leq k-1$, we can color $u_i$ by $1$. Then we can properly color $v$ to get a coloring of $G$, so we can assume that $d(v)$ is even. If $d(v)$ is even, then by Lemma \[extend-coloring\], for all $i$ with $1\leq i\leq k-2$, we can color $u_i$ by $2$. Then if $u_k$ is colored by $1$ we can color $u_{k-1}$ properly and $v$ properly to get a coloring of $G$. If $u_k$ is colored by $2$ or $3$, then it is colored properly and by Lemma \[extend-coloring\] we can color $u_{k-1}$ by $1$. Then we can properly color $v$ to get a coloring of $G$, a contradiction.
\[8-vertex\] If an $8$-vertex $v$ is incident to three incident poor $(3,3,8)$-faces, then it cannot be incident to a semi-poor face, nor two pendant $3$-faces.
Let $v$ be an $8$-vertex in $G$ with $3$ incident poor $(3,3,8)$-faces. Let $u_1, u_2, \cdots, u_6$ be the $3$-vertices in the poor $(3,3,8)$-face and let $u'_1, u'_2, \cdots, u'_6$ be the corresponding pendant neighbors, respectively. We know that for all $i$ with $1\leq i\leq 6$, $d(u'_i)\leq 5$.
\(i) Let $vu_7u_8$ be the incident semi-poor face with $u_7$ being the poor $3$-vertex. Then by the minimality of $G$, $G$\\$\{v, u_1, u_2, \cdots, u_7\}$ is $(3,0,0)$-colorable. By Lemma \[extend-coloring\], $u_1, u_2, \cdots, u_6$ can be colored by $1$. Then if $u_8$ is colored by $1$, we can properly color $u_7$ and then $v$ to get a coloring of $G$. So we may assume that $u_8$ is not colored by $1$, in which case it is nicely colored and we may color $u_7$ with $1$ by Lemma \[extend-coloring\], and then properly color $v$ to get a coloring of $G$, a contradiction.
\(ii) Let $u_7$ and $u_8$ be the bad $3$-vertices adjacent to $v$. Then $G$\\$\{v, u_1, u_2, \cdots, u_7, u_8\}$ is $(3,0,0)$-colorable, by the minimality of $G$. Properly color both $u_7$ and $u_8$. If either $u_7$ or $u_8$ is colored by $1$ or both have the same color, then by Lemma \[extend-coloring\], we may color $u_1, u_2, \cdots, u_6$ by $1$ and then properly color $v$. So we may assume that $u_7$ is colored by $2$ and $u_8$ is colored by $3$. Then we properly color $u_1, u_2, \cdots, u_6$, and it follows that for each $i$ with $1\leq i\leq 3$, $u_{2i-1}$ and $u_{2i}$ must be colored differently. Then $v$ can have at most $3$ neighbors colored by $1$, all properly colored, so $v$ can be colored by $1$, a contradiction.
\[7-vertex\] If a $7$-vertex $v$ is incident to two poor $(3,3,7)$-faces, then it cannot be (i) incident to a semi-poor $(3,6^-,7)$-face and adjacent to a pendant $3$-face, or (ii) adjacent to three pendant $3$-faces.
Let $v$ be a $7$-vertex in $G$ with $2$ incident poor $(3,3,7)$-faces. Let $u_1, u_2, u_3,$ and $u_4$ be the $3$-vertices on the poor $(3,3,7)$-faces and let $u'_1, u'_2, u'_3,$ and $u'_4$ be their corresponding pendant neighbors, respectively. We know that for all $i$ with $1\leq i\leq 4$, $d(u'_i)\leq 5$.
\(i) Let $vu_5u_6$ be a semi-poor face with $u_5$ being a poor $3$-vertex and $d(u_6)\le 6$ and let $u_7$ be a bad $3$-vertex adjacent to $v$. By the minimality of $G$, $G$\\$\{v, u_1, u_2, u_3, u_4, u_5, u_7\}$ is $(3,0,0)$-colorable. Since at this point $u_6$ has only $4$ colored neighbors, if $u_6$ is colored by $1$ then either it is nicely colored or it can be recolored properly. If $u_6$ is not nicely colored, then recolor $u_6$ properly.
Color $u_7$ properly. If $u_7$ is colored by $1$, then by Lemma \[extend-coloring\], we can color $u_1, u_2, \cdots, u_5$ by $1$ and then color $v$ properly, a contradiction. So we may assume w.l.o.g. that $u_7$ is colored by $2$. Color $u_1, u_2, \cdots, u_5$ properly. Then, for each $i$ with $1\leq i\leq 3$, $u_{2i}$ and $u_{2i-1}$ are colored differently and nicely. This leaves $v$ with at most $3$ neighbors colored by $1$, all nicely, so we may color $v$ by $1$ to get a coloring of $G$, a contradiction.
\(ii) Let $u_5$, $u_6$, and $u_7$ be the bad $3$-vertices adjacent to $v$. By the minimality of $G$, $G$\\$\{v, u_1, \ldots, u_7\}$ is $(3,0,0)$-colorable. Properly color $u_5$, $u_6$, and $u_7$. If the set $\{u_5, u_6, u_7\}$ does not contain both colors $2$ and $3$, then by Lemma \[extend-coloring\], we can color $u_1$, $u_2$, $u_3$, and $u_4$ by $1$ and color $v$ properly. So we can assume that $\{u_5, u_6, u_7\}$ contains both colors $2$ and $3$. This implies that at most one vertex is colored by $1$. So we properly color $u_1$, $u_2$, $u_3$, and $u_4$. Then $v$ has at most $3$ neighbors colored by $1$, all nicely, so we can color $v$ by $1$ to get a coloring of $G$, a contradiction.
\[377-face\] Let $uvw$ be a semi-poor $(3,7,7)$-face in $G$ such that $d(v)=d(w)=7$. Then vertices $v$ and $w$ cannot both be $7$-vertices that are incident to two poor $3$-faces, one semi-poor $(3,7,7)$-face, and adjacent to one pendant $3$-face.
![Figure for Lemma \[377-face\][]{data-label="fig9"}](dilemma.pdf "fig:") ![Figure for Lemma \[377-face\][]{data-label="fig9"}](dilemmasolved.pdf "fig:")
Let $uvw$ be a semi-poor $(3,7,7)$-face in $G$ such that $d(v)=d(w)=7$ and both $v$ and $w$ are incident to two poor $3$-faces, one $(3,7,7)$-face, and adjacent to one pendant $3$-face. Let the neighbors of $v$ and $w$ be $t_1, t_2, \cdots, t_5$ and $z_1, z_2, \cdots, z_5$, respectively such that $t_5$ and $z_5$ are bad $3$-vertices (See Figure \[fig9\]).
By the minimality of $G$, $G$\\$\{u, v, w, t_1, t_2, \cdots, t_5, z_1, z_2, \cdots, z_5\}$ is $(3,0,0)$-colorable. By Lemma \[extend-coloring\], we can color $t_1, t_2, t_3,$ and $t_4$ by $1$. Then properly color $t_5$, $v$, and $z_5$ in that order. Vertex $v$ will not be colored by $1$, so w.l.o.g. lets assume that $v$ is properly colored by $2$. If $z_5$ is colored by $1$, then by Lemma \[extend-coloring\], we can color $z_1, z_2, z_3, z_4,$ and $u$ by $1$ and then properly color $w$, to get a coloring of $G$, a contradiction. So we can assume that $z_5$ is not colored by $1$. Then we properly color $z_1, z_2, z_3, z_4$ and $u$, so $w$ can have at most $3$ neighbors colored by $1$, all properly. We can color $v$ by $1$ to get a coloring of $G$, a contradiction.
\
We start the discharging process now. Recall that the initial charge for a vertex $v$ is $\mu(v)=2d(v)-6$ and the initial charge for a face $f$ is $\mu(f)=d(f)-6$.\
We introduce the following discharging rules:
1. Every $4$-vertex gives $1$ to each incident $3$-face.
2. Every $5$ and $6$-vertex gives $2$ to each incident $3$-face.
3. every $6^+$-vertex gives $1$ to each adjacent pendant $3$-face.
4. Each $d$-vertex with $7\le d\le 10$ gives $3$ to each incident poor $(3, 3, *)$-face, $2$ to each incident semi-poor $3$-face, except $7$-vertices give $1$ to special semi-poor $3$-face, where a special semi-poor $(3,7,7+)$-face is a semi-poor $3$-face incident to a $7$-vertex which is also incident to two poor $3$-faces and adjacent to one pendant $3$-face. Each $d$-vertex with $7\le d\le 10$ gives $1$ to all other incident $3$-faces.
5. Every $11^+$-vertex gives $3$ to all incident $3$-faces.
Now let $v$ be a $k$-vertex. By Proposition \[fact\], $k\geq 3$.
When $k=3$, $v$ is not involved in the discharging process, so $\mu^*(v)=\mu(v)=0$.
When $k=4$, by Proposition \[fact\], $v$ can have at most $2$ incident $3$-faces. By (R1), $\mu^*(v)\geq \mu(v)-1\cdot 2=0$.
When $k=5$, by Proposition \[fact\], $v$ can have at most $2$ incident $3$-faces. By (R2), $\mu$\*$(v)\geq \mu(v)-2\cdot 2=0$.
When $k=6$, by Proposition \[fact\], $v$ can have $\alpha\le 3$ incident $3$-faces, and at most $(k-2\alpha)$ pendant $3$-faces. By (R2) and (R3), $\mu^*(v)\geq \mu(v)-2\cdot \alpha -1\cdot (k-2\alpha)=k-6=0$.
When $k=7$, $v$ has an initial charge $\mu(v)=7\cdot 2-6=8$. By Lemma \[incident-poor\], $v$ has at most two poor $3$-faces. If $v$ has less than two incident poor $3$-faces, then by (R3) and (R4), $\mu$\*$(v)\geq \mu(v)-3\cdot 1-1\cdot 5=0$ since $v$ gives at most one charge per vertex excluding vertices in poor $3$-faces. So assume that $v$ has exactly $2$ incident poor $3$-faces. By Lemma \[7-vertex\], $v$ is adjacent to at most two pendant $3$-faces, and if it is incident to a semi-poor $(3,6^-,7)$-face, then $v$ is not adjacent to a pendant $3$-face. So if $v$ is not incident to a semi-poor $(3,7^+, 7)$-face, then by (R3) and (R4), $\mu^*(v)\ge \mu(v)-3\cdot 2-2\cdot 1=0$; If $v$ is incident to a semi-poor $(3,7^+,7)$-face, then by rules (R3) and (R4), $\mu^*(v)\geq \mu(v)-3\cdot 2-1\cdot 1-1\cdot 1=0$.
When $k=8$, $v$ has an initial charge $\mu(v)=8\cdot 2-6=10$. By Lemma \[incident-poor\], $v$ has at most three poor $3$-faces. If $v$ has less than $3$ incident poor $3$-faces, then by (R3) and (R4), $\mu^*(v)\geq \mu(v)-3\cdot 2-1\cdot 4=10-6-4=0$ since $v$ gives at most one charge per vertex excluding vertices in poor $3$-faces. So let $v$ is incident to exactly $3$ poor $3$-faces. By Lemma \[8-vertex\], $v$ cannot be incident to a semi-poor $3$-face or adjacent to two pendant $3$-faces, then $\mu^*(v)\ge \mu(v)-3\cdot 3-1\cdot 1=0$.
When $k=9$, by Lemma \[incident-poor\], $v$ is incident to at most three poor $3$-faces. The worst case occurs when $v$ is incident $3$ poor $(3,3,9)$-faces, incident one semi-poor $(3,3,9)$-face, and pendant one $3$-face. So by (R3) and (R4), $\mu$\*$(v)\geq \mu(v)-1\cdot 1-3\cdot 3-2\cdot 1=12-1-9-2=0$.
When $k=10$, by Lemma \[incident-poor\], $v$ is incident to at most four poor $(3,3,10)$-faces. So by (R3) and (R4), $\mu^*(v)\geq \mu(v)-3\cdot 4-2\cdot 1=14-3\cdot 4-2\cdot 1=0$.
When $k\geq 11$, we assume that $v$ is incident to $\alpha$ $3$-faces, then by Proposition \[fact\], $\alpha\le \lfloor k/2\rfloor$. Thus the final charge of $v$ is $\mu^*\ge 2k-6-3\alpha-1\cdot (k-2\alpha)=k-\alpha-6\ge 0$.
Now let $f$ be a $k$-face in $G$. By the conditions on $G$, $k=3$ or $k\geq 6$. When $k\geq 6$, $f$ is not involved in the discharging procedure, so $\mu*(f)=\mu(f)=k-6\geq 0$. So in the following we only consider $3$-faces.
**Case 1:** $f$ is a $(4^+, 4^+, 4^+)$-face. By the rules, each $4^+$-vertex on $f$ gives at least $1$ to $f$, so $\mu*(f)\geq \mu(f)+1\cdot 3=0$.
**Case 2:** $f$ is a $(3,4^+,4^+)$-face with vertices $u,v,w$ such that $d(u)=3$. If $u$ is not a poor $3$-vertex, then by (R2), $f$ gains $1$ from the pendant neighbor of $u$ and by the other rules, $f$ gains at least $2$ from vertices on $f$, thus $\mu^*(f)\geq \mu(f)+1\cdot 3=0$. If $u$ is a poor vertex (it follows that $f$ is a semi-poor $3$-face), then by Lemma \[3-to-6\], $f$ is a $(3, 4^+, 6^+)$-face. Since $v$ or $w$ is a $6^+$-vertex, it gives at least $2$ to $f$ unless $f$ is a special semi-poor $(3,7,7^+)$-face, and as the other is a $4^+$-vertex, it gives at least $1$ to $f$. Therefore, if $f$ is not a special semi-poor $3$-face, then $\mu^*(f)\geq \mu(f)+2\cdot 1+1\cdot 1=0$; if $f$ is a special semi-poor $(3,7,8^+)$-face, then $f$ receives at least $2$ from the $8^+$-vertex, so $\mu^*(v)\geq \mu(v)+2\cdot 1+1\cdot 1=0$. If $f$ is a special semi-poor $(3, 7, 7)$-face so that both $v$ and $w$ are incident to two poor $3$-faces, one semi-poor $(3, 7, 7)$-face and adjacent to one pendant $3$-face, then by Lemma \[377-face\], is impossible.
**Case 3:** $f$ is a $(3,3,4^+)$-face with $4^+$-vertex $v$. If $d(v)\ge 11$, then by (R5), $\mu^*(f)\geq \mu(f)+3=0$. So assume $d(v)\le 10$. By Lemma \[3-to-6\], if $4\le d(v)\le 6$, then each $3$-vertex has the pendant neighbor of degree $6$ or higher. So by (R1) and (R3) (when $d(v)=4$), $\mu^*(f)\ge \mu(f)+1\cdot 3=0$, or by (R1) and (R2) (when $d(v)>4$), $\mu^*(f)=\mu(f)+2\cdot 1+1\cdot 1=0$.
Let $7\leq d(v)\leq 10$. If $f$ is poor, then by (R4), $\mu^*(f)=\mu(f)+3\cdot 1=0$. If $f$ is semi-poor, then one $3$-vertex on $f$ is adjacent to a $6^+$-vertex and thus by (R3) $f$ gains $1$ from it, together the $2$ that $f$ gains from $v$ by (R4), we have $\mu^*(f)=\mu(f)+2\cdot 1+1\cdot 1=0$. If $f$ is non-poor, then both $3$-vertices on $f$ are adjacent to the pendant neighbors of degrees more than $5$, thus by (R3) and (R4), $\mu^*(f)=\mu(f)+1\cdot 2+1\cdot 1=0$.
**Case 4:** $f$ is a $(3,3,3)$-face. By Lemma \[3-to-6\], each $3$-vertex will have the pendant neighbor of degree $6$ or higher, so by (R3), $\mu^*(f)=\mu(f)+1\cdot 3=0$.
Since for all $x\in V\cup F$, $\mu^*(x)\geq 0$, $\sum_{v\in V}\mu^*(v) +\sum_{f\in F}\mu^*(f) \geq 0$, a contradiction. This completes the proof of Theorem $1.2$.
Acknowledgement {#acknowledgement .unnumbered}
===============
The research is supported in part by NSA grant H98230-12-1-0226 and NSF CSUMS grant. The authors thank Bernard Lidicky for some preliminary discussion.
[99]{}
O.V. Borodin, Colorings of planar graphs: a survey. *Disc. Math.*, to appear.
O. V. Borodin, A. N. Glebov, A. R. Raspaud, and M. R. Salavatipour. Planar graphs without cycles of length from 4 to 7 are 3-colorable. *J. of Comb. Theory, Ser. B*, [**93**]{} (2005), 303–311.
G. Chang, F. Havet, M. Montassier, and A. Raspaud, Steinberg’s Conjecture and near colorings, preprint.
N. Eaton and T. Hull. Defective list colorings of planar graphs. *Bull. Inst. Combin. Appl.*, [**25**]{} (1999), 78–87.
H. Grötzsch, Ein dreifarbensatz fr dreikreisfreienetze auf der kugel. *Math.-Nat.Reihe*, [**8**]{} (1959), 109–120.
R. Škrekovski. List improper coloring of planar graphs. *Comb. Prob. Comp.*, [**8**]{} (1999), 293Ð299.
R. Steinberg, The state of the three color problem. Quo Vadis, Graph Theory?, *Ann. Discrete Math.* [**55**]{} (1993), 211–248.
B. Xu, On $(3,1)^*$-coloring of planar graphs, *SIAM J. Disc. Math.*, [**23**]{} (2008), 205–220.
[^1]: Research supported in part by the NSA grant H98230-12-1-0226 and a NSF CSUMS grant
|
---
abstract: 'We provide adaptive inference methods for linear functionals of $\ell_1$-regularized linear approximations to the conditional expectation function. Examples of such functionals include average derivatives, policy effects, average treatment effects, and many others. The construction relies on building Neyman-orthogonal equations that are approximately invariant to perturbations of the nuisance parameters, including the Riesz representer for the linear functionals. We use $\ell_1$-regularized methods to learn the approximations to the regression function and the Riesz representer, and construct the estimator for the linear functionals as the solution to the orthogonal estimating equations. We establish that under weak assumptions the estimator concentrates in a $1/\sqrt{n}$ neighborhood of the target with deviations controlled by the normal laws, and the estimator attains the semi-parametric efficiency bound in many cases. In particular, either the approximation to the regression function or the approximation to the Riesz representer can be “dense" as long as one of them is sufficiently “sparse". Our main results are non-asymptotic and imply asymptotic uniform validity over large classes of models.'
bibliography:
- 'Post-ML\_Ref.bib'
title: 'Double/De-Biased Machine Learning Using Regularized Riesz Representers'
---
Approximate Sparsity vs. Density, Double/De-biased Machine Learning, Regularized Riesz Representers, Linear Functionals
Introduction
============
Consider a random vector $(Y,X')'$ with distribution $P$ and finite second moments, where the outcome $Y$ takes values in $\Bbb{R}$ and the covariate $X$ taking values $ x \in \mathcal{X}$, a Borel subset of $\Bbb{R}^d$. Denote the conditional expectation function map $x \mapsto {{\mathrm{E}}}[Y \mid X =x]$ by $\gamma^*_0$. We consider a function $\gamma_0$, given by $x \mapsto b(x)'\beta_0$, as a sparse linear approximation to $\gamma^*_0$, where $b$ is a $p$-dimensional vector, a dictionary of basis functions, mapping $\mathcal{X}$ to $\Bbb{R}^p$. The dimension $p$ here can be large, potentially much larger than the sample size.
Our goal is to construct high-quality inference methods for a real-valued linear functional of $\gamma_0$ given by: $$\theta_0 = {{\mathrm{E}}}m(X, \gamma_0) = \int m(x, \gamma_0) d F(x),$$ where $F$ is the distribution of $X$ under $P$, for example the average derivative and other functionals listed below. (See Section 2 below regarding formal requirements on $m$). When the approximation error $\gamma_0 - \gamma^*_0$ is small, our inference will automatically re-focus on a more ideal target – the linear functional of the conditional expectation function, $$\theta_0^* = {{\mathrm{E}}}m(X, \gamma^*_0).$$
Consider an average derivative in the direction $a$: $$\theta_0 = {{\mathrm{E}}}a'\nabla_x \gamma_0(X), \quad a \in \Bbb{R}^{d}.$$ This functional corresponds to an approximation to the effect of policy that shifts the distribution of covariates via the map $x \mapsto x + a$, so that $$\int (\gamma_0(x+ a) - \gamma_0(x)) d F(x) \approx \int a' \nabla_x \gamma_0(x) d F(x).$$
Consider the effect from a counterfactual change of covariate distribution from $F_0$ to $F_1$: $$\theta_0 = \int \gamma_0(x) d (F_1(x) - F_0(x)) = \int \gamma_0(x) [d (F_1(x) - F_0(x))/dF(x)] d F(x).$$
Consider the average treatment effect under unconfoundedness. Here $X = (Z,D)$ and $\gamma_0(X) = \gamma_0(D,Z)$, where $D \in \{0,1\}$ is the indicator of the receipt of the treatment, and $$\theta_0 = \int (\gamma_0(1,z) - \gamma_0(0,z)) d F(z) = \int \gamma_0(d,z) (1(d=1) - 1(d=0)) dF(x).$$
We consider $\gamma \mapsto {{\mathrm{E}}}m(X, \gamma)$ as a continuous linear functional on $\Gamma_b$, the linear subspace of $L^2(F)$ spanned by the given dictionary $x \mapsto b(x)$. Such functionals can be represented via the Riesz representation: $${{\mathrm{E}}}m(X, \gamma) = {{\mathrm{E}}}\gamma_0(X) \alpha_0(X),$$ where the Riesz representer $\alpha_0(X) = b(X)'\rho_0$ is identified by the system of equations: $${{\mathrm{E}}}m(X, b) = {{\mathrm{E}}}b(X) b(X)'\rho_0,$$ where $m(X, b): = \{m(X,b_j)\}_{j=1}^p$ is a componentwise application of linear functional $m(X, \cdot)$ to $b = \{ b_j\}_{j=1}^p$. Having the Riesz representer allows us to write the following “doubly robust" representation for $\theta_0$: $$\theta_0 = {{\mathrm{E}}}[ m(X, \gamma_0) + \alpha_0(X) ( Y- \gamma_0(X)) ].$$ This representation is approximately invariant to small perturbations of parameters $\gamma_0$ and $\alpha_0$ around their true values (see Lemma 2 below for details), a property sometime referred to as the Neyman-type orthogonality (see, e.g., [@DML]), making this representation a good one to use as a basis for estimation and inference in modern high-dimensional settings. (See also Proposition 5 in [@LR] for a formal characterization of scores of this sort as having the double robustness property in the sense of [@robins:dr]).
Our estimation and inference will explore an empirical analog of this equation, given a random sample $(Y_i, X_i')_{i=1}^n$ generated as i.i.d. copies of $(Y,X')$. Instead of the unknown $\gamma_0$ and $\beta_0$ we will plug-in estimators obtained using $\ell_1-$ regularization. We shall use sample-splitting in the form of cross-fitting to obtain weak assumptions on the problem, requiring only approximate sparsity of either $\beta_0$ or $\rho_0$, with some weak restrictions on the sparsity indexes. For example, if both parameter values are sparse, then the product of effective dimensions has to be much smaller than $n$, the sample size. Moreover, one of the parameter values, but not both, can actually be “dense" and estimated at the so called “slow" rate, as long as the other parameter is sparse, having effective dimension smaller than $\sqrt {n}$.
We establish that that the resulting “double" (or de-biased) machine learning (DML) estimator $\hat \theta$ concentrates in a $1/\sqrt{n}$ neigborhood of the target with deviations controlled by the normal laws, $$\sup_{t \in \Bbb{R} }\Big |{{\mathrm{P}}}( \sqrt{n} \sigma^{-1} (\hat \theta - \theta_0) \leq t) - \Phi(t) \Big | \leq \varepsilon_n,$$ where the non-asymptotic bounds on $\varepsilon_n$ are also given.
As the dimension of $b$ grows, suppose that the subspace $\Gamma_b$ becomes larger and approximates the infinite-dimensional linear subspace $\Gamma^* \subseteq L^2(P)$, that contains the true $\gamma^*_0$. We assume the functional $\gamma \mapsto {{\mathrm{E}}}m(X, \gamma)$ is continuous on $\Gamma^*$ in this case. If the approximation bias is small, $$\sqrt{n}(\theta_0 - \theta_0^*) \to 0,$$ our inference will automatically focus on the “ideal" target $\theta^*_0$. Therefore our inference can be interpreted as targeting the functionals of the conditional expectation function in the regimes where we can successfully approximate them. However our approach does not hinge on this property and retains interpretability and good properties under misspecification.
It is interesting to note that in the latter case $\alpha_0$ will approximate the true Riesz representer $\alpha^*_0$ for the linear functionals on $\Gamma^*$, identified by the system of equations: $${{\mathrm{E}}}m(X, \gamma) = {{\mathrm{E}}}\gamma(X) \alpha^*_0(X), \quad \forall \gamma \in \Gamma^*.$$ Note that $\theta_0^*$ has a “doubly robust" representation: $$\label{DR}
\theta^*_0 = {{\mathrm{E}}}[ m(X, \gamma^*_0) + \alpha^*_0(X) ( Y- \gamma^*_0(X)) ],$$ which is invariant to perturbations of $\gamma^*_0$ and $\alpha^*_0$. Hence, our approach can be viewed as approximately solving the empirical analog of these equations, in the regimes where $\gamma_0$ does approximate $\gamma^*_0$. In such cases our estimator attains the semi-parametric efficiency bound, because its influence function is in fact the efficient score for $\theta_0$; see [@vaart:1991; @newey94].
When $\Gamma^*= L^2(F)$ and the functional $\gamma \mapsto {{\mathrm{E}}}m(X, \gamma)$ is continuous on $\Gamma^*$, the Riesz representer $ \alpha^*_0(X)$ belongs to $L^2(F)$ and can be stated explicitly in many examples:
- in Example 1: $\alpha^*_0(x) = -\partial_x \log f(x)$, where $f(x) = d F(x)/dx$,
- in Example 2: $\alpha^*_0(x) = d (F_1(x) - F_0(x))/d F(x)$,
- in Example 3: $\alpha^*_0(x) = (1(d=1) - 1(d= 0) )/P(d \mid z)$.
However, such closed-form solutions are not available in many other examples, or when $\Gamma^*$ is smaller than $L^2(F)$, which is probably the most realistic situation occurring in practice.
Using closed-form solutions for Riesz representers $\alpha^*_0$ in several leading examples and their machine learning estimators, [@DML] defined DML estimators of $\theta^*_0$ in high-dimensional settings and established their good properties. Compared to this approach, the new approach proposed in this paper has the following advantages and some limitations:
1. It automatically estimates the Riesz representer $\alpha_0$ from the empirical analog of equations that implicitly characterize it.
2. It does not rely on closed-form solutions for $\alpha^*_0$, which generally are not available.
3. When closed-form solutions for $\alpha^*_0$ are available, it avoids directly estimating $\alpha^*_0$. For example, it avoids estimating derivatives of densities in Example 1 or inverting estimated propensity scores $P(d \mid z)$ in Example 3. Rather it estimates the projections $\alpha_0$ of $\alpha^*_0$ on the subspace $\Gamma_b$, which is a much simpler problem when the dimension of $X$ is high.
4. Our approach remains interpretable under misspecification – when approximation errors are not small, we simply target inference on $\theta_0$ instead of $\theta^*_0$.
5. While the current paper focuses only on sparse regression methods, the approach readily extends to cover other machine learning estimators $\hat \gamma$ as long as we can find (numerically) dictionaries $b$ that (approximately span) the realizations of $\hat \gamma - \gamma_0$, where $\gamma_0$ is the probability limit of $\hat \gamma$.
6. The current approach is limited to linear functionals, but in an ongoing work we are able to extend the approach to nonlinear functionals by first performing a linear expansion and then applying our new methods to the linear part of the expansion.
The paper also builds upon ideas in classical semi-parametric learning theory with low-dimensional $X$, which focused inference on ideal $\theta_0^*$ using traditional smoothing methods for estimating nuisance parameters $\gamma_0$ and $\alpha_0$ \[[@vaart:1991; @newey94; @bickel:semibook; @robins:dr; @vdV]\], that do not apply to the current high-dimensional setting. Our paper also builds upon and contributes to the literature on the modern orthogonal/debiased estimation and inference \[[@c.h.zhang:s.zhang; @BelloniChernozhukovHansen2011; @belloni2014pivotal; @BCK-LAD; @javanmard2014confidence; @JM:ConfidenceIntervals; @JM2015; @vandeGeerBuhlmannRitov2013; @ning2014general; @CHS:AnnRev; @neykov2015unified; @HZhou; @JV:cov; @JV:eff; @JV:m; @bradic:QR; @zhu:breaking; @zhu2017linear]\], which focused on inference on the coefficients in high-dimensional linear and generalized linear regression models, without considering the general linear functionals analyzed here.\
**Notation.** Let $W = (Y, X')'$ be a random vector with law $P$ on the sample space $\mathcal{W}$, and $W_1^n = (Y_i, X_i)_{i=1}^n$ denote the i.i.d. copies of $W$. All models and probability measure $P$ can be indexed by $n$, a sample size, so that the models and their dimensions can change with $n$, allowing any of the dimensions to increase with $n$. We use the notation from the empirical process theory, see [@VW]. Let ${{\mathbb{E}_I}}f$ denote the empirical average of $f(W_i)$ over $i \in I \subset \{1,..., n\}$: $${{\mathbb{E}_I}}f := {{\mathbb{E}_I}}f(W) = | I | ^{-1} \sum_{i \in I} f(W_i).$$ Let $\Bbb{G}_I$ denote the empirical process over $f \in \mathcal{F}: \mathcal{W} \to \Bbb{R}^p$ and $ i \in I$, namely $$\mathbb{G}_I f := \mathbb{G}_I f (W) := | I | ^{-1/2} \sum_{ i \in I} (f(W_i) - P f),$$ where $Pf := P f(W) := \int f(w) d P(w)$. Denote by the $L^q(P)$ norm of a measurable function $f$ mapping the support of $W$ to the real line and also the $L^q(P)$ norm of random variable $f(W)$ by $\| f \|_{P,q} = \| f(W)\|_{P,q}$. We use $\| \cdot\|_q$ to denote $\ell_q$ norm on $\Bbb{R}^d$.
For a differentiable map $x \mapsto f(x)$, mapping $\mathbb{R}^d$ to $\mathbb{R}^k$, we use $\partial_{x'} f$ to abbreviate the partial derivatives $(\partial/\partial x') f$, and we correspondingly use the expression $\partial_{x'} f(x_0)$ to mean $\partial_{x'} f (x) \mid_{x = x_0}$, etc. We use $x'$ to denote the transpose of a column vector $x$.
The DML with Regularized Riesz Representers
===========================================
Sparse Approximations for the Regression Function and the Riesz Representer
----------------------------------------------------------------------------
We work with the set up above. Consider a conditional expectation function $x \mapsto \gamma^*_0(x) = {{\mathrm{E}}}[Y \mid X = x]$ such that $\gamma_0 \in L^2(F)$ and a $p$-vector of dictionary terms $x \mapsto b(x) = (b_j(x))_{j=1}^p$ such that $b \in L^2(F)$. The dimension $p$ of the dictionary can be large, potentially much larger than $n$.
We approximate $\gamma^*_0$ as $$\gamma^*_0 = \gamma_0 + r_\gamma := b'\beta_0 + r_\gamma,$$ where $r_\gamma$ is the approximation error, and $\gamma_0 := b'\beta_0$ is the “best sparse linear approximation" defined via the following Dantzig Selector type problem ([@candes2007dantzig]).
Let $\beta_0$ be a minimal $\ell_1$-norm solution to the approximate best linear predictor equations $$\beta_0 \in \arg \min \|\beta \|_1 : \| {{\mathrm{E}}}[ b(X) ( Y - b(X)'\beta) ] \|_\infty \leq \lambda^\beta_0.$$ When $\lambda^\beta_0 = 0$, $\beta_0$ becomes the best linear predictor parameter (BLP).
We refer to the resulting approximation as “sparse", since solutions $\beta_0$ often are indeed sparse. Note that since ${{\mathrm{E}}}[Y \mid X] = \gamma^*_0(X)$, the approximation error $r_\gamma$ is approximately orthogonal to $b$: $$\| {{\mathrm{E}}}[ b(X) ( Y - b(X)'\beta_0) \|_\infty = \| {{\mathrm{E}}}[ b(X) ( \gamma^*_0(X) - b(X)'\beta_0) ]\|_\infty =
\| {{\mathrm{E}}}[ b(X) r_\gamma(X) ] \|_\infty \leq\lambda^\beta_0.$$
Consider a linear subspace $\Gamma^* \subset L^2(F)$ that contains $\Gamma_b$, the linear subspace generated by $b$. In some of the asymptotic results that follow, we can have $\Gamma_b \uparrow \Gamma^*$ as $p \to \infty$.
- *For each $x \in \mathcal{X}$, consider a linear map $\gamma \mapsto m(x,\gamma)$ from $\Gamma^*$ to $\Bbb{R}$, such that for each $\gamma \in \Gamma^*$, the map $x \mapsto m(x,\gamma)$ from $\mathcal{X}$ to $\Bbb{R}$ is measurable, and the functional $\gamma \mapsto {{\mathrm{E}}}m(X, \gamma)$ is continuous on $\Gamma^*$ with respect to the $L^2(P)$ norm.*
Under the continuity condition in C, this functional admits a Riesz representer $\alpha^*_0\in L^2(F)$. We approximate the Riesz representer via: $$\alpha^*_0(X) = b(X)'\rho_0 + r_\alpha(X),$$ where $r_\alpha(X)$ is the approximation error, and $b(X)'\rho_0$ is the best sparse linear approximation defined as follows.
Let $\rho_0$ be a minimal $\ell_1$-norm solution to the approximate Riesz representation equations: $$\rho_0 \in \arg \min \| \rho\|_1: \| {{\mathrm{E}}}m(X, b) - {{\mathrm{E}}}b(X) b(X)' \rho \|_\infty \leq \lambda^\rho_0,$$ where $\lambda^\rho_0$ is a regularization parameter. When $\lambda^\rho_0 =0$, we obtain $\alpha_0(X) = b(X)'\rho_0$, a Riesz representer for functionals ${{\mathrm{E}}}m(X, \gamma)$ when $\gamma \in \Gamma_b.$
As before, we refer to the resulting approximation as “sparse", since the solutions to the problem would often be sparse. Since ${{\mathrm{E}}}\alpha^*(X) b(X) = {{\mathrm{E}}}m(X, b)$, we conclude that the approximation error $r_\alpha(X)$ is approximately orthogonal to $b(X)$: $$\| {{\mathrm{E}}}(\alpha^*_0(X) - b(X)'\rho_0) b(X) \|_\infty = \| {{\mathrm{E}}}[r_\alpha (X) b(X) ] \|_\infty \leq \lambda^{\rho}_0.$$
The estimation will be carried out using the sample analogs of the problems above, and is a special case of the following Dantzig Selector-type problem.
Consider a parameter $t \in T \subset \Bbb{R}^p$, such that $T$ is a convex set with $ \|T\|_1:= \sup_{t \in T} \| t\|_1 \leq B$. Consider the moment functions $t \mapsto g(t) $ and the estimated function $t \mapsto \hat g(t) $, mapping $T$ to $\Bbb{R}^p$: $$g(t) = G t + M; \quad \hat g(t) = \hat G t + \hat M,$$ where $G$ and $\hat G$ are $p$ by $p$ non-negative-definite matrices and $M$ and $\hat M$ are $p$-vectors. Assume that $t_0$ is the target parameter that is well-defined by: $$\label{estimand:RMD}
t_0 \in \arg \min \|t\|_1: \| g(t) \|_\infty \leq \lambda_0, \quad t \in T.$$ Define the RMD estimator $\hat t$ by solving $$\hat t \in \arg \min \| t\|_1: \|\hat g(t) \|_\infty \leq \lambda_0 + \lambda_1, \quad t \in T,$$ where $\lambda_1 $ is chosen such that $ \|\hat g(t_0) - g(t_0)\|_\infty \leq \lambda_1,$ with probability at least $1-\epsilon_n$.
We define the estimators of $\beta_0$ and $\rho_0$ over subset of data, indexed by a non-empty subset $A$ of $\{1,..., n\}$.
\[D.1\] Define $\hat \beta_A$ as the RMD estimator with parameters $t_0 = \beta_0$, $T$ a convex set with $\| T\|_1 \leq B$, and $$\hat G= \mathbb{E}_A b b', \quad G = {{\mathrm{E}}}b b', \quad \hat M = \mathbb{E}_A Y b, \quad M = {{\mathrm{E}}}Y b(X).$$
\[D.2\] Define $\hat \rho_A$ as the RMD estimator with parameters $t_0 = \beta_0$, $T$ a convex set with $\| T\|_1 \leq B$, and $$\hat G = \mathbb{E}_A b b', \quad G = {{\mathrm{E}}}b b', \quad \hat M = -\mathbb{E}_A m(X, b), \quad M = -{{\mathrm{E}}}m(X, b).$$
Properties of RMD Estimators
----------------------------
Consider sequences of constants $\ell_{1n} \geq 1$, $\ell_{2n} \geq 1$, $B_n \geq 0$, and $\epsilon_n \searrow 0$, indexed by $n$.
- *We have that $t_0 \in T$ with $\|T\|_1:= \sup_{t \in T} \|t \|_1 \leq B_n$, and the empirical moments obey the following bounds with probability at least $1 - \epsilon_n$: $$\sqrt{n} \| \hat G - G\|_\infty \leq \ell_{1n} \text{ and } \sqrt{n} \|\hat M - M\|_\infty \leq \ell_{2n}.$$*
Note that in many applications the factors $\ell_{1n}$ and $\ell_{2n}$ can be chosen to grow slowly, like $\sqrt{ \log (p \vee n) }$, using self-normalized moderate deviation bounds ([@Shao; @belloni2014pivotal]), under mild moment conditions, without requiring sub-Gaussianity.
Define the identifiablity factors for $t_0 \in T$ as : $$s^{-1}(t_0) := \inf_{\delta \in R(t_0) } | \delta'G \delta|/\| \delta \|^2_1,$$ where $R(t_0)$ is the restricted set: $$R(t_0) := \{\delta: \| t_0+ \delta\|_1 \leq \|t_0\|_1, \quad t_0 + \delta \in T\},$$ where $s^{-1}(t_0) := \infty$ if $t_0 = 0$. The restricted set contains the estimation error $\hat t - t_0$ for RMD estimators with probability at least $1- \epsilon_n$. We call the inverse of the identifiability factor, $s(t_0)$, the “effective dimension", as it captures the effective dimensionality of $t_0$; see remark below.
The identifiability factors were introduced in [@CCK] as a generalization of the restricted eigenvalue of [@BickelRitovTsybakov2009]. Indeed, given a vector $\delta \in \Bbb{R}^p$, let $\delta_A$ denote a vector with the $j$-th component set to $\delta_j$ if $j \in A$ and $0$ if $j \not \in A$. Then $s^{-1} (t_0) \geq s^{-1} k/2$ or $$s(t_0) \leq 2 s/k,$$ where $k$ is the restricted eigenvalue: $
k := \inf |\delta'G\delta|/\|\delta_M\|^2_2: \delta \neq 0, \|\delta_{M^c}\|_1 \leq \|\delta_M\|_1,$ $M = \text{support} (t_0)$, $M^c = \{1,...,p\} \setminus M$, and $s = \| t_0 \|_0.$ The inequality follows since $\| t_0+ \delta\|_1 \leq \|t_0\|_1$ implies $\|\delta_{M^c}\|_1 \leq \|\delta_M\|_1$, so that $ \| \delta\|^2_1 \leq 2 \|\delta_{M}\|^2_1 \leq 2 s \|\delta_M\|_2^2$. Here $s$ is the number of non-zero components of $t_0$. Hence we can view $s(t_0)$ as a measure of effective dimension of $t_0$.
\[lemma:RMD\] Suppose that MD holds. Let $$\bar \lambda := \tilde \ell_n/\sqrt{n}, \quad \tilde \ell_n := (\ell_{1n} B_n + \ell_{2n}).$$ Define the RMD estimand $t_0$ via (\[estimand:RMD\]) with $\lambda_0 \vee \lambda_1 \leq \bar \lambda$. Then with probability $1- 2\epsilon_n$ the estimator $\hat t$ exists and obeys: $$(\hat t-t_0)'G(\hat t- t_0) \leq (s(t_0) (4 \bar \lambda)^2) \wedge (8 B_n \bar \lambda).$$
There are two bounds on the rate, one is the “slow rate" bound $8 B_n\bar \lambda$ and the other one is “fast rate". The “fast rate" bound is tighter in regimes where the “effective dimension" $s(t_0)$ is not too large.
\[cor:rates\] Suppose $\beta_0$ and $\rho_0$ belong to a convex set $T$ with $\|T\|_1 \leq B_n$. Consider a random subset $A$ of $\{1,...,n\}$ of size $m \geq n- n/K$ where $K$ is a fixed integer. Suppose that the dictionary $b(X)$ and $Y$ obey with probability at least $1-\epsilon_n$ $$\max_{j,k \leq p} |\Bbb{G}_{A} b_k(X) b_j(X)| \leq \ell_{1n}, \quad \max_{j \leq p} | \Bbb{G}_{A} (Y b_j(X)) | \leq \ell_{2n}, \quad \max_{j \leq p} | \Bbb{G}_{A} m(X, b_j)| \leq \ell_{2n}.$$ If we set $\lambda^{\beta}_0 \vee \lambda^{\beta}_1 \vee \lambda^{\rho}_0 \vee \lambda^{\rho}_1 \leq \bar \lambda = \tilde \ell_n/\sqrt{n}$ for $\tilde \ell_n = B \ell_{1n} + \ell_{2n}$, then with probability at least $1-4\epsilon_n$, estimation errors $u = \hat \beta_A - \beta_0$ and $v = \hat \rho_A - \rho_0$ obey $$u \in R(\beta_0), \quad [u'G u]^{1/2} \leq r_1:= 4 [(\tilde \ell_n (s (\beta_0) / n)^{1/2}) \wedge (\tilde \ell^{1/2}_n B_n n^{-1/4})],$$$$v \in R(\rho_0), \quad [v'Gv]^{1/2} \leq r_2 := 4 [(\tilde \ell_n (s (\rho_0) / n)^{1/2}) \wedge (\tilde \ell^{1/2}_n B_n n^{-1/4})].$$
The corollary follows because the stated conditions imply condition MD by Holder inequality.
Approximate Neyman-orthogonal Score Functions and the DML Estimator
--------------------------------------------------------------------
Our DML estimator of $\theta_0$ will be based on using the following score function: $$\psi (W_i, \theta; \beta, \rho) = \theta - m(X, b(X))'\beta - \rho' b(X) (Y - b(X)'\beta).$$
\[lemma:score\] The score function has the following properties: $$\partial_\beta \psi (X, \theta; \beta, \rho) = m(X, b(X)) + {\rho}' b(X) b(X), \quad \partial_\rho \psi (X, \theta; \beta, \rho) = -b(X) (Y - b(X)\beta),$$$$\partial^2_{\beta \beta'} \psi (X, \theta; \beta, \rho) = \partial^2_{\rho \rho'} \psi (X, \theta; \beta, \rho) =0,
\quad \partial^2_{\beta \rho'} \psi (X, \theta; \beta, \rho) = b(X) b(X)'.$$ This score function is approximately Neyman orthogonal at the sparse approximation $(\beta_0, \rho_0)$, namely $$\|{{\mathrm{E}}}[\partial_\beta \psi (X, \theta_0; \beta, \rho_0)] \|_{\infty} = \|{{\mathrm{E}}}m(X, b) + {\rho_0}' G\|_\infty \leq \lambda^\rho_0,$$$$\|{{\mathrm{E}}}[\partial_\rho \psi (X, \theta_0; \beta_0, \rho)]\|_{\infty} = \| {{\mathrm{E}}}[ b(X) (Y - b(X)'\beta_0)]\|_{\infty} \leq \lambda^\beta_0.$$
The second claim of the lemma is immediate from the definition of $(\beta_0, \rho_0)$ and the first follows from elementary calculations.
The approximate orthogonality property above says that the score function is approximately invariant to small perturbations of the nuisance parameters $\rho$ and $\beta$ around their “true values" $\rho_0$ and $\beta_0$. Note that the score function is exactly invariant, and becomes the doubly robust score, whence $\lambda^\rho_0 = 0$ and $\lambda^\beta_0 = 0$. This approximate invariance property plays a crucial role in removing the impact of biased estimation of nuisance parameters $\rho_0$ and $\beta_0$ on the estimation of the main parameters $\theta_0$. We now define the double/de-biased machine learning estimator (DML), which makes use of the cross-fitting, an efficient form of data splitting.
Consider the partition of $\{1,...,n\}$ into $K \geq 2$ blocks $(I_k)_{k=1}^K$, with $n/K$ observations in each block $I_k$ (assume for simplicity that $K$ divides $n$). For each $k =1,...,K$, and and $I^c_k = \{1,...,N\} \setminus I_k$, let estimator $\hat \theta_{I_k}$ be defined as the root: $${{\mathbb{E}_I}}\psi (W_i, \hat \theta_{I_k}; \hat \beta_{I^c_k}, \hat \rho_{I^c_k}) = 0,$$ where $ \hat \beta_{I^c_k}$ and $\hat \rho_{I^c_k}$ are RMD estimators of $\beta_0$ and $\rho_0$ that have been obtained using the observations with indices $A=I^c_k$. Define the DML estimator $\hat \theta$ as the average of the estimators $\hat \theta_{I_k}$ obtained in each block $k \in \{1,..., K\}$: $$\hat \theta = \frac{1}{K} \sum_{k=1}^K \hat \theta_{I_k}.$$
Properties of DML with Regularized Riesz Representers {#sec:DML}
-----------------------------------------------------
We will note state some sufficient regularity conditions for DML. Let $n$ denote the sample size. To give some asymptotic statements, let $\ell_{1n} \geq 1$, $\ell_{2n} \geq 1$, $\ell_{3n} \geq 1$, $B_n \geq 0$, and $\delta_n \searrow 0$ and $\epsilon_n \searrow 0$ denote sequences of positive constants. Let $c$, $C$, and $q$ be positive constants such that $q > 3$, and let $K \geq 2$ be a fixed integer. We consider sequence of integers $n$ such that $K$ divides $n$ (to simplify notation). Fix all of these sequences and the constants. Consider $P$ that satisfies the following conditions:
- *Assume condition C holds and that in Definitions 1 and 2, it is possible to set $\lambda^\rho_0$ and $\lambda^\beta_0$ such that (a) $ \sqrt{n} (\lambda^\rho_0 + \lambda^\beta_0) B_n \leq \delta_n$ and (b) the resulting parameter values $\beta_0$ and $\rho_0$ are well behaved with $\rho_0 \in T $ and $\beta_0 \in T$, where $T$ is a convex set with $\|T\|_1 \leq B_n$.*
- *Given a random subset $A$ of $\{1,...,n\}$ of size $n - n/K$, the terms in dictionary $b(X)$ and outcome $Y$ obey with probability at least $1-\epsilon_n$ $$\max_{j,k \leq p} |\Bbb{G}_{A} b_k(X) b_j(X)| \leq \ell_{1n}, \quad \max_{j \leq p} | \Bbb{G}_{A} Y b_j(X) | \leq \ell_{2n}, \quad \max_{j \leq p} | \Bbb{G}_{A} m(X, b_j)| \leq \ell_{2n}.$$*
- *Assume that $ c \leq \| \psi(W, \theta_0; \beta_0, \rho_0)\|_{P,q} \leq C$ for $q =2$ and $3$ and that the following continuity relations hold for all $u \in R(\beta_0)$ and $v \in R(\rho_0)$, $$\sqrt{\operatorname{Var}} ((m(X, b) + {\rho_0}' b(X) b(X)) 'u) \leq \ell_{3n} \| b(X)'u\|_{P,2}$$$$\sqrt{\operatorname{Var}} ( (Y - b(X)'\beta_0) b(X) 'v) \leq \ell_{3n} \| b(X)'v\|_{P,2}$$$$\sqrt{\operatorname{Var}} (u 'b(X) b(X)' v) \leq \ell_{3n} ( \|b(X) 'u\|_{P,2} +
\|b(X)'v\|_{P,2}).$$*
- *For $\tilde \ell_n := \ell_{1n} B_n + \ell_{2n}$, $$\left . \begin{array}{l}
r_1 := 4 [(\tilde \ell_n (s (\beta_0) / n)^{1/2}) \wedge (\tilde \ell^{1/2}_n B_n n^{-1/4})] \\
r_2 := 4 [(\tilde \ell_n ( s(\rho_0) /n)^{1/2} ) \wedge (\tilde \ell^{1/2}_n B_n n^{-1/4})]
\end{array} \right |
\ \text{ we have: } \ n^{1/2} r_1 r_2 + \ell_{3n} (r_1 + r_2) \leq \delta_n.$$*
R.1 requires that sparse approximations of $\gamma^*_0$ and $\alpha^*_0$ with respect to the dictionary $b$ admit well-behaved parameters $\beta_0$ and $\rho_0$. R.2 is a weak assumption that can be satisfied by taking $\ell_{1n} = \ell_{2n} = \sqrt{\log (p \vee n) }$ under weak assumptions on moments, as follows from self-normalized moderate deviation bounds ([@Shao; @belloni2014pivotal]), without requiring subgaussian tails bounds. R.3 imposes a modulus of continuity for bounding variances: if elements of the dictionary are bounded with probability one, $ \| b(X) \|_\infty \leq C$, then we can select $\ell_{3n} = C B_n$ for many functionals of interest, so the assumption is plausible. R.4 imposes condition on the rates $r_1$ and $r_2$ of consistency of $\beta_0$ and $\rho_0$, requiring in particular that $r_1 r_2 n^{1/2}$, an upper bound on the bias of the DML estimator, is small; see also the discussion below.
Consider the oracle estimator based upon the true score functions: $$\bar \theta = \theta_0 + n^{-1} \sum_{i =1}^n \psi_0(W_i), \quad \psi_0(W_i) := \psi(W_i, \theta_0; \beta_0, \rho_0).$$
\[theorem: DML\] Under R.1-R.4, we have the adaptivity property, namely the difference between the DML and the oracle estimator is small: for any $\Delta_n \in (0,1)$, $$|\sqrt{n}(\hat \theta - \bar \theta)| \leq R_n := \sqrt{K} \bar C \delta_n/\Delta_n$$ with probability at least $1- \Pi_n$ for $\Pi_n:= K ( 4 \epsilon_n + \Delta^2_n)$, where $\bar C$ is an absolute constant. As a consequence, $\hat \theta_0$ concentrates in a $1/\sqrt{n}$ neighborhood of $\theta_0$, with deviations approximately distributed according to the Gaussian law, namely: $$\label{eq:normal error}
\sup_{z \in \Bbb{R}}\Big | {{\mathrm{P}}}_P ( \sigma^{-1} \sqrt{n} (\hat \theta_0 - \theta_0) \leq z) - \Phi(z) \Big | \leq \bar C' (n^{-1/2} + R_n) + \Pi_n,$$ where $\sigma^2 = \operatorname{Var}( \psi_0(W_i))$, $\Phi(z) = {{\mathrm{P}}}(N(0,1) \leq z)$, and $\bar C'$ only depends on $(c,C)$.
The constants $\Delta_{n}>0$ can be chosen such that right-hand side of (\[eq:normal error\]) converges to zero as $n \to \infty$, yielding the following asymptotic result.
\[corollary:uniform\] Fix constants and sequences of constants specified at the beginning of Section \[sec:DML\]. Let $\mathcal{P}$ be the set of probability laws $P$ that obey conditions R.1-R.4 uniformly for all $n$. Then DML estimator $\hat \theta$ is uniformly asymptotically equivalent to the oracle estimator $\bar \theta$, that is $|\sqrt{n}(\hat \theta - \bar \theta)| = O_P (\delta_n)$ uniformly in $P \in \mathcal{P}$ as $n \to \infty$. Moreover, $\sqrt{n} \sigma^{-1} (\hat \theta - \theta_0)$ is asymptotically Gaussian uniformly in $P \in \mathcal{P}$: $$\lim_{n \to \infty} \sup_{P \in \mathcal{P}} \sup_{z \in \Bbb{R}}\Big | {{\mathrm{P}}}_P ( \sigma^{-1} \sqrt{n} (\hat \theta_0 - \theta_0) \leq z) - \Phi(z) \Big | = 0.$$
Hence the DML estimator enjoys good properties under the stated regularity conditions. Less primitive regularity conditions can be deduced from the proofs directly.
The key regularity condition imposes that the bounds on estimation errors $r_1$ and $r_2$ are small and that the product $n^{1/2} r_1 r_2$ is small. Ignoring the impact of “slow” factors $\ell_n$’s and assuming $B_n$ is bounded by a constant $B$, this requirement is satisfied if $$\textit{ one of the effective dimensions is smaller than $\sqrt{n}$, either $s (\beta_0) \ll \sqrt{n} \ \text{ or } s (\rho_0) \ll \sqrt{n}$.}$$ The latter possibility allows one of the parameter values to be “dense", having unbounded effective dimension, in which case this parameter can be estimated at the “slow" rate $n^{-1/4}$. These types of conditions are rather sharp, matching similar conditions used in [@JM2015] in the case of inference on a single coefficient in Gaussian sparse linear regression models, and those in [@zhu2017linear] in the case of point testing general linear hypotheses on regression coefficients in linear Gaussian regression models.
Proceeding further, we notice that the difference between our target and the ideal target is: $$|\theta_0 - \theta^*_0| = |{{\mathrm{E}}}r_\alpha(X) r_\gamma(X)| \leq \| r_\alpha\|_{P,2} \| r_\gamma\|_{P,2}.$$ Hence we obtain the following corollary.
Suppose that, in addition to R.1-R.4, the product of approximation errors $r_\gamma = \gamma^*_0 - \gamma_0$ and $r_\alpha = \alpha^*_0 - \alpha_0$ is small, $$\sqrt{n} |{{\mathrm{E}}}r_\alpha(X) r_\gamma(X)| \leq \delta_n.$$ Then conclusions of Theorem 1 and Corollary 1 hold with $\theta_0$ replaced by $\theta_0^*$ and the constants $\bar C$ and $\bar C'$ replaced by $2 \bar C$ and $2 \bar C'$.
The plausibility of $\sqrt{n} |{{\mathrm{E}}}r_\alpha(X) r_\gamma(X)| \leq \sqrt{n} \| r_\alpha\|_{P,2} \| r_\gamma\|_{P,2}$ being small follows from the fact that many rich functional classes admit sparse linear approximations with respect to conventional dictionaries $b$. For instance, [@tsybakov:book] and [@belloni2014pivotal] give examples of Sobolev and rearranged Sobolev balls, respectively, as the function classes and elements of the Fourier basis as the dictionary $b$, in which sparse approximations have small errors.
In the context of vanishing approximation errors, and in the context of Example 3 on Average Treatment Effects, our estimator implicitly estimates the inverse of the propensity score directly rather than inverting a propensity score estimator as in most of the literature. The approximate residual balancing estimator of [@athey2016approximate] can also be thought of as implicitly estimating the inverse propensity score. An advantage of the estimator here is its DML form allows us to tradeoff rates at which the mean and the inverse propensity score are estimated while maintaining root-n consistency. Also, we do not require that the conditional mean be linear and literally sparse with the sparsity index $s(\beta_0) \ll \sqrt{n} $; in fact we can have a completely dense conditional mean function when the approximation to the Rietz representer has the effective dimension $s(\rho_0) \ll \sqrt{n}$. More generally, when the approximation errors don’t vanish, our analysis also explicitly allows for misspecification of the regression function.
Proofs
======
**Proof of Lemma \[lemma:RMD\]**. Consider the event $\mathcal{E}_n$ such that $$\label{En1} \| \hat g(t_0)\| \leq \lambda_0 + \lambda_1 \text{ and } \sup_{t \in T} \| \hat g(t) - g(t) \|_\infty \leq \bar \lambda$$ holds. This event holds with probability at least $1-2 \epsilon_n$. Indeed, by the choice of $\lambda_1$ and $\| g(t_0)\| \leq \lambda_0$, we have with probability at least $1-\epsilon_n$: $$\| \hat g(t_0)\|_\infty \leq \| \hat g(t_0) - g(t_0) \|_\infty + \| g(t_0)\| \leq \lambda_1 + \lambda_0.$$ Hence on the event $\mathcal{E}_n$ we have $$\| \hat t \|_1 \leq \| t_0 \|_1 \quad \| \hat g(\hat t )\|_\infty \leq \lambda_0 + \lambda_1.$$ This implies, by $\| g(t_0)\| \leq \lambda_0$, by $\lambda_0 \wedge \lambda_1 \leq \bar \lambda$, and by (\[En1\]), that $$\| G (\hat t - t_0)\|_\infty = \| g(\hat t) - g(t_0) \|_\infty \leq 2 \lambda_0 + \lambda_1 + \sup_{t \in T} \| \hat g(t) - g(t) \|_\infty \leq 4 \bar \lambda.$$ Then $ \delta = \hat t - t_0$ obeys, by definition of $s(t_0)$, $$\| \delta \|_1^2 \leq s(t_0) \delta' G \delta \leq s(t_0) \| G \delta \|_\infty \| \delta\|_1 \leq s(t_0) 4 \bar \lambda \| \delta\|_1,$$ which implies that $$\| \delta \|_1 \leq s(t_0) 4 \bar \lambda, \quad \delta' G \delta \leq s(t_0) (4 \bar \lambda)^2,$$ which establishes the first part of the bound.
The second bound follows from $\|\delta\|_1 \leq 2B_n$ and $
\delta' G \delta \leq \| G \delta \|_\infty \|\delta\|_1 \leq 4 \bar \lambda 2 B. $
**Proof of Theorem \[theorem: DML\].** **Step 1.** We have a random partition $(I_k,I^c_k)$ of $\{1,...,n\}$ into sets of size $m=n/K$ and $n- n/K$. Omit the indexing by $k$ in this step. Here we bound $|\sqrt{m}(\hat \theta_I - \bar \theta_I)|$, where $$\bar \theta_I = \theta_0 + \mathbb{E}_I \psi_0(W_i).$$ Define $$\partial_\beta \psi_0 (W_i) := \partial_\beta \psi (X, \theta; \beta_0, \rho_0) = m(X, b(X)) + {\rho_0}' b(X) b(X)$$ $$\partial_\rho \psi_0 (W_i):= \partial_\rho \psi (X, \theta; \beta_0, \rho_0) = -b(X) (Y - b(X)'\beta_0)$$$$\partial^2_{\beta \rho'} \psi_0 (W_i) := \partial^2_{\beta \rho'} \psi (X, \theta; \beta_0, \rho_0) = b(X) b(X)'.$$ Define the estimation errors $$u = \hat \beta_{I^c} - \beta_0 \text{ and } v= \hat \rho_{I^c} - \rho_0.$$ Since $
\partial^2_{\beta \beta'} \psi (X, \theta; \beta, \rho) = 0$ and $\partial^2_{\rho \rho'} \psi (X, \theta; \beta, \rho) =0,
$ as noted in Lemma \[lemma:score\], we have by the exact Taylor expansion: $$\hat \theta_I = \bar \theta_I + ({{\mathbb{E}_I}}\partial_\beta \psi_0) u +
({{\mathbb{E}_I}}\partial_\rho \psi_0) v + u'({{\mathbb{E}_I}}\partial^2_{\beta \rho'} \psi_0) v.$$ With probability at least $1- 4\epsilon_n$, by Corollary \[cor:rates\], the following event occurs: $$\mathcal{E}_n = \{ u \in R(\beta_0), \ \ v \in R(\rho_0), \ \ \sqrt{u'Gu} \leq r_1, \ \ \sqrt{v'Gv} \leq r_2 \}.$$ Using triangle and Holder inequalities, we obtain that on this event: $$\begin{aligned}
|\sqrt{m}(\hat \theta_I - \bar \theta_I)| \leq \mathrm{rem}_I & := & |{{\mathbb{G}_I}}\partial_\beta \psi_0 u| +
\sqrt{m} \| P \partial_\beta \psi_0 \|_\infty \|u\|_1 \\
& + & |{{\mathbb{G}_I}}\partial_\rho \psi_0 v| + \sqrt{m} \| P \partial_\rho \psi_0 \|_\infty \|v \|_1 \\
& + & |u ' {{\mathbb{G}_I}}\partial^2_{\beta \rho'} \psi_0v | + \sqrt{m} |u' [P \partial^2_{\beta \rho'} \psi_0] v|.\end{aligned}$$ Moreover, on this event, by $\|R(\beta_0)\|_1 \leq 2B_n$ and $\|R(\rho_0)\|_1 \leq 2B_n$, by Lemma \[lemma:score\], and by R.1: $$\sqrt{m} \| P \partial_\beta \psi_0 \|_\infty \|u \|_1 \leq \sqrt{m} \lambda^\beta_0 2B_n \leq \delta_n,
\sqrt{m} \| P \partial_\rho \psi_0 \|_\infty \|v\|_1 \leq \sqrt{m} \lambda^\rho_0 2B_n \leq \delta_n.$$
Note that $v$ and $u$ are fixed once we condition on the observations $(W_i)_{i \in I^c}$. We have that on the event $\mathcal{E}_n$, by i.i.d. sampling, R.3, and R.4, $$\begin{aligned}
\sqrt{\operatorname{Var}} [ {{\mathbb{G}_I}}\partial_\beta \psi_0 u \mid (W_i)_{i \in I^c}]
& = & \sqrt{\operatorname{Var}} (\partial_\beta \psi_0 u\mid (W_i)_{i \in I^c})
\leq \ell_{3n} \sqrt{u' G u} \leq \delta_n , \\
\sqrt{\operatorname{Var}} [ {{\mathbb{G}_I}}\partial_\rho \psi_0 v \mid (W_i)_{i \in I^c}]
& = & \sqrt{\operatorname{Var}} (\partial_\rho \psi_0 ' v\mid (W_i)_{i \in I^c})
\leq \ell_{3n} \sqrt{v' G v} \leq \delta_n, \\
\sqrt{\operatorname{Var}} [ u' {{\mathbb{G}_I}}\partial^2_{\beta \rho'} \psi_0 v \mid (W_i)_{i \in I^c}]
& = &
\sqrt{\operatorname{Var}} [ u ' b b' v \mid (W_i)_{i \in I^c}]
\leq \ell_{3n} [ ( u' G u)^{1/2} + (v' G v)^{1/2}] \leq \delta_n, \\
\sqrt{m} | u ' [P \partial^2_{\beta \rho'} \psi_0] v|
& \leq & \sqrt{m} | u 'G v | \leq \sqrt{m} ( u 'G u v 'G v)^{1/2}
\leq \delta_n.\end{aligned}$$
Hence we have that for some numerical constant $\bar C$ and any $\Delta_n \in (0,1)$: $${{\mathrm{P}}}( \mathrm{rem}_I >\bar C \delta_n/\Delta_n ) \leq {{\mathrm{P}}}( \mathrm{rem}_I > \bar C \delta_n/\Delta_n \cap \mathcal{E}_n) + {{\mathrm{P}}}(\mathcal{E}_n^c)$$ $$\leq {{\mathrm{E}}}{{\mathrm{P}}}( \mathrm{rem}_I > \bar C \delta_n/\Delta_n \cap \mathcal{E}_n \mid (W_i)_{i \in I^c}) + {{\mathrm{P}}}(\mathcal{E}_n^c) \leq \Delta_n^2 + 4 \epsilon_n.$$
**Step 2.** Here we bound the difference between $\hat \theta = K^{-1} \sum_{k=1}^K \hat \theta_{I_k}$ and $\bar \theta = K^{-1} \sum_{k=1}^K \bar \theta_{I_k}$: $$\sqrt{n}|\hat \theta - \bar \theta| \leq \frac{\sqrt{n}}{\sqrt{m}} \frac{1}{K} \sum_{k=1}^K \sqrt{m} | \hat \theta_{I_k} - \bar \theta_{I_k}| \leq \frac{\sqrt{n}}{\sqrt{m}} \frac{1}{K} \sum_{k=1}^K \mathrm{rem}_{I_k}.$$ By the union bound we have that $${{\mathrm{P}}}\left( \frac{1}{K} \sum_{k=1}^K \mathrm{rem}_{I_k}> \bar C \delta_n/\Delta_n\right) \leq K( \Delta_n^2 + 4 \epsilon_n),$$ and we have that $\sqrt{n/m} = \sqrt{ K}$. So it follows that $$|\sqrt{n}(\hat \theta - \bar \theta)| \leq R_n := \bar C \sqrt{K} \delta_n/\Delta_n$$ with probability at least $1- \Pi_n$ for $\Pi_n:= K ( 4 \epsilon_n + \Delta^2_n)$, where $\bar C$ is an absolute constant.
**Step 3**. To show the second claim, let $Z_n := \sqrt{n} {\sigma}^{-1}(\bar \theta - \theta_0)$. By the Berry-Esseen bound, for some absolute constant $A$, $$\sup_{z \in \Bbb{R}} |{{\mathrm{P}}}( Z_n \leq z) - \Phi(z)| \leq A \| \psi_0/\sigma \|^3_{P,3} n^{-1/2} \leq A (C/c)^{3} n^{-1/2},$$ where $ \| \psi_0/\sigma \|^3_{P,3} \leq (C/c)^3$ by R.3. Hence, using Step 2, for any $z \in \Bbb{R}$, we have $$\begin{aligned}
&& {{\mathrm{P}}}( \sqrt{n} {\sigma}^{-1} (\hat \theta - \theta_0) \leq z) - \Phi(z)\leq {{\mathrm{P}}}( Z_n \leq z + {\sigma}^{-1} R_n) + \Pi_n - \Phi(z) \\
&& \quad \quad \leq A (C/c)^{3} n^{-1/2} + \bar \phi {\sigma}^{-1} R_n + \Pi_n \leq \bar C'(n^{-1/2} + R_n) + \Pi_n,\end{aligned}$$ where $ \bar \phi = \sup_{z} \phi(z)$, where $\phi$ is the density of $\Phi(z) = {{\mathrm{P}}}(N(0,1) \leq z)$, and $\bar C'$ depends only on $(C,c)$. Similarly, conclude that ${{\mathrm{P}}}( \sqrt{n} \sigma^{-1} (\hat \theta - \theta_0) \leq z) -\Phi(z) \geq \bar C' (n^{-1/2} + R_n) - \Pi_n.$
|
---
abstract: 'We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the Witten-Reshetikhin-Turaev invariant of a Seifert manifold is the scalar product of two Lagrangian states, and we estimate this scalar product in the large level limit. The leading order terms of the expansion are naturally given in terms of character varieties, the Chern-Simons invariants and some symplectic volumes. For the analytic part, we establish some singular stationary phase lemma for discrete oscillatory sums.'
author:
- 'Laurent Charles [^1]'
bibliography:
- 'biblio.bib'
title: On the Witten asymptotic conjecture for Seifert manifolds
---
Witten asymptotic conjecture sets up a bridge between quantum and classical invariants. It predicts that the Witten-Reshetikhin-Turaev invariant of a closed 3-manifold $M$ for a given group compact $G$ has an asymptotic expansion in the large level limit, the leading terms of this expansion being function of the Chern-Simons invariant, the Reidemeister torsion and the spectral flow of the representation of $\pi_1(M)$ into $G$. This conjecture has been recently settled for some hyperbolic manifolds [@LJ1], [@LJ2] and [@oim_MCG]. It was also checked for many Seifert manifolds [@FrGo], [@Je1], [@Ro2], [@Ro], [@LaRo], [@Han], [@HaTa], [@hi2], [@BeWi], [@An], [@AnHi]
In this paper, we come back to the Seifert case and propose a new approach, more conceptual than the previous ones. In particular, the character variety of $M$ and the Chern-Simons invariant appear naturally in the proof. Furthermore, we compute the leading terms of the amplitudes corresponding to irreducible representations as symplectic volumes of some moduli spaces, which is a new result. In the companion paper [@LL], it is proved that these symplectic volumes are actually integrals of Reidemeister torsion.
To explain briefly our strategy, assume that $M$ is obtained by gluing some solid tori $T_1$, …, $T_n$ to ${\Sigma}\times S^1$ where ${\Sigma}$ is a compact connected orientable surface with $n$ boundary components. The WRT invariant of ${\Sigma}\times S^1$ is an element of a vector space associated to the boundary $\partial {\Sigma}\times S^1$. This vector space may be viewed as the quantization of the character variety of $ \partial {\Sigma}\times S^1$. We will deduce from Verlinde formula and Riemann-Roch theorem that the WRT invariant of ${\Sigma}\times S^1$ is a Lagrangian state supported by the character variety of ${\Sigma}\times S^1$. Similarly, it is known that the WRT invariant of $T_1 \cup \ldots \cup T_n$ is also a Lagrangian state. So the WRT invariant of $M$ is the scalar product of two Lagrangian states. This can be estimated by using some pairing formula, when the Lagrangian submanifolds supporting the states intersect transversally. This transversality assumption is satisfied when $M$ has a non vanishing Euler number. Unfortunately, some difficulties arise because the character varieties of ${\Sigma}\times S^1$ has singularities, which lead us to estimate some singular oscillatory sums. For the sake of simplicity, we restrict ourselves in this paper to Seifert manifolds with one exceptional fiber and the group ${\operatorname}{SU}(2)$.
### Statement of the main result {#statement-of-the-main-result .unnumbered}
Let $S$ be the oriented Seifert manifold with unnormalized invariant $(g$; $(a,b))$, where $g$ is an integer $\geqslant 2$ and $a$, $b$ are two coprime integers such that $a\neq0 $ and $b \geqslant 1$. Recall that $S$ is obtained from a genus $g$ oriented surface ${\Sigma}$ with boundary a circle $C$, by gluing a solid torus $T$ to ${\Sigma}\times S^1$ in such a way that the homology class of a meridian of the boundary of $T$ is sent to $a [C ] + b [S^1]$.
Let ${\mathcal{M}}(S)$ be the space of conjugacy classes of group morphisms from $\pi_1 (S)$ to ${\operatorname}{SU}(2)$. Let $X = \pi_0 ({\mathcal{M}}(S))$ be the set of connected components. Introduce the functions ${\alpha}, {\beta}: {\mathcal{M}}(S) \rightarrow [0, \pi ]$ given by $${\alpha}( [\rho] ) = \arccos \bigl( \tfrac{1}{2} {\operatorname}{tr} ( \rho (C)) \bigr), \quad {\beta}([ \rho] ) = \arccos \bigl( \tfrac{1}{2} {\operatorname}{tr} ( \rho (S^1)) \bigr)$$ These functions are actually constant on each component of ${\mathcal{M}}(S)$. We denote again by ${\alpha}$ and ${\beta}$ the induced maps from $X$ to $[0, \pi ]$. They allow to distinguish between the components, that is the joint map $({\alpha}, {\beta}) : X \rightarrow [0, \pi ]^2$ is one-to-one.
We say that a component of ${\mathcal{M}}(S)$ is abelian if it contains only abelian representations, and irreducible otherwise. We will divide $X$ into 4 disjoint subsets: $X_1 $ is the set of abelian components, whereas $X_2$, $X_3$, $X_4$ are the sets of irreducible components $x$ such that ${\alpha}(x) \neq 0,1$, ${\alpha}(x) = 0$, ${\alpha}( x) =1$ respectively. All the components in the same $X_i$ have the same dimension. In table \[tab:xi\], we indicate for each $i$ the possible values of ${\alpha}$ and ${\beta}$ on $X_i$, the cardinal of $X_i$ and the dimension of its elements.
${\alpha}$ ${\beta}$ type $\#$ $\frac{1}{2}$dimension
------- --------------- -------------- --------------- ---------------------------------------- ------------------------
$X_1$ 0 $\neq 0,\pi$ abelian ${\operatorname}{E} ( \frac{b-1}{2} )$ $g$
$X_2$ $\neq 0, \pi$ 0 or $\pi$ irreducible $|a|-1$ $3g-2$
$X_3$ 0 0 or $\pi$ irred/abelian 2 if $b$ is even $3g-3$
1 otherwise
$X_4$ 1 0 or $\pi$ irreducible 1 if $b$ is odd $3g-3$
0 otherwise
: Characteristics of the components of ${\mathcal{M}}(S)$[]{data-label="tab:xi"}
For any $\rho \in {\mathcal{M}}(S)$, we denote by ${\operatorname}{CS} ( \rho ) \in {{\mathbb{R}}}/ 2 \pi {{\mathbb{Z}}}$ the Chern-Simons invariant of $\rho$, cf. Equation (\[eq:defrhox\]) for a precise definition. The Chern-Simons function ${\operatorname}{CS}$ is constant on each component of ${\mathcal{M}}(S)$. We denote again by ${\operatorname}{CS}$ the induced map from $X$ to ${{\mathbb{R}}}/ 2 \pi {{\mathbb{Z}}}$
For any $t \in [0,1]$, let ${\mathcal{M}}( {\Sigma}, t)$ denote the space of conjugacy classes of group morphism $\rho$ from $\pi_1 ({\Sigma})$ to ${\operatorname}{SU}(2)$ such that ${\operatorname}{tr} ( \rho (C) ) = \cos ( 2 \pi t)$. This space is a symplectic manifold. We denote by $v_g(t)$ its symplectic volume.
\[theo:main-result\] We have the full asymptotic expansion
[2]{} Z\_k ( S) = & \_[x X]{} e\^[ik [CS]{} (x) ]{} k\^[n(x)]{} \_ k\^[-]{} a\_ (x) + \_[x X\_3]{} k\^[m(x)]{} \_ k\^[-]{} b\_ (x) + ( k\^[-]{})
where the coefficients $a_{\ell} (x)$, $b_{\ell} (x)$ are complex numbers. The exponents $n(x)$ and the leading coefficients $a_0 (x)$ are given in table \[tab:coef\] according to whether $x$ belong to $X_1$, $X_2$, $X_3$ or $X_4$. Furthermore, if $x \in X_3$ $$m(x) = 2g - \tfrac{3}{2}, \qquad b_0 (x) = e^{i \pi/4} i^g b^{g - 3/2} a ^{-g} \pi^{-g +1} \frac{\sqrt 2 ( g-1)!}{ ( 2 ( g-1))!} .$$
$$\renewcommand{1.1}{1.7}
\begin{array}{|l|c|c|}
\hline
& n(x) & a_0 (x) \\
\hline
X_1 & g -1/2 & \frac{2 \pi^{g - 1/2} }{ \sqrt{b}} \bigl[ \sin ( {\alpha}(x) ) \bigr] ^{-2g + 1} \sin \bigl( c {\alpha}(x) \bigr) \\
\hline
X_2 & 3g -2 & \frac{1 }{ \sqrt{a}} v_{g} \bigr( {\beta}(x) /\pi \bigl) \sin \bigl( d {\beta}(x) \bigr) \\
\hline
X_3 & 3g -3 & i(4 \pi)^{-1} a^{-3/2} v_g'(0) \\
\hline
X_4 & 3g -3 & (4 \pi)^{-1} a^{-3/2} v'_{g} (1) \\
\hline
\end{array}$$
The novelty in this result is the expression of the leading coefficients $a_{0} (x)$ and $b_0 (x)$. In the paper [@LL], it is proved that $a_0 (x)$ is actually equal to the integral of a Reidemeister torsion, in agreement with the Witten asymptotic conjecture. Comparing Tables \[tab:xi\] and \[tab:coef\], we also see that the exponent $n(x)$ is equal to half the dimension for an irreducible component, and half the dimension minus $1/2$ for the abelian components, in agreement with the prediction in [@FrGo].
All the components of ${\mathcal{M}}(S)$ are smooth manifolds except the ones in $X_3$ which have a singular stratum of abelian representations. For these singular components, we have an additional term in the asymptotic expansion, the series $\sum k^{-\ell} b_{\ell} (x)$. We don’t know any geometric expression for $b_0 (x)$ or $m_0(x)$. For instance, since the singular strata have the same dimension as the abelian component in $X_1$, we could expect that $m(x)$ is equal to $ n(y)$, $y \in X_1$; but it is not the case. As a last remark, we haven’t tried to determine the signs of the coefficients and to understand the spectral flow contribution.
### Comparison with earlier results {#comparison-with-earlier-results .unnumbered}
The main reference on the subject is certainly the article [@Ro2] by Rozansky, where the author first shows that $Z_k (S)$ has an asymptotic expansion, the contribution of irreducible representations being presented in a residue form. In a second part, by comparing various expansions of path integrals, these residues are formally identified with Riemann-Roch number of moduli spaces, cf. Conjecture 5.1 and Proposition 5.3 of [@Ro2]. The connection with Theorem \[theo:main-result\] is that these Riemann-Roch number are approximated at first order by the symplectic volumes $v_g(t)$.
To compare with the previous articles on Witten asymptotic conjecture for Seifert manifolds, our proof has the advantage that the set $X = \pi_0 ( {\mathcal{M}}(S))$ and the various invariants appear naturally. To the contrary, in most papers on the subject, it is first proved that $Z_k (S)$ is a sum of oscillatory terms with an explicit computation of the phases. Independently, one determines the set $X$ and the corresponding Chern-Simons invariants. After that, a one-to-one correspondence between the components of ${\mathcal{M}}(S)$ and the various terms of the asymptotic expansion is established, such that the Chern-Simons invariant of a component is equal to the corresponding phase.
The identification between the amplitudes and the Reidemeister torsion has been done in a few cases similarly by comparing the results of two independent computations. One exception is the paper [@AnHi], where everything is computed intrinsically, but the Seifert manifolds covered in [@AnHi] have all a vanishing Euler number, so they form a family disjoint to the Seifert manifolds we consider.
The strategy we follow is inspired by our previous works [@oim_MCG] and [@LJ1], [@LJ2] in collaboration with J. Marché , where we proved some generalized Witten conjecture for some manifolds with non empty boundary. In [@LJ1], [@LJ2], we considered the complement of the figure eight knot and our main tool was some q-difference relations. The paper [@oim_MCG] was devoted some mapping cylinders of pseudo Anosov diffeomorphisms and our main tool was the Hitchin connection. In the present paper, we prove a generalized Witten conjecture for the manifold ${\Sigma}\times S^1$. The main ingredients we use are the Verlinde formula and the Riemann-Roch theorem.
### Sketch of the proof {#sketch-of-the-proof .unnumbered}
The Seifert manifold $S$ being obtained by gluing a solid torus ${T}$ to ${\Sigma}\times S^1$, $Z_k (S)$ is the scalar product of two vectors $Z_k ({T})$ and $Z_{k} ({\Sigma}\times S^1)$ of the Hermitian vector space $V_k (\partial {\Sigma}\times S^1)$. This vector space has a canonical orthonormal basis $(e_\ell, \; \ell = 1, \ldots, k-1)$. By [@FrGo], the coefficients of $Z_{k} ( {\Sigma}\times S^1)$ in this basis are the Verlinde numbers $N^{g,k}_\ell$.
These numbers can be computed in several ways. First, $N^{g,k}_\ell$ is the number of admissible colorings of a pants decomposition of ${\Sigma}$. However this elementary way is not very useful to study the large $k$ limit. At least, we learn that $N^{g,k}_{\ell}$ vanishes when $\ell$ is even. Second, the $N^{g,k}_{\ell}$ are Riemann-Roch numbers associated to the symplectic manifolds ${\mathcal{M}}( {\Sigma}, s)$ introduced above. This implies that for any odd $\ell$ satisfying $ 1<\ell <k-1$, we have $$\begin{gathered}
\label{eq:N_RR}
N^{g,k}_{\ell} = \Bigl( \frac{k}{2 \pi } \Bigr)^{3g -2} \sum_{n=0}^{3g-2} k^{-n} Q_{g,n} \Bigl( \frac { \ell}{k} \Bigr) \end{gathered}$$ where the $Q_{g,n}$ are smooth function on $]0,1[$, $Q_{g,0} (t)$ being the symplectic volume $v_g (t)$. Third, by Verlinde formula, we have $$\begin{gathered}
\label{eq:N_Verlinde}
N_{\ell}^{g,k} = \sum _{ m =1}^{k-1} S_{m,1} ^{1 - 2g} S_{m, \ell} \quad \text{ where } \quad S_{m, p}= \Bigl( \frac{2}{k}\Bigr)^{1/2} \sin \Bigl( \frac{\pi mp}{k} \Bigr).\end{gathered}$$
Introduce the Hermitian space $\mathcal{H}_k := {{\mathbb{C}}}^{{{\mathbb{Z}}}/ 2k {{\mathbb{Z}}}}$ and denote by $(\Psi_{\ell}, \ell \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}})$ its canonical basis. The family $(\mathcal{H}_k, \; k \in {{\mathbb{N}}})$ may be viewed as the quantization of a two dimensional torus $M = {{\mathbb{R}}}^2 / {{\mathbb{Z}}}^2 \ni (x,y)$. This has the meaning that $(\mathcal{H}_k)$ may be identified with a space of holomorphic sections of the $k$-th power of a prequantum bundle over $M$. In this context some families $(\xi_k \in \mathcal{H}_k, \; k \in {{\mathbb{N}}})$ concentrating in a precise way on a 1-dimensional submanifold of $M$ are called Lagrangian states.
We will consider $V_k (\partial {\Sigma}\times S^1)$ as a subspace of $\mathcal{H}_k := {{\mathbb{C}}}^{{{\mathbb{Z}}}/ 2k {{\mathbb{Z}}}}$ by setting $e_{\ell} = ( \Psi_{\ell} - \Psi_{-\ell}) /\sqrt 2$. We will prove that $( Z_{k} ({\Sigma}\times S^1), k \in {{\mathbb{N}}})$ is a Lagrangian state supported by a Lagrangian submanifold of $M$.
To do this, we will establish and use the following characterization of Lagrangian states. Let $x_0, x_1 \in {{\mathbb{R}}}$ such that $ x_0< x_1< x_0+1$ and $\varphi$ be a function in $ {\mathcal{C}^{\infty}}(]x_0 , x_1 [, {{\mathbb{R}}})$. Let $U$ be the open set $]x_0,x_1 [ \times {{\mathbb{R}}}/ {{\mathbb{Z}}}$ of $M$ and ${\Gamma}$ be the Lagrangian submanifold $\{ (x, \varphi'(x)); \; x \in ]x_0, x_1[ \}$ of $U$. Then a family $( \xi_k = \sum \xi_k ( \ell) \Psi_\ell, k \in {{\mathbb{N}}})$ is a Lagrangian state over $U$ supported by ${\Gamma}$ if and only if $$\xi_k ( 2 k x ) = k^{-1/2 + N } e^{ 4 i \pi k \varphi (x) } \sum_{n = 0 } ^{ \infty} k^{-n } f_n ( x) + {\mathcal{O}}( k^{-\infty})$$ where the $f_n$ are smooth functions on $]x_0, x_1[$. This formula may be viewed as a discrete analogue of the WKB expression, the function $\varphi$ is a generating function of ${\Gamma}$, the leading order term $f_0$ of the amplitude determines the symbol of the Lagrangian state.
By this characterization, we deduce from (\[eq:N\_RR\]) that $Z_{k} ({\Sigma}\times S^1)$ is the sum of two Lagrangian states over $M \setminus \{ x = 0 \text{ or } 1/2 \}$, supported respectively by $\{ y = 0 \}$ and $\{ y = 1/2 \}$. Indeed, we insert a factor $( 1 - (-1)^{\ell})/2$ in the right hand side of (\[eq:N\_RR\]) so that the equation is valid for even and odd $\ell$, and we use that for $\ell = 2kx$, $(-1)^{\ell} = e^{ 4 i \pi k x/2}$. To complete this description on a neighborhood of $\{ x = 0 \text{ or } 1/2 \}$, we will perform a discrete Fourier transform. By Verlinde formula (\[eq:N\_Verlinde\]), we easily get $$\begin{gathered}
\label{eq:3}
Z_{k} ({\Sigma}\times S^1) = \frac{\sqrt k }{2i } \sum_{m \in ({{\mathbb{Z}}}/2k{{\mathbb{Z}}})\setminus \{ 0,k \}} S_{m,1}^{1-2g} \Phi_m \end{gathered}$$ where $(\Phi_m)$ is the basis of $\mathcal{H}_k$ given by $\Phi_m = (2k)^{-1/2} \sum_\ell e^{i \pi \ell m /k} \Psi_\ell$ for $m \in {{\mathbb{Z}}}/2k {{\mathbb{Z}}}$. A similar characterization of Lagrangian state as above holds, where we exchange $x$ and $y$ and replace the coefficients in the basis $( \Psi_{\ell})$ by the ones in $( \Phi_m)$. We deduce from (\[eq:3\]) that $Z_k ( {\Sigma}\times S^1)$ is Lagrangian on $M \setminus \{ y=0 \text{ or } 1/2 \}$ supported by $\{ x =0 \}$.
Gathering these results, we conclude that $Z_k ( {\Sigma}\times S^1)$ is Lagrangian on the open set $ M \setminus \{ (0,1), ( 0,1/2) \}$ and supported by $\{ x= 0 \} \cup \{ y= 0\} \cup \{ y = 1/2 \}$. There is no similar description on a neighborhood of $(0,0)$ and $(0,1/2)$ because the circle $\{x =0 \}$ intersects with $\{ y= 0 \}$ and $\{ y =1/2 \}$ at these points.
By [@LJ1], the state $Z_k ( {T})$ is Lagrangian supported by the circle $\{ y = ax/b \}$. The scalar product of two Lagrangian states supported on Lagrangian manifolds ${\Gamma}$, ${\Gamma}'$ which intersects transversally, has an asymptotic expansion, and we can compute geometrically the leading order terms, each intersection point of ${\Gamma}\cap {\Gamma}'$ having a contribution [@oim_pol]. From this, we obtain Theorem \[theo:main-result\], except for the contribution of $X_3$ which corresponds to the singular points $(0,0)$, $(0,1/2)$.
Actually, a large portion of this paper will be devoted to the contribution of $X_3$ in Theorem \[theo:main-result\]. On one hand we will establish some singular stationary phase lemma for discrete oscillatory sum. On the other hand we will prove several properties of the functions $Q_{g,n}$ in (\[eq:N\_RR\]): first $Q_{g,n}$ vanishes for odd $n$, second $Q_{g, 2m}$ is a polynomial of degree $2(g -m) -1$, third the even part of $Q_{g, 2m}$ is a multiple of the monomial $ x^{2(g-m-1)}$ and the coefficient of this multiple with be explicitly computed for $m=0$.
To prove these facts, we will study the discrete Fourier transform of the family $( \sin ^{-m}( \pi \ell / k ), \; \ell \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}})$ in the semiclassical limit $k \rightarrow \infty$. These families may be viewed as discrete analogue of the homogeneous distributions and are interesting by themselves. Then using Verlinde formula (\[eq:N\_Verlinde\]), we will recover the expression (\[eq:N\_RR\]) and obtain the above properties of the $Q_{g,n}$. Some of these properties also have symplectic proofs. For instance the fact that the $Q_{g,n}$ are polynomials is a consequence of Duistermaat-Heckman theorem by introducing some extended moduli space as in [@Je2]. The fact that the $Q_{g,n}$ vanish for odd $n$ may be deduced from Riemann Roch theorem by computing the characteristic class of the ${\mathcal{M}}( {\Sigma}, s)$, and expressing the Riemann Roch number in terms of $A$-genus instead of the Todd class.
To finish this overview of the proof of Theorem \[theo:main-result\], let us briefly explain the topological interpretation of the previous computation. For any topological space $T$, let ${\mathcal{M}}( T)$ be the space of conjugacy classes of group morphism $\pi_1 (T) \rightarrow {\operatorname}{SU}(2)$. We have a natural identification between ${\mathcal{M}}( \partial {\Sigma}\times S^1)$ and the quotient $N$ of $M = {{\mathbb{R}}}^2 /{{\mathbb{Z}}}^2$ by the involution $(x,y) \rightarrow (-x, -y)$. This identification restricts to a bijection between:
1. the projection of $\{x = 0\} \cup \{y =0\} \cup \{ y = 1/2 \}$ and the image of the restriction map $r: {\mathcal{M}}( {\Sigma}\times S^1 ) \rightarrow {\mathcal{M}}( \partial {\Sigma}\times S^1)$
2. the projection of $\{ y = ax/b \}$ and the image of the restriction map $r':{\mathcal{M}}( {T}) \rightarrow {\mathcal{M}}( \partial {\Sigma}\times S^1)$.
Finally ${\mathcal{M}}(S)$ may be viewed as the fiber product of $r$ and $r'$, its connected components being the fibers of the projection ${\mathcal{M}}(S) \rightarrow {\mathcal{M}}( \partial {\Sigma}\times S^1)$.
So the different part of the asymptotic expansion of $\langle Z_k ( {T}), Z_k ({\Sigma}\times S^1) \rangle$ are naturally indexed by $X= \pi_0 ( {\mathcal{M}}(S))$. In the decomposition $X = X_1 \cup X_2 \cup X_3 \cup X_4$ used in Theorem \[theo:main-result\], $X_1$ correspond to the points of $M$ such that $x=0$ and $y \neq 0,1/2$, $X_2$ to $ y =0$ or $1/2$ and $x \neq 0,1/2$, $X_3$ to $(0,0)$, $(0,1/2)$ and $X_4$ to $(1/2,0)$, $(1/2,1/2)$. As we will see, the Chern-Simons invariants appears naturally by interpreting the prequantum bundle of $L$ as the Chern-Simons bundle. Furthermore the expressions for the coefficients $a_0$ come from the leading order term $Q_{g,0} (t) = v_g (t)$ in (\[eq:N\_RR\]), the $(1-2g)$-th power of the sinus in (\[eq:N\_Verlinde\]) and the coefficients $a,b$ of the surgery.
### Outline of the paper {#outline-of-the-paper .unnumbered}
In section \[sec:witt-resh-tura\], we recall how we compute the WRT invariants of a Seifert manifold in terms of the Verlinde numbers. In section \[sec:discr-four-transf\], we study the discrete Fourier transform of the negative power of a sinus and we apply this to the Verlinde numbers. In section \[sec:geom-quant-tori\], we recall some basic facts on the quantization of tori and their Lagrangian states. Furthermore we establish a criterion characterizing the Lagrangian states in terms of a generating function of the associated Lagrangian manifold. In section \[sec:asympt-behav-z\_k\], we prove that $Z_k ({\Sigma}\times S^1) $ is a Lagrangian state and deduce the asymptotic behavior of $Z_k (S)$ by using some singular stationary phase lemma proved in the Section \[sec:sing-discr-stat\]. Finally, Section \[sec:geom-interpr-lead\] is devoted to the geometric interpretation of the results.
The Witten-Reshetikhin-Turaev invariant of a Seifert manifold {#sec:witt-resh-tura}
=============================================================
For any integer $k \geqslant 2$ and any closed oriented 3-manifold $M$, we denote by $Z_k (M)$ the Witten-Reshetikhin-Turaev (WRT) invariant of $M$ for the group ${\operatorname}{SU}(2)$ at level $k-2$. We are interested in the large level limit, $k \rightarrow \infty$. Since we haven’t chosen a spin structure on $M$, the sequence $Z_k (M)$, $k \geqslant 2$ is only defined up to multiplication by $\tau_k^n$ where $\tau_k = e^{ \frac{3i \pi}{4} - \frac{3i \pi } { 2k}}$.
In the case $M$ is a Seifert manifold, it is easy to compute $Z_k(M)$ by using a surgery presentation of $M$, cf. Section 1 of [@FrGo]. Let us explain this. Let ${\Sigma}$ be a compact oriented surface with boundary a circle $C$. Let $D$ be a disc and $S^1$ be the standard circle. Consider the Seifert manifold $S$ obtained by gluing the solid torus $ D \times S^{1}$ to ${\Sigma}\times S^1$ along a preserving orientation diffeomorphism $\varphi : \partial D \times S^1 \rightarrow C \times S^1$ $$S = ( {\Sigma}\times S^1 ) \cup_{\varphi} ( D \times S^1)^{-} .$$ The WRT invariant of a manifold obtained by gluing two manifolds along their boundary may be computed as a scalar product in the setting of topological quantum field theory, [@Wi] [@ReTu]. In our particular case, we have $$\begin{gathered}
\label{eq:scalar_product}
Z_k (S) = \bigl\langle Z_k ( {\Sigma}\times S^1) , \rho_k ( \varphi) ( Z_k ( D \times S^1)) \bigr\rangle_{ V_k ( C \times S^1)} \end{gathered}$$ Here $V_{k} ( C \times S^1)$ is the Hermitian vector space associated to the torus $C \times S^1$. It has dimension $k-1$. To any oriented basis $( \mu, {\lambda})$ of $H_1 ( C \times S^1)$ is associated an orthornormal basis of $V_k ( C \times S^1)$, called the Verlinde basis. Let us choose $\mu = [ C]$, ${\lambda}= [S^1]$ and denote by $e_ \ell$, $\ell = 1, \ldots , k-1$ the corresponding basis.
The bracket in Equation (\[eq:scalar\_product\]) denote the scalar product of $V_k ( C \times S_1)$. Furthermore for any compact oriented 3-manifold $M$ with boundary $C \times S^1$, we denote by $Z_k (M) \in V_k ( C \times S^1)$ the corresponding vector defined in the Chern-Simons topological quantum field theory.
\[lem:SiS\] One has $Z_k ( D \times S^1) = e_1$ and $$Z_k ( {\Sigma}\times S^1) = \sum_{\ell =1 }^{ k-1} N _\ell ^{g,k} e_{\ell},$$ where $g$ is the genus of ${\Sigma}$ and $N_{\ell} ^{g,k}$ is the dimension of the vector space associated in Chern-Simons quantum field theory to any genus $g$ surface equipped with one marked point colored by $\ell$.
In our convention, the set of colors is $\mathcal{C}_k = \{ 1, 2, \ldots, k-1 \}$, the color $\ell$ corresponding to the $\ell$-dimensional irreducible representation of ${\operatorname}{SU}(2)$.
Recall first that the Verlinde basis consists of the vectors given by $$e_\ell = Z_k ( D \times S^1, x, \ell)$$ where $x$ is the banded link $[0,1/2] \times S^1$ of $D \times S^1$. Since $\ell =1$ is the trivial color in our convention, $Z_k ( D \times S^1) = e_1$. Let us compute the coefficients of $Z_k ( {\Sigma}\times S^1)$. We have $$\langle Z_{k} ( {\Sigma}\times S^1) , e_{\ell} \rangle = Z_k ( \overline{{\Sigma}} \times S^1, x, \ell)$$ where $\overline {\Sigma}$ is the closed surface $ {\Sigma}\cup_{C} D^-$. Viewing $\overline {\Sigma}\times S^1$ as the gluing of $ \overline{{\Sigma}} \times [0,1]$ with itself, we obtain that $Z_k ( \overline{{\Sigma}} \times S^1, x, \ell) $ is the trace of the identity of $V_k ( \overline{ {\Sigma}}, \ell)$.
There are several ways to compute the numbers $N _\ell ^{g,k}$. First $N_{\ell}^{g,k}$ is the number of admissible colorings of any pants decomposition of ${\Sigma}$. But this elementary way is not very useful to study the large $k$ limit. Alternatively we can use the Verlinde formula.
\[theo:Verlinde\] For any $k \in {{\mathbb{N}}}^*$, $\ell = 1 , \ldots , k-1$ and $g \in {{\mathbb{N}}}^*$, we have $$N_{\ell}^{g,k} = \sum _{ m =1}^{k-1} S_{m,1} ^{1 - 2g} S_{m, \ell}$$ where $S_{m, p}= \bigl( \frac{2}{k}\bigr)^{1/2} \sin ( \pi \frac{mp}{k} )$.
Later we will also use the fact that $N_{\ell}^{g,k}$ can be computed with the Riemann-Roch theorem, cf. Theorem \[theo:RR\].
It remains to explain the meaning of the $\rho_k$ appearing in Equation (\[eq:scalar\_product\]). $\rho_k $ is the representation of the mapping class group of $C \times S^1$ in $V_k ( C \times S^1)$ provided by the topological quantum field theory. It is actually a projective representation. More precisely, $\rho_k ( \varphi)$ is well-defined up to a power of the constant $\tau_k$ defined above. Using the basis $( \mu,{\lambda})$, the mapping class group of $C \times S^1$ is identified with ${\operatorname}{SL} ( 2, {{\mathbb{Z}}})$. Then we have
[3]{} \_k(T)e\_= & e\^e\_, & & T=
1&1\
0&1
\_k(S)e\_= & \_[’=1]{}\^[k-1]{}( ) e\_[’]{}, & & S=
0&-1\
1&0
.
Since ${\operatorname}{SL} ( 2, {{\mathbb{Z}}})$ is generated by the matrices $S$ and $T$, the representation is completely determined by these formulas.
The discrete Fourier transform of a negative power of sinus {#sec:discr-four-transf}
===========================================================
Let ${\mathbb{T}}= {{\mathbb{R}}}/ 2 {{\mathbb{Z}}}$ and $R_k = (\frac{1}{k} {{\mathbb{Z}}}) / 2 {{\mathbb{Z}}}\subset {\mathbb{T}}$. Introduce for any $m \in {{\mathbb{N}}}$, the function $\Xi_m$ from $R_k$ to ${{\mathbb{C}}}$ given by $$\begin{gathered}
\label{eq:defXi}
\Xi_{m,k} (x) = ( i )^{-m} \sum_{ y \in R_k \setminus \{ 0, 1\} } \bigl[ \sin ( \pi y ) \bigr]^{-m} e^{ik \pi y x }, \qquad \forall x \in R_k .\end{gathered}$$ This function may be viewed as the discrete Fourier transform of $y \rightarrow \bigl[ \sin ( \pi y ) \bigr]^{-m}$. We are interested in its behavior as $k$ tends to infinity.
\[theo:dev\_part\] For any $m\in {{\mathbb{N}}}^*$, there exists a polynomial function $P_{m} \in {{\mathbb{Q}}}[k,x]$ such that for any $k \in {{\mathbb{N}}}^*$ and $x \in [0,2 ] \cap \frac{1}{k} {{\mathbb{Z}}}$, we have $$\Xi_{m,k} (x) = \bigl( 1 + ( -1 ) ^{kx + m }\bigr) P_{m} ( k,x) .$$ Furthermore, $P_{m}$ is a linear combination of the monomials $ k^q x^p$ where $p,q$ run over the integers satisfying $0 \leqslant p \leqslant q \leqslant m$.
The proof, given in Section \[sec:proof-theor-refth\], allows to compute inductively the polynomials $P_m$. In particular, we have
[1]{} & P\_1 (k,x ) = k ( - x +1 ),\
& P\_2 (k,x) = k\^2 ( - x\^2 +x - ) + ,\
& P\_3 (k,x ) = k\^3 ( - x\^3 + x\^2 - x ) + k ( x - ).
In Section \[sec:furth-prop-polyn\], we will establish further properties of the polynomials $P_m$. First, each $P_m$ is a linear combination of monomials $k^q x^p$ where $0 \leqslant p \leqslant q \leqslant m$ and $q \equiv m$ modulo 2. Second we will describe the singularity of $\Xi_{m,k}$ at $x =0$ as follows. Write $P_m = P_m^{+} + P_m^{-}$ where $P_m^+$ (resp. $P_m ^{-}$) is a linear combination of the $k^q x^{2 \ell}$ (resp. $k^q x^{2 \ell+1}$). Then $$\frac{1}{2} \bigl( P_m ( k ,x ) - P_m ( k, 2 + x ) \bigr) = \begin{cases} P^-_m ( k, x) \quad \text{ if $m$ is even,} \\ P^+_m ( k, x) \quad \text{ if $m$ is odd.}
\end{cases}$$ Furthermore, if $m \geqslant 1$ is even (resp. odd), $P_m^-$ (resp. $P^+_m$) is a linear combination of the monomials $k^{m-2 \ell} x^{m- 2\ell -1}$, with $\ell = 0 ,1 \ldots$. The coefficient of $k^{m} x^{m-1}$ is $ 1 / (m-1)!$. In Section \[sec:appl-count-funct\], we will apply this to the counting function $N^{g,k}_{\ell}$.
Proof of Theorem \[theo:dev\_part\] {#sec:proof-theor-refth}
-----------------------------------
In the sequel, everything depends on $k$. Nevertheless, to reduce the amount of notation, the subscript $k$ will often be omitted. Introduce the space $\mathcal{H} := {{\mathbb{C}}}^{R_k}$ and its scalar product $$\langle f, g \rangle = \frac{1}{2k} \sum_{x \in R_k } f(x) \overline{g(x)} , \qquad f, g \in \mathcal{H} .$$
### The endomorphisms $T$, $L$ and $\Delta$ {#the-endomorphisms-t-l-and-delta .unnumbered}
Introduce the endomorphisms $T$, $L$ and $\Delta$ of $ \mathcal{H}$ given by $$(Tf ) (x) = ( -1)^{kx} f(x), \qquad ( L f) ( x) := f \Bigl( x + \frac{1}{k} \Bigr), \qquad {\Delta}= \tfrac{1}{2} \bigl( L - L^{-1} \bigr)$$ Observe that $T$ is unitary, $T^2 = {\operatorname}{id}$ and $\mathcal{H} = \mathcal{H}^+ \oplus^{\perp} \mathcal{H}^-$ where $\mathcal{H}^{\pm} = \ker ( T \mp {\operatorname}{id}).$ Furthermore, $\mathcal{H}^+$ (resp. $\mathcal{H}^-$) consists of the functions vanishing on the $x \in R_k$ such that $kx$ is odd (resp. even). The orthogonal projector of $\mathcal{H}$ onto $\mathcal{H}^{\pm}$ is $\frac{1}{2} ( {\operatorname}{id} \pm T)$.
Let $u_0 \in \mathcal{H} $ be the function constant equal to $1$, $u_1 = T u_0$ and $u_0 ^\pm = \frac{1}{2} ( u_0 \pm u_1 ) $. Denote by $\mathcal{H}_0^{\pm}$ the subspace of $\mathcal{H}^\pm$ orthogonal to $u_0 ^\pm$.
The kernel of $\Delta$ is spanned by $u_0$ and $u_1$. Furthermore $\Delta$ restricts to a bijection from $\mathcal{H}_0^{\pm} $ to $ \mathcal{H}_0^{\mp}$.
We easily check that $u_0$ and $u_1$ belong to the kernel of $\Delta$ and that this kernel is 2-dimensional. Furthermore we have that $LT + T L = 0$, so that $\Delta T + T \Delta = 0$ and consequently $ \Delta ( \mathcal{H}^{\pm} ) \subset \mathcal{ H}^{\mp}$. Since $L$ is unitary, $\Delta $ is skew-Hermitian. So the range of $\Delta$ is the subspace of $\mathcal{H}$ orthogonal to $u_0$ and $u_1$.
### Discrete Fourier transform {#discrete-fourier-transform .unnumbered}
Let $(u_y, \; y \in R_k )$ be the orthonormal basis of $\mathcal{H}$ $$u_{y} (x) = e^{ik\pi y x }, \quad \forall x \in R_k$$ When $y=0$ or $1$, we recover the functions $u_0$ and $u_1$ introduced previously. Denote by $\mathcal{F} : \mathcal{H} \rightarrow \mathcal{H}$ the discrete Fourier transform $$\mathcal{F} ( f) ( y) = \langle f, u_y \rangle , \qquad \forall y \in R_k .$$ Using the relations $ T u _y = u_{y+1}$ and $ L u _y = e^{i \pi y} u_y$, we deduce that for any $f \in \mathcal{H}$ and $y \in R_k$ $$\begin{gathered}
\label{eq:fourierTDelta}
\mathcal{F} (T f ) ( y) = \mathcal{F}(f) ( y- 1), \qquad \mathcal{F} ( \Delta f) ( y) = i \sin ( \pi y ) \mathcal{F} ( f) ( y) . \end{gathered}$$ By definition of $\Xi_m$, Equation (\[eq:defXi\]), its discrete Fourier transform is given by $$\mathcal{F} ( \Xi_m ) ( y) = \begin{cases} \bigl[ i\; \sin ( \pi y) \bigr]^{-m} \text{ if } y \neq 0, 1 \\ 0 \qquad \text{ otherwise.} \end{cases}$$ Observe that $\mathcal{F} ( \Xi_m ) ( y -1 ) = (-1)^m \mathcal{F} ( \Xi_m ) ( y)$, so that $T \Xi_m = ( -1)^m \Xi_m$. Furthermore $\mathcal{F} ( \Xi_m ) ( 0) = \mathcal{F} ( \Xi_m ) ( 1) = 0$ implies that $\Xi_m$ is orthogonal to both $u_0$ and $u_1$. Hence, $\Xi_m$ belongs to $\mathcal{H}^{+}_0$ or $\mathcal{H}^{-}_0$ according to whether $m$ is even or odd. Furthermore we have that $\Delta \Xi_{m+1} = \Xi_m$. So we can inductively compute $\Xi_m$ by inverting the operators $\Delta: \mathcal{H}^{\pm}_0 \rightarrow \mathcal{H}^{\mp}_0$.
Let us first compute $\Xi_0$. We have $$\Xi_0 = \Bigl( \textstyle{\sum}_{y \in R_k} u_y \Bigr) - ( u_0 + u_1 ) .$$ Furthermore $\sum_{y \in R_k} u_y = 2k \delta$ where $\delta ( x) = 1$ if $x = 0$ and $\delta ( x) = 0$ otherwise. So we obtain $$\begin{gathered}
\label{eq:xi0}
\Xi_0 (x) = 2 k \delta (x) - 1 - ( -1)^{kx} .\end{gathered}$$
Let us compute the inverse of $\Delta :\mathcal{H}^-_0 \rightarrow \mathcal{H}^+_0 $. Denote by $S_k $ the set of integers $\{1, \ldots, k \}$. Let $\tilde{L} $ and $L$ be the endomorphisms of ${{\mathbb{C}}}^{S_k}$ defined by $$\quad \forall \ell \in S_k, \quad \tilde{L} (f) ( \ell ) = \sum _{m = 1}^{ \ell} f (m), \qquad L ( f) = 2 \tilde{L} (f) - \frac{2}{k} \sum_{1}^{k} \tilde{L} (f)(\ell)$$ Let us identify $\mathcal{H}^+$ and $\mathcal{H}^-$ with ${{\mathbb{C}}}^{S_k}$ by sending $g^{\pm} \in \mathcal{H}^\pm$ into $f^{\pm} \in {{\mathbb{C}}}^{S_k}$ given by $$\begin{gathered}
\label{eq:rel1}
f^{+} ( \ell ) = g^+\Bigl( \frac{2(\ell-1)}{k } \Bigr) , \qquad f^{-} ( \ell ) = g^{-} \Bigl( \frac{2 \ell -1 } { k} \Bigr) .\end{gathered}$$ Observe that the subspaces $ \mathcal{H}^{+}_0$ and $\mathcal{H}^{-}_0$ get identified with the subspace of ${{\mathbb{C}}}^{S_k}$ consisting of function with vanishing average.
The inverse of $\Delta : \mathcal{H}^-_0 \rightarrow \mathcal{H}^+_0$ is $L$.
Let $\tilde{\Delta}$ be the endomorphism of ${{\mathbb{C}}}^{S_k}$ corresponding to $\Delta : \mathcal{H} ^- \rightarrow \mathcal{H} ^+$ through the identifications $\mathcal{H}^- \simeq {{\mathbb{C}}}^{S_k}$ and $\mathcal{H}^+ \simeq {{\mathbb{C}}}^{S_k}$. A straightforward computation shows that for any $f \in {{\mathbb{C}}}^{S_k}$, we have $$\tilde{\Delta} f (\ell) = \frac{1}{2} ( f ( \ell) - f ( \ell -1 )) , \qquad \ell \in S_k$$ with the convention that $f (0) = f(k)$. Assume that the average of $f$ vanishes, so that $\tilde {L} f ( k) = \tilde{L} f ( 0)$. So we have that $2 \tilde{\Delta} \tilde{L} f = f$. Since $\tilde{\Delta} 1 = 0 $, it follows that $ \tilde{\Delta} L f = f$. Furthermore $Lf$ has a vanishing average.
Let us apply this to compute $\Xi_1$. By (\[eq:xi0\]) and (\[eq:rel1\]), the function $f^{+}_0 \in {{\mathbb{C}}}^{S_k}$ corresponding to $\Xi_0$ is given by $f^{+}_0 ( 1) = 2k -2 $ and $f^{+}_0 ( \ell ) = -2$ for $\ell = 2, \ldots, k$. So that $\tilde{L} ( f^{+}_0)(\ell) = 2k - 2\ell$ and $L( f^{+}_0)( \ell) = -4 \ell + 2k + 2 $. Inverting the second relation of (\[eq:rel1\]), we obtain for any $x$ of the form $(2 \ell + 1)/k$, $$\begin{gathered}
\label{eq:rel2}
g^{-} ( x) = f^{-} \Bigl( \frac{kx + 1 }{2} \Bigr) \end{gathered}$$ which leads to $ \Xi_1 ( x ) = 2 k ( - x +1 )$. This proves Theorem \[theo:dev\_part\] for $m=1$.
Assume now that Theorem \[theo:dev\_part\] has been proved for some even $m \geqslant 2$. So the restriction of $\Xi_m $ to $[0,2] \cap \frac{2}{k} {{\mathbb{Z}}}$ is a linear combination of the monomials $ k^ q x^p$ where $ p \leqslant q \leqslant m$. Then, the function $f_m^+ \in {{\mathbb{C}}}^{S_k}$ corresponding to $\Xi_m $ is a linear combination of the monomials $ k^ {q-p} \ell^p$ where $ p \leqslant q \leqslant m$. Now, it is a well-known consequence of Euler-Maclaurin formula that for any $p \in {{\mathbb{N}}}$, there exists a polynomial $Q_p$ with degree $p+1$ and vanishing at $0$ such that $$\begin{gathered}
\label{eq:sompol}
\sum_{m = 1 } ^{\ell} m ^p = Q_p ( \ell ) , \qquad \forall \ell \in {{\mathbb{N}}}\end{gathered}$$ Let $f \in {{\mathbb{C}}}^{S_k}$ be given by $f( \ell) = \ell^p$. Then $\tilde{L}( f) ( \ell) = Q_p ( \ell)$. Applying (\[eq:sompol\]) to the monomials of $Q_p$, we obtain a polynomial $R_p$ of degree $p+ 2$, vanishing at $0$ and such that $$\sum_{m = 1 } ^{\ell} Q_p ( m) = R_{p } ( \ell) , \qquad \forall \ell \in {{\mathbb{N}}}$$ Consequently $ L ( f) ( \ell ) = 2Q_p ( \ell) - 2k^{-1}R_{p } (k) $, so that $$L( k^{q-p} f ) ( p) = 2k^{q-p} Q_p ( \ell ) - 2k^{q-p-1} R_p ( k) .$$ Since $R_p(0) = 0$, $k^{-1} R_p ( k)$ is polynomial in $k$ with degree $p+1$. This proves that $L ( f_m)$ is a linear combination of the monomials $ k^ {q-p} \ell^p$ where $ p \leqslant q \leqslant m +1 $. Applying the relation (\[eq:rel2\]), we obtain that the restriction of $\Xi_{m+2} $ to $[0,2] \cap \frac{1}{k} ( 1+ 2{{\mathbb{Z}}})$ is a linear combination of the monomials $ k^ {q-p + p'} x^{p'}$ where $ p' \leqslant p \leqslant q \leqslant m +1 $. Equivalently it is a linear combination of the $ k^ {q} x^{p} $ where $ p \leqslant q \leqslant m +1 $, which proves Theorem \[theo:dev\_part\] for $m+1$.
Similarly we can show that the result for $m$ odd implies the result for $m+1$. To do this, we identify $\mathcal {H}_0^+$ and $\mathcal{H}_0^-$ with ${{\mathbb{C}}}^{S_k}$ by using the relations $$\begin{gathered}
f^{+} ( \ell ) = g^+\Bigl( \frac{2\ell}{k } \Bigr) , \qquad f^{-} ( \ell ) = g^{-} \Bigl( \frac{2 \ell -1 } { k} \Bigr) .\end{gathered}$$ instead of (\[eq:rel1\]). Then the inverse of $\Delta : \mathcal{H}^+_0 \rightarrow \mathcal{H}^-_0$ is still given by $L$. The remainder of the proof is unchanged.
Further properties of the polynomials $P_{m}$ {#sec:furth-prop-polyn}
---------------------------------------------
For any $m$, let us write $P_m = P_m^{+} + P_m ^{-}$, where $P_m^{+}$ (resp. $P_m ^{-}$) is a linear combination of the monomials $ k^q x^{2 \ell}$ (resp. $ k^q x^{2 \ell + 1}$).
\[eq:sing\_part\] For any $m \in {{\mathbb{N}}}^*$, we have $$\frac{1}{2} \bigl( P_m ( k ,x ) - P_m ( k, 2 + x ) \bigr) = \begin{cases} P^-_m ( k, x) \quad \text{ if $m$ is even,} \\ P^+_m ( k, x) \quad \text{ if $m$ is odd.}
\end{cases}$$
The discrete Fourier transform $\mathcal{F}( \Xi_m)$ has the same parity as $m$, so the same holds for $\Xi_m$. This implies that for any $x \in [0,2] \cap \frac{1}{k} {{\mathbb{Z}}}$ such that $kx $ has the same parity as $m$, we have $$P_m( k, 2- x ) = ( -1)^m P_m( k, x)$$ Since $P_m$ is polynomial in $k$ and $x$, this equality actually holds for any $k$ and $x$. So we have $$P_m ( k, x ) - P_m ( k, 2 + x ) = P_m ( k,x ) - ( -1)^m P_m (k, -x) ,$$ which concludes the proof.
\[prop:calcul\_pm\] For any $m \geqslant 2$, we have $$\begin{gathered}
\label{eq:rec_Pm}
\sum_{\ell =1 }^{\infty} \frac{k^{-2 \ell +1 }}{(2 \ell -1)!} \Bigl( \frac{d}{dx} \Bigr)^{2 \ell - 1 } P_{m} ( k, x) = P_{m-1} ( k, x).\end{gathered}$$ Furthermore, if $m$ is even, $$\begin{gathered}
\label{eq:pair_init}
k \int_0^2 P_m ( k, x ) \; dx = 4 \sum_{\ell=1}^{\infty} \frac{B_{2 \ell} }{ ( 2 \ell ) !} \Bigl( \frac{2}{k} \Bigr)^{ 2 \ell -1 } \Bigl( \frac{d}{dx} \Bigr)^{2 \ell - 1 } P_m^- ( k,0) \end{gathered}$$ where the $B_{\ell}$ are the Bernouilli numbers. If $m$ is odd, $$\begin{gathered}
\label{eq:impair_init} P_{m} (k, \tfrac{1}{k} ) = P_{m-1} ( k,0) .\end{gathered}$$
Since $\Delta \Xi_{m} = \Xi_{m-1}$, we have for any $x \in [0,2] \cap \frac{1}{k} {{\mathbb{Z}}}$ such that $kx$ has the same parity as $m-1$, $$\tfrac{1}{2} \bigl( P_{m} (k, x + \tfrac{1}{k} ) - P_{m} (k, x - \tfrac{1}{k} ) \bigr) = P_{m-1}(k, x ) .$$ $P_{m}$ and $P_{m-1}$ being polynomials the same equality holds for any $x$ and $k$. We obtain Equation (\[eq:rec\_Pm\]) by applying Taylor formula.
Recall that the average of $\Xi_m$ vanishes. For even $m$ this implies that $\sum_{\ell = 1}^{k} P_{m} ( k, \tfrac{2\ell}{k} ) = 0 .$ By Euler-Maclaurin formula, we have
[2]{} \_[= 1]{}\^[k]{} P\_[m]{} ( k, ) = & \_0 \^k P\_m (k, ) dx + ( P\_m ( k, 2 ) - P\_m ( k,0) )\
+ & \_[ 1]{} ( )\^[ 2 -1 ]{}
Using that $P_m ( k,2) = \Xi_{m, k } ( 0 ) = P_m ( k,0)$ and Proposition \[eq:sing\_part\], we obtain Equation (\[eq:pair\_init\]).
If $m$ is odd, $\Delta \Xi_{m} = \Xi _{m-1}$ implies that $$\tfrac{1}{2} \bigl( P_m ( k, \tfrac{1}{k} ) - P_m ( k, - \tfrac{1}{k} ) \bigr) = P_{m-1} (k,0).$$ Since $\Xi_m $ is also odd, we have $P_m (k , - \tfrac{1}{k} ) = - P_m ( k, \tfrac{1}{k})$ which proves Equation (\[eq:impair\_init\]).
Proposition \[prop:calcul\_pm\] allows to compute the $P_m$’s inductively. Indeed, we have the following
\[prop:recurrence\] For any $m \geqslant 2$, $P_{m}$ is the unique solution in ${{\mathbb{C}}}[ k,x]$ of Equations (\[eq:rec\_Pm\]) and (\[eq:pair\_init\]) (resp. (\[eq:impair\_init\])) if $m$ is even (resp. odd).
Write $ P_m ( k,x) = k^{r} Q_r ( x) + k^{r-1} P_{r-1} ( x) + \ldots + Q_0 (x)$. Denote by $D$ the derivation $\frac{d}{dx}$. Then
[2]{} \_[=1 ]{}\^ D\^[2 - 1 ]{} P\_[m]{} ( k, ) = & k\^[r-1]{} D Q\_r + k\^[r-2]{} DQ\_[r-1]{} + k\^[r-3 ]{}( DQ\_[r-2]{} +\
& D\^3 Q\_[r]{} ) + k\^[r-4]{} ( DQ\_[r-3]{} + D\^3 Q\_[r-1]{}) + …\
= & \_[=0]{} \^[r]{} k\^[r --1 ]{}( D Q\_[r -]{} + R\_) + (k\^[-2]{} )
where for any $ 0 \leqslant \ell \leqslant r$, $R_{\ell} \in {{\mathbb{C}}}[x]$ only depends on $Q_{r}, Q_{r-1}, \ldots , Q_{r-\ell +1}$. So Equation (\[eq:rec\_Pm\]) leads to a triangular system of equations for the $DQ_{r-\ell}$’s. We conclude that $P_m$ is the unique solution of Equation (\[eq:rec\_Pm\]) up to some polynomial in $k$. Arguing similarly, we prove that this latter polynomial is uniquely determined by Equation (\[eq:pair\_init\]) or (\[eq:impair\_init\]) according to the parity of $m$.
\[prop:par\_k\] For any $m \in {{\mathbb{N}}}^*$, $P_m $ is a linear combination of the monomials $ k^q x^p$, where $p,q$ run over the integers such that $p \leqslant q \leqslant m$ and $q$ has the same parity of $m$.
We have to prove that $P_m ( -k, x) = ( -1)^m P_m ( k, x)$. Assume the result holds for $m-1$. Then we easily see that $P_m ( -k , x)$ satisfies the Equations of Proposition \[prop:calcul\_pm\]. To check Equation (\[eq:impair\_init\]) in the case $m$ is odd, we use that $P_m ( k, -\frac{1}{k} ) = -P_m ( k, \frac{1}{k} )$. We conclude with Proposition \[prop:recurrence\].
\[prop:sing\_part\] For any even $m \geqslant 2$ (resp. odd $m \geqslant 1$), $P_m^-$ (resp. $P^+_m$) is a linear combination of the monomials $k^{m-2 \ell} x^{m- 2\ell -1}$, with $\ell = 0 ,1 \ldots$. The coefficient of $k^{m} x^{m-1}$ is $ 1 / (m-1)!$.
Again the proof is by induction on $m$. Assume that $m$ is even and write $ A = P_m ^{-}$, $B = P_{m-1} ^{+}$. By Proposition \[prop:par\_k\], we have $$\begin{gathered}
A ( k, x ) = k^{m} A_m ( x) + k^{m-2} A_{m-2} ( x) + \ldots + A_0 ( x) , \\ \qquad B(k, x) = k^{m-1} B_{m-1} (x) + k^{m-3} B_{m-3} (x) + \ldots + k B_1 (x) .\end{gathered}$$ We deduce from Equation (\[eq:rec\_Pm\]) that $$\begin{gathered}
D A_m = B_{m-1} , \qquad DA_{m-2} + \tfrac{1}{3!} D^3 A_m = B_{m-3} , \qquad \ldots \\
D A_2 + \tfrac{1}{3!} D^3 A_4 + \ldots , \tfrac{1}{ (m-1)!} D^{m-1} A_m = B_1 ,\\ DA_0+ \tfrac{1}{3!} D A_2 + \ldots + \tfrac{1}{ ( m +1 ) !} D^{m+1} A_m = 0 \end{gathered}$$ Assume that $ B_{\ell} (x) = b_{\ell} x^{\ell-1}$. Then these equations imply that $A_\ell ( x) = a_{\ell} x^{\ell-1} + a_{\ell}^0$. Since the $A_\ell$’s are odd, the constants $a_{\ell}^{0}$ vanish.
Assume now that $m$ is odd and that the results holds for $m-1$. Arguing as above with Proposition \[prop:par\_k\] and Equation (\[eq:rec\_Pm\]), we prove that
[2]{} P\_m \^+ ( k, x) = & k\^[m ]{} ( c\_m x\^[m-1]{} + d\_m ) + k\^[m-2]{} ( c\_[m-2]{} x\^[m-3]{} + d\_[m-2]{} ) + …\
+ & k\^[3]{} ( c\_3 x\^2 + d\_3) + k c\_1
By Proposition \[prop:par\_k\], $P_{m-1} ( k,0)$ is even. So Equation (\[eq:impair\_init\]) implies that $$P_m ^+ (k, \tfrac{1}{k} ) = 0.$$ We deduce that the $d_{\ell}$’s vanish and $c_m + c_{m-2} + \ldots + c_1 = 0$.
Application to the counting function {#sec:appl-count-funct}
------------------------------------
As a consequence of Verlinde formula, we have the following
\[lem:relat-with-count\] For any $k \in {{\mathbb{N}}}^*$, $\ell = 1 , \ldots , k-1$ and $g \in {{\mathbb{N}}}^*$, we have $$N^{g,k}_{\ell} = C_g k ^{ g -1 } \Xi _{2g -1 } ( \ell / k )$$ with $C_g = (-1)^{g-1} 2 ^{-g}$.
We compute from Theorem \[theo:Verlinde\]
[2]{} N\_\^[g,k]{} = & \_[m= 1 ]{} \^[k-1]{} S\_[m,1]{} \^[1 - 2g]{} S\_[m, ]{} = ( )\^[1/2]{} \_[m=1]{}\^[k-1]{} S\_[m,1]{} \^[1 - 2g]{} ( e\^[ i ]{} - e\^[ -i ]{} )\
= & ( )\^[1-g]{} \_[y R\_k { 0 , 1}]{} \^[1-2g]{} e\^[i y ]{}\
= & 2\^[-g]{}(-k ) \^[g-1]{} i\^[1-2g]{} \_[y R\_k { 0 , 1}]{} \^[1-2g]{} e\^[i y ]{}
which ends the proof.
So we deduce from Theorem \[theo:dev\_part\] and the result of Section \[sec:furth-prop-polyn\] the following
\[theo:counting\_smoot\] Let $g\in {{\mathbb{N}}}^*$. Then there exists a family of polynomial functions $P_{g, m} : [0,1] \rightarrow {{\mathbb{R}}}$, $m =0,1,\ldots, g-1$ such that for any $k \in {{\mathbb{N}}}^*$ and for any odd integer $\ell$ satisfying $1 \leqslant \ell \leqslant k-1$, we have $$N^{g, k }_{\ell} = \Bigl( \frac{k}{2\pi} \Bigr)^{ 3 g -2 } \sum _{m = 0 }^{g-1 } k^{-2m} P_{g, m} \Bigl( \frac{\ell}{k} \Bigr) .$$ Furthermore $P_{g, m}$ has degree $2(g -m) -1 $. The even part of $P_{g, m }$ is of the form ${\lambda}_{g,m} x^{ 2( g -m - 1)}$. For $m=0$, we have $${\lambda}_{g,0} = \frac{2 C_g ( 2 \pi)^{ 3g -2}}{ ( 2 ( g-1))!} .$$
The polynomials $P_{g,m}$ will be expressed in Section \[sec:symplectic-volumes\] as integrals of characteristic classes on some moduli spaces. In particular $P_{g,0} (s)$ is a symplectic volume.
Geometric quantization of tori and semiclassical limit {#sec:geom-quant-tori}
======================================================
The quantum spaces {#sec:quantum-spaces}
------------------
Let $(E,{\omega})$ be a real two-dimensional symplectic vector space. Let $R$ be a lattice of $E$ with volume $4\pi$. Let $L_E = E \times {{\mathbb{C}}}$ be the trivial Hermitian line bundle over $E$. Endow $L_E$ with the connection $d + \frac{1}{i} {\alpha}$ where ${\alpha}\in {\Omega}^1 ( E, {{\mathbb{C}}})$ is given by $${\alpha}_x ( y) = \frac{1}{2} {\omega}( x, y).$$ Consider the Heisenberg group $E \times U(1)$ with the product $$\begin{gathered}
\label{eq:action_prod}
(x, u) . ( y,v) = ( x+ y, uv e^{i {\omega}( x, y) /2})\end{gathered}$$ This groups acts on $L_E$ by preserving the metric and the connection, the action of $(x,u)$ being given by formula (\[eq:action\_prod\]) with $(y,v ) \in L_E$. The group $E \times U(1)$ is actually the group of automorphisms of $(L_E , d + \frac{1}{i} {\alpha})$ lifting the translations of $E$.
Since ${\omega}( R, R) \subset 4 \pi {{\mathbb{Z}}}$, $R \times \{1 \}$ is a subgroup of $E \times U(1)$. Let $M$ be the torus $E /R$ and $L_M $ be the bundle $L_E / R \times \{1\}$. The symplectic form ${\omega}$ and the connection $d + \frac{1}{i} {\alpha}$ descend to $M$ and $L_E$ respectively. Let $k$ be a positive integer. For any $ x \in \frac{1}{2k} R$, the action of $(x,1)$ on $L_E^k$ commutes with the action of $R \times \{1 \}$. This defines an action of $(x, 1)$ on $L^k_M$. Denote by $T^*_x$ the pull-back of the sections of $L^k_M$ by the action of $(x,1)$. Observe that for any $ x, y \in R$, we have $$T_{x/2k}^* T_{y/2k} ^* = e^{i{\omega}( x, y)/4k} T_{y/2k}^* T_{x/2k}^* .$$
Choose a linear complex structure $j$ of $E$ compatible with ${\omega}$. This complex structure descends to $M$. Furthermore, $L_M$ inherits a holomorphic structure compatible with the connection. The space $H^0 ( M , L^k_M)$ of holomorphic sections of $L^k_M$ has dimension $2k$. The operators $T_{x/2k}^*$, $x \in R$ introduced above, preserve $H^0 ( M , L^k_M)$.
Let $( \delta, \varphi)$ be a half-form line, that is $\delta$ is a complex line and $\varphi$ is an isomorphism from $\delta^{\otimes 2} $ to the canonical line $K_j$, $$K_j = \{ {\alpha}\in E^* \otimes {{\mathbb{C}}}/ {\alpha}( j \cdot ) = i {\alpha}\} .$$ Let $\mathcal{H}_k = H^0 ( M , L^k_M) \otimes \delta = H^0 ( M , L^k_M \otimes \delta_M )$ where $\delta_M$ is the trivial line bundle $M \times \delta$. $K_j$ has a natural scalar product given by $( {\alpha}, {\beta}) = i {\alpha}\wedge {\overline}{{\beta}} / {\omega}$. We endow $\delta$ with the scalar product making $\varphi$ a unitary map. $\mathcal{H}_k$ has a scalar product defined by integrating the pointwise scalar product against ${\omega}$.
Let $( \mu , {\lambda})$ be a positive basis of $R$, so that ${\omega}( \mu , {\lambda}) = 4 \pi$. It is a known result that $\mathcal{H}_k$ has an orthonormal basis $(\Psi_{\ell} , \; \ell \in {{\mathbb{Z}}}/ 2 k {{\mathbb{Z}}})$ such that $$T^*_{\mu/2k } \Psi_\ell = e ^{ i \pi \ell / k} \Psi_\ell, \qquad T^* _{{\lambda}/ 2k } \Psi_{\ell} = \Psi_{\ell + 1} .$$ The only indeterminacy in the choice of this basis is the phase of $\Psi_0$. We will often use the following normalization $$\Psi ( 0) = {\lambda}{\sigma}^k \otimes {\Omega}_{\mu} \qquad \text{ with } {\lambda}>0$$ Here ${\sigma}\in L_{M, [0]}$ is the vector send to $1$ by the identification $L_{M, [0]} \simeq L_{E, 0} = \{0 \} \times {{\mathbb{C}}}$. ${\Omega}_{\mu}$ is one of the two vectors in $\delta$ satisfying $ \varphi ( {\Omega}_{\mu}^2) ( \mu ) = 1$. We can explicitly compute the coefficient ${\lambda}$ as an evaluation of the Riemann Theta function. It satisfies as $k$ tends to infinity ${\lambda}= 1 + {\mathcal{O}}( e^{-k/C})$ for some positive $C$.
Let $S$ be the automorphism of $L_E^k$ sending $(x,u)$ into $( -x,u)$. This automorphism descends to an automorphism of $L_M$ that we still denote by $S$. The subspace of alternating sections of $\mathcal{H}_k$ is by definition the eigenspace $\ker ( {\operatorname}{Id}_{\mathcal{H}_k} + S^k \otimes {\operatorname}{id}_{\delta}) $. It has dimension $k-1$ and admits as a basis the family $(\Psi_{\ell} - \Psi_{-\ell}, \ell =1, \ldots, k)$.
Semi-classical notions
----------------------
Consider the same data as above. For any $k\in {{\mathbb{N}}}^*$ and $\xi \in \mathcal{H}_k$, we denote by $|\xi | \in {\mathcal{C}^{\infty}}( M , {{\mathbb{R}}})$ the pointwise norm of $\xi$ and by $\| \xi \| \in {{\mathbb{R}}}$ the norm defined previously, so $\| \xi \|^2 = \int_M | \xi |^2 {\omega}$.
We say that a family $ \xi = ( \xi_k \in \mathcal{H}_k, k \in {{\mathbb{N}}}^*)$ is [*admissible*]{} if there exists $N$ and $C>0$ such that for any $k$, $\| \xi_k \| \leqslant C k^{N}$. Equivalently, $\xi$ is admissible if there exists $N$ and $C>0$ such that for any $k$, $| \xi_k | \leqslant C k^{N}$ on $M$.
The [*microsupport*]{} of an admissible family $( \xi_k)$ is the subset ${\operatorname}{MS} ( \xi )$ of $M$ defined as follows: $x \notin {\operatorname}{MS} ( \xi)$ if and only if there exists a neighborhood $U$ of $x$ and a sequence $(C_N)$ such that for any integer $N $ and $x \in U$, $|\xi ( x) | \leqslant C_N k^{-N}$.
Let $U$ be an open set of $M$. Let ${\Gamma}$ be a one dimensional submanifold of $U$. Let $\Theta$ and ${\sigma}$ be sections of $L_M \rightarrow {\Gamma}$ and $\delta_M \rightarrow {\Gamma}$ respectively. Assume that $\Theta$ is flat and that its pointwise norm is constant equal to 1. Let $\xi$ be an admissible family. We say that the restriction of $\xi$ to $U$ is a [*Lagrangian state*]{} supported by ${\Gamma}$ with associated sections $( \Theta, {\sigma})$ if ${\operatorname}{MS} ( \xi ) \cap U \subset {\Gamma}$ and for any $ x_0 \in {\Gamma}$, there exists an open neighborhood $V$ of $x_0$ such that $$\begin{gathered}
\label{eq:lag_state}
\xi( x) = \Bigl( \frac{k}{2 \pi} \Bigr)^{1/4+N} E^k(x) f( x , k) + {\mathcal{O}}( k^{-\infty}) , \qquad x \in V \end{gathered}$$ where the ${\mathcal{O}}$ is uniform on $V$ and
- $E$ is a section of $L_M \rightarrow V$ such that $ E = \Theta$ on ${\Gamma}\cap V$, $|E| < 1$ outside of ${\Gamma}$, ${\overline}{\partial} E \equiv 0$ modulo a section vanishing to infinite order along ${\Gamma}$,
- $(f( . , k))$ is a sequence of ${\mathcal{C}^{\infty}}( V, \delta_M)$ which admits an asymptotic expansion of the form $f_0 + k^{-1} f_1 + \ldots$ with coefficients $f_0, f_1, \ldots $ in ${\mathcal{C}^{\infty}}(V, \delta_M)$. Furthermore $f_0 = {\sigma}$ on ${\Gamma}\cap V$,
- $N$ is a real number which does not depend on $x_0$.
Let us recall how we can estimate the norm and the scalar product of Lagrangian states in terms of the corresponding sections $\Theta$ and ${\sigma}$. In these statements, we identify $ {\sigma}^2 \in {\mathcal{C}^{\infty}}( {\Gamma}, \delta_M^2)$ with the one-form of ${\Gamma}$ given by $${\sigma}^2 (p)( X) := \varphi ( {\sigma}^2(p) ) ( X), \qquad \forall p \in {\Gamma}\text{ and } X \in T_p {\Gamma}$$ where we see $T_p {\Gamma}$ as a subspace of $T_p M = E$. The normalization for $N$ has been chosen so that for any $\rho \in {\mathcal{C}^{\infty}}( M)$ supported in $U$ $$\int_U | \xi _k | ^2 \rho \; {\omega}= \Bigl( \frac{k}{2\pi} \Bigr)^N \int_{{\Gamma}} \rho (x) |{\sigma}|^2 (x) + {\mathcal{O}}( K^{N-1})$$ Here $| {\sigma}|^2 $ is the density $|{\sigma}^2|$ of ${\Gamma}$, so that it makes sense to integrate it on ${\Gamma}$. For a proof of this formula, cf Theorem 3.2 in [[@oim_demi]]{}.
Consider now two Lagrangian states $\xi$ and $\xi'$ over $U$ with associated data $({\Gamma}, \Theta,{\sigma},N)$ and $( {\Gamma}', \Theta', {\sigma}', N')$. Assume that ${\Gamma}\cap {\Gamma}' = \{ y \}$ and this intersection is transverse. Introduce a function $\rho \in {\mathcal{C}^{\infty}}(M)$ such that ${\operatorname}{supp} \rho \subset U$ and $\rho = 1 $ on a neighborhood of $y$. By Theorem 6.1 in [@oim_pol], we have the following asymptotic expansion $$\begin{gathered}
\label{eq:scalprodlag}
\int_U \bigl( \xi_k , \xi'_k \bigr) \rho {\omega}= \Bigl( \frac{k}{2\pi} \Bigr)^{-1/2 +N +N'} \bigl( \Theta (y) , \Theta' ( y) \bigr)^k_{L_{y}} \sum_{\ell = 0 }^{\infty} k ^{\ell} a_\ell + {\mathcal{O}}( k^{-\infty}) \end{gathered}$$ where the $a_{\ell}$’s are complex numbers and $ a_0 = \langle {\sigma}(y), {\sigma}'(y) \rangle_{T_y {\Gamma}, T_y {\Gamma}'} .$
Here the bracket is defined as follows. For any two distinct lines $\nu$, $\nu'$ of $E$, there exists a unique sesquilinear map $ \langle \cdot, \cdot \rangle_{\nu, \nu'} : \delta \times \delta \rightarrow {{\mathbb{C}}}$ such that for any $u$, $v \in \delta$, $$\begin{gathered}
\label{eq:pairing}
\bigl( \langle u , v \rangle_{\nu, \nu'}\bigr)^2 = i \frac{ u^2(X) \overline{v^2(Y)}}{ {\omega}(X, Y)}, \qquad \forall X \in \nu, Y \in \nu'\end{gathered}$$ where $X$ and $Y$ are any non vanishing vectors in $\nu$ and $\nu'$ respectively. The sign of $ \langle u , v \rangle_{\nu, \nu'}$ is determined by the following condition: the bracket depends continuously on $\nu$, $\nu'$ and $\langle u, u \rangle _{\nu, j \nu } \geqslant 0$.
Assume that $ {\sigma}$ and ${\sigma}'$ vanish at $y$. Then $a_0 =0$ and $a_1$ is computed as follows. Write ${\sigma}= f \tau$ and ${\sigma}'= f' \tau'$ with $f$ and $f'$ smooth functions vanishing at $y$. Then $$\begin{gathered}
\label{eq:subpairing}
a_1 = i\frac{d_x f ( X) \overline{ d_x f'(Y)}}{{\omega}( X, Y)} \langle \tau (y) , \tau' (y) \rangle_{T_y {\Gamma}, T_y {\Gamma}'} . \end{gathered}$$ for any nonvanishing vector $X \in T_x {\Gamma}$ and $Y \in T_x {\Gamma}'$.
Alternative description of Lagrangian states
--------------------------------------------
Choose a basis $(\mu, {\lambda})$ of $R$ such that ${\omega}( \mu, {\lambda}) = 4 \pi$. Denote by $p$ and $q$ the linear coordinates of $E$ dual to this basis. Let $s$ be the section of $L_E$ given by $s = e^{-2i \pi pq}$. We easily compute that $$\begin{gathered}
\label{eq:ders}
\nabla s = \frac{4 \pi}{ i} p dq \otimes s .\end{gathered}$$ Observe that $s( 0) =1$ and $s$ is flat along the lines ${{\mathbb{R}}}\mu$, $x + {{\mathbb{R}}}{\lambda}$ for all $x \in {{\mathbb{R}}}\mu$. These conditions completely determine $s$.
Let $q_0$ and $q_1$ in ${{\mathbb{R}}}$ be such that $q_0 < q_1 < q_0 + 1$. Let $U$ be the open set $ U = \{ [p \mu + q {\lambda}]/ \; q\in ]q_0 , q_1[, \; p \in {{\mathbb{R}}}\} $ of $M$. Let $\phi$ be a smooth real valued function defined on the interval $]q_0, q_1[$. Introduce the submanifold ${\Gamma}$ of $U$ $${\Gamma}= \{ [ \phi'(q) \mu + q {\lambda}] ;\quad q \in ]q_0, q_1[ \}$$ and the section $\Theta$ of $L_M \rightarrow {\Gamma}$ such that for any $q \in ]q_0 , q_1[$, $$\begin{gathered}
\label{eq:theta}
\Theta ( [ p \mu + q {\lambda}] ) = e^{4i\pi \phi ( q) } s ( p \mu + q{\lambda}) \quad \text{ with } p = \phi' ( q)\end{gathered}$$ It follows from (\[eq:ders\]) that $\Theta$ is flat. Let ${\sigma}$ be a section of $\delta_M \rightarrow {\Gamma}$.
Let $( \Psi_\ell , \ell \in {{\mathbb{Z}}}/2k {{\mathbb{Z}}})$ be the basis of $\mathcal{H}_k$ corresponding to $(\mu, {\lambda})$. Let $\xi$ be an admissible family. Denote by $\xi_k( \ell)$, $\ell \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}}$ the coefficients of $\xi_k$ in $( \Psi_{-\ell})$.
\[theo:laginbasis\] The restriction of $\xi$ to $U$ is a Lagrangian state with associated data $({\Gamma}, \Theta, {\sigma}, N)$ if and only if for any $q \in ]q_0 , q_1 [ \cap \frac{1}{2k} {{\mathbb{Z}}}$, $$\begin{gathered}
\label{eq:coeff_laginbasis}
\xi_ k ( 2 k q ) = \Bigl( \frac{k}{2\pi} \Bigr)^{-1/2 +N } e^{ 4 i \pi k \phi (q) }\sum_{\ell =0 }^{\infty} k^{-\ell} f_{\ell} (q) + {\mathcal{O}}( k^{-\infty})\end{gathered}$$ where the ${\mathcal{O}}$ is uniform on any compact set of $]q_0, q_1[$, the $f_{\ell}$’s are smooth functions on $]q_0, q_1[$ and the square of $f_0$ satisfies for any $q$, $$\begin{gathered}
\label{eq:symb_laginbasis}
{\sigma}^2 ( \phi ' (q) \mu + q {\lambda}) = f_0 ^2 ( q) \frac{ {\omega}( \cdot, \mu)}{i} .\end{gathered}$$
By Proposition 3.2 [@LJ1], the family $(\Psi_0 \in \mathcal{H}_k, k \in {{\mathbb{N}}}^*)$ is a Lagrangian state supported by the circle $C= \{ [p \mu] ; \; p \in {{\mathbb{R}}}\} \subset M$ and $$\begin{gathered}
\label{eq:psi0}
\Psi_0 ([ p \mu] ) = \Bigl( \frac{k}{2\pi} \Bigr)^{1/4} s^k ( p \mu) \otimes {\Omega}_\mu+ {\mathcal{O}}( k^{-\infty}).\end{gathered}$$ Assume that $\xi$ is a Lagrangian state. Then we can estimate the scalar product $$\xi_k ( 2kq ) = \langle \xi_k , \Psi_{ -2k q } \rangle = \langle \xi_k , T^*_{-q{\lambda}} \Psi_0 \rangle$$ with formula (\[eq:scalprodlag\]). Actually we need a version with parameter of formula (\[eq:scalprodlag\]) to get a uniform control with respect to $q$. Such a version holds and its proof is not more difficult. Let us explain how we obtain the factor $\exp( 4 i \pi k \phi (q))$ in (\[eq:coeff\_laginbasis\]) and Formula (\[eq:symb\_laginbasis\]) for the leading coefficient. First observe that ${\Gamma}$ intersects $q{\lambda}+ C$ transversally at the point $y_q = [\phi ' (q) \mu + q {\lambda}]$. Translating (\[eq:psi0\]) by $q {\lambda}$, we obtain $$\Psi_{ -kq} ( [q \lambda + p \mu ]) = \Bigl( \frac{k}{2\pi} \Bigr)^{1/4} s^k ( q {\lambda}+ p \mu) \otimes {\Omega}_\mu+ {\mathcal{O}}( k^{-\infty}).$$ By the definition of $\Theta$, cf. Equation (\[eq:theta\]), we have $$( \Theta( y_q)^k , s^k ( y_q) )_{L^k_{M,q}} = e^{4i\pi k \phi ( q)}$$ Equation (\[eq:symb\_laginbasis\]) follows from equation (\[eq:pairing\]) and the fact that ${\Omega}_{\mu}^2 ( \mu ) =1$.
Conversely, assume that the asymptotic expansion (\[eq:coeff\_laginbasis\]) holds. By the first part of the proof, there exists a Lagrangian state $\xi '$ such that the coefficients $\langle \xi ' _k , \Psi _{ - 2kq} \rangle$ satisfy the same asymptotic expansion. The coefficients of the sequence $f( \cdot, k)$ in (\[eq:lag\_state\]) have to be defined by successive approximations so that we recover the same coefficients in (\[eq:coeff\_laginbasis\]). Then we have $$\langle \xi_k - \xi ' _k , \Psi _{ - 2kq} \rangle = {\mathcal{O}}( k^{-\infty})$$ uniformly on any compact set of $]q_0, q_1 [$. This has the consequence that the microsupport of $\xi- \xi'$ does not meet $U$. For more details on this last step, see Proposition 2.2 in [@oim_torus].
Application to the functions $\Xi_{m,k}$ {#sec:appl-funct-xi_m}
----------------------------------------
Choose a positive basis $( \mu, {\lambda})$ of $R$ and denote by $( \Psi_{\ell}$, $\ell \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}})$ the corresponding basis of $\mathcal{H}_k$. Recall the function $\Xi_{m,k}$ of Section \[sec:discr-four-transf\]. Define $$\xi_{m,k} = \sum_{\ell \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}}} \Xi _{m,k} \Bigl( \frac{\ell}{k} \Bigr) \Psi_{\ell} .$$ Introduced the subsets of $M$ $$\begin{gathered}
\label{eq:defA12}
A_1 := \{ [ p \mu ]; \; p \in {{\mathbb{R}}}\} , \quad A_2 := \{ [q {\lambda}] , [ \mu /2 + q {\lambda}] ; \; q \in {{\mathbb{R}}}\}.\end{gathered}$$ Introduce the neighborhoods of $A_1 \setminus A_2$ and $A_2 \setminus A_1$ respectively given by $U_1 := A_1 \setminus A_2$ and $U_2 := A_2 \setminus A_1$. Let $A := A_1 \cup A_2$ and $ \Theta_A$ be the section of $L_M \rightarrow A$, which is flat and equal to $1$ at the origin.
\[theo:asympt-behav-xi\] The restriction of $( \xi_{m,k}, \; k \in {{\mathbb{N}}}^*) $ to $U_1$ (resp. $U_2$) is a Lagrangian state supported by $A_1 \setminus A_2$ (resp. $A_2 \setminus A_1$) with order $-1$ (resp. $1/2+m$) and corresponding section $\Theta_A$.
The symbol can also be computed in terms of the polynomials $P_m $ of Theorem \[theo:dev\_part\].
It is a consequence of Theorem \[theo:laginbasis\] and Theorem \[theo:dev\_part\]. Indeed, denoting by $\xi_{m,k} (\ell)$ the coefficient of $\Psi_{-\ell}$ in $\xi_{m,k}$, we have for any $q \in ]0,1[ \cap \frac{1}{2k} {{\mathbb{Z}}}$,
[2]{} \_[m,k]{} ( 2k q ) = & \_[m]{} (-2 q) = (-1)\^m \_[m]{} (2 q)\
\[eq:singla1\] = & ( (-1)\^m + e\^[2i k q]{} ) P\_[m]{} ( k , 2q)
by Theorem \[theo:dev\_part\]. Recall that $P_{m} (k, q)$ depends smoothly on $q$ (even polynomially) and is polynomial in $k$ with degree $m$. So by Theorem \[theo:laginbasis\], the restriction of $(\xi_{m,k})$ to $U_2$ is a Lagrangian state supported by $A_2 \setminus A_1 $.
To prove the result on $U_1$, introduce the basis of $\mathcal{H}_k$ $$\Phi_{\ell} = \frac{e^{i\pi/4}}{\sqrt{2k}} \sum_{n \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}}} e^{i \pi \ell n/k} \Psi_n , \qquad \ell \in {{\mathbb{Z}}}/2k {{\mathbb{Z}}}$$ We check without difficulty that $$T^*_{\mu/2k} \Phi_{\ell} = \Phi_{\ell+1}, \qquad T^*_{{\lambda}/2k} \Phi_{\ell} = e^{-i \pi \ell/k} \Phi_{\ell+1}.$$ Furthermore, the normalization with the factor $e^{i \pi/4}$ has been chosen so that $\Phi_0 ( 0 ) = {\Omega}_{-{\lambda}}$, where ${\Omega}_{-{\lambda}} \in \delta $ is such that ${\Omega}_{-{\lambda}}^2 ( -{\lambda}) = 1$, cf. Theorem 2.3 of [@LJ1] for a proof of this formula. So $( \Phi_{\ell})$ is the basis associated to $( -{\lambda}, \mu)$. Furthermore, it follows from the definition of $\Xi_{m,k} $ that
[2]{} \_[m,k]{} = & ( i)\^[-m]{} \_[ n , / 2k ]{} \^[-m]{} e\^[i n /k ]{} \_[n]{}\
\[eq:xi\_phi\] = & e\^[-i /4]{} ( i )\^[-m]{} \_[ / 2k ]{} \^[-m]{} \_
where by convention $(0)^{-m} = 0$. So by Theorem \[theo:laginbasis\], the restriction of $(\xi_{m,k})$ to $U_1$ is a Lagrangian state supported by $A_1 \setminus A_2 $.
Asymptotic behavior of $Z_k ( {\Sigma}\times S^1)$ and $Z_k (S)$ {#sec:asympt-behav-z_k}
================================================================
Recall that we introduced in Section \[sec:witt-resh-tura\] a compact oriented surface ${\Sigma}$ with a connected boundary $C$. Let $$\begin{gathered}
\label{eq:def_E}
E = H_1 ( C \times S^1 , {{\mathbb{R}}}), \quad R = H_1 ( C \times S^1) , \qquad M = E/R.\end{gathered}$$ Let ${\omega}$ be the symplectic form of $E$ defined as $4\pi$ times the intersection product. Consider the quantum space $\mathcal{H}_k = H^0 (M, L_M^k ) \otimes \delta$ defined in Section \[sec:quantum-spaces\]. We denote by $\mathcal{H}_k^{{\operatorname}{alt}}$ the subspace of alternating sections.
Let $(\mu, {\lambda})$ be the basis of $R$ given by $\mu =[C]$, ${\lambda}= [S^1]$. Denote by $(e_\ell)$ and $(\Psi_\ell)$ the corresponding basis of $V_k ( C \times S^1)$ and $\mathcal{H}_k$ respectively introduced in Section \[sec:witt-resh-tura\] and Section \[sec:quantum-spaces\]. We identify $V_k ( {\Sigma}\times S^1)$ with $\mathcal{H}_k^{{\operatorname}{alt}}$ by sending $e_{\ell}$ into $2^{-1/2} ( \Psi_{\ell} - \Psi_{-\ell})$. As it was proved in Theorem 2.4 of [@LJ1], this identification depends on the choice of the basis $( \mu, {\lambda})$ only up to a multiplicative factor $\exp (i \pi ( \frac{n}{4} + \frac{n'}{2k}))$, $n $ and $n'$ being two integers independent of $k$.
The state $Z_k ( {\Sigma}\times S^1)$
-------------------------------------
The vector $Z_k ( {\Sigma}\times S^1)$ of $V_k ( C \times S^1)$ is given in the basis $( \Psi_\ell)$ by $$\begin{gathered}
\label{eq:Z_k1}
Z_k ( {\Sigma}\times S^1) = \frac{1}{\sqrt 2} \sum_{\ell =1 }^{ k-1} N^{g,k }_{\ell} \bigl( \Psi_{\ell } - \Psi_{-\ell} \bigr) \end{gathered}$$ Using Lemma \[lem:relat-with-count\] and the fact that $\Xi_{2g -1}$ is odd, we get $$\begin{gathered}
\label{eq:Z_k2}
Z_k ( {\Sigma}\times S^1) = \frac{C_g }{ \sqrt 2 } k^{g-1} \sum _{ \ell \in {{\mathbb{Z}}}/ 2k {{\mathbb{Z}}}} \Xi _{ 2g -1} \Bigl( \frac{\ell}{k} \Bigr) \Psi_{\ell} = \frac{C_g }{ \sqrt 2 } k^{g-1} \xi_{2g-1,k}\end{gathered}$$ where $\xi_{2g-1,k}$ is the vector introduced in Section \[sec:appl-funct-xi\_m\]. By Theorem \[theo:asympt-behav-xi\], $(\xi_{2g-1,k})$ is a Lagrangian state, so the same holds for $\bigl( Z_k ( {\Sigma}\times S^1) \bigr)$. Let us complete this result by computing the symbol. In the following statement, we use the sets $A_1$, $A_2$ introduced in (\[eq:defA12\]), their neighborhoods $U_1$, $U_2$ and the corresponding section $\Theta_A$.
\[theo:asympt-behav-z\_k\] The restriction of $( Z_k ( {\Sigma}\times S^1) , \; k \in {{\mathbb{N}}}^*) $ to $U_1$ is a Lagrangian state with associated data $(A_1 \setminus A_2, \Theta_A, {\sigma}_1, g ) $ where $${\sigma}_1 (p \mu ) \equiv i \sqrt{2} \pi ^g \bigl[ \sin ( 2 \pi p ) \bigr]^{-2g+1} (dp)^{1/2}, \qquad \forall p \in {{\mathbb{R}}}.$$ The restriction of $( Z_k ( {\Sigma}\times S^1) , \; k \in {{\mathbb{N}}}^*) $ to $U_2$ is a Lagrangian state with associated data $(A_2 \setminus A_1, \Theta_A, {\sigma}_2, 3g - 3/2) $ where ${\sigma}_2$ is given in terms of the function $P_{g,0}$ introduced in Theorem \[theo:counting\_smoot\] by $${\sigma}_2 (q {\lambda}) \equiv e^{i \pi/4} (\tfrac{\pi}{2})^{1/2} \; P_{g,0} ( 2q) \; ( dq )^{1/2} \ , \qquad \forall q \in (0,1)$$ and ${\sigma}_2( q {\lambda}+ \tfrac{1}{2} \mu ) = - {\sigma}_2 ( q {\lambda})$.
Introduce the same basis $( \Phi_\ell)$ as in the proof of Theorem \[theo:asympt-behav-xi\]. Denoting by $\eta_k ( \ell)$ the coefficient of $\Phi_{-\ell}$ in $Z_k ( {\Sigma}\times S^1)$, we have by Equations (\[eq:Z\_k2\]) and (\[eq:xi\_phi\]) that
[2]{} \_k (2 k p ) = & k\^[g-1]{} e\^[i /4]{} (-1)\^[g]{} \^[-2g+1]{}\
= & k\^[g-1/2]{} \^[-2g+1]{}
because $C_g = (-1)^{g+1} 2^{-g}$. To conclude the computation of ${\sigma}_1$, we use that ${\omega}( \cdot, - {\lambda}) /i = 4 i \pi dp $ and equation (\[eq:symb\_laginbasis\]).
Let us compute the symbol ${\sigma}_2$. By Theorem \[theo:asympt-behav-xi\] and Equation (\[eq:xi\_phi\]), the coefficients $\zeta_{k} ( \ell)$ of $\Psi_{-\ell}$ in $Z_{k} ( {\Sigma}\times S^1)$ satisfy $$\zeta_k ( 2k q ) = \Bigl( \frac{k}{2 \pi} \Bigr)^{ 3g -2 } \bigl( 1 - e^{2i k \pi q} \bigr) f (q) + {\mathcal{O}}( k^{3g -3})$$ with $f$ a smooth function on $]0,1[$. By Equation (\[eq:symb\_laginbasis\]), we have for any $q \in (0,1)$, $${\sigma}_2( q {\lambda}) = f ( q) \sqrt{ 4 i \pi dq }, \qquad {\sigma}_2( q {\lambda}+ \tfrac{1}{2} \mu ) = - {\sigma}_2( q {\lambda}).$$ On the one hand, by Equation (\[eq:Z\_k1\]), $$\zeta_k ( 2k q) = - 2^{-1/2} N_{2kq}^{g,k}.$$ On the other hand, $1- e^{2i k \pi q} =2$ for odd $2k q$. So we conclude from Theorem \[theo:counting\_smoot\] that $2 f( q) = - \frac{1}{\sqrt 2} P_{g,0} ( 2q) $.
The state $\rho_k ( \varphi ) (Z_k (D \times S^1)) $ {#sec:remplissage}
----------------------------------------------------
Recall that $\varphi $ is a diffeomorphism from $\partial D \times S^1$ to $C \times S^1$. So the homology class $\nu $ of $\varphi ( \partial D )$ is a primitive vector of $R$, that is $\nu = a \mu + b {\lambda}$ where $a,b$ are coprime integers. There is no restriction to assume that $b$ is non negative. Introduce the subset of $M$ $$\begin{gathered}
\label{eq:def_B}
B:= \{ [r \nu] \in M ; \; r \in {{\mathbb{R}}}\} .\end{gathered}$$ Observe that $B$ is a circle and there is a unique flat section $\Theta_B$ of $L \rightarrow B$ such that $\Theta_B ( 0) = 1$. The following result is Theorem 3.3 of [@LJ1].
\[theo:state-tore\_solide\] The family $\bigl( \rho_k ( \varphi ) (Z_k (D \times S^1)) $, $k \in {{\mathbb{N}}}^*$) is a Lagrangian state with associated data $(B, \Theta_B, {\sigma}_B, 0)$ where $${\sigma}_B ( r \nu ) = \sqrt 2 \sin ( 2 \pi r) {\Omega}_\nu$$ with ${\Omega}_\nu \in \delta$ such that ${\Omega}_\nu^2 ( \nu ) = 1$.
For any non vanishing $c$, denote by $I_c$ the interval $$I_c = ]-\tfrac{1}{2|c|}, \tfrac{1}{2|c|} [.$$ Assume that $a$ and $b$ do not vanish. Consider the open set $U = \bigl\{ [ p \mu + q {\lambda}] ; \; p \in I_b, \; q \in I_a \bigr\}$ of $M$. Observe that $$B \cap U = \bigl\{ \bigl[ \tfrac{a}{b} q \mu + q {\lambda}\bigr]; \; q \in I_a\}.$$ Introduce a function $f_U \in {\mathcal{C}^{\infty}}(M )$ with support contained in $U$, which is identically equal to $1$ on a neighborhood of the origin and such that $f_U(-x) = f_U(x)$. Let $Z( \ell)$ be the coefficients $$\begin{gathered}
\label{eq:4}
Z( \ell ) = \bigl\langle f_U \rho_k ( \varphi ) (Z_k (D \times S^1)), e_{\ell} \bigr\rangle, \qquad \ell =1 , \ldots, k-1 \end{gathered}$$ We deduce from Theorem \[theo:laginbasis\] and Theorem \[theo:state-tore\_solide\] the following
\[lem:state-remplissage-e\] We have for any $q \in (0, \tfrac{1}{2} ) \cap \frac{1}{2k } {{\mathbb{Z}}}$, $$Z( 2kq) = \Bigl( \frac{2 \pi}{k}\Bigr)^{1/2} e^{2i \pi k \frac{a}{b} q^2} \sum_{\ell = 0 }^{\infty} k^{-\ell } f_\ell (q)+ {\mathcal{O}}( k^{-\infty})$$ where the $f_\ell$ are smooth odd functions on ${{\mathbb{R}}}$ with support contained in $I_a$. Furthermore $f_0 ( q) = e^{-i \pi/4} \sin ( 2 \pi q /b) / \sqrt{ \pi b}$ on a neighborhood of $0$.
Asymptotics of $Z_k ( S)$
-------------------------
Let us assume that $a \neq 0$ and $b \neq 0$. Under this assumption the intersection of $B$ with $A= A_1 \cup A_2 $ is finite. As we will see, each point of $A \cap B$ contributes in the asymptotic expansion of $Z_k (S)$. Actually, since we work with alternating sections, the relevant set is the quotient $X$ of $A \cap B $ by the involution $-{\operatorname}{id}_M$ $$\begin{gathered}
\label{eq:def_N}
-{\operatorname}{id}_M : M \rightarrow M, \qquad [p \mu + q {\lambda}] \rightarrow [- p \mu - q {\lambda}]. \end{gathered}$$ Let us denote by $N$ the quotient of $M$ by $-{\operatorname}{id}_M$ so that $X$ is a subset of $N$. Introduce the functions ${\alpha}$, ${\beta}$ from $N$ to $[0, \pi]$ satisfying $$\begin{gathered}
\label{eq:def_be}
\begin{split}
{\alpha}( [ p \mu + q {\lambda}] ) = \arccos ( \cos ( 2 \pi p ) ), \\
{\beta}( [ p \mu + q {\lambda}] ) = \arccos ( \cos ( 2 \pi q ) ) .
\end{split}\end{gathered}$$ Here it may be worth to observe that $[0,1/2]$ is a fundamental domain for the action of ${{\mathbb{Z}}}\rtimes {{\mathbb{Z}}}_2$ on ${{\mathbb{R}}}$, where ${{\mathbb{Z}}}$ acts by translation and $-1 \in {{\mathbb{Z}}}_2$ by $- {\operatorname}{id}_{{{\mathbb{R}}}}$. The function $\frac{1}{2\pi} \arccos ( \cos ( 2\pi x))$ induces a section from ${{\mathbb{R}}}/ {{\mathbb{Z}}}\rtimes {{\mathbb{Z}}}_2$ to $[0,1/2]$. So if $x = [p \mu + q {\lambda}]$, then ${\alpha}(x) /2\pi \equiv p$ and ${\beta}(x) /2 \pi \equiv q$ modulo ${{\mathbb{Z}}}\rtimes {{\mathbb{Z}}}_2$
The quotient $N$ is an orbifold with four singular points $$p_1 = [ 0], \quad p_2 = [\mu /2], \quad p_3 = [ {\lambda}/2], \quad p_4 = [ \mu/2 + {\lambda}/2 ].$$ corresponding to the fixed points of $-{\operatorname}{{\operatorname{id}}}_M$. All these points belong to $A_2$ and the first two belong to $A_1$ too, actually $A_1 \cap A_2 = \{ p_1 , p_2 \}$. Since these points play a particular role in the asymptotic expansion of $Z_k (S)$, we divide $X$ into four sets $X_1 = ( A_1 \setminus \{ p_1, p_2 \} ) \cap B$, $X_2 = (A_2 \setminus \{ p_1, p_2, p_3 , p_4 \} ) \cap B$, $ X_3 = \{ p_1, p_2 \} \cap B$ and $ X_4 = \{ p_3, p_4 \} \cap B$.
The sets $X_1$ and $X_2$ consist respectively of ${\operatorname}{E} \bigl( \frac{b-1}{2} \bigr)$ and $|a|-1$ points. $X_3 = \{ p_1 , p_2 \}$ if $b$ is even and $\{ p_1 \}$ otherwise. $X_4 = \{ p_3 \}$ if $a$ is even, $\{ p_4 \}$ if $a$ and $b$ are odd, empty if $b$ is even.
Observe that for any $x \in E$, there exists a unique $r \in [0,1/2]$ such that $x = [r ( a \mu + b {\lambda})]$. Furthermore $x \in \{ p_1, p_2, p_3, p_4 \}$ if and only if $r =0$ or $1/2$, $x \in A_1$ if and only if $ r \in \frac{1}{b} \{ 0, 1, \ldots, {\operatorname}{E} ( b/2) \}$, $x \in A_2 $ if and only if $ r \in \frac{1}{2|a|} \{ 0,1, \ldots , |a|\}$. We conclude easily.
For any $x$ in $X$, introduce a function $f_x \in {\mathcal{C}^{\infty}}(N)$ which is identically equal to $1$ on a neighborhood of $x$. Assume furthermore that these functions have disjoint supports. Consider for any $x \in X$ the quantity $$I_x (k) = \bigl\langle f_x Z_k ( {\Sigma}\times S^1) , \rho_k ( \varphi) (Z_k (D \times S^1)) \bigr\rangle$$ where the bracket is the scalar product of ${\mathcal{C}^{\infty}}( M , L^k \otimes\delta_M )$. Introduce two integers $c$ and $d$ such that $ ac + bd = 1$.
We have for any $x \in X_1 \cup X_2 \cup X_4$ that
[2]{} I\_x (k) = & ( )\^[g - 1/2]{} \_A( x) , \_B (x) \^k \_[=0]{}\^ k\^[-]{} a\_(x) + ( k\^[-]{})
where the $a_\ell (x)$ are complex coefficients. If $x \in X_1$ $$n(x) = g - \tfrac{1}{2}, \qquad a_0 (x) \equiv \frac{2 \pi^{g - 1/2} }{ \sqrt{b}} \bigl[ \sin ( {\alpha}(x) ) \bigr] ^{-2g + 1} \sin \bigl( c {\alpha}(x) \bigr)$$ If $x \in X_2$ $$n(x) = 3g -2 , \qquad a_0 (x) \equiv \frac{1 }{ \sqrt{a}} P_{g,0} \Bigr( \frac{{\beta}(x) }{\pi} \Bigl) \sin \bigl( d {\beta}(x) \bigr)$$ with $P_{g, 0}$ is the function introduced in \[theo:counting\_smoot\]. If $x \in X_4$, $$n(x) = 3g -3 , \qquad a_0 (x) \equiv \frac{i P_{g,0}'(1) }{ 4 \pi a^{3/2}} .$$
This follows from Theorem \[theo:asympt-behav-z\_k\], Theorem \[theo:state-tore\_solide\] and Equation (\[eq:scalprodlag\]). To compute the leading coefficient with equation (\[eq:pairing\]), we write $$\frac{ dp ( \mu) \overline{ {\Omega}_{\nu} ^2 ( \nu) }} { {\omega}( \mu, \nu)} = \frac{1}{ 4 \pi b}, \qquad \frac{ dq ( \lambda ) \overline{ {\Omega}_{\nu} ^2 ( \nu) }} { {\omega}( {\lambda}, \nu)} = \frac{-1}{ 4 \pi a} .$$ Furthermore, to compute $\sin ( 2 \pi r)$, we use that for $x = r \nu$, $$r = rac + rbd \equiv \frac{{\alpha}(x)}{2\pi} c \pm \frac{{\beta}(x)}{2 \pi} d \mod {{\mathbb{Z}}}\rtimes {{\mathbb{Z}}}_2$$ If $x$ belongs to $X_1$, then ${\beta}( x) = 0$ which implies that $\sin ( 2 \pi r) \equiv \sin ( c {\alpha}(x))$ up to sign. If $x $ belongs to $X_2$, then ${\alpha}( x) = 0$ or $\pi$ so that $\sin ( 2 \pi r ) \equiv \sin ( {\beta}(x) d)$ up to sign.
To compute $I_x (k)$ with $x \in X_4$, we use formula (\[eq:subpairing\]).
To estimate $I_x (k)$ with $x \in X_3$, we need the followings results. Let ${\alpha}\in {{\mathbb{R}}}$ and $f \in {\mathcal{C}^{\infty}}_0 ( {{\mathbb{R}}}_{+}, {{\mathbb{C}}})$ with ${{\mathbb{R}}}_+ = [0, \infty)$. Introduce the sum $$S_k^+(f) = \tfrac{1}{2} f( 0 ) + \sum_{\ell =1 }^{\infty} e^{i \frac{{\alpha}}{2} \ell^2/k} f \Bigl( \frac{\ell}{k} \Bigr) .$$ $f$ being with compact support, the sum is finite.
\[theo:pd1\] Let ${\alpha}\in {{\mathbb{R}}}^*$ and $f \in {\mathcal{C}^{\infty}}_0 ({{\mathbb{R}}}_+, {{\mathbb{C}}})$ be such that its support is contained in $[0, \frac{2\pi}{|{\alpha}|} )$. If $f$ is even and $f(x) = {\lambda}x^{2n} + {\mathcal{O}}( x^{2n+1})$, then $$S_k ^+(f) = k^{\frac{1}{2} - n } \Bigl( \frac{\pi}{2|{\alpha}|} \Bigr)^{\frac{1}{2}} e^{ i \frac{\pi}{4} {\operatorname}{sgn} {\alpha}} \sum_{\ell= 0 } ^{\infty} k^{-\ell} c_\ell + {\mathcal{O}}( k^{-\infty}) \quad \text{ with } \quad c_0 = \Bigl( \frac{i}{2 {\alpha}} \Bigr)^n \frac{ ( 2n ) !}{ n!} {\lambda}$$ and $c_\ell \in {{\mathbb{C}}}$ for any positive integer $\ell$. If $f$ is odd and $f(x) = {\lambda}x^{2n+1} + {\mathcal{O}}( x^{2n+2})$, then $$S_k ^+(f) = k^{-n} \sum_{\ell= 0 } ^{\infty} k^{-\ell} c_\ell + {\mathcal{O}}( k^{-\infty}) \quad \text{ with } \quad c_0 = \Bigl( \frac{2i} {{\alpha}} \Bigr)^{n+1} \frac{ n !}{ 2} {\lambda}$$ and $c_\ell \in {{\mathbb{C}}}$ for any positive integer $\ell$.
Here we say that a function of ${\mathcal{C}^{\infty}}( {{\mathbb{R}}}_+)$ is even (resp. odd) if its Taylor expansion at the origin contains only even monomials (resp. odd monomials). Similarly, the sum $$S_k^{-}(f) = \tfrac{1}{2} f( 0 ) + \sum_{\ell =1 }^{\infty} (-1)^\ell e^{i \frac{{\alpha}}{2} \ell^2/k} f \Bigl( \frac{\ell}{k} \Bigr)$$ has the following asymptotic behavior.
\[theo:pd2\] Let ${\alpha}\in {{\mathbb{R}}}^*$ and $f \in {\mathcal{C}^{\infty}}_0 ({{\mathbb{R}}}_+, {{\mathbb{C}}})$ be such that its support is contained in $[0, \frac{\pi}{|{\alpha}|} )$. If $f$ is even, $S^{-} _k (f) = {\mathcal{O}}( k^{-\infty})$. If $f$ is odd and $f( x) = {\mathcal{O}}( x^{n})$, then $$S_k^{-}(f) = k^{-n} \sum_{\ell= 0 } ^{\infty} k^{-\ell} c_\ell + {\mathcal{O}}( k^{-\infty})$$ for some complex coefficients $c_\ell$.
These two theorems are proved in Section \[sec:sing-discr-stat\]. We deduce the following
For any $x \in X_3$, we have the asymptotic expansion
[2]{} I\_x (k) = & ( )\^[3g - 3]{} \_[=0 ]{} \^ a\_(x) k\^[-]{} + k\^[ 2g -3/2]{} \_[= 0 ]{}\^ b\_(x) k\^[-]{} + ( k\^[-]{})
with $a_\ell(x)$ and $b_\ell(x)$ complex coefficients, the leading ones being given by $$a_0 (x) = \frac{ P_{g,0}'(0) }{ 4 \pi a^{3/2}}, \qquad b_0 (x) = e^{i \pi/4} i^g b^{g - 3/2} a ^{-g} \pi^{-g +1} \frac{\sqrt 2 ( g-1)!}{ ( 2 ( g-1))!}$$
$I_{p_1} (k)$ is equal to the scalar product of $Z_k ( {\Sigma}\times S^1)$ with the vector $Z_k$ introduced in (\[eq:4\]) $$Z_k = \sum_{\ell =1 }^{k-1} \bigl\langle f_U \rho_k ( \varphi ) (Z_k (D \times S^1)), e_{\ell} \bigr\rangle e_\ell .$$ The asymptotic behavior of the coefficients of $Z_k$ is given in Proposition \[lem:state-remplissage-e\]. By Lemma \[lem:SiS\] and Theorem \[theo:counting\_smoot\], $Z_k ( {\Sigma}\times S^1)$ is the sum of four terms $Z_k^{+,+}$, $Z_k^{+,-}$, $Z_k^{-,+}$, $Z_k^{-,-}$ whose coefficient in the basis $( e_{\ell})$ are $$Z_k^{+, \pm} ( \ell ) = \frac{1}{2} \Bigl( \frac{k}{2\pi} \Bigr)^{ 3 g -2 } \sum _{m = 0 }^{g-1 } k^{-2m} P^{\pm}_{g, m} \Bigl( \frac{\ell}{k} \Bigr) , \quad Z_k^{-, \pm} ( \ell ) = (-1)^{\ell+1} Z_k^{+, \pm} ( \ell ) .$$ As a consequence of Theorem \[theo:pd2\], we have $$\begin{gathered}
\label{eq:5}
\bigl\langle Z_k , Z_k ^{-, -} \bigr\rangle = {\mathcal{O}}( k^{- \infty}), \qquad \bigl\langle Z_k , Z_k ^{-, +} \bigr\rangle = k^{g - 3/2} \sum_{\ell = 0 }^{\infty} k^{-\ell} c_\ell \end{gathered}$$ for some coefficient $c_{\ell}$. To prove the second formula of Equation (\[eq:5\]), we have to take into account that $$\begin{gathered}
\label{eq:2}
P^{+}_{ g,m} (x) = {\mathcal{O}}( x^{ 2 ( g-m -1)} ).\end{gathered}$$ By Theorem \[theo:pd1\], we have $$\bigl\langle Z_k , Z_k ^{+, -} \bigr\rangle = \Bigl( \frac{ k } { 2 \pi} \Bigr)^{3g - 3} \sum_{\ell =0 } ^{\infty} a_\ell(x) k^{-\ell} , \qquad \bigl\langle Z_k , Z_k ^{+, +} \bigr\rangle = k^{ 2g -3/2} \sum_{\ell = 0 }^{\infty} b_\ell(x) k^{-\ell}$$ where $a_0$ and $b_0$ are given by the formula in the statement. To compute $b_0$, we use the expression for $P_{g,0}^{+}$ given in Theorem \[theo:counting\_smoot\]. Furthermore, Equation (\[eq:2\]) implies that the polynomials $P_{g,m}$ with $m \geqslant 1$ do not enter in the computation. Since $ 2g - 3/2 > g - 3/2$, $\bigl\langle Z_k , Z_k ^{-, +} \bigr\rangle$ does not contribute to the leading order terms. This concludes the proof for $x= p_1$.
Assume that $b$ is even. Then $X_3$ consists of $p_1$ and $p_2$ and by a symmetry argument, we see that the computation of $I_{p_3} (k)$ is the same as the one of $I_{p_1} (k)$. Indeed, we have that $(T_{\mu/2}^{*} + {\operatorname}{id} ) Z_k ({\Sigma}\times S^1)=0$. Furthermore, by Theorem \[theo:state-tore\_solide\], $T^*_{\nu/2} \rho_k( \varphi )Z_k ( D \times S^1) =0 $ is a Lagrangian state with associated data $(B, \Theta_B, -{\sigma}_B, 0)$. Since $b$ is even, $\mu/2 = \nu/2$. Clearly, $ p_1 + \mu/2 = p_2$, which concludes the proof.
Singular discrete stationary phase {#sec:sing-discr-stat}
==================================
In this section, we prove Theorem \[theo:pd1\] and Theorem \[theo:pd2\]. Let ${\alpha}, {\beta}$ be two real numbers. Assume that ${\alpha}\neq 0$. Denote by ${{\mathbb{R}}}_{+}$ the set of non negative real numbers. For any function ${\sigma}\in {\mathcal{C}^{\infty}}_0 ( {{\mathbb{R}}}^{+})$ and positive $\tau$, introduce the sum $$S _{\tau}( {\sigma}) = \frac{{\sigma}(0)}{2} + \sum_{\ell =1 }^{\infty} e^{i ( \frac{{\alpha}}{2} \frac{ \ell^2}{ \tau} - {\beta}\ell)} {\sigma}\Bigl( \frac{\ell}{ \tau} \Bigr)$$ In this appendix we study the asymptotics of $S_{\tau} ( {\sigma})$ as $\tau$ tends to infinity. Our treatment is partly inspired by the paper [@KeKn]. We will adapt the stationary phase method. The relevant variable is $x = \ell / \tau$. As we will see, the set of stationary points is $\frac{{\beta}}{{\alpha}} + 2 \pi {{\mathbb{Z}}}$. The origin also contributes non trivially to the asymptotic because the sum starts at $\ell =0$. In Theorem \[theo:pd1\], we are in the most delicate situation, because ${\beta}=0$, and the origin is both a stationary point and an endpoint of the summation interval. Let us start with the easiest case where the support of ${\sigma}$ does not contain any stationary point.
\[theo:app1\] For any ${\sigma}\in {\mathcal{C}^{\infty}}_0( {{\mathbb{R}}}_+)$ such that ${\operatorname}{Supp} {\sigma}\cap \bigl( \frac{{\beta}}{{\alpha}} + 2 \pi {{\mathbb{Z}}}\bigr) = \emptyset$ and ${\sigma}(x) = {\mathcal{O}}( x^n) $ at the origin, we have the following asymptotic expansion $$S( {\sigma}) = k^{-n} \sum_{\ell =0 }^{ \infty} k^{-\ell} c_{\ell}$$ for some complex numbers $c_{\ell}$.
For the proof, we will have to consider more general sums of the form $S_\tau ( \rho ( \cdot, \tau))$ where $\rho ( \cdot, \tau)$, is a family of functions in ${\mathcal{C}^{\infty}}_0 ( {{\mathbb{R}}}_{+})$ whose supports are contained in a fixed compact subset of ${{\mathbb{R}}}_{+}$ and which admits a complete asymptotic expansion in inverse power of $\tau$, $\rho ( \cdot, \tau) = \rho_0 + \tau ^{-1} \rho_1 + \tau^{-2} \rho_2 + \ldots $, for the ${\mathcal{C}^{\infty}}$ topology. We call such a family $(\rho ( \cdot, \tau))$ a [*symbol*]{}. In particular, for any function $f \in {\mathcal{C}^{\infty}}({{\mathbb{R}}})$, we will denote by $f ( \frac{1}{\tau} \frac{\partial}{\partial x}) {\sigma}$ any symbol with the expansion $$f ( 0 ) {\sigma}+ \tau^{-1} f'(0) {\sigma}' + \tau^{-2} f^{(2)} (0 ) {\sigma}^{(2)} + \ldots$$ We will also use the notation $D = \frac{1}{\tau} \frac{\partial}{\partial x}$.
The sum $ \tilde{S}_\tau ( {\sigma}) = \sum_{\ell =0 }^{\infty} e^{i ( \frac{{\alpha}}{2} \frac{ \ell^2}{ \tau} - {\beta}\ell)} {\sigma}\bigl( \frac{\ell}{ \tau} \bigr) $ satisfies the relation $$\begin{gathered}
\label{eq:1}
\tilde{S}_{\tau} ( \sigma ( \delta -1 ) ) + \tilde{ S}_{\tau} ( \delta ( e ^D - 1 ) {\sigma}\bigr) + {\sigma}( 0) = 0 \end{gathered}$$ where $\delta$ is the symbol $ \delta ( x, \tau) = e^{ i( {\alpha}x - {\beta}) + i {\alpha}/2 \tau }$. To prove this, we apply the summation by part formula $$\sum_{\ell = 0 } ^{n} f_{\ell} ( g_{\ell+1} - g_{\ell} ) + \sum_{\ell=0 }^{n} g_{\ell+1} ( f_{\ell + 1} - f_{\ell} ) = f_{n+1} g_{n+1} - f_0 g_0$$ to the sequences $f_{\ell} = {\sigma}\bigl( \frac{\ell}{\tau} \bigr)$ and $g_{\ell} = \exp \bigl( i \frac{{\alpha}}{2} \frac{\ell^2}{\tau} - i {\beta}\ell \bigr).$ Observe that $$f_{\ell +1} = {\sigma}\Bigl( \frac{\ell}{\tau} \Bigr) + \tau^{-1} {\sigma}' \Bigl( \frac{\ell}{\tau} \Bigr) + \tfrac{1}{2} \tau^{-2} {\sigma}'' \Bigl( \frac{\ell}{\tau} \Bigr) + \ldots = \bigl( e^{D} {\sigma}\bigr) \Bigl( \frac{\ell}{\tau} \Bigr) + {\mathcal{O}}( \tau^{-\infty})$$ so that $f_{\ell+1} - f_{\ell} = \bigl( ( e ^D - 1 ) {\sigma}\bigr) ( \ell / \tau)$. Furthermore $g_{\ell +1} = g_{\ell} \delta( \ell/ \tau , \tau) $ and Equation (\[eq:1\]) follows.
We have $$\delta (x,\tau) - 1= e^{ \frac{i}{2} ( {\alpha}x - {\beta}) } \sin \bigl( \tfrac{1}{2} ( {\alpha}x - {\beta}) \bigr) + {\mathcal{O}}( \tau^{-1}).$$ Observe that the zero set of $ \sin \bigl( \frac{1}{2} ( {\alpha}x - {\beta}) \bigr)$ is $ \frac{{\beta}}{{\alpha}} + 2 \pi {{\mathbb{Z}}}$. So if the support of ${\sigma}$ does not intersect this set, we can write ${\sigma}= {\gamma}(\delta -1) $ for some symbol ${\gamma}$. Let us apply (\[eq:1\]) to ${\gamma}$, we obtain $$\tilde{S}_{\tau} ( {\sigma}) = - {\sigma}(0) ( \delta (0) -1) + \tau^{-1} \tilde{S}_{\tau} ( {\sigma}_1)$$ where ${\sigma}_1$ is the symbol $\tau ( \delta ( e^{D} -1) {\gamma}$. Since the support of ${\sigma}_1$ is smaller than the support of ${\sigma}$, we can do the same computation with $\tilde{S}_{\tau} ( {\sigma}_1)$. In this way, we prove that $S_{\tau} ( {\sigma})$ has a complete asymptotic expansion in power of $\tau^{-1}$. With a careful inspection of this computation, we also get that $S_{\tau} ( {\sigma}) = {\mathcal{O}}( k^{-n})$ if ${\sigma}$ vanishes to order $k$ at the origin.
Choosing ${\beta}= \pi$ in the last result, we obtain Theorem \[theo:pd2\]. For the proof of Theorem \[theo:pd1\], we will use the following relation, which has the advantage to be more symmetric that Equation (\[eq:1\]). In the remainder of the appendix, we assume that ${\beta}=0$.
\[lem:sum\_part\] For any ${\sigma}\in {\mathcal{C}^{\infty}}_0( {{\mathbb{R}}}_+)$, we have $$S_{\tau} \bigl( \sin ( {\alpha}\cdot ) {\sigma}\bigr) = \tfrac{i}{2} {\sigma}(0)+ i e^{- i {\alpha}/ (2 \tau)} \Bigl( S_{\tau} \bigl( \sinh (D) {\sigma}\bigr) + \tfrac{1}{2} ( \cosh (D) {\sigma}) (0) \Bigr)$$ up to a ${\mathcal{O}}( \tau^{-\infty})$.
We will use the following summation by part formula $$\begin{gathered}
\tfrac{1}{2} f_0 \delta_0 (g) + \sum_{\ell =1 }^{n-1}f_\ell \delta_{\ell} (g) + \tfrac{1}{2} f_n \delta_n (g) + \tfrac{1}{2} g_0 \delta_0 (f) + \sum_{\ell =1 }^{n-1}g_\ell \delta_{\ell} (f) + \tfrac{1}{2} g_n \delta_n (f) + \\
\tfrac{1}{2} g_0 ( f_1 + f_{-1}) + \tfrac{1}{2} f_0 ( g_1 + g_{-1}) - \tfrac{1}{2} f_{n} ( g_{n-1} + g_{n+1} ) - \tfrac{1}{2} g_n ( f_{n-1} + f_{n+1}) = 0 \end{gathered}$$ to the same sequences $f_{\ell}$ and $g_{\ell}$ that we used in the proof of Theorem \[theo:app1\]. We have that $$\delta_{\ell} (g) = 2i g_{\ell} \sin \bigl( {\alpha}\ell / \tau \bigr) \exp ( i {\alpha}/ 2 \tau ) , \qquad \tfrac{1}{2} ( g_1 + g_{-1} ) = \exp ( i {\alpha}/ 2 \tau ).$$ Furthermore $$\tfrac{1}{2} \delta_{\ell} f \equiv \bigl( \sinh (D) {\sigma}\bigr) \Bigl( \frac{\ell}{\tau} \Bigr), \qquad \tfrac{1}{2} ( f_{1} + f_{-1} ) \equiv \bigl( \cosh (D) {\sigma}\bigr) ( 0 )$$ up to a ${\mathcal{O}}( \tau^{-\infty})$. Applying these expressions in the summation by part formula with $n$ sufficiently large, we get $$2 i e^{i \frac{{\alpha}}{2\tau}} S_{\tau} ( \sin ( {\alpha}\cdot ) {\sigma}) + 2 S_{\tau} ( \sinh (D) {\sigma}) + \bigl( \cosh (D) {\sigma}\bigr) ( 0) +
{\sigma}(0) e^{i \frac{{\alpha}}{2 \tau}} \equiv 0$$ up to a ${\mathcal{O}}( \tau^{-\infty})$, which was the result to proved.
\[lem:leading\_term\] Let $\rho \in {\mathcal{C}^{\infty}}( {{\mathbb{R}}}_+)$ with support contained in $[ 0, \tfrac{2\pi}{ |{\alpha}|} )$ and such that $\rho \equiv 1$ on a neighborhood of $0$. Then $$\frac{2}{\tau} S_{\tau} ( \rho ) = \Bigl( \frac{2 \pi } { \tau} \Bigr) ^{ 1/2} \frac{ e^{i \frac{\pi}{4} {\operatorname}{sgn} {\alpha}}}{| {\alpha}|^{1/2}} + {\mathcal{O}}( \tau ^{-\infty}).$$
We extend $\rho$ to a smooth even function on ${{\mathbb{R}}}$. Then $$2 S_{\tau} ( \rho) = \sum_{\ell= - \infty}^{ + \infty} e^{i \frac{{\alpha}}{2} \frac{ \ell^2}{ \tau}} \rho \Bigl( \frac{\ell}{ \tau} \Bigr)$$ By Poisson formula, $$2 S_{\tau} ( \rho ) = \tau \sum_{\ell = - \infty}^{\infty} I_{\ell}, \qquad \text{ with } \quad I_{\ell} = \int_{{{\mathbb{R}}}} e^{ i \tau ( \frac{{\alpha}}{2} x^2 - 2 \pi x \ell ) } \rho (x) dx .$$ We can estimate each $I_\ell$ by stationary phase method. For $\ell \neq 0$, the phase $\frac{{\alpha}}{2} x^2 - 2 \pi x \ell$ has a unique critical point $2 \pi \ell / {\alpha}$. This point not belonging to the support of $\rho$, $I_\ell ={\mathcal{O}}( \tau^{-\infty})$. We can actually prove the stronger result that $$\sum _{\ell \neq 0} I_{\ell} = {\mathcal{O}}( \tau^{-\infty} \bigr).$$ Estimating $I_0$ we get the final result.
\[theo:sing-discr-stat\] Let ${\sigma}\in {\mathcal{C}^{\infty}}_0 ( {{\mathbb{R}}}_+)$ with support contained in $[0, \tfrac{2 \pi}{ |{\alpha}|})$. Then $$S_{\tau} ({\sigma}) = \tau^{1/2} \sum_{\ell = 0 } ^{\infty} a_{\ell} \tau^{-\ell} + \sum_{\ell =0 }^{\infty} b_{\ell} \tau^{-\ell}$$ where the leading coefficients are $$a_0 = \Bigl( \frac{\pi}{2} \Bigr)^{1/2} \frac{ e^{i \frac{\pi}{4} {\operatorname}{sgn} {\alpha}}}{| {\alpha}|^{1/2}} {\sigma}(0) , \qquad b_0 = i \frac{{\sigma}' (0)}{{\alpha}} .$$
By Theorem \[theo:app1\], we can assume that the support of ${\sigma}$ is contained in $[ 0, \frac{\pi}{|{\alpha}|} )$. Let $\rho$ be a function satisfying the assumption of Lemma \[lem:leading\_term\]. Write $${\sigma}(x) = {\sigma}(0)\rho(x) - i \sin ({\alpha}x) {\sigma}_1 ( x )$$ where ${\sigma}_1 $ is in ${\mathcal{C}^{\infty}}_0 ( {{\mathbb{R}}}_+)$ with support in $[0, \tfrac{2\pi}{|{\alpha}|})$. We have by Lemma \[lem:sum\_part\] $$S_{\tau} ( {\sigma}) = {\sigma}(0) S_{\tau} ( \rho) + \tfrac{1}{2} {\sigma}_1 (0) + e^{- i \frac{{\alpha}}{ 2 \tau}} \Bigl( S_{\tau} \bigl( \sinh (D) {\sigma}_1 \bigr) + \tfrac{1}{2} ( \cosh (D) {\sigma}_1 ) (0) \Bigr)$$ By lemma \[lem:leading\_term\], ${\sigma}(0) S_{\tau} ( \rho) = \tau^{1/2} a_0$ where $a_0$ is defined as in the statement. Furthermore $\tfrac{1}{2} {\sigma}_1 (0) + \tfrac{1}{2} ( \cosh (D) {\sigma}_1 ) (0) = {\sigma}_1 (0) + {\mathcal{O}}( \tau^{-1})$. We also have ${\sigma}_1 (0) = b_0$. So we obtain $$S_{\tau} ( {\sigma}) =\tau^{1/2} a_0 + b_0 + \tau^{-1} R_\tau$$ where $R_{\tau}$ is given by $$R_\tau= e^{- i {\alpha}/ (2 \tau)} \Bigl( S_{\tau} \bigl( \tau \sinh (D) {\sigma}_1 \bigr) + \tfrac{\tau}{2} \bigl( (\cosh (D) {\sigma}_1 ) (0) - {\sigma}_1 (0) \bigr) \Bigr)$$ Observe that $\tau \sinh (D) {\sigma}_1$ is a symbol, we can apply the same argument to $S_\tau ( \tau \sinh (D) {\sigma}_1 )$. We prove in this way the result by successive approximations.
\[theo:sing-discr-stat++\] Let ${\sigma}\in {\mathcal{C}^{\infty}}_0 ({{\mathbb{R}}}_+, {{\mathbb{C}}})$ with support contained in $[0, \frac{2\pi}{|{\alpha}|} )$. If ${\sigma}$ is even and ${\sigma}(x) = {\lambda}x^{2n} + {\mathcal{O}}( x^{2n+1})$ at the origin, then $$S_\tau (f) = \tau^{\frac{1}{2} - n } \Bigl( \frac{i \pi}{2{\alpha}} \Bigr)^{\frac{1}{2}} \sum_{\ell= 0 } ^{\infty} \tau^{-\ell} c_\ell + {\mathcal{O}}( \tau^{-\infty}) \quad \text{ with } \quad c_0 = \Bigl( \frac{i}{2 {\alpha}} \Bigr)^n \frac{ ( 2n ) !}{ n!} {\lambda}.$$ If ${\sigma}$ is odd and ${\sigma}(x) = {\lambda}x^{2n+1} + {\mathcal{O}}( x^{2n+2})$, then $$S_\tau (f) = \tau^{-n} \sum_{\ell= 0 } ^{\infty} \tau^{-\ell} c_\ell + {\mathcal{O}}( \tau^{-\infty}) \quad \text{ with } \quad c_0 = \Bigl( \frac{2i} {{\alpha}} \Bigr)^{n+1} \frac{ n !}{ 2} {\lambda}.$$
First, by adapting the proof of Theorem \[theo:sing-discr-stat\], we show that if ${\sigma}$ is even, the coefficients $b_\ell$ vanish, whereas if ${\sigma}$ is odd, the coefficient $a_\ell$ vanish. For instance, if ${\sigma}$ is even, ${\sigma}_1$ is odd, so that ${\sigma}_1 ( 0) = 0$ and $\sinh (D) {\sigma}_1$ is even. We conclude by iterating.
To compute the leading coefficients, we use the filtration ${\mathcal{O}}( m)$, $m\in {{\mathbb{N}}}$ of the space of symbols defined as follows: $$f \in {\mathcal{O}}(m) \Leftrightarrow f = \sum_{0 \leqslant \ell \leqslant m/2} \tau^{-\ell} g_{\ell} + {\mathcal{O}}( \tau^{-m/2} )$$ where for any $\ell$, the coefficient $g_{\ell} \in {\mathcal{C}^{\infty}}( {{\mathbb{R}}}_+)$ vanishes to order $m -2\ell$ at the origin. Observe that if $f \in {\mathcal{O}}( m+1)$ then $f ( 0 ) = {\mathcal{O}}( \tau ^{- ( m+1) / 2} )$ and $Df \in {\mathcal{O}}( m+1)$.
Assume that ${\sigma}\in {\mathcal{O}}(m)$ and that we want to compute $S_{\tau} ({\sigma})$ up to a ${\mathcal{O}}(\tau ^{- m/2})$. We consider again the proof of Theorem \[theo:sing-discr-stat\]. Introduce the function $${\gamma}(x) = ({\sigma}(x) - {\sigma}(0) \rho (x) )/x .$$ Then $ {\sigma}_1 = \frac{i}{{\alpha}} {\gamma}+ {\mathcal{O}}( m+1)$ and $\sinh (D) {\sigma}_1 = \frac{i }{{\alpha}} D {\gamma}+ {\mathcal{O}}( m+1)$. From this, we deduce that $$S_{\tau} ( {\sigma}) = S_{\tau} ( \rho ) {\sigma}(0) + \tfrac{i}{{\alpha}} {\gamma}(0) + \tfrac{i }{{\alpha}} S_{\tau} \bigl( D {\gamma}+ {\mathcal{O}}( m+1) \bigr) + {\mathcal{O}}(\tau ^{- m/2})$$ To conclude the proof, we choose ${\sigma}= {\lambda}x^m \rho$ and apply this formula as many times as necessary.
This completes the proof of Theorem \[theo:pd1\].
Geometric interpretation of the leading coefficients {#sec:geom-interpr-lead}
====================================================
Symplectic volumes {#sec:symplectic-volumes}
------------------
For any $t \in [0,1]$, denote by $\mathcal{M} ({\Sigma}, t)$ the moduli space of flat ${\operatorname}{SU}(2)$-principal bundles whose holonomy $g$ of the boundary $C = \partial {\Sigma}$ satisfies $\frac{1}{2}{\operatorname}{tr} (g) = \cos ( 2 \pi t)$. Equivalently, $\mathcal{ M} ( {\Sigma}, t)$ is the space of conjugacy classes of group morphisms $\rho$ from $\pi_1 ( {\Sigma})$ to ${\operatorname}{SU}(2)$ such that for any loop ${\gamma}\in \pi_1({\Sigma})$ isotopic to $C$, $\frac{1}{2}{\operatorname}{ tr} ( \rho ( {\gamma})) = \cos ( 2 \pi t)$.
We say that a morphism $\rho$ from $\pi_1 ( {\Sigma})$ to ${\operatorname}{SU}(2)$ is irreducible if the corresponding representation of $\pi_1 ( {\Sigma})$ in ${{\mathbb{C}}}^2$ is irreducible. The subset ${\mathcal{M}}^{{\operatorname}{irr}} ( {\Sigma}, t)$ of ${\mathcal{M}}( {\Sigma}, t)$ consisting of conjugacy classes of irreducible morphisms is a smooth symplectic manifold. Using the usual presentation of $\pi_1 ( {\Sigma})$, one easily sees that any morphism $\rho : \pi_1 ( {\Sigma}) \rightarrow {\operatorname}{SU}(2)$ such that $\rho ( {\gamma}) \neq {\operatorname}{id} $ for ${\gamma}$ isotopic to $C$, is irreducible. Consequently ${\mathcal{M}}^{{\operatorname}{irr} } ( {\Sigma}, t ) = {\mathcal{M}}( {\Sigma}, t)$ for $t \in ( 0,1]$. The subset of ${\mathcal{M}}( {\Sigma}, 0)$ consisting of non irreducible representation is in bijection with ${\operatorname}{Mor} ( \pi_1 ( {\Sigma}), {{\mathbb{R}}}/ {{\mathbb{Z}}})$, the set of group morphisms from $\pi_1 ( {\Sigma})$ to ${{\mathbb{R}}}/Z$.
Furthermore, for $t \in (0,1)$, ${\mathcal{M}}( {\Sigma}, t)$ is $2 ( 3g -2 )$-dimensional, whereas ${\mathcal{M}}( {\Sigma}, 1) $ and ${\mathcal{M}}^{{\operatorname}{ irr}} ( {\Sigma}, 0)$ have dimension $2 ( 3g -3)$.
\[theo:RR\] For any $ k, \ell \in {{\mathbb{N}}}$ such that $ 0< \ell \leqslant k$ and $\ell$ is even, we have $$N^{g, k + 2 }_{\ell +1} = \int_{ {\mathcal{M}}( {\Sigma}, s)} e^{ \frac{k}{2 \pi} {\omega}_s } {\operatorname}{Todd}_s$$ where $s = \ell / k$, ${\omega}_s$ is the symplectic form of ${\mathcal{M}}( {\Sigma}, s)$ and ${\operatorname}{Todd}_s$ any representant of its Todd class
As a corollary, we can compute the polynomial function $P_{g,0}$ as a symplectic volume.
For any $s \in ( 0,1)$, we have $$P_{g,0} ( s) = \int_{{\mathcal{M}}( {\Sigma}, s)} \frac{ {\omega}_s ^{3g-2}}{ ( 3g -2)!} .$$
We can actually recover partially Theorem \[theo:counting\_smoot\] in this way. Introduce the space ${\mathcal{M}}( {\Sigma})$ of conjugacy classes of morphisms from $\pi_1( {\Sigma})$ to ${\operatorname}{SU}(2)$. Let $f : {\mathcal{M}}( {\Sigma}) \rightarrow {{\mathbb{R}}}$ be the function sending $\rho$ into $\frac{1}{\pi} \arccos ( {\operatorname}{tr }( \rho (C)))$. Then for each $s \in [0,1]$, ${\mathcal{M}}( {\Sigma}, s)$ is the fiber at $s$ of $f$. Furthermore $(0,1)$ is the set of regular values of $f$. So we can identify the ${\mathcal{M}}( {\Sigma}, s)$, $ s\in (0,1)$ to a fixed manifold $F$, by a diffeomorphism uniquely defined up to isotopy. In particular, the homology groups of ${\mathcal{M}}( {\Sigma}, s)$ are naturally identified with the ones of $F$.
In [@Je2], Jeffrey introduced an extended moduli space ${\mathcal{M}}^{ \mathfrak{t}} ( {\Sigma})$. This space is a $2(3g-1)$-dimensional $( {{\mathbb{R}}}/ {{\mathbb{Z}}})$-Hamiltonian space, such that for any $s \in ( 0,1)$, ${\mathcal{M}}( {\Sigma},s )$ is the symplectic reduction of ${\mathcal{M}}^{\mathfrak{t}} ( {\Sigma})$ at level $s$. We recover that the various ${\mathcal{M}}( {\Sigma}, s)$, $s \in (0,1)$ can be naturally identified with a fixed manifold $F$ up to isotopy.
Furthermore, by Duistermaat-Heckman Theorem [@DuHe], the cohomology class of ${\omega}_s$ is an affine function of $s$ with value in $H^2(F)$, that is $[{\omega}_s] = {\Omega}+ s c$, where ${\Omega}$ and $c$ are constant cohomology classes in $H^2(F)$. This implies in particular that $P_{g,0}$ is polynomial with degree $(3g -2)$.
We can explain in this way why the shifts of $\ell$ and $k$ we introduced are natural. Indeed it has been proved by Meinrenken-Woodward [@MeWo2] that the canonical class $c_1$ of ${\mathcal{M}}( {\Sigma}, s)$ is $-4 {\Omega}-2 c$. Using that ${\operatorname}{Todd} = \hat{A} e^{-\frac{1}{2} c_1} $ where $\hat{A}$ is the $A$-genus, we obtain the that for any $k, \ell \in {{\mathbb{N}}}$ such that $ 0< \ell < k$ and $\ell$ is odd, $$N^{g,k}_{\ell} = \int_{{\mathcal{M}}( {\Sigma}, s)} e^{ \frac{k}{2 \pi} {\omega}_s } \hat{A}, \qquad \text{ with } s = \ell / k .$$ Since the $\hat{A}$-genus belong to $\bigoplus_{\ell} H^{4\ell}(F) $, it follows that $$Q( k ,s) = \int_{{\mathcal{M}}( {\Sigma}, s)} e^{ \frac{k}{2 \pi} {\omega}_s } \hat{A}$$ is a linear combination of the monomial $k^{2m} s ^p$ with $0 \leqslant p \leqslant 2m$ and $0 \leqslant m \leqslant g-1$, which was already proved in Theorem \[theo:counting\_smoot\].
Character varieties
-------------------
For any topological space $V$, introduce the character variety $\mathcal{M} ( V) $ defined as the space of group morphisms from $\pi_1 (V)$ to ${\operatorname}{SU}(2)$ up to conjugation. If $W$ is a subspace of $V$, we have a natural map from $\mathcal{M} (V)$ to $\mathcal{M} (W)$, that we call the restriction map.
For the circle $C$, $\pi_1 (C)$ being cyclic, $\mathcal{M} (C)$ identifies with the set of conjugacy classes of ${\operatorname}{SU} (2)$. So $\mathcal{M} (C) \simeq [0,\pi]$ by the map sending the morphism $\rho$ to the number $\arccos ( \frac{1}{2} {\operatorname}{tr} \rho (C) )$. Similarly, ${\mathcal{M}}(S^1) \simeq [0,\pi]$.
For the two-dimensional torus $C \times S^1$, there is a natural bijection between $\mathcal{M}(C \times S^1)$ and the quotient of $H_1 (C \times S^1, {{\mathbb{R}}})$ by $H_1(C \times S^1) \rtimes {{\mathbb{Z}}}_2$ defined as follows. Identify $\pi_1 ( C \times S^1)$ with $H_1 ( C \times S^1)$ and denote by $\cdot$ the intersection product of $H_1( C \times S^1)$. Then to any $x \in H_1 ( C \times S^1, {{\mathbb{R}}})$ we associate the representation $\rho_x$ given by $$\begin{gathered}
\label{eq:defrhox}
\rho _x ( {\gamma}) = \exp ( (x \cdot {\gamma}) D), \qquad \forall {\gamma}\in H_1( C \times S^1)\end{gathered}$$ where $D \in {\operatorname}{SU}(2)$ is the diagonal matrix with entries $2i \pi$, $-2 i\pi$.
Recall that we denote by $M$ the quotient of $H_1 ( C \times S^1 , {{\mathbb{R}}})$ by $H_1 ( C \times S^1)$ and by $N$ the quotient of $M$ by $- {\operatorname}{id}_M$, cf. (\[eq:def\_E\]) and (\[eq:def\_N\]). So the map sending $x$ to $\rho_x$ induces a bijection between $N$ and $\mathcal{M} ( C \times S^1)$. Furthermore the restriction maps from $\mathcal{M} ( C \times S^1)$ to $\mathcal{M} ( C)$ and $\mathcal{M} ( S^1)$ identify respectively with the maps ${\beta}$ and ${\alpha}$ introduced in (\[eq:def\_be\]).
Recall that we introduced subsets $A_1$, $A_2$, $A= A_1 \cup A_2$ and $B$ of $M$. We denote by $\tilde{A}_1$, $\tilde{A}_2$, $\tilde{A}$ and $\tilde B$ their projections in $N$. So $\tilde{A}_1$ and $\tilde{A}_2$ consists respectively of the classes $[\rho] \in {\mathcal{M}}( C\times S^1)$ such that $\rho (C) = {\operatorname{id}}$ or $\rho ( S^1) = \pm {\operatorname{id}}$. In other words $\tilde{A}_1 = \beta^{-1} (0)$ and $\tilde{A}_2 = \alpha^{-1} ( \{ 0,\pi \} )$.
\[lem:mapf\] The image of the restriction map $f$ from ${\mathcal{M}}( \Sigma \times S^1)$ to ${\mathcal{M}}( C \times S^1)$ is $\tilde{A}$. For any $x \in \tilde{A}_1 \setminus \tilde{A}_2$, $f^{-1}(x)$ identifies with ${\operatorname}{Mor} ( \pi_1 ( {\Sigma}), {{\mathbb{R}}}/{{\mathbb{Z}}})$. For any $x \in \tilde{A}_2$, $f^{-1} (x)$ identifies with $\mathcal{M} ( {\Sigma}, {\beta}(x) /\pi)$.
Let $\mathbb{T}$ be the subgroup of ${\operatorname}{SU}(2)$ consisting of diagonal matrices. So $\mathbb{T} \simeq {{\mathbb{R}}}/ {{\mathbb{Z}}}$. We will use that for any $g \in \mathbb{T} \setminus \{ \pm {\operatorname}{id} \}$, the centralizer of $g$ in ${\operatorname}{SU}(2)$ is $\mathbb{T}$.
Let $\rho $ be a morphism from $\pi_1 ( {\Sigma}\times S^1) =\pi_1 ( {\Sigma}) \times \pi ( S^1)$ to ${\operatorname}{SU}(2)$. The restriction $\rho'$ of $\rho $ to ${\Sigma}$ commutes with $\rho (S^1)$. Conjugating $\rho$ if necessary, $ g = \rho ( S^1)$ belongs to $\mathbb {T}$. Consider the following two cases:
- If $g$ is not central, the image of $\rho'$ is contained in $\mathbb{T}$. This implies that $\rho ( C) = {\operatorname}{ {\operatorname{id}}}$ so that $f ( \rho ) \in \tilde{A}_1$. Conversely, any $g \in \mathbb{T} \setminus \{ \pm {\operatorname}{id} \}$ and $\rho ' \in {\operatorname}{Mor} ( \pi_1 ( {\Sigma}), \mathbb{T} )$ determines a unique $\rho \in \mathcal{M} ( C \times S^1)$.
- If $g$ is central, then $f( \rho) \in \tilde{A}_2$. Conversely, any $g = \pm {\operatorname{id}}$ and $\rho' \in \mathcal{M} ( {\Sigma})$ determine a unique $\rho \in \mathcal{M} ( C \times S^1)$.
To end the proof in the second case, we view $\mathcal{M} ( {\Sigma})$ as the union of the $\mathcal {M} ( {\Sigma}, t) $ where $t$ runs over $[0,1]$.
Recall that $S$ is the Seifert manifold obtained by gluing the solid torus $D \times S^1$ to $\Sigma \times S^1$ along the diffeomorphism $\varphi$ of $C \times S^1$. Furthermore, $X = \tilde{A} \cap \tilde{B}$.
The components of ${\mathcal{M}}(S)$ are in bijection with $X$. For any $x \in X$ the corresponding component is homeomorphic with $f^{-1} (x)$.
It follows from Van Kampen Theorem that $\pi_1 (S)$ is the quotient of $\pi_1 ( {\Sigma}\times S^1)$ by the subgroup generated by $\varphi (C) $. So the group morphisms from $\pi_1 (S) $ identify with the group morphisms from $\pi ( {\Sigma}\times S^1)$ sending $\varphi ( C)$ to the identity.
On the other hand, for any $x \in H_1 ( C \times S^1, {{\mathbb{R}}})$ the corresponding representation $\rho_x$ defined in (\[eq:defrhox\]) is trivial on $\varphi(C)$ if and only if $x$ belongs to the line generated by $ \nu = a \mu + b {\lambda}$. So $\tilde{B}$ consists of the conjugacy classes of representations which are trivial on $\varphi(C)$.
This implies that the restriction map from ${\mathcal{M}}(S)$ to ${\mathcal{M}}( {\Sigma}\times S^1)$ is injective, and its image is $f^{-1} (\tilde{B})$. The conclusion follows from Lemma \[lem:mapf\], taking into account that the fibers of $f$ are connected.
Chern-Simons invariant
----------------------
For any three-dimensional closed oriented manifold $V$ and $\rho \in {\mathcal{M}}(V)$ the Chern-Simons invariant of $\rho$ is defined by $$\begin{gathered}
\label{eq:defCS}
{\operatorname}{CS}( \rho) = \int _V \tfrac{2}{3} {\alpha}^3 + {\alpha}\wedge d{\alpha}\in {{\mathbb{R}}}/ 2 \pi {{\mathbb{Z}}}\end{gathered}$$ where ${\alpha}\in {\Omega}^1(V, \mathfrak{su} (2))$ is any connection form whose holonomy representation is $\rho$.
For any $ \rho \in {\mathcal{M}}( S)$, the Chern-Simons invariant of $\rho$ is given by $$e^{ i {\operatorname}{CS} ( \rho ) } = \bigl\langle \Theta_A (x) , \Theta_B (x) \rangle$$ where $x \in {\mathcal{M}}( C \times S^1)$ is the restriction of $\rho $ to $C \times S^1$.
The proof is based on the relative Chern-Simons invariants introduced in [@RaSiWe], cf. also [@Fr].
We can define a relative Chern-Simons invariant for compact oriented 3-manifold $V$ with boundary. To do this we define first a complex line bundle $L \rightarrow \mathcal{M} ( \partial V)$, called the Chern-Simons bundle. Then for any $\rho \in {\mathcal{M}}( V)$, $e^{i {\operatorname}{CS} (\rho)}$ is by definition a vector in $L_{r(\rho)}$ where $r$ is the restriction map from ${\mathcal{M}}( V)$ to ${\mathcal{M}}( \partial V)$. This invariant has the three following properties:
- The fiber of $L$ at the trivial representation has a natural trivialization. If $\rho \in {\mathcal{M}}(V)$ is the trivial representation, then $e^{ i {\operatorname}{CS} ( \rho)} =1$ in this trivialization.
- $L$ has a natural connection, and the section of $r^* L$ sending $\rho$ into $e^{i {\operatorname}{CS} ( \rho)}$ is flat.
- If $V$ is closed and obtained by gluing two manifolds $V_1$ and $V_2$ along the common boundary, then for any $\rho \in {\mathcal{M}}(V)$, $$e^{i {\operatorname}{CS} ( \rho) } = \bigl\langle e^{i{\operatorname}{CS} ( \rho_1)}, e^{i {\operatorname}{CS} ( \rho_2)} \bigr\rangle$$ where $\rho_1$ and $\rho_2$ are the restrictions of $\rho$ to $V_1$ and $V_2$ respectively.
In our case, the pull-back of the Chern-Simons bundle of ${\mathcal{M}}( C \times S^1)$ by the projection $ M \rightarrow {\mathcal{M}}( C \times S^1)$ is the prequantum bundle $L_M$, cf. [@LJ2]. Furthermore the image of the restriction maps from ${\mathcal{M}}( D \times S^1)$ and ${\mathcal{M}}({\Sigma}\times S^1)$ to ${\mathcal{M}}( C \times S^1)$ are respectively $\tilde{A}$ and $\tilde{B}$. We conclude by lifting everything to $M$ and by using that $\Theta_A$ and $\Theta_B$ are flat and satisfy $\Theta_A ( 0) = \Theta_B (0)$.
[^1]: Institut de Math[é]{}matiques de Jussieu-Paris give gauche (UMR 7586), Universit[é]{} Pierre et Marie Curie, Paris, F-75005 France.
|
---
author:
- 'Shi Jin[^1] , Hanqing Lu[^2] and Lorenzo Pareschi[^3]'
bibliography:
- 'ChemotaxisUQr.bib'
title: 'A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs[^4]'
---
In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models.\
[**Key words.**]{} Chemotaxis kinetic model, chemotaxis Keller-Segel model, diffusion limit, uncertainty quantification, asymptotic preserving, generalized polynomial chaos, stochastic Galerkin method, implicit-explicit Runge-Kutta methods.
Introduction
============
Chemotaxis is the movement of an organism in response to a chemical stimulus (called chemoattractant), approaching the regions of highest chemoattractant concentration. This process is critical to the early growth and subsequent development of the organism.
Mathematical study of this chemical system originates from the well-known (Patlak-)Keller-Segel model [@keller1970initiation; @keller1971model; @keller1971traveling; @keller1980assessing; @patlak1953random]. This model describes the drift-diffusion interactions between the cell density and chemoattractant concentration at a macroscopic level:
\[ks1\] $$\begin{aligned}
&\partial_t\rho=\nabla\cdot (D\nabla \rho-\chi\rho\nabla s),\\
&\partial_t s=D_0\Delta s+q(s,\rho),\end{aligned}$$
where $\rho(x,t)\geq 0$ is the cell density at position $x\in \mathbb R^n$ and time $t$, $s(x,t)\geq 0$ is the density of the chemoattractant, $D$ and $D_0$ are positive diffusive constants of the cells and the chemoattractant respectively, and $\chi$ is the positive chemotactic sensitivity constant. In (\[ks1\]) the function $q(s,\rho)$ describes the interactions between the cell density and the chemoattractant such as productions and degradations. In the literature, several modifications and studies of the Keller-Segel model have been conducted during recent years, e.g. [@calvez2006modified; @chertock2012chemotaxis; @hillen2009user; @horstmann20031970; @perthame2004pde; @perthame2006transport]. The one related to our study is the modified Keller-Segel model in [@calvez2006modified]:
\[ks2\] $$\begin{aligned}
&\partial_t\rho=\nabla\cdot (D\nabla \rho-\chi\rho\nabla s),\\
&s=-\frac{1}{n\pi}\log|x|\ast \rho,\end{aligned}$$
where $n$ is the space dimension. Notice that in $2$D, (\[ks1\]) and (\[ks2\]) are exactly the same if $q=0$.
An important property of the Keller-Segel system is the blow up behavior, which depends on the dimension of the system and the initial mass [@brenner1999diffusion; @herrero1996chemotactic; @nagai1997global; @stevens1997aggregation]. For the $2$D Keller-Segel system (when (\[ks1\]) and (\[ks2\]) are equivalent), there exists a critical mass $M_c$ depending on the parameters of the system. When the initial mass $M<M_c$ (subcritical case), global solution exists and presents a self-similar profile in long time; When the initial mass $M>M_c$ (supercritical case), the solution will blow up in finite time; When the initial mass $M=M_c$ (critical case), the solution will blow up in infinite time. This property can be extended to $1$D and $3$D for the modified Keller-Segel system (\[ks2\]). The formula for the critical mass is given by $$\label{1-3}
M_c=\frac{2n^2\pi D}{\chi}.$$
From another perspective, the chemotaxis can be described by a class of Boltzmann-type kinetic equations at a microscopic level. The kinetic description of the phase space cell density was first introduced by Alt [@alt1980orientation; @alt1980biased] via a stochastic interpretation of the “run" and “tumble" process of bacteria movements. Later on Othmer, Dunbar and Alt formulated the following non-dimensionalized chemotaxis kinetic system with parabolic scaling in [@othmer1988models]: $$\label{1-4}
\varepsilon\frac{\partial f}{\partial t}+v\cdot \nabla_x f=\frac{1}{\varepsilon}\int_V(T_\varepsilon f'-T_\varepsilon^*f)dv'.$$ Here $f(t,x,v)$ is the density function of cells at time $t\in \mathbb R^+$, position $x\in\mathbb R^n$ and moving with velocity $v\in V$, $V$ is a finite subset of $\mathbb R^n$. The small parameter $\varepsilon$ is the radio of the mean running length between jumps to the typical observation length scale and $f'$ is the abbreviation for $f(t,x,v')$. $T_\varepsilon=T_\varepsilon[s](t,x,v,v')$ with the property $T^*_\varepsilon[s](t,x,v,v')=T_\varepsilon[s](t,x,v',v)$, is the turning kernel operator depending on the density of chemoattractant $s(t,x)$, which also solves the Poisson equation (\[ks1\]b).
The relationship between the kinetic chemotaxis model (\[1-4\]) and the Keller-Segel model (\[ks1\]) was formally derived by Othmer and Hillen in [@othmer2000diffusion; @othmer2002diffusion] using moment expansions. Then Chalub et al. gave a rigorous proof that the Keller-Segel system (\[ks2\]) (before blow up time in supercritical case and for all time in subcritical case) is the macroscopic limit (as $\varepsilon\to 0$) of the kinetic chemotaxis system (\[1-4\]) coupled with (\[ks2\]b) in three dimensions [@chalub2004kinetic]. For certain type of turning kernel $T_\varepsilon$ (the nonlocal model in Section $2.1$), [@chalub2004kinetic] also proved the global existence of the solution to the kinetic systems (\[1-4\]) for any initial conditions, which behaves completely differently from the Keller-Segel system. For other types of of turning kernel $T_\varepsilon$ (e.g. the local model in Section $2.2$), many questions are unsolved yet. Blow up may happen with supercritical initial mass but the critical mass is different from the Keller-Segel equations [@bournaveas2009critical]. The long time behavior of the subcritical case is unclear yet. Also, theoretic proof of the blow up in the $1$D case is not available [@sharifi2011one].
The microscopic kinetic model, with interesting properties and mysterious behaviors, make it appealing to investigate the system numerically. Moreover, the global existence of the solution with nonlocal turning kernel could help us to understand the behavior of chemotaxis after Keller-Segel solutions blow up. One of the difficulties in solving the kinetic chemotaxis model, as other multi-scale kinetic equations, is the stiffness when $0<\varepsilon\ll1$. Classical algorithms require taking spatial and time step of $O(\varepsilon)$, thus causing unaffordable computational cost. To overcome this difficulty, one has to design an *Asymptotic-Preserving* (AP) scheme, which discretizes the kinetic equations with mesh and time step independent of $\varepsilon$ and preserves a consistent discretization of the limiting modified Keller-Segel equation as $\varepsilon\to 0$. The AP methods were first coined in [@jin1999efficient] and have been applied to a variety of multi-scale kinetic equations. We refer to [@degond2011asymptotic; @degond2017asymptotic; @dimarco2014numerical; @jin2010asymptotic] for detailed reviews on AP schemes. In particular, AP schemes have been designed to solve $1$D and $2$D kinetic chemotaxis model in [@carrillo2013asymptotic; @chertockasymptotic], which are most relevant to our study.
The main issue we want to address in this paper is the uncertainties involved in the kinetic model due to modeling and experimental errors. For example, different turning kernels are proposed as operators that mimic the “run" and “tumble" process of cell movements and thus may contain uncertainties. Moreover, initial and boundary data, or other coefficients in the equations could also be measured inaccurately. In such a system that behaves so sensitively to initial mass and turning kernel, only by quantifying the *intrinsic* uncertainties in the model, could one get a better understanding and a more reliable prediction on the chemotaxis from computational simulations, especially in the situation where many properties are not clarified by theoretic study.
The goal of this paper is to design a high order efficient numerical scheme such that uncertainty quantification (UQ) can be easily conducted. Only recently, studies in UQ begin to develop for kinetic equations [@hu2016stochastic; @jin2016august; @jin2017asymptotic; @jinlu; @jin2015asymptotic; @zhu2016vlasov; @crouseilles2017nonlinear]. To deal with numerical difficulties for uncertainty and multi-scale at the same time, the *stochastic Asymptotic-Preserving* (sAP) notion was first introduced in [@jin2015asymptotic]. Since then, the *generalized Polynomial Chaos* (gPC) based *Stochastic Galerkin* (SG) framework has been developed to a variety of kinetic equations [@jinlu; @jin2015asymptotic; @zhu2016vlasov; @crouseilles2017nonlinear]. In this paper, we are going to conduct UQ under the same gPC-SG framework, which projects the uncertain kinetic equations into a vectorized deterministic equations and thus allowing us to extend the deterministic AP solver in [@carrillo2013asymptotic]. The sAP property is going to be verified formally by showing that the kinetic chemotaxis model with uncertainty after SG projection in fully discrete setting, as $\varepsilon\to 0$, automatically becomes a numerical discretization of the Keller-Segel equations with uncertainty after the SG projection. As realized in [@jin2015asymptotic] and rigorously proved in [@jin2016august; @li2016uniform; @jin2017hypocoercivity], the spectral accuracy is expected using this gPC-SG method as long as the regularity of the solution (which is usually preserved from initial regularity in kinetic equations) behaves well.
In addition, we improve the accuracy and efficiency of the numerical scheme by using the implicit-exlicit (IMEX) Runge-Kutta (RK) methods (see [@boscarino2013implicit; @boscarino2017unified; @pareschi2005implicit] and the references therein) and macroscopic penalization method. A similar approach was utilized in our previous work [@jin2017efficient] for linear transport and radiative heat transfer equations with random inputs. In [@jin2017efficient], we improved the parabolic CFL condition $\Delta t=O((\Delta x)^2)$ in [@jin2015asymptotic] to a hyperbolic CFL condition $\Delta t=O(\Delta x)$, which allows to save the computational time significantly.
The rest of the paper is organized as follows. In section 2, the kinetic models with random inputs of two different turning kernels are described and the macroscopic limits of both models are formally derived. From section 3 to section 5, the numerical scheme for the kinetic chemotaxis equations are designed and the sAP properties are illustrated. In section 6, several numerical tests are presented to illustrate the accuracy and efficiency of our scheme. The sAP property is also verified numerically. Different properties, e.g. blow up, stationary solutions etc., influenced by the introduced randomness of the local and nonlocal model, are explored for the chemotaxis system. The interactions between peaks involved with different sources of uncertainty are compared to show the dynamics. Finally, some conclusions are drawn in section 7.
The Kinetic Descriptions for Chemotaxis
=======================================
The chemotaxis kinetic system with random inputs we are going to study is (\[1-4\]) coupled with (\[ks2\]b) in 1D:
\[2-1\] $$\begin{aligned}
&\varepsilon\frac{\partial f}{\partial t} +v\frac{\partial f}{\partial x}=\frac{1}{\varepsilon}\int_V(T_\varepsilon f'-T_\varepsilon^*f)dv',\\
&s=-\frac{1}{\pi}\log|x|\ast\rho, \ \ \ \rho=\int_Vfdv,\end{aligned}$$
where $x\in \Omega=[-x_{\text{max}},x_{\text{max}}]\subset \mathbb R$, $v\in V=[-v_{\text{max}},v_{\text{max}}]\subset \mathbb R$.
The only difference is now $f=f(t,x,v,z)$ and $s=s(t,x,z)$ have dependence on the random variable $z\in I_z\subset\mathbb R^d(d\geq 1)$ with compact support $I_z$, in order to account for random uncertainties.
Now we specify the turning kernel operator $T_\varepsilon$ in (\[2-1\]). Since the turning kernel $T_\varepsilon[s](t,x,z,v,v')$ measures the probability of velocity jump of cells from $v$ to $v'$, it has the following properties $$\label{2-2}
\begin{aligned}
&T_\varepsilon[s](t,x,z,v,v')\geq 0,\\
&T_\varepsilon[s](t,x,z,v,v')=F(z,v)+\varepsilon T_1+O(\varepsilon^2),
\end{aligned}$$ where $F(z,v)$ is the equilibrium of velocity distribution and $T_1\geq 0$ characterizes the directional preference.
The 1D Nonlocal Model
---------------------
Now considering the nonlinear kernel introduced in [@chalub2004kinetic] with uncertainty, $$\label{2-3}
T_\varepsilon[s](t,x,z,v,v')=\alpha_+(z)\psi(s(t,x,z),s(t,x+\varepsilon v,z))+\alpha_-(z)\psi(s(t,x,z),s(t,x-\varepsilon v',z)).$$ The first term describes the cell movement to a new direction decided by the detection of current environment and probable new location and the second term describes the influence of the past memory on the choice of the new moving direction.
For simplicity, the past memory influence is neglected. Since $\alpha_+$ is an experimental parameter, we introduce the randomness on $\alpha_+(z)>0$ with the probability density function $\lambda(z)$ for the random variable $z$ and take $$\label{2-4}
\psi(s(t,x,z),s(t,x+\varepsilon v,z))=\bar F(v)+\delta^\varepsilon s(x,z,v),$$ where $$\label{2-5}
\delta^\varepsilon s(x,z,v)=(s(t,x+\varepsilon v,z)-s(t,x,z))_+:=\left\{
\begin{aligned}
&s(t,x+\varepsilon v,z)-s(t,x,z)&&\text{if}\ \ s(t,x+\varepsilon v,z)-s(t,x,z)>0\\
&0&&\text{otherwise}
\end{aligned}
\right.,$$ and $\bar F(v)$ satisfies $$\label{2-6}
\left\{
\begin{aligned}
&\int_V\bar F(v)dv=1,\\
&\bar F(v)=\bar F(|v|).
\end{aligned}
\right.$$ Notice that $\delta^\varepsilon s$ is an $O(\varepsilon)$ term which corresponds to $\varepsilon T_1$ in (\[2-2\]).
Then the kinetic system (\[2-1\]) becomes
\[2-7\] $$\begin{aligned}
&\varepsilon \frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}=\frac{\alpha_+(z)}{\varepsilon}\left[(\bar F(v)+\delta^\varepsilon s(v))\rho-\left(1+\int_V\delta^\varepsilon s(v')dv'\right)f\right],\\
&s=-\frac{1}{\pi}\log|x|\ast \rho.\end{aligned}$$
Positive initial conditions and reflection boundary conditions for $f$, reflecting boundary conditions for $s$ are imposed as following:
\[2-8\] $$\begin{aligned}
&f(0,x,z,v)=f^I(x,z,v)\geq 0,\\
&s(0,x,z)=s^I(x,z)\geq 0,\\
&f(t,\pm x_{\text{max}},z,v)=f(t,\pm x_{\text{max}},z,-v),\\
&\partial_xs|_{x=\pm x_{\text{max}}}=0.\end{aligned}$$
The global existence of the solution to (\[2-7\]) for fixed $z$ with any initial mass is proved in [@chalub2004kinetic].
The 1D Local Model
------------------
For the local model, we consider the turning kernel introduced in [@bournaveas2009critical] with uncertainty, $$\label{2-9}
T_\varepsilon=T_\varepsilon[s](t,x,z,v,v')=\alpha_+(z)\left[\bar F(v)+\varepsilon(v\cdot \nabla s (x))_+\right],$$ where $\bar F$ is the equilibrium function satisfying (\[2-6\]) and $\alpha(z)>0$ describes the desire of the cell to change to a favorable direction, which could come with uncertainty. Similarly as in section 2.1, we introduce the randomness on $\alpha_+(z)>0$. Then the kinetic equation (\[2-1\]) in one dimension is
\[2-10\] $$\begin{aligned}
&\varepsilon\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}=\frac{\alpha_+}{\varepsilon}\left[(\bar F(v)+\varepsilon(v\cdot \nabla s)_+)\rho-(1+c_1\varepsilon|\nabla s|)f\right],\\
&s=-\frac{1}{\pi}\log|x|\ast \rho,\end{aligned}$$
with $c_1=\int_V(v\cdot \nabla s/|\nabla s|)_+dv=\frac{1}{2}\int_V|v|dv$. The same initial and boundary conditions in (\[2-8\]) are applied.
The Macroscopic Limits
----------------------
The nonlocal kinetic model (\[2-7\]) and the local one (\[2-10\]) give the same asymptotic limit when $\varepsilon\to 0$. Inserting the Hilbert expansion into (\[2-7\]a) and (\[2-10\]a) and collecting the same order terms, one can derive the classical modified Keller-Segel system for $\rho$ as $\varepsilon\to 0$:
\[2-11\] $$\begin{aligned}
&\partial_t \rho=\partial_x\left(\frac{D}{\alpha_+}\partial_x\rho-\chi\rho\partial_x s\right),\\
&s=-\frac{1}{\pi}\log|x|\ast \rho,\\
&\partial_x\rho|_{x=\pm x_{\text{max}}}=0,\\
&\partial_xs|_{x=\pm x_{\text{max}}}=0,\end{aligned}$$
where $$\label{2-12}
D=\int_V|v|^2\bar F(v)dv, \ \ \chi=\frac{1}{2}\int_V|v|^2dv.$$ We refer to [@chalub2004kinetic] for the details.
The Critical Mass with Random Inputs
------------------------------------
To derive the critical mass for system (\[2-11\]), we show, following [@calvez2006modified], that the second momentum (with respect to $x$) of $\rho$ cannot remain positive for all time.
We use
$$\partial_x s=\partial_x(-\frac{1}{\pi}\log|x|\ast\rho)=-\frac{1}{\pi}\int_\Omega\frac{1}{x-y}\rho(y)dy=-\mathcal H \rho,$$
where $\mathcal H$ denotes the Hilbert transform [@zuazo2001fourier]. Then $$\label{2-13}
\begin{aligned}
\frac{d}{dt}\int_\Omega\frac{1}{2}|x|^2\rho(x,z,t)dx=&\int_\Omega\frac{1}{2}|x|^2\frac{\partial \rho}{\partial t}dx\\
=&\int_\Omega\frac{1}{2}|x|^2\partial_x\left(\frac{D}{\alpha_+(z)}\partial_x\rho-\chi\rho\partial_xs\right)dx\\
=&-\int_\Omega x\left(\frac{D}{\alpha_+(z)}\partial_x\rho-\chi\rho\partial_xs\right)dx\\
=&-\frac{D}{\alpha_+(z)}[x_{\text{max}}\rho(x_{\text{max}})+x_{\text{max}}\rho(-x_{\text{max}})]+\frac{D}{\alpha_+(z)}M\\
&-\frac{\chi}{\pi}\int_\Omega \rho(x)\lim_{\delta \to 0}\int_{|x-y|>\delta}\frac{x}{x-y}\rho(y)dydx\\
=&-\frac{D}{\alpha_+(z)}x_{\text{max}}[\rho(x_{\text{max}})+\rho(-x_{\text{max}})]+\frac{D}{\alpha_+(z)}M\\
&-\frac{\chi}{2\pi}\lim_{\delta \to 0}\int_\Omega\int_{|x-y|>\delta}\rho(x)\rho(y)dxdy\\
=& -\frac{D}{\alpha_+(z)}x_{\text{max}}[\rho(x_{\text{max}})+\rho(-x_{\text{max}})]-\frac{\chi}{2\pi}M^2\left(1-\frac{M_c(z)}{M}\right),
\end{aligned}$$ where $$\label{2-14}
M_c(z)=\frac{2\pi D}{\chi \alpha_+(z)}.$$ Here we assume that the initial data is independent of $z$ and we use the conservation of mass, i.e. $M=\int_\Omega \rho dx$ is a constant independent of $z$.
When $M>M_c(z)$, $\frac{d}{dt}\int_\Omega\frac{1}{2}|x|^2\rho(x,z,t)dx\leq -c<0$, where $c$ is a positive constant. To preserve the positivity of this second moment (with respect to $x$), some singularity has to occur so that the above computation will not hold at certain time. The singularity is rigorously analyzed in [@dolbeault2004optimal; @blanchet2006two] and $\partial_xs$ is unbounded in this case. Thus blow up occurs.
When $M< M_c(z)$, the second moment (with respect to $x$) is locally controlled and global existence of weak solution can be obtained [@calvez2006modified].
When $n\geq 2$, the computation is similar and the general formular for $M_c(z)$ is $$M_c(z)=\frac{2n^2\pi D}{\chi\alpha_+(z)}.$$
In practice, one is more interested in the behavior of $\mathbb{E}[\rho(x,z,t)]$, the expected value of $\rho(x,z,t)$. We have the following theorem analyzing the influence of initial mass on $\mathbb{E}[\rho(x,z,t)]$.
Suppose that the total mass $M$ is independent of $z$. Denote $\bar M_c$ as the critical mass for $\mathbb{E}[\rho(x,z,t)]$, i.e. when $M>\bar M_c$, $\mathbb{E}[\rho(x,z,t)]$ will blow up; when $M<\bar M_c$, $\mathbb{E}[\rho(x,z,t)]$ will be bounded for all time. Then we have $$\label{2-15}
\bar{M}_c=\mathbb{E}[M_c(z)].$$
Following the computations in (\[2-13\]), we show that $$\label{2-16}
\begin{aligned}
\frac{d}{dt}\int_\Omega\frac{1}{2}|x|^2\mathbb{E}[\rho(x,z,t)]dx=&\int_\Omega\int_{I_z}\frac{1}{2}|x|^2\frac{\partial \rho(x,z,t)}{\partial t}\lambda(z)dzdx\\
=&\int_\Omega \int_{I_z}\frac{1}{2}|x|^2\partial_x \left(\frac{D}{\alpha_+(z)}\partial_x \rho-\chi \rho\partial_x s\right)\lambda(z)dzdx\\
=&\int_{I_z}\left[\int_\Omega \frac{1}{2}|x|^2\partial_x \left(\frac{D}{\alpha_+(z)}\partial_x\rho-\chi \rho\partial_xs\right)dx\right]\lambda(z)dz\\
=&\int_{I_z}\left[-\frac{D}{\alpha_+(z)}[x_{\text{max}}\rho(x_{\text{max}})+x_{\text{max}}\rho(-x_{\text{max}})]-\frac{\chi}{2\pi}M^2\left(1-\frac{M_c(z)}{M}\right)\right]\lambda(z)dz\\
=&-\int_{I_z}\frac{D}{\alpha_+(z)}x_{\text{max}}[\rho(x_{\text{max}})+\rho(-x_{\text{max}})]\lambda(z)dz-\frac{\chi}{2\pi}M^2\left(1-\frac{\mathbb{E}[M_c(z)]}{M}\right)\\
\leq&-\frac{\chi}{2\pi}M^2\left(1-\frac{\mathbb{E}[M_c(z)]}{M}\right).
\end{aligned}$$ Thus, $\bar{M}_c=\mathbb{E}[M_c(z)]$ is the critical mass for $\mathbb{E}[\rho(x,z,t)]$.
The same conclusion holds for $n\geq 2$.
The Even-Odd Decomposition
==========================
In this section, we apply the even-odd decomposition to reformulate the problem following the same procedure as [@carrillo2013asymptotic] for deterministic kinetic model for chemotaxis.
The 1D Nonlocal Model
---------------------
For $v>0$, (\[2-7\]a) can be split into two equations:
\[3-2\] $$\begin{aligned}
&\varepsilon \frac{\partial f(v)}{\partial t}+v\frac{\partial f(v)}{\partial x}=\frac{\alpha_+(z)}{\varepsilon}\left[(\bar F(v)+\delta^\varepsilon s(v))\rho-\left(1+\int_V\delta^\varepsilon s(v')dv'\right)f(v)\right],\\
&\varepsilon \frac{\partial f(-v)}{\partial t}-v\frac{\partial f(-v)}{\partial x}=\frac{\alpha_+(z)}{\varepsilon}\left[(\bar F(-v)+\delta^\varepsilon s(-v))\rho-\left(1+\int_V\delta^\varepsilon s(v')dv'\right)f(-v)\right].\end{aligned}$$
Now denote the even and odd parities
\[3-3\] $$\begin{aligned}
&r(t,x,z,v)=\mathcal R[f]=\frac{1}{2}(f(t,x,z,v)+f(t,x,z,-v)),\\
&j(t,x,z,v)=\mathcal J[f]=\frac{1}{2\varepsilon}(f(t,x,z,v)-f(t,x,z,-v)).\end{aligned}$$
Then (\[3-2\]) becomes
\[3-4\] $$\begin{aligned}
&\partial_t r+v\partial_xj=\frac{\alpha_+}{\varepsilon^2}[(\bar F(v)+\mathcal R[\delta^\varepsilon s])\rho-(1+\langle \delta^\varepsilon s\rangle)r],\\
&\partial_t j+\frac{1}{\varepsilon^2}v\partial_x r=\frac{\alpha_+}{\varepsilon^2}(\mathcal J[\delta^\varepsilon s]\rho-(1+\langle \delta^\varepsilon s\rangle)j),\end{aligned}$$
where
\[3-5\] $$\begin{aligned}
&\langle\delta^\varepsilon s\rangle=\int_V\delta^\varepsilon s(x,v')dv',\\
&\rho=\int_Vfdv=2\int_{V^+}rdv, \ V^+=\{v\in V|v\geq 0\}.\end{aligned}$$
Notice that, when $\varepsilon\to 0$, (\[3-4\]) yields
\[3-6\] $$\begin{aligned}
&r=\frac{\bar F(v)+\mathcal R[\delta^\varepsilon s]}{1+\langle \delta^\varepsilon s\rangle}\rho=\rho \bar F(v)+O(\varepsilon),\\
&j=\frac{\mathcal J[\delta^\varepsilon s]\rho-v\frac{\partial_x r}{\alpha_+}}{1+\langle \delta^\varepsilon s\rangle}=v\left(\frac{1}{2}\partial_x s\rho-\frac{\partial_x r}{\alpha_+}\right)+O(\varepsilon).\end{aligned}$$
Substituting (\[3-6\]) into (\[3-4\]a) and integrating over $V^+$, one gets the same limiting Keller-Segel equations with random inputs as (\[2-11\]).
The 1D Local Model
------------------
For the 1D local model, one can follow the same even-odd decomposition and obtain
\[3-7\] $$\begin{aligned}
&\partial_t r+v\partial_x j=\frac{\alpha_+}{\varepsilon}\left[(\bar F(v)+\frac{\varepsilon}{2}|v\partial_x s|)\rho-(1+c_1\varepsilon|\partial_xs|)r\right],\\
&\partial_t j+\frac{1}{\varepsilon^2}v\partial_x r=\frac{\alpha_+}{\varepsilon^2}\left[\frac{1}{2}v\partial_x s\rho-(1+c_1\varepsilon|\partial_xs|)j\right].\end{aligned}$$
The remaining work is the same as section 3.1.
The gPC-SG Formulation
======================
Now we deal with the random inputs using the gPC expansion via an orthogonal polynomial series to approximate the solution. That is, for random variable $z\in \mathbb R^d$, one seeks
\[4-1\] $$\begin{aligned}
&r(t,x,z,v)\approx r_N(t,x,z,v)=\sum_{k=1}^K\hat{r}_k(t,x,v)\Phi_k(z),\\
&j(t,x,z,v)\approx j_N(t,x,z,v)=\sum_{k=1}^K\hat{j}_k(t,x,v)\Phi_k(z),\end{aligned}$$
where $\left\{\Phi_k(z),1\leq k\leq K, K=\begin{pmatrix}
d+N\\
d
\end{pmatrix}\right\}$ are from $\mathbb P_N^d$, the $d$-variate orthogonal polynomials of degree up to $N\geq 1$, and orthonormal $$\label{4-2}
\int_{I_z}\Phi_i(z)\Phi_j(z)\lambda(z)dz=\delta_{ij}, \ 1\leq i,j\leq K=\text{dim}(\mathbb P_N^d).$$ Here $\delta_{i,j}$ the Kronecker delta function (See [@xiu2002wiener]).
Now inserts the approximation (\[4-1\]) into the governing equation (\[3-4\]) and enforces the residue to be orthogonal to the polynomial space spanned by $\{\Phi_1,\cdots, \Phi_K\}$. Thus, we obtain a set of vector deterministic equations for $\hat{\bold r}=(\hat r_1,\cdots, \hat r_K)^T$, $\hat{\bold j}=(\hat j_1,\cdots, \hat j_K)^T$ and $\hat{\bold s}=(\hat s_1,\cdots, \hat s_K)^T$:
\[4-3\] $$\begin{aligned}
&\partial_t \hat{\bold r}+v\partial_x\hat{\bold j}=\frac{1}{\varepsilon^2}[\bar F(v)\bold M \hat{\boldsymbol \rho}+\bold B\hat{\boldsymbol \rho}-\bold M\hat{\bold r}-\bold C\hat{\bold r}],\\
&\partial_t \hat{\bold j}+\frac{1}{\varepsilon^2}v\partial_x \hat{\bold r}=\frac{1}{\varepsilon^2}(\bold E\hat{\boldsymbol \rho}-\bold M\hat{\bold j}-\bold C\hat{\bold j}),\\
&\hat{\bold s}=-\frac{1}{\pi}\log|x|\ast \hat{\boldsymbol\rho},\end{aligned}$$
where $$\label{4-4}
\hat{\boldsymbol \rho}(t,x)=\langle \hat{ \bold r}\rangle=2\int_{V^+}\hat{\bold r}dv,$$ and $\bold M=(m_{ij})_{1\leq i,j\leq K}$, $\bold B(\delta^\varepsilon s_N)=(b_{ij}(x,v))_{1\leq i,j\leq K}$, $\bold C(\langle\delta^\varepsilon s_N\rangle )=(c_{ij}(x))_{1\leq i,j\leq K}$ and $\bold E(\delta^\varepsilon s_N)=(e_{ij}(x,v))_{1\leq i,j\leq K}$ are $K\times K$ symmetric matrices with entries respectively
\[4-5\] $$\begin{aligned}
&m_{ij}=\int_{I_z}\alpha_+(z)\Phi_i(z)\Phi_j(z)\lambda(z)dz,\\
&b_{ij}(x,v)=\int_{I_z}\alpha_+(z)\mathcal R[\delta^\varepsilon s_N]\Phi_i(z)\Phi_j(z)\lambda(z)dz,\\
&c_{ij}(x)=\int_{I_z}\alpha_+(z)\langle\delta^\varepsilon s_N\rangle\Phi_i(z)\Phi_j(z)\lambda(z)dz,\\
&e_{ij}(x,v)=\int_{I_z}\alpha_+(z)\mathcal J[\delta^\varepsilon s_N]\Phi_i(z)\Phi_j(z)\lambda(z)dz.\end{aligned}$$
As $\varepsilon\to 0^+$ in (\[4-3\]), since $\langle \delta^\varepsilon s_N\rangle=O(\varepsilon)$ and the matrices $\bold M$ and $\bold C$ are symmetric positive definite thus invertible,
\[4-6\] $$\begin{aligned}
&\hat{\bold r}=(\bold M+\bold C)^{-1}(\bar F(v)\bold M+\bold B)\hat{\boldsymbol \rho}=\bar{F}(v)\hat{\boldsymbol \rho}+O(\varepsilon),\\
&\hat{\bold j}=(\bold M+\bold C)^{-1}(\bold E\hat{\boldsymbol \rho}-v\partial_x\hat{\bold r})=\bold M^{-1}\bold E\hat{\boldsymbol \rho}-v\bold M^{-1}\partial_x\hat{\boldsymbol r}+O(\varepsilon).\end{aligned}$$
Plugging (\[4-6\]) into (\[4-3\]a) and integrating over $V^+$, one obtains $$\label{4-7}
\partial_x\hat{\boldsymbol \rho}=\partial_x\left(D\bold M^{-1}\partial_x \hat{\boldsymbol \rho}-\chi\bold G\hat{\boldsymbol \rho}\right),$$ where $\bold G=\frac{1}{\chi}\bold M^{-1}\langle \bold E\rangle$.
If one applies the gPC-SG formulation for the limiting Keller-Segel equation (\[2-11\]) directly, one gets $$\label{4-8}
\partial_t \tilde{\boldsymbol \rho}=\partial_x\left(D\tilde{\bold M}\partial_x\tilde{\boldsymbol \rho}-\chi \tilde{\bold G}\tilde{\boldsymbol\rho}\right),$$ where $\tilde{\bold M}=(\tilde m_{ij})_{1\leq i,j\leq K}$ and $\tilde{\bold G}=(\tilde g_{ij})_{1\leq i,j\leq K}$ are $K\times K$ symmetric matrix with entries
\[4-9\] $$\begin{aligned}
&\tilde m_{ij}=\int_{I_z}\frac{1}{\alpha_+(z)}\Phi_i(z)\Phi_j(z)\lambda(z)dz,\\
&\tilde g_{ij}=\int_{I_z}(\partial_xs_N)\Phi_i(z)\Phi_j(z)\lambda(z)dz.\end{aligned}$$
Although $\tilde{\bold M}$ is different from $\bold M^{-1}$, proof in [@ShuHuJin] shows that $\tilde {\bold M}\partial_x\tilde{\boldsymbol \rho}$ and $\bold M^{-1}\partial_x\hat{\boldsymbol \rho}$ are spectrally close to each other. The same property holds between $\tilde {\bold G}\tilde{\boldsymbol \rho}$ and $\bold G\hat{\boldsymbol \rho}$.
An efficient sAP Scheme Based on an IMEX-RK Method
==================================================
One can apply the relaxation method as in [@carrillo2013asymptotic] to the projected system (\[4-3\]), which falls into the sAP framework proposed in [@jin2015asymptotic] . However, the method suffers from the parabolic CFL condition $\Delta t=O((\Delta x)^2)$.
Here we propose an efficient sAP scheme using the idea from [@boscarino2013implicit] to get rid of the parabolic CFL condition. By adding and subtracting the term $\mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol \rho}-\chi\tilde{\bold G}\hat{\boldsymbol \rho})$ in (\[4-3\]a) and the term $\phi v\partial_x \hat{\bold r}$ in (\[4-3\]b), we reformulate the problem into an equivalent form:
\[5-1\] $$\begin{aligned}
\partial_t \hat{\bold r}=&-v\partial_x\hat{\bold j}-\mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol \rho}-\chi\tilde{\bold G}\hat{\boldsymbol \rho})+\frac{1}{\varepsilon^2}\left(\bar F(v)\bold M \hat{\boldsymbol \rho}+\bold B\hat{\boldsymbol \rho}-\bold M\hat{\bold r}-\bold C\hat{\bold r}\right)+\mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol \rho}-\chi\tilde{\bold G}\hat{\boldsymbol \rho})\nonumber\\
=& f_1(\hat{\bold r},\hat{\bold j})+f_2(\hat{\bold r},\hat{\bold s}),\\
\partial_t\hat{\bold j}=&-\phi v\partial_x\hat{\bold r}-\frac{1}{\varepsilon^2}\left[(1-\varepsilon^2\phi)v\partial_x \hat{\bold r}-\bold E\hat{\boldsymbol \rho}+\bold M\hat{\bold j}+\bold C\hat{\bold j}\right]=g_1(\hat{\bold r})+g_2(\hat{\bold r},\hat{\bold j}),\\
\hat{\bold s}=&-\frac{1}{\pi}\log|x|\ast \hat{\boldsymbol\rho}=h(\hat{\bold r}),\end{aligned}$$
where $\bold M,\tilde{\bold M},\bold B, \bold C,\bold E$ and $\tilde{\bold G}$ are the same as defined in (\[4-5\]) and (\[4-9\]) and
\[5-2\] $$\begin{aligned}
&f_1(\hat{\bold r},\hat{\bold j})=-v\partial_x\hat{\bold j}-\mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol \rho}-\chi\tilde{\bold G}\hat{\boldsymbol \rho}),\\
&f_2(\hat{\bold r},\hat{\bold s})=\frac{1}{\varepsilon^2}\left(\bar F(v)\bold M \hat{\boldsymbol \rho}+\bold B\hat{\boldsymbol \rho}-\bold M\hat{\bold r}-\bold C\hat{\bold r}\right)+\mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol \rho}-\chi\tilde{\bold G}\hat{\boldsymbol \rho}),\\
&g_1(\hat{\bold r})=-\phi v\partial_x\hat{\bold r},\\
&g_2(\hat{\bold r},\hat{\bold j})=-\frac{1}{\varepsilon^2}\left[(1-\varepsilon^2\phi)v\partial_x \hat{\bold r}-\bold E\hat{\boldsymbol \rho}+\bold M\hat{\bold j}+\bold C\hat{\bold j}\right].\end{aligned}$$
Here we choose $\mu=\mu(\varepsilon)$ such that $$\label{5-3}
\begin{aligned}
&\lim_{\varepsilon\to 0}\mu=1,\\
&\mu=0 \ \ \ \mbox{if} \ \ \ \varepsilon=O(1);
\end{aligned}$$ and $\phi=\phi(\varepsilon)$ such that $$\label{5-4}
0\leq \phi\leq \frac{1}{\varepsilon^2}.$$
The restriction on $\phi$ guarantees the positivity of $\phi(\varepsilon)$ and $(1-\varepsilon^2\phi(\varepsilon))$ so that the problem remains well-posed uniformly in $\varepsilon$. We make the same simple choice of $\phi$ as in [@jin2000uniformly]: $$\label{5-5}
\phi(\varepsilon)=\min \left\{1,\frac{1}{\varepsilon^2}\right\}.$$
Now we apply an IMEX-RK scheme to system (\[5-1\]) where $(f_1,g_1)^T$ is evaluated explicitly and $(f_2,g_2)^T$ implicitly, then we obtain
\[5-6\] $$\begin{aligned}
&\hat{\bold r}^{n+1}=\hat{\bold r}^n+\Delta t\sum_{k=1}^s\tilde{b}_kf_1(\hat{\bold R}^k,\hat{\bold J}^k)+\Delta t\sum_{k=1}^sb_kf_2(\hat{\bold R}^k,\hat{\bold S}^k),\\
&\hat{\bold j}^{n+1}=\hat{\bold j}^n+\Delta t\sum_{k=1}^s\tilde{b}_kg_1(\hat{\bold R}^k)+\Delta t\sum_{k=1}^sb_kg_2(\hat{\bold R}^k,\hat{\bold J}^k),\\
&\hat{\bold s}^{n+1}=-\frac{1}{\pi}\log|x|\ast\hat{\boldsymbol \rho}^{n+1},\end{aligned}$$
where the internal stages are
\[5-7\] $$\begin{aligned}
&\hat{\bold R}^k=\hat{\bold r}^n+\Delta t\sum_{l=1}^{k-1}\tilde{a}_{kl}f_1(\hat{\bold R}^l,\hat{\bold J}^l)+\Delta t\sum_{l=1}^ka_{kl}f_2(\hat{\bold R}^l,\hat{\bold S}^l),\\
&\hat{\bold J}^k=\hat{\bold j}^n+\Delta t\sum_{l=1}^{k-1}\tilde{a}_{kl}g_1(\hat{\bold R}^l)+\Delta t\sum_{l=1}^ka_{kl}g_2(\hat{\bold R}^l,\hat{\bold J}^l),\\
&\hat{\bold S}^k=-\frac{1}{\pi}\log|x|\ast\hat{\boldsymbol {\mathrm{P}}}^k.\end{aligned}$$
It is obvious that the scheme is characterized by the $s\times s$ matrices $$\label{5-8}
\tilde A=(\tilde a_{ij}),A=(a_{ij})$$ and the vectors $\tilde b, b\in\mathbb R^s$, which can be represented by a double table tableau in the usual Butcher notation
(r,0.5cm,0.5cm)\[5pt\][c|c]{}[c|c]{}$\tilde c$& $\tilde A$\
& $\tilde b^T$\
,
(r,0.5cm,0.5cm)\[5pt\][c|c]{}[c|c]{}$c$& $A$\
& $b^T$\
.
The coefficients $\tilde c$ and $c$ depend on the explicit part of the scheme: $$\label{5-9}
\tilde c_i=\sum_{j=1}^{i-1}\tilde a_{ij}, \ \ c_i=\sum_{j=1}^ia_{ij}.$$
In the literature, there are two main different types of IMEX R-K schemes characterized by the structure of the matrix $A$. We are interested in the IMEX-RK method of type $A$ (see [@boscarino2013implicit]) where the matrix $A$ is invertible, so that the implicit parts become more amenable.
As an example, we report the SSP(3,3,2) scheme, which is a second order IMEX scheme we are going to use in Section \[Secn\]
$$\begin{tabular}{c|c c c}
0&0&0&0\\
1/2&1/2&0&0\\
1&1/2&1/2&0\\
\hline
&1/3&1/3&1/3
\end{tabular},\qquad
\begin{tabular}{c|c c c}
1/4&1/4&0&0\\
1/4&0&1/4&0\\
1&1/3&1/3&1/3\\
\hline
&1/3&1/3&1/3
\end{tabular}.
\label{SSP332}$$
To obtain $\hat{\bold R}^k$ in each internal stage of (\[5-7\]), one needs $\hat{\boldsymbol{\mathrm{P}}}^k$ and $\hat{\bold S}^k$ in the implicit part $f_2(\hat{\bold R}^k,\hat{\bold S}^k)$. These quantities can be obtained explicitly by the following procedure.
Suppose one has computed $\hat{\bold R}^l$ and $\hat{\bold S}^l$ for $l=1,\cdots, k-1$, then according to (\[5-7\]a) $$\label{5-10}
\begin{aligned}
\hat{\bold R}^k=&\hat{\bold r}^n+\Delta t\sum_{l=1}^{k-1}\left(\tilde{a}_{kl}f_1(\hat{\bold R}^l,\hat{\bold J}^l)+a_{kl}f_2(\hat{\bold R}^l,\hat{\bold S}^l)\right)\\
&+\Delta t a_{kk}\left[\frac{1}{\varepsilon^2}(\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^k+\bold B^k\hat{\boldsymbol {\mathrm{P}}}^k-\bold M\hat{\bold R}^k-\bold C^k\hat{\bold R}^k)+ \mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol {\mathrm{P}}}^k-\chi\tilde{\bold G}^k\hat{\boldsymbol {\mathrm{P}}}^k)\right]\\
=&\overline{\hat{\bold R}}^{k-1}+\Delta t a_{kk}\left[\frac{1}{\varepsilon^2}(\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^k+\bold B^k\hat{\boldsymbol {\mathrm{P}}}^k-\bold M\hat{\bold R}^k-\bold C^k\hat{\bold R}^k)+ \mu\bar F(v)\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol {\mathrm{P}}}^k-\chi\tilde{\bold G}^k\hat{\boldsymbol {\mathrm{P}}}^k)\right].\end{aligned}$$
Here $\overline{\hat{\bold R}}^{k-1}$ represents all contributions in (\[5-10\]) from the first $k-1$ stages. Now one takes $\langle \cdot \rangle$ on both sides of (\[5-10\]) so that $[\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^k+\bold B^k\hat{\boldsymbol {\mathrm{P}}}^k-\bold M\hat{\bold R}^k-\bold C^k\hat{\bold R}^k]$ is cancelled out on the right hand side and one can approximate $\tilde{\bold G}^k$ by $\tilde{\bold G}^{k-1}$. Now $\hat{\boldsymbol {\mathrm{P}}}^k$ can be obtained from the following diffusion equation in an implicit form: $$\label{5-11}
\hat{\boldsymbol {\mathrm{P}}}^k-\Delta t a_{kk}\mu\partial_x(D\tilde{\bold M}\partial_x\hat{\boldsymbol {\mathrm{P}}}^k-\chi\tilde{\bold G}^{k-1}\hat{\boldsymbol {\mathrm{P}}}^k)=\langle \overline{\hat{\bold R}}^{k-1}\rangle.$$ Then it is plugged back to (\[5-10\]) in order to compute $\hat{\bold R}^k$.
The Space Discretization
------------------------
Second order accuracy is obtained using an upwind TVD scheme (with minmod slope limiter [@leveque1992numerical]) in the explicit transport part and center difference for other second derivatives. During each internal stage (\[5-7\]),
\[5-12\] $$\begin{aligned}
\hat{\bold R}^k_i=&\hat{\bold r}^n_i+\Delta t\sum_{l=1}^{k-1}\tilde a_{kl}\left\{-\frac{v}{2\Delta x}(\hat{\bold J}^l_{i+1}-\hat{\bold J}^l_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold R}^l_{i+1}-2\hat{\bold R}^l_i+\hat{\bold R}^l_{i-1})\right.-\frac{v\phi^{1/2}}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}+\boldsymbol\beta^l_{i+1}-\boldsymbol\beta^l_i)\nonumber\\
&-\frac{\mu}{(\Delta x)^2}\bar F (v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)\left.+\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right)\right\}\nonumber\\
&+\Delta t \sum_{l=1}^ka_{kl}\left\{\frac{1}{\varepsilon^2}\left(\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^l_i+\bold B_i^l\hat{\boldsymbol {\mathrm{P}}}_i^l-\bold M\hat{\bold R}_i^l-\bold C_i^l\hat{\bold R}_i^l\right)\right.\nonumber\\
&+\frac{\mu}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)\left.-\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right)\right\},\\
\hat{\bold J}^k_i=&\hat{\bold j}^n_i+\Delta t\sum_{l=1}^{k-1}\tilde a_{kl}\left\{-\frac{v\phi}{2\Delta x}(\hat{\bold R}^l_{i+1}-\hat{\bold R}^l_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold J}^l_{i+1}-2\hat{\bold J}^l_i+\hat{\bold J}^l_{i-1})-\frac{v\phi}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}-\boldsymbol\beta^l_{i+1}+\boldsymbol\beta^l_i)\right\}\nonumber\\
&-\Delta t\sum_{l=1}^ka_{kl}\frac{1}{\varepsilon^2}\left\{(1-\varepsilon^2\phi)v\frac{\hat{\bold R}_{i+1}^l-\hat{\bold R}_{i-1}^l}{2\Delta x}-\bold E_i^l\hat{\boldsymbol {\mathrm{P}}}_i^l+\bold M\hat{\bold J}_i^l+\bold C_i^l\hat{\bold J}_i^l\right\},\end{aligned}$$
where
\[5-13\] $$\begin{aligned}
\boldsymbol \gamma^l_i=&\frac{1}{\Delta x}\text{minmod}\left(\hat{\bold R}^l_{i+1}+\phi^{-1/2}\hat{\bold J}^l_{i+1}-\hat{\bold R}^l_i-\phi^{-1/2}\hat{\bold J}^l_i,\right. \\
&\left.\hat{\bold R}^l_{i}+\phi^{-1/2}\hat{\bold J}^l_{i}-\hat{\bold R}^l_{i-1}-\phi^{-1/2}\hat{\bold J}^l_{i-1}\right),\\
\boldsymbol \beta^l_i=&\frac{1}{\Delta x}\text{minmod}\left(\hat{\bold R}^l_{i+1}-\phi^{-1/2}\hat{\bold J}^l_{i+1}-\hat{\bold R}^l_i+\phi^{-1/2}\hat{\bold J}^l_i,\right.\\
&\left.\hat{\bold R}^l_{i}-\phi^{-1/2}\hat{\bold J}^l_{i}-\hat{\bold R}^l_{i-1}+\phi^{-1/2}\hat{\bold J}^l_{i-1}\right).\end{aligned}$$
Since $\hat{\boldsymbol {\mathrm{P}}}^k$ can be obtained explicitly by (\[5-11\]), we can fully discretize $\hat{\boldsymbol {\mathrm{P}}}^k_i$ as following: $$\label{5-14}
\hat{\boldsymbol {\mathrm{P}}}_i^k-\Delta t a_{kk}\frac{\mu}{(\Delta x)^2}\left[D\tilde{\bold M}(\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k-2\hat{\boldsymbol {\mathrm{P}}}_{i}^k+\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k)-\chi\left(\tilde{\bold G}_{i+\frac{1}{2}}^{k-1}(\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\hat{\boldsymbol {\mathrm{P}}}_{i}^k)-\tilde{\bold G}_{i-\frac{1}{2}}^{k-1}(\hat{\boldsymbol {\mathrm{P}}}_{i}^k-\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k)\right)\right]=\langle \overline{\hat{\bold R}}_i^{k-1}\rangle.$$
Then using (\[5-14\]), the fully discretized $\hat{\bold R}^k_i$ is obtained and subsequently $\hat{\bold J}^k_i$ from the following:
\[5-15\] $$\begin{aligned}
&\left(\bold I+\frac{a_{kk}\Delta t}{\varepsilon^2}(\bold M+\bold C_i^k)\right)\hat{\bold R}^k_i\nonumber\\
=&\hat{\bold r}^n_i+\Delta t\sum_{l=1}^{k-1}\tilde a_{kl}\left\{-\frac{v}{2\Delta x}(\hat{\bold J}^l_{i+1}-\hat{\bold J}^l_{i-1})\right.+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold R}^l_{i+1}-2\hat{\bold R}^l_i+\hat{\bold R}^l_{i-1})-\frac{v\phi^{1/2}}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}+\boldsymbol\beta^l_{i+1}-\boldsymbol\beta^l_i)\nonumber\\
&-\frac{\mu}{(\Delta x)^2}\bar F (v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)\left.+\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right)\right\}\nonumber\\
&+\Delta t \sum_{l=1}^{k-1}a_{kl}\left\{\frac{1}{\varepsilon^2}\left[\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^l_i+\bold B_i^l\hat{\boldsymbol {\mathrm{P}}}_i^l-\bold M\hat{\bold R}_i^l-\bold C_i^l\hat{\bold R}_i^l\right]\right.\left.+\frac{\mu}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)\right.\nonumber\\
&\left.-\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right)\right\}+\Delta t a_{kk}\left\{\frac{1}{\varepsilon^2}\left[\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^k_i+\bold B_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k\right]\right.\nonumber\\
&\left.+\frac{\mu}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)\right.\left.-\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k\right)\right\},\\
&\left(1+\frac{a_{kk}\Delta t}{\varepsilon^2}(\bold M +\bold C_i^k)\right)\hat{\bold J}^k_i\nonumber\\
=&\hat{\bold j}^n_i+\Delta t\sum_{l=1}^{k-1}\tilde a_{kl}\left\{-\frac{v\phi}{2\Delta x}(\hat{\bold R}^l_{i+1}-\hat{\bold R}^l_{i-1})\right.+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold J}^l_{i+1}-2\hat{\bold J}^l_i+\hat{\bold J}^l_{i-1})\left.-\frac{v\phi}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}+\boldsymbol\beta^l_{i+1}-\boldsymbol\beta^l_i)\right\}\nonumber\\
&-\Delta t\sum_{l=1}^{k-1}a_{kl}\frac{1}{\varepsilon^2}\left\{(1-\varepsilon^2\phi)v\frac{\hat{\bold R}_{i+1}^l-\hat{\bold R}_{i-1}^l}{2\Delta x}-\bold E_i^l\hat{\boldsymbol {\mathrm{P}}}_i^l+\bold M \hat{\bold J}_i^l+\bold C_i^l\hat{\bold J}_i^l\right\}\nonumber\\
&-\Delta ta_{kk}\frac{1}{\varepsilon^2}\left\{(1-\varepsilon^2\phi)v\frac{\hat{\bold R}_{i+1}^k-\hat{\bold R}_{i-1}^k}{2\Delta x}-\bold E_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k\right\},\end{aligned}$$
In the above $\left(1+\frac{a_{kk}\Delta t}{\varepsilon^2}(\bold M +\bold C_i^k)\right)$ is symmetric positive definite, thus invertible. After calculating all $\hat{\bold R}^k_i$ and $\hat{\bold J}^k_i$ for $k=1,\cdots, s$, we can update $\hat{\bold r}^{n+1}_i$ and $\hat{\bold j}^{n+1}_i$ in (\[5-6\]),
\[5-16\] $$\begin{aligned}
\hat{\bold r}^{n+1}_i=&\hat{\bold r}^n_i+\Delta t\sum_{k=1}^{s}\tilde b_{k}\left\{-\frac{v}{2\Delta x}(\hat{\bold J}^k_{i+1}-\hat{\bold J}^k_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold R}^k_{i+1}-2\hat{\bold R}^k_i+\hat{\bold R}^k_{i-1})\right.\nonumber\\
&-\frac{v\phi^{1/2}}{4}(\boldsymbol\gamma^k_i-\boldsymbol\gamma^k_{i-1}+\boldsymbol\beta^k_{i+1}-\boldsymbol\beta^k_i)-\frac{\mu}{(\Delta x)^2}\bar F (v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)\nonumber\\
&\left.+\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k\right)\right\}\nonumber\\
&+\Delta t \sum_{k=1}^sb_{k}\left\{\frac{1}{\varepsilon^2}\left[\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^k_i+\bold B_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k-\bold M\hat{\bold R}_i^k-\bold C_i^k\hat{\bold R}_i^k\right]\right.\nonumber\\
&\left.+\frac{\mu}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)-\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k\right)\right\},\\
\hat{\bold j}^{n+1}_i=&\hat{\bold j}^n_i+\Delta t\sum_{k=1}^{s}\tilde b_{k}\left\{-\frac{v\phi}{2\Delta x}(\hat{\bold R}^k_{i+1}-\hat{\bold R}^k_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold J}^k_{i+1}-2\hat{\bold J}^k_i+\hat{\bold J}^k_{i-1})\right.\nonumber\\
&\left.-\frac{v\phi}{4}(\boldsymbol\gamma^k_i-\boldsymbol\gamma^k_{i-1}-\boldsymbol\beta^k_{i+1}+\boldsymbol\beta^k_i)\right\}-\Delta t\sum_{k=1}^sb_{k}\frac{1}{\varepsilon^2}\left\{(1-\varepsilon^2\phi)v\frac{\hat{\bold R}_{i+1}^k-\hat{\bold R}_{i-1}^k}{2\Delta x}\right.\nonumber\\
&\left.-\bold E_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k+\bold M\hat{\bold J}_i^k+\bold C_i^k\hat{\bold J}_i^k\right\},\end{aligned}$$
where $\boldsymbol \gamma^k_i$ and $\boldsymbol \beta^k_i$ are defined the same as in (\[5-13\]).
Following [@boscarino2013implicit] we choose $$\label{5-17}
\mu=\exp(-\varepsilon^2/\Delta x).$$ Thus, for large value of $\varepsilon$, (e.g., $\varepsilon=1$), we could avoid the loss of accuracy caused by adding and subtracting the penalty term; for very small value of $\varepsilon$, (e.g., $\varepsilon\to 0$), $\mu\to 1$.
The full discrete scheme is obtained using the Gauss-Legendre quadrature nodes for the velocity discretization. Finally, to get the boundary conditions for $\hat{\bold r},\hat{\bold j}$ and $\hat{\bold s}$, we refer to [@jin2000uniformly] for details.
The sAP property
----------------
Denote
\[5-18\] $$\begin{aligned}
f_1(\hat{\bold R}^l_i,\hat{\bold J}^l_i)=&-\frac{v}{2\Delta x}(\hat{\bold J}^l_{i+1}-\hat{\bold J}^l_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold R}^l_{i+1}-2\hat{\bold R}^l_i+\hat{\bold R}^l_{i-1})\nonumber\\
&-\frac{v\phi^{1/2}}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}+\boldsymbol\beta^l_{i+1}-\boldsymbol\beta^l_i)-\frac{\mu}{(\Delta x)^2}\bar F (v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)\nonumber\\
&+\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right),\\
f_2(\hat{\bold R}^l_i)=&\frac{1}{\varepsilon^2}\left[\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^l_i+\bold B_i^l\hat{\boldsymbol {\mathrm{P}}}_i^l-\bold M\hat{\bold R}_i^l-\bold C_i^l\hat{\bold R}_i^l\right]\nonumber\\
&+\frac{\mu}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)-\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right),\\
g_1(\hat{\bold R}^l_i)=&-\frac{v\phi}{2\Delta x}(\hat{\bold R}^l_{i+1}-\hat{\bold R}^l_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold J}^l_{i+1}-2\hat{\bold J}^l_i+\hat{\bold J}^l_{i-1})-\frac{v\phi}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}-\boldsymbol\beta^l_{i+1}+\boldsymbol\beta^l_i),\\
g_2(\hat{\bold R}^l_i,\hat{\bold J}^l_i)=&\frac{1}{\varepsilon^2}\left[(1-\varepsilon^2\phi)v\frac{\hat{\bold R}_{i+1}^l-\hat{\bold R}_{i-1}^l}{2\Delta x}-\bold E_i^l\hat{\boldsymbol {\mathrm{P}}}_i^l+\bold M\hat{\bold J}_i^l+\bold C_i^l\hat{\bold J}_i^l\right].\end{aligned}$$
From (\[5-15\]) we have
\[5-19\] $$\begin{aligned}
\begin{pmatrix}
\hat{\bold R}_i^1\\
\hat{\bold R}_i^2\\
\vdots\\
\hat{\bold R}_i^s
\end{pmatrix}=&
\begin{pmatrix}
\hat{\bold r}_i^n\\
\hat{\bold r}_i^n\\
\vdots\\
\hat{\bold r}_i^n
\end{pmatrix}+\Delta t\begin{pmatrix}
0\\
\tilde a_{21}f_1(\hat{\bold R}_i^1,\hat{\bold J}_i^1)\\
\vdots\\
\sum_{l=1}^{s-1}\tilde a_{sl}f_1(\hat{\bold R}_i^{l},\hat{\bold J}_i^{l})
\end{pmatrix}+\Delta t\bold A
\begin{pmatrix}
f_2(\hat{\bold R}_i^1)\\
f_2(\hat{\bold R}_i^2)\\
\vdots\\
f_2(\hat{\bold R}_i^{s})
\end{pmatrix},\\
\begin{pmatrix}
\hat{\bold J}_i^1\\
\hat{\bold J}_i^2\\
\vdots\\
\hat{\bold J}_i^s
\end{pmatrix}=&
\begin{pmatrix}
\hat{\bold j}_i^n\\
\hat{\bold j}_i^n\\
\vdots\\
\hat{\bold j}_i^n
\end{pmatrix}+\Delta t\begin{pmatrix}
0\\
\tilde a_{21}g_1(\hat{\bold R}_i^1)\\
\vdots\\
\sum_{l=1}^{s-1}\tilde a_{sl}g_1(\hat{\bold R}_i^{l})
\end{pmatrix}+\Delta t\bold A\begin{pmatrix}
g_2(\hat{\bold R}_i^1,\hat{\bold J}_i^1)\\
g_2(\hat{\bold R}_i^2,\hat{\bold J}_i^2)\\
\vdots\\
g_2(\hat{\bold R}_i^{s},\hat{\bold J}_i^s)
\end{pmatrix},\end{aligned}$$
where $$\label{5-20}
\bold A_{K(i-1)+1:Ki,K(j-1)+1:Kj}=A_{i,j}\bold I_{K\times K}, \ \ \ \bold I_{K\times K} \ \ \text{is} \ K\times K \ \text{identity matrix},$$ and $A$ is defined in (\[5-8\]). Denote $\bold W$ as the inverse matrix of $\bold A$, then we obtain from (\[5-19\])
\[5-21\] $$\begin{aligned}
&\Delta t
\begin{pmatrix}
f_2(\hat{\bold R}_i^1)\\
f_2(\hat{\bold R}_i^2)\\
\vdots\\
f_2(\hat{\bold R}_i^{s})
\end{pmatrix}=\bold W\left[\begin{pmatrix}
\hat{\bold R}_i^1\\
\hat{\bold R}_i^2\\
\vdots\\
\hat{\bold R}_i^s
\end{pmatrix}-
\begin{pmatrix}
\hat{\bold r}_i^n\\
\hat{\bold r}_i^n\\
\vdots\\
\hat{\bold r}_i^n
\end{pmatrix}-\Delta t\begin{pmatrix}
0\\
\tilde a_{21}f_1(\hat{\bold R}_i^1,\hat{\bold J}_i^1)\\
\vdots\\
\sum_{l=1}^{s-1}\tilde a_{sl}f_1(\hat{\bold R}_i^{l},\hat{\bold J}_i^{l})
\end{pmatrix}\right],\\
&\Delta t\begin{pmatrix}
g_2(\hat{\bold R}_i^1,\hat{\bold J}_i^1)\\
g_2(\hat{\bold R}_i^2,\hat{\bold J}_i^2)\\
\vdots\\
g_2(\hat{\bold R}_i^{s},\hat{\bold J}_i^s)
\end{pmatrix}=\bold W\left[\begin{pmatrix}
\hat{\bold J}_i^1\\
\hat{\bold J}_i^2\\
\vdots\\
\hat{\bold J}_i^s
\end{pmatrix}-
\begin{pmatrix}
\hat{\bold j}_i^n\\
\hat{\bold j}_i^n\\
\vdots\\
\hat{\bold j}_i^n
\end{pmatrix}-\Delta t\begin{pmatrix}
0\\
\tilde a_{21}g_1(\hat{\bold R}_i^1)\\
\vdots\\
\sum_{l=1}^{s-1}\tilde a_{sl}g_1(\hat{\bold R}_i^{l})
\end{pmatrix}\right].\end{aligned}$$
Since $\bold W$ has the same structure as $\bold A$, $\bold W$ should be a lower triangular matrix with entries $$\label{51}
\bold W_{K(i-1)+1:Ki,K(j-1)+1:Kj}=\omega_{i,j}\bold I_{K\times K},$$ where $W=(\omega_{i,j})$ is the inverse of the lower triangular matrix $A$ in (\[5-8\]).
Then one can rewrite (\[5-21\]) as
\[5-23\] $$\begin{aligned}
&\Delta tf_2(\hat{\bold R}_i^k)=\sum_{l=1}^k\omega_{kl}\left[\hat{\bold R}_i^l-\hat{\bold r}_i^n-\Delta t\sum_{l=1}^{k-1}\tilde{a}_{kl}f_1(\hat{\bold R}_i^l,\hat{\bold J}_i^l)\right],\\
&\Delta tg_2(\hat{\bold R}_i^k,\hat{\bold J}_i^k)=\sum_{l=1}^k\omega_{kl}\left[\hat{\bold J}_i^l-\hat{\bold j}_i^n-\Delta t\sum_{l=1}^{k-1}\tilde a_{kl}g_1(\hat{\bold R}_i^l)\right].\end{aligned}$$
More explicitly,
\[5-24\] $$\begin{aligned}
&\Delta t\left\{\frac{1}{\varepsilon^2}\left[\bar F(v)\bold M \hat{\boldsymbol {\mathrm{P}}}^k_i+\bold B_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k-\bold M\hat{\bold R}_i^k-\bold C_i^k\hat{\bold R}_i^k\right]\right.\nonumber\\
&\left.+\frac{\mu}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)-\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k\right)\right\}\nonumber\\
=&\sum_{l=1}^k\omega_{kl}\left\{\hat{\bold R}_i^l-\hat{\bold r}_i^n-\Delta t\sum_{l=1}^{k-1}\tilde{a}_{kl}\left[-\frac{v}{2\Delta x}(\hat{\bold J}^l_{i+1}-\hat{\bold J}^l_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold R}^l_{i+1}-2\hat{\bold R}^l_i+\hat{\bold R}^l_{i-1})\right.\right.\nonumber\\
&-\frac{v\phi^{1/2}}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}+\boldsymbol\beta^l_{i+1}-\boldsymbol\beta^l_i)-\frac{\mu}{(\Delta x)^2}\bar F (v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}\right)\nonumber\\
&\left.+\frac{\mu}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l\right)\right\},\\
&\Delta t\left\{\frac{1}{\varepsilon^2}\left[(1-\varepsilon^2\phi)v\frac{\hat{\bold R}_{i+1}^k-\hat{\bold R}_{i-1}^k}{2\Delta x}-\bold E_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k+\bold M\hat{\bold J}_i^k+\bold C_i^k\hat{\bold J}_i^k\right]\right\}\nonumber\\
=&\sum_{l=1}^k\omega_{kl}\left\{\hat{\bold J}_i^l-\hat{\bold j}_i^n-\Delta t \sum_{l=1}^{k-1}\tilde a_{kl}\left[-\frac{v\phi}{2\Delta x}(\hat{\bold R}^l_{i+1}-\hat{\bold R}^l_{i-1})+\frac{v\phi^{1/2}}{2\Delta x}(\hat{\bold J}^l_{i+1}-2\hat{\bold J}^l_i+\hat{\bold J}^l_{i-1})\right.\right.\nonumber\\
&\left.\left.-\frac{v\phi}{4}(\boldsymbol\gamma^l_i-\boldsymbol\gamma^l_{i-1}-\boldsymbol\beta^l_{i+1}+\boldsymbol\beta^l_i)\right]\right\}.\end{aligned}$$
Thus, setting $\varepsilon\to 0$, since $\bold M+\bold C_i^k$ is non-singular, one obtains
\[5-25\] $$\begin{aligned}
\hat{\bold R}_i^k=&(\bold M+\bold C_i^k)^{-1}(\bar F(v)\bold M+\bold B_i^k)\hat{\boldsymbol {\mathrm{P}}}_i^k=\bar F(v)\hat{\boldsymbol {\mathrm{P}}}_i^k+O(\varepsilon),\\
\hat{\bold J}_i^k=&(\bold M+\bold C_i^k)^{-1}(\bold E_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k-v\frac{\hat{\bold R}_{i+1}^k-\hat{\bold R}_{i-1}^k}{2\Delta x})=\bold M^{-1}\bold E_i^k\hat{\boldsymbol {\mathrm{P}}}_i^k-v\bold M^{-1}\frac{\hat{\bold R}_{i+1}^k-\hat{\bold R}_{i-1}^k}{2\Delta x}+O(\varepsilon).\end{aligned}$$
Inserting this back to (\[5-16\]a) and letting $\varepsilon\to 0$, $$\label{5-26}
\hat{\bold r}_i^{n+1}=\hat{\bold r}_i^n+\Delta t\sum_{k=1}^s\tilde{b}_k\hat f_1(\hat{\bold R}_i^k)+\Delta t\sum_{k=1}^sb_k\hat f_2(\hat{\bold R}_i^k),$$ where
\[5-27\] $$\begin{aligned}
\hat f_1(\hat{\bold R}_i^k)=&v^2\bar F(v)\frac{\bold M^{-1}}{4(\Delta x)^2}\left(\hat{\bold R}_{i+2}^k-2\hat{\bold R}_{i}^k+\hat{\bold R}_{i-2}^k\right)-\bar{F}(v)D\frac{\tilde{\bold M}}{(\Delta x)^2}\left(\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-2\hat{\boldsymbol {\mathrm{P}}}_i^k-\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k\right)\nonumber\\
&-\frac{v^2}{4\Delta x}(\bold M^{-1}{\bold E}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\bold M^{-1}{\bold E}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k)+\frac{1}{2\Delta x}\bar{F}(v)\chi(\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k),\\
\hat f_2(\hat{\bold R}_i^k)=&\frac{1}{(\Delta x)^2}\bar F(v)D\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)-\frac{1}{2\Delta x}\bar F(v)\chi\left(\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k\right).\end{aligned}$$
Since the difference between $\bold M^{-1}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)$ and $\tilde{\bold M}\left(\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}\right)$ and the difference between $\frac{1}{\chi}\bold M^{-1}\bold E_i^k\hat{\boldsymbol {\mathrm{P}}}^k_{i}$ and $\tilde{\bold G}_i^k\hat{\boldsymbol {\mathrm{P}}}^k_{i}$ are both spectral small, after integrating over $V^+$, $\hat f_1$ goes to $0$ and one gets $$\label{5-28}
\hat{\boldsymbol \rho}_i^{n+1}=\hat{\boldsymbol \rho}_i^n+\Delta t\sum_{k=1}^sb_k\left[\bar F(v)D\tilde{\bold M}\frac{\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}}{(\Delta x)^2}-\bar F(v)\chi\frac{\tilde{\bold G}_{i+1}^k\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^k\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k}{2\Delta x}\right]+O((\Delta x)^2),$$ where $$\label{5-29}
\begin{aligned}
\hat{\boldsymbol {\mathrm{P}}}_i^k=&\hat{\boldsymbol \rho}_i^n+\Delta t\sum_{l=1}^{k-1}a_{kl}\left[\bar F(v)D\tilde{\bold M}\frac{\hat{\boldsymbol {\mathrm{P}}}^l_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^l_{i}+\hat{\boldsymbol {\mathrm{P}}}^l_{i-1}}{(\Delta x)^2}-\bar F(v)\chi\frac{\tilde{\bold G}_{i+1}^l\hat{\boldsymbol {\mathrm{P}}}_{i+1}^l-\tilde{\bold G}_{i-1}^l\hat{\boldsymbol {\mathrm{P}}}_{i-1}^l}{2\Delta x}\right]\\
&+\Delta ta_{kk}\left[\bar F(v)D\tilde{\bold M}\frac{\hat{\boldsymbol {\mathrm{P}}}^k_{i+1}-2\hat{\boldsymbol {\mathrm{P}}}^k_{i}+\hat{\boldsymbol {\mathrm{P}}}^k_{i-1}}{(\Delta x)^2}-\bar F(v)\chi\frac{\tilde{\bold G}_{i+1}^{k-1}\hat{\boldsymbol {\mathrm{P}}}_{i+1}^k-\tilde{\bold G}_{i-1}^{k-1}\hat{\boldsymbol {\mathrm{P}}}_{i-1}^k}{2\Delta x}\right],
\end{aligned}$$ which is an implicit RK scheme for the projected limiting diffusion equation (\[4-8\]). Thus, the sAP property [@jin2015asymptotic] of the efficient IMEX R-K scheme is formally justified.
Numerical Tests {#Secn}
===============
The 1D Nonlocal Deterministic Model
-----------------------------------
The following numerical tests are carried out with $$x\in\Omega=[-1,1],\ v\in V=[-1,1], \ \alpha=1,$$ $$\bar F(v)=\frac{1}{|V|}\mathbf{1}_V:=\left\{
\begin{aligned}
&\frac{1}{|V|}&&\text{if} \ v\in V\\
&0&&\text{otherwise}
\end{aligned}
\right..$$ The critical mass for the limiting Keller-Segel system given by formula (\[1-3\]) is $$M_c=2\pi.$$ The initial conditions are given by $$\rho_I(x)=Ce^{-80x^2}, \ \ f_I(x,v)=\rho_I(x) F(v),$$ where $C=C(M)$ is a constant determined by the total mass $M$.
For the deterministic case, we compare our results by the second order IMEX-RK method (\[SSP332\]) (denoted by SSP2 in the figures) with the results by [@carrillo2013asymptotic] (denoted by CY in the figures). For both tests, we set $\Delta x=0.005$. In their numerical tests, the CFL condition is $$\Delta t=\max \left\{\frac{\varepsilon\Delta x}{2},\frac{\Delta x^2}{2}\right\}.$$ Obviously, when $\varepsilon$ is small, it suffers from the parabolic CFL condition for the diffusive nature of the Keller-Segel system.
For our IMEX-RK method, the choice of $\Delta t$ is given by $$\Delta t=\lambda \Delta x, \ \lambda=0.02,$$ which is much bigger than ${\Delta x^2}/{2}$.
### A Super-Critical Mass
It has been shown in [@chalub2004kinetic] that the solution of the kinetic system can converge to the Keller-Segel system weakly in a finite time interval $[0, t^*]$, with $t^*<t_b$. Here $t_b$ is the blow up time of the corresponding Keller-Segel system.
For the Super-Critical case, we set $$\label{6-1}
M=4\pi>M_c=2\pi, \ t=0.003<t_b\approx 0.0039.$$
![The 1D nonlocal deterministic model in the super-critical case. Solid lines are numerical results obtained in [@carrillo2013asymptotic] and circles are numerical results obtained by the IMEX-RK method. Dashed line is the numerical solution of the Keller-Segel equations as reference.](sup2.eps)
Figure 1 shows that the solution to the kinetic equation $\rho$ converges to the solution of the Keller-Segel solution $\rho_0$ as $\varepsilon\to 0$ at time $t=0.003<t_b$. Our IMEX-RK results match very well with the results in [@carrillo2013asymptotic].
### A Sub-Critical Mass
For the Sub-Critical case, we set $$\label{6-2}
M=\pi<M_c, \ t=0.1.$$
![The 1D nonlocal deterministic model in the sub-critical case. Solid lines are numerical results obtained in [@carrillo2013asymptotic] and circles are numerical results obtained by the IMEX-RK method. Dashed line is the numerical solution of the Keller-Segel equations as reference.](sub2.eps)
Figure 2 shows similar convergence results as the supercritical case for a relatively long time $t=0.1$. Also, good agreements between our new IMEX-RK solutions and the numerical results from [@carrillo2013asymptotic] can be observed, even in zoomed in area.
The 1D Nonlocal Model with Random Inputs in the Supercritical Case
------------------------------------------------------------------
Now we let $$\alpha=1+0.5z, \ z\sim U[-1,1], M=4\pi>\bar M_c\approx2.197\pi.$$ Using the same mesh size as before, we also employ the stochastic collocation method (using 20 quadrature points) as reference solutions. In stochastic collocation, the deterministic solver can be applied directly to a set of selected sample points and then the solution is approximated by interpolation of all sample solutions (see [@xiu2010numerical] for a review of stochastic collocation methods). The gPC expansion has been considered only up to $4$th order in our numerical tests. The following are the comparisons of the two methods in mean and standard deviation for the super-critical case with the same initial mass and stopping time in (\[6-1\]). Given the gPC coefficients $\hat{\rho}_k$ of $\rho$, the mean and standard deviation are calculated as $$\mathbb{E}[\rho]\approx\hat{\rho}_1, \ \ \mathbb{S}[\rho]\approx\sqrt{\sum_{k=2}^K\hat{\rho}_k^2}.$$
### The sAP property
![The 1D nonlocal random model in the super-critical case. Solid line is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the gPC method and circle is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the collocation method. Dashed line is obtained by the IMEX-RK using gPC and cross is obtained by the IMEX-RK using collocation. Different values of $\varepsilon$ are tested and the two quantities of interests are mean value (left) and standard deviation (right).](6_2_1_1.eps "fig:") ![The 1D nonlocal random model in the super-critical case. Solid line is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the gPC method and circle is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the collocation method. Dashed line is obtained by the IMEX-RK using gPC and cross is obtained by the IMEX-RK using collocation. Different values of $\varepsilon$ are tested and the two quantities of interests are mean value (left) and standard deviation (right).](6_2_1_2.eps "fig:") ![The 1D nonlocal random model in the super-critical case. Solid line is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the gPC method and circle is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the collocation method. Dashed line is obtained by the IMEX-RK using gPC and cross is obtained by the IMEX-RK using collocation. Different values of $\varepsilon$ are tested and the two quantities of interests are mean value (left) and standard deviation (right).](6_2_1_3.eps "fig:") ![The 1D nonlocal random model in the super-critical case. Solid line is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the gPC method and circle is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the collocation method. Dashed line is obtained by the IMEX-RK using gPC and cross is obtained by the IMEX-RK using collocation. Different values of $\varepsilon$ are tested and the two quantities of interests are mean value (left) and standard deviation (right).](6_2_1_4.eps "fig:") ![The 1D nonlocal random model in the super-critical case. Solid line is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the gPC method and circle is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the collocation method. Dashed line is obtained by the IMEX-RK using gPC and cross is obtained by the IMEX-RK using collocation. Different values of $\varepsilon$ are tested and the two quantities of interests are mean value (left) and standard deviation (right).](6_2_1_5.eps "fig:") ![The 1D nonlocal random model in the super-critical case. Solid line is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the gPC method and circle is obtained by combining the deterministic solver [@carrillo2013asymptotic] with the collocation method. Dashed line is obtained by the IMEX-RK using gPC and cross is obtained by the IMEX-RK using collocation. Different values of $\varepsilon$ are tested and the two quantities of interests are mean value (left) and standard deviation (right).](6_2_1_6.eps "fig:")
 
Figure 3 shows that the IMEX-RK solution agrees well with results of [@carrillo2013asymptotic] for all $\varepsilon$ no matter combined with gPC approach or collocation approach to deal with the uncertainty. Small differences between the two methods, especially near the singularity for small $\varepsilon$, are observed due to different orders of accuracy, but the SG solution always matches the collocation solution accurately for the same deterministic solver. Figure 4 shows that the mean and standard deviation of the kinetic chemotaxis solutions both tend to the quantities of the limiting Keller-Segel solution as $\varepsilon\to 0$ for fixed $\Delta t$ and $\Delta x$, which verifies the sAP property.
### Global Existence and Finite Time Blow Up
 
As proved in [@chalub2004kinetic], the solution to the kinetic system (\[2-7\]) with the nonlocal turning kernel is bounded on $[0,T]$, for any time $T$. However, the Keller-Segel solution will blow up in finite time with a supercritical mass. We examine the mean value and standard deviation of $\|\rho\|_\infty$ for relatively long time ($t\gg t_b$) in Figure 5. The uncertain systems show the same properties as the deterministic ones, e.g. the kinetic systems have global bound in the first and second moments for different $\varepsilon$ while the Keller-Segel solution will blow up in expected finite time.
### The Stationary Solution of the Kinetic system
 
The numerical tests in [@carrillo2013asymptotic] suggest that the solution of the deterministic kinetic system with a supercritical initial mass stabilizes toward a stationary state after long time. We also check to see if the same property holds for the kinetic system with random inputs. We plot the mean and standard deviation of $\tilde \rho(x)=\varepsilon\rho(\varepsilon x)$ in Figure 6, which shows that the mean and standard deviation both converge to some stationary state at a long time $t=2$, while the mean agrees with the deterministic stationary solution.
The interaction between peaks: the 1D Nonlocal Model with Random Initial Data
-----------------------------------------------------------------------------
As shown in [@BCC], the interactions between several peaks for the modified Keller-Segel system can be interpreted as optimal transportation. In the following numerical tests, we are going to make some observations of the interaction changes in the kinetic system cased by different types of randomness in initial data.
### Case 1: Two symmetric peaks, without enough mass in each peak
![ Deterministic solution of $\rho(x,t)$ with initial data $f_0=4\sqrt{5\pi}\left(1.5e^{-80(x-0.3)^2}+1.5e^{-80(x+0.3)^2}\right)$, $\varepsilon=0.1$. (Figure 8 in [@carrillo2013asymptotic]).](case1deterministic)
![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((1.5+0.5z)e^{-80(x-0.3)^2}+(1.5+0.5z)e^{-80(x+0.3)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.1$. ](case1mean "fig:") ![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((1.5+0.5z)e^{-80(x-0.3)^2}+(1.5+0.5z)e^{-80(x+0.3)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.1$. ](case1sd "fig:")
![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((1.5+0.5z)e^{-80(x-0.3)^2}+(1.5-0.5z)e^{-80(x+0.3)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.1$.](case1_2_mean "fig:") ![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((1.5+0.5z)e^{-80(x-0.3)^2}+(1.5-0.5z)e^{-80(x+0.3)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.1$.](case1_2_sd "fig:")
In this case, we still have $M_c=2\pi$ and $\bar M_c\approx 2.197\pi$. We reproduced the deterministic attraction between two symmetric peaks with total mass $3\pi$ in Figure 7. Then we input symmetric randomness in each peak, i.e. the total mass follows from uniform distribution from $2\pi$ to $4\pi$, keeping each peak without enough mass. Figure 8 shows that symmetric randomness keeps the attraction behavior exactly as the deterministic case. Symmetric properties are preserved both in mean and standard deviation. However, in Figure 9, we input asymmetric randomness in each peak but keeping total mass fixed as $3\pi$. The two peaks will still be attracted in the center but present different behavior as the deterministic one. The asymmetric randomness in this type will widen the mean range of the center peak after concentration, in the sense that asymmetric initial data push the concentrated peak toward the direction with more initial mass.
### Case 2: Two asymmetric peaks with enough mass in each peak
![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((3+z)e^{-80(x-0.3)^2}+(5-z)e^{-80(x+0.3)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.05$. ](case6mean "fig:") ![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((3+z)e^{-80(x-0.3)^2}+(5-z)e^{-80(x+0.3)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.05$. ](case6sd "fig:")
With $M_c=2\pi$ and $\bar M_c\approx 2.197\pi$, we put asymmetric initial mass both larger than $2\pi$. Figure 10 shows similar results as Figure 10 in [@carrillo2013asymptotic]. The mass in each peak is large enough to concentrate but they will merge into a larger peak which locates closer to larger initial peak due to asymmetry. Figure 11 shows the effect of the asymmetric randomness with total initial mass fixed. It can be observed in mean and standard deviation that the randomness affects the concentration time, location and asymmetry, showing the solution behaves sensitively to initial data.
### Case 3: Two Asymmetric peaks (close), one below critical mass, one above critical mass
![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((1+0.5z)e^{-80(x-0.1)^2}+(5-0.5z)e^{-80(x+0.1)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.05$.](case7mean "fig:") ![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random initial condition $f_0=4\sqrt{5\pi}\left((1+0.5z)e^{-80(x-0.1)^2}+(5-0.5z)e^{-80(x+0.1)^2}\right), z\sim \mathcal U[-1,1]$, $\varepsilon=0.05$.](case7sd "fig:")
![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random $\alpha=1+0.5z, z\sim \mathcal U[-1,1]$ and deterministic initial data, $\varepsilon=0.05$.](case7alpha_mean "fig:") ![Left is the mean and right is the standard deviation of $\rho(x,t,z)$ respectively, with random $\alpha=1+0.5z, z\sim \mathcal U[-1,1]$ and deterministic initial data, $\varepsilon=0.05$.](case7alpha_sd "fig:")
 
From Figure 12 to Figure 15, we conduct a series of experiments with two asymmetric peaks, keeping one peak with enough mass and the other one without enough mass. The deterministic case (Figure 12) shows that the peak with less mass will move towards the other one in a short time and then they continue to aggregate mass. In Figure 13, small amount of randomness exchanging between two peaks will not change this tendence in mean. The standard deviation in Figure 13 is asymmetric due to the asymmetric randomness in initial data. In Figure 14, although mean values show no difference, the standard deviation is symmetric because the source of randomness comes from the diffusion coefficient $\alpha$. Figure 15 shows that the position of the two peaks has significant effects on the aggregation behavior in this case. From mean and standard deviation, one can observe that there exists a critical distance between the two peaks, beyond which the two peaks will not be able to merge. They will be separated to behave independently according to their initial mass.
The 1D local Model with Random Initial Data
-------------------------------------------
 
Although theoretic study of the local model with supercritical mass is still not enough to understand the blow up behavior of the local kinetic chemotaxis system, numerical tests in [@carrillo2013asymptotic] suggested blowing up density by using adapted grids. Instead of studying the blowing up property, we are more interested in the sensitive effect brought up by the randomness around critical mass. In Figure 16, the deterministic solution with subcritical initial data will stay bounded as expected from theory. However, the solution keeps aggregating in Figure 17 if we introduce randomness into initial mass ranging from subcritical mass to supercritical mass with mean less than critical mass. More obviously in Figure 18, the deterministic solution will remain bounded while the mean of the random solution appears increasing in time. This indicates that the introduced randomness will influence the properties of the solution. If the range of the initial data contains supercritical regimes, the solution of the random system will behave quite differently from the deterministic one with average initial mass.
Stochastic collocation method is used in test 6.4 to deal with $|\partial_xs|$ as following: Once $\hat{\bold s}=(\hat s_1,\cdots \hat s_K)^T$ is obtained at each time iteration, $\partial_x\hat{\bold s}=(\partial_x\hat s_1,\cdots \partial_x\hat s_K)^T$ can be obtained using finite difference. Then $|\partial_xs(x,z)|$ can be approximated by $|\sum_{k=1}^K\partial_x\hat s_k(x)\Phi_k(z)|$. According to the probability density function of $z$, one can have a set of collocation points $\{z_j\}_{j=1}^M$ with corresponding weights $\{w_j\}_{j=1}^M$. ($M=20$ points are used in our test.) Project $|\partial_xs(x,z)|$ onto the space $\{\Phi_1(z),\cdots,\Phi_K(z)\}$ in order to get the gPC coefficients $(\xi_1,\cdots,\xi_K)^T$ of $|\partial_x s|$ such that $|\partial_xs|\approx\sum_{k=1}^K\xi_k(x)\Phi_k(z)$, one can get $$\begin{aligned}
\xi_k(x)&=\int_{I_z}|\partial_xs(x,z)|\Phi_k(z)\lambda(z)dz\\
&\approx\sum_{j=1}^M|\partial_xs(x,z_j)|\Phi_k(z_j)w_j\\
&\approx\sum_{j=1}^M|\sum_{i=1}^K\partial_x\hat s_i(x)\Phi_i(z_j)|\Phi_k(z_j)w_j, \ \ \ \ \ \ k=1,\cdots, K.
\end{aligned}$$ Then $(\xi_1,\cdots,\xi_K)^T$ is used in the algorithm.
Conclusion
==========
In this article, a high order efficient stochastic Asymptotic-Preserving scheme is designed for the kinetic chemotaxis system with random inputs. Compared with the previous work [@carrillo2013asymptotic] for the deterministic kinetic chemotaxis equations, our new method, based on generalized Polynomial Chaos Galerkin approach to deal with uncertainty, uses the implicit-explicit Runge-Kutta (IMEX-RK) method to gain high accuracy and utilize a macroscopic penalty to improve the CFL stability condition from parabolic type to hyperbolic type in the diffusive regime. Both efficiency and accuracy are verified in the numerical tests.
There are many remaining work for future study. Since the kinetic description of the chemotaxis system is more microscopic and consistent with the classical Keller-Segel equation with more favorable properties (e.g. global existence for nonlocal turning kernel), it is important to complete the theory as well as conduct efficient numerical simulations comparing with experimental results. On one hand, many properties, which have been explored numerically in this paper and previous work [@carrillo2013asymptotic; @chertockasymptotic], remain to be verified by rigorous theory. On the other hand, the high order efficient method in this paper should be extended to 2D and 3D kinetic chemotaxis system to support the theory in future work. Moreover, some general problems for uncertainty quantification, such as high dimensionality and rigorous sensitive analysis, are to be further studied.
[^1]: Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA (sjin[@]{}wisc.edu) and Institute of Natural Sciences, School of Mathematical Science, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China.
[^2]: Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA (hlu57[@]{}wisc.edu).
[^3]: Department of Mathematics & Computer Science, University of Ferrara, Ferrara, 44121, Italy (lorenzo.pareschi[@]{}unife.it).
[^4]: This research was partially supported by NSF grants DMS-1522184 and DMS-1107291: RNMS KI-Net, by NSFC grant No. 91330203, and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
|
---
abstract: 'We present 237 new spectroscopically confirmed pre-main-sequence K and M-type stars in the young Upper Scorpius subgroup of the Sco-Cen association, the nearest region of recent massive star formation. Using the Wide-Field Spectrograph at the Australian National University 2.3m telescope at Siding Spring, we observed 397 kinematically and photometrically selected candidate members of Upper Scorpius, and identified new members by the presence of Lithium absorption. The HR-diagram of the new members shows a spread of ages, ranging from $\sim$3-20Myr, which broadly agrees with the current age estimates of $\sim$5-10Myr. We find a significant range of Li 6708 equivalent widths among the members, and a minor dependence of HR-diagram position on the measured equivalent width of the Li 6708Å line, with members that appear younger having more Lithium. This could indicate the presence of either populations of different age, or a spread of ages in Upper Scorpius. We also use Wide-Field Infrared Survey Explorer data to infer circumstellar disk presence in 25 of the members on the basis of infrared excesses, including two candidate transition disks. We find that 11.2$\pm$3.4% of the M0-M2 spectral type (0.4-0.8[M$_{\odot}$]{}) Upper Sco stars display an excess that indicates the presence of a gaseous disk.'
author:
- |
\
\
bibliography:
- 'master\_reference.bib'
title: 'New Pre-main-Sequence Stars in the Upper Scorpius Subgroup of Sco-Cen'
---
stars: pre-main-sequence - stars: formation - open clusters and association: individual: Sco-Cen - surveys - protoplanetary disks
Introduction {#intro}
============
The Scorpius-Centaurus-Lupus-Crux Association (Sco OB2, Sco-Cen) is the nearest location to the Sun with recent high-mass star formation [@zeeuw99]. Young OB associations, such as Sco-Cen, provide an incredible laboratory in the form of a primordial group of stars directly after formation, which can be exploited in the study of the output of star formation including searches for young exoplanets. The obvious prerequisite for such study is a level of completeness in the identification of association members that is currently not yet attained in Sco-Cen in any mass regime, other than the most massive B-type stars. Sco-Cen contains approximately 150 B-type stars [@myfirstpaper] which have been typically split into three subgroups: Upper Scorpius, Upper-Centaurus-Lupus (UCL) and Lower-Centaurus-Crux (LCC) with only the B, A and F-type membership of Sco-Cen being considered relatively complete, with some 800 members. Even in this high-mass regime, there is expected to be a $\sim$30% contamination by interlopers in the kinematic membership selections, mainly due to the lack of precision radial velocity measurements for these objects [@myfirstpaper]. Additionally, in light of the upcoming high-precision GAIA proper motions and parallaxes, a well characterised spectroscopically confirmed Sco-Cen membership will be instrumental in illuminating the substructure of the association.
Unfortunately, Sco-Cen is poorly characterised for its proximity, the reason for which is the enormous area of sky the association inhabits at low Galactic latitudes ($\sim80^\circ\times25^\circ$ or $\sim150\times50$pc). IMF extrapolation from the high-mass members implies, with any choice of IMF law, that Sco-Cen is expected to have $\sim 10^4$ PMS G, K and M-type members, most of which are, as yet, undiscovered. This implies that the vast majority of PMS ($<$20Myr) stars in the solar neighbourhood are in Sco-Cen [@preibisch02], making Sco-Cen an ideal place to search for young, massive planetary companions. Although some work has been done in illuminating the lower-mass population of Sco-Cen (see @preibisch08), the late-type membership of Sco-Cen cannot be considered complete in any spectral-type or colour range. A more complete picture of the late-type membership of Sco-Cen is the primary requirement for determining the age spread, structure, and star formation history of the association, for illuminating the properties of star formation, and for embarking on further searches for young exoplanets to better define their population statistics.
The age of the Sco-Cen subgroups has been contentious. Upper Scorpius has long been considered to be $\sim$5Myr old, however recent work has shown that it may be as old as 11Myr [@geus92; @pecaut12]. Similarly, B, A and F-type UCL and LCC members have main-sequence turn off/on ages of $\sim16-18$Myr, while studies of the incomplete sample of lithium-rich G, K and M-type members show a variety of mass-dependent age estimates. The HR-diagram age for the known K-type stars in UCL and LCC is $\sim$12Myr, the few known M-type stars indicate a significantly younger age of $\sim$4Myr, most likely due to a bias produced by a magnitude limited sample, and the G-type members have an age of $\sim$17Myr, which is consistent with the more massive stars [@preibisch08; @song12]. There is also a positional trend in the age of the PMS stars of the older subgroups, with stars closer to the Galactic Plane appearing significantly younger than objects further north. This is almost certainly the result of as yet undiscovered and un-clarified substructure within the older subgroups, which may have a very complex star-formation history.
The above is clear motivation for the identification of the full population of the Sco-Cen association, a task that will require significant observational and computational effort to complete. In this paper, we describe a new search for PMS members of the Upper Scorpius region of the Sco-Cen association. We have used statistical methods to select a sample of likely Upper Scorpius members from all-sky data, and have conducted a spectroscopic survey to determine youth and membership in the Sco-Cen association using the Wide-Field Spectrograph instrument at the Australian National University 2.3m telescope.
Selection of Candidate Members
==============================
We have selected candidate Upper Scorpius members using kinematic and photometric data from UCAC4, 2MASS, USNO-B and APASS [@ucac4; @2mass; @usnob; @apass]. A purely kinematic selection of the low-mass members of Sco-Cen is not sufficient to assign membership to G,K and M-type stars because the quality of the astrometric data available would produce an interloper contamination much higher than would be acceptable for future studies using Sco-Cen as an age benchmark. In order to clearly separate young Upper Scorpius members from field stars, spectroscopic follow-up is needed to identify stellar youth indicators. We employed two separate selection methods to prioritise targets based on kinematic and photometric data.
The first selection used was based on the Bayesian Sco-Cen membership selection of @myfirstpaper, which uses kinematic and spatial information to assign membership probabilities. We further developed this method to apply to K and M-type stars, in order to properly treat the absence of a parallax measurement. We took the proper-motions from the UCAC4 catalog [@ucac4] and photometry from 2MASS and APASS [@2mass; @apass], and used the photometry and a premain-sequence isochrone [@siess00] to estimate each candidate member’s distance. We then treated the proper-motion and estimated distance together to calculate the membership probability. This selection was magnitude limited, and covered all stars in the UCAC4 catalog with 10$<$V$<$16, and comprised of $\sim$2000 candidate members with membership probability greater than 2%. For a more complete explanation of the Bayesian selection, including information from [@myfirstpaper] and the changes adopted for use with the K and M-type star data see Appendix \[bayesapp\].
The second selection was based on the selection used for the Coma-Ber cluster in the study of @kraushillenbrand_comaber, and was designed to select targets for the Upper Scorpius field of the Kepler K2 campaign. Targets which were both placed above the main-sequence based on photometric distance estimates, and had proper-motions consistent with Upper Scorpius membership were deemed to be potential members and included in the observing sample. This selection spanned F to late M-type stars, with targets falling on Kepler silicon prioritized for spectroscopic follow-up. This selection is considerably more conservative than the Bayesian selection, and includes a much larger number of candidates. Where the two selections overlap, we have $>$90% of the Bayesian selected stars included in the sample. Our final, combined sample was then drawn from both of the above selection; we include a candidate in the final target list if it was identified by either method. Figure \[selection\_pm\] displays the proper motions of the selected stars from both samples.
![The proper motions of the candidate Upper Scorpius members selected by both the @kraushillenbrand_comaber selection method (black points) and the Bayesian method (purple circles).[]{data-label="selection_pm"}](sample_pm-eps-converted-to.pdf){width="45.00000%"}
In light of the currently ongoing Galactic Archeology Survey, using the HERMES spectrograph on the Anglo-Ausralian Telescope [@zuckerhermes2012], which will obtain high-resolution optical spectra in the coming years for all stars in the Sco-Cen region of sky, down to V=14, we have decided to primarily observe targets in our sample fainter than this limit. While our selection methods identified candidate Upper Scorpius stars across the entire subgroups $(342^\circ<l<360^\circ, 10^\circ<b<30^\circ)$ in our observations we strongly favored candidate members which fell upon the Kepler K2 field 2 detector regions, which covers the majority of the centre of Upper-Scorpius with rectangular windows. As such, the spatial distribution of this sample will not reflect the true substructure of Upper Scorpius. We observed all the targets in our K2 sample with Kepler interpolated V magnitudes of $(\sim13.5<V_{jk}<15)$, as well as some further brighter targets. In total, we obtained optical spectra for 397 candidate Upper Scorpius K and M-type stars. The full list of observed candidate targets, including both those stars determined to be members and non-members, can be found in Table \[obstable\_lowmass\], along with proper motions, computed Bayesian membership probability, integration time, and SNR in the continuum near H-$\alpha$.
------------- ------------- ------- ------- ------- -------------- -------------- -------- -------------------- ------- ----- ----
R.A. Decl. V K $\mu_\alpha$ $\mu_\delta$ T
(J2000.0) (J2000.0) MJD (mag) (mag) (mas) (mas) Source P$_{\mathrm{mem}}$ (sec) SNR M?
15 39 06.96 -26 46 32.1 56462 12.5 8.7 -35.3 -41.7 a 31 90 131 Y
15 37 42.74 -25 26 15.8 56462 13.5 9.7 -14.6 -26.7 a 85 90 80
15 35 32.30 -25 37 14.1 56462 11.7 8.4 -9.0 -22.9 a 69 90 116
15 41 31.21 -25 20 36.3 56462 10.0 7.2 -16.9 -28.7 a 86 90 151 Y
------------- ------------- ------- ------- ------- -------------- -------------- -------- -------------------- ------- ----- ----
Spectroscopy with WiFeS
=======================
The Wide-Field Spectrograph (WIFES) instrument on the Australian National University 2.3m telescope is an integral field, or imaging, spectrograph, which provides a spectrum for a number of spatial pixels across the field of view using an image slicing configuration. The field of view of the instrument is 38$\times$25 arcseconds, and is made up of 25 slitlets which are each one arc second in width, and 38 arcseconds in length. The slitlets feed two 4096$\times$4096 pixel detectors, one for the blue part of the spectrum and the other for the red, providing a total wavelength coverage of 330 - 900$\mu$m, which is dependent on the specific gratings used for the spectroscopy. Each 15 micron pixel corresponds to 1$\times$0.5 arcseconds on sky.
There are a number of gratings offered to observers for use with WiFeS. For identification of Upper Scorpius members, we required intermediate-resolution spectra of our candidate members, with a minimum resolution of $\sim$3000 at the Li 6708Å line, and so selected the R7000 grating for the red arm and the B3000 grating for the blue arm, which was used solely for spectral-typing. This provided $\lambda / \Delta\lambda\sim7000$ spectra covering the lithium 6708Å and H-$\alpha$ spectroscopic youth indicators. A dichroic, which splits the red and blue light onto the two arms of the detector, can be position either at 4800Å or 5600Å. For the first three successful observing nights we use the dichroic at 4800Å which produced a single joined spectrum from 3600 to 7000AA. For the remaining 7 nights, we position the dichroic at 5600Å which produces two separate spectra, with the the blue arm covering 3600 to 4800Å and the red arm covering 5300 to 7000Å. This change was made to accommodate poor weather backup programs being simultaneously carried out, which will be the subject of future publications. To properly identify members, we required a $3\sigma$-detection of a 0.1Å equivalent width Li line, which corresponds to a signal-to-noise ratio of at least 30 per pixel. In order to achieve this, we took exposures of 5 minutes for R=13 stars (approximately type M3 in Upper Scorpius), and binned by 2 pixels in the y-axis, to create 1$\times$1" spatial pixels and reduce overheads. With overheads we were able to observe 10 targets an hour in bright time, or $\sim$80-90 targets per completely clear night.
In total we obtained 18 nights of time using WiFeS, split over 2013 and 2014, however the majority of the 2013 nights were unusable due to weather. Our first two observing runs, in June 2013, and April 2014 yielded one half-night of observations each, and our final observing run yielded seven partially clear nights. During our first two nights, we positioned the dichroic at 5500Å, and during the June 2014 observing run, positioned the dichroic at 4600Å, which provides more of the red arm, because this mode was deemed better for obtaining radial velocities of B, A and F-type Sco-Cen stars, which we observed as backup targets during poor weather, and will be the subject of a future publication.
Data Reduction
==============
The raw WiFeS data was initially reduced with a pre-existing Python data reduction software package called the “WiFeS PyPeline”, which was provided to WiFeS observers. The purpose of the software is to transform the CCD image, which consists of a linear spectrum for each spatial pixel of the WiFeS field of view, into a data cube. This involves bias subtraction, flat-fielding, bad pixel and cosmic ray removal, sky subtraction, wavelength calibration, flux calibration, reformatting into the cube structure, and interpolation across each pixel to produce a single wavelength scale for the entire image. On each night, we observed at least one flux standard from @bessellflux99, which are included in the data reduction pipeline as flux calibrator objects. Once this process is complete, the user is left with a single cube for each object observed, with dimensions 25”$\times$38”$\times$3650 wavelength units. For the grating resolutions and angles used in our observations, we obtained spectral coverage from $3200-5500$Å in increments of 1.3Å in the blue arm, and $5400-7000$Å in increments of 0.78Å in the red arm.
\
Following the standard WiFeS reduction procedure, we continued with a further custom reduction, the aim of which was to measure the centroid position of the target object in each wavelength, such that the presence of H-$\alpha$ emitting low-mass stellar companions, outflows, and H-$\alpha$-bright planetary mass companion could be detected by the measurement of a wavelength-dependent centroid shift. This consisted of determining a best fit point spread function (PSF) model for the spatial image in a clean section of the spectrum, and then measuring the centroid shift of this PSF at each wavelength along the spectrum. An additional benefit of this is a more accurate sky subtraction, and an integrated spectrum of each object, which can be used to measure equivalent widths of key spectral lines. The results of the centroid measurements and any detected companions will be reported in a further publication.
We first cut out a 10” by 10” wide window (10$\times$10 pixels), centered on the target. The vast majority of the stellar flux is contained within the central 3” by 3” region of the windowed image, and so the adopted width of 10” allows a clear region of background around the target; Figure \[subim\_example\] provides an illustration of the data. We then fit a Moffat point spread function [@racine96] to a region of the spectral continuum which does not include any spectral features, but is close to the H-$\alpha$ line. This region consisted of 400 spectral units, spanning $6368-6544$Å. Figure \[spect\_example\] displays the spectral region used for the initial PSF fit, as well as the H-$\alpha$ and Li 6708Å lines for one target in our sample, 1RXS J153910.3-264633, which shows strong indications of youth.
The particular model that we fit to the spatial image is given by;
$$\mathrm{PSF} = \mathrm{S} + \mathrm{F} \frac{(2^{\frac{1}{\beta}}-1)(\beta-1)}{\pi w^2(1 + (2^{\frac{1}{\beta}}-1)(\theta/w)^2)^{\beta}},
\label{moffat}$$
where S indicates the sky contribution to the flux, $\beta$ is an integer parameter that determines the strength of the wings of the Moffat PSF, $\theta$ is the distance from the centre of the profile, $w$ is the half width of the Moffat PSF, and F is the stellar flux. Given that we have a two dimensional PSF, and that each dimension has a different Moffat function half width, we require two different values of $w$. We create this two dimensional Moffat profile by scaling $\theta$ appropriately;
$$\theta = w_x^2(x-x_0)^2 + w_y^2(y-y_0)^2,
\label{moff_scale}$$
where $w_x$ and $w_y$ are the PSF width parameters in each dimension, $(x,y)$ is the position of a given point on the image, and $(x_0,y_0)$ is the image centroid. Inputting this value of $\theta$ into a Moffat function with width $w=1$ will thus produce the desired asymmetric two dimensional profile.
![The full WiFeS integrated spectrum produced by first processing with the WiFeS Pypeline, and then our spectro-astrometric analysis for the stars USco 48, a known member of the Upper Scorpius subgroup, and 2MASS J16232454-1717270, high probability candidate, and a new member in identified in our survey. The USco 48 spectrum is an example of the data from the 4800Å dichroic setup, and the 2MASS J16232454-1717270 spectrum an example of the 5600Å dichroic setup.[]{data-label="fullspec_example"}](fullspec_examples-eps-converted-to.pdf){width="48.00000%"}
We found that $\beta=4$, a value which describes most telescope PSFs, yielded the closest fit to our data. We also attempted to fit a Gaussian profile to the spatial images, in the same format as the Moffat profile described in equation \[moffat\]; however the Gaussian model produced consistently poorer fits to the data than the Moffat model, particularly in the wings of the PSF, with typical values of $\chi^2_r\sim 4$ for the Gaussian model fit and $\chi^2_r\sim 2$ for the Moffat model. On the basis of the goodness of fit difference, we adopted the Moffat model exclusively in our analysis. For each target observed, we used the continuum spectral region between $6368-6544$Å to determine the parameters of the Moffat PSF that most closely reproduced the spatial images. We then fixed the half width parameters in each dimension, and fit our PSF model to each individual wavelength element image along the spectrum to determine $S$, $F$ and the centroid position for each wavelength. This process provides two useful characteristics, the first of which is the integrated spectrum $(F)$ of the target (see Figure \[fullspec\_example\]), with the sky component $(S)$ subtracted out. Using the cleaned output spectra, we then computed equivalent widths of both the Li 6708Å and H-$\alpha$ lines for each observed star. The second useful characteristic is the centroid position of the star image at each wavelength interval in the spectrum. This can be used to detect accreting stellar and substellar companions by the measurement of a centroid shift in the H-$\alpha$ line image. An analysis of the centroid positions will be presented in a future publication.
Spectral Typing
---------------
We spectral type the reduced spectra created by the centroid-fitting procedure using spectral template libraries as reference. It is also important to incorporate extinction into the spectral-typing procedure for Upper Scorpius, given the typical values of $0.5<$A$_V<2.0$. If an extinction correction is omitted, spectral typing will produce systematically later spectral types for the members. A combination of two template libraries was chosen for the spectral typing, with spectral types earlier than M0 taken from the @pickles98 spectral template library, and the M-type templates taken from the more recent @bochanski_templates.
To carry out the spectral typing, we first computed reduced $\chi^2$ values for each data spectrum on a two-dimensional grid of interpolated template spectra and extinction, with spacing of half a spectral sub-type and 0.1 magnitudes in $E(B-V)$. This was done by first interpolating the template spectra onto the wavelength scale of the data, and then applying the particular amount of extinction according to the @savage_mathis79 extinction law. We also removed the H-$\alpha$ region in the data spectra, because the prevalence of significantly larger H-$\alpha$ emission in young stars will not be adequately reproduced by the templates. The spectral type - extinction point on the grid with the smallest reduced $\chi^2$ was then used as a starting point for least squared fitting with the IDL fitting package MPFIT. The fitting procedure used the same methodology as the grid calculations, with the addition of interpolation between template spectra to produce spectral sub-type models for use in the fitting.
We find the limiting factor in spectral-typing our young Sco-Cen stars to be the fact that the spectral template libraries are built from field stars, and so are not ideal for fitting young, active stars. Hence, while we typically have spectral type fits better than half a spectral sub-type, we report spectral types to the nearest half sub-type, and values of A$_V$ with typical uncertainties of 0.2 magnitudes.
The New Members
===============
------- ------------------- ------------- ------------- -------- ------------------- --------------- ------------------------------------ ------ ------------------
R.A. Decl. EW(Li) $\sigma_{EW(Li)}$ EW(H$\alpha$) $\sigma_{\mathrm{EW(H_{\alpha})}}$ A$_{\mathrm{V}}$
Name 2MASS (J2000.0) (J2000.0) (Å) (Å) (Å) (Å) SpT (mag)
RIK-1 J15390696-2646320 15 39 06.96 -26 46 32.1 0.46 0.02 -1.22 0.03 M0.5 0.2
RIK-2 J15413121-2520363 15 41 31.21 -25 20 36.3 0.40 0.01 -2.70 0.04 K2.5 0.1
RIK-3 J15422621-2247458 15 42 26.21 -22 47 46.0 0.46 0.04 -3.08 0.07 M1.5 0.3
RIK-4 J15450970-2512430 15 45 09.71 -25 12 43.0 0.61 0.02 -2.02 0.04 M1.5 0.4
------- ------------------- ------------- ------------- -------- ------------------- --------------- ------------------------------------ ------ ------------------
Table \[obs\_res\_tab\] lists both the Li 6708Å and H$\alpha$ equivalent widths, and the estimated spectral types and extinction for the new Upper Sco members, and figure \[newmems\_pos\] shows the spatial positions of the new members. We have defined a star as an Upper Scorpius member if the measured equivalent width of the Li 6708Å line was more than $1-\sigma$ above 0.1Å. While this Li threshold is low, it is significantly larger than the field Li absorption, and is in general keeping with previous surveys. The use of this threshold is further justified given the effects of episodic accretion on Li depletion in the latest models [@baraffe10]. In general, the vast majority of the identified members have Li 6708Å equivalent width significantly larger than 0.2Å and so are bonafide young stars. In total we identify 257 stars as members based on their Li 6708Å absorption, 237 of which are new.
The proper-motions of the new members, which were calculated from various all-sky catalogs, or taken from the UCAC4 catalog are shown in Figure \[pm\_all\]. The members have proper motions that overlap the Upper Scorpius B, A and F-type members proper motions (blue crosses), although a significantly large spread is seen. This is consistent with the average uncertainty of $\sim$2-3mas/yr for the K and M-type proper motions.
Figure \[ewli\_all\] displays the Lithium equivalent widths for the identified members as a function of spectral type. The majority of our members are M-type, and we see a sequence of equivalent width with a peak at spectral type M0, and a systematically smaller equivalent width in the M2-M3 range compared to earlier or later M-type members. This is expected as the mid-M range is modelled to show faster Lithium depletion timescale [@dantona94].
Interestingly, we also observe a clear spread in the equivalent width of the Lithium 6708Å line. Figure \[ewli\_zoom\] shows the just the M0 to M5 spectral type range. At each spectral type we see a typical spread of $\sim$0.4Å in Li equivalent width, and a median uncertainty in the equivalent width measurements of $\sim$0.03Å. This implies a $\sim$10-sigma spread in EW(Li) at each spectral type. Wether or not this spread is caused by an age spread in Upper Scorpius is difficult to determine: we have examined the behaviour of EW(Li) as a function of spatial position, both in equatorial and Galactic coordinate frames and found no significant trend. We note that a similar spread of EW(Li) for M-type Upper Scorpius members was observed by @preibisch01. Given the lack of correlation with spatial position, if the EW(Li) spread is caused by an age spread among the members, then the different age populations are overlapping spatially and may not be resolvable without sub-milliarcsecond parallaxes.
In Figure \[ewha\_all\], we display the measured H-$\alpha$ equivalent widths for the members. The majority of the PMS members show some level, of H-$\alpha$ emission, with a clear sequence of increasing emission with spectral type. In combination with the presence of Lithium, this is a further indicator of the youth of these objects. Of our 257 members, $\sim$95% show H-$\alpha$ emission with (1Å$<$EW(H-$\alpha$)$<$10Å), and only 11 of the members do not show emission in H-$\alpha$. All of these 11 members without H-alpha emission are earlier than M0 spectral-type. There are also 35 non-members with H-$\alpha$ emission. Given the values of EW(H$\alpha$) for the M-type members we have identified, the majority of them appear to be weak lined T-Tauri stars and $\sim$10% are Classical T-Tauri stars (CTTS) with EW(H$\alpha$) $>$ 10Å. This proportion agrees with previous studies of Upper-Scorpius members [@walter94; @PZ99; @preibisch01], which find a CTTS fraction of between 4 and 10% for K and M-type Upper-Scorpius stars.
The Efficiency of the Bayesian Selection Algorithm
==================================================
The selection methods we have used to create our target list provide a significant improvement of member detection rate when compared to what can be achieved from simple color-magnitude cuts. We see a large overall identification rate of $\sim$65% for our sample of observed stars. Using the membership probabilities computed for the stars we have observed, we expected that 73$\pm$7% of the observed stars would be members, which agrees with the observed members fraction of 68%. We also find that as a function of computed membership probability, the fraction of members identified among the sample behaves as expected. Figure \[bayes\_hist\] displayed the membership fraction as a function of probability.
Given that our probabilities have been empirically verified to provide a reasonable picture of Upper Scorpius membership, we can derive an estimate for the expected number of M-type members in the subgroup by summation of the probabilities. We find that the total expected number of Upper Scorpius members in the $\sim$0.2 to 1.0[M$_{\odot}$ ]{}range, or late-K to $\sim$M5 spectral type range, is $\sim$2100$\pm$100 members. This agrees with initial mass function estimates which indicate that there are $\sim$1900 members with masses smaller than 0.6[M$_{\odot}$ ]{}in Upper Scorpius [@preibisch02].
![$\mathrm{EW}(\mathrm{H-}\alpha)$ for the new members (black) and the non-members (red). The members follow a clear sequence with H-$\alpha$ increasing with spectral-type. In the K spectal types, we see that non-members show H-$\alpha$ absorption which is generally stronger than that seen in the members, some of which show weak emission.[]{data-label="ewha_all"}](EWHA_all_linear-eps-converted-to.pdf){width="47.00000%"}
The HR-Diagram of the Members
=============================
With the spectral types and extinctions we have determined for the members using the @bochanski_templates and @pickles98 spectral libraries, we can place them on a HR-diagram in the model parameter space. There is significant variability in synthesized photometry between different models for PMS stars, making comparison in the color-magnitude space difficult. Furthermore, the most reliable magnitudes for M-type stars are the near-IR 2MASS photometry, which show minimal variation in the M-type regime where the PMS is near vertical. Instead, we use the spectral types and the empirical temperature scale and J-band bolometric corrections for 5-30Myr stars produced by @pecaut13 and we further correct for extinction using our fitted values of A$_V$ from the spectral typing process, and the @savage_mathis79 extinction law. The resulting HR diagram can be seen in Figure \[cmd\_all\]. We have also superimposed five BT-Settl [@btsettl] isochrones of ages 1, 3, 5, 10 and 20Myr onto the HR-diagram at the typical Upper Scorpius distance of 140pc [@myfirstpaper]. These particular models were chosen because they were used by @pecaut13 in the generation of their temperature scale, and so any relative systematic differences between the models and the temperature scale will most likely be minimized.
Upon initial inspection, it appears that for a given temperature range, the Upper Scorpius members inhabit a significant spread of bolometric magnitudes. This is most likely highly dominated by the distance spread of the Upper Scorpius subgroup, which has members at distances between 100 and 200pc, corresponding to a spread in bolometric magnitude of $\sim$1.5mag between the nearest and furthers reaches of Upper Scorpius. Using the distance distribution of the @myfirstpaper high-mass membership for Upper Scorpius, we find that the expected spread in bolometric magnitude due to distance which encompasses 68% of members is approximately $+0.33$ and $-0.54$ magnitudes. Similarly, unresolved multiple systems can bias the sample towards appearing younger by an increase in bolometric magnitude of up to $\sim$0.7mags for individual stars.
![Fraction of stars identified as members plotted against membership probability computed with our Bayesian selection algorithm. The red line represents the ideal fraction of detected members. We see a very close agreement between the computed membership probability and the fraction of stars which were confirmed as members.[]{data-label="bayes_hist"}](bayes_results_hist-eps-converted-to.pdf){width="45.00000%"}
In the later spectral types, beyond $\log{T_{\mathrm{eft}}}=3.52$ we also begin to see the effects of the magnitude limit of our survey, which operated primarily in the range $13.5<$V$<15$ and so only the brightest, and hence nearest and potentially youngest late M-type members in our original target list were identified, although significant Li depletion at these temperatures is not expected to occur until ages beyond 50Myr. Even with distance spread blurring the PMS in Upper Scorpius, we can see that most of the members appear to be centered around the 5-10Myr age range in the earlier M-type members.
We have also indicated the measured EW(Li) values for the members on the HR-diagram as a color gradient, with darker color indicating a smaller EW(Li). The scale encompasses a range of $0.3<$EW(Li)$<0.7$Å, with values outside this range set to the corresponding extreme color. There is a marginal positional dependence of HR-diagram position with EW(Li): we see that, in particular for the earlier M-type members, the larger values of EW(Li) (light orange) are more clustered around the 3-5Myr position, while the smaller values of EW(Li) (dark red) are clustered closer to 5-10Myr. This could indicate the presence of a spread of ages, or populations of different age in the Upper Scorpius subgroup.
![HR diagram for the Upper Scorpius members we have identified, with bolometric corrections and effective temperatures taken from the @pecaut13 young star temperature-color scale. The blue lines are the BT-Settl isochrones [@btsettl] of ages 1, 3, 5, 10 and 20Myr placed at the typical distance to Upper Scorpius of 140pc. The color of each point indicates the measured EW(Li) for the star, with darker color indicating a lower EW(Li). The color range spans $0.3<$EW(Li)$<0.7$ linearly, with values outside this range set to the corresponding extreme color. The uncertainties are determined by the accuracy of our spectral typing methods, which is typical half a spectral sub-type.[]{data-label="cmd_all"}](mems_hrd-eps-converted-to.pdf){width="48.00000%"}
There is some other evidence of different age populations in the Upper Scorpius subgroup: The existence of very young B-type stars, such as $\tau$-Sco, and $\omega$-Sco which have well measured temperatures and luminosities that indicate an age of $\sim$2-5Myr [@simondiaz06] support a young population in Upper Scorpius. The B0.5 binary star $\delta$-Sco is also likely to be quite young ($\sim$5Myr) [@code76]. @pecaut12 place it on the HR diagram at and age of $\sim$10Myr, however, due to the rapid rotation and possible oblate spheroid nature of the primary, the photometric prescriptions for determining the effective temperature and reddening of the primary used by @pecaut12 are likely to fail for this object. The spectral type is more consistent with a temperature of $\sim$30000K. Additionally, the presence of other evolved B-type stars is evidence for an older population [@pecaut12]. Furthermore, the recent age estimate of 13Myr for the F-type members of Upper Scorpius by @pecaut12 further supports an older population in the subgroup. If the HR diagram position on EW(Li) that we observe among our members is real than this also supports multiple age population in Upper Scorpius.
\
\
Disk Candidates
===============
We have also obtained the Wide-Field Infrared Survey Explorer (WISE) infrared photometry [@wise10], from the ALLWISE version of the catalog, for the observed candidate members in order to determine the prevalence of circumstellar disks among our new members. The identification of new populations of stars bearing disks is valuable because it provides extension to the current samples used in the study of disk property measurements and disk evolution. The AllWISE catalog provides photometry in four bands W1, W2, W3 and W4, with effective wavelengths of 3, 4.5, 12 and 22$\mu$m respectively. The W2 and W3 photometry is effective for tracing the presence of an inner disk, while excess in the W4 band photometry can indicate the presence of a colder, outer disk or transitional disk.
We queried the ALLWISE catalog for the positions of the 397 stars we observed from our sample, including 237 new members, with a search radius of 5". The search returned 395 matches with varying levels of photometric quality. We then placed each star on three spectral type-color diagrams incorporating 2MASS [@2mass] K-band photometry, these were K-W2, K-W3, and K-W4. Past studies have used both K, and W1 as the base photometry for building color-color diagrams [@carpenter06; @carpenter09; @rizzuto12; @luhman10; @luhman2012_disk]. Typically, the presence of a disk within $\sim3$AU of a host star increases the brightness in the IR wavelengths, with $\sim$5$\mu$m being the approximate wavelength where the disk dominates in brightness. Both the W1 and K magnitudes are long enough such that reddening is not a significant issue, but also shorter than the expected point of disk domination. We found that examining the WISE bands relative to the K magnitude produced a better separation of disk bearing stars from photospheric emission, and so we report the analysis in terms of this methodology.
Figure \[excess\_figs\] displays the three spectral type-color diagrams. We excluded any WISE photometry in a given band that was flagged as having a signal to noise ratio of $<4$, as a non-detection, or flagged as being contaminated by any type of image artifact in the catalog. This resulted in the exclusion of 56, 10 and 312 objects in the W2, W3, and W4 bands respectively. The primary source of the exclusions for the W4 band was non-detection or low signal to noise at 22$\mu$m, and most of the exclusions in the W2 and W3 bands were due to contamination by image artifacts. To reduce contamination by extended sources we also excluded any object flagged as being nearby a known extended source or with significantly poor photometry fits, there were eight such objects. The WISE band images for these stars were then inspected visually to gauge the extent of contamination. We found the three of the objects were not significantly effected by the nearby extended source, and so included them in the analysis. After excluding these objects we were left with 333, 379 and 77 objects with photometry of sufficient quality in the W2, W3 and W4 bands respectively.
Due to the age of Upper Scorpius of $\sim$10Myr, the majority of members no longer possess a disk, providing sufficient numbers of stars to clearly identify photospheric emission. Hence the photosphere color can be determined from the clustered sequence in the spectral type-color diagrams. We fit a straight lines in the K-W3 and K-W4 WISE band colors, and a disjointed line in K-W2, and then place a boundary where the photospheric sequence ends. For K-W3 the boundary line is given by the points (K0,0.27) and (M5,0.8) and for K-W4 the points (K0,0.56) and (M5,1.6).The sloped part of the boundary line for K-W2 is defined by the points (M0,0.21) and (M5,0.46), and the flat section by K-W2 $=0.21$, for spectral types earlier than M0. These boundaries are shown as black lines in Figures \[excess\_figs\]. Stars with color redder than these boundaries we deem to display an excess in the particular WISE band. Upon inspection, we find that our placement of the end of the photospheric sequence is closely consistent with that of [@luhman2012_disk]. In the K-W4 color, we find that for stars of spectral type later than $\sim$M2, the photospheric emission in W4 is undetectable by WISE.
For those stars which displayed excesses in any combination of WISE bands, we visually inspected the images to exclude the possibility that the excesses could have been caused by the presence of close companions or nebulosity. We also found that in a few cases background structure in the W4 image could cause the appearance of an excess, although this effect was largely mitigated by our signal-to-noise cutoff. We rejected 23 of the excess detections after inspection, 12 of which were caused by background structure or nearby nebulosity, and 11 of which were due to blending with nearby stars. We further excluded any object which shows an excess in only the K-W2 color as likely being produced by unresolved multiplicity. After these rejections, 27 stars remained with reliable excess detections. Additionally, a single object, 2MASS J16194711-2203112 , displayed an excess in K-W3, but had a W4 detection with signal-to-noise of 3.5. Upon inspection of the corresponding W4 band image, we included it as exhibiting an excess in K-W4.
[ccccccccc]{} & R.A. & Decl. & & & & & &\
2MASS & (J2000.0) & (J2000.0) & M & E & D & W2 & W3 & W4\
\
We can classify the disk types by the amount of excess displayed in different colors compared to the photosphere. We adopt disk type criteria in the E(K-W4), E(K-W3) space consistent with those described in @luhman10 and @luhman2012_disk, which identify four different categories of disk: Full or primordial disks, transition disks, evolved disks, and debris or evolved transition disks. Primordial or full disks exhibit strong emission across the entire IR spectral range. Transition disks are structurally different in that they have a significant cleared inner hole, which is visible as a weaker emission at the shorter IR wavelengths, but still relatively bright in the longer IR wavelengths. Evolved disks do not show a gap in IR emission, but have started to become thinned and appear fainter at all IR wavelengths than unevolved full disks, with a steady decline in IR excess with age [@carpenter09]. Debris disk and evolved transition disk have similar IR SED’s, showing only weak excesses at the longer IR wavelengths. Figure \[wk4wk3\] show both E(K-W4) and E(K-W3) for the stars identified as having displaying an excess. The lines in Figure \[wk4wk3\] bound the different regions populated by the various disk types. We classify all objects with excesses in W3 and W4 beneath the dashed line to be debris or evolved transition disks candidates, and the objects above the solid line to be full disks. Stars with excess between these two lines we classify as evolved disk candidates. Finally, we identify the two objects with a large W4 excess, but W3 excesses too small to be classified as full disks, as transition disk candidates. Table \[excess\_table\] lists the excess status for the stars with detected excesses.
In total, we identify 26 of the Upper Scorpius members as displaying a disk-indicating excess with spectral types later than K0, and one star without significant Lithium absorption that also displays an excess. This latter object is an F4.5 spectral type object,, HD-145778, with $EW(Li)=0.09\pm0.2$. The presence of some Lithium absorption, combined with the disk presence mean that this object can be considered to be a member of Upper Scorpius. We have included it as a member at the end of Table \[obs\_res\_tab\]. HD-145778 is not in this HIPPARCOS catalog [@lindegren97], potentially explaining why it was not included in past memberships.
Due to the WISE detection limit in the W4 band, we are almost certainly not able to identify the vast majority of the evolved transitional and debris disks, which show only a small color excess in K-W4. Indeed, we only detect two such disks in our sample, one of which, USco 41, was previously identified with Spitzer photometry [@carpenter09], when significantly more are expected from previous statistics [@carpenter09; @luhman2012_disk]. Furthermore, it is likely that a number of evolved disks around stars of spectral type later than $\sim$M3 are not detected here. For this reason it is difficult to meaningfully estimate the disk or excess fraction for our entire sample. In the M0 to M2 spectral type range, where we expect the majority of the full, evolved and transitional disks to be detectable by WISE, we have 11 disks, 6 of which are full, 4 evolved and 1 transitional. Excluding all those members flagged for extended emission, confusion with image artifacts, or unreliable excesses, we find and excess fraction of 11.2$\pm$3.4%. @carpenter09 found a primordial disk fraction for M-type Upper Scorpius members of $\sim$17%, and @luhman2012_disk find excess fractions of 12% and 21% for K-type and M0 to M4-type members respectively. Given the strong increase in excess fraction towards the late M-type members and the potential for some missed evolved disks due to the WISE detection limits, we find that our excess fraction estimate is consistent with these past results.
Conclusions
===========
We have conducted a spectroscopic survey of 397 candidate Upper Scorpius association K and M-type members chosen through statistical methods, and revealed 237 new PMS members among the sample based on the presence of Li absorption. We also identify 25 members in our sample with WISE near-infrared excesses indicative of the presence of a circumstellar disk, and classify these disk on the basis of their color excess in different WISE bands. We find that the members show a significant spread in EW(Li), and upon placing the members on a HR diagram, we find that there is a potential age spread, with a small correlation between EW(Li) and HR-diagram position. This could indicate the presence of a distribution of ages, or multiple populations of different age in Upper Scorpius.
|
Optical isolators are fundamental components of many laser systems as they prevent unwanted feedback. Such devices consist of a magneto-optic active medium placed in a magnetic field such that the Faraday effect can be exploited to restrict the transmission of light to one direction. For an applied axial field $B$ along a medium of length $L$, the Faraday effect induces a rotation $\theta$ for an initially linearly polarized light beam, where $\theta = VBL$, and $V$ is the Verdet constant. An optical isolator is realized when such a medium is positioned between two polarizers set at $\pi$/4 to each other, with an induced rotation of $\theta=\pi$/4; this arrangement provides high transmission in one direction and isolation in the other.
The technologies of atomic Micro-Electro-Mechanical Systems (MEMS) will eventually be required to create lighter and more compact components `` for use in free-space laser communications, ocean measurements and telecommunications. Currently there is much interest in small, reliable low power laser systems (VCSEL) ``, fabrication of chip-sized alkali-vapor cells `` and gas atoms in hollow core fibers `` for atomic frequency references `` and magnetometers ``. Other applications include gyroscopes ``, laser frequency stabilization `` and atomic sensors `` for cold-atom devices, accelerometers and gravimeters. Here we show that using similar technologies one can envisage a light, compact, high-performance permanent-magnet isolator.
Isolators require large Verdet constants whilst maintaining a small absorption coefficient $\alpha$, hence the figure of merit (FOM) for an isolator is the ratio $V/\alpha$. Commercial isolators often use terbium gallium garnet (TGG), with yttrium iron garnet (YIG) also used in the IR region. Table $\ref{tableconstants}$ shows the Verdet constants and FOM for TGG, YIG and Rb vapor at 780 nm. Note that although the Verdet constant of YIG is much larger than TGG, the latter is used at 780 nm as the performance of the former is strongly compromised by the poor transparency of YIG below 1100 nm ``. Note also that the Verdet constant and absorption coefficient of the atomic vapor are strongly frequency dependent, unlike the crystal media. The FOM for Rb vapour is less than TGG, the much higher Verdet constant however allows a more compact design, as the same rotation is achieved over a much shorter optical path. The frequency dependence of the dichroic and birefringent properties of atomic vapors have also been exploited to realize narrowband atomic filters; see e.g. `` and dichroic beam splitter ``.
Material $V$ (rad T$^{-1}$ m$^{-1}$) FOM (rad T$^{-1}$)
-------------------- ----------------------------- --------------------- --
TGG `` 82 1 $\times$ 10$^{3}$
YIG `` 3.8 $\times$ 10$^{2}$ 2.5
Rb vapor 1.4 $\times$ 10$^{3}$ 1 $\times$ 10$^{2}$
\[tableconstants\]
: Verdet constants and FOMs for the three magneto-optic materials: TGG, YIG and Rb vapor (this work), at a wavelength of 780 nm.
The absolute susceptibility of an atomic vapor can be accurately modeled allowing one to predict both the absorption `` and rotation ``. Furthermore the model can be extended to include binary-collisions ``, and an axial magnetic field ``. In this Letter we use the model for absolute susceptibility for the Rb D$_{2}$ line to predict the performance of an optical isolator based on a compact magnet producing a field of the order of 0.6 T. At such fields the vapor becomes transparent to light resonant with the D$_{2}$ line due to large Zeeman shifts. We measure rotation and transmission and confirm the theoretical predictions and thereby demonstrate the feasibility of using resonant atomic media for optical isolation. Although here we focus on the Rb D$_{2}$ line, using the same principle, resonant atomic isolators can be implemented for other atomic vapors.
Figure \[MainSetup\] shows a schematic of the experimental apparatus along with details of the neodymium magnet. An external cavity diode laser was used to scan across the Rb D$_{2}$ transition (5$^{2}$S$_{1/2}$ $\rightarrow$ 5$^{2}$P$_{3/2}$) at a wavelength of 780 nm. After passing through a polarization beam splitter (PBS) the output beam was linearly polarized along the horizontal direction with a 1/e$^{2}$ radius of 80 $\mu$m. The method adopted for calibrating the frequency axis is described in ``. A weak probe beam `` traverses a 1 mm heated cell containing isotopically pure $^{87}$Rb and buffer gas with a pressure of several Torr ``. An aluminium holder with the same design as in `` was used to hold a neodymium magnet, where the cell was held in an oven allowing the laser beam to pass through. The field profile of the neodymium magnet was measured with a Hall probe and is shown in figure $\ref{MainSetup}$. Over the length of the cell the field was uniform at the 2 $\%$ level. After traversing the cell and magnet, a second PBS was set to $\pi$/4 allowing high transmission along $+z$ for an induced rotation of $\pi/4$. With this arrangement one would also expect isolation along $-z$.
![Schematic of the experimental apparatus and measured magnetic field profile through the center of the neodymium magnet. The dimensions and mass of a typical magnet used in this investigation are also shown. A beam passes through a polarization beam splitter (PBS) providing linearly polarized light along the horizontal axis. The beam then passes through a heated micro-fabricated cell held in a magnet (M) which provides an axial field. A second PBS is set to $\pi$/4 to allow high transmission, $T_{+}$, along ${+z}$ and low transmission, $T_{-}$, along ${-z}$.[]{data-label="MainSetup"}](MainSetup.eps){width="47.00000%"}
Figure \[energydiagram\] shows the absolute transmission spectra for the Rb D$_{2}$ line through (a) a natural-abundant (72$\%$ $^{85}$Rb, 28$\%$ $^{87}$Rb) cell in the absence of an applied magnetic field and (b) a $^{87}$Rb cell in the presence of an applied magnetic field of 0.576 T. For this field (the hyperfine Paschen-Back regime ``) the Zeeman shift is large compared with the hyperfine splitting of the ground and excited terms, and m$_{J}$ is the good quantum number. In (b) we observe the 16 absorption peaks corresponding to the $\Delta$m$_{J}$ = $\pm$1 transitions. Magnetic fields of such magnitude force a large splitting in the transition frequencies, giving a region of high transmission and large dispersion where we would normally expect absorption on the Rb D$_{2}$ line: this is the basis for our isolator.
![Transmission spectra of atomic vapour (a) without and (b) with an applied field illustrating the opening of a transparency window over resonance. Plot (a) shows the theoretical (solid blue) transmission spectra at a temperature of 60.4 $^{\circ}$C through a natural-abundant cell, highlighting the four absorption peaks of interest for isolation. Plot (b) shows the measured (solid green) line transmission spectra at a magnetic field of 0.576 T and a temperature of (60.4 $\pm$ 0.2) $^{\circ}$C through a $^{87}$Rb cell. Plot (c) shows the energy level splittings for the ground (5$^{2}$S$_{1/2}$) and excited terms (5$^{2}$P$_{3/2}$) of $^{87}$Rb on the D$_{2}$ line. In the hyperfine Paschen-Back regime m$_{J}$ is now a good quantum number, the transitions with $\Delta$m$_{J}$ = $\pm$1 correspond to the 16 absorption peaks we measure in plot (b).[]{data-label="energydiagram"}](Energydiagram.eps){width="47.00000%"}
Figure \[Isolation\] shows absolute transmission spectra for the Rb D$_{2}$ line. Plot (a) shows the theoretical (solid black) transmission through a natural-abundant cell in the absence of field and at a temperature of 60.4 $^{\circ}$C. Plots (b), (c) and (d) show comparison between experiment (solid colored) and theory (dashed black) for the rotation of light describing the high transmission along $+z$ (blue), low transmission along $-z$ (green) and extinction values of the isolator (red), respectively. All three spectra are obtained for an isotopically pure $^{87}$Rb cell with a fixed field of 0.597 T and a temperature value of (135.2 $\pm$ 0.4) $^{\circ}$C, which corresponds to a region where we expect $\pi$/4 rotation. These spectra demonstrate the narrow-band nature of isolators based on atomic vapors as there is only a constant dB over a range of 4 GHz. The increased structure in the $T_{+}$ signal which is present in the theoretical and measured signals is the result of additional weak absorption features owing to the fact that the ground terms are not completely decoupled; a detailed study of these features will be the topic of a future publication ``.
![Forward $T_{+}$ and backward $T_{-}$ transmission illustrating the isolator effect as a function of detuning around resonance. Plot (a) shows the theoretical (solid black) transmission through a natural-abundant cell in the absence of field and at a fixed temperature of 60.4 $^{\circ}$C. The four absorption peaks highlight the required detuning values for isolation. Plots (b), (c) and (d) show comparison between experiment (solid colored) and theory (dashed black) through an isotopically pure $^{87}$Rb cell in the presence of a fixed field of 0.597 T and a temperature value of (135.2 $\pm$ 0.4) $^{\circ}$C, which corresponds to a region where we expect $\pi$/4 rotation.[]{data-label="Isolation"}](Isolation.eps){width="47.00000%"}
Important characteristics for isolators are their: ability to extinguish backscattered light, power threshold and temperature stability. In figure \[Isolation\] we define the isolation of the device as $E = -10 \log(T_{+}/T_{-})$ dB, where $T_{+}$ is the transmitted light along ${+z}$ and $T_{-}$ is the transmission along ${-z}$. Previous examples of crystal isolators have measured extinctions of 47 dB for a single device `` and 60 dB for a back-to-back device ``. To the best of our knowledge this is the first resonant atomic vapor isolator and we achieve a 30 dB suppression, limited by the extinction of our polarizers. The excellent agreement between theory and our model is typically achieved in the weak-probe regime. However, when the power of the probe beam was increased by 6 orders of magnitude the extinction in figure \[Isolation\] changed by less than 8 dB. Owing to the strong temperature dependence of the birefringent properties of the medium a device utilizing this effect would require the vapor cell to be temperature stabilized within $\pm$ 0.2 $^{\circ}$C.
In summary, we have demonstrated the principle of an optical isolator for the Rb D$_{2}$ line by exploiting the magneto-optical properties of an isotopically pure $^{87}$Rb vapor in the hyperfine Paschen-Back regime. We show $\pi$/4 rotation for a linearly polarized light in the vicinity of the D$_{2}$ line and achieve an isolation of 30 dB. This work is supported by EPSRC. We thank James Keaveney for the design of the cell heater.
[10]{}
H. Dong, J. Fang, B. Zhou, J. Qin, and S. Wan, Microsyst. Technol. **16**, 1683 (2010). J. Tatum, Proc. SPIE. **6484**, 648403 (2007). L. A. Liew, S. Knappe, J. Moreland, H. Robinson, L. Hollberg, and J. Kitching, Appl. Phys. Lett. **84**, 2694 (2004). F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, Nature **434**, 488 (2005). S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, Phys. Rev. Lett. **94**, 93902 (2005). S. Knappe, V. Shah, A. Brannon, V. Gerginov, H. G. Robinson, Z. Popovi[ć]{}, L. Hollberg, and J. Kitching, Proc. SPIE. **6673**, 667307 (2007). V. Shah, S. Knappe, P. D. D. Schwindt, and J. Kitching, Nat. Photon. **1**, 649 (2007). E. A. Donley, in *Sensors, IEEE* (2010), p. 17. S. A. Knappe, H. G. Robinson, and L. Hollberg, Opt. Express. **15**, 6293 (2007). C. Lee, G. Z. Iwata, E. Corsini, J. M. Higbie, S. Knappe, M. P. Ledbetter, and D. Budker, Rev. Sci. Instrum. **82**, 043107 (2011). J. G. Bai, G. Q. Lu, and T. Lin, Sensor. Actuat. A-Phys. **109**, 9 (2003). J. A. Zieliñska, F. A. Beduini, N. Godbout, and M. W. Mitchell, Opt. Lett. **37**, 524 (2012). R. P. Abel, U. Krohn, P. Siddons, I. G. Hughes, and C. S. Adams, Opt. Lett. **34**, 3071 (2009). E. G. V[í]{}llora, P. Molina, M. Nakamura, K. Shimamura, T. Hatanaka, A. Funaki, and K. Naoe, Appl. Phys. Lett. **99**, 011111 (2011). P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, J. Phys. B: At. Mol. Opt. Phys. **41**, 155004 (2008). S. L. Kemp, I. G. Hughes, and S. L. Cornish, J. Phys. B: At. Mol. Opt. Phys. **44**, 235004 (2011). P. Siddons, C. S. Adams, and I. G. Hughes, J. Phys. B: At. Mol. Opt. Phys. **42**, 175004 (2009). P. Siddons, N. C. Bell, Y. Cai, C. S. Adams, and I. G. Hughes, Nat. Photon. **3**, 225 (2009). L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, J. Phys. B: At. Mol. Opt. Phys. **44**, 195006 (2011). L. Weller, T. Dalton, P. Siddons, C. S. Adams, and I. G. Hughes, J. Phys. B: At. Mol. Opt. Phys. **45**, 055001 (2012). B. E. Sherlock and I. G. Hughes, Am. J. Phys. **77**, 111 (2009). L. Weller, K. S. Kleinbach, M. A. Zentile, S. Knappe, C. S. Adams, and I. G. Hughes, in preparation . A. Sargsyan, G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, A. Papoyan, and D. Sarkisyan, Opt. Lett. **37**, 1379 (2012). D. J. Gauthier, P. Narum, and R. W. Boyd, Opt. Lett. **11**, 623 (1986). R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, Rev. Sci. Instrum. **63**, 5586 (1992).
Informational Fourth Page {#informational-fourth-page .unnumbered}
=========================
[10]{}
H. Dong, J. Fang, B. Zhou, J. Qin, and S. Wan, “Review of atomic mems: driving technologies and challenges,” Microsyst. Technol. **16**, 1683 (2010). J. Tatum, “Vcsel proliferation,” Proc. SPIE. **6484**, 648403 (2007). L. A. Liew, S. Knappe, J. Moreland, H. Robinson, L. Hollberg, and J. Kitching, “Microfabricated alkali atom vapor cells,” Appl. Phys. Lett. **84**, 2694 (2004). F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, “Compact, stable and efficient all-fibre gas cells using hollow-core photonic crystal fibres,” Nature **434**, 488 (2005). S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant optical interactions with molecules confined in photonic band-gap fibers,” Phys. Rev. Lett. **94**, 93902 (2005). S. Knappe, V. Shah, A. Brannon, V. Gerginov, H. G. Robinson, Z. Popovi[ć]{}, L. Hollberg, and J. Kitching, “Advances in chip-scale atomic frequency references at nist,” Proc. SPIE. **6673**, 667307 (2007). V. Shah, S. Knappe, P. D. D. Schwindt, and J. Kitching, “Subpicotesla atomic magnetometry with a microfabricated vapour cell,” Nat. Photon. **1**, 649 (2007). E. A. Donley, “Nuclear magnetic resonance gyroscopes,” in “Sensors, IEEE,” (2010), p. 17. S. A. Knappe, H. G. Robinson, and L. Hollberg, “Microfabricated saturated absorption laser spectrometer,” Opt. Express. **15**, 6293 (2007). C. Lee, G. Z. Iwata, E. Corsini, J. M. Higbie, S. Knappe, M. P. Ledbetter, and D. Budker, “Small-sized dichroic atomic vapor laser lock,” Rev. Sci. Instrum. **82**, 043107 (2011). J. G. Bai, G. Q. Lu, and T. Lin, “Magneto-optical current sensing for applications in integrated power electronics modules,” Sensor. Actuat. A-Phys. **109**, 9 (2003). J. A. Zieliñska, F. A. Beduini, N. Godbout, and M. W. Mitchell, “Ultra-narrow Faraday rotation filter at the Rb D$_{1}$ line,” Opt. Lett. **37**, 524 (2012). R. P. Abel, U. Krohn, P. Siddons, I. G. Hughes, and C. S. Adams, “Faraday dichroic beam splitter for Raman light using an isotopically pure alkali-metal-vapor cell,” Opt. Lett. **34**, 3071 (2009). E. G. V[í]{}llora, P. Molina, M. Nakamura, K. Shimamura, T. Hatanaka, A. Funaki, and K. Naoe, “Faraday rotator properties of $\{$Tb$_{3}$$\}$\[Sc$_{1. 95}$Lu$_{0. 05}$\](Al$_{3}$) O$_{12}$, a highly transparent terbium-garnet for visible-infrared optical isolators,” Appl. Phys. Lett. **99**, 011111 (2011). P. Siddons, C. S. Adams, C. Ge, and I. G. Hughes, “Absolute absorption on rubidium d lines: comparison between theory and experiment,” J. Phys. B: At. Mol. Opt. Phys. **41**, 155004 (2008). S. L. Kemp, I. G. Hughes, and S. L. Cornish, “An analytical model of off-resonant faraday rotation in hot alkali metal vapours,” J. Phys. B: At. Mol. Opt. Phys. **44**, 235004 (2011). P. Siddons, C. S. Adams, and I. G. Hughes, “Off-resonance absorption and dispersion in vapours of hot alkali-metal atoms,” J. Phys. B: At. Mol. Opt. Phys. **42**, 175004 (2009). P. Siddons, N. C. Bell, Y. Cai, C. S. Adams, and I. G. Hughes, “A gigahertz-bandwidth atomic probe based on the slow-light faraday effect,” Nat. Photon. **3**, 225 (2009). L. Weller, R. J. Bettles, P. Siddons, C. S. Adams, and I. G. Hughes, “Absolute absorption on the rubidium D$_{1}$ line including resonant dipole-dipole interactions,” J. Phys. B: At. Mol. Opt. Phys. **44**, 195006 (2011). L. Weller, T. Dalton, P. Siddons, C. S. Adams, and I. G. Hughes, “Measuring the stokes parameters for light transmitted by a high-density rubidium vapour in large magnetic fields,” J. Phys. B: At. Mol. Opt. Phys. **45**, 055001 (2012). B. E. Sherlock and I. G. Hughes, “How weak is a weak probe in laser spectroscopy?” Am. J. Phys. **77**, 111 (2009). L. Weller, K. S. Kleinbach, M. A. Zentile, S. Knappe, C. S. Adams, and I. G. Hughes, in preparation . A. Sargsyan, G. Hakhumyan, C. Leroy, Y. Pashayan-Leroy, A. Papoyan, and D. Sarkisyan, “Hyperfine paschen–back regime realized in Rb nanocell,” Opt. Lett. **37**, 1379 (2012). D. J. Gauthier, P. Narum, and R. W. Boyd, “Simple, compact, high-performance permanent-magnet faraday isolator,” Opt. Lett. **11**, 623 (1986). R. Wynands, F. Diedrich, D. Meschede, and H. R. Telle, “A compact tunable 60-dB faraday optical isolator for the near infrared,” Rev. Sci. Instrum. **63**, 5586 (1992).
|
---
abstract: 'We investigate the connection between radix representations for $\mathbb{Z}^n$ and self-affine tilings of $\mathbb{R}^n$. We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficie ntly large.'
address: 'Department of Mathematics and Statistics, Acadia University, Wolfville, Nova Scotia, Canad B4P 2R6'
author:
- Eva Curry
date: 'February 28, 2005'
title: 'Radix Representations, Self-Affine Tiles, and Multivariable Wavelets'
---
Introduction {#sec_intro}
============
We investigate the connections between radix representations for $\mathbb{Z}^n$, self-affine tilings of $\mathbb{R}^n$, and Haar-like scaling functions for multiresolution analyses and associated wavelet sets.
In a separate paper, we investigate the idea, also introduced by Jeong [@jeo], of radix representations for vectors in $\mathbb{Z}^n$, or general point lattices $\Gamma = M(\mathbb{Z}^n)$ ($M$ a nondegenerate $n \times n$ matrix) [@cu1]. We wish to consider expanding matrices which preserve $\Gamma$. Without loss of generality, we may assume $\Gamma = \mathbb{Z}^n$. A matrix which preserves $\mathbb{Z}^n$ must have integer entries.
A *dilation matrix* for $\mathbb{Z}^n$ is an $n \times n$ matrix $A$ with integer entries, all of whose eigenvalues $\lambda$ satisfy $|\lambda| > 1$.
Note that for a dilation matrix $A$, $q := |\det{A}|$ is an integer, with $q > 1$. Then $\mathbb{Z}^n / A(\mathbb{Z}^n)$ has nontrivial cokernel. Let $D$ be a complete set of coset representatives of $\mathbb{Z}^n / A(\mathbb{Z}^n)$. We call the elemen ts of $D$ *digits*.
We may associate a sequence of digits with each $x \in \mathbb{Z}^n$ by the Euclidean algorithm, as follows. Each $x \in \mathbb{Z}^n$ is in a unique coset of $\mathbb{Z}^n / A(\mathbb{Z}^n)$, thus there exist unique $x_1 \in \mathbb{Z}^n$ and $r_0 \in D
$ such that $$x = Ax_1 + r_0.$$ Similarly, for each $x_j$, $j \geq 1$, there exist unique $x_{j+1} \in \mathbb{Z}^n$ and $r_{j} \in D$ such that $$x_j = Ax_{j+1} + r_{j}.$$ Formally, we write $$x \sim \sum_{j=0}^{\infty} A^j r_j.$$ If there exists a nonnegative integer $N$ such that $r_j = \mathbf{0}$ for all $j > N$, then the Euclidean algorithm terminates and we say that $x$ has a radix representation with radix $A$ and digit set $D$.
Let $A$ be a dilation matrix. We say that the matrix $A$ *yields a radix representation with digit set $D$* if for every $x \in \mathbb{Z}^n$ there exists a nonnegative integer $N = N(x)$ and a sequence of digits $d_0, d_1, \ldots, d_N
$ in $D$ such that $$x = \sum_{j=0}^{N} A^j d_j.$$
That is, a dilation matrix $A$ yields a radix representation with digit set $D$ if *every* $x \in \mathbb{Z}^n$ has a radix representation with radix $A$ and digit set $D$.
Let $A$ be a dilation matrix, and define $$\mu = \min{\{ \sigma:\ \mbox{$\sigma$ a singular value of $A$} \}}.$$ Let $F$ be a fundamental domain for $\mathbb{Z}^n$, centered at the origin, $$F = \left[ -\frac{1}{2}, \frac{1}{2} \right)^n.$$ In [@cu1], we give the following two results about radix representations.
\[rad\_rep\] Let $A$ be an $n \times n$ dilation matrix. If $\mu > 2$ then $A$ yields a radix representation of $\mathbb{Z}^n$ with digit set $D = A(F) \cap \mathbb{Z}^n$.
\[big\_enough\] For every dilation matrix $A$, there exists a positive integer $\beta \geq 1$ such that $A^{\beta}$ yields a radix representation with digit set $D_{\beta} = A^{\beta}(F) \cap \mathbb{Z}^n$.
Radix Representations and Tilings {#sec_radix}
=================================
Radix representations are closely related to self-affine tilings of $\mathbb{R}^n$.
A measurable set $Q \subset \mathbb{R}^n$ gives a self-affine tiling of $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$ if
1. $\cup_{k \in \mathbb{Z}^n} (Q + k) = \mathbb{R}^n$, and the intersection $(Q+k_1) \cap (Q+k_2)$ has measure zero for any two distinct $k_1, k_2 \in \mathbb{Z}^n$ (tiling); and
2. there is a collection of $q = |\det{A}|$ vectors $k_1, \ldots, k_q \in \mathbb{Z}^n$ that are distinct coset representatives of $\mathbb{Z}^n / A(\mathbb{Z}^n)$ such that $$A(Q) \simeq \cup_{i=1}^{q} (Q + k_i) \quad \mbox{(self-affine)}.$$
Set $$T = T(A,D) :=\{\xi \in \mathbb{R}^n:\ \xi = \sum_{j=1}^{\infty} A^{-j} d_j\}$$ with the digits $d_j \in D$ for some digit set $D$. One can easily check that $T$ is a self-affine set. We would like to be able to think of the elements of $T$ as the fractional parts of vectors in $\mathbb{R}^n$ in the same way that the fractional par ts of real numbers lie in $[0,1]$. This is an accurate interpretation if $T$ is congruent to $\mathbb{R}^n / \mathbb{Z}^n$. Thought of another way, we would like $T$ to tile $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$.
\[radix<=>tiling&0\] Let $A$ be a dilation matrix, and let $D$ be a digit set for $A$. Then $A$ yields a radix representation with digit set $D$ if and only if the set $T(A,D)$ tiles $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$ and the origin $\mathbf{0}$ is in the interior of $T$.
We split the proof of this theorem into a few lemmas.
\[radix=>tiling\] Let $A$ be a dilation matrix, and let $D$ be a digit set for $A$. If $A$ yields a radix representation with a digit set $D$, then the set $T(A,D)$ tiles $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$.
Applying Proposition 5.19 from [@woj] the dilation matrix $A$, digit set $D$, and corresponding set $T = T(A,D)$ satisfy
1. $T$ is a compact subset of $\mathbb{R}^n$;
2. $A(T) = \cup_{d \in D} (T+d)$;
3. $\cup_{x \in \mathbb{Z}^n} (T+x) = \mathbb{R}^n$; and
4. $T$ contains an open set.
To show that $T$ tiles $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$, we must show that $$m((T+x) \bigcap (T+y)) = 0\ \mbox{for all}\ x \neq y, x,y \in \mathbb{Z}^n$$ (where $m(\cdot)$ denotes Lebesgue measure). We extend an idea of Lagarias and Wang ([@lw1], p.31) to show that $m((T+x) \cap (T+y)) = 0$ for any distinct $x,y \in \mathbb{Z}^n$ for which there is a radix representation with radix $A$ and digit set $
D$.
Note that for any subset $Q$ of $\mathbb{R}^n$, $m(A(Q)) = q m(Q)$ (where $q = |\det{A}|$). In particular, $$\begin{aligned}
q m(T) &= m(A(T)) = m\left(\bigcup_{d \in D} (T+d)\right)\ \mbox{(by property $2$ of $T$)}\\
&\leq \sum_{d \in D} m(T+d) = \sum_{d \in D} m(T) = q m(T).\end{aligned}$$ Additionally, property $2$ implies that $A^{k+1}(T) = \cup_{d \in D} (A^k(T) + A^kd)$ for all $k \geq 0$. Then $$q^{k+1} m(T) \leq \sum_{d \in D} m(A^k(T) + A^kd) = \sum_{d \in D} q^k m(T) = q^{k+1} m(T).$$ Thus $$m\left((A^k(T) + A^kd_i) \bigcap (A^k(T) + A^kd_j)\right) = 0$$ for all distinct $d_i, d_j \in D$.
Since $\mathbf{0} \in D$, $A^{k+1}(T) \supset A^{k}(T)$ for all $k \geq 0$. So $$(A^{k+1}(T) + A^{k+1}d) \supset (A^{k}(T) + A^{k+1}d)$$ for all $d \in D$. Now consider $T+x$ for any $x \in \mathbb{Z}^n$. By hypothesis, $$x = A^Nd_N + \sum_{j=0}^{N-1}A^jd_j$$ for some $N\geq 0$, with $d_N \neq \mathbf{0} \in D$ and the $d_j \in D$. Then $$\begin{aligned}
T+x &= \{y \in \mathbb{R}^n:\ y = A^Nd_N + \sum_{j=0}^{N-1}A^jd_j + \sum_{j=-\infty}^{-1}A^jd_j,\ \mbox{all}\ d_j \in D\}\\
&\subset A^{N-1}(T) + A^Nd_N \subset A^{N}(T) + A^Nd_N,\end{aligned}$$ and $$(T \bigcap (T+x)) \subseteq (A^{N}(T) \bigcap (A^{N}(T) + A^Nd_N)).$$ Then $d_N \neq \mathbf{0} \in D$ implies that $$0 = m\left(A^N(T) \bigcap (A^N(T) + A^Nd_N)\right) \geq m\left(T \bigcap (T+x)\right).$$
If the set $T = T(A,D)$ tiles $\mathbb{R}^n$, it is not necessarily true that $A$ yields a radix representation. For example, in the case where $A = 2$, we can find a radix representation for all nonnegative integers with the digit set $D = \{ 0, 1\}$, o r for all non-positive integers with the digit set $D = \{0, -1\}$, but we cannot represent all integers with a radix representation using any digit set [@mat]. Yet $T = [0,1]$ (with digit set $D = \{0, 1\}$), does tile $\mathbb{R}$ under translation by $\mathbb{Z}$. Similarly, if $A$ is the twin dragon matrix [@woj], $$A = {\left[ \begin{array}{rr}{1}&{1}\\{-1}&{1}\end{array} \right]}$$ then $A$ is a dilation matrix. A digit set for $A$ is $$D = \{d_0 = {\left[ \begin{array}{r}{0}\\{0}\end{array} \right]}, d_1 = {\left[ \begin{array}{r}{1}\\{0}\end{array} \right]}\},$$ and the set $T$ generated by the twin dragon matrix with this digit set tiles $\mathbb{R}^2$ under translation by $\mathbb{Z}^n$ [@woj], yet $A$ does not yield a radix representation of $\mathbb{Z}^n$ (for example, the vector $${\left[ \begin{array}{r}{0}\\{-1}\end{array} \right]}$$ does not have a radix representation) [@cu1].
![The tile $T$ for the twin dragon matrix.[]{data-label="fig:td_tile"}](td_tile.eps)
In both of the examples, the origin $\mathbf{0}$ is on the boundary of the tile $T$, so that $\cup_{k =0}^{\infty} A^k(T) \subsetneq \mathbb{R}^n$. We claim in Theorem \[radix<=>tiling&0\] that this must be the case for all dilation matrices $A$ which give tiles $T$ but do not yield radix representations. We first introduce a technical lemma.
\[technical\] Let $A$ be a dilation matrix, and let $D$ be a digit set for $A$. If $\mathbf{0} \not\in T^{\circ}$ then there exists an increasing subsequence $\{\ell_j\}_{j \geq 1}$ of the positive integers and a sequence of vectors $\{
\zeta_j:\ \zeta_j \in A^{-\ell_j}(\mathbb{Z}^n)\}_{j \geq 1}$ converging to $\mathbf{0}$ such that $\zeta_j \not\in T$ for all $j \geq 1$.
In the proof of Lemma \[radix=>tiling\], we noted that the set $T$ is compact. Thus for any $\omega \in \mathbb{R}^n$ such that $\omega \not\in T$, $d(\omega,T) > 0$, and there exists an open ball centered at $\omega$, $B_{\omega}$, such that $\overline{B_{\omega}} \cap T = \emptyset$.
Let $\{y_j\}_{j \geq 1}$ be a sequence of vectors in $\mathbb{R}^n$ converging to $\mathbf{0}$ with $y_j \not\in T$ for all $j \geq 1$. Set $\epsilon_j = \|y_j\|_{l^2}$, and notice that $\epsilon_j \geq d(y_j,T) > 0$, with $\lim_{j \rightarrow \infty} \epsilon_j = 0$.
Set $r_1 = \frac{d(y_1,T)}{2}$ and $$r_j = \min{\left\{ \frac{r_{j-1}}{2}, \frac{d(y_j,T)}{2} \right\}}$$ for $j \geq 2$. Then $\{r_j\}_{j \geq 1}$ is a decreasing sequence of positive numbers, $\lim_{j \rightarrow \infty} r_j = 0$, and the open balls $B_j$ of radius $r_j$ centered at the vectors $y_j$ satisfy $\overline{B_j} \cap T = \emptyset$ for all $j \
geq 1$. By construction, if $y_j^*$ is any point in the ball $B_j$ for each $j \geq 1$, then $\lim_{j \rightarrow \infty} y_j^* = \mathbf{0}$ and $y_j^* \not\in T$ for all $j \geq 1$.
In particular, there exists an increasing subsequence $\{\ell_j\}_{j \geq 1}$ of the positive integers such that $A^{-\ell_j}$ is a fine enough lattice to ensure that $A^{-\ell_j}(\mathbb{Z}^n) \cap B_j \neq \emptyset$ for each $j \geq 1$. We may choose some $\zeta_j \in A^{-\ell_j}(\mathbb{Z}^n) \cap B_j$ for each $j \geq 1$. By construction, $\zeta_j \not\in T$ for each $j \geq 1$, and the sequence $\{\zeta_j\}_{j \geq 1}$ converges to $\mathbf{0}$.
\[radix=>0\] Let $A$ be a dilation matrix, and let $D$ be a digit set for $A$. If $A$ yields a radix representation with digit set $D$, then $\mathbf{0}$ is in the interior of the set $T$ generated by $A$ and $D$.
We prove the desired result by contradiction. Assume that $\mathbf{0} \not\in T^{\circ}$. By Lemma \[technical\], let $\{ \zeta_j \}_{j \geq 1}$ be a sequence of vectors in $\mathbb{R}^n$ converging to $\mathbf{0}$ such that $\zeta_j \i
n A^{-\ell_j}(\mathbb{Z}^n)$ (for some increasing subsequence $\{\ell_j\}_{j \geq 1}$ of the positive integers) but $\zeta_j \not\in T$ for all $j \geq 1$. We may write $\zeta_j = A^{-\ell_j} x_j$ for some $x_j \in \mathbb{Z}^n$ for each $j$. Since $A$ yields a radix representation, there exists an integer $N_j$ for each $x_j$ and digits $d^{(j)}_0, \ldots, d^{(j)}_{N_j} \in D$ such that $$x_j = \sum_{i=0}^{N_j} A^i d^{(j)}_i.$$ Thus $$\zeta_j = \sum_{i=0}^{N_j} A^{i-\ell_j} d^{(j)}_i = k_j + \xi_j$$ with $k_j \in \mathbb{Z}^n$ and $\xi_j \in T$.
Since $T$ is compact, $\| \xi_j \|_{l^2}$ is bounded above by $b$ for some $b > 0$. Then, since $\zeta_j$ converges to $\mathbf{0}$, $\| k_j \|_{l^2}$ is also bounded above for sufficiently large $j$. We use the rough estimate that there exists an integ er $M \geq 1$ such that for all $j \geq M$, $\| k_j \|_{l^2} \leq 2b$. Thus for $j \geq M$, the integer vectors $k_j$ all belong to a finite subset of $\mathbb{Z}^n$. This implies that there exists an integer $N \geq 0$ such that $N_j \leq N$ for all $j
\geq M$. For $j > \max{\{M,N\}}$, $N-\ell_j \leq N-j < 0$, and thus $\zeta_j \in T$ for all sufficiently large $j$. This contradicts the choice of $\zeta_j$, thus our assumption that $\mathbf{0} \not\in T^{\circ}$ must be false.
We have shown that if a dilation matrix $A$ yields a radix representation with digit set $D$, then the set $T = T(A,D)$ tiles $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$, and $\mathbf{0} \in T^{\circ}$. Next we prove the converse.
\[0&tiling=>radix\] Let $A$ be a dilation matrix, and let $D$ be a digit set for $A$. If the set $T = T(A,D)$ tiles $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$ and if $\mathbf{0} \in T^{\circ}$, then $A$ yields a radix representat ion with digit set $D$.
Following the notation of [@lw1], set $$D_{A,k} := \{x \in \mathbb{Z}^n:\ x = \sum_{j=0}^{k-1} A^j d_j, d_j \in D\},$$ the set of vectors that can be expressed with a radix representation of length less than or equal to $k$. Note that $D_{A,1} = D$. By construction, $A(T) = \cup_{d \in D} (T+d)$. Thus $$A^{k}(T) = \bigcup_{x \in D_{A,k}} (T + x).$$ By the tiling hypothesis, $(T^{\circ}+x) \cap (T^{\circ}+y) = \emptyset$ for all distinct $x, y \in \mathbb{Z}^n$. Thus $(T^{\circ} + y) \cap A^{k}(T^{\circ}) = \emptyset$ for all $y \in\mathbb{Z}^n$ with $y \not\in D_{A,k}$, and $$D_{A,k} \supseteq (A^k(T^{\circ}) \bigcap \mathbb{Z}^n).$$
Let $B$ be an open ball centered at the origin such that $B \subseteq T^{o}$. Then $$D_{A,k} \supseteq (A^{k}(B) \cap \mathbb{Z}^n).$$ The sets $A^{k+1}(B)$ are expanding, with $\cup_{k \geq 0} A^{k+1}(B) = \mathbb{R}^n$. Thus $$\bigcup_{k \geq 0} D_{A,k} \supseteq \bigcup_{k \geq 0} (A^k(B) \cap \mathbb{Z}^n) = \mathbb{Z}^n.$$ The opposite containment is true as well, since $D_{A,k} \subset \mathbb{Z}^n$ for each $k$. Thus $\cup_{k \geq 0} D_{A,k} = \mathbb{Z}^n$.
We have now completed the proof of Theorem \[radix<=>tiling&0\].
Recall that $\mu$ is the smallest singular value of the dilation matrix $A$, and that $F$ is our canonical fundamental domain of $\mathbb{Z}^n$, $F = [-\frac{1}{2}, \frac{1}{2})^n$. Combining Theorems \[rad\_rep\] and \[radix=>tiling\], we also have t he following corollary.
Let $A$ be a dilation matrix, and let $D = A(F) \cap \mathbb{Z}^n$. If $\mu > 2$ then the set $$T = \{ x \in \mathbb{R}^n:\ x = \sum_{j=-\infty}^{-1} A^j d_j, d_j \in D\}$$ tiles $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$.
Haar-Like Wavelets
==================
Self-affine tiles allow us to construct multivariable wavelet sets associated with multiresolution analyses. We review some basic definitions from wavelet theory here.
A *multiresolution analysis (MRA)* associated with a dilation matrix $A$ is a nested sequences of subspaces $\cdots \subset V_{-1} \subset V_{0} \subset V_{1} \subset \cdots$ of $L^2(\mathbb{R}^n)$ satisfying: [@woj]
1. $\overline{\cup_{j \in \mathbb{Z}} V_{j}} = L^2(\mathbb{R}^n)$;
2. $\cap_{j \in \mathbb{Z}} V_{j} = \{ 0 \}$;
3. $f(x) \in V_{j}$ if and only if $f(Ax) \in v_{j+1}$ for all $j \in \mathbb{Z}$;
4. $f(x) \in V_0$ if and only if $f(x-k) \in V_{0}$ for all $k \in \mathbb{Z}^n$; and
5. there exists a function $\phi(x) \in V_{0}$, called a *scaling function*, such that $$\{\phi(x-k):\ k \in \mathbb{Z}^n\}$$ is a complete orthonormal basis for $V_{0}$.
The existence of multiresolution analyses in dimension $n>1$ has been studied by a number of authors. In [@grm], Gröchenig and Madych showed that if $\phi = (m(Q))^{1/2} \chi_{Q}$ is a scaling function for a multiresolution analysis, then $Q$ mus t be an affine image of a self-affine tiling of $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$. They showed also that if $T$ is a set of the form $$T = \{ x \in \mathbb{R}^n:\ x = \sum_{j=-\infty}^{-1} A^j d_j, d_j \in D\}$$ with $A$ a dilation matrix and $D$ a digit set for $A$, then $\phi = \chi_{T}$ is the scaling function for a multiresolution analysis (note that $|T| = 1$). Lagarias and Wang noted that all self-affine tiles $T$ which tile $\mathbb{R}^n$ under translatio n by $\mathbb{Z}^n$ must be of this form [@lw2]. A scaling function that is the characteristic function of some measurable set is called a Haar-like scaling function (after the Haar scaling function, which is $\chi_{[0,1]}$).
Lagarias and Wang studied necessary conditions for sets of the form $T(A,D)$ to tile $\mathbb{R}^n$ under translation by $\mathbb{Z}^n$ in a series of papers ([@lw1], [@lw4], [@lw2], [@lw3]). Much of their work in these papers concerned t he question of when a set $T(A,D)$ tiles $\mathbb{R}^n$ under translation by a sublattice of $\mathbb{Z}^n$. In [@lw1] and [@lw4], they showed that if $D$ is a complete set of coset representatives of $\mathbb{Z}^n / A(\mathbb{Z}^n)$, then $T(A,D
)$ is a self-affine tile of $\mathbb{R}^n$ under translation by some sublattice of $\mathbb{Z}^n$. They also studied some properties of the tiling set $T(A,D)$. He and Lau [@hel] studied sets $T(A,D)$ which tile $\mathbb{R}^n$ under translation by a sublattice of $\mathbb{Z}^n$ as well; in particular, they looked at possible digit sets $D$.
In order for $\chi_{T}$ to be a scaling function for a multiresolution analysis, however, we need $T$ to tile $\mathbb{R}^n$ under translation by all of $\mathbb{Z}^n$. Lagarias and Wang gave some necessary conditions for a dilation matrix $A$ to yield a Haar-like scaling function in [@lw2] and [@lw3]. They showed that all dilation matrices in dimensions $n=2$ and $3$ yield Haar-like scaling functions. Our results below give a sufficient condition for a dilation matrix $A$ to yield a Haar-like scaling function, in any dimension. Note that for dilation matrices that yield a Haar-like scaling function, Strichartz has shown that multiresolution analyses and associated wavelet bases with arbitrary regularity can be constructed [@str].
The results of the previous section imply the following two theorems.
Let $A$ be a dilation matrix, and let $D$ be a digit set for $A$ such that $A$ yields a radix representation for $\mathbb{Z}^n$ with digit set $D$. Let $T$ be the set depending on $A$ and $D$ defined above. Then $\phi = \chi_{T}$ is the scaling function for a multiresolution analysis. In particular, if $A$ satisfies $\mu > 2$ and if $D$ is the set $D = A(F) \cap \mathbb{Z}^n$ with $F = [-\frac{1}{2},\frac{1}{2})^n$, then $\phi = \chi_{T}$ is the scaling function for a multiresolution analysis.
Let $A$ be a dilation matrix. Then there exists a positive integer $\beta \geq 1$ such that for all integers $k \geq \beta$ there exists a multiresolution analysis associated with the dilation matrix $A^k$.
Thus Haar-like scaling functions and associated MRAs exist for a large class of dilation matrices.
[99]{}
E. Curry, Radix and Pseudodigit Representations in $\mathbb{Z}^n$, to appear.
Julie Belock, Vladimir Dobric, Random Variable Dilation Equation and Multidimensional Prescale Functions, *Trans. Amer. Math. Soc. *, **353** (2001), 4779-4800.
K. Gröchenig and W.R. Madych, Multiresolution Analysis, Haar Bases, and Self-Similar Tilings of $\mathbb{R}^n$, *IEEE Trans. Info. Thy. * **38** (1992), 556-568.
Xing-Gang He and Ka-Sing Lau, Characterization of tile digit sets with prime determinants, *Appl. Comput. Harmon. Anal. *, **16** (2004), 159-173.
E.-C. Jeong, A Number System in $\mathbb{R}^n$, *J. Korean Math. Soc. * **41** (2004), 945-955.
Jeffrey C. Lagarias and Yang Wang, Self-Affine Tiles in $\mathbf{R}^n$, *Adv. in Math. * **121** (1996), 21-49.
Jeffrey C. Lagarias and Yang Wang, Integral Self-Affine Tiles in $\mathbb{R}^n$ Part II: Lattice Tilings, *J. Fourier Anal. and Appl. * **3** (1997), 83-102.
Jeffrey C. Lagarias and Yang Wang, Haar Bases for $L^2(\mathbb{R}^n)$ and Algebraic Number Theory, *J. Number Thy. * **57** (1996), 181-197.
Jeffrey C. Lagarias and Yang Wang, Corrigendum/Addendum: Haar Bases for $L^2(\mathbb{R}^n)$ and Algebraic Number Theory, *J. Number Thy. * **76** (1999), 330-336.
David W. Matula, Basic Digit Sets for Radix Representation, *J. Assoc. Comp. Machinery* **29** (1982), 1131-1143.
R. Strichartz, Wavelets and self-affine tilings, *Constr. Approx. * **9** (1993), 327-346.
P. Wojtaszczyk, “A Mathematical Introduction to Wavelets.” Cambridge University Press, Cambridge, United Kingdom, 1997.
|
---
abstract: 'In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both $(\nabla\times)^2$ and $(\nabla\times)^4$ operators. The element has a very small number of degrees of freedom and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.'
address:
- 'Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, PA 16802'
- 'LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China.'
- 'Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, PA 16802.'
author:
- Bin Zheng
- Qiya Hu
- Jinchao Xu
date: 'January 29, 2010'
title: 'A Nonconforming Finite Element Method for Fourth Order Curl Equations in $\mathbb{R}^3$'
---
[^1]
[^2]
Introduction
============
The magnetohydrodynamics (MHD) equations describe macroscopic dynamics of electrically conducting fluid that moves in a magnetic field. MHD model is governed by Navier-Stokes equations coupled with Maxwell equations through Ohm’s law and Lorentz force. As an example, a resistive MHD system is described by the following equations: $$\left\{
\begin{array}{rcl}
\rho(\mathbf{u}_t+\mathbf{u}\cdot \nabla \mathbf{u})+\nabla p &=& \frac{1}{\mu_0}(\nabla\times \mathbf{B})\times \mathbf{B} + \mu\Delta \mathbf{u},\\
\nabla\cdot \mathbf{u} & = & 0,\\
\mathbf{B}_t - \nabla\times(\mathbf{u}\times \mathbf{B}) & = & -\frac{\eta}{\mu_0}(\nabla\times)^2 \mathbf{B} - \frac{d_i}{\mu_0}\nabla\times((\nabla\times \mathbf{B})\times\mathbf{B}) \\
&& - \frac{\eta_2}{\mu_0}(\nabla\times)^4 \mathbf{B}, \\
\nabla\cdot \mathbf{B} & = & 0,
\end{array}
\right.$$ where $\rho$ is the mass density, $\mathbf{u}$ is the velocity, $p$ is the pressure, $\mathbf{B}$ is the magnetic induction field, $\eta$ is the resistivity, $\eta_2$ is the hyper-resistivity, $\mu_0$ is the magnetic permeability of free space, and $\mu$ is the viscosity. The primary variables in MHD equations are fluid velocity $\mathbf{u}$ and magnetic field $\mathbf{B}$.
MHD model has widespread applications in thermonuclear fusion, magnetospheric and solar physics, plasma physics, geophysics, and astrophysics. Mathematical modeling and numerical simulations of MHD have attracted much research effort in the past few decades. Various numerical algorithms have been used in MHD simulations; examples include finite difference methods, finite volume methods, finite element methods, and Fourier-based spectral and pseudo-spectral methods [@Toth:1996gd]. In [@Jardin:2004qo; @Jardin:2005dq; @Kang:2008ta; @Krzeminski:2000la; @Ovtchinnikov:2007kb], two-dimensional, incompressible MHD problems are studied in terms of finite element approximations of the stream function-vorticity advection formulation. Since MHD flow often develop sharp interfaces, adaptive $h$-refinement techniques have been applied in MHD simulations [@Lankalapallia:2007ye; @Strauss:1998kc; @Ziegler:2003gf]. Finite element computations of MHD problems in three-dimensions have been reported in [@Codina:2006zv; @Gerbeau:2000lo; @Layton:1997hs; @Salah:2001fh; @Schotzau:2004sw; @Wiedmer:1999pr].
In the existing finite element discreitzations for the above MHD model, a standard pair of stable or stabilized finite element spaces are often used to discretize the velocity and pressure variables in the fluid equations. For the magnetic field variable $\mathbf{B}$, however, at least two approaches are possible when the fourth order term $(\nabla\times)^4\mathbf{B}$ is not presented in the model, namely when electron viscosity $\eta_2=0$. One approach is to use the standard edge element ([@Schotzau:2004sw]) and the other approach is to use the Lagrange element after replacing $(\nabla\times)^2\mathbf{B}$ by $-\Delta \mathbf{B}$ ([@Gerbeau:2000lo; @Salah:2001fh; @Wiedmer:1999pr]). Both these approaches will become more difficult when the fourth order term $(\nabla\times)^4\mathbf{B}$ is presented. We may still replace $(\nabla\times)^4 \mathbf{B}$ by a biharmonic operator $\Delta^2 \mathbf{B}$. But discretizing a biharmonic operator in three dimensions is challenging. It requires $220$ degrees of freedom per element if a conforming finite element is used. One possible way to reduce the number of degrees of freedom is to use nonconforming discretizations which allow weaker inter-element smoothness constraints but still provid convergent approximations. Among the class of nonconforming finite elements for fourth order problems, Morley-type elements are special in the sense that they provide approximations with polynomials of minimal degree [@Morley:1968ve; @Wang:2006qr]. In [@Wang:2006bh], a systematic construction of Morley-type elements is provided for solving $2m$-th order partial differential equations in $\mathbb{R}^n$. In particular, we may apply the element in [@Wang:2006bh] with $n=3$ and $m=2$ consisting of piecewise quadratic elements to our system of biharmonic equations. This amounts to $30$ degrees of freedom on each element. This element provides a reasonable discretization of MHD equations when it is appropriate to replace $(\text{curl})^4$ operator by the biharmonic operator. This approach, however, may lead to difficulty for certain boundary conditions in practical applications. Indeed the treatment of boundary conditions is also an issue for the second order problem if $(\nabla\times)^2\mathbf{B}$ is replaced by $-\Delta\mathbf{B}$ [@Guermond:2003fk].
One more natural approach is to discretize the fourth order curl operator by some generalized higher order edge elements. But such type of edge elements are not available in the literature. The construction of such type of edge element is the main goal of this paper.
Another possible approach to deal with the fourth order term $(\nabla\times)^4\mathbf{B}$ is to use operator splitting technique. Namely, one can introduce an intermediate variable $\sigma = (\nabla\times)^2\mathbf{B}$ and then reduce the original problem to a system of second order equations. However, it is known that for some problems, such a technique cannot be applied. For example, when modeling the bending of simply supported plate on non-convex polygonal domains, the original biharmonic problem is not equivalent to the lower order system of two Poisson equations [@Blum:1980ly; @Rannacher:1979zr]. In view of this, we consider discretizing the fourth order problem directly.
In this paper, we investigate MHD equations that contain both fourth-order term and second-order term. In the literature, the major tool used for performing MHD simulations involving a fourth order equation has been the pseudo-spectral method [@Biskamp:1995it]. By choosing an appropriate formulation, we are able to construct a finite element approximation for this problem. This is a nonconforming finite element that involves only a small number of degrees of freedom.
The rest of this paper is organized as follows. In Section 2, we describe a simplified model problem and the corresponding variational formulation. In Section 3, we construct basis functions and provide the convergence analysis. Finally, in Section 4, we give some concluding remarks.
Model Problem
=============
In the following, we introduce model problems for the fourth-order magnetic induction equations described above. Assume that $\Omega\subset \mathbb{R}^3$ is a bounded polyhedron. By considering a semi-discretization in time and then ignoring the nonlinear terms, we obtain the following equations: $$\left\{
\begin{aligned}
\alpha(\nabla\times)^4 \mathbf{u} + \beta(\nabla\times)^2 \mathbf{u} +\gamma \mathbf{u}&=\mathbf{f},\;\text{in}\;\Omega,\\
\nabla\cdot\mathbf{u} & = 0,\;\text{in}\;\Omega,
\end{aligned}
\right.\label{four_curl_bvp}$$ where $\nabla\cdot\mathbf{f}= 0$, and the parameters $\alpha, \beta, \gamma > 0 $. We consider homogeneous boundary conditions, $$\mathbf{u}\times \mathbf{n}= 0,\;\nabla\times \mathbf{u}= 0,\;\text{on}\;\partial\Omega.$$
The above choice of boundary conditions arise naturally in the variational formulation given below. On the other hand, in the numerical simulations of the problem with pseudo-spectral method, one often uses periodic boundary conditions, e.g., [@Biskamp:1995it; @Germaschewski:1999yq].
It is worth pointing out that the parameter $\alpha$ is usually much smaller than either $\beta$ or $\gamma$. This fact imposes some difficulties in designing robust numerical methods, as have been studied in the context of biharmonic problems, e.g., [@Nilssen:2001zr; @Wang:2006ly].
The above fourth-order curl equations also arise from an interior transmission problem in the study of inverse scattering problems for inhomogeneous medium, e.g., [@Cakoni:2007yq].
In order to provide an appropriate framework for our analysis, we define the following function spaces: $$H(\text{curl};\Omega)=\{\mathbf{v}\in (L^2(\Omega))^3 \; | \;\nabla\times \mathbf{v}\in (L^2(\Omega))^3\},$$ $$H_0(\text{curl};\Omega) = \{\mathbf{v}\in H(\text{curl};\Omega)\; | \;\mathbf{v}\times \mathbf{n} = 0, \text{on}\;\partial\Omega\},$$ $$V=\{\mathbf{v}\in H_0(\text{curl};\Omega)\; |\; \nabla\times \mathbf{v}\in H_0^1(\Omega)\}.$$ $V$ is a Hilbert space with scalar product and norm given by $$(\mathbf{u},\mathbf{v})_{V}\triangleq (\nabla(\nabla\times \mathbf{u}),\nabla (\nabla\times
\mathbf{v}))+(\nabla\times \mathbf{u},\nabla\times \mathbf{v})+(\mathbf{u},\mathbf{v}),$$ $$\norm{\mathbf{u}}_{V}\triangleq \sqrt{(\mathbf{u},\mathbf{u})_{V}}.$$
The following lemma gives a sufficient condition for a piecewisely defined function to be an element in $V$.
If $\mathbf{v}$ is piecewise smooth, $\mathbf{v}\times \mathbf{n}$ and $\nabla\times \mathbf{v}$ are continuous across element interfaces, then $\mathbf{v}\in V$.
Using the following identity: $$(\nabla\times)^2\mathbf{u}=-\Delta\mathbf{u}+
\nabla(\nabla\cdot\mathbf{u})$$ and $\nabla\cdot \mathbf{u}=0$, the first equation in (\[four\_curl\_bvp\]) can be rewritten in the following form: $$-\alpha\nabla\times\Delta(\nabla\times \mathbf{u}) + \beta(\nabla\times)^2 \mathbf{u} +\gamma \mathbf{u}=\mathbf{f}.
\label{delta_curlcurl}$$
Multiplying Equation (\[delta\_curlcurl\]) by the test function $\mathbf{v}$ and using integration by parts, we obtain the following variational formulation: $$\text{Find}\; \mathbf{u}\in V\; \text{such that}\; a(\mathbf{u},\mathbf{v})=(\mathbf{f},\mathbf{v}), \;\forall\;\mathbf{v}\in V,\label{V_1_variational}$$ where the bilinear form $a(\cdot,\cdot)$ defined on $V\times
V$ is given by $$a(\mathbf{u},\mathbf{v})=\alpha (\nabla(\nabla\times \mathbf{u}),\nabla(\nabla\times \mathbf{v}))+ \beta (\nabla\times
\mathbf{u},\nabla\times \mathbf{v}) + \gamma (\mathbf{u},\mathbf{v}).$$
The well-posedness of the above variational problem follows from the Lax-Milgram lemma.
The next lemma indicates that the weak solution satisfies the divergence-free constraint.
Assume $\nabla\cdot \mathbf{f} = 0$, and let $\mathbf{u}$ be the solution of problem (\[V\_1\_variational\]). Then $\nabla\cdot \mathbf{u} = 0$.
Choose test function $\mathbf{v}=\nabla \varphi$ where $\varphi\in C_0^\infty(\Omega)$, then $$(\mathbf{u},\nabla\varphi) = (\mathbf{f},\nabla\varphi),$$ hence, $\nabla\cdot \mathbf{u} =\nabla\cdot \mathbf{f} = 0$.
A Nonconforming Finite Element
==============================
In this section, we construct a nonconforming finite element to solve the fourth-order equation. One of the advantages for using a nonconforming element is that the number of degrees of freedom is small compared to that for conforming elements. The following construction is based on Nédélec elements of the first family that consist of incomplete vector polynomials [@Nedelec:1980xe]. The advantage of using incomplete vector polynomial space is that it provides the same order of convergence in terms of energy norms as the one given by corresponding complete polynomial space. In the following, we define the degrees of freedom in a special way to ensure that the consistency error estimate holds.
The finite element triple $(K,P_K,\Sigma_K)$ is defined by
- $K$ is a tetrahedron;
- $\mathcal{P}_K=R_2(K)=\mathbf{P}_{1}\oplus \{\mathbf{p}\in
(\widetilde{P}_2)^3\;|\;\mathbf{p}\cdot \mathbf{x} = 0\}$, where $\widetilde{P}_2$ is the space of homogeneous multivariate polynomials of degree $2$;
- $\Sigma_K$ is the set of degrees of freedom, see Figure \[element\],
- edge degrees of freedom: $$M_e(\mathbf{u})=\bigg\{\int_e \mathbf{u}\cdot \tau\;q\;d s\;|\;\forall
\;q\in P_1(e),\;\forall\;e\subset K\bigg\},
\label{edge d.o.f}$$ where $\tau$ is the unit tangential vector along the edge $e$,
- face degrees of freedom: $$M_f(\mathbf{u})=\bigg\{\frac{1}{|f|^2}\int_f (\nabla\times \mathbf{u})\times \mathbf{n} \cdot
q \;d A\;|\;\forall\;q\in
(P_0(f))^2,\;\forall\;f\subset K\bigg\},
\label{face d.o.f}$$ where $\mathbf{n}$ is the unit normal vector to the face $f$,
$\Sigma_K=M_e(\mathbf{u})\cup M_f(\mathbf{u})$.
\[fem\_triple\_def\]
![Degrees of freedom of the finite element[]{data-label="element"}](element.png){width="60mm"}
In the above finite element triple, the space $\mathcal{P}_K$ is the same as the second order Nédélec element of the first family for $H(\text{curl})$ problem. The difference is the definition of the second set of degrees of freedom. It is designed specifically to ensure consistency for the fourth-order problems. The total number of the degrees of freedom for this element is $20$, which is the same as the dimension of the polynomial space $R_2(K)$.
It should be pointed out that the scaling factor $1/|f|^2$ in the definition of the second set of degrees of freedom is associated with the construction of basis functions to be given later.
The next lemma given in [@Nedelec:1980xe] describes a relation between edge integrals and face integrals which will be useful in the error analysis.
If $u\in R_2(K)$ is such that the edge degrees of freedom (\[edge d.o.f\]) vanish, then $$\int_f(\nabla\times \mathbf{u})\cdot \mathbf{n}\;dA=0,\;\forall\;\text{face}\;f\subset K.
\label{curl_u_face_integral}$$ \[relation-edge-face\]
Given $\mathbf{u}\in R_2(K)$ satisfies $$\int_e \mathbf{u}\cdot\tau \;q\;ds=0,\;\forall \;\text{edge}\; e\subset K.$$ By Stokes’ Theorem, $$\int_f (\nabla_f\times \mathbf{u}_T)\cdot q \;dA- \int_f (\vec{\nabla}_f\times q)\cdot \mathbf{u}_T\;dA = \int_{\partial f} \mathbf{u}\cdot \tau\; q\;ds,$$ where $\mathbf{u}_T$ is the tangential part of $\mathbf{u}$, and $\vec{\nabla}_f\times$ and $\nabla_f\times$ are surface vector curl and surface scalar curl, respectively. Let $q$ be a constant. Notice that $$\nabla_f \times \mathbf{u}_T = (\nabla\times \mathbf{u})\cdot \mathbf{n},$$ we conclude, $$\int_f(\nabla\times \mathbf{u})\cdot \mathbf{n}\;dA=0.$$
As a direct consequence of Lemma \[relation-edge-face\], if both the edge degrees of freedom (\[edge d.o.f\]) and face degrees of freedom (\[face d.o.f\]) vanish, then $$\int_f (\nabla\times \mathbf{u}) \;dA= 0.$$
The polynomial space $R_2(K)$ has the following property [@Girault:1986fk].
If $\mathbf{u}\in R_2(K)$ satisfies $\nabla\times \mathbf{u} = 0$, then $$\mathbf{u}=\nabla p,\;\text{with}\;p\in P_2.$$ \[R2lemma\]
We recall that the finite element $(K,\mathcal{P}_K,\Sigma_K)$ is said to be unisolvent if a function in $\mathcal{P}_K$ can be uniquely determined by specifying values for degrees of freedom in $\Sigma_K$.
The finite element defined by Definition \[fem\_triple\_def\] is unisolvent.
It is sufficient to prove that, given $\mathbf{u}\in R_2(K)$, $$M_e(\mathbf{u}) = M_f(\mathbf{u}) = 0,\;\forall e\subset K, f\subset K\Rightarrow \mathbf{u} = 0.$$ Obviously, $
\nabla (\nabla\times \mathbf{u})$ is a constant vector. Then using (\[curl\_u\_face\_integral\]) and integration by parts, we obtain $$\nabla(\nabla\times \mathbf{u}) = \frac{1}{|K|}\int_K \nabla(\nabla\times \mathbf{u}) dx = \frac{1}{|K|}\int_{\partial K} (\nabla\times \mathbf{u})\mathbf{n}^T dA = 0.$$ This implies that $$\nabla(\nabla\times \mathbf{u}) = 0 \Rightarrow \nabla\times \mathbf{u} = \text{const}.$$
Using again (\[curl\_u\_face\_integral\]), we have $$\nabla\times \mathbf{u} = 0.$$ By Lemma \[R2lemma\], we have $$\mathbf{u}=\nabla p,\;\text{with}\;p\in P_2(K).$$
Since $M_e(\mathbf{u}) = 0$, we have $$\int_e \frac{\partial p}{\partial \tau}\; q\; ds = 0, \forall q\in P_1(e).$$ This implies $\partial p/\partial \tau = 0$ on each edge $e$. Hence, $p$ is constant and $\mathbf{u}=0$.
In the following, we construct the basis functions. The explicit form of these basis functions not only is useful for implementation, but also instrumental for the interpolation error estimate.
Basis functions
---------------
The main idea of the construction is to consider linear combinations of basis functions of a related Nédélec element. Let $K$ be an arbitrary tetrahedron with four vertices $a_i$, $a_j$, $a_k$ and $a_l$, see Figure \[element2\]. The corresponding barycentric coordinates are given by $\lambda_i$, $\lambda_j$, $\lambda_k$, and $\lambda_l$, respectively.
{width="60mm"}
are two tangential vectors on the face $F_l$. \[element2\]
On each of the four faces, say face $l$ (with vertices $a_i, a_j, a_k$), we choose the following two tangential direction vectors: $$\mathbf{q}_{ij} = \overrightarrow{a_j a_i} = 6|K|(\nabla\lambda_l\times \nabla\lambda_k),$$ $$\mathbf{q}_{ik} = \overrightarrow{a_k a_i} =
6|K|(\nabla\lambda_j\times \nabla\lambda_l).$$
The edge degrees of freedom on edge $e_{ij}$ (with vertices $a_i$ and $a_j$) are defined explicitly by: $$M_{ij}^{(1)}(\mathbf{u}) = \int_{e_{ij}} \mathbf{u}\cdot\tau \;ds,$$ $$M_{ij}^{(2)}(\mathbf{u}) = \int_{e_{ij}} \mathbf{u}\cdot
\tau\left(3-\frac{6}{|e_{ij}|}s\right)\;ds,$$ where $\tau$ is the unit direction vector of edge $e_{ij}$, $s$ is an arc length parameter. The face degrees of freedom are defined as: $$M_{lij}(\mathbf{u}) = \frac{1}{|f_l|^2} \int_{f_{l}}(\nabla \times \mathbf{u})\times \mathbf{n}_l\cdot \mathbf{q}_{ij}\;dA,$$ $$M_{lik}(\mathbf{u}) = \frac{1}{|f_l|^2} \int_{f_{l}}(\nabla \times \mathbf{u})\times \mathbf{n}_l\cdot \mathbf{q}_{ik}\;dA,$$ where $\mathbf{n}_l$ is the unit outward normal vector of the face $f_l$.
We recall that the basis functions of the second order Nédélec element of the first family in barycentric coordinates are (see, e.g., [@Gopalakrishnan:2005eu], [@Sun:2008yq], [@Webb:1999sh]):
\(1) Two basis functions on each edge $e_{ij}$: $$\mathbf{L}_{ij} = \lambda_i\nabla\lambda_j-\lambda_j\nabla\lambda_i,$$ $$\mathbf{L}_{ji} = \lambda_i\nabla\lambda_j+\lambda_j\nabla\lambda_i.$$
\(2) Two basis functions on each face $f_l$: $$\mathbf{L}_{ijk} = \lambda_i(\lambda_j\nabla\lambda_k - \lambda_k\nabla\lambda_j),$$ $$\mathbf{L}_{jik} = \lambda_j(\lambda_i\nabla\lambda_k - \lambda_k\nabla\lambda_i).$$
In the following, we list a few useful facts about the geometry of a tetrahedron.
\(1) The unit outward normal vector of face $f_l$ is given by $$-\frac{\nabla \lambda_l}{\|\nabla\lambda_l\|}.$$
\(2) The two tangential vectors of face $f_l$ are given by $\mathbf{q}_{ij}$ and $\mathbf{q}_{ik}$.
\(3) Let $h_l$ be the height of the tetrahedron corresponding to the face $f_l$, then $$\nabla\lambda_l=\frac{1}{6|K|}\mathbf{q}_{ik}\times \mathbf{q}_{jk},$$ $$|\nabla\lambda_l| = \frac{1}{h_l}.$$
\(4) Let $|K|$ be the volume of the tetrahedron $K$, then $$6|K|=|\mathbf{q}_{il}\cdot(\mathbf{q}_{jl}\times \mathbf{q}_{kl})| = \frac{-1}{(\nabla\lambda_i\times \nabla\lambda_j)\cdot \nabla\lambda_k }.$$
Next, we construct basis functions in barycentric coordinates. They provide a set of dual basis functions with respect to the prescribed degrees of freedom.
[**Step 1**]{}. Construct eight basis functions $\{\phi_{lij}\}$ corresponding to the face degrees of freedom such that $$M_{mn}^{(t)}(\phi_{lij})=0,\label{face_function_edge}$$ and $$M_{mnp}(\phi_{lij})=\delta_{ml}\delta_{ni}
\delta_{pj}.\label{face_function_face}$$ We use the basis functions of the second order Nédélec element as building blocks as they automatically satisfy the first condition (\[face\_function\_edge\]). Using the facts listed above, we find that the basis functions corresponding to the facial degrees of freedom on face $f_l$ are given by the following: $$\phi_{lij} = 3|K|(\mathbf{L}_{lij}-\mathbf{L}_{ljk}),$$ $$\phi_{lik} = 3|K|(\mathbf{L}_{lik}-\mathbf{L}_{ljk}).$$
By direct calculation, we have $$\begin{aligned}
&&\int_{f_l}(\nabla\times \mathbf{L}_{lij})\times \nabla\lambda_l\cdot(\nabla\lambda_l\times\nabla\lambda_k)\;dA\\
& = & \int_{f_l}\left[2\lambda_l(\nabla\lambda_i\times\nabla\lambda_j)
+\lambda_i(\nabla\lambda_l\times\nabla\lambda_j)-\lambda_j
(\nabla\lambda_l\times\nabla\lambda_i)\right]\\
&& \cdot\left[\nabla\lambda_l(\nabla_l\cdot\nabla_l)
-\nabla\lambda_k(\nabla\lambda_l\cdot\nabla\lambda_k)
\right]\;dA\\
& = & -\left(\int_{f_l}\lambda_i \;dA\right)[(\nabla\lambda_l\times\nabla\lambda_j)
\cdot \nabla\lambda_k](\nabla\lambda_l\cdot\nabla\lambda_l)\\
&& + \left(\int_{f_l}\lambda_j \;dA\right)[(\nabla\lambda_l\times\nabla\lambda_i)
\cdot \nabla\lambda_k](\nabla\lambda_l\cdot\nabla\lambda_l)\\
& = & -\frac{2}{3}|f_l|\frac{1}{6|K|h_l^2},\end{aligned}$$ and $$\begin{aligned}
\int_{f_l}(\nabla\times \mathbf{L}_{ljk})\times \nabla\lambda_l\cdot(\nabla\lambda_l\times\nabla\lambda_k)\;dA = \frac{1}{3}|f_l|\frac{1}{6|K|h_l^2}.\end{aligned}$$ Hence, $$\begin{aligned}
M_{lij}(\phi_{lij})&=&\frac{1}{|f_l|^2}\int_{f_l}
(\nabla\times \phi_{lij})\times n_l\cdot \mathbf{q}_{ij}\;dA\\
&=& \frac{1}{|f_l|^2}\int_{f_l}\nabla\times\left[
3|K|(\mathbf{L}_{lij}-\mathbf{L}_{ljk})\right]
\times -\frac{\nabla\lambda_l}{\|\nabla\lambda_l\|}
\cdot(6|K|\nabla\lambda_l\times
\nabla\lambda_k)dA
\\
&=& -\frac{18|K|^2h_l}{|f_l|^2}\int_{f_l}\nabla\times
(\mathbf{L}_{lij}
-\mathbf{L}_{ljk})\times\nabla\lambda_l\cdot (\nabla\lambda_l
\times\nabla\lambda_k)\;dA\\
&=& -\frac{18|K|^2h_l}{|f_l|^2}
\left(-\frac{2}{3}|f_l|\frac{1}{6|K|h_l^2}
-\frac{1}{3}|f_l|\frac{1}{6|K|h_l^2}\right) = 1.\end{aligned}$$
Similarly, $M_{lik}(\phi_{lij}) = 0$, and $M_{f_{l'}}(\phi_{lij})=0$ where $l'\neq l$.
[**Step 2**]{}. Construct twelve basis functions $\{\psi_{ij}^{(t)}|1\leq i<j\leq 4, t=1,2\}$ corresponding to the edge degrees of freedom such that $$M_{mn}^{(t')}(\psi_{ij}^{(t)}) = \delta_{t't}\delta_{mn,ij},\label{edge_function_edge}$$ $$M_{mnp}(\psi_{ij}) = 0.\label{edge_function_face}$$
Here, we use the edge basis functions of the second order Nédélec element as building blocks since they satisfy condition (\[edge\_function\_edge\]). Since $\nabla\times \mathbf{L}_{ji}=0$, $\mathbf{L}_{ji}$ automatically satisfy condition (\[edge\_function\_face\]).
For functions $\mathbf{L}_{ij}$, we need to subtract from them a linear combination of face basis functions so that (\[edge\_function\_edge\]) and (\[edge\_function\_face\]) hold. This can be done because by construction, our face basis functions have no edge moments. This strategy for constructing basis functions can be found in [@Gopalakrishnan:2005eu; @Sun:2008yq].
Finally, we can write the basis functions of the new element as the following:
\(1) Two basis functions on each face $l$ ($1\leq l\leq 4$): $$\phi_{lij} = 3|K|(\mathbf{L}_{lij}-\mathbf{L}_{ljk}),$$ $$\phi_{lik} = 3|K|(\mathbf{L}_{lik}-\mathbf{L}_{ljk}),$$ where $\mathbf{L}_{lij}= \lambda_i(\lambda_j\nabla\lambda_k - \lambda_k\nabla\lambda_j)$.
\(2) Two basis functions on each edge $e_{ij}$ ($1\leq i<j\leq 4$): $$\psi_{ij}^{(1)}=\mathbf{L}_{ji},$$ $$\psi_{ij}^{(2)}=\mathbf{L}_{ij}-\sum M_{mnp} (\mathbf{L}_{ij})\phi_{mnp}.$$
Convergence analysis
--------------------
Let $\mathcal{T}_h=\{K_i\}_{i=1}^{N_h}$ be a triangulation of the domain $\Omega$. On this triangulation we introduce the finite element space $V_h$ and define the discrete norm $\|\cdot\|_h$ by $$\| \mathbf{v}\|_h=\left[\sum_{K\in\mathcal{T}_h}
\left(\|\mathbf{v}\|_{0,K}^2+\|\nabla\times \mathbf{v}\|_{0,K}^2+\|\nabla(\nabla\times \mathbf{v})\|_{0,K}^2\right)\right]^{1/2}.$$
Consider the following discrete bilinear form: $$\begin{split}
a_h(\mathbf{u}_h,\mathbf{v}_h)
=\sum_{K\in\mathcal{T}_h} \alpha(\nabla(\nabla\times \mathbf{u}_h),\nabla(\nabla\times
\mathbf{v}_h))_{L^2(K)}&+\beta (\nabla\times \mathbf{u}_h,\nabla\times
\mathbf{v}_h)_{L^2(K)}\\
&+ \gamma (\mathbf{u}_h,\mathbf{v}_h)_{L^2(K)}.
\end{split}$$ It is straightforward to verify that the bilinear form $a_h$ satisfies $$a_h(\mathbf{v},\mathbf{v})\gtrsim \|\mathbf{v}\|_h^2,\;\forall\; \mathbf{v}\in V_h,$$ $$|a_h(\mathbf{u},\mathbf{v})|\lesssim
\|\mathbf{u}\|_h\|\mathbf{v}\|_h,\;
\forall\;\mathbf{u}\in V+V_h, \mathbf{v}\in V_h.$$
The nonconforming finite element discretization of problem (3.10) is:
Find $\mathbf{u}_h\in V_h$, such that for all $\mathbf{v}_h\in V_h$, $$a_h(\mathbf{u}_h,\mathbf{v}_h)=(\mathbf{f},\mathbf{v}_h).\label{discrete var prob1}$$ The convergence of the above finite element approximation can be analyzed through the following second Strang lemma [@Ciarlet:1978qf].
$$\|\mathbf{u}-\mathbf{u}_h\|_h\lesssim \inf_{\mathbf{v}_h\in V_h}\|\mathbf{u}-\mathbf{v}_h\|_h +
\sup_{\mathbf{w}_h\in V_h}\frac{|a_h(\mathbf{u},\mathbf{w}_h)-
(\mathbf{f},\mathbf{w}_h)|}{\|\mathbf{w}_h\|_h},$$ where the first term on the right-hand side is called the interpolation error and the second term is called the consistency error.
In order to estimate the consistency error we first define an average operator $P_{f}$ on a face $f$ by $$P_{f} \mathbf{w} = \frac{1}{|f|}\int_{f} \mathbf{w} \;dA.$$ Since for any $\mathbf{v}_h\in V_h$, the quantity $\int_f \nabla\times \mathbf{v}_h\;dA$ is continuous, we know that $P_f$ is well-defined for $\nabla\times \mathbf{v}_h$. The following two lemmas are standard results.
Given any face $f\subset K$ and $\mathbf{w}\in (H^1(K))^3$, $$\int_{f} |\mathbf{w}-P_{f}\mathbf{w}|^2 \;dA \lesssim h_K|\mathbf{w}|^2_{1,K}.$$\[face\_estimate\_lemma\]
$$\int_{\partial K}|\mathbf{w}|^2 \;dA \lesssim
h_K^{-1}\norm{\mathbf{w}}_{0,K}^2+h_K|
\mathbf{w}|_{1,K}^2.$$\[trace\_lemma\]
Next, we estimate the interpolation error and consistency error separately.
### Interpolation error estimate
Let $K$ and $K'_f$ be the two tetrahedra sharing a common face $f$, $r_K$ be the local interpolation operator for the second order Nédélec element of the first family, namely, given $\mathbf{u}\in V$, define $r_K \mathbf{u}$ such that $$\int_e r_K \mathbf{u}\cdot \tau\;ds = \int_e \mathbf{u}\cdot\tau\;ds,\;\forall\;\text{edge}\;
e\subset K,$$ and $$\int_f (r_K \mathbf{u}\times \mathbf{n})
\cdot q\;dA = \int_f (\mathbf{u}\times \mathbf{n})\cdot \mathbf{q}\;dA,\;\forall\;\mathbf{q}\in (P_0(f))^2,\;\forall\;\text{face}\;f\subset K.$$ Define $\mathbf{u}_I\in V_h$ such that $$M_e(\mathbf{u}_I)=M_e(r_K \mathbf{u})=M_e(\mathbf{u}),$$ $$M_f(\mathbf{u}_I)=[M_f(r_{K} \mathbf{u})+M_f(r_{K'_f} \mathbf{u})]/2.$$
If $f\subset \partial \Omega$, we set $M_f(\mathbf{u}_I)=M_f(r_{K} \mathbf{u})$.
Given $\mathbf{u}\in V$, let $\mathbf{u}_I$ be defined as above, then $$\|\mathbf{u}-\mathbf{u}_I\|_h \lesssim h(|\mathbf{u}|_2+|\nabla\times \mathbf{u}|_2).$$
Let $r_h \mathbf{u}$ be the global interpolation operator defined by $r_h \mathbf{u}|_K = r_K \mathbf{u}$. By triangle inequality, $$\|\mathbf{u}-\mathbf{u}_I\|_h\leq \|\mathbf{u}-r_h \mathbf{u}\|_h+\|r_h \mathbf{u} - \mathbf{u}_I\|_h.$$ By the interpolation error estimate of Nédélec element, we have $$\|\mathbf{u} - r_h \mathbf{u}\|_h \lesssim h(|\mathbf{u}|_2+|\nabla\times \mathbf{u}|_2).$$
Notice that on each tetrahedron $K$, $$r_K \mathbf{u} - \mathbf{u}_I = \sum_{f\subset K}\sum M_{mnp} (r_K \mathbf{u} - \mathbf{u}_I)\phi_{mnp},$$ where $\{\phi_{mnp}\}$ are basis functions on face $f$, and $\{M_{mnp}(\cdot)\}$ are degrees of freedom on face $f$. Using Lemma \[trace\_lemma\] and $\|q_{np}\|_{L^2(f)} = O(h^2)$, we get $$\begin{aligned}
&&2|M_{mnp}(r_K \mathbf{u} - \mathbf{u}_I)|=|M_{mnp}(r_{K'_f} \mathbf{u})-M_{mnp}(r_{K} \mathbf{u})|\\
&&=\frac{1}{|f|^2}\bigg|\int_f(\nabla\times r_{K'_f} \mathbf{u} - \nabla\times r_{K} \mathbf{u} )\times \mathbf{n}\cdot \mathbf{q}_{np}\;dA\bigg|\\
&&\leq \frac{1}{|f|^2}\bigg|\int_f (\nabla\times(r_{K'_f}\mathbf{u} - \mathbf{u})\times \mathbf{n} \cdot \mathbf{q}_{np}\;dA\bigg| + \frac{1}{|f|^2}\bigg|\int_f (\nabla\times(r_{K}\mathbf{u} - \mathbf{u})\times \mathbf{n} \cdot \mathbf{q}_{np}\;dA\bigg| \\
&&\lesssim \frac{\|\mathbf{q}_{np}\|_{L^2(f)}}{|f|^2}
(\|\nabla\times(r_{K'_f}\mathbf{u} - \mathbf{u})\times \mathbf{n}\|_{L^2(f)} +\|\nabla\times (r_{K}\mathbf{u} - \mathbf{u})\times \mathbf{n}\|_{L^2(f)})\\
&&\lesssim h^{-2}(h^{-1/2}\|\nabla\times(r_{K'_f}\mathbf{u} - \mathbf{u})\|_{0, K\cup K'} + h^{1/2}| \nabla\times (r_{K}\mathbf{u} - \mathbf{u})|_{1,K\cup K'})\\
&&\lesssim h^{-1/2}|\nabla\times \mathbf{u}|_{2,K\cup K'}.\end{aligned}$$
Notice $\nabla\lambda_i=O(h^{-1})$, and $\|\phi_{mnp}\|_{0,K}^2=O(h^7)$, by Cauchy-Schwarz inequality, we have $$\begin{aligned}
\|r_K \mathbf{u} - \mathbf{u}_I\|_{0,K} &\leq &\left(\sum_{f\subset K}\sum |M_{mnp}(r_K \mathbf{u} - r_I \mathbf{u})|^2\right)^{1/2}\left( \sum_{f\subset K}\sum \|\phi_{mnp}\|^2_{0,K}\right)^{1/2}\\
& \lesssim & h^3|\nabla\times \mathbf{u}|_{2,S(K)},\end{aligned}$$ where $S(K)=\cup_{K'\in\mathcal{T}_h, K'\cap K\neq \emptyset} K'$.
Hence, $$\|r_h \mathbf{u} - \mathbf{u}_I\|_{0,\Omega} = (\sum_K \|r_K \mathbf{u} - \mathbf{u}_I\|^2_{0,K})^{1/2}\lesssim h^3|\nabla\times \mathbf{u}|_{2,\Omega}.
\label{ruuI}$$ By inverse inequality, we have $$\|\nabla\times (r_h \mathbf{u} - \mathbf{u}_I)\|_{0,\Omega}\lesssim h^2|\nabla\times \mathbf{u}|_{2,\Omega},$$ $$\|\nabla(\nabla\times (r_h \mathbf{u} - \mathbf{u}_I))\|_{0,\Omega}\lesssim h|\nabla\times \mathbf{u}|_{2,\Omega}.$$ Combining these estimates, we get $$\|r_h \mathbf{u} - \mathbf{u}_I\|_h\lesssim h|\nabla\times \mathbf{u}|_{2,\Omega},$$ and the desired estimate follows.
Remark: We note that the error estimate (\[ruuI\]) indicates that $r_h \mathbf{u}$ and $\mathbf{u}_I$ are super-close. Such type of estimate can not usually be obtained by the standard scaling argument (using Bramble-Hilbert lemma). In our proof, we made use of the detailed information of the basis functions constructed in the previous section.
### Consistency error estimate
Given a tetrahedron $K$, in addition to the local interpolation operator $r_K$, we introduce another local interpolation operator $\tilde{r}_K$ corresponding to the first order Nédélec element of the second family, namely, $\tilde{r}_K \mathbf{u}\in (P_1(K))^3\subset R_2(K)$, and $$\int_e ((\tilde{r}_K \mathbf{u})\cdot \tau ) \;q\;ds = \int_e
(\mathbf{u}\cdot\tau)\; q\;ds,\;\forall\;q\in P_1(e),\;\forall\;\text{edge}\;e\subset K.$$
Consider two tetrahedra $K$ and $K'_f$ that share a common face $f$. Given $\mathbf{v}_h\in V_h$, define $\mathbf{v}_K=\mathbf{v}_h|_K$. By definition, $$\tilde{r}_{K}\mathbf{v}_{K} = \tilde{r}_{K'_f}\mathbf{v}_{K'_f},\;\text{on face}\;f.$$ Hence, $$\sum_K \int_{\partial K} \varphi\cdot [(\tilde{r}_K\mathbf{v}_K)\times \mathbf{n}] \;dA = 0,\;\forall\;\varphi\in H(\text{curl};\Omega),
\label{nedelec face equality}$$ where $\mathbf{n}$ is the unit outward normal vector of $\partial K$.
Consider the decomposition (see [@Girault:1986fk]): $$\mathbf{v}_K = \nabla p_K + \mathbf{w}_K,$$ where $\text{div}\;\mathbf{w}_K = 0$, $\mathbf{w}_K\cdot \mathbf{n}|_{\partial K}=0$, and $p_K\in P_2(K)$. The following Lemma \[hu-zou-lemma\] can be found in [@Hu:2004fp]:
$$\|\tilde{r}_K \mathbf{w}_K - \mathbf{w}_K\|_{0,K}\lesssim h\|\nabla\times \mathbf{v}_K\|_{0,K}.$$ \[hu-zou-lemma\]
As a consequence of Lemma \[hu-zou-lemma\], we have the following estimate.
$$\|\tilde{r}_K \mathbf{v}_K - \mathbf{v}_K\|_{0,K}\lesssim h\|\nabla\times \mathbf{v}_K\|_{0,K}.$$ \[wk lemma\]
Using the interpolation operators defined above, we obtain $$\tilde{r}_K\mathbf{v}_K = \tilde{r}_K\nabla p_K + \tilde{r}_K\mathbf{w}_K = \nabla p_K +\tilde{r}_K\mathbf{w}_K.$$ Hence, $$\tilde{r}_K \mathbf{v}_K- \mathbf{v}_K = \tilde{r}_K \mathbf{w}_K - \mathbf{w}_K.$$
By Lemma \[hu-zou-lemma\], we obtain $$\|\tilde{r}_K \mathbf{v}_K - \mathbf{v}_K\|_{0,K}\lesssim h\|\nabla\times \mathbf{v}_K\|_{0,K}.$$
Now, we can show the following lemma, which is critical for the consistency error estimate.
For $\varphi\in H(\text{curl};\Omega)$, $$|\sum_{K}\int_{\partial K}\varphi \cdot (\mathbf{v}_h\times \mathbf{n})\;dA
\lesssim h(\|\varphi\|_{0,\Omega}+\|\nabla\times \varphi\|_{0,\Omega})\left(\sum_K\|\nabla\times \mathbf{v}_h\|_{1,K}^2\right)^{1/2}.$$ \[critical lemma for consistency\]
By the interpolation error estimates of the Nédélec elements $$\|\nabla\times (\tilde{r}_K\mathbf{v}_K -\mathbf{v}_K)\|_{0,K}\lesssim h \|\nabla\times \mathbf{v}_K\|_{1,K},$$ Lemma \[wk lemma\], and Equation (\[nedelec face equality\]), we have $$\begin{aligned}
&&\bigg|\sum_K \int_{\partial K}\varphi\cdot (\mathbf{v}_K\times \mathbf{n})\;dA\bigg|
= \bigg|\sum_K\int_{\partial K}\varphi\cdot [(\tilde{r}_K \mathbf{v}_K-\mathbf{v}_K)\times \mathbf{n}]\;dA\bigg|\\
&&= \bigg|\sum_K\int_K (\nabla\times \varphi)\cdot (\tilde{r}_K\mathbf{v}_K-\mathbf{v}_K)\;dx
- \varphi\cdot [\nabla\times (\tilde{r}_K\mathbf{v}_K-\mathbf{v}_K)]\;dx\bigg| \\
&&\leq \sum_K(\|\nabla\times\varphi\|_{0,K}\|\tilde{r}_K
\mathbf{v}_K-\mathbf{v}_K\|_{0,K}
+\|\varphi\|_{0,K}\|\nabla\times(\tilde{r}_K
\mathbf{v}_K-\mathbf{v}_K)\|_{0,K})\\
&&\lesssim h(\|\varphi\|_{0,\Omega}+
\|\nabla\times\varphi\|_{0,\Omega})
\left(\sum_K\|\nabla\times \mathbf{v}_h\|_{1,K}^2\right)^{1/2}.\end{aligned}$$
Next, we show the consistency error estimate for the nonconforming finite element approximation defined above.
Assume that $\mathbf{u}\in V$ is sufficiently smooth and $\mathbf{v}_h\in V_h$, then $$\begin{gathered}
|a_h(\mathbf{u},\mathbf{v}_h)-(\mathbf{f},\mathbf{v}_h)|
\lesssim h(\| \nabla\times \Delta(\nabla\times \mathbf{u})\|+|\nabla(\nabla\times \mathbf{u})|_1\\+\|(\nabla\times)^2 \mathbf{u}\|+\|\nabla\times \mathbf{u}\|)\left(\sum_K\|\nabla\times
\mathbf{v}_h\|_{1,K}^2\right)^{1/2}.\end{gathered}$$
By applying integration by parts, we get $$\begin{aligned}
&&\;\;\;\;\left(\nabla(\nabla\times \mathbf{u}),\nabla(\nabla\times \mathbf{v}_h)\right)_K \\&=& -(\Delta(\nabla\times \mathbf{u}),\nabla\times \mathbf{v}_h)_K+(\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n},\nabla\times \mathbf{v}_h)_{\partial K}\\
&=&-(\nabla\times\Delta(\nabla\times \mathbf{u}), \mathbf{v}_h)_K
+(\Delta (\nabla\times \mathbf{u}),\mathbf{v}_h\times \mathbf{n})_{\partial K}\\
&&+(\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n},\nabla\times \mathbf{v}_h)_{\partial K},\end{aligned}$$ and $$(\nabla\times \mathbf{u},\nabla\times \mathbf{v}_h)_K
=((\nabla\times)^2 \mathbf{u},\mathbf{v}_h)_K-(\nabla\times \mathbf{u}, \mathbf{v}_h\times \mathbf{n})_{\partial K}.$$
Hence, $$\begin{aligned}
&&\;\;\;\; a_h(\mathbf{u},\mathbf{v}_h)-(\mathbf{f},\mathbf{v}_h)\\
&&=\sum_{K\in\mathcal{T}_h}[\alpha(\Delta(\nabla\times \mathbf{u}), \mathbf{v}_h \times
\mathbf{n})_{\partial K}+\alpha(\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n},\nabla\times
\mathbf{v}_h)_{\partial K}\\
&& \quad\quad\quad\quad- \beta(\nabla\times \mathbf{u},\mathbf{v}_h\times \mathbf{n})_{\partial K}]\\
&&=\sum_{K\in\mathcal{T}_h}\left[(\alpha \Delta (\nabla\times \mathbf{u})-\beta\nabla\times \mathbf{u},\mathbf{v}_h\times \mathbf{n})_{\partial K}\right]\\
&&\quad\quad+\sum_{K\in\mathcal{T}_h}\left[
\alpha(\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n}, \nabla\times \mathbf{v}_h)_{\partial K}\right].\label{int-by-parts}\end{aligned}$$
By Lemma \[critical lemma for consistency\], we have $$\begin{aligned}
&&\sum_{K\in\mathcal{T}_h}\left[(\alpha \Delta (\nabla\times \mathbf{u})-\beta\nabla\times \mathbf{u},\mathbf{v}_h\times \mathbf{n})_{\partial K}\right]\\
&& \lesssim h(\|\Delta(\nabla\times \mathbf{u})\|_{0,\Omega}+\|\nabla\times \Delta(\nabla\times \mathbf{u})\|_{0,\Omega}+\|\nabla\times \mathbf{u}\|_{0,\Omega}+\|(\nabla\times)^2 \mathbf{u}\|_{0,\Omega}
)\\
&&\hskip 3mm \left(\sum_K\|\nabla\times \mathbf{v}_h\|_{1,K}^2\right)^{1/2}.\end{aligned}$$
By Lemma \[face\_estimate\_lemma\] and the inter-element continuity of $\nabla\times \mathbf{v}_h$, we get $$\begin{aligned}
&&\sum_{K\in\mathcal{T}_h}\left[
\alpha(\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n}, \nabla\times \mathbf{v}_h)_{\partial K}\right]\\
&&\leq \alpha\left| \sum_{K\in\mathcal{T}_h}\sum_{f\subset\partial K}(
\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n}-P_f(\nabla(\nabla\times \mathbf{u})\cdot \mathbf{n}),\nabla\times \mathbf{v}_h-P_f(\nabla\times \mathbf{v}_h))_f\right|\\
&&\lesssim h|\nabla(\nabla\times \mathbf{u})|_{1,\Omega}\left(
|\sum_{K\in\mathcal{T}_h}|\nabla\times \mathbf{v}_h|_{1,K}^2\right)^{1/2}.\end{aligned}$$ The theorem follows by combining the above estimates of the two boundary integrals.
Finally, we have the following convergence result.
Let $\mathbf{u}$ and $\mathbf{u}_h$ be the solutions of the problems (3.10) and (3.16) respectively, then $$\norm{\mathbf{u}-\mathbf{u}_h}_{0,h}+\norm{\nabla\times
(\mathbf{u}-\mathbf{u}_h)}_{0,h}+\norm{\nabla(\nabla\times (\mathbf{u}-\mathbf{u}_h))}_{0,h}\lesssim
h \norm{\mathbf{u}}_{4,\Omega}$$ when $\mathbf{u}\in (H^4(\Omega))^3$.\[convergence-theorem\]
Using the second Strang lemma, $$\begin{aligned}
&&\;\;\;\;\norm{\mathbf{u}-\mathbf{u}_h}_{0,h}+
\norm{\nabla\times
(\mathbf{u}-\mathbf{u}_h)}_{0,h}+
\norm{\nabla(\nabla\times (\mathbf{u}-\mathbf{u}_h))}_{0,h}\\
&&\lesssim \inf_{\mathbf{w}_h\in V_{h}}(\norm{\mathbf{u}-\mathbf{w}_h}_{0,h}+\norm{\nabla\times
(\mathbf{u}-\mathbf{w}_h)}_{0,h}+\norm{\nabla(\nabla\times (\mathbf{u}-\mathbf{w}_h))}_{0,h})
\\
&&\;\;\;\;+ \sup_{\mathbf{w}_h\in V_{h},\mathbf{w}_h\neq
0}\frac{a_h(\mathbf{u},\mathbf{w}_h)-(\mathbf{f},
\mathbf{w}_h)}{\norm{\nabla\times
\mathbf{w}_h}_{1,h}},\end{aligned}$$ and previous lemmas, the desired inequality follows.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Prof. Ludmil Zikatanov and Dr. Luis Chacón for many helpful discussions.
[10]{}
D. Biskamp, E. Schwarz and J.F. Drake, *Ion-controlled collisionless magnetic reconnection*, Phys. Rev. Lett., 75:3850-3853, 1995.
H. Blum and R. Rannacher, *On the boundary value problem of the biharmonic operator on domains with angular corners*, Math. Methods Appl. Sci., 2:556-581, 1980.
F. Cakoni and H. Haddar, *A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media*, Inverse Problems and Imaging, 1:443-456, 2007.
P.G. Ciarlet, *The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam New York, 1978.
R. Codina and N. Hernandez - Silva, *Stabilized finite element approximation of the stationary magneto-hydrodynamics equations*, Comput. Mech., 38:344-355, 2006.
J.-F. Gerbeau, *A stabilized finite element method for the incompressible magnetohydrodynamic equations*, Numer. Math., 87:83-111, 2000.
K. Germaschewski and R. Grauer, *Longitudinal and transversal structure functions in two-dimensional electron magnetohydrodynamic flows*, Phys. Plasmas, 6:3788-3793, 1999.
V. Girault and P. Raviart, *Finite Element Methods for Navier-Stokes Equations*, Springer-Verlag, Berlin Heidelberg, 1986.
J. Gopalakrishnan, L.E. Garcia-Castillo, L.F. Demkowicz, *Nédélec spaces in affine coordinates*, Computers and Mathematics with Applications, 49:1285-1294, 2005.
J.L. Guermond and P.D. Minev, *Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case*, Numer. Methods Partial Differential Equations, 19:709-731, 2003.
Q. Hu and J. Zou, *Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions*, Mathemathics of Computation, 73:35-61, 2004.
S.C. Jardin, *A triangular finite element with first-derivative continuity applied to fusion MHD applications*, J.Comput. Phys., 200:133-152, 2004.
S.C. Jardin and J.A. Breslau, *Implicit solution of the four-field extended magnetohydrodynamic equations using higher-order high-continuity finite elements*, Phys. Plasmas, 12:056101.1-056101.10, 2005.
K.S. Kang and D.E. Keyes, *Implicit symmetrized streamfunction formulations of magnetohydrodynamics*, Int. J. Numer. Meth. Fluids, 58:1201-1222, 2008.
S.K. Krzeminski and M. Smialek and M. Wlodarczyk, *Finite element approximation of biharmonic mathematical model for MHD flow using $\Psi$ - An approach*, IEEE Trans. Magn., 36:1313-1318, 2000.
S. Lankalapallia, J.E. Flahertyb, M.S. Shephard and H. Strauss, *An adaptive finite element method for magnetohydrodynamics*, J.Comput. Phys., 225:363-381, 2007.
W.J. Layton, A.J. Meir and P.G. Schmidt, *A two-level discretization method for the stationary MHD equations*, Electron. Trans. Numer. Anal., 6:198-210, 1997.
L. Morley, *The triangular equilibrium problems in the solution of plate bending problems*, Aero. Quart., 19:149-169, 1968.
J.C. Nédélec, *Mixed finite elements in $\mathbb{R}^3$*, Numer. Math., 35:315-341, 1980.
T.K. Nilssen, X.-C. Cai and R. Winther, *A robust nonconforming $H^2$ element*, Math. Comp., 70:489-505, 2001.
S. Ovtchinnikov, F. Dobrian, X.-C. Cai and D.E. Keyes, *Additive Schwarz-based fully coupled implicit methods for resistive Hall Magnetohydrodynamic problems*, J.Comput. Phys., 225:1919-1936, 2007.
R. Rannacher, *Finite element approximation of simply supported plates and the Babuska paradox*, ZAMM, 59:73-76, 1979.
N.B. Salah, A. Soulaimani and W.G. Habashi, *A finite element method for magnetohydrodynamics*, Comput. Methods Appl. Mech. Engrg., 190:5867-5892, 2001.
D. Schötzau, *Mixed finite element methods for stationary incompressible magnetohydrodynamics*, Numer. Math., 96:771-800, 2004.
H.R. Strauss and D.W. Longcope, *An adaptive finite element method for magnetohydrodynamics*, J.Comput. Phys., 147:318-336, 1998.
D. Sun, *Substructuring preconditioners for high order edge finite element discretizations to Maxwell’s equations in three-dimensions*, Ph.D. Thesis, Chinese Academy of Sciences, 2008.
G. Tóth, *Numerical simulations of magnetohydrodynamic flows*, Invited review at the The Interaction of Stars with their Environment conference, 1996.
M. Wang and J. Xu, *The Morley element for fourth order elliptic equations in any dimensions*, Numer. Math., 103:155-169, 2006.
M. Wang and J. Xu, *Minimal finite element spaces for $2m$-th order partial differential equations in $\mathbb{R}^n$* (submitted), 2006.
M. Wang, J. Xu and Y. Hu, *Modified Morley element method for a fourth elliptic singular perturbation problem*, J. Comput. Math., 24:113-120, 2006.
Jon P. Webb, *Hierarchical vector basis functions of arbitrary order for triangular and tetrahedral finite elements*, IEEE Trans. Antennas Propag., 47:1244-1253, 1999.
M. Wiedmer, *Finite element approximation for equations of magnetohydrodynamics*, Math. Comp., 69:83-101, 1999.
U. Ziegler, *Adaptive mesh refinement in MHD modeling, realization, tests and application*, in Edith Falgarone and Thierry Passot, editors, Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics 614, pages 127-151, Springer-Verlag, Berlin Heidelberg, 2003.
[^1]: The second author was supported by The Key Project of Natural Science Foundation of China G10531080, National Basic Research Program of China No. 2005CB321702 and Natural Science Foundation of China G10771178.
[^2]: The third author was supported by the National Science Foundation under contract DMS-0609727 and DMS-0915153, and Center for Computational Mathematics and Applications, Penn State University.
|
---
author:
- 'Rúben Sousa [^1]'
- 'Manuel Guerra [^2]'
- 'Semyon Yakubovich [^3]\'
date: '\'
title: |
**The hyperbolic maximum principle approach to the construction of generalized convolutions\
**
---
Introduction
============
Given a Sturm-Liouville operator on an interval of the real line, it is well-known that its eigenfunction expansion gives rise to an integral transform which shares many properties with the ordinary Fourier transform [@dunfordschwartz1963; @titchmarsh1962]. Since various standard special functions are solutions of Sturm-Liouville equations, the class of integral transforms of Sturm-Liouville type includes, as particular cases, many common integral transforms (Hankel, Kontorovich-Lebedev, Mehler-Fock, Jacobi, Laguerre, etc.).
The Fourier transform lies at the heart of the classical theory of harmonic analysis. This naturally raises a question: *is it possible to generalize the main facts of harmonic analysis to integral transforms of Sturm-Liouville type?*
Starting from the seminal works of Delsarte [@delsarte1938] and Levitan [@levitan1940] it was noticed that the key ingredient for developing of such a generalized harmonic analysis is the so-called product formula. We say that an indexed family of complex-valued functions $\{w_\lambda\}$ on an interval $I \subset \mathbb{R}$ has a *product formula* if for each $x,y \in I$ there exists a complex Borel measure $\bm{\nu}_{x,y}$ (independent of $\lambda$) such that $$\label{eq:intro_prodform}
w_\lambda(x) \, w_\lambda(y) = \int_I w_\lambda \, d\bm{\nu}_{x,y} \qquad (\lambda \in \Lambda).$$ Product formulas naturally lead to generalized convolution operators. To fix ideas, let $\ell(u) = {1 \over r}\bigl[-(pu')' + qu\bigr]$ be a usual Sturm-Liouville differential expression defined on the interval $I$, and let $(\mathcal{F}h)(\lambda) := \int_I h(x) \, w_\lambda(x) \, d\mathrm{m}(x)$ be a Sturm-Liouville type integral transform, where the $w_\lambda$ are solutions of $\ell(w) = \lambda w$ ($\lambda \in \mathbb{C}$). If $\{w_\lambda\}$ has a product formula, then we can define a generalized (Sturm-Liouville type) convolution operator $*$ by $$\label{eq:intro_convdef}
(f * g)(x) := \int_I \biggl( \int_I f \, d\bm{\nu}_{x,y}\biggr) g(y) \,d\mathrm{m}(y).$$ It is not difficult to show that, under reasonable assumptions, the property $\mathcal{F}(f * g) = (\mathcal{F}f) {\kern-.12em\cdot\kern-.12em}(\mathcal{F}g)$ holds for this convolution operator; this means that the analogue of one of the basic identities in harmonic analysis — the Fourier convolution theorem — is satisfied by the generalized convolution.
Consider now the associated hyperbolic partial differential equation $$\label{eq:intro_hypPDE}
{1 \over r(x)} \Bigl\{-\partial_x\bigl[p(x) \, \partial_x f(x,y)\bigr] + q(x) f(x,y)\Bigr\} = {1 \over r(y)} \Bigl\{-\partial_y\bigl[p(y) \, \partial_y f(x,y)\bigr] + q(y) f(x,y)\Bigr\}.$$ If the kernel of the Sturm-Liouville transform is defined via some initial condition $w_\lambda(a) = 1$, then the product $f(x,y) = w_\lambda(x) w_\lambda(y)$ is a solution of satisfying the boundary condition $f(x,a) = w_\lambda(x)$. Studying the properties of the associated hyperbolic equation is therefore a natural strategy for proving the existence of a product formula and extracting information about the measure $\bm{\nu}_{x,y}$.
An especially interesting case is that where $\{\bm{\nu}_{x,y}\}$ turns out to be a family of probability measures (satisfying appropriate continuity assumptions). Indeed, in this case one can show that the convolution gives rise to a Banach algebra structure in the space of finite complex Borel measures in which various probabilistic concepts and properties can be developed in analogy with the classical theory [@bloomheyer1994; @urbanik1964]. Establishing explicit product formulas, or even proving their existence, has been recognized as a difficult problem [@connett1992; @chebli1995]. Nevertheless, using the maximum principle for hyperbolic equations [@weinberger1956], it was shown by Levitan [@levitan1960] (and, under weakened assumptions, by Chebli [@chebli1974] and Zeuner [@zeuner1992]) that this probabilistic property of the product formula holds for a general family of Sturm-Liouville differential expressions on $I = [0,\infty)$ of the form $\ell(u) = - {1 \over A} (Au')'$. This family of Sturm-Liouville operators includes, as important particular cases, the generators of the Hankel transform and the (Fourier-)Jacobi transform; these cases are noteworthy due to the fact that the explicit expression for the measure in the product formula can been determined using results from the theory of special functions (see Examples \[exam:hankelkingman\]–\[exam:fourjacobi\]).
Various examples show that the probabilistic property of the product formula holds only for a restricted class of Sturm-Liouville operators [@litvinov1987; @rosler1995]; this is connected with the fact that the hyperbolic maximum principle requires rather strong assumptions on the coefficients. Notwithstanding, the recent work [@sousaetal2018a; @sousaetal2018b] of the authors on the index Whittaker transform made it apparent that there is room for generalization of the results of [@chebli1974; @levitan1960; @zeuner1992]. In fact, the family of Sturm-Liouville operators considered in these works only includes operators for which the equation is uniformly hyperbolic on $[0,\infty)^2$; a consequence of this is that, under their assumptions, the support ${\mathrm{supp}}(\bm{\nu}_{x,y})$ of the measures in the product formula is always compact. In contrast, the case of the index Whittaker transform provides an example of a product formula whose measures $\bm{\nu}_{x,y}$ have the probabilistic property and satisfy ${\mathrm{supp}}(\bm{\nu}_{x,y}) = [0,\infty)$ for $x,y > 0$; here the associated hyperbolic equation is parabolically degenerate at the boundaries $x=0$ and $y=0$. (The index Whittaker transform is generated by the Sturm-Liouville expression $x^2 u'' + (1+2(1-\alpha) x)u'$ on $I = [0,\infty)$, and its product formula, which is known in closed form, is given in Example \[exam:whittaker\].)
The goal of this work is to introduce a unified framework for the construction of Sturm-Liouville type convolution operators associated with possibly degenerate hyperbolic equations. We will consider a Sturm-Liouville differential expression of the form $$\label{eq:shypPDE_elldiffexpr}
\ell = -{1 \over r} {d \over dx} \Bigl( p \, {d \over dx}\Bigr), \qquad x \in (a,b)$$ ($-\infty \leq a < b \leq \infty$), where $p$ and $r$ are (real-valued) coefficients such that $p(x), r(x) > 0$ for all $x \in (a,b)$ and $p, p', r$ and $r'$ are locally absolutely continuous on $(a,b)$. Concerning the behavior of the coefficients at the boundaries $x=a$ and $x=b$, we will assume respectively that $$\begin{gathered}
\label{eq:shypPDE_Lop_leftBC}
\int_a^c \int_y^c {dx \over p(x)} \, r(y) dy < \infty \\
\label{eq:shypPDE_Lop_rightBC}
\int_c^b \int_y^b {dx \over p(x)} \, r(y) dy = \int_c^b \int_c^y {dx \over p(x)} \, r(y) dy = \infty\end{gathered}$$ where $c \in (a,b)$ is an arbitrary point. These conditions mean that $a$ is a regular or entrance boundary and $b$ is a natural boundary for the operator $\ell$. The notions of regular, entrance and natural boundary refer to the Feller classification of boundaries, which is recalled in Remark \[rmk:shypPDE\_boundaryclassif\], where we also give some comments on the role of conditions –.
The point of departure is the study of the Cauchy problem for the possibly degenerate hyperbolic equation ${1 \over r(x)} \partial_x \bigl( p(x) \, \partial_x f(x,y)\bigr) = {1 \over r(y)} \partial_y \bigl( p(y) \, \partial_y f(x,y)\bigr)$. Under the assumption that the product $p(x)r(x)$ of the coefficients of is an increasing function, we prove an existence and uniqueness theorem for the Cauchy problem which is based on the spectral theory of Sturm-Liouville operators. We then give a sufficient condition for the maximum principle to hold for the hyperbolic equation is given and, as a corollary, the positivity preserving property of the solution of the Cauchy problem is obtained.
Our existence theorem (and the positivity result) covers many hyperbolic equations with initial data on the parabolic line which are outside the scope of the classical theory, and for which the problem of well-posedness of the Cauchy problem was, to the best of our knowledge, open. In fact, given that our results depend heavily on the assumption that the left boundary $a$ is of entrance type (cf. Remark \[rmk:shypPDE\_boundaryclassif\]), they indicate that the well-posedness of the degenerate problem with initial line $y=a$ depends on the Feller boundary classification of $\ell$ at the endpoint $a$.
If the maximum principle holds for the hyperbolic equation associated with $\ell$, then the solution of the hyperbolic Cauchy problem can be written as $f(x,y) = \int_{[a,b)} h \, d\bm{\nu}_{x,y}$, where $h(x) = f(x,a)$ is the initial condition and $\{\bm{\nu}_{x,y}\}$ is a family of finite positive Borel measures on $[a,b)$. Formally, this suggests that the product formula should hold for the kernel $w_\lambda$ of the Sturm-Liouville transform. It turns out that indeed holds and that the $\bm{\nu}_{x,y}$ are probability measures, but the proof requires some effort, especially when the Cauchy problem is parabolically degenerate [@sousaetalforth]. We then define the generalized convolution by , so that the expected convolution theorem $\mathcal{F}(f * g) = (\mathcal{F}f) {\kern-.12em\cdot\kern-.12em}(\mathcal{F}g)$ holds. Moreover, the Young inequality for the $L_p$-spaces with respect to the weighted measure $r(x)dx$ is valid for the convolution , demonstrating that the mapping properties of the generalized convolution structure resemble those of the ordinary convolution.
A fundamental tool for studying the continuity and mapping properties of the generalized convolution is the extension of the Sturm-Liouville transform to complex measures, defined by $\widehat{\mu}(\lambda) = \int_{[a,b)} \! w_\lambda(x) \mu(dx)$. Actually, if we define the convolution of two Dirac measures by $\delta_x * \delta_y = \bm{\nu}_{x,y}$ and then define the convolution $\mu * \nu$ of two complex measures so that $(\mu,\nu) \mapsto \mu * \nu$ is weakly continuous, then the space $\mathcal{M}_{\mathbb{C}}[a,b)$ of finite complex measures on $[a,b)$ becomes a convolution measure algebra for which the Sturm-Liouville transform is a generalized characteristic function, in the sense that the property $\widehat{\mu * \nu} = \widehat{\mu} \cdot\widehat{\nu}$ holds. The algebra $(\mathcal{M}_{\mathbb{C}}[a,b),*)$ is therefore a natural environment for studying notions from probabilistic harmonic analysis, in particular infinite divisibility, Gaussian-type measures and Lévy-type (additive) stochastic processes. As anticipated above, the study of these concepts leads to analogues of chief results in probability theory such as the Lévy-Khintchine formula or the contraction property of convolution semigroups.
The class of Lévy-type processes with respect to the convolution measure algebra includes the diffusion process generated by the Sturm-Liouville expression $\ell$, as well as many other Markov processes with discontinuous paths. We hope that this work illuminates the role of product formulas and hyperbolic Cauchy problems on a purely probabilistic problem — that of constructing a class of Lévy-type processes which accommodates a given diffusion process — and stimulates further research on this topic.
The remaining sections are organized as follows. In Section \[sec:prelim\], after introducing the basic properties of the solution of the Sturm-Liouville equation $\ell(w) = \lambda w$, we summarize some key facts from the theory of eigenfunction expansions of Sturm-Liouville operators and from the theory of one-dimensional diffusion processes. Section \[sec:hypPDE\] is devoted to the hyperbolic Cauchy problem associated with $\ell$: an existence and uniqueness theorem is proved and, under suitable assumptions, it is shown that the unique solution satisfies a weak maximum principle. In Section \[sec:transl\_conv\], the solution of the hyperbolic Cauchy problem is used to define the generalized convolution of probability measures and the generalized translation of functions; moreover, the Sturm-Liouville transform of finite measures is introduced and an analogue of the Lévy continuity theorem is established, together with some other basic properties. The product formula for the solution of the Sturm-Liouville equation is discussed in Section \[sec:prodform\]. In Section \[sec:Lp\_harmonic\] we establish the basic properties of the generalized convolution as an operator on weighted $L_p$-spaces. Section \[sec:probtheory\] explores the probabilistic properties of the convolution, demonstrating that the main concepts and facts from the classical theory of infinitely divisible distributions and convolution semigroups can be developed, in a parallel fashion, in the framework of the generalized convolutions considered here. The concluding Section \[sec:examples\] presents several examples and shows that various convolutions associated with standard integral transforms constitute particular cases of the general construction presented here.
Preliminaries {#sec:prelim}
=============
We use the following standard notations. For a subset $E \subset \mathbb{R}^d$, $\mathrm{C}(E)$ is the space of continuous complex-valued functions on $E$; $\mathrm{C}_\mathrm{b}(E)$, $\mathrm{C}_0(E)$ and $\mathrm{C}_\mathrm{c}(E)$ are, respectively, its subspaces of bounded continuous functions, of continuous functions vanishing at infinity and of continuous functions with compact support; $\mathrm{C}^k(E)$ stands for the subspace of $k$ times continuously differentiable functions. $\mathrm{B}_\mathrm{b}(E)$ is the space of complex-valued bounded and Borel measurable functions. The corresponding spaces of real-valued functions are denoted by $\mathrm{C}(E,\mathbb{R})$, $\mathrm{C}_\mathrm{b}(E,\mathbb{R})$, etc.
$L_p(E;\mu)$ ($1 \leq p \leq \infty$) denotes the Lebesgue space of complex-valued $p$-integrable functions with respect to a given measure $\mu$ on $E$. The space of probability (respectively, finite positive, finite complex) Borel measures on $E$ will be denoted by $\mathcal{P}(E)$ (respectively, $\mathcal{M}_+(E)$, $\mathcal{M}_{\mathbb{C}}(E)$). The total variation of $\mu \in \mathcal{M}_{\mathbb{C}}(E)$ is denoted by $\|\mu\|$, and $\delta_x$ denotes the Dirac measure at a point $x$.
Solutions of the Sturm-Liouville equation
-----------------------------------------
We begin by collecting some properties of the solutions of the Sturm-Liouville equation $\ell(u) = \lambda u$ ($\lambda \in \mathbb{C}$), where $\ell$ is of the form and satisfies the boundary condition . We shall write $f^{[1]} = p f'$ and $\mathfrak{s}(x) = \int_c^x {d\xi \over p(\xi)}$ (this is the so-called *scale function*, cf. [@borodinsalminen2002]).
If the Sturm-Liouville equation is regular at the left endpoint $a$, it is well-known that there is an entire solution $w_\lambda(x)$ of $\ell(u) = \lambda u$ satisfying the initial conditions $w_\lambda(a) = \cos\theta$, $w_\lambda^{[1]}(a) = \sin\theta$ ($0 \leq \theta < \pi$). When we only require that holds (so that $a$ may be an entrance boundary), the following lemma ensures that the same continues to hold for the boundary condition with vanishing derivative ($\theta = 0$):
\[lem:shypPDE\_ode\_wsol\] For each $\lambda \in \mathbb{C}$, there exists a unique solution $w_\lambda(\cdot)$ of the boundary value problem $$\label{eq:shypPDE_ode_wsol}
\ell(w) = \lambda w \quad (a < x < b), \qquad\;\; w(a) = 1, \qquad\;\; w^{[1]}(a) = 0.$$ Moreover, $\lambda \mapsto w_\lambda(x)$ is, for each fixed $x$, an entire function of exponential type.
The proof is similar to [@kac1967 Lemma 3], but for completeness we give a sketch here. Let $$\label{eq:shypPDE_wsol_powseries_eta}
\eta_0(x) = 1, \qquad \eta_j(x) = \int_a^x \bigl(\mathfrak{s}(x) - \mathfrak{s}(\xi)\bigr) \eta_{j-1}(\xi) r(\xi) d\xi \quad (j=1,2,\ldots).$$ Pick an arbitrary $\beta \in (a,b)$ and define $\mathcal{S}(x) = \int_a^x \bigl(\mathfrak{s}(\beta) - \mathfrak{s}(\xi)\bigr) r(\xi) d\xi$. From the boundary assumption it follows that $0 \leq \mathcal{S}(x) \leq \mathcal{S}(\beta) < \infty$ for $x \in (a,\beta]$. Furthermore, it is easy to show (using induction) that $|\eta_j(x)| \leq {1 \over j!} (\mathcal{S}(x))^j$ for all $j$. Therefore, the function $$w_\lambda(x) = \sum_{j=0}^\infty (-\lambda)^j \eta_j(x) \qquad (a < x \leq \beta, \; \lambda \in \mathbb{C})$$ is well-defined as an absolutely convergent series. The estimate $$|w_\lambda(x)| \leq \sum_{j=0}^\infty |\lambda|^j {(\mathcal{S}(x))^j \over j!} = e^{|\lambda| \mathcal{S}(x)} \leq e^{|\lambda| \mathcal{S}(\beta)} \qquad (a < x \leq \beta)$$ shows that $\lambda \mapsto w_\lambda(x)$ is entire and of exponential type. In addition, for $a < x \leq \beta$ we have $$\begin{aligned}
1 - \lambda \int_a^x {1 \over p(y)} \int_a^y w_\lambda(\xi) \, r(\xi) d\xi\, dy & = 1 - \lambda \int_a^x (\mathfrak{s}(x) - \mathfrak{s}(\xi)) w_\lambda(\xi)\, r(\xi) d\xi \\
& = 1-\lambda \int_a^x (\mathfrak{s}(x) - \mathfrak{s}(\xi)) \biggl( \sum_{j=0}^\infty (-\lambda)^j \eta_j(\xi) \biggr) r(\xi) d\xi \\
& = 1 + \sum_{j=0}^\infty (-\lambda)^{j+1} \int_a^x (\mathfrak{s}(x) - \mathfrak{s}(\xi)) \eta_j(\xi)\, r(\xi) d\xi \\
& = 1 + \sum_{j=0}^\infty (-\lambda)^{j+1} \eta_{j+1}(x) \, = \, w_\lambda(x),\end{aligned}$$ i.e., $w_\lambda(x)$ satisfies $$w_\lambda(x) = 1 - \lambda \int_a^x {1 \over p(y)} \int_a^y w_\lambda(\xi) \, r(\xi)d\xi\, dy$$ This integral equation is equivalent to , so the proof is complete.
Throughout this work, $\{a_m\}_{m \in \mathbb{N}}$ will denote a sequence $b > a_1 > a_2 > \ldots$ with $\lim a_m = a$. Next we verify that the solution $w_\lambda$ for the Sturm-Liouville equation on the interval $(a,b)$ is approximated by the corresponding solutions on the intervals $(a_m,b)$:
\[lem:shypPDE\_ode\_wepslimit\] For $m \in \mathbb{N}$, let $w_{\lambda,m}(x)$ be the unique solution of the boundary value problem $$\label{eq:shypPDE_ode_wsoleps}
\ell(w) = \lambda w \quad (a_m < x < b), \qquad\;\; w(a_m) = 1, \qquad\;\; w^{[1]}(a_m) = 0.$$ Then $$\lim_{m \to \infty} w_{\lambda,m}(x) = w_\lambda(x) \quad \text{pointwise for each } a < x < b \text{ and } \lambda \in \mathbb{C}.$$
In the same way as in the proof of Lemma \[lem:shypPDE\_ode\_wsol\] we can check that the solution of is given by $$w_{\lambda,m}(x) = \sum_{j=0}^\infty (-\lambda)^j \eta_{j,m}(x) \qquad (a_m < x < b, \; \lambda \in \mathbb{C})$$ where $\eta_{0,m}(x) = 1$ and $\eta_{j,m}(x) = \int_{a_m}^x \bigl(\mathfrak{s}(x) - \mathfrak{s}(\xi)\bigr) \eta_{j-1,m}(\xi) r(\xi) d\xi$. As before we have $|\eta_{j,m}(x)| \leq {1 \over j!} (\mathcal{S}(x))^j$ for $a_m < x \leq \beta$ (where $\mathcal{S}$ is the function from the proof of Lemma \[lem:shypPDE\_ode\_wsol\]). Using this estimate and induction on $j$, it is easy to see that $\eta_{j,m}(x) \to \eta_j(x)$ as $m \to \infty$ ($a < x \leq \beta$, $j=0,1,\ldots$). Noting that the estimate on $|\eta_{j,m}(x)|$ allows us to take the limit under the summation sign, we conclude that $w_{\lambda,m}(x) \to w_\lambda(x)$ as $m \to \infty$ ($a < x \leq \beta$).
The following lemma provides a sufficient condition for the solution $w_\lambda(\cdot)$ to be uniformly bounded in the variables $x \in (a,b)$ and $\lambda \geq 0$:
\[lem:shypPDE\_wsolbound\] If $x \mapsto p(x)r(x)$ is an increasing function, then the solution of is bounded: $$|w_\lambda(x)| \leq 1 \qquad \text{for all } \, a < x < b, \; \lambda \geq 0.$$
Let us start by assuming that $p(a)r(a) > 0$. For $\lambda = 0$ the result is trivial because $w_0(x) \equiv 1$. Fix $\lambda > 0$. Multiplying both sides of the differential equation $\ell(w_\lambda) = \lambda w_\lambda$ by $2w_\lambda^{[1]}$, we obtain $-{1 \over pr} [(w_\lambda^{[1]})^2]' = \lambda (w_\lambda^2)'$. Integrating the differential equation and then using integration by parts, we get $$\begin{aligned}
\lambda\bigl(1-w_\lambda(x)^2\bigr) & = \int_a^x {1 \over p(\xi) r(\xi)} \bigl(w_\lambda^{[1]}(\xi)^2\bigr)' d\xi \\
& = {w_\lambda^{[1]}(x)^2 \over p(x) r(x)} + \int_a^x \bigl(p(\xi)r(\xi)\bigr)' \biggl({w_\lambda^{[1]}(\xi) \over p(\xi)r(\xi)}\biggr)^{\!2} d\xi, \qquad a < x < b\end{aligned}$$ where we also used the fact that $w_\lambda^{[1]}(a) = 0$ and the assumption that $p(a)r(a) > 0$. The right hand side is nonnegative, because $x \mapsto p(x) r(x)$ is increasing and therefore $(p(\xi)r(\xi))' \geq 0$. Given that $\lambda > 0$, it follows that $1 - w_\lambda(x)^2 \geq 0$, so that $|w_\lambda(x)| \leq 1$.
If $p(a)r(a) = 0$, the above proof can be used to show that the solution of is such that $|w_{\lambda,m}(x)| \leq 1$ for all $a < x < b$, $\lambda \geq 0$ and $m \in \mathbb{N}$; then Lemma \[lem:shypPDE\_ode\_wepslimit\] yields the desired result.
\[rmk:shypPDE\_tildeell\] We shall make extensive use of the fact that the differential expression can be transformed into the standard form $$\widetilde{\ell} = - {1 \over A} {d \over d\xi} \Bigl(A {d \over d\xi} \Bigr) = -{d^2 \over d\xi^2} - {A' \over A} {d \over d\xi}.$$ This is achieved by setting $$\label{eq:shypPDE_tildeell_A}
A(\xi) := \sqrt{p(\gamma^{-1}(\xi)) \, r(\gamma^{-1}(\xi))},$$ where $\gamma^{-1}$ is the inverse of the increasing function $$\gamma(x) = \int_c^x\! \smash{\sqrt{r(y) \over p(y)}} dy,$$ $c \in (a,b)$ being a fixed point (if $\smash{\sqrt{r(y) \over p(y)}}$ is integrable near $a$, we may also take $c=a$). Indeed, it is straightforward to check that a given function $\omega_\lambda: (a,b) \to \mathbb{C}$ satisfies $\ell(\omega_\lambda) = \lambda \omega_\lambda$ if and only if $\widetilde{\omega}_\lambda(\xi) := \omega_\lambda(\gamma^{-1}(\xi))$ satisfies $\widetilde{\ell}(\widetilde{\omega}_\lambda) = \lambda \widetilde{\omega}_\lambda$.
It is interesting to note that the assumption of the previous lemma ($x \mapsto p(x) r(x)$ is increasing) is equivalent to requiring that the first-order coefficient $A' \over A$ of the transformed operator $\widetilde{\ell}$ is nonnegative. We also observe that if this assumption holds then we have $\gamma(b) = \infty$ (otherwise the left-hand side integral in would be finite, contradicting that $b$ is a natural boundary). We have $\gamma(a) > -\infty$ if $a$ is a regular endpoint (Remark \[rmk:shypPDE\_boundaryclassif\]); if $a$ is entrance, $\gamma(a)$ can be either finite or infinite.
Sturm-Liouville type transforms
-------------------------------
For simplicity, we shall write $L_p(r) := L_p\bigl((a,b); r(x)dx\bigr)$ ($1 \leq p < \infty$), and the norm of this space will be denoted by $\|\cdot\|_p$.
It follows from the boundary conditions – that one obtains a self-adjoint realization of $\ell$ in the Hilbert space $L_2(r)$ by imposing the Neumann boundary condition $\lim_{x \downarrow a}u^{[1]}(x) = 0$ at the left endpoint $a$. We state this well-known fact (cf. [@mckean1956; @linetsky2004]) as a lemma:
The operator $$\mathcal{L}: \mathcal{D}_\mathcal{L}^{(2)} \subset L_2(r) \longrightarrow L_2(r), \qquad\quad \mathcal{L} u = \ell(u)$$ where $$\label{eq:shypPDE_Lop_L2domain}
\mathcal{D}_\mathcal{L}^{(2)} := \Bigl\{ u \in L_2(r) \Bigm| u \text{ and } u' \text{ locally abs.\ continuous on } (a,b), \; \ell(u) \in L_2(r), \; \lim_{x \downarrow a} u^{[1]}(x) = 0 \Bigr\}$$ is self-adjoint.
The self-adjoint realization $\mathcal{L}$ gives rise to an integral transform, which we will call the *$\mathcal{L}$-transform*, given by $$\label{eq:shypPDE_Ltransfdef}
(\mathcal{F} h)(\lambda) := \int_a^b h(x) \, w_\lambda(x) \, r(x) dx \qquad (h \in L_1(r), \; \lambda \geq 0)$$ (this is also known as the generalized Fourier transform or the Sturm-Liouville transform). The $\mathcal{L}$-transform is an isometry with an inverse which can be written as an integral with respect to the so-called *spectral measure* $\bm{\rho}_\mathcal{L}$:
\[prop:shypPDE\_Ltransf\] There exists a unique locally finite positive Borel measure $\bm{\rho}_\mathcal{L}$ on $\mathbb{R}$ such that the map $h \mapsto \mathcal{F} h$ induces an isometric isomorphism $\mathcal{F}: L_2(r) \longrightarrow L_2(\mathbb{R}; \bm{\rho}_\mathcal{L})$ whose inverse is given by $$(\mathcal{F}^{-1} \varphi)(x) = \int_\mathbb{R} \varphi(\lambda) \, w_\lambda(x) \, \bm{\rho}_\mathcal{L}(d\lambda),$$ the convergence of the latter integral being understood with respect to the norm of $L_2(r)$. The spectral measure $\bm{\rho}_\mathcal{L}$ is supported on $[0,\infty)$. Moreover, the differential operator $\mathcal{L}$ is connected with the transform via the identity $$\label{eq:shypPDE_Ltransfidentity}
[\mathcal{F} (\mathcal{L} h)] (\lambda) = \lambda {\kern-.12em\cdot\kern-.12em}(\mathcal{F} h)(\lambda), \qquad h \in \mathcal{D}_\mathcal{L}^{(2)}$$ and the domain $\mathcal{D}_\mathcal{L}^{(2)}$ defined by can be written as $$\label{eq:shypPDE_LtransfidentD2}
\mathcal{D}_\mathcal{L}^{(2)} = \Bigl\{ u \in L_2(r) \Bigm| \lambda {\kern-.12em\cdot\kern-.12em}(\mathcal{F} f)(\lambda) \in L_2\bigl([0,\infty); \bm{\rho}_\mathcal{L}\bigr) \Bigr\}.$$
The existence of a generalized Fourier transform associated with the operator $\mathcal{L}$ is a consequence of the standard Weyl-Titchmarsh-Kodaira theory of eigenfunction expansions of Sturm-Liouville operators (see [@sousayakubovich2018 Section 3.1] and [@weidmann1987 Section 8]).
In the general case the eigenfunction expansion is written in terms of two linearly independent eigenfunctions and a $2 \times 2$ matrix measure. However, from the regular/entrance boundary assumption it follows that the function $w_\lambda(x)$ is square-integrable near $x = 0$ with respect to the measure $r(x)dx$; moreover, by Lemma \[lem:shypPDE\_ode\_wsol\], $w_\lambda(x)$ is (for fixed $x$) an entire function of $\lambda$. Therefore, the possibility of writing the expansion in terms only of the eigenfunction $w_\lambda(x)$ follows from the results of [@eckhardt2013 Sections 9 and 10].
It is worth pointing out that the transformation of the Sturm-Liouville operator $\ell$ into its standard form $\widetilde{\ell}$ (Remark \[rmk:shypPDE\_tildeell\]) leaves the spectral measure unchanged: indeed, it is easily verified that the operator $\widetilde{\mathcal{L}}: \mathcal{D}_{\widetilde{\mathcal{L}}}^{(2)} \subset L_2(A) \longrightarrow L_2(A)$, $\widetilde{\mathcal{L}} u = \widetilde{\ell}(u)$ is unitarily equivalent to the operator $\mathcal{L}$ and, consequently, $\bm{\rho}_{\widetilde{\mathcal{L}}} = \bm{\rho}_\mathcal{L}$.
The following lemma gives a sufficient condition for the inversion integral of the $\mathcal{L}$-transform to be absolutely convergent.
\[lem:shypPDE\_Ltransf\_D2prop\] **(a)** For each $\mu \in \mathbb{C} \setminus\mathbb{R}$, the integrals $$\label{eq:shypPDE_Lresolv_specunif}
\int_{[0,\infty)} {w_\lambda(x)\, w_\lambda(y) \over |\lambda - \mu|^2} \bm{\rho}_\mathcal{L}(d\lambda) \qquad\; \text{and} \qquad\; \int_{[0,\infty)} {w_\lambda^{[1]}(x)\, w_\lambda^{[1]}(y) \over |\lambda - \mu|^2} \bm{\rho}_\mathcal{L}(d\lambda)$$ converge uniformly on compact squares in $(a,b)^2$.\
**(b)** If $h \in \mathcal{D}_{\mathcal{L}}^{(2)}$, then $$\begin{aligned}
\label{eq:shypPDE_Ltransf_D2prop}
h(x) & = \int_{[0,\infty)}\! (\mathcal{F}h)(\lambda) \, w_\lambda(x) \, \bm{\rho}_\mathcal{L}(d\lambda)\\
\label{eq:shypPDE_Ltransf_D2propderiv}
h^{[1]}(x) & = \int_{[0,\infty)}\! (\mathcal{F}h)(\lambda) \, w_\lambda^{[1]}(x) \, \bm{\rho}_\mathcal{L}(d\lambda)\end{aligned}$$ where the right-hand side integrals converge absolutely and uniformly on compact subsets of $(a,b)$.
**(a)** By [@eckhardt2013 Lemma 10.6] and [@teschl2014 p. 229], $$\int_{[0,\infty)} {w_\lambda(x) w_\lambda(y) \over |\lambda - \mu|^2} \bm{\rho}_\mathcal{L}(d\lambda) = \int_a^b G(x,\xi,\mu)G(y,\xi,\mu)\, r(\xi) d\xi = {1 \over \mathrm{Im}(\mu)}\, \mathrm{Im}\bigl(G(x,y,\mu)\bigr)$$ where $G(x,y,\mu)$ is the resolvent kernel (or Green function) of the operator $(\mathcal{L}, \mathcal{D}_\mathcal{L}^{(2)})$. Moreover, according to [@eckhardt2013 Theorems 8.3 and 9.6], the resolvent kernel is given by $$G(x,y,\mu) = \begin{cases}
w_\mu(x) \vartheta_\mu(y), & x < y \\
w_\mu(y) \vartheta_\mu(x), & x \geq y
\end{cases}$$ where $\vartheta_\lambda(\cdot)$ is a solution of $\ell(u) = \lambda u$ which is square-integrable near $\infty$ with respect to the measure $r(x)dx$ and verifies the identity $w_\lambda(x) \vartheta_\lambda^{[1]}(x) - w_\lambda^{[1]}(x) \vartheta_\lambda(x) \equiv 1$. It is easily seen (cf. [@naimark1968 p. 125]) that the functions $\mathrm{Im}\bigl(G(x,y,\mu)\bigr)$ and $\partial_x^{[1]} \partial_y^{[1]} \mathrm{Im}\bigl(G(x,y,\mu)\bigr)$ are continuous in $0 < x,y < \infty$. Essentially the same proof as that of [@naimark1968 Corollary 3] now yields that $$\int_{[0,\infty)} {w_\lambda^{[1]}(x) \, w_\lambda^{[1]}(y) \over |\lambda - \mu|^2} \bm{\rho}_\mathcal{L}(d\lambda) = {1 \over \mathrm{Im}(\mu)}\, \partial_x^{[1]} \partial_y^{[1]} \mathrm{Im}\bigl(G(x,y,\mu)\bigr)$$ and that the integrals converge uniformly for $x,y$ in compacts.\
**(b)** By Proposition \[prop:shypPDE\_Ltransf\] and the classical theorem on differentiation under the integral sign for Riemann-Stieltjes integrals, to prove – it only remains to justify the absolute and uniform convergence of the integrals in the right-hand sides.
Recall from Proposition \[prop:shypPDE\_Ltransf\] that the condition $h \in \mathcal{D}_\mathcal{L}^{(2)}$ implies that $\mathcal{F}h \in L_2\bigl([0,\infty); \bm{\rho}_\mathcal{L}\bigr)$ and also $\lambda \,(\mathcal{F}h)(\lambda) \in L_2\bigl([0,\infty); \bm{\rho}_\mathcal{L}\bigr)$. As a consequence, we obtain $$\begin{aligned}
& \int_{[0,\infty)} \bigl|(\mathcal{F}h)(\lambda) w_\lambda(x)\bigr| \bm{\rho}_\mathcal{L}(d\lambda) \\
& \qquad\qquad \leq \!\int_{[0,\infty)} \! \lambda \, \bigl|(\mathcal{F}h)(\lambda)\bigr| \biggl|{w_\lambda(x) \over \lambda + i}\biggr| \bm{\rho}_\mathcal{L}(d\lambda) + \! \int_{[0,\infty)} \bigl|(\mathcal{F}h)(\lambda)\bigr| \biggl| {w_\lambda(x) \over \lambda + i} \biggr| \bm{\rho}_\mathcal{L}(d\lambda) \\
& \qquad\qquad \leq \bigl(\|\lambda \, (\mathcal{F}h)(\lambda)\|_\rho + \|(\mathcal{F}h)(\lambda)\|_\rho\bigr) \biggl\| {w_\lambda(x) \over \lambda + i} \biggr\|_\rho \\
& \qquad\qquad < \infty\end{aligned}$$ where $\| \cdot \|_\rho$ denotes the norm of the space $L_2\bigl(\mathbb{R}; \bm{\rho}_\mathcal{L}\bigr)$, and similarly $$\int_{[0,\infty)}\! \bigl|(\mathcal{F}h)(\lambda) \, w_\lambda^{[1]}(x) \bigr| \bm{\rho}_\mathcal{L}(d\lambda) \leq \bigl(\|\lambda \, (\mathcal{F}h)(\lambda)\|_\rho + \|(\mathcal{F}h)(\lambda)\|_\rho\bigr) \biggl\| {w_\lambda^{[1]}(x) \over \lambda + i} \biggr\|_\rho < \infty.$$ We know from part (a) that the integrals which define $\bigl\| {w_\lambda(x) \over \lambda + i} \bigr\|_\rho$ and $\bigl\| {w_\lambda^{[1]}(x) \over \lambda + i} \bigr\|_\rho$ converge uniformly, hence the integrals in – converge absolutely and uniformly for $x$ in compact subsets.
Diffusion processes
-------------------
In what follows we write $P_{x_0}$ for the distribution of a given time-homogeneous Markov process started at the point $x_0$ and $\mathbb{E}_{x_0}$ for the associated expectation operator.
By an *irreducible diffusion process* $X$ on an interval $I \subset \mathbb{R}$ we mean a continuous strong Markov process $\{X_t\}_{t \geq 0}$ with state space $I$ and such that $$P_x(\tau_y < \infty) > 0 \; \text{ for any } x \in \mathrm{int} \, I \text{ and } y \in I, \quad\;\; \text{ where } \tau_y = \inf\{t \geq 0 \mid X_t = y\}.$$ The *resolvent* $\{\mathcal{R}_\eta\}_{\eta > 0}$ of such a diffusion (or of a general Feller process) $X$ is defined by $\mathcal{R}_\eta u = \int_0^\infty e^{-\eta t} \mathcal{P}_t u \, dt$, $u \in \mathrm{C}_\mathrm{b}(I,\mathbb{R})$, where $(\mathcal{P}_t u)(x) = \mathbb{E}_x[u(X_t)]$ is the transition semigroup of the process $X$. The *$\mathrm{C}_\mathrm{b}$-generator* $(\mathcal{G}, \mathcal{D}(\mathcal{G}))$ of $X$ is the operator with domain $\mathcal{D}(\mathcal{G}) = \mathcal{R}_\eta\bigl(\mathrm{C}_\mathrm{b}(I,\mathbb{R})\bigr)$ ($\eta > 0$) and defined by $$(\mathcal{G} u)(x) = \eta u(x) - g(x) \qquad \text{ for } u = \mathcal{R}_\eta g, \; g \in \mathrm{C}_\mathrm{b}(I,\mathbb{R}), \; x \in I$$ ($\mathcal{G}$ is independent of $\eta$, cf. [@fukushima2014 p. 295]). A *Feller semigroup* is a family $\{T_t\}_{t \geq 0}$ of operators $T_t: \mathrm{C}_\mathrm{b}(I,\mathbb{R}) \longrightarrow \mathrm{C}_\mathrm{b}(I,\mathbb{R})$ satisfying
1. $T_t T_s = T_{t+s}$ for all $t, s \geq 0$;
2. $T_t \bigl(\mathrm{C}_0(I,\mathbb{R})\bigr) \subset \mathrm{C}_0(I,\mathbb{R})$ for all $t \geq 0$;
3. If $h \in \mathrm{C}_\mathrm{b}(I,\mathbb{R})$ satisfies $0 \leq h \leq 1$, then $0 \leq T_t h \leq 1$;
4. $\lim_{t \downarrow 0} \|T_t h - h\|_\infty = 0$ for each $h \in \mathrm{C}_0(I,\mathbb{R})$.
The Feller semigroup is said to be *conservative* if $T_t \mathds{1} = \mathds{1}$ (here $\mathds{1}$ denotes the function identically equal to one). A *Feller process* is a time-homogeneous Markov process $\{X_t\}_{t \geq 0}$ whose transition semigroup is a Feller semigroup. For further background on the theory of Markov diffusion processes and Feller semigroups, we refer to [@borodinsalminen2002] and references therein.
We now recall a known fact from the theory of (one-dimensional) diffusion processes, namely that the negative of the Sturm-Liouville differential operator generates a diffusion process which is conservative and has the Feller property. The proof can be found on [@fukushima2014 Sections 4 and 6] (see also [@mandl1968 Section II.5]).
\[lem:shypPDE\_Lb\_fellergen\] The operator $$\mathcal{L}^\mathrm{(b)}: \mathcal{D}_{\mathcal{L}}^{(\mathrm{b})} \subset \mathrm{C}_\mathrm{b}([a,b),\mathbb{R}) \longrightarrow \mathrm{C}_\mathrm{b}([a,b),\mathbb{R}), \qquad\quad \mathcal{L}^\mathrm{(b)} u = -\ell(u)$$ with domain $$\mathcal{D}_\mathcal{L}^{(\mathrm{b})} = \bigl\{ u \in \mathrm{C}_\mathrm{b}([a,b),\mathbb{R}) \bigm| \ell(u) \in \mathrm{C}_\mathrm{b}([a,b),\mathbb{R}),\, \lim_{x \downarrow a} u^{[1]}(x) = 0 \bigr\}$$ is the $\mathrm{C}_\mathrm{b}$-generator of a one-dimensional irreducible diffusion process $X = \{X_t\}_{t \geq 0}$ with state space $[a,b)$ whose transition semigroup defines a conservative Feller semigroup on $\mathrm{C}_0([a,b),\mathbb{R})$.
The transition probabilities of the one-dimensional diffusion process from the previous lemma admits an explicit representation as the inverse $\mathcal{L}$-transform of the function $e^{-t\lambda} w_\lambda(x)$:
\[lem:shypPDE\_Lb\_diffusion\_tpdf\] The transition semigroup admits the representation $$(\mathcal{P}_t u)(x) = \int_a^b h(y)\, p(t,x,y)\, r(y)dy \qquad (h \in \mathrm{B}_\mathrm{b}\bigl([a,b),\mathbb{R}\bigr), \; t > 0, \; a < x < b)$$ where $p(t,x,y)$ is a nonnegative function which is called the *fundamental solution* for the parabolic equation ${\partial u \over \partial t} = -\ell_x u$ (the subscript indicates the variable in which the operator $\ell$ acts). The fundamental solution and its derivatives are explicitly given by $$\begin{aligned}
(\partial_t^n p)(t,x,y) & = \int_{[0,\infty)} \lambda^n e^{-t\lambda} \, w_\lambda(x) \, w_\lambda(y)\, \bm{\rho}_\mathcal{L}(d\lambda) \\
(\partial_x^{[1]} \partial_t^n p)(t,x,y) & = \int_{[0,\infty)} \lambda^n e^{-t\lambda} \, w_\lambda^{[1]}(x) \, w_\lambda(y)\, \bm{\rho}_\mathcal{L}(d\lambda)\end{aligned}$$ for $n \in \mathbb{N}_0$, where, for fixed $t > 0$, the integrals converge absolutely and uniformly on compact squares of $(a,b) \times (a,b)$.
These assertions are a consequence of the results of [@linetsky2004 Sections 2–3] and [@mckean1956 Section 4].
We mention also that another consequence of the results of [@linetsky2004 Section 3] is that for $h \in L_2(r)$ the expectation of $h(X_t)$ can be written in terms of the $\mathcal{L}$-transform as $$\mathbb{E}_x[h(X_t)] = \int_{[0,\infty)} e^{-t\lambda} w_\lambda(x) \, (\mathcal{F} h)(\lambda)\, \bm{\rho}_\mathcal{L}(d\lambda) \qquad t > 0, \; a < x < b$$ where the integral converges with respect to the norm of $L_2(r)$.
\[rmk:shypPDE\_boundaryclassif\] Let $X$ be a one-dimensional diffusion process on an interval with endpoints $a$ and $b$, whose $\mathrm{C}_\mathrm{b}$-generator is of the form . Let $$I_a = \int_a^c \int_a^y {dx \over p(x)} \, r(y) dy, \qquad J_a = \int_a^c \int_y^c {dx \over p(x)} \, r(y) dy$$ According to *Feller’s boundary classification* for the diffusion $X$, the left endpoint $a$ is called\
----------- ---- ----------------- -----------------------
*regular* if $I_a < \infty$, $\!\!\!J_a < \infty$;
*exit* if $I_a < \infty$, $\!\!\!J_a = \infty$;
----------- ---- ----------------- -----------------------
------------ ---- ----------------- -----------------------
*entrance* if $I_a = \infty$, $\!\!\!J_a < \infty$;
*natural* if $I_a = \infty$, $\!\!\!J_a = \infty$.
------------ ---- ----------------- -----------------------
(the right endpoint is classified in a similar way).
The probabilistic meaning of this classification is the following [@borodinsalminen2002 Chapter II]: an irreducible diffusion can be started from the boundary $a$ if and only if $a$ is regular or entrance; the boundary $a$ is reached from $x_0 \in (a,b)$ with positive probability by an irreducible diffusion if and only if $a$ is regular or exit.
Our standing assumption on the coefficients of the Sturm-Liouville operator means that $a$ is a regular or an entrance boundary for the diffusion process $X$ generated by $\ell$. It is clear from the preceding remarks that Lemma \[lem:shypPDE\_Lb\_fellergen\] relies crucially on this assumption. The same is true for some of the results of the previous subsections: in fact, Lemma \[lem:shypPDE\_ode\_wsol\] fails if $a$ is exit or natural [@ito2006 Sections 5.13–5.14], and the boundary conditions defining $\mathcal{D}_\mathcal{L}^{(2)}$ differ from those in when $a$ is exit or natural [@mckean1956]. In turn, the assumption means that $b$ is a natural boundary for the diffusion $X$. Since one can show that is automatically satisfied whenever Assumption \[asmp:shypPDE\_SLhyperg\] below holds [@sousaetalforth Proposition 3.5], this boundary assumption at $b$ yields no loss of generality on our results concerning product formulas and generalized convolutions.
The hyperbolic equation $\ell_x f = \ell_y f$ {#sec:hypPDE}
=============================================
In this section we investigate the (possibly degenerate) hyperbolic Cauchy problem $$\label{eq:shypPDE_Lcauchy}
(\ell_x f)(x,y) = (\ell_y f)(x,y) \quad\; (x,y \in (a,b)), \qquad\quad
f(x,a) = h(x), \qquad\quad
(\partial_y^{[1]}\!f)(x,a) = 0$$ where $\partial_{\,}^{[1]} u = pu'$, $\ell$ is the Sturm-Liouville operator , and the subscripts indicate the variable in which the operators act.
Since $\ell_y - \ell_x = {p(x) \over r(x)} {\partial^2 \over \partial x^2} - {p(y) \over r(y)} {\partial^2 \over \partial y^2} + \text{lower order terms}$, the equation $\ell_x f = \ell_y f$ is hyperbolic at the line $y=a$ if ${p(a) \over r(a)} > 0$; otherwise, the initial conditions of the Cauchy problem are given at a line of parabolic degeneracy. If $\gamma(a) = -\int_a^c\! \sqrt{r(y) \over p(y)} dy > -\infty$, then we can remove the degeneracy via the change of variables $x = \gamma(\xi)$, $y = \gamma(\zeta)$ (cf. Remark \[rmk:shypPDE\_tildeell\]), through which the partial differential equation is transformed to the standard form $\widetilde{\ell}_\xi u = \widetilde{\ell}_\zeta u$, with initial condition at the line $\zeta = \gamma(a)$. In the case $\gamma(a) = -\infty$, the standard form of the equation is also parabolically degenerate in the sense that its initial line is $\zeta = -\infty$.
Existence and uniqueness of solution
------------------------------------
We start by proving a result which not only assures the existence of solution for Cauchy problems with well-behaved initial conditions but also provides an explicit representation for the solution as an inverse $\mathcal{L}$-transform:
\[thm:shypPDE\_Lexistence\] Suppose that $x \mapsto p(x)r(x)$ is an increasing function. If $h \in \mathcal{D}_{\mathcal{L}}^{(2)\!}$ and $\ell(h) \in \mathcal{D}_{\mathcal{L}}^{(2)\!}$, then the function $$\label{eq:shypPDE_Lexistence}
f_h(x,y) := \int_{[0,\infty)\!} w_\lambda(x) \, w_\lambda(y) \, (\mathcal{F} h)(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda)$$ solves the Cauchy problem .
For ease of notation, unless necessary we drop the dependence in $h$ and denote by $f(x,y)$.
Let us begin by justifying that $\ell_x f$ can be computed via differentiation under the integral sign. It follows from that $w_\lambda^{[1]}(x) = - \lambda \int_a^x w_\lambda(\xi) \,r(\xi) d\xi$ and therefore (by Lemma \[lem:shypPDE\_wsolbound\]) $|w_\lambda^{[1]}(x)| \leq \lambda \int_a^x r(\xi) d\xi$. Hence $$\label{eq:shypPDE_Lexistence_solineq}
\int_{[0,\infty)\!} \bigl| (\mathcal{F} h)(\lambda) \, w_\lambda^{[1]}(x) \, w_\lambda(y)\bigr| \bm{\rho}_{\mathcal{L}}(d\lambda) \leq \int_a^x r(\xi) d\xi {\kern-.12em\cdot\kern-.12em}\int_{[0,\infty)\!} \lambda \, \bigl| (\mathcal{F} h)(\lambda)\, w_\lambda(y) \bigr| \bm{\rho}_{\mathcal{L}}(d\lambda) < \infty,$$ where the convergence (which is uniform on compacts) follows from and Lemma \[lem:shypPDE\_Ltransf\_D2prop\](b). From the convergence of the differentiated integral we conclude that $\partial_x^{[1]}\!f(x,y) = \int_{[0,\infty)\!} (\mathcal{F} h)(\lambda) \, w_\lambda^{[1]}(x) \, w_\lambda(y) \, \bm{\rho}_{\mathcal{L}}(d\lambda)$. Since $(\ell w_\lambda)(x) = \lambda w_\lambda(x)$, in the same way we check that $\int_{[0,\infty)} (\mathcal{F} h)(\lambda) \, (\ell w_\lambda)(x)\, w_\lambda(y)\, \bm{\rho}_{\mathcal{L}}(d\lambda)$ converges absolutely and uniformly on compacts and is therefore equal to $(\ell_x f)(x,y)$. Consequently, $$\label{eq:shypPDE_Lexistence_ellrepr}
(\ell_x f)(x,y) = (\ell_y f)(x,y) = \int_{[0,\infty)\!} \lambda\, (\mathcal{F} h)(\lambda) \, w_\lambda(x) \, w_\lambda(y) \, \bm{\rho}_{\mathcal{L}}(d\lambda).$$ Concerning the boundary conditions, Lemma \[lem:shypPDE\_Ltransf\_D2prop\](b) together with the fact that $w_\lambda(a) = 1$ imply that $f(x,a) = h(x)$, and from we easily see that $\lim_{y \downarrow a} \partial_y^{[1]}\!f(x,y) = 0$. This shows that $f$ is a solution of the Cauchy problem .
Under the assumptions of the theorem, the solution of the hyperbolic Cauchy problem satisfies $$\begin{aligned}
\label{eq:shypPDE_uniq_cond0} f(\cdot, y) \in \mathcal{D}_\mathcal{L}^{(2)} & \qquad \text{for all } \, a < y < b, \\[2pt]
\label{eq:shypPDE_uniq_cond1} \mathcal{F}[\ell_y f(\cdot,y)](\lambda) = \ell_y [\mathcal{F}f(\cdot,y)](\lambda) & \qquad \text{for all } \, a < y < b, \\[2pt]
\label{eq:shypPDE_uniq_cond2} \lim_{y \downarrow a} [\mathcal{F}f(\cdot,y)](\lambda) = (\mathcal{F}h)(\lambda), & \\
\label{eq:shypPDE_uniq_cond3} \lim_{y \downarrow a} \partial_y^{[1]\!} \mathcal{F}[f(\cdot,y)](\lambda) = 0. &\end{aligned}$$ Indeed, by Proposition \[prop:shypPDE\_Ltransf\] we have $[\mathcal{F}f(\cdot,y)](\lambda) = (\mathcal{F}h)(\lambda) \, w_\lambda(y)$ for all $\lambda \in {\mathrm{supp}}(\bm{\rho}_\mathcal{L})$ and $a < y < b$. Since $h \in \mathcal{D}_{\mathcal{L}}^{(2)\!}$ and $|w_\lambda(\cdot)| \leq 1$ (Lemma \[lem:shypPDE\_wsolbound\]), it is clear from that $f(x,y)$ satisfies . Moreover, it follows from that $\mathcal{F}[\ell_y f_j(\cdot,y)](\lambda) = \lambda \, (\mathcal{F}h)(\lambda) \, w_\lambda(y) = \ell_y [\mathcal{F}f_j(\cdot,y)](\lambda)$, hence holds. The properties – follow immediately from Lemma \[lem:shypPDE\_ode\_wsol\].
Next we show that the solution from the above existence theorem is the unique solution satisfying the conditions –:
Let $h \in \mathcal{D}_{\mathcal{L}}^{(2)}$. Let $f_1, f_2 \in \mathrm{C}^2\bigl((a,b)^2\bigr)$ be two solutions of $(\ell_x f)(x,y) = (\ell_y f)(x,y)$. For $f \in \{f_1,f_2\}$, suppose that holds and that there exists a zero $\bm{\rho}_\mathcal{L}$-measure set $\Lambda_0 \subset [0,\infty)$ such that – hold for each $\lambda \in [0,\infty) \setminus \Lambda_0$. Then $$\label{eq:shypPDE_uniq}
f_1(x,y) \equiv f_2(x,y) \qquad \text{ for all } \; x,y \in (a,b).$$
Fix $\lambda \in [0,\infty) \setminus \Lambda_0$ and let $\Psi_{\!j}(y,\lambda) := [\mathcal{F}f_j(\cdot,y)](\lambda)$. We have $$\ell_y \Psi_{\!j}(y,\lambda) = \mathcal{F}[\ell_y f_j(\cdot,y)](\lambda) = \mathcal{F}[\ell_x f_j(\cdot,y)](\lambda) = \lambda \Psi_{\!j}(y,\lambda), \qquad a < y < b$$ where the first equality is due to and the last step follows from . Moreover, $$\lim_{y \downarrow a}\Psi_{\!j}(y,\lambda) = (\mathcal{F}h)(\lambda) \quad \text{ and } \quad \lim_{y \downarrow a} \partial_y^{[1]}\Psi_{\!j}(y,\lambda) = 0$$ by and , respectively. It thus follows from Lemma \[lem:shypPDE\_ode\_wsol\] that $$[\mathcal{F}f_j(\cdot,y)](\lambda) = \Psi_{\!j}(y,\lambda) = (\mathcal{F}h)(\lambda)\, w_\lambda(y), \qquad a < y < b.$$ This equality holds for $\bm{\rho}_\mathcal{L}$-almost every $\lambda$, so the isometric property of $\mathcal{F}$ gives $f_1(\cdot,y) = f_2(\cdot,y)$ Lebesgue-almost everywhere; since the $f_j$ are continuous, we conclude that holds.
We emphasize that the two previous propositions, in particular, ensure that there exists a unique solution for the Cauchy problem with initial condition $$h \in \mathrm{C}_{\mathrm{c},0}^4 := \Bigl\{u \in \mathrm{C}_\mathrm{c}^4[a,b) \Bigm| \ell(u), \ell^2(u) \in \mathrm{C}_\mathrm{c}[a,b), \;\: \lim_{x \downarrow a} u^{[1]}(x) = \lim_{x \downarrow a} [\ell(u)]^{[1]}(x)= 0 \Bigr\}$$ (clearly, if $h \in \mathrm{C}_{\mathrm{c},0}^4$ then $h, \ell(h) \in \mathcal{D}_{\mathcal{L}}^{(2)}$).
If the hyperbolic equation $\ell_x f = \ell_y f$ (or the transformed equation $\widetilde{\ell}_\xi u = \widetilde{\ell}_y u$) is uniformly hyperbolic, the existence and uniqueness of solution for this Cauchy problem is a standard result which follows from the classical theory of hyperbolic problems in two variables (see e.g. [@couranthilbert1962 Chapter V]); in fact, the existence and uniqueness holds under much weaker restrictions on the initial condition. However, our existence and uniqueness result becomes nontrivial in the presence of a (non-removable) parabolic degeneracy at the initial line.
Indeed, even though many authors have addressed Cauchy problems for degenerate hyperbolic equations in two variables, most studies are restricted to equations where the ${\partial^2 \over \partial x^2}$ term vanishes at an initial line $y=y_0$ (we refer to [@bitsadze1964 §2.3], [@radkevic2009 Section 5.4] and references therein). Much less is known for hyperbolic equations whose ${\partial^2 \over \partial y^2}$ term vanishes at the same initial line: it is known that the Cauchy problem is, in general, not well-posed, and the relevance of determining sufficient conditions for its well-posedness has long been pointed out [@bitsadze1964 §2.4], but as far as we are aware little progress has been made on this problem (for related work see [@mamadaliev2000]). The application of spectral techniques to hyperbolic Cauchy problems associated with Sturm-Liouville operators is by no means new, see e.g. [@chebli1974; @carroll1985] and references therein; however, it seems that such techniques had never been applied to degenerate cases.
It is helpful to know that an existence theorem analogous to Theorem \[thm:shypPDE\_Lexistence\] holds when the initial line is shifted away from the degeneracy, because this will allow us to justify that the solution of the degenerate Cauchy problem is the pointwise limit of solutions of nondegenerate problems.
\[prop:shypPDE\_Lexistence\_eps\] Suppose that $x \mapsto p(x)r(x)$ is an increasing function, and let $m \in \mathbb{N}$. If $h \in \mathcal{D}_{\mathcal{L}}^{(2)}$ and $\ell(h) \in \mathcal{D}_{\mathcal{L}}^{(2)\!}$, then the function $$\label{eq:shypPDE_Lexistence_eps}
f_m(x,y) = \int_{[0,\infty)\!} w_\lambda(x) \, w_{\lambda,m}(y) \, (\mathcal{F} h)(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda)$$ is a solution of the Cauchy problem $$\label{eq:shypPDE_Lcauchy_eps}
\begin{aligned}
(\ell_x f_m)(x,y) = (\ell_y f_m)(x,y), & \qquad a < x < b, \: a_m < y < b \\
f_m(x,a_m) = h(x), & \qquad a < x < b \\
(\partial_y^{[1]}\!f_m)(x,a_m) = 0, & \qquad a < x < b.
\end{aligned}$$
Let us begin by justifying that $\partial_x^{[1]}\!f_m(x,y)$ and $(\ell_x f_m)(x,y)$ can be computed via differentiation under the integral sign. The differentiated integrals are given by $$\begin{gathered}
\label{eq:shypPDE_Lexistence_epssol_pf1} \int_{[0,\infty)\!} w_\lambda^{[1]}(x) \, w_{\lambda,m}(y) \, (\mathcal{F} h)(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda) \\
\label{eq:shypPDE_Lexistence_epssol_pf2} \int_{[0,\infty)\!} w_\lambda(x) \, w_{\lambda,m}(y) \, [\mathcal{F} (\ell (h))](\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda)\end{gathered}$$ (for the latter, we used the identities $(\ell w_\lambda)(x) = \lambda w_\lambda(x)$ and ), and their absolute and uniform convergence on compacts follows from the fact that $h, \ell (h) \in \mathcal{D}_\mathcal{L}^{(2)}$, together with Lemma \[lem:shypPDE\_Ltransf\_D2prop\](b) and the inequality $|w_{\lambda,m}(\cdot)|\leq 1$ (which follows from Lemma \[lem:shypPDE\_wsolbound\] if we replace $a$ by $a_m$). This justifies that $\partial_x^{[1]}\!f_m(x,y)$ and $(\ell_x f_m)(x,y)$ are given by , respectively.
We also need to ensure that $\partial_y^{[1]}\!f_m(x,y)$ and $(\ell_y f_m)(x,y)$ are given by the corresponding differentiated integrals, and to that end we must check that $$\int_{[0,\infty)\!} w_\lambda(x) \, w_{\lambda,m}^{[1]}(y) \, (\mathcal{F} h)(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda)$$ converges absolutely and uniformly. Indeed, it follows from that for $y \geq a_m$ we have $w_{\lambda,m}^{[1]}(y) = \lambda \int_{a_m}^y w_{\lambda,m}(\xi) \,r(\xi) d\xi$ and consequently $|w_{\lambda,m}^{[1]}(y)| \leq \lambda \int_{a_m}^y \,r(\xi) d\xi$; hence $$\label{eq:shypPDE_Lexistence_epssol_pf3}
\int_{[0,\infty)\!} \bigl| w_\lambda(x) \, w_{\lambda,m}^{[1]}(y)\, (\mathcal{F} h)(\lambda)\bigr| \bm{\rho}_{\mathcal{L}}(d\lambda) \leq \int_{a_m}^y r(\xi) d\xi {\kern-.12em\cdot\kern-.12em}\! \int_{[0,\infty)\!} \lambda \bigl| w_\lambda(x) (\mathcal{F} h)(\lambda)\bigr| \bm{\rho}_{\mathcal{L}}(d\lambda)$$ and the uniform convergence in compacts follows from and Lemma \[lem:shypPDE\_Ltransf\_D2prop\](b).
The verification of the boundary conditions is straightforward: Lemma \[lem:shypPDE\_Ltransf\_D2prop\](b) together with the fact that $w_{\lambda,m}(a_m) = 1$ imply that $f_m(x,a_m) = h(x)$, and from we easily see that $\partial_y^{[1]}\!f_m (x,a_m) = 0$. This shows that $f_m$ is a solution of the Cauchy problem .
\[cor:shypPDE\_cauchy\_ptapprox\] Suppose that $x \mapsto p(x)r(x)$ is an increasing function. Let $h \in \mathcal{D}_\mathcal{L}^{(2)}$ and consider the functions $f_m$, $f$ defined by , . Then $$\lim_{m \to \infty} f_m(x,y) = f(x,y) \qquad \text{pointwise for each } x, y \in (a,b).$$
Since $w_{\lambda,m}(y) \to w_\lambda(y)$ pointwise as $m \to \infty$ (Lemma \[lem:shypPDE\_ode\_wepslimit\]), the conclusion follows from the dominated convergence theorem (which is applicable due to Lemmas \[lem:shypPDE\_wsolbound\] and \[lem:shypPDE\_Ltransf\_D2prop\](b)).
Maximum principle and positivity of solution
--------------------------------------------
After having shown that the Cauchy problem is well-posed whenever the function $x \mapsto p(x) r(x)$ is increasing, we introduce a stronger assumption on the coefficients which will be seen to be sufficient for a maximum principle to hold for the hyperbolic equation $\ell_x f = \ell_y f$ and, in consequence, for the solution of the Cauchy problem to preserve positivity and boundedness of its initial condition. We shall rely on the transformation of $\ell$ into the standard form (Remark \[rmk:shypPDE\_tildeell\]); in the following assumption, $A$ is the function defined in .
\[asmp:shypPDE\_SLhyperg\] There exists $\eta \in \mathrm{C}^1(\gamma(a),\infty)$ such that $\eta \geq 0$, $\bm{\phi}_\eta := {A' \over A} - \eta \geq 0$, and the functions $\bm{\phi}_\eta$ and $\bm{\psi}_\eta := {1 \over 2} \eta' - {1 \over 4} \eta^2 + {A' \over 2A} {\kern-.12em\cdot\kern-.12em}\eta$ are both decreasing on $(\gamma(a),\infty)$.
Observe that Assumption \[asmp:shypPDE\_SLhyperg\] allows for $\gamma(a) = -\infty$ (this will enable us to treat the case of non-removable degeneracy), and it does not include the left endpoint in the interval where the conditions on $\eta$ are imposed. This assumption is therefore a generalization of an assumption introduced by Zeuner, cf. Example \[exam:SLhypergr\] below.
The proof of the maximum principle presented in the sequel is based on [@zeuner1992 Proposition 3.7] and on the maximum principles of [@weinberger1956]. The key tool is the integral identity which we now state:
Let $\bm{\ell}^B$ be the differential expression $\bm{\ell}^B v := - v'' - \bm{\phi}_\eta v' + \bm{\psi}_\eta v$. For $\gamma(a) < c \leq y \leq x$, consider the triangle $\Delta_{c,x,y} := \{(\xi,\zeta) \in \mathbb{R}^2 \mid \zeta \geq c, \, \xi + \zeta \leq x+y, \, \xi - \zeta \geq x-y \}$, and let $v \in \mathrm{C}^2(\Delta_{c,x,y})$. Write $B(x):=\exp({1 \over 2} \int_\beta^x \eta(\xi)d\xi)$ (with $\beta > \gamma(a)$ arbitrary) and $A_{\mathsmaller{B}}(x) = {A(x) \over B(x)^2}$. Then the following integral equation holds: $$\label{eq:shypPDE_inteqtriangle}
A_{\mathsmaller{B}}(x) A_{\mathsmaller{B}}(y) \, v(x,y) = H + I_0 + I_1 + I_2 + I_3 - I_4$$ where $$\begin{aligned}
H & := \tfrac{1}{2} A_{\mathsmaller{B}}(c) \bigl[A_{\mathsmaller{B}}(x-y+c) \, v(x-y+c,c) + A_{\mathsmaller{B}}(x+y-c) \, v(x+y-c,c)] \\
I_0 & := \tfrac{1}{2} A_{\mathsmaller{B}}(c) \int_{x-y+c}^{x+y-c} A_{\mathsmaller{B}}(s) (\partial_y v)(s,c)\, ds \\
I_1 & := \tfrac{1}{2} \int_c^y A_{\mathsmaller{B}}(s) A_{\mathsmaller{B}}(x-y+s) \bigl[ \bm{\phi}_\eta(s) + \bm{\phi}_\eta(x-y+s) \bigr] v(x-y+s,s)\, ds \\
I_2 & := \tfrac{1}{2} \int_c^y A_{\mathsmaller{B}}(s) A_{\mathsmaller{B}}(x+y-s) \bigl[ \bm{\phi}_\eta(s) - \bm{\phi}_\eta(x+y-s) \bigr] v(x+y-s,s)\, ds \\
I_3 & := \tfrac{1}{2} \int_{\Delta_{c,x,y}\!} A_{\mathsmaller{B}}(\xi) A_{\mathsmaller{B}}(\zeta) \bigl[\bm{\psi}_\eta(\zeta) - \bm{\psi}_\eta(\xi)\bigr] v(\xi,\zeta)\, d\xi d\zeta \\
I_4 & := \tfrac{1}{2} \int_{\Delta_{c,x,y}\!} A_{\mathsmaller{B}}(\xi) A_{\mathsmaller{B}}(\zeta) \, (\bm{\ell}_\zeta^B v - \bm{\ell}_\xi^B v)(\xi,\zeta)\, d\xi d\zeta.\end{aligned}$$
Just compute $$\begin{aligned}
I_4 - I_3 & = \tfrac{1}{2} \int_{\Delta_{c,x,y}}\biggl({\partial \over \partial \xi} \bigl[ A_{\mathsmaller{B}}(\xi) A_{\mathsmaller{B}}(\zeta) \, (\partial_\xi v)(\xi,\zeta) \bigr] - {\partial \over \partial \zeta} \bigl[ A_{\mathsmaller{B}}(\xi) A_{\mathsmaller{B}}(\zeta) \, (\partial_\zeta v)(\xi,\zeta) \bigr] \biggr) d\xi d\zeta \\
& = I_0 - \tfrac{1}{2} \int_0^y A_{\mathsmaller{B}}(s) A_{\mathsmaller{B}}(x-y+s) \, (\partial_\zeta v + \partial_\xi v)(x-y+s,s) \, ds \\
& \quad\; - \tfrac{1}{2} \int_0^y A_{\mathsmaller{B}}(s) A_{\mathsmaller{B}}(x+y-s) \, (\partial_\zeta v - \partial_\xi v)(x+y-s,s) \, ds \\
& = I_0 + I_1 - \int_c^y {d \over ds} \bigl[ A_{\mathsmaller{B}}(s) A_{\mathsmaller{B}}(x-y+s) \, v(x-y+s,s) \bigr] ds \\
& \quad\; + I_2 - \int_c^y {d \over ds} \bigl[ A_{\mathsmaller{B}}(s) A_{\mathsmaller{B}}(x-y+s) \, v(x-y+s,s) \bigr] ds\end{aligned}$$ where in the second equality we used Green’s theorem, and the third equality follows easily from the fact that $(A_{\mathsmaller{B}})' = \bm{\phi}_\eta A_{\mathsmaller{B}}$.
\[thm:shypPDE\_tildeell\_maxprinc\] Suppose Assumption \[asmp:shypPDE\_SLhyperg\] holds, and let $\gamma(a) < c \leq y_0 \leq x_0$. If $u \in \mathrm{C}^2(\Delta_{c,x_0,y_0})$ satisfies $$\label{eq:shypPDE_tildeell_maxprinc_ineq}
\begin{aligned}
(\widetilde{\ell}_x u - \widetilde{\ell}_y u)(x,y) \leq 0, & \qquad (x,y) \in \Delta_{c,x_0,y_0} \\
u(x,c) \geq 0, & \qquad x \in [x_0-y_0+c,x_0+y_0-c] \\
(\partial_y u)(x,c) + \tfrac{1}{2} \eta(c) u(x,c) \geq 0, & \qquad x \in [x_0-y_0+c,x_0+y_0-c]
\end{aligned}$$ then $u \geq 0$ in $\Delta_{c,x_0,y_0}$.
Pick a function $\omega \in \mathrm{C}^2[c,\infty)$ such that $\bm{\ell}^B \omega < 0$, $\omega(c) > 0$ and $\omega'(c) \geq 0$. Clearly, it is enough to show that for all $\delta > 0$ we have $v(x,y) := B(x) B(y) u(x,y) + \delta \omega(y) > 0$ for $(x, y) \in \Delta_{c,x_0,y_0}$.
Assume by contradiction that there exist $\delta > 0$, $(x, y) \in \Delta_{c,x_0,y_0}$ for which we have $v(x,y) = 0$ and $v(\xi,\zeta) \geq 0$ for all $(\xi,\zeta) \in \Delta_{c,x,y} \subset \Delta_{c,x_0,y_0}$. It is clear from the choice of $\omega$ that $v(\cdot,c) > 0$, thus we have $H \geq 0$ in the right hand side of . Similarly, $(\partial_y v)(\cdot,c) = B(x) B(y) \bigl[(\partial_y u)(\cdot,c) + \tfrac{1}{2} \eta(c) u(\cdot,c)\bigr] + \delta \omega'(c) \geq 0$, hence $I_0 \geq 0$. Since $\bm{\phi}_\eta$ is positive and decreasing and $\bm{\psi}_\eta$ is decreasing (cf. Assumption \[asmp:shypPDE\_SLhyperg\]) and we are assuming that $u \geq 0$ on $\Delta_{c,x,y}$, it follows that $I_1 \geq 0$, $I_2 \geq 0$ and $I_3 \geq 0$. In addition, $I_4 < 0$ because $(\bm{\ell}_\zeta^B v - \bm{\ell}_\xi^B v)(\xi,\zeta) = B(x) B(y) (\widetilde{\ell}_\zeta u - \widetilde{\ell}_\xi u)(\xi,\zeta) + (\bm{\ell}^B\omega)(\zeta) < 0$. Consequently, yields $0 = A_{\mathsmaller{B}}(x) A_{\mathsmaller{B}}(y) v(x,y) \geq -I_4 > 0$. This contradiction shows that $v(x,y) > 0$ for all $(x,y) \in \Delta_{c,x_0,y_0}$.
Naturally, this weak maximum principle can be restated in terms of the operator $\ell = -{1 \over r} {d \over dx} ( p \, {d \over dx})$; this is left to the reader. As anticipated above, the positivity-preserving property of the Cauchy problem is a by-product of the maximum principle.
Suppose Assumption \[asmp:shypPDE\_SLhyperg\] holds, and let $m \in \mathbb{N}$. If $h \in \mathcal{D}_\mathcal{L}^{(2)}$, $\ell(h) \in \mathcal{D}_{\mathcal{L}}^{(2)\!}$ and $h \geq 0$, then the function $f_m$ given by is such that $$\label{eq:shypPDE_positivity_eps}
f_m(x,y) \geq 0 \qquad \text{for } x \geq y > a_m.$$ If, in addition, $h \leq C$ (where $C$ is a constant), then $f_m(x,y) \leq C$ for $x \geq y > a_m$.
It follows from Proposition \[prop:shypPDE\_Lexistence\_eps\] that the function $u_m(x,y) := f_m(\gamma^{-1}(x), \gamma^{-1}(y))$ is a solution of the Cauchy problem $$\begin{aligned}
\label{eq:shypPDE_positivity_eps_pf1} (\widetilde{\ell}_x u_m)(x,y) = (\widetilde{\ell}_y u_m)(x,y), & \qquad x, y > \tilde{a}_m \\
\label{eq:shypPDE_positivity_eps_pf2} u_m(x,\tilde{a}_m) = h(\gamma^{-1}(x)), & \qquad x > \tilde{a}_m \\
\label{eq:shypPDE_positivity_eps_pf3} (\partial_y u_m)(x,\tilde{a}_m) = 0, & \qquad x > \tilde{a}_m\end{aligned}$$ where $\tilde{a}_m = \gamma(a_m)$. Clearly, $u_m$ satisfies the inequalities for arbitrary $x_0 \geq y_0 \geq \tilde{a}_m$ (here $c = \tilde{a}_m$). By Theorem \[thm:shypPDE\_tildeell\_maxprinc\], $u_m(x_0,y_0) \geq 0$ for all $x_0 \geq y_0 > \tilde{a}_m$; consequently, holds.
The proof of the last statement is straightforward: if we have $h \leq C$, then $\widetilde{u}_m(x,y) = C - u_m(x,y)$ is a solution of with initial conditions $\widetilde{u}_m(x,\tilde{a}_m) = C - h(\gamma^{-1}(x)) \geq 0$ and , thus the reasoning of the previous paragraph yields that $C - u_m \geq 0$ for $x \geq y > \tilde{a}_m$.
The previous result gives the positivity-preservingness for the solution of the nondegenerate Cauchy problem . The extension of this property to the possibly degenerate problem is an immediate consequence of the pointwise convergence result of Corollary \[cor:shypPDE\_cauchy\_ptapprox\]:
\[cor:shypPDE\_sol\_positivity\] Suppose Assumption \[asmp:shypPDE\_SLhyperg\] holds. If $h \in \mathcal{D}_\mathcal{L}^{(2)}$, $\ell(h) \in \mathcal{D}_{\mathcal{L}}^{(2)\!}$ and $h \geq 0$, then the function $f$ given by is such that $$f(x,y) \geq 0 \qquad \text{for } x, y \in (a,b).$$ If, in addition, $h \leq C$, then $f(x,y) \leq C$ for $x, y \in (a,b)$.
Note that the conclusion holds for all $x,y \in (a,b)$ because the function $f(x,y)$ is symmetric.
Sturm-Liouville translation and convolution {#sec:transl_conv}
===========================================
Assumption \[asmp:shypPDE\_SLhyperg\] will always be in force throughout this and the subsequent sections.
Definition and first properties
-------------------------------
In view of the comments made in the Introduction, it is natural to define the $\mathcal{L}$-convolution $\mu * \nu$ ($\mu,\nu \in \mathcal{M}_\mathrm{C}[a,b)$) in order that, for sufficiently well-behaved initial conditions, the integral $\int_{[a,b)} h(\xi) \, (\delta_x * \delta_y)(d\xi)$ coincides with the solution of the hyperbolic Cauchy problem. Having this in mind, let us first confirm that the solution of the hyperbolic Cauchy problem can be represented as an integral with respect to a family of positive measures:
\[prop:shypPDE\_sol\_subprobrep\] Fix $x, y \in [a,b)$. There exists a subprobability measure $\bm{\nu}_{x,y} \in \mathcal{M}_+[a,b)$ such that, for all initial conditions $h \in \mathrm{C}_{\mathrm{c},0}^4$, the solution of the hyperbolic Cauchy problem can be written as $$\label{eq:shypPDE_sol_subprobrep}
f_h(x,y) = \int_{[a,b)} h(\xi) \, \bm{\nu}_{x,y}(d\xi) \qquad (h \in \mathrm{C}_{\mathrm{c},0}^4).$$
For each fixed $x, y \in [a,b)$, the right hand side of defines a linear functional $\mathrm{C}_{\mathrm{c},0}^4 \ni h \mapsto f_h(x,y) \in \mathbb{C}$. By Corollary \[cor:shypPDE\_sol\_positivity\], $|f_h(x,y)| \leq \|h\|_\infty$ for $h \in \mathrm{C}_{\mathrm{c},0}^4$. Thus it follows from the Hahn-Banach theorem that this functional admits a linear extension $\mathcal{T}_{x,y}: \mathrm{C}_0[a,b) \to \mathbb{C}$ such that $|\mathcal{T}_{x,y} h| \leq \|h\|_\infty$ for all $h \in \mathrm{C}_0[a,b)$. According to the Riesz representation theorem (cf. [@cohn2013 Theorem 7.3.6]), $\mathcal{M}_\mathbb{C}[a,b)$ is the dual of $\mathrm{C}_0[a,b)$; we thus have $\mathcal{T}_{x,y} h = \int_{[a,b)} h(\xi) \bm{\nu}_{x,y}(d\xi)$, where $\bm{\nu}_{x,y}$ is a finite complex measure with $\|\bm{\nu}_{x,y}\| \leq 1$. Finally, the fact that $\int_{[a,b)} h(\xi) \bm{\nu}_{x,y}(d\xi) \equiv f_h(x,y) \geq 0$ for all $h \in \mathrm{C}_{\mathrm{c},0}^4$, $h \geq 0$ (Corollary \[cor:shypPDE\_sol\_positivity\]) yields that $\bm{\nu}_{x,y} \in \mathcal{M}_+[a,b)$ is a subprobability measure.
Let $\mu, \nu \in \mathcal{M}_{\mathbb{C}}[a,b)$. The measure $$(\mu * \nu)(\cdot) = \int_{[a,b)} \int_{[a,b)} \bm{\nu}_{x,y}(\cdot) \, \mu(dx) \, \nu(dy)$$ is called the *$\mathcal{L}$-convolution* of the measures $\mu$ and $\nu$. The *$\mathcal{L}$-translation* of a function $h \in \mathrm{B}_\mathrm{b}[a,b)$ is defined as $$(\mathcal{T}^y h)(x) = \int_{[a,b)} h(\xi) \, \bm{\nu}_{x,y} (d\xi) \equiv \int_{[a,b)} h(\xi) \, (\delta_x * \delta_y)(d\xi), \qquad x,y \in [a,b).$$
It follows from this definition, together with , that the $\mathcal{L}$-convolution is such that (for $\mu_1,\mu_2,\nu,\pi \in \mathcal{M}_{\mathbb{C}}[a,b)$ and $p_1, p_2 \in \mathbb{C}$):
1. $\mu * \nu = \nu * \mu$ (Commutativity);
2. $(\mu * \nu) * \pi = \mu * (\nu * \pi)$ (Associativity);
3. $(p_1 \mu_1 + p_2 \mu_2) * \nu = p_1 (\mu_1 * \nu) + p_2 (\mu_2 * \nu)$ (Bilinearity);
4. $\|\mu * \nu\| \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\|\nu\|$ (Submultiplicativity);
5. If $\mu, \nu \in \mathcal{M}_{+}[a,b)$, then $\mu * \nu \in \mathcal{M}_{+}[a,b)$ (Positivity).
Summarizing this, we have:
\[prop:shypPDE\_conv\_Mbanachalg\] The space $(\mathcal{M}_{\mathbb{C}}[a,b),*)$, equipped with the total variation norm, is a commutative Banach algebra over $\mathbb{C}$ whose identity element is the Dirac measure $\delta_a$.
Moreover, $\mathcal{M}_{+}[a,b)$ is an algebra cone (i.e. it is closed under $\mathcal{L}$-convolution, addition and multiplication by positive scalars, and it contains the identity element).
Given a measure $\mu \in \mathcal{M}_\mathbb{C}[a,b)$, it is natural to define the $\mathcal{L}$-translation by $\mu$ as $$(\mathcal{T}^\mu h)(x) := \int_{[a,b)} (\mathcal{T}^y h)(x)\, \mu(dy) \equiv \int_{[a,b)} h(\xi) \, (\delta_x * \mu)(d\xi) \qquad\quad (h \in \mathrm{B}_\mathrm{b}[a,b))$$ (so that $\mathcal{T}^x \equiv \mathcal{T}^{\delta_x}$ for $a \leq x < b$). It is easy to see that $\|\mathcal{T}^\mu h\|_\infty \leq \| \mu \| {\kern-.12em\cdot\kern-.12em}\|h\|_\infty$ for all $h \in \mathrm{B}_\mathrm{b}[a,b)$ and $\mu \in \mathcal{M}_\mathbb{C}[a,b)$. Observe also that for $h \in \mathrm{C}_{\mathrm{c},0}^4$ we can write (by and ) $$\label{eq:shypPDE_gentranslmu_spectrep}
(\mathcal{T}^\mu h)(x) = \int_{[0,\infty)} (\mathcal{F} h)(\lambda) \, w_\lambda(x) \, \widehat{\mu}(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda) \qquad (h \in \mathrm{C}_{\mathrm{c},0}^4)$$ or equivalently (cf. Proposition \[prop:shypPDE\_Ltransf\]) $$\label{eq:shypPDE_gentranslmu_spectrep_F} \bigl(\mathcal{F}(\mathcal{T}^\mu h)\bigr)(\lambda) = \widehat{\mu}(\lambda) (\mathcal{F}h)(\lambda) \qquad (h \in \mathrm{C}_{\mathrm{c},0}^4).$$ Due to Lemma \[lem:shypPDE\_Ltransf\_D2prop\], the integral converges absolutely and uniformly for $x$ on compact subsets of $(a,b)$.
Sturm-Liouville transform of measures
-------------------------------------
An important tool for the subsequent analysis is the extension of the $\mathcal{L}$-transform to finite complex measures, defined as follows:
Let $\mu \in \mathcal{M}_{\mathbb{C}}[a,b)$. The *$\mathcal{L}$-transform* of the measure $\mu$ is the function defined by the integral $$\widehat{\mu}(\lambda) = \int_{[a,b)} w_\lambda(x)\, \mu(dx), \qquad \lambda \geq 0.$$
The next proposition contains some basic properties of the $\mathcal{L}$-transform of measures which, as one would expect, resemble those of the ordinary Fourier transform (or characteristic function) of finite measures. We recall that, by definition, the complex measures $\mu_n$ converge weakly to $\mu \in \mathcal{M}_\mathbb{C}[a,b)$ if $\lim_n \int_{[a,b)} g(\xi) \mu_n(d\xi) = \int_{[a,b)} g(\xi) \mu(d\xi)$ for all $g \in \mathrm{C}_\mathrm{b}[a,b)$. We also recall that a family $\{\mu_j\} \subset \mathcal{M}_\mathbb{C}[a,b)$ is said to be uniformly bounded if $\sup_j\|\mu_j\| < \infty$, and $\{\mu_j\}$ is said to be tight if for each ${\varepsilon}> 0$ there exists a compact $K_{\varepsilon}\subset [a,b)$ such that $\sup_j \,|\mu_j|([a,b) \setminus K_{\varepsilon}) < {\varepsilon}$. (These definitions are taken from [@bogachev2007].) In the sequel, the notation $\mu_n {\overset{w}{\longrightarrow}}\mu$ denotes weak convergence of measures.
\[prop:shypPDE\_Ltransfmeas\_props\] The $\mathcal{L}$-transform $\widehat{\mu}$ of $\mu \in \mathcal{M}_\mathbb{C}[a,b)$ has the following properties:
1. $\widehat{\mu}$ is continuous on $[0,\infty)$. Moreover, if a family of measures $\{\mu_j\} \subset \mathcal{M}_\mathbb{C}[a,b)$ is tight and uniformly bounded, then $\{\widehat{\mu_j}\}$ is equicontinuous on $[0,\infty)$.
2. Each measure $\mu \in \mathcal{M}_{\mathbb{C}}[a,b)$ is uniquely determined by $\widehat{\mu}$. In particular, each $f \in L_1(r)$ is uniquely determined by $\mathcal{F}f \equiv \widehat{\mu_f}$, where $\mu_f \in \mathcal{M}_{\mathbb{C}}[a,b)$ is defined by $\mu_f(dx) = f(x) r(x) dx$.
3. If $\{\mu_n\}$ is a sequence of measures belonging to $\mathcal{M}_+[a,b)$, $\mu \in \mathcal{M}_+[a,b)$, and $\mu_n {\overset{w}{\longrightarrow}}\mu$, then $$\widehat{\mu_n} \xrightarrow[\,n \to \infty\,]{} \widehat{\mu} \qquad \text{uniformly for } \lambda \text{ in compact sets.}$$
4. Suppose that $\lim_{x \uparrow b} w_\lambda(x) = 0$ for all $\lambda > 0$. If $\{\mu_n\}$ is a sequence of measures belonging to $\mathcal{M}_+[a,b)$ whose $\mathcal{L}$-transforms are such that $$\label{eq:shypPDE_Ftransf_continuity_hyp}
\widehat{\mu_n}(\lambda) \xrightarrow[\,n \to \infty\,]{} f(\lambda) \qquad \text{pointwise in } \lambda \geq 0$$ for some real-valued function $f$ which is continuous at a neighborhood of zero, then $\mu_n {\overset{w}{\longrightarrow}}\mu$ for some measure $\mu \in \mathcal{M}_+[a,b)$ such that $\widehat{\mu} \equiv f$.
***(a)*** Let us prove the second statement, which implies the first. Set $C = \sup_j \|\mu_j\|$. Fix $\lambda_0 \geq 0$ and ${\varepsilon}> 0$. By the tightness assumption, we can choose $\beta \in (a,b)$ such that $|\mu_j|(\beta,b) < {\varepsilon}$ for all $j$. Since the family $\{w_{(\cdot)}(x)\}_{x \in (a, \beta]}$ is equicontinuous on $[0,\infty)$ (this follows easily from the power series representation of $w_{(\cdot)}(x)$, cf. proof of Lemma \[lem:shypPDE\_ode\_wsol\]), we can choose $\delta > 0$ such that $$|\lambda - \lambda_0| < \delta \quad \implies \quad |w_\lambda(x) - w_{\lambda_0}(x)| < {\varepsilon}\; \text{ for all } a < x \leq \beta.$$ Consequently, $$\begin{aligned}
& \bigl|\widehat{\mu_j}(\lambda) - \widehat{\mu_j}(\lambda_0)\bigr| = \biggl| \int_{(a,b)} \bigl(w_\lambda(x) - w_{\lambda_0}(x)\bigr) \mu_j(dx) \biggr| \\
& \quad \leq\int_{(\beta,b)\!} \bigl|w_\lambda(x) - w_{\lambda_0}(x)\bigr| |\mu_j|(dx) + \int_{(a,\beta]\!} \bigl|w_\lambda(x) - w_{\lambda_0}(x)\bigr| |\mu_j|(dx) \\
& \quad \leq 2{\varepsilon}+ C{\varepsilon}= (C+2){\varepsilon}\end{aligned}$$ for all $j$, provided that $|\lambda - \lambda_0| < \delta$, which means that $\{\widehat{\mu_j}\}$ is equicontinuous at $\lambda_0$.\
***(b)*** Let $\mu \in \mathcal{M}_\mathbb{C}[a,b)$ be such that $\widehat{\mu}(\lambda) = 0$ for all $\lambda \geq 0$. We need to show that $\mu$ is the zero measure. For each $h \in \mathrm{C}_{\mathrm{c},0}^{4}$, by we have $$(\mathcal{T}^\mu h)(x) = \int_{[0,\infty)} (\mathcal{F} h)(\lambda) \, w_\lambda(x) \, \widehat{\mu}(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda) \, = \, 0.$$ Since $h \in \mathrm{C}_{\mathrm{c},0}^{4}$, Theorem \[thm:shypPDE\_Lexistence\] assures that $\lim_{x \downarrow a} (\mathcal{T}^y h)(x) = h(y)$ for $y \geq 0$; therefore, by dominated convergence (which is applicable because $\|\mathcal{T}^y h\|_\infty \leq \|h\|_\infty < \infty$), $$0 = \lim_{x \downarrow a} (\mathcal{T}^\mu h)(x) = \lim_{x \downarrow a} \int_{[a,b)\!} (\mathcal{T}^y h)(x)\, \mu(dy) \\
= \int_{[a,b)\!} h(y)\, \mu(dy)$$ This shows that $\int_{[a,b)\!} h(y)\, \mu(dy) = 0$ for all $h \in \mathrm{C}_{\mathrm{c},0}^{4}$ and, consequently, $\mu$ is the zero measure.\
***(c)*** Since $w_\lambda(\cdot)$ is continuous and bounded, the pointwise convergence $\widehat{\mu_n}(\lambda) \to \widehat{\mu}(\lambda)$ follows from the definition of weak convergence of measures. By Prokhorov’s theorem [@bogachev2007 Theorem 8.6.2], $\{\mu_n\}$ is tight and uniformly bounded, thus (by part (i)) $\{\widehat{\mu_n}\}$ is equicontinuous on $[0,\infty)$. Invoking [@klenke2014 Lemma 15.22], we conclude that the convergence $\widehat{\mu_n} \to \widehat{\mu}$ is uniform for $\lambda$ in compact sets.\
***(d)*** We only need to show that the sequence $\{\mu_n\}$ is tight and uniformly bounded. Indeed, if $\{\mu_n\}$ is tight and uniformly bounded, then Prokhorov’s theorem yields that for any subsequence $\{\mu_{n_k}\}$ there exists a further subsequence $\{\mu_{n_{k_j}}\!\}$ and a measure $\mu \in \mathcal{M}_+[a,b)$ such that $\mu_{n_{k_j}}\!\! {\overset{w}{\longrightarrow}}\mu$. Then, due to part (iii) and to , we have $\widehat{\mu}(\lambda) = f(\lambda)$ for all $\lambda \geq 0$, which implies (by part (ii)) that all such subsequences have the same weak limit; consequently, the sequence $\mu_n$ itself converges weakly to $\mu$.
The uniform boundedness of $\{\mu_n\}$ follows immediately from the fact that $\widehat{\mu_n}(0) = \mu_n[a,b)$ converges. To prove the tightness, take ${\varepsilon}> 0$. Since $f$ is continuous at a neighborhood of zero, we have ${1 \over \delta} \int_0^{2\delta} \bigl(f(0) - f(\lambda)\bigr)d\lambda \longrightarrow 0$ as $\delta \downarrow 0$; therefore, we can choose $\delta > 0$ such that $$\biggl| {1 \over \delta} \int_0^{2\delta} \bigl(f(0) - f(\lambda)\bigr)d\lambda \biggr| < {\varepsilon}.$$ Next we observe that, due to the assumption that $\lim_{x \uparrow b} w_\lambda(x) = 0$ for all $\lambda > 0$, we have $\int_0^{2\delta} \bigl( 1-w_\lambda(x) \bigr) d\lambda \longrightarrow 2\delta$ as $x \uparrow b$, meaning that we can pick $\beta \in (a,b)$ such that $$\int_0^{2\delta} \bigl( 1-w_\lambda(x) \bigr) d\lambda \geq \delta \qquad \text{for all } \beta < x < b.$$ By our choice of $\beta$ and Fubini’s theorem, $$\begin{aligned}
\mu_n\bigl[\beta,b) & = {1 \over \delta} \int_{[\beta,b)} \delta\, \mu_n(dx) \\
& \leq {1 \over \delta} \int_{[\beta,b)} \int_0^{2\delta} \bigl( 1-w_\lambda(x) \bigr) d\lambda\, \mu_n(dx) \\
& \leq {1 \over \delta} \int_{[a,b)} \int_0^{2\delta} \bigl( 1-w_\lambda(x) \bigr) d\lambda\, \mu_n(dx) \\
& = {1 \over \delta} \int_0^{2\delta} \bigl(\widehat{\mu_n}(0) - \widehat{\mu_n}(\lambda)\bigr) d\lambda.\end{aligned}$$ Hence, using the dominated convergence theorem, $$\begin{aligned}
\limsup_{n \to \infty} \mu_n[\beta,b) & \leq {1 \over \delta} \limsup_{n \to \infty}\! \int_0^{2\delta} \bigl(\widehat{\mu_n}(0) - \widehat{\mu_n}(\lambda)\bigr) d\lambda \\
& = {1 \over \delta} \int_0^{2\delta}\!\! \lim_{n \to \infty} \bigl(\widehat{\mu_n}(0) - \widehat{\mu_n}(\lambda)\bigr) d\lambda = {1 \over \delta} \int_0^{2\delta} \bigl(f(0) -
f(\lambda)\bigr) d\lambda < {\varepsilon}\end{aligned}$$ due to the choice of $\delta$. Since ${\varepsilon}$ is arbitrary, we conclude that $\{\mu_n\}$ is tight, as desired.
\[rmk:shypPDE\_Ltransfmeas\_propsrmk\] **I.** Parts (c) and (d) of the proposition above show that (whenever $\lim_{x \uparrow b} w_\lambda(x) = 0$ for all $\lambda > 0$) the $\mathcal{L}$-transform possesses the following important property: *the $\mathcal{L}$-transform is a topological homeomorphism between $\mathcal{P}[a,b)$ with the weak topology and the set $\widehat{\mathcal{P}}$ of $\mathcal{L}$-transforms of probability measures with the topology of uniform convergence in compact sets.*\
**II.** Recall that, by definition [@bauer2001 §30], the measures $\mu_n$ converge vaguely to $\mu$ if $\lim_n \int_{[a,b)} g(\xi) \mu_n(d\xi) = \int_{[a,b)} g(\xi) \mu(d\xi)$ for all $g \in \mathrm{C}_0[a,b)$. Much like weak convergence, vague convergence of measures can be formulated via the $\mathcal{L}$-transform, provided that $\lim_{x \uparrow b} w_\lambda(x) = 0$ for all $\lambda > 0$. Indeed, using ${\overset{v}{\longrightarrow}}$ to denote vague convergence of measures, we have:
1. *If $\{\mu_n\} \subset \mathcal{M}_+[a,b)$, $\mu \in \mathcal{M}_+[a,b)$, and $\mu_n {\overset{v}{\longrightarrow}}\mu$, then $\lim \widehat{\mu_n}(\lambda) = \widehat{\mu}(\lambda)$ pointwise for each $\lambda > 0$;*
2. *If $\{\mu_n\} \subset \mathcal{M}_+[a,b)$, $\{\mu_n\}$ is uniformly bounded and $\lim \widehat{\mu_n}(\lambda) = f(\lambda)$ pointwise in $\lambda > 0$ for some function $f \in \mathrm{B}_\mathrm{b}(0,\infty)$, then $\mu_n {\overset{v}{\longrightarrow}}\mu$ for some measure $\mu \in \mathcal{M}_+[a,b)$ such that $\widehat{\mu} \equiv f$.*
(The first part is trivial; the second follows from the reasoning in the first paragraph of the proof of (d) in the proposition above, together with the fact that any uniformly bounded sequence of positive measures contains a vaguely convergent subsequence [@bauer2001 p. 213].)\
**III.** Concerning the additional assumption in the above remarks, one can state: *a necessary and sufficient condition for the condition $\lim_{x \uparrow b} w_\lambda(x) = 0$ ($\lambda > 0$) to hold is that $\lim_{x \uparrow b} p(x)r(x) = \infty$.* This fact can be proved using the transformation into the standard form (Remark \[rmk:shypPDE\_tildeell\]) and known results on the asymptotic behavior of solutions of the Sturm-Liouville equation $-u'' - {A' \over A} u' = \lambda u$ (see [@fruchtl2018 proof of Lemma 3.7]).
The product formula {#sec:prodform}
===================
We saw in the previous section that the hyperbolic maximum principle allows us to introduce a convolution measure algebra associated with the Sturm-Liouville operator. The next aims are to develop harmonic analysis on $L_p$ spaces and to study notions such as the continuity of the convolution or the divisibility of measures. However, this requires a fundamental tool, namely the trivialization property $\widehat{\delta_x * \delta_y} = \widehat{\delta_x} {\kern-.12em\cdot\kern-.12em}\widehat{\delta_y}$ for the $\mathcal{L}$-transform or, which is the same, the product formula for its kernel.
\[thm:shypPDE\_prodform\] The product $w_\lambda(x) \, w_\lambda(y)$ admits the integral representation $$\label{eq:shypPDE_prodform}
w_\lambda(x) \, w_\lambda(y) = \int_{[a,b)} w_\lambda(\xi)\, (\delta_x * \delta_y)(d\xi), \qquad x, y \in [a,b), \; \lambda \in \mathbb{C}.$$
Here we present the proof only in the special (nondegenerate) case $\gamma(a) > - \infty$. The proof of the general case is longer and relies on a different regularization argument; the details are given in [@sousaetalforth].
Assume first that $\ell = -{1 \over A} {d \over dx} (A {d \over dx})$, $0 < x < \infty$, and that Assumption \[asmp:shypPDE\_SLhyperg\] holds with $a = \gamma(a) = 0$. Fix $\lambda \in \mathbb{C}$, and let $\{{\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}\}_{n \in \mathbb{N}} \subset \mathrm{C}_{\mathrm{c},0}^4$ be a sequence of functions such that $${\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(x) = w_\lambda(x) \quad \text{for } x \in [0,n], \qquad\qquad {\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(x) = 0 \quad \text{for } x \geq n + 1.$$ Let $f^{\langle n\rangle \kern-.1em}(x,y)$ be the unique solution of the hyperbolic Cauchy problem with initial condition $h(x) = {\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(x)$. Since the family of characteristics for the hyperbolic equation $(\ell_x u)(x,y) = (\ell_y u)(x,y)$ is $x \pm y = \mathrm{const.}$, the solution $f^{\langle n\rangle \kern-.1em}(x,y)$ depends only on the values of the initial condition on the interval $[|x-y|,x+y]$. Observing that the function ${\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(x) \, {\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(y)$ is a solution of the hyperbolic equation $(\ell_x u)(x,y) = (\ell_y u)(x,y)$ on the square $(x,y) \in [0,n]^2$, we deduce that $$f^{\langle n\rangle \kern-.1em}(x,y) = {\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(x) \, {\mathbf{w}}_{\lambda}^{\langle n\rangle \kern-.1em}(y) = w_\lambda(x) \, w_\lambda(y), \qquad x,y \in [0,\tfrac{n}{2}].$$ It thus follows from Proposition \[prop:shypPDE\_sol\_subprobrep\] that $$w_\lambda(x) \, w_\lambda(y) = \int_{[0,\infty)\!} w_\lambda(\xi) \, \bm{\nu}_{x,y}(d\xi), \qquad x,y \in [0,\tfrac{n}{2}]$$ (note that ${\mathrm{supp}}(\bm{\nu}_{x,y}) = [|x-y|,x+y]$ because of the domain of dependence of the hyperbolic equation). Since $n$ is arbitrary, the identity holds for all $x,y \in [0,\infty)$, proving that the theorem holds for operators of the form $\ell = -{1 \over A} {d \over dx} (A {d \over dx})$, $0 < x < \infty$.
Now, in the general case of an operator $\ell$ of the form , note that $\gamma(a) > -\infty$ means that $\smash{\sqrt{r(y) \over p(y)}}$ is integrable near $a$, so that we may assume that $\gamma(a) = 0$ (otherwise, replace the interior point $c$ by the endpoint $a$ in the definition of the function $\gamma$). Applying the first part of the proof to the transformed operator $\widetilde{\ell} = -{1 \over A} {d \over d\xi}(A {d \over d\xi})$ defined via , we find that $\widetilde{w}_\lambda(x) \, \widetilde{w}_\lambda(y) = \int_{[0,\infty)\!} \widetilde{w}_\lambda(\xi)\, (\delta_x \kern.1em \widetilde{*} \kern.12em \delta_y)(d\xi)$ for $x, y \in [0,\infty)$, where $\widetilde{w}_\lambda(\xi) := w_\lambda(\gamma^{-1}(\xi))$ and $\widetilde{*}$ is the convolution associated with $\widetilde{\ell}$. We can rewrite this as $$w_\lambda(x) \, w_\lambda(y) = \int_{[a,b)} w_\lambda(\xi) \bigl[\gamma^{-1}(\delta_{\gamma(x)} \kern.1em \widetilde{*} \kern.12em \delta_{\gamma(y)})\bigr] \! (d\xi), \qquad x, y \in [a,b), \; \lambda \in \mathbb{C}$$ where the measure in the right hand side is the pushforward of the measure $\delta_{\gamma(x)} \kern.1em \widetilde{*} \kern.12em \delta_{\gamma(y)}$ under the map $\xi \mapsto \gamma^{-1}(\xi)$. But one can easily check that the convolutions $*$ and $\widetilde{*}$ are connected by $\delta_x * \delta_y = \gamma^{-1}(\delta_{\gamma(x)} \kern.1em \widetilde{*} \kern.12em \delta_{\gamma(y)})$ (this is a simple consequence of the definition of the convolution and the relation between the operators $\ell$ and $\widetilde{\ell}$), so we are done.
\[cor:shypPDEconv\_basicprops\] Let $\mu, \nu, \pi \in \mathcal{M}_\mathbb{C}[a,b)$.
1. We have $\pi = \mu * \nu$ if and only if $$\widehat{\pi}(\lambda) = \widehat{\mu}(\lambda)\, \widehat{\nu}(\lambda) \qquad \text{for all } \lambda \geq 0.$$
2. Probability measures are closed under $\mathcal{L}$-convolution: if $\mu, \nu \in \mathcal{P}[a,b)$, then $\mu * \nu \in \mathcal{P}[a,b)$.
If $\lim_{x \uparrow b} p(x)r(x) = \infty$ holds (cf. Remark \[rmk:shypPDE\_Ltransfmeas\_propsrmk\].III), then the following properties also hold:
1. The mapping $(\mu,\nu) \mapsto \mu * \nu$ is continuous in the weak topology.
2. If $h \in \mathrm{C}_\mathrm{b}[a,b)$, then $\mathcal{T}^\mu h \in \mathrm{C}_\mathrm{b}[a,b)$ for all $\mu \in \mathcal{M}_\mathbb{C}[a,b)$.
3. If $h \in \mathrm{C}_0[a,b)$, then $\mathcal{T}^\mu h \in \mathrm{C}_0[a,b)$ for all $\mu \in \mathcal{M}_\mathbb{C}[a,b)$.
***(a)*** Using , we compute $$\begin{aligned}
\widehat{\mu * \nu}(\lambda) & = \int_{[a,b)\!\!} w_\lambda(x) \, (\mu * \nu)(dx) \\
& = \int_{[a,b)\!} \int_{[a,b)\!} \int_{[a,b)\!} w_\lambda(\xi)\, (\delta_x * \delta_y)(d\xi) \, \mu(dx) \nu(dy)\\
& = \int_{[a,b)\!} \int_{[a,b)\!\!} w_\lambda(x) \, w_\lambda(y) \, \mu(dx) \nu(dy) \: = \: \widehat{\mu}(\lambda) \, \widehat{\nu}(\lambda), \qquad\;\; \lambda \geq 0.\end{aligned}$$ This proves the “only if" part, and the converse follows from the uniqueness property in Proposition \[prop:shypPDE\_Ltransfmeas\_props\](b).\
***(b)*** Due to Proposition \[prop:shypPDE\_conv\_Mbanachalg\], it only remains to prove that $(\mu * \nu)[a,b) = 1$ ($\mu, \nu \in \mathcal{P}[a,b)$). But this follows at once from part (a): $$(\mu * \nu)[a,b) = \widehat{\mu * \nu}(0) = \widehat{\mu}(0) {\kern-.12em\cdot\kern-.12em}\widehat{\nu}(0) = \mu[a,b) {\kern-.12em\cdot\kern-.12em}\nu[a,b) = 1.$$
***(c)*** Since $\widehat{\delta_x * \delta_y}(\lambda) = w_\lambda(x) w_\lambda(y)$, Proposition \[prop:shypPDE\_Ltransfmeas\_props\](d) yields that $(x,y) \mapsto \delta_x * \delta_y$ is continuous in the weak topology. Therefore, for $h \in \mathrm{C}_\mathrm{b}[a,b)$ and $\mu_n, \nu_n \in \mathcal{M}_\mathbb{C}[a,b)$ with $\mu_n {\overset{w}{\longrightarrow}}\mu$ and $\nu_n {\overset{w}{\longrightarrow}}\nu$ we have $$\begin{aligned}
\lim_n \int_{[a,b)} h(\xi) (\mu_n * \nu_n)(d\xi) & = \lim_n \int_{[a,b)\!} \int_{[a,b)\!} \biggl( \int_{[a,b)\!} h(\xi)\, (\delta_x * \delta_y)(d\xi) \biggr) \mu_n(dx) \nu_n(dy) \\
& = \int_{[a,b)\!} \int_{[a,b)\!} \biggl( \int_{[a,b)\!} h(\xi)\, (\delta_x * \delta_y)(d\xi) \biggr) \mu(dx) \nu(dy) \\
& = \int_{[a,b)} h(\xi) (\mu * \nu)(d\xi)\end{aligned}$$ due to the continuity of the function in parenthesis.\
***(d)*** Since $(\mathcal{T}^\mu h)(x) = \int_{[a,b)} h(\xi) \, (\delta_x * \mu)(d\xi)$, this follows immediately from part (c)\
***(e)*** It remains to show that $(\mathcal{T}^\mu h)(x) \to 0$ as $x \uparrow b$. Since $w_\lambda(x) \widehat{\mu}(\lambda) \to 0$ as $x \uparrow b$ ($\lambda > 0$), it follows from Remark \[rmk:shypPDE\_Ltransfmeas\_propsrmk\].II that $\delta_x * \mu {\overset{v}{\longrightarrow}}\bm{0}$ as $x \uparrow b$, where $\bm{0}$ denotes the zero measure; this means that for each $h \in \mathrm{C}_\mathrm{0}[a,b)$ we have $$(\mathcal{T}^\mu h)(x) = \int_{[a,b)\!} h(\xi) (\delta_x * \mu)(d\xi) \longrightarrow \int_{[a,b)\!} h(\xi)\, \bm{0}(d\xi) = 0 \qquad \text{as } x \uparrow b$$ showing that $\mathcal{T}^\mu h \in \mathrm{C}_0[a,b)$.
Harmonic analysis on $L_p$ spaces {#sec:Lp_harmonic}
=================================
For the remainder of this work, the coefficients of $\ell$ will be assumed to satisfy $\lim_{x \uparrow b} p(x)r(x) = \infty$ (cf. Remark \[rmk:shypPDE\_Ltransfmeas\_propsrmk\].III), and Assumption \[asmp:shypPDE\_SLhyperg\] continues to be in place.
In this section, we turn our attention to the basic mapping properties of the $\mathcal{L}$-translation and convolution on the Lebesgue spaces $L_p(r)$ ($1 \leq p \leq \infty$). The first result, whose proof depends on the continuity of the mapping $(\mu,\nu) \mapsto \mu * \nu$, ensures that the $\mathcal{L}$-translation defines a linear contraction on $L_p(r)$:
\[prop:shypPDEwl\_gentransl\_Lpcont\] Let $1 \leq p \leq \infty$ and $\mu \in \mathcal{M}_+[a,b)$. The $\mathcal{L}$-translation $(\mathcal{T}^\mu h)(x) = \int_{[a,b)} h(\xi) \, (\delta_x * \mu)(d\xi),$ is, for each $h \in L_p(r)$, a Borel measurable function of $x$, and we have $$\label{eq:shypPDEwl_gentransl_Lpcont}
\|\mathcal{T}^\mu h\|_{p} \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\|h\|_{p} \qquad \text{ for all } h \in L_p(r)$$ (consequently, $\mathcal{T}^\mu\bigl(L_p(r)\bigr) \subset L_p(r)$).
It suffices to prove the result for nonnegative $h \in L_p(r)$, $1 \leq p \leq \infty$.
The map $\nu \mapsto \mu * \nu$ is weakly continuous (Corollary \[cor:shypPDEconv\_basicprops\](c)) and takes $\mathcal{M}_+[a,b)$ into itself. According to [@jewett1975 Section 2.3], this implies that, for each Borel measurable $h \geq 0$, the function $x \mapsto (\mathcal{T}^\mu h)(x)$ is Borel measurable. It follows that $\int_{[a,b)} g(x) (\mu * r)(dx) := \int_a^b (\mathcal{T}^\mu g)(x) r(x) dx$ ($g \in \mathrm{C}_\mathrm{c}[a,b)$) defines a positive Borel measure. For $a \leq c_1 < c_2 < b$, let $\mathds{1}_{[c_1,c_2)}$ be the indicator function of $[c_1,c_2)$, let $h_n \in \mathrm{C}_{\mathrm{c},0}^4$ be a sequence of nonnegative functions such that $h_n \to \mathds{1}_{[c_1,c_2)}$ pointwise, and write $\mathfrak{C} = \{g \in \mathrm{C}_\mathrm{c}^\infty(a,b) \mid 0 \leq g \leq 1\}$. We compute $$\begin{aligned}
(\mu * r)[c_1,c_2) & = \lim_n \int_{[a,b)} h_n(x) (\mu * r)(dx) \\
& = \lim_n \sup_{g \in \mathfrak{C}} \int_a^b (\mathcal{T}^\mu h_n)(x) \, g(x) \, r(x) dx \\
& = \lim_n \sup_{g \in \mathfrak{C}} \int_{[0,\infty)} \! (\mathcal{F}h_n)(\lambda) \, (\mathcal{F}g)(\lambda) \, \widehat{\mu}(\lambda) \, \bm{\rho}_{\mathcal{L}}(d\lambda) \\
& = \lim_n \sup_{g \in \mathfrak{C}} \int_a^b h_n(x) \, (\mathcal{T}^\mu g)(x) \, r(x) dx \\
& \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\lim_n \int_{[a,b)\!} h_n(x) \, r(x) dx \, = \, \|\mu\| {\kern-.12em\cdot\kern-.12em}\int_{[c_1,c_2)\!} r(x)dx \vspace{-2pt}\end{aligned}$$ where the third and fourth equalities follow from and a change of order of integration, and the inequality holds because $\|\mathcal{T}^\mu g\|_\infty \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\| g \|_\infty \leq \|\mu\|$. Therefore, $\|\mathcal{T}^\mu h\|_1 = \|h\|_{L_1([a,b),\mu * r)} \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\|h\|_1$ for each Borel measurable $h \geq 0$. Since $\delta_x * \mu \in \mathcal{M}_+[a,b)$, Hölder’s inequality yields that $\|\mathcal{T}^\mu h\|_p \leq \|\mu\|^{1/q} {\kern-.12em\cdot\kern-.12em}\|\mathcal{T}^\mu |h|^p\|_1^{1/p} \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\| h \|_p$ for $1 < p < \infty$.
Finally, if $h \in L_\infty(r)$, $h \geq 0$ then $h = h_{\mathbf{b}} + h_{\mathbf{0}}$, where $0 \leq h_{\mathbf{b}} \leq \|h\|_\infty$ and $h_{\mathbf{0}} = 0$ Lebesgue-almost everywhere. Since $\|\mathcal{T}^\mu h_{\mathbf{0}}\|_1 \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\|h_{\mathbf{0}}\|_1 = 0$, we have $\mathcal{T}^y h_{\mathbf{0}} = 0$ Lebesgue-almost everywhere, and therefore $\|\mathcal{T}^y h\|_\infty = \|\mathcal{T}^y h_{\mathbf{b}}\|_\infty \leq \|\mu\| {\kern-.12em\cdot\kern-.12em}\|h\|_\infty$.
It is natural to define the $\mathcal{L}$-convolution of functions so that the fundamental identity $\mathcal{F}(h*g) = (\mathcal{F}h) {\kern-.12em\cdot\kern-.12em}(\mathcal{F}g)$ holds (where $\mathcal{F}$ denotes the $\mathcal{L}$-transform ):
Let $h, g:[a,b) \longrightarrow \mathbb{C}$. If the integral $$(h * g)(x) = \int_a^b (\mathcal{T}^y h)(x)\, g(y)\, r(y) dy = \int_a^b \int_{[a,b)} \! h(\xi) \, (\delta_x * \delta_y)(d\xi)\, g(y)\, r(y) dy \vspace{-1pt}$$ exists for almost every $x \in [a,b)$, then we call it the *$\mathcal{L}$-convolution* of the functions $h$ and $g$.
\[prop:shypPDE\_conv\_Ltransfident\] If $h \in \mathrm{C}_{\mathrm{c},0}^4$ and $g \in L_1(r)$, then $\bigl(\mathcal{F}(h * g)\bigr)(\lambda) = (\mathcal{F}h)(\lambda) (\mathcal{F}g)(\lambda)$ for all $\lambda \geq 0$.
For $h \in \mathrm{C}_{\mathrm{c},0}^4$ and $g \in L_1(r)$ we have $$\begin{aligned}
\bigl(\mathcal{F}(h * g)\bigr)(\lambda) & = \int_a^b \int_a^b (\mathcal{T}^\xi h)(x) g(\xi) \, r(\xi) d\xi \, w_\lambda(x) r(x) dx \\
& = \int_a^b \bigl(\mathcal{F}(\mathcal{T}^\xi h)\bigr)(\lambda) \, g(\xi) r(\xi) d\xi \\[3pt]
& \smash{= (\mathcal{F}h)(\lambda) \int_a^b g(\xi) w_\lambda(\xi) r(\xi) d\xi \, = \, (\mathcal{F}h)(\lambda) (\mathcal{F}g)(\lambda)} \\[-14pt]\end{aligned}$$ where we have used Fubini’s theorem and the identity .
Let $p_1,p_2 \in [1, \infty]$ such that ${1 \over p_1} + {1 \over p_2} \geq 1$. For $h \in L_{p_1}(r)$ and $g \in L_{p_2}(r)$, the $\mathcal{L}$-convolution $h * g$ is well-defined and, for $s \in [1, \infty]$ defined by ${1 \over s} = {1 \over p_1} + {1 \over p_2} - 1$, it satisfies $$\| h * g \|_s \leq \| h \|_{p_1} \| g \|_{p_2}$$ (in particular, $h * g \in L_s(r)$). Consequently, the $\mathcal{L}$-convolution is a continuous bilinear operator from $L_{p_1}(r) \times L_{p_2}(r)$ into $L_s(r)$.
The proof is given for completeness; it is analogous to that of the Young inequality for the ordinary convolution.
Define ${1 \over t_1} = {1 \over p_1} - {1 \over s}$ and ${1 \over t_2} = {1 \over p_2} - {1 \over s}$. Observe that $$|(\mathcal{T}^x h)(y)| \, | g(y)| \leq |(\mathcal{T}^x h)(y)|^{p_1/t_1} \, | g(y)|^{p_2/t_2} \bigl[|(\mathcal{T}^x h)(y)|^{p_1} \, | g(y)|^{p_2}\bigr]^{1/s}.$$ Since ${1 \over s} + {1 \over t_1} + {1 \over t_2} = 1$, we have by Hölder’s inequality and $$\begin{aligned}
& \int_a^b |(\mathcal{T}^x h)(y)| \, | g(y)| r(y) dy \\
& \qquad \leq \biggl( \int_a^b |(\mathcal{T}^x h)(y)|^{p_1} r(y) dy \biggr)^{1/t_1} \biggl( \int_a^b |g(y)|^{p_2} r(y) dy \biggr)^{1/t_2} \biggl( \int_a^b |(\mathcal{T}^x h)(y)|^{p_1} \, | g(y)|^{p_2} r(y)dy \biggr)^{1/s} \\
& \qquad = \|h\|_{p_1}^{p_1/t_1} \|g\|_{p_2}^{p_2/t_2} \biggl( \int_a^b |(\mathcal{T}^x h)(y)|^{p_1} \, | g(y)|^{p_2} r(y)dy \biggr)^{1/s}.\end{aligned}$$ Using again we conclude that $$\| h * g \|_s \leq \|h\|_{p_1}^{p_1/t_1} \, \|g\|_{p_2}^{p_2/t_2} \, \|h\|_{p_1}^{p_1/s} \, \|g\|_{p_2}^{p_2/s} = \|h\|_{p_1} \|g\|_{p_2}. \qedhere$$
A consequence of the Young convolution inequality is that the fundamental identity $\bigl(\mathcal{F}(h * g)\bigr)(\lambda) = (\mathcal{F}h)(\lambda) (\mathcal{F}g)(\lambda)$ (Proposition \[prop:shypPDE\_conv\_Ltransfident\]) extends, by continuity, to $h \in L_1(r) \cup L_2(r)$ and $g \in L_1(r)$. Another consequence is the Banach algebra property of the space $L_1(r)$:
The Banach space $L_1(r)$, equipped with the convolution multiplication $h \cdot g \equiv h * g$, is a commutative Banach algebra without identity element.
The Young convolution inequality shows that the $\mathcal{L}$-convolution defines a binary operation on $L_1(r)$ for which the norm is submultiplicative. The commutativity and associativity of the $\mathcal{L}$-convolution are a consequence of the identity $\mathcal{F}(h * g) = (\mathcal{F}h) {\kern-.12em\cdot\kern-.12em}(\mathcal{F}g)$.
Suppose now that there exists $\mathrm{e} \in L_1(r)$ such that $h * \mathrm{e} = h$ for all $h \in L_1(r)$. Then $$(\mathcal{F}h)(\lambda) (\mathcal{F}\mathrm{e})(\lambda) = \bigl(\mathcal{F}(h * \mathrm{e})\bigr)(\lambda) = (\mathcal{F}h)(\lambda) \qquad \text{for all } h \in L_1(r) \text{ and } \lambda \geq 0.$$ Clearly, this implies that $(\mathcal{F}\mathrm{e})(\lambda) = 1$ for all $\lambda \geq 0$. But we know that $\widehat{\delta_a} \equiv 1$, so it follows from Proposition \[prop:shypPDE\_Ltransfmeas\_props\](b) that $\mathrm{e}(x) r(x) dx = \delta_a(dx)$, which is absurd. This shows that the Banach algebra has no identity element.
Applications to probability theory {#sec:probtheory}
==================================
Infinite divisibility of measures and the Lévy-Khintchine representation
------------------------------------------------------------------------
The set $\mathcal{P} _\mathrm{id}$ of *$\mathcal{L}$-infinitely divisible measures* (or *$\mathcal{L}$-infinitely divisible distributions*) is defined in the obvious way: $$\mathcal{P}_\mathrm{id} = \bigl\{ \mu \in \mathcal{P}[a,b) \bigm| \text{for all } n \in \mathbb{N} \text{ there exists } \nu_n \in \mathcal{P}[a,b) \text{ such that } \mu = \nu_n^{*n} \bigr\}$$ where $\nu_n^{*n}$ denotes the $n$-fold $\mathcal{L}$-convolution of $\nu_n$ with itself.
It is a simple exercise to show that the $\mathcal{L}$-transform of $\mu \in \mathcal{P}_\mathrm{id}$ is of the form $$\widehat{\mu}(\lambda) = e^{-\psi_\mu(\lambda)}$$ where $\psi_\mu$ is continuous, nonnegative and $\psi_\mu(0) = 0$. The function $\psi_\mu$ is called the *$\mathcal{L}$-exponent* of $\mu \in \mathcal{P}_\mathrm{id}$.
As we will see, the exponents of $\mathcal{L}$-infinitely divisible measures admit a representation which is analogous to the well-known Lévy-Khintchine formula for infinitely divisible measures with respect to the ordinary Fourier transform. In the present context, the relevant notions of Poisson and Gaussian measures are defined as follows:
Let $\mu \in \mathcal{M}_+[a,b)$. The measure ${\mathbf{e}}(\mu) \in \mathcal{P}[a,b)$ defined by $${\mathbf{e}}(\mu) = e^{-\|\mu\|} \sum_{k=0}^\infty {\mu^{*k} \over k!}$$ (the infinite sum converging in the weak topology) is said to be the *$\mathcal{L}$-compound Poisson measure* associated with $\mu$.
The $\mathcal{L}$-transform of ${\mathbf{e}}(\mu)$ can be easily deduced using Corollary \[cor:shypPDEconv\_basicprops\](a): $$\widehat{{\mathbf{e}}(\mu)}(\lambda) = e^{-\|\mu\|} \sum_{k=0}^\infty {\widehat{\mu^{*k}}(\lambda) \over k!} = e^{-\|\mu\|} \sum_{k=0}^\infty {\bigl(\widehat{\mu}(\lambda)\bigr)^k \over k!} = \exp\bigl(\widehat{\mu}(\lambda) - \|\mu\|\bigr).$$ Since ${\mathbf{e}}(\mu_1 + \mu_2) = {\mathbf{e}}(\mu_1) * {\mathbf{e}}(\mu_2)$ ($\mu_1, \mu_2 \in \mathcal{M}_+[a,b)$), every $*$-compound Poisson measure belongs to $\mathcal{P}_\mathrm{id}$.
To motivate the following definition, we observe that it follows from classical results in probability theory (see e.g. [@klenke2014 Theorem 16.17] and [@linnikostrovskii1977 §III.1]) that an infinitely divisible probability measure on $\mathbb{R}^d$ is Gaussian if and only if it has no nontrivial divisors of the form $\mathfrak{e}(\nu)$, where $\nu$ is a finite positive measure on $\mathbb{R}^d$ and $\mathfrak{e}(\nu)$ denotes the (ordinary) compound Poisson measure associated with $\nu$.
A measure $\mu \in \mathcal{P}_{\mathrm{id}}$ is called an *$\mathcal{L}$-Gaussian measure* if $$\mu = {\mathbf{e}}(\nu) * \vartheta \quad \bigl(a > 0 ,\, \nu \in \mathcal{M}_+[a,b),\, \vartheta \in \mathcal{P}_\mathrm{id}\bigr) \qquad \implies \qquad {\mathbf{e}}(\nu) = \delta_a.$$
We are now ready to state the analogue of the Lévy-Khintchine representation for infinite divisibility with respect to the $\mathcal{L}$-convolution.
\[thm:shypPDE\_levykhin\] The $\mathcal{L}$-exponent of a measure $\mu \in \mathcal{P}_\mathrm{id}$ can be represented in the form $$\label{eq:shypPDE_levykhin}
\psi_\mu(\lambda) = \psi_\alpha(\lambda) + \int_{(a,b)\!} \bigl( 1 - w_\lambda(x) \bigr) \nu(dx)$$ where $\nu$ is a $\sigma$-finite measure on $(a,b)$ which is finite on the complement of any neighbourhood of $a$ and such that $$\int_{(a,b)\!} \bigl( 1 - w_\lambda(x) \bigr) \nu(dx) < \infty$$ and $\alpha$ is an $\mathcal{L}$-Gaussian measure with $\mathcal{L}$-exponent $\psi_\alpha(\lambda)$. Conversely, each function of the form is an $\mathcal{L}$-exponent of some $\mu \in \mathcal{P}_\mathrm{id}$.
We only give a sketch of the proof, and refer to [@volkovich1988] for details.
Let $\mu \in \mathcal{P}_{\mathrm{id}}$, let $b > a_1 > a_2 > \ldots$ with $\lim a_n = a$, and let $I_n = [a,a_n)$, $J_n = [a_n,b)$. Consider the set $\mathcal{Q}$ of all divisors of $\mu$ of the form ${\mathbf{e}}(\pi)$ such that $\pi(I_1) = 0$. One can prove that the set $\mathrm{D}(\mathfrak{P})$ of all divisors (with respect to the $\mathcal{L}$-convolution) of measures $\nu \in \mathfrak{P}$ is relatively compact whenever $\mathfrak{P} \subset \mathcal{P}[a,b)$ is relatively compact (see [@volkovich1992 Corollary 1]); using this fact, it can be shown that $\sup_{\mu = {\mathbf{e}}(\pi) \in \mathcal{Q}} \bigl[ \int_{[a,b)} \bigl(1-w_\lambda(x)\bigr) \pi(dx) \bigr] < \infty$ and, consequently, there exists a divisor $\mu_1 = {\mathbf{e}}(\pi_1) \in \mathcal{Q}$ such that $\pi_1(J_1)$ is maximal among all elements of $\mathcal{Q}$. Write $\mu = \mu_1 * \alpha_1$ ($\alpha_1 \in \mathcal{P}_\mathrm{id}$). Applying the same reasoning to $\alpha_1$ with $I_1$ replaced by $I_2$, we get $\alpha_1 = \mu_2 * \alpha_2 = {\mathbf{e}}(\pi_2) * \alpha_2$. If we perform this successively, we get $$\mu = \alpha_n * \beta_n, \qquad \text{where } \, \beta_n = \mu_1 * \mu_2 * \ldots \mu_n, \qquad \mu_k = {\mathbf{e}}(\pi_k)$$ with $\pi_k(I_k) = 0$ and $\pi_k(J_k)$ having the specified maximality property. The sequences $\{\alpha_n\}$ and $\{\beta_n\}$ are relatively compact; letting $\alpha$ and $\beta$ be limit points, we have $$\mu = \alpha * \beta \qquad (\alpha, \beta \in \mathcal{P}_\mathrm{id}).$$ Suppose, by contradiction, that $\alpha$ is not $\mathcal{L}$-Gaussian, and let ${\mathbf{e}}(\eta)$, with $\eta \neq \delta_a$, be a divisor of $\alpha$. Clearly $\eta(J_k) > 0$ for some $k$; given that each $\alpha_n$ divides $\alpha_{n-1}$, we have $\alpha_k = {\mathbf{e}}(\eta) * \nu$ ($\nu \in \mathcal{P}_\mathrm{id}$). If we let $\widetilde{\eta}$ be the restriction of $\eta$ to the interval $J_k$, then $$\alpha_{k-1} = {\mathbf{e}}(\pi_k + \widetilde{\eta}) * {\mathbf{e}}(\eta - \widetilde{\eta}) * \nu$$ which is absurd (because $(\pi_k + \widetilde{\eta})(J_k) > \pi_k(J_k)$, contradicting the maximality property which defines $\pi_k$). To determine the $\mathcal{L}$-exponent of $\beta$, note that $\beta_n = {\mathbf{e}}(\varPi_n)$ is the $\mathcal{L}$-compound Poisson measure associated with $\varPi_n := \sum_{k=1}^n \pi_k$, thus $\psi_{\beta_n}(\lambda) = \int_{(a,b)} \bigl(1-w_\lambda(x)\bigr) \varPi_n(dx)$. Since $\{\varPi_n\}$ is an increasing sequence of measures and each ${\mathbf{e}}(\varPi_n)$ dividing $\mu$, there exists a $\sigma$-finite measure $\nu$ such that $$\psi_{\beta}(\lambda) = \lim_n \int_{(a,b)} \bigl(1-w_\lambda(x)\bigr) \varPi_n(dx) = \lim_n \int_{(a,b)} \bigl(1-w_\lambda(x)\bigr) \nu(dx) < \infty$$ ($\mu \in \mathcal{P}_\mathrm{id}$ ensures the finiteness of the integral); from the relative compactness of $\mathrm{D}(\{\mu\})$ it is possible to conclude that $\nu(J_k) < \infty$ for all $k$.
For the converse, let $\nu_n$ be the restriction of $\nu$ to the interval $J_n$ defined as above. It is verified without difficulty that the right-hand side of is continuous at zero, hence by Proposition \[prop:shypPDE\_Ltransfmeas\_props\](d) $\alpha * {\mathbf{e}}(\nu_n) {\overset{w}{\longrightarrow}}\mu \in \mathcal{P}[a,b)$, and $\mu \in \mathcal{P}_\mathrm{id}$ because $\mathcal{P}_\mathrm{id}$ is closed under weak convergence of measures.
Convolution semigroups and their contraction properties
-------------------------------------------------------
A family $\{\mu_t\}_{t\geq 0} \subset \mathcal{P}[a,b)$ is called an *$\mathcal{L}$-convolution semigroup* if it satisfies the conditions
- $\mu_s * \mu_t = \mu_{s+t}$ for all $s, t \geq 0$;
- $\mu_0 = \delta_a$;
- $\mu_t {\overset{w}{\longrightarrow}}\delta_a$ as $t \downarrow 0$.
A direct consequence of this definition is that $$\label{eq:shypPDE_infdivsemigr_corresp}
\{\mu_t\} \longmapsto \mu_1 \in \mathcal{P}_\mathrm{id}$$ defines a one-to-one correspondence holds between the set of $\mathcal{L}$-convolution semigroups and the set of $\mathcal{L}$-infinitely divisible measures. Indeed, if $\{\mu_t\}$ is an $\mathcal{L}$-convolution semigroup, then it is clear that each $\mu_t$ is $\mathcal{L}$-infinitely divisible; and if $\mu \in \mathcal{P}_\mathrm{id}$ has exponent $\psi_\mu(\lambda)$, then $\widehat{\mu_t}(\lambda) = \exp(-t \, \psi_\mu(\lambda))$ defines the unique $\mathcal{L}$-convolution semigroup such that $\mu_1 = \mu$ (the proof of this is analogous to that for the classical convolution, cf. [@bauer1996 Theorem 29.6]).
\[prop:shypPDE\_fellerLpsemigr\] Let $\{\mu_t\}$ be an $\mathcal{L}$-convolution semigroup. Then $$(T_t h)(x) := (\mathcal{T}^{\mu_t}h)(x) = \int_{[a,b)} h(\xi) (\delta_x * \mu_t)(d\xi)$$ defines a strongly continuous Markovian contraction semigroup $\{T_t\}_{t \geq 0}$ on $\mathrm{C}_0[a,b)$ and on the spaces $L_p(r)$ ($1 \leq p < \infty$), i.e., the following properties hold:
1. $T_t T_s = T_{t+s}$ for all $t, s \geq 0$;
2. $T_t \bigl(\mathrm{C}_0[0,\infty)\bigr) \subset \mathrm{C}_0[0,\infty)$ for all $t \geq 0$;
3. $T_t \bigl(L_p(r)\bigr) \subset L_p(r)$ for all $t \geq 0$ ($1 \leq p < \infty$);
4. $T_t \mathds{1} = \mathds{1}$ for all $t \geq 0$, and if $f \in \mathrm{C}_\mathrm{b}[0,\infty)$ satisfies $0 \leq h \leq 1$, then $0 \leq T_t h \leq 1$;
5. $\lim_{t \downarrow 0} \|T_t h - h\|_\infty = 0$ for each $h \in \mathrm{C}_0[0,\infty)$;
6. $\lim_{t \downarrow 0} \|T_t h - h\|_p = 0$ for each $h \in L_p(r)$ ($1 \leq p < \infty$).
Moreover, $\{T_t\}$ is translation-invariant: $T_t \mathcal{T}^\nu f = \mathcal{T}^\nu T_t f$ for all $t \geq 0$ and $\nu \in \mathcal{M}_\mathbb{C}[a,b)$.
Parts (ii), (ii’) and (iii) follow at once from Corollary \[cor:shypPDEconv\_basicprops\] and Proposition \[prop:shypPDEwl\_gentransl\_Lpcont\]. Concerning part (i) and the translation invariance property, notice that by we have $$\mathcal{F}(\mathcal{T}^\mu (\mathcal{T}^\nu h)) = \widehat{\mu} \cdot \mathcal{F}\mathcal({T}^\nu h) = \widehat{\mu} \cdot \widehat{\nu} \cdot \mathcal{F}h = \widehat{\mu* \nu} \cdot \mathcal{F}h = \mathcal{F}(\mathcal{T}^{\mu*\nu} h) \qquad (h \in \mathrm{C}_{\mathrm{c},0}^4)$$ so that $\mathcal{T}^\mu (\mathcal{T}^\nu h) = \mathcal{T}^{\mu*\nu} h$ first for $h \in \mathrm{C}_{\mathrm{c},0}^4$ and then, by continuity, for $h \in \mathrm{C}_0[a,b)$ and $h \in L_p(r)$ ($1 \leq p < \infty$).
To prove part (iv) we just need to show that $\lim_{t \downarrow 0} (T_t h)(x) = h(x)$ for all $h \in \mathrm{C}_0[a,b)$ and $x \in [a,b)$, because it is well-known from the theory of Feller semigroups that for a semigroup satisfying (ii) and (iii) this weak continuity property implies the strong continuity of the semigroup (see e.g. [@bottcher2013 Lemma 1.4]). But for $h \in \mathrm{C}_0[a,b)$ and $x \in [a,b)$ we clearly have $$\lim_{t \downarrow 0} \bigl((T_t h)(x) - h(x)\bigr) = \lim_{t \downarrow 0} \int_{[a,b)\!} \bigl((\mathcal{T}^y h)(x) - h(x)\bigr) \mu_t(dy) = \int_{[a,b)} \bigl((\mathcal{T}^y h)(x) - h(x)\bigr) \delta_a(dy) = 0$$ showing that (iv) holds.
For part (iv’), let $h \in L_p(r)$, ${\varepsilon}> 0$ and choose $g \in \mathrm{C}_\mathrm{c}^\infty(a,b)$ such that $\|h - g\|_p \leq {\varepsilon}$. Then it follows from and part (iv) that $$\begin{aligned}
\limsup_{t \downarrow 0} \|T_t h - h\|_p & \leq \limsup_{t \downarrow 0} \Bigl(\|T_t h - T_t g\|_p + \|h - g\|_p + \|T_t g - g\|_p\Bigr) \\
& \leq 2{\varepsilon}+ C {\kern-.12em\cdot\kern-.12em}\limsup_{t \downarrow 0} \|T_t g - g\|_\infty \\
& = \, 2{\varepsilon}\end{aligned}$$ where $C = [\int_{{\mathrm{supp}}(g)} r(x) dx]^{1/p}$ ($C < \infty$ because the support ${\mathrm{supp}}(g) \subset (a,b)$ is compact). Since ${\varepsilon}$ is arbitrary, (iv’) holds.
The result for the space $\mathrm{C}_0[a,b)$ means that $\{T_t\}$ is an $\mathcal{L}$-translation-invariant conservative Feller semigroup. This semigroup is also symmetric with to the measure $r(x)dx$, that is, $\int_a^b (T_t h)(x) g(x) r(x) dx = \int_a^b h(x) (T_t g)(x) r(x) dx$ for $h,g \in \mathrm{C}_\mathrm{c}[a,b)$. Any such symmetric Feller semigroup extends to a strongly continuous Markovian contraction semigroup $\{T_t^{(p)}\}_{t \geq 0}$ on $L_p(r)$, $1 \leq p < \infty$ [@bottcher2013 Lemma 1.45]. However, the conclusion of Proposition \[prop:shypPDE\_fellerLpsemigr\] is stronger: it also states that the integral with respect to the Feller transition function is well-defined for all $h \in \cup_{1 \leq p < \infty} L_p(r)$ and, accordingly, the extensions $T_t^{(p)}$ are also given by $h \mapsto (\mathcal{T}^{\mu_t} h)(x) = \int_{[a,b)} h(\xi) (\delta_x * \mu_t)(d\xi)$.
On the Hilbert space $L_2(r)$, we can take advantage of the $\mathcal{L}$-transform to obtain a characterization of the generator of the $L_2$-Markovian semigroup $T_t^{(2)} \equiv T_t: L_2(r) \longrightarrow L_2(r)$:
Let $\{\mu_t\}$ be an $\mathcal{L}$-convolution semigroup with exponent $\psi$. Then the infinitesimal generator $(\mathcal{A}^{(2)}, \mathcal{D}_{\!\mathcal{A}^{(2)}})$ of the $L_2$-Markovian semigroup $\{T_t^{(2)}\}$ is given by $$\mathcal{F}(\mathcal{A}^{(2)} h) = -\psi {\kern-.12em\cdot\kern-.12em}(\mathcal{F}h), \qquad h \in \mathcal{D}_{\!\mathcal{A}^{(2)}}$$ where $$\mathcal{D}_{\!\mathcal{A}^{(2)}} = \biggl\{ h \in L_2(r) \biggm| \int_{[0,\infty)} \bigl|\psi(\lambda)\bigr|^2 \bigl|(\mathcal{F}h)(\lambda)\bigr|^2 \bm{\rho}_{\mathcal{L}}(d\lambda) < \infty \biggr\}.$$
We give a proof which follows closely that of the corresponding result for the ordinary convolution, as given in [@bergforst1975 Theorem 12.16].
Let $h \in \mathcal{D}_{\!\mathcal{A}^{(2)}}$, so that $L_2\text{-\!}\lim_{t \downarrow 0}{1 \over t} (T_t h - h) = \mathcal{A}^{(2)}h \in L_2(\mathrm{m})$. Recalling that (by ) $\mathcal{F}(T_t h) = \widehat{\mu_t} {\kern-.12em\cdot\kern-.12em}(\mathcal{F}h) = e^{-t\psi} {\kern-.12em\cdot\kern-.12em}(\mathcal{F}h)$ for all $h \in L_2(r)$, we see that $$L_2\text{-\!}\lim_{t \downarrow 0}{1 \over t} \bigl(e^{-t\,\psi} - 1\bigr) {\kern-.12em\cdot\kern-.12em}(\mathcal{F}h) = \mathcal{F}(\mathcal{A}^{(2)}h)$$ The convergence holds almost everywhere along a sequence $\{t_n\}_{n \in \mathbb{N}}$ such that $t_n \to 0$, so we conclude that $\mathcal{F}(\mathcal{A}^{(2)}h) = -\psi \cdot (\mathcal{F}h) \in L_2(\mathbb{R}; \bm{\rho}_\mathcal{L})$.
Conversely, if we let $h \in L_2(r)$ with $-\psi {\kern-.12em\cdot\kern-.12em}(\mathcal{F}h) \in L_2(\mathbb{R}; \bm{\rho}_\mathcal{L})$, then we have $$L_2\text{-\!}\lim_{t \downarrow 0} {1 \over t}\bigl(\mathcal{F}({T_t h}) - \mathcal{F}h \bigr) = -\psi {\kern-.12em\cdot\kern-.12em}(\mathcal{F}h) \in L_2(\mathbb{R}; \bm{\rho}_\mathcal{L})$$ and the isometry gives that $L_2\text{-\!}\lim_{t \downarrow 0} {1 \over t}\bigl(T_t h - h \bigr) \in L_2(\mathrm{m})$, meaning that $h \in \mathcal{D}_{\!\mathcal{A}^{(2)}}$.
Additive and Lévy processes
---------------------------
An $[a,b)$-valued Markov chain $\{S_n\}_{n \in \mathbb{N}_0}$ is said to be *$\mathcal{L}$-additive* if there exist measures $\mu_n \in \mathcal{P}[a,b)$ such that $$\label{eq:shypPDE_markovadditive_def}
P[S_n \in B | S_{n-1} = x] = (\mu_n * \delta_x)(B), \qquad\;\; n \in \mathbb{N}, \: a \leq x < b, \: B \text{ a Borel subset of } [a,b).$$ If $\mu_n = \mu$ for all $n$, then $\{S_n\}$ is said to be an *$\mathcal{L}$-random walk*.
An explicit construction can be given for $\mathcal{L}$-additive Markov chains, based on the following lemma:
There exists a Borel measurable $\Phi:[a,b) \times [a,b) \times [0,1] \longrightarrow [a,b)$ such that $$(\delta_x * \delta_y)(B) = \mathfrak{m}\{\Phi(x,y,\cdot) \in B\}, \qquad x, y \in [a,b), \; B \text{ a Borel subset of } [a,b)$$ where $\mathfrak{m}$ denotes Lebesgue measure on $[0,1]$.
Let $\Phi(x,y,\xi) = \max\bigl( a, \sup\{ z \in [a,b): (\delta_x * \delta_y)[a,z] < \xi \} \bigr)$. Using the continuity of the $\mathcal{L}$-convolution, one can show that $\Phi$ is Borel measurable, see [@bloomheyer1994 Theorem 7.1.3]. It is straightforward that $\mathfrak{m}\{\Phi(x,y,\cdot) \in [a,c]\} = \mathfrak{m}\{ (\delta_x * \delta_y)[a,c] \geq \xi \} = (\delta_x * \delta_y)[a,c]$.
Let $X_1$, $U_1$, $X_2$, $U_2$, $\ldots$ be a sequence of independent random variables (on a given probability space $(\Omega, \mathfrak{A}, \bm{\pi})$) where the $X_n$ have distribution $P_{X_n} = \mu_n \in \mathcal{P}[a,b)$ and each of the (auxiliary) random variables $U_n$ has the uniform distribution on $[0,1]$. Set $$\label{eq:shypPDE_markovadditive_constr}
S_0 = 0, \qquad S_n = S_{n-1} \oplus_{U_n} X_n$$ where $X \oplus_U Y := \Phi(X,Y,U)$. Then we have $P_{S_n} = P_{S_{n-1}} * \mu_n$ ($n \in \mathbb{N}_0$) and, consequently, $\{S_n\}_{n \in \mathbb{N}_0}$ is an $\mathcal{L}$-additive Markov chain satisfying . The identity $P_{S_n} = P_{S_{n-1}} * \mu_n$ is easily checked: $$\begin{aligned}
P_{S_n}(B) = P\bigl[ \Phi(S_{n-1},X_n,U_n) \in B \bigr] & = \int_{[a,b)} \! \int_{[a,b)} \mathfrak{m}\{\Phi(x,y,\cdot) \in B\} P_{S_{n-1}}(dx) P_{X_n}(dy) \\
& = \int_{[a,b)} \! \int_{[a,b)} (\delta_x * \delta_y)(B) P_{S_{n-1}}(dx) P_{X_n}(dy) \\
& = (P_{S_{n-1}} * \mu_n)(B).\end{aligned}$$
We now define the continuous-time analogue of $\mathcal{L}$-random walks:
An $[a,b)$-valued Markov process $Y = \{Y_t\}_{t \geq 0}$ is said to be an *$\mathcal{L}$-Lévy process* if there exists an $\mathcal{L}$-convolution semigroup $\{\mu_t\}_{t\geq 0}$ such that the transition probabilities of $Y$ are given by $$P\bigl[Y_t \in B | Y_s = x\bigr] = (\mu_{t-s} * \delta_x)(B), \qquad 0 \leq s \leq t,\; a \leq x < b,\; B \text{ a Borel subset of } [a,b).$$
The notion of an $\mathcal{L}$-Lévy process coincides with that of a Feller process associated with the Feller semigroup $T_t f = \mathcal{T}^{\mu_{t\!}} f$. Consequently, the general connection between Feller semigroups and Feller processes (see e.g. [@bottcher2013 Section 1.2]) ensures that for each (initial) distribution $\nu \in \mathcal{P}[a,b)$ and $\mathcal{L}$-convolution semigroup $\{\mu_t\}_{t\geq 0}$ there exists an $\mathcal{L}$-Lévy process $Y$ associated with $\{\mu_t\}_{t \geq 0}$ and such that $P_{Y_0} = \nu$. Any $\mathcal{L}$-Lévy process has the following properties:
- It is stochastically continuous: $Y_s \to Y_t$ in probability as $s \to t$, for each $t \geq 0$;
- It has a càdlàg modification: there exists an $\mathcal{L}$-Lévy process $\{\widetilde{Y}_t\}$ with a.s. right-continuous paths and satisfying $P\bigl[Y_t = \widetilde{Y}_t \bigr] = 1$ for all $t \geq 0$.
(These properties hold for all Feller processes, cf. [@bottcher2013 Section 1.2].)
An analogue of the well-known theorem on appoximation of Lévy processes by triangular arrays holds for $\mathcal{L}$-Lévy processes (below the notation ${\overset{d}{\longrightarrow}}$ stands for convergence in distribution):
Let $X$ be an $[a,b)$-valued random variable. The following assertions are equivalent:
1. $X = Y_1$ for some $\mathcal{L}$-Lévy process $Y = \{Y_t\}_{t \geq 0}$.
2. The distribution of $X$ is $\mathcal{L}$-infinitely divisible;
3. $S_{m_n}^n {\overset{d}{\longrightarrow}}X$ for some sequence of $\mathcal{L}$-random walks $S^1, S^2,\, \ldots$ (with $S_0^j = a$) and some integers $m_n \to \infty$.
The equivalence between (i) and (ii) is a restatement of the one-to-one correspondence between $\mathcal{L}$-infinitely divisible measures and $\mathcal{L}$-convolution semigroups. It is obvious that (i) implies (iii): simply let $m_n = n$ and $S^n$ the random walk whose step distribution is the law of $Y_{1/n}$.
Suppose that (iii) holds and let $\pi_n$, $\mu$ be the distributions of $S_j^n$, $X$ respectively. Choose ${\varepsilon}> 0$ small enough so that $\widehat{\mu}(\lambda) > C_{\varepsilon}> 0$ for $\lambda \in [0,{\varepsilon}]$, where $C_{\varepsilon}> 0$ is a constant. By (iii) and Proposition \[prop:shypPDE\_Ltransfmeas\_props\](c), $\widehat{\pi_n}(\lambda)^{m_n} \to \widehat{\mu}(\lambda)$ uniformly on compacts, which implies that $\widehat{\pi_n}(\lambda) \to 1$ for all $\lambda \in [0,{\varepsilon}]$ and, therefore, by Proposition \[prop:shypPDE\_Ltransfmeas\_props\](d) $\pi_n {\overset{w}{\longrightarrow}}\delta_a$. Now let $k \in \mathbb{N}$ be arbitrary. Since $\pi_n {\overset{w}{\longrightarrow}}\delta_a$, we can assume that each $m_n$ is a multiple of $k$. Write $\nu_n = \pi_n^{*(m_n/k)}$, so that $\nu_n^{*k} {\overset{w}{\longrightarrow}}\mu$. By relative compactness of $\mathrm{D}(\{\pi_n^{*m_n}\})$ (see the proof of Theorem \[thm:shypPDE\_levykhin\]), the sequence $\{\nu_n\}_{n \in \mathbb{N}}$ has a weakly convergent subsequence, say $\nu_{n_j} {\overset{w}{\longrightarrow}}\mu_k$ as $j \to \infty$, and from this it clearly follows that $\mu_k^{*k} = \mu$. Consequently, (ii) holds.
As one would expect, the diffusion process generated by the Sturm-Liouville operator (cf. Lemma \[lem:shypPDE\_Lb\_fellergen\]) is an $\mathcal{L}$-Lévy process:
The irreducible diffusion process $X$ generated by $(\mathcal{L}^{(\mathrm{b})}, \mathcal{D}_\mathcal{L}^{(\mathrm{b})})$ is an $\mathcal{L}$-Lévy process.
For $t \geq 0$, $a \leq x < b$ let us write $p_{t,x}(dy) \equiv P_x[X_t \in dy]$. Recall from Lemma \[lem:shypPDE\_Lb\_diffusion\_tpdf\] that $$p_{t,x}(dy) \equiv p(t,x,y) r(y) dy = \int_{[0,\infty)} e^{-t\lambda} w_\lambda(x) \, w_\lambda(y) \, \bm{\rho}_\mathcal{L}(d\lambda) \, r(y) dy, \qquad t > 0,\: \, a < x < b$$ where the integral converges absolutely. Consequently, by Proposition \[prop:shypPDE\_Ltransf\], $$\widehat{p_{t,x}}(\lambda) = e^{-t\lambda} w_\lambda
(x), \qquad t \geq 0,\: a \leq x < b$$ (the weak continuity of $p_{t,x}$ justifies that the equality also holds for $t = 0$ and for $x = a$). This shows that $p_{t,x} = p_{t,a} * \delta_x$ where $\widehat{p_{t,a}}(\lambda) = e^{-t\lambda}$. It is clear from the properties of the $\mathcal{L}$-transform that $\{p_{t,a}\}_{t \geq 0}$ is an $\mathcal{L}$-convolution semigroup; therefore, $X$ is an $\mathcal{L}$-Lévy process.
An $\mathcal{L}$-convolution semigroup $\{\mu_t\}_{t \geq 0}$ such that $\mu_1$ is an $\mathcal{L}$-Gaussian measure is said to be an *$\mathcal{L}$-Gaussian convolution semigroup*, and an $\mathcal{L}$-Lévy process associated with an $\mathcal{L}$-Gaussian convolution semigroup is called an *$\mathcal{L}$-Gaussian process*.
It actually turns out that the diffusion $X$ generated by $(\mathcal{L}^{(\mathrm{b})}, \mathcal{D}_\mathcal{L}^{(\mathrm{b})})$ is an $\mathcal{L}$-Gaussian process. This is a consequence of the following characterization of $\mathcal{L}$-Gaussian measures:
Let $Y = \{Y_t\}_{t \geq 0}$ be an $\mathcal{L}$-Lévy process, let $\{\mu_t\}_{t \geq 0}$ be the associated $\mathcal{L}$-convolution semigroup and let $(\mathcal{G},\mathcal{D}(\mathcal{G}))$ be the $\mathrm{C}_\mathrm{b}$-generator of the process $Y$. The following conditions are equivalent:
1. $\mu_1$ is a Gaussian measure;
2. $\lim_{t \downarrow 0} {1 \over t} \mu_t\bigl([a,b) \setminus \mathcal{V}_a\bigr) = 0$ for every neighbourhood $\mathcal{V}_a$ of the point $a$;
3. $\lim_{t \downarrow 0} {1 \over t} (\mu_t*\delta_x)\bigl([a,b) \setminus \mathcal{V}_x\bigr) = 0 \;$ for every $x \in [a,b)$ and every neighbourhood $\mathcal{V}_x$ of the point $x$;
4. $Y$ has a modification whose paths are a.s. continuous.
If any of these conditions hold then the $\mathrm{C}_\mathrm{b}$-generator of $Y$ is a local operator, i.e., $(\mathcal{G}h)(x) = (\mathcal{G}g)(x)$ whenever $h, g \in \mathcal{D}(\mathcal{G})$ and $h=g$ on some neighbourhood of $x \in [a,b)$.
***(i)$\!\implies\!$(ii):*** Let $\{t_n\}_{n \in \mathbb{N}}$ be a sequence such that $t_n \to 0$ as $n \to \infty$, and let $\nu_n = {\mathbf{e}}\bigl(\tfrac{1}{t_n} \mu_{t_{n\!}}\bigr)$. We have $$\label{eq:shypPDE_gauss_equivdef_pf1}
\lim_{n \to \infty} \widehat{\nu_n}(\lambda) = \lim_{n \to \infty} \exp\biggl[ {1 \over t_n}\bigl(\widehat{\mu_{1}}(\lambda)^{t_n} - 1\bigr) \biggr] = \widehat{\mu_1}(\lambda), \qquad \lambda > 0$$ and therefore, by Proposition \[prop:shypPDE\_Ltransfmeas\_props\](d), $\nu_n {\overset{w}{\longrightarrow}}\mu_1$ as $n \to \infty$. From this it follows, cf. [@volkovich1988], that if $\pi_n$ denotes the restriction of ${1 \over t_n} \mu_{t_n}$ to $[a,b) \setminus \mathcal{V}_a$, then $\{\pi_n\}$ is relatively compact; if $\pi$ is a limit point, then ${\mathbf{e}}(\pi)$ is a divisor of $\mu_1$. Since $\mu_1$ is Gaussian, ${\mathbf{e}}(\pi) = \delta_a$, hence $\pi$ must be the zero measure, showing that (ii) holds.\
***(ii)$\!\implies\!$(i):*** As in , $$\widehat{\mu_1}(\lambda) = \lim_{n \to \infty} \exp\biggl[ {1 \over t_n} \int_{[a,b)} \bigl(w_\lambda(x) - 1\bigr) \mu_{t_n\!}(dx) \biggr] = \lim_{n \to \infty} \exp\biggl[ {1 \over t_n} \int_{\mathcal{V}_a} \bigl(w_\lambda(x) - 1\bigr) \mu_{t_n\!}(dx) \biggr], \qquad \lambda > 0$$ where the second equality is due to (ii), noting that ${1 \over t_n} \int_{[a,b) \setminus \mathcal{V}_a} (w_\lambda(x) - 1) \mu_{t_n\!}(dx) \leq {2 \over t} \mu_{t_n}\bigl([a,b) \setminus \mathcal{V}_a\bigr)$. Given that $\nu_n = {\mathbf{e}}\bigl(\tfrac{1}{t_n} \mu_{t_{n\!}}\bigr) {\overset{w}{\longrightarrow}}\mu_1$, we have (again, see [@volkovich1988]) $$\widehat{\mu}_1(\lambda) = \exp\biggl[ \int_{(a,b)} \bigl(w_\lambda(x) - 1\bigr) \, \eta(dx) \biggr], \qquad \lambda > 0$$ for some $\sigma$-finite measure $\eta$ on $(a,b)$ which, by the above, vanishes on the complement of any neighbourhood of the point $a$. Therefore, $\mu_1$ is Gaussian.\
***(ii)$\!\iff\!$(iii):*** To prove the nontrivial direction, assume that (ii) holds, and fix $x \in (a,b)$. Let $\mathcal{V}_x$ be a neighbourhood of the point $x$ and write $E_x = [a,b) \setminus \mathcal{V}_x$. Pick a function $h \in \mathrm{C}_{\mathrm{c},0}^4$ such that $0 \leq h \leq 1$, $h = 0$ on $E_x$ and $h = 1$ on some smaller neighbourhood $\mathcal{U}_x \subset \mathcal{V}_x$ of the point $x$.
We begin by showing that $$\label{eq:shypPDE_gauss_equivdef_pf2}
\lim_{y \downarrow a} {1 - (\mathcal{T}^x h)(y) \over 1-w_\lambda(y)} = 0 \qquad \text{for each } \lambda > 0.$$ Indeed, it follows from Theorem \[thm:shypPDE\_Lexistence\] that $\lim_{y \downarrow a} (\mathcal{T}^x h)(y) = 1$, $\lim_{y \downarrow a} \partial_y^{[1]} (\mathcal{T}^x h)(y) = 0$ and $$\ell_y (\mathcal{T}^x h)(y) = \int_{[0,\infty)\!} \lambda\, (\mathcal{F} h)(\lambda) \, w_\lambda(x) \, w_\lambda(y) \, \bm{\rho}_{\mathcal{L}}(d\lambda) = \bigl(\mathcal{T}^x \ell(h)\bigr)(y) \xrightarrow[\,y \downarrow a\,]{} \ell(h)(x) = 0,$$ hence using L’Hôpital’s rule twice we find that $\lim_{y \downarrow a} {1 - (\mathcal{T}^x h)(y) \over 1-w_\lambda(y)} = \lim_{y \downarrow a} {\ell_y(\mathcal{T}^x h)(y) \over \lambda w_\lambda(y)} = 0$ ($\lambda > 0$).
By , for each $\lambda > 0$ there exists $a_\lambda > a$ such that $(\mathcal{T}^x \mathds{1}_{E_x})(y) \leq \bigl(\mathcal{T}^x (\mathds{1} - h)\bigr)(y) \leq 1-w_\lambda(x)$ for all $y \in [a,a_\lambda)$ (here $\mathds{1}_{E_x}$ denotes the indicator function of $E_x$). We then estimate $$\begin{aligned}
{1 \over t} (\mu_t*\delta_x)(E_x) & = {1 \over t} \int_{[a,b)\!} (\mathcal{T}^x \mathds{1}_{E_x})(y) \mu_t(dy) \\
& \leq {1 \over t} \int_{[a,a_\lambda)\!} \bigl( 1 - w_\lambda(y) \bigr) \mu_t(dy) + {1 \over t} \mu_t [a_\lambda,b) \\
& \leq {1 \over t} \int_{[a,b)\!} \bigl( 1 - w_\lambda(y) \bigr) \mu_t(dy) + {1 \over t} \mu_t [a_\lambda,b) \\
& = {1 \over t} \bigl( 1-\widehat{\mu_t}(\lambda) \bigr) + {1 \over t} \mu_t [a_\lambda,b).\end{aligned}$$ Given that we are assuming that (ii) holds and, by the $\mathcal{L}$-semigroup property, $\lim_{t \downarrow 0} {1 \over t} \bigl( 1-\widehat{\mu_t}(\lambda) \bigr) = \lim_{t \downarrow 0} {1 \over t} \bigl( 1-\widehat{\mu_1}(\lambda)^t \bigr) = -\log\widehat{\mu_1}(\lambda)$, the above inequality gives $$\limsup_{t \downarrow 0} {1 \over t} (\mu_t*\delta_x)(E_x) \leq -\log\widehat{\mu_1}(\lambda).$$ This holds for arbitrary $\lambda > 0$. Since the right-hand side is continuous and vanishes for $\lambda = 0$, we conclude that $\lim_{t \downarrow 0} {1 \over t} (\mu_t*\delta_x)(E_x) = 0$, as desired.\
***(iii)$\!\implies\!$(iv):*** This follows from a general result in the theory of Feller processes [@ethierkurtz1986 Chapter 4, Proposition 2.9] according to which $\lim_{t \downarrow 0} {1 \over t} P_x[Y_t \in [a,b) \setminus \mathcal{V}_x] = 0$ is a sufficient condition for a given $[a,b)$-valued Feller process $Y$ to have continuous paths.\
***(iv)$\!\implies\!$(iii):*** This is a consequence of Ray’s theorem for one-dimensional Markov processes, which is stated and proved in [@ito2006 Theorem 5.2.1].\
Finally, it is well-known that Markov processes with continuous paths have local generators (see e.g. [@ito2006 Theorem 5.1.1]), thus the last assertion holds.
To finish this section, it is worth mentioning that analogues of the classical limit theorems — such as laws of large numbers or central limit theorems — can be established for the $\mathcal{L}$-convolution measure algebra. As in the setting of hypergroup convolution structures (cf. Example \[exam:SLhypergr\]), solutions $\{\varphi_k\}_{k \in \mathbb{N}}$ of the functional equation $$(\mathcal{T}^y \varphi_k)(x) = \sum_{j=0}^k \binom{k}{j} \varphi_j(x) \varphi_{k-j}(y) \quad\; \bigl(x,y \in [a,b)\bigr), \qquad\; \varphi_0 = 0,$$ which are called *$\mathcal{L}$-moment functions*, play a role similar to that of the monomials under the ordinary convolution.
For the sake of illustration, let us state some strong laws of large numbers which hold true for the $\mathcal{L}$-convolution: let $\{S_n\}$ be an $\mathcal{L}$-additive Markov chain constructed as in , and define the $\mathcal{L}$-moment functions of first and second order by $\varphi_1(x) = \kappa \eta_1(x)$, $\varphi_2(x) = 2[\kappa \eta_2(x) + \eta_1(x)]$ respectively, where $\kappa := \lim_{\xi \to \infty} {A'(\xi) \over A(\xi)} = \lim_{x \uparrow b} {[(pr)^{1/2}]'\!(x) \over 2r(x)}$ and the $\eta_j$ are given by . Then:\
1. *If $\{r_n\}_{n \in \mathbb{N}}$ is a sequence of positive numbers such that $\lim_n r_n = \infty$ and $\sum_{n=1}^\infty {1 \over r_n} \bigl(\mathbb{E}[\varphi_2(X_n)] - \mathbb{E}[\varphi_1(X_n)]^2\bigr) < \infty$, then $$\lim_n {1 \over \sqrt{r_n}} \bigl( \varphi_1(S_n) - \mathbb{E}[\varphi_1(S_n)] \bigr) = 0 \qquad\;\; \bm{\pi}\text{-a.s.}$$*
2. *If $\{S_n\}$ is an $\mathcal{L}$-random walk such that $\mathbb{E}[\varphi_2(X_1)^{\theta/2}] < \infty$ for some $1 \leq \theta < 2$, then $\mathbb{E}[\varphi_1(X_1)] < \infty$ and $$\lim_n {1 \over n^{1 / \theta}} \bigl(\varphi_1(S_n) - n\mathbb{E}[\varphi_1(X_1)]\bigr) = 0 \qquad\;\; \bm{\pi}\text{-a.s.}$$*
3. *Suppose that $\varphi_1 \equiv 0$. If $\{r_n\}_{n \in \mathbb{N}}$ is a sequence of positive numbers such that $\lim_n r_n = \infty$ and $\sum_{n=1}^\infty {1 \over r_n} \mathbb{E}[\varphi_2(X_n)] < \infty$, then $$\lim_n {1 \over r_n} \varphi_2(S_n) = 0 \qquad\;\; \bm{\pi}\text{-a.s.}$$*
4. *Suppose that $\varphi_1 \equiv 0$. If $\{S_n\}$ is an $\mathcal{L}$-random walk such that $\mathbb{E}[\varphi_2(X_1)^{\theta}] < \infty$ for some $0 < \theta < 1$, then $$\lim_n {1 \over n^{1 / \theta}} \varphi_2(S_n) = 0 \qquad\;\; \bm{\pi}\text{-a.s.}$$*
The above assertions can be proved exactly as in the hypergroup framework: the reader is referred to [@zeuner1992 Section 7].
Examples {#sec:examples}
========
We begin with two simple examples where the Sturm-Liouville operator is regular and nondegenerate, and the kernel of the $\mathcal{L}$-transform can be written in terms of elementary functions.
Consider the Sturm-Liouville operator $$\ell = - {d^2 \over dx^2}, \qquad 0 < x < \infty$$ which is obtained by setting $p = r = \mathds{1}$ and $(a,b) = (0,\infty)$. This operator trivially satisfies assumption \[asmp:shypPDE\_SLhyperg\]. Since the solution of the Sturm-Liouville boundary value problem is $w_\lambda(x) = \cos(\tau x)$ (where $\lambda = \tau^2$), the $\mathcal{L}$-transform is simply the cosine Fourier transform $(\mathcal{F} h)(\tau) = \int_0^\infty h(x) \cos(\tau x) dx$. By elementary trigonometric identities, $w_\tau(x) w_\tau(y) = {1 \over 2} [w_\tau(|x-y|) + w_\tau(x+y)]$, hence the $\mathcal{L}$-convolution is given by $$\delta_x * \delta_y = {1 \over 2}(\delta_{|x-y|} + \delta_{x+y}), \qquad x,y \geq 0.$$ In other words, $*$ is (up to identification) the ordinary convolution of symmetric measures.
If we let $p(x) = r(x) = (1+x)^2$ and $(a,b) = (0,\infty)$, we obtain the differential operator $$\ell = - {d^2 \over dx^2} - {2 \over 1+x} {d \over dx}, \qquad 0 < x < \infty,$$ which satisfies Assumption \[asmp:shypPDE\_SLhyperg\] with $\eta(x) = {2 \over 1+x}$. The function $$w_\lambda(x) = \begin{cases}
{1 \over 1+x} [\cos(\tau x) + {1 \over \tau} \sin(\tau x)], & \tau > 0 \\
1, & \tau = 0
\end{cases} \qquad (\lambda = \tau^2)$$ is the solution of the boundary value problem , thus the $\mathcal{L}$-transform can be expressed as a sum of cosine and sine Fourier transforms. A straightforward computation [@zeuner1992 Example 4.10] shows that the product formula $w_\lambda(x) \, w_\lambda(y) = \int_{[a,b)} w_\lambda\, d(\delta_x * \delta_y)$ holds for $\delta_x * \delta_y$ defined by $$\label{eq:exampl_squarehypgr}
(\delta_x * \delta_y)(d\xi) = {1 \over 2(1+x)(1+y)}\bigl[(1+|x-y|)\delta_{x-y}(d\xi) + (1+x+y)\delta_{x+y}(d\xi) + (1+\xi)\mathds{1}_{[|x-y|,x+y]}(\xi) d\xi \bigr]$$ and therefore (by the uniqueness property, Proposition \[prop:shypPDE\_Ltransfmeas\_props\](b)) the $\mathcal{L}$-convolution is given by . This example, which was introduced in [@zeuner1992 Example 4.10], illustrates that, in general, convolutions associated with regular Sturm-Liouville operators have both a discrete and an absolutely continuous component.
Next we present the chief example of a convolution associated with a singular Sturm-Liouville operator:
\[exam:hankelkingman\] Let $\alpha \geq -{1 \over 2}$. The Bessel operator $$\ell = - {d \over dx^2} - {2\alpha + 1 \over x} {d \over dx}, \qquad 0 < x < \infty$$ has coefficients $p(x) = r(x) = x^{2\alpha+1}$. Clearly, Assumption \[asmp:shypPDE\_SLhyperg\] holds with $\eta = 0$. Here the kernel of the $\mathcal{L}$-transform is $$w_\lambda(x) = \bm{J}_\alpha(\tau x) := 2^\alpha \Gamma(\alpha+1) (\tau x)^{-\alpha} J_\alpha(\tau x) \qquad (\lambda = \tau^2)$$ where $J_\alpha$ is the Bessel function of the first kind (this is easily checked using the basic properties of the Bessel function, cf. [@dlmf Chapter 10]). The Sturm-Liouville type transform associated with the Bessel operator is the *Hankel transform*, $(\mathcal{F}h)(\tau) = \int_0^\infty h(x) \, \bm{J}_\alpha(\tau x) \, x^{2\alpha + 1} dx$. It follows from classical integration formulae for the Bessel function [@watson1944 p. 411] that $\bm{J}_\alpha(\tau x) \, \bm{J}_\alpha(\tau y) = \int_0^\infty \bm{J}_\alpha(\tau \xi) \, (\delta_x *_\alpha \delta_y)(d\xi)$, where $$(\delta_x *_\alpha \delta_y)(d\xi) = {2^{1-2\alpha} \Gamma(\alpha+1) \over \sqrt{\pi} \, \Gamma(\alpha + {1 \over 2})} (xy\xi)^{-2\alpha} \bigl[ (\xi^2 - (x-y)^2) ((x+y)^2 - \xi^2) \bigr]^{\alpha - 1/2} \mathds{1}_{[|x-y|,x+y]}(\xi) \, r(\xi) d\xi$$ for $x, y > 0$; this convolution is known as the *Hankel convolution* [@hirschman1960; @cholewinski1965] or *Kingman convolution* [@kingman1963; @urbanik1988].
This example has motivated the development of the theory of generalized translation and convolution operators back since the pioneering work of Delsarte [@delsarte1938]. It plays a special role in the context of the Sturm-Liouville hypergroups in Example \[exam:SLhypergr\] below; in particular, it appears as the limit distribution in central limit theorems on hypergroups [@bloomheyer1994 Section 7.5]. Moreover, since the diffusion ($\mathcal{L}$-Lévy) process generated by $\ell$ is the Bessel process — a fundamental continuous-time stochastic process [@borodinsalminen2002], which in the case $\alpha = {d \over 2} - 1$ ($d \in \mathbb{N}$) can be defined as the radial part of a $d$-dimensional Brownian motion — the Hankel convolution is a useful tool for the study of the Bessel process, cf. e.g. [@rentzschvoit2000; @vanthu2007].
The Jacobi operator provides another example of a singular Sturm-Liouville operator whose the product formula and convolution can be written in terms of standard special functions.
\[exam:fourjacobi\] The coefficients $p(x) = r(x) = (\sinh x)^{2\alpha + 1} (\cosh x)^{2\beta + 1}$ ($\alpha \geq \beta \geq -{1 \over 2}$, $\alpha \neq {1 \over 2}$) give rise to the Jacobi operator $$\ell = - {d \over dx^2} - [(2\alpha + 1) \coth x + (2\beta + 1) \tanh x] {d \over dx}, \qquad 0 < x < \infty.$$ As in the previous example, Assumption \[asmp:shypPDE\_SLhyperg\] holds with $\eta = 0$. The so-called Jacobi function $$w_\lambda(x) = \phi_\tau^{(\alpha,\beta)\!}(x) := {}_2F_1\Bigl(\tfrac{1}{2}(\sigma - i\tau), \tfrac{1}{2}(\sigma + i\tau); \alpha + 1; -(\sinh x)^2\Bigr) \qquad (\sigma = \alpha + \beta + 1,\; \lambda = \tau^2 + \sigma^2)$$ where ${}_2F_1$ denotes the hypergeometric function [@dlmf Chapter 15], can be shown to be the unique solution of the Sturm-Liouville problem . The associated integral transform is the *(Fourier-)Jacobi transform*, $(\mathcal{F}h)(\tau) = \int_0^\infty h(x) \, \phi_\tau^{(\alpha,\beta)\!}(x) \, (\sinh x)^{2\alpha + 1} (\cosh x)^{2\beta + 1} dx$ (this transformation is also known as Olevskii transform, index hypergeometric transform or, in the case $\alpha = \beta$, generalized Mehler-Fock transform [@yakubovich2006]). By a deep result of Koornwinder [@flenstedjensen1973; @koornwinder1984], the product formula $\phi_\tau^{(\alpha,\beta)\!}(x) \, \phi_\tau^{(\alpha,\beta)\!}(y) = \int_0^\infty \phi_\tau^{(\alpha,\beta)} d(\delta_x *_{\alpha,\beta} \delta_y)$ holds for the *Jacobi convolution*, defined by $$\begin{aligned}
(\delta_x *_{\alpha,\beta} \delta_y)(d\xi) = \, & {2^{-2\sigma} \Gamma(\alpha+1) (\cosh x \, \cosh y \, \cosh \xi)^{\alpha - \beta - 1} \over \sqrt{\pi}\, \Gamma(\alpha + {1 \over 2}) (\sinh x \, \sinh y \, \sinh \xi)^{2\alpha}} \times \\
& \times (1-Z^2)^{\alpha - 1/2} {\,}_2F_1\Bigl( \alpha + \beta, \alpha - \beta; \alpha + \tfrac{1}{2}; \tfrac{1}{2}(1-Z) \Bigr) \mathds{1}_{[|x-y|,x+y]}(\xi) r(\xi) d\xi\end{aligned}$$ where $Z := {(\cosh x)^2 + (\cosh y)^2 + (\cosh \xi)^2 - 1 \over 2\cosh x \, \cosh y \, \cosh \xi}$.
For half-integer values of the parameters $\alpha, \beta$, the Jacobi transform and convolution have various group theoretic interpretations; in particular, they are related with harmonic analysis on rank one Riemannian symmetric spaces [@koornwinder1984]. Moreover, a remarkable property of the Jacobi transform is that it admits a positive dual convolution structure, that is, there exists a family $\{\theta_{\tau_1,\tau_2}\}$ of finite positive measures such that the dual product formula $\phi_{\tau_1}^{(\alpha,\beta)\!}(x) \, \phi_{\tau_2}^{(\alpha,\beta)\!}(x) = \int_0^\infty \phi_{\tau_3}^{(\alpha,\beta)\!}(x) \, \theta_{\tau_1,\tau_2}(d\tau_3)$ holds, and this permits the construction of a generalized convolution which trivializes the inverse Jacobi transform [@bensalem1994].
All the examples presented so far belong to the class of Sturm-Liouville hypergroup convolutions which was introduced by Zeuner [@zeuner1992] as follows:
\[exam:SLhypergr\] Consider a Sturm-Liouville operator on the positive half-line with coefficients $p = r = A$, $$\ell = - {d^2 \over dx^2} - {A'(x) \over A(x)} {d \over dx}, \qquad 0 < x < \infty,$$ where the function $A$ satisfies the following conditions:
1. $A \in \mathrm{C}[0,\infty) \cap \mathrm{C}^1(0,\infty)$ and $A(x) > 0$ for $x > 0$.
2. One of the following assertions holds:
1. $A(0) = 0$ and ${A'(x) \over A(x)} = {\alpha_0 \over x} + \alpha_1(x)$ for $x$ in a neighbourhood of $0$, where $\alpha_0 > 0$ and $\alpha_1 \in \mathrm{C}^\infty(\mathbb{R})$ is an odd function;
2. $A(0) > 0$ and $A \in \mathrm{C}^1[0,\infty)$.
3. There exists $\eta \in \mathrm{C}^1[0,\infty)$ such that $\eta \geq 0$, $\bm{\phi}_\eta \geq 0$ and the functions $\bm{\phi}_\eta$, $\bm{\psi}_\eta$ are both decreasing on $(0,\infty)$ ($\bm{\phi}_\eta$, $\bm{\psi}_\eta$ are defined as in Assumption \[asmp:shypPDE\_SLhyperg\]).
The last condition ensures that $A$ satisfies Assumption \[asmp:shypPDE\_SLhyperg\], hence this is a particular case of the general family of Sturm-Liouville operators considered in the previous sections. It was proved by Zeuner [@zeuner1992] that the convolution measure algebra $(\mathcal{M}_\mathbb{C}[0,\infty),*)$ is a commutative hypergroup with identity involution; this means that the Banach algebra property of Proposition \[prop:shypPDE\_conv\_Mbanachalg\] and properties (b)–(c) of Corollary \[cor:shypPDEconv\_basicprops\] hold, as well as the following axioms:
- $(x,y) \mapsto {\mathrm{supp}}(\delta_x * \delta_y)$ is continuous from $[0,\infty) \times [0,\infty)$ into the space of compact subsets of $[0,\infty)$ (endowed with the Michael topology, see [@jewett1975]);
- $0 \in {\mathrm{supp}}(\delta_x * \delta_y)$ if and only if $x = y$.
Observe that the Sturm-Liouville operator $\ell = -{d^2 \over dx^2} - {A' \over A} {d \over dx}$ is either singular or regular, depending on whether the function $A$ satisfies condition SL1.1 or SL1.2. In any event, the associated hyperbolic equation $\ell_x f = \ell_y f$ is uniformly hyperbolic on $[0,\infty)^2$. The construction of the product formula and convolution presented in the previous sections generalizes that of Zeuner because it is also applicable to parabolically degenerate operators.
The next example shows that the two hypergroup axioms on the (compact) support of $\delta_x * \delta_y$ are generally false for operators associated with degenerate hyperbolic equations:
\[exam:whittaker\] The choice $p(x) = x^{2-2\alpha} e^{-1/x}$ and $r(x) = x^{-2\alpha} e^{-1/x}$, with $\alpha < {1 \over 2}$, leads to the normalized Whittaker operator $$\ell = - x^2 {d^2 \over dx^2} - (1+2(1-\alpha)x) {d \over dx}, \qquad 0 < x < \infty.$$ The standard form of this differential operator (Remark \[rmk:shypPDE\_tildeell\]) is $\widetilde{\ell} = - {d^2 \over dz^2} - (e^{-z} + 1 - 2\alpha){d \over dz}$, where $z = \log x \in \mathbb{R}$, and it is apparent that Assumption \[asmp:shypPDE\_SLhyperg\] holds with $\eta = 0$. As pointed out in Section \[sec:hypPDE\], the fact that the operator $\widetilde{\ell}$ is defined on the whole real line means that the hyperbolic partial differential equation associated with the normalized Whittaker operator has a non-removable parabolic degeneracy at the initial line. The unique solution of the boundary value problem turns out to be given by $$w_\lambda(x) = \bm{W}_{\!\!\alpha,i\tau}(x) := x^\alpha e^{1 \over 2x} W_{\alpha, i\tau}(\tfrac{1}{x}) \qquad \bigl( \lambda = \tau^2 + (\tfrac{1}{2} - \alpha)^2 \bigr)$$ where $W_{\alpha,i\tau}$ is the Whittaker function of the second kind of parameters $\alpha$ and $i\tau$ [@dlmf Chapter 13]. The eigenfunction expansion of the normalized Whittaker operator yields the *index Whittaker transform* [@srivastava1998; @sousayakubovich2018] $(\mathcal{F}h)(\tau) = \int_0^\infty h(x) \bm{W}_{\!\!\alpha,i\tau}(x) x^{-2\alpha} e^{-1/x} dx$. The product formula for the kernel $\bm{W}_{\!\!\alpha,i\tau}$ has recently been established by the authors [@sousaetal2018a; @sousaetal2018b] using techniques from classical analysis and known facts in the theory of special functions; it is given by $\bm{W}_{\!\!\alpha,i\tau}(x) \bm{W}_{\!\!\alpha,i\tau}(y) = \int_0^\infty \bm{W}_{\!\!\alpha,i\tau} \, d(\delta_x *_\alpha \delta_y)$, where $*_\alpha$ is the *Whittaker convolution*, defined by $$(\delta_x *_\alpha \delta_y)(d\xi) = {2^{-1-\alpha} \over \sqrt{\pi}} (xy\xi)^{-{1\over 2}+\alpha} \exp\Bigl( {1 \over x} + {1 \over y} + {1 \over \xi} - {(x+y+\xi)^2 \over 8xy\xi} \Bigr) D_{2\alpha}\Bigl( {x+y+\xi \over \smash{\sqrt{2xy\xi}}} \Bigr) r(\xi) d\xi$$ for $x,y > 0$, with $D_\mu$ denoting the parabolic cylinder function [@erdelyiII1953 Chapter VIII]. Notice in particular that ${\mathrm{supp}}(\delta_x *_\alpha \delta_y) = [0,\infty)$ for every $x, y > 0$.
The particular case $\alpha = 0$ is worthy of special mention, because in this case the index Whittaker transform reduces to $(\mathcal{F}h)(\tau) = \pi^{-1/2} \int_0^\infty h(x) K_{i\tau}({1 \over 2x}) x^{-1/2} e^{-{1 \over 2x}} dx$, which is (a normalized form of) the Kontorovich-Lebedev transform; here $K_{i\tau}$ is the modified Bessel function of the second kind with parameter $i\tau$ [@dlmf Chapter 10]. The Kontorovich-Lebedev transform plays a central role in the theory of index type integral transforms [@yakubovich1996]. The Whittaker convolution of parameter $\alpha = 0$, which can be written in the simplified form $$(\delta_x *_0 \delta_y)(d\xi) = {1 \over 2\sqrt{\pi x y \xi}} \exp\Bigl( {1 \over x} + {1 \over y} - {(x+y+\xi)^2 \over 4xy\xi} \Bigr) d\xi,$$ is identical (up to an elementary change of variables) to the Kontorovich-Lebedev convolution, which was introduced by Kakichev in [@kakichev1967] and has been extensively studied, cf. [@yakubovich1996] and references therein.
Our final example illustrates that the (degenerate) hyperbolic equation approach allows us to generalize the results on the Whittaker product formula and convolution to a much larger class of degenerate operators:
Let $\bm{\zeta} \in \mathrm{C}^1(0,\infty)$ be a nonnegative decreasing function such that $\int_1^\infty \bm{\zeta}(y) {dy \over y} = \infty$, and let $\kappa > 0$. The differential expression $$\ell = - x^2 {d^2 \over dx^2} - \bigl[\kappa + x \bigl(1 + \bm{\zeta}(x)\bigr)\bigr] {d \over dx}, \qquad 0 < x < \infty$$ is a particular case of , obtained by considering $p(x) = x e^{-\kappa/x + I_{\bm{\zeta}}(x)}$ and $r(x) = {1 \over x} e^{-\kappa/x + I_{\bm{\zeta}}(x)}$, where $I_{\bm{\zeta}}(x) = \int_1^x \bm{\zeta}(y) {dy \over y}$. (If $\kappa = 1$ and $\bm{\zeta}(x) = 1-2\alpha > 0$, we recover the normalized Whittaker operator from Example \[exam:whittaker\].) The change of variable $z = \log x \in \mathbb{R}$ transforms $\ell$ into the standard form $\widetilde{\ell} = - {d^2 \over dz^2} - {A'(z) \over A(z)}{d \over dz}$, where ${A'(z) \over A(z)} = \kappa e^{-\kappa z} + \bm{\zeta}(e^z)$. It is clear that $\ell$ satisfies Assumption \[asmp:shypPDE\_SLhyperg\] with $\eta = 0$, and the additional assumption $\lim_{x \uparrow b} p(x)r(x) = \infty$ holds because $I_{\bm{\zeta}}(\infty) = \infty$. Therefore, all the results in the previous sections hold for the Sturm-Liouville operator $\ell$. This shows that the class of Sturm-Liouville operators for which one can construct a positivity-preserving convolution structure includes irregular operators which are simultaneously degenerate (in the sense that the associated hyperbolic equation is parabolic at the initial line) and singular (in the sense that the first order coefficient is unbounded near the left endpoint).
Acknowledgements {#acknowledgements .unnumbered}
================
The first and third authors were partly supported by CMUP (UID/MAT/00144/2019), which is funded by Fundação para a Ciência e a Tecnologia (FCT) (Portugal) with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020, and Project STRIDE – NORTE-01-0145-FEDER-000033, funded by ERDF – NORTE 2020. The first author was also supported by the grant PD/BD/135281/2017, under the FCT PhD Programme UC|UP MATH PhD Program. The second author was partly supported by the project CEMAPRE – UID/MULTI/00491/2013 financed by FCT/MCTES through national funds.
[9]{}
H. Bauer, [*Probability Theory*](http://dx.doi.org/10.1515/9783110814668), Walter De Gruyter, Berlin (1996). H. Bauer, [*Measure and Integration Theory*](http://dx.doi.org/10.1515/9783110866209), Walter De Gruyter, Berlin (2001). N. Ben Salem, [*Convolution semigroups and central limit theorem associated with a dual convolution structure*](http://dx.doi.org/10.1007/BF02214276), J. Theor. Probab. **7**, no. 2, pp. 417–436 (1994). C. Berg, G. Forst, [*Potential Theory on Locally Compact Abelian Groups*](http://dx.doi.org/10.1007/978-3-642-66128-0), Springer, Berlin (1975). A. V. Bitsadze, [*Equations of the Mixed Type*](http://dx.doi.org/10.1016/C2013-0-01727-6), Pergamon Press, Oxford, 1964. W. R. Bloom, H. Heyer, [*Harmonic Analysis of Probability Measures on Hypergroups*](http://dx.doi.org/10.1515/9783110877595), Walter de Gruyter, Berlin (1994). V. I. Bogachev, [*Measure Theory. Vol. II*](http://dx.doi.org/10.1007/978-3-540-34514-5), Springer, Berlin (2007). A. N. Borodin, P. Salminen, [*Handbook of Brownian Motion: Facts and Formulae*](http://dx.doi.org/10.1007/978-3-0348-8163-0), Springer, Basel (2002). B. Böttcher, R. Schilling, J. Wang, [*Lévy-Type Processes: Construction, Approximation and Sample Path Properties*](http://dx.doi.org/10.1007/978-3-319-02684-8), in Springer Lecture Notes in Mathematics vol. 2099 (vol. III of the “Lévy Matters" subseries), Springer, Berlin (2014). R. W. Carroll, [*Transmutation theory and applications*](http://dx.doi.org/10.1016/S0304-0208(08)72271-0), North-Holland, Amsterdam (1985). H. Chebli, [*Opérateus de translation généralisée et semi-groupes de convolution*](http://dx.doi.org/10.1007/BFb0060609), in *Théorie du Potentiel et Analyse Harmonique* (J. Faraut, editor), Springer, Berlin, pp. 35–59 (1974). H. Chebli, [*Sturm-Liouville Hypergroups*](http://dx.doi.org/10.1090/conm/183/02055), in: *Applications of hypergroups and related measure algebras: A joint summer research conference on applications of hypergroups and related measure algebras, July 31-August 6, 1993, Seattle, WA*, American Mathematical Society, Providence RI, pp. 71–88 (1995). F. M. Cholewinski, *A Hankel convolution complex inversion theory*, Mem. Amer. Math. Soc. no. 58, American Mathematical Society (1965) D. L. Cohn, [*Measure Theory*](http://dx.doi.org/10.1007/978-1-4614-6956-8), Second Edition, Birkhäuser, New York (2013). W. C. Connett, C. Markett, A. L. Schwartz, [*Convolution and hypergroup structures associated with a class of Sturm-Liouville systems*](http://dx.doi.org/10.2307/2154037), Trans. Amer. Math. Soc. **332**, no. 1, pp. 365–390 (1992). R. Courant, *Methods of Mathematical Physics – Vol. II: Partial Differential Equations*, Wiley, New York (1962). J. Delsarte, *Sur une extension de la formule de Taylor*, J. Math. Pures Appl. **17**, pp. 213–231 (1938). N. Dunford, J. T. Schwartz, *Linear Operators – Part II: Spectral Theory*, Wiley, New York (1963). J. Eckhardt, G. Teschl, [*Sturm-Liouville operators with measure-valued coefficients*](http://dx.doi.org/10.1007/s11854-013-0018-x), J. Anal. Math. **120**, pp. 151–224 (2013). A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, *Higher Transcendental Functions – Vol. II*, McGraw–Hill, New York (1953). S. N. Ethier, T. G. Kurtz, [*Markov Processes – Characterization and Convergence*](http://dx.doi.org/10.1002/9780470316658), Wiley, New York (1986). M. Flensted-Jensen, T. H. Koornwinder, [*The convolution structure for Jacobi function expansions*](http://dx.doi.org/10.1007/BF02388521), Ark. Mat. **11**, pp. 245–262 (1973). F. Fruchtl, [*Sturm-Liouville hypergroups and asymptotics*](http://dx.doi.org/10.1007/s00605-017-1048-8), Monatsh. Math. **186**, no. 1, pp. 11–36 (2018). M. Fukushima, [*On general boundary conditions for one-dimensional diffusions with symmetry*](http://dx.doi.org/10.2969/jmsj/06610289), J. Math. Soc. Japan **66**, no. 1, pp. 289–316 (2014). I. I. Hirschman, [*Variation diminishing Hankel transforms*](http://dx.doi.org/10.1007/BF02786854), J. Analyse Math. **8**, pp. 307–336 (1960). K. Itô, [*Essentials of Stochastic Processes*](http://dx.doi.org/10.1090/mmono/231), American Mathematical Society, Providence RI (2006). R. I. Jewett, [*Spaces with an Abstract Convolution of Measures*](http://dx.doi.org/10.1016/0001-8708(75)90002-X), Advances in Mathematics **18**(1), pp. 1–101 (1975). I. S. Kac, [*The existence of spectral functions of generalized second order differential systems with boundary conditions at the singular end*](http://dx.doi.org/10.1090/trans2/062/04), Amer. Math. Soc. Transl. (2) **62**, pp. 204–262 (1967). V. A. Kakichev, *On the convolution for integral transforms*, Izv Vyssh Uchebn Zaved Mat. **2**, pp. 53–62 (1967) (in Russian). J. F. C. Kingman, [*Random walks with spherical symmetry*](http://dx.doi.org/10.1007/BF02391808), Acta Math. **109**, pp. 11–53 (1963). A. Klenke, *Probability Theory – A Comprehensive Course*, Second Edition, Springer, London (2014). T. H. Koornwinder, [*Jacobi functions and analysis on noncompact semisimple Lie groups*](http://dx.doi.org/10.1007/978-94-010-9787-1_1), in *Special functions: group theoretical aspects and applications* (R. A. Askey, T. H. Koornwinder, W. Schempp, editors), Reidel, Dordrecht, pp. 1–85 (1984). B. M. Levitan, *Die Verallgemeinerung der Operation der Verschiebung im Zusammenhang mit fastperiodischen Funktionen*, Mat. Sb. **7**, no. 49, pp. 449–478 (1940). B. M. Levitan, *On a class of solutions of the Kolmogorov-Smoluchowski equation*, Vestnik Leningrad. Univ. **15**, no. 7, pp. 81–115 (1960) (in Russian). V. Linetsky, [*The spectral decomposition of the option value*](http://dx.doi.org/10.1142/S0219024904002451), Int. J. Theor. Appl. Finance **7**, no. 3, pp. 337–384 (2004). J. V. Linnik, I. V. Ostrovskiǐ, [*Decomposition of Random Variables and Vectors*](http://dx.doi.org/10.1090/mmono/048), American Mathematical Society, Providence RI (1977). G. L. Litvinov, [*Hypergroups and hypergroup algebras*](http://dx.doi.org/10.1007/BF01088201), J. Soviet Math. **38**, no. 2, pp. 1734–1761 (1987). N. K. Mamadaliev, [*On representation of a solution to a modified Cauchy problem*](http://dx.doi.org/10.1007/BF02674745), Sib. Math. J. **41**, no. 5, pp. 889–899 (2000). P. Mandl, *Analytical Treatment of One-dimensional Markov Processes*, Springer, Berlin (1968). H. McKean, [*Elementary solutions for certain parabolic partial differential equations*](http://dx.doi.org/10.1090/S0002-9947-1956-0087012-3), Transactions of the American Mathematical Society **82**, 519-548 (1956). M. A. Naimark, *Linear differential operators. Part II: Linear differential operators in Hilbert space*, Frederick Ungar Publishing Co., New York, 1968. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (Eds.), *NIST Handbook of Mathematical Functions*, Cambridge University Press, Cambridge, 2010. E. V. Radkevich, [*Equations with nonnegative characteristic form II*](http://dx.doi.org/10.1007/s10958-009-9395-1), J. Math. Sci. **158**, pp. 453-604 (2009). C. Rentzsch, M. Voit, [*Lévy Processes on Commutative Hypergroups*](http://dx.doi.org/10.1090/conm/261/04135), in: *Probability on Algebraic Structures: AMS Special Session on Probability on Algebraic Structures, March 12-13, 1999, Gainesville, Florida*, American Mathematical Society, Providence RI, pp. 83–105 (2000). M. Rösler, [*Convolution algebras which are not necessarily positivity-preserving*](http://dx.doi.org/10.1090/conm/183/02068), in: *Applications of hypergroups and related measure algebras: A joint summer research conference on applications of hypergroups and related measure algebras, July 31-August 6, 1993, Seattle, WA*, American Mathematical Society, Providence RI, pp. 71–88 (1995). R. Sousa, M. Guerra, S. Yakubovich, [*On the product formula and convolution associated with the index Whittaker transform*](https://arxiv.org/pdf/1802.06657.pdf), Preprint, arXiv:1802.06657 (2018). R. Sousa, M. Guerra, S. Yakubovich, [*Lévy processes with respect to the index Whittaker convolution*](https://arxiv.org/pdf/1805.03051.pdf), Preprint, arXiv:1805.03051 (2018). R. Sousa, M. Guerra, S. Yakubovich, *Sturm-Liouville hypergroups without the compactness axiom*, Preprint (2019). R. Sousa, S. Yakubovich, [*The spectral expansion approach to index transforms and connections with the theory of diffusion processes*](http://dx.doi.org/10.3934/cpaa.2018112), Commun. Pure Appl. Anal. **17**, no. 6, pp. 2351–2378 (2018). H. M. Srivastava, Y. V. Vasil’ev, S. Yakubovich, *A class of index transforms with Whittaker’s function as the kernel*, [Quart. J. Math. Oxford **49**(2), 375-394 (1998)](http://dx.doi.org/10.1093/qmathj/49.3.375). G. Teschl, [*Mathematical methods in quantum mechanics. With applications to Schrödinger operators*](http://dx.doi.org/10.1090/gsm/157), Second Edition, American Mathematical Society, Providence RI (2014). E. C. Titchmarsh, *Eigenfunction Expansions Associated with Second-Order Differential Equations*, Second Edition, Oxford University Press, Oxford (1962). K. Urbanik, [*Generalized convolutions*](http://dx.doi.org/10.4064/sm-23-3-217-245), Studia Math. **23**, pp. 217–245 (1964). K. Urbanik, [*Analytical Methods in Probability Theory*](http://dx.doi.org/10.1007/978-94-009-3859-5_11), in: *Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol. A*, Reidel, Dordrecht, pp. 151–163 (1988). N. Van Thu, S. Ogawa, M. Yamazato, [*A convolution approach to multivariate Bessel processes*](http://dx.doi.org/10.1142/9789812770448_0014), in: *Stochastic processes and applications to mathematical finance, Proceedings of the 6th Ritsumeikan International Symposium*, World Scientific, Singapore, pp. 233–244 (2007). V. E. Volkovich, [*Infinitely divisible distributions in algebras with stochastic convolution*](http://dx.doi.org/10.1007/BF01083639), Journal of Soviet Mathematics **40**, no. 4, pp. 459–467 (1988). V. E. Volkovich, [*On Symmetric Stochastic Convolutions*](http://dx.doi.org/10.1007/BF01060427), Journal of Theoretical Probability **5**, no. 3, pp. 417–430 (1992). G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, Cambridge University Press, Cambridge (1944) J. Weidmann, [*Spectral Theory of Ordinary Differential Operators*](http://dx.doi.org/10.1007/BFb0077960), Springer, Berlin (1987). H. Weinberger, [*A maximum property of Cauchy’s problem*](http://dx.doi.org/10.2307/1969598), Ann. Math. **64**, no. 2, pp. 505–513 (1956). S. Yakubovich, [*Index Transforms*](http://dx.doi.org/10.1142/9789812831064), World Scientific, Singapore (1996). S. Yakubovich, *On the Plancherel theorem for the Olevskii transform*, Acta Math. Vietnam. **31**, no. 3, pp. 249–260 (2006). H. Zeuner, [*Moment functions and laws of large numbers on hypergroups*](http://dx.doi.org/10.1007/BF02571436), Mathematische Zeitschrift **211**(1), pp. 369–407 (1992).
[^1]: Corresponding author. CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal. Email: `ruben.sousa@outlook.com`
[^2]: CEMAPRE and ISEG (School of Economics and Management), Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisbon, Portugal. Email: `mguerra@iseg.ulisboa.pt`
[^3]: CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal. Email: `syakubov@fc.up.pt`
|
---
abstract: 'We have observed the intermediate regions of the circumstellar envelope of Mira ([*o*]{} Ceti) in photospheric light scattered by three vibration-rotation transitions of the fundamental band of CO, from low-excited rotational levels of the ground vibrational state, at an angular distance of $\beta\sim2\arcsec\ -7\arcsec\,$ away from the star. The data were obtained with the Phoenix spectrometer mounted on the 4 m Mayall telescope at Kitt Peak. The spatial resolution is approximately 0.5 and seeing limited. Our observations provide absolute fluxes, leading to an independent new estimate of the mass-loss rate of approximately $3\times 10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$, as derived from a simple analytic wind model. We find that the scattered intensity from the wind of Mira for 2 $\beta$ [ ]{} 7 decreases as $\beta^{-3}$, which suggests a time constant mass-loss rate, when averaged over 100 years, over the past 1200 years.'
author:
- 'N. Ryde, B. Gustafsson and K. Eriksson'
- 'K. H. Hinkle'
nocite:
- '[@hipp]'
- '[@karovska:93]'
- '[@karovska:97]'
- '[@bowen; @dorfi; @hoefner:98]'
- '[@woitke]'
- '[@planesas:I]'
- '[@marcs:75]'
- '[@par]'
- '[@goor]'
- '[@kirby]'
- '[@goor_chack]'
- '[@hure]'
title: 'Mira’s wind explored in scattering infrared CO lines'
---
INTRODUCTION
============
Ever since the circumstellar lines of the $\alpha$ Her M supergiant were identified in absorption against the photospheric continuum of the G III secondary [@deutsch], there have been numerous attempts to map circumstellar regions. Three spectroscopic tools have been used extensively. The most extensively employed are microwave emission lines. The IR continuum from circumstellar dust, especially in the observations by the IRAS and ISO satellites, has been used. Finally, approximately a dozen circumstellar shells have been imaged in photospheric light scattered by atomic resonance lines. In this paper we shall extend the latter technique to resonance scattering in molecular lines using the mid-infrared vibration-rotation lines of the ubiquitous CO molecule.
In his review of circumstellar envelopes and AGB stars, @habing noted both the usefulness and the uniqueness of measuring scattered resonance lines, for example of Na[i]{} and K[i]{}, but drew attention to the relatively few cases where this method has been applied. This is certainly because the observations and their interpretation are quite complex. There are uncertainties in relating these observations to the mass and structure of the expanding gas since atoms are depleted by dust grains and the details of ionization in the circumstellar shell are not known. An advantage of microwave measurements is that the molecular species sampled may be much less affected by these uncertainties. This advantage is also present in the study of photospheric light scattered by circumstellar CO molecules, as presented in this paper. The spectral energy distributions (SEDs) of AGB stars have a maximum in the infrared which also suggests infrared CO observations. The infrared vibration-rotation lines of CO, as an alternative to rotational CO lines at millimeter wavelengths, also allow the study of regions close to the star and admit higher spatial resolution in single telescope studies.
Pioneering investigations of atomic resonance line scattering were made of the K[i]{} 7699Å circumstellar emission in $\alpha$ Ori by @bernat:75 [@bernat:76]. Recent work includes the study of circumstellar shells around three N-type carbon stars using K[i]{} 7699 Å [@bg:97] and of circumstellar shells of M-type mira stars using various atomic resonance lines [@plez; @gui].
Imaging off-star emission of CO at $4.6\,\mbox{$\mu$m}$ has previously only been done a few times [@sahai:85; @ryde:letter]. @dyck performed speckle interferometry on a number of CO fundamental vibrational-rotational lines. Sahai & Wannier studied the intermediate regions of the circumstellar envelope of the dust-enshrouded and bright IR star CW Leonis (IRC+10216) by using an annular aperture, and determined a kinetic temperature of its shell at an average radius of $2''$. This technique provided only spatially averaged fluxes but led the authors also to the conclusion that the mass-loss rate in the inner parts of the circumstellar shell is less than that corresponding to the region observed in millimeter CO lines. Since the Sahai and Wannier observations, great advances have been made in infrared detectors. It is, however, not trivial to apply infrared arrays in high resolution spectroscopy since at wavelengths longer than about 1.6 $\mu$m thermal radiation from room temperature spectrometers dominates the stellar signals. Cryogenic spectrometers are therefore required, such as the Phoenix spectrometer [@phoenix]. One of the justifications for building this spectrometer was, in fact, to map circumstellar shells as presented in this paper. In @ryde:letter we demonstrated methods of observation and analysis of the vibration-rotation CO emission lines from the intermediate regions of circumstellar winds. Here, we present results of an application of these methods to the circumstellar envelope of the M4-7IIIe giant [*o*]{} Ceti (Mira).
Section 2 discusses Mira. Sections 3 and 4 describe the observational set-up and the reduction procedures, and the absolute flux calibration, respectively. Section 5 reviews the observational results and Section 6 discusses these results in the light of an analytic model for circumstellar envelopes.
MIRA\[mira\]
============
A study of the circumstellar envelope of [*o*]{} Ceti (Mira or HD 14386) is of interest for several reasons. The star is the prototype of the mira-class of long-period variables characterized by large amplitude visual variations. In the case of [*o*]{} Ceti the V-band amplitude is more than 6 magnitudes and the mean period is 332 days. The miras are cool AGB stars with most of the energy emitted in the infrared. The brightness of Mira reflects its closeness, $(128\pm 18)\,\mbox{pc}$ (ESA 1997). The stellar mass loss produces a circumstellar envelope which, at the distance of Mira, has an angular extension of 2 on the sky [@loup]. Note, however that [*o*]{} Ceti departs from many miras in being a binary system. The angular distance between Mira A and the hot, compact companion star Mira B (VZ Ceti) is 0.6$\arcsec$ (Karovska et al. 1993; 1997). The companion could be a white dwarf with a mass of about 1M$_\odot$, a luminosity of 2L$_\odot$, and a temperature of more than 30000K, which is embedded in an accretion disk, giving rise to an abnormal illumination of Mira A [@danchi:94]. Also, note that a number of other high-resolution, optical and infrared wavelength measurements provide evidence that Mira itself is elongated (see discussion by @ryde:2000 and references therein).
Theoretical model calculations (Bowen 1988; Höfner & Dorfi 1997; Höfner et al. 1998; see Woitke (1998) for a review) indicate that the mass loss of these stars is most probably caused by a combination of radial pulsation and radiation pressure on dust grains and/or molecules. It may seem natural to expect this mass loss to be spherically symmetric, but stellar rotation may introduce a latitude dependence [@dorfi;96]. Also, on the scale of the spatial resolution of our observations, corresponding to a time-scale of about 100 years, one could expect the wind to be homogeneous (observations of Mira show the current pulsational behavior dates back to at least 1638).
However, there are indications from radio observations that the wind of Mira departs from this simple picture. There is an asymmetry and a shoulder clearly visible in the radio-line profiles, see for example @planesas:I. There have been several suggestions in the literature of different multi-wind scenarios or other phenomena introducing dramatic variations in the envelope structure with distance in order to fit the profiles. There exist several combinations of mass-loss rate, expansion and turbulent velocities that are able to reproduce the Mira circumstellar line profiles rather well and as a result there is a lack of consensus regarding the actual expansion velocities of Mira’s wind. All these results are based on microwave and sub-millimeter observations of rotational CO lines. @crosas:apss experiment with v$_\mathrm{exp}\approx2$ and v$_\mathrm{turb}\approx4\,\mbox{km\,s$^{-1}$}$ for their inner wind, while @young arrives at v$_\mathrm{exp}=4.8\,\mbox{km\,s$^{-1}$}$ from a fit. However, he points out that the wings show an expansion velocity of $10\,\mbox{km\,s$^{-1}$}$. @knapp:98 model the circumstellar envelope (CSE) with a fast outer wind with a mass-loss rate resembling single wind miras, ${\rm \dot M} = 4.4\cdot 10^{-7}\,\,\mbox{{\rm M$_\odot$
yr$^{-1}$}}$, and an expansion velocity of $(6.7\pm1.0)\,\,\mbox{km\,
s$^{-1}$}$. The inner wind is supposed to be a resumed wind in analogy with the detached shells found in four carbon stars [@ho:96a]. This slow wind component has a lower mass-loss rate and a much lower expansion velocity than what is found normally in miras; ${\rm \dot M}
= 9.4\cdot 10^{-8}\,\,\mbox{{\rm M$_\odot$ yr$^{-1}$}}$ and v$_\mathrm{exp}=(2.4\pm0.4)\,\,\mbox{km\, s$^{-1}$}$. Planesas et al. (1990a) interpret their observations of CO($J=1-0$) and CO($J=2-1$) lines as partly originating from a dominating, spherically symmetric CSE with v$_\mathrm{exp}= 3\,\mbox{km\,s$^{-1}$}$ and a partial collimation of the stellar wind into an additional bipolar lobe.
Obviously it would be of great interest to study whether the peculiarities observed in the microwave lines are also reflected in the infrared CO vibration-rotation lines, which map the intermediate regions (approximately $100-1000$ stellar radii) of Mira’s circumstellar environment.
The geocentric, radial velocity of Mira is large enough for the stellar CO lines to be shifted out of the telluric lines. This is a key factor since the blending of the telluric and stellar lines makes the analysis of the faint circumstellar emission very difficult.
Multi-wavelength angular diameter measurements of Mira [@haniff:95] combined with Hipparcos trigonometric parallaxes [@leeuwen:97] suggest a mean radius of $(464\pm
80)\,\mbox{R$_\odot$}$. Based on the work of @mahler:97 on radius and luminosity variations of Mira from Wing near-IR photometry, one can derive a bolometric luminosity $\mbox{L$_{\mathrm{tot}}$}=8900\,\mbox{L$_\odot$}$, a temperature $\mbox{T$_{\rm{eff}}$}=2400\,\mbox{K}$, and a radius of $550\,\mbox{R$_\odot$}=3.8\cdot 10^{13}\,\mbox{cm}$ for the phase of [*o*]{} Ceti on the date our observations were made, 1998 October 28-29. These values are consistent with the corresponding values of, for example, @danchi:94 and @haniff:95. They arrive at similar temperatures; a mean (over phases) effective temperature of about T$_{\rm{eff}}\approx2800
\mbox{K}$. We note, however, that this value is low as compared with the values suggested by @perrin for giants in the interval M4III-M7III which is characteristic of miras. Assuming a mass of $ {\cal M }\approx1.0\,\mbox{M$_\odot$}$, we find the logarithmic surface gravity, $\log g = \log(\mathrm{G\,M/R_*^2)}$, to be approximately $-1.0$ \[cgs\].
OBSERVATIONAL SET-UP AND REDUCTIONS\[set-up\]
=============================================
The observations were carried out in 1998 on October 28 and 29 using the 4 meter Mayall telescope at Kitt Peak with the Phoenix spectrometer, mounted at the Cassegrain focus of the equatorially mounted telescope. This cryogenic, single-order echelle spectrometer is a long-slit, high spectral resolution instrument, designed for the $1-5\,\mbox{$\mu$m}$ region [@phoenix] and marks a major achievement. At $4.6\,\mbox{$\mu$m}$ it is now possible to observe objects 5 magnitudes fainter than was possible with the Fourier Transform Spectrometer on the same telescope.
The detector is an Aladdin 512$\times$1024 element InSb array cooled to 35 K. The rest of the spectrometer is cooled to 50 K. For our observations, a $30 \arcsec\,$ long and $0.4 \arcsec\,$ (2 pixels) wide entrance slit was used, resulting in a spectral resolution of around $\rm{R}=60\,000$. The slit width projection onto the sky of 0.4 means that the spectral information is an average over, at least, a region of this size. The visual seeing was around $0.8\arcsec\,$, but we note that the seeing disk in the infrared ($4.6\,\mbox{$\mu$m}$) is 60% of that in the visual [@lena], so that the seeing nearly matched the slit size, which is approximately 30% greater than the diffraction limit of the telescope. The dispersion of the spectrograph leads to an over-sampled observed spectrum and each spectrum covers a very small wavelength range; approximately $11\,\,\mbox{cm$^{-1}$}$.
We observed the circumstellar envelope of [*o*]{} Ceti in a hashed configuration, resembling a perpendicular railway crossing with the star in the middle, see Figure \[planesas\]. The long slit was placed in the off-star position at $2 \arcsec\,$ away from the star. We were able to detect emission to a position corresponding to a maximum distance from the star of $7\arcsec\,$. The four positions will overlap at a distance of approximately $\sqrt{2^2+2^2}\arcsec\,$ away from the star, which enables us to make a relative calibration of the observations, for the different slit positions around the star.
We chose an echelle setting for observing low excitation lines of the vibration-rotation, fundamental R-branch of the electronic ground state of $^{12}$C$^{16}$O. The lines selected, 1-0 R(1) (2150.86 cm$^{-1}$, i.e. v$=1\rightarrow 0$ and $J'=2\rightarrow J''=1$), 1-0 R(2) (2154.60 cm$^{-1}$), and 1-0 R(3) (2158.30 cm$^{-1}$), have minimal interference with telluric lines. The potentially important 1-0 R(0) line has an interfering telluric water line. Thus, we observed the region 2150 to 2160 cm$^{-1}$, the spectral coverage being limited by the array length. The spectrometer works in order 12 at these wavelengths.
The CO fundamental bands are located in the thermal infrared where the sky background radiation is high and somewhat variable. Phoenix is not a sky balanced device, which means that the thermal radiation is recorded. At the $4.6\,\mbox{$\mu$m}$ wavelength of CO, the thermal background radiation at Kitt Peak would limit exposure times at $\mathrm{R}=60\,000$ to about ten minutes. However, the $4.6\,\mbox{$\mu$m}$ spectrum has a contribution of telluric lines, mostly CO and H$_2$O, which are seen in emission. To avoid saturation of the telluric lines the exposure time was limited to 60 seconds.
Our observations consist of two types: on-star and off-star exposures. On-star exposures are required to remove scattered light from the off-star exposures. In order to remove the thermal spectrum from the source spectrum, two exposures are required, one of the source and one of the sky [@joyce]. All spectra must also have array pixel sensitivity differences removed using flats and darks as described by @joyce. Since Phoenix has a long slit, nodding along the slit is possible for point sources. Therefore, on-star spectra were taken by moving the star along the slit by $15 \arcsec$ between exposures. The brightness of Mira limited these exposures to $30\,\,\mbox{s}$. Each set of exposures then gave, after differencing, two background-subtracted, on-star spectra. Since the stellar signal is extincted by the optical depth of the telluric lines, the background-subtracted spectra show in absorption the telluric lines which appear in emission in the raw images.
Off-star frames were taken the first night by alternatively observing $2 \arcsec\,$ south and making an identical background observation $62
\arcsec\,$ south of the star, both with an exposure time of $60\,\,\mbox{s}$. [*o*]{} Ceti has a CO shell with a radius of $57
\arcsec$, as measured by radio observations [@loup]. The total effective observing time in the south position was 66 minutes. Corresponding observations were subsequently made in the north position, $2 \arcsec$ N, also here with an exposure time per nod of 60 s. The total effective observing time in the north position was 80 minutes. The second night a more efficient nodding algorithm was employed. Off-star frames were taken by alternatively observing $2 \arcsec$ east and $2 \arcsec$ west, 60 s each, then a background observation was taken $60 \arcsec$ west off the star. A total of 54 effective minutes were observed in both the west and east positions. Over the few minute (at maximum) interval between the source and sky observations the sky background level is almost constant and the airmass changes only slightly. While the thermal radiation and extinction of well-mixed telluric gases, such as CO, are scaled with airmass and time, water is not, on most nights being in the form of clouds visible in the infrared. As a result the telluric water vapor lines will not cancel cleanly in the subsequent differencing of spectra.
The on-star spectra were used to remove the effects of scattered stellar light in the terrestrial atmosphere and in the spectrometer. Thus, these were normalized to unit level and then scaled to the levels of the off-star spectra. The circumstellar CO vibration-rotation emission was obtained by subtracting the scaled on-star spectra from the off-star ones. The emission we wish to detect is about 50 times weaker than the on-star intensity. The on-star intensity is typically about twenty to thirty times brighter than the sky background.
The reduction of the data was performed using standard routines of the latest version of IRAF (Image Reduction and Analysis Facility). The wavelength calibration was made by using the telluric lines in every frame. The accuracy of the wavelength calibration is of the same order as the resolution. Therefore, the uncertainty in the wavelength scale is probably approximately $0.04\,\mbox{cm$^{-1}$}$ or $6\,\mbox{km\,s$^{-1}$}$.
From the ISO data archive[^1] we have retrieved the spectrum around and the flux at $4.6\,\mbox{$\mu$m}$ of Mira, obtained with the Infrared Space Observatory [@kessler]. The reductions were made using the most recent pipeline basic reduction package OLP (v.7) and the ISO Spectral Analysis Package (ISAP v.1.5). The pipeline processing of the data, such as the flux calibration, is described by @sws; the combined absolute and systematic uncertainties in the fluxes are of the order of $\pm$10%.
ABSOLUTE FLUX CALIBRATION\[marcs\]
==================================
The absolute calibration of the spectra in the $4\,\mu$m region was made by a comparison of the measured on-star flux (in counts) with the absolute flux (in physical units) expected in the broad M-band. The conversion factors obtained are then used for calibrating the off-star spectra. This method contains a number of uncertain steps, related to the variability of Mira, the uncertainties in the calibration of broad-band photometry, the scaling from the broad-band to the narrow wavelength region observed spectroscopically, the question of how much stellar light was caught within the spectrograph slit in the on-star observation, and the question whether the sensitivity of the detector or the transmission of the Earth’s atmosphere might have changed between the on-star and off-star observations. In order to check the calibration we therefore compared with ISO fluxes and made a final test in a comparison between the observed spectrum and a calculated one from a model atmosphere, allowing for the stellar radius and the distance as measured by the Hipparcos satellite.
The amplitude of the Mira light curve is visually large, but decreases rapidly towards longer wavelengths. From the NASA Catalog of Infrared Observations [@NASA] a mean of eleven measurements gives a flux in the Johnson M filter of $\log {\cal F_\nu}
= 3.6\pm 0.2$ \[Jy\]. This M-band mean flux is easily converted to $(5.6^{+3.3}_{-2.1})\times
10^{-7}\,\mbox{erg\,s$^{-1}$\,cm$^{-2}$\,$\mu$m$^{-1}$}$ or, since 1 pixel corresponds to $0.26\,\mbox{\AA\,}$, $(1.5^{+0.8}_{-0.5})\times
10^{-11}\,\mbox{erg\,s$^{-1}$\,cm$^{-2}$\,pixel$^{-1}$}$. In view of the fact that the flux is peaked towards the blue end of the M-band in such a way that the flux in the spectroscopic band is higher than the mean flux in the band (cf. Figure \[Mband\]), one must correct the absolute calibration. From the low-resolution ISO spectrum of this region of Mira we find a correction factor of 1.23. We obtained this correction by computing a mean of the ISO spectrum over the M-band and comparing with the ISO flux at $4.64\,\mu$m. Our model atmosphere (see discussion below) gives a similar result. Thus, for October 28 we arrive at a conversion factor of $(3.3^{+1.9}_{-1.2})\times 10^{-14}\,\mbox{erg\,cm$^{-2}$}$ per detector count and for October 29 we get a factor of $(2.7^{+1.6}_{-1.0})\times 10^{-14}\,\mbox{erg\,cm$^{-2}$}$ per detector count. This leads to a flux in the $4.64\,\mbox{$\mu$m}$ band of $4900\,\mbox{Jy}$ or $6.9\times 10^{-7}\,\mbox{erg\,s$^{-1}$\,cm$^{-2}$\,$\mu$m$^{-1}$}$. This may readily be compared with the ISO flux at the same wavelength, $(4700\pm 400)\,\mbox{Jy}$, cf. Section \[set-up\].
In order to check the calibration further we also used an [os-marcs]{} spherical model-atmosphere for the stellar parameters T$_\mathrm{eff}=2400\,\mbox{K}$, $\log g=-1.0$ \[cgs\], and solar metallicity and allowing for the stellar radius (R$_*=3.8\cdot
10^{13}\,\mbox{cm}$) and a Hipparcos distance $\mathrm{d}=128\,\mbox{pc}$, to generate a synthetic spectrum of the region of interest. The stellar parameters are determined for the phase of [*o*]{} Ceti at October 29, 1998, the date the NIR observations were made, see @ryde:2000. The model atmosphere is a part of a new grid of model atmospheres (Plez et al. 2000) being generated by an extensive update of the [marcs]{} code and its input data (based on Gustafsson et al. 1975). Molecular lines of H$_2$O (with line lists of Partridge and Schwencke 1997), of CO (lists of Goorvitch 1994), and of many other molecules, are taken into account in the spherical radiative transfer for the calculation of a synthetic spectrum.
Figure \[SED\] shows the spectrum generated from the model atmosphere and a comparison with the observed one. The general features are well reproduced and are mainly due to photospheric water vapor and CO. The agreement of the over-all flux level is astonishingly good; the model flux is around 30% too small. We find this agreement between the empirical, calibrated, absolute flux and the flux we calculate from a model atmosphere very satisfactory, in view of all the uncertainties anticipated.
It is worth noting that synthetic spectra from the new generation of spherical [marcs]{} models in the near infrared are able to reproduce observed spectra fairly well, even for a pulsating star like Mira. While the cyclic variations in mira spectra are well known and conspicuous at high resolution (Hinkle et al. 1982; Hinkle et al. 1984), little is know about phase dependent variations of the 4.6 $\mu$m photospheric spectra of mira stars. It is possible that depths of formation of the continuum and lines at $4.6\,\mu$m are in a region of the photosphere similar to that of the visual spectrum where the dynamic behavior does not have large effects on absorption line formation. However, it is also possible that the phase of observation gave a fortuitous match to the model. In either case the stellar atmosphere is extended and the [marcs]{} match to the spectrum is impressive.
Figure \[hastighet\] shows a blow-up of the photospheric CO R(2) absorption line as well as the scattered circumstellar R(2) emission. This emission line is shifted by $0.42\,\mbox{cm$^{-1}$}$ compared to the telluric wavelength scale, corresponding to a radial velocity of $58.6\,\mbox{km\,s$^{-1}$}$. Mira has a v$_\mathrm{LSR}=47\,\mbox{km\,s$^{-1}$}$ deduced from radio data [@loup], which equals a heliocentric velocity of $57\,\mbox{km\,s$^{-1}$}$. At the time of the observations and at Kitt Peak, this corresponds to a geocentric radial velocity of $59\,\mbox{km\,s$^{-1}$}$, in excellent agreement with the observations. Thus, the wavelength of the scattered light is centered on the laboratory wavelength corrected for the stellar radial velocity. The photospheric absorption lines, however, are also shifted due to the velocity of the pulsating photosphere. The bisector of the photospheric absorption line lies less than $2\,\mbox{km\,s$^{-1}$}$ blue-wards of the center of the emission line. The weak emission in the absorption line in the on-star spectrum is due to scattered light from the CSE. This light is included in the on-star measurement since the long-slit will also cover the CSE in two directions away from the star. Also present in the on-star spectrum is additional absorption along the line-of-sight through the circumstellar shell. In our measured, off-star, scattered CO light we do not correct for these components. However, they are of nearly equal intensity and should cancel at the level of the uncertainties of our measurements.
Note that when discussing relative fluxes, such as in Section \[obs\] where the R(1)/R(3) and R(2)/R(3) ratios are calculated, the details in the flat-fielding may introduce uncertainties. The three lines span over the entire range of the detector, and therefore variable effects not taken care of by the flat-fielding will show up as an error in the ratios.
OBSERVATIONAL RESULTS\[obs\]
============================
Figure \[on\_off\] shows the off-star spectrum of the west position. Superimposed is the corresponding, scaled, on-star spectrum, which at least approximately represents the radiation exciting the molecules. The intensity of this spectrum is scaled in order to fit the general features in the off-star spectrum. The off-star spectrum obviously consists partly of stellar light that is scattered in the Earth’s atmosphere, in the telescope, the spectrometer, and/or by dust grains in the circumstellar shell, leading to a spectrum resembling the on-star one. Nearly all features are identified as photospheric CO and H$_2$O lines, cf. Section \[marcs\] and Figure \[SED\]. The on-star spectrum includes tens of CO vibration-rotation lines of various excited vibrational states. The cold off-star spectrum, however, includes only the 1-0 R(1), R(2), and R(3) vibration-rotation emission lines of $^{12}$CO within the observed spectral range, except for the scattered, on-star light. Thus, the circumstellar molecules are radiatively excited by the stellar light, which is re-emitted as emission.
Figure \[em\] shows the resulting CO emission from the observations west, east, north and south of [*o*]{} Ceti, integrated over the long slit. This emission is recovered from the data by subtracting the scaled on-star spectrum from the off-star one. The resulting emission lines are the circumstellar R(1), R(2) and R(3) lines. Variations in the telluric lines during the time between the object frame exposure and the background exposure and/or during the time between the on-star and off-star exposures would result in non-zero residuals. From the amplitudes of the signals of the CO emission lines, measured at different directions from the star, and the noise level as shown in Figure \[SN\], we estimate a signal-to-noise ratio of approximately $5-15$. The noise arises partly from spurious mismatches in intensity between the on- and off-star spectra.
To study the emission as a function of the angular distance from the star, we divided our long-slit spectra into 79 sub-spectra (symmetrically around the maximum intensity representing the closest point to the star), one spectrum per pixel in the spatial direction. In this way we obtained 79 spectra for every off-star slit position, each corresponding to 0.2 on the sky, if seeing is neglected. For every spectrum, the intensity of the three CO vibration-rotation emission lines \[R(1), R(2) and R(3)\] from the wind are measured. These spectra provide data representing a distance range from $2\arcsec\,$ to a maximum of $7\arcsec\,$ away from the star, every slit position giving two series of data with three line-fluxes per spectrum, see Figures \[beta3\], \[beta3\_R1\], \[beta3\_R2\], and \[beta3\_R3\]; for example, the west position will sample the south-west and the north-west regions of the wind. The two sequences in every panel in the Figures represent these two series. In Figure \[beta3\] the decline of [*the added*]{} R(1), R(2) and R(3) intensities for the four slit positions are shown.
Table \[kvoter\] gives the observed intensities of the CO emission lines from the spectra representing the closest points to the star, i.e. $(2\pm 0.5)\arcsec\,$ away from the star, as well as the mean intensity ratios, R(1)/R(3) and R(2)/R(3), for positions from 2.0$\arcsec$ to 3.4$\arcsec$. Since the S/N ratio at the observed spatial resolution is too low, the data will unfortunately not permit a useful plot of the intensity ratios as a function of distance; The scatter is too large. The best-fit slopes of the intensity as a function of angular distance from the star, d$\log$ I/d$\log \beta$, for the R(2) emission line are also given in the Table. The uncertainties quoted are pure measuring uncertainties, i.e. they do not include possible systematic uncertainties. We obtained the best signal for our west position. The intensity decreases by a factor of about 40 from $2\arcsec\,$ to $7\arcsec$. A mean slope of the west, east, and north positions is $\mathrm d\log I /\mathrm d\log \beta = -2.8\pm 0.3$. Here we omit the south measurement in order to lower the uncertainty. Naturally, the signal-to-noise ratios decrease rapidly outwards. The uncertainty quoted is the measuring uncertainty.
The R(1) emission line is situated close to the left (red) edge of the detector. The spectra show a peculiar rise in intensity here which makes the measured values of the R(1) lines subject to a systematic uncertainty. Especially the east measurement of the R(1) line seems to be stronger than expected when compared with the measurements in the other directions. The R(1)/R(3) ratios are also affected by this effect. The east and north spectra in Figure \[em\] show a spurious decline at lower wavenumbers (at the red end). This is due to a mismatch in intensity between the on- and off-star spectra at the edge of the detector array probably not caused by flat-fielding or other reduction procedures. This spurious effect may, unfortunately, affect all frames.
DISCUSSION
==========
We find a power-law dependence of the intensity as a function of angular distance on the sky, $\mbox{I}\propto \beta^{-3}$. Furthermore, in view of various uncertainties, e.g. in the positioning of the slit, we find that the measured emission line intensities in different directions from the star are consistent with a symmetric wind. The wind is at least symmetric to within a factor of two in density. A true west and east position of $1.8\arcsec\,\mathrm{W}$ and $2.2\arcsec\,\mathrm{E}$ from the star would yield our measured values for a symmetric wind. Corresponding values for the north and south positions would be $1.7\arcsec\,\mathrm{N}$ and $2.3\arcsec\,\mathrm{S}$. Thus, a small error in the position of $0.2-0.3\arcsec$ (which is approximately the accuracy of the positioning of the slit) in the measurement from a symmetric wind would yield the ‘asymmetric’ values we measured.
The relative intensities measured on the two nights can be checked at the cross-over points, cf. for example Figure \[beta3\]. The intensities differ by approximately 50%, probably reflecting the uncertainties in the slit positions, but also the accuracy of the absolute flux calibration between the nights. The flux calibrations of the observations on the two nights are based on two different on-star measurements, which are subject to different uncertainties, as discussed in Section\[marcs\].
However, a comparison of the two direction in one slit position may reveal asymmetries in the wind. In Figures \[beta3\_R1\], \[beta3\_R2\], and \[beta3\_R3\], which represent the three lines measured, the increase in intensity at 3-4 in the ‘north-east’ data in the ‘north’-panel is seen in all transitions. This is also the case for the ‘south-east’ data in the ‘south’-panel which all show approximately the same morphology. Thus, this could indicate that there is an east-west asymmetry in the intensity measured, reflecting an asymmetric distribution of CO. This could be due to the additional bipolar outflow detected at larger scales in the aperture synthesis CO maps by @planesas:II. Note that their highest resolution is approximately 6.
We now make a simple analytic model of our observations by assuming a spherically symmetric and homogeneous wind with a constant mass-loss rate and a constant expansion velocity. We also assume the wind to be optically thin in the CO lines along the line-of-sight for rays from regions beyond a certain distance from the star. The adequacy of this latter assumption will be investigated below. Based on Eq.(5) by @bg:97 the ratio of the wavelength-integrated, line-scattered intensity, ${\rm I_{CO,i}}$ (ergs$^{-1}$cm$^{-2}$arcseconds$^{-2}$), and the line-scattering flux $\bar {\rm f}_{\lambda}$ (ergs$^{-1}$cm$^{-2}$cm$^{-1}$) as seen by the scattering molecules averaged across the line width but measured at the distance d, is found to be
$$\begin{aligned}
\label{bg_formel}
\frac {{\rm I_{CO,i}}(\beta)}{\bar {\rm f}_{\lambda}} & = & \frac{206265}{32}\,
\frac{e^2 \,\lambda ^2}{m_e \,c^2\,m_H}\, f_{u\leftarrow l}\, \dot {\rm M} \times \, \nonumber \\
& & \frac{{\mathrm{N_i(CO)}/\mathrm{N(CO)}}\cdot \epsilon_{\mathrm{CO}}}{\mu\,v_e \,{\rm d}}
\left (\frac{1}{\beta}\right )^3,\end{aligned}$$
where $f_{u\leftarrow l}$ is the absorption oscillator strength of the line, $\epsilon_{\mathrm{CO}}$ is the fractional abundance of CO molecules, i.e. \[CO\]/\[H\], and N$_i$(CO) denotes the number density of CO molecules in the lower state, i, of the transition. Furthermore, $\mu$ is the mean molecular weight, d is the distance to the star, $v_{\mathrm e}$ is the terminal expansion wind velocity and $\beta$ is the angular distance from the star on the sky in seconds of arc. A feature of this general expression is the minus third power dependence of the scattered intensity as a function of the impact parameter on the sky, $\beta$, which agrees within the uncertainties with our observation (cf. Figure \[beta3\]). Eq.(1) will now be applied to the data for the R(2) line for which our data have the highest quality.
The oscillator strength $f_{u\leftarrow l}$ of the R(2) vibration-rotation CO-line at 4.6 $\mu$m is $6.1\times 10^{-6}$ (Kirby-Docken and Liu 1978 provide oscillator strengths, which for our transitions are consistent with Goorvitch & Chackerian (1994) and Huré & Roueff (1996)[^2]. See also Table \[einstein\]). Thus, from Eq.(\[bg\_formel\]) we find that the mass-loss rate as deduced from the emitted intensity from the R(2) line of CO, and, assuming $\mu\approx1.2$, is
$$\begin{aligned}
\label{mdot}
\dot {\rm M} & = & 4.20\times 10^{-10}\cdot \frac{v_e \,{\rm
d}\,\beta^3}{\mathrm{N_i(CO)}/\mathrm{N(CO)} \cdot
\epsilon_{\mathrm{CO}}} \times \nonumber \\
& & \frac {{\rm I_{CO,i}}(\beta)}{\bar {\rm
f}_{\lambda}}\,\,\,\,\,\,(\mathrm{M_\odot \,yr^{-1}}),\end{aligned}$$
where $v_\mathrm{e}$ is given in , d in parsecs, $\beta$ in seconds of arc, ${\rm I_{CO,i}}$ in ergs$^{-1}$cm$^{-2}$arcseconds$^{-2}$, and $\bar {\rm
f}_{\lambda}$ in ergs$^{-1}$cm$^{-2}$$\mu$m$^{-1}$. The fractional abundance of $^{12}$CO molecules relative to hydrogen, $\epsilon_{\mathrm{CO}}$, is assumed to be constant throughout the envelope and it is assumed that most oxygen is locked-up as CO molecules. From the literature we find $\mathrm{f}_\mathrm{CO}=[\mathrm{CO}]/[\mathrm{H_2}]=5\times 10^{-4}$ [@knapp:98], which means that $\epsilon_{\mathrm{CO}}=2.5\times 10^{-4}$. We now need to estimate the fraction N$_\mathrm{i}$(CO)/N(CO) of CO molecules that are excited to the ($v''=0, J''=2$)-level. We assume the population to be controlled by radiation transitions between the $v''=0$ and $v''=1$ states. This radiation is originally supplied by the stellar photosphere at roughly the stellar effective temperature, $T_\mathrm{eff}$, but diluted by a factor of (R$_*/r)^2$ at a distance $r$ from the star. Neglecting all loss or addition of photons in the optically thick spectral lines we find a characteristic temperature of the radiation $T_r$ from
$$\label{B}
B(\lambda,T_r) \approx B(\lambda,T_\mathrm{eff}) \times (\mathrm{R}_*/r)^2.$$
This estimate gives a radiation temperature at a distance of $3\arcsec$ of about 340 K, suggesting an $\mathrm{N_{i=2}(CO)}/\mathrm{N(CO)}$-value of approximately 4%. This temperature is several times higher than the kinetic temperature of the gas (the level is super-thermally excited). Detailed numerical simulations of the radiative transfer in an envelope model of [*o*]{} Ceti by Ryde & Schöier (2000) verifies the assumption of radiationally controlled populations and indicates population fractions, $\mathrm{N_{i=2}(CO)}/\mathrm{N(CO)}$, that range from approximately 3% to 10% at distances from $2\arcsec$ to $7\arcsec$. Here, we have adopted a value $\mathrm{N_{i=2}(CO)}/\mathrm{N(CO)}$ of 5%, noting that a numerical modelling of the circumstellar envelope will be necessary for a more detailed and accurate discussion. On the assumption that the terminal wind velocity $v_\mathrm{e}=3\,\mbox{km\,s$^{-1}$}$, the mass-loss rate deduced from the measured intensity at 2 away from the star can be written as
$$\label{mdott}
\dot {\rm M}=0.10\cdot {\rm I_{CO}}(\beta)/\bar {\rm f}_{\lambda}\,\,\,\,\,\,(\mathrm{M_\odot \,yr^{-1}}).$$
The observations at $\beta\approx2\arcsec\,$ of the east and west positions give ${\rm I_{CO}}(\beta)/\bar {\rm f}_{\lambda}=3.1\times
10^{-6}\,\mbox{$\mu$m\,($\arcsec$)$^{-2}$}$ and of the south and north positions give ${\rm I_{CO}}(\beta)/\bar {\rm f}_{\lambda}=2.8\times
10^{-6}\,\mbox{$\mu$m\,($\arcsec$)$^{-2}$}$, where $\bar {\rm f}_{\lambda}$ is estimated from the on-star spectrum. The mean mass-loss rate found by this analytic discussion is then $\dot {\rm M}=3.2\times
10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$ for the east and west positions, and $\dot {\rm M}=2.8\times 10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$ for the north and south positions. The uncertainty in the derived mass-loss rate (which is independent of the flux calibration since we use a ratio of fluxes in the derivation), are due to uncertainties in the position of the slit, the level population, the measured scattered intensity, and the on-star intensity. Considering these sources of uncertainties we estimate a conservative uncertainty in the mass-loss rate of typically a factor of 4. Our derived value of the mass-loss rate is in good agreement with values in the literature. For example, @young arrive at $(3.6\pm 1.2)\times 10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$ from the CO($J=3-2$) line for a distance of 100 pc and a $\epsilon_\mathrm{CO}=3\times 10^{-4}$. The largest uncertainty in the estimates of mass-loss rates for miras has until recently been the distance. The Hipparcos satellite [@hipp] has radically improved this situation. Young’s mass-loss rate corresponds to $(2.8\pm 0.9)\times 10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$ with our values of d and $\epsilon_{\mathrm{CO}}$. Other estimates of the mass-loss rate lie between $(1-5)\times 10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$, see e.g. @danchi:94, @loup, @planesas:I, and @knapp:85. It should be mentioned again that the radio profiles are difficult to model due to their asymmetry and require an additional component apart from for the standard CSE usually used. There may be a bipolar nebula seen pole-on (cf. the discussion in Knapp & Morris 1985).
An assumption made in Eq.(\[bg\_formel\]) is that the circumstellar envelope is [*tangentially*]{} optically thin, i.e. along the line-of-sight, at distances from the star at which the measurements are made, i.e. 2-7. This optical depth is easily found from
$$\begin{aligned}
\label{tang}
{\tau_\mathrm{tang} (p)} & = & \frac{e^2 \lambda ^2}{m_e c^2 m_H}\frac
{f_{u\leftarrow l} \dot {\rm M} } {\delta \lambda}
\frac{ \epsilon_{\mathrm{CO}}}{4 \mu v_e}
\frac{N_i(CO)}{N(CO)} \times \nonumber\\
& & \int^{L}_{-L}{\frac{\mathrm{d}l}{p^2+l^2}},\end{aligned}$$
where $p$ is the ‘impact parameter’ which denotes the closest distance from the star at the location on the sky of the observations and $l$ is the variable of integration (tangential distance along the line-of-sight with the closest point to the star as the zero point). The integration should be calculated over a region of the envelope ($-\mathrm{L}<l<\mathrm{L}$) where the differential Doppler shifts are smaller than the width of the line. Approximating the line profile with a step-function, we find this distance to be $$2 L\approx p \,\frac{\delta \lambda}{\lambda}\frac{c}{v_\mathrm{e}}.$$ An estimate of the tangential optical depth of the wind based on the derived mass-loss rate and the estimated expansion velocity, is obtained by solving the integral (Eq.\[tang\]). We find that $\tau_\mathrm{tang}\approx7.6\times 10^{15}/p$, where $p$ should be in centimeters. Inwards of approximately $4\arcsec\ $ we find that $\tau_\mathrm{tang} (p)
{\small
\raisebox{-0.05cm}{\begin{minipage}{0.2cm}
\raisebox{-0.1cm}{$> $} \\
\raisebox{0.1cm}{$\sim$ }
\end{minipage}} }
\,1$, i.e. that the wind is expected to be optically thick along a line-of-sight. Although the assumption that the optical depth is smaller than 1 is not fulfilled in the inner part of the observed region ($\beta \approx2-7\arcsec$), the CO emission declines smoothly as $\beta^{-3}$ all the way in, indicating that the assumption of the wind being optically thin could nevertheless be correct. The dependence on angular distance of the scattered intensity is not expected to show the same behaviour in an optically thick and an optically thin wind. For the K[i]{} scattering in the wind of R Scl [@bg:97] the same situation is found. The $\beta^{-3}$-dependence of the intensity could be preserved throughout the observed parts of the wind if the circumstellar gas were inhomogeneous, which probably is a reasonable assumption, (cf. the discussion in Gustafsson et al. 1997). This fact could lead to a lowering of the optical depth if the structures (‘clumps’) of the wind are of a suitable characteristic size and have a low filling factor in the line-of-sight. Note, that the measured decline of the intensity could also be achieved if the lines are optically thick and the mass-loss rate is not uniform but vary with time. However, this seems to require a fine-tuning of the mass-loss rate with time which does not seem realistic. A numerical simulation of the Mira wind (Ryde & Schöier 2000) will discuss the optical depths in more detail.
In applying Eq.(\[mdot\]) it was assumed that the line radiation at distance $r$ from the center of the star is represented satisfactorily by the measured on-star flux $\bar {\rm f}_{\lambda}$, corrected by the distance factor $(d/r)^2$. In practise, the observed on-star spectrum will also contain strong contributions from stellar light scattered in the inner envelope in non-radial directions, provided that the gas is not devoid of scatterers there. The velocity divergence then brings the photon out of the core of the line profile at points along the new direction and the chances for it to be re-scattered are minimal. Therefore, a considerable fraction of the photons are lost ‘sideways’ from the envelope due to scattering already in its inner regions, see also the discussion in @ryde:2000.
CONCLUSIONS
===========
The successful detection of scattered photospheric light in the circumstellar CO vibration-rotation lines of Mira has made it possible to explore the structure of its intermediate circumstellar envelope (at distances from the star of approximately $100-1000$ stellar radii).
The variation of the emission with angular distance from the star ($2\arcsec<\beta<7\arcsec$) is found to roughly follow a $\beta^{-3}$ behavior, $\beta$ being the angular distance on the sky. Our data indicate that the mass-loss rate, when averaged over about 100 years, was constant over a period of about 1200 years, assuming that the wind is optically thin along the line-of-sight for the measured angular distances.
Our observations provide absolute fluxes scattered in the circumstellar CO R(1), R(2), and R(3) lines, and provide corresponding line ratios. Based on a simple analytic model, the mass-loss rate is estimated to be approximately $3\times 10^{-7}\,\mbox{M$_\odot$ \,yr$^{-1}$}$ which, in view of the uncertainties, is compatible with earlier estimates in the literature. A numerical modelling of the wind will be necessary for a more detailed and accurate discussion of the circumstellar envelope. Such a modelling will be presented in a forthcoming paper [@ryde:2000].
We have also found that the envelope is approximately spherically-symmetric to within a factor of two in density. Note, however, the asymmetries found by, for example, @planesas:II. We have found indications that a similar asymmetry is also present in the CO vibration-rotation emission intensity at a distance of approximately $3\arcsec$ from the star in the east direction as compared with the west direction. It is interesting to note that the companion of Mira is located eastward from the star at a position angle of $(108.3 \pm 0.1)^o$ [@karovska:97].
The referee is thanked for very valuable comments. We should also like to thank J. Valenti for assistance during the observations, and H. Olofsson for stimulating discussions. We are grateful to J. Barnes, L. Borgonovo, K. Ryde, and N. Piskunov for generous assistance. We also owe a dept of thanks to S. Höfner and M. Asplund for valuable comments on the manuscript. We acknowledge financial support from the Swedish National Space Board and the Royal Swedish Academy of Sciences.
, A. P. and [Lambert]{}, D. L., 1975, , 201, L153
, A. P. and [Lambert]{}, D. L., 1976, , 395
, G. H., 1988, , 299
, P., [Ade]{}, P., [Armand]{}, C., et al., 1996, , L38
, M., [Menten]{}, K. M., [Young]{}, K., and [Phillips]{}, T. G., 1997, , 189
, W. C., [Bester]{}, M., [Degiacomi]{}, C. G., [Greenhill]{}, L. J., and [Townes]{}, C. H., 1994, , 1469
, A. J., 1956, , 210
, E. A. and [Höfner]{}, S., 1996, , 605
, H. M., [Beckwith]{}, S., and [Zuckerman]{}, B., 1983, , L79
ESA, 1997, ESA, The Hipparcos and Tycho Catalogues, ESA SP-1200
, D. Y., [Schmitz]{}, M., and [Mead]{}, J. M., 1984,
, D., 1994, , 535
, D. and [Chackerian]{}, C., 1994, , 483
, T., [van den Ancker]{}, M., [Bauer]{}, O. H., et al., 2000, in [The ISO Handbook; SWS-the Short Wavelength Spectrometer, SAI/2000-008/Dc. Version 1.0]{}, p. 96
, C. and [Mauron]{}, N., 1996, , 585
, B., [Bell]{}, R. A., [Eriksson]{}, K., and [Nordlund]{}, A., 1975, , 407
, B., [Eriksson]{}, K., [Kiselman]{}, D., [Olander]{}, N., and [Olofsson]{}, H., 1997, , 535
, H. J., 1996, , 97
, C. A., [Scholz]{}, M., and [Tuthill]{}, P. G., 1995, , 640
, K. H., [Cuberly]{}, R. W., [Gaughan]{}, N. A., et al., 1998, , 810
, K. H., [Hall]{}, D. N. B., [Ridgway]{}, S. T. 1982, , 697
, K. H., [Scharlach]{}, W. W. G., [Hall]{}, D. N. B. 1984, , 1
, S. and [Dorfi]{}, E. A., 1997, , 648
, S., [Jørgensen]{}, U. G., [Loidl]{}, R., and [Aringer]{}, B., 1998, , 497
, J. M. and [Roueff]{}, E., 1996, , 561
, R. R., 1992, in S. [Howell]{} (ed.), [ASP Conf. Ser. 23: Astronomical CCD Observing and Reduction Techniques]{}, p. 258
, M., [Hack]{}, W., [Raymond]{}, J., and [Guinan]{}, E., 1997, , L175
, M., [Nisenson]{}, P., and [Beletic]{}, J., 1993, , 311
, M. F., [Steinz]{}, J. A., [Anderegg]{}, M. E., et al., 1996, , L27
, K. and [Liu]{}, B., 1978, , 359
, G. R., [Morris]{}, M., 1985, , 640
, G. R., [Young]{}, K., [Lee]{}, E., and [Jorissen]{}, A., 1998, , 209
, P., 1988, in [Observational Astrophysics, Springer Verlag]{}, p. 256
, C., [Forveille]{}, T., [Omont]{}, A., and [Paul]{}, J. F., 1993, , 291
, T. A., [Wasatonic]{}, R., and [Guinan]{}, E. F., 1997, , 1
, H., [Bergman]{}, P., [Eriksson]{}, K., and [Gustafsson]{}, B., 1996, , 587
, H. and [Schwencke]{}, D., 1997, , 4618
, G., [Coude Du Foresto]{}, V., [Ridgway]{}, S. T., et al., 1998, , 619
, P., [Bachiller]{}, R., [Martin-Pintado]{}, J., and [Bujarrabal]{}, V., 1990a, , 263
, P., [Kenney]{}, J .D. P., and [Bachiller]{}, R., 1990b, , L9
, B. et al., 2000, in preparation
, B. and [Lambert]{}, D. L., 1994, , L101
, N., [Gustafsson]{}, B., [Hinkle]{}, K. H., [Eriksson]{}, K., [Lambert]{}, D. L., and [Olofsson]{}, H., 1999, , L35
, N. & [Schöier]{}, F.L., 2000, submitted
, R. and [Wannier]{}, P. G., 1985, , 424
, F., [Feast]{}, M. W., [Whitelock]{}, P. A., and [Yudin]{}, B., 1997, , 955
, P., 1998, in [Cyclical Variability in Stellar Winds, Springer Verlag]{}, p. 278
, K., 1995, , 872
[ccccc]{}
I$_\mathrm{R(1)}$ & $3.3\times 10^{-12}$ & $2.0\times 10^{-12}$ & $2.3\times 10^{-12}$ & $0.9\times 10^{-12}$\
I$_\mathrm{R(2)}$ & $2.6\times 10^{-12}$ & $0.9\times 10^{-12}$ & $1.5\times 10^{-12}$ & $0.8\times 10^{-12}$\
I$_\mathrm{R(3)}$ & $2.0\times 10^{-12}$ & $0.8\times 10^{-12}$ & $1.3\times 10^{-12}$ & $0.8\times 10^{-12}$\
R(1)/R(3) & $1.7\pm 0.2$ & $2.1\pm 0.6$ & $1.7\pm 0.3$& $1.2\pm 0.2$\
R(2)/R(3) & $1.3\pm 0.1$ & $1.2\pm 0.2$ & $1.2\pm 0.2$& $1.1\pm 0.1$\
& $-3.3$ (NW) & $-2.5$ (NE) & $-3.0$ (NW) & $-2.3$ (SW)\
& $-2.8$ (SW) & $-2.5$ (SE) & $-2.7$ (NE) & $-3.2$ (SE)\
[cccc]{} R(1) (1-2) & $2150.856$ & $6.791\times 10^{-6}$ & $12.57$\
R(2) (2-3) & $2154.596$ & $6.127\times 10^{-6}$ & $13.55$\
R(3) (3-4) & $2158.300$ & $5.850\times 10^{-6}$ & $14.14$\
[^1]: Partly based on observations with ISO, an ESA project with instruments funded by ESA Member States and with the participation of ISAS and NASA. http://isowww.estec.esa.nl/
[^2]: Note the missing cube in their Eq.(3)
|
---
abstract: 'We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring $R$ of characteristic $p$ is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an $R$-module via some power of the Frobenius endomorphism of $R$. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under composition and decomposition.'
address:
- 'Mathematics Department, University of Missouri, Columbia, MO 65211 USA'
- 'Department of Mathematics, University of Illinois, 273 Altgeld Hall, 1409 West Green Street, Urbana, IL, 61801 USA'
author:
- Srikanth Iyengar
- 'Sean Sather-Wagstaff'
title: 'G-dimension over local homomorphisms. Applications to the [F]{}robenius endomorphism'
---
[^1]
Introduction
============
The main goal of this article is to develop a theory of Gorenstein dimension over local homomorphisms. More precisely, given a local homomorphism ${\varphi}\colon R\to S$, to each finitely generated (in short: finite) $S$-module $M$, we attach an invariant ${\mathrm{G\text{-}dim}}_{{\varphi}}(M)$, called the *G-dimension of $M$ over ${\varphi}$*. This invariant is defined using the technology of Cohen factorizations, developed by Avramov, Foxby, and B. Herzog [@avramov:solh]. The reader can refer to Section \[sec:gd\] for the details. When $M$ happens to be finite over $R$, for instance when ${\varphi}=\mathrm{id}_R$, this coincides with the G-dimension of $M$ over $R$ as defined by Auslander and Bridger [@auslander:smt]; this is contained in Corollary \[cor:vf-is-finite\].
One of the guiding examples for this work is the Frobenius map ${\varphi}\colon R\to R$, given by $x\mapsto x^p$, where $R$ is a local ring of positive prime characteristic $p$. Since ${\varphi}$ is a ring homomorphism, so is ${\varphi}^n$ for each integer $n>0$, and hence one can view $R$ as a left module over itself via ${\varphi}^n$. Denote this $R$-module ${}^{{\varphi}^n}\!\! R$. Like in the case of the residue field, it is known that certain homological properties of ${}^{{\varphi}^n}\!\!R$ determine and are determined by ring-theoretic properties of $R$. Consider, for instance, regularity. The Auslander-Buchsbaum-Serre theorem says that a local ring is regular if and only if its residue field has finite projective dimension. Compare this with the fact that, when $R$ has characteristic $p$, it is regular if and only if the flat dimension of ${}^{{\varphi}^n}\!\! R$ is finite for some $n\geq 1$; this is proved by Kunz [@kunz:corlrocp (2.1)] and Rodicio [@rodicio:oaroa (2)]. This result may be reformulated as: the local ring $R$ is regular if and only if ${\mathrm{pd}}({\varphi}^n)$ is finite for some integer $n\geq 1$. Here, given any local homomorphism ${\varphi}\colon R\to
S$, we write ${\mathrm{pd}}_{{\varphi}}(\text{-})$ for the *projective dimension over ${\varphi}$*, which is also defined via Cohen factorizations, and ${\mathrm{pd}}({\varphi})={\mathrm{pd}}_{{\varphi}}(S)$; see Section \[sec:pd\].
A key contribution of this paper, Theorem A below, is a similar characterization of the Gorenstein property for $R$. It is contained in Theorem \[thm:gdim(f)-finite\] and is analogous to a classical result of Auslander and Bridger: for any local ring, the residue field has finite G-dimension if and only if the ring is Gorenstein.
**Theorem A.**
In the statement, ${\mathrm{G\text{-}dim}}({\varphi}^n)={\mathrm{G\text{-}dim}}_{{\varphi}^n}(R)$. In the special case where ${\varphi}$ is module-finite, the equivalence of conditions (a) and (b) coincides with a recent result of Takahashi and Yoshino [@takahashi:ccmlrbfm (3.1)]. These are related also to a theorem of Goto [@goto:aponlrocp (1.1)].
The bulk of the article is dedicated to a systematic investigation of the invariant ${\mathrm{G\text{-}dim}}_{{\varphi}}(\text{-})$. Some of the results obtained extend those concerning the classical invariant ${\mathrm{G\text{-}dim}}_R(\text{-})$. Others are new even when specialized to the absolute situation. The ensuing theorem is one such. It is comparable to [@foxby:daafuc (3.2)], which can be souped up to: if ${\mathrm{pd}}_{\sigma}(P)$ is finite, then ${\mathrm{pd}}_{\sigma{\varphi}}(P)={\mathrm{pd}}({\varphi})+{\mathrm{pd}}_{\sigma}(P)$; see Theorem \[thm:stab-pd-tensor\] for a further enhancement.
**Theorem B.** *Let ${\varphi}\colon R\to S$ and $\sigma\colon S\to T$ be local homomorphisms, and let $P$ be a nonzero finite $T$-module. If ${\mathrm{pd}}_{\sigma}(P)$ is finite, then $${\mathrm{G\text{-}dim}}_{\sigma{\varphi}}(P)={\mathrm{G\text{-}dim}}({\varphi})+{\mathrm{pd}}_{\sigma}(P).$$ In particular, ${\mathrm{G\text{-}dim}}_{\sigma{\varphi}}(P)$ and ${\mathrm{G\text{-}dim}}({\varphi})$ are simultaneously finite.*
This result is subsumed by Theorem \[thm:stability-tensor\]. The special case $P=T$, spelled out in Theorem \[thm:gdim-compose\], may be viewed as a composition-decomposition theorem for maps of finite G-dimension. It is expected that the composition part of the result holds even when ${\mathrm{G\text{-}dim}}(\sigma)$ is finite [@avramov:rhafgd (4.8)]. However, as Example \[ex:not-weaken\] demonstrates, the decomposition part cannot extend to that generality.
Theorem B and its counterpart for projective dimension are crucial ingredients in the following theorem that generalizes [@avramov:lgh (4.6.c)] and [@avramov:rhafgd (8.8)] proved by Avramov and Foxby.
**Theorem C.** *Let ${\varphi}\colon (R,{{\mathfrak{m}}})\to (S,{{\mathfrak{n}}})$ and $\sigma\colon (S,{{\mathfrak{n}}})\to (T,{{\mathfrak{p}}})$ be local homomorphisms with ${\mathrm{pd}}(\sigma)$ finite. If $\sigma{\varphi}$ is (quasi-)Gorenstein at ${{\mathfrak{p}}}$, then ${\varphi}$ is (quasi-)Gorenstein at ${{\mathfrak{n}}}$ and $\sigma$ is Gorenstein at ${{\mathfrak{p}}}$.*
This result coincides with Theorem \[thm:fin-fd-descent\]. Section \[sec:descent\] contains other results of this flavor. It is worth remarking that there is an analogue of Theorem C for complete intersection homomorphisms, due to Avramov [@avramov:lci (5.7)].
It turns out that the *finiteness* of ${\mathrm{G\text{-}dim}}_{{\varphi}}(M)$ depends only on the $R$-module structure on $M$, although its value depends on ${\varphi}$; this is the content of Theorem \[thm:indep-of-vf\] and Example \[ex:not-equal\]. One way to understand this result would be to compare Gorenstein dimension over ${\varphi}$ to various extensions of the classical G-dimension to $R$-modules that may not be finite. The last section deals with this problem, where Theorem \[thm:gdvsgfd\] contains the following result; in it ${\mathrm{Gfd}}_R(M)$ is the Gorenstein flat dimension of $M$ over $R$.
**Theorem D.** *Assume $R$ is a quotient of a Gorenstein ring and let ${\varphi}\colon R\to S$ be a local homomorphism. For each finite $S$-module $M$, one has $${\mathrm{Gfd}}_R(M) - {\mathrm{edim}}({\varphi}) \leq {\mathrm{G\text{-}dim}}_{{\varphi}}(M) \leq {\mathrm{Gfd}}_R(M)\,.$$ In particular, ${\mathrm{G\text{-}dim}}_{{\varphi}}(M)$ is finite if and only if ${\mathrm{Gfd}}_R(M)$ is finite.*
Foxby, in an unpublished manuscript, has obtained the same conclusion assuming only that the formal fibres of $R$ are Gorenstein. Specializing $X$ to $S$ yields that ${\mathrm{G\text{-}dim}}({\varphi})$ and ${\mathrm{Gfd}}_RS$ are simultaneously finite. This last result was proved also by Christensen, Frankild, and Holm [@christensen:new (5.2)], and our proof of Theorem D draws heavily on their work.
En route to the proof of Theorem D, we obtain results on G-flat dimension that are of independent interest; notably, the following Auslander-Buchsbaum type formula for the depth of a module of finite G-flat dimension. It is contained in Theorem \[prop:gfd-depth\].
**Theorem E.** *Let $(R,{{\mathfrak{m}}},k)$ be a local ring and $E$ the injective hull of $k$. If $M$ is an $R$-module with ${\mathrm{Gfd}}_R(M)$ finite, then* $${\mathrm{depth}}_R(M) = {\mathrm{depth}}R - \sup(E{\otimes^{\mathbf{L}}}_RM)\,.$$
In the preceding discussion, we have focused on modules. However, most of our results are stated and proved for complexes of $R$-modules. This is often convenient and sometimes necessary, as is the case in Theorem \[thm:stability-tensor\]. Section \[sec:back\] is mainly a catalogue of standard notions and techniques from the homological algebra of complexes required in this work; most of them can be found in Foxby’s notes [@foxby:hacr] or Christensen’s monograph [@christensen:gd].
Background {#sec:back}
==========
Let $R$ be a commutative Noetherian ring. A complex of $R$-modules is a sequence of $R$-module homomorphisms $$X= \cdots\xrightarrow{\partial_{i+1}}X_i\xrightarrow{\partial_{i}}X_{i-1}
\xrightarrow{\partial_{i-1}}\cdots$$ such that $\partial_i\partial_{i+1}=0$ for all $i$. The *supremum*, the *infimum*, and the *amplitude* of a complex $X$ are defined by the following formulas: $$\begin{aligned}
\sup(X) & = \sup\{i\mid {\mathrm{H}}_i(X)\neq 0\} \\
\inf(X) & = \inf\{i\mid {\mathrm{H}}_i(X)\neq 0\} \\
{\mathrm{amp}}(X) & = \sup(X)-\inf(X).\end{aligned}$$ Note that ${\mathrm{amp}}(X)=-\infty$ if and only if ${\mathrm{H}}(X)=0$. The complex $X$ is *homologically bounded* if ${\mathrm{amp}}(X)<\infty$, and it is *homologically degreewise finite* if ${\mathrm{H}}(X)$ is degreewise finite. When ${\mathrm{H}}(X)$ is both degreewise finite and bounded we say that $X$ is *homologically finite*.
Let $X$ and $Y$ be complexes of $R$-modules. As is standard, we write $X{\otimes^{\mathbf{L}}}_R Y$ for the derived tensor product of $X$ and $Y$, and ${\mathbf{R}\mathrm{Hom}}_R(X,Y)$ for the derived homomorphisms from $X$ to $Y$. The symbol “$\simeq$” denotes an isomorphism in the derived category. For details on derived categories and derived functors, the reader may refer to the classics, Hartshorne [@hartshorne:rad] and Verdier [@verdier], or, for a more recent treatment, to Gelfand and Manin [@gelfand:moha].
Let $X$ be a homologically bounded complex of $R$-modules. A *projective* *resolution* of $X$ is a complex of projective modules $P$ with $P_i=0$ for $i\ll
0$ and equipped with an isomorphism $P\simeq X$. Such resolutions exist and can be chosen to be degreewise finite when $X$ is homologically finite. The *projective* *dimension* of $X$ is $${\mathrm{pd}}_R(X):=\inf\{\sup\{n\mid P_n\neq 0\}\mid \text{$P$ a projective resolution of $X$}\}.$$ Thus, if ${\mathrm{H}}(X)=0$, then ${\mathrm{pd}}_R(X)$ is $-\infty$, and hence it is not finite. Flat resolutions and injective resolutions, and the corresponding dimensions ${\mathrm{fd}}_R(X)$ and ${\mathrm{id}}_R(X)$, are defined analogously.
The focus of this paper is G-dimension for complexes. In the next few paragraphs, we recall its definition and certain crucial results that allow one to come to grips with it.
A finite $R$-module $G$ is *totally reflexive* if
1. ${\mathrm{Ext}}^i_R(G,R)=0$ for all $i>0$;
2. ${\mathrm{Ext}}^i_R(G^*,R)=0$ for all $i>0$, where $(\text{-})^*$ denotes ${\mathrm{Hom}}_R(\text{-},R)$; and
3. the canonical map $G\to G^{**}$ is bijective.
Let $X$ be a homologically finite complex of $R$-modules. A *G-resolution* of $X$ is an isomorphism $G\simeq X$ where $G$ is complex of totally reflexive modules with $G_i=0$ for $i\ll 0$. A degreewise finite projective resolution of $X$ is also a G-resolution, since every finite projective module is totally reflexive. The *G-dimension* of $X$ is $${\mathrm{G\text{-}dim}}_R(X):=\inf\{\sup\{n\mid G_n\neq 0\}\mid \text{$G$ is a
G-resolution of $X$}\}.$$
The following paragraphs describe alternative, and often more convenient, ways to detect when a complex has finite G-dimension.
\[para:reflexive\] A homologically finite complex $X$ of $R$-modules is *reflexive* if
1. ${\mathbf{R}\mathrm{Hom}}_R(X,R)$ is homologically bounded; and
2. the canonical biduality morphism below is an isomorphism $$\delta^R_X\colon X\to{\mathbf{R}\mathrm{Hom}}_R({\mathbf{R}\mathrm{Hom}}_R(X,R),R).$$
This notion is relevant to this article because of the next result, based on an unpublished work of Foxby; see [@christensen:gd (2.3.8)] and [@yassemi:gd (2.7)].
\[para:reflexive-2\] *The complex $X$ is reflexive if and only if ${\mathrm{G\text{-}dim}}_R(X)<\infty$. When $X$ is reflexive, ${\mathrm{G\text{-}dim}}_R(X)=-\inf({\mathbf{R}\mathrm{Hom}}_R(X,R))$.*
Using this characterization, it is easy to verify the base change formula below; Christensen [@christensen:sdc (5.11)] has established a much stronger statement.
\[lem:fflat\] *Let $R\to S$ be a flat local homomorphism and $X$ a homologically finite complex of $R$-modules. Then ${\mathrm{G\text{-}dim}}_R(X)={\mathrm{G\text{-}dim}}_S(X\otimes_R S)$.*
Henceforth, $R$ is a local ring, where “local” means “local and Noetherian”.
\[para:auslander\] A *dualizing complex* for $R$ is a homologically finite complex of $R$-modules $D$ of finite injective dimension such that the natural map $R\to{\mathbf{R}\mathrm{Hom}}_R(D,D)$ is an isomorphism. When $R$ is a homomorphic image of a Gorenstein ring, for example, when $R$ is complete, it possesses a dualizing complex.
Assume that $R$ possesses a dualizing complex $D$. The *Auslander category of $R$*, denoted ${\mathcal{A}}(R)$, is the full subcategory of the derived category of $R$ whose objects are the homologically bounded complexes $X$ such that
1. $D{\otimes^{\mathbf{L}}}_R X$ is homologically bounded; and
2. the canonical morphism below is an isomorphism $$\gamma_X\colon X\to{\mathbf{R}\mathrm{Hom}}_R(D,D{\otimes^{\mathbf{L}}}_R X).$$
It should be emphasized that a complex can be in the Auslander category of $R$ without being homologically finite. Those that are homologically finite are identified by the following result; see [@christensen:gd (3.1.10)] for a proof.
\[para:auslander-2\] *Let $X$ be a homologically finite complex. Then $X$ is in ${\mathcal{A}}(R)$ if and only if ${\mathrm{G\text{-}dim}}_R(X)<\infty$.*
The various homological dimensions are related to another invariant: depth.
\[para:depth\] Let $K$ be the Koszul complex on a generating sequence of length $n$ for the maximal ideal of $R$. The *depth* of $X$ is defined to be $${\mathrm{depth}}_R(X)=n-\sup (K\otimes_R X).$$ It is independent of the choice of generating sequence and may be calculated via the vanishing of appropriate local cohomology or ${\mathrm{Ext}}$-modules [@foxby:daafuc (2.1)].
For the basic properties of depth, we refer to [@foxby:daafuc]. However, there seems to be no available reference for the following result.
\[lem:depth-eq\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a complex of $S$-modules. If ${\mathrm{H}}(X)$ is degreewise finite over $R$, then ${\mathrm{depth}}_S(X)={\mathrm{depth}}_R(X)$.
Let $K$ denote the Koszul complex on a set of $n$ generators for the maximal ideal of $R$. Note that ${\mathrm{pd}}_S(K\otimes_R S)=n$. Thus $$\begin{aligned}
{\mathrm{depth}}_S(K\otimes_R X)&={\mathrm{depth}}_S((K\otimes_R S)\otimes_S X)\\
&={\mathrm{depth}}_S(X)-{\mathrm{pd}}_S(K\otimes_R S)\\
&={\mathrm{depth}}_S(X)-n\end{aligned}$$ where the second equality is by the Auslander-Buchsbaum formula [@foxby:daafuc (2.4)]. Now, ${\mathrm{H}}(K\otimes_R X)$ is degreewise finite over $R$ and is annihilated by the maximal ideal of $R$; see, for instance, [@iyengar:dfcait (1.2)]. Hence, each ${\mathrm{H}}_i(K\otimes_R X)$ has finite length over $R$, and, therefore, over $S$. In particular, by [@foxby:daafuc (2.7)] one has ${\mathrm{depth}}_S(K\otimes_R X)=-\sup(K\otimes_R X)$. Combining this with the displayed formulas above yields that $${\mathrm{depth}}_S(X)=n+{\mathrm{depth}}_S(K\otimes_R X)=n-\sup(K\otimes_R X)={\mathrm{depth}}_R(X).$$ This is the desired equality.
In [@foxby:daafuc (3.1)], Foxby and Iyengar extend Iversen’s Amplitude Inequality; we require a slight reformulation of their result.
\[thm:GAI\] Let $S$ be a local ring and let $P$ be a homologically finite complex of $S$-modules with ${\mathrm{pd}}_S(P)$ finite. For each homologically degreewise finite complex $X$ of $S$-modules, one has $${\mathrm{amp}}(X) \leq{\mathrm{amp}}(X{\otimes^{\mathbf{L}}}_S P) \leq{\mathrm{amp}}(X)+{\mathrm{pd}}_S(P)-\inf(P).$$ In particular, ${\mathrm{amp}}(X)$ is finite if and only if ${\mathrm{amp}}(X{\otimes^{\mathbf{L}}}_S P)$ is finite.
The inequality on the left is contained in [@foxby:daafuc (3.1)], while the one on the right is by [@foxby:hacr (7.28), (8.17)].
Here is a corollary; one can give a direct proof when the map $\alpha$ is between complexes that are homologically bounded to the right.
\[prop:tool1\] Let $S$ be a local ring, $P$ a homologically finite complex of $S$-modules with ${\mathrm{pd}}_S(P)$ finite, and let $\alpha$ be a morphism of homologically degreewise finite complexes. Then $\alpha$ is an isomorphism if and only if the induced map $\alpha{\otimes^{\mathbf{L}}}_S P$ is an isomorphism.
Let $\mathrm{C}(\alpha)$ and $\mathrm{C}(\alpha{\otimes^{\mathbf{L}}}_S P)$ denote the mapping cones of $\alpha$ and $\alpha{\otimes^{\mathbf{L}}}_S P$, respectively. The homology long exact sequence arising from mapping cones yields that ${\mathrm{H}}(\mathrm{C}(\alpha))$ is degreewise finite. Observe that $\mathrm{C}(\alpha{\otimes^{\mathbf{L}}}_S P)=\mathrm{C}(\alpha){\otimes^{\mathbf{L}}}_S P$. By the previous theorem, ${\mathrm{H}}(\mathrm{C}(\alpha))=0$ if and only if ${\mathrm{H}}(\mathrm{C}(\alpha{\otimes^{\mathbf{L}}}_S P))=0$.
It is well known that the derived tensor product of two homologically finite complexes is homologically finite when one of them has finite projective dimension. In the sequel we require the following slightly more general result, contained in [@avramov:hdouc (4.7.F)]. The proof is short and simple, and bears repetition.
\[lem:back-finite\] Let $\sigma\colon S\to T$ be a local homomorphism and let $X$ and $P$ be homologically finite complexes of modules over $S$ and $T$, respectively. If ${\mathrm{fd}}_S(P)$ is finite, then the complex of $T$-modules $X{\otimes^{\mathbf{L}}}_S P$ is homologically finite.
Replacing $X$ by a soft truncation, one may assume that $X$ is bounded; see, for example, [@christensen:gd p. 165]. With $F$ a bounded flat resolution of $P$ over $S$, the complex $X{\otimes^{\mathbf{L}}}_S P$ is isomorphic to $X\otimes_S F$, which is bounded. Thus, $X{\otimes^{\mathbf{L}}}_S P$ is homologically bounded. As to its degreewise finiteness: let $Y$ and $Q$ be minimal free resolutions of $X$ and $P$ over $S$ and $T$, respectively. Then $X{\otimes^{\mathbf{L}}}_S P$ is isomorphic to $Y\otimes_S Q$, which is a complex of finite $T$-modules. Therefore, the same is true of its homology, since $T$ is Noetherian.
G-dimension over a local homomorphism {#sec:gd}
=====================================
In this section we introduce the G-dimension over a local homomorphism and document some of its basic properties. We begin by recalling the construction of Cohen factorizations of local homomorphisms as introduced by Avramov, Foxby, and B. Herzog [@avramov:solh].
\[def:factor\] Given a local homomorphism ${\varphi}\colon (R,{{\mathfrak{m}}})\to (S,{{\mathfrak{n}}})$, the *embedding dimension* and *depth of ${\varphi}$* are $${\mathrm{edim}}({\varphi}):={\mathrm{edim}}(S/{{\mathfrak{m}}}S)\qquad\text{and}\qquad
{\mathrm{depth}}({\varphi}):={\mathrm{depth}}(S)-{\mathrm{depth}}(R).$$ A *regular* (respectively, *Gorenstein*) *factorization* of ${\varphi}$ is a diagram of local homomorphisms, $R\xrightarrow{\Dot{{\varphi}}}R'\xrightarrow{{\varphi}'}S$, where ${\varphi}={\varphi}'\Dot{{\varphi}}$, with $\Dot{{\varphi}}$ flat, the closed fibre $R'/{{\mathfrak{m}}}R'$ regular (respectively, Gorenstein) and ${\varphi}'\colon R'\to S$ surjective.
Let ${\widehat{S}}$ denote the completion of $S$ at its maximal ideal and $\iota\colon
S\to{\widehat{S}}$ be the canonical inclusion. By [@avramov:solh (1.1)] the composition $\grave{{\varphi}}=\iota{\varphi}$ admits a regular factorization $R\to R'\to {\widehat{S}}$ with $R'$ complete. Such a regular factorization is said to be a *Cohen factorization* of $\grave{{\varphi}}$.
The result below is analogous to [@avramov:rhafgd (4.3)]. Here, and elsewhere, we write ${\widehat{X}}$ for $X\otimes_S {\widehat{S}}$ when $X$ is a complex of $S$-modules.
\[thm:likeAF4.3\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. If $R\stackrel{\Dot{{\varphi}_1}}{\to} R_1\stackrel{{\varphi}_1'}{\to} {\widehat{S}}$ and $R\stackrel{\Dot{{\varphi}_2}}{\to} R_2\stackrel{{\varphi}_2'}{\to} {\widehat{S}}$ are Cohen factorizations of $\grave{{\varphi}}$, then $${\mathrm{G\text{-}dim}}_{R_1}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}_1})
= {\mathrm{G\text{-}dim}}_{R_2}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}_2}).$$
Theorem [@avramov:solh (1.2)] provides a commutative diagram $$\xymatrix{
& R_1 \ar[dr]^{{\varphi}_1'} \\
R \ar[ur]^{\Dot{{\varphi}}_1} \ar[r]^{\Dot{{\varphi}}} \ar[dr]_{\Dot{{\varphi}}_2} &
R' \ar[u]^{v_1} \ar[r]^{{\varphi}'} \ar[d]_{v_2} & {\widehat{S}} \\
& R_2\ar[ur]_{{\varphi}_2'} }$$ where ${\varphi}' \Dot{{\varphi}}$ is a third Cohen factorization of $\grave{{\varphi}}$, and each $v_i$ is surjective with kernel generated by an $R'$-regular sequence whose elements are linearly independent over $R'/{{\mathfrak{m}}}'$ in ${{\mathfrak{m}}}'/(({{\mathfrak{m}}}')^2+{{\mathfrak{m}}}R')$. Here ${{\mathfrak{m}}}$ and ${{\mathfrak{m}}}'$ are the maximal ideals of $R$ and $R'$, respectively. Let $c_i$ denote the length of a regular sequence generating $\ker(v_i)$. For $i=1,2$ one has that $$\begin{aligned}
{\mathrm{G\text{-}dim}}_{R'}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}})
& = [{\mathrm{G\text{-}dim}}_{R_i}({\widehat{X}})+c_i]-[{\mathrm{edim}}(R_i/{{\mathfrak{m}}}R_i)+c_i] \\
& = {\mathrm{G\text{-}dim}}_{R_i}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}_i})\end{aligned}$$ where [@christensen:gd (2.3.12)] gives the first equality. This gives the desired result.
\[def:gdim\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. Let $R\xrightarrow{\Dot{{\varphi}}} R'\xrightarrow{{\varphi}'} {\widehat{S}}$ be a Cohen factorization of $\grave{{\varphi}}$. The *G-dimension of $X$ over ${\varphi}$* is the quantity $${\mathrm{G\text{-}dim}}_{{\varphi}}(X):=
{\mathrm{G\text{-}dim}}_{R'}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}}).$$ Theorem \[thm:likeAF4.3\] shows that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ does not depend on the choice of Cohen factorization. Note that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)\in\{-\infty\}\cup\mathbb{Z}\cup\{\infty\}$, and also that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)=-\infty$ if and only if ${\mathrm{H}}(X)=0$.
The *G-dimension of ${\varphi}$* is defined to be $${\mathrm{G\text{-}dim}}({\varphi}):={\mathrm{G\text{-}dim}}_{{\varphi}}(S).$$ It is clear from the definitions that the corresponding notion of the finiteness of ${\mathrm{G\text{-}dim}}({\varphi})$ agrees with that in [@avramov:rhafgd].
Here are some properties of the ${\mathrm{G\text{-}dim}}_{{\varphi}}(\text{-})$.
\[props\] Fix a local homomorphism ${\varphi}\colon R\to S$, a Cohen factorization $R\to R'\to {\widehat{S}}$ of $\grave{{\varphi}}$, and a homologically finite complex $X$ of $S$-modules.
\[subprop:1\] Let ${\widehat{{\varphi}}}\colon{\widehat{R}}\to{\widehat{S}}$ denote the map induced on completions. One has $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{G\text{-}dim}}_{\grave{{\varphi}}}({\widehat{X}})
={\mathrm{G\text{-}dim}}_{{\widehat{{\varphi}}}}({\widehat{X}}).$$ More generally, let $I$ and $J$ be proper ideals of $R$ and $S$, respectively, with $IS\subseteq J$, and let ${\widetilde}{R}$ and ${\widetilde}{S}$ denote the respective completions. With ${\widetilde}{{\varphi}}\colon{\widetilde}{R}\to{\widetilde}{S}$ the induced map, one has $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{G\text{-}dim}}_{{\widetilde}{{\varphi}}}({\widetilde}{S}\otimes_S X).$$ This is because the completion of ${\widetilde}{{\varphi}}$ at the maximal ideal of ${\widetilde}{S}$ is ${\widehat{{\varphi}}}$.
\[subprop:3\] If $X\simeq X'\oplus X''$, then ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)=\max\{{\mathrm{G\text{-}dim}}_{{\varphi}}(X'),{\mathrm{G\text{-}dim}}_{{\varphi}}(X'')\}$; this follows from the corresponding property of the classical G-dimension.
\[subprop:5\] If ${\varphi}$ has a regular factorization $R\xrightarrow{{\varphi}_1} R_1\xrightarrow{{\varphi}'} S$, then $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{G\text{-}dim}}_{R_1}(X)-{\mathrm{edim}}({\varphi}_1),$$ because the diagram $R\xrightarrow{\grave{{\varphi}_1}} {\widehat{R_1}}\xrightarrow{{\widehat{{\varphi}'}}}
{\widehat{S}}$ is a Cohen factorization of $\grave{{\varphi}}$.
\[subprop:4\] If ${\varphi}$ is surjective, then $R\xrightarrow{=}R\xrightarrow{{\varphi}} S$ is a regular factorization, so $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{G\text{-}dim}}_R(X).$$ Corollary \[cor:vf-is-finite\] below generalizes this to the case when ${\mathrm{H}}(X)$ is finite over $R$.
The following theorem is an extension of the Auslander-Bridger formula, which is the special case ${\varphi}=\mathrm{id}_R$.
\[thm:AB\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. If ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)<\infty$, then $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{depth}}(R)-{\mathrm{depth}}_S(X)$$
Let $R\to R'\xrightarrow{{\varphi}'}{\widehat{S}}$ be a Cohen factorization of $\grave{{\varphi}}$, and let ${{\mathfrak{m}}}$ be the maximal ideal of $R$. The classical Auslander-Bridger formula gives the first of the following equalities; the flatness of $R\to R'$ and the surjectivity of ${\varphi}'$ imply the second; the regularity of $R'/{{\mathfrak{m}}}R'$ yields the third. $$\begin{aligned}
{\mathrm{G\text{-}dim}}_{R'}({\widehat{X}})
& = {\mathrm{depth}}(R') - {\mathrm{depth}}_{R'}({\widehat{X}}) \\
& = [{\mathrm{depth}}(R)+{\mathrm{depth}}(R'/{{\mathfrak{m}}}R')] - {\mathrm{depth}}_{{\widehat{S}}}({\widehat{X}}) \\
& = {\mathrm{depth}}(R)-{\mathrm{depth}}_S(X)+{\mathrm{edim}}(R'/{{\mathfrak{m}}}R')\end{aligned}$$ This gives the desired equality.
As in the classical case, described in \[para:auslander-2\], when $R$ has a dualizing complex one can detect finiteness of ${\mathrm{G\text{-}dim}}_{{\varphi}}(\text{-})$ in terms of membership in the Auslander category of $R$.
\[prop:likeAF4.3\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $R\xrightarrow{\dot{{\varphi}}}
R'\xrightarrow{{\varphi}'} {\widehat{S}}$ a Cohen factorization of $\grave{{\varphi}}$. The following conditions are equivalent for each homologically finite complex $X$ of $S$-modules.
1. ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)<\infty$.
2. ${\mathrm{G\text{-}dim}}_{R'}({\widehat{X}})<\infty$.
3. ${\widehat{X}}$ is in ${\mathcal{A}}(R')$.
4. ${\widehat{X}}$ is in ${\mathcal{A}}({\widehat{R}})$.
When $R$ possesses a dualizing complex, these conditions are equivalent to:
1. $X$ is in ${\mathcal{A}}(R)$.
Indeed, (a) $\iff$ (b) by definition, while (b) $\iff$ (c) by [@christensen:gd (3.1.10)]. Moving on, (c) $\iff$ (d) is contained in [@avramov:rhafgd (3.7.b)], and, when $R$ has a dualizing complex, the equivalence of (d) and (e) is [@avramov:rhafgd (3.7.a)].
Now we turn to the behavior of G-dimension with respect to localizations. Recall that, given a prime ideal ${{\mathfrak{p}}}$ and a totally reflexive $R$-module $G$, the $R_{{{\mathfrak{p}}}}$-module $G_{{{\mathfrak{p}}}}$ is totally reflexive. From this it is clear that for any homologically finite complex $W$, one has ${\mathrm{G\text{-}dim}}_{R_{{{\mathfrak{p}}}}}(W_{{{\mathfrak{p}}}})\leq{\mathrm{G\text{-}dim}}_R(W)$; see [@christensen:gd (2.3.11)]. For G-dimensions over ${\varphi}$, we know only the following weaker result; see also [@avramov:homolhattfe (10.2)]. Its proof is omitted for it is verbatim that of [@avramov:rhafgd (4.5)], which is the special case $X=S$; only, one uses \[prop:likeAF4.3\] instead of [@avramov:rhafgd (4.3)].
\[prop:localization\] Let ${\varphi}\colon R\to S$ be a local homomorphism, $X$ a homologically finite complex of $S$-modules. Let ${{\mathfrak{q}}}$ be a prime ideal of $S$ and ${\varphi}_{{\mathfrak{q}}}$ the local homomorphism $R_{{{\mathfrak{q}}}\cap R}\to S_{{\mathfrak{q}}}$.
If ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)<\infty$, then ${\mathrm{G\text{-}dim}}_{{\varphi}_{{\mathfrak{q}}}}(X_{{\mathfrak{q}}})<\infty$ under each of the conditions:
1. ${\varphi}$ is essentially of finite type; or
2. $R$ has Gorenstein formal fibres.
The next result shows that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ can be computed via any *Gorenstein* factorization of ${\varphi}$, when such a factorization exists; see Definition \[def:factor\]. When the factorization in the statement is regular, the equation becomes ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{G\text{-}dim}}_{R'}(X)-{\mathrm{edim}}(\Dot{{\varphi}})$; compare this with Definition \[def:gdim\].
\[prop:comp-via-Gor-hom\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. If ${\varphi}$ possesses a Gorenstein factorization $R\xrightarrow{\Dot{{\varphi}}}
R'\xrightarrow{{\varphi}'} S$, then $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)
={\mathrm{G\text{-}dim}}_{R'}(X)-{\mathrm{depth}}(\Dot{{\varphi}}).$$
One may assume that ${\mathrm{H}}(X)\neq 0$. It is straightforward to verify that the diagram $R\to {\widehat{R'}}\to{\widehat{S}}$ is a Gorenstein factorization. It follows from [@avramov:rhafgd (3.7)] that ${\widehat{X}}$ is in ${\mathcal{A}}({\widehat{R}})$ exactly when ${\widehat{X}}$ is in ${\mathcal{A}}({\widehat{R'}})$, and, by Proposition \[prop:likeAF4.3\], this implies that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ is finite exactly when ${\mathrm{G\text{-}dim}}_{R'}(X)$ is finite. So one may assume that both the numbers in question are finite. The Auslander-Bridger formula \[thm:AB\], and the fact that ${\mathrm{depth}}_S(X)={\mathrm{depth}}_{R'}(X)$, give the first of the following equalities: $$\begin{aligned}
{\mathrm{G\text{-}dim}}_{R'}(X)&={\mathrm{G\text{-}dim}}_{{\varphi}}(X)+ [{\mathrm{depth}}(R')-{\mathrm{depth}}(R)] \\
& = {\mathrm{G\text{-}dim}}_{{\varphi}}(X)+{\mathrm{depth}}(\Dot{{\varphi}}).\end{aligned}$$ The second equality is by definition.
Projective dimension {#sec:pd}
====================
In this section we introduce a new invariant: projective dimension over a local homomorphism. To begin with, one has the following proposition. Its proof is similar to that of Theorem \[thm:likeAF4.3\], and hence it is omitted.
\[prop:def-pdim\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. If $R\stackrel{\Dot{{\varphi}_1}}{\to} R_1\stackrel{{\varphi}_1'}{\to} {\widehat{S}}$ and $R\stackrel{\Dot{{\varphi}_2}}{\to} R_2\stackrel{{\varphi}_2'}{\to} {\widehat{S}}$ are Cohen factorizations of $\grave{{\varphi}}$, then
[3]{} && \_[R\_1]{}()-() & = \_[R\_2]{}()-(). &&
This leads to the following:
Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. The *projective dimension of $X$ over ${\varphi}$* is the quantity $${\mathrm{pd}}_{{\varphi}}(X):=
{\mathrm{pd}}_{R'}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}})$$ for some Cohen factorization $R\to R'\to
{\widehat{S}}$ of $\grave{{\varphi}}$. The *projective dimension of ${\varphi}$* is defined to be $${\mathrm{pd}}({\varphi}):={\mathrm{pd}}_{{\varphi}}(S).$$
The first remark concerning this invariant is that there is an “Auslander-Buchsbaum formula”, which can be verified along the lines of its G-dimension counterpart, Theorem \[thm:AB\].
\[prop:pdAB\] If ${\mathrm{pd}}_{{\varphi}}(X)<\infty$, then $${\mathrm{pd}}_{{\varphi}}(X)={\mathrm{depth}}(R)-{\mathrm{depth}}_S(X)$$
Other basic rules that govern the behavior of this invariant can be read from [@avramov:solh], although it was not defined there explicitly. For instance, [@avramov:solh (3.2)], rather, its extension to complexes, see [@avramov:homolhattfe (2.5)], translates to
\[prop:pdvsfd\] There are inequalities: $${\mathrm{fd}}_R(X) - {\mathrm{edim}}{\varphi}\leq {\mathrm{pd}}_{{\varphi}}(X) \leq {\mathrm{fd}}_R(X)\,.$$ In particular, the finiteness of ${\mathrm{pd}}_{{\varphi}}(X)$ is independent of $S$ and ${\varphi}$.
One can interpret the difference between ${\mathrm{fd}}_R(X)$ and ${\mathrm{pd}}_{{\varphi}}(X)$ in terms of appropriate depths:
\[prop:pd-fd\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. Then $${\mathrm{pd}}_{{\varphi}}(X) = {\mathrm{fd}}_R(X)+ {\mathrm{depth}}_R(X)-{\mathrm{depth}}_S(X).$$
Indeed, by Property \[prop:pdvsfd\], we may assume that both ${\mathrm{fd}}_R(X)$ and ${\mathrm{pd}}_{\varphi}(X)$ are finite. Now, the first equality below is given by [@avramov:hdouc (5.5)], and the second is due to [@iyengar:dfcait (2.1)]. $${\mathrm{fd}}_R(X)=\sup(X{\otimes^{\mathbf{L}}}_R k)={\mathrm{depth}}(R)-{\mathrm{depth}}_R(X).$$ The Auslander-Buchsbaum formula \[prop:pdAB\] gives the desired formula.
The G-dimension of a finite module, or a complex, is bounded above by its projective dimension. The same behavior carries over to modules and complexes over ${\varphi}$.
\[prop:pd-gd-ineq\] Let ${\varphi}\colon R\to S$ be a local homomorphism. For each homologically finite complex $X$ of $S$-modules, one has $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)\leq{\mathrm{pd}}_{{\varphi}}(X);$$ equality holds when ${\mathrm{pd}}_{{\varphi}}(X)<\infty$.
Let $R\xrightarrow{\Dot{{\varphi}}} R'\to {\widehat{S}}$ be a Cohen factorization of $\grave{{\varphi}}$. Then $${\mathrm{G\text{-}dim}}_{{\varphi}}(X)={\mathrm{G\text{-}dim}}_{R'}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}})\leq
{\mathrm{pd}}_{R'}({\widehat{X}})-{\mathrm{edim}}(\Dot{{\varphi}}) ={\mathrm{pd}}_{{\varphi}}(X)$$ with equality if ${\mathrm{pd}}_{R'}({\widehat{X}})$ is finite; see [@christensen:gd (2.3.10)].
Further results concerning ${\mathrm{pd}}_{{\varphi}}(\text{-})$ are given toward the end of the next section. One can introduce also Betti numbers and Poincaré series over local homomorphisms; an in-depth analysis of these and related invariants is carried out in [@avramov:homolhattfe].
Ascent and descent of G-dimension {#sec:descent}
=================================
The heart of this section, and indeed of this paper, is the following theorem. It is a vast generalization of a stability result of Yassemi [@yassemi:gd (2.15)], and contains Theorem B from the introduction.
\[thm:stability-tensor\] Let ${\varphi}\colon R\to S$ and $\sigma\colon S\to T$ be local homomorphisms. Let $P$ be a complex of $T$-modules that is homologically finite with ${\mathrm{pd}}_{\sigma}(P)$ finite. For every homologically finite complex $X$ of $S$-modules $${\mathrm{G\text{-}dim}}_{\sigma{\varphi}}(X{\otimes^{\mathbf{L}}}_S P)={\mathrm{G\text{-}dim}}_{{\varphi}}(X)+{\mathrm{pd}}_{\sigma}(P).$$ In particular, ${\mathrm{G\text{-}dim}}_{\sigma{\varphi}}(X{\otimes^{\mathbf{L}}}_S P)$ and ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ are simultaneously finite.
The theorem is proved in \[pf:stability-tensor\], toward the end of the section. It is worth remarking that the displayed formula is *not* an immediate consequence of the finiteness of the G-dimensions in question and appropriate Auslander-Bridger formulas. What is missing is an extension of the Auslander-Buchsbaum formula \[prop:pdAB\]; namely, under the hypotheses of the theorem above $${\mathrm{pd}}_{\sigma}(P)={\mathrm{depth}}_S(X)-{\mathrm{depth}}_T(X{\otimes^{\mathbf{L}}}_SP)\,.$$ It is not hard to deduce this equality from [@foxby:daafuc (2.4)], using Cohen factorizations; see the argument in \[pf:stability-tensor\].
We draw a few corollaries that illustrate the power of Theorem \[thm:stability-tensor\]. The first one is just the special case $X=S$ and $P=T$.
\[thm:gdim-compose\] Let ${\varphi}\colon R\to S$ and $\sigma\colon S\to T$ be local homomorphisms with ${\mathrm{pd}}(\sigma)$ finite. Then $${\mathrm{G\text{-}dim}}(\sigma{\varphi})={\mathrm{G\text{-}dim}}({\varphi})+{\mathrm{pd}}(\sigma).$$ In particular, ${\mathrm{G\text{-}dim}}(\sigma{\varphi})$ is finite if and only if ${\mathrm{G\text{-}dim}}({\varphi})$ is finite.
The following example illustrates that the hypothesis on $\sigma$ cannot be weakened to “${\mathrm{G\text{-}dim}}(\sigma)$ finite”. A similar example is constructed in [@apassov:afm p. 931].
\[ex:not-weaken\] Let $R$ be a local, Cohen-Macaulay ring with canonical module $\omega$. Set $S=R\ltimes\omega$, the “idealization” of $\omega$, and ${\varphi}\colon R\to S$ the canonical inclusion. Let $T=S/\omega\cong R$ with $\sigma\colon S\to T$ the natural surjection.
Now, $\sigma{\varphi}=\mathrm{id}^R$, hence ${\mathrm{G\text{-}dim}}(\sigma{\varphi})=0$, for example, by Proposition \[prop:pd-gd-ineq\]; also, $S$ is Gorenstein [@bruns:cmr (3.3.6)], so ${\mathrm{G\text{-}dim}}(\sigma)$ is finite.
We claim that ${\mathrm{G\text{-}dim}}({\varphi})$ is finite if and only if $R$ is Gorenstein. Indeed, ${\mathrm{G\text{-}dim}}({\varphi})$ and ${\mathrm{G\text{-}dim}}_R(S)$ are simultaneously finite, by Corollary \[cor:vf-is-finite\]. From \[subprop:3\] it follows that ${\mathrm{G\text{-}dim}}_R(S)<\infty$ if and only if ${\mathrm{G\text{-}dim}}_R(\omega)<\infty$. The finiteness of ${\mathrm{G\text{-}dim}}_R(\omega)$ is equivalent to $R$ being Gorenstein [@christensen:gd (3.4.12)].
As noted in the introduction, Theorem \[thm:gdim-compose\] allows one to extend certain results of Avramov and Foxby on (quasi-)Gorenstein homomorphisms. In order to describe these, and because they are required in the sequel, we recall the relevant notions.
\[para:quasi-Gorenstein\] \[para:1\] Let $R$ be a local ring with residue field $k$. The *Bass series of $R$* is the formal power series $I_R(t)=\sum_i\mu^i_R(R)t^i$ where $\mu_R^i(R)={\mathrm{rank}}_k{\mathrm{Ext}}_R^i(k,R)$. An important property of the Bass series is that $R$ is Gorenstein if and only if $I_R(t)$ is a polynomial [@matsumura:crt (18.1)].
Let ${\varphi}\colon (R,{{\mathfrak{m}}})\to (S,{{\mathfrak{n}}})$ be a local homomorphism of finite G-dimension. Let $I_{{\varphi}}(t)$ denote the *Bass series of ${\varphi}$*, introduced in [@avramov:rhafgd Section 7]. The Bass series is a formal Laurent series with nonnegative integer coefficients and satisfies the equality $$\label{eqn:bass}
I_R(t)I_{{\varphi}}(t)=I_S(t). \tag{$\dagger$}$$ Let $\sigma\colon (S,{{\mathfrak{n}}})\to (T,{{\mathfrak{p}}})$ also be a homomorphism of finite G-dimension. Assuming ${\mathrm{G\text{-}dim}}(\sigma{\varphi})$ is finite as well, it follows from (\[eqn:bass\]) that $$\label{eqn:more-bass}
I_{\sigma{\varphi}}(t)=I_{\sigma}(t)I_{{\varphi}}(t). \tag{$\ddagger$}$$
The homomorphism ${\varphi}$ is said to be *quasi-Gorenstein at ${{\mathfrak{n}}}$* if $I_{{\varphi}}(t)$ is a Laurent *polynomial*. When ${\mathrm{pd}}({\varphi})<\infty$ and ${\varphi}$ is quasi-Gorenstein at ${{\mathfrak{n}}}$, one says that ${\varphi}$ is *Gorenstein at ${{\mathfrak{n}}}$*; see [@avramov:rhafgd (7.7.1)].
A noteworthy aspect of the class of such homomorphisms is that it is closed under compositions: if ${\varphi}$ and $\sigma$ are quasi-Gorenstein at ${{\mathfrak{n}}}$ and ${{\mathfrak{p}}}$, respectively, then $\sigma{\varphi}$ is quasi-Gorenstein at ${{\mathfrak{p}}}$. This follows from [@avramov:rhafgd (7.10)] and (\[eqn:more-bass\]) above.
The result below is a decomposition theorem for Gorenstein and quasi-Gorenstein homomorphisms; it is Theorem C announced in the introduction.
\[thm:fin-fd-descent\] Let ${\varphi}\colon (R,{{\mathfrak{m}}})\to (S,{{\mathfrak{n}}})$ and $\sigma\colon (S,{{\mathfrak{n}}})\to (T,{{\mathfrak{p}}})$ be local homomorphisms with ${\mathrm{pd}}(\sigma)$ finite. If $\sigma{\varphi}$ is (quasi-)Gorenstein at ${{\mathfrak{p}}}$, then ${\varphi}$ is (quasi-)Gorenstein at ${{\mathfrak{n}}}$ and $\sigma$ is Gorenstein at ${{\mathfrak{p}}}$.
Assume that $\sigma{\varphi}$ is quasi-Gorenstein at ${{\mathfrak{p}}}$; so, it has finite G-dimension, and $I_{\sigma{\varphi}}(t)$ is a Laurent polynomial. Now, ${\mathrm{G\text{-}dim}}({\varphi})$ is finite, by Theorem \[thm:gdim-compose\], as is ${\mathrm{G\text{-}dim}}(\sigma)$, by hypothesis, so equality (\[eqn:more-bass\]) in \[para:quasi-Gorenstein\] applies to yield an equality of formal Laurent series $$I_{\sigma{\varphi}}(t)=I_{\sigma}(t)I_{{\varphi}}(t)$$ In particular, $I_{\sigma}(t)$ and $I_{{\varphi}}(t)$ are Laurent polynomials as well. Thus, both $\sigma$ and ${\varphi}$ are quasi-Gorenstein at the appropriate maximal ideals. Moreover, $\sigma$ is Gorenstein because ${\mathrm{pd}}(\sigma)$ is finite.
Suppose that $\sigma{\varphi}$ is Gorenstein at ${{\mathfrak{p}}}$. Since ${\mathrm{pd}}(\sigma{\varphi})$ and ${\mathrm{pd}}(\sigma)$ are both finite, [@foxby:daafuc (3.2)], in conjunction with Proposition \[prop:pd-fd\], yields that ${\mathrm{pd}}({\varphi})$ is finite. The already established part of the theorem gives the desired conclusion.
The next theorem generalizes another stability result of Yassemi [@yassemi:gd (2.14)].
\[thm:stability-hom\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $P$ a homologically finite complex of $S$-modules with ${\mathrm{pd}}_S(P)$ finite. For every homologically finite complex $X$ of $S$-modules $$\begin{aligned}
{\mathrm{G\text{-}dim}}_{{\varphi}}({\mathbf{R}\mathrm{Hom}}_S(P,X))&={\mathrm{G\text{-}dim}}_{{\varphi}}(X)-\inf(P).\end{aligned}$$ Thus, ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ and ${\mathrm{G\text{-}dim}}_{{\varphi}}({\mathbf{R}\mathrm{Hom}}_S(P,X))$ are simultaneously finite.
The tensor evaluation morphism $X{\otimes^{\mathbf{L}}}_S{\mathbf{R}\mathrm{Hom}}_S(P,S)\to{\mathbf{R}\mathrm{Hom}}_S(P,X)$ is an isomorphism, as $P$ has finite projective dimension. Thus $$\begin{aligned}
{\mathrm{G\text{-}dim}}_{{\varphi}}({\mathbf{R}\mathrm{Hom}}_S(P,X))&={\mathrm{G\text{-}dim}}_{{\varphi}}(X{\otimes^{\mathbf{L}}}_S{\mathbf{R}\mathrm{Hom}}_S(P,S))\\
&={\mathrm{G\text{-}dim}}_{{\varphi}}(X)+{\mathrm{pd}}_S({\mathbf{R}\mathrm{Hom}}_S(P,S))\\
&={\mathrm{G\text{-}dim}}_{{\varphi}}(X)-\inf(P)\end{aligned}$$ where the second equality follows from Theorem \[thm:stability-tensor\] because ${\mathbf{R}\mathrm{Hom}}_S(P,S)$ has finite projective dimension over $S$.
Next we record the analogue of Theorem \[thm:stability-tensor\] for projective dimension; its proof is postponed to \[pf:stab-pd-hom\].
\[thm:stab-pd-tensor\] Let ${\varphi}\colon R\to S$ and $\sigma\colon S\to T$ be local homomorphisms. Let $P$ be a complex of $T$-modules that is homologically finite with ${\mathrm{pd}}_{\sigma}(P)$ finite. For every homologically finite complex $X$ of $S$-modules $${\mathrm{pd}}_{\sigma{\varphi}}(X{\otimes^{\mathbf{L}}}_S P)={\mathrm{pd}}_{{\varphi}}(X)+{\mathrm{pd}}_{\sigma}(P).$$ In particular, ${\mathrm{pd}}_{\sigma{\varphi}}(X{\otimes^{\mathbf{L}}}_S P)$ and ${\mathrm{pd}}_{{\varphi}}(X)$ are simultaneously finite.
Finally, here is the analogue of Theorem \[thm:stability-hom\]; it can be deduced from \[thm:stab-pd-tensor\] in the same way that \[thm:stability-hom\] was deduced from \[thm:stability-tensor\].
\[thm:stab-pd-hom\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $P$ a homologically finite complex of $S$-modules with ${\mathrm{pd}}_S(P)$ finite. For every homologically finite complex $X$ of $S$-modules $$\begin{aligned}
{\mathrm{pd}}_{{\varphi}}({\mathbf{R}\mathrm{Hom}}_S(P,X))&={\mathrm{pd}}_{{\varphi}}(X)-\inf(P).\end{aligned}$$ In particular, ${\mathrm{pd}}_{{\varphi}}(X)$ and ${\mathrm{pd}}_{{\varphi}}({\mathbf{R}\mathrm{Hom}}_S(P,X))$ are simultaneously finite.
The proof of Theorem \[thm:stability-tensor\] uses a convenient construction, essentially given in [@avramov:solh], of Cohen factorizations of compositions of local homomorphisms.
\[para:big-diagram\] Let $R\xrightarrow{{\varphi}} S\xrightarrow{\sigma} T$ be local homomorphisms, and let $$R\xrightarrow{\Dot{{\varphi}}} R'\xrightarrow{{\varphi}'}S
\qquad\text{and}\qquad
R'\xrightarrow{\Dot{\rho}}R''\xrightarrow{\rho'}T$$ be regular factorizations of ${\varphi}$ and $\sigma{\varphi}'$, respectively. The map $\rho'$ factors through the tensor product $S'=R''\otimes_{R'}S$ giving the following commutative diagram $$\xymatrixrowsep{2.5pc}
\xymatrixcolsep{2.5pc}
\xymatrix {
&& R'' \ar@{->}[dr]^{{\varphi}''} \ar@/^2.5pc/[ddrr]^{\rho'=\sigma'{\varphi}''} \\
& R' \ar@{->}[dr]^{{\varphi}'} \ar@{->}[ur]^{\dot\rho} && S' \ar@{->}[dr]^{\sigma'} \\
R \ar@{->}[rr]^{{\varphi}} \ar@/^2.5pc/[uurr]^{\dot\rho \dot{\varphi}} \ar@{->}[ur]^{\dot{\varphi}} && S
\ar@{->}[rr]^{\sigma} \ar@{->}[ur]^{\dot\sigma} && T }$$ where $\dot\sigma$ and ${\varphi}''$ are the natural maps to the tensor products. Then the diagrams $S\to S'\to T$, $R'\to R''\to S'$, and $R\to R''\to T$ are regular factorizations with $${\mathrm{edim}}(\Dot{\sigma})={\mathrm{edim}}(\Dot{\rho}) \qquad\text{and}\qquad
{\mathrm{edim}}(\Dot{\rho}\Dot{{\varphi}}) ={\mathrm{edim}}(\Dot{\rho})+{\mathrm{edim}}(\Dot{{\varphi}}).$$ Indeed, by flat base change, $\Dot{\sigma}$ is flat and has closed fiber $S'\otimes_S
l=R''\otimes_{R'} l$, which is regular. Here $l$ is the common residue field of $R'$ and $S$. This tells us that $S\to S'\to T$ is a regular factorization and that ${\mathrm{edim}}(\Dot{\sigma})={\mathrm{edim}}(\Dot{\rho})$.
The diagram $R'\to R''\to S'$ is a regular factorization because $\Dot{\rho}$ is flat with a regular closed fibre, by hypothesis, and ${\varphi}''$ is surjective, by base change.
As to the diagram $R\to R''\to T$, let ${{\mathfrak{m}}}$ and ${{\mathfrak{m}}}'$ denote the maximal ideals of $R$ and $R'$, respectively. The induced map $R'/{{\mathfrak{m}}}R'\to R''/{{\mathfrak{m}}}R''$ is flat with closed fibre $R''/{{\mathfrak{m}}}'R''$. Since $R'/{{\mathfrak{m}}}R'$ and $R''/{{\mathfrak{m}}}'R''$ are both regular, the same is true of $R''/{{\mathfrak{m}}}R''$, by [@bruns:cmr (2.2.12)]. Thus, $R\to R''\to T$ is a regular factorization. Furthermore, it is stated explicitly in the proof of *loc. cit.* that ${\mathrm{edim}}(R''/{{\mathfrak{m}}}R'')={\mathrm{edim}}(R'/{{\mathfrak{m}}}R')+{\mathrm{edim}}(R''/{{\mathfrak{m}}}' R'')$, which explains the second formula above.
*Proof of Theorem \[thm:stability-tensor\].* \[pf:stability-tensor\] Note that $X{\otimes^{\mathbf{L}}}_S P$ is homologically finite over $T$ by Lemma \[lem:back-finite\], so one may speak of its G-dimension over $\sigma{\varphi}$. Passing to the completions of $S$ and $T$ at their respective maximal ideals, and replacing $X$ and $P$ by ${\widehat{S}}\otimes_S X$ and ${\widehat{T}}\otimes_T P$, respectively, one may assume that $S$ and $T$ are complete. In doing so, one uses the isomorphism $$({\widehat{S}}\otimes_S X){\otimes^{\mathbf{L}}}_{{\widehat{S}}} ({\widehat{T}}\otimes_T P)
\simeq {\widehat{T}}\otimes_T (X{\otimes^{\mathbf{L}}}_S P).$$
The next step is the reduction to the case where ${\varphi}$ and $\sigma$ are surjective. To achieve this, take Cohen factorizations $R\to R'\to S$ and $R'\to R''\to T$, and expand to a commutative diagram as in \[para:big-diagram\].
Let $X'=S'\otimes_S X$. Since $S'=R''\otimes_{R'}S$, by construction, $X'\cong
R''\otimes_{R'} X$ and hence $X'{\otimes^{\mathbf{L}}}_{S'}P\simeq X{\otimes^{\mathbf{L}}}_S P$. Since $R'\to R''$ is faithfully flat, \[lem:fflat\] yields $${\mathrm{G\text{-}dim}}_{R'}(X)={\mathrm{G\text{-}dim}}_{R''}(X').$$ The preceding equality, in conjunction with those in \[para:big-diagram\], yields $$\begin{aligned}
{\mathrm{pd}}_{\sigma}(P) & = {\mathrm{pd}}_{S'}(P)-{\mathrm{edim}}(\Dot{\rho}) \\
{\mathrm{G\text{-}dim}}_{{\varphi}}(X) & ={\mathrm{G\text{-}dim}}_{R''}(X')-{\mathrm{edim}}(\Dot{{\varphi}}) \\
{\mathrm{G\text{-}dim}}_{\sigma{\varphi}}(X{\otimes^{\mathbf{L}}}_S P) & = {\mathrm{G\text{-}dim}}_{R''}(X'{\otimes^{\mathbf{L}}}_{S'}P)-{\mathrm{edim}}(\Dot{\rho})
-{\mathrm{edim}}(\Dot{{\varphi}})\end{aligned}$$ Therefore, it suffices to verify the identity for the diagram $R''\to S'\to T$ and complexes $X'$ and $P$. This places us in the situation where $R\to S$ is surjective, $P$ is homologically finite over $R$, and then the equality we seek is $${\mathrm{G\text{-}dim}}_R(X{\otimes^{\mathbf{L}}}_S P)={\mathrm{G\text{-}dim}}_R(X)+{\mathrm{pd}}_S(P).$$
It suffices to prove that the G-dimensions over $R$ of $X$ and of $X{\otimes^{\mathbf{L}}}_S P$ are simultaneously finite. For, when they are both finite, one has $$\begin{aligned}
{\mathrm{G\text{-}dim}}_R(X{\otimes^{\mathbf{L}}}_S P)
& = {\mathrm{depth}}(R)-{\mathrm{depth}}_S(X{\otimes^{\mathbf{L}}}_S P) \\
& = {\mathrm{depth}}(R)-{\mathrm{depth}}_S(X)+{\mathrm{pd}}_S(P) \\
& = {\mathrm{G\text{-}dim}}_R(X)+{\mathrm{pd}}_S(P)\end{aligned}$$ where the first and the third equalities are by the Auslander-Bridger formula, while the one in the middle is by [@iyengar:dfcait (2.2)].
The rest of the proof is dedicated to proving that $X$ and $X{\otimes^{\mathbf{L}}}_S P$ have finite G-dimension over $R$ simultaneously. In view of \[para:reflexive-2\], this is tantamount to proving:
1. ${\mathbf{R}\mathrm{Hom}}_R(X,R)$ is homologically bounded if and only if the same is true of ${\mathbf{R}\mathrm{Hom}}_R(X{\otimes^{\mathbf{L}}}_S P,R)$; and
2. the biduality morphisms $\delta^R_X$ and $\delta^R_{X{\otimes^{\mathbf{L}}}_S P}$, defined as in \[para:reflexive\], are isomorphisms simultaneously.
The proofs of (a) and (b) use the following observation: when $U$ and $V$ are complexes of $S$-modules such that $V$ is homologically finite and ${\mathrm{pd}}_S(V)<\infty$, the natural morphism $$\label{eqn:1}
\theta_{UV}\colon{\mathbf{R}\mathrm{Hom}}_R(U,R){\otimes^{\mathbf{L}}}_S{\mathbf{R}\mathrm{Hom}}_S(V,S)\to
{\mathbf{R}\mathrm{Hom}}_R(U{\otimes^{\mathbf{L}}}_S V,R) \tag{$\ast$}$$ is an isomorphism. Indeed, it is the composition of tensor evaluation $${\mathbf{R}\mathrm{Hom}}_R(U,R){\otimes^{\mathbf{L}}}_S{\mathbf{R}\mathrm{Hom}}_S(V,S) \to {\mathbf{R}\mathrm{Hom}}_S(V, {\mathbf{R}\mathrm{Hom}}_R(U, R)),$$ which is an isomorphism for $V$ as above, followed by adjunction $${\mathbf{R}\mathrm{Hom}}_S(V, {\mathbf{R}\mathrm{Hom}}_R(U, R))\xrightarrow{\simeq} {\mathbf{R}\mathrm{Hom}}_R(U{\otimes^{\mathbf{L}}}_S V,R).$$
*Proof of (a)*. Since ${\mathbf{R}\mathrm{Hom}}_S(P,S)$ has finite projective dimension over $S$, one has the isomorphism $${\theta}_{XP}\colon{\mathbf{R}\mathrm{Hom}}_R(X,R){\otimes^{\mathbf{L}}}_S{\mathbf{R}\mathrm{Hom}}_S(P,S)\to
{\mathbf{R}\mathrm{Hom}}_R(X{\otimes^{\mathbf{L}}}_S P,R).$$ Thus, Theorem \[thm:GAI\] implies the desired equivalence.
*Proof of (b)*. Consider the following commutative diagram of morphisms of complexes of $S$-modules. $$\xymatrix{
X{\otimes^{\mathbf{L}}}_S P \ar[r]^(0.3){(\delta_X^R){\otimes^{\mathbf{L}}}_S P} \ar@{=}[dd] &
{\mathbf{R}\mathrm{Hom}}_R({\mathbf{R}\mathrm{Hom}}_R(X,R),R){\otimes^{\mathbf{L}}}_S P \ar[d]_{\simeq}^{\nu} \\
& {\mathbf{R}\mathrm{Hom}}_R({\mathbf{R}\mathrm{Hom}}_R(X,R){\otimes^{\mathbf{L}}}_S{\mathbf{R}\mathrm{Hom}}_S(P,S),R) \\
X{\otimes^{\mathbf{L}}}_S P \ar[r]^(0.3){\delta^R_{(X{\otimes^{\mathbf{L}}}_S P)}} & {\mathbf{R}\mathrm{Hom}}_R({\mathbf{R}\mathrm{Hom}}_R(X{\otimes^{\mathbf{L}}}_S
P,R),R) \ar[u]^{\simeq}_{{\mathbf{R}\mathrm{Hom}}_R(\theta_{X\!P},R)} }$$ The morphism $\nu$ is the composition ${\theta}_{UV}\circ (1{\otimes^{\mathbf{L}}}_S\delta^S_P)$ where $U={\mathbf{R}\mathrm{Hom}}_R(X,R)$ and $V={\mathbf{R}\mathrm{Hom}}_S(P,S)$. Note that $\delta^S_P$, and hence $1{\otimes^{\mathbf{L}}}_S\delta^S_P$, is an isomorphism because ${\mathrm{pd}}_S(P)$ is finite. Furthermore, ${\theta}_{UV}$ is an isomorphism since ${\mathrm{pd}}_S(V)$ is finite. This is why $\nu$ is an isomorphism.
From the diagram one obtains that $\delta_{X{\otimes^{\mathbf{L}}}_S P}^R$ is an isomorphism if and only if $(\delta^R_X){\otimes^{\mathbf{L}}}_S P$ is. By Proposition \[prop:tool1\], the morphisms $(\delta^R_X){\otimes^{\mathbf{L}}}_S P$ and $\delta^R_X$ are isomorphisms simultaneously, as ${\mathrm{pd}}_S(P)$ is finite.
To wrap up this section, we give the
*Proof of Theorem \[thm:stab-pd-tensor\].* \[pf:stab-pd-hom\] Arguing as in the proof of Theorem \[thm:stability-tensor\] one reduces to the case where ${\varphi}$ and $\sigma$ are surjective and ${\mathrm{pd}}_S(P)$ is finite. In this situation, one has to verify that ${\mathrm{pd}}_{R}(X)$ and ${\mathrm{pd}}_{R}(X{\otimes^{\mathbf{L}}}_S P)$ are simultaneously finite. Let $k$ be the residue field of $R$. It suffices to show that ${\mathrm{amp}}(k{\otimes^{\mathbf{L}}}_R
X)$ and ${\mathrm{amp}}(k{\otimes^{\mathbf{L}}}_R(X{\otimes^{\mathbf{L}}}_S P))$ are simultaneously finite. By the isomorphism $$k{\otimes^{\mathbf{L}}}_R(X{\otimes^{\mathbf{L}}}_S P)\simeq(k{\otimes^{\mathbf{L}}}_R X){\otimes^{\mathbf{L}}}_S P\,,$$ this follows from Theorem \[thm:GAI\].
Detecting the Gorenstein property {#sec:apps}
=================================
The theorem below extends the Auslander-Bridger characterization [@auslander:smt (4.20)] of Gorenstein rings.
\[thm:gor-char-1\] Let $R$ be a local ring. The following conditions are equivalent.
1. $R$ is Gorenstein.
2. For every local homomorphism ${\varphi}\colon R\to S$ and for every homologically finite complex $X$ of $S$-modules, ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)<\infty$.
3. There is a local homomorphism ${\varphi}\colon R\to S$ and an ideal $I$ of $S$ such that $I\supseteq {{\mathfrak{m}}}S$, where ${{\mathfrak{m}}}$ is the maximal ideal of $R$, and ${\mathrm{G\text{-}dim}}_{{\varphi}}(S/I)<\infty$.
“(a)$\implies$(b)”. Let $R\to R'\to {\widehat{S}}$ be a Cohen factorization of $\grave{{\varphi}}$. The $R'$-module ${\mathrm{H}}({\widehat{X}})$ is finite, because the $S$-module ${\mathrm{H}}(X)$ is finite. Since $R$ is Gorenstein, so is $R'$ [@bruns:cmr (3.3.15)]. Thus, ${\mathrm{G\text{-}dim}}_{R'}({\widehat{X}})<\infty$, that is to say, ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)<\infty$; see Proposition \[prop:likeAF4.3\].
“(b)$\implies$(c)” is trivial.
“(c)$\implies$(a)”. Let $R\to R'\to {\widehat{S}}$ be a Cohen factorization. Composing with the surjection ${\widehat{S}}\xrightarrow{\pi}{\widehat{S}}/I{\widehat{S}}$ gives a diagram $R\to
R'\to {\widehat{S}}/I{\widehat{S}}$ that is also a Cohen factorization. Since ${\mathrm{G\text{-}dim}}_{R'}({\widehat{S}}/I{\widehat{S}})$ is finite, so is ${\mathrm{G\text{-}dim}}(\pi\grave{{\varphi}})$. The composition $\pi\grave{{\varphi}}$ factors through the residue field $k$ of $R$, giving the commutative diagram: $$\xymatrix{
R \ar[dr] \ar[rr] & & {\widehat{S}}/I{\widehat{S}} \\
& k \ar[ur] }$$ The map $k\to {\widehat{S}}/I{\widehat{S}}$ has finite projective dimension because $k$ is a field. Therefore, Theorem \[thm:gdim-compose\] implies that the surjection $R\to k$ has finite G-dimension. Thus, $R$ is Gorenstein by \[subprop:4\] and [@auslander:smt (4.20)].
When ${\varphi}$ is finite and $X$ is a module of finite projective dimension over both $R$ and $S$, the implication “(c)$\implies$(a)” in the next result was proved by Apassov [@apassov:afm Theorem G’].
\[thm:like-apassov\] Let ${\varphi}\colon R\to S$ be a local homomorphism such that $S$ is Gorenstein. The following conditions are equivalent.
1. $R$ is Gorenstein.
2. ${\mathrm{G\text{-}dim}}({\varphi})$ is finite.
3. There exists a homologically finite complex $P$ of $S$-modules such that ${\mathrm{pd}}_S(P)$ is finite and ${\mathrm{G\text{-}dim}}_{{\varphi}}(P)$ is finite.
The implication “(a)$\implies$(b)” is contained in Theorem \[thm:gor-char-1\], while “(b)$\iff$(c)” is given by Theorem \[thm:stability-tensor\].
“(b)$\implies$(a)”. As $S$ is Gorenstein, $I_S(t)=t^{\dim(S)}$, and hence $$I_R(t) I_{{\varphi}}(t) = I_S(t)=t^{\dim(S)}\,,$$ by equality (\[eqn:bass\]) in \[para:1\]. Now, both $I_R(t)$ and $I_{{\varphi}}(t)$ are Laurent series with nonnegative coefficients, so that $I_R(t)$ is a polynomial. This, as noted in \[para:1\], implies that $R$ is Gorenstein.
The last theorem in this section is a characterization of the Gorenstein property of a local ring in terms of the finiteness of G-dimension of Frobenius-like endomorphisms. In order to describe this, we recall the definition of an invariant introduced by Koh and Lee [@koh:nionlr (1.1)].
\[para:col(R)\] For a finite module $M$ over a local ring $(S,{{\mathfrak{n}}})$, set $${\mathrm{s}}(M)=\inf\{t\geq 1\mid{\mathrm{Soc}}(M)\not\subseteq{{\mathfrak{n}}}^t M\}$$ where ${\mathrm{Soc}}(M)$ is the socle of $M$. Furthermore, let $${\mathrm{crs}}(S)=\inf\{{\mathrm{s}}(S/({\mathbf{x}}))\mid
\text{${\mathbf{x}}=x_1,\ldots,x_r$ is a maximal $S$-sequence}\}.$$
The following is a complex version of Koh-Lee [@koh:srotmimr (2.6)] (see also Miller [@miller:tfeahd (2.2.8)]), which, in turn, generalizes a theorem of J. Herzog [@herzog:rdcpuf (3.1)].
\[prop:koh-lee\] Let ${\varphi}\colon (R,{{\mathfrak{m}}})\to (S,{{\mathfrak{n}}})$ be a local homomorphism for which ${\varphi}({{\mathfrak{m}}})\subseteq{{\mathfrak{n}}}^{{\mathrm{crs}}(S)}$, and $X$ a homologically finite complex of $R$-modules. If there is an integer $t\geq\sup(X)$ such that ${\mathrm{Tor}}_{t+i}^R(X,S)=0$ for $1\leq
i\leq{\mathrm{depth}}(S)+2$, then ${\mathrm{pd}}_R(X)<\infty$.
Replace $X$ with a minimal $R$-free resolution to assume that each $X_i$ is a finite free $R$-module, and $\partial(X)\subseteq {{\mathfrak{m}}}X$. Set $Y=X\otimes_R S$. Then $Y$ is a complex of finite free $S$-modules with $\partial(Y)\subseteq{{\mathfrak{n}}}^{{\mathrm{crs}}(S)}Y$ and ${\mathrm{H}}_i(Y)={\mathrm{Tor}}^R_i(X,S)$.
The desired conclusion is that $X_i=0$ for $i\gg 0$. By the minimality of $X$, it suffices to prove that $X_i=0$, equivalently, $Y_i=0$, for some $i>\sup(X)$. Let $r={\mathrm{depth}}(S)$ and $C={\mathrm{Coker}}(\partial^Y_{t+1})$. The truncated complex $$Y_{t+r+2}\xrightarrow{\partial_{t+r+2}} Y_{t+r+1}
\xrightarrow{\partial_{t+r+1}}\cdots\xrightarrow{\partial_{t+1}} Y_t\to 0$$ is the beginning of a minimal $S$-free resolution of $C$. If ${\mathrm{pd}}_S(C)=\infty$, then [@miller:tfeahd (2.2.5),(2.2.6)] implies that each row of $\partial^Y_{t+r+2}$ has an entry outside ${{\mathfrak{n}}}^{{\mathrm{crs}}(S)}$, a contradiction. Thus, ${\mathrm{pd}}_S(C)<\infty$, and the Auslander-Buchsbaum formula implies that the ${\mathrm{pd}}_S(C)\leq r$. The minimality of the complex above implies that $Y_{t+r+1}=0$, completing the proof.
An arbitrary local homomorphism of finite G-dimension is far from being quasi-Gorenstein. Indeed, when $R$ is Gorenstein, any local homomorphism ${\varphi}\colon R\to S$ has finite G-dimension, see Theorem \[thm:gor-char-1\], whereas, by [@avramov:rhafgd (8.2)], such a ${\varphi}$ is quasi-Gorenstein if and only if $S$ is Gorenstein. Endomorphisms however are much better behaved in this regard.
\[prop:5-6\] Let ${\varphi}\colon (R,{{\mathfrak{m}}}) \to (R,{{\mathfrak{m}}})$ be a local homomorphism. If ${\mathrm{G\text{-}dim}}({\varphi})$ is finite, then ${\varphi}^n$ is quasi-Gorenstein at ${{\mathfrak{m}}}$, for each integer $n\geq1$.
If the finiteness of G-dimension localizes–see Proposition \[prop:localization\]–then one could draw the stronger conclusion that ${\varphi}^n$ is quasi-Gorenstein at each prime ideal.
Suppose that ${\mathrm{G\text{-}dim}}({\varphi})$ is finite. The equality (\[eqn:bass\]) in \[para:1\] yields $I_{{\varphi}}(t)=1$ so that ${\varphi}$ is quasi-Gorenstein at ${{\mathfrak{m}}}$, the maximal ideal of $R$. In the light of the discussion in \[para:quasi-Gorenstein\], the same is true of the $n$-fold composition ${\varphi}^{n}$, for all integers $n\geq 1$.
We are now ready to prove the following theorem that subsumes Theorem A in the introduction.
\[thm:gdim(f)-finite\] Let ${\varphi}\colon (R,{{\mathfrak{m}}})\to (R,{{\mathfrak{m}}})$ be a local homomorphism such that ${\varphi}^i({{\mathfrak{m}}})\subseteq{{\mathfrak{m}}}^2$ for some integer $i\geq 1$. The following conditions are equivalent.
1. The ring $R$ is Gorenstein.
2. ${\mathrm{G\text{-}dim}}({\varphi}^n)$ is finite for some integer $n\geq 1$.
3. There is a homologically finite complex $P$ of $R$-modules with ${\mathrm{pd}}_R(P)$ finite and ${\mathrm{G\text{-}dim}}_{{\varphi}^n}(P)$ finite, for some integer $n\geq 1$.
When these conditions hold, ${\mathrm{G\text{-}dim}}({\varphi}^m)=0$, for all $m\geq 1$.
“(a)$\implies$(b)” is contained in Theorem \[thm:gor-char-1\].
“(b)$\implies$(c)” is trivial.
“(c)$\implies$(a)”. By Theorem \[thm:stability-tensor\], ${\mathrm{G\text{-}dim}}({\varphi}^n)$ is finite. The completion ${\widehat{{\varphi}}}\colon {\widehat{R}}\to{\widehat{R}}$ is an endomorphism of ${\widehat{R}}$ such that ${\widehat{{\varphi}}}^i({\widehat{{{\mathfrak{m}}}}})\subseteq{\widehat{{{\mathfrak{m}}}}}^2$ and ${\widehat{{\varphi}^n}}=({\widehat{{\varphi}}})^n$. Furthermore, $R$ is Gorenstein if and only if ${\widehat{R}}$ is Gorenstein, and by \[subprop:1\], ${\mathrm{G\text{-}dim}}({\varphi}^n)$ is finite if and only if ${\mathrm{G\text{-}dim}}({\widehat{{\varphi}}}^n)$ is finite. Thus, passing to ${\widehat{R}}$, one may assume that $R$ is complete. Hence, $R$ has a dualizing complex $D$.
Since ${\mathrm{G\text{-}dim}}({\varphi}^n)$ is finite, Proposition \[prop:5-6\] implies that the $s$-fold composition ${\varphi}^{sn}$ of ${\varphi}^n$ is also quasi-Gorenstein at ${{\mathfrak{m}}}$ for all integers $s\geq 1$. Thus, $D{\otimes^{\mathbf{L}}}_R {}^{{\varphi}^{sn}}\!R$ is a dualizing complex for $R$, for each $s\geq 1$, by [@avramov:rhafgd (7.8)]. This implies that ${\mathrm{H}}(D{\otimes^{\mathbf{L}}}_R
{}^{{\varphi}^{sn}}\!R)$ is finite; that is to say, ${\mathrm{Tor}}_i^R(D,{}^{{\varphi}^{sn}}\!R)=0$ for all $i\gg 0$. Therefore, ${\mathrm{pd}}_R(D)$ is finite, by Proposition \[prop:koh-lee\]. This is equivalent to $R$ being Gorenstein; see, for example [@christensen:gd (3.4.12)]. This completes the proof that (c) implies (a).
When these conditions hold, the Auslander-Bridger formula \[thm:AB\] gives $${\mathrm{G\text{-}dim}}({\varphi}^m)={\mathrm{depth}}(R)-{\mathrm{depth}}(R)=0.$$ This is the desired formula.
The preceding theorem raises the problem: given a local ring $(R,{{\mathfrak{m}}})$ construct endomorphisms of $R$ that map ${{\mathfrak{m}}}$ into ${{\mathfrak{m}}}^2$. The prototype is the Frobenius endomorphism of a local ring of characteristic $p$. There are many such endomorphisms of power series rings over fields. The following example gives a larger class of complete local rings with nontrivial endomorphisms.
Let $k$ be a field and $X_1,\ldots,X_n$ analytic indeterminates and $F_1,\ldots,F_m\in
k[X_1,\ldots,X_n]$ homogeneous polynomials, and set $$R=k[\![X_1,\ldots,X_n]\!]/(F_1,\ldots,F_m)
=k[\![x_1,\ldots,x_n]\!].$$ Let $g$ be an element in $(x_1,\ldots,x_n)R$. The assignment $x_i\mapsto x_ig$ gives rise to a well-defined ring endomorphism ${\varphi}$ of $R$ such that ${\varphi}({{\mathfrak{m}}})\subseteq {{\mathfrak{m}}}^2$.
One property of the Frobenius endomorphism that is hard to mimic is the finiteness of the length of $R/{\varphi}({{\mathfrak{m}}})R$. Again, over power series rings such endomorphisms abound. The desired property is satisfied by the ring $R$ constructed above, when it is a one-dimensional domain and $g\ne0$. Examples in dimension two or higher can be built from these by considering $R[\![Y_1,\ldots,Y_m]\!]$.
More interesting endomorphisms can be obtained as follows. Let $$R=k[\![X_1,\ldots,X_n]\!]/(G_1-H_1,\ldots,G_m-H_m)$$ where, for each $i$, the elements $G_i$ and $H_i$ are monomials of the same total degree. For each positive integer $t$, the assignment $x_i\mapsto x_i^t$ gives rise to a ring endomorphism ${\varphi}_t$ of $R$ such that ${\varphi}_t({{\mathfrak{m}}})\subseteq{{\mathfrak{m}}}^t$ and $R/{\varphi}_t({{\mathfrak{m}}})R$ has finite length. This method allows one to construct Cohen-Macaulay normal domains of arbitrarily large dimension with nontrivial endomorphisms; consider, for example, the maximal minors of a $2\times r$ matrix of variables.
Finiteness of G-dimension over ${\varphi}$
==========================================
The import of the results of this section is that the finiteness of ${\mathrm{G\text{-}dim}}_{\varphi}(X)$ is intrinsic to the $R$-module structure on $X$; this is exactly analogous to the behavior of ${\mathrm{pd}}_{{\varphi}}(X)$; see \[prop:pdvsfd\]. When $R$ is complete, it is contained in Proposition \[prop:likeAF4.3\]; see also Theorem \[thm:gdvsgfd\] ahead.
\[thm:indep-of-vf\] Let ${\varphi}\colon R\to S$ and $\psi\colon R\to T$ be local homomorphisms. Let $X$ and $Y$ be homologically finite complexes of $S$-modules and $T$-modules, respectively, that are isomorphic in the derived category of $R$. Then ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ is finite if and only if ${\mathrm{G\text{-}dim}}_{\psi}(Y)$ is finite.
One may assume that $X$ and $Y$ are homologically nonzero. First, we reduce to the case where ${{\mathfrak{m}}}$, the maximal ideal of $R$, annihilates ${\mathrm{H}}(X)$ and ${\mathrm{H}}(Y)$. To this end, let $K$ be the Koszul complex on a finite generating sequence for ${{\mathfrak{m}}}$. Since $X\otimes_R K=X\otimes_S(S\otimes_R K)$ and $S\otimes_R K$ is a finite free complex of $S$-modules, Theorem \[thm:stability-tensor\] yields that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X\otimes_R K)$ and ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ are simultaneously finite. Similarly, ${\mathrm{G\text{-}dim}}_{\psi}(Y\otimes_R K)$ and ${\mathrm{G\text{-}dim}}_{\psi}(Y)$ are simultaneously finite. Moreover, $X\otimes_R K$ and $Y\otimes_R K$ are isomorphic in the derived category of $R$. As ${{\mathfrak{m}}}$ annihilates ${\mathrm{H}}(X\otimes_R K)$ and ${\mathrm{H}}(Y\otimes_R K)$–see, for instance, [@iyengar:dfcait (1.2)]–replacing $X$ and $Y$ with $X\otimes_R K$ and $Y\otimes_R
K$, respectively, gives the desired reduction.
Let $\alpha\colon X\to Y$ be an isomorphism. Let ${\widetilde}{{\varphi}}\colon{\widehat{R}}\to{\widetilde}{S}$ and ${\widetilde}{\psi}\colon{\widehat{R}}\to{\widetilde}{T}$ be the ${{\mathfrak{m}}}$-adic completions of ${\varphi}$ and $\psi$, respectively, and $\iota\colon R\to{\widehat{R}}$ the completion map. In the derived category of $R$, one has a commutative diagram: $$\xymatrix{ X=R\otimes_R X \ar[r]^(0.55){\iota\otimes_R 1}_(0.55){\simeq}
\ar[d]^\alpha_{\simeq} & {\widehat{R}}\otimes_R X \ar[r]^{{\widetilde}{{\varphi}}\otimes_{{\varphi}}1}_{\simeq}
\ar[d]^{1\otimes_R \alpha}_{\simeq} & {\widetilde}{S}\otimes_S X \\
Y=R\otimes_R Y \ar[r]^(0.55){\iota\otimes_R 1}_(0.55){\simeq} & {\widehat{R}}\otimes_R Y
\ar[r]^{{\widetilde}{\psi}\otimes_{\psi}1}_{\simeq} & {\widetilde}{T}\otimes_T Y }$$ Both $R\to{\widehat{R}}$ and $S\to{\widetilde}{S}$ are flat, so at the level of homology the top row of the diagram reads ${\mathrm{H}}(X)\to{\widehat{R}}\otimes_R {\mathrm{H}}(X)\to{\widetilde}{S}\otimes_S{\mathrm{H}}(X)$. Since ${\mathrm{H}}(X)$ is annihilated by ${{\mathfrak{m}}}$, these are both bijective, that is, $\iota\otimes_R 1$ and ${\widetilde}{{\varphi}}\otimes_{{\varphi}} 1$ are isomorphisms. A similar reasoning justifies the isomorphisms in the bottom row.
In the biimplications below, the first is by \[subprop:1\], the second is by Proposition \[prop:likeAF4.3\], while the third is due to the fact that, by the diagram above, ${\widetilde}{S}\otimes_S X$ and ${\widehat{R}}\otimes_R X$ are isomorphic. $$\begin{aligned}
\text{${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ is finite}
&\iff \text{${\mathrm{G\text{-}dim}}_{{\widetilde}{{\varphi}}}({\widetilde}{S}\otimes_S X)$ is finite} \\
&\iff \text{${\widetilde}{S}\otimes_S X$ is in ${\mathcal{A}}({\widehat{R}})$} \\
&\iff \text{${\widehat{R}}\otimes_R X$ is in ${\mathcal{A}}({\widehat{R}})$}\end{aligned}$$ By the same token, ${\mathrm{G\text{-}dim}}_{\psi}(Y)$ is finite if and only if ${\widehat{R}}\otimes_R Y$ is in ${\mathcal{A}}({\widehat{R}})$. This gives the desired conclusion, since ${\widehat{R}}\otimes_R X$ and ${\widehat{R}}\otimes_R Y$ are isomorphic.
Here is an example discovered by S. Paul Smith to illustrate that, in the set-up of the theorem above, ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ and ${\mathrm{G\text{-}dim}}_{\psi}(Y)$ need not be equal; see however [@avramov:homolhattfe (8.2.4)].
\[ex:not-equal\] Let $R$ be a field, $S$ the localized polynomial ring $R[X]_{(X)}$, and let $T$ be a field extension of $R$, with ${\mathrm{rank}}_RT={\mathrm{rank}}_RS$; in particular, $S$ and $T$ are isomorphic as $R$-modules. Let ${\varphi}\colon R\to S$ and $\psi\colon R\to T$ be the canonical inclusions. Because $R$ is a field, both ${\mathrm{G\text{-}dim}}_{{\varphi}}(S)$ and ${\mathrm{G\text{-}dim}}_{\psi}(T)$ are finite; see Theorem \[thm:gor-char-1\]. By the Auslander-Bridger formula \[thm:AB\], one has $${\mathrm{G\text{-}dim}}_{{\varphi}}(S) = - 1\qquad\text{and}\qquad {\mathrm{G\text{-}dim}}_{\psi}(T)= 0\,.$$
The corollary below extends \[subprop:4\]. It applies, for instance, when $X$ is homologically finite over $S$ and ${\varphi}$ is module-finite.
\[cor:vf-is-finite\] Let ${\varphi}\colon R\to S$ be a local homomorphism and $X$ a complex of $S$-modules. If ${\mathrm{H}}(X)$ is finite over $R$, then $${\mathrm{G\text{-}dim}}_{{\varphi}}(X) ={\mathrm{G\text{-}dim}}_R(X).$$
Theorem \[thm:indep-of-vf\] applied to the homomorphisms ${\varphi}$ and $\mathrm{id}_R$ says that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ and ${\mathrm{G\text{-}dim}}_R(X)$ are simultaneously finite. When they are finite, the Auslander-Bridger formula and Lemma \[lem:depth-eq\] yield the desired equality.
Comparison with Gorenstein flat dimension {#sec:gfd}
=========================================
Keeping in mind the conclusions of the preceding section, and Proposition \[prop:likeAF4.3\], it is natural to ask how G-dimension over ${\varphi}$ compares with other extensions of G-dimension to the non-finite arena. It turns out that the finiteness of ${\mathrm{G\text{-}dim}}_{\varphi}(X)$ is equivalent to the finiteness of ${\mathrm{Gfd}}_R(X)$, the *G-flat dimension* of $X$ over $R$, at least when $R$ has a dualizing complex. A more precise statement is contained in Theorem \[thm:gdvsgfd\] below; it is analogous to Property \[prop:pdvsfd\] dealing with projective dimensions. We begin by recalling the relevant definitions.
\[para:gfd\] Let $R$ be a commutative Noetherian ring. An $R$-module $G$ is said to be *G-flat* if there exists an exact complex of flat $R$-modules $$F= \cdots\xrightarrow{\partial_{i+1}}F_i\xrightarrow{\partial_{i}}F_{i-1}
\xrightarrow{\partial_{i-1}}\cdots$$ with ${\mathrm{Coker}}(\partial_1)=G$ and $E\otimes_RF$ exact for each injective $R$-module $E$. Note that any flat module is G-flat. Thus, each homologically bounded complex of $R$-modules $X$ admits a G-flat resolution, and one can introduce its G-flat dimension to be the number $${\mathrm{Gfd}}_R(X)\colon=\inf\{\sup\{n\mid G_n\ne 0\}\mid \text{$G$ a G-flat resolution of $X$}\}$$ The reader may consult [@christensen:gd] or the book of Enochs and Jenda [@enochs:gf] for details.
Now we state one of the main theorems of this section; it implies Theorem D from the introduction because when $R$ is a quotient of a Gorenstein ring, it has a dualizing complex. As noted in the introduction, Foxby has derived the inequalities below assuming only that the formal fibres of $R$ are Gorenstein. Also, the simultaneous finiteness of ${\mathrm{G\text{-}dim}}_{{\varphi}}(S)$ and ${\mathrm{Gfd}}_R(S)$ is [@christensen:new (5.2)].
\[thm:gdvsgfd\] Suppose $R$ has a dualizing complex. Let ${\varphi}\colon R\to S$ be a local homomorphism, and $X$ a homologically finite complex of $S$-modules. Then $${\mathrm{Gfd}}_R(X) - {\mathrm{edim}}({\varphi}) \leq {\mathrm{G\text{-}dim}}_{{\varphi}}(X) \leq {\mathrm{Gfd}}_R(X)\,.$$ In particular, ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ is finite if and only if ${\mathrm{Gfd}}_R(X)$ is finite.
Observe that doing away with the hypothesis that $R$ has a dualizing complex would provide us with another proof of Theorem \[thm:indep-of-vf\]. One can obtain useful bounds even when $R$ has no dualizing complex; this is explained in \[para:closing\],
\[proof:outline\] The proof calls for considerable preparation and is given in \[proof:gfdvsrfd\]. Here are the key steps in our argument:
*Step $1$.* We verify that ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ and ${\mathrm{Gfd}}_R(X)$ are simultaneously finite. This is an immediate consequence of [@christensen:new (4.3)] and Theorem \[prop:likeAF4.3\].
*Step $2$.* We prove, in Theorem \[prop:gfd=rfd\], that if ${\mathrm{Gfd}}_R(X)$ is finite, then it coincides with the number ${\operatorname{Rfd}}_R(X)$, whose definition is recalled below. This step constitutes the bulk of work in this section and builds on recent work of Christensen, Frankild, and Holm [@christensen:new]. They have informed us that they can prove the same result by using the methods in [@holm:ghd].
*Step $3$.* The last step consists of verifying that when ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ is finite, it is sandwiched between ${\operatorname{Rfd}}_R(X)-{\mathrm{edim}}({\varphi})$ and ${\operatorname{Rfd}}_R(X)$. The details of this step were worked out in conversations with Foxby, and we thank him for permitting us to present them here.
\[para:rfd\] In the next few paragraphs, $R$ denotes a commutative Noetherian ring, not necessarily local, and $W$ a homologically bounded complex of $R$-modules; we do not assume that ${\mathrm{H}}(W)$ is finite. The *large restricted flat dimension* of $W$ over $R$, as introduced in [@rfd], is the quantity $${\operatorname{Rfd}}_R(W) = \sup\{\sup(F{\otimes^{\mathbf{L}}}_RW)\mid \text{$F$ an $R$-module with ${\mathrm{fd}}_R(F)$
finite}\}$$ This number is finite, as long as ${\mathrm{H}}(W)$ is nonzero and the Krull dimension of $R$ is finite; see [@rfd (2.2)]. It is useful to keep in mind an alternative formula [@rfd (2.4)] for computing this invariant: $${\operatorname{Rfd}}_R(W)=\sup\{{\mathrm{depth}}R_{{\mathfrak{p}}}- {\mathrm{depth}}_{R_{{\mathfrak{p}}}}(W_{{\mathfrak{p}}}) \mid {{\mathfrak{p}}}\in{\mathrm{Spec}}R \}\,.$$
We collect a few simple observations concerning this invariant.
\[lem:rfd\] Let $\psi\colon R\to T$ and $\kappa\colon T\to T'$ be homomorphisms of commutative Noetherian rings, and let $W$ and $Y$ be homologically bounded complexes of $R$-modules and of $T$-modules respectively.
1. If $\psi$ is faithfully flat, then $${\operatorname{Rfd}}_T(T\otimes_RW) = {\operatorname{Rfd}}_R(W)\quad\text{and}\quad {\operatorname{Rfd}}_T(Y) \geq {\operatorname{Rfd}}_R(Y)\,.$$
2. If $\kappa$ is faithfully flat, then $ {\operatorname{Rfd}}_R(Y) = {\operatorname{Rfd}}_R(T'\otimes_TY)\,. $
Let $F$ be an $R$-module and let $G$ be an $T$-module.
Proof of (1). The flatness of $\psi$ implies
1. if ${\mathrm{fd}}_R(F)$ is finite, then so is ${\mathrm{fd}}_T(F\otimes_RT)$;
2. if ${\mathrm{fd}}_T(G)$ is finite, then so is ${\mathrm{fd}}_R(G)$.
Remark (a), combined with the isomorphisms $$(F\otimes_RT){\otimes^{\mathbf{L}}}_T(T\otimes_RW) \simeq (F\otimes_RT){\otimes^{\mathbf{L}}}_RW \simeq T\otimes_R
(F{\otimes^{\mathbf{L}}}_RW)$$ and the faithful flatness of $\psi$, implies ${\operatorname{Rfd}}_T(T\otimes_RW)\geq {\operatorname{Rfd}}_R(W)$. The opposite inequality follows from (b) and the associativity isomorphism $$G{\otimes^{\mathbf{L}}}_T(T\otimes_RW)\simeq G{\otimes^{\mathbf{L}}}_RW\,.$$ This justifies the equality. The inequality is a consequence of (a) and the isomorphism $(F\otimes_RT){\otimes^{\mathbf{L}}}_TY \simeq F{\otimes^{\mathbf{L}}}_RY$.
As to (2): it is an immediate consequence of the isomorphism $$F{\otimes^{\mathbf{L}}}_R(T'\otimes_TY)\simeq (F{\otimes^{\mathbf{L}}}_RY)\otimes_TT'$$ and the faithful flatness of $\kappa$.
The next lemma gives a lower bound for the large restricted flat dimension.
\[lem:rfd1\] If $\psi\colon R\to T$ is a local homomorphism and $Y$ is a complex $T$-modules, then $${\operatorname{Rfd}}_R(Y) \geq {\mathrm{depth}}R - {\mathrm{depth}}_T(Y)\,;$$ equality holds if $Y$ is homologically finite over $R$ and ${\mathrm{G\text{-}dim}}_R(Y)$ is finite.
The inequality is a consequence of \[para:rfd\].1 and the (in)equalities $${\mathrm{depth}}_R(Y) = {\mathrm{depth}}_T({{\mathfrak{m}}}T, Y) \leq {\mathrm{depth}}_T(Y)$$ where the first one is by [@iyengar:dfcait (5.2.1)] and the second is by [@iyengar:dfcait (5.2.2)]. If $Y$ is homologically finite over $R$ and ${\mathrm{G\text{-}dim}}_R(Y)$ is finite, then $$\begin{aligned}
{\mathrm{depth}}R - {\mathrm{depth}}_TY&= {\mathrm{depth}}R - {\mathrm{depth}}_R Y \\
&= {\mathrm{G\text{-}dim}}_R(Y) \\
&\geq {\mathrm{G\text{-}dim}}_{R_{{\mathfrak{p}}}}(Y_{{\mathfrak{p}}}) \\
&={\mathrm{depth}}R_{{\mathfrak{p}}}- {\mathrm{depth}}_{R_{{\mathfrak{p}}}}(Y_{{\mathfrak{p}}})
\end{aligned}$$ where the first equality is by Lemma \[lem:depth-eq\], the second and fourth are by the classical Auslander-Bridger formula, while the inequality is well known; see [@christensen:gd (2.3.11)]. In view of \[para:rfd\].1, this justifies the claimed equality.
The next step towards Theorem \[thm:gdvsgfd\] is the formula below. It may be viewed as an Auslander-Buchsbaum formula for complexes of finite G-flat dimension, for is strikingly similar to one for complexes of finite *flat* dimension: ${\mathrm{depth}}_R(W) = {\mathrm{depth}}R -
\sup(k{\otimes^{\mathbf{L}}}_RW)$ when ${\mathrm{fd}}_R(W)$ is finite; see [@foxby:daafuc (2.4)]. What is more, $E{\otimes^{\mathbf{L}}}_RW\simeq E\otimes_RG$, where is $G$ any G-flat resolution of $W$; this is contained in [@christensen:new (3.15)].
\[prop:gfd-depth\] Let $(R,{{\mathfrak{m}}},k)$ be a local ring and $E$ the injective hull of $k$. If $W$ is a complex of $R$-modules with ${\mathrm{Gfd}}_R(W)$ finite, then $${\mathrm{depth}}_R(W) = {\mathrm{depth}}R - \sup(E{\otimes^{\mathbf{L}}}_RW)$$ In particular, $\sup(E{\otimes^{\mathbf{L}}}_RW)$ is finite if and only if ${\mathrm{depth}}_R(W)$ is finite.
Let ${\widehat{R}}$ denote the ${{\mathfrak{m}}}$-adic completion of $R$. Faithful flatness of the completion homomorphism $R\to {\widehat{R}}$ implies that each injective ${\widehat{R}}$-module is injective also as an $R$-module. This remark and an elementary argument based on the definition of G-flat dimension entail: ${\mathrm{Gfd}}_{{\widehat{R}}}({\widehat{R}}\otimes_R W)\leq
{\mathrm{Gfd}}_R(W)$; see also Holm [@holm:ghd (3.10)]. Moreover $${\mathrm{depth}}_{{\widehat{R}}}({\widehat{R}}\otimes_RW) = {\mathrm{depth}}_RW \quad\text{and}\quad {\mathrm{depth}}{{\widehat{R}}}={\mathrm{depth}}R$$ Finally, $E{\otimes^{\mathbf{L}}}_R W\simeq E{\otimes^{\mathbf{L}}}_{{\widehat{R}}}({\widehat{R}}\otimes_RW)$, since $E$ has the structure of an ${\widehat{R}}$-module. Also, $E$ is the injective hull of $k$ over ${\widehat{R}}$. The upshot of this discussion is that one can replace $R$ and $W$ by ${\widehat{R}}$ and ${\widehat{R}}\otimes_RW$, respectively, and assume that $R$ is complete. In particular, $R$ has a dualizing complex $D$.
The G-flat dimension of $W$ is finite, so it follows from [@christensen:new (4.3)] that $W$ belongs to ${\mathcal{A}}(R)$, the Auslander category of $R$; see \[para:auslander\]. Thus, the canonical morphism $W\to {\mathbf{R}\mathrm{Hom}}_R(D,D{\otimes^{\mathbf{L}}}_RW)$ is an isomorphism, and this starts the chain of isomorphisms $$\begin{aligned}
{\mathbf{R}\mathrm{Hom}}_R(k,W) &\simeq {\mathbf{R}\mathrm{Hom}}_R(k,{\mathbf{R}\mathrm{Hom}}_R(D,D{\otimes^{\mathbf{L}}}_RW)) \\
&\simeq {\mathbf{R}\mathrm{Hom}}_R(D{\otimes^{\mathbf{L}}}_Rk,D{\otimes^{\mathbf{L}}}_RW) \\
&\simeq {\mathbf{R}\mathrm{Hom}}_k(D{\otimes^{\mathbf{L}}}_Rk,{\mathbf{R}\mathrm{Hom}}_R(k,D{\otimes^{\mathbf{L}}}_RW))\,.\end{aligned}$$ The second isomorphism is adjunction, so is the last one, since $D{\otimes^{\mathbf{L}}}_Rk$ is isomorphic to a complex of vector spaces over $k$; this latter fact is clear once we compute it with a free resolution of $D$. The complex $D{\otimes^{\mathbf{L}}}_RW$ is homologically bounded, since $W$ is in ${\mathcal{A}}(R)$, so the isomorphisms above with [@foxby:daafuc (1.5)] yield $$\sup({\mathbf{R}\mathrm{Hom}}_R(k,W)) = \sup({\mathbf{R}\mathrm{Hom}}_R(k,D{\otimes^{\mathbf{L}}}_RW)) - \inf(D{\otimes^{\mathbf{L}}}_Rk)\,.$$ For each complex $X$ of $R$-modules, $\sup({\mathbf{R}\mathrm{Hom}}_R(k,X))=-{\mathrm{depth}}_R(X)$ by [@foxby:daafuc (2.1)], and $\inf(D{\otimes^{\mathbf{L}}}_Rk)=\inf(D)$ since $D$ is homologically finite, so the displayed equality translates to $${\mathrm{depth}}_R(W) = {\mathrm{depth}}_R(D{\otimes^{\mathbf{L}}}_RW) + \inf(D)\,.$$ Here is a crucial swindle: since ${\mathrm{Gfd}}_R(W)$ is finite, so is ${\mathrm{Gfd}}_R({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W))$, where ${\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W)$ is the derived local cohomology of $W$ with respect to ${{\mathfrak{m}}}$; this is by [@christensen:new (5.9)]. Thus, the formula above applies to ${\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W)$ as well, and reads $${\mathrm{depth}}_R({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W)) = {\mathrm{depth}}_R(D{\otimes^{\mathbf{L}}}_R{\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W)) + \inf(D)\,.$$ The homology modules of ${\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W)$ are all ${{\mathfrak{m}}}$-torsion, so [@foxby:daafuc (2.7)] yields the first the equality below, while [@foxby:daafuc (2.1)] provides the second one $${\mathrm{depth}}_R({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W))=-\sup({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W))={\mathrm{depth}}_R(W)\,.$$ Now, ${\mathbf{R}\Gamma}_{{\mathfrak{m}}}(D) \simeq {{\scriptstyle{\Sigma}}}^d E$ with $d=\inf({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(D))$, where ${{\scriptstyle{\Sigma}}}^d(-)$ denotes a shift of $d$ steps to the left, so $$D{\otimes^{\mathbf{L}}}_R{\mathbf{R}\Gamma}_{{\mathfrak{m}}}(W)\simeq {\mathbf{R}\Gamma}_{{\mathfrak{m}}}(D){\otimes^{\mathbf{L}}}_R W \simeq {{\scriptstyle{\Sigma}}}^d (E{\otimes^{\mathbf{L}}}_R W)\,.$$ The first isomorphism may be justified by invoking [@lipman (3.1.2)]. The injective hull $E$ is ${{\mathfrak{m}}}$-torsion, so the homology modules of $E{\otimes^{\mathbf{L}}}_RW$ are ${{\mathfrak{m}}}$-torsion: compute via a free resolution of $W$. Thus, ${\mathrm{depth}}_R(E{\otimes^{\mathbf{L}}}_RW) = -\sup(E{\otimes^{\mathbf{L}}}_RW)$ by [@foxby:daafuc (2.7)]. Combining the preceding equalities gets us $${\mathrm{depth}}_R(W) = -\sup(E{\otimes^{\mathbf{L}}}_RW) + \inf({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(D)) + \inf(D)\,.$$ Since $R$ itself has finite G-flat dimension, this formula with $R$ substituted for $W$ reads: $ {\mathrm{depth}}R = \inf({\mathbf{R}\Gamma}_{{\mathfrak{m}}}(D)) + \inf(D)\,. $ Feeding this back into the formula above completes the proof.
In the case when ${\mathrm{H}}(W)$ is concentrated in a single degree, the next theorem is part of [@holm:ghd (3.19)].
\[prop:gfd=rfd\] Let $R$ be a commutative Noetherian ring, and $W$ a complex of $R$-modules. If ${\mathrm{Gfd}}_R(W)$ is finite, then ${\mathrm{Gfd}}_R(W)={\operatorname{Rfd}}_R(W)$.
The proof is the following sequence of equalities: $$\begin{aligned}
{\mathrm{Gfd}}_R(W) &= \sup\{\sup(I{\otimes^{\mathbf{L}}}_RW)\mid \text{$I$ an injective $R$-module}\} \\
&= \sup\{\sup(E(R/{{\mathfrak{p}}}){\otimes^{\mathbf{L}}}_R W) \mid {{\mathfrak{p}}}\in{\mathrm{Spec}}R\} \\
&= \sup\{\sup(E(R/{{\mathfrak{p}}}){\otimes^{\mathbf{L}}}_{R_{{\mathfrak{p}}}} W_{{\mathfrak{p}}}) \mid {{\mathfrak{p}}}\in{\mathrm{Spec}}R\} \\
&= \sup\{{\mathrm{depth}}R_{{\mathfrak{p}}}- {\mathrm{depth}}_{R_{{\mathfrak{p}}}}W_{{\mathfrak{p}}}\mid {{\mathfrak{p}}}\in{\mathrm{Spec}}R\} \\
&={\operatorname{Rfd}}_R(W)\end{aligned}$$ The first one is [@christensen:new (2.8)]; the second follows from this, given the structure of injective modules over commutative Noetherian rings; the third equality is due to the isomorphism $E(R/{{\mathfrak{p}}}){\otimes^{\mathbf{L}}}_R W \simeq E(R/{{\mathfrak{p}}}){\otimes^{\mathbf{L}}}_{R_{{\mathfrak{p}}}} W_{{\mathfrak{p}}}$, the fourth is by Theorem \[prop:gfd-depth\], as ${\mathrm{Gfd}}_{R_{{\mathfrak{p}}}}(W_{{\mathfrak{p}}})\leq {\mathrm{Gfd}}_R(W)$; see [@christensen:gd (5.2.7)], whilst the last equality is \[para:rfd\].1.
We pause to record a corollary.
\[cor:rfd-basechange\] Let $\psi\colon R\to T$ be a faithfully flat homomorphism of commutative, Noetherian rings, and let $W$ be a complex of $R$-modules. If ${\mathrm{Gfd}}_R(W)$ is finite, then $${\mathrm{Gfd}}_T(T\otimes_RW)={\mathrm{Gfd}}_R(T\otimes_RW) = {\mathrm{Gfd}}_R(W)\,.$$
When ${\mathrm{Gfd}}_R(W)$ is finite, so are ${\mathrm{Gfd}}_R(T\otimes_RW)$ and ${\mathrm{Gfd}}_T(T\otimes_RW)$; the second by [@holm:ghd (3.10)] and the first follows from the easily verifiable remark: if $G$ is a G-flat $R$-module, the so is $F\otimes_RG$ for any flat $R$-module $F$. The desired equalities are now a consequence of Theorem \[prop:gfd=rfd\] and Lemma \[lem:rfd\].
This result prompts us to raise the
\[quest\] Does the conclusion of Corollary \[cor:rfd-basechange\] remain true without assuming *a priori* that ${\mathrm{Gfd}}_R(W)$ is finite?
Here, at last, is the proof of Theorem \[thm:gdvsgfd\]; before jumping into it, the reader may wish to glance at \[proof:outline\], which outlines the basic argument.
*Proof of Theorem \[thm:gdvsgfd\].* \[proof:gfdvsrfd\] To begin with $${\mathrm{G\text{-}dim}}_{{\varphi}}(X) < \infty \iff X \in A(R) \iff {\mathrm{Gfd}}_R(X)<\infty\,,$$ where the first biimplication is by Proposition \[prop:likeAF4.3\], while the second one is contained in [@christensen:new (4.3)]. Thus, one may assume that both ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ and ${\mathrm{Gfd}}_R(X)$ are finite. In this case, thanks to theorems \[thm:AB\] and \[prop:gfd=rfd\], what we need to prove is that $${\operatorname{Rfd}}_R(X) - {\mathrm{edim}}({\varphi}) \leq {\mathrm{depth}}R - {\mathrm{depth}}_S X \leq {\operatorname{Rfd}}_R(X)
\tag{$\dagger$}$$ when ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)$ is finite. The inequality on the right is contained in Lemma \[lem:rfd1\]. That leaves us with the one on the left.
Let ${\widehat{S}}$ be the completion of $S$ at its maximal ideal and set ${\widehat{X}} ={\widehat{S}}\otimes_S X$. By Lemma \[lem:rfd\].2, the faithful flatness of the homomorphism $S\to
{\widehat{S}}$ implies ${\operatorname{Rfd}}_R({\widehat{X}}) = {\operatorname{Rfd}}_R(X)$. The other quantities involved in ($\dagger$) also remain unchanged if we substitute ${\widehat{S}}$ for $S$ and ${\widehat{X}}$ for $X$, so we may do so and thereby assume that $S$ is complete. With $R\to R'\to S$ a minimal Cohen factorization of ${\varphi}$, Lemma \[lem:rfd\] provides the inequality below $$\begin{aligned}
{\operatorname{Rfd}}_R(X) &\leq {\operatorname{Rfd}}_{R'}(X) \\
& = {\mathrm{depth}}R' - {\mathrm{depth}}_{R'}(X) \\
&={\mathrm{depth}}R + {\mathrm{edim}}({\varphi}) - {\mathrm{depth}}_{R'}(X) \\
&={\mathrm{depth}}R + {\mathrm{edim}}({\varphi}) - {\mathrm{depth}}_S(X)\end{aligned}$$ Lemma \[lem:rfd1\] explains the first equality; the second holds as $R\to R'$ is flat and $R'/{{\mathfrak{m}}}R'$ is regular, and the last holds because $R'\to S$ is surjective.
\[para:closing\] Let ${\varphi}\colon (R,{{\mathfrak{m}}},k) \to S$ be a local homomorphism and $X$ a homologically finite complex of $S$-modules. Let ${\widehat{R}}$ denote the ${{\mathfrak{m}}}$-adic completion of $R$, and ${\widetilde}S$ the ${{\mathfrak{m}}}S$-adic completion of $S$. Since ${\widehat{R}}$ has dualizing complex, it follows from \[subprop:1\] and Theorem \[thm:gdvsgfd\] that $${\mathrm{Gfd}}_{{\widehat{R}}}({\widetilde}S\otimes_SX) - {\mathrm{edim}}({\varphi}) \leq {\mathrm{G\text{-}dim}}_{{\varphi}}(X) \leq {\mathrm{Gfd}}_{{\widehat{R}}}({\widetilde}S\otimes_SX)\,.$$
At any rate, one has the consolation of knowing a partial result:
Let ${\varphi}\colon R\to S$ be a local homomorphism. Each homologically finite complex of $S$-modules $X$ satisfies ${\mathrm{G\text{-}dim}}_{{\varphi}}(X)\leq {\mathrm{Gfd}}_R(X)$.
The plan is to reduce to the case where $R$ is complete and then apply Theorem \[thm:gdvsgfd\]; confer the proof of Theorem \[thm:indep-of-vf\]. Let $K$ be the Koszul complex on minimal set of generators for ${{\mathfrak{m}}}$, the maximal ideal of $R$. Thus, ${\mathrm{pd}}_R(K)={\mathrm{edim}}R = {\mathrm{pd}}_S(K\otimes_RS)$. Now, if $G$ is a G-flat resolution of $X$ over $R$, then $K\otimes_RG$ is a G-flat resolution of $K\otimes_RX$. This implies that $${\mathrm{Gfd}}_R(K\otimes_RX) \leq {\mathrm{Gfd}}_R(X) + {\mathrm{edim}}(R)\,.$$ Moreover, since $K\otimes_RX\cong (K\otimes_RS)\otimes_SX$, Theorem \[thm:stability-tensor\] applied to the diagram $R\to S\xrightarrow{=}S$, and with $P=(K\otimes_RS)$, yields $${\mathrm{G\text{-}dim}}_{{\varphi}}(K\otimes_RX) = {\mathrm{G\text{-}dim}}_{{\varphi}}(X) + {\mathrm{edim}}(R)\,.$$ Thus, it suffices to prove the desired inequality for the complex of $S$-modules $K\otimes_RX$; in particular, one may pass to $K\otimes_RX$ and assume ${{\mathfrak{m}}}\cdot {\mathrm{H}}(X)=0$. Now, we adopt the notation of \[para:closing\], where we noted that $${\mathrm{G\text{-}dim}}_{{\varphi}}(X) \leq {\mathrm{Gfd}}_{{\widehat{R}}}({\widetilde}S\otimes_SX)\,.$$ It is elementary to verify that the canonical homomorphism of complexes of ${\widehat{R}}$-modules ${\widehat{R}}\otimes_R X \to {\widetilde}S\otimes_S X$ is a homology isomorphism, since ${{\mathfrak{m}}}\cdot{\mathrm{H}}(X)=0$. This gives us the equality below: $${\mathrm{Gfd}}_{{\widehat{R}}}({\widetilde}S\otimes_SX)={\mathrm{Gfd}}_{{\widehat{R}}}({\widehat{R}}\otimes_RX)\leq {\mathrm{Gfd}}_R(X)\,;$$ the inequality is the version for complexes of [@holm:ghd (3.10)], and may be deduced directly from the definitions. To complete the proof, put together the composed inequality above with the penultimate one.
This proposition leads to analogues of the theorems in Section \[sec:apps\], with ${\mathrm{Gfd}}_R(-)$ playing the role of ${\mathrm{G\text{-}dim}}_{{\varphi}}(-)$. The result below, which parallels Theorem \[thm:gdim(f)-finite\], is one such; in it, for any complex of $R$-modules, we write ${}^{{\varphi}^n}\!X$ to indicate that $R$ acts on $X$ via ${\varphi}^n$.
Let ${\varphi}\colon (R,{{\mathfrak{m}}})\to (R,{{\mathfrak{m}}})$ be a local homomorphism such that ${\varphi}^i({{\mathfrak{m}}})\subseteq{{\mathfrak{m}}}^2$ for some integer $i\geq 1$. The following conditions are equivalent.
1. The ring $R$ is Gorenstein.
2. ${\mathrm{Gfd}}_R({}^{{\varphi}^n}\!R)$ is finite for each integer $n\geq 1$.
3. There is a homologically finite complex $P$ of $R$-modules with ${\mathrm{pd}}_R(P)$ finite and ${\mathrm{Gfd}}_R({}^{{\varphi}^n}\!P)$ finite, for some integer $n\geq 1$.
Over a Gorenstein ring, any module has finite G-flat dimension; see [@christensen:gd (5.2.10)]. This justifies “(a) $\implies$ (b)”, while “(b) $\implies$ (c)” is trivial.
“(c) $\implies$ (a)”. The preceding proposition yields that ${\mathrm{G\text{-}dim}}_{{\varphi}^n}(P)$ is finite, so it remains to invoke the corresponding implication in Theorem \[thm:gdim(f)-finite\].
The hypotheses of the preceding result are satisfied when ${\varphi}$ is the Frobenius endomorphism of a local ring of $R$ of positive prime characteristic. In this case, one can add a fourth equivalent condition to those given above:
1. the $R$-module ${}^{{\varphi}^n}\!R$ is G-flat for one integer $n\geq
1$.
Indeed, it is clear from \[para:rfd\].1 that ${\operatorname{Rfd}}_R({}^{{\varphi}^n}\!R)=0$ for each integer $n\geq0$. Therefore, by [@holm:ghd (3.19)], or by its successor, Theorem \[prop:gfd=rfd\], one obtains that the $R$-module ${}^{{\varphi}^n}\!R$ has finite G-flat dimension if and only if it is already G-flat.
Acknowledgments {#acknowledgments .unnumbered}
===============
While this article was being written, the first author was collaborating with Luchezar Avramov and Claudia Miller on [@avramov:homolhattfe], another study of local homomorphisms. We are grateful to them for numerous discussions.
We thank Lars Christensen, Anders Frankild, and Henrik Holm for making [@christensen:new] available to us and for correspondence concerning Section 8. Our work in that section owes an intellectual debt to Hans-Bjørn Foxby; we thank him for comments and criticisms, and for generously sharing his ideas.
Thanks are also due to S. Paul Smith, Diana White, Yongwei Yau, and the referee for useful suggestions.
[10]{}
D. Apassov, *Almost finite modules*, Comm. Algebra **27** (1999), no. 2, 919–931.
M. Auslander and M. Bridger, *Stable module theory*, Memoirs of the American Mathematical Society **94**, American Mathematical Society, Providence, R.I., 1969.
L. L. Avramov, *Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology*, Ann. of Math. (2) **150** (1999), no. 2, 455–487.
L. L. Avramov and H.-B. Foxby, *Homological dimensions of unbounded complexes*, J. Pure Appl. Algebra **71** (1991), 129–155.
, *Locally [G]{}orenstein homomorphisms*, Amer. J. Math. **114** (1992) no. 5, 1007–1047.
, *Grothendieck’s localization problem*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math. **159**, Amer. Math. Soc., Providence, RI, 1994, 1–13.
, *Ring homomorphisms and finite [G]{}orenstein dimension*, Proc. London Math. Soc. (3) **75** (1997), no. 2, 241–270.
L. L. Avramov, H.-B. Foxby, and B. Herzog, *Structure of local homomorphisms*, J. Algebra **164** (1994), 124–145.
L. L. Avramov, S. Iyengar, and C. Miller, *Homology over local homomorphisms*, preprint 2003. Available at: http://arxiv.org/abs/math.AC/0312412
W. Bruns and J. Herzog, *Cohen-[M]{}acaulay rings*, revised ed., Studies in Advanced Mathematics **39**, University Press, Cambridge, 1998.
L. W. Christensen, *Semi-dualizing complexes and their Auslander categories*, Trans. Amer. Math. Soc. **353** (2001), 1839–1883.
L. W. Christensen, *Gorenstein dimensions*, Lecture Notes in Mathematics **1747**, Springer-Verlag, Berlin, 2000.
L. W. Christensen, H.-B. Foxby, and A. Frankild, *Restricted homological dimensions and [C]{}ohen-[M]{}acaulayness*, J. Algebra **251** (2002), no. 1, 479–502.
L. W. Christensen, A. Frankild, and H. Holm, *On Gorenstein projective, injective and flat dimensions—A functorial description with applications*, preprint (2003).
E. E. Enochs, O. M. G. Jenda, *Relative homological algebra*, de Gruyter Expositions in Math. **30**, Walter de Gruyter & Co., Berlin, 2000.
H.-B. Foxby, *Hyperhomological algebra & commutative rings*, in preparation.
H.-B. Foxby and S. Iyengar, *Depth and amplitude for unbounded complexes*, in: Commutative algebra. Interaction with Algebraic Geometry (Grenoble-Lyon 2001), Contemp. Math. **331**, Amer. Math. Soc., Providence, RI, 2003, 119–137.
S. I. Gelfand and Y. I. Manin, *Methods of homological algebra*, second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
S. Goto, *A problem on [N]{}oetherian local rings of characteristic [$p$]{}*, Proc. Amer. Math. Soc. **64** (1977), no. 2, 199–205.
R. Hartshorne, *Residues and duality*, Lecture Notes in Mathematics **20**, Springer-Verlag, Berlin, 1966.
J. Herzog, *Ringe der [C]{}harakteristik [$p$]{} und [F]{}robeniusfunktoren*, Math. Z. **140** (1974), 67–78.
H. Holm, *Gorenstein homological dimensions*, J. Pure Appl. Algebra, **189** (2004), no. 1, 167–193.
S. Iyengar, *Depth for complexes, and intersection theorems*, Math. Z. **230** (1999), 545–567.
J. Koh and K. Lee, *Some restrictions on the maps in minimal resolutions*, J. Algebra **202** (1998), no. 2, 671–689.
, *New invariants of [N]{}oetherian local rings*, J. Algebra **235** (2001), no. 2, 431–452.
E. Kunz, *Characterizations of regular local rings of characteristic $p$*, Amer. J. Math. **91** (1969), 772–784.
Joseph Lipman, *Lectures on local cohomology and duality*, in: Local cohomology and its applications (Guanajuato, 1999), Lecture Notes in Pure and Appl. Math., vol. 226, Dekker, New York, 2002, pp. 39–89.
H. Matsumura, *Commutative ring theory*, second ed., Studies in Advanced Mathematics, **8**, University Press, Cambridge, 1989.
C. Miller, *The [F]{}robenius endomorphism and homological dimensions*, in: Commutative algebra. Interaction with Algebraic Geometry (Grenoble-Lyon 2001), Contemp. Math. **331**, Amer. Math. Soc., Providence, RI, 2003, 207–234.
A. G. Rodicio, *On a result of [A]{}vramov*, Manuscripta Math. **62** (1988), no. 2, 181–185.
R. Takahashi and Y. Yoshino, *Characterizing [C]{}ohen-[M]{}acaulay local rings by [F]{}robenius maps*, Proc. Amer. Math. Soc., to appear.
J.-L. Verdier, *Des catégories dérivées des catégories abéliennes*, Astérisque **239**, Soc. Math. France, 1997.
S. Yassemi, *G-dimension*, Math. Scand. **77** (1995), no. 2, 161–174.
[^1]: This research was conducted while S.I. was partly supported by a grant from the NSF, and S.S.-W. was an NSF Mathematical Sciences Postdoctoral Research Fellow. Part of the work was done at MSRI during the Spring semester of 2003, when the authors were participating in the program in Commutative Algebra.
|
---
abstract: 'The COMPASS Collaboration at CERN has measured the transverse spin azimuthal asymmetry of charged hadrons produced in semi-inclusive deep inelastic scattering using a 160 GeV $\mu^+$ beam and a transversely polarised NH$_3$ target. The Sivers asymmetry of the proton has been extracted in the Bjorken $x$ range $0.003<x<0.7$. The new measurements have small statistical and systematic uncertainties of a few percent and confirm with considerably better accuracy the previous COMPASS measurement. The Sivers asymmetry is found to be compatible with zero for negative hadrons and positive for positive hadrons, a clear indication of a spin-orbit coupling of quarks in a transversely polarised proton. As compared to measurements at lower energy, a smaller Sivers asymmetry for positive hadrons is found in the region $x > 0.03$. The asymmetry is different from zero and positive also in the low $x$ region, where sea–quarks dominate. The kinematic dependence of the asymmetry has also been investigated and results are given for various intervals of hadron and virtual photon fractional energy. In contrast to the case of the Collins asymmetry, the results on the Sivers asymmetry suggest a strong dependence on the four-momentum transfer to the nucleon, in agreement with the most recent calculations.'
title: 'Experimental investigation of transverse spin asymmetries in $\mu$-p SIDIS processes: Sivers asymmetries'
---
In the late 60’s a simple and powerful description was proposed for the nucleon as a stream of partons each carrying a fraction $x$ of the nucleon momentum in a frame where the nucleon momentum is infinitely large. From the dependence of the deep inelastic lepton-nucleon scattering (DIS) cross section on the energy and momentum transfered to the nucleon it was possible to identify charged partons with the earlier postulated quarks, and assess the existence of gluons as carriers of half of the proton momentum.
Since the 90’s it is well known that in order to fully specify the quark structure of the nucleon at twist-two level in quantum chromodynamics (QCD) three types of parton distribution functions (PDFs) are required: the momentum distributions $q(x)$ (or $f_1^q(x)$), the helicity distributions $\Delta q(x)$ (or $g_1^q(x)$) and the transversity distributions $\Delta_T q(x)$ (or $h_1^q(x)$), where $x$ is the Bjorken variable. For a given quark flavour $q$, $q(x)$ is the number density, $\Delta q(x)$ is the difference between the number densities of quarks with helicity equal or opposite to that of the nucleon for a nucleon polarised longitudinally, i.e. along its direction of motion, and the transversity distribution $\Delta_T q(x)$ is the corresponding quantity for a transversely polarised nucleon. If the quarks are assumed to be collinear with the parent nucleon, i.e. neglecting the intrinsic quark transverse momentum $\vec k_T$, or after integration over $\vec k_T$, the three distributions $q(x)$, $\Delta q(x)$ and $\Delta_T q(x)$ exhaust the information on the internal dynamics of the nucleon. On the other hand, from the measured azimuthal asymmetries of hadrons produced in unpolarised semi-inclusive deep inelastic scattering (SIDIS) and Drell-Yan (DY) processes a sizeable transverse momentum of quarks was derived. Taking into account a finite intrinsic transverse momentum $\vec k_T$, in total eight transverse momentum dependent (TMD) distribution functions are required to fully describe the nucleon at leading twist [@Barone:2010zz]. Presently, PDFs that describe non–perturbative properties of hadrons are not yet calculable in QCD from first principles, but they can already be computed in lattice QCD. In the SIDIS cross section they appear convoluted with fragmentation functions (FFs) [@Kotzinian:1994dv; @Bacchetta:2006tn], so that they can be extracted from the data.
A TMD PDF of particular interest is the Sivers function $\Delta_0^T q$ (or $f_{1T}^{\perp q}$), which arises from a correlation between the transverse momentum $\vec{k}_T$ of an unpolarised quark in a transversely polarised nucleon and the nucleon polarisation vector [@Sivers:1989cc]. In SIDIS this $\vec{k}_T$ dependence gives rise to the “Sivers asymmetry” $A_{Siv}$ which is the amplitude of the $\sin \Phi_{ S}$ modulation in the distribution of the produced hadrons. Here the azimuthal angle $\Phi_{ S}$ is defined as $\Phi_{ S}=\phi_h-\phi_s$ with $\phi_h$ and $\phi_s$ respectively the azimuthal angles of hadron transverse momentum and nucleon spin vector, in a reference system in which the z axis is the virtual photon direction and the xz plane is the lepton scattering plane. Neglecting the hadron transverse momentum with respect to the direction of the fragmenting quark, the Sivers asymmetry can be written as $$\begin{aligned}
A_{Siv} = \frac {\sum_q e_q^2 \cdot \Delta_0^T q \otimes D^h_q} {\sum_q e_q^2
\cdot q \otimes D_q^h} \, ,
\label{eq:sivass}\end{aligned}$$ where $\otimes$ indicates the convolutions over transverse momenta, $e_q$ is the quark charge and $D_q^h$ describes the fragmentation of a quark $q$ into a hadron $h$.
In the very recent years, much attention has been devoted to the Sivers function, which was originally proposed to explain the large single-spin asymmetries observed in hadron-hadron scattering. The Sivers function is T–odd, namely it changes sign under naive time reversal, which is defined as usual time reversal but without interchange of initial and final state. For a long time the Sivers function and the corresponding asymmetry were believed to vanish [@Collins:1992kk] due to T–invariance arguments. However Brodsky et al. [@Brodsky:2002cx] showed by an explicit model calculation that final-state interactions in SIDIS arising from gluon exchange between the struck quark and the nucleon remnant (or initial state in DY) produce a non-zero asymmetry. One of the main theoretical achievements of the recent years was the discovery that the Wilson-line structure of parton distributions, which is necessary to enforce gauge invariance of QCD, provides the possibility for non-zero T–odd transverse momentum dependent (TMD) PDFs. According to factorisation the T–odd PDFs are not universal. The Sivers function can be different from zero but must have opposite sign in SIDIS and DY [@Collins:2002kn]. A lot of interest in the Sivers function arises also from its relation with orbital motion of quarks inside a transversely polarised nucleon. In particular it was shown [@Brodsky:2002cx] that orbital angular momentum must exist if the Sivers function doesn’t vanish. Even though no exact relation between Sivers function and orbital angular momentum was derived yet, work is going on, also because the importance of assessing the role of the orbital angular momentum in the nucleon spin sum rule has grown in time (see e.g. [@Burkardt:2011zz; @Wakamatsu:2010cb; @Leader:2011za; @Ji:2012sj]).
Presently, the measurement of the Sivers asymmetry in SIDIS is the only direct way to assess the Sivers function. It became an important part of the experimental programs of the HERMES and COMPASS experiments, and it will be an important part of future SIDIS experiments at JLab12 [@jlab12]. Furthermore, in the near future several experiments using the DY process will address the Sivers function, in particular its sign, in order to establish the prediction of restricted universality [@compass2; @reimer].
Using a 160 GeV longitudinally polarised $\mu^+$ beam COMPASS measured SIDIS on a transversely polarised deuteron ($^6$LiD) target in 2002, 2003 and 2004. In those data no sizeable Sivers asymmetry was observed within the accuracy of the measurements [@Alexakhin:2005iw; @Ageev:2006da; @Alekseev:2008dn], a fact which is understood in terms of a cancellation between the contributions of u- and d-quarks. By scattering the e$^-$ and e$^+$ beams at HERA off a transversely polarised proton target, HERMES measured in 2004 a non-zero Sivers asymmetry for positively charged hadrons [@Airapetian:2004tw]. A combined analysis of the COMPASS and HERMES data allowed for a first extraction of the Sivers function for u- and d-quarks [@Vogelsang:2005cs; @Efremov:2008vf; @Anselmino:2008sga]. Still, as in the case of the Collins asymmetry, measurements on protons at higher beam energies were needed to disentangle possible higher twist effects.
In 2007 COMPASS measured for the first time SIDIS on a transversely polarised proton (NH$_3$) target. The results [@Alekseev:2010rw] on the Sivers asymmetry for positive hadrons were found to be different from zero and turned out to be somewhat smaller than the final HERMES data [@Airapetian:2009ti]. However the COMPASS results had larger statistical errors and a non-negligible overall scale uncertainty of $\pm 0.01$. A more precise measurement was thus mandatory and the entire 2010 data taking period was dedicated to this purpose.
In this Letter, the results of the 2010 run are presented. They confirm with considerably smaller uncertainties the observation of the 2007 measurements. The higher statistics allow for first studies of the kinematic dependence of the asymmetry in a domain larger than the usual COMPASS DIS phase space.
The COMPASS spectrometer is in operation in the SPS North Area of CERN since 2002. The principle of the measurement and the data analysis were already described in refs. [@Abbon:2007pq; @Alexakhin:2005iw; @Ageev:2006da; @Alekseev:2008dn; @Alekseev:2010rw]. The information on the 2010 run, the amount of data collected, the event reconstruction and selection, the statistics of the final samples, are given in a parallel paper on the Collins asymmetry [@Collinsnew] that was measured using the same data. In order to ensure a DIS regime, only events with photon virtuality $Q^2>1$ (GeV/c)$^2$, fractional energy of the virtual photon $0.1<y<0.9$, and mass of the hadronic final state system $W>5$ GeV/c$^2$ are considered. A charged hadron is required to have at least 0.1 GeV/c transverse momentum $p_T^h$ with respect to the virtual–photon direction and a fraction of the available energy $z>0.2$. This is refered to as “standard sample” in the following.
{width="90.00000%"}
The Collins and Sivers asymmetries are the amplitudes of 2 of the 8 azimuthal modulations, which are theoretically expected to be present in the SIDIS cross section for a transversely polarised target. They are extracted simultaneously from the same data as explained in ref. [@Collinsnew]. The measured amplitude of the modulation in sin$\Phi_S$ is $\epsilon_S = f P_T A_{Siv}$, where $f$ is the dilution factor of the NH$_3$ material, and $P_T$ the magnitude of the proton polarisation. In order to extract $A_{Siv}$, the measured amplitudes $\epsilon_S$ in each period are divided by $f$ and $P_T$. The dilution factor of the ammonia target is calculated for semi-inclusive reactions [@Alekseev:2010hc] and is evaluated in each $x$ bin; it increases with $x$ from 0.14 to 0.17, and it is assumed constant in $z$ and $p^h_T$. The proton target polarisation ($\sim 0.8$) was measured individually for each cell and each period. The results for $A_{Siv}$ from all periods of data taking are found to be statistically compatible and the final asymmetries are obtained by averaging the results from the full available statistics. Extensive studies were performed in order to assess the systematic uncertainties of the measured asymmetries, and it was found that the largest contribution is due to residual acceptance variations within the data taking periods. In order to quantify these effects, various types of false asymmetries are calculated from the final data sample assuming wrong sign polarisation for the target cells. Moreover, the physical asymmetries are extracted splitting the events according to the detection of the scattered muon in the spectrometer (top vs bottom, left vs right). The differences between these physical asymmetries and the false asymmetries are used to quantify the overall systematic point-to-point uncertainties, which are evaluated to be 0.5 times the statistical uncertainties. The only relevant systematic scale uncertainty, which arises from the measurement of the target polarisation, is evaluated to be 3% of the target polarisation.
Figure \[fig:s2010\] shows the Sivers asymmetries for positive and negative hadrons extracted from the 2010 proton data as a function of $x$, $z$ and $p_T^h$, where the other two variables are integrated over. For negative hadrons the asymmetry is compatible with zero, while for positive hadrons it is definitely positive and stays positive down to $x \simeq 10^{-3}$, in the region of the quark sea.
{width="80.00000%"}
There is good agreement with the published results from the COMPASS 2007 run [@Alekseev:2010rw] but with a considerable reduction of more than a factor of two in the statistical and in the point-to-point systematic uncertainties. Also, the asymmetry for positive hadrons is clearly smaller than the corresponding one measured by HERMES [@Airapetian:2009ti]. This fact persists even when considering only events with $x>0.032$, in the same $x$ range as the HERMES experiment. The asymmetries in this restricted $x$ range are shown as open points in fig. \[fig:s2010hx\].
The correlation between the Collins and the Sivers azimuthal modulations introduced by the non-uniform azimuthal acceptance of the apparatus as well as the correlations between the Sivers asymmetries measured when binning the same data alternatively in $x$, $z$ or $p_T^h$ were already given in ref. [@Collinsnew]. All correlation coefficients are found to be smaller than 0.2 and are relevant only in case of simultaneous fits of the various asymmetries.
{width="90.00000%"}
![Left panel: mean value of $y$ vs $W$. Middle panel: mean values of $W$ vs $x$ for the standard sample $0.1<y<0.9$ (closed circles, $\bullet$) and for the samples $0.05<y<0.1$ (closed squares, $\scriptstyle\blacksquare$), $0.1<y<0.2$ (open triangles, $\triangledown$), and $0.2<y<0.9$ (open squares, $\scriptstyle\square$). Right panel: mean values of $Q^2$ vs $x$ for the standard sample $0.1<y<0.9$ (closed circles, $\bullet$) and for the samples $0.05<y<0.1$ (closed squares, $\scriptstyle\blacksquare$), $0.1<y<0.2$ (open triangles, $\triangledown$), and $0.2<y<0.9$ (open squares, $\scriptstyle\square$). []{data-label="fig:meansk"}](figs/s_fig4a.eps "fig:"){width="32.00000%"} ![Left panel: mean value of $y$ vs $W$. Middle panel: mean values of $W$ vs $x$ for the standard sample $0.1<y<0.9$ (closed circles, $\bullet$) and for the samples $0.05<y<0.1$ (closed squares, $\scriptstyle\blacksquare$), $0.1<y<0.2$ (open triangles, $\triangledown$), and $0.2<y<0.9$ (open squares, $\scriptstyle\square$). Right panel: mean values of $Q^2$ vs $x$ for the standard sample $0.1<y<0.9$ (closed circles, $\bullet$) and for the samples $0.05<y<0.1$ (closed squares, $\scriptstyle\blacksquare$), $0.1<y<0.2$ (open triangles, $\triangledown$), and $0.2<y<0.9$ (open squares, $\scriptstyle\square$). []{data-label="fig:meansk"}](figs/s_fig4b.eps "fig:"){width="32.00000%"} ![Left panel: mean value of $y$ vs $W$. Middle panel: mean values of $W$ vs $x$ for the standard sample $0.1<y<0.9$ (closed circles, $\bullet$) and for the samples $0.05<y<0.1$ (closed squares, $\scriptstyle\blacksquare$), $0.1<y<0.2$ (open triangles, $\triangledown$), and $0.2<y<0.9$ (open squares, $\scriptstyle\square$). Right panel: mean values of $Q^2$ vs $x$ for the standard sample $0.1<y<0.9$ (closed circles, $\bullet$) and for the samples $0.05<y<0.1$ (closed squares, $\scriptstyle\blacksquare$), $0.1<y<0.2$ (open triangles, $\triangledown$), and $0.2<y<0.9$ (open squares, $\scriptstyle\square$). []{data-label="fig:meansk"}](figs/s_fig4c.eps "fig:"){width="32.00000%"}
{width="50.00000%"}
In order to further investigate the kinematic dependence of the Sivers asymmetry and to understand the reason of the difference with HERMES, the kinematic domain is enlarged to examine the events with smaller $y$ values (in the interval $0.05<y<0.1$), which correspond to smaller $Q^2$ and $W$ values. Additionally, the standard data sample is divided into two parts, corresponding to $0.1<y<0.2$ and $0.2<y<0.9$. Since at small $y$ there are no low-$x$ data, only events with $x>0.032$ are used. Figure \[fig:s2010\_hx\_y3\] shows the Sivers asymmetries measured in these three bins of $y$ as a function of $x$, $z$, and $p_T^h$ respectively. No particular trend is observed in the case of the asymmetries for negative hadrons (bottom plots), which stay compatible with zero as for the standard sample. A clear increase of the Sivers asymmetry for positive hadrons is visible for the low-$y$ data. This strong effect can not be due to the slightly different mean values of $x$, since the Sivers asymmetry does not exhibit an $x$ dependence for $x>0.032$. On the contrary, it could be associated with the smaller values of $Q^2$ and/or with the smaller values of the invariant mass of the hadronic system $W$. A similar dependence of the asymmetries on $y$ was already noticed in the published results from the 2007 data. As can be seen from fig. \[fig:meansk\] (left panel), there is a strong correlation between the $y$ and $W$ mean values: the mean values of $W$ in the high $x$ bins are about 3 GeV/c$^2$ for the sample $0.05<y<0.1$ and larger than 5 GeV/c$^2$ for the standard sample $0.1<y<0.9$ (middle panel of fig. \[fig:meansk\]). On the other hand, as can be seen in the right panel of fig. \[fig:meansk\], bins at smaller $y$ have smaller values of $\langle Q^2 \rangle$. In particular, in each $x$ bin the $Q^2$ mean value decreases by about a factor of 3 for the sample $0.05<y<0.1$ with respect to the standard sample. Although the situation might be different in the target fragmentation region [@Kotzinian:2011av], in the current fragmentation region the Sivers asymmetry is not expected to depend on $y$ (or on $W$), while some $Q^2$ dependence should exist due to the $Q^2$ evolution of both the FFs and the TMD PDFs.
{width="90.00000%"}
Very recently first attempts to estimate the impact of the $Q^2$ evolution of the Sivers function [@Aybat:2011ge] led to encouraging results. In ref. [@Aybat:2011ta] the Sivers asymmetry was evaluated for the HERMES kinematic region using the Sivers functions of ref. [@Anselmino:2011gs] and then evolved to the COMPASS kinematic region. The measured $z$ dependence of the Sivers asymmetries for $0.1<y<0.9$ is compared with the calculated one in fig. \[fig:aybat\], for the entire $x$ region and for $x>0.032$. The linear trend of the data up to $z \simeq 0.75$ is well reproduced, as well as the small increase of the slope for the high $x$ sample. A very recent fit [@Anselmino:2012aa] of the HERMES asymmetries [@Airapetian:2009ti] and the COMPASS deuteron [@Alekseev:2008dn] and proton [@Bradamante:2011xu] results given here was performed taking into account the $Q^2$ evolution in all $x$ bins. It reproduces all the data well and provides strong support to the current TMD approach, which foresees a strong $Q^2$–dependence of the Sivers function.
We have also investigated the behaviour of the Sivers asymmetries at low $z$. Our standard hadron selection requires $z>0.2$ to stay well separated from the target fragmentation region. In the range $0.1<z<0.2$ no effect on $A_{Siv}$ is visible for negative hadrons, but one observes a clear decrease of the asymmetry for positive hadrons. In fig. \[fig:s2010\_z3\] the data are plotted in 3 different $z$ regions: $0.10<z<0.20$, $0.20<z<0.35$, and $0.35<z<1.00$. While the shape of the asymmetry as a function of $x$ stays the same, the size of the asymmetry shows a clear proportionality with $z$, in qualitative agreement with the expected linear behaviour (see, e.g. [@Anselmino:2011ch]).
All the results given in this Letter are available on HEPDATA [@hepdata]. The asymmetries for the standard sample as functions of $x$, $z$ and $p_T^h$ have also been combined with the already published results from the 2007 run [@Alekseev:2010rw] and are also available on HEPDATA.
In summary, COMPASS has obtained precise results on the Sivers asymmetry in SIDIS using a polarised proton target. A first investigation of its dependence on various kinematic variables shows significant dependences on $z$ and $y$. By now, the Sivers asymmetry for positive hadrons is shown to be different from zero in a broad kinematic range and to exhibit strong kinematic dependences. After two decades of speculations, this is an important new insight into the partonic structure of the nucleon. In the light of the most recent theoretical advances refined combined analyses to evaluate the Sivers function and its dependence on the SIDIS variables are required in order to understand the role of the Sivers function in the various transverse spin phenomena observed in hadron-hadron collisions and in future Drell-Yan measurements.
We acknowledge the support of the CERN management and staff, as well as the skills and efforts of the technicians of the collaborating institutes.
[99]{}
for a review of recent developments see e.g. V. Barone, F. Bradamante and A. Martin, Prog. Part. Nucl. Phys. [**65**]{} (2010) 267.
A. Kotzinian, Nucl. Phys. B [**441**]{} (1995) 234. A. Bacchetta [*et al.*]{}, JHEP [**0702**]{} (2007) 093. D. W. Sivers, Phys. Rev. [**D41**]{} (1990) 83.
J. C. Collins, Nucl. Phys. B [**396**]{} (1993) 161. S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B [**530**]{} (2002) 99. J. C. Collins, Phys. Lett. B [**536**]{} (2002) 43. M. Burkardt, Prog. Theor. Phys. Suppl. [**187**]{} (2011) 229. M. Wakamatsu, Phys. Rev. D [**83**]{} (2011) 014012. E. Leader, Phys. Rev. D [**83**]{} (2011) 096012. X. Ji, X. Xiong and F. Yuan, arXiv:1202.2843 \[hep-ph\]. JLab experiment C12-11-111, Contalbrigo M. [*et al.*]{} (2011); JLab experiment E12-11-006, Gao H. [*et al.*]{} (2011).
The COMPASS Collaboration. “COMPASS-II Proposal”, SPSC-2010-014/P-340, 17 May 2010.
P. E. Reimer, Transversity 2011 proceedings, Nuovo Cimento C 35/2 (2012) 225.
V. Y. Alexakhin [*et al.*]{} \[COMPASS Collaboration\], Phys. Rev. Lett. [**94**]{} (2005) 202002. E. S. Ageev [*et al.*]{} \[COMPASS Collaboration\], Nucl. Phys. B [**765**]{} (2007) 31. M. Alekseev [*et al.*]{} \[COMPASS Collaboration\], Phys. Lett. B [**673**]{} (2009) 127. A. Airapetian [*et al.*]{} \[HERMES Collaboration\], Phys. Rev. Lett. [**94**]{} (2005) 012002. W. Vogelsang and F. Yuan, Phys. Rev. D [**72**]{} (2005) 054028. A. V. Efremov, K. Goeke and P. Schweitzer, Eur. Phys. J. ST [**162**]{} (2008) 1. M. Anselmino [*et al.*]{}, Eur. Phys. J. A [**39**]{} (2009) 89. M. G. Alekseev [*et al.*]{} \[COMPASS Collaboration\], Phys. Lett. [**B692** ]{} (2010) 240. A. Airapetian [*et al.*]{} \[HERMES Collaboration\], Phys. Rev. Lett. [**103**]{} (2009) 152002. P. Abbon [*et al.*]{} \[COMPASS Collaboration\], Nucl. Instrum. Meth. A [**577**]{} (2007) 455. COMPASS Collaboration, “Experimental investigation of transverse spin asymmetries in $\mu$–p SIDIS processes: Collins asymmetriy”, submitted to Phys. Lett. B.
M. G. Alekseev [*et al.*]{} \[COMPASS Collaboration\], Phys. Lett. B [**690**]{} (2010) 466. A. Kotzinian, M. Anselmino and V. Barone, Transversity 2011 proceedings, Nuovo Cimento C 35/2 (2012) 85, arXiv:1110.5256 \[hep-ph\]. S. M. Aybat [*et al.*]{}, Phys. Rev. D [**85**]{} (2012) 034043. S. M. Aybat, A. Prokudin and T. C. Rogers, arXiv:1112.4423 \[hep-ph\]. M. Anselmino [*et al.*]{}, arXiv:1107.4446 \[hep-ph\]. M. Anselmino, M. Boglione and S. Melis, arXiv:1204.1239 \[hep-ph\]. F. Bradamante \[COMPASS Collaboration\], Transversity 2011 proceedings, Nuovo Cimento C 35/2 (2012) 107, arXiv:1111.0869 \[hep-ex\]. M. Anselmino [*et al.*]{}, Phys. Rev. D [**83**]{} (2011) 114019. The Durham HepData Project, http://hepdata.cedar.ac.uk/reaction
|
---
abstract: 'The following analog of Bernstein inequality for monotone rational functions is established: if $R$ is an increasing on $[-1,1]$ rational function of degree $n$, then $$R''(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1).$$ The exponential dependence of constant factor on $n$ is shown, with sharp estimates for odd rational functions.'
author:
- 'Andriy V. Bondarenko, Maryna S. Viazovska'
title: Bernstein type inequality in monotone rational approximation
---
Andriy V. Bondarenko, Maryna S. Viazovska\
Faculty of Mech. and Mathematics,\
Kyiv National Taras Shevchenko university,\
Kyiv, 01033, Ukraine.\
tel: (+38)0442590591\
e-mail: [bonda@univ.kiev.ua, v-marina@ukr.net]{}
Introduction
============
Let $P_n$ be the space of all polynomials of degree at most $n$. Denote by $Q_n$ the set of all continuous rational functions on $[\,-1,1\,]$, $r=\frac pq$, where $p$, $q\in P_n$. Now we state the well-known Bernstein inequality: If $p\in P_n$, then $$\label{i8}
|p'(x)|\le\frac n{\sqrt{1-x^2}}\,\|p\|,\quad x\in (-1,1),$$ where $\|\cdot\|:=\|\cdot\|_{C[-1,1]}$ is the uniform norm on $[-1,1]$. Unfortunately, the direct analog of this inequality in rational approximation impossible to establish. Indeed, if $R(x)=\frac{\delta\,x}{x^2+\delta^2}$, $\delta>0$, then $|R(x)|<
1$, $x\in \mathbb{R}$ and $R'(0)=\delta^{-1}$ can be arbitrary large. What is true, is Pekarskii inequality \[1\], where the norm of $R'$ and of $R$ are taken in different spaces. Our main result is
If $R\in Q_{2n}$ is an odd and increasing function on $[-1,1]$, then $$\label{i1}
R'(0)\le\frac 12\cdot 9^nR(1).$$
Theorem 1 easily implies the following analog of estimate for all increasing on $[-1,1]$ rational functions.\
[**Corollary 1.**]{} If $R\in Q_n$ is an increasing function on $[-1,1]$, then $$R'(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1).$$ A lower estimate for the constant in the right hand side of is provided by
For each $n\in\mathbb{N}$ $$\sup_{R}\frac{R'(0)}{\|R\|}\ge 9^{n-1},$$ where the supremum is taken over the set of all odd increasing on $[-1,1]$ rational function $R\in Q_{2n-1}$.
In Section 2 we prove some auxiliary results, in Section 3 we prove Theorem 1 and in Section 4 we prove Theorem 2.
Auxiliary lemmas
================
Let $u_i<v_i$, $i=\overline{1,n}$, be arbitrary numbers. Put $$\begin{aligned}
\Pi:&=\left\{\vec{y}=(y_1,\ldots,y_n)\in\mathbb{R}^n\,|\,u_i
\le y_i\le v_i, i=\overline{1,n}\right\},\\
\Pi^+_k:&=\left\{\vec{y}=(y_1,\ldots,y_n)\in\Pi\,|\,y_k=v_k\right\},
\quad k=\overline{1,n},\end{aligned}$$ and $$\Pi^-_k:=\left\{\vec{y}=(y_1,\ldots,y_n)\in\Pi\,|\,y_k=u_k\right\},
\quad k=\overline{1,n}.$$ To prove the following Lemma 1 we use the well known Brouwer fixed-point theorem [@N]\
[**Theorem B.**]{} Let A be a closed bounded convex subset of $\mathbb{R}^n$ and $F:A\rightarrow A$ be a continuous mapping on $A$. Then $F(\vec{z})=\vec{z}$, for some $\vec{z}\in A$.\
[**Lemma 1.**]{} Let $f_k:\Pi\rightarrow\mathbb{R}$, $k=\overline{1,n}$, be continuous functions satisfying the following inequalities $$f_k(\vec{y})<0,\quad \vec{y}\in\Pi_k^-,\quad k=\overline{1,n},$$ and $$f_k(\vec{y})>0,\quad \vec{y}\in\Pi_k^+,\quad k=\overline{1,n}.$$ Then, there exists $\vec{z}\in\Pi$ such that $f_k(\vec{z})=0$, $k=\overline{1,n}$.
Without any loss of generality we may assume that $\Pi=[-1;1]^n$. Put $\varphi(\vec{x})=(\varphi_1(\vec{x}),...,\varphi_n(\vec{x}))$, where $$\varphi_k(\vec{x})=\frac{f_k(\vec{x})}{|f_i(\vec{x})|+(1-x_k^2)}.$$ This definition readily implies $$\begin{gathered}
\label{L1}
|\varphi_i(\vec{x})|\le 1,\qquad \vec{x}\in\Pi,\\
\label{L2}
\varphi_i(\vec{x})=1,\quad\vec{x}\in\Pi_k^+,\quad k=\overline{1,n},\\
\label{L3}
\varphi_i(\vec{x})=-1,\quad\vec{x}\in\Pi_k^-,\quad k=\overline{1,n}.\end{gathered}$$ Since each $\varphi_k$, $k=\overline{1,n}$, is a continuous function on $\Pi$, then by and , there exists a number $\mu >0$ small enough, such that $$\label{L4}
\varphi(\vec{x})>0,\quad \vec{x}=(x_1,\ldots, x_n)\in\Pi,\quad 1-\mu\le x_k\le
1,$$ and $$\label{L5}
\varphi(\vec{x})<0,\quad \vec{x}=(x_1,\ldots, x_n)\in\Pi,\quad -1\le x_k\le
-1+\mu.$$ Now we prove that the set $A:=\Pi$ and the mapping $F(\vec{x}):=\vec{x}-\mu\varphi(\vec{x})$ satisfy the conditions of the Theorem B. Since $\varphi$ is a continuous mapping, then $F$ is a continuous mapping as well. Finally, we prove that $F(\vec{x})\in\Pi$, for all $\vec{x}\in\Pi$, that is $$\label{L6}
-1\le x_k-\mu\varphi_k(\vec{x})\le 1,\quad x_k\in [\,-1,1\,],\quad
k=\overline{1,n}.$$ If $x_k\in [\,-1+\mu, 1-\mu\,]$, then the inequality readily follows from . Taking into account and we get for $x_k\in [\,1-\mu,1\,]$ and for $x_k\in [\,-1,-1+\mu\,]$ respectively, $k=\overline{1,n}$, so holds. Thus, by Theorem B, $F(\vec{z})=\vec{z}$, for some $\vec{z}\in\Pi$, whence $f_k(\vec{z})=0$, $k=\overline{1,n}$. Lemma 1 is proved.
[**Lemma 2.**]{} Let $f$ be an increasing continuous function on $[\,0,1\,]$ such that $f(0)=0$, $f(1)=1$ and $f'(0)>\frac 12\cdot
9^n$. Then there exist the numbers $0< z_1<z_2<\ldots<z_n\le 1$ satisfying $$\label{i9}
f(z_s)=\sum_{k=1}^{n}4\cdot9^{k-1}\frac{z_k^2
z_s}{z_k^{2}+3z_s^{2}},\quad s=\overline{1,n}.$$
Since $g(x):=f(x)/x$ is a continuous function on $[\,0,1\,]$ ( $\lim_{x\to 0}g(x)=f'(0)>
\frac 12\cdot 9^n$ ) and $g(1)=1$, then there exist the numbers $0<u_n<v_n<u_{n-1}<v_{n-1}<\ldots<u_1<v_1\le 1$ for which $g(u_i)=3\cdot9^{i-1}$ and $g(v_i)=9^{i-1}$, whence $f(u_i)=
3\cdot 9^{i-1}u_i$ and $f(v_i)=9^{i-1}v_i$, $i=\overline{1,n}$. The fact that $f$ is an increasing function yields $$\label{i10}
v_i<3\cdot 9^{k-i}u_k,\qquad 1\le k<i\le n.$$ For each $s=\overline{1,n}$ put $$f_s(\vec{y})=f_s(y_1,\ldots,y_n):=\sum_{k=1}^{n}4\cdot9^{k-1}\frac{y_k^2
y_s}{y_k^{2}+3y_s^{2}}-f(y_s),\quad \vec{y}\in\Pi.$$ If $\vec{y}\in\Pi_s^+$, then $y_s=v_s$, hence $$f_s(\vec{y})> 4\cdot9^{s-1}\frac{v_s^2
v_s}{v_s^{2}+3v_s^{2}}-f(v_s)=0,\quad s=\overline{1,n}.$$ If $\vec{y}\in\Pi_s^-$, then $y_s=u_s$, hence $$\begin{aligned}
f_s(\vec{y})&=\sum_{k=1}^n4\cdot9^{k-1}\frac{y_k^2
u_s}{y_k^{2}+3u_s^{2}}-f(u_s)=
\sum_{k=1}^{s-1}4\cdot9^{k-1}\frac{y_k^2
u_s}{y_k^{2}+3u_s^{2}}\\
&+9^{s-1}u_s+\sum_{k=s+1}^{n}4\cdot9^{k-1}\frac{y_k^2
u_s}{y_k^{2}+3u_s^{2}}-3\cdot 9^{s-1}u_s\\
&\le\sum_{k=1}^{s-1}4\cdot9^{k-1}u_s+\sum_{k=s+1}^{n}4\cdot9^{k-1}
\frac{v_k^2}{3u_s^2}u_s-2\cdot 9^{s-1}u_s\\
&\le\frac 12\cdot 9^{s-1}u_s+\sum_{k=s+1}^{n}\frac 43\cdot
9^{2s-k}u_s-2\cdot 9^{s-1}u_s<0,\quad s=\overline{1,n},\end{aligned}$$ where in the last line we use . Applying Lemma 1 for the functions $f_s$, $s=\overline{1,n}$ we get that there exists $\vec{z}=(z_1,\ldots z_n)\in\Pi$, such that $f_s(\vec{z})=0$, $s=\overline{1,n}$, which is . Lemma 2 is proved.
Proof of Theorem 1
==================
Let $R\in Q_{2n}$ be an odd and increasing on function on $[\,-1,1\,]$ such that $R'(0)>\frac 12\cdot 9^nR(1)$. Without any loss of generality we may assume that $R(1)=1$. By Lemma 2, there exist the numbers $0<z_n<z_{n-1}<\ldots<z_1\le 1$ such that the function $$L(x):=\sum_{k=1}^{n}4\cdot9^{k-1}\frac{z_k^2
x}{z_k^{2}+3x^{2}}-R(x)$$ satisfies the equalities $L(z_s)=0$, $s=\overline{1,n}$. Further, we have $$L'(0)=\sum_{k=1}^{n}4\cdot9^{k-1}-R'(0)<0,$$ and $$\begin{aligned}
L'(z_s)&=\sum_{k=1}^{n}4\cdot9^{k-1}\frac{z_k^2(z_k^2-3z_s^2)}{(z_k^2+3z_s^2)^2}
-R'(z_s)\\
&<\sum_{k=1}^{s-1}4\cdot9^{k-1}-\frac 12\cdot 9^{s-1}<0.\end{aligned}$$ Thus, each of the open intervals $(0, z_n)$, $(z_n, z_{n-1})$, ...,$(z_2, z_1)$ contains at least one zero of the function $L$. Since $L$ is an odd function, then $L$ has at least $4n+1$ zeroes on $[\,-1,1\,]$. On the other hand $l\in Q_{4n}$, so $L\equiv 0$. This contradiction finished the proof of Theorem 1.\
[**Proof of Corollary 1:**]{} Without any loss of generality we may assume that $x>0$. For each increasing rational function $R\in
Q_n$ and $x\in (0,1)$ put $$H(y):=\frac{R(x+y(1-x))-R(x-y(1-x))}2.$$ Evidently, $H\in Q_{2n}$ is odd increasing rational function with $\|H\|\le \|R\|$. Note that $H'(0)=(1-x)R'(x)$. Thus, applying Theorem 1 for the function $H$ we get $$R'(x)\le\frac{9^n}{2(1-x)}\|R\|\le\frac{ 9^n}{1-x^2}\|R\|.$$
Proof of Theorem 2
==================
Below we use without proof two easy inequalities: If $\gamma$, $\alpha>0$, then $$\label{T21}
\frac{\gamma^2(\gamma^2-x^2)}{(\gamma^2+x^2)^2}\ge
-\frac{\gamma^2}{\alpha^2},\quad |x|\ge\alpha,$$ and $$\label{T22}
\frac{\gamma^2(\gamma^2-x^2)}{(\gamma^2+x^2)^2}\ge
-\frac 18,\quad x\in\mathbb{R}.$$ Let $K_n$ be the set of all odd rational functions $R\in Q_{2n-1}$ with $R'(x)>0$, $x\in [-1,1]$. Evidently, it is sufficient to prove that $$\label{T23}
S_n:=\sup_{R\in K_n}\frac{R'(0)}{\|R\|}\ge 9^{n-1}.$$ If $n=1$, then the function $R(x)\equiv x$ provide . Let $R\in K_n$ be an arbitrary function. Fix ${\varepsilon}\in (0,
R'(0)/2)$. Since $R'$ is a continuous function, then there exists $\alpha$, $\beta>0$ such that $R'(x)>R'(0)-{\varepsilon}$, for $|x|<\alpha$, and $R'(x)>\beta$, for $x\in [-1,1]$. For each $\gamma>0$ put $$G_{\gamma}(x):=R(x)+8(R'(0)-2{\varepsilon})\frac{\gamma^2x}{\gamma^2+x^2},
\quad x\in[-1,1].$$ We have $$\label{T24}
G'_{\gamma}(x)=R'(x)+8(R'(0)-2{\varepsilon})\frac{\gamma^2(\gamma^2-x^2)}{(\gamma^2+x^2)^2},$$ so $$\label{T25}
G'_{\gamma}(0)=9R'(0)-16{\varepsilon}.$$ The inequality implies $G'_{\gamma}(x)>R'(0)-{\varepsilon}-(R'(0)-2{\varepsilon})={\varepsilon}$, for $|x|<\alpha$. Thus, by and $G_{\gamma}\in
K_{n+1}$, for all $\gamma$ small enough, such that $$8(R'(0)-2{\varepsilon})\frac{\gamma^2}{\alpha^2}<\beta.$$ Moreover $$\label{T26}
\|G_{\gamma}\|\le
\|R\|+4(R'(0)-2{\varepsilon})\gamma \to \|R\|,
\quad
\gamma\to 0.$$ Since $R\in K_n$ is an arbitrary function and ${\varepsilon}$ can be arbitrary small, then and yield $S_{n+1}\ge 9S_n$. This gives us . Theorem 2 is proved.
[xx]{} G. G. Lorentz, M. v. Golitschek, Y. Makovoz, Constructive Approximation, Springer Verlag, Berlin, 1996. L. Nirenberg, Topics in nonlinear functional analysis, New York, 1974.
|
---
abstract: |
Quantitative nuclear magnetic resonance imaging (MRI) shifts more and more into the focus of clinical research. Especially determination of relaxation times without/and with contrast agents becomes the foundation of tissue characterization, e.g. in cardiac MRI for myocardial fibrosis. Techniques which assess longitudinal relaxation times rely on repetitive application of readout modules, which are interrupted by free relaxation periods, e.g. the Modified Look-Locker Inversion Recovery = MOLLI sequence. These discontinuous sequences reveal an apparent relaxation time, and, by techniques extrapolated from continuous readout sequences, the real $T_1$ is determined. What is missing is a rigorous analysis of the dependence of the apparent relaxation time on its real partner, readout sequence parameters and biological parameters as heart rate. This is provided in this paper for the discontinuous balanced steady state free precession (bSSFP) and spoiled gradient echo readouts. It turns out that the apparente longitudinal relaxation rate is the time average of the relaxation rates during the readout module, and free relaxation period. Knowing the heart rate our results vice versa allow to determine the real $T_1$ from its measured apparent partner.
[**Keywords**]{}: longitudinal relaxation, T1, T2, Lock Locker, MOLLI, balanced steady state free precession, spoiled gradient echo
author:
- Thomas Kampf
- Theresa Reiter
- 'Wolfgang Rudolf Bauer[^1]'
bibliography:
- 'References.bib'
title: |
An analytical Model which Determines the Apparent $T_1$ for Modified Look-Locker Inversion Recovery (MOLLI) -\
Analysis of the Longitudinal Relaxation under the Influence of Discontinuous Balanced and Spoiled Gradient Echo Readouts -\
---
Introduction {#introduction .unnumbered}
============
Many nuclear magnetic resonance imaging techniques depend on periodic perturbative readouts of nuclear magnetization, the dynamics of which otherwise would be solely determined by thermodynamic forces driving it towards equilibrium. Prominent examples of periodic perturbations of relaxation processes are the repetitive application of - spoiled gradient echo sequences in order to determine quickly $T_1$ (Snapshot Flash) [@Deichmann1992] - or balanced steady state free precession sequences (bSSFP) [@Scheffler2003]. Recently characterization of myocardial pathology by fast determination of $T_1$ by modified Look-Locker-Inversion Recovery (MOLLI) techniques [@Messroghli2004] and its modifications (e.g. [@Piechnik2010]) has shifted into the focus of interest in cardiac MRI. MOLLI differs from the aforementioned examples as the periodic perturbation acts on two time scales. Periods of free longitudinal relaxation, the length of which are determined by the heart beat cycle length $T_{RR}$, are interrupted by readout imaging modules, in which the balanced or spoiled gradient echoes are repeated with the much smaller repetition time $T_R$ . Of course it would be of paramount interest to relate this complex driven relaxation process with its apparent relaxation time $T_{1,Molli}^*$ to the sequence parameters, and the tissue parameters $T_1$ and $T_2$. This dependence, which to our knowledge is still unknown, will be derived in this note.
Longitudinal relaxation in the presence of discontinuous periodic readouts {#longitudinal-relaxation-in-the-presence-of-discontinuous-periodic-readouts .unnumbered}
==========================================================================
Periodic perturbation of relaxation processes consist of modules in which external forces, e.g. radio-frequency pulses, are interleaved with non-disturbed relaxation intervals in which thermodynamic forces act. The latter increase entropy of the spin system which becomes apparent in transverse relaxation, and minimize its free energy in longitudinal relaxation. Pulses act linearly on the magnetization vector $\operatorname{{\bf m}}$, whereas thermodynamic forces on the difference of $\operatorname{{\bf m}}$ to its equilibrium value $\operatorname{{\bf m}}_\mathrm{eq} $, i.e. $\operatorname{{\bf m}}-\operatorname{{\bf m}}_\mathrm{eq}$. For simplification we normalize the magnetization by the magnitude of this equilibrium value ${\bf m}\to {\bf m}/m_\mathrm{eq}$, and align the z-direction parallel to the direction of the external magnetic field, i.e. $m_\mathrm{eq}\to \operatorname{{{\bf e}}_z}$, with $\operatorname{{{\bf e}}_z}$ as the corresponding unit vector.
The above mentioned linear/affine response of the magnetization to rf-pulses and thermodynamic forces has the following consequence: When rf-puls(es) and relaxation period are coupled to one module, and when these modules appear contiguously in series, the magnetization at the end of one module is an affine function of that at its beginning, i.e. that at the end of the preceding module. So, for the magnetization after the $n$-th module follows $$\operatorname{{\bf m}}_{n}=\operatorname{\boldsymbol{U}}\operatorname{{\bf m}}_{n-1}+\bf{v}\;,\label{recursive1}$$ with the transformation matrix $\operatorname{\boldsymbol{U}}$, and some vector $\bf{v}$. The steady state is achieved, when the magnetization after the module is identical with that before, which determines the corresponding steady state magnetization as $$\operatorname{{\bf m}}_\mathrm{ss}=(\mathbf{1}-\operatorname{\boldsymbol{U}})^{-1}\bf{v}\;, \label{ss}$$ with $\mathbf{1}$ as the identity matrix. Recursive application of Eq. (\[recursive1\]) and applying rules for geometric series yields that $$\operatorname{{\bf m}}_{n}=\operatorname{\boldsymbol{U}}^{n}(\operatorname{{\bf m}}(0)-\operatorname{{\bf m}}_\mathrm{ss})+\operatorname{{\bf m}}_\mathrm{ss}\;, \label{Ev1}$$ with $\operatorname{{\bf m}}(0)$ as the initial magnetization. The last equation demonstrates that with respect to the steady state magnetization $\operatorname{\boldsymbol{U}}$ is the generator of evolution on the time scale of a module duration. For practical determination of relaxation rates $\operatorname{\boldsymbol{U}}$ is decomposed spectral, after Eigenvectors and Eigenvalues have been determined. In general this yields a multi-exponential decay of $\operatorname{{\bf m}}$.
In case of discontinuous Lock-Locker sequences as the MOLLI the situation is a bit more complex as the module consists of two sub-modules, the readout, with duration $t_{read}$ and the free relaxation period, lasting $t_{free}=T_{RR}-t_{read}$, with $T_{RR}$ as the cycle length of the heart beat. Within the module the sub-modules follow a time evolution as in Eq. (\[Ev1\]). So when $\operatorname{{\bf m}}_{n-1}$ is the magnetization before the $n$-th readout, it develops towards $$\operatorname{{\bf m}}_{n-1/2} =\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}(\operatorname{{\bf m}}_{n-1}-\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(readout)}}}}) +\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(readout)}}}}\;,\label{ASSFP}$$ with $\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}$ as the transformation matrix -, and $\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(readout)}}}}$ as the steady state magnetization of the readout. Note, that $n-1/2$ symbolizes the magnetization directly after the $n$-th readout, but before the free evolution of the $n$-th sequence cycle, completing only “half” the evolution. The explicit forms of the readouts have been determined in the past, e.g. see Refs. [@Deichmann1992; @Scheffler2003], and will be used later on. Thereafter the magnetization freely decays, which is described by the relaxation rate matrix $$\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}=\begin{pmatrix}
e^{-t_{free}/T_2 } & 0& 0\\
0 & e^{-t_{free}/T_2} & 0 \\
0 & 0& e^{-t_{free}/T_1}
\end{pmatrix}$$ So magnetization after the readout develops to $$\operatorname{{\bf m}}_{n}=\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}(\operatorname{{\bf m}}_{n-1/2}-\operatorname{{{\bf e}}_z})+\operatorname{{{\bf e}}_z}\;.\label{AFree}$$ The Eqs. (\[ASSFP\],\[AFree\]) yield the recursive dependence of magnetizations before and after the whole module as $$\operatorname{{\bf m}}_{n}=\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}(\operatorname{{\bf m}}_{n-1}-\operatorname{{\bf m}}_\mathrm{ss}) +\operatorname{{\bf m}}_\mathrm{ss}\;,\label{Evolutioncomb}$$ with the steady state magnetization of the MOLLI sequence determined according to Eq.(\[ss\]) $$\begin{aligned}
\operatorname{{\bf m}}_\mathrm{ss}&=&\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(readout)}}}}+\frac{1}{\mathbf{1}-\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}}\cr
&& (\mathbf{1}-\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}})(\operatorname{{{\bf e}}_z}-\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(readout)}}}})\cr\cr
&=&(\mathbf{1}-\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}})^{-1}\times\cr\cr
&&\bigg(\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}(1-\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}})\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(readout)}}}}+(1-\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}})\operatorname{{{\bf e}}_z}\bigg)\cr\cr
\label{ssMOLLI}\;.\end{aligned}$$ This implies that the evolution operator on the time scale of the composite module $T_{RR}$ is $\boldsymbol{U}^{(comp)}=\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}$. It is noteworthy that in case that the evolution operators $\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}$ and $\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}$ commute, the relaxation rate(s) of the composite module are the time average of those of the sub-modules. This is easily seen as one can assign the evolution matrices $\boldsymbol{U}^{(i)}$ ($i=$ readout -, free -, composite module) generator matrices $\boldsymbol{R}$ with $\boldsymbol{U}^{(i)}=\exp(\boldsymbol{R}^{(i)}t_i)$. As $t_{read}+t_{free}=T_{RR}$ one obtains the addition theorem for respective generators $$\boldsymbol{R}^{(comp)}=t_{read}/T_{RR} \boldsymbol{R}^{(read)}+t_{free}/T_{RR} \boldsymbol{R}^{(free)}\;.$$ In the next sections the two different readout modules which are commonly used, the traditional Lock-Locker (FLASH) and the bSSFP readout, will be investigated.
![Relative Error (in percent) of analytical and numerical results for the MOLLI sequence with bSSFP readouts as a function of the $T_1$ and the heart cycle length $T_{RR}$. Above: for the apparent $T_1^{\hbox{\tiny{* MOLLI}}}$ (see Eqs. (\[MOLLIT1Stern\],\[MOLLIT1Stern2\] for the analytical values). Below for the steady state magnetization (see Eq. (\[msstim\])).For the numerical approach we straightforwardly applied the evolution matrices (e.g. preparation pulse, bSSFP readouts and free relaxation) on magnetization in series. Acquisition of the center of k-space was obtained at $t_{read}/2$, which was also the value for the imaging time determining the steady state magnetization (Eq. (\[msstim\])). The time course of signal from these centers of k-space was fitted by a single exponential providing the numerical values for the apparent relaxation time and steady state magnetization. The sequence parameters were: $T_R =$ 2.4 ms,$\alpha = 35^\circ$, $t_{read} =86 \text{pulses}\times 2.4 \text{ms} = \text{206.4 ms}$, and $T_2=50$ms. []{data-label="T1MOLLIwithFB"}](T1MOLLIwithFB.png "fig:"){width="8cm"} ![Relative Error (in percent) of analytical and numerical results for the MOLLI sequence with bSSFP readouts as a function of the $T_1$ and the heart cycle length $T_{RR}$. Above: for the apparent $T_1^{\hbox{\tiny{* MOLLI}}}$ (see Eqs. (\[MOLLIT1Stern\],\[MOLLIT1Stern2\] for the analytical values). Below for the steady state magnetization (see Eq. (\[msstim\])).For the numerical approach we straightforwardly applied the evolution matrices (e.g. preparation pulse, bSSFP readouts and free relaxation) on magnetization in series. Acquisition of the center of k-space was obtained at $t_{read}/2$, which was also the value for the imaging time determining the steady state magnetization (Eq. (\[msstim\])). The time course of signal from these centers of k-space was fitted by a single exponential providing the numerical values for the apparent relaxation time and steady state magnetization. The sequence parameters were: $T_R =$ 2.4 ms,$\alpha = 35^\circ$, $t_{read} =86 \text{pulses}\times 2.4 \text{ms} = \text{206.4 ms}$, and $T_2=50$ms. []{data-label="T1MOLLIwithFB"}](MssMOLLIwithFB.png "fig:"){width="8cm"}
MOLLI with bSSFP readouts {#molli-with-bssfp-readouts .unnumbered}
=========================
Evolution under the influence of bSSFP {#evolution-under-the-influence-of-bssfp .unnumbered}
--------------------------------------
In bSSFP, magnetization is excited by an initial preparation $\alpha/2$ pulse. Thereafter it develops gradient induced echoes which are all balanced and driven consecutively by alternating $\alpha$-pulses spaced by repetition time $T_R=2T_E$. The theory of longitudinal relaxation under the influence of this sequence has been studied extensively in the past, e.g. [@Scheffler2003; @Schmitt2004], and only essentials necessary for understanding of the paper are repeated here. The repeated application pulses implies that the evolution operator $\operatorname{\boldsymbol{U}^{\hbox{\tiny{(bSSFP)}}}}$ consists of a sequence of identical operators giving the time evolution within the repetition time $\mathbf{\hat{A}}$, i.e. for $m$ repetitions $$\operatorname{\boldsymbol{U}^{\hbox{\tiny{(bSSFP)}}}}=\mathbf{\hat{A}}^m\;\mathbf{\hat{P}}_\mathrm{pre}\;,$$ where $\mathbf{\hat{P}}_\mathrm{pre}$ is the operator realizing the initial preparation, i.e. for an $\alpha/2$ pulse rotated around the x-axis $$\mathbf{\hat{P}}_\mathrm{pre}=\begin{pmatrix}
1&0 &0\\
0 &\cos(\alpha/2)&\sin(\alpha/2)\\
0 & -\sin(\alpha/2) & \cos(\alpha/2)
\end{pmatrix}\;.\label{Pre}$$ $\mathbf{\hat{A}}$ itself consists of operators describing pulse related rotations $\alpha$ around the x-axis and phase shifts $\Pi$ (due to alternation of rotation direction), precession (due to off-resonance) as well as free relaxation within $T_R$. We focus only on the on-resonant case, which derives $\mathbf{\hat{A}}$ as $$\mathbf{\hat{A}}=\begin{pmatrix}
-e^{-\frac{T_R}{T_2}}&0 &0\\
0 &e^{-\frac{T_R}{T_2}}\cos(\alpha)&e^{-\frac{T_R}{2} \left(\frac{1}{T_1}+\frac{1}{T_2}\right)} \sin(\alpha)\\
0 & e^{-\frac{T_R}{2} \left(\frac{1}{T_1}+\frac{1}{T_2}\right)}\sin(\alpha) & e^{-\frac{T_R}{T_1}}\cos(\alpha)
\end{pmatrix}\;.$$ Note that this real matrix is symmetric (Hermitian), which implies an orthogonal system of Eigenvectors. The evolution of magnetization within $T_R$ is obtained by the affine recursion $$\bf{m}_{m}^{\hbox{\tiny{(bSSFP)}}}=\mathbf{\hat{A}}\;\bf{m}_{m-1}^{\hbox{\tiny{(bSSFP)}}}+\bf{v}$$ with $$\bf{v}= \begin{pmatrix}
0\\
e^{-\frac{T_R}{2 T_2}}\left(1-e^{-\frac{T_R}{2 T_1}}\right)\sin(\alpha)\\
1+e^{-\frac{T_R}{2 T_1}}(\cos(\alpha)-1)-e^{-\frac{T_R}{ T_1}}\cos(\alpha)
\end{pmatrix}$$ For the evaluation of longitudinal relaxation in the MOLLI setup, it is useful to obtain the steady state vector, as well as Eigenvectors and - values of the bSSFP readout matrix $\mathbf{\hat{A}}$. as the repetition time is much smaller than the relaxation time, $T_R\ll T_1,\;T_2$, we get $$\begin{aligned}
{3}
\mathbf{e}_1 &=\begin{pmatrix}
1\\0\\0
\end{pmatrix} & \quad & \quad & \lambda_1 &=-\exp\left(-\frac{T_R}{T_2}\right)\nonumber\\
\label{Eigenall}
\mathbf{e}_2 &=\begin{pmatrix}
0\\ -\cos(\alpha/2)\\ \sin(\alpha/2)
\end{pmatrix} & & & \lambda_2 &=-\exp\left(\frac{-T_R}{T_2^{\hbox{\tiny{*bSSFP}}}}\right)\\
\mathbf{e}_3 &=\begin{pmatrix}
0\\ \sin(\alpha/2)\\ \cos(\alpha/2)
\end{pmatrix} & & & \lambda_3 &=\exp\left(\frac{-T_R}{T_1^{\hbox{\tiny{* bSSFP}}}}\right)\nonumber\end{aligned}$$ for the normalized Eigenvectors above, and corresponding Eigenvalues below. Here $$\begin{aligned}
\frac{1}{T_1^{\hbox{\tiny{* bSSFP}}}}&=&\cos^2(\alpha/2)\frac{1}{T_1}+\sin^2(\alpha/2)\frac{1}{T_2}\cr
\frac{1}{T_2^{\hbox{\tiny{* bSSFP}}}}&=&\cos^2(\alpha/2)\frac{1}{T_2}+\sin^2(\alpha/2)\frac{1}{T_1}\label{bSSFPT}\end{aligned}$$ denote the apparent longitudinal or transverse relaxation times of the bSSFP train. The steady state magnetization derives as $$\operatorname{{{\bf m}}_\mathrm{ss}^{\hbox{\tiny{(bSSFP)}}}}\approx\cos(\alpha/2)\;\frac{T_1^{\hbox{\tiny{* bSSFP}}}}{T_1 }\mathbf{e}_3\;,\label{ssbSSFP}$$ i.e. the steady state vector and 3rd Eigenvector are parallel.
Evaluation under the influence of discontinuous bSSFP readouts
--------------------------------------------------------------
We will now investigate the generator of time evolution in the MOLLI setup. The generator for the bSSFP imaging module is given by $$\begin{aligned}
\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}= \mathbf{\hat{P}}_\mathrm{post}\;\operatorname{\boldsymbol{U}^{\hbox{\tiny{(bSSFP)}}}}\end{aligned}$$ and hence $$\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\operatorname{\boldsymbol{U}^{\hbox{\tiny{(reaodut)}}}}=\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\;\mathbf{\hat{P}}_\mathrm{post}\;\mathbf{\hat{A}}^m\;\mathbf{\hat{P}}_\mathrm{pre}\label{EvMOLLI}$$ where $\mathbf{\hat{P}}_\mathrm{post}$ depends on the details of the imaging module, i.e. if at the end the magnetization is flipped back onto the $z$-axis or not. The free relaxation following the bSSFP readout last long, when compared to transverse relaxation $T_{RR}-t_{read}\gg T_2$. This simplifies the corresponding evolution operator to $$\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\approx e^{-(T_{RR}-t_{read})/T_1}\;\mathbf{\hat{\Pi}}_z \label{Ufrs}$$ where with the unit vectors in z-direction $\operatorname{{{\bf e}}_z}$ $$\mathbf{\hat{\Pi}}_z=\operatorname{{{\bf e}}_z}\operatorname{{{\bf e}}_z}^{\bf{T}}=\begin{pmatrix}
0&0 &0\\
0 &0&0\\
0 & 0 & 1
\end{pmatrix}\;$$ is the projection operator of a vector onto the z-axis. Note that $\operatorname{{{\bf e}}_z}^{\bf T}$ is the transposed vector. Keeping in mind that the initial magnetization is parallel to the z-axis, and that the free relaxation periods just leave a magnetization in z-direction (Eq. ), it is sufficient to consider only the part of $\mathbf{\hat{P}}_\mathrm{pre}$ (see Eq. (\[Pre\])) which rotates the z-component of magnetization, i.e. together with the Eigenvectors in Eqs. (\[Eigenall\]) we may write $$\mathbf{\hat{P}}_\mathrm{pre}=\operatorname{{{\bf e}}_\mathrm{3}}\operatorname{{{\bf e}}_z}^{\bf T}$$
Hence, exploiting the property that $\mathbf{e_3}$ is the 3rd Eigenvector of the bSSFP readout matrix $\mathbf{\hat{A}}$ simplifies Eq. (\[EvMOLLI\]) $$\operatorname{\boldsymbol{U}^{\hbox{\tiny{(free)}}}}\operatorname{\boldsymbol{U}^{\hbox{\tiny{(bSSFP)}}}}=\mathbf{\hat{\Pi}}_z\ \xi\;\lambda_3^m\; e^{-\frac{T_{RR}-t_{read}}{T_1}}\;,$$ with $$\xi=\operatorname{{{\bf e}}_z}^{\bf T}\mathbf{\hat{P}}_\mathrm{post} \operatorname{{{\bf e}}_\mathrm{3}}$$ For the magnetization immediately after the readout module two options are considered. When there is no further manipulation we have simply $\mathbf{\hat{P}}_\mathrm{post} = \mathbf{1}$, with $\mathbf{1}$ as the identity matrix. If the magnetization is flipped back, the bSSFP Eigenvector $\operatorname{{{\bf e}}_\mathrm{3}}$ is rotated back parallel to the z-axis, i.e. one gets $\mathbf{\hat{P}}_\mathrm{post}\operatorname{{{\bf e}}_\mathrm{3}}= \operatorname{{{\bf e}}_z}$. So $$\xi=\begin{cases}
\cos(\alpha/2) &\text{no flip back}\\
1 &\text{with flip back}
\end{cases}$$
The apparent relaxation time $T_1^{\hbox{\tiny{* MOLLI}}}$ gives the relaxation of the z-component within $T_{RR}$ by the factor $e^{-T_{RR}/T_1^{\hbox{\tiny{* MOLLI}}}}$, i.e. after inserting the 3rd Eigenvalue, and taking into account that $t_{read}=m\;T_R$ one obtains $$\label{MOLLIT1Stern}
\frac{1}{T_1^{\hbox{\tiny{* MOLLI}}}}=\underbrace{\frac{T_\mathrm{RR} - t_{read}}{T_\mathrm{RR}}\frac{1}{T_1}+\frac{t_{read}}{T_\mathrm{RR}}\frac{1}{T_1^{\hbox{\tiny{* bSSFP}}}}}_{\hbox{\tiny{= time averaged relaxation rate}}}- \frac{\ln(\xi)}{T_\mathrm{RR}}\;.$$ So the apparent relaxation rate is the sum of the time averaged rate of relaxation within the readout module and free relaxation plus a correction term, which depends on the post preparation. With Eq. (\[bSSFPT\])this leads to $$\label{MOLLIT1Stern2}
\frac{1}{T_1^{\hbox{\tiny{* MOLLI}}}}=\frac{1}{T_1}+\frac{t_{read}}{T_\mathrm{RR}}\sin\left(\frac{\alpha}{2}\right)^2\left(\frac{1}{T_2}-\frac{1}{T_1}\right)-\frac{\ln\left(\xi\right)}{T_\mathrm{RR}}$$ The correction term $\ln\left(\xi\right)/T_{RR}$ either vanishes if the flip back is applied or is otherwise also rather small as the following shosw: $T_{RR}$ is typically in the order of $T_1$ and bSSFP readout mainly operates with angles of $\alpha\approx 35^\circ$, which makes the relative difference of relaxation rates with and without flip back about $\approx 5\%$.
The steady state magnetization derives from Eq. (\[ssMOLLI\]). As at this time point only the z-component is important, one may simplify this Equation and derives $$\begin{aligned}
\label{Mss1}
m_{ss}^{\hbox{\tiny{(MOLLI)}}}&=& \left[e^{-\frac{T_{RR}-t_{read}}{T_1}}\bigg(1-e^{-\frac{t_{read}}{T_1^{\hbox{\tiny{*bSSFP}}}}}\bigg)\xi m_{ss}^{\hbox{\tiny{(bSSFP)}}}\right.\\
&&\left. +\bigg(1-e^{-\frac{T_{RR}-t_{read}}{T_1}}\bigg)\right]\left(1-e^{-\frac{T_{RR}}{T_1^{\hbox{\tiny{*MOLLI}}}}}\right)^{-1}\nonumber\end{aligned}$$
When we assume that $t_{read}/T_1^{*,\hbox{\tiny{bSSFP}}}$ is sufficiently small, and inserting the steady state magnetization under bSSFP readout conditions $m_{ss}^{\hbox{\tiny{(bSSFP)}}}$ from Eq. (\[ssbSSFP\]) one obtains in first order expansion in $t_{read}$ $$\begin{aligned}
m_{ss}^{\hbox{\tiny{(MOLLI)}}}&\approx \frac{1-e^{-(T_{RR}-t_{read}\kappa\sin(\alpha/2)^2)/T_1}}{1-e^{T_{RR}/T^*_{1,Molli}}}\\\end{aligned}$$ with $\kappa = 1$ in the absence, and $\kappa = 1/2$ in the presence of the with flip back of magnetization after the readout. Complete neglection of the $t_{read}$ terms further simplifies these results to $$\begin{aligned}
\label{approxmss}
m_{ss}^{\hbox{\tiny{(MOLLI)}}}&\approx&\frac{1-e^{-T_{RR}/T_1}}{1-e^{-T_{RR}/T_1^{\hbox{\tiny{* MOLLI}}}}}\end{aligned}$$ which has the similar form as the classical Lock Locker (FLASH) experiment hence motivating the commonly used correction to obtain $T_1$ from the apparent relaxation time $T_1^{\hbox{\tiny{* MOLLI}}}$.
Equation (\[Mss1\])gives the steady state magnetization at the end of the free relaxation period, i.e. immediately before the next readout module. However, from this next readout module the relevant signal is that obtained when the center of k-space is acquired, the timing $t_{im}$ of which is (due to the preparation pulses) after half the duration of the readout module. Hence, one must also consider the effect of this bSSFP module and one obtains for the measured steady state magnetization $$\begin{aligned}
\label{msstim}
{\bf m}_{ss,\hbox{\tiny{IM}}}^{\hbox{\tiny{(MOLLI)}}} &= {\bf m}_{ss}^{\hbox{\tiny{(bSSFP)}}}+ (m_{ss}^{\hbox{\tiny{(MOLLI)}}}\operatorname{{{\bf e}}_\mathrm{3}}- \operatorname{{\bf m}}_{ss}^{\hbox{\tiny{(bSSFP)}}})e^{-t_{im}/T_1^{* \hbox{\tiny{(bSSFP)}}}}\cr\cr
& \approx \operatorname{{{\bf e}}_\mathrm{3}}\left(\cos\left(\frac{\alpha}{2}\right)\frac{t_{im}}{T_1} +m_{ss}^{\hbox{\tiny{(MOLLI)}}}e^{-t_{im}/T_1^{* \hbox{\tiny{(bSSFP)}}}}\right)\;.\end{aligned}$$ Note, that $T_1^{* \hbox{\tiny{(bSSFP)}}}$ may be obtained from Eq. .
MOLLI with spoiled gradient echo readouts - discontinuous classical Look-Locker (FLASH) {#molli-with-spoiled-gradient-echo-readouts---discontinuous-classical-look-locker-flash .unnumbered}
=======================================================================================
The theory of longitudinal relaxation under the influence of continously applied spoiled gradient echoes has been studied extensively in the past, e.g. [@Deichmann1992]. The spoiling implies that ideally the hf-pulses solely act on a magnetization in z-direction. So, within the readout module it is sufficient for two consecutive pulses to evaluate solely the interdependence of their z-component. As, in addition, the free relaxation period in between the readout module also only leaves a z-component at its end, it is justified as well to study only the z-component of the composed process. This simplifies the mathematical analysis as matrix operations just reduce to multiplication with numbers.
Evolution under the influence of spoiled gradient echos {#evolution-under-the-influence-of-spoiled-gradient-echos .unnumbered}
-------------------------------------------------------
We will only roughly present the well known results. The z-magnetization before two pulses in sequence, separated by the repetition time $T_\mathrm{R}$, are interrelated by the affine recursion $$m_{m}^{\hbox{\tiny{(LL)}}}=A\;m_{m-1}^{\hbox{\tiny{(LL)}}}+v$$ with $$A=e^{-\frac{T_\mathrm{R}}{T_1}}\cos(\alpha)$$ and $$v= 1-e^{-\frac{T_\mathrm{R}}{T_1}}\;.$$ Note that the equilibrium magnetization is normalized to one. Recursive application directly leads to an apparent relaxation rate $$\begin{aligned}
\label{T1SternLL}
\frac{1}{T_1^{\hbox{\tiny{* (LL)}}}}&=&\frac{1}{T_1}-\frac{\ln\left(\cos\left(\alpha\right)\right)}{T_\mathrm{R}}\;.,\end{aligned}$$ and the steady state magnetization derives as $$\begin{aligned}
\label{mssll}
m_{\tiny{ss}}^{{\hbox{\tiny (LL)}}} = \frac{1-e^{-T_\mathrm{R}/T_1}}{1-e^{-T_\mathrm{R}/T_1^{\hbox{\tiny{* (LL)}}}}}\approx \frac{T_1^{\hbox{\tiny{* (LL)}}}}{T_1}\ ,\end{aligned}$$ where the last approximation is justified, as repetition time of the gradient echoes is small compared with $T_1,\;T_1^{*}$.
Evaluation under the influence of discontinuous spoiled gradient echo readouts {#evaluation-under-the-influence-of-discontinuous-spoiled-gradient-echo-readouts .unnumbered}
------------------------------------------------------------------------------
Time evolution during the free relaxation periods between the readouts is given by the factor $$U^{\hbox{\tiny{(free)}}}= e^{-(T_{RR}-t_{read})/T_1}\label{Ufrs2}\;,$$ where duration of the readout $t_{read}$ is determined by the number $m$ of gradient echoes, i.e. $t_{read}=m\;T_R$ So, the generator of discontinuous relaxation (Eq. (\[Evolutioncomb\])) for one period (readout- free relaxation, with duration $T_{RR}$) in the Lock-Locker setup is $$\begin{aligned}
U^{\hbox{\tiny{(free)}}}U^{\hbox{\tiny{(LL)}}}&=&U^{\hbox{\tiny{(free)}}}\;A^m\label{EvLL}\nonumber\cr\cr
&=&e^{-(T_{RR}-t_{read})/T_1}\;e^{-t_{read}/T_1^{\hbox{\tiny{* LL}}}}\nonumber\cr\cr
&=&e^{-T_\mathrm{RR}/T_1^{\hbox{\tiny{* (NCLL)}}}}\end{aligned}$$ with the apparent relaxation rate of the discontinuous spoiled gradient echo readouts $$\begin{aligned}
\frac{1}{T_1^{\hbox{\tiny{* NCLL}}}}&=\frac{T_\mathrm{RR} - t_{read}}{T_\mathrm{RR}}\;\frac{1}{T_1}+\frac{t_{read}}{T_\mathrm{RR}}\;\frac{1}{T_1^{\hbox{\tiny{* LL}}}}\label{timeavLL}\cr\cr
&=\frac{1}{T_1}-\frac{m \ln\left(\cos\left(\alpha\right)\right)}{T_\mathrm{RR}}\end{aligned}$$ Equation \[timeavLL\] implies that the apparent relaxation rate of the combined/discontinuous Look Locker process is the time average of the free relaxation rate and the of the readout module. The steady state magnetization derives from Eq. (\[ssMOLLI\]) as $$\begin{aligned}
\label{MssNCLL}
m_{ss}^{\hbox{\tiny{(NCLL)}}}&=& \left[e^{-\frac{T_{RR}-t_{read}}{T_1}}\bigg(1-e^{-\frac{t_{read}}{T_1^{\hbox{\tiny{*LL}}}}}
\bigg)m_{\tiny{ss}}^{\tiny{(LL)}}\right.\\
&&\left. +\bigg(1-e^{-\frac{T_{RR}-t_{read}}{T_1}}\bigg)\right]\left(1-e^{-\frac{T_{RR}}{T_1^{\hbox{\tiny{*NCLL}}}}}\right)^{-1}\nonumber\;,\end{aligned}$$ and when we assume that $t_{read}/T_1^{*,\hbox{\tiny{LL}}}$ is sufficiently small, one obtains $$\begin{aligned}
m_{ss}^{\hbox{\tiny{(NCLL)}}}&\approx&\frac{1-e^{-T_{RR}/T_1}}{1-e^{-\frac{T_{RR}}{T_1^{\hbox{\tiny{* NCLL}}}}}}.\end{aligned}$$
As in the case of the bSSFP readout, one has to keep in mind that the above steady state magnetization is that at the end of the free relaxation period, just before the subsequent readout. This readout, or more precisely its timing of the center of k-space $t_{im}$ determines the measured steady state magnetization. Until the center of k-space is reached the steady state magnetization of Eq. (\[ss\]) evolves under the influence of repetitive spoiled gradient echos, i.e. measured steady state magnetization is obtained as $$\begin{aligned}
m_{\hbox{\tiny{ss, IM}}}^{\hbox{\tiny{(NCLL)}}} &= m_{\hbox{\tiny{ss}}}^{\hbox{\tiny{LL}}} + \left(m_{ss}^{\hbox{\tiny{(NCLL)}}} - m_{\hbox{\tiny{ss}}}^{\hbox{\tiny{(LL)}}}\right)e^{-\frac{t_{im}}{T_1^{\hbox{\tiny{* (LL)}}}}}\nonumber \\
&\approx \frac{t_{im}}{T_1} + m_{ss}^{\hbox{\tiny{(NCLL)}}}e^{-\frac{t_{im}}{T_1^{\hbox{\tiny{* LL}}}}}\;,\end{aligned}$$ where Eq. can be used to eliminate $T_1^{\hbox{\tiny{* LL}}}$.
Summary and Discussion {#summary-and-discussion .unnumbered}
======================
We derived analytical expressions for the determination of the apparent longitudinal relaxation time in the presence of discontinuous bSSFP or spoiled gradient echo readouts. It turns out that the corresponding relaxation rates are approximately the time average of the rates during the readouts, for which expressions exist, and the free relaxation period. Figure 1 demonstrates that the analytical results are close to those obtained by numerical approaches. Vice versa our results allow to determine the real $T_1$ from its apparent measured partner and sequence parameters, if the heart rate is known.
Our results also help to evaluate the existing techniques which assess $T_1$. The standard bSSFP MOLLI approaches uses a fitted apparent relaxation time and steady state value to obtain $T_1$ by [@Messroghli2004] $$\label{Messroghli}
T_1=[T_1^{\hbox{\tiny{*MOLLI}}}/m_{ss,\hbox{\tiny{IM}}}^{\hbox{\tiny{(MOLLI)}}}]_{\hbox{fitted}}\;.$$. This approach was an extrapolation of: 1. the results valid for application of spoiled gradient echoes, and 2. the approximation that the repetition time is much shorter than the apparent $T_1$ (see Eq. ). So in fact the bSSFP readout modules were considered as “super” pulses with the heart cycle length as repetition time, i.e, $(T_R)_{\hbox{standard approch}}\to T_{RR}$. Obviously the necessary criteria that $T_{RR}/T_1^{\hbox{\tiny{*MOLLI}}}\ll 1 $ is not fulfilled. As the heart cycle length and apparent $T_1$ are in the same order of magnitude, Equation then gives a steady state magnetization after the free relaxation period which in contrast to Eq. yields $m_{ss}^{\hbox{\tiny{(MOLLI)}}}> T_1^{\hbox{\tiny{* MOLLI}}}/T_1$. The question is, why does the standard MOLLI evaluation despite these obvious wrong presuppositions yield rather acceptable $T_1$ values? The answer lies in the time when the center of k-space is acquired which locates the measured steady state magnetization $m_{ss,\hbox{\tiny{IM}}}^{\hbox{\tiny{(MOLLI)}}}$ of Eq. somewhere beneath that after the free relaxation period and above the steady state magnetization of the continuous bSSFP sequence $m_{ss}^{\hbox{\tiny{(bSSFP)}}}$ (see Eq.), i.e. closer to the assumed $T_1^{\hbox{\tiny{* MOLLI}}}/T_1$. In fact, when we assume that the center of k-space is acquired at half the duration of the readout module, i.e. at $t_{im}=t_{read}/2$, and when we further take into account that $t_{read}\ll T_1$, which is in general a rather generous concession, the first term of the approximation of Eq. , $\cos(\alpha/2)\; t_{tim}/T_1$, may be neglected. We then can write $$\begin{aligned}
m_{ss,\hbox{\tiny{IM}}}^{\hbox{\tiny{(MOLLI)}}}&\approx m_{ss}^{\hbox{\tiny{(MOLLI)}}}e^{-1/2\;t_{read}/T_1^{* \hbox{\tiny{(bSSFP)}}}}\cr
&\approx \frac{\sinh\left(\frac{1}{2}\;\frac{T_{RR}}{T_1}\right)}{\sinh\left(\frac{1}{2}\;\frac{T_{RR}}{T_1^{\hbox{\tiny{*MOLLI}}}}\right)}\;,\end{aligned}$$ where we inserted $m_{ss}^{\hbox{\tiny{(MOLLI)}}}$ from Eq. , and $T_1^{* \hbox{\tiny{(bSSFP)}}}$ from Eq. , and made again use of $t_{read}\ll T_1$. With the weaker presupposition $1/2\;T_{RR}\ll T_1,\;T_1^{\hbox{\tiny{*MOLLI}}}$ instead of $T_{RR}\ll T_1,\;T_1^{\hbox{\tiny{*MOLLI}}}$, it is at least understandable that expansion of the hyperbolic sinus provides the approximation of Messrhogli et al. , i.e. $$m_{ss,\hbox{\tiny{IM}}}^{\hbox{\tiny{(MOLLI)}}}\approx T_1^{\hbox{\tiny{*MOLLI}}}/T_1\;.$$ So the the acceptable quality of the Eq. (\[Messroghli\]) for determination of $T_1$ results rather from serendipity than from a rigorous based foundation.
In our opinion our results will be helpful for analysis of already existing $T_1$ mapping techniques as well as for the design of of new ones. Also numerical approaches and simulation may be validated with these rather simple analytical expressions.
Acknowledgment {#acknowledgment .unnumbered}
--------------
The authors’ research was supported by the Deutsche Forschungsgemeinschaft (SFB688, TBB05 to WB) and the Bundesministerium für Bildung und Forschung (BMBF01 EO1504 to WB).
[^1]: corresponding author, Tel. +49-931-201-39011
|
---
abstract: 'Very long and precise follow-up measurements of the X-ray afterglow of very intense gamma-ray bursts (GRBs) allow a critical test of GRB theories. Here we show that the single power-law decay with time of the X-ray afterglow of GRB 130427A, the record-long and most accurately measured X-ray afterglow of an intense GRB by the Swift, Chandra and XMM Newton space observatories, and of all other known intense GRBs, is that predicted by the cannonball (CB) model of GRBs from their measured spectral index, while it disagrees with that predicted by the widely accepted fireball (FB) models of GRBs.'
author:
- Shlomo Dado
- Arnon Dar
title: 'Critical Test Of Gamma-ray Burst Theories'
---
Introduction
============
Gamma-ray bursts are brief flashes of gamma rays lasting between few milliseconds and several hours \[1\] from extremely energetic cosmic explosions \[2\]. They are usually followed by a longer-lived “afterglow” emitted mainly at longer wavelengths \[3\] (X-ray, ultraviolet, optical, infrared, microwave and radio). Roughly they fall into two classes, long duration ones (GRBs) that last more than $\sim$ 2 seconds, and short hard bursts (SHBs) that typically last less than $\sim$ 2 seconds \[4\]. The GRBs seem to be the beamed radiation emitted by highly relativistic jets \[5\] ejected in broad line supernova explosions of type Ic \[6\], following collapse of rapidly rotating stripped envelope high mass stars to a neutron star, quark star, or black hole. The origin of SHBs is not known yet, but it is widely believed to be highly relativistic jets presumably emitted in a phase transition in/of a compact star (white dwarf, neutron star or quark star) to a more compact state following cooling and loss of angular momentum, or in merger of compact stars in close binaries due to gravitational wave emission \[7\].
In the past two decades, two theoretical models of GRBs and their afterglows, the fireball (FB) model \[8\] and the cannonball (CB) model \[9\], have been used extensively to interpret the mounting observational data on GRBs and their afterglows. Both models were claimed to describe successfully the observational data. But, despite their similar names, the two models were and still are quite different in their basic assumptions and predictions (compare, e.g., \[10\] and \[11\]). Hence, at most, only one of them can provide a correct physical theory of GRBs and their afterglows.
In the CB model \[9\], bipolar jets that are made of a succession of highly relativistic plasmoids (cannonballs) are assumed to be launched in accretion episodes of fall-back matter onto the newly formed compact object in broad-line SNeIc akin to SN1998bw. The gamma-ray pulses in a GRB are produced by inverse Compton scattering of glory light -the light halo formed around the progenitor star by scattered light from pre-supernova ejections- by the electrons enclosed in the CBs. The afterglow is mainly synchrotron radiation emitted from the electrons of the external medium which are swept into the CBs and are accelerated there to very high energies by turbulent magnetic fields.
The FB models of GRBs evolved a long way from the original spherical FB models \[8\] to the current conical models \[11\] which assume that GRBs are produced by bipolar jets of highly relativistic thin conical shells ejected in broad line SNeIc explosions. In these models, the prompt emission pulses are synchrotron radiation emitted in the collisions between overtaking shells, while the continuous collision of the merged shells with the circumburst medium drives a forward shock into the medium and a reverse shock in the merged shells, which produce the synchrotron radiation afterglow.
The claimed success of both models to describe well the mounting observational data on GRBs and their afterglows, despite their complexity and diversity, may reflect the fact that their predictions depend on several choices and a variety of free parameters, which, for each GRB, are adjusted to fit the observational data. As a result, when successful fits to observational data were obtained, it was not clear whether they were due to the validity of the theory or due to the multiple choices and free adjustable parameters. Scientific theories, however, must be falsifiable \[12\]. Hence, only confrontations between accurate observational data and the key predictions of the GRB models, which do not depend on free adjustable parameters, can serve as critical tests of the validity of such models.
Critical tests of the origin of the prompt gamma-rays are provided, e.g., by their measured polarization, correlations between various prompt emission properties, and the GRB prompt emission energy relative to that of its afterglow. While the observations have confirmed the predictions of the CB model they have challenged those of the standard FB models \[13\].
Critical tests of the GRB theories are also provided by the observed GRB afterglow. In the FB model the origin of the afterglow is a forward shock in the circumburst medium driven by the ultra-relativistic jet, while in the CB model the afterglow is produced by the Fermi accelerated electrons which are swept into the jet. That, together with the different jet geometries, result in different falsifiable predictions for the afterglow light-curves. In particular, conical FB models predict a broken power-law decline of the light curve of the afterglow \[14\] where the pre-break temporal decline index $\alpha$ increases by $\Delta =3/4$ for an ISM like density distribution, or by $\Delta=1/2$ for a wind-like density distribution, independent of of the afterglow frequency \[11\]. The observed breaks in GRB afterglows, however, often are chromatic breaks with a break-time and $\Delta$ that depend on frequency and satisfy neither $1/2\leq\Delta \leq 3/4$ (see, e.g., FIG. 1) nor the FB closure relations \[11\]. E.g., an analysis of the Swift X-ray data on the 179 GRBs detected between January 2005 and and January 2007 and the optical AGs of 57 pre- and post-Swift GRBs did not find any burst satisfying all the criteria of a jet break \[15\]. Moreover, many GRBs have afterglows that do not show any break at all. Consequently, it has been suggested that, perhaps, these ’missing jet breaks’ take place at rather late-time, when the observations are not precise enough anymore or after they end \[16\].
Recently, however, the X-ray afterglow of GRB 130427A, the brightest gamma-ray burst detected by Swift \[17\] in the last 30 years, was followed with high precision by the sensitive X-ray observatories Chandra and XMM Newton for a record-breaking baseline longer than 80 million seconds \[18\], which allows a critical test of both the standard FB models and the CB model. Detailed comparison between the observed late-time X-ray afterglow of GRB 130427A, and that predicted by the standard fireball models has already been carried in \[18\]. It was concluded there that the forward shock mechanism of the standard FB models with plausible values for the physical parameters involved cannot explain the data, in both cases of constant density and stellar-wind circumburst media.
In contrast, in this paper we show that the observed X-ray afterglow of the very intense GRB 130427A that decays with time like a single power-law with no visible jet break until the end of the measurements, is that expected from the CB model for very intense GRBs, and its temporal decay index is precisely that expected in the CB model from its measured spectral index. Moreover, we show that, within errors, this is also the case for the late-time X-ray afterglows of all the 28 most intense GRBs with known redshift $z$, whose late-time afterglow was well measured.
The X-ray Afterglow In The CB Model
===================================
The circumburst medium in front of a CB moving with a highly relativistic bulk motion Lorentz factor $\gamma\gg 1$ is completely ionized by the CB’s radiation. In the CB’s rest frame, the ions of the medium that are swept in generate within the CB turbulent magnetic fields whose energy density is assumed to be in approximate equipartition with that of the impinging particles. The electrons that enter the CB with a Lorentz factor $\gamma(t)$ in the CB’s rest frame are Fermi accelerated there and cool by emission of synchrotron radiation (SR), which is isotropic in the CB’s rest frame and has a smoothly broken power-law. In the observer frame, the emitted photons are beamed into a narrow cone along the CB’s direction of motion by its highly relativistic motion, their arrival times are aberrated, and their energies are boosted by its Doppler factor $\delta$ and redshifted by the cosmic expansion during their travel time to the observer. For $\gamma^2\gg 1$ and a viewing angle $\theta^2\ll 1$ relative to the CB direction of motion, the Doppler factor satisfies $ \delta\approx 2\,\gamma/ [1+\theta^2\,\gamma^2]$.
The observed spectral energy density (SED) of the [*unabsorbed*]{} synchrotron X-rays has the form (see, e.g., Eq. (28) in \[10\]) $$F_{\nu} \propto n^{(\beta_x+1)/2}\,[\gamma(t)]^{3\,\beta_x-1}\,
[\delta(t)]^{\beta_x+3}\, \nu^{-\beta_x}\, ,
\label{Fnu}$$ where $n$ is the baryon density of the external medium encountered by the CB at a time $t$ and $\beta_x$ is the spectral index of the emitted X-rays, $E\,dn_x/dE\propto E^{-\beta_x}$.
The swept-in ionized material decelerates the CB motion. Energy-momentum conservation for such a plastic collision between a CB of a baryon number $N_{_B}$, a radius $R$ and an initial Lorentz factor $\gamma(0)\gg 1$, which propagates in a constant density ISM at a redshift $z$, yields the deceleration law (Eq. (4) in \[19\]) $$\gamma(t) = {\gamma_0\over [\sqrt{(1+\theta^2\,\gamma_0^2)^2 +t/t_d}
- \theta^2\,\gamma_0^2]^{1/2}}\,,
\label{goft}$$ where $t$ is the time in the observer frame since the beginning of the afterglow, and $t_d={(1\!+\!z)\, N_{_B}/ 8\,c\,
n\,\pi\, R^2\,\gamma_0^3}$ is the deceleration time-scale.
For a constant-density ISM, Eqs. (1) and (2) yield an afterglow whose shape depends only on three parameters: the product $\gamma_0\,\theta$, the deceleration time scale $t_d$, and the spectral index $\beta_x(t)$. As long as $t{\hbox{\rlap{$^<$}$_\sim$}}t_b=(1+\theta^2\,\gamma_0^2)^2\,t_d$, $\gamma(t)$ and consequently also $\delta(t)$ change rather slowly with $t$ and generate a [*plateau phase*]{} of $F_\nu(t)$, which lasts until $t\approx t_b$. Well beyond $t_b$, Eq. (2) yields $\delta(t)\approx\gamma(t)\propto t^{-1/4}$ and $$F_\nu(t)\propto [\gamma(t)]^{(4\,\beta_x+2)}\,\nu^{-\beta_x}
\propto t^{-\alpha_x}\, \nu^{-\beta_x}
\label{Fnulate}$$ where $$\alpha_x=\beta_x+1/2.
\label{Relation}$$ Such a canonical behavior of the X-ray afterglow of GRBs (which was predicted by the CB model \[20\] long before its empirical discovery with Swift \[21\]), is demonstrated in Figure 1, where the 0.3-10 keV X-ray light-curve of GRB 060729 that was measured with the Swift XRT \[17\] is plotted together with its best-fit CB model light-curve \[10\]. Its late-time afterglow between $1.5 \times 10^{5}-
1.5\times 10^7$ s shows a power-law decline with $\alpha_x=1.46\pm 0.025$ \[17\]. In the the CB model, Eq. (4) and the measured photon index $\beta_x=0.99\pm 0.07$ \[17\] yields $\alpha_x=1.49\pm 0.07$, in good agreement with its observed value. Figure 1 also demonstrates that: (1) the observed fast decline $F_\nu \propto t^{-6}$ of the prompt X-ray emission is much steeper than that expected in the FB model from high latitude emission with $F_x\sim
t^{-(2+\beta_x)}\approx t^{-3}$ \[11\] for an observed $\beta_x\sim 1$, (2) the observed $\alpha_x\sim 0$ during the plateau phase does not satisfy the FB model pre-break closure relations, (3) the increases of $\alpha$ by $\Delta\approx 1.5$ beyond the break does not satisfy $0.5\leq \Delta\leq 0.75$, and (4) the closure relation of the standard FB model \[11\] $\alpha=2\,\beta$ beyond the break is not satisfied.
X-ray Afterglows With Missing Breaks
====================================
In the CB model, the break/bend time of the afterglow in the GRB rest frame satisfies \[22\] $t_b/(1+z)\propto 1/[(1+z)\,Ep\, Eiso]^{1/2}$, where $Eiso$ and $Ep$ are, respectively, the GRB equivalent isotropic gamma-ray energy and the observed peak photon energy. Hence, very intense GRBs with relatively large $Ep$ and $Eiso$ values have a relatively small $t_b$, which can be hidden under the prompt X-ray emission or its fast decline phase \[10\]. Consequently, only the post break temporal decline of the afterglow with a decay index $\alpha_x=\beta_x+1/2$ is observed \[10\]. This is demonstrated in Figure 2 where the light-curve of the 0.3-10 keV X-ray afterglow of the very intense ($Eiso\approx 10^{54}$ erg \[23\]) GRB 061007 that was measured with the Swift XRT \[17\] is plotted together with the best-fit single power-law temporal decay index $\alpha_x=1.50 \pm 0.05$. This temporal index is in good agreement with $\alpha_x=1.51\pm 0.05$ predicted by Eq. (4).
To test further whether relation (4) is satisfied universally by the X-ray afterglow of the most energetic GRBs, we have extended our test to the X-ray afterglows of all GRBs with known redshift and $Eiso > 5\times
10^{53}$ erg, which were followed up with an X-ray space based observatory for at least a few days, assuming a single power-law decline (corresponding to a constant ISM density along the CB trajectory). These GRBs are listed in Table 1 together with their measured redshift $z$, $Eiso$, temporal decay index $\alpha_x$ and spectral index $\beta_x$.
The most energetic GRB listed in Table 1 is GRB 160625B, at redshift z=1.406, with $ Eiso\approx 5\times 10^{54}$ erg measured by KONUS-Wind. In Figure 3, the light curve of its X-ray afterglow that was measured with the Swift XRT \[17\] is compared to its best fit single power-law light curve. The best fit power-law has a temporal decay index $\alpha=1.33\pm 0.04)$ in good agreement with the expected value $1.33\pm 0.12$ from Eq. (4) and the measured spectral index $\beta_x=0.83\pm 0.12$ \[17\].
In Figure 4, the measured values of $\alpha_x$ and $\beta_x$ for the 28 most intense GRBs with known redshift that are listed in Table 1 are compared to the CB model prediction (line) as given by Eq. (4). The best-fit line $\alpha_x=a\,(\beta_x+1/2)$ to the data yields a=1.007.
The X-ray Afterglow Of GRB 130427A
==================================
The most accurate test, however, of the CB model relation $\alpha_x=\beta_x+1/2$ for a single GRB is provided by the follow-up measurements of the X-ray afterglow of GRB 130427A, the most intense GRB ever detected by Swift, with the Swift XRT and with the sensitive X-ray observatories XMM Newton and Chandra up to a record time of 83 Ms after burst \[17\]. The measured light-curve has a single power-law decline with $\alpha_x = 1.309\pm 0.007$ in the time interval 47 ks - 83 Ms. The best single power-law fit to the combined measurements of the X-ray light-curve of GRB 130427A with the Swift-XRT \[17\], XMM Newton and Chandra \[18\], and Maxi \[25\] that is shown in Figure 5 yields $\alpha_x=1.294\pm 0.03$. The CB model prediction as given by Eq. (4,) with the measured spectral index $\beta_x=0.79\pm 0.03$ \[18\], is $\alpha_x=1.29\pm 0.03$ , in remarkable agreement with its best fit value.
No doubt, the assumptions of a constant density circumburst medium is an over simplification: Long duration GRBs are produced in supernova explosions of type Ic of short-lived massive stars, which take place mostly in superbubbles formed by star formation. Such superbubble environments may have a bumpy density, which deviates significantly from the assumed constant-density ISM. Probably it is responsible for the observed deviations from the predicted smooth light-curves and $\chi^2/df$ values slightly larger than 1. Moreover, in a constant-density ISM, the late-time distance of a CB from its launch point is given roughly by, $$x={2 c \int^t \gamma\delta dt\over 1+z}\approx
{8\,c\,\gamma_0^2\sqrt{t_d\,t}\over {1+z}}\,.$$ It may exceed the size of the superbubble and even the scale-height of the disk of the GRB host galaxy. In such cases, the transition of a CB from the superbubble into the Galactic ISM, or into the Galactic halo in face-on disk galaxies, will bend the late-time single power-law decline into a more rapid decline, depending on the density profile above the disk. Such a behavior may have been observed by the Swift XRT \[17\] in a few GRBs, such as 080319B and 110918A, at $t> 3\times 10^6$ s and in GRB 060729 at $t>3\times 10^7$ s by Chandra \[26\].
Discussion and conclusions
==========================
The 83-Ms long follow-up measurements with the Swift XRT and the sensitive Chandra and XMM Newton observatories \[18\] of the X-ray afterglow of GRB 130427A, the brightest gamma-ray burst detected in the last 30 years, , allowed the most accurate test so far of the main falsifiable predictions of the standard FB models for the X-ray afterglow of GRBs. These predictions are a broken power-law light-curve with a late-time achromatic break, a post break temporal decay index larger by $1/2\leq \Delta\leq 3/4$ than its pre-break value, and closure relations between the temporal decay index and the spectral index of the afterglow for both pre-break and post break times. The precise record-long measurements of the X-ray afterglow of GRB 130427A disagree with these predictions of the standard FB models where a conical jet drives a forward shock into the circumburst medium \[18\]. In particular, the closure relations predicted by the fireball model require far-fetched values for the physical parameters involved, in both cases of constant density and a wind-like circumburst medium \[18\].
In contrast, the observed temporal decline like a single unbroken power-law of the light-curve of the 0.3-10 keV X-ray afterglow of GRB 130427A is that predicted by the CB model for the measured spectral index of its afterglow. In the CB model, the X-ray afterglow has a deceleration break that takes place at a time $t_b$ after the beginning of the afterglow (not necessarily the beginning of the GRB), and satisfies the correlation $t_b/(1+z)\propto 1/ [(1+z)\, Ep\, Eiso]^{1/2}$ \[23\]. Consequently, in very intense GRBs, the break is often hidden under the prompt emission or its fast decline phase. For GRB 130425A at z=0.34, with Eiso$\approx
8.5\times 10^{53}$ erg and $Ep\approx 1200$ keV \[27\], the above correlation \[20\] yields a deceleration break at $t<200$ s, which, probably, was hidden under the fast declining phase of the prompt emission (see Figure 5).
Moreover, most of the X-ray afterglows of the 28 most intense GRBs among the GRBs with known redshift that were followed long enough with one or more of the space based X-ray telescopes Beppo-SAX, Swift, Chandra and XMM-Newton, have light-curves $F_\nu(t) \propto t^{-\alpha_x}\,
\nu^{-\beta_x}$ with temporal and spectral indices that satisfy within errors the relation $\alpha_x=\beta_x+1/2$ predicted by the CB model for a constant density circumburst medium.
Furthermore, in the FB models, the predicted achromatic break in the light curve of the X-ray afterglow of GRBs is a direct consequence of the assumed conical geometry of the the highly relativistic jet - a conical shell with a half opening angle $\theta_j\gg 1/\gamma_0$, where $\gamma_0\gg 1$ is the initial bulk motion Lorentz factor of the jet. The failure of the conical fireball models to predict correctly the observed break properties in GRB afterglows, and the absence of a jet break in the X-ray afterglow of very intense GRBs such as 130427A, probably, is due to the assumed conical geometry. This is supported by the fact that, unlike the cannonball model, the conical fireball models have failed to predict other major properties of GRBs which strongly depend on the assumed conical jet geometry. That includes the failure to predict/reproduce the observed canonical shape of the lightcurve of the X-ray afterglow of GRBs \[21\] and the main properties of its various phases: The rapid spectral softening during the fast decline phase of the prompt emission, which was interpreted in the framework of the conical FB models as high latitude emission \[28\], was not expected/predicted. The plateau phase that follows was not reproduced and was interpreted aposteriory by postulating continuous energy injection into the blast wave by hypothetical central GRB engines, such as magnetars \[29\]. Furthermore, unlike the cannonball model, where X-ray flashes (XRFs) and low-luminosity GRBs were successfully explained as GRBs produced by SNeIc akin to SN1998bw and viewed far off axis \[30\], the collimated fireball models could not explain why GRBs such as 130427A and 980425, which were produced by the very similar broad line stripped envelope SN2013c and SN1998bw, respectively \[31\], have isotropic equivalent energies which differ by six orders of magnitude. Moreover, the GRB/SN association and the short lifetime of of the massive stars which produce SNeIc imply that the rates of GRBs and star formation are related. But, while the cannonball model predicted correctly the redshift distribution of the joint population of GRBs and XRFs from the observed dependence of the star formation rate on redshift (32), the conical fireball model did not (33).
--------- -------- -------------- ---------------- ---------------- --
GRB $~~$z $~~$Eiso $~~~~\alpha_x$ $~~~~\beta_x$
$10^{54}erg$
990123 1.6 2.78 $1.46 \pm.06$ $0.96\pm .04$
010222 1.477 1.14 $1.33 \pm .04$ $0.97 \pm .05$
061007 1.26 1.0 $1.55 \pm .05$ $1.03 \pm .05$
070328 2.0627 0.64 $1.44 \pm .03$ $0.93 \pm .07$
080607 3.036 1.87 $1.53 \pm .09$ $1.04 \pm .14$
080721 2.591 1.21 $1.49 \pm .05$ $0.86 \pm .09$
080810 3.35 0.5 $1.42 \pm .08$ $1.00 \pm .15$
080916C 4.35 0.88 $1.31 \pm .14$ $0.80 \pm .20$
090323 3.57 3.98 $1.35 \pm .15$ $0.88 \pm .21$
090423 8.26 0.89 $1.41 \pm .08$ $0.86 \pm .20$
090812 2.452 0.44 $1.32 \pm .04$ $0.86 \pm .14$
090902B 1.822 3.6 $1.40 \pm .06$ $0.74 \pm .14$
090926A 2.1062 2.0 $1.41 \pm .05$ $0.98 \pm .10$
110205A 2.22 1.36 $1.55 \pm .04$ $1.01 \pm .10$
110422A 1.77 0.72 $1.32 \pm .05$ $0.90 \pm .09$
110731A 2.83 0.46 $1.26 \pm .04$ $0.76 \pm .05$
110918A 0.984 2.11 $1.63 \pm .04$ $1.03 \pm .19$
130427A 0.3399 0.85 $1.29 \pm .03$ $0.79 \pm .03$
130505A 2.27 3.8 $1.27 \pm .15$ $0.76 \pm .05$
131108A 2.4 0.58 $1.33 \pm .06$ $0.97 \pm .19$
140419A 3.956 1.9 $1.37 \pm .03$ $0.87 \pm .07$
140206A 2.73 2.4 $1.29 \pm .03$ $0.80 \pm .06$
150206A 2.087 0.6 $1.25 \pm .03$ $0.79 \pm .07$
150314A 1.758 0.69 $1.53 \pm .04$ $0.95 \pm .04$
150403A 3.139 0.6 $1.37 \pm .14$ $0.83 \pm .17$
151021A 2.330 1.0 $1.38 \pm .05$ $1.00 \pm .10$
160131A 0.972 0.83 $1.24 \pm .20$ $0.89 \pm .22$
160625B 1.406 5.0 $1.34 \pm .05$ $0.83 \pm .12$
--------- -------- -------------- ---------------- ---------------- --
: The temporal decay index $\alpha_x$ and the spectral index $\beta_x$ of the late-time 0.3-10 keV X-ray afterglow of the 28 most intense GRBs ($Eiso>0.5\times 10^{54}$ erg) with known redshift and long follow-up afterglow measurements with Beppo-SAX, Swift, Chandra and XMM-Newton.
[**Acknowledgment**]{}: We thank an anonymous referee for useful comments and suggestions.
[999]{}
R. W. Klebesadel, I. B. Strong and R. A. Olson, Astrophys. J. 182, L85 (1973).
C. A. Meegan et al. Nature, 355, 143 (1992); G. J. Fishman and C. A. Meegan, Ann. Rev. Astr. Astrophys. 33 415 (1995).
E. Costa et al. Nature, 38, 783 (1997) \[arXiv:astro-ph/9706065\]; J. van Paradijs et al., Nature, 386, 686 (1997); H. E. Bond, IAU circ. 6665 (1997); D. A. Frail et al., Nature, 389, 261 (1997).
J. P. Norris et al. Nature, 308, 434 (1984); C. Kouveliotou et al. Astrophys. J. 413, L101 (1993).
N. Shaviv and A. Dar, Astrophys. J. 447, 863 (1995) \[arXiv:astro-ph/9407039\]; A. Dar, Astrophys. J. 500, L93 (1998) \[arXiv:astro-ph/9709231\].
T. J. Galama et al. Nature, 395, 670 (1998) \[arXiv:astro-ph/9806175\].
See, e.g., S. Dado, A. Dar, and A. De Rújula, Astrophys. J. 693, 311 (2009) \[arXiv:0807.1962\] and references therein.
B. Paczynski, 1986, ApJ, 308, L43 (1986); J. Goodman,, ApJ, 308, L47 (1986); P. Meszaros and M. J. Rees, ApJ, 418, L59 (1993) \[arXiv:astro-ph/9309011\]. For an early review see, e.g., T. Piran, Phys. Rep. 314, 575 (1999) \[arXiv:astro-ph/9810256\] and references therein.
See, e.g., \[5\]; S. Dado, A. Dar, and A. De Rújula, Astr. & Astrophys. 388, 1079 (2002) \[arXiv:astro-ph/0107367\]; A. Dar and A. De Rújula, Phys. Rep. 405, 203 (2004) \[arXiv:astro-ph/0308248\], and references therein.
S. Dado, A. Dar, and A. De Rújula, Astrophys. J. 696, 964 (2009) \[arXiv:0809.4776\], and references therein.
See, e.g., P. Kumar and B. Zhang, Phys. Rep. 561, 1 (2015) \[arXiv:1410.0679\] and references therein.
K. M. Popper, The Logic of Scientific Discovery, Routledge Classics 1959.
For confrontations of predictions of both the FB model and the CB model with observations, see, e.g., A. Dar, Chin. J. Astron. Astrophys. 6, 301 (2006) \[arXiv:astro-ph/0511622\] (major properties), C. Wigger et al. ApJ, 675, 553, (2008) \[arXiv:0710.2858\] (prompt gamma-ray emission), S. Dado, A. Dar and A. De Rujula ApJ, 681, 1408, (2008) \[arXiv:0709.4307\] (the fast decline phase of the prompt emission), S. Dado and A. Dar, ApJ, 775, 16 (2013) \[arXiv:1203.5886\] (correlations between and among the prompt gamma-ray and the X-ray afterglow emissions), ApJ, 785, 70 (2014) \[arXiv:1307.5556\] (consitency with star formation history), S. Covino and D. Gotz, arXiv:1605.03588 (the large linear polarization of the prompt $\gamma$-rays in all GRBs where the polarization was claimed to be measured: 930131, 960924, 021206, 041291A, 061122, 100826A, 110301A, 110721A, 140206A.
R. Sari, T. Piran, and J. P. Halpern, Astrophys. J. 519, L17 (1999) \[arXiv:astro-ph/9903339\]; J. E. Rhoads, Astrophys. J. 525, 737 (1999) \[arXiv:astro-ph/9903400\]; F. van Eerten, and A. MacFadyen, Astrophys. J. 767, 141 (2013) \[arXiv:1209.1985\].
E.g., E. W. Liang et al. Astrophys. J. 675, L528 (2008) \[arXiv:0708.2942\]; D. N. Burrows and J. Racusin, Il Nuovo Cimento B 121, 1273 (2007) \[astro-ph/0702633\]; P. A. Curran, A. J. Van Der Horst and R. A. M. J. Wijers, Mon. Not. Roy. Astr. Soc. 386, 859 (2008) \[arXiv:0710.5285\]; and Ref. \[16\].
J. L. Racusin et al. AIP Conf. Proc. 1000, 196 (2008) \[arXiv:0801.4749\]; J. L. Racusin et al. Astrophys. J. 698, 43 (2009) \[arXiv:0812.4780\];
P. A. Evans et al., (Swift-XRT GRB light-curve repository http://www.swift.ac.uk/xrt$_-$curves/) Mon. Not. Roy. Astr. Soc. 397, 1177 (2009) \[arXiv:0812.3662\].
M. De Pasquale et al. arXiv:1602.04158.
S. Dado and A. Dar, Astrophys. J. 761, 148 (2012) \[arXiv:1203.1228\].
S. Dado, A. Dar, and A. De Rújula, Astr. & Astrophys. 388, 1079 (2002) \[arXiv:astro-ph/0107367, Figures 26-33\].
J. A. Nousek et al., Astrophys. J. 642, 389, (2006) \[arXiv:astro-ph/0508332\]; G. Chincarini et al., Astrophys. J. 671, 1903 (2007) \[arXiv:astro-ph/0508332\].
S. Dado and A. Dar, Astr. & Astrophys. 558, 115 (2013) \[arXiv:1303.2872\].
S. Golenetskii et al. GCN Circ. 5722 (2006).
C. G. Mundell et al., Astrophys. J. 660, 489 (2007) \[arXiv:astro-ph/0610660\].
A. Maselli et al. Science, 343, 48 (2014) \[arXiv:1311.5254\]
D. Grupe et al. Astrophys. J. 711, 1008 (2010) \[arXiv:0903.1258\].
S. Golenetskii et al. GCN Circ. 14487 (2013).
E. E. Fenimore et al., ApJ 473, 998 (1996)\[arXiv:astro-ph/9607163\]; P. Kumar and A. Panaitescu, ApJ, 541, L51 (2000) \[arXiv:astro-ph/0006317\]; C. D. Dermer, ApJ 614, 284 (2004) \[arXiv:astro-ph/0403508\].
Usov, V. V. 1992, Nature, 357, 472; T. A. Thompson, et al. ApJ, 611, 380 (2004) \[arXiv:astro-ph/0401555\].
S. Dado, A. Dar and A. De Rujula A&A, 422, 381 (2004) \[arXiv:astro-ph/0309294\].
A. Melandri, et al. A&A, 567, A29 (2014) \[arXiv:1404.6654\].
S. Dado and A. Dar, ApJ, 785, 70 (2014) \[arXiv:1307.5556\].
B. E. Robertson and R. S. Ellis ApJ, 744, 95 (2012) \[arXiv:1109.0990\]; J. Wei, et al. MNRAS, 439, 3329 (2014), \[arXiv:1306.4415\].
|
---
abstract: 'Runaway electrons are generated in a magnetized plasma when the parallel electric field exceeds a critical value. For such electrons with energies typically reaching tens of MeV, the Abraham-Lorentz-Dirac (ALD) radiation force, in reaction to the synchrotron emission, is significant and can be the dominant process limiting the electron acceleration. The effect of the ALD-force on runaway electron dynamics in a homogeneous plasma is investigated using the relativistic finite-difference Fokker-Planck codes LUKE \[Decker & Peysson, Report EUR-CEA-FC-1736, Euratom-CEA, (2004)\] and CODE \[Landreman et al, Comp. Phys. Comm. [**185**]{}, 847 (2014)\]. Under the action of the ALD force, we find that a bump is formed in the tail of the electron distribution function if the electric field is sufficiently large. We also observe that the energy of runaway electrons in the bump increases with the electric field amplitude, while the population increases with the bulk electron temperature. The presence of the bump divides the electron distribution into a runaway beam and a bulk population. This mechanism may give rise to beam-plasma types of instabilities that could in turn pump energy from runaway electrons and alter their confinement.'
author:
- 'J. Decker'
- 'E. Hirvijoki'
- 'O. Embreus'
- 'Y. Peysson'
- 'A. Stahl'
- 'I. Pusztai'
- 'T. Fülöp'
bibliography:
- 'bump.bib'
title: Bump formation in the runaway electron tail
---
Introduction {#sec:intro}
============
Runaway electrons are typically generated in plasmas in the presence of large electric fields $E>E_{c}$, where the critical field $E_{c}$ is defined as [@dre59] $$E_{c}=\frac{ne^{3}\ln\Lambda}{4\pi\varepsilon_{0}^{2}mc^{2}},$$ where $n$ is the electron density, $m$ is the electron rest mass, $c$ is the speed of light, $e$ is the elementary charge, and $\ln\Lambda$ is the Coulomb logarithm.
In connection with the sudden cooling in a tokamak disruption, a strong electric field is induced, which leads to the generation of a large number of runaway electrons. In certain cases a significant fraction of the initial toroidal current can be driven by a beam of runaway electrons. The formation of such energetic runaway beams would represent a serious threat for reactor-size machines such as ITER[@hen07]. Consequently, a considerable research effort is currently undertaken to prevent the formation of large runaway beams during tokamak disruptions, or to design a controlled damping scenario for runaway beams if they cannot be avoided [@hen07; @izz11; @leh11; @gra14; @hol15]. The condition $E>E_{c}$ for runaway electron generation can also be met during the plasma start-up or ramp-down. During the flat-top phase, runaways can appear if the density is sufficiently low in Ohmic plasmas (since $E_{c}\propto n$), or if an externally applied source of current is suddenly modified.
Experimental measurements show that the maximum runaway electron energy does not increase indefinitely with time but instead reaches a limit in the tens of MeV range [@hol13]. One of the possible mechanisms that could provide an explanation for this limit is the Abraham-Lorentz-Dirac (ALD) radiation force [@pau58] in reaction to the synchrotron emission due to the particle motion in a magnetic field. For electrons in the MeV range, this force can be significant and contribute to limit the particle acceleration [@and01]. The synchrotron emission, which carries energy away from the electrons, is also used as a diagnostic tool for the runaway population [@sta13]. The ALD force is characterized by the synchrotron radiation reaction time scale $\tau_r$, given by $$\tau_{r}^{-1}=\frac{e^{4}B^{2}}{6\pi\varepsilon_{0}(mc)^{3}},\label{eq:taur}$$ where $B$ is the magnetic field.
In the present paper, the runaway electron dynamics in a homogeneous plasma is investigated using the relativistic finite-difference guiding-center Fokker-Planck codes LUKE [@dec04a; @pey14] and CODE [@lan14; @sta15]. The electron distribution function evolves under the combined influence of Coulomb collisions, electric field acceleration, and the ALD radiation reaction force. Under a constant parallel electric field (with respect to the magnetic field) $E_{\Vert}>E_{c}$, the electron distribution never reaches a steady-state in the absence of ALD radiation reaction force. Conversely, it is shown in Sec. \[sec:evolution\] that when the effect of the ALD force is included, the electron distribution evolves towards a steady-state solution. This solution exists even though the synchrotron emission vanishes for electrons with purely parallel motion in a uniform magnetic field. In fact, the expansion of the electron distribution towards higher energies is limited by collisional pitch-angle scattering, which is enhanced by the strong perpendicular anisotropy arising from the combination of electric field acceleration and synchrotron radiation reaction force. This process is found to limit the runaway electron population to energies far below the value for which the contribution from the magnetic field curvature to the ALD radiation reaction force becomes significant [@and01]. Consequently, it is justified to use the homogeneous plasma limit to study the dynamics of runaway electrons in the core region of tokamaks.
In addition, we show that if the electric field amplitude is sufficiently large, a bump appears in the runaway electron tail of the steady-state distribution function, in accordance with analytical predictions [@hir15]. This bump, which peaks on the parallel axis in momentum space, is entirely located in the runaway region. The steady-state population of electrons in the bump is found to increase with the bulk electron temperature $T_{e}$, while their average energy increases with the electric field amplitude $E_{\Vert}$ and decreases with the amplitude of the ALD radiation reaction force, which is proportional to $B^{2}$. For certain parameters, the bump in the electron distribution tail encompasses almost the entire runaway electron population, thus formally dividing the distribution into a bulk population and a runaway beam.
The implementation of the synchrotron reaction force in the kinetic equation is described in Sec. \[sec:kinetic\_equation\]. The time evolution of the electron distribution calculated by the Fokker-Planck modelling code LUKE is presented in Sec. \[sec:evolution\]. The properties of the steady-state distribution function and the mechanism leading to the formation of a bump are described in Sec. \[sec:Properties\], where a comparison between the codes LUKE and CODE is also presented. The bump is characterized in Sec. \[sec:Parametric\] as a function of the electric field amplitude, ALD radiation reaction, bulk electron temperature, and ion effective charge. Implications of the ALD radiation reaction force and bump-in-tail formation are discussed in the Conclusions, Sec. \[sec:Conclusions\].
Synchrotron reaction force in the kinetic equation\[sec:kinetic\_equation\]
============================================================================
Kinetic equation for charged particles in a magnetized plasma
-------------------------------------------------------------
The kinetic equation for species $a$ with charge $q$ and mass $m$ is given by $$\begin{aligned}
\frac{\partial f_{a}}{\partial t}+\frac{\partial}{\partial\mathbf{x}}\cdot\left(\dot{\mathbf{x}}f_{a}\right)+\frac{\partial}{\partial\mathbf{p}}\cdot\left(\dot{\mathbf{p}}f_{a}\right)=C[f_{a},f_{b}],\label{eq:kinetic}\end{aligned}$$ where $C[f_{a},f_{b}]$ is the collision operator between particle species $a$ and $b$ (including intra-species collisions) and $(\dot{\mathbf{x}},\dot{\mathbf{p}})$ are the equations of motion associated with phase-space coordinates $(\mathbf{x},\mathbf{p})$. Here $\mathbf{x}$ is the particle position and $\mathbf{p}=\gamma m\mathbf{v}$ is the particle momentum, with $\gamma=1/\sqrt{1-v^{2}/c^{2}}=\sqrt{1+p^{2}/(mc)^{2}}$ the relativistic factor. In the Fokker-Planck limit, the Coulomb collision operator is given by $$\begin{aligned}
C_{\textrm{FP}}[f_{a},f_{b}]=-\frac{\partial}{\partial\mathbf{p}}\cdot\left(\mathbf{K}_{\textrm{FP}ab}[f_{b}]f_{a}-\mathbb{D}_{\textrm{FP}ab}[f_{b}]\cdot\frac{\partial f_{a}}{\partial\mathbf{p}}\right),\end{aligned}$$ where $\mathbf{K}_{ab}[f_{b}]$ is the collisional friction vector and $\mathbb{D}_{ab}[f_{b}]$ is the collisional diffusion tensor. The relativistic Braams-Karney collision operator is used in this paper [@bra87; @shk97].
So-called *knock-on* collisions represent a $1/\ln\Lambda$ correction to the collision operators. However, when the runaway population becomes significant, these collisions can play an important role as they give rise to an avalanche effect that can significantly increase the runaway growth rate. This secondary runaway generation is neglected in the present work, which is restricted to situations where the runaway population is sufficiently small for secondary electron generation to be negligible. However, it is possible that the ALD radiation reaction force has a significant effect on the secondary runaway generation. Such considerations will be the subject of future work.
The equations of motion combine the Hamiltonian motion from the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$, and the effect of the ALD radiation reaction $\mathbf{F}_{\textrm{ALD}}:$ $$\begin{aligned}
\dot{\mathbf{x}}= & \mathbf{v},\\
\dot{\mathbf{p}}= & q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)+\mathbf{F}_{\textrm{ALD}}
\equiv\mathbf{F}_{E}+\mathbf{F}_{m}+\mathbf{F}_{\textrm{ALD}}.\end{aligned}$$ The Abraham-Lorentz-Dirac force describes momentum loss in reaction to the synchrotron radiation, and takes the form [@pau58] $$\begin{aligned}
\mathbf{F}_{\textrm{ALD}}=&\frac{q^{2}\gamma^{2}}{6\pi\varepsilon_{0}c^{3}}\left[\ddot{\mathbf{v}}+\frac{3\gamma^{2}}{c^{2}}\left(\mathbf{v}\cdot\dot{\mathbf{v}}\right)\dot{\mathbf{v}}\right.\notag\\
&\left.+\frac{\gamma^{2}}{c^{2}}\left(\mathbf{v}\cdot\ddot{\mathbf{v}}+\frac{3\gamma^{2}}{c^{2}}\left(\mathbf{v}\cdot\dot{\mathbf{v}}\right)^{2}\right)\mathbf{v}\right].\label{eq:reaction_force}\end{aligned}$$
In magnetically confined fusion plasmas, the magnetic force $\mathbf{F}_{m}=q\mathbf{v}\times\mathbf{B}$ characterized by the Larmor frequency $\omega_{c}=qB/m$ typically dominates both the electric force $\mathbf{F}_{E}=q\mathbf{E}$ and the radiation reaction force $\mathbf{F}_{\textrm{ALD}}$ such that $\mathbf{v}\cdot\dot{\mathbf{v}}\simeq0$. In a uniform constant magnetic field, the ALD force (\[eq:reaction\_force\]) thus reduces to $$\mathbf{F}_{\textrm{ALD}}\simeq-\frac{m}{\tau_{r}}\left[\mathbf{v}_{\bot}+\frac{\gamma^{2}v_{\bot}^{2}}{c^{2}}\mathbf{v}\right],\label{eq:K}$$ where $\mathbf{v}_{\bot}=\left(\mathbf{I}-\hat{\mathbf{b}}\hat{\mathbf{b}}\right)\cdot\mathbf{v}$ is the perpendicular velocity with norm $v_{\bot}=\left\Vert
\mathbf{v}_{\bot}\right\Vert $ and $\hat{\mathbf{b}}=\mathbf{B}/B$ is the magnetic field unit vector.
Guiding-center transformation \[sub:gc\]
----------------------------------------
In fusion plasmas, the gyroperiod is short compared to the time scale associated with collisions, the ALD radiation reaction force, and the electric field acceleration. Based on this time-scale separation, the kinetic equation is reduced by eliminating the gyromotion in Eq. (\[eq:kinetic\]) using Lie-transform perturbation methods [@bri04; @dec10b]. The transformation of the dissipative ALD force uses the Lie-transform for non-Hamiltonian dynamics, which has been recently derived in Ref. [@hir14]. In a uniform plasma, the resulting guiding-center distribution function for electrons evolves in the 2-D gyro-angle independent momentum space $(p,\xi)$ as
$$\frac{\partial f}{\partial t}+\boldsymbol{\nabla}_{p,\xi}\cdot\mathbf{S}_{p,\xi}\left[f\right]=I_{\textrm{FP}}[f],\label{eq:gc_kinetic}$$
where $p$ is the guiding-center momentum and $\xi=p_{\parallel}/p$ is the pitch-angle cosine. Components of the guiding-center momentum-space flux $$\mathbf{S}_{p,\xi}\left[f\right]=\left(\mathbf{K}_{\textrm{FP}}+\mathbf{K}_{\textrm{E}}+\mathbf{K}_{\textrm{ALD}}\right)f-\mathbb{D}_{\textrm{FP}}\cdot\boldsymbol{\nabla}_{p,\xi}f$$ include convective contributions from collisional drag $\mathbf{K}_{\textrm{FP}}$, electric field acceleration $\mathbf{K}_{\textrm{E}}$, and the radiation reaction force $\mathbf{K}_{\textrm{ALD}}$, and a collisional diffusion tensor $\mathbb{D}_{\textrm{FP}}$. The integral part of the guiding-center collisional operator is denoted $I_{\textrm{FP}}[f]$. It describes the evolution of the bulk population due to collision with fast electrons. When momentum conservation of the electron-electron collision operator is essential - as for the calculation of electron-driven current - the term $I_{\textrm{FP}}[f]$ must be included [@kar86]. It is generally truncated at the first order in Legendre expansion. The truncation ensures momentum conservation but allows energy dissipation such that it is not necessary to model energy transport to reach a steady-state solution. In the present paper, the term is set to zero unless otherwise specified. Omitting $I_{\textrm{FP}}[f]$ makes it possible to use the stream function to interpret the steady-state fluxes in momentum space, as seen in Sec. \[sub:Steady-state-solution\]. It also allows a comparison between the codes LUKE and CODE, since the integral part of the collision operator is not yet included in CODE. This benchmark is presented in Sec. \[sub:Benchmark\], where it is also shown that the effect of $I_{\textrm{FP}}[f]$ on the tail of the electron distribution can be neglected.
Writing out the momentum space divergence operator explicitly, the kinetic equation (\[eq:gc\_kinetic\]) becomes $$\frac{\partial f}{\partial t}+\frac{1}{p^{2}}\frac{\partial}{\partial p}\left(p^{2}S_{p}\right)-\frac{1}{p}\frac{\partial}{\partial\xi}\left(\sqrt{1-\xi^{2}}S_{\xi}\right)=0,\label{eq:kineticexp}$$ where the guiding-center momentum space flux components are $$\begin{aligned}
\begin{split}
S_{p} & =-D_{pp,\textrm{FP}}\frac{\partial f}{\partial p}+\left(K_{p,\textrm{FP}}+K_{p,\textrm{E}}+K_{p,\textrm{ALD}}\right)f,\\
S_{\xi} & =\sqrt{1-\xi^{2}}D_{\xi\xi,\textrm{FP}}\frac{\partial f}{\partial\xi}+\left(K_{\xi,\textrm{E}}+K_{\xi,\textrm{ALD}}\right)f.
\label{eq:kineticfluxes}
\end{split}\end{aligned}$$ The terms contributing to these fluxes are the convection and diffusion coefficients associated with the Fokker-Planck collision operator (which are independent of $\xi$ for isotropic field particle distributions [@bra87; @shk97]) $$\begin{aligned}
{2}
D_{pp,\textrm{FP}} & = & A_{\textrm{FP}}\left(p\right),\label{eq:dpp}\\
K_{p,\textrm{FP}} & = & -F_{\textrm{FP}}\left(p\right),\\
D_{\xi\xi,\textrm{FP}} & = & \frac{B_{\textrm{FP}}\left(p\right)}{p},\end{aligned}$$ the electric field acceleration $$\begin{aligned}
K_{p,\textrm{E}} & = & \xi E_{\Vert},\\
K_{\xi,\textrm{E}} & = & -\sqrt{1-\xi^{2}}E_{\Vert},\end{aligned}$$ and the synchrotron reaction force $$\begin{aligned}
{2}
K_{p,\textrm{ALD}} & = & -\sigma_{r}\gamma p\left(1-\xi^{2}\right),\\
K_{\xi,\textrm{ALD}} & = & -\sigma_{r}\dfrac{p\xi\sqrt{1-\xi^{2}}}{\gamma}.\label{eq:kxi}\end{aligned}$$
In Eqs. (\[eq:kineticexp\]-\[eq:kxi\]), time is normalized to the collision time for relativistic electrons, $$\tau_{c}=\frac{4\pi\varepsilon_{0}^{2}m^{2}c^{3}}{e^{4}n\ln\Lambda},$$ momentum $p$ is given in units of $mc$, the parallel electric field $E_{\Vert}$ is normalized to the critical field $E_{c}$, and $\sigma_{r}\equiv\tau_{c}/\tau_{r}$ measures the relative strength (compared to collisional forces) of the ALD radiation reaction force $$\sigma_{r}=\frac{2}{3}\frac{1}{\ln\Lambda}\frac{\omega_{c}^{2}}{\omega_{p}^{2}},$$ where $\omega_{p}$ is the electron plasma frequency defined by $\omega_{p}^{2}=e^{2}n/(\varepsilon_{0}m)$. The collisional diffusion coefficients $A_{\textrm{FP}}\left(p\right)$ and $B_{\textrm{FP}}\left(p\right)$ are normalized to $(mc)^{2}/\tau_{c}$ while the friction coefficient $F_{\textrm{FP}}\left(p\right)$ is normalized to $mc/\tau_{c}$.
An explicit form of the momentum-space fluxes (\[eq:kineticfluxes\]) is thus $$\begin{aligned}
\begin{split}
S_{p} & =-A_{\textrm{FP}}\left(p\right)\frac{\partial f}{\partial p}+\left[\xi E_{\Vert}-F_{\textrm{FP}}\left(p\right)-\sigma_{r}\gamma p\left(1-\xi^{2}\right)\right]f,\\
S_{\xi} & =\sqrt{1-\xi^{2}}\left(\frac{B_{\textrm{FP}}\left(p\right)}{p}\frac{\partial f}{\partial\xi}-E_{\Vert}-\sigma_{r}\gamma^{-1}p\xi\right)f.
\label{eq:kineticfluxesexp}
\end{split}\end{aligned}$$ We assume cold and infinitely massive ions, so that the normalized collision coefficients $A_{\textrm{FP}}$, $F_{\textrm{FP}}$ and $B_{\textrm{FP}}$ only depend on $T_{e}$ and $Z_{\textrm{eff}}$ [@shk97]. Collisions with ions only enter the pitch-angle scattering term $B_{\textrm{FP}}\left(p\right)=B_{\textrm{FP},e}\left(p\right)+Z_{\textrm{eff}}/(2v)$, while $A_{\textrm{FP}}\left(p\right)=\beta^{2}F_{\textrm{FP}}\left(p\right)/v$, where the normalized electron temperature is defined as $\beta^{2}\equiv k_BT_{e}/(mc^{2})$. To summarize, the normalized equation (\[eq:kineticfluxesexp\]) depends on the following independent parameters only: the parallel electric field $E_{\Vert}$, the normalized ALD frequency $\sigma_{r}$, the electron temperature $T_{e}$, and the effective charge $Z_{\textrm{eff}}$.
Evolution of the electron distribution function\[sec:evolution\]
================================================================
Force balance and runaway region
--------------------------------
Some preliminary insight into runaway electron dynamics can be extracted from the force balance, $K_{p}\equiv K_{p,\textrm{FP}}+K_{p,\textrm{E}}+K_{p,\textrm{ALD}}=0$, which can be expressed as $$\xi E_{\Vert}-F_{\textrm{FP}}\left(p\right)-\sigma_{r}\gamma p\left(1-\xi^{2}\right)=0.\label{eq:force-balance}$$ In the high velocity limit (see Appendix \[sub:High-velocity-limit\]) and in the absence of the ALD force, the force balance yields $$p^{2}=\frac{1}{\xi E_{\Vert}-1}.\label{eq:pcrit}$$ For particles with purely parallel momentum, this condition determines the critical momentum $p_{c}\equiv(E_{\Vert}-1)^{-1/2}$ above which electrons are continuously accelerated. For particles with very large momentum $p\gg1$, the condition (\[eq:pcrit\]) provides an asymptotic value $\xi_{c}=E_{\Vert}^{-1}$, such that $K_{p}>0$ for particles with $\xi>\xi_{c}$.
In this paper, the runaway region is defined as the region where the momentum force balance is positive, i.e. $K_{p}>0$. Electrons located within this region are considered runaway electrons. In the absence of the ALD force, this definition corresponds to the usual idea of a runaway electron, as the probability for an electron to be continuously accelerated if it enters the region where $K_{p,\textrm{FP}}+K_{p,\textrm{E}}>0$ is very high. The momentum space described by finite-difference Fokker-Planck codes is a limited domain by nature with a high-energy boundary defined by a maximum momentum $p_{\max}$. A proper description of the runaway dynamics clearly requires $p_{\max}\gg p_{c}$. Then, we must distinguish between electrons located in the $K_{p,\textrm{FP}}+K_{p,\textrm{E}}>0$ region within the code simulation domain $p<p_{\max}$, *internal runaways*, and electrons having left the simulation domain, *external runaways*. The total runaway population $n_{r}$ consists of both internal and external runaways, the latter being counted in the simulation, albeit without following their momentum space characteristics.
When the ALD force is included, and for relativistic electrons with $p_{\Vert}\gg1$ and $p_{\Vert}\gg p_{\bot}$, the force balance (\[eq:force-balance\]) is approximately given by $$E_{\Vert}-1-\sigma_{r}p_{\bot}^{2}=0$$ and yields a condition on the perpendicular momentum $(p_{\bot}\equiv p\sqrt{1-\xi^{2}})$ $$p_{\bot0}^{2}=\frac{E_{\Vert}-1}{\sigma_{r}}.\label{eq:plim}$$ The momentum space in the far tail of the distribution is separated into the runaway region $p_{\bot}<p_{\bot0}$ where the electric force dominates over the ALD force and the collisional drag such that the net force is positive, and a region $p_{\bot}>p_{\bot0}$ where the ALD force and collisional drag dominate the electric force such that the net force is negative.
As the calculations in the next sections will show, in the presence of an ALD force, the probability for electrons with $K_{p}>0$ to escape the runaway region at some point is high. Therefore, the concept of a runaway electron in this case is more an extension of the usual definition than a true characteristic. We will also see that electrons labelled as runaways can be entirely kept within the simulation domain such that there are no external runaways.
The Fokker-Planck code LUKE
---------------------------
The Fokker-Plank equation is solved numerically by the relativistic guiding-center Fokker-Planck code LUKE. Equation (\[eq:gc\_kinetic\]) is discretized in momentum space $(p,\xi)$ using a 2-D finite-difference scheme with non-uniform grids and a 9-point differentiation procedure. A total of 1200 points are used for the $p$ grid, with 140 grid points describing the $0<p<3$ region with a constant grid step, and 1060 grid points describing the $3<p<p_{\max}=200$ region using increasing grid steps with cubic dependence. A total of 166 points are used for the $\xi$ grid, with a decreasing step size towards $\xi=\pm1$ for increased resolution near the $p_{\Vert}$ axis. The code LUKE has been benchmarked for the usual runaway problem [@dec04a]. A benchmark including the ALD radiation reaction force is conducted against the Fokker-Planck solver CODE and presented in Sec. \[sub:Benchmark\].
The linearization of the electron-electron collision operator implies that the calculation is valid only as long as the electron distribution is not too distorted from the original Maxwellian, which in practice implies that the runaway fraction $n_{r}/n$ does not exceed a few percent.
Time evolution of the distribution function {#sub:time_evo_of_dist}
-------------------------------------------
The electron distribution evolves from an initial relativistic Maxwellian distribution, which is also the steady-state solution of Eq. (\[eq:kineticexp\]) for $E_{\Vert}=0$ and $\sigma_{r}=0$. A constant electric field $E_{\Vert}=3$ is applied, the effective charge is $Z_{\textrm{eff}}=1$, the temperature is $\beta=0.1$ ($T_e=5.11$ keV), and the ratio $B^{2}/n$ is adjusted such that $\sigma_{r}=0.6$, which corresponds to typical low-density conditions in tokamak plasmas (i.e. $n=10^{19}$ m$^{-3}$ and $B=4$ T in the Tore-Supra tokamak). The evolution of the electron distribution function in the parallel direction ($\xi=1$) is shown in Fig. \[fig:fevol\].
In the absence of the ALD force, a runaway tail progressively extends to the edge of the simulation box (Fig. \[fig:fevol-a\]). The electron distribution does not converge to a steady-state. The runaway rate reaches an asymptotic value (Fig. \[fig:fevol-e\]), and the fraction of runaway electrons increases continuously (Fig. \[fig:fevol-c\]). At first, the runaway rate is related to an increase in the internal runaway electron population. Once the runaway tail reaches the edge of the simulation domain (for $t\sim200$), the runaway rate is related to the population leaving the simulation box and becoming external runaways. Note that, strictly speaking, the linearized collision operator is no longer valid for $t/\tau_{c}>10^{5}$ as the fraction of runaway electrons becomes of order unity. Nevertheless, the evolution is continued to illustrate the absence of a steady-state solution. In summary, the distribution function (Fig. \[fig:fevol-a\]) evolves towards an asymptotic solution with the bulk population depleting at a constant rate.
In the presence of the ALD force, however, we find that the distribution evolves towards a steady-state solution (Fig. \[fig:fevol-b\]) as the runaway rate vanishes (Fig. \[fig:fevol-f\]). No electron leaves the simulation box, and the population of internal runaways reaches an asymptotic value $n_{r}/n=0.002$ (Fig. \[fig:fevol-d\]). In addition, we can observe the formation of a region with positive gradient in parallel momentum, which appears as a high-energy bump in the tail of the distribution function (Fig. \[fig:fevol-b\]). Properties of the time-asymptotic electron distribution function are examined in the next section.
Steady-state solution and bump formation \[sec:Properties\]
===========================================================
Steady-state solution\[sub:Steady-state-solution\]
--------------------------------------------------
If a steady-state solution exists, it satisfies the equation $\boldsymbol{\nabla}_{p,\xi}\cdot\mathbf{S}_{p,\xi}=0$. Given the axisymmetry of the momentum space, the divergence-free steady-state fluxes can be expressed as $$\mathbf{S}_{p,\xi}=\boldsymbol{\nabla}_{p,\xi}\times\left[\frac{A(p,\xi)}{2\pi p\sqrt{1-\xi^{2}}}\hat{\boldsymbol{\varphi}}\right],$$ where $A(p,\xi)$ is called the stream function. Since $\mathbf{S}_{p,\xi}\cdot\boldsymbol{\nabla}_{p,\xi}A=0$, contours of $A(p,\xi)$ indicate the direction of the momentum-space fluxes, or streamlines. The total flux of electrons between two contours is given by the corresponding difference in the value of $A(p,\xi)$, such that narrowing contours indicate regions of stronger flux. The stream function thus provides a very informative graphical representation of the steady-state fluxes in momentum space. While no steady-state solution to the runaway problem exists in the absence of ALD force, it is possible to artificially obtain a steady-state distribution function by adding a source term at $p=0$, which compensates exactly for the external runaway rate. Whereas adding cold electrons does not change the runaway rate or the shape of the distribution function in the tail, it enables us to interpret the stream function as a representation of the steady-state fluxes.
The 2D representation of the steady-state solution corresponding to the simulation parameters from Section \[sub:time\_evo\_of\_dist\] is presented in Fig. \[fig:f2D\]. The distribution function is shown in Figs. \[fig:f2D-a\] and \[fig:f2D-b\] for the case without and with ALD force, respectively. The dashed line delimits the runaway region where $K_{p}>0$. The corresponding contours of the stream function $A(p,\xi)$ are drawn in Figs. \[fig:f2D-c\] and \[fig:f2D-d\].
In the absence of the ALD force, the runaway population peaks near the $p_{\Vert}$ axis but extends quite far in the perpendicular direction into the runaway region, as seen if Fig. \[fig:f2D-a\]. The open streamlines represented by the contours of the stream function in Fig. \[fig:f2D-c\] show that electrons located in the runaway region are indefinitely accelerated and eventually escape the simulation domain.
In the presence of the ALD force, the runaway region consists of a narrow band along the $p_{\Vert}$ axis delimited by $p_{\bot}<p_{\bot0}$ at large energies. As seen by the contours of the stream function in Fig \[fig:f2D-d\], the steady-state electron flux is directed towards higher energies for $p_{\bot}<p_{\bot0}$ and towards lower energies for $p_{\bot}>p_{\bot0}$. The electron tail is thus confined in the region close to $p_{\bot}=0$, as seen in Fig. \[fig:f2D-b\], which creates a strong gradient in $p_{\bot}$. As seen in the next section, this gradient results in strong pitch-angle scattering, which contributes to limiting the energy of runaway electrons and gives rise to a bump in the distribution under certain conditions. This bump is centered on the $p_{\Vert}$ axis. As the minimum between the bulk and the bump population naturally lies in the vicinity of the critical field, the bump population almost coincides with the runaway region, such that the electron population can be formally separated into a bulk and a runaway “beam”.
Perpendicular force balance
---------------------------
The spherical representation $(p,\xi)$ is the natural coordinate system for describing collisions. Thus, it is used for the numerical discretization of the kinetic equation in the Fokker-Planck code LUKE. As seen in Fig. \[fig:f2D-b\], however, the tail of the distribution function is determined by the runaway region $p_{\bot}<p_{\bot0}$ and the natural coordinate system is rather the cylindrical representation $(p_{\bot},p_{\Vert})$ with $p_{\bot}=p\sqrt{1-\xi^{2}}$ and $p_{\Vert}=p\xi$.
The transformation to the $(p_{\bot},p_{\Vert})$ is detailed in Section \[sub:Cylindrical-representation\]. Focusing on the tail region of momentum space near the parallel axis (characterized by $p_{\bot}\ll p_{\Vert}$ and $p_{\Vert}\gg\beta$), the momentum-space fluxes entering (\[eq:kineticfluxescyl\]) yield to leading order in $\beta$ $$\begin{aligned}
\begin{split}S_{\bot} & =-\frac{1+Z_{\textrm{eff}}}{2v}\frac{\partial f}{\partial p_{\bot}}-\frac{p_{\bot}}{p_{\Vert}}\left[\frac{1}{v^{2}}+\sigma_{r}v\left(1+p_{\bot}^{2}\right)\right]f,\\
S_{\Vert} & =\left[E_{\Vert}-\frac{1}{v^{2}}-\sigma_{r}vp_{\bot}^{2}\right]f,
\label{eq:kineticfluxescyltail}
\end{split}\end{aligned}$$ where $v\simeq p_{\Vert}/\gamma\simeq p_{\Vert}/\sqrt{1+p_{\Vert}^{2}}$ is the velocity normalized to the speed of light.
At high energy, $v\simeq1$ and (\[eq:kineticfluxescyltail\]) becomes approximately $$\begin{aligned}
\begin{split}S_{\bot} & =-\frac{1+Z_{\textrm{eff}}}{2}\frac{\partial f}{\partial p_{\bot}}-\frac{p_{\bot}}{p_{\Vert}}\left[1+\sigma_{r}\left(1+p_{\bot}^{2}\right)\right]f,\\
S_{\Vert} & =\sigma_{r}\left[p_{\bot0}^{2}-p_{\bot}^{2}\right]f.
\label{eq:kineticfluxescyltail2}
\end{split}\end{aligned}$$ The parallel flux of electrons $S_{\Vert}$ is positive for particles with $p_{\bot}<p_{\bot0}$ and negative for particles with $p_{\bot}>p_{\bot0}$. The resulting strong perpendicular gradient enhances pitch-angle scattering, which creates a positive flux in the $p_{\bot}$ direction at high energy, which is illustrated by the streamlines in Fig. \[fig:f2D-d\]. This flux limits the extension of the electron distribution to higher energies, as seen in Fig. \[fig:f2D-b\]. The existence of a bump in the distribution is also driven by the perpendicular dynamics. From the expression (\[eq:kineticfluxescyltail2\]) for $S_{\bot}$, we see that the convective component decreases with $p_{\Vert}$ while pitch-angle scattering is independent of $p_{\Vert}$ for a given perpendicular gradient $\partial f/\partial p_{\bot}$. Therefore, on average, electrons in the tail are pushed into the runaway region at lower $p_{\Vert}$, while they are scattered away at higher $p_{\Vert}$. This dynamics is clearly seen in the stream function plot \[fig:f2D-d\]. Naturally, the bump appears at the balance point where $S_{\bot}\simeq0$ and electrons accumulate as a result of the perpendicular dynamics. Given the cylindrical symmetry, and since the perpendicular gradient is created by the parallel force balance, we may assume a parabolic dependence scaled by $p_{\bot0}$ around the bump location $$\frac{1}{f}\frac{\partial f}{\partial p_{\bot}}\propto-\frac{p_{\bot}}{p_{\bot0}^{2}}$$ such that from (\[eq:kineticfluxescyltail2\]) and for $p_{\bot}\rightarrow0$ we obtain an estimate for the parametric dependence of the position of the bump $$p_{\Vert b}\propto\frac{2}{1+Z_{\textrm{eff}}}\frac{1+\sigma_{r}}{\sigma_{r}}\left(E_{\Vert}-1\right),\label{eq:bumploc}$$ which is in agreement with the expression obtained from approximately solving the kinetic equation analytically [@hir15]. It is quite intuitive to expect that the bump energy increases with the electric field amplitude, whereas it decreases with the amplitude of the ALD force and the effective charge. However, the underlying processes are rather complex and involves both the parallel and perpendicular dynamics. The parametric dependence of the bump location predicted by Equation (\[eq:bumploc\]) will be compared to numerical calculations in Section \[sec:Parametric\].
Validity of the uniform plasma approximation
--------------------------------------------
In tokamak plasmas, particles with purely parallel velocity are subject to an ALD force due to the toroidal and poloidal periodic motions. For a safety factor $q\approx1$ and electrons with $p_{\Vert}\gg1$, the contribution from the field line curvature to the ALD force is derived in Appendix \[sub:parallel\] and is expressed as (\[eq:kfinal\]) $$K_{R}=-\sigma_{r}\left(\frac{\rho_{0}}{R}\right)^{2}p_{\Vert}^{4},$$ where $\rho_{0}=mc/(eB)$ is the Larmor radius of relativistic electrons and $R$ is the major radius. The momentum $p_{R}$ for which the toroidal ALD and drag forces compensate the electric force is thus given by $E_{\Vert}-1-\sigma_{r}(\rho_{0}/R)^{2}p_{R}^{4}=0$, which yields $$p_{R}=\left(\frac{E_{\Vert}-1}{\sigma_{r}}\right)^{1/4}\left(\frac{R}{\rho_{0}}\right)^{1/2}.$$ For the Tore-Supra example shown in Sec. \[sec:evolution\], $\rho_{0}/R=1.5\times10^{-4}$ and we find $p_{R}=110$, which corresponds to $55$ MeV electrons. We see in Fig. \[fig:fevol-b\] that the combination of uniform plasma ALD force and pitch-angle scattering limits the distribution to energies much below $55$ MeV. Toroidal effects could thus be neglected in this case.
Benchmark of the solution from the LUKE and CODE codes\[sub:Benchmark\]
-----------------------------------------------------------------------
The simulations presented in this paper were obtained using the code LUKE. While the code is extensively benchmarked for the usual runaway problem [@dec04a], LUKE simulations including the ALD reaction force are presented for the first time in this paper. In order to benchmark the numerical simulations, calculations from LUKE are compared to those from the solver CODE, which solves the same Fokker-Planck equation (\[eq:kineticexp\]) but uses a spectral representation of the pitch-angle dependence [@lan14]. The corresponding steady-state distribution functions are shown in Fig. \[fig:luke-code-a\] for the parameters used in Sec. \[sub:time\_evo\_of\_dist\], and two different values for the effective charge, $Z_{\textrm{eff}}=1$ and $Z_{\textrm{eff}}=4$. Results from the two codes are in excellent agreement. In particular, both codes show the appearance of a bump at the same energy for $Z_{\textrm{eff}}=1$, while they show no bump formation for $Z_{\textrm{eff}}=4$.
In addition, the integral part of the collision operator $I_{\textrm{FP}}$ can be included in the code LUKE. Comparing the cases with and without $I_{\textrm{FP}}$, we find that the distribution functions are very similar, as seen in Fig. \[fig:luke-code-b\]. This is not surprising as $I_{\textrm{FP}}$ mainly affects the bulk population, such that it is appropriate to ignore it in the context of the present paper. However, as $I_{\textrm{FP}}$ ensures momentum conservation in the electron-electron collision operator, it must be included for accurate driven current calculations. Indeed, the current density associated with the distributions shown in Fig. \[fig:luke-code-b\] is $J/(ecn)=0.015$ without $I_{\textrm{FP}}$ while it is $J/(ecn)=0.027$ when $I_{\textrm{FP}}$ is included. The difference arises from a shift of the electron bulk in the parallel direction, which is hardly visible in Fig. \[fig:luke-code-b\]; however, the resulting asymmetry has a strong effect on the corresponding current.
Parametric dependences of the electron distribution\[sec:Parametric\]
=====================================================================
The relevant physical parameters for the runaway electron problem described in this paper are the electric field amplitude, the magnitude of the ALD radiation reaction force, the effective charge, and the electron temperature. In this section, the bump formation is characterized as a function of these parameters. The distribution function is evolved until it reaches a steady-state solution.
In a first set of calculations, the electric field is varied while keeping the other relevant parameters fixed, with $Z_{\textrm{eff}}=1$, $\beta=0.1$, and $\sigma_{r}=0.6$. The results are shown in Fig. \[fig:param\_E\]. We observe that the electric field must reach a certain threshold for the bump to appear in the tail of the distribution function. Above this threshold, the energy corresponding to the bump location increases with $E_{\Vert}$, in accordance with the estimate (\[eq:bumploc\]). In addition we observe that the number of runaway electrons, i.e. the number of electrons with a positive parallel force balance, increases with the amplitude of the electric field. Note that the calculation was restricted to $E_{\Vert}<3.5$, as the linearization of the collision operator fails above this limit since the runaway population becomes of the order of the bulk population.
In a second set of calculations, the electric field is fixed to $E_{\Vert}=3$, while we vary the amplitude of the synchrotron radiation force, which is proportional to $B^{2}$. The results are shown in Fig. \[fig:param\_sigma\]. As expected from (\[eq:bumploc\]), we observe that the bump size and the location of the bump maximum in energy both decrease if $\sigma_{r}$ is increased, to the point where the bump disappears if $\sigma_{r}$ is above a certain threshold.
In a third set of calculations, the temperature is varied while the normalized amplitudes of the electric field and synchrotron radiation force are fixed. The results are shown in Fig. \[fig:param\_beta\]. We observe that the bump existence and energy are not affected by the electron bulk temperature, which is again in accordance with the analytical estimate (\[eq:bumploc\]). However, the number of electrons in the bump increases strongly with $T_{e}$, to the point where the linearization of the collision operator fails for $T_{e}>10$ keV with our choice of parameters. This dependence can be explained by the Dreicer effect, which feeds the runaway population from the bulk via collisional diffusion. As seen in Fig. \[fig:fevol-b\], the energy corresponding to the minimum between the bulk and the runaway bump decreases with time, until it becomes of the order of the critical energy. At this stage, the minimum coincides with the point where forces balance, such that the collisional diffusion in energy comes to a halt. In other words, the bump population increases until the negative diffusive flux associated with the positive energy gradient of the bump is sufficient to compensate for the Dreicer flux. Since the latter strongly depends upon the bulk temperature, the bump population evolves accordingly.
Finally, in a fourth set of calculations, the effective ion charge is varied while all other relevant parameters are fixed. The results are shown in Fig. \[fig:param\_Zi\]. We observe that the bump size and energy decrease with $Z_{\textrm{eff}}$, to the point where the bump disappears for $Z_{\textrm{eff}}\geq3$ with this choice of parameters. The effect of the ion effective charge is predicted by the estimate (\[eq:bumploc\]) and understood via the role of pitch-angle scattering, which is proportional to $1+Z_{\textrm{eff}}$. For a given perpendicular gradient in the tail of the distribution function - which is determined by the parallel force balance - pitch-angle scattering is the dominant mechanism to extract electrons from the runaway region. The bump can exist only if the perpendicular convection due to the synchrotron radiation force and collisional drag dominates over pitch-angle scattering at the lower energies in the runaway region.
Conclusions {#sec:Conclusions}
===========
In this paper, the effect of the Abraham-Lorentz-Dirac force in reaction to the synchrotron emission of runaway electrons is investigated for a homogeneous plasma. Whereas a runaway region - with positive force balance - can still be identified in the presence of the ALD force, the electron distribution decreases with momentum at high energy and evolves towards a steady-state solution. This evolution is a result of the strong pitch-angle scattering associated with large gradients in perpendicular momentum. The distribution of electrons is limited to energies well below the value for which the contribution from the toroidal parallel motion to the ALD radiation reaction force becomes significant in a tokamak plasma [@and01], which justifies the uniform plasma approximation to describe the runaway dynamics in the plasma center.
If the electric force is large compared to the ALD force (proportional to $B^{2}$) and the effective charge (which determines the rate of pitch-angle scattering), a bump centered on the parallel momentum axis is formed in the steady-state electron distribution. It results from the competition between the perpendicular convection due to collisions and the ALD force, and pitch-angle scattering. This bump encompasses almost the entire runaway electron population, thus formally dividing the distribution into a bulk population and a runaway beam. The steady-state population of electrons in the bump is found to increase with the bulk electron temperature $T_{e}$. The bump size and average energy increase with the electric field amplitude $E_{\Vert}$, whereas they decrease with the amplitude of the ALD radiation reaction force and the effective charge.
We can summarize the effect of the ALD radiation reaction force on the electron distribution in three points: first, in accordance with experimental observations it limits the energy gained by runaway electrons to the tens of MeV range; second, it increases the perpendicular anisotropy of the electron distribution, which may give rise to kinetic instabilities such as the EXEL or whistler waves [@pokol; @komar_pop; @pokol2014]; third, it can lead to the formation of a bump in the electron tail, which may give rise to plasma-beam types of kinetic instabilities.
Large amplitude kinetic instabilities generated by the runaway population could pump energy away from electrons and also affect their confinement. Both effects could be beneficial when attempting to limit the threat posed by runaway electrons in tokamaks. Quantifying the effect of kinetic instabilities requires a quasilinear treatment of the kinetic wave-particle interaction, which is beyond the scope of this paper.
More generally, it is interesting to note that any force with a magnitude that increases with the particle energy could play a similar role as the ALD radiation reaction force, resulting in a maximum energy limit for runaways and the possible formation of a bump in the energy distribution.
The work presented in this paper was done while J. D. was invited to work at Chalmers University under the Jubileum professorship award. J. D. would like to express his gratitude to T. Fülöp, the eFT group, and Chalmers University for this opportunity.
Cylindrical representation\[sub:Cylindrical-representation\]
============================================================
The kinetic equation (\[eq:kineticexp\]) can be expressed in the $(p_{\Vert},p_{\bot})$ system, which yields $$\frac{\partial f}{\partial t}+\frac{1}{p_{\bot}}\frac{\partial}{\partial p_{\bot}}\left(p_{\bot}S_{\bot}\right)+\frac{\partial}{\partial p_{\Vert}}\left(S_{\Vert}\right)=0,\label{eq:kineticcyl}$$ with the following expressions for the flux components $$\begin{aligned}
S_{\bot} = & -D_{\bot\Vert,\textrm{FP}}\frac{\partial f}{\partial p_{\Vert}}-D_{\bot\bot,\textrm{FP}}\frac{\partial f}{\partial p_{\bot}}\\
&+K_{\bot,\textrm{FP}}f+K_{\bot,\textrm{E}}f+K_{\bot,\textrm{ALD}}f,\\
S_{\Vert} = & -D_{\Vert\Vert,\textrm{FP}}\frac{\partial f}{\partial p_{\Vert}}-D_{\Vert\bot,\textrm{FP}}\frac{\partial f}{\partial p_{\bot}}\\
&+K_{\Vert,\textrm{FP}}f+K_{\Vert,\textrm{E}}f+K_{\Vert,\textrm{ALD}}f,
\end{aligned}
\label{eq:kineticfluxescyl}$$ $$\begin{aligned}
D_{\bot\bot,\textrm{FP}} & = & \frac{p_{\bot}^{2}}{p^{2}}A_{\textrm{FP}}+\frac{p_{\Vert}^{2}}{p^{2}}B_{\textrm{FP}},\\
D_{\bot\Vert,\textrm{FP}} & = & \frac{p_{\Vert}p_{\bot}}{p^{2}}\left(A_{\textrm{FP}}-B_{\textrm{FP}}\right),\\
D_{\Vert\bot,\textrm{FP}} & = & D_{\bot\Vert,\textrm{FP}},\\
D_{\Vert\Vert,\textrm{FP}} & = & \frac{p_{\Vert}^{2}}{p^{2}}A_{\textrm{FP}}+\frac{p_{\bot}^{2}}{p^{2}}B_{\textrm{FP}},\end{aligned}$$ $$\begin{aligned}
K_{\bot,\textrm{FP}} & = & -\frac{p_{\bot}}{p}F_{\textrm{FP}},\\
K_{\Vert,\textrm{FP}} & = & -\frac{p_{\Vert}}{p}F_{\textrm{FP}},\end{aligned}$$ $$\begin{aligned}
K_{\bot,\textrm{E}}^{\textrm{C}} & = & 0,\\
K_{\Vert,\textrm{E}}^{\textrm{C}} & = & E_{\Vert},\end{aligned}$$ $$\begin{aligned}
K_{\bot,\textrm{ALD}}^{\textrm{C}} & = & -\sigma_{r}\frac{p_{\bot}}{\gamma}\left(1+p_{\bot}^{2}\right),\\
K_{\Vert,\textrm{ALD}}^{\textrm{C}} & = & -\sigma_{r}\frac{p_{\Vert}}{\gamma}p_{\bot}^{2}.\end{aligned}$$
High-velocity limit\[sub:High-velocity-limit\]
==============================================
Properties of the electron distribution function are investigated under the conditions that the thermal electron energy is much smaller than the electron rest mass, namely $\beta\ll1$, and that the electric field is larger than the critical field but much smaller than the Dreicer field $E_{D}=\beta^{-2}$, meaning $$1<E_{\Vert}\ll\beta^{-2}$$ This ordering implies that runaway electrons are located in the tail of the distribution function, with a momentum $p\gg\beta$ where $\beta$ is the normalized thermal momentum. For such electrons it is appropriate to take the high velocity limit of the collision operator, which yields $$\begin{aligned}
A_{\textrm{FP}}\left(p\right) &= \frac{\beta^{2}}{v^{3}},\\
F_{\textrm{FP}}\left(p\right) &= \frac{1}{v^{2}},\\
B_{\textrm{FP}}\left(p\right) &= \frac{1+Z_{\textrm{eff}}}{2v}.\end{aligned}$$
ALD radiation reaction force for purely parallel motion\[sub:parallel\]
=======================================================================
In a non-uniform magnetic field, as is found in tokamaks, the field line curvature affects the ALD radiation reaction force. Whereas this effect is expected to be small compared to the contribution from the cyclotron motion for particles with a significant magnetic moment, it could play a role for particles with $\mathbf{p}_{\bot}\simeq 0$, for which the contribution from the cyclotron motion (\[eq:K\]) vanishes. The combined effects of cyclotron motion and field curvature to the ALD radiation reaction force have been evaluated in a previous work for a purely toroidal magnetic field [@and01]. Whereas a self-consistent calculation of the ALD radiation reaction force in a tokamak geometry requires a proper guiding-center transformation [@hir14], the importance of the contribution from the field line curvature can be approximately evaluated by considering the motion of a particle with $\mathbf{p}_{\bot}=0$. The corresponding guiding center follows the field lines with a velocity $\mathbf{v}=v_{\Vert}\hat{\boldsymbol{b}}$, such that the ALD radiation reaction force (\[eq:reaction\_force\]) reduces to $$\mathbf{K}=\sigma_{r}\gamma^{2}v_{\Vert}^{3}\rho_{0}^{2}
\left[\hat{\boldsymbol{b}}\cdot\nabla\left(\hat{\boldsymbol{b}}\cdot\nabla\hat{\boldsymbol{b}}\right)
+\gamma^{2}v_{\Vert}\hat{\boldsymbol{b}}\cdot
\left[\hat{\boldsymbol{b}}\cdot\nabla\left(\hat{\boldsymbol{b}}\cdot\nabla\hat{\boldsymbol{b}}\right)\right]
\mathbf{v}\right],$$ where the normalization of Sec. \[sub:gc\] is used and with $\rho_{0}\equiv mc/(qB)$. In a tokamak with major radius $R_{0}$ and circular concentric flux-surfaces characterized by the local inverse aspect ratio $\varepsilon=r/R_{0}\ll1$, $\mathbf{K}$ can be expressed as $$\begin{aligned}
\mathbf{K}=&-\sigma_{r}\gamma\left(\frac{\rho_{0}}{R_{0}}\right)^{2}p_{\Vert}^{3}\left[\left(1-v_{\Vert}^{2}\left\{ 1-q^{2}\right\} \right)\varepsilon q^{-3}\hat{\boldsymbol{\theta}}\right.\\
&\left.+\left(1+2\varepsilon\cos\theta\left\{ q^{-2}-1\right\} \right)\hat{\boldsymbol{\phi}}+\mathcal{O}(\varepsilon^{2})\right],
\end{aligned}
\label{eq:Kcirc}$$ where $\hat{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\phi}}$ denote the unit vectors in the poloidal and toroidal directions, respectively, $\theta$ is the poloidal angle, and $q(\varepsilon)$ is the safety factor. In the case of purely toroidal field lines ($q\rightarrow\infty$) the results from Ref.[@and01] are retrieved.
The approximation $\varepsilon\ll1$ is valid near the plasma center, where in addition we typically have $q\simeq1$, in which case (\[eq:Kcirc\]) becomes $$\mathbf{K}=-\sigma_{r}\gamma\left(\frac{\rho_{0}}{R_{0}}\right)^{2}p_{\Vert}^{3}\left[\hat{\boldsymbol{b}}+\mathcal{O}(\varepsilon^{2})\right].\label{eq:kfinal}$$
|
---
abstract: 'We present the results of our investigation of three samples kinematically representative of the thin and thick disks and the Hercules stream using the catalogue of Soubiran & Girard (2005). We have observed abundance trends and age distribution of each component. Our results show that the two disks are chemically well separated, they overlap greatly in metallicity and both show parallel decreasing trends of alpha elements with increasing metallicity, in the interval -0.80 $<$ \[Fe/H\] $<$ -0.30. The thick disk is clearly older than the thin disk with a tentative evidence of an AMR over 2-3 Gyr and a hiatus in star formation before the formation of the thin disk. In order to improve the statistics on the disk’s abundance trends, we have developed an automatic code, TGMET[$\alpha$]{}, to determine (Teff, logg, \[Fe/H\], \[$\alpha$/Fe\]) for thousands of stellar spectra available in spectroscopic archives. We have assessed the performances of the algorithm for 350 spectra of stars bieng part of the abundance catalogue.'
address: 'Observatoire de Bordeaux, BP 89, 33270 Floirac, France'
author:
- 'Girard, P.'
- 'Soubiran, C.$^1$'
title: 'Elemental abundances vs. kinematics in the Milky Way’s disk'
---
Abundances
==========
We have compiled a large catalogue of stars from several studies from the literature presenting determinations of O, Na, Mg, Al, Si, Ca, Ti, Fe, Ni abundances (Soubiran & Girard 2005). Because authors of the different studies do not use the same scales and methods in their spectral analyses, systematics between their results have been investigated (for more details on the construction of the catalogue see Soubiran & Girard 2005). The final catalogue of abundances includes 743 stars.
In order to study kinematical groups of the Milky Way’s disk we need velocities and orbits of stars from the abundance catalogue. We first cross-correlated the catalogue with Hipparcos (ESA 1997), selecting stars with $\pi >$10 mas and $\sigma_{\pi} \over \pi$ $<$0.10. Then we have searched for radial velocities in several sources. Distances, proper motions and radial velocities have been combined to compute the 3 components (U, V, W) of the spatial velocities with respect to the Sun and the orbital parameters for 639 stars. In addition the derivation of ages was kindly done by Frédéric Pont making use of the Bayesian method of Pont & Eyer (2004). Good estimations of ages were obtained for 322 stars from the abundance catalogue.
Kinematical classification
==========================
In order to investigate the abundance trends in the thin and the thick disks separately, we have classified the stars into the 2 populations, using kinematical information. A third component, the Hercules stream recently revisited by Famaey et al. (2004) has kinematical parameters intermediate between the thin disk and the thick disk. Its stars could have polluted previous samples of thick disk stars selected on kinematical criteria and thus must be taken into account. According to the known velocity ellipsoid of these 3 populations, we assign a membership probability to each star and select respectively 428, 84 and 44 stars having a high probability to belong to the thin disk, the thick disk and the Hercules stream respectively. The (U, V) plane of the whole sample is shown in the Fig. 8 in Soubiran & Girard (2005).
Abundance trends and ages: results and discussion
=================================================
The abundance trends for each kinematical group are shown in Fig. \[f:figure\_abu\]. In addition we have represented the distribution of ages vs. \[Fe/H\] for stars having well-defined ages for the thin disk, the thick disk and the Hercules stream (see Fig. 12 in Soubiran & Girard 2005).\
Our results confirm previous well established findings:
- The thin disk and the thick disk overlap in metallicity and exhibit parallel slopes of \[$\alpha$/Fe\] vs \[Fe/H\] in the range -0.80 $<$ \[Fe/H\] $<$ -0.30, the thick disk being enhanced.
- The thick disk is older than the thick disk.
{width="6cm"} {width="6cm"} {width="6cm"} {width="6cm"}
We bring new constraints on more controversial issues:
- The thin disk extends down to \[Fe/H\]=-0.80 and exhibits low dispersions in its abundance trends.
- The thick disk also shows smooth abundance trends with low dispersions. The change of slope which reflects the contribution of the different supernovae to the ISM enrichment is visible in \[Si/Fe\] vs \[Fe/H\] and \[Ca/Fe\] vs \[Fe/H\] at \[Fe/H\] $\simeq$ -0.70, less clearly in \[Mg/Fe\] vs \[Fe/H\].
- Al behaves as an $\alpha$ element.
- $\lbrack$O/Fe$\rbrack$ decreases in the whole metallicity range with a change of slope at \[Fe/H\] = -0.50 for the 3 populations.
- An AMR is visible in the thin disk, the most metal-poor stars having 6.2 Gyr on average, those with solar metallicity 3.9 Gyr.
- Ages in the thick disk range from 7 to 13 Gyr with an average of 9.6 $\pm$ 0.3 Gyr. There is a tentative evidence of an AMR extending over 2-3 Gyr.
- The most metal rich stars assigned to the thin disk do not follow its global trends. They are significantly enhanced in all elements (particularly in Na and Ni) except in O which is clearly depleted. They have also a larger dispersion in age. Half of these stars are probable members of the Hyades-Pleiades supercluster, two others are surprisingly old.
![The \[$\alpha$/Fe\] ratio from TGMET[$\alpha$]{} vs. the same ratio from the reference catalogue of abundances.[]{data-label="figure_alf"}](girard_fig5.eps){width="8.0cm"}
Perspectives
============
The investigation of abundances trends in the disk is currently limited by the low number of stars having known abundances, all being in the close solar neibourhood. In order to improve the statistics and reach larger distances, automatic methods of spectral analysis have to be developed. This is especially crucial in the perspective of GAIA which will produce millions of stellar spectra with substancial information on elemental abundances. We have developped a code, TGMET[$\alpha$]{}, to determine automatically (Teff, logg, \[Fe/H\] and \[$\alpha$/Fe\]) on a criterion of minimum distance with respect to the grid of synthetic spectra of Barbuy et al. (2003). We have assessed the performances of this algorithm on high resolution spectra of 350 stars being part of the abundance catalogue. We compare in Fig. 2 the \[$\alpha$/Fe\] ratio obtained with TGMET[$\alpha$]{} to those from the catalogue, considered as reference values. The low dispersion, 0.05 dex, and the lack of systematic difference ensure that indeed reliable abundance ratios can be obtained automatically, at least at high resolution. Future investigation will concern low S/N or low resolution spectra and other grids of synthetic spectra.
Barbuy, B., Perrin, M.-N., Katz, D., Coelho, P., Cayrel, R., Spite, M., Van’t Veer-Menneret, C., 2003, A&A, 404, 661 ESA, 1997, The HIPPARCOS and TYCHO catalogues. Noordwijk, Netherlands: ESA Publications Division, 1997 Girard, P. & Soubiran, C., 2005, A&A in prep. Famaey, B., Jorissen, A., Luri, X., Mayor, M., Udry, S., Dejonghe, H., Turon, C., 2005, A&A 430, 165 Pont, F., Eyer, L., 2004, MNRAS, 351, 487 Soubiran, C. & Girard, P., 2005, A&A, 438, 139
|
---
abstract: |
We present the X-ray point-source catalog produced from the [ *Chandra*]{} Advanced CCD Imaging Spectrometer (ACIS-I) observations of the combined $\sim3.2$ deg$^2$ DEEP2 (XDEEP2) survey fields, which consist of four $\sim 0.7$–1.1 deg$^2$ fields. The combined total exposures across all four XDEEP2 fields range from $\sim
10$ks–1.1Ms. We detect X-ray point-sources in both the individual ACIS-I observations and the overlapping regions in the merged (stacked) images. We find a total of 2976 unique X-ray sources within the survey area with an expected false-source contamination of $\approx 30$ sources ($\lesssim 1$%). We present the combined log N – log S distribution of sources detected across the XDEEP2 survey fields and find good agreement with the Extended [ *Chandra*]{} Deep Field and [*Chandra*]{}-COSMOS fields to $f_{\rm
X,0.5-2keV} \sim 2 \times 10^{-16} \ergpcmsqps$. Given the large survey area of XDEEP2, we additionally place relatively strong constraints on the log N – log S distribution at high fluxes ($f_{\rm X,0.5-2keV} \sim 3 \times 10^{-14} \ergpcmsqps$), and find a small systematic offset (a factor $\sim 1.5$) towards lower source numbers in this regime, when compared to smaller area surveys. The number counts observed in XDEEP2 are in close agreement with those predicted by X-ray background synthesis models. Additionally, we present a Bayesian-style method for associating the X-ray sources with optical photometric counterparts in the DEEP2 catalog (complete to $R_{\rm AB} < 25.2$) and find that 2126 ($\approx 71.4 \pm
2.8$%) of the 2976 X-ray sources presented here have a secure optical counterpart with a $\lesssim 6$% contamination fraction. We provide the DEEP2 optical source properties (e.g., magnitude, redshift) as part of the X-ray–optical counterpart catalog.
author:
- |
A. D. Goulding, W. R. Forman, R. C. Hickox, C. Jones, R. Kraft, S. S. Murray, A. Vikhlinin,\
A. L. Coil, M. C. Cooper, M. Davis, J. A. Newman
bibliography:
- 'bibtex1.bib'
title: 'The [*Chandra*]{} X-ray point-source catalog in the DEEP2 Galaxy Redshift Survey fields'
---
Introduction {#sec:intro}
============
Understanding the role of active galactic nuclei (AGN) in galaxy evolution is a major focus in present day astrophysics. It is now becoming increasingly clear that, despite their vastly differing size-scales, the evolution of massive host galaxies and the growth of their central supermassive black holes (SMBHs) may not be independent events (e.g., @boyle98 [@hopkins06; @silverman09; @Hopkins08; @smolic09]). Indeed, AGN activity and galaxy properties, such as luminosity, color and morphology, are shown to evolve with time. The redshift range $z \sim
1$–2 is a crucial epoch: (1) galaxies are evolving strongly as a function of stellar mass (e.g., @Zheng09 [@Franceschini99; @Serjeant10]); (2) AGN activity is prevalent (e.g., @ueda03 [@hasinger05; @lafranca05; @barger05; @Richards06]); (3) massive clusters are forming (e.g., @lidman08 [@hilton09; @papovich10; @fassbender11; @bauer11; @mehrtens12; @nastasi11]) and (4) the red sequence is becoming established (e.g., @bell04 [@faber07; @willmer06; @brand05; @dominguez11]). To unambiguously determine the dominant physical processes that are driving the growth and evolution of galaxies and their central black holes requires sensitive, wide-field spectroscopic surveys of AGN.
Sensitive blank-field X-ray surveys arguably provide the most efficient selection of AGN that is unbiased to moderate-to-high obscuration, and in general is not readily contaminated by host-galaxy emission. Indeed, as star-formation is relatively weak at X-ray energies ($L_{\rm X,0.5-8keV} \lesssim 10^{42} \ergps$; @moran99 [@lira02]), selection of AGN at these wavelengths can identify many of the most low-luminosity and/or obscured systems (e.g., @fukazawa01 [@done96; @risaliti99; @matt96; @maiolino98; @georgantopoulos09]). By harnessing the unprecedented angular resolution provided by the [ *Chandra*]{} X-ray Observatory, both deep and wide-field X-ray surveys have been instrumental in our current understanding of AGN evolution (e.g., @kenter05 [@nandra05; @worsley05; @brandt05; @brand06; @hasinger07; @laird09]). To date, the two deepest X-ray surveys are the pencil-beam ($\sim
0.1$ deg$^2$) $\sim 4$ Ms [*Chandra*]{} Deep Field South (CDF-S; @giacconi02 [@Luo08; @xue11]) and the $\sim 2$ Ms [*Chandra*]{} Deep Field North (CDF-N; @dma03b) which have successfully identified AGN across more than 95% of cosmic time (out to $z \sim
7$). Complementary to the highest redshift sources detected in the deep fields, nearby ($z < 0.8$) AGN, identified in the relatively shallow contiguous wide-field surveys, such as the 5 ks $\sim
9.3$ deg$^2$ XBootes field (@murray05 [@kenter05; @brand06]), have provided the ability to measure [*environment*]{}, a key component in galaxy and AGN evolution (e.g., @cooper05 [@cooper06; @georgakakis08; @coil09; @hickox09; @cappelluti10; @gilli09]). Furthermore, these wide-field X-ray surveys serendipitously detect significant numbers of rare, extremely luminous AGN and dozens of extended groups and clusters, which allow for a more complete understanding of the most massive SMBHs and cosmic structures in the Universe.
However, the peak of AGN activity, both in total luminosity and relative abundance is believed to occur at $z \sim 1$–2 (e.g., @hopkins07 [@Zheng09; @Serjeant10]). The $3.6$ deg$^2$ DEEP2 Galaxy Redshift Survey (@davis03 [@madgwick03]) provides one of the most detailed censuses of the $z \sim 1$ Universe. DEEP2 is currently one of the widest area and most complete spectroscopic surveys of $z > 1$ galaxies, making it the ideal survey to target large numbers of AGN at $z \sim 1$–2. Indeed, the fourth data release (DR4) of the survey contains spectra for $\sim 50,300$ distant galaxies (with $R_{\rm AB} < 24.1$) within four $\sim
0.7$–1.1 deg$^2$ fields, which are primarily in the redshift range $z
\sim 0.75$–1.4; these were collected using the DEIMOS spectrograph ($R \sim 5000$ in the wavelength range $6400 < \lambda < 9200$Å) on the Keck II telescope. A complete description of the DEEP2 DR4 spectroscopic catalog is available in @newman12.
We have used the [*Chandra*]{} Advanced CCD Imaging Spectrometer (ACIS-I) to provide high-angular resolution X-ray coverage across almost the entire $\sim 3.6$ deg$^2$ survey area covered by the four DEEP2 fields (Field 1 - PI:K.Nandra; Fields 2, 3 and 4 - PI:S.Murray). Here we present the X-ray source catalog for our [ *Chandra*]{} ACIS-I observations of the combined $\sim 3.2$ deg$^2$ DEEP2 (XDEEP2) survey. The four contiguous XDEEP2 fields have combined total exposures ranging from $\sim 10$ks – 1Ms. In section 2 we present a brief introduction to the construction of the survey fields and the data reduction and processing of the X-ray observations. In section 3 we provide an in-depth methodology for the detection of the point sources and the building of the final XDEEP2 catalog. In section 4, we compare our new catalog of Field 1 (the Extended Groth strip), which now includes three recent $\sim 600$ ks ACIS-I observations, to the previous catalog of Laird et al. (2009), and compare the XDEEP2 catalog to the [*Chandra*]{} Source Catalog (@evans10). We further present the flux band ratio and density of number count distributions for the XDEEP2 catalog. In section 5, we outline the optical–X-ray source matching technique used to compare our new X-ray catalog with the Fourth Data Release of the optical DEEP2 photometric catalog. Finally, in section 6 we present a summary of our findings. Throughout the manuscript we adopt a standard flat $\Lambda$CDM cosmology with $H_0 = 71\kmpspMpc$ and $\Omega_M = 0.3$.
When combined, the redshift and galaxy property information established using the DEEP2 optical spectra and the AGN identified using the new [*Chandra*]{} X-ray observations provide one of the most complete views of AGN activity and the growth of large scale structure at $z \sim 1$–2. In forthcoming papers we will present a statistically complete and obscuration-independent view of the evolution of AGN and their host-galaxies identified across the entire electromagnetic spectrum, in the epoch $z \sim 1.5$ to the present-day.
[ccccccc]{} & & & & & &\
& & & & & &\
1 & 96 & 214.7388 & +52.7838 & 0.66 & 662.3 & 139.2\
2 & 12 & 252.4470 & +34.9300 & 0.74 & 15.8 & 8.1\
3 & 17 & 352.4711 & +0.1869 & 1.13 & 10.1 & 8.1\
4 & 12 & 37.2497 & +0.5916 & 0.75 & 15.9 & 8.2\
$^{a}$XDEEP2 field number\
$^{b}$Number of [*Chandra*]{} pointings within field\
$^{c}$Center co-ordinates of field in degrees as projected onto the sky in J2000 system\
$^{d}$Total projected area of field in square degrees\
$^{e}$Effective exposure in kilo-seconds at 20% of total field area\
$^{f}$Effective exposure in kilo-seconds at 80% of total field area
[*Chandra*]{} X-ray observations {#sec:obs}
================================
Construction of the XDEEP2 fields {#sec:deep2}
---------------------------------
![Median offsets between optical and X-ray source positions with associated rms uncertainties in arc-seconds are plotted for the eight merged sub-fields in XDEEP2 Field 1 and the individual ObsIDs for Fields 2, 3 and 4. These median offsets were used to calculate astrometric corrections for the sub-fields in Field 1 and the individual X-ray observations in Fields 2–4. [*Chandra*]{} ObsIDs containing five or fewer X-ray–optical counterparts within 5 arc-minutes of the observation aim-point are shown with dotted error bars.[]{data-label="fig:astrometry"}](figure1.png){width="0.99\linewidth"}
The XDEEP2 survey region consists of four contiguous $\sim
0.7$–1.1 deg$^2$ fields covered by [*Chandra*]{} ACIS-I observations. The field positions, arrangements and main properties are outlined in Table 1. The total area covered by XDEEP2 is $\approx
3.2$ deg$^2$. X-ray catalogs for the previous 200 ks observations in Field 1, also referred to as the Extended Groth Strip, have been presented in @nandra05 and @laird09. For consistency and ease of reference to the previous Field 1 catalogs, we adopt the same sub-field naming convention defined in @laird09 (see column 2 of Table \[tbl\_obslog\]). Additionally, in this manuscript we include the more recent 600 ks ACIS-I observations within three sub-fields (EGS-3; EGS-4; EGS-5) of the Groth Strip centered at $\alpha=215.0733^o$, $\delta= +53.008^o$; 214.808$^o$, +52.806$^o$; 214.527$^o$, +52.622$^o$, which for distinction between this and the previous catalogs, we rename as AEGIS-1, AEGIS-2 and AEGIS-3, respectively.
The catalog presented here was derived from multi-epoch observations taken during AO3 (PI K. Nandra), AO6 and AO9, combined with Guaranteed Time Observations (PI S. Murray; AO9). All [*Chandra*]{} observations for XDEEP2 are publicly available through the [*Chandra*]{} X-ray Center Archive. XDEEP2 consists of 126 separate pointings with varying individual exposures ($\sim 3$–85 ks). With the exception of three exposures, all XDEEP2 observations were performed in [vfaint]{} mode to allow for the best possible background rejection. ObsIDs 3305, 4537 and 4365 were taken in [faint]{} mode. In Table \[tbl\_obslog\] we provide the individual pointing details for each observation and field.
[lcccrccrc]{}
\
& & & & & & & &\
& & & & & & & &\
[ – continued from previous page]{}\
& & & & & & & &\
& & & & & & &\
\
1 & 3305 & EGS-8 & 2002-08-11 21:43:57 & 29.40 & 214.42932 & 52.47367 & 84.74 & FAINT\
1 & 4357 & EGS-8 & 2002-08-12 22:32:00 & 84.36 & 214.42932 & 52.47367 & 84.74 & FAINT\
1 & 4365 & EGS-8 & 2002-08-21 10:56:53 & 83.75 & 214.42933 & 52.47367 & 84.74 & FAINT\
1 & 5841 & EGS-1 & 2005-03-14 00:04:09 & 44.45 & 215.67386 & 53.43149 & 229.11 & VFAINT\
1 & 5842 & EGS-1 & 2005-03-16 15:54:34 & 46.42 & 215.67348 & 53.43141 & 226.09 & VFAINT\
1 & 5843 & EGS-2 & 2005-03-19 17:13:09 & 44.46 & 215.38301 & 53.22857 & 222.38 & VFAINT\
1 & 5844 & EGS-2 & 2005-03-21 22:37:40 & 45.85 & 215.38274 & 53.22848 & 219.86 & VFAINT\
1 & 5845 & AEGIS-1 (EGS-3) & 2005-03-24 14:33:31 & 48.40 & 215.11277 & 53.03769 & 216.61 & VFAINT\
1 & 5846 & AEGIS-1 (EGS-3) & 2005-03-27 04:51:15 & 49.40 & 215.11244 & 53.03755 & 213.67 & VFAINT\
1 & 5847 & AEGIS-2 (EGS-4) & 2005-04-06 20:01:09 & 44.55 & 214.84408 & 52.84563 & 200.99 & VFAINT\
1 & 5848 & AEGIS-2 (EGS-4) & 2005-04-07 21:03:59 & 44.45 & 214.84409 & 52.84563 & 200.99 & VFAINT\
1 & 5849 & AEGIS-3 (EGS-5) & 2005-10-11 12:47:43 & 49.46 & 214.59049 & 52.64737 & 19.89 & VFAINT\
1 & 5850 & AEGIS-3 (EGS-5) & 2005-10-14 05:15:37 & 45.55 & 214.59075 & 52.64755 & 16.94 & VFAINT\
1 & 5851 & EGS-6 & 2005-10-15 03:03:18 & 35.68 & 214.10808 & 52.33121 & 14.80 & VFAINT\
1 & 5852 & EGS-6 & 2005-12-03 13:00:33 & 10.62 & 214.10950 & 52.33506 & 324.80 & VFAINT\
1 & 5853 & EGS-7 & 2005-10-16 20:16:24 & 42.57 & 213.84975 & 52.13788 & 14.04 & VFAINT\
1 & 5854 & EGS-7 & 2005-09-30 23:52:23 & 50.07 & 213.84816 & 52.13692 & 31.03 & VFAINT\
1 & 6210 & EGS-1 & 2005-10-03 14:56:50 & 45.94 & 215.68094 & 53.42341 & 29.70 & VFAINT\
1 & 6211 & EGS-1 & 2005-10-12 11:43:28 & 35.64 & 215.68196 & 53.42394 & 19.80 & VFAINT\
1 & 6212 & EGS-2 & 2005-10-04 22:56:06 & 46.28 & 215.39109 & 53.22076 & 28.00 & VFAINT\
1 & 6213 & EGS-2 & 2005-10-06 06:52:13 & 47.51 & 215.39126 & 53.22084 & 26.47 & VFAINT\
1 & 6214 & AEGIS-1 (EGS-3) & 2005-09-28 08:09:03 & 47.50 & 215.12071 & 53.02977 & 35.13 & VFAINT\
1 & 6215 & AEGIS-1 (EGS-3) & 2005-09-29 15:58:09 & 48.63 & 215.12088 & 53.02982 & 33.67 & VFAINT\
1 & 6216 & AEGIS-2 (EGS-4) & 2005-09-20 09:35:13 & 49.48 & 214.85259 & 52.83816 & 43.79 & VFAINT\
1 & 6217 & AEGIS-2 (EGS-4) & 2005-09-23 01:34:59 & 49.50 & 214.85259 & 52.83816 & 43.79 & VFAINT\
1 & 6218 & AEGIS-3 (EGS-5) & 2005-10-07 05:31:36 & 40.58 & 214.59005 & 52.64709 & 24.71 & VFAINT\
1 & 6219 & AEGIS-3 (EGS-5) & 2005-09-25 15:57:04 & 49.48 & 214.58864 & 52.64648 & 37.70 & VFAINT\
1 & 6220 & EGS-6 & 2005-09-13 09:17:01 & 37.63 & 214.10431 & 52.32963 & 49.79 & VFAINT\
1 & 6222 & EGS-7 & 2005-08-28 17:20:24 & 34.69 & 213.84478 & 52.13607 & 59.04 & VFAINT\
1 & 6223 & EGS-7 & 2005-08-31 05:06:47 & 49.51 & 213.84477 & 52.13608 & 59.04 & VFAINT\
1 & 6366 & EGS-7 & 2005-09-03 06:30:11 & 14.58 & 213.84476 & 52.13605 & 59.04 & VFAINT\
1 & 6391 & EGS-6 & 2005-09-16 20:43:01 & 8.45 & 214.10440 & 52.32956 & 49.79 & VFAINT\
1 & 7169 & EGS-6 & 2005-12-06 02:29:46 & 16.03 & 214.10951 & 52.33509 & 324.25 & VFAINT\
1 & 7180 & EGS-1 & 2005-10-13 05:16:04 & 20.43 & 215.68191 & 53.42394 & 19.80 & VFAINT\
1 & 7181 & EGS-6 & 2005-10-15 21:17:21 & 15.98 & 214.10803 & 52.33122 & 14.80 & VFAINT\
1 & 7187 & EGS-7 & 2005-10-17 19:07:08 & 6.59 & 213.84962 & 52.13785 & 14.04 & VFAINT\
1 & 7188 & EGS-6 & 2005-12-05 04:50:40 & 2.58 & 214.10909 & 52.33499 & 324.80 & VFAINT\
1 & 7236 & EGS-6 & 2005-11-30 19:29:34 & 20.37 & 214.10952 & 52.33505 & 324.80 & VFAINT\
1 & 7237 & EGS-6 & 2005-12-04 05:26:20 & 16.93 & 214.10947 & 52.33504 & 324.80 & VFAINT\
1 & 7238 & EGS-6 & 2005-12-03 10:02:10 & 9.53 & 214.10956 & 52.33502 & 324.80 & VFAINT\
1 & 7239 & EGS-6 & 2005-12-11 08:31:06 & 16.03 & 214.10932 & 52.33545 & 319.59 & VFAINT\
1 & 9450 & AEGIS-1 & 2007-12-11 04:24:07 & 28.78 & 215.07183 & 53.00951 & 319.80 & VFAINT\
1 & 9451 & AEGIS-1 & 2007-12-16 10:52:06 & 25.21 & 215.07180 & 53.00951 & 319.80 & VFAINT\
1 & 9452 & AEGIS-1 & 2007-12-18 05:45:49 & 13.29 & 215.07001 & 53.01006 & 311.30 & VFAINT\
1 & 9453 & AEGIS-1 & 2008-06-15 21:28:03 & 44.69 & 215.05924 & 52.99529 & 130.79 & VFAINT\
1 & 9454 & AEGIS-2 & 2008-09-11 04:47:10 & 59.35 & 214.81134 & 52.80632 & 49.30 & VFAINT\
1 & 9455 & AEGIS-2 & 2008-09-13 19:38:46 & 99.72 & 214.81134 & 52.80633 & 49.30 & VFAINT\
1 & 9456 & AEGIS-2 & 2008-09-24 08:15:30 & 58.35 & 214.81276 & 52.80818 & 34.80 & VFAINT\
1 & 9457 & AEGIS-2 & 2008-06-27 07:08:38 & 32.74 & 214.79607 & 52.80288 & 124.29 & VFAINT\
1 & 9458 & AEGIS-3 & 2009-03-18 12:20:16 & 6.65 & 214.52536 & 52.62140 & 223.14 & VFAINT\
1 & 9459 & AEGIS-3 & 2008-09-30 19:20:28 & 69.55 & 214.55046 & 52.61607 & 30.30 & VFAINT\
1 & 9460 & AEGIS-3 & 2008-10-10 06:17:49 & 21.36 & 214.55050 & 52.61613 & 29.80 & VFAINT\
1 & 9461 & AEGIS-3 & 2009-06-26 09:30:12 & 23.73 & 214.53241 & 52.61042 & 129.79 & VFAINT\
1 & 9720 & AEGIS-1 & 2008-06-17 05:14:02 & 27.79 & 215.05922 & 52.99527 & 130.79 & VFAINT\
1 & 9721 & AEGIS-1 & 2008-06-12 08:09:14 & 16.55 & 215.05741 & 52.99587 & 139.79 & VFAINT\
1 & 9722 & AEGIS-1 & 2008-06-13 07:02:28 & 19.89 & 215.05735 & 52.99589 & 139.79 & VFAINT\
1 & 9723 & AEGIS-1 & 2008-06-18 13:42:40 & 34.47 & 215.05923 & 52.99528 & 130.79 & VFAINT\
1 & 9724 & AEGIS-1 & 2007-12-22 13:37:26 & 14.08 & 215.07007 & 53.01006 & 311.30 & VFAINT\
1 & 9725 & AEGIS-1 & 2008-03-31 05:21:42 & 31.13 & 215.05145 & 53.00445 & 209.78 & VFAINT\
1 & 9726 & AEGIS-1 & 2008-06-05 08:45:04 & 39.62 & 215.05737 & 52.99587 & 139.79 & VFAINT\
1 & 9727 & AEGIS-2 & 2008-09-12 16:44:12 & 34.94 & 214.81132 & 52.80634 & 49.30 & VFAINT\
1 & 9729 & AEGIS-2 & 2008-07-09 16:47:58 & 48.09 & 214.79710 & 52.80272 & 119.79 & VFAINT\
1 & 9730 & AEGIS-2 & 2008-09-25 16:50:54 & 53.72 & 214.81277 & 52.80817 & 34.80 & VFAINT\
1 & 9731 & AEGIS-2 & 2008-07-03 10:58:47 & 21.38 & 214.79688 & 52.80275 & 120.79 & VFAINT\
1 & 9733 & AEGIS-2 & 2008-09-27 01:15:33 & 58.36 & 214.81275 & 52.80818 & 34.80 & VFAINT\
1 & 9734 & AEGIS-3 & 2008-09-16 11:01:21 & 49.47 & 214.54931 & 52.61415 & 44.80 & VFAINT\
1 & 9735 & AEGIS-3 & 2008-09-19 03:14:15 & 49.47 & 214.54930 & 52.61415 & 44.80 & VFAINT\
1 & 9736 & AEGIS-3 & 2008-09-20 11:07:10 & 49.48 & 214.54930 & 52.61416 & 44.80 & VFAINT\
1 & 9737 & AEGIS-3 & 2008-09-21 17:53:00 & 49.48 & 214.54931 & 52.61415 & 44.80 & VFAINT\
1 & 9738 & AEGIS-3 & 2008-10-02 06:56:22 & 61.39 & 214.55047 & 52.61607 & 30.30 & VFAINT\
1 & 9739 & AEGIS-3 & 2008-10-05 11:28:12 & 42.59 & 214.55049 & 52.61614 & 29.80 & VFAINT\
1 & 9740 & AEGIS-3 & 2009-03-09 22:24:18 & 20.37 & 214.52625 & 52.62221 & 229.78 & VFAINT\
1 & 9793 & AEGIS-1 & 2007-12-19 02:53:51 & 23.83 & 215.07005 & 53.01008 & 311.30 & VFAINT\
1 & 9794 & AEGIS-1 & 2007-12-20 04:27:59 & 10.03 & 215.07009 & 53.01004 & 311.30 & VFAINT\
1 & 9795 & AEGIS-1 & 2007-12-20 21:36:20 & 8.91 & 215.07008 & 53.01009 & 311.30 & VFAINT\
1 & 9796 & AEGIS-1 & 2007-12-21 20:28:33 & 16.33 & 215.07004 & 53.01008 & 311.30 & VFAINT\
1 & 9797 & AEGIS-1 & 2007-12-23 13:12:28 & 12.60 & 215.07007 & 53.01011 & 311.30 & VFAINT\
1 & 9842 & AEGIS-1 & 2008-04-02 21:01:59 & 30.44 & 215.05145 & 53.00445 & 209.78 & VFAINT\
1 & 9843 & AEGIS-1 & 2008-04-02 01:11:09 & 13.48 & 215.05143 & 53.00448 & 209.78 & VFAINT\
1 & 9844 & AEGIS-1 & 2008-04-05 13:07:54 & 19.78 & 215.05147 & 53.00443 & 209.78 & VFAINT\
1 & 9863 & AEGIS-1 & 2008-06-07 00:33:47 & 22.01 & 215.05733 & 52.99587 & 139.79 & VFAINT\
1 & 9866 & AEGIS-1 & 2008-06-03 22:43:14 & 25.83 & 215.05737 & 52.99588 & 139.79 & VFAINT\
1 & 9870 & AEGIS-1 & 2008-06-10 15:11:23 & 11.00 & 215.05736 & 52.99583 & 139.79 & VFAINT\
1 & 9873 & AEGIS-1 & 2008-06-11 14:22:06 & 30.75 & 215.05737 & 52.99588 & 139.79 & VFAINT\
1 & 9875 & AEGIS-1 & 2008-06-23 22:54:14 & 25.20 & 215.05968 & 52.99517 & 128.77 & VFAINT\
1 & 9878 & AEGIS-2 & 2008-06-28 06:03:20 & 15.73 & 214.79613 & 52.80289 & 124.29 & VFAINT\
1 & 9879 & AEGIS-2 & 2008-06-29 03:39:20 & 26.80 & 214.79612 & 52.80288 & 124.29 & VFAINT\
1 & 9880 & AEGIS-2 & 2008-07-05 17:00:17 & 29.50 & 214.79688 & 52.80274 & 120.79 & VFAINT\
1 & 10769 & AEGIS-3 & 2009-03-20 13:38:26 & 26.68 & 214.52497 & 52.62063 & 216.98 & VFAINT\
1 & 10847 & AEGIS-3 & 2008-12-31 05:06:27 & 19.27 & 214.54102 & 52.62566 & 302.79 & VFAINT\
1 & 10848 & AEGIS-3 & 2009-01-01 17:11:57 & 17.91 & 214.54109 & 52.62567 & 302.79 & VFAINT\
1 & 10849 & AEGIS-3 & 2009-01-02 21:25:57 & 15.92 & 214.54106 & 52.62570 & 302.79 & VFAINT\
1 & 10876 & AEGIS-3 & 2009-03-11 01:37:20 & 17.21 & 214.52626 & 52.62222 & 229.78 & VFAINT\
1 & 10877 & AEGIS-3 & 2009-03-12 15:15:57 & 16.22 & 214.52630 & 52.62223 & 229.78 & VFAINT\
1 & 10896 & AEGIS-3 & 2009-06-15 18:46:14 & 23.29 & 214.53123 & 52.61075 & 135.32 & VFAINT\
1 & 10923 & AEGIS-3 & 2009-06-22 07:38:22 & 11.62 & 214.53239 & 52.61039 & 129.79 & VFAINT\
2 & 8631 & - & 2007-11-26 00:59:04 & 8.87 & 253.14712 & 35.06573 & 10.90 & VFAINT\
2 & 8632 & - & 2007-11-26 03:52:42 & 8.60 & 252.85635 & 35.06034 & 10.90 & VFAINT\
2 & 8633 & - & 2007-11-26 06:30:13 & 8.60 & 253.14626 & 34.84466 & 10.90 & VFAINT\
2 & 8634 & - & 2007-11-26 09:07:44 & 8.60 & 252.57252 & 35.05619 & 10.90 & VFAINT\
2 & 8635 & - & 2007-11-26 11:45:15 & 8.60 & 252.29343 & 35.04576 & 10.90 & VFAINT\
2 & 8636 & - & 2007-11-26 14:22:46 & 8.60 & 252.00739 & 35.04026 & 10.90 & VFAINT\
2 & 8637 & - & 2007-11-26 17:00:17 & 8.60 & 251.71914 & 35.03216 & 10.90 & VFAINT\
2 & 8638 & - & 2007-11-26 19:37:58 & 8.60 & 252.86086 & 34.84118 & 10.90 & VFAINT\
2 & 8639 & - & 2007-11-26 22:15:40 & 8.60 & 252.57546 & 34.83892 & 10.90 & VFAINT\
2 & 8640 & - & 2007-11-27 00:53:12 & 8.61 & 252.29954 & 34.82097 & 10.90 & VFAINT\
2 & 8641 & - & 2007-11-28 05:53:06 & 8.92 & 252.01425 & 34.81738 & 10.90 & VFAINT\
2 & 8642 & - & 2007-11-28 08:41:27 & 8.66 & 251.72431 & 34.81683 & 10.90 & VFAINT\
3 & 8601 & - & 2008-08-05 04:20:00 & 9.06 & 353.25281 & 0.24568 & 242.49 & VFAINT\
3 & 8602 & - & 2008-08-05 07:12:25 & 8.93 & 352.66172 & 0.28185 & 242.49 & VFAINT\
3 & 8603 & - & 2008-08-05 09:48:46 & 8.84 & 353.46809 & 0.20782 & 242.49 & VFAINT\
3 & 8604 & - & 2008-08-05 12:24:02 & 8.84 & 351.64001 & 0.25772 & 242.49 & VFAINT\
3 & 8605 & - & 2008-08-05 14:59:49 & 8.84 & 353.37170 & 0.01529 & 242.49 & VFAINT\
3 & 8606 & - & 2008-08-05 17:35:02 & 8.83 & 352.97330 & 0.21938 & 242.49 & VFAINT\
3 & 8607 & - & 2008-08-05 20:09:51 & 8.83 & 351.89874 & 0.25007 & 242.49 & VFAINT\
3 & 8608 & - & 2008-08-05 22:44:39 & 8.84 & 351.72303 & 0.01783 & 242.49 & VFAINT\
3 & 8609 & - & 2008-08-06 01:19:39 & 8.84 & 353.09031 & -0.01102 & 242.49 & VFAINT\
3 & 8610 & - & 2008-08-06 03:54:42 & 8.84 & 352.15938 & 0.30473 & 242.49 & VFAINT\
3 & 8611 & - & 2008-08-06 06:29:27 & 8.83 & 352.01198 & 0.02339 & 242.49 & VFAINT\
3 & 8612 & - & 2008-08-06 09:04:08 & 8.84 & 351.47375 & 0.02736 & 242.49 & VFAINT\
3 & 8613 & - & 2008-08-06 11:38:49 & 8.84 & 352.25372 & 0.06485 & 242.49 & VFAINT\
3 & 8614 & - & 2008-08-06 14:13:30 & 8.84 & 352.80328 & 0.06085 & 242.49 & VFAINT\
3 & 8615 & - & 2008-08-06 16:48:28 & 8.84 & 351.48138 & 0.26156 & 242.49 & VFAINT\
3 & 8616 & - & 2008-08-06 19:23:35 & 8.83 & 352.54266 & 0.05529 & 242.49 & VFAINT\
3 & 8617 & - & 2008-08-06 21:58:23 & 8.84 & 352.42188 & 0.29140 & 242.49 & VFAINT\
4 & 8619 & - & 2007-11-28 13:18:37 & 9.04 & 36.63964 & 0.70242 & 50.76 & VFAINT\
4 & 8620 & - & 2007-11-28 16:09:09 & 8.66 & 36.72970 & 0.48135 & 50.76 & VFAINT\
4 & 8621 & - & 2007-11-29 01:03:58 & 9.07 & 36.88713 & 0.70250 & 50.76 & VFAINT\
4 & 8622 & - & 2007-11-29 03:53:59 & 8.66 & 37.14407 & 0.70447 & 50.76 & VFAINT\
4 & 8623 & - & 2007-11-29 06:32:28 & 8.66 & 37.39158 & 0.70452 & 50.76 & VFAINT\
4 & 8624 & - & 2007-11-29 09:10:59 & 8.66 & 37.63909 & 0.70268 & 50.76 & VFAINT\
4 & 8625 & - & 2007-11-29 11:49:28 & 8.66 & 37.89038 & 0.76129 & 50.76 & VFAINT\
4 & 8626 & - & 2007-12-01 11:55:28 & 8.86 & 36.97907 & 0.48142 & 50.76 & VFAINT\
4 & 8627 & - & 2007-12-01 14:50:45 & 8.46 & 37.22656 & 0.47393 & 50.76 & VFAINT\
4 & 8628 & - & 2007-12-01 17:25:55 & 8.47 & 37.48350 & 0.47209 & 50.76 & VFAINT\
4 & 8629 & - & 2007-12-01 20:01:05 & 8.47 & 37.72722 & 0.47402 & 50.76 & VFAINT\
4 & 8630 & - & 2007-12-01 22:36:15 & 8.47 & 37.88970 & 0.59744 & 50.76 & VFAINT\
Data reduction {#sec:datared}
--------------
Basic processing was carried out using the [*Chandra*]{} X-ray Center (CXC) pipeline software. In addition, further processing of the X-ray data was carried out using the [chav]{} (v4.3)[^1] and [ ciao]{} (v4.3) [^2] software packages combined with custom [idl]{} scripts. Each ACIS-I observation was analyzed separately. Individual ACIS-I pointings were reduced from the Level-1 event file products of the standard [*Chandra*]{} data pipeline. We use the [ciao]{} tool [acis\_process\_events]{} to remove the standard pixel randomization, and [status]{}=0 was used to remove streak events, bad pixels and cosmic ray afterglow features.
{width="95.00000%"}
![Merged source count image of the sub-field AEGIS-3 located in Field 1. Aim points of the individual [*Chandra*]{} ObsIDs within the sub-field are shown with green crosses matched to the roll angle of the space craft. The angular separation of the aim-points is sufficiently small ($\sim 5$ arc-seconds) that they allow for the combining of the individual ObsIDs into stacked images.[]{data-label="fig:merge1"}](figure3.png){width="0.95\linewidth"}
All observations were visually inspected for flaring and periods of high background. The majority of the observations were found to not be significantly contaminated. As also noted in @nandra05 and @laird09, observation 4365 does exhibit an interval ($\approx
25$ ks; $\sim 30$ % of the observation) of elevated background. However, unlike the previous analyses, here we conservatively screen-out this period of high background. Final effective exposures in good-time intervals for each observation were generally found to be $> 90$ % of the “on-time” (see Table \[tbl\_obslog\]).
Creation of individual images & exposure maps {#sec:imgexp}
---------------------------------------------
{width="95.00000%"}
{width="95.00000%"}
{width="95.00000%"}
{width="95.00000%"}
Events files were screened using a standard grade set ([ grade]{}=0,2,3,4,6) to construct images for each individual ObsID. Images were constructed in the Full (FB; 0.5–7 keV), Soft (SB; 0.5–2 keV) and Hard (HB; 2–7 keV) bands at the full ACIS-I spatial resolution, 0.492 arcsec/pixel. Here we limit the photon energy to $E<7$ keV to allow a more direct comparison to sources detected in the XBootes survey. Given the small effective area of the ACIS-I detector at $E>7$ keV, relatively few $E>7$ keV photons are detected, and thus this choice of energy boundary is somewhat arbitrary and will have little effect on our conclusions. The [chav]{} tool [aspecthist]{} was used to create aspect histograms in all three bands. These aspect histograms were used to generate exposure maps by convolving them with the standard ACIS-I chip-map ([ccd\_id=0,1,2,3]{}) and reprojecting to the previously created counts images. Reference spectra in monochromatic bands of $E \sim 1.0$, 4.0 and 2.5 keV (i.e., the median energies of the SB, HB and FB, respectively) were used in the creation of the exposure maps.
{width="95.00000%"}
Astrometric calibration & observation merging {#sec:astrocalib}
---------------------------------------------
Due to differing observing strategies and the sizes of exposure area overlaps between individual [*Chandra*]{} observations within each XDEEP2 field, X-ray observations were combined using separate methods for Field 1 and Fields 2–4. As stated previously, Field 1 contains eight sub-fields (see §\[sec:obs\]), with marginal overlap ($\sim
0.01$–0.02 deg$^2$) between one another. Each of these eight sub-fields consists of several (3–28) individual [*Chandra*]{} ACIS-I exposures with significant overlap between the observations within a particular designated sub-field. We used the [ciao]{} Perl script, [merge\_all]{} to create contiguous raw X-ray images and exposure maps within each of the eight Field 1 sub-fields. Briefly, this script searches for bright X-ray sources within two events tables which spatially overlap and compares the astrometric co-ordinates of the detected sources. By computing the average offset between the sources within the tables, and guarding against rogue outliers, the events table and associated aspect histograms are reprojected to the world co-ordinate system (WCS) of the first reference observation within the sub-field.
Given the limited area overlap (which occurs only at large off-axis radii) between the eight sub-fields, a resultant merged events table and images from a further use of [merge\_all]{} to combine the sub-fields, is likely to be highly uncertain. However, one of the primary goals for this XDEEP2 X-ray catalog is to compare the X-ray detected sources with the previously astrometrically-calibrated optical DEEP2 catalog presented in Coil et al. (2004). Hence, we may consider the WCS astrometry of the DEEP2 optical catalog to be an absolute reference frame. Thus, here we use the DEEP2 optical source positions to correct the X-ray sub-fields for any systematic offsets that may be present in the combined X-ray data. Following Brand et al. (2006), we use a counterpart-matching algorithm (described in detail in § \[sec:XOPT\] of this manuscript) to match X-ray sources detected within 3 arc-minutes of the nominal observation aim-point to optical counterparts. We calculated the median offset between the X-ray and optical positions for the respective sources to identify any necessary translation for the X-ray sub-fields. We present these offsets and their associated rms uncertainties in Figure \[fig:astrometry\]. Typically, $\sim 20$–30 X-ray–optical sources were used to determine the necessary translations; the offsets were generally found to be $< 0.25$ arc-seconds (i.e., $\sim 50$% of the ACIS-I pixel scale). While rotations were also allowed in the calculation of the relative astrometries, the magnitude of the angular rotation was always found to be negligible ($\ll 1$ degree) and consistent with no rotation. Hence, we did not include angular rotations and used only linear transformations for the final corrections of the X-ray WCS to that of DEEP2. The required positional offsets for the merged X-ray images were applied using the [ciao]{} tool [wcsupdate]{}. The [ciao]{} tool [reproject\_aspect]{} was used to reproject the events table and aspect solution files.
{width="90.00000%"}
In Fields 2–4, the relatively shallow 9–10 ks X-ray observations include little or no overlap area between exposures. As such, and similar to the merged sub-fields in Field 1, [merge\_all]{} cannot be used to accurately co-align the relative astrometries within the individual X-ray observations in these three fields. Hence, again we consider the WCS reference frame of the optical DEEP2 catalog to be absolute, and use the optical sources to align individual X-ray observations following the same methodology described above. Given the far shallower depth of the X-ray observations in Fields 2–4, we include all X-ray sources with optical counterparts to a distance of $<5$ arc-minutes from the aim-point. This larger off-axis distance encompasses sufficient X-ray–optical source numbers (5–20 per observation) to accurately constrain any required systematic astrometric correction. Four of the [*Chandra*]{} ObsIDs (8637; 8614; 8604; 8628) included five or fewer X-ray–optical sources, and hence we consider any astrometric corrections for these four observations to be sufficiently uncertain that we subsequently include all detected X-ray sources (at all off-axis distances within the observation) to further constrain any median offset. The calculated median offsets and associated uncertainties are also included in Figure \[fig:astrometry\]. Clearly, using our adopted methodology, we do not account for any possible field-to-field (or intra-field) variations in the astrometric accuracy of the optical DEEP2 catalog. However, given the low number of X-ray sources within individual [*Chandra*]{} observations, further investigation and/or necessary correction to the DEEP2 catalog are beyond the scope of this study. Overall, the required astrometric corrections (average correction of 0.24”) for the whole of XDEEP2 are consistent, if not slightly lower, than those found in previous wide-field X-ray surveys (e.g., XBootes: 0.41”; Brand et al. 2006) and can be considered sufficiently precise for our purposes.
Merged XDEEP2 field maps
------------------------
In Figures \[fig:cntsimage\] and \[fig:merge1\], we show examples of the merged full-band (0.5–7 keV) counts images and in Figures \[fig:expfld1\]–\[fig:expfld4\], we present the merged full-band exposure maps for the four survey fields. As shown in Figure \[fig:expangle\]a, the effective exposure (and hence sensitivity depth; see § \[sec:sens\]) across Field 1 is non-uniform and varies dramatically from $\approx$20 ks–1.1 Ms. The effective exposure in Field 1 is dependent on the number of repeat exposures, the large number of overlapping regions and the varying space-craft roll-angles between separate pointings. We show that at the 80th percentile, the effective exposure in Field 1 is $\approx$140 ks. By contrast, the effective exposures in Fields 2, 3 and 4 are relatively uniform ($\sim
9$ ks at 80%) with constant spacecraft roll angle and only small overlap regions between the individual ACIS-I pointings ($ <
20$%). In Figure \[fig:expangle\]b, we also show the effective exposure time across the combined XDEEP2 area and compare this to the [*Chandra*]{}-COSMOS (@elvis09) and Extended-[*Chandra*]{} Deep Field South fields (@lehmer05). It is clear that XDEEP2 complements these previous surveys: the survey depth of XDEEP2 extends well beyond $\sim 200$ ks (the limiting effective exposure of the E-CDF-S) to $> 600$ ks at similar survey area ($A \sim 0.2$ deg$^2$); and XDEEP2 covers a survey area which is a factor $\approx 4$ greater than that of [*Chandra*]{}-COSMOS.
The raw merged count images for each of the four XDEEP2 fields were adaptively smoothed using custom [idl]{} software based on the kernel-smoothing program, [asmooth]{} [@ebeling06]. Given the wide range in exposure times across Field 1, we include a weighting algorithm based on the average number of counts within binned background images (see §\[sec:sens\]) to account for changes in background count rate in overlapping regions. This background-weight is applied to the calculation of the smoothing radii within our custom version of [asmooth]{}. The smoothing scales, which are calculated from analysis of the merged counts images, are then applied directly to the respective exposure maps. We use these to create false-color exposure-corrected smoothed images in each field (see Figure \[fig:3color\]).
{width="90.00000%"}
Point source detection & spurious sources {#sec:srcdec}
=========================================
In this section we outline the methods used to detect point-like sources throughout the XDEEP2 fields. Following earlier analogous methods for numerous wide-field and deep X-ray surveys, we used wavelet decomposition software to detect sources across XDEEP2. Indeed, previous analyses of Field 1 have used the [ciao]{} tool [wavdetect]{} to detect X-ray source candidates. Here, we chose to use [wvdecomp]{} which is publicly available in the [zhtools]{} package (see @vikhlinin98). In §\[sec:laird\] we perform a comparison of the X-ray sources detected in @laird09 which used [wavdetect]{} and additional signal-to-noise criteria to the sources detected in this work using [wvdecomp]{}. Briefly, we find little or no difference between the number of sources detected in either analyses. We find that $\sim 96$% of the unique X-ray sources found in the previous AEGIS-X catalog are included in our new catalog (presented here) which now includes the more recent longer exposure ACIS-I observations. We find that the majority of the sources which are not included in our new catalog are relatively low significance with few counts ($< 10$) and, in general, are detected in only one energy-band in the Laird et al. catalog. Sources similar to these were conservatively removed as possibly spurious detections in our new catalog based on our extensive MARX simulations (see §\[sec:spursrc\]).
Point source detection in individual [*Chandra*]{} ObsIDs
---------------------------------------------------------
Point sources were detected in the individual (non-merged) counts images for the SB, HB and FB energy ranges. We used a point source detection threshold in [wvdecomp]{} of 4.5$\sigma$ (equivalent to a probability threshold of $1 \times 10^{-6}$). Point sources were detected over wavelet scales of {1, $\sqrt{2}$, 2, 4} $\times
0.492''$. After detection of a source candidate, the event data at the approximate wavelet position was iterated up to five times to accurately determine the final events centroid, and hence, source position. In Figure \[fig:wav\_centroid\], we present the offset distances between the wavelet and event centroid positions. We find that $> 90$% of the X-ray sources have offset distances $\lesssim
0.3$” from the wavelet position. Indeed, the vast majority of the sources are consistent with zero offset. Furthermore, we find that the median offset distance between the wavelet and centroid positions are mildly correlated with the on-chip distance of the source from the observation aim-point. Those sources at $d_{\rm OAX} < 5$’ have $\Delta({\rm wavelet} - {\rm centroid}) \sim 0.2$”, while those sources closer to the edge of the FOV, at $d_{\rm OAX} \gtrsim 10$’, have $\Delta({\rm wavelet} - {\rm centroid}) \sim 0.65$”. These increased offsets at large off-axis distances were most likely due to asymmetries in the ACIS PSF shape.
Source lists, generated from the separate energy bands in the individual observations, were cross-correlated based on their source positions. Two-dimensional Gaussian profiles were used to represent the sources detected in the separate energy bands with full-width half maxima (FWHM) determined by the physical size of the 90% encompassed energy fraction (EEF) within the [*Chandra*]{} energy-band images with the assumption of a spherically symmetric model for the ACIS-I PSF. The centers of the Gaussian profiles were allowed to shift within the 1$\sigma$ centroid error (see §\[sec:src\_extract\]) of the source positions to maximise the statistical likelihood of a source match. A unique source was determined to exist when the summed 2-D Gaussian profile was well-fit at the 90% confidence level by a single (approximately symmetrical) 2-D Gaussian profile with FWHM $< 90$% EEF.[^3] This methodology has the advantage that the ‘matching-radius’ naturally becomes a function of both the off-axis position and the energy-band of the source detection. Hence, it incorporates the size increase and rotation of the ACIS-I PSF radius which, while assumed to be symmetrical about the aim-point, still increases significantly for large off-axis distances and simultaneously changes as a function of both azimuthal angle and effective energy.
Sources in overlapping observations in Field 1 {#sec:src_overlap}
----------------------------------------------
As stated previously, sources were detected in each of the individual ObsIDs. In Fields 2,3 and 4 there are small regions of significant exposure ($> 10$ ks), where individual observations overlap. However, given the large systematic uncertainties brought about by significant differences in [*Chandra*]{} PSF radii, we did not attempt to combine these observations to search for faint sources, which would be detected in the merged deeper exposure regions. Instead, where duplicate sources in these overlap regions appear (see previous section), the source which is radially closest to the aim point in a particular [*Chandra*]{} observation (i.e., the source which has the smallest point spread function), is included as a unique source in the final catalog. By contrast, given the large overlap between the [ *Chandra*]{} observations in the sub-fields of Field 1, it is highly likely that the same physical X-ray source is detected in multiple individual exposures and that many fainter sources would be detected in merged X-ray images. Hence, we have created merged events files of the sub-field regions, which were defined in @laird09 (see Table 1 of @laird09 and Table \[tbl\_obslog\] and Figure \[fig:merge1\] in this work).[^4]
When combined, the Field 1 ‘EGS’ sub-fields show a significant increase in the overall exposure and depth. Each of the observations in these sub-fields have varying space-craft roll angles. However, as shown in Table \[tbl\_obslog\], the pointing co-ordinates are similar ($\lesssim 5$ arc-seconds; see also Figure \[fig:merge1\]). As such, these stacked sub-fields do not suffer from significant sensitivity degradation due to large changes in the [*Chandra*]{} point spread function (i.e., the observational setup was similar to that of the CDF-N and CDF-S; e.g., @dma03a [@xue11]). We used wavelet decomposition to search for additional faint sources in these [ *merged*]{} (stacked) sub-field images which would otherwise not be detected in the individual observations. Candidate source lists for Field 1, which were compiled from each of the individual ObsIDs and those lists derived from the merged sub-field images were compared using the same unique-source detection method outlined in the previous section. The final unique source position and associated centroid errors were determined by averaging and combining in quadrature the previously calculated positions and uncertainties in the individual and merged X-ray observations.
Source extraction {#sec:src_extract}
-----------------
Once the unique source locations were determined across each of the XDEEP2 fields, we counted the number of events ($C_{\rm 50,SB/HB/FB}$, $C_{90,SB/HB/FB}$) within the 50% ($R_{50}$) and 90% ($R_{90}$) encircled energy fraction regions of the merged sub-field images (Field 1) and the individual ACIS-I observations (Fields 2–4) for each of the soft, hard and full bands. Within the sub-fields of Field 1, the radii for circular extraction regions were calculated from the off-axis radial distances in PSF simulations. We used the MARX simulator to model a point-source, within a specific energy-band, at varying roll angles and off-axis distances from an observation aim-point. The modeled energy-band images were combined using the method outlined in §\[sec:astrocalib\], and the spatial extent of the merged point-source was measured using a circular aperture to determine accurate extraction radii for the candidate sources identified in the Field 1 sub-fields.
{width="97.00000%"}
![Source count threshold cut in the 0.5–7 keV band as a function of exposure time in MARX simulated [*Chandra*]{} ACIS-I imaging. For a fixed total number of spurious sources of $N < 0.25$ within simulated observations, we show the dependence of the threshold cut on the off-axis position of the detected spurious sources.[]{data-label="fig:marxsim2"}](figure12.png){width="0.9\linewidth"}
We calculated average effective exposures for each candidate source in the $R_{50}$ and $R_{90}$ extraction regions. Background counts were determined for each source by extracting photon counts in annuli at inner and outer radii $\{1.1,2.5\} \times R_{90}$, respectively, in background images (see §\[sec:sens\]). Background counts were scaled by the ratio of the areas of the EEF extraction region and the background extraction region. Scaled background counts were subtracted from the respective $C_{50}$ and $C_{90}$ to give final net source counts ($C_{\rm 50, net}$; $C_{\rm 90, net}$, respectively). The 50% and 90% encircled energy fraction regions were chosen to match those used in the XBootes survey (Murray et al. 2005; Kenter et al. 2006; Brand et al. 2006) allowing direct comparisons to be made between the catalogs in future publications.
For a source detected in a particular energy band image, we computed the total number of source counts in the other energy band images using the analyses described above. We converted the net count rates in each band (SB; HB; FB) to total fluxes ($F_{\rm SB}$; $F_{\rm HB}$; $F_{\rm FB}$, respectively). To build a homogeneous X-ray catalog, we assumed a single simple absorbed power-law spectrum with $\Gamma=1.7$ (i.e., the typical intrinsic slope of an AGN) for all sources and $N_{\rm H} = \{1.24, 1.75, 3.99, 2.89\} \times 10^{20} \pcmsq$ for those sources in Fields 1, 2, 3 and 4, respectively. Here, we use PIMMS to calculate the count-rate–flux conversion factors assuming the simulated ARFs from AO9 of the [*Chandra*]{} program. The use of the AO9 ARFs compared to AO6 results in a $\sim 4$% decrease in the calculated 0.5–7 keV flux. Total galactic HI column densities were determined using @stark92. Uncertainties on the counts and fluxes were calculated using the formalism of [@gehrels86].
Following Murray et al. (2005), we estimated the 90% uncertainty on the source locations as $X_{\rm err} = R_{50} / (C^{1/2}_{50}
-1)$. For those sources with $C_{50} \le 5$ counts, we set a minimum centroid error of 0.8 arc-seconds (i.e., the 99% positional accuracy on the ACIS-I detector[^5]), which takes into account the systematic uncertainties associated with the space-craft and detector astrometry. Random uncertainties also become negligible for sources with large numbers of counts.
Spurious sources {#sec:spursrc}
----------------
Given the widely varying exposure times, and hence varying background levels of individual observations within XDEEP2, it is important to apply further restrictions to the detected-source lists based on the number of counts for a given source. For those observations with large exposure times, the number of spurious sources with seemingly low numbers of counts increases (see Figure \[fig:marxsim1\]a). To limit the number of spurious sources within our final catalog, we applied a minimum photon count threshold of $n_{\rm counts} > n_{\rm cut}$, where $n_{\rm cut}$ was determined through simulations of sourceless background ACIS-I images. We used the MARX software package to simulate 100,000 [*Chandra*]{} ACIS-I images of the unresolved Cosmic X-ray background (XRB), including instrumental effects for exposure times of 3, 6, 9, 12, 15, 20, 30, 50, 75 and 100ks. To approximate the expected emission from the unresolved CXB, we employed a simple absorbed power-law spectrum with $\Gamma=1.4$ (e.g., @hickox06) and $f_X \sim 8.189 \times 10^{-13} \ergps$ in the 0.5–7 keV band; i.e., the XRB surface brightness measured in the ROSAT all-sky survey in a blank-sky region of XDEEP2 Field 1, which was then scaled to the projected area of ACIS-I. We note that this simplification assumes the CXB emission is homogeneous across an ACIS-I observation. We searched each of the simulated XRB ACIS-I images for spurious sources using the same wavelet detection thresholds defined above (see Figure \[fig:marxsim1\]a). To build source lists which were both relatively complete while limiting the number of spurious sources, we cut the source-lists where the expected total number of spurious sources $\Sigma n$ for a given exposure $i$ was $\langle \Sigma n_i \rangle < 0.25$ (see Figure \[fig:marxsim1\]b). By adopting a threshold of $\langle \Sigma n_i
\rangle < 0.25$, we expect a spurious source detection rate of $< 1$% in the final catalog.
As we show in Figure \[fig:marxsim2\], we find that the spurious net count threshold is both a function of exposure time ($t_{\rm exp}$) and off-axis position ($x_{\rm OAX[']}$) of the source within an ACIS-I observation. This count threshold can be approximated by the empirical formula, $$n_{thresh} = - \frac{5}{3} + \frac{3}{10}x_{\rm OAX} + \frac{{ln}[60x_{\rm OAX}-30]}{2}{\rm log } t_{exp}$$ and we use this to derive $n_{thresh}$ for a given fixed off-axis position and exposure. To verify that this parametrization of the count threshold can be extrapolated to larger exposure times (i.e., for the merged AEGIS-1, 2 and 3 sub-fields in XDEEP2 Field 1), we simulated 100 1 Ms ACIS-I exposures using MARX. On average, we detected $< 1$ spurious source in each 1 Ms simulation by using $n_{thresh} > 20.3$. Hence, within the Poisson error, we detected the same number of spurious sources expected when extrapolating the above equation to $t_{exp} = 1$ Ms. By conservatively adopting a threshold of $\langle \Sigma n_i \rangle < 0.25$ across the 126 XDEEP2 pointings we expect $\lesssim 30$ spurious sources in the final XDEEP2 catalog.
The XDEEP2 catalog {#numcnts}
==================
The XDEEP2 point source catalog contains 2976 unique sources, with 1720, 342, 528 and 386 sources in Fields 1, 2, 3 and 4, respectively. For the purposes of our point source catalog, we do not discuss those sources which are extended (e.g., the galaxy clusters) as these will be the subject of a future publication. In Table \[tbl\_main\_src\] we show a short extract from the main source table, which is available electronically. In Figure \[fig:cntshisto\], we show the distribution of source counts across the four XDEEP2 counts in the soft, hard and full bands. It is clear that both the wide-spatial area of XDEEP2 combined with the smaller regions of sensitive long-exposures, are extremely complementary to one another. A significant cut-off is observed for sources with $C_{\rm
90, SB} \lesssim 9$ in Field 1 since relatively few sources ($\sim
100$) are detected with 5–10 counts within $R_{90}$ due to the long integrated exposures, even in the soft-band. However, many more sources, down to $C_{\rm 90, FB} \sim 5$ are detected when the other three XDEEP2 fields are included. Hence, within the point source catalog we detect sources down to $\sim 4$ net counts in the SB, with a completeness to 20 net counts in the HB and 15 counts in the FB. Furthermore, we detect 70 rare bright sources with $C_{\rm 90, FB}
> 500$, which is due to the advantage of the wide-area across the XDEEP2 survey.
![Distribution of X-ray counts for sources detected in each of the four XDEEP2 fields in the full-band (0.5–7keV; top panel), hard-band (2–7keV; middle panel), and soft-band (0.5–2keV; bottom panel). Median source counts for each energy band in the associated field are shown with vertical lines.[]{data-label="fig:cntshisto"}](figure13.png){width="0.97\linewidth"}
[ccccc]{} & &\
& & & &\
Full & 2849 & - & 661 & 1196\
Soft & 2301 & 111 & - & 1006\
Hard & 1663 & 12 & 372 & -\
[Notes]{}–\
$^{a}$Energy band which a source has been detected in\
$^{b}$Number of sources where there is a non-detection in a particular energy-band when it has been detected in a different band.\
In Table \[tbl\_breakdown\] we show the breakdown of the numbers of sources detected and formally undetected in individual energy bands within the main XDEEP2 source catalog. Those X-ray sources which are not formally detected in a particular energy band are denoted by “-1” in the relevant net count and flux error columns of Table \[tbl\_main\_src\] (e.g., [NET COUNTS ERROR SB/HB/FB]{} and [FLUX ERROR SB/HB/FB]{}). For these ‘non-detections’, we use the formalism of @gehrels86 to dervie $3 \sigma$ upper-limits from the number of counts observed in the background images (see § \[sec:sens\]) within the source region. There upper-limits are given in the appropriate [NET COUNT]{} and [FLUX]{} energy-band columns.
Background & sensitivity analysis {#sec:sens}
---------------------------------
As is clearly evident from the merged exposure maps, many of the XDEEP2 ObsIDs spatially overlap with one another; however, a subset of these observations, specifically in Field 1, were performed up to seven years apart. Hence, care was taken to analyze changes between the overlapping images as a result of the physical changes in the detector and varying background levels. Background images were constructed separately for each ObsID in the SB, HB and FB energies. Source counts for candidates which were identified as being significant in a particular energy-band using [wvdecomp]{} were masked. Background annuli, with inner radii $1.1 \times R_{90}$ and outer radii $5 \times R_{90}$ centered at the source position, were used to calculate the mean local background surrounding the candidate source. The masked source region was re-populated with Poisson noise with a mean distribution equal to that of the local background. The same procedure was additionally used to create background maps of the merged sub-fields in Field 1. While this procedure will remove the count contributions from all point-sources, it will not remove extended emission from sources such as clusters (e.g., @bauer02). Hence, the background count levels derived from this method are somewhat conservative, as they will be slightly over-estimated.
[ccccccc]{}
& & &
& &\
& & & & &\
1 & AEGIS 1 & Full & 0.0841 & 0.2898 & 52.5\
1 & AEGIS 1 & Soft & 0.0242 & 0.1539 & 15.1\
1 & AEGIS 1 & Hard & 0.0599 & 0.2425 & 37.4\
1 & AEGIS 2 & Full & 0.0842 & 0.2900 & 51.6\
1 & AEGIS 2 & Soft & 0.0236 & 0.1524 & 14.5\
1 & AEGIS 2 & Hard & 0.0605 & 0.2448 & 37.1\
1 & AEGIS 3 & Full & 0.0991 & 0.3033 & 61.3\
1 & AEGIS 3 & Soft & 0.0284 & 0.1609 & 17.6\
1 & AEGIS 3 & Hard & 0.0706 & 0.2548 & 43.7\
1 & EGS 1 & Full & 0.0243 & 0.1438 & 13.1\
1 & EGS 1 & Soft & 0.0070 & 0.0773 & 3.8\
1 & EGS 1 & Hard & 0.0165 & 0.1186 & 8.9\
1 & EGS 2 & Full & 0.0235 & 0.1419 & 12.0\
1 & EGS 2 & Soft & 0.0068 & 0.0765 & 3.5\
1 & EGS 2 & Hard & 0.0159 & 0.1167 & 8.1\
1 & EGS 6 & Full & 0.0271 & 0.1531 & 14.8\
1 & EGS 6 & Soft & 0.0077 & 0.0815 & 4.2\
1 & EGS 6 & Hard & 0.0184 & 0.1257 & 10.0\
1 & EGS 7 & Full & 0.0241 & 0.1429 & 13.3\
1 & EGS 7 & Soft & 0.0070 & 0.0769 & 3.9\
1 & EGS 7 & Hard & 0.0163 & 0.1176 & 9.0\
1 & EGS 8 & Full & 0.0332 & 0.1684 & 14.7\
1 & EGS 8 & Soft & 0.0111 & 0.0980 & 4.9\
1 & EGS 8 & Hard & 0.0202 & 0.1310 & 8.9\
& & & & &\
& & & & &\
2 & - & Full & 0.0018 & 0.0428 & 11.1\
2 & - & Soft & 0.0005 & 0.0231 & 3.2\
2 & - & Hard & 0.0013 & 0.0361 & 7.9\
& & & & &\
& & & & &\
3 & - & Full & 0.0018 & 0.0432 & 7.4\
3 & - & Soft & 0.0005 & 0.0234 & 2.1\
3 & - & Hard & 0.0013 & 0.0363 & 5.2\
& & & & &\
& & & & &\
4 & - & Full & 0.0018 & 0.0431 & 7.6\
4 & - & Soft & 0.0005 & 0.0231 & 2.2\
4 & - & Hard & 0.0013 & 0.0364 & 5.4\
& & & & &\
–\
$^{a}$XDEEP2 field number\
$^{b}$XDEEP2 sub-field name\
$^{c}$X-ray energy band of background image: Full 0.5–7keV; Soft 0.5–2keV; Hard 2–7keV\
$^{d}$Mean number of background counts per pixel within the non-zero exposure area of the merged images.\
$^{e}$Standard deviation of the background counts within the merged images.\
$^{f}$Total number of background counts within the merged images.\
The mean background counts, their associated standard deviation and total number of background counts for each field (and sub-field) are shown in Table \[tbl\_bkg\]. As expected, the average background counts are a factor of $\approx 15$–50 greater in Field 1 than those in Fields 2–4, owing to the much longer exposure times in Field 1. We find that the average backgrounds appear to be relatively stable across the deep sub-fields AEGIS-1 and AEGIS-2, with a slightly higher ($\approx 15$%) average background count in AEGIS-3. However, we note that the observations in AEGIS-3 occurred 6–12 months after those observations in AEGIS-1 and AEGIS-2. The background levels in XDEEP2 Fields 2, 3 and 4 are almost identical for each of the three energy-bands.
For the purposes of comparing the X-ray point sources detected within each of the XDEEP2 fields, as well as comparing with previous X-ray surveys, it is important to understand the flux sensitivity limitations of a particular X-ray field. The faintest sources detected in the XDEEP2 fields have $f_{\rm X,SB} \sim 3.1 \times 10^{-17}
\ergpcmsqps$ and $f_{\rm X,HB} \sim 1.2 \times 10^{-16} \ergpcmsqps$. While these fluxes are good indicators of the ultimate sensitivity of the survey, sources similar to these may only be detected in stacked images close to the center of several ObsID aim-points where exposure levels are sufficiently high ($\sim 800$ ks) and the combined PSF is relatively small. Hence, given an observing strategy with varying levels of exposure across the fields, X-ray sources at these low flux levels cannot be uniformly detected across the whole of each field. To quantify the expected number of sources as a function of survey area, we have constructed flux sensitivity maps for each merged field in the 0.5–2 keV, 2–7 keV and 0.5–7 keV bands.
{width="95.00000%"}
{width="97.00000%"}
Maps of the [*Chandra*]{} point spread functions for an enclosed energy fraction of 90% were simulated at $E \sim 1.0$, 4.0 and 2.5 keV (mean SB, HB and FB energies, respectively) for each ObsID using the [ciao]{} tool [mkpsfmap]{}. These maps were then merged for all overlapping fields to calculate the mean $R_{90}$ in each image pixel for a merged counts image in the soft, hard and full-bands. We used the formalism of Lehmer et al. (2005) and employed a Poisson model to calculate the average number of counts ($N$) required to detect a source in a given image pixel for the background counts ($b$) enclosed within the mean $R_{90}$ calculated in the merged PSF model,
$${\rm log} (N) = \alpha + \beta {\rm log} b + \gamma({\rm log} b)^2 +
\delta({\rm log} b)^3
\label{eqn:lehmer}$$
where $\alpha = 0.967$, $\beta = 0.414$, $\gamma = 0.0822$ and $\delta
=0.0051$ (@lehmer05). Using equation \[eqn:lehmer\] we convolve the merged PSF and background images at each image pixel and normalize to the appropriate merged exposure maps to create final fluxed sensitivity images in each energy band (three per field; an example sensitivity image is shown in Figure \[fig:sensmap\]).
We calculate empirical sensitivity curves in the SB, HB and FB for each of the four XDEEP2 fields using the sensitivity images derived above (see Figure \[fig:limflux\]). Due to the small overlapping regions in Fields 2–4, the sensitivity curves are found to be relatively smooth over the entire survey region with relatively sharp cut-offs at $f_{X,SB} \sim 4 \times 10^{-15} \ergpcmsqps$, $f_{X,HB}
\sim 9 \times 10^{-15} \ergpcmsqps$ and $f_{X,FB} \sim 7 \times
10^{-15} \ergpcmsqps$. Hence, the sensitivity limit is approximately uniform across the majority of the survey area in Fields 2, 3 and 4. By contrast, the wedding-cake style observational setup of Field 1 combined with changing roll angles produces small ($\sim 0.1$ deg$^2$) regions of high sensitivity, which combine over the field to produce a much more shallow sensitivity curve (i.e., the sensitivity is non-uniform). However, as the average exposure across Field 1 is $\approx 20$–100 times greater than Fields 2–4, the mean sensitivity to the [*detection*]{} of faint sources is vastly improved in Field 1. We find that the limiting flux in the 0.5–7 keV band for source detection, which includes at least 10% of the survey area, is a factor $\approx 16$ lower in Field 1 ($f_{X,FB} > 2.8 \times 10^{-16}
\ergpcmsqps$) than in Fields 2–4 ($f_{X,FB} > 4.5 \times 10^{-15}
\ergpcmsqps$, $f_{X,FB} > 4.6 \times 10^{-15} \ergpcmsqps$ and $f_{X,FB} > 4.6 \times 10^{-15} \ergpcmsqps$, respectively).
Comparison of X-ray sources in Field 1 to Laird et al. (2009) {#sec:laird}
-------------------------------------------------------------
[ccccccccccccccccccccccccccccccccc]{}
& & & & & & & & &\
& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &\
& & & & & & & & & & & & &\
aeg1\_001 & F1\_AEG1 & CXOJ141907.7+525946 & 214.78246 & 52.99710 & 0.84 & 4.48 & 10.36 & 632.0 & 16.62 & 5.77 & 31.17 & 8.86 & 23.17 & 7.15 & 48.97 & 12.25 & 39.81 & 8.74 & 80.26 & 14.70 & 4.80 & 1.44 & 20.6 & 5.62 & 19.2 & 3.78 & 0.20 & 0.04 & 0.43 & 4.75 & 3.18 & 7.66\
aeg1\_002 & F1\_AEG1 & CXOJ141907.8+530025 & 214.78334 & 53.00712 & 0.32 & 4.45 & 10.30 & 628.4 & 162.4 & 13.99 & 311.9 & 19.55 & 57.06 & 9.42 & 95.73 & 13.98 & 219.1 & 16.46 & 406.0 & 23.62 & 50.6 & 3.21 & 40.6 & 6.48 & 102. & 6.14 & -0.53 & -0.59 & -0.47 & 0.88 & 0.69 & 0.97\
aeg1\_003 & F1\_AEG1 & - & 214.79521 & 52.98033 & 1.18 & 6.91 & 12.42 & 611.5 & 14.31 & -1 & 22.97 & -1 & 35.58 & 9.37 & 54.85 & 13.96 & 46.79 & 10.76 & 71.22 & 16.18 & 3.66 & -1 & 25.1 & 6.30 & 18.3 & 4.09 & 0.53 & 0.30 & 0.84 & 12.37 & 7.12 & -1\
aeg1\_004 & F1\_AEG1 & CXOJ141911.2+530320 & 214.79699 & 53.05600 & 1.24 & 4.48 & 10.33 & 623.2 & 12.51 & 5.10 & 20.42 & 7.36 & 12.09 & -1 & 23.29 & -1 & 21.19 & 6.82 & 27.06 & 10.65 & 474. & 91.2 & 824. & -1 & 1710 & 208. & -0.55 & -1.00 & -0.40 & 1.17 & 0.18 & 3.78\
aeg1\_005 & F1\_AEG1 & CXOJ141919.9+530254 & 214.83506 & 53.04790 & 0.84 & 3.42 & 7.95 & 536.8 & 10.13 & 4.84 & 43.10 & 9.10 & 15.51 & 6.08 & 46.83 & 11.03 & 25.57 & 7.31 & 89.53 & 13.87 & 5.98 & 1.31 & 17.7 & 4.55 & 19.3 & 3.18 & 0.02 & -0.11 & 0.20 & 3.26 & 2.25 & 4.34\
aeg1\_006 & F1\_AEG1 & CXOJ141920.6+530028 & 214.83600 & 53.00792 & 0.29 & 3.03 & 7.11 & 514.7 & 117.3 & 12.04 & 226.4 & 16.77 & 12.68 & -1 & 28.79 & 10.21 & 127.6 & 12.91 & 255.2 & 19.24 & 23.7 & 1.76 & 8.15 & 3.07 & 42.3 & 3.22 & -0.79 & -0.85 & -0.70 & 0.41 & 0.25 & 0.56\
aeg1\_007 & F1\_AEG1 & CXOJ141922.8+530132 & 214.84506 & 53.02555 & 0.21 & 2.86 & 6.75 & 498.7 & 100.1 & 11.24 & 161.1 & 14.59 & 121.0 & 12.44 & 222.9 & 17.61 & 221.2 & 16.32 & 384.7 & 22.44 & 18.4 & 1.66 & 73.1 & 5.75 & 70.0 & 4.07 & 0.16 & 0.10 & 0.21 & 4.05 & 3.54 & 4.44\
aeg1\_008 & F1\_AEG1 & - & 214.85376 & 52.99871 & 2.53 & 3.14 & 5.65 & 477.5 & 5.61 & 4.29 & 7.76 & 5.82 & 4.46 & -1 & 7.09 & -1 & 5.03 & -1 & 8.13 & -1 & 0.76 & 0.59 & 0.62 & -1 & 0.78 & -1 & -0.54 & -1.00 & -0.45 & 2865. & 0.06 & -1\
aeg1\_009 & F1\_AEG1 & - & 214.85694 & 53.00549 & 0.43 & 2.53 & 6.01 & 469.5 & 20.15 & 5.88 & 42.87 & 8.77 & 27.37 & 6.90 & 41.56 & 10.01 & 47.39 & 8.60 & 83.71 & 12.87 & 4.32 & 0.88 & 12.0 & 2.90 & 13.4 & 2.07 & -0.03 & -0.17 & 0.13 & 2.88 & 2.14 & 3.96\
aeg1\_010 & F1\_AEG1 & - & 214.85765 & 53.01971 & 0.96 & 2.53 & 6.06 & 469.6 & 9.02 & -1 & 16.26 & -1 & 12.63 & 5.56 & 34.74 & 9.91 & 13.20 & -1 & 36.99 & 11.10 & 1.63 & -1 & 9.82 & 2.86 & 5.75 & 1.78 & 0.80 & 0.74 & 1.00 & 9117. & 18.87 & -1\
aeg1\_011 & F1\_AEG1 & - & 214.86239 & 53.03122 & 0.54 & 2.56 & 5.98 & 464.7 & 22.14 & 6.08 & 44.39 & 8.83 & 11.39 & -1 & 24.43 & 9.07 & 33.24 & 7.62 & 69.73 & 12.22 & 4.56 & 0.91 & 7.06 & 2.70 & 11.3 & 2.02 & -0.34 & -0.51 & -0.10 & 1.92 & 1.11 & 3.12\
aeg1\_012 & F1\_AEG1 & - & 214.86615 & 53.02515 & 0.56 & 2.44 & 5.75 & 453.7 & 10.17 & 4.71 & 29.25 & 7.81 & 18.13 & 6.08 & 27.73 & 9.26 & 28.58 & 7.23 & 58.53 & 11.67 & 2.91 & 0.78 & 7.87 & 2.67 & 9.24 & 1.86 & -0.06 & -0.23 & 0.20 & 3.14 & 1.96 & 4.95\
aeg1\_013 & F1\_AEG1 & CXOJ141928.0+525840 & 214.86670 & 52.97822 & 0.25 & 2.42 & 5.70 & 461.1 & 76.61 & 9.93 & 149.7 & 13.82 & 37.22 & 7.62 & 58.93 & 10.64 & 113.9 & 12.08 & 208.9 & 17.02 & 15.9 & 1.47 & 18.0 & 3.26 & 35.5 & 2.89 & -0.43 & -0.51 & -0.35 & 1.18 & 0.97 & 1.44\
aeg1\_014 & F1\_AEG1 & - & 214.87337 & 53.03977 & 0.29 & 2.37 & 5.63 & 448.2 & 52.24 & 8.47 & 87.44 & 11.22 & 29.69 & 7.07 & 59.41 & 10.85 & 81.87 & 10.58 & 146.5 & 15.17 & 8.73 & 1.13 & 16.8 & 3.15 & 23.2 & 2.44 & -0.19 & -0.29 & -0.09 & 2.12 & 1.62 & 2.50\
aeg1\_015 & F1\_AEG1 & - & 214.87634 & 53.04383 & 1.07 & 3.21 & 5.78 & 446.2 & 13.05 & 5.34 & 22.20 & 7.27 & 4.52 & -1 & 12.87 & 8.22 & 15.98 & 6.83 & 33.72 & 10.53 & 2.26 & 0.75 & 0.41 & -1 & 5.35 & 1.74 & -0.36 & -0.70 & 0.05 & 0.24 & 0.09 & 0.64\
aeg1\_016 & F1\_AEG1 & - & 214.87842 & 53.00748 & 1.44 & 3.34 & 6.01 & 422.9 & 12.95 & 5.34 & 21.81 & 7.27 & 4.74 & -1 & 7.56 & -1 & 11.05 & 6.47 & 17.48 & 9.83 & 2.11 & 0.71 & 0.64 & -1 & 2.67 & 1.53 & -0.81 & -1.00 & -0.78 & 0.27 & 0.02 & 1.31\
aeg1\_017 & F1\_AEG1 & CXOJ141930.8+525915 & 214.87886 & 52.98781 & 0.42 & 2.11 & 5.00 & 428.3 & 30.90 & 6.81 & 62.29 & 9.67 & 9.90 & -1 & 18.33 & -1 & 36.53 & 7.62 & 79.83 & 11.87 & 6.30 & 0.98 & 5.35 & -1 & 13.0 & 1.91 & -0.61 & -0.75 & -0.43 & 0.86 & 0.53 & 1.58\
aeg1\_018 & F1\_AEG1 & - & 214.88704 & 53.04167 & 1.12 & 2.96 & 5.33 & 421.8 & 10.14 & 4.85 & 14.38 & 6.31 & 4.26 & -1 & 6.75 & -1 & 13.21 & 6.28 & 20.96 & 9.27 & 1.51 & 0.65 & 0.61 & -1 & 3.62 & 1.53 & -0.43 & -1.00 & -0.23 & 0.78 & 0.16 & 4.05\
aeg1\_019 & F1\_AEG1 & - & 214.88706 & 52.99963 & 0.82 & 1.86 & 4.58 & 405.0 & 7.57 & -1 & 12.61 & -1 & 10.46 & 4.84 & 17.61 & 7.37 & 10.76 & 5.10 & 22.31 & 8.48 & 1.23 & -1 & 4.92 & 2.08 & 3.44 & 1.32 & 0.52 & 0.39 & 1.00 & 7412. & 6.61 & -1\
aeg1\_020 & F1\_AEG1 & - & 214.88917 & 53.09005 & 1.37 & 4.92 & 8.88 & 496.5 & 20.36 & 6.37 & 24.09 & 8.12 & 16.40 & -1 & 26.88 & -1 & 21.16 & 8.00 & 31.18 & -1 & 3.20 & 1.29 & 12.2 & -1 & 7.86 & -1 & -0.71 & -1.00 & -0.64 & 2.09 & 0.16 & 6.51\
–\
$^{a}$Unique source identifier\
$^{b}$XDEEP2 ObsID/sub-field name\
$^{c}$Unique source identifier for matched XDEEP2 sources present in the [*Chandra*]{} X-ray Source Catalog (CSC)\
$^{d}$X-ray position in J2000 co-ordinates (degrees) and associated centroid positional error (arc-seconds)\
$^{e}$Aperture radius in arc-seconds at 50% and 90% the effective area of ACIS-I given the off-axis distance of the X-ray source\
$^{f}$Off-axis distance in arc-seconds of X-ray source from aim-point of observation\
$^{g}$Soft-band (0.5–2 keV) net counts and associated errors in the $R_{50}$ and $R_{90}$ apertures\
$^{h}$Hard-band (2–7 keV) net counts and associated errors in the $R_{50}$ and $R_{90}$ apertures\
$^{i}$Full-band (0.5–7 keV) net counts and associated errors in the $R_{50}$ and $R_{90}$ apertures\
$^{j}$Total soft-band (S), hard-band (H) and full-band (F) fluxes and associated errors in units of $10^{-16} \ergpcmsqps$\
$^{k}$Classical hardness ratios ($HR = (C_{\rm H} - C_{\rm S}) /
(C_{\rm H} + C_{\rm S})$) and associated 1$\sigma$ upper and lower limits calculated using the BEHR method\
$^{l}$Flux ratios ($FR = F_{\rm HB} / F_{\rm SB}$) and associated 1$\sigma$ upper and lower limits calculated using the BEHR method\
As stated previously, while the analyses presented here include the recent 600ks observations of AEGIS 1–3, the 200ks X-ray source catalog for Field 1 ([*AEGIS-X*]{}) has been previously presented in Laird et al. (2009). Furthermore, the new 600ks observations will also be presented in a forthcoming paper (Nandra et al. in prep.) using similar detection and Bayesian-style sensitivity analyses to that used for the previous [*AEGIS-X*]{} catalog. Since the source detection and extraction analyses differ significantly between [*AEGIS-X*]{} and the XDEEP2 catalog presented here, we now compare the detection methods and results.
{width="97.00000%"}
Briefly, detection of sources in [*AEGIS-X*]{} was carried out using a custom implementation of the CIAO [wavdetect]{} tool. Laird et al. (2009) perform several runs of the detection software using different probability thresholds to build seed catalogs and to derive multiple estimates of the X-ray background in the observation. The final probability threshold for which a particular candidate is determined to be false in [*AEGIS-X*]{} is comparable to that used in our analyses. Laird et al. detected source candidates separately in the full, soft, hard and ultra-hard band images.[^6] These source candidates were then combined into an individual source catalog using Bayesian techniques to statistically associate the source candidates and calculate the fluxes in the respective energy bands. The [*AEGIS-X*]{} catalog contains 1325 sources. Two sources (EGS4\_258; EGS7\_204, nomenclature adopted from Laird et al. 2009) in the [*AEGIS-X*]{} catalog were only detected in the ultra-hard band, i.e., an energy-band which we do not use due to the relatively small effective area of the telescope at these higher energies. Furthermore, four [*AEGIS-X*]{} sources (EGS4\_240; EGS6\_185; EGS7\_194; EGS8\_127) have 90% effective-area extraction regions which significantly ($> 50$%) overlap with those extraction radii of other sources in the [*AEGIS-X*]{} catalog; from visual inspection we find that these four [*AEGIS-X*]{} sources (and their neighbors) are consistent with being single point sources. Hence, we remove these six sources from further comparison between the XDEEP2 and [*AEGIS-X*]{} catalogs.
We compared the 1319 unique source candidates identified in [ *AEGIS-X*]{} to the 1720 source candidates identified in Field 1 of our XDEEP2 catalog solely on the basis of source position using the same varying matching radius method described in §\[sec:XOPT\]. We find that 1260 ($\approx 96$%) of the source candidates identified in [ *AEGIS-X*]{} are included in our new catalog. We have visually inspected each of the 59 [*AEGIS-X*]{} sources which were not identified in the XDEEP2 catalog. We find that the majority (44/59; $\sim 63$%) of the [*AEGIS-X*]{} sources, which are not included as part of the XDEEP2 catalog, were detected by [wvdecomp]{} as source candidates in one energy band. However, on the basis of our MARX simulations, these 44 non-matched [*AEGIS-X*]{} sources did not meet our ultimate and more conservative count detection threshold and were removed as possibly spurious based on their low net counts ($C_{\rm 90,net} \sim
5$–10). A further seven of the 59 non-matched sources were flagged as ‘non-standard’ and possibly spurious; we discuss these seven sources below. Finally, eight of the 59 non-matched [*AEGIS-X*]{} source candidates are not detected using [wvdecomp]{} after the inclusion of the more recent 600ks data.
We now briefly discuss the seven of the 59 non-matched [*AEGIS-X*]{} source candidates (EGS2\_052; EGS5\_105; EGS6\_073; EGS6\_093; EGS7\_180; EGS8\_093; EGS8\_134) that were initially detected by [ wvdecomp]{} and then highlighted by our routine as ‘possibly spurious’. Visual inspection shows that three of these seven source candidates, EGS6\_093, EGS5\_105 and EGS7\_180) have their expected source PSFs partially blended with secondary brighter sources. Indeed, EGS6\_093, is located between ($<2.5$ arc-seconds) two significantly brighter X-ray sources (EGS6\_164 and EGS6\_165; both these sources are included in the [*AEGIS-X*]{} and XDEEP2 catalogs) causing sufficient detection ambiguity and EGS7\_180 has an X-ray morphology consistent with that of a jet. These three sources, while initially detected in XDEEP2, are not included in our final catalog due to our inability to accurately separate the flux contribution from the neighboring bright source. Furthermore, EGS6\_073 falls on a chip-gap; EGS2\_052 has $C_{\rm FB,net} < 6$; and EGS8\_134 has $\sim 50$% of its low source counts ($C_{\rm FB,net} \sim 7$) in one ACIS-I pixel, and is conservatively removed based on our MARX simulation analyses. Finally, EGS8\_093 is possibly part of an extended source which appears extremely diffuse and only has a marginal detection ($P_{\rm SB} \sim 1.4 \times 10^{-6}$) in the [*AEGIS-X*]{} catalog.
As above, eight of the 59 non-matched [*AEGIS-X*]{} source candidates are not detected using [wvdecomp]{} after the inclusion of the 600ks data; while these sources were detected in the [*AEGIS-X*]{} analyses with $P_{\rm band} > 10^{-6}$, we note here that these sources may still be real, but are no longer detected due to intrinsic variability of the source. Similarly, Nandra et al. (in prep.) find that from a re-analysis of [*AEGIS-X*]{}, 17 of the [*AEGIS-X*]{} sources are no longer detected in the three sub-fields which include the new 600ks data. Assuming a similar number of non-detected source candidates across all of Field 1, this would suggest a false source contamination rate of $\approx 45$ sources ($\approx 3.5$%) in [*AEGIS-X*]{}.
In Figure \[fig:xd2vslaird\] we show a comparison between the soft-band fluxes for isolated and formally-detected point-sources in the [*AEGIS-X*]{} (classically derived flux) and XDEEP2 catalogs. We have converted the fluxes we derived using $\Gamma = 1.7$ in XDEEP2 to $\Gamma = 1.4$, as used in [*AEGIS-X*]{}, using a conversion of 1.031 and we have corrected the [*AEGIS-X*]{} fluxes for galactic absorption (a factor of 1.042; Laird et al. 2009). We find excellent agreement between the fluxes derived in the XDEEP2 and [*AEGIS-X*]{} catalogs with a Spearman’s rank coefficient of $r \sim 0.963$ which is significant at $P > 99.99$% level. Additionally, we have used two-dimensional linear-regression analyses to calculate the $3 \sigma$ uncertainty on the derived correlation between the XDEEP2 and AEGIS-X source fluxes (dotted-lines in Figure \[fig:xd2vslaird\]), and the $3 \sigma$ error region on the photon counts used to derive the source fluxes (dash-dot-lines in Figure \[fig:xd2vslaird\]). As expected, the Poissonian error due to low source counts significantly dominate the uncertainty towards low fluxes. We find that 12 ($\lesssim 0.1$%) of the matched XDEEP2–[*AEGIS-X*]{} sources lie substantially outside the $3\sigma$ error region. These outlying sources have large numbers of counts ($\gg 100$) and/or are significantly extended beyond the expected 90% EEF angular size in the merged ACIS-I images. This suggests that these outlying sources are strong candidates for galaxy clusters and/or moderately variable quasi-stellar objects (QSOs).[^7] Furthermore, variations in extraction radii at large offaxis distances, due to the introduction of the more recent ACIS-I observations (which were performed with substantially different roll-angles) may potentially cause significant differences in measured counts/flux for bright X-ray sources with non-point-like profiles, such as galaxy clusters. Indeed, we find that when considering only the previous 200ks observations studied in Laird et al., with matched extraction apertures, the fluxes for all of the matched XDEEP2–[ *AEGIS-X*]{} sources are consistent to within 1$\sigma$.
In Figure \[fig:xd2vslaird\] we show the source flux distributions in [*AEGIS-X*]{} and XDEEP2 including the 460 new XDEEP2 sources which are detected in the new deeper 600 ks observations. As expected, the majority of these 460 new XDEEP2 sources have $f_{\rm 0.5-2keV} \sim
(8$–$80) \times 10^{-16} \ergpcmsqps$, extending the distribution of the previous catalog to lower source fluxes. We additionally highlight the fluxes of the 59 [*AEGIS-X*]{} source candidates, which we conservatively do not include in XDEEP2. Each of these non-matched sources have $f_{\rm 0.5-2keV} < 1.3 \times 10^{-15} \ergpcmsqps$, with the vast majority at the extreme low-flux end of the main [ *AEGIS-X*]{} source-flux distribution ($f_{\rm 0.5-2keV} \sim (1$–$4)
\times 10^{-16} \ergpcmsqps$). Using a Bayesian counterpart matching algorithm, which we present in § \[sec:XOPT\], we have attempted to assign DEEP2 optical counterparts to the 59 [*AEGIS-X*]{} source candidates. We find that the majority (35/59; $\sim 60$%) of these [*AEGIS-X*]{} source candidates lack secure optical counterparts; this is a factor $\sim 2$ larger than the fraction of X-ray sources which lack counterparts across the entire XDEEP2 sample ($\sim
29$%). However, based on simulations of a purely random set of 59 source positions, we would expect only $\sim 3$–7 spurious counterpart matches using our Bayesian matching algorithm. Hence, the 24 [*AEGIS-X*]{} sources found to coincide with an optical counterpart is a factor $\sim 3$–8 larger than the random expectation of spurious counterparts, suggesting that some of these X-ray sources may be real.
Based on our rigorous comparison of the [*AEGIS-X*]{} catalog and our XDEEP2 catalog, we suggest that the two catalogs appear to be in excellent agreement, despite the use of different detection algorithms ([wavdetect]{} versus [wvdecomp]{}). In general, the small ($\approx 4$%) discrepancy between the catalogs can be attributed to the removal of low significance sources in the XDEEP2 catalog based on our MARX simulations. Additionally, we stress that since 51 of the 59 low significance sources are initially identified by both [ wavdetect]{} and [wvdecomp]{}, we cannot rule out that they are real sources, although they ultimately did not meet our more conservative detection criteria.
Comparison of X-ray sources in Fields 2–4 to the [ *Chandra*]{} Source Catalog {#sec:CSC}
------------------------------------------------------------------------------
{width="97.00000%"}
The Chandra Source Catalog (CSC) is a compilation of all relatively bright X-ray sources detected in single ACIS and HRC imaging observations by the Chandra X-ray Observatory in the first eight years of the mission (@evans10). In principle, the X-ray sources detected in XDEEP2 Fields 2–4 and by the CSC are likely to be equivalent. Similar to the CSC, we have not attempted to merge events in overlapping regions of Fields 2–4 as, in general, these regions occur at large off-axis distances where the [*Chandra*]{} PSF is poor. In this section, we compare the XDEEP2 source properties to those detected in the CSC release 1.1. The current CSC data release contains X-ray data products and information (positions; spatial; temporal multi-band count rates; fluxes) for distinct point sources and compact sources, with observed spatial extents $\lesssim 30$” observed in publicly released data to the end of 2009. Highly extended sources, and sources located in selected fields containing bright, highly extended sources are excluded in the CSC. See <http://cxc.cfa.harvard.edu/csc/index.html> for further information.
We have used the publicly available java-applet, CSCview to associate the XDEEP2 X-ray sources in Fields 2–4 to the CSC Master Catalog. Although we do not attempt to merge the individual ACIS-I observations in Fields 2–4, we find that only $\sim 41.9 \pm 2.2$% (i.e., 150/342; 218/528; 158/386 sources in Fields 2, 3 and 4, respectively) of the XDEEP2 sources are identified in the CSC. In Figure \[fig:csc\_comparison\]a we show a comparison of the flux distributions for all XDEEP2 sources and CSC sources within the area covered by Fields 2, 3 and 4 of XDEEP2. The 90% EEF aperture fluxes produced by the CSC are derived using a simple absorbed powerlaw with $\Gamma = 1.7$ and $N_{\rm H} = 3 \times 10^{20} \pcmsq$. Hence, for the purposes of comparison we convert the field-specific $N_{\rm H}$ used to derive the XDEEP2 fluxes to match the CSC fluxes.
We find that while all CSC sources with $f_{\rm 0.5-2} \gtrsim 6
\times 10^{-15} \ergpcmsqps$ are identified in the XDEEP2 catalog, the vast majority of the lower flux XDEEP2 sources are not included in the CSC. By design, the detected CSC X-ray sources have $C_{\rm net}
\gtrsim 10$ counts for an on-axis source ($\gtrsim 20$–30 counts for an off-axis source), i.e., the CSC catalog only includes sources whose flux estimates are greater than three times their estimated 1$\sigma$ uncertainties. However, as we have shown in Figures \[fig:marxsim1\] and \[fig:marxsim2\], and has been shown conclusively by many other deep and wide-field X-ray surveys (e.g., CDF-N; CDF-S; C-COSMOS; AEGIS-X; XBootes), many X-ray sources can be significantly identified with only $\sim 3$–5 net counts, although the source flux will remain relatively unconstrained due to Poisson uncertainties. Indeed, $\gtrsim 98$% of the XDEEP2 sources not identified in the CSC catalog have $C_{\rm net} < 20$ counts. Furthermore, to within $1 \sigma$, we find excellent agreement for the X-ray fluxes of the sources in common between XDEEP2 and the CSC (see Figure \[fig:csc\_comparison\]b).
Given that all of the CSC sources within the survey area are identified in the XDEEP2 catalog and the non-matched sources have lower counts/flux which lie above the thresholds derived from our extensive simulation analyses, we find that the CSC provides a more conservative identification of X-ray sources within the XDEEP2 fields. For completeness, we have also associated the X-ray sources identified in Field 1 to the CSC catalog, and find there are 689 distinct X-ray sources in common between the catalogs. The faintest CSC sources in Field 1 have $f_{\rm 0.5-2} \gtrsim 5 \times 10^{-16}
\ergpcmsqps$, but with the majority at $f_{\rm 0.5-2} \gtrsim 2 \times
10^{-15} \ergpcmsqps$ (i.e., an average factor $\sim 3$ more sensitive per individual observation than Fields 2–4). For ease of comparison with future surveys, we include the CSC source identifiers as part of the XDEEP2 catalog, for all XDEEP2 sources with CSC counterparts.
Source spectral properties: hardness ratios {#sec:HR}
-------------------------------------------
Using the Bayesian Estimator of Hardness Ratio (BEHR) method (@park06), hardness count ratios (HR), defined as ${\rm HR} =
(C_{\rm HB}-C_{\rm SB})/(C_{\rm HB}+C_{\rm SB})$, where $C_{\rm SB}$ and $C_{\rm HB}$ are the counts in the soft and hard bands respectively, as well as the hardness flux ratios (FR), defined as ${\rm FR} = F_{\rm HB}/F_{\rm SB}$, were calculated for all detected sources in the XDEEP2 catalog. FR and HR and their associated uncertainties calculated using BEHR are available in the main XDEEP2 source table. Briefly, BEHR treats the detected source and background X-ray photons as independent Poisson random variables, and uses a Monte Carlo based Gibbs sampler to select samples from posterior probability count distributions to correctly propagate the non-Gaussian uncertainties, which derive from the calculation of hardness ratios. BEHR is particularly powerful in the low-count Poisson regime, and computes a realistic uncertainty for the HR and FR, regardless of whether the X-ray source is detected in both energy bands. In Table \[tbl\_main\_src\], we include the FR and HR ratios with the associated $1\sigma$ upper and lower limits for all XDEEP2 sources. Sources with unconstrained upper or lower limits due to non-detections are denoted by “-1” in the appropriate uncertainty column.
In Figure \[fig:HR\] we show the FR distribution for the XDEEP2 sources. Typically, the XDEEP2 sources which are detected in both the hard and soft-bands have FR in the range $\sim 0.7$–7, with distribution tails at low and high values of FR. Following previous studies (e.g., @bauer02 [@dma03a; @Luo08]), we divide the X-ray sources with low and high-flux at $f_{\rm FB} \sim 4 \times 10^{-15}
\ergpcmsqps$ (i.e., the 10% flux limit of the shallow exposure XDEEP2 fields). While the choice of cut is somewhat arbitrary, clearly we find the same general trend towards higher values of FR for X-ray sources with low-fluxes as has been observed previously (e.g.,@hasinger93 [@vikhlinin95; @giacconi02; @tozzi06]). We find that the distribution of FR values is moderately peaked at ${\rm FR}
\sim 1.3$ sources with high flux ($f_{\rm FB} \gtrsim 4 \times
10^{-15} \ergpcmsqps$). By contrast, lower flux sources have a more extended distribution, with a median value of ${\rm FR} \sim 2.1$ and tailing to higher values of FR. Using the [ciao]{} spectral analysis package, [sherpa]{}, we have simulated X-ray spectra for AGN populations at $0 < z < 6$ in order to quantify the evolution of X-ray spectral slopes due to the k-correction of the observed AGN spectra towards high-z. Based on these simulations, we find that the two peaks observed in the FR distributions are co-incident with the spectral slopes expected for two separate AGN populations with $\Gamma \sim
1.2$–1.4 and $\Gamma \sim 1.7$–1.8. Further, we find that the majority of the 460 low-flux sources in Field 1, which were not previously identified in AEGIS-X due to insufficient survey depth (see § \[sec:laird\]), have a similarly wide FR distribution ($\sim
0.8$–10) to the AEGIS-X source candidates and the sources identified in Fields 2–4. However, the median FR for the new faint Field 1 sources is shifted slightly higher with FR$\sim$3 (i.e., harder spectral indices), suggesting that these new sources have flatter X-ray spectral slopes, and are likely to be more heavily obscured. Hence, their previous non-detection in the 200ks data is due to the combined result of AGN luminosity, distance and intrinsic obscuration.
![[**Main panel:**]{} Flux band ratio defined as ${\rm FR} =
f_{\rm 2-7keV}/f_{\rm 0.5-2keV}$ as a function of full-band counts ($C_{0.5-7keV}$) for all XDEEP2 sources detected in the soft and hard energy bands. Average spectral slopes for fixed values of FR established from X-ray spectral simulations using [Sherpa]{} are highlighted with horizontal dotted lines. [**Right panel:**]{} FR distributions for all detected sources within XDEEP2 with $f_{\rm
0.5-7keV}\gtrsim 4 \times 10^{-15} \ergpcmsqps$ (dot-dashed) and $f_{\rm 0.5-7keV}\lesssim 4 \times 10^{-15} \ergpcmsqps$ (dotted). []{data-label="fig:HR"}](figure18.png){width="0.97\linewidth"}
{width="90.00000%"}
XDEEP2 source number counts {#sec:lognlogs}
---------------------------
We have calculated the cumulative number of sources in the XDEEP2 catalog ($N(>S)$) detected per square degree that are brighter than a given flux in the soft (0.5–2 keV) band, i.e., the ${\rm log} N -
{\rm log} S$ distribution (see Figure \[fig:lognlogs\]). This provides a good check that the merging of the datasets and the extensive calibrations were performed correctly, as well as an excellent comparison to previous X-ray surveys. We choose to compare in the soft-band as this is the most sensitive energy and the specific energy range definition of the soft-band (0.5–2 keV) is consistent across previous surveys. As a consequence of (1) the changing slope of the ${\rm log} N - {\rm log} S$ distribution towards fainter fluxes, and (2) observationally fainter sources possibly being more obscured and/or lower accretion rate AGN than brighter sources, the so-called ‘Eddington bias’ introduces many statistically low-significance sources at the sensitivity limit of the X-ray survey. Hence, we have empirically restricted our analyses presented in this section to only those sources detected with $f_{\rm SB} > 4.5\sigma_{\rm bkg,field}$, i.e., on-axis 0.5–2 keV fluxes of $f_X \gtrsim 9 \times 10^{-17}
\ergpcmsqps$ in Field 1 and $f_X \gtrsim 4 \times 10^{-15}
\ergpcmsqps$ in Fields 2–4 (equivalent to $C_{\rm SB,net} > 10$ and $C_{\rm SB,net} > 6$, respectively). For the purpose of comparison, we have converted all source and field fluxes to $\Gamma = 1.4$ and use the combined flux limits (see §\[sec:sens\]) to construct the ${\rm
log} N - {\rm log} S$ distribution.
To quantify the uncertainties on the derived ${\rm log} N - {\rm log}
S$, we have used a Monte-Carlo (MC) style simulation. Using the formal $1\sigma$ error on the source flux, we built symmetrical probability flux distributions ($P(f_X)$) for each source to be input to 10,000 realizations of our simulation. Within the MC simulation, we randomly assign fluxes to each source within the sample based on the individual $P(f_X)$, and recompute the ${\rm log} N - {\rm log} S$ distribution. The total 90% uncertainty on the ${\rm log} N - {\rm
log} S$ is then defined as the mean absolute deviation of the 10,000 simulated distributions combined in quadrature with the 90% Poissonian error on the main distribution, defined using the formalism of Gehrels (1986). From our MC simulations, in Figure \[fig:lognlogs\] we show that the XDEEP2 ${\rm log} N - {\rm log} S$ is very well constrained ($\sim 0.12$ dex) in the flux range $f_X \sim
(0.2$–$5) \times 10^{-15} \ergpcmsqps$ owing to the large sensitive area in Field 1 around the ‘knee’ of the ${\rm log} N - {\rm log} S$ at $f_X \sim ($6–8$) \times 10^{-15} \ergpcmsqps$. However, we find that the uncertainty on the distribution increases to $\sim 0.3$ dex towards the bright flux tail ($f_X \gtrsim 10^{-14} \ergpcmsqps$) of the ${\rm log} N - {\rm log} S$. We determined that this is caused by the decrease in the space-density of the far rarer bright sources, combined with the relatively large uncertainties on the fluxes for those sources identified in the more shallow exposure Fields 2–4. For these particular sources, which dominate the distribution within this moderate–high flux regime, the majority are detected with relatively few counts ($\sim 6$–15) and hence, $1 \sigma$ flux errors are $\sim
25$–50% of the overall flux. In turn, these relatively large flux uncertainties cause significant scatter of the sources within the simulated distributions.
In Figure \[fig:lognlogs\], we additionally compare the ${\rm log} N
- {\rm log} S$ derived from XDEEP2 to the distributions found in previous wide and deep [*Chandra*]{} surveys \[CDF-N (Bauer et al. 2004); Extended-CDF-S (Lehmer et al. 2005); [*Chandra*]{}-COSMOS (@puccetti09)\]. In the flux range $f_X \sim (0.09$–$20)
\times 10^{-15} \ergpcmsqps$, we find excellent agreement with these previous surveys. We confirm previous results (e.g., Luo et al. 2008), that the CDF-N field may be subject to mild cosmic variance, as it appears to over-estimate (a factor $\sim 1.5$–4) the number count distribution of sources with $f_X \gtrsim 10^{-14}
\ergpcmsqps$. Furthermore, using the X-ray background (XRB) synthesis models of [@gilli07], we have simulated the expected ${\rm log} N
- {\rm log} S$ distribution of both obscured and unobscured populations of AGN with $N_H \sim 10^{20}$–$10^{25}$ cm$^{-2}$, $L_X
\sim 10^{38}$–$10^{46} \Lsun$ in the redshift range $z \sim 0$–8. In accordance with previous surveys, we consistently underestimate the number counts of AGN with $f_X \lesssim 8 \times 10^{-15} \ergpcmsqps$ in comparison to that expected from the XRB (see upper panel of Figure \[fig:lognlogs\]), suggesting that many heavily obscured sources are still being missed in even the most sensitive surveys. Indeed, multi-wavelength studies of deep and wide field X-ray surveys find a large population of seemingly obscured AGN which remain undetected using X-ray data alone (e.g., @alonso06 [@donley07; @daddi07; @melendez08a; @fiore09; @brusa10; @goulding11; @georgantopoulos11; @dma11]). However, for XDEEP2 sources with $f_X \gtrsim 2.5 \times 10^{-14}
\ergpcmsqps$, we find a mild ($\approx 30$–50%) systematic offset from previous X-ray surveys (e.g., E-CDFS; C-COSMOS), resulting in number counts closer to those predicted by XRB models, although the results from each of these surveys are all consistent at the 90% significance level.
Optical DEEP2 & X-ray XDEEP2 source identification {#sec:XOPT}
==================================================
By design, the XDEEP2 [*Chandra*]{} survey is within the same spatial region as the DEEP2 Galaxy Spectroscopic Redshift survey fields. In this section we identify optical counterparts to the sources in the XDEEP2 catalog using a custom Bayesian style analysis. For DEEP2, optical $B$, $R$ and $I$-band photometry was obtained with the Canada-France-Hawaii Telescope (CFHT) 12k camera. The main photometric catalog contains over $>710$,000 sources with a typical absolute astrometric accuracy of $\sim 0.2$ arc-seconds and is complete to $R_{\rm AB} \sim 25.2$ (see @coil04). In Table \[tbl\_opt\_counter\] we show the breakdown for the approximate number of optical sources within the XDEEP2 survey fields. In DEEP2 Field 1, all galaxies which have magnitudes of $R_{\rm AB} < 24.1$ were targeted for spectroscopy using the DEIMOS spectrograph on Keck (see @davis03 for further information on the observational setup of DEEP2). However in Fields 2–4, only those galaxies which meet both a simple $BRI$ color-cut threshold and have magnitudes of $R_{\rm AB}
< 24.1$ were targeted. The 4th data release of the DEEP2 spectroscopic catalog contains 50,319 unique sources [@newman12].
A Bayesian optical–X-ray matching routine
-----------------------------------------
![Fraction of counterparts found between the DEEP2 optical catalog and 100 randomly simulated X-ray catalogs. We find that the fraction of spurious matches decreases rapidly as a function of the probability threshold ($P({\rm match})$) calculated in our Bayesian-style matching algorithm. For $P({\rm match}) = 0.46$, we expect a spurious matching fraction of $\lesssim 6$% (dashed-line).[]{data-label="fig:optXspurious"}](figure20.png){width="0.97\linewidth"}
Given the unique observational construction of the combined XDEEP2 survey, in that it is both relatively shallow in wide areas, while simultaneously being extremely deep in smaller regions across the fields, we require a method of source matching which will account for changes in both the optical and X-ray source densities and statistically associate bright X-ray sources in the shallow fields with optical counterparts which are likely to contain bright AGN (i.e., QSOs). To this end, we have extended the Bayesian source-matching algorithm of Brand et al. (2006) to now include the X-ray source density and the properties of the candidate optical counterparts. Briefly, this method uses Bayesian-style statistics to calculate the probability of a random association occurring between two counterparts given the angular and magnitude distributions of the optical sources in a specific region of the sky. Simultaneously this algorithm accounts for the distribution of matching radii appropriate for a given off-axis position of the X-ray source in a [*Chandra*]{} observation. Furthermore, we allow modifications to the optical source positions, assuming a Gaussian probability based on the centroid and astrometric error of the DEEP2 data. As stated previously, median offsets between DEEP2 and XDEEP2 have been removed a-priori (see § \[sec:astrocalib\]). We use a Gaussian prior based on the characteristics of the [*Chandra*]{} PSF for the positional uncertainty of the X-ray source to derive the probability, $f$ of an X-ray source having an optical counterpart within the catalog (i.e., the survey mean completeness). We combine these posterior assumptions with information specific to the X-ray source (total counts, background level, proximity to other X-ray sources) and the optical properties (star, normal galaxy, quasar etc.) of possible counterparts to assign likelihood association probabilities between pairs of sources. In our new implementation of the algorithm, the probability of identifying an X-ray source $i$ with optical source $k$ is then,
$$P_{ik,match} = f \frac{M_{ik}}{B_{k}} F_{ik} O_{ik} \left[ (1-f) + f \sum \limits_{l=1}^{n_{i}} \sum \limits_{k=1}^{n_{j}} \frac{M_{il}}{B_l} F_{jk} \right]^{-1}$$
where $M_{ik}$ is the simple Gaussian probability of associating an X-ray source $i$ with an optical counterpart $k$ at a given separation including the X-ray and optical positional uncertainties; $B_{k}$ are the Poisson-idealized number counts as a function of optical magnitude within a region encompassing the X-ray position, in effect, $B_{k}$ accounts for both the changing $R$-band magnitude depth and source density within the optical DEEP2 catalog; $F_{ik}$ is the probability that an X-ray source of a given flux and flux limit has an optical association which is then marginalized over the $R$-band magnitude of the proposed optical counterpart; and $O_{ik}$ is the probability function containing the optical classification of the source, and is essentially a weighting based on the probabilistic galaxy classification of the source ($P({\rm gal})$ of 0 ($=$star) to 1 ($=$galaxy) defined in Coil et al. 2004) derived from the optical photometry and SED fitting. We determine the priors for $F_{ik}$ by randomly selecting from a cumulatively summed set of Poisson distributions in Markov-Chain simulations of the X-ray and optical catalogs. For computation speed, we limit the counterpart selection to only optical sources detected in the $R$-band. This also conforms with the selection method used to determine targets for optical spectroscopy. We note here, that while this method increases our ability to include optical sources with R-band magnitudes fainter than the completeness limit of the DEEP2 survey ($R \sim 25.2$ mags), the identification of X-ray sources with optically-faint counterparts is still incomplete at $R \gtrsim 25.2$ (e.g., @dma01 [@brusa10]).
![Positional offset between optical and X-ray positions for the 2126 XDEEP2 X-ray sources with secure DEEP2 optical counterparts found using our Bayesian-style matching algorithm. The spread in residuals is approximately Gaussian across all four DEEP2 fields with a mean positional offset of $\Delta_{\alpha,\delta} < 0.45$ arc-seconds between the X-ray and optical source catalogs.[]{data-label="fig:radecoffset"}](figure21.png){width="0.97\linewidth"}
To compute the probability threshold required to accept the optical source as a counterpart to the X-ray source and to quantitatively assess the false association fraction, we simulated mock XDEEP2 catalogs and compared them to the optical DEEP2 catalog. Following Brand et al. (2006), we randomized the positions of the XDEEP2 sources by $\pm 30''$ offsets and compared the number of false matches produced. In Figure \[fig:optXspurious\] we show the behaviour of the fraction of spurious counterparts for a given matching probability threshold ($P_{\rm match}$) produced by our association routine. We find that using $P_{\rm match} = 0.46$ produces one spurious optical counterpart for $\lesssim 6$% of the X-ray sources in the randomized catalogs (see Figure \[fig:optXspurious\]). The spurious counterpart fraction of $\lesssim 6$% is chosen specifically to be consistent with that found for the previous [*AEGIS-X*]{} catalog which was matched using the Maximum Likelihood technique (see @civano12, and references there-in); in turn, this also allows for further comparison between the catalogs. In 100 Markov-Chain Monte-Carlo (MCMC) simulations, we find that the spurious fraction remains relatively constant for $P_{\rm match} > 0.46$ across all four XDEEP2 fields with an overall dispersion of $<1$% within the MCMC simulations.
[ccccccc]{} & & & & & &\
1 & 1720 & $\sim 100,200$ & 1183 & 68.8 & -0.02 & 0.01\
2 & 342 & $\sim 119,400$ & 254 & 74.3 & 0.05 & 0.02\
3 & 528 & $\sim 146,100$ & 381 & 72.2 & 0.04 & 0.04\
4 & 386 & $\sim 145,300$ & 308 & 79.8 & 0.04 & -0.04\
–\
$^{a}$XDEEP2 field number\
$^{b}$Number of X-ray sources in the XDEEP2 field\
$^{c}$Approximate number of optical sources in the XDEEP2 field region\
$^{d}$Number of X-ray sources with secure optical counterparts\
$^{e}$Percentage fraction of X-ray sources with secure optical counterparts\
$^{f}$Median positional offsets between the DEEP2 optical source co-ordinates and the X-ray source co-ordinates in arc-seconds
In Figure \[fig:radecoffset\] we present the offset in astrometric co-ordinates between the X-ray source position and that of the optical counterpart from the XDEEP2 catalog. We find that the spread in positional offsets is approximately Gaussian across all four DEEP2 fields with a mean positional offset of $\Delta_{\alpha,\delta} <
0.45$ arc-seconds with an approximately zero systematic offset between the two catalogs. This mean offset is consistent with that found in previous deep-wide surveys (e.g., C-COSMOS with 0.81” for 90% of the sources; @elvis09 [@civano12]) Furthermore, we find that the positional offset between the X-ray source and optical counterpart appears to be a moderately-strong function of the ACIS-I off-axis position with on-axis ($< 1.5'$) and off-axis ($> 6'$) X-ray sources having median offsets of $\sim 0.28''$ and $\sim 0.96''$, respectively.
![R-band (AB) magnitude versus full-band (0.5–7.0 keV) flux for all XDEEP2 sources. X-ray sources are divided between those with galaxy probabilities ($P(gal)$) $> 0.3$ (i.e., optically extended sources; open circles) and $< 0.3$ (i.e., point-like sources; open stars). X-ray sources which lack optical counterparts are shown with upper-limits at $R=25.2$, i.e., the magnitude-limit of the DEEP2 catalog. Additionally, constant X-ray–optical flux ratios ($f_X/f_O$) are shown for log$(f_X/f_O) = \{-2.0;-1.0;0.0;1.0\}$, calculated using the relation of McHardy et al. (2003).[]{data-label="fig:fx_rmag"}](figure22.png){width="0.97\linewidth"}
[ccccccccccccc]{} & & & & & &\
& & & & & & & & & & & &\
& & & & & & & & & &\
aeg1\_001 & 1 & 214.78246 & 52.99710 & - & - & - & - & - & - & - & - & -\
aeg1\_002 & 1 & 214.78334 & 53.00712 & 13036677 & 214.78314 & 53.00728 & 0.72 & 0.5646 & 3 & 21.62 & 20.62 & 20.10\
aeg1\_003 & 1 & 214.79521 & 52.98033 & 13027633 & 214.79494 & 52.97998 & 1.38 & 0.7309 & 0.55 & 26.23 & 23.32 & 21.93\
aeg1\_004 & 1 & 214.79699 & 53.05600 & 13036612 & 214.79712 & 53.05598 & 0.29 & - & 3 & 20.31 & 18.55 & 17.91\
aeg1\_005 & 1 & 214.83506 & 53.04790 & - & - & - & - & - & - & - & - & -\
aeg1\_006 & 1 & 214.83600 & 53.00792 & 13036601 & 214.83597 & 53.00809 & 0.63 & - & -2 & 16.67 & 16.36 & 16.23\
aeg1\_007 & 1 & 214.84506 & 53.02555 & 13035495 & 214.84502 & 53.02559 & 0.20 & - & 3 & 24.37 & 24.62 & 23.94\
aeg1\_008 & 1 & 214.85376 & 52.99871 & 13027346 & 214.85321 & 52.99912 & 1.90 & - & 3 & 24.13 & 23.49 & 23.26\
aeg1\_009 & 1 & 214.85694 & 53.00549 & 13100779 & 214.85669 & 53.00582 & 1.28 & - & 3 & 24.69 & 24.47 & 24.27\
aeg1\_010 & 1 & 214.85765 & 53.01971 & 13035756 & 214.85777 & 53.02011 & 1.45 & - & -2 & 22.79 & 21.16 & 20.41\
aeg1\_011 & 1 & 214.86239 & 53.03122 & 13035995 & 214.86253 & 53.03141 & 0.73 & - & 3 & 23.59 & 21.68 & 21.09\
aeg1\_012 & 1 & 214.86615 & 53.02515 & - & - & - & - & - & - & - & - & -\
aeg1\_013 & 1 & 214.86670 & 52.97822 & 13027372 & 214.86674 & 52.97823 & 0.10 & 0.5608 & 0.81 & 23.00 & 22.87 & 22.61\
aeg1\_014 & 1 & 214.87337 & 53.03977 & 13035981 & 214.87335 & 53.03982 & 0.19 & - & 3 & 25.15 & 23.66 & 22.84\
aeg1\_015 & 1 & 214.87634 & 53.04383 & 13035650 & 214.87601 & 53.04362 & 1.03 & 0.3722 & 3 & 26.80 & 23.31 & 21.95\
aeg1\_016 & 1 & 214.87842 & 53.00748 & 13035444 & 214.87830 & 53.00769 & 0.81 & - & 1.00 & 26.06 & 23.14 & 22.27\
aeg1\_017 & 1 & 214.87886 & 52.98781 & 13027475 & 214.87888 & 52.98786 & 0.20 & - & 0.00 & 23.15 & 20.26 & 18.03\
aeg1\_018 & 1 & 214.88704 & 53.04167 & - & - & - & - & - & - & - & - & -\
aeg1\_019 & 1 & 214.88706 & 52.99963 & 13027149 & 214.88704 & 52.99970 & 0.25 & - & 3 & 24.25 & 23.77 & 22.93\
aeg1\_020 & 1 & 214.88917 & 53.09005 & - & - & - & - & - & - & - & - & -\
–\
$^{a}$XDEEP2 unique source identifier\
$^{b}$XDEEP2 field number\
$^{c}$X-ray source position in J2000 co-ordinates (degrees)\
$^{d}$DEEP2 optical source identifier (Coil et al. 2004)\
$^{e}$Optical source position in J2000 co-ordinates (degrees)\
$^{f}$Angular separation between optical and X-ray source positions (arc-seconds)\
$^{g}$Redshift of optical counterpart\
$^{h}$Bayesian probability of being a galaxy based on $R$-band image ($\leq 0$: star/compact; $\geq 1$: galaxy/extended; see Coil et al. 2004)\
$^{i}$Optical photometry in the $B$, $R$ and $I$-bands (AB-magnitude)
X-ray–optical source properties
-------------------------------
Of the 2976 X-ray sources in XDEEP2, we find that 2126 ($\approx 71.4
\pm 2.8$%) have at least one secure optical counterpart in the DEEP2 optical catalog. Multiple candidate counterparts are found for $\approx 11$% of the X-ray sources in XDEEP2. When multiple optical counterparts are associated with one X-ray source, we accept the DEEP2 optical counterpart with the largest $P_{\rm match}$. Given the cumulative distribution of $P_{\rm match}$ found for the XDEEP2 counterpart catalog, we expect a final spurious counterpart fraction of $\approx 4$%. In Table \[tbl\_opt\_counter\] we show the breakdown by field of the number of X-ray sources with optical counterparts, the percentage identified and the median positional offset between the optical and X-ray source positions. We find that 943 ($\approx
75.1$%) of the X-ray sources in Fields 2–4 have secure optical counterparts compared with 1183 ($\approx 68.8$%) in Field 1. This higher fraction of secure counterparts in Fields 2–4 is, in all likelihood, due to the relatively shallow exposure of the [ *Chandra*]{} observations in Fields 2–4 compared to those in Field 1, and hence, brighter X-ray sources tending towards bright optical host galaxies (i.e., X-ray-to-optical flux ratios $\sim 1$–10) which has been found previously in very shallow wide-field X-ray surveys (e.g., @maccacaro88 [@stocke91; @akiyama00; @lehmann01; @murray05]). Indeed, AGN and QSOs are typically found to have similar ratios of $-1
<$ log$(f_X/f_O) < +1$ (e.g., @schmidt98 [@akiyama00; @lehmann01]). In Figure \[fig:fx\_rmag\] we show the full-band X-ray flux versus the DEEP2 $R$-band magnitude for the sources with secure optical counterparts. We illustrate approximate X-ray-to-optical flux ratios for the sources assuming the relation of @mchardy03, and we divide the sample between those optical sources identified in DEEP2 to be extended/galaxy ($P({\rm
gal}) > 0.3$; see @coil04 [@newman12]) and point-like sources (stellar or QSO; $P({\rm gal}) < 0.3$). Of the 1559 optically extended X-ray-optical sources, $\approx 90$% (1425) have log$(f_X/f_O) > -1$, suggesting a significant fraction are bright AGN. We also find that 77 X-ray sources are also detected with very low X-ray-to-optical flux ratios (i.e., log$(f_X/f_O) < -2$. These X-ray sources generally include normal galaxies, stars, and low-luminosity AGN, and as we show in Figure \[fig:fx\_rmag\], all 77 X-ray–optical sources with $R_{\rm AB} < 18$ are point-like suggesting a stellar origin for the X-ray emission. As is clearly evident from the distribution of galaxies in Figure \[fig:fx\_rmag\], our X-ray–optical source matching becomes incomplete towards optically-faint ($R \gtrsim 25$) systems for $f_X \lesssim 6 \times 10^{-15} \ergpcmsqps$ due to the flux-limit of the optical DEEP2 data when compared to the depth of the X-ray observations within Field 1.
![Redshift histograms for all 510 XDEEP2 galaxies with optical spectroscopic counterparts (solid-line) and all optical DEEP2 galaxies (dashed-line). The optical DEEP2 distribution is divided by a factor for 60 for ease of comparison to the X-ray sources. On the top-axis we show the present-day look-back times as a function of redshift, with $z=0$ equivalent to $\tau_{lb} = 0$.[]{data-label="fig:redshift_hist"}](figure23.png){width="0.97\linewidth"}
We also consider the $\approx 450,000$ optical galaxies identified in the CFHT Legacy Survey Deep 3 (CFHTLSD-3) field, which covers an area roughly coincident with the AEGIS 1–3 sub-fields and is complete to $i'_{\rm AB} < 27.0$ with sources detected down to $i'_{\rm AB} \sim
28.6$ (@ilbert06; i.e., complete to $\sim 2$ magnitudes deeper than DEEP2). We find that 1009 X-ray sources have optical counterparts in the CFHTLSD-3 and 228/1009 were not previously identified in the DEEP2 catalog. Of the 228 X-ray sources, which were not previously identified to have optical DEEP2 counterparts, 163/228 have $i'_{\rm AB} > 24.4$ and 118 have $i'_{\rm AB} > 25.0$. With the subsequent inclusion of the CFHTLSD-3, we find an X-ray–optical counterpart fraction of $\approx 82$% within Field 1.
In Table \[tbl\_xopt\_counter\] we provide the matching optical DEEP2 counterpart information for the entire XDEEP2 catalog (e.g., X-ray name; X-ray position; optical DEEP2 counterpart; positional offset; basic optical properties). Furthermore, to guide future multi-wavelength surveys, we additionally include the spectroscopic redshift information from the recently released DEEP2 DR4 catalog (Newman et al. 2012). Of the 2126 X-ray sources with optical counterparts, 700 are included as part of the spectroscopic catalog and 510/700 have secure extragalactic redshifts, with the majority in the range $0.3 \lesssim z \lesssim 1.4$ and the highest redshift source at $z \sim 3.04$. We show in Figure \[fig:redshift\_hist\] that the X-ray sub-sample follows a similar redshift distribution to the main parent DEEP2 redshift catalog. Hence, in redshift terms, the X-ray sources may be considered a representative sample of the overall galaxy population in DEEP2. Future in-depth analyses of the AGN and galaxy redshift populations will allow us to understand the AGN clustering properties and possible correlations of AGN presence and large-scale structures.
Summary {#sec:summary}
=======
We have presented the X-ray source catalog and basic analyses of sources detected in the $\approx 10$ks–1.1 Ms [*Chandra*]{} ACIS-I observations of the four X-ray DEEP2 (XDEEP2) survey fields. The total area of XDEEP2 is $\sim 3.2$ deg$^2$, and to date is the largest medium-deep [*Chandra*]{} X-ray survey constructed. Using wavelet decomposition software ([wvdecomp]{}), we detected X-ray point sources in the individual (non-merged) events and overlapping merged images in the 0.5–2 keV (soft-band \[SB\]), 2–7 keV (hard-band \[HB\]) and 0.5–7 keV (full-band \[FB\]) energy ranges, complete to a false-source probability threshold of $1 \times 10^{-6}$. When considering the survey regions where at least 10% of the area is sensitive, the flux limits in the merged observations are $f_{X,FB} >
2.8 \times 10^{-16} \ergpcmsqps$, $f_{X,FB} > 4.5 \times 10^{-15}
\ergpcmsqps$, $f_{X,FB} > 4.6 \times 10^{-15} \ergpcmsqps$ and $f_{X,FB} > 4.6 \times 10^{-15} \ergpcmsqps$ in XDEEP2 Fields 1, 2, 3 and 4, respectively. The full XDEEP2 point source catalog contains 2976 sources, with 1720, 342, 528 and 386 sources in Fields 1–4. For the detected sources, we have presented the flux band ratio ($f_{\rm
HB} / f_{\rm SB}$) distributions. Consistent with previous results, we confirm that low flux X-ray sources tend towards higher flux ratios ($f_{\rm HB} / f_{\rm SB} \sim 2$–10), consistent with that expected for flatter spectral slopes with $\Gamma \sim 1.2$–1.4.
We have performed a rigorous comparison between our new catalog of Field 1 and that previously presented in Laird et al. (2009). Our new catalog now contains the more recent 600 ks observations of three sub-fields within Field 1. We find excellent agreement between the two catalogs, and show that 96% of the sources identified in the previous catalog, using a substantially different detection technique, are also identified in the new catalog of Field 1. Through extensive source detection simulations, we suggest that the small $\approx 4$% discrepancy between the catalogs can be mainly attributed to our conservative removal of low-significance and possibly spurious sources. Indeed, with the inclusion of the low significance X-ray sources, we show that $\sim 99$% of the sources identified by Laird et al. would be identified here. Furthermore, we present a comparison between the [*Chandra*]{} Source Catalog (CSC) and the X-ray sources identified in the more shallow 10ks Fields 2, 3 and 4. We find that $\sim 41.9 \pm 2.2$% of the XDEEP2 sources within these fields are included in the CSC. The vast majority ($\approx 90$%) of the XDEEP2 sources not identified in the CSC fall below their conservative detection threshold. We have presented the combined log N – log S distribution of soft-band detected sources identified across the XDEEP2 fields; the distribution shows excellent agreement with the Extended [*Chandra*]{} Deep Field and [*Chandra*]{}-COSMOS fields to $f_{\rm X,0.5-2keV} \sim 2 \times 10^{-16} \ergpcmsqps$. Given the large survey area of XDEEP2, we additionally place relatively strong constraints on the log N – log S distribution at high fluxes ($f_{\rm
X,0.5-2keV} > 2 \times 10^{-14} \ergpcmsqps$), and find a small systematic offset (a factor $\sim 1.5$) towards lower source numbers in the high-flux regime than observed previously in smaller area surveys. The number counts for sources with $f_{\rm 0.5-2keV} > 2
\times 10^{-14} \ergpcmsqps$ are in close agreement with the X-ray background synthesis models of Gilli et al. (2007). However, based on our careful analyses of the uncertainty associated with the log N – log S distribution, derived through the use of a Monte-Carlo simulation, we find that at the 90% level we cannot reject the number count distribution predicted by the previous surveys.
We have additionally built upon a previous Bayesian-style method for associating the X-ray sources with their optical counterparts [@brand06] in the DEEP2 photometric catalog (complete to $R_{\rm
AB} < 25.2$; Coil et al. 2004), and find that 2126 of the X-ray sources presented here ($\approx 71.4 \pm 2.8$%) have at least one secure optical counterpart. However, due to the much deeper X-ray exposure regions, we find a lower fraction of optical counterparts in Field 1 ($\approx 68.8$%) compared with Fields 2–4 ($\approx
75.1$%). We have additionally presented the optical photometric properties of the X-ray sources, the X-ray-to-optical ratios and find that the XDEEP2 sample have a similar redshift distribution to the main optical DEEP2 parent catalog, in the range $0 < z < 3$.
We would like to thank the anonymous referee for their considered and comprehensive report, which has allowed us to greatly improve and qualify many aspects of the X-ray catalog and analysis. We are thankful to K. Nandra, F. Civano and N. Wright for helpful discussions that have allowed us to clarify our analyses throughout the manuscript. We are also grateful to B. Lehmer for kindly providing data from the Extended Chandra Deep Field. This research has made use of data obtained from the Chandra Source Catalog, provided by the Chandra X-ray Center (CXC) as part of the Chandra Data Archive.
[*Facilities:*]{} .
[^1]: [chav]{} is available at http://hea-www.harvard.edu/[\~]{}alexey/CHAV/
[^2]: [ciao]{} is available at http://cxc.harvard.edu/ciao/download/
[^3]: We use the [idl]{} routine [mpfit2dpeak]{}, available in the Markwardt software package, to fit the 2-D Gaussian profiles.
[^4]: We note that the analyses presented here now include the new 600ks observations in the sub-fields EGS-3, EGS-4 and EGS-5.
[^5]: see http:$/$$/$cxc.harvard.edu$/$cal$/$ASPECT$/$celmon$/$
[^6]: In [ *AEGIS-X*]{} the ultra-hard band is defined in the energy range 4–7 keV.
[^7]: As noted previously, the cluster candidates and their properties will be discussed in detail in a forthcoming publication.
|
---
abstract: 'The adiabatic chiral magnetic effect (CME) is a phenomenon by which a slowly oscillating magnetic field applied to a conducting medium induces an electric current in the instantaneous direction of the field. Here we theoretically investigate the effect in a ballistic Weyl semimetal sample having the geometry of a slab. We discuss why in a general situation the bulk and the boundary contributions towards the CME are comparable. We show, however, that under certain conditions the adiabatic CME is dominated by the Fermi arc states at the boundary. We find that despite the topologically protected nature of the Fermi arcs, their contribution to the CME is neither related to any topological invariant nor can generally be calculated within the bulk low-energy effective theory framework. For certain types of boundary, however, the Fermi arcs contribution to the CME can be found from the effective low energy Weyl Hamiltonian and the scattering phase characterising the collision of a Weyl excitation with the boundary.'
author:
- Artem Ivashko
- Vadim Cheianov
- 'Jimmy A. Hutasoit'
bibliography:
- 'references-my.bib'
title: |
Non-universality of the adiabatic chiral magnetic effect\
in a clean Weyl semimetal slab
---
Introduction
============
Weyl semimetals (WSMs) are crystalline materials in which the low-energy electronic excitations are described by the Weyl Hamiltonian originating in the theory of massless relativistic fermions in four-dimensional space-time. Such materials were hypothesised more than three decades ago [@nielsen1983adler], then in the course of the last decade several chemical compounds were investigated as candidates [@murakami2007phase; @wan2011topological; @xu2011chern; @burkov2011weyl; @weng2015weyl; @huang2015weyl] culminating in 2015 in photoemission experiments showing quasi-Weyl dispersion of elementary excitations in a semi-metal [@xu2015discovery; @lv2015experimental; @lv2015observation; @yang2015weyl] (see also Refs. and for recent reviews). A typical WSM features an even number of singular points in its Brillouin zone in whose vicinity the effective single-particle Hamiltonian can be written as [@hasan2017discovery; @armitage2017weyl] $$\label{eq:linear-Hamiltonian-Weyl}
\mathcal{H}_\text{eff} = {\varepsilon_\mathrm{diag}}(\bm p) {\mathbb{1}_{2\times 2}} + \chi (\bm p - \bm p_0) \hat v \bm \sigma,$$ where $ \bm p_0$ is the singular point called the “Weyl node”, $\bm\sigma = (\sigma^x,\sigma^y,\sigma^z)$ is the “pseudospin”, which does not necessarily coincide with the electron’s spin, even though it behaves like a spin under the discrete spacetime symmetries and spatial rotations, and ${\varepsilon_\mathrm{diag}}= {\varepsilon_\textsc{w}}+ \bm {v_{\mathbb{1}}}(\bm p - \bm p_0)$ is a scalar part of the energy. We assume that the tensor $\hat v$ is positive definite which enables us to introduce the chirality number $\chi=\pm 1$ characterising each Weyl node. We shall call the nodes having $\chi=1$ right-chiral and those having $\chi=-1$ left-chiral.
A characteristic macroscopic signature of the Weyl spectrum is the hypothetical Chiral Magnetic Effect (CME). The CME was originally predicted in 1980 [@vilenkin1980equilibrium] for ultra-relativistic plasmas, and later on it was discussed in the context of heavy-ion collisions [@fukushima2008chiral; @kharzeev2011testing; @kharzeev2014chiral], the early Universe [@joyce1997primordial; @boyarsky2012long; @boyarsky2012self; @ivashkoCMEmass], and relativistic magnetohydrodynamics in general [@dvornikov2015magnetic; @sigl2016chiral; @boyarsky2015magnetohydrodynamics]. In its simplest form, the CME is a phenomenon by which an electric current develops in the direction of a static magnetic field applied to a system in thermal equilibrium. The CME requires that the system possesses an additional conserved parity-odd charge, which in the case of the Weyl Hamiltonian is the difference between the number of right-chiral and left-chiral particles. If the plasma is prepared in a thermal state such that the right-handed and the left-handed particles have different chemical potentials, ${\mu_\textsc{r}}$ and ${\mu_\textsc{l}},$ then the application of a magnetic field $\mathbf B$ should result in the current density $\bm j = ({\mu_\textsc{r}}- {\mu_\textsc{l}}) \mathcal C \bm B $ where $\mathcal C = e^2/h^2c.$ [@kharzeev2014chiral] (For simplicity, we restrict ourselves to a model with only two Weyl nodes throughout the paper.)
The newly discovered WSMs seem to be natural test beds for the observation of the CME. However such an experimental program is not without a problem. Indeed, in a realistic sample of a solid-state material the chirality quantum number is neither protected against impurity scattering nor preserved in collisions with the sample boundary. Therefore continuous external driving is required in order to maintain the imbalance ${\mu_\textsc{r}}- {\mu_\textsc{l}}\neq 0$ [@vazifeh2013electromagnetic; @yamamoto2015generalized]. One way to achieve this is to apply an electric field $\bm E$ parallel to the magnetic field, $\bm E \parallel \bm B$. In such a case, the mechanism responsible for the driving is the chiral anomaly [@adler1969axial; @bell1969pcac], and it is believed to be the primary cause of the negative longitudinal magnetoresistance which is observed in transport experiments on WSMs [@nielsen1983adler; @son2013chiral; @burkov2014chiral; @li2016chiral; @zhang2016signatures]. It is worth noting, however, that the intrinsic effect of chiral anomaly can be masked by the other effects, e.g. related to the geometry of the measuring setup or the spatial variations of the sample conductivity, see Refs. and . Moreover, the negative longitudinal magnetoresistance was claimed to be observed in 3D materials *without* any Weyl nodes [@ganichev2001giant; @wiedmann2016anisotropic; @li2016negative; @li2016resistivity; @liang2016pressure; @luo2016anomalous; @assaf2017negative].
Another way to drive the system out of equilibrium is to make the magnetic field itself time-dependent, $\bm B(t) = \bm{B_\textsc{AC}}\cos \omega t$. Recent theoretical studies [@chen2013axion; @goswami2015optical; @chang2015chiral; @ma2015chiral; @zhong2016gyrotropic; @alavirad2016role] converge in their conclusion that in a clean infinite sample such a perturbation will lead to the CME of the form $$\label{eq:CME-AC}
\bm j = {\mathcal{C}_\textsc{cme}}\frac{e^2}{h^2c} b_0 ~ \bm{B_\textsc{AC}}\cos \omega t,$$ where $b_0$ is the energy separation between the right-chiral and left-chiral Weyl nodes, $b_0 = {\varepsilon_\textsc{wr}}- {\varepsilon_\textsc{wl}}$. Note that the proportionality coefficient on the right-hand side of Eq. is frequency-independent, therefore the formula predicts the effect in the adiabatic $\omega \to 0 $ limit. We shall call such a CME adiabatic.
In a realistic sample the applicability of Eq. is limited by a number of factors. Arguably, the most important one is the rate $\Gamma$ of chirality relaxation due to the impurity scattering. In the frequency range $\omega/\Gamma \lesssim 1$ chirality relaxation should dominate therefore the CME should be suppressed. Another, less obvious limiting factor is the geometry of the sample. Any physical sample has a finite cross-section and a boundary. Eq. implies that the total CME current is proportional to the cross-sectional area $S_\perp$ of the sample and therefore, one may be tempted to think that the boundary effects would be irrelevant in samples with large cross-sectional areas. This turns out not to be the case [@pesin2016nonlocal; @baireuther2016scattering; @ivashko2017adiabatic].
In particular, the analysis of Ref. exploiting general symmetry constraints on the structure of the gradient expansion of the polarisation tensor implies that the contribution of the boundary layer to the CME current is always one half of the bulk contribution no matter how big the sample. An alternative approach [@baireuther2016scattering] based on microscopic analysis for a particular model of WSM arrives at a similar conclusion: the boundary contribution to the CME current is on the same order as the bulk contribution albeit the numerical coefficient is two rather than one half. These two results are quite remarkable in both their agreement as to the scale of the boundary effect, and their disagreement in regards to the numerical factor defining the actual value of the boundary current relative to the bulk. What is the reason for the discrepancy? The gradient expansion of kinetic coefficients used in Ref. implicitly assumes that these coefficients are (quasi)local. For ballistic systems, however, the low-frequency response is known to be highly non-local, which can be seen already from the fact that the limits $\omega \to 0$ and $k \to 0$ do not commute, for an unbounded sample [@ma2015chiral; @zhong2016gyrotropic; @pesin2016nonlocal]. ($k$ here is the wavevector of the magnetic field, for more details about the non-locality, see Ref. .) For the gradient expansion to work in a finite-size sample the frequency of the magnetic field has to be much greater than $v/L$ where $v$ is the typical speed of an elementary excitation and $L$ is the typical size of the sample’s cross-section. In contrast, the approach of Ref. is valid in the opposite low-frequency (adiabatic) limit [@ivashko2017adiabatic] outside the applicability range of the gradient expansion theory. In the present paper, we further investigate the boundary contribution to the CME current in the adiabatic limit in order to address the following questions a) Is the coefficient ${\mathcal C}_B=1$ in the boundary current $I_B ={\mathcal C}_B (b_0 e^2/h^2c) S_\perp B$ found in Ref. universal (possibly topologically protected)? b) If it is not, can it be nevertheless expressed in terms of the parameters of the effective low-energy theory including the Weyl Hamiltonian of the elementary excitations and the scattering matrix at the boundary? Our main finding is that the answer to both questions is generally “no” although under certain conditions the answer to question b) can be positive.
The paper is organized as follows. In Sec. \[sec:background\] we discuss the methods that we use for the analysis of the adiabatic CME, and the particular set-up. In Secs. \[sec:bulk-effective-theory\] and \[sec:surface-effective-theory\] we discuss the contributions of the bulk and the boundary to the adiabatic CME in the framework of effective low-energy theory. In Sec. \[sec:surface-microscopic-theory\], we take into account the contribution of boundary that is not captured by the effective theory, by using the same microscopic model as in [@baireuther2016scattering]. In Sec. \[sec:discussion\], we discuss our findings.
Methods and setup {#sec:background}
=================
For definiteness, we consider a sample having the geometry of a slab which is infinite in the $y-z$ plane and has thickness $L_\perp$ in the $x$ direction. We assume that the sample is in the state of thermal equilibrium at temperature $T=0,$ and we denote the Fermi energy ${\varepsilon_\textsc{f}}.$ The oscillating magnetic field is applied along the $z$-axis.
It is worth noting that we consider a sample geometry which is slightly different from the geometry of an infinite cylinder with a compact base investigated in Ref. . The original choice of Ref. was motivated by the considerations of numerical convenience in application of the following heuristic formula for the total electric current $I$ along the cylinder’s axis $$\label{eq:adiabatic-Kubo-current}
I =
\frac{e {B_\textsc{AC}}}{h} \sum\limits_\nu \int\limits_\text{BZ} dp ~ \theta({\varepsilon_\textsc{f}}-\varepsilon_\nu(p)) \frac{{\partial}^2 \varepsilon_\nu(p)}{{\partial}B {\partial}p}.$$ Here $p$ is the quasimomentum along the magnetic field, which runs over the one-dimensional Brillouin zone (BZ) of the cylinder, $\nu$ is an additional index that characterizes the energy levels. In the recent paper , Eq. was derived from the first-principle quantum-mechanical linear-response theory, where it was shown that the formula is applicable only for the *adiabatic* driving, meaning that the driving frequency $\omega$ is much less than the spacing between any pair of energy levels associated with a non-vanishing matrix element of the velocity or the magnetic moment operators.
For the slab geometry considered here, the index $\nu$ comprises the quasimomentum $p^y$ along the $y$-axis and some additional discrete index $n.$ In this case the relevant matrix elements between the states having either different $p^y$ or different $p$ vanish due to the translational invariance in $y$ and $z$-directions. As a result, adiabaticity can be broken only in transitions between different $n$. Note that in the limit of large thickness, $L_\perp \to \infty$ the level spacing between the states of different $n$ collapses, which leads to the breakdown of adiabaticity. One of the ways to restore adiabaticity in such a limit is to apply a large *static* background magnetic field $B_0$, which we choose to be directed along the $z$-axis, such that the total field is $B = B_0 + {B_\textsc{AC}}\cos \omega
t$. While the bulk Landau levels are separated by finite energy gaps on the order $v \sqrt{eB_0 \hbar/c}$ (see Sec. \[sec:surface-effective-theory\] for more details), it is in principle possible that for some surface states there is one or more pair of levels with a significantly smaller energy spacing. However, we expect this to occur very rarely as we change $p^y$ for a fixed $p$, since at the same time, these pairs of states must be close to the Fermi energy, in order to contribute to the current . (This expectation of the rare crossings is confirmed by the numerical calculations for a particular model used below.)
The adiabatic regime has an obvious advantage from both analytical and numerical points of view. Namely, in order to find the current $I$ it is enough to know the single-particle energy spectrum $\varepsilon_\nu(p)$, while in the non-adiabatic regime we need to calculate additionally the off-diagonal matrix elements of the velocity and the magnetic moment operators [@ma2015chiral; @ivashko2017adiabatic].
In order to separate the bulk and the surface components of the current from Eq. , we use the result of Ref. [^1] The surface current is found from the following formula $$\label{eq:surface-current-slab}
I_\text{surf} = {B_\textsc{AC}}\frac{e^2}{2h^2 c} S_\perp \int\limits_\text{BZ} dp
\sum\limits_{n,\pm} \left( v_n^z ~ {\text{Sgn}\,}\frac{{\partial}\varepsilon_n}{{\partial}p^y} \right) \Big|_{\varepsilon_n = {\varepsilon_\textsc{f}}} \rho_n(p),$$ where $S_\perp = L_\perp L_y$ is the area of the sample cross-section, $v_n^z =
{\partial}\varepsilon_n/ {\partial}p$ is the group velocity along the magnetic field, the sum goes over the states localized at the right ($+$) and the left ($-$) boundaries, $\rho_n(p) = 1$ if there exists a solution of $\varepsilon_n(p,p^y) = {\varepsilon_\textsc{f}}$ for given $n$ and $p$, and $\rho_n(p) = 0$ otherwise. (In order to deal with finite $S_\perp$, we have introduced a finite width $L_y$ in the $y$-direction, but we assume that this width is much larger that any other length scales in our problem.) The bulk current is given by the expression $$\label{eq:bulk-current}
I_\text{bulk} = \frac{e^2 {B_\textsc{AC}}}{h^2 c} S_\perp \sum\limits_n \sum\limits_{{p_\textsc{f}}} \frac{{\partial}\varepsilon_n({p_\textsc{f}})}{{\partial}B} {\text{Sgn}\,}v^z_n({p_\textsc{f}}),$$ where $\varepsilon_n(p)$ is the energy of the bulk levels, and we drop $p^y$ here owing to the fact that the bulk energy levels (Landau levels) are degenerate with respect to this quasimomentum. The sum goes over all solutions of the equation $\varepsilon_n({p_\textsc{f}}) = {\varepsilon_\textsc{f}}$. Note that both $I_\text{bulk}$ and $I_\text{surf}$ scale linearly with the area $S_\perp$.
Note that the surface CME contribution is different from the well-studied dia- or para-magnetic surface currents. First, the latter appear even in thermal equilibrium, in the absence of a time-dependent component $\delta \bm B$. Second, the total current through the cross-section calculated from dia-/para-magnetic current density $\bm j_\text{eq} = c ~ \bm\nabla \times \bm (\hat
\chi_\text{magn} \bm B)$ is zero. (Here $\hat \chi_\text{magn}$ is the magnetic susceptibility tensor.)
For our numerical analysis in Sec. , we employ the same microscopic model that was used in Ref. , which is a four-band tight-binding model with the single-particle Hamiltonian $$\label{eq:Vazifeh-Franz-Hamiltonian}
\mathcal{H}_\text{lattice} = \begin{pmatrix} \mathcal{H}_{11} & \mathcal{H}_{12} \\ \mathcal{H}_{12}^\dagger & \mathcal{H}_{22} \end{pmatrix},$$ where $$\begin{aligned}
\mathcal{H}_{11} &=& 2t(\sigma^x \sin p^x + \sigma^y \sin p^y) + \frac{\beta^z}{2} \sigma^z, \\
\mathcal{H}_{22} &=& - 2t(\sigma^x \sin p^x + \sigma^y \sin p^y) + \frac{\beta^z}{2} \sigma^z, \\
\mathcal{H}_{12} &=& - i t \sin p^z + M(\bm p) - i \frac{\beta_0}{2} \sigma^z, \\
M(\bm p) &=& M_0 + t (3 - \cos p^x - \cos p^y - \cos p^z).\end{aligned}$$ Here, $t$ describes the nearest-neighbour hopping, $\beta_0$ and $\beta_z$ are parameters that violate the inversion ${\mathcal{P}}$ and time-reversal ${\mathcal{T}}$ symmetries, respectively, $\sigma^x, \sigma^y,$ and $\sigma^z$ are the pseudospin operators, $\bm p$ is the quasimomentum. Breaking ${\mathcal{P}}$ is required in order to have non-vanishing difference of energies $b_0 = {\varepsilon_\textsc{wr}}- {\varepsilon_\textsc{wl}}$, and we are forced to break the time-reversal symmetry in order to deal with only two Weyl nodes. (The minimal number of nodes in presence of ${\mathcal{T}}$ is four [@armitage2017weyl].) The lattice has cubic unit cell, and for simplicity, we take the lattice spacing equal to 1, so that $\bm p$ is measured in units of $\hbar$.
Adiabatic bulk CME in the effective theory {#sec:bulk-effective-theory}
==========================================
In this Section, we study the adiabatic CME current in the framework of effective theory. First, we recall that in an idealised model of a Weyl semimetal neglecting both the momentum dependence of the scalar part ${\varepsilon_\mathrm{diag}}(\bm p)$ of the effective Hamiltonian and the gradient corrections to the linear spectrum, the bulk contribution to the CME is suppressed in the presence of a background magnetic field $B_0$ [@ivashko2017adiabatic]. This can be seen from inspecting the dispersion relations of Landau levels [@johnson1949motion; @berestetskii1982quantum] that enter Eq. , $$\label{eq:bulk-Landau-level-dispersion}
\varepsilon_n - {\varepsilon_\textsc{w}}= \begin{cases} -\chi v^z \delta p, & (n = 0) \\ {\text{Sgn}\,}n \cdot \sqrt{v_z^2 \delta p^2 + 2 |n v^x v^y|\frac{\hbar^2}{{l_\textsc{b}}^2}}. & (n \neq 0) \end{cases}$$ The index $n$ here is the number of the Landau level, which is the effective low-energy counterpart of the index $n$ introduced earlier in this paper, $\delta p = p - p^z_0,$ ${l_\textsc{b}}= \sqrt{eB_0/\hbar c}$ is the magnetic length. Since the energy of the $n=0$ level does not depend on the magnetic field, this level does not contribute to the current $I_\text{bulk}$, according to Eq. . Although the spectrum of the $n \neq 0$ levels involves the magnetic field, their energies are *even* with respect to the difference $p - p^z_0$, which means that they do not contribute to the bulk current either.
The vanishing of the bulk CME in a simplified model is accidental and it is not protected against various deformations of the Hamiltonian. We identify the following main factors that might lead to a non-vanishing bulk current in a more realistic model. Firstly, Eq. is only valid if the Landau quantization of energy levels is stronger than the finite-size quantization. This implies that the magnetic field $B_0$ needs to be strong enough to ensure the condition ${l_\textsc{b}}\ll L_\perp$. For weak background magnetic field violating this bound, the structure of energy levels becomes different, and an appreciable bulk current may develop, in agreement with Ref. . Secondly, the minimal effective Hamiltonian is applicable only in the long wavelength limit. Gradient corrections to this Hamiltonian will generally modify the dispersion relations in a way which will lead to a finite bulk CME current. We discuss such corrections in App. \[app:higher-order-corrections\], and present arguments as to why they are negligible under realistic assumptions. Finally, a violation of the assumption $\bm{v_{\mathbb{1}}}= \bm 0$ may also lead to a bulk current within the chosen model. This can be easily seen, for instance, in the situation of $\bm {v_{\mathbb{1}}}\parallel \bm B.$ It can, however, be shown that in this situation non-vanishing contributions from different Weyl points cancel if the system as a whole possesses time-reversal symmetry. Our reason for working with a model breaking time-reversal symmetry is that we want to compare our results with Ref. . It is purely accidental that in the parametric range used in Ref. the effective parameter $\bm {v_{\mathbb{1}}}$ turns out to be negligible thus emulating the effect of the time-reversal symmetry protected cancellation.
To conclude, the bulk adiabatic CME current is not related to the oscillating magnetic field in a universal way. Its material part, however, is small and its geometric part is controllable and can in principle be tuned to vanish.
Adiabatic boundary CME in the effective theory {#sec:surface-effective-theory}
==============================================
Next, we turn to the analysis of the surface contribution to the adiabatic CME trying to approach the problem from the bulk low-energy effective theory perspective. In order to describe a bounded system, one has to supplement the effective Hamiltonian in Eq. with the boundary condition on the single-particle wavefunction $\psi$. The generic condition for a boundary located at $x = x_\textsc{b}$ has the form [@witten2015three] $$\label{eq:boundary-condition-Weyl}
(\sigma^y \sin\Delta\phi + \sigma^z \cos\Delta\phi + 1) \psi\Big|_{x=x_\textsc{b}} = 0,$$ where $\Delta\phi$ has the meaning of the scattering phase shift at the surface. The effective low-energy theory offers no constraints on the parameter $\Delta \phi.$ The actual value of the phase shift depends on the microscopic detail of the boundary. In a sample having two boundaries, $x = \pm L_\perp/2$, each boundary is characterised by the condition with its own value of $\Delta\phi$. Moreover, each Weyl node has its own scattering phase shift, which means that in our particular setup we have four independent phase shifts in total. In the rest of the text, we denote them as $\Delta \phi^\pm_\textsc{l,r}$, where the upper index corresponds to the $x = \pm L_\perp/2$ boundaries, while L (R) denotes the left- (right-)chiral node.
In the presence of a constant background field $B_0$ the eigenvalues of the system’s Hamiltonian can be classified by the $z$-projection of quasi-momentum $p$ and the eigenvalue $p^y$ of the operator of magnetic translations in the $y$-direction. As is usual in the theory of Landau quantization [@brown1964bloch], an orbital characterised by a given $p^y$ is localised within a distance ${l_\textsc{b}}$ from the plane $x=p^y {l_\textsc{b}}^2/\hbar.$ The equation which defines the dispersion relation $\varepsilon = \varepsilon_n(p,p^y)$ for the effective Hamiltonian in the presence of a magnetic field and the boundary conditions is quite cumbersome. However, in the limiting case we are interested in, ${l_\textsc{b}}\ll L_\perp$, it can be simplified to the following form $$\begin{gathered}
\label{eq:boundary-state-spectrum-magnetic-field}
(\varepsilon - {\varepsilon_\textsc{w}}+ \chi v^z \delta p) D_{\lambda/2 -1}(\tilde p^y) = \\ = \pm
\sqrt{2} \chi \frac{v_\perp}{{l_\textsc{b}}} \tan\left( \frac{\Delta\phi^\pm}{2}\right) D_{\lambda/2}(\tilde p^y),\end{gathered}$$ where the $\pm$ index is chosen depending on whether the orbital is localised near the $x=L_\perp/2$ or $x=-L_\perp/2$ boundary. Here, we have used the following notation: $\lambda \equiv \left((\varepsilon - {\varepsilon_\textsc{w}})^2 -
v_z^2 \delta p^2 \right) {l_\textsc{b}}^2 / v_\perp^2$, $\tilde p^y \equiv \sqrt{2} p^y {l_\textsc{b}}/ \hbar \pm L_\perp /\sqrt{2} {l_\textsc{b}}$, and $D_\nu(x)$ is the parabolic cylinder function [@NIST:DLMF]. We have also chosen $v^x = v^y = v_\perp$.
Note that Eq. encodes the dispersion relation of *both* bulk and surface modes. For the states that are localized at the surface (within the length $\delta x \ll {l_\textsc{b}}$), in the absence of the magnetic field, turning on magnetic field does not affect the dispersion relation much. Generally, in a Weyl semimetal, there is at least one family of such surface states at the Fermi energy called the Fermi arc [@hasan2017discovery]. By tuning $p^y$ at fixed $p$, such a surface branch continuously transforms into one of the bulk Landau levels showing no energy dependence on $p^y.$ This behaviour is illustrated in Fig. \[fig:boundary-states-dispersion\], where Eq. is solved numerically for the $n=0$ state.
It was discussed in Ref. that the surface contribution to the adiabatic CME originates in the inflow of electric charge from the bulk to the boundary, which for every given $p$ is similar to the Hall effect arising in two-dimensional systems [@girvin9907002quantum]. In this picture, the chirality of the edge mode, which is defined as the sign of ${\partial}\varepsilon_n / {\partial}p^y$ at Fermi level, is linked to a topologically protected characteristic (Chern number) of the Weyl semimetal [@armitage2017weyl] in that both describe the direction of the inflow of charge (towards the boundary or away from it) at given value of $p.$ It is for this reason that the chirality of each edge mode enters as a multiplier in Eq. .
![Dependence of the energy $\varepsilon_{n=0}$ on $\tilde p^y$ for a right-chiral electron in strong static magnetic field $B_0$, computed from Eq. . From top to bottom: $\Delta\phi_\textsc{r}^- = -2.3, -1.49, -0.7$, and $-0.2$. $p = p^z_0$, while the other parameters of the effective theory are the same as in Eq. . The states with $\tilde p^y \lesssim -3.5$ are completely localized within the bulk (Landau levels), the other states are localized at the boundary $x = - L_\perp/2$.[]{data-label="fig:boundary-states-dispersion"}](boundary-states-dispersion){width="45.00000%"}
It follows from the above topological considerations that the support function $\rho_n(p)$ in Eq. is non-vanishing as long as the momentum $p$ is inside the region of the one-dimensional projection of the Brillouin zone where the Chern number is finite (this region is approximately bounded by the positions of the two Weyl points). A significant part of this region lies outside the applicability range of the Weyl Hamiltonian, therefore there is no general reason to believe that the integral on the right-hand side of Eq. can be calculated from the parameters of the effective low-energy theory. There still exists one noteworthy exception which is when the dispersion relation of Fermi arcs is separable, meaning that in the absence of magnetic field $$\label{eq:FA-separability}
\varepsilon_0(p,p^y) = \mathcal{G}_1(p) + \mathcal{G}_2(p^y),$$ where $\mathcal{G}_{1,2}$ are some arbitrary functions. Here and below we choose the index $n=0$ for the branch of topologically non-trivial surface states. Indeed for $B_0 = 0$, the contribution of the surface states is insensitive to what happens outside the vicinity of the Weyl nodes. This is because according to Eq. , we get the integral of total derivative, which reduces to the difference of the boundary values of the function $\mathcal{G}_1$ at the points $p = {p^z_\textsc{wr}}$ and $p =
{p^z_\textsc{wl}}$. These values themselves are fixed by the positions of the energies of the Weyl nodes ${\varepsilon_\textsc{wr}}$ and ${\varepsilon_\textsc{wl}}$, respectively, so that the partial contribution of the Fermi arc is $$\label{eq:Fermi-arc-current}
I^\textsc{fa}_\text{surf}(B_0=0) = \frac{e^2}{h^2 c} {B_\textsc{AC}}S_\perp ({\varepsilon_\textsc{wr}}- {\varepsilon_\textsc{wl}}).$$ This is in agreement with Ref. , where the contribution of these surface states to the total coefficient ${\mathcal{C}_\textsc{cme}}$ is argued to be equal to 1.
In case of non-vanishing $B_0$, the contribution to Eq. beyond the effective theory remains unchanged, therefore the integral again reduces to the contribution from the states at the vicinity of the Weyl nodes. The latter is modified by the magnetic field albeit in a way that is completely defined by the low-energy effective theory. In order to specify Eq. for this case, we note that within the effective theory, in the absence of the magnetic field, the energy of the Fermi arc is linear with respect to both $p$ and $p^y$ [@witten2015three]. It means that even in the case of $B_0 \neq 0$, the separability holds both in the effective theory and in the ultraviolet complete theory, provided that the deviation from the Weyl node is large enough, $\delta p \gtrsim \Lambda_0$. Therefore, one can artificially split the integration region in Eq. into two parts: the vicinities of the Weyl nodes, $\delta p \leq \Lambda$ ($\Lambda \gtrsim \Lambda_0$), and the rest. For the vicinities of the two Weyl nodes, we can still use the effective theory and we denote the result as $I_\text{eff}(\Lambda)$. Due to the linear dispersion, this contribution grows linearly with the cutoff $\Lambda$. In the remaining region, the separable relation holds. The corresponding contribution is a linear function of $\Lambda$, which is equal to at $\Lambda = 0$, and its slope is opposite to that of $I_\text{eff}$. Since the total surface CME current is not sensitive to the choice of the cutoff $\Lambda$, we can formally set it to infinity: $$\label{eq:surface-current-separable}
I_\text{surf} = \lim\limits_{\Lambda \to \infty} \left( I_\text{eff}(\Lambda) - \Lambda \frac{d I_\text{eff}(\Lambda)}{d\Lambda} \right) + I^\textsc{fa}_\text{surf}(B_0=0).$$
For the actual calculation using Eq. we have used the effective parameters of the bulk Hamiltonian $$\begin{aligned}
\label{eq:effective-parameters}
{\varepsilon_\textsc{wr}}= -{\varepsilon_\textsc{wl}}= 4.9 \times
10^{-2} t, \quad {\varepsilon_\textsc{f}}= 0, \\ v^z = 0.69 t/\hbar, \quad v_\perp = 2.0 t/\hbar, \quad {l_\textsc{b}}= 50,\end{aligned}$$ and the two Weyl nodes share the same values of $v^z$ and $v_\perp$. Here $t$ is an arbitrary parameter that has dimension of energy. The specific choice of all these input parameters was done in order to compare them (see below) with the results of a particular microscopic calculation and with the results of Ref. . We have found that even in the case of separable Fermi arcs, the resulting prefactor in depends on the choice of $\Delta \phi$: for $\Delta \phi^\pm_\textsc{r} = -\Delta \phi^\pm_\textsc{l} = \pm 1.49$, ${\mathcal{C}_\textsc{cme}}= 0.53$, while for $\Delta \phi^\pm_\textsc{r} = -\Delta \phi^\pm_\textsc{l} = \pm 2.3$, the result changes to ${\mathcal{C}_\textsc{cme}}= 0.89$. (As before, the $+$ index corresponds to the boundary $x = L_\perp/2$, while the $-$ index corresponds to the other boundary.)
To conclude, the ${\mathcal{C}_\textsc{cme}}$ is not universal in the sense that it depends on both the bulk and the *boundary* parameters of the effective theory, *even* in the case of separable energy of Fermi arcs. As we have noted above, if the separability does not hold, the resulting current $I_\text{surf}$ involves microscopic details of the material beyond the information encoded in the parameters of the effective theory. Therefore, in order to understand whether the separability is a generic property of a WSM, in the following Section we study a microscopic model of such a material.
Surface contribution in a microscopic theory {#sec:surface-microscopic-theory}
============================================
As we have established in the previous sections, the boundary CME is a significant effect, which under certain conditions dominates the longitudinal response of a WSM to the applied magnetic field. Although the Fermi arc states responsible for the effect are topologically protected, the magnitude of the boundary CME cannot be linked to any topological invariant. Moreover, the magnitude of the boundary CME is generally not fully determined by the parameters of the low-energy effective theory in the bulk material unless a special condition on the dispersion relation of the Fermi arc states is met. There is no obvious reason why this condition should hold for an arbitrary material interface therefore it is natural to assume that it is likely to be violated in a given experimental sample. Nevertheless, it is instructive to see how the condition breaks down in a particular microscopic model and what consequences this may have for the boundary CME.
To this end, we turn to the microscopic lattice Hamiltonian . The Bloch spectrum of the model contains one right-handed Weyl node located at $\bm p_0(\text{right}) = (0,0,{p^z_\textsc{wr}})$, and one left-handed Weyl node located at $\bm p_0(\text{left}) = - \bm p_0(\text{right}).$ The Bloch momentum ${p^z_\textsc{wr}}$ can be expressed in terms of the parameters of the Hamiltonian as described in Appendix \[app:appendix\]. By performing a unitary transformation from the original basis to the eigenbasis of the Hamiltonian at $\bm p = \bm p_0(\text{right})$, together with the linearisation with respect to $\bm p - \bm p_0(\text{right})$, the resulting effective Hamiltonian is $$\label{eq:right-chiral-effective-Hamiltonian-VF}
\mathcal{H}_\text{R} = {\varepsilon_\mathrm{diag}}(p^z) {\mathbb{1}_{2\times 2}} + v_\perp(p^x \sigma^x + p^y \sigma^y) + v^z (p^z - {p^z_\textsc{wr}}) \sigma^z.$$ Here ${\varepsilon_\mathrm{diag}}(p^z) = {\varepsilon_\textsc{wr}}+ {v_{\mathbb{1}}}(p^z - {p^z_\textsc{wr}})$ and the relationships between the effective parameters and the parameters of the microscopic theory are listed in Appendix \[app:appendix\]. The Hamiltonian acts on the two-dimensional space which corresponds to the two gapless branches of the Hamiltonian’s Bloch spectrum near the Weyl point. The two remaining gapped branches have been dropped from the effective theory. Note that Eq. has the same form as the effective Hamiltonian .
For the *left*-chiral node, a similar procedure leads to the same effective Hamiltonian , but with the replacement $({p^z_\textsc{wr}},v^z,{\varepsilon_\textsc{wr}}) \to (-{p^z_\textsc{wr}},-v^z,-{\varepsilon_\textsc{wr}})$. Note that the energies of the two Weyl nodes are equal in magnitude and have opposite signs, therefore the “symmetric” choice of the Fermi energy ${\varepsilon_\textsc{f}}= 0$ results in equal density of oppositely charged carriers in the two Weyl pockets. For simplicity and in conformity with the reference study , we limit our considerations to this symmetric situation.
For the actual numerical computations we have used one of the parameter sets from Ref. , namely $\beta_0 =
0.1t$, $\beta^z = 1.2t$, $M_0 = -0.3t$, and ${l_\textsc{b}}= 50$. By using the dictionary in Appendix \[app:appendix\], one can see that this choice leads to the effective paramters listed in Eq. . The smallness of ${\varepsilon_\textsc{wr}}- {\varepsilon_\textsc{f}}$ ensures that the Fermi surfaces of the bulk states are close to the Weyl nodes, so that the effective theory describes the dynamics adequately. In addition, the parameter ${v_{\mathbb{1}}}^z$ turns out to be very small (${v_{\mathbb{1}}}^z = -1.1 \times 10^{-2} t / \hbar$), which results in accidental suppression of $I_\text{bulk}$ in a strong background field $B_0$ (see Sec. \[sec:bulk-effective-theory\]).
We compute the energy spectrum of the bounded system, using the `KWANT` package [@groth2014kwant]. This then serves as the input for the equation . Since we deal with a sample that is infinite in both $y$- and $z$-directions, we perform the dimensional reduction from three to one dimensions by replacing the operators $p$ and $p^y$ with corresponding good quantum numbers. We include the magnetic field by employing the standard Peierls substitution $\bm p
\to \bm
p - e \bm A / c$, and we choose the Landau gauge $\bm A = (0,Bx,0)$, which does not break the one-dimensional character of the problem. We have used 800 lattice sites in the $x$-direction, and checked that the doubling of this number does not change significantly any of the results presented below.
We have checked that the energy dispersion for the Landau levels holds well and the deviations (beyond having non-zero ${v_{\mathbb{1}}}^z$) are in agreement with the dimensional analysis done in Appendix \[app:higher-order-corrections\].
We confirm that the boundary condition for the low-energy excitations that follows from the microscopic Hamiltonian with the hard-wall boundaries indeed has the form . The extracted scattering phase-shifts turn out to be $\Delta\phi^\pm_\textsc{r} = -\Delta\phi^\pm_\textsc{l} = \pm 1.49$, which coincide with one of the two combinations that we used in Sec. \[sec:surface-effective-theory\]. Futhermore, the numerical evaluation of the surface current using `KWANT` gives the *same* value ${\mathcal{C}_\textsc{cme}}= 0.53$ as we found in the effective theory. Surprisingly, we find that for the given set of parameters, the separability condition holds extremely well (within machine precision) in the microscopic theory, which explains the numerical agreement between the two results. Note that for the same set of parameters, the indicated value of the coefficient ${\mathcal{C}_\textsc{cme}}$ is close to the original finding ${\mathcal{C}_\textsc{cme}}\approx 1/2$ of Ref. .
The next step is to see whether the result is robust against microscopic deformation of the boundary. We have modified the boundary by rescaling the $\beta_0$ parameter, $\beta_0 \to 10 \beta_0$, at the boundary sites of the 1D lattice. We have verified that such a modification does not affect the parameters of the bulk effective theory . At the same time, the phase-shifts changed significantly, $\Delta\phi^\pm_\textsc{r} = - \Delta\phi^\pm_\textsc{l} = \pm 2.3$. The resulting drastic change of the Fermi-arc dispersion is illustrated in Fig. \[fig:spectrum-original-deformed-boundary\]. Apart from the quantitative modification of $\Delta\phi$, we have encountered a qualitative change: the separability condition for the Fermi arcs (Eq. ) is no longer satisfied. Thus, we conclude that the separability is rather an *accidental* property of the microscopic theory with a specific boundary condition. The numerical diagonalisation for the modified boundary leads to ${\mathcal{C}_\textsc{cme}}= 1.05$, which is quite different from the result for the original boundary. By recalling the result ${\mathcal{C}_\textsc{cme}}= 0.89$ of the effective theory from Sec. \[sec:surface-effective-theory\] (for the same phaseshifts $\pm 2.3$), we see that it is close to the finding within the microscopic theory, although the agreement between the two approaches is not that good anymore. This is a consequence of violating the separability condition and it makes the prediction of the effective theory unreliable.
![\[fig:spectrum-original-deformed-boundary\] Energy spectrum for the original (left) and the modified boundaries (right) of the microscopic Hamiltonian . In both cases, surface states (Fermi arcs) form continuous lines connecting the Weyl nodes, but the shape of a given line is sensitive to the structure of the boundary. The parameters of the Hamiltonian are taken from Sec. \[sec:surface-microscopic-theory\], $p^y = 10^{-2} \hbar$, $B_0 = 0$. In order to illustrate the finite-size quantization of the energy bands, the number of lattice sites was decreased to 400.](WeylCones-BBLSet2-Original+DeformedBothBoundaries){width="50.00000%"}
Conclusion \[sec:discussion\]
=============================
In this paper, we have considered the adiabatic chiral magnetic effect (CME) in a WSM sample having a boundary. Generally, the contribution of the boundary to the CME current is on the same order as the bulk contribution, in particular both are proportional to the cross-sectional area of the sample. However we find that the boundary current can dominate in a presence of a strong static background magnetic field. This is true if the theory possesses a certain symmetry or if the parameters of the effective Hamiltonian are fine-tuned in a certain way, as is discussed in Sec. \[sec:bulk-effective-theory\].
We have found that there is no topological protection for such a boundary current and in general it cannot even be determined from the parameters of a bulk low energy effective theory. However, under a certain assumption (separability of the Fermi arc energy, Eq. ), the boundary current still can be expressed in terms of the parameters of the bulk effective theory alone (by which we mean the combination of the linearized Hamiltonian and the boundary conditions ). Such an expression can be found in Eq. .
We have investigated the validity of the separability assumption in a particular microscopic model used in Refs. and . This model accidentally has separable energy of the Fermi arcs, and the parameters of the Hamiltonian chosen in Ref. were so that the abovementioned fine-tuning takes place. This results in the surface current being the only source for the adiabatic CME. However, we see that a deformation of the boundary layer in the model both breaks the separability and makes Eq. invalid. In conclusion, the adiabatic CME current in a bounded Weyl semimetal system is non-universal, but depends on the precise way one manufactures the boundaries of the sample.
The authors are grateful to Paul Baireuther, Carlo Beenakker, Mikhail Katsnelson, and Jörg Schmalian for discussions and useful comments. This research was supported by the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Scientific Research (NWO/OCW) through the Delta ITP Consortium and by an ERC Synergy Grant.
Relations between the parameters of the effective and the microscopic Hamiltonians {#app:appendix}
==================================================================================
The effective parameters entering Eq. are expressed via the original parameters of the microscopic Hamiltonian as $$\label{eq:effective-parameters-1}
v_\perp = 2\sqrt{\frac{\beta_z^2 - \beta_0^2}{\beta_z^2 - 4 {\varepsilon_\textsc{wr}}^2 }}$$ $$\label{eq:effective-parameters-2}
v_\parallel = \frac{\sqrt{ \mathcal{K}}}{\beta_z^2 - 4{\varepsilon_\textsc{wr}}^2},$$ $$v_\mathbb{1} = -\frac{4{\varepsilon_\textsc{wr}}(1+M_0) \sin {p^z_\textsc{wr}}+ \beta_0 \beta_z \cos {p^z_\textsc{wr}}}{\beta_z^2 - 4{\varepsilon_\textsc{wr}}^2},$$ where $$\begin{gathered}
\mathcal{K} = (4{\varepsilon_\textsc{wr}}(1+M_0)\sin {p^z_\textsc{wr}}+ \beta_0 \beta_z \cos {p^z_\textsc{wr}})^2 \\ + (\beta_z^2 - 4{\varepsilon_\textsc{wr}}^2)(4 (1+M_0)^2 \sin^2 {p^z_\textsc{wr}}- \beta_0^2 \cos^2 {p^z_\textsc{wr}}).
\nonumber\end{gathered}$$ Here, ${p^z_\textsc{wr}}$ is a positive solution of $$\left( \frac{\beta_0}{\beta_z} \sin {p^z_\textsc{wr}}\right)^2 + 2 (1+M_0) \cos {p^z_\textsc{wr}}= 2 + 2M_0 + M_0^2 - \frac{\beta_z^2 - \beta_0^2}{4},$$ and the energy of right-chiral node is $${\varepsilon_\textsc{wr}}= - \frac{\beta_0}{\beta_z} \sin {p^z_\textsc{wr}}.$$ (For simplicity, we have set the hopping parameter equal to 1, $t=1$.)
Higher-order corrections to the linearized effective Hamiltonian {#app:higher-order-corrections}
================================================================
By using the minimal effective Hamiltonian , we implicitly assume that the coupling to the magnetic field $B$ is captured completely by replacing the quasimomentum $\bm p = -i\hbar\bm\nabla$ with the operator $-i\hbar\bm\nabla - e \bm A /c$. However, since our particles have real spin $\bm s$ (as opposed to the pseudospin $\hbar\bm\sigma/2$), they are expected to have Zeeman coupling, which introduces the correction $\Delta
\mathcal{H}_\text{eff} = - g {\mu_\textsc{b}}\bm s \bm B$, where $g$ is the $g$-factor and ${\mu_\textsc{b}}$ is the Bohr magneton. Then the energy gap between the two neighbouring Landau levels is of order of $v \hbar / {l_\textsc{b}}\sim e^2 / {l_\textsc{b}}$, according to Eq. , while the corrections coming from the Zeeman coupling are expected to be suppressed by an additional factor ${a_\textsc{nn}}/ {l_\textsc{b}}$, which is small for realistic fields $B \lesssim 10^6
{\;{\rm Gauss}\;}$ that can be reached in laboratories in the foreseeable future. Here we have estimated the typical velocity $v$ of an electron and the crystalline lattice spacing ${a_\textsc{nn}}$ to be of order of the corresponding atomic units, $v \sim
e^2/\hbar$, ${a_\textsc{nn}}\sim \hbar^2 / m_e e^2$, which are not quite far from the results of the band-structure calculations for some WSMs [@lee2015fermi] and the direct X-ray diffraction measurements [@boller1963transposition; @xu2015discovery]. This irrelevance of Zeeman coupling is similar to what happens in graphene [@goerbig2011electronic], although contrary to graphene, where the $g$-factor is not very far from the “bare” value $g=2$, see Ref. we can have much larger $g$, which can in principle alter our conclusion. (In the estimate above, we assumed $g \sim 1$. For a similar discussion regarding the importance of the Zeeman coupling to WSMs, see Ref. .) Moreover, in materials with strong spin-orbit coupling such as the transition-metal monopnictides that were the first experimentally discovered WSMs [@hasan2017discovery], additional terms in the effective theory are allowed. One such term is $C_b\bm B \cdot \bm b$, where $\bm b = \bm p_0(\text{right}) - \bm
p_0(\text{left})$ is the momentum separation of the Weyl nodes. Also we neglect the higher-derivative corrections to the dispersion relation of a Weyl fermion, such as the quadratic term $\Delta \mathcal{H}_\text{eff} = C_\text{2} ~ (\bm p -
\bm p_0)^2 / 2m_e$, where $m_e$ is the “bare” electron mass. However, a similar kind of dimensional analysis reveals that in the absence of some “anomalously” large coupling constants $C_b$ and $C_2$, the resulting corrections are expected to be suppressed as well.
[^1]: We note that in Ref. , the derivation was done for the geometry of a cylinder with a *circular* base. However, the generalization to the geometry of a slab is straightforward: one only needs to replace the momentum $p_\parallel$ corresponding to the motion along the perimeter of the circle with the momentum $p^y$, and to take into account that cylinder has only one boundary, while the slab has two. Additional factor $1/2$ in Eq. is due to the fact that the inflow of the charge from the bulk to the boundary is splitted between the two boundaries. (For more details about the inflow mechanism, see Ref. .)
|
---
abstract: 'We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group $S_\infty$, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand–Raikov theorem holds for topological subgroups of $S_\infty$: for all such groups, continuous irreducible representations separate points in the group.'
address: |
Université Paris 7\
UFR de Mathématiques, case 7012\
75205 Paris <span style="font-variant:small-caps;">cedex</span> 13
author:
- Todor Tsankov
bibliography:
- 'mybiblio.bib'
title: Unitary representations of oligomorphic groups
---
Introduction
============
Abstract harmonic analysis is classically restricted to studying representations of locally compact groups, and for a good reason: the Haar measure provides an invaluable tool for constructing and analyzing representations. It gives rise to the left-regular representation (so that every locally compact group has at least one faithful representation) but also allows to define the convolution algebra of the group and various useful topologies on function spaces which are important for understanding the representations. And indeed, many standard theorems of the subject break down for non-locally compact groups: for example, the group of orientation-preserving homeomorphisms of the reals has no non-trivial unitary representations whatsoever (Megrelishvili [@Megrelishvili2001]), while the group of all measurable maps from $[0, 1]$ to the circle has a faithful unitary representation by multiplication on $L^2([0, 1])$ but no irreducible representations (this example is due to Glasner [@Glasner1998a]; see also [@Bekka2008]\*[Example C.5.10]{}). Those rather pathological examples suggest that any attempt to develop a representation theory for non-locally compact groups should be restricted to certain well-behaved classes.
And indeed, a number of interesting classification results have been obtained for the representations of some concrete non-locally compact groups. Lieberman [@Lieberman1972] classified the unitary dual of the infinite symmetric group $S_\infty$ (for us, $S_\infty$ is always the group of *all* permutations of the integers and not only those with finite support). Later, Olshanski developed a rather versatile machine, called the , using which he succeeded to give another proof of Lieberman’s theorem [@Olshanski1985] and also classify the representations of many other non-locally compact groups including the infinite-dimensional unitary, orthogonal, and symplectic groups, and (a variant of) the infinite-dimensional general linear groups over finite fields [@Olshanski1991a] (cf. Remark \[r:Olshanski\]). See the survey [@Olshanski1991a] for an explanation of the method and the references therein for more examples. A classification for the representations of the unitary group had also been announced by Kirillov [@Kirillov1973]. All of the groups above have only countably many irreducible representations and every representation splits as a sum of irreducibles, a situation that very much resembles the case of compact groups.
In this paper, we study the unitary representations of , separable groups (i.e. separable groups that admit a countable basis at the identity consisting of open subgroups), also known as subgroups of $S_\infty$. This property alone allows us to recover one important result valid in the locally compact situation, namely the Gelfand–Raikov theorem. More precisely, we prove the following.
\[th:GelfRaikov\] Let $G$ be a topological subgroup of $S_\infty$. Then for every $x, y \in G$, $x \neq y$, there exists a continuous, irreducible, unitary representation $\pi$ of $G$ such that $\pi(x) \neq \pi(y)$.
The main body of the paper, however, concentrates on groups that have an additional special property, that of (cf. Definition \[df:Roelcke\]). It turns out that this property has a natural translation in the language of permutation groups and model theory. For us, a will be a topological subgroup of the group of all permutations of a countable set $\bX$. The following definition is of central importance for this paper.
\[df:olig\] Let $\bX$ be a countable (finite or infinite) set. A permutation group $G \actson \bX$ is called if the diagonal action $G \actson \bX^n$ has only finitely many orbits for each $n$. A topological group $G$ is if it can be realized as an oligomorphic permutation group.
Closed oligomorphic permutation groups also have a model-theoretic interpretation: they are exactly the automorphism groups of $\omega$-categorical structures (cf. Section \[s:olig-groups\]). A standard way to produce $\omega$-categorical structures is the construction: given a class of finite structures satisfying a certain amalgamation property, there is a way to build a (unique) infinite, homogeneous structure that contains all structures in the class as substructures. We postpone the formal definitions to Section \[s:olig-groups\] and just describe a few examples.
**Examples of $\omega$-categorical limits:**
1. \[i:ex:sinfty\] The limit of all finite sets without structure is a countably infinite set. The corresponding group is $S_\infty$, the group of all permutations of this set.
2. \[i:ex:Q\] The limit of all finite linear orders is the countable dense linear order without endpoints $({{\mathbf Q}}, <)$. We denote the corresponding automorphism group by $\Aut({{\mathbf Q}})$.
3. \[i:ex:Cantor\] The limit of all finite Boolean algebras is the countable atomless Boolean algebra which is isomorphic to the algebra of all clopen subsets of the Cantor space $2^\N$. The corresponding automorphism group is $\Homeo(2^\N)$, the group of all homeomorphisms of $2^\N$.
4. \[i:ex:GL\] The limit of all finite vector spaces over a fixed finite field $\F_q$ is the infinite-dimensional vector space over $\F_q$. The automorphism group is the general linear group $\GL(\infty, \F_q)$.
5. \[i:ex:R\] The limit of all finite graphs is the , the unique countable graph $\bR$ such that for every two finite disjoint sets of vertices $U, V$, there exists a vertex $x$ which is connected by an edge to all vertices in $U$ and to no vertices in $V$. We denote its automorphism group by $\Aut(\bR)$.
There are also many other $\omega$-categorical structures, including, for example, certain groups [@Hodges1993], and a variety of combinatorial structures [@Kechris2005]. There is a rather extensive literature devoted to the subject; we refer the interested reader to the volume [@Kaye1994], or the more recent survey Macpherson [@Macpherson2010p] and the references therein. We also indicate some ways to construct new oligomorphic groups from old ones in Proposition \[p:closure\].
The main theorem describing the unitary representations of oligomorphic groups is the following.
\[th:int:main\] Let $G$ be an oligomorphic group. Then every irreducible unitary representation of $G$ is of the form $\Ind_{C(V)}^G(\sigma)$, where $V \leq G$ is an open subgroup, $C(V)$ is the commensurator of $V$, $V \unlhd C(V)$, and $\sigma$ is an irreducible representation of the *finite* group $C(V)/V$. Moreover, every unitary representation of $G$ is a sum of irreducibles.
We also provide a criterion when two irreducible representations as above are isomorphic (Proposition \[p:ind-reps\]).
As every oligomorphic group has only countably many distinct open subgroups (Corollary \[c:ctbl-open\]), the theorem shows that every oligomorphic group has only countably many irreducible representations.
Our methods are quite different from the approach of Olshanski. In particular, his semigroup method only applies when the group is obtained as the completion of an inductive limit of subgroups, which is not the case for many oligomorphic groups (for example, $\Aut({{\mathbf Q}})$). On the other hand, we have borrowed an important idea from Lieberman: the use of a weak limit point in the proof of Theorem \[th:int:main\].
If one is given a realization of a closed oligomorphic group as the automorphism group of a countable structure, it is possible to give a more concrete description of its representations in terms of the structure. For example, for the automorphism group of the random graph, all irreducible representations can be obtained in the following way. One takes a finite (induced) subgraph $\bA \sub \bR$ and sets $V$ to be the pointwise stabilizer of $\bA$. Then $C(V)$ is the setwise stabilizer of $\bA$ and $C(V)/V \cong \Aut(\bA)$. So in this case, the irreducible representations of $\Aut(\bR)$ are obtained by induction from irreducible representations of automorphism groups of finite graphs (and in fact, this correspondence is one-to-one if one makes the obvious identifications). See Section \[s:examples\] for more details.
As a corollary of the classification of the representations of $\Aut({{\mathbf Q}})$, we obtain that the group $\Homeo^+(\R)$ has no non-trivial unitary representations (this is a special case of a result of Megrelishvili), cf. Corollary \[c:HomeoR\].
As a further application, we establish property (T) for a large class of oligomorphic groups. Our technique is quite similar to the one used by Bekka [@Bekka2003] to prove that the unitary group has property (T) (for which he used Kirillov and Olshanski’s classification of its representations).
All of the examples – above have property (T).
In all of those groups, it is also possible to find explicit finite Kazhdan sets. We also have a more general result (Theorem \[th:propT-gen\]), which requires some additional terminology to state. The question whether all closed oligomorphic groups have property (T) remains open.
The paper is organized as follows. In Section \[s:olig-groups\], we recall the definition of Roelcke precompactness and provide a model-theoretic characterization of Roelcke precompact subgroups of $S_\infty$. In Section \[s:GelfRaikov\], we prove some basic results about representations of non-archimedean groups, including Theorem \[th:GelfRaikov\]. In Section \[s:rep-olig\], we prove the main theorem. Section \[s:examples\] is devoted to some model-theoretic considerations and calculations in specific examples. In Section \[s:propT\], we discuss property (T).
Notation {#notation .unnumbered}
--------
If $G$ is a group and $g, x \in G$, $g^x$ denotes the conjugate $xgx^{-1}$. Note that this is the conjugation action on the left, so that $(g^x)^y = g^{yx}$. If $G$ is the automorphism group of a structure $\bX$ and $\bar a \in \bX^n$ is a tuple, $G_{\bar a}$ denotes the stabilizer of all elements of $\bar a$. If $\bA \sub \bX$ is a substructure, $G_\bA$ is the of $\bA$ (the set of all $g$ such that $g \cdot \bA = \bA$) and $G_{(\bA)}$ is the (the set of all $g$ such that $g \cdot a = a$ for all $a \in \bA$). If $X$ is a set, $X^{[n]}$ denotes the set of all subsets of $X$ of size $n$.
A of a group $G$ is always a unitary representation. If $\pi$ is a representation, $\mcH(\pi)$ denotes its Hilbert space. All Hilbert spaces are complex.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Itaï Ben Yaacov and C. Ward Henson for an important insight for the proof of Lemma \[l:left-right\], Martin Hils for explaining to me some basic model theory, and H. Dugald Macpherson for pointing out an error in a preliminary version of the paper. I am grateful to the Fields Institute in Toronto for its hospitality while this paper was being finished. I am also grateful to the referee for a number of helpful comments.
Oligomorphic groups and Roelcke precompactness {#s:olig-groups}
==============================================
Roelcke precompact topological groups
-------------------------------------
A topological group $G$ is iff the completion of its left uniformity is compact iff the completion of its right uniformity is compact. In that case, the common completion of the left and the right uniformities has the structure of a compact group in which $G$ embeds topologically as a dense subgroup. In particular, if $G$ is Polish, $G$ is precompact iff it is compact. The condition that the left uniformity on a group $G$ is precompact can be written as follows: for every neighborhood $U$ of the identity, there exists a finite set $F \sub G$ such that $FU = G$.
There exists a weaker notion of precompactness that will be central for this paper. A topological group is called if the completion of its (the greatest lower bound of the left and right uniformities) is compact. We will find it, however, more convenient to work with the following direct definition.
\[df:Roelcke\] A topological group $G$ is if for every neighborhood of the identity $U$, there exists a finite set $F \sub G$ such that $G = UFU$.
The notion of Roelcke precompactness is weak enough to include many interesting non-compact examples but still sufficiently powerful to allow the generalisation of some results valid for compact groups. Roelcke precompact groups were introduced by Roelcke–Dierolf [@Roelcke1981], who also gave the first examples, and were later studied by many authors, including [Uspenski]{}and Glasner. Known examples of Roelcke precompact groups include the unitary group of a separable, infinite-dimensional Hilbert space ([Uspenski]{} [@Uspenskii1998]), the isometry group of the bounded Urysohn metric space ([Uspenski]{} [@Uspenskii2008]), and the automorphism group of a standard probability space (Glasner [@Glasner2009p]). We also note that Roelcke precompactness is only interesting for “infinite-dimensional” groups: a locally compact group is Roelcke precompact iff it is compact (to see this, note that if $U$ is a compact neighborhood of the identity, then $G = UFU$ is compact). We suggest the survey [@Uspenskii2002] by [Uspenski]{}as a general reference.
We start by showing that the class of Roelcke precompact groups enjoys most of the closure properties of the class of compact groups with the notable exception of closure under taking closed subgroups. (In fact, as was shown in [@Uspenskii2008], every Polish group embeds as a closed subgroup of a Polish Roelcke precompact group.)
\[p:closure\] The following statements hold:
1. \[i:pcl:quot\] If $\pi \colon G \to H$ is a continuous homomorphism with a dense image and $G$ is Roelcke precompact, then so is $H$.
2. \[i:pcl:prod\] If $G_1$ and $G_2$ are Roelcke precompact, then so is $G_1 \times G_2$.
3. \[i:pcl:inv\] The inverse limit of an inverse system of Roelcke precompact groups is Roelcke precompact. In particular, an arbitrary product of Roelcke precompact groups is Roelcke precompact.
4. \[i:pcl:open\] If $G$ is Roelcke precompact and $H \leq G$ is open, then $H$ is Roelcke precompact.
5. \[i:pcl:ext\] If $N$ is a normal subgroup of $G$ such that both $N$ and $G/N$ are Roelcke precompact, then so is $G$.
. Let $U \sub H$ be an open neighborhood of $1$ . Find an open $V$ such that $1 \in V^2 \sub U$. Let $F \sub G$ be finite such that $G = \pi^{-1}(V)F\pi^{-1}(V)$. We check that $U\pi(F)U = H$. Let $h \in H$. By the density of $\pi(G)$ in $H$, there exists $g \in G$ such that $\pi(g) \in hV^{-1}$. Then $g = v_1 f v_2$ for some $v_1, v_2 \in \pi^{-1}(V)$ and $f \in F$. Finally, $h \in \pi(g)V = \pi(v_1) \pi(f) \pi(v_2) V \sub U\pi(F)U$.
. If $U_1 \times U_2$ is an open neighborhood of the identity in $G_1 \times G_2$ and $F_1 \sub G_1$, $F_2 \sub G_2$ are finite such that $U_1F_1U_1 = G_1$, $U_2F_2U_2 = G_2$, then $G_1 \times G_2 = (U_1 \times U_2)(F_1 \times F_2)(U_1 \times U_2)$.
. Let $G = \varprojlim H_i$. Let $U = \pi_i^{-1}(U_i)$ be an open neighborhood of $1$ in $G$, where $U_i$ is an open neighborhood of $1$ in $H_i$. Then there exists a finite $F \sub H_i$ with $H_i = U_iFU_i$. If $F' \sub G$ is finite with $\pi_i(F') = F$, we check that $UF'U^2 = G$. Indeed, let $x$ in $G$. Then there exist $u_1, u_2 \in G$, $f \in F'$ such that $\pi(x) = \pi(u_1 f u_2)$. Then there is $h \in \ker \pi_i$ such that $x = u_1 f u_2 h \in UF'U^2$ (because $\ker \pi_i \sub U$ by the definition of $U$).
. Let $U \sub H$ be an open neighborhood of $1$. Then $U$ is open in $G$ and there exists a finite $F \sub G$ such that $G = UFU$. Now one easily checks that $H = U(F \cap H)U$.
. Let $U \sub G$ be an open neighborhood of $1$. Let $Q = G/N$ and denote by $\pi \colon G \to Q$ the quotient map. Then $\pi(U)$ is open in $Q$. Find a finite $F \sub G$ such that $\pi(U)\pi(F)\pi(U) = Q$ and assume also that $1 \in F$. Let $V = N \cap \bigcap_{f \in F} U^f$ and note that $V$ is relatively open in $N$. Find a finite $B \sub N$ such that $VBV = N$. We claim that $U^2BFU^2 = G$. Indeed, fix $x \in G$. Find $f \in F$ and $u_1, u_2 \in U$ such that $\pi(u_1 f u_2) = \pi(x)$. Then there exists $h \in N$ with $x = h u_1 f u_2$. Find $v_1, v_2 \in V$ and $b \in B$ such that $u_1^{-1} h u_1 = v_1 b v_2$. Finally, we have $$\begin{split}
x &= h u_1 f u_2 = u_1 (u_1^{-1} h u_1) f u_2 = u_1 v_1 b v_2 f u_2 \\
&= u_1 v_1 b f (f^{-1} v_2 f) u_2 \in U^2 BF U^2,
\end{split}$$ finishing the proof.
We end this subsection with a simple application. In [@Rosendal2009], Rosendal introduced property (OB), intended as a generalization of the well known properties (FA) and (FH). A topological group has if every time it acts (separately) continuously by isometries on a metric space, every orbit is bounded.
\[p:OB\] Every Roelcke precompact group has property (OB).
Let $G$ be a Roelcke precompact group and $G \actson (X, d)$ a separately continuous action on a metric space by isometries. For every point $x_0 \in X$, the function $g \mapsto d(x_0, g \cdot x_0)$ is uniformly continuous in the lower uniformity, as can be seen from the inequality $$\begin{split}
d(x_0, h_1 g h_2 \cdot x_0) &= d(h_1^{-1} \cdot x_0, g h_2 \cdot x_0) \\
&\leq d(h_1^{-1} \cdot x_0, x_0) + d(x_0, g \cdot x_0) + d(g \cdot x_0, gh_2 \cdot x_0) \\
&= d(h_1^{-1} \cdot x_0, x_0) + d(x_0, g \cdot x_0) + d(x_0, h_2 \cdot x_0).
\end{split}$$ It thus extends to the compact completion and must be bounded.
Permutation groups and closed subgroups of $S_\infty$
-----------------------------------------------------
We now concentrate on the main objects of study in this paper, namely infinite permutation groups. Let $S_\infty$ be the group of all permutations of a countable infinite set $\bX$. It becomes naturally a topological group if equipped with the pointwise convergence topology, where $\bX$ is taken to be discrete. A is a topological subgroup of the group of all permutations of $\bX$.
It is well known that the topological groups that can be realized as permutation groups are exactly the separable topological groups that admit a countable basis at the identity consisting of open subgroups (those groups are often called ). The basis of open subgroups is given by the pointwise stabilizers of finite subsets of $\bX$. A natural way in which closed permutation groups arise in practice is as automorphism groups of countable structures in model theory, for example, automorphism groups of countable graphs, countable orders, or various algebraic structures. Of special interest to us will be the oligomorphic groups (see Definition \[df:olig\]) for which it is possible to translate back and forth between model-theoretic and permutation group-theoretic language. For a gentle introduction to the subject of oligomorphic groups, we refer the reader to Cameron [@Cameron1990]. It turns out that there is a close connection between the properties of being Roelcke precompact and oligomorphic. To see this, we first reformulate Definition \[df:Roelcke\] for non-archimedean groups: a topological subgroup $G$ of $S_\infty$ is Roelcke precompact iff for every open subgroup $V \leq G$, the set of double cosets $$V \backslash G / V = \set{VxV : x \in G}$$ is finite.
\[th:char-olig\] For a topological subgroup $G \leq S_\infty$, the following are equivalent:
1. \[i:tco:RC\] $G$ is Roelcke precompact;
2. \[i:tco:olig\] for every continuous action $G \actson \bX$ on a countable, discrete set $\bX$ with finitely many orbits, the induced action $G \actson \bX^n$ has finitely many orbits for each $n$;
3. \[i:tco:inv-limit\] $G$ can be written as an inverse limit of oligomorphic groups.
$\Implies$ . Without loss of generality, we can suppose that the action $G \actson \bX$ is transitive. We use induction on $n$. The case $n = 1$ is given by the hypothesis. For the induction step $n \to n+1$, it suffices to find for every $\bar a \in \bX^n$, a finite set $B(\bar a) \sub \bX$ such that for every $d \in \bX$, there is $h \in G_{\bar a}$ and $b \in B(\bar a)$ such that $d = h \cdot b$. Then if $\set{\bar a_1, \ldots, \bar a_s}$ is a complete set of representatives for the orbits of $G \actson \bX^n$, $\set{(\bar a_i, b) : i \leq s, b \in B(\bar a_i)}$ will be a complete set of representatives for the action $G \actson \bX^{n+1}$. Indeed, let $(\bar c, d) \in \bX^{n+1}$. Using the induction hypothesis, find $g \in G$ such that $g \cdot \bar a_i = \bar c$. Find $h \in G_{\bar a_i}$ and $b \in B(\bar a_i)$ such that $g^{-1} \cdot d = h \cdot b$. Then one has $$gh \cdot (\bar a_i, b) = g \cdot (a_i, h \cdot b) = (\bar c, d).$$
Fix now $\bar a \in \bX^n$, let $c_0$ be an arbitrary element of $\bX$, and let $\set{G_{\bar a c_0} g_0 G_{\bar a c_0}, \ldots, \linebreak G_{\bar a c_0} g_k G_{\bar a c_0}}$ be a complete list of the double cosets of $G_{\bar a c_0}$. Set $B(\bar a) = \set{g_i \cdot c_0 : i = 1, \ldots, k}$. Let now $d \in \bX$ be arbitrary, $d = g \cdot c_0$ (using the transitivity of the action). Let $i, h_1, h_2$ be such that $h_1, h_2 \in G_{\bar a c_0}$ and $g = h_1 g_i h_2$. We have $$d = g \cdot c_0 = h_1 g_i h_2 \cdot c_0 = h_1 g_i \cdot c_0,$$ finishing the proof.
$\Implies$ . Let $\{V_n : n \in \N\}$ be a basis at $1_G$ of open subgroups such that for all $n$, $V_{n+1} \leq V_n$. Then for each $n$, $G$ acts continuously by permutations on the discrete set $G/V_n$ and this gives rise to a continuous homomorphism $\pi_n \colon G \to \mathrm{Sym}(G/V_n)$. The groups $\{\pi_n(G) : n \in \N \}$ form a directed system (there are natural maps $\pi_{n+1}(G) \to \pi_n(G)$) and its inverse limit is isomorphic to $G$. Each $\pi_n(G)$ is oligomorphic by the hypothesis.
$\Implies$ . In view of Proposition \[p:closure\], it suffices to show that if $G$ is oligomorphic, then it is Roelcke precompact. Let $G$ be a group of permutations of $\bX$ such that the action $G \actson \bX$ is oligomorphic. It suffices to show that every stabilizer $G_{\bar a}$ for $\bar a \in \bX^n$ has finitely many double cosets. If $\set{(\bar a, g_1 \cdot \bar a), \ldots, (\bar a, g_k \cdot \bar a)}$ is a complete list of representatives for the $G$-orbits on $\bX^{2n}$ that are subsets of $G \cdot \bar a \times G \cdot \bar a$, then it is easy to check that $\set{G_{\bar a}g_iG_{\bar a} : i = 1, \ldots, k}$ exhausts all double cosets of $G_{\bar a}$.
A proof that every (even approximately) oligomorphic group is Roelcke precompact is essentially contained in [@Rosendal2009]. Also, some special cases of the above theorem had been known before: [Uspenski]{}had shown that $S_\infty$ and $\Homeo(2^\N)$ are Roelcke precompact [@Uspenskii2001; @Uspenskii2002].
A basic application of the theorem is the following corollary which had been noted before by many authors.
\[c:ctbl-open\] Every oligomorphic group has only countably many distinct open subgroups.
Fix a countable basis at the identity of open subgroups $\set{V_i}$. Every open subgroup contains a basic open subgroup and is therefore a union of finitely many double cosets of some $V_i$.
Now we turn to some basic group-theoretic lemmas about oligomorphic groups that will be used later. Recall that two subgroups of a group are if their intersection has finite index in both. If $H \leq G$ is a subgroup, define the to be $$\Comm_G(H) = \set{g \in G : H \text{ and } H^g \text{ are commensurate}}.$$ It is a standard fact that $\Comm_G(H)$ is a subgroup of $G$ containing $H$. If $H_1$ and $H_2 \leq G$ are commensurate, then $\Comm_G(H_1) = \Comm_G(H_2)$. Note also that the number of left cosets in $HgH$ is equal to $[H : H \cap H^g]$ and the number of right cosets is $[H^g : H \cap H^g]$. The following lemma will be particularly useful for studying commensurators in oligomorphic groups. I am grateful for the idea of the proof to Itaï Ben Yaacov and C. Ward Henson.
\[l:left-right\] Let $G$ be a Roelcke precompact group and $V \leq G$ an open subgroup. Then, for every $x \in G$, the double coset $VxV$ contains finitely many left cosets of $V$ iff it contains finitely many right cosets of $V$.
Note first that if $H_1, H_2, H_3 \leq G$ and $H_1H_2$ contains finitely many $H_2$-cosets and $H_2H_3$ contains finitely many $H_3$-cosets, then $H_1H_3 \sub H_1H_2H_3$ also contains only finitely many $H_3$-cosets.
For $H \leq G$, denote $$\begin{split}
\mcF(H) &= \set{yH : HyH \text{ contains finitely many left } H\text{-cosets}} \\
&= \set{yH : HH^y \text{ contains finitely many } H^y\text{-cosets}}.
\end{split}$$ Note that $\mcF(H)$ is exactly the union of double cosets that contain only finitely many left cosets and as $G$ has only finitely many double cosets of $V$, $\mcF(V)$ is finite. Now we show that $$\label{eq:Ftrans}
xV, yV \in \mcF(V) \implies xyV \in \mcF(V).$$ Indeed, we have that $VV^x$ contains finitely many $V^x$ cosets and $VV^y$ contains finitely many $V^y$ cosets. Conjugating the latter by $x$, we obtain that $V^xV^{xy}$ contains finitely many $V^{xy}$ cosets. Now applying the observation in the beginning of the proof, we have that $VV^{xy}$ contains finitely many $V^{xy}$ cosets, i.e. $xyV \in \mcF(V)$.
Suppose now that $x \in G$ is such that $xV \in \mcF(V)$ and consider the map $\Phi \colon G/V \to G/V$ defined by $\Phi(yV) = xyV$. By , $\Phi(\mcF(V)) \sub \mcF(V)$ and as $\mcF(V)$ is finite and $\Phi$ is injective, in fact, $\Phi(\mcF(V)) = \mcF(V)$. Therefore there exists $yV \in \mcF(V)$ such that $\Phi(yV) = V$, i.e. $yV = x^{-1}V$. So we conclude that $x^{-1}V \in \mcF(V)$ or equivalently, that $VxV$ contains finitely many right cosets. The other direction of the statement is obtained by replacing $x$ with $x^{-1}$.
As a corollary, we obtain that we can define the commensurator of an open subgroup in a Roelcke precompact group as $$\begin{split} \label{eq:defcomm}
\Comm_G(V) &= \bigcup \set{VgV : VgV \text{ contains finitely many left cosets of } V} \\
&= \bigcup \set{VgV : VgV \text{ contains finitely many right cosets of } V}.
\end{split}$$
\[l:commen\] Let $G$ be Roelcke precompact and $V \leq G$ be open. Then the following hold:
1. \[i:lcm:fi\] $[\Comm_G(V) : V] < \infty$;
2. \[i:lcm:idemp\] $\Comm_G(\Comm_G(V)) = \Comm_G(V)$;
3. \[i:lcm:2cms\] If $V_1, V_2 \leq G$ are open and $\Comm_G(V_1)$ and $\Comm_G(V_2)$ are commensurate, then $\Comm_G(V_1) = \Comm_G(V_2)$.
. Since $V \leq \Comm_G(V)$, $V$ is open, and $G$ is Roelcke precompact, $\Comm_G(V)$ is a union of finitely many double cosets of $V$. By , every double coset $VxV$ in $\Comm_G(V)$ contains only finitely many left cosets, so the claim follows.
. Follows from .
. We have $$\Comm_G(V_1) = \Comm_G(\Comm_G(V_1)) = \Comm_G(\Comm_G(V_2)) = \Comm_G(V_2).$$
We see that $\Comm_G(V)$ is the maximal subgroup of $G$ containing $V$ in which $V$ is of finite index. In view of , we say that an open subgroup $H$ of a Roelcke precompact, non-archimedean group $G$ is if $\Comm_G(H) = H$.
Examples from model theory {#ss:examples-mth}
--------------------------
A natural class of permutation groups is obtained from model theory. A (or a ) $\mcL$ is a collection of relation and function symbols, where each symbol has a certain $n(\cdot)$. A structure for $\mcL$ is a set $\bX$ together with interpretations for the symbols: for each relation symbol $R$, a relation $R^\bX \sub \bX^{n(R)}$ and for each function symbol $F$, a function $F^\bX \colon \bX^{n(F)} \to \bX$. An of the structure $\bX$ is a permutation that preserves the relations and the functions. A structure is if the signature has no functions symbols.
Every closed permutation group can be obtained as the automorphism group of a relational structure: one just adds a relation for every orbit on $\bX^n$ for all $n$ (see, for example, [@Becker1996] for more details). An important model-theoretic characterization of the structures whose automorphism groups are oligomorphic is given by the following classical theorem (see [@Hodges1993] for a proof).
For a countable structure $\bX$, the following are equivalent:
- $\bX$ is $\omega$-categorical;
- $\Aut(\bX) \actson \bX$ is an oligomorphic permutation group.
A structure $\bX$ is called if $\bX$ is the unique (up to isomorphism) countable model of the first-order theory of $\bX$.
An especially attractive situation is when $\bX$ is in the following strong sense: every isomorphism between finite substructures of $\bX$ extends to a full automorphism of $\bX$ (sometimes those structures are called ). A classical theorem of describes the homogeneous structures as the limits of classes of finite structures satisfying a certain amalgamation property [@Hodges1993]\*[Section 7.4]{}. A homogeneous structure $\bX$ is $\omega$-categorical iff for every $n$ there are only finitely many isomorphism types of substructures of $\bX$ generated by $n$ elements. In particular, every homogeneous structure in a finite, relational signature is $\omega$-categorical. We also see that all examples given in the introduction are $\omega$-categorical. We add one slightly less known, but instructive, example to the list.
**Examples (cont.):**
1. \[i:ex:Cherlin\] (Cherlin and Hrushovski) Consider a signature with infinitely many relation symbols $\set{E_n}_{n \geq 1}$, where $E_n$ is of arity $2n$. Let $\mcX$ be the class of all finite structures in this signature, where each $E_n$ is interpreted as an equivalence relation on *subsets* of the structure of size $n$ with at most $2$ equivalence classes. For every $k \in \N$, there are only finitely many structures in $\mcX$ of size $k$, so the limit $\bX$ of $\mcX$ is $\omega$-categorical. What is remarkable about this structure is that its automorphism group $G$ has a quotient isomorphic to $(\Z / 2\Z)^\N$. The reason for this is that $G$ acts on the sets $\bX^{[n]} / E_n$ all of which have size $2$. Extending this example, Evans and Hewitt [@Evans1990] constructed for every profinite, metrizable group $H$ an oligomorphic group $G$ such that $H$ is a quotient of $G$.
Typical homogeneous structures that are often encountered in the literature which are *not* $\omega$-categorical are the discrete structures that arise as approximations of continuous ones: the rational Urysohn metric space, countable measured Boolean algebras, etc.
Representations of permutation groups {#s:GelfRaikov}
=====================================
A of a topological group $G$ is a homomorphism $\pi \colon G \to U(\mcH)$ to the unitary group of some Hilbert space $\mcH$ (i.e. the map $G \to \mcH$, $g \mapsto \pi(g)\xi$ is continuous for every $\xi \in \mcH$). A permutation group $G \actson \bX$ has some natural representations defined using the action on $\bX$, namely, $G \actson \ell^2(\bX^n)$ for $n \in \N$. Those representations clearly separate the points of $G$. In this section, we show that *irreducible* representations also separate points. (A representation $\pi$ is if $\mcH(\pi)$ does not have non-trivial subspaces invariant under $\pi$.) This can be considered as a version of the classical Gelfand–Raikov [@Gelfand1943] theorem for subgroups of $S_\infty$.
If $\pi$ is a representation of $G$ and $V \leq G$, denote by $\mcH_V(\pi)$ the closed subspace of fixed points of $V$. We start with a simple but key lemma a version of which had been previously used by Lieberman [@Lieberman1972] and Glasner–Weiss [@Glasner2005a].
\[l:dense-fp\] Let $G$ be a subgroup of $S_\infty$ and $\set{V_i : i \in \N}$ be a basis at the identity for $G$ consisting of open subgroups. Then for any continuous unitary representation $\pi$ of $G$, the space $\bigcup_i \mcH_{V_i}(\pi)$ is dense in $\mcH(\pi)$.
Let $\xi_0 \in \mcH(\pi)$ be an arbitrary vector and fix $\eps > 0$. As the representation is continuous, there is $i \in \N$ such that $\pi(V_i) \xi_0$ is contained in the ball with radius $\eps$ around $\xi_0$. Let $C$ be the closure of the convex hull of $\pi(V_i) \xi_0$ and let $\eta$ be the unique element of least norm in $C$. As $\pi(V_i)C = C$ and $\pi$ preserves the norm, $\eta$ is a fixed point of $\pi(V_i)$. Also, by the choice of $V_i$, $\nm{\eta - \xi_0} \leq \eps$.
Recall that a continuous function $\phi \colon G \to \C$ is called if for every $x_1, \ldots, x_n \in G$ and $c_1, \ldots, c_n \in \C$, $$\label{eq:posdef}
\sum_{i, j = 1}^n \phi(x_j^{-1}x_i)c_i \conj{c_j} \geq 0,$$ i.e. the matrix $\big(\phi(x_j^{-1}x_i)\big)_{i, j}$ is positive-definite. If $\pi$ is a representation of $G$ and $\xi \in \mcH(\pi)$, the function $x \mapsto \ip{\pi(x)\xi, \xi}$ is positive definite and conversely, the GNS construction produces from a positive definite function $\phi$ a representation $\pi$ and a $\xi \in \mcH(\pi)$ (i.e. such that the linear span of the orbit $\pi(G)\xi$ is dense in $\mcH(\pi)$) such that $$\phi(x) = \ip{\pi(x)\xi, \xi} \quad \text{for all } x \in G$$ (see [@Bekka2008]\*[Appendix C]{} for more details). In particular, $|\phi(x)| \leq \phi(1)$ for all $x \in G$. We now have the following basic observation.
\[l:const-dblcst\] Let $G$ be a group and $H \leq G$ a subgroup. If $\phi$ is a positive definite function that is constant on $H$, then it is constant on double cosets of $H$.
Let $\pi$ and $\xi \in \mcH(\pi)$ be such that $\phi(x) = \ip{\pi(x)\xi, \xi}$ for $x \in G$. Then by the hypothesis, for any $h \in H$, $\ip{\pi(h)\xi, \xi} = \ip{\pi(1)\xi, \xi} = \nm{\xi}^2$, i.e. $\xi \in \mcH_H(\pi)$. Now we have for any $x \in G$ and $h_1, h_2 \in H$: $$\phi(h_1 x h_2) = \ip{\pi(h_1 x h_2)\xi, \xi} = \ip{\pi(x)\pi(h_2)\xi, \pi(h_1^{-1})\xi}
= \ip{\pi(x)\xi, \xi} = \phi(x),$$ finishing the proof.
Let $$\mcP_1(G) = \set{\phi \colon G \to \C : \phi \text{ is positive definite and } \phi(1) = 1}$$ and if $V \leq G$ is open, let also $$\mcP_V(G) = \set{\phi \in \mcP_1(G) : \phi(v) = 1 \text{ for all } v \in V}.$$ By Lemma \[l:const-dblcst\], we can consider $\mcP_V(G)$ as a subset of $\ell^\infty(V \backslash G / V)$. $\mcP_V(G)$ is convex and bounded and by the definition , it is also closed in the weak$^*$ topology of $\ell^\infty(V \backslash G / V)$ and thus compact.
\[l:extr-points\] If $\phi$ is an extreme point of $\mcP_V(G)$, then it is also an extreme point of $\mcP_1(G)$.
Suppose that $\phi \in \mcP_V(G)$ and $\psi_1, \psi_2 \in \mcP_1(G)$ and $t \in (0, 1)$ are such that $\phi = t \psi_1 + (1 - t)\psi_2$. For every $v \in V$, we have $$1 = \phi(v) = t \Re \psi_1(v) + (1 - t) \Re \psi_2(v) \leq t \psi_1(1) + (1 - t) \psi_2(1) = 1,$$ showing that we must have equality in the middle, i.e. $\Re \psi_1(v) = \Re \psi_2(v) = 1$ for all $v \in V$. As $|\psi_1(v)|, |\psi_2(v)| \leq 1$, this implies that $\psi_1(v) = \psi_2(v) = 1$. Thus $\psi_1, \psi_2 \in \mcP_V(G)$, proving the lemma.
We now see that the classical proof of Gelfand–Raikov extends to our situation.
It suffices to show that for every $x \in G$, $x \neq 1$, there is an irreducible representation $\pi$ such that $\pi(x) \neq \pi(1)$. Recall that a representation $\pi$ with a cyclic unit vector $\xi$ is irreducible iff the corresponding positive definite function is an extreme point of $\mcP_1(G)$ [@Bekka2008]\*[Theorem C.5.2]{}. Let now $1 \neq x \in G$. Let $V \leq G$ be an open subgroup such that $x \notin V$. Consider the positive definite function $\chi_V$ (the characteristic function of $V$) which corresponds to the representation $G \actson \ell^2(G/V)$ with cyclic vector $\delta_V$. We have that $\chi_V \in \mcP_V(G)$ and $\chi_V(x) \neq \chi_V(1)$. Consider now the weak$^*$ closed, convex set $C = \set{\phi \in \mcP_V(G) : \phi(x) = \phi(1)}$. As $C \subsetneq \mcP_V(G)$, by the Krein–Milman theorem, there exists an extreme point $\phi$ of $\mcP_V(G)$ such that $\phi \notin C$. By Lemma \[l:extr-points\], $\phi$ is also an extreme point of $\mcP_1(G)$, producing the required irreducible representation.
Representations of oligomorphic groups {#s:rep-olig}
======================================
Let $G$ be a subgroup of $S_\infty$ and $G \actson Y$ be a continuous action on a discrete, countable set $Y$. There is a natural associated representation of $G$ on $\ell^2(Y)$ and if $Y = \bigsqcup_i Y_i$ is the decomposition of $Y$ into orbits, we have that $\ell^2(Y) = \boplus_i \ell^2(Y_i)$. Therefore we can as well suppose that the action $G \actson Y$ is transitive; in this case, the corresponding representation is just the $\ell^2(G/V)$ for some open subgroup $V$ of $G$. In order to describe those, we recall the notion of induced representation.
Let $G$ be a topological group and $H$ be an subgroup of $G$. Let $\sigma$ be a representation of $H$. The $\Ind_H^G(\sigma)$ is defined as follows. Let $T$ be a complete system of left coset representatives of $H$ in $G$. Let $M$ be the space of all functions $f \colon G \to \mcH(\sigma)$ for which $$\label{eq:induced-cond}
f(gh) = \sigma(h^{-1})f(g) \quad \text{for all } g \in G, h \in H.$$ In particular, for $f \in M$, $\nm{f(x)}$ is constant on left cosets of $H$. For $f \in M$, define $$\label{eq:induced-nm}
\nm{f} = \Big(\sum_{g \in T} \nm{f(g)}^2 \Big)^{1/2}$$ and note that because of the above observation, $\nm{f}$ does not depend on the choice of $T$. Let $\mcH = \set{f \in M : \nm{f} < \infty}$. Then the representation $\Ind_H^G(\sigma)$ on the Hilbert space $\mcH$ is defined by $$\big(\Ind_H^G(\sigma)(g) \cdot f \big)(x) = f(g^{-1}x).$$ As $H$ is open, the representation $\Ind_H^G(\sigma)$ is continuous. For example, the quasi-regular representation $\ell^2(G/H)$ can be written as $\Ind_H^G(1_H)$, where $1_H$ is the trivial one-dimensional representation of $H$.
We note that as we only need to induce from open subgroups $H \leq G$, the homogeneous space $G/H$ always carries the counting measure and we are spared the measure-theoretic complications that occur in the locally compact setting. For more details on induced representations, see, for example, [@Bekka2008]\*[Appendix E]{}.
Suppose now that $G$ is Roelcke precompact and fix an open subgroup $V \leq G$. Let $H$ be a subgroup of $G$ such that $V \unlhd H$. As for normal subgroups double cosets coincide with left cosets, $V$ has finite index in $H$. Denote by $K$ the finite group $H/V$ and by $\lambda_K$ the left-regular representation of $K$, which we will also consider as a representation of $H$. Then using the theorem about induction in stages ([@Bekka2008]\*[Theorem E.2.4]{}), we have $$\label{eq:dbl-ind}
\ell^2(G/V) \cong \Ind_V^G(1_V) \cong \Ind_H^G\big(\Ind_V^H(1_V)\big) \cong \Ind_H^G(\lambda_K).$$ As $\lambda_K$ splits as a sum of irreducible representations of $K$ (and in fact all irreducible representations of $K$ occur as direct summands), we are led to consider representations of the form $\Ind_H^G(\sigma)$, where $H$ is an open subgroup of $G$ and $\sigma$ is an irreducible representation of some finite quotient of $H$. There is a general criterion known as the Mackey irreducibility criterion for determining whether representations of the form $\Ind_H^G(\sigma)$ are irreducible for $H$ an open subgroup of $G$ and $\sigma$ an irreducible *finite-dimensional* representation of $H$. The criterion is usually stated for discrete groups but works equally well in this more general setting. It is due to Mackey [@Mackey1951] when $\sigma$ is one-dimensional and to Corwin [@Corwin1975] in the general case. Below we state and prove a special version of the criterion adapted to our situation.
If $H \leq G$, $g \in G$, and $\sigma$ is a representation of $H$, define the representation $\sigma^g$ of $H^g$ by $$\sigma^g(x) = \sigma(x^{g^{-1}}).$$
\[p:ind-reps\] Let $G$ be a Roelcke precompact subgroup of $S_\infty$. Then the following hold:
1. \[i:pir:irred\] If $H \leq G$ is a commensurator, $V \unlhd H$ is open, and $\sigma$ is a representation of $H/V$, then $\Ind_H^G(\sigma)$ is irreducible iff $\sigma$ is.
2. \[i:pir:equiv\] If $H_1, H_2 \leq G$ are commensurators, $V_1 \unlhd H_1$, $V_2 \unlhd H_2$ are open, and $\sigma_1$, $\sigma_2$ are irreducible representations of $H_1/V_1$, $H_2/V_2$, respectively, then $\Ind_{H_1}^G(\sigma_1) \cong \Ind_{H_2}^G(\sigma_2)$ iff there exists $g \in G$ such that $H_2 = H_1^g$ and $\sigma_2 \cong \sigma_1^g$.
. $(\Rightarrow)$ If $\sigma = \sigma_1 \oplus \sigma_2$, then $\Ind_H^G(\sigma) = \Ind_H^G(\sigma_1) \oplus \Ind_H^G(\sigma_2)$.
$(\Leftarrow)$ Suppose $\sigma$ is irreducible and denote $\pi = \Ind_H^G(\sigma)$. We first show that $$\label{eq:HV}
\mcH_V(\pi) = \set{f \in \mcH(\pi) : f(x) = 0 \text{ for } x \notin H}.$$ Suppose first that $f(x) = 0$ for all $x \notin H$. Let $g \in V$. If $x \notin H$, then $g^{-1}x \notin H$ and $0 = f(x) = f(g^{-1}x) = (\pi(g)f)(x)$. If $x \in H$, then $$(\pi(g)f)(x) = f(g^{-1}x) = f(xx^{-1}g^{-1}x) = \sigma(x^{-1}gx) f(x) = f(x)$$ as $\sigma$ is trivial on $V$. For the other direction, suppose that $f \in \mcH(\pi)$ is $V$-invariant and $x \in G$ is such that $f(x) \neq 0$. If $x \notin H$, then as $H$ is its own commensurator, by , $HxH$ contains infinitely many left cosets of $H$. Since $[H : V] < \infty$, $VxH$ also contains infinitely many left cosets of $H$. As $f$ is $V$-invariant, its norm must be infinite, contradiction. We thus obtain that $$\label{eq:sigma-is}
f \mapsto f(1) \text{ is an isomorphism between } \pi(H)|_{\mcH_V(\pi)} \text{ and } \sigma.$$
Now suppose that $\pi$ is reducible, i.e. $\mcH(\pi) = \mcK \oplus \mcK^\perp$, where $\mcK$ is $\pi(G)$-invariant. As the projection onto $\mcK$ commutes with $\pi(V)$, we have $$\mcH_V(\pi) = (\mcH_V(\pi) \cap \mcK) \oplus (\mcH_V(\pi) \cap \mcK^\perp)$$ and the two parts on the right-hand side are $\pi(H)$-invariant. By and the irreducibility of $\sigma$, either $\mcH_V(\pi) \sub \mcK$ or $\mcH_V(\pi) \sub \mcK^\perp$. Since by , $\mcH_V(\pi)$ is cyclic for $\pi$, we have that $\mcK = \mcH(\pi)$ or $\mcK^\perp = \mcH(\pi)$, proving that $\pi$ is irreducible.
. $(\Leftarrow)$ Let $T \colon \mcH(\sigma_1) \to \mcH(\sigma_2)$ be a unitary operator that realizes the equivalence $\sigma_1^g \cong \sigma_2$ (i.e. $\sigma_2 T = T \sigma_1^g$). Let $\pi_i = \Ind_{H_i}^G(\sigma_i)$ and define the map $U \colon \mcH(\pi_1) \to \mcH(\pi_2)$ by $$U(f)(x) = Tf(x^{g^{-1}})$$ It is not difficult to check that $U$ is a well-defined unitary equivalence between $\pi_1$ and $\pi_2$.
$(\Rightarrow)$ Suppose that $\pi_1$ and $\pi_2$ are equivalent. Then there exists a non-zero $f \in \mcH(\pi_1)$ which is invariant under $\pi_1(V_2)$. By the same argument as in , we obtain that there is $g \in G$ such that $V_2 g H_1$ contains only finitely many left cosets of $H_1$, or, equivalently, $[V_2 : V_2 \cap H_1^g] < \infty$. Symmetrically, we find $h \in G$ such that $[V_1 : V_1 \cap H_2^h] < \infty$. For two subgroups $A, B \leq G$, say that if $[B : A \cap B] < \infty$. As $[H_i : V_i] < \infty$, we have that $$\begin{aligned}
H_1^g &\text{ is large in } H_2, \quad \text{and} \label{eq:H1inH2} \\
H_2^h &\text{ is large in } H_1, \label{eq:H2inH1}\end{aligned}$$ so by conjugating by $h$ and using transitivity, we can conclude that $H_1^{hg}$ is large in $H_1$. Applying Lemma \[l:left-right\], we obtain that $H_1$ and $H_1^{hg}$ are commensurate and therefore equal (by Lemma \[l:commen\] ). Conjugating by $g^{-1}h^{-1}$, we obtain that $H_2^{g^{-1}}$ is large in $H_1^{g^{-1}h^{-1}} = H_1$, while conjugating by $g^{-1}$, we see that $H_1$ is large in $H_2^{g^{-1}}$, so that $H_1$ and $H_2^{g^{-1}}$ are commensurate and therefore equal. So finally, $H_2 = H_1^g$.
Now let $\pi_1' = \Ind_{H_1^g}^G(\sigma_1^g) = \Ind_{H_2}^G(\sigma_1^g)$. By the $(\Leftarrow)$ direction and the hypothesis, $\pi_1' \cong \pi_1 \cong \pi_2$. Let $U \colon \mcH(\pi_1') \to \mcH(\pi_2)$ realize the equivalence. Then we must have $U(\mcH_{V_2}(\pi_1')) = \mcH_{V_2}(\pi_2)$. By and the fact that $U$ commutes with the $H_2$-action, we have that $\sigma_2 \cong \sigma_1^g$.
We are now ready to prove the main theorem.
\[th:main-th\] Suppose that $G$ is a Roelcke precompact subgroup of $S_\infty$. Then every unitary representation of $G$ is a sum of irreducible representations of the form $\Ind_H^G(\sigma)$, where $H$ is a commensurator and $\sigma$ is an irreducible representation of $H$ that factors through a finite quotient of $H$.
Let $\pi$ be a representation of $G$. For $\xi \in \mcH(\pi)$, let $\phi_\xi(x) = \ip{\pi(x)\xi, \xi}$ be the positive definite function on $G$ associated to $\xi$. If $\xi$ is fixed by an open subgroup $V_0 \leq G$, then by Lemma \[l:const-dblcst\], $\phi_\xi$ is constant on double cosets of $V_0$, so the function $\phi_\xi$ takes only finitely many values. By Lemma \[l:dense-fp\], there exists some non-zero $\xi \in \mcH(\pi)$ which is fixed by an open subgroup. Choose now a non-zero $\xi_0 \in \mcH(\pi)$ such that $\phi_{\xi_0}$ takes the *minimum possible number of distinct values*. Let $$V = \set{g \in G : \pi(g)\xi_0 = \xi_0} = \set{g \in G : \phi_{\xi_0}(g) = \nm{\xi_0}^2}$$ and note that as the image of $\phi_{\xi_0}$ is discrete, $V$ is open.
If $g \notin \Comm_G(V)$, then $\phi_{\xi_0}(g) = 0$.
Let $g \notin \Comm_G(V)$ be arbitrary. By , $VgV$ contains infinitely many left cosets of $V$. Towards a contradiction, suppose that $\ip{\pi(g)\xi_0, \xi_0} \neq 0$. Let $h_1gV, h_2gV, \ldots$ be distinct left cosets of $V$ with $h_i \in V$ for all $i$. Set $\xi_i = \pi(h_i g)\xi_0$ and let $\eta$ be a weak limit point of the $\xi_i$s. By passing to a subsequence, we can assume that $\xi_i \to^w \eta$. Since $$\ip{\xi_i, \xi_0} = \ip{\pi(h_i g)\xi_0, \xi_0}
= \ip{\pi(g)\xi_0, \pi(h_i^{-1})\xi_0} = \ip{\pi(g)\xi_0, \xi_0}$$ is bounded away from $0$, we have that $\eta \neq 0$. Next we observe that the set of values of $\phi_\eta$ is a subset of the set of values of $\phi_{\xi_0}$. Indeed, fix $i \in \N$ and note that for all $x \in G$, we have $$\begin{split}
\ip{\pi(x)\eta, \xi_i} &= \ip{\eta, \pi(x^{-1})\xi_i} \\
&= \lim_{j \to \infty} \ip{\xi_j, \pi(x^{-1}h_ig)\xi_0} \\
&= \lim_{j \to \infty} \ip{\pi(h_j g) \xi_0, \pi(x^{-1}h_ig)\xi_0} \\
&= \lim_{j \to \infty} \phi_{\xi_0}(g^{-1}h_i^{-1} x h_j g).
\end{split}$$ Now, taking limits as $i \to \infty$, $$\phi_\eta(x) = \ip{\pi(x)\eta, \eta} = \lim_{i \to \infty} \ip{\pi(x)\eta, \xi_i} = \lim_{i \to \infty} \lim_{j \to \infty } \phi_{\xi_0}(g^{-1}h_i^{-1} x h_j g).$$ As the image of $\phi_{\xi_0}$ is discrete, $\phi_\eta(x)$ is a value of $\phi_{\xi_0}$.
On the other hand, note that $\nm{\eta} < \nm{\xi_0}$. Indeed, if $\nm{\eta} = \nm{\xi_0} = \nm{\xi_i}$, then $\xi_i$ converges to $\eta$ in norm, so for all $\eps > 0$, there exists $N$ such that for $i, j > N$, $\nm{\xi_i - \xi_j} < \eps$. As $\nm{\xi_i - \xi_j}$ can take only finitely many values, the sequence $\xi_i$ is eventually constant. This contradicts the assumption that $h_i g$ and $h_j g$ are in different left cosets of $V$. It follows that the set of values of $\phi_\eta$ is a strict subset of the set of values of $\phi_{\xi_0}$ (as $\nm{\xi_0}^2$ is a value of $\phi_{\xi_0}$ which is not a value of $\phi_\eta$), contradicting the choice of $\xi_0$. This completes the proof of the claim.
Put now $H = \Comm_G(V)$ and $V' = \bigcap_{h \in H} V^h$. Then $V' \unlhd H$ and as $V$ has finite index in $H$, $V'$ also has finite index in $H$. Let $\mcK = \Span\set{\pi(g)\xi_0 : g \in H}$ and note that $\mcK$ is finite-dimensional.
$\pi(x)\mcK \perp \pi(y)\mcK$ if $xH \neq yH$.
Let $g, h \in H$. We have: $$\ip{\pi(xg)\xi_0, \pi(yh)\xi_0} = \ip{\pi(h^{-1}y^{-1}xg)\xi_0, \xi_0} = \phi_{\xi_0}(h^{-1}y^{-1}xg) = 0.$$ The last equality follows from the fact that if $y^{-1}x \notin H$, then $h^{-1}y^{-1}xg \notin H$ and the previous claim.
Note now that $\mcK$ is fixed pointwise by $V'$. Indeed, if $g \in H$ and $h \in V'$, by the definition of $V'$, $hgV = gV$, so there exists $h' \in V$ such that $hg = gh'$ and $\pi(h)\pi(g)\xi_0 = \pi(gh')\xi_0 = \pi(g)\xi_0$. Thus we obtain a representation of $H$ on $\mcK$ that factors through $H/V'$. Denote this representation by $\sigma$. Let $T$ be a system of left coset representatives for $H$. We verify that the partial isometry $U \colon \mcH(\Ind_H^G(\sigma)) \to \mcH(\pi)$ given by $$U(f) = \boplus_{x \in T} \pi(x)f(x)$$ does not depend on the choice of $T$ and is a unitary equivalence between $\Ind_H^G(\sigma)$ and the cyclic subrepresentation of $\pi$ generated by $\xi_0$. That it does not depend on $T$ follows from ; to check that it intertwines the representations, observe that $$\begin{split}
U(\Ind_H^G(\sigma)(g)f)(x) &= \boplus_{x \in T} \pi(x) f(g^{-1}x) \\
&= \boplus_{x \in g^{-1}T} \pi(gx) f(x) \\
&= \pi(g)\big(U(f)(x)\big).
\end{split}$$ So we obtained that $\pi$ contains a subrepresentation that is isomorphic to $\Ind_H^G(\sigma)$, where $\sigma$ factors through a finite quotient of $H$. By passing to a subrepresentation if necessary, we can assume that $\sigma$ is irreducible. Then $\Ind_H^G(\sigma)$ is irreducible by Proposition \[p:ind-reps\]. Now using Zorn’s lemma, we conclude that $\pi$ is actually a sum of such representations.
Open subgroups and imaginaries {#s:examples}
==============================
In the previous section, we saw that in order to describe completely the representations of an automorphism group of an $\omega$-categorical structure, it suffices to understand the lattice of its open subgroups. It is most natural to understand the open subgroups in terms of the structure the group acts on. As we will see below (and as is well known), the open subgroups of the automorphism group correspond precisely to the imaginary elements of the structure. A particularly simple situation is when we can see all the open subgroups already in the structure itself, that is, when the structure in a suitable weak sense. We proceed now with the formal definitions.
Let $\bX$ be an $\omega$-categorical structure and $G = \Aut(\bX)$. Recall that by the Ryll–Nardzewski theorem, a set $A \sub \bX^n$ is (first-order) definable iff it is $G$-invariant. We are going to use the two terms interchangeably. A tuple $\bar b \in \bX^n$ is over $\bar a \in \bX^m$ if the orbit $G_{\bar a} \cdot \bar b$ is finite. The of $\bar a$ is the set of all $b \in \bX$ algebraic over $\bar a$. As $\bX$ is $\omega$-categorical, the algebraic closure of a finite set is always finite. An (or just an imaginary) is an equivalence class of some definable equivalence relation on a definable subset of $\bX^n$. If $\theta$ is a definable equivalence relation and $\bar a \in \bX^n$, $\bar a / \theta$ will denote the $\theta$-equivalence class of $\bar a$. If $\alpha = \bar a / \theta$ is an imaginary, we will denote by $G_\alpha$ the stabilizer of $\alpha$ in $G$. As $G_{\bar a} \leq G_\alpha$, $G_\alpha$ is open in $G$. Conversely, if $V \leq G$ is an open subgroup, there exists $\bar a$ such that $G_{\bar a} \leq V$. If we define the equivalence relation $\theta$ on $G \cdot \bar a$ by $$(g_1 \cdot \bar a) \eqrel{\theta} (g_2 \cdot \bar a) \iff g_1V = g_2V,$$ then $\theta$ is $G$-invariant and $V = G_{\bar a/\theta}$.
Recall that the structure $\bX$ admits if for every imaginary $\alpha$, there exists a first-order formula $\phi(\bar x, \bar y)$ such that the set $$\label{eq:def-im}
D(\phi, \alpha) = \big\{\bar c \in \bX^n : \alpha = \set{\bar x \in \bX^m : \phi(\bar x, \bar c)}\big\}$$ is finite and non-empty (that is, for every imaginary, we can choose finitely many tuples to represent it). The following lemma is folklore but I have not been able to find a suitable reference. I am grateful to Martin Hils for explaining it to me.
\[l:weak-el\] Suppose that $\bX$ is an $\omega$-categorical structure that admits weak elimination of imaginaries. Then for every open subgroup $V \leq G$, there exists a unique finite, algebraically closed substructure $\bA \sub \bX$ such that $G_{(\bA)} \leq V \leq G_{\bA}$ and $G_\bA = \Comm_G(V)$. In particular, every commensurator is of the form $G_\bA$ for some $\bA$.
By the preceding discussion, there exists an imaginary $\alpha$ such that $V = G_\alpha$. Let $\phi$ be a formula such that the set $D(\phi, \alpha)$ defined in is finite and non-empty. Let $\bA$ be the algebraic closure of $D(\phi, \alpha)$. Then clearly $G_{(\bA)} \leq V$. Also, from the definition of $D(\phi, \alpha)$, $V \cdot D(\phi, \alpha) = D(\phi, \alpha)$ and thus $V \cdot \bA = \bA$, so that $V \leq G_\bA$. The group $G_{(\bA)}$ has finite index in $G_\bA$ because it is equal to the kernel of the homomorphism $G_\bA \to \Aut(\bA)$ given by restriction. To prove that $G_\bA = \Comm_G(V)$, it suffices to check that $G_\bA$ has no proper supergroups in which it is of finite index. To see this, note that, by the definition of algebraically closed, if $g \cdot \bA \neq \bA$, then the orbit $G_\bA \cdot (g \cdot \bA)$ is infinite, showing that $G_\bA$ has infinite index in $\langle G_\bA, g \rangle$. The uniqueness of $\bA$ follows from the fact that the commensurator of $V$ is uniquely defined and two different algebraically closed substructures have different stabilizers.
In fact, the converse of Lemma \[l:weak-el\] also holds (see [@Hodges1993]\*[Exercise 7.3.16]{}) and for the examples we consider below, the conclusion of the lemma can easily be verified directly.
If the structure $\bX$ is moreover homogeneous (as defined in Section \[s:olig-groups\]), then for every finite substructure $\bA \sub \bX$, the canonical homomorphism $G_\bA \to \Aut(\bA)$ (that is, the restriction to $\bA$) is surjective and we have the following.
\[c:wk-elim\] Let $\bX$ be an $\omega$-categorical, homogeneous structure that admits weak elimination of imaginaries and $G = \Aut(\bX)$. Then the following is a complete list of the irreducible representations of $G$: $$\begin{gathered}
\set{\Ind_{G_\bA}^G(\sigma) : \bA \sub \bX \text{ is finite, algebraically closed and } \\
\sigma \text{ is an irreducible representation of } \Aut(\bA)}.\end{gathered}$$ Moreover, this list is without repetitions (only one substructure appears of each isomorphism type).
Every representation in the above list is irreducible by Proposition \[p:ind-reps\]. Conversely, let $\pi$ be an irreducible representation of $G$. By Theorem \[th:main-th\], there exist open subgroups $V \unlhd H \leq G$ such that $H = \Comm_G(V)$ and an irreducible representation $\sigma$ of $H/V$ such that $\pi = \Ind_H^G(\sigma)$. By Lemma \[l:weak-el\], there exists a finite, algebraically closed substructure $\bA \sub \bX$ such that $G_{(\bA)} \leq V$ and $G_\bA = H$. As the quotient map $G_\bA \to G_\bA/V$ factors through $G_\bA/G_{(\bA)} \cong \Aut(\bA)$, we can consider $\sigma$ as a representation of $\Aut(\bA)$. Finally, that the list is without repetitions follows from Proposition \[p:ind-reps\] and homogeneity (the groups $G_\bA$ and $G_\bB$ are conjugate iff $\bA$ and $\bB$ are isomorphic).
It is well known and not difficult to check that Examples – from the introduction admit weak elimination of imaginaries (see [@Hodges1993]\*[Section 4.2]{} for a general method to verify this), so in particular all of their irreducible representations are obtained by induction from representations of automorphism groups of finite substructures. In the case of $S_\infty$, we obtain the theorem of Lieberman [@Lieberman1972]. The finite groups that appear in the representations of $S_\infty$ and $\Homeo(2^\N)$ are the symmetric groups, while the ones associated to $\GL(\infty, \F_q)$ are $\GL(n, \F_q)$. As any finite group can be realized as the automorphism group of a finite graph, we see that the representations of $\Aut(\bR)$ encode the representations of all finite groups.
\[r:Olshanski\] In [@Olshanski1991a], Olshanski gives a description of the representations of another completion of the inductive limit $\varinjlim \GL(n, \F_q)$ which is different from our $\GL(\infty, \F_q)$. It is possible to obtain a proof of his result by our methods as follows. Let $\bV$ be the vector space over $\F_q$ with basis $\set{e_1, e_2, \ldots}$ and let $e_1', e_2', \ldots$ be the elements in the dual defined by $\ip{e_i, e_j'} = \delta_{ij}$. Let $\bV'$ be the (proper) subspace of the dual generated by $e_1', e_2', \ldots$. Our structure then consists of the disjoint union of the vector spaces $\bV$ and $\bV'$ together with binary relations for the pairing $\bV \times \bV' \to \F_q$. One checks that it is homogeneous and that its automorphism group is isomorphic to the one considered in [@Olshanski1991a]. The finite substructures of this structure are pairs of the form $(\bA, \bA')$, where $\bA$ is a finite subspace of $\bV$ and $\bA'$ is a finite subspace of $\bV'$.
The structure in Example does not admit weak elimination of imaginaries. Another standard example of an $\omega$-categorical structure that does not eliminate imaginaries is the group $\Gamma = (\Z/4\Z)^{<\N}$; the cosets of $\Gamma/2\Gamma$ are imaginaries that cannot be eliminated.
In $({{\mathbf Q}}, <)$, finite substructures are rigid, so we have the following.
\[c:repsQ\] The irreducible representations of $\Aut({{\mathbf Q}})$ are $$\Aut({{\mathbf Q}}) \actson \ell^2({{\mathbf Q}}^{[n]}), \quad n \in \N.$$
As $\Aut({{\mathbf Q}})$ can be densely embedded in $\Homeo^+(\R)$ (the latter being equipped with any of the uniform convergence, pointwise convergence, or compact-open topologies which coincide on it), we obtain the result of Megrelishvili [@Megrelishvili2001] mentioned in the introduction.
\[c:HomeoR\] The group $\Homeo^+(\R)$ has no non-trivial unitary representations.
Let $G = \Homeo^+(\R)$ and let $\pi$ be a unitary representation of $G$. Let $$H = \set{g \in G : g({{\mathbf Q}}) = {{\mathbf Q}}}$$ (here ${{\mathbf Q}}$ is regarded as a subset of $\R$) and note that $H$ is a continuous homomorphic image of $\Aut({{\mathbf Q}})$. By Corollary \[c:repsQ\], $\pi|_H$ is a direct sum $\boplus_i \ell^2({{\mathbf Q}}^{[n_i]})$. Define $g_k \in H$ by $g_k(x) = x + 1/k$. Then $g_k$ converges to the identity in $G$ but assuming that $n_i \neq 0$ for some $i$ and letting $\delta_\bA$ be the vector in $\ell^2({{\mathbf Q}}^{[n_i]})$ which is $1$ on some subset $\bA \in {{\mathbf Q}}^{[n_i]}$ and $0$ everywhere else, we obtain $\nm{\pi(g_k)\delta_\bA - \delta_\bA} = \sqrt{2}$ for all $k$, a contradiction with the continuity of $\pi$. Hence, $n_i = 0$ for all $i$ and $\pi$ is trivial on $H$. As $H$ is dense in $G$, $\pi$ must be trivial on $G$, too.
In [@Megrelishvili2001], Megrelishvili proves the stronger result that $\Homeo^+(\R)$ does not have any non-trivial representations by isometries on reflexive Banach spaces.
We finally remark that the continuity assumption in our definition of a representation is often not restrictive because of the phenomenon of automatic continuity. Say that a Polish group $G$ satisfies the if every homomorphism $f \colon G \to H$ to a separable group $H$ is continuous. This quite remarkable property has been verified for many non-locally compact groups, including Examples – from the introduction (for $S_\infty$, $\GL(\infty, \F_q)$, and $\Aut(\bR)$ it is due to Kechris–Rosendal [@Kechris2007a] and Hodges–Hodkinson–Lascar–Shelah [@Hodges1993a]; for $\Aut({{\mathbf Q}})$ and $\Homeo(2^\N)$, it is a result of Rosendal–Solecki [@Rosendal2007]).
Let $G$ be an oligomorphic group that satisfies the automatic continuity property. Then the conclusion of Theorem \[th:main-th\] applies to any (not a priori continuous) unitary representation of $G$ on a *separable* Hilbert space.
Of course, the condition that the Hilbert space be separable is necessary: every oligomorphic group considered as discrete has its left-regular representation which is not of the kind described in the theorem.
Property (T) {#s:propT}
============
We recall the definition of property (T) for topological groups.
\[df:propT\] Let $G$ be a group, $Q \sub G$, $\eps > 0$. If $\pi$ is a unitary representation of $G$, we say that a non-zero vector $\xi \in \mcH(\pi)$ is if for all $x \in Q$, $\nm{\pi(x)\xi - \xi} \leq \eps \nm{\xi}$. The topological group $G$ is said to have if there exist a compact $Q \sub G$ and $\eps > 0$ such that every representation $\pi$ of $G$ that has a $(Q, \eps)$-invariant vector, actually has an invariant vector. $G$ has the if $Q$ can be chosen to be finite. The set $Q$ is called a for $G$.
This property has mostly been studied for locally compact groups, where it has found many applications, but by now, there are also some non-locally compact examples. Of course, groups that have no unitary representations trivially have property (T) but there are also some large groups that have property (T) for non-trivial reasons. The first examples of this type were the loop groups over $\SL(n, \C)$ (Shalom [@Shalom1999]), another is the infinite-dimensional unitary group (Bekka [@Bekka2003]).
We note that property (FH), which is equivalent to (T) for locally compact Polish groups (see [@Bekka2008]), is strictly weaker in general. While all Roelcke precompact groups have property (FH) (Proposition \[p:OB\]), additional work is needed to find appropriate Kazhdan sets.
As every representation of a Roelcke precompact subgroup of $S_\infty$ splits as a sum of irreducibles, to verify that such a group has property (T), it suffices to find $(Q, \eps)$ such that $\sup_{g \in Q} \nm{\pi(g)\xi - \xi} \geq \eps$ for every non-trivial, irreducible $\pi$ and unit vector $\xi \in \mcH(\pi)$.
We have seen in the previous section that when an $\omega$-categorical structure $\bX$ admits weak elimination of imaginaries, all irreducible representations of $G = \Aut(\bX)$ can be extracted from the action $G \actson \bX$. As we are concerned with non-trivial representations, it will be convenient to disregard the points in $\bX$ that are fixed by $G$. Denote $\bX_0 = \bX \sminus \set{a \in \bX : G \cdot a = a}$. The next proposition shows that to verify property (T), it suffices to consider only tensor powers of the representation $\ell^2(\bX_0)$.
\[p:subreps\] Let $\bX$ be an $\omega$-categorical structure that admits weak elimination of imaginaries and $G = \Aut(\bX)$. Then every non-trivial, irreducible representation of $G$ is a subrepresentation of $\ell^2(\bX_0^n)$ for some $n > 0$.
Let $\pi$ be an irreducible representation of $G$. In the same way as in the proof of Corollary \[c:wk-elim\], we see that there exists an algebraically closed, finite substructure $\bA \sub \bX$ such that $\pi$ is equivalent to $\Ind_{G_\bA}^G(\sigma)$ for some irreducible representation $\sigma$ of the finite group $K = G_\bA/G_{(\bA)}$. As every irreducible representation of a finite group is contained in its left-regular representation, in the same way as in , we obtain: $$\pi \cong \Ind_{G_\bA}^G(\sigma) \leq \Ind_{G_\bA}^G(\lambda_K) \cong \ell^2(G/G_{(\bA)}).$$ If $k = |\bA|$, then $G/G_{(\bA)}$ is an orbit of the action $G \actson \bX^k$, so $\ell^2(G/G_{(\bA)}) \sub \ell^2(\bX^k)$. To finish the proof, we check by induction on $k$ that $\ell^2(\bX^k)$ splits as a direct sum $\boplus_{i = 1}^s \ell^2(\bX_0^{n_i})$. If $k = 0$, we can take $s = 1$ and $n_1 = 0$. Denote $Y = \bX \sminus \bX_0$ and observe that as the action $G \actson \bX$ has only finitely many orbits, $Y$ is finite. Now we deduce the statement for $k+1$ from the one for $k$: $$\begin{split}
\ell^2(\bX^{k+1}) \cong \ell^2(\bX) \otimes \ell^2(\bX^k)
&\cong \big(\ell^2(\bX_0) \oplus \ell^2(Y)\big) \otimes \big(\bigoplus_{i=1}^s \ell^2(\bX_0^{n_i})\big) \\
&\cong \bigoplus_{i=1}^s \ell^2(\bX_0^{n_i+1}) \oplus |Y| \cdot \bigoplus_{i=1}^s \ell^2(\bX_0^{n_i}).
\end{split}$$ Finally, as $\pi$ is irreducible and non-trivial, it is a subrepresentation of one of the $\ell^2(\bX_0^{n_i})$ for some $n_i > 0$.
If $Y$ is a set, denote by $\ell^1_+(Y)$ the subset of $\ell^1(Y)$ consisting of all non-negative $\ell^1$ functions of norm $1$.
\[l:nonam-prod\] Suppose that a group $G$ acts on a set $Y$ so that there exists a subset $Q \sub G$ and $\eps > 0$ such that for every $f \in \ell^1_+(Y)$, $$\label{eq:def-nonamen}
\sup_{g \in Q} \nm{g \cdot f - f}_1 \geq \eps.$$ Then holds also for every $f \in \ell^1_+(Y^n)$ (with the diagonal action $G \actson Y^n$) for every $n \geq 2$.
The proof is by induction on $n$. Let $f \in \ell^1_+(Y^{n+1})$. Define $\tilde f \in \ell^1_+(Y^n)$ by $\tilde f(\bar x) = \sum_{y \in Y} f(\bar x, y)$. Then for any $g \in G$, $$\begin{split}
\nm{g \cdot \tilde f - \tilde f}_1 &= \sum_{\bar x} |\tilde f(g^{-1} \cdot \bar x) - \tilde f(\bar x)| \\
&= \sum_{\bar x} \Big|\sum_y f(g^{-1} \cdot \bar x, y) - \sum_y f(\bar x, y)\Big| \\
&= \sum_{\bar x} \Big|\sum_y f(g^{-1} \cdot \bar x, g^{-1} \cdot y) - \sum_y f(\bar x, y)\Big| \\
&\leq \sum_{\bar x} \sum_y |f(g^{-1} \cdot \bar x, g^{-1} \cdot y) - f(\bar x, y)| = \nm{g \cdot f - f}_1,
\end{split}$$ showing that holds for $f$ if it holds for $\tilde f$.
\[p:KazhSet\] Suppose that $\bX$ is an $\omega$-categorical structure that admits weak elimination of imaginaries and $G = \Aut(\bX)$. Then if $Q \sub G$ is compact and $$\label{eq:equivT}
\inf_{f \in \ell^1_+(\bX_0)} \sup_{g \in Q} \nm{g \cdot f - f}_1 > 0,$$ $Q$ is a Kazhdan set for $G$. Conversely, if the action $G \actson \bX_0$ has only infinite orbits and $Q$ is a Kazhdan set for $G$, then holds.
Suppose first that holds and set $\eps = \inf_{f \in \ell^1_+(\bX_0)} \sup_{g \in Q} \nm{g \cdot f - f}_1$. By Proposition \[p:subreps\], to see that $Q$ is a Kazhdan set for $G$, it suffices to show that for every $n > 0$ and every $f \in \ell^2(\bX_0^n)$, $\sup_{g \in Q} \nm{g \cdot f - f}_2 \geq (\eps/2)\nm{f}_2$. Suppose that $f \in \ell^2(\bX_0^n)$, $\nm{f}_2 = 1$, $g \in G$. Set $\tilde f(x) = |f(x)|^2$. Then $\tilde f \in \ell^1_+(\bX_0^n)$ and we have $$\begin{split}
\nm{g \cdot \tilde f - \tilde f}_1 &= \sum_{\bar x \in \bX_0^n} \big| |f(g^{-1} \cdot \bar x)|^2 - |f(\bar x)|^2 \big| \\
& = \sum_{\bar x \in \bX_0^n} \big||f(g^{-1} \cdot \bar x)| - |f(\bar x)|\big| \cdot (|f(g^{-1} \cdot \bar x)| + |f(\bar x)|) \\
&= \ip{\big| g \cdot |f| - |f|\big|, g \cdot |f| + |f|} \\
&\leq \nm{g \cdot f - f}_2 \nm{g \cdot |f| + |f|}_2 \leq 2\nm{g \cdot f - f}_2.
\end{split}$$ Combining this with Lemma \[l:nonam-prod\] finishes the proof.
Now suppose that does not hold but there exists $\eps > 0$ such that $(Q, \eps)$ is a Kazhdan pair for $G$. Then there exists $f \in \ell^1_+(\bX_0)$ such that $\sup_{g \in Q} \nm{g \cdot f - f}_1 < \eps^2$. Using the inequality $|a-b|^2 \leq |a^2 - b^2|$, which holds for all non-negative real numbers $a$ and $b$, we see that $f^{1/2}$ is a $(Q, \eps)$-invariant vector for the representation $\ell^2(\bX_0)$. By property (T), $\ell^2(\bX_0)$ has an invariant vector $f_0$ and then $\set{a \in \bX_0 : |f_0(a)| = \max |f_0|}$ is a finite $G$-invariant set in $\bX_0$, which is a contradiction with the hypothesis that the action $G \actson \bX_0$ has infinite orbits.
The next lemma shows that at least for certain well-behaved structures, we can always construct a compact set $Q \sub G$ that satisfies . Recall that an $\omega$-categorical structure has if the algebraic closure of every finite substructure $\bA$ is $\bA$ itself. This is equivalent to the condition that the stabilizer $G_{(\bA)}$ has infinite orbits on $\bX \sminus \bA$.
\[l:consrQ\] Let $\bX$ be a homogeneous, relational structure with no algebraicity and let $G = \Aut(\bX)$. Then there exists a compact set $Q \sub G$ such that for every $f \in \ell^1_+(\bX)$, $$\sup_{g \in Q} \nm{g \cdot f - f}_1 \geq 1/2.$$
We will use a back-and-forth construction. Enumerate $\bX = \set{a_1, a_2, \ldots}$ and set $\bA_n = \set{a_i : i \leq n}$ for $n \geq 0$. We will define inductively finite families $S_1, S_2, \ldots$ of finite partial automorphisms of $\bX$ with the following properties:
1. $S_1 = \set{\emptyset}$;
2. \[i:unique-rest\] if $\phi \in S_{n+1}$, there exists a unique $\phi' \in S_n$ such that $\phi \supseteq \phi'$;
3. \[i:dom-ran\] for every $\phi \in S_{2n}$, $\bA_n \sub \dom \phi$ and for every $\phi \in S_{2n+1}$, $\bA_n \sub \ran \phi$;
4. \[i:inter\] for every $\phi \in S_{2n} \cup S_{2n+1}$, $\dom \phi \cap \ran \phi \sub \bA_n$;
5. \[i:cond-odd\] for every $\phi \in S_{2n-1}$, there exist $\psi_1, \psi_2, \ldots, \psi_{2^{n+1}} \in S_{2n}$ such that $\psi_i \supseteq \phi$ for every $i$ and $\ran \psi_i \cap \ran \psi_j = \ran \phi$ for all $i \neq j$;
6. \[i:cond-even\] for every $\phi \in S_{2n}$, there exist $\psi_1, \psi_2, \ldots, \psi_{2^{n+1}} \in S_{2n+1}$ such that $\psi_i \supseteq \phi$ for every $i$ and $\dom \psi_i \cap \dom \psi_j = \dom \phi$ for all $i \neq j$.
One can view the sets $S_n$ as the levels of a finitely splitting rooted tree.
Recall that for a homogeneous structure the no algebraicity assumption can be reformulated as the following extension property (see Hodges [@Hodges1993]):
> for every pair of finite structures $\bA, \bB$, embeddings $\psi \colon \bA \to \bX$ and $\theta \colon \bA \to \bB$, and finite set $D \sub \bX$, there exists an embedding $\phi \colon \bB \to \bX$ such that $\phi \circ \theta = \psi$ and $\phi(\bB) \cap D \sub \psi(\bA)$.
Start the construction with $S_1 = \set{\emptyset}$. Suppose now that $S_{2n-1}$ has been constructed and proceed to build $S_{2n}$. For every $\phi \in S_{2n-1}$, we will construct a set $E_\phi$ of extensions of $\phi$ to put in $S_{2n}$. If $a_n \in \dom \phi$, set $E_\phi = \set{\phi}$. If $a_n \notin \dom \phi$, let $\bB = \dom \phi \cup \set{a_n}$ and construct $E_\phi = \set{\psi_1, \psi_2, \ldots, \psi_{2^{n+1}}}$ inductively as follows. Suppose $\psi_1, \ldots, \psi_k$ have been constructed and apply the extension property to the inclusion map $\dom \phi \to \bB$ and the embedding $\phi \colon \dom \phi \to \bX$ to obtain a partial automorphism $\psi_{k+1}$ with domain $\bB$ that extends $\phi$ and such that $$\label{eq:disj}
\ran \psi_{k+1} \cap \Big(\bigcup_{i \leq k} \ran \psi_i \cup \bB \Big) \sub \ran \phi.$$ Finally, set $S_{2n} = \bigcup_{\phi \in S_{2n-1}} E_\phi$. To construct $S_{2n+1}$ apply the same procedure symmetrically (replacing $\phi$ with $\phi^{-1}$). Conditions , , , and the existence part of are satisfied by construction. For the uniqueness part of , note that any two distinct elements of $S_n$ are incomparable. Finally, suppose that condition is not verified and let $i$ be the least natural number such that there exists $\phi_i \in S_i$ and $a \in (\dom \phi_i \cap \ran \phi_i) \sminus \bA_{\lfloor i/2\rfloor}$. We will consider the case when $i = 2n$ is even, the other one being similar. Denote by $\phi_{2n-1}$ the element in $S_{2n-1}$ such that $\phi_{2n-1} \sub \phi_{2n}$. By , $\ran \phi_{2n} \cap \dom \phi_{2n} \sub \ran \phi_{2n-1}$, so $a \in \ran \phi_{2n-1}$. On the other hand, by construction, $\dom \phi_{2n} = \dom \phi_{2n-1} \cup \set{a_n}$. Hence $a \in \dom \phi_{2n-1}$, a contradiction with the minimality of $i$.
Now $\set{S_n}_n$ forms naturally an inverse system with the maps $$S_{n+1} \to S_n, \quad \phi \mapsto \text{the unique } \phi' \in S_n \text{ such that } \phi' \sub \phi.$$ Denote by $Q$ the inverse limit, i.e. $$Q = \set{g \in G : \forall n \exists \phi \in S_n \ \phi \sub g}.$$ (Every map in the inverse limit is a full automorphism of $\bX$ because of condition .) As the sets $S_n$ are finite, $Q$ is compact.
We now check that for any $f \in \ell^1_+(\bX)$, there exists $g \in Q$ such that $\nm{g \cdot f - f}_1 \geq 1/2$. We build inductively a sequence $\phi_1, \phi_2, \ldots$ such that $\phi_i \in S_i$ and $\phi_i \sub \phi_{i+1}$. Let $\phi_1 = \emptyset$. Suppose that $\phi_{2n-1}$ is already chosen. If $a_n \in \dom \phi_{2n-1}$, set $\phi_{2n} = \phi_{2n-1}$. If $a_n \notin \dom \phi_{2n-1}$, by and the fact that $\nm{f}_1 = 1$, there exists $\phi_{2n} \in S_{2n}$ such that $$\label{eq:phi-l}
f(\phi_{2n}(a_n)) \leq 2^{-(n+1)}.$$ Now we proceed to choose $\phi_{2n+1}$. If $a_n \notin \dom \phi_{2n-1}$, let $\phi_{2n+1} \in S_{2n+1}$ be an arbitrary extension of $\phi_{2n}$. If $a_n \in \dom \phi_{2n-1}$, then by , $a_n \notin \ran \phi_{2n-1}$, so $a_n \notin \ran \phi_{2n}$ either (as in this case, $\phi_{2n} = \phi_{2n-1}$). Now, using , we can choose $\phi_{2n+1} \in S_{2n+1}$ so that $$\label{eq:phi-r}
f(\phi_{2n+1}^{-1}(a_n)) \leq 2^{-(n+1)}.$$ In summary, we obtained that for every $n$ at least one of the inequalities and holds. Let $g = \bigcup_i \phi_i$. Combining and , we have that for every $n$, $$\min \big( f(a_n), f(g^{-1} \cdot a_n) \big) \leq 2^{-(n+1)}.$$ Now a calculation yields: $$\begin{split}
\nm{g \cdot f - f}_1 &= \sum_{n=1}^\infty |f(g^{-1} \cdot a_n) - f(a_n)| \\
&= \sum_{n=1}^\infty \max \big( f(a_n), f(g^{-1} \cdot a_n) \big)
- \min \big( f(a_n), f(g^{-1} \cdot a_n) \big)\\
&\geq \nm{f}_1 - \sum_{n=1}^\infty \min \big( f(a_n), f(g^{-1} \cdot a_n) \big) \\
&\geq 1 - \sum_{n = 1}^\infty 2^{-(n+1)} = 1/2,
\end{split}$$ finishing the proof.
Combining everything we have so far, we obtain the following.
\[th:propT-gen\] Let $\bX$ be an $\omega$-categorical, relational, homogeneous structure with no algebraicity that admits weak elimination of imaginaries. Then $\Aut(\bX)$ has property (T).
Follows from Proposition \[p:KazhSet\] and Lemma \[l:consrQ\].
As the next theorem shows, in many concrete examples, it is not difficult to find *finite* Kazhdan sets. In fact, I do not know whether, in the setting of Theorem \[th:propT-gen\], it is always possible to do so (cf. Question ).
\[th:Tconcrete\] All of the following groups have a Kazhdan set with two elements: $$S_\infty, \ \Aut({{\mathbf Q}}), \ \GL(\infty, \F_q), \ \Homeo(2^\N), \ \Aut(\bR).$$
In view of Proposition \[p:KazhSet\], to prove that the automorphism group $G$ of an $\omega$-categorical structure $\bX$ with weak elimination of imaginaries admits a finite Kazhdan set, it suffices to find a finitely-generated subgroup $\Gamma \leq G$ such that the action $\Gamma \actson \bX_0$ is (i.e. holds with $Q$ a generating set for $\Gamma$). In fact, for all of the above groups we will find a copy of the free group $\bF_2$ in $G$ that acts freely on $\bX_0$. As it is well-known that any free action of $\bF_2$ is non-amenable, this will complete the proof.
$S_\infty$. Consider the left action of $\bF_2$ on itself. This gives a copy of $\bF_2$ in the group of all permutations of $\bF_2$ that acts freely.
$\Aut({{\mathbf Q}})$. It suffices to find an ordering on the group $\bF_2$ which is left-invariant and isomorphic to ${{\mathbf Q}}$. Then the left action of $\bF_2$ on itself will produce the required embedding $\bF_2 \hookrightarrow \Aut({{\mathbf Q}})$. It is well known that $\bF_2$ admits a (invariant under multiplication on both sides) linear ordering (see, for example, [@Rolfsen2001u]). We now check that any bi-invariant ordering on $\bF_2$ is dense without endpoints. First, if $x > 1$, then $x^2 > x$ and if $x < 1$, $x^2 < x$, showing that the ordering has no endpoints. To see that it is dense, it suffices, for every $x > 1$, to find $z$ such that $1 < z < x$. By bi-invariance, all conjugates of $x$ are $> 1$. Let now $y$ be an arbitrary element that does not commute with $x$. If $x^y < x$, we are done and if $x^y > x$, by conjugating with $y^{-1}$, we obtain that $x^{y^{-1}} < x$.
$\GL(\infty, \F_q)$. Label a basis of the vector space by the elements of $\bF_2$ and let $\bF_2$ act freely on this basis in the natural way. This action extends to an action on the whole vector space. As $\bF_2$ is torsion-free, no finite, non-empty subset of $\bF_2$ is invariant under any non-identity element of the group, so the support of every non-zero vector is moved by every non-trivial element of $\bF_2$, showing that the action is free (when restricted to the non-zero elements of the vector space).
$\Homeo(2^\N)$. Consider the natural shift action $\bF_2 \actson 2^{\bF_2}$ given by $(g \cdot \omega)(h) = \omega(g^{-1}h)$. Since this action is by homeomorphisms, it gives an action $\bF_2 \actson \Clopen(2^{\bF_2})$. Using a similar argument to the one above, we see that its restriction to $\Clopen(2^{\bF_2}) \sminus \set{\emptyset, 2^{\bF_2}}$ is free.
$\Aut(\bR)$. Consider a of the free group constructed in the following way. Let $S$ be a random set of unordered pairs $\set{g, g^{-1}}$ of elements of $\bF_2 \sminus \set{1}$, i.e. each pair is included or not in $S$ independently with probability $1/2$. The vertices of the graph are the elements of $\bF_2$ and two vertices $x, y \in \bF_2$ are connected by an edge iff $x^{-1}y \in S$. By Cameron [@Cameron2000], the random Cayley graph of $\bF_2$ is isomorphic to $\bR$ with probability $1$, so in particular, Cayley graphs of $\bF_2$ isomorphic to $\bR$ exist. As $\bF_2$ acts freely by isomorphisms on any of its Cayley graphs, we obtain the desired embedding $\bF_2 \hookrightarrow \Aut(\bR)$.
It is also possible, using the method of Bekka [@Bekka2003], to find optimal Kazhdan constants for those Kazhdan sets. In fact, the constant is the same as in [@Bekka2003].
We note that not all oligomorphic groups have the strong property (T). Indeed, the group in Example admits $(\Z / 2\Z)^\N$ as a quotient and the latter does not admit a finite Kazhdan set by [@Bekka2003]\*[Proposition 5]{} (this can also easily be seen directly).
As was already noted in [@Bekka2003], there is no special connection between property (T) and amenability for non-locally compact groups. Of the groups in Theorem \[th:Tconcrete\], all are amenable except $\Homeo(2^\N)$. (A topological group is called if every time it acts continuously on a compact space, there is an invariant measure.)
It is an open problem whether there exists a discrete subgroup of $\Aut({{\mathbf Q}})$ with property (T). (A discrete group embeds in $\Aut({{\mathbf Q}})$ iff it acts faithfully by orientation-preserving homeomorphisms on the reals iff it is left-orderable; see Morris [@Morris1994]). I am grateful to the referee for pointing this out.
We conclude with two open problems.
**Questions:**
1. Does every closed oligomorphic subgroup of $S_\infty$ have property (T)? More generally, does every Roelcke precompact Polish group have property (T)?
2. \[qi:non-amen\] Suppose that $\bX$ is an $\omega$-categorical, relational, homogeneous structure with no algebraicity. Is it always the case that the action $\Aut(\bX) \actson \bX$ is non-amenable, i.e. must there always exist a *finite* set $Q \sub \Aut(\bX)$ such that for every $f \in \ell^1_+(\bX)$, $$\sup_{g \in Q} \nm{g \cdot f - f}_1 > 0?$$
|
---
abstract: 'In the framework of the semiclassical approach the universal spectral correlations in the Hamiltonian systems with classical chaotic dynamics can be attributed to the systematic correlations between actions of periodic orbits which (up to the switch in the momentum direction) pass through approximately the same points of the phase space. By considering symbolic dynamics of the system one can introduce a natural ultrametric distance between periodic orbits and organize them into clusters. Each cluster consists of orbits approaching closely each other in the phase space. We study the distribution of cluster sizes for the backer’s map in the asymptotic limit of long trajectories. This problem is equivalent to the one of counting degeneracies in the length spectrum of the [*de Bruijn*]{} graphs. Based on this fact, we derive the probability $\P_k$ that $k$ randomly chosen periodic orbits belong to the same cluster. Furthermore, we find asymptotic behaviour of the largest cluster size $|\Cll_{\max}|$ and derive the probability $P(t)$ that a random periodic orbit belongs to a cluster of the size smaller than $t|\Cll_{\max}|$, $t\in[0,1]$.'
address: ' ${}^\dag$ Faculty of Physics, University Duisburg-Essen, Lotharstr. 1, 47048 Duisburg, Germany; ${}^*$ Institute of Theoretical Physics, Cologne University Zülpicher Str. 77, 50937 Cologne, Germany. '
author:
- 'Boris Gutkin${}^\dag$, Vladimir Osipov${}^*$'
title: 'Counting of Sieber-Richter pairs of periodic orbits'
---
Introduction
============
In their seminal paper [@bgs] Bohigas Giannoni and Schmit conjectured that the local energy spectrum statistics of quantum systems with fully chaotic classical dynamics are universal and can be described by standard ensembles of Random Matrix Theory (RMT). To explore the origin of such universality semiclassical techniques based on the applications of the Gutzwiller trace formula was introduced by Berry in [@berry]. By this approach the correlations between energy levels of a quantum Hamiltonian system can be related to the correlations between periodic orbit actions in the corresponding classical system. Within the diagonal approximation, where only correlations between periodic orbits themselves are taken into account, Berry managed to obtain the leading order of the universal spectral form factor. The diagonal approximation, however, turned out to be insufficient to reproduce full RMT result, whose derivation had remained a distinguished challenge for yet a long time [@ADDKKSS; @haake]. The breakthrough was achieved in 2001 when Sieber and Richter discovered a non-trivial mechanism of correlations between periodic orbit actions [@sr]. They showed that the next to the leading order term of the universal spectral form factor can be obtained by taking into account correlations between long periodic orbits with one self-crossing under a small angle (usually referred as [*encounter*]{}) and the ass! ociated partner orbits, see fig. \[sieber\_richter\]a. Such pairs of orbits have close actions and contribute systematically into the spectral correlations. Later, this approach was extended to include correlations between periodic orbits having an arbitrary number of encounters which culminated in the derivation of the full RMT result [@haake1].
In a nutshell Sieber-Richter pairs and its many-encounter analogs are nothing more then bunches of periodic orbits running through almost the same points of the phase space up to the switch in the momentum direction. Furthermore, in the case of broken time reversal symmetry only orbits passing close to each other with the same momentum direction are of relevance, see fig. \[sieber\_richter\]b. In the present paper we restrict consideration only to the latter case. Due to the hyperbolic nature of the dynamics the action difference between such trajectories is small and determined by the lengths of the encounters. Equally important, the correlation mechanism of Sieber-Richter pairs is robust to perturbations of the dynamical system. In fact, any hyperbolic system contains a large number of long periodic orbits with close actions which are not of Sieber-Richter type. However, it might be expected that, in general, the differences between their actions fluctuate enormously under perturbations of the system. Therefore, the contribution from a generic pair of periodic orbits is washed out after averaging (e.g., over ensemble of systems) and only pairs of Sieber-Richter type contribute systematically to the spectral correlations.
[ a)![ []{data-label="sieber_richter"}](sieber_richter.eps "fig:"){height="3.0cm"}0.5cm b)![ []{data-label="sieber_richter"}](sieber_richterb.eps "fig:"){height="2.2cm"} ]{}
The robustness of Sieber-Richter correlation mechanism can be easily understood by considering symbolic dynamics of the system. Assuming that a finite Markov partition exists, any periodic orbit can be encoded by a finite sequence of symbols $x=x_1 x_2\dots x_n $ from some alphabet [@licht]. The fact that two periodic orbits come close to each other in the phase space has a natural interpretation on the level of the corresponding symbol sequences. Namely, let $x$ and $y$ be two sequences of length $n$ such that any subsequence of length $p$ occurs in $x$ exactly the same number of times as in $y$. This property of sequences $x$ and $y$ is referred below as [*$p$-closeness*]{}. It is straightforward to see that any pair of $p$-close sequences defines in fact a pair of periodic orbits which are close with respect to the Euclidean metric of the phase space. Their metric closeness is controlled by $p$: the larger parameter $p$ is, the closer two periodic orbits approach each other. Accordingly, all periodic orbits of the system can be organized into a disjoint union of [*clusters*]{} of $p$-close orbits. Each cluster is composed of periodic orbits with close actions running through approximately the same points of the phase space in a different time order.
Motivated by the application to quantum chaos, questions regarding the number of periodic trajectories with close actions/lengths were previously addressed both in physics and mathematics literature in the context of several different models of chaotic dynamical systems: geodesics flows on manifolds of negative curvature [@sharp1], billiards [@uzy3; @petkov], quantum maps [@uzy1] and quantum graphs [@uzy2; @sharp2; @tanner; @Berkolajko]. In the present paper we study the phenomenon of clustering of periodic orbits on the level of symbolic dynamics. Our consideration is restricted to the simplest possible grammar assuming a two-letter alphabet, $x_i\in\{0,1\}$, and absence of pruning, i.e. each symbol in the sequence can be followed by any other symbol. These grammar rules is met, for instance, in the baker’s map [@backer]. The main question to be addressed below can be informally stated as follows: given an integer $p$, what is the probability that a randomly picked up periodic sequence (equiv. orbit) of length $n$ has a certain number of $p$-close partners when $n$ is large enough?
Definitions and main results
============================
The baker’s map
---------------
In what follows we consider clustering of periodic orbits within the paradigm model of chaotic system – the baker’s map $\T$. Explicitly, the action of $\T$ on the points $v=(\q,\p)$ of the two-dimensional phase space $V=[0,1)\times [0,1)$ is given by: $$\T\cdot v=\begin{cases}
(2\q,\frac{1}{2}\p)& \text{if $\q\in[0,\frac{1}{2})$},\\
(2\q-2,1-\frac{1}{2}\p)& \text{if $\q\in[\frac{1}{2}, 1)$},
\end{cases}$$ where $\q$ and $\p$ play the role of the coordinate and momentum respectively, see e.g., [@backer] for details. The baker’s map has an advantage of having a particularly simple symbolic dynamics. This allows to avoid cumbersome notation and makes the exposition more transparent.
Symbolic dynamics and periodic orbits
-------------------------------------
Symbolic dynamics is a standard tool widely used in the theory of hyperbolic dynamical systems. By this approach each point of the system phase space is identified with a sequence of symbols from a certain alphabet. Given such a representation the time evolution of the system takes a simple form.
To introduce symbolic dynamics it is necessary first to define Markov partition of the phase space $V$. A standard choice for the baker’s map is $V_0\sqcup V_1$, where $V_0=[0,\frac{1}{2})\times [0,1)$ and $V_1=[\frac{1}{2}, 1)\times [0,1)$. Then any point $v=(\q,\p)\in V$ can be uniquely encoded by a two-sided sequence $x_-.x_+$ of zeros and ones: $x_+=x_1x_2\dots$, $x_- =\dots x_{-2}x_{-1}x_{0}$, $x_i\in\{0,1\}$ using a simple algorithm: $x_i=0$ if $\T^i v\in V_0$ and $x_i=1$ if $\T^i v\in V_1$ for $i\in\mathbb{Z}$. Positive $i$ correspond to the “future” evolution of $v$, which is written in the $x_+$ subsequence. The coordinate $q$ expresses through this subsequence by the formula $ \q=0.x_+$. The “past” history of $v$ and the momentum $\p$ are defined by $x_-$ subsequence: $ \p=0.x_-$.
Within the above symbolic representation the time evolution of the system is given by the shift map $$\sigma:\; [\dots x_{-1}x_{0}.x_1x_2 x_3\dots]\to[\dots x_{-1}x_{0}x_1.x_2 x_3\dots],$$ which moves the separation point “$.$” between the future and the past in the sequence of symbols step by step.
All infinite periodic sequences composed of one and the same finite piece $x\in X_n$ correspond to periodic orbits of the system. Here and below symbol $X_n$ stands for the set of all possible sequences of zeroes and ones having the length $n$. Let $\gamma_x$ denotes the periodic orbit of the backer’s map associated with the sequence $x\in X_n$. Note, that two sequences $x$, $x'$ correspond to the same periodic trajectory if and only if they are related by the cyclic shift, i.e. if $x=x_1 x_2 \dots x_n$ and $x'=x_{i+1}\dots x_n x_1\dots x_i$ for some $i\in\{1,\dots n-1\}$ then correspond to one and the same periodic trajectory $\gamma_x=\gamma_{x'}$. In what follows we will also consider the quotient set $\X_n:=X_n/\sim$ with respect to the cyclic shift $x\sim x'$. It is convenient to think of the elements belonging to the set $\X_n$ as of sequences from $X_n$ with the “glued” ends. Importantly, the elements from $\X_n$ and all periodic orbits having the period $n$ are in one to one correspondence according to the remark made above in this paragraph.
Clusters of periodic orbits
---------------------------
To define the clusters of periodic orbit, firsts, we need to introduce the notion of their closeness. Take two $n$-periodic orbits $\gamma_x$, $\gamma_y$ composed of $n$ points $\{\gamma_x(i)\}_{i=0}^{n-1}$, $\{\gamma_y(i)\}_{i=0}^{n-1}$ in the phase space. It is natural to think about two orbits $\gamma_x$, $\gamma_y$ as of close ones if in a vicinity of any point $\gamma_x(i)$, $i=0,\dots n-1$ one can find some point from the set $\{\gamma_y(i)\}_{i=0}^{n-1}$ and vice versa. In other words, two trajectories pass through approximately the same parts of the phase space but perhaps in a different order. To put this picture on a more solid background, we say $\gamma_x$ is in the $p$-neighborhood of $\gamma_y$ if there exist exactly $n$ pairs $\left(\gamma_x(i_k), \gamma_y(i'_k) \right)$, $k=1,\dots n$ such that for each $k$ distance between points is bounded by $||\gamma_x(i_k), \gamma_y(i'_k)||\leq 2^{-p}$, where the distance $||v,v'||$ in the phase space between $v=(\q,\p)$ and $v'= (\q',\p')$ is defined as $||v,v'||:=\max\{|\q-\q'|, |\p-\p'|\} $.
The above notion of metric closeness between periodic trajectories can be carried over to the topological space $\X_n$. We will say that two sequences $x,y\in \X_n$ are $p$-close if any sequence of $p\leq n$ consecutive symbols $a_1 a_2 \dots a_p$, $a_i \in\{0,1\}$ appears the same number of times (which might be also zero) both in $x$ and $y$. Speaking informally $x$ and $y$ are $p$-close if their local content (of the length $p$) is exactly the same in both sequences. This equivalence relation is denoted below as $x \pclone y$. It is straightforward to see that $\gamma_x$ is in the $p$-neighborhood of $\gamma_y$ whenever $x \pclone y$.
[ ![ []{data-label="fig2"}](hierarhy_ex.eps "fig:"){height="7.8cm"} ]{}
There are two simple but important properties of the equivalence relation $x \pclone y$ which should be emphasized:
- The relations $x \pclone y$ and $x \pclone z$ also imply that $z \pclone y$;
- The relation $x \ppclone y$ implies that $x \pclone y$.
According to the first property all periodic sequences (resp. periodic orbits) can be separated into a number of clusters $\Cll^{(p)}_i$, $i=1,\dots \N_p$ such that two sequences $x$ and $y$ (resp. $\gamma_x$, $\gamma_y$) belong to the same cluster if and only if $x \pclone y$. For instance, for given $p=2$ three sequences $[1101000]$, $[1100010]$, $[1100100]$ belong to the same cluster, see fig. \[fig2\]. In a completely analogous way one can consider clusters $\Cl^{(p)}_i$, $i=1,\dots \N_p$ of sequences from the set $X_n$. The connection between clusters is given by: $\Cll^{(p)}_i= \Cl^{(p)}_i/\sim$. In other words, each $x\in \Cll^{(p)}_i$ corresponds to a set of sequences from $\Cl^{(p)}_i$ which are related to each other by a cyclic shift.
The second property of $x \pclone y$ allows to organize the clusters of periodic sequences in a tree like structure. The $p$-th level of the tree contains clusters of $p$-close periodic sequences, see figs. \[fig2\],\[fig1\]. One can introduce a distance in the space of sequences based on this hierarchical structure. The distance $d(x,y)$ between two elements $x$ and $y$ being proportional to the maximal level of the tree where $x$ and $y$ belong to the same cluster: $d(x,y)=n-\max\{ p |x \pclone y\}$ ($d(x,x)=0$) satisfies the ultrametric property $d(x,y)\leq\max\{ d(x,z), d(z,y)\}$ [@VVZ1994]. By the identification of periodic orbits $\gamma_x$, $\gamma_y$ with the corresponding sequences $x,y$, one can lift the distance $d(x,y)$ to the space of periodic orbits. And we come to the conclusion that the space $\X_n$ (or equivalently the space of periodic orbits) acquires a natural ultrametric structure.
[ ![ []{data-label="fig1"}](hierarhy.eps "fig:"){height="5.8cm"} ]{}
Cluster distribution
--------------------
The primary goal of the present paper is to understand how the distribution of the cluster sizes $|\Cll^{(p)}_i|$ depends on the level $p$ in the limit $n\to\infty$. To this end, we need to estimate the moments of the cluster sizes: $$\Z_k=\sum^{\N_p}_{i=1} |\Cll^{(p)}_i|^{k}. \label{mom1}$$ It turns out, however, that a more convenient object to consider is $$Z_k=\sum^{\N_p}_{i=1} |\Cl^{(p)}_i|^{k}, \label{mom2}$$ where the sum runs over the clusters of sequences from $X_n$ (rather than sequences from $\X_n$). The connection between cluster sizes in $X_n$ and $\X_n$ is particularly simple if $n$ is a prime number. In this case for each $x\in \X_n$ except $x^{(0)}=[00\dots0]$ and $x^{(1)}=[11\dots 1]$, there are exactly $n$ sequences from $X_n$ which are related by the cyclic shift. Two special clusters corresponding to sequences $x^{(0)}$, $x^{(1)}$ contain just one element. For all other clusters: $$|\Cl^{(p)}_i|=n|\Cll^{(p)}_i|.\label{connection}$$ This yields $$\Z_k=\frac{Z_k -2}{n^k} +2, \qquad d_n=\frac{2^n -2}{n} +2,$$ where $d_n$ stands for the total number of elements in $\X_n$. In the case when $n$ is not a prime number, the connection between cluster sizes in $X_n$ and $\X_n$ is not anymore trivial due to the presence of periodic orbits with a period less than $n$. However, if one includes only prime periodic orbits (whose period is exactly $n$) into clusters $\Cll^{(p)}_i$, $\Cl^{(p)}_i$ then the connection (\[connection\]) remains valid. As a matter of fact, the exclusion of non-prime periodic orbits can be justified on the ground of a standard argument that their number is negligible in comparison with the number of prime periodic orbits in the limit $n\to\infty$.[^1] We arrive to the following relationship between (\[mom1\]) and (\[mom2\]): $$\Z_k=\frac{Z_k}{n^k}\left(1+O(n^{-1})\right) \qquad d_n=\frac{2^n}{n}\left(1+O(n^{-1})\right). \label{superconec}$$ Note also that for a typical cluster $|\Cl^{(p)}_i|\left(1+O(n^{-1})\right)=n|\Cll^{(p)}_i|$.
It worth of mentioning that the rescaled moments have a simple interpretation as probabilities of finding a number of periodic orbits in the same cluster. Indeed, let $\gamma_i$, $i=1,\dots k$ be a set of $k\geq 2$ orbits randomly chosen from the total set $\Gamma_n=\{\gamma_x| x\in \X_n\}$ of periodic orbits of length $n$. Then the probability that all $k$ orbits belong to the same cluster is given by $$\P_{k}=
\frac{\Z_{k}}{(d_n)^{k}}. \label{probab}$$ In particular, the probability $ \P_{2}$ that two periodic orbits belong to the same cluster is given by $\Z_{2}/d_{n}^2$.
Main Results
------------
The central result of the present paper is the following asymptotic formula for $Z_{k}$ in the limit $n\to\infty$: $$Z_{k}
= 2^{nk}\left( \frac{1}{k}\right)^{2^{p-2}}\left(\frac{2^p}{\pi n}\right)^{(k-1)2^{p-2}}\left(1+O(n^{-1})\right).\label{moments}$$ Using then eq. (\[superconec\]) and eq. (\[probab\]) we obtain the probability of finding $k$ random orbits in the same cluster $$\P_{k}
= \left( \frac{1}{k}\right)^{2^{p-2}}\left(\frac{2^p}{\pi n}\right)^{(k-1)2^{p-2}}\left(1+O(n^{-1})\right).\label{probfin}$$
In addition, we show that the number of periodic orbits in the largest cluster $\Cll^{(p)}_{\max}$ is asymptotically given by: $$|\Cll^{(p)}_{\max}|
= \left(\frac{2^{n}}{n}\right)\left(\frac{2^p}{\pi n}\right)^{2^{p-2}}\left(1+O(n^{-1})\right).\label{maxcluster}$$ Based on eqs. (\[moments\],\[maxcluster\]) we deduce probability $P(t)$, $t\in [0,1]$ that random periodic orbit from the set $\Gamma_n$ belongs to a cluster with the size less then $t |\Cll^{(p)}_{\max}|$ and show that in the limit of $n\to\infty$ this probability depends only on $p$: $$P(t)=\int_0^t\rho(\tau)d\tau,\qquad \rho(\tau)=\frac{\left(\log \tau\right)^{2^{p-2}-1}}{(2^{p-2}-1)!}.$$
As we show in the body of the paper the problem of counting cluster distribution of $p$-close periodic orbits is in fact equivalent to the one of counting degeneracies in the length spectrum of the so-called [*de Bruijn*]{} graphs [@Bruijn]. In this context the asymptotic behavior of $Z_{2}$ and related questions have been considered previously in [@sharp2; @uzy2; @Berkolajko; @tanner]. The exact connection between our results and the above mentioned works is discussed in the last section of the paper.
Organization of the paper
-------------------------
The paper is organized as follows. In Sec. 3 we show that the problem of counting cluster sizes of $p$-close periodic orbits in the backer’s map can be cast in the form of counting closed paths on a certain graph passing through the same edges (or vertices). Using this connection we express $Z_k$ as a matrix integral of certain type. In Sec. 4 we evaluate these integrals in the saddle point approximation and derive eq. (\[moments\]). In Sec. 5 we obtain asymptotic formula for the size of the largest cluster $\Cll^{(p)}_{\max}$. Using then the results from Sec. 4 we arrive to the probability $P(t)$ of finding a periodic orbit in a cluster of the size smaller than $t |\Cll^{(p)}_{\max}|$. Sec. 6 is devoted to the discussion of uniformity of periodic orbits distribution over the graph. Finally, the concluding remarks are presented in Sec. 7.
Clusters of closed paths on graphs
==================================
As we show below, the counting problem of $p$-close periodic orbits is equivalent to the one of counting closed paths on the de Bruijn graph $G_p$ passing the same number of times through its edges. The graph $G_p$ is constructed in the following way. With each sequences $a=[a_1a_2\dots a_p]$, $a\in X_p$ we associate a directed edge $e_a$ of $G_p$ whose initial and terminal points are denoted by $e_a^{\scriptscriptstyle{(in)}}$ and $e_a^{\scriptscriptstyle{(out)}}$, respectively. The connections between $2^p$ edges are fixed by the rule: for any pair of edges $e_a$, $e_b$, defined by the sequences $a=[a_1a_2\dots a_p]$, $b=[b_1b_2\dots b_p]$ the endpoints $e_a^{\scriptscriptstyle{(in)}}$, $e_b^{\scriptscriptstyle{(out)}}$ belong to the same vertex if and only if $a_i=b_{i+1}$ for all $i=1,\dots p-1$, see fig. \[graphs\].
[ ![ []{data-label="graphs"}](graph.eps "fig:"){height="5.8cm"} 1.2cm ![ []{data-label="graphs"}](graph1.eps "fig:"){height="5.8cm"} ]{}
It is straightforward to see that any closed path $g$ on the graph $G_p$ passing through $n$ edges can be uniquely represented by a sequence $x=[x_1x_2\dots x_n]$ from the set $ X_n$. By such identification $i$’th edge of $G_p$ passed by $g$ corresponds to the segment $[x_ix_{i+1}\dots x_{p-1+i}]$ of the sequence $x$. We will use notation $g_x$ to denote closed paths corresponding to $x\in X_n$. For each closed path $g_x$ let $\bm n (x) = \{n_a, a\in X_p\}$ be the set of integers, such that $n_a$ is the number of times $g_x$ passes through the edge $a$. Then $x\pclone y$ if and only if $g_x$, $g_y$ go through every edge of $G_p$ the same number of times (but in different time order) i.e., $\bm n (x) = \bm n (y)$. Therefore, each cluster $\Cl_{\bm n}$ of $p$-close periodic orbits is uniquely determined by the vector of integers $\bm n=\{n_a, a\in X_p\}$.
a\) By attaching a length to each edge of the graph $G_p$ one can turn it to a metric graph. Note that each cluster $\Cl_{\bm n}$ consists then of trajectories having the same length. Accordingly, counting of cluster sizes is equivalent to counting of degeneracies in the length spectrum of the corresponding metric graph. The last problem have been studied in [@uzy2; @sharp2; @tanner; @Berkolajko] for some classes of metric graphs, see also discussion in Sec. 7.
b\) Counting of closed paths passing the same number of times through the edges of $G_p$, is actually equivalent to counting of closed paths passing the same number of times through the vertices of twice larger graph $G_{p+1}$. Indeed, let us enumerate the $2^p$ vertices of the graph $G_{p+1}$ in the same way as edges of $G_p$, see fig. \[graphs\]. By the identification of each closed path of the length $n$ with a sequence $x\in X_n$ we obtain one-to-one correspondence between closed paths on two graphs. Correspondingly, clusters of closed paths have the same sizes.
To find the size $|\Cl_{\bm n}|$ of $\bm n$’th cluster we need to count the number of closed paths which go through the edges $a\in X_p$ of $G_p$ exactly $n_a$ times. To this end we introduce connectivity matrix $Q$ between edges of the graph. It is convenient to use for this purpose a tensorial representation for the vector space on which $Q$ acts, see [@GO; @Gu]. Let $\H$ be the $2^p$-dimensional linear space spanned by the vectors $|a\rangle= |a_1\rangle\otimes|a_2\rangle\otimes\dots\otimes|a_p\rangle$, $a_i \in\{0,1\}$. The linear operator $Q$ acts on the vectors from $\H$ according to the rule: $$\label{MatrixQ_tensor}
Q|a_1\rangle\otimes|a_2\rangle\dots\otimes|a_p\rangle=|a_2\rangle\otimes\dots\otimes|a_p\rangle\otimes\frac{1}{2}\left(|0\rangle +|1\rangle\right).$$ Note that this definition agrees with the connectivity rules between edges on the graph $G_p$, where each edge $e_a$, $a=[a_1a_2\dots a_p]$ is connected with the edges $e_{a'}$, $a'=[a_2\dots a_p 0]$ and $e_{a''}$, $a''=[a_2\dots a_p 1]$. In addition, with each edge $e_a$ we associate a phase $\phi_a$ and define the diagonal operator $\Lambda(\bm \phi )$, $\bm \phi:=\{\phi_a| a\in X_p\}$: $$\Lambda(\bm \phi )|a_1\rangle\otimes|a_2\rangle\dots\otimes|a_p\rangle=e^{i\phi_a}|a_1\rangle\otimes|a_2\rangle\dots\otimes|a_p\rangle.$$ It is straightforward to see that in the matrix form $Q$, $\Lambda(\bm \phi )$ can be written as [@GO]:
$$Q =
\begin{pmatrix}
1&0 & \dots&0&1&0 & \dots&0\\
1&0 &\dots&0 &1&0 &\dots&0 \\
0&1 &\dots&0 &0&1 &\dots&0\\
0&1 & \dots&0 &0&1 &\dots&0\\
\vdots& \vdots& \ddots &\vdots&\vdots& \vdots& \ddots &\vdots\\
0&0 & \dots&1 & 0&0 & \dots&1\\
0&0 & \dots &1 & 0&0 & \dots &1
\end{pmatrix},
\quad
\Lambda(\bm \phi )=
\begin{pmatrix}
e^{i\phi_1}&0 &0& \dots&0 \\
0&e^{i\phi_2} &0&\dots&0 \\
\vdots& \vdots& \ddots &\vdots& \vdots\\
0&0 &\dots&e^{i\phi_{2^p-1}} & 0\\
0&0 &\dots &0 & e^{i\phi_{2^p}}
\end{pmatrix}.$$
The introduction of matrices $Q$, $ \Lambda(\bm \phi )$ is useful because of the following relationship between traces of their products and the sizes of the clusters $\Cl_{\bm n}$: $$\trace (Q \Lambda(\bm \phi ))^n=\sum_{\bm n} |\Cl_{\bm n}|\exp{\left( i(\bm n, \bm \phi )\right)}, \qquad (\bm n, \bm \phi )=\sum_{a\in X_p} n_a \phi_a,\quad n=\sum_{a\in X_p} n_a,\label{KeyFormula}$$ with the first sum running over all clusters $\Cl_{\bm n}$. Eq. (\[KeyFormula\]) is a key component of our analysis, as it allows to express $|\Cl_{\bm n}|$ through the traces of powers of matrix $Q \Lambda(\bm \phi )$. In particular, the second moment $Z_2$ can be represented in the form of the integral over $\phi_a$: $$Z_2=\sum_{\bm n} |\Cl_{\bm n}|^2=\prod_{a\in X_p}\int_{0}^{2\pi}\frac{d\phi_a}{2\pi} \, |\trace (Q \Lambda(\bm \phi ))^n |^2. \label{SecondMoment}$$ Analogously, higher order moments are given by $$Z_k=\sum_{\bm n} |\Cl_{\bm n}|^k=\prod_{j=1}^k\prod_{a\in X_p}\int_{0}^{2\pi}\frac{d\phi^{(j)}_a}{2\pi} \, \trace \left(Q \Lambda(\bm \phi^{(j)} )\right)^n\delta\left(\sum_{l=1}^k \phi^{(l)}_a\right). \label{Moments}$$
As we show below the number of integration and dimensions of matrices in eqs. (\[SecondMoment\], \[Moments\]) can be actually reduced by the factor of two. Note that the $2^{p}\times 2^{p} $ matrix $Q$ can be represented as the product $Q=R S$ of the matrices $$\label{RS}
R =
\begin{pmatrix}
1&0 &0 & \dots&0\\
1&0 &0 & \dots&0 \\
0&1 &0 & \dots&0 \\
0&1 &0 & \dots&0 \\
\vdots&\vdots & \vdots& \ddots &0\\
0&0 &0 & \dots&1 \\
0&0 &0 & \dots &1
\end{pmatrix}, \qquad
S =
\begin{pmatrix}
1& 0 &0 &0 &\dots& 1&0 &0 &0& \dots\\
0&1&0 & 0 &\dots& 0&1&0 &0& \dots\\
0& 0& 1 &0 &\dots& 0& 0& 1 &0& \dots \\
0& 0& 0 &1 &\dots& 0& 0& 0 &1& \dots \\
\vdots&\vdots& \vdots& \vdots&\ddots& \vdots&\vdots & \vdots& \vdots&\ddots
\end{pmatrix},$$ whose dimensions are $2^{p-1}\times 2^{p} $ and $2^{p}\times 2^{p-1} $, respectively. Changing the order of the matrix product in eq. (\[KeyFormula\]) yields then $$\trace (Q \Lambda(\bm \phi ))^n=\trace (S \Lambda(\bm \phi ) R)^n= \trace (Q' (\bm \phi ) )^n,$$ where $Q' (\bm \phi )= S\Lambda(\bm \phi ) R$ is $2^{p-1}\times 2^{p-1} $ matrix of the form $$\label{MatrixQpsik}
Q '(\bm \phi ) =
\begin{pmatrix}
e^{i\phi_1}&0 & \dots&0&e^{i\phi_{2^{p-1}+1}}&0 & \dots&0\\
e^{i\phi_2}&0 &\dots&0 &e^{i \phi_{2^{p-1}+2}}&0 &\dots&0 \\
0&e^{i\phi_3} &\dots&0 &0&e^{i\phi_{2^{p-1}+3}} &\dots&0\\
0&e^{i\phi_4} & \dots&0 &0&e^{i \phi_{2^{p-1}+4} } &\dots&0\\
\vdots& \vdots& \ddots &0&\vdots& \vdots& \ddots &0\\
0&0 & \dots&e^{i\phi_{2^{p-1}-1}} & 0&0 & \dots&e^{i\phi_{2^{p}}}\\
0&0 & \dots &e^{i\phi_{2^{p-1}}} & 0&0 & \dots &e^{i\phi_{2^{p}}}
\end{pmatrix}.$$ Furthermore, using the invariance of the trace under the transformation $$Q' (\bm \phi )\to \Lambda(\bm \delta )Q' (\bm \phi )\Lambda^{-1}(\bm \delta ),$$ where $\Lambda(\bm \delta )$ is an arbitrary diagonal matrix, we can exclude half of the integration variables in eqs. (\[SecondMoment\], \[Moments\]), see Appendix A. The result is the following expression for the moments $$Z_k=\prod_{j=1}^k\prod_{i=1}^{2^{p-1}}\int_{0}^{2\pi}\frac{d\phi^{(j)}_i}{2\pi} \, \trace \left(\Q (\bm \phi^{(j)} )\right)^n\delta\left(\sum_{l=1}^k \phi^{(l)}_i\right), \label{FinalMoments}$$ with the matrix $\Q (\bm \phi)$ given by $$\label{FinalMatrix}
\Q (\bm \phi ) =
\begin{pmatrix}
e^{i\phi_{1}}&0 & \dots&0&1&0 & \dots&0\\
e^{i \phi_{2}}&0 &\dots&0 &1&0 &\dots&0 \\
0&e^{i\phi_{3}} &\dots&0 &0&1 &\dots&0\\
0&e^{i \phi_{4} } & \dots&0 &0&1&\dots&0\\
\vdots& \vdots& \ddots &0&\vdots& \vdots& \ddots &0\\
0&0 & \dots&e^{i\phi_{2^{p-1}-1}} & 0&0 & \dots&1\\
0&0 & \dots &e^{i\phi_{2^{p-1}}} & 0&0 & \dots &1
\end{pmatrix}.$$ In the next section we will use eq. (\[FinalMoments\]) to obtain large $n$ asymptotic of $Z_k$.
Clustering Probability of $k$ orbits
=====================================
In order to evaluate moments $Z_k$ we will apply saddle point approximation to the integral (\[FinalMoments\]), where $n$ will play the role of a large parameter. To make the exposition more transparent we first consider below the second moment and later extend the result to all $k>2$.
Saddle point approximation for $Z_2$
-------------------------------------
For $k=2$ the integral (\[FinalMoments\]) can be written as
$$Z_2=\left(\prod_{i=1}^{2^{p-1}}\int_{0}^{2\pi}\frac{d\phi_i}{2\pi}\right)\cdot \exp{\F_n(\bm \phi )}, \qquad \F_n(\bm \phi )=
\log|\trace (\Q(\bm \phi ))^n |^2. \label{SecondMoment2}$$
It is easy to see that the global maximum of $\F_n(\bm \phi )$ is attained at $\bm \phi =\bm0$, where $\phi_i =0$, $i=1,\dots 2^{p-1}$ and $\F_n(\bm 0)=n\log 2$. We therefore need to expand $\F_n(\bm \phi )$ around zero up to the second order in $\bm \phi$ and then use saddle point approximation in (\[SecondMoment2\]). To evaluate derivatives of $\F_n(\bm \phi )$ it is convenient to use the following decomposition of the matrix $\Q(\bm \phi )$: $$\Q(\bm \phi )=\Lambda(\bm \phi )Q_0+Q_1.$$ Here $Q_0=[R\; 0]$, $Q_1=[0\; R]$ are matrices composed of two blocks. The first (resp. second) block is given by the matrix $R$ (from eq. (\[RS\])) of the dimension $2^{p-1}\times 2^{p-2}$, while the second (resp. first) one is $2^{p-1}\times 2^{p-2}$ matrix of zeroes (for the element-wise definition of $Q_0,Q_1$, see Appendix B). Straightforward calculations give then: $$\frac{\partial \F_n(\bm \phi )}{\partial\phi_j}= 2n\Re\left[\frac{i\trace\left(\Lambda(\bm \phi )P_j Q_0 (\Q(\bm \phi ))^{n-1}\right)}{\trace(\Q(\bm \phi ))^{n}}\right],
\label{FirstDeriv}$$ where $P_j$ denotes projection matrix on $j$ element of the basis, i.e., $(P_j)_{m,l}=\delta_{m,i}\delta_{l,i}$. It follows immediately that $\frac{\partial \Q(\bm \phi )}{\partial\phi_j}\rvert_{\bm \phi=0}=0$, as it should be for a saddle point. Taking an additional derivative in (\[FirstDeriv\]) yields $$\begin{gathered}
\frac{\partial^2 \F_n(\bm \phi )}{\partial\phi_j \partial\phi_i}\Big{\rvert}_{\bm \phi =0}=2n
\left[
\frac{-\sum_{k=0}^{n-2}\trace\left(P_i Q_0 Q^{k} P_j Q_0 Q^{n-k-2}
\right) +\delta_{i,j}\trace\left( P_i Q_0 Q^{n-1} \right) }{\trace \, Q^{n}} \right] \\
+2n^2\left[ \frac{\trace\left( P_iQ_0 Q^{n-1}\right) \trace\left( P_jQ_0 Q^{n-1}\right) }{ (\trace \, Q^{n})^2 }
\right] ,
\label{SecondtDeriv}\end{gathered}$$ implying that $$\frac{\partial^2 \F_n(\bm \phi )}{\partial\phi_j \partial\phi_i}\Big{\rvert}_{\bm \phi =0}=-n B_{i,j},$$ where the matrix $B$ is defined by: $$B=2^{-p-1}\left(\bar{Q}+\bar{Q}^T +2 I -(1+2p)\left(\frac{Q}{2}\right)^p \right), \qquad \bar{Q}= Q_0\sum_{k=0}^{p-1}
\left(\frac{Q}{2}\right)^k.\label{MatrixB}$$ We can use now (\[SecondMoment2\]) to evaluate $Z_2$ in the large $n$ limit: $$\begin{aligned}
\label{MomentZ2_result}
Z_2& =2^{2n}\left(\prod_{j=1}^{2^{p-1}}\int_{0}^{2\pi}\frac{d\phi_j}{2\pi} \right)\cdot\exp{\left[-\frac{n}{2}\sum_{i,j} B_{i,j}\phi_i \phi_j\right]}\left(1+O\left(\frac{1}{n}\right) \right)\nonumber\\
& =2^{2n}(2\pi n)^{-2^{p-2}} \left(\det B\right)^{\frac{1}{2}} \left(1+O\left(\frac{1}{n}\right) \right). \end{aligned}$$ The determinant of $B$ can be explicitly calculated (see Appendix B) which finally leads to: $$Z_2 (n)=2^{2n}\left(\frac{2^{p-1}}{\pi n}\right)^{2^{p-2}} \left(1+O\left(\frac{1}{n}\right) \right). \label{FinalSecondMoment}$$
Saddle point approximation for $Z_k$, $k>2$
--------------------------------------------
Our starting point is the representation (\[FinalMoments\]) for $Z_k$: $$\begin{gathered}
Z_k=\left(\prod_{j=1}^{k-1}\prod_{i}^{2^{p-1}}\int_{0}^{2\pi}\frac{d\phi^{(j)}_i}{2\pi}\right) \cdot \exp{\F_n(\{\bm \phi^{(j)}\} )},\label{FinalMoments2}\\
\F^{(k)}_n(\{\bm \phi^{(j)}\} )=\log\left[\trace \left(\Q \left(- \bar{\bm \phi} \right)\right)^n \prod_{i=1}^{k-1} \trace \left(\Q \left(\bm \phi^{(i)} \right)\right)^n
\right], \nonumber\end{gathered}$$ where we introduced notation: $$\bar{\bm \phi}:= \sum_{j=1}^{k-1}\bm \phi^{(j)}.$$ As in the case $k=2$, the maximum of $\F^{(k)}_n$ is attained when all phases vanish i.e., for $\bm \phi^{(j)}=\bm 0, j=1,\dots k-1$. After taking the first derivative of $\F^{(k)}_n$ we have $$\begin{gathered}
\frac{\partial \F^{(k)}_n(\bm \phi )}{\partial\phi^{(l)}_j}= n\left[\frac{i\trace\left(\Lambda(\bm \phi^{(l)} )P_j Q_0 (\Q(\bm \phi^{(l)} ))^{n-1}\right)}{\trace(\Q(\bm \phi^{(l)} ))^{n}} \right]\\
- n\left[\frac{i\trace\left(\Lambda(-\bar{\bm \phi} )P_j Q_0 (\Q(- \bar{\bm \phi} ))^{n-1}\right)}{\trace(\Q(- \bar{\bm \phi} ))^{n}}
\right].
\label{FirstDeriv1}\end{gathered}$$ The second derivative of $\F^{(k)}_n$ at $\bm \phi^{(j)}=0, j=1,\dots k-1$ is then: $$\frac{\partial^2 \F^{(k)}_n(\bm \phi )}{\partial\phi^{(l)}_j \partial\phi_i^{(m)}}\Big{\rvert}_{\bm \phi =0}:=-n B^{(k)}_{[i, l;j, m]},$$ with the elements of the $2^{p(k-1)}\times 2^{p(k-1)}$ matrix $B^{(k)}$ given by: $$B^{(k)}_{[i, l;j, m]}=\frac{1}{2} B_{i,j}(\delta_{l,m}+1).$$ Note that $B^{(k)}$ can also be written as the product of two matrices composed of $(k-1)\times(k-1)$ blocks: $$B^{(k)}=
\frac{1}{2}\begin{pmatrix}
2\1& \1&\dots & \1\\
\1& 2\1&\dots &\1\\
\vdots& \vdots& \ddots& \vdots\\
\1& \1 &\dots & 2\1
\end{pmatrix}
\begin{pmatrix}
B& \0 &\dots & \0 \\
\0 & B &\dots & \0 \\
\vdots& \vdots& \ddots& \vdots\\
\0 & \0 &\dots & B
\end{pmatrix},\label{MatrixForm}$$ with $\1$ and $\0$ standing for unit and zero $2^{p}\times 2^{p}$ matrices, respectively.
Applying now saddle point approximation to (\[FinalMoments2\]) we obtain $$\begin{gathered}
Z_k =2^{kn}\left(\prod_{l=1}^{k-1}\prod_{j=1}^{2^{p-1}}\int_{0}^{2\pi}\frac{d\phi^{(l)}_j}{2\pi} \right)\cdot\exp{\left[-\frac{n}{2}\sum_{i,j} B^{(k)}_{[i, l;j, m]}\phi^{(l)}_i \phi^{(m)}_j\right]}\left(1+O\left(\frac{1}{n}\right) \right)\\
=2^{kn}(2\pi n)^{-(k-1)2^{p-2}} \left(\det B^{(k)}\right)^{\frac{1}{2}} \left(1+O\left(\frac{1}{n}\right) \right). \label{AlmostFinal}\end{gathered}$$ By eq. (\[MatrixForm\]) the determinant of $B^{(k)}$ can be explicitly evaluated: $$\det B^{(k)}=\left(\frac{k}{2^{k-1}}\right)^{2^{p}}\left(\det B\right)^{k-1}.$$ Substituting this expression into eq. (\[AlmostFinal\]) gives then $$Z_k=\frac{2^{2k}}{k^{2^{p-2}}}\left(\frac{2^{p}}{\pi n}\right)^{(k-1)2^{p-2}} \left(1+O\left(\frac{1}{n}\right) \right).\label{Final_k_Moment}$$
[ ![ []{data-label="figmoments"}](moments1.eps "fig:"){height="5.8cm"} ]{}
To verify the above asymptotic formula we evaluated the ratio between the leading order term of (\[Final\_k\_Moment\]) and the exact value of $Z_k$ obtained by numerical calculations of $|\Cl_{\bm n}|$. The results for $p=3$ are presented on fig. (\[figmoments\]). As can be observed, this ratio is close to one for large $n$.
Distribution of cluster sizes
=============================
We will use now the results of the previous section to evaluate the probability $P(t)$ that a randomly chosen periodic orbit belongs to a cluster of the size $t|\Cl_{\max}|$, $t\in [0,1]$, where $|\Cl_{\max}|=\max_{\bm n}|\Cl_{\bm n}|$ is the size of the largest cluster. To this end we first need to establish the asymptotic behavior of $|\Cl_{\max}|$.
By eq. (\[KeyFormula\]) the maximal size is given by: $$\begin{gathered}
|\Cl_{\max}|=\max_{\bm n}\left( \prod_{j=1}^{2^{p-1}} \int_{0}^{2\pi} \frac{d\phi_j}{2\pi} \right) \cdot\exp{\F(\bm \phi )},\\
\F(\bm \phi )=-i({\bm n},\bm \phi )+\log\trace\left( \Q\Lambda(\bm \phi )\right)\nonumber.\end{gathered}$$ To evaluate the above expression we can once again use saddle point approximation. Taking the first derivative of $\F(\bm \phi )$ leads to the following equation for the saddle point $${n}_i=n\frac{\trace\left(\Lambda(\bm \phi )P_i Q_0 (\Q(\bm \phi ))^{n-1}\right)}{\trace(\Q(\bm \phi ))^{n}},
\label{FirstDeriv3}$$ with ${n}_i$ being $i$’s component of the vector $\bm n$. It can be expected that the cluster of the maximum size is provided by the most homogeneous vector $\bar{\bm n}$ among all admissible vectors $\bm n$. For the sake of simplicity take $n$, such that $n\mod 2^p=0$. In this case $\bar{\bm n}=2^{-p}(1,1,\dots 1)$ and the saddle point equation (\[FirstDeriv3\]) is satisfied when $\bm \phi=\bm 0$. At this point $\F(\bm 0 )=n\log 2$. Since $|\F(\bm \phi )|\leq n\log 2$ for any $\bm \phi$, it is clear that $\Cl_{\bar{\bm n}}$ is, indeed, has (at least asymptotically) the largest size among all clusters. Repeating then the same calculations for the second derivative of $\F(\bm \phi )$, as in the previous section, we obtain after saddle point approximation: $$|\Cl_{\max}|=Z_2\left({n}/{2} \right)\left(1+O\left( {1}/{n} \right)\right),\label{Finalmaxcluster}$$ where $Z_2(n)$ is given by eq. (\[FinalSecondMoment\]).
To calculate the probability $P(t)$, it is useful to notice that the moments $Z_k$ can be represented as integrals: $$\frac{Z_k}{2^n|\Cl_{\max}|^{k-1}}=\int_{0}^{1}dt \rho(t) t^{k-1}, \label{MomRepresentation}$$ where $\rho(t) \Delta t$ is the probability to find a random periodic orbit in a cluster $\Cl_{\bm n}$ whose size belongs to the interval $|\Cl_{\max}|\cdot t\leq|\Cl_{\bm n}|\leq|\Cl_{\max}|\cdot(t+ \Delta t)$. The probability to find a random periodic orbit in the cluster of the size smaller than $|\Cl_{\max}|t$ is therefore: $$P(t)=\int_{0}^{t}d\tau\rho(\tau).$$
It remains to find $\rho(\tau)$. After substituting into eq. (\[MomRepresentation\]) the asymptotic expressions for $Z_k$ and $|\Cl_{\max}|$ we obtain in the limit $n\to\infty$: $$\int_{0}^{1}dt \rho(t) t^{k-1}=k^{-2^{p-2}}.\label{EqForRho}$$ By taking the Laplace transform on both sides of eq. (\[EqForRho\]) one has $$\rho(t)=\frac{\left(\log t\right)^{2^{p-2}-1}}{(2^{p-2}-1)!}.$$ In particular, for $p=2,3$ this gives: $$P(t)=t \quad \mbox{ (for $p=2$) }; \qquad P(t)=t (\log t -1) \quad \mbox{ (for $p=3$) }.$$ The comparison of the above result with the direct numerical simulation is shown on fig. \[distribution\].
[ ![ []{data-label="distribution"}](distribution.eps "fig:"){height="5.8cm"} ]{}
Anisotropy properties of clusters
=================================
The method used in the previous sections to calculate moments $Z_k$ can be also applied to obtain a more refined information on the distribution of periodic orbits in clusters with regard to each edge individually. One can ask for instance how many times a sequence $a\in X_p$ appears in a random sequence $x$ of length $n$. As one can expect, if all orbits are weighted in the same way the result does not depend on $a$: $$<n_a>:=2^{-n}\sum_{\bm n} n_a |\Cl_{\bm n} |=-2^{-n}i\partial_{\phi_a}\trace\left( Q\Lambda(\bm \phi )\right)^n|_{\bm\phi =0}=n/2^p. \label{anys1}$$ In other words, the periodic orbits are ergodically distributed over the graph $G_p$. However, if only periodic orbits from large clusters (or small clusters) are considered the question regarding homogeneity of their distribution over the edges of the graph, does not seem to have a trivial answer. For instance it is clear that smallest clusters are dominated by periodic orbits with either large number of zeroes or ones. Therefore, among the periodic orbits belonging to small clusters there should be enhanced probability to meet a subsequence consisting of all zeroes (or ones).
The purpose of the present section is to investigate the anisotropy properties of the graph with regard to distribution of periodic orbits belonging to clusters of different size. To this end let us consider the sum $$<n_a>_2:=\frac{\sum_{\bm n} n_a |\Cl_{\bm n} |^2}{\sum_{\bm n} |\Cl_{\bm n} |^2}. \label{anys2}$$ As opposed to $<n_a>$ the above quantity gives a larger weight to the periodic orbits from larger clusters. Thus, in principle, $<n_a>_2$ might depend on $a$. To verify this, we observe that (\[anys2\]) can be written as $$<n_a>_2 :=\frac{1}{Z_2}\left(\prod_{j=1}^{2^{p-1}}\int_{0}^{2\pi}\frac{d\phi_j}{2\pi}\right)\cdot \exp{\F^{(a)}_n(\bm \phi )},\label{AnysSecondMoment}$$ $$\begin{gathered}
\F^{(a)}_n(\bm \phi )=
\log\left[\Re(-i\partial_{\phi_a}\trace (\Q(\bm \phi ))^n\trace (\Q(-\bm \phi ))^n )\right]\nonumber\\
=\log \left[n\Re(\langle a|\Lambda(\bm \phi ) Q_0 Q(\bm \phi )^{n-1}|a\rangle\trace (\Q(-\bm \phi ))^n )\right]
, \nonumber\end{gathered}$$ where $|a\rangle$ is the state corresponding to the sequence (edge) $a$. We can now apply saddle point approximation to eq. (\[AnysSecondMoment\]). Expanding $\F_n(\bm \phi )$ up to the second order in $\phi_j$’s gives $$\F^{(a)}_n(\bm \phi )=\log(n2^{2n-p})-\frac{n}{2}\left(\sum_{i,j}B_{i,j}\phi_i\phi_j +O(\phi_i^4)\right), \label{expansion}$$ with the matrix $B$ given by eq. (\[MatrixB\]). Substituting then (\[expansion\]) into (\[AnysSecondMoment\]) leads to $$<n_a>_2=n/2^p+O(n^0),$$ where only the subleading term (possibly) depends on $a$. This form of $<n_a>_2$ indicates that to the leading order in $n$ the periodic orbits are equidistributed uniformly over the graph. The asymmetry shows up only in the second and higher order terms of the asymptotic expansion. In fig. \[figanysotropy\] we plotted $<n_a>_2$ as a function of $n$ for different edges of the graph. It is clearly visible that existing asymmetry between different edges, essentially, does not grow with $n$.
[ ![ []{data-label="figanysotropy"}](assymetry1.eps "fig:"){height="4.5cm"}9.1cm ![ []{data-label="figanysotropy"}](assymetry1_82.eps "fig:"){height="4.5cm" width="4cm"}0.1cm ![ []{data-label="figanysotropy"}](assymetry2.eps "fig:"){height="4.5cm"} ]{}
In an analogous way one can estimate averages: $$<n_a>_k :=\frac{\sum_{\bm n} n_a |\Cl_{\bm n} |^k}{\sum_{\bm n} |\Cl_{\bm n} |^k}. \label{anysk}$$ for an arbitrary $k$. As in the case of $k=2$, the leading order term of the asymptotic expansion of $<n_a>_k$ is equal to $n/2^p$. This can be interpreted to the extent that periodic orbits uniformly (to the leading order of $n$) pass through the edges of the graph $G_p$, independently of the size of the clusters they belong. The non-uniformity, however, does appear in the next order $n^0$ of the asymptotic expansion. To demonstrate this, we calculate numerically the averages: $$\bar{n}_a(t):=\frac{\sum\limits_{|\Cl_{\bm n} |< t|\Cl_{\max}|} n_a |\Cl_{\bm n} |}{\sum\limits_{|\Cl_{\bm n} |< t|\Cl_{\max}|} |\Cl_{\bm n} |}, \label{anys4}$$ where the sum runs over clusters with sizes less than $t|\Cl_{\max}|$, $t\in [0,1]$. It follows straightforwardly from the definition of $ \bar{n}_a(t)$ and eq. (\[anys1\]) that $$\sum_{a\in X_p} \bar{n}_a(t)=n \mbox{ for $t\in [0,1]$},\qquad \bar{n}_a(1)=n/2^p.$$ In the case $p=3$, the resulting plot of $\bar{n}_a(t)$ is shown on fig. (\[figanysotropy\]) for $n=82$ and $n=60$, respectively. As one can see, there is enhanced probability for periodic orbits from small clusters $t\ll 1$ to pass through the edge $a=[000]$, and suppressed probability to pass through the edge $[011]$. On the other hand this picture depends very little on $n$.
Discussion
==========
To summarize, the problem of counting $p$-close periodic orbits of the backer’s map can be cast in an equivalent form of finding degeneracies in the length spectrum of the de Bruijn graph $G_p$. The latter problem has previously attracted attention in [@sharp2], where an asymptotic expression for $\Z_2$ has been obtained for a generic graph. Related counting problems were also considered in [@uzy2] for fully connected graphs and in [@tanner] for binary directed graphs. In comparison to [@sharp2] we go somewhat further, as we derive asymptotics for all moments $\Z_k$ and explicitly obtain the leading term prefactors depending on $p$. Our considerations have been restricted to a simplest possible symbolic dynamics which occurs in the backer’s map (leading to a specific binary graph). As a matter of fact, the results can be straightforwardly generalized to the alphabets with a larger number of symbols. Moreover, the present approach can be extended to the symbolic dynamics with non-trivial grammar rules, when certain symbolic subsequences are forbidden. In that case the resulting graph $G_p$ would typically have a non-homogeneous structure, where the number of outgoing and incoming edges depend on a specific vertex.
The rescaled moments $\Z_k$ can be interpreted as the probabilities $\P_k$ that $k$ randomly chosen periodic sequences are $p$-close. Making use of information on $\P_k$, we obtain the asymptotics of the probability $P(t)$ to find a random periodic sequence in a cluster of the size smaller than $t|\Cll_{\max}|$, $t\leq 1$, with $\Cll_{\max}$ being the largest cluster. Most significantly, in the large $n$ limit, $P(t)$ does not depend on the length of sequence, but only on $p$. It is worth noting that $P(t)$ is basically determined by the distribution of “large“ clusters, whose size is of the order $|\Cll_{\max}|$. “Small“ clusters $\Cll_i$, whose size is $|\Cll_i|\ll |\Cll_{\max}|$ do not affect (asymptotically) the moments $\Z_k$ and therefore, do not contribute to $P(t)$. A natural question arises: what is the number of “large“ clusters in comparison to the number of “small” clusters? As we show in Appendix C the total number of clusters $\N_p$ is proportional to $n^{2^{p-1}}$. On the other hand, based on eqs. (\[Finalmaxcluster\], \[Final\_k\_Moment\]) a rough estimation of “large“ clusters yields $$\N_p^{(l)}\sim \Z_k/|\Cll_{\max}|^k \sim n^{2^{p-2}}.$$ This shows that the number of “large“ clusters scales as $\N_p^{(l)}\sim\sqrt{\N_p}$. In other words most of the clusters are “small” in the limit $n\to\infty$, although their influence on $\Z_k$ is negligible.
We have also studied the distribution of the periodic orbits over the graph $G_p$. A priori, it might be expected that subsequences consisting of only zeroes (or ones) appear more often than others in periodic sequences belonging to “small“ clusters. As we show, this is indeed so, but such inhomogeneities vanish asymptotically. To the leading order of $n$ periodic orbits cover the graph $G_p$ uniformly. In other words, in long sequences belonging to clusters of size comparable with $|\Cll_{\max}|$ each subsequence of $p$ symbols occurs on average the same number of times.
One of the main motivations for the present study comes from the theory of quantum chaos, where clusters of $p$-close orbits play an important role. It is worth mentioning, however, that the limit considered in the present paper, $n\to \infty$ with a fixed $p$, is somewhat different from the usual semiclassical limit. In the last case spectral correlations are determined by pairs of long periodic orbits whose encounter lengths are proportional to the logarithm of their total lengths [@haake1]. This corresponds to the limit of both $n\to\infty$, $p\to\infty$ such that the ratio $n/2^p$ is fixed. In this limit one might expect completely different behavior for $\P_k$. In particular the number of $p$-close pairs should grow faster than $2^n$. The exact behavior of $\P_k$ in the semiclassical limit is of great interest and we leave it for future investigations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank P. Braun and M. Kieburg for valuable discussions. This work was supported by the Sonderforschungsbereich Transregio 12
Change of integration variables in (\[SecondMoment\]) and (\[Moments\]). {#Appendix1}
========================================================================
Here we demonstrate how the total number of integration variables in the integral (\[SecondMoment\]) can be effectively reduced by half. To perform the change of variables we use the invariance of the trace under the multiplication of each matrix $Q\Lambda(\bm\phi)$ by a diagonal matrix $\Lambda(\bm \xi)={\textrm{diag}{\left\{e^{i\xi_1},\dots,e^{i\xi_{2^p} }\right\}}} $ and its conjugate from the left and from the right side correspondingly: $$Q'\Lambda(\bm\phi)\to e^{-i\xi}\Lambda(\bm \xi) Q'\Lambda(\bm\phi)\Lambda(\bm \xi)^\dag.$$ Any such transformation with an arbitrary $\bm \xi=(\xi_1,\dots \xi_{2^{p-1}})$ and $\xi$ leaves the integrand in (\[SecondMoment\], \[Moments\]) intact. Note also that this transformation does not affect the structure of the matrix (\[MatrixQpsik\]), but only its phases. Using this one can eliminate the phases ${\phi}'_{1}\equiv\phi_{2^{p-1}+1}, \dots {\phi}'_{2^{p-1}}\equiv\phi_{2^{p}} $ on the “right side” of the matrix $Q'$ by imposing the system of linear equations on $\xi_k$, $k=1,\dots,2^{p-1}$ $$\begin{aligned}
\label{Variable_Change1}
\bar{\phi}_{2k-1}-\xi_{2k-1}+\xi_{2^{p-1}+k}-\xi&=&0;\\
\bar{\phi}_{2k}-\xi_{2k}+\xi_{2^{p-1}+k}-\xi&=&0.\nonumber\end{aligned}$$ On the “left side” of the matrix $Q'$ the above transformation induces new phases $$\begin{aligned}
\label{Variable_Change2}
\varphi_{2k-1}&=&\phi_{2k-1}-\xi_{2k-1}+\xi_k-\xi;\\
\varphi_{2k}&=&\phi_{2k}-\xi_{2k}+\xi_k-\xi,\nonumber\end{aligned}$$ where $\xi_{i}$’s are determined by eq. (\[Variable\_Change1\]). After fixing the common phase $\xi$ to be a symmetric combination of other variables $\xi_k$, i.e. $\xi=2^{-p}\sum_k\xi_k$ the system of equations (\[Variable\_Change1\]-\[Variable\_Change2\]) can be cast into the matrix form: $$\begin{aligned}
\label{Variable_Change3}
{\bm\phi'}=F\bm\xi;\\
\bm\varphi=\bm\phi-G \bm\xi,\label{Variable_Change4}\end{aligned}$$ where the matrices $F$ and $G$ are expressed in terms of $Q_0$, $Q_1$, and $Q$: $$\label{Matrices_F_and_G}
G=\1-Q_0+(Q/2)^p ;\qquad F=\1 -Q_1+(Q/2)^p .$$
Combining now eq. (\[Variable\_Change3\]) with eq. (\[Variable\_Change4\]) we define the following linear transformation $$\label{Jacobian}
\left(\begin{array}{c}
\bm\phi\\
\bm{\phi}'
\end{array}\right)=
\left(\begin{array}{cc}
\1&GF^{-1}\\
0&\1
\end{array}\right)
\left(\begin{array}{c}
\bm\varphi\\
{\bm\varphi}'
\end{array}\right),$$ which exists since the determinants of both $F$ and $G$ are different from zero (see proposition \[Proposition1\]). By construction, the resulting matrix $e^{-i\xi}\Lambda(\bm \xi) Q'\Lambda(\bm\phi)\Lambda(\bm \xi)^\dag$ has the required form (\[FinalMatrix\]) with the phases given by $\bm \varphi$. Furthermore, since the Jacobian of the linear transformation (\[Jacobian\]) equals one, the variables ${\bm \varphi}'$ do not enter the integrand at all and can be integrated out. In the case of higher moments $Z_k$, $k>2$ one has to perform the similar transform for each set of variables $\bm\phi^{(j)}$. The resulting expression for the integral is given then by the equations (\[FinalMoments\]) and (\[FinalMatrix\]).
Calculation of the determinant of the matrix (\[MatrixB\]) {#Appendix2}
==========================================================
In this appendix we calculate the determinant of the matrix $B$. We recall that $B$ is defined using $2^p\times 2^p$ matrices $Q_0$, $Q_1$, their sum $Q=Q_0+Q_1$ and $Q_p=\left(\frac{Q}{2}\right)^p$, see (\[MatrixB\]). For the sake of completeness, we give their element-wise definition:
- The entries of the matrices $Q$ and $Q_0$, $Q_1$ are given by ($i,j =1,\dots 2^p$): $$\begin{gathered}
\label{MatrixQ_definition}
[Q]_{i\;j}=\sum_{k=1}^{2^{p-1}}(\delta_{i,2k-1}+\delta_{i,2k})(\delta_{j,k}+\delta_{j,2^{p-1}+k})\\
[Q_0]_{i\;j}=\sum_{k=1}^{2^{p-1}}(\delta_{i,2k-1}+\delta_{i,2k})\delta_{j,k};\quad
{}[Q_1]_{i\;j}=\sum_{k=1}^{2^{p-1}}(\delta_{i,2k-1}+\delta_{i,2k})\delta_{j,2^{p-1}+k}.\end{gathered}$$
- All elements of matrix $Q_p$ are equal to $2^{-p}$.
Bellow we collect a number of matrix relations which are of use in the further analysis. They follow directly from the matrix definitions. For any natural $k$: $$\label{matrix_relations}
Q^kQ_p=Q_pQ^k=2^kQ_p;\qquad Q_0^kQ_p=Q_1^kQ_p=Q_p;\qquad Q_p^k=Q_p;$$ $$\label{Q1_Q0_strange_relation}
Q_0^TQ_0+Q_1^TQ_1=2\cdot\1.$$ For the traces of various matrix products we have: $$\label{matrixQ0_trace}
\trace Q_p Q_{0,1}^k=\trace Q_{0, 1}^k=1, \qquad \trace Q^k=2^k,$$ where (and further) $Q_{0,1}$ stands either for $Q_0$ or $Q_1$.
Two more, useful equalities can be derived by using (\[matrix\_relations\]) $$\begin{aligned}
\label{Matrices_in_power1}
(Q_{0,1}-Q_p)^k&=&(Q_{0,1}-Q_p)Q_{0,1}^{k-1};\\
(Q-Q_p)^k&=&Q^k-(2^k-1)Q_p.\label{Matrices_in_power2}\end{aligned}$$ Both can be proved by induction. To derive (\[Matrices\_in\_power1\]) it is enough to notice that $Q_p(Q_{0,1}-Q_p)=0$. To obtain the coefficient at $Q_p$ in eq. (\[Matrices\_in\_power2\]) we assume that $(Q-Q_p)^k=Q^k-a_kQ_p$ with some yet unknown $a_k$. Then the next iteration gives the following recurrence relation for $a_k$: $a_{k+1}=2^k+a_k$. Its solution is $a_k=2^k-1$ as it is indicated in (\[Matrices\_in\_power2\]).
Having described the properties of the matrices $Q_{0,1}$, $Q$ and $Q_p$ we come to calculation of their determinants. The first axillary statement is: [For any real $\alpha$ $$\det\left[\1-\alpha(Q_{0,1}-Q_p)\right]= 1, \qquad \det \left[\1-\alpha({Q}-{Q_p})\right]=1-\alpha. \label{Useful_determinant}$$ \[Proposition1\]]{}
Consider the logarithm of the determinant: $$\log\det[\1-\alpha(Q_{0,1}-Q_p)]= -\sum_{k=1}^\infty \frac{\alpha^k}{k}\trace(Q_0-Q_p)^k.$$ Applying now the formula (\[Matrices\_in\_power1\]) we come to the matrix $(Q_0-Q_p)Q_0^k$ which trace is zero for all $k>0$ due to the relations (\[matrixQ0\_trace\]). This proves the first statement. In the same way, using the equation (\[Matrices\_in\_power2\]) and the information about traces we derive $$\log\det \left[\1-\alpha({Q}-{Q_p})\right] = - \sum_{k=1}^\infty \frac{\alpha^k}{k}\trace\left(Q-Q_p\right)^k=\log(1- \alpha).$$
We can use the above proposition for $\alpha=1$ to obtain the determinants of the matrices $F$ and $G$ defined in (\[Matrices\_F\_and\_G\]): [ $$\det F=\det G=1$$. \[Corollary 1\]]{}\
In the remaining part of the appendix the determinant of $2^p\times 2^p$ matrix (\[MatrixB\]) $$B=2^{-p-1}\left(2\1 -(1+2p)Q_p+Q_0 \sum_{r=0}^{p-1}\left(\frac{Q}{2}\right)^r+\sum_{r=0}^{p-1} \left(\frac{Q^T}{2}\right)^rQ_0^T \right),$$ is evaluated. The following remark helps to perform the calculations. [In the saddle-point calculations of the integral (\[FinalMoments\]) made after the change of variables one comes to the Gaussian integral (\[MomentZ2\_result\]) involving matrix $B$. However, change of the order leads to another matrix, $2^{-p}M_{p+1}$ . It has twice larger dimensionality and connected to the matrix $B$ by the relation $$\begin{gathered}
\label{Matrices_connection}
2^{-p}M_{p+1}=
\left(\begin{array}{cc}
\1&0\\
-GF^{-1}&\1
\end{array}\right)
\left(\begin{array}{cc}
B&0\\
0&0
\end{array}\right)\left(\begin{array}{cc}
\1&-GF^{-1}\\
0&\1
\end{array}\right)\\=\left(\begin{array}{cc}
B&-BGF^{-1}\\
-\big(GF^{-1}\big)^TB&\big(GF^{-1}\big)^TBGF^{-1}
\end{array}\right).\end{gathered}$$ Explicit form of matrix $M_{p+1}$ is $$\label{Matrix_M}
M_{p+1}=\1+\sum_{r=1}^p\left[ \left(\frac{Q}{2}\right)^r+ \left(\frac{Q^T}{2}\right)^r\right]- (2p+1) Q_{p+1},$$ where all matrices are of the size $2^{p+1}\times 2^{p+1}$. ]{}
Eq. (\[Matrices\_connection\]) allows to cast the problem of calculation of $ \det B$ into a “symmetric” form. Namely, by the following proposition the determinant of $B$ can be expressed through the eigenvalues of two matrices which depend only on $Q$ and $Q^T$. [The determinant of the matrix $B$ can be represented as $$\label{DetB}
\det B=2^{-2^p}\frac{4\prod_{\lambda_j\ne0}\lambda_j}{\det\tilde{M}_p},$$ where $\lambda_j$ are non-zero eigenvalues of $2^{p+1}\times 2^{p+1}$ matrix $M_{p+1}$ and $\tilde{M}_p$ is $2^p\times 2^p$ matrix $\tilde{M}_p:=\1+M_p+3 Q_p$. ]{}
Consider the spectral problem for the matrix $M_{p+1}$: $$M_{p+1}\left(
\begin{array}{c}
\bm X\\
\bm Y
\end{array}
\right)=\lambda \left(
\begin{array}{c}
\bm X\\
\bm Y
\end{array}
\right)\Longrightarrow \begin{array}{l}
B \bm X- BC \bm Y=\lambda \bm X\\
-C^T B \bm X + C^TB C\bm Y=\lambda \bm Y,
\end{array}$$ where $C=GF^{-1}$. The latter system reduces to the equation $$\lambda(C^T\bm X+ \bm Y)=0,$$ meaning that either $\lambda=0$ or $-C^T\bm X=\bm Y$. Backward substitution results in the following eigenvalue problem $$B(\1+C C^T)\bm X=\lambda \bm X$$ where $\lambda$’s are non-zero eigenvalues of the matrix $M$, so that we can write down $$\det B=\frac{\prod_{\lambda_j\ne0}\lambda_j}{\det(\1+ C C^T)}=\frac{\prod_{\lambda_j\ne0}\lambda_j}{\det(F^T F+G^T G)}.$$ In the last expression the result of the proposition \[Proposition1\] has been used.
One can proceed further by evolving the denominator. First we observe that it depends only on the matrix $Q$. Indeed, due to the properties (\[matrix\_relations\]), (\[Q1\_Q0\_strange\_relation\]) one has $$\begin{gathered}
F^T F+G^T G=4\cdot\1-Q-Q^T+2Q_p\\=2\left(\1-\frac{Q}{2}+\frac{Q_p}{2}\right)\left(\frac{1}{\1-\frac{Q}{2}+\frac{Q_p}{2}}+\frac{1}{\1-\frac{Q^T}{2}+\frac{Q_p}{2}}\right)\left(\1-\frac{Q^T}{2}+\frac{Q_p}{2}\right)\end{gathered}$$ Expansion of each of the fractions with the help of equation (\[Matrices\_in\_power2\]), $$\frac{1}{\1-\frac{Q}{2}+\frac{Q_p}{2}}=
\sum_{k=0}\frac{(Q-Q_p)^k}{2^k}=\sum_{k=0}^{p-1}\left(\frac{Q}{2}\right)^k-(p-2)Q_p,$$ yields $$F^T F+G^T G
=2\left(\1-\frac{Q}{2}+\frac{Q_p}{2}\right)\left(\1+M_p+3 Q_p\right)\left(\1-\frac{Q^T}{2}+\frac{Q_p}{2}\right),$$ where all matrices are of the size $2^p\times 2^p$. By taking into account (\[Useful\_determinant\]) we arrive to the final expression for the determinant of $B$.
It remains now to calculate the spectrum of matrices $M_{p+1}$ and $\tilde{M}_{p}$. Note that both matrices have a similar structure and can be treated in the same way. Their spectra is provided by the following lemma. [\[lemma1\]The spectrum of the matrix $M_{p+1}$ contains $2^p$ (half of the total number) zero eigenvalues, the non-zero part of the spectrum has the following structure: $$\lambda_k=k+1\quad \mbox{ with multiplicity } \quad 2^{p-k-1},\qquad 0\le k \le p-1,$$ and $\lambda_p=p+1$ has multiplicity one. The non-trivial eigenvalues of $\tilde{M}_{p}$ are given by: $$\tilde\lambda_k=k+2\quad \mbox{ with multiplicity } \quad 2^{p-k-2},\qquad 0 \le k \le p-2,$$ and $\tilde\lambda_{p-1}=p+1$, $\tilde\lambda_p=4$ having multiplicity one. The remaining $2^{p-1}-1$ eigenvalues are all equal to one. ]{}
To prove this lemma it is convenient to work within the tensorial representation of matrices. According to (\[MatrixQ\_tensor\]) the matrix $Q^r$ is given by $$\label{Operator_Q}
Q^r=\left(\underbrace{s\otimes\dots \otimes s}_{r\;\; \mbox{\scriptsize times}}\otimes \1\otimes\dots\otimes \1\right)\cdot T^r,$$ where $T$ is the following shift operator: $$T|a_1\rangle\otimes|a_2\rangle\dots\otimes|a_p\rangle=|a_2\rangle\otimes\dots\otimes|a_p\rangle\otimes|a_1\rangle,$$ and the projection $s$ acts on $|0\rangle,|1\rangle$ as $$s|0\rangle=0;\qquad
s|1\rangle=|1\rangle.$$ Note that the result of the action of the operator $Q^r$ on the basis vectors $|j_1\rangle\otimes|j_2\rangle\otimes\dots\otimes|j_p\rangle$, $j_k\in\{0,1\}$ essentially depends on the number of consecutive “ones" at the end of the sequence $j_1j_2\dots j_p$. Whenever this number is larger than $r$, the result of the action of $Q^r$ is zero. This property allows to find all eigenvalues of the operator $M_{p+1}$.
Let $\chi^{(k)}$ be a vector having exactly $k$ “ones” at the right end of the encoding sequence, separated by two “zeroes” (if $k< p$) from the rest of the sequence , i.e., $$\chi^{(k)}=|0\rangle\otimes\omega_{p-k-1}\otimes|0\rangle\otimes\underbrace{|1\rangle\otimes\dots\otimes|1\rangle}_{k\;\; \mbox{\scriptsize times}},$$ where $\omega_{p-k-1}=|j_1\rangle\otimes|j_2\rangle\otimes\dots\otimes|j_{p-k-1}\rangle$ is an arbitrary product vector of the length $p-k-1$. By the definition $\chi^{(p+1)}$ is the vector consisting of only “ones”. It is easy to see that $\chi^{(p+1)}$ is an eigenvector of $M_{p+1}$ with zero eigenvalue. All other eigenvectors of $M_{p+1}$ can be constructed as linear combinations of the rotations of $\chi^{(k)}$’s: $$\label{eigenvecors_of_operator_M}
\sum_{j=0}^k\alpha_j\bm T^j\chi^{(k)}.$$ The action of $M_{p+1}$ on each of these combinations ($k\le p$) results in $$\bm M_{p+1}\sum_{j=0}^k\alpha_j\bm T^j\chi^{(k)}=\left(\sum_{j=0}^k\alpha_j\right)\left(\sum_{j'=0}^k\bm T^{j'}\chi^{(k)}\right).$$ Equating the right hand side of this expression with $\lambda\sum_{j=0}^k\alpha_j\bm T^j\chi^{(k)}$ we rewrite the original problem as the eigenvalue problem for the $k\times k$ matrix consisting of all ones: $$\sum_{j=0}^k\alpha_j=\lambda \alpha_k,\qquad k=0,1,\dots,p.$$ The solution is well known – all eigenvalues are zeros except one which is equal to $k+1$. The degeneracy of the eigenvalues is defined by the free part $\omega_{p-k-1}$ of the vector $\chi^{(k)}$ and, therefore, equals to $2^{p-k-1}$ for $0\le k\le p-1$ and to one for $k=p$. The total number of non-zero eigenvalues is $$1+\sum_{k=0}^{p-1}2^{p-k-1}=2^p.$$
The eigenvalues of $\tilde{M}_p$ can be obtained by a general shift of all eigenvalues of $M_{p}$ by $1$ with the only exception of zero eigenvalue corresponding to $\chi^{(p)}$ which should be shifted by $4$.
By Lemma \[lemma1\] one has the following chain of identities $$\prod_{\lambda_j\ne0}\lambda_j=(p+1)\prod_{k=0}^{p-1}(k+1)^{2^{p-k-1}}
=(p+1)\prod_{k=0}^{p-2}(k+2)^{2^{p-k-2}}=\frac{1}{4}\det\tilde{M}_p.$$ Substituting this into (\[DetB\]) gives the final expression for the determinant of $B$: [$$\det B=2^{-2^{p}}.$$ ]{}
Calculation of the total number of clusters {#Appendix3}
===========================================
Here we estimate the total number $\mathcal{N}_p(n)$ of equivalence classes $\Cl_{\bm n}$ of all sequences of the total length $n$ generated by the equivalence relation $x\pclone y$ ($x,y\in \X_n$). Recall that every equivalence class is uniquely parametrized by a vector of integers $\bm n={\left\{n_a, a\in
X_p\right\}}$. The elements of this vector determine the number of times a periodic orbit from $\Cl_{\bm n}$ passes trough the corresponding edge of the graph $G_p$. Since vector $\bm n$ corresponds to a real periodic orbit of the length $n$, its components must satisfy the following constraints:
- The total length of the trajectory is fixed: $$\label{Currents_conservation1}
\sum_{a\in X_p}n_a=n.$$
- The number of times a periodic orbit enters a vertex of the graph $G_p$ must be equal to the number of exits from the same vertex. This balancing condition is represented by the equation $$\label{Currents_conservation}
S\bm n=\bm n^T R.$$
It is easy to see that for any vector $\bm n$ satisfying the above conditions, with $n_a\neq 0$ for all $a$, it is possible to find a closed path on the graph which passes through an edge $a$ exactly $n_a$ times. Note also that the number of solutions with $n_a= 0$ for some $a$ is smaller by a factor $1/n$ than the total number of solutions of (\[Currents\_conservation1\],\[Currents\_conservation\]), see [@Berkolajko]. Therefore, to find the leading asymptotics of $\mathcal{N}_p(n)$ it is sufficient to count vectors of positive integers satisfying the equations (\[Currents\_conservation1\],\[Currents\_conservation\]).
Since the system (\[Currents\_conservation\]) is composed of $2^{p-1}-1$ linearly independent conditions, we can chose $2^{p-1}$ first elements of $\bm n$ freely, while the rest is then uniquely fixed by eqs. (\[Currents\_conservation1\],\[Currents\_conservation\]). In addition, the constraints $n_i\geq 0$, $i=1,\dots2^{p}$ must be satisfied. These constraints define a $2^{p-1}$-polytope $\mathcal{V}_p$ in the $2^{p-1}$-dimensional space of $n_1,\dots n_{2^{p-1}}$. Geometrically the number of clusters $\mathcal{N}_p(n)$ can be interpreted as the total number of points with integer coordinates encompassed by $\mathcal{V}_p$. Accordingly, the leading term of $\mathcal{N}_p(n)$ in the large-$n$ limit is given by the volume of $\mathcal{V}_p$. Therefore, $$\mathcal{N}_p(n)={w}_p n^{2^{p-1}}(1+O(1/n)), \label{c3}$$ where the coefficient ${w}_p$ can be calculated explicitly for low values of $p$. We illustrate this by the following example.
[**Example:**]{} For $p=2$ the conditions (\[Currents\_conservation1\],\[Currents\_conservation\]) take the form: $$n_2=n_3, \qquad n_1 +n_2 +n_3+n_4=n.$$ We chose two independent integer be $n_1=k$, $n_2=m$. Since $n_i\geq 0$ for all $i=1,2,3, 4$, the problem is reduced to calculation of the area of the triangle $$\mathcal{V}_2=\{x\geq 0, y\geq 0, n-2x-y\geq 0\},$$ in the $(x,y)$-plane. As a result, for $p=2$ we obtain $\mathcal{N}_2(n)= n^2/4 + O(n)$.
It worth mentioning that similar problem for non-directed graphs was considered in [@Berkolajko]. It was shown that the number of equivalence classes (equiv. the degeneracy classes in the length spectrum of the graph) in the leading order of $n$ is proportional to $n^{|E|-1}$, where $|E|$ is the total number of edges in the graph. For comparison with (\[c3\]), note that the number of edges in $G_p$ is $2^p$. This reflects the fact that the number of equivalence classes in directed graphs is essentially smaller than in non-directed graphs.
[99]{}
O. Bohigas, M. J. Giannoni, C. Schmit, [*Phys. Rev. Lett.* ]{} [**52**]{} (1) (1984)
M. Berry, Semiclassical theory of spectral rigidity [*Proc. R. Soc. A*]{} [**400**]{}, 229-251 (1985)
N. Argaman, F.M. Dittes, E. Doron, J.P. Keating, A. Kitaev, M. Sieber and U. Smilansky [*Phys. Rev. Lett.*]{} [**71**]{}, 4326-4329 (1993); F.M. Dittes, E. Doron and U. Smilansky [*Phys. Rev. E*]{} [**49**]{}, R963-R966 (1994)
F. Haake, Quantum Signatures of Chaos, 2nd ed. Springer-Verlag, Berlin, (2001)
M. Sieber, K. Richter [*Phys. Scripta.*]{} [**T90**]{}, 128 (2001)
S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, [*Phys. Rev. Lett.*]{} [**93**]{}, 014103 (2004); [*Phys. Rev. E*]{} [**72**]{}, 046207 (2005)
A. J. Lichtenberg M. A. Lieberman 1992 Regular and Chaotic Dynamics (New York: Springer)
Pollicott M., Sharp R.: Correlations for pairs of closed geodesics. [*Invent. Math.*]{} [**163**]{}, 1–24 (2006) D. Cohen, H. Primack and U. Smilansky Quantal- classical duality and the semiclassical trace formula. [*Annals of Physics*]{}, [**264**]{}, 108-170, (1998)
V. Petkov, L. Stoyanov [*Nonlinearity*]{} [**22**]{} 2657 (2009)
U. Smilansky and B. Verdene Action correlations and Random matrix Theory [*J. Phys. A.*]{} [**36** ]{} 3525-3549 (2003)
U. Gavish and U. Smilansky Degeneracies in the length spectra of metric graph [*J. Phys. A: Math. Theor.*]{} [**40**]{} (2007) Sharp R.: Degeneracy in the length spectrum for metric graphs. [*Geometriae Dedicata Volume*]{} [**149**]{}, Number 1, 177-188 (2010) G. Tanner [*J. Phys. A*]{} [**33**]{}, 3567-3586 (2000) Berkolaiko, G.: Quantum star graphs and related systems. PhD Thesis, University of Bristol (2000)
Balazs N L, Voros A [*Ann. Phys.,*]{} [**190**]{} 1–31 (1989); Saraceno M, Voros A 1994 [*Physica D*]{} [**79**]{} 206–68 (1994)
N. G. de Bruijn, “A Combinatorial Problem”. [*Koninklijke Nederlandse Akademie v. Wetenschappen*]{} [**49**]{}, 758–764 (1946) ; I. J. Good, “Normal recurring decimals”. [*Journal of the London Mathematical Society*]{} [**21**]{} (3): 167–169 (1946)
Rammal, R.; Toulouse, G., Virasoro, M. “Ultrametricity for physicists”. [*Reviews of Modern Physics*]{} [**58**]{} (3): 765–788 (1986)
B. Gutkin and V.Al. Osipov Spectral problem of block-rectangular hierarchical matrices [*J. Stat. Phys.*]{} [**143**]{}, 72 (2011) B. Gutkin Entropic bounds on semiclassical measures for quantized one-dimensional maps [*Commun. Math. Phys.*]{} [**294**]{}, 303 (2010)
[^1]: The number of prime and non-prime periodic orbits scales as $2^n$ and $2^{n/r}$, respectively, where $r\geq 2$ is a minimal divisor of $n$
|
---
abstract: 'We address the possible emergence of spin triplet superconductivity in CrO$_{2}$ bilayers, which are half-metals with fully spin-polarized conducting bands. At large doping, the $p+ip$ channel has a sequence of topological phase transitions that can be tuned by gating effects and interaction strength. Among several phases, we find chiral topological phases having a single Majorana mode at the edge.'
author:
- 'Xu Dou, Kangjun Seo, and Bruno Uchoa$^{*}$'
title: 'Possible Chiral Topological Superconductivity in CrO$_{2}$ bilayers'
---
*Introduction.* Half-metals ** such as CrO$_{2}$ [@Yu; @Soulen] are promising materials for the prospect of emergent topological superconductivity. By having a metallic Fermi surface with a single spin, they raise the possibility of chiral superconductivity in the triplet channel [@Pickett], which is believed to occur only in a handful of systems such as Sr$_{2}$RuO$_{4}$ [@Mackenzie], which may have a spinfull triplet state, UPt$_{3}$ and some heavy fermions superconductors [@Kallin; @Maeno]. A distintictive property of spin triplet chiral topological superconductivity is the presence of Majorana fermions propagating at the edges [@Read; @Qi2; @Qi; @Lee; @Alicea; @Sato] and half-flux quantum vortices [@Chung2; @Jang] that can trap Majorana modes [@Jackiw; @Xu]. Majorana edge states were predicted to exist in different heterostructures with strong spin-orbit coupling [@Fu; @Akhmerov; @fu2; @Lutchyn; @Chung; @Li] and may have been recently observed in an anomalous Hall insulator-superconductor structure [@Qi3; @He].
In its most common form, CrO$_{2}$ is a three dimensional bulk material with rutile structure [@Schwartz; @Katsnelson]. It was recently suggested [@Cai] that CrO$_{2}$/TiO$_{2}$ heterostructures have fully spin polarized conduction bands over a wide energy window around the Fermi level, and behave effectively as a two dimensional (2D) crystal. In its simplest 2D form, CrO$_{2}$ will form a bilayer. It is natural to ask if this material could spontaneously develop 2D chiral topological superconducting phases and host Majorana fermions [@Fu].
We start from a lattice model for a single CrO$_{2}$ bilayer to address the formation of spin triplet pairs $p_{x}+ip_{y}$ symmetry, which leads to a fully gapped state. Due to the strong anisotropy of the gap, the superconducting order has a line of quantum critical points as a function of both doping and coupling strength. In the $p+ip$ state, we show that the system has an exotic sequence of topological phase transitions, that could be tuned with gating effects. Different non-trivial topological phases may occur in the vicinity of van-Hove singularities of the band, where the density of states (DOS) diverges, allowing the possibility for both conventional and purely electronic mechanisms. We suggest that this system may provide an experimental realization of intrinsic 2D chiral topological superconductivity in the triplet channel.
\[fig1\] {width="0.92\linewidth"}
*Lattice model.* In a bilayer system, the Cr atoms form two interpenetrating square sublattices, $A$ and $B$, each one sitting on a different layer. From above, the Cr atoms are arranged in a checkerboard pattern, as shown in Fig. 1. Each site on sublattice $A$ ($B$) has two orbitals with $d_{xy}$ and $d_{xz}$($d_{yz}$) symmetry. Nearest neighbors (NN) hopping between a $d_{xy}$ orbital in sublattice $B$ with a $d_{xz}$ orbital in sublattice $A$ has amplitude $t_{1}$ along the the $(1,\bar{1})$ direction and zero along the $(1,1)$ direction by symmetry. In the same way, NN hopping between a $d_{xy}$ orbital in sublattice $A$ and with a $d_{yz}$ orbital in $B$ has amplitude $t_{2}$ along the $(1,1)$ direction and zero along the other diagonal in the $xy$ plane. Intra-orbital NN hopping is finite between $d_{xy}$ orbitals $(t_{3}$) but zero between $d_{xz}$ and $d_{yz}$ orbitals ($t_{4}$), which are othogonal to each other. Among next-nearest neighbors (NNN), the dominant processes are described by intra-orbital hoppings $t_{j}^{\alpha}$, with $\alpha=xy,\,xz$ for sites in sublattice $j=A$ and $\alpha=xy,\,yz$ for $B$ sites.
The Hamiltonian can be described in a four component basis $\Psi=(\psi_{A,xy},\psi_{A,xz},\psi_{B,xy},\psi_{B,yz})$. In momentum space, $\mathcal{H}_{0}=\sum_{\mathbf{q}}\Psi_{\mathbf{q}}^{\dagger}h(\mathbf{q})\Psi_{\mathbf{q}}$, with [@Cai] $$\begin{aligned}
h(\mathbf{q})= & \left(\begin{array}{cc}
h_{A} & h_{AB}\\
h_{AB}^{\dagger} & h_{B}
\end{array}\right),\label{1}\end{aligned}$$ where $$h_{A}=\left(\begin{array}{cc}
\epsilon_{A}^{xy}(\mathbf{q}) & 0\\
0 & \epsilon_{A}^{xz}(\mathbf{q})
\end{array}\right),\,h_{B}=\left(\begin{array}{cc}
\epsilon_{B}^{xy}(\mathbf{q}) & 0\\
0 & \epsilon_{B}^{yz}(\mathbf{q})
\end{array}\right).\label{eq:hAB}$$ The diagonal terms incorporate NNN hopping processes, where $\epsilon_{j}^{\alpha}(\mathbf{q})=E_{j}^{\alpha}+4t_{j}^{\alpha}\mathrm{cos}q_{x}\mathrm{cos}q_{y}$, with $E_{j}^{\alpha}$ a local potential on obital $\alpha$ in sublattice $j$ and $q_{x,y}=\frac{1}{2}(k_{x}\mp k_{y})$ the momentum along the two diagonal directions of the crystal. The off-diagonal terms in (\[1\]) describe the NN hopping terms illustrated in panels a) and b) in Fig. 1, $$\begin{aligned}
h_{AB}= & \left(\begin{array}{cc}
-2t_{3}\sum_{\nu=x,y}\mathrm{cos}q_{\nu} & 2it_{1}\mathrm{sin}q_{y}\\
2it_{2}\mathrm{sin}q_{x} & -2t_{4}\sum_{\nu=x,y}\mathrm{cos}q_{\nu}
\end{array}\right),\label{eq:3}\end{aligned}$$ where $t_{4}=0$ in the absence of spin-orbit coupling.
The energy spectrum is shown in Fig. 1c, and has two sets of Dirac points along the ($1,1$) and $(1,\bar{1})$ directions, respectively. Enforcing the symmetries of the 2D lattice, namely roto-inversion $S_{4}$ symmetry and mirror symmetry $M$ at the diagonal directions of the unit cell, we adopt $t_{1}=-t_{2}\equiv t\sim0.3$eV as the leading energy scale, and the set of parameters $t_{3}\sim t/30$, $t_{j}^{xy}=-t_{A}^{xz}=-t_{B}^{yz}\sim t/3$, and $E_{j}^{xy}=-E_{A}^{xz}=-E_{B}^{yz}\sim t/6$. We also , following ab initio results [@note1; @Cai]. The four band model breaks down near the edge of the band, where states may hybridize with high energy bands. We also assume that the bands are spinless. The resulting band structure has several van Hove singularities at the saddle points, where the density of states (DOS) diverges logarithmically, as depicted in Fig. 1d. In the vicinity of those points (red dots), the system can be unstable towards superconductivity.
*Pairing Hamiltonian.* For spinless fermions, superconductivity is allowed only in the triplet channel. ** The wavefunction of the Cooper pairs is anti-symmetric under inversion, and hence only states with odd angular momentum are allowed. A conclusive assessment of the stability of those states requires taking fluctuations into account [@Furukawa; @Nandkishore; @Kiessel], which will be considered elsewhere.
For NN sites, the effective interaction term has the form $$\mathcal{H}_{\text{int}}=-\frac{1}{2}\sum_{\mathbf{r}\in\text{NN}}g^{\alpha\beta}\hat{n}_{i,\alpha}(\mathbf{r}_{i})\hat{n}_{j,\beta}(\mathbf{r}_{j})\label{eq:Hint}$$ where $\hat{n}_{i,\alpha}=\psi_{i,\alpha}^{\dagger}\psi_{i,\alpha}$ is the density operator in orbital $\alpha$ on sublattice $i=A,\,B$, $g^{\alpha\alpha}\equiv g_{1}>0$ is the intra-orbital coupling, and $g^{xy,yz}=g^{xz,xy}\equiv g_{2}>0$ is the coupling in the inter-orbital channel. The $p+ip$ pairing follows from the Ansatz on the lattice $\Delta^{\alpha\beta}(\delta_{n})=g^{\alpha\beta}\langle\psi_{A,\alpha}(\mathbf{r})\psi_{B,\beta}(\mathbf{r}+\vec{\delta}_{n})\rangle\equiv\Delta^{\alpha\beta}\text{e}^{i\frac{\pi}{2}n},$ where $\vec{\delta}_{1,3}=\pm\frac{a}{2}(\hat{x}+\hat{y})$ and $\vec{\delta}_{2,4}=\pm\frac{a}{2}(\hat{x}-\hat{y})$ describe the four NN vectors, with $a$ the lattice constant. $\Delta^{\alpha\beta}(\pm\delta_{1,2})\equiv\pm\Delta^{\alpha\beta}$.
Defining $\Delta^{\alpha\alpha}\equiv\Delta_{1}$ and $\Delta^{\alpha\beta}\equiv\Delta_{2}$ for intra-orbital and inter-orbital pairing respectively, the order parameter in momentum space $\Delta_{i}^{C}(\mathbf{q})=\Delta_{i}^{C}(\sin q_{y}+i\sin q_{x})$ has chiral $p_{x}+ip_{y}$ symmetry, with $i=1,\,2$ and $q_{x,y}$ defined as above Eq. (\[eq:3\]). $\Delta_{i}^{p}(\mathbf{q})=\Delta_{i}^{p}(\sin q_{y}+\sin q_{x})$ [@note6]. At the mean field level, Hamiltonian (\[1\]) and (\[eq:Hint\]) result in the Bogoliubov-de Gennes (BdG) Hamiltonian $\mathcal{H}_{{\rm BdG}}=\sum_{\mathbf{k}\in BZ}\Phi_{\mathbf{q}}^{\dagger}h_{\text{{\rm BdG}}}(\mathbf{q})\Phi_{\mathbf{q}}$ with $\Phi_{\mathbf{q}}=(\Psi_{\mathbf{q}},\Psi_{-\mathbf{q}}^{\dagger})$, which has the form $$\begin{aligned}
h_{{\rm BdG}}(\mathbf{q})= & \left(\begin{array}{cc}
h(\mathbf{q}) & \hat{\Delta}(\mathbf{q})\\
\hat{\Delta}^{\dagger}(\mathbf{q}) & -h^{T}(-\mathbf{q})
\end{array}\right),\end{aligned}$$ where $$\hat{\Delta}(\mathbf{q})=\left(\begin{array}{cc}
0 & \Delta_{1}(\mathbf{q})\mathbf{1}+\Delta_{2}(\mathbf{q})\sigma_{x}\\
\Delta_{1}(\mathbf{q})\mathbf{1}+\Delta_{2}(\mathbf{q})\sigma_{x} & 0
\end{array}\right)\label{Delta2}$$ is the pairing matrix, with $\sigma_{x}$ a Pauli matrix in the orbital space. Minimization of the free energy $\mathcal{F}(\Delta_{1},\Delta_{2})=-T\text{tr}\sum_{\mathbf{k}}\ln e^{-h_{\text{BdG}}(\mathbf{k})/T}+\sum_{i=1,2}|\Delta_{i}|^{2}/g_{i}$ for a fixed chemical potential $\mu$ gives the zero temperature ($T=0$) phase diagram shown in Fig. 2a, b as a function of the couplings $g_{1}$ and $g_{2}$. The two leading instabilities in the $p$-wave and chiral $p+ip$ states compete with each other and are addressed below.
*$p+ip$ phase.* The inter-orbital channel $g_{2}$ may lead to gapless chiral $p+ip$ superconductivity ($\Delta_{2}^{C}\neq0$) shown in the red region, which is topologically trivial (Fig. 2a). The dashed line around the gapless phase in Fig. 2b describes a first order phase transition and sets the boundary of the gapless $p+ip$ phase with the others at $g_{2}=g_{2c}(\mu)$. The intra-orbital $p+ip$ pairing state ($\Delta_{1}^{C}\neq0$) on the other hand is fully gapped and can be topological.
\[fig2\] {width="0.97\linewidth"}
The gapped state has multiple minima that compete. The dashed vertical line in Fig. 2b indicates a first order phase transition between the weak and strong coupling topological phases (TSC) at $g_{1}=g_{1c}(\mu)$. At this coupling, the superconducting order parameter $\Delta_{1}^{C}$ jumps (see Fig. 2c) and different gapped phases with distinct topological numbers coexist. The resulting gap is very anisotropic around the Fermi surface (Fig. 2d). In the weak coupling phase $\bar{g}_{1c}(\mu)<g<g_{1c}(\mu)$ shown in the light blue region in Fig. 2b, the intra-orbital chiral gap $\Delta_{1}^{C}$ scales as a power law with the coupling for fixed $\mu$, $$\Delta_{1}^{C}(g_{1})\propto(1-\bar{g}_{1c}/g)^{\beta},\label{eq:Delta2}$$ with $\beta\approx2.7\pm0.1$ for $0.2\lesssim\mu\lesssim0.4$ eV (see Fig. 2c inset). $\Delta_{1}^{C}$ vanishes at the critical coupling $\bar{g}_{1c}$, where the system has a second order phase transition to the normal state, indicated by the green arrows in Fig. 2. A qualitatively similar behavior is also observed in the scaling of the intra-orbital $p$-wave gap $\Delta_{1}^{p}$ near the critical coupling $\tilde{g}_{1}(\mu)$ (blue arrows in Fig 2) [@note2].
\[fig4-1\] ![[(Color online) Mean field phase diagram as a function of the chemical potential $\mu$ and intra-orbital pairing coupling $g_{1}$, both in eV units. The phase diagram does not depend on $g_2$ for $g_2\lesssim 0.1$ eV (see Fig 2). a) Normal phase (N), $p_{x}$ superconducting phase (]{}*p*[SC I), chiral $p+ip$ state (CSC) and topological $p+ip$ phase (TSC). b) Possible topological phases in the chiral $p+ip$ channel. The integers indicate the corresponding BdG Chern number ]{}*$\mathcal{N}$*[. For fixed $g_{1}$, the system has a sequence of topological phase transitions near the van-Hove singularities of the band, where the topology of the Fermi surface changes. The blue regions correspond to the weak coupling gapped $p+ip$ phases, which are topological. Gray and maroon regions: strong coupling phases. In mean field, the gapped $p+ip$ state wins over the non-chiral $p$ state in the strong coupling sector. ]{} ](Fig3 "fig:"){width="1\columnwidth"}
\[fig4\] {width="0.97\linewidth"}
When $\mu$ is in the immediate vicinity of the van Hove singularities, $\bar{g}_{1c}$ abruptly drops towards zero. This singular behavior suggests a crossover to exponential scaling when the Fermi surface is nested at the van Hove singularities [@note2]. In that regime, the phase transition is not quantum critical.
In general, all the gapped chiral phases prevail over the gapless one $(\Delta_{2}^{C}$). At small doping, the two critical couplings of the weak and strong coupling phases merge ($\bar{g}_{1c}=g_{1c}$) below $|\mu|\lesssim0.6t$ and the gapped phase has a first order phase transition to the normal state at $g<g_{1c}(\mu)$ (see Fig 3b).
*Topological phase transitions.* In 2D, spinless superconductors with a bulk gap that breaks time reversal symmetry belong to the C class in the ten-fold classification table [@class1; @class2]. The topological number in this class is defined by the BdG Chern number *$\mathcal{N}$,* which corresponds to the number of chiral Majorana modes propagating along the edge [@Read; @Thouless].
In Fig. 3b, we explicitly calculate the Chern number $$\mathcal{N}=(i/2\pi)\int_{BZ}\text{d}^{2}\mathbf{q}\,\Omega_{z}(\mathbf{q})\label{eq:N}$$ in the gapped state as a function of $\mu$ and intra-orbital coupling $g_{1}$, with $\boldsymbol{\Omega}(\mathbf{q})=\nabla_{\mathbf{q}}\times\langle\psi_{n,\mathbf{q}}|\nabla_{\mathbf{q}}|\psi_{n,\mathbf{q}}\rangle$ the Berry curvature from the eigenstates of the BdG Hamiltonian at the Fermi level, $|\psi_{n,\mathbf{q}}\rangle$. By changing the chemical potential, the system shows a sequence of topological phase transitions.
In the weak coupling phase, shown in the blue areas in Fig. 3b, there are up to five transitions separating different topological phases with $\mathcal{N}=-4,\,-5,\,-6,\,-4,\,-5,$ and $-3$, in the range of $-2t\leq\mu\leq2t=0.6$eV. The critical values of the chemical potential where the system has a topological phase transition are close to the energy of the van Hove singularities of the band (see Fig. 1c) and coincide with the energies where the topology of the Fermi surface changes. At those critical values, the superconducting gap closes and the Chern number jumps by an integer number. The line $g_{1}=\bar{g}_{1c}(\mu)$ separates the blue areas from the normal region through continuous phase transitions. As anticipated, when $|\mu|\lesssim0.6t=0.18$eV, $\bar{g}_{1c}=g_{1c}$, and the weak coupling phases are suppressed. The singular behavior of $\bar{g}_{1c}(\mu)$ when $\mu$ is at the van Hove is not captured by the numerics shown in Fig. 3 due to the smallness of the gap.
The solid curve separating the blue regions in Fig. 3b from the strong coupling phases sets $g_{1c}(\mu)$, which describes a line of first order phase transitions between different gapped phases. At this line, the order parameter is discontinuous [@note3], indicating the onset of a topological phase transition as a function of $g_{1}$ for fixed $\mu$. In all cases, the Chern number changes accross the $g_{1c}(\mu)$ line by $\Delta\mathcal{N}=4$. Deep in the strong coupling regime (gray and maroon regions), for fixed $g>g_{1c}(\mu=0)$, there are six topological phase transitions separating the phases $\mathcal{N}=0,\,-1,\,-2,\,0,\,-1,\,1,\,0$ as a function of the chemical potential. At the wide doping window $1.27t\lesssim\mu\lesssim2t=0.6$ eV, the elemental chiral topological superconducting phase with $\mathcal{N}=\pm1$, and hence a single Majorana mode, can emerge at strong coupling.
*Chiral Majorana edge states.* To explicitly verify the Chern numbers for the different phases, we calculate the edge modes of the gapped state in a two dimensional strip geometry with edges oriented along the $(1,0)$ direction.
The plots in Fig. 4a$-$d (top row) show the evolution of the edge modes in the weak coupling regime ($\bar{g}_{1c}<g<g_{c}(\mu)$) for different values of $\mu$. The $\mathcal{N}=-3$ state shown in Fig. 4a has five edge modes in total, but only three modes that are topologically protected, as indicted by the three different colors. The three modes indicated in blue can be adiabatically deformed into a single zero energy crossing at $k=0$, and hence count as a single topologically protected mode. By decreasing the chemical potential into the contiguous $\mathcal{N}=-5$ state (Fig. 4b), two of those modes become topologically protected, raising the number of Majorana modes to five. By reducing $\mu$ further into the $\mathcal{N}=-4$ state, the topology of the Fermi surface changes drastically, forming gapped pockets of charge around four Dirac nodes, indicated in Fig. 1c. Panel d shows the edge modes of the $\mathcal{N}=-6$ state, for $\mu\lesssim-t=-0.3$eV. The corresponding edge modes in the strong coupling regime ($g>g_{1c}(\mu)$) with $\mathcal{N}=1,\,-1,\,0,$ and $-2$ are shown in the bottom row of Fig. 4 (e$-$h).
*Pairing Mechanism.* Although it is difficult to reliably predict a mechanism of superconductivity, at large doping and in the vicinity of the van Hove singularities, where the DOS is very large, both phonons and electronic interactions could be suitable candidates for a pairing mechanism. We will not discuss the phonon mechanism, since it is conventional.
Electronic mechanisms typically provide attraction when the charge susceptibility at the Fermi surface nesting vector $\mathbf{Q}$ satisfies $\chi(\mathbf{Q})>\chi(0)$ [@Kohn]. When the chemical potential $\mu$ is close to a Van Hove singularity, the electronic bands have energy spectrum $\epsilon(\mathbf{q})=-\alpha q_{x}^{2}+\beta q_{y}^{2}$, ($0<\alpha\leq\beta)$ where $\mathbf{q}$ is the momentum away from the saddle point. The susceptibility in the vicinity of the singularity is logarithmic divergent, *$\chi(0)=\frac{1}{2\pi^{2}}/\sqrt{\alpha\beta}\ln\left(\Lambda/\delta\mu\right)$* with $\delta\mu$ the deviation away from the van Hove and $\Lambda\sim t$ an ultraviolet cut-off around the saddle point [@Gonzalez]. At the nesting wavevector $\epsilon(\mathbf{q}+\mathbf{Q})=-\alpha p_{y}^{2}+\beta p_{x}^{2}$ , the susceptibility is $$\chi(\mathbf{Q})=c/(\alpha+\beta)\ln\left(\Lambda/\delta\mu\right),\label{eq:chi}$$ where the constant $c=\frac{1}{\pi^{2}}\ln(\sqrt{\frac{\alpha}{\beta-\alpha}}+\sqrt{\frac{\beta}{\beta-\alpha}})$ is logarithmically divergent at the nesting condition $\alpha=\beta$ [@Pattnaik]. For the particular lattice Hamiltonian parametrization taken from Ref. [@Cai], the fitting of the bands around the van Hove at $\mu=0.312$ eV has $\alpha\approx1.2$ and $\beta\approx1.7$. That gives the ratio $\chi(\mathbf{Q})/\chi(0)\sim1.20$, suggesting that a purely electronic mechanism of superconductivity is possible [@Gonzalez; @Guinea; @Note4]. The high doping regime could in principle be reached with gating effects for CrO$_{2}$ encapsulated in an insulating substrate [@Mayorov] that preserves the roto-inversion symmetry of the lattice.
*Acknowledgements.* BU acknowledges K. Mullen for discussions. XD, KS and BU acknowledge NSF CAREER Grant No. DMR-1352604 for partial support.
[10]{} Yu. S. Dedkov, M. Fonine, C. König, U. Rüdiger, and G. Güntherodt, Appl. Phys. Lett. **80**, 4181 (2002).
R. J. Soulen, J. M. Byers, M. S. Osofsky, B. Nadgorny, T. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J. Nowak, J. S. Moodera, A. Barry, and J. M. D. Coey, Science **282**, 85 (1998).
W. E. Pickett, Phys. Rev. Lett. **77**, 3185 (1996).
A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. **75**, 657 (2003).
C. Kallin and J. Berlinsky, Rep. Prog. Phys. **79**, 54502 (2016).
Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, K. Ishida, J. Phys. Soc. Jpn. **81,** 011009 (2012).
N. Read and D. Green Phys. Rev. B **61**, 10267 (2000).
X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Rev. Lett. **102**, 187001 (2009)
X.-L. Qi, and S.-C. Zhang, Rev. Mod. Phys. **83,** 1057 (2011).
J. Alicea, Rep. Prog. Phys. **75**, 076501 (2012).
P. A. Lee, Science **346**, 545-546 (2014).
M. Sato and Y. Ando, Rep. Prog. Phys. **80**, 076501 (2017).
R. Jackiw and P. Rossi, Nucl. Phys. B 190, 681 (1981).
J. P. Xu *et al.*, Phys. Rev. Lett. **114**, 017001 (2015).
S. B. Chung, H. Bluhm, and E.-A. Kim, Phys. Rev. Lett. **99**, 197002 (2007).
J. Jang, D. G. Ferguson, V. Vakaryuk, R. Budakian, S. B. Chung, P. M. Goldbart, Y. Maeno, Science **331**, 186 (2011).
L. Fu and C. L. Kane Phys. Rev. Lett. **100**, 096407 (2008).
A. R. Akhmerov, Johan Nilsson, and C. W. J. Beenakker, Phys. Rev. Lett. **102**, 216404 (2009).
L. Fu, C. L. Kane, Phys. Rev. Lett. **102**, 216403 (2009).
R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. **105**, 077001 (2010).
S. B. Chung, H.-J. Zhang, X.-L. Qi, and S.-C. Zhang, Phys. Rev. B **84**, 060510(R) (2011).
J. Li, T. Neupert, Z. J. Wang, A. H. MacDonald, A. Yazdani, B. A. Bernevig, Nat. Comm. **7**, 12297 (2016).
X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B **82**, 184516 (2010).
Q. L. He *et al.,* Science **357**, 294 (2017).
K. Schwarz, Journal of Physics F-Metal Physics **16**, L211 (1986).
M.I. Katsnelson, V.Yu. Irkhin, L. Chioncel, A.I. Lichtenstein, R.A. de Groot, Rev. Mod. Phys. **80**, 315 (2008).
While $t_{4}=0$ by the symmetry of the crystal, spin-orbit coupling effects lead to a finite imaginary $t_{4}=it/8\approx\pm(i)0.036$ eV. See Ref. [@Cai]. This term opens a small gap of $\sim4$ meV at the Dirac nodes.
T. Cai, X. Li, F. Wang, S. Ju, J. Feng, and C.-D. Gong, Nano Lett. **15**, 6434 (2015).
N. Furukawa, T. M. Rice, and M. Salmhofer, Phys. Rev. Lett. **81**, 3195 (1998).
R. Nandkishore, L. S. Levitov and A. V. Chubukov, Nat. Phys. **8**, 158 (2012).
M. L. Kiesel, C. Platt, W. Hanke, D. A. Abanin, and R. Thomale, Phys. Rev. B **86**, 020507(R) (2012).
For a more detailed analysis of the quantum critical scaling, see supplementary materials.
For a symmetry analysis, see supplementary materials.
The spectral gap does not close along the line of first order topological phase transitions. The gap closes, however, if $\Delta_{1}$ is virtually changed as a continuous parameter connecting two topologically distinct ground states.
A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B **78**, (2008).
S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig, New J. Phys. **12**, 65010 (2010).
D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. de Nijs, Phys. Rev. Lett. **49**, 405 (1982).
W. Kohn, J. M. Luttinger, Phys. Rev. Lett. **15**, 524 (1965).
J. Gonzalez, Phys. Rev. B **78**, 205431 (2008).
P. C. Pattnaik, C. L. Kane, D. M. Newns, C. C. Tsuei, Phys. Rev. B **45**, 5714 (1992).
F. Guinea, B. Uchoa, Phys. Rev. B **86**, 134521 (2012).
Mayorov *et al.*, Nano Lett. **11**, 2396 (2011).
|
---
author:
- 'F. Kemper[^1]'
- 'R. Stark'
- 'K. Justtanont'
- 'A. de Koter'
- 'A.G.G.M. Tielens'
- 'L.B.F.M. Waters'
- 'J. Cami'
- 'C. Dijkstra'
bibliography:
- 'ciska.bib'
date: 'Received / Accepted'
title: Mass loss and rotational CO emission from Asymptotic Giant Branch stars
---
Introduction {#sec:intro_co}
============
Low and intermediate mass stars ($1 < M < 8 \,M_{\odot}$) end their life on the red giant branch and asymptotic giant branch [AGB; see @H_96_review and references herein]. During the AGB phase, the stars have very extended tenuous atmospheres and shed almost their entire hydrogen-rich envelope through a dense and dusty stellar wind. In case of OH/IR stars, mass-loss rates can be so high that the dust shell completely obscures the central star, and the object is observable only at infrared wavelengths and through molecular line emission at radio wavelengths. The AGB phase is one of the few occasions in stellar evolution when time scales are not driven by nuclear (shell) burning but by surface mass loss. Helped by the low surface gravity and strong stellar pulsations, gas can move away from the star and will gradually cool. When the temperature drops below $\sim$1400 K, dust formation occurs, and a dust driven wind will develop. The mass-loss rates increase from $\dot{M} \approx 10^{-7}$ to a few times $10^{-5}$ $M_{\odot}$ yr$^{-1}$, while the AGB star evolves from the Mira phase to an OH/IR star [@VH_88_IRAScolors]. Recently, it has been suggested that higher mass-loss rates can be achieved for oxygen-rich AGB stars. @JST_96_OH26 find that OH 26.5+0.6 has undergone a recent increase in mass loss, leading to a current rate of $5.5 \cdot 10^{-4} \, M_{\odot}$ yr$^{-1}$, a result recently confirmed by @FJM_02_oh26. Even higher mass-loss rates were found for another oxygen-rich AGB star, , for which the mass-loss rate may be as high as $\sim$10$^{-3}$ $M_{\odot}$ yr$^{-1}$ [@DWK_02_IRAS16342]. A similar rate of a few times 10$^{-3}$ $M_{\odot}$ yr$^{-1}$ is found for the carbon-rich evolved star [@SMB_97_Egg].
AGB stars are important contributors of dust to the interstellar medium (ISM); it is estimated that a substantial fraction of the interstellar dust is produced by oxygen-rich AGB stars [e.g. @G_89_stardust]. In the outflow of evolved stars with an oxygen-rich chemistry the dust composition is dominated by silicates, both amorphous and crystalline [e.g. @SKB_99_ohir; @MWT_02_xsilI]. The appearance of crystalline silicate features in the far-infrared spectra of AGB stars seems to be correlated with a high optical depth in the amorphous silicate resonance at 9.7 $\mu$m and hence a high mass-loss rate [@WMJ_96_mineralogy; @CJJ_98_ohir; @SKB_99_ohir]. This could be interpreted as evidence that a certain threshold value for the density is required to form crystalline silicates. However, @KWD_01_xsilvsmdot showed that observational selection effects may play an important role in detecting crystalline silicates in AGB stars with low mass-loss rates. Therefore, the relation between mass-loss rate and crystallinity remains unclear at present.
In order to further study the correlation between the wind density and the dust composition, reliable mass-loss rates should be determined. Mass-loss rates of AGB stars can be obtained from the thermal emission from dust, predominantly coming from the warm inner regions [e.g. @B_87_dustshells]. They can also be inferred from observations of molecular transitions, in particular from CO [e.g. @KM_85_massloss]. A catalogue compiled by @LFO_93_CO lists observations of the CO $J=1 \rightarrow 0$ and $J = 2 \rightarrow 1$ transitions of both O-rich and C-rich AGB stars. (Hereafter we will use for these rotational transitions the notation CO(1$-$0) etc.) The mass-loss rates of a large number of objects from the catalogue are derived. However, the derived mass-loss rates seem to be underestimated for OH/IR stars, compared to the dust mass loss. @HFO_90_deficiency have studied the correlation between IRAS colours and mass-loss rates derived from CO(2$-$1) and CO(1$-$0) observations. In the case of very massive dust shells, they find that the intensity of the CO(1$-$0) transition is too low compared to the CO(2$-$1) transition, which they suspect to be due to a mass-loss rate increase over time. This then hints towards a superwind phase, which is generally believed to be important in the evolution of a Mira towards an OH/IR star [e.g. @IR_83_AGBevolution and references herein]. The superwind model was initially introduced to explain the amount of mass seen in planetary nebulae assuming that Miras are the progenitors of these nebulae [@R_81_superwind]. Miras are believed to evolve into OH/IR stars when they suddenly increase their mass-loss rate with a factor of $\sim$100.
As the inner regions are warmer they are better probed by higher rotational transitions. Thus a sudden density jump should be detectable in the CO lines. Model calculations by @JST_96_OH26 have demonstrated this effect for , using observations of rotational transitions up to CO(4$-$3). Unfortunately this transition is not sufficiently high to firmly establish the recent onset of a superwind, as its excitation temperature is only 55 K. Nevertheless, @JST_96_OH26 found that the peak intensities of these lines were significantly higher than what could be expected based on the extrapolation of the observed line strength of the CO(2$-$1) transition and the upper limit obtained for the CO(1$-$0) transition, assuming a constant mass-loss rate. Similar results are reported for other AGB stars [e.g. @G_94_OH32OH44; @DKF_97_superwind].
The work presented here aims to determine the mass-loss history of a number of oxygen-rich AGB stars with an intermediate or high optical depth in the near- and mid-infrared. For the first time, observations of rotational transitions up to CO (7$-$6) have been obtained ($T_{\mathrm{ex}} = 155$ K) which probe the more recent mass-loss phases. In Sect. \[sec:obs\] we describe the observations and data analysis. Sect. \[sec:conditions\] describes the model. Our results are discussed in Sect. \[sec:analysis\]. Concluding remarks and an outlook to future work is presented in Sect. \[sec:disc\].
Observations and data reduction {#sec:obs}
===============================
Instrumental set-up {#sec:setup}
-------------------
---------- ----------- --------------- ---------------------- ----------
receiver Frequency CO transition $\eta_{\mathrm{mb}}$ HPBW
(GHz)
A3 215–275 CO(2$-$1) 0.69 $19.7''$
B3 315–373 CO(3$-$2) 0.63 $13.2''$
W/C 430–510 CO(4$-$3) 0.52 $10.8''$
W/D 630–710 CO(6$-$5) 0.30 $8.0''$
E 790–840 CO(7$-$6) 0.24 $6.0''$
---------- ----------- --------------- ---------------------- ----------
: Technical details of the JCMT heterodyne receivers. The columns list the used receivers, the frequency windows at which they operate, the observable CO rotational transition, the beam efficiency $\eta_{\mathrm{mb}}$ and the half power beam width (HPBW).
\[tab:efficiencies\]
Observations of the $^{12}$CO(2$-$1), (3$-$2), (4$-$3), (6$-$5) and (7$-$6) rotational transitions in the outflow of evolved stars were obtained during several observing periods between April 2000 and September 2002 using the *James Clerk Maxwell Telescope* (JCMT) on Mauna Kea, Hawaii. For this purpose, all five different heterodyne receivers available at the JCMT were used, including the new MPIfR/SRON E-band receiver which operates in the frequency range. A description of this new receiver is given in Sect. \[sec:E-band\]. The technical details and beam properties of the JCMT set up with the appropriate heterodyne receivers are summarized in Table \[tab:efficiencies\]. Observations with the B3- and W-receivers were performed in double sideband (DSB) and dual polarization mode. The DSB mode was also used for the observations with the MPIfR/SRON E-band receiver. The bandwidth configuration of the receiver, and hence the spectral resolution was determined by the expected line width of the CO lines. We used bandwidths of at least twice the expected line width to have a sufficiently broad region for baseline subtraction. Estimates for the line width – which is determined by the outflow velocity – were based on published values of line widths of the CO(1$-$0) transition [e.g. @LFO_93_CO and references herein].
We used the beam-switching technique to eliminate the background. The secondary mirror was chopped in azimuthal direction over an angle of 120$''$. Over these small angles the noise from the sky is assumed to be constant. In case of extended sources we used a beam-switch of 180$''$.
The MPIfR/SRON 800 GHz receiver {#sec:E-band}
-------------------------------
The observations of the CO(7$-$6) line were made with the MPIfR/SRON receiver in October 2001. This PI system is in operation at the JCMT Cassegrain focus cabin since spring 2000. The receiver consists of a single-channel fixed-tuned waveguide mixer with a diagonal horn. The mixer consists of a Nb SIS junction with NbTiN and Al wiring layers fabricated at the University of Groningen, The Netherlands. Details on the fabrication of similar devices can be found in @JDL_00_E. Measured receiver temperatures at the cryostat window are DSB. The receiver has an intermediate frequency of $2.5-4$ GHz. System temperatures including atmospheric losses varied between 6000–14000 K (SSB) at the time of the observations. The beam shape and efficiency have been determined through observations of Mars and yield a deconvolved half power beam width (HPBW) of 6$''$ and a main beam efficiency $\eta_{\mathrm{mb}}$ of 24%.
Observations and data reduction {#sec:subobs}
-------------------------------
Our sample of evolved stars is given in Table \[tab:obslist\], which also indicates the distances towards the programme stars. The sample includes AGB stars and red supergiants. In Table \[tab:obsdetails\] an overview of the observed transitions is given, including cumulative integration times and the observing date. The data were obtained over a long period from April 2000 until September 2002 in flexible observing mode, and are part of a larger ongoing programme. During the observations, spectra of CO spectral standards used at the JCMT were also obtained. If necessary, a multiplication factor was applied to the observations of our sample stars, to correct for variations in the atmospheric conditions. These factors are listed in Col. 4 of Table \[tab:obsdetails\] and are based on measured standard spectra. Reliable standards are only available for the transitions observed with the A3-, B3- and W/C-receivers, for which the flux calibration accuracy is around 10%. For the W/D- and MPIfR/SRON E-band reliable standards for our lines of interest are lacking. Therefore we estimate that the absolute flux calibration in these bands has an accuracy of 30%.
Table \[tab:efficiencies\] lists the beam efficiencies $\eta_{\mathrm{mb}}$ for all receivers. The main beam temperatures were calculated according to $T_{\mathrm{mb}} =
T_{\mathrm{A}}^{\ast} / \eta_{\mathrm{mb}}$, where $T_{\mathrm{A}}^{\ast}$ is the measured antenna temperature. These main beam temperatures can directly be compared to observations from other telescopes.
![Correction of the profile of the CO(3$-$2) transition of . The dotted line represents the observation in which the interstellar contribution is clearly visible. Ignoring the interstellar contribution results in the solid line, which is used to obtain the integrated intensity.[]{data-label="fig:correct"}](H4224F1.eps){width="8.5cm"}
The reduced data is presented in Table \[tab:transitions\]. A linear baseline has been subtracted from the raw data, and the spectrum has been rebinned to improve the signal-to-noise ratio. We aimed to cover the line profile with at least $\sim$80 bins, which limits the rebinning factor. The bin sizes after rebinning and the corresponding r.m.s. values are listed in Cols. 4 and 5 of Table \[tab:transitions\]. Emission lines were detected in almost all observations, except for CO(3$-$2) and CO(6$-$5) and (7$-$6), for which we only obtained upper limits on the main beam temperatures. The line profiles of all transitions are shown in Figs. \[fig:wxpsc\]–\[fig:oh104\]. In some cases, interstellar lines are visible in the spectrum, for example in . To determine the integrated intensities we have cut the interstellar lines out of the spectrum, and interpolated both parts of the spectrum, as is demonstrated in Fig. \[fig:correct\]. The resulting profile was integrated to obtain $I$, which is the integrated intensity in K km s$^{-1}$. The system velocity $V_{\mathrm{LSR}}$ and the terminal expansion velocity $v_{\infty}$ are estimated directly from the line profile. The lines show a wide variety of shapes. There are parabolic line profiles, like those of (Fig. \[fig:wxpsc\]), (Fig. \[fig:irc50\]), (Fig. \[fig:gl\]), (Fig. \[fig:crl\]) and (Fig. \[fig:oh104\]). These parabolic line profiles indicate that the lines are optically thick [@M_80_CO]. On the other hand, many objects show signs the double-horned profiles indicative of an optically thin molecular layer. The most illustrative example is , in which the CO(2$-$1) and CO(3$-$2) transitions clearly show a double-peaked profile, although the peak around the central velocity indicates a more complex outflow structure (Fig. \[fig:vycma\]). In addition, some flat-topped profiles are observed, most notably those of (Fig. \[fig:vxsgr\]). These flat-topped line profiles are considered to be characteristic of molecular layers which have $\tau \sim 1$ at these frequencies [@M_80_CO].
@JST_96_OH26 have observed with the JCMT as well and report that they find line intensities $I = 25.8$ and 36.0 K km s$^{-1}$ for the CO(3$-$2) and (4$-$3) transition respectively. In addition they have scaled IRAM observations of the CO(2$-$1) to a 15m dish, to mimic the JCMT. The intensity of this line turned out to be 7.8 K km s$^{-1}$. Their results agree well with our results in case of the CO(2$-$1) and (3$-$2) transition, but they have observed an intensity of a factor of $\sim$2 higher for the CO(4$-$3) transition. The origin of the discrepancy with our results is unknown.
Physical conditions in the outflow: a model {#sec:conditions}
===========================================
The observed line profiles provide information on the physical structure of the outflow of these AGB stars, as the spectral resolution at the observed frequencies is sufficiently high to resolve the velocity structure. The terminal expansion velocity $v_{\infty}$ can be derived directly from the width of the line profile (Table \[tab:transitions\]). The model we use to analyze the CO data is based on a study by @S_88_molspec and was previously used by @JCT_94_redgiantwinds. The interpretation of our observations using this model is discussed in Sect. \[sec:analysis\].
Description of the model {#sec:modelco}
------------------------
The code consists of two parts: The first part solves the radiation transfer equation in the co-moving frame [@MKH_75_twolevel], computes the level populations (in full non-LTE) and iterates until level populations and radiation field are consistent. For solving the level populations, a Newton-Raphson method is used [@SH_86_multilevel]. The calculations take into account (de-)excitation through collisions, of which the rate is defined by the thermal velocity distribution, calculated from the local temperature, as well as (de-)excitation induced by a local radiation field and spontaneous de-excitation. The code treats pure rotational transitions in the ground and first vibrational levels, which are connected through these collisional and radiative transitions. The model can calculate the populations of as many as 50 levels at once, and is also applicable to molecules other than CO. The non-LTE rate equations to determine the level populations are described by
$$\begin{aligned}
& & n_i \sum_{j \neq i} (A_{ij} + C_{ij} + B_{ij} \overline{J}_{ij}) - \nonumber\\
& & \qquad \qquad \sum_{j \neq i} n_j (A_{ji} + C_{ji} + B_{ji} \overline{J}_{ij}) = 0
\label{eq:rateq}\end{aligned}$$
A change from level $i$ to level $j$ can be induced by collisional transitions (with the collisional rate $C_{ij}$) and radiative transitions, including spontaneous emission ($A_{ij}$, where $A_{ij} =
0$ for $i < j$) and stimulated emission and absorption ($B_{ij}\overline{J}_{ij}$). The collisional transition rates $C_{ij}$ are taken from laboratory measurements and potentials calculations [@FL_85_ratecoefficients] and are extended up to $J = 30$.
The line profile integrated mean intensity $\overline{J}_{ij}$ consists of two components:
1. The *continuum* radiation, originating from dust locally present. This radiation field can be switched off, by assuming there is no dust present in the considered part of the outflow.
2. Line radiation originating from a local region. The size of this region is defined by a velocity which @S_88_molspec and also @JCT_94_redgiantwinds have referred to as *stochastic* velocity $v_{\mathrm{sto}}$. The nature of this *stochastic* velocity is not specified, but physically should consist of a thermal component $v_{\mathrm{therm}}$ and a turbulent component $v_{\mathrm{turb}}$, given by
$$v_{\mathrm{sto}} = \sqrt{(v_{\mathrm{therm}})^{2}+ (v_{\mathrm{turb}})^{2}}
\label{eq:velocities}$$
In the outflow, the stochastic velocity is assumed to be constant and in almost all cases dominated by turbulence. The effect of the stochastic velocity is Doppler broadening of the lines, which is taken into account in the radiative transfer.
In the second part of the code, the calculated level populations are used as input to determine the observable line profiles by ray-tracing. Again the *stochastic* velocity is used, this time to determine the width of the interaction region along the line-of-sight to the observer. Integration over the full beam, for which the telescope parameters are required, yields the emergent line profile.
Free parameters {#sec:parameters}
---------------
The model has a number of free parameters (see Table \[tab:standard\]). In this section we will discuss the various parameters and their relevance for the model calculations.
### Density profile
The density profile $\rho(r)$ of the outflow determines the collision probabilities and optical depths required to solve Eqs. (\[eq:rateq\]) and to calculate the line profiles. The density profiles follows from the equation of mass continuity $$\rho(r) = \frac{\dot{M}}{4 \pi r^2 \, v_{\mathrm{exp}}(r)}
\label{eq:masscont}$$ where the expansion (or outflow) velocity profile used in the model is defined by $$v_{\mathrm{exp}} (r) = v_{\infty} \bigg( 1 - \frac{b}{r} \bigg)
\label{eq:vel}$$ In this equation $v_{\infty}$ represents the terminal velocity. Constant $b$ is chosen such that the expansion velocity at the stellar surface is given by $v_{\mathrm{exp}}(R_{\ast}) = 10^{-2} v_{\infty}$. The density structure is set by the following input parameters
1. The gas *mass-loss rate* $\dot{M}$ determines the mass input at the inner radius of the circumstellar shell. Our model allows us to simulate the effect of a time-variable mass-loss rate introducing *one jump* in the mass-loss history at an arbitrary point in the outflow ($r_{\mathrm{superwind}}$), where the density can increase or decrease with a specified factor. Except for this jump the mass-loss rate is constant, and therefore the density profile scales with the current mass-loss rate at and with the past mass-loss rate at $r >
r_{\mathrm{superwind}}$.
2. The density profile also scales with the outflow velocity profile given in Eq. (\[eq:vel\]), which is fixed by the *terminal velocity* $v_{\infty}$.
3. The *stellar radius* $R_{\ast}$ determines the base of the wind. The density $\rho(R_{\ast})$ at the inner radius follows from $R_{\ast}$, $\dot{M}$ and $v_{\exp}(R_{\ast})$ using the equation of mass continuity (\[eq:masscont\]).
4. The *outer radius* $R_{\mathrm{out}}$ denotes the extent of the outflow.
### Temperature profile
The *temperature profile* $T(r)$ is another important parameter that influences the level populations in the circumstellar CO, by means of collisions. The temperature profile may be compiled self-consistently, i.e. based on calculations of realistic heating and cooling processes [e.g. @GS_76_OHIR; @JCT_94_redgiantwinds; @CN_95_water; @ZE_00_WHya]. As a first order estimate we have used a power law of the form $T(r)
\propto r^{-\alpha}$, where the index $\alpha$ depends on the mass-loss rate and is derived from the outer regions of the temperature profiles calculated by @JCT_94_redgiantwinds.
### Dust-to-gas ratio and dust properties
Unfortunately it is difficult to study the gas and dust mass-loss rate completely independent from each other, as continuum emission from dust may have an effect on the rates, as described in Sect. \[sec:modelco\]. In particular, infrared photons at 4.6 $\mu$m pump CO molecules from the ground vibrational state $v=0$ to the first vibrational level $v=1$ [e.g. @M_80_CO; @S_88_molspec]. The molecules will eventually de-excite to the vibrational ground level, but not necessarily to the same rotational ground level. This causes a higher population of the higher CO rotational levels than which reflects the kinetic temperature of the gas and the line radiation field. As the source of the 4.6 $\mu$m radiation is predominantly thermal dust emission, the *dust-to-gas ratio* and the *dust opacity* are important input parameters. For simplicity, we assumed that there was no dust present in the outflows. For some of the calculations we did include dust to study the effect on the line strengths. In those cases we used a dust opacity corresponding to the mixture of solid state components derived for , a typical OH/IR star [@KDW_02_composition]. The same power-law temperature distribution as for the gas is used to calculate the thermal emission from the grains, although this is most likely not true.
### Velocity field
The velocity field has already been mentioned as a constraint for the density structure, but it also plays an important role in the formation of line profiles. The outflow velocity profile (constrained by the *terminal velocity* $v_{\infty}$ and the velocity law given in Eq. (\[eq:vel\])) and the *stochastic velocity* $v_{\mathrm{sto}}$ determine the location and extent of the interaction regions. As said before, the stochastic velocity is assumed to be constant throughout the dust shell.
### Distance and telescope parameters
The resulting main beam temperatures depend on the *distance* towards the object. In addition, the telescope *beam size* is important to determine what part of the object falls inside the beam. In case the circumstellar shell is resolved, the *pointing displacement* (usually 0$''$) should be known as well.
Analysis of the results {#sec:analysis}
=======================
![Overview of integrated intensities for each line observed in our programme stars. The horizontal axis of each panel lists the rotational transitions observed, where the spacing between the tick marks is proportional to the difference in frequency. On the vertical axis the integrated intensity (K km s$^{-1}$) is given. The diamonds represent the measured values; in addition the error bars are shown (data from Table \[tab:transitions\]). Note that only the line strengths of increase with higher rotational transitions. For most of the other stars (except , and ) CO (3$-$2) is the brightest line. In the upper right corner the relative values for the standard model (see Table \[tab:standard\], Fig. \[fig:standard\]) are presented for comparison (indicated with $\times$ symbols).[]{data-label="fig:sed"}](H4224F2.eps){width="8.5cm"}
![Line profiles calculated for a standard AGB model, folded with the JCMT beams. The model parameters are given in Table \[tab:standard\]. []{data-label="fig:standard"}](H4224F3.eps){width="8.5cm"}
Here we will analyze the observations using the model described in Sect. \[sec:conditions\]. Fig. \[fig:sed\] shows the intensities integrated over line width of each observed line for all our sample stars. For all sources, except , the integrated intensity increases from the CO(2$-$1) to (3$-$2) transition, and decreases again for higher transitions. This is also visible in the peak main beam temperatures ${T_{\mathrm{mb}}}$ (Table \[tab:transitions\]). Since most studies concentrate on the lower transitions (up to CO(3$-$2)) this was not noticed before. An exception are the JCMT observations of performed by @JST_96_OH26, where the CO(4$-$3) transition is included as well. However, as pointed out in Sect. \[sec:subobs\], they observed an increasing line strength with increasing rotational transition. This differs from our observations of this object that the CO(3$-$2) is the strongest emission line.
In order to explain the observational trends, we have constructed a standard model assuming physical parameters widely used for AGB outflows (see Table \[tab:standard\]). We used a mass-loss rate of $\dot{M} = 10^{-5}$ $M_{\odot}$ yr$^{-1}$, and calculated the level populations of the CO gas between the stellar radius $R_{\ast}= 4.0
\cdot 10^{13}$ cm and the outer radius . For the terminal velocity we used $v_{\infty} = 15.0$ km s$^{-1}$ and the turbulent velocity was assumed to be $v_{\mathrm{sto}} = 1.0$ km s$^{-1}$. A power-law temperature profile was chosen: $T(r) = 2000 \, (r/R_{\ast})^{-0.7}$ K. We used for the relative abundance of the CO gas with respect to molecular hydrogen \[CO\]/\[H$_2$\] = $3.0\cdot 10^{-4}$, and we ignored the contribution of thermal emission from dust to the local radiation field. Finally, we placed this system at a distance of 1000 pc, and used the JCMT telescope parameters to calculate the emerging line profiles (Table \[tab:standard\], Fig. \[fig:standard\]). The lines show increasing peak and integrated intensities with increasing line strengths, up to CO(6$-$5). The CO(7$-$6) line is again much weaker which can be explained by the relatively narrow HPBW of the E-band (Table \[tab:efficiencies\]). This transition is comparable in strength to the CO(2$-$1) transition, for this standard set of parameters. This is a general characteristic of all other studies calculating the line intensities for commonly used AGB parameters [e.g. @G_94_revisedmodel; @G_94_OH32OH44; @JCT_94_redgiantwinds].
parameter value
-------------------------------- ---------------------------------
distance 1.0 kpc
$v_{\infty}$ 15.0 km s$^{-1}$
$v_{\mathrm{sto}}$ 1.00 km s$^{-1}$
$R_{\mathrm{in}}$ 5 R$_{\ast}$
$R_{\mathrm{out}}$ 6000 R$_{\ast}$
$R_{\ast}$ $4.0 \cdot 10^{13}$ cm
$\dot{M} $ $10^{-5}$ $M_{\odot}$ yr$^{-1}$
$T(r)$ 2000 $(r/R_{\ast})^{-0.7}$ K
$[\mathrm{CO}]/[\mathrm{H}_2]$ $3.0\cdot 10^{-4}$
dust-to-gas ratio 0%
: Parameters of the standard AGB model
\[tab:standard\]
In the following sections we will try to find a set of parameters to explain our observations: in general the CO(3$-$2) is the strongest line, which contradicts the results of the standard model. In order to study as many stars as possible in a systematic way, we will use a line ratio diagram based on the CO(3$-$2)/CO(2$-$1) and CO(4$-$3)/CO(2$-$1) ratios of integrated intensities, rather than trying to fit the intensities and line profiles.
The two low mass-loss rate AGB stars and are added to the sample; these stars are the only ones for which sufficient reliable line ratios of interest can be derived from published JCMT data (see Table \[tab:litvalues\]). For the ratios of the integrated intensities are 1.5 and 2.0 for CO(3$-$2)/CO(2$-$1) and CO(4$-$3)/CO(2$-$1) respectively [@BKO_00_rvboo]. For these numbers are 2.1 and 3.4 respectively [@KYL_98_COsurvey; @KO_99_COcatalogue]. The observations of @KYL_98_COsurvey were obtained with the CalTech Submillimeter Observatory (CSO) and are rescaled to the JCMT observations of @KO_99_COcatalogue such that the line intensities of the CO(3$-$2) transitions both reflect the same dish size and can be compared to our CO(3$-$2) and CO(2$-$1) observations. One should bear in mind however, that and are not representative of AGB stars with spherical outflows. @KJ_96_XHer have mapped in CO lines and conclude that in addition to a slow spherically expanding shell there are indications for bipolar outflows with a higher velocity, which carry a significant fraction of the ejected gas. This result is confirmed by @KOP_03_massloss, who in addition present SiO line observations indicative of a circumstellar rotating disk. Interferometric CO line observations of indicate that this object also has a disk, possibly showing Keplerian rotation [@BKO_00_rvboo]. Therefore, comparison of these stars with our data and analysis should be done with some reservation.
In the literature, we found a sample of six Miras, which were observed in all three lines discussed here, using CSO (see Table \[tab:litvalues\]). It is possible to scale these observations to the JCMT observations by accounting for the dish size. However, we have chosen not to do this, because it is unknown how reliable the rescaled data still is, as little is known about the beam filling factor of the various transitions, while the beam sizes of the telescopes are very different. Instead we chose to compare these observations with our model calculations, as will be discussed in Sect. \[sec:mdotco\].
A constant mass-loss rate? {#sec:mdotco}
--------------------------
![Line ratio diagram. On the horizontal axis the ratio of the integrated intensities of CO(3$-$2)/CO(2$-$1) is given, while the CO(4$-$3)/CO(2$-$1) ratio is plotted on the vertical axis. The diamonds represent the positions of our sample stars, complemented with literature data for and . Equal ratios are indicated with the dashed line. In case of our sample stars, the CO(3$-$2) transition is stronger than the (4$-$3) transition, therefore all observations can be found in the lower right half of the diagram. The only exception is which is found in the upper left half. The asterisks mark the positions of model calculations, where we used the standard parameters (see Table \[tab:standard\]). Only the mass-loss rate was varied and is given in units of $M_{\odot}$ yr$^{-1}$.[]{data-label="fig:mcst"}](H4224F4.eps){width="8cm"}
In a line-ratio diagram (Fig. \[fig:mcst\]), the observed values occupy the lower right half of the diagram, corresponding to the region where the CO(3$-$2) line is stronger than the CO(4$-$3) line. The values corresponding to are an exception and are found in the upper left half. This data point should be treated with care though, as the detected lines suffer from interference with interstellar absorption (Fig. \[fig:oh127\]) and therefore the line intensities are not well known (see Table \[tab:transitions\]). The literature data of and are also located in the upper left half of the diagram.
First, we assume that the mass-loss rate is constant. For five different mass-loss rates (10$^{-8}$, 10$^{-7}$, 10$^{-6}$, 10$^{-5}$ and 10$^{-4}$ $M_{\odot}$ yr$^{-1}$) we have calculated the emerging line profiles, thus covering the full range in $\dot{M}$ from Miras to OH/IR stars [@B_87_dustshells; @VH_88_IRAScolors]. All other input parameters were assumed to have the standard values given in Table \[tab:standard\]. The predicted CO(3$-$2)/CO(2$-$1) and CO(4$-$3)/CO(2$-$1) ratios of the integrated intensities were compared to the observed ratios.
The model calculations (marked with asterisks) are found in the upper left half of the diagram where the CO(4$-$3) line is stronger than the CO(3$-$2) line, and are therefore not consistent with the observed line ratios. All model line ratios are found in a narrow range to one end of the region where the observations are found (see Fig. \[fig:mcst\]). Only the observations of match the modelled line ratios, but this could be merely a coincidence as is not a typical AGB star.
![Line ratio diagram for CSO data. On the horizontal axis the ratio of the integrated intensities of CO(3$-$2)/CO(2$-$1) is given, while the CO(4$-$3)/CO(2$-$1) ratio is plotted on the vertical axis. The diamonds represent the ratios obtained for a sample of six Miras. The CO(4$-$3) and CO(3$-$2) data are obtained from @Y_95_CO3-2, and the CO(2$-$1) data is taken from a study by @KYL_98_COsurvey. For and additional CSO CO(3$-$2) data are available, from @KYL_98_COsurvey and @SKY_95_molecular respectively. In the diagram these measurements are indicated with ’R Hya K’ [@KYL_98_COsurvey] and an ’$\chi$ Cyg S’ [@SKY_95_molecular]. Equal ratios are indicated with the dashed line. The asterisks mark the positions of model calculations, performed for the CSO beam and dish size, where we used the standard parameters (see Table \[tab:standard\]). Only the mass-loss rate was varied and is given in units of $M_{\odot}$ yr$^{-1}$.[]{data-label="fig:cso"}](H4224F5.eps){width="8cm"}
There are not many reports in the literature of AGB stars observed in these three lines with the JCMT, but we compared the results discussed here with observations performed using the CSO. For that purpose, we have recalculated the model line ratios for the CSO beam and dish size. A sample of six Miras is consistently observed with CSO, where the CO(4$-$3) and CO(3$-$2) measurements are obtained by @Y_95_CO3-2, and the CO(2$-$1) observations by @KYL_98_COsurvey. Additional CSO observations of the CO(3$-$2) line in two of these objects were also included [@KYL_98_COsurvey; @SKY_95_molecular]. The results are shown in Fig. \[fig:cso\]. Similar to Fig. \[fig:mcst\], the line ratios derived from the standard calculations are found just above the dashed line, indicating that the CO(4$-$3) line should be stronger than the CO(3$-$2) transition. However, most observations are found well below the dashed line, where the CO(3$-$2) is the strongest line. falls in this region as well. However, if we use the measurement of @SKY_95_molecular for the CO(3$-$2) line, the line ratios become such that it is found in the same region as the model ratios. Possibly this point is unreliable, as it does not come from a consistent data set. seems to be an outlier for both CO(3$-$2) measurements. Another remarkable observation is that the observed CO(2$-$1) lines seem to be weaker than what is expected from the model calculations, given the fact that the calculated ratios are closer to the origin of the plot. We may conclude that in general the CSO observations occupy more or less the same region of the plot with respect to the model ratios as our observations. Therefore, in the remainder of this paper we will limit our detailed analysis to our JCMT data.
Apparently, variations of the mass-loss rate alone do not change the line ratios enough to significantly increase the strength of the CO(3$-$2) line with respect to the CO(4$-$3) line. In the next section, we will investigate to what extent variations in the other parameters can shift the model calculations such that the line strength ratios more closely resemble the observed values.
Exploring parameter space {#sec:parspace}
-------------------------
To further explore parameter space, we opted to vary the input parameters of the standard model (Table \[tab:standard\]) one by one, and compare the line ratios with the observations. Combining the changes in line ratios from variations in the individual parameters then provides a feeling for the range in line ratios that can be covered, and may show whether or not it is possible to explain the observed line ratios at all. Of course, once a satisfactory match in line ratios is achieved by combining the effects of changes in individual parameters, fine tuning should be performed to fit the observed data in detail. This is necessary as some of these parameters might not be completely independent from each other, and the precise combined effect on the line profile is difficult to predict.
![Mosaic of diagrams representing the CO(4$-$3)/CO(2$-$1) ratio on the vertical axis versus the CO(3$-$2)/CO(2$-$1) ratio on the horizontal axis. We used the standard model described in Sect. \[sec:analysis\] and Table \[tab:standard\] and varied for each panel one of the parameters. From the upper left corner turning clockwise the investigated parameters are: the stochastic velocity $v_{\mathrm{sto}}$ (km s$^{-1}$), the outer radius $R_{\mathrm{out}}$ ($R_{\ast}$), the dust-to-gas ratio and the distance $D$ (pc). The line ratios resulting from the model calculations are marked with asterisks and the observed line ratios with diamonds. The dashed lines indicate equal line ratios. Note that the ranges plotted on the axes are smaller than the ranges in Fig. \[fig:mcst\] to improve readability.[]{data-label="fig:mosaic1"}](H4224F6.eps){width="8.5cm"}
In Fig. \[fig:mosaic1\] a mosaic of line-ratio diagrams is shown, in which the effects of changes in the stochastic velocity, the outer radius, the dust-to-gas ratio and the distance are shown. In general the effects due to changes in these parameters are small. To keep the plots readable, only small parts of the original line-ratio diagram (Fig. \[fig:mcst\]) are shown. The modelled line ratios for which these parameters are varied scatter mainly closely around the observed values for . In all these modelled line strengths, the CO(3$-$2) line is still weaker than the CO(4$-$3) line. Of course varying the parameters mentioned here causes changes in the absolute line strengths, but the line ratios are not so much affected.
In the models where the distance was varied, we placed the object progressively closer to the observer, such that the beam filling factor is initially less than unity, but increases with decreasing distance. Although the beam size corresponding to the CO(2$-$1) transition is larger than that corresponding to the CO(3$-$2) transition, the line formation region of the CO(2$-$1) transition is located so much further out that the object is first resolved for the CO(2$-$1) transition. This implies that less emission from this line is received by the telescope. When this happens, the line ratios increase. The CO(4$-$3)/CO(2$-$1) line ratio increases faster for decreasing distance than does CO(3$-$2)/CO(2$-$1), because the CO(3$-$2) line emission is the next to become resolved, as this line is formed more inwards in the circumstellar shell, but still further out than the higher transitions.
The stochastic or turbulent velocity determines the interaction length along the line-of-sight, i.e. the region over which the line is formed (see also Sect. \[sec:modelco\] and \[sec:parameters\]). The effect of a larger turbulent velocity is different for optically thick and optically thin lines. In the optically thin case, a change in profile strength may result from changes in the line source function in the (near and far) parts of the line interaction region, that is added relative to the default case. In the optically thick case the relevant source function is the one at the location where $\tau
\approx 1$, which shifts towards the observer when $v_{\mathrm{sto}}$ is increased. It may therefore differ from the default case. As these effects tend in the same direction for all lines (except possibly when lines change from optically thin to optically thick), the line ratios are found not to change dramatically when varying the turbulent velocity.
Changing the outer radius has a stronger effect on the line ratios, as can be seen in Fig. \[fig:mosaic1\]. When the outer radius is increased, more relatively cold gas will be present. In this gas mostly the lower rotational levels are populated, thus increasing predominantly the CO(2$-$1) transition. The higher the transition, the less it is affected by the outer radius.
The last parameter shown in Fig. \[fig:mosaic1\] is the dust-to-gas ratio. The most important effect of adding dust to the circumstellar shell is in the population of the rotational levels. Continuum emission at 4.6 $\mu$m can be absorbed by CO molecules, exciting them from the ground to the first vibrational level. They will return to the vibrational ground state by spontaneous emission, but preferentially to a higher rotational level than they started from. This has a non-LTE effect on the level populations, leading to variations in both the line strengths and the line ratios.
![Mosaic of diagrams representing the CO(4$-$3)/CO(2$-$1) ratio versus the CO(3$-$2)/CO(2$-$1) ratio. Again, the standard model parameters (Table \[tab:standard\]) were used. Only the temperature profile was varied. From the panel in the upper left corner turning clockwise the adopted temperature profiles are: [**i)**]{} $T(r) \propto r^{-\alpha}$, with $\alpha$ indicated in the plot. [**ii)**]{} $T(r)$ is described with a function consisting of two power-laws. See text for description. [**iii)**]{} $T(r) \propto r^{-\alpha}$, with negative values of $\alpha$, indicated in the plot. [**iv)**]{} A constant temperature throughout the circumstellar shell, where the adopted values are indicated in the plot. Again, the predicted line ratios are marked with asterisks and the observed line ratios with diamonds. The diamonds are not labelled to avoid a crowded plot, but can easily be identified using Fig. \[fig:mcst\]. The dashed lines indicate equal line ratios.[]{data-label="fig:mosaic2"}](H4224F7.eps){width="8.5cm"}
We also investigated the effect of the temperature distribution. The results are shown in Fig. \[fig:mosaic2\]. Various temperature profiles have been used in the different panels of this figure. The simplest approach is to consider a power law $T(r) = T_{0}
(r/R_{\ast})^{-\alpha}$, where $\alpha$ is usually positive and has a value around for realistic profiles [@JCT_94_redgiantwinds]. We considered a much broader range of $\alpha$, including negative values and also a constant temperature, . These cases are of course not a true physical representation of the dust shell, but are just considered to study the effect of extreme conditions. In most cases, we used a temperature at the inner edge of $T_{0} = 2000$ K. However, when the power law is shallow (small $\alpha$), the resulting temperature at the outer radius would be higher than 25 K if we use the same value for $T_{0}$. In that case, we adjusted $T_{0}$ such that . This outer boundary temperature is in the regime of excitation temperatures of the lower rotational transitions. For the negative values of $\alpha$, the temperature $T_{0}$ was assumed to be 25 K. A number of models with $\alpha = 0$ has also been computed, see the lower left panel of Fig. \[fig:mosaic2\]. Adopted temperatures are 10, 25, 50 and 100 K. The models with a constant temperature or an outwards increasing temperature are unrealistic, but we included them in our parameter study, to see if it is possible at all to change the line ratios significantly by changing the run of the temperature.
profile $\alpha_{\mathrm{in}}$ $T_0$ (K) $T_{\mathrm{ex}}$ (K)
--------- ------------------------ ----------- -----------------------
1 1.0 2500 33.1
2 1.5 2500 33.1
3 1.0 2500 16.6
4 1.0 2500 55.2
5 1.0 2200 33.1
: Parameters for the temperature profiles with a change in slope. See text for details.
\[tab:kink\]
To add to the realism of the models, we composed a number of temperature profiles consisting of two power laws with different values for $\alpha$. These profile are inspired by heat balance calculations of @JCT_94_redgiantwinds, and are defined as follows: $$T(r) = \left\{ \begin{array}{ll}
T_0 (r/R_{\ast})^{-\alpha_{\mathrm{in}}} &\mathrm{for} \, T > T_{\mathrm{ex}}\\
T_1 (r/R_{\ast})^{-\alpha_{\mathrm{out}}} &\mathrm{for} \, T < T_{\mathrm{ex}}
\end{array} \right.$$ Five different profiles with a change in slope were constructed, where the excitation temperatures of the CO(4$-$3) ($T_{\mathrm{ex}}=55.2$ K), (3$-$2) (33.1 K) and (2$-$1) (16.6 K) were used to define the position of the change in the slope. In all cases $\alpha_{\mathrm{out}}$ was chosen to be 0.7. For the other parameters, the reader is referred to Table \[tab:kink\]. The resulting line ratios are plotted in the upper right panel of Fig. \[fig:mosaic2\]. All models with a power law with a slope change cluster remarkably close to the CO(4$-$3)/CO(2$-$1) and CO(3$-$2)/CO(2$-$1) ratios observed in and . The only outlier is profile 2 (see Table \[tab:kink\]). Although from the various panels in Fig. \[fig:mosaic2\] it seems to be possible to explain the observed ratios of and , it is not possible to explain the line ratios of other stars of our sample, not even for extreme temperature profiles.
A representative case: WX Psc {#sec:wxpsc}
-----------------------------
In order to investigate the possibilities to explain the integrated intensities of the CO rotational transitions, we will focus on . All transitions are observed and detected. The signal-to-noise ratio is reasonable for all transition, except for the CO(6$-$5) line. The previous section has shown that the line ratios of the lower rotational transitions can be explained using power law temperature profiles. In this section we expand our investigations to the higher rotational transitions. The observed values for the integrated intensities $I$ of the CO(6$-$5) and (7$-$6) transition are much lower than the expected values based on the standard model described in Sect. \[sec:analysis\].
parameter value ref.
-------------------- -------------------------------------- ------
distance 0.74 kpc 1
$v_{\infty}$ 20 km s$^{-1}$
$v_{\mathrm{LSR}}$ +9 km s$^{-1}$
$R_{\mathrm{in}}$ 6.6 R$_{\ast}$ 2
$T_{\mathrm{eff}}$ 2250 K 2
$\alpha$ 0.5 3,4
sp. type M9-10 5
$L_{\ast}$ $(1.22-1.31) \cdot 10^4$ L$_{\odot}$ 6
$M_{\ast}$ $>5$ M$_{\odot}$ 7
: Physical parameters of . The terminal velocity $v_{\infty}$ and system velocity of the object $v_{\mathrm{LSR}}$ are derived from our observations (Table \[tab:transitions\]). The other parameters are extracted from the literature, the references are: $^{1}$@VVV_90_phaselags, $^{2}$@HBB_01_WXPsc, $^{3}$@JCT_94_redgiantwinds, $^{4}$@ZE_00_WHya, $^{5}$[simbad]{}, $^{6}$@LL_96_SED, $^{7}$@LW_00_coolstars.
\[tab:wxpsc\]
is a well studied AGB star with an intermediate mass-loss rate. From recent studies, notably the work of @HBB_01_WXPsc, we have retrieved the physical characteristics of the star and the circumstellar environment (see Table \[tab:wxpsc\]). These values were used as input parameters for our model calculations. For required parameters which are not accurately known, we maintained the values of our standard model (Table \[tab:standard\]). We used a stellar radius of $5.4 \cdot
10^{13}$ cm, implying a luminosity of $1.3 \cdot 10^4$ L$_{\odot}$ for $T_{\mathrm{eff}}=2250$ K.
----------------------------- ------------------------------- ----------------------- -------------- --------------------- -------------
tracer $\dot{M}_{\mathrm{gas}}$ $T_{\mathrm{ex}}$ (K) $R/R_{\ast}$ $R$ (cm) travel time
(M$_{\odot}$ yr$^{-1}$) (yr)
$L - [12\, \mu \mathrm{m}]$ $2.0 \cdot 10^{-5}$ $<600$
CO 7–6 $3.0 (\pm 0.3) \cdot 10^{-7}$ 155 700 $3.8 \cdot 10^{16}$ 600
CO 6–5 $1.4 (\pm 0.1) \cdot 10^{-7}$ 116 900 $4.9 \cdot 10^{16}$ 780
CO 4–3 $1.3 (\pm 0.1) \cdot 10^{-6}$ 55.1 1400 $7.6 \cdot 10^{16}$ 1200
CO 3–2 $6.3 (\pm 0.2) \cdot 10^{-6}$ 33.1 1100 $5.9 \cdot 10^{16}$ 940
CO 2–1 $8.0 (\pm 0.9) \cdot 10^{-6}$ 16.6 1900 $1.0 \cdot 10^{17}$ 1600
----------------------------- ------------------------------- ----------------------- -------------- --------------------- -------------
\[tab:wxmdot\]
Determination of the mass-loss rate from the integrated intensities of the rotational transitions gives an idea of the mass-loss history of the AGB star. Fig. \[fig:ewdmdt\] shows how the integrated line intensities depend on it. In Table \[tab:wxmdot\] the mass-loss rates derived from each observed transition are listed, while all other parameters were kept fixed. In addition, the gas mass-loss rate, derived from the $L - [12\, \mu \mathrm{m}]$ colour [@KDW_02_composition] is given, where a dust-to-gas ratio of 1% is assumed. The dust spectral energy distribution covers a temperature range of $\sim200-800$ K, which corresponds to a region even more inwards than the CO line emission.
We conclude that constant mass-loss rate models cannot explain all of the observed line intensities. Rather, it seems that the mass-loss rate varies with the $J$-level under consideration. Specifically, the mass-loss rate corresponding to the CO(2$-$1) emission is almost comparable in strength to the mass-loss rate derived from the dust emission (Table \[tab:wxmdot\]). For the higher rotational transitions, the derived mass-loss rates go down with increasing line frequency, although it perhaps increases slightly again for the CO(7$-$6) transition. The mass-loss rates determined from the high rotational transitions disagree with the mass-loss rate derived from the infrared dust emission. A difference of at least an order of magnitude occurs although the regions that are traced by the high rotational transitions and the dust emission are closest in temperature, and are therefore spatially close together. In general, a decreasing mass-loss rate with increasing rotational energy level is observed, which is inconsistent with predictions based on the superwind model [e.g. @G_94_OH32OH44; @JST_96_OH26; @DKF_97_superwind]. The results derived here point towards a mass-loss rate decreasing with time, rather than a stratification consistent with the onset of a superwind phase. In the next sections we will try to explain this discrepancy.
![Using the known parameters of (Table \[tab:wxpsc\]) predicted integrated intensities are given for a large range of mass-loss rates. Integrated intensities are plotted in a logarithmic scale on the vertical axis, and mass-loss rates ($M_{\odot}$ yr$^{-1}$) on the horizontal axis, also in logarithmic scale. The integrated intensities have been calculated for all lines observable with the JCMT and these calculated models are indicated with symbols (see legend). The models have been connected with a line. Using the observed integrated intensity for a certain line, the mass-loss rate of WX Psc can be estimated from this plot (see Table \[tab:wxmdot\]).[]{data-label="fig:ewdmdt"}](H4224F8.eps){width="8cm"}
Possible explanations for the inconsistency {#sec:expl}
-------------------------------------------
### Mass-loss variations?
![Normalized intensity ($I(p) p^3$) as a function of impact parameter $p$. The curves for each rotational transition are calculated using the mass-loss rate corresponding to that transition (see Table \[tab:wxmdot\]). []{data-label="fig:reg"}](H4224F9.eps){width="8cm"}
In principle, it should be possible to construct a *combination of a density and temperature profile*, such that the observed line intensity ratios can be explained. This is not possible for a constant mass-loss rate, as becomes apparent from Fig. \[fig:mosaic2\], so apparently there must also have been variations in $\dot{M}$. Using the model calculations, we can derive an impression of the mass-loss history using the mass-loss rates listed in Table \[tab:wxmdot\]. For all these values we have calculated the region where the respective line originates (Fig. \[fig:reg\]). This is not done in terms of radial distance to the central star, but as a function of impact parameter, in which case contributions in line-of-sights due to interactions at various radial distances have been integrated. Therefore, the values on the horizontal axis can not directly be translated to a radial distance towards the central star, but present a lower limit to this distance. In addition, one has to bear in mind that the regions from which the various lines originate are not distinct, but largely overlap. Some overlap in Fig. \[fig:reg\] is due to projection effects along the line-of-sight, but a significant fraction is due to real physical overlap of the line-formation regions. Although all regions are plotted in one figure, they do not arise from the same model but are the calculated for the corresponding mass-loss rate for each line (see Table \[tab:wxmdot\]). Therefore, it is possible that the CO (4$-$3) seems to originate from a region that is more distant from the central star than the region where the CO (3$-$2) line originates, although their excitation temperatures would suggest otherwise in an outwards decreasing temperature profile. Concluding, the mass-loss rates that we have determined are only average values for these line formation regions. Nevertheless, estimates of the distances from the line forming regions towards the central star can be derived for the mass-loss rates traced by the observed transitions. Using a stellar radius of $5.4 \cdot 10^{13}$ cm and an expansion velocity of 20 km s$^{-1}$ the time elapsed since the ejection of the gas from the stellar surface, traced by the various transitions can be calculated. The results are listed in Table \[tab:wxmdot\]. The cycle can be completed by adding the dust mass-loss, mostly originating from the region inwards of the CO(7$-$6) transition, and thus ejected less than 600 years ago. Note that the dust mass-loss rate is transferred into a gas mass-loss rate by assuming a dust-to-gas mass ratio of 0.01. Actual deviations to this ratio imply a different gas mass-loss rate traced by the $L-[12$ $\mu$m$]$ colour. From Table \[tab:wxmdot\] we can determine that the interval between the two maximum mass-loss rates, traced by the CO(2$-$1) transition and the $L-[12$ $\mu$m$]$ colour of the dust emission, is of the order of $\sim$1000 years.
Mass-loss variations on such time scales have in fact been observed in other evolved stars. Circumstellar series of arc-like structures have been interpreted as due to mass-loss modulations, notably for C-rich post-AGB stars, where the separation is a measure for the time scale of these variations. @KSH_98_bipolar derive that the separation between arcs observed around [IRAS 17150$-$3224]{} corresponds to a time scale of . For reasonable numbers for the distance and outflow velocity one can determine that these arcs may be due to mass-loss variations on time scales of 200–1000 yr. A similar time scale (200–800 yr) is derived by @MH_99_IRC10216 for [IRC+10216]{}. The circumstellar arcs around [CRL 2688]{} (Egg Nebula) are believed to be ejected at 75–200 yr intervals, assuming a distance of 1 kpc and an outflow velocity of 20 km s$^{-1}$ [@STW_98_eggnebula]. IRAS LRS spectroscopy has shown that hot dust ($T>500$ K) is absent around a number of AGB stars. This is interpreted as a drop in mass-loss rate which occurred $\sim$100 years ago, consistent with the spacing between the arcs observed around post-AGB stars [@MIZ_01_100yr]. Hydrodynamic calculations considering the gas and dust as partially or completely decoupled outflow components resulted in mass-loss variations of an order of magnitude at intervals of 200–350 year for partially and 400 year for completely decoupled fluids [@SID_01_quasiperiodic]. Moreover, @FMS_03_multiple report on the discovery of multiple shells seen in CO (1$-$0) emission around . These shells are found to have intershell time scales of 1300–2900 year. The circumstellar arcs and molecular shells observed around post-AGB stars and the density enhancements emerging from hydrodynamic calculations have similar time scales to what we derive here for mass-loss variations in the outflow of , indicating that the same phenomenon may perhaps play a role here.
Variations in the mass-loss rate of AGB stars have already been studied for a long time. It is generally accepted that the AGB phase is terminated by the superwind; a phase in which the mass-loss rate rapidly increases [@R_81_superwind; @BH_83_maserstrength]. However, the mass-loss rates inferred from the CO line intensities for decrease with time and are thus opposite to the classical superwind model predictions. The thermal pulses associated with He-shell ignition are also thought to cause mass-loss variations [@VW_93_massloss]. As for the superwind, the behaviour and time scale of these variations do not comply with our model predictions.
### A gradient in the turbulent velocity?
![The influence on the line profile of CO(3$-$2) due to variations of the stochastic velocity. The input parameters of our standard model are used (Table \[tab:standard\]), only the turbulent velocity – which in our model is independent of $r$ – is varied, in the range from 0.05 – 2.0 km s$^{-1}$.[]{data-label="fig:vsto"}](H4224F10.eps){width="8cm"}
Besides a complex density-temperature profile due to periodic mass-loss variations, there may be another way to explain the line intensities of the CO rotational transitions observed in ; a *gradient in stochastic velocity $v_{\mathrm{sto}}$*. The stochastic velocity is an important parameter in the line formation process (see Sect. \[sec:conditions\]). Fig. \[fig:vsto\] illustrates this for the profiles of one of the rotational transitions for various stochastic velocities.
----------- --------------------
tracer $v_{\mathrm{sto}}$
(km s$^{-1}$)
CO(7$-$6) 3.2 $\pm$ 0.4
CO(6$-$5) 8 $\pm$ 1
CO(4$-$3) 1.0 $\pm$ 0.1
CO(3$-$2) 0.24 $\pm$ 0.04
CO(2$-$1) 0.16 $\pm$ 0.05
----------- --------------------
: Stochastic velocities for . The values are derived for each observed transition independently, while all other parameters were kept constant. The stochastic velocities were determined by fitting the integrated intensities.
\[tab:vsto\]
Analogous to the determination of the mass-loss history, it is possible to estimate the variations in the stochastic velocities traced by the integrated intensities of the sequence of rotational transition observed for . For this purpose, the mass-loss rate was assumed to be constant at a rate of 10$^{-6}$ $M_{\odot}$ yr$^{-1}$ throughout the circumstellar outflow. The temperature profile and the other parameters were kept the same as the ones used in the mass-loss history analysis. The results are listed in Table \[tab:vsto\]. Again, the line formation regions and thus the values derived here are not independent and should be seen as averages over the formation regions. The observed line intensities may be explained by a gradient in the stochastic velocity if it is lowest in the outer parts of the outflow, traced by the low rotational transitions, and has its maximum in the gas traced by the CO(6$-$5) transition.
One has to bear in mind that we derived the stochastic velocities for one particular mass-loss rate, namely $10^{-6} \, M_{\odot}$. As pointed out before, the mass-loss rate has a considerable effect on the line strengths as well, however, the negative gradient will be maintained for other choices of $\dot{M}$. To explain the observations $v_{\mathrm{sto}}$ has to increase to an unrealistically high maximum of 8 km s$^{-1}$ in the region of CO(6$-$5) formation and then decrease again to 0.16 km s$^{-1}$ at the CO(2$-$1) formation zone.
The stochastic velocity can be considered as a composition of thermal and turbulent components, according to Eq. (\[eq:velocities\]). The thermal molecular velocities for CO are given by ${v_{\mathrm{therm}}}
= \sqrt{2 k T / m_{\mathrm{CO}}}$, where $T$ is the temperature at the line formation region, which is usually of the order of the excitation temperature. For the lines observed we can determine that $v_{\mathrm{therm}}$ ranges between 0.01 and 0.03 km s$^{-1}$. It is obvious that only a minor fraction of the required total stochastic velocity ${v_{\mathrm{sto}}}$ can be explained by thermal motion, and also that the observed gradient is not sufficiently reproduced by the thermal component.
The nature of the remaining turbulent velocity $v_{\mathrm{turb}}$ is unknown, but could in part be induced by stellar pulsations. These pulsations cause stochastic velocities of 2–5 km s$^{-1}$ in the inner parts of the circumstellar shell required to start the dust formation process. However, these stochastic velocities will damp quickly and are practically absent beyond [@S_01_modulation]. Hence, if variations in the stochastic velocity are important for the CO line intensities, the origin of such variations is presently unclear.
### Other factors
Other factors that could be important include the outflow velocity profile, and the geometry. Our adopted outflow velocity profile is very simple (see Eq. (\[eq:vel\])), but hydrodynamical calculations show that it may be more complex and also time-dependent [@SID_01_quasiperiodic]. The effect of such complex outflow velocity profiles on the CO line profiles has not yet been studied. Perhaps they could serve as a source of turbulence.
Non-spherical winds, e.g. density enhancements in the equatorial region, could also play a role in the observed line strengths. The observed line profiles would reflect such an axi-symmetric geometry, if it exists. Close examination of the observed lines (Figs. \[fig:wxpsc\]–\[fig:oh104\]) shows that their profiles are similar for all transitions (per source), and one can therefore conclude that the regions where the lines originate have almost the same velocity structure. Apparently there is no change in geometry for the regions traced by the various rotational transitions, e.g. a slowly outflowing disk traced by the lower transitions and a fast polar outflow traced by the higher transitions. Thus this possibility most likely can be ruled out as an explanation for the discrepancy between the observations and the model results. Only in case of (Fig. \[fig:vycma\]) the profiles show significant differences between the lower and higher rotational transitions.
Concluding remarks {#sec:disc}
==================
CO rotational transitions as mass-loss indicators {#sec:co}
-------------------------------------------------
In this work, we presented submillimeter observations of various carbon monoxide rotational transitions (CO(7$-$6), (6$-$5), (4$-$3), (3$-$2), (2$-$1)) observed in AGB stars and red supergiants in various evolutionary states. We have attempted to determine the mass-loss history of the programme stars by modelling of the observed transitions. For the first time the CO(7$-$6) and (6$-$5) transitions were used, in addition to lower transitions. In this way the gap between the regions in the outflow traced by the gas and that traced by the dust emission was largely closed. Many studies in the past have focussed on only one or two transitions to determine the gas mass-loss rate [e.g. @KM_85_massloss; @LFO_93_CO; @JCT_94_redgiantwinds; @G_94_OH32OH44; @JST_96_OH26; @DKF_97_superwind]. The extension of the data towards higher rotational transitions clearly demonstrates that determination of a unique gas mass-loss rate from a single CO rotational transition is highly unreliable. We found that the observed line strengths indicate that the outflow has a more complex physical structure than was previously assumed. Not a superwind, but periodic mass-loss variations comparable to the arc-like structures and rings observed around post-AGB stars, may possibly account for the observed line strengths. Part of the discrepancy could be due to a gradient in the stochastic velocity as well.
Independently, another research group has reached the same conclusion during the last year. Initially, @OGK_02_massloss modelled the mass-loss rates of a large sample of irregular and semi-regular M-type variables by fitting 2, 3 or in one case 4 CO rotational transitions by assuming a constant mass-loss rate over the last 1000 years. They derive rates for their sample stars and do not report on problems similar to ours, but their figures 2 and 11 show that the line strength of the higher transitions is overestimated when this model is used. In the same volume of A&A, @SRO_02_mdothistory describe a model that is able to use periodic mass-loss variations to calculate the rotational transitions of CO in C-rich stars. The development of this model is driven by the discovery of mass-loss modulations. However, after thorough analysis, they conclude that mass-loss modulations are not important nor necessary to explain the CO rotational line profiles. The most recent results of @GOK_03_dynamics indicate otherwise, however. When trying to derive the mass loss rate of more evolved Miras (i.e. with higher mass-loss rates than the semi-regulars), @GOK_03_dynamics find that a model assuming a constant mass-loss rate underestimates the strength of the low transitions. This is in principle the same as our result that the high transitions are overestimated.
Future work {#sec:future}
-----------
The work presented here has revealed a much more complex picture of AGB stellar ejecta than previously assumed. Additional research is required, which we plan to do in the near future. Of particular importance are the following issues:
- First, more observational data should be obtained, in particular of high rotational transitions. Our study is the first to include the CO(6$-$5) and (7$-$6) transitions in the mass-loss rate determinations of three evolved stars. In addition, for one object () observations up to CO(6$-$5) were secured. This is not enough to draw firm conclusions on the degree of complexity of the physical structure in the outflows of AGB stars, therefore this sample should be enlarged. It is important to pay attention to the completeness: if all transitions are observed, variations in the important physical parameters can be much better constrained. In that respect it is also worthwhile to extend the data with observations of $^{13}$CO for the lower transitions, which provide additional independent constraints on the physical conditions.
- Second, a more realistic representation of the physical conditions in the outflow of AGB stars should be used. This includes adding a gradient in turbulence and periodic mass-loss variations as we have argued in this study. In addition, the velocity law could also be improved, e.g. following the results of @SID_01_quasiperiodic. Although these adjustments will lead to an increase in the number of free parameters, it is likely that we will be able to use the *line profiles* to constrain the model parameters. This will certainly help in disentangling the physical structure of the outflow.
FK is grateful for the hospitality at the Stockholm Observatory. The support provided by the staff of the JCMT is greatly appreciated. The James Clerk Maxwell Telescope is operated on behalf of the Particle Physics and Astronomy Research Council of the United Kingdom, the Netherlands Organisation for Scientific Research and the National Research Council of Canada. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. FK, AdK and LBFMW acknowledge financial support from NWO ’Pionier’ grant 616-78-333. Support for this work was provided by NASA through the SIRTF Fellowship Program, under award 011 808-001.
Observations — Tables & Figures
===============================
-------- ------------------ ------------------ ------------------
object $\alpha$ (J2000) $\delta$ (J2000) $D$
(kpc)
00 21 46.27 $-$20 03 28.9 0.238$^{\rm{a}}$
01 06 25.99 +12 35 53.4 0.74$^{\rm{b}}$
01 33 51.19 +62 26 53.4 2.90$^{\rm{b}}$
02 19 20.793 $-$02 58 39.51 0.128$^{\rm{a}}$
05 11 19.37 +52 52 33.7 0.820$^{\rm{c}}$
05 55 10.305 +07 24 25.43 0.131$^{\rm{a}}$
07 22 58.33 $-$25 46 03.2 0.562$^{\rm{a}}$
16 29 24.461 $-$26 25 55.21 0.185$^{\rm{a}}$
17 14 39.78 +11 04 10.0
17 44 23.89 $-$31 55 39.11 1.19$^{\rm{d}}$
18 08 04.05 $-$22 13 26.6 0.330$^{\rm{a}}$
18 35 46.9 +05 35 48 2.48$^{\rm{d}}$
18 37 32.52 $-$05 23 59.4 1.37$^{\rm{b}}$
18 48 41.5 $-$02 50 29 1.77$^{\rm{e}}$
18 52 22.19 $-$00 14 13.9 5.02$^{\rm{b}}$
19 21 36.56 +09 27 56.3 1.13$^{\rm{b}}$
19 26 48.09 +11 21 16.7 5$^{\rm{f}}$
20 46 25.7 +40 06 56 1.22$^{\rm{g}}$
21 43 30.461 +58 46 48.17 1.613$^{\rm{a}}$
21 56 58.3 +62 18 43 2.03$^{\rm{d}}$
22 19 27.9 +59 51 22 2.30$^{\rm{b}}$
-------- ------------------ ------------------ ------------------
: Programme stars. Distances are taken from $^{\rm{a}}$Hipparcos [@ESA_97_Hipparcos], $^{\rm{b}}$@VVV_90_phaselags, $^{\rm{c}}$@HBF_72_1612MHz, $^{\rm{d}}$@YUM_99_distance, $^{\rm{e}}$@HBP_86_luminosity, $^{\rm{f}}$@JHG_93_irc+10420, $^{\rm{g}}$@DGH_01_NMLCyg
\[tab:obslist\]
-------- ------------ -------------------- ------ -----------
object transition $t_{\mathrm{int}}$ $f$ obs. date
(s)
CO(2$-$1) 1800 1.07 03-Sep-02
CO(2$-$1) 1800 0.90 22-Mar-01
CO(3$-$2) 1200 1 02-Jul-00
CO(4$-$3) 4800 1 21-Apr-00
CO(6$-$5) 7320 – 10-Oct-01
CO(7$-$6) 7200 – 09-Oct-01
CO(2$-$1) 1200 1.09 03-Sep-02
CO(3$-$2) 2400 1 02-Jul-00
CO(4$-$3) 5600 1 13-Apr-00
CO(2$-$1) 600 1.08 03-Sep-02
CO(3$-$2) 600 1 02-Jul-00
CO(2$-$1) 1800 0.96 06-Dec-00
CO(3$-$2) 2400 1.10 05-Dec-00
CO(3$-$2) 5400 1 02-Jul-00
CO(2$-$1) 3600 1 22-Mar-01
CO(3$-$2) 2400 1.10 05-Dec-00
CO(6$-$5) 8400 – 10-Oct-01
CO(7$-$6) 5400 – 09-Oct-01
CO(3$-$2) 1800 1.10 04-Jul-00
CO(2$-$1) 1800 1.01 03-Sep-02
CO(3$-$2) 1200 1 17-Apr-00
CO(2$-$1) 1860 1 22-Mar-01
CO(3$-$2) 2400 1 18-Apr-00
CO(4$-$3) 2400 1 04-Jul-00
CO(2$-$1) 1800 1 22-Mar-01
CO(3$-$2) 1200 1 17-Apr-00
CO(4$-$3) 8400 1 21-Apr-00
CO(2$-$1) 1800 0.95 22-Mar-01
CO(3$-$2) 1200 1 17-Apr-00
CO(4$-$3) 8400 1 21-Apr-00
CO(3$-$2) 1020 1 06-Jul-00
CO(3$-$2) 1200 1 18-Apr-00
CO(3$-$2) 2400 1 07-Jul-00
CO(2$-$1) 1800 0.95 22-Mar-01
CO(3$-$2) 1200 1 17-Apr-00
CO(4$-$3) 2400 1 21-Apr-00
CO(2$-$1) 1800 0.90 22-Mar-01
CO(3$-$2) 1200 1 17-Apr-00
CO(4$-$3) 1200 1 21-Apr-00
CO(6$-$5) 4800 – 10-Oct-01
CO(2$-$1) 1800 1 20-Sep-02
CO(3$-$2) 1200 1 18-Apr-00
CO(2$-$1) 1920 1 20-Sep-02
CO(3$-$2) 3600 1 07-Jul-00
CO(2$-$1) 2400 0.90 22-Mar-01
CO(3$-$2) 2400 1 18-Apr-00
CO(4$-$3) 4800 1.16 13-Apr-00
2160 1 21-Apr-00
CO(6$-$5) 11400 – 10-Oct-01
CO(7$-$6) 2400 – 09-Oct-01
-------- ------------ -------------------- ------ -----------
: Details of the observations. For each source the observed transitions are listed, together with the integrated observing time in seconds. The correction factor $f$ has been applied to our measurements, derived from standard measurements. The last column lists the observing dates.
\[tab:obsdetails\]
object transition $T_{\mathrm{mb}}$ (K) r.m.s. (K) bin (MHz) $V_{\mathrm{LSR}}$ (km s$^{-1}$) $V_{\infty}$ (km s$^{-1}$) $I$ (K km s$^{-1}$)
----------------- ------------ ----------------------- ------------ ----------- ---------------------------------- ---------------------------- ---------------------
T Cet CO(2$-$1) 0.44 0.056 0.3125 +23.1 $\pm$ 0.5 6.7 $\pm$ 0.5 3.9 $\pm$ 0.4
WX Psc CO(2$-$1) 2.35 0.034 0.6250 +9.0 $\pm$ 0.5 20.2 $\pm$ 0.5 66 $\pm$ 7
CO(3$-$2) 2.91 0.043 1.2500 +9.2 $\pm$ 0.5 20.3 $\pm$ 0.5 82 $\pm$ 8
CO(4$-$3) 1.86 0.090 0.9375 +9.0 $\pm$ 0.5 20.6 $\pm$ 0.5 50 $\pm$ 5
CO(6$-$5) 0.45 0.224 2.5000 +8.4 $\pm$ 1.5 17 $\pm$ 3 10 $\pm$ 5
CO(7$-$6) 0.82 0.378 3.1250 +9.8 $\pm$ 1.5 21 $\pm$ 3 23 $\pm$ 11
OH 127.8+0.0 CO(2$-$1) 0.28 0.048 0.6250 $-$56 $\pm$ 3 13 $\pm$ 2 5.5 $\pm$ 1.6
CO(3$-$2) 0.68 0.050 0.6250 $-$55 $\pm$ 3 13 $\pm$ 2 12 $\pm$ 1
CO(4$-$3) 0.15 0.057 1.5625 complex complex 23 $\pm$ 2
$o$ Cet CO(2$-$1) 13.80 0.170 0.1562 +46.3 $\pm$ 0.1 8 $\pm$ 1 64 $\pm$ 6
CO(3$-$2) 21.61 0.134 0.3125 +46.4 $\pm$ 0.1 8 $\pm$ 2 108 $\pm$ 11
IRC+50137 CO(2$-$1) 1.37 0.033 0.4688 +2.8 $\pm$ 0.5 19.1 $\pm$ 0.5 37 $\pm$ 4
CO(3$-$2) 1.44 0.047 0.9375 +3.2 $\pm$ 0.5 18.5 $\pm$ 0.5 39 $\pm$ 4
$\alpha$ Ori CO(3$-$2) complex 0.043 0.3125 +3.4 $\pm$ 0.5 15.7 $\pm$ 0.5 50 $\pm$ 5
VY CMa CO(2$-$1) complex 0.033 0.6250 +25 $\pm$ 3 47 $\pm$ 3 66 $\pm$ 7
CO(3$-$2) 3.00 0.043 1.2500 +25 $\pm$ 3 47 $\pm$ 3 173 $\pm$ 17
CO(6$-$5) 4.37 0.469 3.7500 +27 $\pm$ 2 48 $\pm$ 3 257 $\pm$ 77
CO(7$-$6) 7.41 0.908 3.7500 +29 $\pm$ 2 44 $\pm$ 3 433 $\pm$ 130
$\alpha$ Sco CO(3$-$2) – 0.036 1.8750 – – –
V438 Oph CO(2$-$1) 0.24 0.058 0.3125 +9.7 $\pm$ 0.5 4.3 $\pm$ 1.0 1.1 $\pm$ 0.2
AFGL 5379 CO(3$-$2) 2.76 0.056 1.2500 $-$22.7 $\pm$ 1.5 24 $\pm$ 2 84 $\pm$ 8
VX Sgr CO(2$-$1) complex 0.048 0.4688 +6.4 $\pm$ 1.0 25 $\pm$ 1 31 $\pm$ 3
CO(3$-$2) 2.37 0.059 0.6250 +6.9 $\pm$ 1.0 26 $\pm$ 1 97 $\pm$ 10
CO(4$-$3) 1.22 0.237 1.2500 +6.4 $\pm$ 0.5 22 $\pm$ 2 42 $\pm$ 4
CRL 2199 CO(2$-$1) 1.16 0.033 0.6250 +33.7 $\pm$ 0.2 17.6 $\pm$ 0.5 26 $\pm$ 3
CO(3$-$2) 1.25 0.034 1.2500 +33.5 $\pm$ 0.2 17.9 $\pm$ 0.5 28 $\pm$ 3
CO(4$-$3) 0.98 0.066 1.2500 +33.2 $\pm$ 0.5 18 $\pm$ 1 22 $\pm$ 2
OH 26.5+0.6 CO(2$-$1) contamin. 0.049 0.3125 contamin. contamin. 9 $\pm$ 3
CO(3$-$2) 1.05 0.046 0.9375 +26.9 $\pm$ 0.5 18 $\pm$ 1 23 $\pm$ 2
CO(4$-$3) 0.86 0.051 0.9375 +27.7 $\pm$ 0.3 15.9 $\pm$ 0.5 19 $\pm$ 2
OH 30.1$-$0.7 CO(3$-$2) contamin. 0.076 0.9375 contamin. contamin. contamin.
OH 32.8$-$0.3 CO(3$-$2) contamin. 0.047 0.9375 contamin. contamin. contamin.
OH 44.8$-$2.3 CO(3$-$2) 0.63 0.047 0.9375 $-$70.3 $\pm$ 0.2 17.7 $\pm$ 0.5 15 $\pm$ 2
IRC+10420 CO(2$-$1) 1.65 0.034 0.4688 +75 $\pm$ 1 43 $\pm$ 3 95 $\pm$ 10
CO(3$-$2) 3.23 0.075 0.9375 +75 $\pm$ 1 45 $\pm$ 3 180 $\pm$ 18
CO(4$-$3) 2.84 0.078 1.8750 +76.1 $\pm$ 0.5 42 $\pm$ 2 150 $\pm$ 15
NML Cyg CO(2$-$1) complex 0.039 0.3125 $-$2 $\pm$ 3 33 $\pm$ 3 99 $\pm$ 10
CO(3$-$2) complex 0.104 0.6250 $-$2 $\pm$ 3 33 $\pm$ 3 210 $\pm$ 21
CO(4$-$3) complex 0.106 1.2500 $-$1 $\pm$ 2 34 $\pm$ 2 133 $\pm$ 13
CO(6$-$5) complex 0.232 2.5000 +2 $\pm$ 2 34 $\pm$ 2 111 $\pm$ 56
$\mu$ Cep CO(2$-$1) complex 0.025 1.2500 +22 $\pm$ 2 33 $\pm$ 3 2.5 $\pm$ 0.3
CO(3$-$2) complex 0.066 0.9375 +21 $\pm$ 3 35 $\pm$ 3 14 $\pm$ 3
IRAS 21554+6204 CO(2$-$1) 0.58 0.041 0.4688 $-$19 $\pm$ 1 19 $\pm$ 2 12 $\pm$ 1
CO(3$-$2) 0.49 0.036 0.9375 $-$19.0 $\pm$ 0.5 18.1 $\pm$ 0.5 10 $\pm$ 1
OH 104.9+2.4 CO(2$-$1) 0.21 0.026 0.6250 $-$26 $\pm$ 1 18.6 $\pm$ 0.5 5.4 $\pm$ 0.5
CO(3$-$2) 0.43 0.043 0.9375 $-$25 $\pm$ 1 18.3 $\pm$ 0.5 11 $\pm$ 1
CO(4$-$3) 0.11 0.042 1.8455 $-$26 $\pm$ 1 18 $\pm$ 1 3.6 $\pm$ 0.5
CO(6$-$5) – 0.224 3.7500 – – –
CO(7$-$6) – 1.011 3.1250 – – –
\[tab:transitions\]
source line $I$(K km s$^{-1}$) telescope ref.
-------- ----------- -------------------- ----------- ------
CO(4$-$3) 8.87 JCMT a
CO(3$-$2) 6.81 JCMT a
CO(2$-$1) 4.47 JCMT a
CO(4$-$3) 42.17 JCMT b
CO(3$-$2) 25.69 JCMT b
CO(3$-$2) 13.3 $\pm$ 1.3 CSO c
CO(2$-$1) 6.4 $\pm$ 0.4 CSO c
CO(4$-$3) 7.1 CSO d
CO(3$-$2) 9.5 CSO d
CO(2$-$1) 2.72 $\pm$ 0.38 CSO c
CO(4$-$3) 46.1 CSO d
CO(3$-$2) 37 CSO d
CO(3$-$2) 22.2 $\pm$ 2.2 CSO c
CO(2$-$1) 4.9 $\pm$ 0.5 CSO c
CO(4$-$3) 2.3 CSO d
CO(3$-$2) 2.5 CSO d
CO(2$-$1) 0.64 $\pm$ 0.25 CSO c
CO(4$-$3) 4.9 CSO d
CO(3$-$2) 9.1 CSO d
CO(2$-$1) 2.53 $\pm$ 0.51 CSO c
CO(4$-$3) 5.7 CSO d
CO(3$-$2) 9.4 CSO d
CO(2$-$1) 2.3 $\pm$ 0.2 CSO c
CO(4$-$3) 52.7 CSO d
CO(3$-$2) 63 CSO d
CO(3$-$2) 41.5 CSO e
CO(2$-$1) 28.8 $\pm$ 0.7 CSO c
: Observed CO line transitions in semi-regular variables and AGB stars, obtained from the literature. The listed stars are observed in CO(4$-$3), CO(3$-$2) and CO(2$-$1). The references are $^{\mathrm{a}}$@BKO_00_rvboo, $^{\mathrm{b}}$@KO_99_COcatalogue, $^{\mathrm{c}}$@KYL_98_COsurvey, $^{\mathrm{d}}$@Y_95_CO3-2, $^{\mathrm{e}}$@SKY_95_molecular.
\[tab:litvalues\]
[^1]: SIRTF Fellow
|
---
abstract: 'Music source separation with deep neural networks typically relies only on amplitude features. In this paper we show that additional phase features can improve the separation performance. Using the theoretical relationship between STFT phase and amplitude, we conjecture that derivatives of the phase are a good feature representation opposed to the raw phase. We verify this conjecture experimentally and propose a new DNN architecture which combines amplitude and phase. This joint approach achieves a better signal-to distortion ratio on the DSD100 dataset for all instruments compared to a network that uses only amplitude features. Especially, the bass instrument benefits from the phase information.'
bibliography:
- 'references.bib'
---
Introduction {#sec:intro}
============
*Music source separation* (MSS) refers to the problem of obtaining instrument estimates $\mathbf{\hat s}_j(n) \in {\mathbb{R}}^I$ from the mixture $$\mathbf{x}(n) = \sum\nolimits_{j\in \mathcal{J}} \mathbf{s}_j(n),$$ where $n$ denotes the discrete time index, $I$ gives the number of channels and $\mathcal{J}$ is the set of instruments. A common setup is the extraction of $\mathcal{J} := \{\text{bass}, \text{drums}, \text{vocals}, \text{other}\}$ from stereo mixtures, i.e., $I = 2$. This setup was used for the last *SiSEC* contests on MSS [@ono20152015; @liutkus20172016; @stoter20182018] and is also the basis of our work.
This paper studies the appropriateness of the *short-time Fourier transform* (STFT) phase as an input feature for MSS systems based on *deep neural networks* (DNNs). Current state-of-the-art approaches perform MSS by only considering the mixture STFT amplitude from which they estimate the target instrument STFT amplitude [@huang2014deep; @huang2014singing; @uhlich2015deep; @nugraha2016multichannel; @uhlich2017improving; @takahashi2017multi; @takahashi2018mmdenselstm]. It is well known that the STFT phase contains useful information for speech enhancement, see e.g. [@gerkmann2015phase] and, therefore, should not be neglected. Recent attempts have been made in order to improve MSS using phase. [@lee2017fully] proposed a fully complex-valued DNN, which predicts the complex STFT of the target instrument from the complex mixture STFT. [@dubey2017does] analyzed whether phase is beneficial as input feature for a DNN compared to a network using only amplitude. Another approach is [@TAKIS2018], which estimates the phase of the instrument from the mixture amplitude and phase by treating the phase retrieval problem as a classification problem. This paper presents another approach where phase is used as an additional input feature to improve the amplitude estimation. In contrast to [@dubey2017does], we propose a special architecture to exploit the information that is present in the STFT phase as a simple concatenation of amplitude and phase at the input of the network yields trained networks that focus only on amplitude information.
-0.3cm
-0.3cm
We first show that expressing the phase through its *instantaneous frequency* (time derivative) and its *group delay* (frequency derivative) greatly improves the efficiency of DNNs compared to networks fed with raw phase inputs. This is done by looking at experimental results as well as studying the theoretical relationship between phase and amplitude of a continuous-time STFT. Moreover, we demonstrate that the discrete-time STFT introduces systematic shifts into the phase and that correcting these shifts improves the efficiency of the DNN to exploit these features.
Finally, we design a network architecture which takes full advantage of this additional feature. It is formed by two independent networks, taking respectively amplitude and phase, whose outputs are concatenated afterwards through a dense layer. Intuitively, each network independently extracts features from amplitude and phase and forwards them to a fusion layer, which reconstructs the spectrum based on these features. With the suggested data pre-processing method and architecture design, our system achieves on average a relative improvement of 2.3% and up to 6% for bass compared to an amplitude-only system.
For clarity, the paper is divided into two parts. Sec. \[sec:phase\_feature\] is dedicated to the properties of the phase as a feature for MSS. In this section, we consider the problem of using only the phase for estimating the instrument amplitude. This allows us to better understand this feature and the development of an appropriate pre-processing method. Sec. \[sec:combination\_of\_amp\_phase\], in contrast, considers both amplitude and phase from the mixture signal in order to produce an improved estimate of the instrument amplitude, which is the ultimate goal of the paper.
0.15in
---------------- --------------------------------------------------------------------------------------------------------------------------------- -- -- --
**Instrument** **DNN$_\textbf{A}$ using & **Upper baseline of** & **Upper baseline of**\
& **approach (a) & **approach (b)** & **approach (c)**\
& & (**A$_\text{IRM}$ & $\boldsymbol{\varphi}_\text{mixture}$**) & (**A$_\text{DNN}$ & $\boldsymbol{\varphi}_\text{oracle}$**)\
Bass & 3.24 & 7.92 & 6.59\
Drums & 4.68 & 8.53 & 6.15\
Other & 3.54 & 8.19 & 5.35\
Vocals & 4.78 & 11.10 & 6.86\
****
---------------- --------------------------------------------------------------------------------------------------------------------------------- -- -- --
: Comparison of upper limits reachable by two different approaches. Results on DSD100 test set (SDR in dB).[]{data-label="tab:approach_comparison"}
-0.1in
Phase as Input Feature {#sec:phase_feature}
======================
Motivation
----------
Fig. \[fig:comparison\_mss\_approaches\] shows three different approaches for MSS, where $\mathbf{A}(k,m) \in {\mathbb{R}}_+^I$ and $\boldsymbol{\varphi}(k,m) \in [-\pi,\pi)^I$ denote the STFT amplitude and phase at frequency bin index $k$ and frame index $m$. $\mathbf{\hat s}_j(n) \in {\mathbb{R}}^I$ is the estimated target instrument signal.
Typically, approach (a) is used where the instrument amplitude $\mathbf{\hat{A}}_j$ is estimated from the mixture amplitude $\mathbf{A}_x$, while the instrument phase is simply approximated by the mixture phase $\boldsymbol{\varphi}_x$. The estimated instrument $\mathbf{\hat{s}}_j(n)$ is produced by applying an inverse STFT with the estimated source amplitude $\mathbf{\hat{A}}_j$ and mixture phase $\boldsymbol{\varphi}_x$.
Approaches (b) and (c) show two different ways to improve upon (a). Approach (b), which was, e.g., used in [@dubey2017does], is similar in all respects except that the mixture phase is used to improve the instrument amplitude estimation. Approach (c), which was, e.g., followed by [@TAKIS2018], estimates the instrument phase $\boldsymbol{\hat{\varphi}}_j$ which can then be used for the inverse STFT.
In order to choose between the two possible improvements, we did a simple experiment shown in Table \[tab:approach\_comparison\]. We compare the upper limits achievable by both strategies: on one side a signal synthesized with the *ideal ratio mask* (IRM) amplitude and the mixture phase; on the other side the oracle phase and the amplitude estimation from the network DNN$_\text{A}$[^1]. We can see that approach (b) has more room for improvement as the upper limit achievable has a relative improvement of 122%. In contrast, the upper limit of approach (c) allows an average relative improvement of 57%, which indicates that currently the amplitude estimation is still the main bottleneck for MSS performance. We therefore investigate approach (b) in this paper.
Theoretical Relationship {#subsec:theory}
------------------------
Interestingly, for the continuous-time STFT $$X(\omega,t) = A(\omega,t)e^{j \varphi(\omega,t)}$$ of a continuous-time signal $x(t)$, there is a theoretical relationship between the amplitude $A(\omega,t)$ and the phase $\varphi(\omega,t)$. The continuous-time STFT is given by $$\begin{aligned}
X(\omega,t) = e^{j\omega t/2}\int\limits_{-\infty}^\infty x(u)h(t-u)e^{-j\omega u} \text{d}u.\end{aligned}$$
Using a Gaussian window $h(t) = \lambda^{-1/2}\pi^{-1/4}e^{-t^2/(2\lambda^2)}$, [@auger2012phase] showed that
$$\begin{aligned}
\frac{\partial}{\partial t}\varphi(\omega,t) &= \hphantom{-}\lambda^{-2}\frac{\partial}{\partial\omega} \log\left( A(\omega,t) \right) + \frac{\omega}{2},\\
\frac{\partial}{\partial\omega}\varphi(\omega,t) &= -\lambda^{2}\frac{\partial}{\partial t} \log\left( A(\omega,t) \right) - \frac{t}{2}.\end{aligned}$$
\[eq:relationship\]
From , we can see that the derivatives of phase and log-magnitude are linked and, therefore, we hope that the amplitude estimation for our target instrument from the mixture phase can be improved by using phase features.
Furthermore, we conjecture from that better results for the amplitude estimation can be obtained if we work with time/frequency derivatives of the phase instead of the raw phase. This intuition will be experimentally confirmed in Sec. \[subsec:experiment\]. As we work with discrete-time signals, we will approximate the derivatives by differences, i.e., in the following we will use
$$\begin{aligned}
\Delta_t\varphi &:= \varphi(k,m) - \varphi(k,m-1),\\
\Delta_f\varphi &:= \varphi(k,m) - \varphi(k-1,m).\end{aligned}$$
Please note that from the phase information, we are able to recover the amplitude up to an unknown scale. This can be seen from considering a signal $s(n) \in {\mathbb{C}}$ and a scaled version $s'(n) = a \cdot s(n)$ with $a > 0$ as $\angle s(n) = \angle s'(n)$ whereas $|s(n)| \neq |s'(n)|$. Hence, the phase only contains information about variations of the amplitude. This property is consistent with which links phase and log-amplitude through their derivatives.
Shifts in discrete Short-Time Fourier Transform {#subsec:shift}
-----------------------------------------------
Fig. \[fig:shift\] shows the distribution of $\Delta_t \varphi = \varphi(k,m) - \varphi(k,m-1)$ for consecutive frequency bins. We can observe a systematic offset in the statistical distribution which can be explained by the shift theorem of the *discrete Fourier transform* (DFT) [@smith2007mathematics]. It states that a delay in the time domain results in a linear phase term in the frequency domain, i.e., $$\begin{aligned}
x(n-n_0) \xrightarrow{DFT} e^{j\frac{2\pi}{N}k n_0} X(k),\end{aligned}$$ where $n_0$ is the shift and $N$ the DFT/FFT size.
Therefore, in the case of a stationary signal transformed by an STFT, with hop size $n_0$, the phase of two consecutive frequency bins is expected to be shifted by a term $$\begin{aligned}
\text{phase shift} = -\frac{2\pi}{N}k n_0 \end{aligned}$$ For example, an overlap of $75\%$ results in a shift of $-k\frac{\pi}{2}$ which can also be seen in Fig. \[fig:shift\].
DNNs are known to be sensitive to the feature distribution and, therefore, this shift should be properly compensated for during the pre-processing stage, as described in Sec. \[subsec:preproc\], in order to ensure a proper training of the DNN.
Pre-processing {#subsec:preproc}
--------------
According to the conclusions drawn in Sec. \[subsec:theory\] and Sec. \[subsec:shift\], we apply the following pre-processing steps to the raw phase:
- The time and frequency derivatives are first approximated by the difference between two consecutive time frames ($\Delta_t \varphi$) and by the difference between two consecutive frequency bins ($\Delta_f \varphi$), respectively.
- A linear term $2 \pi k \frac{n_0}{N}$ is added to the time differences in order to compensate for the effect described in Sec. \[subsec:shift\]. Consequently, for a stationary signal $\Delta_t \varphi = 0$.
- For $\Delta_f \varphi$, we could empirically observe a systematic shift of $\pi$ in its statistical distribution, see Fig. \[fig:distribution\_phase\] (a). We compensate it by subtracting $\pi$ in order to obtain $\mathbb{E}(\Delta_f \varphi) = 0$.
- Finally, all values are wrapped to $[-\pi, \pi)$ using $$\Delta \varphi = \left((\Delta \varphi + \pi) \bmod 2\pi \right)- \pi.$$
The effects of this pre-processing method on feature statistical distribution are illustrated in Fig. \[fig:distribution\_phase\].
Experimental Validation {#subsec:experiment}
-----------------------
In order to see whether our pre-processing is effective, we run two experiments, which we now describe in detail.
The network used to evaluate the suggested pre-processing method is formed by two dense layers of 500 hidden units, intersected by ReLU non-linearities and completed by a dense output layer matching the target dimensions. At the very end, a bias layer initialized with the average amplitude per frequency bin over the training set shifts the output and a ReLU non-linearity ensures non-negative output values. We use a context of five preceding/succeeding frames as temporal context. Fig. \[fig:phase\_only\_arch\] shows the network structure and the overall MSS framework is summarized in Fig. \[fig:framework\].
![Network architecture with phase feature only.[]{data-label="fig:phase_only_arch"}](./images/FNN_phase_horizontal.pdf){width="\columnwidth"}
We now give the results for estimating the STFT amplitude from the phase using a DNN. By these experiments, we show that it is advantageous to consider the time/frequency derivatives instead of the raw phase. Furthermore, the pre-processing described in Sec. \[subsec:preproc\], is also shown to be relevant.
In the first experiment, we reconstruct the instrument amplitude from the instrument phase. Thus, we do not consider a separation problem but instead focus on the ability of a DNN to recover a signal knowing its phase. By this, we can compare different phase feature representations and observe their effects on the DNN learning power. The training MSE curves are shown in Fig. \[fig:train\_MSE\_clean\]. We can observe that the phase derivatives show the best performance as we previously conjectured.
In the second experiment, we estimate the instrument amplitude from the mixture phase. This goes one step further than the previous experiment by involving separation in the comparative analysis of the pre-processing methods. The trained networks are then integrated in the MSS framework illustrated in Fig. \[fig:framework\]. Estimations are scored following SiSEC 2016 policy [@liutkus20172016]. Fig. \[fig:SDR\_phase\_only\] shows the *signal-to-distortion ratio* (SDR) values [@vincent2007first] on the DSD100 dataset where the values are obtained by first averaging the SDR values for each song and then computing the median over all $50$ songs of the train set or test set, respectively. Again, we can observe that phase derivatives are a much better feature representation and that shift correction systematically improves learning power of the system, leading occasionally to overfitting. The best test SDR is achieved by the frequency-derivative representation of the phase which generalizes better than the time-derivative representation.
Note that a network fed with phase features can only estimate the amplitude values up to a scale, meaning that it uses the average amplitude value per frequency bin of the training set as a starting point and estimates the variations from it based on the phase input. The post-processing stage uses a multi-channel Wiener filter [@sivasankaran2015robust; @nugraha2016multichannel; @uhlich2017improving] to recover the correct scale afterwards.
-0.2cm
-0.2cm
Combining Amplitude and Phase Features {#sec:combination_of_amp_phase}
======================================
In the previous section, we have seen that phase features can be used to estimate the instrument STFT amplitude. Therefore, we now turn to the problem of combining phase and amplitude features.
Proposed Architecture {#subsec:combine:arch}
---------------------
The most straight-forward way of combining amplitude and phase is a concatenation of both features at the input of the DNN. However, training such an approach results in networks that only rely on amplitude features as they set all weights in the input layer corresponding to the phase close to zero.[^2] We could observe this if we use the raw phase as well as if we use the phase pre-processing described in Sec. \[subsec:preproc\].
Hence, we have to take special care to exploit the information of the phase features and we use the DNN architecture that is shown in Fig. \[fig:concat\_architecture\]. Instead of concatenating the features directly, we first process both through two dense layers before concatenating them.
The upper part of the network in Fig. \[fig:concat\_architecture\] deals with the amplitude features. The features are first normalized by a bias layer and a scale layer, initialized with the mean and standard deviation per frequency bin over the training set. Two fully connected layers of 500 hidden units perform the feature extraction. The lower part of the network in Fig. \[fig:concat\_architecture\] deals with the phase features. It takes as input both time and frequency derivatives, properly pre-processed as described in Sec. \[subsec:preproc\] and stacked together into an extra dimension. As for amplitude, two fully connected layers of 500 hidden units perform the feature extraction. The concatenation layer stacks the output of both previous networks and produces the amplitude estimates, which are de-normalized with the help of a final bias layer and a ReLU non-linearity, as described in Sec. \[subsec:experiment\]. The training process is similar to the one described in Sec. \[subsec:experiment\]. Time context is, as well, kept to five preceding/succeeding frames.
0.2in -0.2cm
-0.3cm
Results
-------
Fig. \[fig:SDR\_amp\_phase\] shows the results obtained with amplitude and phase combination. For comparison, we also trained a network DNN$_\text{A}$, which uses only amplitude as feature and has the same structure as shown in Fig. \[fig:concat\_architecture\] with the phase branch removed. Therefore, the amplitude information undergoes the same transformations and we can directly compare the two networks.
We use different combinations of pre-processing methods described in \[subsec:preproc\] in order to experience the individual relevance of each step. As expected, applying all proposed phase pre-processing methods together is beneficial for the MSS performance.
Finally, Table \[tab:concat\_results\] shows the SDR obtained on the DSD100 test set. Comparing the baseline system DNN$_\text{A}$ with DNN$_\text{A \& $\varphi$} \ (\Delta_f \varphi_\text{shift}, \Delta_t \varphi_\text{shift})$, we observe that we can improve the SDR for all instruments and that especially the bass instrument improves by $0.2$ dB.
0.15in
**Instrument** **DNN$_\text{A}$** **DNN$_\text{A \& $\boldsymbol\varphi$}$** **Relative improv.**
---------------- -------------------- -------------------------------------------- ---------------------- --
Bass 3.24 3.44 $+ 6.17 \%$
Drums 4.68 4.71 $+ 0.64 \%$
Other 3.54 3.65 $+ 3.11 \%$
Vocals 4.78 4.82 $+ 0.84 \%$
: Results for concatenation architecture on DSD100 test set (SDR in dB).[]{data-label="tab:concat_results"}
Conclusion
==========
In this paper, we proposed to consider the phase as an additional input feature to enhance the amplitude estimation. We studied the relationship between the phase and the amplitude of an STFT and deducted a meaningful pre-processing, which was experimentally confirmed as relevant. We also found that special care must be taken in order to combine phase and amplitude features and, consequently, designed an adequate network architecture. The developed system improved SDRs on DSD100 for all instruments compared to an amplitude-only network with a similar network structure which showed the effectiveness of our system. Perceptually, this results in instruments more clearly separated from each other.
[^1]: DNN$_\text{A}$ is a network which estimates the instrument amplitude from the mixture amplitude. Please refer to Sec. \[sec:combination\_of\_amp\_phase\] for more details about this network.
[^2]: In our opinion, this behaviour is due to the fact that the information in the STFT mixture amplitude is more easily accessible than the information in the STFT phase.
|
---
abstract: 'We study the semileptonic decays of $\Lambda_c^+ \to \Lambda(n)\ell^+ \nu_{\ell}$ in two relativistic dynamical approaches of the light-front constituent quark model (LFCQM) and MIT bag model (MBM). By considering the Fermi statistic between quarks and determining spin-flavor structures in baryons along with the helicity formalism in the two different dynamical models, we calculate the branching ratios (${\cal B}$s) and averaged asymmetry parameters ($\alpha$s) in the decays. Explicitly, we find that ${\cal B}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(3.43\pm0.57,3.48)\%$ and ${\alpha}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(-0.96\pm0.03,-0.83)$ in (LFCQM, MBM), in comparison with the data of ${\cal B}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(3.6\pm0.4)\%$ and ${\alpha}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=-0.86\pm 0.04$ given in the Particle Data Group, respectively. We also predict that ${\cal B}( \Lambda_c^+ \to n e^+ \nu_{e})=(2.15\pm0.41, 2.55)\times 10^{-3}$ and ${\alpha}( \Lambda_c^+ \to n e^+ \nu_{e})=(-0.97\pm0.01,-0.85)$ in (LFCQM, MBM), which could be observed by the ongoing experiments at BESIII, LHCb and BELLEII.'
author:
- 'C.Q. Geng$^{1,2,3,4}$, Chong-Chung Lih$^{5}$, Chia-Wei Liu$^{3}$ and Tien-Hsueh Tsai$^{3}$'
title: ' Semileptonic decays of $\Lambda_c^+$ in dynamical approaches'
---
introduction
============
Recently, the LHCb Collaboration has published the newest precision measurements on the anti-triplet charmed baryon lifetimes [@Aaij:2019lwg], given by $$\begin{aligned}
\label{q0}
\tau_{\Lambda_c^+}&=&203.5\pm 1.0\pm 1.3\pm 1.4 \text{ fs}\,, \nonumber\\
\tau_{\Xi_c^+}&=&456.8\pm 3.5 \pm 2.9 \pm 3.1 \text{ fs}\,, \nonumber \\
\tau_{\Xi_c^0}&=&154.5 \pm 1.7 \pm 1.6 \pm 1.0 \text{ fs}\,.
%\nonumber\end{aligned}$$ Surprisingly, the lifetime of $\Xi_c^0$ measured by LHCb magnificently deviates from the previous value of $\tau_{\Xi_c^0}=112^{+13}_{-10}\text{ fs}$ in PDG [@Tanabashi:2018oca]. Meanwhile, the Belle Collaboration has measured the absolute branching ratios of ${\cal B} (\Xi_c^0 \to \Xi^- \pi^+)=(1.8\pm0.5 )\%$ [@Li:2018qak] and ${\cal B} (\Xi_c^+ \to \Xi^- \pi^+\pi^+)=(2.86 \pm 1.21 \pm 0.38)\%$ [@Li:2019atu], which are the golden modes to determine other $\Xi_c^{0,+}$ decay channels, respectively. It is clear that we are now witnessing a new era of charm physics. One can expect there will be more and more new experimental data and precision measurements in the future, which are also the guiding light for people to explore new physics beyond the standard model.
There have been recently many works discussing the anti-triplet charm baryon decays. Because of the complicated structures of these baryons with large non-perturbative effects of the quantum chromodynamic (QCD), it is very hard to calculate the decay amplitudes from first principles. In the literature, people use the flavor symmetry of $SU(3)_f$ to analyze various charmed baryon decay processes, such as semi-leptonic, two-body and three-body non-leptonic decays, to obtain reliable results [@Geng:2017mxn; @Savage:1989qr; @Savage:1991wu; @zero; @Geng:2018bow; @He:2018joe; @He:2018php; @Wang:2017azm; @Geng:2018plk; @Geng:2018rse; @Geng:2018upx; @Geng:2019awr; @Geng:2019bfz; @Geng:2019xbo; @Grossman:2019xcj; @Lu:2016ogy; @Wang:2017gxe; @Cen:2019ims; @Hsiao:2019yur; @Roy:2019cky; @Jia:2019zxi]. However, the $SU(3)_f$ symmetry is an approximate symmetry, resulting in about $10\%$ error for the predictions naturally. In order to more precision calculations, we need a dynamical QCD model to understand each process. To avoid other complicated problem like the non-factorizable effect, we only discuss the semi-leptonic processes, which are purely factorizable ones. In particular, we focus on the $\Lambda_c^+$ semi-leptonic decays in this work. There are several theoretical analyses and lattice QCD calculations on the charmed baryon semi-leptonic decays with different models in the literature [@Cheng:1995fe; @Zhao:2018zcb; @PerezMarcial:1989yh; @Faustov:2016yza; @Meinel:2016dqj; @Meinel:2017ggx]. In this paper, we will mainly use the light-front (LF) formalism to study the decays and check the results in the MIT bag model (MBM) as comparisons.
The LF formalism is considered as a consistent relativistic approach, which has been very successful in the mesonic and light quark sectors [@Schlumpf:1992ce; @Zhang:1994ti]. Due to this success, it has been extended to other systems, such as those involving the heavy mesons, pentaquarks and so on [@Cheng:2003sm; @Cheng:2004cc; @Chang:2019obq; @Zhao:2018mrg; @Xing:2018lre; @Cheng:2017pcq; @Geng:1997ws; @Lih:1999it; @Geng:2000fs; @Geng:2000if; @Geng:2001de; @Geng:2001vy; @Geng:2003su; @Geng:2012qg; @Geng:2016pyr]. In addition, the bottom baryon to charmed baryon nonleptonic decays in the LF approach have been done in Refs. [@Chua:2018lfa; @Chua:2019yqh]. For a review on the non-perturbative nature in the equation of motion and QCD vacuum structure for the LF constituent quark model (LFCQM), one can refer to the article in Ref. [@Zhang:1994ti]. The advantage of LFCQM is that the commutativity of the LF Hamiltonian and boost generators provide us with a good convenience to calculate the wave-function in different inertial frames because of the recoil effect. In addition, since the AdS/CFT correspondence [@Maldacena:1997re] was proposed by Juan Maldacena in the late of 1997, the LF holography as a feature of the AdS/CFT duality has brought the LF QCD from a phenomenological theory to a more fundamental one [@Brodsky:2011vc].
This paper is organized as follows. We present our formal calculations of the baryonic transition form factors for LFCQM and MBM in Secs. II and III, respectively. We show our numerical results of the form factors, branching ratios and averaged asymmetry parameters in Sec. IV. We also compare our results with those in the literature. In Sec. V, we give our discussions and conclusions.
Baryonic transition form factors in LFCQM
=========================================
Vertex function of baryon
-------------------------
In LFCQM, a baryon with its momentum $P$ and spin $S$ as well the z-direction projection of $S_z$ are considered as a bound state of three constitute quarks. As a result, the baryon state can be expressed by [@Zhang:1994ti; @Schlumpf:1992ce; @Cheng:2004cc; @Ke:2007tg; @Ke:2012wa; @Ke:2019smy] $$\begin{aligned}
\label{eq1}
&|{\bf B}&,P,S,S_z\rangle=\int\{{d^3{\tilde{p}_1}}\}\{{d^3{\tilde{p}_2}}\}\{{d^3{\tilde{p}_3}}\}2(2\pi)^3\frac{1}{\sqrt{P^+}}\delta^3(\tilde{P}-\tilde{p}_1-\tilde{p}_2-\tilde{p}_3) \nonumber \\
&\times& \sum_{\lambda_1,\lambda_2,\lambda_3}\Psi^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda_1,\lambda_2,\lambda_3)C^{\alpha\beta\gamma}F_{abc}|q_{\alpha}^{a}(\tilde{p}_1,\lambda_1) q_{\beta}^{b}(\tilde{p}_2,\lambda_2)q_{\alpha}^{a}(\tilde{p}_3,\lambda_3) \rangle
\label{baryon}\end{aligned}$$ where $\Psi^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda_1,\lambda_2,\lambda_3)$ is the vertex function, which can be formally solved from Bethe-Salpeter equations by the Faddeev decomposition method, $C^{\alpha\beta\gamma}$ and $F_{abc}$ are the color and flavor factors, $\lambda_i$ and $\tilde{p}_i$ with $i=1,2,3$ are the LF helicities and 3-momentum of the on-mass-shell constituent quarks, defined as $$\begin{aligned}
\tilde{p}_{i}=(p_i^+,p_{i\perp})\,, ~p_{i\perp}=(p_i^1,p_i^2) \,,~ p_i^-=\frac{m_i^2+p_{i\perp}^2}{p_i^+} \,,\end{aligned}$$ and $$\begin{aligned}
&&{d^3\tilde{p}_i}\equiv \frac{dp_i^+d^2p_{i\perp}}{2(2\pi)^3}\,, ~ \delta^3(\tilde{p})=\delta(p^+)\delta^2(p_{\perp})\,,
\nonumber\\
&&|q_\alpha^a(\tilde{p},\lambda)\rangle=d^{\dagger a}_{\alpha}(\tilde{p},\lambda)|0\rangle\,,
~\{d_{\alpha'}^{a'}(\tilde{p'},\lambda'),d^{\dagger a}_{\alpha}(\tilde{p},\lambda)\}=2(2\pi)^{3}\delta^3(\tilde{p'}-\tilde{p})\delta_{\lambda'\lambda}\delta_{\alpha'\alpha}\delta^{a'a}\,.
\label{state}\end{aligned}$$ To describe the internal motion of the constituent quarks, we introduce the kinematic variables $(q_{\perp},\xi)$ and $(Q_{\perp},\eta)$ and $P_{tot}$, given by $$\begin{aligned}
P_{tot}&=&\tilde{P}_1+\tilde{P}_2+\tilde{P}_3, \qquad \xi=\frac{p_1^+}{p_1^++p_2^+}, \qquad \eta=\frac{p_1^++p_2^+}{P_{tot}^+}\,,
\nonumber \\
q_{\perp}&=&(1-\xi)p_{1\perp}-\xi p_{2\perp},\quad Q_{\perp}=(1-\eta)(p_{1\perp}+p_{2\perp})-\eta p_{3\perp} \,,
\label{Lkin}
%\nonumber\end{aligned}$$ where $(q_{\perp},\xi)$ characterize the relative motion between the first and second quarks, while $(Q_{\perp},\eta)$ the third quark and other two quarks. The invariant masses of $(q_{\perp},\xi)$ and $(Q_{\perp},\eta)$ systems are represented by [@Schlumpf:1992ce] $$\begin{aligned}
M_3^2=\frac{q_\perp^2}{\xi(1-\xi)}+\frac{m_1^2}{\xi}+\frac{m_2^2}{1-\xi}\,, \nonumber\\
M^2=\frac{Q_\perp^2}{\eta(1-\eta)}+\frac{M_3^2}{\eta}+\frac{m_3^2}{1-\eta}\,,\end{aligned}$$ respectively. Unlike Refs. [@Ke:2007tg; @Ke:2012wa] or Ref. [@Ke:2019smy], which treat the diquark as a point like object or spectator, we consider the three constituent quarks in the baryon independently with suitable quantum numbers satisfying Fermi statistics to have a correct baryon bound state system. The vertex function of $\Psi^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda_1,\lambda_2,\lambda_3)$ in Eq. (\[eq1\]) can be written as [@Lorce:2011dv; @Zhang:1994ti; @Schlumpf:1992ce] $$\begin{aligned}
\Psi^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda_1,\lambda_2,\lambda_3)&=&\Phi(q_\perp,\xi,Q_\perp,\eta)\Xi^{SS_z}(\lambda_1,\lambda_2,\lambda_3)\,,\end{aligned}$$ where $\Phi(q_\perp,\xi,Q_\perp,\eta)$ is the momentum distribution of constituent quarks and $\Xi^{SS_z}(\lambda_1,\lambda_2,\lambda_3)$ represents the momentum-depended spin wave function, given by $$\begin{aligned}
\Xi^{SS_z}(\lambda_1,\lambda_2,\lambda_3)&=&\sum_{s_1,s_2,s_3}\langle\lambda_1|R^{\dagger}_1|s_1\rangle\langle\lambda_2|R^{\dagger}_2|s_2\rangle\langle\lambda_3|R^{\dagger}_3|s_3\rangle \bigg\langle\frac{1}{2}s_1,\frac{1}{2}s_2,\frac{1}{2}s_3\bigg|SS_z\bigg\rangle\,,\end{aligned}$$ with $\big\langle\frac{1}{2}s_1,\frac{1}{2}s_2,\frac{1}{2}s_3\big|SS_z\big\rangle$ the usual $SU(2)$ Clebsch-Gordan coefficient, and $R_i$ the well-known Melosh transformation, which corresponds to the $i$th constituent quark and can be expressed by $$\begin{aligned}
R_M(x,p_{\perp},m,M)&=&\frac{m+xM-i\vec{\sigma}\cdot(\vec{n}\times \vec{q})}{\sqrt{(m+xM)^2+q_\perp^2}}\end{aligned}$$ and $$\begin{aligned}
&&R_1=R_M(\eta,Q_\perp,M_3,M)R_M(\xi,q_\perp,m_1,M_3)\,, \nonumber\\
&&R_2=R_M(\eta,Q_\perp,M_3,M)R_M(1-\xi,-q_\perp,m_2,M_3) \,,\nonumber\\
&&R_3=R_M(1-\eta,-Q_\perp,m_3,M)\,,\end{aligned}$$ where $\sigma_i$ is the Pauli matrix and $\vec{n}=(0,0,1)$. This is the generalization of the Melosh transformation from two-particle systems, which can be derived from the transformation property of angular momentum operators [@Polyzou:2012ut; @Schlumpf:1992ce]. We further represent the LF kinematic variables $(\xi,q_\perp)$ and $(\eta,Q_\perp)$ in the forms of the ordinary 3-momenta ${\bf q}$ and ${\bf Q}$: $$\begin{aligned}
&&E_{1(2)}=\sqrt{{\bf q}^2+m_{1(2)}^2} \,,\quad E_{12}=\sqrt{{\bf Q}^2+M_3^2}\,, \quad E_3=\sqrt{{\bf Q}^2+m_3^2}\,, \nonumber \\
&&q_z=\frac{\xi M_3}{2}-\frac{m_1^2+q_\perp^2}{2M_3\xi}\,, \quad Q_z=\frac{\eta M}{2}-\frac{M_3^2+Q_\perp^2}{2M\eta} \,,
\label{3kin}\end{aligned}$$ to get more clear physical pictures of the momentum distribution wave functions.
It is known that the exact momentum wave function cannot be solved from the QCD first principle currently due to the lake of knowledge in the QCD effective potential in the three-body system. Hence, we choose the phenomenological Gaussian type wave function with suitable shape parameters to including the diquark clustering effects in $\Lambda_c^+$ and $\Lambda$ baryons [@Ke:2019smy; @Schlumpf:1992ce]. The baryon spin-flavor-momentum wave function $F_{abc}\Psi^{SS_z}(\tilde{p}_1,\tilde{p}_2,\tilde{p}_3,\lambda_1,\lambda_2,\lambda_3)$ should be totally symmetric under any permutations of quarks to keep the Fermi statistics. The spin-flavor-momentum wave functions of $\Lambda_c^+$, $\Lambda$ and the neutron are given by $$\begin{aligned}
&&|\Lambda_c\rangle=\frac{1}{\sqrt{6}}[\phi_3\chi^{\rho3}(|duc\rangle-|udc\rangle)+\phi_2\chi^{\rho2}(|dcu\rangle-|ucd\rangle)+\phi_1\chi^{\rho1}(|cdu\rangle-|cud\rangle)]\,,
\nonumber\\
&&|\Lambda\rangle=\frac{1}{\sqrt{6}}[\phi_3\chi^{\rho3}(|duc\rangle-|uds\rangle)+\phi_2\chi^{\rho2}(|dsu\rangle-|usd\rangle)+\phi_1\chi^{\rho1}(|sdu\rangle-|sud\rangle)]\,,
\nonumber\\
&&|n\rangle=\frac{1}{\sqrt{3}}\phi[\chi^{\lambda_3}|ddu\rangle+\chi^{\lambda_2}|dud\rangle+\chi^{\lambda_1}|udd\rangle]\,,\end{aligned}$$ respectively, where $$\begin{aligned}
\chi^{\rho 3}_{\uparrow}&=&\frac{1}{\sqrt{2}}(|\uparrow\downarrow\uparrow\rangle-|\downarrow\uparrow\uparrow\rangle)\,,
\quad \chi^{\lambda 3}_{\uparrow}=\frac{1}{\sqrt{6}}(|\uparrow\downarrow\uparrow\rangle+|\downarrow\uparrow\uparrow\rangle-2|\uparrow\uparrow\downarrow\rangle)\,,
\nonumber\\
\phi_3&=&{\cal N}\sqrt{\frac{\partial q_z}{\partial \xi}\frac{\partial Q_z}{\partial \eta}}e^{-\frac{{\bf Q}^2}{2\beta_{Q}^2}-\frac{{\bf q}^2}{2\beta_q^2}}\,, \end{aligned}$$ and $\phi_{1(2)}$ has the form by replacing $({\bf q},{\bf Q})$ with $({\bf q}_{1(2)},{\bf Q}_{1(2)})$ in $\phi_3$, with ${\cal N}=2(2\pi)^3(\beta_q\beta_{Q}\pi)^{-3/2}$ and $\beta_{q,Q}$ being the normalized constant and shape parameters, respectively. Explicitly, ${\bf q}_{1(2)}$ and ${\bf Q}_{1(2)}$ are given by $$\begin{aligned}
\xi_{1(2)}&=&\frac{p_{2(3)}^+}{p_{2(3)}^++p_{(1)}^+}, \qquad \eta_{1(2)}=1-\frac{p_{1(2)}^+}{P_{tot}^+}\,, \nonumber \\
q_{1(2)\perp }&=&(1-\xi_{1(2)})p_{2(3)\perp}-\xi_{1(2)} p_{3(1)\perp},\nonumber \\
Q_{1(2)\perp}&=&(1-\eta_{1(2)})(p_{2(3)\perp}+p_{3(1)\perp})-\eta_{1(2)} p_{1(2)\perp} \,.\end{aligned}$$ Here, the baryon state is normalized as $$\begin{aligned}
\langle {\bf B}&,P',S',S'_z|{\bf B}&,P,S,S_z\rangle=2(2\pi)^3P^+\delta^3(\tilde{P'}-\tilde{P})\delta_{S_z'S_z}\,,
\label{baryonN}\end{aligned}$$ resulting in the normalization of the momentum wave function, given by $$\begin{aligned}
\frac{1}{2^2(2\pi)^6} \int d\xi_{(1,2)} d\eta_{(1,2)} d^2q_{(1,2)\perp}d^2Q_{(1,2)\perp} |\phi_{3(1,2)}|^2=1\,.\end{aligned}$$ We emphasize that the momentum wave functions of $\phi_i$ with the different shape parameters of $\beta_q$ and $\beta_{Q}$ describe the scalar diquark effect in $\Lambda_{(c)}$. For the neutrons, the momentum distribution functions are the same, $i.e.$ $\phi=\phi_3(\beta_q=\beta_{Q})$, for any spin-flavor state due to the isospin symmetry. Note that there is no $SU(6)$ spin-flavor symmetry in $\Lambda_{(c)}$ even though the forms of these states are similar to those with the $SU(6)$ spin-flavor wave functions.
Transition form factors
-----------------------
The baryonic transition form factors of the $V-A$ weak current are defined by $$\begin{aligned}
&&\langle {\bf B}_f,P',S',S'_z|\bar{q}\gamma^{\mu}(1-\gamma_5)c|{\bf B}_i,P,S,S_z\rangle \nonumber\\
&&=\bar {u}_{{\bf B}_f}(P',S'_z)\bigg[\gamma^{\mu}f_1(k^2)-i\sigma^{\mu\nu}\frac{k_{\nu}}{M_{{\bf B}_i}}f_2(k^2)+\frac{k^{\mu}}{M_{{\bf B}_i}}f_3(k^2)\bigg]u_{{\bf B}_i}(P,S_z)\nonumber\\
&&-\bar {u}_{{\bf B}_f}(P',S'_z)\bigg[\gamma^{\mu}g_1(k^2)-i\sigma^{\mu\nu}\frac{k_{\nu}}{M_{{\bf B}_i}}g_2(k^2)+\frac{k^{\mu}}{M_{{\bf B}_i}}g_3(k^2)\bigg]\gamma_5u_{{\bf B}_i}(P,S_z)\end{aligned}$$ where $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$ and $P'-P=k$. We choose the frame such that $P^+$ is conserved ($k^+=0,k^2=-k_\perp^2$) to calculate the form factors to avoid other $x^{+}$-ordered diagrams in the LF formalism [@Schlumpf:1992ce]. The Matrix elements of the vector and axial-vector currents at quark level correspond to three different lowest-order Feynman diagrams as shown in Fig. 1.
\(a) {width="2in"}
\(b) {width="2in"}.
\(c) {width="2in"}
Since the spin-flavor-momentum wave functions of baryons are totally symmetric under the permutation of quarks, we have that $(a)+(b)+(c)=3(a)=3(b)=3(c)$ [@Schlumpf:1992ce]. As an illustration, we only present the calculations for the diagram (c), which contains simpler and cleaner forms with the notation $(q_{\perp},Q_{\perp},\xi,\eta)$. We can extract the form factors from the matrix elements through the relations $$\begin{aligned}
f_1(k^2)&=&\frac{1}{2P^+}\langle {\bf B}_f,P',\uparrow|\bar{q}\gamma^{+}c|{\bf B}_i,P,\uparrow\rangle\,, \nonumber \\
f_2(k^2)&=&\frac{1}{2P^+}\frac{M_{{\bf B}_i}}{k_{\perp}}\langle {\bf B}_f,P',\uparrow|\bar{q}\gamma^{+}c|{\bf B}_i,P,\downarrow\rangle\,, \nonumber \\
g_1(k^2)&=&\frac{1}{2P^+}\langle {\bf B}_f,P',\uparrow|\bar{q}\gamma^{+}\gamma_5c|{\bf B}_i,P,\uparrow\rangle\,, \nonumber \\
g_2(k^2)&=&\frac{1}{2P^+}\frac{M_{{\bf B}_i}}{k_{\perp}}\langle {\bf B}_f,P',\uparrow|\bar{q}\gamma^{+}\gamma_5|{\bf B}_i,P,\downarrow\rangle \,.\label{fm}\end{aligned}$$ Note that $f_3$ and $g_3$ cannot be obtained when $k^+=0$, but they are negligible because of the suppressions of the $k^2$ factors . With the help of the momentum distribution functions and the Melosh transformation matrix, the transition matrix elements can be expressed as $$\begin{aligned}
&&\langle {\bf B}_f,P',S',S'_z|\bar{q}\gamma^{+}c|{\bf B}_i,P,S,S_z\rangle \nonumber \\
&&=\frac{1}{2^2(2\pi)^6}\int d\xi d\eta d^2q_\perp d^2Q_\perp\Phi(q'_{\perp},\xi,Q'_\perp,\eta)\Phi(q_{\perp},\xi,Q_\perp,\eta)F^{def} F_{abc}\delta_{d}^{a}\delta_{e}^{b}\nonumber\\
&&\times \sum_{s_1,s_2,s_3}\sum_{s'_1,s'_2,s'_3}\langle S',S'_z|s'_1,s'_2,s'_3\rangle\langle s_1,s_2,s_3|S,S_z\rangle
\langle s'_1|R'_1R^{\dagger}_1|s_1\rangle\langle s'_2|R'_2R^{\dagger}_2|s_2\rangle \nonumber\\
&&\times 2P^+\sum_{\lambda'_3\lambda_3}\langle s'_3|R'_3|\lambda'_3\rangle (\delta_{q_{f}q}3\delta_{\lambda'_3\lambda_3}\delta_{cq^c})\langle\lambda_3|R_3^\dagger|s_3\rangle \label{vf}\,,\end{aligned}$$ $$\begin{aligned}
&&\langle {\bf B}_f,P',S',S'_z|\bar{q}\gamma^{+}\gamma_5c|{\bf B}_i,P,S,S_z\rangle \nonumber \\
&&=\frac{1}{2^2(2\pi)^6}\int d\xi d\eta d^2q_\perp d^2Q_\perp\Phi(q'_{\perp},\xi,Q'_\perp,\eta)\Phi(q_{\perp},\xi,Q_\perp,\eta)F^{def} F_{abc}\delta_{d}^{a}\delta_{e}^{b}\nonumber\\
&&\times \sum_{s_1,s_2,s_3}\sum_{s'_1,s'_2,s'_3}\langle S',S'_z|s'_1,s'_2,s'_3\rangle\langle s_1,s_2,s_3|S,S_z\rangle
\langle s'_1|R'_1R^{\dagger}_1|s_1\rangle\langle s'_2|R'_2R^{\dagger}_2|s_2\rangle \nonumber\\
&&\times 2P^+\sum_{\lambda'_3\lambda_3}\langle s'_3|R'_3|\lambda'_3\rangle (\delta_{q_{f}q}3(\sigma_z)_{\lambda'_3\lambda_3}\delta_{cq^c})\langle\lambda_3|R_3^\dagger|s_3\rangle \label{af}\end{aligned}$$ Using Eqs. (\[fm\]), (\[vf\]) and (\[af\]), we find that $$\begin{aligned}
\label{f1}
&&f_1(k^2)=\frac{3}{2^2(2\pi)^6}\int d\xi d\eta d^2q_\perp d^2Q_\perp\Phi(q'_{\perp},\xi,Q'_\perp,\eta)\Phi(q_{\perp},\xi,Q_\perp,\eta)(F^{def} F_{abc}\delta_{q_{f}q}\delta_{cq^c}\delta_{d}^{a}\delta_{e}^{b})\nonumber\\
&&\times \sum_{s_1,s_2,s_3}\sum_{s'_1,s'_2,s'_3}\langle S',\uparrow|s'_1,s'_2,s'_3\rangle\langle s_1,s_2,s_3|S,\uparrow\rangle
\prod_{i=1,2,3}\langle s'_i|R'_iR^{\dagger}_i|s_i\rangle \,,\end{aligned}$$ $$\begin{aligned}
\label{g1}
&&g_1(k^2)=\frac{3}{2^2(2\pi)^6}\int d\xi d\eta d^2q_\perp d^2Q_\perp\Phi(q'_{\perp},\xi,Q'_\perp,\eta)\Phi(q_{\perp},\xi,Q_\perp,\eta)(F^{def} F_{abc}\delta_{q_{f}q}\delta_{cq^c}\delta_{d}^{a}\delta_{e}^{b})\nonumber\\
&&\times \sum_{s_1,s_2,s_3}\sum_{s'_1,s'_2,s'_3}\langle S',\uparrow|s'_1,s'_2,s'_3\rangle\langle s_1,s_2,s_3|S,\uparrow\rangle
\prod_{i=1,2}\langle s'_i|R'_iR^{\dagger}_i|s_i\rangle \langle s'_3|R'_3\sigma_zR^{\dagger}_3|s_3\rangle\,,\end{aligned}$$ $$\begin{aligned}
\label{f2}
&&f_2(k^2)=\frac{3}{2^2(2\pi)^6}\frac{M_{{\bf B}_i}}{k_{\perp}}\int d\xi d\eta d^2q_\perp d^2Q_\perp\Phi(q'_{\perp},\xi,Q'_\perp,\eta)\Phi(q_{\perp},\xi,Q_\perp,\eta)(F^{def} F_{abc}\delta_{q_{f}q}\delta_{cq^c}\delta_{d}^{a}\delta_{e}^{b})\nonumber\\
&&\times \sum_{s_1,s_2,s_3}\sum_{s'_1,s'_2,s'_3}\langle S',\uparrow|s'_1,s'_2,s'_3\rangle\langle s_1,s_2,s_3|S,\downarrow\rangle
\prod_{i=1,2,3}\langle s'_i|R'_iR^{\dagger}_i|s_i\rangle \,,\end{aligned}$$ $$\begin{aligned}
&&g_2(k^2)=\frac{3}{2^2(2\pi)^6}\frac{M_{{\bf B}_i}}{k_{\perp}}\int d\xi d\eta d^2q_\perp d^2Q_\perp\Phi(q'_{\perp},\xi,Q'_\perp,\eta)\Phi(q_{\perp},\xi,Q_\perp,\eta)(F^{def} F_{abc}\delta_{q_{f}q}\delta_{cq^c}\delta_{d}^{a}\delta_{e}^{b})\nonumber\\
&&\times \sum_{s_1,s_2,s_3}\sum_{s'_1,s'_2,s'_3}\langle S',\uparrow|s'_1,s'_2,s'_3\rangle\langle s_1,s_2,s_3|S,\downarrow\rangle
\prod_{i=1,2}\langle s'_i|R'_iR^{\dagger}_i|s_i\rangle \langle s'_3|R'_3\sigma_zR^{\dagger}_3|s_3\rangle\,.
\label{g2} \end{aligned}$$
Baryonic transition form factors in MBM
=======================================
The formalism and other details for MBM can be found in Ref. [@PerezMarcial:1989yh]. In the calculation of MBM, we take the same notations as those in Ref. [@PerezMarcial:1989yh]. In this approach, the current quark masses are used, given by $$m_{u,d}= 0.005~\text{GeV}\,,~~~~m_s= 0.28~\text{GeV}\,,~~~~m_c = 1.5~ \text{GeV}\,,~~~~R=5~\text{GeV}^{-1}\,,$$ where $R$ corresponds to the bag size. Note that the form factors can be only evaluated at $\vec{k}=0$ ($k^2=\Delta M^2$) due to the assumption of the static bag. The form factors are decomposed as follows: $$\begin{aligned}
f_{1}&=&\mathcal{V}_{0}-\mathcal{V}_{M} \Delta M^{2} / M_{12}-\mathcal{V}_{V} \Delta M \,,\nonumber\\
f_{2}&=&\left(-\mathcal{V}_{0}+\mathcal{V}_{M} M_{12}+\mathcal{V}_V \Delta M\right) M_{1} / M_{12}\,, \nonumber\\
f_{3}&=&\mathcal{V}_{V} M_{1}+\mathcal{V}_{M} M_{1} \Delta M / M_{12}\nonumber\\
g_{1}&=&\left(1-\Delta M^{2} / 2 M_{12}^{2}\right) \mathcal{A}_{s}+\left(\mathcal{A}_{T} \Delta M-\mathcal{A}_{0}\right) 4 M_{1} M_{2} \Delta M / M_{12}^{2}\,,\nonumber \\
g_{2}&=&\left(\mathcal{A}_{T} \Delta M-\mathcal{A}_{s} \Delta M / 8 M_{1} M_{2}-\mathcal{A}_{0}\right) 4 M_{1}^{2} M_{2} / M_{12}^{2} \nonumber\\
g_{3}&=&\left(\mathcal{A}_{s} / 2+\mathcal{A}_{T} 4 M_{1} M_{2}\right) M_{1} / M_{12}\,,\end{aligned}$$ with $\Delta M = M_1- M_2$, $M_{12}=M_1 + M_2$ and $$\begin{aligned}
\mathcal{V}_{0}&=&A R^{3}\left(W^{i}_+W^{f}_+I_{00}+W^{i}_-W^{f}_-I_{11}\right)\,, \nonumber\\
\mathcal{V}_{v}&=&A R^{3}\left(W^{i}_-W^{f}_+I_{10}-W^{i}_+W^{f}_-I_{01}\right)(R / 3)\,, \nonumber\\
\mathcal{V}_{M}&=&A R^{3}\left(W^{i}_-W^{f}_+I_{10}+W^{i}_+W^{f}_-I_{01}\right)(R / 3)\,, \nonumber\\
\mathcal{A}_{0}&=&A R^{3}\left(W^{i}_-W^{f}_+I_{10}-W^{i}_+W^{f}_-I_{01}\right)(R / 3)\,, \nonumber\\
\cal{A}_s&=&AR^3\left(W^i_+W^f_+ - W^i_-W^f_-I_{11}/3\right) \,,\nonumber\\
\mathcal{A}_{T}&=&A R^{3} W^{i}_-W^{f}_- J_{11}(-2 R^2 / 15)\,,\end{aligned}$$ where $A$ is the normalized factor for the baryon, corresponding to the baryon spin-flavor structures as given by Table. II in Ref. [@PerezMarcial:1989yh], $W^q_{\pm}$ are associated with the normalized factors for quarks, given by $$W_{\pm}^{q} \equiv\left(\frac{\omega^{q}\pm m_{q}}{\omega^{q}}\right)^{1 / 2}$$ with $\omega^q$ representing the quark energy, and $I$ and $J$ stand for the overlapping factors for the quark wave functions, defined by $$\begin{aligned}
I_{n n} &\equiv& \int_{0}^{1} d t t^{2} j_{n}\left(t x_{0}^{i}\right) j_{n}\left(t x_{0}^{f}\right), \quad n=0,1\nonumber\\
I_{n m} &\equiv &\int_{0}^{1} d t t^{3} j_{n}\left(t x_{0}^{i}\right) j_{m}\left(t x_{0}^{f}\right), \quad n, m=0,1~(n \neq m)\nonumber\\
J_{11} &\equiv& \int_{0}^{1} d t\, t^{4} j_{1}\left(t x_{0}^{i}\right) j_{1}\left(t x_{0}^{f}\right)\,,\end{aligned}$$ with $j_n$ the Bessel function and $x^q_0$ the lowest root of the transcendental equation of $$\begin{aligned}
\tan(x^q)=\frac{x^q}{1-m_qR-[(x^q)^2+(m_qR)^2]^{1/2}}\,.\end{aligned}$$
Numerical Results
=================
In section II, we have derived the baryonic transition form factors in LFCQM. The form factors can be evaluated only in the space-like region ($k^2=-k^2_\perp$) because of the condition $k^+=0$. Thus, we follow the standard procedures in Refs. [@Cheng:2003sm; @Ke:2007tg; @Cheng:2004cc] to extract the information of the form factors in the time-like region. These procedures have widely been tested and discussed in the mesonic sector [@Jaus:1991cy; @Jaus:1996np]. We fit $f_{1(2)}(k^2)$ and $g_{1(2)}(k^2)$ with some analytic functions in the space-like region, which are analytically continued to the physical time-like region $(k^2>0)$. We employ the numerical values of the constituent quark masses and shape parameters in Table. I.
\[sh\]
$m_c$ $m_s$ $m_d$ $m_c$ $\beta_{q\Lambda_c}$ $\beta_{Q\Lambda_c}$ $\beta_{q\Lambda}$ $\beta_{Q\Lambda}$ $\beta_{qn}$ $\beta_{Qn}$
------- ------- ------- ------- ---------------------- ---------------------- -------------------- -------------------- -------------- --------------
1.3 0.4 0.26 0.26 0.89 0.75 0.53 0.52 0.44 0.44
: Values of the constituent quark masses ($m_i$) and shape parameter ($\beta_{q{\bf B}}$ and $\beta_{{Q\bf B}}$) in units of GeV.
The values of the shape parameters can be determined approximately by the calculations in the mesonic sectors [@Chang:2018zjq; @Ke:2019smy]. By assuming that the Coulomb-like potential is dominant in the quark-quark strong interaction, one can deduce the shape parameter of quark pairs to be $\sqrt{2}$ greater than those fro the mesonic sectors because the interaction is about twice stronger between the quark-quark pair than quark-anti-quark one [@Ke:2019smy]. Since the reciprocals of the shape parameters are related to the sizes of systems, we adopt $\beta_{q\Lambda_c}\simeq2(\sqrt{2}\beta_{u\bar{d}})$ and $\beta_{q\Lambda}\simeq1.2(\sqrt{2}\beta_{u\bar{d}})$, where the factors of 2 and 1.2 come from the effects of the diquark clusterings, respectively, which make the light quark pairs to be more compact. By using Eqs. (\[f1\])-(\[g2\]), we compute totally 32 points for all form factors from $k^2=0$ to $k^2=-9.7\text{ GeV}^2$. With the MATLAB curve fitting toolbox, we present our results of $\Lambda_c^+\to \Lambda$ in Figs. \[lclvffit\] and \[lclaffit\] and $\Lambda_c^+\to n$ in Figs. \[lcnvffit\] and \[lcnaffit\] with $95\%$ confidence bounds in Appendix. To fit the $k^2$ dependences of the form factors, we use the form $$\begin{aligned}
F(k^2)=\frac{p_1k^4+p_2k^2+F(0)}{1-q_1k^2+q_2k^4} \,.\end{aligned}$$ In the $\Lambda_c^+\to\Lambda$ transition, we choose $p_1=0$, $p_2=0$ for $f_{1,2}$ and $g_1$, but only c $q_1=0$ for $g_2$. On the other hand, in the $\Lambda_c^+\to n$ transition, we take $p_1=p_2=0$ for all form factors to fit the numerical values in the space-like region. We present our fitting results in Table. \[nf\].
[ccccc]{}\
&$f_1$&$f_2$&$g_1$&$g_2$\
$F(0)$&$0.63\pm0.01$&$0.66\pm0.01$&$0.51\pm0.01$&$-(2.7\pm0.1)\times10^{-3}$\
$q_1$&$0.74\pm0.15$&$0.58\pm0.14$&$0.53\pm0.12$&-\
$q_2$&$0.67\pm0.12$&$0.53\pm0.11$&$0.46\pm0.08$&$1.9\pm0.3$\
$p_1$&-&-&-&$-(1.0\pm0.2)\times10^{-3}$\
$p_2$&-&-&-&$-(6.9\pm0.6)\times10^{-3}$\
\
&$f_1$&$f_2$&$g_1$&$g_2$\
$F(0)$&$0.57\pm0.01$&$0.64\pm0.01$&$0.46\pm0.01$&$(26.5\pm0.6)\times10^{-3}$\
$q_1$&$0.82\pm0.16$&$0.81\pm0.16$&$0.57\pm0.13$&$0.52\pm0.15$\
$q_2$&$0.81\pm0.14$&$0.78\pm0.14$&$0.55\pm0.09$&$0.73\pm0.12$\
\[nf\]
For MBM, we use the Lorentzian type functions for the $k^2$ dependences of the form factors, given by $$\begin{aligned}
f_{i}(k^2)=\frac{(1+d_f)f_{i}(0)}{(1-\frac{k^2}{M^2_V})^2+d_f}\\
g_{i}(k^2)=\frac{(1+d_g)g_{i}(0)}{(1-\frac{k^2}{M^2_A})^2+d_g}\end{aligned}$$ where $M_V=2.112~(2.010)$, $M_A=2.556~(2.423)$, $d_f=0.2~(0.1)$ and $d_g=0.1~(0.05)$ for $c\to s(d)$ processes, respectively. We list $f_{i}(0)=f_i$ and $g_{i}(0)=g_i$ in Table. \[MBM\].
$f_1$ $f_2$ $f_3$ $g_1$ $g_2$ $g_3$
-------------------------- ------- ------- ------- ------- ------- -------
$\Lambda_c^+\to \Lambda$ 0.54 0.22 0.00 0.52 -0.06 -0.50
$\Lambda_c^+\to n$ 0.40 0.22 0.00 0.43 -0.07 -0.53
: Fitting results of the form factors in MBM
\[MBM\]
In order to calculate the decay branching ratios and other physical quantities, we introduce the the helicity amplitudes of $H^{V(A)}_{\lambda_2\lambda_W}$, which give more intuitive physical pictures and simpler expressions when discussing the asymmetries of the decay processes, such as the integrated (averaged) asymmetry, also known as the longitudinal polarization of the daughter baryon. Relations between the helicity amplitudes and form factors are given by $$\begin{aligned}
H^{V}_{\frac{1}{2}1}&=&\sqrt{2K_-}\left(-f_1-\frac{M_{\bf B_{i}}+M_{\bf B_{f}}}{M_{\bf B_{i}}}f_2 \right) \,,\nonumber \\
H^{V}_{\frac{1}{2}0}&=&\frac{\sqrt{K_-}}{\sqrt{k^2}}\left((M_{\bf B_{i}}+M_{\bf B_{f}})f_1+\frac{k^2}{M_{\bf B_{i}}}f_2\right)\,,\nonumber \\
H^{V}_{\frac{1}{2}t}&=&\frac{\sqrt{K_+}}{\sqrt{k^2}}\left((M_{\bf B_{i}}+M_{\bf B_{f}})f_1+\frac{k^2}{M_{\bf B_{i}}}f_3\right)\,,\nonumber \\
H^{A}_{\frac{1}{2}1}&=&\sqrt{2K_+}\left(g_1-\frac{M_{\bf B_{i}}-M_{\bf B_{f}}}{M_{\bf B_{i}}}g_2\right)\,,\nonumber \\
H^{A}_{\frac{1}{2}0}&=&\frac{\sqrt{K_+}}{\sqrt{k^2}}\left(-(M_{\bf B_{i}}-M_{\bf B_{f}})g_1+\frac{k^2}{M_{\bf B_{i}}}g_2\right) \,,\nonumber\\
H^{A}_{\frac{1}{2}t}&=&\frac{\sqrt{K_-}}{\sqrt{k^2}}\left(-(M_{\bf B_{i}}-M_{\bf B_{f}})g_1+\frac{k^2}{M_{\bf B_{i}}}g_3\right)\,,
\label{amp}\end{aligned}$$ where $K_{\pm}=(M_{\bf B_i}-M_{\bf B_f})^2-k^2$. We note that both $f_3$ and $g_3$ have been set to be 0 in LFCQM.
The differential decay width and asymmetries can be expressed in the analytic forms in terms of the helicity amplitudes, which can be found in our previous work of Ref. [@Geng:2019bfz]. In our numerical calculations, we use the center value of $\tau_{\Lambda_c^+}=203.5\times 10^{-15}s$ in Eq. (\[q0\]) [@Aaij:2019lwg]. Our predictions of the decay branching ratios ($Br$s) and asymmetries ($\alpha$s) are listed in Table. \[result\]. In Table. \[com\], we compare our results with the experimental data and those in various calculations in the literature.
---------------------------------------- ----------------- ---------------- ---------- ----------
$Br(\%)$ $\alpha$ $Br(\%)$ $\alpha$
$\Lambda_c^+\to \Lambda e^+ \nu_e$ $3.43\pm0.57$ $-0.96\pm0.03$ $3.48$ $-0.83$
$\Lambda_c^+\to \Lambda \mu^+ \nu_\mu$ $3.30\pm0.56$ $-0.96\pm0.03$ $3.38$ $-0.82$
$\Lambda_c^+\to ne^+\nu_e$ $0.215\pm0.041$ $-0.97\pm0.01$ $0.255$ $-0.85$
$\Lambda_c^+\to n\mu^+\nu_\mu$ $0.209\pm0.041$ $-0.97\pm0.01$ $0.250$ $-0.85$
---------------------------------------- ----------------- ---------------- ---------- ----------
: Predictions of branching ratio and asymmetry parameters
\[result\]
----------------------------------------- --------------- ---------------- ----------------- ----------------
$Br(\%)$ $\alpha$ $Br(\%)$ $\alpha$
LFCQM $3.43\pm0.57$ $-0.96\pm0.03$ $0.215\pm0.041$ $-0.97\pm0.01$
MBM $3.48$ $-0.83$ $0.255$ $-0.85$
Data [@Tanabashi:2018oca] $3.6\pm0.4$ $-0.86\pm0.04$ - -
$SU(3)$ [@Geng:2019bfz] $3.4\pm0.3$ $-0.86\pm0.04$ $0.53\pm0.05$ $-0.89\pm0.04$
HQET [@Cheng:1995fe] 1.42 - - -
LF [@Zhao:2018zcb] 1.63 - 0.201 -
MBM [^1] (NRQM) [@PerezMarcial:1989yh] 2.6 (3.2) - 0.20 (0.30) -
LQCD [@Meinel:2016dqj; @Meinel:2017ggx] $3.80\pm0.22$ - $0.410\pm0.029$ -
RQM [@Faustov:2016yza] 3.25 - 0.268 -
----------------------------------------- --------------- ---------------- ----------------- ----------------
: Our results in comparisons with the experimental data and those in various calculations in the literature.[]{data-label="com"}
In LF [@Zhao:2018zcb] and HQET [@Cheng:1995fe], the authors use a specific spin-flavor structure of $c(ud-du)\chi_{s_z}^{\rho_3}$ for the charmed baryon state, in which only the permutation relation is considered between light quarks. In addition, they assume that the diquarks from the light quark pairs are spectator and structureless. These simplifications in Refs. [@Zhao:2018zcb] and [@Cheng:1995fe] make their results to be not good compared with the experimental data as shown in Table. \[com\]. Based on the Fermi statistics, the overall spin-flavor-momentum structures are determined, from which the parameters like quark masses, baryon masses and shape parameters can recover the spin-flavor symmetry. In LFCQM, we consider the different diquark clustering effect in different baryons. We expect that this effect is stronger if the mass of the third quark is greater than others, which is encoded in the shape parameter of $\beta_{q{\bf B}}$. There is an interesting observation that the shape parameters $\beta_{{Q\bf B}}$ and $\beta_{q{\bf B}}$ in our study are almost the same in each baryon, which implies the totally symmetric momentum distribution of three constituent quarks in the baryon. In addition, the flavor symmetry breaking effect due to the quark masses seems to get canceled due to the clustering effect of the shape parameters in the momentum distribution functions. Our numerical results indicate that the form factors follow the Lorentzian functions of $F(k^2)=F(0)/(1-q_1k^2+q_2k^4)$ except $g_2(k^2)$ in the $\Lambda_c^+ \to \Lambda$ processes. Our results of $f_i(k^2)\neq g_i(k^2)$ show that the lowest order of the heavy quark symmetry is failure because the constituent charm quark mass is not heavy enough.
From Table. \[result\], we predict that ${\cal B}(\Lambda_c^+\to \Lambda e^+\nu_e)=(3.43\pm0.57)\times 10^{-2}$ and ${\cal B}(\Lambda_c^+\to n e^+\nu_e)=(2.15\pm0.41)\times 10^{-3}$, and $\alpha(\Lambda_c^+\to \Lambda e^+\nu_e)=-0.96\pm0.03$ and $\alpha(\Lambda_c^+\to n e^+\nu_e)=-0.97\pm0.01$ in LFCQM, in which the value of ${\cal B}(\alpha)$ for the mode of $ \Lambda_c^+\to \Lambda e^+\nu_e)$ is lower (higher) than but acceptable by the experimental one $(3.6\pm0.4)\times 10^{-2}~ (-0.86\pm0.04)$ in PDG [@Tanabashi:2018oca]. The errors in our results mainly come from the numerical fits of the MATLAB curve fitting toolbox in Appendix, in which the $95\%$ confidence bounds are broadened and tightened in the time-like space-like regions, respectively. Our results are also consistent with those in the Lattice QCD (LQCD) [@Meinel:2016dqj; @Meinel:2017ggx] and relativistic quark model (RQM) [@Faustov:2016yza]. For MBM, Although the semi-leptonic processes have been fully studied in Ref. [@PerezMarcial:1989yh], their results are mismatched with the current data. By using the same formalism with the same input parameters, we are able to get the same values of the form factors at the zero recoil point. By taking the Lorentzian $k^2$ dependences for the form factors, inspired from our LF calculations, we obtain much better results as shown in Table. \[com\]. It is interesting to see that our results for $\Lambda_c^+\to n e^+\nu_e$ are consistent with other calculations except those from $SU(3)_F$ and LQCD.
Conclusions
===========
We have studied the semi-leptonic decays of $\Lambda_c^+ \to \Lambda(n)\ell^+ \nu_{\ell}$ in the two dynamical approaches of LFCQM and MBM. We have used the Fermi statistics to determine the overall spin-flavor-momentum structures and recover the spin-flavor symmetry with the quark and baryon masses and shape parameters. We have found that ${\cal B}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(3.43\pm0.57)\%$ and $3.48\%$ in LFCQM and MBM, which are consistent with the experimental data of $(3.6\pm0.4)\times 10^{-2}$ [@Tanabashi:2018oca] as well as the values predicted by $SU(3)_F$ [@Geng:2019bfz], LQCD [@Meinel:2016dqj; @Meinel:2017ggx] and RQM [@Faustov:2016yza], but about a factor of two larger than those in HQET [@Cheng:1995fe] and LF [@Zhao:2018zcb]. We have also obtained that ${\alpha}( \Lambda_c^+ \to \Lambda e^+ \nu_{e})=(-0.96\pm0.03)$ and $-0.83$ in LFCQM and MBM, which are lower and higher than but acceptable by the experimental data of $-0.86\pm0.04$ [@Tanabashi:2018oca], respectively. We have also predicted that ${\cal B}( \Lambda_c^+ \to n e^+ \nu_{e})=(2.15\pm0.41, 2.55)\times 10^{-3}$ and ${\alpha}( \Lambda_c^+ \to n e^+ \nu_{e})=(-0.97\pm0.01,-0.85)$ in (LFCQM, MBM), in which our results of ${\cal B}( \Lambda_c^+ \to n e^+ \nu_{e})$ in LFCQM and MBM as well as that in RQM [@Faustov:2016yza] are consistent with each other, but about two times smaller than those in $SU(3)_F$ [@Geng:2019bfz] and LQCD [@Meinel:2016dqj; @Meinel:2017ggx]. It is clear that our predicted values for the decay branching ratio and asymmetry in $ \Lambda_c^+ \to n e^+ \nu_{e}$ could be tested in the ongoing experiments at BESIII, LHCb and BELLEII. Finally, we remark that our calculations in LFCQM and MBM can be also extended to the other charmed baryons, such as $\Xi_c^+$, $\Xi_c^0$, and even b baryons.
Appendix
========
We now show our numerical results for the form factors in Eqs. (\[f1\])-(\[g2\]) in LFCQM. In Fig. \[lclvffit\], we plot the vector form factors of $f_{1,2}$ with respect to the transfer momentum $k^2$ in unit of $\text{ GeV}^2$ for $\Lambda_c^+\to \Lambda$, where the symbol of “$\bullet$” denotes the value calculated by Eqs. (\[f1\]) and (\[f2\]) from $k^2=0$ to $-9.7\text{ GeV}^2$ with Mathematica, while the blue line corresponds to the fitted function by the MATLAB curve fitting toolbox and the dashed line represents the $95\%$ confidence bound of the fit. Similarly, we depict the axial-vector form factors of $g_{1,2}$ in Fig. \[lclaffit\]. The corresponding results for $\Lambda_c^+\to n$ are given in Figs. \[lcnvffit\] and \[lcnaffit\].
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work was supported in part by National Center for Theoretical Sciences and MoST (MoST-107-2119-M-007-013-MY3).
[99]{} R. Aaij [*et al.*]{} \[LHCb Collaboration\], Phys. Rev. D [**100**]{}, 032001 (2019) M. Tanabashi [*et al.*]{} \[Particle Data Group\], Phys. Rev. D [**98**]{}, 030001 (2018). Y. B. Li [*et al.*]{} \[Belle Collaboration\], Phys. Rev. Lett. [**122**]{}, 082001 (2019). Y. B. Li [*et al.*]{} \[Belle Collaboration\], Phys. Rev. D [**100**]{}, 031101 (2019). M. J. Savage and R. P. Springer, Phys. Rev. D [**42**]{}, 1527 (1990). M. J. Savage, Phys. Lett. B [**257**]{}, 414 (1991).
C. D. Lu, W. Wang and F. S. Yu, Phys. Rev. D [**93**]{}, 056008 (2016).
C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, JHEP [**1711**]{}, 147 (2017).
W. Wang, Z. P. Xing and J. Xu, Eur. Phys. J. C [**77**]{}, 800 (2017).
C. Q. Geng, Y. K. Hsiao, Y. H. Lin and L. L. Liu, Phys. Lett. B [**776**]{}, 265 (2018).
C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Phys. Rev. D [**97**]{}, 073006 (2018).
D. Wang, P. F. Guo, W. H. Long and F. S. Yu, JHEP [**1803**]{}, 066 (2018).
C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Eur. Phys. J. C [**78**]{}, 593 (2018).
X. G. He and W. Wang, Chin. Phys. C [**42**]{}, 103108 (2018). X. G. He, Y. J. Shi and W. Wang, arXiv:1811.03480 \[hep-ph\].
C. Q. Geng, Y. K. Hsiao, C. W. Liu and T. H. Tsai, Phys. Rev. D [**99**]{}, 073003 (2019).
C. Q. Geng, C. W. Liu and T. H. Tsai, Phys. Lett. B [**790**]{}, 225 (2019)
C. Q. Geng, C. W. Liu and T. H. Tsai, Phys. Lett. B [**794**]{}, 19 (2019).
C. Q. Geng, C. W. Liu, T. H. Tsai and Y. Yu, Phys. Rev. D [**99**]{}, 114022 (2019).
J. Y. Cen, C. Q. Geng, C. W. Liu and T. H. Tsai, Eur. Phys. J. C [**79**]{}, 946 (2019).
Y. K. Hsiao, Y. Yao and H. J. Zhao, Phys. Lett. B [**792**]{}, 35 (2019).
C. Q. Geng, C. W. Liu, T. H. Tsai and S. W. Yeh, Phys. Lett. B [**792**]{}, 214 (2019). Y. Grossman and S. Schacht, JHEP [**1907**]{}, 020 (2019).
S. Roy, R. Sinha and N. G. Deshpande, arXiv:1911.01121 \[hep-ph\].
C. P. Jia, D. Wang and F. S. Yu, arXiv:1910.00876 \[hep-ph\].
H. Y. Cheng and B. Tseng, Phys. Rev. D[**53**]{}, 1457 (1996). S. Meinel, Phys. Rev. Lett. [**118**]{}, 082001 (2017). S. Meinel, Phys. Rev. D[**97**]{}, 034511 (2018). R. N. Faustov and V. O. Galkin, Eur. Phys. J. C [**76**]{}, 628 (2016).
Z. X. Zhao, Chin. Phys. C [**42**]{}, 093101 (2018).
R. Perez-Marcial, R. Huerta, A. Garcia and M. Avila-Aoki, Phys. Rev. D [**40**]{}, 2955 (1989) Erratum: \[Phys. Rev. D [**44**]{}, 2203 (1991)\]. W. M. Zhang, Chin. J. Phys. [**32**]{}, 717 (1994). F. Schlumpf, hep-ph/9211255. C. Q. Geng, C. C. Lih and W. M. Zhang, Phys. Rev. D [**57**]{}, 5697 (1998). C. C. Lih, C. Q. Geng and W. M. Zhang, Phys. Rev. D [**59**]{}, 114002 (1999). C. Q. Geng, C. C. Lih and W. M. Zhang, Phys. Rev. D [**62**]{}, 074017 (2000). C. Q. Geng, C. C. Lih and W. M. Zhang, Mod. Phys. Lett. A [**15**]{}, 2087 (2000). C. Q. Geng, C. W. Hwang, C. C. Lih and W. M. Zhang, Phys. Rev. D [**64**]{}, 114024 (2001). [@Geng:2001vy] C. Q. Geng, C. W. Hwang and C. C. Liu, Phys. Rev. D [**65**]{}, 094037 (2002). C. Q. Geng and C. C. Liu, J. Phys. G [**29**]{}, 1103 (2003). H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D [**69**]{}, 074025 (2004).
H. Y. Cheng, C. K. Chua and C. W. Hwang, Phys. Rev. D [**70**]{}, 034007 (2004). C. Q. Geng and C. C. Lih, Phys. Rev. C [**86**]{}, 038201 (2012) Erratum: \[Phys. Rev. C [**87**]{}, 039901 (2013)\]. C. Q. Geng, C. C. Lih and C. Xia, Eur. Phys. J. C [**76**]{}, 313 (2016). H. Y. Cheng and X. W. Kang, Eur. Phys. J. C [**77**]{}, 587 (2017) Erratum: \[Eur. Phys. J. C [**77**]{}, 863 (2017)\].
Z. X. Zhao, Eur. Phys. J. C [**78**]{}, 756 (2018). Z. P. Xing and Z. X. Zhao, Phys. Rev. D [**98**]{}, 056002 (2018). Q. Chang, L. T. Wang and X. N. Li, JHEP [**1912**]{}, 102 (2019).
C. K. Chua, Phys. Rev. D [**99**]{}, 014023 (2019). C. K. Chua, Phys. Rev. D [**100**]{}, 034025 (2019). J. M. Maldacena, Int. J. Theor. Phys. [**38**]{}, 1113 (1999) \[Adv. Theor. Math. Phys. [**2**]{}, 231 (1998)\]. S. J. Brodsky and G. F. de Teramond, Few Body Syst. [**52**]{}, 203 (2012). H. W. Ke, N. Hao and X. Q. Li, Eur. Phys. J. C [**79**]{}, 540 (2019). H. W. Ke, X. Q. Li and Z. T. Wei, Phys. Rev. D [**77**]{}, 014020 (2008). H. W. Ke, X. H. Yuan, X. Q. Li, Z. T. Wei and Y. X. Zhang, Phys. Rev. D [**86**]{}, 114005 (2012). C. Lorce, B. Pasquini and M. Vanderhaeghen, JHEP [**1105**]{}, 041 (2011). W. N. Polyzou, W. Glockle and H. Witala, Few Body Syst. [**54**]{}, 1667 (2013).
W. Jaus, Phys. Rev. D [**44**]{}, 2851 (1991). W. Jaus, Phys. Rev. D [**53**]{}, 1349 (1996) Erratum: \[Phys. Rev. D [**54**]{}, 5904 (1996)\]. Q. Chang, X. N. Li, X. Q. Li, F. Su and Y. D. Yang, Phys. Rev. D [**98**]{}, 114018 (2018).
[^1]: Although the values of $f_i$ and $g_i$ are the same at the zero recoil point ($\vec{q}=0$), we use the Lorentzian type of the $k^2$ dependences for the form factors instead of the dipole ones in this work.
|
---
abstract: 'We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact Kähler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.'
address:
- 'Serge Cantat, IRMAR, campus de Beaulieu, bâtiments 22-23 263 avenue du Général Leclerc, CS 74205 35042 RENNES Cédex'
- ' Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation (LPSM), F-75005 Paris, France'
author:
- Serge Cantat
- Romain Dujardin
bibliography:
- 'biblio-serge.bib'
title: Random dynamics on real and complex projective surfaces
---
Introduction {#sec:introduction}
============
Random dynamical systems {#par:RDS_intro}
------------------------
Consider a compact manifold $M$ and a probability measure $\nu$ on ${{\mathsf{Diff}}}(M)$. To simplify the exposition we assume throughout this introduction that the support $\operatorname{Supp}(\nu)$ is finite. The data $(M,\nu)$ defines a **random dynamical system**, obtained by randomly composing independent diffeomorphisms with distribution $\nu$. In this paper, these random dynamical systems are studied from the point of view of *ergodic theory*, that is, we are mostly interested in understanding the asymptotic distribution of orbits.
Let us first recall some basic vocabulary. A probability measure $\mu$ on $M$ is [**[$\nu$-invariant]{}**]{} if $f_*\mu=\mu$ for $\nu$-almost every $f\in {{\mathsf{Diff}}}(M)$, and it is [**[$\nu$-stationary]{}**]{} if it is invariant on average: $\int f_*\mu \, d\nu(f)=\mu$. A simple fixed point argument shows that stationary measures always exist. On the other hand, the existence of an invariant measure should hold only under special circumstances, for instance when the group $\Gamma_\nu$ generated by $\operatorname{Supp}(\nu)$ is amenable, or has a finite orbit, or preserves an invariant volume form.
According to Breiman’s law of large numbers, the asymptotic distribution of orbits is described by stationary mesures. More precisely, for every $x\in M$ and $\nu^{\mathbf{N}}$-almost every $(f_j)\in {{\mathsf{Diff}}}(M)^{\mathbf{N}}$, every cluster value of the sequence of empirical measures $$\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f_j\circ \cdots \circ f_0(x)}$$ is a stationary measure. Thus a description of stationary measures gives a complete understanding of the asymptotic distribution of such random orbits, as $n$ goes to $+\infty$.
For a deterministic dynamical system, the space of invariant measures is typically too large to be amenable to a complete description. On the other hand a number of recent works in random dynamics have shown that stationary measures, even if they always exist, tend to satisfy some *rigidity properties*. Our goal in this article is to combine tools from algebraic and holomorphic dynamics together with some recent results in random dynamics to study the case when $M$ is a closed real or complex algebraic surface and the action is by algebraic diffeomorphisms. In this context we will reveal some new measure rigidity phenomena. Before describing in more detail the state of the art and stating a few precise results, let us highlight a nice geometric example to which our techniques can be applied.
Randomly folding pentagons {#par:pentagons_intro}
--------------------------
Let $\ell_0, \ldots, \ell_4$ be five positive real numbers such that there exists a pentagon with side lengths $\ell_i$. Here a pentagon is just an ordered set of points $(a_i)_{i=0, \ldots, 4}$ in the Euclidean plane such that $\operatorname{dist}(a_i,a_{i+1})=\ell_i$ for $i=0,\ldots, 4$ (with $a_5 = a_0$ by definition); pentagons are not assumed to be convex, and two distincts sides $[a_i, a_{i+1}]$ and $[a_j, a_{j+1}]$ may intersect at a point which is not one of the $a_i$’s.
Let $\operatorname{{Pent}}(\ell_0, \ldots, \ell_4)$ be the set of pentagons with side lengths $\ell_i$. Note that $\operatorname{{Pent}}(\ell_0, \ldots, \ell_4)$ may be defined by polynomial equations of the form $\operatorname{dist}(a_i, a_{i+1})^2 = \ell_i^2$, so it is naturally a real algebraic variety. For every $i$, $a_i$ is one of the two intersection points ${\left\{a_i, a'_i\right\}}$ of the circles of respective centers $a_{i-1}$ and $a_{i+1}$ and radii $\ell_{i-1}$ and $\ell_i$. The transformation exchanging these two points $a_i$ and $a_i’$ while keeping the other vertices fixed defines an involution $s_i$ of $\operatorname{{Pent}}(\ell_0, \ldots, \ell_4)$. It commutes with the action of the group ${{\sf{SO}}}_2({\mathbf{R}})\ltimes {\mathbf{R}}^2$ of positive isometries of the plane, hence, it induces an involution $\sigma_i$ on the quotient space $$\operatorname{{Pent}}^0(\ell_0, \ldots, \ell_4)=\operatorname{{Pent}}(\ell_0, \ldots, \ell_4)/({{\sf{SO}}}_2({\mathbf{R}})\ltimes {\mathbf{R}}^2).$$ Each element of $\operatorname{{Pent}}^0(\ell_0, \ldots, \ell_4)$ admits a unique representative with $a_0=(0,0)$ and $a_1=(\ell_0, 0)$, so as before $\operatorname{{Pent}}^0(\ell_0, \ldots, \ell_4)$ is a real algebraic variety, which is easily seen to be of dimension 2 ([@Curtis-Steiner; @Shimamoto-Vanderwaart]). When it is smooth, this is an example of K3 surface, and the five involutions $\sigma_i$ act by algebraic diffeomorphisms on this surface, preserving a canonically defined area form (see §\[par:pentagons\]). It can be shown that for a general choice of lengths, the group generated by these involutions generates a rich dynamics. Now, we naturally define a random dynamical system by starting with some pentagon $P$ and at every unit of time, apply one involution at random among the $\sigma_i$. In this way we obtain a random sequence of pentagons, and our results explain how this sequence is asymptotically distributed on $\operatorname{{Pent}}^0(\ell_0, \ldots, \ell_4)$. (The dynamics of the folding maps acting on plane quadrilaterals was studied for instance in [@esch-rogers; @benoist-hulin].)
Stiffness {#par:intro_stiffness}
---------
Let us present a few landmark results that shape our understanding of these problems. First, suppose that $\nu$ is a finitely supported probability measure on ${{\sf{SL}}}_2({\mathbf{C}})$, which we view as acting by projective linear transformations on $M = {\mathbb{P}}^1({\mathbf{C}})$. Suppose that the group $\Gamma_\nu$ generated by the support of $\nu$ is **non-elementary**, that is, $\Gamma_\nu$ is non-compact and acts strongly irreducibly on ${\mathbf{C}}^2$ (in the non-compact case, this means that $\Gamma_\nu$ does not have any orbit of cardinality $1$ or $2$ in ${\mathbb{P}}^1({\mathbf{C}})$). Then, *there is a unique $\nu$-stationary probability measure $\mu$ on ${\mathbb{P}}^1({\mathbf{C}})$, and this measure is not invariant*. This is one instance of a more general result due to Furstenberg [@Furstenber:1963].
Temporarily leaving the setting of diffeomorphisms, let us consider the semigroup of transformations of the circle ${\mathbf{R}}/{\mathbf{Z}}$ generated by $m_2$ and $m_3$, where $m_d(x) = dx \mod 1$. Since the multiplications by 2 and 3 commute, the so-called Choquet-Deny theorem asserts that any stationary measure is invariant. Furstenberg’s famous “$\times 2\times 3$ conjecture” asserts that any atomless probability measure $\mu$ invariant under $m_2$ and $m_3$ is the Lebesgue measure (see [@furstenberg-disjointness]). This question is still open so far, and has attracted a lot of attention. Rudolph [@rudolph] proved that the answer is positive when $\mu$ is of positive entropy with respect to $m_2$ or $m_3$.
Back to diffeomorphisms, let $\nu$ be a finitely supported measure on ${{\sf{SL}}}_2({\mathbf{Z}})$, and consider the action of ${{\sf{SL}}}_2({\mathbf{Z}})$ on the torus $M={\mathbf{R}}^2/{\mathbf{Z}}^2$. In that case, the Haar measure $dx\wedge dy$ of ${\mathbf{R}}^2/{\mathbf{Z}}^2$, as well as the atomic measures equidistributed on finite orbits $\Gamma_\nu(x,y)$, for $(x,y)\in {\mathbf{Q}}^2/{\mathbf{Z}}^2$, are examples of $\Gamma_\nu$-invariant measures. By using Fourier analysis and additive combinatorics techniques, Bourgain, Furman, Lindenstrauss and Mozes [@BFLM] proved that if $\Gamma_\nu$ is non-elementary, then *every stationary measure $\mu$ on ${\mathbf{R}}^2/{\mathbf{Z}}^2$ is $\Gamma_\nu$-invariant, and furthermore it is a convex combination of the above mentioned invariant measures.* This can be viewed as an affirmative answer to a non-Abelian version of the $\times2\times 3$ conjecture. This property of automatic invariance of stationary measures was called **stiffness** (or more precisely $\nu$-stiffness) by Furstenberg [@furstenberg_stiffness], who conjectured it to hold in this setting. Soon after, Benoist and Quint [@benoist-quint1] gave an ergodic theoretic proof of this result, which allowed them to extend the stiffness property to certain actions of discrete groups on homogeneous spaces. They also derived the following equidistribution result for the action of ${{\sf{SL}}}_2({\mathbf{Z}})$ on the torus: *for every $(x,y)\notin {\mathbf{Q}}^2/{\mathbf{Z}}^2$, the random trajectory of $(x,y)$ determined by $\nu$ almost surely equidistributes towards the Haar measure.*
Finally, Brown and Rodriguez-Hertz [@br], building on the work of Eskin and Mirzakhani [@eskin-mirzakhani], managed to recast these measure rigidity results in terms of smooth ergodic theory to obtain a version of the stiffness theorem of [@BFLM] for general $C^2$ diffeomorphisms of compact surfaces. The precise results of [@br] are not so easy to explain briefly, so for the moment we will content ourselves with a consequence of their work: as before, let $\nu = \sum\alpha_j\delta_{f_j}$ be a finitely supported probability measure on ${{\sf{SL}}}_2({\mathbf{Z}})$ and consider small perturbations ${\left\{f_{i, \varepsilon }\right\}}$ of the $f_i$ in the group ${{\mathsf{Diff}}}^2_{{{\sf{vol}}}}({\mathbf{R}}^2/{\mathbf{Z}}^2)$ of $C^2$ diffeomorphisms of ${\mathbf{R}}^2/{\mathbf{Z}}^2$ preserving the Haar measure. Set $\nu_{\varepsilon}= \sum\alpha_j\delta_{f_j, {\varepsilon}}$. Then if the perturbation is sufficiently small, the stiffness property still holds, that is: *any $\nu_{\varepsilon}$-stationary measure on ${\mathbf{R}}^2/{\mathbf{Z}}^2$ is invariant, and is a combination of the Haar measure and measures supported on finite $\Gamma_{\nu_{\varepsilon}}$-orbits*.
In this paper, we obtain a new generalization of the stiffness theorem of [@BFLM], for algebraic diffeomorphisms of real algebraic surfaces. Before entering into specifics, let us emphasize that the article [@br], by Brown and Rodriguez-Hertz, is our main source of inspiration and is a key ingredient for some of our main results.
Sample results: stiffness, classification, and rigidity
-------------------------------------------------------
Let $X$ be a smooth complex projective surface, or more generally a compact Kähler surface. Denote by ${\mathsf{Aut}}(X)$ its group of holomorphic diffeomorphisms, referred to in this paper as [**[automorphisms]{}**]{}. When $X\subset {\mathbb{P}}^N({\mathbf{C}})$ is defined by polynomial equations with real coefficients, the complex conjugation induces an anti-holomorphic involution $s\colon X\to X$, whose fixed point set is the real part of $X$: $$X({\mathbf{R}})={\mathrm{Fix}}(s)\subset X.$$ We denote by $X_{\mathbf{R}}$ the surface $X$ viewed as an algebraic variety defined over ${\mathbf{R}}$, and by ${\mathsf{Aut}}(X_{\mathbf{R}})$ the group of automorphisms defined over ${\mathbf{R}}$; ${\mathsf{Aut}}(X_{\mathbf{R}})$ coincides with the subgroup of ${\mathsf{Aut}}(X)$ that centralizes $s$. Equivalently, the elements of ${\mathsf{Aut}}(X_{\mathbf{R}})$ are the real-analytic diffeomorphisms of $X({\mathbf{R}})$ admitting a holomorphic extension to $X$. Note that in stark contrast with groups of smooth diffeomorphisms, the groups ${\mathsf{Aut}}(X_{\mathbf{R}})$ and ${\mathsf{Aut}}(X)$ are typically *discrete and countable*.
The group ${\mathsf{Aut}}(X)$ acts on the cohomology $H^*(X;{\mathbf{Z}})$. By definition, a subgroup $\Gamma\subset {\mathsf{Aut}}(X)$ is [**[non-elementary]{}**]{} if its image $\Gamma^*\subset {{\sf{GL}}}(H^*(X;{\mathbf{C}}))$ contains a non-abelian free group; equivalently, $\Gamma^*$ is not virtually abelian. When $\Gamma$ is non-elementary, there exists a pair $(f,g)\in \Gamma^2$ generating a free group of rank $2$ such that the topological entropy of every element in that group is positive (see Lemma \[lem:non-elementary\_free\_groups\]). Pentagon foldings provide examples for which ${\mathsf{Aut}}(X_{\mathbf{R}})$ is non-elementary.
Let $\nu$ be a finitely supported probability measure on ${\mathsf{Aut}}(X)$. As before we denote by $\Gamma_\nu$ the subgroup generated by $\operatorname{Supp}(\nu)$.
\[mthm:stiffness\] Let $X_{\mathbf{R}}$ be a real projective surface and $\nu$ be a finitely supported symmetric probability measure on ${\mathsf{Aut}}(X_{\mathbf{R}})$. If $\Gamma_\nu$ preserves an area form on $X({\mathbf{R}})$, then every ergodic $\nu$-stationary measure $\mu$ on $X({\mathbf{R}})$ is either invariant or supported on a proper $\Gamma_\nu$-invariant subvariety. In particular if there is no $\Gamma_\nu$-invariant algebraic curve, the random dynamical system $(X,\nu)$ is stiff.
This theorem is mostly interesting when $\Gamma_\nu$ is non-elementary and we will focus on this case in the remainder of this introduction.
Stationary measures supported on invariant curves are rather easy to analyse (see §\[par:invariant\_curves2\]). Moreover, it is always possible to contract all $\Gamma_\nu$-invariant curves, creating a complex analytic surface $X_0$ with finitely many singularities. Then on $X_0({\mathbf{R}})$, stiffness holds unconditionally.
This result applies to many interesting examples, because Abelian, K3, and Enriques surfaces, which concentrate most of the dynamically interesting automorphisms on compact complex surfaces, admit a canonical ${\mathsf{Aut}}(X)$-invariant $2$-form. In particular, it applies to the dynamics of pentagon foldings. Note also that linear Anosov maps on ${\mathbf{R}}^2/{\mathbf{Z}}^2$ fall into this category, so Theorem \[mthm:stiffness\] contains the stiffness statement of [@BFLM]. While not directly covered by this article, the character variety of the once punctured torus (or the four times punctured sphere) should be amenable to the same strategy (see [@Cantat:BHPS; @Goldman:1988; @Goldman:2003]).
Once stiffness is established, the next step is to *classify invariant measures*. When $X$ is a K3 surface and $\Gamma_\nu$ contains a **parabolic** automorphism, $\Gamma_\nu$-invariant measures were classified by the first named author in [@cantat_groupes]. A parabolic automorphism acts by translations along the fiber of some genus $1$ fibration with a shearing property between nearby fibers (see below §\[par:parabolic\_automorphisms\] for details). An example is given by the composition of the folding $\sigma_i$ and $\sigma_{i+1}$ of two adjacent vertices in the space of pentagons. In a companion paper [@invariant] we generalize and make more precise the results of [@cantat_groupes]. A nice consequence is that for a non-elementary group of ${\mathsf{Aut}}(X_{\mathbf{R}})$ containing parabolic elements and preserving an area form, any invariant measure is either atomic, or concentrated on a $\Gamma_\nu$-invariant algebraic curve, or is the restriction of the area form on some smoothly bounded open subset in $X({\mathbf{R}})$.
When specialized to random pentagon foldings, these results give a complete answer to the equidistribution problem raised in §\[par:RDS\_intro\]. Indeed, assume that the group generated by the five involutions $\sigma_i$ of $\operatorname{{Pent}}^0(\ell_0, \ldots, \ell_4)$ does not preserve any proper Zariski closed set, and that $\mathrm{Pent}^0(\ell_0, \ldots, \ell_4)$ is connected. Then the stiffness and classification theorems imply that the only stationary measure is the canonical area form. Therefore by Breiman’s law of large numbers, for every initial pentagon $P\in \operatorname{{Pent}}^0(\ell_0, \ldots, \ell_4)$ and almost every random sequence $(m_j)\in \{0, \ldots, 4\}^{\mathbf{N}}$, the random sequence of pentagons $P_n=(\sigma_{m_{n-1}}\circ \cdots \circ \sigma_{m_0})(P)$ equidistributes with respect to the area form. Thus, quantities like the asymptotic average of the diameter are given by explicit integrals of semi-algebraic functions, independently of the starting pentagon $P$.
Another family of examples which was previously studied in the literature is the family of **Wehler surfaces**. These are the smooth surfaces $X\subset {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ defined by an equation of degree $(2,2,2)$. Then for every index $i\in \{1,2,3\}$, one can consider the projection $\pi_{i}\colon X\to {\mathbb{P}}^1\times {\mathbb{P}}^1$ which “forgets the variable $x_i$” and denote by $\sigma_i$ the involution permuting the points in the fibers of this ramified $2$-to-$1$ cover.
Let $X_{\mathbf{R}}\subset {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ be a real Wehler surface such that $X({\mathbf{R}})$ is non empty. If $X_{\mathbf{R}}$ is generic, then:
1. the surface $X$ is a K3 surface and there is a unique (up to choosing an orientation of $X({\mathbf{R}})$) algebraic $2$-form ${{\sf{vol}}}_{X_{\mathbf{R}}}$ on $X({\mathbf{R}})$ such that $\int_{X({\mathbf{R}})}{{\sf{vol}}}_{X_{{\mathbf{R}}}}=1$;
2. the group ${\mathsf{Aut}}(X_{\mathbf{R}})$ is generated by the three involutions $\sigma_i$ and coincides with ${\mathsf{Aut}}(X)$; furthermore it preserves the probability measure defined by ${{\sf{vol}}}_{X_{\mathbf{R}}}$;
3. if $\nu$ is finitely supported and $\Gamma_\nu$ has finite index in ${\mathsf{Aut}}(X_{\mathbf{R}})$ then $(X({\mathbf{R}}), \nu)$ is stiff: the only $\nu$-stationary measures on $X({\mathbf{R}})$ are convex combinations of the probability measures defined by ${{\sf{vol}}}_{X_{\mathbf{R}}}$ on the connected components of $X({\mathbf{R}})$.
Here by generic we mean that the equation of $X$ belongs to the complement of at most countably many hypersurfaces in the set of polynomial equations of degree $(2,2,2)$ (see §\[par:Wehler\] for details). This result follows from Theorem \[mthm:stiffness\] together with Proposition \[pro:Wehler\_generic\] and Corollary \[cor:real\_invariant\]. Actually in Assertion (3) we only show in this paper that the $\nu$-stationary measures are convex combinations of volume forms on components of $X({\mathbf{R}})$, together with measures supported on finite orbits. The generic non-existence of finite orbits will be established in a forthcoming paper [@finite_orbits] dedicated to this topic. Also, the stiffness property holds for a much larger class of measures, including some measures with an infinite support for which $\Gamma_\nu$ has infinite index in ${\mathsf{Aut}}(X)$.
Without assuming the existence of parabolic elements in $\Gamma_\nu$ we establish a measure rigidity result in the spirit of Rudolph’s theorem on the $\times 2\times 3$ conjecture.
\[mthm:rigidity\] Let $X_{\mathbf{R}}$ be a real projective surface and $\Gamma$ a non-elementary subgroup of ${\mathsf{Aut}}(X_{\mathbf{R}})$. If all elements of $\Gamma$ preserve a probability measure $\mu$ supported on $X({\mathbf{R}})$ and if $\mu$ is ergodic and of positive entropy for some $f\in \Gamma$, then $\mu$ is absolutely continuous with respect to the area measure on $X({\mathbf{R}})$.
In particular if $\Gamma$ is a group of area preserving automorphisms, then up to normalization $\mu$ will be the restriction of the area form on some $\Gamma$-invariant set. **Kummer examples** are a generalization of linear Anosov maps of tori to other projective surfaces (see [@Cantat-Zeghib; @cantat-dupont] for more on such mappings). When $\Gamma$ contains a real Kummer example $f$, we can derive an exact analogue of the classification of invariant measures of [@BFLM], that is we can replace the assumption that $\mu$ has positive entropy by the fact that $\mu$ has no atoms (Theorem \[thm:rigidity\_kummer\]). We also obtain a version of Theorem \[mthm:rigidity\] for plane polynomial automorphisms (see Theorem \[thm:rigidity\_henon\]).
Some ingredients of the proofs
------------------------------
The proofs of Theorems \[mthm:stiffness\] and \[mthm:rigidity\] rely on the deep results of Brown and Rodriguez-Hertz [@br]. To be more precise, recall that an ergodic stationary measure $\mu$ on $X$ admits a pair of Lyapunov exponents $\lambda^+(\mu)\geq \lambda^-(\mu)$, and that $\mu$ is said [**[hyperbolic]{}**]{} if $\lambda^+(\mu)> 0 > \lambda^-(\mu)$. In this case the (random) Oseledets theorem shows that for $\mu$-almost every $x$ and $\nu^{\mathbf{N}}$-almost every $\omega = (f_j)_{j\in {\mathbf{N}}}$ in ${\mathsf{Aut}}(X)^{\mathbf{N}}$, there exists a stable direction $E^s_\omega (x)\subset
T_xX_{\mathbf{R}}$. In [@br], stiffness is established for area preserving $C^2$ random dynamical systems on surfaces, under the condition that the stable direction $E^s_\omega(x)\subset T_xX_{\mathbf{R}}$ depends non-trivially on the random itinerary $\omega = (f_j)_{j\in {\mathbf{N}}}$, or equivalently that stable directions do not induce a measurable $\Gamma_\nu$-invariant line field. One of our main contributions is to take care of this possibility in our setting: for this we study the dynamics on the [*complex*]{} surface $X$.
\[mthm:alternative\_stable\] Let $X$ be a compact projective surface and $\nu$ be a finitely supported probability measure on ${\mathsf{Aut}}(X)$. If $\Gamma_\nu$ is non-elementary, then any hyperbolic ergodic $\nu$-stationary measure $\mu$ on $X$ satisfies the following alternative:
1. either $\mu$ is invariant, and its fiber entropy $h_\mu(X;\nu)$ vanishes;
2. or $\mu$ is supported on a $\Gamma_\nu$-invariant algebraic curve;
3. or the field of Oseledets stable directions of $\mu$ is not $\Gamma_\nu$-invariant; in other words, it genuinely depends on the itinerary $(f_j)_{j\geq 0}\in {\mathsf{Aut}}(X)^{\mathbf{N}}$.
As opposed to Theorems \[mthm:stiffness\] and \[mthm:rigidity\], this result holds in full generality, without assuming the existence of an invariant volume form nor an invariant real structure. Understanding this somewhat technical result requires a substantial amount of material from the smooth ergodic theory of random dynamical systems, which will be introduced in due time. When $\mu$ is not invariant, nor supported by a proper Zariski closed subset, Assertion (c) precisely says that the above mentioned condition on stable directions used in [@br] is satisfied. This is our key input towards Theorems \[mthm:stiffness\] and \[mthm:rigidity\].
The arguments leading to Theorem \[mthm:alternative\_stable\] involve an interesting blend of Hodge theory, pluripotential analysis, and Pesin theory. They rely on the following well-known principle in higher dimensional holomorphic dynamics: if $\mu$ is an ergodic hyperbolic stationary measure, $\mu$-almost every point admits a Pesin stable manifold biholomorphic to ${\mathbf{C}}$; then, according to a classical construction going back to Ahlfors and Nevanlinna, to any immersion $\phi:{\mathbf{C}}\to X$ is associated a (family of) positive closed $(1,1)$-current(s) describing the asymptotic distribution of $\phi({\mathbf{C}})$ in $X$, hence also a cohomology class in $H^2(X, {\mathbf{R}})$. These currents provide a basic link between the infinitesimal dynamics along $\mu$ and the action of $\Gamma_\nu$ on $H^2(X;{\mathbf{R}})$, which itself can be analyzed by combining tools from complex algebraic geometry with Furstenberg’s theory of random products of matrices.
\[mthm:currents\] Let $X$ be a compact projective surface and $\nu$ be a finitely supported probability measure on ${\mathsf{Aut}}(X)$, such that $\Gamma_\nu$ is non-elementary. Let $\kappa_0$ be a fixed Kähler form on $X$.
1. If $\kappa$ is any Kähler form on $X$, then for $\nu^{\mathbf{N}}$-almost every $\omega:=(f_j)_{j\geq 0}\in {\mathsf{Aut}}(X)^{\mathbf{N}}$ the limit $$T_{\omega}^s:=\lim_{n\to +\infty} \frac{1}{\int_X \kappa_0 \wedge (f_n\circ \cdots \circ f_0)^*\kappa } (f_n \circ \cdots \circ f_0)^*\kappa$$ exists as a positive closed $(1,1)$-current. Moreover this current $T_\omega^s$ does not depend on $\kappa$ and has Hölder continuous potentials.
2. If the $\nu$-stationary measure $\mu$ is ergodic, hyperbolic and not supported on a $\Gamma_\nu$-invariant proper Zariski closed set, then for $\mu$-almost every $x$ and $\nu^{\mathbf{N}}$-almost every $\omega$, the only Ahlfors-Nevanlinna current of mass $1$ (with respect to $\kappa_0$) associated to the stable manifold $W^s_\omega(x)$ is $T_{\omega}^s$.
It could be expected that the right setting for a statement such as this one is that of a Kähler surface $X$. We actually show in §\[subs:X-is-projective\] that any Kähler surface supporting a non-elementary group of automorphisms is projective (see also Appendix \[par:appendix\_non\_kahler\] for the non-Kähler case). The algebraicity of $X$ is, in fact, a crucial technical ingredient in the proof of assertion (2), because we use techniques of laminar currents which are available only on projective surfaces. Theorem \[mthm:currents\] enters the proof of Theorem \[mthm:alternative\_stable\] as follows: since $\Gamma_\nu$ is non-elementary, Furstenberg’s description of the random action on $H^2(X, {\mathbf{R}})$ implies that the cohomology class $[T_{\omega}^s]$ depends non-trivially on $\omega$; therefore for $\mu$-almost every $x$, $W^s_\omega(x)$ also depends non-trivially on $\omega$.
Beyond finitely supported measures, our results hold under optimal moment conditions on $\nu$, which makes the presentation slightly more technical at some places (notably in Sections \[sec:furstenberg\] and \[sec:currents\]).
Organization of the article
---------------------------
Let $X$ be a compact Kähler surface and $\nu$ be a probability measure on ${\mathsf{Aut}}(X)$.
1. In Section \[par:Hodge\_Minkowski\] we describe the action of ${\mathsf{Aut}}(X)$ on the cohomology group $H^*(X;{\mathbf{Z}})$, and in particular the Dolbeault cohomology group $H^{1,1}(X;{\mathbf{R}})$. The Hodge index theorem endows it with a Minkowski structure, which is essential in our understanding of the dynamical properties of the action of $\Gamma_\nu$ on the cohomology. This section prepares the ground for the analysis of random products of matrices done in Section \[sec:furstenberg\]. A delicate point to keep in mind is that the action of a non-elementary subgroup of ${\mathsf{Aut}}(X)$ on $H^{1,1}(X;{\mathbf{R}})$ may not be strongly irreducible.
2. Section \[par:Examples\_Classification\] describes several classes of examples, including pentagon foldings and Wehler’s surfaces. It is also shown there that a Kähler surface with a non-elementary group of automorphims is necessarily projective.
3. After a short Section \[sec:Glossary\_I\] introducting the vocabulary of random products of diffeomorphisms, Furstenberg’s theory of random products of matrices is applied in Section \[sec:furstenberg\] to the action of $\nu$ on $H^{1,1}(X;{\mathbf{R}})$. This, combined with the theory of closed positive currents, leads to the proof of the first assertion of Theorem \[mthm:currents\] in Section \[sec:currents\]. The continuity of the potentials of the currents $T^s_\omega$, which plays a key role in the subsequent analysis of Section \[sec:nevanlinna\], relies on a recent result of Gouëzel and Karlsson [@Gouezel-Karlsson].
4. Pesin theory enters into play in Section \[sec:Glossary\_II\], in which the basics of the smooth ergodic theory of random dynamical systems (specialized to complex surfaces) are described in some detail. This is used in Section \[sec:nevanlinna\] to relate the Pesin stable manifolds to the currents $T_\omega^s$, using techniques of laminar currents.
5. Theorem \[mthm:alternative\_stable\] is proven in Section \[sec:No\_Invariant\_Line\_Fields\] by combining ideas of [@br], with Theorem \[mthm:currents\] and an elementary fact from local complex geometry inspired by a lemma of Bedford, Lyubich and Smillie [@bls].
6. Theorem \[mthm:stiffness\] is finally established in Section \[sec:stiffness\]. When $\Gamma_\nu$ is non-elementary (Theorem \[thm:stiffness\_real\]) it follows rather directly from [@br], Theorem \[mthm:alternative\_stable\], and a result of Avila and Viana [@avila-viana]. Elementary groups are handled separately by using the classification of automorphism groups of compact Kähler surfaces (see Theorems \[thm:stiffness\_elementary\_groups\] and Proposition \[pro:stiffness\_elementary2\]). Note that the symmetry of $\nu$ is used only in the elementary case.
7. Sections \[sec:parabolic\] and \[sec:rigidity\] are devoted to the classification of invariant measures. In Section \[sec:parabolic\], after recalling the results of [@cantat_groupes; @invariant], we show that when $\Gamma_\nu$ contains a parabolic element, any invariant measure giving no mass to subvarieties is hyperbolic. Our approach is inspired by the work of Barrientos and Malicet [@barrientos-malicet]. This provides an interesting connection with some classical problems in conservative dynamics (see §\[subs:hyperbolicity\] for a discussion). In Section \[sec:rigidity\] we prove Theorem \[mthm:rigidity\], as well as several related results. This relies on a measure rigidity theorem of [@br], together with ideas similar to the ones involved in the proof of Theorem \[mthm:alternative\_stable\].
This article is part of a series of papers in which we study random holomorphic dynamics on compact Kähler surfaces, in particular K3 and Enriques surfaces. The article [@invariant] is focused on the classification of invariant measures in presence of parabolic elements. In [@finite_orbits] we study the existence of finite orbits for non-elementary group actions; tools from arithmetic dynamics are used to study the case where $X$ and its automorphisms are defined over a number field. In a forthcoming work, we plan to extend the techniques of Brown and Rodriguez-Hertz to the complex setting. With Theorem \[mthm:alternative\_stable\] at hand, this would extend Theorem \[mthm:stiffness\] from the real to the complex case.
Conventions
-----------
Throughout the paper $C$ stands for a “constant" which may change from line to line, independently of some asymptotic quantity that should be clear from the context (typically an integer $n$ corresponding to the number of iterations of a dynamical system). Using this convention, we write $a\lesssim b$ if $a\leq Cb$ and $a\asymp b$ if $a\lesssim b \lesssim a$.
All complex manifolds are considered to be connected, so from now on “complex manifold” stands for “connected complex manifold”. For a random dynamical system on a disconnected complex manifold, there is a finite index sugbroup $\Gamma'$ of $\Gamma_\nu$ which stabilizes every connected component, and an induced measure $\nu'$ on $\Gamma'$ with properties qualitatively similar to those of $\nu$ (see §\[subs:inducing\]), so the problem is reduced to the connected case.
Acknowledgments
---------------
We are grateful to Sébastien Gouëzel and François Ledrappier for their insightful comments. The first named author was partially supported by a grant from the French Academy of Sciences (Del Duca foundation), and the second named author by a grant from the Institut Universitaire de France.
Hodge index theorem and Minkowski spaces {#par:Hodge_Minkowski}
========================================
In this section we define the notion of a non-elementary group action on a compact Kähler surface $X$. We study the action of a non-elementary subgroup of ${\mathsf{Aut}}(X)$ on the cohomology of $X$, and in particular the question of (ir)reducibilty. We refer to Appendix \[par:appendix\_non\_kahler\] for a discussion of the non-Kähler case.
Cohomology
----------
### Hodge decomposition {#par:Hodge_decomposition}
Denote by $H^*(X;R)$ the cohomology of $X$ with coefficients in the ring $R$; we shall use $R={\mathbf{Z}}$, ${\mathbf{Q}}$, ${\mathbf{R}}$ or ${\mathbf{C}}$. The group ${\mathsf{Aut}}(X)$ acts on $H^*(X;{\mathbf{Z}})$, and ${\mathsf{Aut}}(X)^*$ will denote the image of ${\mathsf{Aut}}(X)$ in ${{\sf{GL}}}(H^*(X;{\mathbf{Z}}))$. The Hodge decomposition $$H^{k}(X;{\mathbf{C}})=\bigoplus_{p+q=k} H^{p,q}(X;{\mathbf{C}})$$ is ${\mathsf{Aut}}(X)$-invariant. On $H^{0,0}(X;{\mathbf{C}})$ and $H^{2,2}(X;{\mathbf{C}})$, ${\mathsf{Aut}}(X)$ acts trivially. Throughout the paper we denote by $[\alpha]$ the cohomology class of a closed differential form (or current) $\alpha$.
The intersection form on $H^2(X;{\mathbf{Z}})$ will be denoted by $\langle\cdot \, \vert\, \cdot \rangle$; the self-intersection $\langle a \vert a \rangle$ of a class $a$ will also be denoted by $a^2$ for simplicity. This intersection form is ${\mathsf{Aut}}(X)$-invariant. By the Hodge index theorem, it is positive definite on the real part of $H^{2,0}(X;{\mathbf{C}})\oplus H^{0,2}(X;{\mathbf{C}})$ and it is non-degenerate and of signature $(1,h^{1,1}(X)-1)$ on $H^{1,1}(X;{\mathbf{R}})$.
\[lem:unitary\_on \_H20\] The restriction of ${\mathsf{Aut}}(X)^*$ to the subspace $H^{2,0}(X;{\mathbf{C}})$ (resp. $H^{0,2}(X;{\mathbf{C}})$) is contained in a compact subgroup of ${{\sf{GL}}}(H^{2,0}(X;{\mathbf{C}}))$ (resp. ${{\sf{GL}}}(H^{0,2}(X;{\mathbf{C}}))$).
This follows from the fact that $\langle \cdot \vert \cdot \rangle$ is positive definite on the real part of $H^{2,0}(X;{\mathbf{C}})\oplus H^{0,2}(X;{\mathbf{C}})$. An equivalent way to describe this argument it to identify $H^{2,0}(X;{\mathbf{C}})$ with the space of holomorphic $2$-forms on $X$. Then, there is a natural, ${\mathsf{Aut}}(X)$-invariant, hermitian form on this space: given two holomorphic $2$-forms $\Omega_1$ and $\Omega_2$, the hermitian product is the integral $$\int_X\Omega_1\wedge\overline{\Omega_2}.$$ Thus, the image of ${\mathsf{Aut}}(X)$ in ${{\sf{GL}}}(H^{2,0}(X;{\mathbf{C}}))$ is relatively compact.
The **Néron-Severi group** ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ is, by definition, the discrete subgroup of $H^{1,1}(X;{\mathbf{R}})$ defined by ${{\mathrm{NS}}}(X;{\mathbf{Z}}) =H^{1,1}(X;{\mathbf{R}})\cap H^{2}(X;{\mathbf{Z}})$; more precisely, it is the intersection of $H^{1,1}(X;{\mathbf{R}})$ with the image of $H^{2}(X;{\mathbf{Z}})$ in $H^{2}(X;{\mathbf{R}})$, i.e. with the torsion free part of the abelian group $H^{2}(X;{\mathbf{Z}})$. The Lefschetz theorem on $(1,1)$-classes identifies ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ with the subgroup of $H^{1,1}(X;{\mathbf{R}})$ given by Chern classes of line bundles on $X$. The Néron-Severi group is ${\mathsf{Aut}}(X)$-invariant, as well as ${{\mathrm{NS}}}(X;R):={{\mathrm{NS}}}(X;{\mathbf{Z}})\otimes_{\mathbf{Z}}R$ for $R={\mathbf{Q}}$, ${\mathbf{R}}$, or ${\mathbf{C}}$. The dimension of ${{\mathrm{NS}}}(X;{\mathbf{R}})$ is the **Picard number** $\rho(X)$.
### Norm of $f^*$
Let ${\left\vert \cdot \right\vert}$ be any norm on the vector space $H^*(X;{\mathbf{C}})$. If $L$ is a linear transformation of $H^*(X;{\mathbf{C}})$ we denote by ${{\left\VertL\right\Vert}}$ the associated operator nom and if $W\subset H^*(X;{\mathbf{C}})$ is an $L$-invariant subspace of $H^*(X;{\mathbf{C}})$, we denote by ${{\left\Vert L\right\Vert}}_W$ the operator norm of $L{ \arrowvert_{ W}}$.
If $u$ is an element of $H^{1,0}(X;{\mathbf{C}})$, then $u\wedge {\overline{u}}$ is an element of $H^{1,1}(X;{\mathbf{R}})$ such that ${\left\vert u \right\vert}^2 \leq C {\left\vert u\wedge {\overline{u}}\right\vert} $ for some constant $C$ that depends only on the choice of norm on the cohomology; in particular, the norm of $f^*$ acting on $H^{1,0}(X;{\mathbf{C}})$ is controlled by that of $f^*$ acting on $H^{1,1}(X;{\mathbf{C}})$. Using complex conjugation, the same results hold on $H^{0,1}(X;{\mathbf{C}})$; by Poincaré duality we also control ${{\left\Vert f^*\right\Vert}}_{H^{p,q}(X;{\mathbf{C}})}$ for $p+q>2$. Together with Lemma \[lem:unitary\_on \_H20\], we obtain:
\[lem:cohomological\_norm\_estimates\] Let $X$ be a compact Kähler surface. There exists a constant $C_0>1$ such that $$C_0^{-1} {{\left\Vert f^* \right\Vert}}_{H^*(X;{\mathbf{C}})} \leq {{\left\Vert f^*\right\Vert}}_{H^{1,1}(X;{\mathbf{R}})} \leq {{\left\Vert f^*\right\Vert}}_{H^*(X;{\mathbf{C}})}$$ for every automorphism $f\in {\mathsf{Aut}}(X)$.
The Kähler, nef, and pseudo-effective cones {#par:cones_definition}
-------------------------------------------
(See [@Boucksom:ENS; @Lazarsfeld:Book1] for details on the notions introduced in this section.)
Let ${{\mathrm{Kah}}}(X)\subset H^{1,1}(X;{\mathbf{R}})$ be the [**[Kähler cone]{}**]{}, i.e. the cone of classes of Kähler forms. Its closure ${{\overline{{\mathrm{Kah}}}}}(X)$ is a salient, closed, convex cone, and $${{\mathrm{Kah}}}(X)\subset {{\overline{{\mathrm{Kah}}}}}(X)\subset\{ v\in H^{1,1}(X;{\mathbf{R}})\; ; \; \langle v \,\vert\, v \rangle \geq 0\}.$$ The intersection ${{\mathrm{NS}}}(X;{\mathbf{R}})\cap {{\mathrm{Kah}}}(X)$ is the [**[ample cone]{}**]{} ${{\mathrm{Amp}}}(X)$, while ${{\mathrm{NS}}}(X;{\mathbf{R}})\cap {{\overline{{\mathrm{Kah}}}}}(X)$ is the [**[nef cone]{}**]{} ${{\mathrm{Nef}}}(X)$. They are all invariant under the action of ${\mathsf{Aut}}(X)$ on $H^{1,1}(X;{\mathbf{R}})$. We shall also say that the elements of ${{\overline{{\mathrm{Kah}}}}}(X)$ are nef classes, but the notation ${{\mathrm{Nef}}}(X)$ will be reserved for ${{\mathrm{NS}}}(X;{\mathbf{R}})\cap {{\overline{{\mathrm{Kah}}}}}(X)$.
The set of classes of closed positive currents is the [**[pseudo-effective cone]{}**]{} ${{\mathrm{Psef}}}(X)$. This cone is an ${\mathsf{Aut}}(X)$-invariant, salient, closed, convex cone. It is dual to ${{\overline{{\mathrm{Kah}}}}}(X)$ for the intersection form (see [@Boucksom:ENS Lem. 4.1]): $$\label{eq:cone_duality}
{{\overline{{\mathrm{Kah}}}}}(X)=\{ u \in H^{1,1}(X;R)\; ; \; \langle u \,\vert\, v\rangle\geq 0 \quad \forall v \in {{\mathrm{Psef}}}(X)\}$$ and vice-versa. We fix once and for all a reference Kähler form $\kappa_0$ with $[\kappa_0]^2= \int \kappa_0\wedge \kappa_0 =1$. Then we define the **mass** of a pseudo-effective class $a$ by ${{\mathbf{M}}}(a) = \langle a\,\vert\, [\kappa_0] \rangle$, or equivalently the mass of a positive closed current $T$ by ${{\mathbf{M}}}(T) = \int T\wedge \kappa_0$. The compactness of the set of positive closed currents of mass 1 implies that, for any norm ${\left\vert\cdot\right\vert}$ on $H^{1,1}(X, {\mathbf{R}})$, there exists a constant $C$ such that $$\label{eq:comparison_norm_mass}
\text{for every }a\in {{\mathrm{Psef}}}(X), \; \ C{^{-1}}{\left\verta\right\vert}\leq {{\mathbf{M}}}(a)\leq C {\left\verta\right\vert}.$$
If $v$ is an element of ${{\mathrm{Psef}}}(X)$ and $v^2\geq 0$, then by the Hodge index theorem we know that $\langle u \,\vert\, v\rangle\geq 0$ for every class $u\in H^{1,1}(X;{\mathbf{R}})$ such that $u^2\geq 0$ and $\langle u\,\vert\, [\kappa_0]\rangle\geq 0$ (see Equation ). So, in Equation , the most important constraints come from the classes $v\in {{\mathrm{Psef}}}(X)$ with $v^2<0$. If $v$ is such a class, its Zariski decomposition expresses $v$ as a sum $v=p(v)+n(v)$ with the following property (see [@Boucksom:ENS]):
1. this decomposition is orthogonal: $\langle p(v) \,\vert\, n(v)\rangle = 0$;
2. $p(v)$ is a nef class, i.e. $p(v)\in {{\overline{{\mathrm{Kah}}}}}(X)$;
3. $n(v)$ is negative: it is a sum $n(v)=\sum_i a_i [D_i]$ with positive coefficients $a_i\in {\mathbf{R}}_+^*$ of classes of irreducible curves $D_i\subset X$ such that the Gram matrix $(\langle D_i\,\vert\, D_j\rangle)$ is negative definite.
\[pro:extremal\_rays\] If a ray ${\mathbf{R}}_+ v$ of the cone ${{\mathrm{Psef}}}(X)$ is extremal, then either $v^2\geq 0$ or ${\mathbf{R}}_+ v={\mathbf{R}}_+[D]$ for some irreducible curve $D$ such that $D^2<0$. The cone ${{\mathrm{Psef}}}(X)$ contains at most countably many extremal rays ${\mathbf{R}}_+v$ with $v^2<0$.
Let $u$ be an isotropic element of ${{\overline{{\mathrm{Kah}}}}}(X)$. If ${\mathbf{R}}_+ u$ is not an extremal ray of ${{\mathrm{Psef}}}(X)$, then $u$ is proportional to an integral class $u'\in {{\mathrm{NS}}}(X)$.
If ${\mathbf{R}}_+ v$ is extremal, the Zariski decomposition $v=p(v)+n(v)$ involves only one term. If $v=p(v)$ then $v^2\geq 0$. Otherwise $v=n(v)$ and by extremality $n(v)=a[D]$ for some irreducible curve $D$ with $D^2<0$. The countability assertion follows, because ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ is countable.
For the last assertion, multiply $u$ by $\langle u \vert [\kappa_0]\rangle^{-1}$ to assume $\langle u \vert [\kappa_0]\rangle=1$ and write $u$ as a convex combination $u=\int v \, d\alpha(v)$, where $\alpha$ is a probability measure on ${{\mathrm{Psef}}}(X)$ such that $\alpha$-almost every $v$ satisfies
- $\langle v \vert [\kappa_0]\rangle =1$,
- ${\mathbf{R}}_+ v$ is extremal in ${{\mathrm{Psef}}}(X)$ and does not contain $u$.
Since $u$ is nef, $\langle u\,\vert\, v \rangle \geq 0$ for each $v$; and $u$ being isotropic, we get $v\in u^\perp\setminus {\mathbf{R}}u$ for $\alpha$-almost every $v$. By the Hodge index theorem, $v^2<0$ almost surely. Now, the first assertion of this proposition implies that $v\in {\mathbf{R}}_+[D_v]$ for some irreducible curve $D_v\subset X$ with negative self-intersection; there are only countably many classes of that type, thus $\alpha$ is purely atomic, and $u$ belongs to $\operatorname{Vect}([D_v]; \alpha(v)>0)$, which is a subspace of ${{\mathrm{NS}}}(X;{\mathbf{R}})$ defined over ${\mathbf{Q}}$. On this subspace, the intersection form $q_X$ is semi-negative, and by the Hodge index theorem its kernel is ${\mathbf{R}}u$. Since $\operatorname{Vect}([D_v]; \alpha(v)>0)$ and $q_X$ are defined over ${\mathbf{Q}}$, we deduce that $u$ is proportional to an integral class.
Non-elementary subgroups of ${\mathsf{Aut}}(X)$
-----------------------------------------------
When $X$ is a compact Kähler surface, the action of ${\mathsf{Aut}}(X)$ on $H^{1,1}(X, {\mathbf{R}})$ is subject to several constraints: the Hodge index theorem implies that it must preserve a Minkowski structure and in addition it preserves the lattice given by the Neron-Severi group. In this section we review the first consequences of these constraints.
### Isometries of Minkowski spaces
Consider the Minkowski space ${\mathbf{R}}^{m+1}$, endowed with its quadratic form $q$ of signature $(1,m)$ defined by $$q(x)=x_0^2-\sum_{i=1}^mx_i^2.$$ The corresponding bilinear form will be denoted $\langle \cdot \vert \cdot \rangle$. For future reference, note the following reverse Schwarz inequality: $$\label{eq:reverse_CS}
\text{ if } \; q(x)\geq 0 \text{ and } q(x')\geq 0 \; \text{ then } \; \langle{x\,\vert\, x'}\rangle \geq q(x)^{1/2} q(x')^{1/2}$$ with equality if and only if $x$ and $x'$ are collinear. We say that a subspace $W\subset {\mathbf{R}}^{m+1}$ is of [**[Minkowski type]{}**]{} if the restriction $q_{\vert W}$ is non-degenerate and of signature $(1,\dim(W)-1)$.
In this section, we review some well-known facts concerning isometries of ${\mathbf{R}}^{1,m}=({\mathbf{R}}^{m+1},q)$ (see e.g. [@Ratcliffe; @kapovich; @franchi-lejan] for more details). We denote by ${\left\vert\cdot\right\vert}$ the Euclidean norm on ${\mathbf{R}}^{m+1}$, and by ${\mathbb{P}}\colon {\mathbf{R}}^{m+1}\setminus\{ 0\}\to {\mathbb{P}}( {\mathbf{R}}^{m+1})$ the projection on the projective space ${\mathbb{P}}( {\mathbf{R}}^{m+1})={\mathbb{P}}^m({\mathbf{R}})$.
The hyperboloid ${\left\{x \; ; \; q(x)=1\right\}}$ has two components, and we denote by ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$ the subgroup of the orthogonal group ${{\sf{O}}}_{1,m}({\mathbf{R}})$ that preserves the component $\mathcal Q =\{ q(x)=1\; ; \; x_0 >0\}$. It is well known that $\mathcal Q$, endowed with the distance $d_{\mathbb{H}}(x,y)= \cosh{^{-1}}\langle x \,\vert\, y\rangle $ is a model of the real hyperbolic space ${\mathbb{H}}^m$ of dimension $m$. The boundary at infinity of ${\mathbb{H}}^m$ will be identified with $\partial{\mathbb{P}}(\mathcal Q)\subset {\mathbb{P}}({\mathbf{R}}^{m+1})$ and will be denoted by ${\partial}{\mathbb{H}}^m$. It is the set of isotropic lines of $q$.
Any isometry $\gamma$ of ${\mathbb{H}}^m$ is induced by an element of ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$, and extends continuously to ${\partial}{\mathbb{H}}^m$: its action on ${\partial}{\mathbb{H}}^m$ is given by its linear projective action on ${\mathbb{P}}({\mathbf{R}}^{m+1})$. Isometries are classified in three types, according to their fixed point set in ${\mathbb{H}}^m \cup {\partial}{\mathbb{H}}^m$:
- $\gamma$ is [**[elliptic]{}**]{} if $\gamma$ has a fixed point in ${\mathbb{H}}^m$;
- $\gamma$ is [**[parabolic]{}**]{} if $\gamma$ has no fixed point in ${\mathbb{H}}^m$ and a unique fixed point in ${\partial}{\mathbb{H}}^m$;
- $\gamma$ is [**[loxodromic]{}**]{} if $\gamma$ has no fixed point in ${\mathbb{H}}^m$ and exactly two fixed points in ${\partial}{\mathbb{H}}^m$.
A subgroup $\Gamma$ of ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$ is [**[non-elementary]{}**]{} if it does not preserve any finite subset of ${\mathbb{H}}^m\cup \partial{\mathbb{H}}^m$. Equivalently $\Gamma$ is non-elementary if and only if it contains two loxodromic elements with disjoint fixed point sets.
The group ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$ admits a **Cartan** or **KAK decomposition** (see [@franchi-lejan §I.5]). To state it, denote by $e_0=(1,0, \ldots, 0)$ the first vector of the canonical basis of ${\mathbf{R}}^{m+1}$; this vector is an element of ${\mathbb{H}}^m$, and its stabilizer $\mathrm{Stab}(e_0)$ in ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$ is a maximal compact subgroup, isomorphic to ${{\sf{O}}}_{m-1}({\mathbf{R}})$.
\[lem:KAK\] Every $\gamma\in{{\sf{O}}}^+_{1,m}({\mathbf{R}}) $ can be written (non-uniquely) as $\gamma= k_1 a k_2$, where $k_i\in \mathrm{Stab}({{\mathbf{e}}}_0)$ and $a$ is a matrix of the form $$\begin{pmatrix} \cosh r & \sinh r & 0\\ \sinh r & \cosh r & 0 \\ 0 & 0 & \operatorname{id}_{m-1} \end{pmatrix}$$ with $r=d_{\mathbb{H}}(e_0, \gamma e_0)$.
Note that $K:=\mathrm{Stab}(e_0)$ acts transitively on the set of hyperbolic geodesics through $e_0$. Denote by $L$ the hyperbolic geodesic ${\mathbb{H}}^m\cap \operatorname{Vect}(e_0, e_1)$, where $e_1=(0,1, 0, \ldots, 0)$ is the second element of the canonical basis of ${\mathbf{R}}^{m+1}$. If $\gamma(e_0) = e_0$ then $\gamma$ belongs to $K$ and we are done. Otherwise let $k_1,k_2\in K$ such that $k_1{^{-1}}(\gamma(e_0))\in L$, $k_2(\gamma{^{-1}}(e_0))\in L$, and $e_0$ lies in between $k_2(\gamma{^{-1}}(e_0))$ and $k_1{^{-1}}(\gamma(e_0))$; then $e_0$ is in fact the middle point of $[k_2(\gamma{^{-1}}(e_0)), k_1{^{-1}}(\gamma(e_0))]$ because $d_{\mathbb{H}}(e_0, \gamma(e_0)) = d_{\mathbb{H}}(e_0, \gamma{^{-1}}(e_0))>0$. Then $a:=k_1{^{-1}}\gamma k_2{^{-1}}$ maps $k_2(\gamma{^{-1}}(e_0))\in L$ to $e_0$ and $e_0$ to $k_1{^{-1}}(\gamma(e_0))\in L$. It follows that $a$ is a hyperbolic translation along $L$ of translation length $d_{\mathbb{H}}(e_0, k_1{^{-1}}(\gamma(e_0)) = d_{\mathbb{H}}(e_0, \gamma(e_0))$. To conclude, change $a$ into $a\circ k{^{-1}}$ and $k_2$ into $k\circ k_2$ where $k$ is the element of $K$ that preserves $e_1$ and acts like $a$ on the orthogonal complement of $\operatorname{Vect}(e_0, e_1)$.
\[cor:KAK\] If ${{\left\Vert\cdot\right\Vert}}$ denotes the operator norm associated to the euclidean norm in ${\mathbf{R}}^{m+1}$, then ${{\left\Vert\gamma\right\Vert}} = {{\left\Verta\right\Vert}}$, where $\gamma =k_1 a k_2$ is any Cartan decomposition of $\gamma$. In particular $ {{\left\Vert\gamma\right\Vert}} = {{\left\Vert\gamma{^{-1}}\right\Vert}}$ and $${{\left\Vert\gamma\right\Vert}} \asymp \cosh d_{\mathbb{H}}(e_0, \gamma(e_0)) \asymp {\left\vert\gamma e_0\right\vert}.$$ Furthermore for every $e\in {\mathbb{H}}^m$ and any $\gamma\in{{\sf{O}}}^+_{1,m}({\mathbf{R}}) $ $${{\left\Vert\gamma\right\Vert}} \asymp \cosh d_{\mathbb{H}}(e, \gamma(e)),$$ where the implied constant depends only on the base point $e$.
This is an immediate corollary of the previous lemma.
### Irreducibility
A non-elementary subgroup of ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$ does not need to act irreducibly on ${\mathbf{R}}^{m+1}$. Proposition \[pro:invariant\_cohomological\_decomposition\], below, clarifies the possible situations.
\[lem:restriction\_isom\] Let $\Gamma$ be a non-elementary subgroup of ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$ (resp. $\gamma$ be an element of ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$). Let $W$ be a subspace of ${\mathbf{R}}^{1,m}$.
1. If $W$ is $\Gamma$-invariant, then either $(W, q{ \arrowvert_{ W}})$ is a Minkowski space and $\Gamma{ \arrowvert_{ W}}$ is non-elementary, or $q{ \arrowvert_{ W}}$ is negative definite and $\Gamma{ \arrowvert_{ W}}$ is contained in a compact subgroup of ${{\sf{GL}}}(W)$.
2. If $W$ is $\gamma$-invariant and contains a vector $w$ with $q(w)>0$, then $\gamma{ \arrowvert_{ W}}$ has the same type (elliptic, parabolic, or loxodromic) as $\gamma$; in particular, $W$ contains the $\gamma$-invariant isotropic lines if $\gamma$ is parabolic or loxodromic.
The restriction $q{ \arrowvert_{W}}$ is either a Minkowski form or is negative definite. Indeed, it cannot be positive definite, because $W$ would then be a $\Gamma$-invariant line intersecting the hyperbolic space ${\mathbb{H}}^m$ in a fixed point; and it cannot be degenerate, since otherwise its kernel would give a $\Gamma$-invariant point on $\partial {\mathbb{H}}^m$. If $q{ \arrowvert_{ W}}$ is a Minkowski form and $\Gamma{ \arrowvert_{ W}}$ is elementary, then $\Gamma$ preserves a finite subset of $({\mathbb{H}}^m\cup\partial{\mathbb{H}}^m) \cap V$ and $\Gamma$ itself is elementary. This proves the first assertion. The proof of the second one is similar.
Let $\Gamma$ be a non-elementary subgroup of ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$. Let ${{\mathsf{Zar}}}(\Gamma)\subset {{\sf{O}}}_{1,m}({\mathbf{R}})$ be the Zariski closure of $\Gamma$, and $$G={{\mathsf{Zar}}}(\Gamma)^\mathrm{irr}$$ the neutral component of ${{\mathsf{Zar}}}(\Gamma)$, for the Zariski topology. Note that the Lie group $G({\mathbf{R}})$ is not necessarily connected for the euclidean topology.
\[lem:finite-index\] The group $\Gamma\cap G({\mathbf{R}})$ has finite index in $\Gamma$. If $\Gamma_0$ is a finite index subgroup of $\Gamma$, then ${{\mathsf{Zar}}}(\Gamma_0)^\mathrm{irr}=G$.
The index of $G$ in ${{\mathsf{Zar}}}(\Gamma)$ is equal to the number $\ell$ of irreducible components of the algebraic variety ${{\mathsf{Zar}}}(\Gamma)$, and the index of $\Gamma\cap G({\mathbf{R}})$ in $\Gamma$ is at most $ \ell$. Now, let $\Gamma_0$ be a finite index subgroup of $\Gamma$. Then, $\Gamma_0\cap G({\mathbf{R}})$ has finite index in $\Gamma\cap G({\mathbf{R}})$, and we can fix a finite subset $\{\alpha_1, \dots , \alpha_k\}\subset \Gamma\cap G({\mathbf{R}})$ such that $\Gamma\cap G({\mathbf{R}})=\bigcup_j\alpha_j(\Gamma_0\cap G({\mathbf{R}}))$. So $${{\mathsf{Zar}}}(\Gamma\cap G({\mathbf{R}})) \subset
\bigcup_j\alpha_j{{\mathsf{Zar}}}(\Gamma_0\cap G({\mathbf{R}}))\subset G({\mathbf{R}}).$$ Because $\Gamma\cap G({\mathbf{R}})$ is Zariski dense in the irreducible group $G$ we find $G =
{{\mathsf{Zar}}}(\Gamma_0\cap G({\mathbf{R}}))$. So $G\subset
{{\mathsf{Zar}}}(\Gamma_0)$ and the Lemma follows as $G = {{\mathsf{Zar}}}(\Gamma)^\mathrm{irr}$.
\[pro:invariant\_cohomological\_decomposition\] Let $\Gamma\subset {{\sf{O}}}^+_{1,m}({\mathbf{R}})$ be non-elementary.
1. The representation of $\Gamma\cap G({\mathbf{R}})$ (resp. of $G({\mathbf{R}})$) on ${\mathbf{R}}^{1,m}$ splits as a direct sum of irreducible representations, with exactly one irreducible factor of Minkowski type: $${\mathbf{R}}^{1,m}=V_+\oplus V_0;$$ here $V_+$ is of Minkowski type, and $V_0$ is an orthogonal sum of irreducible representations $V_{0,j}$ on which the quadratic form $q$ is negative definite.
2. The restriction $G{ \arrowvert_{ V_+}}$ coincides with ${{\sf{SO}}}(V_+;q{ \arrowvert_{ V_+}})$.
3. The subspaces $V_+$ and $V_0$ are $\Gamma$-invariant, and the representation of $\Gamma$ on $V_+$ is strongly irreducible.
A group $\Gamma$ is non-elementary if and only if any of its finite index subgroups is non-elementary. So, we can apply Lemma \[lem:restriction\_isom\] to $\Gamma\cap G({\mathbf{R}})$: if $W\subset {\mathbf{R}}^{1,m}$ is a non-trivial $(\Gamma\cap G({\mathbf{R}}))$-invariant subspace, $q{ \arrowvert_{ W}}$ is non-degenerate. As a consequence, ${\mathbf{R}}^{1,m}$ is the direct sum $W\oplus W^\perp$, where $W^\perp$ is the orthogonal complement of $W$ with respect to $q$. This implies that the representation of $\Gamma\cap G({\mathbf{R}})$ on ${\mathbf{R}}^{1,m}$ splits as a direct sum of irreducible representations, with exactly one irreducible factor of Minkowski type, as asserted in (1).
The group $G$ preserves this decomposition, and by Proposition 1 of [@Benoist-Harpe], the restriction $G{ \arrowvert_{ V_+}}$ coincides with ${{\sf{SO}}}(V_+;q{ \arrowvert_{ V_+}})$; this group is isomorphic to the almost simple group ${{\sf{SO}}}_{1,k}({\mathbf{R}})$, with $1+k=\dim(V_+)$. This proves the second assertion.
Since $G$ is normalized by $\Gamma$, we see that for any $\gamma\in \Gamma$, $\gamma V^+$ is a $G$-invariant subspace of the same dimension as $V^+$ and on which $q$ is of Minkowski type. Hence $V_+$, as well as its orthogonal complement $V_0$ are $\Gamma$-invariant. By Lemma \[lem:finite-index\], the action of $\Gamma$ on $V_+$ is strongly irreducible; indeed, if a finite index subgroup $\Gamma_0$ in $\Gamma$ preserves a non-trivial subspace of $V_+$ then, by Zariski density of $\Gamma_0\cap G({\mathbf{R}})$ in $G({\mathbf{R}})$, this subspace must be $V_+$ itself. On $V_0$, $\Gamma$ permutes the irreducible factors $V_{0,j}$.
Now, set $V={\mathbf{R}}^{1,m}$ and assume that there is a lattice $V_{\mathbf{Z}}\subset V$ such that
- $V_{\mathbf{Z}}$ is $\Gamma$-invariant;
- the quadratic form $q$ is an integral quadratic form on $V_{\mathbf{Z}}$.
In other words, there is a basis of $V$ with respect to which $q$ and the elements of $\Gamma$ are given by matrices with integer coefficients. In particular, $V$ has a natural ${\mathbf{Q}}$-structure, with $V({\mathbf{Q}})=V_{\mathbf{Z}}\otimes_{\mathbf{Z}}{\mathbf{Q}}$. This situation naturally arises for the action of automorphisms of compact Kähler surfaces on ${{\mathrm{NS}}}(X;{\mathbf{R}})$. The next lemma will be useful in [@finite_orbits].
\[lem:decomposition\_V+V0\_rational\] If $\Gamma$ contains a parabolic element, the decomposition $V_+\oplus V_0$ is defined over ${\mathbf{Q}}$, $\Gamma{ \arrowvert_{ V_0}}$ is a finite group, and $G$ is the subgroup ${{\sf{SO}}}(V_+;q)\times \{\operatorname{id}_{V_0}\}$ of ${{\sf{O}}}(V;q)$.
If $\gamma$ is parabolic, it fixes pointwise a unique isotropic line, therefore this line is defined over ${\mathbf{Q}}$. In addition it must be contained in $V_+$ because $(\gamma^n(u))_{n\geq 0}$ converges to the boundary point determined by this line for every $u\in {\mathbb{H}}^m$. So, $V_+$ contains at least one non-zero element of $V_{\mathbf{Z}}$. Since the action of $\Gamma$ on $V_+$ is irreducible, the orbit of this vector generates $V_+$ and is contained in $V_{\mathbf{Z}}$, so $V_+$ is defined over ${\mathbf{Q}}$. Its orthogonal complement $V_0$ is also defined over ${\mathbf{Q}}$, because $q$ itself is defined over ${\mathbf{Q}}$. As a consequence, $\Gamma{ \arrowvert_{V_0}}$ preserves the lattice $V_0\cap V_{\mathbf{Z}}$ and the negative definite form $q{ \arrowvert_{ V_0}}$; hence, it is finite. Thus $G{ \arrowvert_{ V_0}}$ is trivial and the last assertion follows from the above mentionned equality $G{ \arrowvert_{ V_+}}={{\sf{SO}}}(V_+;q{ \arrowvert_{ V_+}})$.
\[eg:square\_free\] The purpose of this example is to show that the existence of a parabolic element in $\Gamma$ is indeed necessary in Lemma \[lem:decomposition\_V+V0\_rational\], even for a group of automorphisms of a K3 surface.
Let $a$ be a positive square free integer, for instance $a=7$ or $15$. Let $\alpha$ be the positive square root $\sqrt{a}$, $K$ be the quadratic field ${\mathbf{Q}}(\alpha)$, and $\eta$ be the unique non-trivial automorphism of $K$, sending $\alpha$ to its conjugate ${\overline{\alpha}}:=\eta(\alpha)=-\sqrt{a}$. We view $\eta$ as a second embedding of $K$ in ${\mathbf{C}}$. Let ${\mathcal O}_K$ be the ring of integers of $K$.
Let $\ell$ be an integer $\geq 2$. Consider the quadratic form in $\ell+1$ variables defined by $$q_\ell (x_0, x_1, \ldots, x_\ell) = \alpha x_0^2- x_1^2- \cdots - x_\ell^2.$$ It is non-degenerate and its signature is $(1,\ell)$. The orthogonal group ${{\sf{O}}}(q_\ell; {\mathcal O}_K)$ is a lattice in the real algebraic group ${{\sf{O}}}(q_\ell, {\mathbf{R}})$. The conjugate quadratic form ${\overline{q_\ell}} = {\overline{\alpha}} x_0^2- x_1^2- \cdots - x_\ell^2$ is negative definite.
Now, embed ${\mathcal O}_K^{\ell + 1}$ into ${\mathbf{R}}^{2\ell +2}$ by the map $(x_i)\mapsto (x_i, \eta(x_i))$, to get a lattice $\Lambda \subset {\mathbf{R}}^{2\ell +2}$ and consider the quadratic form $Q_\ell:=q_\ell \oplus {\overline{q_\ell}}$. Then embed ${{\sf{O}}}(q_\ell; {\mathcal O}_K)$ into ${{\sf{O}}}(Q_\ell;{\mathbf{R}})$ by the homomorphism $A\in {{\sf{O}}}(q_\ell, {\mathcal O}_K)\mapsto A\oplus \eta(A)$; we denote its image by $\Gamma_\ell^*\subset {{\sf{O}}}(Q_\ell;{\mathbf{R}})$. It can shown that $Q_\ell$ is defined over ${\mathbf{Z}}$ with respect to $\Lambda$, $\Gamma_\ell^*\subset {{\sf{O}}}(Q_\ell; {\mathbf{Z}})$, and the group $G={{\mathsf{Zar}}}(\Gamma_\ell^*)^\mathrm{irr}$ coincides with ${{\sf{SO}}}^0(q_\ell;{\mathbf{R}})\times {{\sf{SO}}}^0(q_\ell;{\mathbf{R}})$ (the group $\eta({{\sf{O}}}(q_\ell; {\mathcal O}_K))$ is dense in the compact group ${{\sf{O}}}({\overline{q_\ell}};{\mathbf{R}})$). We refer to [@Morris:Book-Lattices], Chapter 6.4, for a proof.
Now, assume $2\leq \ell \leq 4$, so that $2\ell + 2 \leq 10$, and change $Q_\ell$ into $4Q_\ell$: it is an even quadratic form on the lattice ${\mathbf{Z}}^{2\ell+2}\simeq \Lambda$. According to Corollary 2.9 of [@Morrison:1984], there is a complex projective K3 surface $X$ for which $({{\mathrm{NS}}}(X;{\mathbf{Z}}), q_X)$ is isometric to $(\Lambda, 4Q_\ell)$. On such a surface, the self-intersection of every curve is divisible by $4$ and consequently there is no $(-2)$-curve. So, by the Torelli theorem for K3 surfaces (see [@BHPVDV]), ${\mathsf{Aut}}(X)^*_{\vert {{\mathrm{NS}}}(X;{\mathbf{Z}})}$ has finite index in ${{\sf{O}}}(4Q_\ell;{\mathbf{Z}})$.
Since ${{\sf{O}}}(4Q_\ell;{\mathbf{Z}})={{\sf{O}}}(Q_\ell;{\mathbf{Z}})$ we can view $\Gamma_\ell^*$ as a subgroup of ${{\sf{O}}}(4Q_\ell;{\mathbf{Z}})$. Set $\Gamma^*={\mathsf{Aut}}(X)^*\cap \Gamma_\ell^*$ and let $\Gamma$ denote its pre-image in ${\mathsf{Aut}}(X)$. Then, $\Gamma$ is a subgroup of ${\mathsf{Aut}}(X)$ for which the decomposition ${{\mathrm{NS}}}(X;{\mathbf{R}})_+\oplus {{\mathrm{NS}}}(X;{\mathbf{R}})_0$ is non-trivial (here, both have dimension $\ell+1$) while the representation is irreducible over ${\mathbf{Q}}$.
### The hyperbolic space ${\mathbb{H}}_X$ {#par:hyp_X}
Let $X$ be a compact Kähler surface. By the Hodge index theorem, the intersection form on $H^{1,1}(X, {\mathbf{R}})$ has signature $(1,h^{1,1}(X)-1)$. The hyperboloid $${\left\{u\in H^{1,1}(X, {\mathbf{R}}), \ \langle u \,\vert\, u\rangle\right\}}=1$$ has two connected components, one of which intersecting the Kähler cone. The hyperbolic space ${\mathbb{H}}_X$ is by definition this connected component, which is thus a model of the hyperbolic space of dimension $h^{1,1}(X)-1$. We denote by $d_{\mathbb{H}}$ the hyperbolic distance, which is defined as before by $\cosh(d_{\mathbb{H}}(u, v)) = \langle u \,\vert\, v\rangle$. From Lemma \[lem:cohomological\_norm\_estimates\] and Corollary \[cor:KAK\] we see that if ${{\left\Vert\cdot\right\Vert}}$ is any norm on $H^*(X,{\mathbf{C}})$, then ${{\left\Vertf^*\right\Vert}}\asymp {{\left\Vert(f^*){^{-1}}\right\Vert}} \asymp \langle [\kappa_0] \,\vert\, f^*[\kappa_0]\rangle$ (here $\kappa_0$ is the fixed Kähler form introduced in Section \[par:cones\_definition\]).
According to the classification of isometries of hyperbolic spaces, there are three types of automorphisms: **elliptic**, **parabolic** and **loxodromic**. An important fact for us is that the type of isometry is related to the dynamics on $X$; for instance, every parabolic automorphism $f$ preserves a genus $1$ fibration; every loxodromic automorphism has positive topological entropy. We refer the reader to [@Cantat:Milnor] for more details. A subgroup $\Gamma$ of ${\mathsf{Aut}}(X)$ is said [**[non-elementary]{}**]{} if its action on ${\mathbb{H}}_X$ is non-elementary. As we shall see below, the existence of such a subgroup forces $X$ to be projective:
\[thm:X-is-projective1\] If $X$ is a compact Kähler surface such that ${\mathsf{Aut}}(X)$ is non-elementary, then $X$ is projective.
For expository reasons, the proof of this result is postponed to §\[subsub:X-is-projective\], Theorem \[thm:X-is-projective2\].
### Automorphisms and Néron-Severi groups {#par:dec_h11_Gamma}
Let $X$ be a compact Kähler surface and $\Gamma$ be a non-elementary subgroup of ${\mathsf{Aut}}(X)$. Let $\Gamma^*_{p,q}$ be the image of $\Gamma$ in ${{\sf{GL}}}(H^{p,q}(X;{\mathbf{C}}))$, and $\Gamma^*$ be its image in ${{\sf{GL}}}(H^2(X;{\mathbf{C}}))$. If we combine Proposition \[pro:invariant\_cohomological\_decomposition\] together with Lemma \[lem:unitary\_on \_H20\] to $\Gamma^*_{1,1}$, we get an invariant decomposition $$H^{1,1}(X;{\mathbf{R}}) = H^{1,1}(X;{\mathbf{R}})_+\oplus H^{1,1}(X;{\mathbf{R}})_0.$$ Denote by $H^2(X;{\mathbf{R}})_0$ the direct sum of $H^{1,1}(X;{\mathbf{R}})_0$ and of the real part of $H^{2,0}(X;{\mathbf{C}})$; then $$H^2(X;{\mathbf{R}}) = H^{1,1}(X;{\mathbf{R}})_+\oplus H^2(X;{\mathbf{R}})_0$$ and $\Gamma^*{ \arrowvert_{H^2(X;{\mathbf{R}})_0}}$ is contained in a compact group (see Lemma \[lem:unitary\_on \_H20\]). The Néron-Severi group is $\Gamma$-invariant, and since $X$ is projective it contains a vector with positive self-intersection. Then Proposition \[pro:invariant\_cohomological\_decomposition\] and Lemma \[lem:restriction\_isom\] imply:
\[pro:NS\_H11\] Let $X$ be a compact Kähler surface and $\Gamma$ be a non-elementary subgroup of ${\mathsf{Aut}}(X)$. Then $H^{1,1}(X;{\mathbf{R}})_+={{\mathrm{NS}}}(X;{\mathbf{R}})_+$ is a Minkowski space, and the action of $\Gamma$ on this space is non-elementary and strongly irreducible.
Since non-elementary groups of isometries of ${\mathbb{H}}^m$ occur only for $m\geq 2$, we get:
\[cor:pic\_number\_3\] Under the assumptions of Proposition \[pro:NS\_H11\], the Picard number $\rho(X)$ is greater than or equal to $3$. If equality holds then ${{\mathrm{NS}}}(X;{\mathbf{R}})_+={{\mathrm{NS}}}(X;{\mathbf{R}})$ and the action of $\Gamma$ on ${{\mathrm{NS}}}(X;{\mathbf{R}})$ is strongly irreducible.
From now on we set: $$\label{eq:piGamma_nu-def}
\Pi_\Gamma:=H^{1,1}(X;{\mathbf{R}})_+={{\mathrm{NS}}}(X;{\mathbf{R}})_+.$$ This is a Minkowski space on which $\Gamma$ acts strongly irreducibly; the intersection form is negative definite on the orthogonal complement $$\Pi_\Gamma^\perp\subset H^{1,1}(X;{\mathbf{R}}).$$ Moreover by Proposition \[pro:invariant\_cohomological\_decomposition\].(2) the group $G={{\mathsf{Zar}}}(\Gamma)^\mathrm{irr}$ satisfies $G({\mathbf{R}}){ \arrowvert_{\Pi_\Gamma}}={{\sf{SO}}}(\Pi_\Gamma)$. If $\Gamma$ contains a parabolic element, then $\Pi_\Gamma$ is rational with respect to the integral structures of ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ and $H^2(X;{\mathbf{Z}})$, and $G({\mathbf{R}})={{\sf{SO}}}(\Pi_\Gamma)\times \{\operatorname{id}_{\Pi_\Gamma^\perp}\}$ (see Lemma \[lem:decomposition\_V+V0\_rational\]).
### Invariant algebraic curves I
Assume that $\Gamma$ is non-elementary and let $C\subset X$ be an irreducible algebraic curve with a finite $\Gamma$-orbit. Then the action of $\Gamma$ on $\operatorname{Vect}_{\mathbf{Z}}{\left\{f^*[C]; f\in \Gamma\right\}}\subset {{\mathrm{NS}}}(X)$ factors through a finite group. From Propositions \[pro:invariant\_cohomological\_decomposition\] and \[pro:NS\_H11\] we deduce that the intersection form is negative definite on $\operatorname{Vect}_{\mathbf{Z}}(\Gamma\cdot [C])$, thus $\operatorname{Vect}_{\mathbf{R}}(\Gamma\cdot [C])$ is one of the irreducible factors of ${{\mathrm{NS}}}(X, {\mathbf{R}})_0$. This argument, together with Grauert’s contraction theorem, leads to the following result (we refer to [@Cantat:Milnor; @Kawaguchi:AJM] for a proof; the result holds more generally for subgroups containing a loxodromic element):
\[lem:periodic\_curves\] Let $X$ be a compact Kähler surface and let Then, there are at most finitely many $\Gamma$-periodic irreducible curves. The intersection form is negative definite on the subspace of ${{\mathrm{NS}}}(X)$ generated by the classes of these curves. There is a compact complex analytic surface $X_0$ and a $\Gamma$-equivariant bimeromorphic morphism $X\to X_0$ that contracts these curves and is an isomorphism in their complement.
The next result follows from [@diller-jackson-sommese].
\[pro:diller-jackson-sommese\] Let $X$ be a compact Kähler surface and $\Gamma$ a non-elementary subgroup of ${\mathsf{Aut}}(X)$. Then any $\Gamma$-periodic curve has arithmetic genus 0 or 1.
Note if $C$ is $\Gamma$-periodic, this result applies to $\widetilde C = \Gamma\cdot C$, which is invariant. If $C$ is smooth and irreducible, this means that the geometric genus equals 0 or 1. If furthermore $X$ is a K3 or Enriques surface, then $C$ must be a rational curve of self-intersection $-2$. For general $C$, this condition implies that the normalization of any component of $C$ has genus 0 or 1, and the incidence graph of the components of $C$ obeys certain restrictions (see [@Cantat:Milnor §4.1] for more details).
### The limit set {#par:limit_set_I}
The [**[limit set]{}**]{} of $\Gamma$ is the closed subset $\operatorname{Lim}(\Gamma)\subset \partial {\mathbb{H}}_X \subset {\mathbb{P}}{\left(H^{1,1}(X;{\mathbf{R}})\right)}$ defined by one of the following equivalent assertions:
1. $\operatorname{Lim}(\Gamma)$ is the smallest, non-empty, closed, and $\Gamma$-invariant subset of ${\mathbb{P}}(\overline{{\mathbb{H}}_X})$;
2. $\operatorname{Lim}(\Gamma)\subset {\partial}{\mathbb{H}}_X$ is the closure of the set of fixed points of loxodromic elements of $\Gamma$ in ${\partial}{\mathbb{H}}_X$ (these fixed points correspond to isotropic lines on which the loxodromic isometry act as a dilation or contraction);
3. $\operatorname{Lim}(\Gamma)$ is the accumulation set of any $\Gamma$-orbit $\Gamma({\mathbb{P}}(v))\subset {\mathbb{P}}(H^{1,1}(X;{\mathbf{R}}))$, for any $v\notin \Pi_\Gamma^\perp$.
We refer to [@kapovich; @Ratcliffe] for a study of such limit sets. From the second characterization we get:
\[lem:limitset\_in\_NS\] The limit set $\operatorname{Lim}(\Gamma)$ is contained in ${\mathbb{P}}(\Pi_\Gamma)\cap \partial{\mathbb{H}}_X$.
From the third characterization, $\operatorname{Lim}(\Gamma)$ is contained in the closure of $\Gamma({\mathbb{P}}([\kappa]))$ for every Kähler form $\kappa$ on $X$. Since $X$ must be projective, we can chose $[\kappa]$ in ${{\mathrm{NS}}}(X;{\mathbf{Z}})$. As a consequence, $\operatorname{Lim}(\Gamma)$ is contained in ${{\mathrm{Nef}}}(X)$:
\[lem:limitset\_in\_nef\] Let $X$ be a compact Kähler surface. If $\Gamma$ is a non-elementary subgroup of ${\mathsf{Aut}}(X)$ its limit set satisfies $\operatorname{Lim}(\Gamma)\subset {\mathbb{P}}({{\mathrm{Nef}}}(X)) \subset {\mathbb{P}}({{\mathrm{NS}}}(X;{\mathbf{R}}))$.
Parabolic automorphisms {#par:parabolic_basics}
-----------------------
In this short paragraph we collect a few basic facts on parabolic automorphisms. This will be used in the next section to describe some explicit examples, and then in Sections \[sec:stiffness\] and \[sec:parabolic\].
Let $f$ be a parabolic automorphism of a compact Kähler surface. Then $f^*$ preserves a unique point on the boundary ${\partial}{\mathbb{H}}_X$, and $f$ preserves a unique genus $1$ fibration $\pi_f\colon X\to B$ onto some Riemann surface $B$. The fixed point of $f^*$ on ${\partial}{\mathbb{H}}_X$ is given by the class $[F]$ of any fiber of $\pi_f$ (see [@Cantat:Milnor]). The fibers of $\pi_f$ are the elements of the linear system $\vert F\vert$, and $\pi$ is uniquely determined by $[F]$, and if $g$ is another automorphism of $X$ that preserves a smooth fiber of $\pi_f$ (resp. the point ${\mathbb{P}}[F]\in {\mathbb{P}}{{\mathrm{NS}}}(X;{\mathbf{R}})$), then $g$ preserves the fibration and is either elliptic or parabolic.
\[lem:charac\_parabolic\_on\_K3\] Let $X$ be a K3 or Enriques surface, and $\pi\colon X\to B$ be a genus $1$ fibration. If $g\in{\mathsf{Aut}}(X)$ maps some fiber $F$ of $\pi$ to a fiber of $\pi$, then $g$ preserves the fibration and either $g$ is parabolic or it is periodic of order $\leq 66$.
Since $g$ maps a fiber $F$ to some fiber $F'$, it maps the complete linear system $\vert F\vert$ to $\vert F'\vert$, but both linear systems are made of the fibers of $\pi$. So $g$ preserves the fibration and it is not loxodromic. If $g$ is not parabolic, then it is elliptic and its action on cohomology has finite order since it preserves $H^2(X, {\mathbf{Z}})$. On a K3 or Enriques surface, the kernel of the homomorphism $f\in {\mathsf{Aut}}(X)\mapsto f^*$ is finite (see Proposition \[prop:discrete\]), so it follows that any elliptic automorphism has finite order. The upper bound on the order of $g$ was obtained in [@Keum:2016].
\[pro:parabolic\_infinite\] Let $X$ be a compact Kähler surface and let $f$ be a parabolic automorphism of $X$, preserving the genus $1$ fibration $\tau\colon X\to B$. Consider the group ${\mathsf{Aut}}(X;\tau):=\{ g\in {\mathsf{Aut}}(X)\; ; \; \exists g_B\in {\mathsf{Aut}}(B), \; \tau\circ g = g_B\circ \tau\}$, and assume that the image of the homomorphism $g\in {\mathsf{Aut}}(X;\tau)\to g_B\in {\mathsf{Aut}}(B)$ is infinite. Then, $X$ is a torus.
This result directly follows from the proof of Proposition 3.6 in [@cantat-favre]. In particular the automorphism $f_B\in {\mathsf{Aut}}(B)$ such that $\pi_f\circ f = f_B\circ \pi_f$ has finite order when $X$ is a K3 or rational surface. The dynamics of these automorphisms is described in Section \[par:parabolic\_automorphisms\].
\[lem:pairs\_of\_twists\] If $\Gamma$ is a subgroup of ${\mathsf{Aut}}(X)$ containing a parabolic automorphism $g$, then $\Gamma$ is non-elementary if and only if it contains another parabolic automorphism $h$ such that the invariant fibrations $\pi_g$ and $\pi_h$ are distinct. Then the tangency locus of the two fibrations is either empty or a complex curve in $X$, and there are positive integers $m$, $n$ such that $g^m$ and $h^n$ generate a free group of rank $2$.
Let $F$ be a fiber of $\pi_g$. If $\Gamma$ is non-elementary, there is an element $f$ in $\Gamma$ that does not fix $[F]$; in particular $f$ does not preserve $\pi_g$. Then, $h:=f^{-1}\circ g\circ f$ is another parabolic automorphism with a distinct invariant fibration, namely $\pi_h=\pi_g\circ f$. Being distinct, $\pi_g$ and $\pi_h$ have a tangency locus of codimension $\geq 1$.
Conversely, if $\Gamma$ contains two parabolic automorphisms with distinct fixed point in ${\partial}{\mathbb{H}}_X$, then the ping-pong lemma proves that there are powers $m$, $n\geq 1$ such that $\langle g^m, h^n\rangle$ is a free group of rank $2$; in particular, $\Gamma$ is non-elementary. (See [@Cantat:Milnor] for more precise results.)
Examples and classification {#par:Examples_Classification}
===========================
This section may be skipped in a first reading. It describes a few examples, and proves that a compact Kähler surface $X$ is projective when its automorphism group is non-elementary.
Wehler surfaces (see [@Cantat-Oguiso; @Reschke; @Wang:1995; @Wehler]) {#par:Wehler_surfaces_I}
---------------------------------------------------------------------
\[par:Wehler\] Consider the variety $M={\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ and let $\pi_1$, $\pi_2$, and $\pi_3$ be the projections on the first, second, and third factor: $\pi_i(z_1,z_2,z_3)=z_i$. Denote by $L_i$ the line bundle $\pi_i^*({\mathcal{O}} (1))$ and set $$L = L_1^2\otimes L_2^2\otimes L_3^2 = \pi_1^*({\mathcal{O}} (2))\otimes \pi_2^*({\mathcal{O}} (2))\otimes \pi_3^*({\mathcal{O}}(2)).$$ Since $K_{{\mathbb{P}}^1} = \mathcal {O}(-2)$, this line bundle $L$ is the dual of the canonical bundle of $K_M$. By definition, $\vert L\vert\simeq {\mathbb{P}}(H^0(M, L))$ is the linear system of surfaces $X\subset M$ given by the zeroes of global sections $P\in H^0(M, L)$. Using affine coordinates $(x_1,x_2,x_3)$ on $M={\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$, such a surface is defined by a polynomial equation $P(x_1,x_2,x_3)=0$ whose degree with respect to each variable is $\leq 2$ (see [@Cantat:Acta; @McMullen:Crelle] for explicit examples). These surfaces will be referred to as [**[Wehler surfaces]{}**]{} or [**(2,2,2)-surfaces**]{}; modulo the action of ${\mathsf{Aut}}(M)$, they form a family of dimension $17$.
Fix $k\in \{1, 2, 3\}$ and denote by $i < j$ the other indices. If we project $X$ to ${\mathbb{P}}^1\times {\mathbb{P}}^1$ by $\pi_{ij}=(\pi_i,\pi_j)$, we get a $2$ to $1$ cover and as soon as $X$ is smooth the involution $\sigma_k$ that permutes the two points in each fiber of $\pi_{ij}$ is an involutive automorphism of $X$ (indeed $X$ is a K3 surface and any birational self-map of such a surface is an automorphism).
\[pro:Wehler\_surface\_Pic\] There is a countable union of proper Zariski closed subsets $(W_i)_{i\geq 0}$ in $\vert L\vert$ such that
1. if $X$ is an element of $\vert L\vert\setminus W_0$, then $X$ is a smooth K3 surface and $X$ does not contain any fiber of the projections $\pi_{ij}$;
2. if $X$ is an element of $\vert L\vert\setminus (\bigcup_i W_i)$, the restriction morphism ${{\mathrm{Pic}}}(M)\to {{\mathrm{Pic}}}(X)$ is surjective. In particular its Picard number is $\rho(X) = 3$.
From the second assertion, we deduce that for a very general $X$, ${{\mathrm{Pic}}}(X)$ is isomorphic to ${{\mathrm{Pic}}}(M)$: it is the free abelian group of rank $3$, generated by the classes $$c_i:=[(L_i)_{\vert X}].$$ The elements of $\vert (L_i)_{\vert X} \vert $ are the curves of $X$ given by the equations $z_i=\alpha$ for some $\alpha \in {\mathbb{P}}^1$. The arithmetic genus of these curves is equal to $1$: in other words the projection $(\pi_i)_{\vert X}\colon X\to {\mathbb{P}}^1$ is a genus $1$ fibration. Moreover, for a general choice of $X$ in $\vert L \vert$, $(\pi_i)_{\vert X}$ has $24$ singular fibers of type ${\sf{I}}_1$, i.e. isomorphic to a rational curve with exactly one simple double point. The intersection form is given by $c_i^2=0$ and $\langle c_i\vert c_j \rangle= 2$ if $i\neq j$, so that its matrix is given by $$\label{eq:intersection_matrix}
\left(\begin{array}{ccc} 0 & 2 & 2\\
2 & 0 & 2\\
2 & 2 & 0\end{array}\right).$$
By Bertini’s theorem, $X$ is smooth as soon as it is in the complement of some proper Zariski closed subset $W_0\subset \vert L \vert$. Now, let us assume that $X$ is smooth. The adjunction formula implies that the canonical bundle $K_X$ is trivial. From the hyperplane section theorem of Lefschetz [@Milnor:Morse_Theory], we know that $X$ is simply connected. So, $X$ is a K3 surface (see [@BHPVDV]). Write the equation of $X$ as $A(x_1,x_2)x_3^2+B(x_1,x_2)x_3+C(x_1,x_2)=0$. Then, $X$ contains a fiber $\pi_{12}^{-1}(a_1,a_2)$ if and only if the three curves given by $A=0$, $B=0$, and $C=0$ contain the point $(a_1,a_2)$. This imposes a non-trivial algebraic condition on $X$; hence, enlarging $W_0$, the first assertion is satisfied.
For the second assertion, we apply a general form of the Noether-Lesfchetz theorem [@Voisin:BookHodge Théorème 15.33]. We know that $L$ is very ample, that $H^{2,0}(X)$ is isomorphic to ${\mathbf{C}}$. Indeed $X$ is a K3 surface, and $H^{2,0}(X)$ is contained in the vanishing cohomology since $X$ may degenerate on six copies of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ (taking the equation $(x_1^2-1)(x_2^2-1)(x_3^2-1)=0$). So, the Noether-Lefschetz theorem says precisely that the restriction morphism is surjective for a very general choice of $X\in \vert L\vert$.
\[lem:wehler\_no\_fiber\_free\] Assume that $X$ does not contain any fiber of the projection $\pi_{ij}$. Then, the involution $\sigma_k^*$ preserves the subspace ${\mathbf{Z}}c_1\oplus {\mathbf{Z}}c_2\oplus {\mathbf{Z}}c_3$ of ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ and $$\sigma_k^*c_i=c_i,\; \sigma_k^*c_j=c_j,\; \sigma_k^*c_k=-c_k+2c_i+2c_j.$$ Equivalently, the action of $\sigma_k^*$ on $\operatorname{Vect}_{\mathbf{R}}(c_1, c_2, c_3)$ preserves the classes $c_i$ and $c_j$ and acts as a reflexion with respect to the hyperplane $\operatorname{Vect}(c_i, c_j)\subset {{\mathrm{NS}}}(X;{\mathbf{R}})$. In other words, $\sigma_k(v)=v+\frac{1}{2} \langle v\vert u_k\rangle u_k$ for all $v$ in ${\mathbf{Z}}c_1\oplus {\mathbf{Z}}c_2\oplus {\mathbf{Z}}c_3$.
Since $\sigma_k$ preserves $\pi_{ij}$ it preserves the fibers of $\pi_i$ and $\pi_j$, hence $\sigma_k^*$ fixes $c_i$ and $c_j$. Now, consider a fiber $C=\{ z_k=w\}\subset X$ of $\pi_k$. Then, $\sigma_k(C)\cup C=\pi_{ij}^{-1}(\pi_{ij}(C))$ because there is no curve in the fibers of $\pi_{ij}$. On the other hand, $\pi_{ij}(C)\subset {\mathbb{P}}^1\times {\mathbb{P}}^1$ is a (2,2)-curve so it is rationally equivalent to the union of two vertical and two horizontal projective lines. This gives $\sigma_k^*c_k=-c_k+2c_i+2c_j$.
Combining this lemma with the previous proposition, we see that a very general Wehler surface has Picard number $3$, ${\mathbb{H}}_X$ has dimension 2, ${{\mathrm{NS}}}(X;{\mathbf{Z}})=\operatorname{Vect}_{\mathbf{Z}}(c_1,c_2,c_3)$ and the matrices of the $\sigma_i^*$ in the basis $(c_i)$ are $$\label{eq:matrices_sigmai_wehler}
\sigma_1^*= \left(\begin{array}{ccc} -1 & 0 & 0\\
2 & 1 & 0\\
2 & 0 & 1\end{array}\right), \;\,
\sigma_2^*= \left(\begin{array}{ccc} 1 & 2 & 0\\
0 & -1 & 0\\
0 & 2 & 1\end{array}\right), \;\,
\sigma_3^*= \left(\begin{array}{ccc} 1 & 0 & 2\\
0 & 1 & 2 \\
0 & 0 & -1\end{array}\right).$$
\[pro:Wehler\_generic\] If $X$ is a very general Wehler surface then:
1. $X$ is a smooth K3 surface with Picard number $3$;
2. ${\mathsf{Aut}}(X)$ is equal to $\langle \sigma_1, \sigma_2, \sigma_3 \rangle$, it is a free product of three copies of ${\mathbf{Z}}/2{\mathbf{Z}}$, and ${\mathsf{Aut}}(X)^*$ is a finite index subgroup in the group of integral isometries of ${{\mathrm{NS}}}(X;{\mathbf{Z}})$;
3. ${\mathsf{Aut}}(X)^*$ acts strongly irreducibly on ${{\mathrm{NS}}}(X;{\mathbf{R}})$;
4. ${\mathsf{Aut}}(X)$ does not preserve any algebraic curve $D\subset X$;
5. the limit set of ${\mathsf{Aut}}(X)^*$ is equal to $\partial {\mathbb{H}}_X$;
6. the compositions $\sigma_i\circ \sigma_j$ and $\sigma_i\circ \sigma_j\circ \sigma_k$ are respectively parabolic and loxodromic for every triple $(i,j,k)$ with ${\left\{ i, j, k \right\}}={\left\{1,2,3\right\}}$.
The first three assertions follow from Proposition \[pro:Wehler\_surface\_Pic\], [@Cantat:Acta §1.5] and [@Cantat-Oguiso Thm 3.6]. For the fourth one, note that any invariant curve $D$ would yield a non-trivial fixed point $[D]$ in ${{\mathrm{NS}}}(X;{\mathbf{Z}})$, contradicting assertion (3). The fifth one follows from the second because the limit set of a lattice in ${\mathsf{Isom}}({{\mathrm{NS}}}(X;{\mathbf{R}}))$ is always equal to $\partial {\mathbb{H}}_X$. To prove the last assertion, it suffices to compute the corresponding product of matrices given in Equation (see [@Cantat:Acta]).
In [@Baragar:2016], Baragar gives examples of smooth surfaces $X\in \vert L\vert$ for which $\rho(X)\geq 4$ and the limit set of ${\mathsf{Aut}}(X)^*$ in $\partial {\mathbb{H}}_X$ is a genuine fractal set.
Pentagons {#par:pentagons}
---------
The dynamics on the space of pentagons with given side lengths, introduced in §\[par:pentagons\_intro\], shares important similarities with the dynamics on Wehler surfaces. A pentagon with side lengths $\ell_0, \ldots, \ell_4$ modulo translations of the plane is the same as the data of a 5-tuple of vectors $(v_i)_{i=0, \ldots , 4}$ in ${\mathbf{R}}^2$ (identified with ${\mathbf{C}}$) with respective length $\ell_i$ and such that $\sum_i v_i = 0$. Write $v_i = \ell_i t_i$ with ${\left\vertt_i\right\vert} = 1$. Then the action of ${{\sf{SO}}}_2({\mathbf{R}})$ can be identified to the diagonal multiplicative action of ${{\sf{U}}}_1=\{\alpha \in {\mathbf{C}}\; ; \; {\left\vert\alpha\right\vert}=1\}$ on the $t_i$: $$\alpha\cdot (t_0, \ldots, t_4)=(\alpha t_0, \ldots \alpha t_4).$$ Now, following Darboux [@darboux], we consider the surface $X$ in ${\mathbb{P}}^4_{\mathbf{C}}$ defined by the equations $$\label{eq:equation_pentagons}
\begin{cases}
\ell_0z_0+ \ell_1 z_1+\ell_2z_2+\ell_3z_3+\ell_4 z_4 = 0\\
\ell_0/z_0+ \ell_1/ z_1+\ell_2/z_2+\ell_3/z_3+\ell_4/ z_4 = 0
\end{cases}$$ where $[z_0:\ldots :z_4]$ is some fixed choice of homogeneous coordinates, and the second equation must be multiplied by $z_0z_1z_2z_3z_4$ to obtain a homogeneous equation of degree $4$.
This surface is isomorphic to the Hessian of a cubic surface (see [@Dolgachev:CAG §9]). More precisely, consider a cubic surface $S\subset {\mathbb{P}}^3_{\mathbf{C}}$ whose equation $F$ can be written in Sylvester’s pentahedral form that is, as a sum $F=\sum_{i=0}^4 \lambda_i F_i^3$ for some complex numbers $\lambda_i$ and linear forms $F_i$ with $\sum_{i=0}^4 F_i=0$. By definition, its Hessian surface $H_F$ is defined by $\det(\partial_i\partial_j F)=0$. Then, using the linear form $F_i$ to embed $H_F$ in ${\mathbb{P}}^4_{\mathbf{C}}$, we obtain the surface defined by the pair of equations $\sum_{i=0}^4 z_i=0$ and $\sum_{i=0}^4\frac{1}{\lambda_iz_i}=0$. Thus, $H_F$ is our surface $X$, for $\ell_i^2=\lambda_i$. We refer to [@Dolgachev-Keum; @Dardanelli-vanGeemen; @Dolgachev:Salem; @Rosenberg] for an introduction to these surfaces and their birational transformations.
For completeness, we prove some of its basic properties.
Let $\ell=(\ell_0, \ldots, \ell_4)$ be an element of $({\mathbf{C}}^*)^5$. The surface $X\subset {\mathbb{P}}^4_{\mathbf{C}}$ defined by the system has $10$ singularities at the points $q_{ij}$ obtained by the equations $\ell_i z_i+\ell_j z_j=0$, $z_k=z_l=z_m=0$ with $i<j$ and $\{i,j,k,l,m\}=\{0,1,2,3,4\}$. In the complement of these ten isolated singularities, $X$ is smooth if and only if $$\label{eq:pentagon_smoothness}
\sum_{i=0}^4 {\varepsilon}_i \ell_i \neq 0\quad \forall {\varepsilon}_i \in {\left\{\pm 1\right\}}.$$
We first look for singularities in the complement of the hyperplanes $z_i=0$, and work in the chart $z_0 = 1$. Then $z_4 = - (\ell_0+ \ell_1x+ \ell_2y + \ell_3z)/\ell_4$ and we replace in the second equation of to obtain an affine equation of $X$ in this chart, namely: $$\label{eq:equation_pentagons_affine}
\frac{\ell_1}{z_1} + \frac{\ell_2}{z_2} + \frac{\ell_3}{z_3} - \frac{\ell_4^2}{\ell_0+ \ell_1z_1+ \ell_2z_2+ \ell_3z_3} + \ell_0 = 0.$$ Singularities are determined by the system of equations $z_1^2 =z_2^2=z_3^2 = \ell_4^{-2}(\ell_0+ \ell_1z_1+ \ell_2z_2 + \ell_3z_3)^2$. So, by symmetry, at a singularity where none of the coordinates vanishes we must have $z_i=\varepsilon _i z$ for some $\varepsilon _i=\pm 1$ and a common factor $z\neq 0$; this is precisely Condition .
Looking for singularities with one coordinate equal to $0$, say $z_1=0$ in the chart $z_0=1$, we obtain the system of equations $$\label{eq:equation_pentagons_singularities}
\begin{cases}
0= (\ell_0 z_2z_3+\ell_3 z_2+\ell_2z_3)(\ell_0 +\ell_2 z_2+\ell_3 z_3)+(\ell_1^2-\ell_4^2)z_2z_3\\
0=\ell_1 z_3(\ell_0+2\ell_2 z_2+\ell_3z_3)\\
0= \ell_1 z_2 (\ell_0+\ell_2z_2+2\ell_3 z_3)
\end{cases}$$ together with $\ell_0+\ell_2 z_2+\ell_3z_3+\ell_4z_4=0$ and $\ell_1 z_2z_3z_4=0$ (in particular, $z_2$, $z_3$ or $z_4$ must vanish). The solutions of this system are given by $z_1=z_2=z_3=0$, which gives the point $q_{04}=[\ell_4:0:0:0:-\ell_0]$, or $z_1=z_2=0$ and $\ell_0+\ell_3z_3=0$, which corresponds to $q_{03}=[\ell_3:0:0:-\ell_0:0]$, or $z_1=z_3=0$ which gives $q_{02}$, or $z_1=z_4=0$ but then either $z_2=0$ or $z_3=0$ and we end up again with $q_{02}$ and $q_{03}$. The result follows by symmetry.
If $\ell\in ({\mathbf{C}}^*)^5$ satisfies Condition , then the ten singularities are simple nodes (Morse singularities) and the surface $X$ is a (singular) K3 surface: a minimal resolution $\hat X$ of $X$ is a K3 surface, which is obtained by blowing-up its ten nodes, thereby creating ten rational $(-2)$-curves.
Working in the chart $z_0=1$ and replacing $z_4$ by $-(\ell_0+\ell_1z_1+\ell_2z_2+\ell_3z_3)/\ell_4$, the quadratic term of the equation of $X$ at the singularity $(z_1,z_2,z_3)=(0,0,0)$ is $(-\ell_0/\ell_4)Q$, where $$\label{eq:pentagons_quadratic_Q}
Q(z_1,z_2,z_3)=\ell_1z_2z_3+\ell_2z_1z_3+\ell_3z_1z_2$$ is a non-degenerate quadratic form (its determinant is $2\ell_1\ell_2\ell_3\neq 0$). So locally $X$ is holomorphically equivalent to the quadratic cone $\{Q=0\}$, hence to a quotient singularity $({\mathbf{C}}^2,0)/\eta$ with $\eta(x,y)=(-x,-y)$. The minimal resolution of such a singularity is obtained by a simple blow-up of the ambient space, the exceptional divisor being a $(-2)$-curve in the smooth surface $\hat{X}$. The adjunction formula shows that there is a holomorphic $2$-form $\Omega_X$ on the regular part of $X$; locally, $\Omega_X$ lifts to an $\eta$-invariant form $\Omega_X'$ on ${\mathbf{C}}^2\setminus \{ 0\}$, which by Hartogs extends at the origin to a non-vanishing $2$-form. To recover $\hat{X}$, one can first blow-up ${\mathbf{C}}^2$ at the origin and then take the quotient by (the lift of) $\eta$: a simple calculation shows that $\Omega_X'$ determines a non-vanishing $2$-form on $\hat{X}$. After such a surgery is done at the ten nodes, $\hat{X}$ is a smooth surface with a non-vanishing section of $K_{\hat{X}}$; since it contains at least ten rational curves, it can not be an abelian surface, so it must be a K3 surface.
Let $L_{ij}$ be the line defined by the equations $z_i=0$, $z_j=0$, $\ell_0z_0+\cdots +\ell_4z_4=0$; each of these ten lines is contained in $X$, each of them contains $3$ singularities of $X$ (namely $q_{kl}$, $q_{lm}$, $q_{km}$ with obvious notations), and each singularity is contained in three of these lines. If one projects them on a plane, the ten lines $L_{ij}$ form a Desargues configuration (see [@Dolgachev-Keum; @Dolgachev:Salem]).
All this works for any choice of complex numbers $\ell_i\neq 0$. Now, since the $\ell_i$ are real, $X$ is endowed with two real structures. First, one can consider the complex conjugation $c\colon [z_i]\mapsto [\overline{z_i}]$ on ${\mathbb{P}}^4({\mathbf{C}})$ and restrict it to $X$: this gives a first antiholomorphic involution $c_X$. Another one is given by $s_X\colon [z_i]\mapsto [1/\overline{z_i}]$. To be more precise, consider first, the quartic birational involution $J\in {{\mathsf{Bir}}}({\mathbb{P}}^4_{\mathbf{C}})$ defined by $J ([z_i]) =[1/z_i]$; $J$ preserves $X$, it determines a birational transformation $J_X\in {{\mathsf{Bir}}}(X)$, and on ${\hat{X}}$ it becomes an automorphism because every birational transformation of a K3 surface is regular. Thus, $s_X=J_X\circ c_X$ determines a second antiholomorphic involution $s_{{\hat{X}}}$ of ${\hat{X}}$. In what follows, we denote by $(X,s_X)$ this real structure (even if it would be better to study it on ${\hat{X}}$); its real part is the fixed point set of $s_X$, i.e. the set of points in $X({\mathbf{C}})$ with coordinates of modulus $1$: the real part does not contain any of the singularities of $X$, this is why we prefer to stay in $X$ rather than lift everything to ${\hat{X}}$. Thus, [*with the real structure defined by $s_X$, the real part of $X$ coincides with $\mathrm{Pent}^0(\ell_0, \ldots , \ell_4)$*]{} if $(\ell_i)\in ({\mathbf{R}}_+^*)^5$.
When $\ell_i>0$ for all indices $i\in \{0,\ldots, 4\}$, a complete description of the possible homeomorphism types for the real locus (in the smooth and singular cases) is given in [@Curtis-Steiner]: [*in the smooth case, it is an orientable surface of genus $g = 0, \ldots, 4$ or the union of two tori*]{}.
The involution $J$ preserves $X$ and the two real structures $(X,c_X)$ and $(X,s_X)$. It lifts to a fixed point free involution ${\hat{J_X}}$ on $\hat{X}$, and $\hat{X}/{\hat{J_X}}$ is an Enriques surface. On pentagons, $J$ corresponds to the symmetry $(x,y)\in {\mathbf{R}}^2\mapsto (x,-y)$ that reverses orientation. Thus we see that the space of pentagons modulo affine isometries is an Enriques surface. When $X$ acquires an eleventh singularity which is fixed by $J_X$, then $\hat{X}/{\hat{J_X}}$ becomes a Coble surface: see [@Dolgachev:Salem §5] for nice explicit examples. This happens for instance when all lengths are $1$, except one which is equal to $2$ (this corresponds to $t=1/4$ in [@Dolgachev:Salem §5.2]).
Finally, let us express the folding transformations in coordinates. Given $i\neq j$ in ${\left\{0, \ldots, 4\right\}}$ (consecutive or not) we define an involution $(t_i, t_j)\mapsto (t_i', t_j')$ preserving the vector $\ell_i t_i+ \ell_j t_j$ by taking the symmetric of $t_i$ and $t_j$ with respect to the line directed by $\ell_i t_i+ \ell_j t_j$. In coordinates, $t'_k = u / t_k$ for some $u$ of modulus 1, and equating $\ell_i t_i+ \ell_j t_j = \ell_i t'_i+ \ell_j t'_j$ one obtains $$(t_i',t_j') = {\left(\frac{u}{t_i}, \frac{u}{t_j}\right)} \text{, with } u = \frac{\ell_i t_i + \ell_j t_j}{\ell_i t_i{^{-1}}+ \ell_j t_j{^{-1}}}.$$ Observe that these computations also make sense when the $\ell_i$ are complex numbers, or when we replace the $t_i$ by the complex numbers $z_i$. This defines a birational involution $\sigma_{ij}: X\dasharrow X$, $$\sigma_{ij}[z_0: \ldots :z_4]=[z'_0:\ldots :z_4']$$ with $z'_k=z_k$ if $k\neq i,j$, $z'_i=vz_j$, and $z_j'=vz_i$ with $v= (\ell_i z_i+\ell_jz_j)/(\ell_i z_j+\ell_j z_i)$. Again, since every birational self-map of a K3 surface is an automorphism, these involutions $\sigma_{ij}$ are elements of ${\mathsf{Aut}}({\hat{X}})$ that commute with the antiholomorphic involution $s_{{\hat{X}}}$; hence, they generate a subgroup of ${\mathsf{Aut}}({\hat{X}}; s_{{\hat{X}}})$. Thus we have constructed a family of projective surfaces ${\hat{X}}$, depending on a parameter $\ell \in {\mathbb{P}}^4({\mathbf{C}})$, endowed with a group of automorphisms generated by involutions. Note that this group can be elementary: for instance when the five lengths are all equal the group is finite because in that case $(z_i', z_j') = (z_j, z_i)$. When $j=i+1$ modulo $5$, $\sigma_{ij}$ corresponds to the folding transformation described in the introduction.
\[rem:sigma\_geometric\_pentagon\] Pick a singular point $q_{ij}$, and project $X$ from that point onto a plane, say the plane $\{z_i=0\}$ in the hyperplane $P=\{\ell_0 z_0+ \cdots +\ell_4z_4=0\}$. One gets a $2$ to $1$ cover $X\to {\mathbb{P}}^2_{\mathbf{C}}$, ramified along a sextic curve (this curve is the union of two cubics, see [@Rosenberg]). The involution $\sigma_{ij}$ permutes the points in the fibers of this $2$ to $1$ cover: if $x$ is a point of $X$, the line joining $q_{ij}$ and $x$ intersects $X$ in the third point $\sigma_{ij}(x)$. The singularity $q_{ij}$ is an indeterminacy point, mapped by $\sigma_{ij}$ to the opposite line $L_{ij}$.
\[pro:pentagons\] For a general parameter $\ell\in {\mathbb{P}}^4({\mathbf{C}})$:
1. $X$ is a K3 surface with ten nodes, with two real structures $c_X$ and $s_X$ when $\ell\in {\mathbb{P}}^4({\mathbf{R}})$;
2. if $i$, $j=i+1$, $k=i+2$ are distinct consecutive indices (modulo $5$), then $\sigma_{ij}\circ\sigma_{jk}$ is a parabolic transformation on ${\hat{X}}$;
3. if $i$, $j$, $k$, and $l$ are four distinct indices (modulo $5$), then $\sigma_{ij}$ commutes to $\sigma_{kl}$.
4. the group $\Gamma$ generated by the involutions $\sigma_{ij}$ is a non-elementary subgroup of ${\mathsf{Aut}}({\hat{X}}; s_{\hat{X}})$ that does not preserve any algebraic curve.
In [@Dolgachev:Salem], Dolgachev computes the action of $\sigma_{ij}$ on ${{\mathrm{NS}}}(\hat{X})$. This contains a proof of this proposition. He also describes, up to finite index, the Coxeter group generated by the $\sigma_{ij}$. The automorphism groups of $\hat{X}$ and of the Enriques surface ${\hat{X}}/{\hat{J_X}}$ are described in [@Dolgachev-Keum] and [@Shimada] respectively.
We already established Assertion (1) in the previous lemmas. For Assertion (2), denote by $l,m$ the indices for which ${\left\{i,j,k,l,m\right\}} = {\left\{0, \ldots, 4\right\}}$, and consider the linear projection $\pi_{lm}\colon {\mathbb{P}}^5({\mathbf{C}})\dasharrow{\mathbb{P}}^1({\mathbf{C}})$ defined by $[z_0:\ldots :z_4]\mapsto [z_l:z_m]$. The fibers of $\pi_{lm}$ are the hyperplanes containing the plane $\{z_l=z_m=0\}$, which intersects $X$ on the line $L_{l m}$. This line is a common component of the pencil of curves cut out by the fibers of $\pi_{l m}$ on $X$, and the mobile part of this pencil determines a fibration $\pi_{l m\vert X}\colon X\to {\mathbb{P}}^1$ whose fibers are the plane cubics $$\label{eq:cubic_pilm}
(\ell_l z_l+\ell_m z_m)(\ell_m z_l+\ell_l z_m)z_i z_j z_k=z_l z_m (\ell_i z_jz_k+\ell_j z_iz_k+\ell_k z_iz_j)(\ell_i z_i+\ell_j z_j+\ell_k z_k),$$ with $[z_l:z_m]$ fixed. The general member of this fibration is a smooth cubic, hence a curve of genus $1$.
Then $\sigma_{ij}$ and $\sigma_{jk}$ preserve $\pi_{l m\vert X}$, and along the general fiber of $\pi_{l m\vert X}$ each of them is described by Remark \[rem:sigma\_geometric\_pentagon\]; for instance, $\sigma_{ij}(x)$ is the third point of intersection of the cubic with the line $(q_{ij}, x)$. Thus, writing such a cubic as ${\mathbf{C}}/\Lambda_{[z_l:z_m]}$, $\sigma_{ij}$ acts as $z\mapsto -z+b_{ij}$, for some $b_{ij}\in {\mathbf{C}}/\Lambda_{[z_l:z_m]}$ that depends on $[z_l:z_m]$ and the parameter $\ell$; it has four fixed points on the cubic curve, which are the points of intersection of the cubic with the hyperplanes $z_i=z_j$ and $z_i=-z_j$; equivalently, the line $(q_{ij},x)$ is tangent to the cubic at these four points.
By Lemma \[lem:charac\_parabolic\_on\_K3\], either $\sigma_{ij}\circ\sigma_{jk}$ is of order $\leq 66$ (in fact of order $\leq 12$ because it preserves $\pi_{l m\vert X}$ fiber-wise), or it is parabolic. Due to this bound on the order, and the fact that there do exist pentagons for which $\sigma_{ij}\circ\sigma_{jk}$ is of infinite order (indeed, this reduces to the corresponding fact for quadrilaterals, see the example below), $\sigma_{ij}\circ\sigma_{jk}$ is parabolic for general $\ell$.
Take $\ell=1$ and $m=2$, and normalize our pentagons to assume that $t_0=1$, which means that the first vertices are $a_0=(0,0)$ and $a_1=(\ell_0,0)$; in homogeneous coordinates this corresponds to the normalization $[1:z_1:z_2:z_3:z_4]$ with $z_i= t_i$. Now, the pentagon in a fiber of $\pi_{12\vert X}$ have three fixed vertices, namely $a_0$, $a_1$ and $a_2$. The remaining vertices $a_3$ and $a_4$ move on the circles centered at $a_2$ and $a_0$ and of respective radii $\ell_2$ and $\ell_4$, with the constraint $a_3a_4=\ell_3$. The circles are two conics, the fiber is a $2$ to $1$ cover of each of these two conics, and the automorphisms $\sigma_{23}$ and $\sigma_{34}$ preserve these fibers. Forgetting the vertex $a_1$, and looking at the quadrilateral $(a_0, a_2, a_3, a_4)$, one recovers the involutions described in [@benoist-hulin]. The fixed points of $\sigma_{23}$ correspond to configurations with tangent circles, i.e. $a_3$ on the segment $[a_2,a_4]$.
Assertion (3) follows directly from the fact that $\sigma_{ij}$ changes the coordinates $z_i$ and $z_j$ but keeps the other three fixed.
Finally, for a general parameter $\ell$, $\Gamma$ contains two such parabolics associated to distinct fibrations $\pi_{lm}$ and $\pi_{l'm'}$ so it is non-elementary (see Lemma \[lem:pairs\_of\_twists\]). In addition $\Gamma$ does not preserve any curve in $\hat{X}$. Indeed, let $E\subset {\hat{X}}$ be a $\Gamma$-periodic irreducible curve, and denote by $F$ its image in ${\mathbb{P}}^4_{\mathbf{C}}$ under the projection ${\hat{X}}\to X$. If $F$ is a point, it is one of the singularities $q_{ij}$, and changing $E$ into its image under (the lift of) $\sigma_{ij}$ the curve $F$ becomes the line $L_{ij}$. So, we may assume that $F$ is an irreducible curve. Now, the orbit of $F$ is periodic under the action of the parabolic automorphisms $g_i=\sigma_{ij}\circ\sigma_{jk}$ with $k=j+1$ and $j=i+1$. Since the invariant curves of a parabolic automorphisms are contained in the fibers of its invariant fibration, we deduce that $F$ is contained in the fibers of each of the projections $\pi_{lm}$; this is obviously impossible.
Enriques surfaces (see [@Cossec-Dolgachev:book; @Dolgachev:Kyoto]) {#par:Enriques}
------------------------------------------------------------------
Enriques surfaces are quotients of K3 surfaces by fixed point free involutions. According to Horikawa and Kondō ([@Horikawa:Enriques1; @Horikawa:Enriques2; @Kondo:1994]), the moduli space ${\mathcal{M}}_E$ of complex Enriques surfaces is a rational quasi-projective variety of dimension 10. An Enriques surface $X$ is nodal if it contains a smooth rational curve; such rational curves have self-intersection $-2$, and are called nodal-curves or $(-2)$-curves. Nodal Enriques surfaces form a hypersurface in ${\mathcal{M}}_E$.
Using standard vocabulary from the theory of unimodular lattices (see e.g. [@Cossec-Dolgachev:book Chap. II]), for any Enriques surface $X$, the lattice $({{\mathrm{NS}}}(X;{\mathbf{Z}}), q_X)$ is isomorphic to the orthogonal direct sum $E_{10}=U \operp E_8(-1)$, ([^1]). Let $W_X\subset {{\sf{O}}}({{\mathrm{NS}}}(X;{\mathbf{Z}}))$ be the subgroup generated by reflexions about classes $u$ such that $u^2=-2$, and $W_X(2)$ be the subgroup of $W_X$ acting trivially on ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ modulo $2$. Both $W_X$ and $W_X(2)$ have finite index in ${{\sf{O}}}({{\mathrm{NS}}}(X;{\mathbf{Z}}))$. The following result is due independently to Nikulin and Barth and Peters (see [@Dolgachev:Kyoto] for details and references). Recall that ${\mathsf{Aut}}(X)^*$ denotes the image of ${\mathsf{Aut}}(X)$ in ${{\sf{GL}}}(H^*(X, {\mathbf{Z}}))$.
If $X$ is an Enriques surface which is not nodal, then ${\mathsf{Aut}}(X)$ is isomorphic to ${\mathsf{Aut}}(X)^*$ and $W_X(2) \subset {\mathsf{Aut}}(X)^* \subset W_X$.
In particular, for any unnodal Enriques surface, ${\mathsf{Aut}}(X)$ is non-elementary, contains parabolic elements, and acts irreducibly on ${{\mathrm{NS}}}(X;{\mathbf{R}})$; thus, it does not preserve any curve.
Examples on rational surfaces: Coble and Blanc {#par:Coble-Blanc}
----------------------------------------------
Closely related to Enriques surfaces are the examples of Coble, obtained by blowing up the ten nodes of a general rational sextic $C_0\subset {\mathbb{P}}^2$. The result is a rational surface $X$ with a large group of automorphisms. To be precise, consider the canonical class $K_X\subset {{\mathrm{NS}}}(X;{\mathbf{Z}})$; its orthogonal complement $K_X^\perp$ is a lattice of dimension $10$ isomorphic to $E_{10}$ and we define $W_X(2)$ exactly in the same way as for Enriques surfaces. Then, ${\mathsf{Aut}}(X)^*$ preserves the decomposition $K_X\oplus K_X^\perp$, and ${\mathsf{Aut}}(X)^*$ contains $W_X(2)$ when $X$ does not contain any smooth rational curve of self-intersection $-2$ (see [@Cantat-Dolgachev], Theorem 3.5). Another similarity with Enriques surfaces comes from the fact that Coble surfaces may be thought of as degeneracies of Enriques surfaces: an interesting difference is that $[K_X]$ is non trivial; in particular, ${{\mathrm{NS}}}(X;{\mathbf{Z}})_0$ is always non-trivial, for any $\Gamma\subset {\mathsf{Aut}}(X)$.
Also, there is a holomorphic section of $-2K_X$ vanishing exactly along the strict transform $C\subset X$ of the rational sextic curve $C_0$; this means that there is a meromorphic section $\Omega_X=\xi(x,y) (dx\wedge dy)^2$ of $K_X^{\otimes 2}$ that does not vanish and has a simple pole along $C$. Thus, the formula $${{\sf{vol}}}_X(U)=\int_U
{\left\vert\xi(x,y)\right\vert} dx\wedge dy\wedge d{\overline{x}}\wedge d{\overline{y}} =\int_U {\left\vert\xi(x,y)\right\vert} ({\mathsf{i}}dx\wedge d{\overline{x}}) \wedge ({\mathsf{i}} dy\wedge d{\overline{y}})$$ determines a finite measure([^2]) ${{\sf{vol}}}_X$ ($={\text{``}}\,\Omega_X^{1/2}\wedge {\overline{\Omega_X^{1/2}}}\,{\text{''}}$), which we may assume to be a probability after multiplying $\Omega_X$ by some adequate constant; this measure is $\Gamma$-invariant (for instance because $\Gamma$ is generated by involutions, see Remark \[rem:pm1\] below).
Another family of examples has been described by Blanc in [@Blanc:Michigan]. One starts with a smooth cubic curve $C_0\subset {\mathbb{P}}^2$. If $q_1$ is a point of $C_0$, there is a unique birational involution $s_1$ of ${\mathbb{P}}^2$ that fixes $C_0$ pointwise and preserves the pencil of lines through $q_1$. The indeterminacy points of $s_1$ are $q_1$ and the four tangency points of $C_0$ with this pencil (one of them may be “infinitely near $q_1$’’ and in that case it corresponds to the tangent direction of $C_0$ at $q_1$); thus the indeterminacies of $s_1$ are resolved by blowing-up points of $C_0$ (or of its strict transform). After such a sequence of blow-ups $s_1$ becomes an automorphism of a rational surface $X_1$ that preserves the strict transform of $C_0$. So, if we blow-up other points of this curve, $s_1$ lifts to an automorphism of the new surface. In particular, we can start with a finite number of points $q_i\in C_0$, $i=1, \ldots, k$, and resolve simultaneously the indeterminacies of the involutions $s_i$ determined by the $q_i$. The result is a surface $X$, with a subgroup $\Gamma:=\langle s_1, \ldots, s_k\rangle$ of ${\mathsf{Aut}}(X)$. Blanc proves that (1) there are no relations between these involutions, that is, $\Gamma$ is a free product $$\langle s_1, \ldots, s_k\rangle \simeq \bigast_{i=1}^{k} {\mathbf{Z}}/2{\mathbf{Z}},$$ (2) the composition of two distinct involutions $s_i\circ s_j$ is parabolic, and (3) the composition of three distinct involutions is loxodromic. Again, there is a meromorphic section $\Omega_X$ of $K_X$ with a simple pole along the strict transform $C$ of $C_0$, but on Blanc’s surfaces the form ${{\sf{vol}}}_X:=\Omega_X\wedge{\overline{\Omega_X}}$ is not integrable.
\[rem:pm1\] If $\Gamma\subset {\mathsf{Aut}}(X)$ is generated by involutions and there is a meromorphic form $\Omega$ such that $f^*\Omega=\xi(f)\Omega$ for every $f\in \Gamma$, then $\xi(f)=\pm 1$: this is the case for Blanc’s examples or general Coble surfaces, since $W_X(2)$ is also generated by involutions (see [@Dolgachev:Kyoto]).
Real forms {#par:Examples-Real-Forms}
----------
For each of the examples described in Sections \[par:Wehler\] to \[par:Coble-Blanc\], we may ask for the existence of an additional real structure on $X$, and look at the group of automorphisms ${\mathsf{Aut}}(X_{\mathbf{R}})$ that preserve the real structure (automorphisms commuting with the anti-holomorphic involution describing the real structure). Note that if $X$ is a smooth projective variety with a real structure, then $X({\mathbf{R}})$ is a compact, smooth, and totally real surface in $X$ (it may be empty). For instance, if $X$ is a Wehler surface defined by a polynomial equation $P(x_1,x_2,x_3)$ with real coefficients the $\sigma_i$ are automatically defined over ${\mathbf{R}}$. If $X$ is a Blanc surface for which $C_0$ is defined over ${\mathbf{R}}$ and the points $q_i$ are chosen in $C_0({\mathbf{R}})$, then again $\langle s_1, \ldots, s_k\rangle\subset {\mathsf{Aut}}(X_{\mathbf{R}})$. Real Enriques and Coble surfaces are harder to describe, but there are examples with non-elementary groups ${\mathsf{Aut}}(X_{\mathbf{R}})$ (see [@Degtyarev-Itenberg-Kharlamov:LNM]).
Surfaces admitting non-elementary groups of automorphisms {#subs:X-is-projective}
---------------------------------------------------------
The surfaces in the previous examples are all projective. This is a general fact, which we prove in this paragraph: we rely on the Kodaira-Enriques classification to describe compact Kähler surfaces which support a non-elementary group of automorphisms and prove Theorem \[thm:X-is-projective1\].
### Minimal models {#par:minimal_models}
We refer to Theorem 10.1 of [@Cantat:Milnor] for the following result:
\[thm:existence\_loxodromic\] If $X$ is a compact Kähler surface with a loxodromic automorphism, then
- either $X$ is a rational surface, and there is a birational morphism $\pi\colon X\to {\mathbb{P}}^2_{\mathbf{C}}$;
- or the Kodaira dimension of $X$ is equal to $0$, and there is an ${\mathsf{Aut}}(X)$-equivariant bimeromorphic morphism $\pi\colon X\to X_0$ such that $X_0$ is a compact torus, a K3 surface, or an Enriques surface.
In particular, $h^{2,0}(X)$ equals $0$ or $1$.
\[rem:volume\_form\] If $X$ is a torus or K3 surface, there is a holomorphic $2$-form $\Omega_X$ on $X$ that does not vanish and satisfies $\int_X\Omega_X\wedge {\overline{\Omega_X}} = 1$. It is unique up to multiplication by a complex number of modulus $1$. A consequence of utmost importance to us is that the volume form $$\Omega_X\wedge {\overline{\Omega_X}}$$ is ${\mathsf{Aut}}(X)$-invariant. Furthermore for every $f$ we can write $f^*\Omega_X=J(f)\Omega_X$, where the Jacobian $f\in {\mathsf{Aut}}(X)\mapsto J(f)\in {\mathbb{U}}_1$ is a unitary character on the group ${\mathsf{Aut}}(X)$. Since $H^{2,0}(X;{\mathbf{C}})$ is generated by $[\Omega_X]$, we obtain $$f^*w=J(f)w \quad \forall w \in H^{2,0}(X;{\mathbf{C}}).$$ If $Y$ is an Enriques surface, and $X\to Y$ is its universal cover, then $X$ is a K3 surface: the volume form $\Omega_X\wedge {\overline{\Omega_X}}$ is invariant under the group of deck transformations, and determines an ${\mathsf{Aut}}(Y)$-invariant volume form on $Y$. So, if $X$ is not rational, the dynamics of ${\mathsf{Aut}}(X)$ is conservative: it preserves a [**[canonical volume form]{}**]{} which is uniquely determined by the complex structure of $X$.
It follows from Theorem \[thm:existence\_loxodromic\] that, in most cases, ${\mathsf{Aut}}(X)$ is countable (see [@Cantat:Milnor Rmk 3.3]).
\[prop:discrete\] Let $X$ be a compact Kähler surface. If ${\mathsf{Aut}}(X)$ is non-elementary, then ${\mathsf{Aut}}(X)$ is discrete unless $X$ is a torus. More precisely, if ${\mathsf{Aut}}(X)$ contains a loxodromic element, then the kernel of the homomorphism ${\mathsf{Aut}}(X)\to {\mathsf{Aut}}(X)^*\subset {{\sf{GL}}}({{\mathrm{NS}}}(X;{\mathbf{Z}}))$ is finite unless $X$ is a torus.
### Projectivity {#subsub:X-is-projective}
\[thm:X-is-projective2\] Let $X$ be a compact Kähler surface and $\Gamma$ be a non-elementary subgroup of ${\mathsf{Aut}}(X)$. Then $X$ is projective, and is birationally equivalent to a rational surface, an abelian surface, a K3 surface, or an Enriques surface.
From the discussion in §§\[par:Wehler\]–\[par:Coble-Blanc\] we see that there exist examples with a non-elementary group of automorphisms for each of these four classes of surfaces. Theorem \[thm:X-is-projective2\] is a direct consequence of the following lemmas together with Theorem \[thm:existence\_loxodromic\].
\[lem:proj\_root\] Let $f$ be a loxodromic automorphism of a compact Kähler surface $X$. The following properties are equivalent:
1. on $H^{2,0}(X;{\mathbf{C}})$, $f^*$ acts by multiplication by a root of unity;
2. $X$ is projective.
If $X$ supports a loxodromic automorphism, then $\dim(H^{2,0}(X;{\mathbf{C}}))\leq 1$; and with notation as in Remark \[rem:volume\_form\], the first assertion is equivalent to
1. [*either $H^{2,0}(X;{\mathbf{C}})=0$ or $J(f)$ is a root of unity*]{}.
The characteristic polynomial $\chi_f$ of $f^*\colon H^2(X;{\mathbf{Z}})\to H^2(X;{\mathbf{Z}})$ is a monic polynomial with integer coefficients. Its decomposition into irreducible factors can be written as $$\chi_f(t)=S_f(t) \times R_f(t) = S_f(t) \times \prod_{i=1}^m C_{f,i}(t)$$ where $S_f(t)$ is a Salem polynomial or a reciprocal quadratic polynomial with a unique root $\lambda(f)>1$, and the $C_{f,i}(t)$ are cyclotomic polynomials. Indeed besides $\lambda(f)$ and $\lambda(f){^{-1}}$, all other roots of $\chi_f$ have modulus 1, so $\lambda(f)$ is a reciprocal quadratic integer or a Salem number (see § 2.4.3 of [@Cantat:Milnor] for more details). The other irreducible factors of $\chi_f$ are monic polynomials with integer coefficients with all their roots on the unit circle, so they must be cyclotomic polynomials. In particular if $\xi$ is an eigenvalue of $f^*$ and a root of unity, we see that $\xi$ is a root of $R_f(t)$ but not of $S_f(t)$.
The subspace $H^{2,0}({\mathbf{C}})\subset H^2(X;{\mathbf{C}})$ is $f^*$-invariant and, by Lemma \[lem:unitary\_on \_H20\], all eigenvalues of $f^*$ on that subspace have modulus $1$; if an eigenvalue of $f^*{ \arrowvert_{H^{2,0}(X;{\mathbf{C}})}}$ is not a root of unity, then it is a root of $S_f$.
Assume that all eigenvalues of $f^*$ on $H^{2,0}(X;{\mathbf{C}})$ are roots of unity. Then ${{\mathrm{Ker}}}(S_f(f^*))\subset H^2(X;{\mathbf{R}})$ is a $f^*$-invariant subspace of $H^{1,1}(X;{\mathbf{R}})$. This subspace is defined over ${\mathbf{Q}}$ and is of Minkowski type; in particular, it contains integral classes of positive self-intersection, and by the Kodaira embedding theorem, $X$ is projective. Conversely, assume that $X$ is projective. The Néron-Severi group ${{\mathrm{NS}}}(X;{\mathbf{Q}})\subset H^{1,1}(X;{\mathbf{R}})$ is $f^*$-invariant and contains vectors of positive self-intersection, so by Proposition \[pro:invariant\_cohomological\_decomposition\] it contains all isotropic lines associated to loxodromic automorphisms. Now any $f^*$ invariant subspace defined over ${\mathbf{Q}}$ and containing the eigenspace associated to $\lambda(f^*)$ contains ${{\mathrm{Ker}}}(S_f(f^*))$, so we deduce that ${{\mathrm{Ker}}}(S_f(f^*))\subset {{\mathrm{NS}}}(X;{\mathbf{Q}})$. In particular, ${{\mathrm{Ker}}}(S_f(f^*))$ does not intersect $H^{2,0}(X;{\mathbf{C}})$, which is invariant, and we conclude that all eigenvalues of $f^*$ on $H^{2,0}(X;{\mathbf{C}})$ are roots of unity.
\[lem:virtually\_cyclic\_non\_projective\] Let $X$ be a compact Kähler surface. If $X$ is not projective the group ${\mathsf{Aut}}(X)^*$ is virtually abelian.
Assume that ${\mathsf{Aut}}(X)^*$ is not virtually abelian, we want to prove that $X$ is projective. According to Theorem 3.2 of [@Cantat:Milnor], ${\mathsf{Aut}}(X)^*$ contains a non-abelian free group $\Gamma$ such that all elements of $\Gamma\setminus \{\operatorname{id}\}$ are loxodromic; from Theorem \[thm:existence\_loxodromic\], we deduce that either $h^{2,0}(X)=0$ or $X$ is the blow-up of a torus or a K3 surface. In the first case, the Hodge index and Kodaira embedding theorems show that $X$ is projective. In the second case, by uniqueness of the minimal model, the morphism $X\to X_0$ onto the minimal model $X$ is ${\mathsf{Aut}}(X)$-equivariant, so we can assume that $X=X_0$ is minimal and that $h^{2,0}(X)=1$. Consider the homomorphism $J\colon {\mathsf{Aut}}(X)\to {\mathbb{U}}_1$, as in Remark \[rem:volume\_form\]. Since ${\mathbb{U}}_1$ is abelian the kernel $\ker(J{ \arrowvert_{\Gamma}})$ contains loxodromic elements: indeed if $f,g\in \Gamma$ and $f\neq g$ then $[f,g] = f g f{^{-1}}g{^{-1}}$ is loxodromic and $J([f,g]) =1$. From Lemma \[lem:proj\_root\] we deduce that $X$ is projective.
Glossary of random dynamics, I {#sec:Glossary_I}
==============================
We now initiate the random iteration by introducing a probability measure on ${\mathsf{Aut}}(X)$. In this section we introduce a first set of ideas from the theory of random dynamical systems, as well as some notation that will be used throughout the paper.
Random holomorphic dynamical systems {#par:Random_holomorphic_dynamical_systems}
------------------------------------
Let $X$ be a compact Kähler surface, such that ${\mathsf{Aut}}(X)$ is non-elementary. Note that ${\mathsf{Aut}}(X)$ is locally compact for the topology of uniform convergence –in many interesting cases it is actually discrete (see Proposition \[prop:discrete\])– so it admits a natural Borel structure. We fix some Riemannian structure on $X$, for instance the one induced by the Kähler form $\kappa_0$. For $f\in {\mathsf{Aut}}(X)$, we denote by ${{\left\Vert f\right\Vert}}_{{{C}}^1}$ the maximum of ${{\left\Vert Df_x\right\Vert}}$ where the norm of $Df_x\colon T_xM\to T_{f(x)}M$ is computed with respect to this Riemannian metric.
We consider a probability measure $\nu$ on ${\mathsf{Aut}}(X)$ satisfying the **moment condition** (or integrability condition) $$\label{eq:moment}
\int {\left(\log{{\left\Vertf\right\Vert}}_{C^1(X)}+ \log{{\left\Vertf{^{-1}}\right\Vert}}_{C^1(X)}\right)} \, d\nu (f) < + \infty.$$ The norm ${{\left\Vert\,\cdot\,\right\Vert}}_{C^1(X)}$ is relative to our choice of Riemannian metric, but the finiteness of the integral in does not depend on this choice. In many interesting situations the support of $\nu$ will be finite, in which case the integrability , as well as stronger moment conditions which will appear later (see Conditions and ), are obviously satisfied.
The measure $\nu$ satisfies the moment condition if and only if it satisfies the higher moment conditions $$\label{eq:moment_Ck}
\int {\left(\log{{\left\Vertf\right\Vert}}_{C^k(X)}+ \log{{\left\Vertf{^{-1}}\right\Vert}}_{C^k(X)}\right)} \, d\nu (f) <\infty,$$ for all $k\geq 1$.
Here the $C^k$ norm is relative to the expression of $f$ in a system of charts (we don’t need to be precise here because only the finiteness in matters). This lemma follows from the Cauchy estimates. In particular, if $\nu$ satisfies , then $\nu$ satisfies an integrability property for the $C^2$ norm, which is required for Pesin’s theory.
Given $\nu$, we consider independent, identically distributed sequences $(f_n)_{n\geq 0}$ of random automorphisms of $X$ with distribution $\nu$, and study the dynamics of random compositions of the form $f_{n-1}\!\circ \cdots \circ \! f_0$. The data $(X, \nu)$ will thus be referred to as a [**[random holomorphic dynamical system]{}**]{} on $X$. Many properties of the random dynamical system $(X, \nu)$ depend on the properties of the subgroup $$\Gamma = \Gamma_\nu:=
\langle \operatorname{Supp}(\nu)\rangle$$ generated by (the support of) $\nu$ in ${\mathsf{Aut}}(X)$. If in addition $\Gamma_\nu$ is non-elementary, we say that $(X, \nu)$ is **non-elementary**.
\[rem:general\_RDS\] We are **not** considering the most general version of random holomorphic dynamical systems: one might consider compositions $f_{\vartheta^{n-1}(\xi)} \circ \cdots \circ f_{\vartheta(\xi)}\circ f_{\xi}$ where $\vartheta:\Sigma\to \Sigma$ is some measure preserving transformation of a probability space and $\Sigma\ni\xi\mapsto f_\xi\in {\mathsf{Aut}}(X)$ is measurable. The methods developed below do not apply to this more general setting.
Invariant and stationary measures
---------------------------------
Let $G$ be a topological group and $\nu$ be a probability measure on $G$. Consider a measurable action of $G$ on some measurable space $(M, \mathcal A)$. Every $f\in G$ determines a push-forward operator $\mu\mapsto f_\varstar\mu$, acting on positive measures $\mu$ on $(M, \mathcal A)$. By definition, a measure $\mu$ on $(M, \mathcal A)$ is [**[$\nu$-stationary]{}**]{} if $$\int f_\varstar\mu \, d\nu(f) = \mu,$$ and it is **$\nu$-almost surely invariant** if $f_\varstar\mu = \mu$ for $\nu$-almost every $f$. Most often we drop the mention to $\nu$ and simply talk about stationary and almost surely invariant measures. A stationary measure is **ergodic** if it is an extremal point of the closed convex set of stationary measures (see [@benoist-quint_book §2.1.3]).
If $\mu$ is almost surely invariant then it is stationary but the converse is generally false. If $M$ is compact (and ${{\mathcal A}}$ is the Borel $\sigma$-algebra), the Kakutani fixed point theorem implies the existence of at least one stationary measure. On the other hand the existence of an invariant measure is a very restrictive property. For instance, proximal, strongly irreducible linear actions on projective spaces have no (almost surely) invariant probability measure (see Sections \[par:intro\_stiffness\] and \[par:furstenberg\_measure\]). Following Furstenberg [@furstenberg_stiffness] we say that an action is **stiff** (or $\nu$-stiff) if any $\nu$-stationary measure is $\nu$-almost surely invariant.
We shall consider several measurable actions of the group ${\mathsf{Aut}}(X)$: its tautological action on $X$, but also its action of the projectivized tangent bundle ${\mathbb{P}}(TX)$, on cohomology groups of $X$ and their projectivizations, on spaces of currents, etc. In all cases, $M$ will be a locally compact space and $\mathcal A$ its Borel $
\sigma$-algebra, which will be denoted by $\mathcal B(M)$. Also $\mu$ will always be a probability measure, so from now on “stationary measure” stands for “stationary probability measure”.
\[rem:discrete\] Suppose $X$ is not a torus and ${\mathsf{Aut}}(X)$ is non-elementary. By Proposition \[prop:discrete\], ${\mathsf{Aut}}(X)$ is a discrete group so $\mu$ is $\nu$-almost surely invariant if and only if it is $\Gamma_\nu$-invariant. If $X$ is a torus ${\mathbf{C}}^2/\Lambda$, the connected component ${\mathsf{Aut}}(X)^\circ$ of $\operatorname{id}$ is the group of translations $({\mathbf{C}}^2/\Lambda,+)$; then, $\Gamma^0_\nu:=\Gamma_\nu\cap {\mathsf{Aut}}(X)^\circ$ acts equicontinuously on $X$ and if $\mu$ is almost surely invariant, it is automatically invariant under the action of the real Lie group ${\overline{\Gamma_\nu^0}}\subset {\mathbf{C}}^2/\Lambda$. Thus, in all cases, if $\mu$ is almost surely invariant, then it is indeed invariant under the action of $\Gamma_\nu$.
Random compositions
-------------------
Set $\Omega = {\mathsf{Aut}}(X)^{\mathbf{N}}$. Endowed with its product topology it is a locally compact space. The associated Borel $\sigma$-algebra coincides with the product $\sigma$-algebra, and it is generated by cylinders (see § \[par:definition\_skew\_products\]). We endow $\Omega$ with the product measure $\nu^{\mathbf{N}}$. Choosing a random element in $\Omega$ with respect to $\nu^{\mathbf{N}}$ is equivalent to choosing an i.i.d. random sequence of automorphisms in ${\mathsf{Aut}}(X)$ with distribution $\nu$. For $\omega\in \Omega$, we let $f_\omega = f_0$ and denote by $f^n_\omega$ the left composition of the $n$ first terms of $\omega$, that is $$f^n_\omega= f_{n-1}\circ \cdots \circ f_0$$ for $n>0$. By definition $f^0_\omega = \mathrm{id}$. Let us record for future reference the following consequence of the Borel-Cantelli lemma.
\[lem:borel-cantelli\_moment\] If $(X, \nu)$ is a random dynamical system satisfying the moment condition , then for $\nu^{\mathbf{N}}$-almost every sequence $\omega=(f_n)\in \Omega$, $${\frac{1}{n}} {\left(\log{{{\left\Vertf_n\right\Vert}}_{ {C}^1}} + \log{{{\left\Vertf_n{^{-1}}\right\Vert}}_{ {C}^1}}\right)}{\underset{n\to\infty}{\longrightarrow}}0.$$
Furstenberg theory in $H^{1,1}(X;{\mathbf{R}})$ {#sec:furstenberg}
===============================================
Consider a non-elementary random holomorphic dynamical system $(X,\nu)$ on a compact Kähler surface, satisfying the moment condition . The main purpose of this section is to analyze the linear action of $(X, \nu)$ on $H^{1,1}(X, {\mathbf{R}})$ by way of the theory of random products of matrices. Basic references for this subject are the books by Bougerol and Lacroix [@bougerol-lacroix] and by Benoist and Quint [@benoist-quint_book].
Moments and cohomology {#subs:moments_cohomology}
----------------------
We start with a general discussion on the dilatation of cohomology classes under smooth transformations. Let $M$ be a compact connected manifold of dimension $m$, endowed with some Riemannian metric ${{\mathrm{g}}}$. If $f\colon M\to M$ is a smooth map, as before we denote by ${{\left\Vertf\right\Vert}}_{C^1}$ the maximum norm of its tangent action, computed with respect to ${{\mathrm{g}}}$ (see Section \[par:Random\_holomorphic\_dynamical\_systems\]). Thus, $f$ is a Lipschitz map with $\operatorname{{Lip}}(f)={{\left\Vertf\right\Vert}}_{C^1}$ for the distance determined by ${{\mathrm{g}}}$; in particular ${{\left\Vertf\right\Vert}}_{C^1}\geq 1$. Fix a norm ${\left\vert \cdot \right\vert}_{H^k}$ on each cohomology group $H^k(M;{\mathbf{R}})$, for $0\leq k\leq m$.
There is a constant $C>0$, that depends only on $M$, ${{\mathrm{g}}}$, and the norms ${\left\vert \cdot \right\vert}_{H^k}$, such that $${\left\vert f^*[\alpha]\right\vert}_{H^k} \leq C^k \operatorname{{Lip}}(f)^k {\left\vert [\alpha]\right\vert}_{H^k}$$ for every class $[\alpha]\in H^k(M;{\mathbf{R}})$ and every map $f\colon M\to M$ of class $C^1$. In other words, the operator norm ${{\left\Vert f^*\right\Vert}}_{H^k}$ is controlled by the Lipschitz constant: $$\label{eq:gromov}
{{\left\Vert f^*\right\Vert}}_{H^k}\leq C^k \operatorname{{Lip}}(f)^k \leq C^k {{\left\Vert f\right\Vert}}_{C^1}^k.$$
Pick a basis of the homology group $H_k(M;{\mathbf{R}})\simeq H^k(M;{\mathbf{R}})^*$ given by smoothly immersed, compact, $k$-dimensional manifolds $\iota_i\colon N_i\to M$, and a basis of $H^k(M;{\mathbf{R}})$ given by smooth $k$-forms $\alpha_j$. Then, the integral $\int_{N_i} \iota_i^*(f^*\alpha_j)$ is bounded from above by $C^k{{\left\Vertf^*\right\Vert}}_{C^1}^k$ for some constant $C$, because $$\vert (f^*\alpha_j)_x(v_1, \ldots, v_k)\vert = \vert \alpha_j(f_*v_1, \ldots, f_* v_k) \vert \leq c_j {{\left\Vertf\right\Vert}}_{C^1}^k \prod_{\ell=1}^k{\left\vertv_\ell\right\vert}_{{{\mathrm{g}}}}$$ for every point $x\in M$ and every $k$-tuple of tangent vectors $v_\ell\in T_xM$; here, $c_j$ is the supremum of the norm of the multilinear map $(\alpha_j)_x$ over $x\in M$.
If $\nu$ is a probability measure on ${{\mathsf{Diff}}}(M)$ satisfying the moment condition , it follows that: $$\label{eq:moment_Hk}
\int_{{{\mathsf{Diff}}}(M)} \log{\left({{\left\Vert f^*\right\Vert}}_{H^k}\right)} +\log {\left({{\left\Vert(f^{-1})^*\right\Vert}}_{H^k}\right)} \, d\nu(f)<+\infty.$$ If we specialize this to automorphisms of compact Kähler surfaces we get $$\label{eq:moment_H1_both_sides}
\int_{{\mathsf{Aut}}(X)} \log{\left({{\left\Vert f^*\right\Vert}}_{H^{1, 1}}\right)} +\log {\left({{\left\Vert(f^{-1})^*\right\Vert}}_{H^{1, 1}}\right)} \; d\nu(f)<+\infty,$$ which is actually equivalent to by Lemma \[lem:cohomological\_norm\_estimates\]. We saw in §\[par:hyp\_X\] that ${{\left\Vertf^*\right\Vert}}_{H^{1, 1}} \asymp {{\left\Vert(f{^{-1}})^*\right\Vert}}_{H^{1, 1}}$, so this last condition is in turn equivalent to $$\label{eq:moment_H1_final}
\int_{{\mathsf{Aut}}(X)} \log{\left({{\left\Vert f^*\right\Vert}}_{H^{1, 1}}\right)} \; d\nu(f)<+\infty.$$
Cohomological Lyapunov exponent {#subs:furstenberg}
-------------------------------
From now on we denote by ${\left\vert\cdot\right\vert}$ a norm on $H^{1,1}(X, {\mathbf{R}})$ and by ${{\left\Vert\cdot\right\Vert}}$ the associated operator norm. The linear action induced by the random dynamical system $(X, \nu)$ on $H^{1,1}(X, {\mathbf{R}})$ defines a random product of matrices. Since the moment condition is satisfied, we can define the **upper Lyapunov exponent** $\lambda_{H^{1,1}}$ (or $\lambda_{H^{1,1}}(\nu)$) by $$\begin{aligned}
\label{eq:def_LE}
\lambda_{H^{1,1}} &= \lim_{n\to\infty} \frac{1}{n}\int \log({{\left\Vert (f^n_\omega)^*\right\Vert}}) d\nu^{\mathbf{N}}(\omega)\\
& = \lim_{n\to +\infty} \frac{1}{n}\log{{\left\Vert (f^n_\omega)^*\right\Vert}}\label{eq:def_LE2}\end{aligned}$$ where the second equality holds almost surely, i.e. for $\nu^{\mathbf{N}}$-almost every $\omega\in \Omega$. This convergence follows from Kingman’s subadditive ergodic theorem, since, ${{\left\Vert\cdot\right\Vert}}$ being an operator norm, $(\omega, n)\mapsto \log({{\left\Vert (f^n_\omega)^*\right\Vert}})$ defines a subadditive cocycle (see [@benoist-quint_book Thm 4.28] or [@bougerol-lacroix Thm I.4.1]). Note that $(f_\omega^n)^* = f_{0}^*\circ \cdots \circ f_{n-1}^*$, so we are dealing with right compositions instead of the usual left composition. However since $f_{0}^*\circ \cdots \circ f_{n-1}^*$ has the same distribution as $f_{n}^*\circ \cdots \circ f_{0}^*$, the Lyapunov exponent in corresponds to the most common definition of the upper Lyapunov exponent of the random product of matrices. We refer to [@bougerol-lacroix; @Ledrappier:SaintFlour] for the definition and main properties of the subsequent Lyapunov exponents (see also [@benoist-quint_book §10.5]).
Let $(X, \nu)$ be a non-elementary holomorphic dynamical system on a compact Kähler surface, satisfying the moment condition , or more generally . Then the cohomological Lyapunov exponent $\lambda_{H^{1,1}}$ is positive and the other Lyapunov exponents of the linear action on $H^{1,1}(X, {\mathbf{R}})$ are $-\lambda_{H^{1,1}}$, with multiplicity $1$, and $0$, with multiplicity $h^{1,1}(X)-2$.
Consider the $\Gamma_\nu$-invariant decomposition $\Pi_{\Gamma_\nu}\oplus \Pi_{\Gamma_\nu}^\perp$ given by Proposition \[pro:NS\_H11\] and Equation . Since the intersection form is negative definite on $\Pi_{\Gamma_\nu}^\perp$, the group $\Gamma_\nu^*{ \arrowvert_{\Pi_{\Gamma_\nu}^\perp}}$ is bounded and all Lyapunov exponents of $\Gamma_\nu^*{ \arrowvert_{\Pi_{\Gamma_\nu}^\perp}}$ vanish. The linear action of $\Gamma_\nu$ on $\Pi_{\Gamma_\nu}$ is strongly irreducible and non-elementary, hence not relatively compact. Therefore Furstenberg’s theorem asserts that $\lambda_{H^{1,1}}>0$ (see e.g. [@bougerol-lacroix Thm III.6.3] or [@benoist-quint_book Cor 4.32]), and the remaining properties of the Lyapunov spectrum on $\Pi_{\Gamma_\nu}$ follow from the KAK decomposition in ${{\sf{O}}}^+_{1,m}({\mathbf{R}})$, with $1+m=\dim(\Pi_{\Gamma_\nu})$ (see Lemma \[lem:KAK\]).
\[lem:growth\] If $a$ is any class such that $a^2>0$, for instance if $a$ is a Kähler class, then $$\lim_{n\to +\infty}\frac{1}{n}\log {\left\vert (f_\omega^n)^*a\right\vert} =\lambda_{H^{1,1}}$$ for $\nu^{\mathbf{N}}$-almost every $\omega$.
Corollary \[cor:KAK\] implies that if $a\in {\mathbb{H}}_X$ then for every $f\in {\mathsf{Aut}}(X)$, ${\left\vertf^*a\right\vert}\asymp {{\left\Vertf^*\right\Vert}}$, where the implied constants depend only on $a$. Thus the result follows from Equation .
\[rem:order\_composition\] It is natural to expect that the result of Lemma \[lem:growth\] holds for any $a\in \Pi_\Gamma\setminus {\left\{0\right\}}$. This can indeed be established under the more stringent moment assumption . See below §\[subsub:exponential\_moments\] for a discussion.
If the order of compositions is reversed (which is less natural from the point of view of iterated pull-backs), then Lemma \[lem:growth\] indeed holds for any $a$ in $\Pi_{\Gamma_\nu}$ (see [@bougerol-lacroix Cor. III.3.4.i]):
\[lem:growth2\] For any $a\in \Pi_{\Gamma_\nu}$ and for $\nu^{\mathbf{N}}$-almost every $\omega=(f_n)_{n\geq 0} \in \Omega$ we have $$\lim_{n\to +\infty}\frac{1}{n}\log {\left\vert f_n^*\cdots f_1^* a\right\vert} =\lambda_{H^{1,1}}.$$
The measure $\mu_\partial$ {#par:furstenberg_measure}
--------------------------
By Furstenberg’s theory the linear projective action of the random dynamical system $(X, \nu)$ on ${\mathbb{P}}\Pi_{\Gamma_\nu} \subset {\mathbb{P}}H^{1,1}(X;{\mathbf{R}})$ admits a unique stationary measure $\mu_{{\mathbb{P}}\Pi_{\Gamma_\nu}}$; this measure does not charge any proper projective subspace of ${\mathbb{P}}\Pi_{\Gamma_\nu}$.
\[lem:definition\_e\] For $\nu^{\mathbf{N}}$-almost every $\omega$ there exists a unique nef class $e(\omega)$ such that ${{\mathbf{M}}}(e(\omega))=1$ and $$\label{eq:definition_e}
\frac{1}{{{\mathbf{M}}}((f_\omega^n)^*a)} (f_\omega^n)^*a \underset{n\to\infty}\longrightarrow e(\omega)$$ for any pseudo-effective class $a$ with $a^2>0$ (in particular for any Kähler class). In addition, the class $e(\omega)$ is almost surely isotropic and ${\mathbb{P}}(e(\omega))$ is a point of the limit set $\operatorname{Lim}(\Gamma_\nu)\subset\partial{\mathbb{H}}_X$.
Before starting the proof, note that $\Gamma_\nu^*{ \arrowvert_{\Pi_{\Gamma_\nu}}}$ is proximal, in the sense of [@benoist-quint_book §4.1]; equivalently, $\Gamma_\nu^*{ \arrowvert_{\Pi_{\Gamma_\nu}}}$ is contracting, in the sense of [@bougerol-lacroix Def III.1.3]. In other words, there are sequences of elements $g_n\in \Gamma_\nu$ such that ${{\left\Vertg_n^*\right\Vert}}^{-1}g_n^*{ \arrowvert_{\Pi_{\Gamma_\nu}}}$ converges to a matrix of rank $1$: for instance one can take $g_n=f^n$, where $f\in \Gamma_\nu$ is any loxodromic automorphism.
For $f\in {\mathsf{Aut}}(X)$, we use the notation $\underline f^*$ for its action on ${\mathbb{P}}H^{1,1}(X;{\mathbf{R}}) $. Since the action of $\Gamma_\nu$ on $\Pi_{\Gamma_\nu}$ is strongly irreducible and proximal, its projective action satisfies the following contraction property (see [@bougerol-lacroix Thm III.3.1]): there is a measurable map $\omega\in \Omega\mapsto \underline{e}(\omega)\in {\mathbb{P}}\Pi_{\Gamma_\nu}$ such that for almost every $\omega$, any cluster value $L(\omega)$ of $${\frac{1}{{{\left\Vertf_0^*\cdots f_n^* \right\Vert}}}}f_0^*\cdots f_n^*$$ in ${{\sf{End}}}(\Pi_{\Gamma_\nu})$ is an endomorphism of rank $1$ whose range is equal to ${\mathbf{R}}\underline{e}(\omega)$.
Let $e(\omega)$ be the unique vector of mass $1$ in the line ${\mathbf{R}}\underline{e}(\omega)$. If $a\in \Pi_{\Gamma_\nu}$ is pseudo-effective and $a^2>0$, then any cluster value of ${{\mathbf{M}}}((f_\omega^n)^*a)^{-1}(f_\omega^n)^*a$ must coincide with $e(\omega)$ because by Corollary \[cor:KAK\] the mass ${{\mathbf{M}}}((f_\omega^n)^*a)$ is comparable to the norm ${{\left\Vertf_0^*\cdots f_n^* \right\Vert}}$. Thus, the convergence of Equation is satisfied. Furthermore $e(\omega)$ is nef, because $a$ is nef and ${\mathsf{Aut}}(X)$ preserves the nef cone, and $e(\omega)$ belongs to $\operatorname{Lim}(\Gamma_\nu)$. In particular, $e(\omega)$ is isotropic.
Now, let $a$ and $a'$ be two classes of ${\mathbb{H}}_X$ with $a\in \Pi_{\Gamma_\nu}$. Since the hyperbolic distance between $ (f_\omega^n)^*(a)$ and $(f_\omega^n)^*(a')$ remains constant and the convergence holds for $a$, it also holds for $a'$. This concludes the proof because every class with positive self-intersection is proportional to a unique class in ${\mathbb{H}}_X$.
As in Remark \[rem:order\_composition\], under the exponential moment condition , the convergence in Equation holds for any $a\in \Pi_\Gamma\setminus {\left\{0\right\}}$ and almost every $\omega\in \Omega$; to be precise, $\frac{1}{{{\mathbf{M}}}((f_\omega^n)^*a)} (f_\omega^n)^*a$ converges towards $e(\omega)$ or its opposite. Then, we actually get the convergence for any $a\in H^{1,1}(X;{\mathbf{R}})\setminus \Pi_\Gamma^\perp$, by writing $a = a_++a_0 $ and using the fact that $\Gamma_\nu$ acts by isometries on $\Pi_\Gamma^\perp$.
Here is a summary of the properties of the stationary measure $\mu_{{\mathbb{P}}\Pi_{\Gamma_\nu}}$; from now on, we view it as a measure on ${\mathbb{P}}H^{1,1}(X;{\mathbf{R}})$ and rename it as $\mu_{\partial}$ because it is supported on ${\partial}{\mathbb{H}}_X$.
\[thm:def\_stationary\] The probability measure $$\label{eq:representation_mufr}
\mu_{\partial}=\int \delta_{{\mathbb{P}}(e(\omega))}\, d\nu^{\mathbf{N}}(\omega)$$ is $\nu$-stationary and ergodic. It is the unique stationary measure on ${\mathbb{P}}H^{1,1}(X;{\mathbf{R}}) $ such that $\mu_\partial({\mathbb{P}}(\Pi_{\Gamma_\nu}^\perp))=0$. The measure $\mu_\partial$ has no atoms and is supported on $\operatorname{Lim}(\Gamma_\nu)$; in particular, if $\Lambda'\subset \operatorname{Lim}(\Gamma_\nu)$ is such that $\mu_\partial(\Lambda')>0$ then $\Lambda'$ is uncountable.
The top Lyapunov exponent satisfies the so-called Furstenberg formula: $$\begin{aligned}
\label{eq:Furstenberg_formula1}\lambda_{H^{1,1}} & = \int \log\left( \frac{{\left\vert f^*{\tilde{u}}\right\vert}}{{\left\vert {\tilde{u}}\right\vert}}\right)\, d\nu(f)\, d\mu_\partial(u), \end{aligned}$$ where $\tilde u$ in $H^{1,1}(X, {\mathbf{R}})$ denotes any lift of $u\in \operatorname{Lim}(\Gamma_\nu) \subset {\mathbb{P}}H^{1,1}(X, {\mathbf{R}})$.
The ergodicity of $\mu_{\partial}=\mu_{{\mathbb{P}}\Pi_{\Gamma_\nu}}$ as well as its representation follow from the properties of the action of $\Gamma_\nu$ on ${\mathbb{P}}(\Pi_\Gamma)$ (see [@bougerol-lacroix Chap. III]). Also, we know that $\lambda_{H^{1,1}}$ is equal to the top Lyapunov exponent of the restriction of the action to ${\mathbb{P}}(\Pi_{\Gamma_\nu})$, so the formula follows from the strongly irreducible case (see [@bougerol-lacroix Cor III.3.4]).
Let now $\mu$ be a stationary measure on ${\mathbb{P}}H^{1,1}(X;{\mathbf{R}})$ such that $\mu({\mathbb{P}}\Pi_{\Gamma_\nu}^\perp)=0$. A martingale convergence argument shows that $(\underline f_\omega^n)^*\mu$ converges to some measure $\mu_\omega$ for almost every $\omega$ (see [@bougerol-lacroix Lem. II.2.1]). Since $\Gamma_\nu$ preserves the decomposition $\Pi_{\Gamma_\nu} \oplus \Pi_{\Gamma_\nu}^\perp$ and ${{\left\Vert (f_\omega^n)^*\right\Vert}}$ tends to infinity while $\parallel (f_\omega^n)^*{ \arrowvert_{\Pi_{\Gamma_\nu}^\perp}} \parallel$ stays uniformly bounded, we get that $(f_\omega^n)^*u$ converges to ${\mathbb{P}}\Pi_{\Gamma_\nu}$ for $\mu$-almost every $u$ and $\nu^{\mathbf{N}}$-almost every $\omega$; thus $\mu_\omega$ is almost surely supported on ${\mathbb{P}}\Pi_{\Gamma_\nu}$. Since by stationarity $\mu = \int \mu_\omega d\nu^{\mathbf{N}}(\omega)$ we conclude that $\mu$ gives full mass to ${\mathbb{P}}(\Pi_{\Gamma_\nu})$, hence $\mu = \mu_{\partial}$.
If $\operatorname{Supp}(\nu)$ generates $\Gamma_\nu$ as a semi-group, then $\operatorname{Supp}(\mu_\partial) = \operatorname{Lim}(\Gamma_\nu)$, otherwise the inclusion can be strict: think of a Schottky group $\Gamma = \langle f,g \rangle \subset \mathsf{PSL}(2, {\mathbf{R}})$ and $\nu = (\delta_f+\delta_g)/2$.
\[rem:lambda\_with\_log\_mass\] Since $\operatorname{Lim}(\Gamma_\nu)\subset {{\mathrm{Psef}}}(X)$, for every $u\in \operatorname{Lim}(\Gamma_\nu)$ there exists a unique $\tilde u$ such that ${\mathbb{P}}\tilde u = u$ and $\langle \tilde{u}\, \vert \, [\kappa_0]\rangle= {{\mathbf{M}}}(\tilde u) = 1$. Then the following formula holds: $$\begin{aligned}
\label{eq:Furstenberg_formula2}
\lambda_{H^{1,1}} = \int \log {\left({{\mathbf{M}}}(f^*{\tilde{u}})\right)} \, d\nu(f) \, d\mu_\partial(u) = \int \log\left( \frac{{{\mathbf{M}}}(f^*{\tilde{u}})}{{{\mathbf{M}}}({\tilde{u}})}\right) \, d\nu(f) \, d\mu_\partial(u). \end{aligned}$$ Indeed set $r(w)={{\mathbf{M}}}(w)/{\left\vert w\right\vert}$. On the limit set this function satisfies $1/C\leq r({\tilde{u}})\leq C$, where $C$ is the positive constant from Equation . Then, the stationarity of $\mu_\partial$ implies that for all $m\geq 1$, $$\int \log\left( \frac{r( f^*{\tilde{u}})}{r({\tilde{u}})} \right) \, d\nu (f) \, d\mu_\partial(u) = \\
\int \log\left( \frac{r( f_m^* \cdots f_0^*{\tilde{u}})}{r(f_{m-1}^* \cdots f_0^*{\tilde{u}})}\right)\, d\nu (f_{m})\cdots d\nu (f_0)\, d\mu_\partial(u).$$ Summing from $m=0$ to $n-1$, telescoping the sum, and dividing by $n$ gives $$\int \log\left( \frac{r( f^*{\tilde{u}})}{r({\tilde{u}})} \right) \, d\nu (f) \, d\mu_\partial(u) =
\frac1n \int \log\left( \frac{r( f_{n-1}^* \cdots f_0^*{\tilde{u}})}{r({\tilde{u}})}\right)\, d\nu (f_{n-1})\cdots d\nu (f_0)\, d\mu_\partial(u).$$ Finally since $1/C\leq r \leq C$, the right hand side tends to zero as $n\to\infty$. Hence the integral of $\log(r\circ f^*/r)$ vanishes, and follows from Furstenberg’s formula.
\[pro:extremality\] The point $e(\omega)$ is $\nu^{\mathbf{N}}$-almost surely extremal in ${\mathbb{P}}({{\overline{{\mathrm{Kah}}}}}(X))$ and in ${\mathbb{P}}({{\mathrm{Psef}}}(X))$.
The class $e(\omega)$ almost surely belongs to the isotropic cone and the boundary of the Kähler cone. Hence, by the Hodge index theorem – more precisely by the case of equality in the reverse Schwarz Inequality –, $e(\omega)$ cannot be a non-trivial convex combination of classes with non-negative intersection, so it is an extremal point of the convex set ${\mathbb{P}}({{\overline{{\mathrm{Kah}}}}}(X))\subset {\mathbb{P}}H^{1,1}(X;{\mathbf{R}})$.
From Proposition \[pro:extremal\_rays\], there are at most countably many points ${\mathbb{P}}(u)$ in ${\mathbb{P}}({{\overline{{\mathrm{Kah}}}}}(X))$ such that $u^2=0$ and ${\mathbb{P}}(u)$ is not extremal in ${\mathbb{P}}({{\mathrm{Psef}}}(X))$. Therefore the second assertion follows from the fact that $\mu_{\partial}$ is atomless.
Some estimates for random products of matrices
----------------------------------------------
### Sequences of good times
We now describe a theorem of Gouëzel and Karlsson, specialized to our specific context. Fix a base point $e_0$ in the hyperbolic space ${\mathbb{H}}_X$, for instance $e_0=[\kappa_0]$ with $\kappa_0$ a fixed Kähler form (as in Section \[par:cones\_definition\]). Consider the two functions of $(n,\omega)\in {\mathbf{N}}\times \Omega$ defined by $$\label{eq:two_cocycles}
T(n,\omega)= d_{\mathbb{H}}(e_0, (f_\omega^n)^*e_0), \quad N(n,\omega)=\log {{\left\Vert (f_\omega^n)^*\right\Vert}}.$$ They satisfy the subadditive cocycle property $$a(n+m, \omega)\leq a(n,\omega)+a(m,\sigma^n(\omega)),$$ where $\sigma$ is the unilateral shift on $\Omega$. Let $a(n,\omega)$ be such a subadditive cocycle; if $a(1,\omega)$ is integrable the asymptotic average is defined to be the limit $$A=\lim_{n\to +\infty} \frac{1}{n}\int a(n,\omega)\, d\nu^{\mathbf{N}}(\omega);$$ it exists in $[-\infty,+\infty)$, and we say it is finite if $A\neq -\infty$. The functions $T$ and $N$ are examples of ergodic subadditive cocycles and from Theorem \[thm:def\_stationary\], Remark \[rem:lambda\_with\_log\_mass\], and Corollary \[cor:KAK\], we deduce that the asymptotic average of each of these cocycles is equal to $ \lambda_{H^{1,1}}$.
Following [@Gouezel-Karlsson], we say that $a(n,\omega)$ is [**[tight along the sequence of positive integers $(n_i)$]{}**]{} if there is a sequence of real numbers $(\delta_\ell) = (\delta_\ell(\omega))_{\ell\geq 0}$ such that
- $\delta_\ell$ converges to $0$ as $\ell$ goes to $+\infty$;
- for every $i$, and for every $0\leq \ell \leq n_i$, $${\left\vert a(n_i,\omega)-a(n_i-\ell,\sigma^\ell(\omega))-A\ell\right\vert} \leq \ell \delta_\ell;$$
- for every $i$ and for every $0\leq \ell \leq n_i$ $$a(n_i,\omega)-a(n_i-\ell,\omega)\geq (A-\delta_\ell)\ell.$$
\[thm:GK\_good\_times\] Let $a(n,\omega)$ be an ergodic subadditive cocycle, with a finite asymptotic average $A$. Then, for almost every $\omega$, the cocycle is tight along a subsequence $(n_i(\omega))$ of positive upper density.
Recall that the (asymptotic) [[upper density]{}]{} of a subset $S$ of ${\mathbf{N}}$ is the non-negative number defined by $${\overline{\mathrm{dens}}}(S)=\limsup_{k\to +\infty} \left(\frac{\vert S\cap [0,k-1]\vert}{k} \right).$$ A sequence $(n_i)_{i\geq 0}$ is said to have positive upper density if its set of values $S=\{ n_i\; ; \; i\geq 0\}$ satisfies ${\overline{\mathrm{dens}}}(S) >0$.
Let us explain how this result follows from [@Gouezel-Karlsson]. First, fix a small positive real number $\rho>0$, and apply Theorem 1.1 and Remark 1.2 of [@Gouezel-Karlsson] to get a set $\Omega_\rho$ of measure $1-\rho$ such that the first two properties (i) and (ii) are satisfied for every $\omega\in \Omega_\rho$ with respect to a sequence $(\delta_\ell)$ that does not depend on $\omega$, and for a sequence of times $(n_i(\omega))$ of upper density $\geq 1-\rho$.
To get (iii), we apply Lemma 2.3 of [@Gouezel-Karlsson] to the sub-additive cocycle $a(n,\omega)$ (not to the cocycle $b(n,\omega)= a(n, \sigma^{-n}(\omega))$ as done in [@Gouezel-Karlsson]). For every ${\varepsilon}>0$, there is a subset $\Omega'_{\varepsilon}\subset \Omega$ and a sequence $(\delta'_\ell)_{\ell \geq 0}$, with the following properties:
- $\nu^{\mathbf{N}}(\Omega'_{\varepsilon})>1-{\varepsilon}$ and $\delta'_\ell$ converges towards $0$ as $\ell$ goes to $+\infty$;
- for every $\omega\in \Omega'_{\varepsilon}$, there is a set of bad times $B(\omega)\subset {\mathbf{N}}$ such that for every $k\geq 0$ ${\left\vert B(\omega)\cap [0,k-1]\right\vert}\leq {\varepsilon}k$ and such that for every $n\notin B(\omega)$ and every $0\leq \ell \leq n$, $$a(n,\omega)-a(n-\ell,\omega)\geq (A-\delta'_\ell) \ell.$$
If $\omega$ belongs to the intersection $\Omega_\rho\cap \Omega'_\varepsilon $, the set of indices $i$ for which $n_i(\omega)\notin B(\omega)$ is infinite. More precisely, the set $S(\omega)=\{ n_j(\omega) \; ; \; n_j(\omega)\notin B(\omega)\}$ has asymptotic upper density $\geq 1-\rho - \varepsilon $. Along this subsequence, the three properties (i), (ii), and (iii) are satisfied. Since this holds for every $\omega \in \Omega'_\varepsilon \cap \Omega_\rho$ and the measure of this set is $\geq 1-\rho-\varepsilon $, this holds for $\nu^{\mathbf{N}}$-almost every $\omega$.
\[cor:GK\_good\_times\] For $\nu^{\mathbf{N}}$-almost every $\omega\in {\mathsf{Aut}}(X)^{\mathbf{N}}$, there is an increasing sequence of integers $(n_i(\omega))$ going to $+\infty$ and a real number $A(\omega)$ such that $$\label{eq:cor_GK}
\sum_{j=0}^{n_i(\omega)} \frac{\big\|{ \big(f_\omega^{j}\big)^*}\big\|}{\big\|{ \big(f_\omega^{n_i(\omega)}\big)^*}\big\|} \leq A(\omega) \; {\text{ and }} \;
\sum_{j=0}^{n_i(\omega)} \frac{\big\|{ \big(f_{\sigma^j(\omega)}^{n_i(\omega)-j}\big)^*}\big\|}{\big\|{ \big(f_\omega^{n_i(\omega)}\big)^*}\big\|} \leq A(\omega)$$for all indices $i\geq 0$.
Apply Theorem \[thm:GK\_good\_times\] to the subadditive cocyle $N(n,\omega)$ and note that $$\sum_{j=0}^{n_i(\omega)} \frac{\big\|{ \big(f_\omega^{j}\big)^*}\big\|}{\big\|{ \big(f_\omega^{n_i(\omega)}\big)^*}\big\|} =
\sum_{\ell=0}^{n_i(\omega)} \frac{\big\|{ \big(f_\omega^{n_i- \ell}\big)^*}\big\|}{\big\|{ \big(f_\omega^{n_i}\big)^*}\big\|} =
\sum_{\ell=0}^{n_i(\omega)} \frac{e^{N(n_i-\ell,\omega) }}{e^{ N(n_i, \omega)} }\leq
\sum_{\ell=0}^{n_i(\omega)} e^{-\ell (\lambda_{H^{1,1}}-\delta_\ell)}$$ which is bounded as $n_i(\omega)\to \infty$. The second estimate in is similar.
### A mass estimate for pull-backs
Assume that $(X, \nu)$ is non-elementary and satisfies the condition . Recall from Lemma \[lem:growth2\] that ${{\mathbf{M}}}((f_\omega^n)^* a){^{-1}}(f_\omega^n)^*a$ converges to the pseudo-effective class $e(\omega)$ for almost every $\omega$ and every Kähler class $a$. Thus, on a set of total $\nu^{\mathbf{N}}$-measure, this convergence holds for all $\sigma^k(\omega)$, $k\geq 0$. Since ${{\mathbf{M}}}(e(\omega)) = 1$, we obtain $$\label{eq:equivariance_e}
f_0^* e(\sigma\omega) = {{\mathbf{M}}}(f_0^*e(\sigma\omega)) e(\omega);$$ more generally, $$(f_\omega^k)^* e(\sigma^k\omega) = {{\mathbf{M}}}((f_\omega^k)^*e(\sigma^k\omega)) e(\omega)$$ for every $k\geq 1$.
\[lem:variant\_furstenberg\] For $\nu^{\mathbf{N}}$-almost every $\omega$, we have $$\frac1{n} \log {{\mathbf{M}}}((f_\omega^n)^*e(\sigma^n\omega)) \underset{n\to\infty}{\longrightarrow}\lambda_{H^{1,1}}.$$
This does *not* follow from Lemma \[lem:growth\] because $e(\sigma^n\omega)$ depends on $n$. Our argument relies on Theorem \[thm:GK\_good\_times\] for convenience but other strategies could certainly be applied.
For almost every $\omega$, for every $k\geq 1$, and for every Kähler class $a$, we have $$e(\sigma^k\omega) = \lim_{n\to\infty} \frac{f_k^*\cdots f_{n-1}^* a}{{{\mathbf{M}}}(f_k^*\cdots f_{n-1}^* a)}.$$ So $$\label{eq:zetakf}
f_0^*\cdots f_{k-1}^* e(\sigma^k(\omega)) = \left(\lim_{n\to\infty} \frac{{{\mathbf{M}}}( f_0^*\cdots f_{n-1}^*a)}{{{\mathbf{M}}}( f_k^*\cdots f_{n-1}^*a)}\right) e(\omega)
=: \zeta(k, \omega) e(\omega)$$ where $\zeta(k, \omega)$ is both equal to ${{\mathbf{M}}}((f_\omega^k)^* e(\sigma^k(\omega)))$ and to the limit $$\label{eq:zetakf2}
\zeta(k, \omega) = \lim_{n\to\infty} \frac{{{\mathbf{M}}}( f_0^*\cdots f_{n-1}^*a)}{{{\mathbf{M}}}( f_k^*\cdots f_{n-1}^*a)}= \lim_{n\to\infty} \frac{{{\mathbf{M}}}( (f_\omega^n)^*a)}{{{\mathbf{M}}}( (f_{\sigma^k(\omega)}^{n-k})^*a)} .$$ We want to show that, $\nu^{\mathbf{N}}$-almost surely, $ ( 1/k)\log \zeta(k,\omega) $ converges to $\lambda_{H^{1,1}}$.
Before starting the proof, note that $\zeta$ is a multiplicative cocycle: $ \zeta(k,\omega)= \prod_{\ell = 1}^k \zeta (1, \sigma^\ell\omega)$; in particular, $\log \zeta(k,\omega) $ is equal to the Birkhoff sum $\sum_{\ell=1}^k \log \zeta (1, \sigma^\ell\omega)$. Since $$C{^{-1}}{{\left\Vert(f_0{^{-1}})^*\right\Vert}}_{H^{1,1}} \leq {{\mathbf{M}}}(f_0^*e(\sigma(\omega))) \leq C{{\left\Vertf_0^*\right\Vert}}_{H^{1,1}} ,$$ our moment condition shows that $\log(\zeta(1,\omega))$ is integrable. So, by the ergodic theorem of Birkhoff, $\lim_{k} \frac1{k} \log \zeta(k,\omega)$ exists $\nu^{\mathbf{N}}$-almost surely.
Pick a sequence $(n_i)$ of good times for $\omega$, as in Theorem \[thm:GK\_good\_times\]. If we compute the limit in Equation along the subsequence $(n_i)$ we see that $\zeta(k,\omega) \geq C \exp((\lambda_{H^{1,1}}-\delta(k))k)$ for some constant $C>0$, and some sequence $\delta(k)$ converging to $0$ as $k$ goes to $+\infty$. This gives $$\label{eq:ineq_zeta_lambda1}
\limsup_{k\to +\infty} \frac1{k} \log \zeta(k,\omega) \geq \lambda_{H^{1,1}}.$$ Now, consider the linear cocycle $\Upsilon: \Omega \times H^{1,1}(X, {\mathbf{R}}) \to \Omega\times H^{1,1}(X, {\mathbf{R}})$ defined by $$\Upsilon (\omega, u) = (\sigma(\omega), (f_\omega^1)_* u)$$ and let $\mathbb P \Upsilon$ be the associated projective cocycle on $ \Omega\times {\mathbb{P}}H^{1,1}(X, {\mathbf{R}})$. The Lyapunov exponents of $\Upsilon$ are $\pm\lambda_{H^{1,1}}$, each with multiplicity $1$, and $0$, with multiplicity $h^{1,1}(X)-2$. Since ${\mathbb{P}}((f_\omega^1)^*e(\sigma(\omega))) = {\mathbb{P}}(e(\omega))$, the measurable section ${\left\{(\omega, {\mathbb{P}}(e(\omega)))\; ; \; \omega\in \Omega\right\}}$ is ${\mathbb{P}}\Upsilon$-invariant. Therefore, by ergodicity of $\sigma$ with respect to $\nu^{\mathbf{N}}$, $m= \int \delta_{{\mathbb{P}}(e(\omega))} \, d\nu^{{\mathbf{N}}}(\omega)$ defines an invariant and ergodic measure for $\mathbb P \Upsilon$. It follows from the invariance of the decomposition into characteristic subspaces in Oseledets’ theorem that $e(\omega)$ is contained in a given characteristic subspace of the cocycle $\Upsilon$; thus, if $\lambda$ denotes the Lyapunov exponent of $\Upsilon$ in that characteristic subspace, we get (as in Remark \[rem:lambda\_with\_log\_mass\]) that $$\begin{aligned}
\lambda = \int \log \frac{{\left\vert(f_\omega^1)_* u\right\vert}}{{\left\vertu\right\vert}} \; dm(\omega, u)
& = \int \log \frac{{{\mathbf{M}}}((f_\omega^1)_*( e(\omega))}{{{\mathbf{M}}}( e(\omega))}\, d\nu^{{\mathbf{N}}} (\omega)\\
& \notag = \int \log \zeta(1, \omega){^{-1}}\, d\nu^{{\mathbf{N}}} (\omega) \end{aligned}$$ (see Ledrappier [@Ledrappier:SaintFlour §1.5]). Birkhoff’s ergodic theorem implies that $\lim_{k} \frac1{k} \log \zeta(k,\omega) = -\lambda$, with $ \lambda\in {\left\{\pm\lambda_{H^{1,1}}, 0\right\}}$, therefore the Inequality concludes the proof.
### Exponential moments {#subsub:exponential_moments}
The result of this section will only be used in Theorem \[thm:holder\] so this paragraph may be skipped on a first reading. Consider the exponential moment condition $$\label{eq:exponential_moment}
\exists \tau>0, \ \int {\left({{{\left\Vert f\right\Vert}}_{ C^1}} + {{{\left\Vertf^{-1}\right\Vert}}_{C^1}}\right)}^\tau \; d\nu(f)<+\infty.$$ As in Section \[subs:moments\_cohomology\], this upper bound implies the cohomological moment condition $$\label{eq:exponential_moment_cohomology}
\exists \tau>0, \ \int {\left({{{\left\Vert f^*\right\Vert}}_{H^{1, 1}}} + {{{\left\Vert(f^{-1})^*\right\Vert}}_{H^{1, 1}}}\right)}^\tau \; d\nu(f)<+\infty.$$
\[pro:gouezel\] Assume that $\nu$ satisfies the exponential moment condition . Let $D\colon {\mathsf{Aut}}(X)\to {\mathbf{R}}_+$ be a measurable function such that $\int { D(f)^{\tau'}}d\nu(f)<\infty$ for some $\tau'>0$. Then, there is a measurable function $B\colon \Omega\to {\mathbf{R}}_+$ satisfying $$\label{eq:moment_gouezel}
\int \log^+(B(\omega)) \, d\nu^{\mathbf{N}}(\omega)<\infty$$ such that for $\nu^{\mathbf{N}}$-almost every $\omega = (f_n)$ and every $n\geq 0$ $$\label{eq:gouezel}
\sum_{j=1}^{n-1} D(f_{j-1})
\frac{\big\| {f_j^*\cdots f_{n-1}^*} \big\| }{{{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}}} \leq B(\omega), \; \text{ and } \;
\sum_{j=1}^{n-1} D(f_j)\frac{\big\| {f_0^*\cdots f_{j-1}^*} \big\| }{{{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}}} \leq B(\omega).$$
This is a refined version of Corollary \[cor:GK\_good\_times\]. The result is stated in our setting, but it holds for more general random products of matrices.
We are grateful to Sébastien Gouëzel for explaining this argument to us. Without loss of generality, we assume $\tau= \tau'$. We temporarily use the notation ${\mathbb{P}}(\cdot)$ for probability with respect to $\nu^n$ or $\nu^{\mathbf{N}}$ (so, here, ${\mathbb{P}}$ does not denote projectivisation).
[**[First Estimate.–]{}**]{} We start with the first estimate in .
[**[Step 1.–]{}**]{} For every $0<{\varepsilon}<\lambda_{H^{1,1}}$ there exists $c, C>0$ such that for every $a\in \Pi_\Gamma$ with ${\left\vert a\right\vert}=1$, $${\mathbb{P}}{\left({\left\vert(f_\omega^n)^*a\right\vert}\leq e^{{\varepsilon}n}\right)} \leq Ce^{-cn}.$$ This estimate follows from condition (see for instance [@benoist-quint_book Prop. 14.6]).
[**[Step 2.–]{}**]{} Let us prove that $$\label{eq:proba_estimate}
{\mathbb{P}}{\left(\frac{\big\| {f_j^*\cdots f_{n-1}^*} \big\| }{{{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}}} > e^{-{\varepsilon}j} \right)}\leq Ce^{-cj}.$$
For this, let us first fix $f_j, \ldots, f_{n-1}$. Then, there is a point $a\in \Pi_\Gamma$ with ${\left\verta\right\vert}=1$ such that ${{\left\Vertf_j^*\cdots f_{n-1}^*\right\Vert}} = {\left\vert f_j^*\cdots f_{n-1}^*a \right\vert}$. Hence, if ${{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}} < \big\| {f_j^*\cdots f_{n-1}^*} \big\| e^{{\varepsilon}j}$, we infer that $${\left\vertf_0^*\cdots f_{n-1}^*a\right\vert} < \big\| {f_j^*\cdots f_{n-1}^*} \big\| e^{{\varepsilon}j} = \big|{f_j^*\cdots f_{n-1}^*a} \big| e^{{\varepsilon}j}.$$ Thus, if we set $$b = {\frac{1}{\big|{f_j^*\cdots f_{n-1}^*a}\big| }} f_j^*\cdots f_{n-1}^*a ,$$ we obtain that ${\left\vertf_0^*\cdots f_{j-1}^*b\right\vert} < e^{{\varepsilon}j}$; this happens with (conditional) probability $\leq Ce^{-cj}$, for the uniform constants given in Step 1. Averaging over $f_j, \ldots, f_{n-1}$, we get the result.
[**[Step 3.–]{}**]{} The moment condition satisfied by $D$ implies that ${\mathbb{P}}(D>K)\leq C_1 K^{-\tau}$ for some constant $C_1>0$. Fix ${\varepsilon}\in {\mathbf{R}}_+^*$ small with respect to $\lambda_{H^{1,1}}$ and $\tau$. Then, on a set $\Omega({\varepsilon},J)$ of measure $\geq 1-C_2(e^{-({\varepsilon}\tau/2) J}+e^{-{\varepsilon}c J})$, we have both $D(f_{j-1})\leq e^{{\varepsilon}j/2}$ and $\frac{{{\left\Vertf_j^*\cdots f_{n-1}^*\right\Vert}}}{{{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}}} \leq e^{-{\varepsilon}j}$ for all $j\geq J$. For $\omega=(f_n)$ in $\Omega({\varepsilon},J)$, we get $$\begin{aligned}
\label{eq:sum_gouezel}
\sum_{j=1}^{n-1} D(f_{j-1})\frac{\big\| {f_j^*\cdots f_{n-1}^*} \big\| }{{{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}}}
&\leq \sum_{j=1}^J D(f_{j-1}) \frac{\big\| {f_j^*\cdots f_{n-1}^*} \big\| }{{{\left\Vertf_0^*\cdots f_{n-1}^*\right\Vert}}} + \sum_{j=J+1}^{n-1} e^{-{\varepsilon}j/2}\\
\notag&\leq \sum_{j=1}^J D(f_{j-1}) {{\left\Vert(f_{j-1}{^{-1}})^* \cdots (f_0{^{-1}})^*\right\Vert}} + C_3 e^{-J} \\
& \notag= C_3+ \sum_{j=0}^{J-1} {{\left\Vertf_0^*\right\Vert}}\cdots {{\left\Vertf_j^*\right\Vert}} D(f_j).
\end{aligned}$$ The moment condition gives ${\mathbb{P}}({{\left\Vertf^*\right\Vert}}>K)\leq C_4K^{-\tau}$ and as already noticed, we also have ${\mathbb{P}}(D(f)>K)\leq C_1 K^{-\tau}$. So, on a set of probability at least $1- C_5JK^{-\tau}$, $$\sum_{j=0}^{J-1} D(f_j) {{\left\Vertf_0^*\right\Vert}}\cdots {{\left\Vertf_j^*\right\Vert}} \leq C_6J K^{J+2}.$$ Taking $K = J^{3/\tau}$, we have $JK^{-\tau}=J^{-2}$, and we obtain $$\label{eq:J}
{\mathbb{P}}{\left( \sum_{j=0}^{J-1} D(f_j){{\left\Vertf_0^*\right\Vert}}\cdots {{\left\Vertf_j^*\right\Vert}} > J^{1+(3 J + 6)/\tau} \right)}\leq C_7J^{-2}.$$ Also, note that $J^{1+(3 J + 6)/\tau} \leq \exp{\left(CJ^{3/2}\right)}$. By the Borel-Cantelli lemma, as $n\to\infty$, the sum in is almost surely bounded by some constant $B(\omega)$ which satisfies ${\mathbb{P}}{\left(\log B > J^{3/2}\right)}\leq CJ^{-2}$; in particular ${\mathbb{E}}{\left(\log^+ B\right)}<\infty$.
[**[Second Estimate.–]{}**]{} To estimate the second term in , we simply apply the above proof to the reversed random dynamical system induced by $\check{\nu}:f\mapsto \nu(f{^{-1}})$. Indeed, the core of the argument is the inequality which is not sensitive to the order of compositions.
Limit currents {#sec:currents}
==============
Our goal in this section is to prove the counterpart of the convergence at the level of positive closed currents on $X$. Throughout this section we fix a non-elementary random holomorphic dynamical system $(X, \nu)$ satisfying the moment condition , so that all results of §\[sec:furstenberg\] apply. We refer the reader to [@Guedj-Zeriahi:Book] (in particular Chapter 8) for basics on pluripotential theory on compact Kähler manifolds.
Potentials and cohomology classes of positive closed currents {#par:Notations_kappa_i_potentials}
-------------------------------------------------------------
Let us fix once and for all a family of Kähler forms $(\kappa_i)_{1\leq i\leq h^{1,1}(X)}$ such that $[\kappa_i]^2 = 1$ and the $[\kappa_i]$ form a basis of $H^{1,1}(X;{\mathbf{R}})$; in addition we require that the $\kappa_i$ satisfy $$\kappa_0= \beta \sum_i\kappa_i$$ for some $\beta >0$, where $\kappa_0$ is the Kähler form chosen in Section \[par:cones\_definition\] (note that necessarily $\beta <1$). By definition the [**[mass]{}**]{} of a current is the quantity ${{\mathbf{M}}}(T) = \int T\wedge \kappa_0$.
We also fix a smooth volume form ${{\sf{vol}}}_X$ on $X$, normalized by $\int_X {{\sf{vol}}}=1$. On tori, K3 and Enriques surfaces, we choose ${{\sf{vol}}}_X$ to be the canonical ${\mathsf{Aut}}(X)$-invariant volume form (see Remark \[rem:volume\_form\]). It is convenient to assume in all cases that ${{\sf{vol}}}_X$ is also the volume form associated with the Kähler metric $\kappa_0$ (up to scaling). On tori, K3 and Enriques surfaces this implies that $\kappa_0$ is the unique Ricci-flat Kähler metric in its Kähler class; its existence is guaranteed by Yau’s theorem (see [@Filip-Tosatti] for the interest of such a choice in holomorphic dynamics).
The action of a current $T$ on a test form $\varphi$ will be denoted by $\langle T, \varphi\rangle$ or $\int T\wedge \varphi$. If $T$ is closed, we denote its cohomology class by $[T]$; so, if $\varphi$ is a closed form, $\langle T, \varphi\rangle=\langle [T] \, \vert\, [\varphi]\rangle$.
### Normalized potentials {#subs:normalized_potentials}
If $a$ is an element of $H^{1,1}(X;{\mathbf{R}})$, we denote by $(c_i(a))_{1\leq i \leq h^{1,1}(X)}$ its coordinates in the basis $([\kappa_i])$, so that $a =\sum_i c_i(a)[\kappa_i]$. Then, we define $$\Theta(a)=\sum_i c_i(a) \kappa_i.$$ Likewise, given a closed $(1,1)$-form $\alpha$ or a closed current of bidegree $(1,1)$, we set $c_i(\alpha)=c_i([\alpha])$ and $\Theta(\alpha)=\Theta([\alpha])$; hence, $[\Theta(\alpha)]=[\alpha]$. It is worth keeping in mind that some coefficients $c_i(\alpha)$ can be negative and $\Theta(\alpha)$ need not be semi-positive, even if $\alpha$ is a Kähler form. If $T$ is a closed positive current of bidegree $(1,1)$ on $X$ we define its **normalized potential** to be the unique function $u_T\in L^1(X)$ such that $$\label{eq:def_uT}
T=\Theta(T)+dd^c(u_T) \; \text{ and } \; \int_X u_T \; {{\sf{vol}}}=0$$ (see [@Guedj-Zeriahi:Book §8.1]). The function $u_T$ is locally given as the difference $v-w$ of a psh potential $v$ of $T$ and a smooth potential $w$ of $\Theta(T)$.
\[lem:uniform\_Ak\_psh\] There is a constant $A>0$ such that the following properties are satisfied for every closed positive current $T$ of mass $1$
1. $-A\leq c_i(T) \leq A$ for all $1\leq i\leq h^{1,1}(X)$, and $-A\kappa_0 \leq \Theta(T)\leq A\kappa_0$.
2. the function $u_T$ is $(A\kappa_0)$-psh.
Since the coefficients $T\mapsto c_i(T)$ are continuous functions on the space of currents and closed positive currents of mass $1$ form a compact set $K$, the functions ${\left\vertc_i\right\vert}$ are bounded by some uniform constant $A'$ on $K$. Setting $A=A'\beta{^{-1}}$, we get $-A\kappa_0 \leq \Theta(T)\leq A\kappa_0$ for all $T\in K$. Then $dd^cu_T=T-\Theta(T)\geq -A \kappa_0$ and (ii) follows.
\[cor:compact\] The set of potentials $${\left\{ u_T \; \vert \; T \, {\text{ is a closed positive current of mass}} \, 1\ \text{on } X\right\}}$$ is compact subset of $L^{1}(X;{{\sf{vol}}})$.
Since this is a set of $(A\kappa_0)$-psh functions which are normalized with respect to a smooth volume form, the result follows from Proposition 8.5 and Remark 8.6 in [@Guedj-Zeriahi:Book].
\[rem:normalization\] Another usual normalization is to impose the condition $\sup_{x\in X} u_T(x)=0$. By compactness this would only change $u_T$ by some uniformly bounded constant. However since many of our dynamical examples preserve a natural volume form it is more convenient for us to normalize as in .
### The diameter of a pseudo-effective class
For a class $a\in {{\mathrm{Psef}}}(X)$ we define $$\operatorname{{Cur}}(a)=\{ T\; ; \; T\, {\text{is a closed positive current with}}\; [T]=a\},$$ This is a compact convex subset of the space of currents. If $S$ and $T$ are two elements of $\operatorname{{Cur}}(a)$, then $\Theta(S)=\Theta(T)$ and $T-S=dd^c(u_T-u_S)$. We set $$\operatorname{dist}(S,T)=\int_X\vert u_S-u_T\vert\, {{\sf{vol}}}\; .$$ This is a distance that metrizes the usual weak topology on $\operatorname{{Cur}}(a)$: this follows for instance from the fact that by Corollary \[cor:compact\] $(\operatorname{{Cur}}(a), \operatorname{dist})$ is compact.
By definition, the **diameter** of the class $a$ is the diameter of $\operatorname{{Cur}}(a)$ for this distance: $$\operatorname{Diam}(a)=\operatorname{Diam}(\operatorname{{Cur}}(a))= \sup\{\operatorname{dist}(S,T)\; ; \; S, \, T \, {\text{in}}\; \operatorname{{Cur}}(a)\},$$ If $a\in {{\mathrm{Psef}}}(X)$, then $\operatorname{Diam}(a)$ is a non-negative real number which is finite by Corollary \[cor:compact\]. If $\operatorname{{Cur}}(a)= \emptyset$, we set $\operatorname{Diam}(a)=-\infty$. Note that $\operatorname{Diam}$ is homogeneous of degree $1$: $\operatorname{Diam}(t a)=t\operatorname{Diam}(a)$ for every $a\in {{\mathrm{Psef}}}(X)$ and $t>0$.
Let $\pi\colon X\to B$ be a fibration of genus $1$. Let $a$ be the cohomology class of any fiber $X_w=\pi{^{-1}}(w)$, $w\in B$. Then, to every probability measure $\mu_B$ on $B$ corresponds a closed positive current $T_{\mu_B}\in \operatorname{{Cur}}(a)$, defined by $\langle T_{\mu_B}, \varphi\rangle = \int_B \int_{X_w} \varphi d\mu_B(w)$, and any closed positive current in $\operatorname{{Cur}}(a)$ is of this form. In this case $\operatorname{Diam}(a)>0$. Now, assume that $f$ is a loxodromic automorphism of $X$, and denote by $\theta_f$ the unique $(1,1)$-class of mass $1$ that satisfies $f^*\theta_f=\lambda_f \theta_f$, where $\lambda_f$ is the spectral radius of $f^*\in {{\sf{GL}}}(H^{1,1}(X;{\mathbf{R}}))$; then $\operatorname{{Cur}}(\theta_f)$ is represented by a unique closed positive current $T_f^+$ and $\operatorname{Diam}(\theta_f)=0$. For generic Wehler surfaces, these two types of classes, given by eigenvectors of loxodromic automorphisms and classes of genus $1$ fibrations, are dense in the boundary of ${\mathbb{H}}_X\cap {{\mathrm{NS}}}(X;{\mathbf{R}})$ (see [@Cantat:Milnor]).
\[lem:diam\_measurable\] The function $a\mapsto \operatorname{Diam}(a)$ is upper semi-continuous, hence measurable, on ${{\mathrm{Psef}}}(X)$.
Let $(a_n)$ be a sequence of pseudo-effective classes converging to $a$. For every $n$ we choose a pair of currents $(S_n,T_n)$ in $\operatorname{{Cur}}(a_n)$ such that $\operatorname{dist}(S_n,T_n)\geq \operatorname{Diam}(a_n)-1/n$. The masses of $S_n$ and $T_n$ are uniformly bounded because they depend only on $a_n$. By Corollary \[cor:compact\], we can extract a subsequence such that (1) $S_n$ and $T_n$ converge towards closed positive currents $S$ and $T$ in $\operatorname{{Cur}}(a)$, and (2) $u_{S_n}$ and $u_{T_n}$ converge towards their respective potentials $u_S$ and $u_T$ in $L^1(X,{{\sf{vol}}})$. Then, $\operatorname{dist}(S,T)=\int_X\vert u_S-u_T\vert {{\sf{vol}}}=\lim_n \operatorname{dist}(S_n,T_n)$, which shows that $ \operatorname{Diam}(a)\geq \limsup_{n}\left( \operatorname{Diam}(a_n)\right)$.
Action of ${\mathsf{Aut}}(X)$
-----------------------------
### A volume estimate
Let $X$ be a compact, complex manifold, and let ${{\sf{vol}}}$ be a $C^0$-volume form on $X$ with ${{\sf{vol}}}(X)=1$. If $f$ is an automorphism of $X$, let $\operatorname{{Jac}}(f)$ denote its Jacobian determinant with respect to the volume form ${{\sf{vol}}}$: $f^*{{\sf{vol}}}= \operatorname{{Jac}}(f){{\sf{vol}}}$. The following lemma is a variation on well-known ideas in holomorphic dynamics (see for instance [@Guedj:Fourier]).
\[lem:volume-estimate\] Let $\kappa$ be a hermitian form on $X$. Let $h$ be a $\kappa$-psh function on $X$ such that $\int_X h\, {{\sf{vol}}}=0$, and let $f$ be an automorphism of $X$. Then, $$\int_X {\left\vert h\circ f\right\vert} \, {{\sf{vol}}}\leq C\log(C{{\left\Vert \operatorname{{Jac}}(f^{-1}) \right\Vert}}_\infty)$$ for some positive constant $C$ that depends on $(X,\kappa)$ but neither on $f$ nor on $h$.
We first observe that there is a constant $c>0$ such that ${{\sf{vol}}}\{\vert h\vert\geq t\}\leq c \exp(-t/c)$. Indeed this follows directly from Lemma 8.10 and Theorem 8.11 in [@Guedj-Zeriahi:Book], together with Chebychev’s inequality (see Remark \[rem:normalization\] for the normalization). Then, we get $$\begin{aligned}
\int_X \vert h\circ f\vert \, {{\sf{vol}}}& = & \int_0^\infty {{\sf{vol}}}\{ \vert h\circ f\vert \geq t\} dt \\
&\notag = & \int_0^\infty {{\sf{vol}}}(f^{-1}\{ \vert h\vert \geq t\}) dt \\
&\notag \leq & \int_0^s {{\sf{vol}}}(X) dt + {{\left\Vert \operatorname{{Jac}}(f^{-1}) \right\Vert}}_\infty \int_s^\infty c\exp(-t/c)dt \\
& \label{eq:vol_lastline} \leq & s \, {{\sf{vol}}}(X)+ {{\left\Vert \operatorname{{Jac}}(f^{-1}) \right\Vert}}_\infty c^2\exp(-s/c)\end{aligned}$$ where the inequality in the third line follows from the change of variable formula. Now, we minimize by choosing $s=c\log (c{{\left\Vert \operatorname{{Jac}}(f^{-1}) \right\Vert}}_\infty/{{\sf{vol}}}(X))$ and we infer that $$\int_X \vert h\circ f\vert \, {{\sf{vol}}}\leq c\, {{\sf{vol}}}(X)\left(1+\log\left( \frac{c{{\left\Vert \operatorname{{Jac}}(f^{-1}) \right\Vert}}_\infty}{{{\sf{vol}}}(X)}\right)\right).$$ Note also that since the total volume is invariant, ${{\left\Vert \operatorname{{Jac}}(f) \right\Vert}}_\infty\geq 1$. Then the asserted estimate easily follows.
### Equivariance
Let us come back to the study of $(X,\nu)$.
If $f$ is an automorphism of $X$, then $f^*\operatorname{{Cur}}(a)=\operatorname{{Cur}}(f^*(a))$ for every class $a\in H^{1,1}(X, {\mathbf{R}})$. If $a\in {{\mathrm{Psef}}}(X)$ and $T\in \operatorname{{Cur}}(a)$, we write $T=\Theta(a)+dd^c(u_T)$ hence $$\label{eq:f*T}
f^*T=f^*\Theta(a)+dd^c(u_T\circ f)= \Theta(f^*a) + dd^c(u_{f^*\Theta(a)} + u_T\circ f).$$ This shows that the normalized potential of $f^*T$ is given by $$\label{eq:ufT_uT}
u_{f^*T}= u_{f^*\Theta(a)} + u_T\circ f +E(f,T)$$ where $E(f,T)\in {\mathbf{R}}$ is the constant for which the integral of $u_{f^*T}$ vanishes; since $u_{f^*\Theta(a)}$ has mean $0$, we get $$E(f,T)=-\int_X {\left(u_{f^*\Theta(a)} + u_T\circ f\right)} \, {{\sf{vol}}}=-\int_X u_T\circ f \, {{\sf{vol}}}.$$
If the volume form is $f$-invariant, for instance if ${{\sf{vol}}}$ is the canonical volume on a K3 or Enriques surface, then $E(f,T)=0$.
\[lem:EfT\] On the set of closed positive currents of mass $1$, the function $(f,T)\mapsto E(f,T)$ satisfies $$\vert E(f,T)\vert \leq C \log\left( C{{\left\Vert \operatorname{{Jac}}(f^{-1})\right\Vert}}_\infty \right)$$ where the implied positive constant $C$ depends neither on $f$ nor on $T$.
From Lemma \[lem:uniform\_Ak\_psh\], the potentials $u_T$ are uniformly $(A\kappa_0)$-psh, so the conclusion follows from Lemma \[lem:volume-estimate\].
\[lem:diam\_f\*a\] There exists a constant $C$ such that if $a$ is any pseudo-effective of mass $1$, and $f$ is any automorphism of $X$, then $$\label{eq:diam_f*a}
\operatorname{Diam}(f^*a)\leq C\log\left( C{{\left\Vert\operatorname{{Jac}}(f^{-1})\right\Vert}}_\infty\right).$$
Indeed, if $S$ and $T $ belong to $\operatorname{{Cur}}(a)$, by Equation we have $$u_{f^*T} - u_{f^*S} = (u_T-u_S)\circ f + E(f,T)- E(f,S)$$ so $$\operatorname{dist}(f^*T,f^*S) \leq \int{\left\vert u_T\circ f \right\vert} \, {{\sf{vol}}}+ \int{\left\vert u_S\circ f \right\vert} \, {{\sf{vol}}}+ {\left\vertE(f,T)\right\vert}+{\left\vertE(f,S)\right\vert},$$ and the result follows from Lemmas \[lem:volume-estimate\] and \[lem:EfT\] and the fact that $u_S$ and $u_T$ are uniformly $(A\kappa_0)$-psh.
### An estimate for canonical potentials
\[lem:utheta\] For any Kähler form $\kappa $ on $X$ there exists a positive constant $C(\kappa)$ such that for every $f\in \mathrm{Aut}(X)$, $$\label{eq:utheta}
{{\left\Vert u_{f^*\kappa}\right\Vert}}_{C^1} \leq C(\kappa) {{\left\Vert f\right\Vert}}_{C^1}^2 .$$ In addition $C(\kappa)\leq C'{{\left\Vert\kappa\right\Vert}}_\infty$, where ${{\left\Vert\kappa\right\Vert}}_\infty$ is sup norm of the coefficients of $\kappa$ in a system of coordinate charts and $C'$ depends only on $X$ (and the choice of these coordinate charts).
Recall the basis of Kähler forms $\kappa_i$ from § \[par:Notations\_kappa\_i\_potentials\] and the definition of $\Theta(a)$ for a psef class $a\in {{\mathrm{Psef}}}(X)$.
\[cor:utheta\] If in Lemma \[lem:utheta\], $\kappa$ is of the form $\kappa= \sum_i c_i\kappa_i$ then the constant $C(\kappa)$ satisfies $C(\kappa)\leq C''{{\mathbf{M}}}(\kappa)$. Likewise, ${{\left\Vertu_{f^*\Theta(a)} \right\Vert}}_{C^1}\leq C'''{{\mathbf{M}}}(a) {{\left\Vertf\right\Vert}}_{C^1}^2$ for all $a\in {{\mathrm{Psef}}}(X)$.
Indeed in the first assertion $C(\kappa)\leq C' {{\left\Vert\kappa\right\Vert}}_\infty \leq C'' \sum_i {\left\vertc_i\right\vert}$ and in the second one $u_{f^*\Theta(a)} = \sum c_i(a) u_{f^*\kappa_i}$.
By definition $f^*\kappa - \Theta(f^*\kappa) = dd^c {\left(u_{f^*\kappa}\right)}$. The estimate will be obtained by constructing a solution $\phi$ to the equation $$\label{eq:ddcphi}
dd^c \phi = f^*\kappa - \Theta(f^*\kappa)$$ which satisfies ${{\left\Vert \phi\right\Vert}}_{C^1} \leq C {{\left\Vert f\right\Vert}}_{C^1}^2 $. Then, since $u_{f^*\kappa}$ and $\phi$ differ by a constant and $u_{f^*\kappa}$ is known to vanish at some point, it follows that $u_{f^*\kappa}$ satisfies the same estimate.
To construct the potential $\phi$, we follow the method of Dinh and Sibony [@dinh-sibony_jams Prop. 2.1] which it itself based on work of Bost, Gillet and Soulé [@bost-gillet-soule] (we closely follow the notation of [@dinh-sibony_jams]). To be specific, let $\alpha $ be a closed $(2,2)$ form in $X\times X$ which is cohomologous to the diagonal $\Delta$. An explicit $(1,1)$ form $K$ on $X\times X$ such that $dd^cK = [\Delta]-\alpha$ is constructed in [@bost-gillet-soule], which is referred to there as a “Green current”. It is $C^\infty$ outside the diagonal, and along $\Delta$, it satisfies the estimates $$\label{eq:K}
K(x,y) = O{\left( \frac{\log {\left\vertx-y\right\vert}}{{\left\vertx-y\right\vert}^2}\right)}\; \text{ and } \; \nabla K(x,y) = O{\left( \frac{\log {\left\vertx-y\right\vert}}{{\left\vertx-y\right\vert}^3}\right)}$$ (here we mean that these estimates hold for the coefficients of $K$ and $\nabla K$ in local coordinates). These estimates are easily deduced from the explicit expression of $K$ as $\pi_*(\widehat \varphi \eta - \beta)$ given in [@dinh-sibony_jams Prop. 2.1], where $\pi:\widehat{X\times X}\to X\times X$ is the blow-up of the diagonal, $\eta$ and $\beta$ are smooth (1,1) forms on $\widehat{X\times X}$ and $\widehat \varphi $ is a function with logarithmic singularities along the proper transform of $\Delta$ in $X\times X$.
It is shown in [@dinh-sibony_jams Prop. 2.1] that a solution to Equation is given by $$\phi(x) = \int_{y\in X} K(x,y) \wedge {\left(f^*\kappa (y) - \Theta(f^*\kappa)(y) \right)}$$ (in the notation of [@dinh-sibony_jams], $f^*\kappa$ and $\Theta(f^*\kappa)$ correspond to $\Omega^+$ and $\Omega^-$ respectively). The coefficients of the smooth $(1,1)$-forms $f^*\kappa$ and $ \Theta(f^*\kappa)$ have their uniform norms bounded by $C {{\left\Vert f\right\Vert}}_{C^1}^2$, where $C = C(\kappa) \leq C'{{\left\Vert\kappa\right\Vert}}_\infty$. The first estimate in implies that the coefficients of $K$ belong to $L^{p}_{\mathrm{loc}}$ for $p<2$, so it follows from the Hölder inequality that ${{\left\Vert\phi\right\Vert}}_{C^0}\leq C'{{\left\Vert\kappa\right\Vert}}_\infty {{\left\Vert f\right\Vert}}_{C^1}^2 $ (with possibly a new constant $C'$ depending only on $X$). Likewise, derivating under the integral sign and using that $ \nabla K \in L^{p}_{\mathrm{loc}}$ for $p<4/3$ yields a similar estimate for $ \nabla \phi$. This concludes the proof.
Convergence and extremality
---------------------------
\[thm:uniq+extremal\] Let $(X,\nu)$ be a non-elementary random holomorphic dynamical system on a compact Kähler surface $X$, satisfying the moment condition . Then for $\mu_{\partial}$-almost every point $\underline a\in \Lambda(\Gamma)$, the following properties hold:
1. there is a unique nef and isotropic class $ {a} \in H^{1,1}(X;{\mathbf{R}})$ of mass $1$ with ${\mathbb{P}}( {a})=\underline a$;
2. the convex set $\operatorname{{Cur}}({a})$ is a singleton $\{ T_{ {a}}\}$;
3. the class $a$ is an extremal point of ${\mathbb{P}}({{\overline{{\mathrm{Kah}}}}}(X))$ and of ${\mathbb{P}}({{\mathrm{Psef}}}(X))$;
4. the current $T_{{a}}$ is extremal in the convex set of closed positive currents of mass $1$.
Combining this result with Lemma \[lem:definition\_e\] and Equation we obtain the first and second assertions of the following corollary; the third assertion follows from the first one and the equivariance relation .
\[cor:Tsomega\] The following properties are satisfied for $\nu^{\mathbf{N}}$-almost every $\omega$:
1. there exists a unique closed positive current $T_\omega^s$ in the cohomology class $e(\omega)$;
2. for every Kähler form $\kappa$, $${\frac{1}{{{\mathbf{M}}}{\left( (f_\omega^n)^* \kappa \right)}}} (f_\omega^n)^* \kappa {\underset{n\to\infty}{\longrightarrow}}T_\omega^s.$$
3. the currents $T^s_\omega$ satisfy the equivariance property $$\label{eq:equivariance_Ts}
(f_\omega)^*T_{\sigma(\omega)}^s=\frac{{{\mathbf{M}}}((f_\omega)^*T_{\sigma(\omega)}^s)}{{{\mathbf{M}}}(T_\omega^s)}T_\omega^s= {{{\mathbf{M}}}((f_\omega)^*T_{\sigma(\omega)}^s)}T_\omega^s.$$
The first and third assertions were already established, respectively in Lemma \[lem:limitset\_in\_NS\] and \[lem:limitset\_in\_nef\] and Proposition \[pro:extremality\]. Property (4) follows from (2) and (3).
It remains to prove (2). For this, we denote by $\underline{f}^*$ the projective action of $f^*$ on ${\mathbb{P}}H^{1,1}(X;{\mathbf{R}})$ and for $\underline a\in \Lambda(\Gamma)$ we set ${\mathrm{diam}}{\left(\underline a\right)}=\operatorname{Diam}(a)$, where $a$ is the unique pseudo-effective class of mass $1$ such that ${\mathbb{P}}( {a})=\underline a$. This defines a measurable function on $\operatorname{Lim}(\Gamma)$ by Lemma \[lem:diam\_measurable\]. Our purpose is to show that ${\mathrm{diam}}{\left(\underline a\right)} =0$ for $\mu_{\partial}$-almost every $\underline{a}$. The stationarity of $\mu_{\partial}$ reads $$\int {\mathrm{diam}}{\left(\underline {a}\right)} \; d\mu_\partial{\left(\underline a\right)} =
\int\!\! \int {\mathrm{diam}}{\left(
{\underline{f}}^*{\left(\underline {a}\right)}\right)} \; d\nu(f) d\mu_\partial{\left(\underline a\right)}$$ and iterating this relation we get $$\int {\mathrm{diam}}{\left(\underline {a}\right)} \; d\mu_\partial{\left(\underline a\right)}
= \int {\mathrm{diam}}{\left(
\underline{f}_n^*\cdots \underline{f}_1^*{\left(\underline {a}\right)}\right)} \; d\nu(f_1)\cdots d\nu(f_n) d\mu_\partial{\left(\underline a\right)}$$ (notice the order of compositions chosen here). Since the diameter is upper-semicontinuous it is uniformly bounded on $\operatorname{Lim}(\Gamma)$. So, if we prove that $$\label{eq:lim_diameters}
\lim_{n\to +\infty} {\mathrm{diam}}\big({
\underline{f}_n^*\cdots \underline{f}_1^*{\left(\underline {a}\right)}}\big) = 0$$ for $\nu^{\mathbf{N}}$-almost every $(f_n)$ and every $\underline{a}$, then we can apply the dominated convergence theorem to infer that ${\mathrm{diam}}{\left(\underline a\right)} =0$ $\mu_{\partial}$-almost surely. To derive the convergence , note that $$\label{eq:diam_jac}
{\mathrm{diam}}{\left(
\underline{f}_n^*\cdots \underline{f}_1^*{\left(\underline {a}\right)}\right)} = \frac{\operatorname{Diam}{\left(f_n^*\cdots f_1^* a\right)}}
{{{\mathbf{M}}}{\left({f_n^*\cdots f_1^* a}\right)}}$$ because $\operatorname{Diam}$ is homogeneous. Applying Lemma \[lem:diam\_f\*a\] and the multiplicativity of the Jacobian we get that $${\mathrm{diam}}{\left(\underline{f}_n^*\cdots \underline{f}_1^*{\left(\underline {a}\right)}\right)}
\leq \frac{C\log {\left(C {{\left\Vert\operatorname{Jac}(f_1\circ \cdots \circ f_n){^{-1}}\right\Vert}}_\infty\right)}}{{{\mathbf{M}}}{\left({f_n^*\cdots f_1^* a}\right)}} \leq C \frac{ \sum_{i=0}^{n-1} \log{{{\left\Vertf_i{^{-1}}\right\Vert}}_{C^1}}}{{{\mathbf{M}}}(f_n^*\cdots f_1^* a)}.$$ We conclude with two remarks. Firstly, the moment condition implies that ${\frac{1}{n}} \sum_{i=0}^{n-1} \log{{{\left\Vertf_i{^{-1}}\right\Vert}}_{C^1}}$ is almost surely bounded. Secondly, Lemma \[lem:growth2\] shows that ${{\mathbf{M}}}(f_n^*\cdots f_1^* a)$ goes exponentially fast to infinity for $\nu^{\mathbf{N}}$-almost every $\omega=(f_n)$ (this is where the order of compositions matters). Thus ${\mathrm{diam}}\big({
\underline{f}_n^*\cdots \underline{f}_1^*{\left(\underline {a}\right)}}\big) \to 0$ almost surely, and we are done.
\[rem:measurability\] The uniqueness of $T_a$ in its cohomology class implies that $T_a$ depends measurably on $a$. Indeed there is a set $E\subset \operatorname{Lim}(\Gamma)$ of full measure along which the map $\underline a\mapsto T_a$ is continuous (recall that the space $\operatorname{{Cur}}_1(X)$ of positive closed currents of mass 1 on $X$ is a compact metrizable space). This implies that $\underline a\mapsto T_a$ is a measurable map from $\operatorname{Lim}(\Gamma)$, endowed with the $\mu_{\partial}$-completion of the Borel $\sigma$-algebra, to $\operatorname{{Cur}}_1(X)$, endowed with its Borel $\sigma$-algebra.
Continuous potentials
---------------------
We now study the limit currents $T^s_\omega$ introduced in Corollary \[cor:Tsomega\].
\[thm:Tsomega\_continuous\] Let $(X,\nu)$ be a non-elementary random holomorphic dynamical system on a compact Kähler surface $X$, satisfying the moment condition . Then for $\nu^{\mathbf{N}}$-a.e. $\omega$ the current $T^s_\omega$ has continuous potentials.
Let $\kappa$ be any Kähler form on $X$. Before proving this theorem, we start with the following lemma.
\[lem:bounded\_potentials\_subsequence\] For $\nu^{\mathbf{N}}$-almost every $\omega$, there exists an increasing sequence of integers $(n_i)_{i\geq 0}=(n_i(\omega))$ such that
1. the potentials ${{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)^{-1}u_{{\left(f_\omega^{n_i}\right)}^*\kappa}$ are uniformly bounded;
2. the potentials ${{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)^{-1}u_{(f_\omega^{n_i})_*\kappa}$ are uniformly bounded too.
If the exponential moment condition holds, these two assertions hold for all $n$ (i.e. extracting a subsequence $(n_i)$ is not necessary); in addition the function $\omega\mapsto \log^+{{\left\Vertu_{T^s_\omega}\right\Vert}}_\infty $ is $\nu^{\mathbf{N}}$-integrable.
Recall the notation $\omega=(f_n)_{n\geq 0}$. First, $$\begin{aligned}
f_{n-1}^*\kappa & = f_{n-1}^*\Theta(\kappa)+dd^c\left( u_\kappa{\!\circ\!}f_{n-1}\right) \\
&\notag = \Theta(f_{n-1}^*\kappa)+dd^c\left( u_{f_{n-1}^*\Theta(\kappa)}+u_\kappa{\!\circ\!}f_{n-1} \right).\end{aligned}$$ Note that we will not introduce the constants $E(f_n; \kappa)$ in this computation. Then, we obtain $$\begin{aligned}
f_{n-2}^*f_{n-1}^*\kappa &= f_{n-2}^*\Theta(f_{n-1}^*\kappa)+dd^c\left( u_{f_{n-1}^*\Theta(\kappa)}{\!\circ\!}f_{n-2}+u_\kappa{\!\circ\!}(f_{n-1}{\!\circ\!}f_{n-2}) \right) \\
&= \Theta(f_{n-2}^*f_{n-1}^*\kappa)+dd^c\left( u_{ f_{n-2}^*\Theta(f_{n-1}^*\kappa)}+u_{f_{n-1}^*\Theta(\kappa)} {\!\circ\!}f_{n-2}+u_\kappa{\!\circ\!}(f_{n-1}{\!\circ\!}f_{n-2}) \right) .\end{aligned}$$ Setting $G_{j,k}=f_{k-1}\!\circ \cdots \circ \! f_j$, for $ j\leq k-1$ (so in particular $G_{0,j} = f_\omega^j$ for all $j\geq 1$) with the additional convention that $G_{j,j}=\operatorname{id}_X$, we get $$\begin{aligned}
(f_\omega^n)^*\kappa &= \Theta((f_\omega^n)^*\kappa) + dd^c\left( u_\kappa{\!\circ\!}f_\omega^n+ \sum_{j=0}^{n-1} u_{f_j^*\Theta(G_{j+1,n}^*\kappa)}\circ G_{0,j} \right).\end{aligned}$$ Let $u_n$ be the function in the parenthesis. We want to estimate the sup-norm ${{\left\Vert u_n\right\Vert}}_\infty$. Lemma \[lem:utheta\] and Corollary \[cor:utheta\] provide successively the following upper bounds $$\begin{aligned}
{{\left\Vert u_{f_j^*\Theta(G_{j+1,n}^*\kappa)}\right\Vert}}_\infty
\leq C {{\left\Vert f_j \right\Vert}}_{C^1}^2{{\mathbf{M}}}(G_{j+1,n}^*\kappa)
\leq C {{\mathbf{M}}}(\kappa) {{\left\Vert f_j\right\Vert}}_{C^1}^2 {{\left\Vert G_{j+1,n}^*\right\Vert}}, \end{aligned}$$ and $$\label{eq:sum_potential}
{{\left\Vert {\frac{1}{{{\mathbf{M}}}((f_\omega^{n})^*\kappa)}}u_{n}\right\Vert}}_\infty
\leq \frac{{{\left\Vert u_\kappa\right\Vert}}_\infty}{{{\mathbf{M}}}((f_\omega^{n})^*\kappa)}+ C{{\mathbf{M}}}(\kappa)
\sum_{j=0}^{n-1}{{\left\Vert f_j\right\Vert}}_{C^1}^2
\frac{{{\left\Vert G_{j+1,n}^*\right\Vert}}}{{{\mathbf{M}}}((f_\omega^{n})^*\kappa)}.$$ To estimate this sum we apply Theorem \[thm:GK\_good\_times\] to the subadditive cocycle $N(n,\omega)=\log {{\left\Vert (f_\omega^n)^*\right\Vert}}$, as we did for Corollary \[cor:GK\_good\_times\]: there exists a sequence $(\delta_j)$ of positive numbers converging to $0$, an increasing sequence $n_i=n_i(\omega)$ of integers, and a constant $C'(\omega)$ such that $$\frac{\big\|{ G_{j+1,n_i}^*}\big\|}{{{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)}\asymp \frac{\big\|{ f_{j+1}^* \cdots f_{n_i-1}^*}\big\|}{{{\left\Vert f_{0}^* \cdots f_{n_i-1}^*\right\Vert}}}\leq C' \exp(-(\lambda_1-\delta_j)j)$$ for all $i\geq 1$ and all $0\leq j\leq n_i$. Fix any real number ${\varepsilon}$ with $0< {\varepsilon}< \lambda_1$. Then from Lemma \[lem:borel-cantelli\_moment\], we know that, for almost every $\omega$, there is a constant $C''(\omega)$ such that ${{\left\Vert f_j\right\Vert}}_{C^1}^2\leq C''\exp({\varepsilon}j)$. So from we get $${{\left\Vert {\frac{1}{{{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)}}u_{n_i}\right\Vert}}_\infty \leq \frac{{{\left\Vert u_\kappa\right\Vert}}_\infty}{{{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)}+
C'''(\omega) {{\mathbf{M}}}(\kappa)\sum_{j=1}^{n_i}\exp(-(\lambda_1-{\varepsilon}-\delta(j))j)$$ This inequality shows that ${{\left\Vert {{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)^{-1}u_{n_i}\right\Vert}}_\infty $ is uniformly bounded.
Now, note that $u_{(f_\omega^n)^*\kappa}=u_n+E_n$ with $E_n = - \int u_n {{\sf{vol}}}$. Since ${{\left\Vert {{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)^{-1}u_{n_i}\right\Vert}}_\infty $ is uniformly bounded, so is ${{\mathbf{M}}}((f_\omega^{n_i})^*\kappa)^{-1} E_{n_i}$, and the first assertion of the lemma is established.
The second assertion is proved exactly in the same way with expressions of the form ${f_j^*\Theta(G_{j+1,n}^*\kappa)}$ replaced by ${(f_{n-j}{^{-1}})^*\Theta( (f_0{^{-1}}{\!\circ\!}\cdots {\!\circ\!}f_{n-j-1}{^{-1}})^*\kappa)}$, using the second estimate in Corollary \[cor:GK\_good\_times\], and the fact that for every $f\in {\mathsf{Aut}}(X)$, ${{\left\Vertf^*\right\Vert}}\asymp {{\left\Vert(f{^{-1}})^*\right\Vert}}$.
If the exponential moment condition holds, we follow the same argument and apply Proposition \[pro:gouezel\] instead of Theorem \[thm:GK\_good\_times\] to , with $D(f) = {{\left\Vertf\right\Vert}}_{C^1} ^2$.
First, we prove that [*the normalized potential $u_{T_\omega^s}$ is bounded, for $\nu^{\mathbf{N}}$-almost every $\omega$*]{}. To see this, recall that ${{\mathbf{M}}}((f_\omega^n)^*\kappa)^{-1} (f_\omega^n)^*\kappa$ converges to $T_\omega^s$ as $n\to \infty$. From Lemma \[lem:bounded\_potentials\_subsequence\], we know that the normalized potential ${{\mathbf{M}}}((f_\omega^{n})^*\kappa)^{-1}u_{{\left(f_\omega^{n}\right)}^*\kappa}$ of the current ${{\mathbf{M}}}((f_\omega^n)^*\kappa)^{-1} {\left(f_\omega^n\right)}^*\kappa$ is uniformly bounded along some subsequence $n_i=n_i(\omega)$. These potentials are $A\kappa_0$-psh functions on $X$ so, by compactness, they converge to $u_{T_\omega^s}$ in $L^1(X;{{\sf{vol}}})$. Thus, $u_{T_\omega^s}$ is essentially bounded. We conclude that $u_{T_\omega^s}$ is bounded because quasi-plurisubharmonic functions have a value (in ${\mathbf{R}}\cup\{-\infty\}$) at every point.
Now, we show that [*$u_{T_\omega^s}$ is continuous*]{}. Here, we use an argument which is similar to that of the proof of Theorem \[thm:uniq+extremal\]. Under a stronger (exponential) moment condition, one can derive better continuity results through a finer study of the sequence ${{\mathbf{M}}}((f_\omega^{n})^*\kappa)^{-1}u_{{\left(f_\omega^{n}\right)}^*\kappa}$ (see Theorem \[thm:holder\] below). If $T$ is a positive closed current with bounded potential on $X$ we define $${\mathrm{Jump}}(T)=\max_{x\in X} \left(\limsup_{y\to x} u_{T}(y)- \liminf_{y\to x} u_{T}(y)\right).$$ Then $0\leq {\mathrm{Jump}}(T)\leq 2{{\left\Vert u_{T}\right\Vert}}_\infty$, and $\mathrm{Jump}(T)=0$ if and only if $u_{T}$ is continuous. In addition $\mathrm{Jump}(f^*T) = \mathrm{Jump}(T)$ for every $f\in {\mathsf{Aut}}(X)$ because $f^*T=\Theta(f^*a) + dd^c(u_{f^*\Theta(a)} + u_T\circ f)$ and $u_{f^*\Theta([T])}$ is continuous (see Equation ). From the equivariance relation $$\label{eq:equivariance_Tsn}
T^s_\omega = {\frac{1}{{{\mathbf{M}}}{\left({{\left(f_\omega^n \right)}^* T^s_{\sigma^n\omega}}\right)}}} T^s_{\sigma^n\omega},$$ which follows from Equation , we get that $${\mathrm{Jump}}{\left(T^s_\omega\right)}= {\frac{1}{{{\mathbf{M}}}{\left({{\left(f_\omega^n \right)}^* T^s_{\sigma^n\omega}}\right)}}} {\mathrm{Jump}}{\left(T^s_{\sigma^n\omega}\right)}.$$ Remark \[rem:measurability\] says that $\omega\mapsto T^s_\omega$ is measurable; therefore $\omega\mapsto u_{T^s_\omega}$ is measurable too. From the first step of the proof, if $C$ is large, then there is a subset $\Omega_C\subset \Omega$ with $\nu(\Omega_C)>0$ such that ${{\left\Vert u_{T^s_\omega}\right\Vert}}_\infty\leq C$ for every $\omega\in \Omega_C$. The ergodicity of the shift implies that $\sigma^n\omega\in \Omega_C$, hence $\big\Vert u_{T^s_{\sigma^n\omega}} \big\Vert_\infty\leq C$, for almost every $\omega$ and infinitely many $n$; thus, ${\mathrm{Jump}}{\left(T^s_{\sigma^n\omega}\right)}\leq 2C$. By Lemma \[lem:variant\_furstenberg\], ${{\mathbf{M}}}{\left({{\left(f_\omega^n \right)}^* T^s_{\sigma^n\omega}}\right)}$ goes to infinity almost surely. So, we conclude that ${\mathrm{Jump}}{\left(T^s_\omega\right)}=0$ and the proof is complete.
\[thm:holder\] Let $(X,\nu)$ be a non-elementary random holomorphic dynamical system on a compact Kähler surface $X$, satisfying the exponential moment condition . Then there exists $\theta>0$ such that for $\nu^{\mathbf{N}}$-almost every $\omega$ the potential $u_{T^s_\omega}$ is Hölder continuous of exponent $\theta$.
The proof is based on the following well-known fact, applied to $u = u_{T_\omega^s}$: let $u_n$ be a sequence of continuous functions converging uniformly to $u:M\to {\mathbf{R}}$ on some metric space $M$. If ${{\left\Vertu_n-u\right\Vert}}_\infty\leq A^n$ and $\operatorname{{Lip}}(u_n)\leq B^n$ with $A<1<B$, then $u$ is a Hölder continuous function of exponent $\alpha = -\log(A)/(\log(B)-\log(A))$.
The proof is based on computations similar (but not identical) to those of of Lemma \[lem:bounded\_potentials\_subsequence\]. Keeping the notation $G_{j,n}=f_{n-1}\circ \cdots \circ f_j$, a descending induction starting from $$f_{n-1}^* T_{\sigma^n \omega}^s = \Theta(f_{n-1}^*T_{\sigma^n \omega}^s)
+dd^c{\left(u_{ f_{n-1}^* \Theta(T_{\sigma \omega}^s)} + u_{T_{\sigma^n \omega}^s} \circ f_{n-1}\right)}$$ yields $$\begin{aligned}
(f_\omega^n)^*T_{\sigma^n \omega}^s &=
\label{eq:enorme} \Theta{\left((f_\omega^n)^*T_{\sigma^n \omega}^s\right)} + dd^c{\left(\sum_{j=0}^{n-1} u_{f_j^*\Theta(G_{j+1, n}^* T_{\sigma^n \omega}^s)} \circ f_\omega^{j}
+ u_{T_{\sigma^n \omega}^s} \circ f_\omega^{n}
\right)}. \end{aligned}$$ Thus, there is a constant of normalization $E=E(\omega; n)$ such that $$\label{eq:uTs_final}
u_{T^s_\omega} = {\frac{1}{{{\mathbf{M}}}((f_\omega^n)^*( T_{\sigma^n\omega}^s))}} {\left(\sum_{j=0}^{n-1} u_{f_j^*\Theta(G_{j+1, n}^* T_{\sigma^n \omega}^s)} \circ f_\omega^{j}
+ u_{T_{\sigma^n \omega}^s} \circ f_\omega^{n}
\right)}+ E.$$ Note that the additional term $E$ does not affect the modulus of continuity of $u_{T^s_\omega}$. By Lemma \[lem:utheta\] and Corollary \[cor:utheta\] we have $\operatorname{{Lip}}(u_{f_j^*\Theta(a)})\leq C {{\left\Vertf_j\right\Vert}}_{C^1}^2 {{\mathbf{M}}}(a)$ for every class $a\in {{\mathrm{Psef}}}(X)$; hence $$\begin{aligned}
\operatorname{{Lip}}{\left(u_{f_j^*\Theta(G_{j+1, n}^* T_{\sigma^n \omega}^s)}\right)}& \leq C {{\left\Vertf_j\right\Vert}}_{C^1}^2 {{\mathbf{M}}}(G_{j+1, n}^* T_{\sigma^n \omega}^s) \leq
C {{\left\Vertf_j\right\Vert}}_{C^1}^2 {{\left\VertG_{j+1, n}^* \right\Vert}} \\&\leq C {{\left\Vertf_j\right\Vert}}_{C^1}^2 \prod_{\ell=j+1}^{n-1} {{\left\Vertf_\ell^*\right\Vert}}_{H^{1, 1}}
\leq C \prod_{\ell=j}^{n-1} {{\left\Vertf_\ell\right\Vert}}^2_{C^1} .\end{aligned}$$ Finally, using that ${{\left\Vertf_j\right\Vert}}_{C^1} \geq \operatorname{{Lip}}(f_j)\geq 1$, we obtain that for every $0\leq j\leq n-1$, $$\begin{aligned}
\operatorname{{Lip}}{\left(u_{f_j^*\Theta(G_{j+1, n}^* T_{\sigma^n \omega}^s)}\circ f_\omega^j \right)}
& \leq \operatorname{{Lip}}{\left(u_{f_j^*\Theta(G_{j+1, n}^* T_{\sigma^n \omega}^s)}\right)} \prod_{\ell=0}^{j-1} \operatorname{{Lip}}(f_\ell) \\
& \leq C \prod_{\ell=0}^n {{\left\Vertf_\ell\right\Vert}}_{C^1}^2.
\end{aligned}$$ Denoting the modulus of continuity by $\omega(u,r) = \sup_{d(x,x')\leq r} {\left\vertu(x) - u(x')\right\vert}$, we infer from Equation that $$\omega( {u_{T^s_\omega}, r} )\leq {\frac{1}{{{\mathbf{M}}}{\left((f_\omega^n)^*( T_{\sigma^n\omega}^s)\right)}}}
{\left( C n \prod_{\ell=0}^{n-1} {{\left\Vertf_\ell\right\Vert}}_{C^1}^2 \cdot r+ \big\|{u_{T_{\sigma^n \omega}^s}}\big\|_\infty\right)}.$$
To ease notation set $\lambda = \lambda_{H^{1,1}}$. Fix a small ${\varepsilon}>0$. By Lemma \[lem:variant\_furstenberg\], for almost every $\omega$ there exists $C = C_{\varepsilon}(\omega)$ such that ${{\mathbf{M}}}{\left((f_\omega^n)^*( T_{\sigma^n\omega}^s)\right)}^{-1} \leq C e^{-n(\lambda-{\varepsilon})}$ for every $n$.
Fix $M$ larger than but close to $\exp{\left({\mathbb{E}}{\left(\log{{\left\Vertf\right\Vert}}_{C^1}\right)}\right)}$. Applied to the $\nu^{\mathbf{N}}$-integrable function $\omega=(f_n)\mapsto \log{{\left\Vertf_0\right\Vert}}_{C^1}$, the Birkhoff ergodic theorem gives $$\prod_{\ell=1}^n {{\left\Vertf_\ell\right\Vert}}_{C^1}^2 \leq CM^n
\; \text{ as well as } \;
n\prod_{\ell=1}^n {{\left\Vertf_\ell\right\Vert}}_{C^1}^2 \leq CM^n$$ for some $C=C_M(\omega)$ (increase $M$ to get from the first to the second inequality). Thus, $$\omega( {u_{T^s_\omega}, r}) \leq C e^{-n(\lambda -{\varepsilon})} {\left(M^n r+
\big\|{u_{T_{\sigma^n \omega}^s}}\big\|_\infty\right)}$$ for some $C>0$. By Lemma \[lem:bounded\_potentials\_subsequence\], $\omega\mapsto \log^+{{\left\Vertu_{T^s_\omega}\right\Vert}}_\infty$ is integrable, so for almost every $\omega$ there exists $C= C_{\varepsilon}(\omega)$ such that $\big\|{u_{T_{\sigma^n \omega}^s}}\big\|_\infty \leq C e^{{\varepsilon}n}$ holds for every $n$, and we infer that $$\omega({u_{T^s_\omega}, r}) \leq C e^{-n (\lambda- {\varepsilon})} (M^nr+e^{{\varepsilon}n}) =
C e^{-n (\lambda- 2{\varepsilon})}{\left((Me^{-{\varepsilon}})^nr+1\right)}.$$ Choosing $n$ so that $r \asymp (Me^{-{\varepsilon}})^{-n}$ we get $\omega({u_{T^s_\omega}, r}) \leq C r^\theta$ with $\theta = \frac{\lambda-2{\varepsilon}}{ \log M+{\varepsilon}}$ and the proof of the theorem is complete.
Glossary of random dynamics, II {#sec:Glossary_II}
===============================
In this section we consider a random holomorphic dynamical system $(X, \nu)$ on a compact Kähler surface, satisfying the moment condition . Here the group $\Gamma_\nu$ may a priori be elementary. Note also that the compactness assumption on $X$ can be dropped in most of these results (in this case should be strengthened to a $C^2$ moment condition).
Our goal is to collect a number of facts from the ergodic theory of random dynamical systems, including the construction of associated skew products, fibered entropy, Lyapunov exponents of stationary measures, stable and unstable manifolds, and various measurable partitions. Since some subsequent arguments will rely on the work [@br] of Brown and Rodriguez-Hertz, we have tried to make notation consistent with that paper as much as possible.
Skew products and stationary measures associated to $(X, \nu)$ {#par:definition_skew_products}
--------------------------------------------------------------
Define:
- $\Omega = {\mathsf{Aut}}(X)^{\mathbf{N}}$, whose elements are denoted by $\omega = (f_n)_{n\geq 0}$. The one-sided shift acting on $\Omega$ is denoted by $\sigma$.
- $\Sigma = {\mathsf{Aut}}(X)^{\mathbf{Z}}$, whose elements are denoted by $\xi = (f_n)_{n\in {\mathbf{Z}}}$. The two-sided shift acting on $\Sigma$ is denoted by $\vartheta\colon \Sigma\to \Sigma$.
- ${\mathscr{X}}= \Sigma \times X$ and ${\mathscr{X}}_+ = \Omega \times X$, whose elements are denoted by ${\mathscr{x}}= (\xi, x)$ and ${\mathscr{x}}= (\omega, x)$ respectively. The natural projections are denoted by $\pi_\Sigma:{\mathscr{X}}\to \Sigma$ (resp. $\pi_\Omega:{\mathscr{X}}_+\to \Omega$) and $\pi_X:{\mathscr{X}}\to X$ (resp. $\pi_X:{\mathscr{X}}_+\to X$, using the same notation).
### Skew products
For $\omega\in \Omega$ and $n\geq 1$, $f^n_\omega$ is the left composition $f^n_\omega= f_{n-1}\circ \cdots \circ f_0$; in particular, $f^1_\omega=f_0$. For $n=0$, we set $f^0_\omega = \mathrm{id}$. This is consistent with the notation used in the previous sections. The same notation $f_\xi^n$ is used for $\xi\in \Sigma$ and $n\geq 0$. When $n<0$, we set $f^n_\xi=(f_{n}){^{-1}}\circ\cdots \circ (f_{-1}){^{-1}}$. With this definition the cocycle formula $f^{n+m}_\xi = f^n_{\vartheta^m\xi}\circ f^m_\xi$ holds for all $(m,n) \in {\mathbf{Z}}^2$.
By definition, the skew products induced by the random dynamical system $(X, \nu)$ are the transformations $ {F}_+\colon {\mathscr{X}}_+ \to {\mathscr{X}}_+$ and ${F}\colon {\mathscr{X}}\to {\mathscr{X}}$ defined by $$\begin{aligned}
F_+\colon (\omega, x) &\longmapsto (\sigma\omega, f^1_\omega(x)) \\
{F}\colon \; (\xi, x) &\longmapsto (\vartheta\xi, f^1_\xi(x)). \end{aligned}$$ Let $\varpi$ denote the natural projection $\varpi\colon \Sigma \to \Omega$; then $\varpi\circ {F}= {F}_+ \circ \varpi$. Note that $F$ is invertible, with $F^{-1}({\mathscr{x}})=(\vartheta^{-1}\xi, f^{-1}_{\theta^{-1}\xi}(x))$, but $F_+$ is not; indeed it can be shown that [*$({\mathscr{X}}, {F})$ is the natural extension of $({\mathscr{X}}_+, {F}_+)$*]{}.
\[lem:stationary\] The measure $\mu$ on $X$ is stationary if and only if the product measure $${\mathscr{m}}_+:= \nu^{\mathbf{N}}\times \mu$$ on ${\mathscr{X}}_+$ is invariant under ${F}_+$.
Recall that the product $\sigma$-algebra on $\Omega$ is generated by [**[cylinders]{}**]{} ([^3]), and that it coincides with the Borel $\sigma$-algebra $\mathcal B(\Omega)$. So the invariance of ${\mathscr{m}}_+$ is equivalent to the equality $$\label{eq:stationary}
{\mathscr{m}}_+({F}_+{^{-1}}(C\times A)) = {\mathscr{m}}_+ (C\times A) = \left( \prod_{j=0}^N \nu(C_j) \right) \cdot \mu(A),$$ for all cylinders $C = C_0\times \cdots \times C_N$ in $\Omega$ and Borel sets $A\subset X$. By definition $${F}_+{^{-1}}(C\times A) = {\left\{(\omega, x)\in \Omega\times X\; ; \ f_N\in C_{N-1} , \ldots , f_1\in C_{0},
f_0(x)\in A\right\}},$$ so clearly it is enough to check for $N=1$. Now by Fubini’s theorem $$\begin{aligned}
\label{eq:fubini}
(\nu\times \mu){\left({\left\{ (f_0, x) \; ; \ f_0(x)\in A \right\}}\right)} &\notag= \int\int \mathbf{1}_{f_0{^{-1}}(A)}(x) \, d\nu(f_0)\, d\mu(x) \\ &= \int\int \mu(f_0{^{-1}}(A))\, d\nu(f_0) \end{aligned}$$ and the result follows.
Recall that a stationary measure is [**[ergodic]{}**]{} if it is an extremal point in the convex set of stationary measures; hence, $\mu$ is ergodic iff ${\mathscr{m}}_+$ is ${F}_+$-ergodic. Actually $\mu$ is ergodic if and only if every $\nu$-almost surely invariant measurable subset $A\subset X$ (that is a measurable subset such that for $\nu$-almost every $f$, $\mu(A\Delta f{^{-1}}(A)) =0$) has measure $\mu(A)=0$ or 1. This is by no means obvious since ${F}_+$-invariant sets have no reason to be of product type. This statement is part of the so-called **random ergodic theorem** (see Propositions 1.8 and 1.9 in [@benoist-quint_book]).
\[pro:m\] There exists a unique ${F}$-invariant probability measure ${\mathscr{m}}$ on ${\mathscr{X}}$ projecting on ${\mathscr{m}}_+$ under the natural projection ${\mathscr{X}}\to {\mathscr{X}}_+$. Moreover,
1. the measure ${\mathscr{m}}$ is equal to the weak-$\star$ limit $${\mathscr{m}}= \lim_{n\to\infty} ({F}^n)_\varstar {\left(\nu^{\mathbf{Z}}\times \mu\right)}.$$
2. the projections $(\pi_\Sigma)_*{\mathscr{m}}$ and $(\pi_X)_*{\mathscr{m}}$ are respectively equal to $\nu^{\mathbf{Z}}$ and $\mu$;
3. the equality ${\mathscr{m}}= \nu^{\mathbf{Z}}\times \mu $ holds if and only if $\mu$ is $f$-invariant for $\nu$-almost every $f$;
4. $({\mathscr{X}}, {F}, {\mathscr{m}})$ is ergodic if and only if $({\mathscr{X}}_+, {F}_+, {\mathscr{m}}_+)$ is.
The existence and uniqueness of ${\mathscr{m}}$, as well as the characterization of its ergodicity, follow from the fact that $({\mathscr{X}}, {F})$ is the natural extension of $({\mathscr{X}}_+, {F}_+)$ (see [@kifer §1.2] for a detailed explanation).
Let us prove directely that the limit in (1) does exist, and show that this limit ${\mathscr{m}}$ satisfies (2) and (3). Since $\varpi_\varstar{\left(\nu^{\mathbf{Z}}\times \mu\right)} = \nu^{\mathbf{N}}\times \mu = {\mathscr{m}}_+$ and $\varpi\circ {F}= {F}_+ \circ \varpi$, the ${F}_+$-invariance of $ {\mathscr{m}}_+$ gives $\varpi_\varstar({F}^n)_\varstar {\left(\nu^{\mathbf{Z}}\times \mu\right)} = {\mathscr{m}}_+$ for every $n\in {\mathbf{Z}}$. So if we prove that the limit $\lim_{n\to\infty} ({F}^n)_\varstar {\left(\nu^{\mathbf{Z}}\times \mu\right)} $ exists, then this limit ${\mathscr{m}}$ will be an ${F}$-invariant probability measure projecting on ${\mathscr{m}}_+$ under $\varpi$; hence it will coincide with the invariant measure ${\mathscr{m}}$. To prove this convergence, we consider a cylinder $C = \prod_{j=-N}^N C_j$ in $\Sigma$ and a Borel set $A\subset X$, and we show that $(\nu^{\mathbf{Z}}\times \mu )({F}^{-n}(C\times A))$ stabilizes for $n>N$. Arguing as in Lemma \[lem:stationary\], we see that the set ${F}^{-n}(C\times A)$ is equal to the set of points ${\mathscr{x}}= (\xi, x)$ satisfying the constraints $(\theta^n\xi)_j\in C_j$ for $-N\leq j\leq N$ and $x\in (f_\xi^n)^{-1}(A)$; for $n>N$, these constraints are independent, and ${\left(\nu^{\mathbf{Z}}\times \mu\right)}({F}^{-n}(C\times A))$ is equal to $$\begin{aligned}
\nu^{\mathbf{Z}}(\theta^{-n}(C))\times {\left( \nu^n\times\mu\right)}{\left(
{\left\{ (f_0, \ldots , f_{n-1}, x) \; ; \; \ f_{n-1}\circ \cdots \circ f_0 (x)\in A \right\}}\right)}.\end{aligned}$$ Then the invariance of $\nu^{\mathbf{Z}}$ under the shift and the the stationarity of $\mu$ give (see Equation ) $$\begin{aligned}
{\left(\nu^{\mathbf{Z}}\times \mu\right)}({F}^{-n}(C\times A)) &= \nu^{\mathbf{Z}}(C) \times \int \mu{\left(f_{n-1}{^{-1}}\cdots f_0{^{-1}}A\right)} \nu(f_0) \cdots \nu(f_{n-1}) \\
&= \notag\nu^{\mathbf{Z}}(C) \times \mu(A).\end{aligned}$$ This proves Assertions (1) and (2). For Assertion (3) it will be enough for us to consider the case where $\Gamma$ is discrete. By Assertion (1) we see that $\nu^{\mathbf{Z}}\times \mu$ is $F$-invariant if and only if ${\mathscr{m}}= \nu^{\mathbf{Z}}\times \mu$. Now assume ${\mathscr{m}}= \nu^{\mathbf{Z}}\times \mu $ and let us show that $\mu$ is $\Gamma_\nu$-invariant. The reverse implication is similar. Fix $f_0 \in \operatorname{Supp}(\nu)$ and consider the cylinder $C = C_0 = {\left\{f_0\right\}}$ (in $0^{\mathrm{th}}$ position). If $A\subset X$ is a Borel subset we have $$\label{eq:CtimesA}
{\left(\nu^{\mathbf{Z}}\times \mu\right)}({F}(C\times A)) = {\left(\nu^{\mathbf{Z}}\times \mu\right)}(C\times A) = \nu{\left(C_0\right)} \times \mu(A).$$ On the other hand ${F}(C\times A) = \vartheta (C) \times f_0(A)$ so the left hand side of is equal to $ \nu{\left(C_0\right)} \times \mu(f_0(A))$. Thus, we obtain that $\mu(f_0(A)) = \mu(A)$, which proves that $\mu$ is $\Gamma_\nu$-invariant.
### Past, future, and partitions
Let $\mathcal F$ denote the $\sigma$-algebra on ${\mathcal{X}}$ obtained by taking the ${\mathscr{m}}$-completion of $\mathcal{B}(\Sigma)\otimes \mathcal{B}(X)$. It will often be important to detect objects depending only on the “future” or “past”. To formalize this, we define two $\sigma$-algebras on $\Sigma$:
- $\hat{ \mathcal{F}}^+$ is the $\nu^{\mathbf{Z}}$-completion of the $\sigma$-algebra generated by the cylinders $C = \prod_{j=0}^N C_j$.
- $\hat{\mathcal {F}}^-$ is the $\nu^{\mathbf{Z}}$-completion of the $\sigma$-algebra generated by the cylinders $C = \prod_{j=-N}^{-1} C_j$.
To formulate it differently, we define [**[local stable and unstable sets]{}**]{} for the shift $\vartheta$: $$\label{eq:definition_sigma_s_u}
\Sigma^s_{\mathrm{loc}}(\xi) = {\left\{\eta \in \Sigma, \ \forall i\geq 0, \ \eta_i = \xi_i\right\}} \; \text{ and } \;
\Sigma^u_{\mathrm{loc}}(\xi) = {\left\{\eta \in \Sigma, \ \forall i< 0, \ \eta_i = \xi_i\right\}}.$$ Then a subset of $ \Sigma$ is $\hat{\mathcal {F}}^+$-measurable (resp. $\hat{\mathcal {F}}^-$ measurable) if up to a set of zero $\nu^{\mathbf{Z}}$-measure it is Borel and saturated by local stable sets $\Sigma^s_{\mathrm{loc}}(\xi)$ (resp. unstable sets $\Sigma^u_{\mathrm{loc}}(\xi)$). The $\sigma$-algebra $\mathcal{F}^+$ on ${\mathscr{X}}$ will be the ${\mathscr{m}}$-completion of $\hat{ \mathcal{F}}^+\otimes \mathcal B(X)$. An $\mathcal{F}^+$-measurable object should be understood as “depending only on the future” thus it makes sense on ${\mathscr{X}}$ and ${\mathscr{X}}_+$. The $\sigma$-algebra $\mathcal F^-$ of “objects depending only on the past” is defined analogously. This $\sigma$-algebra $\mathcal F^-$ is generated, modulo ${\mathscr{m}}$-negligible sets, by the partition into subsets of the form $\Sigma^u_{\mathrm{loc}}(\xi)\times {\left\{x\right\}}$; each of these sets can be naturally identified to $\Omega$.
For $\xi\in \Sigma$ we set $X_\xi = {\left\{\xi\right\}}\times X = \pi_\Sigma{^{-1}}(\xi)$, which can be naturally identified with $X$. The disintegration of the probability measure ${\mathscr{m}}$ with respect to the partition into fibers of $\pi_\Sigma$ gives rise to a family of conditional probabilities ${\mathscr{m}}_\xi$ such that ${\mathscr{m}}=\int {\mathscr{m}}_\xi \, d\nu^{\mathbf{Z}}(\xi)$, because $(\pi_\Sigma)_*{\mathscr{m}}=\nu^{\mathbf{Z}}$.
\[lem:conditionals\] The conditional measure ${\mathscr{m}}_\xi$ on $X_\xi$ satisfies $\nu^{\mathbf{Z}}$-almost surely $$\notag
{\mathscr{m}}_\xi = \lim_{n\to+\infty} (f_{-1}\circ \cdots \circ f_{-n})_\varstar \mu = \lim_{n\to+\infty} (f^{n}_{\vartheta^{-n}\xi})_\varstar \mu.$$ In particular, the family of measures $\xi\mapsto {\mathscr{m}}_\xi$ is $\mathcal{F}^-$-measurable.
It follows from the martingale convergence theorem that the limit $$\tilde\mu_\xi:= \lim_{n\to+\infty} (f_{-1}\circ \cdots \circ f_{-n})_\varstar \mu$$ exists almost surely (see e.g. [@benoist-quint_book §2.5] or [@bougerol-lacroix] II.2). Now ${F}^n$ maps $X_{\vartheta^{-n}\xi}$ to $X_\xi$ and $F^n{ \arrowvert_{X_{\vartheta^{-n}\xi}}} = f_{-1}\circ \cdots \circ f_{-n}$, so $${\left(({F}^n)_\varstar (\nu^{\mathbf{Z}}\times \mu)\right)}(\; \cdot\; \vert X_\xi)
= (f_{-1}\circ \cdots \circ f_{-n})_\varstar \mu.$$ Identify $\tilde\mu_\xi$ with a measure on $X_\xi$. For every continuous function $\phi$ on ${\mathscr{X}}$ the dominated convergence theorem gives $$\begin{aligned}
{\left( ({F}^n)_\varstar (\nu^{\mathbf{Z}}\times \mu)\right)} (\varphi)
& = \int {\left(\int_{X_\xi}\varphi(x) \; d(f_{-1}\circ \cdots \circ f_{-n})_\varstar \mu(x)\right)} d\nu^{\mathbf{Z}}(\xi) \\
& {\underset{n\to\infty}{\longrightarrow}}\int {\left(\int_{X_\xi}\varphi(x) \; d\tilde\mu_\xi(x)\right)}d\nu^{\mathbf{Z}}(\xi).\end{aligned}$$ But ${\left( ({F}^n)_\varstar (\nu^{\mathbf{Z}}\times \mu)\right)} (\varphi)$ converges to ${\mathscr{m}}(\varphi)$, and the marginal of ${\mathscr{m}}$ with respect to the projection $\pi_\Sigma\colon {\mathscr{X}}\to \Sigma$ is $\nu^{\mathbf{Z}}$, so we get the result.
Since $\xi \mapsto {\mathscr{m}}_\xi$ is $\mathcal F^-$-measurable, the conditional measures of ${\mathscr{m}}$ on the atoms ${\mathcal F^-}({\mathscr{x}}):=\Sigma^u_{\mathrm{loc}}(\xi)\times {\left\{x\right\}}$ of the partition $\mathcal F^-$ are induced by the lifts of the conditionals of $\nu^{\mathbf{Z}}$ on the $\Sigma^u_{\mathrm{loc}}(\xi)$ under the natural projection $\pi_\Sigma:{\mathscr{X}}\to \Sigma $; in addition $\Sigma^u_{\mathrm{loc}}(\xi)$ identifies with $\Omega$ and $\nu^{\mathbf{Z}}(\; \cdot\; \vert \; \Sigma^u_{\mathrm{loc}})$ corresponds to $\nu^{\mathbf{N}}$. Thus, we get $$\begin{aligned}
\label{eq:conditional}
{\mathscr{m}}(\; \cdot\; \vert \; \mathcal F^-({\mathscr{x}})) & = {\nu^{\mathbf{Z}}(\; \cdot\; \vert \; \Sigma^u_{\mathrm{loc}}(\xi))}\times \delta_x \\
& \simeq \nu^{\mathbf{N}}\end{aligned}$$ for ${\mathscr{m}}$-almost every ${\mathscr{x}}= (\xi, x)\in {\mathscr{X}}$. This corresponds to Equation (9) in [@br]. By [@br Prop. 4.6], this implies that $\mathcal F^+\cap \mathcal F^-$ is equivalent, modulo ${\mathscr{m}}$-negligible sets, to ${\left\{\emptyset, \Sigma\right\}}\otimes \mathcal B(X)$.
Lyapunov exponents {#subs:lyapunov}
------------------
If $\mu$ is a stationary measure for $(X, \nu)$, the Oseledets theorem respectively applied to the tangent cocycles defined by the fiber dynamics $({\mathscr{X}}_+, F_+, {\mathscr{m}}_+)$ and $({\mathscr{X}}, F, {\mathscr{m}})$ allows us to define Lyapunov exponents. It is useful to state it separately in the invertible and non-invertible cases. Assume that the stationary measure $\mu$ (or equivalently ${\mathscr{m}}$ or ${\mathscr{m}}_+$) is ergodic. The upper and lower Lyapunov exponents $\lambda^+\geq \lambda^-$ are respectively defined by the almost sure limits $$\lambda^{+} = \lim_{n\to\infty} {\frac{1}{n}} \log {{\left\VertD_xf_\omega ^n\right\Vert}} \; \text{ and } \; \lambda^{-} =
\lim_{n\to\infty} {\frac{1}{n}} \log {{\left\Vert{\left(D_xf_\omega^{n}\right)}{^{-1}}\right\Vert}}{^{-1}};$$ and the convergence also holds on average. The existence of these limits is guaranteed by Kingman’s subadditive ergodic theorem, thanks to the moment condition .
### The non-invertible case
Define the tangent bundles $T{\mathscr{X}}_+ := \Omega \times TX$ and $T{\mathscr{X}}:= \Sigma \times TX$, and denote by $DF$ and $DF_+$ the natural tangent maps, that is $D_{(\xi, x)}F: {\left\{\xi\right\}}\times T_xX\to
{\left\{\vartheta \xi\right\}}\times T_{f_\xi(x)}X$ is induced by $D_xf_\xi$: $$D_{(\xi, x)}F(v)=D_xf_\xi^1(v) \quad (\forall v \in T_xX_\xi=T_xX)$$ For the non-invertible dynamics on ${\mathscr{X}}_+$, the Oseledets theorem gives: for ${\mathscr{m}}_+$-almost every $(\omega, x)$, there exists a non-trivial complex subspace $V^-(\omega, x)$ of ${\left\{\omega\right\}}\times T_xX$ such that $$\begin{aligned}
\forall v\in V^-(\omega, x)\setminus\{0\}, \; \; & \lim_{n\to + \infty} {\frac{1}{n}}\log {{\left\VertD_x f^n_\omega (v)\right\Vert}} = \lambda^{-} \\
\forall v\notin V^-(\omega,x), \; \; & \lim_{n\to + \infty} {\frac{1}{n}}\log {{\left\VertD_x f^n_\omega (v)\right\Vert}} = \lambda^{+}. \end{aligned}$$ The field of subspaces $V^-$ is measurable and almost surely invariant. Two cases can occur: either $\lambda^{-}<\lambda^{+}$ and $V^-(\omega, x)$ is almost surely a complex line, or $\lambda^{-} = \lambda^{+}$ and $V^-(\omega, x)={\left\{\omega\right\}}\times T_xX$.
### The invertible case
For the dynamical system ${F}\colon {\mathscr{X}}\to {\mathscr{X}}$, the statement is:
- if $\lambda^{-} = \lambda^{+}$ then for ${\mathscr{m}}$-almost every ${\mathscr{x}}= (\xi,x)$, for every non-zero $v\in T_xX_\xi\simeq T_xX$, $$\lim_{n\to\pm \infty} {\frac{1}{n}}\log {{\left\VertD_x f^n_\xi (v)\right\Vert}} = \lambda^{-};$$
- if $\lambda^{-} < \lambda^{+}$ then for ${\mathscr{m}}$-almost every ${\mathscr{x}}$ there exists a decomposition of $ T_xX_\xi$ $E^-(\xi, x) \oplus E^+(\xi, x)$ such that for $\star\in {\left\{-, +\right\}}$ and every $v\in E ^\star(\xi, x)\setminus {\left\{0\right\}}$, $$\lim_{n\to \pm\infty} {\frac{1}{n}}\log {{\left\VertD_xf^n_\xi(v)\right\Vert}} = \lambda^\star.$$ Furthermore the line fields $E^{\pm}$ are measurable and invariant, and $\log{\left\vert\angle (E^-, E^+)\right\vert}$ is integrable (here, the “angle” $\angle(E^-({\mathscr{x}}),E^+({\mathscr{x}}))$ is the distance between the two lines $E^-({\mathscr{x}})$ and $E^+({\mathscr{x}})$ in ${\mathbb{P}}(T_{{\mathscr{x}}}{\mathscr{X}})$).
### Hyperbolicity
It can happen that $\lambda^{-}$ and $\lambda^{+}$ have the same sign. If $\lambda^-$ and $\lambda^+$ are both negative, the conditional measures ${\mathscr{m}}_\xi$ are atomic: this can be shown by adapting a classical Pesin-theoretic argument (see e.g. [@katok-hasselblatt Cor. S.5.2]) to the fibered dynamics of $F$ on ${\mathcal{X}}$ (see [@lejan Prop. 2] for a direct proof and an example where the ${\mathscr{m}}_\xi$ have several atoms). Such random dynamical systems are called **proximal**. For instance, generic random products of automorphisms of ${{\mathbb{P}^2}}$ (that is, random products of matrices in ${{\sf{PGL}}}(3, {\mathbf{C}})$) are proximal; in such examples the stationary measure is not invariant. Other examples are given by contracting iterated function systems.
When $\lambda^+$ and $\lambda^-$ are both non-negative, we have the so-called **invariance principle** (this terminology is from [@avila-viana]).
\[thm:invariance\_principle\] Let $(X, \nu)$ be a random holomorphic dynamical system satisfying the integrability condition , and let $\mu$ be an ergodic stationary measure. If $\lambda^+(\mu) \geq \lambda^-(\mu)\geq 0$ then $\mu$ is almost surely invariant.
This result was proven by Crauel, building on ideas of Ledrappier which will be described in §\[subs:ledrappier\_invariance\_principle\] below (see Theorem 5.1, Corollary 5.3 and Remark 5.6 in [@crauel], and also Avila-Viana [@avila-viana Thm B]).
\[rem:invariance\_principle\_strict\] If in addition $\lambda^-$ and $\lambda^+$ are positive then $\mu$ is atomic. Indeed, since $\mu$ is almost surely invariant we get ${\mathscr{m}}= \nu^{\mathbf{Z}}\times \mu$. Reversing time, ${\mathscr{m}}$ has two negative Lyapunov exponents for $F{^{-1}}$, so as explained above the measures ${\mathscr{m}}_\xi$ are atomic. But by invariance ${\mathscr{m}}_\xi = \mu$, so $\mu$ is atomic as well.
By definition $\mu$ is [**hyperbolic**]{} if $\lambda^{-} <0<\lambda^{+}$. In this case we rather use the conventional superscripts $s/u$ instead of $-/+$. In the hyperbolic case (or more generally when $\lambda^{-}<\lambda^{+}$), we have $E^s = V^s$ so it follows that the complex line field $E^s$ on $T{\mathscr{X}}$ is $\mathcal F^+$-measurable. Conversely the unstable line field $E^u$ is $\mathcal F^-$-measurable.
Invariant volume forms
----------------------
Let us start with a well-known result.
\[lem:sum\_exponents\] Let $(X, \nu)$ be a random holomorphic dynamical system satisfying the integrability condition , and $\mu$ be an ergodic stationary probability measure. Then $$\lambda^{-}+\lambda^{+} = \int \log{\left\vert\operatorname{Jac}f(x)\right\vert} d\mu(x)d\nu(f),$$ where $\operatorname{Jac}$ denotes the Jacobian determinant relative to any smooth volume form on $X$.
We omit the proof, since this result is contained in Proposition \[pro:sum\_exponents\] below.
\[cor:sum\_exponentsK3\] Assume that $X$ is an Abelian, or K3, or Enriques surface. Let $\nu$ be a probability measure on ${\mathsf{Aut}}(X)$ satisfying the integrability condition , and $\mu$ be an ergodic $\nu$-stationary measure. Then $\lambda^{-}+\lambda^{+}=0$.
Remark \[rem:volume\_form\] provides an ${\mathsf{Aut}}(X)$-invariant volume form on $X$, thus the corollary follows from Lemma \[lem:sum\_exponents\].
Let $\eta$ be a non-trivial meromorphic $2$-form on the surface $X$. Then there is a cocycle $\operatorname{Jac}_\eta$, with values in the multiplicative group ${{\mathcal M}}(X)^\times$ of non-zero meromorphic functions, such that $$f^*\eta=\operatorname{Jac}_\eta(f)\eta$$ for every $f\in {\mathsf{Aut}}(X)$. We say that $\eta$ is [**[almost invariant]{}**]{} if ${\left\vert\operatorname{Jac}_\eta(f)(x)\right\vert}=1$ for every $x\in X$ and $\nu$-almost every $f\in {\mathsf{Aut}}(X)$ (in particular $\operatorname{Jac}_\eta(f)$ is a constant).
\[pro:sum\_exponents\] Let $(X, \nu)$ be a random holomorphic dynamical system satisfying the integrability condition , and $\mu$ be an ergodic stationary measure. Let $\eta$ be a non-trivial meromorphic $2$-form on $X$ such that
1. $\displaystyle \int \log^+\vert \operatorname{Jac}_\eta(f)(x)\vert d\mu(x) d\nu(f)<+\infty$;
2. $\mu$ gives zero mass to the set of zeroes and poles of $\eta$.
Then $$\label{eq:sum_exponent}
\lambda^{-}+\lambda^{+} = \int \log({\left\vert\operatorname{Jac}_\eta f (x)\right\vert}^2) d\mu(x)d\nu(f);$$ in particular $\lambda^{-} + \lambda^{+} = 0$ if $\eta$ is almost invariant.
We refer to the Examples section, in particular §\[par:Coble-Blanc\] for examples with an invariant meromorphic 2-form.
Fix a trivialization of the tangent bundle $TX$, given by a measurable family of linear isomorphisms $L(x)\colon T_xX\to {\mathbf{C}}^2$ such that (a) $\det(L(x))=1$ and (b) $1/C\leq {{\left\VertL(x)\right\Vert}}+{{\left\VertL(x)^{-1}\right\Vert}}\leq C$, for some constant $C>1$; here, the determinant is relative to the volume form ${{\sf{vol}}}$ on $X$ and the standard volume form $dz_1\wedge dz_2$ on ${\mathbf{C}}^2$, and the norm is with respect to the Kähler metric $(\kappa_0)_x$ on $T_xX$ and the standard euclidean metric on ${\mathbf{C}}^2$. For $(\xi, x)\in {\mathscr{X}}$ and $n\geq 0$, the differential $D_xf^n_\xi$ is expressed in this trivialization as a matrix $A^{(n)}(\xi, x)=L(f^n_\xi(x)) \circ D_xf^n_\xi\circ L(x)^{-1}$. Let $\chi^{-}_n(\xi, x)\leq \chi^{+}_n(\xi, x)$ be the singular values of $A^{(n)}(\xi, x)$. Then ${\mathscr{m}}$-almost surely, ${\frac{1}{n}}\log \chi^{\pm}_n(\xi, x) \to \lambda^{\pm}$ as $n\to +\infty$.
The form $\eta\wedge{\overline{\eta}}$ can be written $\eta\wedge {\overline{\eta}}=\varphi(x) {{\sf{vol}}}$ for some function $\varphi\colon X\to [0,+\infty]$. Locally, one can write $\eta=h(x) dx_1\wedge dx_2$ where $(x_1,x_2)$ are local holomorphic coordinates and $h$ is a meromorphic function; then $\varphi(x){{\sf{vol}}}= {\left\verth(x)\right\vert}^2 dx_1\wedge dx_2\wedge d{\overline{x_1}}\wedge d\overline{x_2}$. The jacobian $\operatorname{Jac}_\eta$ satisfies $$\vert \operatorname{Jac}_\eta(f)(x)\vert^2=\frac{\varphi(f(x))}{\varphi(x)}\operatorname{Jac}_{{\sf{vol}}}(f)(x)$$ for every $f\in {\mathsf{Aut}}(X)$ and $x\in X$. Using $\det(L(x))=1$, we get $$\label{eq:determinant}
\det (A^{(n)}(\xi, x)) = \operatorname{Jac}_{{\sf{vol}}}(f^n_\xi)(x),$$ and then $$\label{eq:jacobian}
{\frac{1}{n}}\log \chi^{-}_n(\xi, x) + {\frac{1}{n}}\log \chi^{+}_n(\xi, x) = \frac{2}{n}\log {\left\vert\operatorname{Jac}_\eta f^n_\xi(x)\right\vert} - {\frac{1}{n}}\log(\varphi(f^n_\xi(x))/\varphi(x)).$$ By the Oseledets theorem, the left hand side of converges almost surely to $\lambda^{-}+\lambda^{+}$. Since the Jacobian $\operatorname{Jac}_\eta$ is multiplicative along orbits, i.e. $\operatorname{Jac}_\eta f^n_\xi(x) = \prod_{k=0}^{n-1}
\operatorname{Jac}_\eta f_{\vartheta^k\xi}(f^k_\xi x)$, the integrability condition and the ergodic theorem imply that, almost surely, $$\begin{aligned}
\lim_{n\to\infty} \frac{2}{n} \log{\left\vert\operatorname{Jac}_\eta f_\xi^n(x)\right\vert}
& = 2 \int \log{\left\vert\operatorname{Jac}_\eta f_\xi^1 (x)\right\vert} d{\mathscr{m}}(\xi, x) \\
&\notag = 2 \int \log{\left\vert\operatorname{Jac}_\eta f_\omega^1 (x)\right\vert} d{\mathscr{m}}_+(\omega, x) \\
&\notag = 2 \int \log{\left\vert\operatorname{Jac}_\eta f(x)\right\vert} d\mu(x)d\nu(f).\end{aligned}$$ Since $\mu$ is ergodic and does not charge the zeroes and poles of $\eta$, we deduce that for ${\mathscr{m}}$-almost every $(\xi, x)$, there exists a sequence $(n_j)$ such that $f^{n_j}_\xi(x)$ stays at positive distance from the divisor of zeros and poles of $\eta$; along such a sequence, $\log\vert \varphi(f^{n_j}_\xi(x))/\varphi(x)\vert$ stays bounded, and the right hand side of tends to $\int \log{\left\vert\operatorname{Jac}_\eta f(x)\right\vert} d\mu(x)d\nu(f)$. This concludes the proof.
Intermezzo: local complex geometry {#subs:local}
----------------------------------
Recall that $X$ is endowed with a Riemannian structure, hence a distance, induced by the Kähler metric $\kappa_0$. For $x\in X$, we denote by ${{\sf{euc}}}_x$ the translation-invariant Hermitian metric on $T_xX$ (which is considered here as a manifold in its own right) associated to the Riemannian structure induced by $(\kappa_0)_x$. Given any orthonormal basis $(e_1,e_2)$ of $T_xX$ for this metric, we obtain a linear isometric isomorphism from $T_xX$ to ${\mathbf{C}}^2$, endowed respectively with ${{\sf{euc}}}_x$ and the standard euclidean metric; we shall implicitly use such identifications in what follows.
We denote by ${\mathbb{D}}(z;r)$ the disk of radius $r$ around $z$ in ${\mathbf{C}}$, and set ${\mathbb{D}}(r)={\mathbb{D}}(0;r)$.
### Hausdorff and $C^1$-convergence {#par:Hausdorff_C1}
Let $U\subset {\mathbf{C}}$ be a domain. If $\gamma\colon U\to X$ is a holomorphic curve, we can lift it canonically to a curve $\gamma^{(1)}\colon U\to TX$ by setting $\gamma^{(1)}(z)=(\gamma(z),\gamma'(z))\in T_{\gamma(z)}X$, where $\gamma'(z)$ denotes the velocity of $\gamma$ at $z$. Also, the riemannian metric $\kappa_0$ induces a riemannian metric and therefore a distance $\operatorname{dist}_{TX}$ on $TX$. We say that two parametrized curves $\gamma_1$ and $\gamma_2$ are $\delta$-close in the $C^1$-topology if $\operatorname{dist}_{TX}(\gamma_1^{(1)}(z),\gamma_2^{(1)}(z))\leq \delta$ uniformly on $U$. This implies that $\gamma_1(U)$ and $\gamma_2(U)$ are $\delta$-close in the Hausdorff sense, but the converse does not hold (take $U={\mathbb{D}}(1)$, $\gamma_1(z)=(z,0)$, and $\gamma_2(z)=(z^k, {\varepsilon}z^\ell)$ with $k$ and $\ell$ large while ${\varepsilon}$ is small).
### Good charts {#par:good_charts}
Let $R_0$ be the injectivity radius of $\kappa_0$. We fix once for all a family of charts $\operatorname{\Phi}_x\colon U_x\subset T_xX\to X$ with the following properties (for some constant $C_0$):
(i) $\operatorname{\Phi}_x(0)=x$ and $(D\operatorname{\Phi}_x)_0=\operatorname{id}_{T_xX}$;
(ii) $\operatorname{\Phi}_x$ is a holomorphic diffeomorphism from its domain of definition $U_x$ to an open subset $V_x$ contained in the ball of radius $R_0$ around $x$;
(iii) on $U_x$, the riemannian metrics ${{\sf{euc}}}_x$ and $\operatorname{\Phi}_x^*$ satisfy $C_0{^{-1}}\leq {{\sf{euc}}}_x / \operatorname{\Phi}_x^*\kappa_0 \leq C_0$ on $U_x$;
(iv) the family of maps $\operatorname{\Phi}_x$ depends continuously on $x$.
With $r_0=R_0/C_0$, we can add:
(i) for every orthonormal basis $(e_1,e_2)$ of $T_xX$, the bidisk ${\mathbb{D}}(r_0)e_1+ {\mathbb{D}}(r_0)e_2$ is contained in $U_x$; in particular, the ball of radius $r_0$ centered at the origin for ${{\sf{euc}}}_x$ is contained in $U_x$.
To make assertion (iv) more precise, fix a continuous family of orthonormal basis $(e_1(x),e_2(x))$ on some open set $V$ of $X$: Assertion (iv) means that, if we compose $\operatorname{\Phi}_x$ with the linear isomorphism $(z_1,z_2)\in {\mathbf{C}}^2\mapsto z_1e_1(x)+z_2e_2(x)\in T_xX$ we obtain a continuous family of maps. If needed, we can also add the following property (see [@griffiths-harris pp. 107-109]):
(i) ${{\sf{euc}}}_x$ osculates $\operatorname{\Phi}_x^*\kappa_0$ up to order 2 at $x$.
### Families of disks {#par:disks_size}
A holomorphic disk $\Delta\subset X$ containing $x$ is said to be a disk [**[of size (at least) $r$ at $x$]{}**]{} (resp. [**[of size exactly $r$ at $x$]{}**]{}), for some $r < r_0$, if there is an orthonormal basis $(e_1,e_2)$ of $T_xX$ such that $\operatorname{\Phi}_x^{-1}(\Delta)$ contains (resp. is) the graph $\{ ze_1+\varphi(z)e_2\; ; \; z\in {\mathbb{D}}(r)\}$ for some holomorphic map $\varphi\colon {\mathbb{D}}(r)\to {\mathbb{D}}(r)$. An alternative definition could be that $\Delta$ contains the image of an injective holomorphic map $\gamma\colon {\mathbb{D}}(r)\to X$ such that $\gamma({\partial}{\mathbb{D}}(r))\subset X\setminus B_{X}(x;r)$ and ${{\left\Vert\gamma'\right\Vert}}\leq D$, for some fixed constant $D$. Then, if $\Delta$ contains a disk of size $r$ for one of these definitions, it contains a disk of size ${\varepsilon}_0 r$ for the other definition, for some uniform ${\varepsilon}_0>0$. In particular, there is a constant $C$ depending only on $X$ such that a disk of size $r$ at $x$ contains an embedded submanifold of $B_X(x; Cr)$.
By the Koebe distortion theorem if $\Delta$ has size $r$ at $x$, then its geometric characteristics around $x$ at scale smaller than $r/2$, say, are comparable to that of a flat disk.
Let $(x_n)$ be a sequence converging to $x$ in $X$, and let $r$ be smaller than the radius $r_0$ introduced in Assertion [(v)]{}, § \[par:good\_charts\]. Let $\Delta_n$ be a family of disks of size at least $r$ at $x_n$ and $\Delta$ be a disk of size at least $r$ at $x$. We say that $\Delta_n$ [**[converges towards]{}**]{} $\Delta$ [*as a sequence of disks of size $r$*]{}, if there is an orthonormal basis $(e_1,e_2)$ of $T_xX$ for ${{\sf{euc}}}_x$ such that
(i) $\operatorname{\Phi}_x^{-1}(\Delta)$ contains the graph $\{z e_1+\varphi(z)e_2; z\in {\mathbb{D}}(r)\}$ for some holomorphic function $\varphi\colon {\mathbb{D}}(r)\to {\mathbb{D}}(r)$;
(ii) for every $s<r$, if $n$ is large enough, the disk $\operatorname{\Phi}_x^{-1}(\Delta_n)$ contains the graph $\{ze_1+\varphi_n(z)e_2; z\in {\mathbb{D}}(s)\}$ of a holomorphic function $\varphi_n\colon {\mathbb{D}}(s)\to {\mathbb{D}}(r)$;
(iii) for every ${\varepsilon}>0$, we have ${\left\vert\varphi(z)-\varphi_n(z)\right\vert}<{\varepsilon}$ on ${\mathbb{D}}(s)$ if $n$ is large enough.
By the Cauchy estimates, the convergence then holds in the $C^1$-topology (see § \[par:Hausdorff\_C1\]). It follows from the usual compactness criteria for holomorphic functions that the space of disks of size $r$ on $X$ is compact (for the topology induced by the Hausdorff topology in $X$). Likewise, if a sequence of disks of size $r$ converges in the Hausdorff sense, then it also converges in the $C^1$ sense, at least as disks of size $s<r$, because two holomorphic functions $\varphi$ and $\psi$ from ${\mathbb{D}}(r)$ to ${\mathbb{D}}(r)$ whose graphs are ${\varepsilon}$-close are also ${\varepsilon}(r-s)^{-1}$-close in the $C^1$-topology.
It may also be the case that the $\Delta_n$ are contained in different fibers $X_{\xi_n}$ of ${\mathscr{X}}$. By definition, we say that the sequence $\Delta_n$ converges to $\Delta\subset X_\xi$ if $\xi_n$ converges to $\xi$ and the projections of $\Delta_n$ converge to $\Delta$ in $X$.
### Entire curves
An [**[entire curve]{}**]{} in $X$ is by definition a holomorphic map $\psi\colon {\mathbf{C}}\to X$. If the velocity of $\psi$ does not vanish, we obtain an immersed entire curve. If $\psi_1$ and $\psi_2$ are two immersed entire curves, with the same image then there exists a holomorphic diffeomorphism of ${\mathbf{C}}$, i.e. a non-constant affine map $A\colon z\mapsto az+b$, such that $\psi_2=\psi_1\circ A$. Our main examples of immersed curves will, in fact, be injective and immersed entire curves. If $\psi$ is an immersed entire curve and ${\left\vert\psi'\right\vert}\geq \eta$ on ${\mathbb{D}}(z_0,s)$, its image contains a disk of size $Cs$ at $\psi(z_0)$, for some $C>0$ that depends only on $\eta$ and $\kappa_0$.
Stable and unstable manifolds {#subs:pesin}
-----------------------------
Recall that, by the Cauchy estimates, the moment condition implies similar moment conditions for higher derivatives, so Pesin’s theory applies. The following proposition summarizes the main properties of Pesin local stable and unstable manifolds. Recall that a function $h$ is **${\varepsilon}$-slowly varying** relative to some dynamical system $g$ if $e^{-{\varepsilon}}\leq h(g(x))/h(x)\leq e^{\varepsilon}$ for every $x$. We view the stable manifold of ${\mathscr{x}}= (\xi,x)$ as contained in $X_\xi$; it can also be viewed as a subset of $X$: whether we consider one or the other point of view should be clear from the context. If ${\mathscr{x}}=(\xi,x)$ and ${\mathscr{y}}=(\xi,y)$ are points of the same fiber $X_\xi$, we denote by $\operatorname{dist}_X({\mathscr{x}},{\mathscr{y}})$ the riemannian distance between $x$ and $y$ computed in $X$.
\[pro:pesin\] Let $(X, \nu)$ be a random dynamical system, and $\mu$ be an ergodic and hyperbolic stationary measure. Then, for every $\delta>0$, there exists measurable positive $\delta$-slowly varying functions $r$ and $C$ on ${\mathscr{X}}$ (depending on $\delta$) and, for ${\mathscr{m}}$-almost every ${\mathscr{x}}\in {\mathscr{X}}$, local stable and unstable manifolds $ W^{s}_{r({\mathscr{x}})}({\mathscr{x}})$ and $W^{u}_{r({\mathscr{x}})}({\mathscr{x}})$ in $X_\xi$ such that ${\mathscr{m}}$-almost surely:
1. $W^{s}_{r({\mathscr{x}})}({\mathscr{x}})$ and $W^u_{r({\mathscr{x}})}({\mathscr{x}})$ are holomorphic disks of size at least $ 2 r({\mathscr{x}})$ at ${\mathscr{x}}$ respectively tangent to $E^s({\mathscr{x}})$ and $E^u({\mathscr{x}})$;
2. for every $\mathscr{y}\in W^{s}_{r({\mathscr{x}})}({\mathscr{x}})$ and every $n\geq 0$, $$\operatorname{dist}_X(F^n({\mathscr{x}}), F^n(\mathscr{y})) \leq C({\mathscr{x}}) \exp ((\lambda^s+\delta) n);$$ likewise for every $\mathscr{y}\in W^{u}_{r({\mathscr{x}})}({\mathscr{x}})$ and every $n\geq 0$ $$\operatorname{dist}_X(F^{-n}({\mathscr{x}}), F^{-n}(\mathscr{y})) \leq C({\mathscr{x}}) \exp (-(\lambda^u-\delta) n);$$
3. $F( W^{s}_{r({\mathscr{x}})}({\mathscr{x}})) \subset W^{s}_{r(F({\mathscr{x}}))}(F({\mathscr{x}})) $ and $F{^{-1}}( W^{u}_{r(F({\mathscr{x}}))}(F({\mathscr{x}})) )\subset W^{u}_r({\mathscr{x}}) $.
By Lusin’s theorem, for every ${\varepsilon}>0$ we can select a compact subset $\mathcal R_{\varepsilon}\subset {\mathscr{X}}$ with ${\mathscr{m}}(\mathcal R_{\varepsilon})>0$ on which $r({\mathscr{x}})$ and $C({\mathscr{x}})$ can be replaced by uniform constants (respectively denoted by $r$ and $C$) and the following additional property holds:
1. *on $\mathcal R_{\varepsilon}$ the local stable and unstable manifolds $W^{s/u}_r({\mathscr{x}})$ vary continuously for the $C^1$-topology (in the sense of § \[par:Hausdorff\_C1\] and \[par:disks\_size\]).*
The subsets $\mathcal R_{\varepsilon}$ are usually called [**[Pesin sets]{}**]{}, or regular sets. We shall also denote the local stable or unstable manifolds by $W^{s/u}_{\mathrm{loc}}({\mathscr{x}})$, or by $W^{s/u}_r({\mathscr{x}})$ when ${\mathscr{x}}$ is in a Pesin set on which $r(\cdot)\geq r$.
The global stable and unstable manifolds of ${\mathscr{x}}$ are respectively defined by $$\begin{aligned}
W^s({\mathscr{x}}) & = {\left\{(\xi,y) \in X_\xi \; ; \ \limsup_{n\to\infty}{\frac{1}{n}}\log \operatorname{dist}_X(F^n(\xi,y), F^n(\xi,x)) <0\right\}} \\
W^u ({\mathscr{x}}) & = {\left\{ (\xi, y) \in X_\xi\; ; \ \limsup_{n\to-\infty}{\frac{1}{{\left\vertn\right\vert}}}\log \operatorname{dist}_X(F^n(\xi,y), F^n(\xi,x)) <0\right\}}.\end{aligned}$$ This definition shows that $(\xi,x) \mapsto W^s(\xi,x)$ is $\mathcal F^+$-measurable and $(\xi,x) \mapsto W^u(\xi,x)$ is $\mathcal F^-$-measurable; since stable manifolds are $\mathcal F^+$-measurable, they can naturally be viewed as living in ${\mathcal{X}}_+$. These stable and unstable manifolds can be written as the following increasing unions: $$\label{eq:union}
W^s({\mathscr{x}}) = \bigcup_{n\geq 0} F^{-n} {\left(W^s_{r({\mathscr{x}})}(F^n({\mathscr{x}}))\right)} \; \text{ and } \; W^u({\mathscr{x}}) = \bigcup_{n\geq 0} F^{n} {\left(W^u_{r({\mathscr{x}})}(F^{-n}({\mathscr{x}}))\right)}.$$ In particular, they are injectively immersed holomorphic curves in $X_\xi$.
Under the assumptions of Proposition \[pro:pesin\], $W^s({\mathscr{x}})$ and $W^u({\mathscr{x}})$ are biholomorphic to ${\mathbf{C}}$ for ${\mathscr{m}}$-almost every ${\mathscr{x}}$.
More precisely, $W^s({\mathscr{x}})$ is parametrized by an injectively immersed entire curve $\psi^s_{\mathscr{x}}: {\mathbf{C}}\to X$ such that $\psi^s_{\mathscr{x}}(0) = x$) and this parametrization is unique, up to an homothety $z\mapsto az$ of ${\mathbf{C}}$. Likewise, $W^s({\mathscr{x}})$ is parametrized by such an entire curve $\psi^u_{\mathscr{x}}$.
First, note that $W^s({\mathscr{x}})$ is an increasing union of disks and is therefore a Riemann surface homeomorphic to the plane ${\mathbf{R}}^2$. Let $A\subset {\mathscr{X}}$ be a set of positive measure on which $r\geq r_0$ and $C\leq C_0$. By Proposition \[pro:pesin\].[(ii)]{}, there exists $n_0\in {\mathbf{N}}$ and $m_0>0$ such that if $n\geq n_0$, ${\mathscr{x}}$ and $F^{n}({\mathscr{x}})$ belong to$ A$, then $W^{s}_r(F^{n}( \xi,x))\setminus {\left(F^{n} W^s_{r}( \xi,x)\right)}$ is an annulus of modulus $\geq m_0$. Now for ${\mathscr{m}}$-almost every ${\mathscr{x}}\in {\mathscr{X}}$ there exists an infinite sequence $(k_j)$ such that $F^{k_j}({\mathscr{x}}) \in A$ and $k_{j+1} - k_j> n_0$. For such an ${\mathscr{x}}$, $W^s({\mathscr{x}})\setminus W^s_r({\mathscr{x}})$ contains an infinite nested sequence of annuli $F^{-k_{j+1}}(W^s_r(F^{k_{j+1}}({\mathscr{x}}))\setminus F^{k_{j+1}-k_j}(W^s_r(F^{k_j}({\mathscr{x}}))$, each of which of modulus at least $m_0$. Therefore $W^s({\mathscr{x}})$ is biholomorphic to ${\mathbf{C}}$.
If we are only interested in stable manifolds, there is a simplified version of Proposition \[pro:pesin\] which takes place only on $X$; let us write in detail the statement that we will need.
\[pro:pesin\_stable\] Let $(X, \nu)$ be a random holomorphic dynamical system and $\mu$ an ergodic stationary measure, whose Lyapunov exponents satisfy $\lambda^{-} < 0 \leq \lambda^{+}$. Then for ${\mathscr{m}}_+$-almost every $(\omega, x)$ the stable set $$\label{eq:pesin_stable}
W^s (\omega, x) = {\left\{y \in X, \ \limsup_{n\to\infty} {\left( \log \operatorname{dist}(f_\omega^n(y), f_\omega^n(x)) \right)}< 0\right\}}$$ is an injectively immersed entire curve in $X$.
Fibered entropy
---------------
Here we recall the definition of the **metric fibered entropy** of a stationary measure $\mu$ (see [@kifer §2.1] or [@liu-qian Chap. 0 and I] for more details). If $\eta$ is a finite measurable partition of $X$, its entropy relative to $\mu$ is $H_\mu(\eta) = -\sum_{C\in \eta} \mu(C) \log\mu(C)$. Then, we set $$h_{\mu}(X, \nu; \eta) = \lim_{n\to\infty} {\frac{1}{n}}\int H_\mu{\left(\bigvee_{k=0}^{n-1} {\left(f^n_\xi\right)}{^{-1}}(\eta) \right)} d\nu^{\mathbf{N}}(\xi)$$ and $$h_{\mu}(X, \nu) = \sup{\left\{h_{\mu}(X, \nu; \eta) \; ; \; \eta\text{ a finite measurable partition of }X\right\}}.$$ Actually $h_{\mu}(X, \nu; \eta)$ can be interpreted as a conditional (or fibered) entropy for the skew-product $F_+$ on ${\mathscr{X}}_+$ (resp. $F$ on ${\mathscr{X}}$). More precisely the so-called Abramov-Rokhlin formula holds [@bogenschutz-crauel]: $$\begin{aligned}
h_{\mu}(X, \nu) & = h_{\nu^{\mathbf{N}}\times \mu}(F_+\vert \pi_\Omega) = h_{{\mathscr{m}}_+}(F_+) - h_{\nu^{\mathbf{N}}}(\sigma) \\
& = h_{{\mathscr{m}}}(F\vert \pi_\Sigma) \; = \; h_{\mathscr{m}}(F) - h_{\nu^{\mathbf{Z}}}(\vartheta),\end{aligned}$$ where as before $\pi_\Sigma$ denotes the first projection in ${\mathscr{X}}$, and in the second and fourth equalities we assume $h_{\nu^{\mathbf{N}}} (\sigma)= h_{\nu^{\mathbf{Z}}}(\vartheta)<\infty$. The next result is the fibered version of the **Margulis-Ruelle inequality** in our context.
\[pro:margulis-ruelle\] Let $(X, \nu)$ be a random holomorphic dynamical system satisfying the moment condition and $\mu$ be an ergodic stationary measure. If $h_{\mu}(X, \nu)>0$ then $\mu$ is hyperbolic and $$\min(\lambda^u, - \lambda^s) \geq {\frac{1}{2}} h_{\mu}(X, \nu).$$
See [@bahnmuller-bogenschutz] or [@liu-qian Chap. II] for the direct inequality $\lambda^u \geq {\frac{1}{2}} h_{\mu}(X, \nu)$. For the reverse inequality $- \lambda^s\geq {\frac{1}{2}} h_{\mu}(X, \nu)$, we use the fact that $h_m(F\vert \pi_\Sigma) = h_m(F{^{-1}}\vert \pi_\Sigma)$ (see e.g. [@liu-qian I.4.2]) and apply the Margulis-Ruelle inequality to $F{^{-1}}$. Beware that there is a slightly delicate point here: $F{^{-1}}$ is not associated to a random dynamical system in our sense (that is, it is not i.i.d.); fortunately, the statement of the Margulis-Ruelle inequality in [@bahnmuller-bogenschutz] (see also [@liu-qian Appendix A]) also covers this situation.
Unstable conditionals and entropy {#subs:conditionals}
---------------------------------
Assume $\mu$ is ergodic and hyperbolic. An **unstable Pesin partition** $\eta^u$ on ${\mathscr{X}}$ is by definition a measurable partition of $({\mathscr{X}}, \mathcal{F}, \mu)$ with the following properties:
- $\eta$ is increasing: $F{^{-1}}\eta^u$ refines $\eta^u$;
- for ${\mathscr{m}}$-almost every ${\mathscr{x}}$, $\eta^u({\mathscr{x}})$ is an open subset of $W^u({\mathscr{x}})$ and $$\label{eq:exhaustion}
\bigcup_{n\geq 0} F^n{\left(\eta^u (F^{-n} ({\mathscr{x}}))\right)} = W^u({\mathscr{x}});$$
- $\eta^u$ is a generator, i.e. $\bigvee_{n=0}^\infty F^{-n}(\eta^u)$ coincides ${\mathscr{m}}$-almost surely with the partition into points.
Here, as usual, $\eta^u({\mathscr{x}})$ denotes the atom of $\eta^u$ containing ${\mathscr{x}}$, and $F{^{-1}}\eta^u$ is the partition defined by $(F{^{-1}}\eta^u)({\mathscr{x}}) = F{^{-1}}(\eta^u(F({\mathscr{x}})))$. The definition of a **stable Pesin partition** $\eta^s$ is similar. A neat proof of the existence of such a partition is given by Ledrappier and Strelcyn in [@ledrappier-strelcyn], which easily adapts to the random setting (see [@liu-qian §IV.2]).
\[lem:stable\_partition\] There exists a stable Pesin partition, the atoms of which are $\mathcal F^+$-measurable, that is, saturated by local stable sets $\Sigma^s_{\mathrm{loc}}$.
To justify the existence of such a partition, we briefly review the proof of Ledrappier and Strelcyn [@ledrappier-strelcyn] and show that it can be rendered $\mathcal F^+$-measurable. Let $E$ a be a set of positive measure in ${\mathscr{X}}$ such that (a) $\pi_X(E)$ is contained in a ball of size $r_0$, (b) for every ${\mathscr{x}}= (\xi,x)\in E$, and every $0<r\leq 2r_0$, $W^s({\mathscr{x}})$ contains a disk of size exactly $r$ at ${\mathscr{x}}$, denoted by $\Delta^s({\mathscr{x}}, r)$ and (c) for every $0<r\leq 2r_0$, $E\ni{\mathscr{x}}\mapsto \Delta^s({\mathscr{x}}, r)$ is continuous for the $C^1$ topology. Then for $0<r< r_0$ we define a measurable partition $\eta_r$ whose atoms are the $\Delta^s({\mathscr{x}}, r)$ for $x\in E$ as well as ${\mathscr{X}}\setminus \bigcup_{{\mathscr{x}}\in E} \Delta^s({\mathscr{x}}, r)$. Since stable manifolds are $\mathcal F^+$-measurable, we can further require that for every $\xi' \in \Sigma^s_{\mathrm{loc}}(\xi)$, with ${\mathscr{x}}' = (\xi', x)$, we have $\Delta^s({\mathscr{x}}', r) = \Delta^s({\mathscr{x}}, r)$. The argument of [@ledrappier-strelcyn] shows that for Lebesgue-almost every $r\in [0, r_0]$, the partition $\eta^s = \bigvee_{n=0}^\infty F^{-n}(\eta_r)$ is a Pesin stable partition. Thus with ${\mathscr{x}}$ and ${\mathscr{x}}'$ as above we infer that $$\eta^s({\mathscr{x}}') = \bigcap_{n\geq 0} F^{-n} \eta_r(F^n({\mathscr{x}}')) = \bigcap_{n\geq 0} F^{-n} \eta_r(F^n({\mathscr{x}})) =\eta^s({\mathscr{x}})$$ where the middle equality comes from the fact that $\vartheta^n\xi'\in \Sigma^s_{\mathrm{loc}}(\vartheta^n\xi)$, and we are done.
The existence of unstable partitions enables us to give a meaning to the **unstable conditionals** of ${\mathscr{m}}$. Indeed, first observe that if $\eta^u$ and $\zeta^u$ are two unstable Pesin partitions, then ${\mathscr{m}}$-almost surely ${\mathscr{m}}(\cdot \vert \eta^u)$ and ${\mathscr{m}}(\cdot \vert
\zeta^u)$ coincide up to a multiplicative factor on $\eta^u(x)\cap \zeta^u(x)$. Furthermore, there exists a sequence of unstable partitions $\eta^u_n$ such that for almost every ${\mathscr{x}}$, if $K$ is a compact subset of $W^u({\mathscr{x}})$ for the intrinsic topology (i.e. the topology induced by the biholomorphism $W^u({\mathscr{x}})\simeq {\mathbf{C}}$) then $K\subset \eta_n^u(x)$ for sufficiently large $n$: indeed by , the sequence of partitions $F^{n}\eta^u$ does the job. Hence almost surely the conditional measure of ${\mathscr{m}}$ on $W^u({\mathscr{x}})$ is well-defined up to scale; we define ${\mathscr{m}}^u_{\mathscr{x}}$ by normalizing so that ${\mathscr{m}}^u_{\mathscr{x}}(\Delta^u({\mathscr{x}}, 1)) = 1$. The next proposition is known as the (relative) **Rokhlin entropy formula**, stated here in our specific context.
\[pro:rokhlin\_formula\] Let $(X, \nu)$ be a random holomorphic dynamical system satisfying the moment condition , and $\mu$ be an ergodic and hyperbolic stationary measure. Let $\eta^u$ be an associated Pesin unstable partition. Then $$\label{eq:rokhlin}
h_\mu(X, \nu) = H_{\mathscr{m}}(F{^{-1}}\eta^u\vert \, \eta^u) = \int \log J_{\eta^u}({\mathscr{x}}) d{\mathscr{m}}({\mathscr{x}}),$$ where $J_{\eta^u}({\mathscr{x}})$ is the “Jacobian” of $F$ relative to $\eta^u$, that is $$J_{\eta^u}({\mathscr{x}}) = {\mathscr{m}}\left(F{^{-1}}\left(\eta^u(F({\mathscr{x}}))\right)\vert\, \eta^u({\mathscr{x}})\right){^{-1}}.$$
The argument is based on the following sequence of equalities, in which $\eta_\Sigma $ is the partition into fibers of $\pi_\Sigma$: $$\begin{aligned}
\notag h_\mu(X, \nu) &= h_{\mathscr{m}}(F\vert {\eta_\Sigma} ) = h_{\mathscr{m}}(F{^{-1}}\vert {\eta_\Sigma} )\\ &= h_{\mathscr{m}}(F{^{-1}}\vert \eta^u \vee {\eta_\Sigma} ) \label{eq:entropy_generator}\\
&\notag := H_{\mathscr{m}}(\eta^u\vert F\eta^u \vee \eta_\Sigma)
=H_{\mathscr{m}}(\eta^u\vert F\eta^u) = H_{\mathscr{m}}(F{^{-1}}\eta^u\vert \eta^u)
\end{aligned}$$ The equalities in the first and last line follow from the general properties of conditional entropy: see [@liu-qian Chap. 0] for a presentation adapted to our context (note that the conditional entropy would be denoted by $h^{\eta_\Sigma}_{\mathscr{m}}$ there) or Rokhlin [@rokhlin] for a thorough treatment. On the other hand the equality is non-trivial. If $\eta^u$ were of the form $\bigvee_{n=0}^{+\infty} \eta$, where $\eta$ is a 2-sided generator with finite entropy, this result would indeed follow from the general theory. For a Pesin unstable partition the result was established for diffeomorphisms in [@ledrappier-young1 Cor 5.3] and adapted to random dynamical systems in [@liu-qian Cor. VI.7.1].
As a consequence of this formula we have the well-known:
\[cor:rokhlin\_zero\_entropy\] Under the assumptions of the previous proposition, the following assertions are equivalent:
1. $h_\mu(X, \nu) = 0$;
2. ${\mathscr{m}}(\cdot \vert \eta^u({\mathscr{x}})) = \delta_{\mathscr{x}}$ for ${\mathscr{m}}$-a.e. ${\mathscr{x}}$;
3. ${\mathscr{m}}(\cdot \vert \eta^u({\mathscr{x}}))$ is atomic for ${\mathscr{m}}$-a.e. ${\mathscr{x}}$.
In view of the definition of $J_{\eta^u}$, the entropy vanishes if and only if for ${\mathscr{m}}$-almost every ${\mathscr{x}}$, ${\mathscr{m}}(\cdot \vert \eta^u({\mathscr{x}}))$ is carried by a single atom of the finer partition $F{^{-1}}\eta^u$. Now since $H_{\mathscr{m}}(F{^{-1}}\eta^u\vert \, \eta^u) = {\frac{1}{n}} H_{\mathscr{m}}(F^{-n}\eta^u\vert \, \eta^u)$, the same is true for $F^{-n}\eta^u$, and finally since $(F^{-n}\eta^u)$ is generating, we conclude that (a)$\Leftrightarrow$(b). The fact (c) implies (a) follows from the same ideas but it is slightly more delicate, see [@viana-yang §2.1-2.2] for a clear exposition in the case of the iteration a single diffeomorphism, which readily adapts to our setting.
A further result is that if the fiber entropy vanishes there is a set of full ${\mathscr{m}}$-measure which intersects any global unstable leaf in only one point. This was originally shown for individual diffeomorphisms in [@ledrappier-young1 Thm. B].
Stable manifolds and limit currents {#sec:nevanlinna}
===================================
Let as before $(X, \nu)$ be a non-elementary random holomorphic dynamical system on a compact Kähler (hence projective) surface, and assume $\mu$ is an ergodic stationary measure admitting exactly one negative Lyapunov exponent, as in Proposition \[pro:pesin\_stable\]. Our purpose in this section is to relate the stable manifolds $W^s(\omega,x)$ to the stable currents $T_\omega^s$ constructed in §\[sec:currents\]. The link between these two objects will be given by the well-known Ahlfors-Nevanlinna construction of positive closed currents associated to entire curves.
Ahlfors-Nevanlinna currents {#par:Ahlfors-Nevanlinna-Construction}
---------------------------
We denote by ${\left\{V\right\}}$ the integration current on a (possibly non-closed or singular) curve $V$. Let $\phi:{\mathbf{C}}\to X$ be an entire curve. By definition if $\alpha$ is a test 2-form $\langle {\left\{\phi({\mathbb{D}}(0,t))\right\}}, \alpha\rangle= \int_{{\mathbb{D}}(0,t)} \phi^*\alpha$, which accounts for possible multiplicities coming from the lack of injectivity of $\phi$. For $R>0$ we set $$A(R) = \int_{{\mathbb{D}}(0, R)}\phi^*\kappa_0 \; \text{ and } \; T(R) = \int_0^R A(t) \frac{dt}{t}.$$ When $\phi$ is an immersion the area $A(r)$ is equal to the mass ${{\mathbf{M}}}{\left({\left\{\phi({\mathbb{D}}(0, R))\right\}}\right)}$.
\[pro:brunella\] If $\phi:{\mathbf{C}}\to X$ is a non-constant entire curve, there exist sequences of radii $(R_n)$ increasing to infinity such that the sequence of currents $$\label{eq:brunella}
N(R_n) = {\frac{1}{T(R_n)}} \int_0^{R_n} {\left\{\phi({\mathbb{D}}(0,t))\right\}} \frac{dt}{t}$$ converges to a closed positive current $T$. If furthermore $\phi({\mathbf{C}})$ is Zariski dense, and $T$ is such a closed current, the class $[T]\in H^{1,1}(X, {\mathbf{R}})$ is nef. In particular $\langle [T] \, \vert\, [C] \rangle\geq 0$ for every algebraic curve $C\subset X$.
Such limit currents $T$ will be referred to as **Ahlfors-Nevanlinna currents** associated to the entire curve $\phi\colon {\mathbf{C}}\to X$. Note that if $\phi({\mathbf{C}})$ is not Zariski dense then $\overline{\phi({\mathbf{C}})}$ is a (possibly singular) curve of genus $0$ or $1$; if $\phi$ is injective, then $\overline{\phi({\mathbf{C}})}$ is rational.
Equidistribution of stable manifolds
------------------------------------
If $\mu$ is hyperbolic, or more generally if it admits exactly one negative Lyapunov exponent, then, for ${\mathscr{m}}_+$-almost every ${\mathscr{x}}\in {\mathscr{X}}_+$, the stable manifold $W^s({\mathscr{x}})$, which is viewed here as a subset of $X$ as in Proposition \[pro:pesin\_stable\], is parametrized by an injectively immersed entire curve. Then we can relate the Ahlfors-Nevanlinna currents to the limit currents $T^s_\omega$. Here are the three main results that will be proved in this section.
\[thm:nevanlinna\] Let $(X, \nu)$ be a non-elementary random holomorphic dynamical system on a compact Kähler surface, satisfying . Let $\mu$ be an ergodic stationary measure such that $\lambda^-(\mu)<0\leq \lambda^+(\mu)$. Then exactly one of the following alternative holds.
1. For ${\mathscr{m}}_+$-almost every ${\mathscr{x}}$, the stable manifold $W^s({\mathscr{x}})$ is not Zariski dense. Then $\mu$ is supported on a $\Gamma_\nu$-invariant curve $Y\subset X$ and for ${\mathscr{m}}_+$-almost every ${\mathscr{x}}$, $W^s({\mathscr{x}}) \subset Y$. In addition every component of $Y$ is a rational curve, and the intersection form is negative definite on the subspace generated by the classes of components of $Y$.
2. For ${\mathscr{m}}_+$-almost every ${\mathscr{x}}$ the stable manifold $W^s({\mathscr{x}})$ is Zariski dense and the only normalized Ahlfors-Nevanlinna current associated to $W^s({\mathscr{x}})$ is $T^s_\omega$.
\[cor:nevanlinna\] Under the assumptions of Theorem \[thm:nevanlinna\], if in addition $\mu$ is hyperbolic and non-atomic, then the Alternative *(b)* is equivalent to
1. $\mu$ is not supported on a $\Gamma_\nu$-invariant curve.
In particular *(a)* and *(b)* are mutually exclusive.
\[cor:HD\_stable\] Under the assumptions of Theorem \[thm:nevanlinna\], assume furthermore that $\nu$ satisfies the exponential moment condition . Then in Alternative (b) there exists $\theta>0$ such that for ${\mathscr{m}}_+$-almost every ${\mathscr{x}}\in {\mathscr{X}}_+$ the Hausdorff dimension of $\overline{W^s({\mathscr{x}})}$ equals $2+\theta$.
Proof of Theorem \[thm:nevanlinna\] and its corollaries
-------------------------------------------------------
We work under the assumptions of Theorem \[thm:nevanlinna\].
\[lem:mu\_zariski\] If there exists a proper Zariski closed subset of $X$ with positive $\mu$-measure, then:
- either $\mu$ is the uniform counting measure on a finite orbit of $\Gamma_\nu$;
- or $\mu$ has no atom and it is supported on a $\Gamma_\nu$-invariant algebraic curve; this curve is the $\Gamma_\nu$-orbit of an irreducible algebraic curve.
Consider the real number $\delta_{\mathrm{max}}^0(\mu)=\max_{x\in X} \mu{\left(\{x\}\right)}$. If $\delta_{\mathrm{max}}^0(\mu)>0$, there is a non-empty finite set $F\subset X$ for which $\mu{\left(\{x\}\right)} = \delta_{\mathrm{max}}^0(\mu)$. By stationarity, $F$ is $\Gamma_\nu$-invariant, and by ergodicity $\mu$ is the uniform measure on $F$. Now, assume that $\mu$ has no atom. Let $\delta_{\mathrm{max}}^1(\mu)$ be the maximum of $\mu(D)$ among all irreducible curves $D\subset X$. If $\mu(Z)>0$ for some proper Zariski closed subset $Z\subset X$, then $\delta_{\mathrm{max}}^1(\mu)>0$. Since two distinct irreducible curves intersect in at most finitely many points and $\mu$ has no atom, there are only finitely many irreducible curves $E$ such that $\mu(E)=\delta_{\mathrm{max}}^1(\mu)$. To conclude, we argue as in the zero dimensional case.
Recall that if $V\subset X$ is a smooth curve, possibly with boundary, if $T$ is a positive closed (1,1) current on $X$ with a continuous normalized potential $u_T$ (as in § \[subs:normalized\_potentials\]), then by definition $$\langle T\wedge {\left\{V\right\}}, \varphi\rangle = \int_V \varphi \, \Theta_T + \int_V \varphi \, dd^c(u_T{ \arrowvert_{V}}),$$ for every test function $\varphi$. Here is the key relation between stable manifolds and limit currents:
\[lem:stable\_intersection\_Ts\] For ${\mathscr{m}}_+$-almost every ${\mathscr{x}}$, if $\Delta$ is a disk contained in $W^s({\mathscr{x}})$, then $T_\omega^s \wedge \{ \Delta\} = 0$.
We consider points ${\mathscr{x}}=(\omega, x)\in {\mathscr{X}}_+$ which are generic in the sense that they are regular from the point of view of Pesin’s theory and $T^s_\omega $ satisfies the conclusions of §\[sec:currents\]. Let $\Delta \subset W^s({\mathscr{x}})$ be a disk; with no loss of generality we assume that the boundary of $\Delta$ in $W^s({\mathscr{x}})\simeq {\mathbf{C}}$ is smooth.
By Pesin’s theory, for every ${\varepsilon}>0$, there is a set $A_{\varepsilon}\subset \mathbf N$ of density larger than $1-{\varepsilon}$, such that for $n$ in $A_{\varepsilon}$, the local stable manifold $W^s_{r}(F_+^n({\mathscr{x}}))$, is a disk of size $r=r({\varepsilon})$ at $f_\omega^n(x)$ and $f_\omega^n(\Delta)$ is a disk contained in an exponentially small neighborhood of $f_\omega^n(x)$. We have $$\label{eq:TnsDelta}
{{\mathbf{M}}}(T_{\sigma^n\omega}^s\wedge \{f_\omega^n(\Delta)\})
= \int_{ W^s_{r} (F_+^n({\mathscr{x}}))}\! \mathbf{1}_{f_\omega^n(\Delta)} \, \Theta_{T^s_{\sigma^n\omega}}
+ \int_{ W^s_{r}(F_+^n({\mathscr{x}}))}\! \mathbf{1}_{f_\omega^n(\Delta)}
dd^c u_{T_{\sigma^n\omega}^s}.$$ Since ${{\mathbf{M}}}(T_{\sigma^n\omega}^s)=1$, Lemma \[lem:uniform\_Ak\_psh\] shows that $\Theta_{T^s_{\sigma^n\omega}} $ is bounded by $A\kappa_0$; so the first integral on the right hand side of is bounded by a constant times the area of $f_\omega^n(\Delta)$, which is exponentially small. By ergodicity, there exists $A'_{\varepsilon}\subset A_{\varepsilon}$ of density at least $1-2{\varepsilon}$ such that if $n\in A'_{\varepsilon}$, $\| u_{T_{\sigma^n\omega}^s}\| _\infty$ is bounded by some contant $D_{\varepsilon}>0$. For such an $n$, let $\chi$ be a test function in $W^s_{\mathrm{loc}}(F_+^n({\mathscr{x}}))$ such that $\chi=1$ in $W^s_{r/2} (F_+^n({\mathscr{x}}))$. We write $$\begin{aligned}
\notag\int_{ W^s_{r}(F_+^n({\mathscr{x}}))}\! \mathbf{1}_{f_\omega^n(\Delta)}
dd^c u_{T_{\sigma^n\omega}^s}
&\leq \int_{ W^s_{r}(F_+^n({\mathscr{x}}))}\! \chi dd^c u_{T_{\sigma^n\omega}^s} \\
&= \int_{ W^s_{r}(F_+^n({\mathscr{x}}))}\!
u_{T_{\sigma^n\omega}^s} dd^c\chi \\ \notag
& \leq C(r) {{\left\Vert\chi\right\Vert}}_{C^2} \big\|{u_{T_{\sigma^n\omega}^s} } \big\|_\infty\end{aligned}$$ and this last term is uniformly bounded because $n\in A'_{\varepsilon}$. Thus we conclude that ${{\mathbf{M}}}(T_{\sigma^n\omega}^s\wedge\{f_\omega^n(\Delta)\})$ is bounded along such a subsequence.
On the other hand, the relation $(f_\omega^n)^* T_{\sigma^n\omega}^s = {{\mathbf{M}}}((f_\omega^n)^*T_{\sigma^n\omega}^s) T_{\omega}^s$ gives $$T_{\sigma^n(\omega)}^s\wedge \{ f_\omega^n(\Delta)\} = {{\mathbf{M}}}{\left((f_\omega^n)^*T_{\sigma^n(\omega)}^s\right)}
(f_\omega^n)_* (T_\omega^s\wedge \{ \Delta\}).$$ The mass ${{\mathbf{M}}}((f_\omega^n)_* (T_\omega^s\wedge \{ \Delta\}))$ is constant, equal to the mass of the measure $T_\omega^s\wedge \{ \Delta\}$; so $${{\mathbf{M}}}{\left(T_{\sigma^n(\omega)}^s\wedge \{ f_\omega^n(\Delta)\} \right)} = {{\mathbf{M}}}((f_\omega^n)^*T_{\sigma^n(\omega)}^s) {{\mathbf{M}}}(T_\omega^s\wedge \{\Delta\}).$$ By Lemma \[lem:variant\_furstenberg\], ${{\mathbf{M}}}((f_\omega^n)^*T_{\sigma^n(\omega)}^s) $ goes exponentially fast to infinity. Since the left hand side is bounded, this shows that ${{\mathbf{M}}}(T_\omega^s\wedge \{\Delta\})=0$, as desired.
With Lemma \[lem:periodic\_curves\], the following statement takes care of the first alternative in Theorem \[thm:nevanlinna\].
\[lem:stable\_zariski\_dense\] If there is a measurable subset $A\subset {\mathscr{X}}_+$ of positive measure such that for every ${\mathscr{x}}\in A$, the stable manifold $W^s({\mathscr{x}})$ is contained in an algebraic curve, then $\mu$ is supported on a $\Gamma_\nu$-invariant algebraic curve. In addition, for ${\mathscr{m}}_+$-almost every ${\mathscr{x}}$, $\overline{W^s({\mathscr{x}})}$ is an irreducible rational curve of negative self-intersection.
For ${\mathscr{x}}\in A$, let $D({\mathscr{x}})$ be the Zariski closure of $W^s({\mathscr{x}})$. Discarding a set of measure zero if needed, $W^s({\mathscr{x}})$ is biholomorphic to ${\mathbf{C}}$ so $D({\mathscr{x}})$ is a (possibly singular) irreducible rational curve, and $D({\mathscr{x}})\setminus W^s({\mathscr{x}})$ is reduced to a point. By Lemma \[lem:stable\_intersection\_Ts\], $T^s_\omega\wedge {\left\{\Delta\right\}} = 0$ for every disk $\Delta \subset W^s({\mathscr{x}})$. And because $T^s_\omega$ has continuous potentials, $T^s_\omega\wedge {\left\{D({\mathscr{x}})\right\}}$ gives no mass to points (see e.g. [@cantat-dupont Lem. 10.13] for the singular case). It follows that $T^s_\omega\wedge {\left\{D({\mathscr{x}})\right\}}=0$, hence $\langle e(\omega) \, \vert \, [D({\mathscr{x}})]\rangle= 0$.
By the Hodge index theorem, either $[D({\mathscr{x}})]^2<0$ or $[D({\mathscr{x}})]$ is proportional to $e(\omega)$, however this latter case would contradict the fact that $e(\omega)$ is $\nu^{\mathbf{N}}$-almost surely irrational (see Theorem \[thm:def\_stationary\]; one could also use that $\operatorname{{Cur}}(e(\omega))$ is reduced to $T^s_\omega$). Thus, $[D({\mathscr{x}})]^2<0$.
An irreducible curve with negative self-intersection is uniquely determined by its cohomology class; since ${{\mathrm{NS}}}(X)$ is countable, there are only countably many irreducible curves $(D_k)_{k\in {\mathbf{N}}}$ with negative self intersection. Since $W^s_{\mathrm{loc}}({\mathscr{x}})\subset D_k$ if and only if $D({\mathscr{x}}) = D_k$, and since local stable manifolds vary continuously on the Pesin regular set $\mathcal R_{\varepsilon}$ for every ${\varepsilon}>0$, we infer that ${\left\{{\mathscr{x}}\in A\; ; \; \ D({\mathscr{x}}) = D_k\right\}}$ is measurable for every $k$. Hence there exists an index $k$ such that ${\mathscr{m}}_+ {\left( {\left\{ {\mathscr{x}}\in A\; ; \; [D({\mathscr{x}})]=[D_k]\right\}}\right)}>0$. Since $x$ belongs to $W^s_{\mathrm{loc}}({\mathscr{x}})$, Fubini’s theorem implies that $\mu(D_k)>0$, and Lemma \[lem:mu\_zariski\] shows that $\mu$ is supported on the $\Gamma_\nu$-orbit of $D_k$.
Finally, this argument shows that the property $W^s_{\mathrm{loc}}({\mathscr{x}})\subset \bigcup_{k\in {\mathbf{N}}} D_k$, or equivalently that $W^s_{\mathrm{loc}}({\mathscr{x}})$ is contained in a rational curve of negative self intersection is invariant and measurable, so by ergodicity of ${\mathscr{m}}_+$ it is of full measure. The proof is complete.
We are now ready to conclude the proof of Theorem \[thm:nevanlinna\]. Choose ${\mathscr{x}}\in {\mathscr{X}}_+$ satisfying the previous generic requirements. By Lemma \[lem:stable\_zariski\_dense\], if there is a set of positive ${\mathscr{m}}_+$-measure such that $W^s({\mathscr{x}})$ is not Zariski dense, then Alternative [[(a)]{}]{} holds. Otherwise we can assume that $W^s({\mathscr{x}})$ is almost surely Zariski dense. Pick an Ahlfors-Nevanlinna current $N$ associated to $W^s({\mathscr{x}})$. By Proposition \[pro:brunella\], $[N]$ is a nef class so $[N]^2 \geq 0$. Thus, if we are able to show that $\langle [N] \, \vert \, [T^s_\omega] \rangle = 0$, we deduce from the Hodge index theorem and ${{\mathbf{M}}}(N)=1$ that $[N] =[T^s_\omega] = e(\omega)$, hence $N = T^s_\omega$ by Theorem \[thm:uniq+extremal\]. So, it only remains to prove that $\langle [N] \, \vert \, [T^s_\omega] \rangle = 0$, or equivalently $$\label{eq:NT=0}
N\wedge T^s_\omega = 0.$$ This is intuitively clear because $N$ is an Ahlfors-Nevanlinna current associated to the entire curve $W^s({\mathscr{x}})$ and $T^s_\omega\wedge {\left\{\Delta\right\}}=0$ for every bounded disk $\Delta\subset W^s({\mathscr{x}})$. However, there is a technical difficulty to derive from $T^s_\omega\wedge {\left\{\Delta\right\}}=0$, even if $W^s({\mathscr{x}})$ is an increasing union of such disks $\Delta$.
At least two methods were designed to deal with this situation: the first one uses the geometric intersection theory of laminar currents (see [@bls; @isect]), and the second one was developed by Dinh and Sibony in the preprint version of [@dinh-sibony_jams] (details are published in [@cantat-dupont §10.4]). Unfortunately these papers only deal with the case of currents of the form $\lim_{n} {\frac{1}{A(R_n)}}{\phi({\mathbb{D}}(0, R_n))}$, instead of the Ahlfors-Nevanlinna currents introduced in Section \[par:Ahlfors-Nevanlinna-Construction\], which were designed in order to get the nef property stated in Proposition \[pro:brunella\]. So, we now explain how to adapt the formalism of [@bls; @isect] to the Ahlfors-Nevanlinna currents of Proposition \[pro:brunella\].
Following [@duval] we say that $T$ is an **Ahlfors current** if there exists a sequence $(\Delta_n)$ of *unions* of smoothly bounded holomorphic disks such that $\operatorname{length}({\partial}\Delta_n) = o{\left({{\mathbf{M}}}(\Delta_n)\right)}$ and $N $ is the limit as $n\to\infty$ of the sequence of normalized integration currents ${\frac{1}{{{\mathbf{M}}}(\Delta_n)}}{\left\{\Delta_n\right\}}$ (where the length is computed with respect to the riemannian structure induced by $\kappa_0$). We say furthermore that $T$ is an **injective Ahlfors current** if the disks $\Delta_n$ are disjoint or intersect along relatively open subsets. By discretizing the integral in Equation we see that any Ahlfors-Nevanlinna current is an injective Ahlfors current.
[**[Strongly approximable]{}**]{} laminar currents are a class of positive currents introduced in [@isect] with geometric properties which are well suited for geometric intersection theory. In a nutshell, a current $T$ is a strongly approximable laminar current if for every $r>0$, there exists a uniformly laminar current $T_r$ (non closed in general) made of disks of size $r$, and such that ${{\mathbf{M}}}(T-T_r) = O(r^2)$. This mass estimate is crucial for the geometric understanding of wedge products of such currents. Since these notions have been studied in a number of papers, we refer to [@bls; @isect; @Cantat:Milnor] for definitions, the basic properties of these currents, and technical details. This presentation in terms of disks of size $r$ is from [@fatou §4]. The next lemma is a mild generalization of the methods of [@bls §7], [@Cantat:Acta §4.3] and [@isect §4]. For completeness we provide the details in Appendix \[par:appendix\_ahlfors\].
\[lem:ahlfors\_current\] Any injective Ahlfors current $T$ on a projective surface $X$ is a strongly approximable laminar current: if $T= \lim_n {\frac{1}{{{\mathbf{M}}}(\Delta_n)}}{\left\{\Delta_n\right\}}$ where the disks $\Delta_n$ have smooth boundaries and $\operatorname{length}({\partial}\Delta_n) = o{\left({{\mathbf{M}}}(\Delta_n)\right)}$, one can construct a family of uniformly laminar currents $T_r$, whose constitutive disks are limits of pieces of the $\Delta_n$, such that if $S$ is any closed positive current with continuous potential on $X$, then $S\wedge T_r$ increases to $S\wedge T$ as $r$ decreases to $0$.
With this lemma at hand, let us conclude the proof of Theorem \[thm:nevanlinna\]. Since $X$ is projective, we can apply the previous lemma to any Ahlfors-Nevanlinna current $N$ associated to $W^s({\mathscr{x}})$. In this way we get a family of currents $N_r$ such that $N_{r}\wedge T^s_\omega$ increases to $N\wedge T^s_\omega$ as $r$ decreases to $0$. On the other hand, by Lemma \[lem:stable\_intersection\_Ts\], the intersection of $T^s_\omega$ with all disks contained in $W^s({\mathscr{x}})$ is equal to $0$, so again using the fact that $T^s_\omega$ has continuous potential, we infer that if $\Delta$ is any disk subordinate to $N_r$, $T^s_\omega\wedge {\left\{\Delta\right\}} =0$, hence $N_{r}\wedge T^s_\omega = 0$ for every $r>0$. Finally $N\wedge T^s_\omega=0$, as desired.
It is clear that (b’) and (a) are contradictory so (b’) implies (b). Conversely assume that $\mu$ is hyperbolic, non atomic and supported on a $\Gamma_\nu$-invariant curve $C$. Since $\mu$ has no atom, it gives full mass to the regular set of $C$, hence $\Sigma\times T(\mathrm{Reg}(C))$ defines a $DF$-invariant subbundle of ${\mathcal{X}}$, and by the Oseledets theorem the ergodic random dynamical system $(C, \nu, \mu)$ must either have a positive or a negative Lyapunov exponent. If this exponent were positive then $\mu$ would be atomic, as observed in Remark \[rem:invariance\_principle\_strict\]. Hence, the Lyapunov exponent tangent to $C$ is negative and $W^s({\mathscr{x}})$ is contained in $C$ for ${\mathscr{m}}_+$-almost every ${\mathscr{x}}$. So (b) implies (b’).
Since $\nu$ satisfies an exponential moment condition, Theorem \[thm:holder\] provides a $\theta>0$ such that $u_{T^s_\omega}$ is Hölder continuous of exponent $\theta$ for $\nu^{\mathbf{N}}$-almost every $\omega$. This implies that $T^s_\omega$ gives mass $0$ to sets of Hausdorff dimension $<2 + \theta$ (see [@sibony Thm 1.7.3]). Since for ${\mathscr{m}}_+$-almost every $x$, $\operatorname{Supp}(T^s_\omega)\subset \overline{ W^s({\mathscr{x}})}$, we infer that $\mathrm{HDim}\big({\overline{ W^s({\mathscr{x}})}}\big)\geq 2+\theta$.
To conclude the proof it is enough to show that ${\mathscr{x}}\mapsto \mathrm{HDim}\big({\overline{ W^s({\mathscr{x}})}}\big)$ is constant on a set of full ${\mathscr{m}}_+$-measure. Indeed, ${\mathscr{x}}\mapsto \mathrm{HDim}\big({\overline{ W^s({\mathscr{x}})}}\big)$ defines an $F_+$-invariant function, defined on the full measure set $\mathcal R$ of Pesin regular points. If we show that this function is measurable, then the result follows by ergodicity. This is a consequence of the following two facts:
1. the assignment ${\mathscr{x}}\mapsto {\overline{ W^s({\mathscr{x}})}}$ defines a Borel map from $\mathcal R$ to the space $\mathcal{K}(X)$ of compact subsets of $X$;
2. the function $\mathcal{K}(X)\ni K \mapsto \mathrm{HDim}(K)$ is Borel (see [@mattila-mauldin Thm 2.1]).
In both cases $\mathcal{K}(X)$ is endowed with the topology induced by the Hausdorff metric. For the first point, observe that $\mathcal R$ is the increasing union of the compact sets $\mathcal R_{\varepsilon}$ so it is Borel; then, on a Pesin set $\mathcal R_{\varepsilon}$, ${\mathscr{x}}\mapsto {\overline{ W^s_r({\mathscr{x}})}}$ is continuous, so ${\mathscr{x}}\mapsto F^{-n} \big(\overline{ W^s_r(F^n({\mathscr{x}}))}\big)$ is continuous as well. Since $F^{-n} \big(\overline{ W^s_r(F^n({\mathscr{x}}))}\big)$ converges to ${\overline{ W^s({\mathscr{x}})}}$ in the Hausdorff topology, we infer that ${\mathscr{x}}\mapsto {\overline{ W^s ({\mathscr{x}})}}$ is a pointwise limit of continuous maps on $\mathcal R_{\varepsilon}$, hence Borel, and finally ${\mathscr{x}}\mapsto {\overline{ W^s ({\mathscr{x}})}}$ is Borel on $\mathcal R$, as claimed.
No invariant line fields {#sec:No_Invariant_Line_Fields}
========================
Let as above $(X, \nu)$ be a random holomorphic dynamical system satisfying the moment condition and $\mu$ be an ergodic hyperbolic stationary measure. Recall from §\[subs:lyapunov\] and §\[subs:pesin\] that stable manifolds and stable Oseledets directions are $\mathcal F^+$-measurable. Then for ${\mathscr{m}}_+$-almost every ${\mathscr{x}}\in {\mathscr{X}}_+$, there is a Pesin stable manifold $W^s({\mathscr{x}})$ (resp. an Oseledets direction $E^s({\mathscr{x}})$). Let $V ({\mathscr{x}})=V(\omega,x)$ be such a measurable family of obtjects (stable manifolds, or stable directions, etc); we say that $V ({\mathscr{x}})$ is **non-random** if for $\mu$-almost every $x$, $V(\omega,x)$ does not depend on $\omega$, that is, there exists $V(x)$ such that $V_{\omega}(x) = V(x)$ for $\nu^{\mathbf{N}}$-almost every $\omega$. If $V$ is not non-random, we say that it **depends non-trivially on the itinerary**. Since stable directions depend only on the future, $E^s(\omega, x)$ is naturally identified with $E^s(\xi, x)$ under the natural projection $(\xi, x) \in{\mathscr{X}}\mapsto (\omega,x)\in {\mathscr{X}}_+$; the same property holds for stable manifolds. So the random versus non-random dichotomy can be analyzed in ${\mathscr{X}}$. Our purpose in this section is to establish the following result.
\[thm:alternative\_stable\] Let $(X, \nu)$ be a non-elementary random holomorphic dynamical system on a compact Kähler surface satisfying the moment condition and let $\mu$ be an ergodic and hyperbolic stationary measure, not supported on a $\Gamma_ \nu$-invariant curve. Then the following alternative holds:
1. either the Oseledets stable directions depend non-trivially on the itinerary;
2. or $\mu$ is $\nu$-almost surely invariant and $h_\mu(X, \nu) = 0$.
We will see that [(a)]{} often implies that $\mu$ is invariant (see §\[sec:stiffness\] for instance). In [(b)]{}, the almost-sure invariance implies that $\mu$ is in fact $\Gamma_\nu$-invariant (see Remark \[rem:discrete\]).
It turns out that [(a)]{} and [(b)]{} are mutually exclusive. Indeed it follows from the main argument of [@br] ([^4]) that if Oseledets stable directions depend non-trivially on the itinerary, then the fiber entropy is positive (see [@br Rmk 12.3]). So we get the following:
\[cor:positive\_entropy\] Let $(X, \nu, \mu)$ be as in Theorem \[thm:alternative\_stable\]. If $\mu$ is not $\nu$-almost surely invariant, then its fiber entropy is positive.
Intersection multiplicities
---------------------------
Let us start with some basics on intersection multiplicities for curves. If $V_1$ and $V_2$ are germs of curves at $0\in {\mathbf{C}}^2$, with an isolated intersection at 0, the **intersection multiplicity** $\operatorname{inter}_0(V_1, V_2)$ is, by definition, the number of intersection points of $V_1$ and $V_2+u$ in $N$ for small generic $u\in {\mathbf{C}}^2$, where $N$ is a neighborhood of 0 such that $V_1\cap V_2\cap N = {\left\{0\right\}}$ (see Chirka [@chirka §12]). It is a (finite) positive integer, and $\operatorname{inter}_0(V_1, V_2)=1$ if and only if $V_1$ and $V_2$ are transverse at 0. We extend this definition by setting $\operatorname{inter}_0(V_1, V_2) = 0$ if $V_1$ or $V_2$ does not contain 0 and $\operatorname{inter}_0(V_1, V_2)= \infty$ if 0 is not an isolated point of $V_1\cap V_2$, that is locally $V_1= V_2$. The intersection multiplicity extends to analytic cycles (that is, formal integer combinations of analytic curves).
\[lem:upper\_sc\_inter\] The multiplicity of intersection $\operatorname{inter}_0(\cdot, \cdot)$ is upper semi-continuous for the Hausdorff topology on analytic cycles.
In our situation we will only apply this result to holomorphic disks with multiplicity 1, in which case the topology is just the usual local Hausdorff topology.
Assume $\operatorname{inter}_0(V_1, V_2)=k$ and $V_{1,n} \to V_1$ (resp. $V_{2,n} \to V_2$) as cycles; we have to show that $\limsup \operatorname{inter}_0(V_{1,n}, V_{2,n}) \leq k$. If $k = \infty$ there is nothing to prove. Otherwise, ${\left\{0\right\}}$ is isolated in $V_1\cap V_2$, so we can fix a neighborhood $U$ of $0$ such that $V_1\cap V_2\cap U ={\left\{0\right\}}$; then, the result follows from [@chirka Prop 2 p.141] (stability of proper intersections).
Generic intersection multiplicity of stable manifolds
-----------------------------------------------------
Recall from §\[subs:pesin\] that for ${\mathscr{m}}$-almost every ${\mathscr{x}}= (\xi,x)\in {\mathscr{X}}$ there exists a local stable manifold $W^s_{r({\mathscr{x}})}({\mathscr{x}}) \subset
X_\xi\simeq X$, depending measurably on ${\mathscr{x}}$; we might simply denote by $W^s_{\mathrm{loc}}({\mathscr{x}})$.
Let us cover a subset of full measure in ${\mathscr{X}}$ by Pesin subsets $\mathcal R_{{\varepsilon}_n}$. Take a point $x\in X$, and consider the set of points $((\xi, x), (\zeta, x))\in {\mathcal R_{{\varepsilon}_n}}\times {\mathcal R_{{\varepsilon}_m}}$, for some fixed pair of indices $(n,m)$; Lemma \[lem:upper\_sc\_inter\] shows that the intersection multiplicity $\operatorname{inter}_x {\left(W^s_{\mathrm{loc}}(\xi, x), W^s_{\mathrm{loc}}(\zeta, x)\right)}$ is an upper semi-continuous function of $((\xi, x), (\zeta, x))$ on that compact set. Thus, the intersection multiplicity $\operatorname{inter}_x {\left(W^s_{\mathrm{loc}}(\xi, x), W^s_{\mathrm{loc}}(\zeta, x)\right)}$ is a measurable function of $(\xi,\zeta)$.
Recall that the $\sigma$-algebra $\mathcal F^-$ on ${\mathcal{X}}$ is generated, modulo ${\mathscr{m}}$-negligible sets, by the partition into subsets of the form $\Sigma^u_{\mathrm{loc}}(\xi)\times {\left\{x\right\}}$ (see § \[par:definition\_skew\_products\], Equation ), that $\xi \mapsto {\mathscr{m}}_\xi$ is $\mathcal F^-$-measurable (${\mathscr{m}}_\xi = {\mathscr{m}}_\zeta$ almost surely when $\zeta\in \Sigma^u_{\mathrm{loc}}(\xi)$), and that the conditional measures of ${\mathscr{m}}$ with respect to this partition satisfy (see Equation ) $$\label{eq:conditional_Part_8}
{\mathscr{m}}(\; \cdot\; \vert \; \mathcal F^-({\mathscr{x}})) = {\nu^{\mathbf{Z}}(\; \cdot\; \vert \; \Sigma^u_{\mathrm{loc}}(\xi))}\times \delta_x.$$
The next lemma can be seen as a complex analytic version of [@br Lemma 9.9].
\[lem:alternative\_multiplicity\] Let $k\geq 1$ be an integer. Exactly one of the following assertions holds:
1. for ${\mathscr{m}}$-almost every ${\mathscr{x}}=(\xi,x)$ and for ${\mathscr{m}}(\; \cdot\; \vert \; \mathcal F^-(\xi,x))$-almost every $\eta$ $$\operatorname{inter}_x{\left(W^s_{\mathrm{loc}}(\xi, x), W^s_{\mathrm{loc}}(\eta, x)\right)}\geq k+1;$$
2. for ${\mathscr{m}}$-almost every ${\mathscr{x}}$ and for ${\mathscr{m}}(\; \cdot\; \vert \; \mathcal F^-(\xi,x))$-almost every $\eta$ $$\operatorname{inter}_x{\left(W^s_{\mathrm{loc}}(\xi, x), W^s_{\mathrm{loc}}(\eta, x)\right)}\leq k.$$
The relation defined on ${\mathscr{X}}$ by $(\xi, x)\simeq_{k} (\eta, y)$ if $x=y$ and $W^s_{\mathrm{loc}}(\xi, x)$ and $W^s_{\mathrm{loc}}(\eta, y)$ have order of contact at least $k+1$ at $x$ is an equivalence relation which defines a partition $\mathcal Q_k $ of ${\mathscr{X}}$. We shall see below that $\mathcal Q_k $ is a measurable partition. Since $F\colon {\mathscr{X}}\to {\mathscr{X}}$ acts by diffeomorphisms on the fibers $X$ of ${\mathscr{X}}$, we get that $F(\mathcal Q_k ({\mathscr{x}})) = \mathcal Q_k(F({\mathscr{x}}))$ for almost every ${\mathscr{x}}\in {\mathscr{X}}$. Then, the proof of [@br Lemma 9.9] applies verbatim to show that if $${\mathscr{m}}{\left({\left\{{\mathscr{x}}\; ; \; {\mathscr{m}}(\mathcal Q_k ({\mathscr{x}})\vert \mathcal F^-({\mathscr{x}}))>0 \right\}}\right)} >0,$$ then $${\mathscr{m}}{\left({\left\{{\mathscr{x}}\; ; \; {\mathscr{m}}(\mathcal Q_k ({\mathscr{x}})\vert \mathcal F^-({\mathscr{x}}))=1 \right\}}\right)} =1.$$ This is exactly the desired statement. (Note this assertion says more than the mere ergodicity of ${\mathscr{m}}$, which only implies that ${\mathscr{m}}{\left({\left\{{\mathscr{x}}, \ {\mathscr{m}}(\mathcal Q_k ({\mathscr{x}})\vert \mathcal F^-({\mathscr{x}}))>0 \right\}}\right)} =1$.)
It remains to explain why $\mathcal Q_k$ is a measurable partition. For this, we have to express the atoms of $\mathcal Q_k$ as the fibers of a measurable map to a Lebesgue space. As for the measurability of the intersection multiplicity, we consider an exhaustion of ${\mathscr{X}}$ by Pesin sets; then, it is sufficient to work in restriction to some compact set $\mathcal K \subset {\mathscr{X}}$ on which local stable manifolds have uniform size and vary continuously. Taking a finite cover of $X$ by good charts (see § \[par:good\_charts\]), and restricting $\mathcal K$ again to keep only those local stable manifolds which are graphs over some fixed direction, we can also assume that $\pi_X(\mathcal K)$ is contained in the image of a chart $\operatorname{\Phi}_{x_0}\colon U_{x_0}\to V_{x_0}\subset X$ and there is an orthonormal basis $(e_1,e_2)$ such that for every ${\mathscr{y}}\in \mathcal K$ the local stable manifold $\pi_X(W^s_{\mathrm{loc}}({\mathscr{y}}))$ is a graph $\{z e_1+\psi^s_{\mathscr{y}}(z)e_2\}$ in this chart, for some holomorphic function $\psi^s_{\mathscr{y}}$ on ${\mathbb{D}}(r)$. Now the map from $\mathcal K $ to $ {\mathbf{C}}^2\times {\mathbf{C}}^k$ defined by $${\mathscr{x}}\longmapsto {\left(\operatorname{\Phi}_{x_0}^{-1}(\pi_X({\mathscr{x}})), (\psi^s_{\mathscr{x}})'(0), \ldots , (\psi^s_{\mathscr{x}})^{(k)}(0)\right)}$$ is continuous. Since the fibers of this map are precisely the (intersection with $\mathcal K$ of the) atoms of $\mathcal Q_k$, we are done.
The previous lemma is stated on ${\mathscr{X}}$ because its proof relies on the ergodic properties of $F$. However, since stable manifolds depend only on the future, it admits the following more elementary formulation on $X$:
\[cor:alternative\_multiplicity\] Let $k\geq 1$ be an integer. Exactly one of the following assertions holds:
1. for $\mu$-almost every $x\in X$ and $(\nu^{\mathbf{N}}) ^2$-almost every $(\omega, \omega')$, $$\operatorname{inter}_x{\left(W^s_{\mathrm{loc}}(\omega, x), W^s_{\mathrm{loc}}(\omega', x)\right)}\geq k+1;$$
2. or for $\mu$-almost every $x\in X$ and $(\nu^{\mathbf{N}}) ^2$-almost every $(\omega, \omega')$, $$\operatorname{inter}_x{\left(W^s_{\mathrm{loc}}(\omega, x), W^s_{\mathrm{loc}}(\omega', x)\right)}\leq k.$$
Combining this alternative with the results of the previous sections, we prove now that, for a generic point $x\in X$, there is a generic finite order of contact between stable manifolds when the itinerary $\omega$ varies:
\[lem:generic\_multiplicity\] There exists a unique integer $k_0\geq 1$ such that for $\mu$-almost every $x\in X$ and $(\nu^{\mathbf{N}}) ^2$-almost every pair $(\omega, \omega')$, $$\operatorname{inter}_x{\left(W^s(\omega, x), W^s(\omega', x)\right)}= k_0.$$
Fix a small ${\varepsilon}>0$ and consider a compact set $\mathcal R_{\varepsilon}\subset {\mathscr{X}}_+$ with ${\mathscr{m}}_+(\mathcal R_{\varepsilon})\geq 1-{\varepsilon}$, along which local stable manifolds have size at least $r({\varepsilon})$ and vary continuously. Since by Theorem \[thm:nevanlinna\] for ${\mathscr{m}}_+$-a.e. ${\mathscr{x}}$, the only Nevanlinna current associated to $W^s({\mathscr{x}})$ is $T^s_\omega$, we can further assume that this property holds for every ${\mathscr{x}}\in \mathcal R_{\varepsilon}$. Let $A\subset X$ be a subset of full $\mu$-measure on which the alternative of Corollary \[cor:alternative\_multiplicity\] holds for every $k\geq 1$. In ${\mathscr{X}}_+$, consider the measurable partition into fibers of the form $\Omega\times {\left\{x\right\}}$ (which corresponds to $\mathcal F^-$ in Lemma \[lem:alternative\_multiplicity\]); then, the associated conditional measures ${\mathscr{m}}_+(\, \cdot \, \vert \, \Omega\times {\left\{x\right\}})$ are naturally identified with $\nu^{\mathbf{N}}$. Fix $x\in A$ such that ${\mathscr{m}}_+(\mathcal R_{\varepsilon}\vert \Omega\times {\left\{x\right\}})>0$. Since $(X,\nu)$ is non-elementary, Theorems \[thm:def\_stationary\] and \[thm:uniq+extremal\] provide pairs $(\omega_1, \omega_2)$ in $(\pi_\Omega(\mathcal R_{\varepsilon}))^2$ for which the currents $T^s_{\omega_1}$ and $T^s_{\omega_2}$ are not cohomologous. By Theorem \[thm:nevanlinna\] these currents respectively describe the asymptotic distribution of $W^s(\omega_1, x)$ and $W^s(\omega_2, x)$ so we infer that $W^s(\omega_1, x)\neq W^s(\omega_2, x)$ and by the analytic continuation principle it follows that $W_{\mathrm{loc}}^s(\omega_1, x)\neq W_{\mathrm{loc}}^s(\omega_2, x)$. Let $k_1<\infty$ be the intersection multiplicity of these manifolds at $x$. Since the intersection multiplicity is upper semi-continuous, we infer that for $\omega'_j\in \mathcal R_{\varepsilon}$ close to $\omega_j$, $j=1,2$, $\operatorname{inter}_x(W_{\mathrm{loc}}^s(\omega'_1, x), W_{\mathrm{loc}}^s(\omega'_2, x))\leq k_1$. Thus for $k=k_1$ we are in case (b) of the alternative of Corollary \[cor:alternative\_multiplicity\]. Applying then Corollary \[cor:alternative\_multiplicity\] successively for $k=1, \ldots , k_1$, there is a first integer $k_0$ for which case (b) holds, and since (a) holds for $k_0-1$, we conclude that generically $\operatorname{inter}_x{\left(W_{\mathrm{loc}}^s(\omega, x), W^s_{\mathrm{loc}}(\omega', x)\right)} = k_0$.
Transversal perturbations
-------------------------
The key ingredient in the proof of Theorem \[thm:alternative\_stable\] is the following basic geometric lemma, which is a quantitative refinement of [@bls Lemma 6.4].
\[lem:BLS\_intersection\] Let $k$ be a positive integer. If $r$ and ${\varepsilon}$ are positive real numbers, then there are two positive real numbers $\delta=\delta(k,r,c)$ and $\alpha=\alpha(k, r,c)$ with the following property. Let $M_1$ and $M_2$ be two complex analytic curves in ${\mathbb{D}}(r)\times {\mathbb{D}}(r)\subset {\mathbf{C}}^2$ such that
1. $M_1$ and $M_2$ are graphs $\{(z,f_j(z))\; ; \; w\in {\mathbb{D}}_r\}$ of holomorphic functions $f_j\colon{\mathbb{D}}(r)\to {\mathbb{D}}(r)$;
2. $M_1\cap M_2=\{(0,0)\}$, and $\operatorname{inter}_{(0,0)}(M_1, M_2) = k$;
3. the $k$-th derivative satisfies ${\left\vert(f_1-f_2)^{(k)}(0)\right\vert} \geq c$.
If $M_3\subset {\mathbb{D}}(r)\times {\mathbb{D}}(r)$ is a complex curve that does not intersect $M_1$ but is $\delta$-close to $M_1$ in the $C^1$-topology , then $M_2$ and $M_3$ have exactly $k$ transverse intersection points in ${\mathbb{D}}(\alpha r)\times {\mathbb{D}}(\alpha r)$ (i.e. with multiplicity $1$).
Without loss of generality we may assume that $\delta <1$.
[**Step 1.–** ]{} We claim that there exists $\alpha_1 = \alpha_1 (k,r,c)$ such that for every $\alpha\leq \alpha_1$ and every $z\in {\mathbb{D}}(\alpha r)$ the following estimates hold: $$\begin{aligned}
\label{eq:f12k}
\frac{1}{2 } \frac{{\left\vert(f_1-f_2)^{(k)}(0)\right\vert}}{k!} {\left\vertz\right\vert}^k & \leq {\left\vertf_1(z)- f_2(z)\right\vert} \leq \frac{3}{2 }\frac{ {\left\vert(f_1-f_2)^{(k)}(0)\right\vert} }{k!} {\left\vertz\right\vert}^k \\
\frac{1}{2} \frac{{\left\vert(f_1-f_2)^{(k)}(0)\right\vert} }{(k-1)!}{\left\vertz\right\vert}^{k-1} & \leq {\left\vert f_1'(z) - f_2'(z) \right\vert} \leq \frac{3}{2} \frac{{\left\vert(f_1-f_2)^{(k)}(0)\right\vert} }{(k-1)!}{\left\vertz\right\vert}^{k-1}. \label{eq:derivative_f_12}\end{aligned}$$
Indeed put $g=f_1-f_2=\sum_{m\geq k} g_m z^m$. Assumptions (i) and (iii) give${\left\vertg(z)\right\vert}\leq 2r$ on ${\mathbb{D}}(r)$, and $ g^{(k)}(0)\neq 0$. By the Cauchy estimates, ${\left\vertg_n\right\vert}\leq 2r^{1-n}$ for all $n\geq 0$. Then on ${\mathbb{D}}(\alpha r)$ we get $$\begin{aligned}
\notag {\left\vertg(z)-\frac{g^{(k)}(0)}{k!} z^k\right\vert} &\leq 2 r \left( \frac{{\left\vertz\right\vert}}{r}\right)^{k+1}
\left( 1-\frac{{\left\vertz\right\vert}}{r}\right)^{-1} \leq 2r^{1-k} \frac{\alpha }{1-\alpha} {\left\vertz\right\vert}^k.\end{aligned}$$ There exists $\alpha_1(k,r,c)$ such that as soon as $\alpha \leq \alpha_1$, the right hand side of this inequality is smaller than $c{\left\vertz\right\vert}^k/2$; hence Estimate follows. The same argument applies for because $$\begin{aligned}
\notag {\left\vertg'(z)- \frac{g^{(k)}(0)}{(k-1)!}z^{k-1}\right\vert} &\leq 4 (k+1) \left( \frac{{\left\vertz\right\vert}}{r}\right)^k \left( 1-\frac{{\left\vertz\right\vert}}{r}\right)^{-2}
\leq 4 (k+1) r^{1-k} \frac{\alpha}{ (1-\alpha)^{2}}{\left\vertz\right\vert}^{k-1}. \end{aligned}$$
[**Step 2.–**]{} For every $\alpha\leq \alpha_1$, if $\delta < c (\alpha r)^k/ 2k!$, $M_2$ and $M_3$ have exactly $k$ intersection points, counted with multiplicities, in ${\mathbb{D}}(\alpha r)\times {\mathbb{D}}(\alpha r)$.
Indeed, the intersection points of $M_3$ and $M_2$ correspond to the solutions of the equation $f_3=f_2$. To locate its roots, note that on the circle ${\partial}{\mathbb{D}}(\alpha r)$, the Inequality implies $$\label{eq:rouche}
{\left\vertf_1 - f_2\right\vert} \geq \frac12 \frac{c}{k!} (\alpha r)^k.$$ Since ${\left\vertf_1-f_3\right\vert}< \delta$, the choice $\delta < c (\alpha r)^k/ 2k!$ is tailored to assure that the hypothesis of the Rouché theorem is satisfied in ${\mathbb{D}}(\alpha r)$; so, counted with multiplicities, there are $k$ solutions to the equation $f_3=f_2$ on that disk. Furthermore by the Schwarz lemma ${\left\vertf_2\right\vert}< \alpha r$ on ${\mathbb{D}}(\alpha r)$ so the corresponding intersection points between $M_2$ and $M_3$ are contained in ${\mathbb{D}}(\alpha r)\times {\mathbb{D}}(\alpha r)$.
If $k=1$ the proof is already complete at this stage, so from now on we assume $k\geq 2$.
[**Step 3.–**]{} Set $\delta_0={\left\vertf_3(0)\right\vert}$, and note that $\delta_0\leq \delta$. Then for every $\alpha \leq 1/2$, in ${\mathbb{D}}(\alpha r)$ we have $$\begin{aligned}
\label{eq:harnack}
\delta_0^{\frac{1+\alpha}{1-\alpha}}\leq {\left\vertf_1(z) - f_3(z)\right\vert} &\leq \delta_0^{\frac{1-\alpha}{1+\alpha}} \\
\label{eq:harnack_derivee}
{\left\vertf_1'(z) - f_3'(z)\right\vert} &\leq {\frac{1}{\alpha r}} \delta_0^{\frac{1-2\alpha}{1+2\alpha}}. \end{aligned}$$ For this, recall the Harnack inequality: for any negative harmonic function in ${\mathbb{D}}$ $$\frac{1- {\left\vert\zeta\right\vert}}{1+{\left\vert\zeta\right\vert}}\leq \frac{u(\zeta)}{u(0)}\leq \frac{1+ {\left\vert\zeta\right\vert}}{1-{\left\vert\zeta\right\vert}}.$$ Since $f_1-f_3$ does not vanish and ${\left\vertf_1-f_3\right\vert}\leq \delta<1$ in ${\mathbb{D}}(r)$, the function $\log{\left\vertf_1-f_3\right\vert}$ is harmonic and negative there. Thus for $\alpha \leq 1/2$, the Harnack inequality can be applied to $\zeta\mapsto (f_1-f_3)(r\zeta)$ in ${\mathbb{D}}$: this gives . Likewise, we infer that $$\delta_0^{\frac{1+2\alpha}{1-2\alpha}} \leq {\left\vertf_1(z) - f_3(z)\right\vert} \leq \delta_0^{\frac{1-2\alpha}{1+2\alpha}}$$ in ${\mathbb{D}}(2\alpha r)$, and follows from the Cauchy estimate ${{\left\Vertg'\right\Vert}}_{{\mathbb{D}}(\alpha r)} \leq (\alpha r){^{-1}}{{\left\Vertg\right\Vert}}_{{\mathbb{D}}(2\alpha r)}$.
[**Step 4.–**]{} We now conclude the proof. Fix $\alpha = \alpha (k,r,c)$ such that $\alpha \leq \alpha_1$ and $$\beta(\alpha):=\frac{1-2\alpha }{1+2\alpha } -\frac{k-1}{k}\times \frac{1+ \alpha }{1- \alpha } >0.$$ (This will be our final choice for $\alpha$.) Fix $\delta < c (\alpha r)^k/ 2k!$ and consider a solution $z_0$ of the equation $f_2(z) = f_3(z)$ in ${\mathbb{D}}(\alpha r)$ provided by Step 2. The transversality of $M_2$ and $M_3$ at $(z_0, f_2(z_0))$ is equivalent to $f_3'(z_0)\neq f_2'(z_0)$, so we only need $${\left\vert(f_3-f_1)'(z_0)\right\vert}< {\left\vert(f_2-f_1)'(z_0)\right\vert}.$$ Since $(f_1- f_3)(z_0) = (f_1-f_2)(z_0)$, combining the right hand side of Inequality and the left hand side of Inequality \[eq:harnack\], we get that $$\frac{3}{2}\frac{ {\left\vert(f_1-f_2)^{(k)}(0)\right\vert} }{k!} {\left\vertz_0\right\vert}^k \geq \delta_0^{\frac{1+\alpha}{1-\alpha}} ,$$ thus $${\left\vertz_0\right\vert} \geq \delta_0^{{\frac{1}{k}} \frac{1+\alpha}{1-\alpha}} {\left(\frac{2k!}{3 }\right)}^{{\frac{1}{k}}} {\left\vert(f_1-f_2)^{(k)}(0)\right\vert}^{-{\frac{1}{k}}}$$ Hence by we get that $$\begin{aligned}
{\left\vert(f_2-f_1)'(z_0)\right\vert} &\geq {\frac{1}{2(k-1)!}} {\left(\frac{2k!}{3}\right)}^\frac{k-1}{k} \delta_0^{\frac{k-1}{k} \frac{1+\alpha}{1-\alpha}}
{\left\vert(f_1-f_2)^{(k)}(0)\right\vert}^{{\frac{1}{k}}} \\
&\notag \geq {\frac{1}{2(k-1)!}}{\left(\frac{2k!}{3}\right)}^\frac{k-1}{k} \delta_0^{\frac{k-1}{k} \frac{1+\alpha}{1-\alpha}} c^{{\frac{1}{k}}}.
\end{aligned}$$ On the other hand by Estimate $${\left\vert(f_3-f_1)'(z_0)\right\vert} \leq {\frac{1}{\alpha r}} \delta_0^{\frac{1-2\alpha}{1+2\alpha}}$$ Since $\delta_0 \leq \delta$, we only need to impose one more constraint on $\delta$ (together with $\delta < c (\alpha r)^k/ 2k!$), namely $$\delta^{\beta(\alpha)}< {\frac{1}{2(k-1)!}}{\left(\frac{2k!}{3}\right)}^\frac{k-1}{k} c^{{\frac{1}{k}}} r \alpha,$$ to get the desired inequality ${\left\vert(f_3-f_1)'(z_0)\right\vert} < {\left\vert(f_2-f_1)'(z_0)\right\vert}$.
Let $\Delta_1$ and $\Delta_2$ be two disks of size $r$ at $x\in X$, which are tangent at $x$; let $e_1\in T_xX$ be a unit vector in $T_x\Delta_1=T_x\Delta_2$ and $e_2$ a unit vector orthogonal to $e_1$ for $\kappa_0$. Then, in the chart $\operatorname{\Phi}_x$, $\Delta_1$ and $\Delta_2$ are graphs $\{ze_1+\psi_i(z)e_2\}$ of holomorphic functions $\psi_i\colon {\mathbb{D}}(r)\to {\mathbb{D}}(r)$, $i=1$, $2$, such that $\psi_i(0)=0$ and $\psi_i'(0)=0$. If $\operatorname{inter}_x(\Delta_1, \Delta_2) = k$, then for $j = 1, \ldots , k-1$ one has $\psi^{(j)}_1(0) = \psi^{(j)}_2(0)$ and $\psi^{(k)}_1(0) \neq \psi^{(k)}_2(0)$. We define the **$k$-osculation** of $\Delta_1$ and $\Delta_2$ at $x$ to be $$\operatorname{osc}_{k,x,r}(\Delta_1, \Delta_2) = {\left\vert\psi^{(k)}_1(0) - \psi^{(k)}_2(0)\right\vert}.$$ If $s\leq r$ and we consider $\Delta_1$ and $\Delta_2$ as disks of size $s$, then $\operatorname{osc}_{k,x,s}(\Delta_1, \Delta_2) =\operatorname{osc}_{k,x,r}(\Delta_1, \Delta_2)$. Thus, $\operatorname{osc}_{k,x,r}(\Delta_1, \Delta_2)$ does not depend on $r$, so we may denote this osculation number by $\operatorname{osc}_{k,x}(\Delta_1, \Delta_2)$. With this terminology, Lemma \[lem:BLS\_intersection\] directly implies the following corollary.
\[cor:bls\_quantitative\] Let $k$ be a positive integer, and $r$ and $c$ be positive real numbers. Then, there are two positive real numbers $\delta$ and $\alpha$, depending on $(k,r,c)$, satisfying the following property. Let $\Delta_1$ and $\Delta_2$ be two holomorphic disks of size $r$ through $x$, such that $\operatorname{inter}_x(\Delta_1, \Delta_2)=k$ and $\operatorname{osc}_{k,x}(\Delta_1, \Delta_2))\geq c$. Let $\Delta_3$ be a holomorphic disk of size $r$ such that $\Delta_3$ is $\delta$-close to $\Delta_1$ in the $C^1$-topology but $\Delta_3\cap \Delta_1 = \emptyset$. Then $\Delta_3$ intersects $\Delta_2$ transversely in exactly $k$ points in $B_X(x, \alpha r)$.
The following lemma follows directly from the first step of the proof of Lemma \[lem:BLS\_intersection\].
\[lem:unique\_intersection\] Let $k$ be a positive integer, and $r$ and $c$ be positive real numbers. Then there exists a constant $\beta$ depending only on $(r,k,c)$ such that if $\Delta_1$ and $\Delta_2$ are two holomorphic disks of size $r$ through $x$, such that $k=\operatorname{inter}_x(\Delta_1, \Delta_2)$ and $\operatorname{osc}_{k,x}(\Delta_1, \Delta_2))\geq c$, then $x$ is the only point of intersection between $\Delta_1$ and $\Delta_2$ in the ball $B_X(x,\beta r)$.
Proof of Theorem \[thm:alternative\_stable\]
--------------------------------------------
Before starting the proof, we record the following two facts from elementary measure theory:
\[lem:trivial\] Let $(\Omega, \mathcal F, {\mathbb{P}})$ be a probability space, and $\delta\in (0,1)$.
1. If $\varphi$ is a measurable function with values in $[0, 1]$ and such that $\int \varphi \; d{\mathbb{P}}\geq 1-\delta$, then $${\mathbb{P}}{\left({\left\{ x \; ; \; \varphi(x)\geq 1-\sqrt\delta\right\}} \right)}\geq 1-\sqrt\delta.$$
2. If $A_j$ is a sequence of measurable subsets such that ${\mathbb{P}}(A_j)\geq 1-\delta$ for every $j$, then ${\mathbb{P}}(\limsup A_j)\geq 1-\delta$.
Let us now prove Theorem \[thm:alternative\_stable\]. If the integer $k_0$ of Lemma \[lem:generic\_multiplicity\] is equal to 1, then Pesin stable manifolds corresponding to different itineraries at a $\mu$-generic point $x \in X$ are generically transverse; hence, we are in case [(i)]{} of the theorem –note that the conclusion is actually stronger than mere non-randomness. So, we now assume $k_0>1$ and we prove that $\mu$ is almost surely invariant and that its entropy is equal to zero.
[**[Step 1.–]{}**]{} First, we construct a subset $\mathcal G_{\varepsilon}$ of “good points” in ${\mathscr{X}}$.
As observed in Section \[par:definition\_skew\_products\], the atoms of $\mathcal F^-$ are the sets $\mathcal F^-({\mathscr{x}})=\Sigma^u_{\mathrm{loc}}(\xi)\times {\left\{x\right\}}$ and the measures ${\mathscr{m}}(\, \cdot \, \vert \mathcal F^-({\mathscr{x}}))$ can be naturally identified to $\nu^{\mathbf{N}}$ under the natural projections $\mathcal F^-({\mathscr{x}}) \overset{\sim}\to \Sigma^u_{\mathrm{loc}}(\xi) \overset{\sim}\to \Omega$. For notational simplicity we denote these measures by ${\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}}$. For a small ${\varepsilon}>0$, let $\mathcal R_{\varepsilon}\subset {\mathscr{X}}$ be a compact subset with ${\mathscr{m}}(\mathcal R_{\varepsilon})> 1-{\varepsilon}$, along which local stable manifolds have size at least $2r({\varepsilon})$ and vary continuously. Since $\int {\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}}(\mathcal R_{\varepsilon}) \, d{\mathscr{m}}({\mathscr{x}}) \geq 1-{\varepsilon}$, by Lemma \[lem:trivial\] (i) we can select a compact subset $\mathcal R_{\varepsilon}'\subset \mathcal R_{\varepsilon}$ with ${\mathscr{m}}(\mathcal R'_{\varepsilon})\geq 1-\sqrt {\varepsilon}$ such that for every ${\mathscr{x}}\in \mathcal R'_{\varepsilon}$ one has ${\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}}(\mathcal R_{\varepsilon}) \geq 1-\sqrt {\varepsilon}$.
By assumption, $$\operatorname{inter}_x(W^s_{\mathrm{loc}}({\mathscr{y}}_1), W^s_{\mathrm{loc}}({\mathscr{y}}_2))= k_0$$ for ${\mathscr{m}}$-almost every ${\mathscr{x}}= (\xi, x)\in \mathcal R'_{\varepsilon}$ and for $({\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}}\otimes {\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}})$-almost every pair of points $({\mathscr{y}}_1, {\mathscr{y}}_2)\in (\mathcal F^-({\mathscr{x}})\cap \mathcal R_{\varepsilon})^2$. Then there exists $\mathcal R''_{\varepsilon}\subset \mathcal R'_{\varepsilon}$ of measure at least $1-2\sqrt{{\varepsilon}}$ and a constant $c({\varepsilon})>0$ such that $$\operatorname{osc}_{k_0,x, r({\varepsilon})} (W^s_{\mathrm{loc}}({\mathscr{y}}_1), W^s_{\mathrm{loc}}({\mathscr{y}}_2)) \geq c({\varepsilon})$$ for every ${\mathscr{x}}=(\xi,x)\in \mathcal R''_{\varepsilon}$ and all pairs $({\mathscr{y}}_1,{\mathscr{y}}_2)$ in a subset $A_{{\varepsilon}, {\mathscr{x}}} \subset (\mathcal F^-_{\mathscr{x}}\cap \mathcal R_{\varepsilon})^2$ depending measurably on ${\mathscr{x}}$ and of measure $$({\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}}\otimes {\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}})(A_{{\varepsilon}, {\mathscr{x}}} )\geq 1-4\sqrt{{\varepsilon}}$$ (we just used $({\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}}\otimes {\mathscr{m}}^{\mathcal F^-}_{\mathscr{x}})((\mathcal F^-_{\mathscr{x}}\cap \mathcal R_{\varepsilon})^2) \geq (1-\sqrt {\varepsilon})^2 > 1- 4\sqrt{{\varepsilon}}$). Finally, Fubini’s theorem and Lemma \[lem:trivial\] (i) provide a set $\mathcal G_{\varepsilon}\subset\mathcal R''_{\varepsilon}$ such that
(a) ${\mathscr{m}}(\mathcal G_{\varepsilon})\geq 1-2{\varepsilon}^{1/4}$
(b) for every ${\mathscr{x}}\in \mathcal G_{\varepsilon}$, $W^s_{\mathrm{loc}}({\mathscr{x}})$ has size $2r({\varepsilon})$;
(c) for every ${\mathscr{x}}\in \mathcal G_{\varepsilon}$, there exists a measurable set $\mathcal{G}_{{\varepsilon}, {\mathscr{x}}} \subset \mathcal F^-_{\mathscr{x}}$ with $ {\mathscr{m}}^{\mathcal F^-}(\mathcal{G}_{{\varepsilon}, {\mathscr{x}}}) \geq 1-2{\varepsilon}^{1/4}$ such that for every ${\mathscr{y}}$ in $\mathcal{G}_{{\varepsilon}, {\mathscr{x}}}$, $W^s_{\mathrm{loc}}({\mathscr{y}})$ has size $\geq r({\varepsilon})$ and, viewed as a subset of $X$,
- it is tangent to $W^s_{\mathrm{loc}}({\mathscr{x}})$ to order $k_0$ at $x$,
- $\operatorname{osc}_{k_0,x, r({\varepsilon})} (W^s_{\mathrm{loc}}({\mathscr{x}}), W^s_{\mathrm{loc}}({\mathscr{y}})) \geq c({\varepsilon})$.
Note that ${\mathscr{x}}\notin \mathcal R_{{\varepsilon},{\mathscr{x}}}$: indeed, when the local stable manifolds vary continuously, one can think of $A_{{\varepsilon}, {\mathscr{x}}}$ as the complement of a small neighborhood of the diagonal in $\Omega\times \Omega$.
[**[Step 2.–]{}**]{} To make the argument more transparent, we first show that the fiber entropy vanishes.
Let $\eta^s$ be a Pesin partition subordinate to local stable manifolds in ${\mathcal{X}}$. By Corollary \[cor:rokhlin\_zero\_entropy\] it is enough to show that for ${\mathscr{m}}$-almost every ${\mathscr{x}}$, ${\mathscr{m}}(\cdot \vert \eta^s({\mathscr{x}}))$ is atomic (hence concentrated at $x$). Assume by contradiction that this is not the case. Therefore for ${\varepsilon}>0$ small enough there exists ${\mathscr{x}}= (\xi, x)\in \mathcal G_{\varepsilon}$ such that ${\mathscr{m}}(\cdot \vert \eta^s({\mathscr{x}}))_{\vert \eta^s({\mathscr{x}})\cap \mathcal G_{\varepsilon}}$ is non-atomic, and there exists an infinite sequence of points ${\mathscr{x}}_j=(\xi, x_j)$ in $\mathcal G_{\varepsilon}\cap \eta^s({\mathscr{x}})$ converging to ${\mathscr{x}}$. Then with $\mathcal{G}_{{\varepsilon}, \star}$ as in Property (c) of the definition of $\mathcal G_{\varepsilon}$, for every $j$ we have that ${\mathscr{m}}^{\mathcal F^-}_{{\mathscr{x}}_j}(\mathcal{G}_{{\varepsilon}, {\mathscr{x}}_j})\geq 1-2{{\varepsilon}}^{1/4}$.
Identifying all $\mathcal {F}^-({\mathscr{x}}_j) $ with $\Sigma^u_{\mathrm{loc}}(\xi)$, by Lemma \[lem:trivial\] (ii) we can find $\zeta\in \Sigma^u_{\mathrm{loc}}(\xi)$ such that $(\zeta, x_j)$ belongs to $\mathcal G_{{\varepsilon},(\zeta, x_j)}$ for infinitely many $j$’s. Along this subsequence the local stable manifolds $W^s_{\mathrm{loc}}(\zeta, x_j)$ form a sequence of disks of uniform size $r = 2r({\varepsilon})$ at $x_j$. Two such local stable manifolds are either pairwise disjoint or coincide along an open subset because they are associated to the same itinerary $\zeta$.
Let us now use the notation from Corollary \[cor:bls\_quantitative\] and Lemma \[lem:unique\_intersection\]. We know that $W^s_{r({\varepsilon})}(\zeta, x_j)$ is tangent to $W^s_{r({\varepsilon})}(\xi, x)$ at $x_j$ to order $k_0$, with $\operatorname{osc}_{k_0,x_j, r({\varepsilon})} (W^s_{r({\varepsilon})}({\mathscr{x}}), W^s_{r({\varepsilon})}(\zeta, x_j)) \geq c({\varepsilon})$; so, by Lemma \[lem:unique\_intersection\], $W^s_{r({\varepsilon})}(\zeta, x_j)$ and $W^s_{r({\varepsilon})}(\zeta, x_{j'})$ are disjoint as soon as $\operatorname{dist}_X(x_{j}, x_{j'}) < \beta r({\varepsilon})$. Finally, if $j$ and $j'$ are large enough, then $\operatorname{dist}_X(x_{j}, x_{j'}) < \alpha r({\varepsilon})$ and the $C^1$ distance between $W^s_{r({\varepsilon})}(\zeta, x_j)$ and $W^s_{r({\varepsilon})}(\zeta, x_{j'})$ is smaller than $\delta$; thus, Corollary \[cor:bls\_quantitative\] asserts that $W^s_{r({\varepsilon})}(\zeta, x_j)$ and $W^s_{r({\varepsilon})}(\zeta, x_{j'})$ cannot both be tangent to $W^s_{r({\varepsilon})}(\xi, x)$. This is a contradiction, and we conclude that the fiber entropy of ${\mathscr{m}}$ vanishes.
[**[Step 3.–]{}**]{} We now prove the almost sure invariance.
As in [@br Eq. (11.1)] we consider a measurable partition $\mathcal P$ of ${\mathcal{X}}$ with the property that for ${\mathscr{m}}$-almost every $(\xi, x)$, $$\Sigma^s_{\mathrm{loc}}(\xi) \times W^s_{r(\xi, x)}(\xi, x) \subset \mathcal P (\xi, x) \subset \Sigma^s_{\mathrm{loc}}(\xi) \times W^s (\xi, x).$$ The existence of such a partition is guaranteed for instance by Lemma \[lem:stable\_partition\]. By [@br Prop 11.1]([^5]), to show that $\mu$ is almost surely invariant it is enough to prove that: $$\label{eq:contradict}
\text{for }{\mathscr{m}}\text{ almost every }\xi, \ {\mathscr{m}}(\, \cdot\, \vert \mathcal P(\xi, x)) \text{ is
concentrated on }\Sigma^s_{\mathrm{loc}}(\xi) \times {\left\{x\right\}}.$$ By contradiction, assume that fails. By contraction along the stable leaves, it follows that almost surely $\Sigma^s_{\mathrm{loc}}(\xi) \times {\left\{x\right\}}$ is contained in $$\operatorname{Supp}{\left({\mathscr{m}}(\cdot \vert \mathcal P(\xi, x))_{\vert \mathcal P(\xi, x) \setminus \Sigma^s_{\mathrm{loc}}(\xi) \times {\left\{x\right\}}}\right)}$$ (this is identical to the argument of Corollary \[cor:rokhlin\_zero\_entropy\]). In particular for small ${\varepsilon}$ we can find ${\mathscr{x}}= (\xi,x)\in \mathcal G_{\varepsilon}$ and a sequence of points $ {\mathscr{x}}_j = (\xi_j, x_j)\in \mathcal G_{\varepsilon}$ such that ${\mathscr{x}}_j$ belongs to $\mathcal P({\mathscr{x}}) \cap \mathcal G_{\varepsilon}$, $x_j\neq x$ and $(x_j)$ converges to $x$ in $X$. We can also assume that the $x_j$ are all distinct. By definition of $\mathcal G_{\varepsilon}$ we have that ${\mathscr{m}}^{\mathcal F^-}_{{\mathscr{x}}_j}{\left(\mathcal{G}_{{\varepsilon}, {\mathscr{x}}_j}\right)}\geq 1-2{{\varepsilon}}^{1/4}$ for every $j$.
For $(\xi, \zeta)\in \Sigma^2$, set $$[\xi, \zeta] = \Sigma^u_{\mathrm{loc}}(\xi)\cap \Sigma^s_{\mathrm{loc}}(\zeta);$$ that is, $[\xi, \zeta] $ is the itinerary with the same past as $\xi$ and the same future as $\zeta$. As above, identifying the atoms of the partition $\mathcal F^-$ with $\Omega$, Lemma \[lem:trivial\] (ii) provides an infinite subsequence $(j_\ell)$ and for every $\ell$, an itinerary $\zeta_{j_\ell}\in \Sigma^u_{\mathrm{loc}}(\xi_{j_\ell})$ such that ${\mathscr{y}}_{j_\ell}:=
(\zeta_{j_\ell}, x_{j_\ell})$ belongs to $\mathcal{G}_{{\varepsilon}, {\mathscr{x}}_{j_\ell}}$ and all the $\zeta_{j_\ell}$ have the same future, that is $\zeta_{j_\ell}$ is of the form $[\xi_{j_\ell}, \zeta]$ for a fixed $\zeta$. By definition we have $$\begin{aligned}
\label{eq:osc}
\operatorname{inter}_{x_{j_\ell}}(W^s_{\mathrm{loc}}({\mathscr{x}}_{j_\ell}), W^s_{\mathrm{loc}}({\mathscr{y}}_{j_\ell}) )& = k_0 \\
\operatorname{osc}_{k_0, x_{j_\ell}, r({\varepsilon})} (W^s_{\mathrm{loc}}({\mathscr{x}}_{j_\ell}), W^s_{\mathrm{loc}}({\mathscr{y}}_{j_\ell})) & \geq c({\varepsilon}).\end{aligned}$$ In addition the disks $\pi_X(W^s_{\mathrm{loc}}({\mathscr{y}}_{j_\ell}))$ are pairwise disjoint or locally coincide because the $x_{j_\ell}$ are distinct and the $\zeta_{j_\ell}$ have the same future. Moreover, since ${\mathscr{x}}_{j_\ell}$ belongs to $\mathcal P({\mathscr{x}})$, $W^s({\mathscr{x}}_{j_\ell})$ coincides with $W^s({\mathscr{x}})$.
Therefore, the $\pi_X(W^s_{\mathrm{loc}}({\mathscr{y}}_{j_\ell}))$ form a sequence of disjoint disks of size $2r({\varepsilon})$ at $x_j$, all tangent to $\pi_X(W^s_{\mathrm{loc}}({\mathscr{x}}))$ to order $k_0$, with osculation bounded from below by $c({\varepsilon})$. Since this sequence of disks is continuous and $(x_j)$ converges towards $x$, Lemma \[lem:unique\_intersection\] and Corollary \[cor:bls\_quantitative\] provide a contradiction, exactly as in Step 2. This completes the proof of the theorem.
Stiffness {#sec:stiffness}
=========
In this section we study Furstenberg’s stiffness property for diffeomorphisms of compact Kähler surfaces, thereby concluding the proof of Theorem \[mthm:stiffness\]. Our first results in §\[subs:stiffness\_elementary\] deal with elementary subgroups and only use the classification of such groups and general group-theoretic criteria for establishing stiffness, which are recalled in Sections \[subs:stiffness\_generalities\] and \[subs:inducing\]. Then, Theorem \[thm:stiffness\_real\] concerns the much more interesting case of non-elementary subgroups; its proof relies on the previous sections and the work of Brown and Rodriguez-Hertz [@br].
Stiffness {#subs:stiffness_generalities}
---------
Following Furstenberg [@furstenberg_stiffness], a random dynamical system $(X, \nu)$ is **stiff** if any $\nu$-stationary measure is almost surely invariant; equivalently, every ergodic stationary measure is almost surely invariant. This property can conveniently be expressed in terms of $\nu$-harmonic functions on $\Gamma$. Indeed if $\xi\colon X\to {\mathbf{R}}$ is a continuous function and $\mu$ is $\nu$-stationary, then $\Gamma\ni g\mapsto \int_X \xi(gx) \, d\mu(x)$ is a bounded, continuous, left $\nu$-harmonic function on $\Gamma$; thus proving that $\mu$ is invariant amounts to proving that such harmonic functions are constant. Stiffness can also be defined for group actions: a group $\Gamma$ **acts stiffly** on $X$ if and only if $(X, \nu)$ is stiff for every probability measure $\nu$ on $\Gamma$ whose support generates $\Gamma$; in this definition, the measures $\nu$ can also be restricted to specific families, for instance symmetric finitely supported measures, or measures satisfying a moment condition.
There are general criteria ensuring stiffness of an action just from the properties of $\Gamma$. A first case is when $G$ is a topological group acting continuously on $X$ and $\Gamma\subset G$ is relatively compact. Then $\Gamma$ acts stiffly on $X$: this follows directly from [@furstenberg_stiffness Thm 3.5]. Another important case for us is that of Abelian and nilpotent groups.
\[thm:raugi\] Let $G$ be a locally compact, second countable, topological group, and let $\nu$ be a probability measure on $G$. If $G$ is nilpotent of class $\leq 2$, then any measurable, $\nu$-harmonic and bounded function $\varphi\colon G\to {\mathbf{R}}$ is constant. In particular every measurable action of such a group is stiff.
This is due to Dynkin-Malyutov and to Guivarc’h; we refer to [@Guivarch:1973] for a proof ([^6]). The case of Abelian groups is the famous Blackwell-Choquet-Deny theorem. We shall apply Theorem \[thm:raugi\] to subgroups $A\subset {\mathsf{Aut}}(X)$; what we implicitly do is first replace $A$ by its closure in ${\mathsf{Aut}}(X)$ to get a locally compact group, and then apply the theorem to this group.
Subgroups and hitting measures {#subs:inducing}
------------------------------
A basic tool in this area of research is the [**hitting measure**]{} on a subgroup which we briefly introduce now (see [@benoist-quint_book Chap. 5] for more details on this notion). Let $G$ be a locally compact second countable topological group, $H\subset G$ a finite index subgroup, and $\nu$ a probability measure on $G$. Consider the left random walk governed by $\nu$ on $G$; for $\omega=(g_i)\in G^{\mathbf{N}}$, define the hitting time $$T = T_{H}(\omega) := \min{\left\{n\geq 1\; ; \; g_n\cdots g_1\in H\right\}}.$$ Since $[G:H]<\infty$, $T$ is almost surely finite and the distribution of $g_{T(\omega)}\cdots g_1$ is by definition the [**[hitting measure]{}**]{} of $\nu$ on $H$, which will be denoted $\nu_{H}$. The key property of $\nu_{H}$ is that if $\varphi: G\to {\mathbf{R}}$ is a $\nu$-harmonic function, then $\varphi{ \arrowvert_{H}}$ is also $\nu_H$-harmonic. Therefore, if $\mu$ is a $\nu$-stationary measure, then it is also $\nu_{H}$-stationary. Additionally, $\nu_{H}$ has a finite first moment (resp. a finite exponential moment) if and only if $\nu$ does: this follows from the fact that the stopping time $T$ admits an exponential moment. Conversely, any bounded $\nu_{H}$-harmonic function $h$ on $H$ admits a unique extension $\widetilde h$ to a bounded $\nu$-harmonic function on $G$; this extension is defined by the formula $$\label{eq:doob}
\widetilde h(x) = {\mathbb{E}}_x(h(g_{T_{x,H}(\omega)}\cdots g_1x)) = \int h(g_{T_{x,H}(\omega)}\cdots g_1x)\, d\nu^{\mathbf{N}}(\omega)$$ where the stopping time $T_{x, H}$ is defined by $T_{x, H}(\omega) = \min{\left\{n\geq 0 \; ; \; g_n\cdots g_1x\in H\right\}}$. The uniqueness comes from Doob’s optional stopping theorem, which asserts that if $(M_t)_{t\geq 0}$ is a bounded martingale and $T$ is a stopping time which is almost surely finite then ${\mathbb{E}}(M_T) = {\mathbb{E}}(M_0)$. Thus, any bounded $\nu$-harmonic function $h$ on $G$ satisfies Formula .
On a slightly different note, let us assume that $H$ is a normal subgroup of $G$ with $G/H$ isomorphic to ${\mathbf{Z}}$, and that $\nu$ is symmetric with a finite first moment. Then, the projection ${\overline{\nu}}$ of $\nu$ on $G/H$ is a symmetric measure with a finite first moment, and this implies that the random walk governed by ${\overline{\nu}}$ on $G/H\simeq {\mathbf{Z}}$ is recurrent (see the Chung-Fuchs Theorem in [@Durrett:BookV5 §5.4] or [@Chung-Fuchs]). Then, the hitting measure $\nu_{H}$ is well defined. It satisfies the same properties as in the finite index case, except for the estimate on the moments of $\nu_H$.
\[lem:stiff\_finite\_extension\] Let $\nu$ be a probability measure on ${\mathsf{Aut}}(X)$ and $\Gamma'$ be a closed subgroup which is recurrent relative to the random walk induced by $\nu$. Let $\nu'$ be the induced measure on $\Gamma'$. If $(X, \nu')$ is stiff then $(X, \nu)$ is stiff as well. This holds in particular if:
1. either $[\Gamma_\nu:\Gamma']<\infty$
2. or $\Gamma'$ is a normal subgroup of $\Gamma_\nu$, with $\Gamma_\nu/\Gamma'$ isomorphic to ${\mathbf{Z}}$, and $\nu$ is symmetric with a finite first moment.
Let $\mu$ be a $\nu$-stationary measure on $X$. Then $\mu$ is $\nu'$-stationary, hence by stiffness it is $\Gamma'$-invariant. Therefore for every Borel set $B\subset X$, the function $\Gamma\ni g \mapsto \mu(g{^{-1}}B)$ is a bounded $\nu$-harmonic function which is constant on $\Gamma'$ so by the uniqueness of harmonic extension it is constant, and $\nu$ is $\Gamma$-invariant.
Elementary groups {#subs:stiffness_elementary}
-----------------
Recall that ${\mathsf{Aut}}(X)$ is a topological group for the topology of uniform convergence and is in fact a complex Lie group (with possibly infinitely many connected components). Let ${\mathsf{Aut}}(X)^\circ$ be the connected component of the identity in ${\mathsf{Aut}}(X)$ and $${\mathsf{Aut}}(X)^\# = {\mathsf{Aut}}(X)/{\mathsf{Aut}}(X)^\circ.$$ Let $\rho: {\mathsf{Aut}}(X)\to {{\sf{GL}}}(H^*(X;{\mathbf{Z}}))$ be the natural homomorphism; its image is ${\mathsf{Aut}}(X)^* = \rho({\mathsf{Aut}}(X))$ (see § \[par:Hodge\_decomposition\]); is kernel contains ${\mathsf{Aut}}(X)^\circ$ and a theorem of Lieberman [@lieberman] shows that ${\mathsf{Aut}}(X)^\circ$ has finite index in $\ker(\rho)$. If $\Gamma$ is a subgroup of ${\mathsf{Aut}}(X)$, we set $\Gamma^* = \rho(\Gamma)$.
\[thm:stiffness\_elementary\_groups\] Let $X$ be a compact Kähler surface and $\nu$ be a symmetric probability measure on ${\mathsf{Aut}}(X)$ satisfying the moment condition . Assume that $\Gamma_\nu$ is elementary with $\Gamma_\nu^*$ infinite. Then $(X, \nu)$ is stiff.
Note that stiffness can fail when $\Gamma_\nu^*$ is finite: see Example \[eg:stiffness\_elementary\] below. The proof relies on the classification of elementary subgroups of ${\mathsf{Aut}}(X)$ (see [@Cantat:Milnor Thm 3.2], [@favre_bourbaki]): if $\Gamma_\nu$ is elementary there exists a finite index subgroup $A^*\subset \Gamma_\nu^*$ which is
(a) either cyclic and generated by a loxodromic map;
(b) or a free Abelian group of parabolic transformations possessing a common isotropic line; in that case, there is a genus $1$ fibration $\tau\colon X\to S$, onto a compact Riemann surface $S$, such that $\Gamma_\nu$ permutes the fibers of $\tau$.
Denote by $\rho_{\Gamma_\nu} \colon \Gamma_\nu\to \Gamma_\nu^*$ the restriction of $\rho$ to $\Gamma_\nu$. We distinguish two cases.
Let $A$ be the pre-image of $A^*$ in $\Gamma$. The group $A$ fits into an exact sequence $1\to F\to A\to A^*\to 0$ with $F$ finite, so a classical group theoretic lemma (see Corollary 4.8 in [@cantat-guirardel-lonjou]) asserts that $A$ contains a finite index, free abelian subgroup $A_0$, such that $\rho_{\Gamma_\nu}(A_0)$ is a finite index subgroup of $A^*$. The index of $A_0$ in $\Gamma$ being finite, by Lemma \[lem:stiff\_finite\_extension\] it is sufficient to prove that the action of $(A_0,\nu_{A_0})$ on $X$ is stiff. But since $A_0$ is Abelian, this follows from Theorem \[thm:raugi\]
In case (a), $X$ is a torus ${\mathbf{C}}^2/\Lambda$ and $\ker(\rho_{\Gamma_\nu})$ is a group of translations of $X$ (see Proposition \[prop:discrete\]). Let $A\subset \Gamma_\nu$ be the pre-image of $A^*$; setting $K=\ker(\rho_{\Gamma_\nu})$, we obtain an exact sequence $0\to K\to A\to A^*\to 0$, with $A\subset \Gamma_\nu$ of finite index, $A^*\simeq {\mathbf{Z}}$ generated by a loxodromic element, and $K\subset X$ an infinite group of translations. Since $\nu$ is symmetric, the measure $\nu_A$ is also symmetric; since $\nu_A$ satisfies the moment condition , its projection on $A^*$ has a first moment (note that if $f$ is loxodromic, then $\log({{\left\Vert(f^*)^n\right\Vert}})\asymp {\left\vertn\right\vert}$). Since $K$ is abelian, its action on $X$ is stiff; thus, as in Lemma \[lem:stiff\_finite\_extension\], the action of $A$ on $X$ is stiff. Since $A$ has finite index in $\Gamma$, the action of $\Gamma$ on $X$ is stiff too.
In case (b), we apply Proposition \[pro:parabolic\_infinite\]. So, either $X$ is a torus, or the action of $\Gamma_\nu$ on the base $S$ of its invariant fibration $\tau\colon X\to S$ has finite order. In the latter case, a finite index subgroup $\Gamma_0$ of $\Gamma$ preserves each fiber of $\tau$; then, $\Gamma_0$ contains a subgroup of index dividing $12$ acting by translations on these fibers. This shows that $\Gamma$ is virtually abelian; in particular, $\Gamma$ is stiff. The last case is when the image of $\Gamma$ in ${\mathsf{Aut}}(S)$ is infinite and $X$ is a torus ${\mathbf{C}}^2/\Lambda_X$. Then, $S={\mathbf{C}}/\Lambda_S$ is an elliptic curve and $\tau$ is induced by a linear projection ${\mathbf{C}}^2\to {\mathbf{C}}$, say the projection $(x,y)\mapsto x$. Lifting $\Gamma$ to ${\mathbf{C}}^2$, and replacing $\Gamma$ by a finite index subgroup if necesssary, its action is by affine transformations of the form $${\tilde{f}} \colon (x,y)\mapsto (x+a, y+m x +b)$$ with $m$ in ${\mathbf{C}}^*$, and $(a,b)$ in ${\mathbf{C}}^2$. This implies that $\Gamma$ is a nilpotent group of length $\leq 2$; by Theorem \[thm:raugi\] it also acts stiffly and we are done.
\[eg:stiffness\_elementary\] If $X={\mathbb{P}}^2({\mathbf{C}})$, its group of automorphism is ${{\sf{PGL}}}_3({\mathbf{C}})$ and for most choices of $\nu$ there is a unique stationary measure, which is not invariant; the dynamics is proximal, and this is opposite to stiffness (see [@furstenberg_stiffness]). If $X={\mathbb{P}}^1({\mathbf{C}})\times C$, for some algebraic curve $C$, then ${\mathsf{Aut}}(X)$ contains ${{\sf{PGL}}}_2({\mathbf{C}})\times {\mathsf{Aut}}(C)$ and if $\nu$ is a probability measure on ${{\sf{PGL}}}_2({\mathbf{C}})\times \{\operatorname{id}_C\}$, then in general there is a unique stationary measure, which again is not invariant.
\[pro:stiffness\_elementary2\] Let $X$ be a complex projective surface, and $\Gamma$ be a subgroup of ${\mathsf{Aut}}(X)$ such that $\Gamma^*$ is finite. Assume that $\Gamma$ preserves a probability measure, the support of which is Zariski dense. Then the action of $\Gamma$ on $X$ is stiff.
The main examples we have in mind is when the invariant measure is given by a volume form, or by an area form on the real part $X({\mathbf{R}})$ for some real structure on $X$, with $X({\mathbf{R}})\neq \emptyset$.
Replacing $\Gamma$ by a finite index subgroup we may assume that $\Gamma\subset{\mathsf{Aut}}(X)^\circ$. Denote by $\mu$ the invariant measure. Let $G$ be the closure (for the euclidean topology) of $\Gamma$ in the Lie group ${\mathsf{Aut}}(X)^\circ$; then $G$ is a real Lie group preserving $\mu$.
Let $\alpha_X\colon X\to A_X$ be the Albanese morphism of $X$. There is a homomorphism of complex Lie groups $\rho\colon {\mathsf{Aut}}(X)^\circ\to {\mathsf{Aut}}(A_X)^\circ$ such that $\alpha_X\circ f = \rho(f)\circ \alpha_X$ for every $f$ in ${\mathsf{Aut}}(X)^\circ$.
Pick a very ample line bundle $L$ on $X$, denote by ${\mathbb{P}}^N({\mathbf{C}})$ the projective space ${\mathbb{P}}(H^0(X,L)^\vee)$, where $N+1=h^0(X, L)$, and by $\Psi_L\colon X\to {\mathbb{P}}^N({\mathbf{C}})$ the Kodaira-Iitaka embedding of $X$ given by $L$. By construction, $(\Psi_L)_*\mu$ is not supported by a hyperplane of ${\mathbb{P}}^N({\mathbf{C}})$.
[**[Step 1.]{}**]{}— Suppose $\rho(G)=1$. Then, since ${{\mathrm{Pic}}}^0(X)$ and $A_X$ are dual to each other, $G$ acts trivially on ${{\mathrm{Pic}}}^0(X)$ and $L$ is $G$-invariant, that is $g^*L=L$ for every $g\in G$. Thus there is a homomorphism $\beta\colon G\to {{\sf{PGL}}}_{N+1}({\mathbf{C}})$ such that $\Psi_L\circ g = \beta(g)\circ \Psi_L$ for every $g\in L$. If $G$ is not compact, there is a sequence of elements $g_n\in G$ going to infinity in ${{\sf{PGL}}}_{N+1}({\mathbf{C}})$: in the KAK decomposition $g_n=k_n a_n k_n’$, the diagonal part $a_n$ goes to $\infty$. But then, any probability measure on ${\mathbb{P}}^N({\mathbf{C}})$ which is invariant under all $g_n$ is supported in a proper projective subset of ${\mathbb{P}}^N({\mathbf{C}})$, and this contradicts our preliminary remark. Thus, $G$ is compact in that case.
[**[Step 2.]{}**]{}— Now, assume that $\rho(G)$ is infinite. Then identifying ${\mathsf{Aut}}(A_X)^\circ$ with $A_X$, $\rho({\mathsf{Aut}}(X)^\circ)$ is a complex algebraic subgroup of the torus $A_X$, of positive dimension since it contains $\rho(G)$. If the kernel of $\rho$ is finite, then ${\mathsf{Aut}}(X)^\circ$ is compact and virtually abelian; thus, we may assume $\dim(\ker(\rho))\geq 1$. In particular the fibers of $\alpha_X$ have positive dimension, thus $\dim(\alpha_X(X))\leq 1$ and $\alpha_X(X)$ is a curve, which is elliptic because it is invariant under the action of $\rho({\mathsf{Aut}}(X)^\circ)$. Then by the universal property of the Albanese morphism, we infer that $\alpha_X(X) = A_X$. In particular, $\alpha_X$ is a submersion because its critical values form a proper, $\rho({\mathsf{Aut}}(X)^\circ)$-invariant subset of $A_X$. Thus, $X$ is a ${\mathbb{P}}^1({\mathbf{C}})$-bundle over $A_X$ because the fibers of $\alpha_X$ are smooth, are invariant under the action of $\ker(\rho)$, and can not be elliptic since otherwise $X$ would be a torus. From [@Maruyama Thm 3] (see also [@Loray-Marin; @Potters] for instance), there are two cases:
1. either $X=A_X\times {\mathbb{P}}^1({\mathbf{C}})$, ${\mathsf{Aut}}(X)={\mathsf{Aut}}(A_X)\times {{\sf{PGL}}}_2({\mathbf{C}})$ and we deduce as in the first step that $G$ is a compact group;
2. or ${\mathsf{Aut}}(X)^\circ$ is Abelian.
In both cases stiffness follows, and we are done.
Pushing the analysis a little further, it can be shown that under the assumptions of the proposition $\Gamma$ is always relatively compact. Indeed in the last considered case, if $\Gamma$ is not bounded one deduces from [@Maruyama Thm 3] that there are elements with wandering dynamics, with all orbits in some Zariski open subset converging towards a section of $\alpha_X$. This contradicts the invariance of $\mu$.
Invariant algebraic curves II {#par:invariant_curves2}
-----------------------------
Let us start with an example.
\[eg:mobius\] Consider an elliptic curve $E={\mathbf{C}}/\Lambda$ and the abelian surface $A=E\times E$. The group ${{\sf{GL}}}_2({\mathbf{Z}})$ determines a non-elementary group of automorphisms of $E\times E$ of the form $(x,y) \mapsto (ax+by, cx+dy)$. The involution $\eta=-\operatorname{id}$ generates a central subgroup of ${{\sf{GL}}}_2({\mathbf{Z}})$, hence ${{\sf{PGL}}}_2({\mathbf{Z}})$ acts on the (singular) Kummer surface $A/\eta$. Each singularity gives rise to a smooth ${\mathbb{P}}^1({\mathbf{C}})$ in the minimal resolution $X$ of $A/\eta$, the group $\{ B\in {{\sf{PGL}}}_2({\mathbf{Z}})\; ; \; B\equiv \operatorname{id}\mod 2\}$ preserves each of these $16$ rational curves, and its action on these curves is given by the usual linear projective action on ${\mathbb{P}}^1({\mathbf{C}})$. In particular, it is proximal and strongly irreducible so it admits a unique, non-invariant, stationary measure.
The next result shows that under when $\nu$ is symmetric, every non-invariant stationary measure is similar to the previous example.
\[pro:stiffness\_curves\] Let $(X, \nu)$ be a random holomorphic dynamical system; assume furthermore that $\nu$ is symmetric.Let $\mu$ be an ergodic $\nu$-stationary measure giving positive mass to some proper Zariski closed subset of $X$. Then $\mu$ is supported on a $\Gamma_\nu$-invariant proper Zariski closed subset and
1. either $\mu$ is invariant;
2. or the Zariski closure of $\operatorname{Supp}(\mu)$ is a finite, disjoint union of smooth rational curves $C_i$, the stabilizer of $C_i$ in $\Gamma$ induces an unbounded proximal subgroup of ${\mathsf{Aut}}(C_i)$, and $\mu(C_i)^{-1}\mu{ \arrowvert_{C_i}}$ is the unique stationary measure of this group of Möbius transformations.
Moreover, if $(X,\nu)$ is non-elementary, the curves $C_i$ have negative self-intersection and can be contracted on cyclic quotient singularities.
Before giving the proof, let us briefly discuss the question of stiffness for Möbius actions on ${{\mathbb{P}^1}}({\mathbf{C}})$. Let $\nu$ be a symmetric measure on ${{\sf{PGL}}}_2({\mathbf{C}})$. Note that no moment assumption is assumed here. As already said, by Furstenberg’s theory if $\Gamma_{\nu}$ is strongly irreducible and unbounded, it admits a unique non-invariant stationary measure. Otherwise, any $\nu$-stationary measure is invariant. Indeed:
- either $\Gamma_{\nu}$ is relatively compact and stiffness follows from [@furstenberg_stiffness Thm. 3.5];
- or $\Gamma_{\nu}$ admits an invariant set made of two points, then it is virtually Abelian and stiffness follows from the Choquet-Deny theorem;
- of $\Gamma_{\nu}$ is conjugate to a subgroup of the affine group ${{\sf{Aff}}}({\mathbf{C}})$ with no fixed point.
In the latter case after conjugating to a subgroup of ${{\sf{Aff}}}({\mathbf{C}})$ we can write any $g\in \Gamma_{\nu}$ as $g(z) = a(g)z+b(g)$. If $a(g)\equiv 1$ then $\Gamma_{\nu}$ is Abelian and we are done. Otherwise $\Gamma_{\nu}$ is merely solvable and we apply the following result of Bougerol and Picard (see [@bougerol-picard Thm. 2.4]; for convenience we sketch the proof).
Let $\nu$ be a symmetric probability measure on ${{\sf{Aff}}}({\mathbf{C}})$. Assume that:
- $\nu({\left\{g\; ; \ a(g)\neq 1\right\}})>0$;
- there does not exist a point in ${\mathbf{C}}$ fixed by $\nu$-almost every $g$.
Then the only $\nu$-stationary probability on ${\mathbb{P}}^1({\mathbf{C}})$ is the point mass at $\infty$.
Assume by contradiction that there exists a stationary measure $\mu$ such that $\mu({\mathbf{C}}) = 1$ and $\mu(\{\infty\})=0$. Under our assumptions, $\mu$ cannot be atomic nor $\Gamma_\nu$-invariant.
Let $r_n$ be the right random walk associated to $\nu$ on ${{\sf{Aff}}}({\mathbf{C}})$. Let $\nu^\infty = \sum_{k=0}^\infty 2^{-{k+1}} \nu^{*k}$. A classical martingale convergence argument (see [@bougerol-lacroix Lem. II.2.1]) implies that there exists a measurable set $\Omega_0$ with $\nu^{\mathbf{N}}(\Omega_0) = 1$ such that if $\omega\in \Omega_0$ then
- $r_n(\omega)_* \mu$ converges toward a probability measure $\mu_\omega$ and $\mu = \int \mu_\omega d\nu^{\mathbf{N}}(\omega)$;
- for $ \nu^\infty$-almost every $ \gamma $, $r_n(\omega)_*\gamma_*\mu$ converges towards the same limit $\mu_\omega$.
Since $\mu = \int \mu_\omega d\nu^{\mathbf{N}}(\omega)$, we have $\mu_\omega({\mathbf{C}})= 1$ almost surely. Now, assume that for some $\omega\in \Omega_0$, $r_n(\omega)$ does not go to $\infty$ in ${{\sf{PGL}}}_2({\mathbf{C}})$. Extracting a convergent subsequence $r_{n_j}(\omega)\to r$, we infer that $\gamma_*\mu = \gamma'_*\mu = (r{^{-1}})_*\mu_\omega$ for $(\nu^\infty\times \nu^\infty)$-almost-every $(\gamma, \gamma')$; hence $\mu$ is $\Gamma_\nu$-invariant, a contradiction. Thus for almost every $\omega$, $r_n(\omega)$ goes to $\infty$ in ${{\sf{PGL}}}_2({\mathbf{C}})$.
Suppose first that $(a(r_n(\omega)), b(r_n(\omega)))$ is unbounded in ${\mathbf{C}}^2$ for a subset $\Omega_0'\subset \Omega_0$ of positive measure. Set $$\tilde r_{n} = {\frac{1}{\max({\left\verta(r_{n}(\omega))\right\vert}, {\left\vertb(r_{n}(\omega))\right\vert})}} r_{n}(\omega)$$ and extract a subsequence ${n_{j}}$ so that $\tilde r_{n_{j}} \to \ell_\omega$. If $\ell_\omega(z)\neq 0$ then $r_{n_{j}}(\omega)(z)\to \infty$. Since $r_{n_{j}}(\omega)_*\mu\to \mu_\omega$ and $\mu_\omega({\mathbf{C}})= 1$, we deduce that $\mu(\ell_\omega^{-1}\{0\}) =1$. This is a contradiction because $\mu$ is not atomic.
Thus, we may assume that $(a(r_n(\omega)), b(r_n(\omega)))$ is almost surely bounded. Since $r_n(\omega)$ goes to $\infty$ in ${{\sf{PGL}}}_2({\mathbf{C}})$, $a(r_n(\omega))$ must tend to 0 almost surely, in contradiction with the symmetry of $\nu$. This concludes the proof.
If $\mu$ has an atom then, by ergodicity, it is supported on a finite orbit and is invariant, so we now assume that $\mu$ is atomless. Then, by ergodicity, $\mu$ gives full mass to a $\Gamma_\nu$-invariant curve $D$. Let $C_1, \ldots , C_n$ be its irreducible components. Let $\Gamma'$ be the finite index subgroup of $\Gamma_\nu$ stabilizing each $C_i$ and $\nu'$ be the measure induced by $\nu$ on $\Gamma'$, which is symmetric. Then $\mu$ is $\nu'$-stationary, as well as $\mu{ \arrowvert_{C_i}}$ for each $C_i$.
If the genus of (the normalization of) $C_1$ is positive, then $\Gamma'{ \arrowvert_{C_1}}\subset {\mathsf{Aut}}(C_1)$ is virtually Abelian, hence $\mu{ \arrowvert_{C_1}}$ is $\Gamma'$-invariant. Since $\mu$ is ergodic, $\Gamma_\nu$ permutes transitively the $C_i$, and arguing as in Lemma \[lem:stiff\_finite\_extension\], we see that $\mu$ is $\nu$-invariant as well. Now, assume that the normalization $\hat{C_1}$ is isomorphic to ${\mathbb{P}}^1({\mathbf{C}})$. If $C_1$ is not smooth, or if it intersects another $\Gamma_\nu$-periodic curve, then the image of $\Gamma'$ in ${\mathsf{Aut}}(\hat{C_1})\simeq {{\sf{PGL}}}_2({\mathbf{C}})$ is not strongly irreducible, and the discussion preceding the proof shows that $\mu$ is $\Gamma'$ invariant. Again, this implies that $\mu$ is $\Gamma_\nu$-invariant. The same holds if $\Gamma'$ is a bounded subgroup of ${\mathsf{Aut}}(\hat{C_1})$. The only possibility left is that $C_1$ is smooth, disjoint from the other periodic curves, and $\Gamma'$ induces a strongly irreducible subgroup of ${\mathsf{Aut}}(C_1)$. Since $\Gamma_\nu$ permutes transitively the $C_i$, conjugating the dynamics of the groups $\Gamma'{ \arrowvert_{C_i}}$, the same property holds for each $C_i$.
If $\Gamma_\nu$ is non-elementary, Lemma \[lem:periodic\_curves\] shows that $C_i^2=-m$ for some $m>0$, which does not depend on $i$ because $\Gamma_\nu$ permutes the $C_i$ transitively. Then, the $C_i$ being disjoint, one can contract them simultaneously, each of the contractions leading to a quotient singularity $({\mathbf{C}}^2,0)/\langle \eta\rangle$ with $\eta(x,y)=(\alpha x, \alpha y)$ for some root of unity $\alpha$ of order $m$ (see [@BHPVDV §III.5]).
Non-elementary groups: real dynamics
------------------------------------
We now consider general non-elementary actions. As explained in the introduction, so far our results are restricted to subgroups of ${\mathsf{Aut}}(X)$ preserving a totally real surface $Y$. We further assume that there exists a $\Gamma_\nu$-invariant volume form on $Y$. This is automatically the case if $X$ is a K3 or an Enriques surface (see Lemma \[lem:volume\_Y\]). Note that, *a posteriori*, the results of §\[sec:parabolic\] and \[sec:rigidity\] suggest that measures supported on a totally real surface and invariant under a non-elementary subgroup of ${\mathsf{Aut}}(X)$ tend to be absolutely continuous, unless they are supported by a curve or a finite set.
We saw in Example \[eg:mobius\] that stiffness can fail in presence of invariant rational curves along which the dynamics is that of a proximal and strongly irreducible random product of Möbius transformations. The next theorem shows that for actions preserving a totally real surface, this obstruction to stiffness is the only one.
\[thm:stiffness\_real\] Let $(X, \nu)$ be a non-elementary random holomorphic dynamical system satisfying the moment condition . Assume that $Y\subset X$ is a $\Gamma_\nu$-invariant totally real 2-dimensional smooth submanifold such that the action of $\Gamma$ on $Y$ preserves a probability measure ${{\sf{vol}}}_Y$ equivalent to the Riemannian volume on $Y$. Then, every ergodic stationary measure $\mu$ on $Y$ is:
1. either almost surely invariant,
2. or supported on a $\Gamma_\nu$-invariant algebraic curve.
In particular if there is no $\Gamma_\nu$-invariant curve then $(Y, \nu)$ is stiff. Moreover, if the fiber entropy of $\mu$ is positive, then $\mu$ is the restriction of ${{\sf{vol}}}_Y$ to a subset of positive volume.
Recall from Lemma \[lem:periodic\_curves\] that $\Gamma_\nu$-invariant curves can be contracted. For the induced random dynamical system on the resulting singular surface, stiffness holds unconditionally. If furthermore $\nu$ is symmetric then the result can be made more precise by applying Proposition \[pro:stiffness\_curves\].
Let $\mu$ be an ergodic stationary measure supported on $Y$, giving no mass to curves. Since the action is volume preserving, its Lyapunov exponents satisfy $\lambda^{-}+\lambda^{+}=0$ (see Lemma \[lem:sum\_exponents\]). By the invariance principle (Theorem \[thm:invariance\_principle\]) if $\lambda^{-}\geq 0$, $\mu$ is almost surely invariant; so we may assume $\mu$ be hyperbolic. In this situation Theorem 3.4 in [@br] provides the following trichotomy:
(a) either $\mu$ has finite support, so it is invariant;
(b) or the distribution of Oseledets stable directions is non-random;
(c) or $\mu$ is almost surely invariant and absolutely continuous with respect to ${{\sf{vol}}}_Y$: even more, it is the restriction of ${{\sf{vol}}}_Y$ to a subset of positive volume.
Theorem \[thm:alternative\_stable\] shows that in case (b), $\mu$ is $\nu$-almost surely invariant, so the proof of the first assertion is complete, and the stiffness property follows when $\Gamma$ has no periodic curve.
Now, assume that the fiber entropy of $\mu$ is positive. By the Margulis-Ruelle inequality (Proposition \[pro:margulis-ruelle\]) $\mu$ is hyperbolic. If $\mu$ is supported on an algebraic curve, the proof of Corollary \[cor:nevanlinna\] leads to the following alternative: either $\mu$ is atomic or the Lyapunov exponent along that curve is negative; in the latter case $\mu$ is proximal along that curve and stable conditionals are points. In both cases the fiber entropy would vanish, in contradiction with our hypothesis, so $\mu$ is not supported on an algebraic curve. In the above trichotomy, (a) is now excluded, as well as (b) by Theorem \[thm:alternative\_stable\]. So, we are in case (c), as asserted.
We conclude this section with a variant of Theorem \[thm:stiffness\_real\] for singular volume forms; it may be applied to Blanc’s examples (see § \[par:Coble-Blanc\]).
Let $(X, \nu)$ be a non-elementary random holomorphic dynamical system satisfying the moment condition , and preserving a totally real 2-dimensional submanifold $Y\subset X$. Assume that there exists a meromorphic 2-form $\eta$ which is almost invariant under every $f\in\Gamma_\nu$ (i.e. $f^*\eta=\operatorname{Jac}_\eta(f)\eta$ with ${\left\vert\operatorname{Jac}_\eta(f)\right\vert} = 1$). Then every $\nu$-stationary measure supported on $Y$ is either supported on a $\Gamma_\nu$-invariant algebraic curve or almost surely invariant.
The proof is identical to that of Theorem \[thm:stiffness\_real\], except that we use Proposition \[pro:sum\_exponents\] instead of Lemma \[lem:sum\_exponents\]. Indeed by ergodicity if $\mu$ is not supported on an invariant algebraic curve it gives zero mass to the set of zeros and poles of $\Omega$ so, by Proposition \[pro:sum\_exponents\], we have $\lambda^++\lambda^- = 0$ and we can proceed as in the proof of Theorem \[thm:stiffness\_real\].
Subgroups with parabolic elements {#sec:parabolic}
=================================
We say that a subgroup $\Gamma\subset {\mathsf{Aut}}(X)$ is [**[twisting]{}**]{} if it contains a parabolic automorphism (this terminology will be justified below). In this section we investigate the dynamics of $(X, \nu)$ when $\Gamma_\nu$ is non-elementary and twisting. In particular under this assumption invariant measures can be classified (Theorem \[thm:classification\_invariant\]) and they are hyperbolic when not carried by some proper algebraic subset (Theorem \[thm:hyperbolic\]).
In most examples for which ${\mathsf{Aut}}(X)$ contains a non-elementary group, ${\mathsf{Aut}}(X)$ contains also a parabolic automorphism (see the examples in §§\[par:Wehler\]–\[par:Coble-Blanc\]). So, if we are interested in the study of ${\mathsf{Aut}}(X)$ itself or random dynamical systems for which $\Gamma_\nu$ is of finite index in ${\mathsf{Aut}}(X)$, the twisting assumption is quite natural. Also, if ${\mathsf{Aut}}(X)$ is both twisting and non-elementary, then there are thin subgroups $\Gamma\subset {\mathsf{Aut}}(X)$ with the same property: one can take two parabolics automorphisms $g$ and $h$ generating a non-elementary group, and set $\Gamma=\langle g^m, h^n\rangle$ for large integers $m$ and $n$.
Dynamics of parabolic automorphisms {#par:parabolic_automorphisms}
-----------------------------------
Parabolic automorphisms behave like “complex Dehn twists”. As shown in the next proposition, they preserve a unique genus $1$ fibration, acting by translations along the fibers, with a shearing property in the transversal direction. This twisting property justifies the vocabulary introduced for “twisting groups”. When $X$ is a rational surface, the invariant genus $1$ fibration comes from a Halphen pencil of ${\mathbb{P}}^2_{\mathbf{C}}$ (see [@Cantat:SLC]); this is the reason why parabolic automorphisms are also called [**[Halphen twists]{}**]{}.
Recall from §\[par:parabolic\_basics\] that if $h$ is a parabolic automorphism of a compact Kähler surface $X$, it preserves a unique genus $1$ fibration, given by the fibers of a rational map $\pi_h\colon X\to B$. In particular there is an automorphism $h_B$ of $B$ such that $$\pi\circ h=h_B\circ \pi.$$ Moreover, if $X$ is not a torus there exists an integer $m>0$ such that $h^m$ preserves every fiber of $\pi$ and acts by translation on every smooth fiber ([See Proposition \[pro:parabolic\_infinite\]]{}).
Assume $h$ is a parabolic automorphism with $h_B=\operatorname{id}_B$. The critical values of $\pi$ form a finite subset $\mathrm{Crit}(\pi)\subset B$; we denote its complement by $B^\circ$. Each fiber $X_w:=\pi^{-1}(w)$, $w\in B^\circ$, is a smooth curve of genus $1$, isomorphic to ${\mathbf{C}}/L(w)$ for some lattice $L(w)={\mathbf{Z}}\oplus {\mathbf{Z}}\tau(w)$; and $h$ induces a translation $h_w(z)=z+t(w)$, for some $t(w)\in {\mathbf{C}}/L(w)$. The points $w$ for which $h_w$ is periodic are characterized by the property that $t(w)\in {\mathbf{Q}}\oplus {\mathbf{Q}}\tau(w)$. If $$t(w)-(a+b\tau(w))\in {\mathbf{R}}\times (p+q\tau(w))$$ for some $(a,b)\in {\mathbf{Q}}^2$ and $(p,q)\in {\mathbf{Z}}^2$, the closure ${\mathbf{Z}}t(w)$ in ${\mathbf{C}}/L(w)$ is an Abelian Lie group of dimension $1$, isomorphic to ${\mathbf{Z}}/ k{\mathbf{Z}}\times {\mathbf{R}}/{\mathbf{Z}}$ for some $k>0$; then, the closure of each orbit of $h_w$ is a union of $k$ circles. Locally in $B^\circ$ this occurs along a countable union of analytic curves $(R_j)$. Otherwise, the orbits of $h_w$ are dense in $X_w$, and the unique $h_w$ invariant probability measure is the Haar measure on $X_w$.
Now, assume that $Y\subset X$ is a real analytic subset of $X$ of real codimension $2$, and that $h$ preserves $Y$; for instance $h$ may preserve a real structure on $X$, and $Y$ be a connected component of the real part $X({\mathbf{R}})$. Then, the projection $\pi(Y)$ is (locally) contained in some of these curves $R_j$. The smooth fibers $\pi_{\vert Y}^{-1}(w)$, for $w\in \pi(Y)$, are union of circles along which the orbits of $h_w$ are either dense (for most $w\in \pi(Y)$) or finite (for countably many $w\in \pi(Y)$).
\[lem:halphen\_flat\] Assume that $h_B$ is the identity. Let $U\subset B^\circ$ be a simply connected open subset. There is a countable union of analytic curves $R_j\subset U$, such that
1. $h$ acts by translation on each fiber $\pi^{-1}(w)$, $w \in U$;
2. for $w\in U\setminus\cup_j R_j$, the action of $h$ in the fiber $\pi^{-1}(w)$ is a totally irrational translation (it is uniquely ergodic, and its orbits are dense in $\pi^{-1}(w)$);
3. for $w$ in some countable subset of $U$, the orbits of $h_w$ are finite;
4. if the orbits of $h_w$ are neither dense nor finite, then $w\in \cup_j R_j$ and the closure of each orbit of $h_w$ is dense in a finite union of circles.
5. there is a finite subset $\mathrm{Flat}(h)\subset U$ such that for $x\notin \pi{^{-1}}{\left(
\mathrm{Flat}(h)\right)}$ $$\lim_{n\to \pm \infty} {{\left\VertD_xh^n\right\Vert}}\to +\infty$$ locally uniformly in $x$; more precisely for every $v\in T_xX \setminus{\left\{0\right\}}$, ${{\left\VertDh^{n}_x(v)\right\Vert}}$ grows linearly while $\frac{1}{n}\pi_*(D_xh^{n}(v))$ converges to $0$.
Moreover, if $h$ preserves a $2$-dimensional real analytic subset $Y\subset X$, then
1. $\pi$ induces on $Y$ a singular fibration whose generic leaves are union of circles, and there exists an integer $m>0$ such that $h^m$ preserves these circles and is uniquely ergodic along these circles except countably many of them.
This lemma is proven in [@cantat_groupes; @invariant]; Property (5) is the above mentioned twisting property of $h$.
Classification of invariant measures
------------------------------------
In this paragraph, we review the classification of invariant ergodic probability measures for twisting non-elementary groups of automorphisms; we refer to [@cantat_groupes; @invariant] for details and examples. If $X$ is a real K3 or Abelian surface and $X({\mathbf{R}})$ is non-empty there is a unique section of the canonical bundle of $X$ which, when restricted to $X({\mathbf{R}})$, induces a positive area form of total area $1$; we denote this area form by ${{\sf{vol}}}_{X({\mathbf{R}})}$. The associated probability measure is invariant under the action of ${\mathsf{Aut}}(X_{\mathbf{R}})$, the subgroup of ${\mathsf{Aut}}(X)$ preserving the real structure. Such a smooth invariant probability measure exists also on any totally real invariant surface (see [@invariant §5]):
\[lem:volume\_Y\] Let $X$ be an abelian surface, or a K3 surface, or an Enriques surface with universal cover $X'$. Let $Y\subset X$ be a (real) surface of class $C^1$. Let ${\mathsf{Aut}}(X;Y)$ be the subgroup of ${\mathsf{Aut}}(X)$ preserving $Y$. If $Y$ is totally real, $\Omega_X$ (resp. $\Omega_{X'}$) induces a smooth ${\mathsf{Aut}}(X;Y)$-invariant probability measure ${{\sf{vol}}}_Y$ on $Y$.
Note that there indeed exists examples of subgroups preserving a totally real surface which is not a real form of $X$ (see [@invariant §6]). The classification of invariant measures then reads as follows.
\[thm:classification\_invariant\] Let $X$ be a compact Kähler surface. Let $\Gamma$ be a twisting non-elementary subgroup of ${\mathsf{Aut}}(X)$. Let $\mu$ be a $\Gamma$-invariant ergodic probablity measure on $X$. Then, $\mu$ satisfies one and only one of the following properties.
1. $\mu$ is the average on a finite orbit of $\Gamma$;
2. $\mu$ is supported by a $\Gamma$-invariant curve $D\subset X$;
3. there is a $\Gamma$-invariant proper algebraic subset $Z$ of $X$, and a $\Gamma$-invariant, totally real, real analytic submanifold $Y$ of $X\setminus Z$ such that [*[(1)]{}*]{} $\mu(Z)=0$, [*[(2)]{}*]{} the support of $\mu$ is a union of finitely many connected components of $Y$, [*[(3)]{}*]{} $\mu$ is absolutely continuous with respect to the Lebesgue measure on $Y$, and [*[(4)]{}*]{} the density of $\mu$ with respect to any real analytic area form on $Y$ is real analytic;
4. there is a $\Gamma$-invariant proper algebraic subset $Z$ of $X$ such that [*[(1)]{}*]{} $\mu(Z)=0$, [*[(2)]{}*]{} the support of $\mu$ is equal to $X$, [*[(3)]{}*]{} $\mu$ is absolutely continuous with respect to the Lebesgue measure on $X$, and [*[(4)]{}*]{} the density of $\mu$ with respect to any real analytic volume form on $X$ is real analytic on $X\setminus Z$.
If $X$ is not a rational surface, then in case *(c)* (resp. *(d)*) we can further conclude that the invariant measure is locally proportional to ${{\sf{vol}}}_Y$ (resp. equal to ${{\sf{vol}}}_X$).
The reason why we say that $\mu$ is proportional to ${{\sf{vol}}}_Y$ (and not equal to it) in the last sentence is because $\mu$ may be equal to zero on some components of $Y\setminus Z$. This theorem is a combination of Theorem 1.1 and § 5.3 of [@invariant]. Let us also point out the following corollary of the proof.
\[cor:real\_invariant\] Let $\Gamma\leq {\mathsf{Aut}}(X)$ be as in Theorem \[thm:classification\_invariant\]. Assume furthermore that $X$ and $\Gamma$ are defined over ${\mathbf{R}}$ and $\Gamma$ does not preserve any proper Zariski closed subset of $X$. Then any $\Gamma$-invariant ergodic measure supported on $X({\mathbf{R}})$ is supported by a union $X({\mathbf{R}})'=\cup_j X({\mathbf{R}})_j$ of connected components $X({\mathbf{R}})_j$ of $X({\mathbf{R}})$, and is locally given by positive real analytic $2$-forms on $X({\mathbf{R}})'$. If $X$ is not rational, $\mu$ is equal to the restriction of ${{\sf{vol}}}_{X({\mathbf{R}})}$ to $X({\mathbf{R}})'$, up to some normalizing factor.
Using this classification we can now sharpen the conclusion of Theorem \[thm:stiffness\_real\] in the presence of parabolic automorphisms. When $Y= X({\mathbf{R}})$, the statement can also be combined with Corollary \[cor:real\_invariant\] to get an even more precise result.
\[cor:real\_stationary\] Let $(X, \nu)$ be a random holomorphic dynamical system on a compact Kähler surface, satisfying and such that $\Gamma_\nu$ is twisting and non-elementary. Assume that $Y\subset X$ is a $\Gamma_\nu$-invariant, smooth, totally real surface such that the action of $\Gamma_\nu$ on $Y$ preserves a probability measure ${{\sf{vol}}}_Y$ equivalent to the Riemannian volume on $Y$. Then up to a positive multiplicative factor, every ergodic stationary measure $\mu$ supported on $Y$ is :
- either the counting measure on a finite orbit;
- or supported on a $\Gamma_\nu$-invariant algebraic curve;
- or the restriction of ${{\sf{vol}}}_Y$ to a $\Gamma_\nu$-invariant open subset of $Y$ whose boundary is piecewise smooth.
In the last alternative, the boundary is obtained by intersecting an algebraic curve $D\subset X$ with $Y$; it may have a finite number of singularities.
We just have to repeat the proof of Theorem \[thm:stiffness\_real\], by incorporating the classification given in Theorem \[thm:classification\_invariant\]. Note that $Y$ is automatically real analytic in this case.
Hyperbolicity of the invariant volume {#subs:hyperbolicity}
-------------------------------------
It is a fundamental (and mostly open) problem in conservative dynamics to show the typicality of non-zero Lyapunov exponents on a set of positive Lebesgue measure. In deterministic dynamics, a recent breakthrough is the work of Berger and Turaev [@berger-turaev]. Adding some randomness makes it easier to obtain such a hyperbolicity result: see [@blumenthal-xue-young] for random perturbation of the standard map, and [@barrientos-malicet; @obata-poletti] for random conservative diffeomorphisms on (closed real) surfaces. The results of Barrientos and Malicet or Obata and Poletti [@barrientos-malicet; @obata-poletti] are perturbative in nature and do not give explicit examples. Here the high rigidity of complex algebraic automorphisms will be sufficient to show that twisting, non-elementary, random dynamical systems $(X,\nu)$ automatically satisfy some non-uniform hyperbolicity with respect to the volume.
\[thm:hyperbolic\] Let $X$ be a compact Kähler surface, and let $\Gamma$ be a non-elementary and twisting subgroup of ${\mathsf{Aut}}(X)$. Let $\mu$ be an ergodic $\Gamma$-invariant measure giving no mass to proper Zariski closed subsets of $X$ ([^7]). Then for every probability measure $\nu$ on ${\mathsf{Aut}}(X)$ satisfying the moment condition and such that $\Gamma_\nu=\Gamma$, $\mu$ is hyperbolic and of positive fiber entropy.
The same argument leads to a variant of this result when $\Gamma_\nu$ contains a Kummer example. Before stating our next result, let us recall the definition of classical Kummer examples (see [@cantat-dupont §1.3] for a more general definition). Let $A = {\mathbf{C}}^2/\Lambda$ be a complex torus and let $\eta$ be the involution given by $\eta(z_1, z_2) = (-z_1, -z_2)$, which has 16 fixed points. Then $A/\langle\eta\rangle$ is a surface with $16$ singular points, and resolving these singularities (by a single blow-up) yields a so-called **Kummer surface** $X$: a K3 surface with $16$ disjoint nodal curves. Let $f_A$ be a loxodromic automorphism of $A$ which is induced by a linear transformation of ${\mathbf{C}}^2$ preserving $\Lambda$; then $f_A$ commutes to $\eta$ and descends to an automorphism $f$ of $X$; such automorphisms will be referred to as **classical Kummer examples**. Of course, they preserve the canonical volume ${{\sf{vol}}}_X$. Notice that the Kummer surface $X$ also supports automorphisms which are not coming from automorphisms of $A$ (see [@Keum-Kondo] and [@Dolgachev-Keum] for instance).
\[thm:hyperbolic\_kummer\] Let $(X, \nu)$ be a non-elementary random dynamical system on a Kummer K3 surface satisfying and such that $\Gamma_\nu$ contains a classical Kummer example. Then any ergodic $\Gamma_\nu$-invariant measure giving no mass to proper Zariski closed subsets of $X$ is hyperbolic and has positive fiber entropy.
Note that in this statement we do not assume that $\Gamma_\nu$ contains a parabolic element. In Theorem \[thm:rigidity\_kummer\] below, we classify invariant probability measures which are supported on an invariant, real analytic, and totally real surface $Y$, when the group contains a Kummer example. Theorems \[thm:hyperbolic\] and \[thm:hyperbolic\_kummer\] will be proven in §\[subs:proof\_hyperbolic\].
Ledrappier’s invariance principle and invariant measures on ${\mathbb{P}}T{\mathcal{X}}$ {#subs:ledrappier_invariance_principle}
----------------------------------------------------------------------------------------
This paragraph contains preliminary results for the proof of Theorems \[thm:hyperbolic\] and \[thm:hyperbolic\_kummer\]. Our presentation is inspired by the exposition of [@barrientos-malicet]; in spirit, it is similar to [@obata-poletti], which relies on the “pinching and twisting” formalism of Avila and Viana (see [@viana] for an introduction[^8]). Most of this discussion is valid for a random dynamical system on an arbitrary complex surface (not necessarily compact), satisfying .
Let $\mu$ be an ergodic $\nu$-stationary measure. We introduce the projectivized tangent bundles ${\mathbb{P}}T {\mathcal{X}}_+ = \Omega \times {\mathbb{P}}TX$ and ${\mathbb{P}}T{\mathcal{X}}= \Sigma\times {\mathbb{P}}TX$. Note that the tangent bundles $TX$ and ${\mathbb{P}}TX$ admit measurable trivializations over a set of full measure. Consider any probability measure $\hat\mu$ on ${\mathbb{P}}TX$ that is stationary under the random dynamical system induced by $(X,\nu)$ on ${\mathbb{P}}TX$ and whose projection on $X$ coincides with $\mu$, i.e. $\pi_*\hat\mu= \mu $ where $\pi\colon {\mathbb{P}}TX\to X$ is the natural projection. Such measures always exist. Indeed the set of probability measures on ${\mathbb{P}}TX$ projecting to $\mu$ is compact and convex, and it is non-empty since it contains the measures $\int \delta_{[v(x)]} d\mu(x)$ for any measurable section $x\mapsto [v(x)]$ of ${\mathbb{P}}TX$; thus, the operator $\int {\mathbb{P}}(Df) \,d\nu(f)$ has a fixed point in that set. The stationarity of $\hat\mu$ is equivalent to the invariance of $\nu^{\mathbf{N}}\times \hat\mu$ under the transformation ${\widehat{F}_+}\colon \Omega\times {\mathbb{P}}TX\to \Omega\times {\mathbb{P}}TX$ defined by $${\widehat{F}_+}(\omega, x, [v])= (\sigma(\omega), f^1_\omega(x), {\mathbb{P}}(D_xf^1_\omega)[v])$$ for any non-zero tangent vector $v\in T_xX$. We denote by $\hat\mu_x$ the family of probability measures – on the fibers ${\mathbb{P}}T_x X$ of $\pi$ – given by the disintegration of $\hat\mu$ with respect to $\pi$; the conditional measures of $\nu^{\mathbf{N}}\times\hat\mu$ with respect to the projection ${\mathbb{P}}T{\mathcal{X}}\to X$ are given by $\hat\mu_{\omega,x} = \nu^{\mathbf{N}}\times \hat\mu_x$.
Even when $\mu$ is $\Gamma_\nu$-invariant, this construction only provides a stationary measure on ${\mathbb{P}}TX$. This is exactly what happens for twisting non-elementary subgroups: indeed we will show in §\[subs:proof\_hyperbolic\] that projectively invariant measures do not exist in this case.
The tangent action of our random dynamical system gives rise to a stationary product of matrices in ${{\sf{GL}}}(2, {\mathbf{C}})$. To see this, fix a measurable trivialization $P\colon TX\to X\times {\mathbf{C}}^2$, given by linear isomorphisms $P_x\colon T_xX\to {\mathbf{C}}^2$, which conjugates the action of $DF_+$ to that of a linear cocycle $A:{\mathcal{X}}_+ \times {\mathbf{C}}^2\to {\mathcal{X}}_+ \times {\mathbf{C}}^2$ over $({\mathcal{X}}_+, F_+, \nu^{\mathbf{N}}\times \mu)$. In this context, Ledrappier establishes in [@ledrappier_stationary] the following “invariance principle”.
\[thm:ledrappier\_invariance\_principle\] If $\lambda^{-}(\mu) = \lambda^{+}(\mu)$, then for any stationary measure $\hat \mu$ on ${\mathbb{P}}TX$ projecting to $\mu$, for $\mu$-almost every $x$ and $\nu$-almost every $f$, $${\mathbb{P}}(D_xf)_*\hat\mu_x = \hat\mu_{f(x)}$$
The second main ingredient that we need is a classification of such projectively invariant measures; this is where we follow [@barrientos-malicet]. To explain this result we need a bit of notation. Let $V$ and $W$ be two hermitian vector spaces of dimension 2. Endow the projective lines ${\mathbb{P}}(V)$ and ${\mathbb{P}}(W)$ with their respective Fubini-Study metrics. If $g\colon V\to W$ is a linear isomorphism, set $$\llbracket g \rrbracket= {{\left\Vert {\mathbb{P}}(g) \right\Vert}}_{C^1}$$ where ${\mathbb{P}}(g) \colon {\mathbb{P}}(V)\to {\mathbb{P}}(W)$ is the projective linear map induced by $g$ and ${{\left\Vert\cdot\right\Vert}}_{C^1}$ is the maximum of the norms of $D_z{\mathbb{P}}(g) \colon T_z{\mathbb{P}}(V)\to T_{{\mathbb{P}}gz)}{\mathbb{P}}(W)$ with respect to the Fubini-Study metrics. If we fix orthonormal bases of $V$ and $W$ we obtain isometric isomorphisms $\iota_v\colon V\to {\mathbf{C}}^2$ and $\iota_W\colon W\to {\mathbf{C}}^2$ to the standard hermitian space ${\mathbf{C}}^2$. If we denote by $\iota_W\circ g\circ \iota_V^{-1} = k_1ak_2$ the KAK decomposition of $\iota_W\circ g\circ \iota_V^{-1}$ in ${{\sf{PSL}}}(2, {\mathbf{C}})$, we get $$\llbracket g \rrbracket = {{\left\Verta\right\Vert}}^2 = {{\left\Vert\iota_W\circ g\circ \iota_V^{-1}\right\Vert}}^2$$ where ${{\left\Vert\cdot\right\Vert}}$ is the matrix norm in ${{\sf{PSL}}}_2({\mathbf{C}})={{\sf{SL}}}_2({\mathbf{C}})/\langle \pm \operatorname{id}\rangle$ associated to the Hermitian norm in ${\mathbf{C}}^2$. In particular:
- $\llbracket g \rrbracket=1$ if and only if ${\mathbb{P}}(g)$ is an isometry from ${\mathbb{P}}(V)$ to ${\mathbb{P}}(W)$;
- for a sequence $(g_n)$ of linear maps $V\to W$, $\llbracket g_n\rrbracket$ tends to $+\infty$ as $n$ goes to $+\infty$ if and only if ${\mathbb{P}}(\iota_W\circ g\circ \iota_V^{-1})$ tends to $\infty$ in ${{\sf{PSL}}}_2({\mathbf{C}})$.
We are now ready to state the classification of projectively invariant measures.
\[thm:classification\_proj\_invariant\] Let $(X, \nu)$ be a random dynamical system on a complex surface and let $\mu$ be an ergodic stationary measure. Let $\hat\mu$ be a stationary measure on ${\mathbb{P}}TX$ such that $\pi_*{\hat{\mu}}=\mu$ and $({\mathbb{P}}D_xf)_*\hat\mu_x = \hat\mu_{f(x)}$ for $\mu$-almost every $x$ and $\nu$-almost every $f$. Then, exactly one of the following two properties is satisfied:
1. For $(\nu^{\mathbf{N}}\times \mu)$-almost every ${\mathscr{x}}= (\omega, x)$, the sequence $\llbracket D_xf^n_\omega\rrbracket$ is unbounded and then:
- either there exists a measurable $\Gamma_\nu$-invariant family of lines $E(x)\subset T_xX$ such that $\hat\mu_x=\delta_{[E(x)]}$ for $\mu$-almost every $x$;
- or there exists a measurable $\Gamma_\nu$-invariant family of pairs of lines $E_1(x), E_2(x) \subset T_xX$ and positive numbers $ \lambda_1, \lambda_2$ with $\lambda_1+\lambda_2 =1$ such that $\hat\mu_x=\lambda_1\delta_{[E_1(x)]} + \lambda_2\delta_{[E_2(x)]}$ for $\mu$-almost every $x$;
2. The projectivized tangent action of $\Gamma_\nu$ is reducible to a compact group, that is there exists a measurable trivialization of the tangent bundle $(P_x: T_xX \to {\mathbf{C}}^2)_{x\in X}$, such that for every $f\in \Gamma_\nu$ and every $x$, ${\mathbb{P}}{\left(P_{f(x)}\circ D_xf\circ P_x{^{-1}}\right)}$ belongs to the unitary group $\mathsf{PU}_2({\mathbf{C}})$.
In assertion (1.b), the pair is not naturally ordered, i.e. there is no natural distinction of $E_1$ and $E_2$, the random dynamical system may a priori permute these lines. The proof is obtained by adapting the arguments of [@barrientos-malicet] to the complex case. Details are given in Appendix \[app:barrientos\_malicet\].
Proofs of Theorems \[thm:hyperbolic\] and \[thm:hyperbolic\_kummer\] {#subs:proof_hyperbolic}
--------------------------------------------------------------------
### Proof of Theorem \[thm:hyperbolic\]
Let us prove Theorem \[thm:hyperbolic\]. By Theorem \[thm:classification\_invariant\], $\mu$ is either equivalent to the Lebesgue measure on $X$, or to the $2$-dimensional Lebesgue measure on some components of an invariant totally real surface $Y\subset X$. Let us assume, by contradiction, that $\mu$ is not hyperbolic. Hence its Lyapunov exponents vanish, and by Theorem \[thm:ledrappier\_invariance\_principle\] and Theorem \[thm:classification\_proj\_invariant\], there is a measurable set $X'\subset X$ with $\mu(X') = 1$ such that one of the following properties is satisfied along $X'$: \[alternative\_abc\]
- there is a measurable $\Gamma_\nu$-invariant line field $E(x)$;
- there exists a measurable $\Gamma_\nu$-invariant splitting $E(x)\oplus E'(x)=T_xX$ of the tangent bundle; here, the invariance should be taken in the following weak sense: an element $f$ of $\Gamma_\nu$ maps $E(x)$ to $E(f(x))$ or $E'(f(x))$;
- there exists a measurable trivialization $P_x\colon T_xX\to {\mathbf{C}}^2$ such that in the corresponding coordinates the projectivized differential ${\mathbb{P}}(Df_x)$, $f\in \Gamma_\nu$, takes its values in $\mathsf {PU}_2({\mathbf{C}})$.
Fix a small ${\varepsilon}>0$. By Lusin’s theorem, there is a compact set $K_{\varepsilon}$ with $\mu(K_{\varepsilon})>1-{\varepsilon}$ such that the data $x\mapsto E(x)$, resp. $x\mapsto (E(x), E'(x))$ or $x\mapsto P_x$ in the respective cases (a,b,c) above are continuous on $K_{\varepsilon}$. In particular, in case (c), the norms of $P_x$ and $P_x^{-1}$ are bounded by some uniform constant $C({\varepsilon})$ on $K_{\varepsilon}$; hence, if $g$ is an element of $\Gamma_\nu$ and $x$ and $g(x)$ belong to $K_{\varepsilon}$, $\llbracket Dg_x \rrbracket$ is bounded by $C({\varepsilon})^2$.
Fix a pair of Halphen twists $g$ and $h\in \Gamma_\nu$ with distinct invariant fibrations $\pi_g\colon X\to B_g$ and $\pi_h\colon X\to B_h$ respectively (see Lemma \[lem:pairs\_of\_twists\]). In a first stage assume that $X$ is not a torus: then by Proposition \[pro:parabolic\_infinite\] we may assume that $g$ and $h$ preserve every fiber of their respective invariant fibrations (see Section \[par:parabolic\_automorphisms\]).
First assume that $\mu$ is absolutely continuous with respect to the Lebesgue measure on $X$, with a positive real analytic density on the complement of some invariant, proper, Zariski closed subset. Since the invariant fibration is holomorphic, the disintegration $\mu_b$ of $\mu$ is absolutely continuous on almost every fiber $\pi_h^{-1}(b)$. Thus, there exists a fiber $\pi_h{^{-1}}(b)$ such that (1) the Haar measure of $K_\varepsilon \cap \pi_h{^{-1}}(b)$ is positive, (2) $b \notin \mathrm{Flat}(h))$ and (3) the dynamics of $h$ in $\pi_h{^{-1}}(b)$ is uniquely ergodic (see Lemma \[lem:halphen\_flat\]). Then we can pick $x\in \pi_h{^{-1}}(b)$ such that $(h^{k}(x))_{k\geq 0}$ visits $K_{\varepsilon}$ infinitely many times. The fifth assertion of Lemma \[lem:halphen\_flat\] rules out case (c) because the twisting property implies that the projectivized derivative tends to infinity along those times $n$ for which $h^n(x)\in K_{\varepsilon}$. Case (b) is also excluded: under the dynamics of $h$, tangent vectors projectively converge to the tangent space of the fibers, so the only possible invariant subspace is $\ker(D\pi_h)$. Thus we are in case (a) and moreover $E(x) = \ker D_x\pi_h$ for $\mu$-almost every $x$. But then, using $g$ instead of $h$ and the fact that $\mu$ does not charge the algebraic curve along which the fibrations $\pi_g$ and $\pi_h$ are tangent, we get a contradiction. This shows that alternative (a) does not hold either, and this contradiction proves that $\mu$ is hyperbolic.
If $\mu$ is supported by a $2$-dimensional real analytic subset $Y\subset X$, the same proof applies, except that we disintegrate $\mu$ along the singular foliation of $Y$ by circles induced by $\pi_h$ and use the fact that a generic leaf is a circle along which $h$ is uniquely ergodic (see Assertion (6) in Lemma \[lem:halphen\_flat\]). If $X$ is a torus, then its tangent bundle is trivial and the differential of an automorphism is constant. The differential of a Halphen twist $h$ is of the form $$\begin{pmatrix} 1 & \alpha \\ 0 & 1 \end{pmatrix} \text{ with } \alpha\neq 0$$ in an appropriate basis. Thus we are in case (a) with $E(x) = \ker D_x\pi_h$ for $\mu$-almost every $x$, and using another twist $g$ transverse to $h$ we get a contradiction as before.
Since $\mu$ is invariant then the invariant measure ${\mathscr{m}}$ on ${\mathcal{X}}$ is equal to $\nu^{\mathbf{Z}}\times \mu$. Then in both cases $\mu\ll {{\sf{vol}}}_X$ and $\mu\ll {{\sf{vol}}}_Y$ the absolute continuity of the foliation by local Pesin unstable manifolds implies that the unstable conditionals of ${\mathscr{m}}$ cannot be atomic (this is identical to the classical argument showing that an absolutely continuous invariant measure has the SRB property, see[@ledrappier_sinai]). Thus positivity of the entropy follows from Corollary \[cor:rokhlin\_zero\_entropy\], and the proof of Theorem \[thm:hyperbolic\] is complete.
### Proof of Theorem \[thm:hyperbolic\_kummer\]
The proof is similar to that of Theorem \[thm:hyperbolic\] so we only sketch it. Assume by contradiction that $\mu$ is not hyperbolic; since $X$ is a K3 surface, Corollary \[cor:sum\_exponentsK3\] shows that the sum of the Lyapunov exponents of $\mu$ vanish; thus, each of them is equal to $0$, and one of the alternatives of Theorem \[thm:classification\_proj\_invariant\] holds, referred to as (a), (b), (c) on page . By assumption, $\Gamma_\nu$ contains a map $f$ which is uniformly hyperbolic in some Zariski open set $\Omega$, which is thus of full $\mu$-measure. We denote by $x\mapsto E^u_f(x)\oplus E^s_f(x)$ the associated splitting of $TX{ \arrowvert_{\Omega}}$. Since $f$ is uniformly expanding/contracting on $E^{u/s}_f$, alternative (c) is not possible. If alternative (a) holds, then $E(x)$ being $f$-invariant on a set of full measure, it must coincide with $E^u_f$ or $E^s_f$, say with $E^u_f$. By continuity any $g\in \Gamma_\nu$ preserves $E^u_f$ pointwise.
On the other hand, $E^u_f$ is everywhere tangent to an $f$-invariant (singular) holomorphic foliation $\mathcal F^u$, induced by a linear foliation on the torus $A$ given by the Kummer structure. Every leaf of that foliation, except a finite number of them, is biholomorphically equivalent to ${\mathbf{C}}$, and the Ahlfors-Nevanlinna currents of these entire curves are all equal to the unique closed positive current $T^+_f$ that satisfies ${{\mathbf{M}}}(T^+_f)=1$ and $f^*T^+_f=\lambda(f)T^+_f$ with $\lambda(f)>1$.
Now, pick any element $g$ of $\Gamma_\nu$. Since $g$ preserves the line field $E^u_f$, $g$ preserves $\mathcal F^u$ as well, hence also the ray ${\mathbf{R}}_+[T^+_f]$, contradicting the non-elementary assumption.
Finally, if alternative (b) holds, any $g\in \Gamma_\nu$ preserves $ \{E^u_f(x), E^s_f(x)\}$ on a set of full measure so by the continuity of the hyperbolic splitting it must either preserve or swap these directions. Passing to an index 2 subgroup we can assume that both directions are preserved, and we are back to case (a).
Measure rigidity {#sec:rigidity}
================
In view of the results of Sections \[sec:stiffness\] and \[sec:parabolic\], it is natural to wonder whether a classification of invariant measures is possible without assuming the existence of parabolic elements in $\Gamma$. The results in this section belong to a thread of measure rigidity results starting with Rudolph’s theorem [@rudolph] on Furstenberg’s $\times 2\times 3$ conjecture. If $\mu$ is a probability measure on $X$, we denote by ${\mathsf{Aut}}_\mu(X)$ the group of automorphisms of $X$ preserving $\mu$.
\[thm:rigidity\] Let $f$ be an automorphism of a compact Kähler surface $X$, preserving a totally real and real analytic surface $Y\subset X$. Let $\mu$ be an ergodic $f$-invariant measure on $Y$ with positive entropy. Then
1. either $\mu$ is absolutely continuous with respect to the Lebesgue measure on $Y$;
2. or the group ${\mathsf{Aut}}_\mu(X)$ is virtually cyclic.
If in addition the Lyapunov exponents of $f$ with respect to $\mu$ satisfy $\lambda^s(f,\mu)+ \lambda^u(f,\mu) \neq 0$, then case (a) does not occur, so ${\mathsf{Aut}}_\mu(X)$ is virtually cyclic.
This result and its proof below may be viewed as a counterpart, in our setting, to Theorems 5.1 and 5.3 of [@br], where again the possibility of invariant line fields is ruled out by using the complex structure. As before the typical case to keep in mind is when $X$ is a projective surface defined over ${\mathbf{R}}$ and $Y=X({\mathbf{R}})$. Observe that by ergodicity, if $f$ preserves a smooth volume ${{\sf{vol}}}_Y$, then in case (a) $\mu$ will be the restriction of ${{\sf{vol}}}_Y$ to a Borel set of positive volume invariant under $f$ and $g$.
Since it admits a measure of positive entropy, $f$ is a loxodromic transformation. By the Ruelle-Margulis inequality $\mu$ is hyperbolic with respect to $f$ and it does not charge any point, nor any piecewise smooth curve: indeed, the entropy of a homeomorphism of the circle or the interval is equal to zero. We first assume that $X$ is projective, the case of non-projective surfaces will be studied at the end of the proof.
For $\mu$-almost every $x\in X$, the stable manifold $W^s(f,x)$ is an entire curve in $X$ which is either transcendental or contained in a periodic rational curve (see [@Cantat:Milnor Thm. 6.2]). Since $f$ has only finitely many invariant algebraic curves (see [@Cantat:Milnor Prop. 4.1]) and $\mu$ gives no mass to curves, $W^s(f,x)$ is $\mu$-almost surely transcendental; then, the only Ahlfors-Nevanlinna current associated to $W^s(f,x)$ is $T^+_f$; similarly, the Ahlfors-Nevanlinna currents of the unstable manifolds give $T^-_f$. (This is the analogue in deterministic dynamics of Theorem \[thm:nevanlinna\].)
Fix $g\in {\mathsf{Aut}}_\mu(X)$ and set $\Gamma :=\langle f,g \rangle$. Our first goal is to prove the following:
[**[Alternative]{}**]{}: [*either $\Gamma^*$ is virtually cyclic and preserves $\{ {\mathbb{P}}[T^+_f], {\mathbb{P}}[T^-_f]\}\subset {\partial}{\mathbb{H}}_X$; or $\mu$ is absolutely continuous with respect to the Lebesgue measure on $Y$.*]{}
Let $Y'\subset Y$ be the union of the connected components of $Y$ of positive $\mu$-measure. The measure $\mu$ does not charge any analytic subset of $Y$ of dimension $\leq 1$; thus, by analytic continuation, any $h\in \Gamma$ preserves $Y'$. So, without loss of generality we can replace $Y$ by $Y'$.
We divide the argument into several cases according to the existence or non-existence of certain $\Gamma$-invariant line fields. In the first two cases we will conclude that $\Gamma$ is elementary. In the third case, $\mu$ will be absolutely continuous with respect to the Lebesgue measure on $Y$. Then by the Pesin formula its Lyapunov exponents satisfy $\lambda^u(f, \mu) = -\lambda^s(f, \mu)= h_\mu(f)$ so when $\lambda^u (f, \mu)+ \lambda^s(f, \mu)\neq 0$, Case 3 is actually impossible.
[**Case 1.–** ]{} [*There exists a $\Gamma$-invariant measurable line field.*]{} Specifically, we mean a measurable field of complex lines $x\mapsto [E(x)] \in {\mathbb{P}}(T_xX)$, defined on a set of full $\mu$-measure, such that $D_xh(E(x)) = E(h(x))$ for every $h\in \Gamma$ and almost every $x\in X$; since $\mu$ is supported on the totally real surface $Y$, the field of real lines $[E(x)\cap T_xY]\in {\mathbb{P}}T_xY$ is also invariant, and determines $[E(x)]$. Now, $\mu$ being ergodic and hyperbolic for $f$, the Oseledets theorem shows that either $E(x) = E^s_f(x)$ $\mu$-almost everywhere or $E(x) = E^u_f(x)$ $\mu$-almost everywhere. Changing $f$ into $f^{-1}$ if necessary, we may assume that $E(x) = E^s_f(x)$.
Consider the automorphism $h = g{^{-1}}f g \in {\mathsf{Aut}}_\mu(X)$. Since $h$ is conjugate to $f$, $\mu$ is also ergodic and hyperbolic for $h$. Thus, either $E^s_h(x) = E^s_f(x)$ for $\mu$-almost every $x$ or $E^u_h(x) = E^s_f(x)$ for $\mu$-almost every $x$.
\[lem:equal\_stable\_manifolds\] If there is a measurable set $A$ of positive measure along which $E^s_h(x) = E^s_f(x)$ (resp. $E^u_h(x) = E^s_f(x)$), then $W^s(f,x) = W^s(h, x)$ for almost every $x$ in $A$ (resp. $W^u(h,x) = W^s(f, x)$).
Let us postpone the proof of this lemma and conclude the argument. Suppose first that $E^s_h(x) = E^s_f(x)$ on a subset $A$ with $\mu(A)>0$. Then $T^+_f = T^+_h$ because for $\mu$-almost every $x$, the unique Ahlfors-Nevanlinna current associated to the (complex) stable manifold $W^s(f,x)$ (resp. $W^s(h, x)$) is $T^+_f$ (resp. $T_h^+$). Since $T^+_{h} = {{\mathbf{M}}}(g^*T^+_f){^{-1}}g^* T^+_f$, we see that $g$, and therefore $\Gamma$ itself, preserve the line ${\mathbf{R}}[T^+_f]\subset H^{1,1}(X)$. Since $[T^+_f]^2=0$, $\Gamma$ fixes a point ${\mathbb{P}}[T^+_f]$ of the boundary ${\partial}{\mathbb{H}}_X$, so it is elementary. Since in addition $\Gamma$ contains a loxodromic element, Theorem 3.2 of [@Cantat:Milnor] shows that $\Gamma^*$ is virtually cyclic.
Now, suppose that $E^u_h(x) = E^s_f(x)$ on $A$. Then, $T^-_h=T^+_f$ and the group generated by $f$ and $h$ is elementary. Since it contains a loxodromic element again we deduce from [@Cantat:Milnor Thm 3.2] that $\langle f^*, h^*\rangle$ is virtually cyclic, hence it also fixes ${\mathbb{P}}[T^-_f] \in{\partial}{\mathbb{H}}_X$. This implies that $g$, hence $\Gamma$, preserves the pair of boundary points $\{ {\mathbb{P}}[T^+_f], {\mathbb{P}}[T^-_f]\}\subset {\partial}{\mathbb{H}}_X$. Thus, in both cases $\Gamma^*$ is virtually cyclic and preserves $\{ {\mathbb{P}}[T^+_f], {\mathbb{P}}[T^-_f]\}\subset {\partial}{\mathbb{H}}_X$.
The argument is similar to that of Theorem \[thm:alternative\_stable\], in a simplified setting, so we only sketch it.
For $\mu$-almost every $x$, $W^s(f,x)$ and $W^s(h, x)$ are tangent at $x$. Assume by contradiction that there exists a measurable subset $A'$ of $A$ of positive measure such that $W^s(f,x) \neq W^s(h, x)$ for every $x\in A'$. Then for small ${\varepsilon}>0$ there exists two positive constants $r=r({\varepsilon})$ and $c=c({\varepsilon})$, an integer $k\geq 2$ and a measurable subset $\mathcal G_{\varepsilon}\subset A'$ such that $\mu(\mathcal G_{\varepsilon})>0$ and
- $W_{\mathrm{loc}}^s(f,x)$ and $W_{\mathrm{loc}}^s(h, x)$ are well defined and of size $r$ for every $x\in \mathcal G_{\varepsilon}$,
- $W_{\mathrm{loc}}^s(f,x)$ and $W_{\mathrm{loc}}^s(h, x)$ depend continuously on $x$ on $\mathcal G_{\varepsilon}\subset X$,
- $\operatorname{inter}_x(W^s_{\mathrm{loc}}(f,x), W^u_{\mathrm{loc}}(f,x)) = k $ for every $x\in \mathcal G_{\varepsilon}$,
- and $\operatorname{osc}_{(k, x, r)} (W^s_{r}(f,x), W^s_{r}(h,x))\geq c$ for every $x\in \mathcal G_{\varepsilon}$.
Indeed, to get the first and second properties, one intersects $A'$ with a large Pesin set $\mathcal R_{\varepsilon}$. On $A'\cap \mathcal R_{\varepsilon}$ the multiplicity of intersection $x\mapsto \operatorname{inter}_x(W^s_{\mathrm{loc}}(f,x), W^u_{\mathrm{loc}}(f,x))$ is semi-continuous, so we can find $k\geq 2$ and a subset $\mathcal R'_{\varepsilon}\subset (A'\cap \mathcal R_{\varepsilon})$ of positive measure such that $$\operatorname{inter}_x(W^s_{\mathrm{loc}}(f,x), W^u_{\mathrm{loc}}(f,x)) = k$$ for every $x\in \mathcal R'_{\varepsilon}$. Thus, the $k$-th osculation number is well defined, and the last property holds on a subset $\mathcal G_{\varepsilon}\subset \mathcal R'_{\varepsilon}$ of positive measure if $c$ is small.
Let $\eta^s$ be a Pesin partition subordinate to the local stable manifolds of $f$. Since $h_\mu(f)>0$ the conditional measures $\mu(\cdot\vert \eta^s)$ are non-atomic. Thus there exists $x\in {\mathcal G}_{\varepsilon}$ such that $x$ is an accumulation point of $\operatorname{Supp}\big(\mu(\cdot \vert \eta^s(x)){ \arrowvert_{{\mathcal G}_{\varepsilon}\cap \eta^s(x)}}\big)$. Fix a neighborhood $N$ of $x$ such that $W^s_r(f,x)\cap W^s_r(h,x)\cap N = {\left\{x\right\}}$, and then pick a sequence $(x_j)$ of points in ${\mathcal G}_{\varepsilon}\cap \eta^s(x)\cap N$ converging to $x$. The local stable manifolds $W^s_r(h,x_j)$ form a sequence of disks of size $r$ at $x_j$, each of them tangent to $W^s_r(f,x)$ (at $x_j$), and all of them disjoint from $W^s_r(h,x)$ (because $x_j$ does not belong to $W^s_r(h,x)$). This contradicts Corollary \[cor:bls\_quantitative\] and the proof is complete.
[**Case 2.–** ]{} [*There is a pair of distinct measurable line fields ${\left\{E_1(x), E_2(x)\right\}}$ invariant under $\Gamma$*]{}. Again by the Oseledets theorem applied to $f$, necessarily $\{E_1(x), E_2(x)\} = \{E^s_f(x), E^u_f(x)\}$. For $\mu$-almost every $x$, $g( \{E^s_f(x), E^u_f(x)\}) = \{E^s_f(g(x)), E^u_f(g(x))\}$. As before, consider the automorphism $h = g{^{-1}}f g \in {\mathsf{Aut}}_\mu(X)$. Since $h$ is conjugate to $f$, it is hyperbolic and ergodic with respect to $\mu$, and $\{E^s_f(x), E^u_f(x)\} = \{E^s_h(x), E^u_h(x)\}$ for almost every $x$. Replacing $h$ by $h{^{-1}}$ if necessary, there exists a set $A$ of positive measure for which $E^s_h(x) = E^s_f(x)$, and we conclude as in Case 1.
[**Case 3.–** ]{} [*There is no $\Gamma$-invariant line field or pair of line fields.*]{} In other words, Cases 1 or 2 are now excluded. This part of the argument is identical to the proof of [@br Thm 5.1.a].
First, we claim that there exists $g_1, g_2 \in \Gamma$ and a subset $A$ of positive measure such that $D_xg_1(E^s_f(x)) \notin \{E^s_f(g_1(x)), E^u_f(g_1(x))\}$ and $D_xg_2(E^u_f(x))\notin \{E^s_f(g_2(x)), E^u_f(g_2(x)\}$ for every $x$ in $A$. Indeed since we are not in Case 2 (possibly switching $E^u_f$ and $E^s_f$) there exists $g_1 \in \Gamma $ and a set $A$ of positive measure such that for $x\in A$, $D_xg_1(E^s_f(x)) \nsubset E^s_f(g_1(x))\cup E^u_f(g_1(x))$. Since we are not in Case 1, there exists $g\in \Gamma $ and a set $B$ of positive measure such that for $x\in B$, $D_xg(E^u_f(x))\neq E^u_f(g(x))$. If $D_xg(E^s_f(x))\in \{E^s_f(g(x)), E^u_f(g(x))\}$ on a subset $B'$ of $B$ of positive measure, then choose $k>0$ and $\ell>0$ such that $\mu(f^\ell(A)\cap B')>0$ and $\mu(f^k(g(f^\ell(A)))\cap A)>0$ and define $g_2 = g_1f^k gf^\ell$; otherwise, set $g_2= gf^\ell$ with $\ell$ such that $\mu(f^\ell(A)\cap B)>0$. Then change $A$ into $A=A\cap f^{-\ell}(B')$ (resp. $A\cap f^{-\ell}(B)$).
Denote by $\Delta$ the simplex ${\left\{(a, b,c, d)\in ({\mathbf{R}}_+^*)^4\; ; \; a+b+c+d =1\right\}}$. For $\alpha=(a, b,c, d)$ in $\Delta$, let $\nu_\alpha$ be the probability measure $\nu_\alpha = a \delta_f + b\delta_{f{^{-1}}} + c \delta_g + d \delta_{g{^{-1}}}$. Then $\mu$ is $\nu_\alpha$-stationary and since $\mu$ is $f$-ergodic and $\nu_\alpha({\left\{f\right\}})>0$, it is also ergodic as a $\nu_\alpha$-stationary measure (see [@benoist-quint_book §2.1.3]). Since we are not in Cases 1 or 2 and $\mu$ is hyperbolic for $f$, Theorems \[thm:ledrappier\_invariance\_principle\] and \[thm:classification\_proj\_invariant\] imply that the Lyapunov exponents of $\mu$, viewed as a $\nu_\alpha$-stationary measure, satisfy $\lambda_\alpha^-(\mu)< \lambda_\alpha^+(\mu)$.
There exists a choice of $\alpha\in \Delta$ such that $\mu$ is a hyperbolic $\nu_\alpha$-stationary measure, i.e. $\lambda_\alpha^-(\mu)< 0 < \lambda_\alpha^+(\mu)$
This is automatic when $f$ and $g$ are volume preserving because $\lambda_\alpha^-(\mu)=- \lambda_\alpha^+(\mu)$ in that case. For completeness, let us copy the proof given in [@br §13.2.4]. The assumptions of Case 3 and the strict inequality $\lambda^-(\mu)< \lambda^+(\mu)$ imply that $$\alpha \in \Delta \mapsto ( \lambda^-_\alpha(\mu), \lambda^+_\alpha(\mu))\in {\mathbf{R}}^2$$ is continuous (see [@br Prop. 13.7] or [@viana Chap. 9]). Since $\lambda^-_\alpha(\mu)< \lambda^+_\alpha (\mu)$ for every $\alpha\in \Delta$, one of $\lambda^-_\alpha$ and $\lambda^+_\alpha$ is non zero. Furthermore, $\mu$ being invariant, the involution $(a,b,c,d) \mapsto (b,a,d,c)$ interchanges the Lyapunov exponents. It follows that $P = {\left\{\alpha\in \Delta, \lambda^+_\alpha>0\right\}}$ and $N = {\left\{\alpha\in \Delta, \lambda^-_\alpha<0\right\}}$ are non-empty open subsets of $\Delta$ such that $P\cup N = \Delta$. The connectedness of $\Delta$ implies $P\cap N\neq \emptyset$, as was to be shown.
Fix $\alpha\in \Delta$ such that $\mu$ is hyperbolic as a $\nu_\alpha$-stationary measure. The assumptions of Case 3 imply that the stable directions depend on the itinerary so the main result of [@br] shows that $\mu$ is fiberwise SRB (on the surface $Y$), that is, the unstable conditionals of the measures $\mu_{\mathscr{x}}$ (here $\mu_{\mathscr{x}}= \mu$) are given by the Lebesgue measure (in some natural affine parametrizations of the unstable manifolds by the real line ${\mathbf{R}}$). Since $\mu$ is invariant, we can revert the stable and unstable directions by applying the argument to $F{^{-1}}$, and we conclude that the stable conditionals are given by the Lebesgue measure as well. The absolute continuity property of the stable and unstable laminations then implies that $\mu$ is absolutely continuous with respect to the Lebesgue measure on $Y$.
[**[Conclusion.–]{}**]{} Let us assume that $\mu$ is not absolutely continuous with respect to the Lebesgue measure on $Y$. The above alternative holds for all subgroups $\Gamma= \langle f, g \rangle$, where $g\in {\mathsf{Aut}}_\mu(X)$ is arbitrary. Therefore, if $X$ is projective, we deduce that ${\mathsf{Aut}}_\mu(X)^*$ preserves $\{ {\mathbb{P}}[T^+_f], {\mathbb{P}}[T^-_f]\}\subset {\partial}{\mathbb{H}}_X$, which implies that ${\mathsf{Aut}}_\mu(X)^*$ is virtually cyclic. By Lemma \[lem:virtually\_cyclic\_non\_projective\], ${\mathsf{Aut}}_\mu(X)^*$ is also virtually cyclic when $X$ is not projective. So the only remaining issue is to prove that ${\mathsf{Aut}}_\mu(X)$ itself is virtually cyclic. If this is not the case, then ${\mathsf{Aut}}(X)^\circ$ is infinite, $X$ must be a torus ${\mathbf{C}}^2/\Lambda$ (see Proposition \[prop:discrete\]), and ${\mathsf{Aut}}_\mu(X)\cap {\mathsf{Aut}}(X)^\circ$ is a normal subgroup of ${\mathsf{Aut}}_\mu(X)$ containing infinitely many translations. This group is a closed subgroup of the compact Lie group ${\mathsf{Aut}}(X)^\circ={\mathbf{C}}^2/\Lambda$; thus, the connected component of the identity in ${\mathsf{Aut}}_\mu(X)\cap {\mathsf{Aut}}(X)^\circ$ is a (real) torus $H\subset {\mathbf{C}}^2/\Lambda$ of positive dimension. This torus is invariant under the action of $f$ by conjugacy. Since $X={\mathbf{C}}^2/\Lambda$, $f$ is a complex linear Anosov diffeomorphism of $X$, and it follows that $\dim_{\mathbf{R}}(H)\geq 2$. Being $H$-invariant, $\mu$ is then absolutely continuous with respect to the Lebesgue measure of $Y$; this contradiction completes the proof.
Theorem \[thm:rigidity\] can be extended to the case of singular analytic subsets $Y$, after minor adjustments of the proof, because $\mu$ can not charge its singular locus.
It is natural to expect that the positive entropy assumption in Theorem \[thm:rigidity\] could be replaced by a much weaker assumption, namely, “[*$\mu$ gives no mass to proper Zariski closed subsets*]{}”. In full generality this seems to exceed the scope of techniques of this paper, however we are able to deal with a special case.
\[thm:rigidity\_kummer\] Let $f$ be a Kummer example on a compact Kähler surface $X$. Let $\mu$ be an atomless, $f$-invariant, and ergodic probability measure that is supported on a totally real, real analytic surface $Y\subset X$. If $g\in {\mathsf{Aut}}(X)$ preserves $\mu$, then:
1. either $\mu$ is absolutely continuous with respect to ${{\sf{vol}}}_Y$;
2. or $\langle f, g \rangle$ is virtually isomorphic to ${\mathbf{Z}}$.
Thus, as in the case of subgroups containing parabolic transformations, the stiffness Theorem \[thm:stiffness\_real\] takes a particularly strong form when $\operatorname{Supp}(\nu)$ contains a Kummer example.
Let us start with a preliminary remark. Assume that $\mu(C)>0$ for some irreducible curve $C\subset X$; since $\mu$ does not charge any point the support of $\mu_{\vert C}$ is Zariski dense in $C$, and $C$ is an $f$-periodic curve. But $f$ being a Kummer example, such a curve is a rational curve $C\simeq {\mathbb{P}}^1({\mathbf{C}})$ (obtained by blowing-up a periodic point of a linear Anosov map on a torus), on which $f$ has a north-south dynamics; thus, all $f$-invariant probability measures on $C$ are atomic, and we get a contradiction. This means that the assumption “[*$\mu$ has no atom*]{}” is equivalent to the assumption “[*$\mu$ gives no mass to proper Zariski closed subsets of $X$*]{}”.
Now, we can follow step by step the proof of Theorem \[thm:rigidity\], only insisting on the points requiring modification. Since $\mu$ does not charge any curve, we can contract all $f$-periodic curves, and lift $(f,\mu)$ to $({\tilde{f}}, {\tilde{\mu}})$, where ${\tilde{f}}$ is a linear Anosov diffeomorphism of some compact torus ${\mathbf{C}}^2/\Lambda$ and ${\tilde{\mu}}$ is an ${\tilde{f}}$-invariant probability measure (see [@Cantat-Zeghib] for details on Kummer examples). We deduce that $\tilde{\mu}$ is hyperbolic for ${\tilde{f}}$ and then, coming back to $X$, that $\mu$ is hyperbolic for $f$. Case 3 of the proof of Theorem \[thm:rigidity\] only requires hyperbolicity of $\mu$ so it carries over in this case without modification. In Cases 1 and 2 we have to show that if $\Gamma = \langle f,g \rangle$ preserves a measurable line field or a pair of measurable line fields then $\Gamma^*$ is elementary. In either case we consider $ h = g f g{^{-1}}$ and up to possibly replacing $E^u_f$ by $E^s_f$ and $h$ by $h{^{-1}}$, we have that $E^s_f(x) = E^s_h(x)$ on a set of positive measure. But now $f$ and $h$ are Kummer examples so their respective stable foliations $\mathcal F^{s} _f$ and $\mathcal F^{s} _h $ are (singular) holomorphic foliations. From the previous reasoning $\mathcal F^{s} _f$ and $\mathcal F^{s} _h $ are tangent on a set of positive $\mu$-measure, so, since $\mu$ gives no mass to subvarieties we infer that $\mathcal F^{s} _f = \mathcal F^{s} _h$ and we conclude exactly as in Theorem \[thm:hyperbolic\_kummer\].
Unlike most results in this paper, Theorem \[thm:rigidity\] can be extended to a rigidity theorem for polynomial automorphisms of ${\mathbf{R}}^2$ with essentially the same proof.
\[thm:rigidity\_henon\] Let $f$ be a polynomial automorphism of ${\mathbf{R}}^2$. Let $\mu$ be an ergodic $f$-invariant measure with positive entropy supported on ${\mathbf{R}}^2$. If $g\in{\mathsf{Aut}}({\mathbf{R}}^2)$ satisfies $g_*\mu = \mu$, then:
1. either $f$ and $g$ are conservative and $\mu$ is the restriction of $\operatorname{Leb}_{{\mathbf{R}}^2}$ to a Borel set of positive measure invariant under $f$ and $g$;
2. or the group generated by $f$ and $g$ is solvable and virtually cyclic; in particular, there exists $(n,m)\in {\mathbf{Z}}^2\setminus{{\left\{(0,0)\right\}}}$ such that $f^n = g^m$.
We briefly explain the modifications required to adapt the proof of Theorem \[thm:rigidity\], and leave the details to the reader. We freely use standard facts from the dynamics of automorphisms of ${\mathbf{C}}^2$. Let $f$ and $g$ be as in the statement of the theorem, and set $\Gamma=\langle f, g \rangle$.
Since its entropy is positive, $f$ is of Hénon type in the sense of [@lamy]: this means that $f$ is conjugate to a composition of generalized Hénon maps, as in [@friedland-milnor], Theorem 2.6. Thus, the support of $\mu$ is a compact subset of ${\mathbf{C}}^2$, because the basins of attraction of the line at infinity for $f$ and $f^{-1}$ cover the complement of a compact set; moreover, as in Theorem \[thm:rigidity\], $\mu$ cannot charge any proper Zariski closed subset.
Let $\gamma$ be an arbitrary element of $\Gamma$; then $h:=\gamma{^{-1}}f \gamma$ is also of Hénon type. We run through Cases 1, 2 and 3 as in the proof of Theorem \[thm:rigidity\]. Case 3 is treated exactly in the same way as above and implies that $\mu$ is absolutely continuous. This in turn implies that the Jacobian of $f$, a constant $\operatorname{{Jac}}(f)\in {\mathbf{C}}^*$ since $f\in{\mathsf{Aut}}({\mathbf{C}}^2)$, is equal to $\pm1$; and since $\mu$ is ergodic for $f$, it must be the restriction of $\operatorname{Leb}_{{\mathbf{R}}^2}$ to some $\Gamma$-invariant subset. In Cases 1 and 2, arguing as before and keeping the same notation, we arrive at $W^s(h,x) = W^s(f,x)$ or $W^u(f,x)$ on a set of positive measure. For a Hénon type automorphism of ${\mathbf{C}}^2$, the closure of any stable manifold is equal to the forward Julia set $J^+$, and $J^+$ carries a unique positive closed current $T^+$ of mass $1$ relative to the Fubini Study form in ${\mathbb{P}}^2({\mathbf{C}})$ (see [@sibony]). So we infer that $T^+_h = T^+_f$ or $T^+_h =T^-_f$; as a consequence, the Green functions of $f$ and $h$ satisfy $G^+_h=G^+_f$ or $G^+_h=G^-_f$, respectively.
Automorphisms of ${\mathbf{C}}^2$ act on the Bass-Serre tree of ${\mathsf{Aut}}({\mathbf{C}}^2)$, each automorphism $u\in {\mathsf{Aut}}({\mathbf{C}}^2)$ giving rise to an isometry $u_*$ of the tree. Hénon type automorphisms act as loxodromic isometries; the axis of such an isometry $u_*$ will be denoted $\operatorname{Geo}(u_*)$: it is the unique $u_*$-invariant geodesic, and $u_*$ acts as a translation along its axis. Theorem 5.4 of [@lamy] shows that $G^+_h = G^+_f$ implies $\operatorname{Geo}(h_*)=\operatorname{Geo}(f_*)$; changing $f$ into $f{^{-1}}$, $G^+_h =G^-_f$ gives $\operatorname{Geo}(h_*)=\operatorname{Geo}(f_*{^{-1}})=\operatorname{Geo}(f_*)$ because the axis of $f_*$ and $f_*{^{-1}}$ coincide. Since $\gamma_*$ maps $\operatorname{Geo}(f_*)$ onto $\operatorname{Geo}(h_*)$, we deduce that $\Gamma$ preserves the axis of $f$; as a consequence, all elements $u$ of $\Gamma$ of Hénon type satisfy $\operatorname{Geo}(u_*)=\operatorname{Geo}(f_*)$. From [@lamy Prop. 4.10], we conclude that $\Gamma$ is solvable and virtually cyclic.
With the techniques developed in [@Cantat:BHPS], the same result applies to the dynamics of ${\mathsf{Out}}({\mathbb{F}}_2)$ acting on the real part of the character surfaces of the once punctured torus.
General compact complex surfaces {#par:appendix_non_kahler}
================================
Let $M$ be a compact complex surface; here we do not assume $M$ to be Kähler. Let $\Gamma$ be a subgroup of ${\mathsf{Aut}}(M)$. We say that $\Gamma$ is [**[cohomologically non-elementary]{}**]{} if its image $\Gamma^*$ in ${{\sf{GL}}}(H^*(M;{\mathbf{Z}}))$ contains a non-abelian free subgroup, and that $\Gamma$ is [**[dynamically non-elementary]{}**]{} if it contains a non-abelian free group $\Gamma_0$ such that the topological entropy of every $f\in \Gamma_0\setminus \{ \operatorname{id}\}$ is positive. According to Theorem 3.2 of [@Cantat:Milnor], when $M$ is a compact Kähler surface then $\Gamma$ is non-elementary (in the sense of Section \[par:hyp\_X\]) if and only if it is cohomologically non-elementary, if and only if it is dynamically non-elementary. In the general case one implication remains true:
\[lem:non-elementary\_free\_groups\] If $\Gamma\subset {\mathsf{Aut}}(M)$ is cohomologically non-elementary, then $\Gamma$ is dynamically non-elementary.
We split the proof in two steps, the first concerning groups of matrices, the second topological entropy.
[**[Step 1.-]{}**]{} [*$\Gamma^*$ contains a free subgroup $\Gamma_1^*$, all of whose non-trivial elements have spectral radius larger than $1$*]{}.
The proof uses basic ideas involved in Tits’s alternative, but in the simple case of subgroups of ${{\sf{GL}}}_n({\mathbf{Z}})$. Let $N$ be the rank of $H^*_{t.f.}(M;{\mathbf{Z}})$, where $t.\!f\!.$ stands for “torsion free”. Fix a basis of this free ${\mathbf{Z}}$-module. Then $\Gamma^*$ determines a subgroup of ${{\sf{GL}}}_N({\mathbf{Z}})$. Our assumption implies that the derived subgroup of $\Gamma^*$ contains a non-abelian free group $\Gamma_0^*$ of rank $2$.
If all (complex) eigenvalues of all elements of $\Gamma_0^*$ have modulus $\leq 1$, then by Kronecker’s lemma all of them are roots of unity. This implies that $\Gamma_0^*$ contains a finite index nilpotent subgroup (see Proposition 2.2 and Corollary 2.4 of [@Benoist:Grp_Disc]), contradicting the existence of a non-abelian free subgroup. Thus, there is an element $f^*$ in $\Gamma_0^*$ with a complex eigenvalue of modulus $\alpha>1$. Let $m$ be the number of eigenvalues of $f^*$ of modulus $\alpha$, counted with multiplicities. Consider the linear representation of $\Gamma_0^*$ on $\bigwedge^m H^*(M;{\mathbf{C}})$; the action of $f^*$ on this space has a unique dominant eigenvalue, of modulus $\alpha^m$; the corresponding eigenline determines an attracting fixed point for $f^*$ in the projective space ${\mathbb{P}}(\bigwedge^m H^*(M;{\mathbf{C}}))$; the action of $f^*$ on this topological space is proximal.
Let $$\{0\}=W_0 \subset W_1\subset \cdots \subset W_k \subset W_{k+1}=\bigwedge^m H^*(M;{\mathbf{C}})$$ be a Jordan-Hölder sequence for the representation of $\Gamma^*$: the subspaces $W_i$ are invariant, and the induced representation of $\Gamma^*$ on $W_{i+1}/W_i$ is irreducible for all $0\leq i\leq k$. Let $V$ be the quotient space $W_{i+1}/W_{i}$ in which the eigenvalue of $f^*$ of modulus $\alpha^m$ appears. Since $\Gamma_0^*$ is contained in the derived subgroup of $\Gamma$, the linear transformation of $V$ induced by $f^*$ has determinant $1$; thus, $\dim(V)\geq 2$. Now, we can apply Lemma 3.9 of [@Benoist:Grp_Disc] to (a finite index, Zariski connected subgroup of) $\Gamma^*_{0}{ \arrowvert_{V}}$: changing $f$ is necessary, both $f^*{ \arrowvert_{V}}$ and $(f^{-1})^*{ \arrowvert_{ V}}$ are proximal, and there is an element $g^*$ in $\Gamma^*$ that maps the attracting fixed points $a^+_f$ and $a^-_f\in {\mathbb{P}}(V)$ of $f^*{ \arrowvert_{ V}}$ and $(f^*{ \arrowvert_{V}}){^{-1}}$ to two distinct points (i.e. $\{ a^+_f, a^-_f\}\cap \{ g(a^+_f), g(a^-_f)\} =\emptyset$) ; then, by the ping-pong lemma, large powers of $f^*$ and $g^*\circ f^*\circ (g^*)^{-1}$ generate a non-abelian free group $\Gamma_1^*\subset \Gamma^*$ such that each element $h^*\in \Gamma_1^*\setminus \{\operatorname{id}\}$ has an attracting fixed point in ${\mathbb{P}}(V)$. This implies that every element of $\Gamma_1^*\setminus \{\operatorname{id}\}$ has an eigenvalue of modulus $>1$ in $H^*(M;{\mathbf{C}})$.
[**[Step 2.-]{}**]{} Since $\Gamma_1^*$ is free, there is a free subgroup $\Gamma_1\subset \Gamma$ such that the homomorphism $\Gamma_1\mapsto \Gamma_1^*$ is an isomorphism. By Yomdin’s theorem [@yomdin], all elements of $\Gamma_1\setminus \{\operatorname{id}\}$ have positive entropy.
Let $M$ be a compact complex surface. If ${\mathsf{Aut}}(M)$ contains a cohomologically non-elementary subgroup, then $M$ is projective.
Indeed it was shown in [@Cantat:CRAS] that every compact complex surface possessing an automorphism of positive entropy is Kähler. Then the result follows from Theorem \[thm:X-is-projective1\].
Strong laminarity of Ahlfors currents {#par:appendix_ahlfors}
=====================================
In this appendix, we sketch the proof of Lemma \[lem:ahlfors\_current\], explaining how to adapt arguments of [@bls; @Dujardin:Laminar2003; @isect], written for $X={\mathbb{P}}^2({\mathbf{C}})$, to our context.
Let $(\Delta_n)$ be a sequence of unions of disks, as in the definition of injective Ahlfors currents, such that ${\frac{1}{{{\mathbf{M}}}(\Delta_n)}}{\left\{\Delta_n\right\}}$ converges to $T$. Since $X$ is projective we can choose a finite family of meromorphic fibrations $\varpi_i: X\dasharrow {{\mathbb{P}^1}}$ such that
- the general fibers of $\varpi_i$ are smooth curves of genus $\geq 2$;
- for every $x\in X$, there are at least two of the fibrations $\varpi_{i}$, denoted for simplicity by $\varpi_{1}$ and $\varpi_{2}$, which are well defined in some neighborhood $U_x$ of $x$ ($x$ is not a base point of the corresponding pencils), satisfy $(d\varpi_{1}\wedge d\varpi_{2})(x)\neq 0$ (the fibrations are transverse), and for which the fibers $\varpi_{k}^{-1}(\varpi_{k}(x))$ containing $x$ are smooth.
If we blow-up the base points of $\varpi_{k}$, $k=1,2$, we obtain a new surface $X'\to X$ on which each $\varpi_{k}$ lifts to a regular fibration $\varpi_{k}'$; the open neighborhood $U_x$ is isomorphic to its preimage in $X'$ so, when working on $U_x$, we can do as if the two fibrations $\varpi_{k}$ were local submersions with smooth fibers of genus $\geq 2$.
To construct $T_r$, we follow the proof of [@isect Proposition 4.4] (see also [@Dujardin:Laminar2003 Proposition 3.4]). The construction will work as follows: we fix a sequence $(r_j)$ converging to zero, and for every $j$ we extract from ${\frac{1}{{{\mathbf{M}}}(\Delta_n)}}{\left\{\Delta_n\right\}}$ a current $T_{n,r_j}$ made of disks of size $\approx r_j$ which are obtained from $\Delta_n$ by only keeping graphs of size $r_j$ over one of the projections $\varpi_i$.
By a covering argument, it is enough to work locally near a point $x$, with two projections $\varpi_{1}$ and $\varpi_{2}$ as above. Let $S\subset {\mathbf{C}}$ be the unit square $\{x+{\mathsf{i}} y\; ; \; 0\leq x \leq 1, \; 0\leq y\leq 1\}\simeq [0,1]^2$. To simplify the exposition, we may assume that $$\quad \varpi_{k}(U_x)= S\subset {\mathbf{C}}\subset {\mathbb{P}}^1({\mathbf{C}})\quad ({\text{for }} \; k=1,2).$$ Set $r_j=2^{-j}$ and consider the subdivision ${\mathcal{Q}}_j$ of $S\simeq [0,1]^2$ into $4^j$ squares $Q$ of size $r_j$. A connected component of $\Delta_n\cap \varpi_{k}^{-1}(Q)$, for such a small square $Q$, is called a graph (with respect to $\varpi_{k}$) if it lifts to a local section of the fibration $\varpi_{k}'\colon X'\to {\mathbb{P}}^1({\mathbf{C}})$ above $Q$. Then, we fix $j$, intersect $\Delta_n$ with $\varpi_{k}^{-1}(Q)$, and keep only the components of $\varpi_{k}{^{-1}}(Q\cap \Delta_n)$, $Q\in {\mathcal{Q}}_j$ which are graphs with respect to $\varpi_{k}$. Such a family of graphs is normal because the fibers of $\varpi'_{k}$ have genus $\geq 2$ (compare to Lemma 3.5 of [@Dujardin:Laminar2003]).
This being done, we can copy the proof of [@isect Proposition 4.4]. Letting $n$ go to $+\infty$ and extracting a converging subsequence, we obtain a uniformly laminar current $T_{{\mathcal{Q}}_j, k}\leq T$. Away from the base points of $\varpi_{k}$, $T_{{\mathcal{Q}}_j, k}$ is made of disks of size $\asymp r_j$ which are limits of disks contained in the $\Delta_n$. Combining the two currents $T_{{\mathcal{Q}}_j, k}$, we get a current $T_{r_j}\leq T$ which is uniformly laminar in every cube $\varpi_{1}{^{-1}}(Q)\cap \varpi_{2}{^{-1}}(Q')$, $Q,Q'\in{\mathcal{Q}}_j$, and such that $$\label{eq:strong_laminar}
\langle T-T_{r_j}, \varpi_{1}^* {\kappa_{{\mathbb{P}}^1}}+ \varpi_{1}^* {\kappa_{{\mathbb{P}}^1}}\rangle
\leq \langle T-T_{{\mathcal{Q}}_j, 1}, \varpi_{1}^* {\kappa_{{\mathbb{P}}^1}}\rangle + \langle T-T_{{\mathcal{Q}}_j, 2}, \varpi_{2}^* {\kappa_{{\mathbb{P}}^1}}\rangle,$$ where ${\kappa_{{\mathbb{P}}^1}}$ is the Fubini-Study form. By definition, $T$ will be strongly approximable if locally ${{\mathbf{M}}}(T - T_{r_j})\leq O(r_j^2)$. Using the fact that $\varpi_{1}^* {\kappa_{{\mathbb{P}}^1}}+ \varpi_{1}^* {\kappa_{{\mathbb{P}}^1}}\geq C \kappa_0$ and the Inequality , it will be enough to show that $\langle T - T_{{\mathcal{Q}}_j, k}, \varpi_{k}^*{\kappa_{{\mathbb{P}}^1}}\rangle = O(r_j^2)$ for $k=1,2$. This itself reduces to counting (with multiplicity) the number of “good components” of $\Delta_n$ for the projections $\varpi_{k}: \Delta_n \to {\mathcal{Q}}_j$ that is, the components above the squares $Q$ of $Q_j$ that are kept in the above contruction of $T_{{\mathcal{Q}}_j, k}$ (the graphs relative to $\varpi_{k}$).
The counting argument is identical to [@bls §7], except that we apply the Ahlfors theory of covering surfaces to a union of disks, not just one. For notational ease, set $\varpi = \varpi_{k}$, $r=r_j$ and ${\mathcal{Q}}= {\mathcal{Q}}_j$; ${\mathcal{Q}}$ is a subdivision of $S\simeq [0,1]^2$ by squares of size $2^{-j}$. We decompose ${\mathcal{Q}}$ as a union of four non-overlapping subdivisions ${\mathcal{Q}}^{\ell}$, $\ell = 1, 2, 3, 4$; by this we mean that for each $\ell$, the squares $Q\in {\mathcal{Q}}^\ell$ have disjoint closures $\overline Q$. Fix such an $\ell$ and let $q = \# {\mathcal{Q}}^\ell= 4^{j-1}$. Applying Ahlfors’ theorem to each of the disks constituting $\Delta_n$ and summing over these disks, we deduce that the number of good components $N({\mathcal{Q}}^{\ell})$ satisfies $$\label{eq:good_components}
N({\mathcal{Q}}^{\ell}) \geq (q-2) \operatorname{area}_{{{\mathbb{P}^1}}} (\Delta_n) - h \operatorname{length}_{{{\mathbb{P}^1}}}({\partial}\Delta_n),$$ where $\operatorname{area}_{{{\mathbb{P}^1}}}$ (resp. $\operatorname{length}_{{{\mathbb{P}^1}}}$) is the area of the projection $\varpi(\Delta_n)$ (resp. length of $\varpi({\partial}\Delta_n)$), counted with multiplicity, and $h$ is a constant that depends only on the geometry of ${\mathcal{Q}}^\ell$. Dividing by $\operatorname{area}_{{{\mathbb{P}^1}}} (\Delta_n)$, using $\operatorname{length}_{{{\mathbb{P}^1}}}({\partial}\Delta_n) = o(\operatorname{area}_{{{\mathbb{P}^1}}} (\Delta_n))$, which is guaranteed by Ahlfors’ construction, and letting $n$ go to $+\infty$, we obtain $$\langle T_{{\mathcal{Q}}}{ \arrowvert_{{\mathcal{Q}}^{\ell}}}, \varpi^*{\kappa_{{\mathbb{P}}^1}}\rangle \geq (q-2) r^2 =
\operatorname{area}_{{{\mathbb{P}^1}}}{\left( \bigcup\nolimits_{S\in {\mathcal{Q}}^\ell} S \right)} -2 r^2.$$ Finally, summing from $\ell=1$ to $4$, we see that, relative to $\varpi^*\kappa_{{{\mathbb{P}^1}}}$, the mass lost by discarding the bad components of size $r$ in $T$ is of order $O(r^2)$: this is precisely the required estimate.
Let us now justify the geometric intersection statement, following step by step the proof of [@isect Thm. 4.2]: let $S$ be a current with continuous normalized potential on $X$; we have to show that $S\wedge T_r$ increases to $S\wedge T$ as $r$ decreases to 0. Again the result is local so we work near $x$, use the projections $\varpi_1$ and $\varpi_2$, and keep notation as above. Given squares $Q, Q'\in {\mathcal{Q}}$ and a real number $\lambda<1$, we denote by $\lambda Q$ the homothetic of $Q$ of factor $\lambda$ with respect to its center, and by $C(Q, Q')$ the cube $\varpi_{1}{^{-1}}(Q)\cap \varpi_{2}{^{-1}}(Q')$. Fix ${\varepsilon}>0$. We want to show that for $r\leq r({\varepsilon})$, the mass of $(T-T_r)\wedge S$ is smaller than ${\varepsilon}$. The first observation is that there exists $\lambda({\varepsilon})\in (0,1)$, independent of $r$, such that translating ${\mathcal{Q}}$ if necessary, the mass of $T\wedge S$ concentrated in ${\bigcup_{Q, Q'} C(Q, Q')\setminus C(\lambda Q, \lambda Q')}$ is smaller than $ {\varepsilon}/2$ (see [@isect Lem. 4.5]). Fix such a $\lambda$. It only remains to estimate the mass of $(T-T_r)\wedge S$ in $\bigcup_{Q, Q'} C(\lambda Q, \lambda Q')$. In such a cube $C(\lambda Q, \lambda Q')$ the argument presented in [@isect pp. 123-124], based on an integration by parts, gives the estimate $$\int_{ C(\lambda Q, \lambda Q')} (T-T_r)\wedge S \leq
C(\lambda) \omega(u_S, r) \frac{1}{r^2} {{\mathbf{M}}}{\left((T-T_r){ \arrowvert_{ C( Q, Q')}}\right)},$$ where $\omega(u_S, r)$ is the modulus of continuity of the potential $u_S$ of $S$. To conclude, we sum over all squares $Q, Q'$ and use the estimate $M(T-T_r) = O(r^2)$ to get that $${{\mathbf{M}}}{\left((T-T_r){ \arrowvert_{ \bigcup_{Q, Q'} C(\lambda Q, \lambda Q')}}\right)}\leq C\omega(u_S, r).$$ This is smaller than ${\varepsilon}/2$ if $r\leq r({\varepsilon})$.
Proof of Theorem \[thm:classification\_proj\_invariant\] {#app:barrientos_malicet}
========================================================
Let us consider a random dynamical system $(X, \nu)$ and $\mu$ an ergodic stationary measure, as in Theorem \[thm:classification\_proj\_invariant\]. We keep the notation from §\[subs:ledrappier\_invariance\_principle\].
We say that a sequence of real numbers $(u_n)_{n\geq 0}$ [**[almost converges towards $+\infty$]{}**]{} if for every $K\in {\mathbf{R}}$, the set $L_K={\left\{ n\in {\mathbf{N}}\; ; \; u_n\leq K \right\}}$ has an asymptotic lower density $$\underline\operatorname{dens}(L_K):=\liminf_{n\to +\infty}\left( \frac{\sharp (L_K\cap [0,n])}{n+1}\right)$$ which is equal to $0$: $\underline \operatorname{dens}(L_K)=0$ for all $K$.
\[lem:alternative\_bdd\] The set of points ${\mathscr{x}}=(\omega,x)$ in ${\mathscr{X}}_+$ such that $\llbracket D_xf^n_\omega\rrbracket$ almost converges towards $+\infty$ on ${\mathbb{P}}(T_xM)$ is $F_+$-invariant. In particular, by ergodicity,
1. either $\llbracket D_xf^n_\omega\rrbracket$ almost converges towards $+\infty$ for $(\nu^{\mathbf{N}}\times \mu)$-almost every $(\omega, x)$;
2. or, for $(\nu^{\mathbf{N}}\times \mu)$-almost every $(\omega, x)$, there is a sequence $(n_i)$ with positive lower density along which $\llbracket D_xf^{n_i}_\omega \rrbracket$ is bounded.
The proof is straightforward. We are now ready for the proof of Theorem \[thm:classification\_proj\_invariant\]. Let us emphasize one delicate issue. In Conclusion (1) of the theorem, it is important that the directions $E$ (resp. $E_1$ and $E_2$) only depend on $x\in X$ (and not on ${\mathscr{x}}= (x,\omega)\in {\mathcal{X}}_+$). Likewise in Conclusion (2), the trivialization $P_x$ should depend only on $x$. This justifies the inclusion of a detailed proof of Theorem \[thm:classification\_proj\_invariant\], since in the slightly different setting of [@barrientos-malicet], the authors did not have to check this point carefully.
We fix a measurable trivialization $P\colon TX\to X\times {\mathbf{C}}^2$, given by linear isometries $P_x\colon T_xX\to {\mathbf{C}}^2$, where $T_xX$ is endowed with the hermitian form $(\kappa_0)_x$, and ${\mathbf{C}}^2$ with its standard hermitian form. This trivialization conjugates the action of $DF_+$ to that of a cocycle $A\colon {\mathcal{X}}_+ \times {\mathbf{C}}^2\to {\mathcal{X}}_+ \times {\mathbf{C}}^2$ over $F_+$. We denote by $A_{\mathscr{x}}\colon {\left\{{\mathscr{x}}\right\}}\times {\mathbf{C}}^2\to {\left\{F_+({\mathscr{x}})\right\}}\times {\mathbf{C}}^2$ the induced linear map; observe that $A_{\mathscr{x}}= A_{(\omega, x)}$ depends only on $x$ and on the first coordinate $f_\omega^1 = f_0$ of $\omega$. Using $P$ we transport the measure $\hat\mu$ to a measure, still denoted by $\hat\mu$, on the product space $X\times {\mathbb{P}}^1({\mathbf{C}})$. By our invariance assumption, its disintegrations $\hat\mu_{\mathscr{x}}=\hat\mu_x$ satisfy $({\mathbb{P}}A_{\mathscr{x}})_*\hat\mu_{\mathscr{x}}=\hat\mu_{F_+({\mathscr{x}})}=\hat\mu_{f^1_\omega(x)}$.
**The bounded case. –** In this paragraph we show that in the essentially bounded case (b) of Lemma \[lem:alternative\_bdd\], Conclusion (2) of Theorem \[thm:classification\_proj\_invariant\] holds. We streamline the argument following the proof of [@barrientos-malicet Prop. 4.7] which deals with the more general case of ${{\sf{GL}}}(d, {\mathbf{R}})$-cocycles, and is itself a variation on previously known ideas (see e.g. [@Arnold-Nguyen-Oseledets; @Zimmer:Israel1980]).
Set $G ={{\sf{PGL}}}(2, {\mathbf{C}})$, and define the $G$-extension ${\widetilde{F}}_+$ of $F_+$ on ${\mathscr{X}}_+\times G$ by $${\widetilde{F}}_+({\mathscr{x}}, g)=(F_+({\mathscr{x}}), {\mathbb{P}}(A_{\mathscr{x}}) g)=((\sigma(\omega), f^1_\omega(x)), {\mathbb{P}}(A_{(\omega, x)}) g)$$ for every ${\mathscr{x}}=(\omega,x)$ in ${\mathscr{X}}_+$ and $g$ in $G$; thus ${\widetilde{F}}_+$ is given by $F_+$ on ${\mathscr{X}}_+$ and is the multiplication by ${\mathbb{P}}(A_{\mathscr{x}})$ on $G$. Since ${\mathbb{P}}(A_{(\omega, x)})$ depends on $\omega $ only through its first coordinate, ${\widetilde{F}}_+$ can be interpreted as the skew product map associated to a random dynamical system on $X\times G$. Denote by $\mathcal P$ the convolution operator associated to this random dynamical system; thus $\mathcal P$ acts on probability measures on $X\times G$. Let $\mathrm{Prob}_\mu(X\times G)$ the set of probability measures on $X\times G$ projecting to $\mu$ under the natural map $X\times G \to X$. Since $\mu$ is stationary, $\mathcal P$ maps $\mathrm{Prob}_\mu(X\times G)$ to itself.
Recall that by assumption there is a set $E$ of positive measure in ${\mathscr{X}}_+$, a compact subset $K_G$ of $G$, and a positive real number $\varepsilon_0$ such that $$\underline{\operatorname{dens}}{\left\{ n\; ; \; {\mathbb{P}}(A^{(n)}_{\mathscr{x}})\in K_G \right\}}\geq \varepsilon_0$$ for all ${\mathscr{x}}$ in $E$.
There exists an ergodic, stationary, Borel probability measure $\widetilde{\mu}_G$ on $X\times G$ with marginal measure $\mu$ on $X$.
(See [@barrientos-malicet Prop. 4.13] for details). Let $\widetilde{\mu}_G$ be any cluster value of the sequence of probability measures $${\frac{1}{N}}\sum_{i=0}^{N-1} \mathcal P^i (\mu\times \delta_{1_G}).$$ By the boundedness assumption, $\widetilde{\mu}_G$ has mass $M\geq {\varepsilon}_0$ and is stationary (i.e. $\mathcal P$-invariant). Standard arguments show that its projection on the first factor is equal to $M\mu$. We renormalize it to get a probability measure and using the ergodic decomposition and the ergodicity of $\mu$, we may replace it by an ergodic stationary measure in $\mathrm{Prob}_\mu(X\times G)$.
Denote by $\widetilde {\mathscr{m}}_G = \nu^{\mathbf{N}}\times \widetilde \mu_G$ the ${\widetilde{F}}_+$-invariant measure associated to $\widetilde{\mu}_G$. The action of ${\widetilde{F}}_+$ on ${\mathcal{X}}_+\times G$ (resp. of the induced random dynamical system on $X\times G$) commutes to the action of $G$ by right multiplication, i.e. to the diffeomorphisms $R_h$, $h\in G$, defined by $$R_h({\mathscr{x}}, g)= ({\mathscr{x}}, g h).$$ Slightly abusing notation we also denote by $R_h$ the analogous map on $X\times G$. The next lemma combines classical arguments due to Furstenberg and Zimmer.
Let $\widetilde\mu_G$ be a Borel stationary measure on $X\times G$ with marginal $\mu$ on $X$. Set $$H ={\left\{ h \in G \; ; \; (R_h)_*{\widetilde{\mu}}_G = \widetilde \mu_G\right\}}
= {\left\{ h \in G \; ; \; (R_h)_*{\widetilde{{\mathscr{m}}}}_G =\widetilde {\mathscr{m}}_G \right\}}.$$ Then $H$ is a compact subgroup of $G$ and there is a measurable function $Q\colon X \to G$ such that the cocycle $B_{\mathscr{x}}= Q_{f^1_\omega(x)}^{-1}\times {\mathbb{P}}(A_{\mathscr{x}}) \times Q_x$ takes its values in $H$ for $(\nu^{\mathbf{N}}\times \mu)$-almost every ${\mathscr{x}}$.
Clearly, $H$ is a closed subgroup of $G$. If $H$ were not bounded then, given any compact subset $C$ of $G$, we could find a sequence $(h_n)$ of elements of $H$ such that the subsets $R_{h_n}(C)$ are pairwise disjoint. Choosing $C$ such that $X\times C$ has positive $\widetilde\mu_G$-measure, we would get a contradiction with the finiteness of $\widetilde\mu_G$. So $H$ is a compact subgroup of $G$.
We say that a point $(x,g)$ in $X\times G$ is generic if for $\nu^N$-almost every $\omega$, $$\label{eq:generic_G}
\frac{1}{N}\sum_{n=0}^{N-1} \varphi{\left({\widetilde{F}}^n_+(\omega, x, g) \right)} \underset{N\to\infty}\longrightarrow
\int_{{\mathcal{X}}_+\times G} \varphi \; d \widetilde{\mathscr{m}}_G$$ for every compactly supported continuous function on ${\mathcal{X}}_+\times G$. The Birkhoff ergodic theorem provides a Borel set $\mathcal E$ of full $\widetilde \mu_G$-measure made of generic points. Now if $(x, g_1)$ and $(x, g_2)$ belong to $\mathcal E$, writing $g_2=g_1 h=R_h(g_1)$ for $h=g_1^{-1} g_2$, we get that $h$ is an element of $H$. Given $g\in G$, define ${\mathcal{E}}_x\subset G$ to be the set of elements $g\in G$ such that $(x,g)$ is generic. Then there exists a measurable section $X\ni x\mapsto Q_x \in G$ such that $Q_x\in {\mathcal{E}}_x$ for almost all $x$. By definition of $\mathcal{E}_x$, $(\omega, x, Q_x)$ satisfies for $\nu^N$-almost every $\omega$. Then for $\nu$-almost every $f_0 = f_\omega^1$, by $\widetilde F_+$-invariance of the set of Birkhoff generic points we infer that $(f_\omega^1(x), {\mathbb{P}}(A_{\mathscr{x}}) Q_x)$ belongs to $\mathcal E$. Since $(f_\omega^1(x), Q_{f_\omega^1(x)})$ belongs to $\mathcal E$ as well, it follows that $Q_{f_\omega^1(x)}^{-1} {\mathbb{P}}(A_{\mathscr{x}}) Q_x$ is in $H$. We conclude that the cocycle $B_{\mathscr{x}}= Q_{f^1_\omega(x)}^{-1}\times {\mathbb{P}}(A_{\mathscr{x}}) \times Q_x$ takes its values in $H$ for almost all ${\mathscr{x}}$, as claimed.
Note that the map $x\mapsto Q_x$ lifts to a measurable map $x\mapsto Q'_x\in {{\sf{GL}}}_2({\mathbf{C}})$. Conjugating $H$ to a subgroup of $\mathsf{PU}_2$ by some element $g_0\in G$, we can now readily conclude from the two previous lemmas that when $\llbracket D_xf^{n}_\omega \rrbracket$ is essentially bounded, Conclusion (2) of Theorem \[thm:classification\_proj\_invariant\] holds (the $P_x$ are obtained by composing the $Q'_x$ with a lift of $g_0$ to ${{\sf{GL}}}_2({\mathbf{C}})$).
**The unbounded case. –** Now, we suppose that $\llbracket D_xf^n_\omega\rrbracket$ is essentially unbounded (alternative (a) of Lemma \[lem:alternative\_bdd\]), and adapt the results of [@barrientos-malicet §4.1] to the complex setting to arrive at one of the Conclusions (1.a) or (1.b) of Theorem \[thm:classification\_proj\_invariant\]. The main step of the proof is the following lemma.
\[lem:1/2\] Let $A$ be a measurable ${{\sf{GL}}}(2, {\mathbf{C}})$ cocycle over $({\mathcal{X}}_+, F_+, \nu^{\mathbf{N}}\times \mu)$ admitting a projectively invariant family of probability measures ${\left(\hat \mu_x\right)}_{x\in X}$ such that almost surely $\llbracket A_{{\mathscr{x}}}^{(n)}\rrbracket$ almost converges to infinity. Then for almost every $x$, $\hat\mu_x$ possesses an atom of mass at least $1/2$; more precisely:
- either $\hat\mu_x$ has a unique atom $[w(x)]$ of mass $\geq 1/2$, that depends measurably on $x\in X$;
- or $\hat\mu_x$ has a unique pair of atoms of mass $1/2$, and this (unordered) pair depends measurably on $x\in X$.
For the moment, we take this result for granted and proceed with the proof. By ergodicity, the number of atoms of $\hat\mu_x$ and the list of their masses are constant on a set of full measure. A first possibility is that $\hat\mu_x$ is almost surely the single point mass $\delta_{[w(x)]}$; this corresponds to (1.a). A second possibility is that $\hat\mu_x$ is the sum of two point masses of mass $1/2$; this corresponds to (1.b). In the remaining cases, there is exactly one atom of mass $1/2\leq \alpha <1$ at a point $[w(x)]$. Changing the trivialization $P_x$, we can suppose that $[w(x)]=[w]=[1:0]$. Then we write $\hat\mu_x=\alpha \delta_{[1:0]}+\hat\mu_x'$, and apply Lemma \[lem:1/2\] to the family of measures $\hat\mu_{\mathscr{x}}'$ (after normalization to get a probability measure). We deduce that almost surely $\hat\mu_x'$ admits an atom of mass $\geq (1-\alpha)/2$. Two cases may occur:
- $\hat\mu_x'$ has a unique atom of mass $\beta \geq (1-\alpha)/2$,
- $\hat\mu_x'$ has two atoms of mass $(1-\alpha)/2$.
The second one is impossible, because changing the trivialization, we would have $\hat\mu_x=\alpha \delta_{[1:0]}+\frac{1-\alpha}{2} (\delta_{[-1:1]}+\delta_{[1:1]})$, and the invariance of the finite set ${\left\{ [1:0], [-1:1], [1:1] \right\}}$ would imply that the cocycle ${\mathbb{P}}(A_{\mathscr{x}})$ stays in a finite subgroup of ${{\sf{PGL}}}_2({\mathbf{C}})$, contradicting the unboundedness assumption.
If $\hat\mu_{\mathscr{x}}'$ has a unique atom of mass $\beta \geq (1-\alpha)/2$, we change $P_x$ to put it at $[0:1]$ (the trivialization $P_x$ is not an isometry anymore). We repeat the argument with $\hat\mu_x=\alpha \delta_{[1:0]}+\beta \delta_{[0:1]}+\hat\mu_x''$. If $\beta = 1-\alpha$, i.e. $\hat\mu_x''=0$, then we are done. Otherwise $\hat\mu_x''$ has one or two atoms of mass $\gamma \geq (1-\alpha-\beta)/2$, and we change $P_x$ to assume that one of them is $[1:1]$ and the second one –provided it exists– is $[\tau(x):1]$; here, $x\mapsto \tau(x)$ is a complex valued measurable function. Endow the projective line ${\mathbb{P}}^1({\mathbf{C}})$ with the coordinate $[z:1]$; then ${\mathbb{P}}(A_{\mathscr{x}})$ is of the form $z\mapsto a({\mathscr{x}}) z$. Since ${\mathbb{P}}(A_{\mathscr{x}}) {\left({\left\{1, \tau(x)\right\}}\right)} = {\left({\left\{1, \tau(F_+({\mathscr{x}}))\right\}}\right)}$, we infer that:
- either $a({\mathscr{x}})1 = 1$ and ${\mathbb{P}}(A_{\mathscr{x}})$ is the identity;
- or $a({\mathscr{x}})1 = \tau(\pi_X(F_+({\mathscr{x}}))) $ and $a({\mathscr{x}})\tau(x) =1$ in which case $\tau(\pi_X(F_+({\mathscr{x}}))) = \tau(x){^{-1}}$.
Thus we see that along the orbit of ${\mathscr{x}}$, $a(F^n_+({\mathscr{x}}))$ takes at most two values $\tau(\pi_X(F^n_+({\mathscr{x}})))^{\pm 1}$, and $\llbracket A^{(n)}_{\mathscr{x}}\rrbracket$ is bounded, which is contradictory. This concludes the proof.
Let $r$ and ${\varepsilon}$ be small positive real numbers. Let $\mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))$ be the set of probability measures $m$ on ${\mathbb{P}}^{1}({\mathbf{C}})$ such that $\sup_{x\in {{\mathbb{P}^1}}} m(B(x, r))\leq 1/2-{\varepsilon}$, where the ball is with respect to some fixed Fubini-Study metric. This is a compact subset of the space of probability measures on ${{\mathbb{P}^1}}$. The set $$G_{r, {\varepsilon}} = {\left\{\gamma\in {{\sf{PGL}}}(2, {\mathbf{C}}),\ \exists m_1, m_2 \in \mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}})), \ \gamma_*m_1 =m_2\right\}}$$ is a bounded subset of ${{\sf{PGL}}}(2, {\mathbf{C}})$. Indeed otherwise there would be an unbounded sequence $\gamma_n$ together with sequences $(m_{1, n})$ and $(m_{2, n})$ in $\mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))$ such that $(\gamma_n)_*m_{1, n} = m_{2, n}$. Denote by $\gamma_n=k_n a_n k'_n$ the KAK decomposition of $\gamma_n$ in ${{\sf{PGL}}}(2, {\mathbf{C}})$, with $k_n$ and $k'_n$ two isometries for the Fubini-Study metric; since $\gamma_n$ is unbounded, we can extract a subsequence such that the measures $(k'_n)_*m_{1, n}$ and $(k_n^{-1})_*m_{2, n}$ converge in $\mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))$ to two measures $m_1$ and $m_2$, while the diagonal transformations $a_n$ converge locally uniformly on ${\mathbb{P}}^1({\mathbf{C}})\setminus{\left\{[0:1]\right\}}$ to the constant map $\gamma: {\mathbb{P}}^1({\mathbf{C}}) \setminus{\left\{[0:1]\right\}} \mapsto {\left\{[1:0]\right\}}$. Then $$\gamma_*{\left(m_{1\vert {\mathbb{P}}^1({\mathbf{C}})\setminus{\left\{[0:1]\right\}}}\right)} = m_1({\mathbb{P}}^1({\mathbf{C}})\setminus{\left\{[0:1]\right\}}) \delta_a \leq m_2;$$ since $m_1$ belongs to $\mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))$, $m_1({\mathbb{P}}^1({\mathbf{C}})\setminus{\left\{[0:1]\right\}}) \geq 1/2+{\varepsilon}$, hence $m_2\geq (1/2+{\varepsilon}) \delta_a$, in contradiction with $m_2\in \mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))$. This proves that $G_{r,{\varepsilon}}$ is bounded.
To prove the lemma, let us consider the ergodic dynamical system ${\mathbb{P}}DF_+$, and the family of conditional probability measures $\hat\mu_{\mathscr{x}}$ for the projection $(\omega, x,v)\mapsto {\mathscr{x}}=(\omega,x)$. If there exist $r, {\varepsilon}>0$ such that $\hat \mu_{\mathscr{x}}$ belongs to $\mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))$ for ${\mathscr{x}}$ in some positive measure subset $B$ then, by ergodicity, for almost every ${\mathscr{x}}\in {\mathscr{X}}_+$ there exists a set of integers $L({\mathscr{x}})$ of positive density such that for $n\in L({\mathscr{x}})$, $F_+^n({\mathscr{x}})$ belongs to $B$, hence $A^{(n)}_{{\mathscr{x}}} $ belongs to $G_{r, {\varepsilon}}$ ([^9]). From the above claim we deduce that $\llbracket A^{(n)}_{\mathscr{x}}\rrbracket$ is uniformly bounded for $n\in L({\mathscr{x}})$, a contradiction. Therefore for every $r, {\varepsilon}>0$, the measure of ${\left\{{\mathscr{x}}, \ \hat \mu_{\mathscr{x}}\in \mathrm{Prob}_{r, {\varepsilon}}({\mathbb{P}}^1({\mathbf{C}}))\right\}}$ is equal to $0$; it follows that for almost every ${\mathscr{x}}$, $\hat\mu_{\mathscr{x}}$ possesses an atom of mass at least $1/2$.
If there is a unique atom of mass $\geq 1/2$, this atom determines a measurable map ${\mathscr{x}}\mapsto [w({\mathscr{x}})]\in {\mathbb{P}}T_xX$; since $\hat \mu_{\mathscr{x}}$ does not depend on $\omega$, $[w({\mathscr{x}})]$ depends only on $x$, not on $\omega$. If there are generically two atoms of mass $\geq 1/2$, then both of them has mass $1/2$, and the pair of points determined by these atoms depends only on $x$.
[^1]: Here, $U$ is the standard $2$-dimensional Minkowski lattice, $({\mathbf{Z}}^2, x_1x_2)$, and $E_8$ is the root lattice given by the corresponding Dynkin diagram; so $E_8(-1)$ is negative definite, and $E_{10}$ has signature $(1,9)$. Also, recall that in this paper ${{\mathrm{NS}}}(X;{\mathbf{Z}})$ denotes the torsion free part of the Néron-Severi group, which is sometimes denoted by ${\mathsf{Num}}(X;{\mathbf{Z}})$ in the literature on Enriques surfaces.
[^2]: if locally $C=\{ x=0\}$ then $\xi(x,y)=\eta(x,y)/x$ where $\eta$ is regular; thus, ${\left\vert\xi\right\vert} ={\left\vert\eta\right\vert} {\left\vertx\right\vert}^{-1}$ is locally integrable because $\frac{1}{r^\alpha}$ is integrable with respect to $rdrd\theta$ when $\alpha<2$
[^3]: By definition cylinders are products $C=\prod C_j$ of Borel sets, all of which are equal to ${\mathsf{Aut}}(X)$ except finitely many of them, say $C_0$, $C_2$, $\ldots$, $C_N$. For simplicity, we denote a cylinder by $C=\prod_{j=0}^N C_j$ if $C_k={\mathsf{Aut}}(X)$ for $k>N$.
[^4]: This actually requires checking that the whole proof of [@br] can be reproduced in our complex setting: we will come back to this issue in a forthcoming paper. Since we are just using this remark here in Corollary \[cor:positive\_entropy\] we take the liberty to anticipate on that research.
[^5]: Brown and Rodriguez-Hertz make it clear that this result holds for an arbitrary smooth random dynamical system on a compact manifold.
[^6]: The proof in [@Raugi:2004] is not correct, because Lemma 2.5 there is false. But the proof works perfectly, and is quite short, if the support of $\nu$ is countable or if the nilpotency class of the group is $\leq 2$.
[^7]: Hence by Theorem \[thm:classification\_invariant\], $\mu$ is equivalent to either ${{\sf{vol}}}_X$ or ${{\sf{vol}}}_Y$ for some real analytic invariant surface with boundary.
[^8]: Beware that the word “twisting” has a different meaning there.
[^9]: We are slightly abusing here when the Fubini-Study metric depends on $x$, for instance when $P_x$ is not an isometry; however restricting to subset of large positive measure the metric $(P_x)_*(\kappa_0)_x$ is uniformly comparable to a fixed Fubini-Study metric.
|
---
author:
- 'Klaus M. Frahm'
- 'Dima L. Shepelyansky'
date: 'Dated: 28 April 2018'
title: |
Dynamical decoherence of a qubit\
coupled to a quantum dot or the SYK black hole
---
Introduction
============
The problem of qubit decoherence is crucial for the process of quantum measurement [@braginsky] and the field of quantum information and computation [@chang]. The experimental realization of superconducting qubits [@nakamura; @esteve] extended this problem to a world of large objects due to a macroscopic size of superconducting qubits (see e.g. [@averin; @shnirman; @wendin]). In theoretical considerations the decoherence of a qubit is usually due to the contact with a thermal bath, weak measurements or other statistical systems characterizing a detector (or sensor) being in a contact with the qubit [@averin; @shnirman; @wendin]. A model of a deterministic detector, whose evolution takes place in a regime of quantum chaos, was studied in [@lee2005] demonstrating the emergence of dynamical decoherence of a qubit in absence of any thermal bath, noise and external randomness. We extend this research line [@lee2005] considering as a deterministic detector a quantum dot with interacting fermions or the Sachdev-Ye-Kitaev (SYK) black hole model.
The question about dynamical decoherence is closely related to the problem of quantum dynamical thermalization and random matrix theory (RMT) invented by Wigner [@wigner; @mehta; @guhr] for the description of complex atoms and nuclei. While the properties of one-particle quantum chaos and their link with RMT are now mainly understood (see e.g. [@bohigas1984; @haake; @ullmo]), the analysis of many-body quantum systems is more difficult due to the complexity of quantum many-body systems (QMBS). Furthermore RMT is only an approximation to QMBS since in nature we have only two-body interactions and hence the exponentially large Hamiltonian matrix of QMBS has only a small fraction of non-zero matrix elements. To capture this feature a two-body random interaction model of fermions (TBRIM) was proposed in [@french1; @bohigas1; @french2; @bohigas2] and it was shown that at strong interactions this model is characterized by RMT level spacing statistics. The first numerical results and analytical arguments for a critical interactions strength in TBRIM with a finite level spacing $\Delta$ between one-particle orbitals was proposed by Sven [Å]{}berg in [@aberg1; @aberg2]. For the TBRIM the [Å]{}berg criterion for onset of quantum chaos and dynamical thermalization has the form $$\label{eq:abergcriterion}
\delta E = E - E_g > \delta E_{\rm ch} \approx g^{2/3} \Delta \;, \;\; g = \Delta/U ,$$ where $U$ is a typical strength of two-body interactions, $\Delta$ is an average one-particle level spacing in a finite size quantum dot with interacting fermions, $E_g$ is the ground state energy of the quantum dot when all electrons are below the Fermi energy $E_F$ and $E$ is the energy of an excited eigenstate. The dimensional parameter $g \gg 1$ is assumed to be large playing the role of the conductance of a quantum dot with weakly interacting electrons. The validity of the [Å]{}berg criterion (\[eq:abergcriterion\]) for the emergence of RMT level statistics was confirmed in first numerical simulations [@aberg1; @aberg2] and in independent more extensive analytical and numerical studies for 3 particles in a quantum dot [@sushkov], TBRIM [@jacquod], spin glass shards [@georgeotspin], quantum computers with imperfections [@georgeotqc; @nobel; @benentiqc]. Advanced theoretical arguments developed in [@mirlin1; @mirlin2] confirm the relation (\[eq:abergcriterion\]) for interacting fermions in a quantum dot.
While the validity of the [Å]{}berg criterion for emergence of RMT in TBRIM and other models is satisfactory confirmed by numerical and analytical studies, a dynamical thermalization conjecture (DTC), which is used for the derivation of (\[eq:abergcriterion\]), is more difficult for the numerical verification since it requires the knowledge not only of the eigenvalues but also the computation of eigenstates that is more difficult. The TBRIM numerical results [@flambaum] for the probability distribution over one-particle orbitals, averaged over many random realizations, showed a certain proximity to the Fermi-Dirac distribution expected from the quantum statistical mechanics [@landau]. The validity of the Fermi-Dirac distribution for a single eigenstate was demonstrated numerically for eigenstates of a quantum computer with imperfections and residual inter-qubit couplings [@benentiqc]. We stress that the DTC is proposed for a purely isolated system without any contact to an external thermostat and the dynamical thermalization is only due to internal many-body quantum chaos.
However, for a single eigenstate the fluctuations of probabilities $n_k$ on one-particle orbitals are significant requiring heavy large matrix diagonalizations to obtain a reasonable agreement with the Fermi-Dirac distribution [@benentiqc]. Another method was developed for nonlinear disordered chains described by classical Hamiltonian equations [@mulansky; @ermannnjp]. It is based on the computation of entropy $S$ and energy $E$ tracing the dependence $S(E)$ which is obtained as an implicit function from $S(T)$ and $E(T)$ where $T$ is the system temperature appearing due to dynamical thermalization in a completely isolated system without any contact to an external thermostat. Since the quantities $S$ and $E$ are extensive [@landau] their fluctuations are reduced due to self-averaging. The dependence $S(E)$ for many-body quantum systems was computed for bosons in disordered Bose-Hubbard model in 1D [@schlageck] and for spinless fermions in the TBRIM [@kolovsky2017]. These studies demonstrated the stability and efficiency of $S(E)$-computations confirming validity of the DTC for many-body interacting quantum systems. The dynamical thermalization of an individual eigenstate was also demonstrated in [@schlageck; @kolovsky2017]. At present the interest of many-body interacting quantum systems is also growing in the context of many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH) (see e.g. [@huse; @polkovnikov; @borgonovi; @alet]).
Another bust of interest to the TBRIM type models appeared due to the recent results of Sachdev-Ye-Kitaev for a strange metal and its links to a quantum black hole model in 1+1 dimensions (coordinate plus time) known now as the SYK black hole [@sachdevprl; @kitaev; @sachdevprx]. In fact, the SYK model, in its fermionic formulation, corresponds to the TBRIM considered in the limit of very strong interactions with a conductance close to zero $g \rightarrow 0$. The analogy between physical representations of the SYK model attracted a significant interest of researchers in quantum gravity, many-body systems, RMT and quantum chaos (see e.g. [@rosenhaus; @maldacena; @tezuka]). Recent advanced numerical and analytical results on the validity of RMT for the SYK model with Majorana fermions are reported in [@garcia1; @garcia2; @wettig].
In this work we study the dynamical decoherence of a qubit coupled to the TBRIM model. This is a completely isolated system in absence of noise, thermal bath and external decoherence. At $g \gg 1$ the qubit is coupled to a quantum dot of weakly interacting fermions with our main interest being focused on the regime of dynamical thermalization when the [Å]{}berg criterion (\[eq:abergcriterion\]) is satisfied. At $g \ll 1$ our model becomes equivalent to the SYK black hole model with a qubit coupled to it. We note that the decoherence of a qubit coupled to a quantum black hole is extensively discussed in the context of the black hole problem of information loss for the infalling observer (see [@harlow] and Refs. therein). We expect that the dynamical qubit decoherence considered here will be useful for a better understanding of this problem.
The paper is composed as follows: In Section 2 the TBRIM is introduced and some of its properties are reminded while in Section 3 the additional qubit-fermion coupling is introduced. The qubit relaxation rates are studied in Section 4 and in Section 5 the link to a quantum small-world networks is discussed. In Section 6 results of the residual level of qubit density matrix relaxation at long times are described and Section 7 concludes with the discussion. In Appendix A a rather detailed analytical and numerical study for the approximate Gaussian form of the average density of states of the TBRIM is presented while Appendices B and C deal with the specific issues of weakly excited initial states of the TBRIM, where it is difficult to obtain clear relaxation rates, and initial states with negative temperatures.
TBRIM construction and properties
=================================
As in Ref. [@kolovsky2017] we consider the TBRIM [@jacquod] with $M$ one-particle orbitals and $0\le L\le M$ spinless fermions with the Hamiltonian: $$\label{eq_TBRIM}
H_I=\frac{1}{\sqrt{M}}\sum_{k=1}^M v_k\,c^\dagger_k c^\pdag_k
+\frac{4}{\sqrt{2M^3}}\sum_{i<j,k<l} J_{ij,kl}\,c^\dagger_i c^\dagger_j
c^\pdag_l c^\pdag_k$$ where $c^\dagger_k$, $c^\pdag_k$ are fermion operators for the $M$ orbitals satisfying the usual anticommutation relations. Here $v_k$ ($J_{ij,kl}$) are real Gaussian random variables with zero mean and variance $\langle v_k^2\rangle=V^2$ ($\langle J_{ij,kl}^2\rangle=J^2(1+\delta_{ik}\delta_{jl})$) such that the non-interacting orbital one-particle energies are given by $\epsilon_k=v_k/\sqrt{M}$. The variance of the interaction matrix elements is chosen such they correspond to a GOE-matrix (Gaussian orthogonal ensemble) of size $M_2\times M_2$ with $M_2=M(M-1)/2$. The number of nonzero elements for a column (or row) of $H_I$ is $K = 1 + L(M-L) + L(L-1)(M-L)(M-L-1)/4$ [@jacquod; @flambaum].
As shown in Appendix \[appa\] the density of states (DOS) of the TBRIM Hamiltonian (\[eq\_TBRIM\]) is approximately Gaussian $$\label{eq_DOS_TBRIM}
\rho(E)\approx\frac{d}{\sqrt{2\pi\sigma^2}}\,
\exp\left(-\frac{E^2}{2\sigma^2}\right)
\quad,\quad
\sigma=\sqrt{\frac{L(M-L)}{M(M-1)}}\,V_{\rm eff}$$ which is normalized to $d=M!/(L!(M-L)!)$ being the dimensionality of the Hilbert space for $M$ orbitals and $L$ particles and $$\label{eq_Veff}
V_{\rm eff}=\sqrt{V^2+a(M,L)\,J^2}$$ is a rescaled effective energy scale taking into account the increase of $\sigma$ due to finite values of $J$. The coefficient $a(M,L)$ is computed in Appendix \[appa\] from the average of $\langle\mbox{Tr}(H_I^2)\rangle$ with the result : $$\label{eq_a_coeff}
a(M,L)=\frac{2(M-1)(L-1)}{M^2}
\left(\frac{4}{M-L}+M-L+3\right).$$ The expression (\[eq\_DOS\_TBRIM\]) fits numerically quite well the DOS for sufficiently large values of $M$ and $L$ and even in the SYK-case, i.e. when $J\neq 0$ but $V=0$, it is quite accurate. The corresponding average many body level spacing (at the band center) is $\Delta_{\rm MB}=\sqrt{2\pi}\,\sigma/d$. For later use we also define an effective rescaled average one-particle level spacing by $\Delta_1=\sqrt{2\pi}V_{\rm eff}/M^{3/2}$. At $J\ll V$ we have $V_{\rm eff}\approx V$ and $\Delta_1$ is just the average distance of the one-particle energies $\epsilon_k$ (in the band center). Thus the effective dimensionless conductance of our TBRIM (see [@jacquod]) is $g \approx \Delta_1/U_{s} \approx \sqrt{\pi} V_{\rm eff}/2J
\approx V/J \gg 1$ for $J \ll V$ and $g \approx 1$ for $J \gg V$ at $M \approx L/2$ ($U_s = 2\sqrt{2} J /M^{3/2}$ is an effective interaction strength).
Since we are using only a small number of statistical realizations, we have chosen realizations of $v_k$ such that exactly $\sum_k v_k=0$ and $\sum_k \epsilon_k^2=(1/M)\sum_k v_k^2=V^2$.
We have numerically diagonalized $H_I$ and done further numerical computations described below for the cases $M=12$, $M=14$ and $M=16$ with $L=M/2-1\approx M/2$. In this work we only show the results for the case of largest matrix size $M=16$ and $L=7$ corresponding to $d=11440$ (for this case the coefficient in (\[eq\_Veff\]) and (\[eq\_a\_coeff\]) is just $a(16,7)=8.75$ and the number of nonzero matrix elements per row/column of $H_I$ is $K=820$). Unless stated otherwise, all results presented below, especially in the figures apply to this case. We have, however, verified that the physical interpretation of the results also apply to the cases of smaller matrix size (with some restrictions concerning reduced times scales for the long time behavior, more limited parameter range etc.). We present the results for one specific disorder realization but we checked that, apart from fluctuations, the results remain stable for other realizations.
First we diagonalize numerically one realization of $H_I$ for $M=16$, $L=7$, $V=\sqrt{14}\approx3.74166$, various values of $J$ or the SYK-case (i.e. $V=0$, $J=1$). Similar to [@kolovsky2017] we determine for each many body eigenstate the occupation numbers $n_k=\langle c^\dagger_k c^\pdag_k\rangle$ with the corresponding fermion entropy [@landau] : $$\label{eq_entropy}
S=-\sum_{k=1}^M\Bigl(n_k\,\ln n_k+(1-n_k)\,\ln(1-n_k)\Bigr)$$ and the effective one-particle total energy $$\label{eq_1p_energy}
E_{1p}=\sum_{k=1}^M\,\epsilon_k\,n_k$$ based on the assumption on non- or weakly-interacting fermions. These energies are rather close to the exact many body energies $E_{\rm ex}\approx E_{1p}$ provided $J\ll V$.
 [*Top and center panels:*]{} Dependence of the fermion entropy $S$ given by (\[eq\_entropy\]) on the effective one-particle total energy $E_{1p}$ defined in (\[eq\_1p\_energy\]) (blue cross symbols) and the exact many body energy $E_{\rm ex}$ (red plus symbols). The green curve shows the theoretical Fermi-Dirac thermalization ansatz (\[eq\_fermidirac\]) as explained in the text. All panels correspond to $M=16$ orbitals, $L=7$ particles and Hamiltonian matrix size $d=11440$. Both top and center left panels correspond to $V=\sqrt{14}\approx 3.74166$ and $J=0.025$ (top left), $J=0.25$ (top right) and $J=1$ (center left). Center right panel corresponds to the SYK case at $V=0$ and $J=1$ with the green curve computed from a model of equidistant one-particle energies of non-interacting fermions. [*Bottom panels:*]{} Dependence of the inverse temperature $\beta=1/T$ on energy $E$ (bottom right panel) and chemical potential $\mu$ on $\beta$ (bottom left panel) corresponding to the Fermi-Dirac ansatz for the set of one-particle energies $\epsilon_k$ used for the chosen realization of $H_I$ at $V=\sqrt{14}\approx 3.74166$. ](fig1){width="48.00000%"}
In Fig. \[fig1\] we compare the dependence of $S$ on both energy scales with the theoretical fermionic behavior where $n_k$ in (\[eq\_entropy\]) is replaced by the usual thermal Fermi-Dirac distribution (or ansatz) over one-particle orbitals [@landau]: $$\label{eq_fermidirac}
n_k=1/(1+\exp[\beta(\epsilon_k-\mu)]) \;, \;\; \beta=1/T$$ with the inverse temperature $\beta$ and chemical potential $\mu$ determined by the implicit conditions (\[eq\_1p\_energy\]) and $L=\sum_k n_k$ with the given set of diagonal one-particle energies $\epsilon_k$. For the SYK case with $V=0$ and $J=1$ we choose for the “theoretical” curve the case of one-particles energies equidistant values $\epsilon_k$ such that $\sum_k \epsilon_k=0$ and $\sum_k \epsilon_k^2=V_{\rm eff}^2$ with the effective rescaled energy scale (\[eq\_Veff\]) at $V=0$ and $J=1$ ( a similar procedure was used in [@kolovsky2017] for this SYK case).
At $V=\sqrt{14}\approx 3.74166$ one can observe in Fig. \[fig1\] the onset of thermalization with increasing interaction strength $J$. At very weak interaction $J=0.025$ the entropy is typically below the theoretical behavior indicating that the system is not thermalized. We can also mention that for this case the level spacing distribution of $H_I$ does not obey the Wigner surmise (for the GOE case) and is closer to the Poisson distribution (with some small level-repulsion for very short energy differences). At $J=0.25$ (this value corresponds to the case $J=1$ in [@kolovsky2017] due to a difference in the normalization) the system is well thermalized but the interaction is still sufficiently low so that $E_{1p}\approx E_{\rm ex}$. Here and also for larger values of $J$ the level spacing distribution clearly corresponds to the Wigner surmise (this was also seen in [@kolovsky2017] and we do not show these data here). Thus at $J=0.25$ we have onset of the dynamical thermalization induced by weak many-body interactions. At $J=1$ the data points for $E_{1p}$ coincide very well with the theoretical fermionic curve confirming the onset of dynamical thermalization induced by interactions. However, here due the stronger interaction values the ratio $E_{\rm ex}/E_{1p}$ is considerably larger than unity.
For the SYK case $V=0$, $J=1$ the entropy is close to its maximal value $S\approx 11$ for nearly all eigenstates and the theoretical model of equidistant one-particle energies is not confirmed. This value of $S$ is actually consistent with $n_k\approx 0.5$ for all orbitals $k$ which gives due (\[eq\_entropy\]) $S\approx 16\ln(2)\approx 11.1$. For the SYK case the numerical level spacing distribution also corresponds to the Wigner surmise.
The results of this Section show that at moderate interactions with $g \ll 1$ the DTC is well working (e.g. $J=0.25, V=\sqrt{14}, g \approx 15$) and the dependence $S(E)$ is well described by the thermal Fermi-Dirac distribution (\[eq\_fermidirac\]). Of course, at very small interactions (e.g. $J=0.025,\ V=\sqrt{14},\ g \approx 150$) the DTC is not valid in qualitative agreement with the [Å]{}berg criterion (\[eq:abergcriterion\]). Here we do not investigate the exact numerical values for the [Å]{}berg criterion since our main aim is the investigation of the interaction of a qubit with the TBRIM in the regimes of a thermalized quantum dot (e.g. $g \approx 15$) or SYK black hole (e.g. $g \approx 1, V=0, J=1$). As discussed in [@kolovsky2017] the question about thermal description of quantum chaos via effective hidden modes in the SYK regime remains open.
Qubit interacting with TBRIM
============================
In order to study the decoherence of one qubit coupled to the fermionic system described by the TBRIM Hamiltonian $H_I$ defined in (\[eq\_TBRIM\]) we consider the total Hamiltonian $$\label{eq_ham_qubit}
H=\delta\cdot\,\sigma_x+
\varepsilon \frac{V_{\rm eff}}{V_0}\,\sigma_z\sum_{k=1}^{M-1}
\left(c^\dagger_k c^\pdag_{k+1}+c^\dagger_{k+1} c^\pdag_k\right)
+H_I$$ where $\sigma_x$ and $\sigma_z$ are the usual Pauli matrices in qubit space and $\delta$ is (half) the unperturbed energy separation of the two qubit levels introducing Rabi oscillations with frequency $\omega_R=2\delta$. We typically choose $\delta=\Delta_1/2$ (or a simple multiple of this) with $\Delta_1$ being the effective rescaled one-particle level spacing given above in terms of the effective energy scale $V_{\rm eff}$. In (\[eq\_ham\_qubit\]) we have chosen the orbital indices $k$ such that the one-particle energies are ordered, i.e. : $\epsilon_{k+1}>\epsilon_k$, implying that the qubit-fermion coupling term creates transitions between adjacent orbitals with approximate energy difference $\sim \Delta_1$. The quantity $\varepsilon$ is the coupling parameter which will take various values in the interval $0.005\le \varepsilon\le 1$ and the ratio $V_{\rm eff}/V_0$ (with $V_0=\sqrt{14}$) ensures that at different values of $V$ and $J$ the coupling parameter is measured in units of the overall bandwidth $\sigma\sim V_{\rm eff}$ such that results at different values of $V$ and $J$ at same $\varepsilon$ are indeed comparable. We mention that the Hamiltonian (\[eq\_ham\_qubit\]) is similar in structure to the Hamiltonian studied in Ref. [@lee2005] where the qubit was coupled to a quantum kicked rotor model. As already mentioned we present below results for $M=16$ orbitals and $L=7$ particles corresponding to a combined qubit-fermion Hilbert space dimension of $22880$ but we have also verified the smaller cases at $M=12$ or $M=14$ with $L=M/2-1$ obtaining there similar results.
Explicitly, we compute numerically the exact time evolution of a state ${\left| \psi_m(t) \right\rangle}=\exp(-iHt)\,{\left| \psi_m(0) \right\rangle}$ with the initial vector $$\label{eq_init_state}
{\left| \psi_m(0) \right\rangle}={\left| \phi_m \right\rangle}({\left| 0 \right\rangle}+2\,{\left| 1 \right\rangle})/\sqrt{5}$$ where ${\left| \phi_m \right\rangle}$ is an exact eigenstate of $H_I$ at level number $m$ with many body energy $E_m$, i.e. $H_I\,{\left| \phi_m \right\rangle}=E_m\,{\left| \phi_m \right\rangle}$, and ${\left| 0 \right\rangle}$, ${\left| 1 \right\rangle}$ denote the two qubit states with bottom and upper energies. The time evolution operator $\exp(-iHt)$ is computed exactly by diagonalizing $H$ and expressing the matrix exponential using the exact eigenvalues and eigenvectors of $H$. For $M=16$ and $L=7$ this corresponds to a numerical diagonalization in the combined fermion-qubit Hilbert space of dimension $22880$. As in Ref. [@lee2005] we determine the $2\times 2$ density matrix $\rho_{ij}(t)$, $i,j=0,1$ from the partial trace over the fermionic states by: $\rho_{ij}(t)={\left\langle i \right|}\mbox{Tr}_{\rm ferm.}
\left({\left| \psi_m(t) \right\rangle}{\left\langle \psi_m(t) \right|}\right)\,{\left| j \right\rangle}$. In absence of qubit-fermion coupling, i.e. $\varepsilon=0$, the density matrix $\rho(t)$ does not depend on the choice of ${\left| \phi_m \right\rangle}$ and a simple standard calculation gives the result: $$\begin{aligned}
\label{eq_rho11}
\rho_{11}(t)&=&1-\rho_{00}(t)=
\frac12+\frac{3}{10}\cos(\omega_R t)\quad,\\
\label{eq_rho01}
\rho_{01}(t)&=&\rho_{10}^*(t)=\frac25+\frac{3}{10}\,i\,\sin(\omega_R t)
\quad,\\
\Rightarrow\quad |\rho_{01}(t)| &=&
\frac{1}{2}\left(\frac{41}{50}-\frac{9}{50}\cos(2\omega_R\,t)\right)^{1/2}\quad,\end{aligned}$$ where $\omega_R=2\delta$ is the Rabi frequency.
For practical reasons we compute the density matrix $\rho(t)$ at $t=\tau\,\Delta t$ with integer values of $\tau$ and the elementary time unit $\Delta t=1/(\Delta_1\,M)$ where $\Delta_1$ is the rescaled effective one-body level spacing. This time step corresponds roughly to the inverse one-particle band-width and represents the shortest quantum time scale in the system. We consider the maximal time value $t_{\rm max}=(d/2)\,\Delta t=5720\,\Delta t=
\sqrt{\frac{L(M-L)}{4(M-1)}}\,t_H\approx\,t_H$ with $L\approx M/2$ and $t_H=1/\Delta_{\rm MB}$ being the Heisenberg time.
![\[fig2\] Time dependence of $\rho_{11}(t)$ (red plus symbols) and $|\rho_{01}(t)|$ (green crosses) for the initial state being the ground state of $H_I$ with the level number $m=0$ and qubit state (\[eq\_init\_state\]) for $V=3.74166$, $J=0.25$ ($V=0$, $J=1$) in left (right) panel at coupling strength $\varepsilon=0.01$. The time is measured in units of $\Delta t=1/(\Delta_1\,M)$ where $\Delta_1$ is the rescaled effective one-body level spacing defined in the text. ](fig2){width="48.00000%"}
In Fig. \[fig2\] we show the time dependence of $\rho_{11}(t)$ and $|\rho_{01}(t)|$ for a weak coupling strength $\varepsilon=0.01$, an initial state (\[eq\_init\_state\]) with level number $m=0$, corresponding to the ground state of $H_I$, and two cases for different values of $V$ and $J$. For $\varepsilon=0.01$ the dependence $\rho_{11}(t)$ is close to the analytical result (\[eq\_rho11\]). However for $|\rho_{01}(t)|$ the situation is more complicated with the appearance of a further frequency leading to a quasi-periodic structure. Apparently the ground state ${\left| \phi_0 \right\rangle}$ of $H_I$ is also weakly coupled to the next state ${\left| \phi_1 \right\rangle}$ due to the indirect qubit-fermion coupling leading to an additional frequency. The results of Fig. \[fig2\] show that there is no qubit decoherence when it is coupled with a quantum dot or SYK system when they are in their ground state.
 As in Fig. \[fig2\] but for level number $m=5720$ of the initial state (\[eq\_init\_state\]) corresponding to an energy in the center of the spectrum of $H_I$. The fit functions $f_{11}(t)$ (thin black line) to approximate $\rho_{11}(t)$ and $f_{01}(t)$ (thin blue line) to approximate $|\rho_{01}(t)|$ are given by (\[eq\_f11\]) and (\[eq\_f01\]) with the fit parameters : $A_1=0.49593\pm 0.00005$, $B_1=0.3070\pm 0.0002$, $\Gamma_1=0.002195\pm 0.000003$, $\omega_1=0.063423\pm 0.000003$, $\alpha_1=6.2492\pm 0.0006$ and $A_2=0.0063\pm 0.0001$, $\tilde B_2=0.194\pm 0.001$, $\tilde\Gamma_2=0.00435\pm0.00004$, $\omega_2=0.12641\pm 0.00004$, $\alpha_2=3.232\pm 0.007$, $B_2=0.800\pm 0.001$, $\Gamma_2=0.00713\pm 0.00002$ for $V=3.74166$, $J=0.25$ (left panel) and $A_1=0.5008\pm 0.0001$, $B_1=0.2897\pm 0.0004$, $\Gamma_1=0.000449\pm 0.000002$, $\omega_1=0.062552\pm 0.000002$, $\alpha_1=6.228\pm 0.0002$ and $A_2=0.0391\pm 0.0004$, $\tilde B_2=0.172\pm 0.002$, $\tilde\Gamma_2=0.00094\pm0.00002$, $\omega_2=0.12508\pm 0.00002$, $\alpha_2=3.06\pm 0.02$, $B_2=0.825\pm 0.002$, $\Gamma_2=0.00209\pm 0.00001$ for $V=0$, $J=1$ (right panel). ](fig3){width="48.00000%"}
For higher level numbers $m$ the situation changes and for many eigenstates ${\left| \phi_m \right\rangle}$ of $H_I$ an exponential relaxation is found for $\rho_{00}(t)$ tending to the equilibrium value $1/2$ and $|\rho_{01}(t)|$ tending to a value $\sim 1/\sqrt{n}$ where $n$ is roughly the number of eigenstates of $H_I$ contributing in ${\left| \psi_m(t) \right\rangle}$. Therefore, motivated by the analytic expressions at $\varepsilon=0$, we use the following fit functions for small values $0<\varepsilon\ll 1$ : $$\begin{aligned}
\label{eq_f11}
f_{11}(\tau\Delta t)&=&A_1+B_1\,e^{-\Gamma_1\tau}\,\cos(\omega_1\tau+\alpha_1)
\quad,\\
\nonumber
f_{01}(\tau\Delta t)&=&\frac12 \Bigl(A_2+\tilde B_2\,e^{-\tilde\Gamma_2\tau}\,
\cos(\omega_2\tau+\alpha_2)\\
\label{eq_f01}
&& +B_2\,e^{-\Gamma_2\tau}\Bigr)^{1/2}\quad,\end{aligned}$$ to approximate $\rho_{11}(t)$ by $f_{11}(t)$ and $|\rho_{01}(t)|$ by $f_{01}(t)$. The parameter $\tau=t/\Delta t$ is the rescaled time in units of $\Delta t=1/(\Delta_1\,M)$ where $\Delta_1$ is the rescaled effective one-body level spacing introduced above. These fits work very well for the two cases shown in Fig. \[fig3\] with level number $m=5720$ (corresponding to the band center of $H_I$) and $\varepsilon=0.01$. From (\[eq\_rho11\]), (\[eq\_rho01\]) and for the choice $\delta=\Delta_1$, $M=16$ we expect that $\omega_1=\omega_R\,\Delta t=2\delta/(M\Delta_1)=1/M=0.0625$ and $\omega_2=2\omega_1=0.125$ which is indeed well confirmed by the fits shown in Fig. \[fig3\].
For larger values of the coupling strength $\varepsilon\ge 0.1$ the fits with the oscillatory terms do not work very well and have to be simplified to simple exponential fits, i.e. by omitting the term $\sim \tilde B_2$ in (\[eq\_f01\]) or replacing $\cos(\omega_1\tau+\alpha_1)\to 1$ in (\[eq\_f11\]). In Appendix \[appb\] we discuss certain cases, with low values of the level number $m$ of the initial state (\[eq\_init\_state\]) where the fit procedure is also problematic. However, in global the fits of the relaxation of the density matrix components work well and allow to determine the dependence of the relaxation rates $\Gamma_1, \Gamma_2$ on system parameters.
Qubit relaxation rates {#sec4}
======================
Dependence on coupling strength
-------------------------------
The relaxation rates are computed by the methods described in the previous Section. Here we analyze the dependence of these rates on system parameters. We note that according to usual cases of superconducting qubit relaxation [@averin; @shnirman; @wendin; @lee2005] the rate $\Gamma_2$ describes the dephasing of qubit while $\Gamma_1$ describes the population relaxation.
The obtained dependence of $\Gamma_1$ on the qubit coupling strength $\varepsilon$ is shown in Fig. \[fig4\] for the initial state $m=5720$ taken in the middle of the total energy band and the TBRIM values $J=0.15,\ 0.25,\ 1$ at $V=3.74166$ ($V_{\rm eff}/V_0= 1.0070$, $1.0193$, $1.2747$ and $\Delta_1 = 0.1475$, $0.1494$, $0.1868$ respectively) corresponding to the quantum dot regime and $J=1$ at $V=0$ ($V_{\rm eff}/V_0= 0.7905,\ \Delta_1=0.1158$) corresponding to the SYK black hole regime. For small coupling $\varepsilon < 0.1$ the results are well described by the quadratic dependence on coupling, typical for the Fermi golden rule regime: $$\label{eq_gammafermi1}
\Gamma_1 = C_1 \varepsilon^2 \; .$$ The fit value of the exponent is $p=2.00 \pm 0.02$ being compatible with the quadratic dependence.
The dimensionless constant $C_1$ in (\[eq\_gammafermi1\]) is practically independent of $J$ (at fixed $V=\sqrt{14}$) when the system is in the regime of dynamical thermalization being $C_1 \approx 23$ for $J=0.15,\ 0.25$ and $C_1 \approx 8$ for $J=1$. For the SYK case we find $C_1=4.6 \pm 0.3$ at $J=1,\ V=0$. We consider that this variation of $C_1$ is not significant since it changes only by a factor $5$ while $J^2$ is changed by a factor $44$ and in addition the model is changed from quantum dot to SYK regime. At such changes the total energy band width is also changed by a factor 2 (see Fig. \[fig1\]) but we remind that due to the definition of the model and parameters in Sections 2 and 3 both $\varepsilon$ and the relaxation rates are measured in units of effective energy (or inverse time) scales that take into account the modification of total energy band width due to different values of $V$ and $J$. We note that the dependence (\[eq\_gammafermi1\]) was also found for the dynamical relaxation of a qubit coupled to a deterministic detector described by the quantum Chirikov standard map [@lee2005] with $C_1 \approx 0.5$ corresponding to regime of the phase damping noise channel [@chang; @lee2005]. Here we obtain $C_1$ being by a factor 10 larger but in our model (\[eq\_ham\_qubit\]) the qubit is coupled with several TBRIM states and we assume that this is the reason for the increase of $C_1$.
For $\varepsilon > \varepsilon_c \approx 0.1$ we obtain a decrease of the relaxation rate described by the dependence $$\label{eq_gammazero1}
\Gamma_1 = C_1 \varepsilon^p \; , \; p =-1.15 \pm 0.02$$ with $C_1 \approx 0.002$. As in [@lee2005] we attribute this decrease of $\Gamma_1$ with increase of $\varepsilon$ to the quantum Zeno effect [@qzeno1; @qzeno2]: repeated measurements produced by a coupled detector, represented by TBRIM in the regime of quantum chaos, reduce the relaxation rate. In the so called ohmic relaxation regime it is expected that $\Gamma_1 \sim {\delta}^2/\Gamma_2
\sim B {\delta}^2/\varepsilon^2$ [@shnirman; @lee2005] (here $\delta=\Delta_1/2$). For the model of quantum chaos detector it was found that $B \approx 2.7$ [@lee2005]. Instead, here we find that the exponent $|p| =1.15 \pm 0.02 \approx 1$ being significantly different. We attribute this difference to the fact that in TBRIM the qubit is coupled to many one-particle states represented by a sum over $k$ in (\[eq\_ham\_qubit\]). For the numerical value $C_1 \approx 0.002$ we find that it is still approximately given by the relation $C_1 \approx B {\delta}^2$ with $B \approx 0.4$ being smaller than those in [@lee2005]. A surprising feature of the obtained quantum Zeno regime is that here $\Gamma_1$ is practically independent of parameter choice presented in Fig. \[fig4\] corresponding to DTC for the quantum dot and SYK quantum chaos regimes.
The transition between the Fermi golden mean regime ($ \Gamma_1 \propto \varepsilon^2$) and the quantum Zeno regime ($\Gamma_1 \propto 1/\varepsilon$) takes place at $\varepsilon_c \approx 0.07 - 1$. This corresponds to the relaxation rate $\Gamma_c=\Gamma_1(\varepsilon_c) \approx 0.05$ which remains practically the same for all parameter regimes presented in Fig. \[fig4\]. According to the results and arguments presented in [@lee2005; @lyapunov1; @lyapunov2] it is expected that $\Gamma_c$ is given by the Lyapunov exponent $\Lambda$ of an underlined classical dynamics of the detector coupled to qubit. Indeed, this was the case for the dynamical detector considered in [@lee2005], however, for the TBRIM it is more difficult to establish what is the Lyapunov exponent of the corresponding classical TBRIM dynamics. It would be possible to expect that $\Gamma_c$ can be related to the Breit-Wigner width $\Gamma \sim J^2 \rho_c$ appearing in the TBRIM in the Fermi golden rule for the transition between directly coupled states with the density $\rho_c$ [@georgeot1997]. However, the independence of $\Gamma_c$ of system parameters presented in Fig. \[fig4\] excludes this expectation.
We make the conjecture that for given parameters $\Gamma_c$ is determined by an effective time $T_c$ of spreading over the network of exponentially large size $d$ ($\ln d \sim M$ at large $M, L$ values) with a very small number of links (nonzero transition matrix elements): $N_l=K =820 \ll d = 11440$ (for $M=16$ and $L=7$). Such a network is similar to the small-world networks appearing in many cases of social relations [@milgram; @dorogovtsev]. It is known that a very rapid spreading takes place on such networks for classical [@dorogovtsev] and quantum spreading [@giraud] with a time scale $T_c$ being only logarithmic in system size $d$ (effect of [*six degrees of separation*]{} described in [@milgram; @dorogovtsev]). Thus about six transitions (links) are required to connect on average any pair of nodes on such networks (for the Facebook network there is only four degrees of separation [@vigna2012]). For typical networks like Wikipedia or WWW of universities there are only about $N_\ell \sim 10-20$ nonzero links per row/column in the full matrix of the network of size $d \sim 10^6$ [@rmp2015].
In the TBRIM case we have a much larger number of links per row/column and thus we expect that only about 2-4 transitions are sufficient to connect any two nodes (levels) of the system. Due to this we can expect that in this quantum small-world regime we have $\Gamma_c \sim C_d \Delta_1 $ with a numerical constant $C_d \approx 0.5$. The proportionality $\Gamma_c \propto \Delta_1$ appears since $\Delta_1$ plays a role of oscillator frequency (as for an oscillator) determining the time scale in the regime of explosive spreading over network, $C_d$ is inversely proportional to the degree of separation of the network which is of the order of 2-4 transitions for TBRIM since the number of links per column is much larger than for Wikipedia or Facebook networks. Thus we assume that this kind of explosive spreading, already discussed in [@giraud], is at the origin of the independence of $\Gamma_c$ of system parameters (for the range visible in Fig. \[fig4\]). We note that this kind of explosive spreading, with exponentially many states populated in a finite time, was also observed for the emergence of quantum chaos in a quantum computer core [@georgeotqc2] (see e.g. Fig.6 there).
 Dependence of the relaxation rate $\Gamma_1$ on the coupling strength $\varepsilon$ at level number $m=5720$ for the initial state (\[eq\_init\_state\]) for $V=3.74166$, $J=0.15$ (red plus symbols), $J=0.25$ (green crosses), $J=1$ (dark blue stars) and $V=0$, $J=1$ (pink squares) in a double logarithmic representation. The two lines correspond to the power law fits $\Gamma_1=C_1\,\varepsilon^p$ for $V=0$, $J=1$ with $C_1=4.6\pm 0.3$, $p=2.00\pm 0.02$ for $\varepsilon\le 0.1$ (light blue line) and $C_1=0.00219\pm 0.00006$, $p=-1.15\pm 0.02$ for $\varepsilon>0.1$ (black line). ](fig4){width="48.00000%"}
 Dependence of the relaxation rate $\Gamma_2$ obtained from the fit (\[eq\_f01\]) on the coupling strength $\varepsilon$ at level number $m=5720$ for the initial state (\[eq\_init\_state\]) for $V=3.74166$, $J=0.15$ (red plus symbols), $J=0.25$ (green crosses), $J=1$ (dark blue stars) and $V=0$, $J=1$ (pink squares) in a double logarithmic representation. The black line corresponds to the power law fit $\Gamma_2=C_2\,\varepsilon^p$ for the case $V=3.74166$, $J=1$ with $C_2=24\pm 8$, $p=2.02\pm 0.09$ and fit range $\varepsilon\le 0.1$. The data points with small full circles correspond to the relaxation rate $\tilde\Gamma_2$ of the oscillatory term for $\varepsilon<0.1$ in (\[eq\_f01\]) (same colors as other data points for different cases of $V$ and $J$). For $\varepsilon\ge 0.1$ the relaxation rate $\Gamma_2$ is obtained from a simplified exponential fit without oscillatory term. ](fig5){width="48.00000%"}
The dependence of the dephasing rate $\Gamma_2$ on the coupling strength $\varepsilon$ is shown in Fig. \[fig5\] for the parameters considered in Fig. \[fig4\]. In agreement with the usual expectations [@shnirman; @lee2005] we find $$\label{eq_gammazero2}
\Gamma_2 = C_2 \varepsilon^2 \; , \; C_2 = 24 \pm 8 \; .$$ Indeed, the numerical fit gives the exponent $p=2.02 \pm 0.09$ being very close to the Fermi golden rule value $p=2$. For the range $\varepsilon < \varepsilon_c \approx 0.7$ we have the approximate relation $\Gamma_1 \approx \Gamma_2$ as it was also found in [@lee2005] corresponding to the general results of Ref. [@shnirman]. We note that the fit results give for the other exponential decay rate $\tilde\Gamma_2$ of the oscillatory term in (\[eq\_rho01\]) (see full color circles in Fig. \[fig5\]) comparable values and parameter dependence as for $\Gamma_2$.
Dependence on excitation level number
-------------------------------------
The dependence of decay rates on the initial eigenvalue number $m$ (with eigenstate energy $E_{\rm ex}(m)$) is shown for $\Gamma_1$ in Fig. \[fig6\] and $\Gamma_2$ in Fig. \[fig7\]. All data are given for a weak qubit coupling $\varepsilon=0.01$ corresponding to the Fermi golden rule regime in Fig. \[fig4\]. The independence of $m$ is surprising since we know that the density of coupled states for effectively interacting electrons excited above the Fermi level $\epsilon_F$ on energy $\epsilon \approx T \ll \epsilon_F$ growth with energy as $\rho_c \approx T^3/{\Delta_1}^4$ (number of effectively interacting electrons is $\delta n \sim T/\Delta_1$ and the effective density of interacting two-particle states is $\rho_{2,\rm eff} \sim T/\Delta_1$ with $\rho_c \sim \rho_{2,\rm eff} (\delta n)^2$) and the interaction induced transition rate also grows with energy as $\Gamma_c \sim J^2 \rho_c
\sim J^2 T^3/{\Delta_1}^4$ [@aberg1; @jacquod]. Thus one could expect an increase of $\Gamma_1, \Gamma_2$ with an increase of $m$. The results presented in Figs. \[fig6\],\[fig7\] clearly show no increase with $m$ for the range $500 \leq m \leq 5720$, for the range $100 \leq m < 500$ there is also no increase with $m$ but the data is more fluctuating. These fluctuations become even stronger for the range $0 \leq m < 100$ so that the fits of relaxation decay in this range become not reliable (this is discussed in detail in Appendix Section B). The increase of fluctuations at low excitation numbers $m$ is natural since for lower $m$ values we have a decrease of number of states effectively coupled to the qubit. We note that the values of ${\tilde \Gamma_2}$, shown by full circles in Fig. \[fig7\], show a similar behavior as $\Gamma_2$ (with somewhat larger fluctuations at low $m$ values since the corresponding fit (\[eq\_f01\]) is more sensitive to errors).
 Dependence of the relaxation rate $\Gamma_1$ on the level number $m$ used for the initial state (\[eq\_init\_state\]) at coupling strength $\varepsilon=0.01$ for $V=3.74166$, $J=0.15$ (red plus symbols), $J=0.25$ (green crosses), $J=1$ (dark blue stars) and $V=0$, $J=1$ (pink squares) in a double logarithmic representation. ](fig6){width="48.00000%"}
 Dependence of the relaxation rate $\Gamma_2$ obtained from the fit (\[eq\_f01\]) on the level number $m$ used for the initial state (\[eq\_init\_state\]) at coupling strength $\varepsilon=0.01$ for $V=3.74166$, $J=0.15$ (red plus symbols), $J=0.25$ (green crosses), $J=1$ (dark blue stars) and $V=0$, $J=1$ (pink squares) in a double logarithmic representation. The data points with small full circles correspond to the relaxation rate $\tilde\Gamma_2$ of the oscillatory term in (\[eq\_f01\]) (same colors as other data points for different cases of $V$ and $J$).](fig7){width="48.00000%"}
Thus even if the variation of $m$ is rather large (factor $10$ or $50$) the variation of $\Gamma_1, \Gamma_2$ remains in the same range as in Figs. \[fig4\], \[fig5\] being restricted approximately by a factor $5$. We explain this independence of $m$ in the same manner as in previously arguing that for $m > 100$ the transitions between non-interacting many-body states proceed in an explosive spreading typical on small-world networks in a regime $\Gamma_1 \approx \Gamma_2 \sim 30 \Delta_1 \gg \Delta_1$. In fact for $m \approx 100, J=0.15$ we find $\delta E \approx T^2/\Delta_1 \approx 1.4$ with $\Delta_1 =0.1475$ that gives $T \approx 0.45$ (see Fig. \[fig1\]) and from above estimates we obtain $\Gamma_c/\Delta_1 \approx 30$. This ratio becomes even larger for other parameters of Figs. \[fig4\], \[fig6\].
TBRIM as a quantum small-world network
======================================
 Frequency distributions $N_f(N_l)$ of link number $N_l$ per node (right panels) and $N_f(N_E)$ of Erdös number $N_E$ (left panels) for an effective network constructed from $H_I$ where states/nodes $i$ and $j$ are connected by a link if the condition $|(H_I)_{ij}|>C|(H_I)_{ii}-(H_I)_{jj}|$ with the cut value $C=0.1$ (top panels) or $C=1$ (bottom panels) is met. The Erdös number $N_E$ of a node represents the minimal number of links necessary to connect indirectly this node via other intermediate nodes to the hub $=0$ corresponding to the many-body state where first $7$ out of $16$ orbitals are occupied. The hub itself has $N_E=0$ and the value $N_E=-1$ indicates that a node cannot be indirectly connected to the hub. Color of curves/data points is red (SYK, $V=0$, $J=1$), green ($V=\sqrt{14}$, $J=0.025$), blue ($V=\sqrt{14}$, $J=0.25$) and pink ($V=\sqrt{14}$, $J=1$). For these four cases respectively the mean and the width of the distribution of $N_l$ are: $532\pm 61$, $3.35\pm 1.87$, $33.7\pm 7.3$, $129\pm 20$ ($C=0.1$, top right panel) and $91.8\pm 25.2$, $0.346\pm 0.586$, $3.36\pm 1.89$, $13.4\pm 4.09$ ($C=1$, bottom right panel); also the mean and the width of the distribution of $N_E$ (not counting $N_E=-1$ cases) are: $2.24\pm 0.53$, $0.211\pm 0.464$, $16.1\pm 4.86$, $4.63\pm 1.12$ ($C=0.1$, top left panel) and $3.40\pm 1.09$, $0\pm 0$, $0.469\pm 1.10$, $41.2\pm 13.7$ ($C=1$, bottom left panel). In bottom left panel the green curve is completely hidden by the blue curve and contains only two values $N_f(-1)=11439$ and $N_f(0)=1$ meaning that the hub is not connected to any other node. All curves were obtained from an average of 100 different random realizations of $H_I$ for $M=16$, $L=7$ and $d=11440$. The vertical axis represents the number $N_f$ of nodes having the link number $N_l$ (right panels) or having the Erdös number $N_E$ (left panels). ](fig8){width="48.00000%"}
 Frequency distribution $N_f(N_E)$ of Erdös number $N_E$ for the same effective network of Fig. \[fig8\] for the hub $=5720$ corresponding to the many-body state where orbitals: 3, 4, 8, 9, 12, 14, 15 out of $16$ orbitals are occupied. The signification of $N_E$ is as in Fig. \[fig8\] and the value of $N_E=-1$ corresponds to the case of nodes not connected to the hub. Top (bottom) panels correspond to the cut value $C=0.1$ ($C=1$). Left panels correspond to the range $-2\le N_E\le 500$ and a logarithmic representation for $N_f(N_E)$ and right panels correspond to a zoomed range $-2\le N_E\le 15$ and normal representation for $N_f(N_E)$. Color of curves/data points is red (SYK, $V=0$, $J=1$), green ($V=\sqrt{14}$, $J=0.025$), blue ($V=\sqrt{14}$, $J=0.25$) and pink ($V=\sqrt{14}$, $J=1$). For these four cases respectively the mean and the width of the distribution of $N_E$ (not counting $N_E=-1$ cases) are: $2.24\pm 0.53$, $96.5\pm 74.9$, $6.22\pm 3.03$, $2.90\pm 0.74$ ($C=0.1$, top panels) and $3.42\pm 1.10$, $0.371\pm 0.622$, $91.7\pm 64.7$, $15.1\pm 9.5$ ($C=1$, bottom panels). All curves were obtained from an average over the same 100 different random realizations of $H_I$ ($M=16$, $L=7$, $d=11440$) used in Fig. \[fig8\]. ](fig9){width="48.00000%"}
Above we proposed an analogy between the TBRIM and a quantum small-world networks studied in [@giraud; @giraud2], tracing parallels with the small-world networks in social relations [@milgram; @dorogovtsev]. On a first glance this analogy may look to be strange since for TBRIM the number of nonzero matrix elements per row/column of the Hamiltonian matrix is fixed being $K$ while the small-world networks are characterized by a broader distribution of links [@dorogovtsev]. However, for the TBRIM the physical relevant quantity is not the formal number of nonzero elements but the number of effectively directly coupled states. As was discussed above and in [@aberg1; @jacquod] the density and number of such states depends on energy (this is especially visible in proximity of the Fermi energy). According to quantum perturbation theory we need to count only transitions for which the transition matrix element is at least comparable to the energy detuning between the states involved in the transition. For the states with large energy detunings the effective probabilities (weights) of the transitions become small and their influence can be neglected, at least in a first approximation.
Therefore we construct from the TBRIM Hamiltonian $H_I$ defined in (\[eq\_TBRIM\]) an effective (symmetric undirected) network where two many-body states i and j are coupled by a link if the condition $|(H_I)_{ij}|>C|(H_I)_{ii}-(H_I)_{jj}|$ is met where $C$ is a parameter of order unity which we either choose $C=0.1$ or $C=1$. This can be considered as a numerical selection following the [Å]{}berg criterion [@aberg1; @jacquod]. A similar procedure has been considered in [@prelovsek] for spin chains. We emphasize that the diagonal matrix elements $(H_I)_{ii}$ are constituted of two contributions: the first term in (\[eq\_TBRIM\]) given by the sum of energies of occupied orbitals and certain non-vanishing diagonal contributions from the interaction which have, according to the discussion in Appendix \[appa\], a variance which is $L(L-1)$ larger than the variance of the non-diagonal interaction matrix elements (for the case where two occupied orbitals differ between the two states). For the limit $J\ll V$ the diagonal matrix elements are of course dominated by the orbital energy contribution but even for the SYK case with vanishing orbital energies ($V=0$, $J=1$) the diagonal terms have a considerable size due to the diagonal interactions.
Using this kind of network model we determine the frequency distribution $N_f(N_l)$ of number of links per node $N_l$ for the four cases $V=0$, $J=1$ (SYK case with strongest interactions and quantum chaos), $V=\sqrt{14}$, $J=0.025$ (weak interactions without dynamical thermalization), $V=\sqrt{14}$, $J=0.25$ (moderate interactions with dynamical thermalization) and $V=\sqrt{14}$, $J=1$ (strong interactions with dynamical thermalization). Furthermore we choose our standard parameters $L=7$, $M=16$ giving a matrix dimension $d=11440$ of $H_I$ and the number of non-zero couplings elements per state $K=820$ which is an obvious upper bound for $N_l$. As can be seen in Fig. \[fig8\] the frequency distribution of $N_l$ is not a power law and not scale free. Essentially the criterion in terms of diagonal energy differences implies that the typical link number $N_l$ is a certain fraction of $K$ which does not fluctuate too strongly for different initial states. However, this fraction is smallest for $V=\sqrt{14}$, $J=0.025$ with a maximal value $N_{f,{\rm max}}=16$ (if $C=0.1$) or $7$ (if $C=1$) and largest for the SYK case $V=0$, $J=1$ with $N_{f,{\rm max}}=687$ (if $C=0.1$) or $217$ (if $C=1$). According to Fig. \[fig8\] the frequency distribution of $N_l$ provides largest values for the case of strongest coupling (SYK, $V=0$, $J=1$) and smallest values for the case of weakest coupling ($V=\sqrt{14}$, $J=0.025$). The choice $C=1$ as compared to $C=0.1$ provides a general shift to smaller values. Actually, for $V=\sqrt{14}$ the case $C=1$, $J=0.25$ is rather comparable to $C=0.1$ and $J=0.025$ which is rather obvious since reducing the constant $C$ by a certain factor corresponds to reducing the typical interaction couplings by the same factor. However, these two cases are not perfectly identical and the remaining small differences are due to complications from the diagonal interaction matrix elements in $(H_I)_{ii}$.
In global we see that the frequency distribution of links $N_f(N_l)$ is peaked near a certain average value that can be viewed as a broadening of the delta-function distribution of random graphs introduced by Erdös-Rényi [@erdos], known as the Erdös-Rényi model [@dorogovtsev]. Below we check if our quantum network possesses the small-world property typical for the social networks [@milgram; @vigna2012; @dorogovtsev]
With this aim we compute a more interesting quantity which we call the Erdös number $N_E$. This number represents the minimal number of links necessary to connect indirectly a specific node via other intermediate nodes to a particular node called the hub.
We choose as hub two example states at index values $0$ and $5720$ in the many body Hilbert space of dimension $d=11440$ (at $M=16$ and $L=7$). In our numerical mapping of states (i.e. the way the many-states are enumerated) the hub $=0$ corresponds to the state where the the first $L$ of the $M$ (i.e. first 7 of 16) orbitals are occupied. Since we have chosen the orbital energies ordered with respect to the orbital index number (see text below Eq. (\[eq\_ham\_qubit\])) this state corresponds to the non-interacting ground state, i.e. the Fermi sea, for the case $V>0$ and $J=0$. According to the Gaussian density of states this implies that typical energy differences of this state with the next excited states are rather large and therefore this hub is quite “badly” coupled to other nodes in our network model.
The other hub $=5720$ corresponds roughly to a state in the middle of the non-interacting energy spectrum (for $V>0$ and $J=0$) and in our numerical mapping this corresponds to the state where the 7 orbitals: 3, 4, 8, 9, 12, 14, and 15 are occupied. Here the typical energy differences with respect to neighbor states are quite small.
Therefore, for the three cases with $V=\sqrt{14}$ we expect there will be a considerable difference in the connectivity between both hubs. However, for the SYK case with $V=0$ and $J=1$ the residual diagonal energies in $H_I$ of these states (due to the interaction) are really fully random and both hubs are statistically expected to be equivalent and rather well connected.
The Erdös number corresponds roughly to the ergodic time scale (in units of link-iterations) for the classical stochastic dynamics induced by the network. Depending on the typical coupling strength of the network it is possible that certain or even many nodes are not at all coupled to the hub by indirect links, especially for the hub $=0$ (if $V>0$). In this case we attribute artificially the value $N_E=-1$ to such topologically separated nodes from the hub, while the hub itself has $N_E=0$ and the remaining nodes (indirectly coupled to the hub) have values $N_E>0$.
The frequency distribution $N_f(N_E)$ of the Erdös number $N_E$ for hub $=0$ is shown in the left panels of Fig. \[fig8\]. For the SYK case (strongest coupling) the distribution is strongly peaked with typical values at $\sim 2$ ($\sim 3$) for $C=0.1$ ($C=1$). Then with decreasing coupling (or increasing value of $C$) the width and mean values of the distribution increase provided there is still a sufficient fraction of nodes (indirectly) coupled to the hub. For the cases of weakest coupling $V=\sqrt{14}$, $J=0.025$ (if $C=0.1$) or $J\le 0.25$ (if $C=1$) nearly all nodes are not at all coupled to the hub as can be seen from the strong peaks at $N_E=-1$. The mean and width of the distribution of the few number of remaining nodes (eventually only the hub itself) is very small. For the two cases $V=\sqrt{14}$, $J=0.25$ (if $C=0.1$) or $J=1$ (if $C=1$) there is a large fraction of isolated nodes but there are still enough remaining nodes coupled to the hub providing a non-trivial distribution of largest values $\sim 40$ or $\sim 90$ respectively. Apart from the SYK cases only the case $V=\sqrt{14}$, $J=1$ at $C=0.1$ provides a strongly peaked distribution with typical value at $4.6\pm 1$.
Fig. \[fig9\] shows the frequency distribution $N_f(N_E)$ of the Erdös number $N_E$ for the other hub $=5720$. As expected the two SYK cases are very similar to the first hub $=0$ of Fig. \[fig8\]. However for the cases with $V=\sqrt{14}$ the connectivity is indeed “better” as compared to Fig. \[fig8\], i.e. either the typical values are smaller or there are less isolated nodes (lower or absent peaks at $N_E=-1$). Especially the two cases $J=0.025$ (if $C=0.1$) or $J=0.25$ (if $C=0$), with nearly only isolated nodes in Fig. \[fig8\], provide now a non-trivial rather large distribution for a modest fraction of non isolated nodes. Furthermore, these two cases are actually quite comparable as already discussed above for the frequency distribution of links.
 Frequency distribution $N_f(N_l)$ of link number $N_l$ per node (right panels) and probability distribution $w_f(N_E)$ of Erdös number $N_E$ (left panels) for an effective network constructed from $H_I$ as in Fig. \[fig8\] but only using nodes/states satisfying the energy condition : $|(H_I)_{ii}-(H_I)_{{\rm hub,hub}}|<1.5\Delta_1$ for hub $=5720$. Color of curves/data points is red (SYK, $V=0$, $J=1$), green ($V=\sqrt{14}$, $J=0.025$), blue ($V=\sqrt{14}$, $J=0.25$) and pink ($V=\sqrt{14}$, $J=1$). For these four cases respectively the mean and the width of the distribution of $N_l$ are: $323\pm 67$, $3.50\pm 1.87$, $28.8\pm 7.0$, $69.9\pm 10.4$ ($C=0.1$, top right panel) and $89.3\pm 23.4$, $0.368\pm 0.603$, $3.50\pm 1.90$, $13.4\pm 4.0$ ($C=1$, bottom right panel); the mean and the width of the distribution of $N_E$ (not counting $N_E=-1$ cases) are: $2.18\pm 0.56$, $12.0\pm 6.1$, $2.54\pm 0.63$, $2.23\pm 0.58$ ($C=0.1$, top left panel) and $2.51\pm 0.55$, $0.371\pm 0.622$, $10.2\pm 4.5$, $3.41\pm 0.85$ ($C=1$, bottom left panel); the average effective dimension/reduced network size is: $4107$, $869$, $883$, $1102$ (all panels). Left panels show the probability distribution $w_f(N_E)$ normalized to unity for a better visibility as compared to $N_f(N_E)$ (shown in Figs. \[fig8\] and \[fig9\]) with different normalizations due to different network sizes. As in Fig. \[fig8\] the case $N_E=-1$ represents nodes which cannot be reached by the hub. All curves were obtained from an average over the same 100 different random realizations of $H_I$ ($M=16$, $L=7$, $d=11440$) used in Fig. \[fig8\]. ](fig10){width="48.00000%"}
The last two cases correspond to a (partial) ergodicity but only after a large number of network iterations. This observation may be related to a diffusive dynamics in energy space where it takes some time to explore different energy layers such that the networks are not really of small-world type. Therefore we also consider a reduced network where we keep only nodes/states whose diagonal energies are relatively close to the diagonal energy of the hub, i.e. such that the energy condition $|(H_I)_{ii}-(H_I)_{{\rm hub,hub}}|<1.5\Delta_1$ for the hub $=5720$ is satisfied and where $\Delta_1$ is the effective rescaled average one-particle level spacing introduced in Section 2 (see text below Eq. (\[eq\_a\_coeff\])). We remind that $\Delta_1$ is small compared to the overall energy band width but typically large compared to the many body level spacing and also with respect to the effective level spacing of directly interaction coupled states [@jacquod; @georgeot1997]. As a consequence of this condition the effective dimension or network size of remaining nodes/states is considerably reduced to values $\sim 4000$ for the SYK case or $\sim 1000$ for the three cases with $V=\sqrt{14}$. The modified distributions for this reduced network of link number $N_l$ and Erdös number $N_E$ are shown in Fig. \[fig10\]. The frequency distribution $N_f(N_l)$ is similar as in Fig. \[fig8\] with a clear ordering of typical sizes from strongest coupling (SYK) to weakest coupling ($V=\sqrt{14}$ and $J=0.025$) and an overall shift from $C=0.1$ to $C=1$. The distribution of Erdös numbers for SYK is not changed (apart from the modified normalization) while the cases with $V=\sqrt{14}$ are now generally closer to a small-world situation. Here $J=1$ is now identical (close) to SYK, $J=0.25$ provides typical Erdös numbers $\sim 2-3$ ($\sim 10$), and the case $J=0.025$ corresponds to a typical Erdös number $\sim 12$ (majority of nodes isolated from hub) all for $C=0.1$ ($C=1$). This clearly confirms that large Erdös numbers $\sim 10^2$ of the full network before correspond to diffusion to other energy layers.
The data of Figs. \[fig8\]-\[fig10\] clearly show that the TBRIM is characterized by small-world properties provided the interaction strength is sufficiently large. Especially the SYK case with an average Erdös number $\langle N_E\rangle =2.2\pm 0.5$ ($3.4\pm 1$) for $C=0.1$ ($C=1$) shows very strong small-world properties. However, for modest interaction strength there are some complications due to diffusion in energy space leading to possible Erdös numbers $\sim 10^2$. We think that the further development of the analogy between quantum many-body interacting systems and small-world networks will bring a better understanding of these quantum systems.
We note that the small-world network constructed for an energy layer of a finite width (we use the width of $3\Delta_1$) is more relevant for the qubit relaxation analyzed in previous Sections: the coupling of the qubit with the states inside this layer leads to its rapid relaxation on the time scale related to $\Gamma_c \sim \Delta_1$, while slow transitions from one energy layer to another layer describe the residual level of density matrix relaxation analyzed in the next Section.
Residual level of density matrix relaxation
===========================================
Our TBRIM model contains a finite number of states $d$ and hence the relaxation of density matrix components stops at a certain residual level of density matrix elements $|\rho_{01}|$ determined by quantum deterministic fluctuations and noise. In fact since the spectrum of our system is discrete and the system is bounded we will always have the Poincaré recurrences to the initial state in agreement with the Poincaré recurrence theorem [@poincare]. However, the time $t_r$ of such a recurrence grows exponentially with the system size $\ln t_r \propto d$ being enormously large even for our case with $L=7$ particles. However, depending on the initial state and parameters it is possible that the effective number $d_{\rm eff}$ of excited states contributing in the exact time evolution is much smaller than $d$ implying that for these cases $t_r$ is strongly reduced. Therefore we compute the deterministic residual level of quantum fluctuations given by $|\rho_{01}|$ averaged over long times for $d/4\le t/\Delta t\le d/2$ roughly corresponding to $t_H/2\le t \le t_H$ (or $|\rho_{00}-1/2|$ with similar results).
 Density plot of residual level of quantum fluctuations determined as the time average of $|\rho_{01}(t)|$ at long times for $d/4\le /\Delta t\le d/2$. The horizontal axis corresponds to the level number $m=0,1,2,3,4,7,11,18,29,$ $47,76,122,198,320,517,836,1353,2187,3537,5720$ of the initial state (\[eq\_init\_state\]) and the vertical axis corresponds to the value of the interaction strength $J$ at $V=\sqrt{14}\approx 3.74166$ except for the top row (with symbol “$J=\infty$”) representing the SYK case $J=1$ and $V=0$. The coupling strength is $\varepsilon=0.03$. The colors red, green or blue correspond to maximum $|\rho_{01}(t)|=0.4353$, intermediate or minimum (zero) fluctuation values (they are shown by color bar on top with numbers showing the percentage of maximal value). ](fig11){width="48.00000%"}
The dependence of the residual level of quantum fluctuations on $J, m$ is shown in Fig. \[fig11\]. The lowest level is at the middle of energy band with $m=5720$ corresponding to infinite temperature $T$, The highest level is found for the ground state $m=0$ and first excited states $m=1,2$ with $J<1$ at $V=\sqrt{14}$. The amplitude of residual fluctuations decreases with increase of $J$ but it is difficult to establish a clear border in $(J,m)$ plane. We attribute this to the fact that the [Å]{}berg border (\[eq:abergcriterion\]) works mainly for small $J$ values with $g \gg 1$ so that a special analysis of this region is required that was not the main aim of this work.
We note that the residual fluctuations are rather similar for the SYK regime at $J>10$ and the quantum dot regime above the [Å]{}berg border (\[eq:abergcriterion\]) with $0.15 \leq J < 10$ (except very low excited levels $m<7$ and $J<0.5$). We attribute this to the fact that in this region $\Gamma_c \gg \Delta_1$ leading to the explosive spreading over the quantum small-world.
In analogy with [@lee2005] we expect that in the regime of developed quantum chaos the residual level $R_q$ of quantum fluctuations of qubit drops as a square-root of the states of a detector $R_q \propto 1/\sqrt{d}$. However, the quantum computations for TBRIM detector are more complicated compared to the kicked rotator case and we did not performed detailed numerical checks of this relation which is however in a qualitative agreement with the results of Fig. \[fig11\].
 Time dependence of $|\rho_{01}(t)|$ at level number $m=5720$ for the initial state in (\[eq\_init\_state\]) for various values of the parameter $\delta$ according to: $0.3\le 2\delta/\Delta_1\le 6$. The horizontal axis for the time is shown in logarithmic representation for a better visibility. Top (bottom) panels correspond to coupling strength $\varepsilon=0.1$ ($0.01$). Left (right) panels correspond to $V=3.74166$, $J=1$ ($V=0$, $J=1$). ](fig12){width="48.00000%"}
Finally we make a note on the relaxation dependence of the qubit energy given by $2\delta$. Above we presented results for a fixed value $\delta = \Delta_1$ but we checked that the relaxation of density matrix components goes in a similar manner for other values of the ratio $0.3 \leq 2\delta/\Delta_1 <3$ as it is shown in Fig. \[fig12\] The changes of the decay curves start to be visible for $2\delta/\Delta_1 \ge 3$ but in this range the qubit energy becomes comparable to the energy size of the TBRIM band that corresponds to another physical regime where the qubit cannot be considered as a weak perturbation.
We also mention that our above discussion of the properties of qubit relaxation concern the range of positive temperatures with $m \leq d/2$. The regime of negative temperatures is briefly discussed in Appendix C where we find comparable results for the qubit relaxation as in the regime of positive temperatures. This is also in agreement with spin relaxation at negative temperatures considered in [@abragam].
Discussion
==========
We presented results for a dynamical decoherence of a qubit weakly coupled to the TBRIM system in the regime of dynamical thermalization induced by interactions and quantum many-body chaos, corresponding to the quantum dot of interacting fermions and the SYK black hole model. The relaxation rates of qubit population $\Gamma_1$ and dephasing $\Gamma_2$ are determined as a function of qubit coupling strength $\varepsilon$ with $\Gamma_1 \propto \varepsilon^2$ in the Fermi golden rule regime and $\Gamma_1 \propto 1/\varepsilon$ in the quantum Zeno regime with $\Gamma_2 \propto \varepsilon^2$ for the whole considered range. These results are in a satisfactory agreement with the usual thermal bath qubit decoherence considered in the literature (see e.g. [@shnirman]). The surprising finding of our studies is that the values of $\Gamma_1, \Gamma_2$ remain practically unchanged in a broad range of parameters of the quantum dot or the SYK model. We propose a tentative explanation of this effect by tracing an analogy between TBRIM system and quantum small-wold networks with appearance of explosive spreading over exponential number of sites (states) in a finite time. This explosive spreading appears in both regimes of quantum dot and SYK when the transition rates between directly coupled states become larger than an effective level spacing between one-particle states. We hope that our results will stimulate further investigations of dynamical decoherence in quantum many-body interacting systems and a further development of parallels between these systems and the small-world networks.
This work was supported in part by the Pogramme Investissements d’Avenir ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT (project THETRACOM); it was granted access to the HPC resources of CALMIP (Toulouse) under the allocation 2017-P0110.
Gaussian density of states {#appa}
==========================
Analytical computation of the variance
--------------------------------------
The TBRIM Hamiltonian $H_I$ given by (\[eq\_TBRIM\]) exhibits in the limit $M\to\infty$ at a fixed value of particle number $L$ an average density of states which is obviously Gaussian in absence of interaction ($J=0$) since in this case the many body energy levels are given by $E(\{n_j\})=\sum_j n_j\,\epsilon_j$ with $n_j\in\{0,\,1\}$ which is a sum of random Gaussian variables with vanishing average and variance: $$\begin{aligned}
\nonumber
\sigma_\epsilon^2&=&\left\langle E(\{n_j\})^2\right\rangle
= \sum_{j,j'=1}^M n_j\,n_{j'}\,
\langle \epsilon_j\,\epsilon_{j'}\rangle\\
\label{eq_envar1}
&=&\frac{1}{M}\sum_{j=1}^M n_j^2 V^2=\frac{L}{M}\,V^2\ .\end{aligned}$$ However, the expression (\[eq\_envar1\]) requires to take the ensemble average over the one-particle energies $\epsilon_j$, i.e. the numerical verification of the variance requires an average over many realizations and from a pure mathematical point of view the Gaussian form of the distribution of $E(\{n_j\})$ requires indeed the limit $M\to\infty$ at fixed value $L$ (and of $V^2/M\to$const.) providing a sum of [*independent*]{} Gaussian variables.
On the other hand, we find numerically that the density of states is very close to a Gaussian distribution already for one sample of $H_I$ at the values of $M$ and $L$ we considered. To understand this let us first consider $J=0$ and let $\epsilon_j$ be one sample of one-particle energies initially drawn from a Gaussian distribution (with zero mean and variance $V^2/M$) and then slightly modified by a small universal shift and rescaling factor to ensure [*exactly*]{} that $\sum_j \epsilon_j=0$ and $\sum_j \epsilon_j^2=V^2$. Now we consider this set of one-particle energies fixed and perform the average of $E(\{n_j\})$ not with respect to $\epsilon_j$ but with respect to all configurations $n_j\in\{0,\,1\}$ such that $L=\sum_j n_j$. In this case we have obviously $\langle n_j\rangle =L/M$. Furthermore we find: $$\label{eq_L2average}
L^2=\sum_{j,j'=1}^M \langle n_j\,n_{j'}\rangle=
M\langle n_j\rangle +M(M-1) \langle n_j\,n_{j'}\rangle_{j\neq j'}$$ where we have separated the terms with $j=j'$ from those with $j\neq j'$. From (\[eq\_L2average\]) we immediately find: $$\label{eq_njnjp}
\langle n_j\,n_{j'}\rangle_{j\neq j'}=\frac{L(L-1)}{M(M-1)}$$ and therefore we get, for $J=0$, a different variance with respect to (\[eq\_envar1\]): $$\begin{aligned}
\nonumber
\sigma^2(0)&=&\left\langle E(\{n_j\})^2\right\rangle =
\frac{L}{M}\sum_{j=1}^M \epsilon_j^2
+\frac{L(L-1)}{M(M-1)}\sum_{j\neq j'}^M \epsilon_j\,\epsilon_{j'}\\
\label{eq_envar2}
&=&\frac{L}{M}\left(1-\frac{L-1}{M-1}\right)\sum_{j=1}^M \epsilon_j^2
=\frac{L(M-L)}{M(M-1)}\,V^2\ .\end{aligned}$$ To obtain (\[eq\_envar2\]) we have used that: $$\label{eq_used}
\sum_{j\neq j'}^M \epsilon_j\,\epsilon_{j'}=\left(
\sum_{j=1}^M \epsilon_j\right)^2-\sum_{j=1}^M \epsilon_j^2
=-\sum_{j=1}^M \epsilon_j^2$$ since $\sum_j\epsilon_j=0$ by choice.
Now we consider a non-vanishing interaction strength $J\neq 0$. In the limit for sufficiently small $J$ we expect that the density of states is not affected by $J$. If we assume that the density of states remains Gaussian, also for larger values of $J$, (see below for the numerical confirmation of this) we can compute the variance $\sigma^2$ from the average: $$\begin{aligned}
\nonumber
\sigma^2&=&\frac{1}{d}\int_{-\infty}^\infty E^2\,\langle\rho(E)\rangle\,dE
=\frac{1}{d}\left\langle \sum_{m=0}^{d-1} E_m^2\right\rangle\\
\label{eq_envar3}
&=& \frac{1}{d} \langle\mbox{Tr}(H_I^2)\rangle
=\sigma^2(0)+\frac{1}{d}\langle\mbox{Tr}(H_J^2)\rangle\end{aligned}$$ where $E_m$ are the exact many body energies, $\sigma^2(0)$ is the variance at $J=0$ given in (\[eq\_envar2\]) and $$\label{eq_HJ}
H_J=\frac{4}{\sqrt{2M^3}}\sum_{i<j,k<l} J_{ij,kl}\,c^\dagger_i c^\dagger_j
c^\pdag_l c^\pdag_k$$ is the interaction contribution in (\[eq\_TBRIM\]). In (\[eq\_envar3\]) the average is done at fixed one-particle energies with respect to the different configurations of the occupation numbers $n_j$ (satisfying $\sum_j n_j=L$) and with respect to the Gaussian interaction matrix elements $J_{ij,kl}$. To evaluate $(1/d)\langle\mbox{Tr}(H_J^2)\rangle$ let us consider one particular many body state where exactly $L$ of the $M$ orbitals are occupied. This state is coupled by the interaction to three groups of other states: (i) “itself”, i.e. with identical occupation numbers $n_j$, (ii) $L(M-L)$ states that differ exactly for one particle occupying another orbital, and (iii) $L(L-1)(M-L)(M-L-1)/4$ states that differ exactly for two particles occupying other orbitals. This corresponds to a total number of coupled states $1+L(M-L)+L(L-1)(M-L)(M-L-1)/4$, an expression already given in [@jacquod; @flambaum].
However, in order to evaluate the contributions of the corresponding interaction matrix elements in $\langle\mbox{Tr}(H_j^2)\rangle$ this (global) number is not relevant since the average variance of the interaction matrix element differs between these three groups. The interaction matrix element of the state with itself, corresponding to the group (i), uses $L(L-1)/2$ terms of (\[eq\_HJ\]) since there are $L(L-1)/2$ possibilities to destroy a pair of particles in the set of given $L$ particles and to recreate them afterwards in their same original orbitals. This corresponds to a sum of $L(L-1)/2$ independent Gaussian variables $J_{ij,ij}$ with variance[^1] $2J^2$, giving a contribution in $\langle\mbox{Tr}(H_j^2)\rangle$ being $(8/M^3)\,J^2 L(L-1)$.
Concerning the group (ii), we need to consider in (\[eq\_HJ\]) the index pairs $i<j$ and $k<l$ where one index of the first pair is identical to one index of the other pair and the other one is different. This gives $L-1$ possibilities to destroy the pair of particles and recreate them afterwards such that one of the two particles stays in the same orbital and the other one has changed its orbital. Therefore the total contribution of all states of the group (ii) to $\langle\mbox{Tr}(H_j^2)\rangle$ is $(8/M^3)\,J^2 L(M-L)(L-1)$.
Concerning the group (iii) both indices must be different and there is only one term in (\[eq\_HJ\]) contributing to the interaction matrix element. Hence the total contribution of all states of the group (iii) to $\langle\mbox{Tr}(H_j^2)\rangle$ is $(8/M^3)\,J^2 L(L-1)(M-L)(M-L-1)/4$.
This argumentation does not depend on the choice of initial state giving a factor $d$ canceling the factor $1/d$ in (\[eq\_envar3\]). Putting this all together, we obtain from (\[eq\_envar3\]) the expressions (\[eq\_DOS\_TBRIM\]), (\[eq\_Veff\]) and (\[eq\_a\_coeff\]) given in the main text for $\sigma$ in terms of the effective energy scale $V_{\rm eff}$ and the coefficient $a(M,L)$ which measures the global energy rescaling due to finite values of $J/V$.
Numerical verification
----------------------
In order to verify numerically the Gaussian density of states with the theoretical variance given in (\[eq\_DOS\_TBRIM\]) it is more convenient to determine the integrated density of states: $$\label{eq_IDOS}
P(E)=\frac{1}{d}\int_{-\infty}^E \rho(\tilde E)\,d\tilde E\ .$$ The prefactor $1/d$ assures the limit $\lim_{E\to\infty} P(E)=1$ since $\rho(E)$ is chosen to be normalized to $d$ and not unity. In case of an ideal Gaussian density of states, as in (\[eq\_DOS\_TBRIM\]), we have: $$\label{eq_IDOS_gauss}
P(E)=
P_{\rm gauss}(E)=\frac{1}{2}\left(1+\mbox{erf}\left(\frac{E}{\sqrt{2}\sigma}
\right)\right)$$ with $\mbox{erf}(x)=(2/\sqrt{\pi})\int_0^x \exp(-y^2)\,dy$.
If $E_m$ represent the numerically computed eigenvalues (of one sample of $H_I$ and ordered in increasing order with level number $m=0,\ldots,d-1$) the integrated density of states is simply obtained by drawing the quantity $z_m=(m+0.5)/d$ versus $E_m$ which gives the appearance of a rather smooth curve for a sufficiently large value of $d$ which can be compared to the expression (\[eq\_IDOS\_gauss\]). In order to perform a more sophisticated fit analysis we generalize (\[eq\_IDOS\_gauss\]) to: $$\label{eq_IDOS_gauss2}
P_k(E)=\frac{1}{2}\left(1+\mbox{erf}\left(q_k(E)/\sqrt{2}\right)\right)$$ where $q_k(E)$ is a polynomial of degree $k$. The case $k=1$ with $q_1(E)=(E-E_c)/\sigma_{\rm fit}$ corresponds to a Gaussian density of states with variance $\sigma_{\rm fit}$ and center $E_c$. Choosing larger values of $k>1$ we may analyze deviations with respect to the ideal Gaussian distribution. From the practical point of view a direct fit of $z_m$ with $P_k(E_m)$ is a bit tricky because it is non-linear and it is easier to perform the least-square minimization not in the vertical but in the horizontal axis. To do this explicitly let, for $0<x<1$, the function $\mbox{inverf}(x)$ be defined as the inverse of $\mbox{erf}(x)$ such that $\mbox{erf}(\mbox{inverf}(x))=x$. Then we apply the fit $q_k(E_m)=\sqrt{2}\;\mbox{inverf}(2z_m-1)$ which is linear in the coefficients of the polynomial $q_k(E)$ and provides a unique well defined solution.
We have applied this fit for the two cases $k=1$ and $k=5$, for many different values the ratio $J/V$ covering many orders of magnitude and our standard choice $M=16$, $L=7$ with $d=11440$. In all cases the hypothesis of an approximate Gaussian density of states is well confirmed with a value of $\sigma_{\rm fit}$ confirming the theoretical expression in (\[eq\_DOS\_TBRIM\]) with an error below 1%. As an additional verification, we have also numerically determined the variance from the trace, i.e. the quantity $\sigma_{\rm Tr}^2=(1/d)\mbox{Tr}(H_I^2)=(1/d)\sum_m E_m^2$ (the last equality is valid with numerical precision $\sim 10^{-14}$). In all cases $\sigma_{\rm Tr}$ also coincides with $\sigma_{\rm fit}$ and the theoretical expression with an error below 1%.
 Integrated density of states $P(E)$ of the TBRIM Hamiltonian (\[eq\_TBRIM\]) represented by the curve $z_m=(m+0.5)/d$ versus energy level $E_m$ (red curve) with $m=0,\,\ldots,\,d-1$ being the level number. The Hilbert space dimension is $d=11440$ for $L=7$ particles and $M=16$ orbitals. Shown are the curves for one individual spectrum at $V=3.74166$, $J=0.25$ (top left panel) and the SYK-case $V=0$, $J=1$ (top right panel). The functions $P_k(E)$ correspond to the fit (\[eq\_IDOS\_gauss2\]). Shown are the cases $k=1$ (green curve) and $k=5$ (blue curve). The case $k=1$ corresponds to the (integrated) Gaussian density of states with two fit parameters for the width $\sigma_{\rm fit}$ and center $E_c$. The fits for $k=1$ provide for $V=3.74166$, $J=0.25$ ($V=0$, $J=1$) the values $E_c=-0.008\pm 0.001$ ($-0.032\pm 0.001$) and $\sigma_{\rm fit}=1.951\pm 0.001$ ($1.508\pm 0.001$) giving the ratio $\sigma_{\rm fit}/\sigma=0.9983\pm 0.0006$ ($0.9952\pm 0.0008$) where $\sigma=1.954$ ($1.516$) is the theoretical value obtained from (\[eq\_DOS\_TBRIM\]). For comparison the quantity $\sigma_{\rm Tr}$, obtained numerically from the trace of $H_I^2$, gives for both cases $\sigma_{\rm Tr}=1.947$ ($1.503$). In top panels the blue curves for $P_5(E)$ coincide with the red curves for $P(E)$ on graphical precision while the green curves for $P_1(E)$ are slightly above (below) the red curve for $E>0$ ($E<0$). Bottom panels show the difference $P_k(E)-P(E)$ of the fit functions with respect to the numerical function $P(E)$ for $k=1$ (green curve) and $k=5$ (blue curve) using an increased scale.](fig13){width="48.00000%"}
However, a careful comparison of the numerical curve of $P(E)$ with $P_1(E)$ shows small but systematic deviations which can be significantly reduced by increasing the degree of the fit polynomial $q_k(E)$. For $k=5$ it is already nearly impossible to distinguish the numerical curve from $P_5(E)$ on graphical precision. This is clearly illustrated in Fig. \[fig13\] where we compare the numerical curve $P(E)$ with $P_1(E)$ and $P_5(E)$ for the two cases $=3.74166$, $J=0.25$ and $V=0$, $J=1$ (SYK-case). In order to see the differences between the two fits it is actually necessary to draw the difference of $P(E)-P_k(E)$ in an increased scale as it is done in the lower panels of Fig. \[fig13\].
We attribute the small deviations to the Gaussian density of states to the finite values of $M$ and $L$ and also to the fact that we used only one numerical sample of $H_J$. Actually, for finite values of $J$ it is to our knowledge still an open problem if the average density of states of $H_I$ is indeed Gaussian even for the limit $M\to\infty$ and $L$ finite (previous analytical results [@tezuka; @garcia1] apply to the SYK-case with Majorana fermions that is different from our model at $V=0$ and $J=1$).
Weakly excited initial states {#appb}
=============================
The fit procedure using the fit functions (\[eq\_f11\]), (\[eq\_f01\]) to approximate $\rho_{11}(t)$ and $|\rho_{01}(t)|$ are very often quite problematic. First the non-linear fits with a considerable number of parameters depend rather strongly on “good” initial values, especially for the frequencies $\omega_{1,2}$, for the Levenberg-Marquardt iteration. Furthermore it is typically necessary to attribute stronger weights on the initial times. For this we typically perform a first simple exponential fit of the survival probability $p(t)=|\langle\psi(0)|\psi(t)\rangle|^2$ which provides a smooth simplified decay time which we use to fix exponentially decaying weights in time for the more precise fits using the fit functions (\[eq\_f11\]), (\[eq\_f01\]). For larger values of the couplings strength, typically at $\varepsilon\ge 0.1$, the periodic structure with the frequencies $\omega_{1,2}$ also disappears and the fits have to be simplified accordingly as mentioned in the main text.
Even, taking all this into account, for weakly excited initial states, with small values of the level number $m$ in (\[eq\_init\_state\]), the quality of the fits may be rather poor due to the absence of exponential decay, presence of a quasi-periodic structure or the effect that after an initial decrease $|\rho_{01}(t)|$ re-increases at sufficiently long times.
 Time dependence of $\rho_{11}(t)$ (red plus symbols), $|\rho_{01}(t)|$ (green crosses) and the two fit functions $f_{11}(t)$ (thin black line) and $f_{01}(t)$ (thin blue line), defined in (\[eq\_f11\]) and (\[eq\_f01\]), for level number $m=7$ of the initial state (\[eq\_init\_state\]) and $V=3.74166$, $J=0.25$ ($V=0$, $J=1$) in left (right) panels at coupling strength $\varepsilon=0.01$ ($0.02$) in top (bottom) panels. As in Figs. \[fig2\], \[fig3\] the time is measured in units of $\Delta t$ and the number of particles (orbitals) is $L=7$ ($M=16$). ](fig14){width="48.00000%"}
In Fig. \[fig14\] we show some examples of this type for the level number $m=7$ at our usual standard parameters $V=3.74166$, $J=0.25$ or $V=0$, $J=1$ and the coupling strengths $\varepsilon=0.01$ or $0.02$. The quantity $\rho_{11}(t)$ exhibits a structure with beats introducing a second smaller frequency which is only captured by $f_{11}(t)$ at the initial times and even here the deviations due the non-exponential decay are quite well visible. For $|\rho_{01}(t)|$ there are fluctuations with long correlation times for larger time scales which are not well captured by the periodic saturated form of $f_{01}(t)$ at long times. In one case at $J=0.25$, $V=3.74166$ and $\varepsilon=0.02$, the frequency $\omega_2$ is considerably reduced to fit the long range form of $|\rho_{01}(t)|$ but this effect does not reflect the physical reality and provides poor values of the two decay rates $\Gamma_2$ and $\tilde\Gamma_2$.
Due to these effects, we do not show any fit functions in Fig. \[fig2\], which applies to the level number $m=0$, and in Figs. \[fig4\] and \[fig5\] we show the decay rate for the largest level number $m=5720$ which is not problematic as can be seen in Fig. \[fig3\]. Furthermore in Figs. \[fig6\] and \[fig7\], we only show data points for $m>100$.
Initial state with negative temperature {#appc}
=======================================
In Fig. \[fig15\] we present the results for qubit relaxation in the regime of negative temperature (initial state is above the half of energy band width). Here the dynamical temperature of the initial state is $T=1/\beta; \beta=-0.5424$. We see that the relaxation is practically the same as for the initial state with positive temperature at $\beta=0.5443=1/T$. This effect is due to symmetry between negative (positive derivative of the density of states) and positive energies (negative derivative of the density of states). The former correspond to positive and the latter to negative temperatures as can also be seen in the bottom panels of Fig. \[fig1\].
 Time dependence of $\rho_{11}(t)$ (red plus symbols), $|\rho_{01}(t)|$ (green crosses) and the two fit functions $f_{11}(t)$ (thin black line) and $f_{01}(t)$ (thin blue line), defined in (\[eq\_f11\]) and (\[eq\_f01\]), for coupling strength $\varepsilon=0.01$ and $V=3.74166$, $J=0.25$ ($V=0$, $J=1$) in left (right) panels at level numbers $m=2187$ ($9252$) in top (bottom) panels. The initial state for $V=3.74166$, $J=0.25$ at level number $m=2187$ ($9252$) corresponds to the inverse temperature $\beta=0.5443$ ($-0.5424$). As in Figs. \[fig2\], \[fig3\] the time is measured in units of $\Delta t$ and the number of particles (orbitals) is $L=7$ ($M=16$). ](fig15){width="48.00000%"}
[99]{} V.B. Braginsky and F.Ya. Khalili, [*Quantum measurement*]{}, Cambridge University Press, Cambridge, UK (1992). M.A. Nielson and I.J. Chuang, [*Quantum computation and quantum information*]{}, Cambridge University Press, Cambridge, UK (2000). Y. Nakamura, Yu. Pashkin and J.S. Tsai, [*Coherent control of macroscopic quantum states in a single-Cooperpair box*]{}, Nature [**398**]{}, 786 (1999). D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve and M.H. Devoret, [*Manipulating the quantum state of an electrical circuit*]{}, Science [**296**]{}, 886 (2002). A.N. Korotkov and D.V. Averin, [*Continuous weak measurement of quantum coherent oscillations*]{}, Phys. Rev. B [**64**]{}, 165310 (2001). Yu. Makhlin, G. Schön and A. Shnirman, [*Quantum state engineering with Josephson-junction devices*]{}, Rev. Mod. Phys. [**73**]{}, 357 (2001). G. Wendin, [*Quantum information processing with superconducting circuits: a review*]{}, Rep. Prog. Phys. [**80**]{}, 106001 (2017). J. W. Lee, D. V. Averin, G. Benenti and D. L. Shepelyansky, [*Model of a deterministic detector and dynamical decoherence*]{}, Phys. Rev. A [**72**]{}, 012310 (2005). E. Wigner, [*Random matrices in physics*]{}, SIAM Rev. [**9(1)**]{}, 1 (1967). M.L. Mehta, [*Random matrices*]{}, Elsvier Academic Press, Amsterdam (2004). T. Guhr, A. Müller-Groeling and H.A. Weidenmüller, [*Random-matrix theories in quantum physics: common concepts*]{}, Phys. Rep. [**299**]{}, 189 (1998). O. Bohigas, M.J.Giannoni and C. Schmit, [*Characterization of chaotic quantum spectra and universality of level fluctuation*]{}, Phys. Rev. Lett. [**52**]{}, 1 (1984). F. Haake, [*Quantum signatures of chaos*]{}, Springer, Berlin (2010). D. Ullmo, [*Bohigas-Giannoni-Schmit conjecture*]{}, Scholarpedia [**11(9)**]{}, 31721 (2016). J.B. French, and S.S.M. Wong, [*Validity of random matrix theories for many-particle systems*]{}, Phys. Lett. B [**33**]{}, 449 (1970). O. Bohigas and J. Flores, [*Two-body random Hamiltonian and level density*]{}, Phys. Lett. B [**34**]{}, 261 (1971). J.B. French, and S.S.M. Wong, [*Some random-matrix level and spacing distributions for fixed-particle-rank interactions*]{}, Phys. Lett. B [**35**]{}, 5 (1971). O. Bohigas and J. Flores, [*Spacing and individual eigenvalue distributions of two-body random Hamiltonians*]{}, Phys. Lett. B [**35**]{}, 383 (1971). S. [Å]{}berg, [*Onset of chaos in rapidly rotating nuclei*]{}, Phys. Rev. Lett. [**64**]{}, 3119 (1990). S. [Å]{}berg, [*Quantum chaos and rotational damping*]{}, Prog. Part. Nucl. Phys. [**28**]{}, 11 (1992). D.L. Shepelyansky and O.P. Sushkov, [*Few interacting particles in a random potential*]{}, Europhys. Lett. [**37**]{}, 121 (1997). P. Jacquod and D.L. Shepelyansky, [*Emergence of quantum chaos in finite interacting Fermi systems*]{}, Phys. Rev. Lett. [**79**]{}, 1837 (1997). B. Georgeot and D.L. Shepelyansky, [*Integrability and quantum chaos in spin glass shards*]{}, Phys. Rev. Lett. [**81**]{}, 5129 (1998). B. Georgeot and D.L. Shepelyansky, [*Quantum chaos border for quantum computing*]{}, Phys. Rev. E [**62**]{}, 3504 (2000). D.L. Shepelyansky, [*Quantum chaos and quantum computers*]{}, Physica Scripta [**T90**]{}, 112 (2001). G. Benenti, G. Casati and D.L. Shepelyansky, [*Emergence of Fermi-Dirac thermalization in the quantum computer core*]{}, Eur. Phys. J. D [**17**]{}, 265 (2001). I.V. Gornyi, A.D. Mirlin and D.G. Polyakov, [*Many-body delocalization transition and relaxation in a quantum dot*]{}, Phys. Rev. B [**93**]{}, 125419 (2016). I.V. Gornyi, A.D. Mirlin, D.G. Polyakov and A.L. Burin, [*Spectral diffusion and scaling of many-body delocalization transitions*]{}, Ann. Phys. (Berlin) [**529**]{}, 1600360 (2017) V.B. Flambaum and F.M. Izrailev, [*Distribution of occupation numbers in finite Fermi systems and role of interaction in chaos and thermalization*]{}, Phys. Rev. E [**55**]{}, R13(R) (1997). L.D. Landau and E.M. Lifshitz, [*Statistical mechanics*]{}, Wiley, New York (1976). M. Mulansky, K. Ahnert, A. Pikovsky and D.L. Shepelyansky, [*Dynamical thermalization of disordered nonlinear lattices*]{}, Phys. Rev. E [**80**]{}, 056212 (2009). L. Ermann and D.L. Shepelyansky, [*Quantum Gibbs distribution from dynamical thermalization in classical nonlinear lattices*]{}, New J. Phys. [**15**]{}, 123004 (2013). P. Schlageck and D.L. Shepelyansky, [*Dynamical thermalization in Bose-Hubbard systems*]{}, Phys. Rev. E [**93**]{}, 012126 (2016). A.R. Kolovsky and D.L. Shepelyansky, [*Dynamical thermalization in isolated quantum dots and black holes*]{}, EPL [**117**]{}, 10003 (2017). R. Nandkishore and D.A. Huse, [*Many-body localization and thermalization in quantum statistical mechanics*]{}, Annu. Rev. Condens. Matter Phys. [**6**]{}, 15 (2015). L. D. Alessiom Y. Kafri, A. Polkovnikov and M. Rigol, [*From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics*]{}, Adv. Phys. [**65**]{}, 239 (2016). F. Borgonovi, F.M. Izrailev, L.F. Santos and V.G. Zelevinsky, [*Quantum chaos and thermalization in isolated systems of interacting particles*]{}, Phys. Rep. [**626**]{}, 1 (2016). F. Alet and N. Laflorencie, [*Many-body localization: an introduction and selected topics*]{}, arXiv:1711.03145 \[cond-mat.str-el\] (2017). S. Sachdev and J. Ye, [*Gapless spin-fluid ground state in a random quantum Heisenberg magnet*]{}, Phys. Rev. Lett. [**70**]{}, 3339 (1993). A.Kitaev, [*A simple model of quantum holography*]{}, Video talks at KITP Santa Barbara, April 7 and May 27 (2015). S. Sachdev, [*Bekenstein-Hawking entropy and strange metals*]{}, Phys. Rev. X [**5**]{}, 041025 (2015). J. Polchinski and V. Rosenhaus, [*The spectrum in the Sachdev-Ye-Kitaev model*]{}, JHEP [**04**]{}, 1 (2016). J. Maldacena and D. Stanford, [*Remarks on the Sachdev-Ye-Kitaev model*]{}, Phys. Rev. D [**94**]{}, 106002 (2016). J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker, D. Stanford, A. Streicher and M. Tezuka, [*Black holes and random matrices*]{}, JEHP [**05**]{}, 118 (2017). A.M. Garcia-Garcia and J.J.M. Verbaarschot, [*Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model*]{}, Phys. Rev. D [**94**]{}, 126010 (2016). A.M. Garcia-Garcia and J.J.M. Verbaarschot, [*Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N*]{}, Phys. Rev. D [**96**]{}, 066012 (2017). T. Kanazawa and T. Wettig, [*Complete random matrix classification of SYK models with N= 0, 1 and 2 supersymmetry*]{}, JHEP [**09**]{}, 50 (2017). D. Harlow, [*Jerusalem lectures on black holes and quantum information*]{}, Rev. Mod. Phys. [**88**]{}, 015502 (2016). B. Misra and E.C.G. Sudarshan, [*The Zeno’s paradox in quantum theory*]{}, J. Math. Phys. [**18**]{}, 756 (1977). M.C. Fischer, B. Gutiérrez-Medina and M.G. Raizen, [*Observation of the quantum Zeno and anti-Zeno effects in an unstable system*]{}, Phys. Rev. Lett. [**87**]{}, 040402 (2001). A. Gusev, R.A. Jalabert, H.M. Pastawski and D.A. Wisniacki, [*Loschmidt echo*]{}, Scholarpedia [**7(8)**]{}, 11687 (2012). P. Jacquod, P.G. Silvestrov and C.W.J. Beenakker, [*Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo*]{}, Phys. Rev. E [**64**]{}, 055203(R) (2001). B. Georgeot and D. L. Shepelyansky, [*Breit-Wigner width and inverse participation ratio in finite interacting Fermi systems*]{}, Phys. Rev. Lett. [**79**]{}, (1997), 4365. S. Milgram, [*The small-world problem*]{}, Psychology Today [**1(1)**]{}, 61 (May 1967). S. Dorogovtsev, [*Lectures on complex networks*]{}, Oxford University Press, Oxford (2010). O. Giraud, B. Georgeot and D.L. Shepelyansky, [*Quantum computing of delocalization in small-world networks*]{}, Phys. Rev. E [**72**]{}, 036203 (2005). L. Backstrom, P. Boldi, M. Rosa, J. Ugander and S. Vigna, [*Four degrees of separation*]{}, Proc. 4th ACM Web Sci. Conf., ACM N.Y. p.33 (2012). L. Ermann, K.M. Frahm and D.L. Shepelyansky, [*Google matrix analysis of directed networks*]{}, Rev. Mod. Phys. [**87**]{}, 1261 (2015). B. Georgeot and D.L. Shepelyansky, [*Emergence of quantum chaos in the quantum computer core and how to manage it*]{}, Phys. Rev. E [**62**]{}, 6366 (2000). I. Garcia-Mata, O. Giraud, B. Georget, J. Martin, R. Dubertrand and G. Lemarie, [*Scaling theory of the Anderson transition in random graphs: ergodicity and universality*]{}, Phys. Rev. Lett. [**118**]{}, 166801 (2017). P. Prelovsek, O.S. Barisic and M. Mierzejewski, [*Reduced-basis approach to many-body localization*]{}, Phys. Rev. B [**97**]{}, 035104 (2018). P. Erdös and A. Rényi, [*On random graphs I*]{}, Publicationes Mathematicae [**6**]{}, 290 (1959). H. Poincaré, [*Sur le probleme des trois corps et les équations de la dynamique*]{}, Acta Math. [**13**]{}, 1 (1890). A. Abragam, [*The principles of nuclear magnetism*]{}, Oxford University Press, Oxford UK (1961).
[^1]: It is mathematically also possible to consider other symmetry classes GUE or GSE for the interaction matrix which would imply a variance of $2J^2/\beta$ (with $\beta=1$ for GOE, $2$ for GUE and $4$ for GSE) for the variables $J_{ij,ij}$ if we keep the non-diagonal variance $J^2$ of $J_{ij,kl}$ for $(ij)\neq (kl)$.
|
---
abstract: 'We propose a model to explain how a Gamma Rays Burst can take place days or years after a supernova explosion. Our model is based on the conversion of a pure hadronic star (neutron star) into a star made at least in part of deconfined quark matter. The conversion process can be delayed if the surface tension at the interface between hadronic and deconfined-quark-matter phases is taken into account. The nucleation time ([*i.e.*]{} the time to form a critical-size drop of quark matter) can be extremely long if the mass of the star is small. Via mass accretion the nucleation time can be dramaticaly reduced and the star is finally converted into the stable configuration. A huge amount of energy, of the order of 10$^{52}$–10$^{53}$ erg, is released during the conversion process and can produce a powerful Gamma Ray Burst. The delay between the supernova explosion generating the metastable neutron star and the new collapse can explain the delay proposed in GRB990705 [@Amati00] and in GRB011211 [@Reeves02].'
author:
- 'Z. Berezhiani, I. Bombaci, A. Drago, F. Frontera and A. Lavagno'
title: |
Gamma Ray Bursts from delayed collapse\
of neutron stars to quark matter stars
---
Introduction
============
The discovery of a transient (13 s) absorption feature in the prompt emission of the $\sim 40$ s Gamma Ray Burst (GRB) of July 5, 1999 (GRB990705) [@Amati00] and the evidence of emission features in the afterglow of several GRBs [@Piro99; @Yoshida99; @Piro00; @Antonelli00; @Reeves02] have stimulated the interpretation of these characteristics in the context of the fireball model of GRBs. @Amati00 attribute the transient absorption feature of GRB990705 (energy released $\sim 10^{53}$ erg assuming isotropy) to a redshifted K edge of Iron contained in an environment not far from the GRB site ($\sim 0.1$ pc) and crossed by the GRB emission. They estimate an Iron abundance typical of a supernova (SN) environment ($A_\mathrm{Fe} \sim 75$) and a time delay of about 10 years between the SN explosion and the GRB event. @Lazzati01 give a different interpretation of the absorption feature, in terms of a redshifted resonance scattering feature of H–like Iron (transition 1s–2p, $E_\mathrm{rest} =6.927$ keV) in an inhomogeneous high–velocity outflow, but invoke a Iron rich environment as well, due to a preceding SN explosion, even if a shorter time delay ($\sim 1$ yr) between SN and GRB is inferred. A SN explosion preceding the GRB event is also inferred for explaining the properties of the emission features in the X–ray afterglow spectrum of GRB000214 [@Antonelli00] and GRB991216 [@Piro00]. In the latter case it cannot be excluded that the SN explosion occured days or weeks before the GRB [@Rees00]. @Reeves02, to explain the multiple emission features observed in the afterglow spectrum of GRB011211 (time duration of $\sim 270$ s, isotropic gamma–ray energy of $5 \times 10^{52}$ erg), invoke a SN explosion preceding the GRB event by $\sim 4$ days (in the isotropic limit, a minimum of 10 hrs). Even if other interpretations for the afterglow emission lines are possible which do not involve a previous SN explosion (e.g., [@Rees00; @Meszaros01]), this explosion seems to be the most likely way to explain the transient absorption line observed from GRB990705 [@Bottcher02]. In conclusion, the previous observations suggest that, at least for a certain number of GRBs, a SN explosion happened before the GRB, with a time interval between the two events ranging from a few hours to a few years. In this context, an attractive scenario is that described by the [*supranova*]{} model [@Vietri98] for GRBs. In this model, the GRB is the result of the collapse to a black hole (BH) of a supramassive fast rotating neutron star (NS), as it loses angular momentum. According to this model the NS is produced in the SN explosion preceding the GRB event. The initial barionic mass $M_B$ of the NS is assumed to be above the maximum baryonic mass for non-rotating configurations. However, as also noticed by @Bottcher02, on the basis of realistic calculations of collapsing NS [@fryer1998], in these collapses too much baryonic material is ejected and thus the energy output is expected to be too small to produce GRBs. Even if the introduction of magnetic fields or beaming could overcome this limitation, in any case, the GRB duration from a NS collapse should be very short ($\ll1$ s), much shorter than that observed from GRB990705.
In this Letter, we propose an alternative model to explain the existence of GRBs associated with previous SN explosions. In this model, unlike the supranova model, the NS collapse to BH is replaced by the conversion from a metastable, purely hadronic star (neutron star) into a more compact star in which deconfined quark matter (QM) is present. This possibility has already been discussed in the literature [@cd96; @bd00; @wang; @ouyed]. The new and crucial idea we introduce here, is the metastability of the purely hadronic star due to the existence of a non-vanishing surface tension at the interface separating hadronic matter from quark matter. The mean-life time of the metastable NS can then be connected to the delay between the supernova explosion and the GRB. As we shall see, in our model we can easily obtain a burst lasting tens seconds, in agreement with the observations. The order of magnitude of the energy released is also the appropriate one.
Quark Matter nucleation in compact stars
========================================
Recently various possibilities have been discussed in the literature to get compact stars in which matter is, partially or totally, in a state of deconfined quarks (see [*e.g.*]{} @glenbook [@martino]; @andrea). Concerning the stellar quark content, it is possible to have three different classes of compact stars: a) purely hadronic stars (HS), in which no fraction of QM is present; b) hybrid stars (HyS), in which only at the center of the star QM is present either as a mixed phase of deconfined quarks and hadrons or as a pure phase; c)quark stars (QS), in which the surface of the star is made of matter having a large density, of the order of nuclear matter saturation density or larger, and the bulk of the star is made of deconfined QM. The sizeable amount of observational data collected by the new generations of X-ray satellites, has provided a growing body of evidence for the existence of very compact stars, which could be HyS or QS [@bomb97; @che98; @li99a; @li99b; @xu02; @dra02].
In our scenario, we consider a purely HS whose central density (pressure) is increasing due to spin-down or due to mass accretion (e.g., from fallback of ejected material in the SN explosion). As the central density approaches the deconfinement critical density, a virtual drop of quark matter can be formed in the central region of the star. The fluctuations of a spherical droplet of quark matter having a radius $R$ are regulated by a potential energy of the form [@lif] U(R)=[4 3]{} R\^3 n\_q (\_q-\_h)+4 R\^2 + 8 R where $n_q$ is the quark baryon density, $\mu_h$ and $\mu_q$ are the hadronic and quark chemical potentials at a fixed pressure $P$, and $\sigma$ is the surface tension for the surface separating quarks from hadrons. Finally, the term containing $\gamma$ is the so called curvature energy. The value of the surface tension $\sigma$ is poorly known, and typical values used in the literature range from 10 to 50 MeV/fm$^2$ [@hei; @iida]. Following the work of @iida, we have assumed that the term with $\sigma$ takes into account in an effective way also the curvature energy. The term with $\gamma$ is discussed [*e.g.*]{} by @madsen, while other more complicated terms, connected with the Coulomb energy, are discussed in the literature [@hei; @iida]. We have neglected them in our analysis since they do not dramatically modify both the nucleation time and the energy associated with the transition into the stable quark matter configuration.
If the temperature is low enough, the process of formation of a bubble having a critical radius proceeds through quantum tunnelling and it can be computed using a semiclassical approximation. The procedure is rather straightforward. First one computes, using the semiclassical (WKB) approximation, the ground state energy $E_0$ and the oscillation frequency $\nu_0$ of the virtual QM drop in the potential well $U(R)$. Then it is possible to calculate in a relativistic frame the probability of tunneling as [@iida]: p\_0=where A(E)=[2]{}\_[R\_-]{}\^[R\_+]{} dR . Here $R_\pm$ are the classical turning points and M(R)=4\_h(1-[n\_qn\_h]{})\^2 R\^3 , $\rho_h$ being the hadronic energy density (here and in the following we adopt the so-called “natural units”, in which $\hbar=c=1$). $n_h$ and $n_q$ are the baryonic densities at a same and given pressure in the hadronic and quark phase, respectively. The nucleation time is then equal to = (\_0 p\_0 N\_c)\^[-1]{} , where $N_c$ is the number of centers of droplet formation in the star, and it is of the order of $10^{48}$ [@iida].
Results
=======
The typical mass-radius relations for the three types of stars we are discussing can be found e.g. in Fig. 3 of @andrea, where a relativistic non-linear Walecka-type model [@gm] has been used to describe the hadronic phase. As it appears, stars containing QM (either HyS or QS) are more compact than purely HS. In particular QS can have much smaller radii then HS when they have a small mass. In our scenario a metastable HS having a mass of, e.g., 1.3 $ M_\odot$ and a radius of $\sim 13$ km can collapse into an HyS having a radius of $\sim 10.5$ km or into a QS with radius $\sim 9$ km (with respect to the results of Fig. 3 of @andrea, here we use an equation of state (EOS) which includes hyperonic degrees of freedom). The nature of the stable configuration reached after the stellar conversion ([*i.e.*]{} an HyS or a QS) will depend on the parameters of the quark phase EOS.
The time needed to form a critical droplet of deconfined quark matter can be calculated for different values of the stellar central pressure $P_c$ (which enters in the expression of the energy barrier in eq. (1)) and it can be plotted as a function of the gravitational mass $M_\mathrm{HS}$ of the HS corresponding to that given value of central pressure. The results of our calculations for a specific EOS of hadronic matter (the GM3 model with hyperons of @gm) are reported in Fig. 1, where each curve refers to a different value of the bag constant $B$. If we assume, for example, $B^{1/4}=170$ MeV (which corresponds to $B=109$ MeV/fm$^3$) and the initial mass of the HS to be $M_{HS} = 1.32~ M_\odot$, we find that the “life time” for this star is about $10^{12}$ years. As the star accretes a small amount of matter, the consequential increase of the central pressure lead to a huge reduction of the nucleation time, and, as a result, to a dramatic reduction of the HS life time. For our HS with initial mass of $1.32~M_\odot$ the accretion of about $0.01~ M_\odot$ reduces the star life time to a few years. We would like to stress that in our model the delay between the SN explosion and the GRB is regulated by the mass accretion rate, rather then by the mass and the spinning of the metastable star itself. Since the mass accretion rate is generally larger during the first days after the SN explosion, a delay of a few days will be rather typical in our scenario. However, longer delays are also possible if the material ejected during the SN explosion has a small fallback.
In the model we are presenting, the GRB is due to the cooling of the justly formed HyS or QS via neutrino-antineutrino emission (and maybe also via emission of axion-like particles, see below). The subsequent neutrino-antineutrino annihilation generates the GRB. In our scenario the duration of the prompt emission of the GRB is therefore regulated by two mechanisms: 1) the time needed for the conversion of the HS into a HyS or QS, once a critical-size droplet is formed and 2) the cooling time of the justly formed HyS or QS. Concerning the time needed for the conversion into QM of at least a fraction of the star, the seminal work by @Oli87 has been reconsidered by @HB88. The conclusion of this latter work is that the stellar conversion is a very fast process, having a duration much shorter than 1s. On the other hand, the neutrino trapping time, which provides the cooling time of a compact object, is of the order of a few ten seconds [@ignazio], and it gives the typical duration of the GRB in our model. In Table 1 we give the measured duration and the estimated electromagnetic energy (assuming isotropic emission) of the GRBs associated with Fe emission or absorption lines. All bursts last at least 10 s. According to our model the firsts few ten seconds correspond to a prompt $\gamma$-rays emission, while the subsequent emission should be interpreted as the beginning of the afterglow. Actually it has been found that at least the second half of the prompt emission of long bursts is likely due to afterglow [@frontera]. We would like to remark however that we are not suggesting that all the GRBs should be explained in our model. In particular long and energetic bursts could be originated e.g. by collapsars [@woosley]. On the other hand, the variety of GRB durations could be explained within the QS formation scenario itself making use of the “unstable photon decay” mechanism proposed by @ouyedsannino.
Next we consider the total energy $\Delta E$ released in the transition from a metastable HS (with hyperonic degrees of freedom) to HyS or QS (which final state is reached in this transition depends on the details of the QM EOS and in particular on the value of the bag constant). The energy released is calculated as the difference between the gravitational mass of the metastable HS and that of the final stable HyS (or QS) having the same baryonic mass [@bd00]. In Table 2 we report the energy released for various values of the bag constant $B$ and of the surface tension $\sigma$. Notice that the transition will take place when the nucleation time will be reduced to a value of the order years, due e.g. to mass accretion on the HS (recall the exponential dependence of the nucleation time on the mass of the HS, as shown in Fig. 1). Therefore the total energy released in the collapse will be always of the same order of magnitude, once the parameters of the model have been fixed. As shown in Table 2, the released energy is in the range $(3 - 5)\times 10^{52}$ ergs for all the sensible choices of the EOS parameters. The “critical mass” $M_{\mathrm{cr}}$ of a metastable HS having a lifetime $\tau = 1$ yr is in the range $(0.9 - 1.4)~ M_\odot$. When the mass of the HS reaches a value near $M_{\mathrm{cr}}$, the conversion process takes place. It is worth mentioning that the energy released in the conversion can be larger if a diquark condensate forms inside QM, see e.g. @sannino.
To generate a strong GRB, an efficient mechanism to transfer the energy released in the collapse into an electron-photon plasma is needed. In an earlier work [@fryer1998] it was this difficulty that hampered the possibility to connect GRBs and the hadronic-quark matter phase transition in compact stars. Only more recently it was noticed [@salmonson99] that near the surface of a compact stellar object, due to general relativity effects, the efficiency of the neutrino-antineutrino annihilation into $e^+ e^-$ pairs is strongly enhanced with respect to the Newtonion case. The efficiency of the conversion of neutrinos in $e^+ e^-$ pairs could be as high as 10$\%$. In the computation of the energy associated with the final GRB we must take into account the possibility of a moderate anisotropy of the electron motion, due to the presence of the magnetic field of the star, which will in turn generate a moderate anisotropy of the burst emission [^1]. Other anisotropies in the GRB emission could be generated by the rotation of the star which could affect the efficiency of the neutrino-antineutrino annihilation due to general relativity effects. On the basis of these considerations, the energy deposited in the burst could be sufficient to explain the isotropic energy of the GRBs listed in Table 1. We must also recall that more efficient ways to generate photons and/or $e^+ e^-$ pairs have been proposed in the literature, based on the decay of axion-like particles [@noi2000]. This mechanism would have an extremely high efficiency and would transfer most of the energy produced in the collapse into GRB electromagnetic energy.
There are various specific signatures of the mechanism we are suggesting. First, two classes of stars having similar masses but rather different radii should exist: a) pure (metastable) HS, with radii in the range 12–20 km, as is the case of the compact star 1E 1207.4-5209, assuming $M=1.4M_\odot$ [@sanwal], and b) HyS or QS with radii in the range 6–8 km [@bomb97; @li99a; @li99b; @dra02]. Second, all the GRBs generated by the present mechanism should have approximately the same isotropic energy and a duration of at least 10 s.
Conclusions
===========
We propose the following origin for at least some of the GRBs having a duration of tens of seconds. They can be associated with the transition from a metastable HS to a more compact HyS or a QS. The time delay between the supernova explosion originating the metastable HS and the GRB is regulated by the process of matter accretion on the HS. While most of the stellar objects obtained by a SN explosion will possibly have a mass larger than $M_{\mathrm{cr}}$ and will therefore directly stabilize as HyS or QS at the moment of the SN explosion, in a few cases the mass of the protoneutron star will be low enough not to allow the immediate production of QM inside the star. Only when the star will acquire enough mass, the process of QM formation could take place. Due to the surface tension between the hadronic matter and the QM the star will become metastable. The later collapse into a stable HyS or QS will generate a powerfull GRB. It can be interesting to notice that, in order to have a not too small value for $M_{\mathrm{cr}}$, a relatively large value for the bag constant $B$ has to be choosen, $B^{1/4}\sim$ 170 MeV, which turns out to be the prefered value in many hadronic physics calculations (see e.g. [@thomas]). In this situation the final state is an HyS and not a QS.
It is a pleasure to thank Elena Pian and Luciano Rezzolla for very useful discussions.
Amati, L., et al. 2000, Science, 290, 953. Amati, L., et al. 2002, A&A, 390,81. Antonelli, L.A., et al. 2000, ApJ, 545, L39. Berezhiani, Z., & Drago, A. 2000, Phys. Lett. B, 473, 281. Bloom, J.S., Frail, D.A., & Sari, R. 2001, Astronomical J., 121, 2879. Bombaci, I. 1997, Phys. Rev. C, 55, 1587. Bombaci, I., & Datta, B. 2000, ApJ, 530, L69. Böttcher, M., Fryer, C.L., & Dermer, C.D. 2002, ApJ, 567, 441 Cheng, K.S., & Dai, Z.G. 1996, Phys. Rev. Lett., 77, 1210. Cheng, K.S., Dai, Z.G., Wei, D.M., & Lu, T. 1998, Science, 280, 407. Drago, A. & Lavagno, A. 2001, Phys. Lett. B, 511, 229. Drake, J.J., et al. 2002, ApJ, 572, 996. Frontera, F., et al. 2000, ApJs, 127, 59. Fryer, C.L., & Woosley, S.E. 1998, ApJ, 501, 780. Glendenning, N.K., & Moszkowski, S.A. 1991, Phys. Rev. Lett., 67, 2414. Glendenning, N.K. 2000, Compact Stars, 2nd edition (Springer Verlag). Heiselberg, H., Pethick, C.P. & Staubo, E.F. 1993, Phys. Rev. Lett., 70, 1355. Heiselberg, H. & Hjorth-Jensen, M. 2000, Phys. Rept, 328, 237. Hong, D.K., Hsu, S.D.H., & Sannino, F. 2001, Phys. Lett. B., 516, 362. Horvath, J.E. & Benvenuto, O.G. 1988, Phys. Lett. B, 213, 516. Iida, K., & Sato, K. 1998, Phys. Rev. C, 58, 2538. Kluzniak, W., & Ruderman, M. 1998, ApJ, 505, L113. Lazzati, D., et al. 2001, ApJ, 556, 471 Li, X.D., Bombaci, I., Dey, M., Dey, J. & van den Heuvel, E.P.J. 1999, Phys. Rev. Lett., 83, 3776. Li, X.D., Ray, S., Dey, J., Dey, M. & Bombaci, I. 1999, ApJ, 527, L51. Lifshitz, I.M., & Kagan, Yu. 1972, Zh. Eksp. Teor. Fiz., 62, 385 \[Sov. Phys. JETP, 35, 206\]. MacFadyen, A., & Woosley, S.E. 1999, ApJ, 524, 262. Madsen, J. 1993, Phys. Rev. D, 47, 5156. Mészáros, P., & Rees, M.J. 2001, ApJ, 556, L37. Olinto, A. 1987, Phys. Lett. B, 192, 71. Ouyed, R., Dey, J., & Dey, M. 2002, A&A, 390, L39. Ouyed, R., & Sannino, F. 2002, A&A, 387, 725. Piro, L., et al. 1999, ApJ, 514, L73 Piro, L., et al. 2000, Science, 290, 955 Prakash, M., Bombaci, I., Prakash, M., Ellis, P.J., Lattimer, J.M. & Knorren, R. 1997, Phys. Rept., 280, 1. Rees, M.J., & Mészáros, P. 2000, ApJ, 545, L73. Reeves, J.N., et al. 2002, Nature, 414, 512. Salmonson, J.D., & Wilson, J.R. 1999, ApJ, 517, 859. Sanwal, D., Pavlov, G.G., Zavlin, V.E., & Teter, M.A. 2002, ApJ, 574, L61. Steffens, F.M., Holtmann, H., & Thomas, A.W. 1995, Phys. Lett. B., 358, 139. Vietri, M., & Stella, L. 1998, ApJ, 507, L45. Wang, X.Y., Dai, Z.G., Lu, T., Wei, D.M., & Huang, Y.F. 2000, A&A, 357, 543. Xu, R.X. 2002, ApJ, 570, L65. Yoshida, A., et al. 1999, A&A Suppl., 138, 433
------------------ ---------------- --------------------------------
GRB duration \[s\] E$_\mathrm{iso}$/10$^{51}$ erg
\[0.5ex\] 970508 20 7
970828 160 270
990705 42 210
991216 20 500
000214 10 9
011211 270 50
------------------ ---------------- --------------------------------
: Duration and energy released (assuming isotropy) of the GRB associated with the presence of emission or absorption Fe lines in the spectrum. The data have been estracted from @Amati02 and @bloom.
------------------ ------------------------ ------------------ ------------------------------
B$^{1/4}$\[MeV\] $\sigma$\[MeV/fm$^2$\] $M_{cr}/M_\odot$ $\Delta$ E \[10$^{51}$ erg\]
\[0.5ex\] 170 20 1.25 30.0
170 30 1.33 33.5
170 40 1.39 38.0
165 30 1.15 38.6
160 30 0.91 45.7
------------------ ------------------------ ------------------ ------------------------------
: Critical mass $M_{cr}$ of the metastable hadronic star (in unit of the mass of the sun $M_{\odot} = 1.989\times 10^{33}$ g) and energy released $\Delta E$ in the conversion to hybrid star assuming the hadronic star mean life time $\tau$ equal to 1 year. Results are reported for various choices of the surface tension $\sigma$ and of the bag constant $B$. The strange quark mass is taken equal to 150 MeV. For the hadronic matter EOS the GM3 model with hyperons [@gm] has been used.
[^1]: Dramatic effects of a time-dependent magnetic field have been discussed e.g. by @kluzniak.
|
---
abstract: 'It has been a long-standing problem how to relate Chern-Simons theory to the quantum groups. In this paper we recover the classical $r$-matrix directly from a 3-dimensional Chern-Simons theory with boundary conditions, thus creating a direct link to the quantum groups. It is known that the Jones polynomials can be constructed using an $R$-matrix. We show how these constructions can be seen to arise directly from 3-dimensional Chern-Simons theory.'
author:
- Nanna Havn Aamand
title: 'Chern-Simons Theory and the $R$-Matrix'
---
Introduction
============
It was first shown by Witten in a famous paper [@witten1989quantum] that the expectation value of Wilson loops in 3-dimensional Chern-Simons theory gives rise to certain values of the Jones polynomials of knots. On the other hand Reshetikhin and Turaev [@turaev1988yang; @reshetikhin1988quantized] have given constructions of the Jones polynomials from quantum groups, by using an $R$-matrix representation of the Artin braid group. Until now, it has however been unclear how the constructions of Reshetikhin and Turaev can be seen to arise directly from 3-dimensional Chern-Simons theory. The aim of the present paper is to fill in this gap. Motivated by recent papers by Costello, Witten and Yamazaki [@costello2017gauge; @costello2018gauge] we show, working to leading order in perturbation theory, that the propagator of a 3-dimensional Chern-Simons theory with gauge group $G=SL_2(\mathbb{C})$ has the form of an $R$-matrix when imposing boundary conditions that break the $G$-symmetry of the action. This result allows us to give an explicit construction of the Jones polynomials from the expectation value of Wilson loops in the theory. In fact, by choosing a gauge where interactions through the $R$-matrix only occur at the points where two Wilson lines cross in $\mathbb{R}^2$, we obtain a Hecke algebra representation of the Artin braid group with Wilson lines interpreted as braid strands. The original construction of the Jones two-variable polynomials [@jones1990hecke] comes from a Markov trace due to Ocneanu [@freyd1985new] acting on a Hecke algebra representation of the Artin braid group. We show that the expectation value of Wilson loops obtained from the closure of Wilson lines behaves like Ocneanu’s trace function and thus it can be normalized to give the Jones polynomials for specific values of the variables.\
Guadagnini et al. [@guadagnini1990wilson; @cotta1990quantum; @guadagnini1990chern; @guadagnini1991link] have similarly studied the problem of recovering link polynomials from perturbative Chern-Simons theory. It was argued in [@guadagnini1991link] that, without breaking the $G$-symmetry, one recovers the $R$-matrix of a quasi-triangular quasi-Hopf algebra, and in [@guadagnini1990chern] that Wilson line operators are related to a monodromy representation of the braid group. However, until now no explicit construction of the $R$-matrix has been made. The approach of Guadagnini et al. was further studied by Morozov and Smirnov [@morozov2010chern] using a temporal gauge condition. However, since the $G$-symmetry of the Chern-Simons action is not broken they do not recover the $R$-matrix.
The $R$-Matrix {#sec:YBE}
==============
In this section we briefly review the Yang-Baxter formalism [@yang1967some; @baxter1971eight] and present the solutions of the classical Yang-Baxter equation, $r\in \mathfrak{g}\otimes\mathfrak{g}$, for the Lie algebra $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$ which we will be considering in the rest of the paper.\
For an $n$-dimensional vector space $V$, let $R$ be a bilinear operator, $R:V\otimes V\to V\otimes V$. Furthermore, consider $k$ copies of $V$ labeled by $V_1,\dots,V_k$ and define $R_{\mu\nu}:V^{\otimes k}\to V^{\otimes k}$, $\mu,\nu\in\{1,\dots,k\}$ to be the operator obtained by first acting with $R$ on $V_\mu$ and $V_\nu$ and then acting with the permutation operator $P_{\mu\nu}:V^{\otimes k}\to V^{\otimes k}$ that swaps a vector from $V_\mu$ and a vector from $V_\nu$. For example for $k=3$ we have, $$\begin{aligned}
R_{12}=P_{12}(R\otimes {\mathop{\mathrm{id}}}) :V_1\otimes V_2\otimes V_3\to V_2\otimes V_1\otimes V_3.\end{aligned}$$ $R$ is said to be an $R$-matrix if it satisfies the relation, $$\begin{aligned}
R_{\mu\nu}R_{\mu\lambda}R_{\nu\lambda}=R_{\nu\lambda}R_{\mu\lambda}R_{\mu\nu} \label{eq:YBE}
\end{aligned}$$ known as the Yang-Baxter equation. The Yang-Baxter equation is most easily understood from a graphical representation, as the one given in Figure \[fig:YBE\].
![Graphical representation of the Yang-Baxter equation.[]{data-label="fig:YBE"}](YBE.png){width="0.32\paperwidth"}
In Figure \[fig:YBE\] each line represents a vector space, $V_\mu$, $V_\nu$ and $V_\lambda$, and each crossing between two lines represents an $R$-matrix followed by a permutation acting on the corresponding vector spaces. In this picture, the Yang-Baxter equation tells us that the middle line can be pulled across the crossing between two other lines without changing the total outcome.
The Classical $r$-Matrix {#sec:CYBE}
------------------------
In the following we study solutions of the Yang-Baxter equation in the context of gauge theories. We therefore consider a gauge group $G$ with Lie algebra $\mathfrak{g}$, and take $R$ to be an element of $\mathfrak{g}\otimes\mathfrak{g}$. The vector space $V$ then corresponds to the space of spin states of $\mathfrak{g}$. Since we will be working to leading order in perturbation theory, we furthermore write $R$ as an expansion around the identity in the expansion parameter $\hbar$, as $R=I+\hbar r+\mathcal{O}(\hbar^2)$. Here $r\in \mathfrak{g}\otimes\mathfrak{g}$ is known as the classical $r$-matrix. Inserting this expansion into equation we find from the terms at order $\mathcal{O}(\hbar^2)$ that $r$ must satisfy the following equation, known as the classical Yang-Baxter equation, $$\begin{aligned}
[r_{\mu\nu},r_{\mu\lambda}]+[r_{\mu\nu},r_{\nu\lambda}]+[r_{\mu\lambda},r_{\nu\lambda}]=0 .\label{CYBE}\end{aligned}$$ As previously mentioned, we will in the present paper work with the gauge group $G=SL_2(\mathbb{C})$, with corresponding Lie algebra $\mathfrak{g}=\mathfrak{sl_2(\mathbb{C})}$ consisting of all traceless 2 by 2 matrices. The basis elements $e,f,h$ of $\mathfrak{sl}_2(\mathbb{C})$ in the fundamental representation are given by $$\begin{aligned}
e=\begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix}\hspace{15pt}
f=\begin{pmatrix}
0 & 0 \\ 1 & 0 \end{pmatrix} \hspace{15pt} h=\begin{pmatrix}
1 & 0 \\ 0 & -1 \end{pmatrix}, \label{eq:generators}\end{aligned}$$ from which we can infer the Lie brackets: $$\begin{aligned}
[e,f]=h, \hspace{15pt} [h,e]
=2 e ,\hspace{15pt}[h,f]=-2f . \label{Lie brack}\end{aligned}$$ The solutions of the classical Yang-Baxter equation for $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$ can be found in [@chari1995guide]. We have $$\begin{aligned}
r=e\otimes f +\frac{1}{4}h\otimes h \ . \label{CPsol1}\end{aligned}$$ Another solution can be obtained by interchanging $e$ and $f$ since this only changes the left side of by an overall minus sign. We get $$\begin{aligned}
\tilde{r}=f\otimes e +\frac{1}{4}h\otimes h \ . \label{CPsol2}\end{aligned}$$ Notice that the solutions and do not have full $SL_2(\mathbb{C})$-symmetry. Since we hope to recover these solutions from Chern-Simons theory we must therefore find a way of breaking the $SL_2(\mathbb{C})$-symmetry of the Chern-Simons action. As we shall see in the next section, this can be done by imposing specific boundary conditions to the gauge field.
Chern-Simons Theory and the $R$-Matrix {#sec:CS-theory}
======================================
In this section we show how the classical Yang-Baxter solutions presented in Section \[sec:CYBE\] can be recovered from 3-dimensional Chern-Simons theory. More concretely, we will consider the usual 3-dimensional Chern-Simons theory defined on the manifold $M=\mathbb{R}\times I$, where $I$ is a closed interval. The Chern-Simons action has the form $$\begin{aligned}
S_{\text{CS}}&=\frac{1}{4\pi}\int_{\mathbb{R}^2\times I} \operatorname{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right). \label{CS-action}\end{aligned}$$ Let $x_1$, $x_2$ be coordinates on $\mathbb{R}^2$ and $x_3$ a coordinate on $I$, then $A~=~A_1dx_1+A_2dx_2+A_3dx_3$, where the $A_i$’s are elements of the Lie algebra $\mathfrak{g}$ of the gauge group. $\operatorname{Tr}$ denotes a non-degenerate invariant bilinear form on $\mathfrak{g}$. In the case of $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$ in the fundamental 2 by 2 representation , we can take $\operatorname{Tr}$ to be the usual trace: $\operatorname{Tr}(ef)=1$, $\operatorname{Tr}(hh)=2$.\
We show in the following that by imposing boundary conditions on the gauge field at the endpoints of $I$ consistent with those proposed in [@costello2017gauge], the propagator of the Chern-Simons action gives the classical Yang-Baxter solutions and .
Boundary Conditions {#sec:BC}
-------------------
Since the the Chern-Simons action is only gauge invariant up to a surface term, we must make sure that the this term vanishes with the chosen boundary conditions. Under a gauge transformation $A\to A+\delta A$ where $\delta A$ is an exact one-form, the variation of the Chern-Simons action is given by, $$\begin{aligned}
\label{eq:var}
\delta S_{\text{CS}}=\frac{1}{2\pi}\int_{\mathbb{R}^2\times \partial I}\operatorname{Tr}A \wedge \delta A ,\end{aligned}$$ It was argued in [@costello2017gauge] (in the case of a 4-dimension generalisation of the usual Chern-Simons action) that, in order to make the boundary term of the Chern-Simons action vanish while reproducing a solution of the Yang-Baxter equation, one must choose the boundary conditions as follows: For a given Lie algebra $\mathfrak{g}$, let $\mathfrak{l}_0$ and $\mathfrak{l}_1$ be middle-dimensional subalgebras of $\mathfrak{g}$ on which $\operatorname{Tr}({\cdot,\cdot})$ vanishes and which satisfy $\mathfrak{l}_0\cap\mathfrak{l}_1=0$ (or equivalently $\mathfrak{l}_0\oplus\mathfrak{l}_1=\mathfrak{g}$). Choosing for convenience $I=[0,1]$, we then require $A$ and $\delta A$ to take value in $\mathfrak{l}_0$ on the boundary $\mathbb{R}^2\times \{0\}$ and in $\mathfrak{l}_1$ on the boundary $\mathbb{R}^2\times \{1\}$. Clearly, it is not possible to construct such $\mathfrak{l}_0$ and $\mathfrak{l}_1$ for $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$, since this algebra has odd dimension. We will therefore (following [@costello2017gauge]) extend the dimension of the algebra by 1, adding to $\mathfrak{sl}_2(\mathbb{C})$ another copy $\tilde{h}$ of the Cartan $h$ of $\mathfrak{g}$. The resulting Lie algebra thus becomes $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})\oplus \tilde{h}$. We extend the invariant bilinear form on $\mathfrak{sl}_2(\mathbb{C})$ to $\mathfrak{g}$ by defining $\operatorname{Tr}({\tilde{h},\tilde{h}})=2$ and $\operatorname{Tr}(\tilde{h},a)=0$ for all $a\in\mathfrak{sl}_2(\mathbb{C})$. The required properties for $\mathfrak{l}_0$ and $\mathfrak{l}_1$ can now be satisfied by setting $\mathfrak{l}_0=f\oplus (h-i\tilde{h})$ and $\mathfrak{l}_1=e\oplus (h-i\tilde{h})$. We therefore arrive at the following boundary conditions on the gauge field: $$\begin{aligned}\label{eq:bc}
\mathbb{R}^2\times\{0\}&:\hspace{20pt} A^{e}_{i}=0 \hspace{20pt} A^{h}_{i}+iA_i^{\tilde{h}}=0\\
\mathbb{R}^2\times\{1\}&:\hspace{20pt} A^{f}_{i}=0 \hspace{20pt} A^{h}_{i}-iA_i^{\tilde{h}}=0 \ .
\end{aligned}$$ We will in the following take $\tilde{h}$ to act as the identity in the fundamental representation given in Section \[sec:CYBE\]: $$\begin{aligned}
\label{eq:htilde}
\tilde{h}=\begin{pmatrix}
1&0\\0&1
\end{pmatrix} .\end{aligned}$$ We now proceed to determining the propagator of the theory in the presence of these boundary conditions.
The Propagator {#sec:Prop}
--------------
The easiest way to compute the propagator of the theory with boundary conditions is by first computing the propagator of the free theory (with no boundary conditions) and then modifying it so that the boundary conditions are satisfied. This will be done in the following.
### The Propagator of the Free Theory
The propagator, interpreted as a 2-form in the variables $x$ and $x'$, has the form, $$\begin{aligned}
P^{ab}(x,x')=\sum_{i,j=1,2,3}\braket{A_i^a(x),A_j^b(x')}\mathrm{d}(x^i-{x'}^i)\wedge \mathrm{d}(x^j-{x'}^j),\end{aligned}$$ where $a,b\in\{e,f,h,\tilde{h}\}$ are color indices. The expression becomes particularly simple if we choose as our gauge the following modified version of the Lorentz gauge[^1], $$\begin{aligned}
\partial_{x_3}A_{3}=0. \label{gauge}\end{aligned}$$ In this gauge the propagator 2-form $P^{ab}_{ij}(x,x')\coloneqq\braket{A^a_i(x),A^b_j(x')}$ is defined through the relations, $$\begin{aligned}\label{eq:props}
&\partial_{x_3}P^{ab}_{3j}(x,x')=0 \ , \ j\in\{1,2,3\} \\
&\operatorname{Tr}(t^a t^b)\mathrm{d}P^{ab}(x,x')=4\pi\hspace{1pt}\delta_{x_1=x_1'}\delta_{x_2=x_2'}\delta_{x_3=x_3'},
\end{aligned}$$ along with the anti-symmetry property $P^{ab}(x,x')=-P^{ba}(x',x)$, which follows from the anti-symmetry of the kinetic term in the Chern-Simons action. The second equation in implies that the color dependence of the propagator is given by the quadratic Casimir of the Lie algebra $\mathfrak{g}$. Thus, if we reinterpret the propagator to be an element of $\mathfrak{g}\otimes\mathfrak{g}$ it takes the form $P(x,x')C(\mathfrak{g})$, where $C(\mathfrak{g})$ is the quadratic Casimir of $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})\oplus\tilde{h}$: $$\begin{aligned}
C(\mathfrak{g})=e\otimes f +f\otimes e+\frac{1}{2}h\otimes h+\frac{1}{2}\tilde{h}\otimes\tilde{h}. \end{aligned}$$ It can easily be verified that the conditions in are satisfied if we choose as our ansatz the following expression for the propagator, $$\begin{aligned}
P=2\pi\delta_{x_1=x_1'}\delta_{x_2=x_2'}\big(\delta_{x_3>x_3'}-\delta_{x_3< x_3'}\big) C(\mathfrak{g}). \label{free.prop}\end{aligned}$$ Thus, we have determined the propagator of the free theory and we are ready to impose the boundary conditions discussed in Section \[sec:BC\].
### The Propagator with Boundary Conditions
The chosen set of boundary conditions translates into the following constraints on the propagator: In the case of $x_3 < x_3'$ we have $$\begin{aligned}
P^{ea}(x_1,x_2,x_3=0,x')&=P^{af}(x,x_1',x_2',x_3'=1)=0 \nonumber \\ P^{ha}(x_1,x_2,x_3=0,x')&=-P^{\Tilde{h}a}(x_1,x_2,x_3=0,x') \label{const1}\\
P^{ah}(x,x_1',x_2',x_3'=1)&=P^{a\Tilde{h}}(x, x_1',x_2',x_3'=1)
\nonumber\end{aligned}$$ for any $a\in\{e,f,h,\Tilde{h}\}$, and in the case of $x_3>x_3'$ we have $$\begin{aligned}
P^{fa}(x_1,x_2,x_3=1,x')&=P^{ae}(x,x_1',x_2',x_3'=0)=0 \nonumber \\P^{ah}(x,x_1',x_2',x_3'=0)&=-P^{a\Tilde{h}}(x,x_1',x_2',x_3'=0) \label{const2}\\ P^{ha}(x_1,x_2,x_3=1,x')&=P^{\Tilde{h}a}(x_1,x_2,x_3=1,x') \ .\nonumber\end{aligned}$$ Since the propagator in obviously has translation invariance, the constraints in , previously evaluated at $x_3=0$ and $x_3'=1$, must actually hold for all $x_3$ and $x_3'$ with $x_3<x_3'$. Similarly, the constraints in must hold for all $x_3$ and $x_3'$ with $x_3>x_3'$. Thus we can write the total constraints on the propagator imposed by the boundary conditions as follows: $$\begin{aligned}\label{eq:fullconst}
&x_3<x_3': \hspace{20pt}P^{ea}=P^{af}=0 , \ \ P^{ha}=-P^{\Tilde{h}a} \ , \ \
P^{ah}=P^{a\Tilde{h}}\\
&x_3>x_3': \hspace{20pt}P^{fa}=P^{ae}=0 , \ \ P^{ah}=-P^{a\Tilde{h}} \ , \ \
P^{ha}=P^{\Tilde{h}a}\ .
\end{aligned}$$ Starting from the free propagator $P$ in we can construct a propagator in the presence of boundary conditions by adding a term $P'$ that compensates for the relevant elements of $P$ such that is satisfied. In order for the result to still be a valid propagator, $P'$ must satisfy the gauge condition , have vanishing exterior derivative, and obey the anti-symmetry property $P'_{ab}(x,x')=-P'_{ba}(x',x)$. Going back to the formalism of where the propagator is taken to be an element of $\mathfrak{g}\otimes\mathfrak{g}$, we define $$\begin{aligned}
P'=2\pi\hspace{1pt}\delta_{x_1=x_1'}\delta_{x_2=x_2'}\Big(\delta_{x_3>x_3'}+\delta_{x_3<x_3'}\Big)\Big(e\otimes f-f\otimes e+\frac{i}{2}\Tilde{h}\otimes h -\frac{i}{2}h\otimes\Tilde{h}\Big) ,\end{aligned}$$ which has the required properties. By adding $P'$ to the free propagator we reach the following expression for the propagator in the theory with boundary conditions $$\begin{aligned}
P\to P+P'=4\pi\hspace{1pt}&\Big(e\otimes f+\frac{1}{4}(h+i\tilde{h})\otimes(h-i\tilde{h})\Big)\delta_{ x_1=x_1'}\delta_{x_2=x_2'}\delta_{x_3> x_3'}\\&-4\pi\hspace{1pt}\Big(f\otimes e+\frac{1}{4}(h-i\tilde{h})\otimes(h+i\tilde{h})\Big)\delta _{x_1=x_1'}\delta_{x_2=x_2'}\delta_{x_3<x_3'} . \label{prop}
\end{aligned}$$ Let us compare the color factors in this result with the solutions for the classical $R$-matrix given in , . Since $\Tilde{h}$ commutes with all the generators of $\mathfrak{sl_2}(\mathbb{C})$, one easily finds that the Yang-Baxter equation is still satisfied if we include $\Tilde{h}$ in the solutions $r$ and $\tilde{r}$ as in . Thus, we can rewrite the propagator as, $$\begin{aligned}
P(x,x')=4\pi\hspace{1pt} \delta_{ x_1=x_1'}\delta_{x_2=x_2'}\big(r\hspace{1pt}\delta_{x_3> x_3'}-\Tilde{r}\hspace{1pt}\delta_{x_3<x_3'}\big),\label{eq:propagator}\end{aligned}$$ We have thus managed to recover solutions of classical Yang-Baxter equation from a 3-dimensional Chern-Simons theory by using the approach suggested in [@costello2017gauge; @costello2018gauge]. As mentioned in the introduction, Turaev and Reshitikhin have previously given constructions of the Jones polynomials using an $R$-matrix representation of the Artin braid group. The purpose of the remaining part of this paper will be to explain how these construction arise from Chern-Simons theory. Our first step towards this is to introduce Wilson lines to the theory since they, as we will argue in the following, can be seen as representing braid strands.
Wilson Loops and Knots {#sec:WL}
======================
In this section we study one of the fundamental gauge invariant observables of Chern-Simons theory known as Wilson loops. A Wilson loop is obtained by taking the trace of the holonomy of the gauge field $A$ around a simple, smooth, closed curve $\gamma:[0,1]\to\mathbb{R}^2\times I$, $$\begin{aligned}\label{eq:W1}
W(\gamma)&=\operatorname{Tr}\mathcal{P}\exp\left(\oint_\gamma A\right)\\ &=\operatorname{Tr}(\mathds{1})+\operatorname{Tr}\oint_\gamma dx^i A_i(x)+\operatorname{Tr}\oint_{\gamma}dx^i\int^xdx'^j A_i(x)A_j(x')+\dots
\end{aligned}$$ where $\mathcal{P}$ stands for the path ordering of the exponential. In the context of the present paper, the trace is taken over the fundamental representation of the gauge group $\mathfrak{g}=\mathfrak{sl}_2\oplus \tilde{h}$ with generators $\{e, f, h, \tilde{h}\}$ given in and .\
A single Wilson loop, represented by a closed, oriented curve $\gamma$ in $\mathbb{R}^2\times I$, is called a *knot* and a collection of Wilson loops given by $n$ closed, oriented curves $\{\gamma_1,\dots,\gamma_n\}$ that are non-intersecting but may be linked around each other is called an ($n$-component) *link*. In the following sections we will be concerned with studying the expectation value of such links, which we will write as $\braket{W(L)}=\braket{W(\gamma_1)\dots W(\gamma_n)}$.
Interacting Wilson Lines {#sec:IntWL}
------------------------
We start by studying the interaction of open Wilson lines, which means that we will for now be ignoring the trace in .\
For any set of Wilson lines, the leading order interaction between them is given by the propagator in and thus is only non-vanishing at the points where two lines cross in $\mathbb{R}^2$. We will therefore in the following represent Wilson lines by their planar projection onto $\mathbb{R}^2$. In this representation each line corresponds to a space of spin states $V_\mu$ of the fundamental representation of $\mathfrak{sl}_2\oplus\tilde{h}$, and to each crossing is attached an interaction-matrix, given by the propagator, which acts on the corresponding vector spaces.
![The leading order contributing diagrams for the interaction between Wilson lines corresponding to a gluon exchange at the point of crossing between the lines. There are two types of such crossings: an over-crossing $K_+$ and an under-crossing $K_-$.[]{data-label="fig:Cross"}](Crossings.png){width="0.3\paperwidth"}
Notice that any crossing between two oriented line segments in $\mathbb{R}^2$ can be continuously deformed into one of the two crossing shown in Figure \[fig:Cross\]. We will choose as a convention to read a crossing in the following way: the line element coming in from the top left in Figure \[fig:Cross\] is associated to the left gauge field in the propagator $P(x,x')=\braket{A(x)A(x')}$ and the line element coming in from the top right is associated to the right gauge field of the propagator. The over-crossing $K_+$ therefore corresponds to the case $x_3>x_3'$ which, considering and , means that we should associate to it the $R$-matrix $R_{\mu\nu}={\mathop{\mathrm{id}}}+4\pi\hbar r_{\mu\nu}$. Similarly the under-crossing $K_-$ corresponds the case $x_3<x_3'$ so we should associate to it the $\tilde{R}$-matrix $\tilde{R}_{\mu\nu}={\mathop{\mathrm{id}}}-4\pi\hbar\tilde{r}_{\mu\nu}$. Recall from Section \[sec:YBE\] that the operators $R_{\mu\nu}$ and $\tilde{R}_{\mu\nu}$ swaps the pair of outgoing spin states, which is consistent with the fact that two Wilson lines cross when they interact.
Relation to the Braid Group {#sec:BG}
---------------------------
It is a well known result (see e.g. [@turaev1988yang]) that every $R$-matrix gives rise to a representations of the Artin $n$-strand braid group. Since we have found that interactions in our theory are described by an $R$-matrix, this implies that we can consider Wilson lines to represent braid strands. This will be explained in detail in the present subsection. We start by briefly recalling the concept of braids and the braid group.
![Example of a 4-strand braid.[]{data-label="fig:braid"}](Braid.png){width="0.18\paperwidth"}
Consider two lines in $\mathbb{R}^3$ parallel to the $y$-axis and with $(x,z)$-coordinates $(0,0)$ and $(1,0)$ respectively, and mark $n$ points on each line. An $n$-strand braid is a set of $n$ non-intersecting curves (strands) connecting the points on the line at $x=0$ with the points on the line at $x=1$ while strictly increasing in the $x$-direction. Similarly to knots, we can represent a braid by its planar projection onto $\mathbb{R}^2$ (corresponding in the above description to the $(x,y)$-plane). A simple example of a planar 4-strand braid is shown in Figure \[fig:braid\].\
It holds intuitively that any planar braid diagram can be constructed from a series of over-crossings and under-crossing of adjacent strands. This gives rise to the definition of the Artin $n$-strand braid group: $$\begin{aligned}\label{eq:BG}
B_n=\langle\sigma_1\dots\sigma_{n-1}|\ &\sigma_i\sigma_j=\sigma_j\sigma_i \ \ \text{for} \ |i-j|\geq2,\\ &\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_{i}\sigma_{i+1} \ \ \text{for} \ i=1,\dots,n-1\rangle,
\end{aligned}$$ where the graphical interpretation of the generator $\sigma_i$($\sigma_i^{-1}$) is an over(under)-crossing between the braid strands at the $i$’th and $(i+1)$’th position. The first relation in is then easily interpreted since the crossing of the strands at $i$ and $i+1$ is obviously independent of the crossing of the strands at $j$ and $j+1$ if $|i-j|\geq 2$. Notice that by multiplying the second relation in from the left by $\sigma_i^{-1}$ and from the right by $\sigma_{i+1}^{-1}$ or oppositely we reach the following three equivalent relations: $$\begin{aligned}\label{eq:braid2}
\sigma_{i}\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1} \ ,& \ \ \sigma_{i}^{-1}\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\\ \sigma_{i}\sigma_{i+1}\sigma_i^{-1}&=\sigma_{i+1}^{-1}\sigma_i\sigma_{i+1}.
\end{aligned}$$ The graphical interpretation of these relations is shown in Figure \[fig:3dYBE\].
![Graphical representation of the defining relations for the Artin braid group: $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$ and its two implications.[]{data-label="fig:3dYBE"}](ArtinBG.png){width="0.5\paperwidth"}
Starting from a braid, one can obtain a link by closing the strands of the braid, i.e. by connecting the points directly opposite each other. It is a fundamental result in knot theory, known as Alexander’s Theorem, that any link can be obtained as the closure of a braid.
### Wilson Lines and Braids
In the definition of a braid given above, we now wish to interpret the braid strands as representing Wilson lines. We thus label each strand by a space of spin states $V_\mu$, $\mu~=~1\dots, n$ transforming under $\mathfrak{sl}_2(\mathbb{C})\oplus\tilde{h}$ and we attach to each crossing between two strands $V_\mu$ and $V_\nu$ an interaction matrix $R_{\mu\nu}$ or $\tilde{R}_{\mu\nu}$ in accordance with the formalism developed in Section \[sec:IntWL\].
![The unitarity relation for Wilson lines.[]{data-label="fig:parallel"}](Parallel.png){width="0.35\paperwidth"}
In order to show that this is a valid interpretation, we start by verifying the unitarity relation illustrated in Figure \[fig:parallel\] which states that there is an “inverse crossing”. In other words, we want to show that the situation where two Wilson lines cross and then cross back without winding around each other is equivalent to the situation where the lines do not cross at all. Written out in equations, we want $\tilde{R}_{\nu\mu}{R}_{\mu\nu}={\mathop{\mathrm{id}}}$. We have, $$\begin{aligned}
\label{eq:parallel}
\tilde{R}_{\nu\mu}{R}_{\mu\nu}=\left({\mathop{\mathrm{id}}}-4\pi\hbar \tilde{r}_{\nu\mu}\right)\left({\mathop{\mathrm{id}}}+4\pi\hbar {r}_{\mu\nu}\right)\approx {\mathop{\mathrm{id}}}-4\pi\hbar \tilde{r}_{\nu\mu}+4\pi\hbar {r}_{\mu\nu}={\mathop{\mathrm{id}}}\end{aligned}$$ where the last equality sign follows from the relation $\tilde{r}_{ji}=r_{ij}$. This shows that unitarity is indeed satisfied at leading order in perturbation theory.\
We now consider the defining relations for the Artin braid group. Notice first that $R_{\mu\nu}R_{\lambda\rho}=R_{\lambda\rho}R_{\mu\nu}$ if $\mu,\nu,\lambda,\rho$ are all different, since the $R$-matrices act on different vector spaces. This implies that the first relation in is satisfied for Wilson lines and we thus only have left to check the relations in . However, these relations follows immediately from the 3-dimensional generalization of the Yang-Baxter equation which arises with the concept of over-crossings and under-crossings. In fact, using the above unitarity relation, it is relatively straightforward to check that the Yang-Baxter equation extends to the following three equivalent equations which are the analogues of , $$\begin{aligned}\label{eq:3dYBE}
R_{\mu\nu}R_{\mu\lambda}R_{\nu\lambda}=R_{\nu\lambda}R_{\mu\lambda}R_{\mu\nu}\ ,& \ \ \tilde{R}_{\mu\nu}R_{\mu\lambda}R_{\nu\lambda}=R_{\nu\lambda}R_{\mu\lambda} \tilde{R}_{\mu\nu}, \\ R_{\mu\nu}R_{\mu\lambda}\tilde{R}_{\nu\lambda}&=\tilde{R}_{\nu\lambda}R_{\mu\lambda}R_{\mu\nu},
\end{aligned}$$ The representations of these equations in terms of Wilson line diagrams are exactly the ones given in Figure \[fig:3dYBE\]. Thus we have found that interacting Wilson lines in our theory gives rise to a representation of the Artin braid group. We show in Section \[sec:Jones\] this actually corresponds to a Hecke algebra representation of the braid group similar to the one used in [@jones1990hecke] to construct the Jones polynomials.
The Expectation Value of Links {#sec:Exp}
------------------------------
Having seen in the previous subsection that open Wilson lines behave like braid strands, we will in the present subsection study the expectation value of links obtained from closing the braid strands. Let $\ket{s_1}$, $\ket{s_2}$ denote the basis vectors of $V$ in the fundamental representation, corresponding to the spin up and spin down states. Then a basis for the total space $V_1\otimes\cdots\otimes V_n$ is given by $$\begin{aligned}
\label{basis}
\{\ket{s_{k_1}}\otimes\cdots\otimes\ket{s_{k_n}} | \ {k_i=1,2} \ , \ i=1,\dots n\}.\end{aligned}$$ Furthermore, denote the total interaction matrix corresponding to a given $n$-strand braid $\alpha$ by $\mathcal{M}_\alpha$, i.e. $\mathcal{M}_\alpha$ is a product of $R$ and $\tilde{R}$ matrices acting on $V_1\otimes\cdots\otimes V_n$. Connecting the braid strands then corresponds to tracing over $\mathcal{M}$ in the basis as follows, $$\begin{aligned}
\braket{W(L)}=\sum_{k_i=1,2}\bra{s_{k_1}}\otimes\dots\otimes\bra{s_{k_n}}\mathcal{M_\alpha}\ket{s_{k_1}}\otimes\dots\otimes\ket{s_{k_n}},\end{aligned}$$ where $L$ is the link obtained from $\alpha$ by closing the braid strands.
![Two examples of Wilson loop diagrams as the closure of braids.[]{data-label="fig:Diagrams"}](Diagrams.png){width="0.36\paperwidth"}
In the following, we will determine a general expression for the expectation value of a configuration of Wilson loops at leading order in perturbation theory. As a motivating example, we start by computing the diagrams in Figure \[fig:Diagrams\](a),(b). In the diagrams each line corresponds to a vector space $V_1$ or $V_2$, and each line segment between two crossings is labeled by an $\mathfrak{sl}_2\oplus\tilde{h}$ basis vector. Assigning to each crossing the corresponding $R$-matrix element we obtain, $$\label{eq:exp1}
\begin{aligned}
\braket{W(L_1)}&=\bra{s_{k_1}}\otimes\bra{s_{k_2}}R_{12}\ket{s_{k_1}}\otimes\ket{s_{k_2}}=R^{ij}_{ji}=\delta^i_j\delta^j_i+4\pi\hbar r^{ij}_{ji},
\end{aligned}$$ where the indices are summed over. Similarly, $$\label{eq:exp2}
\begin{aligned}
\braket{W(L_2)}&=\bra{s_{k_1}}\otimes\bra{s_{k_2}}R_{12}R_{21}\ket{s_{k_1}}\otimes\ket{s_{k_2}}={R}^{ij}_{kl}{R}^{lk}_{ji}\\&=\left(\delta^i_k\delta^j_l+4\pi\hbar {r}^{ij}_{kl}\right)\left(\delta^l_j\delta^k_i+4\pi\hbar {r}^{lk}_{ji}\right)=\delta^i_i\delta^j_j+8\pi\hbar {r}^{ij}_{ij}.
\end{aligned}$$ With the explicit expressions for the generator matrices in , we have, $$\label{eq:r-value}
\begin{aligned}
r^{ij}_{ij}&=\tilde{r}^{ij}_{ij}=\frac{1}{4}\operatorname{Tr}(\tilde{h})^2=1 , \\
r^{ij}_{ji}&=\tilde{r}^{ij}_{ji}=\operatorname{Tr}(ef)+\frac{1}{4}\left(\operatorname{Tr}(h^2)+\operatorname{Tr}(\tilde{h}^2)\right)=2 ,
\end{aligned}$$ and inserting this into and we get, $$\begin{aligned}
\braket{W(L_1)}=2+8\pi\hbar, \\ \braket{W(L_2)}=4+8\pi\hbar .
\end{aligned}$$ One can relatively easily convince oneself that, for any $n$-component link of Wilson loops, the term at order zero in $\hbar$ is given by $(\delta^i_i)^n=2^n$. Furthermore, for every crossing between two line segments of the same loop, one must add to the expectation value a term of the form $(2^{n-1})4\pi\hbar r^{ij}_{ji}$ for an over-crossing or $(-2^{n-1})4\pi\hbar \tilde{r}^{ij}_{ji}$ for an under-crossing. Similarly, for every crossing between line segments of distinct loops, one must add a term of the form $(2^{n-2})4\pi\hbar r^{ij}_{ij}$ for an over-crossing or $(-2^{n-2})4\pi\tilde{r}^{ij}_{ij}$ for an under-crossing. Thus, with the values in , the expectation value of a general configuration of $n$ Wilson loops, $\gamma_1,\dots,\gamma_n$, forming a link $L$ takes the form, $$\begin{aligned}
\label{eq:M-corr}
\braket{W(L)}=2^n\Bigg(1+4\pi\hbar\sum_{\alpha=1}^n\big(n^+_{\gamma_\alpha}-n^-_{\gamma_\alpha}\big)+\pi{\hbar}\mathop{\sum_{\alpha,\beta=1}}_{\alpha<\beta}^n\big(n^+_{\gamma_\alpha,\gamma_\beta}-n^-_{\gamma_\alpha,\gamma_\beta}\big)\Bigg), \end{aligned}$$ where $n^+_{\gamma_\alpha}$($n^-_{\gamma_\alpha}$) is the number of over(under)-crossings in a planar projection of $\gamma_\alpha$ and $n^+_{\gamma_\alpha,\gamma_\beta}$($n^-_{\gamma_\alpha,\gamma_\beta}$) is the number of over(under)-crossings between a line segment of $\gamma_\alpha$ and a line segment of $\gamma_\beta$. We identify in the above expression the writhe number $\omega(\gamma)=(n^+_{\gamma}-n^-_{\gamma})$ and the Gauss linking number $\operatorname{lk}(\gamma_\alpha,\gamma_\beta)=\frac{1}{2}(n^+_{\gamma_\alpha,\gamma_\beta}-n^-_{\gamma_\alpha,\gamma_\beta})$. Defining, $$\begin{aligned}
\omega(L)\coloneqq\sum_{\alpha=1}^n\omega(\gamma_\alpha) \ , \ \ \ \ \operatorname{lk}(L)\coloneqq\mathop{\sum_{\alpha,\beta=1}}_{\alpha<\beta}^n\operatorname{lk}(\gamma_\alpha,\gamma_\beta),
\end{aligned}$$ we can write $$\begin{aligned}
\label{eq:exp}
\braket{W(L)}=2^n\Big(1+4\pi\hbar\omega(L)+2\pi{\hbar}\operatorname{lk}(L)\Big)\approx 2^n\exp\left\{4\pi\hbar\left(\omega(L)+\frac{1}{2}\operatorname{lk}(L)\right)\right\}.\end{aligned}$$ The appearance of the linking number in the above equation is in agreement with the well known result (see e.g. [@witten1989quantum]) that in abelian Chern-Simons theory the expectation value at first order is given by the Gauss linking number. Notice that the linking number indeed appears from the abelian part of the Lie algebra. The remaining part of expresses the framing anomaly of Chern-Simons theory as will be discussed in the following subsection.
The Framing Anomaly {#sec:FA}
-------------------
The result in implies that the expectation value of configurations of Wilson loops is not in itself a link invariant. Indeed, a link invariant is defined to be invariant under a set of deformations of the link that one can make without changing its isotopy class. There are three such deformations known as Reidemeister moves. The first Reidemeister move corresponds to twisting a strand of the knot and thereby changing the writhe number by $\omega(L)\to\omega(L)\pm 1$ depending on the type of twist (see Figure \[fig:Twist\]).
![Twisting of a strand with either an under-crossing (left) where $\omega\to\omega-1$, or an over-crossing (right) were $\omega\to\omega+1$.[]{data-label="fig:Twist"}](Twists.png){width="0.3\paperwidth"}
It is evident from the expression that under such a twist the expectation value changes according to $$\begin{aligned}
\label{eq:twist}
\braket{W(L)}\to e^{\pm 4\pi\hbar}\braket{W(L)}.\end{aligned}$$ The remaining two Reidemeister moves are satisfied due to the unitarity relation and the 3-dimensional Yang-Baxter equation which were described in Section \[sec:BG\].\
This discrepancy in the expectation value under twisting a strand expresses the so called framing anomaly of Chern-Simons theory. In [@witten1989quantum] the framing anomaly appears as a consequence of the self-interaction of Wilson loops only being well defined with a choice of framing of the loops and it is found that the expectation value will change under a change of framing. With our method in the present paper we get a well defined expression for the self-interaction without having to introduce a framing. However, as a price, the resulting expression is not a link invariant.
Constructing the Jones Polynomials {#sec:Jones}
==================================
In this section we show in detail how we can recover a specific value of the Jones two-variable polynomials from the expectation value of Wilson loops. The construction that we use is similar to the one originally given by Jones in [@jones1990hecke], namely from Ocneanu’s trace acting on a Hecke algebra representation of the Artin braid group.
Hecke Algebra Relation for Wilson Lines {#sec:J1}
---------------------------------------
The Hecke algebra $H_n(q)$ with generators $\{g_i\}_{i=1}^{n-1}$ is obtained from the Artin $n$-strand braid group by adding to an additional Hecke algebra relation: $$\begin{aligned}
\label{eq:Hecke}
(g_i-q^{1/2})(g_i+q^{-1/2})=0 \Leftrightarrow g_i=(q^{1/2}-q^{-1/2})+g_i^{-1} ,\end{aligned}$$ where $q$ is some scalar. The Jones two-variable polynomials were originally constructed from considering a representation of $B_n$ coming from the Hecke algebra[^2], and we are therefore interested in checking if a relation similar to the Hecke algebra relation is satisfied for Wilson lines.\
In terms of Wilson line diagrams corresponds to the following relation:
{width="0.4\paperwidth"} \[Hecke1\]
where the incoming and outgoing lines are label by spins states. Notice that since the lines in the second term on the right hand side of the equality do not cross, the associated outgoing spin states are swapped relatively to the situation in the other two terms. Thus, the corresponding interaction matrices must differ by a permutation matrix swapping the pair of incoming spin states, i.e. $P^{ij}_{kl}=\delta^i_l\delta^j_k$. As one can easily verify, $P$ can be written in matrix form as $$\begin{aligned}
P=e\otimes f+f\otimes e+\frac{1}{2}\big(h\otimes h+\tilde{h}\otimes\tilde{h}\big).\end{aligned}$$ By recognizing in the above the solutions of the classical Yang-Baxter equation and , we find that $$\begin{aligned}
\label{eq:Skein}
P=r+\tilde{r}=\frac{1}{4\pi\hbar}(R-\tilde{R}). \end{aligned}$$ Equivalently, by multiplying both sides of this relation with $P$, we obtain $$\begin{aligned}
\label{eq:Skein1}
R_{12}=\tilde{R}_{12}+4\pi\hbar {\mathop{\mathrm{id}}}.\end{aligned}$$ It is seen that this expression has the form of equation when we take $q$ to be $q=e^{4\pi\hbar}$ and expand to first order in $\hbar$. Thus, interacting Wilson lines indeed give rise to a Hecke-algebra representation of the Artin braid group.
Normalizing the Expectation Value {#sec:Ocn}
---------------------------------
In [@jones1990hecke] the Jones polynomials are obtained from the trace function, $\operatorname{tr}_z$, due to Ocneanu (see [@freyd1985new]), acting on $\bigcup_{n=1}^\infty H_n(q)$, which is defined for any $z\in\mathbb{C}$ to satisfy a so called Markov property: $\operatorname{tr}_z(xg_n)=z\operatorname{tr}_z(x)$ for $x\in H_n(q)$. In fact, a Jones polynomial is a polynomial in the variables $q$ and $\lambda$, where $q$ corresponds to the scalar appearing in the definition of the Hecke algebra and $\lambda$ is a normalisation factor which ensures that $\lambda^{1/2}\operatorname{tr}_z(g_i)=\lambda^{-1/2}\operatorname{tr}_z(g_i^{-1})$. Since the analogue of equation corresponds to the case $\operatorname{tr}_z(g_i^{-1})=z^{-1}$, we take $\lambda=z^{-2}$. Following the definition in [@jones1990hecke], the Jones polynomial for a link $L$ then takes the form, $$\begin{aligned}
\label{eq:jones}
V_{L}(q,z)=z^{-e}\operatorname{tr}_z(\pi(\alpha)),\end{aligned}$$ where $\alpha\in B_n$ is any braid whose closure is $L$, $e$ is the exponent sum of $\alpha$ as a word on the $\sigma_i$’s and $\pi$ is the representation of $B_n$ in $H_n(q)$, $\pi(\sigma_i)=g_i$.\
We notice from the discussion in Section \[sec:FA\] that the Markov property of Ocneanu’s trace is analogous to the framing anomaly of Wilson loops discussed in Section \[sec:FA\]. Indeed, for a given link $L$, let $x\in H_n(q)$ be the Hecke algebra representation of a braid whose closure is $L$. Then the operation $x\to xg_n$, under which Ocneanu’s trace changes by a factor of $z$, corresponds to twisting a strand of $L$ with a twist that adds an over-crossing to $L$. As we known from equation , this increases the total writhe number by 1, which causes the expectation value of $W(L)$ to change by a factor of $e^{\hbar}$. Thus, if we take $q=e^{4\pi\hbar}$ and $\lambda=z^{-2}=e^{-8\pi\hbar}$, the expectation value of Wilson loops can be seen as Ocneanu’s trace acting on a Hecke algebra representation of the Artin braid group. In line with the construction in , we can therefore obtain a value of the Jones polynomials from the expectation value of Wilson loops by normalising it with a factor that depends on the total writhe number of the link. We define $$\begin{aligned}
\label{inv}
X_L\coloneqq e^{-4\pi\hbar \omega(L)}\braket{W(L)}.\end{aligned}$$ It follows from that $X_L$ only depends on the isotopy class of $L$ and, according to the above discussion, it corresponds to the specific value of the Jones polynomials given by $V_L(e^{4\pi\hbar},e^{-8\pi\hbar})$. Notice however that, by assumption $V_\bigcirc\equiv1$, where $\bigcirc$ denotes the unknotted circle, and thus $X_L$ differs from $V_L$ by a normalisation.
Unknotting Wilson loops
-----------------------
![The three skein related links $L_+$, $L_-$ and $L_0$ are identical outside of the encircled area. Incoming and outgoing lines are labeled by spin states of $\mathfrak{sl}_2(\mathbb{C})\oplus\tilde{h}$.[]{data-label="fig:Skein"}](Skein.png){width="0.4\paperwidth"}
A more common way of constructing the Jones polynomial of a given link, is by using that the Jones polynomials are determined uniquely from a so called skein relation. A skein relation in general is a linear relation between the polynomial invariants of links $L_+$, $L_-$ and $L_0$ which differ only at a single crossing where they are as in Figure \[fig:Skein\]. Using the Hecke algebra relation along with the Markov property of Ocneanu’s trace one finds that the Jones polynomials satisfy the following skein relation $$\begin{aligned}
\label{eq:J-skein}
\lambda^{-1/2}V_{L_+}(q,\lambda)-\lambda^{1/2}V_{L_-}(q,\lambda)=\left(q^{1/2}-q^{-1/2}\right)V_{L_0}(q,\lambda).\end{aligned}$$ Using this relation recursively one can “unknot” any link $L$ thus ending up with a set of unlinked unknotted circles which by definition satisfy $V_\bigcirc\equiv 1$. Considering the results in Section \[sec:J1\] and \[sec:Ocn\] we expect $X_L$ to be satisfy the skein relation $$\begin{aligned}
\label{eq:X-skein}
e^{4\pi\hbar}X_{L_+}-e^{-4\pi\hbar}X_{L_-}=\left(e^{2\pi\hbar}-e^{-2\pi\hbar}\right)X_{L_0}.\end{aligned}$$ Indeed, since the links in Figure \[fig:Skein\] are identical outside of the encircled area, the expectation values corresponding to $L_+$, $L_-$ and $L_0$ are related by letting the matrix element corresponding to the crossing inside the encircled area vary between $R^{ij}_{kl}$, $\tilde{R}^{ij}_{kl}$ and $\delta^i_l\delta^j_k$, respectively. We can therefore expand the Hecke algebra relation to a relation between expectation values as follows $$\begin{aligned}
\braket{W(L_+)}-\braket{W(L_-)}=\left(e^{2\pi\hbar}-e^{-2\pi\hbar}\right)\braket{W(L_0)}.\label{eq:skein1}\end{aligned}$$ By substituting $\braket{W(L)}=e^{4\pi\hbar\omega(L)}\hspace{1pt}X_L$ into this equation and noting that $\omega(L_+)=\omega(L_0)~+~1$ and $\omega(L_-)=\omega(L_0)-1$, we recover as expected. This gives a way of recursively unknotting any set of Wilson loops.
Conclusion
==========
In this paper we have shown that, by imposing boundary conditions in one dimension on the gauge field of a 3-dimensional Chern-Simons theory, the interaction matrix at leading order in perturbation theory takes the form of an $R$-matrix. We argued that this result allows us to recover the Jones two-variable polynomials for specific values of the variables from the expectation value of Wilson loops using a construction analogous to the one originally given by Jones. Our results therefore give new insight into the relation between Chern-Simons theory and knot theory. We have been working only to leading order in perturbation theory and so it would be interesting, as a further investigation, to verify that the constructions can be generalized to higher orders.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank my supervisor Kevin Costello for helpful guidance. I also wish to thank Victor Py for useful comments along the way. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Research and Innovation.
[10]{}
Edward Witten. Quantum field theory and the [Jones]{} polynomial. , 121(3):351–399, 1989.
Vladimir G Turaev. The [Yang-Baxter]{} equation and invariants of links. , 92(3):527–553, 1988.
Nikolai Yu Reshetikhin. Quantized universal enveloping algebras, the [Yang-Baxter]{} equation and invariants of links [I, II. LOMI]{} preprints. Technical report, E-4-87, E-17-87, 1988.
Kevin Costello, Edward Witten, and Masahito Yamazaki. Gauge theory and integrability, [I]{}. , 2017.
Kevin Costello, Edward Witten, and Masahito Yamazaki. Gauge theory and integrability, [II]{}. , 2018.
Vaughan FR Jones. Hecke algebra representations of braid groups and link polynomials. In [*New Developments In The Theory Of Knots*]{}, pages 20–73. World Scientific, 1990.
Peter Freyd, David Yetter, Jim Hoste, WB Raymond Lickorish, Kenneth Millett, and Adrian Ocneanu. A new polynomial invariant of knots and links. , 12(2):239–246, 1985.
Enore Guadagnini, M Martellini, and M Mintchev. . , 330(2-3):575–607, 1990.
P Cotta-Ramusino, Enore Guadagnini, M Martellini, and M Mintchev. Quantum field theory and link invariants. , 330(2-3):557–574, 1990.
Enore Guadagnini, M Martellini, and M Mintchev. . , 235(3-4):275–281, 1990.
Enore Guadagnini, M Martellini, and M Mintchev. . , 18(2):121–134, 1991.
Alexei Morozov and Andrey Smirnov. . , 835(3):284–313, 2010.
Chen-Ning Yang. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. , 19(23):1312, 1967.
Rodney J Baxter. Eight-vertex model in lattice statistics. , 26(14):832, 1971.
Vyjayanthi Chari and Andrew N Pressley. . Cambridge university press, 1995.
[^1]: This gauge condition follows from the Lorentz gauge by rescaling the $x_1$ and $x_2$ components of the gauge field, $A_1'=\lambda^{-1} A_1$, $A_2'=\lambda^{-1} A_2$, $A_3'=A_3$, and then taking the limit $\lambda\to 0$. We are allowed to do this since the theory is metric independent.
[^2]: In the original construction of the Jones polynomials, the generators $\{g_i\}_{i=1}^{n-1}$ was defined to satisfy a slightly modified version of the Hecke algebra relation, given by, $g_i^2=(q-1)g_i+q$. However, the resulting algebra is isomorphic to the one in obtained by sending $g_i\to q^{-1/2}g_i$.
|
---
abstract: 'We prove that the multiple summing norm of multilinear operators defined on some $n$-dimensional real or complex vector spaces with the $p$-norm may be written as an integral with respect to stables measures. As an application we show inclusion and coincidence results for multiple summing mappings. We also present some contraction properties and compute or estimate the limit orders of this class of operators.'
address:
- 'Daniel Carando. Departamento de Matemática - Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina and IMAS-CONICET'
- 'Verónica Dimant. Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires, Argentina and CONICET'
- 'Santiago Muro. Departamento de Matemática - Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina and CONICET'
- 'Damián Pinasco. Departamento de Matemáticas y Estadística, Universidad Torcuato di Tella, Av. F. Alcorta 7350, (1428), Ciudad Autónoma de Buenos Aires, ARGENTINA and CONICET'
author:
- Daniel Carando
- Verónica Dimant
- Santiago Muro
- Damián Pinasco
title: An integral formula for multiple summing norms of operators
---
[^1]
Introduction {#introduction .unnumbered}
============
The rotation invariance of the Gaussian measure on $\mathbb K^N$, which we will denote by $\mu_2^N$, allows us to show the Khintchine equality. It asserts that if $c_{2,q}$ denotes the $q$-th moment of the one dimensional Gaussian measure, and $\ell_2^N$ denotes $\mathbb K^N$ with the euclidean norm, then for any $\alpha\in\mathbb K^N$, $1\le q<\infty$, $$\label{khintchine gaussiano}
c_{2,q}\|\alpha\|_{\ell_2^N}=
\Big(\int_{\mathbb K^N}|\langle\alpha,z\rangle|^qd\mu^N_2(z)\Big)^{1/q}.$$ We may interpret this formula as follows: the norm of a linear functional $\alpha$ on $\ell_2^N$ is a multiple of the $L^q$-norm of the linear functional with respect to the Gaussian measure on $\ell_2^N$. One may ask if there is a formula like (\[khintchine gaussiano\]) for linear functionals on some other space, or even for linear or multilinear operators. For linear functionals, an answer is provided by the $s$-stable Lévy measure (see for example [@DefFlo93 24.4]): for $s< 2$ there exists a measure on $\mathbb K^N$, called the [*$s$-stable Lévy measure*]{} and denoted by $\mu_{s}$, which satisfies that for any $0<q<s$, $\alpha\in\mathbb
K^N$, $$\label{estable}
c_{s,q}\|\alpha\|_{\ell_s^N}=\Big(\int_{\mathbb K^N}|\langle\alpha,z\rangle|^qd\mu^N_{s}(z)\Big)^{1/q},$$ where $$c_{s,q}=\Big(\int_{\mathbb K}|z|^qd\mu_{s}^1(z)\Big)^{1/q}.$$ The question for linear operators is more subtle because there are many norms which are natural to consider on $\mathcal
L(\ell_2^N)$. The first result in this direction is due to Gordon [@Gor69] (see also [@DefFlo93 11.10]), who showed that the formula holds for the identity operator on $\ell_2^N$, considering the absolutely $p$-summing norm of $id_{\ell_2^N}$, that is $$\pi_p(id_{\ell_2^N})=c_{2,q}\Big(\int_{\mathbb K^N}\|z\|_{\ell_2^N}^q \, d\mu^N_2(z)\Big)^{1/q}.$$ Pietsch [@Pie72] extended this formula for arbitrary linear operators from $\ell_{s'}^N\to\ell_s^N$, $s\ge2$ and used it to compute some limit orders (see also [@Pie80 22.4.11]).
To generalize the formula to the multilinear setting there is again a new issue, because there are many natural candidates of classes of multilinear operators that extend the ideal of absolutely $p$-summing linear operators (for instance the articles [@PelSan11; @Per05] are devoted to their comparison). Among those candidates, the ideal of multiple summing multilinear operators is considered by many authors the most important of these extensions and is also the most studied one. Some of the reasons are its connections with the Bohnenblust-Hille inequality [@PerVil04], or the results on the unconditional structure of the space of multiple summing operators [@DefPer08]. Multiple summing operators were introduced by Bombal, Pérez-García and Villanueva [@BomPerVil04] and independently by Matos [@Mat03]. In this note we show that multiple summing operators constitute the correct framework for a multilinear generalization of formula . For this we present integral formulas for the exact value of the multiple summing norm of multilinear forms and operators defined on $\ell_p^N$ for some values of $p$. Moreover, we prove that for some other finite dimensional Banach spaces these formulas hold up to some constant independent of the dimension.
One particularity of the class of multiple summing operators on Banach spaces is that, unlike the linear situation, there is no general inclusion result. In [@BotMicPel10; @Per04; @Pop09] the authors investigate this problem and prove several results showing that on some Banach spaces inclusion results hold, but on some other spaces not. The integral formula for the multiple summing norm, together with Khintchine/Kahane type inequalities will allow us to show some new coincidence and inclusion results for multiple summing operators.
Another application of these formulas deals with unconditionality in tensor products. Defant and Pérez-García showed in [@DefPer08] that the tensor norm associated to the ideal of multiple 1-summing multilinear forms preserves unconditionality on $\mathcal L_r$ spaces. As a consequence of our formulas, we give a simple proof of this fact for $\ell_r$ with $ r \ge 2$. Moreover, we show that vector-valued multiple 1-summing operators also satisfy a kind of unconditionality property in the appropriate range of Banach spaces. Finally, we compute limit orders for the ideal of multiple summing operators.
Our main results are stated in Theorems \[formulita escalar\] and \[formulita\], which give an exact formula for the multiple summing norm, and Proposition \[normas equivalentes\], which gives integral formulas for estimating these norms in a wider range of spaces.
Main results and their applications
===================================
Let $E_1,\dots,E_m,F$ be real or complex Banach spaces. Recall that an $m$-linear operator $T\in\mathcal L(E_1,\dots,E_m;F)$ is [*multiple $p$-summing*]{} if there exists $C>0$ such that for all finite sequences of vectors $(x^1_{j_1})_{j_1=1}^{J_1}\subset E_1,\dots,(x^m_{j_m})_{j_m=1}^{J_m}\subset E_m$ $$\left(\sum_{j_1,\dots,j_m}\|T(x^1_{j_1},\dots,x^m_{j_m})\|_F^p\right)^{\frac1{p}}\le C
w_p((x_{j_1}^1)_{j_1})\dots w_p((x_{j_m}^m)_{j_m}),$$ where $$w_p((y_j)_j)=\sup\left\{\Big(\sum_j|\gamma(y_j)|^p\Big)^{1/p}\,:\,\gamma\in B_{E'}\right\}.$$ The infimum of all those constants $C$ is the multiple $p$-summing norm of $T$ and is denoted by $\pi_p(T)$. The space of multiple $p$-summing multilinear operators is denoted by $\Pi_p(E_1,\dots,E_m;F)$. When $E_1=\dots=E_m=E$, the spaces of continuous and multiple $p$-summing multilinear are denoted by $\mathcal
L(^mE;F)$ and $\Pi_p(^mE;F)$ respectively.
The following theorems are our main results. Their proofs will be given in Section \[sec-proofs\].
\[formulita escalar\] Let $\phi$ be a multilinear form in $\mathcal L(^m\ell_r^N;\mathbb K)$, $p<r'<2$ or $r=2$. Then $$\pi_p(\phi)=\frac{1}{c_{r',p}^m}\ \Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|\phi(z^{(1)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}.$$
Before we state our second theorem, let us recall some necessary definitions and facts. For $1\leq q \leq \infty$ and $1 \leq \lambda < \infty$ a normed space $X$ is called an $\mathcal{L}_{q,\lambda}^g$*-space*, if for each finite dimensional subspace $M \subset X$ and $\varepsilon >0$ there are $R \in \mathcal{L}(M,\ell_q^m)$ and $S \in \mathcal{L}(\ell_q^m, X)$ for some $m \in
\mathbb{N}$ factoring the inclusion map $I_M^X:M\to X$ such that $\|S\| \|R\| \leq \lambda + \varepsilon$: $$\label{facto}
\xymatrix{ M \ar@{^{(}->}[rr]^{I_M^X} \ar[rd]^{R} & & {X} \\
& {\ell_q^m} \ar[ur]^{S} & }.$$ $X$ is called an $\mathcal{L}_{q}^g$*-space* if it is an $\mathcal{L}_{q,\lambda}^g$-space for some $\lambda \geq 1$. Loosely speaking, $\mathcal{L}_{q}^g$-spaces share many properties of $\ell_q$, since they *locally look like $\ell_q^m$*. The spaces $L_q(\mu)$ are $\mathcal{L}_{q,1}^g$-spaces. For more information and properties of $\mathcal{L}_{q}^g$-spaces see [@DefFlo93 Section 23].
\[formulita\] Let $T$ be a multilinear map in $\mathcal L(^m\ell_r^N;X)$, where $X$ is an $\mathcal L_{q,1}^g$-space and suppose $r$, $q$ and $p>0$ satisfy one of the following conditions
- $r=q=2$;
- $r=2$ and either $p<q<2$ or $p=q$;
- $p<r'<2$ and either $p<q\le 2$ or $p=q$.
Then $$\pi_p(T)= \frac{1}{c_{r',p}^m}\ \Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^p \, d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}.$$
It is clear that Theorem \[formulita escalar\] follows from Theorem \[formulita\], but in fact, the proof of Theorem \[formulita\] uses the scalar result, which is much simpler and is interesting on its own. We remark that the formula also holds for any multilinear map in $\mathcal
L(\ell_{r_1}^N,\dots,\ell_{r_m}^N;X)$, where $X$ is an $\mathcal L_{q,1}^g$-space and $r_1,\dots,r_m$, $q$ and $p$ satisfy conditions analogous to those of Theorem \[formulita\]. Moreover, the formula turns into an equivalence between the $\pi_p$ norm and the integral if we take general $\mathcal L_q^g$-spaces.
On the other hand, if we put $\ell_r$ in the domain, since multiple summing operators form a maximal ideal, the formula holds with a limit over $N$ in the right hand side (here we consider $\mathbb K^N$ as a subset of $\ell_r$).
There are situations not covered by the previous theorem where we have an equivalence or, at least, an inequality between the $\pi_p$ and the $L_p(\mu_{s})$ norms.
\[normas equivalentes\] Let $T\in\mathcal L(^m\ell_r^N;X).$
($i$) Suppose either $r=2$ and $p,q<2;$ or $r=2$ and $q\le p$; or $p<r'<2$ and $q\le 2$. If $X$ is an $\mathcal L_{q}^g$-space, then we have $$\pi_p(T)\asymp \left(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^p \, d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\right)^{1/p},$$ that is, the multiple $p$-summing and the $L_{r'}(\mathbb K^N\times\dots\times\mathbb
K^N,\mu_{r'}^N\times\dots\times\mu_{r'}^N)$ norm are equivalent in $\mathcal
L(^m\ell_r^N;X)$, with constants which are independent of $N$.
($ii$) If $r=2$ or $p<r'<2$ then we have, for any Banach space $X$, $$\pi_p(T)\succeq \left(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^p \, d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\right)^{1/p}.$$
Now we describe some applications of these results. The most direct one is an asymptotically correct relationship between the multiple summing norm of a multilinear operator and the usual (supremum) norm. Cobos, Kühn and Peetre [@CobKuhPee99] compared the Hilbert-Schmidt norm, $\pi_2$, with the usual norm of multilinear forms. They showed that if $T$ is any $m$-linear form in $\mathcal L(^m\ell_2^N,\mathbb K)$ then $$\pi_2(T)\le N^{\frac{m-1}{2}}\|T\|.$$ Moreover, the asymptotic bound is optimal in the sense that there exist constants $c_m$ and $m$-linear forms $T$ on $\ell_2^N$ with $\|T\|=1$ and $\pi_2(T)\ge c_mN^{\frac{m-1}{2}}$. It is easy to see from this that the correct exponent for the asymptotic bound for the Hilbert-valued case is $\frac{m}{2}$. The same holds for the multiple $p$-summing norm for any $p$ because all those norms are equivalent to the Hilbert-Schmidt norm in $\mathcal L(^m\ell_2;\ell_2)$, see [@Mat03; @Per04]. We see now that the same optimal exponent holds for multiple $p$-summing operators with values on $\mathcal L_q^g$-spaces.
First, note that passing to polar coordinates we have, in the complex case (the real case follows similarly) $$\begin{aligned}
& &\hspace{-5pt} \int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^p \, d\mu^N_2(z^{(1)})\dots
d\mu^N_2(z^{(m)}) \\
& =&\hspace{-5pt} \frac{1}{\Gamma(N)^m}\int_{(
S^{2N-1})^m} \hspace{-5pt}\|T(\omega^{(1)},\dots,\omega^{(m)})\|_X^p \, d\sigma_{2N-1}(\omega^{(1)})\dots
d\sigma_{2N-1}(\omega^{(m)})\, \Big(\int_{0}^\infty 2\rho^{2N+p-1}e^{-\rho^2}d\rho\Big)^m\\
&\le &\hspace{-5pt} \|T\|^p\Big(\frac{\Gamma(N+p/2)}{\Gamma(N)}\Big)^{m},\end{aligned}$$ where $S^{2N-1}$ denotes the unit sphere in $\mathbb R^{2N}$ and $\sigma_{2N-1}$ the normalized Lebesgue measure defined on it.
As a consequence of Proposition \[normas equivalentes\], we obtain $$\label{desigualdad entre normas}
\pi_p(T) \preceq \left(\frac{\Gamma(N+p/2)}{\Gamma(N)}\right)^{m/p} \|T\| \preceq
N^{\frac{m}{2}}\|T\|$$ for $X$ a $\mathcal L_{q,\lambda}^g$-space and $p\ge q$ or $p,q<2$.
Let us see that for $p,q\le 2$, the exponents are optimal. Since for any $T\in\mathcal
L(^m\ell_2^N;\ell_q)$ we have $$\Big(\sum_{j_1,\dots,j_m=1}^N\|T(e_{j_1},\dots,e_{j_m})\|_{\ell_q}
^p\Big) ^ { \frac1 {p}}\le \pi_p(T)N^{\frac{m}{p}-\frac{m}{2}}\preceq N^{\frac{m}{p}}\|T\|,$$ it suffices to show that the inequality $$\label{triangular}
\Big(\sum_{j_1,\dots,j_m=1}^N\|T(e_{j_1},\dots,e_{j_m})\|_{\ell_q}
^p\Big) ^ { \frac1 {p}}\preceq N^{\frac{m}{p}}\|T\|$$ is optimal. By [@Boa00 Theorem 4], there exist symmetric multilinear operators $\tilde T_N\in\mathcal
L(^m\ell_2^N,\ell_2^N)=\mathcal
L(^{m+1}\ell_2^N)$, such that, $\displaystyle \tilde
T_N=\sum_{j_1,\dots,j_{m+1=1}}^N\varepsilon_{j_1,\dots,j_{m+1}}e_{j_1}\otimes\dots\otimes e_{j_{m+1}}$, with $\varepsilon_{j_1,\dots,j_{m+1}}=\pm 1$ and $\|\tilde T_N\|\asymp \sqrt{N}$.
Let $T_N=i_{2q}\circ \tilde T_N$, where $i_{2q}:\ell_2^N\to\ell_q^N$ is the inclusion. Then, $\displaystyle \|T_N\|\preceq N^\frac1{q}$ and $$\displaystyle\Big(\sum_{j_1,\dots,j_m=1}^N\|T_N(e_{j_1},\dots,e_{j_m})\|_{\ell_q}
^p\Big)^ { \frac1 {p}}N^{\frac1{q}+\frac{m}{p}}\succeq N^{\frac{m}{p}}\|T_N\|.$$ This implies that inequality (\[triangular\]) is optimal and, hence, so is .
Inclusion theorems
------------------
The well-known inclusion theorem for absolutely summing linear operators states that for any Banach spaces $E,F$ we have $$\Pi_s(E,F)\subset\Pi_t(E,F),\quad \textrm{ when }s\le t.$$ Although multiple summing mappings share several properties of linear summing operators, there is no general inclusion theorem in the multilinear case (see [@PerVil04]). It is therefore interesting to investigate in which situations we do have inclusion type theorems. The following theorem summarizes some of the most important known results on this topic.
($i$) If $E$ has cotype $r\ge2$ then $$\Pi_s(^mE,F)=\Pi_1(^mE,F),\quad \textrm{ for }1\le s<r^*.$$ ($ii$) If $F$ has cotype $2$ then $$\Pi_s(^mE,F)\subset\Pi_2(^mE,F),\quad \textrm{ for }2\le s <\infty.$$
The following picture illustrates the above theorem in the particular case where $E=\ell_2$ and $F=\ell_q$, $$\begin{pspicture}(3,3)
\pspolygon[linecolor=green!60!red!60,
fillstyle=hlines](1.5,0)(3,0)(3,3)(1.5,3)
\pspolygon[linecolor=black!60!pink!60,fillcolor=black!60!pink!60,
fillstyle=solid](1.5,3)(0,3)(0,1.5)(1.5,1.5)
\psline(0,-0.2)(0,3.2)\psline(-0.2,0)(3.3,0)
\rput[l](3.4,0){$\frac1{p}$}
\rput[d](0.2,3.2){$\frac1{q}$}
\rput[u](3,-0.2){$1$}
\rput[r](0,3){$1$}
\rput[u](1.5,-0.3){$\frac1{2}$}
\rput[r](-0.1,1.5){$\frac1{2}$}
\psline[linestyle=dashed,linewidth=0.7pt](1.5,0)(1.5,3)
\psline[linewidth=0.7pt,linecolor=green!60!red!60](0.05,1.5)(2.95,1.5)
\end{pspicture}$$ $\smallskip$
In the ruled area we have $\Pi_{p_1}(^m\ell_2;\ell_q)=\Pi_{p_2}(^m\ell_2;\ell_q)$ and in the shaded area we have the reverse inclusion $\Pi_{p_1}(^m\ell_2;\ell_q)\subset\Pi_{2}(^m\ell_2;\ell_q)$ for $p_1\ge 2$.
As a consequence of our integral formula, we obtain the following improvement to the previous result, which will be proved in Section \[sec-proofs\].
\[prop inclusion\] Let $Y$ be a $\mathcal L_2^g$-space and $X$ a $\mathcal L_q^g$-space.
If $p\ge q$, then $\Pi_p(^mY;X)=\Pi_q(^mY;X)$.
If $p\le q$, then $\Pi_p(^mY;X)\subset \Pi_q(^mY;X)$.
With the information given by the above proposition, we have the following new picture. $$\begin{pspicture}(3,3)
\pspolygon[linecolor=green!60!red!60,
fillstyle=hlines](1.5,0)(3,0)(3,3)(0,3)(0,0)(1.5,1.5)
\pspolygon[linecolor=black!60!pink!60,fillcolor=black!60!pink!60,
fillstyle=solid](1.5,1.5)(0,0)(1.5,0)
\psline(0,-0.2)(0,3.2)\psline(-0.2,0)(3.2,0)
\rput[l](3.4,0){$\frac1{p}$}
\rput[d](0.2,3.2){$\frac1{q}$}
\rput[u](3,-0.2){$1$}
\rput[r](0,3){$1$}
\rput[u](1.5,-0.3){$\frac1{2}$}
\rput[r](-0.1,1.5){$\frac1{2}$}
\psline[linestyle=dashed,linewidth=0.7pt](1.5,0)(1.5,1.5)
\end{pspicture}$$ $\smallskip$ ** In the ruled area we have $\Pi_{p_1}(^m\ell_2;\ell_q)=\Pi_{p_2}(^m\ell_2;\ell_q)$ and in the shaded area we have the (direct) inclusion $\Pi_{p}(^m\ell_2;\ell_q)\subset\Pi_{q}(^m\ell_2;\ell_q)$ for $p\le q$.
A contraction result and unconditionality
-----------------------------------------
Let us begin with this contraction result for the $p$-summing norm of multilinear operators.
\[contraction\] Suppose $X$ is a $\mathcal L_q^g$-space and let $r$, $q$ and $p>0$ satisfy one of the conditions in Proposition \[normas equivalentes\] ($i$). Then, there is a constant $K$ (depending on $r$, $q$ and $p$), such that for any finite matrix $(x_{i_1,\dots,i_m})_{i_1,\dots,i_m}\subset X$ and any choice of scalars $\alpha_{i_1,\dots,i_m}$ we have, $$\pi_p\left( \sum _{i_1,\dots,i_m} \alpha_{i_1,\dots,i_m}\ e_{i_1}'\otimes\cdots\otimes e_{i_m}' \
x_{i_1,\dots,i_m} \right) \le K \|(\alpha_{i_1,\dots,i_m})\|_\infty\ \pi_p\left( \sum _{i_1,\dots,i_m}
e_{i_1}' \otimes\cdots\otimes e_{i_m}' \ x_{i_1,\dots,i_m} \right),$$ where the $\pi_p$ norms are taken in $\Pi_p(^m\ell_r;X)$.
If we show the inequality for $\alpha_{i_1,\dots,i_m}=\pm 1$, standard procedures lead to the desired inequality for general scalars, eventually with different constants (see, for example, Section 1.6 in [@DieJarTon95]). We set $$T=\sum _{i_1,\dots,i_m} e_{i_1}' \otimes\cdots\otimes e_{i_m}' \ x_{i_1,\dots,i_m} \quad \text{and}\quad T_\alpha=\sum _{i_1,\dots,i_m} \alpha_{i_1,\dots,i_m} e_{i_1}' \otimes\cdots\otimes e_{i_m}' \ x_{i_1,\dots,i_m}$$ and let $(r_k)_k$ be the sequence of Rademacher functions on $[0,1]$. For any choice of $t_1\dots,t_m\in [0,1]$, we have $$\begin{aligned}
& &
\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T_\alpha(r_1(t_1)z^{(1)},\dots,r_n(t_m)z^{(m)})\|_X^p
d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)}) \\
&=& \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T_\alpha(z^{(1)},\dots,z^{(m)})\|_X^p d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)}).\end{aligned}$$ We integrate on $t_j\in [0,1]$, $j=1,\dots,m$ and use Fubini’s theorem to obtain $$\begin{aligned}
& & \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T_\alpha(z^{(1)},\dots,z^{(m)})\|_X^p d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)}) \label{conalfa}\\
&=& \hspace{-8pt} \int_{\mathbb K^N}\dots\int_{\mathbb K^N} \int_0^1 \dots \int_0^1
\big\|T_\alpha(r_1(t_1)z^{(1)},\dots,r_n(t_m)z^{(m)})\big\|_X^p dt_1\dots dt_m d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)})\nonumber \\
&=& \hspace{-8pt}
\int_{[0;1]^m}\int_{(\mathbb {K^N})^m} \hspace{-3pt}
\big\|\hspace{-8pt} \sum_{i_1,\dots,i_m}\hspace{-6pt} r_{i_1}(t)\dots r_{i_m}(t) \alpha_{i_1,\dots,i_m} z^{(1)}_{i_1} \cdots
z^{(m)}_{i_m} x_{i_1,\dots,i_m} \big\|_X^p \hspace{-1pt} dt_1\ldots dt_m d\mu^N_{r'}(z^{(1)}\hspace{-1pt})\dots
d\mu^N_{r'}(z^{(m)}\hspace{-1pt}).\nonumber\end{aligned}$$ Since $X$ has nontrivial cotype and local unconditional structure, we can apply a multilinear version of Pisier’s deep result [@Pis78 Proposition 2.1] (which follows the same lines as the bilinear result) to show that, for any $z^{(1)},\dots,z^{(m)} \in \mathbb K^N$, we have $$\begin{aligned}
& &
\int_0^1 \dots \int_0^1
\big\| \sum _{i_1,\dots,i_m} r_{i_1}(t)\dots r_{i_m}(t)\ \alpha_{i_1,\dots,i_m}\ z^{(1)}_{i_1} \cdots z^{(m)}_{i_m} \ x_{i_1,\dots,i_m} \big\|_X^2 dt_1\dots dt_m \\
&\le & K_X \ \int_0^1 \dots \int_0^1
\big\| \sum _{i_1,\dots,i_m} r_{i_1}(t)\dots r_{i_m}(t)\ z^{(1)}_{i_1} \cdots z^{(m)}_{i_m} \ x_{i_1,\dots,i_m} \big\|_X^2 dt_1\dots dt_m\end{aligned}$$ Using a multilinear Kahane inequality (which may be proved by induction on $m$), the same holds, with a different constant, if we consider the power $p$ in the integrals. This means that we can take the $\alpha_{i_1,\dots,i_k}$ from , paying the price of a constant $K$. Now, we can go all the way back as before to obtain $$\begin{aligned}
& &
\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T_\alpha(z^{(1)},\dots,z^{(m)})\|_X^p d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)})\\ &\le & K \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^p
d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)}).\end{aligned}$$ The integral formula in Proposition \[normas equivalentes\] gives the result.
Note that, in the scalar valued case, the previous theorem asserts that the monomials form an unconditional basic sequence in $\Pi_1(^m\ell_r)$ for $r\ge 2$. This is a particular case of the result of Defant and Pérez-García in [@DefPer08]. It should be noted that the analogous scalar valued result is much easier to prove: after introducing the Rademacher functions as in the previous proof, we just have to use a multilinear Khintchine inequality and the integral formula from Theorem \[formulita escalar\] to obtain the result (Pisier’s result is, of course, not needed in this case).
Limit orders
------------
As a consequence of the integral formula for the $p$-summing norm, we are able to compute limit orders of multiple summing operators (see definitions below). Limit orders of the ideal of scalar valued multiple 1-summing forms were computed in [@DefPer08] for the bilinear case. In the multlinear case, they were computed in [@CarDimSev07] for $\ell_r$ with $1\le r\le 2$ and in [@Paltesis] for $\ell_r$ with $r\ge 2$. This latter case can be easily obtained from our integral formula for the multiple summing norm. We will actually use the integral formula to compute some limit orders for the vector valued case, the mentioned scalar case being very similar.
A subclass $\mathfrak{A}$ of the class $\mathcal{L}$ of all $m$-linear continuous mappings between Banach spaces is called an [*ideal of $m$-linear mappings*]{} if
1. For all Banach spaces $E_1,\dots,E_m,F$, the component set $\mathfrak{A}(E_1,\dots,E_m;F):=\mathfrak{A}\cap\mathcal(E_1,\dots,E_m;F)$ is a linear subspace of $\mathcal(E_1,\dots,E_m;F)$.
2. If $T_j\in\mathcal(E_j;G_j)$, $\phi\in\mathfrak{A}(G_1,\dots,G_m;G)$ and $S\in\mathcal L(G,F)$, then $S\circ\phi\circ(T_1,\dots,T_m)$ belongs to $\mathfrak{A}(E_1,\dots,E_m;F)$.
3. The application $\mathbb K^m\ni(\lambda_1,\dots,\lambda_m)\mapsto \lambda_1\cdot\ldots\cdot\lambda_m\in\mathbb K$ is in $\mathfrak{A}(\mathbb K,\dots,\mathbb K;\mathbb K)$.
A [*normed ideal of $m$-linear operators*]{} $({\mathfrak A},\|\cdot\|_{{\mathfrak A}})$ is an ideal ${\mathfrak A}$ of $m$-linear operators together with an ideal norm $\|\cdot\|_{{\mathfrak A}}$, that is,
1. $\|\cdot\|_{{\mathfrak A}}$ restricted to each component is a norm.
2. If $T_j\in\mathcal(E_j;G_j)$, $\phi\in\mathfrak{A}(G_1,\dots,G_m;G)$ and $S\in\mathcal L(G,F)$, then $\|S\circ\phi\circ(T_1,\dots,T_m)\|_{{\mathfrak A}}\le\|S\|\|\phi\|_{{\mathfrak A}}\|T_1\|\cdot\dots\cdot\|T_m\|$.
3. $\|\mathbb K^m\ni(\lambda_1,\dots,\lambda_m)\mapsto \lambda_1\cdot\ldots\cdot\lambda_m\in\mathbb K\|_{\mathfrak{A}}=1$.
Given a normed ideal of $m$-linear operators $({\mathfrak A},\|\cdot\|_{{\mathfrak A}})$, the [*limit order*]{} $\lambda_m({\mathfrak A},r,q)$ is defined as the infimum of all $\lambda\ge0$ such that there is a constant $C>0$ satisfying $$\|\Phi_N\|_{{\mathfrak A}}\le CN^\lambda,$$ for every $N\ge1$, where $\Phi_N:\ell_r^N\times\dots\times\ell_r^N\to\ell_q^N$ is the $m$-linear operator, $\Phi_N(x^1,\dots,x^m)=\sum_{j=1}^Nx^1_j\dots x^m_je_j$.
$$\lambda_m(\Pi_1,r,q)=\left\{\begin{array}{lll}
\frac1{q} &\textrm{ if }&q\le r'\le 2\\
\frac1{r'}&\textrm{ if }&r'\le q\le 2\\
\frac1{q}+\frac{m}{2}-\frac{m}{r}&\textrm{ if }& \frac{2mq}{2+mq}< r\le 2\textrm{
and }q\le 2\\
0&\textrm{ if }&1\le r\le \frac{2mq}{2+mq}
\end{array}\right.$$
These values can be represented by the following picture: $$\begin{pspicture}(3,3)
\pspolygon[linecolor=green!60!red!60,linewidth=0.2pt,fillstyle=hlines,hatchangle=71,hatchwidth=0.1pt](1.5,1.5)(2.1,1.5)(2.6,3)(1.5,3)
\pspolygon[linecolor=green!60!red!60,linewidth=0.2pt,fillcolor=gray!40!white!60,
fillstyle=solid ](2.1,0)(3,0)(3,3)(2.585,3)(2.1,1.5)
\pspolygon[linecolor=green!60!red!60,linewidth=0.2pt,fillstyle=hlines,hatchangle=90,hatchwidth=0.1pt](1.5,1.5)(0,3)(0,1.5)
\pspolygon[linecolor=green!60!red!60,linewidth=0.2pt,fillstyle=hlines,hatchangle=0,hatchwidth=0.1pt](1.5,1.5)(0,3)(1.5,3)
\psline(0,-0.2)(0,3.2)\psline(-0.2,0)(3.2,0)
\rput[d](0,3.4){$_{1/{q}}$}
\rput[l](3.4,0){$_{1/{r}}$}
\rput[u](3,-0.2){$_1$}
\rput[r](0,3){$_1$}
\rput[u](1.5,-0.3){$_\frac1{2}$}
\rput[r](-0.1,1.5){$_\frac1{2}$}
\psline[linestyle=dashed,linewidth=0.2pt](1.5,0)(1.5,1.5)
\rput[l](4,1.5){$\lambda_m(\Pi_1,r,q)$}
\rput[c](1.1,2.5){$_{1/q}$}
\rput[c](2.6,1.3){$_0$}
\rput[c](.5,1.9){$_{1/r'}$}
\end{pspicture}$$
The proof will be splitted in several lemmas.
Let $p\le q\le r'\le 2$ then $\lambda_m(\Pi_p,r,q)=\frac1{q}$.
Let $p\le q< r'\le 2$. Then by Theorem \[formulita\], $$\begin{aligned}
c_{r',p}^m\pi_p(\Phi_N)
&=& \Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big(\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^q\Big)^{p/q}d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}\\
&\le & \Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^qd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/q}\\
&= & c_{r',q}^{m}N^{1/q}. \\\end{aligned}$$ Thus, $\lambda_m(\Pi_p,r,q)\le \frac1{q}$. On the other hand, $$\begin{aligned}
c_{r',p}^m\pi_p(\Phi_N) &=& \Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big(\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^q\Big)^{p/q}d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}\\
&\ge & \Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}N^{p/q-1}\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}\\
&= & c_{r',p}^{m}N^{1/q}.\end{aligned}$$ Hence, $\lambda_m(\Pi_p, r, q)\ge \frac{1}{q}$ and the proof is done.
Let $p<r'\le q<2$. Then $\lambda_m(\Pi_p,r,q)= \frac1{r'}$.
Let $1<s<r'\le q<2$. Then, by Theorem \[formulita\], $$\begin{aligned}
c_{r',1}^m\pi_1(\Phi_N)
&=& \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big(\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^q\Big)^{1/q}d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\\
&\le& \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big(\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^s\Big)^{1/s}d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\\
&\le & \Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\sum_j|z^{(1)}_j\dots
z^{(m)}_j|^sd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/s}\\
&= & c_{r',s}^{m}N^{1/s}. \\\end{aligned}$$ Since this is true for every $s<r'$, $\lambda_m(\Pi_1,r,q)\le \frac1{r'}$.
On the other hand, let $\Psi_N:\ell_r^N\times\dots\ell_r^N\times\ell_{q'}^N\to \mathbb C$, the $(m+1)$-linear form induced by $\Phi_N$. By [@PerVil04 Proposition 2.2] or [@Mat03 Proposition 2.5], $\pi_1(\Phi_N)\ge\pi_1(\Psi_N)$. Thus, by Theorem \[formulita escalar\] taking into account the comments after Theorem \[formulita\], we have $$\begin{aligned}
c_{r',1}^mc_{q,1}\pi_1(\Psi_N) &=& \int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|\Psi_N(z^{(1)},\dots,z^{(m+1)})|d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})d\mu^N_{q}(z^{(m+1)})\\
&=& \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big|\sum_jz^{(1)}_j\dots
z^{(m+1)}_j\Big|d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)})d\mu^N_{q}(z^{(m+1)})\\
&=& c_{r',1} \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big(\sum_j|z^{(2)}_j\dots
z^{(m+1)}_j|^{r'}\Big)^{1/r'}d\mu^N_{r'}(z^{(2)})\dots d\mu^N_{r'}(z^{(m)})d\mu^N_{q}(z^{(m+1)})\\
&\ge& c_{r',1}N^{\frac1{r'}-1} \int_{\mathbb K^N}\dots\int_{\mathbb K^N}\sum_j|z^{(2)}_j\dots
z^{(m+1)}_j|d\mu^N_{r'}(z^{(2)})\dots d\mu^N_{r'}(z^{(m)})d\mu^N_{q}(z^{(m+1)})\\
&=& c_{r',1}^mc_{q,1}N^{\frac1{r'}-1}N \,=\, c_{r',1}^mc_{q,1}N^{\frac1{r'}}\end{aligned}$$ Therefore $\lambda_m(\Pi_1,r,q)= \frac1{r'}$.
This proves our assertions for $p=1$. By [@BotMicPel10 Theorem 4.7], $\Pi_p(\ell_r,\ell_q)=\Pi_1(\ell_r,\ell_q)$ for every $1\le p\le 2$, and the lemma follows.
Since $\ell_r$ has cotype 2 for $1\le r\le 2$, given any $m$-linear form $T\in \mathcal
L(\ell_r^N,\dots,\ell_r^N; \mathbb C)$, we know from [@DefPer08 Lemma 4.5] that $$\label{lemadeDP}
\pi_1(T)\asymp \sup \pi_1(T\circ (D_{\sigma_1},\dots,D_{\sigma_m})),$$ where the supremum is taken over the set of norm one diagonal operators $D_{\sigma_j}:\ell_2^N\to \ell_r^N$. The vector-valued version of this result follows the same lines, so holds for any $m$-linear map $T\in \mathcal
L(\ell_r^N,\dots,\ell_r^N; Y)$, for every Banach space $Y$.
Let $1\le p,q,r\le2$. Then
$(i)$ $\lambda_m(\Pi_p,r,q)=0$ for $1\le r\le \frac{2mq}{2+mq}$.
$(ii)$ $\lambda_m(\Pi_p,r,q)=\frac1{q}+\frac{m}{2}-\frac{m}{r}$ for $\frac{2mq}{2+mq}< r\le 2$.
Let $\frac1{t}=\frac1{r}-\frac12$, then for any diagonal operator we have $\|D_\sigma\|_{\mathcal L(\ell_2^N;\ell_r^N)}=\|\sigma\|_{\ell_t^N}$. Since $\Phi_N\circ (D_{\sigma_1},\dots,D_{\sigma_m})\in\mathcal L(^m\ell_2^N;\ell_q^N)$, by Theorem \[formulita\] we have $$\label{pi_1 con diagonales}
\pi_1(\Phi_N\circ (D_{\sigma_1},\dots,D_{\sigma_m}))\asymp
\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\Big(\sum_{j=1}^N |\sigma_1(j)z_j^{(1)}\dots\sigma_m(j)z_j^{(m)}|^q\Big)^{1/q}d\mu^N_{2}(z^{(1)})\dots
d\mu^N_{2}(z^{(m)}).$$
$(i)$ The assumption $1\le r\le \frac{2mq}{2+mq}$ implies $t\le mq$. Then $$\Big(\sum_{j=1}^N |\sigma_1(j)z_j^{(1)}\dots\sigma_m(j)z_j^{(m)})|^q\Big)^{1/q}\le \|\sigma_1\|_{\ell_t^N}\dots\|\sigma_m\|_{\ell_t^N}\sup_{j} |z_j^{(1)}\dots z_j^{(m)}|.$$ Consequently, for any $s\ge1$, we have $$\begin{aligned}
\pi_1(\Phi_N\circ (D_{\sigma_1},\dots,D_{\sigma_m})) &\preceq \int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\sup_{j} |z_j^{(1)}\dots z_j^{(m)}|d\mu^N_{2}(z^{(1)})\dots
d\mu^N_{2}(z^{(m)}) \\
&\le \Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\sum_{j=1}^N |z_j^{(1)}\dots z_j^{(m)}|^s d\mu^N_{2}(z^{(1)})\dots
d\mu^N_{2}(z^{(m)})\Big)^{1/s} = c_{2,s}^mN^\frac1{s},\end{aligned}$$ which implies that $\lambda_m(\Pi_1,r,q)=0$.
$(ii)$ The assumption $\frac{2mq}{2+mq}\le r< 2$ implies $t>mq$. Let $\frac1{q}=\frac{m}{t}+\frac1{s}$. Then $$\Big(\sum_{j=1}^N |\sigma_1(j)z_j^{(1)}\dots\sigma_m(j)z_j^{(m)}|^q\Big)^{1/q}\le \|\sigma_1\|_{\ell_t^N}\dots\|\sigma_m\|_{\ell_t^N}\Big(\sum_{j=1}^N |z_j^{(1)}\dots z_j^{(m)})|^s\Big)^{1/s}.$$ Thus we have, $$\begin{aligned}
\pi_1(\Phi_N\circ (D_{\sigma_1},\dots,D_{\sigma_m})) &\preceq& \int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\Big(\sum_{j=1}^N |z_j^{(1)}\dots z_j^{(m)}|^s\Big)^{1/s}d\mu^N_{2}(z^{(1)})\dots
d\mu^N_{2}(z^{(m)})\\
&\le & \Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\sum_{j=1}^N |z_j^{(1)}\dots z_j^{(m)}|^s d\mu^N_{2}(z^{(1)})\dots
d\mu^N_{2}(z^{(m)})\Big)^{1/s}\\
&=& c_{2,s}^{m} N^{1/s}\quad \asymp\, N^{\frac1{q}+\frac{m}{2}-\frac{m}{r}}.\end{aligned}$$ On the other hand, $$\begin{aligned}
\pi_1(\Phi_N\circ (D_{\sigma_1},\dots,D_{\sigma_m})) &\succeq
N^{-1/q'}\int_{\mathbb K^N}\dots\int_{\mathbb K^N}
\sum_{j=1}^N |\sigma_1(j)z_j^{(1)}\dots\sigma_m(j)z_j^{(m)}|d\mu^N_{2}(z^{(1)})\dots
d\mu^N_{2}(z^{(m)})\\
&= N^{-1/q'}\,c_{2,1}^m\,
\sum_{j=1}^N |\sigma_1(j)\dots\sigma_m(j)|.\end{aligned}$$ Taking supremum over $\sigma_k\in B_{\ell_t^N}$, $k=1,\dots,m$, and using we get that $$\pi_1(\Phi_N) \succeq N^{-1/q'} N^{1-m/t} c_{2,1}^m \asymp\, N^{\frac1{q}+\frac{m}{2}-\frac{m}{r}}.$$ This proves our assertions for $p=1$. Since $\ell_r$ has cotype 2, by [@BotMicPel10 Theorem 4.6], $\Pi_p(\ell_r,\ell_q)$ coincides with $\Pi_1(\ell_r,\ell_q)$ for every $1\le p\le 2$, and the lemma follows.
Proofs of the main results {#sec-proofs}
==========================
The proofs will be splitted in a few lemmas. We will also use the following result, which is [@Per04 Proposition 3.1].
\[perez\] Let $T\in\Pi_p^m(X_1,\dots,X_m;Y)$ and let $(\Omega_j,\mu_j)$ be measure spaces for each $1\le j\le m$. We have $$\begin{gathered}
\Big(\int_{\Omega_1}\dots\int_{\Omega_m}\|T(f_1(w_1),\dots,f_m(w_m))\|_Y^p \, d\mu_1(w_1)\dots
d\mu_m(w_m)\Big)^{1/p} \\ \le \pi_p(T) \prod_{j=1}^m\sup_{x_j^*\in B_{X_j^*}} \Big(\int_{\Omega_j} |\langle
x_j^*,f_j(w_j)\rangle|^pd\mu_j(w_j)\Big)^{1/p},\end{gathered}$$ for every $f_j\in L_p (\mu_j,X_j)$.
A simple consequence of this proposition is the following.
\[formulita desig facil\] Let $T$ be a multilinear operator in $\mathcal L(^m\ell_r^N;Y)$, and $p<r'<2$ or $r=2$. Then $$\Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T(z^{(1)},\dots,z^{(m)})\|_Y^p \, d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)})\Big)^{1/p}\le c_{r',p}^m\pi_p(T).$$
Let $(\Omega_j,\mu_j)=(\mathbb K^N,\mu_{r'})$, $f_j\in L_p((\mathbb K^N,\mu_{r'}),\mathbb K^N)$, $f_j(z)=z$ for all $j$ and $p<r'<2$ or $r=2$. By Proposition \[perez\] and rotation invariance of stable measures, $$\begin{gathered}
\Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\|T(z^{(1)},\dots,z^{(m)})\|_Y^p \, d\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)})\Big)^{1/p} \\
\le \pi_p(T) \prod_{j=1}^m\sup_{w_j\in B_{\ell_{r'}^N}} \Big(\int_{\mathbb K^N} |\langle
z^{(j)},w_j\rangle|^pd\mu^N_{r'}(z^{(j)})\Big)^{1/p}\\
= \pi_p(T)\Big(\int_{\mathbb K^N} |e_1'(z)|^pd\mu^N_{r'}(z)\Big)^{m/p} \; = \; \pi_p(T)c_{r',p}^{m}. \qedhere\end{gathered}$$
Now we are ready to prove Theorem \[formulita escalar\].
*of Theorem \[formulita escalar\]* One inequality is given in the previous Lemma. We prove the reverse inequality by induction on $m$. For $m=1$, we have $\phi\in(\ell_r^N)'=\ell_{r'}^N$ and then $$\begin{aligned}
\pi_p(\phi) & = & \|\phi\|_{\ell_{r'}^N} \; = \;\Big(\sum_{j=1}^N |e_j'(\phi)|^{r'}\Big)^{1/r'} = \;
c_{r',p}^{-1}\Big(\int_{\mathbb K^N}\Big|\sum_{j=1}^N e_j'(\phi) z_j\Big|^pd\mu^N_{r'}(z)\Big)^{1/p} \\
& = & c_{r',p}^{-1}\Big(\int_{\mathbb K^N}|\phi(z)|^pd\mu^N_{r'}(z)\Big)^{1/p} .
\end{aligned}$$ Suppose that for any $k$-linear form $\psi:\ell_r^N\times\dots\times\ell_r^N\to\mathbb K$, with $k<m$, we have, $$\begin{aligned}
\sum_{n_1,\dots,n_k}|\psi(u_{n_1}^{(1)},\dots,u_{n_k}^{(k)})|^p \le c_{r',p}^{-kp}\Big(\int_{\mathbb
K^N}\dots\int_{\mathbb K^N}|\psi(z^{(1)},\dots,z^{(k)})|^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(k)})\Big),\end{aligned}$$ for all sequences $(u_{n_j}^{(j)})\subset \ell_r^N$, with $w_p(u_{n_j}^{(j)})=1$, $j=1,\dots, k$.
Let $\phi$ be an $m$-linear form, and $(u_{n_j}^{(j)})\subset \ell_r^N$, with $w_p(u_{n_j}^{(j)})=1$, $j=1,\dots, m$. Then $$\begin{aligned}
\sum_{n_1,\dots,n_m} & |\phi(u_{n_1}^{(1)},\dots,u_{n_m}^{(m)})|^p =
\sum_{n_1}\sum_{n_2,\dots,n_m}|\phi(u_{n_1}^{(1)},\dots,u_{n_m}^{(m)})|^p \\
& \le c_{r',p}^{-(m-1)p}\sum_{n_1}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|\phi({u_{n_1}^{(1)}},z^{(2)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(2)})\dots d\mu^N_{r'}(z^{(m)})\Big) \\
&= c_{r',p}^{-(m-1)p}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\sum_{n_1}|\phi({u_{n_1}^{(1)}},z^{(2)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(2)})\dots
d\mu^N_{r'}(z^{(m)})\Big) \\
&\le c_{r',p}^{-(m-1)p}\int_{\mathbb K^N}\dots\int_{\mathbb K^N} \Big(c_{r',p}^{-p}\int_{\mathbb
K^N}|\phi(z^{(1)},z^{(2)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(1)})\Big) d\mu^N_{r'}(z^{(2)})\dots d\mu^N_{r'}(z^{(m)}) \\
&=c_{r',p}^{-mp}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N} |\phi(z^{(1)},\dots,z^{(m)})|^p
d\mu_{r'}^N(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big) .
$$ Therefore, $$c_{r',p}^m\pi_p(\phi)\le \Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|\phi(z^{(1)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}. \qedhere$$
Let us continue our way to the proof of Theorem \[formulita\].
Let $T$ be an $m$-linear mapping in $\mathcal L(^mE;\ell_q^M)$, $0<p<q<2$ or $q=2$. Then $$c_{q,p}\pi_p(T)\le \Big(\int_{\mathbb K^M}\pi_p(z\circ T)^pd\mu^M_{q}(z)\Big)^{1/p}.$$ In particular, if $T$ is linear, $$c_{q,p}\pi_p(T)\le \Big(\int_{\mathbb K^M}\|T'(z)\|_{E'}^p \, d\mu^M_{q}(z)\Big)^{1/p}.$$
For $(u_{k_j}^j)\subset \ell_2^N$ with $w_p((u_{k_j}^j))=1$, $j=1, \ldots, m,$ we have $$\begin{aligned}
\sum_{k_1,\dots,k_m} \|T(u_{k_1}^1,\dots,u_{k_m}^m)\|_{\ell_q^M}^p & = & \sum_{k_1,\dots,k_m} \Big(\sum_{j=1}^M |e_j'\circ T(u_{k_1}^1,\dots,u_{k_m}^m)|^q\Big)^{p/q} \\
& = & \sum_{k_1,\dots,k_m} c_{q,p}^{-p}\Big(\int_{\mathbb K^M}\Big|\sum_{j=1}^M e_j'\circ T(u_{k_1}^1,\dots,u_{k_m}^m)
z_j\Big|^pd\mu^M_{q}(z)\Big) \\
& = & \sum_{k_1,\dots,k_m} c_{q,p}^{-p}\Big(\int_{\mathbb K^M}|z\circ
T(u_{k_1}^1,\dots,u_{k_m}^m)|^pd\mu^M_{q}(z)\Big)
\\
& \le & c_{q,p}^{-p}\Big(\int_{\mathbb K^M}\pi_p(z\circ T)^pd\mu^M_{q}(z)\Big) .\end{aligned}$$ Therefore, $$c_{q,p}\pi_p(T)\le \Big(\int_{\mathbb K^M}\pi_p(z\circ T)^pd\mu^M_{q}(z)\Big)^{1/p}.$$
For $m=1$, $z\circ T$ is a linear form, and then we have $\pi_p(z\circ T)=\|z\circ T\|_{E'}=\|T'(z)\|_{E'}$.
By a [*Banach sequence space*]{} we mean a Banach space $X\subset \mathbb K^{\mathbb N}$ of sequences in $\mathbb K$ such that $\ell_1\subset X\subset \ell_\infty$ with norm one inclusions satisfying that if $x\in\mathbb K^{\mathbb N}$ and $y\in X$ are such that $|x_n|\le |y_n|$ for every $n\in\mathbb N$, then $x$ belongs to $X$ and $\|x\|_X\le\|y\|_X$. We will now show that if we consider multilinear mappings whose range are certain Banach sequence spaces, then the norm of the multilinear mapping defined by the integral formula is equivalent to the multiple summing norm.
We will need the following remark, which may be seen as a Khintchine/Kahane type multilinear inequality for the stable measures.
\[khintchine estable\] If $T$ is an $m$-linear form on $\mathbb K^N$ and $q\le p<s<2$, or $q\le p$ and $s=2$, then $$\begin{gathered}
c_{s,p}^{-m}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|T(z^{(1)},\dots,z^{(m)})|^pd\mu^N_{s}(z^{(1)})\dots d\mu^N_{s}(z^{(m)})\Big)^{1/p} \\
\le c_{s,q}^{-m}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|T(z^{(1)},\dots,z^{(m)})|^qd\mu^N_{s}(z^{(1)})\dots d\mu^N_{s}(z^{(m)})\Big)^{1/q}.\end{gathered}$$ For $m=1$ it follows from property of Lévy stable measures, and then we just apply induction on $m$.
Recall that a Banach sequence space $X$ is called [*$q$-concave*]{}, $q\ge1$, if there exists $C>0$ such that for any $x_1,\dots,x_n\in X$ we have $$\Big(\sum_{k=1}^n\|x_k\|_X^q\Big)^\frac1{q}\le C\Big\|\Big(\sum_{k=1}^n|x_k|^q\Big)^{1/q}\Big\|_X.$$
\[lema1\] Let $X$ be a $q$-concave Banach sequence space with constant $C$ and let $T\in\mathcal
L(^mE;X)$ be an $m$-linear operator. Denote by $T_j$ the $j$-coordinate of $T$ ($T_j$ is a scalar $m$-linear form). Then $\pi_q(T)\le C \|(\pi_q(T_j))_j\|_X$.
Just note that for finite sequences $\big(u_{n_k}^{(k)}\big)_{n_k}\subset X$, with $w_q\Big(\big(u_{n_k}^{(k)}\big)_{n_k}\Big)=1$ we have $$\Big(\sum_{n_1,\dots,n_m}\|T(u_{n_1}^{(1)},\dots,u_{n_m}^{(m)})\|_X^q\Big)^{1/q} \le C
\Big\|\Big(\big(\sum_{n_1,\dots,n_m}|T_j(u_{n_1}^{(1)},\dots,u_{n_m}^{(m)})|^q\big)^{1/q}\Big)_j\Big\|_X \le
C\big\|(\pi_q(T_j))_j\big\|_X.$$
\[lema2\] Let $X$ be a Banach sequence space and let $T\in\mathcal
L(^m\ell_r^N;X)$ be an $m$-linear operator, $r\ge2$. Then if either $p,q<r'<2$, or $r=2$, then $$\|(\pi_p(T_j))_j\|_X\le c_{r',1}^{-m} \int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_Xd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\le
(c_{r',q}/c_{r',1})^{m} \pi_q(T).$$
By Theorem \[formulita escalar\], Remark \[khintchine estable\] and Lemma \[formulita desig facil\] we have $$\begin{aligned}
\|(\pi_p(T_j))_j\|_X &= & c_{r',p}^{-m}\Big\|\left(\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|T_j(z^{(1)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}\right)_j\Big\|_X
\\
&\le & c_{r',1}^{-m}\Big\|\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}|T_j(z^{(1)},\dots,z^{(m)})|d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)\Big\|_X \\
&\le & c_{r',1}^{-m} \int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_Xd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\\
&\le & c_{r',1}^{-m} \Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^qd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/q}\\
&\le & (c_{r',q}/c_{r',1})^{m} \pi_q(T).\end{aligned}$$
As a consequence of Lemma \[formulita desig facil\] we obtain one inequality in the following result. For the other inequality, note that if $X$ is $q$-concave, then it is also $s$-concave for any $s\ge q$ and apply the previous two lemmas.
\[formulita q-concave\] Let $X$ be a $q$-concave Banach sequence space and let $T\in\mathcal
L(^m\ell_r^N;X)$. Then for $r=2$ and $q\le s$, or $q\le s<r'<2$, we have $$\begin{aligned}
\pi_s(T) \asymp
\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_X^qd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/q} \\\end{aligned}$$
Standard localization techniques and the previous corollary readily show the coincidence of multiple $s-$summing and multiple $q-$summing operators from $\mathcal L_r^g$-spaces to $q$-concave Banach sequence spaces.
Let $X$ be a $q$-concave Banach sequence space, and let $E$ be an $\mathcal L_r^g$-space. Then $$\Pi_s(^mE;X)=\Pi_q(^mE;X),$$ for $q\le s<r'<2$, or $q\le s$ and $r=2$.
Proceeding as above we may prove the following.
\[pi\_q en L\_q\] Let $X$ be an $\mathcal L_{q,1}^g$-space and let $T\in\mathcal
L(^m\ell_r^N;X)$ be an $m$-linear operator. If $q<r'<2$ or, $r=2$, then we have $$\pi_q(T) = c_{r',q}^{-m}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_{X}^qd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/q}.$$
We have almost finished the proofs of the main results.
It is clearly enough to show the result for operators with range in $\ell_q^M$ for some $M\in\mathbb N$.
One inequality is Lemma \[formulita desig facil\]. For the other inequality, if either $r=q=2$ or; $r=2$ and $p<q<2$ or; $p<r'<2$ and $p<q\le2$, a combination of the previous results gives: $$\begin{aligned}
\pi_p(T) & \le & c_{q,p}^{-1}\Big(\int_{\mathbb K^N}\pi_p(z\circ T)^pd\mu^N_{q}(z)\Big)^{1/p} \\
& \le & c_{r',p}^{-m}c_{q,p}^{-1}\Big(\int_{\mathbb K^N}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}|z\circ
T(z^{(1)},\dots,z^{(m)})|^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)d\mu^N_{q}(z)\Big)^{1/p} \\
& \le & c_{r',p}^{-m}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb K^N}\Big(c_{q,p}^{-p}\int_{\mathbb K^N}|z\circ
T(z^{(1)},\dots,z^{(m)})|^pd\mu^N_{q}(z)\Big)d\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p} \\
& \le & c_{r',p}^{-m}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\pi_p\big(T(z^{(1)},\dots,z^{(m)});\ell_{q'}^M\to\mathbb K\big)^pd\mu^N_{r'}(z^{(1)})\dots
d\mu^N_{r'}(z^{(m)})\Big)^{1/p} \\
& = & c_{r',p}^{-m}\Big(\int_{\mathbb K^N}\dots\int_{\mathbb
K^N}\|T(z^{(1)},\dots,z^{(m)})\|_{\ell_q^M}^pd\mu^N_{r'}(z^{(1)})\dots d\mu^N_{r'}(z^{(m)})\Big)^{1/p}, \\\end{aligned}$$ where by $\pi_p\big(T(z^{(1)},\dots,z^{(m)});\ell_{q'}^M\to\mathbb K\big)$ we denote the absolutely $p$-summing norm of the vector $T(z^{(1)},\dots,z^{(m)})$ thought of as a linear functional on $\ell_{q'}^M$, whose norm is just the $\ell_{q}^M$-norm of the vector.
The cases where $p=q$ follow from Corollary \[pi\_q en L\_q\].
($i$) For $r=2$, the equivalence of norms is a consequence of Theorem \[formulita\] when $p<q\le 2$ or $p=q$ and of Corollary \[formulita q-concave\] for $q\le p$.
For $r>2$, the equivalence of norms is a consequence of Theorem \[formulita\] when $p<r'$ and $p<q\le 2$ and of Corollary \[formulita q-concave\] $q\le p<r'$.
($ii$) This statement follows from Lemma \[formulita desig facil\].
The first assertion follows from Corollary \[formulita q-concave\] and localization. For the second assertion just combine Lemma \[lema1\] with Lemma \[lema2\].
[10]{}
Harold P. Boas. Majorant series. , 37(2):321–337, 2000. Several complex variables (Seoul, 1998).
Fernando Bombal, David P[é]{}rez-Garc[í]{}a, and Ignacio Villanueva. Multilinear extensions of [G]{}rothendieck’s theorem. , 55(4):441–450, 2004.
Geraldo [Botelho]{}, Carsten [Michels]{}, and Daniel [Pellegrino]{}. , 23(1):139–161, 2010.
Daniel Carando, Verónica Dimant, and Pablo Sevilla-Peris. , 11(4):589–607.
Fernando Cobos, Thomas K[ü]{}hn, and Jaak Peetre. On $\mathfrak g_p$-classes of trilinear forms. , 59(03):1003–1022, 1999.
Andreas Defant and Klaus Floret. , 1993.
Andreas Defant and David Pérez-Garc[í]{}a. , 360(6):3287–3306, 2008.
Joe Diestel, Hans Jarchow, and Andrew Tonge. , volume 43 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 1995.
Y. [Gordon]{}. , 7:151–163, 1969.
Mário C. [Matos]{}. , 54(2):111–136, 2003.
Carlos Palazuelos Cabezón. PhD thesis, Universidad Complutense de Madrid, 2009.
Daniel Pellegrino and Joedson Santos. Absolutely summing multilinear operators: a panorama. , 34(4):447–478, 2011.
David Pérez-Garc[í]{}a. , 165(3):275–290, 2004.
David Pérez-Garc[í]{}a. , 85(3):258–267, 2005.
David [Pérez-García]{} and Ignacio [Villanueva]{}. , 42(1):153–171, 2004.
Albrecht [Pietsch]{}. , 31-32:285–315, 1972.
Albrecht Pietsch. . North-Holland Mathematical Library, 20. North-Holland Publishing Co., Amsterdam-New York, 1980.
Gilles [Pisier]{}. , 37:3–19, 1978.
Dumitru [Popa]{}. , 350(1):360–368, 2009.
[^1]: This work was partially supported by CONICET PIP 0624, ANPCyT PICT 2011-1456, ANPCyT PICT 11-0738, UBACyT 1-746 and UBACyT 20020130300052BA
|
---
author:
- Ken Freeman
title: Structure and Evolution of the Milky Way
---
The Thin Disk: Formation and Evolution
======================================
Here are some of the issues related to the formation and evolution of the Galactic thin disk:
- Building the thin disk: its exponential radial structure, and the role of mergers.
- The star formation history: chemical evolution and continued gas accretion.
- Evolutionary processes in the disk: disk heating, radial mixing.
- The outer disk: chemical properties and chemical gradients.
Many of the basic observational constraints on the properties of the Galactic disk are still uncertain. At this time, we do not have reliable information about the star formation history of the disk. We do not know how the metallicity distribution and the stellar velocity dispersions in the disk have evolved with time. One might have expected that these observational questions were well understood by now, but this is not yet so. The basic observational problem is the difficulty of measuring ages for individual stars.
The younger stars of the Galactic disk show a clear abundance gradient of about 0.07 dex kpc$^{-1}$, outlined nicely by the cepheids [@Luck2006]. In the outer disk, for the older stars, the abundance gradient appears to be even stronger: the abundance gradient (and the gradient in the ratio of alpha-elements to Fe) have flattened with time towards the solar values. A striking feature of the radial abundance gradient in the Galaxy is that it flattens for $R > 12$ kpc at an \[Fe/H\] value of about -0.5 [@Carney2005]. A similar flattening of the abundance gradient is seen in the outer regions of the disk of M31 [@Worthey2005].
The relation between the stellar age and the mean metallicity and velocity dispersion are the fundamental observables that constrain the chemical and dynamical evolution of the Galactic thin disk. The age-metallicity relation (AMR) in the solar neighborhood is still uncertain. Different authors find different relations, ranging from a relatively steep decrease of metallicity with age from [@Rocha-Pinto2004] to almost no change of mean metallicity with age from [@Nordstrom2004]. Much of the earlier work indicated that a large scatter in metallicity was seen at all ages, which was part of the motivation to invoke large-scale radial mixing of stars within the disk. This mixing, predicted by [@Sellwood2002], is generated by resonances with the spiral pattern, and is able to move stars from one near-circular orbit to another. It would bring stars from the inner and outer disks, with their different mean abundances, into the solar neighborhood. Radial mixing is potentially an important feature of the evolution of the disk. At this stage, it is a theoretical concept, and it is not known how important it is in the Galactic disk. We are not aware of any strong observational evidence at this stage for the existence of radial mixing. More recent results on the AMR (e.g. Wylie de Boer , unpublished) indicate that there is a weak decrease of mean metallicity with age in the Galactic thin disk, but that the spread in metallicity at any age is no more than about 0.10 dex. If this is correct, then radial mixing may not be so important for chemically mixing the Galactic disk.
The age-velocity dispersion relation (AVR) is also not well determined observationally. The velocity dispersion of stars appears to increase with age, and this is believed to be due to the interaction of stars with perturbers such as giant molecular clouds and transient spiral structure. But there is a difference of opinion about the duration of this heating. One view is that the stellar velocity dispersion $\sigma$ increases steadily for all time, $\sim t^{0.2-0.5}$, based on [@Wielen1977]’s work using chromospheric ages and kinematics for the McCormick dwarfs. Another view [e.g. @Quillen2000], based on the data for subgiants from [@Edvardsson1993] is that the heating takes place for the first $\sim 2$ Gyr, but then saturates when $\sigma \approx 20$ km s$^{-1}$ because the stars of higher velocity dispersion spend most of their orbital time away from the Galactic plane where the sources of heating lie. Data from [@Soubiran2008] support this view. Again, much of the difference in view goes back to the difficulty of measuring stellar ages. Accurate ages from asteroseismology would be very welcome. Accurate ages and distances for a significant sample of red giants would allow us to measure the AMR and AVR out to several kpc from the Sun. This would be a great step forward in understanding the chemical and dynamical evolution of the Galactic disk.
The Formation of the Thick Disk
===============================
Most spiral galaxies, including out Galaxy, have a second thicker disk component. For example, the thick disk and halo of the edge-on spiral galaxy NGC 891, which is much like the Milky Way in size and morphology, has a thick disk nicely seen in star counts from HST images [@Mouhcine2010]. Its thick disk has scale height $\sim 1.4$ kpc and scalelength $\sim 4.8$ kpc, much as in our Galaxy. The fraction of baryons in the thick disk is typically about $10$ to $15$ percent in large systems like the Milky Way, but rises to about $50$% in the smaller disk systems [@Yoachim2008].
The Milky Way has a significant thick disk, discovered by [@Gilmore1983]. Its vertical velocity dispersion is about 40 [km s$^{-1}$]{}; its scale height is still uncertain but is probably about $1000$ pc. The surface brightness of the thick disk is about 10% of the thin disk’s, and near the Galactic plane it rotates almost as rapidly as the thin disk. Its stars are older than 10 Gyr and are significantly more metal poor than the stars of the thin disk; most of the thick disk stars have \[Fe/H\] values between about $-0.5$ and $-1.0$ and are enhanced in alpha-elements relative to Fe. This is usually interpreted as evidence that the thick disk formed rapidly, on a timescale $\sim 1$ Gyr. From its kinematics and chemical properties, the thick disk appears to be a discrete component, distinct from the thin disk. Current opinion is that the thick disk shows no vertical abundance gradient [e.g. @Gilmore1995; @Ivezic2008].
The old thick disk is a very significant component for studying Galaxy formation, because it presents a kinematically and chemically recognizable relic of the early Galaxy. Secular heating is unlikely to affect its dynamics significantly, because its stars spend most of their time away from the Galactic plane.
How do thick disks form ? Several mechanisms have been proposed, including:
- thick disks are a normal part of early disk settling, and form through energetic early star forming events, e.g. in gas-rich mergers [@Samland2003; @Brook2004]
- thick disks are made up of accretion debris [@Abadi2003]. From the mass-metallicity relation for galaxies, the accreted galaxies that built up the thick disk of the Galaxy would need to be more massive than the SMC to get the right mean \[Fe/H\] abundance ($\sim -0.7$). The possible discovery of a counter-rotating thick disk [@Yoachim2008] in an edge-on galaxy would favor this mechanism.
- thick disks come from the heating of the thin disk via disruption of its early massive clusters [@Kroupa2002]. The internal energy of large star clusters is enough to thicken the disk. Recent work on the significance of the high redshift clump structures may be relevant to the thick disk problem: the thick disk may originate from the merging of clumps and heating by clumps [e.g. @Bournaud2009]. These clumps are believed to form by gravitational instability from turbulent early disks: they appear to generate thick disks with scale heights that are radially approximately uniform, rather than the flared thick disks predicted from minor mergers.
- thick disks come from early partly-formed thin disks, heated by accretion events such as the accretion event which is believed to have brought omega Centauri into the Galaxy [@Bekki2003]. In this picture, thin disk formation began early, at $z = 2$ to $3$. The partly formed thin disk is partly disrupted during the active merger epoch which heats it into thick disk observed now, The rest of the gas then gradually settles to form the present thin disk, a process which continues to the present day.
- a recent suggestion is that stars on more energetic orbits migrate out from the inner galaxy to form a thick disk at larger radii where the potential gradient is weaker [@Schonrich2009]
How can we test between these possibilities for thick disk formation? [@Sales2009] looked at the expected orbital eccentricity distribution for thick disk stars in different formation scenarios. Their four scenarios are:
- a gas-rich merger: the thick disk stars are born in-situ
- the thick disk stars come in from outside via accretion
- the early thin disk is heated by accretion of a massive satellite
- the thick disk is formed as stars from the inner disk migrate out to larger radii.
Preliminary results from the observed orbital eccentricity distribution for thick disk stars may favor the gas-rich merger picture [@Wilson2011]. This is a potentially powerful approach for testing ideas about the origin of the thick disk. Because it depends on the orbital properties of the thick disk sample, firm control of selection effects is needed in the identification of which stars belong to the thick disk. Kinematical criteria for choosing the thick disk sample are clearly not ideal.
To summarize this section on the thick disk: Thick disks are very common in disk galaxies. In our Galaxy, the thick disk is old, and is kinematically and chemically distinct from the thin disk. It is important now to identify what the thick disk represents in the galaxy formation process. The orbital eccentricity distribution of the thick disk stars will provide some guidance. Chemical tagging will show if the thick disk formed as a small number of very large aggregates, or if it has a significant contribution from accreted galaxies. This is one of the goals for the upcoming AAT/HERMES survey: see section 5.
The Galactic Stellar Halo
=========================
The stars of the Galactic halo have \[Fe/H\] abundances mostly less than -1.0. Their kinematics are very different from the rotating thick and thin disks: the mean rotation of the stellar halo is close to zero, and it is supported against gravity primarily by its velocity dispersion. It is now widely believed that much of the stellar halo comes from the debris of small accreted satellites [@Searle1978]. There remains a possibility that a component of the halo formed dissipationally during the Galaxy formation process [@Eggen1962; @Samland2003]. Halo-building accretion events continue to the present time: the disrupting Sgr dwarf is an example in our Galaxy, and the faint disrupting system around NGC 5907 is another example of such an event [@Martinez-Delgado2010]. The metallicity distribution function (MDF) of the major surviving satellites around the Milky way is not like the MDF in the stellar halo [e.g. @Venn2008] but the satellite MDFs may have been more similar long ago. We note that the fainter satellites are more metal-poor and are consistent with the Milky Way halo in their \[$\alpha$/Fe\] behaviour.
Is there a halo component that formed dissipationally early in the Galactic formation process? [@Hartwick1987] showed that the metal-poor RR Lyrae stars delineate a two-component halo, with a flattened inner component and a spherical outer component. [@Carollo2010] identified a two-component halo and the thick disk in a sample of 17,000 SDSS stars, mostly with \[Fe/H\] $< -0.5$. They described the kinematics well with these three components:\
Thick disk: ($\bar{V}, \sigma$, \[Fe/H\]) = (182, 51, -0.7)\
Inner halo: ($\bar{V}, \sigma$, \[Fe/H\]) = (7, 95, -1.6)\
Outer halo: ($\bar{V}, \sigma$, \[Fe/H\]) = (-80, 180, -2.2)\
Here \[Fe/H\] is the mean abundance for the component, $\bar{V}$ and $\sigma$ are its mean rotation velocity relative to a non-rotating frame, and velocity dispersion, in [km s$^{-1}$]{}. The outer halo appears to have retrograde mean rotation. As we look at subsamples at greater distances from the Galactic plane, we see that the thick disk dies away and the retrograde outer halo takes over from the inner halo. With the above kinematic parameters, the equilibrium of the inner halo is a bit hard to understand. It may not yet be in equilibrium. From comparison with simulations, [@Zolotov2009] argue that the inner halo has a partly dissipational origin, while the outer halo is made up from debris of faint metal-poor accreted satellites.
Recently [@Nissen2010] studied a sample of 78 halo stars with \[Fe/H\] $>-1.6$ and find that they show a variety of \[$\alpha$/Fe\] enhancement. Their sample shows high and low \[$\alpha$/Fe\] groups, and the low \[$\alpha$/Fe\] stars are mostly in high energy retrograde orbits. The high \[$\alpha$/Fe\] stars could be ancient halo stars born in situ and possible heated by satellite encounters. The low-alpha stars may be accreted from dwarf galaxies.
How much of the halo comes from accreted structures? An ACS study by [@Ibata2009] of the halo of NGC 891 (a nearby edge-on galaxy like the Milky Way) shows a spatially lumpy metallicity distribution, indicating that its halo is made up largely of accreted structures which have not yet mixed away. This is consistent with simulations of stellar halos by [@Font2008], [@Gilbert2009] and [@Cooper2010].
To summarize this section on the Galactic stellar halo: the stellar halo is probably made up mainly of the debris of small accreted galaxies, although there may be an inner component which formed dissipatively.
The Galactic Bar/Bulge
======================
The boxy appearance of the Galactic bulge is typical of galactic bars seen edge-on. These bar/bulges are very common: about 2/3 of spiral galaxies show some kind of central bar structure in the infra-red. Where do these bar/bulges come from ?
Bars can arise naturally from the instabilities of the disk. A rotating disk is often unstable to forming a flat bar structure at its center. This flat bar in turn is often unstable to vertical buckling which generates the boxy appearance. This kind of bar/bulge is not generated by mergers but follows simply from the dynamics of a flat rotating disk of stars. The maximum vertical extent of boxy or peanut-shaped bulges occurs near the radius of the vertical and horizontal Lindblad resonances, i.e. where $$\Omega_b = \Omega - \kappa/2 = \Omega - \nu_z/2.$$ Here $\Omega$ is the circular angular velocity, $\Omega_b$ is the pattern speed of the bar, $\kappa$ is the epicyclic frequency and $\nu_z$ is the vertical frequency of oscillation. We note that the frequencies $\kappa$ and particularly $\nu_z$ depend on the amplitude of the oscillation. Stars in this zone oscillate on 3D orbits which support the peanut shape.
We can test whether the Galactic bulge formed through this kind of bar-buckling instability of the inner disk, by comparing the structure and kinematics of the bulge with those of N-body simulations that generate a boxy/bar bulge [e.g. @Athanassoula2005]. The simulations show an exponential structure and near-cylindrical rotation: do these simulations match the properties of the Galactic bar/bulge?
The stars of the Galactic bulge appear to be old and enhanced in $\alpha$-elements. This implies a rapid history of star formation. If the bar formed from the inner disk, then it would be interesting to know whether the bulge stars and the stars of the adjacent disk have similar chemical properties. This is not yet clear. There do appear to be similarities in the $\alpha$-element properties between the bulge and the thick disk in the solar neighborhood .
The bar-forming and bar-buckling process takes 2-3 Gyr to act after the disk settles. In the bar-buckling instability scenario, the bulge [*structure*]{} is probably younger than the bulge [*stars*]{}, which were originally part of the inner disk. The alpha-enrichment of the bulge and thick disk comes from the rapid chemical evolution which took place in the inner disk before the instability acted. In this scenario, the stars of the bulge and adjacent disk should have similar ages: accurate asteroseismology ages for giants of the bulge and inner disk would be a very useful test of the scenario.
We are doing a survey of about 28,000 clump giants in the Galactic bulge and the adjacent disk, to measure the chemical properties (Fe, Mg, Ca, Ti, Al, O) of stars in the bulge and adjacent disk: are they similar, as we would expect if the bar/bulge grew out of the disk? We use the AAOmega fiber spectrometer on the AAT, to acquire medium-resolution spectra of about 350 stars at a time, at a resolution $R \sim 12,000$.
The central regions of our Galaxy are not only the location of the bulge and inner disk, but also include the central regions of the Galactic stellar halo. Recent simulations [e.g. @Diemand2005; @Moore2006; @Brook2007] indicate that the [*metal-free*]{} (population III) stars formed until redshift $z \sim 4$, in chemically isolated subsystems far away from the largest progenitor. If its stars survive, they are spread throughout the Galactic halo. If they are not found, then it would be likely that their lifetimes are less than a Hubble time which in turn implies a truncated IMF. On the other hand, the [*oldest*]{} stars form in the early rare high density peaks that lie near the highest density peak of the final system. They are not necessarily the most metal-poor stars in the Galaxy. Now, these oldest stars are predicted to lie in the central bulge region of the Galaxy. Accurate asteroseismology ages for metal-poor stars in the inner Galaxy would provide a great way to tell if they are the oldest stars or just stars of the inner Galactic halo. This test would require a $\sim 10$% precision in age.
Our data so far indicate that the rotation of the Galactic bulge is close to cylindrical [see also @Howard2009]. Detailed analysis will be needed to see if there is any evidence for a small classical merger generated bulge component, in addition to the boxy/peanut bar/bulge which probably formed from the disk. We also see a more slowly rotating metal-poor component in the bulge region. The problem now is to identify the [*first*]{} stars from among the expected metal-poor stars of the inner halo.
Galactic Archaeology
====================
The goals of Galactic Archaeology are to find signatures or fossils from the epoch of Galaxy assembly, to give us insight into the processes that took place as the Galaxy formed. A major goal is to identify observationally how important mergers and accretion events were in building up the Galactic disk, bulge and halo of the Milky Way. CDM simulations predict a high level of merger activity which conflicts with some observed properties of disk galaxies, particularly with the relatively common nature of large galaxies like ours with small bulges [e.g. @Kormendy2010].
The aim is to reconstruct the star-forming aggregates and accreted galaxies that built up the disk, bulge, and halo of the Galaxy. Some of these dispersed aggregates can still be recognized kinematically as stellar moving groups. For others, the dynamical information was lost through heating and mixing processes, but their debris can still be recognized by their chemical signatures (chemical tagging). We would like to find groups of stars, now dispersed, that were associated at birth either
- because they were born together and therefore have almost identical chemical abundances over all elements [e.g. @deSilva2009], or
- because they came from a common accreted galaxy and have abundance patterns that are clearly distinguished from those of the Galactic disk .
The galactic disk shows kinematical substructure in the solar neighborhood: groups of stars moving together, usually called moving stellar groups. Some are associated with dynamical resonances (e.g. the Hercules group): in such groups, we do not expect to see chemical homogeneity or age homogeneity [e.g. @Bensby2007]. Others are the debris of star-forming aggregates in the disk (e.g. the HR1614 group and Wolf 630 group). They are chemically homogeneous, and such groups could be useful for reconstructing the history of the galactic disk. Yet others may be debris of infalling objects, as seen in CDM simulations [e.g. @Abadi2003].
The stars of the HR 1614 group appear to be the relic of a dispersed star-forming event. These stars have an age of about 2 Gyr and \[Fe/H\] $= +0.2$, and they are scattered all around us. This group has not lost its dynamical identity despite its age. [@deSilva2007] measured accurate differential abundances for many elements in HR 1614 stars, and found a very small spread in abundances. This is encouraging for recovering dispersed star forming events by chemical tagging.
Chemical studies of the old disk stars in the Galaxy can help to identify disk stars which came in from outside in disrupting satellites, and also those that are the debris of dispersed star-forming aggregates like the HR 1614 group [@Freeman2002]. The chemical properties of surviving satellites (the dwarf spheroidal galaxies) vary from satellite to satellite, but are different in detail from the overall chemical properties of the disk stars.
We can think of a chemical space of abundances of elements: O, Na, Mg, Al, Ca, Mn, Fe, Cu, Sr, Ba, Eu for example. Not all of these elements vary independently. The dimensionality of this space chemical space is probably between about 7 and 9. Most disk stars inhabit a sub-region of this space. Stars that come from dispersed star clusters represent a very small volume in this space. Stars which came in from satellites may have a distribution in this space that is different enough to stand out from the rest of the disk stars. With this chemical tagging approach, we hope to detect or put observational limits on the satellite accretion history of the galactic disk.
Chemical studies of the old disk stars in the Galaxy can identify disk stars that are the debris of common dispersed star-forming aggregates. Chemical tagging will work if
- stars form in large aggregates, which is believed to be true
- aggregates are chemically homogenous
- aggregates have unique chemical signatures defined by several elements or element groups which do not vary in lockstep from one aggregate to another. We need sufficient spread in abundances from aggregate to aggregate so that chemical signatures can be distinguished with accuracy achievable ($\sim 0.05$ dex differentially)
de Silva’s work on open clusters was aimed at testing the last two conditions: they appear to be true. See [@deSilva2009] for more on chemical tagging.
We should stress here that chemical tagging is not just assigning stars chemically to a particular population, like the thin disk, thick disk or halo. Chemical tagging is intended to assign stars chemically to substructure which is no longer detectable kinematically. We are planning a large chemical tagging survey of about a million stars, using the new HERMES multi-object spectrometer on the AAT. The goal is to reconstruct the dispersed star-forming aggregates that built up the disk, thick disk and halo within about 5 kpc of the sun.
HERMES is a new high resolution multi-object spectrometer on the AAT. Its spectral resolution is about 28,000, with a high resolution mode with $R = 50,000$. It is fed by 400 fibers over a 2-degree field, and has 4 non-contiguous wavelength bands covering a total of about 1000Å. The four wavelength bands were chosen to include measurable lines of elements needed for chemical tagging. HERMES is scheduled for first light in late 2012. The HERMES chemical tagging survey will include stars brighter than $V = 14$ and has a strong synergy with Gaia: for the dwarf stars in the HERMES sample, the accurate ($1$%) parallaxes and proper motions will be invaluable for more detailed studies.
The fractional contribution of the different Galactic components to the HERMES sample will be about $78$% thin disk stars, $17$% thick disk stars and about $5$% halo stars. About $70$% of the stars will be dwarfs within about 1000 pc and $30$% giants within about 5 kpc. About $9$% of the thick disk stars and about $14$% of the thin disk stars pass within our 1 kpc dwarf horizon. Assume that all of their formation aggregates are now azimuthally mixed right around the Galaxy, so that all of their formation sites are represented within our horizon. Simulations [@Bland-Hawthorn2004] show that a complete random sample of about a million stars with $V < 14$ would allow detection of about 20 thick disk dwarfs from each of about 4500 star formation sites, and about 10 thin disk dwarfs from each of about 35,000 star formation sites. These estimates depend on the adopted mass spectrum of the formation sites. In combination with Gaia, HERMES will give the distribution of stars in the multi-dimensional{position, velocity, chemical} space, and isochrone ages for about 200,000 stars with $V < 14$. We would be interested to explore further what the HERMES survey can contribute to asteroseismology.
Some authors have argued that the thick disk may have formed from the debris of the huge and short-lived star formation clumps observed in disk galaxies at high redshift [e.g. @Bournaud2009; @Genzel2011]. If this is correct, then only a small number of these huge building blocks would have been involved in the assembly of the thick disk, and their debris should be very easy to identify via chemical tagging techniques.
Chemical tagging in the inner regions of the Galactic disk will be of particular interest. We expect about 200,000 survey giants in the inner region of the Galaxy. The surviving old ($> 1$ Gyr) open clusters are all in the outer Galaxy, beyond a radius of 8 kpc. Young open clusters are seen in the inner Galaxy, but do not appear to survive the disruptive effects of the tidal field and giant molecular clouds in the inner regions. We expect to find the debris of many broken open and globular clusters in the inner disk. These will be good for chemical tagging recovery using the HERMES giants. The radial extent of the dispersal of individual broken clusters will provide an acute test of radial mixing theory within the disk. Another opportunity comes from the the Na/O anomaly, which is unique to globular clusters, and may help to identify the debris of disrupted globular clusters.
|
---
abstract: 'We report a measurement of the longitudinal double-spin asymmetry $A_{LL}$ and the differential cross section for inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production at midrapidity in polarized proton collisions at ${{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$. The cross section was measured over a transverse momentum range of $1 < {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}< 17{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ and found to be in good agreement with a next-to-leading order perturbative [[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation. The longitudinal double-spin asymmetry was measured in the range of $3.7 < {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}< 11{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ and excludes a maximal positive gluon polarization in the proton. The mean transverse momentum fraction of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s in their parent jets was found to be around $0.7$ for electromagnetically triggered events.'
author:
- 'B. I. Abelev'
- 'M. M. Aggarwal'
- 'Z. Ahammed'
- 'A. V. Alakhverdyants'
- 'B. D. Anderson'
- 'D. Arkhipkin'
- 'G. S. Averichev'
- 'J. Balewski'
- 'O. Barannikova'
- 'L. S. Barnby'
- 'S. Baumgart'
- 'D. R. Beavis'
- 'R. Bellwied'
- 'F. Benedosso'
- 'M. J. Betancourt'
- 'R. R. Betts'
- 'A. Bhasin'
- 'A. K. Bhati'
- 'H. Bichsel'
- 'J. Bielcik'
- 'J. Bielcikova'
- 'B. Biritz'
- 'L. C. Bland'
- 'B. E. Bonner'
- 'J. Bouchet'
- 'E. Braidot'
- 'A. V. Brandin'
- 'A. Bridgeman'
- 'E. Bruna'
- 'S. Bueltmann'
- 'I. Bunzarov'
- 'T. P. Burton'
- 'X. Z. Cai'
- 'H. Caines'
- 'M. Calderón de la Barca Sánchez'
- 'O. Catu'
- 'D. Cebra'
- 'R. Cendejas'
- 'M. C. Cervantes'
- 'Z. Chajecki'
- 'P. Chaloupka'
- 'S. Chattopadhyay'
- 'H. F. Chen'
- 'J. H. Chen'
- 'J. Y. Chen'
- 'J. Cheng'
- 'M. Cherney'
- 'A. Chikanian'
- 'K. E. Choi'
- 'W. Christie'
- 'P. Chung'
- 'R. F. Clarke'
- 'M. J. M. Codrington'
- 'R. Corliss'
- 'J. G. Cramer'
- 'H. J. Crawford'
- 'D. Das'
- 'S. Dash'
- 'A. Davila Leyva'
- 'L. C. De Silva'
- 'R. R. Debbe'
- 'T. G. Dedovich'
- 'M. DePhillips'
- 'A. A. Derevschikov'
- 'R. Derradi de Souza'
- 'L. Didenko'
- 'P. Djawotho'
- 'S. M. Dogra'
- 'X. Dong'
- 'J. L. Drachenberg'
- 'J. E. Draper'
- 'J. C. Dunlop'
- 'M. R. Dutta Mazumdar'
- 'L. G. Efimov'
- 'E. Elhalhuli'
- 'M. Elnimr'
- 'J. Engelage'
- 'G. Eppley'
- 'B. Erazmus'
- 'M. Estienne'
- 'L. Eun'
- 'P. Fachini'
- 'R. Fatemi'
- 'J. Fedorisin'
- 'R. G. Fersch'
- 'P. Filip'
- 'E. Finch'
- 'V. Fine'
- 'Y. Fisyak'
- 'C. A. Gagliardi'
- 'D. R. Gangadharan'
- 'M. S. Ganti'
- 'E. J. Garcia-Solis'
- 'A. Geromitsos'
- 'F. Geurts'
- 'V. Ghazikhanian'
- 'P. Ghosh'
- 'Y. N. Gorbunov'
- 'A. Gordon'
- 'O. Grebenyuk'
- 'D. Grosnick'
- 'B. Grube'
- 'S. M. Guertin'
- 'A. Gupta'
- 'N. Gupta'
- 'W. Guryn'
- 'B. Haag'
- 'T. J. Hallman'
- 'A. Hamed'
- 'L-X. Han'
- 'J. W. Harris'
- 'J. P. Hays-Wehle'
- 'M. Heinz'
- 'S. Heppelmann'
- 'A. Hirsch'
- 'E. Hjort'
- 'A. M. Hoffman'
- 'G. W. Hoffmann'
- 'D. J. Hofman'
- 'R. S. Hollis'
- 'H. Z. Huang'
- 'T. J. Humanic'
- 'L. Huo'
- 'G. Igo'
- 'A. Iordanova'
- 'P. Jacobs'
- 'W. W. Jacobs'
- 'P. Jakl'
- 'C. Jena'
- 'F. Jin'
- 'C. L. Jones'
- 'P. G. Jones'
- 'J. Joseph'
- 'E. G. Judd'
- 'S. Kabana'
- 'K. Kajimoto'
- 'K. Kang'
- 'J. Kapitan'
- 'K. Kauder'
- 'D. Keane'
- 'A. Kechechyan'
- 'D. Kettler'
- 'D. P. Kikola'
- 'J. Kiryluk'
- 'A. Kisiel'
- 'S. R. Klein'
- 'A. G. Knospe'
- 'A. Kocoloski'
- 'D. D. Koetke'
- 'T. Kollegger'
- 'J. Konzer'
- 'M. Kopytine'
- 'I. Koralt'
- 'W. Korsch'
- 'L. Kotchenda'
- 'V. Kouchpil'
- 'P. Kravtsov'
- 'K. Krueger'
- 'M. Krus'
- 'L. Kumar'
- 'P. Kurnadi'
- 'M. A. C. Lamont'
- 'J. M. Landgraf'
- 'S. LaPointe'
- 'J. Lauret'
- 'A. Lebedev'
- 'R. Lednicky'
- 'C-H. Lee'
- 'J. H. Lee'
- 'W. Leight'
- 'M. J. LeVine'
- 'C. Li'
- 'L. Li'
- 'N. Li'
- 'W. Li'
- 'X. Li'
- 'X. Li'
- 'Y. Li'
- 'Z. Li'
- 'G. Lin'
- 'S. J. Lindenbaum'
- 'M. A. Lisa'
- 'F. Liu'
- 'H. Liu'
- 'J. Liu'
- 'T. Ljubicic'
- 'W. J. Llope'
- 'R. S. Longacre'
- 'W. A. Love'
- 'Y. Lu'
- 'G. L. Ma'
- 'Y. G. Ma'
- 'D. P. Mahapatra'
- 'R. Majka'
- 'O. I. Mall'
- 'L. K. Mangotra'
- 'R. Manweiler'
- 'S. Margetis'
- 'C. Markert'
- 'H. Masui'
- 'H. S. Matis'
- 'Yu. A. Matulenko'
- 'D. McDonald'
- 'T. S. McShane'
- 'A. Meschanin'
- 'R. Milner'
- 'N. G. Minaev'
- 'S. Mioduszewski'
- 'A. Mischke'
- 'M. K. Mitrovski'
- 'B. Mohanty'
- 'M. M. Mondal'
- 'D. A. Morozov'
- 'M. G. Munhoz'
- 'B. K. Nandi'
- 'C. Nattrass'
- 'T. K. Nayak'
- 'J. M. Nelson'
- 'P. K. Netrakanti'
- 'M. J. Ng'
- 'L. V. Nogach'
- 'S. B. Nurushev'
- 'G. Odyniec'
- 'A. Ogawa'
- 'H. Okada'
- 'V. Okorokov'
- 'D. Olson'
- 'M. Pachr'
- 'B. S. Page'
- 'S. K. Pal'
- 'Y. Pandit'
- 'Y. Panebratsev'
- 'T. Pawlak'
- 'T. Peitzmann'
- 'V. Perevoztchikov'
- 'C. Perkins'
- 'W. Peryt'
- 'S. C. Phatak'
- 'P. Pile'
- 'M. Planinic'
- 'M. A. Ploskon'
- 'J. Pluta'
- 'D. Plyku'
- 'N. Poljak'
- 'A. M. Poskanzer'
- 'B. V. K. S. Potukuchi'
- 'C. B. Powell'
- 'D. Prindle'
- 'C. Pruneau'
- 'N. K. Pruthi'
- 'P. R. Pujahari'
- 'J. Putschke'
- 'R. Raniwala'
- 'S. Raniwala'
- 'R. L. Ray'
- 'R. Redwine'
- 'R. Reed'
- 'J. M. Rehberg'
- 'H. G. Ritter'
- 'J. B. Roberts'
- 'O. V. Rogachevskiy'
- 'J. L. Romero'
- 'A. Rose'
- 'C. Roy'
- 'L. Ruan'
- 'M. J. Russcher'
- 'R. Sahoo'
- 'S. Sakai'
- 'I. Sakrejda'
- 'T. Sakuma'
- 'S. Salur'
- 'J. Sandweiss'
- 'E. Sangaline'
- 'J. Schambach'
- 'R. P. Scharenberg'
- 'N. Schmitz'
- 'T. R. Schuster'
- 'J. Seele'
- 'J. Seger'
- 'I. Selyuzhenkov'
- 'P. Seyboth'
- 'E. Shahaliev'
- 'M. Shao'
- 'M. Sharma'
- 'S. S. Shi'
- 'E. P. Sichtermann'
- 'F. Simon'
- 'R. N. Singaraju'
- 'M. J. Skoby'
- 'N. Smirnov'
- 'P. Sorensen'
- 'J. Sowinski'
- 'H. M. Spinka'
- 'B. Srivastava'
- 'T. D. S. Stanislaus'
- 'D. Staszak'
- 'J. R. Stevens'
- 'R. Stock'
- 'M. Strikhanov'
- 'B. Stringfellow'
- 'A. A. P. Suaide'
- 'M. C. Suarez'
- 'N. L. Subba'
- 'M. Sumbera'
- 'X. M. Sun'
- 'Y. Sun'
- 'Z. Sun'
- 'B. Surrow'
- 'T. J. M. Symons'
- 'A. Szanto de Toledo'
- 'J. Takahashi'
- 'A. H. Tang'
- 'Z. Tang'
- 'L. H. Tarini'
- 'T. Tarnowsky'
- 'D. Thein'
- 'J. H. Thomas'
- 'J. Tian'
- 'A. R. Timmins'
- 'S. Timoshenko'
- 'D. Tlusty'
- 'M. Tokarev'
- 'T. A. Trainor'
- 'V. N. Tram'
- 'S. Trentalange'
- 'R. E. Tribble'
- 'O. D. Tsai'
- 'J. Ulery'
- 'T. Ullrich'
- 'D. G. Underwood'
- 'G. Van Buren'
- 'G. van Nieuwenhuizen'
- 'J. A. Vanfossen, Jr.'
- 'R. Varma'
- 'G. M. S. Vasconcelos'
- 'A. N. Vasiliev'
- 'F. Videbaek'
- 'Y. P. Viyogi'
- 'S. Vokal'
- 'S. A. Voloshin'
- 'M. Wada'
- 'M. Walker'
- 'F. Wang'
- 'G. Wang'
- 'H. Wang'
- 'J. S. Wang'
- 'Q. Wang'
- 'X. Wang'
- 'X. L. Wang'
- 'Y. Wang'
- 'G. Webb'
- 'J. C. Webb'
- 'G. D. Westfall'
- 'C. Whitten Jr.'
- 'H. Wieman'
- 'E. Wingfield'
- 'S. W. Wissink'
- 'R. Witt'
- 'Y. Wu'
- 'W. Xie'
- 'N. Xu'
- 'Q. H. Xu'
- 'W. Xu'
- 'Y. Xu'
- 'Z. Xu'
- 'L. Xue'
- 'Y. Yang'
- 'P. Yepes'
- 'K. Yip'
- 'I-K. Yoo'
- 'Q. Yue'
- 'M. Zawisza'
- 'H. Zbroszczyk'
- 'W. Zhan'
- 'S. Zhang'
- 'W. M. Zhang'
- 'X. P. Zhang'
- 'Y. Zhang'
- 'Z. P. Zhang'
- 'J. Zhao'
- 'C. Zhong'
- 'J. Zhou'
- 'W. Zhou'
- 'X. Zhu'
- 'Y. H. Zhu'
- 'R. Zoulkarneev'
- 'Y. Zoulkarneeva'
date: 'November 14, 2009'
title: 'Longitudinal double-spin asymmetry and cross section for inclusive neutral pion production at midrapidity in polarized proton collisions at $\protect\bm{{{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}}$ '
---
The spin structure of the nucleon is one of the fundamental and unresolved questions in Quantum Chromodynamics ([[<span style="font-variant:small-caps;">QCD</span>]{}]{}). Deep-inelastic scattering ([[<span style="font-variant:small-caps;">DIS</span>]{}]{}) experiments studying polarized leptons scattered off polarized nuclei have found the quark and anti-quark spin contributions to the overall spin of the nucleon to be small, at the level of $25\%$ [@Ashman:1989ig; @Filippone:2001ux], leading to increased interest in the spin contribution from gluons. [[<span style="font-variant:small-caps;">DIS</span>]{}]{} experiments have placed coarse constraints on the polarized gluon distribution function $\Delta g (x)$, based on the scale dependence of polarized structure functions [@Adeva:1998vw; @Anthony:2000fn] and on recent semi-inclusive data [@Airapetian:1999ib; @Adeva:2004dh; @Ageev:2005pq]. Measurements using collisions of longitudinally polarized protons are attractive because they provide sensitivity to the polarized gluon spin distribution at leading order through quark–gluon and gluon–gluon scattering contributions to the .
The sensitivity of inclusive hadron and jet production to the underlying gluon polarization in high-energy polarized proton collisions has been discussed in detail in Refs. [@Jager:2002xm; @Jager:2004jh]. The theoretical framework in the context of next-to-leading order perturbative [[<span style="font-variant:small-caps;">QCD</span>]{}]{} ([[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{}) calculations is very well developed to constrain $\Delta g(x)$. The first global analysis of semi-inclusive and inclusive [[<span style="font-variant:small-caps;">DIS</span>]{}]{} data, as well as results obtained by the [[<span style="font-variant:small-caps;">PHENIX</span>]{}]{} [@Adare:2007dg] and [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Abelev:2007vt] experiments, placed a strong constraint on $\Delta g(x)$ in the gluon momentum-fraction range of $0.05 < x < 0.2$, and suggested that the gluon spin contribution is not large in that range [@deFlorian:2008mr]. This conclusion was driven primarily by data on inclusive hadron and jet production in polarized proton collisions at ${{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$ at [[<span style="font-variant:small-caps;">RHIC</span>]{}]{}.
In this paper, we report on the measurement of the cross section and the longitudinal double-spin asymmetry $A_{LL}$ for inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production at midrapidity in polarized proton collisions at ${{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$ by the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} experiment [@Ackermann:2002ad] at [[<span style="font-variant:small-caps;">RHIC</span>]{}]{}. The cross section is compared to a [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation and the observed agreement provides an important basis to apply [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} for the interpretation of $A_{LL}$. The asymmetry is defined as $$\label {all_def}
A_{LL} \equiv \frac{\sigma^{++} - \sigma^{+-}}{\sigma^{++} + \sigma^{+-}},$$ where $\sigma^{++}$ and $\sigma^{+-}$ are the inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} cross sections for equal ($++$) and opposite ($+-$) beam helicity configurations. The measured longitudinal double-spin asymmetry probes a gluon momentum fraction of approximately $0.03 < x < 0.3$, and is compared to [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculations. In addition, we present the mean transverse momentum fraction of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s in electromagnetically triggered jets. This measurement allows one to relate the spin asymmetry measurements performed with inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s to those using reconstructed jets. It may also help to constrain fragmentation models.
The data for the analyses presented here were collected at [[<span style="font-variant:small-caps;">STAR</span>]{}]{} in 2005 using stored polarized $100{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$ proton beams with an average luminosity of $6 \times 10^{30}{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} \mathrm{cm}^{-2}\,\mathrm{s}^{-1}}}}$[$\mskip -0.5\thinmuskip$]{}. Longitudinal polarization of proton beams in the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} interaction region ([[<span style="font-variant:small-caps;">IR</span>]{}]{}) was achieved by spin rotator magnets upstream and downstream of the [[<span style="font-variant:small-caps;">IR</span>]{}]{} that changed the proton spin orientation from its stable vertical direction to longitudinal [@Alekseev:2003sk]. The helicities were alternated between successive proton bunches in one beam and pairs of successive proton bunches in the other beam. This allowed us to obtain all four helicity combinations of the colliding bunch pairs at the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [[<span style="font-variant:small-caps;">IR</span>]{}]{} in quick succession. Additional reduction of systematic uncertainties was achieved by periodically changing the helicity patterns of the stored beams. The polarization of each beam was measured several times per fill using Coulomb–Nuclear Interference ([[<span style="font-variant:small-caps;">CNI</span>]{}]{}) proton–carbon polarimeters [@Nakagawa:2007zza], which were calibrated using a polarized hydrogen gas-jet target [@Makdisi:2007zz]. The average [[<span style="font-variant:small-caps;">RHIC</span>]{}]{} beam polarizations in the 2005 run were $P_1 = 52 \pm 3\%$ and $P_2 = 48 \pm 3\%$. Non-longitudinal beam polarization components were continuously monitored with local polarimeters at [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Kiryluk:2005gg] and were found to be no larger than $9\%$ in absolute magnitude.
The principal [[<span style="font-variant:small-caps;">STAR</span>]{}]{} detector subsystems for the measurements presented here were the Barrel Electromagnetic Calorimeter ([[<span style="font-variant:small-caps;">BEMC</span>]{}]{}) [@Beddo:2002zx] and the Beam–Beam Counters ([[<span style="font-variant:small-caps;">BBC</span>]{}]{}) [@Kiryluk:2005gg]. In addition, the Time Projection Chamber ([[<span style="font-variant:small-caps;">TPC</span>]{}]{}) [@Anderson:2003ur] was used for vertexing, for measuring the charged component in the reconstructed jets, and as a charged particle veto for the [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} reconstruction. The [[<span style="font-variant:small-caps;">BEMC</span>]{}]{} is a lead–scintillator sampling calorimeter with a granularity of $\Delta \eta \times \Delta \varphi = 0.05 \times 0.05{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{\ensuremath{\textrm{rad}}}}}}}$, where one such cell is referred to as a tower. It contains a shower maximum detector ([[<span style="font-variant:small-caps;">BSMD</span>]{}]{}) that consists of two layers of wire proportional counters with cathode strip readout, one in the azimuthal direction and one in the longitudinal direction, at a depth of about $5$ radiation lengths in each calorimeter module, providing a segmentation of $0.007 \times 0.007{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{\ensuremath{\textrm{rad}}}}}}}$. For the 2005 running period, half of the [[<span style="font-variant:small-caps;">BEMC</span>]{}]{} was instrumented and operational, providing $2\pi$ azimuthal coverage for $0 < \eta < 1$. The [[<span style="font-variant:small-caps;">BBC</span>]{}]{}s are composed of segmented scintillator rings, covering $3.3 < \vert \eta \vert < 5.0$ on both sides of the [[<span style="font-variant:small-caps;">IR</span>]{}]{}. The [[<span style="font-variant:small-caps;">BBC</span>]{}]{}s were used to trigger on collisions, to measure the helicity-dependent relative luminosities, and to serve as local polarimeters. The [[<span style="font-variant:small-caps;">TPC</span>]{}]{} provided charged particle tracking inside a $0.5{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{\ensuremath{\textrm{T}}}}}}}$ solenoidal magnetic field over the full range of azimuthal angles for $\vert \eta \vert < 1.3$.
Proton–proton collisions in the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} detector were identified by a minimum bias trigger ([[<span style="font-variant:small-caps;">MB</span>]{}]{}), defined as a coincidence of hits in both [[<span style="font-variant:small-caps;">BBC</span>]{}]{}s. The cross section for this trigger was $\sigma_{\text{{{\textsc{BBC}}}}} = 26.1 \pm 0.2\,\text{(stat)} \pm 1.8\,\text{(syst)}{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{m}}}}{{\ensuremath{\textrm{b}}}}}}}}$, corresponding to $87 \pm 8\%$ of the non-singly diffractive [[[${{\ensuremath{p}}}+ {{\ensuremath{p}}}$]{}]{}]{} cross section at ${{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$ [@Adams:2003kv]. Rare hard scattering events were selected by two high-tower triggers, [[<span style="font-variant:small-caps;">HT</span>]{}]{}1 and [[<span style="font-variant:small-caps;">HT</span>]{}]{}2, that required a transverse energy deposition in a single [[<span style="font-variant:small-caps;">BEMC</span>]{}]{} tower above thresholds of $2.6$ and $3.5{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$[$\mskip -0.5\thinmuskip$]{}, respectively, in addition to satisfying the [[<span style="font-variant:small-caps;">MB</span>]{}]{} condition.
A data sample with an integrated luminosity of $\mathcal{L} = 0.17{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{n}}}}{{\ensuremath{\textrm{b}}}}}\ensuremath{^{-{\ensuremath{\mskip -0.4\thinmuskip}}1}}}}}}}}}$ for [[<span style="font-variant:small-caps;">MB</span>]{}]{}, $0.16{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{p}}}}{{\ensuremath{\textrm{b}}}}}\ensuremath{^{-{\ensuremath{\mskip -0.4\thinmuskip}}1}}}}}}}}}$ for [[<span style="font-variant:small-caps;">HT</span>]{}]{}1, and $0.66{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{p}}}}{{\ensuremath{\textrm{b}}}}}\ensuremath{^{-{\ensuremath{\mskip -0.4\thinmuskip}}1}}}}}}}}}$ for [[<span style="font-variant:small-caps;">HT</span>]{}]{}2 triggers was analyzed for the inclusive cross section measurement. Data with an integrated luminosity of $0.4\,(2.0){{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{p}}}}{{\ensuremath{\textrm{b}}}}}\ensuremath{^{-{\ensuremath{\mskip -0.4\thinmuskip}}1}}}}}}}}}$ of [[<span style="font-variant:small-caps;">HT</span>]{}]{}1$\,$([[<span style="font-variant:small-caps;">HT</span>]{}]{}2) triggers were used for the $A_{LL}$ determination. The event selection criteria for the asymmetry analysis were identical to those used in a previously published jet measurement [@Abelev:2007vt]. About $22\%$ of [[<span style="font-variant:small-caps;">HT</span>]{}]{}1/[[<span style="font-variant:small-caps;">HT</span>]{}]{}2 triggered events also entered the jet $A_{LL}$ measurement [@Abelev:2007vt], but represented a negligible fraction of the much larger inclusive jet data set. Therefore, the statistical correlation of the present [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} and jet $A_{LL}$ measurements is negligible.
Neutral pions were reconstructed in the decay channel ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}\rightarrow\gamma\gamma$ in an invariant mass analysis of pairs of neutral [[<span style="font-variant:small-caps;">BEMC</span>]{}]{} clusters, i.e., those that did not have a [[<span style="font-variant:small-caps;">TPC</span>]{}]{} track pointing to them, with a cut on the two-particle energy asymmetry of $|E_1-E_2|/(E_1+E_2) \le 0.7$. The tower granularity was insufficient to resolve cluster pairs in [[<span style="font-variant:small-caps;">HT</span>]{}]{}1/[[<span style="font-variant:small-caps;">HT</span>]{}]{}2 data because of the small opening angle between daughter photons of pions that satisfied these triggers. Therefore, the [[<span style="font-variant:small-caps;">BSMD</span>]{}]{} clusters were used to determine the photon coordinates in those data. A fiducial volume cut on the detector pseudorapidity of $0.1 < \eta < 0.9$ was imposed. The reconstructed value of the pion pseudorapidity with respect to the vertex position was required to fall in the range $0 < \eta < 1$. The [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} yield was extracted in ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ bins by integrating the background-subtracted invariant mass distribution in a ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$-dependent window around the ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}$ peak that corresponded to an approximately $\pm\,3\,\sigma$ range. The combinatorial background was determined using the event mixing method with a jet alignment correction [@ref_grebenyuk_thesis; @pi0_pp_dau].
The cross section for [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production is given by $$E \frac{d^{{\ensuremath{\mskip 0.5\thinmuskip}}3} \sigma}{d{\ensuremath{\mskip 0.5\thinmuskip}}{\textbf{p}}^3} = \frac{1}{2\pi {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}\,\Delta {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}\Delta \eta} \, c \, \frac{N}{\mathcal{L}},$$ where $\Delta{{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ and $\Delta \eta$ are the bin widths in ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ and pseudorapidity, $N$ is the [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} yield in a bin, and $c$ is an overall correction factor that accounts for acceptance, reconstruction, and trigger efficiency in that bin, which was determined using a Monte Carlo simulation of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s passed through the [[<span style="font-variant:small-caps;">geant</span>]{}]{} [@Brun:1978fy] model of the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} detector. Figure \[fig:XSect\]
 (a) Cross section for inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production at midrapidity in [[[${{\ensuremath{p}}}+ {{\ensuremath{p}}}$]{}]{}]{} collisions at ${{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$[$\mskip -0.5\thinmuskip$]{}, compared to a [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation [@Jager:2002xm] based on the [[<span style="font-variant:small-caps;">DSS</span>]{}]{} set of fragmentation functions [@deFlorian:2007aj], and to the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{{\ensuremath{\pm}}}}}}}}$ measurement [@Adams:2006nd]. (b) The ratio of measured cross section and the [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation. The scale uncertainty is indicated by the dashed curves ($\mu = 2{{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$, ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}/2$). The error bars are statistical and shaded bands are ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$-correlated systematic uncertainties. The normalization uncertainty is indicated by a shaded band around unity on the right-hand side.](fig1)
shows the differential cross section for inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production. This analysis covered the pion transverse momentum range of $1 < {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}< 17{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$, and data points were scaled to the bin centers using local exponential fits around each bin. The cross sections up to $4{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ were measured using [[<span style="font-variant:small-caps;">MB</span>]{}]{} triggered events; above $4\,$($7$)${{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ the entries were obtained from [[<span style="font-variant:small-caps;">HT</span>]{}]{}1$\,$([[<span style="font-variant:small-caps;">HT</span>]{}]{}2) triggers. The different trigger samples agreed within errors.
The dominant systematic uncertainty ($25\%$ on average) of the measured cross section was due to a $5\%$ uncertainty in the global energy scale of the [[<span style="font-variant:small-caps;">BEMC</span>]{}]{}. The other systematic uncertainties were related to yield extraction ($7\%$), reconstruction efficiency ($6\%$), and relative normalization of [[<span style="font-variant:small-caps;">HT</span>]{}]{}1/[[<span style="font-variant:small-caps;">HT</span>]{}]{}2 and [[<span style="font-variant:small-caps;">MB</span>]{}]{} triggers ($5\%$). An additional uncertainty due to the limited quality of the electromagnetic shower simulation at low photon energies in our [[<span style="font-variant:small-caps;">geant</span>]{}]{} model was assigned to the cross section obtained from [[<span style="font-variant:small-caps;">HT</span>]{}]{}1/[[<span style="font-variant:small-caps;">HT</span>]{}]{}2 data \[$15$($2$)${{\ensuremath{{}}}\%}$ at ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}= 4$($7$)${{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$\].
In Fig. \[fig:XSect\], the measured cross section is compared to a [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation [@Jager:2002xm] performed using the [[<span style="font-variant:small-caps;">CTEQ6M</span>]{}]{} set of unpolarized parton distribution functions [@Pumplin:2002vw] and the [[<span style="font-variant:small-caps;">DSS</span>]{}]{} set of fragmentation functions [@deFlorian:2007aj]. In this calculation, the factorization and renormalization scales were identified with ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ (solid curve), and were varied by a factor of two to estimate the impact of scale uncertainties (dashed curves). The [[<span style="font-variant:small-caps;">DSS</span>]{}]{} analysis included recent measurements of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production at midrapidity by [[<span style="font-variant:small-caps;">PHENIX</span>]{}]{} [@Adare:2007dg] and at forward rapidity by [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Adams:2006uz]. The [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation shows, within errors, good agreement with our data in the fragmentation region ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}> 2{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$. We also compare the cross section for ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}$ production to the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{{\ensuremath{\pm}}}}}}}}$ measurement [@Adams:2006nd]. The [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} and $({{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{+}}}}}}+ {{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{-}}}}}})/2$ cross sections are expected to be equal, and the two [[<span style="font-variant:small-caps;">STAR</span>]{}]{} measurements agree within statistical errors, in spite of using independent detector sub-systems.
The transverse momentum fraction carried by a high-${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} in its parent jet, $z = {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}({{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}})/{{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}(\rm{jet})$, was investigated by associating pions with jets found in the same event [@Abelev:2006uq]. The [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} sample, defined by the invariant mass window, contained $\approx$[$\mskip 0.5\thinmuskip$]{}$8{{\ensuremath{{}}}\%}$ of combinatorial background. An association was made if the pion was within a cone of radius $R = \sqrt{(\Delta\eta)^2 + (\Delta\varphi)^2} = 0.4$ around the jet axis. The analysis was restricted to $0.4 < \eta < 0.6$ in the jet pseudorapidity, so that the reconstructed jets were fully contained in the [[<span style="font-variant:small-caps;">BEMC</span>]{}]{} acceptance. The transverse momentum of the jet was required to exceed $5{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$. The jet was required to have a neutral energy fraction less than $0.95$, in order to minimize contributions from beam background to the reconstructed jet sample.
Figure \[fig:MeanZ\](a)
 (a) Mean transverse momentum fraction of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s in their associated jets, as a function of pion ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$, for electromagnetically triggered events. Systematic errors are shown by the shaded band around the data points. The curves are results from simulations with the [[<span style="font-variant:small-caps;">pythia</span>]{}]{} event generator. The solid curve includes detector effects simulated by [[<span style="font-variant:small-caps;">geant</span>]{}]{}, while the dashed curve uses jet finding at the [[<span style="font-variant:small-caps;">pythia</span>]{}]{} particle level. (b) The distribution of $z$ for one [[[${{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}$]{}]{}]{} bin, compared to [[<span style="font-variant:small-caps;">pythia</span>]{}]{} with a full detector response simulation.](fig2)
shows the mean value of $z$ as a function of pion ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$, combined for [[<span style="font-variant:small-caps;">HT</span>]{}]{}1 and [[<span style="font-variant:small-caps;">HT</span>]{}]{}2 triggers. The data points are plotted at the bin centers in pion ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$. The results were not corrected for detector effects, such as acceptance, efficiency, or resolution of the jet reconstruction. The systematic error band shown includes contributions from the uncertainty of the jet energy scale, the influence of the cut on minimum jet ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$, the contribution of events with $z > 1$, and a variation of other analysis cuts.
The [$\langle$[$z$]{}$\rangle$]{} of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s in electromagnetically triggered jets was found to be around $0.7$ and to rise slightly with pion ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$, consistent with measurements of leading charged hadrons in jets in fixed-target experiments [@Boca:1990rh]. The results also compare well to recent theoretical calculations for charged pions [@deFlorian:2009fw], considering the increase of the measured pion momentum fraction due to energy not reconstructed in the jet. The expectations from a [[<span style="font-variant:small-caps;">pythia</span>]{}]{}-based (version [<span style="font-variant:small-caps;">6.205</span>]{} [@Sjostrand:2001yu] with ‘[[<span style="font-variant:small-caps;">CDF</span>]{}]{} Tune [<span style="font-variant:small-caps;">A</span>]{}’ settings [@Field:2005sa]) Monte Carlo simulation are also shown. The [$\langle$[$z$]{}$\rangle$]{} measured in jets found on the [[<span style="font-variant:small-caps;">pythia</span>]{}]{} particle level, i.e., without any detector effects, is lower than in the data due to resolution effects and losses in the jet reconstruction, indicating the influence of the detector on the measurement. Results from a [[<span style="font-variant:small-caps;">geant</span>]{}]{}-based [[<span style="font-variant:small-caps;">STAR</span>]{}]{} detector simulation show good agreement with the data, demonstrating the reliability of the simulation framework used in the present analysis.
Figure \[fig:MeanZ\](b) shows the distribution of $z$ for one of the bins in pion [[[${{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}$]{}]{}]{} in comparison to [[<span style="font-variant:small-caps;">pythia</span>]{}]{} with a [[<span style="font-variant:small-caps;">geant</span>]{}]{}-based detector simulation. To maximize the statistics in the simulation, the generator-level [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s were used without requiring an explicit reconstruction. This led to a softening of the falling edge of the distribution at high $z$ in simulations, since a full [[<span style="font-variant:small-caps;">geant</span>]{}]{} simulation was used for the containing jets, but did not affect the mean of the distribution. A small fraction of the events had $z > 1$, apparently corresponding to pions that carried more transverse momentum than their containing jet. This excess was caused by corrections applied during jet reconstruction, which in some cases led to an underestimation of the jet energy, and was well reproduced in simulations.
The asymmetry \[Eq. (\[all\_def\])\] was calculated as $$A_{LL} = \frac{1}{P_1 P_2}\frac{(N^{++} - RN^{+-})}{(N^{++} + RN^{+-})},$$ where $N^{++}$ and $N^{+-}$ are the ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}$ yields in equal and opposite beam helicity configurations, respectively, and $R$ is the luminosity ratio for those two helicities. Typical values of $R$, measured with the [[<span style="font-variant:small-caps;">BBC</span>]{}]{}s to a statistical precision of $10^{-3}$–$10^{-4}$ per run, ranged from $0.85$ to $1.2$, depending on fill and bunch pattern. Figure \[fig:All\]
 Longitudinal double-spin asymmetry for inclusive ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}$ production at midrapidity in [[[${{\ensuremath{p}}}+ {{\ensuremath{p}}}$]{}]{}]{} collisions at ${{\ensuremath{\sqrt{{\ensuremath{\mskip -0.5\thinmuskip}}{{\ensuremath{s}}}{\ensuremath{\mskip 0.5\thinmuskip}}}}}}= 200{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}}}}$, compared to [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculations based on the gluon distributions from the [[<span style="font-variant:small-caps;">GRSV</span>]{}]{} [@Gluck:2000dy], [[<span style="font-variant:small-caps;">GS-C</span>]{}]{} [@Gehrmann:1995ag], and [[<span style="font-variant:small-caps;">DSSV</span>]{}]{} [@deFlorian:2008mr] global analyses. The systematic error (shaded band) does not include a $9.4\%$ normalization uncertainty due to the beam polarization measurement.](fig3)
shows the measured longitudinal double-spin asymmetry for ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}$ production. The data points are plotted at the mean pion ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ in each bin. The lowest-${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$ point at $4.17{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ was obtained from [[<span style="font-variant:small-caps;">HT</span>]{}]{}1 triggers only; other points are the [[<span style="font-variant:small-caps;">HT</span>]{}]{}1 and [[<span style="font-variant:small-caps;">HT</span>]{}]{}2 combined results.
The systematic errors shown in the figure include point-to-point contributions from ${{{\ensuremath{{\ensuremath{\pi}}\ensuremath{^{0}}}}}}$ yield extraction \[(4–14)$\times 10^{-3}$\], invariant mass background subtraction \[(6–11)$\times 10^{-3}$\], and remaining beam background \[(1–9)$\times 10^{-3}$\], as well as ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$-correlated contributions from relative luminosity uncertainties ($9\times 10^{-4}$) and from non-longitudinal spin components ($3\times 10^{-4}$). All of the errors above are absolute errors on the measured asymmetry. An evaluation of the effects of non-longitudinal components of the beam polarization was not possible due to the limited statistics of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s in data taken with transversely polarized beams. Instead, the largest value from the jet measurement [@Abelev:2007vt] over the relevant momentum range was taken as an estimate of this systematic error. An overall normalization uncertainty of $9.4\%$ due to the uncertainty in the [[<span style="font-variant:small-caps;">RHIC</span>]{}]{} [[<span style="font-variant:small-caps;">CNI</span>]{}]{} polarimeter calibration is not shown. Studies of parity-violating single spin asymmetries and randomized spin patterns showed no evidence of bunch-to-bunch or fill-to-fill systematics.
In Fig. \[fig:All\], the measured values for $A_{LL}$ are compared to [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculations [@Jager:2002xm] based on various sets of polarized gluon distribution functions. The [[<span style="font-variant:small-caps;">DSSV</span>]{}]{} curve [@deFlorian:2008mr] is the result of the first global analysis that includes semi-inclusive and inclusive [[<span style="font-variant:small-caps;">DIS</span>]{}]{} data, as well as results obtained by the [[<span style="font-variant:small-caps;">PHENIX</span>]{}]{} [@Adare:2007dg] and [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Abelev:2007vt] experiments. The [[<span style="font-variant:small-caps;">GS-C</span>]{}]{} curve [@Gehrmann:1995ag] refers to a polarized gluon distribution function that has a large positive gluon polarization at low $x$, a node near $x \approx 0.1$, and a negative gluon polarization at large $x$. The [[<span style="font-variant:small-caps;">GRSV</span>]{}]{} standard curve is based on the best fit to [[<span style="font-variant:small-caps;">DIS</span>]{}]{} data [@Gluck:2000dy], while the other [[<span style="font-variant:small-caps;">GRSV</span>]{}]{} curves show scenarios of extreme positive ($\Delta g = +g$), extreme negative ($\Delta g = -g$), and vanishing ($\Delta g = 0$) gluon polarization at the starting scale [@Gluck:2000dy; @Jager:2004jh]. A maximal positive gluon polarization scenario, which has a total gluon spin contribution $\Delta G \equiv \int_{0}^{1}\! \Delta g(x)\, dx = 1.26$ at an initial scale of $0.4{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}^2}}}$ [@Gluck:2000dy; @Gluck:1998xa], is excluded by our measurement at $98{{\ensuremath{{}}}\%}$ confidence level, including systematic uncertainties. This is in agreement with the conclusions from the inclusive jet measurements by [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Abelev:2006uq; @Abelev:2007vt] and from the inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} measurement by [[<span style="font-variant:small-caps;">PHENIX</span>]{}]{} [@Adare:2007dg]. The data are consistent with all other gluon polarization scenarios, in particular with the [[<span style="font-variant:small-caps;">DSSV</span>]{}]{} case.
In summary, we report a measurement of the invariant cross section and the longitudinal double-spin asymmetry $A_{LL}$ for inclusive [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production at midrapidity with the [[<span style="font-variant:small-caps;">STAR</span>]{}]{} detector at [[<span style="font-variant:small-caps;">RHIC</span>]{}]{}. The cross section was determined for $1 < {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}< 17{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ and found to be in agreement with a [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculation based on the [[<span style="font-variant:small-caps;">CTEQ6M</span>]{}]{} parton distribution functions and the [[<span style="font-variant:small-caps;">DSS</span>]{}]{} fragmentation functions. This set of fragmentation functions was constrained by data that included measurements of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{} production at midrapidity by [[<span style="font-variant:small-caps;">PHENIX</span>]{}]{} [@Adare:2007dg] and at forward rapidity by [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Adams:2006uz]. The mean transverse momentum fraction of [[[${\ensuremath{\pi}}\ensuremath{^{0}}$]{}]{}]{}’s in electromagnetically triggered jets was found to be approximately $0.7$ and to rise slightly with pion ${{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}$, in agreement with a [[<span style="font-variant:small-caps;">pythia</span>]{}]{}-based Monte Carlo simulation that included detector effects. This measurement has the potential to contribute to future fragmentation function studies. The asymmetry $A_{LL}$ was measured in the hard scattering regime at $3.7 < {{{\ensuremath{{{\ensuremath{p}}}\ensuremath{_{{{\ensuremath{T}}}}}}}}}< 11{{\ensuremath{{\ensuremath{\mskip 1.5\thinmuskip}} {{{\ensuremath{{{{\ensuremath{\textrm{G}}}}{{\ensuremath{\mskip 0.1\thinmuskip}}{\ensuremath{\textrm{e}}}{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{\textrm{V}}}}}}{\ensuremath{\mskip -1\thinmuskip}}/{\ensuremath{\mskip -0.3\thinmuskip}}{{\ensuremath{c}}}}}}}}}}$ and found to be consistent with [[<span style="font-variant:small-caps;">NLO</span>]{}]{} [p[<span style="font-variant:small-caps;">QCD</span>]{}]{} calculations utilizing polarized quark and gluon distributions from inclusive and semi-inclusive [[<span style="font-variant:small-caps;">DIS</span>]{}]{} data and from polarized proton data. Our data exclude a maximal positive gluon polarization in the nucleon, in agreement with results obtained from inclusive jet production in polarized proton collisions by [[<span style="font-variant:small-caps;">STAR</span>]{}]{} [@Abelev:2006uq; @Abelev:2007vt], while being a statistically independent measurement, subject to a different set of systematic uncertainties. With increasing integrated luminosity, the neutral pion channel has the potential to provide additional constraints on the gluon polarization in the polarized proton.
We thank the [[<span style="font-variant:small-caps;">RHIC</span>]{}]{} Operations Group and [[<span style="font-variant:small-caps;">RCF</span>]{}]{} at [[<span style="font-variant:small-caps;">BNL</span>]{}]{}, the [[<span style="font-variant:small-caps;">NERSC</span>]{}]{} Center at [[<span style="font-variant:small-caps;">LBNL</span>]{}]{} and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Offices of [<span style="font-variant:small-caps;">NP</span>]{} and [<span style="font-variant:small-caps;">HEP</span>]{} within the U[$\mskip -0.6\thinmuskip$]{}.[$\mskip 1\thinmuskip$]{}S. [<span style="font-variant:small-caps;">DOE</span>]{} Office of Science, the U[$\mskip -0.6\thinmuskip$]{}.[$\mskip 1\thinmuskip$]{}S. [<span style="font-variant:small-caps;">NSF</span>]{}, the Sloan Foundation, the [<span style="font-variant:small-caps;">DFG</span>]{} cluster of excellence ‘Origin and Structure of the Universe’, [<span style="font-variant:small-caps;">CNRS/IN2P3</span>]{}, [<span style="font-variant:small-caps;">STFC</span>]{} and [<span style="font-variant:small-caps;">EPSRC</span>]{} of the United Kingdom, [<span style="font-variant:small-caps;">FAPESP</span>]{} [<span style="font-variant:small-caps;">CNP</span>]{}q of Brazil, Ministry of Ed. and Sci. of the Russian Federation, [<span style="font-variant:small-caps;">NNSFC</span>]{}, [<span style="font-variant:small-caps;">CAS</span>]{}, [<span style="font-variant:small-caps;">M</span>]{}o[<span style="font-variant:small-caps;">ST</span>]{}, and [<span style="font-variant:small-caps;">M</span>]{}o[<span style="font-variant:small-caps;">E</span>]{} of China, [<span style="font-variant:small-caps;">GA</span>]{} and [<span style="font-variant:small-caps;">MSMT</span>]{} of the Czech Republic, [<span style="font-variant:small-caps;">FOM</span>]{} and [<span style="font-variant:small-caps;">NWO</span>]{} of the Netherlands, [<span style="font-variant:small-caps;">DAE</span>]{}, [<span style="font-variant:small-caps;">DST</span>]{}, and [<span style="font-variant:small-caps;">CSIR</span>]{} of India, Polish Ministry of Sci. and Higher Ed., Korea Research Foundation, Ministry of Sci., Ed. and Sports of the Rep. Of Croatia, Russian Ministry of Sci. and Tech, and RosAtom of Russia.
[35]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
(), ****, ().
, ****, ().
(), ****, ().
(), ****, ().
()[$\mskip -0.5\thinmuskip$]{}, ****, ().
(), ****, ().
, , , , ****, ().
, , , ****, ().
(), ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
(), , ().
(), ****, ().
, Ph.D. thesis, ().
(), .
, , , (), .
, , , ****, ().
(), ****, ().
, ****, ().
(), ****, ().
(), ****, ().
, ****, ().
, ****, ().
, , , , , , , ****, ().
(), .
, , , , ****, ().
, , , ****, ().
|
---
abstract: |
This paper is the continuation of a previous one \[L. [Š]{}amaj and B. Jancovici, 2007 [*J. Stat. Mech.*]{} P02002\]; for a nearly classical quantum fluid in a half-space bounded by a plain plane hard wall (no image forces), we had generalized the Wigner-Kirkwood expansion of the equilibrium statistical quantities in powers of Planck’s constant $\hbar$. As a model system for a more detailed study, we consider the quantum two-dimensional one-component plasma: a system of charged particles of one species, interacting through the logarithmic Coulomb potential in two dimensions, in a uniformly charged background of opposite sign, such that the total charge vanishes. The corresponding classical system is exactly solvable in a variety of geometries, including the present one of a half-plane, when $\beta e^2=2$, where $\beta$ is the inverse temperature and $e$ is the charge of a particle: all the classical $n$-body densities are known. In the present paper, we have calculated the expansions of the quantum density profile and truncated two-body density up to order $\hbar^2$ (instead of only to order $\hbar$ in the previous paper). These expansions involve the classical $n$-body densities up to $n=4$, thus we obtain exact expressions for these quantum expansions in this special case.
For the quantum one-component plasma, two sum rules involving the truncated two-body density (and, for one of them, the density profile) have been derived, a long time ago, by heuristic macroscopic arguments: one sum rule is about the asymptotic form along the wall of the truncated two-body density, the other one is about the dipole moment of the structure factor. In the two-dimensional case at $\beta e^2=2$, we have now explicit expressions up to order $\hbar^2$ of these two quantum densities, thus we can microscopically check the sum rules at this order. The checks are positive, reinforcing the idea that the sum rules are correct.
address:
- ' Laboratoire de Physique Théorique, Université Paris-Sud (Unité Mixte de Recherche no. 8627 - CNRS), 91405 Orsay, France'
- ' Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 845 11 Bratislava, Slovakia'
author:
- 'B. Jancovici and L. [Š]{}amaj'
title: 'Correlations and sum rules in a half-space for a quantum two-dimensional one-component plasma'
---
[**Keywords:**]{} Charged fluids (Theory)
Introduction
============
The model under consideration is the two-dimensional (2D) one-component plasma (also called jellium). This model consists of one species of charged particles in a plane. Each particle has a charge $e$ and a mass $m$. Two particles, at a distance $r$ from each other, interact through the 2D Coulomb interaction $v(r)$. This interaction is determined by the 2D Poisson equation $\enabla^2 v(r)=-2\pi e^2\delta(\mathbf{r})$, the solution of which is $v(r)=-e^2\ln(r/r_0)$, where $r_0$ is an arbitrary length which only fixes the zero of this potential. In addition, there is a charged uniform background of charge density opposite to the particle charge, so that the total system is neutral.
When the inverse temperature $\beta$ is such that the dimensionless coupling constant $\beta e^2=2$, the equilibrium statistical mechanics of the classical (i.e. non-quantum) system is completely solvable in a variety of geometries, in particular when the background and particles are confined into a half-space by an impenetrable rectilinear plain hard wall (there are no image forces): all the classical $n$-body densities are known [@Janco; @CJB].
In its three-dimensional version, the quantum one-component plasma is not only of academic interest. It has been used, in first approximation, as a model for the electrons of a metal [@NP]. In the present geometry of a half-space, this model might be used for describing what happens near the surface of the metal. Since the logarithmic interaction $v(r)$ is the Coulomb potential in two dimensions, the 2D one-component plasma is expected to have general features which mimic those of the 3D one.
Under certain conditions, the equilibrium properties of an infinite quantum fluid, in the nearly classical regime, can be expanded in powers of Planck’s constant $\hbar$: this is the Wigner-Kirkwood expansion [@Wigner; @Kirkwood]. In a previous paper [@WK], we have generalized the Wigner-Kirkwood expansion to the case of a quantum fluid occupying a half space. We have obtained expressions for the first quantum correction of order $\hbar$ to the density profile and to the two-body density, in terms of some $n$-body densities of the classical fluid. In section 3 of the present paper we extend these calculations to order $\hbar^2$. Actually, instead of $\hbar$, we use the thermal de Broglie wavelength proportional to it, $\lambda=\hbar\sqrt{\beta/m}$.
For the quantum one-component plasma in a half space, by heuristic macroscopic methods, some sum rules involving the one-body density profile and the two-body density have been derived. The purpose of the present paper is to check these sum rules at order $\hbar^2$, in the special case of the 2D one-component plasma at $\beta e^2=2$, using the generalized Wigner-Kirkwood expansion, which is a microscopic approach.
The paper is organized as follows. Section 2 brings a recapitulation of the method for constructing the expansion of the quantum Boltzmann density in configuration$\vec{\bf r}$-space for fluids constrained to a half-space [@WK]. This expansion is subsequently used in section 3 to compute the quantum one-body profile and two-body density to order $\hbar^2$ for the studied 2D one-component plasma. In section 4, three sum rules are reviewed: a perfect screening rule, a sum rule about the asymptotic form of the two-body density along the wall, and a dipole sum rule; their expansions to order $\hbar^2$ are given. In section 5, some classical $n$-body correlation functions, which are needed, are studied. In section 6, the perfect screening sum rule is shown to hold, at order $\hbar^2$, for the quantum 2D one-component plasma at $\beta=2$. In sections 7 and 8, the same is done for the asymptotic form and the dipole sum rules, respectively. Section 9 is a Conclusion.
Boltzmann density for the half-space geometry
=============================================
We first consider a general quantum system of $N$ identical particles $j=1,2,\ldots,N$ of mass $m$, formulated in $\nu$ space dimensions. Particle position vectors ${\bf r}_1,{\bf r}_2,\ldots,{\bf r}_N$ are confined to the half-space $\Lambda$ defined by Cartesian coordinates ${\bf r}=(x>0,{\bf r}^{\perp})$, where ${\bf r}^{\perp}\in {\rm R}^{\nu-1}$ denotes the set of $(\nu-1)$ unbounded coordinates normal to $x$. As usual, we start with a finite $N$ and a finite volume $|\Lambda|$, and later we take the thermodynamic limit $N$ and $|\Lambda|$ going to infinity (when $\Lambda$ becomes a half space) with a finite mean number density $n=N/|\Lambda|$. The hard wall in the complementary half-space $\bar{\Lambda}$ of points ${\bf r} = (x<0,{\bf r}^{\perp})$ is considered to be impenetrable to particles, i.e. the wavefunctions of the particle system vanish as soon as one of the particles lies at the wall. For the sake of brevity, we denote the $\nu N$-dimensional position vector in configuration space by $\vec{\bf r}=({\bf r}_1,{\bf r}_2,\ldots,{\bf r}_N)$ and the corresponding gradient by $\bfnabla = (\enabla_1,\enabla_2,\ldots,\enabla_N)$. In the absence of a magnetic field, the Hamiltonian of the particle system is given by $$\label{2.1}
H = \frac{1}{2 m} \left( - {\rm i} \hbar \bfnabla \right)^2
+ V(\vec{\bf r}) ,$$ where $\hbar$ stands for Planck’s constant and $V(\vec{\bf r})$ is the total interaction potential.
For infinite (bulk) quantum fluids of particles interacting via pairwise sufficiently smooth interactions with neglected fermion/boson exchange effects, Wigner [@Wigner] and Kirkwood [@Kirkwood] constructed a semiclassical expansion of the Boltzmann density in configuration space (at inverse temperature $\beta$), $B_{\beta}(\vec{\bf r}) =
\langle \vec{\bf r}\vert {\rm e}^{-\beta H} \vert \vec{\bf r} \rangle$, in even powers of the thermal de Broglie wavelength $\lambda = \hbar (\beta/m)^{1/2}$. Recently [@WK], we have generalized the Wigner-Kirkwood method to quantum fluids constrained to the above defined half-space $\Lambda$. The final result for the Boltzmann density in configuration space was obtained as a series $$\label{2.2}
B_{\beta}(\vec{\bf r}) = \sum_{n=0}^{\infty} B_{\beta}^{(n)}(\vec{\bf r}) ,$$ where the terms $B_{\beta}^{(n)}(\vec{\bf r})$ with $n=0,1,2,\ldots$ can be calculated systematically with the aid of an operator technique.
The result for $B_{\beta}^{(0)}(\vec{\bf r})$ was found in the form $$\label{2.3}
B_{\beta}^{(0)}(\vec{\bf r}) = \frac{1}{(\sqrt{2\pi} \lambda)^{\nu N}}
{\rm e}^{-\beta V} \prod_{j=1}^N \left( 1 - {\rm e}^{-2x_j^2/\lambda^2}
\right) .$$ The bulk counterpart of this term corresponds to the classical Boltzmann density $\propto {\rm e}^{-\beta V}$. Here, each particle gets an additional “boundary” factor $1-\exp(-2 x^2/\lambda^2)$ which goes from $0$ at the $x=0$ boundary to $1$ in the bulk interior $x\to\infty$ on the length scale $\sim\lambda$. The product of boundary factors then ensures that the quantum Boltzmann density vanishes as soon as one of the particles lies on the boundary. The dependence of the boundary factor on the de Broglie wavelength $\lambda$ is non-analytic; this fact prevents one from a simple classification of contributions to the Boltzmann density according to integer powers of $\lambda$ like it is in the bulk case. However, when in the calculation of statistical averages the exponential part $\exp(-2 x^2/\lambda^2)$ of the boundary factor is integrated over the $x$-coordinate, the analyticity of the result in the parameter $\lambda$ is restored. At this stage we only notice that when the product of boundary factors is expanded as follows $$\label{2.4}
\prod_{j=1}^N \left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
= 1 - \sum_{j=1}^N {\rm e}^{-2x_j^2/\lambda^2}
+ \frac{1}{2!} \sum_{j,k=1\atop (j\ne k)}^N {\rm e}^{-2x_j^2/\lambda^2}
{\rm e}^{-2x_k^2/\lambda^2} + \cdots ,$$ the integration of each exponential term $\exp(-2 x^2/\lambda^2)$ over $x$ produces one $\lambda$-factor as the result of the substitution of variables $x=\lambda x'$.
The result for $B_{\beta}^{(1)}(\vec{\bf r})$ reads $$\begin{aligned}
B_{\beta}^{(1)}(\vec{\bf r}) & = &
\frac{1}{(\sqrt{2\pi} \lambda)^{\nu N}} \Bigg\{
\sum_{k=1}^N \prod_{j=1\atop (j\ne k)}^N
\left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
x_k {\rm e}^{-2x_k^2/\lambda^2}
\frac{\partial }{\partial x_k} {\rm e}^{-\beta V} \nonumber \\
& & + {\rm e}^{-\beta V} \prod_{j=1}^N
\left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
\lambda^2 \left[ - \frac{\beta}{4} \bfnabla^2 V +
\frac{\beta^2}{6} \left( \bfnabla V \right)^2 \right] \Bigg\} . \label{2.5}\end{aligned}$$ Here, the dependence on $\lambda$ appears also via the combination $x \exp(-2x^2/\lambda^2)$. This function has a maximum of order $\lambda$ and therefore it is a legitimate expansion parameter. When integrated over the particle coordinate $x$, it gives a contribution of order $\lambda^2$, “weaker” than $\lambda$.
Keeping all contributions up to the order $\lambda^2$ in the Boltzmann term $B_{\beta}^{(2)}(\vec{\bf r})$, one has $$\label{2.6}
\fl B_{\beta}^{(2)}(\vec{\bf r}) =
\frac{1}{(\sqrt{2\pi} \lambda)^{\nu N}} {\rm e}^{-\beta V}
\prod_{j=1}^N \left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
\lambda^2 \left[ \frac{\beta}{6} \bfnabla^2 V
- \frac{\beta^2}{8} \left( \bfnabla V\right)^2 \right]
+ o(\lambda^2) .$$ Note that, with regard to the equality $$\label{2.7}
\bfnabla^2 {\rm e}^{-\beta V} = {\rm e}^{-\beta V} \left[
\beta^2 \left( \bfnabla V \right)^2 - \beta \bfnabla^2 V \right],$$ one can eliminate the squared gradient term in favour of the Laplacian term in equations (\[2.5\]) and (\[2.6\]).
The first three Boltzmann terms (\[2.3\]), (\[2.5\]) and (\[2.6\]) exhibit properties analogous to their bulk counterparts: the maximum of $B_{\beta}^{(n)}(\vec{\bf r})$ is of order $\lambda^n$. We anticipate that this formal structure is also maintained on higher levels. As a consequence, the knowledge of the first three Boltzmann terms $B_{\beta}^{(0)}$, $B_{\beta}^{(1)}$ and $B_{\beta}^{(2)}$ is sufficient in order to obtain the expansion of the quantum Boltzmann density up to the $\lambda^2$ order: $$\label{2.8}
B_{\beta}(\vec{\bf r}) = B_{\beta}^{(0)}(\vec{\bf r}) +
B_{\beta}^{(1)}(\vec{\bf r}) + B_{\beta}^{(2)}(\vec{\bf r}) + o(\lambda^2) .$$
The quantum fluid of present interest is the one-component plasma (jellium) in $\nu=2$ space dimensions. The system is composed of $N$ mobile pointlike charges $e$, neutralized by a uniform oppositely charged fixed background. The total interaction potential $V(\vec{\bf r})$ satisfies for each of the particle coordinates the Poisson differential equation $$\label{2.9}
\enabla_j^2 V(\vec{\bf r}) = - 2\pi e^2 \sum_{k=1\atop(k\ne j)}^N
\delta({\bf r}_j-{\bf r}_k) + 2\pi e^2 n ,
\quad j=1,2,\ldots,N .$$ Here, the second term on the right-hand side (rhs) comes from the particle-background interaction and $n=N/\vert\Lambda\vert$ is the mean number density of the mobile charges. The summation over the particle index $j$ of the set of $N$ Poisson equations (\[2.9\]) results in $$\label{2.10}
\bfnabla^2 V(\vec{\bf r}) = - 2\pi e^2 \sum_{j,k=1\atop (j\ne k)}^N
\delta({\bf r}_j-{\bf r}_k) + 2\pi e^2 N n .$$
The first term on the rhs of (\[2.10\]), when weighted by the classical Boltzmann factor $\propto \prod_{j<k} \vert {\bf r}_j-{\bf r}_k \vert^{\beta e^2}$ which vanishes at zero interparticle distance, does not give any contribution to the Boltzmann density. The application of the equality (\[2.7\]) and the replacement of $\bfnabla^2 V(\vec{\bf r})$ by the particle-background term $2\pi N e^2 n$ in the relations (\[2.5\]) and (\[2.6\]) simplifies substantially the expansion formula for the quantum Boltzmann density (\[2.8\]): $$\begin{aligned}
B_{\beta}(\vec{\bf r}) & = & \frac{1}{(\sqrt{2\pi}\lambda)^{\nu N}} \Bigg\{
{\rm e}^{-\beta V} \left( 1 - \frac{\lambda^2}{24} 2\pi\beta e^2 N n \right)
\prod_{j=1}^N \left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right) \nonumber \\
& & + \sum_{k=1}^N \prod_{j=1\atop (j\ne k)}^N
\left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
x_k {\rm e}^{-2x_k^2/\lambda^2}
\frac{\partial }{\partial x_k} {\rm e}^{-\beta V} \nonumber \\
& & + \prod_{j=1}^N \left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
\frac{\lambda^2}{24} \bfnabla^2 {\rm e}^{-\beta V} \Bigg\} + o(\lambda^2) .
\label{2.11}\end{aligned}$$
Statistical quantities for the half-space geometry
==================================================
According to the standard formalism of statistical quantum mechanics, the partition function of the $N$-particle fluid (with ignored exchange effects) is given by the integration of the quantum Boltzmann density over configuration space: $$\label{3.1}
Z_{\rm qu} = \frac{1}{N!} \int_{\Lambda} {\rm d}\vec{\bf r}\,
B_{\beta}(\vec{\bf r}) .$$ The quantum average of a function $f(\vec{\bf r})$ is defined as follows $$\label{3.2}
\left\langle f \right\rangle_{\rm qu} =
\frac{1}{Z_{\rm qu}} \frac{1}{N!}
\int_{\Lambda} {\rm d}\vec{\bf r}\,
B_{\beta}(\vec{\bf r}) f(\vec{\bf r}) .$$ At the one-particle level, one introduces the particle number density $$\label{3.3}
n_{\rm qu}({\bf r}) = \Bigg\langle
\sum_{j=1}^N \delta({\bf r}-{\bf r}_j) \Bigg\rangle_{\rm qu} .$$ At the two-particle level, one considers the two-body density $$\label{3.4}
n_{\rm qu}^{(2)}({\bf r},{\bf r}') = \Bigg\langle
\sum_{j,k=1\atop (j\ne k)}^N
\delta({\bf r}-{\bf r}_j) \delta({\bf r}'-{\bf r}_k) \Bigg\rangle_{\rm qu} .$$ It will be useful to consider also the truncated two-body density $$\label{3.5}
n_{\rm qu}^{(2){\rm T}}({\bf r}_1,{\bf r}_2) =
n_{\rm qu}^{(2)}({\bf r}_1,{\bf r}_2)
- n_{\rm qu}({\bf r}_1) n_{\rm qu}({\bf r}_2)$$ vanishing at asymptotically large distances $\vert {\bf r}_1-{\bf r}_2\vert\to\infty$. The general multiparticle densities are defined in analogy with (\[3.4\]), i.e. the corresponding product of $\delta$-functions is summed out over all possible multiplets of different particles.
The classical partition function $Z$ and the classical average of a function $f(\vec{\bf r})$ are defined as follows $$\begin{aligned}
Z & = & \frac{1}{N!} \int_{\Lambda}
\frac{{\rm d}\vec{\bf r}}{(\sqrt{2\pi}\lambda)^{\nu N}}\,
{\rm e}^{-\beta V(\vec{\bf r})} , \label{3.6} \\
\left\langle f \right\rangle & = & \frac{1}{Z} \frac{1}{N!}
\int_{\Lambda} \frac{{\rm d}\vec{\bf r}}{(\sqrt{2\pi}\lambda)^{\nu N}}\,
{\rm e}^{-\beta V(\vec{\bf r})} f(\vec{\bf r}) . \label{3.7}\end{aligned}$$ The classical values of statistical quantities will be written without a subscript, like $n({\bf r})$, $n^{(2)}({\bf r},{\bf r}')$, etc. In the calculations which follow, we shall need explicitly truncated forms of the classical three-body density $$\begin{aligned}
\fl n^{(3){\rm T}}({\bf r}_1,{\bf r}_2,{\bf r}_3) & = &
n^{(3)}({\bf r}_1,{\bf r}_2,{\bf r}_3) -
n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) n({\bf r}_3)
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_3) n({\bf r}_2) \nonumber \\ & &
- n^{(2){\rm T}}({\bf r}_2,{\bf r}_3) n({\bf r}_1)
- n({\bf r}_1) n({\bf r}_2) n({\bf r}_3) \label{3.8}\end{aligned}$$ and of the classical four-body density $$\begin{aligned}
\fl n^{(4){\rm T}}({\bf r}_1,{\bf r}_2,{\bf r}_3,{\bf r}_4)
& = & n^{(4)}({\bf r}_1,{\bf r}_2,{\bf r}_3,{\bf r}_4)
- n^{(3){\rm T}}({\bf r}_1,{\bf r}_2,{\bf r}_3) n({\bf r}_4)
\nonumber \\ & &
- n^{(3){\rm T}}({\bf r}_1,{\bf r}_2,{\bf r}_4) n({\bf r}_3)
- n^{(3){\rm T}}({\bf r}_1,{\bf r}_3,{\bf r}_4) n({\bf r}_2)
\nonumber \\ & &
- n^{(3){\rm T}}({\bf r}_2,{\bf r}_3,{\bf r}_4) n({\bf r}_1)
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) n^{(2){\rm T}}({\bf r}_3,{\bf r}_4)
\nonumber \\& &
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_3) n^{(2){\rm T}}({\bf r}_2,{\bf r}_4)
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_4) n^{(2){\rm T}}({\bf r}_2,{\bf r}_3)
\nonumber \\ & &
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) n({\bf r}_3) n({\bf r}_4)
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_3) n({\bf r}_2) n({\bf r}_4)
\nonumber \\ & &
- n^{(2){\rm T}}({\bf r}_1,{\bf r}_4) n({\bf r}_2) n({\bf r}_3)
- n^{(2){\rm T}}({\bf r}_2,{\bf r}_3) n({\bf r}_1) n({\bf r}_4)
\nonumber \\ & &
- n^{(2){\rm T}}({\bf r}_2,{\bf r}_4) n({\bf r}_1) n({\bf r}_3)
- n^{(2){\rm T}}({\bf r}_3,{\bf r}_4) n({\bf r}_1) n({\bf r}_2)
\nonumber \\ & &
- n({\bf r}_1) n({\bf r}_2) n({\bf r}_3) n({\bf r}_4) . \label{3.9}\end{aligned}$$
In what follows, we shall restrict ourselves to the model system of the one-component plasma constrained to the two-dimensional half-space $\Lambda$. Now, the Cartesian coordinates of $\mathbf{r}$ become $(x,y)$, with the origin on the rectilinear plain hard wall, the $y$ axis along the wall, and the system occupying the $x>0$ half-space $\Lambda$.
Partition function
------------------
Substituting the expansion of the Boltzmann density (\[2.11\]) into the definition (\[3.1\]) of the quantum partition function $Z_{\rm qu}$ and performing expansions of type (\[2.4\]) for the products of boundary factors, we obtain $$\begin{aligned}
\frac{Z_{\rm qu}}{Z} & = & 1 - \frac{\lambda^2}{24} 2\pi \beta e^2 N n
- \int_{\Lambda} {\rm d}{\bf r}_1\, {\rm e}^{-2x_1^2/\lambda^2} n({\bf r}_1)
\nonumber \\ & &
+ \frac{1}{2!} \int_{\Lambda}{\rm d}{\bf r}_1 \int_{\Lambda}{\rm d}{\bf r}_2\,
{\rm e}^{-2x_1^2/\lambda^2} {\rm e}^{-2x_2^2/\lambda^2}
n^{(2)}({\bf r}_1,{\bf r}_2) \nonumber \\ & &
+ \int_{\Lambda} {\rm d}{\bf r}_1\, {\rm e}^{-2x_1^2/\lambda^2}
x_1 \frac{\partial}{\partial x_1} n({\bf r}_1)
+ \frac{\lambda^2}{24} \int_{\Lambda} {\rm d}{\bf r}_1\,
\frac{\partial^2}{\partial x_1^2} n({\bf r}_1) + o(\lambda^2) . \label{3.10}\end{aligned}$$ Here, we keep in mind that the integration of an exponential term $\exp(-2x^2/\lambda^2)$ over $x$ produces one $\lambda$-factor.
One-body density
----------------
To calculate the quantum one-body density (\[3.3\]), we take advantage of the invariance of the Boltzmann density (\[2.11\]) with respect to permutations of the particle indices and write down $$\label{3.11}
n_{\rm qu}({\bf r}) = \frac{1}{Z_{\rm qu}} \frac{N}{N!}
\int_{\Lambda} {\rm d}\vec{\bf r}\, B_{\beta}(\vec{\bf r})
\delta({\bf r}-{\bf r}_1) .$$ In each term of the Boltzmann density, we separate the “reference” ${\bf r}_1$-dependent part, which is kept unchanged, and expand in analogy with (\[2.4\]) the remaining part dependent on $({\bf r}_2,\ldots,{\bf r}_N)$ coordinates. Like for instance, $$\begin{aligned}
\prod_{j=1}^N \left( 1 - {\rm e}^{-2x_j^2/\lambda^2} \right)
& = & \left( 1 - {\rm e}^{-2x_1^2/\lambda^2} \right)
\Bigg\{ 1 - \sum_{j=2}^N {\rm e}^{-2x_j^2/\lambda^2} \nonumber \\
& &
+ \frac{1}{2!} \sum_{j,k=2\atop (j\ne k)}^N {\rm e}^{-2x_j^2/\lambda^2}
{\rm e}^{-2x_k^2/\lambda^2} + \cdots \Bigg\} . \label{3.12}\end{aligned}$$ The quantum partition function $Z_{\rm qu}$ in the denominator on the rhs of (\[3.11\]) is substituted by the expansion (\[3.10\]) and subsequently expanded in “virtual” $\lambda$ powers. After simple but lengthy algebra, one obtains $$\begin{aligned}
n_{\rm qu}({\bf r}_1) & = &
\left( 1 - {\rm e}^{-2x_1^2/\lambda^2} \right) \Bigg\{
n({\bf r}_1) - \int_{\Lambda} {\rm d}{\bf r}_2\, {\rm e}^{-2x_2^2/\lambda^2}
n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) \nonumber \\ & &
+ \frac{1}{2!} \int_{\Lambda} {\rm d}{\bf r}_2 \int_{\Lambda}
{\rm d}{\bf r}_3\, {\rm e}^{-2x_2^2/\lambda^2} {\rm e}^{-2x_3^2/\lambda^2}
n^{(3){\rm T}}({\bf r}_1,{\bf r}_2,{\bf r}_3) \nonumber \\ & &
+ \int_{\Lambda} {\rm d}{\bf r}_2\, {\rm e}^{-2x_2^2/\lambda^2}
x_2 \frac{\partial}{\partial x_2} n^{(2){\rm T}}({\bf r}_1,{\bf r}_2)
\nonumber \\ & &
+ \frac{\lambda^2}{24} \left[
\frac{\partial^2}{\partial x_1^2} n({\bf r}_1)
+ \int_{\Lambda} {\rm d}{\bf r}_2\, \frac{\partial^2}{\partial x_2^2}
n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) \right] \Bigg\}
\nonumber \\ & &
+ {\rm e}^{-2x_1^2/\lambda^2} x_1 \frac{\partial}{\partial x_1}
\left[ n({\bf r}_1) - \int_{\Lambda} {\rm d}{\bf r}_2\,
{\rm e}^{-2x_2^2/\lambda^2}
n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) \right] . \label{3.13}\end{aligned}$$
Let us assume that the classical averages under integrations in (\[3.13\]) are analytic functions of the $x$-coordinate at the boundary $x=0$. For instance, the classical density profile $n({\bf r})\equiv n(x)$ is supposed to exhibit the Taylor expansion $n(x) = n(0) + n'(0)x + n''(0)x^2/2!+\cdots$. Then, the integral $$\begin{aligned}
\int_0^{\infty} {\rm d}x\, {\rm e}^{-2x^2/\lambda^2} n(x) & = &
\lambda \int_0^{\infty} {\rm d}x'\, {\rm e}^{-2x'^2} n(\lambda x')
\nonumber \\ & = &
\lambda \frac{1}{2} \sqrt{\frac{\pi}{2}} n(0) +
\lambda^2 \frac{1}{4} n'(0) + O(\lambda^3) . \label{3.14}\end{aligned}$$ Performing an analogous procedure in all integrals on the rhs of (\[3.13\]), we finally arrive at the expansion of the one-body density up to the $\lambda^2$ order: $$\begin{aligned}
n_{\rm qu}({\bf r}_1) & = &
\left( 1 - {\rm e}^{-2x_1^2/\lambda^2} \right) \Bigg\{
n({\bf r}_1) - \lambda \sqrt{\frac{\pi}{8}} \int {\rm d}y_2\,
n^{(2){\rm T}}[{\bf r}_1,(0,y_2)] \nonumber \\ & &
+ \frac{\lambda^2}{2} \left( \frac{\pi}{8} \right)
\int {\rm d}y_2 \int {\rm d}y_3\,
n^{(3){\rm T}}[{\bf r}_1,(0,y_2),(0,y_3)] \nonumber \\ & &
+ \frac{\lambda^2}{24} \left[ \frac{\partial^2}{\partial x_1^2} n({\bf r}_1)
- \int {\rm d}y_2\,
\frac{\partial n^{(2){\rm T}}[{\bf r}_1,(x_2,y_2)]}{\partial
x_2} \Bigg\vert_{x_2=0} \right] \Bigg\}
\nonumber \\ & &
+ {\rm e}^{-2x_1^2/\lambda^2} x_1 \frac{\partial}{\partial x_1}
\left\{ n({\bf r}_1) - \lambda \sqrt{\frac{\pi}{8}}
\int {\rm d}y_2\, n^{(2){\rm T}}[{\bf r}_1,(0,y_2)]
\right\} . \label{3.15}\end{aligned}$$
Two-body density
----------------
To calculate the $\lambda$-expansion of the quantum truncated two-body density given by relations (\[3.4\]) and (\[3.5\]), we proceed as in the previous subsection. We first take advantage of the permutation invariance of the Boltzmann density (\[2.11\]) to write down $$\label{3.16}
n_{\rm qu}^{(2)}({\bf r},{\bf r}') = \frac{1}{Z_{\rm qu}} \frac{N(N-1)}{N!}
\int_{\Lambda} {\rm d}\vec{\bf r}\, B_{\beta}(\vec{\bf r})
\delta({\bf r}-{\bf r}_1) \delta({\bf r}'-{\bf r}_2) .$$ Then we separate the reference part in the Boltzmann density (\[2.11\]) which depends on coordinates ${\bf r}_1, {\bf r}_2$ and expand in analogy with (\[3.12\]) the remaining part. Finally, using the integration procedure of type (\[3.14\]) in all integrals we arrive at the expansion of the two-body density up to the $\lambda^2$ order: $$\begin{aligned}
\fl n_{\rm qu}^{(2){\rm T}}({\bf r}_1,{\bf r}_2)
& = & \left( 1 - {\rm e}^{-2x_1^2/\lambda^2} \right)
\left( 1 - {\rm e}^{-2x_2^2/\lambda^2} \right) \Bigg\{
n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) \nonumber \\ \fl & &
- \lambda \sqrt{\frac{\pi}{8}} \int {\rm d}y_3\,
n^{(3){\rm T}}[{\bf r}_1,{\bf r}_2,(0,y_3)] \nonumber \\ \fl & &
+ \frac{\lambda^2}{2} \left( \frac{\pi}{8} \right)
\int {\rm d}y_3 \int {\rm d}y_4\,
n^{(4){\rm T}}[{\bf r}_1,{\bf r}_2,(0,y_3),(0,y_4)]
\nonumber \\ \fl & &
+ \frac{\lambda^2}{24} \left[ \left( \frac{\partial^2}{\partial x_1^2} +
\frac{\partial^2}{\partial x_2^2} \right)
n^{(2){\rm T}}({\bf r}_1,{\bf r}_2) - \int {\rm d}y_3\,
\frac{\partial n^{(3){\rm T}}[{\bf r}_1,{\bf r}_2,(x_3,y_3)]}
{\partial x_3} \Bigg\vert_{x_3=0} \right] \Bigg\}
\nonumber \\ \fl & &
+ \left( 1 - {\rm e}^{-2x_2^2/\lambda^2} \right)
x_1 {\rm e}^{-2x_1^2/\lambda^2} \frac{\partial}{\partial x_1}
\Bigg\{ n^{(2){\rm T}}({\bf r}_1,{\bf r}_2)
\nonumber \\ \fl & &
\qquad - \lambda \sqrt{\frac{\pi}{8}} \int {\rm d}y_3\,
n^{(3){\rm T}}[{\bf r}_1,{\bf r}_2,(0,y_3)] \Bigg\}
\nonumber \\ \fl & &
+ \left( 1 - {\rm e}^{-2x_1^2/\lambda^2} \right)
x_2 {\rm e}^{-2x_2^2/\lambda^2} \frac{\partial}{\partial x_2}
\Bigg\{ n^{(2){\rm T}}({\bf r}_1,{\bf r}_2)
\nonumber \\ \fl & &
\qquad - \lambda \sqrt{\frac{\pi}{8}} \int {\rm d}y_3\,
n^{(3){\rm T}}[{\bf r}_1,{\bf r}_2,(0,y_3)] \Bigg\} . \label{3.17}\end{aligned}$$
Review of the sum rules
=======================
The following sum rules \[except the perfect-screening one (\[4.1\])\] apply only to the one-component plasma (not to many-component ones); they rely on the facts that, for the one-component plasma, the mass and charge fluctuations are proportional to one another and the static resistivity vanishes. Although the sum rules can be written for a quantum one-component plasma in a $\nu$-dimensional half-space with any $\nu$, here we consider only the case $\nu=2$. Some of the sum rules that we review were originally written more generally for the time-displaced correlations, but here we consider only their static limits.
The perfect screening sum rule expresses that the charge cloud around a particle of the system has a charge opposite to the charge of this particle. This sum rule has the same form as in the classical case: $$\int\mathrm{d}\mathbf{r}_1
n^{(2)\mathrm{T}}_{\mathrm{qu}}(\mathbf{r}_1,\mathbf{r}_2)=
-n_{\mathrm{qu}}(x_2). \label{4.1}$$ Although we have no doubt about the validity of (\[4.1\]), for a check of the calculations in section 3 and of their application to the present 2D one-component plasma at $\beta e^2=2$, we shall verify (\[4.1\]) at order $\lambda^2$ in section 6.
Macroscopic arguments gave the asymptotic form of the quantum truncated two-body density along the wall [@Janco2] $$\label{4.2}
n^{(2)\mathrm{T}}_{\mathrm{qu}}(\mathbf{r}_1,\mathbf{r}_2)
\mathop{\sim}_{|y_1-y_2|\rightarrow\infty}
\frac{f(x_1,x_2)}{(y_1-y_2)^2},$$ (after perhaps an averaging on local oscillations in $y_1-y_2$) with the sum rule $$\label{4.3}
\fl \int_0^{\infty}\mathrm{d}x_1\int_0^{\infty}\mathrm{d}x_2 f(x_1,x_2)
=-\frac{1}{4\pi^2 e^2}\left[2\hbar\omega_{\rm s}
\coth(\beta\hbar\omega_{\rm s}/2)
-\hbar\omega_{\rm p}\coth(\beta\hbar\omega_{\rm p}/2)\right]$$ (here $\beta e^2=2$), where the bulk and surface plasma frequencies, for two dimensions and a plain hard wall, are, respectively, $$\label{4.4}
\omega_{\rm p}=\left(\frac{2\pi ne^2}{m}\right)^{1/2}, \quad
\omega_{\rm s}=\left(\frac{\pi ne^2}{m}\right)^{1/2}.$$ Any microscopic check of (\[4.3\]) would be welcome. The expansion of the r.h.s. of (\[4.3\]) in powers of $\hbar$ starts with the classical value $-1/(4\pi^2)$ and the next term is of order $\hbar^4$. Therefore, at order $\lambda^2$, in section 7 we shall only be able to check that there is no quantum correction.
Finally, another macroscopic argument gave the quantum form of the dipole sum rule [@JLM] $$\label{4.5}
\fl \int_0^{\infty}\mathrm{d}x_2\left[\int_0^{\infty}\mathrm{d}x_1 x_1
\int\mathrm{d}y_1 n^{(2)\mathrm{T}}_{\mathrm{qu}}(\mathbf{r}_1,\mathbf{r}_2)
+ x_2n_{\mathrm{qu}}(x_2)\right]
=-\frac{\hbar\omega_{\mathrm{p}}}{4\pi e^2}
\coth\frac{\beta\hbar\omega_{\mathrm{p}}}{2}$$ (here $\beta e^2=2$). The quantity between square brackets in (\[4.5\]) is the dipole moment of the structure factor. Here too, a check would be welcome. Now, the expansion of the rhs of (\[4.5\]) in powers of $\hbar$ starts with the classical value $-1/(4\pi)$ and the next term is $-\lambda^2/(12\pi a^2)$, where $a$ is the average interparticle distance defined by $n=1/(\pi a^2)$. In section 8 we can check this sum rule at order $\lambda^2$.
The classical densities
=======================
We shall need some information about the classical densities of the 2D one-component plasma at $\beta e^2=2$. We express all lengths in units of the average interparticle distance $a$, thus $n=1/\pi$. The density profile is [@Janco] $$\label{5.1}
n(x)=n\frac{2}{\sqrt{\pi}}\int_0^{\infty}\mathrm{d}t
\frac{\exp[-(t-x\sqrt{2})^2]}{1+\Phi(t)},$$ where $\Phi(t)$ is the probability-integral function. The two-body truncated density is $$\label{5.2}
n^{(2)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2)=-n^2\exp[-2(x_1^2+x_2^2)]
k(\mathbf{r}_1,\mathbf{r}_2)k(\mathbf{r}_2,\mathbf{r}_1),$$ where $$k(\mathbf{r}_1,\mathbf{r}_2)=\frac{2}{\sqrt{\pi}}\int_0^\infty\mathrm{d}t
\frac{\exp\left[-t^2+t(x_1+x_2)\sqrt{2}-\mathrm{i}t(y_1-y_2)\sqrt{2}\right]}
{1+\Phi(t)}. \label{5.3}$$ Higher-order $n$-body truncated densities contain a sum of product of $n$ factors $k$, the sum running on all oriented cycles built with $\mathbf{r}_1,\mathbf{r}_2,...,\mathbf{r}_n$ [@CJB]. One finds for the three-body truncated density $$\begin{aligned}
n^{(3)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3) & = &
n^3\exp[-2(x_1^2+x_2^2+x_3^2)][k(\mathbf{r}_1,\mathbf{r}_2)
k(\mathbf{r}_2,\mathbf{r}_3)k(\mathbf{r}_3,\mathbf{r}_1) \nonumber \\
& & + k(\mathbf{r}_1,\mathbf{r}_3)k(\mathbf{r}_3,\mathbf{r}_2)
k(\mathbf{r}_2,\mathbf{r}_1)], \label{5.4}\end{aligned}$$ and for the four-body truncated density $$\begin{aligned}
\fl n^{(4)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3,\mathbf{r}_4)
& = & -n^4\exp[-2(x_1^2+x_2^2+x_3^2+x_4^2)][k(\mathbf{r}_1,\mathbf{r}_2)
k(\mathbf{r}_2,\mathbf{r}_3)k(\mathbf{r}_3,\mathbf{r}_4)
k(\mathbf{r}_4,\mathbf{r}_1) \nonumber \\
& & + k(\mathbf{r}_1,\mathbf{r}_2)k(\mathbf{r}_2,\mathbf{r}_4)
k(\mathbf{r}_4,\mathbf{r}_3)k(\mathbf{r}_3,\mathbf{r}_1) \nonumber \\
& & + k(\mathbf{r}_1,\mathbf{r}_3)k(\mathbf{r}_3,\mathbf{r}_2)
k(\mathbf{r}_2,\mathbf{r}_4)k(\mathbf{r}_4,\mathbf{r}_1)] \nonumber \\
& & + \mbox{complex conjugate}. \label{5.5}\end{aligned}$$
We shall also need some integrals of these densities. In (\[5.2\]), the product of the $k$ functions contains $\int_0^{\infty}\mathrm{d}t\exp(-\mathrm{i}ty_1\sqrt{2})
\int_0^{\infty}\mathrm{d}t'\exp(\mathrm{i}t'y_1\sqrt{2})$, therefore the integral of this quantity on $y_1$ is $\pi\sqrt{2}\,\delta(t-t')$. One obtains $$\label{5.6}
\fl \int\mathrm{d}y_1n^{(2)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2) =
- n^2 4\sqrt{2}\exp[-2(x_1^2+x_2^2)] \int_0^{\infty}\mathrm{d}t
\frac{\exp\left[-2t^2+t(x_1+x_2)2\sqrt{2}\right]}{[1+\Phi(t)]^2}.$$ By the same method, one finds $$\begin{aligned}
\fl \int\mathrm{d}y_3n^{(3)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3)
& = & n^3\frac{8\sqrt{2}}{\sqrt{\pi}}\exp[-2(x_1^2+x_2^2+x_3^2)] \nonumber \\
& &\times\int_0^{\infty}\mathrm{d}t
\frac{\exp[-t^2+t(x_1+x_2)\sqrt{2}-\mathrm{i}t(y_1-y_2)\sqrt{2}]}
{1+\Phi(t)} \nonumber \\
& &\times\int_0^{\infty}\mathrm{d}t'
\frac{\exp[-2t'^2+t'(x_1+x_2+2x_3)\sqrt{2}
-\mathrm{i}t'(y_2-y_1)\sqrt{2}]}{[1+\Phi(t')]^2} \nonumber \\
& & + \mbox{complex conjugate}, \label{5.7}\end{aligned}$$ $$\begin{aligned}
\fl \int\mathrm{d}y_1\int\mathrm{d}y_3n^{(3)\mathrm{T}}
(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3) & = &
n^3 32\sqrt{\pi}\exp[-2(x_1^2+x_2^2+x_3^2)] \nonumber \\
& & \times\int_0^{\infty}\mathrm{d}t
\frac{\exp[-3t^2+t(x_1+x_2+x_3)2\sqrt{2}]}{[1+\Phi(t)]^3},
\label{5.8}\end{aligned}$$ $$\begin{aligned}
\fl \int\mathrm{d}y_3\int\mathrm{d}y_4n^{(4)\mathrm{T}}
[\mathbf{r}_1,\mathbf{r}_2,(0,y_3),(0,y_4)] =
- n^4 32\exp[-2(x_1^2+x_2^2)] \nonumber \\
\times\left\{2\int_0^{\infty}\mathrm{d}t
\frac{\exp[-t^2+t(x_1+x_2)\sqrt{2}-\mathrm{i}t(y_1-y_2)\sqrt{2}]}{1+\Phi(t)}
\right. \nonumber \\
\times\int_0^{\infty}\mathrm{d}t'\frac{\exp[-3t'^2+t'(x_1+x_2)\sqrt{2}
-\mathrm{i}t'(y_2-y_1)\sqrt{2}]}{[1+\Phi(t')]^3} \nonumber \\
+\int_0^{\infty}\mathrm{d}t\frac{\exp[-2t^2+t(x_1+x_2)\sqrt{2}
-\mathrm{i}t(y_1-y_2)\sqrt{2}]}{[1+\Phi(t)]^2} \nonumber \\
\left.\times\int_0^{\infty}\mathrm{d}t'\frac{\exp[-2t'^2+t'(x_1+x_2)\sqrt{2}
-\mathrm{i}t'(y_2-y_1)\sqrt{2}]}{[1+\Phi(t')]^2}\right\} \nonumber \\
+ \mbox{complex conjugate}, \label{5.9}\end{aligned}$$ $$\begin{aligned}
\fl \int\mathrm{d}y_1\int\mathrm{d}y_3\int\mathrm{d}y_4n^{(4)\mathrm{T}}
[\mathbf{r}_1,\mathbf{r}_2,(0,y_3),(0,y_4)] =
- n^4 192\pi\sqrt{2}\exp[-2(x_1^2+x_2^2)] \nonumber \\
\times \int_0^{\infty}\mathrm{d}t\frac{\exp[-4t^2+t(x_1+x_2)2\sqrt{2}]}
{[1+\Phi(t)]^4}. \label{5.10}\end{aligned}$$
Perfect screening
=================
Omitting some terms which do not contribute to the sum rules at order $\lambda^2$, and using the results in section 5, we find from (3.17) $$\begin{aligned}
\fl \int\mathrm{d}y_1
n_{\mathrm{qu}}^{(2)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2)
&&=[1-\exp(-2x_1^2/\lambda^2)][1-\exp(-2x_2^2/\lambda^2)] \nonumber \\
&&\times\exp[-2(x_1^2+x_2^2)]\left\{-n^2 4\sqrt{2}\int_0^{\infty}
\mathrm{d}t\frac{\exp[-2t^2+t(x_1+x_2)2\sqrt{2}]}{[1+\Phi(t)]^2}\right.
\nonumber \\
&&-\lambda n^3 8\sqrt{2}\,\pi\int_0^{\infty}\mathrm{d}t
\frac{\exp[-3t^2+t(x_1+x_2)2\sqrt{2}]}{[1+\Phi(t)]^3} \nonumber \\
&&-\lambda^2 n^4 12\sqrt{2}\,\pi^2\int_0^{\infty}\mathrm{d}t
\frac{\exp[-4t^2+t(x_1+x_2)2\sqrt{2}]}{[1+\Phi(t)]^4} \nonumber \\
&&\left.-\lambda^2 n^3\frac{8}{3}\sqrt{2\pi}\int_0^{\infty}\mathrm{d}t\,t
\frac{\exp[-3t^2+t(x_1+x_2)2\sqrt{2}]}{[1+\Phi(t)]^3}\right\} \nonumber \\
&&+\left\{-\lambda^2 n^2\frac{\sqrt{2}}{6}\left(\frac{\partial^2}
{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}\right)
-n^2 4\sqrt{2}\exp(-2x_1^2/\lambda^2)x_1\frac{\partial}{\partial x_1}\right.
\nonumber \\
&&\left.-n^2 4\sqrt{2}\exp(-2x_2^2/\lambda^2)x_2\frac{\partial}{\partial x_2}
\right\} \nonumber \\
&&\times\left\{\exp[-2(x_1^2+x_2^2)]
\int_0^{\infty}\mathrm{d}t\frac{\exp[-2t^2+t(x_1+x_2)2\sqrt{2}]}{[1+\Phi(t)]^2}
\right\}+\cdots\; . \label{6.1}\end{aligned}$$ On the other hand, we find from (3.15) $$\begin{aligned}
\fl n_{\mathrm{qu}}(x_2) & = &
[1-\exp(-2x_2^2/\lambda^2)]\left\{\frac{2}{\sqrt{\pi}} n\exp(-2x_2^2)
\int_0^{\infty}\mathrm{d}t\frac{\exp(-t^2+tx_2 2\sqrt{2})}{1+\Phi(t)}
\right.\nonumber \\
\fl & &+\lambda n^2 2\sqrt{\pi}\exp(-2x_2^2)
\int_0^{\infty}\mathrm{d}t\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2}
\nonumber \\
\fl & &+\lambda^2 n^3 2\pi^{3/2}\exp(-2x_2^2)
\int_0^{\infty}\mathrm{d}t\frac{\exp(-3t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^3}
\nonumber \\
\fl & &+\lambda^2 n\frac{1}{12\sqrt{\pi}}\frac{\partial^2}{\partial x_2^2}
\left[\exp(-2x_2^2)
\int_0^{\infty}\mathrm{d}t\frac{\exp(-t^2+tx_2 2\sqrt{2})}{1+\Phi(t)}\right]
\nonumber \\
\fl & &\left.+\lambda^2 n^2\frac{2}{3}\exp(-2x_2^2)
\int_0^{\infty}\mathrm{d}t\,t\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2}
\right\} \nonumber \\
\fl & &+\exp(-2x_2^2/\lambda^2)x_2\frac{\partial}{\partial x_2}
\left\{n\frac{2}{\sqrt{\pi}}\exp(-2x_2^2) \int_0^{\infty}
\mathrm{d}t\frac{\exp(-t^2+tx_2 2\sqrt{2})}{1+\Phi(t)} \right. \nonumber \\
\fl & &\left.+\lambda n^2 2\sqrt{\pi}\exp(-2x_2^2)\int_0^{\infty}\mathrm{d}t
\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2}\right\}+\cdots\; . \label{6.2}\end{aligned}$$ Using [@GR] $$\int_0^{\infty}\mathrm{d}x\exp(-2x^2+tx2\sqrt{2})=\sqrt{\frac{\pi}{8}}
\exp(t^2)[1+\Phi(t)], \label{6.3}$$ we can compute the integral on $x_1$ of (\[6.1\]) at order $\lambda^2$ and check that it is equal to the opposite of (\[6.2\]).
The sum rule (\[4.1\]) is verified.
The sum rule about an asymptotic form
=====================================
For obtaining the asymptotic forms of the classical $n$-body densities as $|y_1-y_2|\rightarrow\infty$, one uses the integration per partes for the Fourier transform of a function $F(t)$ $$\label{7.1}
\int_0^{\infty}\mathrm{d}tF(t)\exp[-\mathrm{i}ty\sqrt{2}]\sim
\frac{F(0)}{\mathrm{i}y\sqrt{2}}.$$ The different classical $n$-body truncated densities or their integrals which appear in (3.17) are all found to have an asymptotic form $\propto
\exp[-2(x_1^2+x_2^2)]/(y_1-y_2)^2$. This gives the asymptotic form (\[4.2\]), where (some terms which do not contribute to the sum rule at order $\lambda^2$ have not been kept) $$\begin{aligned}
f(x_1,x_2) & = & [1-\exp(-2x_1^2/\lambda^2)][1-\exp(-2x_2^2/\lambda^2)]
\nonumber \\ & & \times \exp[-2(x_1^2+x_2^2)]
\left( -n^2\frac{2}{\pi}-\lambda n^3 4 -\lambda^2 n^4 6\pi \right)
+ \cdots\; . \label{7.2}\end{aligned}$$ Now, we note from equation (\[3.14\]) that, at order $\lambda^2$, an integral of the form $\int_0^{\infty}\mathrm{d}x\exp(-2x^2/\lambda^2)F(x)$ becomes $\lambda\sqrt{\pi/8}F(0)+\lambda^2(1/4)F'(0)$. Then, from (\[7.2\]), taking into account that in our units $n=1/\pi$, $$\begin{aligned}
\int_0^{\infty}\mathrm{d}x_1\int_0^{\infty}\mathrm{d}x_2 f(x_1,x_2)
& = &
-\frac{1}{4\pi^2} +
\lambda \left( \frac{1}{4\pi^2}+\frac{1}{4\pi^2}-\frac{1}{2\pi^2} \right)
\nonumber \\
& & +\lambda^2 \left( -\frac{1}{4\pi^2}+\frac{1}{2\pi^2}+\frac{1}{2\pi^2}
-\frac{3}{4\pi^2} \right) + o(\lambda^2) \nonumber \\
& = & -\frac{1}{4\pi^2}+o(\lambda^2) \label{7.3}\end{aligned}$$ where $\lambda$ is in units of $a$.
At order $\lambda^2$, there are no quantum corrections, in agreement with section 4.
Dipole sum rule
===============
Omitting some terms which do not contribute to the sum rule at order $\lambda^2$, using [@GR] $$\int_0^{\infty}\mathrm{d}x\,x\exp(-2x^2+2\sqrt{2}\,tx)=
\frac{1}{4}+\frac{\sqrt{\pi}}{4}t\exp(t^2)[1+\Phi(t)], \label{8.1}$$ and noting that, at order $\lambda^2$, an integral of the form $\int_0^{\infty}\mathrm{d}x\,x\exp(-2x^2/\lambda^2)F(x)$ becomes $(1/4)\lambda^2 F(0)$, one finds from (\[6.1\]), contributing to the sum rule at order $\lambda^2$, $$\begin{aligned}
\fl \int_0^{\infty}\mathrm{d}x_1\,x_1\int\mathrm{d}y_1
n^{(2)\mathrm{T}}_{\mathrm{qu}}(\mathbf{r}_1,\mathbf{r}_2) =
[1-\exp(-2x_2^2/\lambda^2)]\exp(-2x_2^2) \nonumber \\
\times\left\{-n^2\sqrt{2}\int_0^{\infty}\mathrm{d}t
\left[\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2}
+\sqrt{\pi}\,t\frac{\exp(-t^2+tx_2 2\sqrt{2})}{1+\Phi(t)}\right] \right.
\nonumber \\
+\lambda^2 n^2\sqrt{2}\int_0^{\infty}\mathrm{d}t
\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2} \nonumber \\
-\lambda n^3 2\sqrt{2}\pi
\int_0^{\infty}\mathrm{d}t\left[\frac{\exp(-3t^2+tx_2 2\sqrt{2})}
{[1+\Phi(t)]^3}+\sqrt{\pi}\,t\frac{\exp(-2t^2+tx_2 2\sqrt{2})}
{[1+\Phi(t)]^2}\right] \nonumber \\
-\lambda^2 n^4 3\sqrt{2}\pi^2 \int_0^{\infty}\mathrm{d}t
\left[\frac{\exp(-4t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^4}+
\sqrt{\pi}\,t\frac{\exp(-3t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^3}\right]
\nonumber \\
\left.-\lambda^2 n^3 \frac{2}{3}\sqrt{2\pi}\int_0^{\infty}\mathrm{d}t\,t
\left[\frac{\exp(-3t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^3}+
\sqrt{\pi}\,t\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2}\right]\right\}
\nonumber \\
-\lambda^2 n^2 \frac{\sqrt{2}}{6}\exp(-2x_2^2)\int_0^{\infty}\mathrm{d}t
\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2} \nonumber \\
+\left\{-\lambda^2 n^2\frac{\sqrt{2}}{24}\frac{\partial^2}{\partial x_2^2}
-n^2\sqrt{2}\exp(-2x_2^2/\lambda^2)x_2\frac{\partial}{\partial x_2}\right\}
\nonumber \\
\times\left\{\exp(-2x_2^2)\int_0^{\infty}\mathrm{d}t
\left[\frac{\exp(-2t^2+tx_2 2\sqrt{2})}{[1+\Phi(t)]^2}\right.\right.
\nonumber \\
\left.\left.+\sqrt{\pi}\,t\frac{\exp(-t^2+tx_2 2\sqrt{2})}{1+\Phi(t)}
\right]\right\}+\cdots\; . \label{8.2}\end{aligned}$$ On the other hand, $n_{\mathrm{qu}}(x_2)$ is given by (\[6.2\]).
The classical part of the sum rule comes from the second line of (\[8.2\]) and the first line of (\[6.2\]). Since $\exp(-t^2)/[1+\Phi(t)]^2=-(\sqrt{\pi}/2)(\mathrm{d}/\mathrm{d}t)
[1+\Phi(t)]^{-1}$, an integration per partes gives, with $n=1/\pi$, $$\int_0^{\infty}\mathrm{d}x_1x_1\int\mathrm{d}y_1
n^{(2)\mathrm{T}}(\mathbf{r}_1,\mathbf{r}_2)+x_2 n(x_2)=
-\frac{1}{\sqrt{2}\,\pi^{3/2}}\exp(-2x_2^2). \label{8.3}$$ Integrating on $x_2$ gives the classical result $-1/(4\pi)$.
Using (\[6.3\]) and (\[8.1\]), it is straightforward to integrate on $x_2$ most of the terms of (\[8.2\]) and $x_2$ times (\[6.2\]). The term in (\[8.2\]), line $-3$, of the form $\partial^2 F(x_2)/\partial x_2^2$ requires some care for evaluating its integral on $x_2$, which is $\partial F(x_2)/\partial x_2 \left|_0^{\infty}\right.$. The point is that $F(x_2)$ has a term which is not zero at infinity. Indeed $$\begin{aligned}
\fl \exp(-2x_2^2)\int_0^{\infty}\mathrm{d}t\,t\frac{\exp(-t^2+tx_2 2\sqrt{2})}
{1+\Phi(t)} & = & \int_0^{\infty}\mathrm{d}t\,t\frac{\exp[-(t-x_2\sqrt{2})^2]}
{1+\Phi(t)} \nonumber \\ & &
\mathop{\sim}_{x_2\rightarrow\infty}x_2\sqrt{2}
\int_{-\infty}^{\infty}\mathrm{d}t' \frac{\exp(-t'^2)}{2}
= \sqrt{\pi/2}\,x_2 \label{8.4}\end{aligned}$$ where $t'=t-x_2\sqrt{2}$, and the derivative with respect to $x_2$ of (\[8.4\]) is $\sqrt{\pi/2}$ at infinity. The contribution of this derivative at infinity to the sum rule is found to be $-\lambda^2/(24\pi)$. Similarly, the term in (\[6.2\]), line $-2$, gives a contribution of the form $x_2[\partial^2 G(x_2)/\partial x_2^2]$; when integrated per partes on $x_2$, in particular it gives a term $-G(\infty)$. Since $$\label{8.5}
\fl G(x_2)=\int_0^{\infty}\mathrm{d}t\frac{\exp[-(t-x_2\sqrt{2})^2]}{1+\Phi(t)}
\mathop{\sim}_{x_2\rightarrow\infty}
\int_{-\infty}^{\infty}\mathrm{d}t'\frac{\exp(-t'^2)}{2}=\sqrt{\pi}/2,$$ the corresponding contribution to the sum rule is $-\lambda^2/(24\pi)$. All other quantum contributions to the sum rule are found to cancel each other, at order $\lambda^2$. Finally, $$\label{8.6}
\fl \int_0^{\infty}\mathrm{d}x_2\left[\int_0^{\infty}\mathrm{d}x_1x_1
\int\mathrm{d}y_1 n^{(2)\mathrm{T}}_{\mathrm{qu}}(\mathbf{r}_1,\mathbf{r}_2)
+x_2 n_{\mathrm{qu}}(x_2)\right]=-\frac{1}{4\pi}-\frac{\lambda^2}{12\pi}
+o(\lambda^2) .$$
Since $\lambda$ is in units of $a$, the sum rule (\[4.5\]) does have a quantum correction $-\lambda^2/(12\pi a^2)$ at order $\lambda^2$.
Conclusion
==========
That the perfect screening sum rule (\[4.1\]) is satisfied is no surprise. This is rather a check of our calculations. The other sum rules are less straightforward. In their heuristic macroscopic derivations [@Janco2; @JLM], an essential feature of the quantum systems, that the $n$-body densities have to vanish on the wall, was not explicitly taken into account. Thus a check that they are indeed correct is welcome.
The case of the asymptotic-form sum rule (\[4.3\]) is not entirely satisfactory. Although we have checked it at order $\lambda^2$, at this order we could only verify that there is no quantum correction, in agreement with the expansion of the rhs of (\[4.3\]). It would have been more satisfactory to verify the quantum correction of order $\lambda^4$ of this rhs. Unfortunately, this would involve pushing the Wigner-Kirkwood expansion to that order, which would be straightforward but so tedious that we cannot hope for its feasibility in a reasonable time.
The dipole sum rule (\[4.5\]) is more tractable, and we have indeed checked a finite quantum correction at order $\lambda^2$. Moreover, another derivation of this sum rule is feasible. Indeed, the generalization of the classical Stillinger-Lovett sum rule [@SL] for a bulk quantum one-component plasma has been microscopically done [@MO]; adapting the result to two dimensions gives, with $\mathbf{r}=\mathbf{r}_1-\mathbf{r}_2$, $$\int\mathrm{d}\mathbf{r}\,\mathbf{r}^2 n_{\mathrm{qu,\,bulk}}^{(2)\mathrm{T}}
(\mathbf{r}_1,\mathbf{r}_2)=-\frac{1}{\pi e^2}\hbar\omega_{\mathrm{p}}
\coth\frac{\beta\hbar\omega_{\mathrm{p}}}{2}. \label{9.1}$$ Then, we can use the same kind of argument as the one in pages 965 and 966 of ref. [@JLM] for deriving the dipole sum rule (\[4.5\]).
We gratefully acknowledge the support received from the European Science Foundation (ESF “Methods of Integrable Systems, Geometry, Applied Mathematics”) and from the VEGA grant 2/6071/2006 of the Slovak Grant Agency.
References {#references .unnumbered}
==========
[99]{}
Jancovici B, 1982 [*J. Stat. Phys.*]{} [**28**]{} 43
Cornu F, Jancovici B, and Blum L, 1988 [*J. Stat. Phys.*]{} [**50**]{} 1221
Pines D and Nozières P, 1996 [*The theory of quantum liquids. Vol. I : Normal Fermi liquids*]{} (New York: Benjamin)
Wigner E P, 1932 [*Phys. Rev.*]{} [**40**]{} 749
Kirkwood J G, 1933 [*Phys. Rev.*]{} [**44**]{} 31 Kirkwood J G, 1934 [*Phys. Rev.*]{} [**45**]{} 116
Šamaj L and Jancovici B, 2007 [*J. Stat. Mech.*]{} P02002
Jancovici B, 1985 [*J. Stat. Phys.*]{} [**39**]{} 427
Jancovici B, Lebowitz J L, and Martin Ph A, 1985 [*J. Stat. Phys.*]{} [**41**]{} 941
Gradshteyn I S and Ryzhik I M, 1980 [*Tables of integrals, series, and products*]{} (London: Academic)
Stillinger F and Lovett R, 1968 [*J. Chem. Phys.*]{} [**49**]{} 1991
Martin Ph A and Oguey Ch, 1985 [*J. Phys. A*]{} [**18**]{} 1995
|
---
author:
- Angelo Montanari
- Pietro Sala
bibliography:
- 'biblio.bib'
title: 'Interval-based Synthesis'
---
|
---
abstract: 'Regenerating code is a class of code very suitable for distributed storage systems, which can maintain optimal bandwidth and storage space. Two types of important regenerating code have been constructed: the minimum storage regeneration (MSR) code and the minimum bandwidth regeneration (MBR) code. However, in hostile networks where adversaries can compromise storage nodes, the storage capacity of the network can be significantly affected. In this paper, we propose two optimal constructions of regenerating codes through rate-matching that can combat against this kind of adversaries in hostile networks: 2-layer rate-matched regenerating code and $m$-layer rate-matched regenerating code. For the 2-layer code, we can achieve the optimal storage efficiency for given system requirements. Our comprehensive analysis shows that our code can detect and correct malicious nodes with higher storage efficiency compared to the universally resilient regenerating code which is a straightforward extension of regenerating code with error detection and correction capability. Then we propose the $m$-layer code by extending the 2-layer code and achieve the optimal error correction efficiency by matching the code rate of each layer’s regenerating code. We also demonstrate that the optimized parameter can achieve the maximum storage capacity under the same constraint. Compared to the universally resilient regenerating code, our code can achieve much higher error correction efficiency.'
author:
- 'Jian LiåTongtong LiåJian Ren[^1]'
date: 'November 4, 2015'
title: 'Optimal Construction of Regenerating Code through Rate-matching in Hostile Networks'
---
Optimal regenerating code, MDS code, error-correction, adversary.
Introduction
============
Distributed storage is a popular method to store files securely without requiring data encryption. Instead of storing a file and its replications in multiple servers, we can break the file into components and store the components into multiple servers. In this way, both the reliability and the security of the file can be increased. A typical approach is to encode the file using an $(n,k)$ Reed-Solomon (RS) code and distribute the encoded file into $n$ servers. When we need to recover the file, we only need to collect the encoded parts from $k$ servers, which achieves a trade-off between reliability and efficiency. However, when repairing or regenerating the contents of a failed node, the whole file has to be recovered first, which is a waste of bandwidth.
The concept of regenerating code was introduced in [@Dimakis], where a replacement node is allowed to connect to some individual nodes directly and regenerate a substitute of the failed node, instead of first recovering the original data then regenerating the failed component. Compared to the RS code, regenerating code achieves an optimal tradeoff between bandwidth and storage within the minimum storage regeneration (MSR) and the minimum bandwidth regeneration (MBR) points.
However, when malicious behaviors exist in the network, both the regeneration of the failed node or the reconstruction of the original file will fail. The error resilience of the Reed-Solomen code based regenerating code in the network with errors and erasures was analyzed in [@Rashmi-err]. In our previous work, a Hermitian code based regenerating code was proposed to provide better error correction capability compared to the Reed-Solomen code based approach.
Inspired by the nice performance of Hermitian code based regenerating codes, in this paper we step forward to further construct optimal regenerating codes which have similar layered structure like Hermitian code in distributed storage. The main contributions of this paper are:
- We propose an optimal construction of 2-layer rate-matched regenerating code. Both theoretical analysis and performance evaluation show that this code can achieve storage efficiency higher than the universally resilient regenerating code proposed in [@Rashmi-err].
- We propose an optimal construction of $m$-layer rate-matched regenerating code. The $m$-layer code can achieve higher error correction efficiency than the code proposed in [@Rashmi-err] and the Hermitian code based regenerating code proposed in [@jian14]. Furthermore, the $m$-layered code is easier to understand and has more flexibility than the Hermitian based code.
Here we will focus on error correction and malicious node locating in data regeneration and reconstruction in distributed storage. When no error occurs or no malicious node exists, the data regeneration and reconstruction can be processed the same as the existing works.
It it worth to note that although there are two types of regenerating codes: MSR code and MBR code on the MSR point and MBR point respectively, in this paper we will only focus on the optimization of the MSR code for the following two reasons:
1. The processes and results of the optimization for these two codes are similar. The optimization for the MSR code can be directly applied to the MBR code with similar optimization results.
2. The differences between the constructions of MSR code and MBR code have little impact on the optimization proposed in this paper.
The rest of this paper is organized as follows: in Section \[Sec:related\] we introduce the related work. In Section \[Sec:Preliminary\], the preliminary of this paper is presented. In Section \[Sec:rate-matched-MSR\], we propose two component codes for the rate-matched regenerating codes. We propose and analyze the 2-layer rate-matched regenerating code in Section \[sec:2-layer\]. Then we propose and analyze the $m$-layer rate-matched regenerating code in Section \[sec:m\_layer\_msr\]. The paper is concluded in Section \[Sec:Conclusion\].
Related Work {#Sec:related}
============
When a storage node in the distributed storage network that employing the conventional $(n,k)$ RS code (such as OceanStore [@Ocean] and Total Recall [@Total]) fails, the replacement node connects to $k$ nodes and downloads the whole file to recover the symbols stored in the failed node. This approach is a waste of bandwidth because the whole file has to be downloaded to recover a fraction of it. To overcome this drawback, Dimakis *et al*. [@Dimakis] introduced the conception of $\{n, k, d, \alpha, \beta, B\}$ regenerating code based on the network coding. In the context of regenerating code, the contents stored in a failed node can be regenerated by the replacement node through downloading $\gamma$ help symbols from $d$ helper nodes. The bandwidth consumption for the failed node regeneration could be far less than the whole file. A data collector (DC) can reconstruct the original file stored in the network by downloading $\alpha$ symbols from each of the $k$ storage nodes. In [@Dimakis], the authors proved that there is a tradeoff between bandwidth $\gamma$ and per node storage $\alpha$. They found two optimal points: minimum storage regeneration (MSR) and minimum bandwidth regeneration (MBR) points. Currently there are many literatures focusing on the optimal regenerating codes design: [@Daniel09searchingformsr; @Shah10explicitmbr; @Changho10exactmdsia; @Yunnan11constructionsysmdsmbr; @Papailiopoulos12simrc; @El10fracrepetitioninds; @Tamo11mdsarraycodes; @Viveck11optimalrepairviaia; @Papailiopoulos13repairhadamard; @Shah10flexiblercfords; @Shum11existmbrcorc; @Anyu13exactcorcmbr]. In [@Hou13basicrcbinary; @Yuliang09rcbasedp2p] the implementation of the regenerating code were studied.
The regenerating code can be divided into functional regeneration and exact regeneration. In the functional regeneration, the replacement node regenerates a new component that can functionally replace the failed component instead of being the same as the original stored component. [@Ywu] formulated the data regeneration as a multicast network coding problem and constructed functional regenerating codes. [@Duminuco] implemented a random linear regenerating codes for distributed storage systems. [@Shum] proved that by allowing data exchange among the replacement nodes, a better tradeoff between repair bandwidth $\gamma$ and per node storage $\alpha$ can be achieved. In the exact regeneration, the replacement node regenerates the exact symbols of a failed node. [@Ywu2] proposed to reduce the regeneration bandwidth through algebraic alignment. [@Shah] provided a code structure for exact regeneration using interference alignment technique. [@Rashmi] presented optimal exact constructions of MBR codes and MSR codes under product-matrix framework. This is the first work that allows independent selection of the nodes number $n$ in the network.
None of these works above considered code regeneration under node corruption or adversarial manipulation attacks in hostile networks. In fact, all these schemes will fail in both regeneration and reconstruction if there are nodes in the storage cloud sending out incorrect responses to the regeneration and reconstruction requests.
In [@Oggier11byzanfaulttolofrc], the Byzantine fault tolerance of regenerating codes were studied. In [@Pawar], the authors discussed the amount of information that can be safely stored against passive eavesdropping and active adversarial attacks based on the regeneration structure. In [@Han], the authors proposed to add CRC codes in the regenerating code to check the integrity of the data in hostile networks. Unfortunately, the CRC checks can also be manipulated by the malicious nodes, resulting in the failure of the regeneration and reconstruction. In [@Chen12dataintegrityprotection], the authors proposed to add data integrity protection in distributed storage. In [@Cachin06optimalresforerasurecoded], the authors proposed an erasure-coded distributed storage based on threshold cryptography. In [@Abd05lazyver], the authors analyzed the verification cost for both the client read and write operation in workloads with idle periods. In [@Rashmi-err], the authors analyzed the error resilience of the RS code based regenerating code in the network with errors and erasures. They provided the theoretical error correction capability. In [@jian14] the authors proposed a Hermitian code based regenerating code, which could provide better error correction capability. In [@Rashimi-universal] the authors proposed the universally secure regenerating code to achieve information theoretic data confidentiality. But the extra computational cost and bandwidth have to be considered for this code. In [@jian14glo] the authors proposed to apply linear feedback shift register (LFSR) to protect the data confidentiality.
Preliminary and Assumptions {#Sec:Preliminary}
===========================
\[Sec:Preliminary\]
Regenerating Code
-----------------
Regenerating code introduced in [@Dimakis] is a linear code over finite filed $\mathbb{F}_q$ with a set of parameters $\{n, k, d, \alpha, \beta, B\}$. A file of size $B$ is stored in $n$ storage nodes, each of which stores $\alpha$ symbols. A replacement node can regenerate the contents of a failed node by downloading $\beta$ symbols from each of $d$ randomly selected storage nodes. So the total bandwidth needed to regenerate a failed node is $\gamma = d\beta$. The data collector (DC) can reconstruct the whole file by downloading $\alpha$ symbols from each of $k\leq d$ randomly selected storage nodes. In [@Dimakis], the following theoretical bound was derived: $$\label{eq:min_cut}
B \leq \sum_{i=0}^{k-1}\min \{ \alpha, (d-i)\beta \}.$$ From equation (\[eq:min\_cut\]), a trade-off between the regeneration bandwidth $\gamma$ and the storage requirement $\alpha$ was derived. $\gamma$ and $\alpha$ cannot be decreased at the same time. There are two special cases: minimum storage regeneration (MSR) point in which the storage parameter $\alpha$ is minimized; $$\label{eq:MSR_tradeoff}
(\alpha_{MSR},\gamma_{MSR})= \left(\frac Bk, \frac{Bd}{k(d-k+1)}\right),$$ and minimum bandwidth regeneration (MBR) point in which the bandwidth $\gamma$ is minimized: $$\label{eq:MBR_tradeoff}
(\alpha_{MBR},\gamma_{MBR})= \left(\frac{2Bd}{2kd-k^2 + k},\frac{2Bd}{2kd-k^2 + k} \right).$$
System Assumptions and Adversarial Model
----------------------------------------
In this paper, we assume there is a secure server that is responsible for encoding and distributing the data to storage nodes. Replacement nodes will also be initialized by the secure server. DC and the secure server can be implemented in the same computer and can never be compromised. We use the notation $\CH$/$\CL$ to refer to either the full rate/fractional rate MSR code or a codeword of the full rate/fractional rate MSR code. The exact meaning can be discriminated clearly according to the context.
We assume some network nodes may be corrupted due to hardware failure or communication errors, and/or be compromised by malicious users. As a result, upon request, these nodes may send out incorrect responses to disrupt the data regeneration and reconstruction. The adversary model is the same as [@Rashmi-err], We assume that the malicious users can take full control of $\tau$ ($\tau \leq n$ and corresponds to $s$ in [@Rashmi-err]) storage nodes and collude to perform attacks.
We will refer these symbols as *bogus* symbols without making distinction between the corrupted symbols and compromised symbols. We will also use corrupted nodes, malicious nodes and compromised nodes interchangeably without making any distinction.
Component Codes of Rate-matched Regenerating Code {#Sec:rate-matched-MSR}
=================================================
In this section, we will introduce two different component codes for rate-matched MSR code on the MSR point with $d = 2k-2$. The code based on the MSR point with $d > 2k -2$ can be derived the same way through truncating operations. In the rate-matched MSR code, there are two types of MSR codes with different code rates: full rate code and fractional rate code.
Full Rate Code
--------------
### Encoding
The full rate code is encoded based on the product-matrix code framework in [@Rashmi]. According to equation (\[eq:MSR\_tradeoff\]), we have $\alpha_H = d/2$, $\beta_H = 1$ for one block of data with the size $B_H=(\alpha+1)\alpha$. The data will be arranged into two $\alpha \times \alpha$ symmetric matrices $S_1,S_2$, each of which will contain $B_H/2$ data. The codeword $\CH$ is defined as $$\label{eq:encoding_msr_h}
\CH = [\Phi \: \:\: \Lambda\Phi]
\begin{bmatrix}
S_1 \\
S_2
\end{bmatrix}
= \Psi M_H=\begin{bmatrix}\ch_1\\ \vdots\\ \ch_n\end{bmatrix},$$ where $$\label{eq:phi}
\Phi =
\begin{bmatrix}
1 & 1 & 1 & \dots & 1 \\
1 & g & g^2 & \dots & g^{\alpha-1} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & g^{n-1}& (g^{n-1})^2 & \dots & (g^{n-1})^{\alpha-1}
\end{bmatrix}$$ is a Vandermonde matrix and $\Lambda=\mbox{diag}[\lambda_1,\lambda_2,\cdots,\lambda_n]$ such that $\lambda_i\in \mathbb{F}_q$ and $\lambda_i\ne\lambda_j$ for $1\leq i, j\leq n, i\ne j$, $g$ is a primitive element in $\mathbb{F}_q$, and any $d$ rows of $\Psi$ are linearly independent. Then each row $\ch_i=\boldsymbol{\psi}_iM_H$ ($0 \leq i < n$) of the codeword matrix $\CH$ will be stored in storage node $i$, where the encoding vector $\boldsymbol{\psi}_i$ is the $i^{th}$ row of $\Psi$.
### Regeneration {#Sec:Reg_high_rate}
Suppose node $z$ fails, the replacement node $z'$ will send regeneration requests to the rest of $n-1$ helper nodes. Upon receiving the regeneration request, helper node $i$ will calculate and send out the help symbol $p_i = \ch_i \boldsymbol{\phi}_z^T=\boldsymbol{\psi}_iM_H\boldsymbol{\phi}_z^T$, where $\boldsymbol{\phi}_z$ is the $z^{th}$ row of $\Phi$. $z'$ will perform Algorithm \[alg:reg\_h\] to regenerate the contents of the failed node $z$. For convenience, we define $\Psi_{i\to j}=\begin{bmatrix}
\boldsymbol{\psi}_i^T,
\boldsymbol{\psi}_{i+1}^T
\cdots,
\boldsymbol{\psi}_j^T
\end{bmatrix}^T,
$ where $\boldsymbol{\psi}_t$ is the $t^{th}$ row of $\Psi$ $(i\leq t\leq j)$ and $\mathbf{x}^{(j)}$ is the vector containing the first $j$ symbols of $M_H \boldsymbol{\phi}_z^T$.
Suppose $p'_i = p_i + e_i$ is the response from the $i^{th}$ helper node. If $p_i$ has been modified by the malicious node $i$, we have $e_i \in{\mathbb{F}_q}\backslash \{0\}$. We can successfully regenerate the symbols in node $z$ when the number of errors in the received help symbols ${p_i}'$ from $n-1$ helper nodes is less than $\lfloor (n-d-1)/2\rfloor$, where $\left\lfloor \cdot\right\rfloor$ is the floor operation. Without loss of generality, we assume $0 \leq i \leq n-2$.
$z'$ regenerates symbols of the failed node $z$ \[alg:reg\_h\]
[**Step n:**]{}
Decode $\mathbf{p}'$ to $\mathbf{p}_{cw}$, where $\mathbf{{p}}'=[p'_0, p'_1, \cdots, p'_{n-2}]^T$ can be viewed as an MDS code with parameters $(n-1,d,n-d)$ since $\Psi_{0\to (n-2)} \cdot \mathbf{{x}}^{(n-1)} = \mathbf{p}'$.
Solve $\Psi_{0\to (n-2)} \cdot \mathbf{{x}}^{(n-1)} = \mathbf{p}_{cw}$ and compute $\ch_z=\boldsymbol{\phi}_z S_1 + \lambda_z \boldsymbol{\phi}_z S_2$ as described in [@Rashmi].
For regeneration, the full rate code can correct errors from $\left\lfloor (n-d-1)/2 \right\rfloor$ malicious nodes, where $\left\lfloor \cdot\right\rfloor$ is the floor operation.
### Reconstruction {#Sec:Rec_high_rate}
When the DC needs to reconstruct the original file, it will send reconstruction requests to $n$ storage nodes. Upon receiving the request, node $i$ will send out the symbol vector $\mathbf{c}_i$ to the DC. Suppose $\mathbf{c}'_i= \mathbf{c}_i + \mathbf{e}_i$ is the response from the $i^{th}$ storage node. If $\mathbf{c}_i$ has been modified by the malicious node $i$, we have $\mathbf{e}_i \in \mathbb{F}_q^{\alpha} \backslash \{\mathbf{0}\}$.
The DC will reconstruct the file as follows: Let $R' = [{\ch'_{0}}^T, {\ch'_{1}}^T, \cdots, {\ch'_{n-1}}^T]^T$, we have $$\Psi
\begin{bmatrix}
S'_1\\
S'_2
\end{bmatrix}
=[\Phi \: \:\: \Lambda\Phi]
\begin{bmatrix}
S'_1\\
S'_2
\end{bmatrix}
=R',$$ $$\label{eq:rec_eqn}
\Phi S'_1 \Phi^T + \Lambda \Phi S'_2 \Phi^T = {R}'\Phi^T.$$
Let $C=\Phi S'_1 \Phi^T$, $D=\Phi S'_2 \Phi^T$, and ${\widehat{R}}'={R}'\Phi^T$, then $$C + \Lambda D = {\widehat{R}}'.$$ Since $C,D$ are both symmetric, we can solve the non-diagonal elements of $C,D$ as follows: $$\label{eq:recon_recovery}
\left\{\begin{matrix}
C_{i,j} + \lambda_i \cdot D_{i,j}= {\widehat{R}}'_{i,j}\\
C_{i,j} + \lambda_j \cdot D_{i,j}= {\widehat{R}}'_{j,i}.
\end{matrix}\right.$$ Because matrices $C$ and $D$ have the same structure, here we only focus on $C$ (corresponding to $S'_1$). It is straightforward to see that if node $i$ is malicious and there are errors in the $i^{th}$ row of $R'$, there will be errors in the $i^{th}$ row of ${\widehat{R}}'$. Furthermore, there will be errors in the $i^{th}$ row and $i^{th}$ column of $C$. Define $S'_1 \Phi^T=\widehat{S}'_1$, we have $\Phi \widehat{S}'_1 = C$. Here we can view each column of $C$ as an $(n-1, \alpha, n - \alpha)$ MDS code because $\Phi$ is a Vandermonde matrix. The length of the code is $n - 1$ since the diagonal elements of $C$ is unknown. Suppose node $j$ is a legitimate node, we can decode the MDS code to recover the $j^{th}$ column of $C$ and locate the malicious nodes. Eventually $C$ can be recovered. So the DC can reconstructs $S_1$ using the method similar to [@Rashmi; @jian14], For $S_2$, the recovering process is similar.
For reconstruction, the full rate code can correct errors from $\left\lfloor (n-\alpha-1) /2 \right\rfloor$ malicious nodes.
Fractional Rate Code
--------------------
### Encoding
For the fractional rate code, we also have $\alpha_L = d/2$, $\beta_L = 1$ for one block of data with the size $$\label{eq:BL}
B_L\!=\!\left\{
\begin{matrix}
&xd(1+xd)/2, &x \in (0,0.5]\\
&\!\alpha(\alpha \! \!+ \!\!1)/2\! +\!\! (x \!-\!0.5)d(1\!\!+\!\!(x\!-\!0.5)d)/2, &x \in\!\! (0.5,\!1] \end{matrix}\right.,$$ where $x$ is the match factor of the rate-matched MSR code. It is easy to see that the fractional rate code will become the full rate code with $x=1$. The data ${\mathbf{m}} = [m_1,m_2,\dots,m_{B_L}] \in(\mathbb{F}_q)^{B_L}$ will be processed as follows:
When $x \leq 0.5$, the data will be arranged into a symmetric matrix $S_1$ of the size $\alpha \times \alpha$: $$S_1 = \begin{bmatrix}
m_1 & m_2 & \dots & m_{xd} &0 &\dots &0\\
m_2 & m_{xd+1} & \dots & m_{2xd-1} &0 &\dots &0\\
\vdots & \vdots & \ddots & \vdots &\vdots & \ddots & \vdots\\
m_{xd} & m_{2xd-1} & \dots & m_{B_L/2}&0 &\dots &0\\
0 & 0 &\dots & 0 & 0 &\dots & 0\\
\vdots & \vdots & \ddots & \vdots &\vdots & \ddots & \vdots\\
0 & 0 &\dots & 0 & 0 &\dots & 0
\end{bmatrix}.$$ The codeword $\CL$ is defined as $$\label{eq:encoding_msr_l}
\CL = [\Phi \: \:\: \Lambda\Phi]
\begin{bmatrix}
S_1 \\
\mathbf{0}
\end{bmatrix}
= \Psi M_L,$$ where $\mathbf{0}$ is the $\alpha \times \alpha$ zero matrix and $\Phi, \Lambda, \Psi$ are the same as the full rate code.
When $x>0.5$, the first $\alpha(\alpha+1)/2$ data will be arranged into an $\alpha \times \alpha$ symmetric matrix $S_1$. The rest of the data $m_{\alpha(\alpha+1)/2 +1}, \dots, m_{B_L}$ will be arranged into another $\alpha \times \alpha$ symmetric matrix $S_2$: $$S_2 = \begin{bmatrix}
m_{\alpha(\alpha+1)/2 +1} & \dots & m_{\alpha(\alpha+1)/2 +xd} &0 &\dots &0\\
m_{\alpha(\alpha+1)/2 +2} & \dots & m_{\alpha(\alpha+1)/2 +2xd-1} &0 &\dots &0\\
\vdots & \ddots & \vdots &\vdots & \ddots & \vdots\\
m_{\alpha(\alpha+1)/2 +xd} & \dots & m_{B_L/2}&0 &\dots &0\\
0 &\dots & 0 & 0 &\dots & 0\\
\vdots & \ddots & \vdots &\vdots & \ddots & \vdots\\
0 & \dots & 0 & 0 &\dots & 0
\end{bmatrix}.$$ The codeword $\CL$ is defined the same as equation (\[eq:encoding\_msr\_h\]) with the same parameters $\Phi, \Lambda$ and $\Psi$.
Then each row $\cl_i$ ($0 \leq i < n$) of the codeword matrix $\CL$ will be stored in storage node $i$ respectively, in which the encoding vector $\boldsymbol{\psi}_i$ is the $i^{th}$ row of $\Psi$.
The fractional rate code can achieve the MSR point in equation (\[eq:MSR\_tradeoff\]) since it it encoded under the product-matrix MSR code framework in [@Rashmi].
### Regeneration {#sec:reg:frac}
The regeneration for the fractional rate code is the same as the regeneration for the full rate code described in Section \[Sec:Reg\_high\_rate\] with only a minor difference. If we define $\mathbf{x}^{(j)}$ as the vector containing the first $j$ symbols of $M_L \boldsymbol{\phi}_z^T$, there will be only $xd$ nonzero elements in the vector. According to $\boldsymbol{\Psi}_{0\to {n-2}} \cdot \mathbf{{x}}^{(n-1)} = \mathbf{p}'$, the received symbol vector $\mathbf{p}'$ for the fractional rate code in [**[Step 1]{}**]{} of Algorithm \[alg:reg\_h\] can be viewed as an $(n-1,xd,n-xd)$ MDS code. Since $x<1$, we can detect and correct more errors in data regeneration using the fractional rate code than using the full rate code.
For regeneration, the fractional rate code can correct errors from $\left\lfloor (n-xd-1) /2 \right\rfloor$ malicious nodes.
### Reconstruction {#reconstruction}
The reconstruction for the fractional rate code is similar to that for the full rate code described in Section \[Sec:Rec\_high\_rate\]. Let $R' = [{\cl'}_0^T, {\cl'}_1^T, \cdots, {\cl'}_{n-1}^T]^T$.
When the match factor $x>0.5$, reconstruction for the fractional rate code is the same to that for the full rate code.
When $x \le 0.5$, equation (\[eq:rec\_eqn\]) can be written as: $$\label{eq:rec_eqn_low}
\Phi S'_1 = R'.$$ So we can view each column of $R'$ as an $(n,xd,n-xd+1)$ MDS code. After decoding $R'$ to $R_{cw}$, we can recover the data matrix $S_1$ by solving the equation $\Phi S_1 = R_{cw}.$ Meanwhile, if the $i^{th}$ rows of $R'$ and $R_{cw}$ are different, we can mark node $i$ as corrupted.
For reconstruction, when the match factor $x>0.5$, the fractional rate code can correct errors from $\left\lfloor (n-\alpha-1) /2 \right\rfloor$ malicious nodes. When the match factor $x \leq 0.5$, the fractional rate code can correct errors from $\left\lfloor (n-xd) /2 \right\rfloor$ malicious nodes.
2-Layer Rate-matched regenerating Code {#sec:2-layer}
======================================
In this section, we will show our first optimization of the rate-matched MSR code: 2-layer rate-matched MSR code. In the code design, we utilize two layers of the MSR code: the fractional rate code for one layer and the full rate code for the other. The purpose of the fractional rate code is to correct the erroneous symbols sent by malicious nodes and locate the corresponding malicious nodes. Then we can treat the errors in the received symbols as erasures when regenerating with the full rate code. However, the rates of the two codes must match to achieve an optimal performance. Here we mainly focus on the rate-matching for data regeneration. We can see in the later analysis that the performance of data reconstruction can also be improved with this design criterion.
We will first fix the error correction capabilities of the full rate code and the fractional rate code. Then we will derive the optimal rate matching criteria to optimize the data storage efficiency under the fixed error correction capability.
Rate Matching
-------------
From the analysis above, we know that during regeneration, the fractional rate code can correct up to $\left \lfloor (n-xd-1)/2 \right \rfloor$ errors, which are more than $\left \lfloor (n-d-1)/2 \right \rfloor$ errors that the full rate code can correct. In the 2-layer rate-matched MSR code design, our goal is to match the fractional rate code with the full rate code. The main task for the fractional rate code is to detect and correct errors, while the main task for the full rate code is to maintain the storage efficiency. So if the fractional rate code can locate all the malicious nodes, the full rate code can simply treat the symbols received from these malicious nodes as erasures, which requires the minimum redundancy for the full rate code. The full rate code can correct up to $n-d-1$ erasures. Thus we have the following optimal rate-matching equation: $$\label{eq:rate-match}
\left \lfloor (n-xd-1)/2 \right \rfloor = n-d-1,$$ from which we can derive the match factor $x$.
Encoding
--------
To encode a file with size $B_F$ using the 2-layer rate-matched MSR code, the file will first be divided into $\theta_H$ blocks of data with the size $B_H$ and $\theta_L$ blocks of data with the size $B_L$, where the parameters should satisfy $$\label{eq:size_equation}
B_F = \theta_H B_H +\theta_L B_L.$$ Then the $\theta_H$ blocks of data will be encoded into code matrices $\CH_1,\dots,\CH_{\theta_H}$ using the full rate code and the $\theta_L$ blocks of data will be encoded into code matrices $\CL_1,\dots,\CL_{\theta_L}$ using the fractional rate code. To prevent the malicious nodes from corrupting the fractional rate code only, the secure server will randomly concatenate all the matrices together to form the final $n \times \alpha(\theta_H + \theta_L)$ codeword matrix: $$\label{eq:final_codeword_matrix}
\mathsf{CM} = [\mbox{Perm}(\CH_1,\dots,\CH_{\theta_H},\CL_1,\dots,\CL_{\theta_L})],$$ where $\mbox{Perm}(\cdot)$ is the random matrices permutation operation. The secure sever will also record the order of the permutation for future code regeneration and reconstruction. Then each row $\mathbf{c}_i = [\mbox{Perm}(\ch_{1,i}, \dots, \ch_{\theta_H,i},\cl_{1,i}, \dots, \cl_{\theta_L,i})]$ ($0 \leq i \leq n-1$) of the codeword matrix $\mathsf{CM}$ will be stored in storage node $i$, where $\ch_{j,i}$ is the $i^{th}$ row of $\CH_j$ ($1 \leq j \leq \theta_H$), and $\cl_{j,i}$ is the $i^{th}$ row of $\CL_j$ ($1 \leq j \leq \theta_L$). The encoding vector $\boldsymbol{\psi}_i$ for storage node $i$ is the $i^{th}$ row of $\Psi$ in equation (\[eq:encoding\_msr\_h\]). Therefore, we have the following Theorem.
The encoding of 2-layer rate-matched MSR code can achieve the MSR point in equation (\[eq:MSR\_tradeoff\]) since both the full rate code and the fractional code are MSR codes.
Regeneration {#regeneration}
------------
Suppose node $z$ fails, the security server will initialize a replacement node $z'$ with the order information of the fractional rate code and the full rate code in the 2-layer rate-matched MSR code. Then the replacement node $z'$ will send regeneration requests to the rest of $n-1$ helper nodes. Upon receiving the regeneration request, helper node $i$ will calculate and send out the help symbol $p_i = {\bf{c}}_i \boldsymbol{\phi}_z^T$. $z'$ will perform Algorithm \[alg:reg\_rate\_matched\] to regenerate the contents of the failed node $z$. After the regeneration is finished, $z'$ will erase the order information. So even if $z'$ was compromised later, the adversary would not get the permutation order of the fractional rate code and the full rate code.
$z'$ regenerates symbols of the failed node $z$ for the 2-layer rate-matched MSR code \[alg:reg\_rate\_matched\]
[**Step n:**]{}
According to the order information, regenerate all the symbols related to the $\theta_L$ data blocks encoded by the fractional rate code, using Algorithm \[alg:reg\_h\]. If errors are detected in the symbols sent by node $i$, it will be marked as a malicious node.
Regenerate all the symbols related to the $\theta_H$ data blocks encoded by the full rate code, using Algorithm \[alg:reg\_h\]. During the regeneration, all the symbols sent from nodes marked as malicious nodes will be replaced by erasures $\bigotimes$.
It is easy to see that Algorithm \[alg:reg\_rate\_matched\] can correct errors and locate malicious node using the fractional rate code while achieve high storage efficiency using the full rate code. We summarize the result as the following Theorem.
For regeneration, the 2-layer rate-matched MSR code can correct errors from $\left\lfloor (n-xd-1) /2 \right\rfloor$ malicious nodes.
Parameters Optimization {#sec:para_opt_reg}
-----------------------
We have the following design requirements for a given distributed storage system applying the 2-layer rate-matched MSR code:
- The maximum number of malicious nodes $M$ that the system can detect and locate using the fractional rate code. We have $$\label{eq:error_correction}
\left \lfloor (n-xd-1)/2\right \rfloor = M.$$
- The probability $P_{\det}$ that the system can detect all the malicious nodes. The detection will be successful if each malicious node modifies at least one help symbol corresponding to the fractional rate code and sends it to the replacement node. Suppose the malicious nodes modify each help symbol to be sent to the replacement node with probability $P$, we have $$\label{eq:det_suc}
(1-(1-P)^{\theta_L})^M \ge P_{\det}.$$
So there is a trade-off between $\theta_L$ and $\theta_H$: the number of data blocks encoded by the fractional rate code and the number of data blocks encoded by the full rate code. If we encode using too much full rate code, we may not meet the detection probability $P_{\det}$ requirement. If too much fractional rate code is used, the redundancy may be too high.
The storage efficiency is defined as the ratio between the actual size of data to be stored and the total storage space needed by the encoded data: $$\label{eq:efficiency}
\delta_S = \frac{\theta_H B_H + \theta_L B_L}{(\theta_H + \theta_L)n\alpha} = \frac{B_F}{(\theta_H + \theta_L)n\alpha}.$$ Thus we can calculate the optimized parameters $x$, $d$, $\theta_H$, $\theta_L$ by maximizing equation (\[eq:efficiency\]) under the constraints defined by equations (\[eq:rate-match\]), (\[eq:size\_equation\]), (\[eq:error\_correction\]), (\[eq:det\_suc\]).
$d$ and $x$ can be determined by equation (\[eq:rate-match\]) and (\[eq:error\_correction\]): $$\begin{aligned}
d &=& n - M - 1,\\
x &=& (n - 2M -1) / (n - M -1).\end{aligned}$$ Since $B_F$ is constant, to maximize $\delta_S$ is equal to minimize $\theta_H + \theta_L$. So we can rewrite the optimization problem as follows: $$\mbox{Minimize} \:\:\: \theta_H + \theta_L,\:\:\: \mbox{subject to}~(\ref{eq:size_equation})\:\: \mbox{and} \:\:(\ref{eq:det_suc}).$$ This is a simple linear programming problem. Here we will show the optimization results directly: $$\begin{aligned}
\theta_L &=& \log_{(1-P)}(1-P_{\det}^{1/M}),\\
\theta_H &=& (B_F - \theta_L B_L) / B_H.\end{aligned}$$ In this paper we assume that we are storing large files, which means $B_F > \theta_L B_L $. So an optimal solution for the 2-layer rate-matched MSR code can always be found. We have the following theorem:
When the number of blocks of the fractional rate code $\theta_L$ equals to $\log_{(1-P)}(1-P_{\det}^{1/M})$ and the number of blocks of the full rate code $\theta_H$ equals to $(B_F - \theta_L B_L) / B_H$, the 2-layer rate-matched MSR code can achieve the optimal storage efficiency.
Reconstruction {#reconstruction-1}
--------------
When DC needs to reconstruct the original file, it will send reconstruction requests to $n$ storage nodes. Upon receiving the request, node $i$ will send out the symbol vector $\mathbf{c}_i$. Suppose $\mathbf{c}'_i= \mathbf{c}_i + \mathbf{e}_i$ is the response from the $i^{th}$ storage node. If $\mathbf{c}_i$ has been modified by the malicious node $i$, we have $\mathbf{e}_i \in \mathbb{F}_q^{\alpha(\theta_L + \theta_H)} \backslash \{\mathbf{0}\}$. Since DC has the permutation information of the fractional rate code and the full rate code, similar to the regeneration of the 2-layer rate-matched MSR code, DC will perform the reconstruction using Algorithm \[alg:rec\_rate\_matched\].
DC reconstructs the original file for the 2-layer rate-matched MSR code \[alg:rec\_rate\_matched\]
[**Step n:**]{}
According to the order information, reconstruct each of the $\theta_L$ data blocks encoded by the fractional rate code and locate the malicious nodes.
Reconstruct each of the data blocks encoded by the full rate code. During the reconstruction, all the symbols sent from malicious nodes will be replaced by erasures $\bigotimes$.
In Section \[sec:para\_opt\_reg\], we optimized the parameters for the data regeneration, considering the trade-off between the successful malicious node detection probability and the storage efficiency. For data reconstruction, we have the following theorem:
When the number of blocks of the fractional rate code $\theta_L$ equals to $log_{(1-P)}(1-P_{\det}^{1/M})$ and the number of blocks of the full rate code $\theta_H$ equals to $(B_F - \theta_L B_L) / B_H$, the 2-layer rate-matched MSR code can guarantee that the same constraints for data regeneration (equation (\[eq:error\_correction\]), (\[eq:det\_suc\]) ) be satisfied for the data reconstruction.
The maximum number of malicious nodes can be detected for the data reconstruction is no smaller than $M$: if $x>0.5$, the number is $\left \lfloor (n - \alpha - 1) /2 \right \rfloor$. We have $\left \lfloor (n - \alpha - 1) /2 \right \rfloor \ge \left \lfloor (n - xd - 1) /2 \right \rfloor = M$. If $x \le 0.5$, the number is $\left \lfloor (n -xd) /2 \right \rfloor$. We have $\left \lfloor (n -xd) /2 \right \rfloor \ge \left \lfloor (n - xd - 1) /2 \right \rfloor = M$.
The successful malicious node detection probability for the data reconstruction is larger than $P_{\det}$: the probability is $(1-(1-P)^{\alpha\theta_L})^M $, so we have $(1-(1-P)^{\alpha\theta_L})^M > (1-(1-P)^{\theta_L})^M \ge P_{\det}$.
Although the rate-matching equation (\[eq:rate-match\]) does not apply to the data reconstruction, the reconstruction strategy in Algorithm \[alg:rec\_rate\_matched\] can still benefit from the different rates of the two codes. When $x \le 0.5$, the fractional rate code can detect and correct $\left \lfloor (n - xd) /2 \right \rfloor$ malicious nodes, which are more than $\left \lfloor (n - d/2 - 1) /2 \right \rfloor$ malicious nodes that the full rate code can detect. When $x>0.5$, the full rate code and the fractional rate code can detect and correct the same number of malicious nodes: $\left \lfloor (n - \alpha - 1) /2 \right \rfloor$.
From the analysis above we can see that the same optimized parameters, which are obtained for the data regeneration, can maintain the optimized trade-off between the malicious node detection and storage efficiency for the data reconstruction.
Performance Evaluation {#Sec:performance}
----------------------
From the analysis above, we know that for a distributed storage system with $n$ storage nodes out of which at most $M$ nodes are malicious, the 2-layer rate-matched MSR code can guarantee detection and correction of the malicious nodes during the data regeneration and reconstruction with the probability at least $P_{\det}$.
For a distributed storage system with $n=30$, $M=11$ and $P=0.2$, suppose we have a file with the size $B_F=14000M$ symbols to be stored in the system. The number of the fractional rate code blocks $\theta_L$ and the number of the full rate code blocks $\theta_H$ for different detection probabilities $P_{\det}$ are shown in Fig. \[fig:number\_of\_blocks\]. From the figure we can see that the number of fractional rate code blocks will increase when the detection probability becomes larger. Accordingly, the number of full rate code blocks will decrease.
![The number of fractional/full rate code blocks for different $P_{\det}$[]{data-label="fig:number_of_blocks"}](number_of_blocks){width=".8\columnwidth"}
For the universally resilient MSR code constructed in [@Rashmi-err], the efficiency of the code with the same regeneration performance as the 2-layer rate-matched MSR code is defined as $${\delta'}\!\!_S = \frac{\alpha'(\alpha' + 1)}{\alpha' n} = \frac{\alpha'+1}{n} = \frac{xd/2+1}{n}.$$ In Fig. \[fig:efficiency\_comp\] we will show the efficiency ratios $\eta = \delta_S / {\delta'}\!\!_S$ between the 2-layer rate-matched MSR code and the universally resilient MSR code under different detection probabilities $P_{\det}$. From the figure we can see that the 2-layer rate-matched MSR code has higher efficiency than the universally resilient MSR code. When the successful malicious nodes detection probability is $0.999999$, the efficiency of the 2-layer rate-matched MSR code is about $70\%$ higher.
![Efficiency ratios between the 2-layer rate-matched MSR code and the normal error correction MSR code for different $P_{\det}$[]{data-label="fig:efficiency_comp"}](efficiency_ratio){width=".8\columnwidth"}
$m$-Layer Rate-matched regenerating Code {#sec:m_layer_msr}
========================================
In this section, we will show our second optimization of the rate-matched MSR code: $m$-layer rate-matched MSR code. In the code design, we extend the design concept of the 2-layer rate-matched MSR code. Instead of encoding the data using two MSR codes with different match factors, we utilize $m$ layers of the full rate MSR codes with different parameter $d$, written as $d_i$ for layer $L_i$, $1 \leq i \leq m$, which satisfy $$\label{eqn:m_layer_assump}
d_i \leq d_j,\:\: \forall 1 \leq i \leq j \leq m.$$ The data will be divided into $m$ parts and each part will be encoded by a distinct full rate MSR code. According to the analysis above, the code with a lower code rate has better error correction capability.
The codewords will be decoded layer by layer in the order from layer $L_1$ to layer $L_m$. That is, the codewords encoded by the full rate MSR code with a lower $d$ will be decoded prior to those encoded by the full rate MSR code with a higher $d$. If errors were found by the full rate MSR code with a lower $d$, the corresponding nodes would be marked as malicious. The symbols sent from these nodes would be treated as erasures in the subsequent decoding of the full rate MSR codes with higher $d$’s. The purpose of this arrangement is to try to correct as many as erroneous symbols sent by malicious nodes and locate the corresponding malicious nodes using the full rate MSR code with a lower rate. However, the rates of the $m$ full rate MSR codes must match to achieve an optimal performance. Here we mainly focus on the rate-matching for data regeneration. We can see in the later analysis that the performance of data reconstruction can also be improved with this design criterion.
The main idea of this optimization is to optimize the overall error correction capability by matching the code rates of different full rate MSR codes.
Rate Matching and Parameters Optimization {#sec:rate_matching_m}
-----------------------------------------
According to Section \[Sec:Reg\_high\_rate\], the full rate MSR code $\CH_i$ for layer $L_i$ can be viewed as an $(n-1,d_i,n-d_i)$ MDS code for $1 \leq i \leq m$. During the optimization, we set the summation of the $d$’s of all the layers to a constant $d_0$: $$\label{eqn:sumd}
\sum_{i=1}^{m} d_i = d_0.$$ Here we will show the optimization through an illustrative example first. Then we will present the general result.
### Optimization for $m=3$
There are three layers of full rate MSR codes for $m=3$: $\CH_1$, $\CH_2$ and $\CH_3$.
The first layer code $\CH_1$ can correct $t_1$ errors: $$\label{eqn:3_t1}
t_1 = \left \lfloor (n-d_1-1)/2 \right \rfloor = (n-d_1-1 - \varepsilon_1 )/2,$$ where $\varepsilon_1 = 0$ or $1$ depending on whether $ (n-d_1-1)/2 $ is even or odd.
By regarding the symbols from the $t_1$ nodes where errors are found by $\CH_1$ as erasures, the second layer code $\CH_2$ can correct $t_2$ errors: $$\begin{array}{rcl}
t_2 & = & \left \lfloor (n-d_2 - 1 - t_1)/2 \right \rfloor + t_1 \\
& = & (n-d_2 - 1 - t_1 - \varepsilon_2)/2 + t_1 \\
& = & (2(n-d_2) + n - d_1 -2\varepsilon_2 - \varepsilon_1 - 3)/4,
\end{array}$$ where $\varepsilon_2 = 0$ or $1$, with the restriction that $n-d_2-1 \ge t_1$, which can be written as: $$\label{eqn:l1l2}
-d_1 + 2d_2 \leq n + \varepsilon_1 - 1.$$
The third layer code $\CH_3$ also treat the symbols from the $t_2$ nodes as erasures. $\CH_3$ can correct $t_3$ errors: $$\begin{aligned}
\label{eqn:opt_t3}
t_3 & = & \left \lfloor (n-d_3 - 1 - t_2)/2 \right \rfloor + t_2 \nonumber \\
& = & (n-d_3 - 1 - t_2 - \varepsilon_2)/2 + t_2 \\
& = & (4(n-d_3) + 2(n-d_2) + n - d_1\! -\! 4\varepsilon_3\! -\! 2\varepsilon_2 \!-\! \varepsilon_1\! -\! 7)/8, \nonumber
$$ where $\varepsilon_3 = 0$ or $1$, with the restriction that $n-d_3-1 \ge t_2$, which can be written as: $$\label{eqn:l2l3}
-d_1 - 2d_2 + 4d_3 \leq n + \varepsilon_1 + 2\varepsilon_2 - 1.$$
According to the analysis above, the $d$’s of the three layers satisfy: $$\begin{aligned}
d_1 - d_2 & \le & 0, \label{eqn:x1x2} \\
d_2 - d_3 & \le & 0. \label{eqn:x2x3} \end{aligned}$$ And we can rewrite equation (\[eqn:sumd\]) as: $$\begin{aligned}
d_1 + d_2 + d_3 & \le & d_0, \label{eqn:sumd_positive} \\
-d_1 - d_2 - d_3 & \le & -d_0. \label{eqn:sumd_negative} \end{aligned}$$
To maximize the error correction capability of the $m$-layer rate-matched MSR code for $m=3$, we have to maximize $t_3$, the number of errors that the third layer code $\CH_3$ can correct, since $t_3$ has included all the malicious nodes from which errors are found by the codes of first two layers. With all the constraints listed above, the optimization problem can written as:
$$\begin{array}{lll}
\mbox{Maximize} &\ & t_3\:\: \mbox{in}~(\ref{eqn:opt_t3}), \\
\mbox{subject to} && (\ref{eqn:l1l2}),\: (\ref{eqn:l2l3}),\: (\ref{eqn:x1x2}),\: (\ref{eqn:x2x3}),\: (\ref{eqn:sumd_positive}),\: (\ref{eqn:sumd_negative}).
\end{array}$$
Now we have changed this optimization problem into a typical linear programming problem. This linear programming problem has a feasible solution. We solve it using the SIMPLEX algorithm [@IA]. When $d_1 = d_2 = d_3 = \mbox{Round} (d_0/3)=\widetilde{d} $, the $m$-layer rate-matched MSR code can correct errors from at most $$\begin{aligned}
\widetilde{t}_3 &=& (7n - 7\widetilde{d} - 4\varepsilon_3 - 2\varepsilon_2 - \varepsilon_1- 7)/8 \nonumber \\
&\ge& (7n - 7\widetilde{d} -14)/8 \:\:\mbox{(worst case)}\end{aligned}$$ malicious nodes, where Round($\cdot$) is the rounding operation.
### Evaluation of the Optimization for $m=3$ {#sec:eva_3}
Similar to the storage efficiency $\delta_S$ defined in Section \[sec:2-layer\], here we can define the error correction efficiency $\delta_C$ of the $m$-layer rate-matched MSR code as the ratio between the maximum number of malicious nodes that can be found and the total number of storage nodes in the network: $$\delta_C = (7n - 7\widetilde{d} -14)/(8n).$$ The universally resilient MSR code with the same code rate can be viewed as an $(n-1,\widetilde{d}, n - \widetilde{d})$ MDS code which can correct errors from at most $(n - \widetilde{d} - 1) /2$ malicious nodes (best case). So the error correction efficiency ${\delta'}\!\!_C$ is $${\delta'}\!\!_C = ( n - \widetilde{d} - 1) / (2n).$$ The comparison of the error correction capability between $m$-layer rate-matched MSR code for $m=3$ and universally resilient MSR code is shown in Fig. \[fig:delta\_c\_3\]. In this comparison, we set the number of storage nodes in the network $n=30$. From the figure we can see that the $m$-layer rate-matched MSR code for $m=3$ improves the error correction efficiency more than $50\%$.
![Comparison of the error correction capability between $m$-layer rate-matched MSR code for $m=3$ and universally resilient MSR code[]{data-label="fig:delta_c_3"}](error_correction_efficiency){width=".8\columnwidth"}
### General Optimization Result
For the general $m$-layer rate-matched MSR code, the optimization process is similar.
The first layer code $\CH_1$ can correct $t_1$ errors as in equation (\[eqn:3\_t1\]). By regarding the symbols from the $t_{i-1}$ nodes where errors are found by $\CH_{i-1}$ as erasures, the $i^{th}$ layer code can correct $t_i$ errors for $2\le i \le m$: $$\begin{aligned}
t_i & = & \left \lfloor (n-d_i - 1 - t_{i-1})/2 \right \rfloor + t_{i-1} \nonumber \\
& = & (n-d_i - 1 - t_{i-1} - \varepsilon_i)/2 + t_{i-1} \label{eqn:opt_n_ti}\\
&=&\left(\sum_{j=1}^{i} 2^{j-1}(n-d_j) - \sum_{j=1}^i{2^{j-1}\varepsilon_j - 2^i+1}\right)/2^{i},\nonumber\end{aligned}$$ where $\varepsilon_i=0$ or $1$, with the restriction that $n-d_i - 1 \ge t_{i-1} $, which can be written as: $$\label{eqn:restri_n}
-\sum_{j=1}^{i-1} 2^{j-1}d_j + 2^{i-1}d_i \leq n + \sum_{j=1}^{i-1} 2^{j-1}\varepsilon_j - 1.\\$$
Similarly, the parameter $d$ of the $i^{th}$ layer for $2\le i\le m$ must satisfy $$\label{eqn:n_x1x2}
d_{i-1} - d_i \le 0.$$ And equation (\[eqn:sumd\]) can be written as: $$\begin{aligned}
\sum_{j=1}^{m}d_j & \le & d_0, \label{eqn:n_sumd_positive} \\
-\sum_{j=1}^{m}d_j & \le & -d_0. \label{eqn:n_sumd_negative} \end{aligned}$$
We can maximize the error correction capability of the $m$-layer rate-matched MSR code by maximizing $t_m$. With all the constrains listed above, the optimization problem can be written as: $$\label{eqn:opt_original}
\begin{array}{lll}
\mbox{Maximize} &\ & t_i\ \mbox{for}\ i=m \ \mbox{in}~(\ref{eqn:opt_n_ti}), \\
\mbox{subject to} &\ & (\ref{eqn:restri_n}) \ \mbox{and} \ (\ref{eqn:n_x1x2}) \ \mbox{for} \ 2\le i \le m, \ (\ref{eqn:n_sumd_positive}),\ (\ref{eqn:n_sumd_negative}).
\end{array}$$
After verifying that this linear programming problem has a feasible solution, we can use the SIMPLEX algorithm to solve it. The optimization result can be summarized as follows:
\[thm:m-layer-reg\] For the regeneration of $m$-layer rate-matched MSR code, when $$\label{eqn:opt_result_d}
d_i= \rm{Round}(d_0/m)=\widetilde{d}\:\:\:\: \mbox{for}\:\: 1 \le i \le m,$$ it can correct errors from at most $$\begin{aligned}
\widetilde{t}_m &=& ((2^{m}-1)(n-\widetilde{d}) - \sum_{j=1}^m 2^{j-1}\varepsilon_j - 2^m + 1)/2^m\nonumber \\
&\ge&((2^m\!-\!1)(n\!-\!\widetilde{d})\! -\! 2^{m+1}\! +\! 2)/2^m \:\:\mbox{(worst case)}.\end{aligned}$$ malicious nodes.
The error correction efficiency for the $m$-layer rate-matched MSR code is $$\label{eqn:efficiency_m_layer}
\delta_C = ((2^{m}-1)(n-\widetilde{d}) - 2^{m+1} + 2)/(2^{m}n).$$ This is a monotonically increasing function for $m$, so we have:
\[cor:monoincrease\] The error correction efficiency of the $m$-layer rate-matched MSR code increases with m, which is the number of layers.
During the optimization, we set the code rate of the rate-matched MSR code to a constant value and maximize the error correction capability. To optimizing the rate-matched MSR code, we can also set the error correction capability $t_i$ for $i=m$ in (\[eqn:opt\_n\_ti\]) to a constant value $$\label{eqn:dual_eqn}
t_m = t_0$$ and maximize the code rate. The problem can be written as: $$\begin{array}{l}
\mbox{Maximize} \:\:\:\:\:\sum_{j=1}^{m}d_j \\
\mbox{subject to} \:\:\:\:\: (\ref{eqn:restri_n})\:\: \mbox{and} \: (\ref{eqn:n_x1x2})\:\:\mbox{for}\:\:2\le i \le m, \:\: (\ref{eqn:dual_eqn}).
\end{array}$$ The optimization result is the same as that of (\[eqn:opt\_original\]): when all the $d_i's$ for $1 \leq i \leq m$ are the same, the code rate is maximized. $d_i$, $1 \le i \le m$, satisfies the following equation: $$d_i \ge n - \frac{2^mt_0+2^{m+1}-2}{2^m-1} \:\:\mbox{(worst case)}.$$
### Evaluation of the Optimization
Although at the beginning of this section we propose to decode the code with a lower rate first in the $m$-layer rate-matched MSR code, equation (\[eqn:opt\_result\_d\]) shows that we can get the optimized error correction capability when all the rates of the codes in the $m$-layer code are equal. However, this result is not in conflict with our assumption in equation (\[eqn:m\_layer\_assump\]).
#### Comparison with the Hermitian code based MSR code in [@jian14]
The Hermitian code based MSR code (H-MSR code) in [@jian14] has better error correction capability than the universally resilient MSR code. However, because the structure of the underlying Hermitian code is predetermined, the error correction capability might not be optimal. In figure \[fig:comp\_h\], the maximum number of malicious nodes from which the errors can be corrected by the H-MSR code is shown. Here we set the parameter $q$ of the Hermitian code [@Hermitian] from 4 to 16 with a step of 2. In the figure, we also plot the performance of the $m$-layer rate-matched MSR code with the same code rates as the H-MSR code. The comparison result demonstrates that the rate-matched MSR code has better error correction capability than the H-MSR code. Moreover, the rate-matched code is easier to understand and has more flexibility than the H-MSR code.
![Comparison of error correction capability between the $m$-layer rate matched MSR code and the H-MSR code[]{data-label="fig:comp_h"}](compare_with_hermitian_n_rs){width=".8\columnwidth"}
#### Number of layers and error correction efficiency
Since we have seen the advantage of the rate-matched MSR code over the universally resilient MSR code in Section \[sec:eva\_3\], here we will mainly discuss how the number of layers can affect the error correction efficiency. The error correction efficiency of the $m$-layer rate-matched MSR code is shown is Fig. \[fig:delta\_c\_n\], where we set $n=30$ and $d_0=50$. We also plot the error correction efficiency ${\delta'}_C$ of the universally resilient MSR code with same code rates for comparison. From the figure we can see that when $n$ and $d_0$ are fixed, the optimal error correction efficiency will increase with the number of layers $m$ as in Corollary \[cor:monoincrease\].
![The optimal error correction efficiency of the $m$-layer rate-matched MSR code under different m for $2 \le m \le 16$[]{data-label="fig:delta_c_n"}](error_correction_efficiency_m_new){width=".8\columnwidth"}
#### Optimized storage capacity
Moreover, the optimization condition in equation (\[eqn:opt\_result\_d\]) also leads to maximum storage capacity besides the optimal error correction capability. We have the following theorem:
The $m$-layer rate-matched MSR code can achieve the maximum storage capacity if the parameter $d$’s of all the layers are the same, under the constraint in equation (\[eqn:sumd\]).
The code of the $i^{th}$ layer can store one block of data with the size $B_i = \alpha_i(\alpha_i +1) =(d_i/2)(d_i/2 + 1) $. So the $m$-layer code can store data with the size $B=\sum_{i=1}^{m}(d_i/2)(d_i/2 + 1) $. Our goal here is to maximize $B$ under the constraint in equation (\[eqn:sumd\]).
We can use Lagrange multipliers to find the point of maximum $B$. Let $$\label{eqn:largrange_m}
\Lambda_L(d_1,\dots,d_m,\lambda) = \sum_{i=1}^{m}(d_i/2)(d_i/2 + 1) + \lambda(\sum_{i=1}^{m}d_i - d_0).$$ We can find the maximum value of $B$ by setting the partial derivatives of this equation to zero: $$\frac{\partial \Lambda_L}{\partial d_i} = \frac{d_i+1}{2} - \lambda = 0, \:\: \forall 1\le i \le m.$$ Here we can see that when all the parameter $d$’s of all the layers are the same, we can get the maximum storage capacity $B$. This maximization condition coincides with the optimization condition for achieving the goal of this section: optimizing the overall error correction capability of the rate-matched MSR code.
Practical Consideration of the Optimization
-------------------------------------------
So far, we implicitly presume that there is only one data block of the size $B_i = \alpha_i(\alpha_i +1)$ for each layer $i$. In practical distributed storage, it is the parameter $d_i$ that is fixed instead of $d_0$, the summation of $d_i$. However, as long as we use $m$ layers of MSR codes with the same parameter $d=\widetilde{d}$, we will still get the optimal solution for $d_0 = m\widetilde{d}$. In fact, the $m$-layer rate-matched MSR code here becomes a single full rate MSR code with parameter $d=\widetilde{d}$ and $m$ data blocks. And based on the dependent decoding idea we describe at the beginning of Section \[sec:m\_layer\_msr\], we can achieve the optimal performance.
So when the file size $B_F$ is larger than one data block size $\widetilde{B}$ of the single full rate MSR code with parameter $d=\widetilde{d}$, we will divide the file into $\left. \lceil B_F/\widetilde{B} \rceil \right.$ data blocks and encode them separately. If we decode these data blocks dependently, we can get the optimal error correction efficiency.
### Evaluation of the Optimal Error Correction Efficiency
In the practical case, $\widetilde{d}$ in equation (\[eqn:efficiency\_m\_layer\]) is fixed. So here we will study the relationship between the number of dependently decoding data blocks $m$ and the error correction efficiency $\delta_C$, which is shown in Fig. \[fig:delta\_c\_practical\]. We set $n=30$ and $\widetilde{d}=5,10$. From the figure we can see that although $\delta_C$ will become higher with the increasing of dependently decoding data blocks $m$, the efficiency improvement will be negligible for $m\ge8$. Actually when $m=7$ the efficiency has already become $99\%$ of the upper bound of $\delta_C$.
![The optimal error correction efficiency for $2 \le m \le 16$[]{data-label="fig:delta_c_practical"}](error_correction_efficiency_p){width=".8\columnwidth"}
On the other hand, there exist parallel algorithms for fast MDS code decoding [@Dabiri95]. We can decode blocks of MDS codewords parallel in a pipeline fashion to accelerate the overall decoding speed. The more blocks of codewords we decode parallel, the faster we will finish the whole decoding process. For large files that could be divided into a large amount of data blocks ($\theta$ blocks), we can get a trade-off between the optimal error correction efficiency and the decoding speed by setting the number of dependently decoding data blocks $m$ and the number of parallel decoding data blocks $\rho$ under the constraint $\theta = m\rho$.
Encoding
--------
From the analysis above we know that to encode a file with size $B_F$ using the optimal $m$-layer rate-matched MSR code is to encode the file using a full rate MSR code with predetermined parameter $d=2\alpha=\widetilde{d}$. First the file will be divided into $\theta$ blocks of data with size $\widetilde{B}$, where $\theta= \left. \lceil B_F / \widetilde{B} \rceil \right.$. Then the $\theta$ blocks of data will be encoded into code matrices $\CH_1,\dots,\CH_{\theta}$ and form the final $n \times \alpha\theta$ codeword matrix: $CM = [\CH_1,\dots,\CH_{\theta}]$. Each row $\mathbf{c}_i = [\ch_{1,i}, \dots, \ch_{\theta,i}]$, $0 \leq i \leq n-1$, of the codeword matrix $CM$ will be stored in storage node $i$, where $\ch_{j,i}$ is the $i^{th}$ row of $\CH_j$, $1 \leq j \leq \theta$. The encoding vector $\boldsymbol{\psi}_i$ for storage node $i$ is the $i^{th}$ row of $\Psi$ in equation (\[eq:encoding\_msr\_h\]).
The encoding of m-layer rate-matched MSR code can achieve the MSR point in equation (\[eq:MSR\_tradeoff\]) since both the full rate code and the fractional code are MSR codes.
Regeneration {#regeneration-1}
------------
Suppose node $z$ fails, the replacement node $z'$ will send regeneration requests to the rest of $n-1$ helper nodes. Upon receiving the regeneration request, helper node $i$ will calculate and send out the help symbol $p_i = {\bf{c}}_i \boldsymbol{\phi}_z^T$.
![Lattice of received help symbols for regeneration[]{data-label="fig:lattice_arrange"}](lattice){width=".8\columnwidth"}
As we discuss above, combining both dependent decoding and parallel decoding can achieve the trade-off between optimal error correction efficiency and decoding speed. Although all $\theta$ blocks of data are encoded with the same MSR code, $z'$ will place the received help symbols into a 2-dimension lattice with size $m \times \rho$ as shown in Fig. \[fig:lattice\_arrange\]. In each grid of the lattice there are $n-1$ help symbols corresponding to one data block, received from $n-1$ helper nodes. We can view each row of the lattice as related to a layer of an $m$-layer rate-matched MSR code with $\rho$ blocks of data, which will be decoded parallel. We also view each column of the lattice as related to $m$ layers of an $m$-layer rate-matched MSR code with one block of data each layer, which will be decoded dependently. $z'$ will perform Algorithm \[alg:reg\_m\_layer\] to regenerate the contents of the failed node $z$.
Arrange the received help symbols according to Fig. \[fig:lattice\_arrange\]. Repeat the following steps from Layer $1$ to Layer $m$:
$z'$ regenerates symbols of the failed node $z$ for the $m$-layer rate-matched MSR code \[alg:reg\_m\_layer\]
[**Step n:**]{}
For a certain grid, if errors are detected in the symbols sent by node $i$ in previous layers of the same column, replace the symbol sent from node $i$ by an erasure $\bigotimes$.
Parallel regenerate all the symbols related to $\rho$ data blocks using the algorithm similar to Algorithm \[alg:reg\_h\] with only one difference: parallel decode all the $\rho$ MDS codes in [**[Step 1]{}**]{} of Algorithm \[alg:reg\_h\].
The error correction capability of the regeneration is described in Theorem \[thm:m-layer-reg\].
Reconstruction {#reconstruction-2}
--------------
When DC needs to reconstruct the original file, it will send reconstruction requests to $n$ storage nodes. Upon receiving the request, node $i$ will send out the symbol vector $\mathbf{c}_i$. Suppose $\mathbf{c}'_i= \mathbf{c}_i + \mathbf{e}_i$ is the response from the $i^{th}$ storage node. If $\mathbf{c}_i$ has been modified by the malicious node $i$, we have $\mathbf{e}_i \in \mathbb{F}_q^{\alpha\theta} \backslash \{\mathbf{0}\}$. The strategy of combining dependent decoding and parallel decoding for reconstruction is similar to that for regeneration. $DC$ will place the received symbols into a 2-dimension lattice with size $m \times \rho$. The only difference is that in a grid of the lattice there are $n$ symbol vectors $\ch'_{j,0}, \dots, \ch'_{j,n-1}$ corresponding to data block $j$, received from $n$ storage nodes. DC will perform the reconstruction using Algorithm \[alg:rec\_rate\_matched\_m\_rec\].
Arrange the received symbols similar to Fig. \[fig:lattice\_arrange\]. Here we place received codeword matrix $\CH'_j$ into grid $j$ instead of help symbols received from n-1 help nodes. Repeat the following steps from Layer $1$ to Layer $m$:
DC reconstructs the original file for the $m$-layer rate-matched MSR code \[alg:rec\_rate\_matched\_m\_rec\]
[**Step n:**]{}
For a certain grid, if errors are detected in the symbols sent by node $i$ in previous layers of the same column, replace symbols sent from node $i$ by erasures $\bigotimes$.
Parallel reconstruct all the symbols of the $\rho$ data blocks using the algorithm similar to Section \[Sec:Rec\_high\_rate\] with only one difference: parallel decode all the MDS codes in Section \[Sec:Rec\_high\_rate\].
For data reconstruction, we have the following theorem:
For the reconstruction of $m$-layer rate-matched MSR code, when $$\label{eqn:opt_result_d}
d_i= \rm{Round}(d_0/m)=\widetilde{d}\:\:\:\: \mbox{for}\:\: 1 \le i \le m,$$ the number of malicious nodes from which the errors can be corrected is maximized.
From Section \[sec:rate\_matching\_m\] we know that for regeneration of an optimal $m$-layer rate-matched MSR code, the parameter $d$’s of all the layers are the same, which implies the parameter $\alpha$’s of all layers are also the same. Since the optimization of regeneration is derived based on the decoding of $(n-1,d,n-d)$ MDS codes and in reconstruction we have to decode $(n-1,\alpha,n-\alpha)$ MDS codes, if the parameter $\alpha$’s of all the layers are the same, we can achieve the same optimization results for reconstruction.
Conclusion {#Sec:Conclusion}
==========
In this paper, we develop two rate-matched regenerating codes for malicious nodes detection and correction in hostile networks: 2-layer rate-matched regenerating code and $m$-layer rate-matched regenerating code. We propose the encoding, regeneration and reconstruction algorithms for both codes. For the 2-layer rate-matched code, we optimize the parameters for the data regeneration, considering the trade-off between the malicious nodes detection probability and the storage efficiency. Theoretical analysis shows that the code can successfully detect and correct malicious nodes using the optimized parameters. Our analysis also shows that the code has higher storage efficiency compared to the universally resilient regenerating code ($70\%$ higher for the detection probability $0.999999$). Then we extend the 2-layer code to $m$-layer code and optimize the overall error correction efficiency by matching the code rate of each layer’s regenerating code. Theoretical analysis shows that the optimized parameter could also achieve the maximum storage capacity under the same constraint. Furthermore, analysis shows that compared to the universally resilient regenerating code, our code can improve the error correction efficiency more than $50\%$.
[10]{}
A. Dimakis, P. Godfrey, Y. Wu, M. Wainwright, and K. Ramchandran, “Network coding for distributed storage systems,” [*IEEE Transactions on Information Theory*]{}, vol. 56, pp. 4539 – 4551, 2010.
K. Rashmi, N. Shah, K. Ramchandran, and P. Kumar, “Regenerating codes for errors and erasures in distributed storage,” in [*International Symposium on Information Theory (ISIT) 2012*]{}, pp. 1202–1206, 2012.
J. Li, T. Li, and J. Ren, “Beyond the mds bound in distributed cloud storage,” in [*INFOCOM, 2014 Proceedings IEEE*]{}, pp. 307–315, April 2014.
S. Rhea, C. Wells, P. Eaton, D. Geels, B. Zhao, H. Weatherspoon, and J. Kubiatowicz, “Maintenance-free global data storage,” [*IEEE Internet Computing*]{}, vol. 5, pp. 40 – 49, 2001.
R. Bhagwan, K. Tati, Y.-C. Cheng, S. Savage, and G. M. Voelker, “Total recall: System support for automated availability management,” in [*roc. Symp. Netw. Syst. Design Implementation*]{}, pp. 337–350, 2004.
D. Cullina, A. G. Dimakis, and T. Ho, “Searching for minimum storage regenerating codes,” [*Available:arXiv:0910.2245*]{}, 2009.
N. Shah, K. Rashmi, P. Kumar, and K. Ramchandran, “Explicit codes minimizing repair bandwidth for distributed storage,” in [*Information Theory Workshop (ITW), 2010 IEEE*]{}, pp. 1–5, 2010.
C. Suh and K. Ramchandran, “Exact-repair mds codes for distributed storage using interference alignment,” in [*2010 IEEE International Symposium on Information Theory Proceedings (ISIT)*]{}, pp. 161–165, 2010.
Y. Wu, “A construction of systematic mds codes with minimum repair bandwidth,” [*IEEE Transactions on Information Theory*]{}, vol. 57, no. 6, pp. 3738–3741, 2011.
D. Papailiopoulos, J. Luo, A. Dimakis, C. Huang, and J. Li, “Simple regenerating codes: Network coding for cloud storage,” in [*INFOCOM, 2012 Proceedings IEEE*]{}, pp. 2801–2805, 2012.
S. El Rouayheb and K. Ramchandran, “Fractional repetition codes for repair in distributed storage systems,” in [*2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)*]{}, pp. 1510–1517, 2010.
I. Tamo, Z. Wang, and J. Bruck, “Mds array codes with optimal rebuilding,” in [*2011 IEEE International Symposium on Information Theory Proceedings (ISIT)*]{}, pp. 1240–1244, 2011.
V. R. Cadambe, C. Huang, S. A. Jafar, and J. Li, “Optimal repair of mds codes in distributed storage via subspace interference alignment,” [ *Available:arXiv:1106.1250*]{}, 2011.
D. Papailiopoulos, A. Dimakis, and V. Cadambe, “Repair optimal erasure codes through hadamard designs,” [*IEEE Transactions on Information Theory*]{}, vol. 59, no. 5, pp. 3021–3037, 2013.
N. Shah, K. V. Rashmi, and P. Kumar, “A flexible class of regenerating codes for distributed storage,” in [*2010 IEEE International Symposium on Information Theory Proceedings (ISIT)*]{}, pp. 1943–1947, 2010.
K. Shum and Y. Hu, “Existence of minimum-repair-bandwidth cooperative regenerating codes,” in [*2011 International Symposium on Network Coding (NetCod)*]{}, pp. 1–6, 2011.
A. Wang and Z. Zhang, “Exact cooperative regenerating codes with minimum-repair-bandwidth for distributed storage,” in [*INFOCOM, 2013 Proceedings IEEE*]{}, pp. 400–404, 2013.
H. Hou, K. W. Shum, M. Chen, and H. Li, “Basic regenerating code: Binary addition and shift for exact repair,” in [*2013 IEEE International Symposium on Information Theory Proceedings (ISIT)*]{}, pp. 1621–1625, 2013.
Y.-L. Chen, G.-M. Li, C.-T. Tsai, S.-M. Yuan, and H.-T. Chiao, “Regenerating code based p2p storage scheme with caching,” in [*ICCIT ’09. Fourth International Conference on Computer Sciences and Convergence Information Technology, 2009*]{}, pp. 927–932, 2009.
Y. Wu, A. G. Dimakis, and K. Ramchandran, “Deterministic regenerating codes for distributed storage,” in [*45th Annu. Allerton Conf. Control, Computing, and Communication*]{}, 2007.
A. Duminuco and E. Biersack, “A practical study of regenerating codes for peer-to-peer backup systems,” in [*ICDCS ’09. 29th IEEE International Conference on Distributed Computing Systems, 2009*]{}, pp. 376 – 384, June 2009.
K. Shum, “Cooperative regenerating codes for distributed storage systems,” in [*2011 IEEE International Conference on Communications (ICC)*]{}, pp. 1–5, 2011.
Y. Wu and A. G. Dimakis, “Reducing repair traffic for erasure coding-based storage via interference alignment,” in [*IEEE International Symposium on Information Theory, 2009. ISIT 2009.*]{}, pp. 2276–2280, 2009.
N. Shah, K. Rashmi, P. Kumar, and K. Ramchandran, “Interference alignment in regenerating codes for distributed storage: Necessity and code constructions,” [*IEEE Transactions on Information Theory*]{}, vol. 58, pp. 2134 – 2158, 2012.
K. Rashmi, N. Shah, and P. Kumar, “Optimal exact-regenerating codes for distributed storage at the msr and mbr points via a product-matrix construction,” [*IEEE Transactions on Information Theory*]{}, vol. 57, pp. 5227–5239, 2011.
F. Oggier and A. Datta, “Byzantine fault tolerance of regenerating codes,” in [*2011 IEEE International Conference on Peer-to-Peer Computing (P2P)*]{}, pp. 112–121, 2011.
S. Pawar, S. El Rouayheb, and K. Ramchandran, “Securing dynamic distributed storage systems against eavesdropping and adversarial attacks,” [*IEEE Transactions on Information Theory*]{}, vol. 57, pp. 6734 – 6753, 2011.
Y. Han, R. Zheng, and W. H. Mow, “Exact regenerating codes for byzantine fault tolerance in distributed storage,” in [*Proceedings IEEE INFOCOM*]{}, pp. 2498 – 2506, 2012.
H. Chen and P. Lee, “Enabling data integrity protection in regenerating-coding-based cloud storage,” in [*2012 IEEE 31st Symposium on Reliable Distributed Systems (SRDS)*]{}, pp. 51–60, 2012.
C. Cachin and S. Tessaro, “Optimal resilience for erasure-coded byzantine distributed storage,” in [*DSN 2006. International Conference on Dependable Systems and Networks, 2006*]{}, pp. 115–124, 2006.
M. Abd-El-Malek, G. Ganger, G. Goodson, M. Reiter, and J. Wylie, “Lazy verification in fault-tolerant distributed storage systems,” in [*SRDS 2005. 24th IEEE Symposium on Reliable Distributed Systems, 2005*]{}, pp. 179–190, 2005.
N. B. Shah, K. V. Rashmi, K. Ramchandran, and P. V. Kumar, “Privacy-preserving and secure distributed storage codes,” [ *<http://www.eecs.berkeley.edu/~nihar/publications/privacy_security.pdf/>*]{}.
J. Li, T. Li, and J. Ren, “Secure regenerating code,” in [*IEEE GLOBECOM 2014*]{}, pp. 770–774, 2014.
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, [*Introduction to Algorithms*]{}. The MIT Press, 3rd ed., 2009.
J. Ren, “On the structure of hermitian codes and decoding for burst errors,” [*IEEE Transactions on Information Theory*]{}, vol. 50, pp. 2850– 2854, 2004.
D. Dabiri and I. Blake, “Fast parallel algorithms for decoding reed-solomon codes based on remainder polynomials,” [*IEEE Transactions on Information Theory*]{}, vol. 41, pp. 873–885, Jul 1995.
[^1]: The authors are with the Department of ECE, Michigan State University, East Lansing, MI 48824-1226. Email: {lijian6, tongli, renjian}@msu.edu
|
---
abstract: 'We develop and extend a method presented in \[S. Patinet, D. Vandembroucq, and M. L. Falk, Phys. Rev. Lett., 117, 045501 (2016)\] to compute the local yield stresses at the atomic scale in model two-dimensional Lennard-Jones glasses produced via differing quench protocols. This technique allows us to sample the plastic rearrangements in a non-perturbative manner for different loading directions on a well-controlled length scale. Plastic activity upon shearing correlates strongly with the locations of low yield stresses in the quenched states. This correlation is higher in more structurally relaxed systems. The distribution of local yield stresses is also shown to strongly depend on the quench protocol: the more relaxed the glass, the higher the local plastic thresholds. Analysis of the magnitude of local plastic relaxations reveals that stress drops follow exponential distributions, justifying the hypothesis of an average characteristic amplitude often conjectured in mesoscopic or continuum models. The amplitude of the local plastic rearrangements increases on average with the yield stress, regardless of the system preparation. The local yield stress varies with the shear orientation tested and strongly correlates with the plastic rearrangement locations when the system is sheared correspondingly. It is thus argued that plastic rearrangements are the consequence of shear transformation zones encoded in the glass structure that possess weak slip planes along different orientations. Finally, we justify the length scale employed in this work and extract the yield threshold statistics as a function of the size of the probing zones. This method makes it possible to derive physically grounded models of plasticity for amorphous materials by directly revealing the relevant details of the shear transformation zones that mediate this process.'
author:
- Armand Barbot
- Matthias Lerbinger
- 'Anier Hernandez-Garcia'
- 'Reinaldo García-García'
- 'Michael L. Falk'
- Damien Vandembroucq
- Sylvain Patinet
bibliography:
- 'bilbio\_BLHGFVP.bib'
title: Local yield stress statistics in model amorphous solids
---
\[sec:introduction\] Introduction
=================================
Despite numerous advances during the last two decades, a physical description of plasticity in amorphous materials, known to be quantitatively tied to well-characterized atomistic processes, remains a grand challenge [@procaccia_physics_2009; @barrat_heterogeneities_2011; @rodney_modeling_2011]. All the constitutive laws describing the plastic flow of this large class of materials, such as glasses, amorphous polymers or gels, remain based on phenomenological assumptions. This fact is due to the lack of systematic characterization of elementary mechanisms of plasticity at the atomic scale. For the amorphous solids, the absence of crystalline order prevents, by definition, any identification of crystallographic defects such as dislocations. These defects are, however, the quanta of plastic deformation from which it has been possible to derive constitutive equations of crystal plasticity on robust physical grounds. In amorphous materials, plastic deformation manifests as local rearrangements [@argon_1979] exhibiting a broad distribution of sizes and shapes [@lerner_locality_2009], nonaffine displacements [@zaccone_microscopic_2014; @laurati_long-lived_2017] and connectivity changes between particles [@van_doorn_linking_2017] that lead to a redistribution of elastic stresses in the system [@chattoraj_elastic_2013; @lemaitre_structural_2014]. By analogy with dislocations, it therefore appears natural to try to describe the plastic flow from the dynamics of localized “defects” commonly referred to as Shear Transformation Zones (STZs) [@falk_dynamics_1998].
A variety of metrics have been proposed to locate and characterize the defects that control plastic activity including structural properties (free volume [@spaepen_microscopic_1977], packing [@jack_information-theoretic_2014], short-range order [@shi_stress-induced_2007] and internal stress [@tsamados_study_2008]) and linear responses measures (elastic moduli [@tsamados_local_2009] and localized soft vibrational modes [@widmer-cooper_irreversible_2008; @widmer-cooper_localized_2009; @tanguy_vibrational_2010; @ghosh_connecting_2011; @manning_vibrational_2011; @chen_measurement_2011; @mosayebi_soft_2014; @ding_soft_2014; @schoenholz_understanding_2014]). Unfortunately, the definition of these structural properties are often system-dependent and have shown a relatively low predictive power with respect to the plastic activity [@tsamados_local_2009; @jack_information-theoretic_2014]. The approaches based on linear response measures (e.g. soft vibrational mode analysis) have shown that the correlation between these local properties and the location of plastic rearrangements decreases rather quickly as the system is deformed plastically since they are derived from perturbative calculations [@patinet_connecting_2016].
To address these problems, new methods have recently been proposed. They are based either on combinations of static and dynamic properties (atomic volume and vibrations) [@ding_universal_2016], on the non-linear plastic modes [@lerner_micromechanics_2016; @zylberg_local_2017] or on machine learning methods [@cubuk_identifying_2015; @cubuk_structural_2016; @schoenholz_structural_2016]. These approaches allow the calculation of local fields (respectively named by their authors *flexibility volume*, *local thermal energy* and *softness field*) which show abilities to detect plastic defects far superior to previous attempts. Independently of their high degree of correlation, they nevertheless have the disadvantage of giving access only to quantities that are not directly related to local yield criteria that are more commonly used in models of plasticity. Moreover, with the exception of works based on the vibrational soft modes [@rottler_predicting_2014; @smessaert_structural_2014], the vast majority of these previous approaches only gives access to scalar quantities, which by definition also neglect the important orientational aspect inherent in STZ activity.
It is precisely in this context that we have recently developed a numerical technique to systematically measure the local yield stress field of an amorphous solid on the atomic scale [@patinet_connecting_2016]. This new approach responds to some issues raised previously by providing access to a natural quantity in the context of plasticity, i.e. a local yield stress, and shows an extremely strong correlation with the plastic activity. In addition, this method is non-perturbative and can investigate large strains. It also gives access to a tensorial quantity and is thus able to describe several possible directions of flow. It is therefore an ideal candidate to quantitatively characterize the relationship between structure and plasticity.
In this paper, we present in detail the principles of this method. The statistics of local yield stress are calculated in a model glass synthesized from different quench protocols. The correlation between local slip threshold and plastic activity is investigated as a function of the degree of relaxation of the system. The method is subsequently extended to the study of the amplitudes of the plastic relaxations. Additionally, the consequences of the orientation of the mechanical loading are examined. Finally, we address the effect of the length scale over which the local yield stress field is computed.
\[sec:methods\] Simulation methods
==================================
\[sec:preparation\] Sample preparation
--------------------------------------
We performed molecular dynamics and statics simulations with the LAMMPS open software [@plimpton_fast_1995]. The object of study is a two-dimensional binary glass which is known for its good glass formability [@lancon_structural_1984; @widom_quasicrystal_1987]. It has previously been used to study the plasticity of amorphous materials [@falk_dynamics_1998; @shi_structural_2005; @shi_strain_2005; @shi_evaluation_2007]. One hundred samples each containing $10^4$ atoms were obtained by quenching liquids at constant volume. The density of the system is kept constant and equals $10^4/(98.8045)^2 \approx 1.02$. As in [@shi_strain_2005], we choose our composition such that the number ratio of large (L) and small (S) particles equals $N_{L}:N_{S}=(1+\sqrt{5})/4$. The two-types of atoms interact via standard $6-12$ Lennard-Jones interatomic potentials. In the following, all units will therefore be expressed in terms of the mass $m$ and the two parameters describing the energy and length scales of interspecies interaction, $\epsilon$ and $\sigma$, respectively. Accordingly, time will be measured in units of $t_{0}=\sigma\sqrt{m/\epsilon}$. In the present study, these potentials have been slightly modified to be twice continuously differentiable functions. This is done by replacing the Lennard-Jones expression for interatomic distances greater than $R_{in}=2\sigma$ by a smooth quartic function vanishing at a cutoff distance $R_{cut}=2.5\sigma$. For two atoms $i$ and $j$ separated by a distance $r_{ij}$: $$\label{eq:interatomicpotential}
U(r_{ij}) =
\begin{cases}
4\epsilon \left[ \left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]+A, \quad \text{ for } r_{ij}<R_{in}\\
\sum_{k=0}^{4}C_{k}(r_{ij}-R_{in})^{k}, \quad \text{for } R_{in}<r_{ij}<R_{cut}\\
0, \qquad \qquad \qquad \qquad \qquad \qquad \ \ \text{for } r_{ij}>R_{cut},
\end{cases}$$ with
\[eq:interatomicpotentialcoeff\] $$\begin{aligned}
A&=C_{0}-4\epsilon\left[\left(\frac{\sigma}{R_{in}}\right)^{12}-\left(\frac{\sigma}{R_{in}}\right)^{6}\right]\\
C_{0}&=-(R_{cut}-R_{in})(3C_{1}+C_{2}(R_{cut}-R_{in}))/6 \\
C_{1}&=24\epsilon\sigma^6(R_{in}^6-2\sigma^6)/R_{in}^{13} \\
C_{2}&=12\epsilon\sigma^6(26\sigma^6-7R_{in}^6)/R_{in}^{14} \\
C_{3}&=-(3C_{1}+4C_{2}(R_{cut}-R_{in}))/(3(R_{cut}-R_{in})^2) \\
C_{4}&=(C_{1}+C_{2}(R_{cut}-R_{in}))/(2(R_{cut}-R_{in})^3). \\\end{aligned}$$
**(a)** ![\[fig:mesoscopic\] a) Typical stress-strain curves for the three quench protocols: instantaneous quench from a High Temperature Liquid (HTL, continuous line), instantaneous quench from an Equilibrated Supercooled Liquid (ESL, dashed line) and a Gradual Quench (GQ, dash-dotted line). The inset shows a zoom of a stress drop corresponding to one plastic event. b) Plastic event computed between the onset of instability and just after the event: the arrows and the color scale are the displacement $\vec{u}$ and maximum shear strain $\eta_{VM}$ fields, respectively. For the sake of clarity, the arrows are magnified by a factor $200$ and deleted in the core region. The atom with the maximum shear strain gives the location of the plastic rearrangement.](stress_vs_strain "fig:"){width="0.85\columnwidth"}\
**(b)** ![\[fig:mesoscopic\] a) Typical stress-strain curves for the three quench protocols: instantaneous quench from a High Temperature Liquid (HTL, continuous line), instantaneous quench from an Equilibrated Supercooled Liquid (ESL, dashed line) and a Gradual Quench (GQ, dash-dotted line). The inset shows a zoom of a stress drop corresponding to one plastic event. b) Plastic event computed between the onset of instability and just after the event: the arrows and the color scale are the displacement $\vec{u}$ and maximum shear strain $\eta_{VM}$ fields, respectively. For the sake of clarity, the arrows are magnified by a factor $200$ and deleted in the core region. The atom with the maximum shear strain gives the location of the plastic rearrangement.](plastic_event "fig:"){width="0.85\columnwidth"}
Periodic boundary conditions are imposed on square boxes of linear dimensions $L=98.8045 \sigma$. For the non-smoothed version of the interatomic potential, the glass transition temperature $T_{g}$ of this system is known to be approximately $T_{g}=0.325 \epsilon/k$, where $k$ is the Boltzmann constant [@shi_strain_2005]. This temperature corresponds to the mode coupling temperature which is an upper bound of the glass transition temperature [@shi_structural_2005]. In order to highlight the links between the microstructure, the stability of glasses and their mechanical properties, three different quench protocols are considered. The first two kinds of glass are obtained after instantaneous quenches from High Temperature Liquid (HTL) and Equilibrated Supercooled Liquid (ESL) states at $T=9.18 T_{g}$ and $T=1.08 T_{g}$, respectively. The last protocol consists in a Gradual Quench (GQ) in which temperature is continuously decreased from a liquid state, equilibrated at $1.08 T_{g}$, to a low-temperature solid state at $0.092 T_{g}$, over a period of $10^{6} t_{0}$ using a Nose-Hoover thermostat [@nose_unified_1984; @hoover_canonical_1985]. Afterwards, the system is quenched instantaneously as well. All quench protocols are followed by a static relaxation via a conjugate gradient method to equilibrate the system mechanically at zero temperature. The forces on each atom are minimized up to machine precision. The same relaxation algorithm is used hereafter to study the response to mechanical loading.
This approach produces three highly contrasting types of amorphous solids. The greater the temperature from which the system has fallen out of equilibrium, the less relaxed the system [@sastry_signatures_1998; @debenedetti_supercooled_2001]. This fact is clearly reflected in the values of the average potential energies per atom of the generated inherent states, equal to $-2.1015\pm0.0011$, $-2.3248\pm 0.0015$ and $-2.3977\pm0.0019 \epsilon$ for the HTL, ESL and GQ protocols, respectively.
\[sec:loading\] Mechanical loading: generation of plastic events
----------------------------------------------------------------
Beginning from a quenched unstrained configuration, the glasses are deformed in simple shear imposing Lees-Edwards boundary conditions with an Athermal Quasi Static method (AQS) [@malandro_molecular-level_1998; @malandro_relationships_1999; @maloney_universal_2004; @maloney_amorphous_2006] . We apply a series of deformation increments $\Delta\gamma_{xy}$ to the material by moving the atom positions $\vec{r}$ following an affine displacement field such that $r_{x} \rightarrow r_{x}+r_{y}\Delta\gamma_{xy}$ and $r_{y} \rightarrow r_{y}$. After each deformation increment, we relax the system to its mechanical equilibrium.
In order not to miss plastic events, a sufficiently small strain increment equal to $\Delta\gamma_{xy}=10^{-5}$ is chosen. Plastic events are detected when the computed stress $\tau_{xy}$ decreases, a signature of mechanical instability. A reverse step $-\Delta\gamma_{xy}$ is systematically applied after each stress drop to confirm that the strains generated in the solid are irreversible when the stress criterion is satisfied. The observed response is typical for amorphous materials and is characterized by reversible elastic branches interspersed by plastic events as illustrated in Fig. \[fig:mesoscopic\]a. The more relaxed the system, the stiffer and harder the glass. In agreement with [@shi_strain_2005], we observe that the localization of plastic strain, under load, increases with the degree of relaxation of the initial state. Note that in the case of the GQ system a strong localization of the strain is observed around $\gamma_{xy}\sim0.06$ due to shear banding.
The average shear moduli $\mu$ are obtained from the ratio between the stress response of the entire system following a deformation increment $\Delta\tau_{xy}/\Delta\gamma_{xy}$, i.e. the slope at the origin of the stress-strain curves reported in Fig. \[fig:mesoscopic\]a. $\mu$ equals $8.85$, $14.73$ and $19.03$ for HTL, ESL and GQ systems, respectively. As expected, the stiffness of the system increases with its level of relaxation.
\[sec:strain\_field\] Strain field computation
----------------------------------------------
In order to quantify the correlation between deformation thresholds and plastic rearrangements two types of deformation fields are calculated. The first corresponds to the plastic deformation induced by a single plastic event, the second describes the total cumulative deformation. The displacement field of the former is calculated using the difference between the position of atoms after and just before that instability occurs minus the applied affine displacement increment. The displacement field of the latter is merely computed as the difference between the position of atoms in configurations at a given strain and the as-quenched state, that is to say the state that has not yet been deformed mechanically. The Green-Lagrange strain tensor $\eta_{ij}$ is then evaluated from displacement fields $\vec{u}$ of each atom following the coarse-graining method developed in [@hinkle_coarse_2017] based on the atomic gradient tensor evaluation [@zimmerman_deformation_2009]. An octic polynomial coarse-graining function $\phi(r)$ is employed [@lemaitre_structural_2014]. This function has a single maximum and continuously vanishes at $r=R_{CG}$ where $r$ is the distance between the strain evaluation location and the atom positions. It is expressed as: $$\label{eq:coarse-grainingfunction}
\phi(r) =
\begin{cases}
\frac{15}{8\pi R_{CG}^2}(1-2(\frac{r}{R_{CG}})^4+(\frac{r}{R_{CG}})^8), &\text{for } r<R_{coars}\\
0, &\text{otherwise}.
\end{cases}$$ It is desirable to consider a large enough coarse-graining length scale so that continuum mechanics quantities make sense while keeping it as small as possible in order to account for heterogeneity and to preserve spatial resolution. To this aim, we choose $R_{CG}=5\sigma$. On this scale, a continuous description makes sense (Hooke’s law holds) but the solid is still anisotropic and heterogeneous [@goldhirsch_microscopic_2002; @goldenberg_particle_2007; @tsamados_local_2009].
To simplify the analysis, we choose to work with a scalar quantity by computing the maximum of the shear deformation $\eta_{VM}=\sqrt{((\eta_{xx}-\eta_{yy})/2)^2+\eta_{xy}^2}$. The positions of a plastic rearrangement are then defined as the position of the atom having undergone the maximum $\eta_{VM}$ during a plastic event. This approach allows us to obtain the successive positions of localized plastic rearrangements during deformation from the quenched state as exemplified in Fig. \[fig:mesoscopic\]b.
**(a)** ![\[fig:local\_method\] a) Schematic drawing of the local yield stress computation on a regular square grid of mesh size $R_{sampling}$. Region I (of radius $R_{free}$) is fully relaxed while region II (of width $2R_{cut}$) is forced to deform following an affine pure shear deformation in the $\alpha$ direction. b) Typical local stress increment-strain curves of the Region I for different loading directions $\alpha=0$, $45$, $90$ and $-45^{\circ}$. The measurement of the threshold $\Delta\tau_c$ and relaxation $\Delta\tau_r$ are represented for $\alpha=90^{\circ}$. Inset: zoom one the low strain region of the corresponding stress-strain curves, i.e. without subtracting the initial local shear stress within the as-quenched glass $\tau_{0}(\alpha)$.](schematic_local_method "fig:"){width="0.75\columnwidth"}\
**(b)** ![\[fig:local\_method\] a) Schematic drawing of the local yield stress computation on a regular square grid of mesh size $R_{sampling}$. Region I (of radius $R_{free}$) is fully relaxed while region II (of width $2R_{cut}$) is forced to deform following an affine pure shear deformation in the $\alpha$ direction. b) Typical local stress increment-strain curves of the Region I for different loading directions $\alpha=0$, $45$, $90$ and $-45^{\circ}$. The measurement of the threshold $\Delta\tau_c$ and relaxation $\Delta\tau_r$ are represented for $\alpha=90^{\circ}$. Inset: zoom one the low strain region of the corresponding stress-strain curves, i.e. without subtracting the initial local shear stress within the as-quenched glass $\tau_{0}(\alpha)$.](stress_vs_strain_local "fig:"){width="0.85\columnwidth"}
\[sec:local\_method\] Local yield stress measurement method
===========================================================
We used a method developed in [@patinet_connecting_2016] which allows us to sample the local flow stresses of glassy solids for different loading directions. Similar techniques have been employed to sample the local elastic moduli [@mizuno_measuring_2013] or the yield stresses along a single direction in model glasses [@sollichtalks_2011; @Puosi_Probing_2015]. The principle of the numerical method is illustrated in Fig. \[fig:local\_method\]a. It consists in locally probing the mechanical response within an embedded region of size $R_{free}$ (named region I) by constraining the atoms outside of it (named region II) to deform in a purely affine manner. Only the atoms within the region I are relaxed and can deform nonaffinely. Plastic rearrangements are, thus, forced to occur within this region and the local yield stress can be identified.
In [@patinet_connecting_2016], the embedded region I was centered on every atom of the system to test the reliability of the method. Here, to lay the groundwork for an up-scaling strategy, we rather sample the local yield stress on a regular square grid of size $R_{sampling}$. Furthermore, the non-interacting atoms, located farther than a distance $R_{free}+2R_{cut}$ from the center of the probed region, are deleted during the local loading simulations, thereby speeding up the computation. Unless mentioned explicitly, we chose $R_{sampling}=L/39 \approx R_{cut}$ and $R_{free}=5\sigma$, consistently with the coarse-graining scale $R_{CG}$ used for the strain computation. The yield criterion and the AQS incremental method are the same as those described in section \[sec:loading\] to shear the system remotely. At this scale the amorphous system is highly heterogeneous and the yield stress may not be the same for all orientations of the imposed shear. We thus sample the mechanical response using pure shear loading conditions in different loading directions $\alpha$. Eighteen directions, uniformly distributed between $\alpha=-90^{\circ}$ and $\alpha=90^{\circ}$, are investigated. $\alpha=0^{\circ}$ corresponds to the remote simple shear direction $\alpha_l$.
{width="1.9\columnwidth"}
To save computational time, the strain increment is this time equal to $\Delta\gamma=10^{-4}$. The shear stress in the $\alpha$ direction $\tau(\alpha)$ is computed over the atoms that belong to region I using the Irving and Kirkwood formula [@irving_statistical_1950] as a function of the applied strain. The local region is sheared up to the first mechanical instability occurring at a critical stress $\tau_c(\alpha)$. Even at rest, the glasses feature non-zero internal stress due to the frustration inherent to amorphous solids (see inset of Fig. \[fig:local\_method\]b). A more relevant quantity to link the local properties with plastic activity is thus the amount of stress needed to trigger a plastic rearrangement. The initial local shear stress state within the as-quenched glass $\tau_0(\alpha)$ is thus subtracted from the critical stress to get the local stress increase that would trigger an instability $\Delta\tau_c(\alpha)=\tau_c(\alpha)-\tau_0(\alpha)$. Local shear stress-strain curves are exemplified for four different directions in Fig. \[fig:local\_method\]b. It can be observed that the mechanical response depends on the loading orientation. As expected from elasticity theory, we verify that $\tau_{0}(\alpha)=-\tau_{0}(\alpha+90^{\circ})$ and that the local shear moduli $\mu(\alpha)=\mu(\alpha+90^{\circ})$ as shown in the inset of Fig. \[fig:local\_method\]b [^1]. On the other hand, the critical stress increments $\Delta\tau_{c}$ do not show elastic symmetry and depend on the orientation considered. The computation of $\Delta\tau_{c}$ is repeated systematically for all the grid points of the system and for the different loading directions.
We now want to consider the implications of the field of local $\Delta\tau_c(\alpha)$ for a particular direction of remote loading $\alpha_l$. For this purpose, we make the simplifying assumption of homogeneous elasticity within the system or, equivalently, of localization tensor equal to the identity tensor. Of course, the elasticity in this system at this length scale is heterogeneous and leads to non-affine displacements under remote loading as shown in [@tsamados_local_2009]. This assumption is simply used to be able to estimate the stress felt by a local zone due to a remote loading. This assumption will be discussed further in section \[sec:orientation\_effects\]. If the applied shear stress is homogeneous in the glass, plastic rearrangement that would be activated for a given site is the minimum (positive) $\Delta\tau_c(\alpha)$ projected along the remote loading direction. This may be expressed as: $$\label{eq:loctauy}
\Delta\tau_{y}=\min_{\alpha}\frac{\Delta\tau_{c}(\alpha)}{\cos(2[\alpha-\alpha_l])} ~~ \textrm{with} ~~ |\alpha-\alpha_l| < 45^{\circ}.$$ Maps of local $\Delta\tau_{y}$ are shown in the Fig. \[fig:yield\_stress\_map\_quench\_effect\] for the three quench protocols. One distinguishes a correlation length corresponding to the size of region I. Indeed, the same shear transformation zone can be activated for several grid locations if its threshold is smaller than others in its vicinity which leads to the assignment of close $\Delta\tau_{y}$ values on a scale $\sim R_{free}$. The influence of the size of region I will be addressed in section \[sec:length\_scale\].
\[sec:local\_rearrangement\_statistics\] Local rearrangement statistics
=======================================================================
\[sec:distributions\_of\_local\_yield\_stress\] Distributions of local yield stress
-----------------------------------------------------------------------------------
The effect of the quench protocol on the yield stress maps shown in Fig. \[fig:yield\_stress\_map\_quench\_effect\] is remarkable. It is readily apparent that the lower the temperature at which the system falls out of equilibrium during its synthesis, the more mechanically stable the glass. An advantage of our method is that it allows one to assess stability not from the global scale, as in Fig. \[fig:mesoscopic\]a, but locally. The HTL system shows an overabundance of small energy barriers characteristic of systems far from equilibrium. In contrast, the GQ system has a low proportion of soft zones embedded in a hard skeleton [@shi_strain_2005; @shi_does_2006]. As expected, ESL presents an intermediate situation. More quantitatively, the distributions of $\Delta\tau_{y}$ are computed for the three quenching protocols as shown in Fig. \[fig:PDF\_yield\_stress\]. The probability densities $p(\Delta\tau_{y})$ are noticeably shifted toward higher values with increasing system stability, weak areas being depopulated.
![\[fig:PDF\_yield\_stress\] Probability distribution function of the local yield stresses for the three different quench protocols. The corresponding cumulative distribution functions are represented in log-log scale in the inset. The straight line is a power law of exponent $1+\theta=1.6$, i.e. the expected scaling of the integral of the probability distribution function as $\Delta\tau_{y}$ approaches zero.](PDF_yield_stress){width="0.9\columnwidth"}
Through this method we are able to analyse the statistics of the sites that are about to rearrange plastically. Previous work (based on mean-field theoretical approaches [@lin_mean_field_2016], atomistic simulations [@karmakar_statistical_2010; @hentschel_stochastic_2015] and mesoscopic simulations [@lin_density_2014; @lin_criticality_2015]) proposed a scaling for these soft areas such as $\lim_{\Delta\tau_{y} \to 0}
P(\Delta\tau_{y})\sim\Delta\tau_{y}^{\theta}$ where $\theta$ is a non-trivial exponent. For systems at rest, i.e. after the quench, it was shown that $\theta\approx
0.6$ [@karmakar_statistical_2010; @hentschel_stochastic_2015]. From detailed inspections of our results, as shown as inset of Fig. \[fig:PDF\_yield\_stress\], it seems difficult to extract this exponent with the exception of ESL. In any case, it appears that the behavior of $P(\Delta\tau_{y})$ in the limit of zero $\Delta\tau_{y}$ varies with the preparation of the system. In the case of the GQ system, a smaller exponent can even be observed while for HTL a larger one can be fitted for small threshold values. Several reasons may account for this disagreement, including the lack of statistics or the size of the strain increment $\Delta \gamma$. The fixed boundary conditions applied during local probing may also prevent some relaxation of the system. Also, notably, the previous approaches have considered the distribution of critical strains applied to the whole system which is strictly equivalent to our approach for an elastically homogeneous glass, a strong assumption at this length scale [@tsamados_local_2009]. The answer to this question deserves more investigation, which is outside the scope of the present study.
\[sec:correlation\_with\_plastic\_activity\] Correlation with plastic activity
------------------------------------------------------------------------------
The position of the first ten plastic rearrangements during remote loading are illustrated in Fig. \[fig:yield\_stress\_map\_quench\_effect\]. Plastic rearrangements clearly tend to occur in the soft zones, i.e. for areas in which $\Delta\tau_y$ are small. To quantify this correlation, we apply the same method as in [@patinet_connecting_2016].
We propose a correlation coefficient that allows us to relate the order of appearance of zones in which the plastic arrangements appear and a local scalar field, the local yield stress. The aim here is to compute the predictive power of a structural indicator for the location of successive rearrangements from the sole knowledge of the initial state of the system, i.e. before deformation. To achieve this, the correlation coefficient is computed from the value of the cumulative distribution function of $\Delta\tau_y$ corresponding to the point of the grid $i_{max}$, i.e. the closest to the location of the plastic rearrangement (determined according to the method described in the Sec. \[sec:loading\]). The correlation coefficient is defined as: $$\label{eq:correlation_CDF}
C_{\Delta\tau_{y}}=1-2\overline{CDF}[\Delta\tau_{y}(i_{max},\gamma_{xy})],$$ where $\overline{CDF}$ is the disorder average of the cumulative distribution function. $C_{\Delta\tau_{y}} \sim 1$ indicates a perfect correlation, i.e. a localized plastic rearrangement on the lowest yield threshold grid point ($CDF=0$), while $C_{\Delta\tau_{y}} \sim
0$ means an absence of correlation. $C_{\Delta\tau_{y}}$ is calculated for all plastic events as a function of the deformation applied $\gamma_{xy}$ as shown in Fig. \[fig:correlations\]a. Note that relation (\[eq:correlation\_CDF\]) neglects the stress redistribution due to successive rearrangements. Moreover, it only takes into account the rearrangements producing the maximum of local shear strain located at $i_{max}$ and therefore ignores the possibility that a plastic event may be composed of several localized rearrangements. In agreement with [@patinet_connecting_2016], an excellent correlation is observed. Above all, this correlation shows a slow decrease indicating a persistence of weak sites.
We observe that the level of correlation depends on the preparation of the system. The more relaxed the system, the more robust the observed correlation. The first ten plastic rearrangements occur in areas belonging to the softest $23$, $13$ and $8.5\%$ sites for the HTL, ESL and GQ systems, respectively. For slowly quenched glasses GQ, it is interesting to note that the correlation decreases sharply for a deformation corresponding to the softening due to the localization of the deformation. It can be argued that the origin of the best correlation observed in the most relaxed system comes directly from the distribution of local thresholds. The relaxed systems have a much smaller population of low yield threshold zones. They therefore exhibit larger shear susceptibility when compared to other zones or to mechanical noise, making it easier to predict the onset of plastic activity.
**(a)** ![\[fig:correlations\] Correlation between the local yield stresses computed in the quenched state and the locations of the plastic rearrangement as a function of the applied strain for the three quench protocols. The error bars correspond to one standard deviation. a) Correlation computed from individual plastic rearrangements using Eq. (\[eq:correlation\_CDF\]). The arrows correspond to the average strain of the tenth plastic event. The lines are empirical $A+Be^{-(\gamma/\gamma_{d})^2}$ fits from which the decorrelation strain $\gamma_{d}$ is estimated. b) Pearson correlation coefficient computed between the local yield stress fields and the local strain fields using Eq. (\[eq:correlation\_Pearson\]). Error bars are smaller than symbols.](Correlation_CDF_QP_vs_Strain "fig:"){width="0.85\columnwidth"}\
**(b)** ![\[fig:correlations\] Correlation between the local yield stresses computed in the quenched state and the locations of the plastic rearrangement as a function of the applied strain for the three quench protocols. The error bars correspond to one standard deviation. a) Correlation computed from individual plastic rearrangements using Eq. (\[eq:correlation\_CDF\]). The arrows correspond to the average strain of the tenth plastic event. The lines are empirical $A+Be^{-(\gamma/\gamma_{d})^2}$ fits from which the decorrelation strain $\gamma_{d}$ is estimated. b) Pearson correlation coefficient computed between the local yield stress fields and the local strain fields using Eq. (\[eq:correlation\_Pearson\]). Error bars are smaller than symbols.](Correlation_Pearson_QP_vs_Strain "fig:"){width="0.85\columnwidth"}
The problem of the preceding method is that it assumes the existence of individual and well-localized events. However, Refs. [@maloney_subextensive_2004; @dasgupta_microscopic_2012] have shown that if this hypothesis is relatively well satisfied for small deformation levels, it does not hold with the increase in deformation during which avalanches, through system spanning plastic events, are observed. To circumvent this problem, we deal directly with the correlation between the entire local yield stress field of the as-quenched state $\Delta\tau_{y}$ and the cumulative deformation field $\eta_{VM}$ in the same spirit as Ref. [@smessaert_structural_2014]. The cross-correlation, or Pearson’s correlation, is calculated as a function of the applied strain $\gamma_{xy}$ as: $$\label{eq:correlation_Pearson}
\rho_{\Delta\tau_{y},\eta_{VM}}(\gamma_{xy})=-\frac{\sum_{i=1}^{N}(\Delta\tau_{y}^{i}-\overline{\Delta\tau_{y}})(\eta_{VM}^{i}-\overline{\eta_{VM}})}{N\sigma_{\Delta\tau_{y}}\sigma_{\eta_{VM}}},$$ where $N$ is the number of points on the grid on which the thresholds are calculated, $\sigma_{\Delta\tau_{y}}$ and $\sigma_{\eta_{VM}}$ are the standard deviations of $\Delta\tau_{y}$ and $\eta_{VM}$, respectively. The minus sign is added here to obtain a positive value since large $\eta_{VM}$ are expected for locations where $\Delta\tau_{y}$ are small (i.e. anti-correlation). $\overline{A}$ denotes the ensemble average of the quantity $A$. Note that explicit dependence on $\gamma_{xy}$ of $\eta_{VM}$ is omitted in the r.h.s for the sake of simplicity. Fig. \[fig:correlations\]b shows the evolution of $\rho_{\Delta\tau_{y},\eta_{VM}}$ as a function of the imposed deformation. The general trend is qualitatively similar to that of Fig. \[fig:correlations\]a. The correlation between local thresholds and plastic activity is greater for the more relaxed systems. There are, however, some differences. It can be observed that the correlation begins to increase as plastic rearrangements start to accumulate on weak sites. On the other hand, as expected, the decay of the GQ system is more marked upon the formation of shear bands, the latter concentrating the deformation.
The correlation appears smaller than the one expected from the computation based on local rearrangements in Eq. \[eq:correlation\_CDF\], however, we remark that this calculation is based on crude assumptions. For example, we have not dissociated the elastic part from the plastic part when calculating $\eta_{VM}$. Moreover, this approach does not take into account the distribution of amplitudes of plastic rearrangements. Still, we have verified that in both cases - correlation based on the maxima of the strain field (Eq. \[eq:correlation\_CDF\]) and the cross-correlation based on the cumulative deformation (Eq. \[eq:correlation\_Pearson\]) - the correlations between $\Delta\tau_y$ and plastic activity are significantly better, and more persistent with deformation, than those obtained for the local classical structural indicators reviewed in [@patinet_connecting_2016].
\[sec:distributions\_of\_local\_relaxation\] Distributions of local relaxation
------------------------------------------------------------------------------
In this section, we extend our method to study the amplitude of the local plastic relaxations that follow plastic rearrangements. The loading of region-I described in Sec. \[sec:local\_method\] is continued after the instability until the local stress $\tau(\alpha)$ increases again, signaling the end of the plastic rearrangement and the return to mechanical stability of the sheared zone. This final stress $\tau_{f}(\alpha) $ is also computed for all directions. Plastic relaxation in the $\alpha$ direction is then deduced by simply subtracting $\tau_{f}(\alpha)$ from the stress just before the instability $\tau_{c}(\alpha)$. This amplitude of relaxation $\Delta\tau_{r}(\alpha)=\tau_{c}(\alpha)-\tau_{f}(\alpha)$ is exemplified in Fig. \[fig:local\_method\]b. Like the thresholds, it depends on the direction of shear. Note that this method can only give an estimate of the relaxation amplitude, as the frozen boundary conditions constrain some degrees of freedom during relaxation.
**(a)** ![\[fig:relaxations\] a) Probability distribution function of the stress drops for the three different quench protocols in lin-log scale. The same quantities rescaled by the shear moduli, i.e. the slip increments, are represented in the inset. The lines are exponential fits. b) Average stress drop as a function of the average local yield stress for the free different quench protocols. The line is an exponential fit.](PDF_DeltaTaup_QP "fig:"){width="0.85\columnwidth"}\
**(b)** ![\[fig:relaxations\] a) Probability distribution function of the stress drops for the three different quench protocols in lin-log scale. The same quantities rescaled by the shear moduli, i.e. the slip increments, are represented in the inset. The lines are exponential fits. b) Average stress drop as a function of the average local yield stress for the free different quench protocols. The line is an exponential fit.](DeltaTaup_vs_DeltaTauy "fig:"){width="0.85\columnwidth"}
The plastic rearrangements actually observed during the shearing of the system correspond to the thresholds $\Delta\tau_{y}$ calculated in Eq. \[eq:loctauy\], and occur when the patch is loaded in the weakest direction $\alpha_{min}$. To place ourselves in a coarse-graining perspective, we want to derive a scalar indicator that corresponds to mechanical response in the remote loading direction $\alpha_l$ and disregard for now the tensorial aspect of the problem. The amplitude of plastic relaxation is therefore calculated, in turn, by projecting $\Delta\tau_{r}$ in the $\alpha_l$ direction according to the relation: $$\label{eq:tau_p}
\Delta\tau_{p}=\Delta\tau_{r}(\alpha_{min})\cos(2[\alpha_{min}-\alpha_l]).$$ Note that this estimator of the stress relaxation slightly underestimates the plastic relaxations because of the projection. Nevertheless, it gives access to a sufficiently simple scalar indicator. We verified that the absence of projection does not qualitatively change our results (not shown here).
The distributions of stress relaxation amplitudes reported in Fig. \[fig:relaxations\]a in lin-log scale for the three quench protocols show an exponential decay. The average stress drops increases with the relaxation of the system. The mean plastic relaxations calculated from exponential regressions are $\overline{\Delta\tau_{p}}=0.164$, $0.269$ and $0.337$ for HTL, ESL and GQ systems, respectively. The amplitude of plastic deformation, or slip increment, can also be estimated by computing the eigen-deformations of plastic rearrangements as $\gamma_{p}\sim\Delta\tau_{p}/\mu$ where $\mu$ is the average shear modulus of the glass. Remarkably, the distributions $\gamma_{p}$ collapse on a master curve of mean $\overline{\gamma_ {p}}=0.00887$ independently of the quench protocol as reported in the inset of Fig. \[fig:relaxations\]a. These results justify the assumption of a characteristic relaxation commonly used in mesoscopic simulations or in mean-field models. It is also in agreement with previous atomistic computations based on different methods such as the mapping between elastic field and Eshelby inclusion model [@albaret_mapping_2016; @boioli_shear_2017] and automatic saddle point search techniques [@fan_energy_2017].
We also take advantage of this analysis to study the dependence of the relaxation amplitude $\Delta\tau_{p}$ with the distance to thresholds $\Delta\tau_{y}$ as shown in Fig. \[fig:relaxations\]b. The former increases on average according to the latter. Remarkably, the relationship observed does not seem to depend too much on the quench protocol. If it seems reasonable that the amplitude of relaxation increases with the increase of the local yield stress, the stored elastic energy being larger, we have no explanation to derive this relationship at the moment. An exponential dependence is adjusted empirically and gives $\Delta\tau_{p}\sim0.054e^{0.976\Delta\tau_{y}}$.
Let us note finally that if we are sufficiently confident in the capacity of this local method to quantify the thresholds, the measurements of the relaxation amplitudes are more questionable insofar as the frozen boundary conditions prevent some relaxations. A more adequate treatment of this question would require developments that are beyond the scope of this work. One may for instance think about the implementation of quasicontinuum simulation techniques that, by relaxing elastically the surrounding matrix, will provide flexible boundary conditions to the atomistic region. The picture that emerges from Fig. \[fig:relaxations\] is nevertheless interesting and sheds new light on the plastic deformation as it greatly simplifies representation of relaxation in glassy systems.
\[sec:orientation\_effects\] Orientation effects
================================================
\[sec:loading\_direction\] Loading direction
--------------------------------------------
So far, relatively few studies have addressed the issue of the variation of the local yield stress as a function of loading orientation. This is due to the fact that most of the proposed local indicators are scalar quantities. Only work based on soft modes attempted to explore susceptibility to loading orientation by taking advantage of the vectorial aspect of vibrational eigenmodes [@rottler_predicting_2014; @smessaert_structural_2014]. In order to address this issue, we use here another asset of our local method which naturally gives us access to this directional information.
{width="1.9\columnwidth"}
To test the dependence on the direction of the mechanical loading, the quenched glasses are deformed by the same AQS protocol but following different orientations. In addition to the simple shear described in Sec. \[sec:loading\], the systems are deformed in pure shear by applying strain increments $\Delta\gamma/2 =-\Delta\epsilon_{xx}=\Delta\epsilon_{yy}$ and simple shear in the negative direction by applying deformation increments $\Delta\gamma =-\Delta\gamma_{xy}$. These remote loadings correspond in the infinitesimal strain limit to shearing along $\alpha_{l}=45^{\circ}$ and $\alpha_{l}=-90^{\circ}$ directions, respectively. Pure shear thus produces a diagonally-oriented shear. The negative simple shear corresponds to a laterally-oriented shear but in the opposite direction with respect to the positive simple shear remote loading employed so far. The positions of the first ten plastic rearrangements for the different loading directions are exemplified for a GQ glass in the top row of Fig. \[fig:yield\_stress\_map\_orientation\_effect\]. In agreement with Ref. [@gendelman_shear_2015], the location of plastic rearrangements show a strong dependence on the loading protocol. Most plastic events occur in different areas for different loading protocols. Only occasionally rearrangements will appear in the same location.
At the same time, local yield stresses are also calculated using the formula \[eq:loctauy\] with the corresponding loading directions $\alpha_{l}$. $\alpha_{l}$ is thus equal to $0^{\circ}$, $45^{\circ}$ and $90^{\circ}$ for positive simple shear, pure shear and negative simple shear, receptively. The maps of $\Delta\tau_{y}$ are shown in the top row of Fig. \[fig:yield\_stress\_map\_orientation\_effect\]. A strong dependence on the loading orientation is observed. The rotation of the shear results in the appearance (disappearance) of soft (hard) zones. For example, the areas close to the 1st and 3rd plastic rearrangements for positive simple shear in Fig. \[fig:yield\_stress\_map\_orientation\_effect\]a disappear in the case of negative simple shear in Fig. \[fig:yield\_stress\_map\_orientation\_effect\]c. Conversely, soft areas appear as those close to the 5th and 7th rearrangements in Fig. \[fig:yield\_stress\_map\_orientation\_effect\]c. As with the simple shear detailed above, pure shear and negative simple shear show an excellent correlation between the soft zones and the zones where the plastic rearrangements occur. The quantification of correlations through Eq. (\[eq:correlation\_CDF\]) and (\[eq:correlation\_Pearson\]) as described in Sec. \[sec:correlation\_with\_plastic\_activity\] is quantitatively similar (not shown here).
![\[fig:Cross\_Correlation\_QP\_vs\_AlphaL\] Cross-correlation of the local yield stress field as a function of the loading direction shift. Error bars are smaller than symbols.](Cross_Correlation_QP_vs_AlphaL){width="0.85\columnwidth"}
In order to highlight the discrete aspect of the variation of the local yield stress field, we compute the threshold contrast $TC$ existing between two loading directions. This contrast is defined locally as the ratio between their difference and their averages as $$\label{eq:threshold_contrast}
TC(\alpha_{l}^1,\alpha_{l}^2)=\frac{|\Delta\tau_y(\alpha_{l}^1)-\Delta\tau_y(\alpha_{l}^2)|}{(\Delta\tau_y(\alpha_{l}^1)+\Delta\tau_y(\alpha_{l}^2))/2},$$ where $\alpha_{l}^1$ and $\alpha_{l}^2$ are two remote loading directions. Contrast maps are shown in the bottom row of Fig. \[fig:yield\_stress\_map\_orientation\_effect\]. These maps feature the trends described qualitatively above. The change in loading angle clearly shows areas of marked contrasts as a function of the loading orientations considered. The change of loading direction “turns on" or “turns off" the soft areas which gives rise to large local contrasts.
We observe that the greater the difference between the angles $\Delta\alpha_{l}=|\alpha_{l}^1-\alpha_{l}^2|$, the greater the number and intensity of the contrasts. In order to quantify this trend, we compute the cross-correlation of the yield stress field as a function of the difference of the loading angles $\Delta\alpha_{l}$. The result is shown in Fig. \[fig:Cross\_Correlation\_QP\_vs\_AlphaL\]. The trend observed confirms that the correlation of the yield stress field decreases rapidly with the loading angle, regardless of the quench protocol. We note, however, that the correlation is never completely zero, and is still significant even for the largest $\Delta\alpha_{l}=90^{\circ}$ that corresponds to the correlation between a shearing direction and its opposite direction. We attribute this effect to the small correlation existing between stable (unstable) zones and their tendency to have large (small) slip barriers [@rodney_yield_2009; @patinet_connecting_2016]. The decorrelation due to the directional aspect is nevertheless clearly the dominant effect.
This result shows that local stress thresholds are a very sensitive probe of the loading protocol insofar as it is possible to acurately predict the plastic activity as a function of the orientation of the load. Unlike [@gendelman_shear_2015], we thus believe that these results are consistent with a plasticity-based view of shear transformation zones. Indeed, from our point of view, the dependence of the plastic activity upon the loading protocol does not rule out the existence of plastic deformation via discrete units encoded in the structure, and that these discrete units clearly preexist within the material prior to loading. Our results show rather that the plastic deformations of an amorphous solid, at least for the transient regime at small deformations, can be seen as a sequence of activation of discrete shear transformation zones having weak slip orientations.
\[sec:loading\_direction\] Fluctuations
---------------------------------------
The local information given by our method allows us to also study the fluctuations in the direction of plastic rearrangements around the loading direction $\alpha_{l}$. At first, we have verified that the distributions of thresholds do not depend on the angle $\alpha_{l}$. As expected, the glasses are isotropic on average in the as-quenched state. We then consider the angle $\alpha_{min}$ minimizing Eq. (\[eq:loctauy\]) for a given $\alpha_{l}$, i.e. the weakest local direction for a given loading direction. The distributions of $\alpha_{min}$ around $\alpha_{l}$, shown in figure \[fig:orientation\_effect\]a, are well described by a Gaussian function of standard deviation $\sigma_{\alpha_{min}} \sim
12^{\circ}$. The latter decreases slightly with the relaxation of the system.
![\[fig:orientation\_effect\] Probability distribution functions of angles for the three quench protocols for which the projected local yield stress is minimal $\alpha_{min}$. The lines correspond to Gaussian fits which standard deviations are reported in the inset.](PDF_Angle){width="0.85\columnwidth"}
We also examine the consequences of taking into account the different possibilities of rearrangement directions on the correlation between local yield stress and plastic activity. Three types of local indicators can be considered: the minimum $\Delta\tau_{c}(\alpha)$ over all directions, the threshold $\Delta\tau_{c}(\alpha=\alpha_{l})$ only along the loading direction and $\Delta\tau_{y}$ as previously defined in Eq. \[eq:loctauy\]. The correlations calculated for these three quantities from the relation \[eq:correlation\_CDF\] are plotted in Fig. \[fig:correlation\_orientation\_effect\] as a function of the imposed deformation.
![\[fig:correlation\_orientation\_effect\] Same as Fig. \[fig:correlations\]a for the GQ protocol for different local yield stress fields: its minimum over all orientations $\min_{\alpha}\Delta\tau_{c}(\alpha)$, its value along the loading direction $\Delta\tau_{c}(\alpha=\alpha_{l})$ and its minimum once projected along the loading direction $\min_{\alpha}\Delta\tau_{c}(\alpha)/\cos(2(\alpha-\alpha_{l}))$ (as in Fig. \[fig:correlations\]a).](Correlation_CDF_Projection_vs_Strain){width="0.85\columnwidth"}
We observe that the minimum over all angles $\min_{\alpha}\Delta\tau_{c}(\alpha)$ gives the best correlations only for the very first plastic rearrangements and then decreases rapidly with deformation. This indicator corresponds to isotropic excitation, such as fluctuations in thermal energy, and is therefore sensitive to small barriers. Conversely, $\Delta\tau_{c}(\alpha=\alpha_{l})$ shows a poorer correlation with the location of plastic activity for the first rearrangements as it misses the low thresholds which are slightly disoriented with respect to the loading direction of the system. On the other hand, the correlation is better for larger deformations. Finally, $\Delta\tau_{y}$ shows the best correlation for both small and large deformations. Due to the projection, it is sensitive to small thresholds while retaining information specialized for macroscopic loading direction for larger yield stresses. We see here the importance of having access to a directional quantity. The local yield stresses defined in this article are therefore a good compromise between simplicity (purely local and scalar) and performance that justifies our approach. Note that qualitatively similar results have been obtained as a function of the relaxation of the system or when the correlations are computed from Eq. (\[eq:correlation\_Pearson\]) (not shown here).
\[sec:length\_scale\] Length scale of the local probing zone
============================================================
\[sec:optimal\_size\] Optimal size
----------------------------------
We are interested here in the effects of the patch size $R_{free}$ (Region I) on which the local yield stresses are computed. $R_{free}$ is varied from $2.5$ to $15 \sigma$. The procedure is the same as described in Sec. \[sec:local\_method\]. The size of the grid $R_{sampling}$ on which $\Delta\tau_{y}(R_{free})$ is sampled is kept constant and equal to $L/39 \approx R_{cut}$. We first investigate the correlations of the thresholds with the plastic activity using relation (\[eq:correlation\_CDF\]). To quantify the degree of correlation, three kinds of indicators are considered: the correlation of the first plastic rearrangements $C_{\Delta\tau_{y}}(R_{free},\gamma_{xy}\rightarrow 0^{+})$, the characteristic deformation $\gamma_{d}$ on which the correlation decreases with imposed deformation and the average correlation over the investigated strain window $\langle
C_{\Delta\tau_{y}}(R_{free})\rangle$. The variations of these three indicators as a function of $R_{free}$ are shown for the three quench protocols in Fig. \[fig:correlation\_size\_effect\].
![\[fig:correlation\_size\_effect\] Correlation indicators computed as a function of the size of the probing zone $R_{free}$. Top: Correlation with the first plastic rearrangement locations. Middle: decorrelation strain. Bottom: Average correlation.](Corr_vs_Rfree_Strain){width="0.85\columnwidth"}
{width="1.9\columnwidth"}
**(a)** {width="0.59\columnwidth"} **(b)** {width="0.59\columnwidth"} **(c)** {width="0.61\columnwidth"}
The correlation of the first plastic rearrangement $C_{\Delta\tau_{y}}(R_{free},\gamma_{xy}\rightarrow 0^{+})$ increases with the size $R_{free}$. Indeed, the increase of the probing zone makes it possible to progressively integrate the elastic loading heterogeneities. The loading felt by the sheared zones converges with $R_{free}$ toward the effective loading produced by a remote loading, which makes it easier to identify the weak zones. For $R_{free}<5$, we observe a marked drop of the correlation. For this small size, in addition to larger elastic loading heterogeneities, the frozen boundary conditions over-constrain the measurement of local shear stress thresholds.
The decorrelation strain $\gamma_{d}$ is extracted from a fit of the curves $C_{\Delta\tau_{y}}(R_{free},\gamma_{xy})$ with the expression: $A+Be^{-(\gamma_{xy}/\gamma_{d})^2}$ as shown in Fig. \[fig:correlations\]a. This deformation, corresponding to the characteristic deformation on which the glasses lose their memory of the quenched state, decreases with $R_{free}$. Indeed, after the first plastic events, the use of a large $R_{free}$ loses information about the hard zones surrounding the softest ones. A small $R_{free}$ allows us, while still having good spatial resolution, to maintain a significant correlation for higher deformations, the hard zones being simply advected during plastic flow.
The last kind of correlation indicator is the average of $C_{\Delta\tau_{y}}$ computed as: $$\langle C_{\Delta\tau_{y}}(R_{free})\rangle=(1/\gamma_{*})\int_{0}^{\gamma_{*}}C_{\Delta\tau_{y}}(R_{free},\gamma_{xy})d\gamma_{xy}.$$ The upper bound of the interval of integration $\gamma_{*}$ is chosen equal to the largest decorrelation strain $\gamma_{*}=\gamma_{d}(R_{free}=2.5)$, i.e. computed for the smallest $R_{free}$. This is a global indicator that gathers information on the degree of correlation at the origin and during deformation as the glass loses its memory from the quench state. The results reported in Fig. \[fig:correlation\_size\_effect\] show an overall decrease of the average correlation with $R_ {free}$. This decrease is less marked between $R_{free}=2.5$ and $R_{free}=5$. The maximum of average correlation is even found for $R_{free}=5$ for the quench protocols HTL and GQ.
These results show empirically that a patch size of $R_{free}=5$ is a good compromise in terms of correlation between $\Delta\tau_{y}$ and plastic activity. Calculating the stress thresholds over this scale allows one to precisely locate the first plastic events while preserving the spatial resolution and keeping the memory of the initial quenched state. The effect of quench protocols is qualitatively similar to our previous observations. A greater relaxation of the system results in both a greater correlation for the first plastic events as well as a larger characteristic decorrelation strain, resulting in a larger average correlation.
\[sec:statistical\_size\_effects\] Statistical size effects
-----------------------------------------------------------
We are interested here in the effect of the patch size $R_{free}$ on the slip barrier statistic. Several mechanisms such as mechanical and statistical size effects can be anticipated. Mechanical size effects correspond to elastic heterogeneities as well as the influence of frozen boundary conditions. Frozen boundaries affect the simulation in the following way: the closer an atom to the boundary, the more affine its displacement, thus deviating its trajectory with respect to non constrained simulations. Statistical size effects play a role insofar as the local yield stress is primarily controlled by the weakest zones in the patch since its amplitude is given by the smallest threshold contained in the patch. Maps of local $\Delta\tau_{y}$ computed for different $R_{free}$ are shown in Fig. \[fig:yield\_stress\_map\_size\_effect\] (top row) for the quench protocol GQ. We observe that the variation of $R_{free}$ modifies the global statistic of the thresholds. The distribution functions presented in Fig. \[fig:PDF\_DeltaTauy\_vs\_Rfree\] for the three quench protocols show that the increase of $R_{free}$ induces a significant shift of the distributions toward smallest values of $\Delta\tau_{ y}$.
Obviously, the maps obtained for the large $R_{free}$ can be explained by the spatial increase of the zones centered on weak sites. The softest areas tend to “invade” the glass as the radius of the area on which the threshold is computed increases. Hence, the statistical effect seems to be dominant. On the basis of this observation, we try to understand the variations of the distributions of the local yield stresses with $R_{free}$. We choose to work from the observed distributions for a size $R_{free}=5$. We make the simplifying hypothesis that all the thresholds $\Delta\tau_{y}(R_{free})$ of the grid points take the value of the smallest local minima $\Delta\tau_{y}(R_{free}=5)$ located inside a disk of radius $R_{free}$. For comparison, maps deduced by this procedure are given in Fig. \[fig:yield\_stress\_map\_size\_effect\] (bottom row). This purely geometric approach shows a remarkable agreement compared to the local yield stress maps calculated by actually varying $R_{free}$.
This approach allows us to deduce the distribution of the yield stresses as a function of a given patch size $R_{free}>5$ from the distribution obtained for $R_{free}=5$. The comparisons between the distributions computed for the three quench protocols for different $R_{free}$ and those estimated from our procedure reported in Fig. \[fig:PDF\_DeltaTauy\_vs\_Rfree\] show a satisfactory agreement. The variation of the distributions of $\Delta\tau_{y}$ is therefore dominated by statistical effects. The increase of $R_{free}$ plays the role of a low-pass filter for the thresholds, shifting their distributions toward smaller yield stress values. The agreement between the measured distributions and the deduced distributions is nevertheless slightly lower for the less relaxed systems and for the large $R_{free}$ values. We attribute this discrepancy to the larger elastic disorder and to the lower sensitivity of the soft zones due to narrower threshold distributions in these systems.
\[sec:conclusion\] Conclusions
==============================
In this article, we describe a method for sampling local slip thresholds in model amorphous solids. A robust correlation is observed between the zones with small yield stresses and the locations where the plastic rearrangements occur. As expected, the more the state of the glass is relaxed, the more the barrier distributions shift towards the larger values, explaining the strengthening of glasses from their local stability.
This local method has been extended to measure the amplitude of the plastic relaxations. We show that the assumption of a characteristic mean plastic relaxation is reasonable, the relaxation amplitudes following exponential distributions. Interestingly, we have shown that the amplitude of the plastic relaxations increases on average with the yield stresses.
The effects of loading orientation have shown that the variation of the plastic activity with the direction of loading is well captured by the variation of the local yield stress field calculated using our method. Finally, the variation of the threshold statistics with the size of the probing zones can be reproduced with reasonable agreement on the basis of simple geometric arguments. These results reinforce the coherence of the amorphous plasticity modeling based on discrete flow defects that possess weak slip directions and which are encoded in the structure of the material.
The advantages of the method presented in this work are numerous. It allows to probe the local slip thresholds in a non-perturbative way over a well-defined length scale. Moreover, its generalization to other atomic systems does not seem to pose any particular difficulty since it is, in principle, transposable to all glassy solids. Finally, a last advantage of this method is its computational cost. While, for instance, normal mode analysis based-methods scale as the cube of the number of atoms, our method scales linearly with it. Furthermore, as it treats the different parts of the solid independently it is by construction suited for massively parallel simulations. It is therefore possible to handle extended systems.
The implementation of this local method opens up several promising perspectives. It will be interesting to compare quantitatively the predictive power of the plastic activity of this method with other recent works also providing robust indicators of plasticity [@rodney_distribution_2009; @rodney_yield_2009; @fan_how_2014; @fan_crossover_2015; @fan_energy_2017; @ding_universal_2016; @cubuk_identifying_2015; @schoenholz_structural_2016; @cubuk_structural_2016].
Future research could focus on the measurement of quantities on an atomic scale needed for coarse-grained approaches [@hinkle_coarse_2017]. For instance, our method can provide the threshold statistics necessary to take into account the disorder in the mesoscopic models [@BulatovArgon94a; @BVR-PRL02; @Picard-PRE02; @vandembroucq_mechanical_2011; @TPVR-Meso12; @Nicolas-SM14; @tyukodi_depinning_2016; @nicolas_deformation_2017] and could explicitly deal with the tensorial nature of the problem and the effect of the loading geometry [@budrikis_universal_2017].
This work paves the way, for example, to study the correlation between local energy barriers and frequencies of thermally activated rearrangements simulated by molecular dynamics. Hence, an important question left for future work is to study the effect of thermomechanical history on the statistics of local yield stresses. Our method will allow us to test some of the many phenomenological hypotheses upon which continuum models are still based [@rottler_unified_2005; @falk_dynamics_1998; @sollich_rheology_1997; @hebraud_mode-coupling_1998] and thus significantly improve the multi-scale modeling of plasticity of amorphous solids.
M.L. and S.P. acknowledge the support of French National Research Agency through the JCJC project PAMPAS under grant ANR-17-CE30-0019-01. R.G.G. and A.H.G acknowledge the support of CNRS and French National Research Agency under grant ANR-16-CE30-0022-03. M.L.F acknowledges the support of the U.S. National Science Foundation under Grant Nos. DMR1408685/1409560.
[^1]: These symmetries are related to the rotation of the stress tensor and the elasticity tensor in the $xy$-plane through an angle $\alpha$ about the center of the patch. Noting the new coordinates $(x',y')$ of a point $(x,y)$ after rotation, the shear stress along $\alpha$ is equal to $\tau(\alpha)=\tau_{x'y'}=(\sigma_{yy}-\sigma_{xx})\sin(2\alpha)/2+\tau_{xy}\cos(2\alpha)$ and thus $\tau(\alpha)=-\tau(\alpha+90^{\circ})$. The shear modulus along $\alpha$ is equal to $\mu(\alpha)=C_{x'y'x'y'}=(C_{xxxx}+C_{yyyy}-2C_{xxyy})\sin(2\alpha)^2/4+(C_{xxxy}-C_{yyxy})\sin(4\alpha)/2+C_{xyxy}\cos(2\alpha)^2$ and thus $\mu(\alpha)=\mu(\alpha+90^{\circ})$.
|
---
abstract: 'In this paper we study sequences, series, power series and uniform convergence in the $\mathcal{A}$-Calculus. Here $\mathcal{A}$ denotes an associative unital real algebra. We say a function is $\mathcal{A}$-differentiable if it is real differentiable and its differential is in the regular representation of the algebra. We show the theory of sequences and numerical series resembles the usual theory, but, the proof to establish this claim requires modification of the standard arguments due to the submultiplicativity of the norm on $\mathcal{A}$. In contrast, the theorems concerning divergence of power series over $\mathcal{A}$ are modified notably from the standard theory. We study how the ratio, root and geometric series results are modified due to both the submultiplicativity of the norm and the calculational novelty of zero-divisors. We establish the usual calculations with power series transfer nicely to the $\mathcal{A}$-calculus. Power series are used to define sine, cosine, hyperbolic sine, hyperbolic cosine and the exponential. Finally, special functions are introduced and we derive the $N$-Pythagorean Theorem.'
author:
- |
Daniel FreeseJames S. Cook\
djfreese@iu.edu jcook4@liberty.edu
title: '[Theory of Series in the ${\mathcal{A}}$-calculus and the $N$-Pythagorean Theorem]{}'
---
hypercomplex analysis
Introduction and overview
=========================
We use ${\mathcal{A}}$ to denote a real unital associative algebra of finite dimension. Elements of ${\mathcal{A}}$ are known as ${\mathcal{A}}$-numbers. We study calculus where real numbers have been replaced by ${\mathcal{A}}$-numbers. The resulting calculus we refer to as ${\mathcal{A}}$-calculus. Our typical goal is to find theorems which apply to as large a class of real associative algebras as possible. To our knowledge, the results presented in Sections 3 to 7 have not appeared in the literature in the generality which we supply in this work.\
Setting aside complex analysis, calculus over more general number systems have been studied from about 1890 to the present time. There are too many papers to list. To see how our current framework relates to the existing literature please see [@cookAcalculusI].\
The goal of this paper is to study sequences, series and power series over an algebra. This serves as a foundation for an ongoing project to develop ${\mathcal{A}}$-calculus generalizations of the standard calculational techniques. We should mention N. BeDell has written three supplementary algebra papers [@bedellI],[@bedellII] and [@bedellIII] which provide further algebraic discussion of zero-divisors, logarithms and the $N$-Pythagorean Theorem proved in Section \[sec:nthagtheorem\] of this paper. Then in the sequel to this paper [@cookbedell] one of the authors and N. BeDell present the theory of ${\mathcal{A}}$-ordinary differential equations.\
In Section \[sec:reviewofAcalculus\] we review the major developments from [@cookAcalculusI]. In particular, we discuss representations, submultiplicativity of the norm, the definition and theory of differentiation over an algebra and select theorems from integral calculus over ${\mathcal{A}}$. Many further examples, proofs and motivations can be found in [@cookAcalculusI] and we encourage the reader to consult that paper before digesting this current work.\
We begin the presentation of new results in Section \[sec:sequences\] where we develop the theory of sequences over ${\mathcal{A}}$ following [@Rudin]. We show the usual arithmetic of limits transfers nicely to sequences in ${\mathcal{A}}$. The results are quite natural, however, the submultiplicativity complicated the usual proofs from real or complex analysis.\
Numerical series over ${\mathcal{A}}$ are covered in Section \[sec:series\]. We found that the usual elementary convergence and divergence tests are meaningful over an algebra. The $n$-th term test, comparison test, absolute convergence, root and ratio test all naturally generalize over ${\mathcal{A}}$. The Cauchy Criterion is meaningful and the Cauchy product exists to multiply series where at least one is absolutely convergent. Once more, the proofs required significant modification due to the submultiplicativity of the norm.\
In Section \[sec:powerseries\] we study power series in ${\mathcal{A}}$. We attempt to generalize the elementary convergence and divergence theorems of real calculus. We find the Root and Ratio Test need significant modification. One aspect of the modification is that the submultiplicative constant ${m_{{\mathcal{A}}}}$ appears in convergence results. For example, where $R= 1/ \alpha$ in the usual calculus we found $R = 1/ {m_{{\mathcal{A}}}}\alpha$. Also, the geometric series $1+z + z^2+ \cdots $ converges for $\| z \| < 1/ {m_{{\mathcal{A}}}}$. A second, and initially perplexing, modification is seen in the absense of divergence cases for the Root and Ratio Tests for power series over ${\mathcal{A}}$. The multiplicative property of the norm in real or complex analysis is important to obtain the boundary between divergence and convergence of a power series. Submultiplicativity of the norm on ${\mathcal{A}}$ spoils the usual argument for divergence for both the Root and Ratio Tests. However, we also understand the reason for this modifcation in view of the phenomenon seen in Example \[exa:bandconverge\]. Zero divisors allow new domains of convergence which are not seen in single-variate analysis over ${ \mathbb{R} }$ or ${ \mathbb{C} }$. Uniform convergence and Weierstrauss $M$-Test are studied. The standard theorems concerning sequences of uniformly convergent functions hold for ${\mathcal{A}}$-differentiable sequences of functions. The integral of the limiting function is the limit of the integrals, however, the result for derivatives is not as simple. See Theorem \[thm:difflimitfunseries\] which mirrors the usual theorem of real analysis. We show the term-by-term derivative of a power series in ${\mathcal{A}}$ is indeed the derivative of the given power series. Given an entire function on ${ \mathbb{R} }$ we show there exists a unique entire extension to ${\mathcal{A}}$. We show entire functions are absolutely convergent on ${\mathcal{A}}$ and uniformly convergent on any finite ball in ${\mathcal{A}}$. Finally, we show the product of entire functions is again entire. Indeed, the entire functions on ${\mathcal{A}}$ form an algebra.\
Transcendental functions such as the exponential, sine, cosine and hyperbolic sine and cosine are covered in Section \[sec:transfunct\]. Theorem \[thm:entire\] indicates the defintions we offer are inescapable. We find the usual indentities for the elementary functions on ${ \mathbb{R} }$ extend naturally to any commutative unital algebra ${\mathcal{A}}$.\
Section \[sec:nthagtheorem\] reverses the direction of study from that of Section \[sec:transfunct\]. We ask, given a specific choice of ${\mathcal{A}}$, which functions appear naturally? In particular, we study functions which appear as component functions of the exponential. We call these the [*special functions*]{} of ${\mathcal{A}}$. We find a theorem we call the $N$-Pythagorean Theorem which provides an identity which holds for the special functions. This theorem makes the identities $\cos^2 \theta + \sin^2 \theta =1$ and $\cosh^2 \phi - \sinh^2 \phi = 1$ part of a chain of such identities.
Review of ${\mathcal{A}}$ calculus {#sec:reviewofAcalculus}
==================================
In this Section we have two main goals. First, to provide necessary background to understand the new theory developed in the later sections. Second, to alert the reader to some of the major results which are already established in [@cookAcalculusI]. Please consult [@cookAcalculusI] for references and discussion of how our work connects to the existing literature.
Algebra and the regular representations
---------------------------------------
We say[^1] ${\mathcal{A}}$ is an [**algebra**]{} if ${\mathcal{A}}$ is a finite-dimensional real vector space paired with a function $\star: {\mathcal{A}}\times {\mathcal{A}}\rightarrow {\mathcal{A}}$ which is called [**multiplication**]{}. In particular, the multiplication map satisfies the properties below:
1. [**bilinear:**]{} $ (cx + y ) \star z = c( x \star z)+ y \star z$ and $x \star (cy+ z) = c(x \star y)+ x \star z $ for all $x,y,z \in {\mathcal{A}}$ and $c \in { \mathbb{R} }$,
2. [**associative:**]{} for which $x \star (y \star z) = (x \star y) \star z$ for all $x,y,z \in {\mathcal{A}}$ and,
3. [**unital:**]{} there exists $\mathds{1} \in {\mathcal{A}}$ for which $\mathds{1} \star x = x$ and $x \star \mathds{1} = x$.
We say $x \in {\mathcal{A}}$ is an [**${\mathcal{A}}$-number**]{}. If $x \star y = y \star x$ for all $x,y \in {\mathcal{A}}$ then ${\mathcal{A}}$ is [**commutative**]{}.\
The [**left-multiplication by $x$**]{} is the map $L_x: {\mathcal{A}}\rightarrow {\mathcal{A}}$ defined by $L_x(y) = x \star y$ for all $y \in {\mathcal{A}}$. Observe, by associativity of ${\mathcal{A}}$, $$L_x(y) = x \star 1 \star y = L_x(1) \star y.$$
A linear transformation $T: {\mathcal{A}}\rightarrow {\mathcal{A}}$ is [**right ${\mathcal{A}}$ linear**]{} if $T(x \star y) = T(x) \star y$ for all $x, y \in {\mathcal{A}}$. We say the set ${\mathcal{R}_{{\mathcal{A}}}}$ of all right ${\mathcal{A}}$ linear transformations forms the [**regular representation**]{} of ${\mathcal{A}}$. Since ${\mathcal{A}}$ is unital the regular representation is isomorphic[^2]to ${\mathcal{A}}$. The isomorphism from ${\mathcal{A}}$ to ${\mathcal{R}_{{\mathcal{A}}}}$ is denoted ${\textbf{map}}$ and we find it convenient to use $\#$ for ${\textbf{map}}^{-1}$. In particular, $${\textbf{map}}(x) = L_x \qquad \& \qquad \#(T) = T(1).$$ The idea here is that $\#(T)$ provides the ${\mathcal{A}}$ number which corresponds to $T$. If $\beta$ is a basis for ${\mathcal{A}}$ then the [**matrix regular representation**]{} of ${\mathcal{A}}$ with respect to $\beta$ is $${\text{M}_{{\mathcal{A}}}}(\beta) = \{ [T]_{\beta, \beta} \ | \ T \in {\mathcal{R}_{{\mathcal{A}}}}\}$$ where $[T]_{\beta,\beta}$ denotes the matrix of $T$ with respect to the basis $\beta$. In the case ${\mathcal{A}}= { \mathbb{R} }^n$ we may forego the $\beta$ notation and write $${\text{M}_{{\mathcal{A}}}}= \{ [T] \ | \ T \in {\mathcal{R}_{{\mathcal{A}}}}\}$$ for the [**regular representation**]{} of ${\mathcal{A}}$. There is a natural isomorphism of ${\mathcal{A}}$ and ${\text{M}_{{\mathcal{A}}}}$: If $\beta = \{ v_1 , \dots , v_n \}$ is a basis for ${\mathcal{A}}$ where $v_1 = \mathds{1}$ then $${\mathbf{M}}( x) = \left[ [x]_{\beta}| [x \star v_2]_{\beta}| \cdots |
[x \star v_n]_{\beta} \right]$$ where $[x]_{\beta}$ is the coordinate vector of $x$ with respect to $\beta$. In many applications we consider the case ${\mathcal{A}}= { \mathbb{R} }^n$ with $\beta = \{ e_1, \dots , e_n \}$ the usual standard basis such that $e_1 = \mathds{1}$. Given these special choices we obtain much improved formula $${\mathbf{M}}(x) = [x| x \star e_2| \cdots | x \star e_n].$$
\[Ex:number1\] The [**complex numbers**]{} are defined by ${ \mathbb{C} }= { \mathbb{R} }\oplus i{ \mathbb{R} }$ where $i^2=-1$. We denote $a+ib = [a,b]^T$ corresponding to our identifications $e_1=1$ and $e_2=i$. Notice, $(a+ib)e_2 = (a+ib)i =ia-b = [-b,a]^T$. Thus, $\mathbf{M}(a+ib) = \left[ \begin{array}{cc} a & -b \\ b & a \end{array} \right]$ is a typical element of the regular matrix representation of ${ \mathbb{C} }$ which we denote $\text{M}_{{ \mathbb{C} }}$.
\[Ex:number2\] The [**hyperbolic**]{} numbers are given by ${\mathcal{H}}= { \mathbb{R} }\oplus j{ \mathbb{R} }$ where $j^2=1$. Identifying $e_1=1$ and $e_2=j$ we have $a+bj = [a,b]^T$. Moreover, $$(a+bj)e_2 = (a+bj)j = aj+b = [b,a]^T.$$ Therefore, $ \mathbf{M}(a+bj) = \left[ \begin{array}{cc} a & b \\ b & a \end{array} \right]$ is a typical matrix in $\text{M}_{{\mathcal{H}}}$.
We say $x \in {\mathcal{A}}$ is a [**unit**]{} if there exists $y \in {\mathcal{A}}$ for which $x \star y = y \star x = \mathds{1}$. The set of all units is known as the [**group of units**]{} and we denote this by ${\mathcal{A}^{\times}}$. We say $a \in {\mathcal{A}}$ is a [**zero-divisor**]{} if $a \neq 0$ and there exists $b \neq 0$ for which $a \star b = 0$ or $b \star a =0$. Let ${\mathbf{zd}(\mathcal{A})}= \{ x \in {\mathcal{A}}\ | \ x=0 \ \text{or $x$ is a zero-divisor} \}$.
\[prop:isomorphismhyperbolic\] and $\mathbf{zd}( {\mathcal{H}}) = \{ a+bj \ | \ a^2=b^2 \}$ whereas ${\mathcal{H}}^{\times} = \{ a+bj \ | \ a^2 \neq b^2 \}$. The reciprocal of an element in ${\mathcal{H}}^{ \times}$ is simply $$\frac{1}{a+bj} = \frac{a-bj}{a^2-b^2}$$ this follows from the identity $(a+bj)(a-bj) = a^2-b^2$ given $a^2-b^2 \neq 0$. Let ${\mathcal{B}}= { \mathbb{R} }\times { \mathbb{R} }$ with $(a,b)(c,d) = (ac, bd)$ for all $(a,b),(c,d) \in {\mathcal{B}}$. We can show that $$\Psi(a,b) = a\left(\frac{1+j}{2}\right)+b\left(\frac{1-j}{2}\right) \qquad \& \qquad
\Psi^{-1}(x+jy) = (x+y,x-y)$$ provide an isomorphism of ${\mathcal{H}}$ and ${ \mathbb{R} }\times { \mathbb{R} }$. In [@cookAcalculusI] an examples are given which show how this isomorphism can be used to solve the quadratic equation in ${\mathcal{H}}$ and to derive d’Alembert’s solution to the wave equation.
Submultiplicative norms
-----------------------
The division algebras ${ \mathbb{R} }, { \mathbb{C} }$ and ${\mathbb{H}}$ can be given a [**multiplicative norm**]{} where $\| x \star y \| = \| x \| \, \| y \|$. Generally we can only find [**submultiplicative norm**]{}.
If ${\mathcal{H}}$ is given norm $\| x+jy \| = \sqrt{x^2+y^2}$ then $\| zw \| \leq \sqrt{2} \|z \| \, \|w \|$.
If ${\mathcal{A}}$ is an algebra over ${ \mathbb{R} }$ with basis $\{ v_1, \dots , v_n \}$ then define [**structure constants**]{} $C_{ijk}$ by $v_i \star v_j = \sum_{k=1}^n C_{ijk} v_k$ for all $1 \leq i,j \leq n$. For proof of what follows see [@cookAcalculusI].
\[thm:submultiplicative\] **(submultiplicative norm)** If ${\mathcal{A}}$ is an associative $n$-dimensional algebra over ${ \mathbb{R} }$ then there exists a norm $|| \cdot ||$ for ${\mathcal{A}}$ and ${m_{{\mathcal{A}}}}>0$ for which $|| x \star y|| \leq {m_{{\mathcal{A}}}}||x|| ||y||$ for all $x,y \in {\mathcal{A}}$. Moreover, for this norm we find ${m_{{\mathcal{A}}}}= \mathbf{C}(n^2-n+1) \sqrt{n}$ where $\mathbf{C} = \text{max} \{ C_{ijk} \ | \ 1 \leq i,j,k \leq n \}$.
\[prop:ineqnpower\] If $\| x \star y \| \leq {m_{{\mathcal{A}}}}\| x \| \|y \|$ for $x,y \in {\mathcal{A}}$ then $\| z^n \| \leq {m_{{\mathcal{A}}}}^n \| z \|^n$ for each $z \in {\mathcal{A}}$ and $n \in {\mathbb{N}}$.
\[thm:quotientinequality\] Suppose ${m_{{\mathcal{A}}}}>0$ is a real constant such that $|| x \star y|| \leq {m_{{\mathcal{A}}}}||x|| ||y||$ for all $x,y \in {\mathcal{A}}$. If $b \in {\mathcal{A}^{\times}}$ and $a \in {\mathcal{A}}$ then $ \frac{||a||}{||b||} \leq {m_{{\mathcal{A}}}}\, \big{|}\big{|} \frac{a}{b} \big{|}\big{|}$.
Differential calculus on ${\mathcal{A}}$
----------------------------------------
The definition of differentiability with respect to an algebra variable is open to some debate. There seem to be two main approaches:
> 1. define differentiability in terms of an algebraic condition on the differential,
>
> 2. define differentiability in terms of a deleted-difference quotient
>
In [@cookAcalculusI] it is shown that these definitions are interchangeable on an open set in the context of a commutative semisimple algebra. However, it is also shown that in there exist [**D1**]{} differentiable functions which are nowhere [**D2**]{}. Hence, we prefer to use [**D1**]{} as it is more general. Following [@cookAcalculusI] we define differentiability with respect to an algebra variable as follows:
\[defn:Adiff\] Let $U \subseteq {\mathcal{A}}$ be an open set containing $p$. If $f: U \rightarrow {\mathcal{A}}$ is a function then we say $f$ is [**${\mathcal{A}}$-differentiable at $p$**]{} if there exists a linear function $d_pf \in {\mathcal{R}_{{\mathcal{A}}}}$ such that $$\label{eqn:frechetquotAcal}
\lim_{h \rightarrow 0}\frac{f(p+h)-f(p)-d_pf(h)}{||h||} = 0.$$
In other words, $f$ is ${\mathcal{A}}$-differentiable at a point if its differential at the point is a right-${\mathcal{A}}$-linear map. Equivalently, given a choice of basis, $f$ is ${\mathcal{A}}$-differentiable if its Jacobian matrix is found in the matrix regular representation of ${\mathcal{A}}$. If ${\mathcal{A}}$ has basis $\beta = \{ v_1, \dots , v_n\}$ has coordinates $x_1, \dots , x_n$ then $d_pf(e_j) = \frac{\partial f}{\partial x_j}(p)$. Suppose $v_1=\mathds{1}$ then $ d_pf(1) = \frac{\partial f}{\partial x_1}(p)$. Observe right linearity of the differential indicates $d_pf(v_j) = d_pf(\mathds{1} \star v_j) = d_pf(1) \star v_j$ hence for each $p$ at which $f$ is ${\mathcal{A}}$ differentiable we find: $$\frac{\partial f}{\partial x_j} (p)= \frac{\partial f}{\partial x_1}(p) \star v_j.$$ These are the [**${\mathcal{A}}$-Cauchy Riemann Equations**]{}. There are $n-1$ equations in ${\mathcal{A}}$ which amount to $n^2-n$ scalar equations. If the ${\mathcal{A}}$-CR equations hold for a continuously differentiable $f$ at $p$ then we have that $d_pf \in {\mathcal{R}_{{\mathcal{A}}}}$.\
Next we wish to explain how to construct the derivative function $f'$ on ${\mathcal{A}}$. We are free to use the isomorphism between the right ${\mathcal{A}}$ linear maps and ${\mathcal{A}}$ as to define the [*derivative at a point*]{} for via $f'(p) = \#(d_pf)$. This is special to our context. In the larger study of real differentiable functions on an $n$-dimensional space no such isomorphism exists and it is not possible to identify arbitrary linear maps with points.
\[defn:derivative\] Let $U\subseteq {\mathcal{A}}$ be an open set and $f: U \rightarrow {\mathcal{A}}$ an ${\mathcal{A}}$-differentiable function on $U$ then we define $f': U \rightarrow {\mathcal{A}}$ by $f'(p) = \# (d_pf) $ for each $p \in U$.
Equivalently, we could write $f'(p) = d_pf( \mathds{1})$ since $\#(T) = T( \mathds{1})$ for each $T \in {\mathcal{R}_{{\mathcal{A}}}}$. Many properties of the usual calculus hold for ${\mathcal{A}}$-differentiable functions.
For $f$ and $g$ both ${\mathcal{A}}$-differentiable at $p$,
1. $ {\displaystyle}(f+g)'(p) = f'(p)+g'(p)$,
2. for constant $c \in {\mathcal{A}}$, $ {\displaystyle}(c \star f)'(p) = c \star f'(p)$,
3. Given ${\mathcal{A}}$ is commutative, $ (f \star g)'(p) = f'(p) \star g(p)+ f(p) \star g'(p)$.
4. $ (f { \,{\scriptstyle \stackrel{\circ}{}}\, }g)'(p) = f'(g(p)) \star g'(p). $
5. If $f(\zeta) = \zeta^n$ for some $n \in {\mathbb{N}}$ then $f'(\zeta) = n \zeta^{n-1}$.
If ${\mathcal{A}}$ is not commutative then the product of ${\mathcal{A}}$-differentiable functions need not be ${\mathcal{A}}$-differentiable. In [@cookAcalculusI] an example is given where $f,g$ and $f \star g$ are ${\mathcal{A}}$-differentiable yet $g \star f$ is not ${\mathcal{A}}$-differentiable.\
We are also able to find an ${\mathcal{A}}$-generalization of Wirtinger’s calculus. In [@cookAcalculusI] we introduce conjugate variables $\bar{\zeta}_2, \dots, \bar{\zeta}_n$ for ${\mathcal{A}}$ and find for commutative algebras if $f: {\mathcal{A}}\rightarrow {\mathcal{A}}$ is ${\mathcal{A}}$-differentiable at $p$ then ${\displaystyle}\frac{\partial f}{\partial \overline{\zeta}_j} = 0$ for $j=2, \dots , n$. In other words, another way we can look at ${\mathcal{A}}$-differentiable functions is that they are functions of $\zeta$ alone.\
The theory of higher derivatives is also developed in [@cookAcalculusI].
\[defn:higherderivative\] Suppose $f$ is a function on ${\mathcal{A}}$ for which the derivative function $f'$ is ${\mathcal{A}}$-differentiable at $p$ then we define $ f''(p) = (f')'(p)$. Furthermore, supposing the derivatives exist, we define $f^{(k)}(p) = (f^{(k-1)})'(p)$ for $k =2,3, \dots$.
Naturally we define functions $f'', f''', \dots, f^{(k)}$ in the natural pointwise fashion for as many points as the derivatives exist. Furthermore, with respect to $\beta = \{ v_1, \dots , v_n \}$ where $v_1 = \mathds{1}$, we have $f'(p) = d_pf(\mathds{1}) = \frac{\partial f}{\partial x_1}(p)$. Thus, $f' = \frac{\partial f}{\partial x_1}$. Suppose $f''(p)$ exists. Note, $$f''(p) = (f')'(p) = \#( d_p f'(\mathds{1}) )= \frac{\partial f'}{\partial x_1}(p) = \frac{\partial^2f}{\partial x_1^2}(p).$$ Thus, $f'' = \frac{\partial^2 f}{\partial x_1^2}$. By induction, we find if $f^{(k)}$ exists then $f^{(k)} = \frac{\partial^k f}{\partial x_1^k}$. Furthermore, if $f: {\mathcal{A}}\rightarrow {\mathcal{A}}$ is $k$-times ${\mathcal{A}}$-differentiable then $$\frac{\partial^k f}{\partial x_{i_1}\partial x_{i_2} \cdots \partial x_{i_k}} = \frac{\partial^k f}{\partial x_1^k} \star v_{i_{1}} \star v_{i_{2}} \star \cdots \star v_{i_{k}}.$$
The Theorem below gives us license to convert equations in ${\mathcal{A}}$ to partial differential equations which every component of an ${\mathcal{A}}$-differentiable function must solve!
Let $U$ be open in ${\mathcal{A}}$ and suppose $f: U \rightarrow {\mathcal{A}}$ is $k$-times ${\mathcal{A}}$-differentiable. If there exist $B_{i_1i_2\dots i_k} \in { \mathbb{R} }$ for which $\sum_{i_1i_2\dots i_k} B_{i_1i_2\dots i_k}v_{i_1} \star v_{i_2} \star \cdots \star v_{i_k} = 0$ then $$\sum_{i_1i_2\dots i_k} B_{i_1i_2\dots i_k}\frac{\partial^k f}{\partial x_{i_1} \partial x_{i_2} \cdots \partial x_{i_k} }=0.$$
Since $i^2=-1$ in ${ \mathbb{C} }$ it follows for $z=x+iy$ that complex differentiable $f$ have $f_{yy}=-f_{xx}$. We usually see this in notation $f=u+iv$ and the observation $u_{xx}+u_{yy}=0$ and $v_{xx}+v_{yy}=0$. The real and imaginary parts of a complex differentiable function are harmonic because $1+i^2=0$.
Since $j^2=1$ in ${\mathcal{H}}$ it follows for $z=x+jy$ that ${\mathcal{H}}$-differentiable $f$ have $f_{yy}=f_{xx}$. If $f=u+iv$ then $u_{xx}-u_{yy}=0$ and $v_{xx}-v_{yy}=0$. Thinking of $y$ as time and $x$ as position, the partial differential equation $u_{xx}=u_{yy}$ is a unit-speed wave equation. Component functions of hyperbolic differentiable functions are solutions to the wave equation in one dimension!
For complex algebras the Cauchy Integral Formula links differentiation and integration in such a way that one complex derivative’s existence requires all higher complex derivative’s likewise exist. We consider many examples where ${\mathcal{A}}$ does not permit such a simplification. However, if we know $f$ is smooth in the real sense and once ${\mathcal{A}}$-differentiable then it is shown in [@cookAcalculusI] that $f$ allows infinitely many ${\mathcal{A}}$-derivatives. Taylor’s Formula for ${\mathcal{A}}$ is also given:
\[thm:AcalTaylor\] **(Taylor’s Formula for ${\mathcal{A}}$-Calculus:)** Let ${\mathcal{A}}$ be a commutative, unital, associative algebra over ${ \mathbb{R} }$. If $f$ is real analytic at $p \in {\mathcal{A}}$ then $$f(p+h) = f(p)+f'(p) \star h + \frac{1}{2}f''(p) \star h^2+ \cdots + \frac{1}{k!}f^{(k)}(p) \star h^k+ \cdots$$ where $h^2 = h \star h$ and $h^{k+1} = h^{k} \star h$ for $k \in {\mathbb{N}}$.
Integral calculus on ${\mathcal{A}}$
------------------------------------
Integration along curves in ${\mathcal{A}}$ is defined in [@cookAcalculusI] in much the same fashion as ${ \mathbb{C} }$. If $\zeta: [t_o,t_1] \rightarrow {\mathcal{A}}$ is differentiable parametrization of a curve $C$ and $f$ is continuous near $C$ then $${\displaystyle}\int_C f ( \zeta) \star d\zeta = \int_{t_o}^{t_f} f( \zeta (t)) \star \frac{d\zeta}{dt} \, dt.$$
\[thm:subML\] Let $C$ be a rectifiable curve with arclength $L$. Suppose $||f(\zeta) || \leq M$ for each $\zeta \in C$ and suppose $f$ is continuous near $C$. Then $$\bigg{|}\bigg{|} \int_C f( \zeta) \star d\zeta \bigg{|}\bigg{|} \leq {m_{{\mathcal{A}}}}ML$$ where ${m_{{\mathcal{A}}}}$ is a constant such that $|| z \star w || \leq {m_{{\mathcal{A}}}}||z|| \, ||w||$ for all $z,w \in {\mathcal{A}}$.
Let us conclude with a list of notable results given in [@cookAcalculusI]. The order in which these results are derived is perhaps surprising. In fact, the next result is last:
\[thm:FTCIforalg\] **(Fundamental Theorem of Calculus Part I:)** Let $C$ be a differentiable curve from $\zeta_o$ to $\zeta$ in $U \subseteq {\mathcal{A}}$ where $U$ is an open simply connected subset of ${\mathcal{A}}$. Assume $f$ is ${\mathcal{A}}$ differentiable on $U$ then $$\frac{d}{d\zeta}\int_C f( \eta) \star d \eta = f(\zeta).$$
**(Fundamental Theorem of Calculus Part II:)** \[thm:FTCforalg\] Suppose $f = \frac{dF}{d\zeta}$ near a curve $C$ which begins at $P$ and ends at $Q$ then $$\int_C f( \zeta) \star d \zeta = F( Q) - F(P).$$
\[thm:topologicalAintegral\] Let $f: U \rightarrow {\mathcal{A}}$ be a function where $U$ is a connected subset of ${\mathcal{A}}$ then the following are equivalent:
1. $\int_{C_1} f \star d\zeta = \int_{C_2} f \star d\zeta$ for all curves $C_1,C_2$ in $U$ beginning and ending at the same points,
2. $\int_C f \star d\zeta =0$ for all loops in $U$,
3. $f$ has an antiderivative $F$ for which $\frac{dF}{d\zeta} = f$ on $U$.
\[coro:adiffloopszero\] **(Cauchy’s Integral Theorem for ${\mathcal{A}}$:)** If $U \subseteq {\mathcal{A}}$ is simply connected then $\int_C f \star d \zeta = 0$ for all loops $C$ in $U$ if and only if $f$ is ${\mathcal{A}}$-differentiable on $U$.
Sequences {#sec:sequences}
=========
In this section we will discuss sequences and the concept of limits and convergence in for a real associative algebra of finite dimension. To do this, we will generalize many of the theorems from real analysis to our context. Much of our generalization parallels Rudin’s *Principles of Mathematical Analysis* [@Rudin].
\[defn:sequence\] A function $f: {\mathbb{N}}\rightarrow {\mathcal{A}}$ is called a [**sequence**]{} in ${\mathcal{A}}$. If $f(n) = x_n$, for $n \in {\mathbb{N}}$, then it is customary to denote the sequence $f$ by the symbol $\{x_n\}$. The values of $f$, that is, the elements $x_n$, are called the [**terms**]{} of the sequence.
Convergence of sequences is measured in terms of the norm $|| \cdot ||$ on ${\mathcal{A}}$.
\[defn:convergence\] Let ${\mathcal{A}}$ be a real associative algebra with a norm $\|\cdot \|$. A sequence $\{p_n\}$ in ${\mathcal{A}}$ is said to converge if there is a point p $\in{\mathcal{A}}$ with the following property: For every $\varepsilon>0$ there is an integer M such that $n \geq M$ implies that $$\|p_n- p\|<\varepsilon.$$ In this case we also say that $\{p_n\}$ converges to p, or that p is the limit of $\{p_n\}$, and we write $p_n\to$p, or $ {\displaystyle}\lim_{n\to\infty} p_n=p$. If $\{p_n\}$ does not converge, then it is said to diverge.
Consider the sequence $\{p_n\}$ in ${ \mathbb{C} }$ defined by $p_n=\frac{in+1+i}{n}$ for all $n \in {\mathbb{N}}.$ Recall $| \cdot |$ defined by $|x+iy| = \sqrt{x^2+y^2}$ provides a norm on ${ \mathbb{C} }$. Consider, $$|p_n-i|=\left|\frac{in+1+i}{n}-i\right|=\frac{|in+1+i-in|}{n}=\frac{|1+i|}{n}=\frac{\sqrt{2}}{n}.$$ But, given $\varepsilon>0$, we know from the Archimedian property of real numbers that there exists $M \in {\mathbb{N}}$ such that $\frac{1}{M}<\frac{\varepsilon}{\sqrt{2}}$. Thus, for all $n\geq M$, we have: $$|p_n-i|=\frac{\sqrt{2}}{n}\leq\frac{\sqrt{2}}{M}<\sqrt{2}\left(\frac{\varepsilon}{\sqrt{2}}\right)=\varepsilon.$$ Thus $\{ p_n \}$ converges to $i$.
Since ${\mathcal{A}}$ is a vector space, a sequence in ${\mathcal{A}}$ is a sequence of vectors over ${ \mathbb{R} }$. Part (iii.) of the Theorem below explains that the convergence of a vector sequence is tied to the convergence of its component sequences relative to a basis. In contrast, Parts (i.), (ii.) and (iv.) of the Theorem below directly resemble the usual results for real sequences,the linearity and multiplicativity of limits, following Theorem 3.3 of [@Rudin].
\[thm:limit laws\] Suppose ${\mathcal{A}}$ is an associative algebra paired with a submultiplicative norm $\|\cdot\|$ and a basis $\{v_1, \dots ,v_N \}$ such that $\| v_i \|=1$ and for all $x=\sum_{i=1}^{N}x^iv_i, |x^i|\leq\|x\|$ for all $i$. Suppose also that $\{s_n\}, \{t_n\}$ are sequences in ${\mathcal{A}}$ with $s_n=\sum_{i=1}^{N} s_n^i v_i, \ t_n = \sum_{j=1}^{N} t_n^j v_j$, and $\lim_{n\to\infty} s_n=s, \lim_{n\to\infty} t_n=t$ where $s=\sum_{i=1}^{N}s^i, t=\sum_{j=1}^{N}t^j$. Then:
> 1. $\lim_{n\to\infty} (s_n+t_n)=s+t$,
>
> 2. $ \lim_{n\to\infty} (\alpha\star s_n)=\alpha\star s$, for any number $\alpha\in{\mathcal{A}}$,
>
> 3. $ \lim_{n\to\infty}s_n=s$ if and only if $lim_{n\to\infty}s_n^i=s^i$ for all $ i =1,2,\dots , N$,
>
> 4. $ \lim_{n\to\infty} (s_n\star t_n)=s\star t$.
>
**Proof:** begin with [**(i.)**]{} and [**(ii.)**]{} suppose $s_n \rightarrow s$ and $t_n \rightarrow t$ with respect to $|| \cdot ||$ on ${\mathcal{A}}$ as described in the Theorem. For $\alpha\neq 0$, given $\varepsilon>0$, there exist integers $N_1, N_2$ such that $$\begin{aligned}
&n\geq N_1\implies \|s_n-s\|<\frac{\varepsilon}{2{m_{{\mathcal{A}}}}\|\alpha\|},\\ \notag
&n\geq N_2\implies \|t_n-t\|<\frac{\varepsilon}{2}.\end{aligned}$$ If $N_3= \text{max} \{ N_1, N_2 \}$ then $n \geq N_3$ implies $$\begin{aligned}
\|(\alpha\star s_n+t_n)-(\alpha\star s+t)\|&\leq\|\alpha\star(s_n-s)\|+\|t_n-t\|\\ \notag
&\leq {m_{{\mathcal{A}}}}\|\alpha\|\|(s_n-s)\|+\|t_n-t\|\\ \notag
&<{m_{{\mathcal{A}}}}\|\alpha\|\frac{\varepsilon}{2{m_{{\mathcal{A}}}}\|\alpha\|}+\frac{\varepsilon}{2}\\ \notag
&=\varepsilon.\end{aligned}$$ Let $\alpha=1$ to obtain (i.) and let $t_n=0$ for all $n\in{\mathbb{N}}$ to obtain (ii.) for $\alpha \neq 0$. If $\alpha=0$, then for any $\varepsilon >0$ we note $|| \alpha\star s_n - \alpha \star s|| = ||0-0||= 0$ hence $ \lim_{n\to\infty} (\alpha\star s_n)=\alpha\star s$.\
Next we give the proof of [**(iii.):**]{}\
($\Rightarrow$) Assume $s_n \rightarrow s$. For $\varepsilon>0$, suppose there exists $M\in{\mathbb{N}}$ such that $n\geq M$ implies $\|s_n-s\|<\varepsilon$. Thus $n\geq M$ implies $ |s_n^i-s^i|\leq \| s_n-s \|<\varepsilon$ and we find $s_n^i \rightarrow s^i$ for all $i=1, \dots N$.\
($\Leftarrow$) Assume $s_n^i \rightarrow s^i$ for all $i=1,2,\dots , N$. Given $\varepsilon>0$, choose $ M_1, M_2,...M_N \in {\mathbb{N}}$ such that $n\geq M_i$ implies $|s_n^i-s^i|<\frac{\varepsilon}{N}$. If $M=\text{max}\{M_1, \dots , M_N \}$ and $n \geq M$ then we find $$\begin{aligned}
\|s_n-s\|&=\left\|\sum_{i=1}^{N}(s_n^i-s^i)v_i \right\|\\ \notag
&\leq\sum_{i=1}^{N}\|(s_n^i-s^i)v_i\|\\ \notag
&=\sum_{i=1}^{N}|s_n^i-s^i|\| v_i\|\\ \notag
&<\sum_{i=1}^{N} \frac{\varepsilon}{N}(1) \\ \notag
&=\varepsilon.\end{aligned}$$ Therefore, $s_n \rightarrow s$ and this completes the proof of [**(iii.)**]{}.\
The proof [**(iv.)**]{} requires some calculation. Let $C_{ijk}$ be constants such that $v_i \star v_j = \sum_k C_{ijk} v_k$ and $\mathbf{C} = \text{max} \{ |C_{ijk}| \ | \ 1 \leq i,j,k \leq n \}$. Applying the triangle inequality we find: $$\label{eqn:ncestimate}
\| v_i \star v_j \| \leq \sum_{k=1}^N \| C_{ijk}v_k \| \leq \sum_{k=1}^N \mathbf{C} \| v_k \| =\mathbf{C} \| v_k \|\sum_{k=1}^N 1 = \mathbf{C}N$$ as $\| v_k \| = 1$ and $\sum_{k=1}^N 1 = N$. Calculate: $$\begin{aligned}
\label{eqn:stackoestimatesI}
\|s_n\star t_n-s\star t\|
&=\bigg{\|}\sum_{i,j}(s_n^i v_i)\star (t_n^j v_j)-\sum_{i,j}(s^i v_i)\star (t^j v_j) \bigg{\|}\\ \notag
&= \bigg{\|}\sum_{i,j}(s_n^it_n^j-s^it^j)(v_i\star v_j) \bigg{\|}\\ \notag
&\leq \sum_{i,j}\left|s_n^it_n^j-s^it^j\right| \| v_i\star v_j \| \\ \notag
&\leq \sum_{i,j}\left|s_n^it_n^j-s^it^j\right|\mathbf{C}N \end{aligned}$$ where we have used Equation \[eqn:ncestimate\] in the last step. Observe that $$s_n^it_n^j-s^it^j=(s_n^i-s^i)(t_n^j-t^j)+s^i(t_n^j-t^j)+t^j(s_n^i-s^i)$$ and apply it to Equation \[eqn:stackoestimatesI\] as to obtain: $$\begin{aligned}
\label{eqn:stackoestimates}
\|s_n\star t_n-s\star t\|
&= \mathbf{C}N\sum_{i,j} \left|(s_n^i-s^i)(t_n^j-t^j)+s^i(t_n^j-t^j)+t^j(s_n^i-s^i)\right|\\ \notag
&\leq \mathbf{C}N\biggl(\sum_{i,j}|s_n^i-s^i||t_n^j-t^j|+\sum_{i,j}|s^i||t_n^j-t^j|+\sum_{i,j}|t^j||s_n^i-s^i|\biggr)\\ \notag
&\leq \mathbf{C}N\biggl(\sum_{i,j}|s_n^i-s^i||t_n^j-t^j|+\sum_{i,j}M_s|t_n^j-t^j|+\sum_{i,j}M_t|s_n^i-s^i|\biggr) \\ \notag
&\leq \mathbf{C}N\biggl(\sum_{i,j}|s_n^i-s^i||t_n^j-t^j|+\sum_{j}NM_s|t_n^j-t^j|+\sum_{i}NM_t|s_n^i-s^i|\biggr)\end{aligned}$$ where $M_s=\text{max}\{s^1, \dots, s^N\}$ and $M_t=\text{max}\{t^1, \dots, t^N\}$. Let $\varepsilon>0$. Assume $s_n \rightarrow s$ and $t_n \rightarrow t$ thus by ([**iii.**]{}) we know the component sequences $s_n^i \rightarrow s^i$ and $t_n^i \rightarrow t^i$. It follows we may choose $N_1^i, N_2^i, N_3^i, N_4^i \in {\mathbb{N}}$ for $i = 1, \dots , N$ such that $$\begin{aligned}
\label{eqn:estimotto}
&n\geq N_1^i \ \ \Rightarrow \ \ |s_n^i-s^i|<\frac{\sqrt{\varepsilon N}}{N^2\sqrt{2\mathbf{C}}}, \\ \notag
&n\geq N_2^i\ \ \ \Rightarrow \ \ |t_n^i-t^i|<\frac{\sqrt{\varepsilon N}}{N^2\sqrt{2\mathbf{C}}},\\ \notag
&n\geq N_3^i\ \ \ \Rightarrow \ \ |s_n^i-s^i|<\frac{\varepsilon}{4N^2M_s\mathbf{C}}, \\ \notag
&n\geq N_4^i\ \ \ \Rightarrow \ \ |t_n^i-t^i|<\frac{\varepsilon}{4N^2M_t\mathbf{C}}.
\end{aligned}$$ If $N_5=\text{max}\{ N_j^i \ | \ j=1,2,3,4, i = 1, \dots , N \}$ then following Equation \[eqn:stackoestimates\] and \[eqn:estimotto\] we find $$\begin{aligned}
\|s_n\star t_n-s\star t\|&\leq \mathbf{C}N\biggl( \sum_{i,j=1}^{N}\frac{\sqrt{\varepsilon N}}{N^2\sqrt{2\mathbf{C}}} \cdot \frac{\sqrt{\varepsilon N}}{N^2\sqrt{2\mathbf{C}}}+\sum_{j=1}^{N}\frac{NM_s\varepsilon}{4N^2M_s\mathbf{C}}+\sum_{i=1}^{N}\frac{NM_t\varepsilon}{4N^2M_t\mathbf{C}}\biggr) \\ \notag
&= \frac{\varepsilon}{2N^2}\sum_{i,j=1}^{N}1+\frac{\varepsilon}{4N}\sum_{j=1}^{N}1+\frac{\varepsilon}{4N}\sum_{i=1}^{N}1 \\ \notag
&=\varepsilon.\end{aligned}$$ Therefore, $s_n \star t_n \rightarrow s \star t$. $\square$\
Continuity of functions on ${\mathcal{A}}$ can be described sequentially.
\[thm:continuity\] The following are equivalent:
1. For all $\varepsilon>0$, there exists $ \delta>0$ such that $0<\|z-z_0\|<\delta$ implies $\|f(z)-f(z_0)\|<\varepsilon$.
2. For all $\{z_n\}$ such that $z_n\rightarrow z_0\ as\ n\rightarrow\infty,\ \lim_{n \to \infty}f(z_n)=f(z_0)$.
[**Proof:**]{} [**(i.)**]{} $\Rightarrow$ [**(ii.)**]{}: Suppose [**(i.)**]{} and let $\{z_n\}$ be a sequence in ${\mathcal{A}}$ such that $\lim_{n\to\infty}z_n=z_0$. Let $\varepsilon>0$. Choose $\delta>0$ such that $\|z-z_0\|<\delta$ implies $\|f(z)-f(z_0)\|<\varepsilon$. As $\{z_n\}$ converges to $z_0$, we know there exists $ M \in {\mathbb{N}}$ such that $n>M\ \Rightarrow \|z_n-z_0\|<\delta$. Observe, if $n>M$ then $\|z_n-z_0\|<\delta$ and thus $\|f(z_n)-f(z)\|<\varepsilon$. Consequently, $\lim_{n\to\infty}f(z_n)=f(z_0)$.\
[**(ii.)**]{} $\Rightarrow$ [**(i.)**]{}: Suppose [**(ii.)**]{} and assume towards a contradiction that there exists $\varepsilon_0>0$ such that for any $\delta>0$, there exists $z^*$ such that $\|z^*-z_0\|<\delta$ and $\|f(z^*)-f(z_0)\|\geq\varepsilon_0$. In particular, this holds for $\delta=1/n$. For each $n\in{\mathbb{N}}$ we choose an element $z_n$ such that $\|z_n-z_0\|<1/n$ and $\|f(z_n)-f(z_0)\|\geq\varepsilon_0$ and thus construct the sequence $\{z_n\}$. Therefore, $z_n\rightarrow z_0$ and $\lim_{n \to \infty}f(z_n)\neq f(z_0)$. But this is a contradiction to [**(ii.)**]{} and conclude that [**(i.)**]{} is true. $\square$
On any algebra ${\mathcal{A}}$ with a submultiplicative norm described above, the function $f:\ {\mathcal{A}}\to{\mathcal{A}}$ defined by $f(z)=z^2 = z \star z$ for all $z\in{\mathcal{A}}$ is continuous. Indeed, for any $z_0\in{\mathcal{A}}$ and any sequence $\{z_n\}$ which converges to $z_0$, we have: using (iv.) of Theorem \[thm:limit laws\], $$\lim_{n\to\infty}f(z_n)=\lim_{n\to\infty}z_n \star z_n=(\lim_{n\to\infty}z_n) \star (\lim_{n\to\infty}z_n)=z_0\star z_0=z_0^2=f(z_0)$$ which, by Theorem 3.5, shows continuity at every point of ${\mathcal{A}}$ and thus continuity on ${\mathcal{A}}$.
We close this section with a valuable concept, based on Definition 3.8 of [@Rudin].
\[def:Cauchy sequence\] A sequence $\{p_n\}$ in an algebra ${\mathcal{A}}$ with a norm $\|\cdot\|$ is said to be a [**Cauchy sequence**]{} if for every $\varepsilon>0$ there is an integer $M$ such that if $n,\ m\geq M$ then $\|p_n-p_m\|<\varepsilon$.
\[thm:Complete\] **(Cauchy Criterion)** If ${\mathcal{A}}$ is an associative algebra paired with a submultiplicative norm $\|\cdot\|$ and a basis $\{ v_1, \dots , v_N\}$ such that for all $x=\sum_{i=1}^{N}x^i v_i, |x^i|<\|x\| $ for each $i=1, \dots, N$, then a sequence $\{p_n\}$ in ${\mathcal{A}}$ is is convergent if and only if it is Cauchy.
**Proof:** ($\Rightarrow$) Suppose the sequence $\{p_n\}$ converges to p. Given $\varepsilon>0$, there exists an integer $ M>0$ such that $n\geq M$ implies $\|p_n-p\|<\frac{\varepsilon}{2}$. Suppose $m,\ n\geq N$ $$\|p_n-p_m\|=\|p_n-p+(p-p_m)\|\leq\|p_n-p\|+\|p_m-p\|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.$$ Thus $\{p_n\}$ is a Cauchy sequence.\
($\Leftarrow$) Suppose the sequence $\{p_n\}$ is Cauchy. Thus, given $\varepsilon>0$, there exists and integer $M>0$ such that $n,\ m>N$ implies $\|p_n-p_m\|<\varepsilon$. Consider, $p_n-p_m=\sum_{i=1}^{n}(p_n^i-p_m^i)v_i$. Thus, given the properties of the basis of ${\mathcal{A}}$, $|p_n^i-p_m^i|<\|p_n-p_m\|<\varepsilon$ for $i=1, \dots , N$. We have shown all the component sequences $\{p_n^i\}$ are Cauchy. We know from real analysis that a real sequence converges if and only if it is Cauchy. Thus, for each $i=1, \dots , N$, $\{p_n^i\}$ is a real Cauchy sequence and there exists $p^i\in{ \mathbb{R} }$ for which $p_n^i \rightarrow p^i$. Thus, defining $p = \sum_{i=1}^N p^i v_i$ we find $p_n \rightarrow p$ by part (iii.) of Theorem \[thm:limit laws\]. $\square$\
If ${\mathcal{A}}$ meets the Cauchy Criterion, we say that it is a complete algebra.
${ \mathbb{R} }^{n\times n}$ with $\| A \| = \sqrt{ \text{trace}( A^TA) }$ is a complete algebra since its norm has the necessary property with respect to the basis of unit-matrices $\{ E_{ij} \}$ defined by $(E_{ij})_{kl} = \delta_{ik}\delta_{jl}$. It is not hard to show $|A_{ij}| \leq \| A \|$ and $\| E_{ij} \| =1$ for all $i,j$.
All the examples we study are complete since it is known that any finite dimensional vector space over ${ \mathbb{R} }$ is complete. For example, see Theorem 2.4-2 on page 73 of [@Kreyszig].
Series {#sec:series}
======
We now consider sequences of sums and the limits of these sequences. The definition and the two theorems that follow mirror Rudin’s 3.21-3.23 in [@Rudin]
\[def:series\] Given a sequence $\{a_n\}$ in an algebra ${\mathcal{A}}$, we call the sequence $\{s_n\}$ where $ {\displaystyle}s_n=\sum_{k=1}^{n}a_k$ the [**sequence of partial sums**]{} of $\{a_n\}$. We denote ${\displaystyle}\lim_{n\to\infty}s_n=\sum_{k=1}^{\infty}a_k$ or $\sum a_k$ and call this limit an infinite series. If $\{s_n\}$ converges to s, we say that the [**series converges**]{} and write $\sum_{k=1}^{\infty}a_k=s$. If $\{s_n\}$ diverges, we say that the [**series diverges**]{}.
Notice if $m \geq n$ and $s_n=\sum_{k=1}^{n}a_k$ then $$s_m-s_n = \sum_{k=1}^{m}a_k - \sum_{k=1}^{n}a_k = \sum_{k=m}^{n}a_k.$$ Hence the Cauchy criterion (Theorem \[thm:Complete\]) applied to the sequence of partial sums provides the following useful test for convergence of a series:
\[thm:Cauchy series\] ( **Cauchy criterion for series**) The series $\sum a_k$ in ${\mathcal{A}}$ converges if and only if for every $\varepsilon>0$ there exists $M \in {\mathbb{N}}$ such that $m\geq n\geq M$ implies $ {\displaystyle}\bigg{\|} \sum_{k=n}^{m}a_k \bigg{\|} \leq\varepsilon$.
In particular, if $\sum a_k$ converges then by taking $m=n$, we find $\|a_n\|\leq\varepsilon$ for all $n \geq M$. Therefore, we find:
\[thm:term test\] **($n$-th Term Test)** If $\sum a_n$ converges then $\lim_{n\to\infty} a_n=0$.
We now develop some theorems to test for convergence of series. The next theorem follows 3.25 of [@Rudin], with a modified second part.
\[thm:comparison\] **(Comparison Test)** If $\|a_n\|\leq c_n$ for $ n\geq N_0$, where $N_0$ is some fixed integer, and if $\sum c_n$ converges, then $\sum a_n$ converges. Likewise, if there exists $b_n \in [0, \infty)$ for which $\sum_n b_n$ diverges and $b_n \leq \| a_n \|$ then $\sum \| a_n \|$ diverges.
[**Proof:**]{} Suppose $\varepsilon>0$. There exists $M\geq N_0$ such that $m\geq n\geq M$ implies $\sum_{k=n}^{m}c_k\leq\varepsilon$, by the Cauchy criterion. Hence $$\bigg{\|} \sum_{k=n}^{m}a_k\bigg{\|} \ \leq\ \sum_{k=n}^{m}\|a_k\|\ \leq\ \sum_{k=n}^{m}c_k\ \leq\ \varepsilon.$$ Therefore $\sum a_k$ converges as we have shown it satisfied Cauchy criterion for series. For the divergent case, since $\sum b_n$ diverges it follows its sequence of partial sums are unbounded and hence $\| a_n \|$ also has a unbounded sequence of partial sums hence $\sum_n \| a_n \|$ diverges. $\square$
In ${ \mathbb{R} }^{2 \times 2}$ where $\left \| \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\right\| = \sqrt{a^2+b^2+c^2+d^2}$, the series ${\displaystyle}\sum_{n=1}^\infty \left[ \begin{array}{rr} 1/n^2 & 2/n^2 \\ 3/n^2 & 4/n^2 \end{array}\right]$ converges. For every $n$, we have $\left\|\left( \begin{array}{rr} 1/n^2 & 2/n^2 \\ 3/n^2 & 4/n^2 \end{array}\right)\right\|=\frac{\sqrt{1+4+9+16}}{n^2}=\frac{\sqrt{30}}{n^2}$. But, we know from real analysis that $\sum_{n=1}^\infty\frac{
\sqrt{30}}{n^2}$ converges, so the result follows by the Comparison Test.
If the series $\sum \|a_n\|$ converges then $\sum a_n$ is said to [**converge absolutely**]{}.
Naturally, we recover the standard meaning of absolute convergence for real series given the choice ${\mathcal{A}}= { \mathbb{R} }$ with $ \| x \| = \sqrt{x^2}$.
In ${ \mathbb{C} }$, the series $\sum\frac{i^n}{n}$ converges, as can be shown in complex analysis, but it does not converge absolutely. Indeed, $\sum|\frac{i^n}{n}|=\sum\frac{1}{n}$ does not converge.
If $\sum \| a_n \|$ converges then $\sum a_n$ converges. In other words, absolute convergence implies ordinary convergence for series in ${\mathcal{A}}$.
[**Proof:**]{} Suppose $\sum \| a_n \|$ converges. Let $n>m$ $$0 \leq \| \sum^n a_k - \sum^m a_k \| = \| \sum_{k=m}^n a_k \| \leq \sum_{k=m}^n \| a_k \|$$ Since $\sum \| a_n \|$ converges we know by Cauchy Criterion $\sum_{k=m}^n \| a_k \| \rightarrow 0$. Hence, $$\| \sum^n a_k - \sum^m a_k \| \rightarrow 0$$ and by Theorem \[thm:Cauchy series\] we find $\sum a_k$ converges. $\Box$\
The next two theorems mirror 3.33 and 3.34 of [@Rudin].
\[thm:numericalRoottest\] **(Root Test)** Given $\sum a_n$, put $\alpha=\limsup_{n\to\infty}\sqrt[n]{\|a_n\|}$. Then:
> 1. if $\alpha<1$, $\sum a_n$ converges absolutely;
>
> 2. if $\alpha>1$, $\sum a_n$ diverges;
>
> 3. if $\alpha=1$, the test gives no information.
>
**Proof:** If $\alpha<1$, we can choose $\beta$ such that $\alpha<\beta<1$, and an integer $M$ such that $$\sqrt[n]{\|a_n\|}<\beta$$ for $n\geq M$. That is, $n\geq M$ implies $$\|a_n\|<\beta^n.$$ Since $0<\beta<1$ we recognize the geometric series $\sum\beta^n$ converges. Convergence of $\sum \| a_n \|$ follows from the comparison test (Theorem \[thm:comparison\]). To prove [**(ii.)**]{} supppose $\alpha >1$ then by definition of limsup there exists a subsequence $\sqrt[n_k]{\|a_{n_k}\|} \rightarrow \alpha >1$ thus $a_n \nrightarrow 0$ hence $\sum a_n$ diverges. [**(iii.)**]{} the usual examples from real calculus suffice. $\Box$\
In ${ \mathbb{C} }$, the series $\sum(\frac{i \cos(n)}{2})^n$ converges by the Root Test. Indeed, we have: $$\alpha=\limsup_{n\to\infty}\sqrt[n]{\left|\frac{i\cos(n)}{2}\right|^n}=\limsup_{n\to\infty}\left|\frac{i \cos(n)}{2}\right|=\limsup_{n\to\infty}\frac{\cos(n)}{2}=\frac{1}{2}<1.$$
\[thm:Ratio test\] **(Ratio Test)** The series $\sum a_n$
> 1. converges absolutely if $\limsup_{n\to\infty}\frac{\|a_{n+1}\|}{\|a_n\|}<1$,
>
> 2. diverges if $\frac{\|a_{n+1}\|}{\|a_n\|}\geq1$ for all $n\geq n_0$, where $n_0$ is some fixed integer.
>
[**Proof:**]{} If condition [**(i.)**]{} holds, we can find $\beta<1$, and an integer $M$, such that $$\frac{\|a_{n+1}\|}{\|a_n\|}<\beta$$ for $n\geq M$. In particular, $$\|a_{M+1}\| <\beta \|a_M\| \ \Rightarrow \
\|a_{M+2}\|<\beta \|a_{M+1}\|<\beta^2 \|a_M\| \ \Rightarrow \ \cdots \ \Rightarrow \ \|a_{M+p}\|<\beta^p \|a_M\|.$$ Thus, $\|a_n\|<\|a_M\|\beta^{-M}\cdot \beta^n$ for $n\geq M$, and it follows from the comparison test that $\sum \|a_n\|$ converges, since $\sum \beta^n$ converges, and thus we obtain [**(i.)**]{}.\
To understand [**(ii.)**]{} suppose $\|a_{n+1}\|\geq \|a_n\| \neq 0$ for $n\geq n_0$, it is easily seen that the condition $a_n\rightarrow 0$ fails and thus [**(ii.)**]{} follows by the $n$-th term test. $\Box$\
Care should be taken with part (ii.), the absence of a limiting process is significant. The knowledge that lim $a_{n+1}/a_n=1$ implies nothing about the convergence of $\sum a_n$. The series $\sum 1/n$ and $\sum 1/n^2$ demonstrate this.
In the quaternions, the series $\sum\frac{(1+i+j+k)^n}{n!}$ converges absolutely by the Ratio Test. Note, $$\alpha=\limsup_{n\to\infty}\frac{\left\|\frac{(1+i+j+k)^{n+1}}{(n+1)!}\right\|}{\left\|\frac{(1+i+j+k)^n}{n!}\right\|}=\limsup_{n\to\infty}\frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}}=\limsup_{n\to\infty}\frac{2}{n+1} = 0 < 1.$$
Finally, we examine the convergence of sums and products of series.
\[thm:Sum of series\] If $\sum a_n$ and $\sum b_n$ converge and $c \in {\mathcal{A}}$ then $$\sum (a_n+b_n) = \sum a_n+\sum b_n \qquad \& \qquad \sum c \star a_n = c \star \sum a_n.$$
[**Proof:**]{} Suppose there exist $A,B \in {\mathcal{A}}$ for which $\sum a_n = A$ and $\sum b_n = B$. Partial sums $$A_n=\sum_{k=0}^{n}a_k \qquad \& \qquad B_n=\sum_{k=0}^{n}b_k$$ have $A_n \rightarrow A$ and $B_n \rightarrow B$. By Theorem \[thm:limit laws\] we calculate for $c \in {\mathcal{A}}$, $$c\star A_n+B_n \rightarrow c \star A+B.$$ This proves the theorem. $\Box$\
The multiplication of two series is understood in terms of the Cauchy product.
Given $\sum a_n=A$, and $\sum b_n=B$, we set $$c_n=\sum_{k=0}^{n}a_k \star b_{n-k}$$ for $n\in{\mathbb{N}}$ and call $\sum c_n$ the [**product**]{} of the two given series.
Absolute convergence is useful to study the existence of the product of two series. In particular, we show that the product of a convergent series with an absolutely convergent series converges to the Cauchy product:
\[thm:productofseries\] Suppose
> 1. $\sum_{n=0}^{\infty}a_n\ converges\ absolutely,$
>
> 2. $ \sum_{n=0}^{\infty}a_n=A \in {\mathcal{A}}$ and $\sum_{n=0}^{\infty}b_n=B \in {\mathcal{A}}$,
>
> 3. $ c_n=\sum_{k=0}^{n}a_k \star b_{n-k}$ for all $n\in{\mathbb{N}}$.
>
Then $\sum_{n=0}^{\infty}c_n=A \star B$.
[**Proof:**]{} assume [**(i.)**]{}, [**(ii.)**]{} and [**(iii.)**]{} from the statement of the Theorem are given. Let $$A_n=\sum_{k=0}^{n}a_k,\ \ \ \ B_n=\sum_{k=0}^{n}b_k,\ \ \ \ C_n=\sum_{k=0}^{n}c_k,\ \ \ \ \beta_n=B_n-B.$$ Then notice we can rearrange the terms in the finite sum $C_n$ as follows: $$\begin{aligned}
C_n&=a_0 \star b_0+(a_0 \star b_1+a_1 \star b_0)+\dots+(a_0 \star b_n+a_1 \star b_{n-1}+\dots +a_n \star b_0)\\ \notag
&=a_0 \star B_n+a_1 \star B_{n-1}+\dots+a_n \star B_0\\ \notag
&=a_0 \star (B+\beta_n)+a_1 \star (B+\beta_{n-1})+\dots+a_n \star \beta_0\\ \notag
&=A_n \star B+\underbrace{a_0 \star \beta_n+a_1 \star \beta_{n-1}+\dots+a_n \star \beta_0}_{ \gamma_n}.\end{aligned}$$ We wish to show that $C_n\rightarrow A\star B$. Since $A_n \star B\rightarrow A\star B$, it suffices to show that $\gamma_n \rightarrow 0$. Since $\sum a_n$ converges absolutely, we know there exists $\alpha \in [0,\infty)$ for which $$\alpha=\sum_{n=0}^{\infty}\|a_n\|.$$ Let $\varepsilon>0$ be given. By [**(ii.)**]{}, $\beta_n\rightarrow 0$. Hence we can choose $M \in {\mathbb{N}}$ such that $\|\beta_n\|\leq\frac{\varepsilon}{\alpha{m_{{\mathcal{A}}}}}$ for $n\geq M$, in which case $$\begin{aligned}
\|\gamma_n\|&\leq\|a_n \star \beta_0+\dots+a_{n-M} \star \beta_M\|+\|a_{n-M-1}\star \beta_{M+1}+\dots+a_0 \star \beta_n\|\\ \notag
&\leq\|a_n\star\beta_0+\dots+a_{n-M}\star \beta_M\|+{m_{{\mathcal{A}}}}\|a_{n-M-1} \|\| \beta_{M+1} \|+\dots+{m_{{\mathcal{A}}}}\| a_0 \| \| \beta_n\|\\ \notag
&\leq\|a_n \star \beta_0+\dots+a_{n-M}\star \beta_M\|+{m_{{\mathcal{A}}}}\|a_{n-M-1} \|\frac{\varepsilon}{\alpha{m_{{\mathcal{A}}}}}+\dots+{m_{{\mathcal{A}}}}\| a_0 \| \frac{\varepsilon}{\alpha{m_{{\mathcal{A}}}}} \\ \notag
&\leq\|a_n \star\beta_0+\dots+a_{n-M}\star\beta_M\|+\left( \|a_{n-M-1} \| +\dots+\| a_0 \| \right) \frac{\varepsilon}{\alpha} \\ \notag
&\leq\|a_n\beta_0+\dots+a_{n-M}\beta_M\|+\varepsilon.\end{aligned}$$ Fix $M$ and let $n\rightarrow\infty$, we find $$\limsup_{n\to\infty}\|\gamma_n\|\leq \varepsilon$$ since $a_k\rightarrow 0$ as $k\rightarrow\infty$. Since $\varepsilon$ is arbitrary, we find $\gamma_n \rightarrow 0$ and the Theorem follows. $\square$\
If ${\mathcal{A}}$ is not commutative then it is possible that $A \star B \neq B \star A$. However, it is clear that a similar argument could be given if we were instead given the absolute convergence of $\sum b_n$. Consequently, the product of two convergent series converges to the product of their sums if at least one of the two series converges absolutely.
Power series {#sec:powerseries}
============
Suppose there exists $z_0 \in {\mathcal{A}}$ and $c_0,c_1, c_2, \dots \in {\mathcal{A}}$ and $$f(z) = \sum_{n=0}^{\infty}c_n \star (z-z_0)^n$$ for each $z \in {\mathcal{A}}$ for which the series converges. Then we say $f(z)$ is a [**power series centered at $z_0$ in ${\mathcal{A}}$**]{} with [**coefficients**]{} $c_n$.
The domain of a power series is controlled by both its coefficients and its center. If we change the center while holding the coefficients fixed then the domain is modified by translation.
\[lem:centersub\] If $f(z) = \sum_{n=0}^{\infty} c_n \star (z-z_1)^n$ converges on $U \subseteq {\mathcal{A}}$ then $g(z) = \sum_{n=0}^{\infty} c_n \star (z-z_2)^n$ converges on $z_2-z_1+U = \{ z_2-z_1+u \ | \ u \in U \}$. If $\sum_{n=0}^{\infty} c_n \star z^n$ converges on $U$ then $\sum_{n=0}^{\infty} c_n \star (z-z_0)^n$ converges on $z_0+U$.
[**Proof:**]{} Suppose $x = z_2-z_1+z$ for $z \in U$ then $x-z_2 = z-z_1$ hence $$g(x) = \sum_{n=0}^{\infty} c_n \star (z-z_1)^n = f(z).$$ Since and $f(z)$ exists for $z \in U$ we find $g(x)$ exists for each $x \in z_2-z_1+U$. Finally, set $z_1=0$ and $z_2=z_0$ to obtain the last claim of the Lemma. $\Box$\
The result below differs from the usual Root Test of real or complex series in that the test does not guarantee divergence for $ \| z \| > R$.
**(Root Test for Power Series)** \[thm:roottestpowerseries\] Given an algebra ${\mathcal{A}}$ with $ \| x \star y \| \leq {m_{{\mathcal{A}}}}\|x \| \|y \|$ for all $x,y \in {\mathcal{A}}$ and power series $\sum c_n \star (z-z_o)^n$ in ${\mathcal{A}}$, let $$\alpha = \limsup_{n\to\infty}\sqrt[n]{\|c_n\|} \qquad \& \qquad \ R=\frac{1}{ {m_{{\mathcal{A}}}}\alpha }.$$ Then $\sum c_n \star (z-z_o)^n$ is absolutely convergent for $\|z-z_o\|<R$. Moreover, if $\alpha=0$ then $\sum c_n \star (z-z_o)^n$ converges absolutely on ${\mathcal{A}}$.
[**Proof:**]{} Let us study power series $\sum c_n \star z^n$ in ${\mathcal{A}}$ centered at $0$. Let $a_n=c_n \star z^n$ and seek to apply Theorem \[thm:numericalRoottest\]. Consider: $$\begin{aligned}
\limsup_{n\to\infty}\sqrt[n]{\|a_n\|}&=\limsup_{n\to\infty}\sqrt[n]{\|c_n \star z^n \|} \\ \notag
&\leq\limsup_{n\to\infty}\sqrt[n]{m_{\mathcal{A}}\|z^n\| \| c_n\|}\\ \notag
&\leq\limsup_{n\to\infty}\sqrt[n]{m_{\mathcal{A}}^n\|z\|^n\| c_n\|} \ \ \ \text{(applied Proposition \ref{prop:ineqnpower})}\\ \notag
&=m_{\mathcal{A}}\|z\|\limsup_{n\to\infty}\sqrt[n]{\|c_n\|} \\ \notag
&=\frac{\|z\|}{R}.\end{aligned}$$ Hence, the series is absolutely convergent if $\| z \| < R$. If $\alpha=0$ then Theorem \[thm:numericalRoottest\] provides absolute convergence at each $z \in {\mathcal{A}}$. Finally, Lemma \[lem:centersub\] completes the proof for $z_o \neq 0$. $\Box$\
In the theory of power series over the real or complex numbers the root test provides a boundary between points of convergence and divergence. However, we will see in Example \[exa:bandconverge\] the appearance of zero divisors makes it is possible to find additional points of convergence beyond those indicated by the root test.
\[thm:norootcriterianeeded\] Suppose that $\sum c_n \star (z-z_o)^n$ converges for all $\|z-z_o\|<R$ for some $R>0$. Then $\alpha=\limsup_{n\to\infty}\sqrt[n]{\|c_n\|} \leq \frac{\| 1 \|}{R}$.
[**Proof:**]{} Consider $z = z_o+\frac{\mathds{1}}{\| \mathds{1} \|}b$ where $0< b < R$ and $\mathds{1}$ denotes the unity in the algebra. Notice $\| z-z_o \| = b < R$ . Note, if $\alpha=\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}$ then $$\limsup_{n\to\infty}\sqrt[n]{\|c_n\star (z-z_o)^n\|}
= \frac{b}{\| \mathds{1}\|}\limsup_{n\to\infty}\sqrt[n]{\left\|c_n \right\|} = \frac{b \alpha}{\| \mathds{1} \|} \leq 1$$ where we used Theorem \[thm:numericalRoottest\] to obtain the above inequality. Thus $\alpha \leq \frac{\| 1 \|}{b}$ for arbitrary $b \in (0,R)$ and it follows that $\alpha \leq \frac{\| 1 \|}{R}$. $\Box$
\[exa:bandconverge\] Consider $f(z) = \sum_{n=0}^{ \infty} (1+j)z^n$ over the hyperbolic numbers ${\mathcal{H}}= { \mathbb{R} }\oplus j { \mathbb{R} }$ where $j^2=1$. Observe $c_n= 1+j$ for all $n$ hence $\| c_n \| = \sqrt{2}$ and the root test applies: $$\alpha = \limsup_{n\to\infty}\sqrt[n]{\|c_n\|} = \lim_{ n \to \infty} 2^{1/2n} = 1.$$ Therefore, the Root Test provides for absolute convergence of this series where $\| z \| < 1/{m_{{\mathcal{A}}}}\alpha = 1 /\sqrt{2}$ as $\| zw \| \leq \sqrt{2} \| z \| \, \|w \|$ for hyperbolic numbers. Yet, consider $z = a(1-j)$ where $a \in { \mathbb{R} }$, $$f( a(1-j) ) = (1+j)(1)+ a(1+j)(1-j) + a^2(1+j)(1-j)^2+ \cdots = 1+j$$ as $(1+j)(1-j)=0$. Therefore, $f(z)$ converges along the line $y=-x$ in the hyperbolic numbers where $z= x+jy$. Moreover, we can show this result extends to a whole band of nearby hyperbolic numbers. Consider points near $y=-x$ of the form $w=a+\varepsilon+j(\varepsilon-a)$. Such a point is distance $\sqrt{2} |\varepsilon|$ from $y=-x$. It is helpful to note the following identity: $$(1+j)(x+yj) = (1+j)(x+y) \ \ \& \ \ (1+j)(x+yj)^n = (1+j)(x+y)^n$$ for all $n \in {\mathbb{N}}$. Thus, for $\varepsilon, a \in { \mathbb{R} }$, $$\|(1+j)w^n\| = \|(1+j)(a+\varepsilon + \varepsilon-a)^n\| = \sqrt{2}\, |2\varepsilon|^n.$$ Using the root test for the numerical series $\sum (1+j)w^n$, we obtain $$\alpha = \limsup_{n\to\infty}\sqrt[n]{\|c_n\|} = \lim_{ n \to \infty} |2\varepsilon| 2^{1/2n} = 2|\varepsilon|.$$ This series converges for $|\varepsilon|<\frac12$ and thus $f(z$) converges for all $z$ at a distance of $\sqrt{2}|\varepsilon| < \frac1{\sqrt{2}}$ from the line $y=-x$. Observe, this domain of convergence includes the disk $\|z\|<\frac1{\sqrt{2}}$ which we obtained via the root test, and, an infinite band of width $\sqrt{2}$ about the $y=-x$ line.
Note, ${m_{{\mathcal{A}}}}= 1$ for ${\mathcal{A}}= { \mathbb{R} }$ with $|x| = \sqrt{x^2}$ or ${\mathcal{A}}= { \mathbb{C} }$ with $\| z \| = \sqrt{z \bar{z}}$ hence the result below is a natural extension of the usual geometric series from real or complex analysis; the radius of the domain for a geometric series in ${\mathcal{A}}$-calculus depends inversely on ${m_{{\mathcal{A}}}}$.
\[thm:geometric\] For any commutative[^3] algebra ${\mathcal{A}}$, the geometric series $\sum z^n$ converges to $(1 - z)^{-1}$ for all $z\in{\mathcal{A}}$ such that $1-z\in{\mathcal{A}}^\times$ and $\|z\|< \frac1{m_{{\mathcal{A}}}}$.
**Proof:** The fact that the geometric series converges follows from the root test. To show that it converges to $(1-z)^{-1}$, let $S_n = \sum_{k=0}^n z^k$ and $z$ meeting the conditions in the theorem. Observe, $$\begin{aligned}
S_n(1-z) &= S_n-zS_n \\ \notag
&=1+z+z^2+...+z^n-z-z^2-...-z^n-z^{n+1}\\ \notag
&=1-z^{n+1}\end{aligned}$$ and hence $S_n = (1-z^{n+1})(1 - z)^{-1}$ for all $n$ for which $(1-z)^{-1} \in {\mathcal{A}}^{\times}$. Thus, $$\begin{aligned}
\|S_n-(1 - z)^{-1}\| &= \|(1-z^{n+1})(1 - z)^{-1} - (1 - z)^{-1}\|\\ \notag
&=\|z^{n+1}(1 - z)^{-1}\|\\ \notag
&\leq {m_{{\mathcal{A}}}}\|z^{n+1}\|\|(1 - z)^{-1}\|\\ \notag
&\leq {m_{{\mathcal{A}}}}^{n+1}\|z\|^{n+1}\|(1 - z)^{-1}\| \ \ \ \text{(applied Proposition \ref{prop:ineqnpower})}\end{aligned}$$ which can be made sufficiently small for $\|z\|<\frac1{m_{{\mathcal{A}}}}$. $\square$\
However, the domain defined in the theorem above imay fail to be maximal. Consider:
\[exa:directprodgeom\] Consider ${\mathcal{A}}= { \mathbb{R} }\times { \mathbb{R} }$ where $(a,b) \star (x,y) = (ax,by)$ hence $(1,0) \star (0,1) = (0,0)$ are zero-divisors. Observe ${\mathcal{A}}^{ \times} = \{ (a,b) \ | \ ab \neq 0 \}$. Consider $z = (x,y)$ and $$\sum_{n=0}^{ \infty} z^n = \sum_{n=0}^{ \infty}(x^n,y^n) = \left( \sum_{n=0}^{ \infty}x^n,\sum_{n=0}^{ \infty}y^n\right).$$ The component series are geometric series with radii $x$ and $y$ respective. The series $\sum_{n=0}^{ \infty} z^n$ converges when both its component series converge. In particular, we need $|x|<1$ and $|y|<1$. We find the geometric series in ${ \mathbb{R} }\times { \mathbb{R} }$ converges on the square $(-1,1)^2$. If we set $\| (x,y) \| = \sqrt{x^2+y^2}$ then it can be shown that $\| z \star w \| \leq \| z \| \|w \|$ hence ${m_{{\mathcal{A}}}}= 1$ for ${\mathcal{A}}= { \mathbb{R} }\times { \mathbb{R} }$ thus Theorem \[thm:geometric\] only provides convergence on the disk $\| z \| <1$.
In what follows we use the result of Example \[exa:bandconverge\] to derive a result which is directly related to Example \[exa:directprodgeom\] via the isomorphism of Proposition \[prop:isomorphismhyperbolic\].
In Example \[exa:bandconverge\] we saw $f(z) =\sum_{n=0}^{\infty}(1+j)z^n$ converged on the infinite band $$B_+ = \{ x+jy \ | \ -1-x \leq y \leq 1-x \}.$$ By entirely similar arguments, $g(z) = \sum_{n=0}^{\infty}(1-j)z^n$ will converge on $$B_- = \{ x+jy \ | \ -1+x \leq y \leq 1+x \}.$$ Note, $h(z) = f(z)+g(z)$ is defined for each $z \in B_+ \cap B_-$. In particular, $$h(z) = 2 \sum_{n=0}^{ \infty} z^n$$ hence we find the geometric series in the hyperbolic numbers converges on a diamond with vertices $\pm 1, \pm j$. Notice, Theorem \[thm:geometric\] merely provides convergence on the inscribed disk; $\| z \| < \frac{1}{{m_{{\mathcal{A}}}}} = \frac{1}{\sqrt{2}}$. Furthermore, the diamond with vertices $\pm 1$ and $\pm j$ is the image of the square $[-1,1]^2$ under the linear isomorphism $\Psi$ of Proposition \[prop:isomorphismhyperbolic\].
The interplay between ${ \mathbb{R} }\times { \mathbb{R} }$ with the direct product and the hyperbolic numbers ${ \mathbb{R} }\oplus j { \mathbb{R} }$ is illustrative of an important calculational technique. It is often wise to exchange a problem in analysis in one algebra for a more lucid problem in an isomorphic algebra.\
Up to this point many of our theorems are likely true for noncommutative algebras. However, to discuss fractions in noncommutative algebras we would need to consider left and right divisors. We leave the noncommuting case to a future work.
(**Ratio Test for Series with Unit-Coefficients** \[thm:algebraratiotest\] Suppose ${\mathcal{A}}$ is an algebra with $ \| x \star y \| \leq {m_{{\mathcal{A}}}}\|x \| \|y \|$ for all $x,y \in {\mathcal{A}}$ and power series $\sum c_ n \star (z-z_o)^n$ where $c_n \in {\mathcal{A}}^\times$ for all $n$. Let $$\alpha = \limsup_{n\to\infty}\left\|\frac{c_{n+1}}{c_n} \right\| \qquad \& \qquad R=\frac{1}{m_{\mathcal{A}}^2 \alpha}.$$ Then $\sum c_n \star (z-z_o)^n$ is absolutely convergent for $z-z_o \in {\mathcal{A}}^\times$ with $\|z-z_o\|<R$. Moreover, if $\alpha=0$ then $\sum c_n \star (z-z_o)^n$ converges absolutely on ${\mathcal{A}}$.
[**Proof:**]{} Suppose $z_o=0$ and set $a_n=c_n \star z^n$ where $c_n,z \in {\mathcal{A}^{\times}}$ and note $a_n =c_n \star z^n \in {\mathcal{A}^{\times}}$ hence we are free to apply the ratio test for numerical series in ${\mathcal{A}}$: $$\begin{aligned}
\limsup_{n\to\infty}\frac{\|a_{n+1}\|}{\|a_n\|} &=
\limsup_{n\to\infty}\frac{\|c_{n+1} \star z^{n+1}\|}{\|c_n \star z^n\|} \\ \notag
&\leq \limsup_{n\to\infty}{m_{{\mathcal{A}}}}\left\|\frac{c_{n+1} \star z^{n+1}}{c_n \star z^n} \right\| \ \ \ \text{(by Corollary \ref{thm:quotientinequality})} \\ \notag
&= {m_{{\mathcal{A}}}}\limsup_{n\to\infty} \left\|\frac{c_{n+1}}{c_n}\star z \right\| \\ \notag
&= {m_{{\mathcal{A}}}}\limsup_{n\to\infty}{m_{{\mathcal{A}}}}\left\|\frac{c_{n+1}}{c_n} \right\| \| z \| \\ \notag
&= {m_{{\mathcal{A}}}}^2\| z \| \limsup_{n\to\infty} \left\|\frac{c_{n+1}}{c_n} \right\| \\ \notag
&= {m_{{\mathcal{A}}}}^2 \| z \| \alpha.\end{aligned}$$ Since $R =\frac{1}{m_{\mathcal{A}}^2 \alpha}$ the condition ${m_{{\mathcal{A}}}}^2 \| z \| \alpha<1$ is equivalent to $\|z\| < R$ and Theorem \[thm:Ratio test\] allows us to conclude that the series converges absolutely for all $z \in \mathcal{A}^\times$ such that $\|z\| < R$. Finally, apply Lemma \[lem:centersub\] to extend the proof to $z_o \neq 0$. $\Box$
Consider ${\mathcal{H}}= { \mathbb{R} }\oplus j { \mathbb{R} }$ where $j^2=1$. Form the hyperbolic power series $$f(z) = \sum_{n=0}^{ \infty} (1+j)n! z^n.$$ Observe, $$f( c(1-j)) = \sum_{n=0}^{ \infty} (1+j)n! c^n(1-j)^n = 1+j$$ as $(1-j)(1+j)=0$ implies the terms with $n \geq 1$ all vanish. Thus the power series converges on $S=\{ c(1+j) \ | \ c \in { \mathbb{R} }\}$. However, if $z \notin S$ then $(1+j)n! z^n \in {\mathcal{H}}^{ \times}$ and it can be shown $\lim_{n \rightarrow \infty} (1+j)n! z^n \neq 0$ thus $f(z)$ diverges outside $S$.
The Example above generalizes to other algebras. We can construct power series which converge on the zero-divisors or some subset of the zero-divisors and yet diverge everywhere else. Zero-divisors are simply beyond the scope of the ratio test for general power series in ${\mathcal{A}}$. Furthermore, Example \[exa:bandconverge\] illustrates that the domain of convergence is not governed by root test alone. Interesting things can happen along zero divisors in the algebra. For example, we suspect $f(\zeta) = \sum_n (1+j+j^2)\zeta^n$ where $\zeta = x+yj+zj^2$ and $j^3=1$ converges on the infinite slab of thickness $2$ centered about the plane $x+y+z=0$.\
In contrast to Theorem \[thm:algebraratiotest\], the Theorem below can give us information about the convergence of the series at zero-divisors of the algebra.
(**Ratio Test for Series with Real Coefficients** )\[thm:algebraratiotestII\] Let ${\mathcal{A}}$ be an algebra with $ \| x \star y \| \leq {m_{{\mathcal{A}}}}\|x \| \|y \|$ for all $x,y \in {\mathcal{A}}$. power series $\sum c_n \star (z-z_o)^n$ where $0 \neq c_k \in \mathbb{R}$ for all $k \in \mathbb{N}$, put $$\alpha = \limsup_{n\to\infty}\frac{|c_{n+1}|}{|c_n|},\ \ \ R=\frac{1}{m_{\mathcal{A}}\alpha}.$$ Then $\sum c_n \star (z-z_o)^n$ converges absolutely for $\|z-z_o\|<R$. Moreover, if $\alpha=0$ then $\sum c_n \star (z-z_o)^n$ converges absolutely on ${\mathcal{A}}$.
[**Proof:**]{} Set $z_o=0$ to begin. If $c_n \in { \mathbb{R} }$ then $ \| c_n \star z^n \| = |c_n| \| z^n \|$ . Put $a_n=c_n \star z^n$, and work towards applying the ratio test (Theorem \[thm:Ratio test\]): $$\begin{aligned}
\limsup_{n\to\infty}\frac{\|a_{n+1}\|}{\|a_n\|} &= \limsup_{n\to\infty}\frac{\|c_{n+1} z^{n+1}\|}{\|c_n z^n\|} \\ \notag
&= \limsup_{n\to\infty}\frac{|c_{n+1}|}{|c_n|} \frac{\| z^{n+1} \|}{\| z^n \|} \\ \notag
&\leq \limsup_{n\to\infty}\frac{|c_{n+1}|}{|c_n|} \frac{ {m_{{\mathcal{A}}}}\|z\|\| z^{n} \|}{\| z^n \|} \\ \notag
&\leq {m_{{\mathcal{A}}}}\|z\| \limsup_{n\to\infty}\frac{|c_{n+1}|}{|c_n|} \\ \notag
&= {m_{{\mathcal{A}}}}\|z\| \alpha. \end{aligned}$$ Since $R=\frac{1}{m_{\mathcal{A}}\alpha}$ we find ${m_{{\mathcal{A}}}}\|z\| \alpha < 1$ provides $ \| z \| < R$. Thus, applying Theorem \[thm:Ratio test\], the series converges for all $z \in \mathcal{A}^\times$ such that $\|z\| < R$. To conclude we apply Lemma \[lem:centersub\] to extend the proof to $z_o \neq 0$. $\Box$\
Next we compare and constrast the convergence indicated by Theorem \[thm:roottestpowerseries\], \[thm:algebraratiotest\] and \[thm:algebraratiotestII\].
Consider $\sum_{n=1}^{\infty} \frac{3^n}{n} z^n$ for $z \in {\mathcal{H}}= { \mathbb{R} }\oplus j { \mathbb{R} }$ with $j^2=1$. Notice, $c_n = \frac{3^n}{n}$ are real and recall ${m_{{\mathcal{A}}}}= \sqrt{2}$ for the hyperbolic numbers we consider here. Use $R_x$ to denote the radius of convergence suggested by Theorem $x$. We find: $$\left \| \frac{c_{n+1}}{c_n} \right \| = \frac{3n}{n+1} \rightarrow 3 \ \ \Rightarrow \ \ R_{\ref{thm:algebraratiotestII}} = \frac{1}{6} \ \ \& \ \ R_{\ref{thm:algebraratiotest}} = \frac{1}{3\sqrt{2}}$$ and, $$\sqrt[n]{\| c_n \|} = \sqrt[n]{\frac{3^n}{n}} = \frac{3}{n^{1/n}} \rightarrow 3
\ \ \Rightarrow \ \ R_{\ref{thm:roottestpowerseries}} = \frac{1}{3\sqrt{2}}.$$ Naturally, the root test Theorem \[thm:roottestpowerseries\] and the real-coefficient ratio test Theorem \[thm:algebraratiotestII\] both provide convergence of the series for $ \| z \| < \frac{1}{3 \sqrt{2}}$. However, Theorem \[thm:algebraratiotest\] only indicates convergence for $ \| z \| < \frac{1}{6}$. There is nothing illogical about this as Theorem \[thm:algebraratiotest\] is silent concerning the divergence of the series.
Next we study sequences and series of functions. In particular, we apply these results to power functions to gain insight into the theory of power series in ${\mathcal{A}}$.
Suppose $\{f_n\}$ is a sequence of functions $f_n: E \rightarrow {\mathcal{A}}$ where $E \subseteq {\mathcal{A}}$, and suppose the sequence of numbers $\{f_n(z)\}$ converges for every $z \in E$. Then $f(z)=\lim_{n\to\infty}f_n(z) $ defines $f: E \rightarrow {\mathcal{A}}$. We say that $\{f_n\}$ converges to $f$ [**pointwise**]{}.
Pointwise convergence does not always guarantee properties of the sequence transfer to the limit. For example, it is possible to have the limit function of a sequence of continuous functions which is discontinuous. To remedy this shortcoming of pointwise convergence we the stronger criteria of uniform convergence:
We say that a sequence of functions $\{f_n\}$ on $E \subseteq {\mathcal{A}}$ [**converges uniformly to $f: E \rightarrow {\mathcal{A}}$**]{} if for every $\varepsilon>0$ there is an integer $M$ such that $n\geq M$ implies $\|f_n(z)-f(z)\| < \varepsilon$ for all $z\in E$.
Similar terminology is given for series of functions on ${\mathcal{A}}$. We say that the [**series $\sum f_n(z)$ converges uniformly**]{} on $E$ if the sequence $\{s_n\}$ of partial sums defined by $$\sum_{i=1}^{n}f_i(z)=s_n(z)$$ converges uniformly on E. There is also a Cauchy criterion for uniform convergence:
\[thm:cauchycriterionfnct\] A sequence of functions $\{f_n\}$ defined on $E \subseteq {\mathcal{A}}$ converges uniformly on $E$ if and only if for every $\varepsilon>0$ there exists $M \in {\mathbb{N}}$ such that $m,n \geq M$ and $z\in E$ implies $$\|f_n(z)-f_m(z)\| < \varepsilon.$$
[**Proof:**]{} If $f_n \rightarrow f$ uniformly on $E$ then there exists an integer $M$ such that $n\geq M$ and $z\in E$ imply $$\|f_n(z)-f(z)\| < \frac{\varepsilon}{2}.$$ Suppose $n, m \geq M$ and $z\in E$ and consider that $$\|f_n(z)-f_m(z)\|\leq\|f_n(z)-f(z)\|+\|f_m(z)-f(z)\| < \varepsilon/2+\varepsilon/2 = \varepsilon.$$ Conversely, suppose the Cauchy condition holds. As ${\mathcal{A}}$ is complete, the sequence $\{f_n(z)\}$ converges for every $z$, to a limit we may call $f(z)$. Thus $f_n \rightarrow f$ on $E$. It remains to show this convergence is uniform. Let $\varepsilon >0$ be given, and choose $M \in {\mathbb{N}}$ such that $\|f_n(z)-f_m(z)\|< \varepsilon$. Fix $n$ and let $m \rightarrow \infty$. Since $f_m(z)\rightarrow f(z)$ as $m\rightarrow\infty$, this gives $$\|f_n(z)-f(z) \| < \varepsilon$$ for every $n\geq M$ and $z\in E$ hence $f_n \rightarrow f$ uniformly on $E$. $\square$\
Weierstrauss’ taught us that uniform convergence of series of functions on $E \subseteq { \mathbb{C} }$ can be derived from the existence of a [*majorizing series*]{}. In particular, if a convergent numerical series bounds the values of the function on $E$ then the series of functions converges uniformly on $E$. This is often known as Weierstrauss $M$-test.
\[thm:weierstraussM\] ( [**Weierstrauss $M$-Test for ${\mathcal{A}}$**]{} ) Suppose $\{f_n(z)\}$ is a sequence of functions defined on $E$. If $\sum M_n$ is a convergent series in ${ \mathbb{R} }$ and $ \|f_n(z) \|\leq M_n$ for all $z\in E$ and $n\in{\mathbb{N}}$ then $\sum f_n$ converges uniformly on $E$.
[**Proof:**]{} Assume $ \|f_k(z) \|\leq M_k$ for each $k \in {\mathbb{N}}$ and note for $m \geq n$: $$\left\|\sum_{i=1}^{m}f_i(z) - \sum_{i=1}^{n}f_i(z) \right\| = \left\|\sum_{i=n}^{m}f_i(z) \right\| \leq \sum_{i=n}^{m} \left\|f_i(z) \right\| \leq \sum_{i=n}^{m} M_n$$ for each $z \in E$. Furthermore, by convergence of $\sum M_n$, for each $\varepsilon>0$ we may select $M \in {\mathbb{N}}$ for which $m,n \geq M$ imply $\sum_{i=n}^{m} M_n < \varepsilon$. Consequently, the conditions of Theorem \[thm:cauchycriterionfnct\] are met and we conclude $\sum f_n$ converges uniformly on $E$. $\square$
\[thm:pwrseriesnormalconvergence\] If $\sum c_n \star (z-z_o)^n$ is a power series on ${\mathcal{A}}$ and $$\alpha = \limsup_{n\to\infty}\sqrt[n]{\|c_n\|} \qquad \& \qquad \ R=\frac{1}{ {m_{{\mathcal{A}}}}\alpha }.$$ then $\sum_n c_n \star (z-z_o)^n $ is uniformly absolutely convergent for $\|z-z_o\| \leq R- \varepsilon$ for any $\varepsilon \in (0, R)$. If R is infinite, then the series is uniformly absolutely convergent for $\|z\|\leq L$ for any $L>0$.
[**Proof:**]{} absolute convergence is given by Theorem \[thm:roottestpowerseries\]. Let $\varepsilon \in (0, R)$ and choose $z_1$ such that $z_1-z_o = {m_{{\mathcal{A}}}}(R- \varepsilon) > 0$. Note $z_1-z_o$ is by construction real and $z_1-z_o < {m_{{\mathcal{A}}}}R = 1/\alpha$ hence: $$\limsup_{n\to\infty}\sqrt[n]{\|c_n \star (z_1-z_o)^n\|} = (z_1-z_o)\limsup_{n\to\infty}\sqrt[n]{\|c_n\|} < \alpha /\alpha = 1.$$ Therefore, $\sum_n \| c_n \star (z_1-z_o)^n \|$ converges by Theorem \[thm:numericalRoottest\]. For, $z$ with $\| z-z_o \| \leq R- \varepsilon$, $$\begin{aligned}
\| c_n \star (z-z_o)^n \| &\leq {m_{{\mathcal{A}}}}^n \|c_n\| \|z-z_o\|^n \\ \notag
&\leq \|c_n \| {m_{{\mathcal{A}}}}^n(R- \varepsilon)^n \\ \notag
&= \|c_n \| (z_1-z_o)^n \\ \notag
&=\|c_n\star (z_1-z_o)^n\|
\end{aligned}$$ for each $n$ thus by the Weierstrauss $M$-Test (\[thm:weierstraussM\]) the theorem follows $\Box$.\
Integration in ${\mathcal{A}}$ and uniform limits can be interchanged:
\[thm:uniformcontinuityintegral\] Suppose $C$ be a piecewise smooth curve of length $L< \infty$ in ${\mathcal{A}}$ and suppose $U$ is an open set containing $C$. If $\{ f_j \}$ is a sequence of continuous ${\mathcal{A}}$-valued functions on $U$, and if $\{ f_j \}$ converges uniformly to $f$ on $U$ then $\int_{C} f_j(z) \star dz$ converges to $\int_{C} f(z) \star dz$.
[**Proof:**]{} suppose $\epsilon >0$. Note, uniform convergence of $\{ f_j \}$ to $f$ implies there exists $N \in {\mathbb{N}}$ for which $n \geq N$ implies $ ||f_n(z)-f(z)|| < \frac{\epsilon}{{m_{{\mathcal{A}}}}L} $ for all $z \in U$. Since $C \subset U$ we have the same estimate for points on $C$. Furthermore, by the treatment of integration in [@cookAcalculusI], $$\begin{aligned}
\bigg{|}\bigg{|} \int_C f_n(z) \star dz - \int_C f(z) \star dz\bigg{|}\bigg{|} &= \bigg{|}\bigg{|} \int_C \left( f_n(z)-f(z) \right) \star dz\bigg{|}\bigg{|} \\ \notag
&\leq {m_{{\mathcal{A}}}}\frac{\epsilon}{{m_{{\mathcal{A}}}}L}L = \epsilon. \ \ \Box \end{aligned}$$
Observe: if $f_n \rightarrow f$ uniformly near $C$ then $\lim_{n \rightarrow \infty} \int_C f_n \star dz = \int_C \left( \lim_{n \rightarrow \infty} f_n \right) \star dz$.\
In complex analysis we learn one consquence of Cauchy’s Integral Formula is that a uniformly convergent sequence of complex-differentiable functions has a limit function which is likewise complex-differentiable. However, in the absense of Cauchy’s Integral Formula, no such luxury is available in the study of differentiability of the limit function. We face the usual difficulty of real analysis which is nicely addressed by Dieudonn’ e in result 8.6.3 of [@Dmaster]. We show Dieudonn’ e’s result extends naturally to ${\mathcal{A}}$-calculus: we provide sufficient conditions for a sequence of ${\mathcal{A}}$-differentiable functions to have an ${\mathcal{A}}$-differentiable limit function:
\[thm:difflimitfunseries\] Let $U$ be an open connected subset of ${\mathcal{A}}$, $f_n: U \rightarrow {\mathcal{A}}$ an ${\mathcal{A}}$-differentiable mapping of $U$ for each $n \in {\mathbb{N}}$. Suppose that:
1. there exists one point $z_0 \in U$ such that the sequence $\{ f_n(z_0) \}$ converges in ${\mathcal{A}}$,
2. for every point $a \in U,$ there is a ball $B(a)$ of center $a$ contained in $U$ and such that in $B(a)$ the sequence $\{ f'_n \}$ converges uniformly.
Then for each $a \in U,$ the sequence $\{f_n \}$ converges uniformly in $B(a)$; moreover, if, for each $z \in U,$ $f(z)=\lim_{n \rightarrow \infty} f_n(z)$ and $g(z) = \lim_{n \rightarrow \infty} f'_n(z),$ then $g(z)=f'(z),$ for each $z \in U.$
To be clear, when we write $f'(z)$ this indicates the ${\mathcal{A}}$-derivative of $f$. Hence, in part, the Theorem asserts $f_n \rightarrow f$ where $f$ is ${\mathcal{A}}$ differentiable. Furthermore, for each $z \in U$: $$\frac{d}{dz} \left(\lim_{n \rightarrow \infty}f_n(z) \right) = \lim_{n \rightarrow \infty} \left( \frac{df_n}{dz}(z) \right).$$
[**Proof:**]{} suppose $f_n: U \rightarrow {\mathcal{A}}$ is an ${\mathcal{A}}$-differentiable mapping on an open connected $U \subseteq {\mathcal{A}}$. In addition, suppose conditions (i.) and (ii.) hold. Notice that ${\mathcal{A}}$-differentiable implies Frechet differentiable. Hence, by result 8.6.3 in [@Dmaster] we find uniform convergence of $\{ f_n \}$ as described in the Theorem. Dieudonn’ e uses the notation $f'(x)$ for the Frechet derivative of $f$ at $x$. We change notation and write $Df(x)$ for the Frechet derivative of $f$ at $x$. Hence, by $(8.6.3)$ in [@Dmaster], if for each $z \in U,$ $f(z)=\lim_{n \rightarrow \infty} f_n(z)$ and $g(z) = \lim_{n \rightarrow \infty} Df_n(z),$ then $g(z)=Df(z),$ for each $z \in U$. Let $\{ v_1, v_2, \dots , v_N \}$ serve as a basis for ${\mathcal{A}}$ with coordinates $x_1, x_2, \dots , x_n$. By the definition of partial derivative, for each $i=1,2, \dots , N$, $$Df(z)(v_i) = \frac{\partial f}{\partial x_i}(z) \qquad \& \qquad Df_n(z)(v_i) = \frac{\partial f_n}{\partial x_i}(z)$$ thus [@Dmaster] provides the existence of the Frechet derivative as well as the following identity for the partial derivatives: $$\label{eqn:exchangelimits}
\frac{\partial }{\partial x_i} \left( \lim_{n \rightarrow \infty} f_n(z) \right) = \lim_{n \rightarrow \infty} \left( \frac{\partial f_n}{\partial x_i}(z) \right).$$ It remains to show $f = \lim_{n \rightarrow \infty} f_n$ is ${\mathcal{A}}$-differentiable on $U$. From (ii.) we know $f_n$ is ${\mathcal{A}}$-differentiable hence $f_n$ satisfy the symmetric CR-equations[^4] $$\frac{\partial f_n}{\partial x_i} \star v_j = \frac{\partial f_n}{\partial x_j} \star v_i.$$ Hence, using the symmetric ${\mathcal{A}}$-CR equations and Equation \[eqn:exchangelimits\] we derive: $$\begin{aligned}
\frac{\partial f}{\partial x_i} \star v_j &= \frac{\partial}{\partial x_i}\left[\lim_{n \rightarrow \infty} f_n \right] \star v_j \\ \notag
&= \lim_{n \rightarrow \infty} \left[ \frac{\partial f_n}{\partial x_i} \right] \star v_j \\ \notag
&= \lim_{n \rightarrow \infty} \left[ \frac{\partial f_n}{\partial x_i} \star v_j \right] \\ \notag
&= \lim_{n \rightarrow \infty} \left[ \frac{\partial f_n}{\partial x_j} \star v_i \right].
\end{aligned}$$ Consequently, $\frac{\partial f}{\partial x_i} \star v_j = \frac{\partial f}{\partial x_j} \star v_i$ and thus $f$ is ${\mathcal{A}}$-differentiable with $g(z) = f'(z)$. $\Box$\
Theorem \[thm:difflimitfunseries\] allows us to establish the ${\mathcal{A}}$-differentiability of power series in ${\mathcal{A}}$. In particular, if an ${\mathcal{A}}$-series converges on an open ball about its center then we find the derivative of the series exists and can be obtained by term-wise differentiation.
\[coro:derivativepowseries\] If $\sum c_n \star (z-z_o)^n$ is a power series on ${\mathcal{A}}$ which converges for $\| z - z_o \| < R$. Then $ \frac{d}{dz} \sum c_n \star (z-z_o)^n = \sum nc_n \star (z-z_o)^{n-1}$ for each $z \in {\mathcal{A}}$ with $\| z-z_o \| < \frac{1}{{m_{{\mathcal{A}}}}\alpha}$ where $\alpha = \limsup_{n\to\infty}\sqrt[n]{\|c_n\|}$.
[**Proof:**]{} We begin with $z_o =0$. Assume the series $\sum c_n \star z^n$ converges for $\| z \| < R$. Let $U = \{ z \in {\mathcal{A}}\ | \ \| z \| <R \}$ and note $U$ is an open set. Define $f_n(z) = \sum_{k=0}^n c_k \star z^k$ for $n=0,1,\dots $ and $z \in U$. Observe $\frac{d f_n}{dz} = \sum_{k=1}^n kc_k \star z^{k-1}$ for $z \in U$ and $0 \in U$ with $\{ f_n (0) \} = \{ c_0 \}$ is convergent. If $a \in U$ then note $B_{r}(a) = \{ z \ | \ \| z-a \| <r \} \subseteq U$ where $r = \text{min}\{ \| a \|/2, |R-\| a \||/2 \}$. We need to show $\{ f_n' \}$ converges uniformly on $B_r(a)$. Let $\alpha =\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}$ then Theorem \[thm:norootcriterianeeded\] provides $\alpha \leq \frac{\mathds{1}}{R}$ which shows $\alpha$ is finite. Since $\sqrt[n]{n}\rightarrow 1\ as\ n\to\infty$, we have $$\limsup_{n\to\infty}\sqrt[n]{n\|c_n\|}=\limsup_{n\to\infty}\sqrt[n]{\|c_n\|} = \alpha < \infty$$ hence Theorem \[thm:roottestpowerseries\] provides that $\sum nc_n \star z^{n-1}$ converges for $\| z \| < \frac{1}{{m_{{\mathcal{A}}}}\alpha}$. Therefore, by Theorem \[thm:pwrseriesnormalconvergence\], we find $\{ f_n' \}$ is uniformly convergent for $\| z \| < R- \varepsilon$ for any $\varepsilon \in (0, R)$. Thus $\{ f_n' \}$ converges uniformly on $B_r(a)$ as we are free to adjust $\varepsilon$ such that $B_r(a) \subset B_{R-\epsilon}(0)$. In summary, we have satisfied conditions (i.) and (ii.) of Theorem \[thm:difflimitfunseries\] and the Corollary follows from the identifications $f(z) = \sum c_n \star z^n$ and $g(z) = \sum nc_n \star z^{n-1}$ for which $f'(z)=g(z)$ on $U$. Finally, we apply Lemma \[lem:centersub\] to shift the $z_o=0$ result to $z_o \neq 0$. $\Box$\
A similar result is available for higher derivatives:
\[coro:coeffbyderivatives\] If $f(z) =\sum c_n \star (z-z_o)^n$ is a power series on ${\mathcal{A}}$ which converges for $\| z-z_o \| < R$ then $f$ has derivatives of all orders in $\|z-z_o\|<\frac{1}{{m_{{\mathcal{A}}}}\alpha}$ where $\alpha = \limsup_{n\to\infty}\sqrt[n]{\|c_n\|}$. Moreover, the higher-derivative functions are obtained by term-wise differentiation: $$f^{(k)}(z)=\sum_{n=k}^\infty n(n-1)\cdot \cdot \cdot (n-k+1)c_n \star (z-z_o)^{n-k}$$ and $f^{(k)}(z_o)=k!c_k$ for $k=0, 1, 2, \dots$.
[**Proof:**]{} observe $
\lim_{n \rightarrow \infty}\sqrt[n]{n(n-1)\cdot \cdot \cdot (n-k+1)} =1$ for any $k \in {\mathbb{N}}$. Thus the argument given for Corollary \[coro:derivativepowseries\] naturally extends to the $k$-th derivative. $\square$\
We close this section with a discussion of entire functions on an ${\mathcal{A}}$.
A function $f:{\mathcal{A}}\to{\mathcal{A}}$ is called entire if it can be written as a power series $\sum a_n \star z^n$ which converges on all of ${\mathcal{A}}$.
\[thm:entire\] If $f(x) = \sum a_n x^n$ is an entire function on the reals, then there exists a unique entire extension to ${\mathcal{A}}$. The extension has the form $\tilde{f}(z) = \sum a_n z^n$ for each $z \in {\mathcal{A}}$.
**Proof:** Let $f$ be an entire function on the reals. Thus, its radius of convergence is infinite, and by the real root test we have: $$\limsup_{n\to\infty} \sqrt[n]{|a_n|} = 0$$ Let $\tilde{f}(z) = \sum a_n z^n$ for each $z \in {\mathcal{A}}$. Since the coefficients of the extended function $\tilde{f}(z)$ are the same as those of $f(x)$ we find $\tilde{f}(z)$ is entire by Theorem \[thm:roottestpowerseries\]. If $g(z) = \sum b_n \star z^n$ is entire function on ${\mathcal{A}}$ for which $g|_{{ \mathbb{R} }} = f= \tilde{f}|_{{ \mathbb{R} }}$ then $g^{(n)}(0) = \tilde{f}^{(n)}(0)$ for $n=0,1,2,\dots $. Hence, $ b_n = a_n$ for $n=0,1,2,\dots $ by Corollary \[coro:coeffbyderivatives\]. Thus the extension $\tilde{f}(z) = \sum a_nz^n$ is the unique extension to ${\mathcal{A}}$. $\square$\
\[thm:entirefnctuniformabs\] If f is an entire function on ${\mathcal{A}}$, then we have $$\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}=0$$ and thus f is uniformly absolutely convergent for $\|z\|<L$ for all $L>0$.
**Proof:** Since $f$ is entire, it follows that $\sum c_n z^n$ converges on ${\mathcal{A}}$ and hence, by Theorem \[thm:term test\], we have $$\lim_{n\to\infty} c_n z^n = 0 \Rightarrow \lim_{n\to\infty} \|c_n z^n\| = 0 \qquad \forall z\in{\mathcal{A}}.$$ Given $\varepsilon>0$, there exist $N\in{\mathbb{N}}$ such that for each $n\geq N$, $$\left\|c_n\left(\frac1\varepsilon\right)^n \right\|< 1 \ \
\Rightarrow \ \ \|c_n\|<\varepsilon^n \ \ \Rightarrow \ \ \sqrt[n]{\|c_n\|}<\varepsilon.$$ Thus, $\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}=0$ and by Theorem \[thm:pwrseriesnormalconvergence\] we reach the desired result.$\ \square$
The set of entire functions on ${\mathcal{A}}$ is an algebra, and the product of two entire functions $\sum a_n \star z^n$ and $\sum b_n \star z^n$ is equal to $\sum c_n \star z^n$ where $c_n=\sum_{k=0}^{n}a_k \star b_{n-k}$.
**Proof:** It is clear that this set is a real vector space, so all we must show is that the product of two entire functions $\sum a_n \star z_n$ and $\sum b_n \star z_n$ is also entire. By Theorem \[thm:pwrseriesnormalconvergence\], these functions are absolutely convergent on all of ${\mathcal{A}}$, and thus by Theorem \[thm:productofseries\], their product will converge on all of ${\mathcal{A}}$ and will be $$\begin{aligned}
\left(\sum a_n \star z^n\right) \star \left(\sum b_n \star z^n\right)&= \sum_{n=0}^\infty\left(\sum_{k=0}^n a_k \star z^k\star b_{n-k} \star z^{n-k}\right)\\ \notag
&=\sum_{n=0}^\infty\left(\sum_{k=0}^n a_k \star b_{n-k} \star z^n\right)\\ \notag
&=\sum_{n=0}^\infty\left(\sum_{k=0}^n a_k \star b_{n-k}\right) \star z^n\\ \notag
&=\sum_{n=0}^\infty c_n\star z^n \ where\ c_n=\sum_{k=0}^{n}a_k \star b_{n-k}.\end{aligned}$$ Thus, we have the desired result. $\square$
Transcendental functions {#sec:transfunct}
========================
In this section we let ${\mathcal{A}}$ denote a real associative finite dimensional commutative algebra ${\mathcal{A}}$. Theorem \[thm:entire\] encourages us to take the power series formulation of elementary functions as fundamental. If we want to recover standard elementary functions by restriction to ${ \mathbb{R} }\subseteq {\mathcal{A}}$ then our definitions must be given. We provide concrete definitions for the exponential, sine, cosine, hyperbolic sine and hyperbolic cosine over any ${\mathcal{A}}$. We also provide proofs for identities of these functions which equally well apply to a myriad of choices for ${\mathcal{A}}$.
Exponential
-----------
For each z $\in{\mathcal{A}}$, we define $$exp(z)=\sum_{n=0}^\infty \frac{z^n}{n!}.$$
\[thm:propofexp\] Let ${\mathcal{A}}$ be a commutative, unital, associative algebra over ${ \mathbb{R} }$.
> 1. $exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}$ is entire on ${\mathcal{A}}$,
>
> 2. $exp(z)\star exp(w) = exp(z+w) $ for all $z,w \in {\mathcal{A}}$,
>
> 3. $exp(0)=1$ and $exp(-z) =exp(z)^{-1}$ hence $exp(z) \in {\mathcal{A}^{\times}}$.
>
[**Proof:**]{} [**(i.)**]{} Identify the coefficients of the exponential are $c_n=\frac{1}{n!}$, which gives us $$\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}=\limsup_{n\to\infty}\frac1{\sqrt[n]{n!}}=0.$$ Thus, by Theorem \[thm:pwrseriesnormalconvergence\], this series uniformly converges absolutely for all $z\in{\mathcal{A}}$.\
[**(ii.)**]{} By Theorem \[thm:productofseries\], we have for each $z, w \in{\mathcal{A}}$: $$\begin{aligned}
exp(z)\star exp(w)&=\biggl{(}\sum_{n=0}^\infty \frac{z^n}{n!}\biggr{)}\star \biggl{(}\sum_{n=0}^\infty \frac{w^n}{n!}\biggr{)}\\ \notag
&=\sum_{n=0}^\infty \sum_{k=0}^n \frac{z^k}{k!}\star\frac{w^{n-k}}{(n-k)!}\\ \notag
&=\sum_{n=0}^\infty \sum_{k=0}^n \frac{n!}{k!(n-k)!}\frac{z^n\star w^{n-k}}{n!}\\ \notag
&=\sum_{n=0}^\infty \frac{1}{n!}\sum_{k=0}^n {n\choose k}z^n\star w^{n-k}\\ \notag
&=\sum_{n=0}^\infty\frac{(z+w)^n}{n!}\\ \notag
&=exp(z+w).\end{aligned}$$ [**(iii.)**]{} Clearly $exp(0)=1$ and since for each $z\in{\mathcal{A}}$ we have $$exp(z) \star exp(-z)=exp(z-z)=exp(0)=1,$$ thus $exp(z)\in{\mathcal{A}^{\times}}$ for each $z \in {\mathcal{A}}$. Moreover, $exp(-z)=exp(z)^{-1}$. $\Box$\
The following Theorem should not be surprising.
For each $z \in {\mathcal{A}}$, $\frac{d}{dz}exp(z) = exp(z)$.
[**Proof:**]{} From Corollary \[coro:derivativepowseries\], we know that $\frac{d}{dz}exp(z)$ exists and $$\frac{d}{dz}exp(z)=\sum_{n=1}^\infty \frac{nz^{n-1}}{n!}=\sum_{n=0}^\infty \frac{z^{n}}{n!}=exp(z)$$ for all $z\in{\mathcal{A}}$. $\Box$\
To appreciate the content of this section we expand the exponential into component functions of several interesting algebras.
Let ${\mathcal{A}}= { \mathbb{R} }$ then $exp(x)=e^x$ is the usual exponential of real calculus.
Let ${\mathcal{A}}= { \mathbb{C} }$ then $exp(x+iy)= exp(x)exp(iy) = e^x \cos y+ ie^x \sin y$. Setting $u+iv = exp(x+iy)$ we find $u= e^x \cos y$ and $v=e^x \sin y$. The component functions of the exponential are solutions to both the Cauchy Riemann equations $u_x=v_y$ and $v_x = -u_y$ and Laplace’s Equation $\phi_{xx}+\phi_{yy}=0$.
Let ${\mathcal{A}}= {\mathcal{H}}$ then $exp(x+jy)= exp(x)exp(jy) = e^x( \cosh y+ j \sinh y)$. Setting $u+jv = exp(x+jy)$ provides $u = e^x \cosh y$ and $v=e^x\sinh y$. These functions satisfy the hyperbolic Cauchy Riemann equations $u_x=v_y$ and $u_y = v_x$ and both are solutions to the wave equation $\phi_{xx}-\phi_{yy}=0$.
Let ${\mathcal{A}}= { \mathbb{R} }\oplus {\epsilon}{ \mathbb{R} }$ with ${\epsilon}^2=0$. Then $exp(x+{\epsilon}y)= exp(x)exp({\epsilon}y)$. Calculate $$exp( {\epsilon}y) = 1+ {\epsilon}y+ \frac{1}{2}{\epsilon}^2 y^2+ \cdots = 1+ {\epsilon}y$$ hence $exp(x+{\epsilon}y) = e^x+ {\epsilon}ye^x $. The component functions of the exponential are $e^x$ and $ye^x$ for this nilpotent algebra.
Hyperbolic sine and cosine
--------------------------
As in the usual calculus, the hyperbolic sine and cosine appear as the odd and even pieces of the exponential function.
\[def:hyperbolictrig\] For each $z \in {\mathcal{A}}$, we define $$cosh(z)=\sum_{n=0}^\infty \frac{z^{2n}}{(2n)!} \qquad \& \qquad sinh(z)=\sum_{n=0}^\infty \frac{z^{2n+1}}{(2n+1)!}.$$
\[thm:convhyperbolic\] The series defining $cosh(z)$ and $sinh(z)$ uniformly converge absolutely on ${\mathcal{A}}$.
[**Proof:**]{} For $cosh(z)$ we have coefficents $c_n = 1/(2n)!$ hence calculate $$\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}=\limsup_{n\to\infty}\frac1{\sqrt[n]{(2n)!}}=0$$ and for $sinh(z)$ we have coefficients $c_n = 1/(2n+1)!$ hence calculate: $$\limsup_{n\to\infty}\sqrt[n]{\|c_n\|}=\limsup_{n\to\infty}\frac1{\sqrt[n]{(2n+1)!}}=0$$ Thus, by the Theorem \[thm:pwrseriesnormalconvergence\] we find both series converge absolutely for each $ z \in {\mathcal{A}}$. $\Box$
\[thm:hyperbolicpropsI\] Hyperbolic sine and cosine have the following properties:
> 1. $cosh(0)=1$ and $sinh(0)=0$,
>
> 2. $cosh(-z) = cosh(z)$ and $sinh(-z) = -sinh(z)$ for each $z \in {\mathcal{A}}$,
>
> 3. $\frac{d}{dz} cosh(z) = sinh(z)$ and $\frac{d}{dz} sinh(z) = cosh(z)$ for each $z \in {\mathcal{A}}$.
>
[**Proof:**]{} Observe [**(i.)** ]{} and [**(ii.)**]{} follow immediately from Definition \[def:hyperbolictrig\]. Using Corollary \[coro:derivativepowseries\] we derive [**(iii.)**]{} as follows: $$\begin{aligned}
\frac{d}{dz}cosh(z)&=\sum_{n=1}^\infty \frac{2nz^{2n-1}}{(2n)!}=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}=\sum_{n=0}^\infty \frac{z^{2n+1}}{(2n+1)!}=sinh(z) \\ \notag
\frac{d}{dz}sinh(z)&=\sum_{n=0}^\infty \frac{(2n+1)z^{2n}}{(2n+1)!}=\sum_{n=0}^\infty \frac{z^{2n}}{(2n)!}=cosh(z). \ \ \Box \end{aligned}$$
Sums and products of these series will also be entire, giving us the following:
\[thm:hyperbolicpythago\] $cosh^2(z)-sinh^2(z)=1$ for all $z\in{\mathcal{A}}$.
[**Proof:**]{} Let $g(z)=cosh^2(z)-sinh^2(z)$ for all $z\in{\mathcal{A}}$. By Theorems \[thm:hyperbolicpropsI\], \[thm:convhyperbolic\] and the chain-rule, $$g'(z)=2cosh(z) \star sinh(z)-2sinh(z) \star cosh(z)=0$$ for all $z \in{\mathcal{A}}$. Since ${\mathcal{A}}$ is a connected it follows $g(z)$ is constant. Moreover, $$g(0)=cosh^2(0)-sinh^2(0)=1$$ hence the Theorem follows. $\square$
\[thm:hyperbolicidentI\] For all $z\in{\mathcal{A}}$,
> 1. $e^z=cosh(z)+sinh(z)$,
>
> 2. $cosh(z)=\frac{1}{2}(e^z+e^{-z})$,
>
> 3. $sinh(z)=\frac{1}{2}(e^z-e^{-z})$.
>
[**Proof:**]{} item (i.) is verified directly from the definitions: $$cosh(z)+sinh(z)=\sum_{n=0}^\infty \frac{z^{2n}}{(2n)!}+\sum_{n=0}^\infty \frac{z^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty \frac{z^{n}}{n!}=e^z.$$ Hence (ii.) follows from (i.) by simple calculation: $$\begin{aligned}
\frac{1}{2}(e^z+e^{-z}) &= \frac{1}{2}(cosh(z)+sinh(z)+cosh(-z)+sinh(-z))\\ \notag
&= \frac{1}{2}(cosh(z)+sinh(z)+cosh(z)-sinh(z)) \\ \notag
&= cosh(z).\end{aligned}$$ Likewise (iii.) follows from (i.) $$\begin{aligned}
\frac{1}{2}(e^z-e^{-z})&= \frac{1}{2}(cosh(z)+sinh(z)-cosh(-z)-sinh(-z))\\ \notag
&= \frac{1}{2}(cosh(z)+sinh(z)-cosh(z)+sinh(z)) \\ \notag
&= sinh(z).\end{aligned}$$ Alternatively, we could have differentiated (ii.) to obtain (iii.). $\Box$
For all $z, w\in{\mathcal{A}}$ $${\bf (i.)} \ \ cosh(z+w)=cosh(z)\star cosh(w)+sinh(a)\star sinh(w),$$ $${\bf (ii.)} \ \ sinh(z+w)=sinh(z)\star cosh(w)+cosh(z)\star sinh(w).$$
[**Proof:**]{} For all $z, w\in{\mathcal{A}}$ apply Theorems \[thm:hyperbolicidentI\] and \[thm:propofexp\] part (ii.) $$\begin{aligned}
cosh(z+w)&=\frac{1}{2}(e^{z+w}+e^{-(z+w)})\\ \notag
&=\frac{1}{2}(e^z\star e^w+e^{-z}\star e^{-w})\\ \notag
&=\frac{1}{2}(e^z \star e^w+e^z \star e^{-w}-e^z \star e^{-w}+e^{-z}\star e^{-w})\\
\notag
&=\frac{1}{2}(e^z\star (e^w+e^{-w})-(e^z-e^{-z}) \star e^{-w})\\
\notag
&=\frac{1}{2}(2e^z \star cosh(w)-2sinh(z) \star e^{-w})\\
\notag
&= c(z) \star c(w)+s(z) \star c(w)-s(z) \star c(-w)-s(z) \star s(-w)\\
\notag
&=cosh(z)\star cosh(w)+sinh(z)\star sinh(w)\end{aligned}$$ where we used the abbreviated notation $c(z) =cosh(z)$ and $s(z)=sinh(z)$ in the next to last line. Fix $w$ and differentiate with respect to $z$ to find $$sinh(z+w)=sinh(z) \star cosh(w)+cosh(z) \star sinh(w).$$ Thus the adding angles formulas for hyperbolic functions exist for ${\mathcal{A}}$. $\Box$
Sine and cosine
---------------
In certain contexts we could use the imaginary unit $i$ for which $i^2=-1$ to aid in the definition of sine and cosine. However, there are many algebras without such an imaginary unit hence we provide a treatment which only requires the theory of power series to establish the structure of trigonometric functions. Once more, we find identities for trigonometric function which transcend the choice of ${\mathcal{A}}$. Our arguments here are parallel those found in Section 68 of [@JandP].
\[def:sinecosine\] For each z $\in{\mathcal{A}}$, we define $$cos(z)=\sum_{n=0}^\infty (-1)^n\frac{z^{2n}}{(2n)!}
\qquad \& \qquad sin(z)=\sum_{n=0}^\infty (-1)^n\frac{z^{2n+1}}{(2n+1)!}.$$
\[thm:convtrig\] The series defining $cos(z)$ and $sin(z)$ converge absolutely on ${\mathcal{A}}$.
[**Proof:**]{} For $cos(z)$ note $c_n = (-1)^n/(2n)!$ and for $sin(z)$ the coefficient $c_n = (-1)^n /(2n+1)!$ consequently the proof for Theorem \[thm:convhyperbolic\] equally well applies here. $\Box$
\[thm:trigpropsI\] Sine and cosine over ${\mathcal{A}}$ have the following properties:
> 1. $cos(0)=1$ and $sin(0)=0$,
>
> 2. $cos(-z) = cos(z)$ and $sin(-z) = -sin(z)$ for each $z \in {\mathcal{A}}$,
>
> 3. $\frac{d}{dz} cos(z) = -sin(z)$ and $\frac{d}{dz} sin(z) = cos(z)$ for each $z \in {\mathcal{A}}$.
>
[**Proof:**]{} follows from argument nearly identical to those given for Theorem \[thm:hyperbolicpropsI\]. $\Box$\
The following identity is well-known for ${ \mathbb{R} }$ or ${ \mathbb{C} }$, but it just as well applies to sine and cosine over any ${\mathcal{A}}$:
\[thm:pythagcosinesine\] $cos^2(z)+sin^2(z)=1$ for all $z\in{\mathcal{A}}$.
[**Proof:**]{} Let $g(z)=cos^2(z)+sin^2(z)$ for all $z\in{\mathcal{A}}$. By Theorems \[thm:convtrig\] and \[thm:trigpropsI\] and the chain-rule, we have for all $z \in{\mathcal{A}}$ $$g'(z)=-2cos(z)\star sin(z)+2sin(z)\star cos(z)=0.$$ Thus $g(z)$ is constant on all of ${\mathcal{A}}$. Since $g(0)=cos^2(0)+sin^2(0)=1$, the result follows. $\square$
\[thm:icsine\] If $f$ is a function on a connected subset $E$ of ${\mathcal{A}}$, satisfying $f''(z)=-f(z)$ for all $z \in E$ and $f(0)=0$, $f'(0)=b \in{\mathcal{A}}$, then $f(z)=b \star sin(z)$ for all $z \in E$.
[**Proof:**]{} Let $$U(z)=f(z) \star sin(z)+f'(z) \star cos(z) \ \ \ \& \ \ \ V(z)=f(z) \star cos(z)-f'(z) \star sin(z).$$ Apply the product rule to obtain: $$\begin{aligned}
U'(z) &=f'(z) \star sin(z)+f(z) \star cos(z)+f''(z) \star cos(z)-f'(z) \star sin(z) \\ \notag
&=f(z) \star cos(z)-f(z) \star cos(z)=0.\end{aligned}$$ and $$\begin{aligned}
V'(z) &=f'(z) \star cos(z)-f(z) \star sin(z)-f''(z) \star sin(z)-f'(z) \star cos(z) \\ \notag
&=-f(z) \star sin(z)+f(z) \star sin(z)=0.\end{aligned}$$ Thus, as $E$ is connected, $U$ and $V$ are constant. Thus $U(z)=U(0)=b$ and $V(z)=V(0)=0$ for all z in E. Hence, $$\label{eqn:forb}
b=f(z) \star sin(z)+f'(z) \star cos(z)$$ also $$\label{eqn:forcossin}
f(z) \star cos(z)=f'(z) \star sin(z).$$ Combining Equations \[eqn:forcossin\] and \[eqn:forb\] with Theorem \[thm:pythagcosinesine\] we derive: $$\begin{aligned}
b \star sin(z) &=[f(z) \star sin(z)+f'(z) \star cos(z)] \star sin(z) \\ \notag
&=f(z)\star sin^2(z)+f'(z) \star sin(z) \star cos(z) \\ \notag
&=f(z)\star sin^2(z)+f(z)\star cos(z) \star cos(z) \\ \notag
&=f(z) \star [sin^2(z)+cos^2(z)] \\ \notag
&= f(z). \ \ \Box\end{aligned}$$
\[thm:icsincos\] If $f$ is a function on a connected subset $E$ of ${\mathcal{A}}$, satisfying $f''(z)=-f(z)$ for all $z\in E$ and $f(0)=a$, $f'(0)=b\in{\mathcal{A}}$, then $f(z)=a \star cos(z)+b \star sin(z)$ for all $z\in E$.
[**Proof:**]{} Let $g(z)=f(z)-a\star cos(z)$. Then $g''(z)=-g(z)$ for all $z \in E$ and $g(0)=0$, $g'(0)=b$, so by Theorem \[thm:icsine\], we find $g(z)=b \star sin(z)$ thus $f(z)=a \star cos(z)+b\star sin(z). \, \square$\
We now arrive at the angle addition formulas for trigonometric functions on ${\mathcal{A}}$:
For $z,w \in{\mathcal{A}}$, $${\bf (i.)} \ \ sin(z+w)=sin(z)\star cos(w)+sin(w) \star cos(z)$$ $${\bf (ii.)} \ \ cos(z+w)=cos(z) \star cos(w)-sin(z)\star sin(w)$$
[**Proof:**]{} Fix $w \in {\mathcal{A}}$ and let $f(z)=sin(z+w)$ for all $z \in{\mathcal{A}}$. Then $f''(z)=-f(z)$ for all $z \in{\mathcal{A}}$ and $f(0)=sin(w)$ and $f'(0)=cos(w)$, so by Theorem \[thm:icsincos\], we have $$f(z)=sin(w) \star cos(z)+cos(w) \star sin(z)$$ for all $z,w \in{\mathcal{A}}$ which proves [**(i.)**]{}. Continue to hold $w$ fixed and differentiate [**(i.)**]{} with respect to $z$ to find: $$cos(z+w)=cos(z) \star cos(w)-sin(z) \star sin(w)$$ thus [**(ii.)**]{} holds true. $\square$\
These are helpful to uncover the component function content of sine or cosine over ${\mathcal{A}}$.
Consider ${\mathcal{H}}= { \mathbb{R} }\oplus j { \mathbb{R} }$ where $j^2=1$. For $x+jy \in {\mathcal{H}}$ we calculate: $$cos( x+jy) = cos(x) cos(jy) - sin(x) sin(jy).$$ But, as $(jy)^{2n} = j^{2n}y^{2n} = y^2$ and $(jy)^{2n+1} = (j)^{2n+1} y^{2n+1} = jy^{2n+1}$ hence $\cos(jy) = \cos(y)$ and $ sin(jy) = j sin(y)$. Consequently, $$cos( x+jy) = cos(x) cos(y) - j sin(x) sin(y).$$ Differentiate with respect to $x$ holding $y$ fixed and find $$sin( x+jy) = sin(x) cos(y) + j cos(x) sin(y).$$ You might recognize the products of sine and cosine as well-known solutions to the unit-speed wave-equation $\phi_{xx}=\phi_{yy}$. This is no accident, the unit-speed wave equation is the generalized Laplace equation for ${\mathcal{H}}$ and we know the component functions of an ${\mathcal{H}}$-differentiable function solve the generalized Laplace equation of ${\mathcal{H}}$.
The N-Pythagorean theorem {#sec:nthagtheorem}
=========================
In Section \[sec:transfunct\] we studied functions whose properties were not tied to a particular choice of algebra. We saw how exponentials, cosine, sine, cosh and sinh all enjoy properties which hold in a multitude of algebras. The direction of the current section is quite the opposite. We now consider a method of obtaining new functions which are particular to our choice of algebra. We call such functions the [*special functions*]{} of ${\mathcal{A}}$.
Special functions of an algebra
-------------------------------
The exponential function on an algebra is naturally defined by $$e^z = 1+z+\frac{1}{2} z^2+ \cdots = \sum_{k=0}^{\infty} \frac{z^k}{k!}.$$ We say the component functions of the exponential are the [*special functions*]{} of the algebra. We explain how to calculate the special functions of a particular type of algebra in this section.
If $z \in { \mathbb{C} }$ then $z = x+iy$ where $i^2=-1$. Notice the map $t \mapsto e^{it}$ is the composite of the complex exponential and the path $t \mapsto it$ in ${ \mathbb{C} }$. We calculate, $$\begin{aligned}
e^{it}=\sum_{n=0}^\infty \frac{(it)^n}{n!}
&=\sum_{n=0}^\infty i^{2n}\frac{t^{2n}}{(2n)!}+\sum_{n=0}^\infty i^{2n+1}\frac{t^{2n+1}}{(2n+1)!}\\ \notag
&=\sum_{n=0}^\infty (-1)^n\frac{t^{2n}}{(2n)!}+i\sum_{n=0}^\infty (-1)^n\frac{t^{2n+1}}{(2n+1)!}\\ \notag
&=cos(t)+isin(t).\end{aligned}$$ The real and imaginary parts of this exponential are the real sine and cosine functions. Extending $t \in { \mathbb{R} }$ to $z \in { \mathbb{C} }$ provides the special functions $z \mapsto \cos(z)$ and $z \mapsto \sin(z)$ for ${ \mathbb{C} }$. These extended functions are the [*special functions of ${ \mathbb{C} }$*]{}
Let $\mathcal{H}$ denote the hyperbolic numbers. If $z \in \mathcal{H}$ then $z = x+jy$ where $j^2=1$ and $x,y \in { \mathbb{R} }$. Let $t \in { \mathbb{R} }$ and calculate: $$\begin{aligned}
e^{jt}=\sum_{n=0}^\infty \frac{(jt)^n}{n!}
&=\sum_{n=0}^\infty j^{2n}\frac{t^{2n}}{(2n)!}+\sum_{n=0}^\infty j^{2n+1}\frac{t^{2n+1}}{(2n+1)!}\\ \notag
&=\sum_{n=0}^\infty \frac{t^{2n}}{(2n)!}+j\sum_{n=0}^\infty \frac{t^{2n+1}}{(2n+1)!}\\ \notag
&=cosh(t)+jsinh(t).\end{aligned}$$ Extending hyperbolic cosine and sine to ${\mathcal{H}}$ we obtain the [*special functions of ${\mathcal{H}}$*]{} are given by $z \mapsto \cosh(z)$ and $z \mapsto \sinh(z)$ for $z \in {\mathcal{H}}$.
Let us generalize the observations above for a unital algebra with generator $\varepsilon$. If each $z \in {\mathcal{A}}$ has the form $z = x_1+x_2\varepsilon+ \cdots + x_N \varepsilon^{N-1}$ for $x_1, x_2, \dots , x_N \in { \mathbb{R} }$ then we say ${\mathcal{A}}$ is generated by $\varepsilon$.
\[thm:specialfunctions\] If ${\mathcal{A}}$ is an $N$-dimensional real algebra generated by $\varepsilon$ then there exist unique functions $f_1,\ f_2,\dots ,f_N: { \mathbb{R} }\rightarrow { \mathbb{R} }$ for which $$e^{\varepsilon t}=f_1(t)+\varepsilon f_2(t)+\dots +\varepsilon^{N-1}f_N(t)$$ for all $t \in { \mathbb{R} }$. Moreover, for each $i=1,\dots , n$, there exist real constants $c_{ij}$ such that $f_i(t) = \sum_{j=0}^{\infty} c_{ij}t^j$ for all $t \in { \mathbb{R} }$. That is, $f_1, \dots , f_n$ are entire on ${ \mathbb{R} }$. Furthermore, the functions $f_1, \dots , f_n$ uniquely extend to ${\mathcal{A}}$ via $f_i(z) = \sum_{j=0}^{\infty} c_{ij}z^j$ for each $z \in {\mathcal{A}}$.
[**Proof:**]{} the map $z \mapsto \text{exp}(z)$ is an entire function on ${\mathcal{A}}$. It follows that $f(t) = exp(\varepsilon t) = \sum_{k=0}^{\infty} \frac{(\varepsilon t)^k}{k!}$ converges for each $t \in { \mathbb{R} }$. We define components of $f$ by: $$f(t) = exp( \varepsilon t) = f_1(t)+ \varepsilon f_2(t)+ \cdots + \varepsilon^{N-1} f_{N}(t)$$ for each $t \in { \mathbb{R} }$. Similarly, we let $f_i^m: { \mathbb{R} }\rightarrow { \mathbb{R} }$ be the component functions of the map $t \mapsto \sum_{k=0}^{m} \frac{(\varepsilon t)^k}{k!}$. In particular, $$\sum_{k=0}^m \frac{(\varepsilon t)^k}{k!} = f_1^m(t)+ \varepsilon f_2^m(t)+ \cdots + \varepsilon^{N-1} f_{N}^m(t).$$ Since $\sum_{k=0}^m \frac{(\varepsilon t)^k}{k!} \rightarrow exp( \varepsilon t)$ as $m \rightarrow \infty$ we find $f_i^m(t) \rightarrow f_i(t)$ as $m \rightarrow \infty$ for each $t \in { \mathbb{R} }$. It follows $f_1, f_2, \dots , f_N$ are entire on ${ \mathbb{R} }$; there exist real constants $c_{ij} \in { \mathbb{R} }$ for which $f_i(t) = \sum_{j=0}^{\infty} c_{ij}t^j$ for all $t \in { \mathbb{R} }$. We conclude by applying Theorem \[thm:entire\] which provides $f_i$ extends uniquely to an entire function on ${\mathcal{A}}$ for $i=1,2,\dots , N$. $\Box$
Given ${\mathcal{A}}$ and $f_1, \dots , f_N: {\mathcal{A}}\rightarrow {\mathcal{A}}$ as discussed in Theorem \[thm:specialfunctions\] we say $f_1, f_2, \dots , f_N$ are the [*special functions*]{} of ${\mathcal{A}}$.
If ${\mathcal{H}}_3$ has basis $\{1,j,j^2\}$, where $j^3=1$. We say ${\mathcal{H}}_3$ are the $3$-hyperbolic numbers. Let $t \in { \mathbb{R} }$, $$\begin{aligned}
e^{jt}&=\sum_{n=0}^\infty \frac{(jt)^n}{n!}\\ \notag
&=\sum_{n=0}^\infty j^{3n}\frac{t^{3n}}{(3n)!}+\sum_{n=0}^\infty j^{3n+1}\frac{t^{3n+1}}{(3n+1)!}+\sum_{n=0}^\infty j^{3n+2}\frac{t^{3n+2}}{(3n+2)!}\\ \notag
&=\sum_{n=0}^\infty \frac{t^{3n}}{(3n)!}+j\sum_{n=0}^\infty\frac{t^{3n+1}}{(3n+1)!}+j^2\sum_{n=0}^\infty\frac{t^{3n+2}}{(3n+2)!}.\end{aligned}$$ Therefore, the special functions of the $3$-hyperbolic numbers are defined by: $$\label{eqn:larrymoeandcurly}
f_1(z)=\sum_{n=0}^\infty \frac{z^{3n}}{(3n)!}, \ \
f_2(z)=\sum_{n=0}^\infty\frac{z^{3n+1}}{(3n+1)!}, \ \
f_3(z)=\sum_{n=0}^\infty\frac{z^{3n+2}}{(3n+2)!}.$$
Suppose $\Gamma_3$ is generated by $\varepsilon$ with $\varepsilon^3=0$. The series for the exponential truncates nicely: $$e^{\varepsilon x}=\sum_{n=0}^\infty \frac{(\varepsilon x)^n}{n!}=1+\varepsilon x+\frac{1}{2}\varepsilon^2 x^2.$$ Thus we find special functions for the $3$-null numbers are $$f_1(z)=1,\ \ \ \ f_2(z)=z, \ \ \ \ f_3(z) = \frac{1}{2}z^2$$ for all $z \in { \mathbb{R} }\oplus \varepsilon { \mathbb{R} }\oplus \varepsilon^2 { \mathbb{R} }$.
In similar fashion,the $\Gamma_N$ generated by $\varepsilon$ with $\varepsilon^N=0$ produces monomials $\frac{1}{k!}z^k$ for $k=0,1,\dots, N-1$.
N-trigonometric and N-hyperbolic functions
==========================================
Our goal in this section is to describe the special functions of ${ \mathbb{C} }_N$ and ${\mathcal{H}}_N$. If $z \in { \mathbb{C} }_N$ then we say $z$ is an $N$-complex number and $$z = x_1+x_2j+\cdots + x_N j^{N-1}$$ where $j^N=-1$ and $x_1, \dots , x_N \in { \mathbb{R} }$. We also define ${\mathcal{H}}_N$ to be the set of $N$-hyperbolic numbers which have the form $$z = x_1+x_2j+\cdots + x_N j^{N-1}$$ where $j^N=1$ and $x_1, \dots , x_N \in { \mathbb{R} }$. Observe, 2-complex numbers are the ordinary complex numbers and 2-hyperbolic numbers form the hyperbolic numbers.
For a given $N\in{\mathbb{N}}$, the $N$-trigonometric functions are defined on an algebra ${\mathcal{A}}$ as follows: $$cos_N(z)=\sum_{k=0}^\infty (-1)^k\frac{z^{Nk}}{(Nk)!} \ \ \& \ \
sin_{N,p}(z)=\sum_{k=0}^\infty (-1)^k\frac{z^{Nk+p}}{(Nk+p)!}$$ for $p=1,2,\dots , N-1$. Likewise, the $N$-hyperbolic functions are defined by: $$cosh_N(z)=\sum_{k=0}^\infty\frac{z^{Nk}}{(Nk)!} \ \ \& \ \ sinh_{N,p}(z)=\sum_{k=0}^\infty\frac{z^{Nk+p}}{(Nk+p)!}$$ for $p=1,2,\dots , N-1$.
The series which define the $N$-trigonometric and $N$-hyperbolic functions converge for each $z \in {\mathcal{A}}$ for any algebra. For example, in complex analysis the hyperbolic functions are [*entire*]{} functions which have interesting applications. We pause to note the differential relations, $$\begin{aligned}
&\frac{d}{dz} \cos_N(z) = -\sin_{N,N-1}(z),
&\frac{d}{dz} \cosh_N(z) = \sinh_{N,N-1}(z),\\ \notag
&\frac{d}{dz} \sin_{N,1}(z) = \cos_{N}(z),
&\frac{d}{dz} \sinh_{N,1}(z) = \cosh_{N}(z), \\ \notag
&\frac{d}{dz} \sin_{N,p}(z) = \sin_{N,p-1}(z),
&\frac{d}{dz} \sinh_{N,p}(z) = \sinh_{N,p-1}(z). \end{aligned}$$ for $p=2, \dots , N-1$ hold independent of our choice of ${\mathcal{A}}$.
We observe that:
1. the N-trigonometric functions are the special functions of ${ \mathbb{C} }_N$,
2. the N-hyperbolic functions are the special functions of ${\mathcal{H}}_N$.
[**Proof:**]{} Begin with [**(i.)**]{}. Suppose $j^N=-1$. We have, for all $t \in { \mathbb{R} }$: $$\begin{aligned}
\notag
e^{jt}&=\sum_{k=0}^\infty \frac{(jt)^{Nk}}{(Nk)!}+\sum_{k=0}^\infty \frac{(jt)^{Nk+1}}{(Nk+1)!}+\dots +\sum_{k=0}^\infty \frac{(jt)^{Nk+N-1}}{(Nk+N-1)!}\\ \notag
&=\sum_{k=0}^\infty \frac{(j^N)^kt^{Nk}}{(Nk)!}+\sum_{k=0}^\infty \frac{j(j^N)^kt^{Nk+1}}{(Nk+1)!}+\dots +\sum_{k=0}^\infty \frac{j^{N-1}(j^N)^kt^{Nk+N-1}}{(Nk+N-1)!}\\ \notag
&=\sum_{k=0}^\infty \frac{(-1)^kt^{Nk}}{(Nk)!}+j\sum_{k=0}^\infty \frac{(-1)^kt^{Nk+1}}{(Nk+1)!}+\dots +j^{N-1}\sum_{k=0}^\infty \frac{(-1)^kt^{Nk+N-1}}{(Nk+N-1)!}\\ \notag
&=\cos_N(t)+j\sin_{N,1}(t)+\dots+j^{N-1}\sin_{N,N-1}(t).\end{aligned}$$ Thus $\cos_N,\ \sin_{N,1},\dots, \sin_{N,N-1}$ form the special functions of ${ \mathbb{C} }_N$. In contrast, for ${\mathcal{H}}_N$ we have $j^N=1$ thus $j^{Nk+p} = (j^N)^kj^p = j^p$ and it follows we find the special functions of ${\mathcal{H}}_N$ are $\cosh_N,\ \sinh_{N,1},\dots, \sinh_{N,N-1}$. $\Box$
Pythagorean functions
=====================
We may study a finite dimensional associative algebras over ${ \mathbb{R} }$ by instead studying the associated [*regular representation*]{} of ${\mathcal{A}}$. When ${\mathcal{A}}= { \mathbb{R} }^N$ the matrix corresponding to $z \in {\mathcal{A}}$ is the standard matrix of the left multilication by $z$ map; $L_z: {\mathcal{A}}\rightarrow {\mathcal{A}}$ is defined by $L_z(x) = zx$ for all $x \in {\mathcal{A}}$ and $$\mathbf{M}(z) = [L_z] = \left[ ze_1 | ze_2| \cdots | ze_N \right].$$ When we use $e_1=1 \in {\mathcal{A}}$ the matrix representation of $z$ is further simplified.
If $z = x+iy \in { \mathbb{C} }$ then note $e_1=1$ and $e_2=i$ hence $$\mathbf{M}(z) = [z|zi] = [x+iy|xi-y] = \left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]$$
If $z \in {\mathcal{H}}_N$ where $e_1=1, e_2=j, \dots , e_N = j^{N-1}$ and $j^N=1$ then $$\mathbf{M}(z) = [z|zj| \cdots | zj^{N-1}]$$ To be explicit, in the $N=3$ case we have: $$\mathbf{M}(x+jy+zj^2) = \left[ \begin{array}{ccc} x & z & y \\
y & x & z \\
z & y & x\end{array}\right]$$
Combining the exponential on ${\mathcal{A}}$ generated by $j$ with the determinant function allows us to create a new function on ${\mathcal{A}}$ which we define below:
For ${\mathcal{A}}= { \mathbb{R} }^N$ with basis $1, j , j^2, \dots , j^{N-1}$ we define the [*real Pythagorean Function*]{} by $$\mathcal{P}_{{\mathcal{A}}}(t) = \text{det}\left[ e^{j t}| j e^{j t} | \cdots | j^{N-1} e^{j t} \right]$$ for each $t \in { \mathbb{R} }$. The [*Pythagorean function of ${\mathcal{A}}$*]{} is the unique extension of the real Pythagorean function to ${\mathcal{A}}$.
Notice Pythagorean function is manifestly a formula which involves the special functions of ${\mathcal{A}}$. If one is willing to study the matrices with components taken from an algebra then we may express $$\mathcal{P}_{{\mathcal{A}}}(z) = \text{det}\left[ e^{j z}|j e^{j z} | \cdots | j^{N-1} e^{j z} \right]$$ for each $z \in {\mathcal{A}}$. Once more it is instructive to examine how this construction unfolds for complex and hyperbolic numbers.
In ${ \mathbb{C} }$, as $e^{it} = \cos( t) + i \sin( t)$ and $ie^{it} = i \cos( t)-\sin( t)$ we find the Pythagorean function: $$\mathcal{P}_{{ \mathbb{C} }}(z) =det\left[ \begin{array}{rr} \cos (z)& -\sin (z) \\ \sin( z) & \cos (z)\end{array}\right]
= \cos^2 (z) +\sin^2 (z).$$ Observe $\mathcal{P}_{{ \mathbb{C} }}(z) = 1$ for all $z \in { \mathbb{C} }$.
In ${\mathcal{H}}$ which is generated by $j$ with $j^2=1$ we found $e^{jt} = \cosh(t) + j \sinh(t)$ and $je^{jt} = j\cosh(t)+\sinh(t)$. Thus, the Pythagorean function is: $$\mathcal{P}_{{\mathcal{H}}}(z) =det\left[ \begin{array}{rr} cosh(z)& sinh(z)\\ sinh(z) & cosh(z)\end{array}\right]
=cosh^2(z)-sinh^2(z).$$ It can be shown $\mathcal{P}_{{\mathcal{H}}}(z) = 1$ for all $z \in {\mathcal{H}}$.
These examples are part of a larger pattern:
**(N-Pythagorean Theorem)** If ${\mathcal{A}}={ \mathbb{R} }^N$ is generated by $j$ where $j^N = c \in { \mathbb{R} }$ then $\mathcal{P}_{{\mathcal{A}}}(z) = 1$ for all $z \in {\mathcal{A}}$.
We prove this result in Section \[sec:proofofNpythag\]. Our examples are based on the choices $c = \pm 1$ or $c=0$. However, the choice $c=0$ is not especially interesting:
Consider the dual numbers ${\mathcal{A}}= { \mathbb{R} }\oplus \eta { \mathbb{R} }$ with $\eta^2=0$. Observe, $e^{\eta t} = 1+\eta t$ hence $\eta e^{\eta t} = \eta$ and so $$\mathcal{P}_{{\mathcal{A}}}(z) = \text{det} \left[ \begin{array}{cc} 1 & 0 \\ z & 1 \end{array}\right] = 1.$$
In contrast, the identities found below are surprising.
In ${\mathcal{H}}_3$ we have basis $1,j,j^2$ with $j^3=1$ and we calculate: $$\begin{aligned}
\mathcal{P}_{{\mathcal{H}}_3}(z) &=\text{det} \left[ \begin{array}{ccc} cosh_3(z) & sinh_{3,2}(z)& sinh_{3,1}(z)\\ sinh_{3,1}(z)& cosh_3(z)& sinh_{3,2}(z)\\ sinh_{3,2}(z)& sinh_{3,1}(z)& cosh_3(z)\end{array} \right]\\ \notag
&=cosh_3^3(z)+sinh_{3,1}^3(z)+sinh_{3,2}^3(z)-3cosh_3(z)sinh_{3,1}(z)sinh_{3,2}(z).\end{aligned}$$ The $N$-Pythagorean Theorem indicates $\mathcal{P}_{{\mathcal{H}}_3}(z) = 1$ for each $z \in {\mathcal{H}}_3$.
Consider ${ \mathbb{C} }_3$ generated by $j$ with $j^3=-1$. Then $e^{jz} = \cos_3(z)+j\sin_{3,1}(z)+ j^2 \sin_{3,2}(z)$ and we calculate: $$\begin{aligned}
\mathcal{P}_{{ \mathbb{C} }_3}(z) &=\text{det} \left[ \begin{array}{ccc}
\cos_3(z) & -\sin_{3,2}(z) & -\sin_{3,1}(z) \\
\sin_{3,1}(z) & \cos_3(z) & -\sin_{3,2}(z) \\
\sin_{3,2}(z) & \sin_{3,1}(z) & \cos_3(z)
\end{array} \right]\\ \notag
&=cos_3^3(z)-sin_{3,1}^3(z)+sin_{3,2}^3(z)+3cos_3(z)sin_{3,1}(z)sin_{3,2}(z).\end{aligned}$$ The $N$-Pythagorean Theorem indicates $\mathcal{P}_{{ \mathbb{C} }_3}(z) = 1$ for each $z \in { \mathbb{C} }_3$.
The relations implied by $\mathcal{P}_{{\mathcal{A}}}(z)=1$ for $N \geq 4$ are rather lengthy.
Proof of N-Pythagorean theorem {#sec:proofofNpythag}
==============================
[**Proof:**]{} Assume ${\mathcal{A}}= { \mathbb{R} }^N$ is generated by $j$ with $j^N=c$ for some given $c \in { \mathbb{R} }$. Begin by writing an explicit formula for the real Pythagorean function: $$\mathcal{P}_{{\mathcal{A}}}(t) = \text{det} ( \mathbf{M}(e^{jt}) )= \text{det} \left[ e^{jt}|je^{jt}| \cdots | j^{N-1}e^{jt} \right].$$ If $f: \text{dom}(f) \subseteq {\mathcal{A}}\rightarrow {\mathcal{A}}$ is ${\mathcal{A}}$-differentiable and $g: { \mathbb{R} }\rightarrow {\mathcal{A}}$ is real differentiable then there is a chain rule for the composite $f { \,{\scriptstyle \stackrel{\circ}{}}\, }g: { \mathbb{R} }\rightarrow {\mathcal{A}}$. In particular, see Theorem 6.7 of [@cookAcalculusI], $$\frac{d}{dt} (f (g(t)) = \frac{df}{dz}(g(t)) \frac{dg}{dt}.$$ Here $g = g_1+ jg_2+ \cdots + j^{N-1}g_N$ has $\frac{dg}{dt} = \frac{dg_1}{dt}+ j\frac{dg_2}{dt}+ \cdots + j^{N-1}\frac{dg_{N-1}}{dt}$. Consider $f(z) = e^z$ and $g(t) = jt$ then we know $\frac{df}{dz} = e^z$ and it is simple to calculate $\frac{dg}{dt} = j$. Hence, $$\frac{d}{dt} (e^{jt}) = je^{jt}.$$ Multiply by $j^{p-1}$ and note: $$j^{p-1}\frac{d}{dt} (e^{jt}) = j^{p-1}je^{jt} \ \ \Rightarrow \ \ \frac{d}{dt} \left(j^{p-1}e^{jt} \right) = j^pe^{jt}.$$ In particular, $\frac{d}{dt} \text{Col}_{p}(\mathbf{M}(e^{jt}) ) = \text{Col}_{p+1}(\mathbf{M}(e^{jt}) )$ for $p=1, \dots , N-1$. However, $$\frac{d}{dt} (j^{N-1}e^{jt}) = j^Ne^{jt}$$ provides $\frac{d}{dt} \text{Col}_{N}(\mathbf{M}(e^{jt}) ) = c\, \text{Col}_{1}(\mathbf{M}(e^{jt}) )$ where $j^N = c$. Hence observe every column in the regular representation of $e^{jt}$ has a derivative which is proportional to another column in the representation.\
In order to compress the notation a bit let us set $\mathbf{M}(e^{jt}) = B = [B_1|\cdots |B_N]$ which gives $\mathcal{P}_{{\mathcal{A}}}(t) = \text{det}(B)$. The formula for the determinant below makes manifest the fact the determinant is a multilinear function of its columns: $$\text{det}(B) = \sum_{i_1\dots i_N=1}^{N} \epsilon_{i_1i_2\dots i_N}B_{i_11}B_{i_22} \cdots B_{i_NN}$$ here $\epsilon_{i_1\dots i_N}$ is the completely antisymmetric symbol where $\epsilon_{12\dots N}=1$. By the $N$-fold product rule we find: $\frac{d}{dt}( \text{det}(B)) =$ $$\begin{aligned}
&= \sum_{i_1\dots i_N} \epsilon_{i_1\dots i_N}\frac{dB_{i_11}}{dt}B_{i_22} \cdots B_{i_NN}+ \sum_{i_1\dots i_N} \epsilon_{i_1\dots i_N} B_{i_11}\frac{dB_{i_22}}{dt} \cdots B_{i_NN} \\ \notag
& \qquad + \cdots + \sum_{i_1\dots i_N} \epsilon_{i_1\dots i_N} B_{i_11}B_{i_22} \cdots \frac{dB_{i_NN}}{dt} \\ \notag
&= \text{det}[B_2|B_2|\cdots |B_N]+
\text{det}[B_1|B_3|B_3|\cdots |B_N]+ \cdots + \text{det}[B_1|B_2|\cdots |c B_1] \\ \notag
&=0.\end{aligned}$$ Thus $t \mapsto \text{det} ( \mathbf{M}(e^{jt}) )$ is a constant function on ${ \mathbb{R} }$. Notice $t=0$ maps to $\text{det}(I)=1$ hence $\mathcal{P}_{{\mathcal{A}}}(t) = 1$. for each $t \in { \mathbb{R} }$. Finally, we extend to all of ${\mathcal{A}}$ using Theorem \[thm:entire\]. $\Box$
Acknowledgements
================
The authors are thankful to N. BeDell for helpful comments on a rough draft of this article. We should mention that W.S. Leslie provided an alternate, purely algebraic, proof of the $N$-Pythagorean Theorem in private communication. Finally, we thank Khang Nguyen for finding the improved Theorem 5.4 as well as reformulations of Corollary 5.20 and 5.21.
[99.]{} N. BeDell, [*Doing Algebra in Associative Algebras*]{}, in preparation.
N. BeDell, [*Logarithms over Associative Algebras*]{}, in preparation.
N. BeDell, [*Generalized Trigonometric Functions over Associative Algebras*]{}, in preparation.
J. S. Cook, W. S. Leslie, M. L. Nguyen, B. Zhang, [*Laplace Equations for Real Semisimple Associative Algebras of Dimension 2, 3 or 4*]{}, Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics $\&$ Statistics Vol. 64, pp 67-83 (2013)
J. S. Cook, [*Introduction to ${\mathcal{A}}$-Calculus*]{}, in preparation.
J. S. Cook, N. BeDell, [*Introduction to the Theory of ${\mathcal{A}}$-ODEs*]{}, in preparation.
J. Dieudonn’ e, [*Foundations of Modern Analysis*]{}, Academic Press Inc. (1960)
R. Johnsonbaugh, W.E. Pfaffenberger, [*Foundations of Mathematical Analysis*]{}, Dover Edition, (2002)
E. Kreyszig, [*Introductory Functional Analysis with Applications*]{}, John Wiley $\&$ Sons Inc. (1978)
W. Rudin, [*Principles of Mathematical Analysis*]{}, McGraw-Hill, (1964)
[^1]: we write $({\mathcal{A}}, \star)$ to emphasize the pairing where helpful
[^2]: isomorphic as associative real algebras, we say $({\mathcal{A}}, \star)$ and $({\mathcal{B}}, { \,{\scriptstyle \stackrel{\circ}{}}\, })$ are isomorphic if there is a linear bijection $\Psi: {\mathcal{A}}\rightarrow {\mathcal{B}}$ for which $\Psi( x \star y) = \Psi(x) { \,{\scriptstyle \stackrel{\circ}{}}\, }\Psi(y)$ for all $x,y \in {\mathcal{A}}$.
[^3]: this theorem is likely true in the noncommutative context as well
[^4]: if $v_1 = \mathds{1}$ then symmetric CR-equations reduce to the usual CR-equations $\frac{\partial f}{\partial x_j} = \frac{\partial f}{\partial x_1} \star v_j$.
|
---
abstract: 'Let $M$ be a smooth manifold, and let $\mathcal{O}(M)$ be the poset of open subsets of $M$. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) $F \colon {\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ from ${\mathcal{O}(M)}$ to the category of spaces. Weiss showed that polynomial cofunctors of degree $\leq k$ are determined by their values on ${\mathcal{O}_k(M)}$, where ${\mathcal{O}_k(M)}$ is the full subposet of ${\mathcal{O}(M)}$ whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. Afterwards Pryor showed that one can replace ${\mathcal{O}_k(M)}$ by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from ${\mathcal{O}(M)}$ to any simplicial model category ${\mathcal{M}}$. If $F_k(M)$ stands for the unordered configuration space of $k$ points in $M$, we also show that the category of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ is weakly equivalent to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$ provided that ${\mathcal{M}}$ has a zero object. Using a new approach, we also show that if ${\mathcal{M}}$ is a general model category and $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor, then the homotopy right Kan extension of $F$ along the inclusion ${\mathcal{O}_k(M)}{\hookrightarrow}{\mathcal{O}(M)}$ is also an isotopy cofunctor.'
author:
- |
Paul Arnaud Songhafouo Tsopméné\
Donald Stanley
title: ' **Polynomial functors in manifold calculus**'
---
Introduction
============
Let $M$ be a smooth manifold, and let $\mathcal{O}(M)$ be the poset of open subsets of $M$. *Manifold calculus* is a calculus of functors suitable for studying cofunctors [^1] $F : {\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ from ${\mathcal{O}(M)}$ to the category of spaces (of which the embedding functor $\text{Emb}(-, W)$ for a fixed manifold $W$ is a prime example). So manifold calculus belongs to the world of calculus of functors, and therefore it definitely has a notion of polynomial cofunctor. Roughly speaking, a *polynomial cofunctor* is a contravariant functor ${\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ that satisfies an appropriate higher-order excision property, similar to the case of [@good03] (see Definition \[poly\_defn\]). In [@wei99 Theorems 4.1, 5.1] Weiss characterizes polynomial cofunctors. More precisely, he shows that polynomial cofunctors of degree $\leq k$ are determined (up to equivalence of course) by their values on ${\mathcal{O}_k(M)}$, the full subposet of ${\mathcal{O}(M)}$ whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. Many examples of polynomial and homogeneous cofunctors are also provided in [@wei99]. Another good reference where the reader can find an introduction to manifold calculus is [@mun10].
Weiss’ characterization of polynomial cofunctors was generalized by Pryor in [@pryor15] as follows. Let ${\mathcal{B}}$ be a basis for the topology of $M$. We assume that ${\mathcal{B}}$ is *good*, that is, every element of ${\mathcal{B}}$ is a subset of $M$ diffeomorphic to an open ball. For instance, if $M = {\mathbb{R}}^m$, we can take ${\mathcal{B}}$ to be the collection of genuine open balls (with respect to the euclidean metric), or cubes, or simplices, or convex $d$-bodies more generally. For $k \geq 0$, we let ${\mathcal{B}_k(M)}\subseteq {\mathcal{O}_k(M)}$ denote the full subposet whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}$. So one possible choice of ${\mathcal{B}_k(M)}$ is ${\mathcal{O}_k(M)}$ itself. In [@pryor15 Theorem 6.12] Pryor shows, in the same spirit as Weiss, that any polynomial cofunctor ${\mathcal{O}(M)}{\longrightarrow}\text{Spaces}$ of degree $\leq k$ is determined by its restriction to ${\mathcal{B}_k(M)}$. So one can replace ${\mathcal{O}_k(M)}$ by ${\mathcal{B}_k(M)}$ without losing any homotopy theoretic information when forming the polynomial approximation to a cofunctor.
In this paper we generalize the aforementioned results of Weiss-Pryor to cofunctors from ${\mathcal{O}(M)}$ to any simplicial model category ${\mathcal{M}}$. Specifically we have the following theorem, which is our first result.
\[main\_thm\] Let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$, and let ${\mathcal{B}_k(M)}\subseteq {\mathcal{O}(M)}$ be the full subposet whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}$. Consider a simplicial model category ${\mathcal{M}}$ and a cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$. Then $F$ is good (see Definition \[good\_defn\]) and polynomial of degree $\leq k$ (see Definition \[poly\_defn\]) if and only if the restriction $F|{{\mathcal{B}}_k}(M)$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]) and the canonical map $F {\longrightarrow}(F|{{\mathcal{B}}_k}(M))^!$ is a weak equivalence. Here $(F|{{\mathcal{B}}_k}(M))^! \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is the cofunctor defined as $$(F|{{\mathcal{B}}_k}(M))^!(U):= \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V).$$
Notice that Theorem \[main\_thm\] implies that the category of good polynomial cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $\leq k$ is *weakly equivalent*, in the sense of Definition \[we\_defn\], to the category of isotopy cofunctors ${{\mathcal{B}}_k}(M) {\longrightarrow}{\mathcal{M}}$. Also notice that our definition of *good cofunctor* is slightly different from the classical one (see [@wei99 Page 71] or [@mun10 Definition 1.3.4] or [@pryor15 Definition 3.1]) as we add an extra axiom: our cofunctors are required to be also objectwise fibrant. We need that extra axiom to be able to use the homotopy invariance theorem (see Theorem \[fib\_cofib\_thm\]) and the cofinality result (see Theorem \[htpy\_cofinal\_thm\]). If one works with a category ${\mathcal{M}}$ in which every object is fibrant, the extra axiom becomes a tautology. This is the case in Weiss’ paper [@wei99] where ${\mathcal{M}}=$ Spaces. For the main ingredients in the proof of Theorem \[main\_thm\], see Outline of the paper below.
As mentioned earlier, our result generalizes those of Weiss. In fact, from Theorem \[main\_thm\] with ${\mathcal{M}}=\text{Spaces}$ and ${\mathcal{B}}= {\mathcal{O}}$, the maximal good basis, one can easily deduce the main results of [@wei99], which are Theorems 4.1, 5.1 and 6.1.
The following conjecture says that Theorem \[main\_thm\] still holds when ${\mathcal{M}}$ is replaced by a general model category. We believe in that conjecture, which could be handled by using the same approach as that we use to show Theorem \[main\_thm\]. The issue with that approach is the fact that some important results/properties regarding homotopy limits in a general model category (for example Theorem \[holim\_tot\_thm\] and Proposition \[comm\_prop\]) are available nowhere in the literature. So the proof of the conjecture may turn into a matter of homotopy limits. A good reference, where the reader can find the definition and several useful properties of homotopy limits (in a general model category of course), is [@hir03 Chapter 19]. Another good reference is [@dhks04].
\[main\_conj\] Theorem \[main\_thm\] remains true if one replaces ${\mathcal{M}}$ by a general model category.
Now we state our second result. Given a good cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ one can define its $k$th polynomial approximation, denoted $T_kF \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$, as the homotopy right Kan extension of the restriction $F|{\mathcal{O}_k(M)}$ along the inclusion ${\mathcal{O}_k(M)}{\hookrightarrow}{\mathcal{O}(M)}$. In order words $T_kF(U) := \underset{V \in {\mathcal{O}_k(M)}}{\text{holim}} \; F(V)$. The difference between $T_kF$ and $T_{k-1}F$ belongs to a nice class of cofunctors called *homogeneous cofunctors* of degree $k$ (see Definition \[hc\_defn\]). When $k =1$ we talk about *linear cofunctors*. Thanks to the fact that Theorem \[main\_thm\] holds for any good basis ${\mathcal{B}}$ we choose, one can prove the following result, which roughly states that the category of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ is weakly equivalent to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$. Here $F_k(M)$ stands for the unordered configuration space of $k$ points in $M$.
\[main2\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Assume that ${\mathcal{M}}$ has a zero object (that is, an object which is both terminal an initial).
1. Then the category ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$ of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ (see Definition \[hc\_defn\]) is weakly equivalent (in the sense of Definition \[we\_defn\]) to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$. That is, $${\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}}) \simeq {\mathcal{F}}_1 ({\mathcal{O}}(F_k(M)); {\mathcal{M}}).$$
2. For $A \in {\mathcal{M}}$ we have the weak equivalence $${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}}) \simeq {\mathcal{F}}_{1A} ({\mathcal{O}}(F_k(M)); {\mathcal{M}}),$$ where ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}})$ stands for the category of homogeneous cofunctors $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ such that $F(U) \simeq A$ for any $U$ diffeomorphic to the disjoint union of exactly $k$ open balls.
The second part of this result will be used in [@paul_don17-3]. A similar result (but with a different approach) to Theorem \[main2\_thm\] was obtained by the authors in [@paul_don17 Corollary 3.31] for very good homogeneous functors. Note that neither [@paul_don17 Corollary 3.31] nor Theorem \[main2\_thm\] was known before, even for ${\mathcal{M}}= \text{Spaces}$. Theorem \[main2\_thm\] is interesting in the sense that it reduces the study of homogeneous cofunctors of degree $k$ to the study of linear cofunctors, which are easier to handle. In [@paul_don17-3] we use it (Theorem \[main2\_thm\]) as the starting point in the classification of homogeneous cofunctors of degree $k$.
Our third result is a partial answer to Conjecture \[main\_conj\].
\[iso\_cof\_thm\] Let ${\mathcal{M}}$ be a model category. Let $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor $F^{!} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ defined as $$\begin{aligned}
\label{fsrik_defn}
F^{!}(U) := \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V)\end{aligned}$$ is an isotopy cofunctor as well.
The method we use to prove Theorem \[iso\_cof\_thm\] is completely different from that we use to prove Theorem \[main\_thm\] essentially because of the following. First note that in Theorem \[main\_thm\] ${\mathcal{M}}$ is a simplicial model category, while in Theorem \[iso\_cof\_thm\] ${\mathcal{M}}$ is a general model category. To prove Theorem \[main\_thm\], we use several results/properties of homotopy limits in simplicial model categories such as the Fubini theorem (see Theorem \[fubini\_thm\]), and Proposition \[comm\_prop\]. However, Proposition \[comm\_prop\] involves the notion of totalization of a cosimplicial object, which a priori does not make sense in a general model category.
The key concept we introduce to prove Theorem \[iso\_cof\_thm\] is called *admissible family* of open subsets. Roughly speaking, a sequence $B = B_0, \cdots, B_n$ of open balls is said to be *admissible* if $B_i \cap B_{i+1} \neq \emptyset$ for all $i$. One can extend that definition to sequences $V=V_0, \cdots, V_n$ of objects of ${\mathcal{O}_k(M)}$ (see Definition \[pw\_adm\_defn\]). Such sequences yield zigzags of isotopy equivalences of ${\mathcal{O}_k(M)}$ between $V$ and $V_n$, and the collection of those form a category denoted ${{\mathcal{D}}(V)}$ (see Definition \[dv\_defn\]). This latter category plays a crucial role in Section \[iso\_cof\_section\]. Indeed, one can deduce Theorem \[iso\_cof\_thm\] by applying the homotopy limit functor to appropriate diagrams in ${\mathcal{M}}$ indexed by ${{\mathcal{D}}(V)}$.
**Outline of the paper**
This paper is subdivided into two detailed and almost disconnected parts. The first one covers Sections \[notation\_section\], \[holim\_simpl\_section\], \[sos\_good\_section\], \[poly\_section\] and \[hc\_section\] where we prove Theorems \[main\_thm\] and \[main2\_thm\], while the second covers Section \[iso\_cof\_section\] where we prove Theorem \[iso\_cof\_thm\].
1. In Section \[notation\_section\] we fix some notation. We also give a table that plays the role of a dictionary between our notation and that of Weiss-Pryor. The purpose of that table is to help the exposition of certain proofs, especially in Subsections \[cof\_fsp\_subsection\], \[sos\_good\_subsection\].
2. Section \[holim\_simpl\_section\] deals with homotopy limits in simplicial model categories. We follow Hirschhorn’s style [@hir03 Chapters 18-19]. Since the homotopy limit is so ubiquitous in this work, we first give its definition in Subsection \[holim\_subsection\]. Next, in the same subsection, we recall some of its basic properties including the homotopy invariance (see Theorem \[fib\_cofib\_thm\]), the cofinality theorem (see Theorem \[htpy\_cofinal\_thm\]), and the Fubini theorem (see Theorem \[fubini\_thm\]). All these properties are indeed used in many places in this work. Subsection \[cr\_subsection\] deals with cosimplicial replacement of a diagram. We prove Proposition \[comm\_prop\], which is the main new result of the section. It says that the canonical isomorphism $\underset{{\mathcal{D}}}{\text{holim}} \; F \cong {\text{Tot}}\; {\Pi^{\bullet}}F$ between the homotopy limit of a diagram $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ and the totalization of its cosimplicial replacement is natural in the following sense. If $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is a functor between small categories, then the obvious square involving the isomorphisms $\underset{{\mathcal{D}}}{\text{holim}} \; F \cong {\text{Tot}}\; {\Pi^{\bullet}}F$ and $\underset{{\mathcal{C}}}{\text{holim}} \; F \theta \cong {\text{Tot}}\; {\Pi^{\bullet}}(F \theta)$ must commutes. Proposition \[comm\_prop\] will be used in the proof of Theorem \[sos\_thm\].
3. Section \[sos\_good\_section\] proves two important results: Theorem \[sos\_thm\] and Theorem \[good\_thm\]. The first, which is the crucial ingredient in the proof of Theorem \[main\_thm\], roughly says that the homotopy limit ${F^{!}_{{\mathcal{B}}}}(U) := \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V)$ does not depend on the choice of the basis ${\mathcal{B}}$. Specifically, it says that for any good basis ${\mathcal{B}}$ for the topology of $M$, for any isotopy cofunctor $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$, the canonical map $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V)$ is a weak equivalence for all $U \in {\mathcal{O}(M)}$. The proof of Theorem \[sos\_thm\] goes through two big steps. The first step (see Subsection \[cof\_fsp\_subsection\]) consists of splitting ${F^{!}_{{\mathcal{B}}}}$ into smaller pieces ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}= \{{\tilde{F}^{!p}_{{\mathcal{B}}}}\}_{p \geq 0}$, and show that ${\tilde{F}^{!p}_{{\mathcal{B}}}}$ is independent of the choice of the basis ${\mathcal{B}}$ for all $p$ (see Proposition \[sosp\_prop\]). This idea of splitting comes from the paper of Weiss [@wei99], and the nice thing is that the collection ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}$ turns out to be a cosimplicial object in the category of cofunctors from ${\mathcal{O}(M)}$ to ${\mathcal{M}}$. The second step, inspired by Pryor’s work [@pryor15], is to connect $\underset{[p] \in \Delta}{\text{holim}} \; {\tilde{F}^{!p}_{{\mathcal{B}}}}(U)$ and ${F^{!}_{{\mathcal{B}}}}(U)$ by a zigzag of natural weak equivalences (see Subsection \[sos\_good\_subsection\]). It is very important that every map of that zigzag is natural in both variables $U$ and ${\mathcal{B}}$. This is one of the reasons we really need Section \[holim\_simpl\_section\] where all those maps are carefully inspected, especially the map that appears in Theorem \[holim\_tot\_thm\]. Regarding Theorem \[good\_thm\], it says that ${F^{!}_{{\mathcal{B}}}}$ is good provided that $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor. This result is a part of the proof of Theorem \[main\_thm\], and its proof is based on Theorem \[sos\_thm\] and the Grothendieck construction (see Subsection \[gro\_const\_subsection\]).
4. In Section \[poly\_section\] we prove the main result of the first part: Theorem \[main\_thm\]. To do this we use Theorem \[sos\_thm\] and Theorem \[good\_thm\] as mentioned earlier. We also use Lemmas \[cofinal\_lem\], \[poly\_lem\], \[charac\_lem\]. The first lemma says that a certain functor is right cofinal. The second (which is a generalization of [@wei99 Theorem 4.1]) states that if $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor, then ${F^{!}_{{\mathcal{B}}}}$ is polynomial of degree $\leq k$. The proof of this result also uses the Grothendieck construction. The third lemma (which is a generalization of [@wei99 Theorem 5.1]) is a characterization of polynomial cofunctors. Note that Lemma \[poly\_lem\] and Lemma \[charac\_lem\] are important themselves.
5. Section \[hc\_section\] deals with homogeneous cofunctors, and is devoted to the proof of Theorem \[main2\_thm\]. The key ingredient we need is Lemma \[homo\_lem\], which roughly says that homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ are determined by their values on subsets diffeomorphic to the disjoint union of exactly $k$ open balls provided that ${\mathcal{M}}$ has a zero object. So Lemma \[homo\_lem\] is also a useful result in its own right since it characterizes homogeneous cofunctors. Note that the proof of Lemma \[homo\_lem\] is based on the results we obtained in Section \[sos\_good\_section\] and Section \[poly\_section\].
6. Section \[iso\_cof\_section\] proves Theorem \[iso\_cof\_thm\]. To do this we use a completely different method (but rather lengthy) from that we used in previous sections. As mentioned earlier, the key concept here is that of *admissible family* (see Definition \[pw\_adm\_defn\]) introduced in [@paul_don17]. In Subsection \[holim\_subsectiong\] we recall some useful properties for homotopy limits in general model categories. Subsections \[dv\_subsection\], \[hp\_subsection\] are preparatory subsections dealing with technical tools needed for the proof of Theorem \[iso\_cof\_thm\]. Lastly, Subsection \[iso\_cof\_subsection\] proves Theorem \[iso\_cof\_thm\].
**Acknowledgements.** This work has been supported by Pacific Institute for the Mathematical Sciences (PIMS) and the University of Regina, that the authors acknowledge. We are also grateful to P. Hirschhorn, J. Scherer, and W. Chacholski for helpful conversations (by emails) about homotopy limits and homotopy colimits.
Notation {#notation_section}
========
In this section we fix some notation.
1. We let $M$ denote a smooth manifold. If $U$ is a subset of $M$, we let ${\mathcal{O}}(U)$ denote the poset of open subsets of $U$, morphisms being inclusions of course. In particular one has the poset ${\mathcal{O}(M)}$.
2. For $k \geq 0$, and $U \in {\mathcal{O}(M)}$, we let ${\mathcal{O}_k}(U) \subseteq {\mathcal{O}}(U)$ denote the full subposet whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. In particular one has the poset ${\mathcal{O}_k(M)}$.
3. We write ${\mathcal{O}}$ for the collection of all subsets of $M$ diffeomorphic to an open ball. Certainly ${\mathcal{O}}$ is a full subposet of ${\mathcal{O}(M)}$.
4. We let ${\mathcal{B}}$ denote a good basis (see Definition \[gb\_defn\]) for the topology of $M$. Clearly, one has ${\mathcal{B}}\subseteq {\mathcal{O}}$ for any good basis ${\mathcal{B}}$.
5. We write ${\mathcal{M}}$ for a simplicial model category unless stated otherwise.
6. If $\beta \colon F {\longrightarrow}G$ is a natural transformation, the component of $\beta$ at $x$ will be denoted $\beta[x] \colon F(x) {\longrightarrow}G(x)$.
7. We use the notation $x := \text{def}$ to state that the left hand side is defined by the right hand side.
Since the proofs of some important results in this paper are based on [@pryor15] and [@wei99], we need a dictionary of notations which is provided by the following table. The purpose of that table is then to help the exposition of certain proofs, especially in Subsections \[cof\_fsp\_subsection\], \[sos\_good\_subsection\] as we said before. The first column gives the notation that we use in this paper, while the second and the third regard the notation used in [@pryor15] and [@wei99] respectively. The notations that appear in the same row have the same meaning. The word nothing means that there is no notation with the same meaning in the corresponding paper. For instance, in the first row we have the notation ${\mathcal{O}}$ in this paper, which stands for the maximal good basis for the topology of $M$. However there is no notation in [@pryor15] and [@wei99] that has the same meaning as ${\mathcal{O}}$.
In this paper In Pryor’s paper [@pryor15] In Weiss’ paper [@wei99]
------------------------------------------------------------------------ --------------------------------------------- --------------------------------------------
${\mathcal{O}}$ nothing nothing
${\mathcal{O}(M)}$ ${\mathcal{O}(M)}$ or just ${\mathcal{O}}$ ${\mathcal{O}(M)}$ or just ${\mathcal{O}}$
${\mathcal{O}_k(M)}$ ${\mathcal{O}_k}$ ${\mathcal{O}}k$
${\mathcal{B}_k(M)}$ (see Definition \[fsb\_defn\]) ${{\mathcal{B}}_k}$ nothing
${\widetilde{{\mathcal{O}}}_{k, p}}(M)$ (see Definition \[bkp\_defn\]) nothing ${\mathcal{I}}k {\mathcal{O}}k_p (M)$
${\widetilde{{\mathcal{B}}}_{k, p}}(M)$ (see Definition \[bkp\_defn\]) ${\mathcal{A}}_k ({{\mathcal{B}}_k})_p (M)$ nothing
${\widehat{{\mathcal{B}}}_{k, q}}(M)$ $({\mathcal{A}}_k)_q {{\mathcal{B}}_k}(M)$ nothing
${F^{!}_{{\mathcal{O}}}}$ (see Example \[fso\_expl\]) nothing $E^{!}$
${F^{!}_{{\mathcal{B}}}}$ (see Definition \[fsb\_defn\]) $F^{!}$ nothing
${\tilde{F}_{{\mathcal{B}}}}^p$ (see Definition \[fsbp\_defn\]) $F_p$ nothing
${\tilde{F}_{{\mathcal{B}}}}^{!p}$ (see Definition \[fsbp\_defn\]) $F^{!}_p$ nothing
${\tilde{F}_{{\mathcal{O}}}}^{!p}$ (see Definition \[fsbp\_defn\]) nothing $E^{!}_p$
${\hat{F}_{{\mathcal{B}}}}^q$ (see (\[fhq\])) $\widehat{F}_q$ nothing
${\hat{F}_{{\mathcal{B}}}}^{!q}$ (see (\[fhsq\])) $\widehat{F}^{!}_q$ nothing
For the meaning of ${\widehat{{\mathcal{B}}}_{k, q}}(M)$ we refer the reader to the beginning of Subsection \[sos\_good\_subsection\].
Homotopy limits in simplicial model categories {#holim_simpl_section}
==============================================
In this section we recall some useful definitions and results about homotopy limits in simplicial model categories. We also prove Corollary \[hir\_coro\] and Proposition \[comm\_prop\], which will be used in Section \[sos\_good\_section\]. The main reference here is Hirschhorn’s book [@hir03 Chapter 18].
Let us begin with the following remark and notation.
For the sake of simplicity, all the functors $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ in this section are covariant unless stated otherwise. However, in next sections our functors will be contravariant since manifold calculus deals with contravariant functors. This is not an issue of course since all the statements of this section hold for contravariant functors as well: it suffices to replace everywhere ${\mathcal{C}}$ by its opposite category ${{\mathcal{C}}^{\text{op}}}$.
The following standard notations will be used only in this section.
1. We let $\Delta$ denote the category whose objects are $[n] = \{0, \cdots, n\}, n \geq 0$, and whose morphisms are non-decreasing maps. For $n \geq 0$, we let $\Delta[n]$ denote the simplicial set defined as $(\Delta[n])_p = \underset{\Delta}{\text{hom}} ([p], [n])$.
2. If ${\mathcal{C}}$ is a category, we write $N({\mathcal{C}})$ for the nerve of ${\mathcal{C}}$. If $c \in {\mathcal{C}}$, we let ${\mathcal{C}}\downarrow c$ denote the over category. An object of ${\mathcal{C}}\downarrow c$ consists of a pair $(x, f)$, where $x \in {\mathcal{C}}$ and $f \colon x {\longrightarrow}c$ is a morphism of ${\mathcal{C}}$. A morphism from $(x, f)$ to $(x', f')$ consists of a morphism $g \colon x {\longrightarrow}x'$ of ${\mathcal{C}}$ such that the obvious triangle commutes.
Homotopy limits {#holim_subsection}
---------------
Here we recall the definition of the homotopy limit of a diagram in a simplicial model category. Next we recall some useful results due to P. Hirschhorn [@hir03].
\[holim\_defn\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{C}}$ be a small category. Consider a covariant functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$. The *homotopy limit* of $F$, denoted $\underset{{\mathcal{C}}}{\text{holim}} \; F$, is the object of ${\mathcal{M}}$ defined to be the equalizer of the maps $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar@<1ex>[r]^-{\phi} \ar@<-1ex>[r]_-{\psi} & \underset{(f \colon c {\rightarrow}c') \in {\mathcal{C}}}{\prod} (F(c'))^{N({\mathcal{C}}\downarrow c)}. }$$ [^2] Here $\phi$ and $\psi$ are defined as follows. Let $f \colon c {\longrightarrow}c'$ be a morphism of ${\mathcal{C}}$.
1. The projection of $\phi$ on the factor indexed by $f$ is the following composition where the first map is a projection $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[rr]^-{(F(f))^{N({\mathcal{C}}\downarrow c)}} & & (F(c'))^{N({\mathcal{C}}\downarrow c)}.}$$
2. The projection of $\psi$ on the factor indexed by $f$ is the following composition where the first map is again a projection. $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c'))^{N({\mathcal{C}}\downarrow c')} \ar[rr]^-{(F(c'))^{N({\mathcal{C}}\downarrow f)}} & & (F(c'))^{N({\mathcal{C}}\downarrow c)}.}$$
Let ${\mathcal{M}}$ be a category. Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be small categories, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor. If $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ is an ${\mathcal{D}}$-diagram in ${\mathcal{M}}$, then the composition $F \circ \theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is called the *${\mathcal{C}}$-diagram in ${\mathcal{M}}$ induced by $F$*, and it is denoted $\theta^* F$. That is, $$\begin{aligned}
\label{theta_starx}
\theta^* F := F \circ \theta. \end{aligned}$$
The following proposition will be used in many places in this paper. Especially, we will use it to define morphisms between homotopy limits of diagrams of different shape. Also it will be used to show that certain diagrams commute. That proposition regards the change of the indexing category of a homotopy limit.
\[induced\_holim\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between two small categories. If $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ is an ${\mathcal{D}}$-diagram, then there is a canonical map $$\begin{aligned}
\label{thetax}
[\theta; F] \colon \underset{{\mathcal{D}}}{\text{holim}}\; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F.\end{aligned}$$ Furthermore, this map is natural in both variables $\theta$ and $F$. The naturality in $\theta$ says that if $\beta \colon \theta {\longrightarrow}\theta'$ is a natural transformation, then the following square commutes. $$\begin{aligned}
\label{nat_theta}
\xymatrix{\underset{{\mathcal{D}}}{\text{holim}}\; F \ar[rr]^-{[\theta; F]} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F \\
\underset{{\mathcal{D}}}{\text{holim}}\; F \ar[rr]_-{[\theta'; F]} \ar[u]^-{id} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta'^{*}F \ar[u]_-{\text{holim}(F\beta)} }\end{aligned}$$ Here we have assumed $F$ contravariant (in the covariant case, one has to reverse the righthand vertical map). Regarding the naturality in $F$, it says that if $\eta \colon F {\longrightarrow}F'$ is a natural transformation, then the following square commutes. $$\begin{aligned}
\label{theta_square}
\xymatrix{ \underset{{\mathcal{D}}}{\text{holim}}\; F \ar[rr]^-{[\theta; F]} \ar[d]_-{\text{holim}(\eta)} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F \ar[d]^-{\text{holim}(\theta^*\eta)} \\
\underset{{\mathcal{D}}}{\text{holim}}\; F' \ar[rr]_-{[\theta; F']} & & \underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F'.}\end{aligned}$$
The construction of $[\theta; F]$ comes from the following observation, which provides a nice way to define a map between two equalizers. This observation will be also used in Subsection \[cr\_subsection\].
Consider the following diagrams in ${\mathcal{M}}$. $$\xymatrix{A \ar@<1ex>[r]^-{\alpha} \ar@<-1ex>[r]_-{\beta} & B} \qquad \xymatrix{A' \ar@<1ex>[r]^-{\alpha'} \ar@<-1ex>[r]_-{\beta'} & B'}.$$ If $\Psi \colon A {\longrightarrow}A'$ is a map that satisfies the property $$\begin{aligned}
\label{eq_cond}
(\text{for all $g \colon E {\longrightarrow}A$}) \left((\alpha g = \beta g) \Rightarrow (\alpha'\Psi g = \beta' \Psi g)\right),\end{aligned}$$ then we have an induced map $$\widetilde{\Psi} \colon \text{eq}\left(\xymatrix{A \ar@<1ex>[r]^-{\alpha} \ar@<-1ex>[r]_-{\beta} & B}\right) {\longrightarrow}\text{eq}\left(\xymatrix{A' \ar@<1ex>[r]^-{\alpha'} \ar@<-1ex>[r]_-{\beta'} & B'} \right).$$ Now we define $[\theta; F] \colon \underset{{\mathcal{D}}}{\text{holim}}\; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F$. By Definition \[holim\_defn\], the homotopy limit of $\theta^* X$ is the equalizer of the maps $$\xymatrix{\underset{c \in {\mathcal{C}}}{\prod} (F(\theta(c)))^{N({\mathcal{C}}\downarrow c)} \ar@<1ex>[r] \ar@<-1ex>[r] & \underset{(c {\longrightarrow}c') \in {\mathcal{C}}}{\prod} (F(\theta(c')))^{N({\mathcal{C}}\downarrow c)}. }$$ Define $$\Psi \colon \underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}\downarrow d)} {\longrightarrow}\underset{c \in {\mathcal{C}}}{\prod} (F(\theta(c)))^{N({\mathcal{C}}\downarrow c)}$$ as follows. For $c \in {\mathcal{C}}$ the map from $\underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}\downarrow d)}$ to the factor indexed by $c$ is defined to be the composition $$\begin{aligned}
\label{fd_fc}
\xymatrix{\underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}\downarrow d)} \ar[r] & (F(\theta(c)))^{N({\mathcal{D}}\downarrow \theta(c))} \ar[r] & (F(\theta(c)))^{N({\mathcal{C}}\downarrow c)}},\end{aligned}$$ where the first map is the projection onto the factor indexed by $\theta(c)$, and the second one is induced by the canonical functor ${\mathcal{C}}\downarrow c {\longrightarrow}{\mathcal{D}}\downarrow \theta(c)$. It is straightforward to see that $\Psi$ satisfies condition (\[eq\_cond\]). We thus obtain $[\theta; F] := {\widetilde{\Psi}}$.
It is also straightforward to check that the squares (\[nat\_theta\]) and (\[theta\_square\]) commute.
We end this subsection with the following three important properties of homotopy limits. The first (see Theorem \[fib\_cofib\_thm\]) is known as the homotopy invariance for homotopy limits. The second (see Theorem \[htpy\_cofinal\_thm\]) is the cofinality theorem. And the last (see Theorem \[fubini\_thm\]) is the so-called Fubini Theorem for homotopy limits. Before we state those properties, we need the recall the following three definitions.
\[uc\_defn\] Let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor, and let $d \in {\mathcal{D}}$. The *under category* $d \downarrow \theta$ is defined as follows. An object of $d \downarrow \theta$ is a pair $(c, f)$ where $c$ is an object of ${\mathcal{C}}$ and $f$ is a morphism of ${\mathcal{D}}$ from $d$ to $\theta(c)$ . A morphism from $(c, f)$ to $(c', f')$ consists of a morphism $g \colon c {\longrightarrow}c'$ of ${\mathcal{C}}$ such that the obvious triangle commutes (that is, $\theta(g) f = f'$). In similar fashion, one has the *over category* $\theta \downarrow d$.
[@hir03 Definition 19.6.1] \[cofinal\_defn\] A functor $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is *homotopy right cofinal* (respectively *homotopy left cofinal*) if for every $d \in {\mathcal{D}}$, the under category $d \downarrow \theta$ (respectively the over category $\theta {\downarrow}d$) (see Definition \[uc\_defn\]) is contractible.
\[owf\_defn\] If ${\mathcal{C}}$ is a category and ${\mathcal{M}}$ is a model category, a functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is said to be an *objectwise fibrant* functor if the image of every object under $F$ is fibrant.
[@hir03 Theorems 18.5.2, 18.5.3] \[fib\_cofib\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{C}}$ be a small category.
1. If a functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is objectwise fibrant (see Definition \[owf\_defn\]), then the homotopy limit $\underset{{\mathcal{C}}}{\text{holim}} \; F$ is a fibrant object of ${\mathcal{M}}$.
2. Let $\eta \colon F {\longrightarrow}G$ be a map of ${\mathcal{C}}$-diagrams in ${\mathcal{M}}$. Assume that both $F$ and $G$ are objectwise fibrant. If for every object $c$ of ${\mathcal{C}}$ the component $\eta[c] \colon F(c) {\longrightarrow}G(c)$ is a weak equivalence, then the induced map of homotopy limits $\underset{{\mathcal{C}}}{\text{holim}} \; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}} \; G$ is a weak equivalence of ${\mathcal{M}}$.
[@hir03 Theorem 19.6.7] \[htpy\_cofinal\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. If $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is homotopy right cofinal (respectively homotopy left cofinal), then for every objectwise fibrant contravariant (respectively covariant) functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$, the natural map $[\theta; F]$ from Proposition \[induced\_holim\_prop\] is a weak equivalence.
\[fubini\_thm\] Let ${\mathcal{M}}$ be simplicial model category, and let ${\mathcal{C}}$ and ${\mathcal{D}}$ be small categories. Let $F \colon {\mathcal{C}}\times {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ be a bifunctor. Then there exists a natural weak equivalence $$\underset{{\mathcal{C}}}{\text{holim}} \; \underset{{\mathcal{D}}}{\text{holim}}\; F \stackrel{\sim}{{\longrightarrow}} \underset{{\mathcal{D}}}{\text{holim}} \; \underset{{\mathcal{C}}}{\text{holim}}\; F.$$
This is the dual of [@cha_sch01 Theorem 24.9].
Cosimplicial replacement of a diagram {#cr_subsection}
-------------------------------------
The goal of this subsection is to prove Corollary \[hir\_coro\] and Proposition \[comm\_prop\]. As we said before those two results will be used in Section \[sos\_good\_section\].
\[cr\_defn\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{C}}$ be a small category. For a covariant functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$, we define its *cosimplicial replacement*, denoted ${\Pi^{\bullet}}F \colon \Delta {\longrightarrow}{\mathcal{M}}$, as $$\Pi^n F := \prod_{(c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})} F ({c_n}).$$ For $0 \leq j \leq n-1$ the codegeneracy map $s^j \colon \Pi^n F {\longrightarrow}\Pi^{n-1} F$ is defined as follows. The projection of $s^j$ onto the factor indexed by $c_0 {\rightarrow}\cdots {\rightarrow}c_{n-1}$ is defined to be the projection of $\Pi^n F$ onto the factor indexed by $c_0 {\rightarrow}\cdots {\rightarrow}c_j \stackrel{id}{{\longrightarrow}} c_j {\rightarrow}\cdots {\rightarrow}c_{n-1}$. Cofaces are defined in a similar way.
\[betax\_rmk\] Let ${\mathcal{M}}$ be a model category, and let ${\mathcal{C}}$ and ${\mathcal{D}}$ be small categories. Consider a covariant functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$. Also consider a functor $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$. Recall the notation $\theta^*(-)$ from (\[theta\_starx\]). Then there exists a canonical map $$\beta^{\bullet}_F \colon \Pi^{\bullet} F {\longrightarrow}\Pi^{\bullet} (\theta^* F)$$ defined as follows. The map from $\Pi^n F$ to the factor of $\Pi^n (\theta^* F)$ indexed by $c_0 {\rightarrow}\cdots {\rightarrow}c_n$ is just the projection of $\Pi^n F$ onto the factor $F(\theta(c_n))$ indexed by $\theta(c_0) {\rightarrow}\cdots {\rightarrow}\theta(c_n)$.
The following proposition is stated (without any proof) in [@bous_kan72 Chapter XI, Section 5] for diagrams of simplicial sets.
\[rf\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category, and let $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant covariant functor. Then the cosimplicial replacement $\Pi^{\bullet} F$ is *Reedy fibrant* (see the definition of Reedy fibrant in the proof).
First we recall the definition of *Reedy fibrant*. Let ${Z^{\bullet}}\colon \Delta {\longrightarrow}{\mathcal{M}}$ be a cosimplicial object in ${\mathcal{M}}$. For $n \geq 0$, we let ${\mathcal{E}}_n$ denote the category whose objects are maps $[n] {\longrightarrow}[p]$ of $\Delta$ such that $p < n$. A morphism from $[n] {\longrightarrow}[p]$ to $[n] {\longrightarrow}[q]$ consists of a map $[p] {\longrightarrow}[q]$ of $\Delta$ such that the obvious triangle commutes. The *matching object* of ${Z^{\bullet}}$ at $[n] \in \Delta$, denoted $M^n {Z^{\bullet}}$, is defined to be the limit of the ${\mathcal{E}}_n$-diagram that sends $[n] {\longrightarrow}[p]$ to $Z^p$. That is, $$M^n {Z^{\bullet}}:= \underset{([n] {\rightarrow}[p]) \in {\mathcal{E}}_n}{\text{lim}} \; Z^p.$$ By the universal property, there exists a unique map $\alpha^n \colon Z^n {\longrightarrow}M^n {Z^{\bullet}}$ that makes certain triangles commutative. That map is induced by all codegeneracies $s^j \colon Z^n {\longrightarrow}Z^{n-1}, 0 \leq j \leq n-1$. We say that ${Z^{\bullet}}$ is *Reedy fibrant* if $\alpha^n$ is a fibration for all $n \geq 0$.
We come back to the proof of the proposition. Let $n \geq 0$. Since each codegeneracy map $s^j \colon \Pi^n F {\longrightarrow}\Pi^{n-1} F$ is a projection (see Definition \[cr\_defn\]), it follows that $\alpha^n$ is also a projection. This implies (by the assumption that $F(c)$ is fibrant for any $c \in {\mathcal{C}}$) that $\alpha^n$ is a fibration, which completes the proof.
\[tot\_defn\] Let ${\mathcal{M}}$ be a simplicial model category. Let $Z^{\bullet} \colon \Delta {\longrightarrow}{\mathcal{M}}$ be a cosimplicial object in ${\mathcal{M}}$. The *totalization* of $Z^{\bullet}$, denoted $\text{Tot} \; Z^{\bullet}$, is defined to be the equalizer of the maps $$\xymatrix{\underset{[n] \in \Delta}{\prod} (Z^n)^{\Delta[n]} \ar@<1ex>[r]^-{\phi'} \ar@<-1ex>[r]_-{\psi'} & \underset{(f \colon [n] {\rightarrow}[p]) \in \Delta}{\prod} (Z^p)^{\Delta[n]}. }$$ Here the maps $\phi'$ and $\psi'$ are defined in the similar way as the maps $\phi$ and $\psi$ from Definition \[holim\_defn\].
[@hir14 Theorem 12.5] \[holim\_tot\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category, and let $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ be a covariant functor. Then there exists an isomorphism $$\begin{aligned}
\label{phic_map}
\Phi_{{\mathcal{C}}} \colon \underset{{\mathcal{C}}}{\text{holim}} \; F \stackrel{\cong}{{\longrightarrow}} \text{Tot} \; \Pi^{\bullet} F,\end{aligned}$$ which is natural in $F$.
This is well detailed in [@hir14 Theorem 12.5]. However, for our purposes, specifically for the proof of Proposition \[comm\_prop\] below, we will recall only the construction of $\Phi_{{\mathcal{C}}}$. The map $\Phi_{{\mathcal{C}}}$ is in fact the composition of three isomorphisms (each obtained by using the observation we made at the beginning of the proof of Proposition \[induced\_holim\_prop\]): $$\xymatrix{\underset{{\mathcal{C}}}{\text{holim}} \; F \ar[rr]^-{{\widetilde{\Psi}}_{1{\mathcal{C}}}}_-{\cong} & & X \ar[rr]^-{{\widetilde{\Psi}}_{2{\mathcal{C}}}}_-{\cong} & & X' \ar[rr]^-{{\widetilde{\Psi}}_{3{\mathcal{C}}}}_-{\cong} & & {\text{Tot}}\; {\Pi^{\bullet}}F, }$$ where
1. $X$ is the equalizer of a diagram $$\xymatrix{ \underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} \ar@<1ex>[rr] \ar@<-1ex>[rr] & & Y }$$
2. $X'$ is the equalizer of a diagram $$\xymatrix{ \underset{n \geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(c_n))^{\Delta[n]} \ar@<1ex>[rr] \ar@<-1ex>[rr] & & Y' }.$$
3. By the definition of the cosimplicial replacement (see Definition \[cr\_defn\]), and by the the definition of the totalization (see Definition \[tot\_defn\]), one can easily see that ${\text{Tot}}\; {\Pi^{\bullet}}F$ is the equalizer of a diagram $$\xymatrix{ \underset{n \geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(c_n))^{\Delta[n]} \ar@<1ex>[rr] \ar@<-1ex>[rr] & & Y''}.$$ Since we are only interested in the definition of maps, it is not important here to know the definition of $Y$, $Y'$, and $Y''$.
Recalling $\underset{{\mathcal{C}}}{\text{holim}} \; F$ from Definition \[holim\_defn\], the map ${\widetilde{\Psi}}_{1{\mathcal{C}}}$ is induced by the map $$\begin{aligned}
\Psi_{1 {\mathcal{C}}} \colon \underset{ c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} {\longrightarrow}\underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} , \end{aligned}$$ which is defined as follows. The projection of $\Psi_{1 {\mathcal{C}}}$ onto the factor indexed by $(c \in {\mathcal{C}}, n \geq 0, \Delta[n] \stackrel{\sigma}{{\longrightarrow}} N({\mathcal{C}}\downarrow c))$ is the composition $$\xymatrix{ \underset{ c \in {\mathcal{C}}}{\prod} (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c))^{N({\mathcal{C}}\downarrow c)} \ar[r] & (F(c))^{\Delta[n]},}$$ where the first map is the projection onto the factor indexed by $c$, and the second is the canonical map induced by $\sigma \colon \Delta[n] {\longrightarrow}N({\mathcal{C}}\downarrow c)$.
Regarding the map ${\widetilde{\Psi}}_{2{\mathcal{C}}}$, it is induced by the map $$\begin{aligned}
\Psi_{2 {\mathcal{C}}} \colon \underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} {\longrightarrow}\underset{n \geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(c_n))^{\Delta[n]},\end{aligned}$$ which is defined as follows. The projection of $\Psi_{2 {\mathcal{C}}}$ onto the factor indexed by $(n \geq 0, c_0 {\rightarrow}\cdots {\rightarrow}c_n)$ is just the projection $$\underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(c))^{\Delta[n]} {\longrightarrow}(F(c_n))^{\Delta[n]}$$ onto the factor indexed by $(c_n, c_0 {\rightarrow}\cdots {\rightarrow}c_n, c_n \stackrel{id}{{\longrightarrow}} c_n)$.
Lastly, the map ${\widetilde{\Psi}}_{3 {\mathcal{C}}}$ is induced by the identity map. So $$\begin{aligned}
\label{psi3_eq}
\Psi_{3{\mathcal{C}}} := id.\end{aligned}$$
[@hir03 Theorem 19.8.7] \[hir\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let $Z^{\bullet} \colon \Delta {\longrightarrow}{\mathcal{M}}$ be a cosimplicial object in ${\mathcal{M}}$. If $Z^{\bullet}$ is Reedy fibrant then the Bousfield-Kan map ${\text{Tot}}\; Z^{\bullet} {\longrightarrow}\underset{\Delta}{\text{holim}} \; Z^{\bullet}$ (see [@hir03 Definition 19.8.6]) is a weak equivalence, which is natural in $Z^{\bullet}$.
We end this section with the following corollary and proposition. These results will be used in the course of the proof of Theorem \[sos\_thm\] and Theorem \[good\_thm\], which will be done at the end of Subsection \[sos\_good\_subsection\]
\[hir\_coro\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category, and let $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant covariant functor. Then the Bousfield-Kan map $\text{Tot} \; \Pi^{\bullet} F {\longrightarrow}\underset{\Delta}{\text{holim}} \; \Pi^{\bullet} F$ (see [@hir03 Definition 19.8.6]) is a weak equivalence, which is natural in ${\Pi^{\bullet}}X$.
This follows directly from Proposition \[rf\_thm\] and Theorem \[hir\_thm\].
\[comm\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between small categories. Consider a covariant functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$. Also consider the maps $[\theta; F], \beta^{\bullet}_F$, and $\Phi_{{\mathcal{C}}}$ from Proposition \[induced\_holim\_prop\], Remark \[betax\_rmk\], and Theorem \[holim\_tot\_thm\] respectively. Then the following square commutes. $$\xymatrix{ \underset{{\mathcal{D}}}{\text{holim}} \; F \ar[rr]^-{\Phi_{{\mathcal{D}}}}_-{\cong} \ar[d]_-{[\theta; F]} & & \text{Tot} \; {\Pi^{\bullet}}F \ar[d]^-{\text{Tot}\; \beta^{\bullet}_F} \\
\underset{{\mathcal{C}}}{\text{holim}} \; \theta^* F \ar[rr]_-{\Phi_{{\mathcal{C}}}}^-{\cong} & & \text{Tot} \; {\Pi^{\bullet}}(\theta^*F).}$$
Warning! Proposition \[comm\_prop\] does not follow from the naturality of the map $\Phi_{{\mathcal{C}}}$ from Theorem \[holim\_tot\_thm\]. This is because $F$ and $\theta^* F$ does not have the same domain. So to prove Proposition \[comm\_prop\] we really have to use the definition of $\Phi_{{\mathcal{C}}}$.
Recall the maps $\Psi_{i (-)}, 1 \leq i \leq 3,$ from the proof of Theorem \[holim\_tot\_thm\]. To prove the proposition, it suffices to see that the three squares induced by the pairs $(\Psi_{i {\mathcal{D}}}, \Psi_{i {\mathcal{C}}}), 1 \leq i \leq 3,$ are all commutative. Let us begin with the following square induced by $(\Psi_{1 {\mathcal{D}}}, \Psi_{1 {\mathcal{C}}})$. $$\begin{aligned}
\label{sq1_eqn}
\xymatrix{ \underset{d \in {\mathcal{D}}}{\prod} (F(d))^{N({\mathcal{D}}{\downarrow}d)} \ar[rr]^-{\Psi_{1{\mathcal{D}}}} \ar[d]_-{\alpha} & & \underset{d \in {\mathcal{D}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{D}}\downarrow d)}{\prod} (F(d))^{\Delta[n]} \ar[d]^-{\lambda} \\
\underset{c \in {\mathcal{C}}}{\prod} (F(\theta(c)))^{N({\mathcal{C}}{\downarrow}c)} \ar[rr]_-{\Psi_{1{\mathcal{C}}}} & & \underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow c)}{\prod} (F(\theta(c)))^{\Delta[n]}. }\end{aligned}$$ Here $\alpha$ is the composition from (\[fd\_fc\]), while the projection of $\lambda$ onto the factor indexed by $(c, n, \sigma \colon \Delta[n] {\rightarrow}N({\mathcal{C}}{\downarrow}c))$ is the projection $$\underset{d \in {\mathcal{D}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{D}}\downarrow d)}{\prod} (F(d))^{\Delta[n]} {\longrightarrow}F(\theta(c))^{\Delta[n]}$$ onto the factor indexed by $(\theta(c), n, \Delta[n] \stackrel{\sigma}{{\longrightarrow}} N({\mathcal{C}}{\downarrow}c) \stackrel{f}{{\longrightarrow}} N({\mathcal{D}}{\downarrow}\theta(c)))$, where $f$ is induced by the obvious functor ${\mathcal{C}}{\downarrow}c {\longrightarrow}{\mathcal{D}}{\downarrow}\theta(c)$. Using the definitions, it is straightforward to check that the square (\[sq1\_eqn\]) commutes.
It is also straightforward to see that the following square, induced by the pair $(\Psi_{2{\mathcal{D}}}, \Psi_{2{\mathcal{C}}})$, is commutative. $$\xymatrix{ \underset{d \in {\mathcal{D}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{D}}\downarrow d)}{\prod} (F(d))^{\Delta[n]} \ar[d]_-{\lambda} \ar[rr]^-{\Psi_{2{\mathcal{D}}}} & & \underset{n\geq 0, (d_0 {\rightarrow}\cdots {\rightarrow}d_n) \in N_n({\mathcal{D}})}{\prod} (F(d_n))^{\Delta[n]} \ar[d] \\
\underset{c \in {\mathcal{C}}, n \geq 0, \Delta[n] {\rightarrow}N({\mathcal{C}}\downarrow d)}{\prod} (F(\theta(c)))^{\Delta[n]} \ar[rr]_-{\Psi_{2 {\mathcal{C}}}} & & \underset{n\geq 0, (c_0 {\rightarrow}\cdots {\rightarrow}c_n) \in N_n({\mathcal{C}})}{\prod} (F(\theta(c_n)))^{\Delta[n]} }$$ Lastly, the square induced by $(\Psi_{3{\mathcal{D}}}, \Psi_{3{\mathcal{C}}})$ is clearly commutative since $\Psi_{3{\mathcal{D}}} = id$ and $\Psi_{3{\mathcal{C}}} = id$ by (\[psi3\_eq\]).
Special open sets and good cofunctors {#sos_good_section}
=====================================
The goal of this section is to prove Theorem \[sos\_thm\] and Theorem \[good\_thm\] below. The first result is a key ingredient, which will be used in many places throughout Sections \[sos\_good\_section\], \[poly\_section\], \[hc\_section\]. It roughly says that a certain homotopy limit $\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V)$ is independent of the choice of the basis ${\mathcal{B}}$. The second theorem is a part of the proof of the main result of this paper (Theorem \[main\_thm\]). Note that the subsections \[cof\_fsp\_subsection\], \[sos\_good\_subsection\] are influenced by the work of Pryor [@pryor15].
From now on, we let $M$ denote a smooth manifold, and we let ${\mathcal{O}(M)}$ to be the poset of open subsets of $M$. Also, for $k \geq 0$, we let ${\mathcal{O}_k(M)}\subseteq {\mathcal{O}(M)}$ to be the full subposet whose objects are open subsets diffeomorphic to the disjoint union of at most $k$ balls. Below (see Example \[fso\_expl\]) we will see that ${\mathcal{O}_k(M)}$ can be obtained in another way. In [@wei99] Weiss calls objects of ${\mathcal{O}_k(M)}$ *special open sets*.
\[sos\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{B}}$ and ${\mathcal{B}}'$ be good bases (see Definition \[gb\_defn\]) for the topology of $M$ such that ${\mathcal{B}}\subseteq {\mathcal{B}}'$. Let ${\mathcal{B}}'_k(M) \subseteq {\mathcal{O}_k(M)}$ denote the full subcategory whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}'$. Consider an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]) $F \colon {\mathcal{B}}'_k(M) {\longrightarrow}{\mathcal{M}}$. Also consider the cofunctors ${F^{!}_{{\mathcal{B}}}}, {F^{!}_{{\mathcal{B}}'}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ from Definition \[fsb\_defn\]. Then the natural map $$[\theta; F] \colon {F^{!}_{{\mathcal{B}}'}}{\longrightarrow}{F^{!}_{{\mathcal{B}}}}$$ induced by the inclusion functor $\theta \colon {{\mathcal{B}}_k}(M) {\longrightarrow}{\mathcal{B}}'_k(M)$ is a weak equivalence. Here the notation comes from Proposition \[induced\_holim\_prop\].
\[good\_thm\] Let ${\mathcal{M}}$ be a simplicial model category, and let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$. Consider the poset ${\mathcal{B}_k(M)}$ from Definition \[fsb\_defn\], and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor ${F^{!}_{{\mathcal{B}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ from Definition \[fsb\_defn\] is good (see Definition \[good\_defn\]).
The proof of Theorem \[sos\_thm\] and Theorem \[good\_thm\] will be done in Subsection \[sos\_good\_subsection\] after some preliminaries results.
Isotopy cofunctors
------------------
The goal of this subsection is to prove Proposition \[fsrik\_prop\], which will be used in Sections \[poly\_section\], \[hc\_section\]. Note that this result is well known in the context of topological spaces.
We begin with several definitions. The first one is the notion of isotopy equivalence, which is well known in differential topology, manifold calculus, and other areas. Nevertheless we need to recall it for our purposes in Section \[iso\_cof\_section\].
\[iso\_eq\_defn\] A morphism $U {\hookrightarrow}U'$ of ${\mathcal{O}(M)}$ is said to be an *isotopy equivalence* if there exists a continuous map $$L \colon U \times [0, 1] {\longrightarrow}U', \quad (x, t) \mapsto L_t(x) := L(x, t)$$ that satisfies the following three conditions:
1. $L_0 \colon U {\hookrightarrow}U'$ is the inclusion map;
2. $L_1(U) = U'$;
3. for all $t$, $L_t \colon U {\longrightarrow}U'$ is a smooth embedding.
Such a map $L$ is called an *isotopy from $U$ to $U'$*.
\[isotopy\_cof\_defn\] Let ${\mathcal{C}}\subseteq {\mathcal{O}(M)}$ be a subcategory of ${\mathcal{O}(M)}$, and let ${\mathcal{M}}$ be a model category. A cofunctor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is called *isotopy cofunctor* if it satisfies the following two conditions:
1. $F$ is objectwise fibrant (see Definition \[owf\_defn\]);
2. $F$ sends isotopy equivalences (see Definition \[iso\_eq\_defn\]) to weak equivalences.
\[good\_defn\] Let ${\mathcal{M}}$ be a simplicial model category. A cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is called *good* if it satisfies the following two conditions:
1. $F$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]);
2. For any string $U_0 {\rightarrow}U_1 {\rightarrow}\cdots$ of inclusions of ${\mathcal{O}(M)}$, the natural map $$F\left(\bigcup_{i=0}^{\infty} U_i\right) {\longrightarrow}\underset{i}{\text{holim}} \; F(U_i)$$ is a weak equivalence.
In order words, a cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is *good* if it satisfies three conditions: (a), (b) from Definition \[isotopy\_cof\_defn\], and (b) from Definition \[good\_defn\]. As we said in the introduction, this definition is slightly different from the classical one [@wei99 Page 71] (see the comment we made right after Theorem \[main\_thm\]).
\[gb\_defn\] A basis for the topology of $M$ is called *good* if each element in there is diffeomorphic to an open ball.
\[fsb\_defn\] Let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$.
1. For $k \geq 0$, we define ${\mathcal{B}_k(M)}\subseteq {\mathcal{O}(M)}$ to be the full subposet whose objects are disjoint unions of at most $k$ elements from ${\mathcal{B}}$.
2. If $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ is a cofunctor, we define $F^{!}_{{\mathcal{B}}} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ as $${F^{!}_{{\mathcal{B}}}}(U) := \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V).$$
\[fso\_expl\] Let ${\mathcal{O}}$ be the collection of all subsets of $M$ diffeomorphic to an open ball. Certainly this is a good basis (see Definition \[gb\_defn\]) for the topology of $M$. So, by Definition \[fsb\_defn\], one has the poset ${\mathcal{O}_k(M)}$, which is exactly the same as the poset ${\mathcal{O}_k(M)}$ we defined just before Theorem \[sos\_thm\]. If $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is a cofunctor, one also has the cofunctor ${F^{!}_{{\mathcal{O}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ defined as $${F^{!}_{{\mathcal{O}}}}(U) := \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V).$$ Clearly ${\mathcal{O}}$ is the biggest (with respect to the inclusion) good basis for the topology of $M$.
As we said before, the following proposition will be used in Section \[poly\_section\] and Section \[hc\_section\].
\[fsrik\_prop\] Let ${\mathcal{B}}$ and ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant cofunctor.
1. Then there is a natural transformation $\eta$ from $F$ to the restriction ${F^{!}_{{\mathcal{B}}}}| {\mathcal{B}_k(M)}$, which is an objectwise weak equivalence.
2. If in addition $F$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]), then so is the restriction of ${F^{!}_{{\mathcal{B}}}}$ to ${\mathcal{B}_k(M)}$.
<!-- -->
1. Let $U \in {\mathcal{B}_k(M)}$, and let $\theta, \theta' \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{{\mathcal{B}}_k}(U)$ be functors defined as $\theta(V) = V$ and $\theta'(V) = U$. Certainly there is a natural transformation $\beta \colon \theta {\longrightarrow}\theta'$. This induces by (\[nat\_theta\]) the following commutative square. $$\xymatrix{\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(V) \ar[rr]^-{[\theta; F]} & & \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(V) \\
\underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(V) \ar[rr]_-{[\theta'; F]} \ar[u]^-{id} & & \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(U). \ar[u]_-{\text{holim}(F\beta)} }$$ Clearly one has $F(U) \simeq \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}}\; F(U)$. This allows us to define $\eta[U] := \text{holim} (F \beta)$. Since $\theta$ is the identity functor, it follows that the map $[\theta; F]$ is a weak equivalence (in fact it is the identity functor as well). The map $[\theta'; F]$ is also a weak equivalence (by Theorem \[htpy\_cofinal\_thm\]) since $\theta'$ is homotopy right cofinal. Indeed, for every $V \in {{\mathcal{B}}_k}(U)$ the under category (see Definition \[uc\_defn\]) $V \downarrow \theta'$ has a terminal object, namely $(U, V {\hookrightarrow}U)$. Now, applying the two-out-of-three axiom we deduce that the map $\text{holim} (F\beta)$ is a weak equivalence. Regarding the naturality of $\eta[U]$ in $U$, it follows easily from (\[theta\_square\]).
2. Certainly the functor ${F^{!}_{{\mathcal{B}}}}|{{\mathcal{B}}_k}(M)$ satisfies condition (a) from Definition \[isotopy\_cof\_defn\] because of Theorem \[fib\_cofib\_thm\]. Condition (b) from the same definition is also satisfied by ${F^{!}_{{\mathcal{B}}}}|{{\mathcal{B}}_k}(M)$ (this follows directly from part (i)).
The cofunctors $F^{!p}$ {#cof_fsp_subsection}
-----------------------
In this subsection we consider a good basis ${\mathcal{B}}$ and the poset ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Also we consider the basis ${\mathcal{O}}$ from Example \[fso\_expl\]. The main results here are Proposition \[sosp\_prop\] and Proposition \[isop\_prop\] whose proofs are inspired by Pryor’s work [@pryor15]. Those propositions are one of the key ingredients in proving Theorem \[sos\_thm\] and Theorem \[good\_thm\].
\[bkp\_defn\] Let $k, p\geq 0$.
1. Define ${\widetilde{{\mathcal{B}}}_{k, p}}(M)$ to be the poset whose objects are strings $V_0 {\rightarrow}\cdots {\rightarrow}V_p$ of $p$ composable morphisms in ${\mathcal{B}_k(M)}$. A morphism from $V_0 {\rightarrow}\cdots {\rightarrow}V_p$ to $W_0 {\rightarrow}\cdots {\rightarrow}W_p$ consists of a collection $\{f_i \colon V_i {\hookrightarrow}W_i\}_{i=0}^p$ of isotopy equivalences such that all the obvious squares commute.
2. Taking ${\mathcal{B}}$ to be ${\mathcal{O}}$, we have the poset ${\widetilde{{\mathcal{O}}}_{k, p}}(M)$.
The following remark claims that the collection ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(M) = \{{\widetilde{{\mathcal{B}}}_{k, p}}(M)\}_{p \geq 0}$ is equipped with a canonical simplicial object structure.
\[bkp\_rmk\] Let $U \in {\mathcal{O}(M)}$. For $0 \leq i \leq p+1$ define $d_i \colon \widetilde{B}_{k, p+1}(U) {\longrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U)$ as $$d_i(V_0 {\rightarrow}\cdots {\rightarrow}V_{p+1}) = \left\{ \begin{array}{ccc}
V_1 {\rightarrow}\cdots {\rightarrow}V_{p+1} & \text{if} & i =0 \\
V_0 {\rightarrow}\cdots V_{i-1} {\rightarrow}V_{i+1} {\rightarrow}\cdots {\rightarrow}V_{p+1} & \text{if} & 1 \leq i \leq p \\
V_0 {\rightarrow}\cdots {\rightarrow}V_p & \text{if} & i = p+1.
\end{array} \right.$$ For $0 \leq j \leq p$ define $s_j \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\longrightarrow}\widetilde{B}_{k, p+1}(U)$ as $$s_j(V_0 {\rightarrow}\cdots {\rightarrow}V_p) = V_0 {\rightarrow}\cdots {\rightarrow}V_j \stackrel{\text{id}}{{\rightarrow}} V_j {\rightarrow}\cdots {\rightarrow}V_p.$$ One can easily check that $d_i$ and $s_j$ satisfy the simplicial relation. So ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(U)$ is a simplicial object in Cat, the category of small categories.
\[fsbp\_defn\] Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be a cofunctor.
1. Define a cofunctor ${\tilde{F}^{!p}_{{\mathcal{B}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ as $${\tilde{F}^{!p}_{{\mathcal{B}}}}(U) := \underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U)}{\text{holim}} \; {\tilde{F}^p_{{\mathcal{B}}}},$$ where $${\tilde{F}^p_{{\mathcal{B}}}}\colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\longrightarrow}{\mathcal{M}}, \quad V_0 {\rightarrow}\cdots {\rightarrow}V_p \mapsto F(V_0).$$
2. Taking again ${\mathcal{B}}$ to be ${\mathcal{O}}$, we have the cofunctor ${\tilde{F}^{!p}_{{\mathcal{O}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$.
The following remark will be used in Subsection \[sos\_good\_subsection\].
\[fsp\_rmk\] Let $U \in {\mathcal{O}(M)}$, and let ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(U)$ be the simplicial object from Remark \[bkp\_rmk\]. Using the simplicial structure on ${\widetilde{{\mathcal{B}}}_{k, \bullet}}(U)$, one can endow the collection ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}(U) = \{{\tilde{F}^{!p}_{{\mathcal{B}}}}(U) \}_{p \geq 0}$ with a canonical cosimplicial structure as follows. First recall the notation $\theta^{*}(-)$ from (\[theta\_starx\]). Also recall the notation $[-;-]$ introduced in Proposition \[induced\_holim\_prop\]. Let $d_i$ and $s_j$ as in Remark \[bkp\_rmk\], and consider $d_0 \colon \widetilde{{\mathcal{B}}}_{k, p+1}(U) {\longrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U)$. Also consider the natural transformation $\beta \colon d_0^* {\tilde{F}^p_{{\mathcal{B}}}}{\longrightarrow}\tilde{F}^{p+1}_{{\mathcal{B}}}$ defined as $$\beta[V_0 \stackrel{f}{{\rightarrow}} V_1 {\rightarrow}\cdots {\rightarrow}V_{p+1}] := F(V_1) \stackrel{F(f)}{{\longrightarrow}} F(V_0).$$ Now define $d^i \colon {\tilde{F}^{!p}_{{\mathcal{B}}}}(U) {\longrightarrow}\tilde{F}^{!(p+1)}_{{\mathcal{B}}} (U)$ as $$d^i = \left\{ \begin{array}{ccc}
\text{holim}(\beta) \circ [d_0; {\tilde{F}^p_{{\mathcal{B}}}}] & \text{if} & i =0 \\
\left[d_i; {\tilde{F}^p_{{\mathcal{B}}}}\right] & \text{if} & 1 \leq i \leq p+1.
\end{array} \right.$$ Also define $s^j \colon \tilde{F}^{!(p+1)}_{{\mathcal{B}}} (U) {\longrightarrow}{\tilde{F}^{!p}_{{\mathcal{B}}}}(U), 0 \leq j \leq p+1$, as $$s^j = [s_j; \tilde{F}^{p+1}_{{\mathcal{B}}}].$$ Certainly the maps $d^i$ and $s^j$ satisfy the cosimplicial relations. So ${\tilde{F}^{!\bullet}_{{\mathcal{B}}}}(U)$ is a cosimplicial object in ${\mathcal{M}}$ for any $U \in {\mathcal{O}(M)}$.
\[loc\_defn\] Let ${\mathcal{M}}$ be a category with a class of weak equivalences, and let ${\mathcal{C}}$ be any other category. A functor ${\mathcal{C}}{\longrightarrow}{\mathcal{M}}$ is called *locally constant* if it sends every morphism of ${\mathcal{C}}$ to a weak equivalence.
\[loc\_expl\] Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor ${\tilde{F}^p_{{\mathcal{B}}}}$ from Definition \[fsbp\_defn\] is locally constant. This follows directly from the definition of a morphism of ${\widetilde{{\mathcal{B}}}_{k, p}}(M)$ (see Definition \[bkp\_defn\]) and the definition of an isotopy cofunctor.
Now we state and prove the main results of this subsection (Proposition \[sosp\_prop\] and Proposition \[isop\_prop\]). First we need three preparatory lemmas.
\[cisinski\_lem\] Let ${\mathcal{M}}$ be a simplicial model category. Let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between small categories. Consider a functor $G \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$, and assume that it is locally constant (see Definition \[loc\_defn\]). Also assume that the nerve of $\theta$ is a weak equivalence. Then the canonical map $$[\theta; G] \colon \underset{{\mathcal{D}}}{\text{holim}} \; G {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}} \; \theta^{*} G$$ (see Proposition \[induced\_holim\_prop\]) is a weak equivalence.
This is just the dual of Proposition 1.17 from [@cis09].
\[pryor1\_lem\] For any $U \in {\mathcal{O}(M)}$ the nerve of the inclusion functor ${\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{O}}}_{k, p}}(U)$ is a homotopy equivalence for all $k, p \geq 0$.
This is done in the course of the proof of Theorem 6.12 from [@pryor15].
[@pryor15 Lemma 6.8] \[pryor2\_lem\] Let $f \colon U {\hookrightarrow}U'$ be a morphism of ${\mathcal{O}(M)}$. If $f$ is an isotopy equivalence, then the nerve of the inclusion functor ${\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U')$ is a homotopy equivalence.
\[sosp\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Let $U$ be an object of ${\mathcal{O}(M)}$. Consider $\theta \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{O}}}_{k, p}}(U)$, the inclusion functor. Also consider ${\tilde{F}^p_{{\mathcal{O}}}}\colon {\widetilde{{\mathcal{O}}}_{k, p}}(U) {\longrightarrow}{\mathcal{M}}$, the cofunctor from Definition \[fsbp\_defn\]. Then the canonical map $[\theta; {\tilde{F}^p_{{\mathcal{O}}}}] \colon {\tilde{F}^{!p}_{{\mathcal{O}}}}(U) {\longrightarrow}{\tilde{F}^{!p}_{{\mathcal{B}}}}(U)$ is a weak equivalence for all $p \geq 0$. Furthermore that map is natural in $U$.
Set ${\mathcal{C}}:= {\widetilde{{\mathcal{B}}}_{k, p}}(M), {\mathcal{D}}:= {\widetilde{{\mathcal{O}}}_{k, p}}(M),$ and $G := {\tilde{F}^p_{{\mathcal{O}}}}$. Since $G$ is locally constant by Example \[loc\_expl\], and since the nerve of $\theta$ is a weak equivalence by Lemma \[pryor1\_lem\], it follows that $[\theta; {\tilde{F}^p_{{\mathcal{O}}}}]$ is a weak equivalence by Lemma \[cisinski\_lem\]. The naturality in $U$ comes directly from the definitions.
\[isop\_prop\] Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then for any $p \geq 0$ the cofunctor ${\tilde{F}^{!p}_{{\mathcal{B}}}}$ (see Definition \[fsbp\_defn\]) is an isotopy cofunctor as well.
Let $U {\hookrightarrow}U'$ be an isotopy equivalence, and let $\theta \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{B}}}_{k, p}}(U')$ denote the inclusion functor. Consider the cofunctor ${\tilde{F}^p_{{\mathcal{B}}}}\colon {\widetilde{{\mathcal{B}}}_{k, p}}(U') {\longrightarrow}{\mathcal{M}}$ from Definition \[fsbp\_defn\]. Since ${\tilde{F}^p_{{\mathcal{B}}}}$ is locally constant by Example \[loc\_expl\], and since the nerve of $\theta$ is a weak equivalence by Lemma \[pryor2\_lem\], the desired result follows by Lemma \[cisinski\_lem\].
Grothendieck construction {#gro_const_subsection}
-------------------------
In this subsection we recall the Grothendieck construction, and we give some examples that will be used further. We also recall an important result (see Theorem \[cha\_sch\_thm\]), which regards the homotopy limit of a diagram indexed by the Grothendieck construction.
\[intcf\_defn\] Let ${\mathcal{C}}$ be a small category, and let ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ be a covariant functor from ${\mathcal{C}}$ to the category Cat of small categories. Define $\int_{{\mathcal{C}}} {\mathcal{F}}$ to be the category whose objects are pairs $(c, x)$ where $c \in {\mathcal{C}}$ and $x \in {\mathcal{F}}(c)$. A morphism $(c, x) {\longrightarrow}(c', x')$ consists of a pair $(f, g)$, where $f \colon c {\longrightarrow}c'$ is a morphism of ${\mathcal{C}}$, and $g \colon {\mathcal{F}}(f)(x) {\longrightarrow}x'$ is a morphism of ${\mathcal{F}}(c')$. The construction that sends ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ to $\int_{{\mathcal{C}}} {\mathcal{F}}$ is called the *Grothendieck construction*.
Here are two examples of the Grothendieck construction. The first one will be used in Subsection \[sos\_good\_subsection\], while the second will be used in Section \[poly\_section\].
\[intcf\_expl1\] Let ${\mathcal{D}}_0 {\hookrightarrow}{\mathcal{D}}_1 {\hookrightarrow}{\mathcal{D}}_2 \cdots $ be an increasing inclusion of small categories. Define ${\mathcal{C}}$ to be the category $\{0 {\rightarrow}1 {\rightarrow}2 {\rightarrow}\cdots \}$, and ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ as ${\mathcal{F}}(i) = {\mathcal{D}}_i$. Then one can see that $$\int_{{\mathcal{C}}} {\mathcal{F}}= {\mathcal{C}}\times \left(\bigcup_{i=0}^{\infty} {\mathcal{D}}_i\right).$$
\[intcf\_expl2\] Let $k \geq 0$, and let ${\mathcal{C}}$ be the category defined as $${\mathcal{C}}= \left\{S \subseteq \{0, \cdots, k\}\ \ \text{such that} \ \ S \neq \emptyset \right\}.$$ Given two objects $S, T \in {\mathcal{C}}$, there exists a morphism from $S$ to $T$ if $T \subseteq S$. If $U$ is an open subset of $M$, and $A_0, \cdots, A_k$ are pairwise disjoint closed subsets of $U$, we let $\Omega$ denote the basis (for the topology of $U$) where an element is a subset $B$ diffeomorphic to an open ball such that $B$ intersects at most one $A_i$. In other words, if one introduces the notation $U(S):= U \backslash \cup_{i \in S} A_i$, then $$\Omega := \left\{B \subseteq U| \; \text{$B$ is diffeomorphic to an open ball and $B \subseteq U\left(\{0, \cdots, \hat{i}, \cdots, k \}\right)$} \right\},$$ where the hat means taking out. Certainly $\Omega$ is a good basis (see Definition \[gb\_defn\]). Now, for $V \in {\mathcal{O}}(U)$ we let $\Omega_k(V) \subseteq {\mathcal{O}_k}(V)$ denote the full subposet whose objects are disjoint unions of at most $k$ elements from $\Omega$. Define $${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat} \quad \text{as} \quad {\mathcal{F}}(S) := \Omega_k(U(S)).$$ Clearly, one has ${\mathcal{F}}(S) \subseteq {\mathcal{F}}(T)$ whenever $T \subseteq S$. So ${\mathcal{F}}(S {\rightarrow}T)$ is just the inclusion functor. One can then consider the category $\int_{{\mathcal{C}}} {\mathcal{F}}$ (see Definition \[intcf\_defn\]), which can be described as follows. An object of that category is a pair $(S, V)$ where $\emptyset \neq S \subseteq \{0, \cdots, k\}$, and $V \subseteq U \backslash \cup_{i \in S} A_i$ is the disjoint union of at most $k$ elements from $\Omega$. There exists a morphism $(S, V) {\longrightarrow}(T, W)$ if and only if $T \subseteq S$ and $V \subseteq W$.
[@cha_sch01] \[cha\_sch\_thm\] Let ${\mathcal{M}}$ be a simplicial model category. Let ${\mathcal{C}}$ be a small category and let ${\mathcal{F}}\colon {\mathcal{C}}{\longrightarrow}\text{Cat}$ be a covariant functor. Consider a collection $\{G_c \colon \colon {\mathcal{F}}(c) {\longrightarrow}{\mathcal{M}}\}_{c \in {\mathcal{C}}}$ of functors such that for any $f \colon c {\longrightarrow}c'$ in ${\mathcal{C}}$ the following triangle commutes. $$\xymatrix{{\mathcal{F}}(c') \ar[rr]^-{G_{c'}} \ar[d]_-{{\mathcal{F}}(f)} & & {\mathcal{M}}\\
{\mathcal{F}}(c) \ar[rru]_-{G_c} & & }$$ Then the canonical map $$\beta \colon \underset{(c, x) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; G_c(x) {\longrightarrow}\underset{c \in {\mathcal{C}}}{\text{holim}} \; \underset{x \in {\mathcal{F}}(c)}{\text{holim}} \; G_c(x)$$ is a weak equivalence (see Definition \[intcf\_defn\]).
This is the dual of [@cha_sch01 Theorem 26.8].
Special open sets and good cofunctors {#sos_good_subsection}
-------------------------------------
The goal of this subsection is to prove Theorem \[sos\_thm\] and Theorem \[good\_thm\] announced at the beginning of Section \[sos\_good\_section\].
To prove Theorem \[sos\_thm\] we will need Lemma \[fqf\_lem\] below. First we need to introduce some notation. For $U \in {\mathcal{O}(M)}, q \geq 0$ we let ${\widehat{{\mathcal{B}}}_{k, q}}(U)$ denote the poset whose objects are strings $W_0 {\rightarrow}\cdots {\rightarrow}W_q$ of $q$ composable morphisms in ${{\mathcal{B}}_k}(U)$ such that $W_i {\rightarrow}W_{i+1}$ is an isotopy equivalence for all $i$. A morphism from $W_0 {\rightarrow}\cdots {\rightarrow}W_q$ to $W'_0 {\rightarrow}\cdots {\rightarrow}W'_q$ consists of a collection $f = \{f_i \colon W_i {\longrightarrow}W'_i\}_{i=0}^{q}$ of morphisms of ${{\mathcal{B}}_k}(U)$ such that all the obvious squares commute. Now let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant cofunctor. Define a new cofunctor ${\hat{F}_{{\mathcal{B}}}}^q \colon {\widehat{{\mathcal{B}}}_{k, q}}(U) {\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{fhq}
{\hat{F}_{{\mathcal{B}}}}^q (W_0 {\rightarrow}\cdots {\rightarrow}W_q) := F(W_0).\end{aligned}$$ Also define ${\hat{F}_{{\mathcal{B}}}}^{!q} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{fhsq}
{\hat{F}_{{\mathcal{B}}}}^{!q} (U) := \underset{{\widehat{{\mathcal{B}}}_{k, q}}(U)}{\text{holim}} \; {\hat{F}_{{\mathcal{B}}}}^q. \end{aligned}$$
\[fsp\_bkp\_rmk\]
1. As in Remark \[fsp\_rmk\], the collection ${\hat{F}_{{\mathcal{B}}}}^{!\bullet} (U) = \{{\hat{F}_{{\mathcal{B}}}}^{!q} (U)\}_{q \geq 0}$ is a cosimplicial object in ${\mathcal{M}}$ for all $U \in {\mathcal{O}(M)}$.
2. Recall the poset ${\widetilde{{\mathcal{B}}}_{k, p}}(U)$ and the functor ${\tilde{F}_{{\mathcal{B}}}}^p$ from Definition \[bkp\_defn\] and Definition \[fsbp\_defn\] respectively. Also recall ${\Pi^{\bullet}}(-)$ from Definition \[cr\_defn\]. Then one can easily see that $$\Pi^q {\tilde{F}_{{\mathcal{B}}}}^p = \Pi^p {\hat{F}_{{\mathcal{B}}}}^q$$ since the set of $q$-simplices of the nerve $N({\widetilde{{\mathcal{B}}}_{k, p}}(U))$ is equal to the set of $p$-simplices of the nerve $N({\widehat{{\mathcal{B}}}_{k, q}}(U))$.
\[fqf\_lem\] Let ${\mathcal{M}}$ be a simplicial model category. For $U \in {\mathcal{O}(M)}$, consider the functor $\theta \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{\widehat{{\mathcal{B}}}_{k, q}}(U)$ defined as $\theta(V) := V {\rightarrow}\cdots {\rightarrow}V$. Then the canonical map $$\begin{aligned}
\label{fqf_eq}
[\theta; {\hat{F}_{{\mathcal{B}}}}^q] \colon {\hat{F}_{{\mathcal{B}}}}^{!q} (U) {\longrightarrow}F^{!}_{{\mathcal{B}}} (U)\end{aligned}$$ is a weak equivalence. Furthermore this map is natural in $U$.
Let $W_0 {\rightarrow}\cdots {\rightarrow}W_q \in {\widehat{{\mathcal{B}}}_{k, q}}(U)$. The under category $(W_0 {\rightarrow}\cdots {\rightarrow}W_q) \downarrow \theta$ is contractible since it has an initial object, namely $(W_q, f)$ where $f$ is the obvious map from $W_0 {\rightarrow}\cdots {\rightarrow}W_q$ to $W_q {\rightarrow}\cdots {\rightarrow}W_q$. So $\theta$ is homotopy right cofinal, and therefore the map $[\theta; {\hat{F}_{{\mathcal{B}}}}^q]$ is a weak equivalence by Theorem \[htpy\_cofinal\_thm\]. The naturality of that map in $U$ is readily checked.
We are now ready to prove Theorem \[sos\_thm\].
In the following proof we will work with ${\mathcal{B}}'={\mathcal{O}}$. Notice that one can perform exactly the same proof with any good basis ${\mathcal{B}}'$ containing ${\mathcal{B}}$.
Let $U \in {\mathcal{O}(M)}$. Recall ${\tilde{F}^{!p}_{{\mathcal{B}}}}$ from Definition \[fsbp\_defn\]. We will show that the objects $\underset{[p] \in \Delta}{\text{holim}} \; {\tilde{F}_{{\mathcal{B}}}}^{!p} (U)$ and $F^{!}_{{\mathcal{B}}} (U)$ are connected by a zigzag of natural weak equivalences. Consider the following diagram. $$\xymatrix{{\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U)}{\text{holim}}}\; {\tilde{F}_{{\mathcal{B}}}}^p \ar[r]^-{\cong} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\tilde{F}_{{\mathcal{B}}}}^p \ar[r]^-{\sim} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{[q] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}_{{\mathcal{B}}}}^p \ar[r]^-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}_{{\mathcal{B}}}}^p \\
{\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{{\widetilde{{\mathcal{O}}}_{k, p}}(U)}{\text{holim}}}\; {\tilde{F}^p_{{\mathcal{O}}}}\ar[u] \ar[r]_-{\cong} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\tilde{F}^p_{{\mathcal{O}}}}\ar[u] \ar[r]_-{\sim} & {\underset{[p] \in \Delta}{\text{holim}}}\; {\underset{[q] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}^p_{{\mathcal{O}}}}\ar[u] \ar[r]_-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}^p_{{\mathcal{O}}}}\ar[u]_-{\lambda} }$$
1. In the first row
1. the first map is the homotopy limit of the map (\[phic\_map\]),
2. the second is the homotopy limit of the Bousfield-Kan map from Corollary \[hir\_coro\], and
3. the third is provided by the Fubini Theorem \[fubini\_thm\].
2. The maps in the second row are obtained in the similar way since ${\mathcal{B}}$ is a subposet of ${\mathcal{O}}$.
3. The lefthand vertical map is nothing but the homotopy limit of $[\theta; {\tilde{F}^p_{{\mathcal{O}}}}]$, where $\theta \colon {\widetilde{{\mathcal{B}}}_{k, p}}(U) {\hookrightarrow}{\widetilde{{\mathcal{O}}}_{k, p}}(U)$ is just the inclusion functor.
4. The three others are the canonial ones induced by the map $\beta^{\bullet}_{{\widetilde{{\mathcal{O}}}_{k, p}}(U)}$ from Remark \[betax\_rmk\].
Certainly the lefthand square commutes by Proposition \[comm\_prop\]. The middle one commutes by the fact that the Bousfield-Kan map ${\text{Tot}}\; Z^{\bullet} {\longrightarrow}\underset{\Delta}{\text{holim}} \; Z^{\bullet}$ is natural in $Z^{\bullet}$. The third square commutes since the map in the Fubini Theorem \[fubini\_thm\] is also natural. So the above diagram is commutative. Therefore, since the first vertical map is a weak equivalence by Proposition \[sosp\_prop\] and Theorem \[fib\_cofib\_thm\], it follows that the last one, $\lambda$, is also a weak equivalence.
Similarly the following diagram (in which the equality comes from Remark \[fsp\_bkp\_rmk\]) is commutative as well. $$\xymatrix{ {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}^p_{{\mathcal{B}}}}\ar@{=}[r] & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^p {\hat{F}_{{\mathcal{B}}}}^q & {\underset{[q] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\hat{F}_{{\mathcal{B}}}}^q \ar[l]_-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{{\widehat{{\mathcal{B}}}_{k, q}}(U)}{\text{holim}}}{\hat{F}_{{\mathcal{B}}}}^q \ar[l]_-{\cong} \\
{\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^q {\tilde{F}_{{\mathcal{O}}}}\ar@{=}[r] \ar[u]_-{\lambda}^-{\sim} & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{[p] \in \Delta}{\text{holim}}}\; \Pi^p {\hat{F}_{{\mathcal{O}}}}^q \ar[u] & {\underset{[q] \in \Delta}{\text{holim}}}\; {\text{Tot}}\; {\Pi^{\bullet}}{\hat{F}_{{\mathcal{O}}}}^q \ar[l]^-{\sim} \ar[u] & {\underset{[q] \in \Delta}{\text{holim}}}\; {\underset{{\widehat{{\mathcal{O}}}_{k, q}}(U)}{\text{holim}}}\; {\hat{F}_{{\mathcal{O}}}}^q \ar[l]^-{\cong} \ar[u]_-{\varphi}. }$$ So the map $\varphi$ is a weak equivalence.
Now consider the following square. $$\xymatrix{{\underset{[q] \in \Delta}{\text{holim}}}\; {\hat{F}_{{\mathcal{B}}}}^{!q} (U) \ar[rr]^-{\sim} & & {F^{!}_{{\mathcal{B}}}}(U) \\
{\underset{[q] \in \Delta}{\text{holim}}}\; {\hat{F}_{{\mathcal{O}}}}^{!q} (U) \ar[rr]_-{\sim} \ar[u]^-{\varphi}_-{\sim} & & {F^{!}_{{\mathcal{O}}}}(U), \ar[u] }$$ where the top horizontal arrow is the homotopy limit of the map (\[fqf\_eq\]), which is itself a weak equivalence by Lemma \[fqf\_lem\]. In similar fashion the bottom horizontal map is a weak equivalence. The lefthand vertical map is the above map $\varphi$, which is a weak equivalence. So, since the square commutes, it follows that the righthand vertical map is also a weak equivalence. We thus obtain the desired result.
To prove Theorem \[good\_thm\] we will need the following lemma.
\[gd\_lem\] Let $U \in {\mathcal{O}(M)}$, and let ${\overline{{\mathcal{B}}}_k}(U) \subseteq {{\mathcal{B}}_k}(U)$ be the full subposet defined as $$\begin{aligned}
\label{bkb_eq}
{\overline{{\mathcal{B}}}_k}(U) = \{V \in {{\mathcal{B}}_k}(U)| \; \overline{V} \subseteq U\}.\end{aligned}$$ Here $\overline{V}$ stands for the closure of $V$. Let ${\mathcal{M}}$ be a simplicial model category. Consider an isotopy cofunctor $F \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{\mathcal{M}}$. Then the canonical map $$\begin{aligned}
\label{good_eq}
[\theta; F] \colon \underset{V \in {{\mathcal{B}}_k}(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{V \in {\overline{{\mathcal{B}}}_k}(U)}{\text{holim}} \; F(V),\end{aligned}$$ induced by the inclusion functor $\theta \colon {\overline{{\mathcal{B}}}_k}(U) {\longrightarrow}{{\mathcal{B}}_k}(U)$, is a weak equivalence.
Let $\overline{{\mathcal{B}}}$ be the following basis for the topology of $U$. $$\overline{{\mathcal{B}}} = \{B \subseteq U| \; \text{$B \in {\mathcal{B}}$ and $\overline{B} \subseteq U$}\}.$$ Certainly $\overline{{\mathcal{B}}}$ is a good basis (see Definition \[gb\_defn\]). One can easily see that each object of ${\overline{{\mathcal{B}}}_k}(U)$ is the disjoint union of at most $k$ elements from $\overline{{\mathcal{B}}}$. So by Theorem \[sos\_thm\], the map $[\theta; F]$ is a weak equivalence.
We begin with part (a) of goodness. Let $U, U' \in {\mathcal{O}(M)}$ such that $U \subseteq U'$. Assume that the inclusion map $U {\hookrightarrow}U'$ is an isotopy equivalence. Then the canonical map $$\underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U')}{\text{holim}} \; {\tilde{F}_{{\mathcal{B}}}}^p {\longrightarrow}{\underset{{\widetilde{{\mathcal{B}}}_{k, p}}(U)}{\text{holim}}}\; {\tilde{F}_{{\mathcal{B}}}}^p$$ is a weak equivalence by Proposition \[isop\_prop\]. Now, by replacing
1. ${\widetilde{{\mathcal{O}}}_{k, p}}(U)$ by ${\widetilde{{\mathcal{B}}}_{k, p}}(U')$, $${\tilde{F}_{{\mathcal{O}}}}^p \colon {\widetilde{{\mathcal{O}}}_{k, p}}(U) {\longrightarrow}{\mathcal{M}}\qquad \text{by} \qquad {\tilde{F}^p_{{\mathcal{B}}}}\colon {\widetilde{{\mathcal{B}}}_{k, p}}(U') {\longrightarrow}{\mathcal{M}},$$ $${\hat{F}_{{\mathcal{O}}}}^q \colon {\widehat{{\mathcal{O}}}_{k, q}}(U) {\longrightarrow}{\mathcal{M}}\qquad \text{by} \qquad {\hat{F}_{{\mathcal{B}}}}^q \colon {\widehat{{\mathcal{B}}}_{k, q}}(U') {\longrightarrow}{\mathcal{M}},$$
2. ${\widehat{{\mathcal{O}}}_{k, q}}(U)$ by ${\widehat{{\mathcal{B}}}_{k, q}}(U')$, ${\hat{F}_{{\mathcal{O}}}}^{!q} (U)$ by ${\hat{F}_{{\mathcal{B}}}}^{!q} (U')$, and ${F^{!}_{{\mathcal{O}}}}(U)$ by ${F^{!}_{{\mathcal{B}}}}(U')$,
in the proof of Theorem \[sos\_thm\], we deduce that the canonical map ${F^{!}_{{\mathcal{B}}}}(U') {\longrightarrow}{F^{!}_{{\mathcal{B}}}}(U)$ is a weak equivalence. This proves part (b) from Definition \[isotopy\_cof\_defn\]. Part (a) from the same definition follows immediately from the fact that $F \colon {{\mathcal{B}}_k}(M) {\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor by assumption, and from Theorem \[fib\_cofib\_thm\].
Now we show part (b) of goodness. Let $U_0 {\rightarrow}U_1 {\rightarrow}\cdots$ be a string of inclusions of ${\mathcal{O}(M)}$. Consider the following commutative square. $$\xymatrix{ \underset{V \in {{\mathcal{B}}_k}(\cup_i U_i)}{\text{holim}} \; F(V) \ar[rr] \ar[d]_-{\sim} & & \underset{i}{\text{holim}} \; \underset{V \in {{\mathcal{B}}_k}(U_i)}{\text{holim}} \; F(V) \ar[d]^-{\sim} \\
\underset{V \in {\overline{{\mathcal{B}}}_k}(\cup_i U_i)}{\text{holim}} \; F(V) \ar[rr]_-{\sim} & & \underset{i}{\text{holim}} \; \underset{V \in {\overline{{\mathcal{B}}}_k}(U_i)}{\text{holim}} \; F(V). }$$
1. Both vertical maps come from (\[good\_eq\]), and therefore are weak equivalences by Lemma \[gd\_lem\].
2. The bottom horizontal map is a weak equivalence by the following reason. Consider the data from Example \[intcf\_expl1\], and set ${\mathcal{D}}_i := {\overline{{\mathcal{B}}}_k}(U_i)$. Then it is straightforward to see that $$\begin{aligned}
\label{int_eq}
\int_{{\mathcal{C}}} {\mathcal{F}}= {\mathcal{C}}\times \left(\bigcup_i {\mathcal{D}}_i\right) = {\overline{{\mathcal{B}}}_k}(\cup_i U_i). \end{aligned}$$ The first equality is obvious, while the second one comes from the definition of ${\overline{{\mathcal{B}}}_k}(-)$ (see (\[bkb\_eq\])). Furthermore the canonical map $$\underset{(i, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; F(V) {\longrightarrow}\underset{i \in {\mathcal{C}}}{\text{holim}} \; \underset{V \in {\mathcal{D}}_i}{\text{holim}} \; F(V)$$ is a weak equivalence by Theorem \[cha\_sch\_thm\]. But, by (\[int\_eq\]), this latter map is nothing but the map we are interested in.
Hence the top horizontal map is a weak equivalence, and this completes the proof.
Polynomial cofunctors {#poly_section}
=====================
The goal of this section is to prove Theorem \[main\_thm\] announced in the introduction. We will need three preparatory lemmas: Lemma \[cofinal\_lem\], Lemma \[poly\_lem\], and Lemma \[charac\_lem\]. The two latter ones are important themselves.
Let us begin with the following definition.
\[poly\_defn\] A cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is called *polynomial of degree $\leq k$* if for every $U \in {\mathcal{O}(M)}$ and pairwise disjoint closed subsets $A_0, \cdots, A_k$ of $U$, the canonical map $$F(U) {\longrightarrow}\underset{S \neq \emptyset}{\text{holim}} \; F(U \backslash \cup_{i \in S} A_i)$$ is a weak equivalence. Here $S \neq \emptyset$ runs over the power set of $\{0, \cdots, k\}$.
\[cofinal\_lem\] Consider the data from Example \[intcf\_expl2\]. Then the functor $\theta \colon \int_{{\mathcal{C}}} {\mathcal{F}}{\longrightarrow}\Omega_k(U)$, defined as $\theta (S, V) = V$, is homotopy right cofinal (see Definition \[cofinal\_defn\]).
First of all, let us consider the notation (${\mathcal{C}}, U(S), \Omega, \Omega_k(U), {\mathcal{F}}$) introduced in Example \[intcf\_expl2\]. One has the following properties.
1. ${\mathcal{F}}(S \cup T) = {\mathcal{F}}(S) \cap {\mathcal{F}}(T)$ for any $S, T \in {\mathcal{C}}$;
2. for any $X \in \Omega_k(U)$ there exists $j \in \{0, \cdots, k\}$ such that $X \cap A_j = \emptyset$.
The first property follows directly from the definitions. The second comes from the following three facts: (i) By definition, each element of $\Omega$ intersects at most one of the $A_i$’s. (ii) $X$ is the disjoint union of at most $k$ elements from $\Omega$. (iii) The cardinality of the set $\{A_0, \cdots, A_k\}$ is $k+1$, which is greater than the number of components of $X$. This property is nothing but the pigeonhole principle.
Now let $V \in \Omega_k(U)$. We have to prove that the under category (see Definition \[uc\_defn\]) $V \downarrow \theta$ is contractible. It suffices to show that it admits an initial object. Consider the pair $(S, V)$ where $$S = \left\{i \in \{0, \cdots, k\} | \ V \cap A_i = \emptyset \right\}.$$ Certainly $S \neq \emptyset$ by the property (b). So $S$ is an object of ${\mathcal{C}}$. Moreover one can see that $V \in \cap_{i \in S} {\mathcal{F}}(\{i\})$. This amounts to saying that $V \in {\mathcal{F}}(S)$ since $\cap_{i \in S} {\mathcal{F}}(\{i\}) = {\mathcal{F}}(\cup_{i \in S} \{i\})$ by (a). So $(S, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}$. Hence the pair $((S, V), id_V)$ is an object of $V \downarrow \theta$. We claim that this latter object is an initial object of $V \downarrow \theta$. To prove the claim, let $((T, W), V {\hookrightarrow}W)$ be another object of $V \downarrow \theta$. Since $V \subseteq W$, it follows that $\{i | \ V \cap A_i \neq \emptyset \}$ is a subset of $\{i | \ W \cap A_i \neq \emptyset\}$. This implies that $$\{i | W \cap A_i = \emptyset\} \subseteq \{i | V \cap A_i = \emptyset\} = S.$$ Furthermore, $T$ is a subset of $\{i | W \cap A_i = \emptyset\} $ since $W \in {\mathcal{F}}(T) = \Omega_k(U \backslash (\cup_{i \in T} A_i))$. So $T \subseteq S$, and therefore there is a unique morphism from $((S, V), id_V)$ to $((T, W), V {\hookrightarrow}W)$ in the under category $V \downarrow \theta$. This completes the proof.
\[poly\_lem\] Let ${\mathcal{M}}$ be a simplicial model category.
1. Let ${\mathcal{O}}$ and ${\mathcal{O}_k(M)}$ as in Example \[fso\_expl\]. Let $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]). Then the cofunctor $F^{!}_{{\mathcal{O}}} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ (see Example \[fso\_expl\]) is polynomial of degree $\leq k$.
2. Let ${\mathcal{B}}$ and ${\mathcal{B}}_k(M)$ as in Definition \[fsb\_defn\]. Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor. Then the cofunctor $F^{!}_{{\mathcal{B}}} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ (see Definition \[fsb\_defn\]) is polynomial of degree $\leq k$.
We begin with the first part. First let us consider again the notation (${\mathcal{C}}, U(S), \Omega, \Omega_k(U), {\mathcal{F}}$) introduced in Example \[intcf\_expl2\]. We will first show that the canonical map $$\Phi_{\Omega} \colon \underset{V \in \Omega_k(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{F}}(S)}{\text{holim}} \; F(V)$$ is a weak equivalence. Next, by using the fact that $\Omega$ is another basis (for the topology of $U$) contained in ${\mathcal{O}}|U$, we will deduce that the canonical map $$\Phi_{{\mathcal{O}}} \colon \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; F(V) {\longrightarrow}\underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{O}_k}(U(S))}{\text{holim}} \; F(V)$$ is also a weak equivalence.
One can see that the map $\Phi_{\Omega}$ factors through $\underset{(S, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; F(V)$. That is, there is a commutative triangle $$\xymatrix{ \underset{V \in \Omega_k(U)}{\text{holim}} \; F(V) \ar[rr]^-{\Phi_{\Omega}} \ar[rd]_-{\alpha} & & \underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{F}}(S)}{\text{holim}} \; F(V) \\
& \underset{(S, V) \in \int_{{\mathcal{C}}} {\mathcal{F}}}{\text{holim}} \; F(V) , \ar[ru]_-{\beta} }$$ where
1. $\alpha$ is nothing but $[\theta; F]$ (see (\[thetax\])), where $\theta \colon {\int_{{\mathcal{C}}} {\mathcal{F}}}{\longrightarrow}\Omega_k(U)$ is the map from Lemma \[cofinal\_lem\].
2. $\beta$ is the canonical map from Theorem \[cha\_sch\_thm\].
Since $\alpha$ is a weak equivalence by Lemma \[cofinal\_lem\] and Theorem \[htpy\_cofinal\_thm\], and since $\beta$ is a weak equivalence by Theorem \[cha\_sch\_thm\], it follows that $\Phi_{\Omega}$ is a weak equivalence as well.
Now consider the following commutative square induced by the inclusion $\Omega_k(U) {\hookrightarrow}{\mathcal{O}_k}(U)$. $$\xymatrix{ \underset{V \in \Omega_k(U)}{\text{holim}} \; F(V) \ar[rr]^-{\Phi_{\Omega}}_-{\sim} & & \underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{F}}(S)}{\text{holim}} \; F(V) \\
\underset{V \in {\mathcal{O}}_k(U)}{\text{holim}} \; F(V) \ar[u]^-{\sim} \ar[rr]_-{\Phi_{{\mathcal{O}}}} & & \underset{S \in {\mathcal{C}}}{\text{holim}}\; \underset{V \in {\mathcal{O}_k}(U(S))}{\text{holim}} \; F(V) \ar[u]_-{\sim} }$$ Since the lefthand vertical map is a weak equivalence by Theorem \[sos\_thm\], and since the righthand vertical map is also a weak equivalence by Theorems \[sos\_thm\], \[fib\_cofib\_thm\] (remember that $U(S) := U\backslash \cup_{i \in S}A_i$ and ${\mathcal{F}}(S) := \Omega_k(U(S))$), it follows that $\Phi_{{\mathcal{O}}}$ is a weak equivalence as well. And this proves part (i).
Now we prove part (ii). Let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an isotopy cofunctor. We have to show that ${F^{!}_{{\mathcal{B}}}}$ is polynomial of degree $\leq k$. First, consider the cofunctor $G \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ defined as $G(U) := {F^{!}_{{\mathcal{B}}}}| {\mathcal{O}_k(M)}$, the restriction of ${F^{!}_{{\mathcal{B}}}}$ to ${\mathcal{O}_k(M)}$. Certainly $G$ is an isotopy cofunctor since ${F^{!}_{{\mathcal{B}}}}$ is good by Theorem \[good\_thm\]. This implies by the first part that the cofunctor ${G^{!}_{{\mathcal{O}}}}\colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is polynomial of degree $\leq k$. Moreover the canonical map ${G^{!}_{{\mathcal{O}}}}{\longrightarrow}(G | {{\mathcal{B}}_k}(M))^{!}_{{\mathcal{B}}}$ is a weak equivalence by Theorem \[sos\_thm\]. So $(G | {{\mathcal{B}}_k}(M))^{!}_{{\mathcal{B}}}$ is also polynomial of degree $\leq k$. Now, since the canonical map $F {\longrightarrow}G | {{\mathcal{B}}_k}(M)$ is a weak equivalence by Proposition \[fsrik\_prop\], it follows that the induced map ${F^{!}_{{\mathcal{B}}}}{\longrightarrow}(G | {{\mathcal{B}}_k}(M))^{!}_{{\mathcal{B}}}$ is a weak equivalence as well. And therefore ${F^{!}_{{\mathcal{B}}}}$ is polynomial of degree $\leq k$. This proves the lemma.
\[charac\_lem\] Let ${\mathcal{M}}$ be a simplicial model category.
1. Let $F, G \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be good and polynomial cofunctors of degree $\leq k$. Let $\eta \colon F {\longrightarrow}G$ be a natural transformation such that for any $U \in {\mathcal{O}_k(M)}$ the component $\eta[U] \colon F(U) {\longrightarrow}G(U)$ is a weak equivalence. Then for any $U \in {\mathcal{O}(M)}$ the map $\eta[U]$ is a weak equivalence.
2. Let $F, G \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be good and polynomial cofunctors of degree $\leq k$. Let ${\mathcal{B}}$ and ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Consider a natural transformation $\eta \colon F {\longrightarrow}G$ such that for any $U \in {\mathcal{B}_k(M)}$ the component $\eta[U] \colon F(U) {\longrightarrow}G(U)$ is a weak equivalence. Then for any $U \in {\mathcal{O}(M)}$ the map $\eta[U]$ is a weak equivalence.
The first part can be proved by following exactly the same steps as those of the proof of Theorem 5.1 from [@wei99]. Now we prove the second part. Let $U \in {\mathcal{O}_k(M)}$. Since ${\mathcal{B}}$ is a basis for the topology of $M$, there exists $V \in {\mathcal{B}_k(M)}$ contained in $U$ and such that the inclusion $V {\hookrightarrow}U$ is an isotopy equivalence. Applying $\eta$ to $V {\hookrightarrow}U$, we get the following commutative square. $$\xymatrix{F(U) \ar[r]^-{\sim} \ar[d]_-{\eta[U]} & F(V) \ar[d]^-{\sim}_-{\eta[V]} \\
G(U) \ar[r]_-{\sim} & G(V).}$$ The top and the bottom maps are weak equivalences since $F$ and $G$ are good by hypothesis. The righthand vertical map is a weak equivalence by assumption. So the lefthand vertical map is also a weak equivalence. Hence $\eta[U]$ is a weak equivalence for every $U \in {\mathcal{O}_k(M)}$. Now the desired result follows from the first part.
We are now ready to prove the main result of the paper.
Assume that $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is good and polynomial of degree $\leq k$. Define $G$ to be the restriction of $F$ to ${\mathcal{B}_k(M)}$. That is, $G:= F|{{\mathcal{B}}_k}(M)$. Since $F$ is good, and then in particular an isotopy cofunctor, it follows that $G$ is an isotopy cofunctor as well. Now we want to show that the canonical map $\eta \colon F {\longrightarrow}G^{!}$ is a weak equivalence. Let $U \in {\mathcal{O}(M)}$. One can rewrite $\eta[U] \colon F(U) {\longrightarrow}G^{!}(U)$ as the composition $$\eta[U] \colon \xymatrix{ F(U) \ar[r]^-{f} & \underset{V \in {\mathcal{O}}(U)}{\text{holim}} \; F(V) \ar[r]^-{g} & G^{!} (U), }$$ where $f$ comes from the fact that $U$ is the terminal object of ${\mathcal{O}}(U)$. The same fact allows us to conclude that $f$ is a weak equivalence. The map $g$ is nothing but $[\theta; F]$, where $\theta \colon {{\mathcal{B}}_k}(U) {\longrightarrow}{\mathcal{O}}(U)$ is just the inclusion functor. Now assume $U \in {{\mathcal{B}}_k}(M)$. Then for every $V \in {\mathcal{O}}(U)$ the under category $V \downarrow \theta$ is contractible since it has a terminal object, namely $(U, V {\hookrightarrow}U)$. Therefore, by Theorem \[htpy\_cofinal\_thm\], the map $g$ is a weak equivalence. This implies that $\eta[U]$ is a weak equivalence when $U \in {\mathcal{B}_k(M)}$. So, by Lemma \[charac\_lem\], $\eta[U]$ is also a weak equivalence for any $U \in {\mathcal{O}(M)}$.
Conversely, assume that $G:=F|{{\mathcal{B}}_k}(M)$ is an isotopy cofunctor and that the canonical map $\eta \colon F \stackrel{\sim}{{\longrightarrow}} G^{!}$ be a weak equivalence. By Theorem \[good\_thm\] and Lemma \[poly\_lem\] the cofunctor $G^{!}$ is good and polynomial of degree $\leq k$, which proves the converse. We thus obtained the desired result.
Homogeneous cofunctors {#hc_section}
======================
The goal of this section is to prove Theorem \[main2\_thm\] (announced in the introduction), which roughly says that the category of homogeneous cofunctors ${\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ of degree $k$ is weakly equivalent to the category of linear cofunctors ${\mathcal{O}}(F_k(M)) {\longrightarrow}{\mathcal{M}}$. We begin with three definitions. Next we prove Lemma \[homo\_lem\], which is the key lemma here, and which roughly states that homogeneous cofunctors of degree $k$ are determined by their values on open subsets diffeomorphic to the disjoint union of exactly $k$ balls. Note that this lemma is also a useful result in its own right, and its proof is based on the results we obtained in Section \[sos\_good\_section\] and Section \[poly\_section\].
Let ${\mathcal{M}}$ be a simplicial model category, and let $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be a cofunctor. The *$k$th polynomial approximation* to $F$, denoted $T_kF$, is the cofunctor $T_k F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ defined as $$T_k F (U) := \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} F(V).$$
\[hc\_defn\] Let ${\mathcal{M}}$ be a simplicial model category that has a terminal object denoted $0$.
1. A cofunctor $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is called *homogeneous of degree* $k$ if it satisfies the following three conditions:
1. $F$ is a good cofunctor (see Definition \[good\_defn\]);
2. $F$ is polynomial of degree $\leq k$ (see Definition \[poly\_defn\]);
3. The unique map $T_{k-1} F (U) {\longrightarrow}0$ is a weak equivalence for every $U \in {\mathcal{O}(M)}$.
2. A *linear cofunctor* is a homogeneous cofunctor of degree $1$.
The category of homogeneous cofunctors of degree $k$ and natural transformations will be denoted by ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$.
\[we\_defn\] Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be categories both equipped with a class of maps called weak equivalences.
1. We say that two functors $F, G \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ are *weakly equivalent*, and we denote $F \simeq G$, if they are connected by a zigzag of objectwise weak equivalences.
2. A functor $F \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is said to be a *weak equivalence* if it satisfies the following two conditions.
1. $F$ preserves weak equivalences.
2. There is a functor $G \colon {\mathcal{D}}{\longrightarrow}{\mathcal{C}}$ such that $FG$ and $GF$ are both weakly equivalent to the identity. The functor $G$ is also required to preserve weak equivalences.
3. We say that ${\mathcal{C}}$ is weakly equivalent to ${\mathcal{D}}$, and we denote ${\mathcal{C}}\simeq {\mathcal{D}}$, if there exists a zigzag of weak equivalences between ${\mathcal{C}}$ and ${\mathcal{D}}$.
By Definition \[we\_defn\], it follows that if two categories ${\mathcal{C}}$ and ${\mathcal{D}}$ are weakly equivalent, then their localizations with respect to weak equivalences are equivalent in the classical sense. Note that no model structure is required on ${\mathcal{C}}$ and ${\mathcal{D}}$. So our notion of weak equivalences between categories is not comparable, in general, with the well known notion of Quillen equivalence.
As mentioned earlier the following lemma is the key ingredient in proving Theorem \[main2\_thm\].
\[homo\_lem\] Let ${\mathcal{B}}$ be a good basis (see Definition \[gb\_defn\]) for the topology of $M$. Let ${\mathcal{B}^{(k)}}(M) \subseteq {\mathcal{O}(M)}$ denote the subposet whose objects are disjoint unions of exactly $k$ elements from ${\mathcal{B}}$, and whose morphisms are isotopy equivalences. Let ${\mathcal{M}}$ be a simplicial model category. Assume that ${\mathcal{M}}$ has a zero object $0$ (that is, an object which is both terminal an initial).
1. Then the category ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$ of homogeneous cofunctors of degree $k$ (see Definition \[hc\_defn\]) is weakly equivalent (in the sense of Definition \[we\_defn\]) to the category ${\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ of isotopy cofunctors ${\mathcal{B}^{(k)}}(M) {\longrightarrow}{\mathcal{M}}$ (see Definition \[isotopy\_cof\_defn\]). That is, $${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}\simeq {\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}).$$
2. For $A \in {\mathcal{M}}$ we have the weak equivalence $${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}}) \simeq {\mathcal{F}}_A({\mathcal{B}^{(k)}}(M); {\mathcal{M}}),$$ where ${\mathcal{F}}_{kA}({\mathcal{O}(M)}; {\mathcal{M}})$ is the category from Theorem \[main2\_thm\] and ${\mathcal{F}}_A({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ denotes the category of isotopy cofunctors $F \colon {\mathcal{B}^{(k)}}(M) {\longrightarrow}{\mathcal{M}}$ such that $F(U) \simeq A$ for every $U \in {\mathcal{B}^{(k)}}(M)$.
We will prove the first part; the proof of the second part is similar. The idea of the proof is to define a new category and show that it is weakly equivalent to both ${\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ and ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$. To define that category, let us first recall the notation ${\mathcal{B}_k(M)}$ from Definition \[fsb\_defn\]. Define ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$ to be the category whose objects are isotopy cofunctors $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ such that the restriction to ${\mathcal{B}}_{k-1} (M)$ is weakly equivalent to the constant functor at $0$. That is, $$\begin{aligned}
\label{wd_cond}
\text{for all $U \in {\mathcal{B}}_{k-1}(M)$,} \quad F(U) \simeq 0. \end{aligned}$$ Now consider the following diagram $$\begin{aligned}
\label{psii_phii}
\xymatrix{{\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}) \ar@<1ex>[r]^-{\psi_1} & {\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}}) \ar@<1ex>[l]^-{\phi_1} \ar@<1ex>[r]^-{\psi_2} & {{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}\ar@<1ex>[l]^-{\phi_2} }\end{aligned}$$ where the maps are defined as follows.
1. $\phi_1$ is the restriction functor. That is, $\phi_1(F) = F|{\mathcal{B}^{(k)}}(M)$.
2. $\phi_2$ is also the restriction functor. To see that it is well defined, let $F \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ be an object of ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$. We have to check that $F$ satisfies condition (\[wd\_cond\]). So let $U \in {\mathcal{B}}_{k-1}(M)$. Recalling the notation $[-; -]$ from Proposition \[induced\_holim\_prop\], we have the following commutative diagram $$\xymatrix{\underset{V \in {\mathcal{O}}_{k-1}(U)}{\text{holim}} \; F(V) \ar[rr]^-{\sim} \ar[d]_-{[\theta; F]}^-{\sim} & & 0 \\
\underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; F(V) \ar[rru]^-{\sim} & & F(U). \ar[ll]^-{\sim} \ar[u] }$$ Here $\theta \colon {\mathcal{B}}_{k-1}(U) {\hookrightarrow}{\mathcal{O}}_{k-1}(U)$ is the inclusion functor. The bottom horizontal map is a weak equivalence since $U$ is the terminal object of ${\mathcal{B}}_{k-1}(U)$ (see Proposition \[fsrik\_prop\]). The top one is a weak equivalence since $F$ is homogeneous of degree $k$. Since the lefthand vertical map is also a weak equivalence (by Theorem \[sos\_thm\]), it follows that $F(U)$ is weakly equivalent to $0$.
3. $\psi_1$ is defined as $$\psi_1(F)(U) := \left\{ \begin{array}{cc}
F(U) & \text{if $U \in {\mathcal{B}^{(k)}}(M)$ } \\
0 & \text{otherwise,}
\end{array} \right.$$ Certainly $\psi_1(F)$ satisfies (\[wd\_cond\]) and is an isotopy cofunctor. This latter assertion comes from the fact that if $U \subseteq U'$ is an isotopy equivalence, then $U$ and $U'$ definitely have the same number of connected components. On morphisms $\psi_1$ is defined in the obvious way.
4. $\psi_2$ is defined as $\psi_2(F) := {F^{!}_{{\mathcal{B}}}}$ (see Definition \[fsb\_defn\]). On morphisms $\psi_2$ is defined by the fact that the homotopy right Kan extension is functorial. By Theorem \[good\_thm\] and Lemma \[poly\_lem\], it is clear that $\psi_2(F)$ is good and polynomial of degree $\leq k$. To see that $\psi_2(F)$ satisfies condition (c) from Definition \[hc\_defn\], let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an object of ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$. Consider the following commutative diagram $$\xymatrix{\underset{V \in {\mathcal{O}}_{k-1}(U)}{\text{holim}} \; {F^{!}_{{\mathcal{B}}}}(V) \ar[r]^-{[\theta; {F^{!}_{{\mathcal{B}}}}]}_-{\sim} \ar[d] & \underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; {F^{!}_{{\mathcal{B}}}}(V) \ar@{=}[r] & \underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; \underset{W \in {{\mathcal{B}}_k}(V)}{\text{holim}} \; F(W) \ar[lld]^-{\sim} \\
0 & & \underset{V \in {\mathcal{B}}_{k-1}(U)}{\text{holim}} \; F(V), \ar[ll]^-{\sim} \ar[u]_-{\sim}}$$ where the righthand vertical map is induced by the canonical map $F(V) {\longrightarrow}\underset{W \in {{\mathcal{B}}_k}(V)}{\text{holim}} \; F(W)$. Since $V$ belongs to ${\mathcal{B}}_{k-1}(U)$ it follows that $V$ is the terminal object of ${{\mathcal{B}}_k}(V)$, and therefore this latter map is a weak equivalence (by Proposition \[fsrik\_prop\]). The bottom horizontal map is a weak equivalence since $F$ belongs to ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$, and then satisfies (\[wd\_cond\]). Regarding the map $[\theta; {F^{!}_{{\mathcal{B}}}}]$, it is a weak equivalence by Theorem \[sos\_thm\]. All this implies that the lefthand vertical map is a weak equivalence as well. So ${F^{!}_{{\mathcal{B}}}}$ satisfies condition (c).
Certainly $\phi_1, \psi_1$ and $\phi_2$ preserve weak equivalences. The functor $\psi_2$ preserves weak equivalences as well by Theorem \[fib\_cofib\_thm\] and condition (a) from Definition \[isotopy\_cof\_defn\]. Moreover, it is clear that $\phi_1\psi_1 =id$ and $\psi_1 \phi_1 \simeq id$. So the category ${\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}})$ is weakly equivalent to the category ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$. Furthermore, by using Propsosition \[fsrik\_prop\], one can easily come to $\phi_2\psi_2 \simeq id$ and $\psi_2\phi_2 \simeq id$. So the categories ${\mathcal{F}}_k({\mathcal{B}_k(M)}; {\mathcal{M}})$ and ${{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}$ are also weakly equivalent. This proves the lemma.
We are now ready to prove Theorem \[main2\_thm\].
We will prove the first part; the proof of the second part is similar. Recall the notation $F_k(M)$, which is that of the space of unordered configuration of $k$ points in $M$. The proof of part (i) follows from the following three weak equivalences: (\[we1\]), (\[we2\]), and (\[we3\]). The first one $$\begin{aligned}
\label{we1}
{{\mathcal{F}}_k({\mathcal{O}(M)}; {\mathcal{M}})}\simeq {\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}),\end{aligned}$$ is nothing but Lemma \[homo\_lem\] -(i). The second $$\begin{aligned}
\label{we2}
{\mathcal{F}}({\mathcal{B}^{(k)}}(M); {\mathcal{M}}) \cong {\mathcal{F}}({\mathcal{B}}'^{(1)} (F_k(M)); {\mathcal{M}}),\end{aligned}$$ is actually an isomorphism where ${\mathcal{B}}'$ is the basis for the topology of $F_k(M)$ whose elements are products of exactly $k$ elements from ${\mathcal{B}}$. This isomorphism comes from the fact that ${\mathcal{B}^{(k)}}(M) \cong {\mathcal{B}}'^{(1)} (F_k(M))$. The last weak equivalence $$\begin{aligned}
\label{we3}
{\mathcal{F}}({\mathcal{B}}'^{(1)} (F_k(M)); {\mathcal{M}}) \simeq {\mathcal{F}}_1 ({\mathcal{O}}(F_k(M)); {\mathcal{M}}),\end{aligned}$$ is again Lemma \[homo\_lem\] -(i).
Isotopy cofunctors in general model categories {#iso_cof_section}
==============================================
This section is independent of previous ones, and its goal is to prove Theorem \[iso\_cof\_thm\] (announced in the introduction), which says that the cofunctor $F^{!}={F^{!}_{{\mathcal{O}}}}$ from Example \[fso\_expl\] is an isotopy cofunctor provided that $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor (here ${\mathcal{M}}$ is a general model category). This result is proved in Theorem \[good\_thm\] when ${\mathcal{M}}$ is a simplicial model category. To prove Theorem \[good\_thm\] we used several results/properties (about homotopy limits in simplicial model categories) including Theorem \[fubini\_thm\], Proposition \[comm\_prop\]. This latter result involves the notion of totalization of a cosimplicial object, which does not make sense in a general model category. So the method we used before do not work here anymore. In this section we present a completely different approach, but rather lengthy, that uses only two properties of homotopy limits (see Theorem \[fib\_cofib\_thmg\] and Theorem \[htpy\_cofinal\_thmg\]). That approach is inspired by our work in [@paul_don17]. For the plan of this section, we refer the reader to the table of contents and the outline given at the introduction. For a faster run through the section, the reader could, after reading the Introduction, jump directly to the beginning of Section \[iso\_cof\_subsection\] to get a better idea of the proof of Theorem \[iso\_cof\_thm\].
Homotopy limits in general model categories {#holim_subsectiong}
-------------------------------------------
This subsection recalls some useful properties of homotopy limits in general model categories. We also recall two results (Proposition \[induced\_holim\_propg\] and Proposition \[fsrik\_propg\]) that will be used in next subsections.
Homotopy limits and colimits in general model categories are constructed in [@hir03; @dhks04] by W. Dwyer, P. Hirschhorn, D. Kan, and J. Smith. They use the notion of *frames* that we now recall briefly. Let ${\mathcal{M}}$ be a model category, and let $X$ be an object of ${\mathcal{M}}$. A *cosimplicial frame* on $X$ is a cofibrant replacement (in the Reedy model category of cosimplicial objects in ${\mathcal{M}}$) of the constant cosimplicial object at $X$ that satisfies certain properties. A *simplicial frame* on $X$ is the dual notion. For a more precise definition we refer the reader to [@hir03 Definition 16.6.1]. A *framing* on ${\mathcal{M}}$ is a functorial cosimplicial and simplicial frame on every object of ${\mathcal{M}}$. A *framed model category* is a model category endowed with a framing (see also [@hir03 Definition 16.6.21]). A typical example of a framed model category is any simplicial model category as we considered in previous sections.
In [@hir03 Theorem 16.6.9] it is proved that there exists a framing on any model category. It is also proved that two any framings are weakly equivalent [@hir03 Theorem 16.6.10]. Throughout this section, ${\mathcal{M}}$ is a model category endowed with a fixed framing.
Using the notion of framing, one can define the homotopy limit and colimit of a diagram in ${\mathcal{M}}$. We won’t give that definition here since it is not important for us (the reader who is interested in that definition can find it in [@hir03 Definition 19.1.2 and Definition 19.1.5]). All we need are some properties of that homotopy limit and colimit (see Theorem \[fib\_cofib\_thmg\] and Theorem \[htpy\_cofinal\_thmg\] below).
[@hir03 Theorem 19.4.2] \[fib\_cofib\_thmg\] Let ${\mathcal{M}}$ be a model category, and let ${\mathcal{C}}$ be a small category. Let $\eta \colon F {\longrightarrow}G$ be a map of ${\mathcal{C}}$-diagrams in ${\mathcal{M}}$.
1. If for every object $c$ of ${\mathcal{C}}$ the component $\eta[c] \colon F(c) {\longrightarrow}G(c)$ is a weak equivalence of cofibrant objects, then the induced map of homotopy colimits $\text{hocolim} \; F {\longrightarrow}\text{hocolim} \; G$ is a weak equivalence of cofibrant objects of ${\mathcal{M}}$.
2. If for every object $c$ of ${\mathcal{C}}$ the component $\eta[c] \colon F(c) {\longrightarrow}G(c)$ is a weak equivalence of fibrant objects, then the induced map of homotopy limits $\text{holim} \; F {\longrightarrow}\text{holim} \; G$ is a weak equivalence of fibrant objects of ${\mathcal{M}}$.
[@hir03 Theorem 19.6.7] \[htpy\_cofinal\_thmg\] Let ${\mathcal{M}}$ be a model category. If $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ is homotopy left cofinal (respectively homotopy right cofinal) (see Definition \[cofinal\_defn\]), then for every objectwise fibrant covariant (respectively contravariant) functor $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$, the natural map $[\theta; F]$ from Proposition \[induced\_holim\_propg\] is a weak equivalence.
We will also need the following two propositions. The first is a generalization of Proposition \[induced\_holim\_prop\], while the second is a generalization of Proposition \[fsrik\_prop\].
[@hir03 Proposition 19.1.8] \[induced\_holim\_propg\] Let ${\mathcal{M}}$ be a model category, and let $\theta \colon {\mathcal{C}}{\longrightarrow}{\mathcal{D}}$ be a functor between two small categories. If $F \colon {\mathcal{D}}{\longrightarrow}{\mathcal{M}}$ is an ${\mathcal{D}}$-diagram, then there is a canonical map $$\begin{aligned}
\label{thetaxg}
[\theta; F] \colon \underset{{\mathcal{D}}}{\text{holim}}\; F {\longrightarrow}\underset{{\mathcal{C}}}{\text{holim}}\; \theta^*F.\end{aligned}$$ (see (\[theta\_starx\])) Furthermore, this map is natural in both variables $\theta$ and $F$ as in Proposition \[induced\_holim\_prop\].
\[fsrik\_propg\] Let ${\mathcal{B}}$ and ${\mathcal{B}_k(M)}$ as in Definition \[fsb\_defn\]. Let ${\mathcal{M}}$ be a model category, and let $F \colon {\mathcal{B}_k(M)}{\longrightarrow}{\mathcal{M}}$ be an objectwise fibrant cofunctor.
1. There is a natural transformation $\eta$ from $F$ to the restriction ${F^{!}_{{\mathcal{B}}}}| {\mathcal{B}_k(M)}$ (see Definition \[fsb\_defn\]), which is an objectwise weak equivalence.
2. If $F$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]), then so is the restriction of ${F^{!}_{{\mathcal{B}}}}$ to ${\mathcal{B}_k(M)}$.
This is very similar to the proof of Proposition \[fsrik\_prop\].
The category ${{\mathcal{D}}(V)}$ {#dv_subsection}
---------------------------------
Consider the following data:
1. $U, U' \in {\mathcal{O}(M)}$ such that $U \subseteq U'$;
2. $V \in {\mathcal{O}_k}(U)$;
3. $L \colon U \times I {\longrightarrow}U', (x, t) \mapsto L_t(x):= L(x, t)$ is an isotopy from $U$ to $U'$ (see Definition \[iso\_eq\_defn\]).
The aim of this subsection is to define an important category ${{\mathcal{D}}(V)}$ (see Definition \[dv\_defn\]) out of these data. The definition of ${{\mathcal{D}}(V)}$ is rather technical. Roughly speaking, an object of ${{\mathcal{D}}(V)}$ is a zigzag $x$ of isotopy equivalences between $W$ and $L_1(W)$, where $W$ is a object of ${\mathcal{O}_k}(V)$. Morphisms of ${{\mathcal{D}}(V)}$ are inclusions. There are two types of morphisms from $x$ to $y$ depending on the fact that $x$ and $y$ have the same length or not. If $x$ and $y$ have the same length, a morphism from $x$ to $y$ is just an inclusion. Otherwise a morphism is still an inclusion, but more subtle. We also prove Proposition \[dv\_contractible\_prop\], which says that ${{\mathcal{D}}(V)}$ is contractible.
Let us begin with the following notation, and some technical definition.
\[ssie\_not\] Given two objects $W, T \in {\mathcal{O}(M)}$, we use the notation $W {\subseteq_{ie}}T$ to mean that $W$ is a subset of $T$ and the inclusion map $W {\hookrightarrow}T$ is an isotopy equivalence.
In [@paul_don17 Section 3.2] we introduced the concept of *admissible family* $x=\{a_0, \cdots, a_{n+1}\}$ with respect to $L$ and a compact subset $K \subseteq U$. If one has different compact for each interval $[a_i, a_{i+1}]$, the family $x$ is said to be *piecewise admissible*. More precisely, we have the following definition.
\[pw\_adm\_defn\] Let $W \in {\mathcal{O}_k}(U)$, and let $[a, b] \subseteq [0, 1]$. Let $x=\{a_0, \cdots, a_{n+1}\} \subseteq [0, 1]$ be a family such that $a_0 = a, a_{n+1} = b,$ and $a_i \leq a_{i+1}$ for all $i$. Let $K = \{K_0, \cdots, K_n\}$ be a family of nonempty compact subsets $K_i \subseteq L_{a_i}(W)$ such that $\pi_0(K_i {\hookrightarrow}L_{a_i}(W))$ is surjective. The family $x$ is said to be *piecewise admissible* with respect to $\{K, L\colon W \times I {\longrightarrow}U'\}$ (or just *piecewise admissible*) if for every $i$ there exists an object $W_{i(i+1)}$ of ${\mathcal{O}_k(M)}$ such that for all $s \in [a_i, a_{i+1}]$, $$\begin{aligned}
L_s(L_{a_i}^{-1}(K_i)) \subseteq W_{i(i+1)} {\subseteq_{ie}}L_s(W),\end{aligned}$$ and $$\begin{aligned}
\label{vii_eqn}
\overline{W_{i(i+1)}} \subseteq L_s(W),\end{aligned}$$ where $L_{a_i} \colon W {\longrightarrow}L_{a_i}(W)$ is the canonical homeomorphism induced by $L$, and $\overline{W_{i(i+1)}}$ stands for the closure of $W_{i(i+1)}$.
The following proposition, which will be used in the proof of Proposition \[dv\_contractible\_prop\], can be deduced easily from [@paul_don17 Proposition 3.10 ].
\[existence\_adm\_prop\] Let $[a, b] \subseteq I,$ and let $t \in [a, b]$. Let $W \in {\mathcal{O}_k}(U)$, and let $K \subseteq L_t(W)$ be a nonempty compact subset such that $\pi_0(K {\hookrightarrow}L_a(W))$ is surjective.
1. If $t = a$ (respectively $t=b$), there exists $t' > t$ (respectively $t'' < t$) such that the family $\{t, t'\}$ (respectively $\{t'', t\}$) is admissible with respect to $\{K, L\}$.
2. If $t \in (a, b)$, there exists $\epsilon_t >0$ such that the family $\{t-\epsilon_t, t+\epsilon_t\}$ is admissible with respect to $\{K, L\}$.
\[mcale\_defn\] Define ${\mathcal{E}}$ to be the category whose objects are finite subsets $A= \{a_0, \cdots, a_{n+1}\}$ of the interval $[0, 1]$ such that $a_0 = 0, a_{n+1} = 1$ and $a_i \leq a_{i+1}$ for all $i$. Morphisms of ${\mathcal{E}}$ are inclusions.
\[ia\_defn\] Let $A = \{a_0, \cdots, a_{n+1}\}$ be an object of ${\mathcal{E}}$. Define ${\mathcal{I}}_A$ to be the poset whose objects are $$\{a_0\}, \{a_1\} \cdots, \{a_n\}, \{a_0, a_1\}, \{a_1, a_2\}, \cdots, \{a_{n-1}, a_n\},$$ and whose morphisms are inclusions $\{a_i\} {\longrightarrow}\{a_i, a_{i+1}\}$ and $\{a_{i+1}\} {\longrightarrow}\{a_i, a_{i+1}\}, 0 \leq i \leq n$.
The category ${\mathcal{I}}_A$ looks like a zigzag starting at $\{a_0\} = \{0\}$ and ending at $\{a_{n+1}\} =\{1\}$. For instance, if $n=2$, then $${\mathcal{I}}_A = \left\{\xymatrix{\{a_0\} \ar[r] & \{a_0, a_1\} & \{a_1\} \ar[l] \ar[r] & \{a_1, a_2\} & \{a_2\} \ar[l] } \right\}.$$
\[theta\_ab\_prop\] The construction that sends $A$ to ${\mathcal{I}}_A$ is a contravariant functor ${\mathcal{E}}{\longrightarrow}\text{Cat}$ from ${\mathcal{E}}$ to the category Cat of small categories.
Given $A, B \in {\mathcal{E}}$ such that $A \subseteq B$ with $B = \{b_0, \cdots, b_{m+1}\}$, we need to define a morphism $$\begin{aligned}
\label{theta_ab}
\theta_{AB} \colon {\mathcal{I}}_B {\longrightarrow}{\mathcal{I}}_A.\end{aligned}$$ Let us begin with an example. Take $A = \{a_0, a_1, a_2\}$ and $B = \{b_0, b_1, b_2, b_3\}$ such that $b_2 = a_1$ as shown Figure \[ab\_fig\].
The idea of the definition of $\theta_{AB}$ is as follows. First consider the elements of $A \cap B = \{b_0, b_2, b_3\}$, and define $\theta_{AB}(\{b_0\}) = \{a_0\}, \theta_{AB}(\{b_2\}) = \{a_1\}$, and $\theta_{AB}(\{b_3\}) = \{a_2\}$. Next consider $B \backslash A = \{b_1\}$. Since $D =[a_0, a_1]$ is the smallest closed interval containing $b_1$ such that $\text{Inf} D \in A, \text{Sup} D \in A$, and $D \cap A = \{a_0, a_1\}$, we have $\theta_{AB}(\{b_1\}) := \{a_0, a_1\}$. A similar observation gives $
\theta_{AB}(\{b_0, b_1\}) := \{a_0, a_1\}, \theta_{AB}(\{b_1, b_2\}):= \{a_0, a_1\}, \text{ and } \theta_{AB}(\{b_2, b_3\}) = \{a_1, a_2\}.
$ The following diagram summarizes the definition of $\theta_{AB}$. $$\xymatrix{\{a_0\} \ar[rr] & & \{a_0, a_1\} & & \{a_1\} \ar[ll] \ar[r] & \{a_1, a_2\} & \{a_2\} \ar[l] \\
\{b_0\} \ar[r] \ar[u] & \{b_0, b_1\} \ar[ru] & \{b_1\} \ar[u] \ar[l] \ar[r] & \{b_1, b_2\} \ar[lu] & \{b_2\} \ar[l] \ar[r] \ar[u] & \{b_2, b_3\} \ar[u] & \{b_3\}. \ar[l] \ar[u]}$$
Now we give a precise definition of $\theta_{AB}$. For $b \in B$, define $$c(b) := \text{max}\{x \in A | \ x \leq b\} \quad \text{and} \quad d(b) := \text{min}\{x \in A| \ x \geq b\}.$$ Now define $\theta_{AB}$ as $$\theta_{AB}(\{b\}) = \left\{ \begin{array}{ccc}
\{b\} & \text{if} & b \in A \\
\{c(b), d(b)\} & \text{if} & b \notin A,
\end{array} \right.$$ and $$\theta_{AB}(\{b_i, b_{i+1}\}) = \{c(b_i), d(b_{i+1})\}.$$ On morphisms of ${\mathcal{I}}_B$, $\theta_{AB}$ is defined in the most obvious way. Regarding the composition, if $A, B, C \in {\mathcal{E}}$ such that $A \subseteq B \subseteq C$, then one obviously has $$\begin{aligned}
\label{theta_abc}
\theta_{AC} = \theta_{AB}\theta_{BC}, \end{aligned}$$ which completes the proof.
We are now ready to define ${{\mathcal{D}}(V)}$.
\[dv\_defn\] Recall the posets ${\mathcal{E}}$ and ${\mathcal{I}}_A$ from Definition \[mcale\_defn\] and Definition \[ia\_defn\] respectively. Also recall the isotopy $L$ from the beginning of this subsection. The category ${{\mathcal{D}}(V)}$ is defined as follows.
1. An object is a triple $(W, A, {\mathcal{X}_A})$ (or just a pair $(W, {\mathcal{X}_A}$)) where $W \in {\mathcal{O}_k}(V)$, $A = \{a_0, \cdots, a_{n+1}\} \in {\mathcal{E}}$, and ${\mathcal{X}_A}\colon {\mathcal{I}}_A {\longrightarrow}{\mathcal{O}_k(M)}$ is a contravariant functor that satisfies the following three conditions:
1. ${\mathcal{X}_A}(\{a\}) = L_a(W)$ for all $a \in A$.
2. For every $i \in \{0, \cdots, n\}$, for every $s \in [a_i, a_{i+1}]$, $${\mathcal{X}_A}(\{a_i, a_{i+1}\}) {\subseteq_{ie}}L_s(W).$$ (See Notation \[ssie\_not\] for the meaning of ${\subseteq_{ie}}$.)
3. For every $i \in \{0, \cdots, n\}$, for every $s \in [a_i, a_{i+1}]$, $$\begin{aligned}
\label{xai_eqn}
\overline{{\mathcal{X}_A}(\{a_i, a_{i+1}\})} \subseteq L_s(W).
\end{aligned}$$
2. A morphism from $(W, A, {\mathcal{X}_A})$ to $(T, B, {\mathcal{Y}_B})$ consists of a triple $(f, g, \Lambda_{AB})$ (or just $\Lambda_{AB}$) where $f \colon W {\hookrightarrow}T$ and $g \colon A {\hookrightarrow}B$ are both the inclusion maps, and $\Lambda_{AB} \colon {\mathcal{X}_A}\theta_{AB} {\longrightarrow}{\mathcal{Y}_B}$ is a natural transformation.
In other words, an object of ${{\mathcal{D}}(V)}$ is a zigzag of isotopy equivalences between $L_0(W) = W$ and $L_1(W)$, where $W \in {\mathcal{O}_k}(V)$. For instance, when $A = \{a_0, a_1, a_2\}$, an object looks like (\[xzig\]). $$\begin{aligned}
\label{xzig}
(W,{\mathcal{X}_A}) = \left\{ \xymatrix{W=X_0 & X_{01} \ar[l]_-{\simeq} \ar[r]^-{\simeq} & X_1 & X_{12} \ar[l]_-{\simeq} \ar[r]^-{\simeq} & X_2 = L_1(W)} \right\}.\end{aligned}$$ There are two kind of morphisms from $(W,{\mathcal{X}_A})$ to $(T, {\mathcal{Y}_B})$ depending on the fact that $A =B$ or $A$ is a proper subset of $B$. These morphisms are illustrated by (\[mor1\]) and (\[mor2\]). $$\begin{aligned}
\label{mor1}
\xymatrix{X_0 \ar[d] & X_{01} \ar[l]_-{\simeq} \ar[r]^-{\simeq} \ar[d] & X_1 \ar[d] & X_{12} \ar[l]_-{\simeq} \ar[r]^-{\simeq} \ar[d] & X_2 \ar[d] \\
Y_0 & Y_{01} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_1 & Y_{12} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_2. }\end{aligned}$$
$$\begin{aligned}
\label{mor2}
\xymatrix{X_0 \ar[d] & & X_{01} \ar[ll]_-{\simeq} \ar[rr]^-{\simeq} \ar[d] \ar[ld] \ar[rd] & & X_1 \ar[d] & X_{12} \ar[l]_-{\simeq} \ar[r]^-{\simeq} \ar[d] & X_2 \ar[d] \\
Y_0 & Y_{01} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_1 & Y_{12} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_2 & Y_{23} \ar[l]^-{\simeq} \ar[r]_-{\simeq} & Y_3.}\end{aligned}$$
\[associated\_rmk\] To any piecewise admissible family $A$ (see Definition \[pw\_adm\_defn\]), one can associate a canonical object $(W, {\mathcal{X}_A})$ of ${{\mathcal{D}}(V)}$ by letting ${\mathcal{X}_A}(\{a_i, a_{i+1}\}) := W_{i(i+1)}.$
\[dv\_contractible\_prop\] The category ${{\mathcal{D}}(V)}$ is contractible.
It suffices to show that ${{\mathcal{D}}(V)}$ is *filtered*, that is, it satisfies the following two conditions:
1. For every pair of objects $(W,{\mathcal{X}_A})$ and $(T, {\mathcal{Y}_B})$ there are morphisms to a common object $(W, {\mathcal{X}_A}) {\longrightarrow}(S, {\mathcal{Z}_C})$ and $(T, {\mathcal{Y}_B}) {\longrightarrow}(S, {\mathcal{Z}_C})$;
2. For every pair of parallel morphisms $\Lambda_{AB}, \Lambda'_{AB} \colon (W, {\mathcal{X}_A}) {\longrightarrow}(T, {\mathcal{Y}_B})$, there is some morphism $\Lambda_{BC} \colon (T, {\mathcal{Y}_B}) {\longrightarrow}(S, {\mathcal{Z}_C})$ such that $\Lambda_{BC}\Lambda_{AB} = \Lambda_{BC} \Lambda'_{AB}$.
Since ${{\mathcal{D}}(V)}$ is a poset by definition, it clearly satisfies (2). To check (1), let $(W, {\mathcal{X}_A}), (T, {\mathcal{Y}_B}) \in {{\mathcal{D}}(V)}$. Set $D = A \cup B$. Certainly $D$ is a finite subset, denoted $\{d_0, \cdots, d_{p+1}\}$, of $I$ such that $d_0 = 0, d_{p+1} =1$, and $d_i \leq d_{i+1}$ for all $i$. One can write the intervals $[d_i, d_{i+1}], 0 \leq i \leq p,$ as $$[d_i, d_{i+1}] = \big[a_{r(i)}, a_{r(i)+1}\big] \cap \big[b_{s(i)}, b_{s(i)+1}\big],$$ where $$a_{r(i)} := \text{max}\left\{x \in A| \ x \leq d_i \right\} \quad \text{and} \quad b_{s(i)} := \text{max}\left\{y \in B| \ y \leq d_i\right\}.$$ Of course, $a_{r(i)+1}$ (respectively $b_{s(i)+1}$) is the successor of $a_{r(i)}$ in $A$ (respectively the successor of $b_{s(i)}$ in $B$). Note that the interior of $[d_i, d_{i+1}]$ does not intersect either $A$ or $B$.
Take $S = V$. The idea of the construction of ${\mathcal{Z}_C}$ is to subdivide each $[d_i, d_{i+1}]$ into small intervals $[a, b]$ such that there exists $Z_{ab} \in {\mathcal{O}_k(M)}$ that is contained in $L_a(V) \cap L_b(V)$ and that contains both ${\mathcal{X}_A}\left(\{a_{r(i)}, a_{r(i)+1}\}\right)$ and ${\mathcal{Y}_B}\left(\{b_{s(i)}, b_{s(i)+1}\}\right)$. So let $i \in \{0, \cdots, p\}$, and let $t \in [d_i, d_{i+1}]$. Thanks to (\[xai\_eqn\]) one can consider the compact subset ${\mathcal{K}}_i \subseteq L_t(V)$ defined as $${\mathcal{K}}_i = \overline{{\mathcal{X}_A}\left(\left\{a_{r(i)}, a_{r(i)+1} \right\} \right)} \bigcup \overline{{\mathcal{Y}_B}\left(\left\{b_{s(i)}, b_{s(i)+1} \right\} \right)}.$$ If $t \in (d_i, d_{i+1})$ then by Proposition \[existence\_adm\_prop\] there exist $\epsilon_t > 0$ and $Z_{i(i+1)} \in {\mathcal{O}_k(M)}$ such that for all $u \in [t-\epsilon_t, t+\epsilon_t]$, $$L_u(L_t^{-1}({\mathcal{K}}_i)) \subseteq Z_{i(i+1)} {\subseteq_{ie}}L_u(V) \quad \text{and} \quad \overline{Z_{i(i+1)}} \subseteq L_u(V).$$ Clearly one has $${\mathcal{X}_A}\left(\{a_{r(i)}, a_{r(i)+1}\}\right) \subseteq L_{t-\epsilon_t}(V) \cap Z_{i(i+1)} \cap L_{t+\epsilon} (V)$$ and $${\mathcal{Y}_B}\left(\{b_{s(i)}, b_{s(i)+1}\}\right) \subseteq L_{t-\epsilon_t}(V) \cap Z_{i(i+1)} \cap L_{t+\epsilon} (V).$$ If $t= d_i$ (respectively $t=d_{i+1}$) there is an admissible family $\{d_i, t'\}$ (respectively $\{t'', d_{i+1}\}$) again by Proposition \[existence\_adm\_prop\]. Letting $t$ vary in $[d_i, d_{i+1}]$ one obtains an open cover, $[d_i, t') \cup \{(t-\epsilon_t, t+\epsilon_t)\}_t \cup (t'', d_{i+1}])$, of $[d_i, d_{i+1}]$. Now, applying the compactness we get an ordered finite subset $C^i = \{c^i_0, \cdots, c^i_{n_i}\}$ of $[d_i, d_{i+1}]$ such that $c^i_0 = d_i$, $c^i_{n_i} = d_{i+1}$, and for every $j$ the interval $[c^i_j, c^i_{j+1}]$ is contained in one of the open subsets from the cover. This implies that $C:= \cup_{i=0}^p C^i$ is piecewise admissible and contains both $A$ and $B$. Moreover, it is clear that the associated object $(V, {\mathcal{Z}_C})$ of ${{\mathcal{D}}(V)}$ (as in Remark \[associated\_rmk\]) has the desired property. This ends the proof.
\[theta\_zo\_lem\] The functors $$\theta_0 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{O}_k}(V) \quad \text{and} \quad \theta_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{O}_k}(L_1(V))$$ defined as $\theta_0(W, {\mathcal{X}_A}) = W$ and $\theta_1(W, {\mathcal{X}_A}) = L_1(W)$ are both homotopy right cofinal (see Definition \[cofinal\_defn\]).
For every $W \in {\mathcal{O}_k}(V)$ the under category $W \downarrow \theta_0$ is contractible. This works exactly as the proof of Proposition \[dv\_contractible\_prop\]. Similarly, one can show that $\theta_1$ is homotopy right cofinal.
\[dv\_functor\_prop\] The construction ${\mathcal{D}}\colon {\mathcal{O}_k}(U) {\longrightarrow}\text{Cat}$ that sends $V$ to ${{\mathcal{D}}(V)}$ is a covariant functor.
It is very easy to establish. For a morphism $V {\hookrightarrow}V'$ of ${\mathcal{O}_k}(U)$, we define $\theta \colon {{\mathcal{D}}(V)}{\longrightarrow}{{\mathcal{D}}(V')}$ as $\theta(W, {\mathcal{X}_A}) = (W, {\mathcal{X}_A})$. Certainly this defines a functor from ${{\mathcal{D}}(V)}$ to ${{\mathcal{D}}(V')}$.
The functors $H, P_0, P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ {#hp_subsection}
----------------------------------------------------------------------------------
The goal of this subsection is to define three important functors, $H, P_0, P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$, and two natural weak equivalences, $\eta_0 \colon H {\longrightarrow}P_0$ and $\eta_1 \colon H {\longrightarrow}P_1$. We will use them in the proof of Theorem \[iso\_cof\_thm\], which will be done at Subsection \[iso\_cof\_subsection\]. In this subsection ${\mathcal{M}}$ is a model category, $F \colon {\mathcal{O}_k(M)}{\longrightarrow}{\mathcal{M}}$ is an isotopy cofunctor (see Definition \[isotopy\_cof\_defn\]), and $F^{!} \colon {\mathcal{O}(M)}{\longrightarrow}{\mathcal{M}}$ is the cofunctor defined by (\[fsrik\_defn\]). We continue to use the same data as those provided at the beginning of Subsection \[dv\_subsection\].
### The functor $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$
Before we define $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$, we need to first recall a certain model category of diagrams, and next define $\Psi \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$ (for the categories ${{\mathcal{D}}(V)}$ and ${\mathcal{E}}$, see Definition \[dv\_defn\] and Definition \[mcale\_defn\] respectively), which is functorial in each argument.
For $E \in {\mathcal{E}}$, consider the category ${\mathcal{M}}^{{\mathcal{I}}_E}$ of ${\mathcal{I}}_E$-diagrams in ${\mathcal{M}}$ (recall the poset ${\mathcal{I}}_E$ from Definition \[ia\_defn\]). In the literature there exist many model structures on ${\mathcal{M}}^{{\mathcal{I}}_E}$. But for our purposes we endow it with the one described by Dwyer and Spalinski in [@dwyer_spa95 Section 10]. First recall that this model structure is only defined for diagrams indexed by *very small categories* (see the paragraph just after 10.13 from [@dwyer_spa95]), which is the case for ${\mathcal{I}}_E$. Next recall that this model structure states that weak equivalences and cofibrations are both objectwise. A map ${\mathcal{X}}{\longrightarrow}{\mathcal{Y}}$ is a fibration if certain explicit morphisms in ${\mathcal{M}}$ associated with ${\mathcal{X}}{\longrightarrow}{\mathcal{Y}}$ are fibrations. (See for example (10.9), (10.10), and Proposition 10.11 from [@dwyer_spa95].) One of the advantages of this model structure is the fact that any diagram admits an explicit fibrant replacement as shown the following illustration.
\[fib\_repl\_expl\] Consider the following objectwise fibrant diagram in ${\mathcal{M}}$. $${\mathcal{X}}= \left\{ \xymatrix{X_0 \ar[r]^-{f_0} & X_{01} & X_1 \ar[l]_-{f_1} \ar[r]^-{f_2} & X_{12} & X_2 \ar[l]_-{f_3}} \right\}$$ Then its fibrant replacement, $R{\mathcal{X}}$, is the second row of the following commutative diagram $$\xymatrix{X_0 \ar[r]^-{f_0} \ar@{>->}[d]^-{\sim}_-{g_0} & X_{01} \ar@{>->}[d]^-{\sim}_-{id} & X_1 \ar[l]_-{f_1} \ar[r]^-{f_2} \ar@{>->}[d]^-{\sim}_-{g_1} & X_{12} \ar@{>->}[d]^-{\sim}_-{id} & X_2 \ar[l]_-{f_3} \ar@{>->}[d]^-{\sim}_-{g_2} \\
{\widetilde{X}}_{0} \ar@{->>}[r] & X_{01} & {\widetilde{X}}_1 \ar@{->>}[l] \ar@{->>}[r] & X_{12} & {\widetilde{X}}_2. \ar@{->>}[l] }$$ To get $R{\mathcal{X}}$, first we take a fibrant replacement ${\widetilde{X}}_{i(i+1)}, 0 \leq i \leq n$ (here $n=1$), of $X_{i(i+1)}$ in ${\mathcal{M}}$. Since ${\mathcal{X}}$ is objectwise fibrant, we then take ${\widetilde{X}}_{i(i+1)} = X_{i(i+1)}$. Next the functorial factorization of the composition $idf_0$ (respectively $idf_3$) provides ${\widetilde{X}}_0$ (respectively ${\widetilde{X}}_{n+1} = {\widetilde{X}}_2$). Lastly, ${\widetilde{X}}_1$ comes from the functorial factorization $$\xymatrix{X_1 \ar[rr]^-{(idf_1, idf_2)} \ar@{>->}[rd]^-{\sim}_-{g_1} & & X_{01} \times X_{12} \\
& \widetilde{X}_1 \ar@{->>}[ru] }$$
\[trx\_rmk\] Let $\theta \colon {\mathcal{I}}{\longrightarrow}{\mathcal{J}}$ be a functor between small categories, and let ${\mathcal{X}}\colon {\mathcal{J}}{\longrightarrow}{\mathcal{M}}$ be an ${\mathcal{J}}$-diagram in ${\mathcal{M}}$. Then $\theta^*(R{\mathcal{X}})$ is not equal to $R \theta^*({\mathcal{X}})$ in general, but there is a natural map $\theta^*(R{\mathcal{X}}) {\longrightarrow}R \theta^*({\mathcal{X}})$. This map comes directly from the way we construct our fibrant replacements.
Now we define $\Psi \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$. First recall the covariant functor $\theta_{AB} \colon {\mathcal{I}}_B {\longrightarrow}{\mathcal{I}}_A$ defined in the course of the proof of Proposition \[theta\_ab\_prop\]. For $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$ such that $A \subseteq E$, one can consider the composition $$\begin{aligned}
\label{ieae_comp}
\xymatrix{{\mathcal{I}}_E \ar[rr]^-{\theta_{AE}} & & {\mathcal{I}}_A \ar[rr]^-{{\mathcal{X}_A}} & & {\mathcal{O}_k(M)}\ar[rr]^-{F^{!}} & & {\mathcal{M}}}, \end{aligned}$$ which is nothing but an ${\mathcal{I}}_E$-diagram in ${\mathcal{M}}$. Define $\Psi((W,{\mathcal{X}_A}), E) \in {\mathcal{M}}$ as $$\begin{aligned}
\label{psi_defn}
\Psi((W,{\mathcal{X}_A}), E) = \left\{ \begin{array}{ccc}
\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta_{AE}^*({\mathcal{X}_A}) & \text{if} & A \subseteq E \\
\emptyset & \text{if} & \text{$A$ is not contained in $E$},
\end{array} \right.\end{aligned}$$ where $\emptyset$ stands for the initial object of ${\mathcal{M}}$.
The construction $\Psi \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$ that sends $((W,{\mathcal{X}_A}), E)$ to $\Psi((W,{\mathcal{X}_A}), E)$ is contravariant in the first variable and covariant in the second one.
Let $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$. We have to prove two things.
1. Functoriality in the first variable. Let $(T,{\mathcal{Y}_B}) \in {{\mathcal{D}}(V)}$ such that $A \subseteq B$, and let $\Lambda_{AB}$ be a morphism in ${{\mathcal{D}}(V)}$ from $(W,{\mathcal{X}_A})$ to $(T,{\mathcal{Y}_B})$. Then, by Definition \[dv\_defn\], $\Lambda_{AB} \colon {\mathcal{X}_A}\theta_{AB} {\longrightarrow}{\mathcal{Y}_B}$ is a natural transformation. If $B \subseteq E$, then one has the composition $F^{!}\Lambda_{AB} \theta_{BE} \colon F^{!}{\mathcal{X}_A}\theta_{AB}\theta_{BE} {\longleftarrow}F^{!}{\mathcal{Y}_B}\theta_{BE}$ (remember that $F^{!}$ is contravariant, and that $\theta_{BE}$ is covariant), which is the same as $$\begin{aligned}
\label{fab}
F^{!}\Lambda_{AB} \theta_{BE} \colon F^{!}\theta_{AE}^*({\mathcal{X}_A}) {\longleftarrow}F^{!}\theta_{BE}^*({\mathcal{Y}_B})\end{aligned}$$ since $\theta_{AB}\theta_{BE} = \theta_{AE}$ by (\[theta\_abc\]). This induces a morphism $$\Psi(\Lambda_{AB}, id) := \text{lim}(RF^{!}\Lambda_{AB}\theta_{BE}) \colon \Psi((W,{\mathcal{X}_A}), E) {\longleftarrow}\Psi((T,{\mathcal{Y}_B}), E).$$ If $E$ does not contain $B$, then $\Psi((T,{\mathcal{Y}_B}), E)$ is the initial object by definition, and therefore $\Psi(\Lambda_{AB}, id)$ is the unique morphism from $\emptyset$ to $\Psi((W,{\mathcal{X}_A}), E)$.
2. Functoriality in the second variable. Let $E' \in {\mathcal{E}}$ such that $E \subseteq E'$. If $A \subseteq E$, then we have $$\theta^*_{EE'}\left(F^{!}\theta^*_{AE}({\mathcal{X}_A}) \right) = F^{!}\theta^*_{AE'}({\mathcal{X}_A})$$ by (\[theta\_abc\]) and (\[theta\_starx\]). The map $$\Psi(id, E {\hookrightarrow}E') \colon \Psi((W,{\mathcal{X}_A}), E) {\longrightarrow}\Psi((W,{\mathcal{X}_A}), E')$$ is then defined to be the composition $$\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta_{AE}^* ({\mathcal{X}_A}) {\longrightarrow}\underset{{\mathcal{I}}_{E'}}{\text{lim}} \; \theta^*_{EE'} \left(RF^{!} \theta_{AE}^* ({\mathcal{X}_A})\right) {\longrightarrow}\underset{{\mathcal{I}}_{E'}}{\text{lim}} \; R\theta^*_{EE'} \left(F^{!} \theta_{AE}^* ({\mathcal{X}_A})\right).$$ Here the first arrow is the canonical map induced by $\theta_{EE'} \colon {\mathcal{I}}_{E'} {\longrightarrow}{\mathcal{I}}_E$, and the second is the natural map that comes directly from the way fibrant replacements of ${\mathcal{I}}_{E'}$-diagram are constructed (see Example \[fib\_repl\_expl\] and Remark \[trx\_rmk\]). As before, the case where $A$ is not contained in $E$ is obvious.
This proves the proposition.
Before we define $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$, we need to equip the category ${\mathcal{M}}^{{\mathcal{E}}}$ of ${\mathcal{E}}$-diagrams in ${\mathcal{M}}$ with a nice model structure. Thanks to the fact that the category ${\mathcal{E}}$ is a *direct category* (see [@hovey99 Definition 5.1.1]), and therefore a *Reedy category* (see [@hovey99 Definition 5.2.1]), we can endow ${\mathcal{M}}^{{\mathcal{E}}}$ with a Reedy model structure that we now recall. For $E \in {\mathcal{E}}$, we define the *latching space* functor $L_E \colon {\mathcal{M}}^{{\mathcal{E}}} {\longrightarrow}{\mathcal{M}}$ as follows. Let ${\mathcal{E}}_E$ be the category of non-identity maps in ${\mathcal{E}}$ with codomain $E$, and define $L_E$ to be the composite $$L_E \colon \xymatrix{{\mathcal{M}}^{{\mathcal{E}}} \ar[r] & {\mathcal{M}}^{{\mathcal{E}}_E} \ar[rr]^-{\text{colim}} & & {\mathcal{M}}}$$ where the first arrow is restriction. Clearly there is a natural transformation $L_E{\mathcal{X}}{\longrightarrow}{\mathcal{X}}(E)$.
[@hovey99 Theorem 5.1.3] \[struc\_me\_thm\] There exists a model structure on the category ${\mathcal{M}}^{{\mathcal{E}}}$ of ${\mathcal{E}}$-diagrams in ${\mathcal{M}}$ such that weak equivalences and fibrations are objectwise. Furthermore, a map ${\mathcal{X}}{\longrightarrow}{\mathcal{Y}}$ is a (trivial) cofibration if and only if the induced map ${\mathcal{X}}(E) \coprod_{L_E{\mathcal{X}}} L_E{\mathcal{Y}}{\longrightarrow}{\mathcal{Y}}(E)$ is a (trivial) cofibration for all $E$.
Note that any object ${\mathcal{X}}$ of ${\mathcal{M}}^{{\mathcal{E}}}$ has an explicit cofibrant replacement $Q{\mathcal{X}}\colon {\mathcal{E}}{\longrightarrow}{\mathcal{M}}$ obtained by induction as follows. (Recall that by definition $\{0, 1 \}$ is the initial object of ${\mathcal{E}}$). First take the cofibrant replacement $Q{\mathcal{X}}(\{0, 1\})$ of ${\mathcal{X}}(\{0, 1\})$. Next, for any other object $E \in {\mathcal{E}}$, $Q{\mathcal{X}}(E)$ comes from the functorial factorization of the obvious map $$\underset{E' \subset E}{\text{colim}} \; Q{\mathcal{X}}(E') {\longrightarrow}{\mathcal{X}}(E),$$ where $E' \in {\mathcal{E}}$ runs over the set of proper subsets of $E$. As an example, the cofibrant replacement of $\Psi((W,{\mathcal{X}_A}), -)$ is an ${\mathcal{E}}$-diagram on the form $$\begin{aligned}
\label{qpsi_shape}
Q\Psi((W,{\mathcal{X}_A}), -) = \xymatrix{ & & \bullet \cdots \ar@{.}[d] \\
\emptyset \ar[r] & Q\Psi((W,{\mathcal{X}_A}), A) \ar@{.>}[ru] \ar@{.>}[r] \ar@{.>}[rd] & \bullet \cdots \ar@{.}[d]\\
& & \bullet \cdots } \end{aligned}$$
\[cofibrant\_rmk\] By construction, every object of the diagram $Q{\mathcal{X}}$ is cofibrant in ${\mathcal{M}}$.
\[hoco\_co\_prop\] The natural map $$\underset{{\mathcal{E}}}{\text{hocolim}} \; Q\Psi((W, {\mathcal{X}_A}), -) {\longrightarrow}\underset{{\mathcal{E}}}{\text{colim}} \; Q\Psi((W, {\mathcal{X}_A}), -)$$ is a weak equivalence.
This follows from [@hir03 Theorem 19.9.1].
We come to the definition of $H$.
Recall $\Psi$ from (\[psi\_defn\]), and define the functor $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{functor_h}
H(W, {\mathcal{X}_A}) = \underset{E \in {\mathcal{E}}}{\text{colim}} \; Q\Psi((W, {\mathcal{X}_A}), E). \end{aligned}$$
Let ${\mathcal{E}}_A \subseteq {\mathcal{E}}$ denote the full subcategory whose objects are $E$ containing $A$, and let ${\widetilde{\Psi}}((W,{\mathcal{X}_A}), -)$ denote the restriction of $\Psi((W,{\mathcal{X}_A}), -)$ to ${\mathcal{E}}_A$. Then one has $$\begin{aligned}
\label{hxa_formula}
H(W,{\mathcal{X}_A}) = \underset{ {\mathcal{E}}}{\text{colim}} \; Q\Psi((W,{\mathcal{X}_A}), -) \cong \underset{ {\mathcal{E}}_A}{\text{colim}} \; Q{\widetilde{\Psi}}((W,{\mathcal{X}_A}), -). \end{aligned}$$ The isomorphism $\cong$ follows from the fact that the diagram (\[qpsi\_shape\]) contains the initial object, $\emptyset$, of ${\mathcal{M}}$.
Now we define another map (the map $h$ below) which will be used in the next subsection. Recalling the data provided at the beginning of Subsection \[dv\_subsection\], and using definitions, we can easily see that for every $E \in {\mathcal{E}}_A$, for every $x \in {\mathcal{I}}_E$, one has ${\mathcal{X}_A}\theta_{AE}(x) \subseteq U'$. Applying the contravariant functor $F^{!}$ to this latter inclusion, we get maps $F^{!}(U') {\longrightarrow}F^{!} {\mathcal{X}_A}\theta_{AE}(x)$. This induces a natural transformation $F^{!}(U') {\longrightarrow}{\widetilde{\Psi}}({\mathcal{X}_A}, -)$ between two ${\mathcal{E}}_A$-diagrams in ${\mathcal{M}}$, the first one being the constant diagram (recall that $RF^{!}(U') = F^{!}(U')$ by the assumption that $F$ is objectwise fibrant and by Theorem \[fib\_cofib\_thm\]). Now taking the cofibrant replacement of this latter map, passing to the colimit, and using (\[hxa\_formula\]) we have a map $$\begin{aligned}
\label{map_h}
h \colon QF^{!}(U') {\longrightarrow}H(W,{\mathcal{X}_A}), \end{aligned}$$ which is natural in $(W,{\mathcal{X}_A})$.
### The functors $P_0, P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$
To define $P_0$ and $P_1$, we will first define $$\Phi_0 \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}\quad \text{and} \quad \Phi_1 \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}.$$ To do this, we need to introduce some notation. If $E= \{a_0, \cdots, a_{n+1}\}$ is an object of ${\mathcal{E}}$ and ${\mathcal{X}}\colon {\mathcal{I}}_E {\longrightarrow}{\mathcal{M}}$ is a functor, we define two objects $\phi_0 {\mathcal{X}}$ and $\phi_1 {\mathcal{X}}$ of ${\mathcal{M}}$ as $$\phi_0 {\mathcal{X}}:= {\mathcal{X}}(\{a_0\}) \quad \text{and} \quad \phi_1 {\mathcal{X}}:= {\mathcal{X}}(\{a_{n+1}\}).$$ In other words, $\phi_0 {\mathcal{X}}$ is the first object of the zigzag ${\mathcal{X}}$, while $\phi_1 {\mathcal{X}}$ is the last one.
Let $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$. If $A \subseteq E$, then one can consider the composition $F^{!}{\mathcal{X}_A}\theta_{AE} \colon {\mathcal{I}}_E {\longrightarrow}{\mathcal{M}}$ (from (\[ieae\_comp\])), which is an object of ${\mathcal{M}}^{{\mathcal{I}}_E}$. Let $RF^{!}{\mathcal{X}_A}\theta_{AE}$ denote its fibrant replacement with respect to the Dwyer-Spalinski model structure we described before (see Example \[fib\_repl\_expl\] for an illustration of what we call fibrant replacement). Define $\Phi_0((W,{\mathcal{X}_A}), E)$ as $$\begin{aligned}
\label{phiz_defn}
\Phi_0((W,{\mathcal{X}_A}), E) = \left\{ \begin{array}{ccc}
\phi_0R F^{!} {\mathcal{X}_A}\theta_{AE} & \text{if} & A \subseteq E \\
\emptyset & \text{if} & \text{$A$ is not contained in $E$}.
\end{array} \right.\end{aligned}$$
Replacing $\phi_0$ by $\phi_1$ in (\[phiz\_defn\]), we have the definition of $\Phi_1((W,{\mathcal{X}_A}), E)$. The following remark about $\Phi_0$ and $\Phi_1$ is important.
\[phixa\_rmk\] By inspection, for every $((W,{\mathcal{X}_A}),E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$, one has $$\Phi_0((W,{\mathcal{X}_A}), E) = \Phi_0((W,{\mathcal{X}_A}), A) \quad \text{and} \quad \Phi_1((W,{\mathcal{X}_A}), E) = \Phi_1((W,{\mathcal{X}_A}), A),$$ provided that $A \subseteq E$. This easily comes from three things: the definition of ${{\mathcal{D}}(V)}$, that of $\theta_{AE}$, and the way fibrant replacements of ${\mathcal{I}}_E$-diagrams in ${\mathcal{M}}$ are constructed (see Example \[fib\_repl\_expl\]).
The construction $\Phi_i \colon {{\mathcal{D}}(V)}\times {\mathcal{E}}{\longrightarrow}{\mathcal{M}}, i=0, 1$, that sends $((W,{\mathcal{X}_A}), E)$ to $\Phi_i((W,{\mathcal{X}_A}), E)$ is contravariant in the first variable and covariant in the second one.
For the functoriality in the first variable, let $\Lambda_{AB}$ be a morphism of ${{\mathcal{D}}(V)}$ from $(W,{\mathcal{X}_A})$ to $(T,{\mathcal{Y}_B})$, and consider the map $F^{!}\Lambda_{AB}\theta_{BE}$ from (\[fab\]). Its fibrant replacement gives $$\Phi_0(\Lambda_{AB}, id) \colon \Phi_0((W,{\mathcal{X}_A}), E) {\longleftarrow}\Phi_0((T,{\mathcal{Y}_B}), E).$$ The functoriality in the second variable is obvious by Remark \[phixa\_rmk\]. In fact, if $i \colon E {\hookrightarrow}E'$ is a morphism of ${\mathcal{E}}$ then $\Phi_0((W,{\mathcal{X}_A}), i) = id$ when $A \subseteq E$. A similar proof can be performed with $\Phi_1$ in place of $\Phi_0$.
Recall $\Phi_0$ and $\Phi_1$ from (\[phiz\_defn\]), and define $P_0 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ and $P_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ as $$\begin{aligned}
\label{functor_po}
P_0(W,{\mathcal{X}_A}) = \underset{{\mathcal{E}}}{\text{colim}} \; Q\Phi_0((W,{\mathcal{X}_A}), -) \quad \text{and} \quad P_1(W,{\mathcal{X}_A}) = \underset{{\mathcal{E}}}{\text{colim}} \; Q\Phi_1((W,{\mathcal{X}_A}), -),\end{aligned}$$ where $Q\Phi_i((W, {\mathcal{X}_A}), -), i \in \{0, 1\}$, is the cofibrant replacement of the ${\mathcal{E}}$-diagram $\Phi_i((W, {\mathcal{X}_A}), -)$ with respect to the model structure given by Theorem \[struc\_me\_thm\].
Now we define two important maps ($p_0$ and $p_1$ below) that will be also used in the next subsection. First, recalling the definition of ${{\mathcal{D}}(V)}$ (from Definition \[dv\_defn\]) and that of $\theta_{AB}$ (from (\[theta\_ab\])), one can see that the functor $F^{!}{\mathcal{X}_A}\theta_{AE}$ from (\[ieae\_comp\]) is nothing but a zigzag in ${\mathcal{M}}$ starting at $F^{!}(W)$ and ending at $F^{!}(L_1(W))$. If ${\widetilde{\Phi}}_i({\mathcal{X}_A}, -), i \in \{0, 1 \}$ denotes the restriction of $\Phi_i({\mathcal{X}_A}, -)$ to ${\mathcal{E}}_A$, by the definition of the fibrant replacement, (\[phiz\_defn\]) immediately implies the existence of two natural weak equivalences: $$F^{!}(W) \stackrel{\sim}{{\longrightarrow}} {\widetilde{\Phi}}_0((W,{\mathcal{X}_A}), -) \quad \text{and} \quad F^{!} (L_1(W)) \stackrel{\sim}{{\longrightarrow}} {\widetilde{\Phi}}_1((W,{\mathcal{X}_A}), -).$$ Of course the functors involves are viewed as ${\mathcal{E}}_A$-diagrams. Taking the cofibrant replacement with respect to the model structure described in Theorem \[struc\_me\_thm\] of those maps, we get $QF^{!}(W) \stackrel{\sim}{{\longrightarrow}} Q{\widetilde{\Phi}}_0((W,{\mathcal{X}_A}), -)$ and $QF^{!}(L_1(W)) \stackrel{\sim}{{\longrightarrow}} Q{\widetilde{\Phi}}_1((W,{\mathcal{X}_A}), -)$. Passing to the colimit, and using the observation (\[hxa\_formula\]) we did for $H(W,{\mathcal{X}_A})$, we have weak equivalences $$\begin{aligned}
\label{qfv_colim}
p_0 \colon QF^{!}(W) \stackrel{\sim}{{\longrightarrow}} P_0(W,{\mathcal{X}_A}) \quad \text{and} \quad p_1 \colon QF^{!}(L_1(W)) \stackrel{\sim}{{\longrightarrow}} P_1(W,{\mathcal{X}_A}).
\end{aligned}$$ Notice that these maps are both natural in $(W,{\mathcal{X}_A})$. Also notice that by Remark \[phixa\_rmk\] the diagram ${\widetilde{\Phi}}_i((W,{\mathcal{X}_A}), -)$ is in fact the constant diagram whose value is ${\widetilde{\Phi}}_i((W,{\mathcal{X}_A}), A)$.
### The maps $\eta_i \colon H {\longrightarrow}P_i, i=0, 1$
The aim here is to show that the definitions of $H$ (\[functor\_h\]), $P_0,$ and $P_1$ (\[functor\_po\]) imply the existence of natural weak equivalences $$\begin{aligned}
\label{etai}
\eta_0 \colon \xymatrix{H \ar[r]^-{\sim} & P_0} \quad \text{and} \quad \eta_1 \colon \xymatrix{H \ar[r]^-{\sim} & P_1}. \end{aligned}$$ We will show the existence of $\eta_0$; the existence of $\eta_1$ is similar. The idea is to define for every $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$ a natural map $$\alpha[(W,{\mathcal{X}_A}), E] \colon \Psi((W,{\mathcal{X}_A}), E) {\longrightarrow}\Phi_0((W,{\mathcal{X}_A}), E),$$ and show that it is a weak equivalence. Applying the cofibrant replacement functor, and then the colimit functor to $\alpha[(W,{\mathcal{X}_A}), -]$, and using Remark \[cofibrant\_rmk\], Theorem \[fib\_cofib\_thmg\], and Proposition \[hoco\_co\_prop\] we will then deduce that $\eta_0$ is a weak equivalence. So let $((W,{\mathcal{X}_A}), E) \in {{\mathcal{D}}(V)}\times {\mathcal{E}}$. If $A$ is not contained in $E$, then $\Psi((W,{\mathcal{X}_A}), E) = \emptyset = \Phi_0((W, {\mathcal{X}_A}), E)$ by definition, and therefore $\alpha[(W,{\mathcal{X}_A}), E]$ is the obvious map. Assume $A \subseteq E$. Then $\Psi((W,{\mathcal{X}_A}), E) = \underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!}\theta^*_{AE}({\mathcal{X}_A})$ by definition. Define $\alpha[(W,{\mathcal{X}_A}), E]$ to be obvious map $$\begin{aligned}
\label{lim_phi}
\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta^*_{AE}({\mathcal{X}_A}) {\longrightarrow}\phi_0RF^{!} \theta^*_{AE}({\mathcal{X}_A}) = \Phi_0((W,{\mathcal{X}_A}), E).\end{aligned}$$ This map is so obvious because $\phi_0RF^{!} \theta^*_{AE}({\mathcal{X}_A})$ is nothing but one piece from the diagram $RF^{!} \theta^*_{AE}({\mathcal{X}_A})$. It is straightforward to check the naturality of $\alpha[(W,{\mathcal{X}_A}), E]$ in both variables. One can also see that (\[lim\_phi\]) is a weak equivalence essentially by the following reason. First, since ${\mathcal{X}_A}$ is a zigzag of isotopy equivalences by the condition (b) from Definition \[dv\_defn\], and since $F^{!} | {\mathcal{O}_k(M)}$ is an isotopy cofunctor by Proposition \[fsrik\_propg\], it follows that every morphism of the diagram $F^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is a weak equivalence. This implies that every morphism of $RF^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is a weak equivalence as well. Moreover $RF^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is fibrant. So every morphism of the diagram $RF^{!}\theta^*_{AE} ({\mathcal{X}_A})$ is a weak equivalence which is also a fibration. Thanks to the shape of this diagram, one can compute its limit by taking successive pullbacks. An illustration of this is given by (\[pb\_diag\]). $$\begin{aligned}
\label{pb_diag}
\xymatrix{{\widetilde{X}}_0 \ar@{->>}[r]^-{\sim} & X_{01} & {\widetilde{X}}_1 \ar@{->>}[l]_-{\sim} \ar@{->>}[r]^-{\sim} & X_{12} & {\widetilde{X}}_2 \ar@{->>}[l]_-{\sim} \\
& Y_1 \ar[lu] \ar[u] \ar[ru] & & Y_2 \ar[lu] \ar[u] \ar[ru] & \\
& & Z \ar[lu] \ar[ru] & & }\end{aligned}$$ Now applying the fact that the pullback of a fibration is again a fibration, and the fact that the pullback of a weak equivalence along a fibration is again a weak equivalence, we deduce that the map from $\underset{{\mathcal{I}}_E}{\text{lim}} \; RF^{!} \theta^*_{AE}({\mathcal{X}_A})$ to each piece of the diagram is a weak equivalence.
Proof of the main result of the section {#iso_cof_subsection}
---------------------------------------
The goal here is to prove Theorem \[iso\_cof\_thm\], which is the main result of this section.
Let $U {\hookrightarrow}U'$ be an isotopy equivalence of ${\mathcal{O}(M)}$, and let $L \colon U \times I {\longrightarrow}U', (x, t) \mapsto L_t(x)$, be an isotopy from $U$ to $U'$. Our aim is to show that the canonical map $F^{!}(U') {\longrightarrow}F^{!}(U)$ is a weak equivalence. The idea is to first consider the commutative diagram (\[big\_diag\]), which will be defined below (for $V \in {\mathcal{O}_k}(U)$). Next we will show that the map $$\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; (f_1) \colon F^{!}(U') {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; B_1(V)$$ is a weak equivalence. By the two-out-of-three axioms, we will deduce successively that the maps $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; ({\widetilde{f}}_1)$, $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; (g)$, $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; ({\widetilde{f}}_0),$ and $\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}}\; (f_0)$ are weak equivalences. Using the fact that this latter morphism is a weak equivalence, we will deduce the theorem.
$$\begin{aligned}
\label{big_diag}
\xymatrix{ & & {\widetilde{B}}_0(V) \ar[rrd]^-{k_0}_-{\sim} \ar[d]^-{\sim} & & \\
QF^{!}(U') \ar[rru]^-{{\widetilde{f}}_0} \ar[rrd]^{g} \ar[d]_-{\sim} & & B_0(V) & & A_0(V) \\
F^{!}(U') \ar[rru]_{f_0} \ar[rrd]^{f_1} & & G(V) \ar[rru]^-{\sim}_-{l_0} \ar[rrd]^-{l_1}_-{\sim} & & \\
QF^{!}(U') \ar[rru]_{g} \ar[rrd]_-{{\widetilde{f}}_1} \ar[u]^-{\sim} & & B_1(V) & & A_1(V) \\
& & {\widetilde{B}}_1(V) \ar[u]_-{\sim} \ar[rru]^-{\sim}_-{k_1} & & }\end{aligned}$$
Now we explain the construction of the diagram (\[big\_diag\]).
1. Q(-) is the cofibrant replacement functor in ${\mathcal{M}}$.
2. The objects $B_i(V), {\widetilde{B}}_i(V), i \in \{0, 1\}$, are defined as $$B_i(V) = \underset{(W, {\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; F^{!}(L_i(W)) \quad \text{and} \quad {\widetilde{B}}_i(V) = \underset{(W, {\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; QF^{!}(L_i(W)).$$ (Recall that $L_0 \colon U {\longrightarrow}U'$ is the inclusion functor; so $L_0(W) = W$.) Clearly these objects are functorial in $V$. Indeed, if $V {\hookrightarrow}V'$ is a morphism of ${\mathcal{O}_k}(U)$ then we have the inclusion functor $\theta \colon {{\mathcal{D}}(V)}{\longrightarrow}{{\mathcal{D}}(V')}$ defined in the course of the proof of Proposition \[dv\_functor\_prop\]. This latter functor and Proposition \[induced\_holim\_propg\] allow us to get the desired functoriality.
3. G(V) is defined as $G(V) = \underset{(W, {\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; H(W, {\mathcal{X}_A})$, where $H \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ is the functor from (\[functor\_h\]). As before the construction that sends $V$ to $G(V)$ is a contravariant functor.
4. The maps $f_i, {\widetilde{f}}_i, i \in \{0, 1\}$, are defined as $f_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (h_i)$ and ${\widetilde{f}}_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (Qh_i)$ where $h_i \colon F^{!}(U') {\longrightarrow}F^{!}(L_i(W))$ is the obvious map obtained by applying $F^{!}$ to the inclusion $L_i(W) \subseteq U'$.
5. The map $g$ is defined as $g = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (h)$, where $h \colon QF^{!}(U') {\longrightarrow}H(W, {\mathcal{X}_A})$ is the map from (\[map\_h\]).
6. The objects $A_i(V), i \in \{0, 1\}$, are defined as $A_i(V) := \underset{(W,{\mathcal{X}_A}) \in {{\mathcal{D}}(V)}}{\text{holim}} \; P_i(W, {\mathcal{X}_A}),$ where $P_i \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{M}}$ comes from (\[functor\_po\]). Again as before $A_i(V)$ is functorial in $V$.
7. The maps $k_i, i \in \{0, 1\}$, are defined as $k_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (p_i)$, where $p_i \colon QF^{!}(W) \stackrel{\sim}{{\longrightarrow}} P_i(W, {\mathcal{X}_A})$ comes from (\[qfv\_colim\]). Since $p_i$ is a weak equivalence, and since by assumption $F$ is objectwise fibrant, it follows that $k_i$ is a weak equivalence as well by Theorem \[fib\_cofib\_thmg\] [^3].
8. Lastly, the maps $l_i, i \in \{0, 1\}$, are defined as $l_i = \underset{{{\mathcal{D}}(V)}}{\text{holim}} (\eta_i)$, where $\eta_i \colon H \stackrel{\sim}{{\longrightarrow}} P_i$ is the natural transformation from (\[etai\]). As before, $l_i$ is a weak equivalence.
Using definitions, one can see that every map from the diagram (\[big\_diag\]) is natural in $V$. One can also check that the squares containing $f_i, {\widetilde{f}}_i, k_i$ and $l_i, i \in \{0, 1\}$, are both commutative. Now applying the homotopy limit functor (when $V$ runs over ${\mathcal{O}_k}(U)$) to each morphism of (\[big\_diag\]), we get a new diagram, denoted $\mathbb{D}$, in which the map $\text{holim}(f_1) \colon F^{!}(U') {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; B_1(V)$ is a weak equivalence because of the following. First consider the following commutative diagram constructed as follows. $$\begin{aligned}
\label{small_diag}
\xymatrix{\underset{W \in {\mathcal{O}_k}(L_1(V))}{\text{holim}}\; F^{!}(W) \ar[rr]^-{\sim} & & B_1(V) \\
F^{!}(L_1(V)) \ar[u]^-{\sim} & & F^{!}(U'). \ar[u]_-{f_1} \ar[ll]^-{h_1} \ar[llu]_-{q} }\end{aligned}$$
1. The top map is nothing but $[\theta_1; F^{!}]$ (see Proposition \[induced\_holim\_propg\] for the notation $[-;-]$), where $\theta_1 \colon {{\mathcal{D}}(V)}{\longrightarrow}{\mathcal{O}_k}(L_1(V))$ is defined as $\theta_1(W, {\mathcal{X}_A}) = L_1(W)$, and $F^{!} \colon {\mathcal{O}_k}(L_1(V)) {\longrightarrow}{\mathcal{M}}$ is just the restriction of $F^{!}$ to ${\mathcal{O}_k}(L_1(V))$. By Corollary \[theta\_zo\_lem\] the functor $\theta_1$ is homotopy right cofinal, and therefore the map $[\theta_1; F^{!}]$ is a weak equivalence by Theorem \[htpy\_cofinal\_thmg\].
2. The maps $f_1$ and $h_1$ have been defined before, while $q$ is induced by the canonical map $F^{!}(U') {\longrightarrow}F^{!}(W)$.
3. The lefthand vertical map is the map $\text{holim} (\eta)$, where $ \eta[W] \colon F(W) \stackrel{\sim}{{\longrightarrow}} F^{!}(W)$ is the map from Proposition \[fsrik\_propg\].
Applying the homotopy limit functor (when $V$ runs over ${\mathcal{O}_k}(U)$ of course) to each morphism of (\[small\_diag\]), we get a new commutative diagram, denoted $\mathbb{S}$, in which the map $\text{holim} (h_1) \colon F^{!}(U') {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} F^{!}(L_1(V))$ is a weak equivalence because of the following reason. Consider the functor $\theta \colon {\mathcal{O}_k}(U) {\longrightarrow}{\mathcal{O}_k}(U')$ defined as $\theta(V) = L_1(V)$. Also consider $F^{!} \colon {\mathcal{O}_k}(U') {\longrightarrow}{\mathcal{M}}$. Clearly $\theta$ is an isomorphism since $L_1 \colon U {\longrightarrow}U'$ is a homeomorphism. So for any $W \in {\mathcal{O}_k}(U')$, the pair $(\theta^{-1}(W), id)$ is the initial object of the under category $W \downarrow \theta$. This shows that $\theta$ is homotopy right cofinal, and therefore the map $[\theta; F^{!}] \colon \underset{V \in {\mathcal{O}_k}(U')}{\text{holim}} \; F^{!}(V) {\longrightarrow}\underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; (\theta^{*}F^{!})(V)$ is a weak equivalence by Theorem \[htpy\_cofinal\_thmg\]. By inspection, the map $\text{holim}(h_1)$ is nothing but the composition $$\xymatrix{\underset{V \in {\mathcal{O}_k}(U')}{\text{holim}} \; F(V) \ar[rr]^-{\text{holim}(\eta)}_-{\sim} & & \underset{V \in {\mathcal{O}_k}(U')}{\text{holim}} \; F^{!}(V) \ar[rr]^-{[\theta; F^{!}]}_-{\sim} & & \underset{V \in {\mathcal{O}_k}(U)}{\text{holim}} \; (\theta^{*}F^{!})(V),}$$ where $\eta[V] \colon F(V) \stackrel{\sim}{{\longrightarrow}} F^{!}(V)$ is again the map from Proposition \[fsrik\_propg\]. Now, applying the two-out-of-three axiom to the diagram $\mathbb{S}$ we deduce that the map $\text{holim}(f_1)$ is a weak equivalence.
We come back to the diagram $\mathbb{D}$. As we said before, the two-out-of-three axiom shows successively that the maps $\text{holim}({\widetilde{f}}_1)$, $\text{holim}(g), \text{holim} ({\widetilde{f}}_0), $ and $\text{holim} (f_0)$ are weak equivalences. Now, replacing $1 $ by $0$ in the diagram (\[small\_diag\]), one can see that the map $\text{holim}(h_0) \colon F^{!}(U') {\longrightarrow}F^{!}(U)$ is a weak equivalence by the two-out-of-three axiom. But this is what we had to show.
[99]{} A.K. Bousfield and D. M. Kan, *Homotopy limits, completion and localizations*, Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 304. W. Chacholski and J. Scherer, *Homotopy theory of diagrams*, Mem. Amer. Math. Soc. 155 (2002), no. 736, x+90 pp. D.C. Cisinski, *Locally constant functors*, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 593–614. W. Dwyer, P. Hirschhorn, D. Kan, and J. Smith, *Homotopy limit functors on model categories and homotopical categories*, Mathematical Surveys and Monographs, 113, American Mathematical Society, Providence, RI, 2004. viii+181 pp. W. Dwyer, J. Spalinski, *Homotopy theories and model categories*, Handbook of algebraic topology, 73–126, North-Holland, Amsterdam, 1995. T. Goodwillie, Calculus III, *Taylor series*, Geom. Topol. 7 (2003) 645–711. P. Hirschhorn, *Models categories and their localizations*, Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, 2003. P. Hirschhorn, *Notes on homotopy colimits and homotopy limits*, Work in progress available online at P. Hirschhorn’s homepage: $http://www-math.mit.edu/~psh/\#hocolim$. M. Hovey, *Model categories*, Mathematical Surveys and Monographs, vol. 63, *American Mathematical Society, Providence, RI*, 1999. B. A. Munson, *Introduction to the manifold calculus of Goodwillie-Weiss*, Morfismos, Vol. 14, No. 1, 2010, pp. 1–50. D. Pryor, *Special open sets in manifold calculus*, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 89–103. P. A. Songhafouo Tsopméné, D. Stanley, *Very good homogeneous functors in manifold calculus*, Preprint. 2017, arXiv:1705.0120. P. A. Songhafouo Tsopméné, D. Stanley, *Classification of homogeneous functors in manifold calculus*, Preprint. 2018, arXiv:1807.06120. M. Weiss, *Embedding from the point of view of immersion theory I*, Geom. Topol. 3 (1999), 67–101.
*E-mail address: pso748@uregina.ca*
*E-mail address: donald.stanley@uregina.ca*
[^1]: In this paper the word cofunctor means contravariant functor
[^2]: One of the axioms of the definition of a *simplicial model category* ${\mathcal{M}}$ [@hir03 Definition 9.1.6] says that for an object $Y \in {\mathcal{M}}$ and every simplicial set $K$, there exists an object $Y^K$ of ${\mathcal{M}}$
[^3]: If $P_i(W, {\mathcal{X}_A})$ is not fibrant, one can always substitute it by its fibrant replacement
|
---
abstract: 'Direct imaging of exoplanets requires both high contrast and high spatial resolution. Here, we present the first scientific results obtained with the newly commissioned Apodizing Phase Plate coronagraph (APP) on VLT/NACO. We detected the exoplanet $\beta$ Pictoris b in the narrow band filter centered at 4.05 $\mu$m (NB4.05). The position angle ($209.13^{\circ}\pm2.12^{\circ}$) and the projected separation to its host star ($0.''''354\pm0''''.012$, i.e., 6.8 $\pm$ 0.2 AU at a distance of 19.3 pc) are in good agreement with the recently presented data from @lagrange2010. Comparing the observed NB4.05 magnitude of 11.20 $\pm$ 0.23 mag to theoretical atmospheric models we find a best fit with a 7–10 M$_{\rm Jupiter}$ object for an age of 12 Myr, again in agreement with previous estimates. Combining our results with published $L''$ photometry we can compare the planet’s $[L''-{\rm NB4.05}]$ color to that of cool field dwarfs of higher surface gravity suggesting an effective temperature of $\sim$1700 K. The best fit theoretical model predicts an effective temperature of $\sim$1470 K, but this difference is not significant given our photometric uncertainties. Our results demonstrate the potential of NACO/APP for future planet searches and provides independent confirmation as well as complementary data for $\beta$ Pic b.'
author:
- 'Sascha P. Quanz, Michael R. Meyer'
- Matthew Kenworthy
- 'Julien H. V. Girard'
- Markus Kasper
- 'Anne-Marie Lagrange'
- Daniel Apai
- Anthony Boccaletti
- 'Mickaël Bonnefoy, Gael Chauvin'
- 'Philip M. Hinz'
- Rainer Lenzen
title: |
First results from VLT NACO Apodizing Phase Plate:\
4–micron images of the exoplanet $\beta$ Pictoris b
---
Introduction
============
The first images of exoplanets around stars were published in the last two years (HR8799 b, c, d – @marois2008; Fomalhaut b – @kalas2008; 1RXS J1609-2105 – @lafreniere2008 [@lafreniere2010]) following the discovery of a planetary mass companion to the brown dwarf 2MASS1207-3932 in 2005 [@chauvin2005]. For the planets around HR8799 follow-up observations at 3–5$\mu$m revealed evidence for non-equilibrium chemistry in the planetary atmospheres [@janson2010; @hinz2010] highlighting the potential of multi-wavelength imagery of extrasolar planetary systems to constrain physical properties in comparison to theoretical atmospheric models.
@lagrange2009a detected a planetary mass candidate around the young A-type star $\beta$ Pictoris [A5V, 12$^{+8}_{-4}$ Myr, 19.3$\pm$0.2 pc, 1.75 M$_{\sun}$; @zuckerman2001; @crifo1997] in $L'$ images from 2003. After a non-detection based on data obtained in early 2009 [@lagrange2009b], @lagrange2010 recently re-detected the planet $\beta$ Pictoris b which had moved on the other side of the star. The authors constrained the semi-major axis to 8–15 AU (the smallest of all imaged exoplanets today) and derived a mass estimate of 9$\pm$3 M$_{\rm Jupiter}$.
Here, using the newly commissioned apodizing phase plate (APP) on VLT/NACO [@kenworthy2010; @codona2006], we confirm the detection of $\beta$ Pictoris b with an independent data set and provide complementary photometry.
Observations and data reduction
===============================
The data were obtained on 2010-04-03, during the commissioning of the APP with the AO high-resolution camera NACO [@lenzen2003; @rousset2003] mounted on ESO’s UT4 on Paranal. The APP is designed to work in the 3–5 $\mu$m range where it enhances the contrast capabilities between $\sim$2–7 $\lambda/D$ on one side of the PSF. We used the L27 camera ($\sim$ 27 mas pixel$^{-1}$) with the visible wavefront sensor. All images were taken in pupil stabilized mode with the NB4.05 filter ($\lambda_{c}=4.05\,\mu$m, $\Delta\lambda=0.02\,\mu$m)[^1]. We used the “cube mode” where all data frames are saved individually. To ensure that no frames were lost we only read out a 512$\times$512 pixel sub-array of the detector. As we knew the position angle where the planet was to be expected [@lagrange2009a; @lagrange2009b] we rotated the camera by $140^{\circ}$ so that the planet would appear in the “clean”, high-contrast side of the APP PSF (Fig. \[finalimage\], left panel).
We obtained six data cubes of $\beta$ Pictoris and directly thereafter six data cubes of the PSF reference star HR2435. HR2435 has already been used in previous studies as reference star for $\beta$ Pictoris [e.g., @lagrange2009a] as one can obtain data at matching parallactic angles. Also, the star is close in the sky and has a similar brightness providing comparable AO correction. For both sources, each data cube was taken at a slightly different dither position following a 3–point dither pattern which was repeated twice. The on-source integration time was 20 minutes each. Table \[observations\] summarizes the observations and the observing conditions. We chose to saturate the core of the stellar PSFs but we note that the APP reduces the peak flux in the PSF core by roughly 40% so that the comparatively long exposure time of 1 sec did not lead to saturation effects outside of the inner $\sim$5 pixels.
For the photometric calibration we also obtained unsaturated images of $\beta$ Pictoris ($\sim$50% Full Well) prior to the science observations described above. We used the same observing strategy but decreased the detector integration time to 0.2 sec.
The data reduction was done using self-developed IDL routines. The following steps were applied to all three datasets (i.e., $\beta$ Pictoris unsaturated and saturated images, HR2435 saturated images). First, in order to eliminate the sky background emission, we subtracted from each individual frame the corresponding frame from another cube taken at a different dither position (e.g., Cube 2 frame 10 - Cube 3 frame 10) and vice versa. As the first two frames in each cube always showed a higher detector noise level they were disregarded. We also disregarded frames where the AO correction was poor (flux measured in the PSF core less than 50% compared to the previous frame). Then, bad pixels and cosmic ray hits deviating by more than 3-$\sigma$ in a 5$\times$5 pixels box were replaced with the mean of the surrounding pixels. After this we continued as follows: The unsaturated $\beta$ Pictoris images were aligned and cube-wise median combined, yielding six final images on which we performed photometry (Fig. \[finalimage\], left panel). The alignment of the images was done using cross-correlation where the optimal shift between two images was determined with an accuracy of 0.1 pixel. The same procedure was applied to the saturated HR2435 images yielding six individual reference PSF images (i.e., one per cube; Fig. \[finalimage\], middle panel). The individual frames were not de-rotated to the same field orientation before the combination, so that all static telescope aberrations remained constant throughout the cubes and in the final images. For the saturated frames of $\beta$ Pictoris we determined the parallactic angle for each individual frame by linear interpolation between the first frame and the final frame in each cube[^2]. Using cross-correlation, all frames from all cubes were then aligned to the same reference image for which we used the final, median-combined image of the first cube of the unsaturated $\beta$ Pictoris exposures. Thereafter we aligned, scaled and subtracted one of the six final HR2435 reference PSFs from each individual saturated $\beta$ Pictoris frame. The choice of the reference PSF was a trade-off between matching parallactic angle and observing conditions in the individual cubes. The best results (i.e., strongest signal of the planet, least residuals) were obtained using reference PSF 2 for the frames in cubes one and two, PSF 3 for cubes three and four, and PSF 5 for cubes five and six. The scaling was done in the bright, righthand side of the PSF where the diffraction rings were clearly visible. We scaled the reference PSF by minimizing the mean in the subtracted images in a semi-annulus centered on the star with an inner radius of 15 pixels, an outer radius of 27 pixels (i.e., covering the 3$^{\rm rd}$ and 4$^{\rm th}$ Airy ring) and an opening angle of 150$^\circ$. Finally, we derotated all PSF-subtracted frames to match the parallactic angle of the first frame and created a median combined final image.
Results
=======
The detection of $\beta$ Pic b
------------------------------
The right-hand panel in Fig. \[finalimage\] shows the final image of the data reduction process. The exoplanet is clearly detected in the left side of the image. We conducted several tests to exclude the possibility of a false detection: (1) We created final images for each individual cube by derotating and combining the respective PSF subtracted images. The planet was detected in the final images of all six cubes. (2) Comparing the position angle of the exoplanet in the final images of the individual cubes revealed the expected rotation introduced by the pupil tracking mode. (3) Speckles can appear along the spider arms holding the secondary mirror of the telescope, but the nearest arm of the telescope spider was $\ga 20^{\circ}$ away from the position of the planet. (4) We implemented a second independent data reduction pipeline based on the LOCI algorithm [@lafreniere2007b]. As the small field rotation in our data set did not allow us to construct a reference PSF directly from the $\beta$ Pic images [Angular Differential Imaging; e.g., @marois2006] we used the HR2435 data set. For each $\beta$ Pic frame we constructed a reference PSF based on a linear combination of the final HR2435 images so that the residuals in the high-contrast side of the PSF were minimized. With this approach we also recovered the signal of the exoplanet at the same position.
The projected separation between star and planet is $0.''354\pm0''.012$ (i.e., 6.8 $\pm$ 0.2 AU at a distance of 19.3 pc) and the position angle is $209.13^{\circ}\pm2.12^{\circ}$ (East of North). The values are mean values derived from the six final PSF subtracted cube images (see, test (2) above) and the errors include the corresponding standard deviation of the mean and systematic uncertainties from the alignment of the images. The position of the star in the unsaturated reference image used for the alignment and the position of the planet in the six final images were determined by fitting a 2-dimensional Gaussian to the respective source. For our final astrometric numbers we had to rely on the plate scale and field orientation provided in the header of the images as we did not observe an astrometric calibrator. However, potential deviations are expected to be small compared to our errors [@lagrange2009a]. Comparing the final numbers to the results from @lagrange2010 we find that the position angle is in very good agreement and the separation appears to have increased as expected from the planet’s orbital motion. However, the short time baseline between our data and the data from @lagrange2010 does not allow us to put any new constraints on the planet’s actual orbit.
Photometry and color of $\beta$ Pic b
-------------------------------------
To derive the relative photometry between the planet and its star we used an aperture with a radius of 2 pixels and computed the mean flux of $\beta$ Pictoris in the six final unsaturated images and the mean flux of the planet in the six final PSF subtracted cube images. The flux derived from the unsaturated images was scaled to account for the difference in the integration time. Each flux measurement was corrected for residual background emission by measuring and subtracting the mean flux per pixel in a semi-annulus centered on the planet as well as on the star. The semi-annulus covered only the high-contrast side of the objects and had an inner and outer radius of 5 and 8 pixel for the planet and of 18 and 25 pixel for the star. The contrast between the star and the planet in the NB4.05 filter amounts to $\Delta m_{\rm{4.05}}= 7.75\pm0.23$ mag. The error was derived from error propagation taking into account the standard deviations of the mean of both the flux of the planet and the flux of the unsaturated $\beta$ Pictoris images.
In the $L'$ filter the contrast between the planet and the star is $\Delta m_{\rm{L'}}= 7.7\pm0.3$ mag[^3] which, combined with the stars apparent magnitude of $L'=3.454\pm0.003$ mag [@bouchet1991], translates into an apparent magnitude for the exoplanet of $m_{\rm{L'}}=11.15\pm0.3$ mag. Since an A5V star has an $[L'-M]$ color of $\sim$0.01 mag we can assume that the intrinsic $[L'-{\rm NB4.05}]$ color of $\beta$ Pic is negligible. Based on the observed contrast in the NB4.05 filter the exoplanet’s apparent magnitude is then $m_{\rm{NB4.05}}=11.20\pm0.23$ and its color $[L'-{\rm NB4.05}] = -0.05\pm0.38$ mag.
Color comparison to cool field dwarfs
-------------------------------------
@cushing2008 published NIR spectra for cool L and T field dwarfs that cover the NB4.05 filter but terminate before the long wavelengths cut-off of ESO’s $L'$ filter. However, for some of these objects @leggett2002 published $L'$ photometry. To obtain the magnitudes in the NB4.05 filter we used the filter transmission curve and convolved it with the spectra from @cushing2008. The zero point of the filter was derived from a Kurucz model of an A0V star. Using the $L'$ filter transmission curve we first scaled the Kurucz model to the observed flux density of Vega in the $L'$ filter. For this we assumed $m_{\rm L'}^{{\rm Vega}}=m_{\rm V}^{{\rm Vega}}=0.03$ mag and $F_\nu^{{\rm Vega}}=246.105$ Jy or $F_\lambda^{{\rm Vega}}=5.219\cdot10^{-11}$ W m$^{-2}$ ${\mu}$m$^{-1}$ in the ESO $L'$ filter. Then we measured the flux density of the A0V star in the NB4.05 filter and derived the zero point assuming the star has the same magnitude here. The derived flux densities of the field objects could then be converted into magnitudes and we determined the $[L'-{\rm NB4.05}]$ color for the objects. The error in the magnitude in the NB4.05 filter was derived from computing a minimum and a maximum magnitude by either subtracting or adding the mean error of the spectra in that wavelength range. Whatever resulted in a bigger deviation from the mean magnitude was used as (conservative) error bar. Finally, as the $L'$ photometry from @leggett2002 was done in the MKO photometric system and not in the ESO system we applied a first order correction by subtracting the zero point offset of 0.02 mag[^4] from the quoted photometric measurements.
In Fig. \[field\_objects\] we plot the spectral type of the field objects against their color (black points). Data were available for spectral types L1, L3.5, L7.5, T2 and T5[^5]. Fitting a straight line to the data points we find that the color of $\beta$ Pic b is most similar to that of a field object with a spectral type of L4. However, spectral types from L0 to L7 are consistent with the observed color given the large error bars. For field objects around spectral type L4, @cushing2008 found effective temperatures of 1700 K which is the same temperature one obtains using the spectral type – temperature relation derived by @golimowski2004. We note, however, that the surface gravity of the young exoplanet is expected to be much lower than that of the field objects [log $g\approx4.0$ rather than 4.5–5.5 for field L/T dwarfs; @cushing2008].
Comparison to atmospheric/evolutionary models
---------------------------------------------
In Fig. \[model\] we compare the observed magnitudes to theoretical isochrones for low-mass objects based on the DUSTY models from @chabrier2000. These evolutionary models are based on a spherical collapse model where the objects start initially with very large radii and their internal energy is dominated by contraction. Using a distance of 19.3 pc and assuming an age of 12 Myr we find a mass of 7–10 M$_{\rm Jupiter}$ which agrees with the 6–12 M$_{\rm Jupiter}$ derived from the $L'$ filter. Here, the mass range is only determined by the photometric uncertainties. The most likely mass based on the NB4.05 magnitude is 8 M$_{\rm Jupiter}$ which corresponds to $T_{\rm eff}\approx1470$ K and log $g\approx4.0$. Assuming an age of 20 Myr this mass would increase to 11 M$_{\rm Jupiter}$ with $T_{\rm eff}\approx1425$ K and log $g\approx4.1$, while for 8 Myr we would estimate 6 M$_{\rm Jupiter}$ with $T_{\rm eff}\approx1380$ K and log $g\approx3.9$.
Contrast curve analysis
-----------------------
Using the final image (right panel, Fig. \[finalimage\]) we computed a contrast curve showing the detection limit for potential additional companions as a function of radius (Fig. \[contrast\_curve\]). To derive the curve we computed the standard deviation $\sigma$ of the pixel values in semi-annuli (5 pixel width) as a function of radial distance from the center in the high-contrast side of the image. The signal and the noise of a hypothetical companion can be written as $$S=n\cdot\eta\cdot\sigma\quad$$ and $$N\approx\sqrt{n}\cdot\sigma$$ with $n$ being the number of pixels in the chosen photometric aperture (here: $n=\pi\cdot2^2\approx12.56$) and $\eta$ being a factor that depends of the chosen S/N ratio one wants to achieve. For a S/N of 5, $\eta$ can be derived from $$\frac{S}{N}=5\approx\sqrt{n}\cdot\eta$$ The ratio between the signal of a planet with a S/N of 5 (equation (1)) and the mean flux of beta Pictoris computed in a 2 pixel aperture relates to a difference in magnitude which we compute as a function of $\sigma(r)$ (Fig. \[contrast\_curve\]). As the noise of the residuals is not perfectly Gaussian in all annuli [see also, e.g., @kasper2007] we chose to plot the conservative 5-$\sigma$ limit only. A more sophisticated analysis, i.e., the insertion of fake planets, may result in a somewhat different contrast curve.
Discussion
==========
$\beta$ Pictoris b is an exoplanet whose estimated mass, age and separation are in agreement with predictions from core accretion planetary formation models [see, @lagrange2010 and references therein]. It is interesting to note that dynamical studies analyzing the sub-structure of the remnant debris disk around $\beta$ Pic predicted the mass and the orbit of the planetary companion before it was imaged [e.g., @mouillet1997; @augereau2001]. It was shown that an object with a mass significantly higher than $\sim$10 M$_{\rm Jupiter}$ would create much larger disk asymmetries. Thus, the dynamical analysis provides further support for the mass estimates derived from the observed photometry and the evolutionary models. This in turn led @lagrange2010 to the conclusion that the “cold start” evolutionary models of young, giant planets [@fortney2008] are not in agreement with $\beta$ Pic b. These models predict the planet to be significantly less luminous for its age and dynamically predicted mass. Our new flux point at 4.05 $\mu$m confirms the brightness of the source and supports the argument from @lagrange2010. However, given the comparatively large uncertainties in the observed fluxes we refrain from undertaking a more comprehensive comparison to additional sets of evolutionary and atmospheric models [e.g., @burrows2006]. For instance for an assumed age of 12 Myr the models from @chabrier2000 predict \[$L'-{\rm NB4.05}$\] $\le$ 0.3 mag for $\emph{all}$ objects between 4 and 100 M$_{\rm Jupiter}$. Also, the apparent difference in effective temperature of $\beta$ Pic b derived in sections 3.3. and 3.4. is not statistically significant. Given the error bars in Fig. \[field\_objects\], $T_{\rm eff}$ as high as $\sim$1700 K is still consistent with the observed magnitudes within 2–$\sigma$.
Future observations at different wavelengths will help us to better constrain the atmospheric parameters and composition of $\beta$ Pic b and to check whether the preliminary result, that the models predict a lower $T_{\rm eff}$ than the comparison to field dwarfs, persists. A frequent monitoring of the exoplanet’s position will eventually allow us to determine the orbital parameters with higher accuracy.
This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. We are indebted to Michael Cushing, Isabelle Baraffe, Udo Wehmeier and the ESO staff on Paranal, in particular Jared O’Neal, for their support. We also thank the referee for useful comments and suggestions.
[*Facilities:*]{}
[32]{} natexlab\#1[\#1]{}
, J. C., [Nelson]{}, R. P., [Lagrange]{}, A. M., [Papaloizou]{}, J. C. B., & [Mouillet]{}, D. 2001, , 370, 447
, A. S., [Santerne]{}, A., [Alonso]{}, R., [Gazzano]{}, J., [Havel]{}, M., [Aigrain]{}, S., [Auvergne]{}, M., [Baglin]{}, A., [Barbieri]{}, M., [Barge]{}, P., [Benz]{}, W., [Bord[é]{}]{}, P., [Bouchy]{}, F., [Bruntt]{}, H., [Cabrera]{}, J., [Cameron]{}, A. C., [Carone]{}, L., [Carpano]{}, S., [Csizmadia]{}, S., [Deleuil]{}, M., [Deeg]{}, H. J., [Dvorak]{}, R., [Erikson]{}, A., [Ferraz-Mello]{}, S., [Fridlund]{}, M., [Gandolfi]{}, D., [Gillon]{}, M., [Guenther]{}, E., [Guillot]{}, T., [Hatzes]{}, A., [H[é]{}brard]{}, G., [Jorda]{}, L., [Lammer]{}, H., [Lanza]{}, A. F., [L[é]{}ger]{}, A., [Llebaria]{}, A., [Mayor]{}, M., [Mazeh]{}, T., [Moutou]{}, C., [Ollivier]{}, M., [P[ä]{}tzold]{}, M., [Pepe]{}, F., [Queloz]{}, D., [Rauer]{}, H., [Rouan]{}, D., [Samuel]{}, B., [Schneider]{}, J., [Tingley]{}, B., [Udry]{}, S., & [Wuchterl]{}, G. 2010, ArXiv e-prints
, P., [Schmider]{}, F. X., & [Manfroid]{}, J. 1991, , 91, 409
, A., [Sudarsky]{}, D., & [Hubeny]{}, I. 2006, , 640, 1063
, G., [Baraffe]{}, I., [Allard]{}, F., & [Hauschildt]{}, P. 2000, , 542, 464
, G., [Lagrange]{}, A., [Dumas]{}, C., [Zuckerman]{}, B., [Mouillet]{}, D., [Song]{}, I., [Beuzit]{}, J., & [Lowrance]{}, P. 2005, , 438, L25
, J. L., [Kenworthy]{}, M. A., [Hinz]{}, P. M., [Angel]{}, J. R. P., & [Woolf]{}, N. J. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6269, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
, F., [Vidal-Madjar]{}, A., [Lallement]{}, R., [Ferlet]{}, R., & [Gerbaldi]{}, M. 1997, , 320, L29
, M. C., [Marley]{}, M. S., [Saumon]{}, D., [Kelly]{}, B. C., [Vacca]{}, W. D., [Rayner]{}, J. T., [Freedman]{}, R. S., [Lodders]{}, K., & [Roellig]{}, T. L. 2008, , 678, 1372
, S., [Bond]{}, I. A., [Gould]{}, A., [Koz[ł]{}owski]{}, S., [Miyake]{}, N., [Gaudi]{}, B. S., [Bennett]{}, D. P., [Abe]{}, F., [Gilmore]{}, A. C., [Fukui]{}, A., [Furusawa]{}, K., [Hearnshaw]{}, J. B., [Itow]{}, Y., [Kamiya]{}, K., [Kilmartin]{}, P. M., [Korpela]{}, A., [Lin]{}, W., [Ling]{}, C. H., [Masuda]{}, K., [Matsubara]{}, Y., [Muraki]{}, Y., [Nagaya]{}, M., [Ohnishi]{}, K., [Okumura]{}, T., [Perrott]{}, Y. C., [Rattenbury]{}, N., [Saito]{}, T., [Sako]{}, T., [Sato]{}, S., [Skuljan]{}, L., [Sullivan]{}, D. J., [Sumi]{}, T., [Sweatman]{}, W., [Tristram]{}, P. J., [Yock]{}, P. C. M., [The MOA Collaboration]{}, [Bolt]{}, G., [Christie]{}, G. W., [DePoy]{}, D. L., [Han]{}, C., [Janczak]{}, J., [Lee]{}, C., [Mallia]{}, F., [McCormick]{}, J., [Monard]{}, B., [Maury]{}, A., [Natusch]{}, T., [Park]{}, B., [Pogge]{}, R. W., [Santallo]{}, R., [Stanek]{}, K. Z., [The [$\mu$]{}FUN Collaboration]{}, [Udalski]{}, A., [Kubiak]{}, M., [Szyma[ń]{}ski]{}, M. K., [Pietrzy[ń]{}ski]{}, G., [Soszy[ń]{}ski]{}, I., [Szewczyk]{}, O., [Wyrzykowski]{}, [Ł]{}., [Ulaczyk]{}, K., & [The OGLE Collaboration]{}. 2009, , 698, 1826
, J. J., [Marley]{}, M. S., [Saumon]{}, D., & [Lodders]{}, K. 2008, , 683, 1104
, D. A., [Leggett]{}, S. K., [Marley]{}, M. S., [Fan]{}, X., [Geballe]{}, T. R., [Knapp]{}, G. R., [Vrba]{}, F. J., [Henden]{}, A. A., [Luginbuhl]{}, C. B., [Guetter]{}, H. H., [Munn]{}, J. A., [Canzian]{}, B., [Zheng]{}, W., [Tsvetanov]{}, Z. I., [Chiu]{}, K., [Glazebrook]{}, K., [Hoversten]{}, E. A., [Schneider]{}, D. P., & [Brinkmann]{}, J. 2004, , 127, 3516
, P. M., [Rodigas]{}, T. J., [Kenworthy]{}, M. A., [Sivanandam]{}, S., [Heinze]{}, A. N., [Mamajek]{}, E. E., & [Meyer]{}, M. R. 2010, , 716, 417
, M., [Bergfors]{}, C., [Goto]{}, M., [Brandner]{}, W., & [Lafreni[è]{}re]{}, D. 2010, , 710, L35
, P., [Graham]{}, J. R., [Chiang]{}, E., [Fitzgerald]{}, M. P., [Clampin]{}, M., [Kite]{}, E. S., [Stapelfeldt]{}, K., [Marois]{}, C., & [Krist]{}, J. 2008, Science, 322, 1345
, M., [Apai]{}, D., [Janson]{}, M., & [Brandner]{}, W. 2007, , 472, 321
, M. A., [Codona]{}, J. L., [Hinz]{}, P. M., [Angel]{}, J. R. P., [Heinze]{}, A., & [Sivanandam]{}, S. 2007, , 660, 762
, M. A., [Quanz]{}, S. P., [Meyer]{}, M. R., [Kasper]{}, M. E., [Lenzen]{}, R., [Codona]{}, J. L., [Girard]{}, J. H. V., & [Hinz]{}, P. M. 2010, ArXiv e-prints
, D., [Jayawardhana]{}, R., & [van Kerkwijk]{}, M. H. 2008, , 689, L153
—. 2010, ArXiv e-prints
, D., [Marois]{}, C., [Doyon]{}, R., [Nadeau]{}, D., & [Artigau]{}, [É]{}. 2007, , 660, 770
, A., [Gratadour]{}, D., [Chauvin]{}, G., [Fusco]{}, T., [Ehrenreich]{}, D., [Mouillet]{}, D., [Rousset]{}, G., [Rouan]{}, D., [Allard]{}, F., [Gendron]{}, [É]{}., [Charton]{}, J., [Mugnier]{}, L., [Rabou]{}, P., [Montri]{}, J., & [Lacombe]{}, F. 2009, , 493, L21
, A., [Kasper]{}, M., [Boccaletti]{}, A., [Chauvin]{}, G., [Gratadour]{}, D., [Fusco]{}, T., [Ehrenreich]{}, D., [Apai]{}, D., [Mouillet]{}, D., & [Rouan]{}, D. 2009, , 506, 927
Lagrange, A.-M., Bonnefoy, M., Chauvin, G., Apai, D., Ehrenreich, D., Boccaletti, A., Gratadour, D., Rouan, D., Mouillet, D., Lacour, S., & Kasper, M. 2010, Science, science.1187187
, S. K., [Golimowski]{}, D. A., [Fan]{}, X., [Geballe]{}, T. R., [Knapp]{}, G. R., [Brinkmann]{}, J., [Csabai]{}, I., [Gunn]{}, J. E., [Hawley]{}, S. L., [Henry]{}, T. J., [Hindsley]{}, R., [Ivezi[ć]{}]{}, [Ž]{}., [Lupton]{}, R. H., [Pier]{}, J. R., [Schneider]{}, D. P., [Smith]{}, J. A., [Strauss]{}, M. A., [Uomoto]{}, A., & [York]{}, D. G. 2002, , 564, 452
, R., [Hartung]{}, M., [Brandner]{}, W., [Finger]{}, G., [Hubin]{}, N. N., [Lacombe]{}, F., [Lagrange]{}, A., [Lehnert]{}, M. D., [Moorwood]{}, A. F. M., & [Mouillet]{}, D. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4841, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. [M. Iye & A. F. M. Moorwood]{}, 944–952
, C., [Lafreni[è]{}re]{}, D., [Doyon]{}, R., [Macintosh]{}, B., & [Nadeau]{}, D. 2006, , 641, 556
, C., [Macintosh]{}, B., [Barman]{}, T., [Zuckerman]{}, B., [Song]{}, I., [Patience]{}, J., [Lafreni[è]{}re]{}, D., & [Doyon]{}, R. 2008, Science, 322, 1348
, D., [Larwood]{}, J. D., [Papaloizou]{}, J. C. B., & [Lagrange]{}, A. M. 1997, , 292, 896
, G., [Lacombe]{}, F., [Puget]{}, P., [Hubin]{}, N. N., [Gendron]{}, E., [Fusco]{}, T., [Arsenault]{}, R., [Charton]{}, J., [Feautrier]{}, P., [Gigan]{}, P., [Kern]{}, P. Y., [Lagrange]{}, A., [Madec]{}, P., [Mouillet]{}, D., [Rabaud]{}, D., [Rabou]{}, P., [Stadler]{}, E., & [Zins]{}, G. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4839, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. [P. L. Wizinowich & D. Bonaccini]{}, 140–149
, S. & [Santos]{}, N. C. 2007, , 45, 397
, B., [Song]{}, I., [Bessell]{}, M. S., & [Webb]{}, R. A. 2001, , 562, L87
[lll]{} UT start & 00:39:31.1 & 01:34:21.3\
NDIT $\times$ DIT & 200 $\times$ 1 s & 200 $\times$ 1 s\
NINT & 6 & 6\
Parallactic angle start & 69.469$^{\circ}$ & 69.451$^{\circ}$\
Parallactic angle end & 74.492$^{\circ}$ & 74.499$^{\circ}$\
Airmass & 1.38–1.44 & 1.42–1.48\
Typical DIMM seeing & 0.70–0.90$''$ & 0.60–0.75$''$\
$\langle EC \rangle$ & 29.5–44.4 % & 22.7–42.9 %\
$\langle \tau_0 \rangle$ & 4.3–7.9 ms& 4.0–7.8 ms\
[^1]: The APP has been optimized to work with the NB4.05 and IB4.05 filter but it can also be used with the $L'$ filter [@kenworthy2010].
[^2]: Per default, only the parallactic angle at the beginning and at the end of each cube are saved in the fits header in NACO’s cube mode.
[^3]: This figure was obtained from the October 2009 dataset in @lagrange2010 as well as from the November 2003 dataset in @lagrange2009a.
[^4]: see, http://www.gemini.edu/sciops/instruments/midir-resources/imaging-calibrations/fluxmagnitude-conversion
[^5]: Optical spectral types differ slightly from NIR spectral types for 3 sources. However, the outcome of this analysis remains unchanged if we use the NIR spectral types instead.
|
8.0in 5.8in
.3in
\
1.0cm
A. N. Das[^1] and Jayita Chatterjee[^2]
0.50cm
[*Saha Institute of Nuclear Physics\
1/AF Bidhannagar, Calcutta 700064, India*]{}\
1.0cm
PACS No.71.38. +i, 63.20.kr 1.0cm
[**Abstract**]{}
0.5cm
A convergent perturbation method using modified Lang Firsov transformation is developed for a two-site single-polaron system. The method is applicable for the entire range of the electron-phonon coupling strength from the antiadiabatic limit to the intermediate region of hopping. The single-electron energies, oscillator wave functions and correlation functions, calculated using this method, are in good agreement with the exact results.
[**1. Introduction**]{}
0.3cm
The interaction of conduction electrons with lattice vibrations is described by the so called electron-phonon problem. The Holstein model [@Hol] is one of the fundamental models which has been studied widely in this context. The model consists of a one-electron hopping term, Einstein phonons at each site and a site-diagonal interaction term which couples the electron density and ionic displacements at a given site. For weak electron-phonon (e-ph) coupling the frequency of the phonons and the effective mass of the the electron are renormalized, which are described by the Migdal approximation [@Mig]. For large e-ph coupling the electrons are self-trapped in the lattice deformation producing a small polaron. The motion of the electron is then accompanied by the lattice deformation. This results in a large effective mass or reduced effective hopping of the dressed electrons (polarons). The Lang-Firsov (LF) method based on the LF canonical [@LF] transformation works in this strong coupling region in the antiadiabatic limit. For weak coupling the Migdal approximation is satisfactory. However, no conventional analytical method exists at present which is beleived to describe a Holstein model for the entire range of the coupling strength. So, it would be useful to develop or identify an analytic method which could be applied to both the strong and weak coupling cases. Ranninger and Thibblin [@RT] made an exact diagonalization study of a two-site polaron problem and showed that the behavior of the polaron differs very much from that predicted by the classical LF method. Marsiglio [@Mar] extended those calculations to the bulk limit in one dimension by studying the Holstein model with one electron up to 16 site lattices. He concluded that for intermediate coupling strength neither the Migdal nor the small-polaron approximation is in quantitative agreement with the exact results. Kabanov and Ray [@KR] and Alexandrov $et~al.$ [@Alex] noted that for $t> \omega_0$ the adiabatic small-polaron approximation describes the ground state energy accurately except for intermediate coupling strength. Ranninger and Thibblin [@RT] and Marsiglio [@Mar] studied also the correlation functions using exact diagonalization technique with the finite size Holstein model and found that the results are non trivial and cannot be described by any conventional analytical method. Recently, de Mello and Ranninger [@MR] emphasized this point further.
The objective of the present work is to search for an analytical method which may be applicable reasonably well for the major range of e-ph coupling strength. For our study we consider a two-site one-electron Holstein model for which exact results are available [@RT]. Previously, we [@DC; @CD] investigated the ground state energy and the nature of polarons in a two-site and a four-site Holstein model using the modified Lang Firsov (modified LF) transformation and two-phonon coherent states and found that the energy obtained within such method is very close to the exact result. In this work we develop a perturbation expansion within the modified LF transformation and show that this expansion converges for the entire range of the coupling strength from the antiadiabatic limit to the intermediate region of hopping. The energy and the correlation functions calculated within our approach are almost identical with the exact results.
1.0cm
[**2. Formalism** ]{}
0.3cm
The Hamiltonian of a two-site one-electron Holstein model reads as $$\begin{aligned}
H = \sum_{i,\sigma} \epsilon n_{i \sigma} - \sum_{\sigma}
t (c_{1 \sigma}^{\dag} c_{2 \sigma} + c_{2 \sigma}^{\dag} c_{1 \sigma})
+ g \omega_0 \sum_{i,\sigma} n_{i \sigma} (b_i + b_i^{\dag})
+ \omega_0 \sum_{i} b_i^{\dag} b_i \end{aligned}$$ where $i$ =1 or 2, denotes the site. $c_{i\sigma}$ ($c_{i\sigma}^{\dag}$) is the annihilation (creation) operator for the electron with spin $\sigma$ at site $i$ and $n_{i \sigma}$ (=$c_{i\sigma}^{\dag} c_{i\sigma}$) is the corresponding number operator, $g$ denotes the e-ph coupling strength, $b_i$ and $b_{i}^{\dag}$ are the annihilation and creation operators, respectively, for the phonons corresponding to interatomic vibrations at site $i$, $\omega_0$ is the phonon frequency. In Hamiltonian (1) there is no spin dependent or spin reversal term so for the study of one-electron case the spin index is redundant. In the following we shall not use the spin index.
Introducing new phonon operators $a=~(b_1+b_2)/ \sqrt 2$ and $d=~(b_1-b_2)/\sqrt 2 $ the Hamiltonian is separated into two parts ($H=H_d + H_a)$ : $$\begin{aligned}
H_d = \sum_{i} \epsilon n_{i} - t (c_{1}^{\dag} c_{2} + c_{2}^{\dag} c_{1})
+ \omega_0 g_{+} (n_1-n_2) (d + d^{\dag})
+ \omega_0 d^{\dag} d \end{aligned}$$ and $$H_a = \omega_0 \tilde{a}^{\dag}\tilde{a} - \omega_0 n^2 g_{+}^2$$ where $g_{+}=g/\sqrt 2$, $\tilde{a}=a +ng_{+}$ and $\tilde{a}^{\dag}=a^{\dag} +ng_{+}$.
$H_a$ describes a shifted oscillator which couples only with the total number of electrons $n(=n_1+n_2)$, which is a constant of motion. The last term in Eq.(3) represents lowering of energy achieved through the lattice deformations of sites 1 and 2 by the total number of electrons.
$H_d$ represents an effective e-ph system where phonons directly couple with the electronic degrees of freedom. $H_d$ cannot be solved exactly by any analytical method. We now use the modified LF transformation where the lattice deformations are treated as variational parameters [@DC; @DS; @LS]. For the present system, $$\tilde{H_d} = e^R H_d e^{-R}$$ where $R =\lambda (n_1-n_2) ( d^{\dag}-d)$, $\lambda$ is a variational parameter and linearly related to the displacement of the d oscillator.
The transformed Hamiltonian is then obtained as $$\begin{aligned}
\tilde{H_d} &=& \omega_0 d^{\dag} d + \sum_{i} \epsilon_p n_{i} -
t [c_{1}^{\dag} c_{2}~ \rm{exp}(2 \lambda (d^{\dag}-d)) \nonumber\\
&+& c_{2}^{\dag} c_{1}~\rm{exp}(-2 \lambda (d^{\dag}-d))]
+ \omega_0 (g_{+} -\lambda) (n_1-n_2) (d + d^{\dag}) \end{aligned}$$ where $$\epsilon_p = \epsilon - \omega_0 ( 2 g_{+} - \lambda) \lambda$$
It may be mentioned that with an ordinary LF transformation, where one chooses a phonon basis of the oscillator with a fixed displacement ($\lambda= g_{+}$ for this case), one can diagonalize the Hamiltonian in absence of hopping. The hopping term, containing off-diagonal matrix elements in the new phonon basis, may then be treated within the perturbation approach in the strong coupling and antiadiabatic limit [@FK]. However, in order to develop a perturbation theory to be valid for the entire range of coupling strength one should consider a variational phonon basis such that the major part of the hamiltonian could be diagonalized for different values of the coupling strength. The modified LF transformation, where the phonon basis are formed by the oscillator with variable displacement, would serve this purpose.
For the single polaron problem we choose the basis set (for $\tilde{H_d}$) $$\begin{aligned}
|+,N \rangle = \frac{1}{\sqrt 2} (c_{1}^{\dag} + c_{2}^{\dag})
|0\rangle_e |N\rangle \nonumber\\ \\
|-,N \rangle = \frac{1}{\sqrt 2} (c_{1}^{\dag} - c_{2}^{\dag})
|0\rangle_e |N\rangle \nonumber\end{aligned}$$ where $|+\rangle$ and $|-\rangle$ are the bonding and antibonding electronic states and $|N\rangle$ denotes the $N$th excited oscillator state.
Note that the last term in Eq.(5) has only off-diagonal matrix elements connecting bonding and antibonding states with the change in phonon number by $\pm 1$, $$\begin{aligned}
\langle M , \pm| \omega_0 (g_{+} -\lambda) (n_1-n_2) (d + d^{\dag})
|\mp , N \rangle \nonumber \\
= (\sqrt N~ \delta_{M,N-1} +\sqrt{N+1}~ \delta_{M,N+1})\omega_0(g_{+}-\lambda)\end{aligned}$$ while the hopping term $H_t$= $- t [c_{1}^{\dag} c_{2}~ \rm{exp}(2 \lambda (d^{\dag}-d)) + c_{2}^{\dag}
c_{1}~\rm{exp}(-2 \lambda (d^{\dag}-d))]$ has both diagonal and off-diagonal elements in the chosen basis. The diagonal part of $H_t$ is given by, $$\begin{aligned}
\langle N, \pm|H_t|\pm , N\rangle=\mp t_e\sum_{i=0}^{N}\left[
\frac{(2\lambda)^{2i}}{i!}(-1)^i N_{C_i}\right]\end{aligned}$$ where $t_e=t~\rm{ exp}{(-2\lambda^2)}$ and $N_{C_i}=\frac{N!}{i!(N-i)!}$.
The diagonal part of the Hamiltonian $\tilde{H_d}$ (in the chosen basis) is considered as the unperturbed Hamiltonian ($H_0$) and the remaining part of the Hamiltonian $H_{1}= \tilde{H_{d}}-H_0$ is treated as a perturbation.
The unperturbed energy of the state $| \pm,N\rangle$is given by $$\begin{aligned}
E_{\pm,N}^{(0)}= \langle N,\pm|H_0|\pm, N \rangle=
N\omega_0 +\epsilon_p \mp t_e \left[ \sum_{i=0}^{N}
(\frac{(2\lambda)^{2i}}{i!} (-1)^i N_{C_i}\right]
\end{aligned}$$ The general off-diagonal matrix elements of $H_1$ between the two states $|\pm,N \rangle$ and $|\pm,M \rangle$ are calculated as (for $(N-M)>0$) $$\begin{aligned}
\langle N,\pm|H_1|\pm, M \rangle &=& P(N,M)~~~\rm{for}~\rm{even}~(N-M) \\
\nonumber \\
\langle N,\pm|H_1|\mp,M \rangle &=& P(N,M)+ \sqrt{N}\omega_0
(g_{+}-\lambda) \delta_{N,M+1}~~
\rm{for}~\rm{odd}~ (N-M). \end{aligned}$$\
where $$\begin{aligned}
P(N,M) = \mp t_e (2\lambda)^{N-M}
\sqrt{\frac{N!}{M!}} \left[ \frac{1}{(N-M)!}+\sum_{R=1}^{M}
[(-1)^R \right. \nonumber \\
\left. \frac{(2\lambda)^
{2R}}{(N-M+R)! R!}M(M-1)...(M-R+1) ] \right] \nonumber \end{aligned}$$ In the following we present the perturbation corrections to the energies and the correlation functions for the ground and the first excited state.
1.0cm
[**3. The energies and the correlation functions**]{}
0.3cm *[A.]{} Ground state :*
For the system considered, the state $|+\rangle|0\rangle$ has the lowest unperturbed energy, $ E_0^{(0)}=\epsilon_p-t_e$. The matrix element connecting this ground state and an excited state $|e,N\rangle$ is given by $$\begin{aligned}
\langle N,e|H_{1}|+,0\rangle&=&\left[ -t_e\frac{(2\lambda)^N}
{\sqrt{N!}}+\omega_0(g_{+}-\lambda)\delta_{N,1} \right] \delta_{e,-}
\hspace{.4cm} \rm{for~odd ~ N }\nonumber \\
&=& \left[ -t_e\frac{(2\lambda)^N}{\sqrt{N!}} \right] \delta_{e,+}
\hspace{.4cm} \rm{for~ even~ N }\end{aligned}$$
The first order correction to the ground state wave function is obtained as,
$$\begin{aligned}
|\psi_0^{(1)}\rangle &=&-\frac{\omega_0(g_{+}-\lambda)-2\lambda t_e}
{[\omega_0+2t_e(1-2\lambda^2)]}~|-,1\rangle +\sum_{N=2,4,..}
\frac{t_e(2\lambda)^N}{\sqrt{N!} (E_{+,N}^{(0)}- E_0^{(0)})}~|+,N\rangle \nonumber\\
&+& \sum_{N=3,5,..}\frac{t_e(2\lambda)^N}{\sqrt{N!}(E_{-,N}^{(0)}-
E_0^{(0)})}~|-,N\rangle \end{aligned}$$
where $E_{\pm,N}^{(0)}$ is the unperturbed energy of the state $|\pm,N\rangle$ as given in Eq.(10).
The second order correction to the ground state energy is given by $$\begin{aligned}
E_0^{(2)} &=& -\frac{t_e^2(2\lambda-\frac{\omega_0}{t_e}(g_{+}-
\lambda))^2} {\left[\omega_0+2t_e(1-2\lambda^2)\right]}
-\sum_{N=2}^{\infty}\frac{t_e^2 (2\lambda)^{2N}}
{N!(E_{e,N}^{(0)}-E_0^{(0)})} \end{aligned}$$ where e=+ or - for even and odd N, respectively.
Now, one has to make a proper choice of $\lambda$, hence choice for the displaced phonon basis, so that the perturbative expansion becomes convergent. In the usual modified LF method $\lambda$ is found out by minimizing the ground state energy of the system. Here we adopt that method and check whether it gives satisfactory results. From our previous studies [@DC; @CD; @DS] we know that $\lambda$ remains small as long as $g_{+}<1$, while for large values of $g_{+}$ (in the strong coupling limit) it approaches or attains the full LF value of $g_{+}$ (see Table I). For small values of $\lambda$ the perturbation series involving linear Frohlich type (polaron-phonon) interaction term ($\propto (g_{+}-\lambda)$) converges automatically, while that involving hopping, containing powers of $2 \lambda$, would converge provided $t<\omega_0$ or $t\sim$ $\omega_0$. For strong coupling, as $\lambda$ approaches to $g_{+}$, the Frohlich (polaron-phonon) interaction term almost vanishes as well as $t_e$ becomes very small so the perturbation series converges. Thus, it is expected that the perturbation method following the modified LF transformation would work satisfactorily in both the weak and strong coupling limits. In this work we have shown that it works reasonably well for whole range of the coupling strength for $t \le \omega_0$.
Following the spirit of the modified LF method the value of $\lambda$ is found out as $\lambda=\omega_0g_{+}/(\omega_0+2t_e)$ from minimization of the unperturbed ground state energy. It is interesting to note that for this particular choice of $\lambda$, the coefficient of $|-,1\rangle$ in Eq. (14) as well as the first term in the r. h. s. of Eq. (15) vanishes. In other words, the off-diagonal matrix element between the states $|+,0\rangle$ and $|-,1\rangle$ becomes zero for the modified LF choice of phonon basis which leads to small perturbation correction to the ground state within this method.
To check whether the perturbation series is converging properly we have calculated and computed the third order correction to the energy for the ground state. The third order correction ($ E_0^{(3)}$) to the ground state energy is given by,
$$\begin{aligned}
E_0^{(3)} &=& \sum_{k\neq 0}[ (H_1)_{0k}\sum_{m\neq 0}[ \frac{(H_1)_{km}
(H_1)_{m0}}{(E_{0}^{(0)}-E_k^{(0)})(E_{0}^{(0)}-E_m^{(0)})} ]~]\end{aligned}$$
where the subscript k,m denote the states $|\pm,N\rangle$ with the unperturbed energy $E_{\pm,N}^{(0)}$ and the subscript 0 refers to the ground state $|+,0\rangle$. The off-diagonal matrix elements of $H_1$ are calculated using Eqs.(11) and (12).
The ground state wave function of $\tilde{H_d}$ considering up to the second order corrections in perturbation is given by, $$\begin{aligned}
|\tilde {G} \rangle \equiv |+,0\rangle+|\psi_0^{(1)}
\rangle + |\psi_0^{(2)}\rangle \nonumber\end{aligned}$$\
which can be alternatively written as, $$\begin{aligned}
|\tilde{G} \rangle = |+,0\rangle +\sum_{N=2,4,..}
a_{N}|+,N \rangle +\sum_{N=1,3,..}b_{N} |-,N \rangle \end{aligned}$$\
The coefficients $a_{N}$ and $b_{N}$ are determined from Eq.(14) and the second order correction to the wave function. The normalized ground state wave function $|\tilde{G} \rangle_{N}$ is $$|\tilde{G} \rangle_{N}=\frac{1}{\sqrt{N_G}}|\tilde{G} \rangle$$ where $N_G$ is obtained as $$\begin{aligned}
N_G \equiv \langle \tilde{G}|\tilde{G}\rangle =1
+\sum_{N=2,4,..}a_{N}^2+\sum_{N=1,3..}b_{N}^2 \end{aligned}$$\
Within the modified LF method the ground state wave function for the $d$ oscillators is a displaced Gaussian $$\begin{aligned}
\phi(x)=\frac{1}{\pi^{\frac{1}{4}}}\rm{exp}[-(x-x_0)^2] \end{aligned}$$ where, $x_{0}= - (n_1-n_2) \sqrt{2}~\lambda$. Including the corrections due to the perturbation the ground state wave function for the $d$ oscillator is obtained as, $$\begin{aligned}
G(x)&\equiv& \tilde{G}(x-x_{0})= \langle x-x_{0}|0\rangle +\sum_{N=2,4..}a_N
\langle x-x_{0}|N\rangle \nonumber\\
&+& \sum_{N=1,3...}b_N\langle x-x_{0}|N\rangle \end{aligned}$$ Note that $G(x)$ and $\tilde{G}(x)$ are the ground state oscillator wave functions for $H_d$ and $\tilde{H_d}$, respectively. If the electron is located on site 1 then $x_{0}= -\sqrt{2}~\lambda$.
- Correlation function calculation:
The static correlation function $\langle n_1 u_{1}\rangle_{0}$ and $\langle n_1 u_{2}\rangle_{0}$, where $u_1$ and $u_2$ are the lattice deformations at site 1 and 2 respectively, produced by an electron at site 1, indicates the strength of polaron induced lattice deformation and their spread. These correlation functions are determined as $$\begin{aligned}
\langle n_{1} u_{1} \rangle_{0}&=&\frac{1}{2}
\left[-(g_{+} +\lambda) + \frac{A_0}{N_G}\right] \\
\langle n_{1} u_{2} \rangle_{0}&=&\frac{1}{2}
\left[-(g_{+}-\lambda)- \frac{A_0}{N_G}\right] \nonumber\end{aligned}$$\
where $$\begin{aligned}
A_0\equiv\langle \tilde{G} |n_1(d+d^{\dag})|\tilde{G}\rangle=\sum_
{N=1,3,..}b_N\left[
\sqrt{N}~a_{N-1}+\sqrt{N+1}~a_{N+1}\right] \nonumber\end{aligned}$$
0.7 cm [*B. The First Excited State:*]{} 0.3 cm
The unperturbed energies of the states $|+,1\rangle$ and $|-,0\rangle$ are ($\epsilon_p +\omega_0 -t_e(1-4\lambda^2)$) and ($\epsilon_p +t_e$), respectively. For $2t > \omega_0$, the energy of the state $|+,1\rangle$ is lower than that of $|-,0\rangle$ for $g_{+}=0$, while it is higher for large values of $g_{+}$ when $t_e$ becomes negligible. The off-diagonal matrix element of $\tilde{H_{d}}$ between these two states is nonzero. Crossing of the unperturbed energies of these two states at an intermediate value of $g_{+}$ and nonzero off-diagonal matrix elements requires that one should follow the degenerate perturbation theory. So, linear combinations of the states $|+,1\rangle$ and $|-,0\rangle$ are formed to obtain two new elements of basis states so that $\tilde{H_{d}}$ becomes diagonal in the sub-space spanned by these two states. The first excited state of $\tilde{H_{d}}$ is described by one of the linear combinations which has lower energy. The unperturbed first excited state is given by $$\begin{aligned}
|\psi_1^{(0)}\rangle= a |-,0\rangle + b|+,1\rangle\end{aligned}$$ The ratio ($c$) of the coefficients $a$ and $b$ and the unperturbed energy ($\alpha$) of the first excited state may be found out from the relation $$\begin{aligned}
c = \frac{\alpha-H_{11}}{H_{12}}
=\frac{H_{12}}{\alpha-H_{22}}\end{aligned}$$ where $H_{11}$, $H_{22}$, $H_{12}$ are the matrix elements of $\tilde{H_{d}}$ in the subspace of $|-,0\rangle$ and $|+,1\rangle$ and are given in the matrix form in the following,
$$\begin{aligned}
\begin{tabular}{c|c c}
{$\tilde{H_{d}}$} & {$|-,0\rangle$} & {$|+,1\rangle$}\\ \hline\\
{$\langle 0,-|$} & {$(\epsilon_p+t_e)$} &
{$2\lambda t_e+\omega_0(g_{+}-\lambda)$}\\ \\
{$\langle 1,+|$} & {$2\lambda t_e+\omega_0(g_{+}-\lambda)$} &
{$\omega_0+\epsilon_p-t_e(1-4\lambda^2)$}\\
\end{tabular} \end{aligned}$$
Eq.(23) gives two roots of $\alpha$, the lower value of $\alpha$ (say, $\alpha_1$) corresponds to the first excited state.
The first order correction to the first excited state wave function is obtained as, $$\begin{aligned}
|\psi_1^{(1)}\rangle &=&\frac{1}{\sqrt{1+c^2}} \left[ \sum_{N=2,4,..}
\frac{W_e}{(\alpha_1-E_{-,N}^{(0)})}~~ |-,N\rangle \right. \nonumber\\
&+&\left. \sum_{N=3,5..}\frac{W_o}{( \alpha_1-E_{+,N}^{(0)})}
~~|+,N\rangle \right]\end{aligned}$$ where $W_o=t_e\frac{(2\lambda)^N}{\sqrt{N!}}(1+2\lambda c-\frac{cN}
{2\lambda})$\
and $W_e= W_o +\sqrt{2} \omega_0 c(g_{+}-\lambda)\delta_{N,2}$
Second order correction to the first excited state energy is given by, $$\begin{aligned}
E_0^{(2)} = \frac{1}{1+c^2}\left[ \sum_{N=2,4,..}
\frac{|W_e|^2}{(\alpha_1-E_{-,N}^{(0)})}
+ \sum_{N=3,5..}\frac{|W_o|^2}{( \alpha_1-E_{+,N}^{(0)})} \right]\end{aligned}$$
1.0cm
[**4. Results and discussions**]{}
0.3cm
In this paper we report mainly the results of $t$=1.1 (in a scale of $\omega_0$ =1) for which exact results [@RT] are available. For the ground state the wave function and the energy have been calculated up to the second order and the third order in perturbation, respectively. For the numerical calculation we consider up to 25 phonon states in the series of Eqs. (14), (15) and (16). It is found that except for very high values of $g_{+}$ cosideration of 20 phonon states is more than sufficient, while for large values of $g_{+}$ (1.8-2.2) consideration of 25 phonon states is enough.
In Table-I we have shown the unperturbed energy, the second and third order corrections to the ground state energy. It is seen that the magnitude of the higher order corrections decreases rapidly which clearly indicates the convergence of the series and reasonability of our approach. The second and third order perturbation corrections to the energy are small in both the weak and strong coupling limits and appreciable only in the intermediate coupling limit $1.0\leq g_+\leq 1.3$ where higher order corrections may be necessary.
It may be noted that for lower values of $t$ (results for $t=1.1$ are shown here) the perturbation series converges more rapidly with smaller perturbation corrections. So, the present method based on modified LF transformation is expected to work very satisfactorily for $t < \omega_0$.
In Fig. 1 we have shown the single electron energies as a function of $g_{+}$ for the ground state (calculated up to the third order) and the first excited state (calculated up to the second order). The results are found to be almost identical with the exact results by Ranninger and Thibblin (within the resolution of Fig. 1 of Ref. 4). It should be mentioned that in a range $1.2<g_{+}<1.4$ the second order correction to the energy for the first excited state is $\sim 10-12\%$ of the unperturbed energy and so third order correction may be necessary in this region. For other regions the second order correction to the energy of the first excited state is small.
In Fig. 2 we have shown the ground state wavefunction for the $d$ oscillator as a function of position $x$ for different values of the e-ph coupling when the electron is located on site 1. For weak coupling ($g_{+}<1)$ the wave function shows displaced Gaussian like single peak where the displacement is given by the modified LF value, $x_{0}= - \sqrt{2}~\lambda$. However, for $g_{+}$=1.3 an additional prominent shoulder appears. For higher values of $g_{+}$ this shoulder takes the form of a broad peak. These results are completely consistent with the results obtained by Ranninger and Thibblin by exact diagonalization study [@RT].
In Fig. 3 we have plotted the variation of the correlation functions $\langle n_{1} u_{1} \rangle_{0}$ and $\langle n_{1} u_{2} \rangle_{0}$ with $g_{+}$. Our perturbation results are found to be very close to the exact results of Ref. [@RT]. It may be mentioned that the second order correction to the ground state wave function becomes very important in determining the shape of the correlation function within our method. In our method the ground state has no component of $| -,1\rangle$ up to the first order correction, but it appears in the second order correction to the wavefunction. Presence of $|-,1\rangle$ in the ground state has a significant contribution to the correlation function. The correlation functions are found to be very sensitive to any small correction to the wave function unlike the single elctron energies. We find a slight departure of our results from the exact results of Ref. cite[RT]{} at intermediate coupling strength. This is due to the finite series (up to the second order in perturbation for the wave function) that we have considered.
It may be mentioned that recently de Mello and Ranninger concluded from their study of a two site one polaron problem that it would be very difficult for any analytical method to describe the Holstein model except in the extreme adiabatic or nonadiabatic limit. We have shown here that even for $t/\omega_0=1.1$ which is in between the above two limits the exact results are fairly reproducible by our analytical method based on the modified LF transformation and perturbation expansion. Comparisons of the energies and the correlation functions in the ground state with exact results show that this method, used here, with the second order perturbation in the wave function could match the exact results to a desirable accuracy except in a narrow range of $g_+$ from 1 to 1.3. For this narrow range of $g_+$ reasonable results are obtained but one should include higher order corrections to obtain results to match with the exact one. It should be mentioned that as one decreases the value of $t$ and move towards the antiadiabatic limit the convergence becomes better and the region of $g_+$, where the perturbation corrections are appreciable, becomes narrower. In that case the perturbation method up to the second order correction in the wave function would describe the system for a wider range of $g_+$.
1.0cm
[**5. Conclusion**]{}
0.3cm
In the present work we develop an analytical perturbation method within the modified LF approach to deal with an electron-phonon system for the whole range of e-ph coupling strength. This method is applicable from the antiadiabatic limit ($t< \omega_0$) to the intermediate region of hopping ($t\sim \omega_0$). Considering a two-site one electron system we have calculated the single electron energies, the ground state oscillator wave functions and the correlation functions (up to the second order perturbation correction to the wave function) and find that the results are in the good agreement with the exact results. The perturbation series converges quite rapidly for all values of $g_{+}$.
Table I. Variational parameter ($\lambda$), unperturbed single electron energy (measured with respect to the bare site energy $\epsilon$), second order correction and third order correction to the energy for different values of the coupling strength ($g_{+}$) for the ground state of two-site one polaron problem.
[$g_{+}$]{} [$\lambda$]{} [$E_0^{(0)}-\epsilon$]{} [$E_0^{(2)}$]{} [$E_0^{(3)}$]{}
------------- --------------- -------------------------- ----------------- -----------------
[0.2]{} [.0628]{} [-1.1125]{} [-.00007]{} [.00000]{}
[0.5]{} [.1619]{} [-1.1795]{} [-.00275]{} [.00001]{}
[0.8]{} [.2771]{} [-1.3099]{} [-.01767]{} [.00062]{}
[1.0]{} [.3763]{} [-1.4397]{} [-.04564]{} [.00522]{}
[1.1]{} [.4421]{} [-1.5212]{} [-.07266]{} [.01408]{}
[1.2]{} [.5363]{} [-1.6183]{} [-.12168]{} [.03805]{}
[1.3]{} [1.0202]{} [-1.7489]{} [-.28799]{} [.09973]{}
[1.4]{} [1.3047]{} [-1.9874]{} [-.20805]{} [.02331]{}
[1.7]{} [1.6875]{} [-2.8935]{} [-.11787]{} [.00094]{}
[1.9]{} [1.8969]{} [-3.6108]{} [-.09082]{} [.00014]{}
[2.2]{} [2.1997]{} [-4.8401]{} [-.06240]{} [.00002]{}
[999]{}
Figure captions :
FIG. 1. Single electron energies (in units of $\omega_0=1$) as a function of the coupling strength ($g_{+}$). Dashed curve: ground state, solid curve: first excited state.
0.5 cm
FIG. 2. Ground state oscillator wave function $G(x)$ as a function of x for different values of the coupling strength when the electron is located on site 1. Solid: $g_{+}=0.1$, short-dashed: $g_{+}=0.7$, long-dashed: $g_{+}=1.3$ and dot-dashed: $g_{+}=2.0$.
0.5 cm
FIG. 3. Plot of the correlation functions (a): $\langle n_1u_1 \rangle_{0}$ and (b): $\langle n_1u_2 \rangle_{0}$ versus $g_{+}$. To compare the results of Ref. (4) we use a unit of $\frac{1}{2} \sqrt{\frac{\hbar}{M\omega}}$ for the correlation functions.
[^1]: e-mail: atin@cmp.saha.ernet.in
[^2]: e-mail: moon@cmp.saha.ernet.in
|
---
abstract: 'We describe the idea of [*adaptive shrinkage*]{} (), a general purpose Empirical Bayes (EB) method for shrinkage estimation, and demonstrate its application to several signal denoising problems. The method takes as input a set of estimates and their corresponding standard errors, and outputs shrinkage estimates of the underlying quantities (“effects"). Compared with existing EB shrinkage methods, a key feature of is its use of a flexible family of [*unimodal*]{} distributions to model the distribution of the effects. The approach is not only flexible and self-tuning, but also computationally convenient because it results in a convex optimization problem that can be solved quickly and reliably. Here we demonstrate the effectiveness and convenience of by applying it to several signal denoising applications, including smoothing of Poisson and heteroskedastic Gaussian data. In both cases consistently produces estimates that are as accurate – and often more accurate – than other shrinkage methods, including both simple thresholding rules and purpose-built EB procedures. We illustrate the potential for Poisson smoothing to provide an alternative to “peak finding" algorithms for sequencing assays such as Chromatin Immunoprecipitation (ChIP-Seq). The methods are implemented in an R package, [smashr]{} (SMoothing by Adaptive SHrinkage in R), available from <http://www.github.com/stephenslab/smashr>.'
author:
- |
Zhengrong Xing[^1]\
Department of Statistics, University of Chicago\
zhengrong@galton.uchicago.edu\
and\
Matthew Stephens\
Department of Human Genetics, Department of Statistics, University of Chicago\
mstephens@uchicago.edu
bibliography:
- 'smash.bib'
title: '**Smoothing via Adaptive Shrinkage (smash): denoising Poisson and heteroskedastic Gaussian signals**'
---
[*Keywords:*]{} Empirical Bayes, wavelets, non-parametric regression, mean estimation, variance estimation
Introduction
============
Shrinkage and sparsity play a key role in many areas of modern statistics, including, for example, high-dimensional regression [@Tibshirani1996Regression], covariance or precision matrix estimation [@Bickel2008Covariance], multiple testing [@Efron2004] and signal denoising [@Donoho1994Ideal; @donoho95]. One attractive way to achieve shrinkage and sparsity is via Bayesian or Empirical Bayes (EB) methods [e.g. @Efron2002Empirical; @Johnstone2005Empirical; @Clyde2000Flexible; @Daniels2001Shrinkage]. However, such methods are usually perceived to require context-specific implementations, and this overhead can limit their use in practice. Here we consider a flexible EB approach to shrinkage, which we call [*adaptive shrinkage*]{} (), whose goal is to provide a generic shrinkage method that could be useful for a range of applications. This method takes as input a vector of estimated “effects" and their corresponding standard errors, and outputs shrunken estimates of the effects (both point and interval estimates). It is fast, stable, and self-tuning, requiring no additional user input: the appropriate amount of shrinkage is learned from the input data. Here we demonstrate the flexibility, efficacy, and convenience of by applying it to several signal denoising problems in which shrinkage plays a central role.
Shrinkage methods are widely-used in signal denoising applications, because signal denoising can be accurately and conveniently achieved by shrinkage in a transformed (e.g. wavelet) domain [@Donoho1994Ideal]. Commonly-used shrinkage methods include both simple thresholding rules [@Coifman1995Translationinvariant; @Donoho1994Ideal] and EB methods [@Johnstone2005Empirical; @Clyde2000Flexible]. Being an EB method, has much in common with these previous EB methods, but generalizes them in two ways. First, allows more flexibility in the underlying distribution of effects ($g$) than the Laplace or spike-and-slab distributions used in @Johnstone2005Empirical [@Clyde2000Flexible]; indeed, at its most general allows $g$ to be any unimodal distribution. Second, allows for variations in precision in the transformed observations (e.g. wavelet coefficients); this plays an important role in the Poisson and heteroskedastic Gaussian settings we consider here.
In addition to demonstrating the benefits of , an important contribution of our work is to provide convenient software implementations for some commonly-encountered denoising problems. Specifically, we provide methods for smoothing Gaussian means in the presence of heteroskedastic variance, smoothing of Gaussian variances, and smoothing Poisson means. These are all settings that are relatively underserved by existing implementations. Indeed, we are unaware of any existing EB implementation for smoothing either the mean or the variance in the heteroskedastic Gaussian case. Consistent with previous studies [@Antoniadis2001Wavelet; @Besbeas2004Comparative] we find our EB methods to be more accurate than commonly-used thresholding rules, and, in the Poisson case, competitive with a purpose-built EB method [@Kolaczyk1999Bayesian]. Smoothing methods are widely used in scientific applications, and here we illustrate the potential for Poisson smoothing to provide an alternative to “peak finding" algorithms for sequencing assays such as Chromatin Immunoprecipitation (ChIP-Seq). Our methods are available in the R packages [ashr]{} (Adaptive SHrinkage in R) and [smashr]{} (SMoothing by Adaptive SHrinkage in R), available from <http://www.github.com/stephens999/ashr> and <http://www.github.com/stephenslab/smashr> respectively.
Methods
=======
Adaptive shrinkage
------------------
Here we briefly outline : full details are provided in a companion paper, @Stephens038216, which applies to estimate false discovery rates in multiple testing settings. Our work here complements this, by demonstrating that provides the adaptive shrinkage estimates that its name promises.
Adaptive shrinkage is an EB method for estimating quantities $\beta=(\beta_1,\dots,\beta_n)$ from noisy estimates $\bhat=(\bhat_1,\dots,\bhat_n)$ and their corresponding standard errors $\shat = (\shat_1,\dots,\shat_n)$. In its simpest form it assumes the hierarchical model $$\begin{aligned}
\label{eq:ash_model_1}\beta_j \| \shat_j & \sim g(\cdot) \\
\bhat_j \| \beta_j, \shat_j & \sim N(\beta_j, \shat_j^2), \label{eqn:blik}\end{aligned}$$ where the distribution $g$ is modelled using a mixture of zero-centered normal distributions. That is, $$\label{eqn:gnorm}
g(\cdot)=\sum_{k=0}^K \pi_k N(\cdot;0,\sigma_k^2),$$ where the mixture weights $\pi_0,\dots,\pi_K$ are non-negative and sum to 1, and $N(\cdot; \mu,\sigma^2)$ denotes the density of a normal distribution with mean $\mu$ and variance $\sigma^2$. A key idea, which substantially simplifies inference, is to take $\sigma_0,\dots,\sigma_K$ to be a fixed grid of values ranging from very small (e.g. $\sigma_0=0$, in which case $g$ includes a point mass at 0) to very large. Estimating $g$ then boils down to estimating the mixture weights $\pi$, which is done by maximum likelihood. Maximizing this likelihood is a convex optimization problem, and can be performed very efficiently using interior point methods [@koenker2014convex], or more simply (though less efficiently for large problems) using an accelerated EM algorithm [@Varadhan2004].
Given an estimate $\hat{g}$ for $g$, the conditional distributions $p(\beta_j \| \bhat, \shat, \hat{g})$ are analytically tractable, and the posterior mean $E(\beta_j \| \bhat, \shat, \hat{g})$ provides a shrinkage point estimate for $\beta_j$.
The representation (\[eqn:gnorm\]) provides a flexible family of unimodal and symmetric distributions $g$. Specifically, with a sufficiently large and dense grid of $\sigma_0,\dots,\sigma_K$ the distribution $g$ in (\[eqn:gnorm\]) can arbitrarily accurately approximate any scale mixture of normals. This family includes, as a special case, many distributions used in shrinkage estimation, such as the “spike and slab", double-exponential (Laplace), and $t$ distributions [@Clyde2000Flexible; @Johnstone2005Empirical]. In this sense is more flexible than previous EB approaches. Furthermore, in many ways this more flexible approach actually [*simplifies*]{} inference, because the only parameters to be estimated are the mixture proportions (the variances are fixed).
@Stephens038216 also introduces various embellishments that are implemented in the [ashr]{} package, including generalizing the normal likelihood to a $t$ likelihood, and dropping the symmetric constraint on $g$ by replacing the mixture of normals with a more flexible (though less smooth) mixture of uniforms. We do not use these embellishments here.
Smoothing with Adaptive Shrinkage
---------------------------------
We consider estimating a “spatially-structured" mean $\bmu=(\mu_1,\dots,\mu_T)$ (and perhaps also the corresponding variance $\bm{\s}=(\s_1^2,\dots,\s_T^2)$), from observations $\bY = (Y_1,\dots,Y_T)$, where $t=1,\dots,T$ indexes location in a one-dimensional space, such as time, or, as in our example later, location along the genome. By “spatially-structured" we mean that $\mu_t$ will often be similar to $\mu_t'$ for small $|t-t'|$, though we do not rule out occasional abrupt changes in $\mu_t$.
The best studied version of this problem is the homoskedastic Gaussian case; that is, the $Y_t$ have Gaussian noise and constant variance. Here we consider the more general case of Gaussian data with spatially-structured mean [*and*]{} spatially-structured (possibly non-constant) variance, and our methods provide estimates for both the mean and variance. We also consider denoising Poisson data (where the variance depends on the mean, so a spatially-structured mean implies spatially-structured variance).
In both cases we make use of “multi-scale" denoising methods, which essentially involve i) transforming the data using a multi-scale transform; ii) performing shrinkage estimation with in the transformed space; and iii) inverting the multi-scale transformation to obtain estimates in the original space. This strategy exploits the fact that a multi-scale transform will map smooth signals in the original space to sparse signals in the transformed space. Thus performing shrinkage in the transformed space induces smoothness in the original space.
For convenience we assume that $T=2^J$ for some integer $J$, as is common in multi-scale analyses.
### Gaussian data {#gaussian-data .unnumbered}
Suppose $$\label{eq:1d gaussian model}
\bY =\bmu+\be$$ where $\be \sim N_T({\bm 0},D)$ with $D$ the diagonal matrix with diagonal entries $(\s_1^2,\dots,\s_T^2)$. We consider, in turn, the problems of estimating $\bmu$ when $\bm{\s}$ is known; estimating $\bm{\s}$ when $\bmu$ is known; and finally the typical case of estimating $\bmu$ and $\bm{\s}$ when both are unknown.
[*Estimating $\bmu$ with $\bm{\s}$ known*]{}\
We first transform the data using a multi-scale transformation, specifically a discrete wavelet transform. This involves pre-multiplying $\bY$ by an orthogonal $T\times T$ matrix $W$ that depends on the wavelet basis chosen. Pre-multiplying (\[eq:1d gaussian model\]) by $W$ yields $$W\bY=W\bmu + W\be$$ which we write $$\label{eq:wc}
\tbY=\tbmu + \tbe,$$ where $\tbe \sim N_T(0,W D W')$.
Next we use to obtain (posterior mean) shrinkage estimates $\hat{\tbmu}$ for $\tbmu$, which we then reverse transform to estimates for $\mu$: $$\hat{\mu} := W^{-1} \hat{\tbmu}.$$
Applying requires only point estimates for the elements of $\tbmu$, and corresponding standard errors, which we obtain from the marginals of (\[eq:wc\]): $$\label{eq:normlik}
\tY_j | \tmu_j, s^2_j \sim N(\tmu_j, \omega^2_j),$$ where $$\omega^2_j:=\sum_{t=1}^T \s^2_t W_{jt}^2.$$ Thus to obtain $\hat{\tbmu}$ we apply to the estimates $\bhat_j:= \tY_j$ with standard errors $\shat_j = \omega_j$. (In practice it is important to group the wavelet-transformed observations $\tY_j$ by their resolution level before shrinking; see note below.)
In focusing on the marginals \[eq:normlik\] we are ignoring correlations among the $\tY_j$, and this is the primary simplification here. We are not alone in making this simplification; see @Silverman1999Wavelets for example.
The above outlines the basic strategy, but there are some important additional details:
1. Rather than use a single wavelet transform, we use the “non-decimated" (“translation invariant") wavelet transform, which averages results over all $T$ possible rotations of the data (effectively treating the observations as coming from a circle, rather than a line). Although not always necessary, this is a standard trick to reduce artifacts that can occur near discontinuities in the underlying signal [e.g. @Coifman1995Translationinvariant]).
2. The non-decimated wavelet transform yields $T$ wavelet coefficients (transformed values of $\bY$) at each of $J=\log_2(T)$ resolution levels. We apply separately to the $T$ wavelet coefficients at each resolution level, so that a different distribution $g$ for the $(\tmu_j)$ is estimated for each resolution. This is the usual way that EB approaches are applied in this context [e.g. @Johnstone2005Empirical] and indeed is crucial because the underlying distribution $g$ will vary with resolution (because smoothness of $\bmu$ will vary with resolution).
3. Although we have presented the wavelet transform as a matrix multiplication, which is an $o(T^2)$ operation, in practice both the wavelet transform and the inverse transform are implemented using efficient algorithms [@Beylkin1992Ontherepresentation; @Coifman1995Translationinvariant], implemented in the [wavethresh]{} package [@Nason2013], taking only $O(T\log_2 T)$ operations.
[*Estimating $\bm{\s}$ with $\bmu$ known*]{}\
To estimate the variance $\bm{\s}$ we apply wavelet shrinkage methods to the squared deviations from the mean, similar to the strategies of @Delouille2004Smooth and @Cai2008Adaptive. Specifically, define $$\begin{aligned}
\label{eq:varobs1}
Z_t^2=(Y_t-\mu_t)^2\end{aligned}$$ and note that $\mathbb{E}(Z_t^2)=\s_t^2$, so estimating $\bm{\s}$ is now a mean estimation problem. We tackle this using the mean estimation procedure above, effectively treating the wavelet-transformed values $(W\bZ^2)_t$ (where $\bZ^2\equiv (Z_1^2,...,Z_T^2)$) as Gaussian when really they are linear combinations of $\chi_1^2$ random variables. This requires an estimate of the variance of $Z_t^2$: we use $\frac{2}{3}Z_t^4$, which is an unbiased estimator of the variance. (If $Z^2 \sim \s^2 \chi_1^2$, then $\mathbb{E}(Z^4)= 3\s^4$, and $\mathbb{V}(Z^2)=2\s^4$.)
Despite the approximations made here, we have found this procedure to work well in practice in most cases, perhaps with a tendancy to oversmooth quickly-varying variance functions.
[*Estimating $\bmu$ and $\bm{\s}$ jointly*]{}\
We iterate the above procedures to deal with the (typical) case where both mean and variance are unknown. To initialize we esitmate the variance $\bm{\s}^2$ using $$\begin{aligned}
\label{eq:initial var est}
\hat{\s}_t^2=\frac{1}{2}\left((Y_t-Y_{t-1})^2+(Y_t-Y_{t+1})^2\right)\end{aligned}$$ where $Y_0\equiv Y_n$ and $Y_{T+1}\equiv Y_1$ (equivalent to putting the observations on a circle).
Then we iterate:
1. Estimate $\bm{\mu}$ as if $\bm{\s}^2$ is known (with the value obtained from the previous step).
2. Estimate $\bm{\s}^2$ as if ${\bm{\mu}}$ is known (with the value obtained by the previous step); return to 1.
We cannot guarantee that this procedure will converge, but in our simulations we found that two iterations of steps 1-2 reliably yielded accurate results (so the full procedure consists of initialize + Steps 1-2-1-2).
### Poisson data {#poisson-data .unnumbered}
Consider now $$Y_t \sim \Poi(\mu_t), \qquad (t=1,\dots,T).$$ To estimate $\bm{\mu}$ we apply to the Poisson multiscale models from @Kolaczyk1999Bayesian [@Timmermann1999Multiscale; @Nowak2000Statistical], which are analogues of wavelet methods for Poisson data.
To explain, first recall the following elementary distributional result: if $Y_1,Y_2$ are independent, with $Y_j \sim \Poi(\mu_j)$ then $$\begin{aligned}
Y_1 + Y_2 & \sim \Poi(\mu_1 + \mu_2) \\
Y_1 | (Y_1+Y_2) & \sim \Bin(Y_1+Y_2, \mu_1/(\mu_1+\mu_2)).\end{aligned}$$
To extend this to $T=4$, introduce the notation $v_{i:j}$ to denote, for any vector $v$, the sum $\sum_{t=i}^j v_t$. Then $$\begin{aligned}
Y_{1:4} & \sim \Poi(\mu_{1:4}) \label{eqn:y14} \\
Y_{1:2} | Y_{1:4} & \sim \Bin(Y_{1:4}, \mu_{1:2}/\mu_{1:4}) \label{eqn:y12} \\
Y_1 | Y_{1:2} & \sim \Bin(Y_{1:2}, \mu_1/\mu_{1:2}) \label{eqn:y1} \\
Y_3 | Y_{3:4} & \sim \Bin(Y_{3:4}, \mu_3/\mu_{3:4}). \label{eqn:y3}\end{aligned}$$ Together these models are exactly equivalent to $Y_j \sim \Poi(\mu_j)$, and they decompose the overall distribution $Y_1,\dots,Y_4$ into parts involving aspects of the data at increasing resolution: (\[eqn:y14\]) represents the coarsest resolution (the sum of all data points), whereas (\[eqn:y1\]) and (\[eqn:y3\]) represent the finest resolution, with (\[eqn:y12\]) in between. Further, this representation suggests a reparameterization, from $\bmu=(\mu_1,\mu_2,\mu_3,\mu_4)$ to $\mu_{1:4}$ plus the binomial parameters $\bp=(\mu_{1:2}/\mu_{1:4}, \mu_1/\mu_{1:2}, \mu_3/\mu_{3:4})$, where $p_1$ controls lower-resolution changes in the mean vector $\mu$ and $p_2,p_3$ control higher resolution changes.
This idea extends naturally to $T=2^J$ for any $J$, reparameterizing $\bmu$ into its sum $\mu_{1:T}$ and a vector $\bp$ of $T-1$ binomial probabilities that capture features of $\bmu$ at different resolutions. This can be thought of as an analogue of the Haar wavelet transform of $\bmu$ for Poisson data.
Note that, in this reparameterization, $p_j=0.5 \, \forall j$ corresponds to the case of a constant mean vector, and values of $p_j$ far from 0.5 correspond to large changes in $\mu$ (at some scale). Thus estimating a spatially-structured $\bmu$ can be achieved by shrinkage estimation of $\bp$, with shrinkage towards $p_j=0.5$. Both @Kolaczyk1999Bayesian and @Timmermann1999Multiscale use purpose-built Bayesian models to achieve this shrinkage, by introducing a prior distribution on elements of $\bp$ that is a mixture of a point mass at 0.5 (creating shrinkage toward 0.5) and a Beta distribution. Here we take a different approach, reparameterizing $\alpha_j = \log(p_j/(1-p_j))$, and then using to shrink $\alpha_j$ towards 0.
To apply we need an estimate $\hat{\alpha}_{j}$ and corresponding standard error $\hat{s}_j$ for each $j$. This involves estimating a log-odds ratio, and its standard error, which is a well-studied problem [e.g. @Gart1967Bias]. The main challenge is in dealing satisfactorily with cases where the maximum likelihood estimator for $\alpha_j$ is infinite. We use estimates based on results from @Gart1967Bias; see Appendix \[app:reconstruction\].
The simplest way to estimate $\bmu$ is to estimate $\alpha_j$ by its posterior mean, as output by , and then reverse the above reparameterization. The resulting estimate of $\mu_t$ is the exponential of the posterior mean for $\log(\mu_t)$ (because each $\log(\mu_t)$ is a linear combination of $\alpha_j$). Alternatively we can estimate $\mu_t$ by approximating its posterior mean using the Delta method; see Appendix \[app:reconstruction\]. Both methods are implemented in our software; for the results here we use the latter method to be more comparable with previous approaches that estimate $\mu$ on the raw scale rather than $\log$ scale.
As for the Gaussian case, we actually use a translation invariant version of the above procedure; see Appendix \[app:TI\]
Results
=======
Illustration
------------
Figure \[fig:simple\_eg\] illustrates the key features of in the context of smoothing a Gaussian signal. The data (panel (a)) is first transformed into wavelet coefficients (WCs) at different scales. Each such “observed" WC can be thought of as a noisy estimate of some “true" WC for the unknown mean that we wish to estimate. Each WC also has an associated standard error that depends on the variance of the data about the mean. The idea behind wavelet denoising is to “shrink" the observed WCs towards 0, which produces a smoother estimate of the mean than the observed data.
A crucial question is, of course, how much to shrink. A key idea behind is that the shrinkage is “adaptive", in that it determined by the data, in two distinct ways. First, if at a particular scale many observed WCs are “large" (compared with their standard errors) then infers that, at this scale, many of the true WCs are large - that is, the estimated distribution $g$ in (\[eq:ash\_model\_1\]) will have a long tail. Consequently will shrink less at this scale than at scales where few observed WCs are large, for which the estimated $g$ will have a short tail. This is illustrated in panels (b)-(c): at scale 1, many observed WCs are large (b), and so very little shrinkage is applied to these estimates (c). In contrast, at scale 7, few observed WCs are large (b), and so stronger shrinkage is applied (c). Second, because the likelihood \[eqn:blik\] incorporates the standard error of each observation, shrinkage is adaptive to the standard error: at a given scale, WCs with larger standard error are shrunk more strongly than WCs with small standard error. This is illustrated in panel (d). (The standard errors vary among WCs because of the heteroskedastic variance.)
The end result is that i) data that are consistent with a smooth signal, get smoothed more strongly; ii) smoothing is stronger in areas of the signal with larger variance. In this example the smoothed signal from is noticeably more accurate than using TI-thresholding (with variance estimated by running median absolute deviation, RMAD; see @Gao1997Wavelet) (panel (e)).
[0.48]{} ![Illustration of both and SMASH. Panel (a) shows the “Spikes" mean function (black) $\pm$2 standard deviations (red), and corresponding simulated data. Panel (b) contrasts the distributions of the (true) wavelet coefficients (WCs) at two different scales - one coarse (scale 1) and one fine (scale 7). The scale 7 WCs are much more peaked around 0, because the signal is much smoother at that scale. Panel (c) contrasts the shrinkage for these two scales: the scale 7 WCs are strongly shrunk, whereas the scale 1 WCs are not. This is because infers from the data that the scale 7 WCs are peaked around 0, and consequently shrinks them more strongly. Panel (d) illustrates that shrinks WCs differently depending on their precision. Specifically WCs that are less precise are shrunk more strongly. Panel (e) plots the estimated mean functions from SMASH and TI-thresh against the true mean function; TI-thresh has some noteable artifacts.[]{data-label="fig:simple_eg"}](simple_eg_1.pdf "fig:"){width="\textwidth"}
[0.48]{} ![Illustration of both and SMASH. Panel (a) shows the “Spikes" mean function (black) $\pm$2 standard deviations (red), and corresponding simulated data. Panel (b) contrasts the distributions of the (true) wavelet coefficients (WCs) at two different scales - one coarse (scale 1) and one fine (scale 7). The scale 7 WCs are much more peaked around 0, because the signal is much smoother at that scale. Panel (c) contrasts the shrinkage for these two scales: the scale 7 WCs are strongly shrunk, whereas the scale 1 WCs are not. This is because infers from the data that the scale 7 WCs are peaked around 0, and consequently shrinks them more strongly. Panel (d) illustrates that shrinks WCs differently depending on their precision. Specifically WCs that are less precise are shrunk more strongly. Panel (e) plots the estimated mean functions from SMASH and TI-thresh against the true mean function; TI-thresh has some noteable artifacts.[]{data-label="fig:simple_eg"}](simple_eg_2.pdf "fig:"){width="\textwidth"}
[0.48]{} ![Illustration of both and SMASH. Panel (a) shows the “Spikes" mean function (black) $\pm$2 standard deviations (red), and corresponding simulated data. Panel (b) contrasts the distributions of the (true) wavelet coefficients (WCs) at two different scales - one coarse (scale 1) and one fine (scale 7). The scale 7 WCs are much more peaked around 0, because the signal is much smoother at that scale. Panel (c) contrasts the shrinkage for these two scales: the scale 7 WCs are strongly shrunk, whereas the scale 1 WCs are not. This is because infers from the data that the scale 7 WCs are peaked around 0, and consequently shrinks them more strongly. Panel (d) illustrates that shrinks WCs differently depending on their precision. Specifically WCs that are less precise are shrunk more strongly. Panel (e) plots the estimated mean functions from SMASH and TI-thresh against the true mean function; TI-thresh has some noteable artifacts.[]{data-label="fig:simple_eg"}](simple_eg_3.pdf "fig:"){width="\textwidth"}
[0.48]{} ![Illustration of both and SMASH. Panel (a) shows the “Spikes" mean function (black) $\pm$2 standard deviations (red), and corresponding simulated data. Panel (b) contrasts the distributions of the (true) wavelet coefficients (WCs) at two different scales - one coarse (scale 1) and one fine (scale 7). The scale 7 WCs are much more peaked around 0, because the signal is much smoother at that scale. Panel (c) contrasts the shrinkage for these two scales: the scale 7 WCs are strongly shrunk, whereas the scale 1 WCs are not. This is because infers from the data that the scale 7 WCs are peaked around 0, and consequently shrinks them more strongly. Panel (d) illustrates that shrinks WCs differently depending on their precision. Specifically WCs that are less precise are shrunk more strongly. Panel (e) plots the estimated mean functions from SMASH and TI-thresh against the true mean function; TI-thresh has some noteable artifacts.[]{data-label="fig:simple_eg"}](simple_eg_4.pdf "fig:"){width="\textwidth"}
[0.48]{} ![Illustration of both and SMASH. Panel (a) shows the “Spikes" mean function (black) $\pm$2 standard deviations (red), and corresponding simulated data. Panel (b) contrasts the distributions of the (true) wavelet coefficients (WCs) at two different scales - one coarse (scale 1) and one fine (scale 7). The scale 7 WCs are much more peaked around 0, because the signal is much smoother at that scale. Panel (c) contrasts the shrinkage for these two scales: the scale 7 WCs are strongly shrunk, whereas the scale 1 WCs are not. This is because infers from the data that the scale 7 WCs are peaked around 0, and consequently shrinks them more strongly. Panel (d) illustrates that shrinks WCs differently depending on their precision. Specifically WCs that are less precise are shrunk more strongly. Panel (e) plots the estimated mean functions from SMASH and TI-thresh against the true mean function; TI-thresh has some noteable artifacts.[]{data-label="fig:simple_eg"}](simple_eg_5.pdf "fig:"){width="\textwidth"}
Simulations
-----------
We conducted extensive simulation studies to compare the performance of our method, SMASH (SMoothing with Adaptive SHrinkage), with existing approaches. In particular, simulations with Gaussian noise were conducted within a [Dynamical Statistical Comparison](https://github.com/stephens999/dscr) (DSC) framework, which was designed to facilitate comparisons among statistical methods. The code for this DSC is available at [dscr-smash](https://github.com/zrxing/dscr-smash), which also contains instructions for reproducing and extending the results presented here.
### Gaussian mean estimation {#gaussian-mean-estimation .unnumbered}
We focus initially on the homoskedastic case, modelling our simulation study after @Antoniadis2001Wavelet. Specifically we used many of the same test functions, a variety of signal lengths ($T$), and two different signal to noise ratios (SNRs). We compare SMASH with Translation Invariant (TI) thresholding [@Coifman1995Translationinvariant], which was one of the best-performing methods in @Antoniadis2001Wavelet, and with the Empirical Bayes shrinkage procedure Ebayesthresh [@Johnstone2005Empirical]. All methods were applied using the Symmlet8 wavelet basis [@Daubechies1992Ten].
Figure \[fig:gaus\_homo\] compares the mean integrated squared errors (MISEs) of the methods for the “Spikes” mean function, with SNR=3 and $T$=1024. We applied SMASH in three ways, the first (SMASH) estimating the variance function allowing for heteroskedasticity, the second estimating the variance assuming homoskedasticity (SMASH-homo) and the third using the true variance function (SMASH true variance), which could be viewed as a “gold standard". All three versions of SMASH outperform both EbayesThresh and TI-thresholding. The different SMASH versions perform similarly, demonstrating that in this case there is little cost in allowing for heteroskedasticity when the truth is homoskedastic. We obtained similar results for other mean functions, SNRs and sample sizes (see Supplementary Materials).
![Comparison of accuracy (MISE) of estimated mean curves for data simulated with homoskedastic Gaussian errors. Results are for the “Spikes” mean function shown in Figure \[fig:simple\_eg\]. The figure shows violin plots of MISEs for (from bottom to top) SMASH, SMASH with homoskedastic assumption, TI-thresholding with homoskedastic assumption, Ebayesthresh with homoskedastic assumption, and SMASH with known true variance. Colors highlight certain features of a method: yellow for methods that assume homoskedastic variance; dark purple for methods that are provided the true variance; magenta for the default SMASH method that estimates both mean and variance. Smaller MISE implies better performance; dashed green line indicates the median MISE for SMASH. SMASH outperforms both TI-thresholding and Ebayesthresh, and SMASH with estimated variance performs nearly as well as with true variance.[]{data-label="fig:gaus_homo"}](violin_gaus_homo.pdf){width="65.00000%"}
Turning to heteroskedastic errors, we again compare SMASH with EbayesThresh (which assumes homoskedastic variance) and TI-thresh. For TI-thresh we considered three different ways of estimating the heteroskedastic variance: RMAD [@Gao1997Wavelet], the SMASH estimated variance, and the true variance. (TI-thresh with homoskedastic variance performed very poorly; Supplementary Results.) Figure \[fig:gaus\_hetero\] shows results for two sets of test functions: the “Spikes” mean function with the “Clipped Blocks” variance function and the “Corner” mean function with “Doppler” variance function, both with SNR=3 and $T=1024$.
To summarize the main patterns in Figure \[fig:gaus\_hetero\]:
1. SMASH outperforms all TI-thresh variants (including TI with the true variance).
2. SMASH performs almost as well when estimating the variance as when given the true variance.
3. Allowing for heteroskedasticity within SMASH can substantially improve accuracy of mean estimation (compare SMASH with SMASH-homo and EbayesThresh).
4. TI-thresh performs considerably better when used with the SMASH variance estimate than with the RMAD variance estimate.
These main patterns hold for a variety of different mean and variance functions, SNRs, and sample sizes (see Supplementary Materials). Some variance functions are harder to estimate than others (e.g. the “Bumps" function), and in these cases providing methods the true variance can greatly increase accuracy compared with estimating the variance. As might be expected, the gain in allowing for heteroskedastic variance tends to be greatest when the variance functions are more volatile.
[0.65]{} ![Comparison of accuracy (MISE) of estimated mean curves, for data simulated with heteroskedastic Gaussian errors. Panels (a) and (c) show violin plots of MISEs for various methods on two sets of mean-variance functions, shown as mean $\pm 2$ standard deviations in panels (b) and (d). In (a) and (c) TI-based methods are shown in red, while other colors are as in Figure \[fig:gaus\_homo\]. Dashed green line indicates SMASH median MISE.[]{data-label="fig:gaus_hetero"}](violin_gaus_hetero_1.pdf "fig:"){width="\textwidth"}
[0.3]{}
[0.65]{} ![Comparison of accuracy (MISE) of estimated mean curves, for data simulated with heteroskedastic Gaussian errors. Panels (a) and (c) show violin plots of MISEs for various methods on two sets of mean-variance functions, shown as mean $\pm 2$ standard deviations in panels (b) and (d). In (a) and (c) TI-based methods are shown in red, while other colors are as in Figure \[fig:gaus\_homo\]. Dashed green line indicates SMASH median MISE.[]{data-label="fig:gaus_hetero"}](violin_gaus_hetero_2.pdf "fig:"){width="\textwidth"}
[0.3]{}
### Gaussian variance estimation {#sec:mfvb}
One unusual feature of SMASH is that it performs joint mean and variance estimation. Indeed, we found no existing R packages for doing this. The only work we have found on wavelet-based variance estimation is @Cai2008Adaptive, which applied a wavelet thresholding approach to first order differences. Previous related non-wavelet-based work includes @Fan1998Efficient, which estimates the variance by smoothing the squared residuals using local polynomial smoothing, @Brown2007Variance, which uses difference-based kernel estimators, and @Menictas2015Variational, which introduces a Mean Field Variational Bayes (MFVB) method for both mean and variance estimation. In no case could we find publicly-available software implementations of these methods. However, we did receive code implementing MFVB by email from M. Menictas, which we use in our comparisons here.
The MFVB method is based on penalized splines, and so is not well suited to many standard test functions in the wavelet literature, which often contain “spiky" local features not well captured by splines. Hence, we compared SMASH and MFVB on some smoother mean and variance (standard deviation) functions, specifically scenario A in Figure 5 from @Menictas2015Variational (Figure \[fig:mfvb\_fn\]), using scripts kindly provided by M. Menictas.
We simulated data under two different scenarios:
1. We generated $T=500$ independent $(X_t,Y_t)$ pairs, with $X_t \sim$ Uniform(0,1), and $Y_t | X_t=x_t \sim N(m(x_t),s(x_t)^2)$ where $m(\cdot)$ and $s(\cdot)$ denote the mean and standard deviation functions (Figure \[fig:mfvb\_fn\]). We measured performance by the MSE evaluated at 201 equally spaced points on $(X_{min},X_{max})$ for both the mean and the standard deviation.
2. We generated $T=1024$ independent $(X_t,Y_t)$ pairs, with the $X_t$’s (deterministically) equally spaced on (0,1), and $Y_t|X_t$ as above. Performance is measured by MSE evaluated at the 1024 $X_t$’s for both the mean and the standard deviation.
![The mean function $m(x)$ $\pm$2 standard deviations $s(x)$ used in simulations comparing SMASH and MFVB. These functions correspond to mean and standard deviation functions (A) in Figure 5 from @Menictas2015Variational.[]{data-label="fig:mfvb_fn"}](mfvb_eg.pdf){width="\textwidth"}
The first scenario presents some issues for SMASH because the number of data points is not a power of two, nor are the points equally-spaced. To deal with the first issue, following standard ideas in the wavelet literature, we first mirrored the data about the right edge and extract the first $2^{\lfloor\log_2(2T)\rfloor}$ sample points, so that the number of data points in the new “dataset” is a power of two, and the mean curve is continuous at the right edge of the original data. To further ensure that the input to SMASH is periodic, we reflected the new dataset about its right edge and used this as the final input. To deal with the second issue we follow the common practice of treating the observations as if they were evenly spaced (see @Sardy1999Wavelet for discussion). To estimate the original mean and variance functions, we extract the first $T$ points from the SMASH estimated mean and variance. To evaluate MSE at the 201 equally spaced points (Scenario 1) we use simple linear interpolation between the estimated points.
Table \[table:mfvb\_comp\] shows mean MSEs over 100 independent runs for each scenario. Despite the fact that these simulation scenarios – particularly Scenario 1 – seem better suited to MFVB than SMASH, SMASH performs comparably or better than MFVB for both mean and variance estimation in both Scenarios.
------- ---------------- -------------- ---------------- --------------
MSE (for mean) MSE (for sd) MSE (for mean) MSE (for sd)
MFVB 0.0330 0.0199 0.0172 0.0085
SMASH 0.0334 0.0187 0.0158 0.0065
------- ---------------- -------------- ---------------- --------------
: Comparison of accuracy (MSE) of SMASH and MFVB for two simulation scenarios. True mean and sd functions are shown in Figure \[fig:mfvb\_fn\]. In Scenario 1 the data are not equally spaced and not a power of 2; here SMASH is comparable to MFVB in mean estimation and more accurate for sd estimation. In Scenario 2 the data are equally spaced and a power of 2; here SMASH outperforms MFVB in both mean and sd estimation.[]{data-label="table:mfvb_comp"}
### Poisson Data
For Poisson data we compared SMASH with six other methods, but we focus here on the comparisons with the best-performing other methods, which are Haar-Fisz (HF) [@Fryzlewicz2004HaarFisz] and BMSM [@Kolaczyk1999Bayesian]. The latter, like SMASH, is an EB method, but with a less flexible prior distribution on the multi-scale coefficients. The HF method involves first performing a transformation on the Poisson counts, and applying Gaussian wavelet methods to the transformed data. There are many choices for Gaussian wavelet methods, and performance depends on these choices, with different choices being optimal for different data sets. The settings we used here for HF are documented in Supplementary Information, and were chosen by us to optimize (average) performance through moderately extensive experimentation on a range of simulations.
To compare the methods we simulated data from several different test functions from @Timmermann1999Multiscale, @Fryzlewicz2004HaarFisz and @Besbeas2004Comparative. We varied the minimum and maximum intensity of each test function, using (min,max) intensities of (0.01,3), (1/8,8) and (1/128,128). For each test function and intensity level we simulated 100 datasets, each with $T=1024$ data points. We focus on results for the first two intensity settings, which have smaller average intensities. These settings produce smaller average counts, making them more challenging, and also more representative of the kinds of genomic application that we consider below. Complete results are included in Supplementary Materials.
In summary, SMASH outperformed both HF and BMSM in the majority of simulations, with the gain in accuracy being strongest for the more challenging lower-intensity scenarios. A typical result is shown in Figure \[fig:pois\_sim\]. Among the other two methods, BMSM is the more consistent performer. HF, with the settings used here, performs quite variably, being worse than the other methods in most scenarios, but occasionally performing the best (specifically for the Angles, Bursts and Spikes test functions, with (min,max) intensity (1/128,128)). As noted above, the HF transform can be used with many settings, so results here should be viewed only as a guide to potential performance.
[0.75]{} ![Comparison of methods for denoising Poisson data for the “Bursts” test function. Panel (a) shows the (unscaled) test function. The violin plots in (b) and (c) show distributions of MISE for each method over 100 datsets, with smaller values indicating better performance. Panel (b) is for simulations with a (min,max) intensity of (0.01,3), and panel (c) is for simulations with a (min,max) intensity of (1/8,8). The dashed green line indicates the median MISE for SMASH. []{data-label="fig:pois_sim"}](bursts_pois.pdf "fig:"){width="\textwidth"}
[0.45]{} ![Comparison of methods for denoising Poisson data for the “Bursts” test function. Panel (a) shows the (unscaled) test function. The violin plots in (b) and (c) show distributions of MISE for each method over 100 datsets, with smaller values indicating better performance. Panel (b) is for simulations with a (min,max) intensity of (0.01,3), and panel (c) is for simulations with a (min,max) intensity of (1/8,8). The dashed green line indicates the median MISE for SMASH. []{data-label="fig:pois_sim"}](violin_pois_1.pdf "fig:"){width="\textwidth"}
[0.45]{} ![Comparison of methods for denoising Poisson data for the “Bursts” test function. Panel (a) shows the (unscaled) test function. The violin plots in (b) and (c) show distributions of MISE for each method over 100 datsets, with smaller values indicating better performance. Panel (b) is for simulations with a (min,max) intensity of (0.01,3), and panel (c) is for simulations with a (min,max) intensity of (1/8,8). The dashed green line indicates the median MISE for SMASH. []{data-label="fig:pois_sim"}](violin_pois_8.pdf "fig:"){width="\textwidth"}
One disadvantage of the HF transform is that, to achieve translation invariance, the transform has to be done explicitly for each shift of the data: the tricks usually used to do this efficiently [@Coifman1995Translationinvariant] do not work here. Thus, making HF fully translation invariant increases computation by a factor of $T$, rather than the factor of $\log(T)$ for the other methods. Here we follow advice in @Fryzlewicz2004HaarFisz to reduce the computational burden by averaging over 50 shifts of the data rather than $T$. Even so, HF was substantially slower than the other methods. A direct comparison of computational efficiency between SMASH and BMSM is difficult, as they are coded in different programming environments. Nevertheless, similarities between the two methods suggest that they should have similar computational cost. Both SMASH and BMSM took, typically, less than a second per dataset in our simulations.
Applications
============
Motorcycle Acceleration Data {#motorcycle-acceleration-data .unnumbered}
----------------------------
![Results from fitting SMASH to the motorcycle acceleration data discussed in @silverman1985some. The figure shows the estimate mean curve (solid black line) with $\pm2$ the estimated standard deviation curve (dashed red line).[]{data-label="fig:motorcycle"}](motorcycle.pdf){width="65.00000%"}
To further illustrate the heteroskedastic Gaussian version of SMASH, we apply it to the motorcycle acceleration dataset from @silverman1985some. The data consist of 133 observations measuring head acceleration in a simulated motorcycle accident that is used to test crash helmets. The dependent variable is [*acceleration*]{} (in [*g*]{}), and the independent variable is [*time*]{} (in [*ms*]{}). To deal with repeated measurements, we take the median of the measurements for acceleration for any given [*time*]{} value. As in Section \[sec:mfvb\] we treat the data as if they are equally spaced although they are not. The fitted mean and variance curves (Figure \[fig:motorcycle\]) provide a visually appealing fit to the data, and were achieved without hand tuning of any parameters. This contrasts with results in @Delouille2004Smooth – which also uses a wavelet-based approach for heteroskedastic variance, but accounts for the unequal spacing of the data – which required the [*ad hoc*]{} removal of high-resolution wavelet coefficients to produce a visual appealing fit.
ChIP-Seq Data {#chip-seq-data .unnumbered}
-------------
Chromatin immunoprecipitation sequencing (ChIP-seq) is used to measure transcription factor binding along the genome. After pre-processing (read mapping), the data consist of counts of sequencing reads mapping to each location in genome. These can be treated as arising from an inhomogeneous Poisson process, whose intensity at base $b$ is related to the strength of the binding of the transcription factor near $b$. Because binding tends to be quite localized, the intensity is low on average (the vast majority of counts are 0), but has a small number of intense “peaks". Identifying these peaks can help discover regions where binding occurs, which is an important component of understanding gene regulation. Consequently there are many methods published for “peak detection" in ChIP-Seq data [@Wilbanks2010Evaluation]. Our goal here is to briefly outline how Poisson SMASH could provide an alternative approach to the analysis of ChIP-seq data. The idea is simply to estimate the underlying intensity function, and then identify “peaks" as regions where the estimated intensity exceeds some threshold.
To illustrate, we applied SMASH to a small example ChIP-seq dataset. The resulting estimated intensity is shown in Figure \[fig:seq\_peak\_est\], overlaid on peaks called by the popular peak calling software MACS [@Zhang2008Modelbased]. The locations with the strongest SMASH intensity estimates corresponds to peaks found by MACS. However, the intensity estimate also suggests the presence of several additional weaker peaks not identified by MACS.
The reliable calling of peaks in ChIP-seq data is a multi-faceted problem, and a full assessment lies outside the scope of this paper. Nonetheless, these results suggest that this approach could be worth pursuing. One nice feature of our multi-scale Poisson approach is that it deals well with a range of intensity functions, and could perform well even in settings where peaks are broad and/or not especially well defined. In contrast, the performance of different peak-finding algorithms is often reported to be quite sensitive to the “kinds" of peak that are present, so an algorithm that performs well in one setting may perform poorly in another.
[0.85]{} ![Illustration of the potential for SMASH to identify peaks in ChIP-seq data. The data are ChIP-seq for the transcription factor [*YY1*]{} in cell line GM12878 collected by the ENCODE (**Enc**yclopedia **O**f **D**NA **E**lements) project, from a region of length $2^{17} (\approx 131k)$ basepairs from chromosome 1 (hg19 chr1:880001-1011072). Panel (a) shows counts (summed across two replicate experiments). Due to over-plotting, darker regions of the plot correspond to higher concentrations of data points. Panel (b) shows the estimated intensity function from SMASH (black solid line) and location of peaks called by MACS (red markers beneath the estimated intensity).[]{data-label="fig:seq_peak"}](peaks_comp_a.png "fig:"){width="\textwidth"}
[0.85]{} ![Illustration of the potential for SMASH to identify peaks in ChIP-seq data. The data are ChIP-seq for the transcription factor [*YY1*]{} in cell line GM12878 collected by the ENCODE (**Enc**yclopedia **O**f **D**NA **E**lements) project, from a region of length $2^{17} (\approx 131k)$ basepairs from chromosome 1 (hg19 chr1:880001-1011072). Panel (a) shows counts (summed across two replicate experiments). Due to over-plotting, darker regions of the plot correspond to higher concentrations of data points. Panel (b) shows the estimated intensity function from SMASH (black solid line) and location of peaks called by MACS (red markers beneath the estimated intensity).[]{data-label="fig:seq_peak"}](peaks_comp_b.pdf "fig:"){width="\textwidth"}
Discussion
==========
We have demonstrated “smoothing via adaptive shrinkage" (SMASH) for smoothing Gaussian and Poisson data using multi-scale methods. The method is built on the EB shrinkage method, , whose two key features are i) using a flexible family of unimodal distributions to model the multi-scale coefficients; and ii) to account for varying precision in each coefficient. The first feature allows to flexibly adapt the amount of shrinkage to the data in hand, so that data that “look smooth" are smoothed more strongly than data that do not. The second feature allows to deal effectively with heteroskedastic variances, and has the consequence that the mean gets smoothed more strongly in regions where the variance is bigger.
Notably, and unlike many wavelet shrinkage approaches, SMASH is “self-tuning", and requires no specification of a “primary resolution level" [e.g. @Nason2002Choice] or other tuning parameters. This feature is due to the “adaptive" nature of noted above: when a particular resolution level shows no strong signal in the data, learns this and adapts the amount of shrinkage/smoothing appropriately. This self-tuning capability is important for two reasons. First it makes the method easy to use by non-experts, who may find appropriate specification of tuning parameters challenging. Second, it means that the method can be safely applied “in production" to large numbers of datasets in settings, like genomics, where it is impractical to hand-select appropriate tuning parameters separately for every dataset.
Our results here demonstrate that SMASH provides a flexible, fast and accurate approach to smoothing and denoising. We illustrated the flexibility by applying it to two challenging problems - Gaussian heteroskedastic regression, and smoothing of Poisson signals. In both cases our method is consistently competitive in accuracy with existing approaches. And although SMASH requires more computation than a simple thresholding rule, it is fast enough to deal with large problems. This is partly because optimizing over the unimodal distribution in is a convex optimization problem that can be solved very stably and quickly [@Stephens038216]. For example, our implementation, which exploits the [mosek]{} convex optimization library [@mosek], typically takes less than 30s for 100,000 observations. SMASH requires multiple applications of (specifically, because we apply at each resolution level, it takes $\log_2(T)$ applications of in the Poisson case), but is still fast enough to be practical for moderately large problems. For example, smoothing a signal of length 32,768 ($2^{15}$) takes less than a minute for the Poisson case, and less than 2 minutes for the Gaussian, on a modern laptop. It is likely these times could be further improved by more efficient implementations.
Besides its accuracy for point estimation, SMASH also has the advantage that it naturally provides measures of uncertainty in estimated wavelet coefficients, which in turn provide measures of uncertainty (e.g. credible bands) for estimated mean and variance functions, which may be useful in some applications.
Although we have focussed here on applications in one dimension, there is nothing to stop being similarly applied to multi-scale approaches in higher dimensions, such as image denoising, as in @Nowak1999Multiscale for example. For some types of images alternatives to wavelets, such as curvelets [@Cande00], may produce better results, and extending our work to those settings could be an interesting area for future work. More generally, provides a generic approach to shrinkage estimation that could be useful in a range of applications beyond the signal-denoising and smoothing applications considered here.
Reference
=========
{#app:var estimation}
**Variance estimation for Gaussian denoising**
With $\bm{Z}$ as defined in , we apply the wavelet transform $W$ to $\bm{Z}^2$, and obtain the wavelet coefficients $\bm{\Gd}=W\bm{Z}^2$. Note that $\mathbb{E}(\bm{\Gd})=(\bm{\Gg})$, where $\bm{\Gg}=W\bm{\s}^2$. We treat the likelihood for $\bm{\Gg}$ as if it were independent, resulting in $$\begin{aligned}
L(\bm{\Gg}|\bm{\Gd})=\prod_{j=0}^J\prod_{k=0}^{T-1}P(\Gd_{jk}|\Gg_{jk}).\end{aligned}$$ The likelihoods $L(\Gg_{jk}|\Gd_{jk})$ are not normal, but we approximate the likelihood by a normal likelihood through matching the moments of a normal distribution to the distribution $P(\Gd_{jk}|\Gg_{jk})$. That is, $$\begin{aligned}
P(\Gd_{jk}|\Gg_{jk})\approx N(\Gg_{jk},\hat{\mathbb{V}}(\Gd_{jk}))\end{aligned}$$ so that $$\begin{aligned}
\label{eq:gaus approx}
L(\Gg_{jk}|\Gd_{jk})\approx \phi(\Gd_{jk};\Gg_{jk},\mathbb{V}(\Gd_{jk}))\end{aligned}$$ where $\phi$ is the normal density function, and $\mathbb{V}(\Gd_{jk})$ is the variance of the empirical wavelet coefficients. Since these variances are unknown, we estimate them from the data and then proceed to treat them as known. More specifically, since $Z_t\sim N(0,\s_t^2)$, we have that $$\begin{aligned}
&\mathbb{E}(Z_t^4)\approx 3\s_t^4\notag\\
\label{eq:varvarest}&\mathbb{V}(Z_t^2)\approx 2\s_t^4\end{aligned}$$ and so we simply use $\frac{2}{3}Z_t^4$ as an unbiased estimator for $\mathbb{V}(Z_t^2)$. It then follows that $\hat{\mathbb{V}}(\Gd_{jk})$ is given by $\sum_{l=1}^T \frac{2}{3}Z_l^4W_{jk,l}^2$, and is unbiased for $\mathbb{V}(\Gd_{jk})$. These will be the inputs to , which then produces shrunk estimates in the form of posterior means for the corresponding parameters. Although this works well in most cases, there are variance functions for which the above procedure tends to overshrink the wavelet coefficients at the finer levels. This is likely because the distribution of the wavelet coefficients is extremely skewed, especially when the true coefficients are small (at coarser levels the distributions are much less skewed since we are dealing a linear combination of a large number of data points). One way around this issue is to employ a procedure that jointly shrinks the coefficients $\bm{\Gg}$ and their variance estimates (implemented in the `jash` option in our software). The final estimate of the variance function is obtained from the posterior means via the average basis inverse across all the shifts.
Poisson denoising {#app:reconstruction}
=================
First summarize the data in a recursive manner by defining: $$\begin{aligned}
Y_{Jk}\equiv Y_k\end{aligned}$$ for $k=1,...,T$, with $T=2^J$, and $$\begin{aligned}
Y_{jk}=Y_{j+1,2k}+Y_{j+1,2k+1}\end{aligned}$$ for resolution $j=0,...,J-1$ and location $k=0,...,2^j-1$. Hence, we are summing more blocks of observations as we move to coarser levels.
This recursive scheme leads to: $$\begin{aligned}
Y_{jk}=\sum_{l=k2^{J-j}+1}^{(k+1)2^{J-j}}Y_l\end{aligned}$$ for $j=0,...,J$ and $k=0,...,2^j-1$.
Similarly, define the following: $$\begin{aligned}
\mu_{Jk}\equiv \mu_k\end{aligned}$$ for $k=1,...,T$, and $$\begin{aligned}
\mu_{jk}=\mu_{j+1,2k}+\mu_{j+1,2k+1}\end{aligned}$$ for $j=0,...,J-1$ and $k=0,...,2^j-1$. And define $$\begin{aligned}
\label{eq:poisson wc}\Ga_{jk}=\log(\mu_{j+1,2k})-\log(\mu_{j+1,2k+1})\\\end{aligned}$$ for $s=0,...,J-1$ and $l=0,...,2^j-1$. The ${\Ga}$’s defined this way are analogous to the (true) Haar wavelet coefficients for Gaussian signals.
Using this recursive representation, the likelihood for $\bm{\Ga}$ factorizes into a product of likelihoods, where $\bm{\Ga}$ is the vector of all the $\Ga_{sl}$’s. See @Kolaczyk1999Bayesian for example. Specifically, $$\begin{aligned}
L(\bm{\Ga}|\mathbf{Y})&=&P(\mathbf{Y}|\bm{\Ga})\\
&=&P(Y_{0,0}|\mu_{0,0})\prod_{j=0}^{J-1}\prod_{k=0}^{2^j-1}P(Y_{j+1,2k}|Y_{j,k},\Ga_{j,k})\\
&=&L(\mu_{0,0}|Y_{0,0})\prod_{j=0}^{J-1}\prod_{k=0}^{2^j-1}L(\Ga_{j,k}|Y_{j+1,2k},Y_{j,k}).\end{aligned}$$ Note that $Y_{00}|\mu_{00}\sim \textrm{Pois}(\mu_{00})$. For any given $j,k$, $Y_{jk}$ is a sum of two independent Poisson random variables, and is itself a Poisson random variable. Hence $$Y_{j+1,2k}|Y_{jk},\Ga_{jk}\sim \textrm{Bin}({Y_{jk},\frac{1}{1+e^{-\Ga_{jk}}}\equiv\frac{\mu_{j+1,2k}}{\mu_{jk}}})$$
Estimates and standard errors for $\alpha_j$
--------------------------------------------
Each $\alpha_{j}$ is a ratio of the form $\log(\mu_{a:b}/\mu_{c:d})$ whose maximum likelihood estimate (mle) is $\log(Y_{a:b}/Y_{c:d})$. The main challenge here is that the mle is not well behaved when either the numerator or denominator of $Y_{a:b}/Y_{c:d}$ is 0. To deal with this, when either is 0 we use Tukey’s modification [@Gart1967Bias]. Specifically, letting $S$ denote $Y_{a:b}$, $F$ denote $Y_{c:d}$ and $N=S+F$ (corresponding to thinking of these as successes and failures in a binomial experiment, given $Y_{a:b}+Y_{c:d}$), we use $$\begin{aligned}
\label{eq:pseudoMLE1}
&&\hat{\Ga}=\left\{
\begin{array}{lll}
\log\{(S+0.5)/(F+0.5)\}-0.5&\ \ \ S=0\\
\log\{S/F\}&\ \ \ S=1,2,...,N-1\\
\log\{(S+0.5)/(F+0.5)\}+0.5&\ \ \ S=N\\
\end{array}
\right.\\ \label{eq:pseudoMLE1se}
&&se(\hat{\Ga})=\sqrt{V^*(\hat{\Ga})-\frac{1}{2}\{V_3(\hat{\Ga})\}^2\left\{V_3(\hat{\Ga})-\frac{4}{N}\right\}}\end{aligned}$$ where $$\begin{aligned}
&&V_3(\hat{\Ga})=\frac{N+1}{N}\left(\frac{1}{S+1}+\frac{1}{F+1}\right)\ \ \ S=0,...,N\\
\label{eq:pseudoMLE2}&&V^*(\hat{\Ga})=V_3(\hat{\Ga})\left\{1-\frac{2}{N}+\frac{V_3(\hat{\Ga})}{2}\right\}\end{aligned}$$ The square of the standard error in corresponds to $V^{\ast\ast}$ from @Gart1967Bias [pp. 182], and is chosen because it is less biased for the true variance of $\hat{\Ga}$ (when $N$ is small) as compared to the asymptotic variance of the MLE [see @Gart1967Bias]. The other two variance estimators from @Gart1967Bias, $V_1^{++}$ and $V^{++}$, were also considered in simulations and gave similar results, but $V^{\ast\ast}$ was chosen for its simple form.
Signal reconstruction
---------------------
The first step to reconstructing the signal is to find the posterior means of $p_{jk}:=\frac{\mu_{j+1,2k}}{\mu_{jk}}$ and $q_{jk}:=\frac{\mu_{j+1,2k+1}}{\mu_{jk}}$ (for $j=0,...,J-1$ and $k=0,...,2^j-1$). Specifically, for each $j$ and $k$, we require $$\begin{aligned}
\label{eq:pfromwc1}
&&E(p_{jk})\equiv E\left(\frac{e^{\Ga_{jk}}}{1+e^{\Ga_{jk}}}\right)\\
\label{eq:pfromwc2}&&E(q_{jk})\equiv E\left(\frac{e^{-\Ga_{jk}}}{1+e^{-\Ga_{jk}}}\right).\end{aligned}$$ Given the posterior means and variances for $\Ga_{jk}$ from , we can approximate - using the Delta method. First, define $$\begin{aligned}
\label{eq:ff}\ff(x)=\frac{e^x}{1+e^x}\end{aligned}$$ and consider the Taylor expansion of $\ff(x)$ about $\ff(E(x))$: $$\begin{aligned}
\label{eq:delta}\ff(x)\approx \ff(E(x))+\ff'(E(x))(x-E(x))+\frac{\ff''(E(x))}{2}(x-E(x))^2\end{aligned}$$ where $$\begin{aligned}
\label{eq:fderiv}&&\ff'(x)=\frac{e^x}{(1+e^x)^2}\\
\label{eq:sderiv}&&\ff''(x)=\frac{e^x(1-e^{x})}{(1+e^x)^3}\end{aligned}$$ Thus $$\begin{aligned}
&&E(p_{jk})\approx \ff(E(\Ga_{jk}))+\frac{\ff''(E(\Ga_{jk}))}{2}Var(\Ga_{jk}),\\
\label{eq:Ep}&&E(q_{jk})\approx \ff(-E(\Ga_{jk}))+\frac{\ff''(-E(\Ga_{jk}))}{2}Var(\Ga_{jk}),\end{aligned}$$ which can be computed by plugging in $E(\Ga)$ and $Var(\Ga)$ from .
Finally, we approximate the posterior mean for $\mu_t$ by noting that $\mu_t$ can be written as a product of the $p$’s and $q$’s for any $i=1,2,...,T$. Specifically, let $\{c_1,...,c_J\}$ be the binary representation of $i-1$, and $d_m=\sum_{j=1}^m c_j2^{m-j}$ for $j=1,...,J-1$. Then $$\begin{aligned}
\label{eq:product}\mu_k=\mu_{00}p_{00}^{1-c_1}p_{1,d_1}^{1-c_2}...p_{J-1,d_{J-1}}^{1-c_J}q_{00}^{c_1}q_{1,d_1}^{c_2}...q_{J-1,d_{J-1}}^{c_J}\end{aligned}$$ where we usually estimate $\mu_{00}$ by $\sum_l Y_l$ (see Kolaczyk (1999)). Using the independence of the $p$’s and $q$’s from different scales, we have: $$\begin{aligned}
\label{eq:Eproduct}E(\mu_t)=\mu_{00}E(p_{00})^{1-c_1}E(p_{1,d_1})^{1-c_2}...E(p_{J-1,d_{J-1}})^{1-c_J}\notag\\
\cdot E(q_{00})^{c_1}E(q_{1,d_1})^{c_2}...E(q_{J-1,d_{J-1}})^{c_J}.\end{aligned}$$
We can also approximate the posterior variance of $\mu_t$. (This allows creation of an approximate credible interval by making a normal approximation.) From we have: $$\begin{aligned}
\label{eq:E2product}E(\mu_t^2)=\mu_{00}^2E(p_{00}^2)^{1-c_1}E(p_{1,d_1}^2)^{1-c_2}...E(p_{J-1,d_{J-1}}^2)^{1-c_J}\notag\\
\cdot E(q_{00}^2)^{c_1}E(q_{1,d_1}^2)^{c_2}...E(q_{J-1,d_{J-1}}^2)^{c_J}.\end{aligned}$$ To compute this we again use the Delta method (with $\ff(x)=(\frac{e^x}{1+e^x})^2$) to obtain: $$\begin{aligned}
&E(p_{jk}^2)\approx \left(\ff(E(\Ga_{jk}))+\frac{\ff''(E(\Ga_{jk}))}{2}Var(\Ga_{jk})\right)^2+\notag\\
& \hspace{1.5 in}\{\ff'(E(\Ga_{jk}))\}^2Var(\Ga_{jk})\\
&E(q_{jk}^2)\approx \left(\ff(-E(\Ga_{jk}))+\frac{\ff''(-E(\Ga_{jk}))}{2}Var(\Ga_{jk})\right)^2+\notag\\
& \hspace{1.5 in}\{\ff'(E(-\Ga_{jk}))\}^2Var(\Ga_{jk}).\end{aligned}$$ Finally we combine and to find $Var(\mu_k)$.
Translation Invariance {#app:TI}
----------------------
It is common in multi-scale analysis to perform analyses over all $T$ circulant shifts of the data, because this is known to consistently improve accuracy. (The $t$-th circulant shift of the signal $\bm{Y}$ is created from $\bm{Y}$ by moving the first $T-t$ elements of $\bm{Y}$ $t$ positions to the right and then putting the last $t$ elements of $\bm{Y}$ in the first $t$ locations.)
To implement this in practice, we begin by computing the $\Ga$ coefficients, and their corresponding standard errors, for all $T$ circulant shifts of the data. This is done efficiently (in $O(\log_2 T)$ operations), using ideas from @Coifman1995Translationinvariant. Details are as in @Kolaczyk1999Bayesian; indeed our software implementation benefited from Matlab code provided by @Kolaczyk1999Bayesian for the TI table construction, which we ported to C++ and integrated with R using Rcpp [@eddelbuettel2011rcpp].
This yields a table of $\Ga$ coefficients, with $T$ coefficients at each of $\log_2(T)$ resolution levels, and a corresponding table of standard errors. As in the Gaussian case we then apply separately to the $T$ coefficients at each resolution level, to obtain a posterior mean and posterior variance for each $\Ga$. We then use the methods above to compute quantities of interest averaged over all $T$ shifts of the data. For example, our final estimate of the mean signal is given by $$\begin{aligned}
\label{eq:TIapproxexp}\frac{1}{T}\sum_{t=1}^T \hat{\mu}_k^{(t)}\end{aligned}$$ where $\hat{\mu}_k^{(t)}$ denotes the posterior mean of $\mu_k$ computed from the $t$-th circulant shift of the data. Again, using idea from @Coifman1995Translationinvariant this averaging can be done in $O(\log_2 T)$ operations.
[^1]: The authors gratefully acknowledge *please remember to list all relevant funding sources in the unblinded version*
|
---
abstract: 'We report the experimental demonstration of four-photon quantum interference using telecom-wavelength photons. Realization of multi-photon quantum interference is essential to linear optics quantum information processing and measurement-based quantum computing. We have developed a source that efficiently emits photon pairs in a pure spectrotemporal mode at a telecom wavelength region, and have demonstrated the quantum interference exhibiting the reduced fringe intervals that correspond to the reduced de Broglie wavelength of up to the four photon ‘NOON’ state. Our result should open a path to practical quantum information processing using telecom-wavelength photons.'
address:
- 'Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan'
- 'Center for Frontier Science and Engineering, University of Electro-Communications, Tokyo 182-8585, Japan'
- 'Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan'
- 'Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan'
- 'Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan'
author:
- Masahiro Yabuno
- Ryosuke Shimizu
- Yasuyoshi Mitsumori
- Hideo Kosaka
- Keiichi Edamatsu
title: 'Four-Photon Quantum Interferometry at a Telecom Wavelength'
---
A variety of novel quantum optical technologies have been proposed for use in quantum information processing and quantum metrology [@nielsen0521635039; @Nature390_575]. Photons are the most promising and practical media to demonstrate such novel quantum technologies. Indeed, linear optical quantum computing [@Nature409_46] and measurement-based quantum computing [@PhysRevA68_022312] have attracted much attention. However, use of the latest quantum technologies requires a large number of multiple photons at the same time. Furthermore, the photons must be indistinguishable from each other because the quantum operations rely on quantum interference between photons. Thus, we need practical and efficient sources that can provide many, indistinguishable photons. To date, quantum processing including up to six photons has been demonstrated [@NaturePhys3_91; @PhysRevLett103_20504]. These demonstrations used near infrared (800-nm band) photons, for which efficient photon sources and reliable single-photon detectors are available. However, use of telecom-band (1.5-$\mu$m band) photons is desired for practical purposes. Here we report an experiment with four-photon quantum interference using telecom-wavelength photons, in which the photons exhibited reduced ($\propto N^{-1}$) fringe intervals corresponding to the number ($N$) of photons [@PhysRevLett82_2868; @PhysRevLett89_213601; @Nature429_161; @Nature429_158; @PhysRevA74_33812; @Science316_726; @Science328_879]. Combined with the recent developments of novel photon-detecting devices [@NaturePhoton3_12], our result should open a path to practical quantum information processing using telecom-wavelength photons.
The most popular photon sources so far used for the demonstration of quantum information processing are based on spontaneous parametric down-conversion (SPDC), which generates twin (signal and idler) photons that can be used as entangled photons [@PhysRevLett75_4337] or heralded single photons [@OptComm246_545]. The linear optical quantum computing [@Nature409_46] and measurement-based quantum computing [@PhysRevA68_022312] both rely on quantum interference between photons to carry out quantum operations and quantum measurements. Photons generated by SPDC must be indistinguishable from each other to make them interfere. To do so, spectrotemporal purity of the photons is essential [@PhysRevA56_1627]. However, in general, photons generated by SPDC have spectrotemporal correlation [@OptExpress17_16385] that destroys the purity of each photon. Spectral filtering has often been used to purify the spectrotemporal modes of photons; however, such filtering inevitably reduces the generation efficiency. Thus, efficient generation of spectrotemporal-correlation-free photons is indispensable for multi-photon quantum interference.
The recent development of group-velocity-matched parametric down-conversion (GVM-SPDC) has made it possible to generate spectrotemporal-correlation-free photons with high efficiency [@PhysRevA64_063815; @NJPhysics10_093011; @PhysRevLett100_133601; @OptExpress18_3708; @PhysRevLett105_253601; @PhysRevLett106_013603; @PhysRevLett94_083601]. In the telecom wavelength region, GVM-SPDC with periodically-poled $\rm KTiOPO_4$ (PPKTP) has been demonstrated [@PhysRevLett105_253601; @PhysRevLett106_013603; @PhysRevLett94_083601; @OptExpress17_16385], and two-photon interference between the generated twin photons has been examined. For further demonstration of multi-photon interference, one can generate spectrotemporal-correlation-free photons by controlling the pump bandwidth and the phase-matching bandwidth [@PhysRevLett105_253601; @PhysRevLett106_013603; @PhysRevLett94_083601]. Furthermore, in the GVM-SPDC, one can use longer crystals than those used in standard SPDC, and thereby realize the efficient generation of spectrotemporal-correlation-free photons. Thus, the GVM-SPDC with PPKTP is a promising source for multi-photon quantum interferometry in the telecom wavelength region.
Using a photon source based on the GVM-SPDC, we examined a multi-photon (up to four photons) quantum interference that exhibited fringe intervals inversely proportional to the number of photons concerned. The reduced fringe intervals correspond to the photonic “de Broglie wavelength" [@PhysRevLett74_4835]; an ensemble of multiple photons exhibits the effective de Broglie wavelength of $\lambda/N$, where $\lambda$ and $N$ are the classical wavelength of light and the number of the constituent photons, respectively. This unique phenomenon can be applied to, for instance, super-resolution [@PhysRevLett104_123602] and super-sensitivity [@Science316_726] in the phase measurement beyond the classical or the standard quantum limit (SQL). It is known that the reduced fringe interval is observed in the multi-photon entangled state called the “NOON" state $({|N, 0\rangle}_{\rm AB} - {|0, N\rangle}_{\rm AB}) / \sqrt{2}$, where ${|N, 0\rangle}_{\rm AB}$ $\left({|0, N\rangle}_{\rm AB}\right)$ represents the state in which there are $N$ (zero) photons in the optical mode A with zero ($N$) photons in the mode B. Thus far, the reduced de Broglie wavelength with the NOON state has been demonstrated for $N=2$ [@PhysRevLett82_2868; @PhysRevLett89_213601], $3$ [@Nature429_161], $4$ [@Nature429_158; @PhysRevA74_33812; @Science316_726] and $5$ [@Science328_879]. Note that all these demonstrations were carried out using 800-nm-band photons, mainly because efficient photon sources and reliable single-photon detectors are available in this wavelength region. Our experiment is, to the best of our knowledge, the first demonstration of the four photon quantum interference using telecom band photons.
![\[fig:MZI\] (a) Path-mode and (b) polarization-mode Mach-Zehnder interferometer. BS1, BS2: beam splitters; PS: phase shifter; LCVR: liquid crystal variable retarder. ](PolarizationMZI_v5){width="0.7\hsize"}
In the following, we explain the principle of our quantum interferometry using the conventional Mach-Zehnder interferometer (MZI) as shown in Fig. \[fig:MZI\](a). In the actual experiment, as described later, we used the polarization-mode MZI as shown in Fig. \[fig:MZI\](b). First, we consider the input quantum state ${|1, 0\rangle}_{\rm AB}$, which contains a single photon in the path mode A and a vacuum in the mode B. After BS1 and PS, the state becomes the single photon NOON state: $({|1, 0\rangle}_{\rm CD} - e^{i\phi}{|0, 1\rangle}_{\rm CD}) / \sqrt{2}$. After BS2, the probability of detecting the photon in the mode F is $(1 -\cos\phi)/2$, which exhibits the normal fringe interval as in the classical interference. Next, we consider the state ${|1, 1\rangle}_{\rm AB}$ as the input. Twin photons generated from degenerate SPDC can be used as this state. After BS1 and PS, this state becomes the two photon NOON state: $({|2, 0\rangle}_{\rm CD} - e^{2i\phi}{|0, 2\rangle}_{\rm CD}) / \sqrt{2}$. Note that the state ${|1, 1\rangle}_{\rm CD}$ is missing because of the destructive Hong-Ou-Mandel (HOM) interference [@PhysRevLett59_2044] at BS1, and that the phase difference is enhanced by a factor of two because the two photons experience the same phase shift together. As a result, the probability of detecting the two photons together in the mode E (or F) is $(1 - {\rm cos}2\phi)/4$, which exhibits a fringe interval that is reduced by a factor of 2 [@PhysRevLett89_213601]. Finally, we consider the state ${|2, 2\rangle}_{\rm AB}$ as the input. This state can be generated from a couple of simultaneously generated twin photons from degenerate SPDC, provided that all photons are indistinguishable from each other. After BS1 and PS, the state becomes the NOON-like state: $\sqrt{3/8}({|4, 0\rangle}_{\rm CD} + {|0, 4\rangle}_{\rm CD}) - {|2, 2\rangle}_{\rm CD}/ 2$. This state contains the four photon NOON state as well as the unwanted ${|2, 2\rangle}_{\rm CD}$ term. However, since the unwanted term does not produce the state ${|3, 1\rangle}_{\rm EF}$ after BS2 while the NOON state does, one can observe the interference that originates only from the NOON state by the post selective detection of the final state ${|3, 1\rangle}_{\rm EF}$. The detection probability of the state ${|3, 1\rangle}_{\rm EF}$ is $3(1 - {\rm cos}4\phi) / 16$; we can observe a fourfold reduction of the fringe interval [@PhysRevA65_33820].
![\[fig:Experimental\_setup\] Experimental layout. LCVR: liquid crystal variable retarder; HWP: half-wave plate; PBS: polarizing beam splitter; SPCM: fiber-coupled single-photon counting module; C.C.: coincidence counter.](Experimental_setup_v4){width="0.7\hsize"}
In the experiment, we used a polarization-mode MZI with a liquid crystal variable retarder (LCVR) as shown in Fig. \[fig:MZI\](b). The function of the polarization-mode MZI is the same as that of the path-mode MZI; the polarization modes H, V, $\rm H'$, $\rm V'$, $\rm H''$ and $\rm V''$ in Fig. \[fig:MZI\](b) correspond to the path modes A, B, C, D, E and F in Fig. \[fig:MZI\](a), respectively. The LCVR whose slow and fast axes ($\rm H'$ and $\rm V'$) are rotated by $45^\circ$ with respect to the original H and V polarizations functions as the beamsplitters (BS1 and BS2) as well as the phase shifter (PS) in the path-mode MZI. The polarization-mode MZI provides superior phase stability during the long-time measurement required in our multi-photon experiment. Figure \[fig:Experimental\_setup\] sketches the experimental setup. A mode-locked Ti:sapphire laser operating at a center wavelength of 792 nm and a repetition rate of 80 MHz was used as a pump source of the SPDC. A 30-mm-long PPKTP crystal with a poling period of 46.1 $\rm\mu m$ was used for the type-II GVM-SPDC. The pump light was focused (beam waist: 50 $\rm\mu m$) onto the center of the PPKTP crystal and generated twin photons having orthogonal polarizations at a center wavelength of 1584 nm. The GVM-SPDC generates the twin photons with a positive spectral correlation [@OptExpress17_16385]. Combined with the negative spectral correlation originating from the pump spectral bandwidth, one can control the joint spectral distribution of the produced twin photons [@PhysRevLett105_253601; @PhysRevLett106_013603]. In our experiment, the pump bandwidth (full width at half maximum) was adjusted to be 0.4 nm (corresponding to a pulse duration of 2.3 ps) to generate the photons with eventually no spectrotemporal correlation. The joint spectral distribution was observed by a couple of tunable bandpass filters followed by the coincidence detection of the twin photons (not shown in Fig. \[fig:Experimental\_setup\] ). A 15-mm-long KTP crystal whose crystallographic axes are rotated by 90$^\circ$ with respect to the PPKTP crystal compensates for the temporal retardation difference between photons having horizontal (H) and vertical (V) polarizations, and thus makes the photons indistinguishable except for their polarizations.
![\[fig:Exp2photonspec\] Measured joint spectral distribution of photon pairs generated by GVM-SPDC with a PPKTP crystal.](2spe_3_shimizu){width="0.7\hsize"}
Then the photons having H and V polarizations were led into the polarization-mode MZI. After passing through the MZI, the photons were led into single mode fibers and then detected by the four single-photon detectors (id Quantique id201), one of which was for the $\rm V''$ polarization mode and the other three of which were for the $\rm H''$ mode. The generated twin photon state in the lowest order was ${|1, 1\rangle}_{\rm HV}$. For the single-photon interference, we prepared the state ${|1, 0\rangle}_{\rm HV}$ by blocking the V-polarized photon with a polarizing beam splitter, and observed the photon in the $\rm V''$ mode. For the two-photon interference, we used the state ${|1, 1\rangle}_{\rm HV}$ as the input and observed the photons in the state ${|2, 0\rangle}_{\rm H''V''}$ by the twofold coincidence between the two detectors in the $\rm H''$ mode. For the four-photon interference, the state ${|2, 2\rangle}_{\rm HV}$, a pair of simultaneously generated twin photons, was used. This was done by selecting the events with fourfold coincidence detection. We observed the photons in the final state ${|3, 1\rangle}_{\rm H''V''}$ by the fourfold coincidence between the three detectors in the $\rm H''$ mode and one detector in the $\rm V''$ mode.
![\[fig:4photon\_interference\] Measured interference patterns on (A) one-, (B) two- and (C) four-photon interference.](4PhotonInterference_v2){width="0.75\hsize"}
Figure \[fig:Exp2photonspec\] shows the measured joint spectral distribution of the photon pairs generated from our photon source. The signal and idler photons have almost identical spectral shapes with a bandwidth of 1.7 nm centered at 1583.9 nm. More importantly, the joint spectral distribution exhibits practically no spectral correlation between the signal and idler, indicating that the photons are generated in a pure, indistinguishable spectrotemporal mode. In fact, the Schmidt number [@NJPhysics10_093011] obtained from the measured joint spectral distribution was $K_{\rm exp}$=1.01, indicating almost no spectral correlation. This value was in good agreement with the expected Schmidt number ($K_{\rm calc}$=1.01) for the joint spectral distribution $| \sigma(\omega_s,\omega_i) |^2 = | f(\omega_s,\omega_i) g(\omega_s+\omega_i) |^2$ calculated from the amplitudes of phase-matching function $f(\omega_s,\omega_i)$ and pump spectrum $g(\omega_p)= g(\omega_s+\omega_i) $, where $\omega_p$, $\omega_s$, and $\omega_i$ are the frequencies of pump, signal, and idler photons, respectively. The Schmidt number expected for the joint spectral amplitude $\sigma(\omega_s,\omega_i) = f(\omega_s,\omega_i) g(\omega_s+\omega_i)$ was $K$=1.21, which corresponds to a good spectral purity of signal or idler photons, i.e, $P$=$K^{-1}$=0.83. Thus, this source allows us to conduct the multi-photon interference experiment without spectral filtering. The photon pair production rate of our source was as high as 48,000 pairs/mW/sec. In the following experiments, we used a pump power of $\sim$50 mW, from which we expect the mean photon-pair number of $3.0\times10^{-2}$ pairs/pulse. Detailed characteristics of our photon source will be discussed elsewhere.
Figure \[fig:4photon\_interference\] shows the measured interference fringes for (a) one-, (b) two- and (c) four-photon NOON states, as a function of the single-photon phase difference from 0 to $\pi$. We see that the fringe interval of the $N$-photon interference was $2\pi/N$, exhibiting the $1/N$ reduction expected from the theory. Also note that the fringe visibilities ($V$) are quite high: $V$=0.98, 0.88 and 0.74 for one-, two-, and four-photon interferences, respectively. Our four-photon interference clearly exhibited the reduced fringe interval with high visibility that demonstrated the super resolution beyond the classical limit ($V$=0.20) [@PhysRevLett104_123602]. However, it did not reach the threshold ($V$=0.816) [@Science316_726] required to beat the SQL in the phase sensitivity. The degraded fringe visibility is in part attributable to the imperfect purity ($P$=0.83) of our photon pair source. The rest of the degradation might have originated from uncorrelated background counts and the slight spectral difference between the signal and idler photons. In conclusion, we have generated spectrotemporal-correlation-free photons via GVM-SPDC at telecom wavelength. Using the intrinsically indistinguishable photons thus obtained, we demonstrated the quantum multi-photon interference up to the four-photon NOON state. To our knowledge, this is the first demonstration of four-photon quantum interference using photons all in a telecom wavelength. To date, the major problems with photonic quantum information processing in the telecom wavelength region were the lack of efficient photon sources and photon detectors. Recently, we have seen steep progress in photon-detecting devices such as transition edge sensors and superconducting nanowire single photon detectors working in the telecom wavelength region [@NaturePhoton3_12]. Our results, combined with the latest advancements in novel photon-detecting devices, will open a path to practical quantum information processing using telecom-wavelength photons.
This work was supported by a Grant-in-Aid for Creative Scientific Research (17GS1204) and a Grant-in-Aid for JSPS Fellows (22$\cdot$7182) from the Japan Society for the Promotion of Science, and JST PRESTO program.
[99]{}
M. A. Nielsen, and I. L. Chuang, , (Cambridge Univ. Press, Cambridge, 2000).
D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, , 575–579 (1997).
E. Knill, R. Laflamme, and G. J. Milburn, , 46–52 (2001).
R. Raussendorf, D. E. Browne, and H. J. Briegel, , 022312 (2003).
C. Y. Lu, X.Q. Zhou, O. G[ü]{}hne, W.B. Gao, J. Zhang, Z. S. Yuan, A. Goebel, T. Yang, and J. W. Pan, , 91–95 (2007).
W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. T[ó]{}th, and H. Weinfurter, , 20504 (2009).
E. J. S. Fonseca, C. H. Monken, and S. P[á]{}dua, , 2868–2871 (1999).
K. Edamatsu, R. Shimizu, and T. Itoh, , 213601 (2002).
M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, , 161–164 (2004).
P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, , 158–161 (2004).
F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, , 33812 (2006).
T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, , 726 (2007).
I. Afek, O. Ambar, and Y. Silberberg, , 879 (2010).
R. H. Hadfield, , 696–705 (2009).
P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, , 4337–4341 (1995).
T. B. Pittman, B. C. Jacobs, and J. D. Franson, , 545–550 (2005).
W. P. Grice, and I. A. Walmsley, , 1627 (1997).
R. Shimizu, and K. Edamatsu, , 16385–16393 (2009).
W. P. Grice, A. B. U’Ren, and I. A. Walmsley, , 063815 (2001).
P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, , 093011 (2008).
P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, , 133601 (2008).
Z. H. Levine, J. Fan, J. Chen, A. Ling, and A. Migdall, , 3708–3718 (2010).
P. G. Evans, R. S. Bennink, W. P. Grice, T. S. Humble, and J. Schaake, , 253601 (2010).
A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn, , 13603 (2011).
O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, , 083601 (2005).
J. Jacobson, G. Bj$\ddot{o}$rk, I. Chuang, and Y. Yamamoto, , 4835–4838 (1995).
I. Afek, O. Ambar, and Y. Silberberg, , 123602 (2010).
C. K. Hong, Z. Y. Ou, and L. Mandel, , 2044–2046 (1987).
O. Steuernagel, , 33820 (2002).
|
---
abstract: 'Bike-sharing systems are becoming an urban mode of transportation. In such systems, users arrive at a station, take a bike and use it for a while, then return it to another station of their choice. Each station has a finite capacity: it cannot host more bikes than its capacity. A stochastic model is proposed to study the effect of users random choices on the number of problematic stations, *i.e.*, stations that host zero bikes or that have no available spots at which a bike can be returned. The influence of the stations’ capacities is quantified and the fleet size that minimizes the proportion of problematic stations is computed. Even in a homogeneous city, the system exhibits a poor performance: the minimal proportion of problematic stations is to the order of (but not lower than) the inverse of the capacity. We show that simple incentives, such as suggesting users to return to the least loaded station among two stations, improve the situation by an exponential factor. We also compute the rate at which bike has to be redistributed by trucks to insure a given quality of service. This rate is to the order of the inverse of the stations capacity. For all cases considered, the optimally reliable fleet size is a little more than half of the station capacity, the value of the *little more* depends on the system parameters.'
author:
- Christine Fricker
- Nicolas Gast
bibliography:
- 'ref.bib'
title: 'Incentives and Redistribution in Bike-Sharing Systems with Stations of Finite Capacity'
---
Introduction
============
Bike-sharing systems (BSS) are becoming a public mode of transportation devoted to short trips. A few BSS have been launched since Copenhagen laucnhed theirs in 1995. BSS were widely deployed in the 2000s after Paris launched the large-scale program called Velib, in July 2007. Velib consists of 20000 available bikes and 1500 stations. Nowadays, there are more than $400$ cities equipped with BSS around the world (see [@demaio2009bike] for a history of BSS). This gives rise to a recent research activity.
The concept of BSS is simple: A user arrives at a station, takes a bike, uses it for a while and then returns it to any station. A lack of resources is one of the major issues: a user can arrive at a station that host no bike, or can want to return her bike at a station with no empty spot. The allocation of resources, bikes and empty places, has to be managed by the operator in order to offer a reliable alternative to other transportation modes.
The strength of such a system is its ability to meet the demand, in bikes and empty spots. This demand is complex. It depends on the time of the day, the day of the week (week or week-end), the season and the weather, but also the location: housing or working areas generate going-and-coming flows; flows are also generated from up-hill to down-hill stations. This creates unbalanced traffic during the day. Moreover, the system is stochastic due to the arrivals at the stations, the origin-destination pairs and the trip lengths. The lack of resources also generates random choices from the users, who must search for another station. These facts are supported by several data analyses, *e.g.*, [@Borgnat-1; @Froehlich-1; @Nair-1].
When building a bike-sharing system, a first strategic decision is the planning of the number of stations, their locations and their size. Other long-term operation decisions involve static pricing and deciding on the number of bikes in the system. Several papers have studied these issues. See [@katzev2003car; @shaheen2007growth; @demaio2004will; @demaio2009bike; @Nair-2]. They study the static planning problem based on economical aspects and growth trend. Another important research direction concerns bike repositioning: to solve the problem of unbalanced traffic, bikes can be moved by the operator, either during the night when the traffic is low (static repositioning) or during the day (dynamic repositioning). [@Raviv-1] study the optimal placement of bikes at the beginning of a day. [@Chemla-1] develop an algorithm that minimizes the distance traveled by trucks to achieve a given bike positioning, assuming that bikes do not move (*e.g.*, during the night). The dynamic aspects are studied by [@Contardo-1] but their model ignores bike moves between periodical updates of the system, or by [@Hampshire-1; @Raviv-2]; they model the station states. These papers mainly rely on optimization techniques and model the arrival and departure of users at each station as stochastic processes.
Redistribution can also been done by users. In most bike-sharing systems, users have access to the real-time state of the stations, *e.g.*, by using a smartphone. They can choose to take or to return their bikes to neighboring stations of their destinations. Moreover they can be encouraged to do so by the system. For example, Paris, via the Velib+ system, offers a static reward by giving free time slots to bring more bikes to up-hill stations. It is estimated that the number of people that obtain rewards via Velib+ during the day is equivalent to the number of bikes redistributed by truck. To compensate for real-time congestion problems, an alternative is to use real-time pricing mechanisms. This type of congestion control mechanism is widely applied in the transportation or car-rental industry, *e.g.*, [@guerriero2012revenue; @waserhole2012vehicle]. Nevertheless, its application to BSS is unclear, as the price paid per trip for using BSS is usually low.
A few papers tackle the stochasticity of BSS or more generally of vehicle rental systems (see [@Godfrey-1; @George-2] and literature therein). Their first idea is to obtain an simplified asymptotic behavior when the system gets large. This approximation is valid as BSS are large systems and can give qualitative and quantitative properties for the model. In models with product-form steady-state distribution (see [@George-1; @George-2]), the asymptotic expansion of the partition function can be obtained via complex analysis (saddle point method), see [@Malyshev-6], or probabilistic tools, see [@Fayolle-7]. One of the main limitations of these papers is that they ignore that stations have finite capacities and therefore neglect the saturation effect.
**Contributions** – In this paper, we present a model of bike-sharing systems and we analyze its steady-state performance. The system is composed of a large number of stations and a fleet of bikes. Each station can host up to a finite number of bikes $K$, called its *capacity*. We measure the performance in terms of proportion of so-called [*problematic stations*]{}, *i.e.*, stations with no bikes or no empty spot.
The framework is a simple scenario in which each station has the same parameters. In a sense, it is a best-case analysis of a system in which the flow of bikes between two stations is, on average, identical in both directions. We investigate the effects of imbalance due to the random choices of users and we characterize the influence of the station capacity on the performance. The main point is that the model is the simplest model to deal with. It enables us to compare incentives and redistribution mechanisms and obtain closed-form characteristics of their performance. Moreover, this model can be straightforwardly extended to model non-homogeneous cities in order to take into account the difference of attractivity among the stations. The paper by [@velib-aofa] provides analytical results when the stations can be grouped into clusters of stations that have the same characteristics. These results will be briefly presented and discussed in this paper. Incentive mechanisms and redistribution in an heterogeneous setting are not studied by [@velib-aofa] but will be included in their upcoming paper [@velib-2choices-2clusters].
Our first contribution is to study the simplest model without any incentives or redistribution mechanisms. Our main argument is to use mean-field methods, which enables us to obtain the asymptotic behavior for our model and its variants, as the system size becomes large. This methods works even if close form (product-form) expressions are not available for the original model. This method has some similarities with stochastic networks appearing in other applications as communication networks. This asymptotic dynamics leads to simple expressions that give qualitative and quantitative results. The proportion of problematic stations depends on the fleet size and decreases slowly with the capacity $K$. The optimally reliable fleet size is given in closed form. It is equal to $K/2+\lambda/\mu$ bikes per station, where $\lambda$ is the arrival rate of users at a station and $1/\mu$ is the average trip time. This answers the fleet sizing problem. The term $\lambda/\mu$ quantifies the quite intuitive idea that the greater the demand is, the more bikes must be put in the system. For this fleet size, the proportion of problematic stations is $2/(K+1)$.
To improve the situation, we investigate two different directions: incentives and redistribution. A practical implementation of incentives is not discussed. We rather assume that users have access to real-time information on the system and follow the rules. The improvement that is obtained in that case is quantified. We show that returning bikes to a non-saturated station does not change significantly the behavior of the stations and the performance with our metric. The situation improves dramatically when users return their bikes to the least loaded station among two, even if only a fraction of the users do this. We show that in this case, the proportion of problematic stations can be as low as $\sqrt{K}2^{-K/2}$. These results are confirmed by simulations in which users choose among two neighbooring stations. Again, the optimal fleet size is a little more than $K/2$.
We then study what we call the redistribution rate in Section \[sec:regulation\]. We define the redistribution rate as the ratio of the number of bikes that have to be moved manually by trucks over the number of bikes that are taken by users. It is proved that the redistribution rate which optimizes performance depends on the fleet size and the station capacity. The redistribution rate needed to suppress problematic stations is minimal when the fleet size is $K/2+\lambda/\mu$ bikes per station and is equal to $1/(K-1)$.
Finally, we discuss in Section \[sec:validation\] the limitations of the model. We describe briefly the differences that will occur when considering a time- or space-inhomogeneous model. We mainly refer to the paper by [@velib-aofa], which main result is an extension of the expression of minimal proportion of problematic stations in a cluster at some fleet size, which generalizes $s=K/2$. We also show simulations results of a more realistic model that takes the geometry into account. In all cases, this model behaves closely to the original mean-field model.
**Organization of the paper** – Section \[sec:model\] presents the model description and the mean field techniques. Section \[sec:symmetric\] deals with the basic model for the homogeneous system where stations have finite capacity. Incentive mechanisms where a fraction of users choose the least loaded of two stations to return their bike are studied in Section \[sec:incentives\]. The rate of redistribution by trucks which optimizes performance is computed in Section \[sec:regulation\]. Section \[sec:validation\] deals with simulation validations and the conclusion is presented in Section \[sec:conclusion\].
System Model and Mean-Field Analysis {#sec:model}
====================================
Basic Model and Optimal Fleet Size {#sec:symmetric}
==================================
Incentives and the Power of Two Choices {#sec:incentives}
=======================================
Optimal Redistribution Rate {#sec:regulation}
===========================
Validation of the Model: Discussion and Simulations {#sec:validation}
===================================================
Conclusion and Future Work {#sec:conclusion}
==========================
In this paper, we investigate the influence of the station capacities on the performance of homogeneous bike-sharing systems. Using a stochastic model and a fluid approximation, we provide analytical expressions for the performance. They are summarized in Table \[tab:capacity\]. The optimal fleet size is approximately $K/2$ for all models. Without using incentives, the capacity has only a linear effect on the performance or on the optimal redistribution rate. For this purpose, an incentive to return bikes to the least loaded station among two improves dramatically the performance, even if a small proportion of users accept to do this. Moreover, even if this model does not take into account any geographic aspect of the system, simulations show that these results also hold when considering simple geometric models with local interactions.
[|c|c|c|]{} &
-------------------------
Minimal proportion
of problematic stations
-------------------------
: Summary of the main results: influence of the station capacity $K$ on the proportion of problematic stations.
&Optimal fleet size $s$\
Original model & $2/(K+1)$ &$s=K/2+\lambda/\mu$\
Two-choice & $\sqrt{K}2^{-K/2}$ & $s-\lambda/\mu\in[K/2;K-\log_2(K)]$\
Regulation& 0 if $\gamma\ge\lambda/(K-1)$ & $s=K/2+\lambda/\mu$\
\[tab:capacity\]
Our results prove that the mismatch of performance due to random choices should not be neglected when studying the performance of a bike-sharing system. Even in a completely balanced system, they dramatically affect the performance. A natural extension of this work is to consider stations with different parameters. The steady-state performance of such a system is given by [@velib-aofa]. It proves that, without repositioning via incentives or trucks, the performance is very poor. One interesting question is whether the steady-state performance can be used as a metric in a system with varying operation conditions, such as peak-hours and non-peak hours. Our work can serve as a building block for studying the effect of incentives and redistribution mechanisms. Studying practical implementations of these mechanisms in real-world systems is postponed for future work. Morevoer, the transient behavior of such mechanisms in a city where the attractiveness of stations varies over time could be studied.
|
---
abstract: 'We consider the problem of finding the resonances of the Laplacian on truncated Riemannian cones. In a similar fashion to Cheeger–Taylor, we construct the resolvent and scattering matrix for the Laplacian on cones and truncated cones. Following Stefanov, we show that the resonances on the truncated cone are distributed asymptotically as $Ar^{n} + o(r^{n})$, where $A$ is an explicit coefficient. We also conclude that the Laplacian on a non-truncated cone has no resonances.'
address:
- 'Department of Mathematics, Texas A&M University'
- 'Department of Mathematics, Northwestern University'
author:
- Dean Baskin
- Mengxuan Yang
title: Scattering resonances on truncated cones
---
Introduction {#sec:introduction}
============
In this note, we consider the resonances on truncated Riemannian cones and establish a Weyl-type formula for their distribution. To fix notation, we let $(Y,h)$ be a compact $(n-1)$-dimensional Riemannian manifold (with or without boundary) and let $C(Y)$ denote the cone over $Y$. In other words, $C(Y)$ is diffeomorphic to the product $(0,\infty)_{r}\times Y$ and is equipped with the incomplete Riemannian metric $g = dr^{2} + r^{2}h$. We refer the reader to the foundational work of Cheeger–Taylor [@CT1; @CT2] for more details on the geometric set-up. We also introduce the *truncated* Riemannian cone $C_{a}(Y)$ formed by introducing a boundary at $r=a$, i.e., $C_{a}(Y)$ is diffeomorphic to $[a, \infty)_{r} \times Y$ and equipped with the same metric.
The (negative-definite) Laplacian on $C(Y)$ (or $C_{a}(Y)$ with a choice of boundary conditions) has the form $${\partial_{r}}^{2} + \frac{n-1}{r}{\partial_{r}} + \frac{1}{r^{2}}{\Delta}_{h},$$ where ${\Delta}_{h}$ denotes the Laplacian of $(Y,h)$. Its resolvent $R(\lambda)$ is given by $$R(\lambda) = ({\Delta}+ \lambda^2)^{-1}.$$ We consider the *cutoff resolvent* $\chi R(\lambda)\chi$, where $\chi$ is a (fixed) smooth compactly supported function on $C(Y)$ (or $C_a(Y)$). One consequence of the resolvent formula of Theorem \[thm:resolvent\] is that the cutoff resolvent extends meromorphically to the logarithmic cover of ${\mathbb{C}}\setminus \{ 0 \}$.
More precisely, we identify elements $\lambda$ of the logarithmic cover of ${\mathbb{C}}\setminus \{0 \}$ by a magnitude $|\lambda|$ and a phase $\arg \lambda \in {\mathbb{R}}$. We identify the “physical half-plane” as those $\lambda$ with $\arg \lambda \in (0, \pi)$. These $\lambda$ correspond to the resolvent set ${\mathbb{C}}\setminus [0, \infty)$ via the map $\lambda \mapsto |\lambda|^{2}e^{2i\arg\lambda}$. The cutoff resolvent then extends to be meromorphic as a function of $\lambda$ on this logarithmic cover.
The poles of the cutoff resolvent consist of possibly finitely many $L^2$-eigenvalues lying in the upper half-plane (which do not appear with Dirichlet boundary conditions) and poles lying on other sheets of the cover. The latter poles are called the *resonances* of ${\Delta}$.
The main theorem of this paper counts the most physically relevant resonances for the truncated cone. In particular, we count those resonances $\lambda$ nearest to the physical half-plane, i.e., those with $\arg \lambda \in (-\frac{\pi}{2}, 0)$ and $\arg \lambda \in (\pi, \frac{3\pi}{2})$. The resonances on other “sheets” of the cover remain more mysterious and are related to the zeros of Hankel functions near the real axis. We consider the resonance counting function on these sheets, defined by $$N(r) = \# \left\{ \lambda : \lambda \text{ is a resonance and
}|\lambda| \leq r\right\}.$$ The following theorem provides an asymptotic formula for $N(r)$.
\[thm:main-thm\] Suppose either that the set of periodic geodesics of $(Y,h)$ has Liouville measure zero or that $Y = {\mathbb{S}}^{n-1}$ equipped with a constant rescaling of the standard metric. Consider the truncated cone $C_1(Y)$ equipped with the Dirichlet Laplacian and let $N(r)$ denote its resonance counting function on the neighboring sheets as above. We then have, as $r\to \infty$, $$N(r) = A_{n}\operatorname{Vol}(Y,h) r^n + {o}(r^n),$$ where $A_{n}$ is an explicit constant (defined below in equation ) and $\operatorname{Vol}(Y,h)$ denotes the volume of the Riemannian manifold $(Y,h)$.
The constant $A_{n} \operatorname{Vol}(Y,h)$ in Theorem \[thm:main-thm\] is the same constant as computed by Stefanov [@Stefanov] for the resonance counting function on the domain exterior to a ball in ${\mathbb{R}}^{n}$. When $Y = \mathbb{S}^{n-1}$ is equipped with its standard metric, the truncated cone $C_{1}(Y)$ can can be thought of as the exterior of the unit ball in Euclidean space. Theorem \[thm:main-thm\] recovers Stefanov’s result. (When $Y =
\mathbb{S}^{n-1}$, $n$ odd, is equipped with its standard metric, the cutoff resolvent in fact continues to the complex plane; this can be seen in the resolvent formulae below.)
We also state the following theorem, which is known to the community but does not seem to be in the literature.
\[thm:full-cone\] If $(Y,h)$ is a compact Riemannian manifold (with or without boundary) then the cone $C(Y)$ has no resonances.
In fact, Theorem \[thm:resolvent\] below shows that $\lambda$ is a resonance of the truncated cone $C_{1}(Y)$ if and only if $\lambda /
a$ is a resonance of the truncated cone $C_{a}(Y)$. Sending $a$ to $0$ then pushes all resonances out to infinity and provides evidence for Theorem \[thm:full-cone\].
The proof of Theorem \[thm:main-thm\] has two main steps. We first separate variables and obtain an explicit resolvent formula in Theorem \[thm:resolvent\] to characterize the resonances as zeros of a Hankel function. In Section \[sec:reson-count-trunc\] we consider the asymptotic distribution of the zeros of each Hankel function appearing in the resolvent formula. The hypothesis on the link $(Y,h)$ is used to control the error terms when synthesizing the result. Theorem \[thm:full-cone\] is an immediate corollary of the resolvent formula in Theorem \[thm:resolvent\].
The proof of Theorem \[thm:main-thm\] follows an argument of Stefanov [@Stefanov] very closely. Stefanov established a Weyl-type law for the distribution of resonances for the exterior of a ball in odd-dimensional Euclidean space. The main contribution of this paper is the observation that, after some natural modifications, the core of Stefanov’s argument applies to the setting of cones. Borthwick [@borthwick:2010; @borthwick:2012] and Borthwick–Philipp [@Borthwick-Philipp] showed that a similar approach works in the asymptotically hyperbolic setting.
We further remark that we have specialized to the Dirichlet Laplacian in Theorem \[thm:main-thm\] only for simplicity. For Neumann or Robin boundary conditions, the resolvent formula of Theorem \[thm:resolvent\] has an analogous expression. The resonance counting problem then involves counting zeros of $H^{(2)'}_{\nu} + C\nu H^{(2)}_{\nu}$, which can be handled with similar arguments.
Resolvent construction
======================
In this section we write down an explicit formula (via separation of variables) for the resolvent and then show that the cut-off resolvent has a meromorphic continuation to the logarithmic cover $\Lambda$ of the complex plane. The construction is essentially contained in the work of Cheeger–Taylor [@CT1; @CT2], but the resolvent is not explicitly written there.
Suppose that $\phi_j$ form an orthonormal family of eigenfunctions for $-{\Delta}_h$ with corresponding eigenvalues $\mu_j^2$. We decompose $L^{2}(C(Y))$ into a direct sum in terms of the eigenspaces of $-{\Delta}_{h}$, i.e., $$L^{2}(C_{a}(Y); {\mathbb{C}}) = \bigoplus_{j=0}^{\infty}L^{2}((a,
\infty); E_{j}), \quad f(r,y) = \sum_{j=0}^{\infty}f_{j}(r) \phi_{j}(y),$$ where the first space is defined with respect to the volume form induced by the metric and the latter spaces can be identified (via the identification $f(r)\phi_{j}(y) \mapsto f(r)$) with the space $L^{2}((a,\infty); {\mathbb{C}})$ equipped with the volume form $r^{n-1}\,dr$.
For $\arg \lambda \in (0,\pi)$, the resolvent $R(\lambda)$ splits as a direct sum of operators $R_{j}(\lambda)$ acting on $L^{2}((a,\infty), E_{j})$, with measure $r^{n-1}\,dr$. $$R(\lambda)\left( \sum_{j=1}^\infty f_j(r) \phi_j(y)\right) = \bigoplus_{j=1}^\infty \left( R_j(\lambda)f_j \right)\phi_j (y).$$
In this section, we prove the following explicit formula for the $j$-th piece of the resolvent. For the cone $C(Y)$ (i.e., for $a=0$), we use the Friedrichs extension of the Laplacian to guarantee self-adjointness (though in high enough dimension the Laplacian is essentially self-adjoint):
\[thm:resolvent\] The piece of the resolvent corresponding to the $j$-th eigenvalue has the following explicit expression on the truncated cone $C_{a}(Y)$ or the cone $C(Y)$ ($a=0$): $$(R_{j}(\lambda) f)(r) = \int_{a}^{\infty}K_{a,j}(r,\tilde{r}) f(\tilde{r}) \tilde{r}^{n-1}\,d\tilde{r}$$ where $K_{a,j}(r,\tilde{r})$ is given by $$K_{a,j}(r,\tilde{r}) = \frac{\pi}{2i}\left(
\tilde{r}r\right)^{-\frac{n-2}{2}}
\begin{cases}
H^{(1)}_{\nu_{j}}(\lambda \tilde{r}) J_{\nu_{j}}(\lambda r) -
\frac{J_{\nu_{j}}(\lambda a)}{H^{(1)}_{\nu_{j}}(\lambda
a)}H^{(1)}_{\nu_{j}}(\lambda \tilde{r})
H^{(1)}_{\nu_{j}}(\lambda r) & r < \tilde{r} \\
J_{\nu_{j}}(\lambda \tilde{r}) H^{(1)}_{\nu_{j}} (\lambda r) -
\frac{J_{\nu_{j}}(\lambda a)}{H^{(1)}_{\nu_{j}}(\lambda
a)}H^{(1)}_{\nu_{j}}(\lambda \tilde{r})
H^{(1)}_{\nu_{j}}(\lambda r) & r > \tilde{r}
\end{cases}$$ Here $J_{\nu}$ are the standard Bessel functions of the first kind and $H^{(1)}_{\nu}$ are the Hankel functions of the first kind. The second term in both expressions should be interpreted as $0$ when $a=0$.
After separating variables, we may assume that $f = f_j(r)\phi_j(y)$. We construct the resolvent for $\Im \lambda > 0$ and then meromorphically continue the expression.
Writing $u = u_j (r) \phi_j(y)$, the equation $( {\Delta}+ \lambda^2) u = f$ induces the following differential equation for $u_j$: $$\label{eq:inhomog}
{\partial_{r}}^2 u_j+ \frac{n-1}{r}{\partial_{r}}u_j - \frac{\mu_j^2}{r^2}u_j + \lambda^2 u_j = f_j.$$ We solve this equation by showing it is equivalent to a Bessel equation.
Changing variables to $\rho = \lambda r$ and writing $\tilde{u}(\rho) = u(\rho / \lambda)$ yields $${\partial_{\rho}}^2\tilde{u} + \frac{n-1}{\rho}{\partial_{\rho}}\tilde{u} +
\left( 1 - \frac{\mu_j^2}{\rho^2}\right) \tilde{u} = \frac{1}{\lambda^{2}}\tilde{f}(\rho).$$ Writing $v = \rho ^{(n-2)/2}\tilde{u}$, we obtain a Bessel equation for $v$: $$\label{eq:Bessel}
v '' + \frac{1}{\rho}v' + \left( 1 - \frac{\nu_j^2}{\rho^2}\right) v =
g(\rho),$$ where $\nu_j^2 = \mu_j^2 + \left( \frac{n-2}{2}\right)^2$ and $g(\rho ) = \frac{\rho^{(n-2)/2}}{\lambda^2}\tilde{f}(\rho)$.
We now proceed by the standard ODE technique of variation of parameters. One basis for the space of solutions of the homogeneous version of this Bessel equation is $\{ J_{\nu_{j}}(\rho), H^{(1)}_{\nu_{j}}(\rho)\}$, where $J_{\nu}$ is the Bessel function of the first kind and $H^{(1)}_{\nu}$ is the Hankel function of the first kind. We thus may use the following basis for the space of solutions of the homogeneous equation: $$\label{eq:basis}
w_{1}(r) = r^{-(n-2)/2}J_{\nu_{j}}(\lambda r), \quad w_{2}(r) = r^{-(n-2)/2}H^{(1)}_{\nu_{j}}(\lambda r)$$
For $\Im \lambda > 0$, $R_{j}(\lambda) f_{j}$ must lie in $L^{2}((a,\infty), r^{n-1}\,dr)$. If $f_{j}$ is compactly supported, this means that $u_{j} = R_{j}(\lambda)f_{j}$ must be a multiple of $r^{-(n-2)/2}H^{(1)}_{\nu_{j}}(\lambda r)$ near infinity. When $a >0$, $u_{j}$ must satisfy the boundary condition at $r=a$. When $a=0$, the choice of the Friedrichs extension requires that both $u_{j}$ and $u_{j}'$ lie in the the weighted $L^{2}$ space near $0$ and so $u_{j}$ must be a multiple of $r^{-(n-2)/2}J_{\nu_{j}}(\lambda r)$ near $r=0$ as any nonzero multiple of $w_{2}$ will not have this property.
We may thus write $$u_{j}(r) = \left( \int_{r}^{\infty}
\frac{w_{2}(\tilde{r})f_{j}(\tilde{r})}{W(w_{1}, w_{2})(\tilde{r})}\, d\tilde{r} \right) w_{1}(r) +
\left( C + \int_{a}^{r} \frac{w_{1}(\tilde{r})f_{j}(\tilde{r})}{W(w_{1},
w_{2})(\tilde{r})}\,d\tilde{r}\right) w_{2}(r),$$ where $C$ is a yet-to-be-determined constant, the functions $w_{1}$ and $w_{2}$ are as in equation , and $W(w_{1}, w_{2})$ is their Wronskian. The Wronskian $W$ can be easily computed in terms of the Wronskian of the Bessel and Hankel functions and seen to be $$W(w_{1}, w_{2})(r) = r^{-(n-1)}\cdot \frac{2i}{\pi}.$$
We now turn our attention to the boundary condition. For $a = 0$, the requirement that the solution and its derivative live in $L^{2}$ forces $C = 0$, yielding the result. For $a \neq 0$, we require that $u_{j}(a) = 0$, i.e., $$\left( \frac{\pi}{2i} \int_{a}^{\infty} H^{(1)}_{\nu_{j}}(\lambda
\tilde{r}) \tilde{r}^{\frac{n}{2}}f(\tilde{r}) \,d\tilde{r}\right) a^{-(n-2)/2} J_{\nu_{j}} (\lambda
a) + C a^{-(n-2)/2}H^{(1)}_{\nu_{j}}(\lambda a) = 0,$$ and so we must have $$C = - \frac{\pi}{2i} \frac{J_{\nu_{j}}(\lambda
a)}{H^{(1)}_{\nu_{j}}(\lambda a)} \int_{a}^{\infty}
H^{(1)}_{\nu_{j}}(\lambda \tilde{r}) \tilde{r}^{\frac{n}{2}}f(x) \, dx,$$ finishing the proof.
We now claim that $\chi R(\lambda)\chi$ has a meromorphic continuation:
Given a fixed $\chi \in C^\infty_c({\mathbb{R}}_+ \times Y)$, $\chi R(\lambda) \chi$ meromorphically continues from $$\{ \lambda \in \mathbb{C} : \Im \lambda > 0 \}$$ to the logarithmic cover $\Lambda$ of the complex plane.
We first prove the statement for the full cone; the statement for the truncated cone will follow by an appeal to the analytic Fredholm theorem.
Fix $\chi \in C^{\infty}_{c}((0,\infty))$ and regard $\chi (r)$ as a compactly supported smooth function on $C(Y)$. We let $R(\lambda)$ denote the resolvent on the non-truncated cone (i.e., $a=0$) and $K(\lambda; r, y, {\widetilde{r}}, {\widetilde{y}})$ denote its integral kernel. In order to show that $\chi R(\lambda)\chi$ meromorphically continues, it suffices to show that for any $f, g \in L^{2}(C(Y))$, the function $$\lambda \mapsto \langle \chi R(\lambda) \chi f, g\rangle$$ meromorphically continues to $\Lambda$.
Fix two such functions $f, g\in L^{2}(C(Y))$ and let $f_{j}(r)$ and $g_{j}(r)$ denote their coefficients in the expansion in terms of eigenfunctions of ${\Delta}_{h}$, i.e., $$f(r,y) = \sum_{j=0}^{\infty}f_{j}(r) \phi_{j}(y).$$ We observe that because $f$ and $g$ are square-integrable, the sum and the integral commute, i.e., $${\| f\|_{L^{2}(C(Y))}}^{2} = \int_{0}^{\infty}
\sum_{j=0}^{\infty}|f_{j}(r)|^{2} r^{n-1}\,dr =
\sum_{j=0}^{\infty} \int_{0}^{\infty}|f_{j}(r)|^{2} r^{n-1}\,dr.$$
From Theorem \[thm:resolvent\], we may write $$\begin{aligned}
\label{eq:pair-expansion}
\langle \chi R(\lambda) \chi f, g \rangle &=
\sum_{j=0}^{\infty} \left(
\int_{0}^{\infty}\int_{0}^{r}
({\widetilde{r}}r)^{-\frac{n-2}{2}}
\chi(r)\chi({\widetilde{r}})
f_{j}({\widetilde{r}}) g_{j}(r)
J_{\nu_{j}}(\lambda
{\widetilde{r}})
H_{\nu_{j}}^{(1)}(\lambda
r) {\widetilde{r}}^{n-1} r^{n-1}
\,d{\widetilde{r}}\,dr \right. \notag\\
&\quad \left. + \int_{0}^{\infty}\int_{r}^{\infty} ({\widetilde{r}}r)^{-\frac{n-2}{2}} \chi(r) \chi({\widetilde{r}}) f_{j}({\widetilde{r}}) g_{j}(r)
J_{\nu_{j}}(\lambda r) H_{\nu_{j}}^{(1)}(\lambda {\widetilde{r}}) {\widetilde{r}}^{n-1}
r^{n-1}\, d{\widetilde{r}}\,dr \right),
\end{aligned}$$ where $J_{\nu}$ and $H_{\nu}^{(1)}$ are as above. Because each term in equation meromorphically continues to the Riemann surface $\Lambda$, it suffices to show that the partial sums of the series converge locally (in $\lambda$) uniformly (in $j$).
By the asymptotic expansions of Bessel functions for large order, we know [@DLMF 10.19] that, locally in $\lambda \in \Lambda$, and for $r\in \operatorname{supp}\chi$, $$\begin{aligned}
J_{\nu}(\lambda r) &= \frac{1}{\sqrt{2\pi \nu}} \left(
\frac{e \lambda r}{2\nu}\right)^{\nu} + {o}\left( \frac{1}{\sqrt{\nu}}
\left( \frac{e \lambda r}{2\nu}\right)^{\nu}\right), \\
H_{\nu}^{(1)} (\lambda r) &= \frac{1}{i} \sqrt{\frac{2}{\pi \nu}}
\left( \frac{e \lambda
r}{2\nu}\right)^{-\nu} +
{o}\left( \frac{1}{\sqrt{\nu}}
\left( \frac{e \lambda r}{2\nu}\right)^{-\nu}\right),
\end{aligned}$$ as $\nu \to \infty$ through the positive reals. In particular, for $j$ large enough, each term in equation can be bounded by $$\begin{aligned}
&C \int_{0}^{\infty}\int_{0}^{r} \frac{1}{\pi \nu_{j}} \chi(r)
\chi({\widetilde{r}}) f_{j}({\widetilde{r}}) g_{j}(r) \left[ \left(
\frac{{\widetilde{r}}}{r}\right)^{\nu_{j}} (1 + o (1))\right] ({\widetilde{r}}r)^{\frac{n}{2}}\, d{\widetilde{r}}\, dr \\
&\quad \quad+ C\int_{0}^{\infty}\int_{r}^{\infty} \frac{1}{\pi \nu_{j}} \chi(r)
\chi({\widetilde{r}}) f_{j}({\widetilde{r}}) g_{j}(r) \left[ \left(
\frac{r}{{\widetilde{r}}}\right)^{\nu_{j}} (1 + o (1))\right] ({\widetilde{r}}r)^{\frac{n}{2}}\, d{\widetilde{r}}\, dr.
\end{aligned}$$ Observe that in the first integral, ${\widetilde{r}}/ r$ is bounded by $1$, while $r/{\widetilde{r}}$ is bounded by $1$ in the second.
Because $\chi$ is compactly supported, we may therefore bound each term (for $j$ large enough) by $$\frac{C_{\chi}}{\nu_{j}} {\| f_{j}\|_{L^{2}}} {\| g_{j}\|_{L^{2}}}.$$ This sequence is absolutely summable, so the partial sums of the series in equation converge locally uniformly. This establishes that the cut-off resolvent on the full cone ($a=0$) meromorphically extends to the logarithmic cover $\Lambda$ of the complex plane.
We now proceed to the case of the truncated cone ($a > 0$). We proceed by an appeal to the analytic Fredholm theorem.
Fix $\chi_{0}, \chi_{\infty} \in C^{\infty}((a,\infty))$ so that $\chi_{0}(r)$ is supported near $r=a$, $\chi_{\infty}(r)$ is identically zero near $r=a$, and $\chi_{0} + \chi_{\infty} = 1$. We let $R_{\infty}(\lambda)$ denote the resolvent on the non-truncated cone and $R_{0}(\lambda)$ denote the resolvent on a compact manifold with boundary into which the support of $\chi_{0}$ embeds isometrically. We define the parametrix $$Q(\lambda) = \tilde{\chi}_{0} R_{0}(\lambda) \chi_{0} +
\tilde{\chi}_{\infty} R_{\infty}(\lambda) \chi_{\infty},$$ where $\tilde{\chi}$ have similar support properties and are identically $1$ on the support of their counterparts. Applying ${\Delta}+ \lambda^{2}$ yields a remainder of the form $I + \sum [{\Delta},
\tilde{\chi}_{i}] R_{i}(\lambda) \chi_{i}$. Both terms are compact and the operator is invertible for large $\Im \lambda$ by Neumann series, so applying $R_{a}(\lambda)$ to both sides and inverting the remainder shows that it has a meromorphic continuation.
Proof of Theorem \[thm:main-thm\] {#sec:reson-count-trunc}
=================================
By the formula for the resolvent in Theorem \[thm:resolvent\], the resonances of $R_{a}(\lambda)$ correspond to those $\lambda$ for which $H_{\nu_{j}}^{(1)}(\lambda a) = 0$ for some $j$. For simplicity we will discuss only the case $a=1$ as the other cases can be found by rescaling. As mentioned in the introduction, we consider only those resonances nearest to the upper half-plane, i.e., those with $$\label{eq:arg-restriction}
-\frac{\pi}{2} < \arg \lambda < 0 \quad \text{or} \quad \pi <
\arg \lambda < \frac{3\pi}{2}.$$
Because $\nu_{j}$ is real, we may relate the zeros of $H_{\nu_{j}}^{(1)}(\lambda)$ in the region given by equation to zeros of $H^{(2)}_{\nu_{j}}(\lambda)$ in the quadrant $0 < \arg \lambda < \frac{\pi}{2}$ via analytic continuation formulae. Indeed, it is well-known [@DLMF 10.11.5, 10.11.9] that $$\begin{aligned}
\label{eq:connection-formulae}
H^{(1)}_{\nu}(z e^{\pi}) &= -e^{-\nu \pi \imath} H^{(2)}_{\nu}(z),
\\
H^{(1)}_{\nu}(\overline{z}) &= \overline{H^{(2)}_{\nu}(z)}. \notag\end{aligned}$$ The first of these equations identifies zeros of $H^{(1)}_{\nu}$ in $\pi <
\arg\lambda < \frac{3\pi}{2}$ to zeros of $H^{(2)}_{\nu}$ in the first quadrant; the second equation does the same for zeros of $H^{(1)}_{\nu}$ with $-\frac{\pi}{2}< \arg \lambda < 0$. In particular, each zero of $H^{(2)}_{\nu}$ with $0 \leq \arg \lambda
\leq \pi / 2$ corresponds to exactly two resonances.
For large enough $\nu$, the zeros of the Hankel function $H^{(2)}_{\nu}$ in the first quadrant lie near the boundary of (a scaling of) an “eye-like” domain $K\subset {\mathbb{C}}$. The domain $K$ is symmetric about the real axis and is bounded by the following curve and its conjugate: $$z = \pm (t \coth t - t^{2})^{1/2} + i (t^{2} - t \tanh t)^{1/2},
\quad 0 \leq t \leq t_{0},$$ where $t_{0}$ is the positive root of $t = \coth t$. We refer to the piece of the boundary of $K$ lying in the upper half-plane by ${\partial_{}}K_{+}$.
The constant $A_{n}$ given above is given by the following: $$\label{eq:defn-ofAn}
A_{n} = \frac{2(n-1) \operatorname{Vol}(B_{n-1})}{n (2\pi )^{n}} \int_{{\partial_{}}K_{+}} \frac{|1-z^{2}|^{1/2}}{|z|^{n+1}}\,d|z|,$$ where $B_{n-1}$ is the $(n-1)$-dimensional unit ball. Observe that, up to a factor of the volume of the unit sphere (which is replaced by the volume of $Y$ in the theorem statement), the constant $A_{n}$ is the same constant computed by Stefanov [@Stefanov].
We use below two different parametrizations of the piece of ${\partial_{}}K_{+}$ lying the in the quadrant $0 \leq \arg z \leq \pi / 2$. The first parametrization is by the argument of $z$, i.e., by the map $$\left[ 0 , \frac{\pi}{2}\right] \to {\partial_{}}K_{+}, \quad \theta = \arg z
\mapsto z = z(\theta).$$
For the second parametrization, we introduce the function $\rho$, defined by $$\label{eq:rho-defn}
\rho(z) =\frac{2}{3} \zeta^{3/2} = \log \frac{1 + \sqrt{1-z^{2}}}{z} - \sqrt{1-z^{2}}, \quad
|\arg z | < \pi,$$ where (following Stefanov [@Stefanov Section 4] and Olver [@Olver-book Chapter 10]) the branches of the functions above are chosen so that $\zeta$ is real when $z$ is. Another characterization is that the principal branches are chosen when $0 < z
< 1$ and continuity is demanded elsewhere.
The boundary ${\partial_{}}K$ is the vanishing set of $\Re \rho$. This yields a parametrization of the part of ${\partial_{}}K_{+}$ lying in $0\leq \arg z
\leq \pi / 2$: $$\left[0, \frac{\pi}{2}\right] \to {\partial_{}}K_{+}, \quad t \mapsto \rho
^{-1}(- i t) = z.$$ The transition between the two parametrizations is given by $$\frac{dt}{d\theta} = \frac{dt}{dz} \frac{dz}{d\theta} = (i \rho' (z))
(iz) = \sqrt{1-z^{2}}.$$
The function $\zeta$ defined in equation is the solution of the ODE $$\left( \frac{d\zeta}{dz}\right) ^{2} = \frac{1-z^{2}}{\zeta z^{2}}$$ that is infinitely differentiable on the positive real axis (including at $z=1$). As is implicit in equation , it can be analytically continued to the complex plane with a branch cut along the negative real axis.
Because the resonances correspond to zeros of $H^{(2)}_{\nu_{j}}$, we must also consider the asymptotic distribution of the $\nu_{j}$. In what follows, we consider only the case when the periodic geodesics of $(Y,h)$ have measure zero.[^1] The eigenvalues $\mu_{j}^{2}$ of ${\Delta}_{h}$ obey Weyl’s law: $$\begin{aligned}
N_{h}(\mu) &= \# \{ \mu_{j} : \mu_{j} \leq \mu \text{ with
multiplicity }\} \\
&= \frac{\operatorname{Vol}{B_{n-1}}}{(2\pi)^{n-1}}\operatorname{Vol}(Y,h) \mu^{n-1} + R(\mu).\end{aligned}$$ Here $\operatorname{Vol}(B_{n-1})$ denotes the volume of the unit ball in ${\mathbb{R}}^{n-1}$ and $\operatorname{Vol}(Y,h)$ is the volume of $Y$ equipped with the metric $h$. In general, $R(\mu) = {O}(\mu^{n-2})$, but if we now impose the dynamical hypothesis (that the set of periodic geodesics of $(Y,h)$ has Liouville measure zero), then a theorem of Duistermaat–Guillemin [@DG] (in the boundaryless case) and Ivrii [@Ivrii1; @Ivrii2] (in the boundary case) shows that $$R(\lambda) = {o}(\mu^{n-2}).$$ The non-periodicity assumption then allows us to count eigenvalues on intervals of length one: $$\begin{aligned}
N_{h}(\mu, \mu+1) &=\#\{ \mu_{j} : \mu \leq \mu_{j} \leq \mu + 1
\text{ with multiplicity }\} \\
&= (n-1) \frac{\operatorname{Vol}(B_{n-1})}{(2\pi)^{n-1}}\operatorname{Vol}(Y,h)\mu^{n-2} + {o}(\mu^{n-2}).\end{aligned}$$ As $\nu_{j}^{2} = \mu_{j}^{2} + (n-2)^{2} / 4$, the same counting formula holds for $\nu_{j}$, i.e., $$\begin{aligned}
\label{eq:nu-counting}
N_{\nu} (\rho, \rho + 1) &= \# \{ \nu_{j} : \rho \leq \nu_{j} \leq
\rho + 1 \text{ with multiplicity }\} \notag\\
&= (n-1) \frac{\operatorname{Vol}(B_{n-1})}{(2\pi)^{n-1}}\operatorname{Vol}(Y,h)\rho^{n-2} + {o}(\rho^{n-2}).\end{aligned}$$
We now turn our attention to the zeros of the Hankel function $H_{\nu}^{(2)}(z)$ with $\arg z \in [0 , \pi / 2]$. An argument from Watson [@Watson pages 511–513] is easily adapted to give a precise count of the number of zeros of $H^{(2)}_{\nu}$ in this sector. Indeed, that argument shows that the number of zeros is given by the closest integer to $\nu / 2 - 1/4$ (when $\nu - 1/2$ is an integer, there is a zero on the imaginary axis and so rounds up).
As $\nu \to \infty$ through positive real values, we have an asymptotic expansion [@DLMF 10.20.6] relating the Hankel function to the Airy function $$\label{eq:h2-airy}
H^{(2)}_{\nu}(\nu z) \sim 2e^{i\pi / 3} \left(
\frac{4\zeta}{1-z^{2}}\right)^{1/4} \left( \frac{\operatorname{Ai}(e^{-2\pi
i/3}\nu^{2/3}\zeta)}{\nu^{1/3}} \sum_{k=0}^{\infty}
\frac{A_{k}(\zeta)}{\nu^{2k}} + \frac{\operatorname{Ai}'(e^{-2\pi
i/3}\nu^{2/3}\zeta)}{\nu^{5/3}} \sum_{k=0}^{\infty}\frac{B_{k}(\zeta)}{\nu^{2k}}\right).$$ Here $A_{k}$ and $B_{k}$ are real and infinitely differentiable for $\zeta \in {\mathbb{R}}$. This expansion is uniform in $|\arg z| \leq \pi - \delta$ for fixed $\delta > 0$. In particular, for large enough $\nu$, the zeros of the Hankel function are well-approximated by zeros of the Airy function and we may identify each zero $h_{\nu, k}$ of the Hankel function $H^{(2)}_{\nu}$ with a zero of the Airy function $\operatorname{Ai}(-z)$.
Let $a_{k}$ denote the $k$-th zero of the Airy function $\operatorname{Ai}(-z)$; all $a_{k}$ are positive and $$a_{k} = \left[ \frac{3}{2} \left(k \pi -
\frac{\pi}{4}\right)\right]^{2/3} + {O}(k^{-4/3}).$$
We now define $\lambda_{\nu, k}$ and $\widetilde{\lambda}_{\nu,k}$ via the Airy zeros and their leading approximations: $$\begin{aligned}
\lambda _{\nu, k} &= \nu \zeta ^{-1}(\nu^{-2/3} e^{-i
\frac{\pi}{3}}a_{k}) = \nu \rho^{-1}\left(
-i\frac{2}{3} a_{k}^{3/2} \nu^{-1}\right) \\
\widetilde{\lambda}_{\nu,k} &= \nu \rho^{-1} \left( - i \left( k -
\frac{1}{4}\right)\pi \nu^{-1}\right), \end{aligned}$$ where $k = 1, \dots, \floor{\nu /2 + 1/4}$. By the Hankel expansion , $|h_{\nu, k} -
\lambda_{\nu, k}| \leq C / \nu$ for large enough $\nu$ while $|h_{\nu,
k} - \widetilde{\lambda}_{\nu, k}| \leq C / \nu$ for large enough $\nu$ and $k$. As we have identified $\floor{\nu / 2 + 1 /4}$ approximate zeros, we can conclude that these account for all $h_{\nu,
k}$.
We now divide our attention into those zeros with small argument and those with large argument. We introduce the auxiliary counting function $$N(r, \theta_{1}, \theta_{2}) = \# \{ \sigma : \sigma \text{ is a
resonance with } |\sigma | \leq r, \arg \sigma \in [ \theta_{1}, \theta_{2}]\}.$$
We first address those with small argument. Fix $\epsilon > 0$ and consider those zeros with $|z| < r$ and $\arg z \in [0,\epsilon]$. We need count those $\lambda_{\nu,k}$ with $\arg \lambda _{\nu, k} \in [ 0, \epsilon]$ and $|\lambda _{\nu,
k} | \leq r$. As $|\lambda_{\nu,k}|$ is comparable to $\nu$, we can overcount these zeros by counting all $\lambda_{\nu, k}$ with argument in $[0, \epsilon]$ and $\nu \leq Cr$.
Because $|\rho| \leq C \epsilon^{3/2}$ for those $\lambda_{\nu, k}$ with $\arg \lambda _{\nu, k} \in [0,\epsilon]$, we must only count those $a_{k}$ with $a_{k} \leq C \nu^{2/3}\epsilon$. The leading order asymptotic [@DLMF 9.9.6] for the zeros of the Airy function shows that this number is ${O}(\nu \epsilon^{3/2})$.
We now count those resonances with argument in $[0,\epsilon]$. Putting together the asymptotic for $\nu_{j}$ in equation with the previous two paragraphs, we have (with $m(\nu_{j})$ denoting the multiplicity of $\nu_{j}$) $$\begin{aligned}
\label{eq:small-arg-est}
N(r, 0, \epsilon) &= \sum_{j = 1}^{\infty}m(\nu_{j}) \#\left\{ h_{\nu_{j}, k} :
|h_{\nu_{j},k}| \leq r , \arg h_{\nu_{j},k} \in
[ 0, \epsilon]\right\} \notag\\
&\leq \sum_{j=1}^{Cr} m(\nu_{j}) C\nu_{j} \epsilon^{3/2} \notag\\
&\leq C\epsilon^{3/2} \sum_{\rho = 0}^{Cr}\sum_{\nu_{j} \in [\rho,
\rho+1]}m(\nu_{j}) \rho \leq C \epsilon^{3/2}r^{n}. \end{aligned}$$
We now consider those resonances with argument in $[\epsilon , \pi /
2]$. For large enough $\nu$, the approximations $\widetilde{\lambda}_{\nu,k}$ are valid for these resonances. We count those approximate resonances with $\nu_{j} \in [ \rho, \rho +
1)$ and $\arg \lambda_{\nu,k} \in [\theta, \theta + \Delta\theta]$. We start by introducing, for fixed $\nu$, the number $\Delta k_{\nu}$ of $\widetilde{\lambda}_{\nu, k}$ with argument lying in $[\theta, \theta
+ \Delta \theta]$. Observe that the definition of $\widetilde{\lambda}_{\nu,k}$ relates $\Delta k_{\nu}$ with $\Delta t$ by $$\Delta k_{\nu} = \frac{\nu}{\pi}\Delta t + {O}(1),$$ where $\Delta t$ denotes the change in $t$ corresponding to $\Delta
\theta$ in the parametrizations above. Note that $\Delta t$ is *independent* of the choice of $\nu$. We can then write $$\begin{aligned}
\# \left\{ \widetilde{\lambda}_{\nu,k} : \nu_{j} \in [\rho, \rho+1), \arg
\widetilde{\lambda}_{\nu,k} \in [\theta , \theta + \Delta \theta]\right\} & =
\sum_{\rho
\leq
\nu_{j}
\leq
\rho+1}
m(\nu_{j})
\Delta
k_{\nu}
\\
&=
\sum_{\rho
\leq
\nu_{j}
<
\rho
+ 1}
m(\nu_{j})
\left(
\frac{\nu_{j}}{\pi}
\Delta
t +
{O}(1)\right) \end{aligned}$$
By the definition of the approximate zeros $\widetilde{\lambda}_{\nu,k}$, we can estimate their size $|\widetilde{\lambda}_{\nu,k}|$ in terms of $|z(\theta)|$, provided that $\arg \widetilde{\lambda}_{\nu,k}\in [\theta, \theta + \Delta
\theta]$, yielding $$|\widetilde{\lambda}_{\nu,k}| = \nu \left( |z(\theta)| + {O}( \Delta \theta)\right).$$ In particular, if $\nu_{j}|z(\theta)| \geq r$ but $|\lambda _{\nu, k}
| \leq r$, then $\nu_{j} \in \left[ \frac{r}{|z(\theta)|}(1 - c\Delta
\theta), \frac{r}{|z(\theta)|}\right]$. We may thus rewrite our counting function as follows: $$\begin{aligned}
\#\left\{
\widetilde{\lambda}_{\nu,k} :
|\widetilde{\lambda}_{\nu,k}|
\leq r , \arg
\widetilde{\lambda}_{\nu, k}
\in [\theta, \theta
+ \Delta \theta]\right\}
&=
\sum_{\substack{|\widetilde{\lambda}_{\nu, k}| \leq r \\ \arg \widetilde{\lambda}_{\nu, k} \in
[\theta, \theta + \Delta \theta]}} m(\nu_{j}) \\
&= \sum_{\substack{\nu_{j}|z(\theta)| \leq r \\ \arg \widetilde{\lambda}_{j,k}
\in [\theta, \theta + \Delta \theta]}} m(\nu_{j}) +
\sum_{\substack{\nu_{j} \in \left[ \frac{r}{|z(\theta)|} (1 - c
\Delta \theta) , \frac{r}{|z(\theta)|}\right] \\ \arg \widetilde{\lambda}_{\nu,
k} \in [\theta, \theta + \Delta \theta]}} m(\nu_{j}).\end{aligned}$$ By our improved Weyl’s law , the second term is ${O}(r^{n-2})$.
We now focus our attention on the first term (here $\floor{\cdot}$ denotes the “floor” function): $$\begin{aligned}
\sum_{\substack{\nu_{j}|z(\theta)| \leq r \\ \arg \widetilde{\lambda}_{j,k}
\in [\theta, \theta + \Delta \theta]}} m(\nu_{j}) &= \sum_{\rho =
0}^{\floor{r/|z|
-
1}}\sum_{\nu_{j}\in[\rho,
\rho + 1)}
\sum_{\arg
\widetilde{\lambda}_{\nu,k}\in
[\theta, \theta
+ \Delta
\theta]}
m(\nu_{j}) +
\sum_{\nu_{j}
\in
[\floor{r/z},
r/z]}
\sum_{\arg\widetilde{\lambda}_{\nu,k}\in
[\theta, \theta
+ \Delta
\theta]}
m(\nu_{j}) \\
&= \sum_{\rho = 0}^{\floor{r/|z| - 1}} \sum_{\nu_{j} \in [\rho,
\rho+1)}m(\nu_{j}) \Delta k_{\nu} + \sum_{\nu_{j}
\in
[\floor{r/z},
r/z]}
\sum_{\arg\widetilde{\lambda}_{\nu,k}\in
[\theta, \theta
+ \Delta
\theta]}
m(\nu_{j}) .\end{aligned}$$ Again by Weyl’s law, we observe that the second term is ${O}(r^{n-2})$. By relating $\Delta t$ and $\Delta k_{\nu}$ we can rewrite the first term: $$\begin{aligned}
\sum_{\rho = 0}^{\floor{r/|z| - 1}} \sum_{\nu_{j} \in [\rho,
\rho+1)}m(\nu_{j}) \Delta k_{\nu} &= \sum_{\rho =
0}^{\floor{r/|z|-1}}
\sum_{\nu_{j} \in [\rho, \rho
+ 1)}m(\nu_{j})
\frac{\nu_{j}}{\pi} \Delta t
+ \sum_{\nu_{j} \leq
\floor{r/|z|}} m(\nu_{j}) O(1).\end{aligned}$$ By Weyl’s law , the second term is ${O}(r^{n-1})$, so we again consider the first term.
As $\Delta t$ is independent of $\nu_{j}$, we may use Weyl’s law as well on the first term: $$\begin{aligned}
\sum_{\rho =0}^{\floor{r/|z|-1}} \sum_{\nu_{j} \in [\rho, \rho +
1)}m(\nu_{j}) \frac{\nu_{j}}{\pi} \Delta t &= \sum_{\rho =
0}^{\floor{r/|z|-1}}\left[\frac{n-1}{2^{n-1}\pi^{n}}\operatorname{Vol}(B_{n-1})\operatorname{Vol}(Y,h)\rho^{n-1}
\Delta t + {O}(\rho
^{n-2}) +
{o}(\rho^{n-1}) \Delta
t\right] \\
&= \frac{2(n-1)}{(2\pi)^{n}}\operatorname{Vol}(B_{n-1}) \operatorname{Vol}(Y,h) \Delta t \sum_{\rho
= 0}^{\floor{r/|z|-1}} \rho^{n-1} + {O}(r^{n-1}) +
{o}(r^{n})\Delta t \\
&= \frac{2(n-1)}{(2\pi)^{n}n}\operatorname{Vol}(B_{n-1})\operatorname{Vol}(Y,h) \frac{1}{n}\left(
\frac{r}{|z(\theta)|}\right)^{n} \Delta t + {O}(r^{n-1}) +
{o}(r^{n}) \Delta t .\end{aligned}$$
We finally introduce a Riemann sum in $t$ to understand this main term: $$\begin{aligned}
\label{eq:main-est}
\# \{ \widetilde{\lambda}_{\nu,k} &: |\widetilde{\lambda}_{\nu,k}|\leq r , \arg
\widetilde{\lambda}_{\nu,k} \in [\epsilon, \pi/2]\} \\
&=
\int_{t^{-1}(\epsilon)}^{\pi
/ 2} \left(
\frac{2(n-1)\operatorname{Vol}(B_{n-1})}{(2\pi)^{n}
n} \operatorname{Vol}(Y, h) \right)
\frac{r^{n}}{|z(\theta)|^{n}}
\, dt + {O}(r^{n-1}) + {o}(r^{n}) \notag \\
&= \frac{(n-1) \operatorname{Vol}(B_{n-1})}{(2\pi)^{n}n }\operatorname{Vol}(Y,h) r^{n} \int_{{\partial_{}}K_{+}} \frac{1}{|z(\theta)|^{n}} \,dt + {O}(\epsilon r^{n}) +
{o}(r^{n}) \notag \\
&= \left(\frac{(n-1)\operatorname{Vol}(B_{n-1})}{(2\pi)^{n} n} \operatorname{Vol}(Y,h) \int_{{\partial_{}}K_{+}} \frac{|1-z^{2}|^{1/2}}{|z|^{n+1}} \, d|z|\right) r^{n} +
{O}(\epsilon r^{n}) + {o}(r^{n})\notag \\
&= A_{n} \operatorname{Vol}(Y,h) r^{n} + {O}(\epsilon r^{n}). + {o}(r^{n}) \notag \end{aligned}$$ Here the prefactor of $2$ disappeared because the first integral parametrizes only half of ${\partial_{}}K_{+}$. It reappears in the statement of Theorem \[thm:main-thm\] because each zero here corresponds to two resonances (one on each sheet). We further observe that the constant $A_{n}\operatorname{Vol}(Y,h)$ agrees with the leading term found in the Euclidean case found by Stefanov [@Stefanov].
Sending $\epsilon$ to $0$ establishes the theorem for the approximate zeros $\lambda_{\nu,k}$. Because each $\lambda_{\nu,k}$ is in a $C/\nu$ neighborhood of a zero $h_{\nu,k}$, this finishes the proof of the theorem.
Acknowledgments {#sec:acknowledgements .unnumbered}
===============
Part of this research formed the core of the second author’s Master’s project at Texas A&M University. DB acknowledges partial support from NSF grants DMS-1500646 and DMS-1654056. The authors also thank David Borthwick, Tanya Christiansen, Colin Guillarmou, and Jeremy Marzuola for helpful conversations.
[10]{}
David Borthwick. Sharp upper bounds on resonances for perturbations of hyperbolic space. , 69(1-2):45–85, 2010.
David Borthwick. Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends. , 5(3):513–552, 2012.
David Borthwick and Pascal Philipp. Resonance asymptotics for asymptotically hyperbolic manifolds with warped-product ends. , 90(3-4):281–323, 2014.
Jeff Cheeger and Michael Taylor. On the diffraction of waves by conical singularities. [I]{}. , 35(3):275–331, 1982.
Jeff Cheeger and Michael Taylor. On the diffraction of waves by conical singularities. [II]{}. , 35(4):487–529, 1982.
. http://dlmf.nist.gov/, Release 1.0.21 of 2018-12-15. F. W. J. Olver, A. B. [Olde Daalhuis]{}, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds.
J. J. Duistermaat and V. W. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. , 29(1):39–79, 1975.
V. Ja. Ivriĭ. The second term of the spectral asymptotics for a [L]{}aplace-[B]{}eltrami operator on manifolds with boundary. , 14(2):25–34, 1980.
V. Ya. Ivriĭ. Exact spectral asymptotics for elliptic operators acting in vector bundles. , 16(2):30–38, 96, 1982.
F. W. J. Olver. . Academic Press \[A subsidiary of Harcourt Brace Jovanovich, Publishers\], New York-London, 1974. Computer Science and Applied Mathematics.
Plamen Stefanov. Sharp upper bounds on the number of the scattering poles. , 231(1):111–142, 2006.
G. N. Watson. . Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.
[^1]: When $(Y,h)$ is a sphere, the analysis is simplified slightly. In that case, one replaces the use of the Weyl formula with explicit formulae for the eigenvalues $\mu_{j}^{2}$ and their multiplicities.
|
---
abstract: 'Herbig Ae/Be stars lie in the mass range between low and high mass young stars, and therefore offer a unique opportunity to observe any changes in the formation processes that may occur across this boundary. This paper presents medium resolution VLT/X-Shooter spectra of six Herbig Ae/Be stars, drawn from a sample of 91 targets, and high resolution VLT/CRIRES spectra of five Herbig Ae/Be stars, chosen based on the presence of CO first overtone bandhead emission in their spectra. The X-Shooter survey reveals a low detection rate of CO first overtone emission (7 per cent), consisting of objects mainly of spectral type B. A positive correlation is found between the strength of the CO $v=$ 2–0 and Br$\gamma$ emission lines, despite their intrinsic linewidths suggesting a separate kinematic origin. The high resolution CRIRES spectra are modelled, and are well fitted under the assumption that the emission originates from small scale Keplerian discs, interior to the dust sublimation radius, but outside the co-rotation radius of the central stars. In addition, our findings are in very good agreement for the one object where spatially resolved near-infrared interferometric studies have also been performed. These results suggest that the Herbig Ae/Be stars in question are in the process of gaining mass via disc accretion, and that modelling of high spectral resolution spectra is able to provide a reliable probe into the process of stellar accretion in young stars of intermediate to high masses.'
author:
- '\'
date: 'Accepted 2014 September 12. Received 2014 September 11; in original form 2014 August 14'
title: 'Investigating the inner discs of Herbig Ae/Be stars with CO bandhead and Br$\gamma$ emission[^1]'
---
\[firstpage\]
stars: early-type – stars: pre-main sequence – stars: formation – stars: circumstellar matter – accretion, accretion discs
Introduction {#sec:intro}
============
Circumstellar discs surrounding young stellar objects (YSOs) have been the focus of much research because not only do they provide the location for possible planet formation to occur, but they play an essential role in the regulation and evolution of the accretion that takes place during the star formation process (see the review of @turner_2014). Pre-main sequence Herbig Ae and Be stars (HAeBes, see @waters_1998) lie in the mass range between lower mass T Tauri stars ($M_{\star}<2$M$_{\odot}$) and short-lived, obscured massive young stellar objects (MYSOs, $M_{\star}>8$M$_{\odot}$). Thus, they offer a unique opportunity to observe and characterise any similarities or differences between low- and high-mass star formation processes (see @larson_2003 and @mckee_2007 for reviews). For example, across the mass range between T Tauri stars and MYSOs, there is evidence for a change in the mechanism that transfers material from the surrounding natal cloud, through the disc, and on to the central protostar. The mechanism is thought to switch from T Tauri-like magnetospheric accretion - in which the disc is truncated at radial distances no larger than the co-rotation radius and accretion proceeds to the stellar surface via magnetically channelled accretion funnels [@bertout_1989; @bouvier_2007] - to some other, as yet uncharacterised phenomenon [@vink_2003; @vink_2005]. The reason for this possible change in accretion mechanism is because the interior envelopes of HAeBes are thought to be mostly radiative in nature [@hubrig_2009]. Therefore, they lack the interior convection currents required to power strong magnetic fields - a requirement for such magnetospheric accretion to occur. Recent observations show a low detection rate of magnetic fields ($\sim$7 per cent) across a large sample of HAeBes, supporting this scenario [@alecian_2013a]. It is possible that the lower mass Herbig Ae stars undergo similar magnetospheric accretion to that of T Tauri stars [@muzerolle_2004; @mottram_2007], but the situation for the higher mass Herbig Be stars is not known.
An alternative to magnetospheric accretion is direct accretion (also called boundary layer accretion), where material from the disc is accreted directly onto the stellar surface, along the ecliptic plane [@lynden-bell_1974; @bertout_1988; @blondel_2006]. For boundary layer accretion to occur, a disc-like geometry would have to be present on scales smaller than the co-rotation radius of the star - the location at which any magnetospheric accretion funnels would likely begin to operate [@shu_1994; @muzerolle_2003]. Therefore, in order to investigate the accretion mechanisms of these young stars, information on the geometry of inner regions of the circumstellar disc, close to the central star is required.
While it is believed that dust is responsible for most of the thermal emission from circumstellar discs, it is likely that gas is responsible for the majority of their mass. HAeBes possess strong stellar radiation fields compared to that of their lower mass counterparts. Because of this, regions of their circumstellar discs close to the central star are likely to be heated to high temperatures. If this temperature exceeds the dust sublimation temperature, then the dust in the disc is destroyed. This gives rise to an inner disc consisting of only gas, out to the location of the dust sublimation radius, on scales of a few astronomical units [see the review of @dullemond_2010].
Direct observations of these regions of HAeBes are complicated by the fact that most objects lie at relatively large distances. Thus, the small sizes of the regions involved mean that imaging is only possible using interferometry. However, this is an observationally complex task limited to bright targets [@tatulli_2008; @kraus_2008b; @wheelwright_2012]. Therefore, there is much interest in determining more indirect observational techniques that can probe the conditions close to the central star.
The most abundant molecule in circumstellar discs is molecular hydrogen (H$_{2}$). However, the large energies required to excite H$_{2}$, low transition probabilities, and atmospheric absorption across the relevant wavelength range mean that thermal emission from this molecule is difficult to observe, and it is therefore not an efficient tracer of these regions. Coupled rotational and vibrational emission of the next most abundant molecule, CO, offers an alternative diagnostic. CO bandhead emission (also called overtone emission) is excited in warm ($T =$ 2500–5000K) and dense (n $>$ $10^{15}$cm$^{-3}$) neutral gas - exactly the conditions expected in the inner parts of accretion discs, making this emission a valuable probe of these regions [@glassgold_2004]. Several previous investigations have been successful in fitting the CO bandhead spectra of young stars under the assumption that the emission originates from a gaseous circumstellar disc [@carr_1989; @blum_2004; @bik_2004; @thi_2005; @wheelwright_2010; @cowley_2012; @ilee_2013].
The $n = 7$–4 transition of atomic hydrogen (H[i]{}) in the Brackett series (Br$\gamma$) occurs at $\lambda$ = 2.16, and is also excited at high temperatures ($T\gtrsim 10^{4}$K). The origin of such hydrogen recombination emission is still unknown. Several theories have been proposed, including; magnetospheric accretion phenomena [@muzerolle_1998a], inner disc regions [@muzerolle_2004], stellar winds [@strafella_1998] and disk winds [@ferreira_1997]. @muzerolle_1998b found that the Br$\gamma$ line luminosity in a sample of low mass (0.2–0.8M$_{\odot}$) T Tauri stars was tightly correlated with the accretion luminosity as measured from blue continuum excess. @calvet_2004 extended this investigation to YSOs with masses up to 4M$_{\odot}$, and find good agreement with the previous study, and the relationship was used to examine the accretion rates of 36 Herbig Ae stars by @garcia-lopez_2006. More recently, @mendigutia_2011 determined accretion luminosities from 38 Herbig Ae and Be stars by examining the UV excess in the Balmer discontinuity, and found a correlation with Br$\gamma$ luminosity similar to @calvet_2004.
This paper utilises a collection of Very Large Telescope (VLT) X-Shooter and CRIRES observations of several Herbig Ae/Be objects based on the detection of CO first overtone bandhead emission in their spectra. The observations, sample selection and determination of stellar parameters are described in Section \[sec:obs\]. The measured observable quantities are presented and analysed in Section \[sec:obsres\]. Modelling of the CO spectra is discussed, and comments on individual objects are given in Section \[sec:modres\]. Discussion of the results from both sets of observations is presented in Section \[sec:discussion\], and finally conclusions are outlined in Section \[sec:conclusions\].
Observations & sample selection {#sec:obs}
===============================
The observations for this investigation were obtained using two instruments on the ESO VLT at Cerro Paranal. High resolution $2.3\,\micron$ spectra of 5 Herbig Ae/Be stars, targeted because of previous detection of CO first overtone bandhead emission, were obtained using the cryogenic spectrograph CRIRES [@kaufl_2008]. The CRIRES observations of HD 36917, HD 259431 and HD 58647 were taken on 26 and 27 October 2010. Using a slit width of 0.2 arcsec, the observations achieved spectral resolution of approximately 80000. The observations of PDS 37 were taken on 06 June 2007 and originally published in @ilee_2013. Using a slit width of 0.6 arcsec, a spectral resolution of approximately 30000 was achieved. The archival observations of HD 101412 were taken on 5 April 2011, and were originally published by @cowley_2012. Using a slit width of 0.2 arcsec, this achieved a spectral resolution of over 90000. Telluric line removal for all CRIRES observations was performed using standard stars at comparable airmasses, obtained during the same observing run as the science observations.
In addition to the targeted CRIRES observations, a medium resolution spectroscopic survey was performed using the cross-dispersed wide band spectrograph X-Shooter [@vernet_2011]. A total of 91 objects were observed in service mode between October 2009 and March 2010 (@oudmaijer_2011, Fairlamb et al., in prep). X-Shooter provides simultaneous wavelength coverage from 300–2480nm using three spectrograph arms - UVB, VIS, and NIR. The original sample of 91 Herbig Ae/Be stars were taken from the catalogues of @the_1994 and @vieira_2003, and were selected based on sky co-ordinates appropriate for the observing semester. A small number were discarded due to insufficient brightness or ambiguous assignment as a HAeBe star. This sample is larger than most other published studies by a factor of 2–5. In addition to the large sample size, the use of X-Shooter allows comparison of many spectral features from a single observation, which is important given that HAeBes have been shown to be both photometrically and spectrally variable [@oudmaijer_2001]. This paper utilises data from the NIR arm, and deals mainly with the subset of six objects from the full sample that showed a detection of CO first overtone bandhead emission at 2.3.
The observations using X-Shooter achieved a spectral resolution of $R
\sim 8\,000$ ($\Delta \lambda = 0.28$nm at $\lambda = 2.3$) using a slit width of 0.4arcsec. A single pixel element covered 0.06nm, while a resolution element covered 4.3 pixels. The atmospheric seeing conditions in the optical varied from 1.1–1.6arcsec between observations. The exposure times ranged from several minutes for the brightest sources, up to 30 minutes for the faintest ones. Nodding along the slit was performed to allow background subtraction. The data were reduced with version 0.9.7 of the ESO pipeline software [@modigliani_2010], and verified with manually reduced data for a handful of objects to ensure consistency. The data were of high quality, with signal-to-noise ratios of 100–140 in most cases across the entire sample of HAeBes.
To correct telluric absorption features within the X-Shooter spectra, the ESO software [molecfit]{} was used (Smette et al. 2014, in prep., Kausch et al. 2014, in prep.). The [molecfit]{} program models the atmospheric absorption above the telescope using temperature, pressure and humidity profiles for the observing site, a radiative transfer code, and a database of molecular parameters. We used the code to accurately model the atmospheric absorption features in the telluric observations themselves. This then produced model telluric spectra tuned to the exact atmospheric conditions measured on the night of the observation, but free from the effects of noise. These model spectra were then used to remove telluric features from the science observations, which resulted in a better correction than was possible using the standard stars alone.
A log of the observations of these objects is shown in Table \[tab:haebe\_obs\], their spectra around the CO first overtone and Br$\gamma$ region are shown in Figures \[fig:crires\_spectra\] and \[fig:xshooter\_spectra\], and their astrophysical parameters are given in Table \[tab:haebe\_stellar\_params\].
----------- ------------- ------------- --------------- ------------ ----- -------------------- ------------
Object Other name RA Dec Instrument S/N t$_{\mathrm{exp}}$ Date
(J2000) (J2000) (h)
HD 36917 05:34:47.00 $-$05:34:10.5 CRIRES 270 0.2 2010-10-26
HD 259431 MWC 147 06:33:04.90 $+$10:19:20.3 CRIRES 172 0.2 2010-10-26
HD 58647 07:25:56.10 $-$14:10:45.8 CRIRES 208 0.3 2010-10-27
PDS 37 Hen 3$-$373 10:10:00.32 $-$57:02:07.3 CRIRES 114 0.1 2007-06-06
HD 101412 V1052 Cen 11:39:44.46 $-$60:10:27.9 CRIRES 150 0.2 2011-04-05
HD 35929 05:27:42.79 $-$08:19:38.6 X-Shooter 68 0.03 2009-12-17
PDS 133 SPH 6 07:25:04.95 $-$25:45:49.7 X-Shooter 51 0.30 2010-02-24
HD 85567 V596 Car 09:50:28.53 $-$60:58:03.0 X-Shooter 123 0.02 2010-03-06
PDS 37 Hen 3$-$373 10:10:00.32 $-$57:02:07.3 X-Shooter 115 0.03 2010-03-31
HD 101412 V1052 Cen 11:39:44.46 $-$60:10:27.9 X-Shooter 72 0.06 2010-03-30
PDS 69 Hen 3$-$949 13:57:44.12 $-$39:58:44.2 X-Shooter 48 0.03 2010-03-29
----------- ------------- ------------- --------------- ------------ ----- -------------------- ------------
----------- ----------------- -------- -------------- -------------------- ------------------ ------------------------- ------------------------ ------------------------ -------------------- ----------- --------------------
Object Spectral K $d$ $T_{\mathrm{eff}}$ $A_{\mathrm{V}}$ $\log L_{\mathrm{bol}}$ $M_{\star}$ $R_{\star}$ $R_{\mathrm{sub}}$ $v\sin i$ $R_{\mathrm{cor}}$
Type (mags) (pc) (K) (mags) ($\mathrm{L}_{\odot}$) ($\mathrm{M}_{\odot}$) ($\mathrm{R}_{\odot}$) (au) () (au)
HD 36917 B9.5e$^{a}$ 5.7 470$^{a}$ 10000$^{a}$ 0.5$^{b}$ 2.20$^{b}$ 2.5$^{a}$ 1.8$^{a}$ 0.4 125$^{b}$ 0.02
HD 259431 B6e$^{c}$ 5.7 800$^{c}$ 14125$^{c}$ 1.2$^{c}$ 3.19$^{c}$ 6.6$^{c}$ 6.63$^{c}$ 3.3 100$^{d}$ 0.07
HD 58647 B9e$^{e}$ 5.4 277$^{e}$ 10500$^{f}$ 1.0 2.95$^{f}$ 3.0$^{e}$ 2.8$^{e}$ 0.8 118$^{g}$ 0.03
PDS 37 B2e$^{h}$ 7.0 720$^{h}$ 22000$^{h}$ 5.66 3.27 7.0 3.0 3.8 $\dots$ $\dots$
HD 101412 A0III/IVe$^{i}$ 7.5 $395\pm65$ $9\,750\pm250$ 0.39 1.58 2.3 2.2 0.5 8$^{l}$ 0.15
HD 35929 F2IIIe$^{o}$ 6.7 $325\pm60$ $7\,000\pm250$ 0.10 1.67 2.7 4.6 0.5 70$^{o}$ $\dots$
PDS 133 B6e$^{h}$ 9.3 $2270\pm500$ $13\,250\pm1000$ 1.61 2.16 3.2 2.4 1.0 $\dots$ $\dots$
HD 85567 B7–8Ve$^{m}$ 5.8 $470\pm220$ $12\,500\pm1000$ 0.76 2.48 3.8 3.9 1.5 50$^{q}$ $\dots$
PDS 69 B4Ve$^{n}$ 7.2 $645\pm120$ $16\,500\pm750$ 1.49 2.81 4.7 3.2 2.2 $\dots$ $\dots$
----------- ----------------- -------- -------------- -------------------- ------------------ ------------------------- ------------------------ ------------------------ -------------------- ----------- --------------------
\
, , , , , . , , , , , , , , , , .
Determining stellar parameters {#sec:stellarparams}
------------------------------
Stellar parameters for the targets observed with CRIRES were taken from various sources in the literature (see Table \[tab:haebe\_stellar\_params\]). Stellar parameters for the X-Shooter sample were determined directly using two methods that will be described in detail by Fairlamb et al. (in prep.), but here we briefly summarise the approach. The first method involved calculating the temperature and surface gravity, $\log g$, of the objects by adopting a similar method to that of @montesinos_2009. A best fit was performed across the hydrogen recombination lines of the Balmer series (H$\beta$, H$\gamma$ and H$\delta$) between the observed X-Shooter spectra and a grid of Kurucz-Castelli models [@kurucz_1993; @castelli_2004]. The fit was made specifically to the wings of the recombination lines, as their broadness is sensitive to changes in temperature and surface gravity. Only the flux measurements above a level of 0.8 of the normalised continuum were included, to avoid contamination of the fit by any emission component. These temperature and surface gravity values were then compared against the [parsec]{} pre-main sequence evolutionary tracks models [@bressan_2012], which provided a corresponding stellar mass, radius and luminosity.
However, only half of the objects in the sample could be constrained using this method, due to extreme emission of the Balmer series which eclipses even the broad wings, and often other strong emission lines were present complicating any temperature estimate. For these objects, known photometry from the literature was used to determine a luminosity for the objects. Then the Kurucz-Castelli models were fit to the photometry by reddening each model until a best-fitting slope was found, providing a temperature and surface gravity. A range of distances were then tested to provide a luminosity and radius, and from the surface gravity a mass was determined. A cross comparison of these parameters was then made with the [parsec]{} tracks, where for each temperature and luminosity pair there was a unique mass and radius. This then gave a luminosity that with matching parameters between both the [parsec]{} tracks and the photometry fit. Both of the methods described above were tested for consistency on a handful of objects, and produced very similar stellar parameters. The stellar parameters of the object PDS 37 proved difficult to determine using the methods above, so the distance and luminosity were fixed to literature values [@vieira_2003], which allowed a mass, radius, temperature and luminosity to be determined.
Estimation of the typical sizes of important physical regions was also carried out for each object - specifically the location of the dust sublimation radii, $R_{\mathrm{sub}}$, and the co-rotation radii, $R_{\mathrm{cor}}$. One of the most simple approaches in estimating $R_{\mathrm{sub}}$ is an analytic prescription, such as those described in @tuthill_2001 and @monnier_2002. Such approaches are based on the assumptions about the absorption efficiencies of the dust in the disc, and using these assumptions to calculate the radius at which dust would survive given a certain stellar luminosity. However, these calculations neglect second order effects, such back-warming of the disc material through re-radiation of the stellar heating from the dust grains, or the effect of non-homogenously sized grains. Addressing such effects requires a proper treatment of radiative transfer. @whitney_2004 calculated a series of two-dimensional radiative transfer simulations of discs around young stars, with effective temperatures up to 3$\times 10^{4}$K. Their models include the effect of re-radiation, and use a distribution of dust grain sizes based on the description in [@wood_2002], with sizes between 0.01–1000. From the results of these simulations, the authors determined an analytic relationship between the dust sublimation radius and the stellar effective temperature of $$R_{\mathrm{sub}} = R_{\star} \left( \frac{T_{\mathrm{sub}}}{T_{\star}} \right)^{-2.085},
\label{eqn:rsub}$$ where $T_{\mathrm{sub}}$ is the temperature at which the dust sublimates. By assuming $T_{\mathrm{sub}}$ is 1500K, we have used this relation along with the stellar effective temperatures in order to calculate the dust sublimation radii of our objects, which are shown in Table \[tab:haebe\_stellar\_params\]. However, it seems worthwhile to note that it is difficult to assign a single value to $R_{\mathrm{sub}}$ - different dust species will likely survive to different temperatures based on their composition, and in reality the dust sublimation will take place across a range of radii in the disc. Nonetheless, as we are simply estimating the size of typical regions within the discs, such treatment is beyond the scope of this investigation.
Where available in the literature, measured $v\sin i$ values were obtained and used together with the derived disc inclinations from the CO bandhead fitting (see Section \[sec:modres\]), in order to determine the stellar angular velocity $\omega$. This was then used to determine the co-rotation radii, $$R_{\mathrm{cor}} = \left( \frac{GM_{\star}}{\omega^{2}} \right)^{1/3},
\label{eqn:rcor}$$ of the objects, which are shown in Table \[tab:haebe\_stellar\_params\].
Observational results {#sec:obsres}
=====================
![VLT/CRIRES spectra of the sample of objects showing CO first overtone bandhead emission. Spectra have been normalised to the continuum and shifted vertically for clarity. Vertical ticks mark the rest wavelengths of the CO $v=$ 2–0 and 3–1 transitions, respectively. The observations of PDS 37 were conducted using an alternative wavelength setting to the rest of the observations, therefore no data is available beyond 2.311$\micron$.[]{data-label="fig:crires_spectra"}](figure1.eps){width="\columnwidth"}
{width="\textwidth"}
From the sample of 91 Herbig Ae/Be stars taken with X-Shooter, we investigate two near infrared emission features - Br$\gamma$ and the CO first overtone bandheads. The equivalent widths ($W$) and full width at half maximum (FWHM) of both lines were measured using the [iraf noao/onedspec]{} package. For each object, ten measurements were taken, and an average of these was reported as the final result, with the error in this value given as the standard deviation of these measurements.
Because many A- and B-type stars exhibit photospheric Br$\gamma$ absorption, the equivalent widths for the Br$\gamma$ lines needed to be corrected for this effect. We adopted a method similar to @garcia-lopez_2006, with the expression $$W(\mathrm{Br}\,\gamma)_{\mathrm{circ}} = W(\mathrm{Br}\,\gamma)_{\mathrm{obs}} - W(\mathrm{Br}\,\gamma)_{\mathrm{phot}}\,10^{-0.4\Delta K},
\label{eqn:bry_corr}$$ where $W(\mathrm{Br}\,\gamma)_{\mathrm{obs}}$ is the observed equivalent width, $W(\mathrm{Br}\,\gamma)_{\mathrm{phot}}$ is the equivalent width of the corresponding Kurucz-Castelli stellar spectra based on spectral type of the object, and $\Delta K$ is the disc continuum emission, computed by subtracting the observed $K$ magnitude from the photospheric $K$ magnitude of the corresponding model at the same given distance. After performing the correction for photospheric absorption, Br$\gamma$ emission was detected in 64 objects in the sample, absorption was detected in 6, and 21 sources did not show detections (where we define non detection for the Br$\gamma$ line as sources which show $-1\,\AA < W_{\mathrm{circ}} < 1\,\AA$).
Six objects exhibited CO first overtone emission (HD 35929, PDS 133, HD 85567, PDS 37, HD 101412 and PDS 69), and the remaining 85 objects did not show detections above a 3-sigma level across the $v=$ 2–0 transition. This subset of six stars showing detections of CO overtone emission in the X-Shooter sample forms the basis of the subsequent analysis in this work, and their spectra across the wavelength range including Br$\gamma$ and CO overtone are shown in Figure \[fig:xshooter\_spectra\]. The objects PDS 133, HD 85567, PDS 37 and PDS 69 all show emission from the $v=$ 2–0, 3–1, 4–2 and 5–3 vibrational transitions, while the objects HD 35929 and HD 101412 only show emission across the $v=$ 2–0 and 3–1 transitions.
The detection rate of Br$\,\gamma$ emission is 70 per cent, which is similar to other studies of lower mass T Tauri stars (74 per cent, @folha_2001). The detection rate of CO first overtone emission is 7 per cent, which is lower than other studies of CO bandhead emission in young stellar objects, where detection rates of around 20 per cent have been reported in low to intermediate mass YSOs [e.g. @carr_1989; @connelley_2010] and 17 per cent for massive YSOs [e.g. @cooper_2013].
The line flux is calculated from the product of the equivalent width of the emission line (in the case of Br$\gamma$ we use the circumstellar component $W_{\mathrm{circ}}$) and the extinction corrected flux of the object in the K-band (where we take $A_{\mathrm{K}} = 0.11\,A_{\mathrm{V}}$, @cardelli_1989). Examination of the near- to far-infrared continuum fluxes of each object suggested that a extrapolation of the K-band magnitude to $2.16\,\micron$ and $2.3\,\micron$ is appropriate to determine the line fluxes for both Br$\gamma$ and CO. Line luminosities were then calculated using the distances determined in Section \[sec:stellarparams\], where all measurements and corresponding errors are shown in Table \[tab:obs\_measured\].
For the CO first overtone emission, the strongest emission relative to the continuum is detected in PDS 69, while the weakest is in HD 35929 (the only star of spectral type F with a CO detection in our sample). The strongest Br$\gamma$ emission relative to the continuum is observed in PDS 37, while the weakest emission is detected in HD 101412. While most of the Br$\gamma$ emission is single peaked, we also observe double peaked emission in the spectrum of HD 101412. The objects displaying single peaked Br$\gamma$ emission (PDS 133, HD 85567, PDS 367 and PDS 69) also show other Hydrogen recombination lines with similar lineshapes - specifically single peaked H$\alpha$ and Pa$\gamma$ emission. HD 101412 shows a weak double peak in H$\alpha$ and Pa$\gamma$, located within a region of broad absorption. In contrast to the other objects, the Br$\gamma$ line in HD 32929 appears to be in absorption with a very broad extent in Figure \[fig:xshooter\_spectra\], but after correction for the effect of photospheric absorption, we determine this line to be in emission. It should be noted that this object has also previously been classified as a post-main sequence star [@miroshnichenko_2004]. A full study of the other atomic hydrogen lines mentioned here, and their diagnostic power, will be performed in a subsequent publication (Fairlamb et al. in prep.).
Similarly to @brittain_2007, the lineshapes of the Br$\gamma$ emission are broad (130–220) and do not exhibit a blue-shifted absorption component often seen in Herbig Ae/Be stars, which suggests the emission does not originate in a wind. The Br$\gamma$ emission of HD 101412 shows a double peaked line profile, with a separation of 50. As the source of this double peaking is unknown, the FWHM is measured across the full width of the line, and should be considered a maximal value. The FWHM of each lobe of the emission corresponds to approximately 110kms$^{-1}$. The FWHMs of the Br$\gamma$ emission are approximately 5–10 times the thermal linewidth expected for hydrogen gas at the effective temperatures of the central protostars (20–30). This suggests the hydrogen recombination lines are broadened by non-thermal mechanisms, such as rotation, turbulence or in-fall toward the central star.
------------------- --------------------- ---------------------------- --------------- ------------------ --------------------------------- ------------------------------- ----------------- --------------------
Object $W$(CO) $F$(CO) $L$(CO) FWHM(Br$\gamma$) $W$(Br$\gamma_{\mathrm{circ}}$) $F$(Br$\gamma$) $L$(Br$\gamma$) $\dot{M}$
(Å) (Wm$^{-2}$) (L$_{\odot}$) () (Å) (Wm$^{-2}$) (L$_{\odot}$) ()
[**CRIRES**]{}
HD 36917 $-7.6\pm0.2$ $1.9\pm0.3\times10^{-15}$ $-1.9\pm0.3$ $\dots$ $\dots$ $\dots$ $\dots$
HD 259431 $-4.4\pm0.1$ $1.2\pm0.2\times10^{-15}$ $-2.2\pm0.4$ $\dots$ $-3.9\pm0.3^{\ast}$ $1.1\pm0.1\times10^{-15\ast}$ $-2.1\pm0.4$ $3.2\times10^{-6}$
HD 58647 $-2.4\pm0.2$ $7.9\pm0.8\times10^{-15}$ $-2.2\pm0.1$ $\dots$ $-3.3\pm0.3^{\ast}$ $9.8\pm0.1\times10^{-16\ast}$ $-1.7\pm0.1$ $4.8\times10^{-7}$
HD 101412 $-2.8\pm0.3$ $1.3\pm0.4\times10^{-16}$ $-3.2\pm0.4$ $\dots$ $\dots$ $\dots$ $\dots$
PDS 37 $-2.0\pm0.4^{\dag}$ $2.4\pm0.8\times10^{-16}$ $-2.4\pm0.3$ $\dots$ $\dots$ $\dots$ $\dots$
[**X-Shooter**]{}
HD 35929 $-2.4\pm0.4$ $2.2\pm 0.4\times10^{-16}$ $-1.7\pm0.2$ $570\pm90$ $-1.3\pm0.9$ $1.2\pm0.2\times10^{-16}$ $-2.6\pm0.2$ $1.5\times10^{-7}$
PDS 133 $-4.0\pm0.7$ $4.0\pm 0.7\times10^{-17}$ $-2.2\pm0.9$ $170\pm5$ $-5.4\pm0.5$ $5.3\pm0.6\times10^{-17}$ $-2.2\pm0.9$ $1.3\times10^{-6}$
HD 85567 $-4.7\pm0.2$ $1.1\pm 0.1\times10^{-15}$ $-1.6\pm0.2$ $130\pm2$ $-4.3\pm0.1$ $9.8\pm0.3\times10^{-16}$ $-1.7\pm0.2$ $1.2\times10^{-6}$
PDS 37 $-3.0\pm0.2$ $3.7\pm 0.3\times10^{-16}$ $-2.9\pm0.3$ $190\pm6$ $-9.1\pm0.9$ $1.2\pm0.1\times10^{-15}$ $-3.2\pm0.5$ $1.3\times10^{-6}$
HD 101412 $-5.1\pm0.5$ $2.4\pm 0.2\times10^{-16}$ $-3.1\pm0.4$ $220\pm20$ $-2.9\pm0.4$ $1.3\pm0.5\times10^{-16}$ $-3.4\pm0.4$ $1.3\times10^{-7}$
PDS 69 $-6.7\pm0.8$ $4.6\pm 0.6\times10^{-16}$ $-2.2\pm0.5$ $205\pm15$ $-8.4\pm0.6$ $5.8\pm0.6\times10^{-16}$ $-2.1\pm0.5$ $9.1\times10^{-7}$
------------------- --------------------- ---------------------------- --------------- ------------------ --------------------------------- ------------------------------- ----------------- --------------------
\
\
A relationship between CO bandhead and Br$\gamma$ emission?
-----------------------------------------------------------
![A comparison of the line luminosities measured from the CO $v=$ 2–0 and Br$\gamma$ lines (after correction for photospheric absorption) for objects displaying such lines in emission or with an upper limit for non-detection. Filled circles represent measurements from the simultaneous X-Shooter observations, open circles represent non-simultaneous measurements from the CRIRES observations and literature data, and arrows represent measured upper limits for non-detections. A best fit to the detections, $\log L({\mathrm{CO}}) = (0.6\pm0.2)\,\log L({\mathrm{Br}\,\gamma})
- (0.8\pm0.5)$, is shown with a red line.[]{data-label="fig:haebe_linelum"}](figure3.eps){width="\columnwidth"}
The relationship between the Br$\gamma$ and CO first overtone is of interest, as both emission lines originate in circumstellar environments. In addition to the simultaneous X-Shooter observations, the objects HD 259431 and HD 58647 for which we present CRIRES data were also shown to posses Br$\gamma$ emission by @brittain_2007. These fluxes are included in Table \[tab:obs\_measured\]. However, it should be noted that because these data are not simultaneous, any effect of spectroscopic variability may alter the values slightly.
Figure \[fig:haebe\_linelum\] shows the luminosities of both lines measured from the X-Shooter and CRIRES datasets. The detections (filled and open circles) occupy the upper ranges of the relationship, showing that detection of CO overtone emission is associated with the detection of Br$\gamma$ emission. The detections show a positive correlation, best fitted with the relationship $\log L({\mathrm{CO}})
= (0.6\pm0.2)\,\log L({\mathrm{Br}\,\gamma}) - (0.8\pm0.5)$. While this correlation does not imply a direct dependence on both of the emission lines, it suggests that similar factors may affect the strength of both lines when they are present.
As the calculation of upper limits for CO first overtone emission is complicated by the non-Gaussian shape of the overall feature, here we describe the process adopted to determine them. Using the modelling routine described in Section \[sec:modres\], a synthetic CO first overtone spectra is created, along with a spectrum of random noise. The strength of random noise is altered to produce a series of spectra that span the range of SNR shown by the observations (50–260). The strength of the CO emission is decreased until the peak of the $v=$ 2–0 bandhead drops below the 3-sigma level in each of these spectra. The equivalent width is then measured across the 2.29–2.3$\micron$ wavelength range for the corresponding model (with no random noise). The relationship between the true equivalent width and the signal-to-noise ratio was then determined, and then used to calculated the upper limit of equivalent width for each non-detection, based on the SNR of the spectrum. The upper limits for non-detections were then taken as the product of the K-band flux density and this equivalent width, $F_{\mathrm{UL}} = F_{\mathrm{K}}W_{\mathrm{UL}}$. To ensure consistency, we also perform an identical procedure for the non detections of Br$\gamma$ emission. However in this case, we use a Gaussian lineshape centered at 2.1655$\micron$, rather than a model bandhead spectra.
Analysis of the non-detections shows that the upper limits for the line luminosities of CO first overtone emission are between 1 and 1.5 dex below the luminosities calculated from the detections. This suggests that our observations are not limited by noise in the spectra, and that more sensitive observations with longer integration times would not allow the detection of weaker CO first overtone emission following the same trend.
Given that the line luminosities in both axis of Figure \[fig:haebe\_linelum\] were calculated by multiplying the equivalent widths by the same continuum fluxes and the same square distances to the corresponding objects (Section \[sec:stellarparams\]), we tested the possibility that the correlation is spurious. However, the Spearman’s probability of false correlation does not significantly increase when both the continuum and distance values are considered through the partial correlation technique [@wall_2003]. This suggests that the correlation between line luminosities is not spurious, although we note that more data is necessary to provide any statistical significance.
This correlation between the line luminosities is in agreement with several previous studies of both emission lines in young stars. @carr_1989 studied a sample of 40 YSOs, in which they find a positive correlation between the line luminosity of the CO $v=$ 2–0 emission and the Br$\gamma$ emission in the 10 objects with such emission. @connelley_2010 examined NIR spectra of 110 Class I YSOs, and found a positive correlation between the equivalent widths of the CO $v=$ 2–0 and the Br$\gamma$ lines, where detected in their sample.
Determining accretion rates {#sec:mdot}
---------------------------
As mentioned previously, the luminosity of Br$\gamma$ has been shown to correlate well with the accretion luminosity of intermediate mass YSOs, giving rise to several correlations between the two quantities (see, e.g. @calvet_2004). Recently, @mendigutia_2011 looked at a large sample of Herbig Ae stars and determined a correlation following
$$\log \frac{L_{\mathrm{acc}}}{L_{\odot}} = (0.91\pm0.27) \times \log \frac{L_{\mathrm{\mathrm{Br}\,\gamma}}}{L_{\odot}} + (3.55 \pm 0.80).
\label{eqn:mdot}$$
We have used this correlation between accretion luminosity and Br$\gamma$ line luminosity to determine the accretion luminosities of the objects possessing Br$\gamma$ emission. Once the accretion luminosity has been determined, it is then used to calculate the mass accretion rate $\dot{M}$ via
$$\dot{M} = \frac{L_{\mathrm{acc}}\, R_{\star}} {G M_{\star}}.$$
Table \[tab:obs\_measured\] shows the mass accretion rates that were determined for each object using this procedure (including the two objects where measurements were taken from @brittain_2007). We also performed measurements of strength of Br$\gamma$ emission for the remaining objects in the X-Shooter observations.
The mass accretion rates of the all objects exhibiting Br$\gamma$ emission span the range of 10$^{-9}$–10$^{-4}$, with the majority of objects possessing rates of approximately $1\times10^{-7}$. The mass accretion rates of the objects exhibiting both Br$\gamma$ and CO first overtone emission span a smaller range of 10$^{-7}$–10$^{-6}$, with an average of $6\times10^{-7}$. Therefore, while it seems objects possessing CO first overtone emission span a small range in mass accretion rate, their average mass accretion rate does not seem substantially different to objects without such CO emission.
The accretion rates are somewhat higher than measured by @calvet_2004, who examined the spectra of nine intermediate mass T Tauri stars and find an average mass accretion rate of $3\times10^{-8}$. However, the HAeBes studied here are located at larger distances, are more luminous, and are thus (on average) likely younger than the T Tauri stars described above. If mass accretion rates decrease with stellar age, then this may explain why the mass accretion rates of the HAeBes are higher than those of the T Tauri stars. When compared with similar objects, such as the study of 38 Herbig Ae stars by @mendigutia_2011, the accretion rates determined here are in agreement with their reported median mass accretion rate of $2\times10^{-7}$.
However, it should be noted that the relation in Equation \[eqn:mdot\] was determined from examination of Herbig Ae objects (with $T_{\mathrm{eff}} < 1.2\times10^{4}\,$K), and the applicability of this magnetospheric shock model in relation to the Herbig Be objects presented here has not yet been proven [@mendigutia_2012]. Other emission lines have been shown to accurately correlate with various emission excesses, allowing them to be used as accretion tracers (e.g. He[i]{}, @oudmaijer_2011). This will be investigated in detail for the remainder of the X-Shooter dataset in a subsequent publication (Fairlamb et al. in prep.).
Modelling the CO bandheads {#sec:modres}
==========================
In order to determine the origin of the CO emission, we fitted the spectra using a model describing the circumstellar environment of CO as a thin disc in Keplerian rotation, previously utilised in @wheelwright_2010 [@ilee_2013] and @murakawa_2013. The program is briefly described below.
The population of the CO rotational levels, to a maximum of $J=100$, for each $\Delta v = 2$ vibrational transition considered are determined in each cell according to the Boltzmann distribution, which assumes local thermodynamic equilibrium, and a CO/H$_{2}$ ratio of 10$^{-4}$. The disc is divided into 75 radial and 75 azimuthal cells. Each transition is assumed to follow a Gaussian with a width of $\Delta \nu$. The intensity of emission from each cell of the disc is calculated from $I_{\nu} = B_{\nu}(T)\left(1-e^{-\tau_{\nu}}\right)$. The emission is then assigned a weight determined from the solid angle subtended by the cell on the sky. The emission from each cell is wavelength shifted to account for the line-of-sight velocity due to the rotational velocity of the disc. The emission from all cells is then summed together, smoothed to the instrumental resolution, and then shifted in wavelength to account for the radial velocity of the object to produce the entire CO bandhead profile for the disc.
The excitation temperature and surface number density of the disc are described analytically as decreasing power laws, $$\begin{aligned}
T(r) = T_{\mathrm{i}} \left( \frac{r}{R_{\mathrm{i}}} \right)^{p} \\
N(r) = N_{\mathrm{i}} \left( \frac{r}{R_{\mathrm{i}}} \right)^{q},\end{aligned}$$ where $T_{\mathrm{i}}$ and $N_{\mathrm{i}}$ are the excitation temperature and surface density at the inner edge of the disc $R_{\mathrm{i}}$, and $p$ and $q$ are the exponents describing the temperature and surface density gradient, respectively. The optical depth, $\tau$, is taken to be the product of the absorption coefficient per CO molecule, and the CO column density. Since we are considering a geometrically thin disc, the column density is given by the surface number density $N$. The outer radius of the CO emission region is taken to be the radius at which $T$ falls below 1000K, the temperature at which we assume CO overtone emission can no longer be excited sufficiently to be detected.
The best fitting model is determined using the downhill simplex algorithm, implemented by the <span style="font-variant:small-caps;">amoeba</span> routine of the IDL distribution. The input spectra are first continuum subtracted, and then normalised to the peak of the $v=$ 2–0 bandhead. Model fits are compared to the data using the reduced chi-squared statistic, $\chi_{r}^{2}$, and the error in the data is taken to be the standard deviation of the flux in the pre-bandhead portion of the spectra. Free parameters of the fit are the inner surface number density, the inner temperature, the inner radius, the intrinsic linewidth, the temperature and density exponents and the inclination. The fitting routine is repeated with six starting positions spread across the parameter space to avoid recovering only local minima in $\chi^{2}_{\mathrm{r}}$, and the final best fitting model is determined from these six runs. The range of parameter space searched is shown in Table \[tab:ranges\]. Errors on the best fitting parameters are calculated by holding all other fitting variables at their best fitting values, and altering the parameter of interest until the difference in reduced chi-squared, $\Delta \chi^{2}_{r}$, increases by unity.
Parameter Range
---------------------------------------- -----------------------------------------------
Inclination $i$ $0 < i < 90\,\degr$
Intrinsic linewidth $\Delta \nu$ $1 < \Delta \nu < 30$
Inner radius $R_{\mathrm{i}}$ $1 < R_{\mathrm{i}} < 100$R$_{\star}$
Inner temperature $T_{\mathrm{i}}$ $1000 < T_{\mathrm{i}} < 5000$K
Inner surface density $N_{\mathrm{i}}$ $10^{12} < N_{\mathrm{i}} < 10^{25}$cm$^{-2}$
Temperature exponent $p$ $-4 < p < 0$
Surface density exponent $q$ $-4 < q < 0$
: Parameter space that is searched during the model fitting procedure[]{data-label="tab:ranges"}
Fitting the X-Shooter observations
----------------------------------
The object HD 101412 provides useful test case for our analysis, as it has been observed with both X-Shooter at medium resolution and CRIRES at high resolution. Figure \[fig:reso\] shows the comparison of the $v=$ 2–0 bandhead for both sets of observations.
![Comparison of CRIRES and X-Shooter data for HD 101412. The CRIRES spectrum at $R \sim 90\,000$ shows much detail that is not distinguishable when considering data of a lower spectral resolution of $R \sim 8\,000$.[]{data-label="fig:reso"}](figure4.eps){width="1.0\columnwidth"}
The CRIRES spectrum at $R \sim 90\,000$ shows much detail that is not seen when the considering data of a lower spectral resolution of $R
\sim 8\,000$. In particular, the individual rotational transitions in the blue shoulder of the bandhead and the double-peaked rotational transitions are almost entirely lost in the X-Shooter spectrum. When the CRIRES spectrum is convolved with a Gaussian corresponding to the spectral resolution of the X-Shooter data, re-binned the appropriate amount, and given a similar level of random noise, the data appears qualitatively similar to the X-Shooter spectrum. However, measurement of the equivalent width of the original and degraded CRIRES spectra yielded very similar results (2.8Å), while measurement of the X-Shooter spectrum gave a larger equivalent width of 5.1Å, suggesting that the CO first overtone emission in HD 101412 may be variable. Nonetheless, it is clear that much information is lost when the spectrum is degraded to the resolution of X-Shooter.
We performed the fitting routine on both the high resolution CRIRES data and the lower resolution X-Shooter data for HD 101412. The fitting routine performed poorly on the X-Shooter data, recovering multiple best fit solutions at similar $\chi^{2}_{\mathrm{r}}$ values with very different parameters (which we do not show here). We attribute this to the low resolution data not showing important features in the spectra, such as the blue shoulder of the bandhead and the narrow, double peaked rotational transitions mentioned earlier. However, when the fitting routine was performed on the high resolution CRIRES spectra for HD 101412, a single, unambiguous best fitting model was obtained.
Fitting the CRIRES observations
-------------------------------
Given the issues of fitting the X-Shooter spectra described previously, we chose to restrict our modelling of the CO bandheads to the 5 objects observed with CRIRES. The results of this fitting are shown in Table \[tab:disco\_fits\] and Figure \[fig:disco\_fits\]. Below we discuss the fitting results on an object-by-object basis.
[lllllll]{} Output Parameter & &HD 36917 & HD 259431 & HD 58647 & HD 101412 & PDS 37\
CO inner radius & $R_{\mathrm{i}}$(au) & 0.1$^{+0.01}_{-0.02}$ & $0.89^{+0.1}_{-0.1}$ & $0.18 \pm 0.01$ & $1.0^{+0.2}_{-0.1}$ & 1.6$^{+0.1}_{-0.5}$\
CO outer radius & $R_{\mathrm{o}}$(au) & 1.5 & 4.3 & 0.62 & 1.1 & 3.8\
Inclination & $i$() &51$^{+7}_{-2}$ & 52$^{+5}_{-3}$ & $75^{+\ast}_{-10}$ & $87^{+\ast}_{-20}$ & 89$^{+1\ast}_{-35}$\
Inner surface number density & $N_{\mathrm{i}}$(cm$^{-2}$) &$6.0^{+2}_{-3}\times10^{20}$ &$0.16^{+5.6}_{-\ast}\times10^{20}$ & $0.62^{+12}_{-\ast}\times10^{20}$ & $14^{+75}_{-4}\times10^{21}$ & 0.1$^{+6.3}_{-\ast}\times10^{20}$\
Inner temperature & $T_{\mathrm{i}}$(K) & 3400$^{+800}_{-250}$ & 3200$^{+10}_{-200}$ & $2800^{+10}_{-300}$ & $1000^{+50}_{-\ast}$ & $5000^{+\ast}_{-1200}$\
Intrinsic linewidth & $\Delta \nu$(km s$^{-1}$) & 5.4$^{+3}_{-2}$ & 10.7$\pm \ast$ & $10.8 \pm \ast$ & $5.2^{+3}_{-2}$ & $18\pm \ast$\
Temperature exponent & $p$ & $-0.5^{+0.1}_{-0.1}$ & $-0.74^{+0.4}_{-1.1}$ & $-0.89^{+0.3}_{-0.9}$ & $ -0.46^{+\ast}_{-2.4}$ & $-1.9^{+0.7}_{-\ast}$\
Surface number density exponent & $q$ & $-1.8^{+0.2}_{-0.2}$ & $-3.3^{+1.2}_{-\ast}$ & $-1.6^{+1.5}_{-\ast}$ & $-0.5 \pm \ast$ & $-1.7 \pm \ast$\
Reduced chi-squared & $\chi_{\mathrm{r}}^{2}$ & 3.0 & 5.3 & 6.2 & 3.8 & 2.5\
{width="0.97\columnwidth"} {width="0.97\columnwidth"}
{width="0.97\columnwidth"} {width="0.97\columnwidth"}
{width="0.97\columnwidth"} {width="0.97\columnwidth"}
{width="0.97\columnwidth"} {width="0.97\columnwidth"}
{width="0.97\columnwidth"}\
{width="0.97\columnwidth"}
### HD 36917
HD 36917 is assumed to be a B9.5 type, 2.5M$_{\odot}$ star with a stellar radius of 1.8R$_{\odot}$, an effective temperature of 10$^{4}$K and located at a distance of 470pc [@manoj_2002; @brittain_2007]. The bolometric luminosity has been determined to be 245L$_{\odot}$, with a visual extinction of 0.5 mags [@hamaguchi_2005].
Our modelling of the CO bandheads indicates a best fitting disc model extending from 0.1–1.5au, at an inclination of 51. The inner edge of the CO emitting region reaches a temperature of 3400K, at a density of $6\times 10^{20}$cm$^{-2}$. The temperature and surface number density exponents are well constrained at $-0.5$ and $-1.8$ respectively, and the temperature exponent agrees well with the value of $-0.5$ expected from a flat disc in radiative equilibrium [@chiang_1997]. The location of the CO emission region crosses inside the dust sublimation radius of 0.4au, as calculated from Equation \[eqn:rsub\], but lies beyond the co-rotation radius of 0.02au. The intrinsic linewidths of the individual transitions in the CO bandhead correspond to 5.4km$^{-1}$, which are approximately 2–5 times the thermal linewidths for CO at temperatures between 1000–5000K, indicating broadening by non-thermal mechanisms.
Our results are in contrast to the study of @berthoud_2008, who are unable to fit their observations of the CO $v=$ 2–0 bandhead using a disc and/or ring model, due to the fitting procedure returning solutions that converge with unphysical values (such as temperatures higher than the dissociation temperature of CO). The authors therefore model the emission using an expanding shell of CO, which produces satisfactory fits to the spectrum, in particular the rounded, convex-shaped blue shoulder of the bandhead in their data. Our observations of HD 36917 do not exhibit such a rounded blue shoulder, but rather a traditional concave-shaped blue shoulder, traditionally attributed to emission from a disc. This allows us to obtain a satisfactory fit to the spectrum using our disc model. As our data is of higher spectral resolution than the observations presented in @berthoud_2008, it is unlikely that the differences in the spectrum presented here are due to a resolution effect. It is possible that the source of the CO emission in HD 36917 is variable, and that overtones are excited both within a disc and shell like geometry around the central star. However, time monitoring of any spectral variability of the source would be required to confirm this.
### HD 259431
HD 259431 (MWC 147) is taken to be a 6.6M$_{\odot}$ star with a radius of 6.6R$_{\odot}$, having a spectral type of B6, located at a distance of 800pc. It has a high bolometric luminosity of 1550L$_{\odot}$, and an effective temperature of 14125K [@kraus_2008a].
The best fitting disc model for HD 259431 extends from 0.89–4.3au at an inclination of 52. It is interesting to note that the spectro-interferometric study of @kraus_2008a determine an inclination of approximately 50 for this object, which agrees very well our derived disc inclination. The temperature and surface number density of the inner disc are 3000K and $2 \times
10^{20}$cm$^{-2}$, with exponents of $-0.7$ and $-3.3$ respectively. The temperature exponent is in agreement with the expected value of $-0.75$ from a flat blackbody disc [@chiang_1997]. The inner edge of the disc extends to within the dust sublimation radius of 3.3au, however this lies outside the co-rotation radius of 0.07au as calculated from $v \sin i = 100$[@hillenbrand_1992]. The linewidth of the individual rotational transitions of CO is 10.7, however this is not well constrained. Nevertheless, this is a factor of 3–8 times the thermal linewidth for CO at 1000–5000K.
@brittain_2007 measured the Br$\gamma$ emission of HD 259431 and determine a full width at zero intensity (FWZI) of 350, with a luminosity of $38.1\times10^{-4}$$L_{\odot}$ which they calculate to correspond to a mass accretion rate of $4.1\times10^{-7}$using a relationship based upon UV veiling. Using these measurements with the relationship described in @mendigutia_2012, we calculate a higher mass accretion rate $3.2\times10^{-6}$. @hillenbrand_1992 also find a higher accretion rate of $1.01\times10^{-5}$, determined from fitting the SED of HD 259431. This discrepancy may be explained by the fact that the accretion rate calibrations used have not been proven to be valid for Herbig Be stars as hot as HD 259431.
### HD 58647
The stellar mass of HD 58647 is assumed to be 3.0M$_{\odot}$ with a radius of 2.8M$_{\odot}$, located at a distance of 277pc [@brittain_2007]. It has an effective temperature of 10500K and a bolometric luminosity of 910L$_{\odot}$ [@montesinos_2009].
Minor issues were encountered while fitting HD 58647, which showed a decrease in signal-to-noise across the final detector chip containing the $v=$ 3–1 bandhead. For this reason, the fitting routine was restricted to data from the first and second detector chip and extrapolated across the remaining data. Though this data in not included in the formal fitting process, it can be seen that the model qualitatively reproduces the features across this region of the spectrum very well.
The best fitting model extends from 0.18–0.62au, at an inclination of 75. This is entirely within the dust sublimation radius of 1.1au as calculated from Equation \[eqn:rsub\]. The inner edge of the CO emitting region reaches a temperature of 2800K, at a density of $6.2\times 10^{19}$cm$^{-2}$. The temperature and surface number density exponents are not well constrained, but give best fitting values of $-0.89$ and $-1.6$ respectively. The intrinsic linewidths of the individual transitions in the CO bandhead correspond to 10.8, a factor of 3–8 times the thermal linewidth for CO at 1000–5000K.
@berthoud_2008 fitted their observations of the CO overtone emission in HD 58647 with an optically thick ring at a temperature of 2380K, a surface density of $1.6\times10^{20}$cm$^{-2}$, and an intrinsic linewidth of 7.7, seen at an high inclination to the line of sight. These values agree with the fit obtained using our observations and disc model to within approximately 1–1.5$\sigma$.
@brittain_2007 measure the Br$\gamma$ emission line from HD 58647 to be double peaked, with a full width at zero intensity (FWZI) of 400, and a luminosity of $21.8\times10^{-4}$L$_{\odot}$ They calculate this to correspond to an accretion rate of $3.5\times10^{-7}$, which is consistent with the accretion rate of $4.8\times10^{-7}$determined using Equation \[eqn:mdot\].
### HD 101412 {#sec:hd101412}
We determine HD101412 to be a 2.3$\mathrm{M}_{\odot}$ star with a radius of 2.2$\mathrm{R}_{\odot}$, and an effective temperature of 9750K. It has a relatively low bolometric luminosity of 38L$_{\odot}$, and a visual extinction of $A_{\mathrm{V}} = 0.39$ mags. Our modelling of the CO overtone indicates a best fitting disc model extending from 1.0–1.1au, at an inclination of 87. The inner edge of the CO emitting region reaches a relatively low temperature of 1000K, at a relatively high density of $1.4\times
10^{22}$cm$^{-2}$. The temperature and surface number density exponents are $-0.46$ and $-0.5$ respectively. These values are not well constrained, likely due to the fact that the emitting region is very narrow, and thus determination of a gradient for the temperature and density is difficult. The intrinsic linewidths of the individual transitions in the CO bandhead correspond to 5.2kms$^{-1}$, which is 2–4 times the thermal linewidth. In contrast to all other best fitting disc models, the CO emission region for HD 101412 lies beyond the dust sublimation radius of 0.5au calculated from Equation \[eqn:rsub\]. Additionally, HD 101412 exhibits the relatively weak double-peaked Br$\gamma$ emission, contrary to the strong single peaked Br$\gamma$ emission of other objects studied in this work.
HD 101412 was the subject of a similar investigation involving CO bandhead emission by @cowley_2012. The authors find fits to their spectra assuming a disc of CO that is at most 0.8–1.2au in extent, at a temperature of 2500K, and assuming the disc is edge on, following the inclination of 80 determined from @fedele_2008. Our results only differ slightly from @cowley_2012, but it should be noted that their model and fitting routine were different to the methods presented here. For instance, they assume an isothermal ring of CO with fixed parameters such as inclination, and the fitting was performed visually with no systematic $\chi_{r}^{2}$ minimisation. However, the results here still suggest a narrow ring of CO, at approximately distance from the central protostar, at a slightly cooler temperature.
High spectral resolution observations of the \[O[i]{}\] emission line at 6300[Å]{} were examined by @vanderplas_2008, who determined that this emission originates from a region in a disc from 0.15–10au, viewed at an inclination of 30, and corresponds to a $v\sin\,i$ of 8kms$^{-1}$. The authors suggest that HD101412 is in transition between a flaring and self shadowed disc. The authors also note a drop in the radial \[O[i]{}\] emission of 50 per cent at approximately 0.5au (corresponding to their calculated dust sublimation radius), and a re-brightening shortly afterwards at approximately 0.8au. The initial drop is attributed to the self shadowing of a puffed up inner rim at the dust sublimation radius, but the authors note that the observed re-brightening is unexpected. HD 101412 was one of the subjects of a study of CO fundamental ro-vibrational emission by @vanderplas_2010. The comparable linewidths of CO and \[O[i]{}\] led this author to suggest that HD 101412 has a disc with strongly flared gas, but mostly settled dust.
This interpretation could explain why we apparently detect CO bandhead emission beyond the dust sublimation radius in HD 101412, in contrast to the other objects studied here. Our best fitting model for the spectrum of HD 101412 suggests a relatively cool ($1000$K) but high density ($10^{22}$cm$^{-2}$) emission environment - if there is a sufficient amount of dense gas located above the highly settled dust in the disc, then CO first overtone emission may originate from these regions. The X-Shooter spectrum of HD 101412 only exhibits the CO $v=$ 2–0 and 3–1 bandheads, suggesting there is not sufficient energy in the origin environment to excite the higher vibrational transitions.
It is not clear how our disc model would perform when attempting to fit CO emission that is not from an axisymmetric disc geometry, which may explain why our reported high inclination is in contrast to the low measured $v\sin\,i$ of 8kms$^{-1}$ and low inclination reported in @vanderplas_2008. In addition, as there is an inversely proportional degeneracy between the location of the emission and the disc inclination in our CO modelling procedure, adopting a lower inclination would mean the corresponding emission region is closer to the star, likely inside the dust sublimation radius. As the inclination recovered for HD 101412 is almost exactly edge on, then the reported location of the CO emission represents an upper limit to the radial distance of this region.
### PDS 37
PDS 37 (aka G282.2988$-$00.7769) was previously investigated as a massive young stellar object in @ilee_2013, where a stellar mass, radius and effective temperature of 11.8M$_{\odot}$ and 4.7R$_{\odot}$, 26100K were calculated from the bolometric luminosity and adopted for the fitting of the CO emission. This lead to a best fitting disc model 1.7–9au in extent, at an inclination of 80. The inner temperature of the disc was 4800K, and the surface density was $1\times10^{20}$cm$^{-2}$, varying with a slope of -0.97 and -1.4 respectively. The intrinsic linewidth of the transitions was determined to be 16.3.
Here we calculate PDS 37 to have a mass of 7.0M$_{\odot}$, a radius of 3.0R$_{\odot}$, a bolometric luminosity of 1860L$_{\odot}$ and an effective temperature of 22000K. Modelling the CO bandhead emission using these stellar parameters indicates a best fitting disc model extending from 1.6–3.8au, at an inclination of $89^{+1}_{-35}$. The inner edge of the CO emitting region reaches a temperature of 5000K, at a density of $1.0\times
10^{19}$cm$^{-2}$. The temperature and surface number density exponents are not well constrained at $-1.9$ and $-1.7$ respectively. The location of the CO emission region coincides with dust sublimation radius of 1.5au, as calculated from Equation \[eqn:rsub\]. The intrinsic linewidths of the individual transitions in the CO bandhead correspond to 18, which is approximately 6–14 times the thermal linewidths for CO at temperatures between 1000–5000K. Altering the stellar parameters changes the best fitting model parameters slightly, however still indicates a relatively large CO emission region, at a high temperature, viewed almost edge-on.
PDS 37 is also the subject of a spectropolarimetric study by Ababakr et al. (in prep.), where strong polarisation signatures are seen across the H$\alpha$ and doubly peaked Fe[ii]{} emission lines, indicating the presence of a gaseous disc viewed at a high inclination to the line of sight.
Discussion {#sec:discussion}
==========
The detection rate of CO first overtone emission
------------------------------------------------
From an initial sample of 90 targets obtained with X-Shooter (the most complete spectroscopic sample of Herbig Ae/Be stars to date), we find a low detection rate of CO first overtone bandhead emission of approximately 7 per cent. While a low detection rate in itself is in agreement with previous studies, our detection rate is substantially lower than studies of low mass T Tauri and Herbig Ae stars [20 per cent, @carr_1989; @connelley_2010], and also of higher mass MYSOs [17 per cent, @cooper_2013].
It is also striking that although our full X-Shooter sample contains many A-type stars, we have only detected CO overtone emission in one A-type star. In contrast to this, although our sample contains few B- and F-type stars, we have detected CO in a total of 7 B-type stars and one F-type star. So, also in our sample there may be evidence that the detection rate for CO first overtone emission is lower for intermediate-mass young stars than for low- and high-mass young stars. Below we discuss possible explanations for this.
High temperatures are required to excite the CO sufficiently in order for CO bandhead emission to become detectable. Several of the objects in our study are B-type stars (and HD 101412 is of type HA0, having also been classified as B-type by @manoj_2006). Therefore, these objects are hotter and more massive than their A-type counterparts. It could be that many T Tauri and Herbig Ae stars are not hot enough to continually excite the CO overtones in a circumstellar disc environment. In such cases, variable CO emission in lower mass YSOs could be explained by bursts of active accretion [@biscaya_1997] or by originating from a different environment (e.g. magnetic funnel flows, @martin_1997). However, modelling of high spectral resolution observations of a large number of T Tauri stars would be required to confirm the origin of the emission.
It is also possible that the majority HAeBes do not have enough gas in their close circumstellar environments to allow sufficient excitation of the CO bandheads. In addition to high temperatures, high densities (n $>$ $10^{15}$cm$^{-3}$) are required before this ro-vibrational emission becomes sufficiently excited to be detectable. While direct measurements of the amount of gas within these young stellar systems is difficult, there is evidence of cleared gaps around many HAeBe stars (a recent example being HD 142527, @casassus_2012). If the gas within these inner regions is cleared efficiently, then it will not be possible to reach densities high enough to allow overtone emission to occur. Our modelling of the CO overtone spectra indicates the emission originates from environments with a surface density of at least $10^{20}$cm$^{-2}$. In addition, @muzerolle_2004 present models of the inner regions of discs around HAeBe stars, and show that for accretion rates greater than $10^{-8}$, the inner gaseous disc becomes optically thick. The accretion rates determined from our analysis of the Br$\gamma$ in these objects are above this level, further suggesting that these objects possess a large amount of gas of a high density close to the central protostar.
One unresolved issue with this interpretation is that there are a handful of objects possessing high accretion rates that do not exhibit CO first overtone emission. The work of @calvet_1991 showed that CO in absorption could be expected from high mass accretion rates (observed in FU Ori objects), however we do not detect such absorption in our observations.
The location and orientation of the emitting regions {#sec:location}
----------------------------------------------------
The location of the detected emission is of interest, as it determines which regions of the circumstellar environment can be probed. We find that four out of five of the objects possess best fitting disc models with inner radii located interior to the corresponding dust sublimation radii. This suggests that the CO emission originates from a gaseous disc, close to the central protostar. The one object for which this is not the case, HD 101412, exhibits features which are not typical when compared to the other objects studied here (see Section \[sec:hd101412\]).
The co-rotation radii lie between 0.02–0.23au, and are interior to the dust sublimation radius in all objects. The CO emission is also shown to originate from beyond the co-rotation radius, in objects where $v\sin i$ measurements are available in the literature. Therefore, our modelling suggests that while CO first overtone emission is a valuable probe of the inner gaseous disc component around young stars, other spectral tracers are required to trace regions close to the co-rotation radius, where any deviations from magnetospheric accretion geometry are likely to occur.
The inclinations of the best fitting disc models range from $51$–$72\degr$, suggesting a preference for moderate to high inclinations. While the number of objects modelled in this paper is too low to accurately determine the statistical significance of this, it is nonetheless possible that a geometric selection effect is at work. One possible explanation for a preference toward more inclined discs could be that in addition to a inner disc, the CO emission may also trace the vertical inner wall of the dust disc, located at the dust sublimation radius. However, further investigation using models that are able to include such emission geometry would be required to confirm this.
A preference for moderate to high inclinations is in contrast to the study of CO emission of massive YSOs by @ilee_2013, which found an essentially random orientation of disc inclinations. The masses of the objects studied in @ilee_2013 were determined from the bolometric luminosity of the objects, which may have included contributions from accretion, and could therefore be overestimates of the true stellar masses. In such cases, an overestimate of the stellar mass can lead to a lower inclination being recovered. This effect can be seen in our modelling of PDS 37 - in this work, we recover a an inclination of 87 using a stellar mass of 7.0M$_{\odot}$, while in @ilee_2013 we recover an inclination of 80 from a stellar mass of 12M$_{\odot}$. While this effect is small, it may explain why no such preference for moderate to high inclination angles was found for MYSOs.
A positive correlation between the line luminosities of the CO bandhead and Br$\gamma$ is found, and while this does not imply a direct dependence (and the number of objects with emission is too low to attribute a statistical significance to the correlation), it does suggest that similar factors affect the strength of both emission lines. However, analysis of the linewidths shows that the Br$\gamma$ emission is approximately 20 times larger than the corresponding linewidths obtained from the fitting of the CO bandheads. This difference in linewidth suggests that both lines do not originate in the same kinematic environment, and are therefore likely not co-spatial. This is in contrast to the recent interferometric study of @eisner_2014, who find a near-coincidence of CO overtone, Br$\gamma$ and continuum emission in 5 YSOs. The spectro-interferometric study of @kraus_2008b suggests two possible origins for Br$\gamma$ emission - compact regions, or more extended regions possibly tracing stellar or disc winds. Further analysis on the precise location of the Br$\gamma$ emission will be required in order to study any possible connections between these two emission lines.
Conclusions {#sec:conclusions}
===========
This paper presents medium resolution VLT/X-Shooter and high resolution VLT/CRIRES near-infrared spectra of several Herbig Ae/Be stars, in an investigation of the inner regions of their circumstellar discs. Below we summarise the main findings:
- From a large spectroscopic survey of over 90 HAeBe targets, we detect only six objects exhibiting CO first overtone bandhead emission, corresponding to a detection rate of approximately 7 per cent. Analysis of the upper limits suggests that the majority of non-detections are not due to the sensitivity of the X-Shooter instrument.
- The objects displaying CO overtone emission are mainly of spectral type B, and are thus hotter and more massive than their A-type counterparts.
- In all objects that display CO bandhead emission, we also find Br$\gamma$ emission of varying strengths. We find a positive correlation between the strength of the CO $v=$ 2–0 bandhead and the Br$\gamma$ line, in agreement with previous investigations [@carr_1989; @connelley_2010], showing this correlation extends to YSOs of higher masses.
- The high resolution spectra of 5 objects exhibiting CO first overtone emission are fitted with a model of a thin disc undergoing Keplerian rotation, and good fits are obtained to all spectra. It was determined that the spectral resolution of the X-Shooter instrument was insufficient to obtain reliable model fits using this procedure.
- The linewidths of the Br$\gamma$ emission are between 10–40 times larger than the intrinsic linewidths of the CO overtone emission, suggesting that they originate in different kinematic environments.
- The location of the CO overtone emission in these best fitting models is consistent with the hypothesis that it originates from a small scale gaseous disc, interior to the dust sublimation radius, but beyond the co-rotation radius of the central star.
It is important to note that for the object where spatially resolved observations have also been performed, HD 259431 (MWC 147, @kraus_2008a), we obtain a remarkably similar value to the inclination of the disc based on our fitting technique ($\sim
50$). While this comparison can currently only be made in one object, it does suggest that high spectral resolution observations can be used as an alternative to interferometric observations to investigate the sub-au scale regions around young stars.
We plan to investigate this with further observations using VLTI/AMBER, which will enable direct measurements of the spatial extent of the CO-emitting gas, and allow comparison with our spectral fitting technique. This, alongside more sophisticated modelling that can include the vertical structure of inner discs, will provide much information on the nature of the inner regions around Herbig Ae/Be stars.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank the referee Wing-Fai Thi for comments that improved the clarity of the manuscript. In addition, we also thank Peter Woitke, Rens Waters and the members of the FP7 DIANA team for useful discussions. JDI gratefully acknowledges funding from the European Union FP7-2011 under grant agreement no. 284405. JRF gratefully acknowledges a studentship from the Science and Technology Facilities Council of the UK. SK acknowledges support through an STFC Ernest Rutherford fellowship.
[83]{} E. [et al.]{}, 2013, , 429, 1001
M. G., 2008, PhD thesis, Cornell University
C., 1989, , 27, 351
C., [Basri]{} G., [Bouvier]{} J., 1988, , 330, 350
A., [Thi]{} W. F., 2004, , 427, L13
A. M., [Rieke]{} G. H., [Narayanan]{} G., [Luhman]{} K. L., [Young]{} E. T., 1997, , 491, 359
P. F. C., [Tjin A Djie]{} H. R. E., 2006, , 456, 1045
R. D., [Barbosa]{} C. L., [Damineli]{} A., [Conti]{} P. S., [Ridgway]{} S., 2004, , 617, 1167
J., [Alencar]{} S. H. P., [Harries]{} T. J., [Johns-Krull]{} C. M., [Romanova]{} M. M., 2007, Protostars and Planets V, 479
A., [Marigo]{} P., [Girardi]{} L., [Salasnich]{} B., [Dal Cero]{} C., [Rubele]{} S., [Nanni]{} A., 2012, , 427, 127
S. D., [Simon]{} T., [Najita]{} J. R., [Rettig]{} T. W., 2007, , 659, 685
N., [Muzerolle]{} J., [Brice[ñ]{}o]{} C., [Hern[á]{}ndez]{} J., [Hartmann]{} L., [Saucedo]{} J. L., [Gordon]{} K. D., 2004, , 128, 1294
N., [Patino]{} A., [Magris]{} G. C., [D’Alessio]{} P., 1991, , 380, 617
J. A., [Clayton]{} G. C., [Mathis]{} J. S., 1989, , 345, 245
J. S., 1989, , 345, 522
S., [Perez M.]{} S., [Jord[á]{}n]{} A., [M[é]{}nard]{} F., [Cuadra]{} J., [Schreiber]{} M. R., [Hales]{} A. S., [Ercolano]{} B., 2012, , 754, L31
F., [Kurucz]{} R. L., 2004, ArXiv Astrophysics e-prints
E. I., [Goldreich]{} P., 1997, , 490, 368
M. S., [Greene]{} T. P., 2010, , 140, 1214
H. D. B. [et al.]{}, 2013, , 430, 1125
C. R., [Hubrig]{} S., [Castelli]{} F., [Wolff]{} B., 2012, , 537, L6
P. T., [Hoogerwerf]{} R., [de Bruijne]{} J. H. J., [Brown]{} A. G. A., [Blaauw]{} A., 1999, , 117, 354
C. P., [Monnier]{} J. D., 2010, , 48, 205
J. A., [Hillenbrand]{} L. A., [Stone]{} J. M., 2014, , 443, 1916
D. [et al.]{}, 2008, , 491, 809
J., 1997, , 319, 340
D. F. M., [Emerson]{} J. P., 2001, , 365, 90
R., [Natta]{} A., [Testi]{} L., [Habart]{} E., 2006, , 459, 837
A. E., [Najita]{} J., [Igea]{} J., 2004, , 615, 972
M. M., [Alencar]{} S. H. P., [Corradi]{} W. J. B., [Vieira]{} S. L. A., 2006, , 457, 581
K., [Yamauchi]{} S., [Koyama]{} K., 2005, , 618, 360
L. A., [Strom]{} S. E., [Vrba]{} F. J., [Keene]{} J., 1992, , 397, 613
S., [Sch[ö]{}ller]{} M., [Savanov]{} I., [Yudin]{} R. V., [Pogodin]{} M. A., [[Š]{}tefl]{} S., [Rivinius]{} T., [Cur[é]{}]{} M., 2009, Astronomische Nachrichten, 330, 708
J. D. [et al.]{}, 2013, , 429, 2960
H. U. [et al.]{}, 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
S. [et al.]{}, 2008, , 489, 1157
S., [Preibisch]{} T., [Ohnaka]{} K., 2008, , 676, 490
R. L., 1993, [SYNTHE spectrum synthesis programs and line data]{}. Smithsonian Astrophysical Observatory
R. B., 2003, Reports on Progress in Physics, 66, 1651
D., [Pringle]{} J. E., 1974, , 168, 603
P., [Bhatt]{} H. C., [Maheswar]{} G., [Muneer]{} S., 2006, , 653, 657
P., [Maheswar]{} G., [Bhatt]{} H. C., 2002, , 334, 419
S. C., 1997, , 478, L33
C. F., [Ostriker]{} E. C., 2007, , 45, 565
I., [Calvet]{} N., [Montesinos]{} B., [Mora]{} A., [Muzerolle]{} J., [Eiroa]{} C., [Oudmaijer]{} R. D., [Mer[í]{}n]{} B., 2011, , 535, A99
I., [Mora]{} A., [Montesinos]{} B., [Eiroa]{} C., [Meeus]{} G., [Mer[í]{}n]{} B., [Oudmaijer]{} R. D., 2012, , 543, A59
A. S., [Gray]{} R. O., [Klochkova]{} V. G., [Bjorkman]{} K. S., [Kuratov]{} K. S., 2004, , 427, 937
A. S., [Levato]{} H., [Bjorkman]{} K. S., [Grosso]{} M., 2001, , 371, 600
A. [et al.]{}, 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7737, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series
J. D., [Millan-Gabet]{} R., 2002, , 579, 694
B., [Eiroa]{} C., [Mora]{} A., [Mer[í]{}n]{} B., 2009, , 495, 901
A. [et al.]{}, 2001, , 378, 116
J. C., [Hoare]{} M. G., [Lumsden]{} S. L., [Oudmaijer]{} R. D., [Urquhart]{} J. S., [Sheret]{} T. L., [Clarke]{} A. J., [Allsopp]{} J., 2007, , 476, 1019
K., [Lumsden]{} S. L., [Oudmaijer]{} R. D., [Davies]{} B., [Wheelwright]{} H. E., [Hoare]{} M. G., [Ilee]{} J. D., 2013, , 436, 511
J., [Calvet]{} N., [Hartmann]{} L., 1998, , 492, 743
J., [Calvet]{} N., [Hartmann]{} L., [D’Alessio]{} P., 2003, , 597, L149
J., [D’Alessio]{} P., [Calvet]{} N., [Hartmann]{} L., 2004, , 617, 406
J., [Hartmann]{} L., [Calvet]{} N., 1998, , 116, 2965
R. D. [et al.]{}, 2001, , 379, 564
R. D. [et al.]{}, 2011, Astronomische Nachrichten, 332, 238
B., [Zinnecker]{} H., 1993, , 278, 81
F., [Najita]{} J., [Ostriker]{} E., [Wilkin]{} F., [Ruden]{} S., [Lizano]{} S., 1994, , 429, 781
M. F. [et al.]{}, 2006, , 131, 1163
F., [Pezzuto]{} S., [Corciulo]{} G. G., [Bianchini]{} A., [Vittone]{} A. A., 1998, , 505, 299
E. [et al.]{}, 2008, , 489, 1151
P. S., [de Winter]{} D., [Perez]{} M. R., 1994, , 104, 315
W.-F., [van Dalen]{} B., [Bik]{} A., [Waters]{} L. B. F. M., 2005, , 430, L61
N. J., [Fromang]{} S., [Gammie]{} C., [Klahr]{} H., [Lesur]{} G., [Wardle]{} M., [Bai]{} X.-N., 2014, ArXiv e-prints
P. G., [Monnier]{} J. D., [Danchi]{} W. C., 2001, , 409, 1012
M. E., [de Winter]{} D., [Tjin A Djie]{} H. R. E., 1998, , 330, 145
G., 2010, PhD thesis, Astronomical Institute Anton Pannekoek, University of Amsterdam
G., [van den Ancker]{} M. E., [Fedele]{} D., [Acke]{} B., [Dominik]{} C., [Waters]{} L. B. F. M., [Bouwman]{} J., 2008, , 485, 487
F., 2010, , 151, 209
J. [et al.]{}, 2011, , 536, A105
S. L. A., [Corradi]{} W. J. B., [Alencar]{} S. H. P., [Mendes]{} L. T. S., [Torres]{} C. A. O., [Quast]{} G. R., [Guimar[ã]{}es]{} M. M., [da Silva]{} L., 2003, , 126, 2971
J. S., [Drew]{} J. E., [Harries]{} T. J., [Oudmaijer]{} R. D., [Unruh]{} Y., 2005, , 359, 1049
J. S., [Drew]{} J. E., [Harries]{} T. J., [Oudmaijer]{} R. D., [Unruh]{} Y. C., 2003, , 406, 703
J. V., [Jenkins]{} C. R., 2003, [Practical Statistics for Astronomers]{}. Cambridge University Press
L. B. F. M., [Waelkens]{} C., 1998, , 36, 233
H. E., [de Wit]{} W. J., [Weigelt]{} G., [Oudmaijer]{} R. D., [Ilee]{} J. D., 2012, , 543, A77
H. E., [Oudmaijer]{} R. D., [de Wit]{} W. J., [Hoare]{} M. G., [Lumsden]{} S. L., [Urquhart]{} J. S., 2010, , 408, 1840
B. A., [Indebetouw]{} R., [Bjorkman]{} J. E., [Wood]{} K., 2004, , 617, 1177
K., [Lada]{} C. J., [Bjorkman]{} J. E., [Kenyon]{} S. J., [Whitney]{} B., [Wolff]{} M. J., 2002, , 567, 1183
\[lastpage\]
[^1]: Based on observations made with the ESO Very Large Telescope at the Cerro Paranal Observatory under programme IDs 079.C-0725, 084.C-0952A, 087.C-0124A and 279.C-5031A
|
---
abstract: 'The origin and composition of ultra-high-energy cosmic rays (UHECRs) remain a mystery. The proton dip model describes their spectral shape in the energy range above $10^9$ GeV by pair production and photohadronic interactions with the cosmic microwave background. The photohadronic interactions also produce cosmogenic neutrinos peaking around $10^9$ GeV. We test whether this model is still viable in light of recent UHECR spectrum measurements from the Telescope Array experiment, and upper limits on the cosmogenic neutrino flux from IceCube. While two-parameter fits have been already presented, we perform a full scan of the three main physical model parameters: source redshift evolution, injected proton maximal energy, and spectral index. We find qualitatively different conclusions compared to earlier two-parameter fits in the literature: a mild preference for a maximal energy cutoff at the sources instead of the Greisen–Zatsepin–Kuzmin (GZK) cutoff, hard injection spectra, and strong source evolution. The predicted cosmogenic neutrino flux exceeds the IceCube limit for any parameter combination. As a result, the proton dip model is challenged at more than 95% C.L. This is strong evidence against this model independent of mass composition measurements.'
author:
- 'Jonas Heinze, Denise Boncioli, Mauricio Bustamante, & Walter Winter'
title: Cosmogenic Neutrinos Challenge the Cosmic Ray Proton Dip Model
---
Introduction
============
Ultra-high-energy cosmic rays (UHECRs) are charged particles of astrophysical origin with energies above $10^9$ GeV, the highest observed. Their sources are unknown, but their energy spectrum has been measured with increasing precision [@Valino:2015; @Ivanov:2015]. It exhibits a hardening at about $5 \times 10^9$ GeV – the “ankle” – and a strong suppression at the topmost energies, around $5 \times 10^{10}$ GeV.
If UHECRs above $10^9$ GeV are mainly protons of extragalactic origin, the spectral features can be attributed to interactions with the cosmic microwave background (CMB) and the infrared/optical photon background (CIB). In this “proton dip” model, energy losses due to electron-positron pair production on CMB photons are responsible for the ankle [@DeMarco:2003ig; @Berezinsky:2005cq; @Berezinsky:2002nc; @Aloisio:2006wv; @Aloisio:2007rc]. The UHECR spectrum is consistent with a power law in energy, with spectral index of $2.4 - 2.8$, where lower values imply a stronger redshift evolution of the number density of UHECR sources [@DeMarco:2005ia]. Photopion production on the CMB creates a high-energy cutoff – the “GZK cutoff” [@Greisen:1966jv; @Zatsepin:1966jv]. This is effectively a cosmic ray horizon: protons detected with energy above the GZK cutoff were necessarily born in the local universe. Photopion interactions also create “cosmogenic neutrinos”, with $\sim 10^9$ GeV.
An alternative to the proton dip model posits that the transition to a flux dominated by extragalactic cosmic rays occurs at the ankle. Additionally, if UHECRs are a mixture of nuclei, the interpretation of spectral features is more intricate[^1]; the flux suppression at the highest energies is due to the photodisintegration of nuclei on the photon backgrounds. Presently, the proton dip, ankle, and mixed composition models all remain ostensibly viable alternatives. We will test whether the former still is.
The largest UHECR observatories – the Pierre Auger Observatory [@ThePierreAuger:2015rma] and the Telescope Array (TA) [@AbuZayyad:2012kk] – aim to settle the issue. They detect UHECR-initiated extensive air showers via surface Cherenkov water tanks or scintillators, fluorescence detectors, or a combination of both techniques. Measurements of the UHECR mass composition, [[*i.e.*]{}]{}, the relative abundance of lighter versus heavier nuclei, could in principle test the validity of the proton dip model. Composition is determined chiefly by measuring the column depth in the atmosphere at which the particle content of a shower is maximal.
While TA finds consistency with a light primary composition above $10^9$ GeV [@Belz:2015], Auger finds that the mass of the primary reaches a minimum around $10^{9.3}$ GeV before rising with energy [@Porcelli:2015]. Measurements of the correlation between the depth of the shower maximum and the number of muons can contribute to the determination of the primary composition. A similar idea is used within the Auger analysis in the energy range of the ankle, finding that the results are in favor of a mixed composition [@Yushkov:2015]. Taken at face value, the Auger results could be interpreted as evidence against the proton dip model [@Allard:2005ha; @Allard:2008gj; @Aloisio:2009sj]. However, the depths of the shower maximum of the two experiments agree within systematic uncertainties [@Unger:2015], and strong conclusions about composition are unattainable due to uncertainties in the interaction of different primaries and shower development [@Aab:2014kda; @Aab:2014aea].
A self-consistent interpretation of spectrum and composition results would require propagating a mix of nuclei that interact with the CMB and CIB [@Allard:2008gj; @Taylor:2011ta; @Fang:2013cba; @Aloisio:2013hya; @Taylor:2013gga; @Taylor:2015rla; @Peixoto:2015ava; @Globus:2015xga; @Unger:2015laa; @diMatteo:2015]. We do not attempt to discuss such a mixed-composition interpretation.
In this paper, we instead ask whether the simplest UHECR model – the proton dip model – is still viable in light of two recent results: the UHECR spectrum measurements from the TA Collaboration, comprising 7 years of data [@TAcombined; @Ivanov:2015], and the recent upper bound on the flux of cosmogenic neutrinos from the IceCube Collaboration [@Ishihara:2015; @icecubeupdate]. We do not use mass composition measurements.
Assuming a population of generic extragalactic proton sources, we scan simultaneously over the three key model parameters: proton spectral injection index, maximal injected energy, and source redshift evolution. While two-parameter fits have been performed before by @DeMarco:2005ia [@Ahlers:2010fw; @Kido:2015], this is the first reported three-parameter fit.
We find a compelling conclusion: the high cosmogenic neutrino fluxes implied by TA data challenge the proton dip model at $>95\%$ C.L., for any parameter combination.
This paper is organized as follows. We present our proton injection and propagation model, and the fitting procedure, in [Section]{} \[sec.propagation\]. In [Section]{} \[sec.fit\] we show the results of 2D and 3D scans of the parameter space. In [Section]{} \[sec.neu\] we calculate the associated cosmogenic neutrinos. We summarize and conclude in [Section]{} \[sec.conclusions\]. [Appendix]{} \[sec.appA\] shows the result of the fit performed using the same assumptions related to the sources as TA, while [Appendix]{} \[sec.appB\] shows that our conclusions are robust to model variations.
Model and methods {#sec.propagation}
=================
3D fits of UHECR parameters {#sec.fit}
===========================
Cosmogenic neutrinos {#sec.neu}
====================
Summary and conclusions {#sec.conclusions}
=======================
We thank Dmitri Ivanov and Peter Tinyakov for useful communications; and John Beacom, Shirley Li, Kohta Murase and Alan Watson for valuable discussion. MB was partially supported by NSF Grant PHY-1404311 to JFB. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant No. 646623).
[93]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
() (, ).
() (, ).
, , , ****, (), .
, , , ****, (), .
, , , ****, (), .
, , , , , , ****, (), .
, , , , ****, (), .
, ****, (), .
, ****, ().
, ****, (), .
, ****, (), .
(), ****, (), .
(), ****, (), .
() (, ).
() (, ).
() (, ).
, , , , , ****, (), .
, , , , , ****, (), .
, , , ****, (), .
() (, ).
(), ****, (), .
(), ****, (), .
, , , ****, (), .
, , , ****, (), .
, , , ****, (), .
, ****, (), .
, , , ****, (), .
, , , ****, (), .
, , , ****, (), .
, , , (), .
() (, ).
(), .
() (, ).
(), .
, , , , , ****, (), .
() (, ).
, ****, (), .
, , , ****, (), .
, ****, (), .
, , , ****, (), .
, , , ****, (), .
, , , , ****, (), .
, , , , , ****, (), .
, , , ****, (), .
(), ****, (), .
, , , ****, (), .
, ****, (), .
, ****, (), .
, ****, (), .
, ****, (), .
(), in ** ().
, ****, ().
, , , ****, (), .
, , , , ****, (), .
, ****, (), .
, , , , , , , , ****, (), .
, , , , ****, (), .
, , , ****, (), .
, , , , ****, (), .
, , , ****, (), .
, ****, (), .
, , , ****, (), .
, ****, (), .
, , , , ****, (), .
, ****, ().
, , , , , , ****, (), .
, ****, (), .
, ****, (), .
(), (), .
(), ****, (), .
(), ****, (), .
, ****, ().
, ****, (), .
(), ****, ().
, ****, (), .
(), ****, (), .
(), in ** ().
(), ****, (), .
() (, ).
, , (), ****, (), .
(), ****, (), .
(), ****, (), .
(), ****, (), .
(), ****, (), .
(), ****, (), .
(), ****, (), .
, ****, (), .
, ****, ().
, , , ****, (), .
(), ****, (), .
(), (), .
, ****, (), .
, , , ****, (), .
Fits using the astrophysical TA assumptions {#sec.appA}
===========================================
Varying the model and fitting procedure {#sec.appB}
=======================================
[^1]: For a review on the influence of the extragalactic propagation of UHECRs on their energy spectrum and composition, see @Allard:2011aa.
|
---
abstract: 'In this paper, we propose a novel generative network (SegAttnGAN) that utilizes additional segmentation information for the text-to-image synthesis task. As the segmentation data introduced to the model provides useful guidance on the generator training, the proposed model can generate images with better realism quality and higher quantitative measures compared with the previous state-of-art methods. We achieved Inception Score of 4.84 on the CUB dataset [@WahCUB_200_2011] and 3.52 on the Oxford-102 dataset [@Nilsback08]. Besides, we tested the self-attention SegAttnGAN which uses generated segmentation data instead of masks from datasets for attention and achieved similar high-quality results, suggesting that our model can be adapted for the text-to-image synthesis task.'
author:
- Yuchuan Gou
- Qiancheng Wu
- Minghao Li
- Bo Gong
- Mei Han
bibliography:
- 'egbib.bib'
title: 'SegAttnGAN: Text to Image Generation with Segmentation Attention'
---
Introduction
============
The task of generating high fidelity, realistic-looking images based on semantic description is central to many applications. A lot of research has been focused on the text-to-image synthesis task, which takes in natural language descriptions to generate images matching the text.
Many models for this task use generative adversarial networks (GANs) [@han2017stackgan; @Tao18attngan; @Han17stackgan2; @reed2016generative; @qiao2019mirrorgan], conditioned on the text input rather than Gaussian noise for image generation. Although models like [@Tao18attngan] achieve satisfactory visual quality while maintaining the image-text consistency, there is little control over the layout of the generated images except for specific keywords which uniquely constrain the shape of the objects. Frequently these models would generate objects with deformed shapes or images with unrealistic layouts (see Figures \[fig:intro\_result\] and \[fig:result1\]).
Recent work in [@park2019SPADE] has shown that decent results can be achieved for image synthesis task when spatial attention from segmentation data is utilized to guide image generation. To solve the deformed shapes and unrealistic layouts problems, we designed SegAttnGAN, which utilizes the segmentation to add global spatial attention in addition to text input. We hope that the spatial information would regulate the layout of generated images thus create more realistic images. Experimentation has shown promising results when additional segmentation information is used to guide image generation.
![Sample results generated from AttnGAN [@Tao18attngan], our proposed SegAttnGAN and self-attention SegAttnGAN.[]{data-label="fig:intro_result"}](images/intro_result.png){width="\linewidth"}
Our contributions can be summarized as the following:
1. We proposed a novel generative network that uses both text and spatial attention to generate realistic images.
2. We verified that the addition of spatial attention mechanism to GAN could substantially increase visual realism by regulating object shapes and image layouts.
3. We built a self-attention network to generate segmentation masks first and then use it for image generation. Based on the qualitative results, self-attention model can also constrain the object shapes very well.
Related Work
============
As text-to-image synthesis played an important role in many applications, different techniques have been proposed for text-to-image synthesis task. Reed [@reedpixelcnn] utilized PixelCNN to generate image from text description. Mansimov [@mansimov16_text2image] proposed a model iteratively draws patches on a canvas, while attending to the relevant words in the description and Nguyen [@nguyen] used an approximate Langevin sampling approach to generate image conditioned on text.
Since Goodfellow [@goodfellow] introduced Generative Adversarial Networks (GANs), extensive research has been conducted on image generation task with different types of GANs and high-quality results have been achieved [@radford2015; @conditionalgan; @pggan; @biggan; @stylegan; @zhu2017unpaired; @pix2pix2017; @park2019SPADE]. At the same time, researchers have also started to apply GAN techniques on text-to-image synthesis tasks. Reed [@reed2016generative] proposed a conditional GAN to generate images of birds and flowers from detailed text descriptions and in [@reed2016learning] they added object location control to the conditional GAN. Zhang [@han2017stackgan] proposed StackGAN to generate images from text. StackGAN consists of Stage-I and Stage-II GANs where the Stage-I GAN generates low-resolution images and the Stage-II GAN generates high-resolution images. Compared with StackGAN which is conditioned on sentence level, AttnGAN proposed by Xu [@Tao18attngan] develops conditioning on both sentence level and word level aiming at generating fine-grained high-quality images from text descriptions. Zhang [@zhang2018hdgan] proposed a hierarchically-nested GAN for text-to-image synthesis. Qiao [@qiao2019mirrorgan] proposed MirrorGAN in order to achieve both visual realism and semantic consistency. Hong [@hong18] and Li [@objgan19] are both concentrating on text-to-image synthesis task in a coarse-to-fine way. But their focus is the word embedding module and object-level discrimination by designing a bidirectional LSTM on either global or object level. While our focus lies on the generator with attention mechanism to effectively constrain the object boundary given segmentation maps.
Semantic information provides useful guidance in image generation. It has been introduced as input in different formats. Works in [@huang2018munit; @pix2pix2017; @zhu2017multimodal] used edge map as guidance in image to image translation. Karacan [@Karacan2016LearningTG] and Park [@park2019SPADE] used semantic layout as guidance in image generation. In [@gu2019maskguided], facial masks have been provided as guidance to generate faces. Our work differs from these works as we apply semantic masks in text-to-image synthesis task while theirs are dealing with image-to-image translation or image generation.
SegAttnGAN for text-to-image synthesis
======================================
{width="0.9\linewidth"}
SegAttnGAN architecture
-----------------------
Text-to-image generation models usually encode the whole sentence text description into a conditional vector. AttnGAN [@Tao18attngan] has also proposed a word attention model that helps model generate different images conditioned on words. As shown in Figure \[fig:generator\], we adapt this mechanism and an LSTM text encoder is used in our SegAttnGAN to extract word features and sentence features. The sentence feature is concatenated with random latent vector, and the word features are used as word-level attention.
Segmentation attention module
-----------------------------
The segmentation attention module is used to enhance image synthesis by persevering the spatial constraint of the input semantic maps. Park [@park2019SPADE] has shown its efficacy and we use the same mechanism for segmentation attention module.
Mathematically, we define $F$ as features from the previous layer and $S$ as the input segmentation maps. The output of this attention module, referred from [@park2019SPADE], to preserve spatial constrain could be expressed as in Equation \[attention\_math\]:
$$\label{attention_math}
\begin{aligned}
F\prime = BN(F) * Conv(S) + Conv(S)
\end{aligned}$$
where $BN()$ is the batch normalization function while $Conv()$ is the convolution function.
The core of this function is its property to preserve spatial information of segmentation masks. It’s closely similar to the attention module in the Super Resolution task [@wang2018sftgan]. By introducing the semantic map attention into each upsampling layer in a coarse-to-fine strategy, this model promisingly avoids the semantics being eliminated by pure upsampling layers.
Segmentation mask strategies
----------------------------
We have two different models when we apply different strategies for segmentation masks. The first model, named SegAttnGAN, uses pre-existing masks in the datasets as attention input. The other model, named self-attention SegAttnGAN, uses masks generated by the self-attention generator.
The self-attention generator generates segmentation masks and trained with the corresponding discriminator. Same as SegAttnGAN, it utilizes coarse-to-fine training strategy, with resolutions from 64\*64, 128\*128 to 256 \* 256. The self-attention generator takes the same z vector and text embedding vector from SegAttnGAN as input. And at each resolution level, there is a discriminator for training.
Objective
---------
For the generative adversarial network, the classic objective function with conditional inputs is a min-max game between generator and discriminators defined in Equation \[conditional GAN\]:
$$\label{conditional GAN}
\begin{aligned}
\min_G \max_D V(G,D) &= E_{x\sim P_{data}(x)}[\log D(x,t)] \\
& + E_{z\sim P_{z}(z)}[\log(1-D(G(z, t, s),t))]
\end{aligned}$$
where $x$ refers to images from real data distribution and $z$ represents the random latent vector which drives the fake data generation. And $t$ and $s$ respectively refer to the text and segmentation input.
Therefore, the loss function for generators is defined in Equation \[G loss\]:
$$\label{G loss}
\begin{aligned}
L_{G_i} &= -E_{z \sim P_{z}(z)}[\log(D_i(G_i(z,t,s)))] / 2 \\
& -E_{z \sim P_{z}(z)}[\log(D_i(G_i(z,t,s),t))] / 2
\end{aligned}$$
where the first term is an unconditional loss determining whether the image is real or fake while the second term, the conditional loss, determines whether the generated image matches the text description.
The loss function for discriminator $D_i$ is defined as in Equation \[D condition\]:
$$\label{D condition}
\begin{aligned}
L_{D_i} &= -E_{x \sim P_{data}(x)}[\log(D_i(x))] / 2 \\
& -E_{z \sim P_{z}(z)}[\log (1-D_i(G_i(z, t, s)))] / 2 \\
& -E_{x \sim P_{data}(x)}[\log(D_i(x,t))] / 2 \\
& -E_{z \sim P_{z}(z)}[\log(1-D_i(G_i(z,t,s),t))] / 2
\end{aligned}$$
where the first two terms are corresponding to the unconditional loss for optimizing discriminator while the last two terms are conditional losses.
For self-attention SegAttnGAN, we define self-attention generator as $G_{s}$. We use $G_{s}(z, t)$ instead of $s$ in Equation \[G loss\] and \[D condition\] to define G loss and D loss. The overall loss is defined in Equation \[All loss for self attention\]:
$$\label{All loss for self attention}
\begin{aligned}
L = L_{G} + L_{G_s} + \lambda L_{DAMSM}
\end{aligned}$$
where $L_{G} = \sum_{i=0}^{m-1} L_{G_{i}}$, $L_{G_s} = \sum_{i=0}^{m-1} L_{G_{s_i}}$ and $L_{DAMSM}$ follows the DAMSM loss in [@Tao18attngan].
Implementation details
----------------------
As shown in Figure \[fig:generator\], the generator in SegAttnGAN outputs $64*64$, $128*128$, $256*256$ images. First, we processed the segmentation mask into label maps (each channel contains different objects). And at each upsampling layer of the generator, we downsampled the segmentation label maps into the same resolution tensors as the current hidden features in the generator. Then we applied the attention module after the previous upsampling operations. The text and image encoders are following the same implementation from AttnGAN. For self-attention SegAttnGAN, there is no word features for the self-attention generator. The text embedding dimension is set to $256$, and loss weight $\lambda$ is set to $5.0$. ADAM solver with $beta_1=0.5$ and a learning rate of $0.0002$ are used for generator and discriminators.
Experiments
===========
Dataset
-------
We use CUB and Oxford-102 datasets to evaluate our proposed method. CUB dataset contains images of different birds in 200 categories. We use 8841 images from this dataset for training and 2947 images for testing. Oxford-102 is another dataset consists of flower images. From this dataset, we choose 6141 images for training and 2047 images for testing.
Evaluation metrics
------------------
We use two quantitative measurements to evaluate generated images. The first metric is Inception Score [@salimans2016improved], which has been widely used to evaluate the quality of generated images. Another metric is R-precision, which has been proposed in [@Tao18attngan] as a complimentary evaluation metric for the text-to-image synthesis task to determine whether the generated image is well conditioned on the given text description.
Quantitative results
--------------------
**Inception Scores:** We computed Inception Score with our generated images and compared it with those from other state-of-art methods [@reed2016generative; @reed2016learning; @han2017stackgan; @Han17stackgan2; @Tao18attngan; @qiao2019mirrorgan]. The comparisons on both CUB and Oxford-102 datasets are shown in Table \[tab:IS\]. Our model SegAttnGAN achieves the highest Inception Score on both CUB and Oxford-102 datasets. Compared with the baseline model AttnGAN, our SegAttnGAN boosts Inception Score from $4.36$ to $4.82$ on CUB dataset. Also, our self-attention SegAttnGAN gets a good Inception Score of $4.44$ on CUB and $3.34$ on Oxford-102.
Model CUB Oxford-102
------------------------------------- --------------- ---------------
GAN-INT-CLS 2.88$\pm$0.04 2.66$\pm$0.03
GAWWN 3.62$\pm$0.07 $-$
StackGAN 3.70$\pm$0.04 3.20$\pm$0.01
StackGAN++ 3.82$\pm$0.06 3.26$\pm$0.01
AttnGAN (baseline) 4.36$\pm$0.03 $-$
MirrorGAN 4.56$\pm$0.05 $-$
[**SegAttnGAN (self-attention)**]{} 4.44$\pm$0.06 3.36$\pm$0.08
[**SegAttnGAN**]{} 4.82$\pm$0.05 3.52$\pm$0.09
: Inception Score of state-of-art models and our models (in bold) on CUB and Oxford-102 datasets.[]{data-label="tab:IS"}
**R-precision scores:** As shown in Table \[tab:R\], our SegAttnGAN and self-attention SegAttnGAN also get a good R-precision score compared to AttnGAN. SegAttnGAN’s score is almost the same as AttnGAN’s score, indicating that SegAttnGAN can generate images consistent with input text descriptions. MirrorGAN gets the highest R-precision score as it contains a module especially for improving semantics consistency.
Model CUB
------------------------------------- -----------------
AttnGAN (baseline) $53.31$
MirrorGAN $57.67$
[**SegAttnGAN (self-attention)**]{} [**$52.29$**]{}
[**SegAttnGAN**]{} [**$52.71$**]{}
: R-precision (%) of state-of-art models and our models.[]{data-label="tab:R"}
**Segmentaion attention on other models:** We also applied our segmentation attention module on StackGAN++, and the Inception scores are shown in Table \[tab:IS\_2\]. These results indicate that our segmentation attention module can help constrain the training of different GAN models by extra semantics information and get better image generation quality.
Model CUB
--------------------------------------------- -----------------------
AttnGAN $4.36\pm0.03$
[**AttnGAN + segmentation attention**]{} [**$4.82\pm0.05$**]{}
StackGAN++ $3.82\pm0.06$
[**StackGAN++ + segmentation attention**]{} [**$4.31\pm0.04$**]{}
: Inception Score comparisons of models with (in bold) and without Segmentation Attention.[]{data-label="tab:IS_2"}
Qualitative results
-------------------
In Figure \[fig:result1\](a), we show some samples generated by AttnGAN and our models. As shown in the figure, compared to the baseline model AttnGAN [@Tao18attngan], our SegAttnGAN generates results with better object shape. Although self-attention SegAttnGAN uses generated segmentation masks, it can constrain the object shapes and generate better images than AttnGAN.
![(a) Example results of our models compared to AttnGAN. (b) SegAttnGAN results with text and segmentation. []{data-label="fig:result1"}](images/segattngan_results.png){width="\linewidth"}
Figure \[fig:result1\](b) shows samples illustrating how the shape and text constrain output images of SegAttnGAN on CUB and Oxford-102 datasets. As shown in the figure, words related to color such as red and purple lead to results with different colors. The object shapes in generated images matching the input masks demonstrates that the segmentation map provides very good control over object shapes.
Limitation and discussion
-------------------------
SegAttnGAN performs well and gets the highest Inception Score compared to other methods, but this model needs segmentation input during the inference phase. Our self-attention SegAttnGAN only needs segmentation data during the training phase, and it gets better visual results compared to other models with the help of object shape constrain. But its Inception Score showed that its results get a similar image objectiveness and diversity compared to AttnGAN.
Conclusion
==========
In this paper, we propose SegAttnGAN for text-to-image synthesis tasks, which uses segmentation attention to constrain the GAN training and successfully generates better quality images compared to other state-of-art methods. With the segmentation masks from datasets as input, our SegAttnGAN achieves the highest Inception Scores on both CUB and Oxford-102 datasets. When the masks are generated via our self-attention generator, our self-attention SegAttnGAN also generates results with better visual realism compared to other state-of-art methods.
Acknowledgement
===============
We thank Jui-Hsin Lai and Jinghong Miao for excellent comments.
|
---
author:
- 'G. Lanzuisi[^1]'
- 'I. Delvecchio'
- 'S. Berta[^2]'
- 'M. Brusa'
- 'A. Comastri'
- 'R. Gilli'
- 'C. Gruppioni'
- 'S. Marchesi'
- 'M. Perna'
- 'F. Pozzi'
- 'M. Salvato'
- 'M. Symeonidis'
- 'C. Vignali'
- 'F. Vito'
- 'M. Volonteri'
- 'G. Zamorani.'
date: 'Received 25 October 2016; Accepted 22 February 2017'
title: 'AGN vs. host galaxy properties in the COSMOS field'
---
Introduction {#sec:intro}
============
Super massive black hole (SMBH) growth and galaxy build up follow a similar evolution through cosmic history, with a peak at $z\sim2-3$ and a sharp decline toward the present age (see Madau & Dickinson 2014 for a review). Furthermore, at $z=0$, SMBH and their hosts sit on tight relations that link the SMBH mass and the bulge properties of the host, such as luminosity, stellar mass and velocity dispersion (Kormendy & Richstone 1995, Magorrian et al. 1998, Kormendy & Ho 2013). Therefore the SMBH growth and the star formation history are likely related in some way during the cosmic time.
The key parameter that regulates both processes seems to be cold/molecular gas supply (Lagos et al. 2011, Vito et al. 2014, Delvecchio et al. 2015, Saintonge et al. 2016). Star formation-related processes (e.g. supernova and stellar wind) are known to produce galaxy-wide outflows that can regulate the in-fall of gas and therefore the star formations itself (e.g. Genzel et al. 2011). But more powerful mechanisms, globally identified as ‘AGN feedback’, have been invoked in numerical and semi-analytic models of galaxy evolution (e.g. Granato et al. 2004; Di Matteo, Springel & Hernquist 2005, Menci et al. 2008, Sijacki et al. 2015, Dubois et al. 2016, Pontzen et al. 2016) in order to reproduce the observed galaxy population, and particularly the high mass end of the galaxy mass function.
Observationally, the role of AGN activity in influencing the evolution of the global galaxy population is not clear yet. This issue has been investigated, in the past, looking for correlations between average AGN and host properties, such as BH accretion rates (BHAR) or AGN luminosity (typically in the $2-10$ keV band, hereafter) on one side, and SF rates (SFR) or IR luminosity (in the $8-1000~\mu m$, hereafter) on the other side, taking advantage of the wealth of multi-wavelength information collected in deep extragalactic surveys. However, different, somewhat contradictory results have been reported in the past years.
Several studies have found a flat distribution computing average in bins of of X-ray selected sources (or SFR and BHAR, respectively, e.g. Shao et al. 2010, Rovilos et al. 2012, Rosario et al. 2012) at low redshift and luminosities. A significant positive correlation instead appears for luminous AGN ($>10^{43-44}$ ) and high redshifts ($z>1-2$), suggesting two different triggering mechanisms at high and low luminosities, via merger and secular evolution, respectively.
Other groups have found a linear correlation at all z and , when computing the average in bins of (in log-log space) of IR selected sources (Chen et al. 2013, Delvecchio et al. 2015, Dai et al. 2015), with the ratio Log(SFR/BHAR)$\sim3$, roughly consistent with the local M$_{bulge}$/SMBH value (Magorrian et al. 1998, Marconi & Hunt 2003). Finally, other authors have found no correlation at all, regardless of the and z range (e.g. Mullaney et al. 2012, Stanley et al. 2015).
Looking at AGN obscuration, Rovilos et al. (2012) explored for the first time the possible relation between AGN column density (), as measured from the X-ray spectra, and host properties, finding no correlation on a sample of 65 sources in the XMM-CDFS survey (Comastri et al. 2011). Rosario et al. (2012) found similar result from hardness ratios (HR) on a larger sample in COSMOS, while Rodighiero el al. (2015) found a positive correlation between and , again based on HR, on a sample of $z\sim2$ AGN hosts in the same field.
From a technical point of view, these differences may partly arise from different biases and analysis methods. For example, given that only a small fraction of the X-ray detected sources are FIR detected, and vice-versa, most of these studies rely on X-ray or FIR staking in order to recover the average properties of large samples of AGN/host systems, or are limited to small subsamples. Mullaney et al. (2015) pointed out that modeling the SFR distribution of X-ray selected hosts as a log-normal distribution, and including upper-limits, gives different results than computing the linear mean of the distribution (i.e. via staking), that is instead driven upwards by the bright outliers.
Another issue was raised by Symeonidis et al. (2016), showing that the intrinsic AGN SED in the FIR is cooler than usually assumed. Therefore in some cases there is no ‘safe’ photometric band which can be used in calculating the SFR, without subtracting the AGN contribution. On the other hand, several of the works cited above, take directly the FIR photometry (typically at $60\mu m$) in order to estimate SFR, thus potentially introducing a spurious correlation at high AGN luminosities.
Recently, from theoretical studies, a physical mechanism has been proposed to explain part of these contradictory results: Volonteri et al. (2015a,b) explain these different observations with the way we analyze the data: the bivariate distribution of AGN and SF luminosities gives two very different results, depending on the binning axis. Hickox et al. (2014) reached similar conclusions starting from the simple assumptions that long term AGN activity and SFR are perfectly correlated, that the observed SFR is the average over $\sim100$ Myr, while the AGN activity, traced by X-ray emission, varies on a much shorter time scales. In these models the different time scales involved in AGN and SF variability wash out the linear dependency between the two quantities, if the rapidly variable AGN luminosity is used to build the subsamples to be studied ‘on average’. This result was also confirmed observationally by Dai et al. (2015) using shallow data from XMM-LSS.
Furthermore, Volonteri et al. (2015a) suggest that spatial scales are important: the BH accretion rate should be correlated with the nuclear ($<100$ pc) SFR, while it is less correlated with the total ($<5$ kpc) SFR, except for the most intense merger episodes, that are able to affect the whole host galaxy. Of course, the SFR that can be inferred from the FIR luminosity is the global, galaxy-scale SFR (with the exception of the local Universe, see e.g. Diamond-Stanic & Rieke 2012), and this introduce another source of uncertainty in the observational comparison between BHAR and SFR.
Here we explore the possible correlations between AGN and host properties for a large sample of X-ray and FIR detected sources thanks to the extensive , and coverage on the COSMOS field (Scoville et al. 2007, Hasinger et al. 2007, Elvis et al. 2009, Lutz et al. 2011, Oliver et al. 2012). This approach avoids the uncertainties related to the staking, and allow for a proper SED deconvolution, source by source. This of course limits the significance of our findings to the brightest, most accreting and most star forming systems. These are, however, the most interesting ones: the ones for which there is less agreement in the literature on the presence of a correlation between AGN and SF, and also the ones for which theoretical models predict that the correlation should be stronger.
The paper is organized as follows: section 2 describes the sample and source properties; section 3 presents the analysis of and distributions; in section 4 we compare our results with a set of hydrodynamical simulations; in section 5 we discuss correlations between nuclear obscuration and host properties and in section 6 we discuss our results. Throughout the paper, we assume a standard $\Lambda$CDM cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_\Lambda=0.73$ and $\Omega_M=0.27$ (Bennett et al 2013).
The sample
==========
{width="8cm"} {width="8cm" height="8cm"}
We performed X-ray spectral fitting for all the and detected sources (from the catalogs of Brusa et al. 2010 and Civano et al. 2015 respectively) with more than 30 counts, in Marchesi et al. (2016) and Lanzuisi et al. (2013, 2015), respectively. This sample consist of 2333 individual sources (1949 and 1187 sources, with 803 source in common[^3]).
For all the detected sources in the COSMOS field (Lutz et al. 2011, $\sim17000$ with at least a detection at $>3\sigma$ in one of the FIR bands, from 100 to 500 $\mu m$), an SED deconvolution with 3 components — stellar emission, AGN torus emission and SF-heated dust emission — performed using photometric points from the UV to sub-mm, is available from Delvecchio et al. (2014, 2015, D15 hereafter), following the recipe described in Berta et al. (2013).
We then selected all the and detected sources, having at least one FIR detection (and therefore SED deconvolution). The final sample comprises 692 sources X-ray and FIR detected (the ’X-FIR sample’ hereafter), all of them with an available redshift, 459 spectroscopic and 233 photometric (Civano et al. 2012, Brusa et al. 2010, Salvato et al. 2009, Marchesi et al. 2015). This is to date the largest sample of AGN/host systems for which X-ray spectral parameters, such as column density and absorption-corrected 2-10 keV luminosity, are known in combination with host properties such as and SFR.
AGN properties
--------------
Figure 1 (left) shows the distribution of vs redshift for the X-FIR sample. The average $1\sigma$ error bar on is shown in the upper left corner. The absorption-corrected is affected by uncertainties related to both the number of net counts (observed flux uncertainties) and the spectral shape of each source (uncertainties on and spectral slope). Therefore, the errors have been derived, for each source, using the equivalent in [*Sherpa*]{} (Fruscione et al. 2006) of the [*cflux*]{} model component in [*Xspec*]{} (Arnaud 1996), applied to the best-fit unabsorbed powerlaw. The flux and errors are then computed in the observed band corresponding to 2-10 keV rest-frame, and converted into luminosity.
![SFR vs. distribution for the entire sample of detected sources ($\sim17000$ sources, gray points) and for the 692 sources with X-ray spectral analysis (X-FIR sample, red circles), divided in the five redshift bins defined in Sec. 2.1. The dashed lines in each panel mark the redshift-dependent MS of Withaker et al. (2012). The average $1\sigma$ error-bars are shown in the top left as a black cross. []{data-label="fig:mstardet"}](graf_mstar_sfr_new.ps){width="8.5cm"}
The redshift bins that will be used in the following analysis are shown with vertical dashed lines. The intervals have been chosen with the aim of having a fairly large number of sources in each bin ($\sim80-160$) with reasonably narrow redshift interval. The bins that will be used in the following (1 bin per dex) are shown as horizontal dashed lines.
{width="6cm"}{width="6cm"} {width="6cm"}{width="6cm"}
Figure 1 (right) shows the column density distribution for the X-FIR sample. Arrows show sources for which the obscuration is constrained only by an upper-limit. The average $1\sigma$ error bar on is shown in the upper left corner. The distribution of from spectral analysis has a clear upper boundary around CT column densities[^4] due to the strong flux decrement associated with CT obscuration in the 2-10 keV band. Also, the minimum measurable increases with redshift, as the low energy cut-off due to obscuration move outside the observing band.
The global fraction of X-ray obscured sources (those with $>10^{22} cm^{-2}$) in the X-FIR sample is $\sim50\%$, higher than the typical obscured fraction (30-40%) of the X-ray samples in the - and -COSMOS (Lanzuisi et al. 2015, Marchesi et al. 2016). Indeed, the FIR luminosity (and therefore detection rate) of type-2 AGN seems to be higher than for type-1 QSO (Chen et al. 2015).
Host properties
---------------
The host properties (SFR vs. ) of the 692 sources in the X-FIR sample are shown in figure \[fig:mstardet\] (red circles) divided in five redshift bins as described above. The values are taken from D15: the SFR has been derived by converting the IR luminosity (rest $8-1000~\mu m$) of the best-fitting galaxy SED (i.e. subtracting the AGN emission when present) with the SF law of Kennicutt (1998), scaled to a Chabrier (2003) initial mass function (IMF). The is derived from the SED decomposition itself, and based on Bruzual & Charlot (2003) models, with a consistent IMF. Table 1 (full version available on-line) summarizes the multi-wavelength properties of the sources in the X-FIR sample.
[lccccccccccc]{}\
ID & RA & DEC & z & Log() & Log() & SFR & Log() & Log() & Log() & XID & CID\
& deg & deg & & erg/s & & /yr & cm$^{-2}$ & erg/s & erg/s & &\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11)& (12)\
\
1846545 & 150.500 & 2.862 & 0.102s & $44.77\pm0.07$ & $10.20\pm0.13$ & 15.8 & $<20.77 $ & $41.05\pm0.27$ & 41.96 & 60095 & lid2100\
1883498 & 150.065 & 2.929 & 0.102s & $43.69\pm0.16$ & $10.55\pm0.09$ & 1.3 & $<20.42 $ & $42.64\pm0.06$ & 43.8 & 5617 & lid385\
89570 & 150.372 & 1.609 & 0.104s & $44.03\pm0.08$ & $10.80\pm0.09$ & 2.8 & $22.58_{-0.03}^{+0.03}$ & $43.04\pm0.04$ & 44.29 & 2021 & cid1678\
1612003 & 150.550 & 2.628 & 0.113s & $43.37\pm0.28$ & $10.58\pm0.09$ & 0.4 & $<21.36 $ & $41.63\pm0.36$ & 42.6 & — & lid3189\
1197519 & 150.335 & 2.304 & 0.123s & $44.24\pm0.06$ & $10.58\pm0.09$ & 4.7 & $21.40_{-0.29}^{+0.62}$ & $41.2 \pm0.29$ & 42.11 & 1533 & cid967\
... & & & & & & & & & & &\
[**Notes.**]{} Catalog entries are as follows: (1) Source ID from Capak et al. (2007); (2) and (3) right ascension and declination of the optical/IR counterpart; (4) redshift ([*s*]{} for spectroscopic or [*p*]{} photometric); (5) Log() with $1\sigma$ errors; (6) Log() with $1\sigma$ errors; (7) SFR derived from ; (8) Log() with $1\sigma$ errors or upper-limits; (9) Log() with $1\sigma$ errors; (10) Log() computed from using Marconi et al. (2004); (11) and (12) XMM-COSMOS and Chandra-COSMOS IDs (from Brusa et al. 2010 and Marchesi et al. 2016 respectively). The full table will be available in electronic form at the CDS via <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/> .
The host properties of the sample of detected sources (from D15, $\sim17000$ sources) are shown for comparison with gray dots. The average of the statistical $1\sigma$ error-bars resulting from the SED fit[^5] are shown in the top left corner. The errors on follow a log-normal distribution, with average $\langle err($$) \rangle=0.14$ dex and standard deviation $\sigma=0.09$. The mean error on SFR is $\langle err($SFR$) \rangle=0.10$ dex, and standard deviation of $\sigma=0.07$ as for (see sec. 3.1), since the SFR is derived from adopting a Kennicutt (1998) law. The redshift-dependent MS of star forming galaxies as described in Whitaker et al. (2012) is also shown in each panel. The FIR selected sources broadly follow the MS relation. However, the -based selection is sensitive to the most star forming systems, introducing a cut in SFR that moves towards higher values with increasing redshift. (e.g. Rodighiero et al. 2011, D15).
X-ray detected AGN are preferentially found at the highest , i.e. the fraction of X-ray detected sources increases as a function of , in the first three redshift bins at least. This is a well known effect (Kauffmann et al. 2003, Bundy et al. 2008, Brusa et al. 2009, Silverman et al. 2009, Mainieri et al. 2011, Santini et al. 2012, Delvecchio et al. 2014). Aird et al. (2012) suggested that it is the result of an observational bias, such that more massive galaxies (i.e. more massive BHs), can be detected, at a given X-ray flux limit, with a variety of accretion rates, while lower mass systems can be detected only if they have a high accretion rate. This, combined with a steep Eddington ratio distribution (i.e. sources with low Eddington ratio are much more common than sources with high Eddington ratio) can explain the observed distribution (see also Bongiorno et al. 2012).
In our case there is a threshold at around $\log$$\sim$10.5 in the first 3 redshift bins. A simple calculation shows that this value can be roughly derived from the X-ray flux limit of the and surveys, using standard values for bolometric corrections ($k_{Bol}=10-30$), Eddington ratios ($\sim0.05$) and BH-host mass ratios ($/M_{BH}=1000-3000$). A more detailed study of the Eddington ratio distribution that can be derived from the and distributions, will be presented in Suh et al. (2017 submitted).
Several studies in the local Universe suggest that the fraction of galaxies hosting an AGN increases also with IR luminosity (e.g. Lutz et al. 1998, Imanishi et al. 2010, Alsonso-Herrero et al. 2012, Pozzi et al. 2012). We also tested that the observed threshold in mass is not driven by our requirement of detection: also using the -SFR distribution of Bongiorno et al. 2012, computed for the full XMM-COSMOS catalog, a drop in the number of X-ray detected AGN below $\log$=10.2-10.4 is visible up to $z=2.5$.
The consequence of this selection effect is that the X-FIR sample has a distribution shifted toward higher with respect to the global sample (Fig. \[fig:isto\] top right). The distribution of SFR for the X-FIR sample, instead, is roughly consistent with that of the global sample (Fig. \[fig:isto\] top left). This have important implications when measuring, e.g., sSFR and MS offset (Fig. \[fig:isto\] bottom left and right): due to this selection effect the X-FIR sample has lower sSFR with respect to the MS of star forming galaxies (or to the sample), if the two samples are not properly mass-matched (Silverman et al. 2009, Xue et al. 2010).
![ vs. for the X-FIR sample. Different colors represent different redshift bins: blue for $0.1<z<0.4$, cyan for $0.4<z<0.8$, green for $0.8<z<1.2$, red for $1.2<z<2$ and magenta for $2<z<4$. The average $1\sigma$ errors on and are shown in the upper left corner. []{data-label="fig:lumlum1"}](graf_lir_lx_single.ps){width="8cm"}
vs. distributions
===================
Partial correlation analysis
----------------------------
The two quantities that have been more often used in order to look for BHAR-SFR correlations are the AGN luminosity, often represented by the , and the SF luminosity in the form of (or L$_{60 \mu m}$, Santini et al. 2012, Rosario et al. 2012, Chen et al. 2013). It is generally assumed that the total FIR luminosity is not significantly affected by any contamination from the AGN emission. However, recent studies have shown that the AGN may contribute significantly to the IR emission and in some case even in the FIR band (Symeonidis et al. 2016). Therefore, the SFR derived directly from FIR photometry can be overestimated, especially in high luminosity AGN hosts. Thanks to the SED decomposition available, we will use in the following the computed for the SF component only ( hereafter), after subtracting the AGN contribution, modeled with the SED templates of Fritz et al. (2006, see also Feltre et al. 2012). This will allow us to avoid introducing a spurious correlation between AGN and SF luminosity, especially at the highest luminosities.
Clearly two luminosities are always correlated in any sample that is flux limited in both directions, due to the combination of the luminosity-distance effect and of the fact that the sources tend to cluster at the flux limit (Malmquist bias, e.g. Feigelson & Berg 1983). Figure \[fig:lumlum1\] shows the distribution of vs. for the X-FIR sample.
The $1\sigma$ errors on follow a log-normal distribution with average value $\langle err($$)\rangle=0.10$ dex, and standard deviation of $\sigma=0.07$. As mentioned in sec. 2.1, the errors on the absorption corrected luminosity follow a much broader distribution, depending both on the number of counts available and on the spectral shape. They range from $\simlt0.1-0.2$ dex for bright, unobscured sources, to $\sim0.5-1.0$ dex for faint and highly obscured sources. The average value of the $1\sigma$ error is $\langle err($$)\rangle \sim0.23$ dex, with standard deviation $\sigma=0.18$. We show the average errors with a black cross in the left panel of Fig. 4, while the specific value for each source is used in the following analysis.
{width="8cm"}{width="8cm"}
In order to look for intrinsic correlations between these two quantities, one possibility is to compute the [*partial*]{} Spearman rank correlation between two variables in presence of a third, and to assess the statistical significance of such correlation (e.g. Macklin 1982). To derive the correlation coefficient between and , conditioned by the distance, $\rho$(, , $\dot{z}$), we evaluate the Spearman coefficient $\rho$ related to each couple of parameters and then combined them according to the expression: $$\rho(a,b,\dot{c})=\frac{\rho_{ab}-\rho_{ca}\rho_{bc}}{\sqrt{(1-\rho_{ca}^2)(1-\rho_{bc}^2)}}$$ (Conover 1980) which returns the partial correlation between $a$ and $b$, corrected for the dependency on $c$. The resulting $\rho$ is 0.15, and the associated confidence level, in terms of standard deviations, that the first two variables are correlated, independently of the influence of the third, is $\sim3.7\sigma$, following eq. 6 of Macklin (1982). Therefore, the two quantities appear to be significantly correlated, after the effect of redshift on both of them is taken into account.
Redshift bins
-------------
The second approach, often used in the literature, is to define as narrow as possible redshift bins, to minimize the distance effect, and look for correlations between the two quantities. Thanks to the large sample collected in this work, we can divide the sample in five redshift bins. For every redshift bin, a large distribution in both luminosities can be observed, with the typical luminosity increasing with redshift (Fig. \[fig:lumlum1\]).
Most of the observational works mentioned in Sec. \[sec:intro\] looked for the distribution of [*average*]{} in bins, or [*average*]{} in bins (but see Gruppioni et al. 2016). Both hydrodynamical simulations (e.g. Volonteri et al. 2015a,b) and semi-analytic model (e.g. Neistein & Netzer 2014) show that, in the - plane, there may be the superimposition of a weak correlation for the bulk of the population, and a strong correlation only for the most extreme merger phases, corresponding to the highest and . If the underlying distribution shows such a complex shape, the results of the two approaches (average in bins or average in bins) may be very different.
In Fig. \[fig:lumlum2\] left, we show the result of plotting [*average*]{} in bin of (both in log scale), in five redshift bins. As can be seen, there is no correlation at all and, as expected, there are no sources below the relation computed for a pure AGN template in Mullaney et al. (2011), similar to the one of Netzer et al. (2009). Following this approach, we are therefore able to reproduce the results of Shao et al. (2010), Rosario et al. (2012) and others, that claim no correlation between AGN activity and SF over several orders of magnitudes in luminosity.
On the other hand, computing [*average*]{} in bins (in log scale), from the same bivariate distribution, gives different results. Fig. \[fig:lumlum2\], right, shows that, at all redshifts, the average correlates with the and the binned points are close to the SFR/BHAR$\sim500$ ratio found in Chen et al. (2013, C13 hereafter).
In both panels, we computed the error on the average and through a bootstrap re-sampling procedure, as done in several previous works. For each bin with N sources, we randomly extract N sources, allowing repetitions, and computed the mean value. The process is iterated $10^4$ times, and the standard deviation of the mean is taken as error on the average SFR.
[lccccccc]{}\
z-bin & & & &\
& & $\alpha$ & $\beta$ & $\alpha$ & $\beta$ & $\alpha$ & $\beta$\
$0.1\leq z<0.4$ & & $0.07\pm0.06$ & $-0.61\pm0.09$ & $ 0.44\pm0.13 $ & $-1.24\pm0.21$ & $ 1.28\pm0.45$ & $-0.65\pm0.20$\
$0.4\leq z<0.8$ & & $0.20\pm0.10$ & $0.06\pm0.04$ & $ 0.80\pm0.17 $ & $-0.68\pm0.05$ & $ 1.21\pm0.34$ & $-0.67\pm0.11$\
$0.8\leq z<1.2$ & & $0.12\pm0.09$ & $0.25\pm0.03$ & $ 0.61\pm0.15 $ & $-0.56\pm0.04$ & $ 1.30\pm0.25$ & $-0.75\pm0.09$\
$1.2\leq z<2.0$ & & $0.16\pm0.09$ & $0.62\pm0.04$ & $ 0.48\pm0.12 $ & $-0.34\pm0.08$ & $ 1.29\pm0.15$ & $-0.86\pm0.15$\
$2.0\leq z<4.0$ & & $0.01\pm0.08$ & $1.02\pm0.04$ & $ 0.29\pm0.18 $ & $0.10\pm0.20$ & $ 1.16\pm0.57$ & $-0.85\pm0.17$\
\
The two approaches described above are the equivalent of computing the forward and inverse linear regression of one variable over the other. Table 2 reports the slopes $\alpha$, intercept $\beta$ and associated error, for each redshift bin, in the log-log space, of the least square (LS) fit[^6] of as a function of (hereafter | ), and as a function of (hereafter | ), respectively[^7]. Indeed the slopes in the left panel are all consistent with 0 within $\sim2\sigma$ c.l.. On the other hand, LS fits of ( | ) give steeper correlations at all z bins, and slopes not consistent with 0 at $\sim3\sigma$ c.l.
The SFR/BHAR$\sim500$ ratio plotted in Fig. 5 is the one found in C13, for a sample of 121 FIR selected AGN-hosts at $0.25<z<0.8$. To compare with their results we should look at our first two z-bins: While the z-bin $0.1\leq z<0.4$ has a very flat ( | ) slope, possibly due to the small volume sampled, the $0.4\leq z<0.8$ interval shows a correlation with slope consistent with 1 at $\sim1\sigma$, therefore in broad agreement with the C13 findings. Interestingly, we can extend up to $0.8\leq z<1.2$ the redshift range for which a correlation roughly consistent with SFR/BHAR$\sim500$ can be found. Above this redshift interval, the slopes become flatter. Therefore, we found a strong (almost linear) correlation between $\log$ and $\log$, for ( | ) at redshifts lower than the peak of the SF and AGN activity, i.e. between 4 and 8 Gyr ago, while at higher redshift the correlation is still present but weaker.
The exact value of the ratio SFR/BHAR in terms of and depends strongly on the assumptions made to scale between these quantities, i.e. the accretion efficiency and bolometric correction in the first case, and the SF law and initial mass function (IMF) in the second. C13 derived the SFR from using the Kennicutt (1998) relation, modified for a Chabrier IMF (Chabrier 2003) and the BHAR from using a constant $k_{bol}=22.4$ and accretion efficiency of 0.1. They use as reference the value of SFR/BHAR$\sim500$ derived from the M$_{\rm Bulge}$/M$_{\rm BH}$ ratio observed in Marconi et al. 2004. The authors suggest that the fact that the detected sources sit on the SFR/BHAR$\sim500$ ratio is a coincidence, due to the ratio between the X-ray and FIR flux limits in the Boötes field.
In the X-FIR sample in COSMOS we have a factor of $\sim10$ deeper X-ray data (taking into account the flux limit corresponding to our spectral analysis requirements), while the data are only a factor 2-3 deeper ($\sim8$ mJy at $110\mu m$ and 8 mJy at $250\mu m$) than in Boötes. Nonetheless, our X-FIR detected sample sit close to the C13 relation. We underline that, in both cases, the X-ray and FIR detected sources are a small minority of both the original X-ray and FIR samples (few % up to $\sim20$%), and as discussed also in C13, the flux limit has an important role in the observed properties of the detected sources alone.
![Linear regression for and computed for each redshift bin with the bisector estimator in BCES. The color code is the same as Fig. 4. []{data-label="fig:lumlum3"}](graf_bin_tot_new.ps){width="8cm"}
As discussed in Hickox et al. (2014) and Volonteri et al. (2015), a possible physical explanation for this behavior is that, when looking at left panel of fig. 5, we are averaging a slowly changing quantity, such as the host SFR, of galaxies grouped on the basis of the rapidly changing AGN . In the right panel, instead, the average of a large sample of sources grouped on the basis of the slowly changing SFR, allows us to recover the underlying, long term correlation between AGN activity and SFR. In the same way, from a statistical point of view, it may be reasonable to interpret the as the dependent variable, in this context, as it has larger uncertainties with respect to (Hogg et al. 2010), both in terms of measurement errors (see sec. 3.1) and [*noise*]{} (i.e. variability).
If we instead assume that in this case there is no “dependent” and “independent” variables (see e.g. Tremaine et al. 2002, Novak et al. 2006), the two variables may need to be treated symmetrically. We used again the BCES code, to derive slope and intercept, and their standard deviation, using a symmetric estimator such as the bisector regression[^8] (Isobe et al. 1990). The results are shown in figure \[fig:lumlum3\], while the slopes and intercepts are reported in Table 2. At all redshift bins, the slopes of the linear regression, although always larger than 1, are consistent with 1 within $1\sigma$ c.l.
Effect of Contamination
-----------------------
Since the (Pilbratt et al. 2010) PACS and SPIRE point spread functions are much larger than the one in the optical and NIR bands, going from $\sim5\arcsec$ to $\sim36\arcsec$ FWHM (Poglitsch et al. 2010; Griffin et al. 2010), there is the possibility that the FIR flux of our sources is contaminated by unresolved neighbors (see e.g. Scudder et al. 2016).
We verified the effect of contamination by excluding from the X-FIR sample all sources with a second HST catalog entry, from the ACS F814W (I-band) catalog (28.6 AB limiting magnitude, Scoville et al. 2007, Koekemoer et al. 2007). We choose a circular area of diameter $8\arcsec$ around the optical position. While this distance is not enough to ensure negligible contamination, it has been chosen in order to retain a sufficient number of sources to allow an analysis in all the five redshift bins. The 146 “isolated” sources obtained in this way show the same behavior described above, with a flat distribution of [*average*]{} computed in bin of , and an almost linear correlation of [*average*]{} computed in bin of .
We also verified that sources with a single PACS or SPIRE detections (more subject to contamination) do not affect our results. Indeed, excluding the 154 (out of 692) sources with only one detection (at $3\sigma$) either in PACS or SPIRE photometry, does not change the results presented in sec. 3.2 and in the following paragraphs.
, , sSFR and MS offset
----------------------
![Average Log() in bin of , for the X-FIR sample. The dashed line is the relation found in Netzer et al. (2009) for AGN dominated systems. Error-bars computed as in Fig. 5. []{data-label="fig:lirlbol"}](graf_lir_lbol_new.ps){width="8cm"}
{width="8cm"}{width="8cm"}
Several authors have used the AGN bolometric luminosity (), instead of the , to look for correlation with the or SFR. The is generally derived from the through a luminosity dependent bolometric correction (e.g. Marconi et al. 2004, Lusso et al. 2012). The net effect of this procedure is to stretch the horizontal axis of Fig. 5, left (the high sources have a higher X-ray bolometric correction than the low ones), while keeping the fixed. In Fig. \[fig:lirlbol\] we show the result of this approach (here we used the Marconi et al. 2004 luminosity dependent bolometric correction, but the Lusso et al. 2012 relation would have the same effect): in each redshift bin, the sources populating the highest bin are now spread in two bins and the last bin at each redshift is now populated by a smaller number of more extreme sources. The relation found locally for AGN-dominated systems in Netzer et al. (2009) is also shown. Once again, we are able to reproduce results obtained in other works (Shao et al. 2010, Rosario et al. 2012). However, we are now confident that this result is not in disagreement with what shown in Fig. 5 (right), and the apparent contradiction is only dependent on the way the data are analyzed and grouped, as shown in e.g. Volonteri et al. (2015) and Dai et al. (2015).
Finally, we found a flat distribution when computing [*average*]{} in bins of , sSFR and MS offset, and [*average*]{} , sSFR and MS offset, in bins of , in all the five redshift bins. Indeed, no significant partial correlation is found, between any pair of these quantities, following the approach described in sec. 3.1 to take into account the redshift effect, that affects also , SFR and sSFR ($\sigma<<1$ in all cases). We stress however that the range of covered by our sample is limited to the very high mass end, $10<Log($$)<12$ (to be compared with the underlining galaxies distribution shown e.g. in Laigle et al. 2016, $7<$Log()$<12$ in the same redshift interval). Deeper X-ray surveys are needed to investigate the dependency of with this crucial quantity.
Comparison with simulations
===========================
Here we compare our results with predictions from the simulations of galaxy mergers presented in Volonteri et al. (2015a). They are based on very high spatial and temporal resolution simulations, covering a large range of initial mass ratios (1:1 to 1:10), several orbital configurations, and gas fraction (defined as $M_{gas}/$) in the range $f_{gas}=0.3-0.6$. The very high resolution imposes a limit on the mass of the simulated galaxies, that typically have $\sim(2-8)\times10^{9}$ , i.e. much smaller than the typical mass of our observed galaxies (see Fig. 3). The process is divided into three phases: the [*stochastic*]{} phase, in which the galaxies behave as they do in isolation, that lasts until the second pericenter; the [*merger*]{} phase characterized by strong dynamical torques and angular momentum loss; the [*remnant*]{} phase, that starts when the angular momentum returns to be constant in time. While the stochastic and remnant phases have the same duration (by construction), the merger phase is much shorter (typically 1/10 of the total).
To compare our data with this set of simulated galaxies, we converted the AGN bolometric luminosity into a BH mass accretion rate (BHAR), by assuming an efficiency of $\eta=0.1$ (e.g. Fabian & Iwasawa 1999) and dividing it by the host stellar mass, to obtain a specific BHAR (sBHAR) relative to the host mass, rather than to the BH mass. We choose to do so because, from an observational point of view, the determination of the (from SED fitting) is much less uncertain (see sec. 2.2 for the error budget in our sample) than that of , and is available for both type-1 and type-2 AGN. This value is then compared with the sSFR for each source. The contours of global (within 5 kpc) sSFR vs. sBHAR, obtained from the simulations for the three different phases (stochastic, merger and remnant), are color coded in Fig. \[fig:simul\] (left) with red, yellow and black, respectively.
The results from the X-FIR sample are shown in Fig. \[fig:simul\] (right) for the five redshift bins. As can be seen the observed contours in the low redshift bins span a similar range of physical properties, with respect to simulations, with the bulk of the population concentrated between $5\times10^{-11}$ and $5\times10^{-9}$ yr$^{-1}$ in sSFR, and between $10^{-14}$ and $10^{-11}$ yr$^{-1}$ in sBHAR, and with a tail at higher sSFR and sBHAR, possibly produced by sources in the merger phase as in the simulations (yellow contours). Interestingly, the importance of this tail grows with increasing redshift, even if the selection effect in both directions must be taken into account.
We also exploited the deep HST ACS coverage in the COSMOS field to identify sources in the merger phase. We selected only sources that appear to be in a clear major merger phase, and over-plotted them in Fig. \[fig:simul\] (right) as black stars, in the first three redshift bins (above $z\sim1$ it becomes difficult to assess the AGN host morphology). This selection is not meant to be complete: not all the sources are covered by ACS, and not for all of them it is possible to recognize the host morphology, due to bright point-like AGN contribution, for example. However, it is interesting that AGN hosts clearly in merger state tend to cover the highest sSFR and sBHAR range, as predicted by simulations.
{width="8cm"}{width="8cm"}
Caveat
------
One caveat to be considered here is the fact that the simulations are performed at high-z, starting at $z=3$ and ending after $1-3$ Gyr depending on the merger dynamics (see Capelo et al. 2015 for details). By construction, the simulations have a relatively low gas fraction: 30% of the disc stellar mass. This is probably a low value for SF galaxies at these redshifts. Only one set of simulations has been performed with a higher gas fraction, i.e. 60% , and, as expected, these simulated galaxies move toward higher sSFR and sBHAR as the contours of the observed high redshift sample do.
Another caveat is the fact that the simulations are done for low mass galaxies. The typical for these galaxies is in the range Log()$= 9-9.5$ (), i.e. in the low mass tail of the mass distribution even for the lowest redshift bin of the observed sample. Since the efficiency of SFR and BHAR is most probably mass-dependent, the comparison between different mass ranges may not be straightforward. Volonteri et al. (2015a) argue, however, that SFR and BHAR are self-similar, on the basis of the mass sequence of star forming galaxies and of the possible power-law dependence of the specific BHAR (Aird et al. 2012; Bongiorno et al. 2013, but see Kauffmann & Heckman 2009, Lusso et al. 2012 and Schulze et al. 2015).
Finally, the simulations are not cosmological, in the sense that the gas mass is not replenished by cosmic inflows and gas accretion, as it is the case for real galaxies. This leads to a possible underestimate of SFR and BHAR towards the end of the simulation, when galaxies have converted a large fraction of their gas in stellar and BH mass (see also Vito et al. 2014).
and host properties
====================
![$\log$ vs. $\log M_{gas}$ as derived from the eq. 1 of Scoville et al. (2016). The sources are color coded on the basis of their gas fraction.[]{data-label="fig:gasfr"}](graf_mstar_gasmass.eps){width="8.7cm" height="7.8cm"}
Here we discuss the possible correlations between the column density through the AGN line of sight, as measured by the X-ray , and the host galaxy properties, such as , SFR and sSFR and MS offset. The partial correlation analysis described in sec. 3.1, gives a significant positive correlation (at $>4\sigma$ c.l.) between and , in the entire sample, once the distance effect (both and tend to increase with redshift in two different ways, due to two different selection effects) is removed. We also find a significant negative correlation (at $>5\sigma$ c.l.) between and sSFR, while we do not find any significant correlation of with SFR and MS offset.
As in the case of vs. , the binning direction (or the variable chosen as independent) is relevant for the final distribution of as a function of host properties and vice-versa: computing average SFR, , sSFR and MS offset in bin of we found a remarkably flat distribution of all these quantities, in agreement with results from Shao et al. (2010), Rovilos et al. (2012), Rosario et al. (2012), where the authors do not find any evolution of the average host properties in bins of .
On the other hand, computing average values in bins of gives a positive trend in each redshift bin, while computing the average in sSFR bins gives a negative trend, in agreement with partial correlation analysis. The situation in this case is however complicated by the presence of upper-limits in , that makes the problem inherently asymmetric. We therefore performed the linear regression of (Y|X) with a Bayesian approach using the [linmix]{} code (Kelly et al. 2007) that is able to properly take into account the upper-limits on .
The result is shown in Fig. \[fig:nh\_y\]: the linear regression gives a clear positive correlation of with the host stellar mass, increasing by one-two dex from low to high masses, at all redshifts (slopes in the range $\alpha=0.42$ – $0.88$). An opposite result is found for the sSFR: the average decreases typically by one order of magnitude or more, going from low to high sSFR (slopes in the range $\alpha=-0.35$ – $-0.82$). Given that there is no trend of with SFR, and that the sSFR is defined as SFR/, the two relations are clearly connected.
{width="8cm"}{width="8cm"}
A similar result between and was found in Rodighiero et al. (2015) for a sample of $z\sim2$ AGN hosts. In their analysis however, the average is globally $\sim1$ dex higher (see gray squares in Fig. 9, left), due to the fact that they derive from hardness ratios of the X-ray stacking, which includes also highly obscured and CT, undetected AGN.
Interestingly, a recent study on the distribution of the obscuration observed in X-ray spectra of GRB, as a function of the host galaxy mass, found a similar trend, in the redshift range $1\simlt z\simlt 5$ (Buchner et al. 2017, orange line in fig. 9 left). Since for these sources the from the GRB spectra is probing only the host obscuration, the authors conclude that a large fraction of the obscuration observed in AGN, at least in the Compton thin regime, is not due to the nuclear torus, but to the galaxy-scale gas in the host.
These dependencies imply that at increasing galaxy mass there are more chances to have an additional component to the amount of gas and dust along the line of sight through the AGN. It is well established that the gas fraction is a strong decreasing function of the galaxy mass (e.g. Santini et al. 2014; Peng, Maiolino & Cochrane 2015). However, it is possible to show that the [*total*]{} amount of gas is driven mainly by the total galaxy mass, and not by the gas fraction. To this end, we computed gas mass for all our galaxies, following the empirical relation found in Scoville et al. 2016 (their eq. 1), that links , sSFR offset from the MS, and molecular gas mass. This is shown in Fig. \[fig:gasfr\], where the sources are color-coded on the basis of their gas fraction. Even if at increasing the gas fraction is smaller, the total amount of gas still increases with .
The well-known mass-metallicity relation (e.g. Tremonti et al. 2004, Mannucci et al. 2010) goes in the direction of having more metals (responsible for X-ray absorption) with increasing . In particular, going from Log()=9.5 to 11.5, there is an increase of a factor $\sim2$ in the metallicity, up to $z\sim2$ (Erb et al. 2006). This fact is however not enough to explain the increase in average observed here: Measuring the with fixed metallicity (as done here) for sources with such a range in metallicity, translates into a factor $\sim2$ difference in measured , for a given input obscuration.
Obscured Fraction
-----------------
To compare our results with the literature, we also looked at the fraction of obscured sources as a function of host properties. In Fig. \[fig:obsfrac\] we show the fraction of obscured sources, defined as N$_{Obs}$/N$_{Tot}$ where N$_{Obs}$ is the number of sources with a detection of and $>1\times10^{22}$ cm$^{-2}$. As expected from what shown in the previous section, the fraction of obscured sources increases with increasing , and decreases with sSFR (for sSFR$>1$ Gyr$^{-1}$). The decrease in sSFR is partly washed out by the fact that we are considering the full redshift interval ($z=0.1-4$) while Fig. \[fig:nh\_y\] (right) shows that the range covered by the different subsamples shifts toward higher sSFR with redshift. For this reason we also show in Fig. \[fig:obsfrac\] the results for the first and forth bins (blue and red dashed points, respectively) as an example.
Merloni et al. (2014) found a flat relation between the fraction of obscured sources and in a sample of X-ray detected AGN from the XMM-COSMOS catalog. However, they limited their analysis to a narrow range in (in order to cover a wide range redshift), while the obscured fraction is known to evolve strongly with (e.g. Ueda et al. 2015).
Another group, instead, have found an increasing fraction of obscured sources as a function of sSFR and MS offset, in a sample of $70\mu m$ selected galaxies at $0.3<z<1$, interpreted as an indication of increasing gas fraction or density in the host, that in turn would sustain the increased sSFR.(e.g. Juneau et al. 2013, J13 hereafter).
We note that the definition of obscured AGN adopted here and in J13 are different, and in the latter, mostly based on the lack of X-ray detection: there are 64 sources (out of 99 AGN) classified as obscured AGN on the basis of the Mass-Excitation diagram selection (MEX, Juneau et al. 2011), and the X-ray non detection. If these objects are indeed highly obscured, Compton-thick AGN, this population is mostly missed in our X-ray based sample.
Another possibility is that a fraction of the MEX-selected AGN are not actively/strongly accreting SMBHs. Indeed, a sizable fraction ($\sim30\%$) of the AGN selected in J13 through the MEX diagram, has a host below $Log($$)=10.5$. As shown in Sec. 2.2, however, X-ray detected AGN are rare at low . Therefore all the sources that are X-ray undetected for reasons different from obscuration (variability, intrinsic weakness, contaminant non-AGN etc.) would appear as obscured, low host (hence high sSFR) AGN, possibly affecting the observed trends.
Discussion
==========
We collected a large sample of X-ray and FIR detected AGN and host systems in the COSMOS field, spanning $\sim4$ orders of magnitudes in , , , , and covering the redshift range $0.1<z<4$. We applied X-ray spectral analysis down to very low counts, ($>30$ net counts) and adopted the SED decomposition results derived in D15, to recover both AGN and SF properties of each source. With this data-set in hand, we demonstrated that it is possible to reproduce both the flat distribution of [*average*]{} in bins of and the steeper correlation of [*average*]{} in bins of reported in the literature in the latest years (e.g. Shao et al. 2010, Rosario et al. 2012, Mullaney et al. 2012, C13, Stanley et al. 2015).
The apparently contradictory results found in the literature, and reproduced in Sec. 3.2, are due to the different results that are obtained when binning along one axis or the other, the equivalent of a forward or inverse linear regression (i.e. | vs. | ), as proposed in Hickox et al. (2014) and Volonteri et al. (2015), and found in Dai et al. (2015) on shallow XMM-LSS data.
Both from a physical and a statistical point of view, it seems more appropriate to consider the results from | , given the larger measurement uncertainties on , and the shorter time scale variability of , with respect to , that adds a further term of intrinsic scatter. Doing so, we found a linear correlation between and with slope consistent with 1, at least in the redshift range 0.4-1.2, i.e. below the peak of the SF and BH accretion history. Beyond that and up to $z=4$, the slope becomes significantly flatter, $\alpha=0.3-0.5$.
The other possibility is to adopt a symmetrical approach, even if there is no general agreement on this (see Hogg et al. (2010) on the bisector method). In this case the result is a correlation with slope consistent with $\sim1$, at all redshifts. This would point toward an [*average*]{} one-to-one correlation between SF and BH accretion, in the last 12 Gyr of cosmic history.
Even more interesting is the full distribution of BH and host properties, such as and or sBHAR and sSFR, that can be only qualitatively compared, for the moment, with predictions from galaxy merger simulations, resulting in interesting similarities between observations and models.
We stress again that these results apply to the small subsample of AGN/host systems detected in both X-ray and FIR, that represents only $\sim20$% of the full X-ray sample and $\sim10$% of the AGN FIR sample. Indeed, one of the main reasons why it is so difficult for present observations to probe the AGN-SF connection, is the fact that (X-ray and/or FIR) detected systems span a limited range in AGN and SF activity, sampling only the high /SFR tail of the possible correlation, (e.g. Sijacki et al. 2015).
It is, however, interesting that we are able to reproduce the results obtained via stacking of samples where the vast majority of the sources are not detected (e.g. 20% of FIR detected AGN selected in X-ray in Shao et al. 2010). As suggested in Mullaney et al. (2015), the stacking analysis, being the equivalent of a linear mean, may be dominated by the brightest sources.
A crucial next step in the comparison between theory and observations will be to select the observed systems in different evolutionary stage, to reach a similar level of detail as in the current simulations. This will be feasible for large samples only at low redshift, while detailed and complete morphological studies in COSMOS (and other deep fields) data are very difficult already at $z\simgt1$. From the theoretical point of view, more demanding galaxy merger simulations will be required, in order to cover, with the same high resolution, a mass range comparable to the one of observed systems, and to possibly move toward a high redshift environment.
Finally, a positive correlation between and , and a similar negative correlation with sSFR, have been found at all redshift bins. A similar result was found by Rodighiero et al. (2015) in a large sample of high redshift galaxies, computing HR of stacked X-ray images. A recent study on GRB hosts has found a similar behavior (Buchner et al. 2017), implying that an important fraction (up to 40%) of the Compton thin obscuration found in AGN can be ascribed to galaxy scale gas (Buchner & Bauer 2017).
Several studies have found no correlation between column density and host properties (Rovilos et al. 2012, Rosario et al. 2012), while others (e.g. J13) have found a positive correlation of the fraction of obscured sources with sSFR. Further investigation in this direction will help to shed light on the role of the host in contributing to the obscuration through the AGN line of sight.
The authors thank the anonymous referee for valuable comments. GL, MB, and MP acknowledge financial support from the CIG grant “eEASY” n. 321913. GL acknowledges financial support from ASI-INAF 2014-045-R.0. ID acknowledges the European Union’s Seventh Framework programme under grant agreement 337595 (ERC Starting Grant, “CoSMas”). We acknowledge the contributions of the entire COSMOS collaboration consisting of more than 100 scientists. More information on the COSMOS survey is available at <http://cosmos.astro.caltech.edu/>. Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA’, and data obtained from the Data Archive.
Aird, J., Coil, A. L., Moustakas, J., et al. 2012, , 746, 90 Akritas, M. G., & Bershady, M. A. 1996, , 470, 706 Alonso-Herrero, A., Pereira-Santaella, M., Rieke, G. H., & Rigopoulou, D. 2012, , 744, 2 Arnaud, K. A. 1996, Astronomical Data Analysis Software and Systems V, 101, 17 Berta, S., Lutz, D., Santini, P., et al. 2013, , 551, AA100 Bennett, C. L., Larson, D., Weiland, J. L., et al. 2013, , 208, 20 Bongiorno, A., Merloni, A., Brusa, M., et al. 2012, , 427, 3103 Brusa, M., Fiore, F., Santini, P., et al. 2009, , 507, 1277 Brusa, M., Civano, F. Comastri, A., et al. 2010, , 716, 348 Bruzual, G., & Charlot, S. 2003, , 344, 1000 Buchner, J., Schulze, S., & Bauer, F. E. 2017, , 464, 4545 Buchner, J., & Bauer, F. E. 2017, , 465, 4348 Bundy, K., Georgakakis, A., Nandra, K., et al. 2008, , 681, 931-943 Cai, Z.-Y., Lapi, A., Xia, J.-Q., et al. 2013, , 768, 21 Capak, P., Aussel, H., Ajiki, M., et al. 2007, , 172, 99 Capelo, P. R., Volonteri, M., Dotti, M., et al. 2015, , 447, 2123 Chabrier, G. 2003, , 586, L133 Chen, C.-T. J., Hickox, R. C., Alberts, S., et al. 2013, C13, , 773, 3 Chen, C.-T. J., Hickox, R. C., Alberts, S., et al. 2015, , 802, 50 Civano, F., Elvis, M., Brusa, M., et al. 2012, , 201, 30 Civano, F., Marchesi, S., Elvis, M., et al. 2015, submitted to Comastri, A., Ranalli, P., Iwasawa, K., et al. 2011, , 526, L9 Cresci, G., Hicks, E. K. S., Genzel, R., et al. 2009, , 697, 115 Dai, Y. S., Wilkes, B. J., Bergeron, J., et al. 2015, arXiv:1511.06761 Delvecchio, I., Gruppioni, C., Pozzi, F., et al. 2014, , 439, 2736 Delvecchio, I., Lutz, D., Berta, S., et al. 2015, arXiv:1501.07602 Diamond-Stanic, A. M., & Rieke, G. H. 2012, , 746, 168 Di Matteo, T., Springel, V., & Hernquist, L. 2005, , 433, 604 Dubois, Y., Peirani, S., Pichon, C., et al. 2016, arXiv:1606.03086 Elvis, M., Civano, F., Vignali, C., et al. 2009, , 184, 158 Erb, D. K., Shapley, A. E., Pettini, M., et al. 2006, , 644, 813 Fabian, A. C., & Iwasawa, K. 1999, , 303, L34 Feigelson, E. D., & Berg, C. J. 1983, , 269, 400 Feltre, A., Hatziminaoglou, E., Fritz, J., & Franceschini, A. 2012, , 426, 120 Fritz, J., Franceschini, A., & Hatziminaoglou, E. 2006, , 366, 767 Fruscione, A., et al. 2006, , 6270 Genzel, R., Newman, S., Jones, T., et al. 2011, , 733, 101 Granato, G. L., De Zotti, G., Silva, L., Bressan, A., & Danese, L. 2004, , 600, 580 Griffin, M. J., Abergel, A., Abreu, A., et al. 2010, , 518, L3 Gruppioni, C., Berta, S., Spinoglio, L., et al. 2016, , 458, 4297 Hasinger, G., Cappelluti, N., Brunner, H., et al. 2007, , 172, 29 Hickox, R. C., Mullaney, J. R., Alexander, D. M., et al. 2014, , 782, 9 Hogg, D. W., Bovy, J., & Lang, D. 2010, arXiv:1008.4686 Imanishi, M., Maiolino, R., & Nakagawa, T. 2010, , 709, 801 Isobe, T., Feigelson, E. D., Akritas, M. G., & Babu, G. J. 1990, , 364, 104 Juneau, S., Dickinson, M., Alexander, D. M., & Salim, S. 2011, , 736, 104 Juneau, S., Dickinson, M., Bournaud, F., et al. 2013, J13, , 764, 176 Kauffmann, G., Heckman, T. M., Tremonti, C., et al. 2003, , 346, 1055 Kauffmann, G., & Heckman, T. M. 2009, , 397, 135 Kelly, B. C. 2007, , 665, 1489 Kennicutt, R. C., Jr. 1998, , 498, 541 Koekemoer, A. M., Aussel, H., Calzetti, D., et al. 2007, , 172, 196 Kormendy, J., & Richstone, D. 1995, , 33, 581 Kormendy, J., & Ho, L. C. 2013, , 51, 511 Lagos, C. D. P., Baugh, C. M., Lacey, C. G., et al. 2011, , 418, 1649 Lanzuisi, G., Civano, F., Elvis, M., et al. 2013, , 431, 978 Lanzuisi, G., Ranalli, P., Georgantopoulos, I., et al. 2015a, , 573, AA137 (L15) Lanzuisi, G., Perna, M., Delvecchio, I., et al. 2015b, , 578, A120 Lusso, E., Comastri, A., Simmons, B. D., et al. 2012, , 425, 623 (L12) Lutz, D., Spoon, H. W. W., Rigopoulou, D., Moorwood, A. F. M., & Genzel, R. 1998, , 505, L103 Lutz, D., Poglitsch, A., Altieri, B., et al. 2011, , 532, A90 Macklin, J. T. 1982, , 199, 1119 Madau, P., & Dickinson, M. 2014, , 52, 415 Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, , 115, 2285 Mainieri, V., Bongiorno, A., Merloni, A., et al. 2011, , 535, A80 Maiolino, R., Shemmer, O., Imanishi, M., et al. 2007, , 468, 979 Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., & Gnerucci, A. 2010, , 408, 2115 Marchesi, S., Civano, F., Elvis, M., et al. 2016, , 817, 34 Marchesi, S., Lanzuisi, G., Civano, F., et al. 2016, , 830, 100 Marconi, A., & Hunt, L. K. 2003, , 589, L21 Marconi, A., Risaliti, G., Gilli, R., et al. 2004, , 351, 169 Menci, N., Fiore, F., Puccetti, S., & Cavaliere, A. 2008, , 686, 219 Merloni, A., Bongiorno, A., Brusa, M., et al. 2014, , 437, 3550 Mullaney, J. R., Alexander, D. M., Goulding, A. D., & Hickox, R. C. 2011, , 414, 1082 Mullaney, J. R., Pannella, M., Daddi, E., et al. 2012, , 419, 95 Mullaney, J. R., Alexander, D. M., Aird, J., et al. 2015, , 453, L83 Neistein, E., & Netzer, H. 2014, , 437, 3373 Novak, G. S., Faber, S. M., & Dekel, A. 2006, , 637, 96 Oliver, S. J., Bock, J., Altieri, B., et al. 2012, , 424, 1614 Peng, Y., Maiolino, R., & Cochrane, R. 2015, , 521, 192 Pilbratt, G. L., Riedinger, J. R., Passvogel, T., et al. 2010, , 518, L1 Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, , 518, L2 Pontzen, A., Tremmel, M., Roth, N., et al. 2016, arXiv:1607.02507 Pozzi, F., Vignali, C., Gruppioni, C., et al. 2012, , 423, 1909 Rodighiero, G., Daddi, E., Baronchelli, I., et al. 2011, , 739, L40 Rodighiero, G., Brusa, M., Daddi, E., et al. 2015, , 800, L10 Rosario, D. J., Santini, P., Lutz, D., et al. 2012, , 545, A45 Rovilos, E., Comastri, A., Gilli, R., et al. 2012, , 546, A58 Saintonge, A., Catinella, B., Cortese, L., et al. 2016, arXiv:1607.05289 Salvato, M., Hasinger, G., Ilbert, O., et al. 2009, , 690, 1250 Santini, P., Rosario, D. J., Shao, L., et al. 2012, , 540, A109 Santini, P., Maiolino, R., Magnelli, B., et al. 2014, , 562, A30 Schulze, A., Bongiorno, A., Gavignaud, I., et al. 2015, , 447, 2085 Scoville, N., Aussel, H., Brusa, M., et al. 2007, , 172, 1 Scoville, N., Sheth, K., Aussel, H., et al. 2016, , 820, 83 Scudder, J. M., Oliver, S., Hurley, P. D., et al. 2016, , 460, 1119 Shao, L., Lutz, D., Nordon, R., et al. 2010, , 518, L26 Sijacki, D., Vogelsberger, M., Genel, S., et al. 2015, , 452, 575 Silverman, J. D., Lamareille, F., Maier, C., et al. 2009, , 696, 396 Suh, H. Civano, F., Hasinger, G., et al. 2017, submitted Symeonidis, M., Giblin, B. M., Page, M. J., et al. 2016, , 459, 257 Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, , 574, 740 Tremonti, C. A., Heckman, T. M., Kauffmann, G., et al. 2004, , 613, 898 Ueda, Y., Akiyama, M., Hasinger, G., Miyaji, T., & Watson, M. G. 2014, , 786, 104 Vito, F., Maiolino, R., Santini, P., et al. 2014, , 441, 1059 Volonteri, M., Capelo, P. R., Netzer, H., et al. 2015a, , 449, 1470 Volonteri, M., Capelo, P. R., Netzer, H., et al. 2015b, , 452, L6 Whitaker, K. E., van Dokkum, P. G., Brammer, G., & Franx, M. 2012, , 754, LL29 Xue, Y. Q., Brandt, W. N., Luo, B., et al. 2010, , 720, 368
[^1]:
[^2]: Visiting scientist
[^3]: For sources in common the data from Marchesi et al. (2016) are used, given the deeper coverage.
[^4]: Lanzuisi et al. (2015a,b) present the CT sources detected by , while Lanzuisi et al. 2017 in prep. will present the ones detected by .
[^5]: Systematic errors like, for example, uncertainties related to the adopted IMF or SF law, are not included in the error budget.
[^6]: The LS fit is performed with the BCES code (Akritas & Bershady 1996), adopting $10^4$ bootstrap re-samplings. Similar results are obtained using the LINMIX code (Kelly et al. 2007).
[^7]: In the first case slopes and intercepts refer to a relation in the form $Log~$$=45+\alpha\times(Log~$$ -44)+\beta$, while in the second in the form $Log~$$=44+\alpha\times(Log~$$ -45)+\beta$.
[^8]: We recall that the BCES estimators, both the LS and the symmetric ones, are not immune from biases that arise from data truncation, which is the case for flux-limited samples (see Akritas & Bershady 1996).
|
---
abstract: |
In a previous paper \[Phys. Rev. D [**68**]{}, 034017 (2003)\], we investigated the inclusive production of prompt $J/\psi$ mesons in polarized hadron-hadron, photon-hadron, and photon-photon collisions in the factorization formalism of nonrelativistic quantum chromodynamics providing compact analytic results for the double longitudinal-spin asymmetry $\mathcal{A}_{LL}$. For convenience, we adopted a simplified expression for the tensor product of the gluon polarization four-vector with its charge conjugate, at the expense of allowing for ghost and anti-ghosts to appear as external particles. While such ghost contributions cancel in the cross section asymmetry $\mathcal{A}_{LL}$ and thus were not listed in our previous paper, they do contribute to the absolute cross sections. For completeness and the reader’s convenience, they are provided in this addendum.
PACS: 12.38.Bx, 13.60.Le, 13.85.Ni, 14.40.Gx
author:
- |
M. Klasen$^{a,}$[^1], B. A. Kniehl$^{b,}$[^2], L. N. Mihailă$^{c,}$[^3], M. Steinhauser$^{c,}$[^4]\
\
[*$^a$ Laboratoire de Physique Subatomique et de Cosmologie,*]{}\
[*Université Joseph Fourier/CNRS-IN2P3/INPG,*]{}\
[*53 Avenue des Martyrs, 38026 Grenoble, France*]{}\
\
[*$^b$ II. Institut für Theoretische Physik, Universitä t Hamburg,*]{}\
[*Luruper Chaussee 149, 22761 Hamburg, Germany*]{}\
\
[*$^c$ Institut für Theoretische Teilchenphysik, Universität Karlsruhe,*]{}\
[*Engesserstraße 7, 76131 Karlsruhe, Germany*]{}
title: |
-3cm
DESY 08-021ISSN 0418-9833
TTP08-10
LPSC 08-020
February 2008
1.5cm **Ghost contributions to charmonium production in polarized high-energy collisions**
---
The factorization formalism of nonrelativistic QCD (NRQCD) [@Caswell:1985ui] provides a rigorous theoretical framework for the description of heavy-quarkonium production and decay. This formalism implies a separation of process-dependent short-distance coefficients, to be calculated perturbatively as expansions in the strong-coupling constant $\alpha_s$, from supposedly process-independent long-distance matrix elements (MEs), to be extracted from experiment, and takes into account the complete structure of the $Q\overline{Q}$ Fock space, which is spanned by the states $n={}^{2S+1}L_J^{(C)}$ with definite spin $S$, orbital angular momentum $L$, total angular momentum $J$, and color multiplicity $C=1,8$. By velocity scaling rules, the MEs are predicted to scale with a definite power of the heavy-quark ($Q$) velocity $v\ll1$, so that a small number of these non-perturbative parameters should allow for meaningful predictions in practice.
In Ref. [@Klasen:2003zn], we applied the NRQCD factorization formalism to the inclusive production of prompt $J/\psi$ mesons in polarized hadron-hadron, photon-hadron, and photon-photon collisions and provided compact analytic results for the double longitudinal-spin asymmetry $\mathcal{A}_{LL}$, defined in Eq. (2.1) of Ref. [@Klasen:2003zn]. Specifically, we considered inclusive $J/\psi$ production in polarized $pp$, $\gamma d$, and $\gamma\gamma$ collisions, appropriate for the RHIC-Spin experiments at the BNL Relativistic Heavy Ion Collider (RHIC), the SLAC fixed-target experiment E161, and the TeV-Energy Superconducting Linear Accelerator (TESLA) operated in the $e^+e^-$ and $\gamma\gamma$ modes, respectively. We took the $J/\psi$ mesons to be unpolarized.
There is a technical subtlety related to the definition of the polarization four-vector $\varepsilon(p,\xi)$ of an external gluon, with four-momentum $p$ and helicity quantum number $\xi=\pm1$, which is potentially prone to create confusion. As for the tensor product of $\varepsilon(p,\xi)$ with its charge conjugate, a natural choice, which avoids the introduction of unphysical degrees of gluon polarization, is $$\varepsilon_\mu(p,\xi)\varepsilon_\nu^*(p,\xi)
=\frac{1}{2}\left(-g_{\mu\nu}+\frac{p_\mu\eta_\nu+p_\nu\eta_\mu}{k\cdot\eta}
+i\xi\epsilon_{\mu\nu\rho\sigma}\frac{p^\rho \eta^\sigma}{p\cdot \eta} \right),
\label{eq:1}$$ where $\eta$ is an arbitrary light-like four-vector orthogonal to $p$, with $\eta^2=0\ne p\cdot\eta$. An obvious disadvantage of Eq. (\[eq:1\]) is that it introduces a host of terms involving $\eta$ in intermediate results. In practical calculations such as the one performed in Ref. [@Klasen:2003zn], it is therefore advantageous to omit the second term on the right-hand side of Eq. (\[eq:1\]) and to identify $\eta$ with the four-momentum $p^\prime$ of another external parton [@Bojak:1998bd], so that $$\varepsilon_\mu(p,\xi)\varepsilon_\nu^*(p,\xi)
=\frac{1}{2}\left(-g_{\mu\nu}
+i\xi\epsilon_{\mu\nu\rho\sigma}\frac{p^\rho p^{\prime\sigma}}{p\cdot p^\prime}
\right),
\label{eq:2}$$ at the expense of endowing the gluon with unphysical degrees of polarization, which must be eliminated by subtracting contributions arising from the presence of its ghost $h$ and anti-ghost $\overline{h}$ as external particles. Since such ghost contributions cancel in the cross section differences appearing in the numerator of $\mathcal{A}_{LL}$, as illustrated below, we did not list them in Ref. [@Klasen:2003zn]. However, they are necessary to recover the well-known expressions for the unpolarized cross sections entering the denominator of $\mathcal{A}_{LL}$, as we did. In this sense, they were included in our numerical analysis.
Recently, there has been renewed interest in charmonium production by polarized hadron-hadron and photon-hadron collisions [@Nayak:2005ty; @Meijer:2007eb]. In Ref. [@Nayak:2005ty], the $J/\psi$ and $\psi^\prime$ polarizations were predicted for polarized $pp$ collisions at RHIC-Spin. In Ref. [@Meijer:2007eb], the squares of the helicity amplitudes $\mathcal{M}(a,b,c)$ of the partonic subprocesses $\gamma(a)+g(b)\to Q\overline{Q}[n]+g(c)$ and $g(a)+g(b)\to Q\overline{Q}[n]+g(c)$ were listed for $n={}^1S_0^{(C)},{}^3S_1^{(C)},{}^1P_1^{(C)},{}^3P_J^{(C)}$ with $J=0,1,2$ and $C=1,8$. The longitudinally-polarized differential cross sections evaluated from these helicity amplitudes were found to agree with our results [@Klasen:2003zn] after properly subtracting the ghost contributions mentioned above, which we had provided to the authors of Ref. [@Meijer:2007eb] via private communication. Since these contributions may be useful for applications by other authors as well, we decided to publish them in this addendum to Ref. [@Klasen:2003zn].
In the following, we present the differential cross sections $\mathrm{d}\sigma/\mathrm{d}t$ of the partonic subprocesses $$\{\gamma,g\}h\to c\overline{c}[n]h.$$ Here and in the following, $s$, $t$, and $u$ denote the usual Mandelstam variables. The results for $\{\gamma,g\}\overline{h}\to c\overline{c}[n]\overline{h}$ are identical by charge-conjugation invariance, while those for $h\{\gamma,g\}\to c\overline{c}[n]h$ and $h\overline{h}\to c\overline{c}[n]\{\gamma,g\}$ are related by crossing symmetry, as indicated below. As usual, $\mathrm{d}\sigma/\mathrm{d}t$ is evaluated from the absolute square of the transition matrix element $\mathcal{M}$ through multiplication with factors for flux, phase space, spin, and color, as $$\frac{\mathrm{d}\sigma}{\mathrm{d}t}=
\frac{1}{2s}\,\frac{1}{8\pi s}\,\frac{1}{4}\left(\frac{1}{8}\right)^i
|\mathcal{M}|^2,
\label{eq:3}$$ where $i=1,2$ is the number of color-octet partons (gluons or ghosts) in the initial state.
The only non-vanishing ghost contributions read $$\begin{aligned}
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h)&=&
\frac{24e^2g_s^4\langle\mathcal{O}[{}^1S_0^{(8)}]\rangle Q_c^2su}{Mt(s+u)^2},
\nonumber\\
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^3P_0^{(8)}]h)&=&
\frac{32e^2g_s^4\langle\mathcal{O}[{}^3P_0^{(8)}]\rangle Q_c^2su}{M^3t(s+u)^4}
[(2t+3u)^2+6s(2t+3u)+9s^2],
\nonumber\\
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^3P_1^{(8)}]h)&=&
\frac{32e^2g_s^4\langle\mathcal{O}[{}^3P_1^{(8)}]\rangle Q_c^2}{M^3(s+u)^4}
[u^2(t+u)-su^2+s^2(t-u)+s^3],
\nonumber\\
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^3P_2^{(8)}]h)&=&
\frac{32e^2g_s^4\langle\mathcal{O}[{}^3P_2^{(8)}]\rangle Q_c^2}{5M^3t(s+u)^4}
[3tu^2(t+u)+su(8t^2+21tu+12u^2)
\nonumber\\
&&{}+3s^2(t^2+7tu+8u^2)+3s^3(t+4u)],
\nonumber\\
|\mathcal{M}|^2(gh\to c\overline{c}[{}^1S_0^{(1)}]h)&=&
\frac{4g_s^2\langle\mathcal{O}[{}^1S_0^{(1)}]\rangle}
{3e^2Q_c^2\langle\mathcal{O}[{}^1S_0^{(8)}]\rangle}
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h),
\nonumber\\
|\mathcal{M}|^2(gh\to c\overline{c}[{}^3P_J^{(1)}]h)&=&
\frac{4g_s^2\langle\mathcal{O}[{}^3P_J^{(1)}]\rangle}
{3e^2Q_c^2\langle\mathcal{O}[{}^3P_J^{(8)}]\rangle}
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^3P_J^{(8)}]h),
\nonumber\\
|\mathcal{M}|^2(gh\to c\overline{c}[{}^1S_0^{(8)}]h)&=&
\frac{5g_s^2}{12e^2Q_c^2}
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h),
\nonumber\\
|\mathcal{M}|^2(gh\to c\overline{c}[{}^3S_1^{(8)}]h)&=&
\frac{3g_s^6\langle\mathcal{O}[{}^3S_1^{(8)}]\rangle}{4M^5stu(s+u)^2}
[tu^2(t+u)^2(3t-u)+su(-2t^4+2t^3u
\nonumber\\
&&{}+7t^2u^2+4tu^3+u^4)
+s^2(3t^4+t^3u-4t^2u^2-3tu^3+u^4)
\nonumber\\
&&{}+s^3(4t^3+7t^2u-2tu^2-u^3)+s^4(t^2+6tu-u^2)],
\nonumber\\
|\mathcal{M}|^2(gh\to c\overline{c}[{}^1P_1^{(8)}]h)&=&
\frac{g_s^2\langle\mathcal{O}[{}^1P_1^{(8)}]\rangle}
{e^2Q_c^2\langle\mathcal{O}[{}^1S_0^{(8)}]\rangle M^2}
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h),
\nonumber\\
|\mathcal{M}|^2(gh\to c\overline{c}[{}^3P_J^{(8)}]h)&=&
\frac{5g_s^2}{12e^2Q_c^2}
|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^3P_J^{(8)}]h),
\label{eq:4}\end{aligned}$$ where $e=\sqrt{4\pi\alpha}$, with $\alpha$ being Sommerfeld’s fine-structure constant, and $g_s=\sqrt{4\pi\alpha_s}$ are the electromagnetic and strong gauge couplings, $Q_c$ and $m_c$ are the fractional electric charge and mass of the $c$ quark, and $M=2m_c$. By four-momentum conservation, we have $s+t+u=M^2$.
We now explain how the unpolarized and polarized results of Refs. [@Meijer:2007eb; @Yuan:1999eb] may be recovered from the results of Ref. [@Klasen:2003zn] in combination with Eqs. (\[eq:3\]) and (\[eq:4\]), considering $\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g$ as an example. The unpolarized and polarized results of Eqs. (A4) and (A5) in Ref. [@Yuan:1999eb] are obtained from Eq. (\[eq:3\]) by inserting $$\begin{aligned}
|\mathcal{M}|_\mathrm{unpol}^2(\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g)&=&
\sum_{\xi_a,\xi_b=\pm1}
|\mathcal{M}|_{\xi_a,\xi_b}^2(\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g)
-|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h)
\nonumber\\
&&{}-|\mathcal{M}|^2(\gamma\overline{h}\to c\overline{c}[{}^1S_0^{(8)}]
\overline{h}),
\nonumber\\
|\mathcal{M}|_{LL}^2(\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g)&=&
\sum_{\xi_a,\xi_b=\pm1}(-1)^{\xi_a\xi_b}
|\mathcal{M}|_{\xi_a,\xi_b}^2(\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g),\end{aligned}$$ respectively, where $|\mathcal{M}|_{\xi_a,\xi_b}^2(\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g)$ may be gleaned from Eq. (A5) of Ref. [@Klasen:2003zn] and $|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h)
=|\mathcal{M}|^2(\gamma\overline{h}\to c\overline{c}[{}^1S_0^{(8)}]
\overline{h})$ is given in Eq. (\[eq:4\]) above. As mentioned above, all ingredients entering $|\mathcal{M}|_{LL}^2(\gamma g\to c\overline{c}[{}^1S_0^{(8)}]g)$ are contained in Ref. [@Klasen:2003zn]. By crossing symmetry, we have $$\begin{aligned}
|\mathcal{M}|^2(h\gamma\to c\overline{c}[{}^1S_0^{(8)}]h)&=&
\left.|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h)\right|_{t
\leftrightarrow u},
\nonumber\\
|\mathcal{M}|^2(h\overline{h}\to c\overline{c}[{}^1S_0^{(8)}]\gamma)&=&
\left.|\mathcal{M}|^2(\gamma h\to c\overline{c}[{}^1S_0^{(8)}]h)\right|_{s
\leftrightarrow t}.\end{aligned}$$ Similar relationships hold for the other partonic subprocesses involving two external gluons considered in Ref. [@Klasen:2003zn].
In conclusion, we complemented the partonic cross sections for the inclusive production of prompt $J/\psi$ mesons in polarized hadron-hadron, photon-hadron, and photon-photon collisions listed in the Appendix of Ref. [@Klasen:2003zn] by providing the ghost contributions, which cancel in the cross section differences entering $\mathcal{A}_{LL}$, but contribute to absolute cross sections, including the unpolarized ones.
We thank Jack Smith for carefully comparing the results of Ref. [@Meijer:2007eb] with ours [@Klasen:2003zn]. The work of B.A.K. was supported in part by the German Federal Ministry for Education and Research BMBF through Grant No. 05 HT6GUA, by the German Research Foundation DFG through Grant No. KN 365/6–1, and by the Helmholtz Association through Grant No. HA 101.
[00]{}
W. E. Caswell and G. P. Lepage, Phys. Lett. B [**167**]{}, 437 (1986); G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D [**51**]{}, 1125 (1995); [**55**]{}, 5853(E) (1997) \[arXiv:hep-ph/9407339\].
M. Klasen, B. A. Kniehl, L. N. Mihaila, and M. Steinhauser, Phys. Rev. D [**68**]{}, 034017 (2003) \[arXiv:hep-ph/0306080\]. I. Bojak and M. Stratmann, Phys. Lett. B [**433**]{}, 411 (1998) \[arXiv:hep-ph/9804353\]; Nucl. Phys. B [**540**]{}, 345 (1999) \[arXiv:hep-ph/9807405\]. G. C. Nayak and J. Smith, Phys. Rev. D [**73**]{}, 014007 (2006) \[arXiv:hep-ph/0509335\].
M. M. Meijer, J. Smith, and W. L. van Neerven, arXiv:0710.3090 \[hep-ph\]. F. Yuan, H.-S. Dong, L.-K. Hao, and K.-T. Chao, Phys. Rev. D [**61**]{}, 114013 (2000) \[arXiv:hep-ph/9909221\].
[^1]: klasen@lpsc.in2p3.fr
[^2]: kniehl@desy.de
[^3]: luminita@particle.uni-karlsruhe.de
[^4]: matthias.steinhauser@uka.de
|
---
abstract: |
Neutron and x-ray scattering studies on relaxor ferroelectric systems Pb(Zn$_{1/3}$Nb$_{2/3}$)O$_3$ (PZN), Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$ (PMN), and their solid solutions with PbTiO$_3$ (PT) have shown that inhomogeneities and disorder play important roles in the materials properties. Although a long-range polar order can be established at low temperature - sometimes with the help of an external electric field; short-range local structures called the “polar nano-regions” (PNR) still persist. Both the bulk structure and the PNR have been studied in details. The coexistence and competition of long- and short-range polar orders and how they affect the structural and dynamical properties of relaxor materials are discussed.
keywords: relaxors, neutron scattering, x-ray diffraction, skin effect, polar nano-region
author:
- Guangyong Xu
title: 'Competing orders in PZN-$x$PT and PMN-$x$PT relaxor ferroelectrics'
---
Introduction
============
Relaxors, also called “relaxor ferroelectrics”, are a special class of ferroelectric material. Relaxors have a highly frequency dependent dielectric response ($\epsilon$) with a broad maximum in temperature. Unlike normal ferroelectrics, the dielectric response in relaxors remains relatively large for a wide temperature range near T$_{max}$ where $\epsilon$ peaks. Because of their unusual dielectric and piezoelectric response, relaxors have great potentials for applications [@PZT1; @Uchino; @Service] and have attracted much attention after they were first discovered in 1960’s [@discovery]. Pb(Zn$_{1/3}$Nb$_{2/3}$)O$_3$ (PZN) and Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$ (PMN) are two prototypical lead based perovskite relaxors taking the form of Pb(B’B”)O$_3$. Here the B-site is normally occupied by two cations with different valencies. For example, in PZN and PMN, Zn$^{2+}$/Mg$^{2+}$ and Nb$^{5+}$ take a $1:2$ ratio on B-site to achieve an average valence of $4+$ for charge conservation. The frustration between charge neutrality and lattice strain which does not favor a $1:2$ order makes it unlikely for any long-range cation/chemical order to form in these relaxor systems. The charge imbalance due to the randomness of B-site occupancy creates local random fields, which is the key in understanding many relaxor properties [@Random_Field_Ori; @Random_Field_FE; @Random_Field0; @Random_Field1; @Random_Field2].
Long-range polar order
----------------------
The formation of long-range polar order in relaxors can sometimes be suppressed by the random field. In fact, in PMN, no long-range ferroelectric order can be established under zero external field cooling [@Bonneau; @Husson] and the lattice structure remains cubic down to very low temperatures. Only when cooled under an external electric field [@PMN_field] PMN exhibits a long-range ferroelectric phase with rhombohedral (R) structure below its Curie temperature T$_C \sim 210$ K.
The situation for PZN is slightly different. Pure PZN was first believed to go into a rhombohedral ferroelectric phase at T$_C \sim 410$ K with zero field cooling (ZFC) [@PZN_phase2; @Lebon]. However, recent work using high energy (67 keV) x-ray diffraction suggested that the rhombohedral distortion observed before was only limited in a “skin” layer with thickness of tens of microns in these samples. Instead, the bulk of unpoled single crystal PZN actually should have a near cubic lattice without detectable rhombohedral distortion [@PZN_Xu]. With electric field poling, PZN can also exhibit a long-range polar order with clear rhombohedral distortion. This finding implies that pure PZN and PMN have more similarities than previously realized. Nevertheless, the low temperature phase of (unpoled) PZN is not a real cubic paraelectric phase. It was initially called “phase X” because of its anomalous properties [@PZN_Xu; @PZN_Xu2], many of these can be attributed to strains and local inhomogeneities.
Doping with the conventional ferroelectric PbTiO$_3$ (PT) to form solid solutions of PZN-$x$PT and PMN-$x$PT also helps stabilizing the ferroelectric order. As shown in Fig. \[fig:1\], with relatively small PT doping, the low temperature phase of PMN-$x$PT and PZN-$x$PT is stabilized as rhombohedral. However, the charge imbalance due to frustration is still important and many anomalous features such as large strain, “skin” effect (where the outer and inner parts of the crystal have different structures/lattice parameters) are still present [@PMN-10PT; @Xu1; @Xu_apl; @Conlon; @Skin_Review]. Higher PT doping gradually suppresses relaxor properties and eventually the system becomes more like a normal ferroelectric with a tetragonal (T) low temperature phase. The phase boundary separating the relaxor/rhombohedral side and the ferroelectric/tetragonal side is called the “Morphortropic Phase Boundary” (MPB). It is near the MPB where the piezoelectric response - which tells us how good the material is in converting between mechanical and electrical forms of energy - reaches its maximum [@PZT1; @PZN_phase1; @PZN_phase2]. Many studies have been therefore focused on the compositions near the MPB, and a narrow region of monoclinic phases has been discovered [@Kiat; @Singh; @PZN_phase; @Universal_phase; @PMN_phase; @Noheda; @PMN_mphase]. In a monoclinic (M) phase, the polarization of the system is restricted to a plane [@Devonshire; @Polarization] rather than being strictly restricted to one direction in the case of a rhombohedral (R) or tetragonal (T) phase. It has been argued that this freedom in the M phase helps facilitate the “polarization rotation” process and is responsible for the high piezoelectric response from these materials [@Cohen]. Indeed later studies have shown that various M phases can also be induced by an external electric field for compositions near or slightly away from the MPB [@PZN_field; @PZN_efield; @Feiming; @Cao], indicating that the long-range lattice structure here can be rather unstable and easily modified by external conditions.
Short-range polar order
-----------------------
In addition to the chemical disorder/short-range order, short-range polar order is also present in relaxor systems. The concept of “polar nano-regions” (PNR) was initially proposed by Burns and Dacol [@Burns], to explain their results in optical measurements from a series of relaxor systems including PZN and PMN. It was found that their diffraction indices deviate from linear temperature behavior at a temperature T$_d$, later called the “Burns temperature”, which is far above T$_{max}$. It was suggested that local polar clusters start to form at this temperature while the majority of the lattice still remain unpolarized. The existence of PNR makes a clear distinction between the paraelectric phase of normal ferroelectrics and the high temperature phase of relaxors. The large frequency dependency of $\epsilon$ can also be naturally explained with the relaxation process of PNR [@Cross]. Although it is almost certain that these local structures are associated with the frustration and charge imbalance in relaxor systems, the origin of the PNR and how they are formed is however, still not entirely understood. There are both experimental [@Hiro_diffuse] and theoretical [@Burton] implications that they may be formed based on the chemical short-range order, but more studies are clear required.
Since then there have been numerous studies to probe PNR in relaxor systems including Raman and dielectric measurements [@Raman1; @Raman2], high resolution piezoelectric force microscopy [@Piezo_micro2; @Piezo_micro1], x-ray [@PMN_xraydiffuse; @PMN_xraydiffuse2; @Xu_3D] and neutron [@PZN_diffuse; @PZN_diffuse2; @PMN_diffuse; @PMN_diffuse2; @PMN_neutron3; @Hiro_diffuse; @PZN_diffuse3; @Vakhrushev_Diffuse; @Xu_diffuse] diffuse scattering measurements, as well as pair density function (PDF) measurements [@Egame_PDF]. Contrary to the initial expectations, the PNR do not disappear or grow into large macroscopic ferroelectric domains when the systems goes into a long-range ordered phase. Instead, they are found to persist and coexist with the long-range polar order [@Xu_new; @Xu_coexist]. The PNR also respond to external electric fields - instead of being completely suppressed, their behavior under field has been very interesting, and largely depends on the direction of the field. For instance, diffuse scattering from PNR in PZN-8%PT has only been partially suppressed by an external field along the tetragonal (T) \[001\] direction [@PZN-8PT]. With an external field along the rhombohedral (R) \[111\] direction, a redistribution effect has been found between PNR with different polarizations [@Xu_nm1] in PZN-$x$PT single crystals. Similar effect has been observed in PMN-$x$PT systems as well [@PMN-cubic]. Most of these results suggest that the short-range polar order in PNR appears to be an essential part of the relaxor phase - even in the low temperature phase where long-range ferroelectric polar order is established. The two orders can coexist, sometimes compete with each other, but can also develop at the same time.
In addition to understanding the structures and polarizations of PNR, it is more important to find out how these local structures can affect bulk materials properties. Studies have shown that many anomalous properties in the long-range polar structures in PZN-$x$PT and PMN-$x$PT relaxors are closely related with the PNR. For example, diffuse scattering from the PNR have been shown to have contributions from both a polar component and a strain component [@PMN_diffuse; @Xu_coexist]. The former comes from optic type atomic shifts; while the latter arises from acoustic type atomic displacements which is directly associated with the large strain in the lattice. Moreover, recent work indicates that the PNR can interact with various phonon modes [@Xu_NM2; @Stock_couple] and could be responsible for the phase instability in relaxor compounds which is essential for achieving the high piezoelectric response.
In the following sections, I will discuss our findings on PZN-$x$PT and PMN-$x$PT relaxor systems using neutron and x-ray scattering. The anomalous behavior of various long-range polar order/lattice structures will be discussed first, followed by diffuse scattering measurements on PNR. These results show that local disorder due to frustration is responsible for many special properties of relaxor systems and it is extremely important to obtain a better understanding of these local inhomogeneities and how they interact/affect the bulk.
Long-range order: structural studies
====================================
Phase “X”
---------
The concept of phase “X” was first raised by Ohwada [*et al.*]{} in their work on structural properties of PZN-8%PT [@PZN_efield]. A near cubic phase, instead of the rhombohedral phase according to previously known phase diagrams, was discovered upon zero-field-cooling (ZFC). More definitive evidence of this phase “X” was found in unpoled single crystals of PZN [@PZN_Xu]. Using high energy x-ray (67 keV), we were able to probe deeply inside the bulk PZN single crystal and look for the rhombohedral distortion at room temperature (T$_C \sim 410$ K for PZN). In a rhombohedrally distorted lattice, the d-spacings of four {111} planes become different. We have therefore performed mesh scans around the four {111} Bragg peaks. The results obtained from the prepoled (field cooling to room temperature with E=20 kV/cm along \[111\] direction) and unpoled PZN single crystals are clearly different. For the poled crystal, Bragg peaks at (111) and ($\bar{1}$11) appear at different [**Q**]{} lengths due to the rhombohedral distortion (see Fig. \[fig:2\]). The rhombohedral angle obtained from the measurements is $\alpha = 89.935^\circ$, consistent with previous reports [@Lebon]. However, the four mesh scans for the unpoled PZN single crystal all give the same d-spacing at T=300 K (see the bottom frame of Fig. \[fig:2\]), showing no evidence of rhombohedral distortion.
This near cubic low temperature phase was later also discovered in PMN-10%PT [@PMN-10PT] and PMN-20%PT [@Xu1]. Unlike the case in pure PMN where the low temperature phase under ZFC is really cubic, the symmetry of phase “X” is likely not cubic, evidenced by the increase of Bragg peak intensities at T$_C$ due to release of extinction [@Stock1] - which is usually a sign of transition into a lower symmetry phase. There are other signs suggesting that even the unit cell does not show a (detectable) rhombohedral distortion, a structural phase transition has indeed occurred at T$_C$. For example, as shown in Fig. \[fig:3\], a sudden increase of lattice strain along the \[110\] direction occurs at T$_C \sim 300$ K for PMN-20%PT by high $q$-resolution neutron scattering measurements [@Xu1]. Similar effect has been observed in pure PZN with high energy x-ray diffraction measurements as well [@PZN_Xu2]. There in addition to the change of lattice strain, a broadening of the (200) Bragg peak in the transverse direction has also been observed, indicating a sudden increase of crystal mosaic at T$_C\sim 410$ K.
The lack of lattice distortion in phase “X” is quite unusual. Our current understanding of phase “X” is that this is a phase where the lattice prefers to go rhombohedral because of the tendency toward a long-range ordered ferroelectric phase below T$_C$. Ferroelectric polarizations are actually realized by local atomic displacements. However, the unit cell shape still remains nearly cubic. In other words, this can also be called a “less-rhombohedral” phase where the unit cell distortion is much smaller than one would expect. The large strain in this phase is an indication of structural inhomogeneity. It is our belief that the interactions between local inhomogeneities - possibly the PNR - and the bulk lattice, become strong enough and “locks” the long-range lattice structure into this unusual configuration of phase “X”. The phase itself is quite unstable and can be easily driven into a full rhombohedral phase with an external field along \[111\] direction.
The “skin effect”
-----------------
The discovery of phase “X” was actually accompanied with the discovery of another interesting effect - the “skin effect”. The findings of a near cubic phase inside the bulk of PZN (and later PMN-10%PT and PMN-20%PT) single crystals are surprising and not consistent with previous results [@Lebon; @PMN_diffuse2]. In order to resolve the inconsistency, measurements probing different depths into the single crystal sample have been carried out. In Fig. \[fig:4\], longitudinal intensity profiles near the (111) Bragg peak of the same single crystal PZN using x-ray diffraction with different x-ray energies are shown. With 67 keV x-rays, the intensity profile shows a sharp single peak for temperatures both above and below T$_C \sim 410~$K, i.e. no rhombohedral splitting for the low temperature phase. With x-ray energy of 10.7 keV, the situation is drastically different. The profile only has one peak in the high temperature cubic phase; while at low temperature, the (111) Bragg peak splits into two. The answer to the different results lies in the penetration depth of photons into the sample. For 67 keV x-rays, the x-ray penetration depth into the sample (the sample geometry is already taken into consideration) is about 400 $\mu$m, and the measurements are performed in a transmission mode. In other words, the bulk of the sample is being measured and a near cubic phase (phase “X”) is observed. For 10.7 keV x-rays, the penetration depth is much smaller ($\sim 10~\mu$m) and the measurements had to be performed in a reflection mode since x-rays can not penetrate the sample which has a thickness of about 1 mm. Therefore, the rhombohedral distortion is actually only limited to a outer-most layer, or “skin” in the sample. Based on the penetration lengths of the x-ray beams, we can obtain an estimate of thickness of the outer-layer to be between 10 $\mu$m and 100 $\mu$m.
In addition to having different lattice structures, the thermal expansion of the outer-layer and the inside can also be quite different. In Fig. \[fig:5\], the lattice parameters of the inside and outer-layer of the unpoled PZN single crystal are plotted. The behavior of the outer-layer is what one would expect from a normal ferroelectric oxide [@Gen]. The inside of the crystal however, behaves quite differently. The lattice parameter almost remains constant for the whole temperature range, not being affected by the phase transition at T$_C \sim 410$ K. Similar results is also seen in PMN-20%PT (see the bottom panel of Fig. \[fig:3\]). This is another important feature of phase “X” which differs from a normal ferroelectric phase.
The “skin-effect” naturally explains the discrepancy between recent high-energy x-ray and neutron diffraction measurements on single crystal relaxor samples [@PZN_Xu; @Xu1; @PMN-10PT] and previous work where the rhombohedral distortion can be measured from the samples of similar compositions [@Lebon; @PMN_diffuse2; @PMN_Ye]. In previous work, either a lab x-ray source and/or powder samples were used. A lab x-ray source usually gives photons at the energy of Cu K$_\alpha$ line ($\sim 8$ keV) which can only penetrate into the outer most $\sim 10~\mu$m of these lead based relaxor samples. When powder samples are used, the normal grain sizes would also be in the order of tens of $\mu$m, the same as the thickness of the outer layer. It is for this reason that powder and/or lab x-ray measurements only probes the outer-layer but not the inside of the crystal.
The “skin-effect” is not limited to compositions where phase “X” exists (in the bulk). It has also been observed in systems with a rhombohedral lattice structure for the bulk part of the crystal. For example, in PZN-4.5%PT and PZN-8%PT single crystal samples, high energy x-ray measurements confirm that the inside of the crystals are rhombohedrally distorted [@Xu_apl]. The structures obtained with lower energy x-ray (10.7 keV) measurements are also rhombohedral, but the lattice parameter and rhombohedral angles between the outer region and the inside are different. The outer layers have larger rhombohedral distortions (rhombohedral angles further away from 90$^\circ$) and smaller lattice parameters [@Xu_apl]. These results provide more evidence that the strain inside the crystal - most likely due to interaction between the lattice that tends to become polar and the PNR - is the key that prevents or reduces the rhombohedral distortion. When going near the surface, the strain is reduced - which is probably the opposite to many other systems where surface strains can actually affect surface properties, and the rhombohedral distortion is restored.
Interestingly, the “skin effect” is also present in pure PMN, which is believed to be an exception to PZN-$x$PT and PMN-$x$PT systems since it always remains cubic with ZFC. Stock [*et al.*]{} have performed strain measurements using a very narrow neutron beam [@Conlon] on a single crystal PMN sample. The sample can be translated so that one can use the narrow beam to directly probe the lattice structures of different depths into the sample. As shown in Fig. \[fig:6\], similar to what has been observed in PZN-$x$PT samples, in pure PMN there is an outer-layer of about $\alt 100~\mu$m thick near the surface with lattice strain significantly smaller than that of the inside.
The “skin-effect” is discussed in more details in Ref. . It is most likely that the the large strain for the inside structure is associated with the PNR and is unique for relaxor compounds. In addition, with a different “outer-layer” structure, it is important for one to be extra careful when interpreting measurements that may only probe the surface region of these materials.
Monoclinic phases {#dilemma}
-----------------
Another important finding in the research on structural properties of relaxor materials is the discovery of monoclinic phases. In PZN-$x$PT and PMN-$x$PT systems, monoclinic (M) phases were first discovered experimentally [@Noheda; @PMN_phase; @PZN_phase; @Kiat; @Singh; @Universal_phase] for compositions near the morphortropic phase boundary (MPB) that separates the rhombohedral relaxor and the tetragonal ferroelectric phases (see Fig. \[fig:1\]). The M phases are also predicted by theoretical works - while the original Devonshire theory to the sixth-order only supports rhombohedral (R), tetragonal (T) and orthorhombic (O) phases, a further expansion of the theory to the eighth-order [@Devonshire] does predict three different monoclinic phases, M$_A$, M$_B$, and M$_C$ (see Fig. \[fig:7\]). Compared to the low PT doping R phase, where the polarization is confined to the \[111\] direction; and the high PT doping T phase, where the polarization is confined to the \[001\] direction; in the M phases the polarizations are confined in plane [@Polarization] - (1$\bar{1}$0) plane for M$_A$ and M$_B$ phases \[see Fig. \[fig:7\] (a)\], and (010) plane for M$_C$ phases \[see Fig. \[fig:7\] (b)\]. As the polarization is rotated away from \[111\] toward \[001\] with higher PT doping, these M phases act as bridging phases where the polarizations lie in between R and T.
The situation is similar when an external field along \[001\] direction is applied. For compositions on the left side of the MPB with rhombohedral ground state, the polarization can then be rotated by the field toward the \[001\] direction, inducing intermediate monoclinic phases. This is the “polarization rotation mechanism” proposed by Fu and Cohen [@Cohen] to explain the enhanced piezoelectric response near the MPB. A smooth rotation from R (\[111\]) to T (\[001\]) would give a M$_A$ phase, while with higher field, a jump to the M$_C$ phase can also occur. Experimental evidence on field induced M phases have been reported by various neutron and x-ray diffraction measurements on a number of compositions of PZN-$x$PT [@PZN_efield; @PZN_field] and PMN-$x$PT [@Feiming; @Cao] samples. The M$_B$ phase can only be induced with an external field along the \[011\] (orthorhombic) direction [@Cao], as the M$_B$ phase can be considered as a bridge between R and O \[see Fig. \[fig:7\] (a)\].
One dilemma still remains, however. Intuitively, as shown in Fig. \[fig:7\], the bridge phase between R and T should be M$_A$. While in reality, in both PZN-$x$PT and PMN-$x$PT systems, only M$_C$ phase has been observed between the R and T regions without external electric field (in PZN-$x$PT systems, the zero field orthorhombic (O) phase is a special case of M$_C$ phase). This cannot be easily explained by only looking at the long-range structures. In fact, by the end of the next section, I will discuss a possible explanation to this problem considering strain induced by polar nano-regions and its implications.
Short-range order: polar nano-regions
=====================================
Structures and polarizations of the PNR
---------------------------------------
The concept of polar nano-regions is unique to relaxor systems, where local polar orders appear before any long-range polar order is established in the system. Since these PNR represent local structures different from the average lattice, diffuse scattering is one of the most direct tools to probe inside the bulk for these inhomogeneities. In general, diffuse scatterings from PZN-$x$PT and PMN-$x$PT samples with compositions on the left side of the MPB are very similar [@Xu_3D; @Matsuura]. They appear above T$_C$, and increase monotonically with cooling. The distribution of diffuse scattering intensities in the reciprocal space has also been measured for different compositions (with small $x$). An example of these measurements is shown in Fig. \[fig:8\]. Here a mesh scan from neutron diffuse scattering measurements taken at 200 K from a single crystal PMN, around the (100) Bragg peak in the (H0L) plane is plotted. The intensity is highly anisotropic in the reciprocal space, taking a “butterfly” shape in the (H0L) plane. Measurements have also been taken on the right side of the MPB [@PMN-60PT; @Matsuura] and the “butterfly” diffuse scattering disappears, indicating that no PNR exists in the ferroelectric side of the phase diagram.
More detailed measurements probing the three-dimensional (3-D) distribution of diffuse scattering intensities [@Xu_3D] using high energy x-ray beam were performed on single crystals of PZN-$x$PT for x=0, 4.5% and 8%. It was found that the diffuse scattering is dominated by rod type intensities along various $\langle110\rangle$ directions. Although there are totally six different $\langle110\rangle$ rods, they do not always show up with the same intensity across different Bragg peaks. A sketch of the intensity distribution in the 3-D reciprocal space is plotted in Fig. \[fig:9\]. Based on how these $\langle110\rangle$ intensity rods changes, we propose that they come from independent local structures. In other words, the diffuse rod along each $\langle110\rangle$ direction comes from PNR of a certain orientation.
Then the problem becomes relatively simple. Since rod type intensities in reciprocal space must correspond to planar correlations/structures in real space, we can conclude that the short-range polar order in the PNR must take a planar shape in real space. The polarization of these PNR can then be derived from analyzing the “extinction condition” where some $\langle110\rangle$ rods becomes absent near certain Bragg peaks. A simple model called the “pancake model” (see Fig. \[fig:10\]) shows that the there are six possible orientations/polarizations of PNR, with $\langle1\bar{1}0\rangle$ type polarizations, correlated in {110} planes, that give rise to $\langle110\rangle$ diffuse rods. The in-plane and out-of-plane correlation lengths (or, the diameter and thickness of the “pancake” PNR, respectively) can be estimated from the broadness of diffuse scattering perpendicular and along the intensity rod directions. A rough estimate will give a in-plane correlation length of 10 to 20 nm (20 to 40 lattice spacings) while the out-of-plane correlation length is about four times smaller [@Xu_diffuse; @Xu_3D].
The “pancake model” is a simplified model. For example, in this model, the atomic displacements within the PNR are assumed to be all collinear, along the same direction. The possibility of more than a single source to the diffuse scattering is also not considered (e.g. it is possible that the diffuse scattering comes from combination of a polar core plus surrounding lattice strain induced by the core). However, it does provide some important information for the local structures in these relaxor systems. Using this model, most previous diffuse scattering measurements on these systems can be easily explained - for instance, the “butterfly” intensity around (100) peak in the (H0L) plane is simply the cross-section of the \[110\] and \[1$\bar{1}$0\] intensity rods on the (H0L) plane. In addition, it is found that the polarizations/local atomic displacements in the PNR are not along the rhombohedral $\langle111\rangle$ directions as previously believed. This suggests that the PNR are not simply precursors of the macroscopic ferroelectric domains with $\langle111\rangle$ polarizations in the low temperature phase. Instead, the PNR still persist into the low temperature.
Quantitative studies on diffuse scattering intensities across different Bragg peaks [@PMN_neutron3; @PMN_diffuse; @Xu_coexist] can be used to obtain the magnitude of atomic displacements within a unit cell. It is shown that in both pure PMN and PZN-$x$PT crystals, the local atomic shifts in the PNR responsible for the diffuse scattering are always composed of two components: one optic component which gives rise to local polarizations; and one acoustic component, which is related to strains in the system. The former is always expected since these are “polar” structures, and is likely due to the condensation of the ferroelectric transverse optic phonon, which softens significantly below T$_d$ [@Waki1; @Stock1]. Having also the acoustic component is surprising but it helps explain the large strain in these lead-based relaxor systems. The interaction between the PNR and the bulk lattice is a clear example of frustration between lattice strain and charge suppressing/reducing long-range polar order in the system.
Electric field response
-----------------------
Because of their “polar” nature, one would expect the PNR to respond to the application of an external electric field. Indeed, previous studies [@PMN_efield; @PZN-8PT] have shown that neutron diffuse scattering measured in transverse directions to the Bragg vectors can be partially suppressed. Intuitively, if an external electric field can enhance the long-range polar order in the lattice, it should also be able to suppress the short-range polar order in the PNR. Since PZN-$x$PT and PMN-$x$PT relaxors with small $x$ values all have rhombohedral type ground states with \[111\] lattice polarization, we have designed more experiments monitoring the 2-D and 3-D diffuse scattering intensity distribution under an electric field applied along the \[111\] direction. In Fig. \[fig:11\], diffuse scattering intensities from a single crystal sample of PZN-8%PT (T$_C\sim 450$ K) are plotted [@Xu_new]. The measurements have been carried out in the (HKK) plane which is defined by the two vectors along \[100\] and \[011\]. The cross-section of the $\langle110\rangle$ diffuse scattering rods on this plane also takes a “butterfly” shape, as shown by the zero field cooled (ZFC) measurements plotted in Fig. \[fig:11\] (a). With the application of E=2 kV/cm along \[111\] and doing FC, surprisingly, even when the long-range rhombohedral structure is stabilized below T$_C$, the diffuse scattering still persists. The symmetric “butterfly” shape, however, is changed to an asymmetric “butterfly” as shown in Fig. \[fig:11\] (b), suggesting a redistribution of diffuse scattering intensities between different “butterfly” wings. Apparently, the PNR in PZN-8%PT do not simply diminish with the external E-field along \[111\]. Instead, there appears to be a redistribution of PNR with different polarizations.
The \[111\] E-field redistribution of PNR is studied in more details and confirmed with 3-D x-ray diffuse scattering measurements on the single crystal of PZN [@Xu_nm1]. It is found that with the field greater than a threshhold field, the diffuse scattering intensities can be redistributed among the six different $\langle110\rangle$ rods. Those (3 out of 6) diffuse rods coming from PNR with polarizations perpendicular to the E-field are enhanced, while the other 3 are suppressed. Increasing the magnitude of the E-field does not have any further effect on the diffuse scattering. Reducing and eventually reversing the field, however, results in a hysteresis loop very similar to that measured for the polarization vs. E loop on the same single crystal PZN sample.
These results show that cooling in an E-field, and the application of a large enough E-field along \[111\] direction in the low temperature ferroelectric (R) phase, both induce a redistribution of PNR with different polarizations. Instead of suppressing the PNR or aligning their polarizations to be more along the field direction, the field seems to enhance those PNR with polarizations perpendicular to the \[111\] direction. This is quite contrary to what one would naturally expect since the energy of the PNR (dipole moments) alone in the electric field does not favor such a configuration. On the other hand, the similarities between the field dependence of diffuse scattering intensities and that of the polarization [@Xu_nm1] suggest that this redistribution of PNR is likely associated with the rotation of ferroelectric domains by the field. In the paraelectric high temperature phase, there is no long-range ferroelectric domains, and PNR with different $\langle110\rangle$ polarizations are equivalent under the cubic symmetry. When the system goes into the low temperature phase, long-range polar order is established and ferroelectric domains with $\langle111\rangle$ polarizations are formed. Within these domains, the symmetry is lowered to R, and the different $\langle110\rangle$ directions are not equivalent any more. If, within a certain ferroelectric domain, the PNR would prefer to exist in a configuration where their polarizations are perpendicular to that of the domain, all our previous results can be explained easily. As shown in Fig. \[fig:13\] (b), under ZFC, after multi-domain averaging, in the crystal there is no macroscopic preferred $\langle110\rangle$ polarization of the PNR, resulting in the symmetric “butterfly” diffuse scattering shown in Figs. \[fig:8\], \[fig:9\], and \[fig:11\] (a). However, with FC, the volume of the ferroelectric domain polarized along the field \[111\] direction is greatly enhanced \[see Fig. \[fig:13\] (c)\], and our measurements thus provide direct information on how the PNR reside in a \[111\] polarized ferroelectric lattice - they tend to have polarizations perpendicular to that of their surrounding lattice.
The case for pure PMN is an exception. Once an external electric field along \[111\] direction is applied, in addition to the redistribution of diffuse scattering intensities between different $\langle110\rangle$ directions, there is also an overall suppression of the diffuse scattering by the field [@PMN-cubic]. In the mean time, Bragg peak intensities increase, indicating an enhancement of long-range order in the system. Note that in pure PMN, no long-range polar order is established without external field and therefore no macroscopic ferroelectric domains exist at low temperature. However, there could be polar-orders developing in the system at low temperature in the mesoscopic range (e.g. $\alt 1~\mu$m) that provides local $\langle111\rangle$ polarized lattice environment for the PNR. With an external electric field, the re-arrangements of these mesoscopic polar lattice can give rise to the redistribution of PNR and therefore diffuse scattering. On the other hand, not having a long-range ferroelectric order seems to also make the short-range polar order in the PNR less stable and more sensitive to external fields.
This situation where the two competing orders can co-exist, and having the long-range order helps stabilizing the short-range polar order, is quite unusual. In addition, as the temperature decreases, both orders will develop (with the exception of pure PMN) - the long-range polar order develops as evidenced by the increase of rhombohedral distortion with cooling; and the short-range order develops shown by the increase of diffuse scattering intensities. Even an external electric field along \[111\] direction can not change this configuration. Instead, the \[111\] E-field only re-arranges the ferroelectric domains and removes the ambiguity caused by a multi-domain state. The robustness of these local polar orders within the long-range polarized lattice is yet another indication of charge-lattice frustration in relaxor systems.
Coupling to phonons
-------------------
In addition to learning how the PNR exist in relaxor systems as discussed in the previous two subsections, a more important question is that how they affect bulk properties. As discussed before, the long-range lattice structure are affected by the PNR, showing large strains and other anomalous behaviors [@PZN_Xu2; @Xu1]. In addition to static long-range structures, the PNR also affect the lattice dynamics in PZN-$x$PT and PMN-$x$PT relaxor systems.
In ordinary ferroelectrics, there is usually a transverse optic phonon (TO) mode that is associated with the phase transition. The TO mode softens and the zone-center energy goes toward zero at T$_C$ [@PbTiO3]. In relaxors a similar softening of the ferroelectric TO mode is also observed at high temperature, but this mode becomes anomalously broad for small $q$ values in a large temperature range, between the Burns temperature T$_d$ and the Curie temperate T$_C$. With further cooling, the TO mode recovers again below T$_C$. This is called the “water-fall” effect, and has been observed in both pure PMN and PZN, as well as a number of compositions of PZN-$x$PT and PMN-$x$PT on the left side of the phase diagram [@PMN_softmode; @PZN_waterfall1; @PZN_waterfall2; @Stock1; @Cao_phonon]. Because of the temperature range it is observed, and the belief that the PNR are results of TO phonon condensations, the “water-fall” effect has been interpreted as a result of interactions between the PNR and the TO phonon mode. However, in recent work by Stock [*et al.*]{} on a single crystal PMN-60%PT sample, the zone-center TO mode was also found to become heavily damped in a broad temperature range, just like the “water-fall” effect observed in relaxors. Note that PMN-60%PT is located on the right side of the PT doping phase diagram (see Fig. \[fig:1\]), with a first-order phase transition from cubic to tetragonal at T$_C \sim 550$ K [@PMN-60PT]. There is also no “butterfly” shaped diffuse scattering from this material. Having a “water-fall” effect in this ferroelectric material with the absence of PNR suggests that the “water-fall” effect could have other origins.
Although the coupling between PNR and the soft TO phonon mode is becoming controversial, there appears to be strong evidence suggesting that the PNR interact strongly with transverse acoustic (TA) phonon modes in these relaxor systems. Neutron scattering measurements were performed on lattice dynamics and diffuse scattering in different Brillouin zones from single crystal PMN [@Stock_couple]. A strong influence of the diffuse component was observed on TA phonons in the system. In another experiment carried out on a single crystal sample of PZN-4.5%PT, we have used an external E-field along \[111\] direction to help demonstrate the coupling [@Xu_NM2]. The schematic of the measurements is shown in Fig. \[fig:14\] (a). Phonon and diffuse scattering measurements are performed around (220) and ($\bar{2}$20) Bragg peaks. These two peaks are equivalent in the cubic phase. In the low temperature rhombohedral phase, with ZFC, they should also give similar results due to multidomain averaging. With FC for E along \[111\] direction, the two Bragg peaks become different. The diffuse scattering intensity is enhanced near ($\bar{2}$20) and suppressed near (220), as shown in Fig. \[fig:14\] (b). The transverse acoustic phonon mode measured across the two Bragg peaks are clearly affected. Near ($\bar{2}$20) where diffuse scattering is strong, the TA mode becomes very soft and heavily damped. Near (220) where diffuse scattering is weak, the TA mode becomes relatively well defined.
These results suggest that there is a strong coupling between the diffuse scattering and TA phonon modes propagating along different $\langle110\rangle$ directions (the TA2 mode). The interaction with the PNR makes the TA2 mode particularly soft. This marks a structural instability in the system, which is necessary for a system with high piezoelectric response [@Cohen; @Cohen2; @PMN_Critical; @PT_pressure; @Budimir]. In other words, the coupling between PNR and acoustic phonons may be related to the high electromechanical properties of PZN-$x$PT and PMN-$x$PT systems. Furthermore, the soft TA2 mode propagating along $\langle110\rangle$ directions suggest a tendency toward a orthorhombic (O) phase, which in a sense is the dynamical response of the lattice to the orthorhombic type $\langle110\rangle$ strain in the PNR. This orthorhombic strain can help explain the dilemma raised in Section \[dilemma\]: although the only direct bridging phase between R and T is M$_A$, in reality only M$_C$ phases exist in ZFC PZN-$x$PT and PMN-$x$PT?! While compositions on the low PT doping side of the phase diagram is supposed to have R type structures, the presence of PNR induces orthorhombic strains in the system. To bridge structures with orthorhombic strains which are on the left side of the phase diagram, and those with tetragonal strain, on the right side of the phase diagram, a M$_C$ phase is a natural choice [@Xu_NM2] (see Fig. \[fig:7\]).
Summary and future work
=======================
Our neutron and x-ray scattering studies on PZN-$x$PT and PMN-$x$PT relaxor compounds have shown that these materials have complex local structures due to lattice-charge frustration. The short-range polar orders, namely, the polar nano-regions (PNR) can affect the long-range polar order in various ways. They can reduce or even suppress the long-range polar order (phase “X”), and induce large lattice strains. The PNR also interact strongly with acoustic phonon modes, and therefore create a phase instability that may be related to the high piezoelectric response in these materials. On the other hand, the short- and long-range polar orders can still coexist in most of the compositions studied, and there are even implications that the long-range order can help make the short-range polar order more stable.
These complex local structures are far from being fully understood. Currently there are many unanswered questions, and unsolved problems. Here I list a few related topics that would be of interest for future studies:
\(i) [*The origin of the PNR.* ]{} As discussed in earlier parts of the article, the relaxor properties in PZN-$x$PT and PMN-$x$PT systems are related to the random field created by the B-site disorder. Short-range polar order, or the PNR, develops at the Burns temperature T$_d$ as a result of frustration in the system. It is therefore natural to consider the relationship between the PNR and the short-range chemical/cation order in the system. There have been theoretical considerations for this aspect [@Burton], as well as hints from experimental results [@Hiro_diffuse]. In Fig. \[fig:15\], we show diffuse scattering intensity contours below and above T$_d$ from PMN. One can see that at T above T$_d$, where the “butterfly” diffuse scattering already disappears, there is still a weak residue diffuse scattering intensity around both the (110) and (100) Bragg peaks. The residue diffuse scattering intensity does not change much with temperature and should be due to a short-range chemical ordering. What is interesting about the results is that this residue diffuse has shapes that seem to be a conjugate to that of the “butterfly” diffuse from the PNR. This may be a simple coincidence but could also be a hint that those two are related. Further studies are clearly required to clarify this problem. In fact, there has been work done on another lead perovskite system Pb(In$_{1/2}$Nb$_{1/2}$)O$_3$ (PIN) where the B-site disorder can be tuned [@Hirota_PIN]. By increasing the chemical order on the B-site, one is able to tune the system toward a more relaxor type phase with PNR present [@Hirota_PIN].
\(ii) [*How do the PNR respond to E-field along other directions?*]{} The response of PNR for E-field along \[111\] direction as been extensively studied. However, very little work has yet been done on the response of diffuse scattering to E-field along other high symmetry directions such as \[001\] and \[110\]. Preliminary work [@Wen_PMN] has indicated that an \[001\] field does not directly affect the “butterfly” diffuse. Nevertheless, there are indications that diffuse scattering intensities measured along $q\parallel \langle001\rangle$ away from Bragg peaks can be partially suppressed by the \[001\] E-field [@PZN-8PT; @Zhijun]. The “butterfly” diffuse scattering is clearly the dominant part of the diffuse scattering intensity being measured. But it is still possible, as previously mentioned, that there can be other sources to the diffuse scattering intensities than the $\langle110\rangle$ type atomic shifts. These sources may contribute to a portion of the diffuse scattering intensity that behaves differently than the “butterfly” diffuse. This is certainly an issue that requires more attention. Also as \[001\] and \[110\] are the directions along which the piezoelectric response from these relaxor systems are high, it will be interesting to understand if the PNR play any roles in facilitating the “polarization rotation” process under these conditions.
\(iii) [*PNR in other relaxor systems.*]{} In addition to the extensively studied lead perovskite relaxors, there are other relaxor systems such as the lead-free relaxor K$_{1-x}$Li$_x$TaO$_3$ (KLT). In KLT, instead of the B-site disorder, it is the Li displacements that induces local polarization. It would be extremely interesting to explore the properties and response of PNR in this and other materials where he underlying mechanism of having the PNR is completely different.
The work discussed in this article has been carried out with many collaborators. I would first like to give my special acknowledgment to Dr. Gen Shirane, who started our work on relaxor systems in the late 1990’s. I would also like to thank all other collaborators including: F. Bai, Y. Bing, H. Cao, W. Chen, K. H. Conlon, J. R. D. Copley, J. S. Gardner, P. M. Gehring, M. J. Gutmann, H. Hiraka, K. Hirota, S.-H. Lee, J.-F. Li, H. Luo, M. Matsuura, K. Ohwada, C. Stock, I. Swainson, D. Viehland, S. Wakimoto, T. R. Welberry, J. Wen, H. Woo, Z.-G. Ye, Z. Xu, X. Zhao, and Z. Zhong. Financial support from the U.S. Department of Energy under contract No. DE-AC02-98CH10886 is also gratefully acknowledged.
[82]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ** (, ).
, ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , ****, ().
, , , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , , ****, ().
, , , , , , , , **** ().
, , , **** ().
, , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, ****, ().
, , , , , , ****, ().
, ****, ().
, , , , , , ****, ().
, , , , , ****, ().
, , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, , , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , , , , , **** ().
, , , , ****, ().
, ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , , , , ****, ().
, , , , , , ****, ().
, , , ****, ().
, , , , , , , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , , , , , , ****, ().
, , , , ****, ().
, , , , , , , , ****, ().
, , , , , , , ****, ().
, ** (, ).
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , , , , , ****, ().
, , , , , , , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
, , , ****, ().
, , , , , , , , , ****, ().
, , , , ****, ().
, , , , , (), .
![Schematic phase diagrams of PZN-$x$PT and PMN-$x$PT solid solutions. The notations C, R, T, O and M stand for cubic, rhombohedral, tetragonal, orthorhombic, and monoclinic phases, respectively.[]{data-label="fig:1"}](fig1.ps){width="0.6\linewidth"}
![(Color online) High energy x-ray (67 keV) diffraction mesh scans taken around pseudocubic {111} positions of the poled (top frame) and unpoled (bottom frame) PZN single crystals at T=300 K. The intensity is plotted in log scale as shown by the scale bar on the right side. Units of axes are multiples of the pseudocubic reciprocal lattice vector (111) $|\tau_{111}|=\sqrt{3}\cdot2\pi/a_0$ (see Ref.). []{data-label="fig:2"}](fig2.ps){width="0.7\linewidth"}
![Top panel: the resolution corrected longitudinal width of the (220) Bragg peak vs T for PMN-20%PT (open circles), compared with data from PMN-27%PT (close circle) and PMN-10%PT (square) by Gehring [*et al*]{} [@PMN-10PT]. Bottom panel: lattice parameter $a$ vs T for PMN-20%PT. The dotted line represents thermal expansion behavior typical of normal ferroelectrics. The solid lines are guides to the eye (see Ref. )[]{data-label="fig:3"}](fig3.ps){width="0.8\linewidth"}
![(Color online) Longitudinal scans through the (111) Bragg peak, measured at temperatures above and below $T_C \sim 410$ K for the unpoled PZN single crystal. The top panel are diffraction results using 67 keV x-rays, and the bottom panel with 10.2 keV x-rays. The inset shows the rhombohedral distortion angle derived from the 10.2 keV x-ray results. The horizontal bars indicate the instrument resolutions (see Ref. ).[]{data-label="fig:4"}](fig4.ps){width="\linewidth"}
![Lattice parameter $a=Volume^{1/3}$ for the unpoled PZN single crystal, measured by 67 keV x-rays (inside) and 10.2 keV x-rays (outer-layer) (see Ref. ). []{data-label="fig:5"}](fig5.ps){width="\linewidth"}
![Strain measurements using narrow neutron beams on a single crystal of PMN. The upper panel plots the (2,0,0) Bragg peak intensity as a function of translation. The lower panel displays the lattice constant (and hence the strain) as a function of distance into the sample. The vertical dashed line indicates the position of the sample surface. (see Ref. ). []{data-label="fig:6"}](fig6.eps){width="0.6\linewidth"}
![(Color online) A smoothed logarithmic plot of the neutron elastic diffuse scattering intensity measured at 200 K from a single crystal PMN near the (100) Bragg peak in the (H0L) scattering plane (see Ref. ).[]{data-label="fig:8"}](fig8.ps){width="\linewidth"}
![(Color online) Sketch of the diffuse scattering distribution in the 3-D reciprocal space around (100), (110), (111), (010), and (011) reciprocal lattice points from PZN-$x$PT single crystals for x=0, 4.5% and 8% (see Ref. ). []{data-label="fig:9"}](fig9.ps){width="\linewidth"}
![(Color online) PNR in the real space and their contributions to the diffuse scattering in the reciprocal space. A “pancake” shaped PNR in real space corresponds to rod type diffuse scattering in reciprocal space. From (a) to (f), we show PNR with in-plane polarizations along the \[01$\bar{1}$\], \[10$\bar{1}$\], \[1$\bar{1}$0\], \[011\], \[101\], and \[110\] directions, correlated in the (011), (101), (110), (01$\bar{1}$) , (10$\bar{1}$), and (1$\bar{1}$0) planes, and contributing to the diffuse rods along \[011\], \[101\], \[110\], \[01$\bar{1}$\] , \[10$\bar{1}$\], and \[1$\bar{1}$0\] directions, respectively (see Ref.).[]{data-label="fig:10"}](fig10.ps){width="0.6\linewidth"}
![(Color online) Diffuse scattering from PZN-8%PT under an external electric field along the \[111\] direction. The top frame is a schematic of the (HKK) reciprocal scattering plane, in which neutron diffuse scattering measurements were performed close to the (300) Bragg peak. The bottom frames show data measured at $T=300$ K after the sample was (a) zero-field cooled (ZFC), and (b) field-cooled (FC) with E=2 kV/cm along \[111\] through $T_C~\sim 450$ K. The solid green (gray) lines are guides to the eye to help emphasize the symmetric (a) and asymmetric (b) “butterfly” shapes of the diffuse scattering (see Ref. ).[]{data-label="fig:11"}](fig11.ps){width="\linewidth"}
![(Color online) Sketch of the three-dimensional diffuse scattering distribution from single crystal PZN. They are plotted in the 3-D reciprocal space around (100), (110), (111), (010), and (011) reciprocal lattice points for (a) E=0, and (b) E along \[111\]. In (b), the diffuse rods along the \[110\], \[101\], and \[011\] directions are enhanced, while the diffuse rods along the \[1$\bar{1}$0\], \[10$\bar{1}$\], and \[01$\bar{1}$\] are suppressed (see Ref. ). []{data-label="fig:12"}](fig12.ps){width="0.6\linewidth"}
![(Color online) A schematic showing the PNR configurations in a relaxor system in (a) the paraelectric phase, (b) ZFC into the ferroelectric phase, and (c) FC into the ferroelectric phase. The large arrows indicate the polarization of the ferroelectric domains separated by domain walls (solid lines). The small squares represent the PNR (see Ref. ).[]{data-label="fig:13"}](fig13.ps){width="\linewidth"}
![(Color online) Neutron scattering measurements performed on a PZN-4.5PT single crystal. (a) A schematic diagram of the neutron scattering measurements, performed near the ($\bar{2}$20) and (220) Bragg peaks. The blue and red ellipsoids represent the FC diffuse scattering intensity distributions for E along \[111\]. The polarization and propagation vectors for the phonons are also noted. (b) Profile of the diffuse scattering intensity measured along (H 2.1 0) \[dashed line in (a)\] under ZFC and FC conditions. (c) Intensity contours measured near ($\bar{2}$20). (d) Intensity contours measured near (220) (see Ref. ).[]{data-label="fig:14"}](fig14.eps){width="\linewidth"}
![(Color online) Neutron diffuse scattering intensity contours measured from a single crystal PMN, at T below (left column) and above (right column) T$_d$ (see Ref. ).[]{data-label="fig:15"}](fig15.eps){width="0.8\linewidth"}
|
---
author:
- |
Slava Voloshynovskiy[^1]\
`svolos@unige.ch`\
Mouad Kondah$^*$\
`Mouad.Kondah@etu.unige.ch`\
Shideh Rezaeifar$^*$\
`Shideh.Rezaeifar@unige.ch`\
Olga Taran$^*$\
`Olga.Taran@unige.ch`\
Taras Holotyak$^*$\
`Taras.Holotyak@unige.ch`\
Danilo Jimenez Rezende[^2]\
`danilor@google.com`\
bibliography:
- 'egbib.bib'
title: Information bottleneck through variational glasses
---
Abstract
========
Information bottleneck (IB) principle [@tishby2015deep] has become an important element in information-theoretic analysis of deep models. Many state-of-the-art generative models of both Variational Autoencoder (VAE) [@KingmaVAE; @rezende2014stochastic] and Generative Adversarial Networks (GAN) [@goodfellow2014generative] families use various bounds on mutual information terms to introduce certain regularization constraints [@zhao2017infovae; @chen2016infogan; @zhao2018information; @alemi2018information; @alemi2017fixing; @alemi2016deep]. Accordingly, the main difference between these models consists in add regularization constraints and targeted objectives.
In this work, we will consider the IB framework for three classes of models that include supervised, unsupervised and adversarial generative models. We will apply a variational decomposition leading a common structure and allowing easily establish connections between these models and analyze underlying assumptions.
Based on these results, we focus our analysis on unsupervised setup and reconsider the VAE family. In particular, we present a new interpretation of VAE family based on the IB framework using a direct decomposition of mutual information terms and show some interesting connections to existing methods such as VAE [@KingmaVAE; @rezende2014stochastic], $\beta-$VAE [@higgins2017beta], AAE [@makhzani2015adversarial], InfoVAE [@zhao2017infovae] and VAE/GAN [@larsen2015autoencoding]. Instead of adding regularization constraints to an evidence lower bound (ELBO) [@KingmaVAE; @rezende2014stochastic], which itself is a lower bound, we show that many known methods can be considered as a product of variational decomposition of mutual information terms in the IB framework. The proposed decomposition might also contribute to the interpretability of generative models of both VAE and GAN families and create a new insights to a generative compression [@agustsson2018generative; @santurkar2018generative; @tschannen2018deep; @blau2019rethinking]. It can also be of interest for the analysis of novelty detection based on one-class classifiers [@pidhorskyi2018generative] with the IB based discriminators.
[**Notations**]{}: We will denote a joint generative distribution as $p_{\boldsymbol \theta}({\x},{\z}) =p_{\boldsymbol \theta}({\z}) p_{\boldsymbol \theta}({\x}|{\z})$, whereas marginal $p_{\boldsymbol \theta}({\z})$ is interpreted as a targeted distribution of latent space and marginal $p_{\boldsymbol \theta}({\x}) = \mathbb{E}_{p_{\boldsymbol \theta}({\z})} \left[ p_{\boldsymbol \theta}({\x}|{\z})\right] = \int_{\z} p_{\boldsymbol \theta}({\x}|{\z})p_{\boldsymbol \theta}({\z}) \mathrm{d}{\z}$ as a generated data distribution with a generative model described by $p_{\boldsymbol \theta}({\x}|{\z})$. A joint data distribution $q_{\boldsymbol \phi}({\x},{\z}) = \pdx q_{\boldsymbol \phi}({\z}|{\x})$, where $\pdx$ denotes an empirical data distribution and $q_{\boldsymbol \phi}({\z}|{\x})$ is an inference or encoding model and marginal $q_{\boldsymbol \phi}({\z})$ denotes a “true” or “aggregated” distribution of latent space data.
Information bottleneck for different models
===========================================
In this section, we consider the IB framework and summarize some known results for supervised and unsupervised models. Having introduced a common base, we will also extend these results to generative adversarial models. Along this analysis, we will introduce several interesting bounds that will be used to develop a proposed bounded IB auto-encoding.
Information bottleneck for supervised models {#IB_supervised}
--------------------------------------------
We consider a true joint distribution $p({\bf c},{\bf x})$ from which the training set $\{ {\bf x}_m, {\bf c}_m\}^{N}_{m=1}$ is sampled from, where each data sample is ${\bf x} \in \mathbb{R}^n$, $n$ denotes the dimensionality of data and $N$ stands for the number of training samples. We will use ${\bf c} \in \mathcal{M}$, with $ \mathcal{M} = \{1,\cdots,M_c\}$, to denote a class label. We use a vector notation for $\bf c$ to highlight that each label can be encoded according to some representation. The number of classes is denoted as $M_c$. The labeling of $N$ sequences into $M_c$ classes is shown in Figure \[Fig:labeling\],a. It should be noted that many sequences might be assigned to the same class according to a set of chosen common features. We use different colors to reflect this labeling. At the same time, one can consider a “binning” organization principle shown in the bottom part of Figure \[Fig:labeling\],a, where $N$ training sequences are allocated into $M_c$ bins representing $M_c$ classes.
The supervised IB framework is considered based on Figure \[Fig:models\],a. A sample $\bf x$ from a class $\bf c$ is generated by a mapping $p({\bf x}, {\bf c})=p({\bf c})p({\bf x}|{\bf c})$. The supervised IB can be formulated according to [@tishby2015deep] as: $$\label{eq:IB_supervised}
\min_{\boldsymbol \phi: I({\bf Z}; {\bf C}) \geq I_c} I_{\boldsymbol \phi}({\bf X};{\bf Z}).$$ The supervised IB framework assumes an existence of a parametrized probabilistic mapping $q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x})$ with a controllable set of parameters $\boldsymbol \phi$, where $\z$ is considered to be a latent or bottleneck representation with dimensionality and statistical properties different of those of $\x$. It is assumed that three concerned vectors form a Markov chain ${\bf C} \rightarrow {\bf X} \rightarrow {\bf Z}$ and the objective is to find such a mapping $\boldsymbol \phi$, when $\z$ is a minimal sufficient statistic for task $\bf c$. The term $I_{\boldsymbol \phi}({\bf X};{\bf Z})$ denotes the mutual information between $\X$ and $\Z$ considering the above parametric mapping and $I({\bf Z}; {\bf C})$ corresponds to the mutual information between $\Z$ and $\C$.
The main idea behind the supervised IB (\[eq:IB\_supervised\]) consists in a search of parameters $\boldsymbol \phi$ that ensures the preservation of the information $I_c$ about the class $\bf c$ in the latent or bottleneck representation $\bf z$, while filtering out all irrelevant information from $\bf x$ that corresponds to the minimisation of $ I_{\boldsymbol \phi}({\bf X};{\bf Z})$ over $\boldsymbol \phi$. It should be pointed out that the minimization of mutual information can be obtained in different ways that include but are not limited to dimensionality reduction, compression that might include both clustering and quantization, additional of noise or sparsification of $\z$. All these techniques are well known and often used in practical deep net mappers implementing $q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x})$.
Tishby [*et. al.* ]{} [@tishby2000information] also proposed the Langrangian of IB optimization (\[eq:IB\_supervised\]) defined as: $$\label{eq:IB_supervised_Langangian_Tishby}
\mathcal{L}^{\mathrm{S}}(\boldsymbol \phi) = I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) -\beta I({\bf Z}; {\bf C}),$$ where $\mathrm{S}$ stands for the supervised setup and $\beta$ is a regularization parameter corresponding to $I_c$ that leads to an optimization formulation: $$\label{eq:IB_supervised_Langangian_Tishby_solution}
{\boldsymbol {\hat \phi}} = {\operatornamewithlimits{argmin}}_{\boldsymbol \phi}
\mathcal{L}^{\mathrm{ S}}(\boldsymbol \phi).$$
In the following part, we will consider both terms of mutual information in (\[eq:IB\_supervised\_Langangian\_Tishby\]) and establish some useful bounds on them.
### Decomposition of the first term
The first mutual information term $I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) $ in (\[eq:IB\_supervised\_Langangian\_Tishby\]) is defined as:
$$\label{MI_encoder_proof_definition}
\begin{aligned}
I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) & = \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x}) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x} ) }{ q_{\boldsymbol \phi}(\mathbf{z})p_{\mathcal D}(\mathbf{x})} \right]
= \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x}) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x} ) }{ q_{\boldsymbol \phi}(\mathbf{z})} \right]
\\ & = H_{\boldsymbol \phi}({\bf Z}) - H_{\boldsymbol \phi}({\bf Z}|{\bf X}),
\end{aligned}$$
where $p_{\mathcal D}(\mathbf{x})$ denotes the data distribution and $H_{\boldsymbol \phi}({\bf Z}) = - \mathbb{E}_{ q_{\boldsymbol \phi}(\mathbf{z}) } \left[\log q_{\boldsymbol \phi}(\mathbf{z}) \right]$ denotes the entropy of distribution $q_{\boldsymbol \phi}(\mathbf{z}) = \mathbb{E}_{p_{\mathcal D}(\mathbf{x}) } \left[ q_{\boldsymbol \phi}(\mathbf{z}|{\bf x}) \right]$ and $ H_{\boldsymbol \phi}({\bf Z}|{\bf X}) = - \mathbb{E}_{ q_{\boldsymbol \phi}(\mathbf{z},{\bf x}) } \left[ \log q_{\boldsymbol \phi}(\mathbf{z}|{\bf x}) \right]$ denotes the conditional entropy defined by $q_{\boldsymbol \phi}(\mathbf{z}|{\bf x})$. In (\[MI\_encoder\_proof\_definition\]), we used the decomposition of the joint distribution $q_{\boldsymbol \phi}(\mathbf{z},{\bf x}) = q_{\boldsymbol \phi}(\mathbf{z}|{\bf x}) p_{\mathcal D}(\mathbf{x})$. At the moment, we will not address technical details of computing $q_{\boldsymbol \phi}(\mathbf{z})$ and focus on them along the unsupervised setup analysis.
### Decomposition of the second term
The second mutual information term $I({\bf Z}; {\bf C})$ in (\[eq:IB\_supervised\_Langangian\_Tishby\]) can be defined via $\pc_z$ as: $$\label{MI_2nd_unsupervised_def1}
I({\bf Z} ;{\bf C}) = \mathbb{E}_{\pcz} \left[ \log \frac{\pcz}{\pdc {p({\z})}} \right] =\mathbb{E}_{\pcz} \left[ \log \frac{\pc_z}{\pdc} \right].$$ We show in Appendix A, that this mutual information can be lower bounded by $I({\bf Z};{\bf C}) \geq I^{\mathrm{S}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf C})$, where: $$\begin{aligned}
\label{IB_S_lower_bound}
I^{\mathrm{S}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf C}) & \triangleq - \mathbb{E}_{p({\bf c})} \left[ \log p({\bf c}) \right] + \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z}) \right]\right]
\\ & = H({\bf C}) - H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf C}|{\bf Z}),
\end{aligned}$$ with $H({\bf C}) = - \mathbb{E}_{p({\bf c})} \left[ \log p({\bf c}) \right]$ and $H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf C}|{\bf Z}) = - \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z}) \right]\right]$.
Therefore, the corresponding IB Lagrangian is redefined as: $$\label{eq:IB_supervised_Langangian_bounded}
\mathcal{L}^{\mathrm{S}}(\boldsymbol \phi, \boldsymbol \theta) = I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) -\beta I^{\mathrm{S}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf C}),$$ that leads to the optimization problem: $$\label{eq:IB_supervised_optimization}
({\boldsymbol {\hat \theta},\boldsymbol {\hat \phi}}) = {\operatornamewithlimits{argmin}}_{\boldsymbol {\theta}, \boldsymbol \phi}
\mathcal{L}^{\mathrm{S}}(\boldsymbol \phi, \boldsymbol \theta).$$
[**Remark:**]{} since $H({\bf C})$ in (\[IB\_S\_lower\_bound\]) is constant and does not depend on the parameters $\boldsymbol \theta, \boldsymbol \phi$, the supervised IB Lagrangian (\[eq:IB\_supervised\_Langangian\_bounded\]) can be rewritten in yet another commonly know form of supervised IB: $$\label{eq:IB_supervised_Langangian_bounded_form2}
\mathcal{L}^{\mathrm{S}}(\boldsymbol \phi, \boldsymbol \theta) \propto I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) +\beta H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf C}|{\bf Z}).$$ In turns, it can be considered as finding a trade-off between the reduction of mutual information between $\X$ and $\Z$ according to the first term and the prediction accuracy of class $\c$ based on $\z$ according to the second term.
Information bottleneck for unsupervised models {#IB_unsupervised}
----------------------------------------------
In the case of unsupervised setup, the data samples are not labelled by the classes $\bf c$. We will consider a true data distribution $p_{\mathcal D}({\bf x})$ from which the training set $\{ {\bf x}_m \}^{N}_{m=1}$ is sampled from. The data samples can be considered as belonging to a common class with the same label ${\bf c} = 1$ as shown in Figure \[Fig:labeling\],b. Each sequence $\bf x$ is indexed by its proper index $m$. It means that the mapping between $m$ and $\bf x$ is unique $m \leftrightarrow {\bf x}$ in contrast to the supervised setup, where knowing $\bf c$ does not automatically imply that one knows a sample $\bf x$ but rather a set or bin to which it belongs to.
Alternatively, one can interpret the unsupervised setup as the supervised one with $M_c = N$ classes, i.e., when each class is represented by just one sequence as shown in Figure \[Fig:labeling\]b. Therefore, by the direct analogy with the supervised setup, one can replace each class $\bf c$ by its proper representative sequence $\bf x$ as depicted in Figure \[Fig:models\],b. Therefore, the generative process can be considered to start directly from $\bf x$ as shown by a gray circle.
Thus, the unsupervised IB can be considered as a “compression” of $\bf x$ to $\bf z$ via the parametrized mapping $\qzx$ leading to a bottleneck representation $\bf z$ yet preserving a certain level of information $I_x$ in $\bf z$ about $\bf x$. Accordingly, the unsupervised IB problem can be formulated as: $$\label{eq:IB_unsupervised}
\min_{\boldsymbol \phi: I({\bf Z}; {\bf X}) \geq I_x} I_{\boldsymbol \phi}({\bf X};{\bf Z}),$$ and in the Lagrangian formulation as a minimization of: $$\label{eq:IB_unsupervised_Langangian_Tishby}
\mathcal{L}^{\mathrm{U}}(\boldsymbol \phi) = I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) -\beta I({\bf Z}; {\bf X}),$$ where we use the same $\beta$ as for the supervised setup for the sake of simplicity and $\mathrm{U}$ denotes the unsupervised case.
In the following sections, we will consider decompositions of both mutual information terms.
### Decomposition of the first term
The first term $\IXZ_phi$ in (\[eq:IB\_unsupervised\_Langangian\_Tishby\]) can be defined similarly to the supervised case (\[MI\_encoder\_proof\_definition\]) using entropies. The conditional entropy $H_{\boldsymbol \phi}({\bf Z}|{\bf X})$ is computable, since $\qzx$ is defined. However, the entropy $H_{\boldsymbol \phi}({\bf Z}) = - \mathbb{E}_{ q_{\boldsymbol \phi}(\mathbf{z}) } \left[\log q_{\boldsymbol \phi}(\mathbf{z}) \right]$ requires computation of marginal distribution $q_{\boldsymbol \phi}(\mathbf{z}) = \mathbb{E}_{p_{\mathcal D}(\mathbf{x}) } \left[ q_{\boldsymbol \phi}(\mathbf{z}|{\bf x}) \right]$ that might be a computationally expensive task in practice. Therefore, we will proceed with a variational approximation of $\qz$ by a distribution $\pthetaz$[^3]: $$\label{First_term_VA_unsupervised}
\begin{aligned}
I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) & = \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x}) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{x}, \mathbf{z} )}{ q_{\boldsymbol \phi}(\mathbf{z})p_{\mathcal{D}}(\mathbf{x})} \right] =
\mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x}) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x} ) }{ q_{\boldsymbol \phi}(\mathbf{z})} \frac{\pthetaz}{\pthetaz} \right]
\\ & = \underbrace{\mathbb{E}_{p_{\mathcal D}(\mathbf{x})}\left[D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{X=x}) \| p_{\boldsymbol \theta}(\mathbf{z} )\right) \right]}_\text{A} -
\underbrace{ D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z})\right)}_\text{B},
\end{aligned}$$ where the term (A) denotes the KL-divergence $\mathbb{E}_{p_{\mathcal D}(\mathbf{x})}\left[D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{X=x}) \| p_{\boldsymbol \theta}(\mathbf{z} )\right) \right] = \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x}) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x} ) }{ p_{\boldsymbol \theta}( \mathbf{z} ) } \right]= \mathbb{E}_{p_{\mathcal D}(\mathbf{x})} \left[ \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x} ) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x} ) }{ p_{\boldsymbol \theta}( \mathbf{z} ) } \right] \right]$ and the term (B) denotes the KL-divergence $ D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z})\right) = \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} , \mathbf{x}) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} ) }{ p_{\boldsymbol \theta}( \mathbf{z} ) } \right]
= \mathbb{E}_{q_{\boldsymbol \phi}(\mathbf{z} ) } \left[ \log \frac{ q_{\boldsymbol \phi}(\mathbf{z} ) }{ p_{\boldsymbol \theta}( \mathbf{z} ) } \right]$.
### Decomposition of the second term
The second mutual information term $I({\bf Z}; {\bf X})$ in (\[eq:IB\_unsupervised\_Langangian\_Tishby\]) is defined as: $$\label{MI_2nd_unsupervised_def}
I({\bf Z} ;{\bf X}) = \mathbb{E}_{\pzx} \left[ \log \frac{\px_z}{\pdx} \right].$$
To find a variational approximation to the unknown $\px_z$, one can proceed in the same way as with the supervised model. However, one can also directly obtain a variational lower bound on $ I({\bf Z} ;{\bf X})$ by assuming $\bf c \equiv \bf x$ in (\[IB\_S\_lower\_bound\]). This leads to $I({\bf Z};{\bf X}) \geq I^{\mathrm{U}}_{\boldsymbol \theta, \phi}({\bf Z};{\bf X})$, where: $$\begin{aligned}
\label{IB_U_lower_bound}
I^{\mathrm{U}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf X}) & \triangleq
- \mathbb{E}_{\pdx} \left[ \log \pdx \right]
+ \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right]
\\ & = H_{\mathcal D}({\bf X}) - H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf X}|{\bf Z}),
\end{aligned}$$ with $H_{\mathcal D}({\bf X}) = - \mathbb{E}_{\pdx} \left[ \log \pdx \right]$ and $H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf X}|{\bf Z}) = - \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right]$.
Therefore, the corresponding IB Lagrangian is defined as: $$\label{eq:IB_unsupervised_Langangian_bounded}
\mathcal{L}^{\mathrm{U}}(\boldsymbol \phi, \boldsymbol \theta) = I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) -\beta I^{\mathrm{U}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf X}),$$ thus leading to the minimization problem: $$\label{eq:IB_unsupervised_optimization}
({\boldsymbol {\hat \theta},\boldsymbol {\hat \phi}}) = {\operatornamewithlimits{argmin}}_{\boldsymbol {\theta}, \boldsymbol \phi}
\mathcal{L}^{\mathrm{U}}(\boldsymbol \phi, \boldsymbol \theta).$$
In should be pointed out that similarly to the supervised case (\[eq:IB\_supervised\_Langangian\_bounded\_form2\]), the term $H_{\mathcal D}({\bf X})$ in (\[IB\_U\_lower\_bound\]) does not depend on the encoder and decoder parameters $\boldsymbol \phi, \boldsymbol \theta$ and can be skipped from the further consideration, if one is only concerned about the reconstruction task.
Nevertheless, the same model can also be considered for a generative task, which will also be considered below, when a trained encoder-decoder pair or just a sole decoder can be used for the generation of new samples from the latent space distribution. For these reasons, it is of interest to ensure that newly generated samples closely follow the statistics of original data. That is why one can also consider a decomposition of (\[IB\_U\_lower\_bound\]) as: $$\hspace{-0.185cm}
\begin{aligned}
\label{IB_U_lower_bound_KLD}
I^{\mathrm{U}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf X}) & = \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log \frac{\pthx_z}{\pdx} \right] \right]
\\ & = \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log \frac{\pthx_z}{\pdx} \frac{\pthetax}{\pthetax} \right] \right]
\\ & = - \mathbb{E}_{\pdx} \left[ \log \pthetax \right] - \mathbb{E}_{\pdx} \left[ \log \frac{\pdx}{\pthetax} \right] + \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right]
\\ & = H(\pdx; \pthetax) - D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) + \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right].
\end{aligned}$$ where $H(\pdx; \pthetax) = - \mathbb{E}_{\pdx} \left[ \log \pthetax \right] $ denotes a cross-entropy. Since $H(\pdx; \pthetax) \geq 0$, one can lower bound (\[IB\_U\_lower\_bound\_KLD\]) as $I^{\mathrm \bf {U}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf X}) \geq I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} )$, where[^4]:
$$\label{MI_decoder1}
\begin{aligned}
I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ) & \triangleq
\underbrace{ \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right]}_\text{C}
-
\underbrace{D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) }_\text{D}.
\end{aligned}$$
[**Remark:**]{} The term (D) in (\[MI\_decoder1\]) can be implemented based on the density ratio estimation [@GoodfellowGAN] that will be addressed below. The term (C) can be defined explicitly using Gaussian or Laplacian priors. In the Laplacian case, one can define $p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \propto \exp(-\lambda\| {\x} - g_{\boldsymbol \theta}({\z})\|_1)$ with a scale parameter $\lambda$, which leads to $\ell_1$-norm, and $g_{\boldsymbol \theta}({\z})$ denotes the decoder. It also corresponds to the model ${\x} = g_{\boldsymbol \theta}({\z}) + {\bf e}_x$, where ${\bf e}_x$ is a reconstruction error vector following the Laplacian pdf. Therefore, (\[MI\_decoder1\]) reduces to: $$\label{MI_decoder11}
\begin{aligned}
I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ) & =
\underbrace{ - \lambda \mathbb{E}_{\pdx} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})} \left[ \| {\x} - g_{\boldsymbol \theta}({\z})\|_1) \right]\right]}_\text{C}
-
\underbrace{D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) }_\text{D}.
\end{aligned}$$
### Comparison of supervised and unsupervised IB
Having considered the supervised and unsupervised IB formulations, it should be remarked several differences.
The main origin of these differences is in the entropy of classes $H({\bf C})$ and entropy of data $H_{\mathcal D}({\X})$, i.e., $H_{\mathcal D}({\X}) \gg H({\bf C})$. The supervised IB describing the classification task only needs to ensure that the latent space data $\Z$, representing the sufficient statistics for $\C$, should preserve just $\log_2(M_c)$ bits to uniquely encode and recognize each class. In the unsupervised setup, the IB suggests to compress $\X$ to the encoding representation $\Z$ such that each sequence $\X$ is uniquely decodable or identifiable from $\Z$. It means that the entropy of latent space should correspond to the entropy of observation space, i.e., it should encode at least $\log_2 (N)$ bits to uniquely distinguish all $N$ sequences, unless some tolerance is allowed in terms of reconstruction error[^5].
Naturally, this difference also leads to different encoding strategies. In the supervised setup, all common information within the same labeled class is “compressed” or disregarded and only the “differences” between the classes are encoded. With the increase of the number of classes, the differences might be minor that could be a potential source of vulnerability to adversarial attacks. An “informed” attacker knowing how these features are selected, that can be learned having an access to the same training data, might change only several of them to achieve a flipping between the classes. In contrast, the entropy of latent data for the unsupervised setup should be considerably higher than those for the supervised setup.
Finally, the nature of encoding is also different. In the unsupervised encoding, the classes are encoded to satisfy the reconstruction on average, i.e., the sequences close in the observation space might be close or even collude in the latent space, and the features of data contributing the most to the chosen metric of fidelity are preserved while less significant features are compressed or disregarded. As pointed above, all features that are irrelevant to a given classification task will be disregarded in the supervised setup. Using different re-labeling, new class-relevant features will be extracted while class irrelevant information will be filtered out. In the unsupervised case, there is no labeling and the encoding solely depends on statistics of data.
A link to generative adversarial models {#IB_generative}
---------------------------------------
The generative adversarial models can be considered as in Figure \[Fig:models\]c, i.e., the latent representation $\bf z$ of these models is not derived from the input of the network. Instead, it is assumed that the randomly assigned pairs $\{{\bf x}_m,{\bf z}_m\}_{m=1}^N$ are generated from $\pdx$ and $\pthetaz$.
Hence, the samples $\bf z$ are not produced by mapping $\pdx$ via $\qzx$ but directly from ${\z} \sim \pthetaz$ and thus the term $\IXZ_phi = 0$. Therefore, the unsupervised setup ([\[eq:IB\_unsupervised\_Langangian\_bounded\]]{}) reduces to the minimization of: $$\label{eq:IB_generative_Langangian_bounded}
\hat{\boldsymbol \theta} = \min_{\boldsymbol \theta} \mathcal{L}^{\mathrm{G}}( \boldsymbol \theta),$$ where $\mathcal{L}^{\mathrm{G}}( \boldsymbol \theta)= -\beta I^{\mathrm{G}}_{\boldsymbol \theta}({\bf Z};{\bf X})$ and: $$\begin{aligned}
\label{IB_G_def}
I^{\mathrm{G}}_{\boldsymbol \theta}({\bf Z};{\bf X}) & \triangleq \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \log \frac{\pthx_z}{\pdx} \right] \right],
\end{aligned}$$ corresponds $ I^{\mathrm{U}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf X})$ in ([\[eq:IB\_unsupervised\_Langangian\_bounded\]]{}) due to the fact that the sole link between $\Z$ and $\X$ is via $p_{\boldsymbol \theta}({\x}|{\z})$ and the latent vectors are generated from $\pthetaz$ and there is no dependence on $\boldsymbol \phi$.
Equivalently, the minimization problem (\[eq:IB\_generative\_Langangian\_bounded\]) can be reformulated as: $$\label{eq:IB_generative_Langangian_bounded_max}
\hat{\boldsymbol \theta} = \max_{\boldsymbol \theta} I^{\mathrm{G}}_{\boldsymbol \theta}({\bf Z};{\bf X}).$$
Accordingly, using the factorization with respect to the marginal distribution of generated data $\pthetax$ similarly to the unsupervised case (\[IB\_U\_lower\_bound\_KLD\]), one can define $I^{\mathrm{G}}_{\boldsymbol \theta}({\bf Z};{\bf X})$ as:
$$\begin{aligned}
\label{IB_G}
I^{\mathrm{G}}_{\boldsymbol \theta}({\bf Z};{\bf X}) & \triangleq \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \log \frac{\pthx_z}{\pdx} \right] \right]
\\ & = \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \log \frac{\pthx_z}{\pdx} \frac{\pthetax}{\pthetax} \right] \right]
\\ & = - \mathbb{E}_{\pdx} \left[ \log \pthetax \right] - \mathbb{E}_{\pdx} \left[ \log \frac{\pdx}{\pthetax} \right] + \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right]
\\ & = H(\pdx; \pthetax) - D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) + \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right].
\end{aligned}$$
Since $H(\pdx; \pthetax) \geq 0$, one can lower bound (\[IB\_G\]) as $I^{\mathrm{G}}_{\boldsymbol \theta}({\bf Z};{\bf X}) \geq I^{\mathrm{G}_L}_{\boldsymbol \theta}({\bf Z};{\bf X})$ where: $$\begin{aligned}
\label{IB_G:lower_bound}
I^{\mathrm{G}_L}_{\boldsymbol \theta}({\bf Z};{\bf X}) \triangleq - D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) + \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \log p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \right]\right].
\end{aligned}$$
Similarly to (\[MI\_decoder11\]), one can further develop (\[IB\_G:lower\_bound\]) using $ p_{\boldsymbol \theta}(\mathbf{x}| \mathbf{z}) \propto \exp(-\lambda\| {\x} - g_{\boldsymbol \theta}({\z})\|_1)$ with a scale parameter $\lambda$ that results in:
$$\begin{aligned}
\label{IB_G:lower_bound_final}
\hat{\boldsymbol \theta} = \max_{\boldsymbol \theta} I^{\mathrm{G}_L}_{\boldsymbol \theta}({\bf Z};{\bf X}) = \min_{\boldsymbol \theta} D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) + \lambda \mathbb{E}_{\pdx} \left[ \mathbb{E}_{\pthetaz} \left[ \| {\x} - g_{\boldsymbol \theta}({\z})\|_1) \right]\right].
\end{aligned}$$
[**Remark :**]{} Vanilla GANs use only an approximation to the first term for the generator optimization. However, GANs might face a mode collapse and the likelihood term can at least theoretically regularize it.
Bounded information bottleneck AE formulation
=============================================
Having considered the unsupervised and adversarial generative models, we can proceed with the formulation of a new auto-encoding framework. More particularly, we will use the results (\[First\_term\_VA\_unsupervised\]) and (\[MI\_decoder1\]) to propose a new type of unsupervised auto-encoder that combines the elements of VAE and GAN families and is built on the IB principle. We will refer to this auto-encoder as a [*bounded information bottleneck AE*]{} (BIB-AE) and link it to the VAE family of auto-encoders, generative compression and one-class classification. It should also be pointed out that the BIB-AE framework is rather considered as a conceptual generalization then as practical implementation. However, we will comment how to implement the BIB-AE components in practice using known techniques of KL-divergence approximation.
The BIB-AE Lagrangian is based on (\[eq:IB\_unsupervised\_Langangian\_bounded\]) and is defined as: $$\label{BIBN_unsupervised}
\mathcal{L}_{\mathrm{BIB-AE}}(\boldsymbol \theta, \boldsymbol \phi) = I_{\boldsymbol \phi}({\bf X} ; {\bf Z}) -\beta I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ),$$ where $I_{\boldsymbol \phi}({\bf X} ; {\bf Z})$ and $I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} )$ correspond to (\[First\_term\_VA\_unsupervised\]) and (\[MI\_decoder1\]) that we summarize below for the convenience of analysis: $$\label{MI_encoder_sum}
\begin{aligned}
I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) & = \underbrace{\mathbb{E}_{p_{\mathcal D}(\mathbf{x})}\left[D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{X=x}) \| p_{\boldsymbol \theta}(\mathbf{z} )\right) \right]}_\text{A} -
\underbrace{ D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z})\right)}_\text{B},
\end{aligned}$$ $$\label{MI_decoder_sum}
\begin{aligned}
I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ) & =
\underbrace{ \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right]}_\text{C}
-
\underbrace{D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) }_\text{D}.
\end{aligned}$$
The BIB-AE parameters are found according to the following minimization problem: $$\label{eq:BIB_optimization}
({\boldsymbol {\hat \theta},\boldsymbol {\hat \phi}}) = {\operatornamewithlimits{argmin}}_{\boldsymbol {\theta}, \boldsymbol \phi}
\mathcal{L}_{\mathrm{BIB-AE}}(\boldsymbol \theta, \boldsymbol \phi).$$
The diagram explaining the BIB-AE setup is shown in Figure \[Fig:set\_up\]. The reconstruction fidelity is ensured jointly by the terms (C) and (D), while the minimization of mutual information between $\X$ and $\Z$ is guided by the targeted distribution of the latent space $p_{\boldsymbol \theta}({\bf z})$ according to the terms (A) and (B). The “stochasticity” of the encoder will determine to which extend the mappings of data points from the observation space will “overlap” in the latent space yet satisfying the correspondence between the marginal posterior and the prior.
More particularly, as shown in Figure \[Fig:3\], the data distribution $\pdx$ is mapped to the latent space marginal distribution $\qz$ via the stochastic mapping $\qzx$. According to the variational approach, the targeted distribution of latent space is $\pthetaz$ and the encoder tries to optimize the parameters of encoder $\boldsymbol \phi$ according to (\[eq:BIB\_optimization\]) to meet both the constraints on the latent space and the reconstruction fidelity by satisfying the targeted $\beta I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ) $. One can imagine several forms of stochastic encoding: (i) ${\z} = f_{\boldsymbol \phi}({\x}) + {\boldsymbol \epsilon}$, where ${\boldsymbol \epsilon}$ follows the distribution defying the properties of conditional distribution $\qzx$, (ii) ${\z} = f_{\boldsymbol \phi}({\x} + {\boldsymbol \epsilon})$ or (iii) ${\z} = f_{\boldsymbol \phi}( [{\x}, {\boldsymbol \epsilon}])$. However, in practice depending on a chosen way of computing KL-divergence, one might be interested in a tractable density. In this case, the encoding of the first type is used as for example in the VAE family. Disregarding a particular form of randomness injecting mechanism, the green circles in the latent space of Figure \[Fig:3\] denote the resulting stochastic mappings of each point from the observable space.
Connections to the prior art AEs
================================
Generative models of VAE family
-------------------------------
Lagrangian is defined as: $$\label{VAE}
\mathcal{L}_{\mathrm{VAE}}(\boldsymbol \theta, \boldsymbol \phi) = \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[ D_{\mathrm{KL}}(q_{\boldsymbol \phi}(\mathbf{z}|{\bf X= x} ) \| p_{\boldsymbol \theta}(\mathbf{z}))\right] - \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right],$$ and contains only 2 terms (A) and (C) in (\[BIBN\_unsupervised\]) with $ \beta =1$. It can be shown that the VAE is based on an upper bound on $ I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) \leq I^{U}_{\boldsymbol \phi}({\bf X} ;{\bf Z}) = \mathbb{E}_{p_{\mathcal D}(\mathbf{x})}\left[D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{X=x}) \| p_{\boldsymbol \theta}(\mathbf{z} )\right) \right]$, since $
D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z})\right) \geq 0$. Similarly, since $D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right) \geq 0$, and denoting $I^{\text{VAE}} _{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ) = \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right]$, one obtains $ I^{\text{VAE}} _{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} ) \geq I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} )$.
The VAE encoder can be considered as a stochastic mapping with a particular form of parametrization [@KingmaVAE] ${\z} = {\boldsymbol \mu}({\bf x}) + {\boldsymbol \sigma}({\bf x})\odot{\boldsymbol \epsilon}$, where ${\boldsymbol \mu}({\bf x})$ and ${\boldsymbol \sigma}({\bf x})$ are outputs of the network $f_{\boldsymbol \phi} ({\bf x})$ and ${\boldsymbol \epsilon}$ is assumed to be a zero mean unit variance vector, i.e., ${\boldsymbol \epsilon} \sim \mathcal{N}({\bf 0},{\bf I})$, and $\odot$ denotes element wise product. As a result, the conditional distribution of $\Z$ given an input variable $\X$ follows a Gaussian distribution $q_{\boldsymbol \phi} ({\z}|{\x}) = \mathcal{N}({\boldsymbol \mu}({\bf x}), \text{diag}({\boldsymbol \sigma}({\bf x})))$. The VAE also assumes a prior on the latent space to be $p_{\boldsymbol \theta}({\z}) = \mathcal{N}({\bf 0},{\bf I})$. Under these conditions the KL-term (A) can be computed analytically.
It should be pointed out that the VAE encoder maps a point from the observation space into a probabilistic output of Gaussian cloud with mean ${\boldsymbol \mu}({\bf x})$ and “ellipsoid” orientation determined by the diagonal covariance matrix $\text{diag}({\boldsymbol \sigma}({\bf x}))$. This is schematically shown in a form of green ellipsoids for different samples ${\x}_i$, $i=1,\cdots, N$, in the latent space according to Figure \[Fig:4\]. Moreover, since the targeted marginal prior is $p_{\boldsymbol \theta}({\z}) = \mathcal{N}({\bf 0},{\bf I})$, and the KL term for all mappings of ${\x}$’s via $q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x})$ should match this $p_{\boldsymbol \theta}({\z})$ in $\mathbb{E}_{p_{\mathcal D}(\mathbf{x})}\left[D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{x}) \| p_{\boldsymbol \theta}(\mathbf{z} )\right) \right]$, the encoder optimized in such a way will target to make the mean of all mappings close to zero and whiten the ellipsoids.
Without a special guidance, these mappings will converge to the zero mean unit variance Gaussian marginal shape under asymptomatically many input mappings. Obviously, there is a little control on this process but the final goal of stochastic minimization of the upper bound on the mutual information $ I_{\boldsymbol \phi}({\bf X} ;{\bf Z})$ considered as a “compression” is achieved according to the IB framework.
[**$\beta$-VAE [@higgins2017beta]**]{} is linked to (\[BIBN\_unsupervised\]) in the same way as the VAE but with a varying relaxation parameter $\beta$: $$\label{b-VAE}
\mathcal{L}_{\beta-\mathrm{VAE}}(\boldsymbol \theta, \boldsymbol \phi) = \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[ D_{\mathrm{KL}}(q_{\boldsymbol \phi}(\mathbf{z}|{\bf X=x} ) \| p_{\boldsymbol \theta}(\mathbf{z}))\right] - \beta\mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right].$$
The main advantage of $\beta$-VAE over VAE is a possibility to relax the described above stochastic “compression” via mapping everything to a big Gaussian “heap” by applying the relaxation parameter $\beta$ that might give more preference to the reconstruction cost. By increasing $\beta$, one might achieve a sort of “disentangliation”, yet weakly controllable by one global parameter, by allowing Gaussian clouds in the latent space to be far away from each other by less satisfying the KL-term constraint to fit the marginally Gaussian distribution. The semantically similar inputs might be mapped closer thus creating a sort of clusters that might be interpreted as a disentangled representation. Surely, it is only an interpretation of such a relaxed stochastic mapping and the process of “semantic clustering” highly depends on statistics of data. It seems to be quite difficult to achieve a semantically meaningful encoding and interepretability of the latent space without either at least some weak supervision or specially constructed latent space.
[**AAE [@makhzani2015adversarial]**]{} can be defined according to the equivalent Lagrangian cost: $$\label{AAE}
\mathcal{L}_{\mathrm{AAE}}(\boldsymbol \theta, \boldsymbol \phi) = D_{\mathrm{KL}}(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z})) - \beta \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right],$$ where we do not explicitly consider the technical details of KL-divergence approximation and computation whereas one can use adversarial discriminator for this purpose or the maximum mean discrepancy (MMD) [@gretton2012kernel] based discriminator.
It should be pointed out that (\[AAE\]) contains the term (C) which origin can be explained in the same way as for the VAE. Despite of the fact that the term (B) indeed appears in (\[AAE\]) with the opposite sign, it cannot be interpreted either as an upper bound on $I_{\boldsymbol \phi}({\bf X} ;{\bf Z})$ similarly to the VAE or as a lower bound. The goal of AAE is to minimize the reconstruction loss or to maximize the log-likelihood by ensuring that the latent space marginal distribution $q_{\boldsymbol \phi} ({\bf z}) $ matches the prior $p_{\boldsymbol \theta}({\z})$. The latter corresponds to the minimization of $D_{\mathrm{KL}}\left(q_{\boldsymbol \phi} ({\bf z}) \| p_{\boldsymbol \theta}(\mathbf{z})\right)$.
It is interesting to point out that the original AAE paper considers as a potential encoding all options that include: a [*deterministic encoding*]{}, i.e., ${\bf z} = f_{\boldsymbol \phi}({\x})$, as well as the considered in section 3 [*stochastic encodings*]{}. A nice flexibility of AAE comes from a possibility to match the observed marginal distribution $\qz$ to a desired targeted distribution $
p_{\boldsymbol \theta}({\z})$ without the need to have explicitly defined distributions in contrast to the VAE.
An actual implementation of AAE is based on the deterministic encoding. We can imagine this sort of mapping by considering Figure \[Fig:4\]. A point of the observation space is mapped just to one point in the latent space. Under the deterministic encoder the mutual information $I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) = H_{\boldsymbol \phi}({\bf Z})$ since $ H_{\boldsymbol \phi}({\bf Z}|{\bf X}) = 0$[^6].
That is why the ability to compress the observation space to the latent space or to generate from the latent space comes from the relationship between the entropy of observation space distribution $\pdx$ and targeted latent space distribution $p_{\boldsymbol \theta}({\z})$. If the entropy of the observation space is large, i.e., the data are on a complex distributed manifold with a large variance, and the latent space is characterized by a small variance, many samples from the observation space will be mapped very closely to meet the KL-term constraint on the marginal latent space distribution. Naturally, it is a form of “deterministic” compression leading to the reduction of entropy by a “collusion” of many samples from the observation space in the latent space. It should be noticed that in this case, the centroids typically used in quantization based compression are not even used. At the same time, the “continuity” of latent space filling is determined by the randomness of $\pdx$ with respect to $p_{\boldsymbol \theta}({\z})$. If for some reason $p_{\boldsymbol \theta}({\z})$ is chosen to be relatively “broad”, it is not excluded that one might observe some “holes” in the latent space as a result of such a mapping.
Nevertheless, as shown in Figure \[Fig:3\], one can impose any constraint on $p_{\boldsymbol \theta}({\z})$ like Gaussian, Laplacian or even sparsifying prior. Moreover, one can predefine some centroids or clusters and target that the closest samples in the observation space to be mapped into the same centroids. In this sense, the AAE can also implement a form of deterministic compression by clustering.
At the same time, one can relax the quantization requirement to map an input to exactly one closest centroid and instead to envision some relaxation within the allowed KL-term. These options are not directly implemented in the AAE but can be envisioned. We mention and consider them in view of a link to InfoVAE and generative compression that will be addressed in the next section.
[**InfoVAE [@zhao2017infovae]**]{} consists of 3 terms obtained by adding the regularisation term $ I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) $ to an alternative form of the VAE. Since this original way of deriving InfoVAE is not straightforward and does not naturally comes from the IB framework, we will show that the InfoVAE has its BIB-AE counterpart with the terms (A), (B) and (C) and can be defined according to the Lagrangian[^7]: $$\begin{aligned}
\label{InfoVAE}
\mathcal{L}_{\mathrm{InfoVAE}}(\boldsymbol \theta, \boldsymbol \phi) & =\mathbb{E}_{p_{\mathcal D}(\mathbf{x})}\left[D_{\mathrm{KL}}\left(q_{\boldsymbol \phi}(\mathbf{z} | \mathbf{X=x}) \| p_{\boldsymbol \theta}(\mathbf{z} )\right) \right] - D_{\mathrm{KL}}(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z}))
\\ & - \beta \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right].
\end{aligned}$$
In fact, the terms (A) and (B) correspond to $I_{\boldsymbol \phi}({\bf X} ;{\bf Z}) $ while the term (C) corresponds to $ I^{\text{VAE}} _{\boldsymbol \theta, \boldsymbol \phi}({\bf Z}; {\bf X} )$. Besides, it should also be pointed out that in the original paper [@zhao2017infovae] the above three terms have not been used jointly in the reported simulations. Instead, the original InfoVAE uses 2 terms depending on the VAE form, i.e., the terms (A) and (C), or the terms (B) and (C), i.e., the AAE form.
The InfoVAE can also be considered as yet another form of compression by the minimization of $I_{\boldsymbol \phi}({\bf X} ;{\bf Z})$. Since it contains both KL-terms (A) and (B) in $I_{\boldsymbol \phi}({\bf X} ;{\bf Z})$, the encoder can minimize $I_{\boldsymbol \phi}({\bf X} ;{\bf Z})$ by seeking an equality between the terms (A) and (B) since both terms are non-negative. One can consider the presence of term (B) with the regularization parameter $\beta$ as a regularization of VAE term (A). As a result, it will relax the condition to map all conditional distributions to one Gaussian heap how it is done in the VAE case.
Having considered all these connections, it should be pointed out that the interpretability of the latent space in all considered methods is a quite complex task unless special supervised constraints are imposed how it was finally suggested in a semi-supervised AAE framework. For this reason, we will also consider other possibilities of controllable latent space encoding and generation using generative compression. However, it should be noted that the initial goal of this type of encoding has different roots and requires the selection of optimal distribution to meet a rate-distortion trade-off.
[**GANs [@goodfellow2014generative]**]{}: not pretending to consider the whole GAN family, we can mention that the IB considered for the generative adversarial models in section \[IB\_generative\] makes it possible to link GAN with BIB-AE. Considering the generation from the targeted latent space distribution $p_{\boldsymbol \theta}({\z})$ via the generator $\pthx_z$ one uses (\[IB\_G:lower\_bound\]) that corresponds to the terms (D) and (C) in (\[MI\_decoder\_sum\]), respectively. Therefore, the BIB-AE is linked to GANs via the IB framework.
It should be remarked that the original GAN does not include the likelihood term (C). However, according to the BIB-AE analysis, this regularizer naturally follows from the IB framework. It is interesting to mention that Rosca [*et. al.*]{} [@rosca2017variational] have considered this option as a potential solution to the GAN mode collapse problem.
[**VAE/GAN [@larsen2015autoencoding]**]{}: an option to use jointly the VAE represented by term (A) and (C) and the GAN represented by term (D) was envisioned in VAE/GAN model. An equivalent VAE/GAN Lagrangian is formulated as: $$\begin{aligned}
\label{VAE_GAN}
\mathcal{L}_{\mathrm{VAE/GAN}}(\boldsymbol \theta, \boldsymbol \phi) & = \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[ D_{\mathrm{KL}}(q_{\boldsymbol \phi}(\mathbf{z}|{\bf X=x} ) \| p_{\boldsymbol \theta}(\mathbf{z}))\right] - \beta \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\mathbb{E}_{q_{\boldsymbol \phi}({\bf z} | {\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf z})\right]\right] \\ & + \beta D_{\mathrm{KL}}\left(p_{\mathcal D}(\mathbf{x} ) \| p_{\boldsymbol \theta}(\mathbf{x})\right).
\end{aligned}$$ In the original paper, the log-likelihood term was replaced by a special metric in the latent space[^8].
In conclusion, many existing variations of VAE and GAN families can be considered directly from the BIB-AE framework perspectives. The main difference between these approaches, where either the VAE based on ELBO or GAN are taken as a basis and then some regularization terms are added, and the proposed one is in a fact that we proceed directly with the IB formulation and impose the corresponding bounds on the mutual information components of the IB.
Extending the same methodology, next we consider a compression formulation of IB from the Shannon’s rate-distortion perspectives and link it with generative models.
Shannon’s rate-distortion and generative compression AEs
--------------------------------------------------------
In the previous analysis, the targeted latent space distribution was assumed to be any manifold specified by $p_{\boldsymbol \theta}({\bf z})$. However, if one wants additionally to have a latent space with a bounded rate below the entropy $H_{\boldsymbol \theta}({\Z}) = - \mathbb{E}_{p_{\boldsymbol \theta}({\bf z})}[\log p_{\boldsymbol \theta}({\bf z})]$, i.e., targeting some compression, yet providing the best reconstruction and possibly generation from the latent space samples, it is of interest to link the considered analysis to the Shannon’s rate-distortion theory.
Since the latent space of compression AE should be limited to some rate $R_Q$, we will assume that the latent space consists of a codebook $\mathcal{C} = \{ {\bf c}_1, {\bf c}_2, \cdots, {\bf c}_L \}$, containing the codewords ${\bf c}_i \in {\mathbb R}^{n_z}$ of dimension $n_z$ with probabilities $\{ p_j\}_{j=1}^L$ such that $R_Q=- \sum_{j=1}^Lp_j \log p_j$. The codewords of $\mathcal{C} $ can be considered as realizations or centroids generated from $p_{\boldsymbol \theta}(\bf z)$ that makes it conceptually similar to the AAE. This is conceptually shown in Figure \[Fig:3\] as “compressed” latent space.
At the same time, an essential simplification comes from the fact that the encoder is deterministic and maps the input to one of the above centroids. This can be achieved by a vector quantizer ${ \bf \hat z} = Q(f_{\boldsymbol \phi} ({\bf x})) := {\operatornamewithlimits{argmin}}_{1 \leq j \leq L} || f_{\boldsymbol \phi} ({\bf x)} - {\bf c}_j ||_2$, where $f_{\boldsymbol \phi} (\bf x)$ denotes a deterministic encoder and $Q(.)$ a vector quantizer (VQ). Hence, the distribution of the quantized latent space is $p_{\boldsymbol \theta}({\bf \hat z})= \sum_{j=1}^L p_j{\boldsymbol \delta}({ \bf \hat z} - {\bf c}_j)$ that defines the rate $R_Q$.
[**Shannon’s rate-distortion [@cover2012elements]**]{} can be expressed as a special case of (\[BIBN\_unsupervised\]) with (\[MI\_encoder\_sum\]) and (\[MI\_decoder\_sum\]): $$\label{Shannon}
\mathcal{L}_{\mathrm{Shannon-AE}}(\boldsymbol \theta, \boldsymbol \phi) = I^Q_{\boldsymbol \phi}({\bf X} ; {\bf \hat Z}) -\beta I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf \hat Z}; {\bf X} ).$$ It is easy to show that $I_{\boldsymbol \phi}({\bf X} ; {\bf \hat Z}) = I^Q_{\boldsymbol \phi}({\bf X} ; {\bf \hat Z}) = H_{\boldsymbol \phi}({\bf \hat Z})$ due to the deterministic encoding with quantization, while $ I^{\mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf \hat Z}; {\bf X} )$ is reduced to the term (C) that under the deterministic decoding further reduces to $ \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\log p_{\boldsymbol \theta}({\bf x} | {\bf \hat z})\right]$. This term corresponds to the reconstruction distortion that is often expressed as the $\ell_2$-norm that in turns corresponds to the Shannon’s lower bound on rate-distortion function. Therefore, the classical compression schemes satisfy the trade-off between the rate $I^Q_{\boldsymbol \phi}({\bf X} ; {\bf \hat Z}) = R_Q$ and distortion $ \mathbb{E}_{p_{\mathcal D}({\bf x})}\left[\log (- p_{\boldsymbol \theta}({\bf x} | {\bf \hat z}))\right] = D$. Finally, the latent space distribution $p_{\boldsymbol \theta}({\bf \hat z})$ is optimized to ensure the achievability of rate-distortion limit. This is a fundamental difference with the AAE, where the latent space distribution is chosen in advance for the technical reasons.
It is important to note that the Shannon’s rate distortion framework in the considered interpretation is closely linked with the AAE, when the targeted distribution latent space is represented by the compression codebook Figure \[Fig:4\]. The difference is in the practical implementation. The VQ implementation assumes a hard assignment of the input to one of centroids[^9], whereas the AAE proceeds with the optimization of the KL-term to fit the targeted latent space distribution. It means that some deviation from the centroids is still possible. However, in both cases to proceed with the generation from the latent compressed space, one needs to ensure a proper randomness. Otherwise, the space of reconstructed signals will correspond to the number of centroids in the latent space. For this reason, we will consider a generative compression and link it with the IB framework.
[**Generative compression [@agustsson2018generative; @santurkar2018generative; @tschannen2018deep; @blau2019rethinking]**]{} can be considered as an “extension” of Shannon’s rate distortion with the Lagrangian: $$\label{CG_AE}
\mathcal{L}_{\mathrm{GC-AE}}(\boldsymbol \theta, \boldsymbol \phi) = I^Q_{\boldsymbol \phi}({\bf X} ; {\bf \hat Z}) -\beta I^{ \mathrm{U}_L}_{\boldsymbol \theta, \boldsymbol \phi}({\bf \hat Z} + {\bf U}; {\bf X}).$$ The first term is the same as in the classical compression setup, while the second one contains a stochastic component achieved by the addition of the permutation ${\bf U} \sim p_{\bf u}({\bf u})$ to the centroids[^10]. At the same time, it contains both equivalent terms (C) and (D) in (\[MI\_decoder\_sum\]). In practice, the KL-divergence is lower bounded by $f$-divergence that is implemented in a form of adversarial loss based on a density ratio estimation [@nowozin2016; @mohamed2016learning] or its Wasserstein’s counterpart [@arjovsky2017wasserstein]. In the original generative compression papers, the origin of the $f$-divergence term interpreted as a perceptual loss was only explained from the heuristic point of view to make highly compressed fragments of images under a low compression rate to look more naturally but not necessarily to be close to the original fragments. However, we can trace the origin of this term as an outcome of the IB factorization.
Novelty detection AEs
---------------------
The novelty detection problem aims at detecting outliers with respect to some manifold represented by the training data set. It assumed that similarly to the unsupervised setup, the training set consisting of $N$ samples is given. One can use different techniques to measure the relevance of a test sample to the training set or even to train a one class classifier for this purpose.
Alternatively, one can consider a novelty detection problem from the position of unsupervised IB framework in the BIB-AE formulation. It is interesting to note that [@pidhorskyi2018generative] proposed the architecture similar to the BIB-AE presented in Figure \[Fig:set\_up\] and trained with the terms (B), (C) and (D) for the novelty detection. The AE trained in this way might use several metrics such as output of term (D) to detect outliers. This also corresponds to a one-class classification problem. Therefore, the mechanism of novelty detection can be seen from the perspective of using the BIB-AE architecture.
Conclusions
===========
In this paper, we considered the IB for several practical tasks covering supervised, unsupervised, generative adversarial, generative compressive and novelty detection models. We show that the IB for all these models reduces to four terms in the Lagrangian cost. We call this formulation as BIB-AE. This formulation is closely linked with many models ranging from the VAE to VAE/GAN.
Besides this remarkable similarity, we note that this connection is seen via the IB framework with application of variational approach to the decomposition of mutual information terms in contrast to the VAE family that is based on various attempts to regularize the ELBO. As a result, the interpretability of obtained results and connection between methods leads to different conclusions.
Along the same line, we consider the new framework of generative compression in a close link to the IB framework whereas the original works on the generative compression considered it from the “perceptual” perspectives by adding the regularizer similar to the ELBO.
Finally, we also show that the novelty detection problem in the recent interpretability of AE encoding with the adversarial loss can be linked to the BIB-AE interpretation. Altogether the performed analysis gives new insights on the connections between different problems and methods and creates an interesting basis for the interpretability of the latent space.
[**Acknowledgement**]{}
The research was supported by the SNF project No. 200021-182063. The authors are thankful to Behrooz Razeghi for his feedback and fruitful discussions.
[**Appendix A**]{}
In this part, we derive a lower bound on $I({\bf Z};{\bf C})$. According to the definition (\[MI\_2nd\_unsupervised\_def1\]), this mutual information can be further decomposed as: $$\begin{aligned}
\label{MI_2nd_unsupervised_def_copy}
I({\bf Z} ;{\bf C}) & = \mathbb{E}_{\pcz} \left[ \log \frac{\pc_z}{\pdc} \right]
\\ & = - \mathbb{E}_{p({\bf c})} \left[ \log p({\bf c})\right] + \mathbb{E}_{\pcz} \left[ \log p(\mathbf{c}| \mathbf{z})\right].
\end{aligned}$$ The first term of this decomposition corresponds to the entropy of classes $H({\bf C}) = - \mathbb{E}_{p({\bf c})} \left[ \log p({\bf c}) \right]$.
We consider the second term since the transition probability $p(\mathbf{c}| \mathbf{z})$ is unknown. At the same time, it can be written as: $$\begin{aligned}
\label{MI_2nd_unsupervised_cond_entropy1}
\mathbb{E}_{\pcz} \left[ \log p(\mathbf{c}| \mathbf{z})\right] & = \int_{\bf c}\int_{\bf z} p({\bf c,z}) \log p(\mathbf{c}| \mathbf{z}) \ \mathrm{d}{\bf c} \ \mathrm{d}{\bf z}.
\end{aligned}$$ The expectation is with respect to the joint distribution $\pcz$ that can also be defined via the marginalization $\pcz = \int_{\bf x} p({\bf c,z,x})d{\bf x} = \int_{\bf x} p({\bf c,x})\qzx \mathrm{d}{\bf x}$. Therefore, combing these results, one can obtain: $$\begin{aligned}
\label{MI_2nd_unsupervised_cond_entropy2}
\mathbb{E}_{\pcz} \left[ \log p(\mathbf{c}| \mathbf{z})\right] & = \int_{\bf c}\int_{\bf z}\int_{\bf x} p({\bf c,x})\qzx \log p(\mathbf{c}| \mathbf{z}) \ \mathrm{d}{\bf c} \ \mathrm{d}{\bf z} \ \mathrm{d}{\bf x}
\\ & = \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p(\mathbf{c}| \mathbf{z}) \right]\right].
\end{aligned}$$ To overcome the problem of unknown $p(\mathbf{c}| \mathbf{z})$, we will apply a variational distribution $p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z})$ parametrized via a set of parameters $\boldsymbol \theta$ to approximate $p(\mathbf{c}| \mathbf{z})$. This can be considered as a bypass network and formulated as: $$\begin{aligned}
\label{MI_2nd_unsupervised_cond_entropy3}
\mathbb{E}_{\pcz} \left[ \log p(\mathbf{c}| \mathbf{z})\right] & = \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p(\mathbf{c}| \mathbf{z}) \frac{ p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z})}{p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z})} \right]\right]
\\ & = \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z}) \right]\right]
+ \mathbb{E}_{p({\bf c},{\bf z} )} \left[ \log \frac{p(\mathbf{c}| \mathbf{z})}{p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z})} \right],
\end{aligned}$$ where in the second term we used the expectation defined in (\[MI\_2nd\_unsupervised\_cond\_entropy2\]).
At the same time, we can re-write $ p({\bf c},{\bf z} ) = p({\bf z})p( {\bf c}|{\bf z})$ that leads to[^11]: $$\begin{aligned}
\label{MI_2nd_unsupervised_cond_entropy4}
\mathbb{E}_{p({\bf c},{\bf z} )} \left[ \log \frac{p(\mathbf{c}| \mathbf{z})}{p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z})} \right] & = \mathbb{E}_{p({\bf z} )} \left[ \mathbb{E}_{p({\bf c}|{\bf z} )} \left[ \log \frac{p(\mathbf{c}| \mathbf{z})}{p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z})} \right]\right] \\ & = \mathbb{E}_{p({\bf z} )} \left[ D_{\mathrm{KL}}\left(p(\mathbf{c}|{\bf Z=z} ) \| p_{\boldsymbol \theta}(\mathbf{c}|{\bf Z=z})\right) \right] = D_{\mathrm{KL}}\left(p(\mathbf{c}|{\bf z} ) \| p_{\boldsymbol \theta}(\mathbf{c}|{\bf z})\right).
\end{aligned}$$
Since the KL-divergence $D_{\mathrm{KL}}\left(p(\mathbf{c}|{\bf z} ) \| p_{\boldsymbol \theta}(\mathbf{c}|{\bf z})\right) \geq 0$, we can lower bound (\[MI\_2nd\_unsupervised\_cond\_entropy3\]) as: $$\begin{aligned}
\label{MI_2nd_unsupervised_cond_entropy5}
\mathbb{E}_{\pcz} \left[ \log p(\mathbf{c}| \mathbf{z})\right] & \geq \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z}) \right]\right].
\end{aligned}$$
Therefore, the mutual information (\[MI\_2nd\_unsupervised\_def\_copy\]) can be lower bounded as $I({\bf Z};{\bf C}) \geq I^{\mathrm{S}}_{\boldsymbol \theta, \phi}({\bf Z};{\bf C})$, where we define a lower bound as: $$\begin{aligned}
\label{IB_S_lower_bound1}
I^{\mathrm{S}}_{\boldsymbol \theta, \boldsymbol \phi}({\bf Z};{\bf C}) & \triangleq H({\bf C}) + \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z}) \right]\right]
\\ & = H({\bf C}) - H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf C}|{\bf Z}),
\end{aligned}$$ where $H_{\boldsymbol \theta, \boldsymbol \phi} ({\bf C}|{\bf Z}) = - \mathbb{E}_{p({\bf c},{\bf x} )} \left[ \mathbb{E}_{q_{\boldsymbol \phi}({\bf z}|{\bf x})} \left[ \log p_{\boldsymbol \theta}(\mathbf{c}| \mathbf{z}) \right]\right]$.
[^1]: Department of Computer Science, University of Geneva, Carouge 1227, Switzerland
[^2]: DeepMind
[^3]: Technically, the same factorization can be applied to the supervised counterpart (\[MI\_encoder\_proof\_definition\]). However, since in practice it is rarely of interest to generate labels $\bf c$ from $\z$, we only consider it in the scope of unsupervised generative and compression models.
[^4]: The cross-entropy computation requires knowledge of model $\pthetax$, whereas the KL-divergence is based on the ratio of two distributions and can be computed without an explicit knowledge of distributions but only from the training samples. For this reason, we proceed further with the KL-term.
[^5]: The total number of samples in the training set is upper limited by $2^{nH(X)}$ under the i.i.d. assumption, whereas the training set is assumed to contain only $N$ sequences.
[^6]: One can use a variational decomposition $H_{\boldsymbol \phi}({\bf Z}) = - \mathbb{E}_{ q_{\boldsymbol \phi}(\mathbf{z}) } \left[\log q_{\boldsymbol \phi}(\mathbf{z})\frac{p_{\boldsymbol \theta}({\z})}{p_{\boldsymbol \theta}({\z})} \right] = H(q_{\boldsymbol \phi}(\mathbf{z});p_{\boldsymbol \theta}(\mathbf{z})) - D_{\mathrm{KL}}(q_{\boldsymbol \phi}(\mathbf{z} ) \| p_{\boldsymbol \theta}(\mathbf{z}))$. Thus, if one wants to reduce the entropy of latent space to the entropy $H_{\boldsymbol \theta}({\bf Z})$ of targeted distribution $ p_{\boldsymbol \theta}(\mathbf{z})$, one should ensure that the encoder targets $q_{\boldsymbol \phi} ({\bf z}) \rightarrow p_{\boldsymbol {\theta}}({\bf z})$ leading to $H(q_{\boldsymbol \phi} ({\bf z}); p_{\boldsymbol {\theta}}({\bf z})) \rightarrow H_{\boldsymbol \theta}({\bf Z})$. Therefore, the term (B) in the AAE follows from the minimization of $D_{\mathrm{KL}}\left(q_{\boldsymbol \phi} ({\bf z}) \| p_{\boldsymbol \theta}(\mathbf{z})\right)$.
[^7]: The original InvoVAE contains different multipliers in front of KL-terms.
[^8]: One can use both encoded-reconstructed samples and samples generated from $p_{\boldsymbol \theta}({\z})$ in the third term for the adversarial discrimination.
[^9]: The multiple assignments are also possible that is know as [*soft-encoding*]{}.
[^10]: We use another variable $\bf u$ for the randomization of centroids to reflect a fact that it is assigned to the decoder part in contrast to the randomization based on the encoder randomization using $\boldsymbol \epsilon$.
[^11]: One can note that $p({\bf z}) = \qz$.
|
---
abstract: 'The $m$-Tamari Lattices ${\mathcal{T}_{n}^{(m)}}$ were recently introduced by Bergeron and Pr[é]{}ville-Ratelle as posets on $m$-Dyck paths, and it was later shown by Bousquet-M[é]{}lou, Fusy and Pr[é]{}ville-Ratelle that these form intervals in the classical Tamari lattice. It follows from a theorem by Björner and Wachs and a basic property of EL-shellable posets, that the $m$-Tamari lattices are EL-shellable. In this article, we give a new proof of this property, completely in terms of $m$-Dyck paths. Moreover, we derive properties of the M[ö]{}bius function of ${\mathcal{T}_{n}^{(m)}}$, and we characterize the intervals of ${\mathcal{T}_{n}^{(m)}}$ according to their topological properties.'
address: 'Fak. für Mathematik, Universität Wien, Garnisongasse 3, 1090 Wien, Austria'
author:
- Henri Mühle
bibliography:
- '../../literature.bib'
title: 'On the EL-Shellability of the $m$-Tamari Lattices'
---
Introduction {#sec:introduction}
============
The classical Tamari lattices $\mathcal{T}_{n}$ as introduced in [@tamari62algebra], are a well-studied member of the large group of Catalan objects. Bergeron and Pr[é]{}ville-Ratelle [@bergeron11higher] have recently generalized this class of lattices to $m$-Tamari lattices ${\mathcal{T}_{n}^{(m)}}$ in order to calculate the graded Frobenius characteristic of the space of higher diagonal harmonics. Along with this generalization, they proposed a realization of these lattices via $m$-Dyck paths, analogously to the realization of the classical Tamari lattices via classical Dyck paths. In [@bjorner97shellable]\*[Theorem 9.2]{} it was shown that the classical Tamari lattices are EL-shellable, and it is the statement of [@bousquet11number]\*[Proposition 4]{} that ${\mathcal{T}_{n}^{(m)}}$ is an interval in $\mathcal{T}_{mn}$. Since intervals of EL-shellable posets are again EL-shellable (see [@bjorner80shellable]\*[Proposition 4.2]{}) the following theorem is immediate.
\[thm:shellability\_m\_tamari\] The $m$-Tamari lattices ${\mathcal{T}_{n}^{(m)}}$ are EL-shellable for every positive integer $m$ and $n$.
It is the purpose of this article to give a new, independent proof of this theorem, which works completely within the framework of $m$-Dyck paths. It shall be remarked, that another, independent proof of Theorem \[thm:shellability\_m\_tamari\] can be deduced from [@liu99left]. Liu gave a different EL-labeling of $\mathcal{T}_{n}$, which implies with [@mcnamara06poset]\*[Theorem 3]{} that $\mathcal{T}_{n}$ is left-modular. Since left-modularity is a property which is inherited to intervals, the $m$-Tamari lattices are left-modular as well, and thus admit a natural EL-labeling, as pointed out in [@liu99left].
We recall the construction of $m$-Tamari lattices, as well as the definition of EL-shellability of a poset in Section \[sec:preliminaries\]. In Section \[sec:m\_tamari\_shellability\] we give an EL-labeling of ${\mathcal{T}_{n}^{(m)}}$ by generalizing a construction from [@bjorner97shellable]. We conclude this article by giving some applications of Theorem \[thm:shellability\_m\_tamari\] in Section \[sec:applications\], and we characterize the intervals of ${\mathcal{T}_{n}^{(m)}}$ according to their topological properties.
Preliminaries {#sec:preliminaries}
=============
Generalized Tamari Lattices {#sec:tamari}
---------------------------
Let $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ be a sequence of integers which satisfies the conditions $$\begin{aligned}
\label{eq:mdyck_one}
& a_{1}\leq a_{2}\leq\cdots\leq a_{n}\quad\text{and}\\
\label{eq:mdyck_two}
& a_{i}\leq m(i-1),\quad 1\leq i\leq n,\end{aligned}$$ and denote by $\mathcal{D}_{n}^{(m)}$ the set of all these sequences.
The sequences defined before have the following combinatorial interpretation. An *$m$-Dyck path of height $n$* is a path from $(0,0)$ to $(mn,n)$ in $\mathbb{N}\times\mathbb{N}$ which stays above the line $x=my$, and which consists only of steps of the form $(0,1)$, so-called *right-steps*, or steps of the form $(1,0)$, so-called *up-steps*. Given a sequence $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ satisfying conditions and , we can associate an $m$-Dyck path to $\mathbf{a}$ in which the $i$-th up-step is followed by $a_{i+1}-a_{i}$ right-steps.
It is well-known (see for instance [@dvoretzky47problem]) that the number of $m$-Dyck paths of height $n$ is counted by the *Fuss-Catalan number* $$\begin{aligned}
C(m,n)=\frac{1}{mn+1}\binom{(m+1)n}{n},\end{aligned}$$ and thus it follows that $\bigl\lvert\mathcal{D}_{n}^{(m)}\bigr\rvert=C(m,n)$.
For every $i\in\{1,2,\ldots,n\}$ there is a unique subsequence $(a_{i},a_{i+1},\ldots,a_{k})$ of $\mathbf{a}$, called *primitive subsequence* that satisfies $$\begin{aligned}
& a_{j}-a_{i}<m(j-i),\quad i<j\leq k\quad\text{and}\\
& \text{either}\quad k=n\quad\text{or}\quad a_{k+1}-a_{i}\geq m(k+1-i).\end{aligned}$$ The dotted line in Figure \[fig:good\_example\] indicates that $(3,4,4)$ is the unique primitive subsequence at position $3$ in the $4$-Dyck path given there.
We construct a covering relation $\lessdot$ on $\mathcal{D}_{n}^{(m)}$ in the following way: let $\mathbf{a},\mathbf{a}'\in\mathcal{D}_{n}^{(m)}$ with $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$, and let $i\in\{1,2,\ldots,n-1\}$ with $a_{i}<a_{i+1}$. Define $$\begin{gathered}
\mathbf{a}\lessdot\mathbf{a}'\quad\text{if and only if}\\
\mathbf{a}' = (a_{1}\ldots,a_{i},a_{i+1}-1,a_{i+2}-1\ldots,a_{k}-1,a_{k+1},\ldots,a_{n}),\end{gathered}$$ where $(a_{i+1},a_{i+2},\ldots,a_{k})$ is the unique primitive subsequence of $\mathbf{a}$ at position $i+1$. Denote by $\leq$ the reflexive and transitive closure of $\lessdot$. Proposition 4 in [@bousquet11number] implies that the poset $\bigl(\mathcal{D}_{n}^{(m)},\leq\bigr)$ is a lattice, the *$m$-Tamari lattice* ${\mathcal{T}_{n}^{(m)}}$.
Figure \[fig:example\_m32\] shows the lattice of all $2$-Dyck paths of height $3$ and the associated lattice of sequences as defined in the previous paragraph.
; ; (3\*,1\*) node(p0); (2\*,2\*) node(p1); (1\*,3\*) node(p2); (3\*,3\*) node(p3); (2\*,4\*) node(p4); (6\*,4\*) node(p5); (3\*,5\*) node(p6); (5\*,5\*) node(p7); (7\*,5\*) node(p8); (4\*,6\*) node(p9); (6\*,6\*) node(p10); (5\*,7\*) node(p11); (p0) – (p1) – (p2) – (p4) – (p6) – (p9) – (p11); (p0) – (p5) – (p7) – (p10) – (p11); (p1) – (p3) – (p4); (p3) – (p7) – (p9); (p5) – (p8) – (p10); (8\*,4\*) node[$\longleftrightarrow$]{}; (11\*,1\*) node(v0)[024]{}; (10\*,2\*) node(v1)[014]{}; (9\*,3\*) node(v2)[004]{}; (11\*,3\*) node(v3)[013]{}; (10\*,4\*) node(v4)[003]{}; (14\*,4\*) node(v5)[023]{}; (11\*,5\*) node(v6)[002]{}; (13\*,5\*) node(v7)[012]{}; (15\*,5\*) node(v8)[022]{}; (12\*,6\*) node(v9)[001]{}; (14\*,6\*) node(v10)[011]{}; (13\*,7\*) node(v11)[000]{}; (v0) – (v1) – (v2) – (v4) – (v6) – (v9) – (v11); (v0) – (v5) – (v7) – (v10) – (v11); (v1) – (v3) – (v4); (v3) – (v7) – (v9); (v5) – (v8) – (v10);
EL-Shellability of Posets {#sec:shellability}
-------------------------
Initially, this property was introduced for graded posets, in order to create an order-theoretic tool to investigate shellability of posets [@bjorner80shellable]. Shellability of a poset implies several topological and order theoretical properties, for example Cohen-Macaulayness of the associated order complex. More implications of shellability can for instance be found in [@bjorner80shellable; @bjorner83lexicographically; @bjorner96shellable; @bjorner97shellable]. In [@bjorner96shellable], Björner and Wachs generalized EL-shellability to non-graded posets. This is the property of interest in the present article.
Let $(P,\leq)$ be a poset. We call a poset *bounded* if it has a unique and a unique maximal element, denoted by $\hat{0}$ and $\hat{1}$, respectively. Denote by $\mathcal{E}(P)$ the set of all covering relations $p\lessdot q$ in $P$. Hence, $\mathcal{E}(P)$ corresponds to the set of edges in the Hasse diagram of $P$. Consider a non-singleton interval $[x,y]$ in $P$ and a chain $c:x=p_{0}<p_{1}<\cdots<p_{s}=y$. A chain is called *maximal in $[x,y]$* if there are no $q\in P$ and no $i\in\{0,1,\ldots,s-1\}$ such that $p_{i}<q<p_{i+1}$. For some poset $\Lambda$, call a map $\lambda:\mathcal{E}(P)\rightarrow\Lambda$ an *edge-labeling of $P$* and let $\lambda(c)$ denote the sequence $\bigl(\lambda(p_{0},p_{1}),\lambda(p_{1},p_{2}),\ldots,\lambda(p_{s-1},p_{s})\bigr)$ of edge-labels of $c$ with respect to $\lambda$. The chain $c$ is called *rising* if $\lambda(c)$ is a strictly increasing sequence. Moreover, $c$ is called *lexicographically smaller* than another maximal chain $\tilde{c}$ in the same interval if $\lambda(c)<\lambda(\tilde{c})$ with respect to the lexicographic order on $\Lambda^{\ast}$, the set of words over the alphabet $\Lambda$. More precisely, the lexicographic order on $\Lambda^{\ast}$ is defined as $(p_{1},p_{2},\ldots,p_{s})\leq(q_{1},q_{2},\ldots,q_{t})$ if and only if either $$\begin{aligned}
p_{i}=q_{i}, &\quad\text{for}\;1\leq i\leq s\;\text{and}\;s\leq t\quad\text{or}\\
p_{i}<q_{i}, &\quad\text{for the least}\; i\;\text{such that}\;p_{i}\neq q_{i}.\end{aligned}$$ An edge-labeling of $P$ is called *EL-labeling* if for every interval $[x,y]$ in $P$ there exists a unique rising maximal chain in $[x,y]$, which is lexicographically first among all maximal chains in $[x,y]$. A bounded poset that admits an EL-labeling is called *EL-shellable*.
EL-Shellability of ${\mathcal{T}_{n}^{(m)}}$ {#sec:m_tamari_shellability}
============================================
Björner and Wachs have shown in [@bjorner97shellable]\*[Section 9]{} that the classical Tamari lattices are EL-shellable, and [@bousquet11number]\*[Proposition 4]{} and [@bjorner80shellable]\*[Proposition 4.2]{} imply the same for ${\mathcal{T}_{n}^{(m)}}$. We will reprove this property, using the edge-labeling $$\label{eq:labeling}\begin{aligned}
\lambda:\mathcal{E}\bigl({\mathcal{T}_{n}^{(m)}}\bigr) & \rightarrow\mathbb{N}\times\mathbb{N},\quad
(\mathbf{a},\mathbf{b}) & \mapsto (j,a_{j}),
\end{aligned}$$ where $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n}),\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})$, as well as $j=\min\bigl\{i\in\{1,2,\ldots,n\}\mid a_{i}\neq b_{i}\bigr\}$.
Figure \[fig:tam\_32\_labeled\] shows the Hasse diagram of ${\mathcal{T}_{3}^{(2)}}$ together with the edge-labeling defined in .
; ; (3\*,1\*) node(p0)[024]{}; (2\*,2\*) node(p1)[014]{}; (1\*,3\*) node(p2)[004]{}; (3\*,3\*) node(p3)[013]{}; (2\*,4\*) node(p4)[003]{}; (6\*,4\*) node(p5)[023]{}; (3\*,5\*) node(p6)[002]{}; (5\*,5\*) node(p7)[012]{}; (7\*,5\*) node(p8)[022]{}; (4\*,6\*) node(p9)[001]{}; (6\*,6\*) node(p10)[011]{}; (5\*,7\*) node(p11)[000]{}; (p0) – (p1) node at (2.33\*,1.33\*)[(2,2)]{}; (p0) – (p5) node at (3.67\*,1.33\*)[(3,4)]{}; (p1) – (p2) node at (1.33\*,2.33\*)[(2,1)]{}; (p1) – (p3) node at (2.67\*,2.33\*)[(3,4)]{}; (p2) – (p4) node at (1.33\*,3.67\*)[(3,4)]{}; (p3) – (p4) node at (2.33\*,3.33\*)[(2,1)]{}; (p3) – (p7) node at (3.67\*,3.33\*)[(3,3)]{}; (p4) – (p6) node at (2.33\*,4.67\*)[(3,3)]{}; (p5) – (p7) node at (5.33\*,4.33\*)[(2,2)]{}; (p5) – (p8) node at (6.67\*,4.33\*)[(3,3)]{}; (p6) – (p9) node at (3.33\*,5.67\*)[(3,2)]{}; (p7) – (p9) node at (4.33\*,5.33\*)[(2,1)]{}; (p7) – (p10) node at (5.67\*,5.33\*)[(3,2)]{}; (p8) – (p10) node at (6.67\*,5.67\*)[(2,2)]{}; (p9) – (p11) node at (4.33\*,6.67\*)[(3,1)]{}; (p10) – (p11) node at (5.67\*,6.67\*)[(2,1)]{};
The main result of this section is stated in the following theorem.
\[thm:shelling\_m\_tamari\] Let ${\mathcal{T}_{n}^{(m)}}$ be the $m$-Tamari lattice of order $n$. The edge-labeling given in is an EL-labeling of ${\mathcal{T}_{n}^{(m)}}$ with respect to the following order on $\mathbb{N}\times\mathbb{N}$ $$\begin{aligned}
\label{eq:ordering}
(i,a_{i})\leq(j,a_{j})\quad\text{if and only if}\quad
i<j,\quad\text{or}\quad i=j\;\text{and}\;a_{i}\geq a_{j}.
\end{aligned}$$ Moreover, there is at most one falling chain in each interval of ${\mathcal{T}_{n}^{(m)}}$.
We need to show that for every interval $[\mathbf{a},\mathbf{b}]$ in ${\mathcal{T}_{n}^{(m)}}$ there exists exactly one rising chain that is lexicographically first among all maximal chains in $[\mathbf{a},\mathbf{b}]$. Let $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ and $\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})$ and consider the set $D=\{j \mid a_{j}\neq b_{j}\}=\{j_{1},j_{2},\ldots,j_{s}\}$. Let the elements of $D$ be listed in increasing order, namely $j_{1}<j_{2}<\cdots<j_{s}$. Let $\mathbf{r}^{(0)}=\mathbf{a}$ and construct $\mathbf{r}^{(i+1)}$ from $\mathbf{r}^{(i)}$ by decreasing the values in the primitive subsequence at position $j_{k}$ of $\mathbf{r}^{(i)}$ by one, where $j_{k}$ is the smallest element of $D$ such that the $j_{k}$-th entry of $\mathbf{r}^{(i)}$ is larger than $b_{j_{k}}$. By the minimality of $j_{k}$ it is ensured that $\mathbf{r}^{(i+1)}\leq\mathbf{b}$. Since ${\mathcal{T}_{n}^{(m)}}$ is a finite lattice, we obtain $\mathbf{r}^{(t)}=\mathbf{b}$ after a finite number, say $t$, of steps. By construction, it is clear that the chain $$\label{eq:rising_chain}
\mathbf{a}=\mathbf{r}^{(0)}\lessdot\mathbf{r}^{(1)}\lessdot\cdots\lessdot\mathbf{r}^{(t)}
=\mathbf{b}$$ is rising in the interval $[\mathbf{a},\mathbf{b}]$.
Let $\mathbf{r}^{(i)}=(r_{1}^{(i)},r_{2}^{(i)},\ldots,r_{n}^{(i)})$. It is guaranteed by construction that $\lambda\bigl(\mathbf{r}^{(i)},\mathbf{r}^{(i+1)}\bigr)=\bigl(j,r_{j}^{(i)}\bigr)
=\min\left\{\lambda\bigl(\mathbf{r}^{(i)},\bar{\mathbf{r}}\bigr)
\mid\bigl(\mathbf{r}^{(i)},\bar{\mathbf{r}}\bigr)\in\mathcal{E}\bigl({\mathcal{T}_{n}^{(m)}}\bigr)\right\}$. Since the primitive subsequence of $\mathbf{r}^{(i)}$ at position $j$ is unique, any other covering relation yields a label $\bigl(j',r_{j'}^{(i)}\bigr)$, with $j<j'$. By following such a chain upwards, we will eventually encounter an edge $(\mathbf{s},\mathbf{t})$ such that $\lambda(\mathbf{s},\mathbf{t})=(j,s_{j})$, where $\mathbf{s}=(s_{1},s_{2},\ldots,s_{n})$. This creates a descent in such a chain. Hence, the chain in is the unique rising chain in $[\mathbf{a},\mathbf{b}]$ and is lexicographically first.
Now consider the set $D'=\{j\mid a_{j}\neq b_{j}\;\text{and}\;a_{j}\geq a_{j-1}+m\}=\{j_{1},j_{2},\ldots,j_{t}\}$. This means that $k\in D$ implies $a_{k}\neq b_{k}$ and there is no primitive subsequence of $\mathbf{a}$ at position $i<k$, which contains $a_{k}$. Similarly to the previous paragraph, we can see that if there exists a falling chain $\mathbf{a}^{(0)}<\mathbf{a}^{(1)}<\cdots<\mathbf{a}^{(t)}$ in $[\mathbf{a},\mathbf{b}]$, it must have the sequence of edge-labels $$\begin{aligned}
\bigl(j_{t},a_{j_{t}}\bigr),\bigl(j_{t-1},a_{j_{t-1}}\bigr),\ldots,\bigl(j_{1},a_{j_{1}}\bigr),
\end{aligned}$$ since each of the values $a_{j_{1}},a_{j_{2}},\ldots,a_{j_{t}}$ must be decreased along any maximal chain in $[\mathbf{a},\mathbf{b}]$ at least once. Hence, the given chain is the only possible falling chain.
This follows by definition from Theorem \[thm:shelling\_m\_tamari\].
Applications {#sec:applications}
============
According to [@bjorner96shellable], the EL-shellability of ${\mathcal{T}_{n}^{(m)}}$ has some consequences for the Möbius function of ${\mathcal{T}_{n}^{(m)}}$ and the structure of the topological space associated to the intervals of ${\mathcal{T}_{n}^{(m)}}$. For an introduction on how to associate a topological space to a poset, we refer to [@wachs07poset].
Let $[\mathbf{a},\mathbf{b}]$ be an interval of ${\mathcal{T}_{n}^{(m)}}$ and let $\mu$ the Möbius function of ${\mathcal{T}_{n}^{(m)}}$. Then, $\mu(\mathbf{a},\mathbf{b})\in\{-1,0,1\}$.
This is a consequence of [@bjorner97shellable]\*[Theorem 9.6]{}, [@bousquet11number]\*[Proposition 4.2]{} and [@bjorner96shellable]\*[Theorem 5.7]{}.
Every open interval in ${\mathcal{T}_{n}^{(m)}}$ has the homotopy type of either a point or a sphere.
This is a consequence of [@bjorner97shellable]\*[Theorem 9.6]{}, [@bousquet11number]\*[Proposition 4.2]{} and [@bjorner96shellable]\*[Theorem 5.9]{}.
We remark, that our Theorem \[thm:shelling\_m\_tamari\] also reproves these results. We can specify the previous results even more and characterize the spherical intervals, analogously to [@bjorner97shellable]\*[Theorem 9.3]{}. For that purpose, consider $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})\in{\mathcal{T}_{n}^{(m)}}$ as well as $D\subseteq\{2,3,\ldots,n\}$ such that $a_{j}>a_{j-1}$ for all $j\in D$. Define the numbers $$\begin{gathered}
\label{eq:counter}
{\text{ps}_{\mathbf{a}}(j)}=\bigl\lvert\{i\in D\mid i<j\;\text{and}\;
a_{j}-1-a_{i}<m(j-i)\;\text{and}\\
\;a_{k}-a_{i}<m(k-i)\;\text{for all}\;i<k<j\}\bigr\rvert\end{gathered}$$ for all $j\in D$. Consider the sequence $\mathbf{a}_{j}^{\uparrow}\in{\mathcal{T}_{n}^{(m)}}$ that arises from $\mathbf{a}$ by decreasing the primitive subsequence of $\mathbf{a}$ at position $j\in D$ by one. Clearly, $(\mathbf{a},\mathbf{a}_{j}^{\uparrow})\in\mathcal{E}\bigl({\mathcal{T}_{n}^{(m)}}\bigr)$ and ${\text{ps}_{\mathbf{a}}(j)}$ counts the primitive subsequences of $\mathbf{a}{j}^{\uparrow}$ at some $i\in D$ with $i<j$ that contain the $j$-th entry of $\mathbf{a}_{j}^{\uparrow}$.
\[thm:topology\_intervals\] Let $\mathbf{a}\leq\mathbf{b}$ in ${\mathcal{T}_{n}^{(m)}}$, where $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ and $\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})$. Let $D=\{j\mid a_{j}\neq b_{j}\;\text{and}\;a_{j}>a_{j-1}\}$. The open interval $(\mathbf{a},\mathbf{b})$ has the homotopy type of a $(\lvert D\rvert-2)$-sphere if $$\begin{aligned}
\label{eq:sph_1}
a_{j}-1-a_{j-1}<m & \quad\mbox{implies}\quad b_{j}-b_{j-1}<m,
\quad\mbox{and}\\
\label{eq:sph_2}
&\;b_{j}=a_{j}-1-{\text{ps}_{\mathbf{a}}(j)},
\end{aligned}$$ for all $j\in D$. Otherwise, $(\mathbf{a},\mathbf{b})$ has the homotopy type of a point.
Let $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ and $\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})$ such that $\mathbf{a}\leq\mathbf{b}$. We need to show that a falling chain exists in $[\mathbf{a},\mathbf{b}]$ if and only if the conditions and are satisfied.
Write the set $D$ in the form $D=\{j_{1},j_{2},\ldots,j_{s}\}$, where $j_{1}<j_{2}<\cdots<j_{s}$. Let $\mathbf{r}^{(0)}=\mathbf{a}$ and construct $\mathbf{r}^{(i+1)}$ from $\mathbf{r}^{(i)}$ by decreasing the primitive subsequence at position $j_{s-i}$ of $\mathbf{r}^{(i)}$ by one. It is clear that the chain $\mathbf{r}^{(0)}<\mathbf{r}^{(1)}<\cdots<\mathbf{r}^{(s)}$ is falling. We also notice that ${\text{ps}_{\mathbf{r}^{(i)}}(j_{k})}={\text{ps}_{\mathbf{a}}(j_{k})}$ if $k<s-i$.
First we show that is equivalent to $\mathbf{r}^{(k)}\leq\mathbf{b}$ for all $k\in\{1,2,\ldots,s\}$. Assume that there exists a $k\in\{1,2,\ldots,s\}$ such that $a_{j_{k}}-1-a_{j_{k}-1}<m$ and $b_{j_{k}}-b_{j_{k}-1}\geq m$ and $j_{k}$ is maximal in $D$ with respect to this property. Consider the element $\tilde{\mathbf{a}}^{(0)}=
\bigl(\tilde{a}_{1}^{(0)},\tilde{a}_{2}^{(0)},\ldots,\tilde{a}_{n}^{(0)}\bigr)\in
{\mathcal{T}_{n}^{(m)}}$ that arises from $\mathbf{r}^{(s-k)}$ by decreasing the primitive subsequence of $\mathbf{r}^{(s-k)}$ at position $j_{k}$ by one. Thus, $\tilde{\mathbf{a}}^{(0)}=\mathbf{r}^{(s-k+1)}$. Construct elements $\tilde{\mathbf{a}}^{(i+1)}$ from $\tilde{\mathbf{a}}^{(i)}$ by decreasing the value of the primitive subsequence of $\tilde{\mathbf{a}}^{(i)}$ at position $j_{k}-1$ by one. By assumption, we know that $\tilde{a}_{j_{k}}^{(i)}$ is contained in the primitive subsequence of $\tilde{\mathbf{a}}^{(i)}$ at position $j_{k}-1$. Hence, in each such step, we decrease the value of $\tilde{a}_{j_{k}-1}^{(i)}$ and $\tilde{a}_{j_{k}}^{(i)}$ (and possibly some subsequent entries). After a finite number, say $t$, of steps, we obtain an element $\tilde{\mathbf{a}}^{(t)}=(\tilde{a}_{1}^{(t)},\tilde{a}_{2}^{(t)},\ldots,\tilde{a}_{n}^{(t)})$, such that $\tilde{a}_{j_{k}-1}^{(t)}=b_{j_{k}-1}$. Since $b_{j_{k}}$ is not contained in the primitive subsequence of $\mathbf{b}$ at position $j_{k}-1$, we have $\tilde{a}_{j_{k}}^{(t)}<b_{j_{k}}$, and thus $\mathbf{b}\leq\tilde{\mathbf{a}}^{(t)}$. Certainly, there is an $u\in\{0,1,\ldots,t-1\}$ such that $\tilde{a}_{j_{k}}^{(u)}=b_{j_{k}}$, and hence $\tilde{a}_{j_{k}-1}^{(u)}>b_{j_{k}-1}$. This implies that $\tilde{a}_{j_{k}-1}^{(i)}>b_{j_{k}-1}$ for all $i\in\{0,1,\ldots,u\}$, and we can conclude that $\tilde{\mathbf{a}}^{(i)}\not\leq\mathbf{b}$. Thus, $\mathbf{r}^{(s-k+1)}\not\leq\mathbf{b}$. The reverse implication is trivial.
Now we show that is equivalent to the fact that $(\mathbf{r}^{(k-1)},\mathbf{r}^{(k)})\in\mathcal{E}\bigl({\mathcal{T}_{n}^{(m)}}\bigr)$ for all $k\in\{1,2,\ldots,s\}$. The number ${\text{ps}_{\mathbf{a}}(j_{k})}$ corresponds to the number of primitive subsequences of $\mathbf{r}^{(1)}=\bigl(r_{1}^{(1)},r_{2}^{(1)},\ldots,r_{n}^{(1)}\bigr)$ at position $i\in D$, where $i<j_{k}$ that contain $r_{j_{k}}^{(1)}$. Hence, along the chain $\mathbf{r}^{(1)}<\mathbf{r}^{(2)}<\cdots<\mathbf{r}^{(s)}$, the value of $r_{j_{k}}^{(1)}$ is decreased exactly ${\text{ps}_{\mathbf{a}}(j_{k})}$-times and hence $b_{j_{k}}=r_{j_{k}}^{(1)}-{\text{ps}_{\mathbf{a}}(j_{k})}$. By constructing $\mathbf{r}^{(1)}$, we obtain $r_{j_{k}}^{(1)}=a_{j_{k}}-1$ which implies the claim.
By combining both properties, we obtain that $\mathbf{r}^{(0)}<\mathbf{r}^{(1)}<\cdots<\mathbf{r}^{(s)}$ is indeed a falling maximal chain from $\mathbf{a}$ to $\mathbf{b}$.
Let $\mathbf{a}\leq\mathbf{b}$ in ${\mathcal{T}_{n}^{(m)}}$, where $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ and $\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})$. Let $D=\{j\mid a_{j}\neq b_{j}\;\text{and}\;a_{j}>a_{j-1}\}$ and let $\mu$ denote the Möbius function of ${\mathcal{T}_{n}^{(m)}}$. Then, $$\begin{aligned}
\mu(\mathbf{a},\mathbf{b})=\begin{cases}
(-1)^{\lvert D\rvert}, &\;\text{if conditions}\;
\eqref{eq:sph_1}\;\text{and}\;
\eqref{eq:sph_2}\;\text{hold,}\\
0, &\;\text{otherwise}.
\end{cases}
\end{aligned}$$
\[cor:mobius\_zero\] Let $\mathbf{a}\in{\mathcal{T}_{n}^{(m)}}$, where $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$. Let $D=\{j\mid a_{j}\neq (j-1)m\}$. Let $D_{j}=\{i\in D\mid i<j\}$, and let $\mu$ denote the Möbius function of ${\mathcal{T}_{n}^{(m)}}$. Then, $$\begin{aligned}
\mu(\hat{0},\mathbf{a})=\begin{cases}
(-1)^{\lvert D\rvert}, &\text{if}\;a_{j}=(j-1)m-1-\lvert D_{j}\rvert\;
\text{for all}\;j\in D,\\
0, &\text{otherwise}.
\end{cases}
\end{aligned}$$
By construction, $\hat{0}=(0,m,2m,\ldots,(n-1)m)$. Hence, the premise in condition corresponds to $(j-1)m-1-(j-2)m=m-1<m$ and is always satisfied. Moreover, we have $$\begin{aligned}
{\text{ps}_{\hat{0}}(j)}=\lvert\{i\in D\mid i<j\}\rvert=\lvert D_{j}\rvert,
\end{aligned}$$ for all $j\in D$. If $\mathbf{a}$ satisfies , then $a_{j}=(j-1)m-1-{\text{ps}_{\hat{0}}(j)}$. There are two possibilities: either $j-1\in D$, or $j-1\notin D$.
Consider the case that $j-1\in D$. Then, $$\begin{aligned}
a_{j}-a_{j-1} & =(j-1)m-1-{\text{ps}_{\hat{0}}(j)}-(j-2)m+1+{\text{ps}_{\hat{0}}(j-1)}\\
& =m-{\text{ps}_{\hat{0}}(j)}+{\text{ps}_{\hat{0}}(j-1)}\\
& =m-\lvert D_{j}\rvert+\lvert D_{j-1}\rvert,
\end{aligned}$$ which yields $\lvert D_{j}\rvert-\lvert D_{j-1}\rvert>0$ for the conclusion of . Since $j-1\in D$, we know that $D_{j-1}\subsetneq D_{j}$ and the conclusion of is immediately satisfied if $a_{j}=(j-1)m-1-\lvert D_{j}\rvert$ for all $j\in D$.
Now let $j-1\notin D$. Then, $a_{j-1}=(j-2)m$, and we have $$\begin{aligned}
a_{j}-a_{j-1} & =(j-1)m-1-{\text{ps}_{\hat{0}}(j)}-(j-2)m\\
& =m-1-{\text{ps}_{\hat{0}}(j)}\\
& =m-1-\lvert D_{j}\rvert,
\end{aligned}$$ which yields $\lvert D_{j}\rvert+1>0$ for the conclusion of . Since $\lvert D_{j}\rvert\geq 0$, the conclusion of is immediately satisfied if $a_{j}=(j-1)m-1-\lvert D_{j}\rvert$ for all $j\in D$, which completes the proof.
\[cor:mobius\_one\] Let $\mathbf{a}\in{\mathcal{T}_{n}^{(m)}}$, where $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$. Let $D=\{j\mid a_{j}>a_{j-1}\}$ and let $\mu$ denote the Möbius function of ${\mathcal{T}_{n}^{(m)}}$. Then $$\begin{aligned}
\mu(\mathbf{a},\hat{1})=\begin{cases}
(-1)^{\lvert D\rvert}, &\;\text{if}\;
a_{j}={\text{ps}_{\mathbf{a}}(j)}+1\;
\text{for all}\;j\in D\\
0, &\;\text{otherwise}.
\end{cases}
\end{aligned}$$
It follows from the definition of ${\mathcal{T}_{n}^{(m)}}$ that $\hat{1}=(0,0,\ldots,0)$. Hence, for an interval $[\mathbf{a},\hat{1}]$ in ${\mathcal{T}_{n}^{(m)}}$ condition is trivially true for all $j\in D$ and condition reduces to $a_{j}={\text{ps}_{\mathbf{a}}(j)}+1$ for all $j\in D$.
In the remainder of this section, we prove that the number of spherical intervals of ${\mathcal{T}_{n}^{(m)}}$ involving $\hat{0}$ or $\hat{1}$ is $2^{n-1}$ respectively.
\[prop:number\_spherical\_intervals\] Let $m,n\in\mathbb{N}$. Let $\mathcal{S}_{n}^{(m)}(\hat{0})=\{\mathbf{a}\in{\mathcal{T}_{n}^{(m)}}\mid \mu(\hat{0},\mathbf{a})\neq 0\}$ and $\mathcal{S}_{n}^{(m)}(\hat{1})=\{\mathbf{a}\in{\mathcal{T}_{n}^{(m)}}\mid \mu(\mathbf{a},\hat{1})\neq 0\}$. Then, $$\begin{aligned}
\label{eq:number_sphericals}
\lvert\mathcal{S}_{n}^{(m)}(\hat{0})\rvert=2^{n-1}
=\lvert\mathcal{S}_{n}^{(m)}(\hat{1})\rvert.
\end{aligned}$$
Consider $\mathbf{a},\mathbf{b}\in{\mathcal{T}_{n}^{(m)}}$, with associated sequences $\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})$ and $\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})$. Define $$\begin{aligned}
D(\mathbf{a},\mathbf{b})=\bigl\{i\in \{2,\ldots,n\}\mid a_{i}\neq b_{i}\bigr\}.\end{aligned}$$ Now, for $\mathbf{a}\in{\mathcal{T}_{n}^{(m)}}$ and $D\subseteq\{2,3,\ldots,n\}$, and define $$\begin{aligned}
\text{diff}_{D}(\mathbf{a})=\{\mathbf{b}\in{\mathcal{T}_{n}^{(m)}}\mid D(\mathbf{a},\mathbf{b})=D\}.\end{aligned}$$ It is immediately clear that for every $\mathbf{a}\in{\mathcal{T}_{n}^{(m)}}$ $$\begin{aligned}
{\mathcal{T}_{n}^{(m)}}=\bigcup_{D\subseteq\{2,3,\ldots,n\}}{\text{diff}_{D}(\mathbf{a})},\end{aligned}$$ and $\text{diff}_{D_{1}}(\mathbf{a})\cap\text{diff}_{D_{2}}(\mathbf{a})=\emptyset$ if and only if $D_{1}\neq D_{2}$.
\[lem:zero\_prep\] Let $D\subseteq\{2,3,\ldots,n\}$. Then, $\text{diff}_{D}(\hat{0})\neq\emptyset$.
Let $D\subseteq\{2,3,\ldots,n\}$. By construction, $\hat{0}=(0,m,2m,\ldots,(n-1)m)$. Consider the indicator function $$\begin{aligned}
\chi_{D}:\{2,3,\ldots,n\}\rightarrow\{0,1\},\quad i\mapsto\begin{cases}
1, & \text{if}\;i\in D,\\
0, & \text{otherwise}.\end{cases}
\end{aligned}$$ Since $m\geq 1$, it is clear that the sequence $\chi_{D}(\hat{0})=
\bigl(0,m-\chi_{D}(2),2m-\chi_{D}(3),\ldots,(n-1)m-\chi_{D}(n)\bigr)$ corresponds to an $m$-Dyck path again.
Let $[i,j]$ denote the interval $\{i,i+1,\ldots,j\}$ for $1\leq i\leq j\leq n$.
Let $i\in\{2,3,\ldots,n\}$. Then, $\text{diff}_{D}(\hat{1})\neq\emptyset$ if and only if $D=[i,n]$ or $D=\emptyset$.
By construction, $\hat{1}=(0,0,\ldots,0)$ is the sequence associated to $\hat{1}$. If $D\subseteq\{2,3,\dots,n\}$ and $D\neq[i,n]$ for some $2\leq i\leq n$, then any element in $\text{diff}_{D}(\hat{1})$ has to correspond to a sequence of the form $(0,b_{2},b_{3},\ldots,b_{n-1},0)$. By definition it is clear that this encodes an $m$-Dyck path only in the case $b_{2}=b_{3}=\cdots=b_{n-1}=0$. Hence, $D=\emptyset$.
Now let $D=[i,n]$ for some $2\leq i\leq n$. Consider for instance the sequence $$\begin{aligned}
\mathbf{a}=(\underbrace{0,0,\ldots,0}_{i-1},\underbrace{1,1,\ldots,1}_{n-i+1}).
\end{aligned}$$ This sequence clearly encodes an $m$-Dyck path, and hence $\text{diff}_{D}(\hat{1})\neq\emptyset$.
First we show that $\bigl\lvert\mathcal{S}_{n}^{(m)}(\hat{0})\bigr\rvert=2^{n-1}$. Let $D=\{j_{1},j_{2},\ldots,j_{t}\}\subseteq\{2,3\ldots,n\}$ with $j_{1}<j_{2}<\cdots<j_{t}$. Let $\mathbf{a}^{(0)}=\hat{0}$, and construct $\mathbf{a}^{(i+1)}$ from $\mathbf{a}^{(i)}$ by reducing the primitive subsequence of $\mathbf{a}^{(i)}$ at position $j_{t-i}$ by one. Let $\mathbf{a}^{(i)}=(a_{1}^{(i)},a_{2}^{(i)},\ldots,a_{n}^{(i)})$. It is clear by construction that $a_{k}^{(i)}=a_{k}^{(0)}$ for all $k<j_{t-i+1}$. Hence, we obtain a falling chain $\mathbf{a}^{(0)}\lessdot\mathbf{a}^{(1)}\lessdot\cdots\lessdot\mathbf{a}^{(t)}$, where $\mathbf{a}^{(t)}\in\text{diff}_{D}(\hat{0})$. (In fact, $\mathbf{a}^{(t)}$ corresponds to $\chi_{D}(\hat{0})$ as defined in the proof of Lemma \[lem:zero\_prep\].) It follows from the proof of Theorem \[thm:shelling\_m\_tamari\] that if $\mathbf{a}\in\text{diff}_{D}(\hat{0})$ and there exists a falling maximal chain from $\hat{0}$ to $\mathbf{a}$, then $\mathbf{a}=\mathbf{a}^{(t)}$. Hence, for every $D\subseteq\{2,3,\ldots,n\}$ we obtain exactly one $\mathbf{a}\in\text{diff}_{D}(\hat{0})$, which yields $\lvert\mathcal{S}_{n}^{(m)}(\hat{0})\rvert=2^{n-1}$.
Now we show that $\bigl\lvert\mathcal{S}_{n}^{(m)}(\hat{1})\bigr\rvert=2^{n-1}$. Let $D=[i,n]$ for some $2\leq i\leq n$, and let $\{j_{1},j_{2},\ldots,j_{t}\}\subseteq \{i+1,i+2,\ldots,n\}$ with $i<j_{1}<j_{2}<\cdots<j_{t}$, and define $j_{0}=i$. Consider $\mathbf{a}\in\text{diff}_{D}(\hat{1})$ with $$\begin{aligned}
\mathbf{a}=(\underbrace{0,0,\ldots,0}_{j_{0}-1},
\underbrace{1,1\ldots,1}_{j_{1}-j_{0}},
\underbrace{2,2,\ldots,2}_{j_{2}-j_{1}},\ldots,
\underbrace{t+1,t+1,\ldots,t+1}_{n-j_{t}+1}).
\end{aligned}$$ Let $\mathbf{a}^{(0)}=\mathbf{a}$ and construct $\mathbf{a}^{(i+1)}$ from $\mathbf{a}^{(i)}$ by reducing the value of the primitive subsequence of $\mathbf{a}^{(i)}$ at position $j_{t-i}$ by one. We obtain a chain $\mathbf{a}^{(0)}\lessdot\mathbf{a}^{(1)}\lessdot\cdots\lessdot\mathbf{a}^{(t+1)}$, of length $t+1$ with label sequence $\bigl((j_{t},t+1),(j_{t-1},t),\ldots,(j_{0},1)\bigr)$. Moreover, $\mathbf{a}^{(t+1)}=\hat{1}$. This implies that $\mu(p,\hat{1})=(-1)^{t+1}$. Hence, the total number of elements $\mathbf{a}\in\text{diff}_{D}(\hat{1})$ satisfying $\mu(\mathbf{a},\hat{1})\neq 0$ is $2^{n-i}$. This implies $$\begin{aligned}
\lvert\mathcal{S}_{n}^{(m)}(\hat{1})\rvert & =1+\sum_{i=2}^{n}{2^{n-i}}\\
& = 1+\sum_{i=0}^{n-2}{2^{i}}\\
& = 1+2^{n-1}-1\\
& = 2^{n-1}.
\end{aligned}$$
Acknowledgements {#acknowledgements .unnumbered}
================
The author is very grateful to two anonymous referees for their suggestions and helpful comments.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.