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abstract: 'We show, using quasi-exact numerical simulations, that Anderson localization in a disordered one-dimensional potential survives in the presence of attractive interaction between particles. The localization length of the particles center of mass - computed analytically for weak disorder - is in good agreement with the quasi-exact numerical observations using the Time Evolving Block Decimation algorithm. Our approach allows for simulation of the entire experiment including the final measurement of all atom positions.'
address:
- 'Laboratoire Kastler Brossel, UPMC-Paris6, ENS, CNRS; 4 Place Jussieu, F-75005 Paris, France'
- ' Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30-059 Kraków, Poland'
- ' Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30-059 Kraków, Poland'
- ' Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30-059 Kraków, Poland'
- 'Laboratoire Kastler Brossel, UPMC-Paris6, ENS, CNRS; 4 Place Jussieu, F-75005 Paris, France'
- 'ARC Centre for Antimatter-Matter Studies, Curtin University of Technology, GPO Box U1987, Perth, Western Australia 6845, Australia'
- ' Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30-059 Kraków, Poland'
- ' Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30-059 Kraków, Poland'
author:
- Dominique Delande
- Krzysztof Sacha
- Marcin Płodzień
- 'Sanat K. Avazbaev'
- Jakub Zakrzewski
title: 'Many-body Anderson localization in one dimensional systems'
---
Introduction
============
Anderson localization (AL) has been widely investigated in the last 50 years [@Anderson1958; @Lagendijk2009]. The possibility of observing directly localization of the wavefunction in cold atomic gases lead to a recent revival of the interest in localization properties in general, and in AL in particular. AL is characterized by the inhibition of transport in a quantum system, whose classical counterpart behaves diffusively. It is accompanied by an exponential localization of eigenstates in the configuration space, $|\psi(r)|^2 \propto \exp(-|r|/L),$ where $L$ is the localization length. AL is due to the interference between various multiple scattering paths which favors the return of the particle to its initial position and thus decreases its probability to travel a long distance. As the geometry of these paths depends on the system dimension, so AL features depend on the dimension too. For one-dimensional (1D) systems, AL is a generic single particle behavior even for very small disorder when the particle “flies” above the potential fluctuations.
A fundamental question is to understand how interaction between particles affects AL. Presently, there is no consensus on the possible existence and properties of many-body localization. Some results suggest that AL survives for few-body systems, although with a modified localization length [@Shepelyansky1994]; studies of cold bosonic systems in the mean-field regime show that AL is destroyed and replaced by a sub-diffusive behavior [@Pikovsky2008; @Flach2011], but the validity of the mean-field approximation at long times is questionable. There are even predictions that AL survives at finite temperature in the thermodynamic limit, in the presence of interactions [@Aleiner2010]. In this paper, we show, using a specific example, that 1D AL survives in the presence of attractive interactions and is even a rather robust phenomenon. The novelty of our approach is that it uses a quasi-exact numerical scheme to solve the full many-body problem in the presence of disorder. Here, quasi-exact means that all numerical errors can be controlled and reduced below an arbitrary value, just at the cost of increased computational resources. The big advantage of this approach is not to rely on neglecting *a priori* any physical process.
Atomic matter waves have several advantages that made possible an unambiguous demonstration of single particle AL in 1D [@Billy2008; @Roati2008]: atom-atom interaction can be reduced either by diluting the atomic gas or using Feshbach resonances, ensuring a very long coherence time of the atomic matter wave; the spatial and temporal orders of magnitudes are very convenient allowing a direct spatio-temporal visualization of the dynamics; all microscopic ingredients are well controlled; a disordered potential can be created by using the effective potential induced by a far detuned optical speckle.
The model and its solution
==========================
We consider $N$ identical bosonic atoms in a 1D system, in the regime of attractive two-body interactions. We assume the dilute regime where the atom-atom average distance is larger than the scattering length and take the low-energy limit where the interaction can be modeled by a (negative) Dirac-delta potential. The many-body Hamiltonian can be written, using the standard second quantization formalism (assuming unit mass for the particles and taking $\hbar=1$): $$\begin{aligned}
\hat{H} = \int{dz\ \hat{\psi}^{\dagger}(z) \left[ -\frac{1}{2}\ \frac{\partial^2}{\partial z^2} + V(z) \right] \hat{\psi}(z)} \nonumber \\
+ \frac{g}{2} \int{dz\ \hat{\psi}^{\dagger}(z)\ \hat{\psi}^{\dagger}(z)\ \hat{\psi}(z)\ \hat{\psi}(z)}
\label{eq:h}\end{aligned}$$ where $g<0$ is the strength of the atom-atom interaction and $V(z)$ an external potential.
For large $N$, in the absence of an external potential, the ground state of this system is described - within the mean field approach - as a bright soliton, a composite particle with two external degrees of freedom: an irrelevant phase $\theta$, and a classical parameter: the position $q$ of the center of mass. The particle density – normalized to the number $N$ of particles – is given by $|\phi_0(z-q)|^2$ where $$\phi_0(z) = \sqrt{\frac{N}{2\xi}} \frac{{\mathrm e}^{-i\theta}}{\cosh{z/\xi}}.
\label{eq:shape}$$ $\xi=-\frac{2}{Ng}$ is the characteristic size of the soliton. The associated chemical potential is $\mu=-N^2g^2/8.$
This mean-field approach does not describe properly the many-body ground state of the system. Indeed, in the absence of an external potential, the many-body ground state is known exactly, thanks to the Bethe ansatz [@McGuire1964] which predicts, e.g., uniform atomic density. The source of discrepancy lies in a classical treatment of the center of mass $q$ of the system. In a proper description, $q$ must be thought of as the quantum position operator of the soliton, the composite particle formed by the $N$ particles. In the presence of an external potential, it is possible to construct an effective one-body (EOB) Hamiltonian describing the $q$ dynamics quantum mechanically [@Weiss09; @Sacha2009a; @Sacha2009b; @Mueller2011]. Assuming a fixed soliton shape, the EOB Hamiltonian is: $$H_q = \frac{p_q^2}{2N} + \int{\ dz \ |\phi_0(z-q)|^2\ V(z)},
\label{eobpot}$$ where $p_q$ is the momentum conjugate to $q$ [@Weiss09; @Sacha2009a; @Sacha2009b; @Mueller2011]. It describes a composite particle with mass $N$ evolving in a potential that is the convolution of the bare potential with the soliton envelope. The key point of the EOB approach is that the internal degrees of freedom of the soliton are hidden in the reduction of the many-body wavefunction to a single one-body wavefunction $\varphi(q,t)$ describing the evolution of the soliton center of mass. This is possible because the internal degrees of freedom of the bright soliton are gaped, with an energy gap equal to $-\mu=N^2g^2/8,$ so that a weak external perturbation cannot populate internal excited states of the bright soliton, in contrast with the dark soliton case [@Mochol2012].
Because the EOB Hamiltonian (\[eobpot\]) describes a one-dimensional system exposed to a disordered potential, it displays Anderson localization. Within the EOB approximation, the soliton center of mass is localized with a localization length depending on the energy. As an example, we will use - as in real experiments [@Billy2008] - the disorder created by a light speckle shone on a cold atomic gas. The localization length of the EOB model has been calculated (in the weak disorder limit) in [@Sacha2009a; @Sacha2009b; @Mueller2011]: $$\frac{1}{L_N(k)} = \frac{N^4 \xi^2 \pi^3 \sigma_0 V_0^2 (1-k\sigma_0)}{\sinh^2 \pi k\xi}\ \Theta(1-k\sigma_0)
\label{eq:locN}$$ where $k$ is the wave-vector of the soliton, $V_0$ the r.m.s amplitude of the disordered potential, $\sigma_0$ its correlation length of the speckle and $\Theta$ the Heaviside function.
What is the validity of the EOB theory at long time? The answer is far from obvious. Any many-body effect not taken into account within the EOB, could break the reduction of the problem to an EOB wavefunction evolving under the effective Hamiltonian (\[eobpot\]). Especially, it could easily spoil the phase coherence of the EOB wavefunction and consequently Anderson localization. It is the goal of this paper to perform a quasi-exact many-body numerical test of the EOB approach. In order to be as close as possible to a realistic experiment [@Billy2008], we follow the temporal evolution of an initially localized many-body wavepacket. In a first step, we compute the ground state of $N$ interacting particles in the presence of an harmonic trap, but without disorder: this produces a bright soliton localized near the trap center. In a second step, the harmonic trap is abruptly switched off and the disordered speckle potential abruptly switched on, leaving the many-body system expand in the presence of disorder, and eventually localize thanks to many-body AL.
In the presence of an external potential, the many-body problem cannot be solved exactly. One must rely on quasi-exact numerical approaches. A convenient way is to discretize the continuous Hamiltonian, Eq. (\[eq:h\]), over a discrete lattice [@Schmidt2007; @Glick2012]. The discretization of space to a chain of sites located at equally spaced positions $z_l=l\delta$ together with a 3-point discretization of the Laplace operator allows us to write the Hamiltonian in a tight-binding Bose-Hubbard form [@Schmidt2007]: $$H=\sum_l{\left[-J(a_l^{\dagger}a_{l+1} + h.c.) + \frac{U}{2} a_l^{\dagger}a_l^{\dagger}a_{l}a_{l} + V_l\ a_l^{\dagger}a_{l}\right]}
\label{eq:h_discrete}$$ with $J=\frac{1}{2\delta^2}$, $U=\frac{g}{\delta}$ and $V_l=V(z_l)$ (an additional trivial constant term $1/\delta^2$ has been dropped).
For numerical purposes the infinite space is restricted to the $[-K\delta,K\delta]$ interval leading to a 1D chain of $M=2K+1$ discrete sites. $N$ identical bosons are distributed on these $M$ sites. A basis of the total Hilbert space can be built using the direct product of Fock states on each site $|i_1,i_2\ldots i_M\rangle$ with the constraint that the sum of occupation numbers $\sum_{l=1..M}i_l$ is equal to $N.$ The size of this basis increases exponentially with the system size, making its use unpractical for large many-body problems. Instead, we use a variational set of Matrix Product States (MPSs). An MPS is a state which can be written as: $$|\psi\rangle\! = \!\!\!\!\!\!\sum\limits_{\alpha_1,\ldots,\alpha_M;\ i_1,\ldots,i_M}\!\!\!\!\!
\Gamma_{1\alpha_1}^{[1]
,i_1}\lambda^{[1]}_{\alpha_1}\Gamma^{[2],i_2}_{\alpha_1\alpha_2}\ldots\Gamma^{[M],i_M}_{\alpha_{n-1}1} |i_1,\dots,i_M\rangle
\label{eq:MPS}$$ where $\Gamma^{[l],i_l}$ ($\lambda^{[l]}$) are site (bond) dependent matrices (vectors). To describe exactly a generic state in terms of MPS, a large number (exponentially increasing with $M$) of $\alpha_l$ values is needed. However, typical low energy states are only slightly entangled so that $\lambda^{[l]}_{\alpha_l=1,2\ldots}$ are rapidly decaying numbers, which allows for introduction of a cutoff $\chi$ in the sum over Greek indices above, resulting in tractable numerical computations [@Vidal2003]. For a ground state protected by a gap, the area theorem [@Eisert2010] ensures that an efficient MPS representation exists. In the physical situation discussed by us, the area law has no direct applicability.
The $i_1,..i_M$ indices are in principle restricted to the $[0,N]$ interval. In practice, since it is highly unlikely that all the bosons occupy a single site of the system, we lower the cutoff in the sums assuming some $N_{\textrm{max}}<N$. While the maximum average occupation number (at sites near the center of mass of the soliton) is $N\delta/2\xi$=2.5, we found surprisingly that a relatively high $N_{\textrm{max}}=14$ is needed for convergent results, while the $\chi$ value may be kept relatively low.
The ground state as well as the dynamics may be quasi-exactly studied using the Time Evolving Block Decimation (TEBD) algorithm [@Vidal2003; @Vidal2004], essentially equivalent to the time-dependent Density Matrix Renormalization Group approach [@White2004; @Schollwock2011]. The TEBD algorithm describes how the $\Gamma^{[l]}$ and $\lambda^{[l]}$ evolve in time under the influence of an Hamiltonian containing simple terms local on each site as well as hoping terms of the type $a_l a_{l+1}^{\dagger}$ which transfer one particle from site $l$ to site $l+1.$ A maximum of $N=$25 particles could be included in our calculations; similar results are obtained for 10 particles.
We use the soliton size $\xi$ as the unit of length; consequently the time unit is $\xi^2.$ We use the EOB as a guide to choose the parameters of the many-body numerical experiment For example, the trap must be shallow enough not to distort the soliton shape (\[eq:shape\]), but still strong enough to confine its center of mass $q$ over a distance only slightly larger than its size. We chose $\Delta q = \sqrt{\frac{8}{5}} \xi;$ The frequency, $\omega$, of the trapping harmonic potential $\omega^2z^2/2$ is thus such that $N\omega= 5/8\xi^2,$ i.e. $\omega=0.025/\xi^2.$ In order for the localization length to be reasonably short, we choose the strength of the external potential comparable to the initial average energy of the soliton $\omega/4$, that is $V_0=2.5\times10^{-4}.$ We also choose the correlation length of the speckle potential $\sigma_0=0.4\xi$ to be significantly shorter than the soliton size, so that the EOB potential in Eq. (\[eobpot\]) is free of the peculiarities of the speckle potential [@Lugan2009].
Several sources of errors exist in the TEBD algorithm and must be controlled. The first one is due to the spatial discretization. Getting accurate results requires the discretization unit, $\delta$, to be much smaller than both the soliton size $\xi$ and the de Broglie wavelength $2\pi/k$, where $k$ is the typical wave-vector contained in the initial wavepacket for the center of mass in the EOB description. We use $\delta=\xi/5;$ a twice smaller step produces slightly different quantitative results, but the difference is practically invisible on the scale of the figures shown in our work. In order to avoid reflections from the boundaries, the number of lattice points must be sufficiently large; 1921 points are used, but only the central 1201 ones are shown in the plots. A second source of error is the temporal discretization of the evolution operator. We use the standard Trotter expansion [@Vidal2003], whose error can be controlled by varying the time step ($\delta t=0.008\xi^2$ is used). A third source of error is the truncation of the MPS at each step. This error is monitored through the so-called “discarded weight”, that is the weight of the components which have to be discarded from the time evolved many-body state to keep it in MPS form with a fixed parameter $\chi$ – the number of bonds between sites. This can be a serious problem when the entropy of entanglement grows as a function of time. It may even prevent calculation to be ran beyond some rather short time, especially when the system is significantly excited above the ground state [@Daley2004].
The entropy of entanglement is defined as the supremium over all possible bipartitions of the system. Explicitly we compute $$S=\sup_l S_l =\sup_l \left[ - \sum_{\alpha}{ (\lambda^{[l]}_{\alpha})^2 \ln(\lambda^{[l]}_{\alpha})^2 }\right]
\label {entropydef}$$ with $l$ running over all bonds. Typically, the maximum is reached for a link close to the center of the system but it can depend on the disorder and fluctuate in time. Quite surprizingly, we have not observed any significant growth of the entropy of entanglement when AL sets in, see Figure \[entrop\], a result quite opposite to that observed in [@Bardarson2012]. This may be attributed to the fact that the energy of our many-body state is quite small (see the discussion above). The converged calculations reported here use $N_{\textrm{max}}$=14, $\chi$=30 yielding the internal Hilbert space dimension per site being 450. The results has been compared with those with lower $N_{\textrm{max}}$ as well as those for $\chi$=40 (for shorter times) to check that the results presented are fully converged. All in all we were able to run the fully controlled calculation up to time $t=4000$.
Figure \[fig:density\] shows the particle density in configuration space obtained at increasing times averaged over 96 realizations of the disorder. In the absence of disordered potential, the initial wavepacket is expected to spread. The EOB physics gives the characteristic time for this spreading, $t=1/\omega=40,$ much shorter than the time scale in the figure. At $t$=500 and 1000, one clearly sees that the central part of the wavepacket is already more or less localized while the ballistic front for $|z|/\xi>20$ ($|z|/\xi>40$ for $t$=1000) has not yet been scattered and keeps a Gaussian shape similarly as in the EOB description. This corresponds to the wavepacket components with the highest energy and consequently the longest localization length. AL has already setup at $t$=2000 and does no longer evolve further, compare with $t$=4000. Therefore, Fig. \[fig:density\] provides an evidence for many-body AL taking place in a quasi-exact full many-body numerical simulation. At the final time (100 times the characteristic spreading time), we do not observe any indication that AL could be destroyed.
The description of the final state as a MPS makes it possible to compute easily more complicated quantities, such as correlation functions. The simplest one is the one-body density matrix $\langle \psi(z) \psi^{\dagger}(z')\rangle,$ shown in Fig. \[fig:one-body\]. It clearly displays extremely strong correlations between positions $z$ and $z',$ reinforcing the observation that AL probably survives far beyond $t=$4000. The interpretation is simple in terms of bright soliton: all atoms are grouped in a soliton of size $\xi,$ but the center of mass of the soliton itself is widely spread. This has an important consequence: the largest eigenvalue of the one-body density matrix – a value often used as a quantitative criterion for Bose-Einstein condensation [@Pethick] – is here 0.14, much smaller than unity; in contrast, the value at $t=0$ is 0.84. Thus, while the initial state can be considered as a true condensate, the temporal dynamics destroys condensation; any description of our many-body system using the mean field theory via the Gross-Pitaevskii equation (which by construction describes a 100% condensate) *must* fail. In other words, our many-body AL is necessarily beyond the mean field description.
Simulation of a measurement
===========================
Being able to write the many-body state as a MPS has considerable further advantages, especially if it is – like in our calculations – in the so-called canonical form [@Schollwock2011]. For example, expectation values of local operators such as $a_l^{\dagger}a_l$ or $a_l^{\dagger}a_{l+1}$ involves only simple contractions on the local $\Gamma^{[l]}$ tensors and $\lambda^{[l]}$ vectors. It makes it also possible to mimic the measurement process of particle positions as follows. The reduced density matrix $\rho^{[l]}$ on site $l$ is easily constructed by contracting the $\Gamma^{[l]}$ tensor with the neighboring $\lambda^{[l-1]}$ and $\lambda^{[l]}$ vectors: $$\rho^{[l]}_{i,j} = \sum_{\alpha_{l-1},\alpha_l}{[\lambda^{[l-1]}_{\alpha_{l-1}}]^2\ \Gamma^{[l],i}_{\alpha_{l-1}\alpha_l}
\ [\Gamma^{[l],j}_{\alpha_{l-1}\alpha_l}]^* \
[\lambda^{[l]}_{\alpha_{l}}]^2}$$
We then chose randomly the number of particles “measured” on site $l$ following the statistical populations, diagonal elements of the on-site reduced density matrix. Once a given occupation number $i$ is chosen, we project the MPS state onto the subspace with $i_l=i$ and normalize it. This involves only simple contractions on the local $\Gamma$ tensors and $\lambda$ vectors, producing another MPS. The process can be iterated on all sites, and is particularly simple if sites are scanned consecutively starting from one edge and propagating toward the other edge. It is simple to prove that the probability distribution of the measurements is independent of the order used for scanning the various sites.
An individual “measurement” produces a single set of occupation numbers $(i_1,i_2...i_M)$ (whose sum is of course $N$) whose probability is exactly $|\langle i_1,i_2...i_M|\psi\rangle|^2.$ By performing a series of “measurements”, we can sample interesting physical quantities, such as the position of the soliton center of mass, $\langle q\rangle = \sum_l{l\delta i_l}/N,$ and the particle density with respect to this center of mass. Note that these quantities are hard to measure by other means as they involve correlation functions of high order (typically up to $N$) [@Dziarmaga2010; @Mishmash2010].
Examples of our procedure are given in the inset of Fig. \[fig:center-of-mass\]. In this way, we extract both the position of the center of mass of the soliton and the atomic density relative to the center of mass. The later quantity is shown in Fig. \[fig:center-of-mass\] for time $t=4000$ in comparison with the analytic prediction, Eq. (\[eq:shape\]). The agreement is excellent, showing that the internal structure of the soliton is fully preserved for a long time, even after AL has set in. The small difference is a $1/N$ finite size effect.
Comparison with effective one body approach
===========================================
The EOB theory is able to quantitatively predict the long time limit for the spatial density probability of the soliton center of mass, see the detailed derivation and calculations in [@Sacha2009a]. Initially, the center of mass is in the ground state of the harmonic trap (a Gaussian wavepacket) that, after the trap is switched off, expands over a range of energies, each energy component being characterized by its own localization length. Each component displays approximate exponential localization in the long time limit (in a 1D system, Anderson localized eigenstates do not strictly decay exponentially, see e.g. [@gogolin; @muellerdel]). Their superposition displays approximate algebraic localization at long distance, as discussed in [@Sacha2009a].
In Fig. \[fig:comparison\], we show comparison between the full many-body calculation and the EOB corresponding numerical simulation with Hamiltonian (\[eobpot\]). Note that the many-body result is here plotted for the center of mass position, which can slightly differ from the atomic density; the latter, in the EOB approach, is the convolution of the former by the soliton shape. At the scale of the figure, the soliton is extremely narrow so that the result of the convolution is almost equal the center of mass density, compare with Fig. \[fig:density\]. The agreement between the many-body and the EOB calculations is clearly excellent. In Fig. \[fig:comparison\], we also show the $1/|q|$ leading behavior predicted by the EOB theory, Eq. (13) of Ref. [@Sacha2009a]. It predicts quite well the observed behavior but does not aim at being quantitative, because of the existence of a sub-leading logarithmic term. Namely, at very large distance, the exponential term $\exp[-\beta \ln^2(\gamma|q|)]$, where $\gamma$ and $\beta$ are constants, present in Eq. (13) of Ref. [@Sacha2009a] becomes important, leading to a faster decrease of the distribution and eventually to a finite rms displacement $\langle q^2\rangle$ of the soliton. Note also that the formula, Eq. (13) of Ref. [@Sacha2009a], assumes a weak disorder (Born approximation), an assumption not fully satisfied here.
Finally, we compare the localization length of the center of mass of attractively interacting particles with the localization length of a single particle in the same disordered potential. A meaningful comparison must be performed for the same total energy per particle, or equivalently for the same wave-vector per particle; this thus corresponds to a wavector $N$ times larger for the soliton, composed of $N$ individual particles. Within the EOB approach, the ratio is, for $k\sigma_0<1$ and weak disorder: $$\frac{L_{1}(k/N)}{L_{N}(k)} = N^2 \left[\frac{\pi k \xi}{\sinh \pi k \xi}\right]^2\ \frac{1-k\sigma_0}{1-k\sigma_0/N}$$ The physical interpretation is simple and interesting. The $N^2$ factor strongly favors localization of the soliton and reflects the collective behavior of the $N$ attractive bosons when placed in the disordered potential. The second factor – and to a lesser extent, the third one – is smaller than unity and favors delocalization of the soliton. It reflects the fact that the center of mass of the soliton does not feel the raw potential, but rather its convolution with the soliton shape, see eq. (\[eobpot\]); being smoother, the convoluted disordered scatters less efficiently than the raw one, leading to an increase of the localization length. It is ultimately due to the dispersion of the atom positions around the center of mass of the soliton. Whether the localization or the delocalization effect wins depends on the parameter values. For the parameters used here, if $k\xi>1.8,$ the localization length of the soliton is longer than the single atom localization length, shorter otherwise. Thus, no general statement on whether attractive interactions favor or not Anderson localization can be made.
Summary
=======
To summarize, we have shown the existence of many-body Anderson localization for attractive bosons in the presence of a disordered potential. The claim is based on quasi-exact many-body numerical simulations using the TEBD algorithm, which incorporate all complicated phenomena that could spoil the internal phase coherence of the many-body composite object, a bright soliton, displaying Anderson localization. Moreover, we obtain excellent agreement between the many-body calculation and a one-body effective theory, which goes beyond standard mean field theories such as the Gross-Pitaevskii equation. Our quasi-exact many-body approach allows for simulation of the entire experiment starting from the initial state in a harmonic trap till the destructive measurement of all atom positions.
Ancknowledgements
=================
Computing resources have been provided by GENCI and IFRAF. This work was performed within Polish-French bilateral programme POLONIUM No.27742UE. Support of Polish National Science Center via projects DEC-2011/01/N/ST2/00418 (MP) and DEC-2012/04/A/ST2/00088 (KS and JZ) is acknowledged.
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hep-th/0412183\
KCL-MTH-04/16\
ITP–UU–04/51\
SPIN–04/33\
UG-04/04
[**Non-extremal instantons and wormholes in string theory**]{}\
E. Bergshoeff$^1$, A. Collinucci$^1$, U. Gran$^2$, D. Roest$^2$ and S. Vandoren$^3$\
[ *$^1$ Centre for Theoretical Physics, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands\
[e.a.bergshoeff, a.collinucci@phys.rug.nl]{}\
$^2$ Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom\
[ugran,droest@mth.kcl.ac.uk]{}\
$^3$ Institute for Theoretical Physics *and* Spinoza Institute\
Utrecht University, 3508 TD Utrecht, The Netherlands\
[s.vandoren@phys.uu.nl]{}*]{}
------------------------------------------------------------------------
**Abstract**
We construct the most general non-extremal spherically symmetric instanton solution of a gravity-dilaton-axion system with $SL(2,R)$ symmetry, for arbitrary euclidean spacetime dimension $D\geq 3$. A subclass of these solutions describe completely regular wormhole geometries, whose size is determined by an invariant combination of the $SL(2,R)$ charges.
Our results can be applied to four-dimensional effective actions of type II strings compactified on a Calabi-Yau manifold, and in particular to the universal hypermultiplet coupled to gravity. We show that these models contain regular wormhole solutions, supported by regular dilaton and RR scalar fields of the universal hypermultiplet.
------------------------------------------------------------------------
The action
==========
This paper contains a summary of [@Bergshoeff:2004fq]. In addition, we demonstrate the existence of regular wormhole solutions in the universal hypermultiplet, which is present in the low-energy effective action of type II string theories compactified on a Calabi-Yau manifold. Wormhole solutions from Calabi-Yau compactifications were also found in [@Giddings:1989bq]. The main difference with our solution is that in our case we also include RR fields, and furthermore, our solution is regular in the complete domain of the wormhole geometry.
We start with the action of a gravity-dilaton-axion system in $D$-spacetime dimensions. In Minkowski space, the Lagrangian is $$\mathcal{L}_M = \tfrac{1}{2} \sqrt{|g|} \,
[ {R} -\tfrac{1}{2} (\partial {\phi})^2
-\tfrac{1}{2} e^{b {\phi}} (\partial {\chi})^2 ] \, ,
\label{10DIIB}$$ where $b$ is an arbitrary dilaton coupling parameter. This Lagrangian has an $SL(2,R)$ group of symmetries. They can be realized as modular transformations on the complex field $$\tau \equiv \frac{b}{2}\, \chi + i\, e^{-b\phi/2}\ ,\qquad
\tau \rightarrow \frac{\alpha \tau +\beta}{\gamma\tau +\delta}\ ,\qquad
\alpha\delta-\beta\gamma=1\ ,\label{tau}$$ and is valid for any nonzero value of the dilaton coupling $b$.
This theory occurs for example as the scalar section of IIB supergravity in $D=10$ Minkowski space-time with dilaton-coupling parameter $b=2$. Other values of $b$ can arise when considering (truncations of) compactifications of II supergravity. The main example we will discuss here is that of the universal hypermultiplet, that arises after compactifying type IIA strings on a (rigid) Calabi-Yau threefold down to $D=4$. This hypermultiplet contains four scalars, $\phi$ and $\sigma$ coming from the NS sector, and $\psi$ and $\varphi$ coming from the RR sector. The four-dimensional Lagrangian can be written as $$\label{UHM}
\mathcal{L}_M = \tfrac{1}{2} \sqrt{|g|} \,
[ {R} -\tfrac{1}{2} (\partial {\phi})^2
-\tfrac{1}{2} e^{\phi} \big((\partial {\psi})^2+(\partial {\varphi})^2\big)
-\tfrac{1}{2} e^{2\phi}\big(\partial \sigma +\psi \partial \varphi\big)^2] \, .$$ The scalar symmetry group is now $SU(2,1)$, but contains various inequivalent $SL(2,R)$ subgroups. For instance, if we set both $\sigma=\psi=0$, we get (\[10DIIB\]) whith $b=1$ whereby $\varphi$ is identified with $\chi$. If we set $\psi=\varphi=0$, we have $b=2$ and $\sigma$ is identified with $\chi$. By writing down the field equations for (\[UHM\]), it is easy to see that these truncations are consistent. Extremal instantons in the universal hypermultiplet have been discussed in detail in [@Theis:2002er; @Davidse:2003ww; @Davidse:2004gg], and correspond to wrapped (euclidean) membranes along three-cycles, or wrapped NS5-branes along the entire Calabi-Yau. These two cases correspond to $b=1$ and $b=2$ respectively. Using the results obtained in [@Bergshoeff:2004fq], we will here generalize this to the non-extremal case, and show that there are interesting and new solutions that have the spacetime geometry of a wormhole.
To discuss instantons, we first have to perform a Wick rotation. This rotation is best understood by dualizing the axion into a $(D-2)$-form potential. One then finds that under a Wick rotation, $\chi \rightarrow i \chi $. The Euclidean Lagrangian corresponding to (\[10DIIB\]) is then $$\mathcal{L}_E
= \tfrac{1}{2} \sqrt{g} \,
[ {R} -\tfrac{1}{2} (\partial {\phi})^2
+\tfrac{1}{2} e^{b {\phi}} (\partial {\chi})^2 ] \, ,
\label{EuclideanAction}$$ with all fields real. Notice that in the scalar formulation, as opposed to the formulation with the $(D-1)$-form field strength, the contribution to the action coming from the scalar sector is not positive definite. For $b=2$ and $D=10$ this is the gravity-scalar part of Euclidean IIB supergravity, in which the D-instanton can easily be found as a solution of the Euclidean equations of motion [@Gibbons:1996vg; @Green:1997tv]. The non-extremal solutions were found in [@Bergshoeff:2004fq], and we repeat them in the next section.
As already explained, compactifications of string theory can give rise to other values of $b$. The Euclidean version of the universal hypermultiplet Lagrangian (\[UHM\]) can best be understood in terms of the double-tensor multiplet formulation, in which $\varphi$ and $\sigma$ are dualized into two antisymmetric tensors [@Theis:2002er]. After a Wick rotation, $\varphi \rightarrow i\varphi, \sigma \rightarrow i\sigma$, and the Euclidean Lagrangian for the universal hypermultiplet becomes $$\label{EUHM}
\mathcal{L}_E = \tfrac{1}{2} \sqrt{|g|} \,
[ {R} -\tfrac{1}{2} (\partial {\phi})^2
-\tfrac{1}{2} e^{\phi} \big((\partial {\psi})^2-(\partial {\varphi})^2\big)
+\tfrac{1}{2} e^{2\phi}\big(\partial \sigma +\psi \partial \varphi\big)^2] \, .$$ Notice that the two truncations, $\psi = \sigma =0$ and $\psi=\varphi=0$, both fall into the class of (\[EuclideanAction\]), in which we have $b=1$ and $b=2$ respectively.
There are three conserved currents for the $SL(2,R)$ transformations in the Euclidean model, satisfying $\nabla_\mu j^\mu=0$. The corresponding charges are denoted by $q_3,q_+$ and $q_-$, and are normalized as specified in [@Bergshoeff:2004fq]. They transform under $SL(2,R)$ in such a way that the combination $${\bf \mathsf{q}}^2 \equiv q_3^2 - q_+q_-\ ,$$ is invariant [@Bergshoeff:2002mb; @Bergshoeff:2004fq]. The three conjugacy classes of $SL(2,R)$ then correspond to ${\bf \mathsf{q}}^2 < 0, {\bf \mathsf{q}}^2=0$ and ${\bf \mathsf{q}}^2 > 0$. The extremal solutions will have ${\bf \mathsf{q}}^2=0$, the non-extremal ${\bf \mathsf{q}}^2
\neq 0$. The wormhole solutions will have ${\bf \mathsf{q}}^2 < 0$. For later convenience, it is useful to define the quantity $$c\equiv {\sqrt {\frac{2(D-1)}{D-2}}}\ ,$$ which will appear explicitly in the instanton solutions below.
Instanton solutions
===================
We search for generalised instanton solutions with manifest $SO(D)$ symmetry, $$\begin{aligned}
{ds}^2 & = e^{2\,B(r)} (dr^2 + r^2 d\Omega_{D-1}^2) \,, \qquad
\phi=\phi(r) \,, \qquad \chi=\chi(r)\,.
\label{instanton}\end{aligned}$$ The standard D-instanton solution [@Gibbons:1996vg] is obtained for the special case that $B(r)$ is constant. Other references on generalised instantons and wormholes that are related to our work are [@Giddings:1989bq; @Rey:1989xj; @Coule:1989xu; @Kim:1997hq; @Bergshoeff:1998ry; @Einhorn:2002am; @Gutperle:2002km; @Einhorn:2002sj; @Kim:2003js; @Maldacena:2004rf]. To obtain an $SO(D)$ symmetric generalised instanton solution, we allow for a non-constant $B(r)$ and solve the field equations following from the Euclidean action (\[EuclideanAction\]). This was done in detail in [@Bergshoeff:2004fq]. Here we summarise the result.
The solution can be written in a compact form by using a harmonic function $H(r)$ over a conformally flat space with metric as given in (\[instanton\]), $$\begin{aligned}
H(r)= \frac{b\,c}{2}\,\log(f_{+}(r)/f_{-}(r))\,, \quad B(r)=\frac{1}{D-2}
\log(f_{+}f_{-})\,,\quad
f_{\pm}(r) = 1\pm\frac{\mathsf{q}}{r^{D-2}}\,,\end{aligned}$$ The general instanton solution can then be written as $$\label{SolEq}
\boxed{
\begin{aligned}
ds^2 & = \left(1-\frac{\mathsf{q}^2}{r^{2\,(D-2)}}\right)^{2/(D-2)}\,(dr^2
+ r^2 d
\Omega_{D-1}^2) \,, \\
e^{b\,\phi(r)} & =
\left(\frac{q_{-}}{\mathsf{q}}\,\sinh(H(r)+C_1)\right)^2\,,\\
\chi(r) & =
\frac{2}{b\,q_{-}}\,(\mathsf{q}\,\coth(H(r)+C_1)-q_3)\,.
\end{aligned}
}$$ This solution is valid for any value [^1] of $b\neq 0$. The integration constant $C_1$ can be traded for the asymptotic value of the dilaton that we will later identify with the string coupling constant. Notice also the explicit dependence on the $Sl(2,R)$ charges $q_3,q_-$ and $q_+$. The solutions are valid both for $\mathsf{q}^2\equiv
q_3^2-q_-q_+$ positive, negative and zero, corresponding to the three conjugacy classes of $SL(2,R)$. We now discuss these three cases separately:
- $\bf \mathsf{q}^2 >0$: Black Holes
In this case $\mathsf{q}$ is real and the solution is given by with all constants real. However, the metric becomes imaginary below a critical radius $$\label{rcritical}
r^{D-2} < r_c^{D-2} = \mathsf{q} \, .$$ One can check that there is a curvature singularity at $r=r_c$, which happens at strong string coupling: $e^{\phi(r)} \rightarrow \infty$ as $r \rightarrow r_c$.
Between $r=r_c$ and $r=\infty$, $H$ varies between $\infty$ and $0$, and with an appropriate choice of $C_1$, i.e. a positive value of $C_1$, the scalars have no further singularities in this domain. Thus one might hope to have a modification of this solution by higher-order contributions to the effective action of IIB string theory [@Einhorn:2002am]. Alternatively, one can consider the possible resolution of this singularity upon uplifting to one higher dimension. In [@Bergshoeff:2004fq], we showed that this indeed happens for the special case of $$b \geq \sqrt{\frac{2(D-2)}{D-1}} \,,$$ equivalent to $bc \geq 2$. Upon uplifting, this becomes a non-extremal dilatonic black hole. The case when $bc=2$ lifts up to a (non-dilatonic) Reissner-Nordström black hole with mass and charge given by $$\begin{aligned}
Q&= -2\,q_{-} \,, \qquad M=2\,\sqrt{\mathsf{q}^2+q_{-}^2}\qquad
\Rightarrow \qquad \mathsf{q}^2 = \frac{M^2-Q^2}{4}\, .
\label{RN-instanton-relation}\end{aligned}$$ Hence, the ${\bf \mathsf{q}}^2>0$ solutions with $bc\geq 2$ are spatial sections of a higher-dimensional (Lorentzian) black hole solution. The case of $bc < 2$ cannot be uplifted and remain singular instanton solutions in $D$-dimensions. In Einstein frame, these geometries are singular wormholes that are pinched at the selfdual radius $r_{{\rm sd}}=r_c$ [@Bergshoeff:2004fq].
In the case of $\mathsf{q}^2 > 0$, there is an interesting limit in which $q_- \rightarrow 0$. This yields a solution with only two independent integration constants, $q_+$ and $\mathsf{q}^2$. The range of validity of this solution is equal to that of the above solution with $q_- \neq 0$: it is well-defined for $r>r_c$, while at $r =
r_c$ the metric has a singularity and the dilaton blows up. The singularity can be resolved upon uplifting for all values of $bc \geq 2$ to Schwarzschild black holes, with mass $M=2{\bf \mathsf{q}}$. More details can be found in [@Bergshoeff:2004fq].
- $\bf \mathsf{q}^2 =0$: Extremal instantons
We now consider the limit $\mathsf{q}^2 \rightarrow 0$ of the general solution , after rescaling the constant $C_1$ with a factor $\mathsf{q}$ to make the limit well-defined. Taking the limit yields the extremal solution: $$\boxed{
\begin{aligned}
ds^2 = dr^2+r^2\,d\Omega_{D-1}^2 \,, \qquad
e^{b\,\phi(r)/2} = h \qquad \chi(r) = \frac{2}{b}\,(h^{-1} -
\frac{q_3}{q_-}) \,, \label{instlimEq}
\end{aligned}
}$$ where $h(r)$ is the harmonic function: $$\begin{aligned}
h(r) = g_s^{b/2} + \frac{b\,c\,q_-}{r^{D-2}} \, ,\end{aligned}$$ and $g_s$ is the asymptotic value of the dilaton at infinity.
This is the extremal D-instanton solution of [@Gibbons:1996vg]. This solution is regular over the range $0 < r < \infty$ provided one takes both $g_s$ and $b\,c\,q_-$ positive; at $r=0$ however, the harmonic function blows up and the scalars are singular. Similar to the case of $\bf \mathsf{q}^2 >0$, these singular solutions can be lifted to higher dimensions where, e.g. for $bc=2$, they become extremal Reissner-Nordström black holes.
- $\bf \mathsf{q}^2 <0$: Wormholes
In this case $\mathsf{q}$ is imaginary. To obtain a real solution we must take $C_1$ to be imaginary. We therefore redefine $$\mathsf{q}\rightarrow i\,\mathsf{\tilde{q}} \hskip 2truecm C_1
\rightarrow i\,\tilde{C_1}\, ,$$ such that $\mathsf{\tilde{q}}$ and $\tilde{C_1}$ are real. One can now rewrite the solution by using the relation $$\log(f_{+}/f_{-}) = 2 \,{\rm arctanh}(\mathsf{q}/r^{D-2})\, ,$$ and, next, replacing the hyperbolic trigonometric functions by trigonometric ones in such a way that no imaginary quantities appear. We thus find that, for $\mathsf{q}^2 <0$, the general solution takes the following form: $$\boxed{
\begin{aligned}
ds^2 & =
(1+\frac{\mathsf{\tilde{q}}^2}{r^{2\,(D-2)}})^{2/(D-2)}\,(dr^2+r^2\,d\Omega_
{D-1}^2)\,,\\
e^{b \phi(r)} & = \left(\frac{q_{-}}{\mathsf{\tilde{q}}}\,
\sin(b\,c\,\arctan(\frac{\mathsf{\tilde{q}
}}{r^{D-2}})+\tilde{C_1}) \right)^2 \,,\\
\chi(r) & =\frac{2}{b\,q_{-}}\,(\mathsf{\tilde{q}}\,
\cot(b\,c\,\arctan(\frac{\mathsf{\tilde{q}}}
{r^{D-2}})+\tilde{C_1})-q_3)\,.
\label{qminussol}
\end{aligned}
}$$ The metric and curvature are well behaved over the range $0<r<\infty$. However, the scalars can only be non-singular over the same range by an appropriate choice of $\tilde{C}_1$ provided that $bc < 2$. This can be seen as follows. The $\arctan$ varies over a range of $\pi/2$ when $r$ goes from $0$ to $\infty$. It is multiplied by $bc$ and thus the argument of the sine varies over a range of more than $\pi$ if $bc > 2$. Therefore, for $bc>2$ there is always a point $r_c$ such that $\chi \rightarrow \infty$ as $r \rightarrow r_c$. Note that the breakdown of the solution occurs at weak string coupling: $e^{\phi} \rightarrow 0$ as $r \rightarrow r_c$. This singularity is not resolved upon uplifting and corresponds to a black hole with a naked singularity (in the case of Reissner-Nordström, $M^2<Q^2$. The same holds for the liming case of $bc=2$. Therefore the case $\mathsf{q}^2 <0$ only yields regular instanton solutions for $bc < 2$, together with the condition that ${\tilde C}_1$ and ${\tilde C}_1+bc\pi/2$ are on the same branch of the cotangent.
The metric in (\[qminussol\]) has a $Z_2$ isometry corresponding to the reflection $r^{D-2}\rightarrow \mathsf{\tilde{q}} r^{2-D}$ which interchanges the two asymptotically flat regions. This reflection has a fixed point, corresponding to the selfdual radius $$\begin{aligned}
r_{\text{sd}}^{D-2}=\mathsf{\tilde{q}} \,.\end{aligned}$$ Furthermore, the thickness of the neck was in [@Bergshoeff:2004fq] computed to be $$\begin{aligned}
\rho_{\text{sd}}^{D-2} = 2\mathsf{\tilde{q}} \,.
\end{aligned}$$ We have summarised this in the following figure:
Instanton action
================
The value of the action, evaluated on the instanton solution, is a key ingredient in the semiclassical approximation of the euclidean path integral. In [@Bergshoeff:2004fq] we computed the instanton action for the three cases, corresponding to $\mathsf{q}^2 >0, \mathsf{q}^2 =0$, and $\mathsf{q}^2 <0$. This was done by specifying the additional surface term added to (\[EuclideanAction\]), which solely determines the instanton action. This surface term can be found from the dual description in terms of the $(D-1)$-form field strength formulation. We here summarise the results.
For the case when $\mathsf{q}^2 \geq 0$, the contribution to the action coming from infinity is given by $$\begin{aligned}
\label{nonextr-inst-act-infty}
\mathcal{S}^{\infty}_{inst} = \frac{4}{b^2}\,(D-2)\,
\mathcal{V}ol(S^{D-1})\,b\,c\,\Big(\sqrt{\mathsf{q}^2+\frac{q_-^2}{g_s^b}}
\Big)\ .\end{aligned}$$ Here we have used the relation between $C_1$ and the asymptotic value of the dilaton, $g_s^{b}=(q_-/\mathsf{q})^2\,\sinh^2 C_1$. Notice that the instanton action is proportional to the mass of the black hole to which the solution uplifts in one dimension higher. Furthermore, the result (\[nonextr-inst-act-infty\]) also hols for $\mathsf{q}^2=0$, which gives the lowest value of the action. The resulting instanton action is then inversely proportional to $g_s^{b/2}$. The D-instanton of ten-dimensional IIB corresponds to taking $b=2$. The extremal instantons for the universal hypermultiplet action (\[UHM\]) [@Theis:2002er; @Davidse:2003ww] also fall into this class: the membrane instantons correspond to $b=1$ whereas the NS-fivebrane instantons correspond [^2] to $b=2$. We have here given only the contribution from infinitiy. The non-extremal instantons also contribute to the action at the other boundary, where $r=r_c$. Since the solution is singular at this point, it is however not clear that the supergravity approximation is still valid in this region.
The case when $\mathsf{q}^2 < 0$ is very different. For $bc<2$, these are regular wormhole solutions with two asymptotic boundaries at $r=0$ and $r=\infty$ that are related by a reflection symmetry. The wormhole action gets contributions from both these boundaries, and the result is $$\begin{aligned}
\label{tildeq-inst-act}
\mathcal{S}_{wormhole} = \frac{4}{b^2}\,(D-2) \mathcal{V}ol(S^{D-1})\,{b\,c}\,
\mathsf{\tilde {q}}\,\Big(\cot \tilde{C}_1
-\cot (\tilde{C}_1 + bc \frac{\pi}{2})\Big)\,.\end{aligned}$$ Due to the fact that $\tilde{C}_1$ and $\tilde{C}_1 + bc\pi/2$ are on the same branch of the cotangent, the total instanton action is manifestly positive definite. One can rewrite the above result in terms of the string coupling constant, using $g_s^{b/2}\equiv e^{b\phi_{\infty}/2}=(q_-/\mathsf{\tilde {q}})
\sin {\tilde C_1}$. In the neighborhood of $bc\approx 2$, the instanton action becomes very large, and in the limit to the critical point where $bc=2$, it diverges. At that point, the wormhole solution is no longer regular.
Wormholes in string theory
==========================
We have seen that the condition for regular wormholes is that there exist models for which $bc<2$. In type IIB in ten dimensions, this is not satisfied. Toroidal compactifications of string theory only lead to values of $b$ for which $bc\geq 2$, so no wormholes exist for these cases. However, we have seen that for the universal hypermultplet, which descends from a Calabi-Yau compactification of type II strings, one can have the value $b=1$ in $D=4$, and so $bc={\sqrt 3}<2$. The solution is then characterized by the dilaton and the RR scalar $\varphi$ that descends from the RR three-form gauge potential in IIA in ten dimensions. Since the extremal case $\mathsf{q}^2 =0$ corresponds to a wrapped type IIA euclidean membrane over a (supersymmetric) three-cycle, it is natural to suggest that the wormhole, with $\mathsf{q}^2 < 0$, corresponds to a wrapped non-extremal euclidean D2 brane.\
[**Acknowledgement**]{}
It is a pleasure to thank Mathijs de Vroome for stimulating discussions.
This work is supported in part by the Spanish grant BFM2003-01090 and the European Community’s Human Potential Programme under contract HPRN-CT-2000-00131 Quantum Spacetime, in which E.B. and D.R. are associated to Utrecht University. The work of U.G. is funded by the Swedish Research Council.
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[^1]: The case $b=0$ is treated in [@Myers:1988sp].
[^2]: To compare with [@Theis:2002er; @Davidse:2003ww], one has to redefine the string coupling constant by taking the square root. We correct here a minor mistake in [@Bergshoeff:2004fq], in which the membrane and fivebrane instantons were written to correspond to $b=2$ and $b=4$.
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abstract: 'We investigate the 3D structure and drying dynamics of complex mixtures of emulsion droplets and colloidal particles, using confocal microscopy. Air invades and rapidly collapses large emulsion droplets, forcing their contents into the surrounding porous particle pack at a rate proportional to the square of the droplet radius. By contrast, small droplets do not collapse, but remain intact and are merely deformed. A simple model coupling the Laplace pressure to Darcy’s law correctly estimates both the threshold radius separating these two behaviors, and the rate of large-droplet evacuation. Finally, we use these systems to make novel hierarchical structures.'
author:
- 'Lei Xu (徐磊)'
- Alexis Bergès
- 'Peter J. Lu (陸述義)'
- 'André R. Studart'
- 'Andrew B. Schofield'
- Hidekazu Oki
- Simon Davies
- 'David A. Weitz'
title: Drying of complex suspensions
---
[UTF8]{}
The drying of suspensions of colloidal particles gives rise a plethora of fascinating phenomena, from the “coffee-ring” effect[@Deegan] to episodic crack propogation[@Dufresne] and the fractal patterns arising from invasion percolation[@Xu; @Wilkinson; @Page; @Shaw; @Robbins1]. Drying of colloidal suspensions is also important technologically: paints and other coatings depend on colloidal particles for many of their key properties, many ceramics go through a stage of particle drying, and cosmetics often exploit the unique properties of colloidal-scale particles, particularly for such beneficial properties as screening the harmful effects of the sun. However, for many of these technological applications, the colloidal particles are but one of many different components, and drying of the colloids is accompanied by many other phase changes. While these mixtures can become highly complex, a simpler, yet still rich system that embodies many of the complex phenomena of these technological suspensions is a mixture of immiscible fluids with a colloidal suspension; a simple example is a mixture of an emulsion and colloidal particles. The behavior of the emulsion embodies many of the archetypal phenomena of such systems, while still remaining sufficiently tractable to enable it to be fully understood. However, emulsions themselves typically scatter light significantly, and when mixed with a colloidal suspension, this scattering is only enhanced. As a result, it is very difficult to image this mixture, precluding optical studies of its behavior, and knowledge of the actual behavior is woefully missing.
In this Letter, we explore the drying of mixtures of aqueous emulsion droplets and spherical colloidal particles with confocal microscopy, which allows us to resolve the full 3D structure of these mixtures and their temporal dynamics. We find that the particles first jam into a solidified pack, throughout which emulsion drops are dispersed; a front of air then passes through the entire system. When this drying front reaches large emulsion droplets, the droplets unexpectedly collapse and their internal contents are forced into the pore space between the surrounding colloids, driven by an imbalance of pressures at the droplets’ interfaces with air and with the solvent. By contrast, small droplets are deformed by the drying front, yet remain intact without bursting. By coupling the Laplace pressure with Darcy’s law for flow through a porous medium, we predict the duration of large droplet invasion, and show that the threshold size between bursting and deformation is comparable to the size of the colloidal particles. We use this technique to create novel hierarchical materials.
{width="2.9in"}
We suspend sterically-stabilized colloidal spheres of polymethylmethacrylate (PMMA) with radius $r_\mathrm{p}=1$ $\mu$m in decahydronaphthalene (DHN). Separately, we create an emulsion of an aqueous phase, comprising equal volumes of water and glycerol, and PGPR-90 surfactant, required for droplet stability, in the nonpolar DHN; our homogenizer creates polydisperse droplets ranging from microns to tens of microns. We combine particle suspension and emulsion to create our particle-droplet mixtures. These particular components ensure that the refractive indices of the particles, droplets and background solvent are all sufficiently matched that we can image the entire bulk of the 3D structure with confocal fluorescence microscopy, with single-particle resolution[@Peter]. To distinguish particles from droplets, we use different dyes: particles are dyed with nitrobenzoxadiazole (NBD) and appear green; droplets are dyed with rhodamine-B and appear red.
We deposit the particle-droplet mixture on a clean glass coverslip, and image with an inverted confocal microscope (Leica SP5). The droplets are coated immediately by colloids, reminiscent of a pickering emulsion[@Velev; @Dinsmore]. As in the case of particle-only systems, evaporation proceeds in two stages[@Xu]. First, particles and droplets are driven toward the edge of the drying sample, where they jam, analogous to the coffee-ring effect in particle-only systems[@Deegan]. Then, a drying air front invades the jammed system and displaces the DHN, which ultimately evaporates completely; we do not observe cracks during drying, as expected for particles of this size[@Xu]. We use confocal microscopy to observe the particle-droplet mixture after jamming, as the drying front passes through, which allows us to determine the exact size and position of each emulsion droplet and colloidal particle[@Peter]. A 3D reconstruction of a typical jammed mixture before drying is shown in Fig. 1(a), with droplets and particles shown in red and green, respectively. Collecting this full 3D data takes several seconds, far too slow to observe the rapid dynamics that occur during air invasion; instead, to capture these dynamics, we fix the focal plane in the bulk of the sample and collect high-speed 2D images every 70 ms. By imaging the same regions in 3D first, then rapidly in 2D as the air invades, we can observe the drying process with good spatial and temporal resolution.
{width="2.9in"}
In the jammed configurations before air invasion, the emulsion droplets are distributed throughout the colloidal particles, as shown in Fig. 1(a). As the sample dries, more and more particle regions turn black; the invading air displaces the solvent around the particles, destroying the refractive index match, as illustrated in Figs. 1(b)-(d). The droplets also turn black as drying proceeds, but whether they are replaced by air or solvent cannot be resolved with fluorescence, where both appear black. We therefore observe droplet invasion with bright-field transmission microscopy, where the solvent remains transparent but air appears black. We observe that droplets are replaced by material that appears black in bright-field, which therefore must be air, as shown in Figs. 1(e)-(g). For a more detailed understanding of droplet behavior, we observe large isolated emulsion droplets, with radii $R_\mathrm{d}$ greater than a few microns, during drying. Because it is energetically unfavorable to displace the pre-existing organic solvent, which wets the PMMA particles, with the aqueous droplet fluid that does not, we should expect that the aqueous fluid remains in the droplet and slowly evaporates upon contact with air, with very little fluid motion. Contrary to this expectation, however, we instead observe significant fluid motion: air rapidly forces the contents of the droplet into the pore space between the surrounding particles, leaving an empty, spherical void, as shown in Figs. 2(a)-(c). This flow can not occur spontaneously; instead, a strong driving force must exist.
To understand this driving force, we analyze the distribution of pressures inside and around an isolated droplet. The total pressure near an interface is determined by a combination of the external pressure and the interface’s Laplace pressure. The magnitude of the Laplace pressure is usually estimated as the ratio of the interfacial tension to interface’s curvature radius; its sign positive for convex interfaces, negative for concave. Before air invasion, the droplet is surrounded by particles and solvent, as shown in Fig. 2(d). The air protrudes into the solvent, making a concave profile with a negative Laplace pressure, as shown in Fig. 2(e)[@Dufresne]. We estimate $P_\mathrm{solv}$, the pressure of the solvent, as the difference between atmospheric pressure $P_\mathrm{atm}$, and the Laplace pressure of the air-solvent interface, as shown in Fig. 2(e): $P_\mathrm{solv} \cong P_\mathrm{atm} - \sigma_\mathrm{air|solv}/a$, where $\sigma_\mathrm{air|solv}$ is the interfacial tension of the air-solvent interface, and $a$ is the typical size of the pores between colloidal particles. The tiny pore size produces large negative Laplace pressures, and hence a low $P_\mathrm{solv}$. As a result, the droplet is surrounded by a low-pressure environment.
At the interface between droplet and solvent, the aqueous droplet protrudes into the non-polar solvent around the colloids, as shown in Fig. 2(f); the Laplace pressure here is therefore positive. We estimate the pressure inside the droplet as $P_1 \cong P_\mathrm{solv} + \sigma_\mathrm{drop|solv} /a$, where $\sigma_\mathrm{drop|solv}$ is the interfacial tension of the droplet-solvent interface.
As soon as air touches the droplet, however, the pressure distribution changes dramatically. In particular, part of the droplet is now in contact with the much-higher atmospheric air pressure, as illustrated in Fig. 2(g). Near the air-droplet interface, the pressure inside the drop is estimated as $P_2 \cong
P_\mathrm{atm} - \sigma_\mathrm{air|drop} / R_\mathrm{d}$, where we have estimated the radius of air-droplet interface by the droplet size $R_d$. The pressure near the droplet-solvent interface remains $P_1$, as illustrated in Fig. 2(g). If $P_2 > P_1$, then the pressure difference will force droplet fluid into the surrounding pore space between colloidal particles. We estimate this pressure difference: $$\Delta P = P_2 - P_1 \cong \frac{\sigma_\mathrm{air|solv}}{a} -
\frac{\sigma_\mathrm{air|drop}}{R_\mathrm{d}}
- \frac{\sigma_\mathrm{drop|solv}}{a} \label{eqn:Delta_P}$$ where any change in solvent pressure across the droplet is negligible. Interestingly, the $\Delta P$ depends not on atmospheric pressure, but rather on the competition between the Laplace pressures at the various interfaces. We measure the corresponding surface tensions with the pendant drop method: $\sigma_\mathrm{air|solv} = 26 \pm$ 2 mN/m, $\sigma_\mathrm{air|drop} = 51\pm 3$ mN/m, and $\sigma_\mathrm{drop|solv} = 3.8
\pm 0.3$ mN/m. Because of the presence of surfactant at the interface, the droplet-solvent surface tension $\sigma_\mathrm{drop|solv}$ is so much lower than the other two that its contribution to the overall pressure difference is negligible. The radii that we measure for the large droplets, $R_d \cong 10 - 50$ $\mu$m, is orders of magnitude larger than the size of the inter-particle pore space, $a \cong 0.36 r_\mathrm{p} = 0.36$ $\mu$m for random close packed particles[@Frost]. Consequently, the contribution from a pressure drop across the air-droplet interface is also small, and the flow is essentially driven by the low pressure in the solvent evacuating the large drops, as shown in Figs. 2(a)-(c). Since the solvent strongly wets the particles, the menisci of the air evaporating the solvent from the pores creates a low pressure in the solvent. It is this, reflected in the first term in Eqn. \[eqn:Delta\_P\], which establishes the large pressure difference that drives the flow. We estimate $\Delta P \cong 0.6$ atm for these large drops.
![ (a) Measurement of droplet evacuation duration time, $t_\mathrm{ev}$, as a function of droplet radius, $R_\mathrm{d}$, for a typical drying experiment. For the measured half decade, the data are consistent with a power-law of slope 2, confirming the prediction of eqn. \[eqn:evac\_time\]. (b) Schematic of 2D images representing slices through a droplet at different heights. (c) Confocal microscope image at the top of the droplet, showing the hexagonal crystalline ordering of the particles that suggests $\phi \approx 0.74$. (d) Confocal microscope image through the droplet center, showing the dense packing around the droplet, consistent with high $\phi$. In these images, the brightness of the particles in contact with the droplet has been enhanced for clarity.](figure3_091204.eps){width="3in"}
The high time resolution afforded by the rapid collection of 2D image sequences in the confocal microscope allows us to measure $t_\mathrm{ev}$, the evacuation time for air to invade the large droplets and force their contents into the surrounding particles; $t_\mathrm{ev}$ varies for droplets of different sizes. We measure the size of each droplet from the initial 3D confocal data, then quantify $t_\mathrm{ev}$ using fast 2D images of droplet evacuation. Collected from deep in the bulk of the sample, our images are large enough to contain a number of droplets to obtain good statistics, yet are small enough relative to the sample size to achieve a uniform sampling environment.We find that the variation of $t_\mathrm{ev}$ with $R_\mathrm{d}$ is consistent with a power-law, albeit over less than a decade. The exponent is approximately 2, as shown by the comparison of the solid line with the data points on the log-log plot in Fig. 3(a).
{width="3.2in"}
We estimate this relationship theoretically with a simple model based on Darcy’s Law for fluid flow in a porous medium[@Batchelor] to determine a characteristic evacuation velocity, $v_\mathrm{ev} = \kappa \nabla P / \mu$, where $\kappa$ is the permeability of the porous medium, and $\mu$ is the dynamic viscosity of the fluid. We estimate $\nabla P$, the pressure gradient that drives the aqueous fluid into the porous medium, as the characteristic pressure difference divided by the droplet diameter, $\nabla P = \Delta P / 2 R_\mathrm{d}$. We then estimate $t_\mathrm{ev} \cong 2R_\mathrm{d} / v_\mathrm{ev}$, yielding: $$t_\mathrm{ev} = \frac{4\mu}{\kappa \Delta P} R_\mathrm{d}^2
\label{eqn:evac_time}$$
The model predicts that $t_\mathrm{ev}$ data follow an $R_\mathrm{d}^2$ dependence; indeed, the experimental data closely conform to this particular power-law scaling, as shown on the log-log plot in Fig. 3(a). Moreover, we can further test the model by estimating the prefactor $4 \mu / (\kappa \Delta P)$ in Eqn. \[eqn:evac\_time\]. We estimate $\kappa$ using Kozeny-Carman equation for flow through a porous medium[@Carman], $\kappa \cong r_\mathrm{p}^2(1-\phi)^3
/ (45\phi^2)$, where the pore size is determined by the particle volume fraction $\phi$. From the 3D particle positions in the bulk, we measure $\phi =
0.63 \pm 0.03$, consistent with random close-packing; however, around the large droplets, the particles are hexagonally close-packed[@Dinsmore], and the next layer tightly packs the interstices, as illustrated in Figs. 3(b)-(d). We think these well-ordered particles predominantly determine the pore size through which the droplets evacuate. We therefore estimate the permeability using the higher $\phi=0.74$. We measure $\mu = 8.9 \pm 0.1$ mPa-s for the H$_2$O/glyercol mixture, yielding our rough estimate of $4 \mu / (\kappa \Delta P) \approx 0.8 \times 10^9$ s/m$^2$; this value is of the same order as the value of $t_\mathrm{ev} /R_\mathrm{d}^2= 1.2 \times 10^9$ s/m$^2$ from the fit to the experimental data in Fig. 3(a), and provides support that our model correctly captures the proper physics.
Our model explains the air invasion of large droplets, whose contents are forced into the surrounding pore space. It also implies a completely different behavior for sufficiently small droplets: as $R_\mathrm{d}$ decreases, $\Delta P$ must also decrease, as can be seen from Eqn. (\[eqn:Delta\_P\]). Eventually, for sufficiently small drop size, $\Delta P$ will become zero and will no longer drive any flow; the high Laplace pressure of these small droplets makes them so stiff that they can not be invaded by air. Indeed, when we observe small droplets prepared by ultrasonic homogenization, we find that the small droplets are not invaded by air but instead are simply deformed and pushed into pore space between the surrounding particles, as shown in Figs. 4(a)-(b).
We estimate the threshold radius $R_\mathrm{th}$, beneath which droplets will not be invaded by air by using the previous values for the parameters in eqn. (\[eqn:Delta\_P\]) and solving for $\Delta P = 0$; our model predicts that $R_\mathrm{th}=2.30a=0.83 r_\mathrm{p}$. To test this prediction, we measure $R_\mathrm{th}$ for droplets mixed with particles of several different radii, varying $r_\mathrm{p}$ by more than half an order of magnitude. We find the dependence $R_\mathrm{th} = (0.88\pm0.16)*r_\mathrm{p} + (0.1\pm0.1)$, in excellent agreement with the prediction of our model, as shown in Fig. 4c.
We observe two qualitatively different behaviors: droplets that evacuate and collapse, creating large voids; and droplets that remain intact during drying, yielding void-free particle packs. We use thisdichotomy to produce hierarchical materials with several different controllable length scales, by varying droplet and particle sizes. One such structure, using droplets with $R_\mathrm{d} \cong 20$ $\mu$m and particles with $r_\mathrm{p} =
1$ $\mu$m, is shown in Figs. 4(d)-(f). The resulting hierarchical porous material has voids of two length scales: 20 $\mu$m, from droplets, and 0.5 $\mu$m, from the pore space between particles. Hierarchical materials may be useful in making low-density porous materials or to mimic hierarchical natural structures[@Andre; @bone]; our evacuation results demonstrate drying as a general low-energy method to drive desired materials into a porous medium.
We gratefully acknowledge support from the NSF (DMR-0602684), Harvard MRSEC (DMR-0213805), RGC Direct Allocation (2060395), NASA (NNX08AE09G), Pixar and ICI.
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|
---
author:
- 'C.Kramer'
- 'B.Mookerjea'
- 'E.Bayet'
- 'S.Garcia-Burillo'
- 'M.Gerin'
- 'F.P.Israel'
- 'J.Stutzki'
- 'J.G.A.Wouterloot'
bibliography:
- 'aamnem99.bib'
- 'p\_galaxies\_bib.bib'
date: 'Received 3 May 2005 / Accepted '
title: Photon dominated regions in the spiral arms of M83 and M51
---
Introduction
============
Neutral atomic carbon is thought to form predominantly in surface layers of molecular clouds where recombines and CO is dissociated due to the far-UV photons governing the chemical reactions. FUV photons (6eV$<h\nu<13.6$eV) are primarily responsible for the heating of the surface regions via photoelectric effect on dust grains while at larger depths cosmic-ray induced heating will dominate. These regions are referred to as photo dissociation regions or, more generally, as photon dominated regions [(PDRs)]{} [@tielens1985; @stoerzer1996; @kaufman1999]. PDR models take into account the relevant physical processes, solve simultaneously for the chemistry (using an extensive chemical network) and the thermal balance, as a function of cloud depth. It is found that the ratio of [\[\]]{}/[\[\]]{} is an accurate tracer of the FUV field [@gerin_phillips2000], parametrized by $G_0$ in units of the Habing-field $1.6\,10^{-3}$erg s$^{-1}$cm$^{-2}$ [@habing1968]. Another important parameter governing the depth at which forms is the ratio of density over FUV field $n/G_0$ [@tielens1985]. This ratio also determines the efficiency of converting FUV photons to gas heating, i.e. the photoelectric heating efficiency $\epsilon$ [@bakes_tielens1994].
While the Milky Way survey of FIR lines conducted with COBE/FIRAS [@fixsen1999] showed that [\[\]]{} is the dominant cooling line, it also showed the importance of the two finestructure lines of [\[\]]{}. Both lines are ubiquitous and the two lines together amount to 75% of the total cooling of all rotational CO lines in the inner galaxy. This picture has also emerged from extragalactic observations of the [\[\]]{} 1–0 line. These show again that the cooling due to and CO are of the same order of magnitude for most galaxies [@bayet2004; @israel_baas2003; @israel_baas2002; @israel_baas2001]. is found to be a good tracer of molecular gas, possibly more reliable than CO [@gerin_phillips2000].
Several coordinated mapping studies of nearby galaxies have been started during the past years. The BIMA SONG survey [@regan2001] has aimed at obtaining the $^{12}$CO emission of 1–0 and 2–1 rotational lines at high spatial resolutions. There exist velocity-integrated [\[\]]{} observations of large samples of galaxies with the KAO [@stacey1991] and with ISO [@malhotra2001], hereafter MKH01, and [@leech1999; @negishi2001] at $\sim1'$ resolution. The SINGS Spitzer Legacy Project [@kennicutt2003] has started imaging 75 galaxies in the infrared, including M51. In the coming years, both SOFIA and the Herschel Space Observatory are expected to provide velocity-resolved [\[\]]{} data at resolutions of $\sim10''$, complementary to many current single dish observations of CO and [\[\]]{}.
In external galaxies where a large number of clouds or even GMCs fill the beam it is difficult to seperate the different contributions and judge their importance. A long standing problem is that a substantial fraction of the [\[\]]{} emission may originate from the diffuse ionized and neutral medium. Comparison with [\[\]]{} helps to estimate the fraction originating from PDRs, but usually with large uncertainties due to the varying chemical and excitation conditions in different galactic environments [e.g. @contursi2002]. The present study is part of the preparatory work for future airborne and space missions like SOFIA and Herschel. In addition, it may serve as template for studies of e.g. [\[\]]{} and CO in high-$z$ galaxies which have recently become possible [@weiss2005; @walter2004; @pety2004; @neri2003; @weiss2003].
Most extragalactic observations of atomic carbon have so far concentrated on the bright galactic nuclei or enhanced emission of edge-on galaxies. Here, we compare observations of the two nuclei of M83 and M51 with pointed observations at spiral arm positions which show enhanced star forming activity. The galacto-centric distances of the selected outer positions lie between 1.8 and 5.8kpc. We combine observations of atomic carbon with low and mid-$J$ CO and $^{13}$CO data, as well as FIR [\[\]]{}, [\[\]]{}($63\,\mu$m), and [\[\]]{}($122\,\mu$m) data from the ISO data-base and thus include the brightest gas cooling lines of the far-infrared and submillimeter regime.
[lrrrrr]{} & M83 & M51\
RA(2000) & 13:37:00.5 & 13:29:52.7\
DEC(2000) & $-29$:51:55.3 & 47:11:43\
Type & SAB(s)c$^{(1)}$ & SA(s)bc pec$^{(1)}$\
Distance \[Mpc\] & 3.7$^{(5)}$ & 9.6 $^{(4)}$\
$10''$ correspond to & 179pc & 465pc\
Heliocentric velocity \[kms$^{-1}$\] & 516$^{(1)}$ & 463$^{(1)}$\
Position Angle \[deg\] & 45 $^{(2)}$ & 170\
Inclination \[deg\] & 24 $^{(2)}$ & $20$\
$D_{25}\times d_{25}$ \[$'$\] & $12.9\times11.5$$^{(1)}$ & $11.2\times6.9$$^{(1)}$\
$L_{\rm FIR}$ \[$10^9$[L$_{\odot}$]{}\]& $7.1^{(3)}$ & $14^{(3)}$\
$F_{60}/F_{100}$ & $0.43^{(3)}$ & $0.44^{(3)}$\
$F_{60}$ \[Jy\] & $286^{(3)}$ & $85^{(3)}$\
M83
---
M83 (NGC5236) is the most nearby CO-rich grand-design spiral galaxy, seen almost face-on (Table\[tab-the-sample\]). It has a pronounced bar, with two well-defined spiral arms connected to the starburst nucleus. In this paper, we adopt a distance of 3.7Mpc [@devaucouleurs1991] though recent observations of Cepheids indicate a slightly larger distance of 4.5Mpc [@thim2003].
Low-$J$ CO maps were obtained by @crosthwaite2002 [@lundgren2004; @lundgren2004_2; @dumke2001; @sakamoto2004]. [\[\]]{} observations of the center were conducted by @israel_baas2002 and @petitpas_wilson1998. Pointed KAO observations report strong FIR fine-structure lines towards the nucleus with a rapid fall-off towards the arms [@crawford1985]. Here, we present new [\[\]]{} data of the center and two spiral arm positions on the north-eastern arm and south-western bar-spiral transition zone. The emission of [\[\]]{}, [\[\]]{}(122), and [\[\]]{}(88) observed with ISO/LWS (Brauher 2005, priv. com.) is strongly enhanced in these interface regions indicating greatly enhanced star formation rates. ISO/LWS emission from the center was analyzed by @negishi2001. The [**north-eastern arm**]{} was previously studied by @lord_kenney1991 and @rand1999 who presented OVRO interferometric $^{12}$CO 1–0 maps. The eastern position at ($89'',38''$) presented here corresponds to the bright feature \#6 compiled by @rand1999 [Table4]. About $15''$ to the east of the CO arm newly formed stars form the optical arm and an HI ridge. At \#6, the CO and dust arms coincide while they are offset further to the south. Position ($-80'',-72''$) studied here corresponds to a CO 1–0 peak in the [**south-western bar-spiral transition zone**]{} which exhibits a massive GMC complex and luminous regions [@kenney_lord1991]. Note that only less than 5% of the single dish flux is recovered by the interferometric maps [@rand1999]. Thus, relatively smoothly distributed diffuse molecular gas is completely missed.
M51
---
The nearby grand-design spiral galaxy M51 (NGC5194) seen almost face on (Table\[tab-the-sample\]) is interacting with its small companion NGC5195, which lies $4.5'$ to the north. M51 is a Seyfert 2 galaxy [@ho1997]. The central AGN is surrounded by a $\sim100$pc disk [@kohno1996] of dense and warm gas [@matsushita1998].
A large amount of observational data are available for this object, including an extended KAO map of [\[\]]{} [@nikola2001]. @garnett2004 used ISO/LWS data of the M51 region CCM10, to study the ionized gas, finding that abundances are roughly solar. Several single-dish studies mapped the low-lying rotational $^{12}$CO and $^{13}$CO transitions. CO 1–0 and 2–1 was mapped by @gb1993a [@gb1993b] and @nakai1994 [@tosaki2002]. In this grand-design spiral, CO is tightly confined to the spiral arms. Maps of CO 3–2 and 4–3 were obtained at the HHT by @nieten1999 [@wielebinski1999; @dumke2001] who show that warm molecular gas is extended in M51 at galacto-centric distances of at least upto $100''$, resp. 5kpc. Single-dish observations of neutral carbon were so far obtained only in the center region by @gerin_phillips2000 [@israel_baas2002], and Israel, Tilanus, Baas, 2005, in prep.).
Aperture synthesis maps were obtained by @aalto1999 [@sakamoto1999] and @regan2001 in CO 1–0 at resolutions of $4''-6''$. Recently, @matsushita2004 mapped the inner region in $^{12}$CO 3–2. Here, we selected two positions at the spiral arms lying in the northeastern and the southwestern zones, i.e. at $72'',84''$ and $-84'',-84''$, of enhanced [\[\]]{} emission tracing enhanced star formation [@nikola2001]. ISO/LWS data are available for these positions, and for the center @negishi2001 and Brauher (2005, priv. com.). The region studied by @garnett2004 using ISO/LWS, CCM10, lies at about $+148''$,$+45''$ [@carranza1969], $1.5'$ to the north-west of $72''$,$84''$.
[rrrrrcl]{} & Line & $\theta_{\rm{b}}$ & Telescope & $\eta_{\rm{mb}}$ & Refs.\
& & ($''$)\
\
& [\[\]]{} 1–0 & 10 & JCMT 15m & 0.52 & 1 & point.\
& CO 1–0 & 21 & IRAM 30m & 0.78 & 1 & point.\
& CO 1–0 & 55 & NRAO 12m & 0.88 & 2 & map\
& CO 2–1 & 10 & IRAM 30m & 0.57 & 1 & point.\
& CO 2–1 & 28 & NRAO 12m & 0.56 & 2 & map\
& CO 3–2 & 25 & CSO 10m & 0.75 & 3 & map\
& $^{13}$CO 1–0 & 22 & IRAM 30m & 0.79 & 1 & point.\
& $^{13}$CO 2–1 & 10 & IRAM 30m & 0.60 & 1 & point.\
\
& [\[\]]{} 1–0 & 10 & JCMT 15m & 0.52 & 1 & point\
& CO 1–0 & 21 & IRAM 30m & 0.65 & 5 & map\
& CO 2–1 & 10 & IRAM 30m & 0.46 & 5 & map\
& CO 3–2 & 22 & HHT 10m & 0.50 & 4 & map\
& $^{13}$CO 1–0 & 22 & IRAM 30m & 0.79 & 1 & point.\
& $^{13}$CO 2–1 & 10 & IRAM 30m & 0.60 & 1 & point.\
![Spectra of M83 at the central and two spiral arm positions (Table\[tab-data-sets\]). Offsets are given in arcseconds relative to the (0,0) position (Table\[tab-the-sample\]). Velocities are relative to LSR. All spectra are on the $T_{\rm mb}$ scale. The $^{12}$CO data are at a common resolution of $80''$. The [\[\]]{} and $^{13}$CO data are at their original resolutions listed in Table\[tab-data-sets\]. \[fig-m83-spec\] ](m83spectra2.eps){height="10cm"}
![Spectra of M51 at the central and two spiral arm positions (cf. Fig.\[fig-m83-spec\]). \[fig-m51-spec\] ](m51spectra2.eps){height="10cm"}
Data sets
=========
We present here observations of [\[\]]{}, CO, and $^{13}$CO spectra at four spiral arm positions and the centers of M51 and M83 (Table\[tab-data-sets\]). We combine these with ISO/LWS FIR spectral line data at all six positions together with the FIR continuum derived from HIRES/IRAS 60$\,\mu$m and 100$\,\mu$m maps.
[\[\]]{} data taken at the JCMT
-------------------------------
We have observed the fine structure transition of atomic carbon ([\[\]]{}) at 492GHz (609 $\mu$m, $^3$P$_1{{-}}^3$P$_0$; hereafter $1{{-}}0$) in M51 (3 positions) and M83 (3 positions) using the JCMT 15m telescope. We used receiver RxW with a single mixer and the DAS autocorrelator. Observations were carried out during 35 hours in May and June 2003. We used the double-beamswitch observing mode with a wobbler throw of $\pm3'$ in the direction of the major axis, i.e. in the direction of the largest velocity gradient. Pointing was checked using SCUBA after an initial alignment with RxW and Jupiter at the start of each shift. It was found to be accurate to within $2-3''$. The atmospheric zenith opacity at 225GHz varied slowly between 0.1 and 0.05, corresponding to a $\tau$ of 2 and 1 at 492GHz. After merging the DAS autocorrelator spectra using the SPECX software, further data analysis was done using the CLASS/GILDAS package of IRAM.
CO data taken at the IRAM 30m MRT
---------------------------------
We have observed the $^{12}$CO and $^{13}$CO 1–0 and 2–1 rotational transitions at all six positions in M51 and M83 using the IRAM 30m telescope. These observations were carried out in double beam switch mode with a wobbler throw of $\pm4'$ using the filterbank of 1MHz resolution for the 3mm band and the 4MHz filterbank for the 1mm band. Observations were carried out during 15 hours on July, 23rd and 26th, and on September, 10th, 2004. Pointing and focus were checked and corrected every $\sim2$ hours. The pointing accuracy was better than $4''$. The amount of precipitable water vapour varied slowly between 10 and $\sim20$mm. Telescope parameters are listed in Table\[tab-data-sets\].
----------------------------------- ----------- ----------- ----------- ----------
Positions HIRES
[$(\Delta\alpha,\Delta\delta$)]{} FIR [\[\]]{} [\[\]]{} [\[\]]{}
[$('','')$]{} $158\mu$m $122\mu$m $63\mu$m
[**M83:**]{}
(0,0) 32$10^3$ 81.8 14.1 86.7
(-80,-72) 7.2$10^3$ 38.6 5.3 27.3
(89,38) 6.6$10^3$ 33.4 5.9 26.6
[**M51:**]{}
(0,0) 17$10^3$ 44.1 12.3 32.2
(72,84) 4.6$10^3$ 16.7 $<2.3$ 13
(-84,-84) 3.5$10^3$ 15.4 2.1 13.4
----------------------------------- ----------- ----------- ----------- ----------
: \[tab-isodata\] FIR continuum and line intensities in units of $10^{-6}$ergs$^{-1}$cm$^{-2}$sr$^{-1}$. The absolute calibration error is assumed to be 15% [@gry2003].
FIR line fluxes from ISO/LWS {#sec-isolws_data}
----------------------------
The central area of M83 covering $\sim5'\times4'$ was mapped on a fully-sampled grid of 61 positions with ISO/LWS. M51 was observed at 13 positions, mainly along a cut through the center and the two prominent [\[\]]{} lobes in the north-east and south-west seen in the KAO map by @nikola2001. The ISO/LWS line emission data was uniformly processed by Brauher (2005, priv. com.) to derive line fluxes in Wm$^{-2}$. To convert to intensities, we use a LWS beam size of $80''$ ($\Omega_{\rm LWS}=1.2\,10^{-7}$sr), the mean value published in the latest LWS Handbook, and extended source corrections factors [@gry2003]. Resulting ISO intensities of [\[\]]{} (158$\mu$m, [\[\]]{} (122$\,\mu$m), and [\[\]]{} (63$\,\mu$m) at the positions observed in [\[\]]{} are listed in Table\[tab-isodata\]. Since the [\[\]]{} (146$\,\mu$m) line was detected only at the center positions (S.Lord, priv. comm.), we did not include it in the present analysis.
FIR continuum maps taken with IRAS {#sec-iras_data}
----------------------------------
To derive the far-infrared continuum at all selected positions, we obtained high-resolution (HIRES) 60$\,\mu$m and 100$\,\mu$m IRAS maps from the IPAC data center[^1]. Enhanced resolution images were created after 200 iterations using the maximum correlation method [MCM @aumann1990]. These data were smoothed to an effective common circular beam of $80''$ and then combined to create maps of the far-infrared flux. The FIR flux is defined as in @helou1988: FIR $= 1.26\,10^{-14}
[2.58\,I(60\,\mu\rm{m})+I(100\,\mu\rm{m})]$ where FIR is in Wm$^{-2}$ and $I$ is in Janskys. FIR is a good estimate of the flux contained between 42.5 and 122.5$\mu$m [@helou1988]. Table\[tab-isodata\] lists the FIR flux at the selected positions. To derive the total infrared flux TIR, we follow the procedure introduced by @dale2001 who derived an analytical expression for the ratio of total infrared flux TIR to the observed FIR flux, i.e. the bolometric correction, as a function of the $60\,\mu$m/100$\,\mu$m flux density ratio from modelling the infrared SEDs of 69 normal galaxies [^2] For M83 and for M51, the total infrared flux is a factor 2.3 larger than FIR given the global $60\,\mu$m/100$\,\mu$m ratio of 0.43 (Table\[tab-the-sample\]).
Spectra and line ratios
=======================
The main aim of this work is to use the combined FIR ISO/LWS, HIRES/IRAS, and the millimeter/submillimeter line data coherently and to investigate, to what degree these give a consistent fit within the framework of a simple model scenario such as PDR excitation. We therefore smoothed the $^{12}$CO maps (Table\[tab-data-sets\]) to the ISO/LWS angular resolution of $80''$ using Gaussian kernels. Spectra of [\[\]]{}, CO, and $^{13}$CO are displayed in Figures\[fig-m83-spec\],\[fig-m51-spec\] and integrated intensities in Table\[tab-intensities\]. Ratios with [\[\]]{} and $^{13}$CO for which no maps exist, were corrected for beam filling (Table\[tab-ratios\]).
M83 {#sec_m83_spectra}
---
The [\[\]]{} lines are widest at the center position with 130kms$^{-1}$ FWHM and drop to about 30kms$^{-1}$ at the two outer positions. In addition, peak line temperatures drop strongly, leading to a pronounced drop of [\[\]]{} integrated intensities and area integrated [\[\]]{} luminosities by factors of 10 to 18 at galacto-centric distances of less than 2kpc (Table\[tab-intensities\]).
The CO 3–2 transition traces warm and dense gas since its upper level energy corresponds to $2.8\,J(J+1)=33.6\,$K and the critical density needed to thermalize this line is $4\,10^3\,J^3\sim1\,10^5\,$cm$^{-3}$, only weakly dependent on the kinetic temperature. Trapping typically reduces the critical densities by up to an order of magnitude, depending on the optical depth of the lines. The CO 3–2/1–0 line ratio of integrated intensities (Table\[tab-ratios\]) thus is a sensitive tracer of local densities for densities of less than $\sim10^5$cm$^{-3}$. Here, both line maps were smoothed to $80''$ resolution, beam filling factors thus cancel out to first order. The estimated calibration error is 21%. While the 2–1/1–0 ratio is $\sim1$ at all positions indicating that the $J=2$ state is thermalized, the 3–2/1–0 ratio drops slightly from 0.60 in the center to 0.46 at the spiral arm positions. These ratios indicate that densities are lower than $1\,10^5$ needed for thermalization of the $J=3$ level.
The $^{12}$CO 2–1 vs. $^{13}$CO 2–1 line ratio, as well as the corresponding 1–0 ratio, trace the total column densities of the $^{13}$CO line. The ratios remain constant at $\sim10$ for 2–1 and $\sim9$ for the 1–0 transition.
The above ratios found at the center position agree within the quoted errors with the ratios presented in @israel_baas2001.
In M83, we find a significant drop of the CI vs. $^{13}$CO 2–1 ratio from about 4 in the center to $\sim1.4$ at the two outer positions.
M51 {#sec_m51_spectra}
---
Again, the [\[\]]{} line is widest at the center, $\sim100$kms$^{-1}$ FWHM, and drops to $\sim30$kms$^{-1}$ at the two outer positions. The $^{13}$CO lines show similar line widths at all three positions. Peak temperatures of [\[\]]{} hardly drop between the center and the two outer positions. Resulting integrated intensities and luminosities drop by a factor of $\sim4$ only.
The observed CO 2–1/1–0 ratios lie between 0.6 and 0.8 for all three positions and do not peak at the center. This indicates that not even the 2–1 line is thermalized in M51. However, the CO 3–2/2–1 ratios do not drop as would be expected but equal the CO 2–1/1–0 ratio or even exceed them, while staying below 0.8. This is difficult to explain with a single component model as we will show below. The center CO 3–2/1–0 ratio of 0.55 is in agreement with ratios previously found with single-dish telescopes which range between $0.5-0.8$ at beam sizes of $\sim14''$ to $24''$ [@matsushita1999; @mauersberger1999; @wielebinski1999]. Interferometric observations at $\sim4''$ resolution tracing the dense nuclear gas show a high 3–2/1–0 ratio of 1.9 @matsushita2004.
High central column densities are indicated in this work by the rather low $^{12}$CO/$^{13}$CO 2–1 ratio of 4.6. In contrast, the outer postions show ratios between 6 and 14.
We find CI vs. $^{13}$CO 2–1 ratios of $\sim1$ at the center and at ($-84,-84$) while (72,84) exhibits a high ratio of 3.2.
Comparison
----------
The gradients of [\[\]]{} luminosities with galacto-centric distances are strikingly different in M51 and M83. M83 is much more centrally peaked, [\[\]]{} luminosities drop by a factor of 18 at only 1.8kpc distance in M83. In contrast, luminosities in M51 drop by only a factor of 4 at galacto-centric distances which are more than a factor of 3 larger, i.e. at 5.8kpc. However, the central [\[\]]{} luminosities of M51 and M83 agree within 40%.
The line widths observed at the outer positions of M51 and M83 are typical for the disks of these two galaxies [@handa1990; @gb1992]. See Table2 of @gb1993b for a compilation of CO line widths found in these and several other galaxies.
The CI vs. $^{13}$CO 2–1 ratio in the Milky Way is often found to be 1 [e.g. @keene1995] while @israel2004 [@israel_baas2002] find a strong variation of this ratio for 15 galactic nuclei. The [\[\]]{} line is stronger than the $^{13}$CO 2–1 line for all but three galaxy centers. The highest ratios are about 5. Here, we find a variation between 1.3 and 4.
For galaxy centers, @israel_baas2002 found that this ratio is well correlated with the [\[\]]{} luminosity covering a range of 160Kkms$^{-1}$ kpc$^2$ in the active nucleus of NGC3079 down to $\sim1$Kkms$^{-1}$ kpc$^2$ in the quiescent center of Maffei2. Here, we increase the range down to 0.11Kkms$^{-1}$ kpc$^2$ (Table\[tab-intensities\]) at the same resolution of $10''$. In contrast to the galaxy centers, the spiral arm positions observed here do not show a systematic correlation between the [\[\]]{}/$^{13}$CO line ratio and [\[\]]{} luminosity.
----------------------------------- --------------- ------- -------- -------- -------- --------------- --------------- ------- -- --
($\Delta \alpha$,$\Delta \delta$) $R_{\rm gal}$ CO 1–0 CO 2–1 CO 3–2 $^{13}$CO 1–0 $^{13}$CO 2–1
$80''$ $80''$ $80''$ $22''$ $10''$
(1) (2) (3) (4) (5) (6) (7) (8) (9)
[**M83:**]{}
(0,0) 0 79.55 (2) 50.63 47.03 30.15 19.36 21.33
(-80,-72) 1.93 7.57 (0.19) 18.17 18.06 8.35 4.6 5.1
(89,38) 1.76 4.53 (0.11) 17.03 15.93 7.84 4.28 3.27
[**M51:**]{}
(0,0) 0 16.36 (2.78) 43.42 31.53 23.68 7.34 12.65
(72,84) 5.4 4.59 (0.78) 15.32 12.21 9.19 2.27 1.43
(-84,-84) 5.77 4.03 (0.69) 17.29 9.85 7.41 4.01 3.09
----------------------------------- --------------- ------- -------- -------- -------- --------------- --------------- ------- -- --
--------------------------------- ------------------ ------------------ -------- -- -------- -- -------- -- --------------- -- --------------- -- --------------- -- -------------- -- -- -- -- -- -- -- -- -- -- -- -- -- --
[$\Delta\alpha,\Delta\delta$]{} $\Phi_B^{80/10}$ $\Phi_B^{80/21}$ CO 3–2 CO 3–2 CO 2–1 CO 1–0 CO 2–1 [\[\]]{} 1–0 [\[\]]{} 1–0
CO 1–0 CO 2–1 CO 1–0 $^{13}$CO 1–0 $^{13}$CO 2–1 $^{13}$CO 2–1 CO 3–2
[**M83:**]{}
(0,0) 0.22 0.28 0.6 0.64 0.93 9.34 10.02 3.73 0.58
(-80,-72) 0.37 0.45 0.46 0.46 0.99 8.77 9.58 1.48 0.34
(89,38) 0.48 0.47 0.46 0.49 0.93 8.46 10.14 1.38 0.28
[**M51:**]{}
(0,0) 0.56 0.68 0.55 0.76 0.73 8.31 4.62 1.4 0.4
(72,84) 0.62 0.73 0.6 0.76 0.8 9.27 13.81 3.19 0.3
(-84,-84) 0.52 0.72 0.43 0.75 0.57 6.04 6.09 1.25 0.27
--------------------------------- ------------------ ------------------ -------- -- -------- -- -------- -- --------------- -- --------------- -- --------------- -- -------------- -- -- -- -- -- -- -- -- -- -- -- -- -- --
[crrrrrrrr]{} $\Delta\alpha/\Delta\delta$ & $\chi_{\rm min}^2$ & $n_{\rm loc}$ & $T_{\rm kin}$ & $N$(CO)$/\Delta$v & $N$(CO) & $M$ & $n_{\rm av}$\
$['','']$ & & \[cm$^{-3}$\] & \[K\] & \[$10^{16}$cm$^{-2}$/kms$^{-1}$\] & \[$10^{16}$cm$^{-2}$\] & \[$10^6\,$[M$_{\odot}$]{}\] & \[cm$^{-3}$\]\
\
($ 0$,$ 0$) & 1.7 & 3000. & 15.0 & 3.2 & 15.79 & 65.16 & 0.42 &\
($-80$,$-72$) & 3.5 & 3000. & 12.5 & 3.2 & 7.58 & 31.29 & 0.20 &\
($ 89$,$ 38$) & 2.1 & 3000. & 12.5 & 3.2 & 6.69 & 27.60 & 0.18 &\
\
($ 0$,$ 0$) & 3.6 & 30000. & 12.5 & 3.2 & 17.41 & 484.67 & 0.18 &\
($ 72$,$ 84$) & 4.0 & 3000. & 15.0 & 3.2 & 2.07 & 57.76 & 0.02 &\
($-84$,$-84$) & 7.4 & 10000. & 12.5 & 3.2 & 5.46 & 151.89 & 0.06 &\
Physical Conditions
===================
Simple homogeneous models {#sec-lte}
-------------------------
As shown above, the observed CO line ratios cannot be explained with a simple LTE analysis. As a first step in order to estimate the kinetic temperatures and local densities of the CO emitting gas, we present in this section the results of slightly more realistic escape probability radiative transfer calculations of homogeneous spherical clumps [@stutzki_winnewisser1985]. We assumed a $^{12}$CO/$^{13}$CO abundance ratio of 40 [@mauersberger_henkel1993]. As input, we use the four ratios of integrated intensities (Table\[tab-ratios\]): $^{12}$CO 3–2/2–1, $^{12}$CO 2–1/1–0, $^{12}$CO/$^{13}$CO 1–0, and $^{12}$CO/$^{13}$CO 2–1. We calculated model intensities of the three transitions for column densities $10^{14}<N$(CO)/$\Delta$v/(cm$^{-2}$/(Kkms$^{-1}$)$<10^{22}$, local H$_2$ densities $10<n_{\rm loc}/$cm$^{-3}<40$, and kinetic temperatures $10<T_{\rm kin}$/K$<40$. The modelled ratios were compared with the observed ratios taking into account the observational error of 21% to derive the $\chi^2$. The best fitting $N$(CO)/$\Delta$v, $n$, and $T_{\rm kin}$ together with the corresponding minimum reduced $\chi^2$ are listed in Table\[tab-one-comp-model\].
Below, we first describe the results of the one-component fits for the positions observed in M83 and M51. Much more complete physical models of the emitting regions are presented in the next section\[sec-pdr-analysis\].
#### M83: {#m83-1}
At $80''$ resolution, the $^{12}$CO line ratios do not vary significantly between the center and the two bright spiral arm positions observed here. A $^{12}$CO 3–2/1–0 ratio of 0.5 indicates an excitation temperature of $\sim10$K assuming optically thick thermalized emission and simply using the detection equation. The escape probability analysis leads to a similar result (Table\[tab-one-comp-model\]). At all three positions, the ratios are well modelled by a one-component model with a rather low kinetic temperature of only $12-15\,$K and a density of $n_{\rm
loc}=3\,10^3$cm$^{-3}$. This result does not exclude the existence of a warmer and denser gas phase as would be traced by higher CO transitions or the 63$\mu$m [\[\]]{} line as discussed below. For the center, @israel_baas2001 have in fact deduced a warmer phase by including observations of the CO 4–3 line in their radiative transfer analysis.
The $^{12}$CO and $^{13}$CO line ratios found in M83 are characteristic for dynamically active or starburst regions in the classification scheme of @papadopoulos2004. In this scheme, extreme starbursts would show similar $^{12}$CO 2–1/1–0 and 3–2/1–0 ratios but much higher $^{12}$CO/$^{13}$CO ratios.
#### M51: {#m51-1}
In the escape probability analysis of CO and $^{13}$CO ratios in M51, we discarded solutions leading to temperatures below 12.5K and densities below $10^3$cm$^{-3}$ as unphysical. Since the CO $J=$3–2 is strong at all positions, densities and temperatures must be higher.
The CO 3–2/2–1 ratios are $\sim0.8$ at all positions in M51, significantly higher than in M83. This indicates higher densities than found in M83. Indeed, the escape probability analysis finds densities between $3\,10^3$ and $3\,10^4$cm$^{-3}$. The $^{12}$CO/$^{13}$CO ratios agree with this solution for low temperatures of $\sim 12\,$K. However, the observed CO 2–1/1–0 ratios are too low to agree with this solution. Neither the temperatures nor the densities are well constrained, and minimum chi squared values are high. This shows the short coming of a one-component model even when trying to model only the three lowest rotational CO transitions. And it is in fact in agreement with the finding of @gb1993a who used the lowest two transitions of $^{12}$CO and $^{13}$CO and couldn’t find a set of $T_{\rm kin}$ and $n_{\rm loc}$ fitting simultaneously the line ratios for the arms and for the central position.
![Comparison of the observed intensity ratios [\[\]]{}(63 )/[\[\]]{}$_{\rm PDR}$ and ([\[\]]{}(63 )+[\[\]]{}$_{\rm
PDR}$)/TIR$_c$ with PDR models [@kaufman1999] at the six positions in M83 and M51. \[fig\_stand\]](stand_epsilon_oivscii.eps){width="8cm"}
{width="15cm"}
PDR Analysis {#sec-pdr-analysis}
------------
To further constrain the physical conditions at the observed positions in M83 and M51, we compare the observed line intensity ratios with the results of the model for Photon Dominated Regions (PDRs) by @kaufman1999 [@tielens1985]. The physical structure is represented by a semi-infinite slab of constant density, which is illuminated by FUV photons from one side. The model takes into account the major heating and cooling processes and incorporates a detailed chemical network. Comparing the observed intensities with the steady-state solutions of the model, allows for the determination of the gas density of H nuclei, $n$, and the FUV flux, G$_0$, where G$_0$ is measured in units of the @habing1968 value for the average solar neighborhood FUV flux, $1.6\times10^{-3}$ ergs cm$^{-2}$ s$^{-1}$. As has been pointed out in detail by @kaufman1999 in their Section 3.5.1 and several other authors, the application of these models to extragalactic observations is not straightforward since individual molecular clouds are not resolved in single-dish observations and several phases of the ISM are therefore observationally coexistent within each beam. The additional many degrees of freedom in the parameter space for more complex models, however, are ill constrained by the few observed, beam-averaged line ratios. Hence, simplistic models, e.g. with only a single source component, are used to at least derive average properties of the complex sources. Nevertheless, it is possible to obtain some insight into the spatial structure and the local excitation conditions, as we will show.
--------------------------------- --------------- --------------------- ------------ ------------- ---------- ------------------ ----------------- ------------------- -- -- --
($\Delta\alpha$,$\Delta\delta$) $R_{\rm gal}$ $n$ log(G$_0$) T$_{\rm s}$ $\chi^2$ log(G$_{0,obs}$) $\phi_{\rm UV}$ $\phi_A^{\rm CI}$
$['','']$ \[kpc\] \[10$^4$cm$^{-3}$\] \[K\]
(1) (2) (3) (4) (5) (6) (7) (8) (9)
[**M83:**]{}
($ 0, 0$) 0.00 4.0 1.50 76. 2.8 2.16 4.57 0.019
($-80,-72$) 1.93 4.0 1.50 76. 12.9 1.51 1.03 0.011
($ 89, 38$) 1.76 4.0 1.50 76. 15.3 1.47 0.94 0.010
[**M51:**]{}
($ 0, 0$) 0.00 4.0 1.25 66. 3.2 1.88 4.31 0.013
($ 72, 84$) 5.40 4.0 1.25 66. 5.2 1.32 1.17 0.016
($-84,-84$) 5.77 4.0 1.25 66. 14.1 1.19 0.88 0.009
--------------------------------- --------------- --------------------- ------------ ------------- ---------- ------------------ ----------------- ------------------- -- -- --
### [\[\]]{} emission from the ionized and neutral medium {#sec-cii-ion}
Carbon has a lower ionization potential (11.26eV) than hydrogen, so that [\[\]]{} emission arises not only from photon dominated regions, but also from the ionized phases of the ISM and from the diffuse neutral medium traced by .
Analyzing the Milky Way FIR line data obtained with FIRAS/COBE [@fixsen1999], @petuchowski1993 argue that slightly more than half the [\[\]]{} emission in the Milky Way arises from PDRs, the remainder from the extended low density warm ionized medium or diffuse ionized medium (DIM), and an insignificant portion from the ordinary cold neutral medium (CNM).
#### [\[\]]{} emission from the ionized medium. {#sec_cii_di}
Here, we use the observed [\[\]]{}(122$\mu$m) and [\[\]]{} lines, to estimate the fraction of [\[\]]{} originating from the ionized medium. However, a thorough analysis would need more FIR emission line data from the ionized medium, in particular the [\[\]]{}(205$\mu$m) line, to discriminate the relative importance of the different phases of the ionized medium [@bennett1994]. Extragalactic observations of the [\[\]]{}(205$\mu$m) are however very rare to date [@petuchowski1994]. The components of the ionized phase of the ISM which contribute to the [\[\]]{} emission are dense HII regions ($n_{\rm e}\ge~100$cm$^{-3}$) and the diffuse ionized medium (DIM) ($n_e\sim~2$cm$^{-3}$) [@heiles1994].
The fraction of [\[\]]{} stemming from regions depends strongly on the electron density of the ionized medium. @carral1994 showed that upto 30% of [\[\]]{} stems from regions when electron densities exceed 100cm$^{-3}$. For dense HII regions ($n_{\rm e}\gg n_{\rm cr}$), model calculations suggest ${[\ion{C}{ii}]}/{[\ion{N}{ii}]}\ = 0.28 ({\rm C/N})_{\rm dense}$ [@rubin1985] where (C/N)$_{\rm dense}$ is the abundance ratio. The Galactic abundance ratio found in dense regions is 3.8 [@rubin1988; @rubin1993]. We thus expect to find an intensity ratio of $${[\ion{C}{ii}]}/{[\ion{N}{ii}]}_{\rm ion,{\ion{H}{ii}}} = 1.1.
\label{eq-ciinii-hii}$$
Observations of radio free-free emission suggest that the DIM has typical volume densities of n$_{\rm e}\sim2~{\rm cm}^{-3}$ [@mezger1978] and temperatures of T$_{\rm e} = 4000$ K [@mueller1987]. Densities are thus considerably lower than the critical densities of 310 cm$^{-3}$ for the [\[\]]{}(122) line and 50cm$^{-3}$ for the [\[\]]{}(158) line [@genzel1991]. In this limit, the total power emitted in the collisionally excited fine-structure lines is simply the photon energy times the upward collision rate, as given by @heiles1994. The resulting ratio of intensities is: ${[\ion{C}{ii}]}/{[\ion{N}{ii}]}(122)\ = 3.05 {\rm (C/N)}_{\rm diff}$. Galactic absorption line measurements of diffuse gas give (C/N)$_{\rm
DIM} = 1.87$ [@sofia1997; @meyer1997], which implies an expected intensity ratio of $$({[\ion{C}{ii}]}/{[\ion{N}{ii}]}(122))_{\rm ion,DIM}=5.7.
\label{eq-ciinii-dim}$$
The preceeding section follows basically the arguments of MKH01 and @contursi2002.
The metallicities, parametrized by the Oxygen abundances, have been found to be only slightly supersolar in M83 and M51. @zaritsky1994 find (O/H) abundances of $1.5\,10^{-3}$ and $1.9\,10^{-3}$ respectively, at 3kpc galacto-centric distance, from visual spectra of regions, which is about a factor 3 higher than the solar metallicity of $0.46\,10^{-3}$ [@asplund2004]. Abundance gradients with radius are found to be shallow in M83 and M51 [@zaritsky1994]. More recent observations with ISO/LWS and modelling of the CCM10 region of M51 by @garnett2004 indicate instead that the (O/H) abundances are about a factor of 2 less, i.e. roughly solar. In addition, [@garnett1999] showed that the (C/N) abundance ratio, which is of interest for the [\[\]]{}/[\[\]]{} ratio discussed here, is independent of metallicity in both normal and irregular galaxies. We therefore use Galactic abundances to estimate the intensity ratios in M83 and M51. For the Milky Way, @heiles1994 have estimated that [\[\]]{} originates predominantly from the DIM, contributing $\sim70$%. This was derived from the observed [\[\]]{}(122)/[\[\]]{}(205) ratio using photo ionization models. However, the [\[\]]{}(205) line has not yet been observed in M83 and M51. We therefore use Eqs.\[eq-ciinii-dim\] and \[eq-ciinii-hii\], assuming that [\[\]]{} stems solely from regions, or alternatively, solely from the DIM.
Next, we can then derive the fraction of [\[\]]{} emission originating from PDRs: $$\begin{aligned}
{[\ion{C}{ii}]}_{\rm PDR} & = & {[\ion{C}{ii}]}_{\rm obs} - \Bigl(\frac{{[\ion{C}{ii}]}}{{[\ion{N}{ii}]}}\Bigr)_{\rm ion} \times {[\ion{N}{ii}]}(122)_{\rm obs}
\label{eq-ciipdr}\end{aligned}$$
The ratios of observed [\[\]]{} versus [\[\]]{} intensities vary between 3.6 and 7.4 at the six positions observed in M83 and M51 (Table\[tab-ciicorr\]). @garnett2004 found a higher ratio of 11.8 at the CCM10 position in M51.
The ratios found in M83 and M51 lie at the low end of the ratios MKH01 found in the sample of 60 unresolved normal galaxies studied. They show a mean ratio of 8 and a scatter between 4.3 and 29. @contursi2002 observed ratios of more than 7.7 in NGC6946 and ratios of greater than 4 and 10 in NGC1313. @higdon2003 found ratios between $\ge2.6$ and 20 in M33.
If the [\[\]]{} emission originates only from the diffuse ionized medium, then the major fraction of [\[\]]{} arises from this phase, and only a small fraction from PDRs (Table\[tab-ciicorr\]). The observed [\[\]]{}/[\[\]]{} is $\le$5.8 at 3 positions, including the two nuclei, which would indicate that no [\[\]]{} emission at all arises from PDRs. This is however unrealistic, since the emission of the [\[\]]{}(63$\mu$m) line, stemming from warm, dense PDRs, is strong compared to the [\[\]]{} lines, especially in the nuclei (Table\[tab-isodata\]). For this reason, we discard this solution.
Assuming on the other hand, that the fraction of [\[\]]{} from the ionized medium and all [\[\]]{} emission stem only from the dense regions, then a fraction of only 15% to 30% of [\[\]]{} originates from this phase (Eq.\[eq-ciipdr\]), while 70% to 85% of the observed [\[\]]{} emission then stems from PDRs. We prefer this solution and use it in the PDR analysis discussed below. In an ISO/LWS study of star-forming regions in M33, @higdon2003 have recently used FIR lines of the ionized medium, i.e. [\[\]]{}(88$\mu$m), [\[\]]{}($52\mu$m) and others to estimate the electron densities and other parameters of the emitting gas, estimating that between 7% and 47% of [\[\]]{} stems from regions, for their sample of positions. They conclude that the DIM is not needed to explain the observations.
[crrrrrrrrr]{} & [\[\]]{}$_{\rm obs}$ & & [\[\]]{}$_{\rm PDR}$ & & [\[\]]{}(63) & $\epsilon$\
& [\[\]]{}(122) & & [\[\]]{}$_{\rm obs}$ & & [\[\]]{}$_{\rm PDR}$ & \[%\]\
\
[**M83:**]{}\
$ 0, 0$ & 5.8 & & 0.02 & & 57.41 & 0.12\
$-80,-72$ & 7.3 & & 0.22 & & 3.18 & 0.22\
$ 89, 38$ & 5.7 & & 0.00 & & – & 0.17\
[**M51:**]{}\
$ 0, 0$ & 3.6 & & 0.00 & & – & 0.08\
$ 72, 84$ & 7.3 & & 0.22 & & 3.59 & 0.16\
$-84,-84$ & 7.4 & & 0.23 & & 3.78 & 0.21\
\
[**M83:**]{}\
$ 0, 0$ & 5.8 & & 0.81 & & 1.31 & 0.21\
$-80,-72$ & 7.3 & & 0.85 & & 0.83 & 0.36\
$ 89, 38$ & 5.7 & & 0.81 & & 0.99 & 0.35\
[**M51:**]{}\
$ 0, 0$ & 3.6 & & 0.69 & & 1.06 & 0.16\
$ 72, 84$ & 7.3 & & 0.85 & & 0.91 & 0.25\
$-84,-84$ & 7.4 & & 0.85 & & 1.02 & 0.33\
#### [\[\]]{} emission from the diffuse neutral medium.
The predicted [\[\]]{} emission from the atomic gas has in general, for many galactic nuclei, been found to be far too weak to account for the observed [\[\]]{} emission [@stacey1991] because the density is not high enough to appreciably excite the [\[\]]{} emission at the measured column densities. This view was confirmed by [@carral1994] who conducted a detailed study of FIR cooling lines of NGC253 and NGC3256. Both in M51 and in M83, no large-scale correlation between emission and that of [\[\]]{} is seen, indicating again that [\[\]]{} does not trace the diffuse neutral medium [@nikola2001; @crawford1985].
@nikola2001 used column densities [@tilanus_allen1991; @rots1990] to derive the contribution to the [\[\]]{} emission in M51, assuming the same range of temperatures, densities, and ionization fractions for the warm and cold neutral medium (WNM, CNM) as have been found for the Milky Way. They find that the contribution of the WNM is negligible for most of the M51 disk except the northwest, which was not studied here. The contribution of the CNM is estimated to be less than 10%$-$20% in all regions but the northwest.
We have thus not corrected the [\[\]]{} emission for a possible contribution from the diffuse neutral medium.
### The infrared continuum and the FUV field \[sec\_firfuvcomp\]
The stellar FUV photons heat the molecular gas and dust which subsequently cools via the FIR dust continuum and, with a fraction of less than $\sim1$% [@stacey1991],MKH01 via [\[\]]{}, [\[\]]{}(63), and other cooling lines. To the extent that filling factors are 1 and other heating mechanisms like cosmic ray heating can be neglected, the observed TIR continuum intensity should equal the modelled FUV field. This is also expected if a constant fraction of FUV photons escape without impinging on cloud surfaces.
The PDR model of @kaufman1999 assumes a semi-infinite slab illuminated from one side only. For the extragalactic observations described here, we however have several PDRs within one beam and the clouds are illuminated from all sides. Hence, the optically thin total IR intensity stems from the near and far sides of clouds. Here, this is taken into account by dividing the observed TIR by 2 [@kaufman1999]: ${\rm TIR}_{\rm c} = \rm{TIR}/2 = 2.3\times\rm{FIR}/2$ (cf. Sec.\[sec-iras\_data\]). While this correction holds exactly only for finite plane parallel slabs illuminated from both sides, it is a good first approximation. The corrected TIR can then be used to derive the corresponding FUV intensity via $G_{0,{\rm obs}}$ = TIR$_c$$4\pi/(2\times1.6\,10^{-3})$ergs cm$^{-2}$ s$^{-1}$. Following the arguments of @kaufman1999, the additional factor 2 takes into account equal heating of the grains by photons outside the FUV band, i.e. by photons of $h\nu<6\,$eV. We find a variation by one order of magnitude, $15<G_{0,{\rm obs}}<144$ (Table\[tab\_pdrfit\], Fig. \[fig\_pdrplotm83m51\]).
### First estimates of FUV field and density
The two intensity ratios [\[\]]{}(63)/[\[\]]{}$_{\rm PDR}$ and ([\[\]]{}(63)$+$[\[\]]{}$_{\rm PDR}$)/TIR$_c$, of the two major PDR cooling lines and the continuum, have been used extensively to derive the density and FUV field of the emitting regions (e.g. MKH01). Since [\[\]]{}(63) and [\[\]]{} are the dominant coolants, the latter ratio is a good measure of the photoelectric heating efficiency $\epsilon$ [e.g. @kaufman1999]. The former ratio measures the relative importance of [\[\]]{} vs [\[\]]{}(63) cooling. For high FUV fields and high densities, the ratio becomes larger than one [@kaufman1999].
In M83 and M51, the intensity ratio [\[\]]{}(63)/[\[\]]{}$_{\rm PDR}$ varies only slightly between 0.8 and 1.3 (Table\[tab-ciicorr\]b). The heating efficiency varies between $\sim0.25$ and 0.36% at the outskirt positions while it drops to below 0.21% in the centers.
The values which we find in M83 and M51 lie within the range covered by MKH01, who find heating efficiencies ranging between $\sim0.3$% and $\sim0.05$% for the 60 galaxies studied, while the [\[\]]{}/[\[\]]{} ratios range between 0.3 and $\sim10$. Though the scatter is large, the heating efficiency tends to be high $>0.15\%$ for [\[\]]{}/[\[\]]{} ratios of less than 2. The [\[\]]{}(63)/[\[\]]{}$_{\rm PDR}$ ratios found in M83 and M51 agree roughly with the average value found by MKH01, while the heating efficiencies in M83 and M51 span the average value of MKH01 upto the highest efficiencies found by them.
MKH01 corrected the observed [\[\]]{} emission by roughly 50% when taking into account the contribution from the ionized medium, based on the Milky Way results. Here, we corrected by only 15% to 30% (Table\[tab-ciicorr\]b). This uncertainty in how best to correct the [\[\]]{} fluxes, needs to be considered when comparing the derived heating efficiencies and [\[\]]{}/[\[\]]{} ratios.
The small scatter of the observed two ratios at the 6 positions in M83 and M51 indicates that the emitting gas has similar physical properties. Comparison with the results of the Kaufman PDR model shows that two solutions exist (Figure\[fig\_stand\]). The data can be reproduced either by high FUV fields at low densities or by low FUV fields and high densities. The high-$G_0$ solution indicates $2.5\le{\rm log}(n/{\rm cm}^{-3})\le3.2$ and $2.4\le{\rm log}G_0\le3$. As we will show, the low-$G_0$ solution is less plausible. It indicates rather high densities of $4.3\le{\rm log}(n/{\rm
cm}^{-3})\le4.5$ and low FUV fields of $0.1\le{\rm log}G_0\le0.5$. In this case, the observed [\[\]]{} intensities are more than three orders of magnitude larger than the modelled intensities which would indicate that many PDR slabs along the lines of sight. Since the optical depth of the [\[\]]{} line in the line centers is expected to be about one [@kaufman1999], this scenario is discarded. This argument also holds when velocity filling is taken into account, since the velocity filling factor is $<40$ at all positions, as discussed below. We note that @higdon2003 in their analysis of ISO/LWS data of M33, also discussed the two possible PDR solutions, and, following a different line of reasoning, also preferred the high-$G_0$ solution.
The $n,G_0$ values we find for the high-$G_0$ solution, agrees with the average values found by MKH01 who also exclude the low-$G_0$ solution. Their sample of 60 unresolved galaxies covers a slightly larger range of values: $2<{\rm log}(n/{\rm cm}^{-3})<4.25$ and $2.5<{\rm log}G_0<5$. Our values also agree with the average value found in NGC6946 by @contursi2002.
### PDR model fitting of five line ratios
In order to determine with greater confidence the densities and UV fluxes which can explain the intensity ratios, we have performed a $\chi^2$ fitting of the observed 5 line intensity ratios [\[\]]{}$_{\rm
PDR}$/[\[\]]{}(1–0), [\[\]]{}(1–0)/CO(3–2), CO(3–2)/CO(1–0), [\[\]]{}/CO(3–2), and, [\[\]]{}(63)/[\[\]]{}$_{\rm PDR}$ relative to the predictions of the PDR model by @kaufman1999: $$\chi^2 = \frac{1}{4-2}\sum_{i=1}^{5} \Bigl( \frac{R^{\rm
obs}_i-R^{\rm mod}_i}{\sigma_i}\Bigr)^2$$ with the observed and modelled ratios $R^{\rm obs}_i$ and $R^{\rm mod}_i$, and the error $\sigma_i$ which is assumed to be 20% for all ratios[^3]. In Fig. \[fig\_pdrplotm83m51\] each panel shows the observed line intensity ratios and the calculated reduced $\chi^2$ in greyscale for each observed position in M83 and M51. Table \[tab\_pdrfit\] summarizes the local density ($n$) and FUV radiation field ($G_0$) at the position of minimum reduced $\chi^2$ corresponding to the best fit models thus identified. The [\[\]]{}(1–0)/CO(3–2) and the CO(3–2)/(1–0) ratios are excellent tracers of the local densities, almost independent of $G_0$ for $G_0>10$. The sensitivity of [\[\]]{} on the FUV field is reflected by the line ratios which include [\[\]]{}. The combination of all these ratios therefore allows, in principle, to deduce $n$ and $G_0$.
#### Densities.
Though the quality of the fit varies strongly between the six positions, the best fitting FUV field and density are almost identical: $n=10^4$${\rm cm}^{-3}$ and $18<G_0<32$ (Table\[tab\_pdrfit\]). Model surface temperatures are $\sim70$K. This is slightly lower than the upper energy level of [\[\]]{}(158), $E_{\rm up}/k=92$K, and much lower than the corresponding level of [\[\]]{}(63), $E_{\rm up}/k=228$K. Since the critical densities of the two [\[\]]{} lines for collisions with H are also high, $>10^5$cm$^{-3}$, the [\[\]]{} lines are subthermally excited. Nevertheless, gas cooling is dominated by [\[\]]{} and [\[\]]{}, contributions from H$_2$ or Si, covered by ISO/SWS, are negligible at these low temperatures. The best fitting density agrees within a factor of 3 with the local densities derived above via simple radiative transfer analysis, using only the $^{12}$CO and $^{13}$CO line ratios (Table\[tab-one-comp-model\]).
#### UV filling factor.
At the four spiral arm positions, the total infrared continuum agrees perfectly with the best fitting FUV field (Table\[tab\_pdrfit\]), which is an independent confirmation of the validity of the PDR analysis using the five line ratios. The active center regions of M83 and M51 show filling factors of $\sim4$, indicating that other sources than PDRs heat the dust leading to the high observed TIR intensities, e.g. massive protostars.
#### Quality of the fits.
The minimum reduced chi squared of the 4 independent ratios lie between 3 and 15 at all positions (Table\[tab\_pdrfit\]). While the two nuclei and the NE-spiral arm position of M51 show $\chi^2$ of better than 5, the other three spiral arm positions show rather poor $\chi^2$ values of 13–15.
Inspection of Figure\[fig\_pdrplotm83m51\] shows that, at the latter three positions, the [\[\]]{}$_{\rm PDR}$/[\[\]]{} ratio indicates higher FUV fields than the best fitting solution. In addition, Figure\[fig\_pdrplotm83m51\] also shows that the CO 3–2/1–0 line ratios indicates systematically lower densities than the observed [\[\]]{}/CO 3–2 ratios. This holds to varying degrees for all positions and for $G_0>10$. By assuming [\[\]]{} intensities which are a factor 2 higher, the quality of the fit is considerably improved, while the best fitting $G_0$ and density stay constant at all positions. At ($-80$,$-72$) in M83 for example, the $\chi^2$ is improved from 12.9 to 3.4. This indicates that the beam filling factors of the [\[\]]{} emission, derived from the CO 2–1 data (Table\[tab-ratios\]), are too small, i.e. [\[\]]{} is more extended than CO. We also conclude that our results are consistent with the assumption that only the dense ionized medium contributes to the [\[\]]{} emission. There is no need for an extended diffuse component.
Overall, the observed five ratios cannot be well fitted with a single plane-parallel PDR model of constant density at any of the positions. This is not surprising as the excitation requirements of the various tracers collected here vary widely. Especially, the critical densities vary between $5\,10^2$cm$^{-3}$ for the lower [\[\]]{} transition and $5\,10^5$cm$^{-3}$ for the [\[\]]{} 63$\mu$m transition. Any density gradients in the emitting medium may thus lead to the above discrepancies with a single PDR model depending also on the chemical and temperature structure. In addition, the large difference between local densities derived here and beam averaged densities of more than three orders of magnitude (Table\[tab-one-comp-model\]) shows that the emitting volume must be filled with very small but dense structures. From many Galactic observations, it is expected that these structure show a spectrum of masses, adding to the complexity ignored here.
#### Absolute intensities.
When comparing the observed intensities with the model results, the velocity filling has also to be taken into account. The observed [\[\]]{} line widths $\Delta{\rm v}_{\rm obs}$ range between 30 and 130kms$^{-1}$ FWHM (cf.Sec.\[sec\_m83\_spectra\] and \[sec\_m51\_spectra\]) measuring the dispersion of clouds within the beam for these extragalactic observations. On the other hand, the microturbulent velocity dispersion $\delta_v$ of the gas of one PDR model is set to 1.5kms$^{-1}$ [@kaufman1999], corresponding to a Gaussian FWHM $\Delta{\rm v}_{\rm mod}$ of 3.5kms$^{-1}$ as is typical for individual Galactic clouds. To calculate the [\[\]]{} area filling factors (Table\[tab\_pdrfit\]), we divided the observed [\[\]]{} intensities by the predicted [\[\]]{} intensity from the best fitting model, corrected for beam (Table\[tab-ratios\]) and velocity filling, viz., $$\Phi_A^{\rm CI}=\frac{({{\rm [CI]}_{\rm obs}}\times\Phi_B^{80/10})}
{{\rm [CI]}_{\rm mod}}\times\frac{\Delta{\rm v}_{\rm mod}}{\Delta{\rm
v}_{\rm obs}}.$$ [\[\]]{} emission fills only a few percent of the $80''$ beam, the derived area filling factors vary between $(1-2)\,10^{-2}$ (Table\[tab\_pdrfit\]), consistent with the low volume filling factors described above.
Summary
=======
We have studied all major submillimeter and far infrared cooling lines together with the dust total infrared continuum at the center positions of the two galaxies M83 and M51 and at four spiral arm positions. We observed [\[\]]{} 1–0 at the six positions at $10''$ resolution. Complementary [\[\]]{}, [\[\]]{}(63), and [\[\]]{}(122) data were obtained from ISO/LWS at $80''$ resolution. CO maps of the lowest three transitions were obtained from the literature and smoothed to the ISO/LWS resolution. We also obtained pointed $^{13}$CO 1–0 and 2–1 data at all positions. In order to allow a comparison of all these data, the [\[\]]{} and $^{13}$CO data were scaled with beam filling factors derived from the $^{12}$CO data. For completeness, we also obtained the total far-infrared continuum intensities from HIRES/IRAS $60\,\mu$m and $100\,\mu$m data.
- Integrated intensities peak in the two centers. However, M83 is much more centrally peaked than M51 as seen in CO and [\[\]]{} as is already seen in the spectra (Figs.\[fig-m83-spec\],\[fig-m51-spec\]). This is seen even more drastically in the drop of [\[\]]{} luminosities with galacto-centric distance. In M83, luminosities drop by more than one order of magnitude over $\sim2$kpc, while in M51, they drop by only a factor of $\sim$4 over $\sim6$kpc. Obviously, this analysis should be refined by more observations of [\[\]]{} at different radii.
- The $^{12}$CO 3–2/1–0 line ratios lie below 0.6 at all positions, indicating subthermal excitation of the 3–2 line, i.e. densities are less than $10^5$cm$^{-3}$. This is confirmed by more detailed analysis using escape probability and PDR models. The former homogeneous models indicate densities between $3\,10^3$ and $3\,10^4$cm$^{-3}$ from CO and $^{13}$CO line ratios.
- We estimated the fraction of observed [\[\]]{} emission originating from the ionized medium by using the [\[\]]{} 122$\,\mu$m data which solely traces this medium. In the absence of additional data, which would allow to seperate the contributions from the different phases of the ionized medium, we argue that dense regions are the primary source of [\[\]]{} emission. These emit 15% to 30% of the observed [\[\]]{} emission while the remainder stems from PDRs.
- The gas heating efficiency was calculated from the ratio of the two major gas coolants [\[\]]{} at 63$\mu$m and [\[\]]{}$_{\rm PDR}$ at 158$\mu$m versus the total infrared intensity. The efficiency is low in the centers showing ratios of (1.6–2.1)$10^{-3}$, while the outer positions show higher ratios of (2.5–3.6)$10^{-3}$. The latter efficiencies lie at the high end of efficiencies observed by MKH01 in a sample of 60 unresolved normal galaxies.
- We fitted the observed line intensity ratios [\[\]]{}$_{\rm
PDR}$/[\[\]]{}(1–0), [\[\]]{}(1–0)/CO(3–2), CO(3–2)/CO(1–0), [\[\]]{}/CO(3–2), and, [\[\]]{}(63)/[\[\]]{}$_{\rm PDR}$ to the predictions of the PDR model of @kaufman1999. The best fits yield densities of $10^4$cm$^{-3}$ and FUV fields of $\sim\,G_0=20-30$ times the average interstellar, almost constant at the six positions studied here at $80''$ resolution. This finding may be a selection effect, since we selected the four outer positions for their high star forming activity. More observations of other less prominent positions are needed to study variation of physical parameters over the galaxy surfaces.
- Filling factors vary significantly between the center positions and the outer positions in both galaxies. The filling factor $\phi_{\rm UV}$ derived from the FUV field calculated from the observed total infrared intensity versus the fitted $G_0$ lies at 1 at the spiral arm positions, with a scatter of less than 20%. In contrast, the nuclei show ratios of $\sim4-5$. The density contrasts within the emitting gas must be high, given the local densities of $10^4$cm$^{-3}$ and the average densities, derived from CO column densities, of less than 1cm$^{-3}$, leading to very low volume filling factors. In accordance, the area filling factors of the [\[\]]{} emission are less than 2% at all positions.
- The reduced chi squared lie between 3 and 5 at three positions, including the two nuclei, while the fits are worse at the three other positions, which show $\chi^2$ = 13–15. It is shown that the fits can be significantly improved by assuming that [\[\]]{} emission is more extended.
While the present analysis led to the results listed above, it needs to be refined with new data and improved modelling. The assumption that all gas tracers have the same filling factors may not be justified. We find indications that [\[\]]{} is more extended than CO 2–1. Clearly, maps of [\[\]]{} at $10''$ resolution filling the ISO/LWS $80''$ beam would be needed to verify this conclusion. A related question is the nature of the interclump medium and whether it can be ignored when interpreting the submm/FIR emission. The fraction of [\[\]]{} emission from the different phases of the ionized medium has been addressed here by assuming that all emission stems from the dense ionized medium. While this is consistent with the PDR modelling, it probably is an oversimplification. Observations of the [\[\]]{} line at $205\,\mu$m would allow to refine this analysis. Such observations will become possible with a new generation of submm telescopes and receivers operating at high altitudes. In this regard, modelling neads to include the ionized medium. Improved PDR models need to consider a distribution of clouds within each beam, following mass and size distribution laws.
We thank Steve Lord for valuable discussions. We would like to thank the JCMT and the IRAM 30m staff for providing excellent support during several long runs at the Mauna Kea and the Pico Veleta. We are greatful to Michael Dumke for providing us with HHT CO 3–2 and 4–3 M51 data and Lucian Crosthwaite to provide NRAO CO 1–0 and 2–1 M83 data. And we thank Jim Brauher, Steve Lord, and Alessandra Contursi for providing us with the ISO/LWS line fluxes of both galaxies. The James Clerk Maxwell Telescope is operated by the Joint Astronomy Centre 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 work has benefited from research funding from the European Community’s Sixth Framework Programme. We made use of the NASA IPAC/IRAS/HiRes data reduction facilities.
[^1]: A description of IRAS HiRes reduction is available at [ http://irsa.ipac.caltech.edu/IRASdocs/hires\_over.html]{}
[^2]: Note that TIR corresponds to the bolometric FIR dust continuum emission I(FIR) used in the PDR models of @kaufman1999.
[^3]: There are two degrees of freedom since we use four independent line ratios to fit two parameters.
|
---
abstract: 'Recently, Kostelecky \[V.A. Kostelecky, Phys. Lett. B [**701**]{}, 137 (2011)\] proposed that the spontaneous Lorentz invariance violation (sLIV) is related to Finsler geometry. Finsler spacetime is intrinsically anisotropic and induces naturally Lorentz invariance violation (LIV). In this paper, the electromagnetic field is investigated in locally Minkowski spacetime. The Lagrangian is presented explicitly for the electromagnetic field. It is compatible with the one in the standard model extension (SME). We show the Lorentz–violating Maxwell equations as well as the electromagnetic wave equation. The formal plane wave solution is obtained for the electromagnetic wave. The speed of light may depend on the direction of light and the lightcone may be enlarged or narrowed. The LIV effects could be viewed as influence from an anisotropic media on the electromagnetic wave. In addition, the birefringence of light will not emerge at the leading order in this model. A constraint on the spacetime anisotropy is obtained from observations on gamma–ray bursts (GRBs).'
author:
- 'Zhe Chang$^{1,2}$[^1]'
- 'Sai Wang$^{1}$[^2][^3] [^4]'
title: Lorentz invariance violation and electromagnetic field in an intrinsically anisotropic spacetime
---
1. Introduction
===============
At experimentally attainable energy scales, Einstein’s special relativity (SR) is compatible with the present observations. However, the SR is believed to be modified at higher energy scales, such as the Planck scale which involves the effects of quantum gravity [@DSR01; @DSR02; @DSR03; @DSR04; @DSR05; @spacetime; @foam01; @spacetime; @foam02; @spacetime; @foam03; @spacetime; @foam04; @VSR; @Pavlopoulos; @KosteleckyS001; @SME01; @SME02]. The study on the string theory reveals that the Lorentz symmetry could be broken spontaneously in the perturbative framework [@KosteleckyS001; @SME01; @SME02]. The spontaneous Lorentz invariance violation (sLIV) involves nonzero vacuum expectation values of certain tensor fields. It characterizes the anisotropy of spacetime since nonzero vacuum expectations of tensor fields are related to certain preferred directions.
To demonstrate spirit of the sLIV, we review shortly the spontaneous symmetry breaking in the electroweak theory. The electroweak theory involves a Higgs field acquiring a nonzero vacuum expectation value, which leads to the mass terms of other particles. Similarly, certain tensor fields acquire nonzero vacuum expectation values in the sLIV framework. However, these expectation values take along the spacetime indices, which are different from the scalar one in the standard model (SM). Therefore, the velocities of particles and fields may influence propagations and interactions, respectively.
Actually, the sLIV terms are added into the Lagrangian of fields by considering the gauge invariance, renormalizability, etc. The vacuum expectation values of tensor fields become the coupling constants in the sLIV terms. This approach to introduce the sLIV effects is called the standard model extension (SME) [@SME01; @SME02], which is an effective field theory irrelative to the ultimately underlying theory. Obviously, the spacetime background is still Minkowskian in the SME. However, Minkowski spacetime should be amended together with Lagrangian of particles and fields if the Lorentz symmetry is violated (no matter spontaneously or not).
Recently, Kostelecky [@Kostelecky_Finsler] proposed that the SME is closely related to Finsler spacetime which is intrinsically anisotropic. The coupling constants in the sLIV terms could be related to certain fixed preferred directions in the Finsler structure. The most fundamental reason is that Finsler geometry [@Book; @by; @Rund; @Book; @by; @Bao; @Book; @by; @Shen] gets rid of the quadratic restriction on the spacetime structure such that the Finsler metric depends on directions of the spacetime. In addition, the isometric transformations reveal that non–Riemannian Finsler spacetime possesses fewer symmetries than Riemann spacetime [@Finsler; @isometry; @by; @Wang; @Finsler; @isometry; @by; @Rutz; @Finsler; @isometry; @Finsler; @isometry; @LiCM]. These characters imply that Finsler spacetime is intrinsically anisotropic.
Einstein’s special relativity resides in a flat Riemann spacetime, namely Minkowski spacetime. Similarly, the special relativity with LIV effects may reside in a flat Finsler spacetime. In fact, it is found that the SME–related Finsler spacetime is indeed flat in the sense of Finsler geometry [@Kostelecky_Finsler]. For instance, the simplest SME model, with only one nonvanishing coupling constant $a_{\mu}$ in the sLIV terms, leads to a flat spacetime of Randers–Finsler geometry [@Randers]. Actually, the flat Finsler spacetime is called locally Minkowski spacetime [@Book; @by; @Bao], which could be viewed as a generalization of Minkowski spacetime. In addition, doubly special relativity (DSR) [@DSR01; @DSR02; @DSR03; @DSR04; @DSR05] was found to be incorporated into Finsler spacetime [@DSR; @in; @Finsler], as well as very special relativity (VSR) [@VSR; @VSR; @in; @Finsler].
As the LIV corresponds to new spacetime, it is valuable to investigate physics compatible with the LIV effects. In this paper, we try to set up equations of motion for the electromagnetic field in locally Minkowski spacetime. A Lagrangian is proposed for the electromagnetic field in such a spacetime. The LIV effects are induced into the Lagrangian in a natural way. The amended Maxwell equations are obtained via the variation of action. A formal plane wave solution is obtained for the electromagnetic wave. The dispersion relation is modified for the electromagnetic wave. We also study the electromagnetic field at the first order of LIV effects. We compare these perturbative results with those in the SME framework. Relations and differences are discussed between Finsler spacetime and the SME. An interpretation is proposed for the LIV effects as influence of an anisotropic media. In addition, a constraint on the spacetime anisotropy could be obtained from astrophysical observations on gamma-ray bursts (GRBs).
The rest of the paper is arranged as follows. In section 2, we briefly discuss the spacetime in Finsler spacetime, especially the locally Minkowski spacetime. We propose an electromagnetic field model in locally Minkowksi spacetime in section 3. In section 4, we study this model at the first order of LIV effects and compare it with the SME. The anisotropic media is invoked to interpret the LIV effects on the electromagnetic field. In section 5, a constraint on the LIV effects is obtained from the *Fermi*–observations of GRBs in a specific locally Minkowski metric. Conclusions and remarks are listed in section 6.
2. Spacetime anisotropy {#Spacetime anisotropy}
=======================
Finsler spacetime is defined on the tangent bundle $TM:=\bigcup_{x\in M}T_{x}M$ instead of the manifold $M$. Each element of $TM$ is denoted by $(x,y)$, where $x\in M$ and $y\in T_{x}M$. Finsler geometry originates from the integral of the form [@Book; @by; @Rund; @Book; @by; @Bao; @Book; @by; @Shen] $$\label{Finsler geometry}
\int^b_a F\left(x, y\right)d\tau\ ,$$ where $x$ denotes a position and $y:=dx/d\tau$ denotes a so–called 4–velocity. The integrand $F(x,y)$ is called a Finsler structure, which is a smooth, positive and positively 1–homogeneous function defined on the slit tangent bundle $TM\backslash \{0\}$. The positive 1–homogeneity denotes the character $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$. The Finsler metric is defined as $$\label{Finsler metric}
g_{\mu\nu}(x,y):=\frac{\partial}{\partial y^\mu}\frac{\partial}{\partial y^\nu}\left(\frac{1}{2}F^2\right)\ .$$ Together with its inverse tensor, it is used for raising and lowering indices of tensors. Note that the Finsler metric becomes Riemannian if it does not depend on $y$.
A Finsler spacetime ($M$,$F$) is called locally Minkowski spacetime [@Book; @by; @Bao] if there is no dependence on $x$ for Finsler structure $F$, namely $F=F(y)$. Therefore, the Finsler metric $g_{\mu\nu}$ only depends on $y$ according to (\[Finsler metric\]). In such a spacetime, connections and curvatures vanish. Therefore, it is flat and maximally symmetric [@Finsler; @isometry; @LiCM; @constant; @flag; @curvature]. The vanishment of the connections implies that a free particle follows a straight line. It also implies that locally Minkowski spacetime belongs to Berwald spacetime [@Book; @by; @Bao]. All tangent spaces of Berwald spacetime are linearly isomorphic to one common Minkowski–normed linear space. Physically, this character implies that the laws of physics are common at each position in such a spacetime.
In Finsler spacetime, 4–velocity of a free particle is given by the Finsler geodesic equation [@Book; @by; @Bao] $$\label{geodesic}
\frac{d^{2} x^{\mu}}{d\tau^{2}}+\Gamma^{\mu}_{\rho\sigma}(x,\frac{dx}{d\tau})\frac{d x^{\rho}}{d \tau}\frac{d x^{\sigma}}{d \tau}=0\ ,$$ where $\Gamma$ denotes the connection. The Finsler geodesic originates from variation of an integral of the Finsler line element of the form (\[Finsler geometry\]). In locally Minkowski spacetime, the connections vanish, particularly. The Finsler geodesic equation (\[geodesic\]) becomes $$\label{geodesic equation}
\frac{d^2 x^{\mu}}{d\tau^{2}}=0\ .$$ Its solution gives a constant vector to $y$, which means that $y$ is independent on $x$. In this paper, $y$ denotes 4–velocity of a free photon along the Finsler geodesic. For a charged particle, such as electron, it would interact with the electromagnetic field. The Finsler geodesic equation should be modified. An extra term related to electromagnetic force $F^{\mu}(x)$ should be added to the right hand side (r.h.s.) of the Finsler geodesic equation. The velocity of the charged particle is given by the solution of the modified geodesic equation. Thus, it depends on $x$.
3. Electromagnetic field in locally Minkowski spacetime {#Electromagnetic field in locally Minkowski spacetime}
=======================================================
An advantage of studying the LIV in Finsler spacetime is that the principle of relativity is preserved automatically. As in Minkowski spacetime, we define the 4–potential 1–form of electromagnetic field in locally Minkowski spacetime $$\label{1--form}
A:=A_{\mu}(x)dx^{\mu}\ .$$ It preserves the internal U(1) gauge symmetry. The electromagnetic 4–potential is chosen as such a form that its 2–form excludes the terms $dx^{\mu}\wedge \delta y^{\nu}$ and $\delta y^{\mu}\wedge \delta y^{\nu}$ whose physical meaning is unclear. Therefore, the field strength 2–form is given by $$F:=dA=\frac{1}{2!}F_{\mu\nu}(x)dx^{\mu}\wedge dx^{\nu}\ ,$$ where $$\label{2--form}
F_{\mu\nu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}\$$ is invariant under the U(1) gauge group.
One of the Maxwell equations is given by the Bianchi identity $dF=0$, $$\label{fisrt Maxwell's equation}
\frac{\partial F_{\mu\nu}}{\partial x^{\lambda}}+\frac{\partial F_{\nu\lambda}}{\partial x^{\mu}}+\frac{\partial F_{\lambda\mu}}{\partial x^{\nu}}=0\ .$$ It is similar to the one in Minkowski spacetime. In addition, the contravariant field strength $F^{\mu\nu}$ is given via raising the indices of the covariant 2–form (\[2–form\]) by the Finsler metric $g^{\mu\nu}(y)$ of locally Minkowski spacetime, namely $F^{\mu\nu}=g^{\mu\sigma}g^{\nu\lambda}F_{\sigma\lambda}$. In this way, the covariant character is preserved in locally Mikowski spacetime.
We follow the form of the Lagrangian for the electromagnetic field but replace the spacetime metric $\eta_{\mu\nu}$ by the Finsler metric $g_{\mu\nu}(y)$ [@Stueckelberg; @method01; @Stueckelberg; @method02; @Massive; @photons__Stueckelberg; @method]. In this way, the Lagrangian could reduce back to the one in Minkowski spacetime when locally Minkowski spacetime reduces into Minkowski spacetime. The Lagrangian takes the form $$\label{Lagrangian}
L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\ .$$ The LIV effects are introduced via contracting the spacetime indices by the Finsler metric. The Lagrangian is invariant under coordinate transformations.
In locally Minkowski spacetime, an orthogonal base is given by $\{\frac{\partial}{\partial x^{\mu}}\}$ and its dual base is $\{dx^{\mu}\}$. The action of the electromagnetic field could be given by $$\label{action 1}
I=\int \left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right) d\Omega\ ,$$ where $d\Omega=\sqrt{-\det{g_{\mu\nu}(y)}}d^{4}x$ denotes the invariant volume element at each position $x$. The variation of action (\[action 1\]) with respect to $A_{\mu}$ is given by $$\label{variation of first term}
\int \left(\frac{\partial L}{\partial A^{\mu}}-\frac{\partial}{\partial x^{\sigma}}\frac{\partial L}{\partial \left(\frac{\partial A^{\mu}}{\partial x^{\sigma}}\right)}\right)\delta A^{\mu}d\Omega=0\ .$$ It gives the familiar Euler–Lagrangian equation $$\label{Euler--Lagrangian equation}
\frac{\partial L}{\partial A^{\mu}}-\frac{\partial}{\partial x^{\sigma}}\frac{\partial L}{\partial \left(\frac{\partial A^{\mu}}{\partial x^{\sigma}}\right)}=0\ .$$ The Euler–Lagrange equation supplements the Maxwell equations $$\label{second Maxwell's equation}
g^{\mu\nu}\frac{\partial F_{\mu\sigma}}{\partial x^{\nu}}=0\ .$$ The equations (\[fisrt Maxwell’s equation\]) and (\[second Maxwell’s equation\]) form a complete set of equations of motion for the electromagnetic field in locally Minkowski spacetime.
The Maxwell’s equation (\[second Maxwell’s equation\]) could be rewritten in terms of $A_{\sigma}$ as $$\label{electromagnetic wave equation}
g^{\mu\nu}\frac{\partial^{2}A_{\sigma}}{\partial x^{\mu}\partial x^{\nu}}=0\ ,$$ under the Lorentz gauge $$\label{Lorentz gauge}
g^{\mu\nu}\frac{\partial A_{\mu}}{\partial x^{\nu}}=0\ .$$ The above equation is the so–called electromagnetic wave equation. It has a formal plane wave solution $$\label{plane wave solution}
A_{\sigma}\propto \epsilon_{\sigma}e^{-ik_{\mu}x^{\mu}}=\epsilon_{\sigma}e^{-ig_{\mu\nu}k^{\mu}x^{\nu}}\ ,$$ where $\epsilon_{\sigma}$ denotes a polarization and $k^{\mu}$ denotes a wavevector of the electromagnetic plane wave. By substituting (\[plane wave solution\]) into (\[electromagnetic wave equation\]), we obtain a dispersion relation for the electromagnetic plane wave $$\label{dispersion relation of light}
k_{\mu}k^{\mu}=g_{\mu\nu}k^{\mu}k^{\nu}=0\ .$$ Its form is as similar as the one in the Lorentz invariant electrodynamics. However, it is modified by the Finsler metric $g_{\mu\nu}$ since the contraction of spacetime indices is implicated via this metric.
4. Lorentz invariance violation {#Lorentz invariance violation}
===============================
Observations do not show signals of the LIV effects at the present attainable energy scales [@Data; @tables; @for; @Lorentz; @and; @CPT; @violation]. This fact implies that the LIV effects should be very tiny. We could extract the LIV effects by expanding the Finsler metric into $$\label{Finsler metric series}
g^{\mu\nu}(y)=\eta^{\mu\nu}+h^{\mu\nu}(y)\ .$$ In this way, the first–order LIV effects are extracted and characterized completely by $h_{\mu\nu}$.
At the leading order, the Lagrangian (\[Lagrangian\]) of the electromagnetic field could be expanded into $$\begin{aligned}
L&=&-\frac{1}{4}\eta^{\mu\rho}\eta^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}
-\frac{1}{2}\eta^{\mu\rho}h^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\ ,\\
&:=&L_{LI}+L_{LIV}\ ,\end{aligned}$$ where $L_{LI}=-\frac{1}{4}\eta^{\mu\rho}\eta^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}$ denotes the Lorentz invariant term while the LIV term is given as $$\label{sLIV in locally Minkowski spacetime}
L_{LIV}=-\frac{1}{8}\left(\eta^{\mu\rho}h^{\nu\sigma}-\eta^{\nu\rho}h^{\mu\sigma}-\eta^{\mu\sigma}h^{\nu\rho}+\eta^{\nu\sigma}h^{\mu\rho}\right)
F_{\mu\nu}F_{\rho\sigma}\ .$$ In the above equation, we have anti–symmetrized the indices $\mu\nu$ and $\rho\sigma$. In the SME framework, meanwhile, the CPT–even sLIV term in the Lagrangian of the electromagnetic field is given by [@SME02] $$\label{sLIV in SME}
L_{SME}=-\frac{1}{4}k^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\ ,$$ where $k^{\mu\rho\nu\sigma}$ denotes a dimensionless parameter which characterizes the level of the sLIV effects. In the SME, the parameter $k^{\mu\rho\nu\sigma}$ is given by hand.
In locally Minkowski spacetime, however, we could relate this parameter to deformation parameter $h^{\mu\nu}$ of the spacetime from Minkowski to locally Minkowski. By comparing (\[sLIV in locally Minkowski spacetime\]) and (\[sLIV in SME\]), we obtain a relation $$\label{SME_Finsler}
k^{\mu\nu\rho\sigma}=\frac{1}{2}\left(\eta^{\mu\rho}h^{\nu\sigma}-\eta^{\nu\rho}h^{\mu\sigma}-\eta^{\mu\sigma}h^{\nu\rho}
+\eta^{\nu\sigma}h^{\mu\rho}\right)\ .$$ In the SME, the parameter $k$ is a constant since the energy and momentum are conserved [@SME02]. In locally Minkowski spacetime, the geodesic equation (\[geodesic equation\]) of photon gives $y$ a constant vector along the geodesic. Thus, $h$ is a constant and the r.h.s. of equation (\[SME\_Finsler\]) is also a constant. These inferences mean that the LIV electromagnetic field model obtained in locally Minkowski spacetime is compatible with the perturbative results in the SME. In addition, there are ten independent components for $h^{\mu\nu}$ while nineteen for $k^{\mu\nu\rho\sigma}$ [@SME02]. Only components of the form $k^{\mu\nu\mu\sigma}$ are possibly nonvanishing in locally Minkowski spacetime. Furthermore, the birefringence of light will not emerge at the leading order in this Finsler model of electromagnetic field. The reason is that all Weyl components of $k^{\mu\nu\rho\sigma}$ vanish at the leading order [@Electrodynamics; @with; @Lorentz-violating; @operators; @of; @arbitrary; @dimension]. These predictions distinguish the electromagnetic field model in locally Minkowski spacetime from the SME–based one.
As $L_{SME}$ does in the SME, the LIV term $L_{LIV}$ also denotes the Lorentz–violating interactions at first order for the electromagnetic field in locally Minkowski spacetime. Traditionally, the observations on the electromagnetic field give rise to the most stringent tests of the Lorentz symmetry. An incomplete list includes: the LIV could lead to the anisotropy of the speed of light which is tested by the Michelson–Morley experiment [@Michelson--Morley01; @Michelson--Morley02; @Michelson--Morley03; @Michelson--Morley04; @Michelson--Morley05]; there is an atomic clock experiment named as the Hughes–Drever experiment [@Hughes--Drever; @exp01; @Hughes--Drever; @exp02] which is used to test the variation of the SME coefficients with the movement of the Earth; the observations from distant galaxies give severe limits on the birefringence of light [@Birefiringence01; @Birefiringence02], and etc. For a more detailed summarization on the observations of the (s)LIV effects, see for example citations [@Data; @tables; @for; @Lorentz; @and; @CPT; @violation; @Overview; @of; @the; @SME; @by; @Bluhm] and references therein.
The Maxwell equations (\[fisrt Maxwell’s equation\]) and (\[second Maxwell’s equation\]) could be rewritten as $$\begin{aligned}
\label{Maxwell equations in series}
\frac{\partial F_{\mu\nu}}{\partial x^{\lambda}}+\frac{\partial F_{\nu\lambda}}{\partial x^{\mu}}+\frac{\partial F_{\lambda\mu}}{\partial x^{\nu}}&=&0\ ,\\
\eta^{\mu\nu}\frac{\partial F_{\mu\sigma}}{\partial x^{\nu}}+h^{\mu\nu}\frac{\partial F_{\mu\sigma}}{\partial x^{\nu}}&=&0\ .\end{aligned}$$ The second equation includes the LIV effects while the first one is not related to dynamics. The electromagnetic wave equation (\[electromagnetic wave equation\]) could be expanded as $$\label{electromagnetic wave in series}
\eta^{\mu\nu}\frac{\partial^{2}A_{\sigma}}{\partial x^{\mu}\partial x^{\nu}}+h^{\mu\nu}\frac{\partial^{2}A_{\sigma}}{\partial x^{\mu}\partial x^{\nu}}=0\ ,$$ where the first term denotes the electromagnetic wave equation in Minkowski spacetime and the second term denotes the terms related to the LIV effects.
There is a solution for this wave equation (\[electromagnetic wave in series\]) at first order $$A_{\mu}=A_{0\mu}+A_{1\mu}\ ,$$ where $A_{0\mu}$ and $A_{1\mu}$ denote the zero–order solution and the first–order solution, respectively. The zero–order solution $A_{0\mu}$ satisfies the electromagnetic wave equation in Minkowski spacetime, namely $$\eta^{\mu\nu}\frac{\partial^{2}A_{0\sigma}}{\partial x^{\mu}\partial x^{\nu}}=0\ .$$ It has a plane wave solution $A_{0\sigma}\propto\epsilon_{\sigma} e^{-\eta_{\mu\nu}k^{\mu}x^{\nu}}$. Therefore, we could obtain an equation for the first–order solution $A_{1\mu}$ as $$\begin{aligned}
\eta^{\mu\nu}\frac{\partial^{2}A_{1\sigma}}{\partial x^{\mu}\partial x^{\nu}}&=&-h^{\mu\nu}\frac{\partial^{2}A_{0\sigma}}{\partial x^{\mu}\partial x^{\nu}}\nonumber\\
&=& h^{\mu\nu}\eta_{\mu\rho}\eta_{\nu\kappa}k^{\rho}k^{\kappa}A_{0\sigma}\nonumber\\
&=& h_{\mu\nu}k^{\mu}k^{\nu}A_{0\sigma}\ ,\end{aligned}$$ where we have contracted indices with $\eta$ in the third equal. The r.h.s. of the above equation behaves like a source of the electromagnetic field, which could be viewed as influence from a slightly anisotropic media on the electromagnetic wave. Furthermore, the dispersion relation (\[dispersion relation of light\]) could be expanded into $$\label{dispersion relation in series}
\eta_{\mu\nu}k^{\mu}k^{\nu}=-h_{\mu\nu}k^{\mu}k^{\nu}\ .$$ It is also called the lightcone. The lightcone is enlarged if the r.h.s. of the above equation is negative, while narrowed if the r.h.s. of the above equation is positive. It depends on concrete characters of the LIV effects $h_{\mu\nu}$. The spatial speed of light could be superluminal if the lightcone is enlarged while it is subluminal if the lightcone is narrowed, and vice versa [@FSR01; @FSR02; @FSR03]. In addition, the speed of light could depend on its direction since there could be of direction–dependence for $h_{\mu\nu}$. These could be tested by observations, such as the Michelson–Morley experiment.
5. Constraint from Gamma-Ray Bursts {#Constraint from Gamma-Ray Bursts}
===================================
In this section, a specific locally Minkowksi metric is postulated and investigated. The speed of light is obtained to be subluminal and the lightcone is found to be squeezed. In addition, a constraint on the level of the LIV effects is gained from the *Fermi*–observations of GRBs.
To discuss detailed predictions on the LIV effects, we postulate the locally Minkowski spacetime as $$\label{specific locally Minkowski metric}
g^{\mu\nu}=\rm{diag}(1+ay^{0},-1,-1,-1)\ ,$$ where $|ay^{0}|\ll 1$ is assumed and $a$ is positive. $F$ has been normalized $F(\tilde{y})=0$, and $y^{\mu}=\tilde{y}^{\mu}/F(\tilde{y})$. The 4–velocity of a particle is related to 4–momentum of this particle. Thus, $y$ could be characterized by $k$. In the simplest case, $y$ is a linear function of $k$, namely $y\propto k$ as similar as that in quantum mechanics. In this way, the metric could be written as $$\label{specific locally Minkowski metric}
g^{\mu\nu}=\rm{diag}\left(1+\frac{k^{0}}{M},-1,-1,-1\right)\ ,$$ where the constant $M$ is a high–energy scale into which $a$ has been absorbed. The perturbative metric deviation is given by $$\label{metric perturbation in specific locally Minkowski}
h^{00}=-h_{00}=\frac{k^{0}}{M}\ ,$$ and other components vanish. The energy scale $M$ implies a scale for possible occurrence of the LIV effects. Meanwhile, it reveals that the LIV effects are suppressed severely by this scale. Therefore, the LIV effects are expected to be most possibly observed in the ultra–high energy physics, such as the Planck scale.
With the spatially isotropic metric (\[specific locally Minkowski metric\]), the electromagnetic wave equation (\[electromagnetic wave equation\]) (or (\[electromagnetic wave in series\])) becomes $$\label{electromagnetic wave equation in specific}
\left[\left(1+\frac{k^{0}}{M}\right)\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}\right]A_{\sigma}=0\ ,$$ where $\nabla$ denotes the 3D divergence. Comparing this equation with that in the Lorentz–invariant electrodynamics, we obtain the speed of light as $$\label{speed of light}
c=\left(1+\frac{k^{0}}{M}\right)^{-\frac{1}{2}}\approx1-\frac{k^{0}}{2M}\ .$$ It implies that a photon with energy $k^{0}>0$ would propagate subluminally in such a spacetime. Meanwhile, higher the photon energy is, slower it propagates. On the other hand, the dispersion relation (\[dispersion relation of light\]) (or (\[dispersion relation in series\])) becomes $$\eta_{\mu\nu}k^{\mu}k^{\nu}=\frac{k^{0}}{M}(k^{0})^{2}>0\ .$$ It implies that the lightcone is squeezed. Higher the photon energy is, more severely its lightcone is squeezed. These are consistent with the prediction that the speed of light (\[speed of light\]) is “subluminal”.
The above predictions could be tested by the astrophysical observations on GRBs. The reason is that the above LIV effects could be accumulated after photons traveling a cosmological distance. The *Fermi* satellite has observed several GRBs with photon energy larger than $100~\rm{MeV}$ in recent years. It has been shown that $\rm{GeV}$ photons arrive several seconds later than $\rm{MeV}$ photons [@GRB080916C; @GRB090902B; @limit; @on; @variation; @of; @the; @speed; @of; @light; @GRB090926A]. The observed time lag for two photons with energy $k^{0}_{high}$ and $k^{0}_{low}$ consists of two parts [@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs] $$\label{t obs}
\Delta t_{obs}=\Delta t_{LIV}+\Delta t_{int}\ ,$$ where $\Delta t_{int}$ denotes the intrinsic emission time delay, and $\Delta t_{LIV}$ represents the flying time difference induced by the LIV effects. According to the magnetic jet model [@Magnetic; @jet; @model; @for; @GRBs], $\Delta t_{int}$ could be evaluated. In such a model, photons with energy less than $10~\rm{MeV}$ can escape when the jet radius is beyond the Thomson photosphere radius, i.e., the optical depth is $\tau_{T}\sim 1$. Nevertheless, $\rm{GeV}$ photons will be converted into electron–positron pairs at this radius, but can escape later when the pair–production optical depth $\tau_{\gamma\gamma}(k^{0})$ drops below unity. One can calculate the time delay $\Delta t(k^{0})$ for the emissions of $\rm{GeV}$ and $100~\rm{MeV}$ photons relative to $\rm{MeV}$ photons (see detailed discussions in Ref.[@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs]). The intrinsic time delay is $\Delta t_{int}=\Delta t(k^{0}_{high})-\Delta t(k^{0}_{low})$. Therefore, we could obtain the LIV–induced time delay $\Delta t_{LIV}$ according to (\[t obs\]), see Table \[tab:1\].
[ccccccc]{} GRB & k$^{0}$$_{\rm{low}}$ & k$^{0}$$_{\rm{high}}$ & $\triangle$ t$_{\rm{obs}}$ & $\triangle$ t$_{\rm{LIV}}$ & K(z) & 2M\
& MeV & GeV & s & s & s$\cdot$GeV & GeV\
080916c & 100 & 13.22 & 12.94 & 0.24 & 4.50$\times10^{18}$ & 10.02$\times10^{19}$\
090510 & 100 & 31 & 0.20 & 0.14 & 7.02$\times10^{18}$ & 9.73$\times10^{19}$\
090902b & 100 & 11.16 & 9.5 & 0.10 & 3.38$\times10^{18}$ & 9.94$\times10^{19}$\
090926 & 100 & 19.6 & 21.5 & 0.20 & 6.20$\times10^{18}$ & 9.59$\times10^{19}$\
Consider two photons emitted at the same spacetime point, the arrival time delay between them could be written as [@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs; @high; @energy; @photons; @from; @Fermidetected; @GRBs; @LIV-induced; @arrival; @delays; @of; @cosmological; @particles] $$\label{t LIV}
\Delta t_{LIV}=\frac{\Delta k^{0}}{2M}D(z)\ ,$$ where we have used the equation (\[speed of light\]). The cosmological distance $D$ is defined as [@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs; @high; @energy; @photons; @from; @Fermidetected; @GRBs; @LIV-induced; @arrival; @delays; @of; @cosmological; @particles] $$D(z):=H_{0}^{-1}\int_{0}^{z}\frac{(1+z')dz'}{\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}}}\ ,$$ where $H_{0}\approx 72\rm{km\cdot sec^{-1}\cdot Mpc^{-1}}$denotes the Hubble constant, $\Omega_{M}\approx0.3$ and $\Omega_{\Lambda}\approx0.7$ are densities of matter and cosmological constant, respectively. In this way, the LIV energy scale is given by $$\label{LIV energy scale}
2M=\frac{\Delta k^{0}}{\Delta t_{LIV}}D(z)\ .$$ To reveal the LIV effects, one depicts the $\Delta t_{LIV}/(1+z)~\rm{vs.}~K(z)$ plot, where $K(z)$ is defined as [@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs; @Lorentz; @violation; @from; @cosmological; @objects] $$K(z):=\frac{\Delta k^{0}}{1+z}D(z)\ .$$ The slope of this plot denotes the inverse of the level of the LIV effects, i.e., $(2M)^{-1}$.
In Ref.[@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs], Chang [*et al.*]{} took advantage of the *Fermi*–observations of four GRBs to estimate the level of the LIV effects. The four GRBs are GRB 080916c [@GRB080916C], GRB 090902b [@GRB090902B], GRB 090510 [@limit; @on; @variation; @of; @the; @speed; @of; @light] and GRB 090926 [@GRB090926A], respectively. Their LIV–induced time lags $\Delta t_{LIV}$ and $K(z)$ were calculated and listed in Table \[tab:1\]. Their $\Delta t_{LIV}/(1+z)~\rm{vs.}~K(z)$ plot was given by Fig.\[fig1\].
![Figure taken from Ref.[@A; @unified; @constraint; @on; @the; @Lorentz; @invariance; @violation; @from; @GRBs]. The $\Delta t_{LIV}/(1+z)~\rm{vs.}~K(z)$ plot for the four GRBs observed by the *Fermi* satellite. The slope of the fit line was shown to be $(2M)^{-1}\sim 10^{-20}~\rm{GeV}^{-1}$.[]{data-label="fig1"}](001.eps){width="8"}
The slope of the fit line was obtained $(2M)^{-1}\sim 10^{-20}~\rm{GeV}^{-1}$. Correspondingly, the LIV energy scale was shown to be $2M\sim 10^{20}~\rm{GeV}$, which is consistent with the Planck energy scale. Therefore, we would expect to observe the spacetime anisotropy near the planck scale in future astrophysical and cosmological observations.
6. Conclusions and remarks {#Conclusions and remarks}
==========================
Conclusions and remark are listed as follows. Finsler geometry gets rid of the quadratic restriction on the form of the spacetime structure. It is intrinsically anisotropic. The SME–related sLIV effects on the classical point–like particles have been related to this kind of intrinsically anisotropic spacetime. In principle, the laws of physics should be studied in the intrinsically anisotropic spacetime if the Lorentz symmetry is violated. In this paper, we proposed that locally Minkowski spacetime could be a suitable platform to characterize the possible LIV effects. The reason is that locally Minkowski spacetime is the flat and maximally symmetric non–Riemannian Finsler spacetime.
We studied the electromagnetic field in the locally Minkowski spacetime. The Lagrangian with LIV effects was constructed for the electromagnetic field via replacing the spacetime metric with the Finsler metric. It was found that the obtained Lagrangian is invariant under the coordinate transformations, which preserves validation of the principle of relativity. We obtained the Maxwell equations via the Bianchi identity and the variation of action. We presented a formal plane wave solution. The dispersion relation is modified for the electromagnetic wave. The lightcone might be enlarged or narrowed, depending on concrete characters of the LIV effects. The approach proposed in the paper could be generalized straightforward to study the non–Abelian gauge fields with the LIV effects in locally Minkowski spacetime.
To demonstrate the LIV effects clearly in locally Minkowski spacetime, we expanded the Lagrangian of the electromagnetic field around the Minkowski background. The explicit LIV term was especially extracted from the Lagrangian. It is noteworthy that this LIV term could be reduced back to the sLIV one in the SME formally at first order. It reveals that our results are compatible with the previous works in the framework of SME. However, the LIV effects originate in departure from Minkowski spacetime to locally Minkowski spacetime. There are fewer independent parameters for the LIV effects in locally Minkowski spacetime. The birefringence of light would not appear in our model, which is consistent with the astronomical observations. In addition, the LIV influence on the electromagnetic wave was found to behave like a source of the electromagnetic field. It could be interpreted as influence from a slightly anisotropic media on the electromagnetic field.
To discuss phenomenological predictions on the LIV effects, we investigate a specific locally Minkowski metric. The electromagnetic wave equation was studied and the light was found to propagate subluminally. On the other hand, we obtained a squeezed lightcone. Both characters are consistent with each other. Another important feature of this metric was that the lightcone becomes more severely squeezed as increase of the photon energy. These features were tested by the *Fermi*–observations on the GRBs. The LIV effects accumulate when the light propagates from distant GRBs. The GeV photons were found to arrive at the Earth later than the MeV photons. This observation gave a severe constraint on the LIV energy scale, i.e., $10^{20}~\rm{GeV}$. We would expect to observe the spacetime anisotropy near this energy scale in future astrophysical and cosmological observations.
We thank useful discussions with Yunguo Jiang, Ming-Hua Li, Xin Li, Hai-Nan Lin. The author (S. Wang) thanks useful discussions with Jian-Ping Dai, Dan-Ning Li, and Xiao-Gang Wu. This work is supported by the National Natural Science Fund of China under Grant No. 11075166.
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[^1]: E-mail: changz@ihep.ac.cn
[^2]: E-mail: wangsai@ihep.ac.cn
[^3]: E-mail: saiwangihep@gmail.com
[^4]: Corresponding author at IHEP, CAS, 100049 Beijing, China
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Astro2020 Science White Paper
The US Program in Ground-Based Gravitational Wave Science: Contribution from the LIGO Laboratory
**Thematic Areas:** = $\square$ Planetary Systems = $\square$ Star and Planet Formation\
$\blacksquare$ Formation and Evolution of Compact Objects $\blacksquare$ Cosmology and Fundamental Physics\
$\square$ Stars and Stellar Evolution $\square$ Resolved Stellar Populations and their Environments\
$\square$ Galaxy Evolution $\blacksquare$ Multi-Messenger Astronomy and Astrophysics
**Principal Author:**
Name: David Reitze Institution: LIGO Laboratory, California Institute of Technology Email: dreitze@caltech.edu Phone: +1–626–395–6274
**Co-authors:**\
LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA\
LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\
LIGO Hanford Observatory, Richland, Washington 99352, USA\
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
**Abstract:** Recent gravitational-wave observations from the LIGO and Virgo observatories have brought a sense of great excitement to scientists and citizens the world over. Since September 2015, 10 binary black hole coalescences and one binary neutron star coalescence have been observed. They have provided remarkable, revolutionary insight into the “gravitational Universe” and have greatly extended the field of multi-messenger astronomy. At present, Advanced LIGO can see binary black hole coalescences out to redshift and binary neutron star coalescences to redshift . This probes only a very small fraction of the volume of the observable Universe. However, current technologies can be extended to construct “3[rd]{} Generation” (3G) gravitational-wave observatories that would extend our reach to the very edge of the observable Universe. The event rates over such a large volume would be in the hundreds of thousands per year ([*i.e.*]{} tens per hour). Such 3G detectors would have a 10-fold improvement in strain sensitivity over the current generation of instruments, yielding signal-to-noise ratios of 1000 for events like those already seen. Several concepts are being studied for which engineering studies and reliable cost estimates will be developed in the next 5 years.
Introduction {#sec:intro}
============
Long-baseline laser interferometry [@Bond2017; @Adhikari:2013kya] has given life to gravitational-wave astronomy and expanded the vision of multi-messenger observation. When the current generation of interferometric detectors are brought to their full strain sensitivities, they will see binary neutron star coalescences out to redshift $z \simeq \num[round-mode=figures,round-precision=1]{{0.184}{}}$ and binary black hole coalescences out to $z \simeq \num[round-mode=figures,round-precision=1]{{2.7}{}}$ [@T1800042], with nearby events having signal-to-noise ratios of order 100. With a future generation of more sensitive detectors in new, larger facilities, it will be possible to see these binary systems out to redshifts $z > 10$, with nearby events having signal-to-noise ratios in excess of 1000. This paper presents a synopsis of the scientific opportunities afforded by 3G detectors; it also outlines needed technical developments to enable them.
A brief history of the field {#sec:history}
============================
LIGO consists of two NSF-funded [@NSF] US facilities that host long, interferometric gravitational-wave detectors, one in Livingston, Louisiana, and the other in Hanford, Washington. The facilities were constructed from 1992 to 1999, and the initial LIGO detectors (iLIGO) made observations from 2002 to 2010, ultimately reaching a sensitivity to neutron stars out to redshift $z \simeq \num[round-mode=figures,round-precision=1]{{0.01}{}}$ and black holes to redshift $z \simeq \num[round-mode=figures,round-precision=1]{{0.116}{}}$—though no detections were made [@Abbott:2007kv; @LIGO:2012aa].
The second-generation detectors, known as Advanced LIGO (aLIGO), were funded [@NSF; @MPS; @STFC; @ARC] starting in 2008 and installed at both sites from 2011 to 2014. aLIGO was designed to achieve a 10-fold increase in sensitivity over iLIGO—providing a 1000-fold increase in sensitive volume—by reducing the instrument’s high-frequency noise floor and extending the sensitive band to lower frequency [@TheLIGOScientific:2014jea]. Observations began in September 2015 with about four times the sensitivity of iLIGO [@TheLIGOScientific:2016agk]. On September 14, 2015, the first gravitational waves were detected: a chirp signal (GW150914) from the merger of two black holes from Earth [@Abbott:2016blz]. Since then, aLIGO’s sensitivity has been improved and more observations have been made.
The first two aLIGO observing runs (O1, from Sep 2015 to Jan 2016, and O2, from Nov 2016 to Aug 2017) totaled a calendar year of observing time. Near the end of O2, the Virgo observatory (a detector in Cascina, Italy) [@TheVirgo:2014hva] joined LIGO, and critically contributed to the first observation of a binary neutron star merger (GW170817) by reducing the sky location area from a few hundred to a few tens of square degrees [@TheLIGOScientific:2017qsa]. This enabled approximately 90 optical and radio astronomy telescopes to identify and study electromagnetic counterparts to the gravitational waves [@GBM:2017lvd]. In O1 and O2 combined, 10 black hole coalescences and one neutron star collision were observed [@LIGOScientific:2018mvr]. Observing run O3 will begin in April 2019 and run for about one year. KAGRA, an underground detector in Japan [@Akutsu:2019rba], is planning to join O3 toward the end of 2019 at a reduced sensitivity. By the middle of the next decade, LIGO will install and commission an upgrade (referred to as “A+”) to significantly improve the aLIGO detectors beyond their design sensitivity target (by a factor of 5 in detection rate for compact binary mergers compared to aLIGO at design sensitivity). Additionally, a third LIGO detector will be brought online in a new facility in India.
Science case for an international network of 3G detectors {#sec:science}
=========================================================
The worldwide gravitational-wave, high energy astrophysics, and nuclear physics communities are highly energized by the recent discoveries and the prospects for an extremely rich science program ahead. The Gravitational Wave International Committee (GWIC) [@GWIC] is developing a set of science white papers detailing 3G gravitational-wave astronomy based on a 3G gravitational-wave detector network consisting of “Cosmic Explorer” (CE) [@CE; @Evans:2016mbw] and “Einstein Telescope” (ET) [@ET; @Punturo:2010zz] detectors, as described in the next section. Here we summarize the science case for Cosmic Explorer, which delivers two significant advances over the current generation of gravitational-wave detectors: (a) nearby sources will be detected with very high signal-to-noise ratio (SNR); and (b) sources will be detectable to redshifts greater than 10 (see Fig. \[fig:strain\_horizon\]) [@Vitale:2016aso], allowing studies of their evolution and providing access to the first stars in the Universe.
Black holes
-----------
**Physics:** 3G detectors will enable unprecedented tests of the strong-field dynamics of gravity. Some specific effects include the ringdown of a black hole merger [@Gossan:2011ha; @Berti:2016lat], detailed measurements of spin-orbit gravitational interactions [@Kidder:1992fr; @PhysRevD.49.6274; @Vitale:2016icu], the non-linear memory effects in the metric that accompany gravitational radiation [@Christodoulou:1991cr; @Yang:2018ceq], and direct tests of the Hawking area theorem [@Cabero:2017avf]. Many of these tests rely on obtaining very high SNR signals, which is possible with the superior noise performance of the 3G detectors (Fig. \[fig:strain\_horizon\], left side).
**Astronomy:** The right side of Fig. \[fig:strain\_horizon\] shows how Cosmic Explorer will be able to detect black hole mergers throughout cosmic history, thus exploring both the importance of different formation channels (field formation versus dynamical capture) and how these evolved with time. Evidence for black holes from population III stars might help in understanding the formation of supermassive black holes and the galaxies they seed [@Sesana:2009wg]. The mass and spin distribution of black holes will reveal their origin and key properties of their progenitors, such as the magnitude of supernovae kicks [@OShaughnessy:2017eks]. If primordial stellar-mass black hole binaries exist, Cosmic Explorer will detect them if they merge at $z \lesssim 100$. A network of Cosmic-Explorer-class detectors will provide high-SNR signals from all epochs of stellar-mass binary black hole collisions in the Universe.
[0.525]{} ![*Left side:* Cosmic Explorer projected strain noise for Stage 1 (during the 2030s) and Stage 2 (2040s), compared with the strain noise achieved by Advanced LIGO during observing run O2, as well as designed noise performance for Advanced LIGO, and LIGO A+. Less strain noise indicates better strain sensitivity. *Right side:* Astrophysical response distance [@Chen:2017wpg] of Advanced LIGO at O2 sensitivity, LIGO A+, and Cosmic Explorer (Stages 1 and 2), plotted on top of a population of 1.4–1.4$M_\odot$ neutron star mergers (yellow) and 30–30$M_\odot$ black hole mergers (gray), assuming a Madau–Dickinson star formation rate [@Madau:2014bja] and a typical merger time of . The radial distribution of points accounts for the detector-frame merger rate per unit redshift. []{data-label="fig:strain_horizon"}](o2_aligo_aplus_ce1_ce2.pdf "fig:"){width="\textwidth"} \[fig:strain\]
[0.475]{} ![*Left side:* Cosmic Explorer projected strain noise for Stage 1 (during the 2030s) and Stage 2 (2040s), compared with the strain noise achieved by Advanced LIGO during observing run O2, as well as designed noise performance for Advanced LIGO, and LIGO A+. Less strain noise indicates better strain sensitivity. *Right side:* Astrophysical response distance [@Chen:2017wpg] of Advanced LIGO at O2 sensitivity, LIGO A+, and Cosmic Explorer (Stages 1 and 2), plotted on top of a population of 1.4–1.4$M_\odot$ neutron star mergers (yellow) and 30–30$M_\odot$ black hole mergers (gray), assuming a Madau–Dickinson star formation rate [@Madau:2014bja] and a typical merger time of . The radial distribution of points accounts for the detector-frame merger rate per unit redshift. []{data-label="fig:strain_horizon"}](horizon_donut.pdf "fig:"){width="\textwidth"} \[fig:horizon\]
Neutron stars
-------------
**Physics:** Capturing high-SNR binary neutron star coalescences is the most promising way of measuring the equation of state of nuclear matter through precise measurements of the phase evolution of the gravitational-wave signal [@Flanagan:2007ix]. More detailed information about the composition of neutron stars may be available from the gravitational-wave emission from the post-merger collision remnant [@Stergioulas:2011gd]. Furthermore, measurements of the masses and spins of the component neutron stars, and of the remnant, will lead to a better understanding of the the yield of heavy metals produced by these systems [@Metzger:2016pju], and the maximum mass of neutron stars[@Margalit:2017dij]. As shown in Fig. \[fig:strain\_horizon\] (right side), Advanced LIGO and A+ will only detect a very small fraction of binary neutron star coalescences; Cosmic Explorer will have access to the entire coalescing population.
Gravitational waves can also be produced by isolated spinning neutron stars, as long as they are not perfectly spherical [@Zimmermann:1979ip]. Such gravitational waves have not yet been detected, and the amplitude of emission from these sources is not known. Such a signal is expected to be continuous and nearly monochromatic, and would provide information about neutron star ellipticity, moment of inertia, and again nuclear equation of state.
**Astronomy:** 3G detectors can detect binary neutron star coalescences to redshifts of unity and above [@Mills:2017urp]. Many of these events will be accompanied by detectable electromagnetic emission, as in GW170817. With hundreds of thousands of sources per year, 3G detectors will enable precise measurements of the mass function of neutron stars and their merger rate. Routine detections of electromagnetic counterparts will allow us to more fully understand the details of the electromagnetic emission, including the nuclear-decay-powered kilonova [@Metzger:2016pju]. Localizing the hosts will also provide information about the galaxies in which the systems were formed, such as their age and metallicity [@TheLIGOScientific:2016htt].
**Cosmology:** Gravitational waves provide an independent way to measure cosmological parameters. Each signal provides a direct measurement of the luminosity distance to the source. If the source redshift can be obtained by other means ([*e.g.*]{} an electromagnetic counterpart, as for GW170817 [@Abbott:2017xzu], or observation of tidal effects in the waveform [@Messenger:2011gi]), one can then solve for the cosmological parameters. Advanced LIGO will continue to detect binary neutron star coalescences in the local universe, and hence will improve constraints on the Hubble constant [@Chen:2017rfc]; it will not, however, have access to other fundamental quantities, such as $\Omegaup_\Lambdaup$, $\Omegaup_\text{m}$, or the equation of state of dark energy. Cosmic Explorer opens the door to the measurement of the full set of cosmological parameters. Furthermore, it will reveal hundreds of thousands of sources per year up to high redshifts, vastly improving the precision of cosmological parameters and providing insights on the isotropy of the universe at different epochs.
Other science targets
---------------------
3G detectors will improve the prospects for the detection of gravitational waves from core-collapse supernovae [@Powell:2018isq], and a 3G network might detect a stochastic gravitational-wave relic from the early Universe [@Regimbau:2016ike]. Additionally, 3G detectors may reveal gravitational-wave sources that are entirely unanticipated.
Pathways for future development of the field {#sec:future}
============================================
A global 3G network and the US vision for 2020–2030 and beyond
--------------------------------------------------------------
3G detector concepts generally aim at achieving a factor of 10 increase in strain sensitivity compared to aLIGO and Advanced Virgo. To enable multi-messenger astronomy and high-precision science, it is critically important to determine the position of gravitational-wave sources on the sky. This requires a network of 3 or more 3G detectors with site locations around the world [@Mills:2017urp].
The NSF is funding a study of ways in which the US can contribute to an international 3G effort. In this study, coordinated with GWIC, various detector concepts will be evaluated and costed at a preliminary level. Cosmic Explorer [@CE] is a concept for a new facility with much longer arm lengths to boost sensitivity ([*e.g.*]{} over the the current LIGO arm lengths). At present, two Cosmic Explorer detectors, with locations to be determined, are envisioned as part of an international network that would include the European Einstein Telescope [@ET] as a third observatory. Second-generation detectors will play a supporting role in the global network as the first 3G detectors come online.
Cosmic Explorer: a 40 km detector
---------------------------------
![A plausible timeline for the development of Cosmic Explorer, with major milestones for funding and for the R&D program to support the CE Stage 2 upgrade to new technologies. Periods of observations are represented in green, while periods of fabrication and of down-time of the instruments for installation and commissioning are shown in gray.[]{data-label="fig:timeline"}](CosmicExplorerTimeline_E.pdf){width="\textwidth"}
Cosmic Explorer will use longer arm lengths and new technologies to achieve a factor of 10 or more strain sensitivity improvement over the current generation of gravitational-wave detectors [@Evans:2016mbw]. As with LIGO, we envision a staged program for Cosmic Explorer (CE):
- **CE Stage 1** (CE1, targeting operations in the late 2030s) relies on extensions of technologies demonstrated and being developed for A+.
- **CE Stage 2** (CE2, starting operations in the mid-2040s) consists of a major upgrade of CE that will exploit the full potential of the new facility by reducing thermal noise through the use of cryogenics and new materials for test masses and coatings. A promising route currently under investigation consists of employing silicon test masses and amorphous silicon coatings operating at , with or laser light; these are major changes from existing technologies and will require long-term R&D to develop.
This tentative timeline is captured by Fig. \[fig:timeline\], as well as elements of the R&D program necessary to support the construction of the detector and its upgrade to new lasers and new mirror materials. Fig. \[fig:strain\_horizon\] (left side) shows the sensitivity progression of the detectors, from the most recent Advanced LIGO observing run (O2) to CE2. Improving the strain sensitivity beyond what can be done in existing facilities is critical to delivering the science goals discussed above. Running second-generation detectors for a longer period of time would increase linearly the number of detected sources; however, these sources would still lie in the local universe and have low to moderate SNRs. On the other hand, increasing the strain sensitivity extends the reach of the detectors to a redshift of many. This gives access to remote sources that cannot possibly be detected with second-generation detectors, while also measuring nearby sources with SNRs of thousands. Cosmic Explorer will not only collect a much greater number of signals compared to Advanced LIGO; it will also shed light on a gravitational-wave universe that is currently inaccessible to us. Beside requiring new facilities, Cosmic Explorer will require a series of technical developments in several areas. These include larger test masses, suspension systems able to sustain heavier masses, improved vibration isolation, improved angular alignment control, introduction of greater optical power, higher levels of frequency-dependent squeezing, and mitigation of gravity gradients. A significant issue is the reduction of thermal noise in the mirror coatings, which may require multiple approaches, including changes in laser wavelength, new mirror materials, and cryogenic operation. An active research program in the next decade will define the best strategy to maximize the scientific output of Cosmic Explorer. Engineering studies will be needed to establish designs and formal cost estimates, and upgrades to laboratory prototyping facilities are required to carry out the necessary R&D.
Outlook {#sec:funding}
=======
The construction cost of the envisioned 3G detectors is not well known, as full-fledged designs for 3G detectors are not yet available. To move forward we will request funding for engineering and costing studies, and funds to support a basic research program with identified technical goals that must be reached to finalize the design.
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abstract: 'In analogy with the classical theory of filters, for finitely complete categories, we provide the concepts of filter, $\mathfrak{G}$-neighborhood (short for “Grothendieck-neighborhood") and cover-neighborhoodof a point, with the aim of studying convergence, cluster point and closure of sieves on objects of that kind of categories.'
author:
- 'Joaquín Luna-Torres'
title: 'FILTERS AND $\mathfrak{G}$-CONVERGENCE IN CATEGORIES'
---
Introduction
=============
Convergence theory offers a versatile and effective framework for some areas of mathematics. Let us start by saying a few words about the history of this concept.
It was defined for the first time probably by Henri Cartan [@HC]. Although the notion of a limit along a filter was defined in this work, in the maximal possible generality – the considered filter could be a filter on an arbitrary set and the limit was defined for any map from this set to a topological space – the attention of mathematicians in the following years was mostly focused to two special cases.
- In general topology the notion of limit of a filter on a topological space $X$ became one of the two basic tools used to describe the convergence in general topological spaces together with the notion of a net. Some authors studied also the convergence of a sequence along a filter.
- The definition of the limit along a filter can be reformulated using ideals – the dual notion to the notion of filter. This type of limit of sequences was introduced independently by P. Kostyrko et alt., [@KM] and F. Nuray and W. H. Ruckle [@NR] and studied under the name [*[“ I-convergence"]{}*]{}. The motivation for this direction of research was an effort to generalize some known results on statistical convergence.
In category theory, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in general topology.
In this paper, we use the concept of sieve to build filters in categories and establish a link between filters and grothendieck topologies.
The paper is organized as follows: We describe, in section $2$, the notion of sieve as in S. MacLane and I. Moerdijk [@MM] . In section $3$, we present the concepts of filters, base of filters, we study the lattice structure of all filters on a category and we present the concept of ultrafilter; after that, in section $4$, we introduce the concepts of Systems of neighborhood,$\mathfrak{G}$-neighborhood of a point, cover-neighborhood, convergence, cluster point and closure of a sieve and some theorems about them.
Theoretical Considerations
==========================
Throughout this paper, we will work within an ambient category $\mathscr{C} $ which is finitely complete.
Remember that if $\mathscr{C} $ is a category and $C$ is an object of $\mathscr{C} $, [**a sieve**]{} $S: \mathscr{C}^{op} \rightarrow Set$ on $C$ is a subfunctor of $Hom_{\mathscr{C}} (-,C)$
If we denote by $Sieve_{\mathscr{C}}(C)$ (or $Sieve(C)$ for short) the set of all sieves on $C$, then $Sieve(C)$ becomes a partially ordered under inclusion. It is easy to see that the union or intersection of any family of sieves on $C$ is a sieve on $C$, so $Sieve(C)$ is a complete lattice.
From S. MacLane and I. Moerdijk [@MM], Chapter III, we have the following:
- Let $\mathscr{C} $ be a category, and let $Sets^{\mathscr{C}^{op}}$ be the corresponding functor category and let $$\begin{aligned}
y: &\mathscr{C} \rightarrow Sets^{\mathscr{C}^{op}}\\
&C\mapsto \mathscr{C}(-,C)\end{aligned}$$ then a sieve $\mathcal S$ on $\mathscr{C}$ is simply a subobject $\mathcal S\subseteq y(C)$ in $Sets^{\mathscr{C}^{op}}$.
Alternatively, $\mathcal S$ may be given as a family of morphisms in $\mathscr{C}$, all with codomain $C$, such that $$f \in \mathcal S \Longrightarrow f\circ g \in \mathcal S$$ whenever this composition makes sense; in other words, $\mathcal S$ is a right ideal.
- If $\mathcal S$ is a sieve on $\mathscr C$ and $h: D\rightarrow C$ is any arrow to $C$, then $h^{*}(\mathcal S) = \{g \mid cod(g) = D, h\circ g \in \mathcal S\}$ is a sieve on $D$.
Filters on a category
=====================
A filter on a category $\mathscr{C}$ is a function $\mathfrak F$ which assigns to each object $C$ of $\mathscr{C}$ a collection $\mathfrak F(C) $ of sieves on $\mathscr{C}$, in such a way that
1. If $S \in \mathfrak F(C)$ and $ R$ is a sieve on $C$ such that $ S \subseteq R$, then $ R \in \mathfrak F(C)$;
2. every finite intersection of sieves of $\mathfrak F(C)$ belongs to $\mathfrak F(C)$;
3. the empty sieve is not in $\mathfrak F(C)$.
The pair $(C, \mathfrak F(C))$ is called [**a filtered object**]{}.
From the definition of a Grothendieck topology $J$ on a category $\mathscr{C}$ it follows that for each object $C$ of $\mathscr{C}$ and that
- for $S\in J(C)$ any larger sieve $R$ on $C$ is also a member of $J(C)$.
- It is also clear that if $R; S\in J(C)$ then $R\cap S \in J(C)$,
- consequently a Grothendieck topology produces a filter in the same category $\mathscr{C}$; it is enough to remove the empty sieve of $J(C)$.
A filter subbase on a category $\mathscr{C}$ is a function $\mathfrak S$ which assigns to each object $C$ of $\mathscr{C}$ a collection $\mathfrak S(C) $ of sieves on $\mathscr{C}$, in such a way that no finite subcollection of $\mathfrak S(C) $ has an empty intersection.
An immediate consequence of this definition is
A sufficient condition that there should exist a filter $\mathfrak S^{'}$ on a category $\mathscr{C}$ greater than or equal to a function $\mathfrak S$ (as above) is that $\mathscr{C}$ should be a filter subbase on $\mathscr{C}$.
Observe that $\mathfrak S^{'}$ is the coarset filter greater than $\mathfrak S$.
A basis of a filter on a category $\mathscr{C}$ is a function $\mathfrak B$ which assigns to each object $C$ of $\mathscr{C}$ and collection $\mathfrak B(C) $ of sieves on $\mathscr{C} $, in such a way that
1. The intersection of two sieves of $\mathfrak B(C)$ contains a sieve of $\mathfrak B(C)$;
2. $\mathfrak B(C)$ is not empty, and the empty sieve is not in $\mathfrak B(C)$.
If $\mathfrak B$ is a basis of filter on a category $\mathscr{C} $, then $\mathfrak B$ generates a filter $\mathfrak F$ by $$S \in \mathfrak F(C) \Leftrightarrow \exists R\in \mathfrak B(C)\,\ \text{such that}\,\ R\subseteq S$$ for each object $C$ of $\mathscr{C} $.
It is easy to check that this, indeed, defines a filter from a basis $\mathfrak B$.
The ordered set of all filters on a category
--------------------------------------------
Given two filters $\mathfrak F_1$, $\mathfrak F_2$ on the same category $\mathscr{C}$, $\mathfrak F_2$ is said to be finer than $\mathfrak F_1$, or $\mathfrak F_1$ is coarser than $\mathfrak F_2$, if $$\mathfrak F_1(C) \subseteq \mathfrak F_2(C)$$ for all $C$ object of $\mathscr{C}$.
In this way, the set of all filters on a category $\mathscr{C}$ is ordered by the relation [**[ “$\mathfrak F_1$ is coarser than $\mathfrak F_2$"]{}**]{}.
Let $(\mathfrak F_i)_{i\in I}$ be a nonempty family of filters on a category $\mathscr{C}$; then the function $\mathfrak F$ which assigns to each object $C$ the collection $$\mathfrak F(C)=\bigcap_{i\in I}\mathfrak F_i(C)$$ is manifestly a filter on $\mathscr{C}$ and is obviously the greatest lower bound of the family $(\mathfrak F_i)_{i\in I}$ on the ordered set of all filters on $\mathscr{C}$.
An ultrafilter on a category $\mathscr{C}$ is a filter $\mathfrak U$ such that there is no filter on $\mathscr{C}$ which is strictly finer than $\mathfrak U$..
Using the Zorn lemma, we deduce that
If $\mathfrak F$ is any filter on a category $\mathscr{C}$, there is an ultrafilter finer than $\mathfrak F$ on $\mathscr{C}$.
Let $\mathfrak{U}$ be an ultrafilter on a category $\mathscr{C}$, and let $C$ be an object of $\mathscr{C}$. Let $ S,T$ be sieves on $C)$ such that $S\cup T \in \mathfrak{U}(C)$ then either $S \in \mathfrak{U}(C)$ or $T \in \mathfrak{U}(C)$.
If the affirmation is false, there exist sieves $ S,T$ on $C$ that do not belong to $ \mathfrak{U}(C)$, but $S\cup T \in \mathfrak{U}(C)$. Consider a function $\mathfrak{T}:\mathscr{C}\rightarrow Set$ and let us build the collection of sieves $R$ on $C$ such that $ S\cup R\in \mathfrak{U}(C)$. Let us verify that $\mathfrak{T}$ is a filter on a $\mathscr{C}$: Let $C$ be an object of the category $\mathscr{C}$, and let $\mathfrak{T}(C)$ be the image of $C$ under $\mathfrak{T}$; in other words, $\mathfrak{T}(C)=\{ R\in \,\ Sieve(C)\mid R\cup S \in \mathfrak{U}(C)\}$, then
1. if $R^{'} \in \mathfrak T(C)$ then $R^{'} \cup S\in \mathfrak{U}(C)$; and if $ R^{''}$ is a sieve on $C$ such that $ R{'}\subseteq R{''}$, then $ R^{''}\cup S \in \mathfrak U(C)$. Consequently $R^{''} \in \mathfrak T(C)$.
2. We must show that every finite intersection of sieves of $\mathfrak T(C)$ belongs to $\mathfrak T(C)$; indeed, let $$(R_i)_{i=1,\cdots,n}$$ be a finite collection of sieves on $C$ such that $$R_i\cup S\in \mathfrak{U}(C),\,\ \text{for all}\,\ i = 1. . . n,$$ then $$(R_1\cup S)\cap (R_2\cup S)\cap \cdots \cap (R_n\cup S) = \left(\bigcap_{i=1}^{n}R_i\right)\cup S\in \mathfrak{U}(C).$$ which is equivalent to saying that $$\left(\bigcap_{i=1}^{n}R_i\right) \in \mathfrak T(C).$$
3. Evidently, the empty sieve is not in $\mathfrak T(C)$.
Therefore $\mathfrak{T}$ is a filter finer than $\mathfrak{U}$, since $T\in \mathfrak T(C)$; but this contradicts the hypothesis than $\mathfrak{U}$ is an ultrafilter.
If the union of a finite sequence $(S_i)_{i=1,\cdots,n}$ of sieves on $C$ belongs to the image, $\mathfrak{U}(C)$, of an object $C$ under an ultrafilter $\mathfrak{U}$, then at least one of the $S_i$ belongs to $\mathfrak{U}(C)$.
The proof is a simple use of induction on $n$.
Systems of Neighborhoods
========================
Remember that a [**point**]{} of an object $C$ of a category $\mathscr{C}$ is a morphism $p:1\rightarrow C$, where $1$ is a terminal object of $\mathscr{C}$.
Let $(\mathscr C, J)$ be a category endowed with a Grothendieck topology, and let $C$ be an object of $\mathscr{C}$.
1. A sieve $V$ in $J(C)$, is said to be a $\mathfrak{G}$-neighborhood (short for “Grothendieck-neighborhood") of a point $p_{\scriptscriptstyle C}$ of $C$ if there exist a morphism $\phi:D\rightarrow C$ in $V$ and a point $q:1 \rightarrow D $ such that the diagram
D & C\
1\[swap\][p]{}&
is commutative.
2. A sieve $V$ in $J(C)$, is said to be a $\mathfrak{G}$-neighborhood of a sieve $T$ on $C$ if $T\subseteq V$.
Let $(\mathscr C, J)$ be a category endowed with a Grothendieck topology. A cover-neighborhood of $(\mathscr C, J)$ is a function $\mathcal N$ which assigns to each object $(C, J(C))$ of $(\mathscr{C}, J)$ and to each point $p_{\scriptscriptstyle C}$ of $C$, a collection $$\mathcal N_{\scriptstyle p_{_{\scriptscriptstyle C}}}(C)\,\ \text{of sieves of}\,\ \mathscr C$$ such that each sieve in $\mathcal N_{\scriptstyle p_{_{\scriptscriptstyle C}}}(C)$ contains a $\mathfrak{G}$-neighborhood of $p_{\scriptscriptstyle C}$.
Let $\mathscr{C}$ be a category, and let $C$ be an object of $\mathscr{C}$. The pair $(C,\mathscr N_p(C))$, where $\mathscr N_p(C)$ is the collection of all cover-neighborhoods of a point $p$ of $C$, is [**a filtered object**]{}
1. If $S\in\mathscr N_p(C)$ and $R$ is a sieve on $C$ such that $S \subseteq R$, then $R\in\mathscr N_p(C)$, because there is a $\mathfrak{G}$-neighborhood $V$ of $p_{\scriptscriptstyle C}$ such that $V \subseteq S\subseteq R$;
2. let $\{S_1, S_2,\cdots,S_n\}$ be a finite collection of sieves of $\mathscr N_p(C)$, then there exists a collection $\{V_1, V_2,\cdots, V_n\}$ of $\mathfrak{G}$-neighborhood of $p_{\scriptscriptstyle C}$ such that $V_i\subseteq S_i$ for $I=1,2,\cdots n$, therefore $$\bigcap_{i=1}^n V_n \subseteq\bigcap_{i=1}^n S_n \,\ \text{and}\,\ \bigcap_{i=1}^n S_n\in\in\mathscr N_p(C)$$;
3. the empty sieve is not in $\mathscr N_p(C)$ (each sieve contains a point).
In this case, we say that the point $p$ of $C$ is a [**limit point**]{} of $\mathscr N_p(C)$.
\[converges\] Let $(\mathscr C, J)$ be a category endowed with a Grothendieck topology; let $\mathfrak F$ be a filter on $\mathscr{C}$ and let $C$ be an object of $\mathscr{C}$.
1. We shall say that $\mathfrak F(C) $ [**converges**]{} to a point $p$ of $C$ if$\mathscr N_p(C)\subseteq \mathfrak F(C) $.
2. The closure of a sieve $A$ on $C$ is the collection of all points $p$ of $C$ such that every cover-neighborhood of $p$ meets $A$.
3. A point $p$ of $C$ is a cluster point of $\mathfrak B(C)$, the image under the filter base $\mathfrak B$ of $C$, if it lies in the closure of all the sieves on $\mathfrak B(C)$.
4. A point $p$ of $C$ is a cluster point of $\mathfrak F(C)$, the image under the filter $\mathfrak F$ of $C$, if it lies in the closure of all the sieves on $\mathfrak F(C)$.
Let $(\mathscr C, J)$ be a category endowed with a Grothendieck topology; let $\mathfrak F$ be a filter on $\mathscr{C}$ and let $C$ be an object of $\mathscr{C}$. The point $p$ of $C$ is a cluster point of $\mathfrak F(C)$ iif there is a filter $\mathscr G$ finer than $\mathfrak F$ such that $\mathscr G(C) $ [**converges**]{} to $p$.
Let us begin by assuming that the point $p$ of $C$ is a cluster point of $\mathfrak F(C)$; from definition \[converges\], it follows that for each sieve $A$ in $\mathfrak F(C)$, every $\mathfrak{G}$-neighborhood $V$ of $p$ meets $A$. We need to show that the collection $$\mathscr B(C) =\{A\cap V\mid V \,\ \text{is a $\mathfrak{G}$-neighborhood of}\,\ p\}$$ define a base for a filter $\mathscr G$ finer than $\mathfrak F$.in such a way that $\mathscr G(C) $ [**converges**]{} to $p$.
Indeed,
1. Let $A\cap V$, $A\cup W$ two elements of the collection $\mathscr B(C)$, since $$(A\cap V) \cap (A\cup W)= A\cup (V\cap W)$$ and $V\cap W$ is a $\mathfrak{G}$-neighborhood of $p$, there exists $U$, a $\mathfrak{G}$-neighborhood of $p$ such that $$U\subseteq V\cap W,$$ and clearly $A\cap U \in \mathscr B(C)$;
2. Obviously $\mathscr B(C)$ is not empty, and the empty sieve is not in $\mathscr B(C)$.
Now, if $\mathscr G$ is the filter generated by $\mathscr B$ then $\mathscr G$ is finer than $\mathfrak F$, and $\mathscr G(C)$ naturally converges to $p$.
Conversely, if there is a filter $\mathscr G$ finer than $\mathfrak F$ such that $\mathscr G(C) $ [**converges**]{} to $p$ then each sieve $R$ in $\mathfrak F(C)$ and each $\mathfrak{G}$-neighborhood $U$ of $p$ belongs to $\mathscr G$ and hence meet, so the point $p$ of $C$ is a cluster point of $\mathfrak F(C)$.
Let $(\mathscr C, J)$ be a category endowed with a Grothendieck topology; let $C$ be an object of $\mathscr{C}$ and let $A$ be a sieve on $C$. The point $p$ of $C$ lies in the closure of $A$ iif there is a filter $\mathscr G $ such that $A\in\mathscr G(C) $ and $\mathscr G(C) $ [**converges**]{} to $p$.
Let us begin by assuming that The point $p$ of $C$ lies in the closure of $A$; from definition \[converges\], it follows that every $\mathfrak{G}$-neighborhood $V$ of $p$ meets $A$. Then $$\mathscr B(C) =\{A\cap V\mid V \,\ \text{is a $\mathfrak{G}$-neighborhood of}\,\ p\}$$ is a base for a filter $\mathscr G$, in such a way that $\mathscr G(C) $ [**converges**]{} to $p$.
Conversely, if $A \in \mathscr G(C)$ and $\mathscr G(C) $ [**converges**]{} to $p$ then $p$ is a cluster point of $\mathscr G(C) $ and hence $p$ lies in the closure of $A$.
Let $\mathcal A$ be a complete Heyting algebra and regard $\mathcal A$ as a category in the usual way.
- [@MM] Then $\mathcal A$ can be equipped with a base for a Grothendieck topology $K$, given by $$\{a_i\mid i\in I\}\in K(c)\,\ \text{if and only if}\,\ \bigvee_{i\in I}=c,$$ where $\{a_i\mid i\in I\}\subseteq \mathcal A$ and $c\in \mathcal A$.
- A sieve $S$ on an element $c$ of $\mathcal A$ is just a subset of elements $b\leqslant c$ such that $a\leqslant b\in S$ implies $a\in S$.
- In the Grothendieck topology $J$ with basis $K$, a sieve $S$ on $c$ covers $c$ iff $$\bigvee S=c.$$
- A filter on t$\mathcal A$ is a function $\mathfrak F$ which assigns to each element $c$ of $\mathcal A$ a collection $\mathfrak F(c) $ of sieves, such that
1. If $S \in \mathfrak F(c)$ and $ R$ is a sieve on $c$ such that $ S \subseteq R$, then $ R \in \mathfrak F(c)$;
2. every finite intersection of sieves of $\mathfrak F(c)$ belongs to $\mathfrak F(c)$;
3. the empty sieve is not in $\mathfrak F(c)$.
- An immediate consequence of the previous construction of a Grothendieck topology and a fiter on $\mathcal A$ is that
$$\mathfrak F(c)\,\ \text{converges to $c$ iff}\,\ \bigvee S=c, \,\ \text{for each}\,\ S \in \mathfrak F(c).$$
Conclusions and Comments {#conclusions-and-comments .unnumbered}
========================
We proposed the construction of the concepts of filter, $\mathfrak{G}$-neighborhood of a point and cover-neighborhood with the aim of studying convergence, cluster point and closure of sieves on objects of some kind of categories. The properties of these objects (similar to those general topology) are postponed to future works.
Acknowledgments {#acknowledgments .unnumbered}
===============
. The author wish to thank *Fundación Haiko* of Colombia for its constant encouragement during the development of this work.
[10]{}
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abstract: |
A partial wave analysis is presented of $J/\psi \to \phi \pi ^+\pi ^-$ and $\phi K^+K^-$ from a sample of 58M $J/\psi$ events in the BESII detector. The $f_0(980)$ is observed clearly in both sets of data, and parameters of the Flatt' e formula are determined accurately: $M = 965 \pm 8$ (stat) $\pm 6$ (syst) MeV/c$^2$, $g_1 = 165 \pm 10 \pm 15 $ MeV/c$^2$, $g_2/g_1 = 4.21 \pm 0.25 \pm 0.21$. The $\phi \pi \pi$ data also exhibit a strong $\pi \pi$ peak centred at $M = 1335$ MeV/c$^2$. It may be fitted with $f_2(1270)$ and a dominant $0^+$ signal made from $f_0(1370)$ interfering with a smaller $f_0(1500)$ component. There is evidence that the $f_0(1370)$ signal is resonant, from interference with $f_2(1270)$. There is also a state in $\pi \pi$ with $M = 1790 ^{+40}_{-30}$ MeV/c$^2$ and $\Gamma = 270
^{+60}_{-30}$ MeV/c$^2$; spin 0 is preferred over spin 2. This state, $f_0(1790)$, is distinct from $f_0(1710)$. The $\phi K\bar K$ data contain a strong peak due to $f_2'(1525)$. A shoulder on its upper side may be fitted by interference between $f_0(1500)$ and $f_0(1710)$.
[*PACS:*]{} 13.25.Gv, 14.40.Gx, 13.40.Hq
title: 'Resonances in $J/\psi \to \phi \pi ^+\pi ^-$ and $\phi K^+K^-$'
---
M. Ablikim$^{1}$, J. Z. Bai$^{1}$, Y. Ban$^{11}$, J. G. Bian$^{1}$, D. V. Bugg$^{20}$, X. Cai$^{1}$, J. F. Chang$^{1}$, H. F. Chen$^{17}$, H. S. Chen$^{1}$, H. X. Chen$^{1}$, J. C. Chen$^{1}$, Jin Chen$^{1}$, Jun Chen$^{7}$, M. L. Chen$^{1}$, Y. B. Chen$^{1}$, S. P. Chi$^{2}$, Y. P. Chu$^{1}$, X. Z. Cui$^{1}$, H. L. Dai$^{1}$, Y. S. Dai$^{19}$, Z. Y. Deng$^{1}$, L. Y. Dong$^{1}$$^a$, Q. F. Dong$^{15}$, S. X. Du$^{1}$, Z. Z. Du$^{1}$, J. Fang$^{1}$, S. S. Fang$^{2}$, C. D. Fu$^{1}$, H. Y. Fu$^{1}$, C. S. Gao$^{1}$, Y. N. Gao$^{15}$, M. Y. Gong$^{1}$, W. X. Gong$^{1}$, S. D. Gu$^{1}$, Y. N. Guo$^{1}$, Y. Q. Guo$^{1}$, Z. J. Guo$^{16}$, F. A. Harris$^{16}$, K. L. He$^{1}$, M. He$^{12}$, X. He$^{1}$, Y. K. Heng$^{1}$, H. M. Hu$^{1}$, T. Hu$^{1}$, G. S. Huang$^{1}$$^b$, X. P. Huang$^{1}$, X. T. Huang$^{12}$, X. B. Ji$^{1}$, C. H. Jiang$^{1}$, X. S. Jiang$^{1}$, D. P. Jin$^{1}$, S. Jin$^{1}$, Y. Jin$^{1}$, Yi Jin$^{1}$, Y. F. Lai$^{1}$, F. Li$^{1}$, G. Li$^{2}$, H. H. Li$^{1}$, J. Li$^{1}$, J. C. Li$^{1}$, Q. J. Li$^{1}$, R. Y. Li$^{1}$, S. M. Li$^{1}$, W. D. Li$^{1}$, W. G. Li$^{1}$, X. L. Li$^{8}$, X. Q. Li$^{10}$, Y. L. Li$^{4}$, Y. F. Liang$^{14}$, H. B. Liao$^{6}$, C. X. Liu$^{1}$, F. Liu$^{6}$, Fang Liu$^{17}$, H. H. Liu$^{1}$, H. M. Liu$^{1}$, J. Liu$^{11}$, J. B. Liu$^{1}$, J. P. Liu$^{18}$, R. G. Liu$^{1}$, Z. A. Liu$^{1}$, Z. X. Liu$^{1}$, F. Lu$^{1}$, G. R. Lu$^{5}$, H. J. Lu$^{17}$, J. G. Lu$^{1}$, C. L. Luo$^{9}$, L. X. Luo$^{4}$, X. L. Luo$^{1}$, F. C. Ma$^{8}$, H. L. Ma$^{1}$, J. M. Ma$^{1}$, L. L. Ma$^{1}$, Q. M. Ma$^{1}$, X. B. Ma$^{5}$, X. Y. Ma$^{1}$, Z. P. Mao$^{1}$, X. H. Mo$^{1}$, J. Nie$^{1}$, Z. D. Nie$^{1}$, S. L. Olsen$^{16}$, H. P. Peng$^{17}$, N. D. Qi$^{1}$, C. D. Qian$^{13}$, H. Qin$^{9}$, J. F. Qiu$^{1}$, Z. Y. Ren$^{1}$, G. Rong$^{1}$, L. Y. Shan$^{1}$, L. Shang$^{1}$, D. L. Shen$^{1}$, X. Y. Shen$^{1}$, H. Y. Sheng$^{1}$, F. Shi$^{1}$, X. Shi$^{11}$$^c$, H. S. Sun$^{1}$, J. F. Sun$^{1}$, S. S. Sun$^{1}$, Y. Z. Sun$^{1}$, Z. J. Sun$^{1}$, X. Tang$^{1}$, N. Tao$^{17}$, Y. R. Tian$^{15}$, G. L. Tong$^{1}$, G. S. Varner$^{16}$, D. Y. Wang$^{1}$, J. Z. Wang$^{1}$, K. Wang$^{17}$, L. Wang$^{1}$, L. S. Wang$^{1}$, M. Wang$^{1}$, P. Wang$^{1}$, P. L. Wang$^{1}$, S. Z. Wang$^{1}$, W. F. Wang$^{1}$$^d$, Y. F. Wang$^{1}$, Z. Wang$^{1}$, Z. Y. Wang$^{1}$, Zhe Wang$^{1}$, Zheng Wang$^{2}$, C. L. Wei$^{1}$, D. H. Wei$^{1}$, Y. M. Wu$^{1}$, X. M. Xia$^{1}$, X. X. Xie$^{1}$, B. Xin$^{8}$$^b$, G. F. Xu$^{1}$, H. Xu$^{1}$, S. T. Xue$^{1}$, M. L. Yan$^{17}$, F. Yang$^{10}$, H. X. Yang$^{1}$, J. Yang$^{17}$, Y. X. Yang$^{3}$, M. Ye$^{1}$, M. H. Ye$^{2}$, Y. X. Ye$^{17}$, L. H. Yi$^{7}$, Z. Y. Yi$^{1}$, C. S. Yu$^{1}$, G. W. Yu$^{1}$, C. Z. Yuan$^{1}$, J. M. Yuan$^{1}$, Y. Yuan$^{1}$, S. L. Zang$^{1}$, Y. Zeng$^{7}$, Yu Zeng$^{1}$, B. X. Zhang$^{1}$, B. Y. Zhang$^{1}$, C. C. Zhang$^{1}$, D. H. Zhang$^{1}$, H. Y. Zhang$^{1}$, J. Zhang$^{1}$, J. W. Zhang$^{1}$, J. Y. Zhang$^{1}$, Q. J. Zhang$^{1}$, S. Q. Zhang$^{1}$, X. M. Zhang$^{1}$, X. Y. Zhang$^{12}$, Y. Y. Zhang$^{1}$, Yiyun Zhang$^{14}$, Z. P. Zhang$^{17}$, Z. Q. Zhang$^{5}$, D. X. Zhao$^{1}$, J. B. Zhao$^{1}$, J. W. Zhao$^{1}$, M. G. Zhao$^{10}$, P. P. Zhao$^{1}$, W. R. Zhao$^{1}$, X. J. Zhao$^{1}$, Y. B. Zhao$^{1}$, Z. G. Zhao$^{1}$$^e$, H. Q. Zheng$^{11}$, J. P. Zheng$^{1}$, L. S. Zheng$^{1}$, Z. P. Zheng$^{1}$, X. C. Zhong$^{1}$, B. Q. Zhou$^{1}$, G. M. Zhou$^{1}$, L. Zhou$^{1}$, N. F. Zhou$^{1}$, K. J. Zhu$^{1}$, Q. M. Zhu$^{1}$, Y. C. Zhu$^{1}$, Y. S. Zhu$^{1}$, Yingchun Zhu$^{1}$$^f$, Z. A. Zhu$^{1}$, B. A. Zhuang$^{1}$, X. A. Zhuang$^{1}$, B. S. Zou$^{1}$\
(BES Collaboration)\
\[att\] $^{1}$ Institute of High Energy Physics, Beijing 100049, People’s Republic of China\
$^{2}$ China Center for Advanced Science and Technology(CCAST), Beijing 100080, People’s Republic of China\
$^{3}$ Guangxi Normal University, Guilin 541004, People’s Republic of China\
$^{4}$ Guangxi University, Nanning 530004, People’s Republic of China\
$^{5}$ Henan Normal University, Xinxiang 453002, People’s Republic of China\
$^{6}$ Huazhong Normal University, Wuhan 430079, People’s Republic of China\
$^{7}$ Hunan University, Changsha 410082, People’s Republic of China\
$^{8}$ Liaoning University, Shenyang 110036, People’s Republic of China\
$^{9}$ Nanjing Normal University, Nanjing 210097, People’s Republic of China\
$^{10}$ Nankai University, Tianjin 300071, People’s Republic of China\
$^{11}$ Peking University, Beijing 100871, People’s Republic of China\
$^{12}$ Shandong University, Jinan 250100, People’s Republic of China\
$^{13}$ Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China\
$^{14}$ Sichuan University, Chengdu 610064, People’s Republic of China\
$^{15}$ Tsinghua University, Beijing 100084, People’s Republic of China\
$^{16}$ University of Hawaii, Honolulu, Hawaii 96822, USA\
$^{17}$ University of Science and Technology of China, Hefei 230026, People’s Republic of China\
$^{18}$ Wuhan University, Wuhan 430072, People’s Republic of China\
$^{19}$ Zhejiang University, Hangzhou 310028, People’s Republic of China\
$^{20}$ Queen Mary, University of London, London E1 4NS, UK\
$^{a}$ Current address: Iowa State University, Ames, Iowa 50011-3160, USA.\
$^{b}$ Current address: Purdue University, West Lafayette, IN 47907, USA.\
$^{c}$ Current address: Cornell University, Ithaca, New York 14853, USA.\
$^{d}$ Current address: Laboratoire de l’Acc[é]{}l[é]{}ratear Lin[é]{}aire, F-91898 Orsay, France.\
$^{e}$ Current address: University of Michigan, Ann Arbor, Michigan 48109, USA.\
$^{f}$ Current address: DESY, D-22607, Hamburg, Germany.\
The processes $J/\psi \to \phi \pi ^+\pi ^-$ and $\phi K^+K^-$ have been studied previously in the Mark III \[1\] and DM2 \[2\] experiments. Here we report BESII data on these channels with much larger statistics from a sample of 58 million $e^+e^- \to J/\psi$ interactions. The $f_0(980)$, $f_0(1370)$ and a state with mass at 1790 MeV/c$^2$ and with spin 0 preferred over spin 2, called the $f_0(1790)$ throughout this paper, are studied here. A particular feature is that $f_0(1790) \to \pi \pi$ is strong, but there is little or no corresponding signal for decays to $K\bar K$. This behavior is incompatible with $f_0(1710)$, which is known to decay dominantly to $K\bar K$; this indicates the presence of two distinct states, $f_0(1710)$ and $f_0(1790)$.
A detailed description of the BESII detector is given in Ref. [@bes]. It has a cylindrical geometry around the beam axis. Trajectories of charged particles are measured in the vertex chamber (VC) and main drift chamber (MDC); these are surrounded by a solenoidal magnet providing a field of 0.4T. Photons are detected in a Barrel Shower Counter (BSC) comprized of a sandwich array of lead and gas chambers. Particle identification is accomplished using time-of-flight (TOF) information from the TOF scintillator array located immediately outside the MDC and the $dE/dx$ information from the MDC.
Events must have four charged tracks with total charge zero. These tracks are required to lie well within the MDC acceptance with a polar angle $\theta$ satisfying $|\cos \theta | < 0.80$ and to have their point of closest approach to the beam within 2 cm of the beam axis and within 20 cm of the centre of the interaction region along the beam axis. Further, events must satisfy a four-constraint (4C) kinematic fit with $\chi ^2 < 40$.
Kaons, pions, and protons are identified by time-of-flight, $dE/dx$, and also by kinematic fitting. The $\sigma$ of the TOF measurement is 180 ps. Kaons may be identified by TOF and $dE/dx$ up to a momentum of 800 MeV/c. The 4C kinematic fit provides additional good separation between $\phi \pi \pi$ and $\phi KK$; residual crosstalk between these channels is negligible.
The $K^+K^-$ invariant mass distributions for $J/\psi \to K^+K^-\pi
^+\pi ^-$ and $J/\psi \to K^+K^-K^+K^-$ are shown in Figs. 1(a) and (c); in the latter case, the $K^+K^-$ pair with invariant mass closest to the $\phi$ is plotted. The peaks of the $\phi$ lie at 1019.7 $\pm$ 0.2 and 1020.0 $\pm$ 0.2 MeV/c$^2$ in (a) and (c), in reasonable agreement with the value of the Particle Data Group (PDG) \[4\]. In both cases, there is a clear $\phi$ signal over a modest background of events due to $K^+K^-\pi ^+\pi ^-$ or $K^+K^-K^+K^-$ without a $\phi$. The curves in (a) and (c) show the background, assuming it follows a phase space dependence on $M(K^+K^-)$. The resulting background is $(19.0\pm1.5)$% in (a) and $(6.2\pm1.6)$% in (c). Events containing a $\phi$ are selected by requiring at least one kaon identified by TOF or $dE/dx$ and $|M_{K^+K^-} - M_{\phi }| < 15$ MeV/c$^2$.
Before discussing the main physics results, it is necessary to deal with an important background arising in $J/\psi \to K^+K^-\pi ^+\pi
^-$. Events for the study of this background channel are selected in a sidebin having $M(K^+K^-) = 1.045$–1.09 GeV/c$^2$. Fig. 2 shows Dalitz plots and mass projections for these sidebin events; Dalitz plots for $\phi \pi ^+\pi ^-$ and $\phi K^+K^-$ data are shown in Fig. 3. For the $K^+K^-\pi ^+\pi ^-$ sidebin, there is a strong peak in the $\phi
\pi$ mass distribution of Fig. 2(b) centred at 1500 MeV/c$^2$ with a full-width of 200 MeV/c$^2$. This $\phi \pi$ peak is of interest because of an earlier report of a possible exotic state close to this mass with quantum numbers $J^P = 1^-$ \[5\]. The reflection of this peak produces a horizontal band at the bottom of Fig. 2(a); it projects to a broad peak centred at 2450 MeV/c$^2$ in Fig. 2(b). For $K^+K^-K^+K^-$ sidebin events of Fig. 2(c), there is no corresponding peak at low mass in $\phi K$, Fig. 2(d).
In order to investigate the nature of this peak, we select events in the mass range 1400–1600 MeV/c$^2$ from Fig. 2(b). Mass distributions of $K^+\pi ^-$ and $K^-\pi ^+$ pairs are shown in Fig. 2(e) and corresponding distributions for $K^+\pi ^+$ and $K^-\pi ^-$ in Fig. 2(f). There is a strong $K^*(890)$ peak visible in Fig. 2(e) but only a broad peak in Fig. 2(f). It can be shown that the presence of $K^*(890)$ in the background, combined with kinematic selection in a narrow range of $K^+K^-$ masses, can generate the peak position and width of the spurious peak in $\phi \pi$.
A similar effect arises in selected $\phi \pi \pi$ events. Fig. 3(b) shows $M(\phi \pi)$ for events selected as $\phi
\pi^+\pi ^-$ by requiring $M(K^+K^-)$ within $\pm 15$ MeV/c$^2$ of the $\phi$ mass. There is again a $\phi \pi$ peak, centred now at 1460 MeV/c$^2$. Again it can be shown that the peak is consistent entirely with background. There is no significant evidence for an exotic $\phi \pi$ state. If it were misinterpreted as a $\phi \pi$ state, fits show that it requires a $\phi$ combined with an $L = 1$ pion coming from the $K^*_1(890)$, hence quantum numbers $J^P = 1^-$, $0^-$, or $2^-$.
We have carried through a full partial wave analysis in three alternative ways: (a) making a cut in order to exclude events lying within $\pm 80$ MeV/c$^2$ of the central mass of $K^*(890)$, which is slightly narrower than the selection of Fig. 1; (b) including into the fit an incoherent background from $K^*(890)K\pi $; and (c) making a background subtraction which allows for the shift in mass and width between sidebin and data for the background peak in $\phi \pi$ at 1500 MeV/c$^2$. Results of these three approaches agree within errors. We regard the first method as the most reliable, since it is independent of any modelling of the background. Figs. (4) and (5) show the fit from this approach. The cut against $K^*(890)$ eliminates the $\phi \pi$ peak at 1460 MeV/c$^2$, as shown in Fig. 4(d). It also eliminates backgrounds due to channels $K^*(1430)K^*(890)$, observed in the final state $K^+K^-\pi ^+\pi ^-$. It reduces the background under the $\phi$ in Fig. 1(e) to $(13.5\pm1.4)$%; after the background subtraction, the number of $\phi \pi ^+\pi ^-$ events falls to 4180.
The branching fractions for production of $\phi \pi \pi$ and $\phi K\bar
K$ are determined allowing for the efficiencies for detecting the two channels and correcting for unobserved neutral states. Results are: $B(J/\psi \to \phi \pi \pi ) = (1.63 \pm 0.03 \pm 0.20 ) \times
10^{-3}$ and $B(J/\psi \to \phi K \bar K ) = (2.14 \pm 0.04 \pm 0.22
) \times 10^{-3}$. The main contributions to the systematic errors come from differences between data and Monte Carlo simulation for the $\phi$ selection, $K^*(890)$ cut, and particle identification; from uncertainties in the MDC wire resolution; and the total number of $J/\psi$ events.
We turn now to the physics revealed by diagonal bands in the Dalitz plots of Fig. 3 and mass projections of Figs. 4 and 5. There is a strong $f_0(980) \to \pi ^+\pi ^-$ signal in Fig. 4(c) and a low mass peak in Fig. 5(c) due to $f_0(980) \to K^+K^-$. Secondly, the $\phi \pi ^+\pi ^-$ data exhibit in Fig. 4(c) the clearest signal yet observed for $f_0(1370) \to \pi ^+\pi ^-$. Several authors have previously expressed doubts concerning the existence of $f_0(1370)$, but present data cannot be fitted adequately without it. Both Mark III and DM2 groups observed a similar peak with lower statistics. There have been earlier reports of similar but less conspicuous peaks in $\pi \pi \to K\bar K$ from experiments at ANL \[6,7\] and BNL \[8\]. A third feature in the $\phi \pi ^+\pi ^-$ data in Fig. 4(c) is a clear peak at around 1775 MeV/c$^2$. The $\phi K^+K^-$ data of Fig. 5(c) contain a strong $f_2'(1525)$ peak. However, it is asymmetric and may only be fitted by including on its upper side $f_0(1710)$ interfering with other components.
We now describe the maximum likelihood fit to the data. Amplitudes are fitted to relativistic tensor expressions which are documented in Ref. [@pwa]. The full angular dependence of decays of the $\phi$ and $\pi ^+\pi ^-$ or $K^+K^-$ resonances is fitted, including correlations between them. The line-shape of the $\phi$ is not fitted, because the $\phi$ is much narrower than the experimental resolution. We include production of $J^P = 0^+$ resonances with orbital angular momentum $\ell = 0$ and 2 in the production process $J/\psi \to \phi f_0$. For production of $f_2$, there is one amplitude with $\ell = 0$ and three with $\ell = 2$, where $\ell$ and the spin of $f_2$ may combine to make overall spin $S = 0$, 1 or 2. The one possible $\ell = 4$ amplitude makes a negligible contribution. The acceptance, determined from a Monte Carlo simulation, is included in the maximum likelihood fit. All figures shown here are uncorrected for acceptance, which is approximately uniform across Dalitz plots except for the effect of the $K^*(890)$ cut.
The background subtraction is made by giving data positive weight in log likelihood and sidebin events negative weight; the sidebin events (suitably weighted by $K^+K^-$ phase space) then cancel background in the data sample.
The $\phi \pi ^+\pi ^-$ and $\phi K^+K^-$ data are fitted simultaneously, constraining resonance masses and widths to be the same in both sets of data. Table 1 shows branching fractions of each component, as well as the changes in log likelihood when each component is dropped from the fit and remaining components are re-optimised.
We begin the discussion with $\phi K^+K^-$ data. There is a conspicuous peak due to $f_2'(1525)$. The shoulder on its upper side is fitted mostly by $f_0(1710)$ interfering with $f_0(1500)$, but there is also a possible small contribution from $f_0(1790)$ interfering with $f_0(1500)$. The overall contributions to $\phi
K^+K^-$ are shown by the upper histograms in Figs 5(c) and (d).
The $f_2(1270)$ signal reported below in $\phi \pi \pi$ data allows a calculation of the $f_2(1270) \to K^+K^-$ signal expected in $\phi K^+K^-$, using the branching fraction ratio between $K\bar K$ and $\pi \pi$ of the PDG. Its contribution is negligibly small.
We discuss next the fit to $f_0(980)$. In $\phi \pi ^+\pi ^-$ data, it interferes with a broad component well fitted by the $\sigma$ pole [@opipi]. This component interferes constructively with the lower side of the $f_0(980)$ in Fig. 4(c). Its magnitude is shown by the lower curve in Fig. 4(e).
The $f_0(980)$ amplitude has been fitted to the Flatt' e form: $$f=\frac {1}{M^2 - s - i(g_1\rho _{\pi \pi } +
g_2\rho _{K\bar K})}.$$ Here $\rho$ is Lorentz invariant phase space, $2k/\sqrt {s}$, where $k$ refers to the $\pi$ or $K$ momentum in the rest frame of the resonance. The present data offer the opportunity to determine the ratio $g_2/g_1$ accurately. This is done by determining the number of events due to $f_0(980) \to \pi \pi$ and $\to K^+K^-$ and comparing with the prediction from the Flatt' e formula, as follows. After making the best fit to the data, the fitted $f_0(980) \to \pi ^+\pi ^-$ signal is integrated over the mass range from 0.9 to 1.0 GeV/c$^2$. The fitted $f_0(980) \to K^+K^-$ signal is integrated over the mass range 1.0–1.2 GeV/c$^2$, so as to avoid sensitivity to the tail of the $f_0(980)$ at high mass. The latter integral is given by $$0.5 \int ds |f(980)|^2\rho(K^+K^-)\epsilon(K^+K^-)$$ and the former by $$\frac {2}{3} \int ds |f(980)|^2\rho(\pi \pi )\epsilon(\pi ^+\pi ^-).$$ Here $\epsilon(K^+K^-)$ and $\epsilon(\pi ^+\pi ^-)$ are detection efficiencies. The numerical factors at the beginning of each expression take into account (a) there are equal numbers of decays to $K^+K^-$ and $K^0\bar K^0$ and (b) two-thirds of $\pi \pi $ decays are to $\pi ^+\pi ^-$ and one third to $\pi ^0 \pi
^0$.
By an iterative process which converges rapidly, the ratio $g_2/g_1$ is adjusted until the ratio of these two integrals reproduces the fitted numbers of events for $\phi K^+K^-$ and $\phi \pi ^+\pi ^-$. The result is $g_2/g_1 = 4.21 \pm 0.25$ (stat) $\pm 0.21$ (syst). The systematic error arises from (i) varying the choice of side bins and the magnitude of the background under the $\phi$ peak, (ii) changes in the fit when small amplitudes such as $f_0(1500)$ and $f_2(1270) \to K^+K^-$ and $\sigma\to K^+K^-$ are omitted from the fit, (iii) changing the mass and width of other components within errors and different choices of $\sigma$ parameterization from Ref. [@opipi]. The result is a considerable improvement on earlier determinations. The mass and $g_1$ are adjusted to achieve the best overall fit to the peak in $\phi \pi ^+\pi ^-$ data. Values are $M = 965 \pm 8 \pm 6$ MeV/c$^2$, $g_1 = 165 \pm 10 \pm 15$ MeV/c$^2$.
The ratio $g_2/g_1$ is only weakly correlated with $M$ and $g_1$. However, $g_2 $ is rather strongly correlated with $M$. This arises because the term $ig_2\rho _{KK}(s)$ in the Breit-Wigner denominator, eqn. (1), continues analytically below the $KK$ threshold to $-g_2\sqrt {(4M^2_K/s) - 1}$. It then contributes to the real part of the Breit-Wigner amplitude and interacts with the term $(M^2 - s)$. We find that the correlation is given by $dg_2/dM = -5.9$; the mass goes down as $g_2 $ goes up. Other correlations are weak: $dg_1/dM =
-0.75$ and $dr/dg_1 = -0.068$, where $r = g_2/g_1$.
We consider next the peak in $\phi \pi \pi$ centred at a mass of 1335 MeV/c$^2$. An initial fit was made to $f_2(1270)$ and one $f_0$. The $f_0$ optimizes at $M=1410 \pm 50$ MeV/c$^2$, $\Gamma = 270 \pm 45$ MeV/c$^2$, where errors cover systematic variations when small ingredients in the fit are changed. However, both $f_0(1500)$ and $f_0(1370)$ can contribute. Adding $f_0(1500)$, log likelihood improves by 51: an 8.5 standard deviation improvement for four degrees of freedom. Also the fit to the $\pi \pi$ mass distribution improves visibly. Therefore three components are required in the 1335 MeV/c$^2$ peak: $f_2(1270)$, $f_0(1370)$ and $f_0(1500)$. Removing $f_0(1370)$ makes log likelihood worse by 83, a 10.8 standard deviation effect.
Angular correlations between decays of $\phi $ and $f_2$ are very sensitive to the presence of $f_2(1270)$, which is accurately determined. It optimizes at $M = 1275 \pm 15$ MeV/c$^2$, $\Gamma = 190 \pm 20$ MeV/c$^2$, values consistent with $f_2(1270)$. The fact that its mass and width agree well with PDG values rules out the possibility that the remainder of the signal in this mass range is due to spin 2; otherwise the fit to $f_2(1270)$ would be severely affected. Angular distributions for the remaining components are indeed consistent with isotropic decay angular distributions from spin 0.
The $f_0(1370)$ interferes with $f_0(1500)$ and $f_2(1270)$. This helps to make $f_0(1370)$ more conspicuous than in other data. However, because of the interferences, its mass and width are not accurately determined. The mass of $f_0(1370)$ is $1350 \pm 50$ MeV/c, where the error is the quadratic sum of the systematic and statistical errors.
The width of $f_0(1370)$ is somewhat more stable. It is determined essentially by the full width of the peak in $\phi \pi ^+ \pi ^-$ of 270 MeV/c$^2$; interferences with $f_2(1270)$ and $f_0(1500)$ affect this number only by small amounts and the fitted width is $265 \pm 40$ MeV/c$^2$. If both $f_0(1370)$ and $f_0(1500)$ are removed, log likelihood is worse by 595. Removing $f_0(1500)$ from the fit perturbs the mass fitted to $f_0(1370)$ upwards to 1410 $\pm$ 50 MeV/c$^2$; this is obviously due to the fact that $f_0(1370)$ is trying to simulate the missing $f_0(1500)$ component.
The presence of a peak due to $f_0(1370)$ is strongly suggestive of a resonance. In order to check for resonant phase variation, we have tried replacing the amplitude by its modulus, without any phase variation. In this case, log likelihood is worse by 39, nearly a 9 standard deviation effect for a change of one degree of freedom. The conclusion is that the $f_0(1370)$ peak is resonant. It is not possible to display the phase directly, since it is determined by interferences between two $f_0(1370)$ and four $f_2(1270)$ amplitudes.
The magnitude of the signal due to $f_0(1370) \to K^+K^-$ in the fit gives a branching fraction ratio $$\frac {B[f_0(1370) \to K\bar K]}{B[f_0(1370) \to \pi \pi]} = 0.08\pm 0.08.$$ This value is somewhat lower than reported by the Particle Data Group \[4\]. The reason is the conspicuous signal in $\pi \pi$ but absence of any corresponding peak in $K^+ K^-$.
Next we consider the peak in $\pi ^+\pi ^-$ at 1775 MeV/c$^2$ in Fig. 4(c). It fits well with $J^P = 0^+$ with $M = 1790 ^{+40}_{-30}$ MeV/c$^2$, $\Gamma = 270 ^{+60}_{-30}$ MeV/c$^2$. The fitted mass is in reasonable accord with the $f_0(1770)$ reported in Crystal Barrel data on $\bar pp \to (\eta \eta )\pi ^0$ [@crystal]: $M = 1770 \pm
12$ MeV/c$^2$, $\Gamma = 220 \pm 40$ MeV/c$^2$. Allowing for the number of fitted parameters, $f_0(1790)$ is more than a $15\sigma$ signal. It cannot arise from $f_0(1710)$, since the magnitude of $f_0(1710) \to K^+K^-$ is small (see Table 1), and it is known that the branching fraction ratio of $f_0(1710)$ between $\pi \pi$ and $K\bar K$ is $<0.11$ at the $95\%$ confidence level [@okk]; accordingly, the $f_0(1710) \to \pi \pi$ signal in present data should be negligibly small.
We now consider possible fits with an $f_2$ instead. The decay angular distribution in this mass range is consistent with isotropy. So there is no positive evidence for spin 2. However, four spin 2 amplitudes are capable of simulating a flat angular distribution. In consequence, spin 2 gives a log likelihood which is worse than spin 0 by only 4.5 after re-optimising its mass and width. If $f_0(1710)$ is then added with PDG mass and width, it improves log likelihood by a further 2.0; this confirms the result from $\omega K^+K^-$ data that $f_0(1710)$ has a negligible decay to $\pi \pi$. Our experience elsewhere is that using four helicity amplitudes instead of 2 adds considerable flexibility to the fit. The spin 2 amplitude with $\ell = 0$ has a distinctive term $3\cos ^2 \alpha _\pi - 1$, where $\alpha _\pi$ is the decay angle of the $\pi^+$ in the resonance rest frame, with respect to the direction of the recoil $\phi$. Simulation of spin 0 requires large $J = 2$, $\ell = 2$ and 4 amplitudes to produce compensating terms in $\sin ^2\alpha_{\pi}$ and hence a flat angular distribution. Large contributions from $\ell = 2$ are unlikely in view of the low momentum available to the resonance and the consequent $\ell = 2$ centrifugal barrier. If the $J = 2$ hypothesis is fitted only with $\ell = 0$, log likelihood is worse by 95 than for spin 0. We conclude that the state is most likely spin zero.
It is not possible to fit the shoulder in $\phi K^+K^-$ at 1650 MeV/c$^2$ accurately by interference between $f_0(1500)$ and $f_0(1790)$, using the $f_0(1790)$ mass and width found in $\phi \pi \pi$ data. Even if one accepts the poor fit this gives, the branching fraction ratio $K\bar K$/$\pi \pi$ assuming only one $f_0$ resonance here is $0.55 \pm 0.10$. This is a factor 14 lower than that reported in Ref. [@okk] for $f_0(1710)$. For a resonance, branching fractions must be independent of production mechanism. The large discrepancy in branching fractions implies the existence of two distinct states at 1710 and 1790 MeV/c$^2$, the former decaying dominantly to $K\bar K$ and the latter dominantly to $\pi
\pi$. The $f_0(1790)$ is a natural candidate for the radial excitation of $f_0(1370)$. There is earlier evidence for it decaying to $4\pi$ in $J/\psi \to \gamma (4\pi)$ data \[13,14\], with mass and width close to those observed here. There, spin 0 was preferred strongly over spin 2.
The shoulder in $\phi K^+K^-$ at 1650 MeV/c$^2$ is fitted with interference between $f_0(1500)$ and $f_0(1710)$, which is known to decay strongly to $K\bar K$. If both $f_0(1710)$ and $f_0(1790)$ are included in the fit, there is only a small improvement from $f_0(1790)$.
Masses, widths and branching fractions are given in Table I. The errors arise mainly from (i) varying the choice of side bins and the magnitude of the background under the $\phi$ peak, (ii) adding or removing small components such as $f_0(1500)$, $f_2(1270) \to K^+K^-$, and $\sigma\to K^+K^-$ and (iii) varying the mass and width of every component within errors and using different $\sigma$ parameterizations reported in Ref. [@opipi]. It also includes the uncertainty in the number of $J/\psi$ events and the difference between two alternative choices of MDC wire resolution simulation.
-------------- -------------------- ------------------- ---------------------- ---------------------- ------------
Channel Mass Width $B(J/\psi\to\phi X,$ $B(J/\psi\to\phi X,$ $\Delta S$
(MeV/c$^2$) (MeV/c$^2$) $X\to\pi\pi)$ $X\to K \bar{K})$
$(\times 10^{-4})$ $(\times 10^{-4})$
$f_0(980)$ $965 \pm 10 $ see text $5.4\pm0.9$ $4.5\pm0.8$ 1181
$f_0(1370)$ $1350\pm 50$ $265\pm40$ $4.3\pm1.1$ $0.3\pm0.3$ 83
$f_0(1500)$ PDG PDG $1.7\pm0.8$ $0.8\pm0.5$ 51
$f_0(1790)$ $1790^{+40}_{-30}$ $270^{+60}_{-30}$ $6.2\pm1.4$ $1.6\pm 0.8$ 488
$f_2(1270)$ $1275\pm15$ $190\pm20$ $2.3\pm0.5$ $0.1\pm0.1$ 241
$\sigma$ $1.6\pm0.6$ $0.2\pm0.1$ 120
$f_2'(1525)$ $1521 \pm 5$ $77\pm15$ - $7.3\pm1.1$ 440
$f_0(1710)$ PDG PDG - $2.0\pm0.7$ 64
-------------- -------------------- ------------------- ---------------------- ---------------------- ------------
: Parameters of fitted resonances and branching fractions for each channel; improvements in $S=$ log likelihood when the channel is added. PDG means that the mass and width are fixed to the PDG value. For the $f_0(980)$, see the parameterization in the text. The errors are the statistical and systematic errors added in quadrature.
Finally, angular distributions for both production and decay have been examined for each resonance peak. There are no significant discrepancies between data and fit. A fit is shown for the $f_0(1790)$ peak in Fig. 6 to the decay angle $\alpha _\pi$ of the $\pi\pi$ pair, with respect to the recoil $\phi$; the deep dip at $\cos \alpha _\pi = \pm 0.75$ is due to the $K^*(890)$ cut. The remaining angular distribution fits well to spin 0.
It is remarkable that $\phi \pi \pi$ data contain large signals due to several states which are predominantly non-strange: $f_2(1270)$, $f_0(1370)$, $f_0(1500)$ and $f_0(1790)$; direct production with the $\phi$ should favour $s\bar s$ states. There is no agreed explanation. In summary, the data reported here have three important features. Firstly, the parameters of $f_0(980)$ are all well determined. Secondly, there is the clearest signal to date of $f_0(1370) \to \pi
^+\pi ^-$; a resonant phase variation is required, from interference with $f_2(1270)$. Thirdly, there is a clear peak in $\pi
\pi$ at 1775 MeV/c$^2$, consistent with $f_0(1790)$; spin 2 is less likely than spin 0. If the $f_0(1790)$ resonance is used to fit the shoulder at 1650 MeV/c$^2$ in $\phi K^+K^-$, the branching fraction to pions divided by that to kaons is inconsistent with the upper limit for the ratio observed in Ref. [@okk] for $f_0(1710)$, this requires two distinct resonances $f_0(1790)$ and $f_0(1710)$.
The BES collaboration thanks the staff of BEPC for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 19991480, 10225524, 10225525, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, and the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP); by the National Natural Science Foundation of China under Contract No.10175060 (USTC), No.10225522 (Tsinghua University); and the Department of Energy under Contract No.DE-FG03-94ER40833 (U Hawaii). We wish to acknowledge financial support from the Royal Society for collaboration between the BES group and Queen Mary, London under contract Q771.
[99]{}
L. Kopke in : Proc. XXIIIrd Int. Conf. on High Energy Physics (Berkeley, 1986), ed. S. Loken (World Scientific, Singapore, 1987) and Santa Cruz preprint SCIPP 86/74. A. Falvard et al., Phys. Rev. D38 (1988) 2706. J.Z. Bai et al., (BES Collaboration), Nucl. Instr. Meth., A458 (2001) 627. S. Eidelman et al. (Particle Data Group), Phys. Lett. B 592 (2004) 1. S. Bitjokov et al., Yad. Fiz. 38 (1983) 1205 and Sov. J. Nucl. Phys. 38 (1983) 727. V.A. Polychronakos et al., Phys. Rev. D19 (1979) 1317; A.D. Martin and E.N. Ozmutlu, Nucl. Phys. B158 (1979) 520. D. Cohen et al., Phys. Rev. D22 (1980) 2595. A. Etkin et al., Phys. Rev. D25 (1982) 1786.
B.S. Zou and D.V. Bugg, Euro. Phys. J. A16 (2003) 537. M. Ablikim et al., (BES collaboration) , Phys. Letts. B598 (2004) 149. A. Anisovich et al., Phys. Lett. B449 (1999) 154. M. Ablikim et al., (BES collaboration), Phys. Lett. B603 (2004) 138 and hep-ex/0409007. D.V. Bugg et al., Phys. Lett. B353 (1995) 378. J.Z. Bai et al., Phys. Lett. 472 (2000) 207.
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---
bibliography:
- 'sample.bib'
title: ing in IJBC Style
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Usage
=====
\documentclass{ws-ijbc}
\begin{document}
Sample output using ws-ijbc bibliography style file ...
...text.\cite{best,pier,jame} ...
\bibliographystyle{ws-ijbc}
\bibliography{sample}
\end{document}
Database Entries
=================
[@ll@]{}\
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article & ... [@best03; @pier02; @jame02]\
proceedings & ... [@weis94]\
inproceedings & ... [@gupt97]\
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editor & ... [@benh93]\
series & ... [@bake72]\
tech report & ... [@hobb92; @bria84]\
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Some more options:
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---
author:
- Andreas Breslau
- Manuel Steinhausen
- Kirsten Vincke
- Susanne Pfalzner
date:
title: 'Sizes of protoplanetary discs after star-disc encounters'
---
Introduction {#sec:intro}
============
Stars form through the gravitational collapse of dense cores within molecular clouds. Due to the angular momentum conservation during the collapse, most young stars are initially surrounded by a disc consisting of gas and dust. These protoplanetary discs are usually treated as unperturbed by their surrounding. However, as most stars do not form in isolation but in star cluster environments ([[Lada]{} & [Lada]{}]{} 2003; [[Porras]{} [et al.]{}]{} 2003) discs are prone to be affected by external processes like photoevaporation from massive stars (e.g. [[Johnstone]{} [et al.]{}]{} 1998; [[Scally]{} & [Clarke]{}]{} 2001) or gravitational interactions with other cluster members (e.g. [[Moeckel]{} & [Bally]{}]{} 2006; [[Craig]{} & [Krumholz]{}]{} 2013). In this paper we only concentrate on the effect of the latter.
The consequences of gravitational interactions - or encounters - on disc parameters, like the disc’s mass, angular momentum and energy, have been intensively investigated analytically and numerically in the past (e.g. [[Clarke]{} & [Pringle]{}]{} 1993; [Ostriker]{} 1994; [Heller]{} 1995; [[Hall]{} [et al.]{}]{} 1996; [[Kobayashi]{} & [Ida]{}]{} 2001; [Pfalzner]{} 2003; [[Olczak]{} [et al.]{}]{} 2006; [[Pfalzner]{} & [Olczak]{}]{} 2007[a]{}; [[Lestrade]{} [et al.]{}]{} 2011; [[Steinhausen]{} [et al.]{}]{} 2012). By contrast, the encounter induced alteration of the size of a protoplanetary disc has so far been investigated in less detail. The reason is that only for the order of a few hundred objects, disc sizes have so far been determined (for an overview see e.g. [[Williams]{} & [Cieza]{}]{} 2011 and references therein). Deriving correlations between the disc size and the stellar environment seemed so far out of reach. However, with the advent of ALMA in the near future the sample size is likely to increase considerably and the influence of the environment on the disc size becomes testable.
In the past, theoretical investigations have shown that encounters mainly affect the outer disc material. [[Clarke]{} & [Pringle]{}]{} (1993) investigated the effect of parabolic encounters of equal-mass stars with a periastron distance of $1.25$ times the initial radius of the disc, $r_{\mathrm{init}} $, and varying inclinations. They found that in the most destructive encounter (prograde, coplanar) disc material down to $\approx 0.5$ times the periastron distance, $r_{\mathrm{peri}} $, becomes unbound. By varying the disc-size to periastron-distance ratio, [[Hall]{} [et al.]{}]{} (1996) found, that in parabolic, prograde, coplanar encounters of equal mass stars the disc inside of $0.3\,r_{\mathrm{peri}} $ is almost unperturbed. [[Kobayashi]{} & [Ida]{}]{} (2001) focused on the encounter induced increase of eccentricity and inclination of the disc material and the size of the largely unperturbed inner part of the disc. They found that many particles become unbound outside of $\approx 1/3$ of periastron distance after a prograde encounter of equal-mass stars on a parabolic orbit.
Almost all previous investigations focused on a small parameter space, which was mostly restricted to encounters between equal-mass stars. Nevertheless, the above results are often generalised as encounters truncating discs to $1/2\text{--}1/3$ of the periastron distance (e.g. [[Brasser]{} [et al.]{}]{} 2006; [[Adams]{} [et al.]{}]{} 2006; [Adams]{} 2010; [[Jim[é]{}nez-Torres]{} [et al.]{}]{} 2011; [[Malmberg]{} [et al.]{}]{} 2011; [Pfalzner]{} 2013). There the dependence of the disc truncation on the mass of the perturbing star and other geometrical properties of the perturber orbit are disregarded.
One typical context where the disc size after an encounter plays a key role is the formation of the solar system. The size of the solar system, with a density drop at $\approx 30$AU, is attributed to an encounter of the young solar system with another star at a periastron distance of $\approx 100$AU. This distance is used to determine the typical cluster environment in which the solar system might have developed.
In addition, the disc size is a crucial parameter for investigations of the typical types of planetary systems in a given cluster environment. Therefore, [[de Juan Ovelar]{} [et al.]{}]{} (2012) tried to estimate the sizes of discs after encounters considering both, the periastron distance and the mass ratio of the stars. They suggested two different approaches. The first one is based on the assumption that disc material is removed in an encounter at least to the point of gravitational force equilibrium between the stars at the time of periastron passage. In the other approach they suggested a transformation of the results for the disc-mass loss in encounters by [[Olczak]{} [et al.]{}]{} (2006) into a reduction of the disc size. Both approaches assume that disc size change and the removal of disc material are strongly correlated.
However, already [Hall]{} (1997) showed that during an encounter disc material can be moved inward due to loss of angular momentum. That way, the disc size can be reduced even when no mass is lost. More recently, it was demonstrated that a $3\text{--}5\,\%$ loss of disc angular momentum is common in ONC-like star clusters and this holds not only for the central high-density regions but even at the outskirts of clusters ([[Pfalzner]{} & [Olczak]{}]{} 2007[b]{}). Therefore it is very likely, that changes of the disc size are a more common effect in clusters than the removal of disc material.
Using an extended data base of star-disc encounters we show here that mass loss based approaches are not really suitable to determine the resulting disc size. After a short description of our numerical method in Sect. \[sec:method\], a disc-size definition is given, which is comparable to observational size determinations. In Sect. \[sec:results\] we present the numerical results as well as a simple fit formula. We compare our results to previous work in Sect. \[sec:comparison\] and give a brief summary and conclusion in Sect. \[sec:summary\].
Method {#sec:method}
======
Numerical method
----------------
We performed numerical simulations to investigate the effect of stellar encounters on the size of protoplanetary discs. The mass of a protoplanetary disc is usually much lower than the mass of its host star (on average , e.g. [[Andrews]{} [et al.]{}]{} 2013). Because of the resulting low density, especially in the here relevant outer parts of the disc, and the also relatively low temperature, the viscous timescale is long enough that the discs survive at least some million years (e.g. [[Haisch]{} [et al.]{}]{} 2001). Since this is typically much longer than the timescale of encounters, we can mostly neglect hydrodynamical effects as well as self-gravity within the disc. The exception are very close penetrating encounters, which we exclude anyway (see below). We also disregard photoevaporation from the stars as well as radiation transport within the disc. Thus, we only consider gravitational forces from and onto the stars.
For pure gravitational numerical investigations, particle based simulations are most suitable. Since we investigate low-mass discs and neglect the self-gravity, we can even go one step further and use mass-less tracer particles. This has the advantage that the particle distribution can be chosen independently from the mass distribution within the disc. To obtain a higher spatial resolution at the outskirt of the disc, we use a constant particle surface density ($\Sigma(r) = \mathrm{const.}$). The particle masses are associated to the tracer particles in the diagnostic step, making it also possible to investigate the effect of different initial mass distributions with one single simulation ([[Steinhausen]{} [et al.]{}]{} 2012).
In dimensionless units, our discs have an initial outer radius of $r_{\mathrm{init}} = 1$ around a star of mass $M_{\mathrm{1}} = 1$. Because the test particles are only affected by gravitational forces, they initially follow keplerian orbits around their host star. We assume that all particles are initially on circular orbits (eccentricity $e = 0$).
Even though hydrodynamical forces are neglected, we set up our discs with a certain thickness for consistency with previous work (e.g. [[Pfalzner]{} [et al.]{}]{} 2005[b]{}; [[Olczak]{} [et al.]{}]{} 2006; [[Pfalzner]{} & [Olczak]{}]{} 2007[a]{}; [[Steinhausen]{} [et al.]{}]{} 2012). Thus the initial particle distribution in the disc is given by $$\begin{aligned}
\rho(r,z) \propto \Sigma(r) \exp{\left( - \frac{z^2}{2 H^2(r)} \right)},\end{aligned}$$ where $H(r) = 0.05\,r$ the vertical half-thickness of the disc (see also [Pringle]{} 1981). Since this results in inclinations $i \lesssim 5^{\circ}$ for nearly all particles the disc can still be regarded as thin.
A standard method when investigating star-disc encounters is to exclude an inner disc region to avoid unnecessary small time steps. Here, this region extends to $r_{\mathrm{hole}} = 0.1\,r_{\mathrm{init}} $. Therefore, encounters resulting in very small final discs with $r_{\mathrm{final}} \lesssim 0.2\,r_{\mathrm{init}} $ are excluded from further diagnostics, as they might be influenced by the missing disc mass within the hole. Particles which approach one of the stars closer than $0.1\,r_{\mathrm{hole}}$ are treated as accreted. As a compromise between spatial resolution and computation time needed for the parameter study, our discs are modelled with $10\,000$ particles (see also [[Kobayashi]{} & [Ida]{}]{} 2001; [Pfalzner]{} 2003; [[Steinhausen]{} [et al.]{}]{} 2012).
Simulations were performed for varying perturber-mass to host-mass ratios ${m_{\mathrm{12}}} = M_{\mathrm{2}}/M_{\mathrm{1}}$ and periastron-distances $r_{\mathrm{peri}} $. The trajectories of the particles were integrated by using a Runge-Kutta Cash-Karp scheme with adaptive time step size.
As parameter space for our study, we have chosen the mass-ratio and periastron-distance range typical for young dense clusters in the solar neighbourhood. A prime example is the Orion Nebula Cluster (ONC), for which the effect of encounters on the protoplanetary discs was already investigated by [[Pfalzner]{} & [Olczak]{}]{} (2007[b]{}). The mass range in the ONC is given by the mass ratio of the stars at helium burning limit, $M \approx 0.08\,\mbox{M$_{\odot}$} $, and the most massive system $\theta^1$ Ori C with $M \approx 40\,\mbox{M$_{\odot}$} $. Previous studies (e.g. [[Pfalzner]{} [et al.]{}]{} 2005[b]{}; [[Steinhausen]{} [et al.]{}]{} 2012) have shown that encounters with very low mass ratios (${m_{\mathrm{12}}} \lesssim 0.1$) only have an effect on the discs for very small periastron distance ($r_{\mathrm{peri}} \lesssim r_{\mathrm{init}} $). Additionally, hydrodynamical effects may play a role in these encounters, and the mass of the disc is not much smaller than the mass of the perturber, which would make the application of the here used method questionable. Therefore, we treat only the range ${m_{\mathrm{12}}} = 0.3\text{--}500$.
We restrict the periastron distances to the range where the discs are perturbed significantly but not fully destroyed. Cases where the resulting disc sizes are $\lesssim 0.2\,r_{\mathrm{init}} $ are excluded from further diagnostics, as they might be influenced by the inner cut-off of the initial discs. For an equal mass perturber the distances range, for example, from $0.7\,r_{\mathrm{init}} $ to $7\,r_{\mathrm{init}} $. The actual periastron distances depend on the mass ratio and can be found in Table \[tab:sizes\] in the Online Material.
In this work, we only consider prograde, coplanar and parabolic ($e = 1$) encounters as they are the most destructive ones. Therefore our resulting disc sizes represent the lower limits for the disc sizes after encounters on inclined and/or hyperbolic orbits (e.g. [[Clarke]{} & [Pringle]{}]{} 1993; [[Pfalzner]{} [et al.]{}]{} 2005[b]{}). For this parameter space several hundreds of simulations were performed.
At the onset of our simulations the perturber is placed at a distance $d_{\mathrm{init}}$, where its gravitational force onto the closest disc particle is only $1\%$ relative to the force from the disc-hosting star. This condition is satisfied, when $$\begin{aligned}
d_{\mathrm{init}} \gtrsim 10\,r_{\mathrm{init}} \sqrt{m_{12}} +r_{\mathrm{init}} .\end{aligned}$$ We end our simulations, when the perturber has again at least a distance of $d_{\mathrm{init}}$ to the disc-hosting star or a particle at $r_{\mathrm{init}} $ fulfilled at least $1.5$ orbits around the host star after periastron passage.
We investigate here the simplified case of only one star being surrounded by a disc to limit computational costs. This approach is valid, as long as only small fractions of the disc material are captured by the perturbing star ([[Pfalzner]{} [et al.]{}]{} 2005[a]{}). For a realistic modelling of close disc-disc encounters hydrodynamical effects would also have to be considered.
Size determination {#sec:method_size}
------------------
During the diagnostic step masses are attributed to the tracer particles so that the initial disc matches $$\begin{aligned}
\rho(r) = \rho_0 r^{-p}\label{eq:densitydistribution},\end{aligned}$$ where $\rho_0$ is the mass density in the equatorial plane at the inner rim ($r = 0.1\,r_{\mathrm{disc}} $) and $p$ the parameter for the slope of the mass density distribution (see also [[Steinhausen]{} [et al.]{}]{} 2012).
A problem in the determination of the size of a protoplanetary disc after an encounter is, that there is no general definition of a disc size at hand. Taking observations as a guide is only of limited help, as even in the unperturbed case there are several methods used for disc size determination. One common method is to fit the observed SED to disc models with radial density and temperature profiles which follow mostly a truncated power law (e.g. [[Andrews]{} & [Williams]{}]{} 2007). The radius of truncation of the density is then taken as size. In case of resolved images, the size corresponds often to the radius where the luminosity falls below a certain limit (e.g. [[Vicente]{} & [Alves]{}]{} 2005; [O’dell]{} 1998). This drop in luminosity is equivalent to a drop in the disc’s surface density. Since both methods rely somehow on a drop in the disc’s volume or surface density we decided to mimic this methods by defining our disc size as the point of strongest contrast in the surface density.
In the determination of the disc’s surface density after an encounter from the simulations, a problem arises from the particles on highly eccentric orbits. While in the viscosity-free case the semi-major axis $a$ and eccentricity $e$ of the particles do not change any more after the end of the encounter, their radial distances to the disc-hosting star change with time. Therefore, global quantities determined from a snapshot of the particle distribution at a certain time would not necessarily be representative.
For better comparison of our disc size values with observational data, we use therefore a temporally averaged surface density distribution for the disc size determination. This can be obtained directly from the data of the last time step by applying the radial probability functions of the orbits defined by $a$ and $e$. The sum of the radial probability functions of all particles yields the time-averaged [*particle*]{} surface density distribution. When weighting the radial probability functions in this sum with the previously associated particle masses according to Eq. one obtains the [*mass*]{} surface density distribution. For our diagnostics we used particle masses according to an initially $\propto r^{-1}$ ($p=1$) mass surface density distribution.
The disc size is then determined as the point of strongest mass surface density gradient. Here, the steep inner part of the initial $r^{-1}$ mass distribution is excluded. Due to the statistical nature of our data, the surface density distributions have to be smoothed before the size determination algorithm is applied.
![[**a)**]{} Face on view of a disc after an encounter with ${m_{\mathrm{12}}} = 1$ and $r_{\mathrm{peri}} = 2\,r_{\mathrm{init}} $. For clarity, the particle angles have been decorrelated to destroy the spiral arms. The vertical lines mark the disc diameter ($= 2 r_{\mathrm{disc}} $) as obtained with our disc size definition. Corresponding initial (thin) and final (thick) particle (solid) and mass (dashed) surface density distribution (all lines are smoothed). The vertical line shows the final disc size as obtained with our disc size definition.[]{data-label="fig:size"}](decorrelated_disc){width="\textwidth"}
![[**a)**]{} Face on view of a disc after an encounter with ${m_{\mathrm{12}}} = 1$ and $r_{\mathrm{peri}} = 2\,r_{\mathrm{init}} $. For clarity, the particle angles have been decorrelated to destroy the spiral arms. The vertical lines mark the disc diameter ($= 2 r_{\mathrm{disc}} $) as obtained with our disc size definition. Corresponding initial (thin) and final (thick) particle (solid) and mass (dashed) surface density distribution (all lines are smoothed). The vertical line shows the final disc size as obtained with our disc size definition.[]{data-label="fig:size"}](discsize){width="\textwidth"}
Figure \[fig:decorrelated\] shows a disc perturbed by a ${m_{\mathrm{12}}} = 1$ perturber on a parabolic orbit with $r_{\mathrm{peri}} = 2\,r_{\mathrm{init}} $. Figure \[fig:disc\_size\] shows the corresponding initial (thin) and final (thick) time averaged surface density distributions for in initially constant particle (solid) and $\propto r^{-1}$ mass (dashed) distribution. The vertical line shows the final disc size as obtained with our disc size definition. It can be seen, that the vertical line matches the points of steepest gradient in both final density distributions.
To determine the statistical deviations of our disc sizes we performed $\approx 100$ simulations of one encounter with different random seeds for the initial particle distribution. Because smaller radial bins for the surface density computation require a longer smoothing length (as mentioned above) and vice versa, we found a bin width of $0.01\,r_{\mathrm{init}} $ as a good compromise between resolution and smoothing. The statistical deviations are then also on the order of $0.01\,r_{\mathrm{init}} $.
Results {#sec:results}
=======
![Final disc sizes versus perturber periastron for some perturber mass ratios (values shown by the numbers at the lines) from our simulations (black squares for initially $100$AU discs and diamonds for initially $200$AU discs) compared to our fit formula (Eq. \[eq:fitformula\], grey, dotted lines).[]{data-label="fig:disc_sizes"}](disc_sizes_vs_periastron_absolute){width="\hsize"}
Figure \[fig:disc\_sizes\] gives an overview of the results of our parameter study. (The actual values can also be found in Table \[tab:sizes\] in the Online Material.) For clarity and better comparability to observations we present the numerical results in absolute values, assuming an initial disc size of $100$AU (black squares). For comparison, the diamonds represent some exemplary results for discs with an initial size of $200$AU. As long as an encounter changes the sizes of the discs with both initial sizes significantly the resulting sizes are approximately the same. Since the discs do not become bigger[^1] in our simulations, the final sizes are limited by the initial disc sizes. Therefore, the final disc sizes of initially $100$AU and $200$AU discs have to deviate after encounters, where the final size of the $200$AU disc is $\gtrsim 100$AU. For example an encounter with a mass ratio of ${m_{\mathrm{12}}} = 0.3$ and a periastron distance of $500$AU reduces the size of an initially $200$AU disc to $\approx 150$AU while the initially $100$AU disc is unchanged.
There exists a relatively simple dependence of the disc size on the mass ratio between the encountering and the disc-bearing star (${m_{\mathrm{12}}} $) and the periastron distance ($r_{\mathrm{peri}} $) which can be described by a fit formula of the form $$\begin{aligned}
r_{\mathrm{final}} = 0.28 \cdot {m_{\mathrm{12}}} ^{-0.32} \cdot r_{\mathrm{peri}} \label{eq:fitformula}.\end{aligned}$$ Again, the final disc sizes are limited by the initial disc sizes, as the discs do not grow. In the plot the fit function is limited by the initial size of the initially $200$AU discs. Close to the sharp upper cut off of the fit function, the numerical values deviate slightly from the fit since the curves bend smoothly.
The deviations of the final disc sizes obtained with Eq. from the results of the simulations are $\lesssim 5\,\%$ of the initial disc size (i.e. $5$AU for an initially $100$AU disc) for mass ratios ${m_{\mathrm{12}}} = 5\text{--}500$ and $\lesssim 10\,\%$ for mass ratios ${m_{\mathrm{12}}} = 0.3\text{--}5$. These deviations are still on the order of the uncertainty of the size definition. The validity of our fit formula is also restricted to periastra where $r_{\mathrm{final}} > 0.2\,r_{\mathrm{init}} $.
Discussion {#sec:discussion}
==========
Some approximations have been made in the model described above. First, we treated the discs by pure N-body methods while neglecting viscous forces. Viscous forces only play a role in the central areas of the discs. For typical viscosity values the effected area is the region within $ \lesssim 0.2\,r_{\mathrm{init}} $. Only in the most violent interactions the disc size is reduced to such low values. For these cases viscous forces would have to be in principle included. As can be seen in Table \[tab:sizes\], this is, for example, the case for equal-mass stars in penetrating encounters closer than $ 0.7\,r_{\mathrm{init}} $. For typical disc sizes on the order of some $100$AU (e.g. [[Bally]{} [et al.]{}]{} 2000; [[Andrews]{} [et al.]{}]{} 2009), such encounters are relatively rare in ONC-Type clusters ([[Olczak]{} [et al.]{}]{} 2006). In addition, for such destructive encounters the material that remains bound is usually $<20\,\%$ of its initial mass and its structure does not resemble a disc as such. In these cases the definition of a disc size is anyway highly questionable.
An additional approximation is that the case where only one of the stars was surrounded by a disc was considered. This has been done for simplicity of description. In principle, for disc-disc encounters the sizes could be larger because disc material could be transferred from the perturbing star to the disc hosting star and replenish the disc. [[Pfalzner]{} [et al.]{}]{} (2005[a]{}) showed that in case both stars are surrounded by discs, mostly an additive approach can be used as usually captured material is deposited close to the star. In addition, the amount of captured mass is very small compared to the disc mass as such. Therefore in most cases the final disc size is little influenced by captured material. The exception is again the case where the encounter is very violent and the remnant disc mass very low.
Furthermore, the outcome of a disc-disc encounter could be influenced by the dependence of the disc mass and size on the stellar mass. Observations give no conclusive answer to this question. If the disc size scales in some way with the stellar mass, the situation can occur that the perturber has a much higher mass, and therefore has a bigger and possibly more massive disc (see e.g. [[Andrews]{} [et al.]{}]{} 2013). In this case the amount of material captured by the primary could be comparable to or even higher than that of its remaining disc.
We have chosen the point of highest surface density gradient as definition of the disc size (see Sect. \[sec:method\_size\]). However, after an encounter the region where the disc’s surface density drops significantly spans some range (see Fig. \[fig:disc\_size\]). The inner boundary of this region would correspond better to the disc sizes obtained by SED fits, where the size is usually defined as the point, where the surface density transits from a power-law to something steeper. If we had defined our disc sizes as the inner boundary of the density-drop region, the results would be smaller than with our actual definition. Conversely, if defining the size as the point, where the surface density falls below a certain threshold (compare Sect. \[sec:method\_size\]), the sizes would be somewhat bigger than our results. However, results obtained with each of these definitions would usually not differ from our results by more than $\approx 0.1\,r_{\mathrm{init}} $. Our definition is therefore a robust mean value between several possible disc size definitions for perturbed discs.
Since the problem of star disc encounters scales with the periastron ratio, one would expect that the results can be normalised to this ratio $r_{\mathrm{peri}} /r_{\mathrm{init}} $. This is indeed the case, but for more intuitive comparison with observations we have chosen the absolute presentation of the results. The actual values in Table \[tab:sizes\] in the Online Material are normalised to the periastron ratio.
The differences between the simulation results and the fit formula increase for decreasing mass ratios, becoming significant ($\gtrsim 5\%$) for ${m_{\mathrm{12}}} \lesssim 5$. This is because in the close encounters needed to change the disc size for low mass ratios the redistribution of the disc material is non-linear. Therefore the fit formula with linear dependence on the periastron distance can describe this effect only approximately.
It is also important to note, that in contrast to close, penetrating encounters, the disc size change in distant encounters is not mainly caused by a real truncation of the disc. Mostly, the size change is dominated by the redistribution of disc material towards the host star. This redistribution results in smaller disc sizes even when no material is lost at all.
![The parameter space of encounters occurring in a star cluster like the ONC (full height and open to the right) compared to the parameter space of our simulations (grey area) and the validity area of our fit formula (grey, not ruled area). For further details see text.[]{data-label="parameter_space"}](parameter_space_overview){width="\hsize"}
Figure \[parameter\_space\] shows where our fit formula can be applied. The axes are chosen to cover the encounter parameter space typical for an ONC-like star cluster. The grey area shows the parameter space covered by this study. Right from the solid bent line, encounters have almost no effect on the mass, angular momentum or size of a disc. Right from the dashed bent line, it can be assumed that encounters have also a negligible effect. The horizontal, dashed line depicts the parameter space of [[Hall]{} [et al.]{}]{} (1996) while the black dot marks the simulations by [[Kobayashi]{} & [Ida]{}]{} (2001). In the black shaded area the remaining discs have sizes below $0.2\,r_{\mathrm{init}} $ or have less than $10\,\%$ of their initial particles. Since the simulations in this area may be influenced by low resolution or the hole in our discs, we excluded them. The remaining grey area depicts the parameter range for which our fit formula is valid.
[[Kobayashi]{} & [Ida]{}]{} (2001) developed an analytical estimate for the size of the inner disc region, where the velocity dispersion of the disc material after an encounter is still low enough to form planets (see their Eq. (31)). They obtained $$\begin{aligned}
r_{\mathrm{planet}} \propto \left( \frac{{m_{\mathrm{12}}} +1}{{m_{\mathrm{12}}} ^2} \right)^{1/4} \left(\frac{r_{\mathrm{peri}} }{r_{\mathrm{init}} } \right)^{5/4}.\end{aligned}$$ Even though the size of this planet forming region is defined differently than our disc size, the mass dependence is similar to the ${m_{\mathrm{12}}} ^{-0.32}$ dependence of Eq. . However, the periastron dependence of their analytical approximation is stronger than that of our numerical results.
Comparison with previous approximations {#sec:comparison}
=======================================
![Disc sizes from our simulations (black lines) versus perturber periastron for [**a)**]{} an equal mass perturber compared to $1/2$ (dotted grey line) and $1/3$ (short dashed grey line) of the periastron distance as well as our new fit formula (long dashed grey line). [**b)**]{} perturbers with $0.3$ (solid), $9.0$ (dashed) and $90.0$ (dotted) solar masses compared to the disc sizes obtained with Eq. (grey lines). [**c)**]{} perturbers with $1.0$ (solid) and $90.0$ (dotted) solar masses compared to the disc sizes obtained with Eq. (grey lines).[]{data-label="fig:cmp_previous_work"}](cmp_one_third){width="\textwidth"}
![Disc sizes from our simulations (black lines) versus perturber periastron for [**a)**]{} an equal mass perturber compared to $1/2$ (dotted grey line) and $1/3$ (short dashed grey line) of the periastron distance as well as our new fit formula (long dashed grey line). [**b)**]{} perturbers with $0.3$ (solid), $9.0$ (dashed) and $90.0$ (dotted) solar masses compared to the disc sizes obtained with Eq. (grey lines). [**c)**]{} perturbers with $1.0$ (solid) and $90.0$ (dotted) solar masses compared to the disc sizes obtained with Eq. (grey lines).[]{data-label="fig:cmp_previous_work"}](cmp_equal_force){width="\textwidth"}
![Disc sizes from our simulations (black lines) versus perturber periastron for [**a)**]{} an equal mass perturber compared to $1/2$ (dotted grey line) and $1/3$ (short dashed grey line) of the periastron distance as well as our new fit formula (long dashed grey line). [**b)**]{} perturbers with $0.3$ (solid), $9.0$ (dashed) and $90.0$ (dotted) solar masses compared to the disc sizes obtained with Eq. (grey lines). [**c)**]{} perturbers with $1.0$ (solid) and $90.0$ (dotted) solar masses compared to the disc sizes obtained with Eq. (grey lines).[]{data-label="fig:cmp_previous_work"}](cmp_mass_loss){width="\textwidth"}
Figure \[fig:cmp\_one\_third\] shows a comparison of the numerical results (solid line) to the often used approximations (see Sect. \[sec:intro\]) $$r_{\mathrm{final}} = \frac{1}{3} r_{\mathrm{peri}} \mathrm{\hspace{1em} and \hspace{1em}}r_{\mathrm{final}} = \frac{1}{2} r_{\mathrm{peri}} \label{eq:approximations}$$ ($1/3\,r_{\mathrm{peri}} $ - short dashed line, $1/2\, r_{\mathrm{peri}} $ - dotted line). These approximations obviously can only be applied to encounters with $r_{\mathrm{peri}} /r_{\mathrm{final}} < 3$ and $r_{\mathrm{peri}} /r_{\mathrm{final}} < 2$, respectively. Therefore in Fig. \[fig:cmp\_one\_third\] the disc sizes for wider perturber periastra are truncated at $r_{\mathrm{final}} /r_{\mathrm{disc}} =1$.
The $1/2\,r_{\mathrm{peri}} $-approach is always much bigger than the numerical results and never a good approximation of the disc size after an encounter. The approach of using a third of the periastron distance works when the size after the encounter lies in the range of $0.2\text{--}0.4\,r_{\mathrm{init}} $, equivalent to penetrating or grazing encounters ($r_{\mathrm{peri}} /r_{\mathrm{init}} = 0.6\text{--}1.2$). This means, that we confirm the result of [[Kobayashi]{} & [Ida]{}]{} (2001) for solar system forming encounters with solar-type stars. However, a generalisation to more distant encounters common in cluster environments yields much too big disc sizes. The difference between the simulation results and the $1/3$ approximation is up to $\gtrsim 20\,\%$, the difference to the $1/2$ approximation up to $\approx 80\,\%$. For equal mass stars our fit formula Eq. \[eq:fitformula\] simplifies to $$\begin{aligned}
r_{\mathrm{final}} = 0.28 \cdot r_{\mathrm{peri}} .\end{aligned}$$ This formula can reproduce the numerical results slightly better with deviations $\lesssim 10\%$ (see also the long dashed grey line in Fig. \[fig:cmp\_one\_third\]). Nevertheless, a generalisation of any of these mass-ratio independent approximations to other than equal-mass encounters can obviously lead to tremendous errors.
Recently, [[de Juan Ovelar]{} [et al.]{}]{} (2012) derived an upper limit of the disc size after an encounter as the distance $d_{eq}$ at which the gravitational force of the perturbing star dominates at the time of pericentre passage. This is given by $$\begin{aligned}
d_{eq} = \frac{r_{\mathrm{peri}} }{1+{m_{\mathrm{12}}} ^{1/2}}. \label{eq:equal_force}\end{aligned}$$ Figure \[fig:cmp\_equal\_force\] shows the comparison of the sizes approximated by Eq. (thin, grey lines) with the simulation results (thick, black lines). For clarity we show only the curves for three mass ratios ($0.3$ (solid), $9.0$ (dashed) and $90.0$ (dotted)). Evidently, at least for prograde, coplanar encounters, this analytical approximation can be only regarded as upper limit, because the actual disc sizes are typically between a factor of one and two smaller than this estimate.
In addition, [[de Juan Ovelar]{} [et al.]{}]{} (2012) suggested to use the mass loss in an encounter to estimate the resulting disc size. Using the mass-loss expression by [[Olczak]{} [et al.]{}]{} (2006) and assuming that the encounter strips the outer disc layers, they argue that $\Delta M/M$ should be equal to $\Delta r/r$ for a mass surface density within the disc $\propto r^{-1}$. With Eq. (4) from [[Olczak]{} [et al.]{}]{} (2006) one obtains for the disc size after an encounter:
$$\begin{aligned}
\frac{r_{\mathrm{final}} }{r_{\mathrm{init}} } = &\left(1 -\left( \frac{M_2}{M_2 + 0.5 M_1} \right)^{1.2} \ln{\left[ 2.8 \left( {r_{\mathrm{p}}} \right)^{0.1} \right]} \right. \notag\\
&\left.\times \exp \left\{ -\sqrt{\frac{M_1}{M_2 + 0.5 M_1}} \left[ \left( {r_{\mathrm{p}}} \right)^{3/2} - 0.5 \right] \right\}\right).\label{eq:mass_loss}\end{aligned}$$
Figure \[fig:cmp\_mass\_loss\] shows that Eq. is also only an upper limit for the disc sizes after encounters. On the one hand, this can be explained by the fact, that in distant encounters, where no mass is removed from the disc, the disc can shrink by angular momentum removal. On the other hand, our size definition via the point of steepest surface density makes it possible, that a significant amount of bound mass is located in the low density regions outside of our defined disc size. However, this low density is not likely to be detected by observations.
Additionally, both approaches of [[de Juan Ovelar]{} [et al.]{}]{} (2012) suffer from the fact, that the disc size change is dominated by the redistribution of disc material (see also Sec. \[sec:discussion\]) and not by the truncation of the outer disc material as assumed.
In summary, previous approximations and fit formulae are largely unsuitable to describe the disc size after an encounter other than for a narrow range for equal-mass encounter partners.
Summary {#sec:summary}
=======
In the dense stellar environments of young clusters, tidal interactions can change the sizes of protoplanetary discs. Performing N-body simulations of such encounters, we confirm earlier results by [[Kobayashi]{} & [Ida]{}]{} (2001) that close ($\le 100$AU) encounters between equal-mass solar-type stars lead to the shrinking of initially $100$AU-sized discs to $30\text{--}50$AU. Naturally this represents only a special case, in clusters the wide spectrum of encounter partners and periastra has to be taken into account.
In this paper we investigated the disc-size change by encounters for the entire parameter space spanned by mass ratio and periastron distance typically covered in clusters. The central result of our extensive numerical parameter study is that the disc size after a prograde, parabolic encounter is a simple function of the periastron distance $r_{\mathrm{peri}} $ and the mass ratio ${m_{\mathrm{12}}} $ of the two stars of the form $$\begin{aligned}
r_{\mathrm{final}} = 0.28 \cdot r_{\mathrm{peri}} \cdot {m_{\mathrm{12}}} ^{-0.32}. \notag\end{aligned}$$
Prograde, parabolic encounters are the most destructive type of encounters. Inclined and/or hyperbolic encounters lead to less mass loss and therefore larger disc sizes. However, the parameter dependencies in these types of encounters span a wide parameter range. We will investigate this extended parameter range in a follow up study (in preparation).
The disc sizes after a star-disc encounter as obtained with the here presented disc-size definition would not necessarily correspond to the final size of a potentially developing planetary system. The reasons are that on long time scales (several Myr) viscosity leads to an increase in disc size. Simultaneously highly eccentric particles probably become recircularised through viscous processes as they pass the inner parts of the disc while being close to their periastron.
Previous work on disc sizes after encounters was often motivated by the search for the solar birth environment. Here the sudden density drop in the mass distribution at $30\text{--}50$AU is interpreted as the result of a close fly-by of another star during the formation phase of the solar system. Considering only encounters between equal mass stars an encounter distance of $100\text{--}150$AU was deduced. However, recent results show that encounters of the early solar system with less or more massive stars are at least just as likely as with another solar-mass star. Our results show now, that any parabolic, prograde encounter which fulfils the relation $$\begin{aligned}
0.28 \cdot r_{\mathrm{peri}} \cdot {m_{\mathrm{12}}} ^{-0.32} = 30\text{--}50~\mathrm{AU}\end{aligned}$$ can lead to a solar-system size disc.
So far the dependencies of protoplanetary disc sizes on parameters like stellar mass etc. are observationally not well constrained. With the advent of ALMA this will quickly change. Thus it will be also possible to determine whether dense stellar environments have a significant influence on the disc sizes and the forming planetary systems. The here derived dependencies will be a useful tool to determine the corresponding encounter events.
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Relative disc size changes
==========================
Here we present the full numerical results from our parameter study. In contrast to the presentation of the results in the main part of the paper, the values are normalised to the initial disc sizes to allow easier application to arbitrary initial disc sizes. ${r_{\mathrm{p}}}$ is the encounter periastron distance normalised to the initial disc size and ${m_{\mathrm{12}}} $ is the mass of the perturbing star normalised to the mass of the disc hosting star (${m_{\mathrm{12}}} = m_2/m_1$). The final disc size is obtained by multiplying the initial disc size with the respective value from the table. In the lower left part of the table, where no values are given, the disc size change is $\lesssim 1\%$, in the upper right part, no values are given, since they may not be reliable (see also discussion in main part of the paper).
${r_{\mathrm{p}}}$
-------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------- ------- ------- --------
0.30 0.50 1.00 1.50 2.00 3.00 4.00 9.00 20.00 50.00 90.00 500.00
0.4 0.21 - - - - - - - - - - -
0.5 0.23 - - - - - - - - - - -
0.6 0.27 0.20 - - - - - - - - - -
0.7 0.32 0.24 0.20 - - - - - - - - -
0.8 0.34 0.28 0.23 0.20 0.20 - - - - - - -
0.9 0.39 0.30 0.27 0.23 0.21 0.18 - - - - - -
1.0 0.42 0.34 0.28 0.26 0.23 0.23 0.18 0.17 - - - -
1.5 0.59 0.49 0.41 0.38 0.34 0.32 0.28 0.24 0.18 - - -
2.0 0.76 0.65 0.54 0.47 0.45 0.41 0.38 0.29 0.23 0.20 - -
2.5 0.86 0.79 0.66 0.60 0.56 0.49 0.46 0.38 0.28 0.21 - -
3.0 0.94 0.90 0.79 0.70 0.65 0.58 0.54 0.43 0.35 0.24 0.20 -
3.5 0.99 0.96 0.89 0.81 0.77 0.69 0.62 0.49 0.39 0.28 0.26 -
4.0 1.00 0.99 0.95 0.92 0.86 0.79 0.71 0.57 0.45 0.34 0.27 -
4.5 1.00 0.99 0.99 0.97 0.94 0.86 0.81 0.64 0.49 0.38 0.32 -
5.0 0.99 0.99 0.99 0.99 0.98 0.94 0.89 0.70 0.55 0.41 0.35 0.20
5.5 1.00 0.99 0.99 1.00 0.99 0.98 0.94 0.78 0.59 0.46 0.39 0.23
6.0 0.99 0.99 0.99 1.00 1.00 0.99 0.98 0.83 0.66 0.48 0.41 0.24
6.5 1.00 0.99 1.00 0.99 0.99 1.00 0.99 0.90 0.70 0.53 0.45 0.26
7.0 0.99 0.99 0.99 1.00 1.00 1.00 1.00 0.94 0.76 0.56 0.47 0.28
7.5 - - - - 0.99 1.00 0.99 0.97 0.81 0.60 0.51 0.30
8.0 - - - - - 1.00 1.00 0.99 0.85 0.65 0.54 0.30
8.5 - - - - - - 0.99 1.00 0.91 0.68 0.56 0.34
9.0 - - - - - - 1.00 1.00 0.94 0.72 0.59 0.35
9.5 - - - - - - - 1.00 0.97 0.77 0.62 0.38
10.0 - - - - - - - 1.00 0.98 0.80 0.67 0.40
10.5 - - - - - - - 1.00 0.99 0.83 0.69 0.41
11.0 - - - - - - - 1.00 1.00 0.88 0.72 0.42
11.5 - - - - - - - - 0.99 0.91 0.76 0.43
12.0 - - - - - - - - 1.00 0.93 0.79 0.46
12.5 - - - - - - - - - 0.95 0.82 0.47
13.0 - - - - - - - - - 0.97 0.85 0.49
13.5 - - - - - - - - - 0.98 0.88 0.52
14.0 - - - - - - - - - 0.99 0.91 0.54
14.5 - - - - - - - - - 0.99 0.94 0.56
15.0 - - - - - - - - - 0.99 0.94 0.56
15.5 - - - - - - - - - 0.99 0.96 0.58
16.0 - - - - - - - - - 0.99 0.98 0.59
16.5 - - - - - - - - - 0.99 0.99 0.61
17.0 - - - - - - - - - 0.99 0.99 0.64
17.5 - - - - - - - - - 0.99 0.99 0.66
18.0 - - - - - - - - - 0.99 0.99 0.67
18.5 - - - - - - - - - 0.99 0.99 0.69
19.0 - - - - - - - - - 1.00 0.99 0.70
19.5 - - - - - - - - - 1.00 0.99 0.72
20.0 - - - - - - - - - 1.00 0.99 0.74
22.0 - - - - - - - - - - 0.99 0.81
24.0 - - - - - - - - - - 0.99 0.88
26.0 - - - - - - - - - - 0.99 0.94
28.0 - - - - - - - - - - 0.99 0.98
30.0 - - - - - - - - - - 1.00 1.00
32.0 - - - - - - - - - - 0.99 1.00
34.0 - - - - - - - - - - - 1.00
[^1]: according to our size definition
|
---
abstract: 'In this work we present an analysis of production and signature of neutral Higgs bosons $H_2^0$ in the version of the 3-3-1 model containing heavy leptons at the ILC (International Linear Collider) and CLIC (Cern Linear Collider). The production rate is found to be significant for the direct production of $e^{-} e^{+} \rightarrow H_{2}^{0} Z$. We also studied the possibility to identify it using their respective branching ratios.'
author:
- 'J. E. Cieza Montalvo$^1$'
- 'C. A. Morgan Cruz, R. J. Gil Ramírez, G. H. Ramírez Ulloa, A. I. Rivasplata Mendoza$^2$'
- 'M. D. Tonasse$^{3}$[^1]'
title: 'Neutral 3-3-1 Higgs Boson Through $e^{+}e{-}$ Collisions '
---
=5000 =1000
INTRODUCTION \[introd\]
=======================
The Higgs sector still remains one of the most indefinite part of the standard model (SM) [@wsg], but it still represents a fundamental rule by explaining how the particles gain masses by means of a isodoublet scalar field, which is responsible for the spontaneous breakdown of the gauge symmetry, the process by which the spectrum of all particles are generated. This process of mass generation is the so called [*Higgs mechanism*]{}, which plays a central role in gauge theories.
The SM provides a very good description of all the phenomena related to hadron and lepton colliders. This includes the Higgs boson which appears as elementary scalar and which arises through the breaking of electroweak symmetry. The Higgs Boson is an important prediction of several quantum field theories and is so crucial to our understanding of the Universe. So on 4 July 2012, was measured the discovered 126 GeV Higgs boson [@atlas1; @atlas11]. In this model, the Higgs field receives a vacuum expectation value (VEV), $v \simeq 246$ GeV, which breaks the electroweak gauge symmetry and gives masses to the fundamental fermions and gauge bosons.
However, the standard model does not predict the number of scalar multiplets of the theory, for that reason, there are several extensions of the standard model containing neutral and charged Higgs bosons. Since the standard model leaves many questions open, there are several well motivated extensions of it. For example, if the Grand Unified Theory (GUT) contains the standard model at high energies, then the Higgs bosons associated with GUT symmetry breaking must have masses of order $M_{X} \sim {\cal O} (10^{15})$ GeV. Supersymmetry [@supers] provides a solution to this hierarchy problem through the cancellation of the quadratic divergences via the contributions of fermionic and bosonic loops [@cancell]. Moreover, the Minimal Supersymmetric extension of the Standard Model (MSSM) can be derived as an effective theory from supersymmetric Grand Unified Theories [@sgut]. Another promissory class of models is the one based on the $SU(3)_{C}\otimes SU(3)_{L} \otimes U(1)_{N}$ (3-3-1 for short) semisimple symmetry group [@PT93]. In this model the new leptons do not require new generations, as occur in most of the heavy-lepton models [@FH99]. This ones is a chiral electroweak model whose left-handed charged heavy-leptons, which we denote by $P_a$ $=$ $E$, $M$ and $T$, together with the associated ordinary charged leptons and its respective neutrinos, are accommodated in SU(3)$_L$ triplets.
These models emerge as an alternative solution to the problem of violation of unitarity at high energies in processes such as $e^-e^- \to W^-V^-$, induced by right-handed currents coupled to a vector boson $V^-$. The usual way to circumvent this problem is to give particular values to model parameters in order to cancel the amplitude of the process [@PP92], but in this work was proposed an elegant solution assuming the presence of a doubly charged vector boson. The simplest electroweak gauge model is able to realize naturally a double charge gauge boson based on the SU(3)$\otimes$U(1) symmetry [@PP92]. As a consequence of the extended gauge symmetry, the model is compelled to accommodate a much richer Higgs sector.
The main feature of the 3-3-1 model is that it is able to predicts the correct number of fermions families. This is because, contrary to the standard model, the 3-3-1 model is anomalous in each generation. The anomalies are cancelled only if the number of families is a multiple of three. In addition, if we take into account that the asymptotic freedom condition of the QCD is valid only if the number of generations of quarks is to be less than five, we conclude that the number of generations is three [@LS01]. Another good feature is that the model predicts an upper bound for the Weinberg mixing angle at $\sin^{2} {\theta_W} < 1/4$. Therefore, the evolution of $\theta_W$ to high values leads to an upper bound to the new mass scale between 3 TeV and 4 TeV [@JJ97].
In this work we are interested in a version of the 3-3-1 model, whose scalar sector has only three Higgs triplets [@PT93]. The text is organized as follow. In Sect.\[sec2\] we give the relevant features of the model. In Sect.\[sec3\] we compute the total cross sections of the process $e^{-} e^{+} \rightarrow H_{2}^{0} Z$ and the Sect.\[sec4\] contains our results and conclusions.
Basic facts about the 3-3-1 model {#sec2}
==================================
The three Higgs triplets of the model are $$\begin{aligned}
\eta & = & \left(\begin{array}{c} \eta^0 \\ \eta_1^- \\ \eta_2^+ \end{array}\right) \quad \rho = \left(\begin{array}{c} \rho^+ \\ \rho^0 \\ \rho^{++}
\end{array}\right) \quad \chi = \left(\begin{array}{c} \chi^- \\
\chi^{--} \\ \chi^0 \end{array}\right)
$$ transforming as $\left({\bf 3}, 0\right)$, $\left({\bf 3}, 1\right)$ and $\left({\bf 3}, -1\right)$, respectively.
The neutral scalar fields develop the vacuum expectation values (VEVs) $\langle\eta^0\rangle \equiv v_\eta$, $\langle\rho^0\rangle \equiv v_\rho$ and $\langle\chi^0\rangle \equiv v_\chi$, with $v_\eta^2 + v_\rho^2 = v_W^2 = (246 \mbox{ GeV})^2$. The pattern of symmetry breaking is $\mbox{SU(3)}_L \otimes\mbox{U(1)}_N \stackrel{\langle\chi\rangle}{\longmapsto}\mbox{SU(2)}_L\otimes\mbox{U(1)}_Y\stackrel{\langle\eta, \rho\rangle}{\longmapsto}\mbox{U(1)}_{\rm em}$ and so, we can expect $v_\chi \gg v_\eta, v_\rho$. The $\eta$ and $\rho$ scalar triplets give masses to the ordinary fermions and gauge bosons, while the $\chi$ scalar triplet gives masses to the new fermions and new gauge bosons. The most general, gauge invariant and renormalizable Higgs potential is $$\begin{aligned}
V\left(\eta, \rho, \chi\right) & = & \mu_1^2\eta^\dagger\eta + \mu_2^2\rho^\dagger\rho + \mu_3^2\chi^\dagger\chi + \lambda_1\left(\eta^\dagger\eta\right)^2 + \lambda_2\left(\rho^\dagger\rho\right)^2 + \lambda_3\left(\chi^\dagger\chi\right)^2 + \nonumber \\
&& \left(\eta^\dagger\eta\right)\left[\lambda_4\left(\rho^\dagger\rho\right) + \lambda_5\left(\chi^\dagger\chi\right)\right] +
+ \lambda_6\left(\rho^\dagger\rho\right)\left(\chi^\dagger\chi\right) + \lambda_7\left(\rho^\dagger\eta\right)\left(\eta^\dagger\rho\right) + \nonumber \\
&& \lambda_8\left(\chi^\dagger\eta\right)\left(\eta^\dagger\chi\right) + \lambda_9\left(\rho^\dagger\chi\right)\left(\chi^\dagger\rho\right) + \lambda_{10}\left(\eta^\dagger\rho\right)\left(\eta^\dagger\chi\right) + \nonumber \\
&& \frac{1}{2}\left(f\epsilon^{ijk}\eta_i\rho_j\chi_k + {\mbox{H. c.}}\right).
\label{pot}\end{aligned}$$
Here $\mu_i$ $\left(i = 1, 2, 3\right)$, $f$ are constants with dimension of mass and the $\lambda_i$, $\left(i = 1, \dots, 10\right)$ are dimensionless constants. $f$ and $\lambda_3$ are negative from the positivity of the scalar masses. The term proportional to $\lambda_{10}$ violates lepto-barionic number, therefore it was not considered in the analysis of the Ref. [@TO96] (another analysis of the 3-3-1 scalar sector are given in Ref. [@AK] and references cited therein). We can notice that this term contributes to the mass matrices of the charged scalar fields, but not to the neutral ones. However, it can be checked that in the approximation $v_\chi \gg v_\eta, v_\rho$ we can still work with the masses and eigenstates given in Ref. [@TO96]. Here this term is important to the decay of the lightest exotic fermion. Therefore, we will keep it in the Higgs potential (\[pot\]).
As usual, symmetry breaking is implemented by shifting the scalar neutral fields $\varphi = v_\varphi + \xi_\varphi + i\zeta_\varphi$, with $\varphi$ $=$ $\eta^0$, $\rho^0$, $\chi^0$. Thus, the physical neutral scalar eigenstates $H^0_1$, $H^0_2$, $H^0_3$ and $h^0$ are related to the shifted fields as
$$\begin{aligned}
\left(\begin{array}{c} \xi_\eta \\ \xi_\rho \end{array}\right) \approx
\frac{1}{v_W}\left(\begin{array}{cc} v_\eta & v_\rho \\ v_\rho & -v_\eta
\end{array}\right)\left(\begin{array}{c} H^0_1 \\ H^0_2 \end{array}\right),&& \\
\xi_\chi \approx H^0_3, \qquad \zeta_\chi \approx h^0,&&
\label{eign}\end{aligned}$$
and in the charge scalar sector we have $$\begin{aligned}
\eta^+_1 \approx \frac{v_\rho}{v_W}H^+_1, \qquad \rho^+ \approx \frac{v_\eta}{v_W}H_2^+, && \\ \chi^{++} \approx \frac{v_\rho}{v_\chi}H^{++}, &&
\label{eigc}\end{aligned}$$\[eig\] with the condition that $v_\chi \gg v_\eta, v_\rho$ [@TO96].
The content of matter fields form the three SU(3)$_L$ triplets
$$\begin{aligned}
\psi_{aL} = \left(\begin{array}{c} \nu^\prime_{\ell a} \\ \ell^\prime_a \\ P^\prime_a \end{array}\right), \nonumber && \\ Q_{1L} = \left(\begin{array}{c} u^\prime_1 \\ d^\prime_1 \\ J_1 \end{array}\right), \qquad Q_{\alpha L} = \left(\begin{array}{c} J^\prime_\alpha \\ u^\prime_\alpha \\ d^\prime_\alpha \end{array}\right), &&
\label{fer}\end{aligned}$$
transform as $\left({\bf 3}, 0\right)$, $\left({\bf 3}, 2/3\right)$ and $\left({\bf 3}^*, -1/3\right)$, respectively, where $\alpha = 2, 3$. In Eqs. (\[fer\]) $P_a$ are heavy leptons, $\ell^\prime_a = e^\prime, \mu^\prime, \tau^\prime$. The model also predicts the exotic $J_1$ quark, which carries $5/3$ units of elementary electric charge and $J_2$ and $J_3$ with $-4/3$ each. The numbers $0$, $2/3$ and $-1/3$ in Eqs. (\[fer\]) are the U$_N$ charges. We also have the right-handed counterpart of the left-handed matter fields, $\ell^\prime_R \sim \left({\bf 1}, -1\right)$, $P^\prime_R \sim \left({\bf 1}, 1\right)$, $U^\prime_R \sim \left({\bf 1}, 2/3\right)$, $D^\prime_R \sim \left({\bf 1}, -1/3\right)$, $J^\prime_{1R} \sim \left({\bf 1}, 5/3\right)$ and $J^\prime_{2,3R} \sim \left({\bf 1}, -4/3\right)$, where $U = u, c, t$ and $D = d, s, b$ for the ordinary quarks.
The Yukawa Lagrangians that respect the gauge symmetry are $$\begin{aligned}
{\cal L}^Y_\ell & = & -G_{ab}\overline{\psi_{aL}}\ell^\prime_{bR} - G^\prime_{ab}\overline{\psi^\prime_{aL}}P^\prime\chi + {\mbox{H. c.}}, \\
{\cal L}^Y_q & = & \sum_a\left[\overline{Q_1{L}}\left(G_{1a}U^\prime_{aR}\eta + \tilde{G}_{1a}D^\prime_{aR}\rho\right) + \sum_\alpha\overline{Q_{\alpha L}}\left(F_{\alpha a}U^\prime_{aR}\rho^* + \tilde{F}_{\alpha a}D^\prime_{aR}\eta^*\right)\right] + \cr && +\sum_{\alpha\beta}F^J_{\alpha\beta}\overline{Q_{\alpha J}}J^\prime_{\beta R}\chi^* + G^J\overline{Q_{1L}}J_{1R} + {\mbox{ H. c.}}.
\label{yuk}\end{aligned}$$
Here, the $G$’s, $\tilde{G}$’s, $F$’s and $\tilde{F}$’s are Yukawa coupling constants with $a, b = 1, 2, 3$ and $\alpha = 2, 3$.
It should be noticed that the ordinary quarks couple only through $H^0_1$ and $H^0_2$. This is because these physical scalar states are linear combinations of the interactions eigenstates $\eta$ and $\rho$, which break the SU(2)$_L$ $\otimes$U(1)$_Y$ symmetry to U(1)$_{\rm em}$. On the other hand the heavy-leptons and quarks couple only through $H^0_3$ and $h^0$ in scalar sector, [*i. e.*]{}, throught the Higgs that induces the symmetry breaking of SU(3)$_L$$\otimes$U(1)$_N$ to SU(2)$_L$$\otimes$U(1)$_Y$. The Higgs particle spectrum consists of ten physical states: three scalars ($H_{1}^{0}, H_{2}^{0}, H_3^{0}$), one neutral pseudoscalar $h^0$ and six charged Higgs bosons, $H_{1}^{\pm}, H_{2}^{\pm}, H^{\pm\pm}$.
In this work we study the production of a neutral Higgs boson $H_2^0$, which can be radiated from a $Z^{'}$ boson at $e^{+} e^{-}$ colliders such as the International Linear Collider (ILC) ($\sqrt{s} = 1500$ GeV) and CERN Linear Collider (CLIC) ($\sqrt{s} = 3000$ GeV).
CROSS SECTION PRODUCTION {#sec3}
========================
We begin with the direct production of Higgs ($H_{2}^{0}$), that is $e^{-} e^{+} \rightarrow H_{2}^{0} Z$. This process take place via the exchange of a virtual $Z^{\prime}$ boson in the s channel and it can also take place through the $H_{1}^{0}$ and $H_{2}^{0}$, but the contribution of these channels are small due to the small coupling of the Higgs $H_{2}^{0}$ to the electrons. The term involving the $Z$ boson is absent, because there is no coupling between the $Z$ and $H_{2}^{0} Z$. Then using the interaction Lagrangian Eqs. ($2$) and ($10$) we obtain the differential cross section.
$$\begin{aligned}
\left (\frac{d \hat{\sigma}}{d\cos \theta} \right )_{H_{2}^{0} Z} & = &\frac{\beta_{H_{2}^{0}} \alpha^{2} \pi}{32 \sin^{4}_{\theta_{W}} \cos^{2}_{\theta_{W}} s} \ \frac{\Lambda_{ZZ^{\prime} H_{2}^{0}}^2}{(s- M_{Z'}^{2}+ iM_{Z'} \Gamma_{Z'})^{2}}
\Biggl \{ (2M_{Z}^{2}+ \frac{2tu}{M_{Z}^{2}}- 2t- 2u + 2s) \nonumber \\
&& (g_{V'}^{e^{2}}+ g_{A'}^{e^{2}}) \Biggr \} , \nonumber \\
\label{DZZ'H}\end{aligned}$$
the $\beta_{H_{2}^{0}}$ is the Higgs velocity in the c.m. of the subprocess which is equal to $$\beta_{H_{2}^{0}} = \frac{ \left [\left( 1- \frac{(m_{Z}+ m_{H_{2}^{0}})^{2}}{\hat{s}} \right) \left(1- \frac{(m_{Z}- m_{H_{2}^{0}})^{2}}{\hat{s}} \right) \right ]^{1/2}}{1-\frac{m_{Z}^{2}-m_{H_{2}^{0}}^{2}}{\hat{s}}} \ \ ,$$
and $t$ and $u$ are
$$t = m_{Z}^{2} - \frac{s}{2} \Biggl \{ \left(1+ \frac{m_{Z}^{2}- m_{H}^{2}}{s}\right)- \cos \theta \left [\left( 1- \frac{(m_{Z}+ m_{H})^{2}}{s} \right) \left(1- \frac{(m_{Z}- m_{H})^{2}}{s} \right) \right ]^{1/2}\Biggr \},$$
$$u = m_{H}^{2} - \frac{s}{2} \Biggl \{ \left(1- \frac{m_{Z}^{2}- m_{H}^{2}}{s}\right)+ \cos \theta \left [\left( 1- \frac{(m_{Z}+ m_{H})^{2}}{s} \right) \left(1- \frac{(m_{Z}- m_{H})^{2}}{s} \right) \right ]^{1/2}\Biggr \},$$ where $\theta$ is the angle between the Higgs and the incident quark in the CM frame.
The primes $\left(^\prime\right)$ are for the case when we take a $Z'$ boson, $\Gamma_{Z'}$ [@ct2005; @cieto02], are the total width of the $Z'$ boson, $g_{V', A'}^{e}$ are the 3-3-1 lepton coupling constants, $s$ is the center of mass energy of the $e^{-} e^{+}$ system, $g= \sqrt{4 \ \pi \ \alpha}/\sin \theta_{W}$ and $\alpha$ is the fine structure constant, which we take equal to $\alpha=1/128$. For the $Z^\prime$ boson we take $M_{Z^\prime} = \left(1.5 - 3\right)$ TeV, since $M_{Z^\prime}$ is proportional to the VEV $v_\chi$ [@TO96; @PP92; @fra92]. For the standard model parameters, we assume Particle Data Group values, [*i. e.*]{}, $M_Z = 91.19$ GeV, $\sin^2{\theta_W} = 0.2315$, and $M_W = 80.33$ GeV [@Nea10], $\it{t}$ and $\it{u}$ are the kinematic invariants. We have also defined the $\Lambda_{ZZ^{\prime} H_{2}^{0}}$ as the coupling constants of the $Z^{\prime}$ boson to Z boson and Higgs $H_{2}^{0}$, and the $\Lambda_{e \bar{e} Z^{\prime}}$ are the coupling constants of the $Z^{\prime}$ to $e \bar{e}$.
$$\begin{aligned}
\left(\Lambda_{e\bar{e}Z^\prime}\right)_\mu & \approx & -i\frac{g}{2 \sqrt{1-s_w^2}} \gamma_\mu\left[g_{V^\prime}^{e} - g_{A^\prime}^{e} \gamma_5\right], \\
\left(\Lambda_{ZZ^\prime H_2^0}\right)_{\mu\nu} & \approx & \frac{g^2}{\sqrt{3}\left(1 - 4s_W^2\right)}\frac{v_\eta v_\rho}{v_W}g_{\mu\nu},
\label{eigc}\end{aligned}$$
\[eigthen\]
RESULTS AND CONCLUSIONS {#sec4}
=======================
Here we present the cross section for the process $e^+ e^- \rightarrow H_2^0 Z$ for the ILC ($1.5$) TeV and CLIC ($3$ TeV). All calculations were done according to [@TO96; @cnt2] from which we obtain for the parameters and the VEV, the following representative values: $\lambda_{1} =0.3078$, $\lambda_{2}=1.0$, $\lambda_{3}= -0.025$, $\lambda_{4}= 1.388$, $\lambda_{5}=-1.567$, $\lambda_{6}= 1.0$, $\lambda_{7} =-2.0$, $\lambda_{8}=-0.45$, $v_{\eta}=195$ GeV, and $\lambda_{9}=-0.90(-0.76,-0.71)$ correspond to $v_\chi= 1000(1500,2000)$ GeV these parameters and VEV are used to estimate the values for the particle masses which are given in table \[tab1\].
Differently from what we did in the paper [@ct2005], where was taken arbitrary parameters, in this work we take for the parameters and the VEV the following representative values given above and also the fact that the mass of $m_{H_1^0}$ is already defined [@atlas1; @atlas11]. It is remarkable that the cross sections were calculated in order to guarantee the approximation $-f \simeq v_\chi$ [@TO96; @cnt2]. It must be taken into consideration that the branching ratios of $H_2^0$ are dependent on the parameters of the 3-3-1, which determines the size of several decay modes.
$f$ $v_{\chi}$, $m_{J_1}$ $m_E$ $m_M$ $m_{H_3^0}$ $m_{h^0}$ $m_{H_1^0}$ $m_{H_2^0}$ $m_{H^\pm_2}$ $m_V$ $m_U$ $m_{Z^\prime}$ $m_{J_{2, 3}}$
--------- ----------------------- ------- -------- ------------- ----------- ------------- ------------- --------------- -------- -------- ---------------- ---------------- -- -- --
-1008.3 1000 148.9 875 2000 1454.6 126 1017.2 183 467.5 464 1707.6 1410
-1499.7 1500 223.3 1312.5 474.34 2164.32 125.12 1525.8 387.23 694.12 691.76 2561.3 2115
-1993.0 2000 297.8 1750 632.45 2877.07 125.12 2034.37 519.39 922.12 920.35 3415.12 2820
: \[tab1\] Values for the particle masses used in this work. All the values in this Table are given in GeV. Here, $m_{H^{\pm\pm}} =
500$ GeV and $m_T = 2v_\chi$.
The Higgs $H_{2}^{0}$ in 3-3-1 model is not coupled to a pair of standard bosons, it couples to quarks, leptons, Z $Z^{\prime}$, $Z^{\prime}$ $Z^{\prime}$ gauge bosons, $H_{1}^{-} H_{1}^{+}$, $H_{2}^{-} H_{2}^{+}$, $h^{0} h^{0}$, $H_{1}^{0} H_{3}^{0}$ Higgs bosons, $V^{-}V^{+}$ charged bosons, $U^{--} U^{++}$ double charged bosons, $H_{1}^{0} Z$, $H_{1}^{0} Z'$ bosons and $H^{--} H^{++}$ double charged Higgs bosons [@ct2005]. The Higgs $H_{2}^{0}$ can be much heavier than $ 1017.2$ GeV for $v_\chi = 1000 \ $GeV, $1525.8$ GeV for $v_\chi = 1500 \ $GeV, and $2034.37$ GeV for $v_\chi = 2000$ GeV, so the Higgs $H_2^{0}$ is a heavy particle.
In Table \[tab1\] the masses of the exotic boson $Z^{\prime}$, taken above, is in accord with the estimates of the Tevatron, which probes the $Z^{\prime}$ masses in the 923-1023 GeV range, [@tait], while the reach of the LHC is superior for higher masses, that is $1 \ TeV <M_{Z^{\prime}} \leq 5$ TeV [@freitas] and for ATLAS at $8$ TeV with an integrated luminosity approximately of $20$ fb$^{-1}$ the mass range is $2 \ TeV \leq m_{Z^{\prime}} \leq 3$ TeV, [@atlas1; @atlas2014].
ILC - Events
------------
Considering that the expected integrated luminosity for ILC collider will be of order of $500$ fb$^{-1}$, then the statistics we are expecting are the following, the ILC gives a total of $ \simeq 1.68 \times 10^5 (4.83 \times 10^4)$ events per year, if we take the mass of the Higgs boson $m_{H_2^0}= 1100(1300)$ GeV ($\Gamma_{H_2^0} = 878.25, 1091.33 GeV$) and $v_{\chi}=1000$ GeV, see Fig. \[fig1\]. These values are in accord with the Table \[tab1\].
![Total cross section for the process $e^+ e^- \rightarrow H_2^0 Z$ as a function of $m_{H^{0}_{2}}$ for the ILC at 1.5 TeV and $v_{\chi}=1.5$ TeV. []{data-label="fig1"}](Figure1.eps)
To obtain event rates we multiply the production cross sections by the respective branching ratios. Considering that the signal for $H_{2}^{0}Z$ production for $m_{H_{2}^{0}}= 1100(1300)$ GeV and $v_{\chi}=1000$ GeV will be $H_{2}^{0} Z \rightarrow Z H_{1}^{0} Z$, and taking into account that the branching ratios for these particles would be $BR(H^{0}_{2} \to Z H_{1}^{0}) = 39.5 (43.4) \ \% $ [@ctrg2013], and $\operatorname{BR}(Z \to b \bar{b}) = 15.2 \ \% $, and that the particles $H_{1}^{0}$ decay into $W^{+} W^{-}$, and taking into account that the branching ratios for these particles would be $BR(H^{0}_{1} \to W^{+} W^{-}) = 23.1 \ \% $ followed by leptonic decay of the boson $W^{+}$ into $\ell^{+} \nu$ and $W^{-}$ into $\ell^{-} \bar{\nu}$ whose branching ratios for these particles would be $BR(W \to \ell \nu) = 10.8 \ \%$, then we would have approximately $ \simeq 4 (1))$ events per year for ILC for the signal $b\bar{b} b \bar{b} \ell^{+} \ell^{-} X$.
Statistics for $v_{\chi}=1500 (2000)$ gives no result because there is not enough energy to produce the Higgs boson $H_2^0$. That is, we can see that the number of events for the signal for ILC is insignificant.
CLIC - Events
-------------
Considering that the expected integrated luminosity for CLIC collider will be of order of $3000$ fb$^{-1}$/yr, then we obtain a total of $ \simeq 3.1 \times 10^5 (2.8) \times 10^5$ events per year if we take the mass of the Higgs boson $m_{H_2^0}= 1100(1300)$ GeV and $v_{\chi}=1000$ GeV, see Fig.\[fig2\]. Considering the same signal as above for $H_2^0 Z$ production, that is $H_{2}^{0} Z \rightarrow Z H_{1}^{0} Z$, and taking into account that the branching ratios for these particles would be $BR(H^{0}_{2} \to Z H_{1}^{0}) = 39.5 (43.4) \ \% $, [@ctrg2013], and $\operatorname{BR}(Z \to b \bar{b}) = 15.2 \ \% $, and that the particles $H_{1}^{0}$ decay into $W^{+} W^{-}$, and taking into account that the branching ratios for these particles would be $\operatorname{BR}(H^{0}_{1} \to W^{+} W^{-}) = 23.1 \ \% $ followed by leptonic decay of the boson $W^{+}$ into $e^{+} \nu$ and $W^{-}$ into $e^{-} \bar{\nu}$ whose branching ratios for these particles would be $BR(W \to e \nu) = 10.8 \ \%$, then we would have approximately $ \simeq 8(8)$ events per year for CLIC for the signal $b\bar{b} b \bar{b} \ell^{+} \ell^{-} X$.
The statistics for $v_{\chi}=1500$ gives a total of $\simeq 9.8 \times 10^5(8.1 \times 10^5)$ events per year for CLIC, if we take the mass of the Higgs boson $m_{H_{2}^{0}}= 1600(1800)$ GeV, respectively. These values are in accord with Table \[tab1\]. Taking into account the same signal as above, that is $H_{2}^{0} Z \rightarrow Z H_{1}^{0} Z$, and taking into account that the branching ratios for these particles would be $\operatorname{BR}(H^{0}_{2} \to Z H_{1}^{0}) = 44.2 (45.9) \ \% $, [@ctrg2013], $BR(Z \to b \bar{b}) = 15.2 \ \% $, $BR(H^{0}_{1} \to W^{+} W^{-}) = 23.1 \ \% $, $BR(W \to e \nu) = 10.8 \ \%$, we would have approximately $ \simeq 27(23))$ events per year for CLIC for the same signal $b\bar{b} b \bar{b} e^{+} e^{-} X$.
![Total cross section for the process $e^+ e^- \rightarrow H_2^0 Z$ as a function of $m_{H^{0}_{2}}$ for the CLIC at 3.0 TeV and $v_{\chi}=1.0$ TeV (solid line), $v_\chi = 1.5$ TeV (dash-dot line), $v_\chi = 2.0$ TeV (dashed line).[]{data-label="fig2"}](Figure2.eps){width="1.1\columnwidth"}
With respect to vacuum expectation value $v_{\chi}=2000$ GeV, for the masses of $m_{H_{2}^{0}}= 2100(2300)$ it will give a total of $\simeq 2.9 \times 10^5(2.0 \times 10^5 )$ events per year to produce $H_{2}^{0}$. Taking into account the same signal as above, that is $b\bar{b} b \bar{b} \ell^{+} \ell^{-} X$ and considering that the branching ratios for $H_{2}^{0}$ would be $\operatorname{BR}(H^{0}_{2} \to Z H_{1}^{0}) = 46.4 (47.3) \ \% $, [@ctrg2013], $\operatorname{BR}(Z \to b \bar{b}) = 15.2 \ \% $, $\operatorname{BR}(H^{0}_{1} \to W^{+} W^{-}) = 23.1 \ \% $, $BR(W \to e \nu) = 10.8 \ \%$, we will have approximately $ \simeq 8(6)$ events per year
The main background to this signal is $Z W^{+} W^{-} Z$, which cross section is $1.17 \times 10^{-3}$ pb for $\sqrt{s}=3$ TeV. Considering that the $Z Z$ particles decay into $b \bar{b}$, whose branching ratios for these particles would be $\operatorname{BR}(Z \rightarrow b \bar{b}) = 15.2 \%$ followed by leptonic decay of the boson W, that is $\operatorname{BR}(W \rightarrow e \nu) = 10.8 \%$ then we would have approximately a total of $ \simeq 1$ event for the background and $\simeq 8(8)$ events for the signal for $m_{H_{2}^{0}}= 1100(1300)$ GeV and $v_{\chi}=1000$.
Therefore we have that the statistical significance is $\simeq 2.66(2.66) \sigma$ for $m_{H_{2}^{0}}= 1100(1300)$ and $v_{\chi}=1000$ GeV, that is a low probability to detect signals. On the other hand, for $v_{\chi}=1500$ GeV and $m_{H_{2}^{0}}= 1600(1800)$ GeV we have $\simeq 5.10(4.70) \sigma$ discovery in the $b\bar{b} b \bar{b} e^{+} e^{-} X$ final state , for $v_{\chi}=2000$ GeV and $m_{H_{2}^{0}}= 2100(2300)$ GeV which corresponds to $\simeq 2.66(2.27) \sigma$, we have that the signals are too small to be observed.
To extract the signal from the background we must select the $b \bar{b}$ channel using the techniques of b-flavour identification. Later, the Z that comes together with the $H_{2}^{0}$ and the other Z that comes from the decay of $H_{2}^{0}$ would appear as a peak in the invariant mass distribution of b-quark pairs. The charged lepton track from the $W$ decay and the cut on the missing transverse momentum ${p\!\!\slash}_{T} >$ 20 GeV allows for a very strong reduction of the backgrounds.
The $H_{2}^{0} Z$ will also decay into $t \bar{t} \ \ell^{+} \ell^{-}$, and consider that the branching ratios for these particles would be $\operatorname{BR}(H^{0}_{2} \to t \bar{t}) = 5.1 (4.1) \ \% $, , [@ctrg2013], and $BR(Z \to \ell^{+} \ell^{-}) = 10.2 \ \% $ for the mass of the Higgs boson $m_{H_{2}^{0}}= 1100(1300)$ GeV and $v_{\chi}=1000$ GeV and that the particles $t \bar{t}$ decay into $ b \bar{b} W^{+} W^{-}$, whose branching ratios for these particles would be $BR(t \to b W) = 99.8 \ \% $, followed by leptonic decay of the boson W, that is $BR(W \to e \nu) = 10.75 \ \% $, then we would have approximately $\simeq 19 (13)$ events per year for CLIC for the signal $b\bar{b} e^{-} e^{+} \ell^{+} \ell^{-} X$. Considering the vacuum expectation value $v_{\chi}=1500$ GeV and the branching ratios $\operatorname{BR}(H_{2}^{0} \rightarrow t \bar{t}) = 2.8 (2.3) \ \% $, , [@ctrg2013], and taking the same parameters and branching ratios for the same particles given above, then we would have for $m_{H_{2}^{0}}= 1600(1800)$ a total of $ \simeq 32(22)$ events per year for CLIC for the same signal. With respect to vacuum expectation value $v_{\chi}=2000$ GeV, for the masses of $m_{H_{2}^{0}}= 2100(2300)$ and taking into account the same signal as above, that is $b\bar{b} e^{-} e^{+} \ell^{+} \ell^{-} X$ and considering that the branching ratios $\operatorname{BR}(H_{2}^{0} \rightarrow t \bar{t}) = 1.7 (1.5) \ \% $, [@ctrg2013],we will have approximately $ \simeq 6(4)$ events per year.
Taking again the irreducible background for the process $t \bar{t}Z\rightarrow b \bar{b} e^{+} e^{-} \ell^{+} \ell^{-} X$, and using CompHep [@pukhov] we have that a cross section of $1.67 \times 10^{-3}$ pb, which gives $ \simeq 6$ events. So we will have a total of $\simeq 19 (13)$ events per year for the signals for $m_{H_{2}^{0}}= 1100(1300)$ GeV and $v_{\chi}=1000$, which corresponds to have $\simeq 3.80(2.98) \sigma$, then we have an evidence for $\simeq 3.80 \sigma$ discovery in the $b\bar{b} e^{-} e^{+} \ell^{+} \ell^{-} X$ final state. On the other hand, for $v_{\chi}=1500$ we have $\simeq 32(22)$ events for $m_{H_{2}^{0}}= 1600(1800)$ GeV and which corresponds to $\simeq 5.19(4.16) \sigma$, then we have a discovery for $\simeq 5.19 \sigma$ in the $b \bar{b} e^{+} e^{-} \ell^{+} \ell^{-} X$ final state. For $v_{\chi}=2000$ we have $\simeq 6(4)$ events for $m_{H_{2}^{0}}= 2100(2300)$ GeV and which corresponds to $\simeq 1.73(1.27) \sigma$, that is a low probability to detect the signals. We impose the following cuts to improve the statistical significance of a signal, i. e. we isolate a hard lepton from the $W$ decay with $p_{T}^{\ell}>$ 20 GeV, put the cut on the missing transverse momentum ${p\!\!\slash}_{T} >$ 20 GeV and apply the Z window cut $|m_{\ell^{+} \ell^{-}} - m_{Z}| >$ 10 GeV, which removes events where the leptons come from Z decay [@aguila]. However, all this scenarios can only be cleared by a careful Monte Carlo work to determine the size of the signal and background.
We still mention that the initial state radiation (ISR) and beamstrahlung (BS) strongly affects the behaviour of the production cross section around the resonance peaks, modifying as the shape as the size [@nicro], so Fig. $3$ shows the cross section with and without ISR + BS around the resonance point $m_{Z^\prime} = 2561.3$ GeV for CLIC. As can be seen the peak of the resonance shifts to the right and is lowered as a result of the ISR + BS effects.
![Total cross section for the process $e^+ e^- \rightarrow H_2^0 Z$ as a function of center of mass energy ($\sqrt{s}$) with and without ISR + BS (dashed line and solid line respectively) for CLIC at $v_\chi=1.5$ TeV.[]{data-label="fig3"}](Figure3.eps){width="1.1\columnwidth"}
In summary, we showed in this work that in the context of the 3-3-1 model the signatures for neutral Higgs boson $H_2^0$ can be significant in CLIC collider if we take $v_{\chi}=1500$, $m_{H_{2}^{0}}=1600(1800)$ GeV and a luminosity of 3000 $fb^{-1}$, we have $\simeq 5.10(4.70) \sigma$ discovery in the $b\bar{b} b \bar{b} e^{+} e^{-} X$ and $\simeq 5.19(4.16) \sigma$ in the $b \bar{b} e^{+} e^{-} \ell^{+} \ell^{-} X$ final state.
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[^1]: Permanent address: Universidade Estadual Paulista, [*Campus*]{} Experimental de Registro, Rua Nelson Brihi Badur 430, 11900-000 Registro, SP, Brazil
|
---
abstract: |
Let $\manif$ be a smooth compact surface, orientable or not, with boundary or without it, $\Psp$ either the real line $\RRR^1$ or the circle $\aCircle$, and $\DiffM$ the group of diffeomorphisms of $\manif$ acting on $C^{\infty}(\manif,\Psp)$ by the rule $\difM\cdot\mrsfunc=\mrsfunc\circ\difM^{-1}$ for $\difM\in\DiffM$ and $\mrsfunc\in C^{\infty}(\manif,\Psp)$. Let $\mrsfunc:\manif\to\Psp$ be an arbitrary Morse mapping, $\singf$ the set of critical points of $\mrsfunc$, $\DiffMcr$ the subgroup of $\DiffM$ preserving $\singf$, and $\Stabf$, $\Stabfcr$, $\Orbf$, and $\Orbfcr$ the stabilizers and the orbits of $\mrsfunc$ with respect to $\DiffM$ and $\DiffMcr$. In fact $\Stabf=\Stabfcr$.
In this paper we calculate the homotopy type of $\Stabf$, $\Orbf$ and $\Orbfcr$. It is proved that except for few cases the connected components of $\Stabf$ and $\Orbfcr$ are contractible, $\pi_k\Orbf=\pi_k\manif$ for $k\geq 3$, $\pi_2\Orbf=0$, and $\pi_1 O(f)$ is an extension of $\pi_1\DiffM\oplus\ZZZ^{k}$ (for some $k\geq0$) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of $\mrsfunc$.
We also generalize the methods of F. Sergeraert to prove that a finite codimension orbit of a tame smooth action of a tame Lie group on a tame manifold is a tame manifold itself. In particular, this implies that $\Orbf$ and $\Orbfcr$ are tame manifolds.
address: 'Topology Department, Institute of Mathematics, Ukrainian National Academy of Science, Tereshchenkivska str. 3, 01601 Kyiv, Ukraine'
author:
- Sergey Maksymenko
title: |
Homotopy types of stabilizers and orbits\
of Morse functions on surfaces
---
[**Keywords**]{}: surface, Morse function, diffeomorphism, flow, homotopy type\
[**MSC 2000**]{}: 37C05, 57S05, 57R45
|
---
abstract: 'We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let $G=(V,E)$ be a tree with stationary distribution $\pi$. For a vertex $v \in V$, let $H(v,\pi)$ denote the expected length of an optimal stopping rule from $v$ to $\pi$. The *best mixing time* for $G$ is $\min_{v \in V} H(v,\pi)$. We show that among all trees with $|V|=n$, the best mixing time is minimized uniquely by the star. For even $n$, the best mixing time is maximized by the uniquely path. Surprising, for odd $n$, the best mixing time is maximized uniquely by a path of length $n-1$ with a single leaf adjacent to one central vertex.'
author:
- 'Andrew Beveridge[^1] and Jeanmarie Youngblood[^2]'
bibliography:
- 'bestmixref.bib'
title: The best mixing time for random walks on trees
---
=+1pt
\[section\] \[theorem\] [Lemma]{} \[theorem\] [Corollary]{} \[theorem\] [Claim]{} \[theorem\] [Proposition]{} \[theorem\] [Definition]{} \[theorem\] [Remark]{} \[theorem\] [Conjecture]{} \[theorem\] [Question]{} \[theorem\] [Example]{}
§
Ł[L]{}
Introduction
============
We resolve the following extremal question for random walks on trees: what tree structure minimizes/maximizes the expected length of an optimal stopping rule to the stationary distribution $\pi$, given that we start at the most advantageous vertex? Naturally, the star $S_n$ is the minimizing tree structure, but the maximization problem has an unexpected twist. The path $P_{n}$ is the maximizing structure when $n$ is even, but for odd $n$ the maximizing structure is the near-path $Y_n$ consisting of a path on $n-1$ vertices with a single leaf adjacent to one of the two central vertices. We refer to this graph $Y_n$ as the *wishbone*. This choice of name is suggested by the layout of $Y_n$ in Figure \[fig:menagerie\](c) below.
A *random walk* on an undirected graph $G=(V,E)$ consists of a sequence of vertices $(w_0, w_1, \ldots , w_n, \ldots)$ such that $\Pr[w_{t+1} = w \mid w_t = v]$ is $1/ \deg(v)$ if $(v,w) \in E$ and $0$ otherwise. The *hitting time* $H(v,w)$ from vertex $v$ to vertex $w$ is the expected number of steps before a random walk started at $v$ visits $w$ for the first time. We also define $H(v,v)=0$, while $\Ret(v)$ denotes the expected number of steps before a random walk started at $v$ first returns to $v$.
When $G$ is not bipartite, the distribution of $w_t$ converges to the *stationary distribution* $\pi$, where $\pi_v = \deg(v) /2|E|.$ Inconveniently, we do not have convergence for bipartite graphs (including trees), but we can rectify this at the cost of doubling the expected length of any walk. Indeed the distribution $w_t$ converges to $\pi$ when we follow a *lazy random walk* in which we remain at the current state with probability $1/2$ at each time step. A *mixing measure* of a graph $G$ captures the rate of convergence to the stationary distribution $\pi$. Let $\sigma=\sigma_0$ denote our initial distribution and $\sigma_t$ denote the distribution for the $t$-th step of a random walk. For a fixed constant $0 < \epsilon < 1$, the (approximate) mixing time of $G$ is given by $\Tmix(\epsilon) = \max_{\sigma} \min \{T \geq 0 : \, \| \sigma_t - \pi \| < \epsilon \mbox{ for all } t \geq T \},$ where the maximum is taken over all possible starting distributions and $\| \cdot \|$ is the given metric. This mixing time depends upon the choice of the parameter $\epsilon$.
Lovász and Winkler [@lovasz+winkler] studied a class of parameterless mixing measures by using more sophisticated *stopping rules* to drive the random walk to a desired distribution. Suppose that we are given a *starting distribution* $\sigma$ and a *target distribution* $\tau$. A *$(\sigma,\tau)$ stopping rule* halts a random walk whose initial state is drawn from $\sigma$ so that the final state is governed by the distribution $\tau$. The *access time* $H(\sigma, \tau)$ denotes the minimum expected length for a such a $(\sigma,\tau)$ stopping rule to halt. We say that a stopping rule is *optimal* if it achieves this minimum expected length. Using access times, we have three natural mixing measures, the *mixing time*, the *reset time*, and the *best mixing time* given respectively by $$\Tmix = \max_{v \in V} H(v, \pi), \quad \mbox{and} \quad \Tres = \sum_{v \in V} \pi_v H(v, \pi),
\quad \mbox{and} \quad \Tbest = \min_{v \in V} H(v, \pi).$$ These are the worst-case, average-case and best-case mixing measures. These quantities are called *exact mixing measures*, as opposed to approximate mixing measures like $\Tmix(\epsilon)$ above. See [@ALW] for a taxonomy that compares exact and approximate mixing measures for Markov chains. We also note that exact stopping rules converge to $\pi$ on a bipartite graph, even for non-lazy walks. Indeed, stopping rules can employ randomness in deciding when to stop, which has the same periodicity-breaking effect as using a lazy walk.
It is natural to wonder which graph structures lead to slow or rapid mixing, and the study of extremal graphs for random walks is well-established, cf. [@aldous+fill; @bright; @feige]. A natural line of inquiry is to restrict our attention to trees. The extremal tree structures on $n$ vertices for $\Tmix$ and $\Tfor$ were characterized in [@beveridge+wang]. Not surprisingly, the unique minimal structure is the star $S_n=K_{1,n-1}$ and the unique maximal structure is the path.
\[thm:mix-reset\] If $G$ is a tree on $n \geq 3$ vertices then $$\frac{3}{2} \leq \Tmix \leq \frac{2n^2 - 4n + 3}{6}$$ and $$1 \leq \Tres \leq \left\{
\begin{array}{cc}
\frac{1}{4}(n^2 - 2n +2) & \mbox{if $n$ is even}, \\
\frac{1}{4}(n-1)^2 & \mbox{if $n$ is odd}. \\
\end{array}
\right.$$ In each instance, the lower bound is achieved uniquely by the star $S_n$ and the upper bound is achieved uniquely by the path $P_n$.
We add to these extremal tree characterizations by studying the best mixing time $\Tbest$.
\[thm:bestmix\] Let $G$ be a tree on $n \geq 3$ vertices. Then $$1/2 \leq \Tbest(G) \leq
\left\{
\begin{array}{rl}
\frac{1}{12}(n^2 +4n -6) & \mbox{if $n$ is even}, \\
\frac{1}{12}(n^2+4n-15) & \mbox{if $n$ is odd}. \\
\end{array}
\right.$$ The star $S_n$ is the unique minimizing tree. For $n$ even, the path $P_n$ is the unique maximizing tree. For $n$ odd, the wishbone $Y_n$ is the unique maximizing tree.
Below, we assume that $n \geq 4$ since there is a unique tree for $n=3$ (and the formula is correct in that case). The appearance of the wishbone is quite unexpected. At the end of the next section, we discuss the characteristics of $Y_n$ that lead to its maximization of the best mixing time for odd $n$.
The proof of Theorem \[thm:bestmix\] is more involved than the proof of Theorem \[thm:mix-reset\] in [@beveridge+wang]. The reason is that a minor change to the tree can abruptly shift the location of the initial vertex achieving $\tbm$, whereas the behavior of $\Tmix$ and $\Tres$ are more stable under minor structural alterations. Here is an overview of our proof. First, we replace the given tree with a caterpillar having a larger best mixing time. We then follow a prescribed algorithm, carefully moving one or two leaves at a time via a *tree surgery* $S_i$. These surgeries monotonically increase $\Tbest$, until we arrive at an even path or an odd wishbone. Figure \[fig:examples\] shows two sequences of surgeries, the first resulting in $P_{10}$ and the second resulting in $Y_{11}$. The surgeries and the guiding algorithm are described in Section \[sec:proof\].
(1,0) – (7,0);
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background
Preliminaries
=============
For an introduction to the theory of exact stopping rules, see [@lovasz+winkler]. To prove Theorem \[thm:bestmix\], we need the formula for access times from singleton distributions to the stationary distribution. Therefore, we limit ourselves to describing results from [@lovasz+winkler] needed to calculate these values.
Pessimal vertices {#sec:pessimal}
-----------------
A *$v$-pessimal vertex* $v'$ is a vertex that satisfies $H(v',v)=\max_{w\in V}{H(w,v)}.$ Note that pessimal vertices are not necessarily unique; for example, every leaf of the star $S_n$ is pessimal for the central vertex. We use ${v''}$ to denote a pessimal vertex for ${v'}$, and we employ the notation $$\hp(v) = H({v'},v) = \max_{w \in V} H(w,v),$$ so that we can refer to the pessimal hitting time to $v$ in a manner agnostic of the choice of pessimal vertex ${v'}$. This allows us to define the lightweight notation $$\label{eqn:pessdiff}
\pdelt(v)=\hp_{\nG}(v)-\hp_G(v)$$ which shields us from the fact that the $v$-pessimal vertices in $G$ and $\nG$ may be distinct.
Calculating mixing times on trees
---------------------------------
Next we address the calculation of $H(v, \pi)$, the expected length of an optimal stopping rule from the singleton distribution on $v$ to the stationary distribution $\pi$. We refer to $H(v,\pi)$ as the *mixing time from $v$*. The following result holds for any graph $G$.
\[thm:mix\] The expected length of an optimal mixing rule starting from vertex $v$ is $$\label{eqn:mix}
H(v, \pi)
= H({v'}, v) - \sum_{u \in V} \pi_u H(u,v)
= \hp(v) - \sum_{j \in V} \pi_u H(u,v).$$
The very useful formula allows us to calculate the access time $H(v, \pi)$ via a linear combination of vertex-to-vertex hitting times. In other words, if we know all the pairwise hitting times, it is easy to calculate the access time $H(v, \pi)$ for each starting vertex $v \in V$.
For a given graph, we can determine every hitting time $H(u,v)$ by solving a system of linear equations. In the special case of trees, we have an explicit formula in terms of the distances and degrees of the graph. Variations on this formula appear in the literature (cf. [@beveridge; @beveridge+wang; @dtw; @gw] and Chapter 5 of [@aldous+fill]). This paper builds on the results in [@beveridge; @beveridge+wang], so we opt for that formulation. From here forward, we assume that $G=(V,E)$ is a tree on $n \geq 4$ vertices.
We start with a well-known result about the hitting time between adjacent vertices in trees. If $uv \in E$, then removing this edge breaks $G$ into two disjoint trees $G_u$ and $G_v$ where $u \in V(G_u)$ and $v \in V(G_v)$. We define $V_{u:v} = V(G_u)$ and $V_{v:u} = V(G_v)$. We think of $V_{u:v}$ as the set of vertices that are closer to $u$ than to $v$. For these adjacent vertices, let $F$ denote the induced tree on $V_{u:v} \cup \{ v \}$. We have $$\label{eq:adjhtime}
H(u,v)= \Ret_{F} (v) - 1 = \sum_{w\in V_{u:v}}{\deg(w)} = 2|V_{u:v}|-1,$$ where the first equality holds by the well-known equality $\Ret(v) = 2|E|/\deg(v)$. Equation encodes a very useful property of trees: the hitting time from a vertex $u$ to an adjacent vertex $v$ only depends on $|V_{u:v}|$, independent of the particular structure of the tree. This equation can be used to reveal that when $G=P_n$ is the path ($v_1,v_2,\dots,v_n$), the hitting times are $$\label{eq:htimepath2}
H_{P_n}(v_i,v_j)=
\left\{
\begin{array}{rl}
(i-1)^2-(j-1)^2 & i\leq j,\\
(n-j)^2-(n-i)^2 & i>j.
\end{array}
\right.$$ For example, the first of these formulas is calculated by repeatedly using equation to evaluate $H(v_i, v_j) = H(v_i, v_{i+1}) + H(v_{i+1}, v_{i+2}) + \cdots + H(v_{j-1}, v_{j})$. A similar argument produces a formula for the hitting times of an arbitrary tree, but first we need some additional notation. Let $u,v,w \in V$. Define $d(u,v)$ to be the distance between these two vertices and define $$\ell(u,v;w)=\frac{1}{2}(d(u,w)+d(v,w)-d(u,v))$$ to be the length of the intersection of the $(u,w)$-path and the $(v,w)$-path.
\[thm:htime\] For any pair of vertices $v_i, v_j$, we have $$\label{eq:htime}
H(v_i,v_j)=\sum_{v\in V}{\ell(v_i,v;v_j)\deg(v)}.$$
Theorem \[thm:mix\] and Lemma \[thm:htime\] are the fundamental tools in our proof of Theorem \[thm:bestmix\]. Together they provide a simple way to calculate state-to-state hitting times and mixing times.
The foci of a tree
------------------
The center and the barycenter are two well-established notions of centrality. The center of a tree $G$ is the vertex (or two adjacent vertices) that achieves $\min_{u \in V} \max_{v} d(v,u)$. The center does not appear to have any central properties with respect to random walks on trees. The barycenter is the vertex (or two adjacent vertices) that achieves $\min_{u \in V} \sum_{v \in V} d(v,u)$. Proposition 1 of [@beveridge] shows that that the barycenter is the “average” center for random walks on trees: this vertex also achieves $\min_{u} \sum_{v \in V} \pi_v H(v,u)$. A third type of centrality for trees was introduced in [@beveridge]: the *focus* is the “extremal” center for random walks.
\[def:focus\] [**[([@beveridge])]{}**]{} A vertex $v$ achieving $\hp(v) = \min_{w \in V} \hp(w)$ is called a *primary focus* of $G$. If all $v$-pessimal vertices are contained in a single subtree of $G-v$, then the unique $v$-neighbor $u$ in that subtree is also a focus of $G$. If $\hp(u)=\hp(v)$, then $u$ is also a primary focus. Otherwise $u$ is a *secondary focus*.
Every tree has either one focus, in which case it is *focal*, or two adjacent foci, in which case it is *bifocal*. We have the following two theorems, which relate the foci to mixing walks on $G$.
\[thm:tifocus\] If $G$ is focal with focus $u$, then for all $v$ $
H(v, \pi) = H(v,u) + H(u,\pi).
$ If $G$ is bifocal with foci $u$ and $w$, then for $v \in V_{u:w}$, $
H(v, \pi) = H(v,u) + H(u,\pi)
$ and for $v \in V_{w:u}$, $
H(v, \pi) = H(v,w) + H(w,\pi).
$
\[thm:tbmvertex\] If $H(w,\pi) = \tbm(G)$ then $w$ is a focus of $G$. For bifocal $G$ with foci $u$ and $v$, if $H({v'},u) < H({u'},v)$ then $v$ is the unique vertex achieving $\tbm(G)$. If $H({v'},u) > H({u'},v)$ then $u$ is the unique vertex achieving $\tbm(G)$. If $H({v'},u) = H({u'},v)$ then both vertices achieve $\tbm$.
If $H(w,\pi) = \tbm(G)$ then we say that $w$ is a *best mix focus*. Note that a tree can have one or two best mix foci. In the former case, the best mix focus could be either the primary focus or the secondary focus: it depends on the relative sizes of $H({v'},u)$ and $H({u'},v)$. This unusual criterion is what makes handling the best mixing time so delicate. A small change to the tree structure can move the location of the best mix focus, even when the foci do not change.
The best mixing time for stars, paths and odd wishbones {#sec:bestmix-extreme}
-------------------------------------------------------
As our first application of these theorems and lemmas, let us calculate the best mixing time for the star, the path (both even and odd $n$) and the wishbone (for odd $n$). This will justify the expressions that appear in Theorem \[thm:mix\]. Clearly, the central vertex $v$ of the star $S_n=K_{1,n-1}$ is its unique focus. Let $w \neq v$ be any other vertex. We have $\pi_v = 1/2$ and $\pi_w = 1/2(n-1)$. Furthermore, $H(w,v) = \hp(v) = 1$ and obviously $H(v,v)=0$. Therefore $$\tbm (S_n) = \hp(v) - \sum_{w \in V} \pi_w H(w,v) = 1 - \frac{1}{2(n-1)} (n-1) = \frac{1}{2}.$$ In fact, it is easy to see that $S_n$ is the unique tree on $n$ vertices that achieves this value. Let $G \neq S_n$ and let $v$ be a best mix focus. Then $\pi_v = \deg(v)/2(n-1)< 1/2$ because the star is the unique tree with a vertex of degree $n-1$. An optimal mixing rule started at $v$ must exit with probability at least $1 - \pi_v > 1/2$, otherwise the ending distribution will weight vertex $v$ too heavily. This proves that the star is the unique minimizing structure for Theorem \[thm:bestmix\].
Next, we calculate $\tbm(P_n)$. Label the vertices as $(v_1, v_2, \ldots , v_n)$. We first consider odd $n=2r+1$. By symmetry, the unique focus is $v_{r+1}$. By Theorem \[thm:mix\] and equation , $$\begin{aligned}
\Tbest(P_{2r+1}) &=& \hp(v_{r+1}) - \sum_{i=1}^{2r+1} \pi_i H(v_i, v_{r+1})
\, = \, \hp(v_{r+1}) - 2 \sum_{i=1}^{r} \pi_i H(v_i, v_{r+1}) \\
&=& r^2 - \frac{1}{2r} r^2 - \frac{1}{r} \sum_{i=2}^r \left( r^2 - (i-1)^2 \right)
\,=\, \frac{2r^2+1}{6} \, = \, \frac{n^2-2n+3}{12},\end{aligned}$$ where the second equality follows from the symmetry of the odd path. The calculation for the even path is a bit tougher. Setting $n=2r$, we have two best mix foci $v_r, v_{r+1}$ by symmetry. We will take $v_{r+1}$ as the best mix focus in our calculation. We have $$\begin{aligned}
\Tbest(P_{2r})
&=& \hp(v_{r+1}) - \sum_{i=1}^{r} \pi_i H(v_i, v_{r+1}) - \sum_{i=r+2}^{2r} \pi_i H(v_i, v_{r+1}) \\
&=& \hp(v_{r+1}) - 2\sum_{i=1}^{r-1} \pi_i H(v_i, v_{r}) - \sum_{i=1}^{r} \pi_i H(v_r, v_{r+1}) \\
&=& r^2 - \frac{2}{4r-2} (r-1)^2 - \frac{4}{4r-2} \sum_{i=2}^{r-1} \left( (r-1)^2 - (i-1)^2 \right) - \frac{1}{2} (2r-1) \\
&=& \frac{2r^2 + 4r-3}{6} \, = \, \frac{n^2 +4n - 6}{12}\end{aligned}$$ where we use the symmetry of the path and $H(v_i, v_{r+1}) = H(v_i, v_r) + H(v_r,v_{r+1})$ for $1 \leq i < r$ in the third equality.
Finally, we determine $\tbm(Y_{2r+1})$. Let the spine be $(w_1, w_2, \ldots, w_{2r})$ with a leaf $x$ adjacent to $w_{r+1}$. Arguing similarly to the even path, we have $$\begin{aligned}
\tbm(Y_{2r+1}) &=& \hp(w_{r+1}) - \sum_{i=1}^{r} \pi_i H(w_i, w_{r+1}) - \sum_{i=r+2}^{2r} H(w_i, w_{r+1}) - \pi_x H(x,w_{r+1}) \\
&=&
\hp(w_{r+1}) - 2\sum_{i=1}^{r} \pi_i H(w_i, w_r) - \sum_{i=1}^{r} H(v_r, w_{r+1}) - \pi_x H(x,w_{r+1}) \\
&=& r^2 - \frac{1}{2r} (r-1)^2 - \frac{1}{r} \sum_{i=2}^{r-1} ((r-1)^2 - (i-1)^2) - \frac{2r-1}{4r} (2r-1) - \frac{1}{4r} \\
&=& \frac{n^2 + 4n - 15}{12}.\end{aligned}$$ These calculations show that for odd $n=2r+1 \geq 5$, we have $\tbm(Y_n) = \tbm(P_n)+ (n-3)/2$. Furthermore, we can identify precisely what leads to the wishbone’s advantage in equation . The pessimal hitting times to the best mix foci of $Y_{2r+1}$ and $P_{2r+1}$ are both equal to $r^2.$ So the average hitting time to the best mix focus makes the difference: this quantity is smaller for $Y_n$ We will see below that this effect can be attributed to the fact that $P_{2n+1}$ has a single focus. This special balance to the tree actually makes it *easier* to mix from the focus. With these calculations in hand, we have proven the lower bound of Theorem \[thm:bestmix\], and calculated the two expressions that appear in the upper bound. The remainder of the paper proves that these upper bounds are correct.
Caterpillars and Tree Surgeries
===============================
In this section, we show that for a fixed $n \geq 4$, the tree on $n$ vertices that maximizes the best mixing time is a caterpillar. Next, we describe our basic techniques, called *tree surgeries*, for making incremental changes to a caterpillars. We introduce some notation and useful lemmas for showing that the best mixing time monotonically increases after each tree surgery.
Tree to Caterpillar
-------------------
We begin with a lemma showing that $\Tbest$ is maximized by a caterpillar. Recall that a *caterpillar* is a tree such that every vertex is distance at most one from a fixed central path $W = \{ w_1, w_2, \ldots , w_t \}$, called the *spine*. We employ the following notation for talking about sections of the caterpillar. For $2 \leq i \leq t-1$, let $U_i \subset V \backslash W$ denote the set of pendant leaves adjacent to $w_i$. Note that the spinal leaves are not in $V \backslash W$, so $w_1 \notin U_2$ and $w_t \notin U_{t-1}$. Finally, we also define $V_k = U_k \cup \{w_k \}$. See Figure \[fig:treetocat\] for an example of $w_i$ and $U_i \subset V_i$.
\[lem:cat\] Let $G$ be a tree on $n$ vertices that is not a caterpillar. There exists a caterpillar $\nG$ on $n$ vertices such that $\tbm(\nG) > \tbm(G)$.
We give a simple construction of the caterpillar $\nG$; an example is shown in Figure \[fig:treetocat\].
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Choose any $v\in V(G)$. Recall from Section \[sec:pessimal\] that ${v'}$ is a $v$-pessimal vertex, and that ${v''}$ is a ${v'}$-pessimal vertex. Let $
W=\{{v'}=w_1,w_2,\dots,w_t={v''}\}
$ be the vertices on the unique $({v'},{v''})$ path $P \subset G$. These vertices will be the spine of the caterpillar $\nG$. (Note that $v$ is not necessarily a member of $W$.) We replace the subforest attached to each spine vertex $w_i$ with a set of leaves adjacent to $w_i$. Define $\nG$ to be the caterpillar with vertex set $V(G)$ and edge set $
E(\nG)= E(P) \cup \{uw_i\mid u\in U_i, 1<i<t\}.
$
The foci of $\nG$ are contained in $W$ because leaves cannot be foci when $n \geq 3$. Without loss of generality, let $w_r$ be the best mix focus of $\nG$ and let ${w_r'} = w_1$. If $G$ is bifocal, the other focus must be $w_{r-1}$ and ${w_{r-1}'} = w_t$. For $1\leq i\leq t-1$, the quantity $$\sum_{v \in V_{w_i:w_{i+1}}} \deg_G(v) = 2 |V_{w_i:w_{i+1}}| - 1 = \sum_{v \in V_{w_i:w_{i+1}}} \deg_{\nG}(v)$$ does not change. By equation we have $$\label{eqn:nochange}
H_G(w_i, w_j) = H_{\nG}(w_i,w_j)$$ for all $1 \leq i,j \leq t$. Considering any non-spine vertex $v \in U_i$, we have $$H_{\nG}(v, w_r) - H_G(v,w_r)
=
1 + H_{\nG}(w_i, w_r) - \big( H_{G}(v, w_i) + H_{G}(w_i, w_r) \big)
= 1 - H_G(v,w_i) \leq 0.$$ We conclude that $w_1$ is $v_r$-pessimal for both $G$ and $\nG$. Likewise, $w_t$ is $w_{r-1}$-pessimal for both $G$ and $\nG$. By equation and Theorem \[thm:tbmvertex\], the vertex $w_r$ is the best mix focus of $\nG$ and $\hp_G(w_r) = \hp_{\nG}(w_r)$. Finally, Theorem \[thm:mix\] yields $$\begin{aligned}
\lefteqn{ \Tbest(\nG) - \Tbest(G) \, = \,
H_{\nG}(w_r, \npi) - H_{G}(w_r, \pi)
}
\\
&=& \sum_{i=1}^{t} \sum_{v \in V_i} \left( \pi_{v} (H_G (v, w_i)+ H_G (w_i, w_r)) - \npi_{v} (H_{\nG} (v, w_i) + H_{\nG} (w_i, w_r)) \right) \\
&=& \sum_{i=1}^{t} \sum_{v \in V_i} \left( \pi_{v} H_G (v, w_i) - \npi_{v} H_{\nG} (v, w_i) \right)
\, = \,
\sum_{i=2}^{t-1} \sum_{v \in U_i} \left( \pi_{v} H_G (v, w_i) - \npi_{v} H_{\nG} (v, w_i) \right)
\, \geq \, 0\end{aligned}$$ because each $ \sum_{v \in V_i} \pi_{v} H_G (v, w_i) - \npi_{v} H_{\nG} (v, w_i) \geq 0$, with equality holding if and only if every $v \in U_i$ is already adjacent to $w_i$. Since $G$ was not a caterpillar, there must be at least one exceptional $v$ in some $U_i$, so this inequality is strict.
Leaf Transplants and Tree Surgeries
-----------------------------------
For the remainder of the paper, $G$ is a caterpillar on $n$ vertices with spine $W= \{ w_1, w_2, \ldots , w_t \}$. We choose to view $w_1$ as the leftmost vertex and $w_t$ as the rightmost vertex. For $v \in V$, we define the *left pessimal hitting time* $\L(v) = H(w_1,v)$ and the *right pessimal hitting time* $\R(v) = H(w_t,v)$. Of course, $\hp(v) = \max \{ \L(v), \R(v) \}$. When $G$ is bifocal, we always label the foci as $w_{r-1}$ and $w_r$ where $w_r$ achieves $H(w_r,\pi)=\Tbest(G)$. In this case, we view $\{ w_1, \ldots , w_{r-2} \}$ as the *left spine* and $\{ w_{r+1}, \ldots , w_{t} \}$ as the *right spine*. The foci $\{ w_{r-1}, w_r \}$ are considered the *central spine*. Note that this arrangement guarantees that ${w_r'}=w_1$ and ${w_{r-1}'}=w_t$. We collect some basic results about caterpillar foci in the next lemma.
\[lemma:focuscat\] Let $G$ be a caterpillar with spine $W= \{ w_1, w_2, \ldots , w_t \}$. Then
1. The vertex $w_r$ is the unique focus of $G$ if and only if $\L(w_r)=\R(w_r)$.
2. The foci of $G$ are $w_{r-1}, w_r$ if and only if $\L(w_r)>\R(w_r)$ and $\R(w_{r-1})>\L(w_{r-1}).$
3. If $G$ has foci $w_{r-1}, w_r$ then vertex $w_r$ is a best mix focus if and only if $\L(w_{r-1})\leq \R(w_{r}).$
4. If $L(w_r) > R(w_r)$ and $\L(w_{r-1})\leq \R(w_{r})$ then $w_r$ is a best mix focus and $w_{r-1}$ is also a focus of $G$.
Parts (a) and (b) are a direct consequence of Definition \[def:focus\]. Part (c) is restatement of Theorem \[thm:tbmvertex\] for caterpillars. For part (d), we observe that $\L(w_{r-1})\leq \R(w_r) < \R(w_{r-1})$, so that $w_{r-1}$ and $w_r$ satisfy (b) and (c).
When $U_i \neq 0$, we define the *leaf transplant* ${\tau(i,j)}$ as the relocation of $x \in U_i$ so that this leaf is now adjacent to $w_j$ where $1 \leq j \leq t$. Usually, we have $2 \leq j \leq t-1$, so that the leaf $x$ becomes an element of $U_j$. Our most common operation will be an *elementary leaf transplant* where $j=i \pm 1$. Occasionally, we will take $j \in \{1, t\}$, so that the leaf transplant actually increases the length of the spine. Finally, we need one special transplant that reduces the length of the spine by relocating the leaf $w_t$ to become adjacent to the best mix focus $w_r$: we use ${\sigma(t,r)}$ to denote this rare spinal leaf transplant. The more general term *tree surgery* denotes either a single leaf transplant (either ${\tau(i,j)}$ or ${\sigma(t,r)}$), or a pair of leaf transplants ${\tau(i_1,j_1)} \wedge {\tau(i_2,j_2)}$ performed simultaneously. We use $\S(G)$ to denote the caterpillar that results from applying tree surgery $\S$ to caterpillar $G$. Our algorithm uses eleven different tree surgeries. They are enumerated in Tables \[table:phase1\], \[table:phase2\] and \[table:phase3\] below.
The remainder of this section is devoted to proving some fundamental results about the effects of tree surgeries on the best mixing time. Let $G=(V,E)$ be a caterpillar and let $\nG=\S(G) = (V, \nE)$ be the caterpillar obtained by applying tree surgery $\S$, with stationary distribution $\npi = \pi_{\nG}$. We introduce notation for the comparison of various caterpillar measurements. Just as with the pessimal notation introduced in Section \[sec:pessimal\], we are motivated to shield ourselves from the particular locations of important vertices (like foci and pessimal vertices) which may be different in $G$ and $\nG$.
As with equation , we introduce $\Delta$-notation to compactly represent the changes in various quantities due to a tree surgery. For vertices $v, v_1, v_2 \in V$, we define: $$\begin{array}{ccc}
\begin{array}{rcl}
\Delta \deg(v) &=& \deg_{\nG}(v) - \deg_G(v), \\
\Delta \pi(v) &=& \npi(v) - \pi(v) , \\
\Delta H(v_1, v_2) &=& H_{\nG}(v_1, v_2) - H_G(v_1, v_2) , \\
\Delta H(\pi, v) &=& H_{\nG}(\npi, v) - H_G(\pi, v), \\
\end{array}
&\quad&
\begin{array}{rcl}
\Delta \hp(v) &=& \hp_{\nG}(v) - \hp_G(v), \\
\Delta \L(v) &=& \L_{\nG}(v) - \L_G(v), \\
\Delta \R(v) &=& \R_{\nG}(v) - \R_G(v), \\
\Delta \Tbest &=& \Tbest(\nG) - \Tbest(G). \\
\end{array}
\end{array}$$
Let $G$ be a caterpillar and let $\nG=\S(G)$ be the result of a leaf transplant ${\tau(i,j)}$. We give formulas for $\Delta H(v,w_r)$ for use in later sections. We only consider leaf transplants ${\tau(i,j)}$ where either $1 \leq i,j \leq r$ or where $r+1 \leq i,j \leq t$ (so that we never transplant leaves from the left spine to the right spine, or vice versa). The formulas below cover two qualitatively different cases for ${\tau(i,j)}$. When $2 \leq j \leq t-1$, the spine length is unaffected. However, when $j \in \{ 1, t \}$, the spine length increases. The formulas below still hold, though the arguments are slightly different. All four formulas below follow quickly from equation , and we leave these short calculations to the reader. Figure \[fig:transplants\] shows the effect of two example transplants on hitting times. The arguments for $j=i \pm 1$ are straight forward, and the other cases follow inductively.
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Suppose that the leaf transplant ${\tau(i,j)}$ moves $x$ from $U_i$ to $U_j$. For $v \in V_k$, the value of $\Delta H (v,w_r)$ depends on the relative locations of $i,j$ and $k$: $$\begin{aligned}
\label{eq:leftleft2}
\mbox{If } 1\leq j<i\leq r \mbox{ then } &
\Delta H(v,w_r)=
\left\{
\begin{array}{cl}
H_G(w_j,w_i)+2(i-j) & \quad v=x,\\
2(i-j) & \quad k\leq j,\\
2(i-k) & \quad j<k<i,\\
0 & \quad k \leq i \leq r \mbox{ and } v \neq x.
\end{array}
\right.
\\
\label{eq:leftright2}
\mbox{If } 1\leq i<j\leq r \mbox{ then } &
\Delta H(v,w_r)=
\left\{
\begin{array}{cl}
-H_G(w_i,w_j) & \quad v=x,\\
-2(j-i) & \quad k\leq i \mbox{ and } v \neq x,\\
-2(j-k) & \quad i<k<j,\\
0 & \quad j \leq k \leq r.
\end{array}
\right.
\\
\label{eq:rightleft2}
\mbox{If } r \leq j<i\leq t \mbox{ then } &
\Delta H(v,w_r)=
\left\{
\begin{array}{cl}
-H_G(w_i,w_j) & \quad v=x,\\
-2(i-j) & \quad i \leq k \mbox{ and } v \neq x,\\
-2(i-k) & \quad j < k < i,\\
0 & \quad r < k \leq j.
\end{array}
\right.
\\
\label{eq:rightright2}
\mbox{If } r \leq i<j\leq t \mbox{ then } &
\Delta H(v,w_r)=
\left\{
\begin{array}{cl}
H_G(w_j,w_i)+2 & \quad v=x,\\
2(j-i) & \quad j \leq k,\\
2(j-k) & \quad i < k < j,\\
0 & \quad r < k \leq i \mbox{ and } v \neq x.
\end{array}
\right.\end{aligned}$$ We get similar equations for hitting times to the second focus $w_{r-1}$, with the appropriate change to the bounds on the indices $i,j$. This completes our list of useful hitting time changes due to common leaf transplants.
Next, we turn our attention to tracking the change in the best mixing time after a tree surgery. The lemmas that follow will be used to analyze all eleven surgeries used in our algorithm. If $w_r$ is the best mix focus of $G$ and $w_s$ is the best mix focus of $\nG$, then $$\label{eq:tbmchange}
\Delta \tbm = \hp_{\nG}(w_s) - \hp_G(w_r) - \big( H_{\nG}(\npi, w_s) - H_G(\pi, w_r) \big).$$ As we alter our caterpillar, we strive to maintain $w_r$ as the best mix focus and the leftmost spinal vertex as the $w_r$-pessimal vertex. In this case, equation simplifies to $
\Delta \tbm = \Delta \hp(w_r) - \Delta H(\pi, w_r) = \Delta L(w_r) - \Delta H(\pi, w_r).
$ This can be rephrased as the simple criterion: $$\label{eq:tbmchange2}
\Delta \tbm \geq 0 \quad \Longleftrightarrow \quad \pdelt(w_r) \geq \Delta H(\pi,w_r).$$
The next two results track the effect of a surgery on the location of the foci.
\[lem:criteria\] Let $G$ be a caterpillar. Let $\S$ be a tree surgery and let $\nG=\S(G)$. If $$\begin{aligned}
\label{eq:lchange}
L_G(w_r)-R_G(w_r) &>& \Delta\R(w_r)-\Delta\L(w_r) \\
\mbox{and} \quad
\label{eq:newtbmfoc}
R_G(w_r)-L_G(w_{r-1}) &\geq& \Delta\L(w_{r-1})-\Delta\R(w_r),\end{aligned}$$ then $\nG=\S(G)$ has best mix focus $w_{r}$ and $w_{r-1}$ is also a focus.
These criteria are equivalent to the conditions of Lemma \[lemma:focuscat\] (d) for $\nG$.
\[cor:criteria\] Let $G$ be a bifocal caterpillar with focus $w_{r-1}$ and best mix focus $w_r$. Let $\S$ be a tree surgery and let $\nG=\S(G)$. If $\Delta\R(w_r)-\Delta\L(w_r) \leq 0$ and $\Delta\L(w_{r-1})-\Delta\R(w_r)\leq 0$ then $\nG=\S(G)$ also has focus $w_{r-1}$ and best mix focus $w_r$.
This follows directly from Lemma \[lem:criteria\] and the inequalities $L(w_r)-R(w_r)\geq 1$ and $R(w_r)-L(w_{r-1})\geq 0$.
Lemma \[lem:criteria\] and Corollary \[cor:criteria\] are key tools of our methodology. We use them repeatedly to verify the foci and best mix focus of our caterpillar after a surgery has been applied. Once we know that $w_r$ is still the best mix focus, we then verify that equation holds. Next, we give a final test for $\Delta \Tbest \geq 0$. This lemma will be used very frequently in subsequent sections.
\[lem:slack\] Let $G$ be a caterpillar and let $\S$ be a tree surgery such that $w_r$ is a best mix focus of $G$ and $\nG$. Let $A=\{v\in V\mid \Delta H(v,w_r)>\pdelt(w_r)\}$ and let $C = \{ v \in \overline{A} \mid \Delta(v) > 0 \}$. If there exists $C \subset B \subset \overline{A}$ for which $$\begin{aligned}
\label{eqn:slack}
&\sum_{v\in A}{ \Big(\deg_{\nG}(v)(\Delta H(v,w_r)-\pdelt(w_r))+\Delta\deg_G(v)H_G(v,w_r) \Big)} \nonumber \\
\leq&\sum_{u\in B}{ \Big( \deg_{\nG}(u)(\pdelt(w_r)-\Delta H(u,w_r))-\Delta\deg_G(u)H_G(u,w_r) \Big)}
\end{aligned}$$ then $\Delta \tbm \geq 0$.
Before proving this lemma, we make a few comments its use in later sections. First, the set $A$ will always be small: it will only contain one or both of the leaves moved during the surgery. Second, if the lemma holds for $B$, then it holds for any superset of $B$, including $\overline{A}$. However, there is no need to calculate the right hand side for $\overline{A}$ when a small subset will do. For our earlier surgeries, the set $B$ will be small, typically consisting of a handful of spine vertices near the transplant locations. In later surgeries, $B$ will consist of half or all of the spine.
We decompose $\Delta H(\pi,w_r)$ as follows: $$\begin{aligned}
\Delta H(\pi,w_r) &=\sum_{v\in V}{\big(\npi(v)H_{\nG}(v,w_r)-\pi(v)H_G(v,w_r)\big)}\nonumber \\
&=\sum_{v\in V}\big(\npi(v)(\Delta H(v,w_r) + H_G(v,w_r))-\pi(v)H_G(v,w_r)\big)\nonumber \\
&=\sum_{v\in V}{\big(\npi(v)\Delta H(v,w_r)+\Delta\pi(v)H_G(v,w_r)\big)}. \nonumber
\end{aligned}$$ Let $g(v) = \deg_{\nG}(v)(\pdelt(w_r) - \Delta H(v,w_r)) - \Delta\deg_G(v)H_G(v,w_r)$. We can rewrite $$\begin{aligned}
\pdelt(w_r)-\Delta H(\pi,w_r)
& = \pdelt(w_r)-\sum_{v\in V}{\left(\npi(v)\Delta H(v,w_r)+\Delta\pi(v)H_G(v,w_r)\right)}\\
&= \frac{1}{2|E|} \left( \sum_{v\in A} g(v) + \sum_{u\in \overline{A}} g(u) \right) \,
\geq \,\frac{1}{2|E|} \left( \sum_{v\in A} g(v) + \sum_{u\in B} g(u) \right).
\end{aligned}$$ The final inequality holds because $g(u) \geq 0$ for every $u \in \overline{A} \backslash B \subset \overline{A} \backslash C$. Therefore, if $ - \sum_{v\in A} g(v) \leq \sum_{u\in B} g(u)$ then $ \pdelt(w_r)-\Delta H(\pi,w_r) \geq 0$.
We have an immediate corollary for the special case where $A$ is empty.
\[cor:slack\] Given a tree $G$ and any tree surgery $\S$ such that $w_r$ is the best mix focus of $G$ and $\nG$, if $
\Delta H(v,w_r)\leq\pdelt(w_r),
$ for all $v\in V$, then $\Delta \tbm \geq 0$.
Finally, we note that if the $w_r$-pessimal vertices in $G$ and $\nG$ are the leftmost spinal vertices in these graphs, then we can replace $\pdelt(w_r)$ with $\Delta\L(w_r)$ in Lemma \[lem:slack\] and Corollary \[cor:slack\].
Proof of the upper bound in Theorem \[thm:bestmix\] {#sec:proof}
===================================================
In Section \[sec:bestmix-extreme\], we proved the lower bound of Theorem \[thm:bestmix\], and calculated the best mix time for the even path and the odd wishbone. In this section, we prove the upper bound of Theorem \[thm:bestmix\], leaving the details to the lemmas in the accompanying subsections. Two examples of the algorithm in practice are shown in Figure \[fig:examples\] above.
[Theorem \[thm:bestmix\]]{} Given a tree $G$ that is not an even path or an odd wishbone, we must show that it does not maximize the best mixing time. We may assume that $G$ is not a star since that is clearly not the maximizing structure. Starting from our tree $G$, we let $G_0$ be the caterpillar constructed by Lemma \[lem:cat\], so that $\Tbest(G) \leq \Tbest(G_0)$. Next, we apply a sequence of tree surgeries to produce a sequence of caterpillars $G_0,G_1, G_2, \ldots, G_m$ such that $\Tbest(G_{i-1}) \leq \Tbest(G_i)$, where the final caterpillar $G_m$ is either $P_n$ or $Y_n$. This occurs in an algorithmic manner, divided into three phases.
at (0,-2.5) [(c)]{};
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(1.2,-1) circle (3pt); (.8,-1) circle (3pt);
In order to define these phases, we define two special subfamilies of caterpillars which mark milestones in our graph sequence. Examples of these graphs and a wishbone are shown in Figure \[fig:menagerie\]. A *twin broom* is a bifocal caterpillar whose only non-spine leaves are adjacent to the foci $w_{r-1}, w_r$. In other words, $V= W \cup U_{r-1} \cup U_r$. A *seesaw* is a twin broom that has at most one additional leaf on each side of the spine. In other words, a seesaw satisfies $| \cup_{i=2}^{r-2} U_i| \leq 1$ and $| \cup_{i=r+1}^{t-1} U_i| \leq 1$. Figure \[fig:examples\] includes examples of each. In that figure, the output of $S_4$ and the initial tree on 11 vertices are both seesaws. Meanwhile, the outputs of $S_6$ and $S_8$ are both twin brooms (as are all graphs that follow them).
We start with a caterpillar $G$ on $n$ vertices. In Phase One, we convert the caterpillar into a seesaw. Phase Two converts the seesaw into a twin broom. Phase Three converts the twin broom into one of $P_{n}$ (when $n$ is even) or $Y_n$ (when $n$ is odd). Lemmas \[lemma:cat-to-seesaw\], \[lemma:seesaw-to-twinbroom\], and \[lemma:twinbroom-to-end\] below show that the best mixing time monotonically increases in every phase of this process. This proves Theorem \[thm:bestmix\].
Our caterpillar transformation consists of incremental steps that move one or two leaves at a time. This allows us to monitor the delicate balance maintained by the best mix focus. In particular, there may be a critical step at which we change the focus that attains the best mixing time. This happens in one of two ways. Usually, we make a small change that keeps the current best mix focus, but also causes a neighbor to also become a best mix focus. The other change is more abrupt: Surgery $S_5$ below changes the location of the unique best mix focus to a neighbor of the current one. In $S_5$, the prescribed caterpillar structure during that surgery makes this crucial transition manageable. The remainder of this paper is devoted to proving Lemmas \[lemma:cat-to-seesaw\], \[lemma:seesaw-to-twinbroom\], and \[lemma:twinbroom-to-end\].
Phase One: Caterpillar to Seesaw
--------------------------------
In this subsection, we prove that Phase One is successful: we can transform any caterpillar into a seesaw while also increasing the best mixing time. Let $G$ be a caterpillar with spine $W=\{w_1, w_2, \ldots, w_t \}$ where $w_r$ is the best mix focus and ${w_r'}=w_1$.
\[lemma:cat-to-seesaw\] Let $G$ be a caterpillar $G$ on $n$ vertices. Phase One creates a seesaw $\nG$ such that $\tbm(\nG)\geq\tbm(G)$.
If $G$ is already a seesaw then $\nG=G$. Table \[table:phase1\] shows the five surgery types employed during Phase One. We defer the proofs that $\Delta \Tbest \geq 0$ for each of these surgeries to the lemmas that follow.
Figure \[fig:phase1\] shows the workflow for Phase One. First, if the caterpillar has a unique focus then we use $\S_1$ to create a caterpillar with two foci. From here forward, the caterpillar will remain bifocal. Let $w_{r-1}, w_r$ be the foci of a bifocal caterpillar with $w_r$ achieving $\Tbest$. A leaf $x \in V \backslash W$ is *good* when $x \in U_{2} \cup U_{r-1} \cup U_{r} \cup U_{t-1}$. All other leaves in $V \backslash W$ are *bad*. First, we repeatedly use $\S_2$ to move pairs of bad leaves on the same side of the spine (one towards the end and the other towards the center). This loop terminates whether there is at most one bad leaf on each side of the spine. At this point, we use $S_3$ to extend the left spine and to transplant a leaf to $U_r$. This requires there are at least two left leaves (which can only happen when at least one is in $U_2$ since we are done with $\S_2$). After extending the spine, the previously good leaves at $U_2$ become bad. This throws us back into the $\S_2$ loop.
We exit the $S_3$ loop when there is at most one left leaf. At this point, we deal with the right leaves. When $L(w_r) > R(w_r) + 1$, we apply $S_4$, the right-hand surgery analogous to $S_3$. However, if $L(w_r)= R(w_r)+1$ then applying $S_4$ would create a focal caterpillar, which we choose to avoid. Instead, we apply $S_5$, which transplants a single right leaf to the end of the right spine. Applying either $S_4$ or $S_5$ might create bad right leaves, which puts us back into the $S_2$ loop. Surgeries $S_3, S_4, S_5$ reduce the number of non-spinal vertices, so Phase One must terminate. Ultimately, we create a caterpillar with at most one left leaf and at most one right leaf, while $U_{r-1} \cup U_{r}$ may contain many leaves. This is a seesaw graph, as desired.
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Surgery & Initial Conditions & Illustration\
$\S_1$ & $G$ is focal, so $L(w_r) = R(w_r)$. The transplant depends on whether $U_{t-1} = \emptyset$. &
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$\S_2$ & $G$ has two (or more) bad left leaves or has two (or more) bad right leaves; $R(w_r) \geq L(w_{r-1})$. &
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$\S_3$ & $G$ has $x \in U_2$ and another left leaf $y$; $R(w_r) > L(w_{r-1})$ &
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$\S_4$ & $G$ has $y\in U_{t-1}$ and another right leaf $x$; $R(w_r) \geq L(w_{r-1})$ and $L(w_r)>R(w_r)+1$ &
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$\S_5$ & $G$ has $y\in U_{t-1}$ and another right leaf $x$; $R(w_r) \geq L(w_{r-1})$ and $L(w_r)=R(w_r)+1$ &
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![Phase One of the algorithm, turning a caterpillar into a seesaw.[]{data-label="fig:phase1"}](flowchart_phase_I){width="5in"}
Next, we prove that surgeries $S_1, S_2, S_3, S_4, S_5$ each result in $\Delta \Tbest \geq 0$. If we start with a focal caterpillar $G$, we use surgery $\S_1$ to create a bifocal caterpillar $\nG = \S_1(G)$. Depending on the structure of $G$, we use one of two leaf transplants: ${\tau({t-1},r)}$ or ${\sigma(t,r)}$. We note that ${\sigma(t,r)}$ is the only transplant that removes a leaf from the spine.
\[lem:S1\] Let $G$ be a caterpillar with a single focus $w_r$ and with the spine indexed such that $d(w_1,w_r)\geq d(w_t,w_r)$. If $U_{t-1} \neq \emptyset$, then let $S_1 = {\tau({t-1},r)}$. If $U_{t-1} = \emptyset$ then let $\S_1 = {\sigma(t,r)}$. The caterpillar $\nG = \S_1(G)$ is bifocal and $\Delta \tbm > 0$.
We use Lemma \[lem:criteria\] for our proof. The spine is indexed so that $r-1 \geq t-r$ and the unique focus means that $L_G(w_r) = R_G(w_r).$ For both surgeries under consideration, $\Delta R(w_r) < 0$ and $\Delta L(w_r) =0$, so equation is satisfied. Next, we observe that $R(w_r) - L(w_{r-1}) = L(w_r) - L(w_{r-1}) = H(w_{r-1}, w_r) = \sum_{v \in V_{w_{r-1}:w_r}} \deg(v) \geq 2(r-1)-1$ by equation , so it suffices to show that $\Delta L(w_{r-1}) - \Delta R(w_r) = - \Delta R(w_r) \leq 2(r-1) -1$ to verify equation . There are two cases. First, suppose that $U_{t-1} \neq \emptyset$, so that $\S_1={\tau(t-1,r)}$ does not alter the spine. By equation , we have $ -\Delta R(w_r) = 2(t-1-r) < 2(r-1)-1.$ Next, suppose that $U_{t-1} = \emptyset$, so that $\S_1={\sigma(t,r)}$ does alter the spine. We have $$\begin{aligned}
-\Delta R(w_r) &=& -H_G(w_t,w_r) + H_{\nG}(w_{t-1},w_r) \,= \, -1 - \Delta H(w_{t-1},w_r) \\
&=& -1+2(t-1-r) \, = \, 2(t-r)-3 \, < \, 2(r-1)-1.\end{aligned}$$ In either case, the conditions of Lemma \[lem:criteria\] are satisfied, so the foci do not change. The result now follows from Corollary \[cor:slack\] since $\Delta L(w_r)=0 \geq \Delta H(v,w_r)$ for all $v \in V$.
Next, we discuss surgery $\S_2$ which transplants a pair of bad left-hand leaves, moving one toward $U_2$ and one toward $U_{r-1}$, as shown in Table \[table:phase2\]. We repeat $\S_2$ until the left side of the caterpillar has at most one bad leaf.
\[lem:S2\] Let $G$ be a bifocal caterpillar with leaves $x\in U_i$ and $y\in U_j$ where either $2 < i \leq j < r-1$ or $r<i\leq j <t-1$. If $\S_2={\tau(i,i-1)}\wedge{\tau(j,j+1)}$, then $\Delta \tbm > 0$.
We consider the left-hand spine case $2<i\leq j<r-1$. The right-hand spine proof is analogous, switching the roles of $i$ and $j$. Suppose that there are bad leaves $x \in U_i$ and $y \in U_j$. First, we observe that $\Delta \L (w_r) = 0 = \Delta \L(w_{r-1})$: these net hitting time changes are $2-2=0$ as per equations and . The right hand spine is unaffected, so $\Delta \R (w_r) = 0 = \Delta \R(w_{r-1})$. Corollary \[cor:criteria\] guarantees that $\nG$ also has foci $w_{r-1}$ and best mix focus $w_r$.
Having established that the foci do not change, we show that $\tbm$ increases. We must show that $
\Delta\L(w_r)=\Delta H(w_1,w_r)>\Delta H(\pi,w_r).
$ We only argue the case $i<j$, as the case $i=j$ is a straight-forward adaptation. Equations and give $$\begin{array}{rclcrcll}
\Delta H(x,w_r) &=& H_G(w_{i-1},w_i), & &
\Delta H(v,w_r)&=& -2 &\mbox{ for } v \in (V_i \cup \cdots \cup V_j) \backslash \{x, y \}, \\
\Delta H(y,v_r) &=& 2-H_G(w_j,w_{j+1}), &&
\Delta H(v,w_r) &=&0 & \mbox{ for all other } v.
\end{array}$$ The only degree changes are $\Delta \deg(w_i) = -1 = \Delta \deg(w_j)$ and $\Delta \deg(w_{i-1}) = 1 = \Delta \deg(w_{j+1})$. (When $i=j$, the only negative degree change is $\Delta \deg(w_i)=-2$.) We use Lemma \[lem:slack\] with $A= \{x \}$ to show that $\tbm$ does not decrease. The left side of inequality simplifies to $\Delta H(x,w_r) = H_G(w_{i-1},w_i)$. It remains to show that this value is a lower bound for the right side of . Taking $B= \{w_{i-1}, w_i, w_j, w_{j+1}, y \}$, we obtain $$\begin{aligned}
\MoveEqLeft -H_G(w_{i-1},w_r) + \left(2\deg_{\nG}(w_i)+H_G(w_i,w_r) \right)+ \left(2\deg_{\nG}(w_j)+H_G(w_j,w_r)\right)\\
&\qquad -H_G(w_{j+1},w_r)-(2-H_G(w_j,w_{j+1}))\\
&= 2(\deg_G(w_i)+\deg_G(w_j)-2)-2+H_G(w_i,w_r) -H_G(w_{i-1},w_r) \\
&\qquad +H_G(w_j,w_r) -H_G(w_{j+1},w_r)+H_G(w_j,w_{j+1})\\
&> 2H_G(w_j,w_{j+1})-H_G(w_{i-1},w_i)
>H_G(w_{i-1},w_i),\end{aligned}$$ where the last inequality uses equation twice to justify $H_G(w_{i-1},w_i)<H_G(w_j,w_{j+1})$. Thus the condition of Lemma \[lem:slack\] is satisfied, so $\Delta \tbm > 0$.
Once there is at most one bad left leaf, we increase the spine length, starting with the left side. Surgery $\S_3$ requires at least two left leaves, one of which must be in $U_2$. We also require $R(w_r)>L(w_{r-1})$, meaning that $w_r$ is the unique best mix focus. If this is not the case (that is, $R(w_r) = L(w_{r-1})$), then we will take $w_{r-1}$ to be the best mix focus, reverse the labeling of the spine, and then apply $\S_4$ below.
\[lem:S3\] Let $G$ be a bifocal caterpillar with $R(w_r) > L(w_{r-1})$ and with distinct vertices $x\in U_2$ and $y \in U_i$ where $2\leq i\leq r-2$. If $\S_3={\tau(2,1)}\wedge{\tau(i,i+1)}$, then $\Delta \tbm \geq 0$.
First, we use Lemma \[lem:criteria\] to show that the foci do not change. This surgery extends the left-hand side of the spine. The left-pessimal hitting times to $w_{r-1}$ and $w_r$ increase by $\Delta L(w_r) = \Delta L(w_{r-1}) = 3-2=1$ (using equation for the effect of ${\tau(2,1)}$ and equation for ${\tau(i,i+1)}$). The right-pessimal hitting times do not change, $\Delta R(w_r) = \Delta(w_{r-1})=0$. Therefore, equation is satisfied. By assumption, $R( w_r) - L(w_{r-1}) \geq 1 = \Delta\L(w_{r-1})-\Delta\R(w_r)$ so inequality holds. We have, $\Delta H(v,w_r)\leq\Delta\L(w_r)$ for all $v\in V$ holds for all $v \in V$ since $H(x,w_r)$ is the only hitting time to $w_r$ that increases; all others decrease or are constant. By Corollary \[cor:slack\], we have $\Delta \tbm \geq 0$.
Surgery $\S_4$ is the right-hand version of $\S_3$. However, if $
\L(w_r)=\R(w_r)+1
$ then applying $\S_4$ would lead to $
\L_{\nG}(w_r)=\R_{\nG}(w_r)
$ which indicates that $\nG$ has one focus by Lemma \[lemma:focuscat\](a). We choose to avoid this situation, So we require that $\L(w_r) > \R(w_r)+1$ and handle the case $\L(w_r)=\R(w_r)+1$ with $S_5$ below.
\[lem:S4\] Let $G$ be a bifocal caterpillar with $
\L(w_r)>\R(w_r)+1
$ that contains distinct vertices $y \in U_{t-1}$ and $x \in U_i$ where $r < i < t$. If $\S_4={\tau(t-1,t-2)}\wedge{\tau(i,i-1)}$, then $\Delta \tbm > 0$.
This surgery extends the spine on the right-hand side. Analgous to the previous proof, this time the left-pessimal hitting times have $\Delta L(w_{r-1})= 0 = \Delta L(w_r)$ and the right-pessimal hitting times have $\Delta R(w_{r-1}) = 1 = \Delta R(w_r)$. Inequality holds because $L(w_r) - R(w_r) > 1 = \Delta\R(w_r)-\Delta\L(w_r),$ and inequality holds because $\Delta\L(w_{r-1})-\Delta\R(w_r)<0$. By Lemma \[lem:criteria\], $\nG$ has focus $w_{r-1}$ and best mix focus $w_r$.
Next, we show that $\Delta \Tbest > 0$ using Lemma \[lem:slack\] with $A=\{ y \}$. We have $\Delta L(w_r) < \Delta H(y, w_r) = \Delta R(w_r)$, while $\Delta H(v,w_r) \leq 0 = \Delta L(w_r)$ for $v \neq y$. The left hand side of inequality equals $\Delta H(y,w_r) = 1$. We take $ B= \{ w_{i-1}, w_i, w_{t-1}, w_t \}$ because the contribution from each vertex in $\overline{A} \backslash B$ is positive. Using equation , this sum is $$\begin{aligned}
\MoveEqLeft
-H_G(w_{i-1},w_r) + \left( 2\deg_{\nG}(w_i)+H_G(w_i,w_{r}) \right)
+ \left(2\deg_{\nG}(w_{t-1})+H_G(w_{t-1},w_r) \right) \\
&\qquad
+ \left(-2\deg_{\nG}(w_t) -H_G(w_t,w_r) \right)\\
&=
H_G(w_i,w_{i-1}) -H_G(w_t, w_{t-1})
+ 2 \left( \deg_{\nG}(w_i) + \deg_{\nG}(w_{t-1}) -\deg_{\nG}(w_t) \right)
\\
& > H_G(w_i,w_{i-1}) > 1.\end{aligned}$$ By Lemma \[lem:slack\], $\Delta \Tbest > 0$.
We now consider surgery $\S_5$, which is only used when $\L(w_r)=\R(w_r)+1$. This is one of the crucial moments in our algorithm: a minor change threatens the balance described in Lemma \[lemma:focuscat\]. In fact, surgery $\S_5={\tau(t-1,t)}$ is the first surgery that shifts the location of the foci. The resulting graph is bifocal with best mix focus $w_r$, but the second focus moves from $w_{r-1}$ to $w_{r+1}$.
\[lem:S5\] Let $G$ be a bifocal caterpillar with $y\in U_{t-1}$ such that $
L(w_r)= R(w_r)+1.
$ If $\S_5={\tau(t-1,t)}$, then $\Delta \tbm > 0$.
We note that during Phase One, we will also have a second right leaf (otherwise we are done with the right spine). However, our proof does not require the existence of such a leaf.
Let $\nG = \S_5(G)$ and let $y \in V_{t-1}$ be the transplanted vertex. The changes in the left and right hitting times to the foci are $\Delta\L(w_{r-1})=0= \Delta\L(w_{r})$ and $\Delta\R(w_{r-1})= 3 = \Delta\R(w_r).$ Crucially, inequality is not satisfied. We now verify that $w_r$ is the best mix focus of $\nG$ and that $w_{r+1}$ is the other focus. We use Lemma \[lemma:focuscat\] (d), replacing $r$ with $r+1$ and swapping left for right. In other words we must show that $R(w_r) > L(w_r)$ and $R(w_{r+1}) \leq L(w_r)$. Observe that $
\R_{\nG}(w_r)
=\R_G(w_r)+3
>\L_G(w_r)
=\L_{\nG}(w_r),
$ and $$\R_{\nG}(w_{r+1})
=\R_G(w_{r+1})+3 < \R_G(w_{r+1}) + H_{\nG} ( w_{r+1}, w_r)
= \R_G(w_r)
< \L_G(w_r)
=\L_{\nG}(w_r).$$ Next, we show that $\tbm(\nG)>\tbm(G)$ using Lemma \[lem:slack\]. Note that the surgery shifts the foci, so $w_1$ is the $G$-pessimal vertex for $w_r$, while $y$ is the $G'$-pessimal vertex for $w_r$. We find that $\pdelt(w_r)=2$ because $$\hp_{\nG}(w_r) = H_{\nG}(y,w_r) = 3+H_G(w_t,w_r) =2+H_G(w_1,w_r) = 2 + \hp_G(w_r).$$ We will use Lemma \[lem:slack\] with $A=\{y\}$. The left side of equation is $
\deg_{\nG}(y)(\Delta H(y,w_r)-2)=3-2=1.
$ We take $B = \{y, w_{t-1},w_t\}$ since $v \notin B$ means that $\Delta\deg(v)=0$ and $ \Delta H(v,w_r)\leq 2=\pdelt(w_r).$ The right hand side of equation is $$\left( 2\deg_{\nG}(w_{t-1})+H_G(w_{t-1},w_r) \right) -H_G(w_t,w_r)
= 2\deg_{\nG}(w_{t-1})-H_G(w_t,w_{t-1})
> 1,$$ so by Lemma \[lem:slack\], $\Delta \tbm > 0$.
Phase Two: Seesaw to Twin Broom
-------------------------------
In this section, we discuss Phase Two, which inputs a seesaw and outputs a twin broom.
[|c|p[2.75in]{}|@c@|]{}
Surgery & Initial Conditions & Illustration\
$\S_6$ & $G$ has exactly one left leaf and $2r-1 \leq t$ &
[c]{}
(2, .5) circle (1pt);
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$\S_7$ & $2r-2 \geq t$; $G$ has exactly one left leaf and one right leaf &
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(4,0) – (4,-1); (4,-1) circle (4pt);
(2,0) – (11,0);
(2,0) – (2.5,0); (3.5,0) – (4.5,0); (5.5,0) – (7.5,0); (8.5,0) – (9.5,0); (10.5,0) – (11,0);
iin [2,4,9,11]{} [ (i,0) circle (4pt); ]{}
(6,0) circle (4pt);
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(9.25, -1.1) to (11.8,-.25);
at (4.35,-1.25) [$x$]{}; at (8.65,-1.25) [$y$]{};
\
$\S_8$ & $G$ has exactly one right leaf and no left leaves &
[c]{}
(2, .5) circle (1pt);
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(0,0) – (7,0);
(0,0) – (.5,0); (1.5,0) – (3.5,0); (4.5,0) – (5.5,0); (6.5,0) – (7,0);
iin [0,5,7]{} [ (i,0) circle (4pt); ]{}
(5,0) – (5,-1); (5,-1) circle (4pt);
at (5.35,-1.25) [$x$]{};
(4.75, -1.25) to (3.25,-1.25);
(2,0) circle (4pt);
(3,0) circle (6pt); (3,0) circle (4pt);
\
![Phase Two of the algorithm, turning a seesaw into a twin broom.[]{data-label="fig:phase2"}](flowchart_phase_II){width="4.5in"}
\[lemma:seesaw-to-twinbroom\] Let $G$ be a seesaw on $n$ vertices. Phase Two creates a twin broom $\nG$ such that $\tbm(\nG)\geq\tbm(G)$.
If $G$ is already a twin broom then $\nG=G$. The three surgery types $\S_6, \S_7, \S_8$ employed in Phase Two are shown in Table \[table:phase2\] and the workflow is shown in Figure \[fig:phase2\]. The lemmas that follow show that $\Delta \Tbest \geq 0$ for all three surgeries.
Suppose that $G$ has a left leaf. If $2r-1 \leq t$ then we use $\S_6$ to transplant the left leaf onto the end of the left spine. If $2r-2 \geq t$, or equivalently $r-2 \geq t-r$, then there must also be a right leaf because $w_r$ is the best mix focus (see Lemma \[lemma:focuscat\](c)). In this case, we use $S_7$ to simultaneously transplant both of these leaves to the spine. Next, if $G$ still has a right leaf (perhaps we just applied $\S_6$), then we apply $\S_8$ to move this leaf to the end of the right spine. The resulting graph is a twin broom with best mix focus $w_r$ and second focus $w_{r-1}, w_r$.
First, we prove that $\S_6$ moves the final left leaf to the spine while also increasing $\Tbest$.
\[lem:S6\] Let $G$ be a seesaw with $2r-1 \leq t$ and exactly one left leaf $x\in U_i$, where $2 \leq i \leq r-2$. If $\S_6={\tau(i,1)}$ then $\Delta \tbm > 0$.
The surgery extends the left-hand spine, so $L(w_{r-1}) = H_{\nG} (x,w_{r-1}) = H_{\nG} (x,w_1) + H_{\nG} (w_1, w_{r-1}) = 1 + H_G(w_1 + w_{r-1}) + 2(i-1)$ by equation . Therefore $\Delta\L(w_{r-1})=2i-1$, and likewise $\Delta\L(w_r)=2i-1,$ while $\Delta\R(w_{r-1})=0=\Delta\R(w_r).$
We show that the foci have not changed using Lemma \[lemma:focuscat\] (d). We have $\L_{\nG}(w_r) > \L_G(w_r) \geq R_G(w_r) = R_{\nG}(w_r)$. Equation and the assumption $2r-1 \leq t$ yield $
L_{\nG}(w_{r-1}) = (r-1)^2 \leq (t-r)^2 \leq R_{\nG}(w_r).
$ Therefore, $\nG$ has focus $w_{r-1}$ and best mix focus $w_r$.
Next, we use Lemma \[lem:slack\] with $A=\{x\}$. The left-hand side of inequality is $$\Delta H(x,w_r)-\Delta\L(w_r) =(i^2-1)-(2i-1)=i^2-2i.$$ Let $B=\{w_{r-1},w_r,\dots,w_{t-1}\}$. For all $v \in B$, we have $\deg_{\nG}(v) \geq 2$ and $\Delta H(v,w_r)=0=\Delta\deg(v)$. The right-hand side of inequality is $$\sum_{v\in B}{2(2i-1)}
=2(2i-1)(t-r)
\geq 2(2i-1)(r-2)
> i(i-2 )=i^2-2i.$$ Lemma \[lem:slack\] gives $\Delta \tbm > 0$.
We employ surgergy $S_7$ is the particular case where $2r-2 \geq t$ and $G$ has both a left leaf and a right leaf. This surgery removes both leaves simultaneously.
Let $G$ be a seesaw with exactly one left leaf $x\in U_i$ and exactly one right leaf $y\in U_j$, such that $2r-2 \geq t$. If $\S_7={\tau(i,1)}\wedge{\tau(j,t)}$, then $\Delta \tbm > 0$.
First, we show that in fact, $2r-2 = t$, meaning that the left and right spines are the same length. If $r-2 > t-r$ then $L(w_{r-1}) = H(w_1, w_{r-1}) > (r-2)^2 \geq (t-r+1)^2 > H(w_t, w_r) = R(w_r)$, which contradicts Lemma \[lemma:focuscat\](c) because $w_r$ is a best mix focus. Next, we observe that $i-1 \geq t-j$, meaning that $i$ is further from the left endpoint than $j$ is from the right endpoint. Indeed, when the left and right spines are the same length, this is necessary for $L(w_{r-1}) \leq R(w_r)$.
By equations and , we have $\Delta\L(w_{r-1})=2i-1 = \Delta\L(w_r)$ and $\Delta\R(w_{r-1})=2(t-j)+1=\Delta\R(w_r)$. After the surgery, the left and right spines are equal length and leaf-free, so the conditions of Lemma \[lemma:focuscat\](d) hold for $\nG$. In fact, both $w_{r-1}, w_r$ are now best mix foci because $L_{\nG}(w_{r-1}) = R_{\nG}(w_r)$. To prove that $\tbm$ increases, we use Lemma \[lem:slack\]. Taking $A=\{x,y\}$, the left hand side of inequality is $$\big( i^2-1-(2i-1) \big) + \big((t+1-j)^2-1-(2i-1) \big) \leq 2 \big( i^2-1-(2i-1) \big) = 2i^2 - 4i.$$ For the right hand side of inequality , we take $B=W=\{ w_1, w_2, \ldots , w_t \}$, in other words, we ignore any central leaves. This right hand side is at least $$\begin{aligned}
\MoveEqLeft 2t \, \pdelt(w_r) -2 \sum_{k=1}^t \Delta H(w_i, w_r) \\
& \qquad
-H_G(w_1, w_r) + H_G(w_i, w_r) + H_G(w_j, w_r) - H_G(w_t,w_r) \\
& = 2t (2i-1) -4 \sum_{k=1}^{i-1} (i-k) -4 \sum_{k=j+1}^{t} (k-j)
- (i-1)^2- (t-j)^2 \\
& \geq 2t (2i-1) - 4 i(i-1) - 2 ( i-1)^2 \, =\, 4(r-1)(2i-1) - 6i^2 + 8i -2 \\
& > 4(i-1)(2i-1) - 6i^2 + 8i -2 \, =\, 2i^2 - 4i +2.\end{aligned}$$ We have satisfied the conditions of Lemma \[lem:slack\], so $\Delta \tbm > 0$.
Finally, we consider surgery $\S_8$, which moves the final right leaf to the spine.
Let $G$ be a seesaw with exactly one right leaf $x \in U_i$, where $r+1 \leq i \leq t-1$ and no left leaves. If $\S_8={\tau(i,r)}$, then $\Delta \tbm > 0$.
First, we use Lemma \[lem:criteria\], to show that $\nG$ has focus $w_{r-1}$ and best mix focus $w_r$. By equation , $\Delta\L(w_{r-1})=0 =\Delta\L(w_r)=\Delta H(w_1,w_r)$ and $\Delta\R(w_{r-1})=-2(i-r)=\Delta\R(w_r)$. Inequality is clearly satisfied. Next, we verify inequality . By equation , we have $
R_G(w_r)-L_G(w_{r-1})=(t-r)^2+2(i-r)-(r-2)^2
$ and we claim that $(t-r)^2 - (r-2)^2 >0$. Indeed, since $w_r$ is a best mix focus, Lemma \[lem:criteria\] (c) ensures that $L(w_{r-1}) \leq R(w_r)$, or in other words $(r-2)^2 \leq (t-r)^2 + 2(r-i) < (t-r+1)^2.$ Since $r-2$ and $t-r$ are both integers, we must have $r-2 \leq t-r$. This means that $R_G(w_r)-L_G(w_{r-1}) \geq 2(i-r) = \Delta\L(w_{r-1})-\Delta\R(w_r).$ This confirms that the foci do not change. Finally, for all $v\in V$, we have $\Delta H(v,w_r)\leq 0=\L(w_r)$. By Corollary \[cor:slack\], $\Delta \tbm > 0$.
Phase Three: Twin Broom to Path or Wishbone
-------------------------------------------
Phase Three converts a twin broom into either $P_n$ or $Y_n$. The three surgeries are shown in Table \[table:phase3\] and the workflow is shown in Figure \[fig:phase3\].
\[lemma:twinbroom-to-end\] Let $G$ be a twin broom on $n$ vertices. If $n$ is even then Phase Three turns $G$ into the path $P_n$. If $n$ is odd, then Phase Three turns $G$ into the wishbone $Y_n$. Moreover, if $n$ is even then $\Tbest(G) \leq \Tbest(P_n)$ and if $n$ is odd then $\Tbest(G) \leq \Tbest(Y_n)$. Furthmore, equality holds if and only if $G=P_n$ for $n$ even, and $G=Y_n$ for $n$ odd.
The Phase Three surgeries $\S_9, \S_{10}, \S_{11}$ are shown in Table \[table:phase3\] and the workflow is shown in Figure \[fig:phase3\]. The lemmas that follow show that $\Delta \Tbest > 0$ for all three surgeries.
Suppose that $r-2 < t-r$, or equivalently $t < 2r-2$. Since $G$ is bifocal, there must be enough leaves in $U_{r-1}$ so that $(r-1)^2 + 2|U_{r-1}| = L(w_r) > R(w_r) = (t-r)^2$. We apply $\S_9$ until $r-2 = t-r$. At this point (and henceforth), both $w_{r-1}$ and $w_r$ are best mix foci by Lemma \[lemma:focuscat\] (d). We may assume that $|U_{r-1}| \leq |U_{r}|$. If both $U_{r-1}$ and $U_r$ are nonempty, we use $\S_{10}$ to simultaneously extend the left and right spine by taking one vertex from each of these sets. We repeat this until $U_{r-1} = \emptyset$. Next, if there are multiple leaves remaining in $U_r$, we use $\S_{11}$ to extend both ends of the spine. We repeat $\S_{11}$ until there are 0 or 1 leaves left in $U_r$. At this point, we either have a path $P_n$ or a wishbone $Y_n$. We started this phase with a twin broom, which is bifocal by definition. Therefore, if $n$ is even we have constructed a path and if $n$ is odd we have constructed a wishbone.
[|c|p[2.5in]{}|@c@|]{}
Surgery & Initial Conditions & Illustration\
$\S_9$ & $r-2<t-r$, which forces $U_{r-1} \neq 0$ &
[c]{}
(2, .5) circle (1pt);
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(2,0) – (2.5,0); (3.5,0) – (5.5,0); (6.5,0) – (7,0);
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(4,0) circle (4pt);
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(3.75, -1.1) to (1.2,-.25);
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\
$\S_{10}$ & $r-2=t-r$, and there are leaves $x\in U_{r-1}$ and $y\in U_r$ &
[c]{}
(2, .5) circle (1pt);
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(5,0) – (5,-1); (5,-1) circle (4pt);
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(4,0) – (4,-1); (4,-1) circle (4pt);
(2,0) – (7,0);
(2,0) – (2.5,0); (3.5,0) – (5.5,0); (6.5,0) – (7,0);
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$\S_{11}$ & $r-2=t-r$, $U_{r-1} = \emptyset$ and there are leaves $x,y\in U_r$ &
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\
![Phase three of the algorithm, turning a twin broom into a path or a wishbone.[]{data-label="fig:phase3"}](flowchart_phase_III){width="5.5in"}
We first consider $\S_9$, which we apply to equalize the lengths of the left and right spine.
\[lem:S9\] Let $G$ be a twin broom with leaf $x\in U_{r-1}$ such that $2r-1 \leq t.$ If $\S_9={\tau({r-1},1)}$, then $\Delta \Tbest > 0$.
First, it is routine to show that the simultaneous inequalities $2r-1 \leq t$ and $\L_G(w_r) > \R_G(w_r)$ require that $\deg(w_{r-1}) \geq 4$. By equation , we have $\Delta\L(w_{r-1})=2r-3=\Delta\L(w_r)$ and $\Delta\R(w_{r-1})=0=\Delta\R(w_r).$ We use the two criteria of Lemma \[lemma:focuscat\] (d) to show that the foci roles do not change. Clearly $\L_{\nG}(w_r) > \L_G(w_r) \geq \R_G (w_r) = \R_{\nG}(w_r)$. Since $U_i=\emptyset$ for all $i\notin\{r-1,r\}$, we can use equation to calculate $\L_{\nG}(w_{r-1})=(r-1)^2\leq (t-r)^2=\R_{\nG}(w_r).$ Next, we show that the best mixing time increases using Lemma \[lem:slack\] with $A= \{ x \}$. The left-hand side of inequality is $$\Delta H(x,w_r) - \pdelt (w_r) =( (r-1)^2-1) - (2r-3) = r^2 -4r+3.$$ For the right hand side, we take $B=W$, ignoring any other central leaves. We obtain $$\begin{aligned}
\MoveEqLeft
(2t-1) \, \pdelt (w_r) -2 \sum_{i=1}^{r-1} \Delta H(w_i,w_r) - H_G(w_1, w_r) + H_G(w_{r-1},w_r) \\
& = (2t-1) (2r-3) - 4 \sum_{i=1}^{r-1} (r-1-k) - (r-2)^2 \\
& \geq 2(2r-2) (2r-3) -3 r^2 +10 r -8 \, = \, 5r^2 -10r +4, \end{aligned}$$ which is clearly larger than the left hand side, so $\Delta \Tbest > 0$.
Next, we discuss transplanting one leaf from each of $U_{r-1}$ and $U_{r}$.
\[lem:S10\] Let $G$ be a twin broom where $t=2r-2$ and with vertices $x\in U_{r-1}$ and $y\in U_r$. If $\S_{10}={\tau({r-1},1)}\wedge{\tau(r,t)}$, then $\Delta \Tbest > 0$.
The right-hand and left-hand spines have equal lengths before and after this surgery. Corollary \[cor:criteria\] holds because each of $\Delta\L(w_{r-1})$, $\Delta\L(w_{r})$, $\Delta\R(w_{r-1})$, and $\Delta\R(w_{r})$ are equal to $2(r-2)-1= 2r-3$. So $\nG$ has focus $w_{r-1}$ and best mix focus $w_r$. We now show that the best mixing time increases. Setting $A=\{x,y\}$, the left hand side of inequality is $$\Delta H(x,w_r)+\Delta H(y,w_r)-2(2r-3)=
2 \big((r-1)^2-1 \big) - 2(2r-3) = 2r^2 -8r + 6.$$ where we use equations and . As for the right-hand side, we take $B=W$, disregarding the remaining central leaves. This right hand side is at least $$\begin{aligned}
\MoveEqLeft 2t \, \pdelt(w_r) - 2 \sum_{k=1}^t \Delta H(w_k,w_r) -H_G(w_1, w_r) +H(w_{r-1}, w_r) - H_G(w_t, w_r) \\
& = 2(2r-2) (2r-3) -2 \cdot 2 \sum_{k=1}^{r-1}2 (r-1-k) - 2 (r-2)^2 \, = \, 2r^2 -4.\end{aligned}$$ By Lemma \[lem:slack\], $\Delta \Tbest > 0$.
Finally, we discuss transplanting pairs of leaves from $U_r$ to the ends of the spine.
\[lem:S10\] Let $G$ be a twin broom with $t = 2r-2$ where $U_{r-1} = \emptyset$ and with distinct vertices $x,y \in U_r$. If $\S_{11}={\tau(r,1)}\wedge{\tau(r,t)}$, then $\Delta \Tbest > 0$.
This proof is similar to the previous one: the left and right spine lengths are equal before and after the surgery. We have $\Delta\L(w_{r})=2r-1$ while $\Delta\L(w_{r-1})=\Delta\R(w_{r-1})=\Delta \R(w_{r})=2r-3,$ where the value for $\Delta \R(w_{r-1})$ takes into account the removal of two leaves from $U_r$. We have $ \Delta\L(w_{r-1})-\Delta\R(w_r)=0$ and $\Delta\R(w_r)-\Delta\L(w_r)=-2,$ so Corollary \[cor:criteria\] ensures that $\nG$ has focus $w_{r-1}$ and best mix focus $w_r$.
Setting $A= \{ x, y \}$, the left hand side of inequality is $$ (r^2 -1) + ((r-1)^2 -1) - 2(2r-1) = 2r^2 -2r + 1.$$ As for the right-hand side of inequality , we take $B=W$ and obtain $$\begin{aligned}
\MoveEqLeft 2t \, \pdelt(w_r) - 2\sum_{k=1}^t \Delta H(w_k, w_r) -H_G(w_1, w_r) - H_G(w_t, w_r) \\
&= 2(2r-2) (2r-1) - 2\sum_{k=1}^{r-1} 2(r-k) - 2 \sum_{k=r+1}^{2r-2} 2(k-r) - (r-1)^2 - (r-2)^2 \\
&=2r^2+2r-5.\end{aligned}$$ Thus Lemma \[lem:slack\] ensures that $\Delta \Tbest > 0$.
Conclusion
==========
We have characterized the tree structures on $n$ vertices that minimize and maximize $\Tbest = \min_{v \in V} H(v,\pi)$. The star $S_n$ is the unique minimizing structure, but the maximization problem depends on the parity of $n$. For even $n$, the maximizing structure is the path $P_n$, and for odd $n$, it is the wishbone $Y_n$. It is a bit strange that the odd path is not the maximizing structure for $\Tbest$. But all is not lost: we believe that $P_n$ is the maximizing structure for a slightly different quantity. For any graph $G$, the *forget distribution* $\mu$ is the distribution achieving $\max_{v \in V} H(v,\mu) = \min_{\tau} \max_{v \in V} H(v, \tau)$. Lovász and Winkler [@lovasz+winkler-forget] shows that $\mu$ is unique, and they give a general formula. For a tree $G$, the forget distribution is concentrated on its foci [@beveridge]. When $G$ is focal, $\mu$ is a singleton distribution on the unique focus. When $G$ is bifocal, $\mu$ is given by $$\mu_u = \frac{H(v',v) - H(u',v)}{2|E|} \quad \mbox{and} \quad \mu_v = \frac{H(u',u) - H(v',u)}{2|E|}$$ where $u,v$ are the foci of the tree. Instead of $\Tbest=\min_w H(w,\pi)$, we could instead consider the similar quantity $H(\mu, \pi)$. It is easy to see that $S_n$ minimizes $H(\mu, \pi)$ among all trees on $n$ vertices. We conjecture that $P_n$ maximizes this quantity for both even and odd $n$. Indeed, letting $G=P_n$ and $\nG=Y_n$, calculations show that for odd $n$, we have $H_{P_n}(\mu,\pi) = H_{Y_n}(\widetilde{\mu}, \widetilde{\pi}) + (2n-3)/(2n-2)$, and for even $n$, we have $H_{P_n}(\mu,\pi) = H_{Y_n}(\widetilde{\mu}, \widetilde{\pi}) + 1/(2n-2)$. Our tree surgery methods should be a fruitful line of attack, though tracking the changes in $H(\mu, \pi)$ will require a new set of lemmas. We leave this problem for future work.
[^1]: Department of Mathematics, Statistics and Computer Science, Macalester College, Saint Paul, MN 55105
[^2]: Department of Mathematics, University of Minnesota, Minneapolis, MN 55455
|
---
abstract: 'Using one channel to simulate another exactly with the aid of quantum no-signalling correlations has been studied recently. The one-shot no-signalling assisted classical zero-error simulation cost of non-commutative bipartite graphs has been formulated as semidefinite programms \[Duan and Winter, IEEE Trans. Inf. Theory 62, 891 (2016)\]. Before our work, it was unknown whether the one-shot (or asymptotic) no-signalling assisted zero-error classical simulation cost for general non-commutative graphs is multiplicative (resp. additive) or not. In this paper we address these issues and give a general sufficient condition for the multiplicativity of the one-shot simulation cost and the additivity of the asymptotic simulation cost of non-commutative bipartite graphs, which include all known cases such as extremal graphs and classical-quantum graphs. Applying this condition, we exhibit a large class of so-called *cheapest-full-rank graphs* whose asymptotic zero-error simulation cost is given by the one-shot simulation cost. Finally, we disprove the multiplicativity of one-shot simulation cost by explicitly constructing a special class of qubit-qutrit non-commutative bipartite graphs.'
author:
- 'Email: xin.wang-8@student.uts.edu.au, runyao.duan@uts.edu.au'
title: |
On the quantum no-signalling assisted\
zero-error classical simulation cost of\
non-commutative bipartite graphs
---
Introduction
============
Channel simulation is a fundamental problem in information theory, which concerns how to use a channel $\cN$ from Alice (A) to Bob (B) to simulate another channel $\cM$ also from A to B [@KretschmannWerner:tema]. Shannon’s celebrated noisy channel coding theorem determines the capability of any noisy channel $\cN$ to simulate an noiseless channel [@Shannon1948] and the dual theorem “reverse Shannon theorem” was proved recently [@BSST2003]. According to different resources available between A and B, this simulation problem has many variants and the case when A and B share unlimited amount of entanglement has been completely solved [@BSST2003]. To optimally simulate $\cM$ in the asymptotic setting, the rate is determined by the entanglement-assisted classical capacity of $\cN$ and $\cM$ [@BDHS+2009; @QRST-simple]. Furthermore, this rate cannot be improved even with no-signalling correlations or feedback [@BDHS+2009].
In the zero-error setting [@Shannon1956] , recently the quantum zero-error information theory has been studied and the problem becomes more complex since many unexpected phenomena were observed such as the super-activation of noisy channels [@Duan2009; @DS2008; @CCH2009; @CS2012] as well as the assistance of shared entanglement in zero-error communication [@CLMW2010; @LMMO+2012].
Quantum no-signalling correlations (QNSC) are introduced as two-input and two-output quantum channels with the no-signalling constraints. And such correlations have been studied in the relativistic causality of quantum operations [@BGNP2001; @ESW2001; @PHHH2006; @OCB2012]. Cubitt et al. [@CLMW2011] first introduced classical no-signalling correlations into the zero-error classical communication problem. They also observed a kind of reversibility between no-signalling assisted zero-error capacity and exact simulation [@CLMW2011]. Duan and Winter [@Duan2014] further introduced quantum non-signalling correlations into the zero-error communication problem and formulated both capacity and simulation cost problems as semidefinite programmings (SDPs) [@SDP] which depend only on the non-commutative bipartite graph $K$. To be specific, QNSC is a bipartite completely positive and trace-preserving linear map $\Pi: \cL(\cA_i)\otimes \cL(\cB_i)\rightarrow \cL(\cA_o)\otimes \cL(\cB_o)$, where the subscripts $i$ and $o$ stand for input and output, respectively. Let the Choi-Jamiołkowski matrix of $\Pi$ be $\Omega_{\cA_i'\cA_o\cB_i'\cB_o}= (\1_{\cA_i'}\otimes \1_{\cB_i'}\otimes \Pi) (\Phi_{\cA_i\cA_i'}\otimes \Phi_{\cB_i\cB_i'})$, where $\Phi_{\cA_i\cA_i'}= \ketbra{\Phi_{\cA_i\cA_i'}}{\Phi_{\cA_i\cA_i'}}$, and $\ket{\Phi_{\cA_i\cA_i'}}=\sum_k \ket{k_{\cA_i}}\ket{k_{\cA_i'}}$ is the un-normalized maximally-entangled state.The following constraints are required for $\Pi$ to be QNSC [@Duan2014]: $$\begin{aligned}
\Omega_{\cA_i'\cA_o\cB_i'\cB_o}\geq 0, \ \tr_{\cA_o\cB_o}{\Omega_{\cA_i'\cA_o\cB_i'\cB_o}}=\1_{\cA_i'\cB_i'},&\\
\tr_{\cA_o\cA_i'}{\Omega_{\cA_i'\cA_o\cB_i'\cB_o} X^{T}_{\cA_i'}}=0, \forall \tr{X}=0,&\\
\tr_{\cB_o\cB_i'}{\Omega_{\cA_i'\cA_o\cB_i'\cB_o} Y^{T}_{\cB_i'}}=0, \forall \tr{Y}=0.&\end{aligned}$$ The new map $\cM^{A_i\rightarrow B_o}=\Pi^{A_i\ox B_i\rightarrow A_o\ox B_o}\circ \cE^{A_o\rightarrow B_i}$ by composing $\cN$ and $\Pi$ can be constructed as illustrated in Figure \[fig:QNSC\].
![Implementing a channel $\cM$ using another channel $\cE$ with QNSC $\Pi$ between Alice and Bob. []{data-label="fig:QNSC"}](ns.png){width="5.1cm"}
The simulation cost problem concerns how much zero-error communication is required to simulate a noisy channel exactly. Particularly, the *one-shot* zero-error classical simulation cost of $\cN$ assisted by $\Pi$ is the least noiseless symbols $m$ from $A_o$ to $B_i$ so that $\cM$ can simulate $\cN$. In [@Duan2014], the one-shot simulation cost of a quantum channel $\cN$ is given by $$\S(\cN)= \min \tr T_B,\ {\rm s.t. }\ J_{AB}\leq \1_A\ox T_B.$$ Its dual SDP is $$\S(\cN) = \max \tr({J_{AB}}{U_{AB}}) ,\ {\rm s.t. }\ {{U_{AB}} \ge 0,\tr_A {U_{AB}} = {\1_B}},$$ where $J_{AB}$ is the Choi-Jamiołkowski matrix of $\cN$. By strong duality, the values of both the primal and the dual SDP coincide. The so-called “non-commutative graph theory” was first suggested in [@DSW2010] as the non-commutative graph associated with the channel captures the zero-error communication properties, thus playing a similar role to confusability graph. Let $\cN(\rho)=\sum_k E_k\rho E_k^\dag$ be a quantum channel from $\cL(A)$ to $\cL(B)$, where $\sum_k E_k^\dag E_k=\1_{A}$ and $K=K(\cN)=\operatorname{span}\{E_k\}$ denotes the Choi-Kraus operator space of $\cN$. The zero-error classical capacity of a quantum channel in the presence of quantum feedback only depends on the Choi-Kraus operator space of the channel [@DSW2015]. That is to say, the Choi-Kraus operator space plays a role that is quite similar to the bipartite graph. Such Choi-Kraus operator space $K$ is alternatively called “non-commutative bipartite graph” since it is clear that any classical channel induces a bipartite graph and a confusability graph, while a quantum channel induces a non-commutative bipartite graph together with a non-commutative graph [@Duan2014].
Back to the simulation cost problem, since there might be more than one channel with Choi-Kraus operator space included in $K$, the exact simulation cost of the “cheapest” one among these channels was defined as the one-shot zero-error classical simulation cost of $K$ [@Duan2014]: $\S(K)=\min \{\S(\cN): \cN \text{ is quantum channel and } K(\cN)<K\}$, where $K(\cN)<K$ means that $K(\cN)$ is a subspace of $K$. Then the one-shot zero-error classical simulation cost of a non-commutative bipartite graph $K$ is given by [@Duan2014] $$\begin{split}
\label{eq:Sigma}
\S(K) &= \min \tr T_B \ \text{ s.t. }\ 0 \leq V_{AB} \leq \1_A \ox T_B, \\
&\phantom{= \min \tr T_B \text{ s.t. }} \tr_B V_{AB} = \1_A, \\
&\phantom{= \min \tr T_B \text{ s.t. }} \tr (\1-P)_{AB}V_{AB} = 0.
\end{split}$$ Its dual SDP is $$\begin{split}\label{maxSDP-K}
\S(K) &= \max \tr S_A \ \text{ s.t. }\ 0 \leq U_{AB},\ \tr_A U_{AB} = \1_B, \\
&\phantom{= \max \tr S_A \text{ }} P_{AB}(S_A \ox \1_B - U_{AB})P_{AB} \leq 0,
\end{split}$$ where $P_{AB}$ denotes the projection onto the support of the Choi-Jamiołkowski matrix of $\cN$. Then by strong duality, values of both the primal and the dual SDP coincide. It is evident that $\S(K)$ is sub-multiplicative, which means that for two non-commutative bipartite graphs $K_1$ and $K_2$, $\S({K_1} \otimes {K_2}) \le \S({K_1})\S({K_2})$. Furthermore, the multiplicativity of $\S(K)$ for classical-quantum (cq) graphs as well as extremal graphs were known but the general case was left as an open problem [@Duan2014]. By the regularization, the no-signalling assisted zero-error simulation cost is $$S_{0,NS}(K) = \inf_{n\geq 1} \frac1n \log \S\left(K^{\ox n}\right).$$
As noted in previous work [@Duan2014; @DSW2015], $$C_{0,NS}(K)\le C_{\text{minE}}(K)\le S_{0,NS}(K),$$ where $C_{0,NS}(K)$ is the QSNC assisted classical zero-error capacity and $C_{\text{minE}}(K)$ is the minimum of the entanglement-assisted classical capacity [@BSST2003; @Bennett1999] of quantum channels $\cN$ such that $K(\cN) < K$.
Semidefinite programs [@SDP] can be solved in polynomial time in the program description [@Khachiyan1980] and there exist several different algorithms employing interior point methods which can compute the optimum value of semidefinite programs efficiently [@Alizadeh1995; @DeKlerk2002]. The CVX software package [@CVX] for MATLAB allows one to solve semidefinite programs efficiently.
In this paper, we focus on the multiplicativity of $\S(K)$ for general non-commutative bipartite graph $K$. We start from the simulation cost of two different graphs and give a sufficient condition which contains all the known multiplicative cases such as cq graphs and extremal graphs. Then we consider about the simulation cost $\S(K)$ when the “cheapest” subspace is full-rank and prove the multiplicativity of one-shot simulation cost in this case. We further explicitly construct a special class of non-commutative bipartite graphs $K_\alpha$ whose one-shot simulation cost is non-multiplicative. We also exploit some more properties of $K_\alpha$ as well as cheapest-low-rank graphs. Finally, we exhibit a lower bound in order to offer an estimation of the asymptotic simulation cost.
Main results
============
A sufficient condition of the multiplicativity of simulation cost
-----------------------------------------------------------------
\[sufficient\] Let ${K_1}$ and ${K_2}$ be non-commutative bipartite graphs of two quantum channels $\cN_1:\cL(A_1) \to \cL(B_1)$ and ${\cN_2}:\cL(A_2) \to \cL(B_2)$ with support projections $P_{{A_1}{B_1}}$ and $P_{{A_2}{B_2}}$, respectively. Suppose the optimal solutions of SDP(\[maxSDP-K\]) for $\S({K_1})$ and $\S(K_2)$ are $\left\{ {{S_{{A_1}}},{U_1}} \right\}$ and $\left\{ {{S_{{A_2}}},{U_2}} \right\}$. If at least one of $S_{{A_1}}$ and $S_{{A_2}}$ satisfy $$\label{sufficient condition}
{P_{{A_i}{B_i}}}({S_{{A_i}}} \otimes {\1_{{B_i}}}){P_{{A_i}{B_i}}} \ge 0, i=1~\text{or}~2,$$ then $$\S({K_1} \otimes {K_2}) = \S({K_1}) \S({K_2}).$$ Furthermore, $$S_{0,NS}(K_1 \ox K_2)=S_{0,NS}(K_1) + S_{0,NS}(K_2).$$
It is obvious that ${U_1} \otimes {U_2} \ge 0 \text{ and } \tr_{{A_1}{A_2}}({U_1} \otimes {U_2}) = {\1_{{B_1}{B_2}}}$. For convenience, let $P_{{A_1}{B_1}}=P_1$ and $P_{{A_2}{B_2}}=P_2$. Without loss of generality, we assume that ${P_2}({S_{{A_2}}} \otimes {\1_{{B_2}}}){P_2} \ge 0$. From the last constraint of SDP(2), we have that ${P_1}\left( {{S_{{A_1}}} \otimes {\1_{{B_1}}}} \right){P_1} \le {P_1}{U_1}{P_1}$ and ${P_2}({S_{{A_2}}} \otimes {\1_{{B_2}}}){P_2} \le {P_2}{U_2}{P_2}$. Note that ${P_1}\left( {{S_{{A_1}}} \otimes {\1_{{B_1}}}} \right){P_1} \otimes {P_2}({S_{{A_2}}} \otimes {\1_{{B_2}}}){P_2} \le {P_1}{U_1}{P_1} \otimes {P_2}({S_{{A_2}}} \otimes {\1_{{B_2}}}){P_2}$. It is easy to see that $$\begin{split}
&{P_1} \otimes {P_2}\left( {{S_{{A_1}}} \otimes {S_{{A_2}}} \otimes {\1_{{B_1}{B_2}}} - {U_1} \otimes {U_2}} \right){P_1} \otimes {P_2}\\
\le &{P_1}{U_1}{P_1} \otimes [{P_2}({S_{{A_2}}} \otimes {\1_{{B_2}}}){P_2} - {P_2}{U_2}{P_2}] \le 0.
\end{split}$$
Hence, $\left\{ {{S_{{A_1}}} \otimes {S_{{A_2}}}, {U_1} \otimes {U_2}} \right\}$ is a feasible solution of SDP(\[maxSDP-K\]) for $\S({K_1} \otimes {K_2})$, which means that $\S({K_1} \otimes {K_2}) \ge \S({K_1})\S({K_2})$. Since $\S(K)$ is sub-multiplicative, we can conclude that $\S({K_1} \otimes {K_2}) = \S({K_1})\S({K_2})$.
Furthermore, for $K_2^{\ox n}$, it is easy to see that $\{S_{A_2}^{\ox n}, U_2^{\ox n} \}$ is a feasible solution of SDP(\[maxSDP-K\]) for $\S(K_2^{\ox n})$ and ${P_2^{\ox n}}({S_{{A_2}}^{\ox n}} \otimes {\1_{{B_2}}}^{\ox n}){P_2^{\ox n}} \ge 0$. Therefore, $\S(K_2^{\ox n})=\S(K_2)^n$ and $$\S[(K_1 \ox K_2 )^{\ox n}] =\S(K_1^{\ox n} \ox K_2^{\ox n} ) =\S(K_1^{\ox n})\S(K_2^{\ox n}).$$ Hence, $$\begin{aligned}
S_{0,NS}(K_1 \ox K_2)&= \inf_{n\geq 1} \frac1n \log \S\left(K_1^{\ox n} \ox K_2^{\ox n}\right)\\
&=\inf_{n\geq 1} \frac1n \log\S(K_1^{\ox n})\S(K_2^{\ox n})\\
&=S_{0,NS}(K_1) + S_{0,NS}(K_2).\end{aligned}$$
In [@DW2015], the activated zero-error no-signalling assisted capacity has been studied. Here, we consider about the corresponding simulation cost problem.
For any non-commutative bipartite graph K, let $\Delta_\ell=\sum_{k=1}^\ell \ketbra{kk}{kk}$ be the non-commutative bipartite graph of a noiseless channel with $\ell$ symbols, then $${\S(K \ox \Delta_\ell)}={\ell}\S(K),$$ which means that noiseless channel cannot reduce the simulation cost of any other non-commutative bipartite graph.
It is evident that $\Delta_\ell$ satisfies the condition in Theorem \[sufficient\]. Then, ${\S(K \ox \Delta_\ell)}=\ell\S(K)$.
Simulation cost of the cheapest-full-rank non-commutative bipartite graph
-------------------------------------------------------------------------
Given a non-commutative bipartite graph $K$ with support projection $P_{AB}$. Assume the “cheapest channel” in this space is $\cN_c$ with Choi-Jamiołkowski matrix $J_{\cN_c}$. $K$ is said to be **cheapest-full-rank** if there exists $\cN_c$ such that $rank(J_{\cN_c})=rank(P_{AB})$. Otherwise, $K$ is said to be **cheapest-low-rank**.
\[lemma of PAP\] For a quantum channel $\cN$ with Choi-Jamiołkowski matrix $J_{AB}$ and support projection $P_{AB}$, if $P_{AB}CP_{AB}=P_{AB}DP_{AB}$, then $\tr (CJ_{AB})=\tr (DJ_{AB})$.
It is easy to see that $$\begin{aligned}
\tr (CJ_{AB}) &=\tr(CP_{AB}J_{AB}P_{AB})=\tr(P_{AB}CP_{AB}J_{AB})\\
&=\tr(P_{AB}DP_{AB}J_{AB}) =\tr (DJ_{AB}).
$$
\[lemma JW\] For any non-commutative bipartite graph K with support projection $P_{AB}$, suppose that the cheapest channel is $\cN_c$ and the optimal solution of SDP (\[maxSDP-K\]) is $\{S_A, U_{AB}\}$. Assume that $$P_{AB}(S_A \ox \1_B - U_{AB})P_{AB} = -W_{AB} \text{ and } W_{AB} \ge 0.$$ Then, we have that $$\label{lemma2}
\tr W_{AB}J_{AB}=0,$$ and $U_{AB}$ is also the optimal solution of $\S(\cN_c)$, where $J_{AB}$ is the Choi-Jamiołkowski matrix of $\cN_c$.
On one hand, since $\cN_c$ is the cheapest channel, $\S(K)$ will equal to $\S(\cN_c)$, also noting that $\{S_A, U_{AB}\}$ is the optimal solution, we have $$\begin{split}\label{eq1-lemma2}
\tr{S_A} &=\S(K) =\S(\cN_c) \\
&= \max \tr {J_{AB}}{V_{AB}} ,\ {\rm s.t. }\ {{V_{AB}} \ge 0,\tr_A {V_{AB}} = {\1_B}},\\
&\ge \tr {J_{AB}}{U_{AB}}.
\end{split}$$
On the other hand, it is evident that $W_{AB}= P_{AB}WP_{AB} $, then $P_{AB}U_{AB}P _{AB} = P _{AB} (W_{AB}+ S_A \ox \1_B)P _{AB}$. From Lemma \[lemma of PAP\], we can conclude that $\tr U_{AB}J_{AB} = \tr (W_{AB}+ S_A \ox \1_B)J_{AB}=\tr W_{AB}J_{AB}+\tr(S_A \ox \1_B)J_{AB}$.
For Choi-Jamiołkowski matrix $J_{AB}$, we have that $$\begin{split}
\tr({S_A} \otimes {\1_B}){J_{AB}} &= \tr_A\tr_B[({S_A} \otimes {\1_B}){J_{AB}}] \\
&=\tr_A[{S_A}(\tr_B{J_{AB}}) ]= \tr{S_A},
\end{split}$$ then $$\label{eq2-lemma2}
\tr U_{AB}J_{AB}= \tr W_{AB}J_{AB}+\tr{S_A}.$$ Combining (\[eq1-lemma2\]) and (\[eq2-lemma2\]), and noting that $W_{AB}, J_{AB} \ge 0$, we can conclude that $\tr W_{AB}J_{AB}=0$ and $U_{AB}$ is also the optimal solution of $\S(\cN_c)$.
\[full-rank-S\] For any cheapest-full-rank non-commutative bipartite graph $K$, we have $$\begin{split}
\label{eq:Sigma-dual-hat}
\S(K) &= \max \tr S_A \ \text{ s.t. }\ 0 \leq U_{AB},\ \tr_A U_{AB} = \1_B, \\
&\phantom{= \max \tr S_A \text{ s.t. }} P_{AB}(S_A \ox \1_B - U_{AB})P_{AB} = 0.
\end{split}$$ Also, $\S({K} \otimes {K}) = \S({K}) \S({K})$. Consequently, $S_{0,NS}(K)= \log \S(K)$.
And for any other non-commutative bipartite graph $K'$, $S_{0,NS}(K\ox K')=S_{0,\NS}(K)+S_{0,NS}(K')$.
We first assume that $W \ne 0$. Notice $rank(J_{AB})=rank(P_{AB})$, it is easy to see that $\tr WJ_{AB}>0$, which contradicts Eq. (\[lemma2\]). Hence the assumption is false, and we can conclude that $P_{AB}(S_A \ox \1_B - U_{AB})P_{AB} = 0$.
Then by Theorem \[sufficient\], it is easy to see that $\S(K \ox K) = \S(K)\S(K)$. Therefore, $$S_{0,\NS}(K) = \inf_{n\geq 1} \frac1n \log \S(K^{\ox n})= \log \S(K).$$ Furthermore, for any other non-commutative bipartite graph $K'$, $S_{0,NS}(K\ox K')= S_{0,\NS}(K)+S_{0,NS}(K')$.
Noting that any rank-2 Choi-Kraus operator space is always cheapest-full-rank, we have the following immediate corollary.
For any rank-2 Choi-Kraus operator space $K$, $S_{0,NS}(K)= \log \S(K)$. And for any other non-commutative bipartite graph $K'$, $S_{0,NS}(K\ox K')= S_{0,NS}(K)+S_{0,NS}(K')$.
The one-shot simulation cost is not multiplicative
--------------------------------------------------
We will focus on the non-commutative bipartite graph $K_\alpha$ with support projection $P_{AB}=\sum \limits_{j=0}^{2} \ketbra {\psi_j} {\psi_j}$, where $\ket {\psi_0} =\frac{1}{\sqrt 3}(\ket {00}+\ket {01}+\ket {12}),
\ket {\psi_1} =\cos \alpha \ket {02}+\sin \alpha \ket {11},
\ket {\psi_2} =\ket {10}$.
To prove that $K_\alpha$ ($0<\cos^2 \alpha<1$) is feasible to be a class of feasible non-commutative bipartite graphs, we only need to find a channel $\cN$ with Choi-Jamiołkowski matrix $J_{AB}$ such that $P_{AB}J_{AB}=J_{AB}$ and $\text{rank}(P_{AB})=\text{rank}(J_{AB})$ . Assume that $J_{AB}=\sum \limits_{j=0}^{2} a_j\ketbra {\psi_j} {\psi_j}$, then it is equivalent to prove that $\tr_B J_{AB}=\1_A$ and $J_{AB} \ge 0$ has a feasible solution. Therefore, $$\frac{2}{3}a_0+\cos^2\alpha a_1=1,
a_0+a_1+a_2=2,
a_0, a_1, a_2 >0.$$ Noting that when we choose $0< a_1<\frac{1}{2}$, $a_0=\frac{3}{2}(1-\cos^2\alpha a_1)$ and $a_2=\frac{1-(2-3\cos^2\alpha )a_1}{2}$ will be positive, which means that there exists such $J_{AB}$. Hence, $K_\alpha$ is a feasible noncommutative bipartite graph.
There exists non-commutative bipartite graph K such that $\S(K\ox K)<\S(K)^2$.
As we have shown above, it is reasonable to focus on $K_\alpha$. Then, by semidefinite programming assisted with useful tools CVX [@CVX] and QETLAB [@QETLAB], the gap between one-shot and two-shot average no-signalling assisted zero-error simulation cost of $K_\alpha (0.25 \leq \cos^2 \alpha \leq 0.35)$ is presented in Figure \[fig:sc eg\].
To be specfic, when $\alpha=\pi/3$, it is clear that $\cos^2 \alpha=1/4$ and $\ket {\psi_1} =\frac{1}{2} \ket {02}+\frac{\sqrt 3}{2} \ket {11}$. Assume that $S=3.1102\proj{0}-0.5386\proj{1}$ and $U=\frac{99}{50}\proj{u_1}+\frac{51}{50}\proj{u_2}$, where $\ket {u_1}=\frac{10}{3\sqrt {33}}\ket{00}+\frac{5}{3}\sqrt\frac{2}{33}\ket {01}+\frac{7}{3\sqrt {11}}\ket{12}$ and $\ket {u_2}=\frac{1}{\sqrt {51}}\ket{02}-\frac{5}{3}\sqrt\frac{2}{17}\ket{10}+\frac{10}{3\sqrt {17}}\ket{11}$, and it can be checked that $U\ge 0$, $\tr_A U=\1_B$ and $P_{AB}(S_A \ox \1_B - U_{AB})P_{AB} \le 0$. Then $\{S, U\}$ is a feasible solution of SDP (\[maxSDP-K\]) for $\S(K_{\pi/3})$, which means that $\S(K_{\pi/3})\ge \tr S= 2.5716$. Similarily, we can find a feasible solution of SDP (\[eq:Sigma\]) for $\S(K_{\pi/3}\ox K_{\pi/3})$ through Matlab such that $\S(K_{\pi/3}\ox K_{\pi/3})^{1/2}\le 2.57$. (The code is available at [@sc; @code].) Hence, there is a non-vanishing gap between $\S(K_{\pi/3})$ and $\S(K_{\pi/3}\ox K_{\pi/3})^{1/2}$.
![The one-shot (red) and two-shot average (blue) no-signalling assisted zero-error simulation cost of $K_\alpha$ over the parameter $\alpha$.[]{data-label="fig:sc eg"}](sceg.pdf){width="41.00000%"}
We have shown that one-shot simulation cost of cheapest-full-rank non-commutative bipartite graphs is multiplicative while there are counterexamples for cheapest-low-rank ones. However, not all cheapest-low-rank graphs have non-multiplicative simulation cost. Here is one trivial counterexample. Let $K=\text{span}\{\ket 0 \bra 0 ,\ket1 \bra 0 ,\ket 1 \bra 1\}$, the cheapest channel is a constant channel $\cN$ with $E_0=\ket 1 \bra 0$ and $E_1=\ket 1 \bra 1$. In this case, $\S(K\otimes K) = \S(K)\S(K)=1.$ Actually, the simulation cost problem of cheapest-low-rank non-commutative bipartite graphs is complex since it is hard to determine the cheapest subspace under tensor powers. Therefore, it is difficult to calculate the asymptotic simulation cost of non-multiplicative cases.
In [@DSW2015], $K$ is called non-trivial if there is no constant channel $\cN_0: \rho \to \proj \beta$ with $K(\cN_0) < K$, where $\ket \beta$ is a state vector. It was known that $K$ is non-trivial if and only if the no-signalling assisted zero-error capacity is positive, say $C_{0,NS}(K)>0$. Clearly we have the following result.
For any non-commutative bipartite graph $K$, $S_{0,NS}(K)>0$ if and only if $K$ is non-trivial.
If $K$ is non-trivial, it is obvious that $S_{0,NS}(K)\ge C_{0,NS}(K)>0$. Otherwise, $0\le S_{0,NS}(K)\le S_{0,NS}(\cN_0)=0$, which means that $S_{0,NS}(K)=0$.
A lower bound
-------------
Let us introduce a revised SDP which has the same simplified form in cq-channel case: $$\begin{split}
\label{eq:Sigma2 dual}
\S^-(K) &= \max \tr S_A \ \text{ s.t. }\ S_A\ge 0, U_{AB}\ge 0\ \tr_A U_{AB} = \1_B, \\
&\phantom{= \max \tr S_A \text{ s.t. }} P_{AB}(S_A \ox \1_B - U_{AB})P_{AB} \leq 0,
\end{split}$$
\[lowerbound\] For any non-commutative bipartite graphs $K_1$ and $K_2$, $$\S^-(K_1 \ox K_2) \ge \S^-(K_1)\S^-(K_2).$$ Consequently, $\S^-({K_1}) \S^-({K_2}) \le \S({K_1} \otimes {K_2}) \le \S({K_1}) \S({K_2})$.
From SDP (\[eq:Sigma2 dual\]), noting that $P_{AB}(S_A \ox \1_B)P_{AB} \ge 0$, it is easy to prove $\S^-(K_1 \ox K_2) \ge \S^-(K_1)\S^-(K_2)$ by similar technique applied in Theorem 3. Therefore, $\S^-({K_1}) \S^-({K_2}) \le \S^-({K_1} \otimes {K_2}) \le \S({K_1} \otimes {K_2})\le \S({K_1}) \S({K_2})$.
For a general non-commutative bipartite graph $K$, $$\log {\S^ - }(K) \le {S_{0,NS}}(K) \le \log \S(K).$$
By Lemma \[lowerbound\], it is easy to see that ${\S^ - }{(K)^n} \le \S(K^{ \otimes n}) \le \S{(K)^n}$. Then, $\log {\S^ - }(K) \le {S_{0,NS}}(K) \le \log \S(K)$. Also, it is obvious that ${S_{0,NS}}(K)$ will equal to $\log \S(K)$ when $\S^-(K)=\S(K).$
Conclusions
===========
In sum, for two different non-commutative bipartite graphs, we give sufficient conditions for the multiplicativity of one-shot simulation cost as well as the additivity of the asymptotic simulation cost. The case of cheapest-full-rank non-commutative bipartite graphs has been completely solved while the cheapest-low-rank graphs have a more complex structure. We further show that the one-shot no-signalling assisted classical zero-error simulation cost of non-commutative bipartite graphs is not multiplicative. We provide a lower bound of $\S(K)$ such that the asymptotic zero-error simulation cost can be estimated by $\log\S^-(K)\le S_{0,NS}(K) \le \log\S(K)$.
It is of great interest to know whether the sufficient condition of multiplicativity in Theorem \[sufficient\] is also necessary. It also remains unknown about the additivity of the asymptotic simulation cost of general non-commutative bipartite graphs and whether it equals to $\log\S^-(K)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Andreas Winter for his interest on this topic and for many insightful suggestions. XW would like to thank Ching-Yi Lai for helpful discussions on SDP. This work was partly supported by the Australian Research Council (Grant No. DP120103776 and No. FT120100449) and the National Natural Science Foundation of China (Grant No. 61179030).
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abstract: 'The generalized mapping transformation technique is used to obtain the exact solution for the transverse Ising model on decorated planar lattices. Within this scheme, the basic thermodynamic quantities are calculated for different planar lattices with arbitrary spins of decorating atoms. The particular attention has been paid to the investigation of transverse-field effects on magnetic properties of the system under investigation. The most interesting numerical results for the phase diagrams, compensation temperatures and several thermodynamic quantities are discussed in detail for the ferrimagnetic version of the model.'
author:
- |
Jozef Strečka and Michal Jaščur\
Department of Theoretical Physics and Astrophysics, Institute of Physics\
P. J. Šafárik University, Moyzesova 16, 041 54 Košice, Slovak Republic\
E-mail:jozkos@pobox.sk, jascur@kosice.upjs.sk
title: Thermodynamic properties of the exactly solvable transverse Ising model on decorated planar lattices
---
[*Keywords:*]{} exact solution; mapping transformation; transverse field, Ising model\
[*PACS:*]{} 75.10.Hk, 05.50.+q
Introduction
============
For many years magnetic properties of decorated Ising spin systems consisting of spin-$1/2$ and spin-$S$ ($S \geq 1/2$) atoms, have been very intensively studied both theoretically and experimentally. In particular, the behaviour of various decorated models has been explored by variety of mathematical techniques, with some exact results [@1], even in the presence of the single-ion anisotropy [@2]. The strong interest in these models arises partly on account of the rich critical phenomena they display and partly due to the fact that they represent more complicated but simultaneously exactly solvable systems than the simple, undecorated ones. Moreover, the decorated planar models belong to the simplest exactly solvable theoretical models of ferrimagnetism. In this respect, they may exhibit under certain conditions the compensation phenomenon, i. e. the compensation point at which the resultant magnetization vanishes bellow the critical temperature. All these properties of decorated systems make them very interesting also from the experimental point of view, first of all in connection with many possible technological application of ferrimagnets to practice. On the other hand, much work is currently being done in the area of molecular-based magnetic materials and so molecular magnetism has become an important topic of scientific interest. In fact, a rapid progress in molecular engineering brought the possibility to control and to design the magnetic structure and properties of molecular systems. Thus, a number of bimetallic network assemblies have been synthesized which perfectly fulfil the claim of sufficiently small interplane coupling. Nevertheless, the molecular magnets which would possess the decorated network structure have been prepared only a decade ago. From this family, the most frequently prepared compounds are the bimetallic assemblies having the decorated honeycomb sheet structure [@4] or the decorated square sheet structure [@5] (see Fig. 1). Obviously, the experimental discovery of this wide class of compounds has stimulated the renewed interest in studying the mixed spin-$1/2$ and spin-$S$ ($S \geq 1/2$) decorated planar models theoretically.
Owing to these facts, the present article will be devoted to the investigation one of the simplest mixed-spin quantum model. Namely, we will study the decorated model in the presence of a transverse field that can potentially be useful for understanding some of the above mentioned experimental systems. The transverse Ising model has been originally introduced by de Gennes as a pseudo-spin model of hydrogen bonded ferroelectrics [@6], however, during the last decade it has found a wide application in the description of diverse physical systems, for instance, cooperative Jahn-Teller systems [@7], strongly anisotropic magnetic materials in the transverse field [@8], or many other systems [@9]. Although the transverse Ising model is one of the simplest quantum models, the complete exact solution has been obtained in the one-dimensional case only [@10]. For two-dimensional systems there are exactly known only the initial transverse susceptibilities on regular lattices [@11]. On the other hand, over the last few years a simple straightforward method has been developed for obtaining exact results to the transverse Ising model, by assuming that it is composed of quantal and ’classical’ (Ising-type) spins [@12; @13]. The method is based on the generalized mapping transformation (the decoration-iteration or star-triangle one) introduced into the system with the transverse field affecting only one kind of spins (quantal).
The primary purpose of this work is to provide further extension of the exact solution from the reference [@12], by the use of the same approach. Adopting the basic ideas of the transformation techniques, we examine the influence of applied transverse field on the thermodynamic properties of decorated lattices with different planar topology and arbitrary spin values of decorating atoms.
The outline of this paper is as follows. In Section 2, the fundamental framework of the transformation method as applied to the present model, are briefly reviewed. This is followed by a presentation of the numerical results for the spin-$\mu$ ($\mu = 1/2$) and spin-$S_B$ ($S_B \geq 1/2$) decorated transverse Ising ferrimagnet on several planar lattices, in Section 3. Finally, some concluding remarks are given in Section 4.
Formulation
===========
In this work we will study the transverse Ising model on decorated planar lattices. A typical example of the system under investigation is depicted for the case of the square lattice in Fig. 1. As one can see, the system consists of two interpenetrating sublattices and we will further assume that the sites of the original lattice that constitute the sublattice $A$ are occupied by atoms with the fixed spin $\mu = 1/2$. The second sublattice $B$ is occupied by decorating atoms with an arbitrary spin value $S_B$. Then the system is described by the Hamiltonian $$\hat {\cal H}_d = - \frac12 \sum_{i, k} J \hat S_i^z \hat \mu_k^z -
H_A \sum_{k=1}^{N} \hat \mu_k^z - H_B \sum_{i=1}^{Nq/2} \hat S_i^z -
\Omega \sum_{i=1}^{Nq/2} \hat S_i^x,
\label{eq1}$$ where $\hat \mu_k^z, \hat S_i^z$ and $\hat S_i^x$ represent the relevant components of standard spin operators and $J$ is the exchange integral that specifies the exchange interaction between nearest-neighboring atoms. The last three terms describe the interaction of $A$ atoms with an external longitudinal magnetic field $H_A$, as well as the interaction of $B$ atoms with an external longitudinal ($H_B$) and transverse ($\Omega$) magnetic field, respectively. Furthermore, $N$ and $q$ denote the total number of atoms and the coordination number of the original lattice, respectively. As we have already mentioned, the transverse Ising model in which all the atoms interact with the transverse field is not exactly solvable for two- and three-dimensional lattices. This fact is closely related to the mathematical complexities appearing due to the noncomutability of the spin operators in the relevant Hamiltonian. However, in our recent work [@12] we have succeeded to solve exactly the simplified version of this model (described by the Hamiltonian (\[eq1\])) in which only the decorating ($B$) atoms interact with the transverse field. In this section, we at first briefly repeat the main formulae derived in the Ref. [@12] and then we further extend the formalism for the calculation of several interesting thermodynamic quantities of this model.
Following the same steps as in Ref. [@12], one obtains a simple relation between the partition function of the decorated transverse Ising model $({\cal Z}_d)$ and that of the spin-1/2 Ising model on the original lattice $({\cal Z}_0)$. Namely, $${\cal Z}_d(\beta, J, \Omega, H_A, H_B) = A(\beta, J, \Omega, H_B)^{Nq/2}
{\cal Z}_0(\beta, R, H),
\label{eq2}$$ where $\beta = 1/k_B T$, $k_B$ is Boltzmann constant and $T$ is the absolute temperature. The parameters $A, R$ and $H$ represent so-called decoration-iteration parameters that specify the corresponding original (undecorated) system as well as decorating system. All these parameters can be obtained from the decoration-iteration transformation (see Ref. [@12]) and they are, respectively, given by $$\begin{aligned}
A &=& (V_1 V_2 V_3^2)^{1/4}, \nonumber \\
\beta R &=& \ln \frac{V_1 V_2}{V_3^2}, \nonumber \\
\beta H &=& \beta H_A + \frac{q}{2} \ln \frac{V_1}{V_2},
\label{eq3}\end{aligned}$$ where we have defined the functions $V_1, V_2$ and $V_3$ as follows: $$\begin{aligned}
V_1 &=& \sum_{n = -S_B}^{S_B} \cosh \Bigl(
\beta n \sqrt{(J+H_B)^2 + \Omega^2}
\Bigr), \nonumber \\
V_2 &=& \sum_{n = -S_B}^{S_B} \cosh \Bigl(
\beta n \sqrt{(J-H_B)^2 + \Omega^2}
\Bigr), \nonumber \\
V_3 &=& \sum_{n = -S_B}^{S_B} \cosh \Bigl(
\beta n \sqrt{H_B^2 + \Omega^2}
\Bigr).
\label{eq4}\end{aligned}$$ It is worth noticing that the foregoing equations express the exact mapping relationship between the decorated model under investigation and the spin-$1/2$ model on the corresponding original lattice. One should also notice that this transformation is valid for arbitrary values of the decorating spin $S_B$ in the presence of the external longitudinal fields. Thus, the relevant thermodynamic quantities can be calculated even for the nonzero longitudinal fields. Unfortunately, after deriving the final equations, we have to set $H_A = H_B = 0$ since the original planar Ising models are exactly solvable for $H=0$ only.
In order to investigate the magnetic properties of the system, we employ the well-known relations from the statistical mechanics and thermodynamics. Indeed, after a simple calculation one derives the following equations for the Gibbs free energy, internal energy, enthalpy, entropy and specific heat: $${\cal G}_d = {\cal G}_0 - \frac{Nqk_BT}{2} \ln A,
\label{eq5}$$ $${\cal U}_d = \Bigl( \frac{2 {\cal U}_0}{R} - \frac{Nq}{4} \Bigr)
\frac{J^2}{\sqrt{J^2 + \Omega^2}} K_0,
\label{eq6}$$ $$H_d = \frac{2 {\cal U}_0}{R}
\Bigl( \sqrt{J^2 + \Omega^2}K_0 - \Omega K_1 \Bigr) -
\frac{Nq}{4} \Bigl( \sqrt{J^2 + \Omega^2}K_0 + \Omega K_1 \Bigr),
\label{eq7}$$ $$\begin{aligned}
{\cal S}_d = {\cal S}_0 \! \! &+& \! \! \frac{Nqk_B}{2} \ln A
- \frac{Nq}{4T} \Bigl( \sqrt{J^2 + \Omega^2} K_0 + \Omega K_1 \Bigr), \nonumber \\
\qquad {\cal S}_0 \! \! &=& \! \! k_B \ln {\cal Z}_0 + \frac{2 {\cal U}_0}{R T}
\Bigl( \sqrt{J^2 + \Omega^2} K_0 - \Omega K_1
\Bigr),
\label{eq8}\end{aligned}$$ and $${\cal C}_{\Omega} = {\cal C}_0
+ \frac{Nq}{4 k_B T^2}
\biggl [ (J^2 + \Omega^2)(K_2 - K_0^2)
+ \Omega^2 (K_3 - K_1^2)
\biggr ],
\label{eq9}$$ where ${\cal G}_d$, ${\cal U}_d$, $H_d$, ${\cal S}_d$ and ${\cal C}_{\Omega}$ denote, respectively, the Gibbs free energy, internal energy, enthalpy, entropy and specific heat of the decorated lattice. Similarly, ${\cal G}_0$, ${\cal U}_0$, ${\cal S}_0$ and ${\cal C}_0$ (${\cal C}_0 = T (\partial {\cal S}_0 / \partial
T)_{\Omega}$) represent the relevant quantities of the original lattice that are well-known for two-dimensional systems [@14]. Finally, the coefficients $K_0-K_3$ depend on the temperature and transverse field and they are are given by $$K_0 = F_1(J,\Omega), \;
K_1 = F_1(0,\Omega), \;
K_2 = F_2(J,\Omega), \;
K_3 = F_2(0,\Omega),
\label{eq10}$$ where $$\begin{aligned}
F_1(x,y) = \frac{\displaystyle \sum_{n = -S_B}^{S_B}
n \sinh (\beta n \sqrt{x^2 + y^2})
}
{\displaystyle \sum_{n = -S_B}^{S_B} \cosh (\beta n \sqrt{x^2 +
y^2})
},
\label{eq11}\end{aligned}$$ $$\begin{aligned}
F_2(x,y) = \frac{\displaystyle \sum_{n = -S_B}^{S_B}
n^2 \cosh (\beta n \sqrt{x^2 + y^2})
}
{\displaystyle \sum_{n = -S_B}^{S_B} \cosh (\beta n \sqrt{x^2 +
y^2})
}.
\label{eq12}\end{aligned}$$
Next, we turn to the calculation of the magnetization. The spontaneous longitudinal magnetization, as well as transverse one can be directly obtained by the differentiation of the Gibbs free energy (\[eq5\]) with respect to the relevant longitudinal and transverse magnetic fields. Consequently, the spontaneous longitudinal magnetization per spin can be written for both sublattices in the following compact form: $$\begin{aligned}
m_A^z &=& m_0, \nonumber \\
m_B^z &=& m_0 \frac{2J}{\sqrt{J^2+ \Omega^2}} K_0,
\label{eq13}\end{aligned}$$ where $m_A^z$ $(m_B^z)$ means the spontaneous longitudinal magnetization of sublattice $A$ $(B)$ and $m_0$ stands for the spontaneous magnetization of the original lattice ($m_0$ depends only on the exchange interaction $R$ and temperature). Similarly, the same procedure leads to the simple relation for the reduced transverse magnetization of sublattice $B$. Namely, $$m_B^x = \frac12 \biggl (
\frac{\Omega}{\sqrt{J^2+ \Omega^2}} K_0 + K_1
\biggr ) +
2 \varepsilon_0 \biggl (
\frac{\Omega}{\sqrt{J^2+ \Omega^2}} K_0 - K_1
\biggr ).
\label{eq14}$$ Here, $\varepsilon_0$ denotes the two-spin correlation function between nearest-neighboring spins of the original lattice (depending again only on the exchange parameter $R$ and temperature). To complete the analysis of the system under investigation, one has to investigate the phase boundaries, as well as a possibility of compensation phenomena in the case of the ferrimagnetic ordering ($J<0$). The structure of equations for the spontaneous sublattice magnetization implies that the phase transition temperature $T_c$ (or $\beta_c= 1/(k_B T_c)$) can be directly found from the transformation formula after setting $H_A = H_B = H = 0$ and substituting the inverse critical temperature $\beta_c R$ of the relevant original lattice into (\[eq3\]). In this way one obtains the relation $$\beta_c R =
2 \displaystyle \ln \frac{\displaystyle \sum_{n = -S_B}^{S_B} \cosh \Bigl(
\beta_c n \sqrt{J^2 + \Omega^2} \Bigr) }
{\displaystyle \sum_{n = -S_B}^{S_B} \cosh \Bigl(
\beta_c n \Omega \Bigr) },
\label{eq15}$$ which is valid for any planar lattice with the decorating spin $S_B$ and for the special case of $S_B = 1/2$ naturally recovers Eq.(13) in the Ref. [@12].
Moreover, in the case of ferrimagnetic system the compensation temperature $T_k$ (or $\beta_k=1/(k_B T_k) $) can be found from the condition that the total magnetization $M$ of the system vanishes bellow the critical temperature. Since in our case we have $M= N( m_A +
qm_B/2)$, then from the condition $M=0$ one easily finds the following relation for the compensation temperature $$\frac{q|J|}{\sqrt{J^2 + \Omega^2}} K_0 (\beta_k) = 1,
\label{eq16}$$ where the inverse compensation temperature $\beta_k$ is also included in the coefficient $K_0$.
Finally, we determine the transverse susceptibility of the decorated transverse Ising system. For this aim, we differentiate the formula (\[eq14\]) for the transverse magnetization with respect to the transverse field and after a straightforward but a little bit tedious algebra, we can write the transverse susceptibility $\chi_T$ in the form $$\begin{aligned}
\chi_T &=& \frac12 \biggl \{
\frac{J^2}{(J^2 + \Omega^2)^{3/2}} K_0 +
\frac{\beta \Omega^2}{J^2 + \Omega^2} (K_2 - K_0^2)
+ \beta(K_3 - K_1^2)
\biggr \} \nonumber \\
&+& 2 \varepsilon_0 \biggl \{
\frac{J^2}{(J^2 + \Omega^2)^{3/2}} K_0 +
\frac{\beta \Omega^2}{J^2 + \Omega^2} (K_2 - K_0^2)
- \beta(K_3 - K_1^2)
\biggr \} \nonumber \\
&+& 4 \frac{\partial \varepsilon_0}{\partial R}
\biggl \{
\frac{\Omega}{\sqrt{J^2 + \Omega^2}} K_0 - K_1
\biggr \}^2.
\label{eq17}\end{aligned}$$ In above, $\varepsilon_0$ means the two-spin correlation function between nearest-neighboring atoms of the original lattice
Numerical results
=================
In this section we will illustrate the effect of the transverse field, as well as the influence of the decorating spin $S_B$ on magnetic properties of the system under investigation. Moreover, the role of the lattice topology will be also examined. Although we will restrict our numerical calculation to the ferimagnetic case ($J<0$) only, the detailed investigation reveals that all dependences (excepting those for the longitudinal magnetization) remain unchanged also for the ferromagnetic version of the model. This observation follows from the fact that the relevant equations are independent under transformation $J\to -J$.
We start our discussion with the analysis of the critical and compensation temperatures. At first, the variations of the critical (dashed lines) and compensation temperatures (solid lines) with the transverse field are shown in Fig. 2. Here, we have selected the system with $q=4$ (i.e. the decorated square lattice), taking different spin values of decorating atoms ($S_B$). On the other hand, in Fig. 3 we have illustrated the influence of the lattice topology (of different $q$) on the critical and compensation temperature for the system with the fixed decorating spin ($S_B =1$). In both figures, the ordered ferrimagnetic phase is stable bellow dashed lines, and the disordered paramagnetic one becomes stable above the relevant boundary. A closer mathematical analysis reveals that the phase transition between these two phases is of the second order and belongs to the same universality class as that of the usual spin-1/2 Ising model. As one can see, the qualitative features of the results do not significantly depend neither on the lattice topology nor the spin value of atoms of sublattice $B$. In fact, the critical temperature monotonically decreases with increasing in the transverse field, but only in the limit of the infinity strong transverse field tends to zero (due to the fact that the transverse field affects only one sublattice). One also observes here that the value of transition point increases with the coordination number of the original lattice, as well as with the spin value of decorating atoms. Contrary to this behavior, the compensation temperature seems to be independent of the transverse field strength (with the accuracy of twelve orders), although it changes both with the coordination number $q$ and the spin value $S_B$. In general, on basis of our numerical calculation one can state that for arbitrary but fixed $q$ and $S_B$ the compensation points appear only for $\Omega_k/|J| = \sqrt{q^2S_B^2 - 1}$. It is easy to find that this characteristic value of $\Omega$ can be obtained from the Eq. (\[eq16\]) by taking the limit $T_k\to 0$ (or $\beta_k \to \infty$). Physically, the independence of the compensation temperature on the transverse field comes from the fact that the compensation effects appear at relatively strong transverse fields where the relevant transition temperature is rather low and this fact significantly influences the behavior of sublattice magnetization.
Now, we turn to the discussion of the internal energy and enthalpy. Owing to the fact, that all decorated lattices behave similarly, we will further present numerical results for one representative lattice, namely, the decorated square lattice. In Fig. 4 we have depicted the thermal variations of the internal energy and enthalpy for the spin case $S_B = 1/2$ ($N_t$ denotes a total number of atoms). From these dependences one finds that both quantities tend monotonically to zero with increasing the temperature. Nevertheless, if we compare both dependences, we can conclude that the enthalpy is more sensitive (changes the shape of the curve more rapidly) to the transverse field than the internal energy. It is also clear that the thermal dependences of the internal energy (as well enthalpy) exhibit a typical weak energy-type singularity behaviour irrespective of the strength of transverse field.
Further, we have also examined the temperature and transverse field dependences of the spontaneous longitudinal magnetization. In Figs. 5 and 6 we report some typical results for the total spontaneous longitudinal magnetization, when the value of the transverse field is changed (we choose the magnetization curves at temperatures $k_BT/J = 0.1$ and $k_BT/J = 0.2$ and different spin values of decorating atoms from $1/2$ until $2$). It follows from these dependences that the total magnetization may exhibit two possible shapes of the magnetization curves. Namely, the magnetization curve with one compensation point (at lower temperatures) and the downward magnetization curvature without any compensation points (at higher temperatures). Both types of the magnetization curves are closely related to the fact that the transverse field does not directly act on the atoms of original lattice. Hence, the spontaneous magnetization of sublattice $A$ varies very smoothly with the transverse field, whereas the spontaneous magnetization of sublattice $B$ is rapidly destroyed with the transverse field increasing. On the other hand, the temperature dependences of the spontaneous magnetization of both sublattices are standard, regardless of the transverse field strength. The only exceptional case arises as the transverse field reaches the value $\Omega_c$ at which the considered system behaves similarly as an ordinary antiferromagnet.
In contrast to the standard temperature variations of the longitudinal magnetization, the transverse magnetization may display very interesting and unexpected thermal behaviour. To illustrate the case, we have depicted in Figs. 7 and 8 some typical temperature dependences of the transverse magnetization. As shown in Fig. 7 for the spin case $S_B = 2$, the transverse magnetization at relatively small transverse field ($\Omega / J = 1.5$) firstly gradually decreases to its local minimum value and then nearby the transition point increases in the narrow temperature region. However, the transverse magnetization by the stronger transverse fields ($\Omega / J = 2.0$ and $3.0$) remains almost constant and again in the vicinity of the Curie point gradually increases, too. Far beyond the transition temperature, the transverse magnetization monotonically decreases with increasing in temperature, regardless of the transverse field strength. The possible explanation of the temperature-induced increase of the transverse magnetization can be related to the spin release from the spontaneous magnetization direction and spin-ordering towards the transverse field direction. In fact, the stronger the transverse field, the smaller the transition temperature and therefore, by sufficiently strong transverse field the transverse magnetization increases from its initial value, since the thermal reshuffling is relatively small in this temperature region. On the other hand, when the transverse field is smaller, the relevant spin reorientation takes place at higher temperatures, thus the transverse magnetization firstly decreases to its local minimum due to the strong thermal fluctuations in this temperature region. Apparently, also the increase in the transverse magnetization which is connected with the spin reorientation is then smaller, since it is overlapped with the stronger thermal fluctuation. Before proceeding further, we have depicted in Fig. 8 the transverse magnetization against the temperature that illustrate the influence of the different values of spin variable $S_B$.
Furthermore, let us now look more closely at the thermal variations of the specific heat For this purpose, we have studied the specific heat of the simplest spin case $S_B = 1/2$. As one can see from Fig. 9, the logarithmical singularity in the specific heat dependence indicates the second order phase transition towards the paramagnetic state. Moreover, it can be also clearly seen that the broadening of the maximum in the paramagnetic region of the specific heat arises due to the transverse field effect. It turns out that the observed maximum may be thought as a Schottky-type maximum, which has its origin in the thermal excitation of the paramagnetic spins inserted into the transverse field. Next, in order to illustrate the effect of increasing spin $S_B$ we have shown in Fig. 10 the thermal dependences of specific heat for several values of the spin variable $S_B$. As one can ascertain, the lower the spin value $S_B$ of decorating atoms, the stronger the influence of the transverse field on the excitation of these spins in the paramagnetic region. To complete our analysis of the specific heat, we have plotted in Fig. 11 the transverse-field dependences of the specific-heat for $S_B = 1/2$. As one can expect, the depicted behaviour strongly depends on the temperature and above the critical point, the dependence changes suddenly due to the paramagnetic character of the system.
Moreover, in Fig. 12 we display the transverse susceptibility against the transverse field for the same spin case as in Fig. 11. Here one observes, that for sufficiently high temperature (see the dashed line), the transverse susceptibility falls down monotonically with the increasing transverse field. Contrary to this behavior, for the lower temperature (see the solid line), the transverse susceptibility exhibits the standard singularity due to the second order phase transition from the ferrimagnetic state to the paramagnetic one. For completeness, in the insert of Fig. 12, we compare the variations of the transverse susceptibilities for different spin values of decorating atoms. Apparently, there is no essential difference in the behaviour of the transverse susceptibilities, although it is shifted to higher temperatures, as the spin value of the decorating atom is increased. Finally, in Fig. 13 we have depicted the temperature dependence of transverse susceptibility for some selected values of the transverse field. Here one can see, that for the smaller transverse fields, for instance ($\Omega / J = 0.5$), the transverse susceptibility decreases with increasing the temperature, then diverges at $T_c$ and repeatedly decreases. However, in the case of the stronger transverse fields ($\Omega / J = 1.0$ and $1.5$), the transverse susceptibility remains at its initial value, exhibits a divergence at phase transition and afterwards whether gradually decreases (the case $\Omega / J = 1.0$) or exhibits a broad maximum (the case $\Omega / J = 1.5$). The existence of this broad maximum in the paramagnetic region arises evidently on account of the thermal excitation of the paramagnetic spins inserted into the transverse field and therefore, can be observed for relatively strong transverse fields only.
Conclusion
==========
In this work we have presented the exact results (the phase diagrams, compensation temperatures, spontaneous longitudinal magnetization, transverse magnetization, internal and free energy, enthalpy, entropy, specific heat and transverse susceptibility) for the ferrimagnetic transverse Ising model on decorated planar lattices. We have illustrated that the magnetic properties of the system under investigation exhibit the characteristic behaviour depending on the strength of the applied transverse field and the spin of the decorating atoms. In particular, we have found that the considered ferrimagnetic system does not exhibit more than one compensation temperature that is surprisingly completely independent of the transverse field, though it depends on the coordination number of the original lattice and also on the spin of the decorating atoms. As far as we know, such a finding has not been reported in the literature before. Perhaps, the most interesting result to emerge here is the temperature-induced increase of the transverse magnetization in the vicinity of the transition temperature. We have found a strong evidence that this increase arises due to the spin release from the spontaneous magnetization direction, since in the vicinity of the transition temperature spins tending to align into the transverse field direction.
Finally, we would like to emphasize that the presented method can by applied to more complex and realistic models, for example, the transverse Ising models with a crystal field anisotropy, or those with next-nearest-neighbor and multispin interactions. Further generalizations are possible by increasing the spin of atoms on the sublattice $A$ or introducing more realistic Heisenberg interactions. In addition to the above mentioned generalizations, one can also obtain very accurate results for this model on three-dimensional lattice. This can be done by combining the present method with other accurate methods, such as series expansion technique, Monte Carlo simulations or renormalization group methods.
Acknowledgement: This work has been supported by the Ministry of Education of Slovak Republic under VEGA grant No. 1/9034/02.
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[**Figure captions**]{}
- Part of the decorated square lattice. The black circles denote the spin-1/2 atoms of sublattice $A$ (referred to as the atoms of original lattice) and gray circles represent the spin-$S_B$ atoms of sublattice $B$ (decorating atoms).
- Phase boundaries (dashed lines) and the compensation temperatures (solid lines) in the $\Omega$ - $T$ plane for the decorated square lattice ($q=4$) and different spin values of the decorating atoms.
- Critical (dashed lines) and compensation (solid lines) temperatures as a function of the transverse field for various decorated planar lattices with the fixed decorating spin $S_B = 1$.
- Thermal variations of the internal energy (dashed lines) and enthalpy (solid lines) when the transverse field is changed.
- Total longitudinal magnetization per one site versus transverse field for $k_B T / |J| = 0.1$ and different spins of decorating atoms.
- Total longitudinal magnetization versus transverse field for $k_B T / |J| = 0.2$ and different spins of decorating atoms.
- Transverse magnetization as a function of temperature for $S_B = 2$ and different transverse fields.
- Thermal dependences of the transverse magnetization for several spin values of decorating atoms, when the transverse field is fixed ($\Omega/|J| = 0.5$).
- Specific-heat variations with the temperature for selected values of the transverse field.
- Thermal variations of the specific heat for different spin cases of decorating atoms and the fixed transverse field value ($\Omega / |J| = 1.5$).
- Specific heat dependence on the transverse field when the temperature is changed.
- Transverse susceptibility against the transverse field for selected temperatures. In the insert, the comparison between different spin cases is shown.
- Transverse susceptibility dependence on the temperature for the spin $S_B = 2$, when the transverse field is changed.
|
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abstract: 'We demonstrate that a X-ray spectrum of a converging inflow (CI) onto a black hole is the sum of a thermal (disk) component and the convolution of some fraction of this component with the Comptonization spread (Green’s) function. The latter component is seen as an extended power law at energies much higher than the characteristic energy of the soft photons. We show that the high energy photon production (source function) in the CI atmosphere is distributed with the characteristic maximum at about the photon bending radius, $1.5r_{\rm S}$, independently of the seed (soft) photon distribution. We show that high frequency oscillations of the soft photon source in this region lead to the oscillations of the high energy part of the spectrum but not [of]{} the thermal component. The high frequency oscillations of the inner region are not significant in the thermal component of the spectrum. We further demonstrate that Doppler and recoil effects (which are responsible for the formation of the CI spectrum) are related to the hard (positive) and soft (negative) time lags between the soft and hard photon energy channels respectively.'
author:
- 'Philippe Laurent, Lev Titarchuk'
title: 'Timing and Spectral Properties of X-ray Emission from the Converging Flows onto Black hole: Monte-Carlo Simulations'
---
0.5 truecm
Introduction
============
Accreting stellar-mass black holes (BH) in Galactic binaries exhibit so called high-soft and low-hard spectral states (e.g. Borozdin et al. 1999, hereafter BOR99). An increase in the soft blackbody luminosity component leads to the appearance of an extended power law. An important observational fact is that this effect is seen as a persistent phenomenon only in BH candidates, and thus it is apparently a [*unique*]{} black hole signature. Although in Neutron star (NS) systems similar power law components are detected in the intermediate stages (Strickman & Barret 1999; Iaria et al. 2000; Di Salvo et al. 2001), they are of a transient nature, disappearing with increasing luminosity (Di Salvo et al.). It thus seems a reasonable assumption that the unique spectral signature of the soft state of BH binaries is directly tied to the black hole event horizon. This is the primary motivation for the Bulk Motion Comptonization Model (BMC) introduced in several previous papers, and recently applied with striking success to a substantial body of observational data (Shrader & Titarchuk 1998; BOR99; Titarchuk & Shrader 2002, hereafter TS02). A complete theory of BH accretion must, however, be also able to accommodate in a natural manner a growing number of observational traits exhibited in the temporal domain. For example, it is now well established that BH X-ray binaries exhibit quasi-periodic oscillation (QPO) phenomena in three frequency domains: $\sim 0.1$ Hz, ($\sim 1-10$ Hz) and ($\sim 10^2$ Hz). The $\sim 10^2-$Hz QPOs seem to occur during periods of flaring, and when the spectra (although in the high-soft state) tend to be relatively hard, i.e. the relative importance of the power-law component with respect to the thermal. Furthermore, the QPO amplitudes increase with energy, that is there is a higher degree of modulation of the signal in the hard-power law than in the thermal excess component. In addition to the QPO phenomena, it has been noted by Nowak et al. (2000) (also see Cui et al. 2000) that measures of the coherence between the intensity variations in the hard power law and thermal components is negligible.
We argue that the BH X-ray spectrum in the high-soft state is formed in the relatively cold accretion flow with a subrelativistic bulk velocity $v_{bulk}\lax c$ and with a temperature of a few keV and less ($v_{th}\ll c$). In such a flow the effect of the bulk Comptonization is much stronger than the effect of the thermal Comptonization which is a second order with respect to $v_{th}/c$. In this [*Letter*]{} we present results of Monte Carlo simulations probing the spatial, spectral, timing and time-lag properties of X-ray radiation in CI atmosphere in §§2-4. Comparisons are drawn to the recent X-ray observations of BH sources. Summary and conclusions follow in §5.
Emergent spectrum and hard photon spatial distribution
======================================================
The geometry used in these simulations is similar to the one used in the Laurent & Titarchuk (1999), hereafter LT99, consisting of a thin disk with an inner radius of 3 $r_{\rm S}$, merged with a spherical CI cloud harboring a BH in its center, where $r_{\rm S}=2GM/c^2$ is the Sc[h]{}warzchild radius and $M$ is a BH mass. The cloud outer radius is $r_{out}$. The disk is assumed always to be optically thick.
In addition to free-fall into the central BH, we have also taken into account the thermal motion of the CI electrons, simulated at an electron temperature of 5 keV. This is likely to be a typical temperature of the CI in the high-soft state of galactic BHs \[see Chakrabarti & Titarchuk (1995) and BOR99 for details\]. The seed X-ray photons were generated uniformly and isotropically at the surface of the border of the accretion disk, from $r_{d,in}= {3} r_{\rm S}$ to $r_{d, out} = 10r_s$. These photons were generated according to a thermal spectrum with a single temperature of 0.9 keV, similar to the ones measured in Black Hole binary systems (see for example BOR99). In fact, BOR99 also demonstrate that the multicolor disk spectrum can be fitted by the effective single temperature blackbody spectrum in the energy range of interest (for photon energies higher than 2 keV). The parameters of our simulations are the BH mass $m$ in solar units, the CI electron temperature, $T_e=5$keV, the mass accretion rate, $\dot m=4$ in Eddington units (see LT99), and the cloud outer radius, $r_{out}=10r_{\rm S}$. It is worth noting that, for these parameters, the bulk motion effects are not significant at $r> 10r_{\rm S}$.[^1] In Figure 1 we present the simulated spectrum of the X-ray emission emerging from the CI atmosphere. The spectrum exhibits three features: a soft X-ray bump, an extended power law and a sharp exponential turnover near 300 keV. As demonstrated previously (Titarchuk & Zannias 1998, hereafter TZ98; LT99) the extended power law is a result of soft photon upscattering off CI electrons. The qualitative explanation of this phenomena was given by Ebisawa et al. (1996), hereafter ETC and later by Papathanassiou & Psaltis (2001). The exponential turnover is formed by a small fraction of those photons which undergo scatterings near the horizon where the strong curvature of the photon trajectories prevent us from detecting most them. In Figure 1 we indicate, with arrows the places in the CI atmosphere where photons of a particular energy mainly come from.[^2]
A [precise]{} analysis of the X-ray photon distribution in CI atmosphere can be made through calculation of the source function, either by semi-analytical methods solving the relativistic kinetic equation (RKE) (TZ98) or by Monte Carlo simulations (LT99). TZ98 calculate the photon kinetics in the lab frame demonstrating that the source function has a strong peak near the photon bending radius, $1.5r_s$ (Fig. 3 there). In Figure 2 we show the Monte Carlo simulated source functions for four energy bands: 2-5 keV (curve a), 5-13 keV (curve b), 19-29 keV (curve c) and 60-150 keV (curve d). The peak at around $(1.5-2)r_{\rm S}$ is clearly seen for the second and the third bands which is in a good agreement with TZ98’s source function. [^3]
TZ98 calculated the upscattering part of spectrum neglecting the recoil effect. Our Monte Carlo simulations, not limited by TZ98’s approximation, reproduce the source function spatial distribution for high energy bands (curve [*d*]{} in Figure 2). We confirm that the density of the highest energy X-ray photons is concentrated near the black hole horizon. Comparison of TZ98’s semi-analytical calculations and our Monte-Carlo simulations leads us to conclude that the source function really follows the RKE first eigenfunction distribution until very high energies. At that point the upscattering photon spatial distribution also becomes a function of energy. As seen from Figure 2 the source function in the soft energy band (curve [*a*]{}) has two maxima: one is at 2.2 $r_{\rm S}$ related to the photon bending radius and another (wide one) is at 5 $r_{\rm S}$ affected by the disk emission area.
An Illumination Effect of CI Atmosphere and High Frequency QPO Phenomena
========================================================================
We [remind]{} the reader that the emergent spectrum is a result of integration of the product of the photon escape probability and the source function distribution along a line of sight (e.g. Chandrasekhar 1960). The normalization of the upscattering spectrum is determined by the fraction of the soft (disk) photons which illuminate the inner region of the CI atmosphere below $3r_{\rm S}$, i.e. around the maximum of the upscattered photon source function. From the Monte Carlo simulations we extract the fraction of the soft photons emitted at some particular disk radius $r$ which form the high energy part of spectrum (for $E>10$ keV). This distribution is presented in the upper panel of Figure 3. As seen from this plot, the strength of the high energy tail is mostly determined by the photons emitted at the inner edge of the disk. Thus any perturbation in the disk should be immediately translated to the oscillation of the hard tail with a frequency related to the inner disk edge. The CI atmosphere manifests the high frequencies QPO of the innermost part of the disk. In the lower panel of Figure 3 we present the power density spectrum for the simulated X-ray emission coming from CI atmosphere in the energies higher than 10 keV. We assume that the PDS is a sum of the red noise component (where the power law index equals to 1) and QPOs power proportional to the illumination factor for a given disk radius, $r$ (see the upper panel). We also assume that the perturbation frequency at $r$ is related to the Keplerian frequency $\nu_{\rm K}=2.2(3r_{\rm s}/r)^{1.5}/m$ kHz.
There is a striking similarity between the high frequency QPO and spectral energy distribution of the Monte Carlo results and real observations of BHC in their soft-high state. For example, in XTE J1550-564 the $\sim 200-$Hz QPO phenomenon tends to be detected in the high state at times when the bolometric luminosity surges and the hard-power-law spectral component is prominent (TS02). The noted lack of coherence between intensity variations of the high-soft-state low energy bands is also in a good agreement with the our simulations where the high energy tail intensity [*correlates*]{} with the supply of the soft photons from the inner disk edge but [*it does not correlate*]{} with the production of the disk photons at large.
Positive and Negative Time Lags
===============================
Additional important information related to the X-ray spectral-energy distribution can be extracted from the time lags between different energy bands. The hard and soft lags have been observed for several sources (e.g. Reig et al. 2000, hereafter R00: Tomsick & Kaaret 2000 for GRS 1915+105 and Wijnands et al. 1999, Remillard et al. 2001 for XTE J1550-564). It is natural to expect the positive time lags in the case of the unsaturated thermal Comptonization. The primary soft photons gain energy in process of scattering off hot electrons; thus the hard photons spend more time in the cloud than the soft ones. Nobili at al. (2000) first suggested that in a corona with a temperature stratification (a hot core and a relatively cold outer part) the thermal Comptonization can account for the positive and negative time lags as well as the observed colors (see R00).[^4] In the bulk-motion Comptonization case, the soft disk photons at first gain energy in the deep layers of the converging inflow, and then in their subsequent path towards the observer lose energy in the relatively cold outer layers. If the overall optical depth of the converging inflow atmosphere (or the mass accretion rate) is near unity, we would detect only the positive lags as in the thermal Comptonization case, because relatively few photons would lose energy in escaping. But with an increase of the optical depth, the soft lags appear because more hard photons lose their energy in the cold outer layers. In Figure 4 we present the calculations of the of time lags for two energy bands 2-13 keV and 13-900 keV. We first compute the time spent by the photon, drawn in a given direction, to cross the whole system without being scattered. Then, we compute the real time of flight of the photon, taking into account its scattering and its complete trajectory in the system. The general relativistic gravitational time dilatation was also taken into account in these computations. The time lag is then determined as a difference between these two times. In this time-lag definition, a photon which undergoes no scattering has a zero lag. Thus the time lags between two energy bands can be defined a difference of these two time lags, i.e. $\delta t =t_2-t_1$ where $t_1$ and $t_2$ are time lags for 2-13-keV and 13-900-keV bands respectively. The time lags for 2-13-keV band are distributed over the interval between 1 and 600 Schwarzschild times, $t_{\rm S}=r_{\rm S}/c$ and the time lags for 13-900-keV band are distributed over the interval between 200 and $10^3$ ${t}_{\rm S}$. For a ten solar mass BH $t_{\rm S}=0.1$ ms. Thus, for $t_1<15$ ms all time lags $\delta t$ are positive, i.e. high energy photons are produced later than the soft energies photons. The $t_2-$ distribution has a broad peak at 30 ms and $t_1-$ distribution has a broad peak at 8 ms. Consequently, the positive lags $\delta t$ are mostly higher than 20 ms. which is in a good agreement with the time lags detection in GRS 1915+105 (R00). The values of the positive time lags for this source are consistent with the BH mass being around 20 solar masses (see BOR99 for the BH mass determination in GRS 1915+105). We show that the absolute value of the time (phase) lags get higher with the energy. In fact, Wijnands et al. (1999) for XTE J1550-564 (see also Remillard et al. 2001) and Tomsick & Kaaret (2000) for GRS 1915+105 show that the phase lags of both signs increase from 0 to 0.5 radians as the photon energy varies from 3 keV to 100 keV. These phase lag changes correspond to the time lag changes from 0 to 15 ms and from 0 to 50 ms for XTE J1550-564 and GRS 1915+105 respectively. These values are very close to what we obtain in our simulations.
For $t_1>15$ ms we get the time lags of both signs. Positive values of $\delta t$ should be of order 30 ms and higher. Negative values should not be higher than 20 ms for a 10 solar mass BH. Our simulations demonstrate that for high accretion rates time lags of both signs are present, i.e. the upscattering and down-scattering are equally important in the formation of CI spectrum, whereas for the low mass accretion rates the upscattering process leads to the CI spectral formation and only the positive times lags are present. Our simulations are in a good agreement with the R00’s values of $\delta=30-60$ ms. The results for the simulated time lags can be understood using the simple upscattering model (see ETC). We assume that at any scattering event the relative mean energy change is constant, $\eta$, i.e. $\eta=<\delta E>/E$. The number of scatterings $k$ which are needed for the soft photon of energy $E_0$ to gain energy E is $k(E)=\ln(E/E_0)/\ln(1+\eta)$. Then $\delta t= k\times l(1-r_{\rm S}/r)^{-1}/c$ where $l$ is the photon mean free path, $l=[b(E)\sigma_{\rm T}n]^{-1}$, $(1-r_{\rm S}/r)$ is the relativistic dilatation factor (e.g. Landau & Lifshitz 1971) and $b(E)$ is an energy dependent factor less than unity (e.g. Pozdnyakov et al, 1983). The free path $l$ is estimated as $l\sim 2(r/r_{\rm S})^{1.5}r_{\rm S}/b(E)\dot m$ with an assumption that the number density $n$ is calculated for the free fall distribution, (see LT99, Eq. 2). With $r/r_{\rm S}\approx 1.18$ (see Fig. 2, curve [*d*]{}), $\dot m=4$ and $b\approx0.5$ for $E=100$ keV we get $l\approx 1.2 r_{\rm S}$. $b(E)\approx 1$ for $E=10$ keV and $r/r_{\rm S}\gtorder 2$ and consequently $l\gtorder 1.5 r_{\rm S}$.[^5]
Summary and Conclusions
=======================
Using Monte Carlo simulations we have presented a detailed timing analysis of the X-ray radiation from a CI atmosphere. (1) We confirm TZ98’s results that the high energy photons are produced predominantly in the deep layers of the CI atmosphere from 1 to 2 Schwarzchild radii. (2) We also confirm that the $~$ 100-Hz phenomena should be seen in the high energy tail of the spectrum rather than in the soft spectral component because the inner part of the converging inflow is mostly fed by soft photons from the innermost part of the disk but the contribution of this disk area to the disk emission is small. (3) We find that the characteristic time lags in the converging inflow are order of a few tens of ms at 10 keV and the hard (positive) time lags increase logarithmically with energy. (4) We demonstrate that soft (negative) along with hard (positive) lags are present in the X-ray emission from the converging inflow for the mass accretion higher than Eddington. In the end we can also conclude that that almost all of the X-ray timing and spectral properties of the high-soft state in BHs tends to indicate that [*the high energy tail of spectrum is produced within the CI region, $(1-3)\times r_{\rm S}$ where the bulk inflow (gravitational attraction) is a dominant process*]{}.
We acknowledge the fruitful discussions with Chris Shrader and Paul Ray. We also acknowledge the useful suggestions and the valuable corrections by the referee which improve the paper presentation.
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Wijnands, A.D., Homan, J. & van der Klis, M. 1999, ApJ, 526, L33
[^1]: In general, the bulk motion Comptonization is effective when $\tau v_{bulk}/c
=\dot m r_{\rm S}/r$ [*is not smaller than unity*]{} (e.g. Rees, 1978; Titarchuk et al. 1997).
[^2]: Recently, Reig et al. (2001) show that the extended power law can be also formed as a result of the bulk motion Comptonization when the rotational component is dominant in the flow. The photons gain energy from the rotational motion of the electrons.
[^3]: TZ98 implemented the method of separation of variables to solve the RKE Green’s function. They then showed that the source function of the upscattered photons is distributed according to the first eigenfunction of the RKE space operator defined by equations (21-24) in TZ98.
[^4]: This Comptonization model with a hot core (for the similar models, see also Skibo & Dermer 1995 and Lehr et al. 2000) may reproduce the intrinsic spectral and timing properties of the converging inflow.
[^5]: It is worth noting that the emergent high energy photon gains its energy being predominantly close to horizon, $r=(1-2)r_{\rm S}$ where the photon trajectory are close to circular (e.g. TZ98). Thus our estimated values of the free path $l\sim (1-1.5) r_{\rm S}$ are consistent with the trajectory length. With $\eta=0.1$ extracted from our simulations for $E<100$ keV where the extended power law is seen in the spectrum (see also Papathanassiou & Psaltis 2001, Fig. 3) the time lags are $\delta t= 7.6$ ms and $50$ ms for $E=10$ keV and $100$ keV respectively.
|
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abstract: 'We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As the main result we establish for nonnegative weak supersolutions of such problems a weak Harnack inequality with optimal critical exponent. The proof relies on new a priori estimates for time fractional problems and uses Moser’s iteration technique and an abstract lemma of Bombieri and Giusti, the latter allowing to avoid the rather technically involved approach via $BMO$. As applications of the weak Harnack inequality we establish the strong maximum principle, continuity of weak solutions at $t=0$, and a Liouville type theorem.'
---
[**A weak Harnack inequality for fractional evolution equations with discontinuous coefficients**]{}
Rico Zacher
[[**address:**]{} Martin-Luther University Halle-Wittenberg, Institute of Mathematics, Theodor-Lieser-Strasse 5, 06120 Halle, Germany, E-mail: rico.zacher@mathematik.uni-halle.de]{}
[**AMS subject classification:**]{} 45K05, 47G20
[**Keywords:**]{} weak Harnack inequality, Moser iterations, fractional derivative, weak solutions, maximum principle, subdiffusion equations, anomalous diffusion
Introduction and main result
============================
Let $T>0$ and $\Omega$ be a bounded domain in ${\mathbb{R}}^N$. In this paper we are concerned with linear partial integro-differential equations of the form $$\label{MProb}
\partial_t^\alpha (u-u_0)-\mbox{div}\,\big(A(t,x)Du\big)=0,\quad t\in (0,T),\,x\in
\Omega.$$ Here $u_0=u_0(x)$ is a given initial data for $u$, $A=(a_{ij})$ is ${\mathbb{R}}^{N\times N}$-valued, $Du$ denotes the spatial gradient of $u$, and $\partial_t^\alpha$ stands for the Riemann-Liouville fractional derivation operator with respect to time of order $\alpha\in (0,1)$; it is defined by $$\partial_t^\alpha v(t,x)=\partial_t\int_0^t
g_{1-\alpha}(t-\tau)v(\tau,x)\,d\tau,\quad t>0,\,x\in \Omega,$$ where $g_\beta$ denotes the Riemann-Liouville kernel $$g_\beta(t)=\,\frac{t^{\beta-1}}{\Gamma(\beta)}\,,\quad
t>0,\;\beta>0.$$
As to applications, equation (\[MProb\]) is a special case of problems arising in mathematical physics when describing dynamic processes in materials with memory, e.g. in the theory of heat conduction with memory, see [@JanI] and the references therein. Time fractional diffusion equations are also used to model anomalous diffusion, see e.g. [@Metz]. In this context, equations of the type (\[MProb\]) are termed [*subdiffusion equations*]{} (the time order $\alpha$ lies in $(0,1)$; in the case $\alpha\in(1,2)$, which is not considered here, one speaks of [*superdiffusion equations*]{}. Time fractional diffusion equations of time order $\alpha\in (0,1)$ are closely related to a class of Montroll-Weiss continuous time random walk models where the waiting time density behaves as $ t^{-\alpha-1}$ for $t\to \infty$, see e.g. [@Hilfer1], [@Hilfer2], [@Metz]. Problems of the type (\[MProb\]) are further used to describe diffusion on fractals ([@Metz], [@Roma]), and they also appear in mathematical finance, see e.g. [@Sca].
Letting $\Omega_T=(0,T)\times \Omega$ we will assume that
- $A\in L_\infty(\Omega_T;{\mathbb{R}}^{N\times
N})$, and $$\sum_{i,j=1}^N|a_{ij}(t,x)|^2\le \Lambda^2,\quad \mbox{for
a.a.}\;(t,x)\in \Omega_T.$$
- There exists $\nu>0$ such that $$\big(A(t,x)\xi|\xi\big)\ge \nu|\xi|^2,\quad\mbox{for a.a.}\;
(t,x)\in\Omega_T,\; \mbox{and all}\;\xi\in {\mathbb{R}}^N.$$
- $u_0\in L_2(\Omega)$.
We say that a function $u$ is a [*weak solution (subsolution, supersolution)*]{} of (\[MProb\]) in $\Omega_T$, if $u$ belongs to the space $$\begin{aligned}
Z_\alpha:=\{&\,v\in
L_{\frac{2}{1-\alpha},\,w}([0,T];L_2(\Omega))\cap
L_2([0,T];H^1_2(\Omega))\;
\mbox{such that}\;\\
&\;\;g_{1-\alpha}\ast v\in C([0,T];L_2(\Omega)),
\;\mbox{and}\;(g_{1-\alpha}\ast v)|_{t=0}=0\},\end{aligned}$$ and for any nonnegative test function $$\eta\in {\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^{1,1}_2(\Omega_T):=H^1_2([0,T];L_2(\Omega))\cap
L_2([0,T];{\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(\Omega)) \quad\quad
\Big({\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(\Omega):=\overline{C_0^\infty(\Omega)}\,{}^{H^1_2(\Omega)}\Big)$$ with $\eta|_{t=T}=0$ there holds $$\label{BWF}
\int_{0}^{T} \int_\Omega \Big(-\eta_t [g_{1-\alpha}\ast (u-u_0)]+
(ADu|D \eta)\Big)\,dxdt= \,(\le,\,\ge )\,0.$$ Here $L_{p,\,w}$ denotes the weak $L_p$ space and $f_1\ast f_2$ the convolution on the positive halfline with respect to time, that is $(f_1\ast f_2)(t)=\int_0^t f_1(t-\tau)f_2(\tau)\,d\tau$, $t\ge 0$.
Weak solutions of (\[MProb\]) in the class $Z_\alpha$ have been constructed in [@ZWH]. Notice also that the function $u_0$ plays the role of the initial data for $u$, at least in a weak sense. In case of sufficiently smooth functions $u$ and $g_{1-\alpha}\ast(u-u_0)$ the condition $(g_{1-\alpha}\ast
u)|_{t=0}=0$ implies $u|_{t=0}=u_0$, see [@ZWH].
To formulate our main result, let $B(x,r)$ denote the open ball with radius $r>0$ centered at $x\in {\mathbb{R}}^N$. By $\mu_N$ we mean the Lebesgue measure in ${\mathbb{R}}^N$. For $\delta\in(0,1)$, $t_0\ge 0$, $\tau>0$, and a ball $B(x_0,r)$, define the boxes $$\begin{aligned}
Q_-(t_0,x_0,r)&=(t_0,t_0+\delta\tau r^{2/\alpha})\times B(x_0,\delta r),\\
Q_+(t_0,x_0,r)&=(t_0+(2-\delta)\tau r^{2/\alpha},t_0+2\tau
r^{2/\alpha})\times B(x_0,\delta r).\end{aligned}$$
\[localweakHarnack\] Let $\alpha\in(0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1)–(H3) are satisfied. Let further $\delta\in(0,1)$, $\eta>1$, and $\tau>0$ be fixed. Then for any $t_0\ge 0$ and $r>0$ with $t_0+2\tau r^{2/\alpha}\le
T$, any ball $B(x_0,\eta r)\subset\Omega$, any $0<p<\frac{2+N\alpha}{2+N\alpha-2\alpha}$, and any nonnegative weak supersolution $u$ of (\[MProb\]) in $(0,t_0+2\tau
r^{2/\alpha})\times B(x_0,\eta r)$ with $u_0\ge 0$ in $B(x_0,\eta
r)$, there holds $$\label{localwHarnackF}
\Big(\frac{1}{\mu_{N+1}\big(Q_-(t_0,x_0,r)\big)}\,\int_{Q_-(t_0,x_0,r)}u^p\,d\mu_{N+1}\Big)^{1/p}
\le C \operatorname*{ess\,inf}_{Q_+(t_0,x_0,r)} u,$$ where the constant $C=C(\nu,\Lambda,\delta,\tau,\eta,\alpha,N,p)$.
Theorem \[localweakHarnack\] states that nonnegative weak supersolutions of (\[MProb\]) satisfy a weak form of Harnack inequality in the sense that we do not have an estimate for the supremum of $u$ on $Q_-(t_0,x_0,r)$ but only an $L_p$ estimate. We also show that the critical exponent $\frac{2+N\alpha}{2+N\alpha-2\alpha}$ is optimal, i.e. the inequality fails to hold for $p\ge
\frac{2+N\alpha}{2+N\alpha-2\alpha}$.
Theorem \[localweakHarnack\] can be viewed as the time fractional analogue of the corresponding result in the classical parabolic case $\alpha=1$, see e.g. [@Lm Theorem 6.18] and [@Trud]. Sending $\alpha\to 1$ in the expression for the critical exponent yields $1+2/N$, which is the well-known critical exponent for the heat equation. We would like to point out that the statement of Theorem \[localweakHarnack\] remains valid for (appropriately defined) weak supersolutions of (\[MProb\]) on $(t_0,t_0+2\tau
r^{2/\alpha})\times B(x_0,\eta r)$ which are nonnegative on $(0,t_0+2\tau r^{2/\alpha})\times B(x_0,\eta r)$. Here the global positivity assumption cannot be replaced by a local one, as simple examples show, cf. [@Za3]. This significant difference to the case $\alpha=1$ is due to the non-local nature of $\partial_t^\alpha$. The same phenomenon is known for integro-differential operators like $(-\Delta)^\alpha$ with $\alpha\in (0,1)$, see e.g. [@Kass1].
As a simple consequence of the weak Harnack inequality we derive the strong maximum principle for weak subsolutions of (\[MProb\]), see Theorem \[strongmax\] below. The weak maximum principle has been proven in [@Za2], even in a more general setting.
In the classical parabolic case boundedness and the weak (or full) Harnack inequality imply an Hölder estimate for weak solutions, cf. [@DB], [@LSU], [@Lm], [@Moser64]. We also refer to [@GilTrud] and [@Mosell] for the elliptic case. In the present situation one cannot argue anymore as in the classical parabolic case, due to the global positivity assumption in Theorem \[localweakHarnack\]. The same problem arises for the fractional Laplacian, see [@Silv]. However, in our case it is possible to establish at least continuity at $t=0$. This is done in Theorem \[Hoeldert=0\] in the case $u_0=0$. It is shown that in this case any bounded weak solution $u$ of (\[MProb\]) is continuous at $(0,x_0)$ for all $x_0\in \Omega$ and $\lim_{(t,x)\to
(0,x_0)}u(t,x)=0$. Thus for such weak solutions the initial condition $u|_{t=0}=0$ is satisfied in the classical sense.
As a further consequence of the weak Harnack inequality we obtain a theorem of Liouville type, see Corollary \[Liouville\] below. It states that any bounded weak solution of (\[MProb\]) on ${\mathbb{R}}_+\times {\mathbb{R}}^N$ with $u_0=0$ vanishes a.e. on ${\mathbb{R}}_+\times
{\mathbb{R}}^N$.
The proof of Theorem \[localweakHarnack\] relies on new a priori estimates for time fractional problems, which are derived by means of the fundamental identity (\[fundidentity\]) (see below) for the regularized fractional derivative. It further uses Moser’s iteration technique and an elementary but subtle lemma of Bombieri and Giusti [@BomGiu], which allows to avoid the rather technically involved approach via $BMO$-functions. This simplification is already of great significance in the classical parabolic case, see Moser [@Moser71] and Saloff-Coste [@SalCoste].
One of the technical difficulties in deriving the desired estimates in the weak setting is to find an appropriate time regularization of the problem. In the case $\alpha=1$ this can be achieved by means of Steklov averages in time. In the time fractional case this method does not work anymore, since Steklov average operators and convolution do not commute. It turns out that instead one can use the Yosida approximation of the fractional derivative, which leads to a regularization of the kernel $g_{1-\alpha}$. This method has already been used in [@Grip1], [@VZ], [@ZWH], and [@Za2].
We point out that the results obtained in this paper can be easily generalized to quasilinear equations of the form $$\label{QProb}
\partial_t^\alpha (u-u_0)-\mbox{div}\,a(t,x,u,Du)=b(t,x,u,Du),\;t\in (0,T),\,x\in \Omega,$$ with suitable structure conditions on the functions $a$ and $b$. This is possible, as also known from the elliptic and the classical parabolic case, since the test function method used in the proof of Theorem \[localweakHarnack\] does not depend so much on the linearity of the differential operator w.r.t. the spatial variables but on a certain nonlinear structure, cf. [@GilTrud], [@Lm], [@Trud], and [@Za2].
In the literature there exist many papers where equations of the type (\[MProb\]), as well as nonlinear or abstract variants of them are studied in a [*strong*]{} setting, assuming more smoothness on the coefficients and nonlinearities, see e.g. [@Ba], [@CLS], [@CP2], [@Koch], [@Grip1], [@JanI], [@ZEQ], [@ZQ]. Concerning the [*weak*]{} setting described above one finds only a few results. Existence of weak solutions has been shown in [@ZWH] in an abstract setting for a more general class of kernels. Boundedness of weak solutions has been obtained in [@Za2] in the quasilinear case by means of the De Giorgi technique. With the weak Harnack inequality, the present paper establishes a key result towards a De Giorgi-Nash-Moser theory for time fractional evolution equations in divergence form of order $\alpha\in(0,1)$.
We further remark that in the purely time-dependent case, that is for scalar equations of the form $$\partial_t^\alpha (u-u_0)+\sigma u=0,\quad t\in (0,T),$$ with $\sigma\ge 0$, a weak Harnack inequality with optimal exponent $1/(1-\alpha)$ has been proven in [@Za] for nonnegative supersolutions. Recently, the full Harnack inequality for nonnegative solutions has been established in [@Za3]. This, together with the above results, indicates that the full Harnack inequality should also hold for nonnegative solutions of (\[MProb\]), which is still an open problem, even in the case $A(t,x)\equiv Id$.
The paper is organized as follows. In Section 2 we collect the basic tools needed for the proof of Theorem \[localweakHarnack\]. These include two abstract lemmas on Moser iterations and the lemma of Bombieri and Giusti. We further explain the approximation method for the fractional derivation operator and state the fundamental identity (\[fundidentity\]), which is frequently used in Section 3, where we give the proof of the main result. In Section 4 we show that the critical exponent in Theorem \[localweakHarnack\] is optimal. Finally, Section 5 is devoted to applications of the weak Harnack inequality.
Preliminaries
=============
Moser iterations and an abstract lemma of Bombieri and Giusti
-------------------------------------------------------------
Throughout this subsection $U_\sigma$, $0<\sigma\le 1$, will denote a collection of measurable subsets of a fixed finite measure space endowed with a measure $\mu$, such that $U_{\sigma'}\subset
U_\sigma$ if $\sigma'\le \sigma$. For $p\in (0,\infty)$ and $0<\sigma\le 1$, $L_p(U_\sigma)$ stands for the Lebesgue space $L_p(U_\sigma,d\mu)$ of all $\mu$-measurable functions $f:U_\sigma\rightarrow {\mathbb{R}}$ with $|f|_{L_p(U_\sigma)}:=(\int_{U_\sigma}|f|^p\,d\mu)^{1/p}<\infty$.
The following two lemmas are basic to Moser’s iteration technique. The arguments in their proofs have been repeatedly used in the literature (see e.g. [@GilTrud], [@Lm], [@Mosell], [@Moser64], [@SalCoste], [@Trud]), so it is worthwhile to formulate them as lemmas in abstract form, also for future reference. We provide proofs for the sake of completeness.
The first Moser iteration result reads as follows, see also [@CZ Lemma 2.3].
\[moserit1\] Let $\kappa>1$, $\bar{p}\ge 1$, $C\ge 1$, and $\gamma>0$. Suppose $f$ is a $\mu$-measurable function on $U_1$ such that $$\label{mositer1}
|f|_{L_{\beta\kappa}(U_{\sigma'})}\le
\Big(\frac{C(1+\beta)^{\gamma}}{(\sigma-\sigma')^{\gamma}}\Big)^{1/\beta}\,|f|_{L_{\beta}(U_{\sigma})},
\quad 0<\sigma'<\sigma\le 1,\;\beta>0.$$ Then there exist constants $M=M(C,\gamma,\kappa,\bar{p})$ and $\gamma_0=\gamma_0(\gamma,\kappa)$ such that $$\operatorname*{ess\,sup}_{U_{\delta}}{|f|} \le
\Big(\frac{M}{(1-\delta)^{\gamma_0}}\Big)^{1/p}
|f|_{L_{p}(U_1)}\quad \mbox{for all}\;\;\delta\in(0,1),\;p\in
(0,\bar{p}] .$$
[*Proof:*]{} For $q>0$ and $0<\sigma\le 1$, let $$\Phi(q,\sigma)=(\int_{U_\sigma} |f|^q\,d\mu)^{1/q}.$$ Let $0<p\le \bar{p}$ and $\delta\in (0,1)$. Set $p_i=p\kappa^{i}$, $i=0,1,\ldots$ and define the sequence $\{\sigma_i\}$, $i=0,1,\ldots$, by $\sigma_0=1$ and $\sigma_i=1-\sum_{j=1}^i 2^{-j}
(1-\delta)$, $i=1,2,\ldots$; observe that $1=\sigma_0>\sigma_1>\ldots>\sigma_i>\sigma_{i+1}>\delta$ as well as $\sigma_{i-1}-\sigma_{i}=2^{-i}(1-\delta)$, $i\ge 1$. Suppose now $n\in {\mathbb{N}}$. By using (\[mositer1\]) with $\beta=p_i$, $i=0,1,\ldots,n-1$, we obtain $$\begin{aligned}
\Phi(p_n,\delta)& \;\le
\Phi(p_n,\sigma_n)\;=\;\Phi(p_{n-1}\kappa,\sigma_n)\;\le\;
\Big(\frac{C(1+p\kappa^{n-1})^{\gamma}}{[2^{-n}(1-\delta)]^{\gamma}}\Big)^
{\frac{1}{p}\,\kappa^{-(n-1)}}\Phi(p_{n-1},\sigma_{n-1})\\ & \;\le
\Big(\frac{C(2\bar{p}\kappa^{n-1})^{\gamma}}{[2^{-n}(1-\delta)]^{\gamma}}\Big)^
{\frac{1}{p}\,\kappa^{-(n-1)}}\Phi(p_{n-1},\sigma_{n-1})\\ &\;\le
\Big(\frac{\tilde{C}(C,\bar{p},\gamma)^n
\kappa^{\gamma(n-1)}}{(1-\delta)^{\gamma}}\Big)^
{\frac{1}{p}\,\kappa^{-(n-1)}}\Phi(p_{n-1},\sigma_{n-1})\;\le\;\ldots\\
& \;\le \Big(\tilde{C}^{\sum_{j=0}^{n-1}
(j+1)\kappa^{-j}}\kappa^{\gamma\sum_{j=0}^{n-1}
j\kappa^{-j}}(1-\delta)^{-\gamma\sum_{j=0}^{n-1}
\kappa^{-j}}\Big)^{1/p}\,\Phi(p_0,\sigma_0)\\ & \; \le
\Big(\frac{M(C,\bar{p},\gamma,\kappa)}{(1-\delta)^{\frac{\gamma\kappa}{\kappa-1}}}\Big)^
{1/p}\,\Phi(p,1).\end{aligned}$$ We let now $n$ tend to $\infty$ and use the fact that $$\lim_{n\to\infty}\Phi(p_n,\delta)=\operatorname*{ess\,sup}_{U_{\delta}}{|f|}$$ to get $$\operatorname*{ess\,sup}_{U_{\delta}}{|f|} \le
\Big(\frac{M(C,\bar{p},\gamma,\kappa)}{(1-\delta)^{\frac{\gamma\kappa}{\kappa-1}}}\Big)^
{1/p}\,|f|_{L_{p}(U_1)}.$$ Hence the proof is complete. $\square$ $\mbox{}$
The second Moser iteration result is the following, see also [@CZ Lemma 2.5].
\[moserit2\] Assume that $\mu(U_1)\le 1$. Let $\kappa>1$, $0<p_0<\kappa$, and $C\ge 1,\,\gamma>0$. Suppose $f$ is a $\mu$-measurable function on $U_1$ such that $$\label{mositer2}
|f|_{L_{\beta\kappa}(U_{\sigma'})}\le
\Big(\frac{C}{(\sigma-\sigma')^{\gamma}}\Big)^{1/\beta}\,|f|_{L_{\beta}(U_{\sigma})},
\quad 0<\sigma'<\sigma\le 1,\;0<\beta\le \frac{p_0}{\kappa}<1.$$ Then there exist constants $M=M(C,\gamma,\kappa)$ and $\gamma_0=\gamma_0(\gamma,\kappa)$ such that $$|f|_{L_{p_0}(U_{\delta})}\le
\Big(\frac{M}{(1-\delta)^{\gamma_0}}\Big)^{1/p-1/p_0}
|f|_{L_{p}(U_1)}\quad \mbox{for all}\;\;\delta\in(0,1),\;p\in
(0,\frac{p_0}{\kappa}] .$$
[*Proof:*]{} Set $p_i=p_0 \kappa^{-i}$, $i=1,2,\ldots$. Given $\delta\in(0,1)$ we take again the sequence $\{\sigma_i\}$, $i=0,1,2,\ldots$, defined by $\sigma_0=1$ and $\sigma_i=1-\sum_{j=1}^i 2^{-j} (1-\delta)$, $i\ge 1$. Suppose now $n\in {\mathbb{N}}$. By using (\[mositer2\]) with $\beta=p_i$, $i=1,\ldots,n$, we obtain $$\begin{aligned}
\Phi(p_0,\delta) & \le \; \Phi(p_0,\sigma_n)\;=\;\Phi(p_1
\kappa,\sigma_n)\;\le
\;\frac{C^{\kappa/p_0}}{[2^{-n}(1-\delta)]^{\gamma
\kappa/p_0}}\;\Phi(p_1,\sigma_{n-1})\\
& \le \; \frac{C^{\kappa/p_0}}{[2^{-n}(1-\delta)]^{\gamma
\kappa/p_0}}\; \frac{C^{\kappa^2/p_0}}{[2^{-(n-1)}(1-\delta)]^{\gamma
\kappa^2/p_0}}\;\Phi(p_2,\sigma_{n-2})\;\le\;\dots\\
& \le \; \frac{C^{\frac{1}{p_0}\,(\kappa+\kappa^2+\ldots+\kappa^n)}}
{2^{-\frac{\gamma}{p_0}\,(n\kappa+(n-1)\kappa^2+\ldots+2\kappa^{n-1}+\kappa^n)}
(1-\delta)^{\frac{\gamma}{p_0}\,(\kappa+\kappa^2+\ldots+\kappa^n)}}\;\Phi(p_n,\sigma_0).\end{aligned}$$ Since $p_i=p_0 \kappa^{-i}$, we have $$\frac{1}{p_0}\,\sum_{j=1}^n \kappa^j = \frac{\kappa(\kappa^n-1)}{p_0(\kappa-1)}
= \frac{\kappa}{p_0(\kappa-1)}\;(\frac{p_0}{p_n}-1)=
\frac{\kappa}{\kappa-1}\;(\frac{1}{p_n}-\frac{1}{p_0}).$$ Employing the formula $$\sum_{j=1}^n j
\kappa^{j-1}=\frac{1-(n+1)\kappa^n+n\kappa^{n+1}}{(\kappa-1)^2}$$ we have further $$\begin{aligned}
\sum_{j=1}^n (n+1-j)\kappa^j & = \; (n+1)\sum_{j=1}^n
\kappa^j-\sum_{j=1}^n j \kappa^j\\ & =
\;(n+1)\kappa\,\frac{\kappa^n-1}{\kappa-1}-\kappa\;
\frac{1-(n+1)\kappa^n+n\kappa^{n+1}}{(\kappa-1)^2}\\ & = \;
\kappa\;\frac{\kappa^{n+1}-(n+1)\kappa+n}{(\kappa-1)^2}\;\le \;
\frac{\kappa}{(\kappa-1)^2}\kappa^{n+1}\\ & \le \;
\frac{\kappa^3}{(\kappa-1)^3}\;(\kappa^n-1)\;\le\;\frac{\kappa^3}{(\kappa-1)^3}\;
(\frac{p_0}{p_n}-1),\end{aligned}$$ which yields $$\frac{1}{p_0}\,\sum_{j=1}^n (n+1-j)\kappa^j \le \frac{\kappa^3}{(\kappa-1)^3}\;
(\frac{1}{p_n}-\frac{1}{p_0}).$$ Therefore $$\Phi(p_0,\delta) \le \Big[
\frac{2^{\frac{\gamma
\kappa^3}{(\kappa-1)^3}}C^{\frac{\kappa}{\kappa-1}}}
{(1-\delta)^{\frac{\gamma\kappa}{\kappa-1}}}\Big]^
{\frac{1}{p_n}-\frac{1}{p_0}} \Phi(p_n,\sigma_0).$$ Given $p\in(0,p_0/\kappa]$ there exists $n\ge 2$ such that $p_n<p\le
p_{n-1}$. We then have $$\begin{aligned}
\frac{1}{p_n}-\frac{1}{p_0} & \; =\frac{\kappa^n-1}{p_0}\;\le
\;\frac{\kappa^n+\kappa^{n-1}-\kappa-1}{p_0}\;=\;
\frac{(1+\kappa)(\kappa^{n-1}-1)}{p_0}\\ & \; =
(1+\kappa)(\frac{1}{p_{n-1}}-\frac{1}{p_0})\;\le\;(1+\kappa)(\frac{1}{p}-\frac{1}{p_0}),\end{aligned}$$ as well as $$\Phi(p_n,\sigma_0)=\Phi(p_n,1)\le \Phi(p,1),$$ by Hölder’s inequality and the assumption $\mu(U_1)\le 1$. All in all, we obtain $$\Phi(p_0,\delta)\le \Big[
\frac{2^{\frac{\gamma
\kappa^3}{(\kappa-1)^3}}C^{\frac{\kappa}{\kappa-1}}}
{(1-\delta)^{\frac{\gamma\kappa}{\kappa-1}}}\Big]^
{(1+\kappa)(\frac{1}{p}-\frac{1}{p_0})}\Phi(p,1),$$ which proves the lemma. $\square$ $\mbox{}$
The following abstract lemma is due to Bombieri and Giusti [@BomGiu]. For a proof we also refer to [@SalCoste Lemma 2.2.6] and [@CZ Lemma 2.6]
\[abslemma\] Let $\delta,\,\eta\in(0,1)$, and let $\gamma,\,C$ be positive constants and $0<\beta_0\le \infty$. Suppose $f$ is a positive $\mu$-measurable function on $U_1$ which satisfies the following two conditions:
\(i) $$|f|_{L_{\beta_0}(U_{\sigma'})}\le
[C(\sigma-\sigma')^{-\gamma}\mu(U_1)^{-1}]^{1/\beta-1/\beta_0}|f|_{L_{\beta}(U_{\sigma})},$$ for all $\sigma,\,\sigma',\,\beta$ such that $0<\delta\le
\sigma'<\sigma\le 1$ and $0<\beta\le \min\{1,\eta\beta_0\}$.
\(ii) $$\mu(\{\log f>\lambda\})\le C\mu(U_1)\lambda^{-1}$$ for all $\lambda>0$.
Then $$|f|_{L_{\beta_0}(U_{\delta})}\le M \mu(U_1)^{1/\beta_0},$$ where $M$ depends only on $\delta,\,\eta,\,\gamma,\,C$, and $\beta_0$.
The Yoshida approximation of the fractional derivation operator {#SecYos}
---------------------------------------------------------------
Let $0<\alpha<1$, $1\le p<\infty$, $T>0$, and $X$ be a real Banach space. Then the fractional derivation operator $B$ defined by $$B u=\,\frac{d}{dt}\,(g_{1-\alpha}\ast u),\;\;D(B)=\{u\in L_p([0,T];X):\,g_{1-\alpha}\ast u\in \mbox{}_0 H^1_p([0,T];X)\},$$ where the zero means vanishing at $t=0$, is known to be $m$-accretive in $L_p([0,T];X)$, cf. [@Phil1], [@CP], and [@Grip1]. Its Yosida approximations $B_{n}$, defined by $B_{n}=nB(n+B)^{-1},\,n\in {\mathbb{N}}$, enjoy the property that for any $u\in D(B)$, one has $B_{n}u\rightarrow Bu$ in $L_p([0,T];X)$ as $n\to \infty$. Further, one has the representation $$\label{Yos}
B_n u=\,\frac{d}{dt}\,(g_{1-\alpha,n}\ast u),\quad u\in
L_p([0,T];X),\;n\in {\mathbb{N}},$$ where $g_{1-\alpha,n}=n s_{\alpha,n}$, and $s_{\alpha,n}$ is the unique solution of the scalar-valued Volterra equation $$s_{\alpha,n}(t)+n(s_{\alpha,n}\ast g_\alpha)(t)=1,\quad
t>0,\;n\in{\mathbb{N}},$$ see e.g. [@VZ]. Let $h_{\alpha,n}\in L_{1,\,loc}({\mathbb{R}}_+)$ be the resolvent kernel associated with $ng_\alpha$, that is $$\label{hndef}
h_{\alpha,n}(t)+n(h_{\alpha,n}\ast g_\alpha)(t)=ng_\alpha(t),\quad
t>0,\;n\in{\mathbb{N}}.$$ Convolving (\[hndef\]) with $g_{1-\alpha}$ and using $g_\alpha\ast
g_{1-\alpha}=1$, we obtain $$(g_{1-\alpha}\ast h_{\alpha,n})(t)+n([g_{1-\alpha}\ast
h_{\alpha,n}]\ast g_\alpha)(t)=n,\quad t>0,\;n\in{\mathbb{N}}.$$ Hence $$\label{gnprop}
g_{1-\alpha,n}=ns_{\alpha,n}=g_{1-\alpha}\ast h_{\alpha,n},\quad
n\in {\mathbb{N}}.$$ The kernels $g_{1-\alpha,n}$ are nonnegative and nonincreasing for all $n\in{\mathbb{N}}$, and they belong to $H^1_1([0,T])$, cf. [@JanI] and [@VZ]. Note that for any function $f\in L_p([0,T];X)$, $1\le
p<\infty$, there holds $h_{\alpha,n}\ast f\to f$ in $L_p([0,T];X)$ as $n\to \infty$. In fact, setting $u=g_\alpha\ast f$, we have $u\in
D(B)$, and $$B_n u=\,\frac{d}{dt}\,(g_{1-\alpha,n}\ast
u)=\,\frac{d}{dt}\,(g_{1-\alpha}\ast g_\alpha\ast h_{\alpha,n}\ast
f)=h_{\alpha,n}\ast f\,\to\,Bu=f\quad\mbox{in}\;L_p([0,T];X)$$ as $n\to \infty$. In particular, $g_{1-\alpha,n}\to g_{1-\alpha}$ in $L_1([0,T])$ as $n\to \infty$.
We next state a fundamental identity for integro-differential operators of the form $\frac{d}{dt}(k\ast u)$, cf. also [@Za2]. Suppose $k\in H^1_1([0,T])$ and $H\in C^1({\mathbb{R}})$. Then it follows from a straightforward computation that for any sufficiently smooth function $u$ on $(0,T)$ one has for a.a. $t\in (0,T)$, $$\begin{aligned}
\label{fundidentity}
H'(u(t))&\frac{d}{dt}\,(k \ast u)(t) =\;\frac{d}{dt}\,(k\ast
H(u))(t)+
\Big(-H(u(t))+H'(u(t))u(t)\Big)k(t) \nonumber\\
& +\int_0^t
\Big(H(u(t-s))-H(u(t))-H'(u(t))[u(t-s)-u(t)]\Big)[-\dot{k}(s)]\,ds,\end{aligned}$$ where $\dot{k}$ denotes the derivative of $k$. In particular this identity applies to the Yosida approximations of the fractional derivation operator. We remark that an integrated version of (\[fundidentity\]) can be found in [@GLS Lemma 18.4.1]. Observe that the last term in (\[fundidentity\]) is nonnegative in case $H$ is convex and $k$ is nonincreasing.
The subsequent two lemmas are also obtained by simple algebra.
\[comm\] Let $T>0$ and $\alpha\in (0,1)$. Suppose that $v\in
{}_0H^1_1([0,T])$ and $\varphi\in C^1([0,T])$. Then $$\big(g_\alpha\ast(\varphi \dot{v}))(t)=\varphi(t)(g_\alpha\ast
\dot{v})(t)+\int_0^t
v(\sigma)\partial_\sigma\big(g_\alpha(t-\sigma)
[\varphi(t)-\varphi(\sigma)]\big)\,d\sigma,\;\;\mbox{a.a.}\;t\in
(0,T).$$ If in addition $v$ is nonnegative and $\varphi$ is nondecreasing there holds $$\big(g_\alpha\ast(\varphi \dot{v}))(t)\ge \varphi(t)(g_\alpha\ast
\dot{v})(t)-\int_0^t g_\alpha(t-\sigma)
\dot{\varphi}(\sigma)v(\sigma)\,d\sigma,\;\;\mbox{a.a.}\;t\in
(0,T).$$
\[comm2\] Let $T>0$, $k\in H^1_1([0,T])$, $v\in L_1([0,T])$, and $\varphi\in
C^1([0,T])$. Then $$\varphi(t)\,\frac{d}{dt}\,(k\ast v)(t)=\,\frac{d}{dt}\,\big(k\ast
[\varphi v]\big)(t)+\int_0^t
\dot{k}(t-\tau)\big(\varphi(t)-\varphi(\tau)\big)v(\tau)\,d\tau,\;\;\mbox{a.a.}\;t\in
(0,T).$$
An embedding result and a weighted Poincaré inequality
------------------------------------------------------
Let $T>0$ and $\Omega$ be a bounded domain in ${\mathbb{R}}^N$. For $1<p\le
\infty$ we define the space $$\label{Vdef}
V_p:=V_p([0,T]\times \Omega)=L_{2p}([0,T];L_2(\Omega))\cap
L_2([0,T];H^1_2(\Omega)),$$ endowed with the norm $$|u|_{V_p([0,T]\times \Omega)}:=|u|_{L_{2p}([0,T];L_2(\Omega))}
+|Du|_{L_2([0,T];L_2(\Omega))}.$$ Set $$\label{kappa}
\kappa:=\kappa_p:=\,\frac{2p+N(p-1)}{2+N(p-1)}$$ with $\kappa_\infty=1+2/N$. Then $V_p\hookrightarrow
L_{2\kappa}([0,T]\times\Omega)$, and $$\label{Vembedding}
|u|_{L_{2\kappa}([0,T]\times\Omega)}\le C(N,p)|u|_{V_p([0,T]\times
\Omega)},$$ for all $u\in V_p\cap L_2([0,T];{\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(\Omega))$. This is a consequence of the Gagliardo-Nirenberg and Hölder’s inequality. The case $p=\infty$ is contained, e.g., in [@LSU p. 74 and 75]. The proof given there easily extends to the general case. For a more general embedding result (without proof) we also refer to [@Za2 Section 2].
The following result can be found in [@Moser64 Lemma 3], see also [@Lm Lemma 6.12].
\[WeiPI\] Let $\varphi\in C({\mathbb{R}}^N)$ with non-empty compact support of diameter $d$ and assume that $0\le \varphi\le 1$. Suppose that the domains $\{x\in{\mathbb{R}}^N:\varphi(x)\ge a\}$ are convex for all $a\le 1$. Then for any function $u\in H^{1}_2({\mathbb{R}}^N)$, $$\int_{{\mathbb{R}}^N} \big(u(x)-u_\varphi\big)^2 \varphi(x)\,dx \le \,\frac{2
d^2\mu_N(\mbox{{\em supp}}\,\varphi)}{|\varphi|_{L_1({\mathbb{R}}^N)}}\,
\int_{{\mathbb{R}}^N} |Du(x)|^2 \varphi(x)\,dx,$$ where $$u_\varphi=\frac{\int_{{\mathbb{R}}^N} u(x)\varphi(x)\,dx}{\int_{{\mathbb{R}}^N}
\varphi(x)\,dx}.$$
Proof of the main result
========================
The regularized weak formulation, time shifts, and scalings {#SSS}
-----------------------------------------------------------
The following lemma is basic to deriving [*a priori*]{} estimates for weak (sub-/super-) solutions of (\[MProb\]). It provides an equivalent weak formulation of (\[MProb\]) where the singular kernel $g_{1-\alpha}$ is replaced by the more regular kernel $g_{1-\alpha,n}$ ($n\in{\mathbb{N}}$) given in (\[gnprop\]). In what follows the kernels $h_n:=h_{\alpha,n}$, $n\in{\mathbb{N}}$, are defined as in Section \[SecYos\].
\[LemmaReg\] Let $\alpha\in (0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1)–(H3) are satisfied. Then $u\in
Z_\alpha$ is a weak solution (subsolution, supersolution) of (\[MProb\]) in $\Omega_T$ if and only if for any nonnegative function $\psi\in {\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(\Omega)$ one has $$\label{LemmaRegF}
\int_\Omega \Big(\psi \partial_t[g_{1-\alpha,n}\ast
(u-u_0)]+(h_n\ast [ADu]|D\psi)\Big)\,dx\nonumber\\
=\,(\le,\,\ge)\,0,\quad\mbox{a.a.}\;t\in (0,T),\,n\in {\mathbb{N}}.$$
For a proof we refer to Lemma 3.1 in [@Za2], where a more general situation is considered with a slightly different function space for the solution. The proof of Lemma \[LemmaReg\] is analogous.
Let $u\in Z_\alpha$ be a weak supersolution of (\[MProb\]) in $\Omega_T$ and assume that $u_0\ge 0$ in $\Omega$. Then Lemma \[LemmaReg\] and positivity of $g_{1-\alpha,n}$ imply that $$\label{u0weg}
\int_\Omega \Big(\psi \partial_t(g_{1-\alpha,n}\ast u)+(h_n\ast
[ADu]|D\psi)\Big)\,dx \ge \,0,\quad\mbox{a.a.}\;t\in (0,T),\,n\in
{\mathbb{N}},$$ for any nonnegative function $\psi\in {\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(\Omega)$.
Let now $t_1\in (0,T)$ be fixed. For $t\in (t_1,T)$ we introduce the shifted time $s=t-t_1$ and set $\tilde{f}(s)=f(s+t_1)$, $s\in
(0,T-t_1)$, for functions $f$ defined on $(t_1,T)$. From the decomposition $$(g_{1-\alpha,n}\ast u)(t,x)=\int_{t_1}^t
g_{1-\alpha,n}(t-\tau)u(\tau,x)\,d\tau+\int_{0}^{t_1}
g_{1-\alpha,n}(t-\tau)u(\tau,x)\,d\tau,\quad t\in (t_1,T),$$ we then deduce that $$\label{shiftprop}
\partial_t(g_{1-\alpha,n}\ast u)(t,x)=\partial_s(g_{1-\alpha,n}\ast
\tilde{u})(s,x)+\int_0^{t_1}\dot{g}_{1-\alpha,n}(s+t_1-\tau)u(\tau,x)\,d\tau.$$ Assuming in addition that $u\ge 0$ on $(0,t_1)\times \Omega$ it follows from (\[u0weg\]), (\[shiftprop\]), and the positivity of $\psi$ and of $-\dot{g}_{1-\alpha,n}$ that $$\label{shiftprob}
\int_\Omega \Big(\psi \partial_s(g_{1-\alpha,n}\ast
\tilde{u})+\big((h_n\ast [ADu])\,\tilde{}\;|D\psi\big)\Big)\,dx \ge
\,0,\quad\mbox{a.a.}\;s\in (0,T-t_1),\,n\in {\mathbb{N}},$$ for any nonnegative function $\psi\in {\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(\Omega)$. This relation will be the starting point for all of the estimates below.
We conclude this section with a remark on the scaling properties of equation (\[MProb\]). Let $t_0,r>0$ and $x_0\in {\mathbb{R}}^N$. Suppose $u\in Z_\alpha$ is a weak solution (subsolution, supersolution) of (\[MProb\]) in $(0,t_0 r^{2/\alpha})\times B(x_0,r)$. Changing the coordinates according to $s=t/r^{2/\alpha}$ and $y=(x-x_0)/r$ and setting $v(s,y)=u(sr^{2/\alpha},x_0+yr)$, $v_0(y)=u_0(x_0+yr)$, and $\tilde{A}(s,y)=A(sr^{2/\alpha},x_0+yr)$, the problem for $u$ is transformed to a problem for $v$ in $(0,t_0)\times B(0,1)$, namely there holds with $D=D_y$ (also in the weak sense) $$\label{MProbScal}
\partial_s^\alpha (v-v_0)-\mbox{div}\,\big(\tilde{A}(s,y)Dv\big)=\,(\le,\,\ge)\,0,\quad
s\in (0,t_0),\,y\in B(0,1).$$
Mean value inequalities
-----------------------
For $\sigma>0$ we put $\sigma B(x,r):=B(x,\sigma r)$. Recall that $\mu_N$ denotes the Lebesgue measure in ${\mathbb{R}}^N$.
\[superest1\] Let $\alpha\in(0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1)–(H3) are satisfied. Let $\eta>0$ and $\delta\in (0,1)$ be fixed. Then for any $t_0\in(0,T]$ and $r>0$ with $t_0-\eta r^{2/\alpha}\ge 0$, any ball $B=B(x_0,r)\subset\Omega$, and any weak supersolution $u\ge
\varepsilon>0$ of (\[MProb\]) in $(0,t_0)\times B$ with $u_0\ge 0$ in $B$ , there holds $$\operatorname*{ess\,sup}_{U_{\sigma'}}{u^{-1}} \le \Big(\frac{C \mu_{N+1}(U_1)^{-1}
}{(\sigma-\sigma')^{\tau_0}}\Big)^{1/\gamma}
|u^{-1}|_{L_{\gamma}(U_\sigma)},\quad \delta\le \sigma'<\sigma\le
1,\; \gamma\in (0,1].$$ Here $U_\sigma=(t_0-\sigma\eta r^{2/\alpha},t_0)\times \sigma B$, $0<\sigma\le 1$, $C=C(\nu,\Lambda,\delta,\eta,\alpha,N)$ and $\tau_0=\tau_0(\alpha,N)$.
[*Proof:*]{} We may assume that $r=1$ and $x_0=0$. In fact, in the general case we change coordinates as $t\rightarrow
t/r^{2/\alpha}$ and $x\rightarrow (x-x_0)/r$, thereby transforming the equation to a problem of the same type on $(0,t_0/r^{2/\alpha})\times B(0,1)$, cf. Section \[SSS\].
Fix $\sigma'$ and $\sigma$ such that $\delta\le \sigma'<\sigma\le
1$ and put $B_1=\sigma B$. For $\rho\in (0,1]$ we set $V_\rho=U_{\rho\sigma}$. Given $0<\rho'<\rho\le 1$, let $t_1=t_0-\rho\sigma\eta $ and $t_2=t_0-\rho'\sigma\eta
$. Then $0\le t_1<t_2<t_0$. We introduce further the shifted time ${s}=t-t_1$ and set $\tilde{f}(s)=f(s+t_1)$, $s\in
(0,t_0-t_1)$, for functions $f$ defined on $(t_1,t_0)$. Since $u_0\ge 0$ in $B$ and $u$ is a positive weak supersolution of (\[MProb\]) in $(0,t_0)\times B$, we have (cf. (\[shiftprob\])) $$\label{sup0}
\int_B \Big(v \partial_s(g_{1-\alpha,n}\ast \tilde{u})+\big((h_n\ast
[ADu])\,\tilde{}\;|Dv\big)\Big)\,dx \ge \,0,\quad\mbox{a.a.}\;s\in
(0,t_0-t_1),\,n\in {\mathbb{N}},$$ for any nonnegative function $v\in {\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(B)$. For $s\in
(0,t_0-t_1)$ we choose the test function $v=\psi^2
\tilde{u}^{\beta}$ with $\beta<-1$ and $\psi\in C^1_0(B_1)$ so that $0\le \psi\le 1$, $\psi=1$ in $\rho'B_1$, supp$\,\psi\subset \rho
B_1$, and $|D \psi|\le 2/[\sigma (\rho-\rho')]$. By the fundamental identity (\[fundidentity\]) applied to $k=g_{1-\alpha,n}$ and the convex function $H(y)=-(1+\beta)^{-1}y^{1+\beta}$, $y>0$, there holds for a.a. $(s,x)\in (0,t_0-t_1)\times B$ $$\begin{aligned}
-\tilde{u}^{\beta}\partial_{s}(g_{1-\alpha,n}\ast \tilde{u}) & \ge -\,\frac{1}{1+\beta}\,\partial_{s}
(g_{1-\alpha,n}\ast\tilde{u}^{1+\beta})+\Big(\frac{\tilde{u}^{1+\beta}}{1+\beta}\,-\tilde{u}^{1+\beta}\Big)g_{1-\alpha,n}\nonumber\\
& =
-\,\frac{1}{1+\beta}\,\partial_{s}
(g_{1-\alpha,n}\ast\tilde{u}^{1+\beta})-\,\frac{\beta}{1+\beta}\,\tilde{u}^{1+\beta}
g_{1-\alpha,n}. \label{sup1}\end{aligned}$$ We further have $$Dv=2\psi D\psi \,\tilde{u}^{\beta}+\beta\psi^2 \tilde{u}^{\beta-1}D \tilde{u}.$$ Using this and (\[sup1\]) it follows from (\[sup0\]) that for a.a. $s\in (0,t_0-t_1)$ $$\begin{aligned}
-\,\frac{1}{1+\beta}\,& \int_{B_1}\psi^2\partial_{s}
(g_{1-\alpha,n}\ast\tilde{u}^{1+\beta})\,dx+|\beta|\int_{B_1}\big((h_n\ast
[ADu])\,\tilde{}\;|\psi^2 \tilde{u}^{\beta-1}D
\tilde{u}\big)\,dx \nonumber\\
\le & \,2\int_{B_1}\big((h_n\ast [ADu])\,\tilde{}\;|\psi D\psi
\,\tilde{u}^{\beta}\big)\,dx+\,\frac{\beta}{1+\beta}\,\int_{B_1}\psi^2\tilde{u}^{1+\beta}
g_{1-\alpha,n}\,dx. \label{sup2}\end{aligned}$$ Next, choose $\varphi\in C^1([0,t_0-t_1])$ such that $0\le
\varphi\le 1$, $\varphi=0$ in $[0,(t_2-t_1)/2]$, $\varphi=1$ in $[t_2-t_1,t_0-t_1]$, and $0\le \dot{\varphi}\le 4/(t_2-t_1)$. Multiplying (\[sup2\]) by $-(1+\beta)>0$ and by $\varphi(s)$, and convolving the resulting inequality with $g_\alpha$ yields $$\begin{aligned}
\int_{B_1} & g_\alpha\ast
\big(\varphi\partial_{s}(g_{1-\alpha,n}\ast
[\psi^2\tilde{u}^{1+\beta}])\big)\,dx+\beta(1+\beta)\,g_\alpha\ast\int_{B_1}\big((h_n\ast
[ADu])\,\tilde{}\;|\psi^2 \tilde{u}^{\beta-1}D
\tilde{u}\big)\varphi\,dx \nonumber\\
\le \, & \,2|1+\beta|\,g_\alpha\ast\int_{B_1}\big((h_n\ast
[ADu])\,\tilde{}\;|\psi D\psi
\,\tilde{u}^{\beta}\big)\varphi\,dx+|\beta|\,g_\alpha\ast\int_{B_1}\psi^2\tilde{u}^{1+\beta}
g_{1-\alpha,n}\varphi\,dx, \label{sup3}\end{aligned}$$ for a.a. $s\in (0,t_0-t_1)$. By Lemma \[comm\], $$\begin{aligned}
\int_{B_1} g_\alpha\ast &
\big(\varphi\partial_{s}(g_{1-\alpha,n}\ast
[\psi^2\tilde{u}^{1+\beta}])\big)\,dx \ge \int_{B_1} \varphi
g_\alpha\ast \big(\partial_{s}(g_{1-\alpha,n}\ast
[\psi^2\tilde{u}^{1+\beta}])\big)\,dx\nonumber\\
& -\int_0^s g_\alpha(s-\sigma)\dot{\varphi}(\sigma)
\big(g_{1-\alpha,n}\ast
\int_{B_1}\psi^2\tilde{u}^{1+\beta}\,dx\big)(\sigma)\,d\sigma.
\label{sup4}\end{aligned}$$ Furthermore, by virtue of $$g_{1-\alpha,n}\ast [\psi^2\tilde{u}^{1+\beta}]\in
{}_0H^1_1([0,t_0-t_1];L_1(B_1))$$ and $g_{1-\alpha,n}=g_{1-\alpha}\ast h_n$ as well as $g_\alpha\ast
g_{1-\alpha}=1$ we have $$\label{sup5}
g_\alpha\ast \partial_{s}(g_{1-\alpha,n}\ast
[\psi^2\tilde{u}^{1+\beta}])=\partial_s(g_\alpha\ast
g_{1-\alpha,n}\ast [\psi^2\tilde{u}^{1+\beta}])=h_n\ast
(\psi^2\tilde{u}^{1+\beta}).$$ Combining (\[sup3\]), (\[sup4\]), and (\[sup5\]), sending $n\to \infty$, and selecting an appropriate subsequence, if necessary, we thus obtain $$\begin{aligned}
& \int_{B_1}\varphi\psi^2\tilde{u}^{1+\beta}\,dx+
\beta(1+\beta)\,g_\alpha\ast\int_{B_1}\big(\tilde{A}D\tilde{u}|\psi^2
\tilde{u}^{\beta-1}D \tilde{u}\big)\varphi\,dx\nonumber\\
\le \, &
\,2|1+\beta|\,g_\alpha\ast\int_{B_1}\big(\tilde{A}D\tilde{u}|\psi
D\psi
\,\tilde{u}^{\beta}\big)\varphi\,dx+|\beta|\,g_\alpha\ast\int_{B_1}\psi^2\tilde{u}^{1+\beta}
g_{1-\alpha}\varphi\,dx \nonumber\\
& +\int_0^s g_\alpha(s-\sigma)\dot{\varphi}(\sigma)
\big(g_{1-\alpha}\ast
\int_{B_1}\psi^2\tilde{u}^{1+\beta}\,dx\big)(\sigma)\,d\sigma,
\;\;\mbox{a.a.}\;s\in(0,t_0-t_1).
\label{sup6}\end{aligned}$$ Put $w=\tilde{u}^{\frac{\beta+1}{2}}$. Then $Dw=\frac{\beta+1}{2}
\tilde{u}^{\frac{\beta-1}{2}} D\tilde{u}$. By assumption (H2), we have $$\begin{aligned}
\beta(1+\beta)\,g_\alpha\ast\int_{B_1}\big(\tilde{A}D\tilde{u}|\psi^2
\tilde{u}^{\beta-1}D \tilde{u}\big)\varphi\,dx & \,\ge \nu
\beta(1+\beta)\,g_\alpha\ast\int_{B_1} \varphi
\psi^2\tilde{u}^{\beta-1}|D\tilde{u}|^2\,dx \nonumber\\
& \, = \,\frac{4\nu \beta}{1+\beta}\,g_\alpha\ast\int_{B_1}\varphi
\psi^2|Dw|^2\,dx. \label{sup7}\end{aligned}$$ Using (H1) and Young’s inequality we may estimate $$\begin{aligned}
2\big|\big(\tilde{A}D\tilde{u}|\psi D\psi
\,\tilde{u}^{\beta}\big)\varphi\big| & \le 2\Lambda\psi|D\psi|\,|D
\tilde{u}|\tilde{u}^\beta \varphi=2\Lambda\psi|D\psi|\,|D
\tilde{u}|\tilde{u}^{\frac{\beta-1}{2}}\tilde{u}^{\frac{\beta+1}{2}}\varphi\nonumber\\
& \le \,\frac{\nu |\beta|}{2}\, \psi^2\varphi |D \tilde{u}|^2
\tilde{u}^{\beta-1}+\,\frac{2}{\nu |\beta|}\,\Lambda^2
|D\psi|^2\varphi \tilde{u}^{\beta+1}\nonumber\\
& = \,\frac{2\nu |\beta|}{(1+\beta)^2}\,
\psi^2\varphi|Dw|^2+\,\frac{2}{\nu |\beta|}\,\Lambda^2
|D\psi|^2\varphi w^2. \label{sup8}\end{aligned}$$ From (\[sup6\]), (\[sup7\]), and (\[sup8\]) we conclude that $$\label{sup9}
\int_{B_1}\varphi\psi^2w^2\,dx+\,\frac{2\nu
|\beta|}{|1+\beta|}\,g_\alpha\ast\int_{B_1}\varphi \psi^2|Dw|^2\,dx
\le g_\alpha\ast F,\quad\mbox{a.a.}\;s\in(0,t_0-t_1),$$ where $$\begin{aligned}
F(s) =\, & \,\frac{2\Lambda^2|1+\beta|}{\nu |\beta|}\, \int_{B_1}
|D\psi|^2\varphi w^2\,dx
+|\beta|\varphi(s)g_{1-\alpha}(s)\int_{B_1}\psi^2 w^2
\,dx \\
& \,+\dot{\varphi}(s) \big(g_{1-\alpha}\ast \int_{B_1}\psi^2
w^2\,dx\big)(s)\ge 0,\quad\mbox{a.a.}\;s\in(0,t_0-t_1).\end{aligned}$$ We may drop the second term in (\[sup9\]), which is nonnegative. By Young’s inequality for convolutions and the properties of $\varphi$ we then infer that for all $p\in(1,1/(1-\alpha))$ $$\label{sup10}
\Big(\int_{t_2-t_1}^{t_0-t_1} (\int_{B_1}
[\psi(x)w(s,x)]^2\,dx)^p\,ds\Big)^{1/p} \,\le
|g_\alpha|_{L_p([0,t_0-t_1])} \int_0^{t_0-t_1} \!\!\!\!F(s)\,ds,$$ where $$\label{sup11}
|g_\alpha|_{L_p([0,t_0-t_1])}=
\,\frac{(t_0-t_1)^{\alpha-1+1/p}}{\Gamma(\alpha)[(\alpha-1)p+1]^{1/p}}\,\le
\,\frac{\eta^{\alpha-1+1/p}}{\Gamma(\alpha)[(\alpha-1)p+1]^{1/p}}\,
=:C_1(\alpha,p,\eta).$$ We choose any of these $p$ and fix it.
Returning to (\[sup9\]), we may also drop the first term, convolve the resulting inequality with $g_{1-\alpha}$ and evaluate at $s=t_0-t_1$, thereby obtaining $$\label{sup12}
\int_{t_2-t_1}^{t_0-t_1}\int_{B_1}\psi^2|Dw|^2\,dx\,ds \le
\,\frac{|1+\beta|}{2\nu |\beta|}\,\int_0^{t_0-t_1} \!\!\!\!F(s)\,ds.$$ Using $$\int_{t_2-t_1}^{t_0-t_1}\int_{B_1} |D(\psi w)|^2\,dx\,ds\le
2\int_{t_2-t_1}^{t_0-t_1}\int_{B_1}
\big(\psi^2|Dw|^2+|D\psi|^2w^2\big)\,dx\,ds$$ we infer from (\[sup10\])–(\[sup12\]) that $$\begin{aligned}
|\psi w|^2_{V_p([t_2-t_1,t_0-t_1]\times B_1)}\le &\;
2\Big(C_1(\alpha,p,\eta)+\,\frac{|1+\beta|}{\nu|\beta|}\Big)\int_0^{t_0-t_1}
\!\!\!\!F(s)\,ds\nonumber\\
&+4\int_{0}^{t_0-t_1}\int_{B_1} |D\psi|^2w^2\,dx\,ds.
\label{vpest}\end{aligned}$$
We will next estimate the right-hand side of (\[vpest\]). By the assumptions on $\psi$ and $\varphi$, and since $|\beta|>1$, we have $$\int_0^{t_0-t_1}\!\!\!\int_{B_1} |D\psi|^2
w^2\,dx\,ds\le
\,\frac{4}{\sigma^2(\rho-\rho')^2}\,\int_0^{t_0-t_1}
\!\!\!\int_{\rho B_1}w^2\,dx\,d{s}$$ and $$\begin{aligned}
F(s)\le &\;\Big(\,\frac{8\Lambda^2|1+\beta|}{\nu
\sigma^2(\rho-\rho')^2}\,+|\beta|g_{1-\alpha}((t_2-t_1)/2)\Big)\int_{\rho
B_1}w^2\,dx \nonumber\\ &
+\,\frac{4}{t_2-t_1}\,\big(g_{1-\alpha}\ast \int_{\rho B_1}
w^2\,dx\big)(s),\quad \mbox{a.a.}\;s\in (0,t_0-t_1).\end{aligned}$$ Recall that $\sigma\ge \delta>0$. So we have $$\begin{aligned}
\int_0^{t_0-t_1} \!\!\!\!F(s)\,ds \;\le &\;
\Big(\,\frac{8\Lambda^2|1+\beta|}{\nu
\sigma^2(\rho-\rho')^2}\,+\,\frac{2^\alpha|\beta|}
{\Gamma(1-\alpha)(\rho-\rho')^\alpha(\sigma\eta)^\alpha}
\Big)\int_0^{t_0-t_1} \!\!\!\int_{\rho B_1}w^2\,dx\,ds\\
&
\;+\,\frac{4}{(\rho-\rho')\sigma\eta}\,\int_0^{t_0-t_1}g_{2-\alpha}(t_0-t_1-\tau)
\int_{\rho B_1}w(\tau,x)^2\,dx\,d\tau \\
\le & \;
C(\nu,\Lambda,\delta,\eta,\alpha)\,\frac{1+|1+\beta|}{(\rho-\rho')^2}
\int_0^{t_0-t_1} \!\!\!\int_{\rho B_1}w^2\,dx\,ds.\end{aligned}$$ Combining these estimates and (\[vpest\]) yields $$|\psi w|_{V_p([t_2-t_1,t_0-t_1]\times B_1)}\le
C(\nu,\Lambda,\delta,\eta,\alpha,p)\,\frac{1+|1+\beta|}
{\rho-\rho'}\, |w|_{L_{2}([0,t_0-t_1]\times\rho B_1)}.$$ We apply next the interpolation inequality (\[Vembedding\]) to the function $\psi w$ and make use of $\psi=1$ in $\rho'B_1$ to deduce that $$\label{sup13}
|w|_{L_{2\kappa}([t_2-t_1,t_0-t_1]\times \rho'B_1)}\le
C(\nu,\Lambda,\delta,\eta,\alpha,p,N)\,\frac{1+|1+\beta|}{\rho-\rho'}
\,|w|_{L_{2}([0,t_0-t_1]\times \rho B_1)},$$ where the number $\kappa>1$ is given in (\[kappa\]). Since $w=\tilde{u}^{\frac{\beta+1}{2}}$ and by transforming back to the time $t$, we see that (\[sup13\]) is equivalent to $$(\int_{V_{\rho'}}
u^{-|1+\beta|\kappa}\,d\mu_{N+1})^{\frac{1}{2\kappa}}\le
\frac{\tilde{C}(1+|1+\beta|)}{\rho-\rho'}\,(\int_{V_{\rho}}
u^{-|1+\beta|}\,d\mu_{N+1})^{\frac{1}{2}}$$ with $\tilde{C}=\tilde{C}(\nu,\Lambda,\delta,\eta,\alpha,p,N)$. Hence, with $\gamma=|1+\beta|$, $$|u^{-1}|_{L_{\gamma\kappa}(V_{\rho'})}\le
\Big(\frac{\tilde{C}^2(1+\gamma)^2}{(\rho-\rho')^2}\Big)^{1/\gamma}
|u^{-1}|_{L_{\gamma}(V_{\rho})},\quad 0<\rho'<\rho\le
1,\;\gamma>0.$$ Employing the first Moser iteration, Lemma \[moserit1\] (with $\bar{p}=1$), it follows that there exist constants $M_0=M_0(\nu,\Lambda,\delta,\eta,\alpha,p,N)$ and $\tau_0=\tau_0(\kappa)$ such that $$\operatorname*{ess\,sup}_{V_{\theta}}{u^{-1}} \le
\Big(\frac{M_0}{(1-\theta)^{\tau_0}}\Big)^{1/\gamma}
|u^{-1}|_{L_{\gamma}(V_1)}\quad \mbox{for
all}\;\;\theta\in(0,1),\;\gamma\in (0,1] .$$ Thus if we take $\theta=\sigma'/\sigma$ and notice that $$\frac{1}{1-\theta}\,=\,\frac{\sigma}{\sigma-\sigma'}\,\le
\frac{1}{\sigma-\sigma'},$$ we obtain $$\operatorname*{ess\,sup}_{U_{\sigma'}}{u^{-1}} \le
\Big(\frac{M_0}{(\sigma-\sigma')^{\tau_0}}\Big)^{1/\gamma}
|u^{-1}|_{L_{\gamma}(U_\sigma)},\quad \gamma\in (0,1].$$ Hence the proof is complete. $\square$ ${}$
We put $$\tilde{\kappa}:=\kappa_{1/(1-\alpha)}=\,\frac{2+N\alpha}{2+N\alpha-2\alpha}.$$
\[superest2\] Let $\alpha\in(0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1)–(H3) are satisfied. Let $\eta>0$ and $\delta\in (0,1)$ be fixed. Then for any $t_0\in [0,T)$ and $r>0$ with $t_0+\eta r^{2/\alpha}\le T$, any ball $B=B(x_0,r)\subset\Omega$, any $p_0\in(0,\tilde{\kappa})$, and any nonnegative weak supersolution $u$ of (\[MProb\]) in $(0,t_0+\eta
r^{2/\alpha})\times B$ with $u_0\ge 0$ in $B$, there holds $$|u|_{L_{p_0}(U_{\sigma'}')}\le \Big(\frac{C\mu_{N+1}(U'_1)^{-1}
}{(\sigma-\sigma')^{\tau_0}}\Big)^{1/\gamma-1/p_0}
|u|_{L_{\gamma}(U'_\sigma)},\quad \delta\le \sigma'<\sigma\le 1,\;
0<\gamma\le p_0/\tilde{\kappa}.$$ Here $U'_\sigma=(t_0,t_0+\sigma\eta r^{2/\alpha})\times \sigma B$, $C=C(\nu,\Lambda,\delta,\eta,\alpha,N,p_0)$, and $\tau_0=\tau_0(\alpha,N)$.
[*Proof:*]{} We proceed similarly as in the previous proof. Without restriction of generality we may assume that $p_0>1$ and $r=1$. By replacing $u$ with $u+\varepsilon$ and $u_0$ with $u_0+\varepsilon$ and eventually letting $\varepsilon \to 0+$ we may further assume that $u$ is bounded away from zero.
Fix $\sigma'$, $\sigma$ such that $\delta\le \sigma'<\sigma\le 1$ and put $B_1=\sigma B$. For $\rho\in (0,1]$ we set $V'_\rho=U'_{\rho\sigma}$. Given $0<\rho'<\rho\le 1$, let $t_1=t_0+\rho'\sigma\eta$ and $t_2=t_0+\rho\sigma\eta$, so $0\le
t_0<t_1<t_2$. We shift the time by means of ${s}=t-t_0$ and set $\tilde{f}(s)=f(s+t_0)$, $s\in (0,t_2-t_0)$, for functions $f$ defined on $(t_0,t_2)$.
We then repeat the first steps of the preceding proof, the only difference being that now we take $\beta\in (-1,0)$. Note that, as a consequence of this, (\[sup1\]) simplifies to $$-\tilde{u}^{\beta}\partial_{s}(g_{1-\alpha,n}\ast \tilde{u}) \ge
-\,\frac{1}{1+\beta}\,\partial_{s}
(g_{1-\alpha,n}\ast\tilde{u}^{1+\beta}),\quad
\mbox{a.a.}\;(s,x)\in (0,t_2-t_0)\times B,$$ hence we obtain with $\psi\in C_0^1(B_1)$ as above $$\begin{aligned}
-\,\frac{1}{1+\beta}\,& \int_{B_1}\psi^2\partial_{s}
(g_{1-\alpha,n}\ast\tilde{u}^{1+\beta})\,dx+|\beta|\int_{B_1}\big((h_n\ast
[ADu])\,\tilde{}\;|\psi^2 \tilde{u}^{\beta-1}D
\tilde{u}\big)\,dx \nonumber\\
\le & \,2\int_{B_1}\big((h_n\ast [ADu])\,\tilde{}\;|\psi D\psi
\,\tilde{u}^{\beta}\big)\,dx,\quad\quad
\mbox{a.a.}\;s\in(0,t_2-t_0). \label{L1}\end{aligned}$$
Next, choose $\varphi\in C^1([0,t_2-t_0])$ such that $0\le
\varphi\le 1$, $\varphi=1$ in $[0,t_1-t_0]$, $\varphi=0$ in $[t_1-t_0+(t_2-t_1)/2,t_2-t_0]$, and $0\le -\dot{\varphi}\le
4/(t_2-t_1)$. Multiplying (\[L1\]) by $1+\beta>0$ and by $\varphi(s)$, and applying Lemma \[comm2\] to the first term gives $$\begin{aligned}
-\int_{B_1} &
\partial_{s}(g_{1-\alpha,n}\ast
[\varphi\psi^2\tilde{u}^{1+\beta}]\big)\,dx+|\beta|(1+\beta)\,
\int_{B_1}\big(\tilde{A}D\tilde{u}|\psi^2 \tilde{u}^{\beta-1}D
\tilde{u}\big)\varphi\,dx \nonumber\\
\le & \,\int_0^s
\dot{g}_{1-\alpha,n}(s-\sigma)\big(\varphi(s)-\varphi(\sigma)\big)
\big(\int_{B_1}\psi^2\tilde{u}^{1+\beta}\,dx\big)(\sigma)\,d\sigma\nonumber\\
& \;\,+2(1+\beta)\,\int_{B_1}\big(\tilde{A}D\tilde{u}|\psi D\psi
\,\tilde{u}^{\beta}\big)\varphi\,dx+\mathcal{R}_n(s) ,\quad
\mbox{a.a.}\;s\in(0,t_2-t_0), \label{L2}\end{aligned}$$ where $$\begin{aligned}
\mathcal{R}_n(s)= &\,\,-|\beta|(1+\beta)\, \int_{B_1}\big((h_n\ast
[ADu])\,\tilde{}\;-\tilde{A}D\tilde{u}|\psi^2 \tilde{u}^{\beta-1}D
\tilde{u}\big)\varphi\,dx\\
&\,+2(1+\beta)\,\int_{B_1}\big((h_n\ast
[ADu])\,\tilde{}\;-\tilde{A}D\tilde{u}|\psi D\psi
\,\tilde{u}^{\beta}\big)\varphi\,dx,\quad
\mbox{a.a.}\;s\in(0,t_2-t_0).\end{aligned}$$ We set again $w=\tilde{u}^{\frac{\beta+1}{2}}$ and estimate exactly as in the preceding proof, using (H1), (H3) and (\[sup8\]), to the result $$\begin{aligned}
-\int_{B_1} &
\partial_{s}(g_{1-\alpha,n}\ast
[\varphi\psi^2w^2]\big)\,dx+\,\frac{2\nu
|\beta|}{1+\beta}\,\int_{B_1}\varphi
\psi^2|Dw|^2\,dx \nonumber\\
\le & \,\int_0^s
\dot{g}_{1-\alpha,n}(s-\sigma)\big(\varphi(s)-\varphi(\sigma)\big)
\big(\int_{B_1}\psi^2w^2\,dx\big)(\sigma)\,d\sigma\nonumber\\
& \;\,+\,\frac{2\Lambda^2(1+\beta)}{\nu |\beta|}\, \int_{B_1}
|D\psi|^2\varphi w^2\,dx+\mathcal{R}_n(s) ,\quad
\mbox{a.a.}\;s\in(0,t_2-t_0). \label{L3}\end{aligned}$$ Recall that $g_{1-\alpha,n}=g_{1-\alpha}\ast h_n$. Putting $$W(s)=\int_{B_1}\varphi(s)\psi(x)^2w(s,x)^2\,dx$$ and denoting the right-hand side of (\[L3\]) by $F_n(s)$, it follows from (\[L3\]) that $$G_n(s):=\partial_s^\alpha (h_n\ast W)(s)+F_n(s)\ge 0,\quad\quad
\mbox{a.a.}\;s\in(0,t_2-t_0).$$ By (\[sup5\]) and positivity of $h_n$, we have $$0\le h_n\ast W =g_\alpha\ast \partial_s^\alpha (h_n\ast W)\le
g_\alpha\ast G_n+g_\alpha\ast [-F_n(s)]_+$$ a.e. in $(0,t_2-t_0)$, where $[y]_+$ stands for the positive part of $y\in {\mathbb{R}}$. For any $p\in (1,1/(1-\alpha))$ and any $t_*\in[t_2-t_0-(t_2-t_1)/4,t_2-t_0]$ we thus obtain by Young’s inequality $$\label{L4}
|h_n\ast W|_{L_p([0,t_*])}\le
|g_\alpha|_{L_p([0,t_*])}\big(|G_n|_{L_1([0,t_*])}+
|[-F_n]_+|_{L_1([0,t_*])}\big).$$ Since $t_*\le t_2-t_0\le \eta$, we have $|g_\alpha|_{L_p([0,t_*])}\le C_1(\alpha,p,\eta)$ with the same constant as in (\[sup11\]). By positivity of $G_n$, $$|G_n|_{L_1([0,t_*])}=(g_{1-\alpha,n}\ast
W)(t_*)+\int_0^{t_*}\!\!\!F_n(s)\,ds.$$ Observe that $\mathcal{R}_n\rightarrow 0$ in $L_1([0,t_2-t_0])$ as $n\to \infty$. Hence $|[-F_n]_+|_{L_1([0,t_*])}\to 0$ as $n\to\infty$. Further, $$\begin{aligned}
\int_0^{t_*}\!\!&\!\int_0^s
\dot{g}_{1-\alpha,n}(s-\sigma)\big(\varphi(s)-\varphi(\sigma)\big)
\big(\int_{B_1}\psi^2w^2\,dx\big)(\sigma)\,d\sigma\,ds\\
=&\,\int_0^{t_*}
{g}_{1-\alpha,n}(t_*-\sigma)\big(\varphi(t_*)-\varphi(\sigma)\big)
\big(\int_{B_1}\psi^2w^2\,dx\big)(\sigma)\,d\sigma\\
&\,-\int_0^{t_*}\!\!\!\dot{\varphi}(s)\int_0^s
{g}_{1-\alpha,n}(s-\sigma)
\big(\int_{B_1}\psi^2w^2\,dx\big)(\sigma)\,d\sigma\,ds\\
\le&\,-\int_0^{t_*}\!\!\!\dot{\varphi}(s)\int_0^s
{g}_{1-\alpha,n}(s-\sigma)
\big(\int_{B_1}\psi^2w^2\,dx\big)(\sigma)\,d\sigma\,ds,\end{aligned}$$ since $\varphi$ is nonincreasing. We also know that $g_{1-\alpha,n}\ast W\to g_{1-\alpha}\ast W$ in $L_1([0,t_2-t_0])$. Hence we can fix some $t_*\in[t_2-t_0-(t_2-t_1)/4,t_2-t_0]$ such that for some subsequence $(g_{1-\alpha,n_k}\ast W)(t_*)\to (g_{1-\alpha}\ast
W)(t_*)$ as $k\to \infty$. Sending $k\to \infty$ it follows then from (\[L4\]), the preceding estimates, and from $\varphi=1$ in $[0,t_1-t_0]$ that $$\label{L5}
\Big(\int_{0}^{t_1-t_0} (\int_{B_1}
[\psi(x)w(s,x)]^2\,dx)^p\,ds\Big)^{1/p}\le
C_1(\alpha,p,\eta)\Big((g_{1-\alpha}\ast
W)(t_*)+|F|_{L_1([0,t_2-t_0])}\Big),$$ with $$F(s)=\,\frac{2\Lambda^2(1+\beta)}{\nu |\beta|}\, \int_{B_1}
|D\psi|^2\varphi w^2\,dx-\dot{\varphi}(s)\big(g_{1-\alpha}\ast
\int_{B_1}\psi^2w^2\,dx\big)(s).$$
On the other hand, we can integrate (\[L3\]) over $(0,t_*)$ and take the limit as $k\to \infty$ for the same subsequence as before, thereby getting $$\label{L6}
\int_{0}^{t_1-t_0}\!\!\!\int_{B_1}
\psi^2|Dw|^2\,dx\,ds\le\,\frac{1+\beta}{2\nu
|\beta|}\,\Big((g_{1-\alpha}\ast
W)(t_*)+|F|_{L_1([0,t_2-t_0])}\Big).$$ Arguing as above (cf. the lines before (\[vpest\])), we conclude from (\[L5\]) and (\[L6\]) that $$\begin{aligned}
&|\psi w|^2_{V_p([0,t_1-t_0]\times B_1)}\le \;
4\int_{0}^{t_2-t_0}\int_{B_1} |D\psi|^2w^2\,dx\,ds\nonumber\\
&+2\Big(C_1(\alpha,p,\eta)+\,\frac{1+\beta}{\nu|\beta|}\Big)
\Big((g_{1-\alpha}\ast W)(t_*)+|F|_{L_1([0,t_2-t_0])}\Big).
\label{L7}\end{aligned}$$
Since $\varphi=0$ in $[t_1-t_0+(t_2-t_1)/2,t_2-t_0]$ and $t_*\in[t_2-t_0-(t_2-t_1)/4,t_2-t_0]$, we have $$\begin{aligned}
(g_{1-\alpha}\ast W)(t_*)&\,\le
g_{1-\alpha}\big((t_2-t_1)/4\big)\int_0^{t_2-t_0} \!\!\!\int_{
B_1}\psi^2w^2\,dx\,d{s}\\
&\,=\,\frac{4^\alpha}{\Gamma(1-\alpha)(\rho-\rho')^\alpha(\sigma\eta)^\alpha}\,
\int_0^{t_2-t_0} \!\!\!\int_{\rho B_1}w^2\,dx\,d{s}.\end{aligned}$$ Further, $$\int_{0}^{t_2-t_0}\int_{B_1} |D\psi|^2w^2\,dx\,ds\le\,\frac{4}{\sigma^2(\rho-\rho')^2}
\,\int_0^{t_2-t_0}
\!\!\!\int_{\rho B_1}w^2\,dx\,d{s}.$$ The term $|F|_{L_1([0,t_2-t_0])}$ is estimated similarly as in the proof of Theorem \[superest1\] (cf. the lines that follow (\[vpest\])). We obtain $$\begin{aligned}
|F|_{L_1([0,t_2-t_0])}\le
C(\nu,\Lambda,\delta,\eta,\alpha)\,\frac{1+(1+\beta)}{|\beta|(\rho-\rho')^2}
\int_0^{t_2-t_0} \!\!\!\int_{\rho B_1}w^2\,dx\,ds.\end{aligned}$$ Notice the additional factor $|\beta|$ in the denominator. Combining these estimates we deduce from (\[L7\]) that $$|\psi w|_{V_p([0,t_1-t_0]\times B_1)}\le
C(\nu,\Lambda,\delta,\eta,\alpha,p)\,\frac{1+(1+\beta)}
{|\beta|(\rho-\rho')}\, |w|_{L_{2}([0,t_2-t_0]\times\rho B_1)}.$$ By the interpolation inequality (\[Vembedding\]) and since $\psi=1$ in $\rho'B_1$, this implies for all $\beta\in(-1,0)$ $$\label{supL8}
|w|_{L_{2\kappa}([0,t_1-t_0]\times \rho'B_1)}\le
C(\nu,\Lambda,\delta,\eta,\alpha,p,N)\,\frac{1+|1+\beta|}{|\beta|(\rho-\rho')}
\,|w|_{L_{2}([0,t_2-t_0]\times \rho B_1)},$$ where $$\kappa=\kappa_p=\,\frac{2p+N(p-1)}{2+N(p-1)}\,\in
(1,\tilde{\kappa}).$$ We now fix $1<p<1/(1-\alpha)$ such that $\kappa_p=(p_0+\tilde{\kappa})/2$. This is possible because $\kappa_p\nearrow \tilde{\kappa}$ as $p \nearrow 1/(1-\alpha)$.
Next, we set $\gamma=1+\beta\in (0,1)$ and transform back to $u$ to get $$\label{L9}
|u|_{L_{\gamma\kappa}(V'_{\rho'},d\mu)}\le
\Big(\frac{\tilde{C}}{(\rho-\rho')^{2}}\Big)^{1/\gamma}
|u|_{L_{\gamma}(V'_{\rho},d\mu)},\quad 0<\rho'<\rho\le
1,\;0<\gamma\le p_0/\kappa.$$ Here, $\mu=(\eta \omega_N)^{-1}\mu_{N+1}$, $\omega_N$ the volume of the unit ball in ${\mathbb{R}}^N$, and $\tilde{C}=\tilde{C}(\nu,\Lambda,\delta,\eta,\alpha,N,p_0)$ is independent of $\gamma\in(0,p_0/\kappa]$, since $|\beta|$ is bounded away from zero. Note that $\mu(V'_1)\le 1$.
Finally, we employ the second Moser iteration scheme, Lemma \[moserit2\], to conclude from (\[L9\]) that there are constants $M_0=M_0(\nu,\Lambda,\delta,\eta,\alpha,N,p_0)$ and $\tau_0=\tau_0(\kappa)$ such that $$\label{L10}
|u|_{L_{p_0}(V'_{\theta},d\mu)}\le
\Big(\frac{M_0}{(1-\theta)^{\tau_0}}\Big)^{1/\gamma-1/p_0}
|u|_{L_{\gamma}(V'_{1},d\mu)},\quad 0<\theta< 1,\;0<\gamma\le
p_0/\kappa.$$ If we take $\theta=\sigma'/\sigma$ and translate (\[L10\]) back to the measure $\mu_{N+1}$, we obtain $$\label{L11}
|u|_{L_{p_0}(U_{\sigma'}')}\le \Big(\frac{M_0 (\eta\omega_N)^{-1}
}{(\sigma-\sigma')^{\tau_0}}\Big)^{1/\gamma-1/p_0}
|u|_{L_{\gamma}(U'_\sigma)},\quad 0<\gamma\le p_0/\kappa.$$ Since $\kappa<\tilde{\kappa}$, (\[L11\]) holds in particular for all $\gamma\in (0,p_0/\tilde{\kappa}]$. This finishes the proof. $\square$
Logarithmic estimates
---------------------
\[logest\] Let $\alpha\in(0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1)–(H3) are satisfied. Let $\tau>0$ and $\delta,\,\eta\in(0,1)$ be fixed. Then for any $t_0\ge
0$ and $r>0$ with $t_0+\tau r^{2/\alpha}\le T$, any ball $B=B(x_0,r)\subset\Omega$, and any weak supersolution $u\ge
\varepsilon>0$ of (\[MProb\]) in $(0,t_0+\tau r^{2/\alpha})\times
B$ with $u_0\ge 0$ in $B$, there is a constant $c=c(u)$ such that $$\label{logestleft}
\mu_{N+1}\big(\{(t,x)\in K_-: \log u(t,x)>c+\lambda\}\big)\le C
r^{2/\alpha} \mu_N(B)\lambda^{-1},\quad \lambda>0,$$ and $$\label{logestright}
\mu_{N+1}\big(\{(t,x)\in K_+: \log u(t,x)<c-\lambda\}\big)\le C
r^{2/\alpha} \mu_N(B)\lambda^{-1},\quad \lambda>0,$$ where $K_-=(t_0,t_0+\eta \tau r^{2/\alpha})\times \delta B$ and $K_+=(t_0+\eta \tau r^{2/\alpha},t_0+\tau r^{2/\alpha})\times
\delta B$. Here the constant $C$ depends only on $\delta, \eta,
\tau, N, \alpha, \nu$, and $\Lambda$.
[*Proof:*]{} Since $u_0\ge 0$ in $B$ and $u$ is a positive weak supersolution we may assume without loss of generality that $u_0=0$ and $t_0=0$. In fact, in the case $t_0>0$ we shift the time as $t\to
t-t_0$, thereby obtaining an inequality of the same type on the time-interval $J:=[0,\tau r^{2/\alpha}]$. Observe that the property $g_{1-\alpha}\ast u\in C([0,t_0+\tau r^{2/\alpha}];L_2(B))$ implies $g_{1-\alpha}\ast \tilde{u}\in C(J;L_2(B))$ for the shifted function $\tilde{u}(s,x)=u(s+t_0,x)$. So we have $$\label{log1}
\int_B \Big(v \partial_t(g_{1-\alpha,n}\ast u)+(h_n\ast
[ADu]|Dv)\Big)\,dx \ge \,0,\quad\mbox{a.a.}\;t\in J,\,n\in {\mathbb{N}},$$ for any nonnegative test function $v\in {\hspace*{0.39em}\raisebox{0.6ex}{\textdegree}\hspace{-0.72em}H}^1_2(B)$.
For $t\in J$ we choose the test function $v=\psi^2 u^{-1}$ with $\psi\in C^1_0(B)$ such that supp$\,\psi\subset B$, $\psi=1$ in $\delta B$, $0\le \psi \le 1$, $|D \psi|\le 2/[(1-\delta)r]$ and the domains $\{x\in B:\psi(x)^2 \ge b\}$ are convex for all $b\le
1$. We have $$Dv=2\psi D\psi \,u^{-1}-\psi^2 u^{-2}Du,$$ so that by substitution into (\[log1\]) we obtain for a.a. $t\in
J$ $$\begin{aligned}
\label{log1a}
-\int_B \psi^2 u^{-1}\partial_t & (g_{1-\alpha,n}\ast
u)\,dx+\int_B\big(ADu|u^{-2}Du\big)\psi^2\,dx\nonumber\\ &\le
2\int_B\big(ADu|u^{-1}\psi D\psi\big)\,dx+\mathcal{R}_n(t),\end{aligned}$$ where $$\mathcal{R}_n(t)=\int_B\big(h_n\ast [ADu]-ADu|Dv\big)\,dx.$$ By (H1) and Young’s inequality, $$\big|2\big(ADu|u^{-1}\psi D\psi\big)\big|\le 2\Lambda \psi
|D\psi|\,|Du| u^{-1}\le \frac{\nu}{2}\,\psi^2|Du|^2
u^{-2}+\frac{2}{\nu}\,\Lambda^2|D\psi|^2.$$ Using this, (H2) and $|D \psi|\le 2/[(1-\delta)r]$, we infer from (\[log1a\]) that for a.a. $t\in J$ $$\label{log2}
-\int_B \psi^2 u^{-1}\partial_t(g_{1-\alpha,n}\ast
u)\,dx+\frac{\nu}{2}\,\int_B |Du|^2 u^{-2} \psi^2\,dx \le \frac{8
\Lambda^2 \mu_N(B)}{\nu (1-\delta)^2 r^2}\,+\mathcal{R}_n(t).$$ Setting $w=\log u$ we have $Dw=u^{-1} Du$. The weighted Poincaré inequality of Proposition \[WeiPI\] with weight $\psi^2$ yields $$\label{log3}
\int_B (w-W)^2 \psi^2 dx \le \frac{8 r^2 \mu_N(B)}{\int_B \psi^2
dx}\,\int_B |Dw|^2 \psi^2 dx,\quad \mbox{a.a.}\;t\in J,$$ where $$W(t)=\,\frac{\int_B w(t,x) \psi(x)^2 dx}{\int_B \psi(x)^2
dx}\,,\quad\quad \mbox{a.a.}\;t\in J.$$ From (\[log2\]) and (\[log3\]) we deduce that $$-\int_B \psi^2 u^{-1}\partial_t(g_{1-\alpha,n}\ast u)
\,dx+\,\frac{\nu \int_B \psi^2 dx}{16r^2 \mu_N(B)}\,\int_B (w-W)^2
\psi^2 dx \le \frac{8 \Lambda^2 \mu_N(B)}{\nu (1-\delta)^2
r^2}\,+\mathcal{R}_n(t),$$ which in turn implies $$\label{log4}
\frac{-\int_B \psi^2 u^{-1}\partial_t(g_{1-\alpha,n}\ast u)
\,dx}{\int_B \psi^2 dx}+\,\frac{\nu}{16r^2 \mu_N(B)}\,\int_{\delta
B} (w-W)^2 dx \le \frac{C_1}{r^2}\,+S_n(t),$$ for a.a. $t\in J$, with some constant $C_1=C_1(\delta,N,\nu,\Lambda)$ and $S_n(t)=\mathcal{R}_n(t)/\int_B \psi^2 dx$.
The fundamental identity (\[fundidentity\]) with $H(y)=-\log y$ reads (with the spatial variable $x$ being suppressed) $$\begin{aligned}
-u^{-1}\partial_t(g_{1-\alpha,n} & \ast
u)=-\partial_t(g_{1-\alpha,n}\ast \log u)+(\log
u-1)g_{1-\alpha,n}(t)\nonumber\\
&+\int_0^t \Big(-\log u(t-s)+\log
u(t)+\frac{u(t-s)-u(t)}{u(t)}\Big)[-\dot{g}_{1-\alpha,n}(s)]\,ds.\end{aligned}$$ In terms of $w=\log u$ this means that $$\begin{aligned}
\label{log5}
-u^{-1}\partial_t(g_{1-\alpha,n} \ast u)= & \,
-\partial_t(g_{1-\alpha,n}\ast
w)+(w-1)g_{1-\alpha,n}(t)\nonumber\\
& \,\,+\int_0^t
\Psi\big(w(t-s)-w(t)\big)[-\dot{g}_{1-\alpha,n}(s)]\,ds,\end{aligned}$$ where $\Psi(y)=e^y-1-y$. Since $\Psi$ is convex, it follows from Jensen’s inequality that $$\frac{\int_B \psi^2 \Psi\big(w(t-s,x)-w(t,x)\big)\,dx}{\int_B
\psi^2 dx} \ge \Psi \Big( \frac{\int_B \psi^2
\big(w(t-s,x)-w(t,x)\big)\,dx}{\int_B \psi^2 dx}\Big).$$ Using this and (\[log5\]) we obtain $$\begin{aligned}
\frac{-\int_B \psi^2 u^{-1}\partial_t(g_{1-\alpha,n}\ast u)
\,dx}{\int_B \psi^2 dx} & \ge -\partial_t(g_{1-\alpha,n}\ast
W)+(W-1)g_{1-\alpha,n}(t)\nonumber\\
&\quad +\int_0^t
\Psi\big(W(t-s)-W(t)\big)[-\dot{g}_{1-\alpha,n}(s)]\,ds \nonumber\\
& = -e^{-W} \partial_t(g_{1-\alpha,n}\ast e^W), \label{log6}\end{aligned}$$ where the last equals sign holds again by (\[log5\]) with $u$ replaced by $e^W$. From (\[log4\]) and (\[log6\]) we conclude that $$\label{log7}
\frac{\nu}{16r^2 \mu_N(B)}\,\int_{\delta B} (w-W)^2 dx \le e^{-W}
\partial_t(g_{1-\alpha,n}\ast
e^W)+\,\frac{C_1}{r^2}\,+S_n(t),\quad \mbox{a.a.}\;t\in J.$$
We choose $$\label{cwahl}
c(u)=\log \Big(\frac{(g_{1-\alpha}\ast e^W)(\eta\tau
r^{2/\alpha})}{g_{2-\alpha}(\eta\tau r^{2/\alpha})}\Big).$$ This definition makes sense, since $g_{1-\alpha}\ast e^W\in C(J)$. The latter is a consequence of $g_{1-\alpha}\ast u\in C(J;L_2(B))$ and $$e^{W(t)}\le \,\frac{\int_B u(t,x) \psi(x)^2 dx}{\int_B \psi(x)^2
dx}\,,\quad\quad \mbox{a.a.}\;t\in J,$$ where we apply again Jensen’s inequality.
To prove (\[logestleft\]) and (\[logestright\]), one of the key ideas is to use the inequalities $$\begin{aligned}
\mu_{N+1}(\{(t,x) & \in K_-:\; w(t,x)>c(u)+\lambda\})\nonumber\\
\le &\;\mu_{N+1}(\{(t,x)\in K_-:
w(t,x)>c(u)+\lambda\;\,\mbox{and}\,\;W(t)\le c(u)+\lambda/2 \})\nonumber\\
&\; +\mu_{N+1}(\{(t,x)\in K_-:\,W(t)> c(u)+\lambda/2
\})=:I_1+I_2,\quad \lambda>0,\label{mainleft}\\
\mu_{N+1}(\{(t,x) & \in K_+:\; w(t,x)<c(u)-\lambda\})\nonumber\\
\le &\;\mu_{N+1}(\{(t,x)\in K_+:
w(t,x)<c(u)-\lambda\;\,\mbox{and}\,\;W(t)\ge c(u)-\lambda/2 \})\nonumber\\
&\; +\mu_{N+1}(\{(t,x)\in K_+:\,W(t)< c(u)-\lambda/2
\})=:I_3+I_4,\quad \lambda>0,\label{mainright}\end{aligned}$$ and to estimate each of the four terms $I_j$ separately.
We begin with the estimates for $W$. To estimate $I_2$ and $I_4$ we adopt some of the ideas developed in [@Za]. We set $J_-:=(0,\eta
\tau r^{2/\alpha})$, $J_+:=(\eta\tau r^{2/\alpha},\tau
r^{2/\alpha})$, and introduce for $\lambda>0$ the sets $J_-(\lambda):=\{t\in J_-:\,W(t)> c(u)+\lambda \}$ and $J_+(\lambda):=\{t\in J_+:\,W(t)< c(u)-\lambda \}$.
Interestingly, positivity and integrability of the function $e^W$ are sufficient to derive the desired estimate for $I_2$, cf. also [@Za Theorem 2.3]. In fact, with $\rho=\tau r^{2/\alpha}$ we have $$\begin{aligned}
e^\lambda \mu_1\big(J_-(\lambda)\big) & =
e^\lambda\mu_1\big(\{t\in J_-:\,
e^{W(t)}>e^{c(u)}e^{\lambda}\}\big)=\int_{J_-(\lambda)}e^\lambda \,dt\\
& \le \int_{J_-(\lambda)}e^{W(t)-c(u)} \,dt\le
\int_{J_-}e^{W(t)-c(u)} \,dt\\
& = \,\frac{g_{2-\alpha}(\eta\rho)}{(g_{1-\alpha}\ast
e^W)(\eta\rho)}\,\int_0^{\eta\rho}e^{W(t)}\,dt\\
& \le \,\frac{g_{2-\alpha}(\eta\rho)}{(g_{1-\alpha}\ast
e^W)(\eta\rho)}\,\cdot\,\frac{1}{g_{1-\alpha}(\eta\rho)}\,
\int_0^{\eta\rho}g_{1-\alpha}(\eta\rho-t)e^{W(t)}\,dt\\
& =
\,\frac{\Gamma(1-\alpha)}{\Gamma(2-\alpha)}\,\eta\rho=\,\frac{\eta\tau
r^{2/\alpha}}{1-\alpha},\end{aligned}$$ and therefore $$\label{I2est}
I_2=\mu_1\big(J_-(\lambda/2)\big)\mu_N(\delta B)\le\,\frac{2\eta\tau
\delta^N}{(1-\alpha)\lambda}\,r^{2/\alpha}\mu_N(B),\quad \lambda>0.$$
We come now to $I_4$. For $m>0$ define the function $H_m$ on ${\mathbb{R}}$ by $H_m(y)=y$, $y\le m$, and $H_m(y)=m+(y-m)/(y-m+1)$, $y\ge m$. Then $H_m$ is increasing, concave, and bounded above by $m+1$. Further, we have $H_m\in C^1({\mathbb{R}})$, and so by concavity $$\label{Hmprop}
0\le yH_m'(y)\le H_m(y)\le m+1,\quad y\ge 0.$$ Multiplying (\[log7\]) by $e^W H_m'\big(e^W\big)$ and employing (\[Hmprop\]) as well as the fundamental identity (\[fundidentity\]), we infer that $$\label{West1}
\partial_t\Big(g_{1-\alpha,n}\ast
H_m\big(e^W\big)\Big)+\,\frac{C_1}{r^2}\,H_m\big(e^W\big)\ge - S_n
e^W H_m'\big(e^W\big),\quad \mbox{a.a.}\;t\in J.$$ For $t\in J_+$ we shift the time by setting $s=t-\eta\tau
r^{2/\alpha}=t-\eta \rho$ and put $\tilde{f}(s)=f(s+\eta \rho)$, $s\in (0,(1-\eta)\rho)$, for functions $f$ defined on $J_+$. By the time-shifting identity (\[shiftprop\]), (\[West1\]) implies that for a.a. $s\in (0,(1-\eta)\rho)$ $$\label{West2}
\partial_s\Big(g_{1-\alpha,n}\ast
H_m\big(e^{\tilde{W}}\big)\Big)+\,\frac{C_1}{r^2}\,H_m\big(e^{\tilde{W}}\big)\ge
\Upsilon_{n,m}(s)- \tilde{S}_n e^{\tilde{W}}
H_m'\big(e^{\tilde{W}}\big),$$ with the history term $$\Upsilon_{n,m}(s)=\int_0^{\eta\rho}\big[-\dot{g}_{1-\alpha,n}(s+\eta\rho-\sigma)\big]
H_m\big(e^{W(\sigma)}\big)\,d\sigma.$$
For $\theta\ge 0$ define the kernel $r_{\alpha,\theta}\in
L_{1,loc}({\mathbb{R}}_+)$ by means of $$r_{\alpha,\,\theta}(t)+\theta (r_{\alpha,\,\theta}\ast g_\alpha)(t)=
g_\alpha(t),\quad t>0.$$ Observe that $r_{\alpha,\,0}=g_\alpha$. Since $g_\alpha$ is completely monotone, $r_{\alpha,\,\theta}$ enjoys the same property (cf. [@GLS Chap. 5]), in particular $r_{\alpha,\,\theta}(s)>0$ for all $s>0$. Moreover, we have (see e.g. [@Za]) $$r_{\alpha,\,\theta}(s) =\Gamma(\alpha)
g_\alpha(s)\,E_{\alpha,\alpha}(-\theta s^\alpha),\quad s>0,$$ where $E_{\alpha,\beta}$ denotes the generalized Mittag-Leffler-function defined by $$E_{\alpha,\beta}(z)=\sum_{n=0}^\infty \;\frac{z^n}
{\Gamma(n\alpha+\beta)}\;,\quad z\in {\mathbb{C}}.$$
We put $\theta=C_1/r^2$ and convolve (\[West2\]) with $r_{\alpha,\,\theta}$. We have a.e. in $(0,(1-\eta)\rho)$ $$\begin{aligned}
r_{\alpha,\,\theta}\ast \partial_s\Big(g_{1-\alpha,n} & \ast
H_m\big(e^{\tilde{W}}\big)\Big)\,=\partial_s\Big(r_{\alpha,\,\theta}\ast
g_{1-\alpha,n}\ast H_m\big(e^{\tilde{W}}\big)\Big)\\
&\,=\partial_s\Big([g_\alpha-\theta (r_{\alpha,\,\theta}\ast
g_\alpha)]\ast g_{1-\alpha,n}\ast H_m\big(e^{\tilde{W}}\big)\Big)\\
&\,=h_n\ast H_m\big(e^{\tilde{W}}\big) -\theta
r_{\alpha,\,\theta}\ast h_n\ast H_m\big(e^{\tilde{W}}\big),\end{aligned}$$ and so we obtain a.e. in $(0,(1-\eta)\rho)$ $$\begin{aligned}
\label{West3}
h_n\ast H_m\big(e^{\tilde{W}}\big)\ge &
\,\,r_{\alpha,\,\theta}\ast \Upsilon_{n,m}-
r_{\alpha,\,\theta}\ast\big[\tilde{S}_n
e^{\tilde{W}} H_m'\big(e^{\tilde{W}}\big)\big]\nonumber\\
& \,\,+\theta h_n\ast r_{\alpha,\,\theta}\ast
H_m\big(e^{\tilde{W}}\big) -\theta r_{\alpha,\,\theta}\ast
H_m\big(e^{\tilde{W}}\big).\end{aligned}$$ Sending $n\to \infty$ and selecting an appropriate subsequence, if necessary, it follows that $$\label{West4}
H_m\big(e^{\tilde{W}}\big)\ge r_{\alpha,\,\theta}\ast
\Upsilon_{m},\quad\quad \mbox{a.a.}\;s\in(0,(1-\eta)\rho),$$ where $$\Upsilon_{m}(s)=\int_0^{\eta\rho}\big[-\dot{g}_{1-\alpha}(s+\eta\rho-\sigma)\big]
H_m\big(e^{W(\sigma)}\big)\,d\sigma.$$
Observe that for $s\in (0,(1-\eta)\rho)$ we have $$0\le \theta s^\alpha\le \,\frac{C_1}{r^2}\,(1-\eta)^\alpha \big(\tau
r^{2/\alpha}\big)^\alpha=C_1(1-\eta)^\alpha\tau^\alpha=:\omega,$$ and thus by continuity and strict positivity of $E_{\alpha,\alpha}$ in $(-\infty,0]$, $$r_{\alpha,\,\theta}(s)\ge \Gamma(\alpha) g_\alpha(s)\min_{y\in
[0,\omega]} E_{\alpha,\alpha}(-y)=:C_2(\alpha,\omega)\Gamma(\alpha)
g_\alpha(s),\quad s\in (0,(1-\eta)\rho).$$ We may then argue as in [@Za Section 2.1] to obtain $$H_m\big(e^{\tilde{W}(s)}\big)\ge C_2(\alpha,\omega)\,\frac{\alpha
(s/[\eta\rho])^\alpha}{1+(s/[\eta\rho])}\,(\eta\rho)^{\alpha-1}\big(g_{1-\alpha}\ast
H_m\big(e^{W}\big)\big) (\eta\rho),\quad \mbox{a.a.}\;s\in
(0,(1-\eta)\rho).$$ Evidently, $H_m(y)\nearrow y$ as $m\to \infty$ for all $y\in {\mathbb{R}}$. Thus by sending $m\to \infty$ and applying Fatou’s lemma we conclude that $$\label{West5}
e^{\tilde{W}(s)}\ge C_2(\alpha,\omega)\,\frac{\alpha
(s/[\eta\rho])^\alpha}{1+(s/[\eta\rho])}\,(\eta\rho)^{\alpha-1}\big(g_{1-\alpha}\ast
e^{W}\big) (\eta\rho),\quad \mbox{a.a.}\;s\in (0,(1-\eta)\rho).$$
We then employ (\[West5\]) to estimate as follows. $$\begin{aligned}
e^\lambda \mu_1\big(J_+(\lambda)\big) & = e^\lambda\mu_1\big(\{t\in
J_+:\,
e^{W(t)}<e^{c(u)}e^{-\lambda}\}\big)=\int_{J_+(\lambda)}e^\lambda \,dt\\
& \le \int_{J_+(\lambda)}e^{c(u)-W(t)} \,dt\le
\int_{J_+}e^{c(u)-W(t)} \,dt\\
& = \,\frac{(g_{1-\alpha}\ast
e^W)(\eta\rho)}{g_{2-\alpha}(\eta\rho)}\,\int_0^{(1-\eta)\rho}
e^{-\tilde{W}(s)}\,ds\\
& \le \,\frac{C_2(\alpha,\omega)^{-1}(\eta\rho)^{1-\alpha}}{\alpha
g_{2-\alpha}(\eta\rho)}\,\int_0^{(1-\eta)\rho}(1+s/\eta\rho)(s/\eta\rho)^{-\alpha}\,ds\\
& = \,\frac{\Gamma(2-\alpha)\eta\rho}{\alpha
C_2(\alpha,\omega)}\,\int_0^{\frac{1-\eta}{\eta}}
\sigma^{-\alpha}(1+\sigma)\,d\sigma=C_3(\alpha,\eta,\omega)\rho.\end{aligned}$$ Hence $$\label{I4est}
I_4=\mu_1\big(J_+(\lambda/2)\big)\mu_N(\delta
B)\le\,\frac{2C_3(\alpha,\eta,\omega)
\delta^N}{\lambda}\,r^{2/\alpha}\mu_N(B),\quad \lambda>0.$$
We come now to $I_1$. Set $J_1(\lambda)=\{t\in
J_-:\,c-W(t)+\lambda/2\ge 0\}$ and $\Omega^-_t(\lambda)=\{x\in
\delta B:\,w(t,x)>c+\lambda\},\,t\in J_1(\lambda)$, where $c=c(u)$ is given by (\[cwahl\]). For $t\in J_1(\lambda)$, we have $$w(t,x)-W(t)>c-W(t)+\lambda\ge \lambda/2,\quad x\in
\Omega^-_t(\lambda),$$ and thus we deduce from (\[log7\]) that a.e. in $J_1(\lambda)$ $$\label{log8}
\frac{\nu}{16r^2
\mu_N(B)}\,\,\mu_N\big(\Omega^-_t(\lambda)\big)\le
\frac{1}{(c-W+\lambda)^2}\,\Big(e^{-W}
\partial_t(g_{1-\alpha,n}\ast
e^W)+\,\frac{C_1}{r^2}\,+S_n\Big).$$ Set $\chi(t,\lambda)=\mu_N\big(\Omega^-_t(\lambda)\big)$, if $t\in
J_1(\lambda)$, and $\chi(t,\lambda)=0$ in case $t\in J_-\setminus
J_1(\lambda)$. Let further $H(y)=(c-\log y+\lambda)^{-1},\,0<y\le
y_*:=e^{c+\lambda/2}$. Clearly, $H'(y)= (c-\log
y+\lambda)^{-2}y^{-1}$ as well as $$H''(y)=\,\frac{1}{(c-\log y+\lambda)^2 y^2}\,\Big(\frac{2}{c-\log
y+\lambda}-1\Big),\quad 0<y\le y_*,$$ which shows that $H$ is concave in $(0,y_*]$ whenever $\lambda\ge 4$. We will assume this in what follows.
We next choose a $C^1$ extension $\bar{H}$ of $H$ on $(0,\infty)$ such that $\bar{H}$ is concave, $0\le\bar{H}'(y)\le
\bar{H}'(y_*),\,y_*\le y \le 2 y_*$, and $\bar{H}'(y)=0,\,y\ge 2
y_*$. Then $$\label{log8a}
0\le y\bar{H}'(y)\le
\,\frac{2}{\lambda},\quad y>0.$$ In fact, for $y\in(0,y_*]$ we have $$\label{log8aa}
y\bar{H}'(y)=\,\frac{1}{(c-\log y+\lambda)^2}\,\le
\,\frac{1}{(c-\log y_*+\lambda)^2}\,\le \,\frac{4}{\lambda^2}\,\le
\,\frac{1}{\lambda},$$ while in case $y\in[y_*,2y_*]$ we may simply estimate $$y\bar{H}'(y)\le 2y_*\bar{H}'(y_*)\le \,\,\frac{2}{\lambda}.$$
It is clear that $\bar{H}$ is bounded above. There holds $$\label{log8b}
\bar{H}(y)\le \,\frac{3}{\lambda},\quad y>0.$$ To see this, note that since $\bar{H}$ is nondecreasing with $\bar{H}'(y)=0$ for all $y\ge 2 y_*$, the claim follows if the inequality is valid for all $y\in [y_*,2y_*]$. For such $y$ we have by (\[log8aa\]) and by concavity of $\bar{H}$ $$\bar{H}(y)\le \bar{H}(y_*)+\bar{H}'(y_*)(y-y_*)\le
\bar{H}(y_*)+y_*\bar{H}'(y_*)\le \,\frac{3}{\lambda}.$$
Observe also that $$e^{W(t)} H'(e^{W(t)})=\,\frac{1}{(c-W(t)+\lambda)^2
},\quad \mbox{a.a.}\;t\in J_1(\lambda).$$ Since $\bar{H}'\ge 0$, and $e^{-W}
\partial_t(g_{1-\alpha,n}\ast
e^W)+C_1 r^{-2}+S_n\ge 0$ on $J_-$ by virtue of (\[log7\]), we infer from (\[log8\]) and (\[log8a\]) that $$\begin{aligned}
\label{log9}
\frac{\nu}{16r^2 \mu_N(B)}\,\,\chi(t,\lambda) & \le
e^W\bar{H}'(e^{W})\Big(e^{-W}\partial_t(g_{1-\alpha,n}\ast
e^W)+\,\frac{C_1}{r^2}\,+S_n\Big)\nonumber\\
& \le \bar{H}'(e^{W})\partial_t(g_{1-\alpha,n}\ast
e^W)+\,\frac{2C_1}{\lambda r^2}\,+\,\frac{2|S_n(t)|}{\lambda},
\quad\mbox{a.a.}\; t\in J_-.\end{aligned}$$ Since $\bar{H}$ is concave, the fundamental identity (\[fundidentity\]) yields $$\begin{aligned}
\bar{H}'(e^{W})\partial_t(g_{1-\alpha,n}\ast e^W) & \le
\partial_t\Big(g_{1-\alpha,n}\ast \bar{H}\big(e^W\big)\Big)
+ \Big(-\bar{H}(e^{W})+\bar{H}'(e^{W})
e^W\Big)g_{1-\alpha,n}\\
& \le \partial_t\Big(g_{1-\alpha,n}\ast
\bar{H}\big(e^W\big)\Big)+\,\frac{2}{\lambda}\,g_{1-\alpha,n},\quad
\mbox{a.a.}\; t\in J_-,\end{aligned}$$ which, together with (\[log9\]), gives a.e. in $J_-$ $$\label{log10}
\frac{\nu}{16r^2 \mu_N(B)}\,\,\chi(t,\lambda) \le
\partial_t\Big((g_{1-\alpha,n}\ast \bar{H}\big(e^W\big)\Big)
+\,\frac{2}{\lambda}\,g_{1-\alpha,n}+ \,\frac{2C_1}{\lambda
r^2}\,+\,\frac{2|S_n(t)|}{\lambda}\,.$$ We then integrate (\[log10\]) over $J_-=(0,\eta \rho)$ and employ (\[log8b\]) for the estimate $$\Big(g_{1-\alpha,n}\ast
\bar{H}\big(e^W\big)\Big)(\eta\rho)\le
\,\frac{3}{\lambda}\,\int_0^{\eta\rho}g_{1-\alpha,n}(t)\,dt.$$ By sending $n\to \infty$, this leads to $$\begin{aligned}
\int_{J_1(\lambda)}\mu_N & \big(\Omega^-_t(\lambda)\big) \,dt
= \int_0^{\eta\rho} \chi(t,\lambda)\,dt
\le\,\frac{16r^2\mu_N(B)}{\nu}\,\Big(\frac{5}{\lambda}\,g_{2-\alpha}(\eta\rho)+
\,\frac{2C_1\eta\rho}{\lambda
r^2}\Big)\\
& =\,\frac{16r^{2/\alpha}\mu_N(B)}{\nu\lambda}\,
\big(5g_{2-\alpha}(\eta\tau)+
2C_1\eta\tau\big)=:C_4\,\frac{r^{2/\alpha}\mu_N(B)}{\lambda},\quad\lambda\ge
4.\end{aligned}$$ Hence with $C_5=\max\{4\tau,C_4\}$ we find that $$\label{I1est}
I_1\le \,\frac{C_5 r^{2/\alpha}\mu_N(B)}{\lambda},\quad\lambda>0.$$
It remains to derive the desired estimate for $I_3$. To this purpose we shift again the time by putting $s=t-\eta\rho$, and denote the corresponding transformed functions as above by $\tilde{W}$, $\tilde{w}$, ... and so forth. Set further $\tilde{J}_+:=(0,(1-\eta)\rho)$. By the time-shifting property (\[shiftprop\]) and by positivity of $e^W$, relation (\[log7\]) then implies $$\label{log10a}
\frac{\nu}{16r^2 \mu_N(B)}\,\int_{\delta B}
(\tilde{w}-\tilde{W})^2 dx \le e^{-\tilde{W}}
\partial_s(g_{1-\alpha,n}\ast
e^{\tilde{W}})+\,\frac{C_1}{r^2}\,+\tilde{S}_n(s),\quad
\mbox{a.a.}\;s\in \tilde{J}_+.$$ Next, set $J_2(\lambda)=\{s\in
\tilde{J}_+:\tilde{W}(s)-c+\lambda/2\ge 0\}$ and $\Omega_{s}^+(\lambda)=\{x\in \delta B:
\tilde{w}(s,x)<c-\lambda\},\,s\in J_2(\lambda)$. For $s\in
J_2(\lambda)$, we have $$\tilde{W}(s)-\tilde{w}(s,x)\ge
\tilde{W}(s)-c+\lambda\ge \lambda/2,\quad
x\in\Omega_{s}^+(\lambda),$$ and thus (\[log10a\]) yields that a.e. in $J_2(\lambda)$ $$\label{log12}
\frac{\nu}{16r^2
\mu_N(B)}\,\,\mu_N\big(\Omega^+_s(\lambda)\big)\le
\frac{1}{(\tilde{W}-c+\lambda)^2}\,\Big(e^{-\tilde{W}}
\partial_s(g_{1-\alpha,n}\ast
e^{\tilde{W}})+\,\frac{C_1}{r^2}\,+\tilde{S_n}\Big).$$
We proceed now similarly as above for the term $I_1$. Set $\chi(s,\lambda)=\mu_N\big(\Omega^+_{s}(\lambda)\big)$, if $s\in
J_2(\lambda)$, and $\chi(s,\lambda)=0$ in case $s\in
\tilde{J}_+\setminus J_1(\lambda)$. We consider this time the convex function $H(y)=(\log y-c+\lambda)^{-1}$ for $y\ge
y_*:=e^{c-\lambda/2}$ with derivative $H'(y)=-(\log
y-c+\lambda)^{-2} y^{-1}<0$. We define a $C^1$ extension $\bar{H}$ of $H$ on $[0,\infty)$ by means of $$\bar{H}(y)=\left\{ \begin{array}{l@{\;:\;}l}
H'(y_*)(y-y_*)+H(y_*) & 0\le y< y_* \\
H(y) & y\ge y_*.
\end{array} \right.$$ Evidently, $-\bar{H}$ is concave in $[0,\infty)$ and $$\label{log13}
0\le-\bar{H}'(y)y \le \,\frac{1}{(\log
y_*-c+\lambda)^2}\,\le\,\frac{1}{(\lambda/2)^2}\,\le
\,\frac{4}{\lambda},\quad y \ge 0,\;\lambda\ge 1.$$ We will assume $\lambda\ge 1$ in the subsequent lines.
Observe that $$-e^{\tilde{W}(s)} H'(e^{\tilde{W}(s)})=\,\frac{1}{(\tilde{W}(s)-c+\lambda)^2
},\quad \mbox{a.a.}\;s\in J_2(\lambda).$$ Since $-\bar{H}'\ge 0$, and $e^{-\tilde{W}}
\partial_s(g_{1-\alpha,n}\ast
e^{\tilde{W}})+C_1 r^{-2}+\tilde{S_n}\ge 0$ on $\tilde{J}_+$ due to (\[log10a\]), it thus follows from (\[log12\]) and (\[log13\]) that $$\begin{aligned}
\label{log14}
\frac{\nu}{16r^2 \mu_N(B)}\,\,\chi(s,\lambda) & \le
-e^{\tilde{W}}\bar{H}'(e^{\tilde{W}})\Big(e^{-\tilde{W}}\partial_s(g_{1-\alpha,n}\ast
e^{\tilde{W}})+\,\frac{C_1}{r^2}\,+\tilde{S}_n\Big)\nonumber\\
& \le -\bar{H}'(e^{\tilde{W}})\partial_s(g_{1-\alpha,n}\ast
e^{\tilde{W}})+\,\frac{4C_1}{\lambda
r^2}\,+\,\frac{4|\tilde{S}_n(s)|}{\lambda}, \quad\mbox{a.a.}\;
s\in \tilde{J}_+.\end{aligned}$$ By concavity of $-\bar{H}$, the fundamental identity (\[fundidentity\]) provides the estimate $$\begin{aligned}
-\bar{H}'(e^{\tilde{W}}) & \partial_s(g_{1-\alpha,n}\ast
e^{\tilde{W}}) \le -\partial_s\Big(g_{1-\alpha,n}\ast
\bar{H}\big(e^{\tilde{W}}\big)\Big)+\Big(\bar{H}(e^{\tilde{W}})-
\bar{H}'(e^{\tilde{W}})e^{\tilde{W}}\Big)g_{1-\alpha,n} \\
& \le -\partial_s\Big(g_{1-\alpha,n}\ast
\bar{H}\big(e^{\tilde{W}}\big)\Big)+\bar{H}(0)g_{1-\alpha,n} \le
-\partial_s\Big(g_{1-\alpha,n}\ast
\bar{H}\big(e^{\tilde{W}}\big)\Big)
+\,\frac{6}{\lambda}\,g_{1-\alpha,n},\end{aligned}$$ a.e. in $\tilde{J}_+$, which when combined with (\[log14\]) leads to $$\frac{\nu}{16r^2 \mu_N(B)}\,\,\chi(s,\lambda) \le
-\partial_s\Big(g_{1-\alpha,n}\ast
\bar{H}\big(e^{\tilde{W}}\big)\Big)
+\,\frac{6}{\lambda}\,g_{1-\alpha,n}+\,\frac{4C_1}{\lambda
r^2}\,+\,\frac{4|\tilde{S}_n(s)|}{\lambda},$$ for a.a. $s\in \tilde{J}_+$. We integrate this estimate over $\tilde{J}_+$ and send $n\to \infty$ to the result $$\begin{aligned}
\int_{J_2(\lambda)}\mu_N\big( & \Omega^+_s(\lambda)\big) \,ds
= \int_0^{(1-\eta)\rho} \!\!\!\!\chi(s,\lambda)\,ds\le
\,\frac{16r^2\mu_N(B)}{\nu}\,\Big(\,\frac{6}{\lambda}\,g_{2-\alpha}\big((1-\eta)\rho\big)+\,\frac{4C_1(1-\eta)\rho}{\lambda
r^2}\,\Big)\\
&
=\,\frac{16r^{2/\alpha}\mu_N(B)}{\nu\lambda}\big(6g_{2-\alpha}\big((1-\eta)\tau\big)+4C_1(1-\eta)\tau\big)=:C_6\,
\frac{r^{2/\alpha}\mu_N(B)}{\lambda},\quad \lambda\ge 1.\end{aligned}$$ Hence with $C_7=\max\{\tau,C_6\}$ we obtain that $$\label{I3est}
I_3\le \,\frac{C_7 r^{2/\alpha}\mu_N(B)}{\lambda},\quad\lambda>0.$$
Finally, combining (\[mainleft\]), (\[mainright\]), and (\[I2est\]), (\[I4est\]), (\[I1est\]), (\[I3est\]) establishes the theorem. $\square$
The final step
--------------
We are now in position to prove Theorem \[localweakHarnack\]. Without loss of generality we may assume that $u\ge \varepsilon$ for some $\varepsilon>0$; otherwise replace $u$ by $u+\varepsilon$, which is a supersolution of (\[MProb\]) with $u_0+\varepsilon$ instead of $u_0$, and eventually let $\varepsilon\to 0+$.
For $0<\sigma\le 1$, we set $U_\sigma=(t_0+(2-\sigma)\tau
r^{2/\alpha},t_0+2\tau r^{2/\alpha})\times \sigma B$ and $U'_\sigma=(t_0,t_0+\sigma\tau r^{2/\alpha})\times \sigma B$. Clearly, $Q_-(t_0,x_0,r)=U'_\delta$ and $Q_+(t_0,x_0,r)=U_\delta$.
By Theorem \[superest1\], $$\operatorname*{ess\,sup}_{U_{\sigma'}}{u^{-1}} \le \Big(\frac{C \mu_{N+1}(U_1)^{-1}
}{(\sigma-\sigma')^{\tau_0}}\Big)^{1/\gamma}
|u^{-1}|_{L_{\gamma}(U_\sigma)},\quad \delta\le \sigma'<\sigma\le
1,\; \gamma\in (0,1].$$ Here $C=C(\nu,\Lambda,\delta,\tau,\alpha,N)$ and $\tau_0=\tau_0(\alpha,N)$. This shows that the first hypothesis of Lemma \[abslemma\] is satisfied by any positive constant multiple of $u^{-1}$ with $\beta_0=\infty$.
Consider now $f_1=u^{-1}e^{c(u)}$ where $c(u)$ is the constant from Theorem \[logest\] with $K_-=U'_1$ and $K_+=U_1$. Since $\log
f_1=c(u)-\log u$, we see from Theorem \[logest\], estimate (\[logestright\]), that $$\mu_{N+1}(\{(t,x)\in U_1:\;\log f_1(t,x)>\lambda\})\le
M\mu_{N+1}(U_1)\lambda^{-1},\quad \lambda>0,$$ where $M=M(\nu,\Lambda,\delta,\tau,\eta,\alpha,N)$. Hence we may apply Lemma \[abslemma\] with $\beta_0=\infty$ to $f_1$ and the family $U_\sigma$; thereby we obtain $$\operatorname*{ess\,sup}_{U_\delta} f_1\le M_1$$ with $M_1=M_1(\nu,\Lambda,\delta,\tau,\eta,\alpha,N)$. In terms of $u$ this means that $$\label{HH1}
e^{c(u)}\le M_1\, \operatorname*{ess\,inf}_{U_\delta} u.$$
On the other hand, Theorem \[superest2\] yields $$|u|_{L_{p}(U_{\sigma'}')}\le \Big(\frac{C\mu_{N+1}(U'_1)^{-1}
}{(\sigma-\sigma')^{\tau_1}}\Big)^{1/\gamma-1/p}
|u|_{L_{\gamma}(U'_\sigma)},\quad \delta\le \sigma'<\sigma\le 1,\;
0<\gamma\le p/\tilde{\kappa}.$$ Here $C=C(\nu,\Lambda,\delta,\tau,\alpha,N,p)$ and $\tau_1=\tau_1(\alpha,N)$. Thus the first hypothesis of Lemma \[abslemma\] is satisfied by any positive constant multiple of $u$ with $\beta_0=p$ and $\eta=1/\tilde{\kappa}$. Taking $f_2=u
e^{-c(u)}$ with $c(u)$ from above, we have $\log f_2=\log u-c(u)$ and so Theorem \[logest\], estimate (\[logestleft\]), gives $$\mu_{N+1}(\{(t,x)\in U'_1:\;\log f_2(t,x)>\lambda\})\le
M\mu_{N+1}(U'_1)\lambda^{-1},\quad \lambda>0,$$ where $M$ is as above. Therefore we may again apply Lemma \[abslemma\], this time to the function $f_2$ and the sets $U'_\sigma$, and with $\beta_0=p$ and $\eta=1/\tilde{\kappa}$; we get $$|f_2|_{L_p(U'_\delta)}\le M_2 \mu_{N+1}(U'_1)^{1/p},$$ where $M_2=M_2(\nu,\Lambda,\delta,\tau,\eta,\alpha,N,p)$. Rephrasing then yields $$\label{HH2}
\mu_{N+1}(U'_1)^{-1/p}|u|_{L_p(U'_\delta)}\le M_2 e^{c(u)}.$$
Finally, we combine (\[HH1\]) and (\[HH2\]) to the result $$\mu_{N+1}(U'_1)^{-1/p}|u|_{L_p(U'_\delta)}\le M_1 M_2\, \operatorname*{ess\,inf}_{U_\delta}
u,$$ which proves the assertion. $\square$
Optimality of the exponent $\frac{2+N\alpha}{2+N\alpha-2\alpha}$ in the weak Harnack inequality
===============================================================================================
In this section we will show that the exponent $\frac{2+N\alpha}{2+N\alpha-2\alpha}$ in Theorem \[localweakHarnack\] is optimal.
To this purpose consider the nonhomogeneous fractional diffusion equation on ${\mathbb{R}}^N$ $$\label{opt1}
\partial_t^\alpha u-\Delta u=f,\quad t\in (0,T],\,x\in {\mathbb{R}}^N,$$ with initial condition $$\label{opt2}
u(0,x)=0,\quad x\in {\mathbb{R}}^n.$$ Following [@Koch], we say that a function $u\in C([0,T]\times
{\mathbb{R}}^N)\cap C((0,T];C^2({\mathbb{R}}^N))$ with $g_{1-\alpha}\ast u\in
C^1((0,T];C({\mathbb{R}}^N))$ is a classical solution of the problem (\[opt1\]), (\[opt2\]) if $u$ satisfies (\[opt1\]) and (\[opt2\]). For any bounded continuous function $f$ that is locally Hölder continuous in $x$, there exists a unique classical solution $u$ of the problem (\[opt1\]), (\[opt2\]), and it is of the form $$\label{opt3}
u(t,x)=\int_0^t \int_{{\mathbb{R}}^N} Y(t-\tau,x-y)f(\tau,y)\,dy\,d\tau,$$ where $$Y(t,x)=c(N)|x|^{-N}t^{\alpha-1}
H^{20}_{12}\Big(\frac{1}{4}\,t^{-\alpha}|x|^2\Big|{}^{(\alpha,\alpha)}_{(N/2,1),\,(1,1)}\Big),$$ cf. [@Koch]. Here $H^{20}_{12}(z|{}^{(\alpha,\alpha)}_{(N/2,1),\,(1,1)})$ denotes a special $H$ function (also termed Fox’s $H$ function), see [@KST Section 1.12] and [@Koch] for its definition. It is differentiable for $z>0$, the asymptotic behaviour for $z\to \infty$ and $z\to +0$, respectively, is described in [@Koch formulae (3.9) and (3.14)]. It has been also proved in [@Koch] that $Y$ is nonnegative.
We choose a smooth and nonnegative approximation of unity $\{\phi_n(t,x)\}_{n\in {\mathbb{N}}}$ in ${\mathbb{R}}_+\times {\mathbb{R}}^N$ such that each $\phi_n$ is bounded. Put $f=\phi_n$ in (\[opt1\]) and denote the corresponding classical solution of (\[opt1\]), (\[opt2\]) by $u_n$. Evidently, $u_n$ is nonnegative and satisfies $$\partial_t^\alpha u_n-\Delta u_n=\phi_n\ge 0,\quad t\in (0,T],\,x\in
{\mathbb{R}}^N.$$ Hence $u_n$ is a nonnegative supersolution of (\[opt1\]) with $f=0$ for all $n\in {\mathbb{N}}$.
Suppose the weak Harnack inequality (\[localwHarnackF\]) holds for some $p\ge \frac{2+N\alpha}{2+N\alpha-2\alpha}$. Then, by taking $Q_-=(0,1)\times B(0,1)$ and $Q_+=(2,3)\times B(0,1)$ it follows that $$\label{opt4}
\big(\int_{Q_-}u_n^p\,d\mu_{N+1}\big)^{1/p}\le C \inf_{Q_+}
u_n,\quad n\in {\mathbb{N}},$$ where the constant $C$ is independent of $n$. Since $u_n\to Y$ in the distributional sense as $n\to \infty$, we have $$\inf_{Q_+} u_n\le \,\frac{1}{\mu_{N+1}(Q_+)}\,\int_{Q_+}u_n
\,d\mu_{N+1}\le 1+ \,\frac{1}{\mu_{N+1}(Q_+)}\,\int_{Q_+}Y
\,d\mu_{N+1}<\infty,\quad n\ge n_0,$$ for a sufficiently large $n_0$. On the other hand, the left-hand side of (\[opt4\]) cannot stay bounded, since $Y\notin L_p(Q_-)$ for $p\ge \frac{2+N\alpha}{2+N\alpha-2\alpha}$. In fact, writing $H^{20}_{12}(z)=H^{20}_{12}(z|{}^{(\alpha,\alpha)}_{(N/2,1),\,(1,1)})$ for short, we have $$\begin{aligned}
|Y|_{L_p(Q_-)}^p & = \int_0^1 \int_{B(0,1)} c(N)^p
|x|^{-Np}t^{(\alpha-1)p}
H^{20}_{12}\big(t^{-\alpha}|x|^2/4\big)^p\,dx\,dt\\
& = c_1 \int_0^1 \int_0^1
r^{N-1-Np}t^{(\alpha-1)p}H^{20}_{12}\big(t^{-\alpha}r^2/4\big)^p\,dr\,dt\\
& = c_1 \int_0^1 \int_0^{t^{-\alpha/2}}\big(\rho
t^{\alpha/2})^{N-1-Np}t^{(\alpha-1)p+\alpha/2}H^{20}_{12}\big(\rho^2/4\big)^p\,d\rho\,dt\\
& \ge c_1 \int_0^1 t^{\alpha(N-Np)/2+(\alpha-1)p}\,dt \,\int_0^1
\rho^{N-1-Np}H^{20}_{12}\big(\rho^2/4\big)^p\,d\rho\\
& \ge c_2 \int_0^1 t^{\alpha(N-Np)/2+(\alpha-1)p}\,dt,\end{aligned}$$ with some positive constant $c_2$. The last integral diverges for all $p\ge \frac{2+N\alpha}{2+N\alpha-2\alpha}$. Hence (\[opt4\]) yields a contradiction.
Applications of the weak Harnack inequality
===========================================
The strong maximum principle for weak subsolutions of (\[MProb\]) may be easily derived as a consequence of the weak Harnack inequality.
\[strongmax\] Let $\alpha\in(0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1)–(H3) are satisfied. Let $u\in
Z_\alpha$ be a weak subsolution of (\[MProb\]) in $\Omega_T$ and assume that $0\le \operatorname*{ess\,sup}_{\Omega_T}u<\infty$ and that $\operatorname*{ess\,sup}_{\Omega}
u_0\le \operatorname*{ess\,sup}_{\Omega_T}u$. Then, if for some cylinder $Q=(t_0,t_0+\tau r^{2/\alpha})\times B(x_0,r)\subset \Omega_T$ with $t_0,\tau,r>0$ and $\overline{B(x_0,r)}\subset \Omega$ we have $$\label{strrel}
\operatorname*{ess\,sup}_{Q}u \,=\,\operatorname*{ess\,sup}_{\Omega_T}u,$$ the function $u$ is constant on $(0,t_0)\times \Omega$.
[*Proof:*]{} Let $M=\operatorname*{ess\,sup}_{\Omega_T}u$. Then $v:=M-u$ is a nonnegative weak supersolution of (\[MProb\]) with $u_0$ replaced by $v_0:=M-u_0\ge 0$. For any $0\le t_1< t_1+\eta r^{2/\alpha}<t_0$ the weak Harnack inequality with $p=1$ applied to $v$ yields an estimate of the form $$r^{-(N+2/\alpha)}\int_{t_1}^{t_1+\eta
r^{2/\alpha}}\int_{B(x_0,r)}(M-u)\,dx\,dt\le C\,\operatorname*{ess\,inf}_Q (M-u)\,=\,0.$$ This shows that $u=M$ a.e. in $(0,t_0)\times B(x_0,r)$. As in the classical parabolic case (cf. [@Lm]) the assertion now follows by a chaining argument. $\square$ $\mbox{}$
We next apply the weak Harnack inequality to establish continuity at $t=0$ for weak solutions.
\[Hoeldert=0\] Let $\alpha\in(0,1)$, $T>0$, and $\Omega\subset {\mathbb{R}}^N$ be a bounded domain. Suppose the assumptions (H1) and (H2) are satisfied. Let $u\in Z_\alpha$ be a bounded weak solution of (\[MProb\]) in $\Omega_T$ with $u_0=0$. Then $u$ is continuous at $(0,x_0)$ for all $x_0\in \Omega$ and $\lim_{(t,x)\to (0,x_0)}u(t,x)=0$. Moreover, letting $\eta>0$ we have for any cylinder $Q(x_0,r_0):=(0,\eta
r_0^{2/\alpha})\times B(x_0,r_0)\subset \Omega_T$ and $r\in (0,r_0]$ $$\label{oscest}
\operatorname*{ess\,osc}_{Q(x_0,r)} u \le C\Big(\,\frac{r}{r_0}\,\Big)^\delta
|u|_{L_\infty(\Omega_T)},$$ with $\operatorname*{ess\,osc}_{Q(x_0,r)}=\operatorname*{ess\,sup}_{Q(x_0,r)}-\operatorname*{ess\,inf}_{Q(x_0,r)}$ and constants $C=C(\nu,\Lambda,\eta,\alpha,N)>0$ and $\delta=\delta(\nu,\Lambda,\eta,\alpha,N)\in (0,1)$.
[*Proof:*]{} Let $u\in Z_\alpha$ be a bounded weak solution of (\[MProb\]) in $\Omega_T$ with $u_0=0$. Set $u(t,x)=0$ and $A(t,x)=Id$ for $t<0$ and $x\in \Omega$. For $T_0>0$ we shift the time by setting $s=t+T_0$ and put $\tilde{f}(s)=f(s-T_0)$, $s\in
(0,T+T_0)$, for functions $f$ defined on $(-T_0,T)$. Since $Du(t,\cdot)=0$ for $t<0$ and $$\partial_t(g_{1-\alpha,n}\ast u)(t,x)=\partial_t\int_{-T_0}^t
g_{1-\alpha,n}(t-\tau)u(\tau,x)\,d\tau=\partial_s(g_{1-\alpha,n}\ast
\tilde{u})(s,x),$$ the function $\tilde{u}$ is a bounded weak solution of $$\partial_s^\alpha \tilde{u}-\mbox{div}\,\big(\tilde{A}(s,x)D\tilde{u}\big)=0,\quad s\in (0,T+T_0),\,x\in
\Omega.$$
Next, assuming $r\in (0,r_0/2]$ we introduce the cylinders $$\begin{aligned}
Q_*(x_0,r) & =\big(-\eta r^{2/\alpha},\eta r^{2/\alpha}\big)\times
B(x_0,r),\\
Q_-(x_0,r) & =\big(-\eta (2r)^{2/\alpha},-\eta
(3r/2)^{2/\alpha}\big)\times B(x_0,r),\end{aligned}$$ and denote by $\tilde{Q}_*(x_0,r)$ resp. $\tilde{Q}_-(x_0,r)$ the corresponding cylinders in the $(s,x)$ coordinate system. Let us write $M_i=\operatorname*{ess\,sup}_{\tilde{Q}_*(x_0,ir)}\tilde{u}$ and $m_i=\operatorname*{ess\,inf}_{\tilde{Q}_*(x_0,ir)}\tilde{u}$ for $i=1,2$. Choosing $T_0\ge\eta (2r)^{2/\alpha}$, we may apply Theorem \[localweakHarnack\] with $p=1$ to the functions $M_2-\tilde{u}$, $\tilde{u}-m_2$, which are nonnegative in $(0,\eta
(2r)^{2/\alpha}+T_0)\times B(x_0,2r)$, thereby obtaining $$\begin{aligned}
r^{-N+2/\alpha}\int_{\tilde{Q}_-(x_0,r)}(M_2-\tilde{u})\,d\mu_{N+1}
& \le C(M_2-M_1),\\
r^{-N+2/\alpha}\int_{\tilde{Q}_-(x_0,r)}(\tilde{u}-m_2)\,d\mu_{N+1}
& \le C(m_1-m_2),\end{aligned}$$ where $C>1$ is a constant independent of $u$ and $r$. By addition, it follows that $$M_2-m_2\le C(M_2-m_2+m_1-M_1).$$ Writing $\omega(x_0,r)=\operatorname*{ess\,sup}_{\tilde{Q}_*(x_0,ir)}\tilde{u}-\operatorname*{ess\,inf}_{\tilde{Q}_*(x_0,ir)}\tilde{u}$, this yields $$\label{Hc1}
\omega(x_0,r)\le \theta \omega(x_0,2r),\quad r\le r_0/2,$$ where $\theta=1-C^{-1}\in (0,1)$. Iterating (\[Hc1\]) as in the proof of [@GilTrud Lemma 8.23] we obtain $$\omega(x_0,r)\le \,\frac{1}{\theta}\,\Big(\frac{r}{r_0}\Big)^{\log
\theta/\log(1/2)}\omega(x_0,r_0),\quad r\le r_0.$$ The estimate (\[oscest\]) then follows by transforming back to the function $u$ and using that $u=0$ for negative times. In particular, we also see that $u$ is continuous at $(0,x_0)$ for all $x_0\in
\Omega$ and that $\lim_{(t,x)\to (0,x_0)}u(t,x)=0$. $\square$ $\mbox{}$
The last application is a theorem of Liouville type. We say that a function $u$ on ${\mathbb{R}}_+\times {\mathbb{R}}^N$ is a [*global weak solution*]{} of $$\label{GlE}
\partial_t^\alpha u-\mbox{div}\,\big(A(t,x)Du\big)=0,$$ if it is a weak solution of (\[GlE\]) in $(0,T)\times B(0,r)$ for all $T>0$ and $r>0$.
\[Liouville\] Let $\alpha\in(0,1)$. Assume that $A\in L_\infty({\mathbb{R}}_+\times
{\mathbb{R}}^N;{\mathbb{R}}^{N\times N})$ and that there exists $\nu>0$ such that $$\big(A(t,x)\xi|\xi\big)\ge \nu|\xi|^2,\quad\mbox{for a.a.}\;
(t,x)\in{\mathbb{R}}_+\times {\mathbb{R}}^N,\; \mbox{and all}\;\xi\in {\mathbb{R}}^N.$$ Suppose that $u$ is a global bounded weak solution of (\[GlE\]). Then $u=0$ a.e. on ${\mathbb{R}}_+\times {\mathbb{R}}^N$.
[*Proof:*]{} For $r>0$ and $x_0=0$ it follows from the proof of Theorem \[Hoeldert=0\] that $$\label{Hc2}
\omega(0,r)\le \theta \omega(0,2r),\quad r>0,$$ where $\theta\in(0,1)$ is independent of $r$ and $u$. By induction, (\[Hc2\]) yields $$\omega(0,r)\le \theta^n \omega(0,2^nr)\le 2\theta^n
|u|_{L_\infty({\mathbb{R}}_+\times {\mathbb{R}}^N)},\quad r>0,\,n\in{\mathbb{N}}.$$ Sending $n\to \infty$ shows that $u$ is constant. The claim then follows by Theorem \[Hoeldert=0\]. $\square$ $\mbox{}$
[**Acknowledgements:**]{} This paper was initiated while the author was visiting the Technical University Delft (NL) in 2003/2004. The author is greatly indebted to Philippe Clément for many fruitful discussions and valuable suggestions.
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|
---
author:
- |
Aviv Ofir\
für Astrophysik, Georg-August-Universität, Göttingen, Germany.\
title: 'KeSeF - Kepler Self Follow-up Mission\'
---
Introduction
============
The failure of two of Kepler’s four reaction wheels made the spacecraft unsuitable for the survey which it was designed to perform. Additionally, some of Kepler most important discoveries are the large number of candidates, including the large number of multi-transiting systems. Continued monitoring of these systems is scientifically very desirable, allowing the detection of long-period planets as well as an increased time baseline for transit timing variation (TTVs) signals among other things (see details on §\[Rational\]). Continued observations of all of Kepler’s $~150,000$ targets, even at the reduced precision possible with the crippled spacecraft, is unfeasible. However, it appears possible (see §\[Implementing\]) that observing a relatively small subset of the stars – primarily the already-identified Kepler Objects of Interest (KOIs) – at precision not dramatically different than before will be possible. Here we propose the Kepler Self Follow-up (KeSeF) Mission to use Kepler itself for its own follow up, aiming to maximize the scientific return [^1] on the original Kepler Survey Mission.
Science Rational {#Rational}
================
**Target stars:** It is very fortunate that the number of foreseen targets for the KeSeF mission (minimum of 5000 targets, see §\[Implementing\]) is larger than the current number of KOI (Kepler Object of Interest) host stars, and having more KeSeF targets may allow also the inclusion of all known Kepler eclipsing binaries (that may also include circumbinary planets). We therefore label the two populations above as primary and secondary target populations, respectively, that the proposed KeSeF mission aims to observe and perform the self follow-up on. Remaining target resources, if available, may be given to other investigation such as astroseismology, guest observer programs, high-quality non-candidate stars, etc. A small number (few per Kepler channel) of hand-picked particularly bright and stable targets may be added as a calibration set.
**Extended baseline:** Continuing the Kepler Survey Mission with KeSeF will allow to explore the thousands of candidate planetary systems to the fullest extent possible. It is difficult to give exact science deliverables from the proposed KeSeF Mission as the achievable photometric precision is yet unknown. Still, all the basic reasons that caused the Kepler Extended Mission to be a continued monitoring of the same Field of View (FOV) are still valid here, among them:
- long-period planets, primarily giant planets but perhaps smaller too (depending on available precision)
- Smaller planets may become detectable with the addition of new data points.
- Increased time baseline for transit timing variations (TTVs). We note that the contribution of extending the baseline to TTVs signals is much more than linear ($\propto t^{5/2}$ when uniformly sampled), making the continued monitoring of the same FOV particularly valuable.
A good example is KOI 1574 (see Fig. \[KOI1574\]): this system contain a giant inner planet in a 114d period and an interesting outer super-Earth-mass and low-density planet in a 191d orbit where both masses, and hence densities, were derived from TTVs. Importantly, the system includes one of the largest TTV signals known (amplitude of $\sim8$ hr) but the long periods, coupled with the inactive module on Kepler, caused the 191d planet to be observed in transit only 4 times during the entire mission (no more events for this planet beyond those shown). Such a small number of events make dynamical model fitting prone to over fitting and systematic errors as the number of free parameters is similar to the number of data points - so every new data point makes a very significant contribution to long-period planet studies. Also, one can see that the model points for the outer planet (red $\triangle$ on Fig. \[KOI1574\]) actually have a different mean period than the observed one (manifested the upwards trend in (O-C) figure) - again showing the limitation of small number of data points.
![Top: Observed transit timing variations of two planets around KOI 1574 through quarter 13 (black ’$+$’ symbols for TTV of the inner planet due to perturbations of the outer, black ’$\ast$’ symbols for TTV of the outer planet due to perturbations of the inner planet) compared to the best fit simulated ones ($\Diamond$ and $\triangle$ respectively). Error bars are indicated with thick lines. Bottom: Zoom-in showing the low amplitude TTVs of the inner planet (figure extracted from \[6\]).[]{data-label="KOI1574"}](koi1574_ttv_symbols2.ps){width="50.00000%"}
- Also related to the above - the continued monitoring of eclipsing binaries (EBs) may allow the detection of non-transiting giant planets via binary eclipse time variations. Again, extending the baseline to such timing signals has a more than linear effect on the sensitivity to such signals.
- Extending the time baseline for EBs may allow also the detection of new transiting circumbinary planets, as this already well-aligned population of targets is a particularly rich sample of targets \[2\]. Triply-eclipsing systems, such as KOI-126 \[3\] will also become more detectable.
Thus the proposed KeSeF mission has a high potential to continue and extend the great achievements from the Kepler Survey Mission.
Implementing KeSeF {#Implementing}
==================
General requirements
--------------------
The KeSeF proposal in unlike regular proposals in that it must be executed by a crippled spacecraft. Importantly, currently there are only a few and blurry details available about Kepler’s expected photometric performance (e.g. a factor $>30$ in predicted performance for the same target, depending on yet unknown behavior \[1\]). We therefore turn to general past experience in astronomy for guidance: if high raw precision is available, and enough external data in given in order to attenuate external effects significantly, a large fraction of the original raw precision can be later recovered in post-processing. We therefore aim to minimize the effects of Kepler’s coarse pointing ability while maximize the power of post processing. Furthermore, we try to limit ourselves only by hard limits that cannot be changed (e.g., volume of downloaded data) and not by softer limits like software-imposed limits.
The key enabling idea is that if one has to deal with targets that drift across the CCDs, one should at least be able to track the targets well enough to correct for some – if not most – of the problems caused by this drift. Such tracking can be achieved by observing only a small subset of the targets, but observing all of them at short cadance (1 minute). Our proposal is therefore made out of two main modifications to Kepler’s flight software:
1. Adding the ability to specify targets (and apertures) in celestial coordinates so that the on-board software will do the coordinates-to-pixels translation on the fly and before each and every exposure. This will allow the number of pixels allotted to a given target to be maintained relative to the corresponding number during the Kepler Survey Mission.
2. The current limit of 512 simultaneous SC (short cadance) targets would be lifted, or at least significantly increased to $\sim 5000$ (required) or $\sim 10000$ (goal) simultaneous SC targets.
Importantly, significant testing of the expected results of KeSeF can be made by implementing target tracking only, which we believe to be the easier of the two proposed modifications, on the 512 available SC targets. The light curves produced in this manner can give very good idea as for the expected performance of the full KeSeF Mission.
Some details
------------
We note that some details can be given on how (and why) KeSeF may be implemented:
**Memory, computing and telemetry budgets:** The factor of x30 in the number of pixels read by changing a LC (long cadance) target to SC must by compensated by a corresponding reduction in the number of targets to keep the computational/telemetry load roughly as before, hence we require that at least 5000 LC targets will be available. Furthermore, one may be able to use the fact that the pixel data is now much more self-similar (more pixels of the same targets) to achieve better compression, enabling to transmit a larger number of SC targets with the current telemetry envelope, hence the 10000 SC targets goal. Also, by observing all the current KOIs the typical proposed target will be similar to the current typical target, hense the total pixel budget will not be significantly changed.
The current photometric apertures already include the required flexibility in post-processing since all targets already have one or two pixel halos beyond the optimal aperture. In KeSeF operation the used pixels for the apertures will always be no more than 1 pixel away from optimal ones (since they are tracked) and the target apertures will therefore need not be changed.
**Mission duration:** The observing mode will be as similar to the Kepler Survey Mission as possible: a fixed FOV (field of view) up to the final drift limit, followed by a momentum reset and re-pointing to the same starting point as before – for as long as possible. Doing this may also be very repeatable and stable in some sense once perfected.
**Enabling significant post processing:** The pixel crossing time will be roughly 5 minutes, while transits are typically much longer (several hours). The former means that there will be several data samples during the crossing of even a single pixel - and these very pixels will be crossed again and again every nodding (=momentum reset) cycle, helping post-processing. The significantly different time scale mean that correcting the drift effects will likely have a small residual impact on transit detection.
**Post processing:** It is impossible, at this stage, to give details of the best post-processing technique. We do note that some techniques (e.g., SARS \[4\], TFA-EPD \[5\]) already allow the inclusion of external information, such as the pixel- and sub-pixel positions, in the decorrelation process. We therefore believe that an effective post-processing technique can be quickly identified.
**Other notes:** On-board targets tracking and pixel allocation must be able to account for sources that go on- and off- any of the CCDs. If a nodding limit will be greater than the size of the CCD is chosen - all targets would fall off- and on- the CCDs (and maybe on- or off- the next CCD in the same nodding cycle).
Additional goals
----------------
Below we give some incentives to explore the full range of possible implementation scenarios for the KeSeF mission, beyond the basic requirements:
- Having more target resources will allow inclusion and follow-up on all Kepler objects that were referred to in scientific publication (incl. pulsating stars , non stellar objects and continued guest observer program).
- Even shorter integration times (shorter than 1 minute) should be explored. This will limit the number of targets, but may allow for better photometric performance.
- One may allow including targets that were not previously observed (e.g., those that happen to fall outside or in gaps between CCDs on the nominal FOV). This will be possible since some of the nominal target will be falling off the CCDs in the same time.
- We foresee that all targets for the KeSeF Mission would be in SC, but some users may opt to use LC data in order to maximize their number of targets - so maybe LC targets may not be completely deprecated.
- Since the mode of operation we propose for KeSeF is very similar to the nominal one, we propose to keep many of the previous facilities as well, such as: guest observer programs, continuous pixel table up keeping, etc.
Conclusions {#Conclusions}
===========
Above we presented an alternate science investigation for the Kepler spacecraft - follow-up observations of at least all the KOIs, and possibly many more interesting targets, that were detected during the Kepler Survey Mission. We described an operational mode that may allow regaining much of Kepler’s lost photometric precision, allowing the follow-up observation proposed to be truly unobtainable in any other way. Such observations can contribute significantly to the understanding of these systems and to the science result of Kepler in General. We gave a real example that very closely resembles the most optimistic case desired: TTVs of a large inner planet allow to deduce the mass of an outer and small planet near- or in- the habitable zone. It is easy to see in this example how continued monitoring would improve the understanding of such high-value systems. This example is also from a rather faint target (KepMag=14.6) - one which would probably not be selected if the number of SC targets would remain 512, or $\sim10\%$ of the *minimal* number we propose here.
Kepler was an outstanding exoplanets mission because it was carefully designed to do just one thing very well. We proposed the KeSeF Mission that will keep Kepler doing this exact thing and in a manner as similar as possible to the original mission: same field of view, same targets, same photometric apertures, same volume of data, same integration times, some environment (e.g. Solar radiation pressure) and so on. KeSeF does require some adaptations to flight software, but we believe these are both doable and worthwhile. Doing so will keep Kepler closely aligned with the original Kepler Mission goals, as well as with the Kepler Extended Mission goals, in the most natural way.
Acknowledgments
===============
I would like to thank Eric Ford for reading and improving this proposal.
References {#references .unnumbered}
==========
[^1]: Literaly, “kesef” means “money” in Hebrew, again stressing the maximizing return-on-investment theme.
|
---
abstract: 'Deeply embedded low-mass protostars can be used as testbeds to study the early formation stages of solar-type stars, and the prevailing chemistry before the formation of a planetary system. The present study aims to characterise further the physical and chemical properties of the protostellar core Orion B9–SMM3. The Atacama Pathfinder EXperiment (APEX) telescope was used to perform a follow-up molecular line survey of SMM3. The observations were done using the single pointing (frequency range 218.2–222.2 GHz) and on-the-fly mapping methods (215.1–219.1 GHz). These new data were used in conjunction with our previous data taken by the APEX and Effelsberg 100 m telescopes. The following species were identified from the frequency range 218.2–222.2 GHz: $^{13}$CO, C$^{18}$O, SO, *para*-H$_2$CO, and E$_1$-type CH$_3$OH. The mapping observations revealed that SMM3 is associated with a dense gas core as traced by DCO$^+$ and *p*-H$_2$CO. Altogether three different *p*-H$_2$CO transitions were detected with clearly broadened linewidths ($\Delta v\sim8.2-11$ km s$^{-1}$ in FWHM). The derived *p*-H$_2$CO rotational temperature, $64\pm15$ K, indicates the presence of warm gas. We also detected a narrow *p*-H$_2$CO line ($\Delta v=0.42$ km s$^{-1}$) at the systemic velocity. The *p*-H$_2$CO abundance for the broad component appears to be enhanced by two orders of magnitude with respect to the narrow line value ($\sim3\times10^{-9}$ versus $\sim2\times10^{-11}$). The detected methanol line shows a linewidth similar to those of the broad *p*-H$_2$CO lines, which indicates their coexistence. The CO isotopologue data suggest that the CO depletion factor decreases from $\sim27\pm2$ towards the core centre to a value of $\sim8\pm1$ towards the core edge. In the latter position, the N$_2$D$^+$/N$_2$H$^+$ ratio is revised down to $0.14\pm0.06$. The origin of the subfragments inside the SMM3 core we found previously can be understood in terms of the Jeans instability if non-thermal motions are taken into account. The estimated fragmentation timescale, and the derived chemical abundances suggest that SMM3 is a few times $10^5$ yr old, in good agreement with its Class 0 classification inferred from the spectral energy distribution analysis. The broad *p*-H$_2$CO and CH$_3$OH lines, and the associated warm gas provide the first clear evidence of a molecular outflow driven by SMM3.'
author:
- 'O. Miettinen'
title: Characterising the physical and chemical properties of a young Class 0 protostellar core embedded in the Orion B9 filament
---
Introduction
============
Low-mass stars have main-sequence masses of $M_{\star}\simeq0.08-2$ M$_{\sun}$, and are classified with spectral types of M7–A5 (e.g. [@stahler2005]). The formation process of these types of stars begins when the parent molecular cloud core undergoes gravitational collapse (e.g. [@shu1987]; [@mckee2007]). In the course of time, the collapsing core centre heats up due to compression, and ultimately becomes a protostar. The youngest low-mass protostars, characterised by accretion from the much more massive envelope ($M_{\rm env}\gg M_{\star}$), are known as the Class 0 objects ([@andre1993], 2000).
A curious example of a Class 0 protostellar object is SMM3 in the Orion B9 star-forming filament. This object was first uncovered by Miettinen et al. (2009; hereafter Paper I), when they mapped Orion B9 using the Large APEX BOlometer CAmera (LABOCA) at 870 $\mu$m. In Paper I, we constructed and analysed a simple mid-infrared–submillimetre spectral energy distribution (SED) of SMM3, and classified it as a Class 0 object. The physical and chemical properties of SMM3 (e.g. the gas temperature and the level of N$_2$H$^+$ deuteration) were further characterised by Miettinen et al. (2010, 2012; hereafter referred to as Papers II and III, respectively) through molecular line observations. In Paper III, we also presented the results of our Submillimetre APEX BOlometer CAmera (SABOCA) 350 $\mu$m imaging of Orion B9. With the flux density of $S_{350\,{\rm \mu m}}\simeq 5.4$ Jy, SMM3 turned out to be the strongest 350 $\mu$m emitter in the region. Perhaps more interestingly, the 350 $\mu$m image revealed that SMM3 hosts two subfragments (dubbed SMM3b and 3c) on the eastern side of the protostar, where an extension could already be seen in the LABOCA map at 870 $\mu$m. The projected distances of the subfragments from the protostar’s position, 0.07–0.10 pc[^1], were found to be comparable to the local thermal Jeans length. This led us to suggest that the parent core might have fragmented into smaller units via Jeans gravitational instability.
The Orion B or L1630 molecular cloud, including Orion B9, was mapped with *Herschel* as part of the *Herschel* Gould Belt Survey (HGBS; André et al. 2010)[^2]. The *Herschel* images revealed that Orion B9 is actually a filamentary-shaped cloud in which SMM3 is embedded (see Fig. 2 in [@miettinen2013b]). Miettinen (2012b) found that there is a sharp velocity gradient in the parent filament (across its short axis), and suggested that it might represent a shock front resulting from the feedback from the nearby expanding H[II]{} region/OB cluster NGC 2024 ($\sim3.7$ pc to the southwest of Orion B9). Because SMM3 appears to lie on the border of the velocity gradient, it might have a physical connection to it, and it is possible that the formation of SMM3 (and the other dense cores in Orion B9) was triggered by external, positive feedback ([@miettinen2012b]). Actually, the OB associations to the west of the whole Orion B cloud have likely affected much of the cloud area through their strong feedback in the form of ionising radiation and stellar winds (e.g. [@cowie1979]). The column density probability distribution function of Orion B, studied by Schneider et al. (2013), was indeed found to be broadened as a result of external compression.
The Class 0 object SMM3 was included in the Orion protostellar core survey by Stutz et al. (2013, hereafter S13; their source 090003). Using data from *Spitzer*, *Herschel*, SABOCA, and LABOCA, S13 constructed an improved SED of SMM3 compared to what was presented in Paper I. The bolometric temperature and luminosity – as based on the Myers & Ladd (1993) method – were found to be $T_{\rm bol}=36.0\pm0.8$ K and $L_{\rm bol}=2.71\pm0.24$ L$_{\sun}$. They also performed a modified blackbody (MBB) fit to the SED ($\lambda \geq70$ $\mu$m) of SMM3, and obtained a dust temperature of $T_{\rm dust}=21.4\pm0.4$ K, luminosity of $L=2.06\pm0.15$ L$_{\sun}$, and envelope mass of $M_{\rm env}=0.33\pm0.06$ M$_{\sun}$ (see their Fig. 9). The derived SED properties led S13 to the conclusion that SMM3 is likely a Class 0 object, which supports our earlier suggestion (Papers I and III).
Tobin et al. (2015) included SMM3 in their Combined Array for Research for Millimetre Astronomy (CARMA) 2.9 mm continuum imaging survey of Class 0 objects in Orion. This was the first high angular resolution study of SMM3. With a 2.9 mm flux density of $S_{\rm 2.9\, mm}=115.4\pm3.9$ mJy (at an angular resolution of $2\farcs74 \times 2\farcs56$), SMM3 was found to be the second brightest source among the 14 target sources. The total (gas$+$dust) mass derived for SMM3 by Tobin et al. (2015), $M=7.0\pm0.7$ M$_{\sun}$, is much higher than that derived earlier by S13 using a MBB fitting technique, which underpredicted the 870 $\mu$m flux density of the source (see [@tobin2015] and Sect. 4.1 herein for further discussion and different assumptions used). Tobin et al. (2015) did not detect 2.9 mm emission from the subfragments SMM3b or 3c, which led the authors to conclude that they are starless.
Kang et al. (2015) carried out a survey of H$_2$CO and HDCO emission towards Class 0 objects in Orion, and SMM3 was part of their source sample (source HOPS403 therein). The authors derived a HDCO/H$_2$CO ratio of $0.31\pm0.06$ for SMM3, which improves our knowledge of the chemical characteristics of this source, and strongly points towards its early evolutionary stage from a chemical point of view.
Finally, we note that SMM3 was part of the recent large Orion protostellar core survey by Furlan et al. (2016; source HOPS400 therein), where the authors presented the sources’ panchromatic (1.2–870 $\mu$m) SEDs and radiative transfer model fits. They derived a bolometric luminosity of $L_{\rm bol}=2.94$ L$_{\sun}$ (a trapezoidal summation over all the available flux density data points), total (stellar$+$accretion) luminosity of $L_{\rm tot}=5.2$ L$_{\sun}$, bolometric temperature of $T_{\rm bol}=35$ K (following [@myers1993] as in S13), and an envelope mass of $M_{\rm env}=0.30$ M$_{\sun}$, which are in fairly good agreement with the earlier S13 results. We note that the total luminosity derived by Furlan et al. (2016) from their best-fit model is corrected for inclination effects, and hence is higher than $L_{\rm bol}$. Moreover, the aforementioned value of $M_{\rm env}$ refers to a radius of 2500 AU ($=0.012$ pc), which corresponds to an angular radius of about $6\arcsec$ at the distance of SMM3, while a similar envelope mass value derived by S13 refers to a larger angular scale as a result of coarser resolution of the observational data used (e.g. $19\arcsec$ resolution in their LABOCA data).
In the present study, we attempt to further add to our understanding of the physical and chemical properties of SMM3 by means of our new molecular line observations. We also re-analyse our previous spectral line data of SMM3 in a uniform manner to make them better comparable with each other. This paper is outlined as follows. The observations and the observational data are described in Sect. 2. The immediate observational results are presented in Sect. 3. The analysis of the observations is described in Sect. 4. The results are discussed in Sect. 5, and the concluding remarks are given in Sect. 6.
Observations, data, and data reduction
======================================
New spectral line observations with APEX
----------------------------------------
### Single-pointing observations
A single-pointing position at $\alpha_{2000.0}=05^{\rm h}42^{\rm m}45\fs24$, and $\delta_{2000.0}=-01\degr16\arcmin14\farcs0$ (i.e. the *Spitzer* 24 $\mu$m peak of SMM3) was observed with the 12-metre APEX telescope[^3] ([@gusten2006]) in the frequency range $\sim218.2-222.2$ GHz. The observations were carried out on 20 August 2013, when the amount of precipitable water vapour (PWV) was measured to be 1.3 mm, which corresponds to a zenith atmospheric transmission of about 93%.
As a front end we used the APEX-1 receiver of the Swedish Heterodyne Facility Instrument (SHeFI; [@belitsky2007]; [@vassilev2008a],b). The APEX-1 receiver operates in a single-sideband (SSB) mode using sideband separation mixers, and it has a sideband rejection ratio better than 10 dB. The backend was the RPG eXtended bandwidth Fast Fourier Transfrom Spectrometer (XFFTS; see [@klein2012]) with an instantaneous bandwidth of 2.5 GHz and 32768 spectral channels. The spectrometer consists of two units, which have a fixed overlap region of 1.0 GHz. The resulting channel spacing, 76.3 kHz, corresponds to 104 m s$^{-1}$ at the central observed frequency of 220196.65 MHz. The beam size (Half-Power Beam Width or HPBW) at the observed frequency range is $\sim28\farcs1-28\farcs6$.
The observations were performed in the wobbler-switching mode with a $100\arcsec$ azimuthal throw between two positions on sky (symmetric offsets), and a chopping rate of $R=0.5$ Hz. The total on-source integration time was 34 min. The telescope focus and pointing were optimised and checked at regular intervals on the planet Jupiter and the variable star R Leporis (Hind’s Crimson Star). The pointing was found to be accurate to $\sim3\arcsec$. The typical SSB system temperatures during the observations were in the range $T_{\rm sys}\sim130-140$ K. Calibration was made by means of the chopper-wheel technique, and the output intensity scale given by the system is the antenna temperature corrected for the atmospheric attenuation ($T_{\rm A}^{\star}$). The observed intensities were converted to the main-beam brightness temperature scale by $T_{\rm MB}=T_{\rm A}^{\star}/\eta_{\rm MB}$, where $\eta_{\rm MB}=0.75$ is the main-beam efficiency at the observed frequency range. The absolute calibration uncertainty is estimated to be about 10%.
The spectra were reduced using the Continuum and Line Analysis Single-dish Software 90 ([CLASS90]{}) program of the GILDAS software package[^4]. The individual spectra were averaged, and the resulting spectra were Hanning-smoothed to a velocity resolution of 208 m s$^{-1}$ to improve the signal-to-noise (S/N) ratio. Linear (first-order) baselines were determined from the velocity ranges free of spectral line features, and then subtracted from the spectra. The resulting $1\sigma$ rms noise levels at the smoothed velocity resolution were $\sim6.3-19$ mK on a $T_{\rm A}^{\star}$ scale, or $\sim8.4-25.3$ mK on a $T_{\rm MB}$ scale.
The line identification from the observed frequency range was done by using [Weeds]{}, which is an extension of [CLASS]{} ([@maret2011]), and the JPL[^5] and CDMS[^6] spectroscopic databases. The following spectral line transitions were detected: $^{13}$CO$(2-1)$, C$^{18}$O$(2-1)$, SO$(5_6-4_5)$, *para*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$, *para*-H$_2$CO$(3_{2,\,2}-2_{2,\,1})$, *para*-H$_2$CO$(3_{2,\,1}-2_{2,\,0})$, and E$_1$-type CH$_3$OH$(4_2-3_1)$. Selected spectroscopic parameters of the detected species and transitions are given in Table \[table:lines\]. We note that the original purpose of these observations was to search for glycolaldehyde (HCOCH$_2$OH) line emission near 220.2 GHz (see [@jorgensen2012]; [@coutens2015]). However, no positive detection of HCOCH$_2$OH lines was made.
### Mapping observations
The APEX telescope was also used to map SMM3 and its surroundings in the frequency range $\sim215.1-219.1$ GHz. The observations were done on 15 November 2013, with the total telescope time of 2.9 hr. The target field, mapped using the total power on-the-fly mode, was $5\arcmin \times 3\farcm25$ ($0.61\times0.40$ pc$^2$) in size, and centred on the coordinates $\alpha_{2000.0}=05^{\rm h}42^{\rm m}47\fs071$, and $\delta_{2000.0}=-01\degr16\arcmin33\farcs70$. At the observed frequency range, the telescope HPBW is $\sim28\farcs5-29\arcsec$. The target area was scanned alternately in right ascension and declination, i.e. in zigzags to ensure minimal striping artefacts in the final data cubes. Both the angular separation between two successive dumps and the step size between the subscans was $9\farcs5$, i.e. about one-third the HPBW. We note that to avoid beam smearing, the readout spacing should not exceed the value HPBW/3. The dump time was set to one second. The front end/backend system was composed of the APEX-1 receiver, and the 2.5 GHz XFFTS with 32768 channels. The channel spacing, 76.3 kHz, corresponds to 105 m s$^{-1}$ at the central observed frequency of 217104.98 MHz.
The focus and pointing measurements were carried out by making CO$(2-1)$ cross maps of the planet Jupiter and the M-type red supergiant $\alpha$ Orionis (Betelgeuse). The pointing was found to be consistent within $\sim3\arcsec$. The amount of PWV was $\sim0.6$ mm, which translates into a zenith transmission of about 96%. The data were calibrated using the standard chopper-wheel method, and the typical SSB system temperatures during the observations were in the range $T_{\rm sys}\sim120-130$ K on a $T_{\rm A}^{\star}$ scale. The main-beam efficiency needed in the conversion to the main-beam brightness temperature scale is $\eta_{\rm MB}=0.75$. The absolute calibration uncertainty is about 10%.
The [CLASS90]{} program was used to reduce the spectra. The individual spectra were Hanning-smoothed to a velocity resolution of 210 m s$^{-1}$ to improve the S/N ratio of the data, and a third-order polynomial was applied to correct the baseline in the spectra. The resulting $1\sigma$ rms noise level of the average smoothed spectra were about 90 mK on a $T_{\rm A}^{\star}$ scale. The visible spectral lines, identified by using [Weeds]{}, were assigned to DCO$^+(3-2)$ and *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ (see Table \[table:lines\] for details). The latter line showed an additional velocity component at $v_{\rm LSR}\simeq1.5$ km s$^{-1}$, while the systemic velocity of SMM3 is about 8.5 km s$^{-1}$. The main purpose of these mapping observations was to search for SiO$(5-4)$ emission at 217104.98 MHz, but no signatures of this shock tracer were detected.
The spectral-line maps were produced using the Grenoble Graphic ([GreG]{}) program of the GILDAS software package. The data were convolved with a Gaussian of 1/3 times the HPBW, and hence the effective angular resolutions of the final DCO$^+(3-2)$ and *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ data cubes are $30\farcs7$ and $30\farcs4$, respectively. The average $1\sigma$ rms noise level of the completed maps was $\sigma(T_{\rm MB})\sim100$ mK per 0.21 km s$^{-1}$ channel.
Previous spectral line observations
-----------------------------------
In the present work, we also employ the *para*-NH$_3(1,\,1)$ and $(2,\,2)$ inversion line data obtained with the Effelsberg 100 m telescope[^7] as described in Paper II. The angular resolution (full-width at half maximum or FWHM) of these observations was $40\arcsec$. The original channel separation was 77 m s$^{-1}$, but the spectra were smoothed to the velocity resolution of 154 m s$^{-1}$. We note that the observed target position towards SMM3 was $\alpha_{2000.0}=05^{\rm h}42^{\rm m}44\fs4$, and $\delta_{2000.0}=-01\degr16\arcmin03\farcs0$, i.e. $\sim16\farcs7$ northwest of the new target position (Sect. 2.1.1).
In Paper III, we presented the C$^{17}$O$(2-1)$, DCO$^+$(4-3), N$_2$H$^+(3-2)$, and N$_2$D$^+(3-2)$ observations carried out with APEX towards the aforementioned NH$_3$ target position. Here, we will employ these data as well. The HPBW of APEX at the frequencies of the above transitions is in the range $21\farcs7-27\farcs8$, and the smoothed velocity resolution is 260 m s$^{-1}$ for N$_2$H$^+$ and DCO$^+$, and 320 m s$^{-1}$ for C$^{17}$O and N$_2$D$^+$. For further details, we refer to Paper III. Spectroscopic parameters of the species and transitions described in this subsection are also tabulated in Table \[table:lines\].
--------------------------------------------------- ------------- ----------------------- -------------------- ------------------ ---------------------------------------------------
Transition $\nu$ $E_{\rm u}/k_{\rm B}$ $\mu$ $n_{\rm crit}$ Rotational constants and
\[MHz\] \[K\] \[D\] \[cm$^{-3}$\] Ray’s parameter ($\kappa$)
*p*-NH$_3(J,\,K=1,\,1)$ 23694.4955 23.26 1.4719 ($=\mu_C$) $3.9\times10^3$ $A=B=298\,192.92$ MHz,
$C=186\,695.86$ MHz;
$\kappa=+1\Rightarrow$ oblate symmetric top
*p*-NH$_3(J,\,K=2,\,2)$ 23722.6333 64.45 1.4719 ($=\mu_C$) $3.08\times10^3$ …
DCO$^+(J=3-2)$ 216112.5766 20.74 3.888 ($=\mu_A$) $2.0\times10^6$ $B=36\,019.76$ MHz; linear molecule
*p*-H$_2$CO$(J_{K_a,\,K_c}=3_{0,\,3}-2_{0,\,2})$ 218222.192 20.96 2.331 ($=\mu_A$) $2.8\times10^6$ $A=281\,970.5$ MHz, $B=38\,833.98$ MHz,
$C=34\,004.24$ MHz;
$\kappa=-0.961\Rightarrow$ prolate asymmetric top
CH$_3$OH-E$_1(J_{K_a,\,K_c}=4_{2,\,2}-3_{1,\,2})$ 218440.050 45.46 0.899 ($=\mu_A$) $4.7\times10^6$ $A=127\,523.4$ MHz, $B=24\,690.2$ MHz,
$-1.44$ ($=\mu_B$) $C=23\,759.7$ MHz;
$\kappa=-0.982\Rightarrow$ prolate asymmetric top
*p*-H$_2$CO$(J_{K_a,\,K_c}=3_{2,\,2}-2_{2,\,1})$ 218475.632 68.09 2.331 ($=\mu_A$) $1.2\times10^6$ …
*p*-H$_2$CO$(J_{K_a,\,K_c}=3_{2,\,1}-2_{2,\,0})$ 218760.066 68.11 2.331 ($=\mu_A$) $2.6\times10^6$ …
C$^{18}$O$(J=2-1)$ 219560.3568 15.81 0.11079 ($=\mu_A$) $2.0\times10^4$ $B=54\,891.42$ MHz; linear molecule
SO$(N_J=5_6-4_5)$ 219949.442 34.98 1.55 ($=\mu_A$) $2.4\times10^6$ $B=21\,523.02$ MHz; linear molecule
$^{13}$CO$(J=2-1)$ 220398.7006 15.87 0.11046 ($=\mu_A$) $2.0\times10^4$ $B=55\,101.01$ MHz; linear molecule
C$^{17}$O$(J=2-1)$ 224714.199 16.18 0.11034 ($=\mu_A$) $2.1\times10^4$ $B=56\,179.99$ MHz; linear molecule
N$_2$D$^+(J=3-2)$ 231321.912 22.20 3.40 ($=\mu_A$) $1.9\times10^6$ $B=38\,554.71$ MHz; linear molecule
N$_2$H$^+(J=3-2)$ 279511.832 26.83 3.40 ($=\mu_A$) $3.3\times10^6$ $B=46\,586.86$ MHz; linear molecule
DCO$^+(J=4-3)$ 288143.855 34.57 3.888 ($=\mu_A$) $1.9\times10^7$ $B=36\,019.76$ MHz; linear molecule
--------------------------------------------------- ------------- ----------------------- -------------------- ------------------ ---------------------------------------------------
Submillimetre dust continuum data
---------------------------------
In the present study, we use our LABOCA 870 $\mu$m data first published in Paper I. However, we have re-reduced the data using the Comprehensive Reduction Utility for SHARC-2 (Submillimetre High Angular Resolution Camera II) or CRUSH-2 (version 2.12-2) software package[^8] ([@kovacs2008]), as explained in more detail in the paper by Miettinen & Offner (2013a). The resulting angular resolution was $19\farcs86$ (FWHM), and the $1\sigma$ rms noise level in the final map was 30 mJy beam$^{-1}$. Measuring the flux density of SMM3 inside an aperture of radius equal to the effective beam FWHM, we obtained a value of $S_{\rm 870\,\mu m}=1.58\pm0.29$ Jy, where the uncertainty includes both the calibration uncertainty ($\sim10\%$) and the map rms noise around the source (added in quadrature).
The SABOCA 350 $\mu$m data published in Paper III are also used in this study. Those data were also reduced with CRUSH-2 (version 2.03-2). The obtained angular resolution was $10\farcs6$ (FWHM), and the $1\sigma$ rms noise was $\sim60$ mJy beam$^{-1}$. Again, if the flux density is calculated using an aperture of radius $10\farcs6$, we obtain $S_{\rm 350\,\mu m}=4.23\pm1.30$ Jy, where the quoted error includes both the calibration uncertainty ($\sim30\%$) and the local rms noise. This value is about 1.3 times lower than the one reported in Paper III ($5.4\pm1.6$ Jy, which was based on a clumpfind analysis above a $3\sigma$ emission threshold). The APEX dust continuum flux densities of SMM3 are tabulated in Table \[table:photometry\].
Far-infrared and millimetre data from the literature
----------------------------------------------------
For the purpose of the present study, we use the far-infrared (FIR) flux densities from S13, and the 2.9 mm flux density of $S_{\rm 2.9\, mm}=115.4\pm3.9$ mJy from Tobin et al. (2015). Stutz et al. (2013) employed the *Herschel*/Photodetector Array Camera & Spectrometer (PACS; [@pilbratt2010]; [@poglitsch2010]) observations of SMM3 at 70 and 160 $\mu$m. Moreover, they used the *Herschel*/PACS 100 $\mu$m data from the HGBS. The aperture radii used for the photometry at the aforementioned three wavelengths were $9\farcs6$, $12\farcs8$, and $9\farcs6$, respectively, and the flux densities were found to be $S_{\rm 70\,\mu m}=3.29\pm0.16$ Jy, $S_{\rm 100\,\mu m}=10.91\pm2.79$ Jy, and $S_{\rm 160\,\mu m}=16.94\pm2.54$ Jy (see Table 4 in S13). We note that the *Spitzer*/MIPS (the Multiband Imaging Photometer for *Spitzer*; [@rieke2004]) 70 $\mu$m flux density we determined in Paper I, $3.6\pm0.4$ Jy, is consistent with the aforementioned *Herschel*-based measurement (see Table \[table:photometry\] for the flux density comparison).
-------------- --------------------- --------------------- ---------------------- ---------------------- ---------------------- ---------------------- -------------------
Reference $S_{\rm 24\,\mu m}$ $S_{\rm 70\,\mu m}$ $S_{\rm 100\,\mu m}$ $S_{\rm 160\,\mu m}$ $S_{\rm 350\,\mu m}$ $S_{\rm 870\,\mu m}$ $S_{\rm 2.9\,mm}$
\[mJy\] \[Jy\] \[Jy\] \[Jy\] \[Jy\] \[Jy\] \[mJy\]
This work … … … … $4.23\pm1.30$ $1.58\pm0.29$ …
Paper I $5.0\pm0.2$ $3.6\pm0.4$ … … … $2.5\pm0.4$ …
Paper III … … … … $5.4\pm1.6$ … …
[@stutz2013] $4.74\pm0.3$ $3.29\pm0.16$ $10.91\pm2.79$ $16.94\pm2.54$ 3.63 2.2/1.9 …
[@tobin2015] … … … … … … $115.4\pm3.9$
-------------- --------------------- --------------------- ---------------------- ---------------------- ---------------------- ---------------------- -------------------
Observational results
=====================
Images of continuum emission
----------------------------
In Fig. \[figure:images\], we show the SABOCA and LABOCA submm images of SMM3, and *Spitzer* 4.5 $\mu$m and 24 $\mu$m images of the same region. We note that the latter two were retrieved from a set of Enhanced Imaging Products (SEIP) from the *Spitzer* Heritage Archive (SHA)[^9], which include both the Infrared Array Camera (IRAC; [@fazio2004]) and MIPS Super Mosaics.
The LABOCA 870 $\mu$m dust continuum emission is slightly extended to the east of the centrally concentrated part of the core. From this eastern part the SABOCA 350 $\mu$m image reveals the presence of two subcondensations, designated SMM3b and 3c (Paper III). The *Spitzer* 24 $\mu$m image clearly shows that the core harbours a central protostar, while the 4.5 $\mu$m feature slightly east of the 24 $\mu$m peak is probably related to shock emission. In particular, the 4.5 $\mu$m band is sensitive to shock-excited H$_2$ and CO spectral line features (e.g. [@smith2005]; [@ybarra2009]; [@debuizer2010]). As indicated by the plus signs in Fig. \[figure:images\], our previous line observations probed the outer edge of SMM3, i.e. the envelope region. In contrast, the present single pointing line observations were made towards the 24 $\mu$m peak position. This positional difference has to be taken into account when comparing the chemical properties derived from our spectral line data.
{width="\textwidth"}
Spectral line maps
------------------
In Fig. \[figure:linemaps\], we show the zeroth moment maps or integrated intensity maps of DCO$^+(3-2)$ and *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ plotted as contours on the SABOCA 350 $\mu$m image. The DCO$^+(3-2)$ map was constructed by integrating the line emission over the local standard of rest (LSR) velocity range of \[7.4, 11.8\] km s$^{-1}$. The *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ line showed two velocity components. The line emission associated with SMM3 was integrated over \[7.5, 11\] km s$^{-1}$, while that of the lower-velocity component ($v_{\rm LSR}\simeq1.5$ km s$^{-1}$) was integrated over \[-0.27, 2.49\] km s$^{-1}$. The aforementioned velocity intervals were determined from the average spectra. The final $1\sigma$ noise levels in the zeroth moment maps were in the range 0.08–0.16 K km s$^{-1}$ (on a $T_{\rm MB}$ scale).
With an offset of only $\Delta \alpha=-2\farcs6,\, \Delta \delta=3\farcs3$, the DCO$^+(3-2)$ emission maximum is well coincident with the 350 $\mu$m peak position of the core. The corresponding offset from our new line observation target position is $\Delta \alpha=-1\farcs7,\, \Delta \delta=5\farcs3$. Moreover, the emission is extended to the east (and slightly to the west), which resembles the dust emission morphology traced by LABOCA.
The *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ emission, shown by black contours in Fig. \[figure:linemaps\], is even more elongated in the east-west direction than that of DCO$^+$. The emission peak is located inside the $7\sigma$ contour of DCO$^+(3-2)$ emission. We note that the 350 $\mu$m subcondensations SMM3b and 3c lie within the $3\sigma$ contour of both the line emissions.
The low-velocity component of *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$, with a radial velocity of about 1.5 km s$^{-1}$, is concentrated on the east and northeast parts of the mapped region. This is exactly where the $^{13}$CO$(2-1)$ and C$^{18}$O$(2-1)$ line emissions at $\sim1.3$ km s$^{-1}$ were found to be concentrated ([@miettinen2012b]). As discussed by Miettinen (2012b), several other high-density tracer lines at a radial velocity of 1.3–1.9 km s$^{-1}$ have been detected towards other cores in Orion B9 (Papers I–III). Hence, the detection of *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ emission at this low velocity comes as no surprise.
Spectra and spectral line parameters
------------------------------------
The previously observed spectra are shown in Fig. \[figure:spectra1\]. The target position of these measurements is shown by the northwestern plus sign in Figs. \[figure:images\] and \[figure:linemaps\].
The new spectra, observed towards the 24 $\mu$m peak of SMM3, are presented in Fig. \[figure:spectra2\]. The DCO$^+(3-2)$ spectrum shown in the top panel of Fig. \[figure:spectra2\] was extracted from the line emission peak, and, as mentioned above, that position is well coincident with the 24 $\mu$m and 350 $\mu$m peaks (Fig. \[figure:linemaps\]). The *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ line shown in Fig. \[figure:spectra2\] can be decomposed into two components, namely a narrow line at the systemic velocity, and a much broader one with non-Gaussian line-wing emission. The narrow line is probably originating in the quiescent envelope around the protostar, while the broad component is probably tracing the dense ambient gas swept up by an outflow (e.g. [@yildiz2013]). However, the mapped *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ data did not show evidence of line wings (and hence we could not separately image the blue and redshifted parts of the line emission). The other two formaldehyde lines ($3_{2,\,1}-2_{2,\,0}$ and $3_{2,\,2}-2_{2,\,1}$) and the CH$_3$OH line shown in Fig. \[figure:spectra2\] are also broad, and hence likely originate in the swept-up outflow gas. A hint of an outflow wing emission is also visible in the SO spectrum.
Two velocity components are also seen in the $^{13}$CO spectrum, one at the systemic velocity, and the other at $\sim1.5$ km s$^{-1}$, the velocity at which *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ emission was seen in the line maps. The C$^{18}$O and $^{13}$CO spectra exhibit absorption features next to the emission lines. These are caused by emission in the OFF beam positions when chopping between two positions on sky (wobbling secondary). This problem has been recognised in our previous papers on Orion B9, and is difficult to avoid when observing the abundant CO isotopologues. In fact, the detected $^{13}$CO line at the systemic velocity suffers so badly from the subtraction of the off-signal that the line shape and intensity are deformed. For example, the intensity of the C$^{18}$O line appears to be higher than that of the more abundant $^{13}$CO isotopologue. Hence, the $^{13}$CO data are not used in the present study.
The hyperfine structure of the ammonia lines were fitted using the [CLASS90]{}’s methods NH3$(1,\,1)$ and NH3$(2,\,2)$. The former method could be used to derive the optical thickness of the main hyperfine group ($\tau_{\rm m}$; see Sect. 4.2.1). The remaining lines shown in Fig. \[figure:spectra1\] are also split into hyperfine components, and hence were fitted using the [CLASS90]{}’s hyperfine structure method.
Of the newly observed lines, only DCO$^+(3-2)$ (cf. [@vandertak2009]) and $^{13}$CO$(2-1)$ (Cazzoli et al. 2004) exhibit hyperfine structure. In Fig. \[figure:spectra2\], the fits to the $^{13}$CO lines are shown, but, as mentioned above, we do not study the lines further in the present paper. Single-Gaussian fits to the remaining lines were performed using [CLASS90]{}. The obtained line parameters are listed in Table \[table:lineparameters\]. Columns (2)–(5) in this table give the LSR velocity ($v_{\rm LSR}$), FWHM linewidth ($\Delta v$), peak intensity ($T_{\rm MB}$), and the integrated line intensity ($\int T_{\rm MB} {\rm d}v$). Besides the formal $1\sigma$ fitting errors, the errors in the last two quantities also include the calibration uncertainty (15% for the Effelsberg/NH$_3$ data, and 10% for our APEX data). We note that rather than using a Gaussian fit, the integrated intensity of the C$^{17}$O line was computed by integrating over the velocity range \[5.87, 10.14\] km s$^{-1}$ to take the non-Gaussian shape of the line into account.
\[table:lineparameters\]
------------------------------------- ----------------- ----------------- --------------- ---------------------------- -------------------------------- -------------- ---------------
Transition $v_{\rm LSR}$ $\Delta v$ $T_{\rm MB}$ $\int T_{\rm MB} {\rm d}v$ $\tau$ $T_{\rm ex}$ $T_{\rm rot}$
\[km s$^{-1}$\] \[km s$^{-1}$\] \[K\] \[K km s$^{-1}$\] \[K\] \[K\]
*p*-NH$_3(1,\,1)$ $8.40\pm0.01$ $0.40\pm0.01$ $2.46\pm0.40$ $1.91\pm0.30$ $2.01\pm0.11\,(=\tau_{\rm m})$ $6.8\pm0.7$ $10.6\pm0.5$
*p*-NH$_3(2,\,2)$ $8.42\pm0.02$ $0.45\pm0.08$ $0.37\pm0.06$ $0.23\pm0.04$ $0.10\pm0.02\,(=\tau_0)$ $6.8\pm0.7$ …
DCO$^+(3-2)$ $8.48\pm0.02$ $0.60\pm0.04$ $1.51\pm0.19$ $0.98\pm0.11$ $0.84\pm0.30\,(=\tau_0)$ $6.8\pm0.7$ …
*p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ $8.46\pm0.01$ $0.42\pm0.02$ $0.38\pm0.04$ $0.17\pm0.02$ $0.06\pm0.01\,(=\tau_0)$ $11.2\pm0.5$ …
$9.26\pm0.08$ $8.22\pm0.21$ $0.14\pm0.02$ $1.21\pm0.12$ $\ll 1$ … $64\pm15$
CH$_3$OH-E$_1(4_{2,\,2}-3_{1,\,2})$ $9.93\pm0.29$ $10.98\pm0.80$ $0.05\pm0.01$ $0.54\pm0.06$ $0.0009\pm0.0002\,(=\tau_0)$ $64\pm15$ …
*p*-H$_2$CO$(3_{2,\,2}-2_{2,\,1})$ $8.99\pm0.47$ $10.07\pm1.23$ $0.03\pm0.01$ $0.34\pm0.05$ $\ll 1$ … $64\pm15$
*p*-H$_2$CO$(3_{2,\,1}-2_{2,\,0})$ $9.45\pm0.35$ $10.92\pm0.83$ $0.03\pm0.01$ $0.31\pm0.04$ $\ll 1$ … $64\pm15$
C$^{18}$O$(2-1)$ $8.66\pm0.01$ $0.82\pm0.01$ $1.33\pm0.16$ $1.17\pm0.12$ $0.23\pm0.02\,(=\tau_0)$ $11.2\pm0.5$ …
SO$(5_6-4_5)$ $8.67\pm0.01$ $0.68\pm0.02$ $0.41\pm0.05$ $0.29\pm0.03$ $0.06\pm0.01\,(=\tau_0)$ $11.2\pm0.5$ …
$^{13}$CO$(2-1)$ $8.38\pm0.15$ $1.51\pm0.35$ $0.57\pm0.18$ $0.93\pm0.27$ … … …
$1.53\pm0.03$ $0.43\pm0.03$ $1.32\pm0.26$ $0.89\pm0.15$ … … …
C$^{17}$O$(2-1)$ $8.68\pm0.06$ $0.59\pm0.11$ $0.34\pm0.05$ $0.54\pm0.07$ $0.05\pm0.01\,(=\tau_0)$ $11.2\pm0.5$ …
N$_2$D$^+(3-2)$ $8.39\pm0.04$ $0.53\pm0.11$ $0.21\pm0.03$ $0.17\pm0.03$ $0.09\pm0.02\,(=\tau_0)$ $6.8\pm0.7$ …
N$_2$H$^+(3-2)$ $8.57\pm0.03$ $0.85\pm0.09$ $0.62\pm0.10$ $0.67\pm0.08$ $0.36\pm0.11\,(=\tau_0)$ $6.8\pm0.7$ …
DCO$^+(4-3)$ $8.54\pm0.03$ $0.42\pm0.18$ $0.20\pm0.02$ $0.09\pm0.02$ $0.11\pm0.03\,(=\tau_0)$ $6.8\pm0.7$ …
------------------------------------- ----------------- ----------------- --------------- ---------------------------- -------------------------------- -------------- ---------------
Analysis and results
====================
Spectral energy distribution of SMM3 – modified blackbody fitting
-----------------------------------------------------------------
The SED of SMM3, constructed using the *Herschel*/ PACS 70, 100, and 160 $\mu$m, SABOCA 350 $\mu$m, LABOCA 870 $\mu$m, and CARMA 2.9 mm flux densities (see Sects. 2.3 and 2.4, and Table \[table:photometry\]), is show in Fig. \[figure:SED\]. The *Spitzer* 24 $\mu$m data point, which represents a flux density of $4.74\pm0.3$ mJy from S13, is also shown in the figure, but it was excluded from the fit (see below). We note that the S13 24 $\mu$m flux density is close to a value of $5.0\pm0.2$ mJy we determined in Paper I ($13\arcsec$ aperture; see Table \[table:photometry\]). The 24 $\mu$m emission originates in a warmer dust component closer to the accreting central protostar, while the longer wavelength data ($\lambda \geq70$ $\mu$m) are presumable tracing the colder envelope.
The solid line in Fig. \[figure:SED\] represents a single-temperature MBB function fitted to the aforementioned data points. The fit was accomplished with optimisation ($\chi^2$ minimisation) by simulated annealing (Kirkpatrick et al. 1983), which, although more time-consuming, can work better in finding the best fit solution than the most commonly-used standard (non-linear) least-squares fitting method that can be sensitive to the chosen initial values (see also [@bertsimas1993]; [@ireland2007]). The original version of the fitting algorithm was written by J. Steinacker (M. Hennemann, priv. comm.). It was assumed that the thermal dust emission is optically thin ($\tau \ll 1$). We note that this assumption is probably good for the wavelengths longward of 70 $\mu$m, but it gets worse at shorter wavelengths. This, together with the fact that 24 $\mu$m emission originates in a warmer dust component closer to the accreting central protostar than the longer wavelength emission ($\lambda \geq70$ $\mu$m) arising from the colder envelope, is the reason why we excluded the 24 $\mu$m flux density from the fit (e.g. [@ragan2012]). The model fit takes into account the wavelength-dependence of the dust opacity ($\kappa_{\lambda}$). As the dust model, we employed the widely used Ossenkopf & Henning (1994, hereafter OH94) model describing graphite-silicate dust grains that have coagulated and accreted thin ice mantles over a period of $10^5$ yr at a gas density of $10^5$ cm$^{-3}$. For the total dust-to-gas mass ratio we adopted a value of $\delta_{\rm dg}\equiv M_{\rm dust}/M_{\rm gas}=1/141$. This mass ratio is based on the assumption that the core’s chemical composition is similar to the solar mixture, i.e. the mass fractions for hydrogen, helium, and heavier elements were assumed to be $X=0.71$, $Y=0.27$, and $Z=0.02$, respectively[^10].
As can be seen in Fig. \[figure:SED\], the PACS data are reasonably well fitted although the 160 $\mu$m flux density is slightly overestimated. The SABOCA data point is not well fitted, which could be partly casused by the spatial filtering owing to the sky-noise removal. Hence, a ground-based bolometer flux density can appear lower than what would be expected from the *Herschel* data. On the other hand, our LABOCA data point is well matched with the MBB fit. Finally, we note that the CARMA 2.9 mm flux density, which is based on the highest angular resolution data used here, is underestimated by the MBB curve. Radio continuum observations would be needed to quantify the amount of free-free contribution at 2.9 mm (cf. [@wardthompson2011]).
The dust temperature, envelope mass, and luminosity obtained from the SED fit are $T_{\rm dust}=15.1\pm0.1$ K, $M_{\rm env}=3.1\pm0.6$ M$_{\sun}$, and $L=3.8\pm0.6$ L$_{\sun}$. However, we emphasise that these values should be taken with some caution because clearly the fit shown in Fig. \[figure:SED\] is not perfect. In principle, while the 24 $\mu$m emission is expected to trace a warmer dust component than those probed by $\lambda_{\rm obs} \geq70$ $\mu$m observations (e.g. [@ragan2012]), it is possible that our poor single-$T_{\rm dust}$ fit reflects the presence of more than one cold dust components in the protostar’s envelope, and would hence require a multi-$T_{\rm dust}$ fit. However, following S13, and to allow an easier comparison with their results, we opt to use a simplified single-$T_{\rm dust}$ MBB in the present study.
We note that $M_{\rm env}\propto (\kappa_{\lambda}\delta_{\rm dg})^{-1}$, and hence the choice of the dust model (effectively $\kappa_{\lambda}$) and $\delta_{\rm dg}$ mostly affect the envelope mass among the SED parameters derived here (by a factor of two or more; OH94). The adopted dust model can also (slightly) influence the derived values of $T_{\rm dust}$ and $L$ because of the varying dust emissivity index ($\beta$) among the different OH94 models ($\kappa_{\lambda}\propto \lambda^{-\beta}$). The submm luminosity, $L_{\rm submm}$, computed by numerically integrating the fitted SED curve longward of 350 $\mu$m, is about 0.23 L$_{\sun}$, i.e. about $6\%$ of the total luminosity. For Class 0 protostellar cores, the $L_{\rm submm}/L$ ratio is defined to be $>5\times10^{-3}$, which reflects the condition that the envelope mass exceeds that of the central protostar, i.e. $M_{\rm env}\gg M_{\star}$ ([@andre1993], 2000). With a $L_{\rm submm}/L$ ratio of about one order of magnitude higher than the definition limit, SMM3 is clearly in the Class 0 regime.
Our $T_{\rm dust}$ value is by a factor of 1.4 lower than that obtained by S13 through their MBB analysis, while the values of $M_{\rm env}$ and $L$ we derived are higher by factors of about 9.4 and 1.8, respectively (see Sect. 1). We note that similarly to the present work, S13 fitted the data at $\lambda \geq 70$ $\mu$m, but they adopted a slightly different OH94 dust model (coagulation at a density of $10^6$ cm$^{-3}$ rather than at $10^5$ cm$^{-3}$ as here), and a slightly higher gas-to-dust ratio than we ($1.36\times110=149.6$, which is $6\%$ higher than our value of 141). Hence, we attribute the aforementioned discrepancies to the different SABOCA and LABOCA flux density values used in the analysis (e.g. S13 used the peak surface brightness from our SABOCA map, and their fit underestimated the LABOCA flux density), and to the fact that we have here used the new CARMA 2.9 mm data from Tobin et al. (2015) as well.
Given that Class 0 objects have, by definition, $M_{\rm env}\gg M_{\star}$, an envelope mass of $\sim3$ M$_{\sun}$ derived here might be closer to the true value than a value of $\sim0.3$ M$_{\sun}$ derived by S13. Also, as was already mentioned in Sect. 1, SMM3 was found to be a very bright 2.9 mm-emitter by Tobin et al. (2015), and hence they derived a high mass of $7.0\pm0.7$ M$_{\sun}$ under the assumption that $T_{\rm dust}=20$ K and $\delta_{\rm dg}=1/100$ (their mass is $2.3\pm0.5$ times higher than the present estimate, but a direct comparison with a single-flux density analysis is not feasible). In the context of stellar evolution, if the core star formation efficiency is $\sim30$% (e.g. [@alves2007]), and the central SMM3 protostar has $M_{\star} \ll M_{\rm env}$, this source could evolve into a near solar-mass star if $M_{\rm env}\sim3$ M$_{\sun}$ as estimated here, while an envelope mass of $\sim0.3$ M$_{\sun}$ would only be sufficient to form a very low-mass single star (near the substellar–stellar limit of $\sim0.1$ M$_{\sun}$). Moreover, the dust temperature we have derived here is closer to the gas kinetic temperature in SMM3 (the ratio between the two is $1.35\pm0.06$; see Sect. 4.2.1) than the value $T_{\rm dust}=21.4\pm0.4$ K from S13. In a high-density protostellar envelope, the gas temperature is indeed expected to be similar to $T_{\rm dust}$ (e.g. the dust–gas coupling occurs at $\sim10^5$ cm$^{-3}$ in the [@hollenbach1989] prescription). Finally, the physical implication of the higher luminosity we have derived here – $1.8\pm0.3$ times the S13 value – is that the mass accretion rate of the SMM3 protostar is higher by a similar factor.
Analysis of the spectral line data
----------------------------------
### Line optical thicknesses, and the excitation, rotational, and kinetic temperatures
The optical thickness of the main *p*-NH$_3(1,\,1)$ hyperfine group, $\tau_{\rm m}$, could be derived by fitting the hyperfine structure of the line. The main hyperfine group ($\Delta F=0$) has a relative strength of half the total value, and hence the total optical thickness of *p*-NH$_3(1,\,1)$ is given by $\tau_{\rm tot}=2\tau_{\rm m}$ ($=2\times(2.01\pm0.11)$; see [@mangum1992]; Appendix A1 therein). The strongest hyperfine component has a relative strength of $7/30$, which corresponds to a peak optical thickness of $\tau_0\simeq0.94$. The excitation temperature of the line, $T_{\rm ex}$, was calculated from the antenna equation ($T_{\rm MB}\propto (1-e^{-\tau})$; see e.g Eq. (1) in Paper I), assuming that the background temperature is equal to that of the cosmic microwave background radiation, i.e. $T_{\rm bg}\equiv T_{\rm CMB}=2.725$ K ([@fixsen2009]). The obtained value, $T_{\rm ex}=6.8\pm0.7$ K[^11], was also adopted for the *p*-NH$_3(2,\,2)$ line because its hyperfine satellites were not detected. Using this assumption and the antenna equation, the peak *p*-NH$_3(2,\,2)$ optical thickness was determined to be $0.1\pm0.02$. To calculate $\tau_{\rm tot}$, this value should be scaled by the relative strength of the strongest hyperfine component which is $8/35$. The value $T_{\rm ex}=6.8\pm0.7$ K was also adopted for the N$_2$H$^+$, N$_2$D$^+$, and DCO$^+$ lines, although we note that they might originate in a denser gas than the observed ammonia lines. Another caveat is that the $J=3-2$ line of DCO$^+$ was extracted from a position different from the ammonia target position, but, within the errors, the aforementioned $T_{\rm ex}$ value is expected to be a reasonable choice (e.g. [@anderson1999]). The values of $\tau_0$ were then derived as in the case of the $(2,\,2)$ transition of ammonia (see Col. (6) in Table \[table:lineparameters\]).
Using the $\tau_{\rm m}[p-{\rm NH_3(1,\,1)}]$ value and the intensity ratio between the $(2,\,2)$ and $(1,\,1)$ lines of *p*-NH$_3$, we derived the rotational temperature of ammonia ($T_{\rm rot}$; see Eq. (4) in [@ho1979]). This calculation assumed that the $T_{\rm ex}$ values, and also the linewidths, are equal between the two inversion lines. The latter assumption is justified by the observed FWHM linewidths. The derived value of $T_{\rm rot}$, $10.6\pm0.5$ K, was converted into an estimate of the gas kinetic temperature using the $T_{\rm kin}-T_{\rm rot}$ relationship from Tafalla et al. (2004; their Appendix B), which is valid in the low-temperature regime of $T_{\rm kin}\in [5,\,20]$ K. The value we derived, $T_{\rm kin}=11.2\pm0.5$ K[^12], was adopted as $T_{\rm ex}$ for the observed CO isotopologue transitions, SO, and the narrow *p*-H$_2$CO line. The choice of $T_{\rm ex}=T_{\rm kin}$ means that the level populations are assumed to be thermalised, and this is often done in the case of C$^{18}$O (e.g. [@hacar2011]), while in the cases of SO and H$_2$CO it should be taken as a rough estimate only.
The three broad *p*-H$_2$CO lines we detected allowed us to construct a rotational diagram for *p*-H$_2$CO. The rotational diagram technique is well established, and details of the method can be found in a number of papers (e.g. [@linke1979]; [@turner1991]; [@goldsmith1999]; Anderson et al. 1999; Green et al. 2013). When the line emission is assumed to be optically thin, the integrated intensity of the line is related to $T_{\rm rot}$ and the total column density of the species, $N$, according to the equation
$$\label{eq:rot}
\ln \left[\frac{\int T_{\rm MB}{\rm d}v}{\nu Sg_Kg_I}\right]=\ln \left(\frac{2\pi^2\mu^2}{3k_{\rm B}\epsilon_0}\frac{N}{Z_{\rm rot}}\right)-\frac{1}{T_{\rm rot}}\frac{E_{\rm u}}{k_{\rm B}}\,,$$
where $S$ is the line strength, $g_K$ is the $K$-level degeneracy, $g_I$ is the reduced nuclear spin degeneracy, $\epsilon_0$ is the vacuum permittivity, and $Z_{\rm rot}$ is the rotational partition function. The values of $S$ were adopted from the Splatalogue database[^13]. Because H$_2$CO is an asymmetric top molecule, there is no $K$-level degeneracy, and hence $g_K=1$. For the *para* form of H$_2$CO ($K_a$ is even), the value of $g_I$ is $1/4$ ([@turner1991]). The H$_2$CO molecule belongs to a $C_{2v}$ symmetry group (two vertical mirror planes), and its partition function at the high-temperature limit ($hA/k_{\rm B}T_{\rm ex} \ll 1$, where $h$ is the Planck constant) can be approximated as ([@turner1991])
$$\label{eq:part}
Z_{\rm rot}(T_{\rm rot})\simeq\frac{1}{2}\sqrt{\frac{\pi(k_{\rm B}T_{\rm rot})^3}{h^3ABC}}\,.$$
The derived rotational diagram, i.e. the left-hand side of Eq. (\[eq:rot\]) plotted as a function of $E_{\rm u}/k_{\rm B}$, is shown in Fig. \[figure:rot\]. The red solid line represents a least-squares fit to the three data points. The fit provides a value of $T_{\rm rot}$ as the reciprocal of the slope of the line, and $N$ can be calculated from the $y$-intercept. We note that two of the detected *p*-H$_2$CO transitions have almost the same upper-state energy, i.e. they lie very close to each other in the direction of the $x$-axis in Fig. \[figure:rot\], which makes the fitting results rather poorly constrained. We also note that the *ortho*-H$_2$CO$(2_{1,\,1}-1_{1,\,1})$ line detected by Kang et al. (2015) refers to the narrow-line component ($\Delta v=0.45$ km s$^{-1}$), and hence cannot be employed in our rotational diagram for the broad-line component. The value of $T_{\rm rot}$ we derived is $64\pm15$ K, which in the case of local thermodynamic equilibrium (LTE) is equal to $T_{\rm kin}$. Owing to the common formation route for formaldehyde and methanol (Sect. 5.2.3), the aforementioned $T_{\rm rot}$ value was adopted as $T_{\rm ex}$ for the detected CH$_3$OH line (which then appears to be optically thin). The molecular column density calculations are described in the next subsection.
### Molecular column densities and fractional abundances
As described above, the beam-averaged column density of *p*-H$_2$CO for the broad component was derived using the rotational diagram method. The column densities of the species other than NH$_3$ (see below) were calculated by using the standard LTE formulation
$$\label{eq:N}
N=\frac{3h\epsilon_0}{2\pi^2}\frac{1}{\mu^2S}\frac{Z_{\rm rot}(T_{\rm ex})}{g_Kg_I}e^{E_u/k_{\rm B}T_{\rm ex}}F(T_{\rm ex})\int \tau(v){\rm d}v \, ,$$
where $F(T_{\rm ex})\equiv \left(e^{h\nu/k_{\rm B}T_{\rm ex}}-1\right)^{-1}$. Here, the electric dipole moment matrix element is defined as $\left|\mu_{\rm ul} \right|\equiv \mu^2S/g_{\rm u}$, where $g_{\rm u}\equiv g_J=2J+1$ is the rotational degeneracy of the upper state ([@townes1975]). The values of the product $\mu^2S$ were taken from the Splatalogue database, but we note that for linear molecules $S$ is simply equal to the rotational quantum number of the upper state, i.e. $S=J$ (the SO molecule, which possesses a $^3\Sigma$ (electronic spin is 1) electronic ground state, is an exception; [@tiemann1974]). For linear molecules, $g_K=g_I=1$ for all levels, while for the E-type CH$_3$OH, $g_K=2$ and $g_I=1$ ([@turner1991]).
The partition function of the linear molecules was approximated as
$$\label{eq:Z1}
Z_{\rm rot}(T_{\rm ex}) \simeq \frac{k_{\rm B}T_{\rm ex}}{hB}+\frac{1}{3}\,.$$
Equation (\[eq:Z1\]) is appropriate for heteropolar molecules at a high-temperature limit of $hB/k_{\rm B}T_{\rm ex} \ll 1$. For SO, however, the rotational levels with $N\geq1$ are split into three sublevels (triplet of $N=J-1$, $N=J$, and $N=J+1$). To calculate the partition function of SO, we used the approximation formulae from Kontinen et al. (2000; Appendix A therein). For CH$_3$OH, which has an internal rotor, the partition function is otherwise similar to that in Eq. (\[eq:part\]) but with a numerical factor of 2 instead of 1/2 ([@turner1991]).
When the spectral line has a Gaussian profile, the last integral term in Eq. (\[eq:N\]) can be expressed as a function of the FWHM linewidth and peak optical thickness of the line as
$$\label{eq:tau}
\int \tau(v){\rm d}v=\frac{\sqrt{\pi}}{2\sqrt{\ln 2}}\Delta v \tau_0 \simeq1.064\Delta v \tau_0 \,.$$
We note that for the lines with hyperfine structure the total optical thickness is the sum of peak optical thicknesses of the different components. Moreover, if the line emission is optically thin ($\tau \ll 1$), $T_{\rm MB}\propto \tau$, and $N$ can be computed from the integrated line intensity. The values of $\tau$ listed in Col. (6) in Table \[table:lineparameters\] were used to decide whether the assumption of optically thin emission is valid (in which case the column density was calculated from the integrated intensity). To derive the total column density of NH$_3$, we first calculated that in the $(1,\,1)$ state, which, by taking into account both parity states of the level, is given by (e.g. [@harju1993])
$$\label{eq:ammonia1}
N({\rm NH_3})_{(1,\,1)}=N_++N_-=N_+(1+e^{h\nu_{(1,\,1)}/k_{\rm B}T_{\rm ex}})\,.$$
The latter equality follows from the Boltzmann population distribution, and the fact that the two levels have the same statistical weights ($J$ and $K$ do not change in the inversion transition). Because $N_+$ represents the column density in the upper state, its value was calculated from a formula that can be derived by substituting Eq. (\[eq:tau\]) into Eq. (\[eq:N\]), and dividing by the term $Z_{\rm rot}/(g_Kg_I)e^{E_u/k_{\rm B}T_{\rm ex}}$. The value of $S$ for a $(J,\,K)\rightarrow(J,\,K)$ transition is $S=K^2/[J(J+1)]$. Finally, making the assumption that at the low temperature of SMM3 only the four lowest metastable ($J=K$) levels are populated, the value of $N({\rm NH_3})_{(1,\,1)}$ was scaled by the partition function ratio $Z_{\rm rot}/Z_{\rm rot}(1,\,1)$ to derive the total (*ortho*+*para*) NH$_3$ column density as
$$\begin{aligned}
N({\rm NH_3}) &=& N({\rm NH_3})_{(0,\,0)}+N({\rm NH_3})_{(1,\,1)}\nonumber \\
& & +N({\rm NH_3})_{(2,\,2)}+ \,N({\rm NH_3})_{(3,\,3)}\nonumber \\
&=& N({\rm NH_3})_{(1,\,1)}\times \nonumber \\
& & \left(\frac{1}{3}e^{\frac{23.4}{T_{\rm rot}}}+1+\frac{5}{3}e^{-\frac{41.5}{T_{\rm rot}}}+\frac{14}{3}e^{-\frac{101.2}{T_{\rm rot}}} \right)\,.\end{aligned}$$
The column density analysis presented here assumes that the line emission fills the telescope beam, i.e. that the beam filling factor is unity. As can be seen in Fig. \[figure:linemaps\], the DCO$^+(3-2)$ and *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ emissions are somewhat extended with respect to the 350 $\mu$m-emitting core whose size is comparable to the beam size of most of our line observations. Moreover, the detected N-bearing species are often found to show spatial distributions comparable to the dust emission of dense cores (e.g. [@caselli2002a]; [@lai2003]; [@daniel2013]). It is still possible, however, that the assumption of unity filling factor is not correct. The gas within the beam area can be structured in a clumpy fashion, in which case the true filling factor is $<1$. The derived beam-averaged column density is then only a lower limit to the source-averaged value.
The fractional abundances of the molecules were calculated by dividing the molecular column density by the H$_2$ column density, $x=N/N({\rm H_2})$. To be directly comparable to the molecular line data, the $N({\rm H_2})$ values were derived from the LABOCA data smoothed to the resolution of the line observations (cf. Eq. (3) in Paper I). For this calculation, we adopted the dust temperature derived from the SED fit ($T_{\rm dust}=15.1 \pm 0.1$ K), except for the broad component of *p*-H$_2$CO and CH$_3$OH for which $T_{\rm dust}$ was assumed to be $64\pm15$ K ($=T_{\rm rot}(p-{\rm H_2CO})$). The mean molecular weight per H$_2$ molecule we used was $\mu_{\rm H_2}=2.82$, and the dust opacity per unit dust mass at 870 $\mu$m was set to $\kappa_{\rm 870\,\mu m}=1.38$ cm$^2$ g$^{-1}$ to be consistent with the OH94 dust model described earlier. The beam-averaged column densities and abundances with respect to H$_2$ are listed in Table \[table:chemistry\].
### Deuterium fractionation and CO depletion
The degree of deuterium fractionation in N$_2$H$^+$ was calculated by dividing the column density of N$_2$D$^+$ by that of N$_2$H$^+$. The obtained value, $14\%\pm6\%$, is about $40\%$ of the value derived in Paper III (i.e. $0.338\pm0.09$ based on a non-LTE analysis).
To estimate the amount by which the CO molecules are depleted in SMM3, we calculated the CO depletion factors following the analysis presented in Paper III with the following modifications. Recently, Ripple et al. (2013) analysed the CO abundance variation across the Orion giant molecular clouds. In particular, they derived the $^{13}$CO fractional abundances, and found that in the self-shielded interiors ($3<A_{\rm V}<10$ mag) of Orion B, the value of $x({\rm ^{13}CO})$ is $\simeq3.4\times10^{-6}$. On the other hand, towards NGC 2024 in Orion B the average \[$^{12}$C\]/\[$^{13}$C\] ratio is measured to be about 68 ([@savage2002]; [@milam2005]). These two values translate into a canonical (or undepleted) CO abundance of $\simeq2.3\times10^{-4}$. We note that this is 2.3 times higher than the classic value $10^{-4}$, but fully consistent with the best-fitting CO abundance of $2.7_{-1.2}^{+6.4}\times10^{-4}$ found by Lacy et al. (1994) towards NGC 2024. Because we derived the C$^{18}$O and C$^{17}$O abundances towards the core centre and the envelope, respectively, the canonical abundances of these two species had to be estimated. We assumed that the \[$^{16}$O\]/\[$^{18}$O\] ratio is equal to the average local interstellar medium value of 557 ([@wilson1999]), and that the \[$^{18}$O\]/\[$^{17}$O\] ratio is that derived by Wouterloot et al. (2008) for the Galactic disk (Galactocentric distance range of 4–11 kpc), namely 4.16. Based on the aforementioned ratios, the canonical C$^{18}$O and C$^{17}$O abundances were set to $4.1\times10^{-7}$ and $9.9\times10^{-8}$, respectively. With respect to the observed abundances, the CO depletion factors were derived to be $f_{\rm D}=27.3\pm1.8$ towards the core centre (C$^{18}$O data), and $f_{\rm D}=8.3\pm0.7$ in the envelope (C$^{17}$O data). The deuteration level and the CO depletion factors are given in the last two rows in Table \[table:chemistry\]. We note that the non-LTE analysis presented in Paper III yielded a value of $f_{\rm D}=10.8\pm2.2$ towards the core edge, i.e. a factor of $1.3\pm0.3$ times higher than the present value.
Assuming that the core mass we derived through SED fitting, $3.1\pm0.6$ M$_{\sun}$, is the mass within an effective radius, which corresponds to the size of the largest photometric aperture used, i.e. $R_{\rm eff}=19\farcs86$ or $\simeq0.04$ pc, the volume-averaged H$_2$ number density is estimated to be $\langle n({\rm H_2})\rangle=1.7\pm0.3\times10^5$ cm$^{-3}$ (see Eq. (1) in Paper III). Following the analysis presented in Miettinen (2012a, Sect. 5.5 therein), the CO depletion timescale at the aforementioned density (and adopting a $\delta_{\rm dg}$ ratio of 1/141) is estimated to be $\tau_{\rm dep}\sim3.4\pm0.6\times10^4$ yr. This can be interpreted as a lower limit to the age of SMM3.
\[table:chemistry\]
Species $N$ \[cm$^{-2}$\] $x$
--------------------------------------- -------------------------- ---------------------------
NH$_3$ $1.5\pm0.2\times10^{15}$ $6.6\pm0.9\times10^{-8}$
\[1ex\] *p*-H$_2$CO $1.0\pm0.3\times10^{12}$ $2.0\pm0.6\times10^{-11}$
\[1ex\] *p*-H$_2$CO $2.2\pm0.4\times10^{13}$ $3.1\pm1.0\times10^{-9}$
\[1ex\] CH$_3$OH $6.8\times10^{14}$ $9.4\pm2.5\times10^{-8}$
\[1ex\] C$^{18}$O $7.1\pm0.8\times10^{14}$ $1.5\pm0.1\times10^{-8}$
\[1ex\] SO $8.1\pm1.2\times10^{12}$ $1.6\pm0.2\times10^{-10}$
\[1ex\] C$^{17}$O $3.2\pm0.4\times10^{14}$ $1.2\pm0.1\times10^{-8}$
\[1ex\] N$_2$D$^+$ $1.7\pm0.5\times10^{12}$ $4.8\pm1.4\times10^{-11}$
\[1ex\] N$_2$H$^+$ $1.2\pm0.4\times10^{13}$ $2.9\pm0.9\times10^{-10}$
\[1ex\] DCO$^+$ $1.3\pm0.5\times10^{13}$ $2.6\pm1.0\times10^{-10}$
\[1ex\] DCO$^+$ $6.2\pm2.9\times10^{11}$ $2.0\pm0.9\times10^{-11}$
\[1ex\] Core centre Envelope
\[1ex\] \[N$_2$D$^+$\]/\[N$_2$H$^+$\] … $0.14\pm0.06$
\[1ex\] $f_{\rm D}({\rm CO})$ $27.3\pm1.8$ $8.3\pm0.7$
\[1ex\]
: Molecular column densities, fractional abundances with respect to H$_2$, and the degrees of deuteration and CO depletion.
Discussion
==========
Fragmentation and protostellar activity in SMM3
-----------------------------------------------
Owing to the revised fundamental physical properties of SMM3, we are in a position to re-investigate its fragmentation characteristics. At a gas temperature of $T_{\rm kin}=11.2\pm0.5$ K, the isothermal sound speed is $c_{\rm s}=197.5\pm4.4$ m s$^{-1}$, where the mean molecular weight per free particle was set to $\mu_{\rm p}=2.37$. The aforementioned values can be used to calculate the thermal Jeans length
$$\lambda_{\rm J}=\sqrt{\frac{\pi c_{\rm s}^2}{G \langle \rho \rangle}}\, ,$$
where $G$ is the gravitational constant, the mean mass density is $\langle \rho \rangle=\mu_{\rm H_2}m_{\rm H}\langle n({\rm H_2})\rangle$, and $m_{\rm H}$ is the mass of the hydrogen atom. The resulting Jeans length, $\lambda_{\rm J}\simeq0.05$ pc, is a factor of 1.4 shorter than our previous estimate (0.07 pc; Paper III), where the difference can be mainly attributed to the higher gas density derived here. We note that the uncertainty propagated from those of $T_{\rm kin}$ and $\langle n({\rm H_2})\rangle$ is only 1 mpc.
If we use the observed *p*-NH$_3(1,\,1)$ linewidth as a measure of the non-thermal velocity dispersion, $\sigma_{\rm NT}$ ($=169.9\pm4.2$ m s$^{-1}$), the effective sound speed becomes $c_{\rm eff}=(c_{\rm s}^2+\sigma_{\rm NT}^2)^{1/2}=260.5\pm1.6$ m s$^{-1}$. The corresponding effective Jeans length is $\lambda_{\rm J}^{\rm eff}\simeq0.06$ pc. Although not much different from the purely thermal value, $\lambda_{\rm J}^{\rm eff}$ is in better agreement with the observed projected distances of SMM3b and 3c from the protostar position (0.07–0.10 pc). Hence, the parent core might have fragmented as a result of Jeans-type instability with density perturbations in a self-gravitating fluid having both the thermal and non-thermal motions (we note that in Paper III we suggested a pure thermal Jeans fragmentation scenario due to the aforementioned longer $\lambda_{\rm J}$ value). Because information in the core is transported at the sound speed (being it thermal or effective one), the fragmentation timescale is expected to be comparable to the crossing time, $\tau_{\rm cross}=R/c_{\rm eff}$, where $R=0.07-0.10$ pc. This is equal to $\tau_{\rm cross}\sim2.6-3.8\times10^5$ yr, which is up to an order of magnitude longer than the estimated nominal CO depletion timescale (Sect. 4.2.3).
The present SED analysis and the previous studies (see Sect. 1) suggest that SMM3 is in the Class 0 phase of stellar evolution. Observational estimates of the Class 0 lifetime are about $\sim1\times10^5$ yr ([@enoch2009]; [@evans2009]; Maury et al. 2011). In agreement with observations, Offner & Arce (2014) performed radiation-hydrodynamic simulations of protostellar evolution including outflows, and obtained Stage 0 lifetimes of $1.4-2.3\times10^5$ yr, where the Stage 0 represents a theoretical counterpart of the observational Class 0 classification. These observational and theoretical lifetime estimates are comparable to the fragmentation timescale of SMM3, which supports a scenario of the age of SMM3 being a few times $10^5$ yr.
In the present paper, we have presented the first signatures of an outflow activity in SMM3. These are *i)* the broad lines of *p*-H$_2$CO and CH$_3$OH; *ii)* the warm gas ($64\pm15$ K) associated with the broad-line component; and *iii)* the protrusion-like feature seen at 4.5 $\mu$m (Fig. \[figure:images\], bottom right panel), which is likely related to the shock emission near the accreting protostar. Outflow activity reasserts the Class 0 evolutionary stage of SMM3 (e.g. [@bontemps1996]).
The 350 $\mu$m flux densities of the subcondensations SMM3b and 3c are $250\pm60$ mJy and $240\pm60$ mJy, respectively (Paper III). Assuming that the dust temperature is that resulting from the SED of SMM 3 ($15.1\pm0.1$ K), and adopting the same dust model as in Sect. 4.1, in which case the dust opacity per unit dust mass at 350 $\mu$m is $\kappa_{\rm 350\, \mu m}=7.84$ cm$^2$ g$^{-1}$, the condensation masses are only $\sim0.06\pm0.01$ M$_{\sun}$. If we instead use as $T_{\rm dust}$ the gas temperature derived from ammonia, the mass estimates will become about $0.16\pm0.05$ M$_{\sun}$, i.e. a factor of $2.7\pm0.9$ higher. As discussed in the case of the prestellar core Orion B9–SMM6 by Miettinen & Offner (2013b), these types of very low-mass condensations are likely not able to collapse to form stars without any additional mass accretion. Instead, they could represent the precursors of substellar-mass objects or brown dwarfs (e.g. [@lee2013]). Alternatively, if the condensations are gravitationally unbound structures, they could disperse away in the course of time, an issue that could be solved by high-resolution molecular line observations. Finally, mechanical feedback from the protostellar outflow could affect the future evolution of the condensations (cf. the proto- and prestellar core system IRAS 05399-0121/SMM1 in Orion B9; [@miettinen2013a]).
Chemical properties of SMM3
---------------------------
### NH$_3$ and N$_2$H$^+$ abundances
The fractional abundances of the N-bearing species NH$_3$ and N$_2$H$^+$ we derived are $6.6\pm0.9\times10^{-8}$ and $2.9\pm0.9\times10^{-10}$. The value of $x({\rm NH_3})$ in low-mass dense cores is typically found to be a few times $10^{-8}$ (e.g. [@friesen2009]; [@busquet2009]). Morgan et al. (2010) derived a mean $x({\rm NH_3})$ value of $2.6\times10^{-8}$ towards the protostars embedded in bright-rimmed clouds. Their sources might represent the sites of triggered star formation, and could therefore resemble the case of SMM3 – a core that might have initially formed as a result of external feedback. More recently, Marka et al. (2012) found that the average NH$_3$ abundance in their sample of globules hosting Class 0 protostars is $3\times10^{-8}$ with respect to H$_2$.[^14] Compared to the aforementioned reference studies, the ammonia abundance is SMM3 appears to be elevated by a factor of about two or more, although differences in the assumptions of dust properties should be borne in mind. The chemical modelling of the Class 0 sources performed by Marka et al. (2012), which included reactions taking place on dust grain surfaces, predicted that an NH$_3$ abundance exceeds $\sim10^{-8}$ after $10^5$ yr of evolution (see also [@hilyblant2010] for a comparable result). This compares well with the fragmentation timescale in SMM3 estimated above. For their sample of low-mass protostellar cores, Caselli et al. (2002b) found a mean N$_2$H$^+$ abundance of $3\pm2\times10^{-10}$, which is very similar to the one we have derived for SMM3.
The \[NH$_3$\]/\[N$_2$H$^+$\] ratio in SMM3, derived from the corresponding column densities, is $125\pm45$. The abundance ratio between these two species is known to show different values in starless and star-forming objects. For example, Hotzel et al. (2004), who studied the dense cores B217 and L1262, both associated with Class I protostars, found that the above ratio is $\sim140-190$ in the starless parts of the cores, but only about $\sim60-90$ towards the protostars. Our value, measured towards the outer edge of SMM3, lies in between these two ranges, and hence is consistent with the observed trend. A similar behaviour is seen in IRAS 20293+3952, a site of clustered star formation ([@palau2007]), and clustered low-mass star-forming core Ophiuchus B ([@friesen2010]). In contrast, for their sample of dense cores in Perseus, Johnstone et al. (2010) found that the *p*-NH$_3$/N$_2$H$^+$ column density ratio is fairly similar in protostellar cores ($20\pm7$) and in prestellar cores ($25\pm12$). Their ratios also appear to be lower than found in other sources (we note that the statistical equilibrium value of the NH$_3$ *ortho*/*para* ratio is unity; e.g. [@umemoto1999]).
The chemical reactions controlling the \[NH$_3$\]/\[N$_2$H$^+$\] ratio were summarised by Fontani et al. (2012; Appendix A therein). In starless cores, the physical conditions are such that both the CO and N$_2$ molecules can be heavily depleted. If this is the case, N$_2$H$^+$ cannot be efficiently formed by the reaction between H$_3^+$ and N$_2$. On the other hand, this is counterbalanced by the fact that N$_2$H$^+$ cannot be destroyed by the gas-phase CO, although it would serve as a channel for the N$_2$ production (${\rm CO}+{\rm N_2H^+}\rightarrow {\rm HCO^+}+{\rm N_2}$). Instead, in a gas with strong CO depletion, N$_2$H$^+$ is destroyed by the dissociative electron recombination. The absence of N$_2$ also diminishes the production of N$^+$, the cations from which NH$_3$ is ultimately formed via the reaction NH$_4^++{\rm e}^-$. However, the other routes to N$^+$, namely ${\rm CN}+{\rm He}^+$ and ${\rm NH_2}+{\rm He}^+$, can still operate. We also note that H$_3^+$, which also cannot be destroyed by CO in the case of strong CO depletion, is a potential destruction agent of NH$_3$. However, the end product of the reaction ${\rm NH_3}+{\rm H_3^+}$ is NH$_4^+$, the precursor of NH$_3$. For these reasons, the NH$_3$ abundance can sustain at the level where the \[NH$_3$\]/\[N$_2$H$^+$\] ratio is higher in starless cores (strong depletion) than in the protostellar cores (weaker depletion). It should be noted that the study of the high-mass star-forming region AFGL 5142 by Busquet et al. (2011) showed that the \[NH$_3$\]/\[N$_2$H$^+$\] ratio behaves opposite to that in low-mass star-forming regions. The authors concluded that the higher ratio seen towards the hot core position is the result of a higher dust temperature, leading to the desorption of CO molecules from the grains mantles. As a result, the gas-phase CO can destroy the N$_2$H$^+$ molecules, which results in a higher \[NH$_3$\]/\[N$_2$H$^+$\] ratio. Because SMM3 shows evidence for quite a strong CO depletion of $f_{\rm D}=27.3\pm1.8$ towards the core centre, the chemical scheme described above is probably responsible for the much higher abundance of ammonia compared to N$_2$H$^+$.
### Depletion and deuteration
As mentioned above, the CO molecules in SMM3 appear to be quite heavily depleted towards the protostar position, while it becomes lower by a factor of $3.3\pm0.4$ towards the outer core edge. A caveat here is that the two depletion factors were derived from two different isotopologues, namely C$^{17}$O for the envelope zone, and C$^{18}$O towards the core centre. This brings into question the direct comparison of the two depletion factors. Indeed, although the critical densities of the detected CO isotopologue transitions are very similar, the C$^{18}$O linewidth is $1.4\pm0.3$ times greater than that of C$^{17}$O. Although this is not a significant discrepancy, the observed C$^{18}$O emission could originate in a more turbulent parts of the core.
For comparison, for their sample of 20 Class 0 protostellar cores, Emprechtinger et al. (2009) derived CO depletion factors of $0.3\pm0.09-4.4\pm1.0$. These are significantly lower than what we have derived for SMM3. The depletion factor in the outer edge of SMM3 we found is more remisnicent to those seen in low-mass starless cores (e.g. [@bacmann2002]; [@crapsi2005]), but the value towards the core’s 24 $\mu$m peak position stands out as an exceptionally high.
The deuterium fractionation of N$_2$H$^+$, or the N$_2$D$^+$/N$_2$H$^+$ column density ratio, is found to be $0.14\pm0.06$ towards the core edge. This lies midway between the values found by Roberts & Millar (2007) for their sample of Class 0 protostars ($0.06\pm0.01-0.31\pm0.05$). Emprechtinger et al. (2009) found N$_2$D$^+$/N$_2$H$^+$ ratios in the range $<0.029-0.271\pm0.024$ with an average value of 0.097. Among their source sample, most objects had a deuteration level of $<0.1$, while 20% of the sources showed values of $>0.15$. With respect to these results, the deuterium fractionation in SMM3 appears to be at a rather typical level among Class 0 objects. For comparison, in low-mass starless cores the N$_2$D$^+$/N$_2$H$^+$ ratio can be several tens of percent ([@crapsi2005]), while intermediate-mass Class 0-type protostars show values that are more than ten times lower than in SMM3 ([@alonso2010]). A visual inspection of Fig. 3 in Emprechtinger et al. (2009) suggests that for a N$_2$D$^+$/N$_2$H$^+$ ratio we have derived for SMM3, the dust temperature is expected to be $\lesssim25$ K. This is qualitatively consistent with a value of $15.1\pm0.1$ K we obtained from the MBB SED fit. On the other hand, the correlation in the middle panel of Fig. 4 in Emprechtinger et al. (2009; see also their Fig. 10) suggests that the CO depletion factor would be $\sim3$ at the deuteration level seen in SMM3, while our observed value in the envelope is $2.8\pm0.2$ times higher. The fact that CO molecules appear to be more heavily depleted towards the new line observation target position suggests that the degree of deuterium fractionation there is also higher. A possible manifestation of this is that the estimated DCO$^+$ abundance is higher by a factor of $13.0\pm7.7$ towards the core centre than towards the core edge, but this discrepancy could be partly caused by the different transitions used in the analysis ($J=3-2$ and $J=4-3$, respectively).
Recently, Kang et al. (2015) derived a deuterium fractionation of formaldehyde in SMM3 (towards the core centre), and they found a HDCO/H$_2$CO ratio of $0.31\pm0.06$, which is the highest value among their sample of 15 Class 0 objects. This high deuteration level led the authors to conclude that SMM3 is in a very early stage of protostellar evolution.
### H$_2$CO, CH$_3$OH, and SO – outflow chemistry in SMM3
Besides the narrow ($\Delta v=0.42$ km s$^{-1}$) component of the *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ line detected towards SMM3, this line also exhibits a much wider ($\Delta v=8.22$ km s$^{-1}$) component with blue- and redshifted wing emission. The other two transitions of *p*-H$_2$CO we detected, $(3_{2,\,1}-2_{2,\,0})$ and $(3_{2,\,2}-2_{2,\,1})$, are also broad, more than 10 km s$^{-1}$ in FWHM, and exhibit wing emission. The methanol line we detected, with a FWHM of 10.98 km s$^{-1}$, is also significantly broader than most of the lines we have detected. The similarity between the FWHMs of the methanol and formaldehyde lines suggests that they originate in a common gas component. The rotational temperature derived from the *p*-H$_2$CO lines, $64\pm15$ K, is considerably higher than the dust temperature in the envelope and the gas temperature derived from ammonia. The large linewidths and the relatively warm gas temperature can be understood if a protostellar outflow has swept up and shock-heated the surrounding medium.
The H$_2$CO and CH$_3$OH molecules are organic species, and they can form on dust grain surfaces through a common CO hydrogenation reaction sequence (${\rm CO}\rightarrow {\rm HCO}\rightarrow {\rm H_2CO}\rightarrow {\rm CH_3O}$ or H$_3$CO or CH$_2$OH or H$_2$COH $\rightarrow {\rm CH_3OH}$; e.g. Watanabe & Kouchi 2002; [@hiraoka2002]; [@fuchs2009], and references therein). The intermediate compound, solid formaldehyde, and the end product, solid methanol, have both been detected in absorption towards low-mass young stellar objects (YSOs; Pontoppidan et al. 2003; [@boogert2008]). A more recent study of solid-phase CH$_3$OH in low-mass YSOs by Bottinelli et al. (2010) suggests that much of the CH$_3$OH is in a CO-rich ice layer, which conforms to the aforementioned formation path. We note that H$_2$CO can also be formed in the gas phase (e.g. [@kahane1984]; [@federman1991]), and the narrow *p*-H$_2$CO line we detected is likely tracing a quiescent gas not enriched by the chemical compounds formed on dust grains. The estimated *p*-H$_2$CO abundance for this component is very low, only $2.0\pm0.6\times10^{-11}$. We note that the total H$_2$CO column density derived by Kang et al. (2015), $N({\rm H_2CO})=3.3\pm0.4\times10^{12}$ cm$^{-2}$ at $T_{\rm ex}=10$ K, is in good agreement with our *p*-H$_2$CO column density if the *ortho*/*para* ratio is 3:1 as assumed by the authors.
In contrast, the fractional *p*-H$_2$CO abundance is found to be $155\pm70$ times higher for the broad component than for the narrow one. The origin of the H$_2$CO abundance enhancement in low-mass protostars can be understood in terms of the liberation of the ice mantles ([@schoier2004]). We note that there are also gas-phase formation routes for CH$_3$OH, which start from the reaction between CH$_3^+$ and H$_2$O or between H$_3$CO$^+$ and H$_2$CO. The resulting protonated methanol, CH$_3$OH$_2^+$, can recombine with an electron and dissociate to produce CH$_3$OH ([@garrod2006]; [@geppert2006]). However, the gas-phase syntheses are not able to produce the high fractional abundances like observed here towards SMM3 ($9.4\pm2.5\times10^{-8}$). The thermal desorption of CH$_3$OH requires a dust temperature of at least $\sim80$ K ([@brown2007]; [@green2009]). Although highly uncertain, the H$_2$CO rotational temperature we derived does not suggest the dust temperature to be sufficiently high for CH$_3$OH molecules to sublimate. Hence, it seems possible that an outflow driven by SMM3 has sputtered the icy grain mantles (in impacts with gas-phase H$_2$ and He) so that H$_2$CO and CH$_3$OH were released into the gas phase. On the other hand, the high CO depletion factors we derived suggest that the grain ices are rich in CO, and if CH$_3$OH molecules are embedded in CO-rich ice layers, their thermal evaporation temperature can be considerably lower ($\sim30$ K; see [@maret2005]).
The *p*-H$_2$CO/CH$_3$OH column density ratio for the broad line component is found to be $0.03\pm0.005$. This value represents a lower limit to the total H$_2$CO/CH$_3$OH ratio, which depends on the *o*/*p* ratio. Based on the observed abundances of both the *ortho* and *para* forms of H$_2$CO in low-mass dense cores, J[ø]{}rgensen et al. (2005) derived a *o*/*p* ratio of $1.6\pm0.3$. The authors interpreted this to be consistent with thermalisation at 10–15 K on dust grains. If we assume that the *o*/*p* ratio is $\simeq1.6$, we obtain a total H$_2$CO/CH$_3$OH column density ratio of $\simeq0.08\pm0.01$, while for a *o*/*p* ratio, which is equal to the relative statistical weights of 3:1, the total H$_2$CO/CH$_3$OH ratio becomes $0.13\pm0.02$. The H$_2$CO/CH$_3$OH ice abundance ratio in low-mass YSOs is found to be in the range $\sim0.2-6$ ([@boogert2008]), which is higher than the gas-phase abundance ratio towards SMM3. Hence, it is possible that the ices are not completely sublimated into the gas phase. Interestingly, if the total H$_2$CO/CH$_3$OH ratio for SMM3 is $\sim0.1$, and $x({\rm CH_3OH})\sim10^{-7}$, then these properties would resemble those derived for the Galactic centre clouds where shocks (caused by expanding bubbles, cloud-cloud collisions, etc.) are believed to have ejected the species from the grain mantles ([@requena2006]). In contrast, for the hot interiors of Class 0 sources, i.e. hot corinos, the H$_2$CO/CH$_3$OH ratio is found to be higher, in the range $>0.3-4.3$ ([@maret2005]; their Table 3), which are comparable to the aforementioned ice abundance ratios. In hot corinos the dust temperature exceeds 100 K, and the evaporation of ice mantles is the result of radiative heating by the central protostar. Moreover, in the Horsehead photodissociation region (PDR) in Orion B, the H$_2$CO/CH$_3$OH ratio is found to be $2.3\pm0.4$ ([@guzman2013]). Guzm[á]{}n et al. (2013) concluded that in the UV-illuminated PDR both H$_2$CO and CH$_3$OH are released from the grain mantles through photodesorption.
The SO line we detected is narrow ($\Delta v=0.68$ km s$^{-1}$), but low-intensity wing emission can be seen on both sides of it (see [@codella2002] for similar spectra towards the CB34 globule, which harbours a cluster of Class 0 objects). The derived fractional abundance of SO, $1.6\pm0.2\times10^{-10}$, is very low, as for example the average abundance derived by Buckle & Fuller (2003) for their sample of Class 0 objects is $3.1\pm0.9\times10^{-9}$, and in the starless TMC-1 cloud the abundance is found to be $\sim10^{-8}$ ([@lique2006]).
While our narrow SO line is probably originating in the quiescent envelope, where SO is formed through the reactions ${\rm S}+{\rm OH}$ and ${\rm S}+{\rm O_2}$ (e.g. [@turner1995]), the weak line wings provide a hint of an outflowing SO gas. The SO emission is indeed known to be a tracer of protostellar outflows (e.g. [@chernin1994]; [@lee2010]; [@tafalla2013]). Outflow shocks can first release H$_2$S molecules from dust grains, and subsequent hydrogenation reactions produce HS molecules and S atoms (${\rm H_2S}+{\rm H}\rightarrow {\rm HS}+{\rm H_2}$; ${\rm HS}+{\rm H}\rightarrow {\rm S}+{\rm H_2}$; [@mitchell1984]; [@charnley1997]). The oxidation reactions ${\rm HS}+{\rm O}$ and ${\rm S}+{\rm O_2}$ can then lead to the formation of SO (see [@bachiller2001]). For example, Lee et al. (2010) derived an SO abundance of $\sim2\times10^{-6}$ towards the HH211 jet driven by a Class 0 protostar, which shows that a significant abundance enhancement can take place in low-mass outflows. Some of the evolutionary models of the sulphur chemistry by Buckle & Fuller (2003) suggest that, after $\sim10^5$ yr, the abundance of H$_2$S starts to drop, which leads to a rapid decrease in the SO abundance. This could explain the very weak SO wing emission seen towards SMM3, and agrees with the observational estimates of the Class 0 lifetime of about $\sim1\times10^5$ yr (e.g. [@evans2009]; see also [@offner2014] for a comparable result from simulations). Interestingly, some of the Buckle & Fuller (2003) models, for example the one with a gas temperature of 10 K, H$_2$ density of $10^5$ cm$^{-3}$, and a cosmic-ray ionisation rate of $\zeta_{\rm H}=1.3\times10^{-16}$ s$^{-1}$, which is ten times the standard $\zeta_{\rm H}$ (their Fig. 7, bottom left), predict SO abundances comparable to that observed in SMM3 (a few times $10^{-10}$) after $10^5$ yr, so perhaps the narrow-line component could also be (partly) tracing a gas component that was affected by outflows in the past
Summary and conclusions
=======================
We used the APEX telescope to carry out follow-up molecular line observations towards the protostellar core SMM3 , which is embedded in the filamentary Orion B9 star-forming region. The new data were used in conjunction with our earlier APEX data (including SABOCA and LABOCA continuum data), and NH$_3$ observations from the Effelsberg 100 m telescope. The main results are summarised as follows.
1. From the observed frequency range $\sim218.2-222.2$ GHz, the following chemical compounds were identified: $^{13}$CO, C$^{18}$O, SO, *p*-H$_2$CO, and CH$_3$OH-E$_1$. The last two species play a key role in the synthesis of more complex organic molecules and prebiotic chemistry, which makes them particularly interesting compounds in the gas reservoir of a solar-type protostar like SMM3. Our new mapping observations of SMM3 were performed in the frequency range $\sim215.1-219.1$ GHz, from which DCO$^+(3-2)$ and *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ lines were identified.
2. Our revised SED analysis of SMM3 supports its Class 0 classification. The dust temperature, envelope mass, and luminosity were derived to be $15.1\pm0.1$ K, $3.1\pm0.6$ M$_{\sun}$, and $3.8\pm0.6$ L$_{\sun}$. The NH$_3$-based gas kinetic temperature was derived to be $T_{\rm kin}=11.2\pm0.5$ K. The revised analysis of the subfragments seen in our SABOCA 350 $\mu$m map suggests that SMM3 went through a Jeans-type fragmentation phase, where the initial density perturbations might have had contributions from both thermal and non-thermal motions.
3. The CO depletion factor derived from the new C$^{18}$O data towards the core centre is very high, $27.3\pm1.8$., while that re-computed from our previous C$^{17}$O data towards the core edge is clearly lower, $8.3\pm0.7$. We also recalculated the degree of deuterium fractionation in the latter position, in terms of the N$_2$D$^+$/N$_2$H$^+$ ratio, and found a value of $0.14\pm0.06$. Even higher deuteration is to be expected towards the new line observation target position because of the stronger CO freeze out.
4. The new spectral-line mapping observations revealed that SMM3 is associated with extended DCO$^+$ and *p*-H$_2$CO emission (as compared with the 350 $\mu$m-emitting region), and both the line emissions appear to be elongated in the east-west direction. Besides the systemic velocity of $\sim8.5$ km s$^{-1}$, emission from *p*-H$_2$CO$(3_{0,\,3}-2_{0,\,2})$ was also detected at a radial velocity of 1.5 km s$^{-1}$, which concentrates to the east and northeast of SMM3, similarly to the spatial distributions of $^{13}$CO$(2-1)$ and C$^{18}$O$(2-1)$ seen earlier by Miettinen (2012b).
5. The single-pointing observations showed that the $3_{0,\,3}-2_{0,\,2}$ line of *p*-H$_2$CO exhibits two components, a narrow one and a broad one. The other two *p*-H$_2$CO lines we detected, $3_{2,\,1}-2_{2,\,0}$ and $3_{2,\,2}-2_{2,\,1}$, are also broad. Hence, a rotational diagram was constructed for the broad component of *p*-H$_2$CO, which yielded a rotational temperature of $64\pm15$ K. The detected methanol line has a width comparable to those of the broad formaldehyde lines, and is hence likely tracing the same warm gas component.
6. We interpret the broad *p*-H$_2$CO and CH$_3$OH lines, and the elevated gas temperature, to be the first clear evidence of shock processing and outflow activity in SMM3. The abundance of *p*-H$_2$CO in the broad component is enhanced by two orders of magnitude with respect to the quiescent gas component. Additionally, the protrusion-like emission feature seen in the *Spitzer* 4.5 $\mu$m image is likely related to shock emission.
7. The detected SO line shows a narrow component at the systemic velocity, and weak wings on both sides of it. The wing emission points towards a weak SO outflow, while the narrow component is probably tracing the quiescent envelope.
8. The estimated fragmentation timescale of SMM3, and the observed chemical characteristics all suggest that the age of SMM3 is a few times $10^5$ yr, in agreement with its inferred Class 0 evolutionary stage. A dedicated chemical modelling would be useful in setting tighter constraints on the source age.
Putting the results from the previous studies and the present one together, we are in a position to place SMM3 in the wider context of Class 0 objects. Stutz et al. (2013) classified SMM3 as a so-called PACS Bright Red source, or PBRs. This source population is composed of extreme, red Class 0 objects with presumably high-density envelopes and high mass infall rates, and the median values of their MBB-based dust temperature, envelope mass, luminosity, and $L_{\rm submm}/L$ ratio are 19.6 K, 0.6 M$_{\sun}$, 1.8 L$_{\sun}$, and 2.7% (see Table 8 in S13).[^15] Although the physical properties of SMM3 we have derived in the present work are more extreme than the typical PBRs’ properties (it is colder, more massive, and more luminous), it can still be classified as a PBRs in agreement with S13 because this population was also found to contain sources with properties comparable to those we have derived. We note that the Orion B cloud appears to contain a relatively high fraction of PBRs-type objects (17% of the known protostars in Orion B) compared to that in Orion A (1%; S13).
We can also draw a conclusion that SMM3 exhibits a rich chemistry. It is possible that this Class 0 protostellar core hosts a so-called hot corino where the gas-phase chemistry can be as rich as in the hot molecular cores associated with high-mass star formation. This can be tested through high resolution interferometric multi-line observations. Such observations would also be useful to examine whether SMM3 drives a chemically rich/active molecular outflow, as our detection of the broad formaldehyde and methanol lines already suggest. In a more general context of low-mass star formation, SMM3 has the potential to become a useful target source of chemical evolution in a triggered star-forming region (feedback from NGC 2024, which could be ultimately linked to the nearby Ori OB1 association within the Ori-Eri superbubble). By comparing its properties with those of Class 0 objects in more isolated, quiescent regions, it could be possible to investigate whether its (chemical) evolution could have been accelerated as a result of more dynamic environment. The observed fragmentation of the SMM3 core indeed suggests that it has had a dynamical history, and is a fairly atypical object compared to the general Class 0 population in the Galaxy.
I would like to thank the referee for providing helpful, constructive comments and suggestions that improved the content of this paper. This publication is based on data acquired with the Atacama Pathfinder EXperiment (APEX) under programmes [079.F-9313(A)]{}, [084.F-9304(A)]{}, [084.F-9312(A)]{}, [092.F-9313(A)]{}, and [092.F-9314(A)]{}. APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. I would like to thank the staff at the APEX telescope for performing the service-mode heterodyne and bolometer observations presented in this paper. The research for this paper was financially supported by the Academy of Finland, grant no. 132291. This research has made use of NASA’s Astrophysics Data System, and the NASA/IPAC Infrared Science Archive, which is operated by the JPL, California Institute of Technology, under contract with the NASA. This study also made use of APLpy, an open-source plotting package for Python hosted at [http://aplpy.github.com]{}.
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[^1]: In the present work, we have adopted a distance of $d=420$ pc to the source to be consistent with the most recent studies of SMM3 ([@stutz2013]; [@tobin2015]; [@furlan2016]). We note that in Papers I–III, we assumed a distance of $450$ pc, which is a factor of 1.07 larger than used here.
[^2]: The HGBS is a *Herschel* key programme jointly carried out by SPIRE Specialist Astronomy Group 3 (SAG 3), scientists of several institutes in the PACS Consortium (CEA Saclay, INAF-IFSI Rome and INAF-Arcetri, KU Leuven, MPIA Heidelberg), and scientists of the *Herschel* Science Center (HSC). For more details, see [http://gouldbelt-herschel.cea.fr]{}
[^3]: [http://www.apex-telescope.org/]{}
[^4]: Grenoble Image and Line Data Analysis Software (GILDAS) is provided and actively developed by Institut de Radioastronomie Millimétrique (IRAM), and is available at [http://www.iram.fr/IRAMFR/GILDAS]{}
[^5]: Jet Propulsion Laboratory (JPL) spectroscopic database ([@pickett1998]); see [http://spec.jpl.nasa.gov/]{}
[^6]: Cologne Database for Molecular Spectroscopy (CDMS; [@muller2005]); see [http://www.astro.uni-koeln.de/cdms]{}
[^7]: The 100 m telescope at Effelsberg/Germany is operated by the Max-Planck-Institut für Radioastronomie on behalf of the Max-Planck-Gesellschaft (MPG).
[^8]: [http://www.submm.caltech.edu/$\sim$sharc/crush/index.htm]{}
[^9]: [http://sha.ipac.caltech.edu/applications/Spitzer/SHA/]{}
[^10]: In this case, the ratio between the total mass (H+He+metals) to hydrogen mass is $1/X\simeq1.41$.
[^11]: We note that in Paper II we determined a value of $T_{\rm ex}({\rm NH_3})=6.1\pm0.5$ K from a unsmoothed *p*-NH$_3(1,\,1)$ spectrum, while the present value was derived from a smoothed spectrum.
[^12]: The quoted value of $T_{\rm kin}$ differs slightly from the one derived in Paper II ($11.3\pm0.8$ K) because of the smoothed ammonia spectra employed in the analysis in the present work.
[^13]: [http://www.cv.nrao.edu/php/splat/]{}
[^14]: The authors reported the abundances with respect to the total hydrogen column density, which is here assumed to be $N_{\rm H}=2N({\rm H_2})$.
[^15]: We note that these median values were calculated by including the SMM3 values derived by S13, but if they are omitted, the median values are essentially the same. The median envelope mass reported here was scaled to the presently assumed dust-to-gas ratio.
|
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abstract: 'A cosmological approach based on considering a cosmic background with non-zero torsion is shown in order to give an option of explaining a possible phantom evolution, not ruled out according to the current observational data. We revise some aspects of the formal schemes on torsion and, according them, we develop a formalism which can be an interesting alternative for exploring Cosmology.'
author:
- |
Fernando Izaurieta$^{1}$ and Samuel Lepe$^{2}$\
$^{1}$*Departamento de Física, Universidad de Concepción*,\
Casilla 160-C, Concepción, Chile,\
`fizaurie@udec.cl`\
$^{2}$*Instituto de Física, Pontificia Universidad Católica de Valparaíso*,\
Av. Brasil 2950, Valparaíso, Chile,\
`samuel.lepe@pucv.cl`
title: Cosmological Dark Matter Amplification through Dark Torsion
---
Introduction {#introduction .unnumbered}
============
The current knowledge of the nature of dark matter is scarce. However, the cumulative evidence seems to favor the scenario of dark matter as a non-interacting form of matter instead of some modified gravity theory. This is true in particular when considering phenomena as cluster collisions (e.g. Refs. [@Ref-DM-Bullet; @Ref-DM-Cluster2]).
Our ignorance of dark matter physical behavior implies a lack of knowledge of its features as a source of gravity. In particular, it is unknown whether or not the spin tensor of dark matter vanishes. The issue is relevant since a non-vanishing spin tensor is a source of torsion, and torsion requires to go beyond General Relativity (GR). The closest framework to General Relativity is the Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity. There are many other alternatives theories of gravity with torsion, but ECSK is probably the simplest one that includes spinning matter and torsion.
The ignorance regarding the physical nature of dark matter is in sharp contrast with the knowledge on the behavior of the Standard Model matter. For instance, from the Yang-Mills (YM) Lagrangian $$\mathcal{L}_{\mathrm{YM}}=-\frac{1}{4}F^{A}{}_{\mu\nu}F^{B\mu\nu}\,
\mathrm{tr}\left( \boldsymbol{T}_{A}\boldsymbol{T}_{B}\right) ,$$ it is straightforward to see that the spin tensor of Yang-Mills bosons vanishes. Therefore, Yang-Mills bosons are not a source of torsion, and there is no Yang-Mills bosons-torsion interaction. In contrast, the Standard Model fermions have a non-vanishing spin tensor, and therefore they are a source of torsion. They should interact with torsion, but the effect is so feeble that it is hard to foresee any particle-physics experiment capable of detecting it (See Chap. 8.4 of Ref. [@Ref-SUGRA-Van-Proeyen]).
Torsion does not interact with matter at a classical level (see Ref. [@Ref-Hehl-GravProbeB]), and neither do with electromagnetic phenomena. For instance, regardless of background torsion, classical point particles should follow torsionless geodesics, and electromagnetic waves should travel trough torsionless null geodesics. To Standard Model matter, torsion is dark. Perhaps, the only realistic way of detecting torsion could be through precise measurement of the polarization of gravitational waves (see Refs. [@Ref-Nos-2019-GW-Polarization] and [@Ref-Nos-2019-GW-Torsion]).
Even further, due to interaction and decoherence, Standard Model baryons are highly localized, and they form astrophysical structures. In the context of ECSK theory, torsion is not able to propagate in a vacuum (in glaring contrast to the behavior of Riemannian curvature). Therefore, given both the granular nature of the baryonic matter in the current epoch of the Universe and that torsion vanishes in the vacuum, it seems incorrect to associate an effective non-vanishing spin tensor to Standard Model matter in cosmological scales in modern times. In other words, it seems unrealistic (see Ref. [@Ref-Anti-Weyssenhof]) to consider Standard Model baryons as a spin fluid on a cosmological scale: the effective spin tensor of a gas of galaxies vanishes in long scales.
In contrast, the spin tensor of Standard Model matter is a relevant source of torsion in a Universe filled with a high-density plasma of Standard Model fermions. That is the case of bounce models at the very early Universe (see Ref. [@Ref-Poplawski-Big-Bounce]). In this model, the torsion created by high-density fermion plasma gives rise to inflation-like behaviors at very early times.
The situation is arguably different for dark matter. Its lack of interaction with Standard Model matter and its incapability to create dark matter structures lead to the conjecture that the decoherence effects could be feeble for dark matter. Even more, this picture of dark matter fits well with its distribution being broader and more unlocalized than the one of Standard Model matter. Therefore, if dark matter has a non-vanishing spin tensor, it seems natural to expect that it could give rise to torsion in cosmological scales.
The torsion created through this mechanism would be as dark as its source: Standard Model matter would not be able to interact with it. When moving these dark torsion terms to the right-hand side of the field equations, they behave just as an extra (and dark) source of standard torsionless Riemannian curvature.
From an observational point of view, it is possible to measure only the Riemannian curvature and not the torsion. Therefore, in this scenario, the observed gravitational dark matter effects correspond to the ones created by the “bare” dark matter plus the torsional dark dress it creates through its spin tensor.
The current article explores the idea of how dark torsion could amplify the effects of a small amount of dark matter in a cosmological setting. Given the disparity between the amount of dark matter and Standard Model baryons in the Universe, a mechanism as this one may seem of interest. The Section 1 briefly reviews ECSK gravity and shows how torsion amplifies the effects of bare dark matter, creating a higher effective energy density. This total torsion-dressed density would correspond to the observed dark matter density instead of the bare piece. In the case of Standard Model fermions, the canonical approach is to describe their spin tensor as a Weyssenhof fluid. However, given the lack of dark matter self-interaction, this Ansatz does not seem correct. In this Section, we offer a different Ansatz for the spin tensor of dark matter using symmetry and dimensional analysis arguments. The Sections 1.1 and 2 uses the generalized Friedmann equations to analyze the cosmological consequences of torsion and its dark matter amplification effect. The Section 3 studies the thermodynamical effects of the torsional dress of dark matter. Finally, in Section \[Sec\_TheEnd\] we present some conclusions and possible further works.
\[Sec\_DM-DT\]
Dark Matter and Dark Torsion
============================
There are many works in the context of cosmology using alternative theories of gravity involving a non-vanishing torsion ([@Ref-Poplawski-Big-Bounce; @Ref-Pasmatsiou; @Ref-Kranas; @Ref-Cabral; @Ref-Magueijo; @Ref-Alexander; @Ref-Nos-2018-CosmoHorndsk]). The present work focuses on the most straightforward approach, i.e., ECSK theory. It also the closest to standard GR. The idea is to have a taste of some of the consequences of non-vanishing spin tensor for dark matter in the simplest context before considering more exotic approaches.
Let us consider a four-dimensional spacetime with $\left( -,+,+,+\right) $ signature described by the Einstein–Cartan geometry, i.e., the metric $g_{\mu\nu}$ and the connection $\Gamma_{\mu\nu}^{\lambda}$ are independent degrees of freedom. The ECSK action principle corresponds to $$\mathcal{S}=\int\sqrt{\left\vert g\right\vert }\mathrm{d}^{4}x\left(
\mathcal{L}_{\mathrm{G}}+\mathcal{L}_{\mathrm{b}}+\mathcal{L}_{\mathrm{DM}}\right) , \label{Eq_Action}$$ where we are using units $c=8\pi G=k_{\mathrm{B}}=1$. In Eq. (\[Eq\_Action\]) $\mathcal{L}_{\mathrm{b}}$ stands for the Lagrangian for baryonic matter and $\mathcal{L}_{\mathrm{DM}}$ corresponds to an unknown Lagrangian for dark matter. The gravity Lagrangian $\mathcal{L}_{\mathrm{G}}$ corresponds to the standard Einstein–Hilbert term a la Palatini, i.e., without imposing the torsionless condition (and therefore with the metric and the connection as independent degrees of freedom),$$\mathcal{L}_{\mathrm{G}}\left( g,\Gamma,\partial\Gamma\right) =\frac{1}{2}R\left( g,\Gamma,\partial\Gamma\right) -\Lambda.$$ Here $R=g^{\sigma\nu}R^{\mu}{}_{\sigma\mu\nu}$ is the generalization of the Ricci scalar constructed from the generalized Riemann tensor (or Lorentz curvature) $$R^{\rho}{}_{\sigma\mu\nu}=\partial_{\mu}\Gamma_{\nu\sigma}^{\rho}-\partial_{\nu}\Gamma_{\mu\sigma}^{\rho}+\Gamma_{\mu\lambda}^{\rho}\Gamma
_{\nu\sigma}^{\lambda}-\Gamma_{\nu\lambda}^{\rho}\Gamma_{\mu\sigma}^{\lambda},$$ where $\Gamma_{\nu\sigma}^{\rho}$ is a general connection (not necessarily the Christoffel one).
The action principle, Eq. (\[Eq\_Action\]), may seem general. However, it is fair to remark that the Lagrangian choice Eq. (\[Eq\_Action\]) assumes the minimal coupling between dark matter, baryons, and gravity, and it does not include torsional terms as the Holst term. Non-minimal couplings with gravitational terms are sources of torsion, even for scalar bosonic fields [@Ref-Nos-2017-Horndeski] and in cosmological settings [@Ref-Nos-2018-CosmoHorndsk]. Similarly, non-minimal couplings within the Standard Model piece give rise to axions, which may be a promising dark matter candidate. Therefore it is worth remembering that Eq. (\[Eq\_Action\]) corresponds to the simplest ECSK case and there are many other more exotic choices.
The antisymmetric part of the connection $\Gamma_{\mu\nu}^{\lambda}$ defines the torsion tensor as$$T^{\lambda}{}_{\mu\nu}=\Gamma_{\mu\nu}^{\lambda}-\Gamma_{\nu\mu}^{\lambda},$$ and the difference between the general connection $\Gamma_{\mu\nu}^{\lambda}$ and the canonical Christoffel connection $\mathring{\Gamma}_{\mu\nu}^{\lambda
}=\left( 1/2\right) g^{\lambda\rho}\left( \partial_{\mu}g_{\nu\rho
}+\partial_{\nu}g_{\mu\rho}-\partial_{\rho}g_{\mu\nu}\right) $ is given by$$\Gamma_{\mu\nu}^{\lambda}-\mathring{\Gamma}_{\mu\nu}^{\lambda}=K^{\lambda}{}_{\nu\mu},$$ where the right-hand side corresponds to the contorsion[^1] tensor $$K_{\mu\nu\lambda}=\frac{1}{2}\left( T_{\nu\mu\lambda}-T_{\mu\nu\lambda
}+T_{\lambda\mu\nu}\right) . \label{Eq_K=T}$$ It is possible to decompose the generalized curvature in terms of the contorsion as$$R^{\alpha\beta}{}_{\mu\nu}=\mathring{R}^{\alpha\beta}{}_{\mu\nu}+\mathring{\nabla}_{\mu}K^{\alpha\beta}{}_{\nu}-\mathring{\nabla}_{\nu
}K^{\alpha\beta}{}_{\mu}+K^{\alpha}{}_{\lambda\mu}K^{\lambda\beta}{}_{\nu
}-K^{\alpha}{}_{\lambda\nu}K^{\lambda\beta}{}_{\mu}, \label{Eq_R+DK+K2}$$ where $\mathring{R}^{\alpha\beta}{}_{\mu\nu}$ is the canonical torsionless Riemann tensor in terms of the Christoffel connection $\mathring{\Gamma}_{\mu\nu}^{\lambda}$, and $\mathring{\nabla}_{\mu}$ is the standard torsionless covariant derivative in terms of it.
The metric equations of motion are given by$$R_{\mu\nu}^{+}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=\mathcal{T}_{\mu\nu
}^{\left( \mathrm{b}\right) }+\mathcal{T}_{\mu\nu}^{\left( \mathrm{DM}\right) }, \label{Eq_Field_Metric}$$ where $R_{\mu\nu}^{+}$ is the symmetric part of the generalized Ricci tensor[^2] and $\mathcal{T}_{\mu\nu}^{\left( \mathrm{b}\right)
}$ and $\mathcal{T}_{\mu\nu}^{\left( \mathrm{DM}\right) }$ are the stress-energy tensors associated with $\mathcal{L}_{\mathrm{b}}$ and $\mathcal{L}_{\mathrm{DM}}$. The affine equations of motion are given by$$T_{\lambda\mu\nu}-g_{\lambda\mu}T^{\rho}{}_{\rho\nu}+g_{\lambda\nu}T^{\rho}{}_{\rho\mu}=\sigma_{\lambda\mu\nu}^{\left( \mathrm{b}\right) }+\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) },
\label{Eq_Field_Affine}$$ where $\sigma_{\lambda\mu\nu}^{\left( \mathrm{b}\right) }=-\sigma
_{\lambda\nu\mu}^{\left( \mathrm{b}\right) }$ and $\sigma_{\lambda\mu\nu
}^{\left( \mathrm{DM}\right) }=-\sigma_{\lambda\nu\mu}^{\left(
\mathrm{DM}\right) }$ are the spin tensors[^3] associated with $\mathcal{L}_{\mathrm{b}}$ and $\mathcal{L}_{\mathrm{DM}}$.
We would like to end this brief review of ECSK pointing that in general $\nabla^{\mu}\left( R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right) \neq0$ and therefore the right-hand side of Eq. (\[Eq\_Field\_Metric\]) is not longer conserved. It is possible to write down a genuine conservation law using Eq. (\[Eq\_R+DK+K2\]) to move all the torsional terms to the right-hand side$$\mathring{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathring{R}+\Lambda g_{\mu\nu
}=\mathcal{T}_{\mu\nu}^{\left( \mathrm{b}\right) }+\mathcal{T}_{\mu\nu
}^{\left( \mathrm{DM}\right) }+\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }, \label{Eq_TorsionRightHandSide}$$ where $\mathring{R}_{\mu\nu}$ is the standard torsionless Ricci tensor and $\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }$ is the effective stress-energy tensor for torsion given by$$\begin{aligned}
\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) } & =g_{\mu\nu}\left(
\mathring{\nabla}_{\alpha}K^{\alpha\rho}{}_{\rho}+\frac{1}{2}\left[
K^{\alpha}{}_{\lambda\alpha}K^{\lambda\rho}{}_{\rho}-K^{\alpha}{}_{\lambda
\rho}K^{\lambda\rho}{}_{\alpha}\right] \right) +\nonumber\\
& +\frac{1}{2}\left( \mathring{\nabla}_{\nu}K^{\alpha}{}_{\mu\alpha
}+\mathring{\nabla}_{\mu}K^{\alpha}{}_{\nu\alpha}+K^{\alpha}{}_{\lambda\mu
}K^{\lambda}{}_{\nu\alpha}+K^{\alpha}{}_{\lambda\nu}K^{\lambda}{}_{\mu\alpha
}-\left[ \mathring{\nabla}_{\lambda}+K^{\alpha}{}_{\lambda\alpha}\right]
\left[ K^{\lambda}{}_{\mu\nu}+K^{\lambda}{}_{\nu\mu}\right] \right) .
\label{Eq_T_eff_torsion}$$ Doing this, the conservation law takes the form$$\mathring{\nabla}^{\mu}\left( \mathcal{T}_{\mu\nu}^{\left( \mathrm{b}\right) }+\mathcal{T}_{\mu\nu}^{\left( \mathrm{DM}\right) }+\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }\right) =0.$$ As mentioned in the Introduction, the baryons spin tensor $\sigma_{\lambda
\mu\nu}^{\left( \mathrm{b}\right) }$ and the torsion associated with it could have been relevant under the extremely high fermion densities of the very early Universe [@Ref-Poplawski-Big-Bounce]. However, in current times $\sigma_{\lambda\mu\nu}^{\left( \mathrm{b}\right) }=0$ should be an excellent approximation for any cosmological purpose.
Considering $\sigma_{\lambda\mu\nu}^{\left( \mathrm{b}\right) }=0$ and tracing the affine equation of motion (\[Eq\_Field\_Affine\]), it is clear that in cosmological scales we should have$$T_{\lambda\mu\nu}=\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }+\frac{1}{2}\left[ g_{\lambda\nu}\sigma^{\rho}{}_{\rho\mu}^{\left(
\mathrm{DM}\right) }-g_{\lambda\mu}\sigma^{\rho}{}_{\rho\nu}^{\left(
\mathrm{DM}\right) }\right] , \label{Eq_T=DM}$$ which means that torsion vanishes in the absence of dark matter. In the context of ECSK, torsion cannot propagate in a vacuum. To have a propagating torsion, we must have a different action choice than Eq. (\[Eq\_Action\]), for instance, the Holst action or the Horndeski generalization of Refs. [@Ref-Nos-2017-Horndeski; @Ref-Nos-2018-CosmoHorndsk].
Since torsion is dark for baryonic matter, Eq. (\[Eq\_Field\_Metric\]) can be regarded as$$\mathring{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathring{R}+\Lambda g_{\mu\nu
}=\mathcal{T}_{\mu\nu}^{\left( \mathrm{b}\right) }+\mathcal{T}_{\mu\nu
}^{\left( \mathrm{eff-DM}\right) },$$ where the effective dark matter stress-energy tensor $\mathcal{T}_{\mu\nu
}^{\left( \mathrm{eff-DM}\right) }=\mathcal{T}_{\mu\nu}^{\left(
\mathrm{DM}\right) }+\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }$ causes the observed effects of dark matter. Here $\mathcal{T}_{\mu\nu
}^{\left( \mathrm{DM}\right) }$ corresponds to the stress-energy tensor of bare dark matter. In the next sections, we show that torsion amplifies its weight through the effective torsional stress-energy tensor $\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }$ from Eq. (\[Eq\_T\_eff\_torsion\]). Since the interaction of torsion with baryonic matter is negligible in current times, torsion would be as dark as its source. Since current observations are only able to detect the Riemannian gravity piece of the geometry, they would be sensitive to the combined or dressed effect of $\mathcal{T}_{\mu\nu
}^{\left( \mathrm{eff-DM}\right) }=\mathcal{T}_{\mu\nu}^{\left(
\mathrm{DM}\right) }+\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }$, but they won’t be able to distinguish bare dark matter from torsion. Perhaps, only a careful measurement of the propagation of polarization of gravitational waves could distinguish bare dark matter from its torsional dress [@Ref-Nos-2019-GW-Polarization].
At this point, the lack of precise knowledge of the nature of dark matter creates what may seem like an insurmountable problem when trying to model its spin tensor. There are some usual Ansätze for the spin tensor, as the Weyssenhof fluid [@Ref-Weyssenhof; @Ref-Obukhov-Weyssenhof]. However, given our ignorance on the physics of dark matter, any Ansatz for $\sigma
_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }$ may seem excessive. The problem is that since we do not have any information on the spin tensor of dark matter $\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }$, we cannot use the field equation (\[Eq\_T=DM\]). Without this field equation we do not have information on torsion and it seems impossible to solve the system.
In what follows this problem is treated in a cosmological setting. The general idea is that whatever dark matter is, it is possible to use symmetry arguments and dimensional analysis to arrive at a general Ansatz for an effective $\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }$ in cosmological scales. Using this Ansatz, it becomes possible to study the effects of the torsion created by dark matter in cosmic evolution.
Let us start by considering the canonical FLRW metric with a homogeneous, isotropic and Riemannian flat spatial section$$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a^{2}\left( t\right) \left( \mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\right) . \label{Eq_FLRW_Metric}$$ Despite not knowing the Lagrangian $\mathcal{L}_{\mathrm{DM}}$, we know that the only stress-energy tensor compatible with the cosmological symmetries $\pounds _{\zeta}\mathcal{T}_{\mu\nu}^{\left( \mathrm{DM}\right) }=0$ is the canonical$$\mathcal{T}_{\mu\nu}^{\left( \mathrm{DM}\right) }=\left( \rho_{\mathrm{DM}}+p_{\mathrm{DM}}\right) U_{\mu}U_{\nu}+p_{\mathrm{DM}}g_{\mu\nu},$$ where $\rho_{\mathrm{DM}}$ and $p_{\mathrm{DM}}$ are the dark matter density and pressure. With the spin tensor it is possible to do the same. The most general spatially isotropic and homogeneous spin tensor $\pounds _{\zeta
}\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }=0$ for dark matter must have the Cartan staircase form$$\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }=-2\left( g_{\mu\lambda
}g_{\nu\rho}-g_{\mu\rho}g_{\nu\lambda}\right) h^{\rho}\left( t\right)
-2\sqrt{\left\vert g\right\vert }\epsilon_{\lambda\mu\nu\rho}f^{\rho}\left(
t\right) , \label{Ec_Spin_Tensor_Escalera_Cartan}$$ with the 4-vectors $h^{\rho}\left( t\right) $ and $f^{\rho}\left( t\right)
$ having the form $h^{\rho}\left( t\right) =-h\left( t\right) U^{\rho}$ and $f^{\rho}\left( t\right) =-f\left( t\right) U^{\rho}$. In terms of components$$\begin{aligned}
\sigma_{0\mu\nu}^{\left( \mathrm{DM}\right) } &
=0,\label{Ec_Spin_Tensor-0}\\
\sigma_{ij0}^{\left( \mathrm{DM}\right) } & =2g_{ij}h\left( t\right)
,\label{Ec_Spin_Tensor-h}\\
\sigma_{ijk}^{\left( \mathrm{DM}\right) } & =2\sqrt{\left\vert
g\right\vert }\epsilon_{ijk}f\left( t\right) , \label{Ec_Spin_Tensor-f}$$ where $i,j,k,=1,2,3$ and $\lambda,\mu,\nu,=0,1,2,3$. From Eqs. (\[Ec\_Spin\_Tensor-0\]-\[Ec\_Spin\_Tensor-f\]) it is already clear that dark matter spatial isotropy and homogeneity are not fully compatible with usual models as the Weyssenhoff spin fluid. At this point, we start to notice how different it is to model a high-density fermionic plasma as a source of spin and torsion and non-interacting dark matter. A high-density fermionic plasma in the early universe is well modeled as a Weyssenhoff spin fluid because their strong interactions create a rapidly changing spin tensor in short scales. It respects the Copernican principle because in longer cosmological scales only matters the average of these local spin anisotropies. The same arguments do not seem to hold for dark matter, considering it as a non-interacting fluid extended over cosmological distances in the current epoch. That is why the Ansatz Eq. (\[Ec\_Spin\_Tensor\_Escalera\_Cartan\]) for the spin tensor of dark matter may be a far better choice than the standard Weyssenhoff spin fluid.
The spin tensor, torsion and contorsion are all algebraically related through equations (\[Eq\_T=DM\]) and (\[Eq\_K=T\]). From them, it is straightforward to conclude that whatever $h\left( t\right) $ and $f\left( t\right) $ are$$\begin{aligned}
T_{\lambda\mu\nu} & =\left( g_{\mu\lambda}g_{\nu\rho}-g_{\mu\rho}g_{\nu\lambda}\right) h^{\rho}\left( t\right) -2\sqrt{\left\vert
g\right\vert }\epsilon_{\lambda\mu\nu\rho}f^{\rho}\left( t\right)
,\label{Eq_FLRW_T}\\
K_{\mu\nu\lambda} & =\left( g_{\mu\lambda}g_{\nu\rho}-g_{\mu\rho}g_{\nu\lambda}\right) h^{\rho}\left( t\right) +\sqrt{\left\vert
g\right\vert }f^{\rho}\left( t\right) \epsilon_{\rho\mu\nu\lambda},
\label{Eq_FLRW_K}$$ i.e., the same (still unknown) functions $h\left( t\right) $ and $f\left(
t\right) $ describe torsion and contorsion.
Using the expressions (\[Eq\_FLRW\_Metric\],\[Eq\_FLRW\_T\],\[Eq\_FLRW\_K\]), it is possible to calculate the Lorentz curvature components (\[Eq\_R+DK+K2\]) and from it the field equations (\[Eq\_Field\_Metric\]) lead to the generalized Friedmann relations$$\begin{aligned}
3\left[ \left( H+h\right) ^{2}-f^{2}\right] & =\rho_{\mathrm{DM}},\label{Eq_Friedmann_Gen_density}\\
2\left( \dot{H}+\dot{h}\right) +\left( 3H+h\right) \left( H+h\right)
-f^{2} & =-p_{\mathrm{DM}}. \label{Eq_Friedmann_Gen_p}$$ To solve the Eqs. (\[Eq\_Friedmann\_Gen\_density\]) and (\[Eq\_Friedmann\_Gen\_p\]), we need to know the dependence of $f$ and $h$ on other physical variables, like dark matter density and pressure. The next section shows how to find an Ansatz for these torsional equations of state, and to solve the system.
Torsional dressing of Dark Matter and Dark Energy
-------------------------------------------------
At this point, instead of making some standard conjecture (Weyssenhoff fluid, Frenkel condition, Tulczyjew condition, etc.) on the physical nature of the dark matter spin tensor $\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right)
}$, we adopted a simpler approach. We may not have an understanding of $\sigma_{\lambda\mu\nu}^{\left( \mathrm{DM}\right) }$ from first principles, but we have some clues about its form. On the one hand, replacing Eq. (\[Eq\_FLRW\_K\]) in Eq. (\[Eq\_T\_eff\_torsion\]) we can get an effective stress-energy tensor $\mathcal{T}_{\mu\nu}^{\left( \mathrm{T}\right) }$ in terms of $f$ and $h$. Using dimensional analysis on it, it is clear that at least in what concerns units we have$$\begin{aligned}
f & \sim\sqrt{\mathrm{energy}\text{ }\mathrm{density}},\\
h & \sim\sqrt{\mathrm{energy}\text{ }\mathrm{density}},\\
\mathring{\nabla}_{\mu}h^{\mu} & \sim\mathrm{energy}\text{ }\mathrm{density}.\end{aligned}$$ On the other hand, it is clear that in a dark matter vacuum ($\rho
_{\mathrm{DM}}=0$) its spin tensor vanishes and $h\left( t\right) =f\left(
t\right) =0$. Similarly, it seems reasonable to expect $h\left( t\right) $ and $f\left( t\right) $ to grow for higher values of $\rho_{\mathrm{DM}}$. For this reason, it seems natural to propose an Ansätze of barotropic relations between $h\left(
t\right) $ and $f\left( t\right) $ and the dark matter energy density $\rho_{\mathrm{DM}}$ of the form$$\begin{aligned}
f & \sim\sqrt{\rho_{\mathrm{DM}}},\label{Eq_Protobaro_f}\\
h & \sim\sqrt{\rho_{\mathrm{DM}}}. \label{Eq_Protobaro_Dh}$$ Of course, much more complex relationships are possible, but these seem to be the simplest torsional equations of state. Let us consider a standard barotropic relation for the dark matter pressure $p_{\mathrm{DM}}=\omega_{\mathrm{DM}}\rho_{\mathrm{DM}}$ and let us write the barotropic Ansatz for $f$ as$$f=\alpha_{f}\sqrt{\frac{\rho_{\mathrm{DM}}}{3}},$$ where $\alpha_{f}$ is a constant. In terms of $\alpha_{f}$ it proves practical to define the semi-dressed dark matter energy density and pressure$$\begin{aligned}
\rho_{f} & =\rho_{\mathrm{DM}}+3f^{2}=\left( 1+\alpha_{f}^{2}\right)
\rho_{\mathrm{DM}},\label{Eq_Semidressed_Density}\\
p_{f} & =p_{\mathrm{DM}}-f^{2}=\left( \omega_{\mathrm{DM}}-\frac{1}{3}\alpha_{f}^{2}\right) \rho_{\mathrm{DM}}. \label{Eq_Semidressed_Pressure}$$ In terms of $\rho_{f}$ and $p_{f}$, the Eqs. (\[Eq\_Friedmann\_Gen\_density\]) and (\[Eq\_Friedmann\_Gen\_p\]) take the simpler form$$\begin{aligned}
3\left( H+h\right) ^{2} & =\rho_{f},
\label{Eq_Friedmann_Density_Seminaked}\\
2\left( \dot{H}+\dot{h}\right) +\left( 3H+h\right) \left( H+h\right) &
=-p_{f}, \label{Eq_Friedmann_Pressure_Seminaked}$$ where the semi-dressed pressure $p_{f}$ obeys the barotropic relation $p_{f}=\omega_{f}\rho_{f}$ and $$\omega_{f}=\frac{\omega_{\mathrm{DM}}-\alpha_{f}^{2}/3}{1+\alpha_{f}^{2}}.$$ In short, the $f$-component of the spin tensor has the effect of replacing the original bare dark matter density $\rho_{\mathrm{DM}}$ by an amplified semi-dressed energy density $\rho_{f}$, Eq. (\[Eq\_Semidressed\_Density\]). The pressure $p_{\mathrm{DM}}$ is replaced by an smaller semi-dressed $p_{f}$ pressure, Eq. (\[Eq\_Semidressed\_Pressure\]). It is worth to notice that in the case of cold dark matter $\omega_{\mathrm{DM}}=0$, it leads us to an effective negative pressure $-1/3<\omega_{f}\leq0$. This way, torsion can easily produce an effective non-particle negative pressure $p_{f}$ from canonical cold dark matter $\omega
_{\mathrm{DM}}=0$.
From Eqs. (\[Eq\_Friedmann\_Density\_Seminaked\]) and (\[Eq\_Friedmann\_Pressure\_Seminaked\]), we may feel compelled to define a generalized Hubble parameter $H+h$. However, it is important to remember that our observations describe the behavior of classical particles (i.e., galaxies). Classical particles are sensitive only to the Riemannian piece of the geometry and oblivious to torsion, and therefore observations measure $H$ and not $H+h$. For this reason, it is convenient to write down the Eqs. (\[Eq\_Friedmann\_Density\_Seminaked\]) and (\[Eq\_Friedmann\_Pressure\_Seminaked\]) as$$\begin{aligned}
3H^{2} & =\rho_{f}+\rho_{h},\label{Eq_Friedmann_rho_h}\\
2\dot{H}+3H^{2} & =-\left( p_{f}+p_{h}\right) , \label{Eq_Friedmann_p_h}$$ where $\rho_{h}$ and $p_{h}$ are the effective density and pressure originated when moving all the $h\left( t\right) $ terms to the right-hand side of the field equations$$\begin{aligned}
\rho_{h} & =-3\left( h+2H\right) h,\label{Eq_Def_rho_h}\\
p_{h} & =h^{2}+4Hh+2\dot{h}.\end{aligned}$$ The total dark matter and torsion weight is described by the effective dressed density and pressure$$\begin{aligned}
\rho_{\mathrm{dressed}} & =\rho_{f}+\rho_{h},\\
p_{\mathrm{dressed}} & =p_{f}+p_{h}.\end{aligned}$$ At this point, using the relation (\[Eq\_Protobaro\_h\]) we propose the barotropic Ansatz$$h=\alpha_{h}\sqrt{1+\alpha_{f}^{2}}\sqrt{\rho_{\mathrm{DM}}}=\alpha_{h}\sqrt{\rho_{f}},$$ and from here the behavior of the dark matter spin tensor, and its torsion becomes more clear. The two functions $f$ and $h$ parametrize the dark matter spin tensor, and they create an effective torsional dress for $\rho_{\mathrm{DM}}$ and $p_{\mathrm{DM}}$. For instance, a small $\rho_{\mathrm{DM}}$ may be amplified for torsion and create a much bigger $\rho_{\mathrm{dressed}}=\rho_{f}+\rho_{h}$. Current observations would measure the effective $\rho_{\mathrm{dressed}}$ and not the original dark matter density $\rho_{\mathrm{DM}}$.
On the other hand, the torsional-dressed density $\rho_{\mathrm{dressed}}=\rho_{f}+\rho_{h}$ has more complex behavior. From Eq. (\[Eq\_Friedmann\_Density\_Seminaked\]), it is clear that $6\left(
H+h\right) \left( \dot{H}+\dot{h}\right) =\dot{\rho}+\dot{\rho}_{f}$. Replacing this in Eq. (\[Eq\_Friedmann\_Pressure\_Seminaked\]) and considering that dark matter does not interact with SM matter Eq. (\[Eq\_Conservation\_SM\]), it is possible to prove that the two dark matter-torsion modes, the $f$-dressed density $\rho_{f}$ and the $h$-dressed density $\rho_{h}$, interchange energy among them$$\begin{aligned}
\dot{\rho}_{f}+3H\left( \rho_{f}+p_{f}\right) & =-Q,\label{Eq_Q1}\\
\dot{\rho}_{h}+3H\left( \rho_{h}+p_{h}\right) & =Q, \label{Eq_Q2}$$ where$$Q=\left( 1+3\omega_{f}\right) h\rho_{f}, \tag{46}$$ and therefore the effective density $\rho_{\mathrm{dressed}}=\rho_{f}+\rho
_{h}$ obeys the canonical conservation relation$$\dot{\rho}_{\mathrm{dressed}}+3H\left( \rho_{\mathrm{dressed}}+p_{\mathrm{dressed}}\right) =0, \tag{47}$$ with $p_{\mathrm{dressed}}$ obeying a non-trivial equation of state $p_{\mathrm{dressed}}=p_{\mathrm{dressed}}\left( \rho_{f},\rho_{h}\right) $. The next Section analyses the phenomenology of this system for some important particular cases.
\[Sec\_Cosmo\_DM\_DT\]
Cosmological consequences of Dark Torsion
=========================================
The two torsional modes $h$ and $f$ create very distinctive phenomenology in the context of cosmic evolution. The simplest case is $h=\alpha_{h}=0$, leading us to a system of the canonical form$$\begin{aligned}
3H^{2} & =\rho_{f},\tag{48}\\
\dot{\rho}_{f}+3H\left( 1+\omega_{f}\right) \rho_{f} & =0. \tag{49}$$ When $h=0$, the effective density $\rho_{f}$ packs dark matter and torsion altogether. The only difference with the standard $\mathrm{\Lambda CDM}$ torsionless case is that for cold dark matter $\omega_{\mathrm{DM}}=0$, the effective barotropic constant $\omega_{f}=\left( \omega_{\mathrm{DM}}-\alpha_{f}^{2}/3\right) /\left( 1+\alpha_{f}^{2}\right) $ has the allowed range $-1/3<\omega_{f}\leq0$. Since $\rho_{f}=\left( 1+\alpha_{f}^{2}\right)
\rho_{\mathrm{DM}}$, it means that for large values of $\alpha_{f}$ a small quantity of dark matter can get significantly amplified.
The case $h\neq0$. From Eq. (\[Eq\_Friedmann\_Density\_Seminaked\]) we can obtain $$H\left( t\right) =\left( \sqrt{\frac{1}{3}}\frac{\mathrm{s}_{H+h}}{\left\vert \alpha_{h}\right\vert }-\mathrm{s}_{h}\right) \left\vert
h\left( t\right) \right\vert , \tag{50}$$ where we are using the shortcut notation $\mathrm{s}_{X}=\mathrm{sign}\left(
X\right) $. Therefore the $\rho_{h}$ density corresponds to $$\rho_{h}=3\left( \left\vert \alpha_{h}\right\vert -\frac{2}{\sqrt{3}}\mathrm{s}_{h}\mathrm{s}_{H+h}\right) \frac{h^{2}}{\left\vert \alpha
_{h}\right\vert }>0, \tag{51}$$ but $$\begin{aligned}
\mathrm{s}_{h} & =-1\text{ \ },\text{ \ }\mathrm{s}_{H+h}=1\Longrightarrow
H\left( t\right) =\left( \sqrt{\frac{1}{3}}\frac{\mathrm{1}}{\left\vert
\alpha_{h}\right\vert }+\mathrm{1}\right) \left\vert h\left( t\right)
\right\vert \rightarrow\frac{\left\vert h\left( t\right) \right\vert
}{H\left( t\right) }\text{ }<1\text{\ },\text{ \ }\rho_{h}=3\left(
\left\vert \alpha_{h}\right\vert +\frac{2}{\sqrt{3}}\right) \frac{h^{2}}{\left\vert \alpha_{h}\right\vert },\tag{52}\\
\mathrm{s}_{h} & =-1\text{ \ },\text{ \ }\mathrm{s}_{H+h}=-1\Longrightarrow
H\left( t\right) =\left( 1-\sqrt{\frac{1}{3}}\frac{\mathrm{1}}{\left\vert
\alpha_{h}\right\vert }\right) \left\vert h\left( t\right) \right\vert
\Longrightarrow\left\vert \alpha_{h}\right\vert >\sqrt{\frac{1}{3}}\rightarrow\rho_{h}=3\left( \left\vert \alpha_{h}\right\vert -\frac{2}{\sqrt{3}}\right) \frac{h^{2}}{\left\vert \alpha_{h}\right\vert },\tag{53}\\
\mathrm{s}_{h} & =1\text{ \ },\text{ \ }\mathrm{s}_{H+h}=1\Longrightarrow
H\left( t\right) =\left( \sqrt{\frac{1}{3}}\frac{\mathrm{1}}{\left\vert
\alpha_{h}\right\vert }-\mathrm{1}\right) \left\vert h\left( t\right)
\right\vert \Longrightarrow\left\vert \alpha_{h}\right\vert <\sqrt{\frac{1}{3}}\rightarrow\rho_{h}=3\left( \left\vert \alpha_{h}\right\vert -\frac
{2}{\sqrt{3}}\right) \frac{h^{2}}{\left\vert \alpha_{h}\right\vert }<0,
\tag{54}$$ and the last two cases lead to $\left\vert h\left( t\right) \right\vert
/H\left( t\right) >1$. Additionally we note that the case $\mathrm{s}_{h}=-1$, $\mathrm{s}_{H+h}=1$ leads directly to $\rho_{h}>0$ and we are well with the weak energy condition. But, is it reasonable to expect $\left\vert
h\left( t\right) \right\vert /H\left( t\right) <1$ or $\left\vert h\left(
t\right) \right\vert /H\left( t\right) <<1$ during the cosmic evolution? This is an open question.
Now, after replacing $H$ and $h=\alpha_{h}\sqrt{\rho_{f}}$ into $\dot{\rho
}_{f}+3H\left( 1+\omega_{f}\right) \rho_{f}=-\left( 1+3\omega_{f}\right)
h\rho_{f}$, we obtain the following solution for $h$, with $\mathrm{s}_{h}=-1$ and $\mathrm{s}_{H+h}=1$, $$\frac{h\left( t\right) }{h\left( t_{0}\right) }=\left[ 1+\Delta\left(
t-t_{0}\right) \right] ^{-1}, \tag{55}$$ where $t_{0}$ is today and $$\Delta=\left[ \left\vert \alpha_{h}\right\vert -\frac{\sqrt{3}}{2}\left(
1+\omega_{f}\right) \right] \frac{\left\vert h\left( t_{0}\right)
\right\vert }{\left\vert \alpha_{h}\right\vert }. \tag{56}$$ Recalling that $\omega_{\mathrm{DM}}=0\rightarrow\omega_{f}=-\alpha_{f}^{2}/3\left( 1+\alpha_{f}^{2}\right) $, we write $$\Delta=\left[ \left\vert \alpha_{h}\right\vert -\frac{\sqrt{3}}{2}\left(
\frac{1+2\alpha_{f}^{2}/3}{1+\alpha_{f}^{2}}\right) \right] \frac{\left\vert
h\left( t_{0}\right) \right\vert }{\left\vert \alpha_{h}\right\vert },
\tag{57}$$ so that $$\begin{aligned}
\mathrm{s}_{\Delta} & =1\Longrightarrow\Delta>0\rightarrow\text{standard
scheme},\tag{58}\\
\mathrm{s}_{\Delta} & =-1\Longrightarrow\Delta<0\rightarrow\text{phantom
evolution!} \tag{59}$$ But $$\Delta<0\longleftrightarrow\left\vert \alpha_{h}\right\vert <\frac{\sqrt{3}}{2}\left( \frac{1+2\alpha_{f}^{2}/3}{1+\alpha_{f}^{2}}\right) <1, \tag{60}$$ and so $$h\left( t\right) =\frac{h\left( t_{0}\right) }{\left\vert \Delta
\right\vert }\left( t_{s}-t\right) ^{-1}\text{ \ },\text{ \ }t_{s}=t_{0}+\frac{1}{\left\vert \Delta\right\vert }, \tag{61}$$ and all the components explode at $t_{\mathrm{s}}$: the Hubble parameter, the densities $\rho_{\mathrm{DM}}$, $\rho_{f}$, $\rho_{\mathrm{dressed}}$ and $Q$. This a Big Rip singularity.
The $Q$-function becomes$$Q\left( t\right) =-\frac{1}{\left\vert a_{h}\right\vert ^{2}\left(
1+\alpha_{f}^{2}\right) }\left\vert h\right\vert ^{3}, \tag{62}$$ and so, there is energy transference from $\rho_{h}$ to $\rho_{f}$.
Thermodynamics
==============
We inspect two thermodynamics aspects in presence of torsion, adiabaticity and dark matter temperature. We start with the Gibb’s relation $$T\mathrm{d}S=\mathrm{d}\left( \frac{\rho_{\mathrm{DM}}}{n}\right)
+p_{_{\mathrm{DM}}}\mathrm{d}\left( \frac{1}{n}\right) , \tag{63}$$ implying$$nT\frac{\mathrm{d}S}{\mathrm{d}t}=-\left( \rho_{_{\mathrm{DM}}}+p_{_{\mathrm{DM}}}\right) \frac{\dot{n}}{n}+\dot{\rho}_{\mathrm{DM}},
\tag{64}$$ where $T$ is the temperature, $S$ the entropy and $n$ the number particle density. Using the integrability condition ($S$ is a function of state) we have $\partial^{2}S/\partial T\partial n=\partial^{2}S/\partial n\partial T$ and therefore $$n\frac{\partial T}{\partial n}+\left( p_{\mathrm{DM}}+\rho_{\mathrm{DM}}\right) \frac{\partial T}{\partial\rho_{\mathrm{DM}}}=T\frac{\partial
p_{_{\mathrm{DM}}}}{\partial\rho_{\mathrm{DM}}}. \tag{65}$$ Since $\rho_{f}=\left( 1+\alpha_{f}^{2}\right) \rho_{\mathrm{DM}}$ and $\rho_{f}$ satisfies Eq. (\[Eq\_Q1\]), we have that the bare dark matter density obeys the conservation law $$\dot{\rho}_{\mathrm{DM}}+3H\left( 1+\omega_{f}\right) \rho_{\mathrm{DM}}=-\frac{Q}{1+\alpha_{f}^{2}}, \tag{66}$$ and making the hypothesis of $\dot{n}+3Hn=0$ (conservation of the number of dark matter particles) we have that $$nT\frac{\mathrm{d}S}{\mathrm{d}t}=3H\left( p_{_{\mathrm{DM}}}-\omega_{f}\rho_{\mathrm{DM}}\right) -\frac{Q}{1+\alpha_{f}^{2}}. \tag{67}$$ In the case of cold dark matter $\omega_{\mathrm{DM}}=0$ it implies that $$nT\frac{\mathrm{d}S}{\mathrm{d}t}=\frac{1}{1+\alpha_{f}^{2}}\left(
H\alpha_{f}^{2}\rho_{\mathrm{DM}}-Q\right) , \tag{68}$$ and there is not adiabaticity. Given that $Q<0$, then $dS/dt>0$ and the second law of thermodynamics is guaranteed.
As we know, $\Lambda CDM$ [@Ref-PLANCK] has been quite successful in describing the current state of cosmic evolution even when it is not free of problems. As a consequence of this, the idea of dark energy emerged as a more physical alternative to $\Lambda$ and, moreover, dark matter-dark energy interaction is a fact does not ruled out by observation [@Ref-B.Wang]. In this interaction framework, the non-adiabaticity is manifest [@Ref-Victor]. So, we conjecture that torsion is cause of non-adiabaticity!
Using the integrability condition, the temperature can be obtained from $$\frac{\dot{T}}{T}=-3H\omega_{\mathrm{DM}}\left( 1+\frac{\left(
1+\omega_{\mathrm{DM}}\right) \left( 1+\alpha_{f}^{2}\right) }{1+\omega_{\mathrm{DM}}+2\alpha_{f}^{2}/3+Q/3H\rho_{\mathrm{DM}}}\right)
^{-1}, \tag{69}$$ and it is clear that $\omega_{\mathrm{DM}}=0\Longrightarrow T=\mathrm{const}$., consistent with “orthodoxy” which tells us that the dark matter temperature is constant during the cosmic evolution. Thus, torsion does not affect the dark matter temperature.
Final remarks {#Sec_TheEnd}
=============
We have found a phantom scheme originated after considering torsion coupled to the dark matter. As a consequence of this, we would not need dark energy (phantom dark energy described by $\omega_{ph}<-1$) in order to explain such late evolution. This is an alternative that seeks to explain the phantom scheme, not ruled out by the current observational information.
Another interesting fact that we have found is an interaction scheme between the torsional components $h$ and $f$. The significance of this interaction is not clear to us yet. If this interaction is such, is there any way to detect any observational consequence from this? If torsion effects cannot be detected with the current observations, perhaps it can be done in the future if the polarization of gravitational waves is measured and from this observational fact we can also have indications of $\left\vert h\left( t\right)
\right\vert /H\left( t\right) $. This is an interesting conjecture to explore.
From the thermodynamic point of view, the cosmic evolution turns out to be non-adiabatic ($\mathrm{\Lambda CDM}$ is an adiabatic scheme) and since $h<0$, the second law of thermodynamics is guaranteed. The interesting thing is also that the dark matter temperature, even in the presence of torsion, remains constant through the cosmic evolution.
Finally, and as we have already said, future observations could shed some light on the role, if any, of torsion in the cosmic evolution.
FI acknowledges financial support from the Chilean government through FONDECYT grant 1180681 of the Government of Chile.
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[^1]: It seems there is no agreement in the literature on the name of this tensor. Some authors call it contortion, while others use contorsion. We have chosen to use the later one because it sounds closer to torsion. The word contortion may also be confused with a twisting motion.
[^2]: In the case of non-vanishing torsion, the generalized Ricci tensor has an antisymmetric part given by $R_{\mu\nu}^{-}=\frac{1}{2}\left(
R_{\mu\nu}-R_{\nu\mu}\right) =-\frac{1}{2}\left( \nabla_{\lambda}T^{\lambda
}{}_{\mu\nu}+\nabla_{\nu}T^{\lambda}{}_{\lambda\mu}-\nabla_{\mu}T^{\lambda}{}_{\lambda\nu}+T^{\lambda}{}_{\rho\lambda}T^{\rho}{}_{\mu\nu}+T^{\rho}{}_{\lambda\mu}T^{\lambda}{}_{\rho\nu}-T^{\rho}{}_{\lambda\nu}T^{\lambda}{}_{\rho\mu}\right) .$
[^3]: The spin tensor is the variation of the matter Lagrangian with respect to the connection, in the same way as the stress-energy tensor is the variation of the matter Lagrangian with respect to the metric. The spin tensor of classical matter (e.g., dust) vanishes, but the spin tensor of a fermionic particle does not. For instance, the spin tensor of an electron is proportional to its axial current.
|
---
abstract: 'We study sampling from a target distribution $\nu_* \propto e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the target $\nu_*$ satisfies the Poincaré inequality, and the potential $f$ is first-order smooth and dissipative. Under an opaque *uniform warmness* condition on the LMC iterates, we establish that $\widetilde{\mathcal{O}}(\epsilon^{-1})$ steps are sufficient for LMC to reach $\epsilon$ neighborhood of the target in Chi-square divergence. We hope that this note serves as a step towards establishing a complete convergence analysis of LMC under Chi-square divergence.'
author:
- 'Murat A. Erdogdu[^1]'
- 'Rasa Hosseinzadeh[^2]'
title: |
A Brief Note on the Convergence of Langevin Monte Carlo\
in Chi-Square Divergence
---
Acknowledgements {#acknowledgements .unnumbered}
================
Authors would like to thank Andre Wibisono for pointing out to an error in the proof of Lemma 1 in an early version of this note.
[^1]: Department of Computer Science and Department of Statistical Sciences at the University of Toronto, and Vector Institute, `erdogdu@cs.toronto.edu`
[^2]: Department of Computer Science at the University of Toronto, and Vector Institute, `rasa@cs.toronto.edu`
|
---
abstract: 'Internet of Things (IoT) devices communicate using a variety of protocols, differing in many aspects, with the channel access method being one of the most important. Most of the transmission technologies explicitly designed for IoT and Machine-to-Machine (M2M) communication use either an ALOHA-based channel access or some type of Listen Before Talk (LBT) strategy, based on carrier sensing. In this paper, we provide a comparative overview of the uncoordinated channel access methods for IoT technologies, namely ALOHA-based and LBT schemes, in relation with the ETSI and FCC regulatory frameworks. Furthermore, we provide a performance comparison of these access schemes, both in terms of successful transmissions and energy efficiency, in a typical IoT deployment. Results show that LBT is effective in reducing inter-node interference even for long-range transmissions, though the energy efficiency can be lower than that provided by ALOHA methods. The adoption of rate-adaptation schemes, furthermore, lowers the energy consumption while improving the fairness among nodes at different distances from the receiver. Coexistence issues are also investigated, showing that in massive deployments LBT is severely affected by the presence of ALOHA devices in the same area.'
author:
- |
Daniel Zucchetto, Andrea Zanella\
Department of Information Engineering, University of Padova, Italy\
E-mail: {*zucchett,zanella*}@dei.unipd.it
bibliography:
- 'main\_submitted.bib'
title: 'Uncoordinated access schemes for the IoT: approaches, regulations, and performance'
---
[17cm]{}(1.7cm,0.5cm) The final version of this paper has been published in the IEEE Communications Magazine vol. 55, no. 9, pp. 48–54, September 2017. DOI: 10.1109/MCOM.2017.1600617\
**Copyright Notice**: 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, including reprinting/republishing this material for advertising or promotional purposes, collecting new collected works for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Introduction
============
A key element to enable the full realization of the Internet of Things (IoT) vision is the ubiquitous connectivity of end devices, with minimal configuration, as for the so-called *place-&-play* paradigm [@Biral20151]. Today, the main three approaches to provide connectivity to the IoT devices are the following.
*Cellular systems.* The existing cellular networks are a natural and appealing solution to provide connectivity to IoT end-devices, thanks to their world-wide established footprint and the capillary market penetration. Unfortunately, current cellular network technologies have been designed targeting wideband services, characterized by few connections that generate a large amount of data, while most IoT services are expected to generate a relatively small amount of traffic, but from a very large number of different devices. This shift of paradigm challenges the control plan of current cellular standards, which can become the system bottleneck. For these reasons, the IoT and Machine-to-Machine (M2M) scenarios are considered as major challenges for next generation wireless cellular systems, commonly referred to as 5G.
*Short-range multi-hop technologies.* This family collects a number of popular technologies specifically designed for M2M communications or Wireless Personal Area Networks (WPANs). These systems usually operate in the frequency bands centered around 2.4 GHz, 915 MHz and 868 MHz, though the 2.4 GHz is the most common choice. They are characterized by high energy efficiency and medium/high bitrates (order of hundreds of kbit/s or higher), but limited single-hop coverage area. To cover larger areas, most WPAN technologies provide the possibility to relay data in a multihop fashion, realizing a so-called *mesh network*. Examples of standards in this category are IEEE 802.15.4 [@15.4], Bluetooth Low Energy [@BLE], and Z-Wave, the latter having its physical and data link layers specified in ITU-T G.9959 [@g.9959].
*Low-Power Wide-Area (LPWA) networks.* A third relevant class in the arena of IoT-enabling wireless technologies consists in the LPWA solutions. According to [@cisco], LPWA technologies will account for 28% of M2M connections by 2020. These technologies, specifically designed to support M2M connectivity, provide low bitrates, low energy consumption, and wide geographical coverage. Almost all LPWA technologies operate at frequencies around 800 or 900 MHz, though there are also solutions working in the classic 2.4 GHz ISM band or exploiting white spaces in TV frequencies. Some relevant LPWA technologies are LoRaWAN, Sigfox, Ingenu [@CVZZ16].
While cellular systems entail centralized access schemes over dedicated frequency bands, which provide high efficiency, robustness, security, and performance predictability, most of WPAN and LPWA technologies operate on unlicensed radio bands, adopting uncoordinated access schemes. The use of unlicensed bands yields the obvious advantage of lowering the operational costs of the network, while the adoption of uncoordinated channel access schemes makes it possible to simplify the hardware of the nodes, thus reducing the manufacturing costs and the energy consumption. The downside is that the lack of coordination in channel access may yield performance losses in terms of throughput and energy efficiency when the number of contending nodes increases.
To alleviate the problem of channel congestion in the unlicensed bands, radio spectrum regulators have imposed limits on the channel occupation of each device, in terms of bandwidth, time, and on the maximum transmission power. However, the Federal Communications Commission (FCC) in the USA and the Conference of Postal and Telecommunications Administrations (CEPT) in Europe have taken different approaches to limit channel congestion: the first imposes very strict limits on the emission power and favors the use of spread spectrum techniques but do not restrict the number of access attempts that can be performed by the nodes [@fcc-part15], while the second limits the fraction of on-air time of a device to be lower than a given *duty cycle*, or imposes the use of *Listen Before Talk* (LBT) techniques, which are also referred to as *Carrier Sense Multiple Access* (CSMA) protocols [@etsi-2012].[^1]
These precautions are actually effective when the coverage range of the wireless transmitters is relatively small (few meters), as was indeed the case for the first commercial products operating in the ISM frequency bands. However, this condition does no longer hold for LPWA solutions, which have coverage ranges in the order of 10–15 km in rural areas, and 2–5 km in urban areas, with a star-like topology that can exacerbate the mutual interference and hidden node problems. Furthermore, while short-range communication systems usually support a single, or just a few modulation schemes and transmit rates, LPWA technologies usually provide multiple transmit rates to optimize the transmission based on the distance to be covered.
Despite these quite radical changes in the transmit characteristics of the recent LPWA technologies with respect to the previous generation of the so-called Short Range Devices (SRD), the channel access methods and the regulatory constraints are still the same. The objective of this study is hence to investigate the performance of well established uncoordinated channel access schemes in this new scenario, characterized by a huge number of devices with large coverage ranges and multi-rate capabilities. To this end, we first provide a quick overview of the main uncoordinated access schemes used by most common wireless communication technologies considered for the IoT and we discuss the regulatory frameworks, with particular focus to the European case. We then compare the performance achieved by two popular uncoordinated access schemes in a typical LPWA network scenario, considering the limits imposed by the regulations. The paper is then closed with some final considerations and recommendations.
Uncoordinated access techniques for the IoT {#unc}
===========================================
Channel access schemes can be roughly divided in two main categories: coordinated and uncoordinated (or contention-based). Coordinated access schemes require time synchronization among the nodes and, hence, are more suitable for small networks (e.g., Bluetooth) or centrally controlled systems (e.g., cellular), with predictable and/or steady traffic flows (e.g., voice or bulk data transfer). Uncoordinated access strategies, instead, are usually considered for networks with a variable number of devices and unpredictable traffic patterns. In the following we provide a quick overview of the two main uncoordinated access schemes that are widely adopted by the transmission technologies typically associated to the IoT scenarios.
ALOHA-based schemes
-------------------
Many protocols for M2M communication are based on pure ALOHA access schemes, according to which a transmission is attempted whenever a new message is generated by the device. This form of channel access may be coupled with a retransmission scheme, according to which a packet is retransmitted until acknowledged by the receiver. However, some IoT services (e.g., environmental monitoring) can tolerate a certain amount of lost messages. In these cases, a retransmission scheme is not needed, allowing for a simplification of the device firmware and enabling a significant reduction in the energy consumption. For these reasons, ALOHA schemes are widely adopted in M2M communication as, for example, LoRaWAN and Sigfox. Furthermore, some standards that adopt LBT access techniques optionally provide an ALOHA mode of operation, as for the IEEE 802.15.4.
More sophisticated ALOHA-based protocols can be enabled when nodes are time synchronized, e.g., by means of beacons periodically broadcasted by coordinator nodes (e.g., gateways in LoRaWAN). For example, slotted-ALOHA divides the time in intervals of equal size, called slots, and allows transmissions only within slots, thus avoiding packet losses due to partially overlapping transmissions. Framed slotted ALOHA (FSA), instead, organizes the slots in groups, called frames, and allow each node to transmit only once per frame. The limit of these schemes is that packet transmission time should not exceed the slot duration. A common solution to accommodate uneven packet transmission times is to adopt a hybrid access scheme (HYB) that splits the frame in two parts: the first $k$ slots are used by the nodes to send resource reservation messages to the controller, using a FSA access scheme, while the remaining slots in the frame are allocated by the controller to the nodes, according to the amount of resources required in the accepted reservation messages. The nodes get notified about the allocated resources by a control message that is broadcasted by the controller right after the end of the reservation phase. Variants of these basic mechanism are currently used in many different protocols as, e.g., GSM, 802.11e. However, to the best of our knowledge, the HYB approach has not yet been studied in the M2M scenario.
Carrier sensing schemes
-----------------------
When using carrier sensing techniques, each device listens to the channel before transmitting (from which the wording “Listen-Before-Talk”). The channel sensing operation is typically called *Clear Channel Assessment* (CCA) and aims at checking the occupancy of the channel by other transmitters, in which case the channel access will be delayed to avoid mutual interference that may result in the so-called *packet collisions*. The LBT schemes can differ in the way the CCA is performed and in the adopted behavior in case the channel is sensed busy.
The three most common methods to perform the CCA are the following.
- *Energy detection* (ED). The channel is detected as busy if the electromagnetic energy on the channel is above a given ED threshold.
- *Carrier sense* (CS). The channel is reported as busy if the device detects a signal with modulation and spreading characteristics compatible with those used for transmission, irrespective of the signal energy.
- *Carrier sense with energy detection* (CS+ED). In this case, a logical combination of the above methods is used, where the logical operator can be AND or OR.
The IEEE 802.15.4 standard supports all these CCA methods, along with pure ALOHA and two other modes specific for ultra-wideband communications. In an unslotted system, the backoff procedure for the IEEE 802.15.4 CCA mechanism tries to adapt to the channel congestion by limiting the rate at which subsequent CCAs are performed for the same message. If the number of consecutive backoffs exceeds a given threshold, the message is discarded. Details about the CCA procedure in IEEE 802.15.4 networks can be found in [@15.4], together with recommendations about the ED threshold and CCA detection time.
The regulatory framework {#reg}
========================
The use of unlicensed frequency bands by radio emitters is subject to regulations that are intended to favor the coexistence of a multitude of heterogeneous radio transceivers in the same frequency bands, limiting the mutual interference and avoiding any monopolization of the spectrum by single devices. The radio emitters operating in the ISM frequency bands are typically referred to as “Short Range Devices.” However, the ERC Recommendation 70-03, emanated by the CEPT, specifies that *The term Short Range Device (SRD) is intended to cover the radio transmitters which provide either uni-directional or bi-directional communication which have low capability of causing interference to other radio equipment.* Despite the name, there is no explicit mention of the actual coverage range of such technologies. Therefore, long-range technologies operating in the ISM bands, such as Sigfox or LoRa, are still subject to the same regulatory constraints that apply to the actual short range technologies, as IEEE 802.15.4, Bluetooth, IEEE 802.11, and so on.
In the European Union, the European Commission designated the CEPT to define technical harmonization directives for the use of the radio spectrum. In 1988, under the patronage of the CEPT, the European Telecommunications Standards Institute (ETSI) was created to develop and maintain Harmonized Standards for telecommunications.
In the unlicensed radio spectrum at 868 MHz, the ETSI mandates a duty cycle limit between 0.1% and 1% over a 1 hour interval for devices that do not adopt LBT [@etsi-2012]. Only very specific applications, such as wireless audio, are allowed to ignore the duty cycle limitation. The duty cycle constraint can be relaxed by employing an LBT access scheme together with the Adaptive Frequency Agility (AFA), i.e., the ability to dynamically changing channel [@etsi-2012]. Devices with LBT and AFA capabilities, in fact, are only subject to a 2.8% duty cycle limitation for any 200 kHz spectrum. An example of technology that adopts the LBT approach is the IEEE 802.15.4 that, however, does not perfectly match the ETSI specifications, since its channel sensing period is shorter than that mandated by ETSI, which is between $5$ ms and $10$ ms, depending on the used bandwidth [@etsi-2012]. Instead, the recommendations on the LBT sensitivity, which shall be between $-102$ dBm and $-82$ dBm, are usually satisfied by commercial transceivers.
Due to the adoption by the European Union of a new set of rules for the radio equipments, called Radio Equipment Directive (RED) [@red], ETSI is reviewing the related Harmonized Standards. However, devices that are compliant with the previous Radio and Telecommunication Terminal Equipment (R&TTE) Directive [@rtte] can be placed on the market until June 17, 2017. Furthermore, devices that do not satisfy the constraints imposed by the Harmonized Standards can still be commercialized, but subject to a more comprehensive certification procedure attesting that the device meets the essential requirements of the European Directives [@red]. The latest draft version of the ETSI Harmonized Standards [@etsi-2016] includes some changes on the medium access procedures. In particular, the LBT technique is generalized as a *polite spectrum access* technique, while AFA is no more required. Furthermore, the LBT ED threshold has been relaxed, while the minimum CCA listening period has been increased.
The agency designated to regulate radio communications in the USA is the FCC, which also grants permits for the use of licensed radio spectrum and emanates regulations for wired communications. The FCC regulation does not impose any duty cycle restrictions to emitters operating in the 902–928 MHz band, but limits the maximum transmit power, for non-frequency hopping systems, to $-1.25$ dBm [@fcc-part15], which is significantly lower than the 14 dBm allowed by ETSI.
Performance analysis {#perf}
====================
ALOHA schemes and channel sensing techniques have been comprehensively modeled and their performance limits in terms of throughput and capacity are well understood (see, e.g., [@stochastic; @kaynia], just to cite few). However, the use of different spreading techniques and/or modulation-&-coding-schemes to cope with the interference and to trade transmission speed for reliability, the large coverage range enabled by the LPWA technologies, the total reuse of the same frequency bands by different technologies, and the limitations imposed by the regulations to the channel access, raise the question on how effective are the classical uncoordinated channel access techniques to adequately support the expected growth of the IoT services.
In this section we shed some light on these aspects by presenting a simulation analysis of the performance achieved by ALOHA-based (specifically, pure ALOHA and HYB) and LBT access schemes in the simplest IoT scenario sketched in Figure \[fig:scenario\]: a gateway (GW) receiving packets from a multitude of peripheral devices randomly spread over a wide area. Despite its simplicity, this scenario embodies most of the problems that can be expected in a real IoT deployment based on long-range technologies. In particular, we are interested in investigating how the distance from the gateway may impact the performance experienced by the node, with and without multirate capability and using either ALOHA or LBT techniques. ALOHA-based access schemes, in fact, allow the maximum energy saving in light traffic conditions, since they avoid the (even small) energy cost involved in carrier sensing. On the other hand, nodes farther away from the gateway are likely more prone to transmission failure due to interference, which however can potentially be mitigated by the use of LBT. Furthermore, the adoption of rate adaptation techniques is expected to increase the system capacity by reducing the transmit time of nodes closer to the gateway that not only will experience a lower interference probability, but will also have the chance to transmit more packets within the duty cycle limitations. It is hence interesting to investigate how much of such a performance gain will be transferred to the more peripheral nodes, and whether the LBT techniques can further improve performance in a significant manner.
Simulation scenario
-------------------
[llr]{} &\
Spatial node density & $\lambda_s$ & $10^{-3}$ nodes/m$^2$\
Packet generation rate & $\lambda_t$ & 0.01 packets/s\
Transmission power & $P_\mathrm{TX}$ & 14 dBm\
Transmission frequency & $f$ & 868 MHz\
Path loss coefficient & $A$ & 36.36 m$^{-1}$\
Path loss exponent & $\beta$ & 3.5\
Packet length & $L$ & 240 bit\
Transmission bitrates & $\mathcal{R}$ & $\{0.5, \ldots,100\}$ kbit/s\
Bandwidth & $B_W$ & 400 kHz\
Noise spectral density & $N_0$ & $2\cdot10^{-20}$ W/Hz\
Duty cycle & $\delta_T$ & 1%\
Circuit power & $P_c$ & 16 dBm\
Sensing time & $T_s$ & 0.4 ms\
& & 3.98 $\mu$J (LBT)\
& & 0.2 mJ (LBT+ETSI)\
Smoothing parameter& $\alpha$ &0.1\
Target outage probability for RA & $p^*$ & 0.05\
\
Frame duration& $T_W$ & 60 s\
Number of reservation slots in a frame & $N_{RM}$ & 80\
Reservation message size & $L_{RM}$ & 24 bits\
Reservation message transmit rate & $R_{RM}$ & 500 bit/s\
Beacon duration & $T_B$ & 0.12 s\
Resource notification message duration& $T_{RA}$ & 3.84 s\
![Above: simulation scenario, with multiple transmitters scattered around the common receiver (GW). Below: example of signal transmissions by nodes A and B, using different bitrates, and of received signal power at the gateway.[]{data-label="fig:scenario"}](scenario.pdf){width="0.75\linewidth"}
In our simulations we consider a propagation model given by the product of the channel gain, $\gamma(d)=(Ad)^{-\beta}$, which accounts for the power decay with the distance $d$ from the transmitter through the model parameters $A$ and $\beta$, and the Rayleigh fading gain, which is modelled as an exponential random variable with unit mean.
We consider a limited set of possible transmission rates, namely $\mathcal{R}=\{0.5,1,5,10,50,100\}$ kbit/s, and assume that a packet transmitted at rate $r \in \mathcal{R}$ is correctly decoded if the received signal energy over the total noise energy plus interference energy collected by the receiver during the packet reception time (i.e., the Signal-to-Interference-and-Noise Ratio, SINR) is above a certain threshold $\Gamma_{th}(r)$, which is determined from the Shannon channel capacity as $$\Gamma_{th}(r) = 2^{r/W}-1
\label{th}$$ where $W$ is the signal bandwidth.
For the single rate case (SR), we suppose that all nodes transmit with the lowest bitrate of 500 bit/s. For the multirate scenario, instead, we consider a simple rate-adaptation mechanism that keeps a moving-average estimate of the SINR (using a smoothing factor $\alpha$) and selects the rate $R$ so that the expected outage probability is not larger than $p^*=0.05$. To improve the energy efficiency, furthermore, we assumed no acknowledgement or retransmission mechanism is implemented, so that packets that are not successfully received are definitely lost.
The LBT scheme has been implemented based on the IEEE 802.15.4 specifications. The ED CCA threshold has been chosen to match the minimum signal power required to correctly receive a packet transmitted at the basic rate of $500$ bit/s. This value is compatible with the limits on the LBT threshold imposed by ETSI [@etsi-2012].
As exemplified in Figure \[fig:scenario\], transmitting nodes are distributed as for a spatial Poisson process of rate $\lambda_s$ \[devices/m$^2$\] over a circle with radius equal to the maximum coverage distance at the basic rate of $500$ bit/s. Each device generates messages of length $L$ according to a Poisson process of rate $\lambda_t$ \[packets/s\]. All messages are addressed to the gateway that is placed at the center of the circle.
The setting of all the simulation parameters is reported in Table \[tab:param\].
![$p_\mathrm{fail}$ for ALOHA and LBT, for single rate (SR) and rate adaptive (RA) cases, with 95% confidence intervals.[]{data-label="fig:pfail"}](pfail.pdf){width="1\linewidth"}
Transmission failure probability
--------------------------------
We define $p_\mathrm{fail}$ as the probability that a transmitted message (including reservation messages in case of HYB) is received with SINR below threshold and, hence, is not correctly decoded. For HYB we also include in the $p_\mathrm{fail}$ the transmission requests that are not accepted because of lack of slots in the transmission part of the frame. Note that, while we consider both the Single rate (SR) and Rate Adaptation (RA) versions of the pure-ALOHA and LBT schemes, for the HYB protocol we only consider the RA version, since this access scheme is more effective when packet transmissions have uneven duration. In Figure \[fig:pfail\] we report the failure probability for target nodes placed at increasing distances from the gateway. Red curves with circle markers refer to ALOHA, blue plain curves to LBT, and green dashed line with diamond markers to HYB. Solid and dashed lines have been associated to the SR and RA case, respectively.
For the SR case, we can see that the failure probability grows with the distance from the gateway, since nodes farther away have less SINR margin for successful decoding and are hence less robust to the interference produced by overlapping transmissions. In this case, carrier sense can indeed improve performance, even if the sensing range does not prevent the hidden node problem.
The downside of using LBT (not reported here for space constraints) is that up to 55% of the transmission attempts are aborted, in high traffic conditions, because the maximum number of CCAs is reached without finding an idle channel. The adoption of RA changes significantly the performance, smoothing out the differences between the two access protocols. Indeed, higher bitrates allow the nodes near the receiver to occupy the channel for a lower period of time, thus reducing the probability of overlapping with other transmissions and improving the performance of both access schemes. Note that the change of rate with the distance is reflected by the oscillation in the failure probability that, however, remains approximately below $1-p^*$.
Rather interestingly, HYB performs worse than the other schemes. The reason is that, in the considered scenario, the transmit time of reservation messages, always sent at the basic rate, is comparable to that of data packets sent at higher rates. Therefore, the reservation channel can become the system bottleneck. The overall channel occupancy of HYB is thus significantly higher than that of the other two schemes, yielding higher failure probability.
Energy efficiency
-----------------
Another key performance index in the IoT scenario is the *energy efficiency*, which is here defined as the ratio of the total number of bits successfully delivered to the gateway over the entire energy consumed by the node (including channel sensing and failed transmissions).
We modelled the power consumed during a transmission as the sum of a constant term, named circuit power, that represents the power used by the radio circuitry, and a term that accounts for the radiated power, which is called transmission power. When using LBT, we also add the power required to perform the ED CCA. Referring to the data-sheets of some off-the-shelf modules,[^2] we set the circuit power to 16 dBm, the transmit power to 14 dBm, the receive power to 13 dBm, and the CCA power to 10 dBm [@power-ed; @negri2005flexible].
[1]{} {width="1\linewidth"}
[1]{} {width="1\linewidth"}
\[fig:energy\]
In Figure \[fig:energy-sr\] we show the energy efficiency for ALOHA and LBT access schemes when varying the distance of the target node from the gateway, in the SR case. We can observe that peripheral nodes exhibit lower energy efficiency because of the larger number of failure transmissions, and that the carrier sensing mechanism can alleviate this problem. The black curve marked with crosses shows the results obtained when using the parameters imposed by ETSI in the CCA procedure. As it can be seen, the energy efficiency is slightly lower than that obtained with the parameters adopted by commercial technologies, which may suggest that ETSI recommendations in this regard are possibly too conservative.
The adaptive rate case is shown in Figure \[fig:energy-ra\], where we also show the performance achieved by HYB. We can observe that both ALOHA and LBT can reach very high efficiency for nodes near the receiver, since the higher bitrates that decrease the transmit energy and the failure probability. It is worth to note that the first factor is dominant for the energy efficiency. The benefit transfers to the nodes farther away from the gateway, though the performance gain progressively reduces with the distance from the transmitter.
We also observe that, for nodes closer to the gateway, LBT shows a non-negligible energy efficiency loss with respect to ALOHA, which is even more marked when adopting the ETSI parameters. This is clearly due to the energy cost of the carrier sense mechanism, which takes a time comparable with the packet transmission time when using high bitrates. Furthermore, as revealed by the analysis of the failure probability, the carrier sense mechanism is not really worth for nodes close to the gateway when using RA, considering also that it may yield packet drops due to the impossibility of finding the channel idle within the maximum number of carrier sensing attempts. This problem would be further exacerbated in case of overlapping cells. Therefore, the use of CCA appears to be fruitless, if not detrimental, for nodes close to the gateway when RA is enabled.
Finally, we observe that the energy efficiency of HYB is the worst, being affected by both the higher failure probability observed in Figure \[fig:pfail\] and the higher energy consumption due to the transmission of resource messages and the reception of beacons. This inefficiency is more marked for nodes near the receiver, where the energy spent on control messages is actually greater than that used for the high-rate transmissions of small data packets.
Coexistence issues
------------------
![Aggregated throughput for each channel access method in the single and adaptive rate scenarios, with 95% confidence intervals.[]{data-label="fig:coexistence"}](mixed-throughput.pdf){width="1\linewidth"}
![Succesfully received bits per unit of consumed energy for each channel access method, in the single and adaptive rate scenarios, with 95% confidence intervals.[]{data-label="fig:coexistence-energy"}](mixed-energy.pdf){width="1\linewidth"}
Another important question regards the coexistence in the same area of nodes using LBT and ALOHA access schemes.
Figure \[fig:coexistence\] and Figure \[fig:coexistence-energy\] report the throughput of the two access methods, defined as the overall rate of successful packet transmissions, and the energy efficiency. Curves for ALOHA (respectively LBT) have been obtained by fixing the spatial density of this type of nodes to 0.001 nodes/$\mathrm{m}^2$ and increasing the spatial density of LBT (respectively ALOHA) nodes from $10^{-5}$ to $10^{-2}$ nodes/$\mathrm{m}^2$.
Results in Figure \[fig:coexistence\] show that the performance of ALOHA nodes is not impacted by an increase in the number of LBT nodes, while the latter suffer strong performance degradation due to the CCA mechanism that aborts a transmission attempt when the channel is sensed busy for a given number of successive attempts. We can also see that the use of multiple transmission rates can only slightly alleviate the problem, but the fragility of the LBT mechanism in presence of ALOHA traffic still remains. Similar observations can be drawn for the energy efficiency results. In both cases, the use of RA improves the energy efficiency quite significantly.
Conclusions {#con}
===========
In this work, we presented an overview of the three main uncoordinated channel access sensing schemes, namely pure ALOHA, HYB, and LBT, in an IoT scenario. We compared the performance of these schemes in terms of probability of successful transmission and energy efficiency, by considering the duty-cycle limitation for ALOHA, the control packets for HYB, and the CCA procedure for LBT as mandated by the international regulation frameworks.
From this analysis, it appears clear that adding rate adaptation capabilities is pivotal to maintain reasonable level of performance when the coverage range and the cell load increase. Moreover, we observed that LBT generally yields lower transmission failure probability, though packet dropping events may occur because the channel is sensed busy for a certain number of consecutive CCA attempts. This impacts on the actual energy efficiency of the LBT access scheme, which may turn out to be even smaller than that achieved by ALOHA schemes. Furthermore, we also observed that LBT performance undergoes severe degradation when increasing the number of ALOHA devices in the same cell, again because of the channel-blockage effect caused by the other transmitters. Finally, the HYB scheme proves ineffective in the considered scenario, since the reservation channel becomes the system bottleneck with short data packets. Nonetheless, hybrid solutions that adopt LBT for peripheral nodes and ALOHA for nodes closer to the receiver, or apply rate adaptation also to the reservation phase, can potentially lead to a general performance improvement of the system. This analysis, however, is left to future work.
[Daniel Zucchetto]{} received the Bachelor degree in Information Engineering in 2012 and the Master degree in Telecommunication Engineering in 2014, both from the University of Padova, Italy. Since October 2015 he is a Ph.D. student at the Department of Information Engineering of the University of Padova, Italy. His research interests include Low-Power Wide-Area Network technologies and next generation cellular networks (5G), with particular focus on their application to the Internet of Things.
[Andrea Zanella]{} (S’98-M’01-SM’13) is Associate Professor at the University of Padova, Padova, Italy. He has authored more than 130 papers, four books chapters and three international patents in multiple subjects related to wireless networking and Internet of Things. Moreover, he serves as Editor for many journals, included the IEEE Internet of Things Journal, and the IEEE Transactions on Cognitive Communications and Networking.
[^1]: The two terms will be used interchangeably in this paper.
[^2]: Atmel AT86RF212B, Texas Instruments CC1125 and CC1310, and Semtech SX1272 modules.
|
---
abstract: 'Consider a wireless cellular network consisting of small, densely scattered base stations. A user $u$ is [*uniquely covered*]{} by a base station $b$ if $u$ is the only user within distance $r$ of $b$. This makes it possible to assign the user $u$ to the base station $b$ without interference from any other user $u''$. We investigate the maximum possible proportion of users who are uniquely covered. We solve this problem completely in one dimension and provide bounds, approximations and simulation results for the two-dimensional case.'
author:
- 'M. Haenggi'
- 'A. Sarkar'
title: Unique coverage in Boolean models
---
Introduction
============
Consider a wireless cellular network consisting of small, densely scattered base stations, each with limited processing capability. (In [@net:Caire16arxiv] and the related engineering literature, the small base stations are called [*remote radio heads*]{}.) In such a network, a user $u$ is [*uniquely covered*]{} by a base station $b$ if $u$ is the only user within distance $r$ of $b$. This makes it possible to assign the user $u$ to the base station $b$ without interference from any other user $u'$. Ideally, we would like to assign a base station to every user. However, the underlying stochastic geometry will prevent this. In this paper, we investigate the maximum possible proportion of users who can be uniquely assigned base stations, as the communication range $r$ varies, for each pair of densities of both users and base stations.
Although we have just referred to [*two*]{} densities, only their ratio is significant; in other words, the model can be scaled so that we expect one user per unit area. Accordingly, we set the intensity of users to be one. Thus the only parameters we need to consider are the density $\mu$ of base stations, and the range $r$. Moreover, we note that our analysis also solves the problem, considered in [@net:Caire16arxiv], of uniquely assigning users to base stations (so as to avoid [*pilot contamination*]{}); to see this, simply interchange the roles of users and base stations.
All logarithms in this paper are to base $e$.
Model
=====
Our model is as follows. Fix $r>0$, and let ${{\mathcal P}}$ and ${{\mathcal P}}'$ be independent Poisson processes, of intensities $\mu$ and 1 respectively, in ${\mathbb{R}}^d$. The main case of interest is $d=2$. The points of ${{\mathcal P}}$ represent the base stations, and the points of ${{\mathcal P}}'$ represent the users. A user $u\in{{\mathcal P}}'$ is uniquely covered by a base station $b\in{{\mathcal P}}$ if firstly $||b-u||<r$, and secondly $||b-u'||\ge r$ for every other user $u'\in{{\mathcal P}}'$. We wish to calculate (or estimate) the proportion $p^d(\mu,r)$ of users who are uniquely covered by base stations; note that this proportion is also the probability that an arbitrary user is uniquely covered by a base station.
A general result
================
In order to state our main result, we need some notation. First, for simplicity, we will initially consider just the case $d=2$. Next, let $D=D(O,r)$ be the fixed open disc of radius $r$, centered at the origin $O$. Write $f_r(t)$ for the probability density function of the fraction $t$ of $D$ which is left uncovered when discs of radius $r$, whose centers are a unit intensity Poisson process, are placed in the entire plane ${\mathbb{R}}^2$. There is in general no closed-form expression for $f_r(t)$; however, the function is easy to estimate by simulation.
In two dimensions, we have $$\label{basic}
p^2(\mu,r)=\int_0^1(1-e^{-\mu\pi r^2t})f_r(t)\,dt.$$
The main idea of the proof is to put down the users first, and then, for a fixed user $u$, calculate the probability that a base station $b$ “lands" in such a way that $u$ is uniquely covered by $b$. To this end, place a disc $D(u,r)$ of radius $r$ around each user $u$, and then a fixed user $u$ is uniquely covered if there is a base station $b\in D(u,r)$ such that $b\not\in D(u',r)$ for all other users $u'\not= u$. Let $X$ be the random variable representing the uncovered area fraction of $D(u,r)$ when all the other discs $D(u',r)$ are placed randomly in the plane. Then $${\mathbb{P}}(u{\rm \ is\ covered}\mid X=t)=1-e^{-\mu\pi r^2t},$$ since for $u$ to be covered we require that some base station $b$ lands in the uncovered region in $D(u,r)$, which has area $\pi r^2t$. (Here, by “uncovered", we mean “uncovered by the union of all the other discs $\bigcup_{u'\not= u}D(u',r)$".) Consequently, $$p^2(\mu,r)=\int_0^1 {\mathbb{P}}(u{\rm \ is\ covered}\mid X=t)f_r(t)\,dt=\int_0^1(1-e^{-\mu\pi r^2t})f_r(t)\,dt,$$ as required.
The same argument yields the following result for the general case. For $d\ge 1$, write $D^d(O,r)$ for the $d$-dimensional ball of radius $r$ centered at the origin $O$, and $f^d_r(t)$ for the probability density function of the fraction $t$ of $D^d(O,r)$ which is left uncovered when balls of radius $r$, whose centers are a unit intensity Poisson process, are placed in ${\mathbb{R}}^d$. Finally, let $V_d$ be the volume of the unit-radius ball in $d$ dimensions.
In $d$ dimensions, we have $$p^d(\mu,r)=\int_0^1(1-e^{-\mu V_d r^dt})f^d_r(t)\,dt.$$
The case $d=1$
==============
Unfortunately, $f^d_r(t)$ is only known exactly when $d=1$. The result is summarized in the following lemma, in which for simplicity we consider the closely related function $g_r(s):=f^1_r(s/2r)$, which represents the total uncovered length in $(-r,r)$.
In one dimension, we have $$g_r(s):=f^1_r(s/2r)=
\begin{cases}
1-e^{-2r}(1+2r)&\text{\rm point mass at $s=0$}\\
(2+2r-s)e^{-(2r+s)}&0<s<2r\\
e^{-4r}&\text{\rm point mass at $s=2r$}.\\
\end{cases}$$
Consider the interval $I_r:=D^1(O,r)=(-r,r)$. The uncovered length $U$ of $I_r$ is determined solely by the location of the closest user $u_l$ to the left of the origin $O$, and the closest user $u_r$ to the right of $O$. Suppose indeed that $u_l$ is located at $-x$ and that $u_r$ is located at $y$. Then it is easy to see that if $x+y\le 2r$, we have $U=0$; in other words, all of $I_r$ is covered by $D(u_l,r)\cup D(u_r,r)$ when $x+y\le 2r$. At the other extreme, if both $x\ge 2r$ and $y\ge 2r$, then $U=2r$; in this case the entire interval $I_r$ is left uncovered by $D(u_l,r)\cup D(u_r,r)$, and so by the union $\bigcup_{u}D(u,r)$. In general, a lengthy but routine case analysis gives $$U=
\begin{cases}
0&x+y\le 2r\\
x+y-2r&x+y\ge 2r,x\le 2r,y\le 2r\\
x&0\le x\le 2r,y\ge 2r\\
y&0\le y\le 2r,x\ge 2r\\
2r&x\ge 2r,y\ge 2r.\\
\end{cases}$$ This immediately yields the point masses of $g_r(s)$, since $x+y$ has a gamma distribution of mean 2, and $x$ and $y$ are each exponentially distributed with mean 1. For $0<s<2r$ we find, using the above expression, that $$g_r(s)=2e^{-2r}\cdot e^{-s}+\int_{s}^{2r}e^{-x}e^{-(2r+s-x)}\,dx=(2+2r-s)e^{-(2r+s)},$$ completing the proof of the lemma.
Using this lemma, we obtain the following expression for $p^1(\mu,r)$.
In one dimension, we have $$p^1(\mu,r)=\frac{\mu e^{-2r}(\mu+2r+2r\mu)-\mu^2e^{-2r(2+\mu)}}{(1+\mu)^2}.$$
From Theorem 2 and Lemma 3 we have $$\begin{aligned}
p^1(\mu,r)&=\int_0^{2r}g_r(s)(1-e^{-\mu s})\,ds\\
&=e^{-4r}(1-e^{-2r\mu})+\int_0^{2r}(2+2r-s)e^{-(2r+s)}(1-e^{-\mu s})\,ds\\
&=\frac{\mu e^{-2r}(\mu+2r+2r\mu)-\mu^2e^{-2r(2+\mu)}}{(1+\mu)^2}.\end{aligned}$$
$p^1(\mu,r)$ is illustrated in Fig. \[fig:one\_dim\]. The value of $r$ that maximizes $p^1$ is $$r_{\rm opt}(\mu)=\frac{1+\mathcal{W}(\mu(\mu+2)e^{-1})}{2\mu+2} ,
\label{r_opt}$$ where $\mathcal{W}$ is the (principal branch of the) Lambert W-function. It is easily seen that $r_{\rm opt}(0)=1/2$ and that $r_{\rm opt}$ decreases with $\mu$.
![Fraction of users that are uniquely covered in one dimension. The circle indicates the maximum $p^1(\mu,r_{\rm opt})$, where $r_{\rm opt}$ is given in .[]{data-label="fig:one_dim"}](one_dim){width="8cm"}
The case $d=2$
==============
In two dimensions, although the function $f^2_r(t)$ is currently unknown, it can be approximated by simulation, and then the integral can be computed numerically. While this still involves a simulation, it is more efficient than simulating the original model itself, since $f^2_r$ can be used to determine the unique coverage probability for many different densities $\mu$ (and the numerical evaluation of the expectation over $X$ is very efficient). The resulting unique coverage probability $p^2(\mu,r)$ is illustrated in Fig. \[fig:two\_dim\]. The maxima of $p^2(\mu,r)$ over $r$, achieved at $p^2(\mu,r_{\rm opt}(\mu))$, are highlighted using circles. Interestingly, $r_{\rm opt}(\mu)\approx 4/9$ for a wide range of values of $\mu$; the average of $r_{\rm opt}(\mu)$ over $\mu\in [0,10]$ appears to be about $0.45$.
The simulated $f_r(t)$ is shown in Fig. \[fig:pdf\_fr\] for $r=3/9,4/9,5/9$. Remarkably, the density $f_{4/9}(t)$ is very close to uniform (except for the point masses at $0$ and $1$). If the distribution were in fact uniform, writing $v={\mathbb{E}}(X)=e^{-\pi r^2}=e^{-16\pi/81}\approx 0.538$, we would have $$\hat f_{4/9}(t)=\begin{cases}
1+v^4-2v\approx 0.008&\text{\rm point mass at $t=0$}\\
2(v-v^4)\approx 0.908 &0<t<1\\
v^4 \approx 0.084 &\text{\rm point mass at $t=1$}.\\
\end{cases}
\label{f49}$$ Here, $v^4=e^{-4\pi r^2}$ is the probability that no other user is within distance $2r$, in which case the entire disc $D(O,r)$ is available for base stations to cover $O$. The constant $2(v-v^4)$ is also shown in Fig. \[fig:pdf\_fr\] (dashed line). Substituting in yields the following approximation to $p^2(\mu,4/9)$ and to $p^2(\mu,r_{\rm opt})$: $$p^2(\mu,r_{\rm opt}(\mu))\approx 2(v-v^4)\left(1-\frac{1-e^{-c}}{c}\right)+v^4 (1-e^{-c}), \quad c=\mu \pi (4/9)^2=-\mu\log v.
\label{p2_approx}$$ This approximation is shown in Fig. \[fig:p2\_opt\], together with the exact numerical result. For $\mu\in [3,7]$, the curves are indistinguishable.
For small $\mu$, $p^2(\mu,r)\approx e^{-\pi r^2}(1-e^{-\mu\pi r^2})$ (see [Theorem \[t:p2\_bound\]]{} immediately below), and so $$r_{\rm opt}(\mu)\approx \sqrt{\frac{\log(1+\mu)}{\mu\pi}}\to \pi^{-1/2}$$ as $\mu\to 0$.
![Fraction of users that are uniquely covered in two dimensions. The circles indicate the maxima $p^2(\mu,r_{\rm opt})$.[]{data-label="fig:two_dim"}](two_dim){width="8cm"}
![Simulated densities $f_r(t)$ for $r=3/9,4/9,5/9$ in two dimensions. The vertical lines near $0$ and $1$ indicate the point masses. The dashed line is the uniform approximation for $r=4/9$.[]{data-label="fig:pdf_fr"}](pdf_fr){width="8cm"}
![Maximum fraction of users that are uniquely covered in two dimensions. The dashed line is the approximation in .[]{data-label="fig:p2_opt"}](p2_opt){width="8cm"}
Next, we turn to bounds and approximations. It is straightforward to obtain a simple lower bound for $p^2(\mu,r)$.
\[t:p2\_bound\] $p^2(\mu,r)\ge e^{-\pi r^2}(1-e^{-\mu\pi r^2})$.
A given user is covered if there is a base station within distance $r$ (this event has probability $1-e^{-\mu\pi r^2}$), and if there is no other user within distance $r$ of that base station (this event has probability $e^{-\pi r^2}$). These last two events are independent.
This bound should become tight as $\mu\to 0$ (with $r$ fixed), or as $r\to 0$ (with $\mu$ fixed), since, in those limiting scenarios, if there is a base station within distance $r$ of a user, it is likely to be the only such base station.
Finally, here is an approximation for $p^2(\mu,r)$ when $r$ is large. We use standard asymptotic notation, so that $f(x)\sim g(x)$ as $x\to\infty$ means $f(x)/g(x)\to1$ as $x\to\infty$. In our case, we will have $r\to\infty$ with $\mu$ fixed.
\[t:p2\_asymptote\] As $r\to\infty$ with $\mu$ fixed, $p^2(\mu,r)\sim \mu\pi r^2e^{-\pi r^2}$.
(Sketch) We recall Theorem 1, which states that $$p^2(\mu,r)=\int_0^1(1-e^{-\mu\pi r^2t})f_r(t)\,dt,$$ and attempt to approximate $f_r(t)$ as $r\to\infty$.
To this end, it is convenient to describe the geometry of the union of discs $\bigcup_{u\in{{\mathcal P}}'}D(u,r)$ in some detail. Such coverage processes have been studied extensively in the mathematical literature [@net:Gilbert65; @net:Hall88; @net:Janson86; @net:Meester96]; our approach follows that in [@net:Balister09springer; @net:Balister10aap]. The main idea is to consider the boundaries $\partial D(u,r)$ of the discs $D(u,r)$, rather than the discs themselves. Consider a fixed disc boundary $\partial D(u,r)$. This boundary intersects the boundaries $\partial D(u',r)$ of all discs $D(u',r)$ whose centers $u'$ lie at distance less than $2r$ from $u$. There are an expected number $4\pi r^2$ of such points $u'\in{{\mathcal P}}'$, each contributing two intersection points $\partial D(u,r)\cap\partial D(u',r)$, and each intersection is counted twice (once from $u$ and once from $u'$). Therefore we expect $4\pi r^2$ intersections of disc boundaries per unit area over the entire plane; note that these intersections [*do not*]{} form a Poisson process, since they are constrained to lie on various circles.
The next step is to move from intersections to regions. The disc boundaries partition the plane into small “atomic" regions. Drawing all the disc boundaries in the plane yields an infinite plane graph, each of whose vertices (disc boundary intersections) has four curvilinear edges emanating from it. Each such edge is counted twice, once from each of its endvertices, so there are almost exactly twice as many edges as vertices in any large region $R$. It follows from Euler’s formula $V-E+F=2$ for plane graphs [@net:Bollobas98] that the number of atomic regions in $R$ is asymptotically the same as the number of intersection points in $R$. Moreover, each vertex borders four atomic regions, so that the average number of vertices bordering an atomic region is also four. Note that this last figure is just an average, and that many atomic regions will have less than, or more than, four vertices on their boundaries.
The third step is to return to the discs themselves and calculate the expected number of [*uncovered*]{} atomic regions per unit area. It is most convenient to calculate this in terms of uncovered intersection points. A fixed intersection point is uncovered by $\bigcup_{u\in{{\mathcal P}}'}D(u,r)$ with probability $e^{-\pi r^2}$ (using the independence of the Poisson process), so we expect $4\pi r^2 e^{-\pi r^2}$ uncovered intersections, and so $\pi r^2 e^{-\pi r^2}$ uncovered regions, per unit area in $R$. Therefore the expected number of uncovered regions in $D(u,r)$, which has area $\pi r^2$, is $\alpha=(\pi r^2)^2 e^{-\pi r^2}\to 0$.
How large are these uncovered atomic regions? To answer this, recall that the expected uncovered area in $D(u,r)$ is $\pi r^2 e^{-\pi r^2}$. The uncovered atomic regions form an approximate Poisson process, so that the probability of seeing two uncovered regions in $D(u,r)$ is negligible. Now let $X_r$, with density function $f_r(t)$, be the uncovered area fraction in $D(u,r)$. We have $E(X_r)=e^{-\pi r^2}$, but ${\mathbb{P}}(X_r=0)\sim e^{-\alpha}\sim 1-\alpha$. Writing now $Y_r$ for the expected uncovered area fraction in $D(u,r)$ conditioned on $X_r>0$, and $h_r(t)$ for the density of $Y_r$, we see that ${\mathbb{E}}(Y_r)\sim\alpha^{-1}{\mathbb{E}}(X_r)=(\pi r^2)^{-2}$. In other words, if there is uncovered area in $D(u,r)$, it occurs in one atomic region of expected area $(\pi r^2)^{-1}$. Consequently, we have $$\begin{aligned}
p^2(\mu,r)&=\int_0^1(1-e^{-\mu\pi r^2t})f_r(t)\,dt
\sim\alpha\int_0^1(1-e^{-\mu\pi r^2t})h_r(t)\,dt\\
&\sim\alpha\mu\pi r^2\int_0^1th_r(t)\,dt
=\alpha\mu\pi r^2{\mathbb{E}}(Y_r)
\sim\alpha\mu(\pi r^2)^{-1}
=\mu\pi r^2e^{-\pi r^2}.\end{aligned}$$
Note that this is the same result that we would have obtained from the incorrect argument that $X_r$ is concentrated around its mean, whereas in fact its density $f_r(t)$ has a large point mass at $t=0$. Indeed, the thrust of the above argument is that, for the relevant range of $t$ (namely, for $t=O((\pi r^2)^{-2})$), $1-e^{\mu\pi r^2t}-\mu\pi r^2 t=O(r^4t^2)=O(r^{-4})$, which is asymptotically negligible compared to the remaining terms.
Fig. \[fig:p2\_bounds\] shows $p^2(\mu,r)$, together with the lower bound from [Theorem \[t:p2\_bound\]]{} and the asymptote from [Theorem \[t:p2\_asymptote\]]{}. As predicted, [Theorem \[t:p2\_bound\]]{} is close to the truth when $r$ is small, while [Theorem \[t:p2\_asymptote\]]{} is more accurate for large values of $r$.
![$p^2(\mu,r)$ for $\mu=0.05,0.5,5$, with the lower bound from [Theorem \[t:p2\_bound\]]{} and the approximation from [Theorem \[t:p2\_asymptote\]]{}. (Left) linear scale. (Right) logarithmic scale.[]{data-label="fig:p2_bounds"}](p2_bounds_linear){width="\textwidth"}
![$p^2(\mu,r)$ for $\mu=0.05,0.5,5$, with the lower bound from [Theorem \[t:p2\_bound\]]{} and the approximation from [Theorem \[t:p2\_asymptote\]]{}. (Left) linear scale. (Right) logarithmic scale.[]{data-label="fig:p2_bounds"}](p2_bounds_log2){width="\textwidth"}
Both these last two results generalize to the $d$-dimensional setting in the obvious way; for simplicity we omit the details.
Conclusions
===========
In this paper, we have investigated a natural stochastic coverage model, inspired by wireless cellular networks. For this model, we have studied the maximum possible proportion of users who can be uniquely assigned base stations, as a function of the base station density $\mu$ and the communication range $r$. We have solved this problem completely in one dimension and provided bounds, approximations and simulation results for the two-dimensional case. We hope that our work will stimulate further research in this area.
Acknowledgements
================
We thank Giuseppe Caire for bringing this problem to our attention. This work was supported by the US National Science Foundation \[grant CCF 1525904\].
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---
abstract: 'Using a sample of nearly 140,000 red clump stars selected from the LAMOST and Gaia Galactic surveys, we have mapped mean vertical velocity $\overline{V_{z}}$ in the $X$–$Y$ plane for a large volume of the Galactic disk (6 $< R < 16$kpc; $-20 <\phi<50^{\circ}$ ; $|Z| < 1$kpc). A clear signature where $\overline{V_{z}}$ increases with $R$ is detected for the chemically thin disk. The signature for the thick disk is however not significant, in line with the hot nature of this disk component. For the thin disk, the warp signature shows significant variations in both radial and azimuthal directions, in excellent agreement with the previous results of star counts. Fitting the two-dimensional distribution of $\overline{V_z}$ with a simple long-lived static warp model yields a line-of-node angle for this kinematic warp of about $12.5^{\circ}$, again consistent with the previous results.'
author:
- 'X.-Y. Li'
- 'Y. Huang'
- 'B.-Q. Chen'
- 'H.-F. Wang'
- 'W.-X. Sun'
- 'H.-L. Guo'
- 'Q.-Z. Li'
- 'X.-W. Liu'
title: |
Mapping the Galactic disk with the LAMOST and Gaia Red clump sample:\
IV: the kinematic signature of the Galactic warp
---
Introduction
============
Disk warping in the outer regions of spiral galaxies ($>50\%$) is a very common phenomenon (e.g. Saha et al. 2009). In general, the inner disk of a spiral galaxy is largely flat whereas the outskirts show significant warp signature. The warp amplitude increases strongly with radius, reaching as large as a few times of the inner disk scale height. Being a typical spiral galaxy, the Milky Way (hereafter MW) also shows clear warp in the outer disk. The Galactic warp was first detected by Kerr (1957) with H [i]{} 21-cm line observation. This was confirmed later by Weaver & Williams (1974) and Henderson (1979). Not only the neutral gas, other components of the Galactic disk also show that the Galactic outer disk is strongly warped, including the stars (Efremov et al. 1981; Reed 1996; L[ó]{}pez-Corredoira et al. 2002b), the molecular clouds (Wouterloot et al. 1990) and the interstellar dust grains . Studies show that one part of the Galactic outer disk bends up from the Galactic plane to the north Galactic pole, whereas the other part bends down. Further studies indicate that the warp amplitude not only increases strongly with radius but also changes with azmithual angle. The line-of-node angle with respect to the Sun-Galactic Centre line is estimated to range between $-$5 and 26$^{\circ}$ by different groups using different tracers (e.g. L[ó]{}pez-Corredoira et al. 2002b; Momany et al. 2006; Chen et al. 2019b; Skowron et al. 2019).
Theoretically, warping of a spiral galaxy is generally interpreted as the response of the disk to perturbations. Specially, for our MW, the perturbations may come from: i) the interactions of the Galactic disk with nearby satellite galaxies (e.g. the Large and Small Magellanic Clouds or the Sagittarius dwarf galaxy; Weinberg 1995; Garc[í]{}a-Ruiz et al. 2002; Bailin 2003); ii) effects of the triaxial dark-matter halo [@1988MNRAS.234..873S; @1999ApJ...513L.107D]; or iii) the accretion of infalling intergalactic gas . While many scenarios have been proposed, the exact origin of the Galactic warp remains unclear. Further information of the kinematic signature of the Galactic warp would be invaluable to clarify the situation.
Prior to the first Gaia data release, several studies (e.g. Miyamoto et al. 1988; L[ó]{}pez-Corredoira et al. 2014) have attempted to unravel the kinematic signature of the Galactic warp, using catalogs of ground-based proper motion measurements. The results are inconclusive due to the limited accuracy of proper motions employed. With the release of Gaia DR1 , accurate measurements of proper motions and parallaxes for over two million stars become available. With the data, Poggio et al. (2017) have detected signature of kinematic warp by using nearby OB stars. With the Gaia DR1 and the spectroscopic information from the RAVE and LAMOST surveys, Sch[ö]{}nrich & Dehnen (2018) and Huang et al. (2018) have calculated values of vertical velocity $V_{z}$, azimuthal velocity $V_{\phi}$ and vertical angular momentum $L_{z}$ for stars in the Solar neighborhood. They find that mean vertical velocity $\overline{V_{z}}$ increases with $V_{\phi}$, $L_{z}$ and guiding center radius $R_{g}$. The trends are consistent with the predictions of long-lived Galactic warp model. Recently, the Gaia DR2 has been released, providing accurate parallax and proper motion measurements of about 1.3 billion stars. With the new data, accurate kinematics of the Galactic disk has been mapped by Gaia Collaboration et al. (2018c). Poggio et al. (2018; hereafter P18) use two samples, one of stars of the upper main sequence and another of red giant stars, and study the kinematic signature of the Galactic warp in the $X$–$Y$ plane out to a distance of 7kpc from the Sun. However, for their giant sample, only 24 per cent of the stars have line-of-sight velocities. The distances, estimated from the Gaia parallaxes, for more distant stars may also suffer from serious systematics (e.g. Sch[ö]{}nrich et al. 2019). Recently, Huang et al. (2020; hereafter Paper I), based on data from the LAMOST and Gaia surveys, have published a sample of about 140,000 red clump (RC) stars with accurate measurements of distance, proper motions and stellar atmospheric parameters (effective temperature $T_{\rm eff}$, surface gravity log$g$ and metallicity \[Fe/H\]), line-of-sight velocity $V_{\rm los}$ and $\alpha$-element to iron abundance ratio \[$\alpha$/Fe\]. The sample allows one to study the warp signature over a large volume for both the chemically thin and thick populations.
The paper is organized as follows. In Section 2, we define the coordinate systems and describe the data used. The results are presented and discussed in Section 3. Finally, a summary is presented in Section 4.
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![The R–$\overline{V_{z}}$ diagram of 94,028 thin disk sample stars. The blue, orange, green, red, purple and magenta solid lines represent the mean vertical velocities of stars in azimuthal angle ranges $\phi$ $\in$ \[-15.0, -5.0\], \[-5.0, 0.0\], \[0.0, 5.0\], \[5.0, 10.0\], \[10.0, 17.0\] and \[17.0, 35.0\] deg, respectively, with the shaded areas representing the $\pm1\sigma$ uncertainties (estimated with a bootstrapping procedure) of the mean vertical velocities.](phic_bt.eps){width="3.5in"}
{width="6.in"}
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coordinate systems and data
===========================
Coordinate systems
------------------
In this paper, we use two coordinate systems. A Galactocentric cylindrical coordinate system ($R$,$\phi$,$z$), along with associated velocity components ($V_{R}$,$V_{\phi}$,$V_{z}$), is defined with $R$ the projected Galactocentric distance, $\phi$ the azimuthal angle increasing in the direction of Galactic rotation and $z$ the height above the Galactic plane in the direction of the north Galactic pole. The velocity components are calculated from the sky positions, distances, line-of-sight velocities and proper motions, using the standard transformations from Johnson & Soderblom (1987). We adopt Galactocentric distance of the Sun $R_{0}$ of 8.34 kpc [@2014ApJ...783..130R] and circular velocity at the Solar radius of $V_{c} (R_{0})=238$kms$^{-1}$ [@2016MNRAS.463.2623H]. We take Solar motions with respect to the Local Standard of Rest ($U_{\odot}$,$V_{\odot}$,$W_{\odot}$)$=$$(11.10,\,12.24,\,7.25)$kms$^{-1}$ [@2010MNRAS.403.1829S]. Other values of the Solar motions (e.g. Huang et al. 2015) are also tried and the results obtained are essentially the same. Also used is a right-handed Cartesian Galactocentric coordinate system ($X$,$Y$,$Z$), with $X$ pointing towards the Galactic center, $Y$ in the direction of Galactic rotation and $Z$ towards the north Galactic pole.
Data
----
LAMOST is a 4-metre quasi-meridian reflecting Schmidt telescope equipped with 4000 fibers distributed in a field of view of about 20 sq.deg. It can simultaneously collect spectra per exposure of upto 4000 objects, covering the wavelength range 3800–9000Å at a resolving power $R$ of about 1800 [@2012RAA....12.1197C]. The five-year Phase-I LAMOST Regular Surveys started in the fall of 2012 and completed in the summer of 2017. The scientific motivations and target selections of the surveys are described in detail in Deng et al. (2012), Zhao et al. (2012) and Liu et al. (2014). Atmospheric parameters ($T_{\rm eff}$, log$g$, \[Fe/H\]), line-of-sight velocity $V_{\rm los}$ and $\alpha$-element to iron abundance ratio \[$\alpha$/Fe\] of the targeted stars are derived with the LAMOST Stellar Parameter Pipeline at Peking University (LSP3; Xiang et al. 2015, 2017). The second data release of Gaia have been made available to the community since April 2018, providing accurate parallax and proper motion measurements for about 1.3 billion stars (Gaia Collaboration et al. 2018a). Typical uncertainties of the parallaxes are 0.04 mas for bright sources ($G< 14$mag), 0.1 mas at $G = 17$mag and 0.7 mas at $G = 20$mag. For the proper motions, typical uncertainties are 0.05, 0.2 and 1.2 mas yr$^{-1}$ at $G < 14$mag, $G = 17$mag and $G = 20$mag, respectively.
In the current work, a sample of nearly 140,000 RC stars has been used. The sample is described in Paper I, constructed with data from the LAMOST and Gaia surveys. Given the standard candle nature of RCs, distances of those stars have been measured with a typical accuracy of 5-10 per cent, preciser even than values yielded by the Gaia parallax measurements for stars beyond 3-4kpc. With the derived distances, line-of-sight velocities, proper motions, \[Fe/H\] and \[$\alpha$/Fe\] values for the LAMOST and Gaia RC sample stars, we have derived 3D positions and velocities for all the sample stars, and examine the velocity field of disk stars. The current work concentrates on the vertical velocity field of disk stars of different populations in a large disk volume. The spatial distribution of our sample stars is presented in Fig.1. The sample covers a large volume of the Galactic disk of $-16 \leq X \leq -4$kpc, $-3 \leq Y \leq 6$kpc and $|Z| \leq 3$kpc.
Results and discussion
======================
The mean vertical velocity field of the (outer) Galactic disk can be significantly perturbed in the long-lived warp model (e.g. Drimmel et al. 2000). With the current RC sample, we explore how the mean vertical velocity field varies with $R$ and in the disk plane (i.e. $X$-$Y$ plane) for the different stellar populations.
The kinematic warp of the Galactic disks
----------------------------------------
Before mapping the mean vertical velocity field, we first exclude sample stars with vertical velocity uncertainties $e_{V_{z}}$ (estimated for the individual stars with a Monte Carlo method) larger than 15kms$^{-1}$ and $|V_{z} - \overline{V_{z}}| > 3\sigma_{z}$. The latter cut is used to remove significant outliers in the vertical velocity distribution of our sample. The distribution of the remaining 133,061 stars in the \[Fe/H\]-\[$\alpha$/Fe\] plane is presented in the left panel of Fig.2. As the plot shows, a bimodal distribution is clearly seen. As in the previous studies (e.g. Bensby et al. 2005; Lee et al. 2011; Haywood et al. 2013), we define cuts to separate the two populations, one of the chemically thin disk and another of the thick disk in the plane. The cuts result in 94,028 and 5,212 chemically thin and thick disk stars within $|Z| < 1$kpc (close to the Galactic plane), respectively. The mean values of vertical velocity $\overline{V_z}$ as a function of $R$ for the chemically thin and thick disk stars are presented in the right panel of Fig.2. The plot shows that $\overline{V_z}$ of the chemically thin disk stars increases with $R$, from $-1.5$kms$^{-1}$ at $R \sim 8$kpc to $4.5$kms$^{-1}$ at $R \sim 13$kpc. The trend is similar to that reported in P18 (see their Fig.3).
In addition to what found for the chemically thin disk population, we have tried to detect the warp signature for the chemically thick disk population. Fig.2 shows that $\overline{V_z}$ of the chemically thick stars present a very weak positive trend with $R$, from $-1$kms$^{-1}$ at $R = 8$-10kpc to $2.5$kms$^{-1}$ at $R = 12$-13kpc. Due to the limited number of thick disk stars, the uncertainties of mean vertical velocities are large (the typical uncertainty is about $1.6$kms$^{-1}$). According to above analysis, only 2.2$\sigma$ detection is found for the thick disk population on the kinematic warp. In the future, more thick disk stars are required to reduce the random errors of mean $V_z$ to clarify whether there is a clear kinematic warp for the thick disk population. On the other hand, a weak/insignificant kinematic warp signature for the thick disk population is in line with the hot nature of orbits of the thick disk stars (i.e. of large velocity dispersions in all directions; e.g. Chiba & Beers 2000; Bensby et al. 2003; Parker et al. 2004). Because of their hot nature, the thick disk stars are less sensitive to the warp perturbations than the thin disk stars. Secondly, the large velocity dispersions (especially in the radial direction) of the thick disk population can smooth the warp signature along $R$ direction. To fully understand the weak/insignificant warp signature of the thick disk population, both observational efforts (by obtaining more thick disk stars) and theoretical dynamical modeling are required.
For the thin disk population, we extract the kinematic warp signature shown in Fig.2 in different azimuthal slices, from $-15$ to $35$ deg. The results are presented in Fig.3. The positive trend of $\overline{V_z}$ increasing with $R$, i.e. the kinematic warp signature, is seen in almost all the azimuthal slices. The amplitude of warp signature increases with azimuthal angle $\phi$ and reaches a maximum in slice $\phi \in$\[10, 17\] deg, and then decreases slightly in the last azimuthal slice. In the scenario of a long-lived static Galactic warp model, the variations of warp amplitude with $\phi$ presented here indicate an angle of line-of-node between 10 and 17 deg. To obtain a preciser estimate, we fit the distribution of $\overline{V_z}$ in the $X$-$Y$ plane with a naive long-lived static warp model in the next Subsection.
Fitting the $\overline{V_z}$ distribution with a simple model
-------------------------------------------------------------
To fit the kinematic warp with a theoretical model, the distribution of mean vertical velocity in the Galactic plane is presented in the panel (a) of Fig.4. The general trend of variations with $R$ and $\phi$ are similar to those found in P18. Theoretically, the Galactic warp can be simulated with either by a transient model (e.g. Christodoulou et al. 1993; Debattista & Sellwood 1999) or a long-lived one (e.g. Smart et al. 1998; L[ó]{}pez-Corredoira et al. 2002b; Poggio et al. 2017). Observationally, the results from star counting prefer the later (e.g. Djorgovski & Sosin 1989; L[ó]{}pez-Corredoira et al. 2002b). To fit the distribution of $\overline{V_z}$ found here for the chemically thin disk population, we have adopted the long-lived static warp model proposed by Poggio et al. \[2017; see their Eq.(3)\]. We have simply assumed that mean azimuthal velocity $V_{\phi}$ is constant, and $R_{\omega}$ is zero (i.e. the warp starts at the Galactic center; e.g. L[ó]{}pez-Corredoira et al. 2002b). We have also add a constant $c$, corresponding to mean vertical velocity at $R=0$ or $|\phi-\phi_{0}|=\frac{\pi}{2}$ to the warp equation. The final simplified model adopted here is then, $$\overline{V_{z}} = aR^{b}\cos(\phi-\phi_{0})+c,$$ where $\phi_0$ is angle of line-of-node of the warp.
To fit the observational data, values of $\overline{V_z}$ are calculated for the individual bins of $R$ and $\phi$, the uncertainties of the mean vertical velocities are estimated with a bootstrapping procedure. The binsize in $R$ is 0.2kpc and that in $\phi$ is allowed to vary but no less than 1$^{\circ}$ such that there are at least 40 stars in a bin. This results in total of 776 bins. Values of $\overline{V_z}$ of those bins are then fitted with the model described by Eq.(1), using a MCMC method. The best-fit model yields parameters,
$$\begin{aligned}
a &= 3.97^{+1.77}_{-1.24},\\
b &= 0.64^{+0.10}_{-0.09},\\
c &= -15.31^{+2.40}_{-2.97} \rm \,km \ s^{-1} and\\
\phi_{0} &= 12.5^{+2.0}_{-1.8} \rm \,degree.\end{aligned}$$
The angle of line-of-node obtained above agrees very well with recent estimates using Pulsars (Yusifov 2004), RCs and red giants (Momany et al. 2006) and Cepheids (Chen et al. 2019b) as tracers. We note that this is the first estimate of angle of line-of-node of the Galactic warp using kinematic data. However, the other best-fit value of the warp amplitude $a$ found here is not consistent with the results reported by previous star count analysis (e.g., L[ó]{}pez-Corredoira et al. 2002b; Yusifov 2004; Momany et al. 2006; Chen et al. 2019b). This may indicate more additional kinematic parameters are required for explaining this kinematic warp (like a precession; e.g. Poggio et al. 2020). Also, we note that the very large negative value for the parameter $c$ is unconvincing. It’s probably a consequence of assuming that the warp starts at the center of the Galaxy. In Fig.5, we show two typical examples of the best-fit for constant $\phi = 0^\circ$ and for constant $R = 9$kpc. Generally, the model fits the observational data quite well. The model predicted $\overline{V_z}$ distribution, the uncertainties of the mean vertical velocities and the fit residuals are also present in Fig.4. The residuals are largely within 1kms$^{-1}$.
The variations of angle of line-of-node as a function of $R$ is explored by Chen et al. (2019b). Using over one thousand classical Cepheid stars, they find that the angle of line-of-node first decreases with $R$ for $R$ between 8 and 12kpc and then increases with $R$ for $R$ between 12 and 15kpc, and tends to twist near $R = 15.5$kpc. They claim that the increase of the angle of line-of-node between 12 and 15kpc is evidence that the warp in the outer disk is predominately induced by torques associated with the massive inner disk. Our current data of mean vertical velocities do not show clear variations of angle of line-of-node with $R$. This might largely be due to i) the relative large uncertainties of the mean vertical velocities (see Fig.3 and panel (b) of Fig.4); and ii) the limited azimuthal angle coverage of the data. In the near future, this issue could be solved by adding more RC stars to the sample, selected from new LAMOST observations and the planned SDSS V surveys. Moreover, the additional data could allow one to explore the dynamical evolution of the Galactic warp (Poggio et al. 2020).
Summary
=======
In this paper, using a sample of nearly 140,000 RCs with accurate 3D position and 3D velocity measurements, constructed with data from the LAMOST and Gaia surveys, we have explored the kinematic warp signature of the Galactic disk(s). With cuts in the \[Fe/H\]-\[$\alpha$/Fe\] plane, 94,028 and 5,212 chemically thin and thick disk stars of $|Z| < 1$kpc are selected from the sample. Kinematic signature of warp is clearly detected in the data for the chemically thin disk population, but the signal is not significant for the chemically thick disk population. For the thin disk population, a clear positive gradient of mean vertical velocity as a function of $R$ is found for $R$ between 8 and 13kpc. The trend agrees with the recent results from the Gaia DR2 and is also consistent with the prediction of the long-lived large-scale Galactic warp model. The warp signature for the thick disk population is much weaker, largely due to the hot nature of orbits of thick disk stars. For the thin disk stars, we further explore the variations of mean vertical velocity (as a function of $R$) for the different azimuthal slices and find the amplitude of warp increases with $\phi$ and reaches a maximum in slice $\phi \in$ \[10, 17\] deg. To quantitively determine the angle of line-of-node of the warp, we fit the distribution of mean vertical velocities of the thin disk stars with a long-lived static warp model and find an angle around 12.5$^{\circ}$, in excellent agreement with the previous estimates from star counting.
Based on the current study alone, it is still difficult to constrain the exact origin of the Galactic warp. However, with more data expected from the on-going and forthcoming LAMOST, SDSS and Gaia surveys, vital clues about the origin and evolution of the Galactic warp should become available in the near future.
Acknowledgements {#acknowledgements .unnumbered}
================
The Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. The LAMOST FELLOWSHIP is supported by Special fund for Advanced Users, budgeted and administrated by Center for Astronomical Mega-Science, Chinese Academy of Sciences (CAMS). R.S. is supported by a Royal Society University Research of Fellowship.
This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).
[This work is supported by National Natural Science Foundation of China grants 11903027, 11973001, 11833006, 11811530289, U1731108, and U1531244, and National Key R & D Program of China No. 2019YFA0405503. B.Q.C. and Y.H. are supported by the Yunnan University grant No. C176220100006 and C176220100007, respectively. HFW is supported by the LAMOST Fellow project and funded by China Postdoctoral Science Foundation via grant 2019M653504, Yunnan province postdoctoral Directed culture Foundation and the Cultivation Project for LAMOST Scientific Payoff and Research Achievement of CAMS-CAS.]{}
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abstract: 'Recent years have seen a boom in interest in machine learning systems that can provide a human-understandable rationale for their predictions or decisions. However, exactly what kinds of explanation are truly human-interpretable remains poorly understood. This work advances our understanding of what makes explanations interpretable in the specific context of verification. Suppose we have a machine learning system that predicts X, and we provide rationale for this prediction X. Given an input, an explanation, and an output, is the output consistent with the input and the supposed rationale? Via a series of user-studies, we identify what kinds of increases in complexity have the greatest effect on the time it takes for humans to verify the rationale, and which seem relatively insensitive.'
author:
- 'Menaka Narayanan\*'
- 'Emily Chen\*'
- 'Jeffrey He\*'
- Been Kim
- Sam Gershman
- 'Finale Doshi-Velez'
bibliography:
- 'main.bib'
title: |
How do Humans Understand Explanations from Machine Learning Systems?\
An Evaluation of the Human-Interpretability of Explanation
---
Introduction {#sec:intro}
============
Interpretable machine learning systems provide not only decisions or predictions but also explanation for their outputs. Explanations can help increase trust and safety by identifying when the recommendation is reasonable when it is not. While interpretability has a long history in AI [@michie1988machine], the relatively recent widespread adoption of machine learning systems in real, complex environments has lead to an increased attention to interpretable machine learning systems, with applications including understanding notifications on mobile devices [@mehrotra2017interpretable; @wang2016bayesian], calculating stroke risk [@letham2015interpretable], and designing materials [@raccuglia2016machine]. Techniques for ascertaining the provenance of a prediction are also popular within the machine learning community as ways for us to simply understand our increasingly complex models [@lei2016rationalizing; @selvaraju2016grad; @adler2016auditing].
The increased interest in interpretability has resulted in many forms of explanation being proposed, ranging from classical approaches such as decision trees [@breiman1984classification] to input gradients or other forms of (possibly smoothed) sensitivity analysis [@selvaraju2016grad; @ribeiro2016should; @lei2016rationalizing], generalized additive models [@caruana2015intelligible], procedures [@singh2016programs], falling rule lists [@wang2015falling], exemplars [@kim2014bayesian; @frey2007clustering] and decision sets [@lakkaraju2016interpretable]—to name a few. In all of these cases, there is a face-validity to the proposed form of explanation: if the explanation was not human-interpretable, clearly it would not have passed peer review.
That said, these works provide little guidance about when different kinds of explanation might be appropriate, and within a class of explanations—such as decision-trees or decision-sets—what are the limitations of human reasoning. For example, it is hard to imagine that a human would find a 5000-node decision tree as interpretable as 5-node decision tree, for any reasonable notion of interpretable. The reason the explanation is desired is also often left implicit. In @doshi2017roadmap, we point to a growing need for the interpretable machine learning community to engage with the human factors and cognitive science of interpretability: we can spend enormous efforts optimizing all kinds of models and regularizers, but that effort is only worthwhile if those models and regularizers actually solve the original human-centered task of providing explanation. Carefully controlled human-subject experiments provide an evidence-based approach to identify what kinds of regularizers we should be using.
In this work, we make modest but concrete strides toward this large goal of quantifying what makes explanation human-interpretable. We shall assume that there exists some explanation system that generates the explanation—for example, there exist a variety of approaches that use perturbations around a point of interest to produce a local decision boundary [@ribeiro2016should; @singh2016programs]. Our question is: What kinds of explanation can humans most easily utilize? Below, we describe the kind of task we shall explore, as well as the form of the explanation.
#### Choice of Task
The question of what kinds of explanation a human can utilize implies the presence of a downstream task. The task may be intrinsic—relying on just the explanation alone—or extrinsic—relying on the explanation and other facts about the environment.[^1] Intrinsic tasks include problems such as verification—given an input, output, and explanation, can the human user verify that the output is consistent with the input and provided explanation?—and counterfactual reasoning—given an input, output, and explanation, can the human subject identify what input changes would result in a different output? In contrast, extrinsic tasks include goals such as safety—given the input, output, explanation, and observations of the world, does the explanation help the human user identify when the agent is going to make a mistake?—and trust—given the input, output, explanation, and observations of the world, does the explanation increase the human user’s confidence in the agent? Evaluation on extrinsic tasks, while ultimately what we care about, require careful experimental design that ensures all subjects have similar knowledge and assumptions about the environment. One must also tease apart conflations between human perception of knowledge with actual knowledge: for example, it may be possible to manipulate trust arbitrarily separately from prediction accuracy. Thus, evaluating extrinsic tasks is more challenging than intrinsic ones. In this work, we will focus on the simplest intrinsic setting: verification. Given a specific input, explanation, and output, can we quickly determine whether the output is consistent with the input and explanation? Such a setting may arise in consumer recommendation scenarios—for example, a salesperson may wish to ensure that a specific product recommendation is consistent with a consumer’s preferences. Starting simple also provides an opportunity to explore aspects relevant to the experimental design.
#### Choice of Explanation Form
As mentioned above, there have been many forms of explanation proposed, ranging from decision trees to gradients of neural networks. In this work, we consider explanations that are in the form of *decision sets*. Decision sets are a particular form of procedure consisting of a collection of cases, each mapping some function of the inputs to a collection of outputs. An example of a decision set is given below
![Example of a decision set explanation.[]{data-label="fig:rule_set"}](explanation.png){width="3in"}
where each line contains a clause in disjunctive normal form (an or-of-ands) of the inputs, which, if true, provides a way to verify the output (also in disjunctive normal form). As argued in @lakkaraju2016interpretable, decision sets are relatively easy for humans to parse given a specific instance, because they can scan for the rule that applies and choose the accompanying output. Decision sets (also known as rule sets) also enjoy a long history of optimization techniques, ranging from @frank1998generating [@cohen1995fast; @clark1991rule] to @lakkaraju2016interpretable.
However, we can also already see that there are many factors that could potentially make the decision set more or less challenging to follow: in addition to the number of lines, there are ways in which terms interact in the disjunctions and conjunctions, and, more subtly, aspects such as how often terms appear and whether terms represent intermediate concepts. Which of these factors are most relevant when it comes to a human’s ability to process and utilize an explanation, and to what extent? The answer to this question has important implications for the design of explanation systems in interpretable machine learning, especially if we find that our explanation-processing ability is relatively robust to variation in some factors but not others.
#### Contributions
The core contribution of this work is to provide an empirical grounding for what kinds of explanations humans can utilize. We find that while almost all increases in the complexity of an explanation result in longer response times, some types of complexity—such as the number of lines, or the number of new concepts introduced—have a much bigger effect than others—such as variable repetition. We also find some unintuitive patterns, such as how participants seem to prefer an explanation that requires a single, more complex line to one that spans multiple simpler lines (each defining a new concept for the next line). While more work is clearly needed in this area, we take initial steps in identifying what kinds of factors are most important to optimize for when providing explanation to humans.
Related Work
============
Interpretable machine learning methods aim to optimize models for both succinct explanation and predictive performance. Common types of explanation include regressions with simple, human-simulatable functions [@caruana2015intelligible; @kim2015ibcm; @ruping2006learning; @buciluǎ2006model; @ustun2016supersparse; @doshi2014graph; @kim2015mind; @krakovna2016increasing; @hughes2016supervised; @jung2017simple], various kinds of logic-based methods [@wang2015falling; @lakkaraju2016interpretable; @singh2016programs; @liu2016sparse; @safavian1991survey; @bayesian2017wang], techniques for extracting local explanations from black-box models [@ribeiro2016should; @lei2016rationalizing; @adler2016auditing; @selvaraju2016grad; @smilkov2017smoothgrad; @shrikumar2016not; @kindermans2017patternnet; @ross2017right], and visualization [@wattenberg2016attacking]. There exist a range of technical approaches to derive each form of explanation, whether it be learning sparse models [@mehmood2012review; @chandrashekar2014survey], monotone functions [@canini2016fast], or efficient logic-based models [@rivest1987learning]. Related to our work, there also exists a history of identify human-relevant concepts from data, including disentangled representations [@chen2016infogan] and predicate invention in inductive logic programming [@muggleton2015meta]. While the algorithms are sophisticated, the measures of interpretability are often not—it is common for researchers to simply appeal to the face-validity of the results that they find (i.e., “this result makes sense to the human reader”) [@caruana2015intelligible; @lei2016rationalizing; @ribeiro2016should].
In parallel, the literature on explanation in psychology also offers several general insights into the design of interpretable AI systems. For example, humans prefer explanations that are both simple and highly probable [@lombrozo07]. Human explanations typically appeal to causal structure [@lombrozo2006structure] and counterfactuals [@keil2006explanation]. @miller1956magical famously argued that humans can hold about seven items simultaneously in working memory, suggesting that human-interpretable explanations should obey some kind of capacity limit (importantly, these items can correspond to complex *cognitive chunks*—for example, ‘CIAFBINSA’ is easier to remember when it is chunked as ‘CIA’, ‘FBI’, ‘NSA.’). Orthogonally, @kahneman2011thinking notes that humans have different modes of thinking, and larger explanations might push humans into a more careful, rational thinking mode. Machine learning researchers can convert these concepts into notions such as sparsity or simulatability, but the work to determine answers to questions such as “how sparse?” or “how long?” requires empirical evaluation.
Existing studies evaluting the human-interpretability of explanation often fall into the A-B test framework, in which a proposed model is being compared to some competitor, generally on an intrinsic task. For example, @kim2014bayesian showed that human subjects’ performance on a classification task was better when using examples as representation than when using non-example-based representation. @lakkaraju2016interpretable performed a user study in which they found subjects are faster and more accurate at describing local decision boundaries based on decision sets rather than rule lists. @subramanian1992comparison found that users prefer decision trees to tables in games, whereas @huysmans2011empirical found users prefer, and are more accurate, with decision tables rather than other classifiers in a credit scoring domain. @hayete2004gotrees found a preference for non-oblique splits in decision trees (see @freitas2014comprehensible for more detailed survey). These works provide quantitative evaluations of the human-interpretability of explanation, but rarely identify what properties are most essential for what contexts—which is critical for generalization.
Specific application areas have also evaluated the desired properties of an explanation within the context of the application. For example, @tintarev2015explaining provides a survey in the context of recommendation systems, noting differences between the kind of explanations that manipulate trust [@cosley2003seeing] and the kind that increase the odds of a good decision [@bilgic2005explaining]. In many cases, these studies are looking at whether the explanation has an effect, sometimes also considering a few different kinds of explanation (actions of similar customers, etc.). @horsky2012interface describe how presenting the right clinical data alongside a decision support recommendation can help with adoption and trust. @bussone2015role found that overly detailed explanations from clinical decision support systems enhance trust but also create over-reliance; short or absent explanations prevent over-reliance but decrease trust. These studies span a variety of extrinsic tasks, and again given the specificity of each explanation type, identifying generalizable properties is challenging.
Closer to the objectives of the proposed work, @kulesza2013too performed a qualitative study in which they varied the soundness (nothing but the truth) and the completeness (the whole truth) of an explanation in a recommendation system setting. They found completeness was important for participants to build accurate mental models of the system. @allahyari2011user [@elomaa2017defense] also find that larger models can sometimes be more interpretable. @schmid2016does find that human-recognizable intermediate predicates in inductive knowledge programs can sometimes improve simulation time. @sangdeh2017manipulating manipulate the size and transparency of an explanation and find that longer explanations and black-box models are harder to simulate accurately (even given many instances) on a real-world application predicting housing prices. Our work fits into this category of empirical study of explanation evaluation; we perform controlled studies on a pair of synthetic application to assess the effect of a large set of explanation parameters.
Methods
=======
Our main research question is to determine what properties of decision sets are most relevant for human users to be able to utilize the explanations for verification. In order to carefully control various properties of the explanation and the context, in the following we shall present human subjects with explanations that *could* have been machine-generated, but were in fact generated by us. Before describing our experiment, we emphasize that while our explanations are not actually machine-generated, our findings provide suggestions to designers of interpretable machine learning systems about what parameters affect the usability of an explanation, and which should be optimized when producing explanations.
Factors Varied
--------------
Even within decision sets, there are a large number of ways in which the explanations could be varied. Following initial pilot studies (see Appendix), we chose to focus on the three main kinds of variation (described below). We also tested on two different domains—a faux recipe recommendation domain and a faux clinical decision support domain—to see if the context would result in different explanation processing while other factors were held constant.
#### Explanation Variation
We explored the following sources of explanation variation:
- **V1: Explanation Size.** We varied the size of the explanation across two dimensions: the *total number of lines* in the decision set, and the *maximum number of terms within the output clause*. The first corresponds to increasing the vertical size of the explanation—the number of cases—while the second corresponds to increasing the horizontal size of the explanation—the complexity of each case. We focused on output clauses because they were harder to parse: input clauses could be quickly scanned for setting-related terms, but output clauses had to be read through and processed completely to verify an explanation. We hypothesized that increasing the size of the explanation across either dimension would increase the time required to perform the verification task.
- **V2: Creating New Types of Cognitive Chunks.** In Figure \[fig:interface\], the first line of the decision set introduces a new cognitive chunk: if the alien is ‘checking the news’ or ‘coughing,’ that corresponds to a new concept ‘windy.’[^2] On one hand, creating new cognitive chunks can make an explanation more succinct. On the other hand, the human must now process an additional idea. We varied two aspects related to new cognitive chunks. First, we simply adjusted the number of new cognitive chunks present in the explanation. All of the cognitive chunks were necessary for verification, to ensure that the participant had to traverse all of concepts instead of skimming for the relevant one.
Second, we tested whether it was more helpful to introduce a new cognitive chunk or leave it implicit: for example, instead of introducing a concept ‘windy’ for ‘checking the news or coughing,’ (explicit) we could simply include ‘checking the news or coughing’ wherever windy appeared (implicit). We hypothesized that adding cognitive chunks would increase the time required to process an explanation, because the user would have to consider more lines in the decision list to come to a conclusion. However, we hypothesized that it would still be more time-efficient to introduce the new chunk rather than having long clauses that implicitly contained the meaning of the chunk.
- **V3: Repeated Terms in an Explanation.** Another factor that might affect humans’ ability to process an explanation is how often terms are used. For example, if input conditions in the decision list have little overlap, then it may be faster to find the appropriate one because there are fewer relevant cases to consider. We hypothesized that if an input condition appeared in several lines of the explanation, this would increase the time it took to search for the correct rule in the explanation. (Repeated terms was also a factor used by [@lakkaraju2016interpretable] to measure interpretability.)
#### Domain Variation
Below we describe the two contexts, or domains, that we used in our experiments: recipe recommendations (familiar) and clinical decision support (unfamiliar). The domains were designed to feel very different but such that the verification tasks could be made to exactly parallel each other, allowing us to investigate the effect of the domain context in situations when the form of the explanation was exactly the same. We hypothesized that these trends would be consistent in both the recipe and clinical domains.
*Alien Recipes.* In the first domain, study participants were told that the machine learning system had studied a group of aliens and determined each of their individual food preferences in various settings (e.g., snowing, weekend). Each task involved presenting participants with the setting (the input), the systems’s description of the current alien’s preferences (the explanation), and a set of recommended ingredients (the output). The user was then asked whether the ingredients recommendation was a good one. This scenario represents a setting in which customers may wish to know why a certain product or products were recommended to them. Aliens were introduced in order to avoid the subject’s prior knowledge or preferences about settings and food affecting their responses; each task involved a different alien so that each explanation could be unique. All non-literals (e.g. what ingredients were spices) were defined in a dictionary so that all participants would have the same cognitive chunks.
*Alien Medicine* In the second domain, study participants were told that the machine learning system had studied a group of aliens and determined personalized treatment strategies for various symptoms (e.g. sore throat). Each task involved presenting participants with the symptoms (the input), the system’s description of the alien’s personalized treatment strategy (the explanation), and a set of recommended drugs (the output). The user was then asked whether the drugs recommended were appropriate. This scenario closely matches a clinical decision support setting in which a system might suggest a treatment given a patient’s symptoms, and the clinician may wish to know why the system chose a particular treatment.
As before, aliens were chosen to avoid the subject’s prior medical knowledge from affecting their responses; each task involved personalized medicine so that each explanation could be unique. We chose drug names that corresponded with the first letter of the illness (e.g. antibiotic medications were Aerove, Adenon and Athoxin) so as to replicate the level of ease and familiarity of food names. Again, all drug names and categories were provided in a dictionary so that participants would have the same cognitive chunks.
In our experiments, we maintained an exact correspondence between inputs (setting vs. symptoms), outputs (foods vs. drugs), categories (food categories vs. drug categories), and the forms of the explanation. These parallels allowed us to test whether changing the domain from a relatively familiar, low-risk product recommendation setting to a relatively unfamiliar, higher-risk decision-making setting affected how subjects utilized the explanations for verification.
Experimental Design and Interface
---------------------------------
The three kinds of variation and two domains resulted in six total experiments. In the recipe domain, we held the list of ingredients, food categories, and possible input conditions constant. Similarly, in the clinical domain, we held the list of symptoms, medicine categories, and possible input conditions constant. The levels were as follows:
- Length of the explanation (V1). We manipulated the length of the explanation (varying between 2, 6, and 10 lines) and the length of the output clause (varying between 2 and 5 terms). Each combination was tested twice within a experiment, for a total of 12 questions.
- Introducing new concepts (V2): We manipulated the number of cognitive chunks introduced (varying from 1 to 5), and whether they were embedded into the explanation or abstracted out into new cognitive chunks. Each combination was tested once within a experiment, for a total of 10 questions.
- Repeated terms (V3): We manipulated the number of times the input conditions appeared in the explanation (varying from 1 to 5) and held the number of lines and length of clauses constant. Each combination was tested twice within a experiment, for a total of 10 questions.
The outputs were consistent with the explanation and the input 50% of the time, so subjects could not simply learn that the outputs were always (in)consistent.
Participants were recruited via Amazon Mechanical Turk. Before starting the experiment, they were given a tutorial on the verification task and the interface. Then they were given a set of three practice questions. Following the practice questions, they started the core questions for each experiment. They were told that their primary goal was accuracy, and their secondary goal was speed. While the questions were the same for all participants, the order of the questions was randomized for every participant. Each participant only participated in one of the experiments. For example, one participant might have completed a 12-question experiment on the effect of varying explanation length in the recipe domain, while another would have completed 10-question experiment on the effect of repeated terms in the clinical domain. Experiments were kept short to avoid subjects getting tired.
#### Metrics
We recorded three metrics: response time, accuracy, and subjective satisfaction. Response time was measured as the number of seconds from when the task was displayed until the subject hit the submit button on the interface. Accuracy was measured if the subject correctly identified whether the output was consistent with the input and the explanation (a radio button). After each submitting their answer for each question, the participant was also asked to subjectively rate the quality of the explanation on a scale from one to ten.
[.7]{} ![Screenshots of our interface for a task. In the Recipe Domain, the supposed machine learning system has recommended the ingredients in the lower left box, based on its observations of the alien (center box). The top box shows the system’s explanation. In this case, the recommended ingredients are consistent with the explanation and the inputs: The input conditions are weekend and windy (implied by coughing), and the recommendation of fruit and grains follows from the last line of the explanation. In the Clinical Domain, the supposed machine learning system has recommended the medication in the lower box, based on its observations of the alien’s symptoms (center box). The top box shows the system’s reasoning. The interface has exactly the same form as the recipe domain.[]{data-label="fig:interface"}](interface.png "fig:"){width="\textwidth"}
[.7]{} ![Screenshots of our interface for a task. In the Recipe Domain, the supposed machine learning system has recommended the ingredients in the lower left box, based on its observations of the alien (center box). The top box shows the system’s explanation. In this case, the recommended ingredients are consistent with the explanation and the inputs: The input conditions are weekend and windy (implied by coughing), and the recommendation of fruit and grains follows from the last line of the explanation. In the Clinical Domain, the supposed machine learning system has recommended the medication in the lower box, based on its observations of the alien’s symptoms (center box). The top box shows the system’s reasoning. The interface has exactly the same form as the recipe domain.[]{data-label="fig:interface"}](clinical-explanation.png "fig:"){width="\textwidth"}
#### Experimental Interface
Figure \[fig:interface\] shows our interfaces for the Recipe and Clinical domains. The *observations* section (middle) refers to the inputs into the algorithm. The *recommendation* section refers to the output of the algorithm. The *preferences* section (top) contains the explanation—the reasoning that the supposed machine learning system used to suggest the output (i.e., recommendation) given the input, presented as a procedure in the form of a decision set. Finally, the *ingredients* section in the Recipe domain (and the *disease medications* section in the Clinical domain) contained a dictionary of *cognitive chunks* relevant to the experiment (for example, the fact that bagels, rice, and pasta are all grains). Including this list explicitly allowed us to control for the fact that some human subjects may be more familiar with various concepts than others.
The choice of location for these elements was chosen based on pilot studies—while an ordering of input, explanation, output might make more sense for an AI expert, we found that presenting the information in the format of Figure \[fig:interface\] seemed to be easier for subjects to follow in our preliminary explorations. We also found that presenting the decision set as a decision set seemed easier to follow than converting it into paragraphs. Finally, we colored the input conditions in blue and outputs in orange within the explanation. We found that this highlighting system made it easier for participants to parse the explanations for input conditions.
Results
=======
We recruited 100 subjects for each our six experiments, for a total of 600 subjects all together. Table \[tab:demographics\] summarizes the demographics of our subjects across the experiments. Most participants were from the US or Canada (with the remainder being almost exclusively from Asia) and were less than 50 years old. A majority had a Bachelor’s degree. There were somewhat more male participants than female. We note that US and Canadian participants with moderate to high education dominate this survey, and results may be different for people from different cultures and backgrounds.
Feature Category : Proportion
----------- ----------------------- -------------------- --------------------------
Age 18-34 : 59.0% 35-50 : 35.1% 51-69 : 5.9%
Education High School : 28.5% Bachelor’s : 52.4% Beyond Bachelor’s: 14.9%
Gender Male : 58.8% Female : 41.2%
Region US/Canada : 87.1% Asia : 11.4%
: Participant Demographics. There were no patients over 69 years old. 4.2% of participants reported “other” for their education level. The rates of participants from Australia, Europe, Latin America, and South America were all less than 0.5%. (All participants were included in the analyses, but we do not list specific proportions for them for brevity.)
\[tab:demographics\]
All participants completed the full task (each survey was only 10-12 questions). In the analysis below, however, we exclude participants who did not get all three initial practice questions or all two of the additional practice questions correct. While this may have the effect of artificially increasing the accuracy rates overall—we are only including participants who could already perform the task to a reasonable extent—this criterion helped filter the substantial proportion of participants who were simply breezing through the experiment to get their payment. We also excluded one participant in the clinical version of the cognitive chunks experiment who did get sufficient practice questions correct but then took more than ten minutes to answer a single question. Table \[tab:counts\] describes the total number of participants that remained in each experiment out of the original 100 participants.
Recipe Domain Clinical Domain
---------------------- --------------- -----------------
Explanation Size N=88 N=73
New Cognitive Chunks N=77 N=73
Variable Repetition N=70 N=71
: Number of participants who met our inclusion criteria for each experiment.
\[tab:counts\]
Figures \[fig:accuracy\] and \[fig:time\] present the accuracy and response time across all six experiments, respectively. Response time is shown for subjects who correctly answered the questions. Figure \[fig:subjective\_evaluation\] shows the trends in the participants’ subjective evaluation—whether they thought the explanation was easy to follow or not. We evaluated the statistical significance of the trends in these figures using a linear regression for the continuous outputs (response time, subjective score) and a logistic regression for binary outputs (accuracy). For each outcome, one regression was performed for each of the experiments V1, V2, and V3. If an experiment had more than one independent variable—e.g. number of lines and terms in output—we performed one regression with both variables. Regressions were performed with the statsmodels library [@seabold2010statsmodels] and included an intercept term. Table \[tab:pvalues\] summarizes these results.
-- -- --
-- -- --
-- -- --
-- -- --
-- -- --
-- -- --
[|l|l|l|l|l|]{}\
& &\
Factor & weight & p-value & weight & p-value\
Explanation Length (V1) & -0.0116 & 0.00367 & -0.0171 & **0.000127**\
Number of Output Terms (V1) & -0.0161 & 0.0629 & 0.00685 & 0.48\
Number of Cognitive Chunks (V2) & 0.0221 & 0.0377 & 0.0427 & **0.00044**\
Implicit Cognitive Chunks (V2) & 0.0147 & 0.625 & 0.0251 & 0.464\
Number of Variable Repetitions (V3) & -0.017 & 0.104 & -0.0225 & 0.0506\
\
& &\
Factor & weight & p-value & weight & p-value\
Explanation Length (V1) & 3.77 & **2.24E-34** & 3.3 & **5.73E-22**\
Number of Output Terms (V1) & 1.34 & 0.0399 & 1.68 & 0.0198\
Number of Cognitive Chunks (V2) & 8.44 & **7.01E-18** & 4.6 & **1.71E-05**\
Implicit Cognitive Chunks (V2) & -15.3 & **2.74E-08** & -11.8 & **0.000149**\
Number of Variable Repetitions (V3) & 2.4 & **0.000659** & 2.13 & 0.0208\
\
& &\
Factor & weight & p-value & weight & p-value\
Explanation Length (V1) & -0.165 & **5.86E-16** & -0.186 & **1.28E-19**\
Number of Output Terms (V1) & -0.187 & **2.12E-05** & -0.0335 & 0.444\
Number of Cognitive Chunks (V2) & -0.208 & **1.93E-05** & -0.0208 & 0.703\
Implicit Cognitive Chunks (V2) & 0.297 & 0.0303 & 0.365 & 0.018\
Number of Variable Repetitions (V3) & -0.179 & **5.71E-05** & -0.149 & **0.000771**\
\[tab:pvalues\]
#### In general, greater complexity results in higher response times and lower satisfaction.
Increasing the number of lines, the number of terms within a line, adding new concepts, and repeating variables all increase the complexity of an explanation in various ways. In figure \[fig:time\], we see that all of these changes increase response time. The effect of adding lines to scan results in the biggest increases in response time, while the effect of increasing the number of variable repetitions is more subtle. Making new concepts explicit also consistently results in increased response time. This effect may have partly been because adding a new concept explicitly adds a line, while adding it implicitly increases the number of terms in a line—and from V1, we see that the effect of the number of lines is larger than the effect of the number of terms. However, overall this effect seems larger than just adding lines (note the scales of the plots). Subjective scores appear to correlate inversely with complexity and response time.
Perhaps most unexpected was that participants both took longer and seemed less happy when new cognitive chunks were made explicit rather than implicitly embedded in a line—we might have expected that even if the explanation took longer to process, it would have been in some senses easier to follow through. It would be interesting to unpack this effect in future experiments, especially if participant frustration came from there now being multiple relevant lines, rather than just one. Future experiments could also highlight terms from the input in the explanation, to make it easier for participants to find the lines of potential relevance.
#### Different explanation variations had little effect on accuracy.
While there were strong, consistent effects from explanation variation on response time and satisfaction, these variations seemed to have much less effect on accuracy. There existed general decreasing trends in accuracy with respect to explanation length and the number of variable repetitions, and potentially some increasing trends with the number of new concepts introduced. However, few were statistically significant. This serves us as an interesting controlled comparison. In other words, we can now observe effects of different factors, holding the accuracy constant. The lack of effect may be because subjects were told to perform the task as quickly as they could without making mistakes, and our filtering also removed subjects prone to making many errors in the practice questions. Thus, the effect of a task being harder would be to increase response time, rather than decrease accuracy.
The differences in direction of the trends—some increases in complexity perhaps increasing accuracy, others decreasing it—are consistent with findings that sometimes more complex tasks force humans into slower but more careful thinking, while in other cases increased complexity can lead to higher errors [@kahneman2011thinking]. Is it the case that the factors that resulted in the largest increases in response time (new concepts) also force the most concentration? While these experiments cannot differentiate these effects, future work to understand these phenomena may help us identify what kinds of increased complexities in explanation are innocuous or even useful and which push the limits of our processing abilities.
#### Trends are consistent across recipe and clinical domains
In all experiments, the general trends in the metrics are consistent across both the recipe and clinical domains. Sometimes an effect is much weaker or unclear, but never is an effect clearly reversed. We believe this bodes well for there being a set of general principles for guiding explanation design, just as there exist design principles for interfaces and human-computer interaction. However, one small pattern can be noted in Figure 5, which shows lower satisfaction for the clinical domain than the recipe domain. This could be due to the fact that people felt more agitated about performing poorly in the medical domain than the clinical domain.
Discussion and Conclusion
=========================
Identifying how different factors affect a human’s ability to utilize explanation is an essential piece for creating interpretable machine learning systems—we need to know what to optimize. What factors have the largest effect, and what kind of effect? What factors have relatively little effect? Such knowledge can help us expand to faithfulness of the explanation to what it is describing with minimal sacrifices in human ability to process the explanation.
In this work, we investigated how the ability of humans to perform a simple task—verifying whether an output is consistent with an explanation and input—varies as a function of explanation size, new types of cognitive chunks and repeated terms in the explanation. We tested across two domains, carefully controlling for everything but the domain.
We summarized some intuitive and some counter intuitive discoveries—as any increase in explanation complexity increases response time and decreases subjective satisfaction with the explanation—some patterns were not so obvious. We had not expected that embedding a new concept would have been faster to process and more appealing than creating a new definition. We also found that new concepts and the number of lines increase response time more than variable repetition or longer lines. It would be interesting to verify the magnitude of these sensitivities on other tasks, such as forward simulation or counterfactual reasoning, to start building a more complete picture of what we should be optimizing our explanations for.
More broadly, there are many interesting directions regarding what kinds of explanation are best in what contexts. Are there universals that make for interpretable procedures, whether they be cast as decision sets, decision trees, or more general pseudocode; whether the task is verification, forward simulation, or counterfactual reasoning? Do these universals also carry over to regression settings? Or does each scenario have its own set of requirements? When the dimensionality of an input gets very large, do trade-offs for defining intermediate new concepts change? A better understanding of these questions is critical to design systems that can provide rationale to human users.
#### Acknowledgements
The authors acknowledge PAIR at Google and the Harvard Berkman Klein Center for their support.
Description of Pilot Studies {#description-of-pilot-studies .unnumbered}
============================
We conducted several pilot studies in the design of these experiments. Our pilot studies showed that asking subjects to respond quickly or within a time limit resulted in much lower accuracies; subjects would prefer to answer as time was running out rather than risk not answering the question. That said, there are clearly avenues of adjusting the way in which subjects are coached to place them in fast or careful thinking modes, to better identify which explanations are best in each case.
The experiment interface design also played an important role. We experimented with different placements of various blocks, the coloring of the text, whether the explanation was presented as rules or as narrative paragraphs, and also, within rules, whether the input was placed before or after the conclusion (that is, ‘if A: B’’ vs. “B if A”). All these affected response time and accuracy, and we picked the configuration that had the highest accuracy and user satisfaction.
Finally, in these initial trials, we also varied more factors: number of lines, input conjunctions, input disjunctions, output conjunctions, output disjunctions and global variables. Upon running preliminary regressions, we found that there was no significant difference in effect between disjunctions and conjunctions, though the number of lines, global variables, and general length of output clause—regardless of whether that length came from conjunctions or disjunctions—did have an effect on the response time. Thus, we chose to run our experiments based on these factors.
[^1]: As noted in @herman2017promise, explanation can also be used to persuade rather than inform; we exclude that use-case here.
[^2]: One can imagine this being akin to giving names to nodes in a complex model, such as a neural network.
|
---
abstract: 'Efficient energy transduction is one driver of evolution; and thus understanding biomolecular energy transduction is crucial to understanding living organisms. As an energy-orientated modelling methodology, bond graphs provide a useful approach to describing and modelling the *efficiency* of living systems. This paper gives some new results on the efficiency of metabolism based on bond graph models of the key metabolic processes: glycolysis.'
author:
- 'Peter J. Gawthrop[^1]'
- 'Edmund J. Crampin'
title: Biomolecular System Energetics
---
INTRODUCTION {#sec:introduction}
============
As noted by @Pay93a, @OstPerKat71 used bond graphs in their seminal paper *Network Thermodynamics* to describe and analyse systems of coupled chemical reactions. This work was extended by @Kar90, @Cel91, @ThoMoc06 and @GreCel12. These ideas were introduced to the Systems Biology community by @GawCra14 [@GawCra16]. As noted by @Kar90 the bond graph approach is particularly appropriate to electrochemical systems and therefore can be used to model the bioenergetics of excitable membranes [@GawSieKam17; @PanGawTra18X], redox reactions and chemiosmotic energy transduction in mitochondria [@Gaw17a].
Organisms need energy to drive essential organs including the brain [@SteLau15; @Niv16], heart [@Neu07; @Kat11] and muscles [@SmiBarLoi05]. As discussed in the text books [@BerTymGat15; @AlbJohLew15], this energy is derived from metabolism involving glycolysis[^2], the TCA cycle[^3] and the mitochondrial[^4] respiratory chain [@NicFer13]. Both glycolysis and the mitochondrial respiratory chain produce energy storage molecule ATP (adenosine triphosphate) Energy plays a key role in evolution [@NivLau08; @Lan14]. In particular, evolutionary pressure would be expected to lead to organisms with both efficient energy production and consumption.
Efficiency of production has been experimentally investigated in the context of glycolysis and the TCA cycle by @ParRubXu16 and in the context of the mitochondrial respiratory chain by @LarTorLin16. Efficiency of energy consumption has been considered in the context of neurons by @Niv16, in the context of the heart by @LopDha14, and in the context of muscle by [@SmiBarLoi05]. Feedback systems regulate metabolism and its efficiency @Har08 [@TraLoiCra15; @DonSanLin17].
In the context of living systems, efficiency has been defined in a number of ways including ATP/O ratio [@LarTorLin16] and thermodynamic definitions consistent with engineering practice [@SmiBarLoi05; @Nat16]. The latter approach is used here.
As discussed by @Bea11 meaningful numerical simulation of living systems requires, *inter alia*, a firm thermodynamic foundation. Such a foundation is especially important in the context of investigating efficiency. As an energy-based modelling approach, the bond graph provides a firm foundation for studying living systems in general and biomolecular system energetics in particular.
§ \[sec:modelling\] is an introduction to bond graph modelling of biomolecular systems based on the specific system analysed in the paper. § \[sec:efficiency\] gives a bond graph approach to the efficiency of biomolecular systems illustrated by the example of glycolysis and § \[sec:conclusion\] concludes the paper.
MODELLING {#sec:modelling}
=========
This section introduces the modelling of biomolecular systems using bond graphs using the example of the first stage of human metabolism, glycolysis, which converts the high-energy molecule glucose () to the high-energy molecule pyruvate (), and also generating adenosine triphosphate () and (reduced) nicotinamide adenine dinucleotide (). § \[sec:chemical-equations\] looks at a single reaction: hydrolysis, § \[sec:numerical-values\] discusses how the numerical values were obtained and § \[sec:redox-reactions\] examines *redox* (reduction-oxidation) reactions. § \[sec:glycolysis\] discusses the modular bond graph modelling of glycolysis: the first stage of aerobic respiration.
Chemical reactions {#sec:chemical-equations}
------------------
[ \[subfig:ATP0\_abg\] ]{}
The reaction of adenosine triphosphate with water to form adenosine diphosphate , inorganic phosphate and a proton is known as *hydrolysis* and is given by @BerTymStr12 [§ 18.4, p.564] as $$\label{eq:ATP}
\ch{ATP + H2O <>[hyd] ADP + HPO4 + H}$$ In bond graph terms, each chemical (, etc.) can be regarded as a component accumulating each particular chemical and the hydrolysis reaction `hyd` can be regarded as an component driven by the chemical potentials $\mu_{\ch{ATP}}, \mu_{\ch{H2O}}$ etc. with units of generating the molar flow $v$ with units of . Reaction stoichiometry implies that the molar flow $v$ is out of and and into , and . As the the product $\mu
\times v$ has units of , $\mu$ and $v$ are covariables. Hence the reaction (\[eq:ATP\]) may be modelled by the bond graph of Figure \[fig:ATP\].
As discussed by @Gaw17a, it is helpful (to engineers) to measure quantity in Coulombs rather than moles and the corresponding conversion factor is Faraday’s constant $F=\SI{96485}{C.mol^{-1}}$. Noting that has the special unit Volt () and has the special unit Ampere (), the effort covariable becomes the *Faraday-equivalent potential* $\phi = F\mu\si{V}$ and the flow covariable becomes the *Faraday-equivalent flow* $f = \frac{1}{F}v\si{A}$.
Using the standard formula for chemical potential as a function of quantity [@AtkPau11], the *Constitutive Relationship* (CR) of a chemical component associated with substance A gives the potential $\phi_A$ in terms of the amount $x_A$ in terms of the potential $\phi^\Std_A$ and amount $x_A^\Std$ at standard conditions $$\begin{aligned}
\phi_A &= \phi_A^\Std + V_N \ln \frac{x_A}{x_A^\Std}\label{eq:phi_A}\\
&= V_N \ln K_A x_a \label{eq:phi_A_1}\\
\where V_N &= \frac{RT}{F} \approx \SI{26}{mV}
\text{ and } K_A = \frac{\exp \frac{\phi_A^\Std}{V_N}}{x_A^\Std}\end{aligned}$$ The Faraday-equivalent potential $\phi^\std_A$ at any other operating point can be computed from Equation as $$\begin{aligned}
\phi^\std_A &= \phi^\Std_A + V_N \ln \rho_A \label{eq:stdStd}\\
\text{where } \rho_A &= \frac{x_A^\std}{x_A^\Std} = \frac{c_A^\std}{c_A^\Std}\end{aligned}$$ and $c_A^\std$ and $c_A^\Std$ are the concentrations at the relevant conditions.
Whereas each *species* is associated with a *potential* $\phi_A$, each *reaction* $r$ is also associated with a *reaction potential* (which is denoted $\Phi$) split into two components: the *forward reaction potential* $\Phif$ and the *reverse reaction potential* $\Phir$. The net reaction potential, which drives the reaction, is given by $\Phi = \Phif - \Phir$. In the case of the reaction (\[eq:ATP\]) $$\begin{aligned}
\Phif &= \phi_{\ch{ATP}} + \phi_{\ch{H2O}} \\
\Phir &= \phi_{\ch{ADP}} + \phi_{\ch{HPO4}} + \phi_{\ch{H}}\\
\Phi &= \phi_{\ch{ATP}} + \phi_{\ch{H2O}}
- \lb \phi_{\ch{ADP}} + \phi_{\ch{HPO4}} + \phi_{\ch{H}} \rb\end{aligned}$$
The CR of a chemical component (assuming mass-action kinetics) is $$\begin{aligned}
f &= \kappa \lb \exp \frac{\Phif}{V_N} - \exp \frac{\Phir}{V_N} \rb\label{eq:f}\end{aligned}$$ This CR requires the forward ($\Phif$) and reverse ($\Phir$) potentials separately; as discussed by @Kar90 this requires either an implicit modulation or a two-port component.
Numerical Values {#sec:numerical-values}
----------------
Perhaps surprisingly, values for standard potentials and typical cellular concentrations can be hard to find in the biochemical literature. The values used in this paper come from two sources. Chemical potentials $\mu^\Std$ at standard conditions are taken from @LiWuQi11 [Table 5] and converted to Faraday-equivalent potentials $\phi^\Std=F\mu^\Std$. Concentrations are taken @ParRubXu16 [Table 5] and are used in conjunction with Equation to compute potentials at typical cellular conditions.
Reconciling experimental data is a big issue that is beyond the scope of this paper – see, for example, @TumKli18. Using the aforementioned data, some reactions discussed in § \[sec:glycolysis\] were found to have small negative potentials; these are not thermodynamically feasible and so the data from @LiWuQi11 [Table 5] was modified to give small positive potentials. In particular, the potentials of and were adjusted by , about 0.1%. In general, reaction potentials are the difference of large species potentials and thus small percentage changes in the latter can give large percentage changes in the former.
Under such typical cellular conditions, reaction (\[eq:ATP\]) is associated with a Faraday-equivalent potential $\Phi$ of about ; for this reason the reaction can be used to pump chemical reactions against an adverse potential gradient.
Redox reactions {#sec:redox-reactions}
---------------
[ \[subfig:NADH0\_abg\] ]{}
Redox (reduction/oxidation) reactions are the key to aerobic life; the bond graph modelling of such reactions is described by @Gaw17a. Redox reactions can be split into two half reactions each of which explicitly contains the electrons donated or accepted by the reaction. As an example of this in the first stage of the mitochondrial electron transport chain, (reduced Nicotinamide Adenine Dinucleotide) donates two (electrons) in forming (oxidised Nicotinamide Adenine Dinucleotide) which are accepted by [Q]{} (oxidised Ubiquinone) to form (reduced Ubiquinone). $$\label{eq:NADH}
\ch{ NADH <>[ red ] NAD+ + H+ + 2 e-}$$ In Figure \[fig:NADH\], the chemical species and proton are modelled as in § \[sec:chemical-equations\]; the is associated with the *redox potential* of the reaction and can be modelled by an electrical capacitor [@Gaw17a]. Using the Faraday-equivalent potential discussed in § \[sec:chemical-equations\], the potentials of the chemical species are commensurate with the potentials of the . In particular, under typical cellular conditions (§ \[sec:numerical-values\]), reaction (\[eq:NADH\]) is associated with a Faraday-equivalent potential $\Phi$ of about ; once again, for this reason the reaction can be used to pump chemical reactions.
Glycolysis {#sec:glycolysis}
----------
[ ]{} [ ]{}
The enzyme hexokinase is involved in a reaction which converts glucose () to glucose 6-phosphate (G6P): $$\label{eq:G6P}
\ch{ ATP + GLC <> ADP + H + G6P}$$ This can be rewritten as the combination of two reactions: hydrolysis reaction (\[eq:ATP\]) and $$\label{eq:HK}
\ch{ GLC + HPO4 <>[HK] G6P + H2O}$$ As in § \[sec:chemical-equations\], the HK reaction (\[eq:HK\]) has the bond graph representation of Figure \[subfig:hk\_abg\] where the bond graph *source-sensor* component provides a port to connect to the hydrolysis reaction (\[eq:ATP\]). As the HK reaction (\[eq:HK\]) is to be embedded in a larger model, a modular version is obtained by replacing and by the ports and respectively.
[ ]{} [ ]{} [ ]{}
The bond graph models of the two stages of glycolysis are given in Figures \[subfig:GlyA\_abg\] and \[subfig:GlyB\_abg\] and are combined in Figure \[subfig:Gly\_abg\].
Figure \[subfig:GlyA\_abg\] shows the modular version of HK embedded in the first stage of glycolysis. This clearly shows the two key features of the HK catalysed reaction: it converts converts GLC to G6P and it is pumped by ATP hydrolysis. In Figure \[subfig:GlyA\_abg\], the two pathways diverging from the reaction converge on indicate that stage 1 of glycolysis converts each molecule of to two molecules of glyceraldehyde 3-phosphate () and is pumped by two ATP hydrolysis reactions . Figure \[subfig:GlyB\_abg\] shows that stage 2 of glycolysis converts each molecule of to one molecule of and, as indicated by the bond arrow direction, pumps two reverse ATP hydrolysis reactions and a reverse reaction .
The modular bond graph of Figure \[fig:Gly\] is equivalent to the biomolecular system where the associated reaction potentials $\Phi$ correspond to typical cellular conditions (§ \[sec:numerical-values\]).
[2]{} [ ]{} & &()\
[ ]{} & &()\
[ ]{} & &()\
[ ]{} & &()\
[ ]{} & &()\
[ ]{} &&\
&[ ]{} &&()\
[ ]{} & &()\
[ ]{} & &()\
[ ]{} & &()\
[ ]{} & &()
EFFICIENCY {#sec:efficiency}
==========
[ \[subfig:GlyPump\_abg\] ]{}
The biomolecular network implementing glycolysis discussed in § \[sec:glycolysis\] converts glucose () to pyruvate () as well as driving hydrolysis and the reduction of in reverse to store chemical energy. In particular, the overall reaction is:
[2]{} & & &()
At the standard conditions discussed in § \[sec:numerical-values\] and denoted by the $^\std$ symbol, this reaction is associated with a reaction potential $$\Phi^\std_{all} = \SI{837}{ mV}$$ and the corresponding power dissipation is: $$P^\std_{diss} = \Phi^\std_{all}f \si{pW}$$ where $f\si{mA}$ is the reaction flow rate per unit volume.
As in Figure \[fig:GlyPump\], this reaction can be split into three parts:
[2]{} & &()\[eq:GLC-PYR\]\
& &()\
& &()
The latter two reactions represent two reverse hydrolysis reaction and two reverse reduction reactions respectively; the first reaction represents the remainder of the reaction converting to (and ).
The first reaction is associated with a reaction *driving potential* $$\label{eq:Phi_0}
\Phi^\std_0 = \phi^\std_{GLC} - 2\phi^\std_{PYR} - 6\phi^\std_{H} = \SI{2712}{ mV}$$ Because the latter two reactions are being pumped by the first, define the *pumping potential* $\Phi_{pump}$ as: $$\begin{aligned}
\Phi^\std_{pump} &= 2 \lb \Phi^\std_{ATP} + 2 \Phi^\std_{NADH}\rb\notag\\
&= 1184 + 690 = \SI{1874}{ mV}\end{aligned}$$ These potentials are associated with a corresponding *driving power* $P_{0}$ and *pumping power* $P_{pump}$: $$\begin{aligned}
P_{0} &= \Phi_{0}f\\
P_{pump} &= \Phi_{pump}f\end{aligned}$$
With this example in mind define the *pumping efficiency* as the ratio of pumping power to driving power $$\begin{aligned}
\eta &= \frac{P_{pump}}{P_0}\notag\\
&= \frac{\Phi_{pump}}{\Phi_0}\label{eq:eta}\end{aligned}$$ At standard conditions $$\begin{aligned}
\eta^\std= \frac{\Phi^\std_{pump}}{\Phi^\std_0} = \frac{1874}{2712} = 69.1\% \label{eq:eta_std}\end{aligned}$$
[ ]{} [ ]{}
The efficiency computed in Equation corresponds to the nominal values discussed in § \[sec:numerical-values\], which in turn correspond to the nominal flow of $f^\std=\SI{2.3}{mM/min}$. If the concentration of glucose () is varied, the corresponding potential $\phi_{GLC}$ varies according to Equation . then so will $\Phi_0$ of Equation and the pumping efficiency . $$\begin{aligned}
\eta &= \frac{\Phi^\std_{pump}}{\Phi^\std_0 + \tilde{\phi}_{GLC}}\notag\\
\where \tilde{\phi}_{GLC} &= \Phi_0 - \Phi^\std_0 = \phi_{GLC} - \phi^\std_{GLC} \end{aligned}$$ Figure \[subfig:GLC\_eta\] shows how $\eta$ varies with $\tilde{\phi}_{GLC}$. Note that $\tilde{\phi}_{GLC}=\Phi^\std_{pump}-\Phi^\std_0 = -\Phi^\std_{all}$ corresponds to $\eta=100\%$. As this value of $\tilde{\phi}_{GLC}$ corresponds to $\Phi^\std_{all}=0$, the flow $f=0$ at this point.
Figure \[subfig:GLC\_X\_eta\] shows efficiency $\eta$ plotted against the normalised concentration of .
[ \[subfig:gly\_eta\] ]{}
The computation generating the data for Figure \[fig:eta\] does not involve the Faraday-equivalent flow $f$. However, efficiency as a function of flow is of interest. As a approximation to this, the steady-state flow $f$ was computed as $\phi_{\ch{GLC}}$ was varied assuming that all reactions have the mass-action kinetics of using the method of @Gaw18. This was used to generate the data for Figure \[fig:eta\_flow\]. In fact, the reaction kinetics are more complicated than the mass-action representation hence the computations are more challenging than those used to generate Figure \[fig:eta\_flow\].
These results indicate that, under these particular conditions, the pumping efficiency of glycolysis is around 70% except for very low flow rates associated with low concentrations of Glucose () and thus $\Phi_0$ being only slightly larger that $\Phi_{pump}$.
CONCLUSION {#sec:conclusion}
==========
The basic ideas of modelling biomolecular systems using bond graphs and the Faraday-equivalent potential have been outlined and illustrated using the example of glycolysis: the first stage of aerobic respiration.
The concept of *pumping efficiency* has been introduced and illustrated using glycolysis and experimental numerical values drawn from the recent paper of @ParRubXu16. These ideas are currently being extended to mitochondrial metabolism: the TCA cycle and the electron transport chain.
[34]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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[^1]: Corresponding author. **peter.gawthrop@unimelb.edu.au**
[^2]: Glycolysis is the metabolic process converting the sugar Glucose to the intermediate high-energy molecule pyruvate.
[^3]: The tricarboxylic acid (TCA) cycle, also known as the citric acid cycle or the Krebs cycle, converts the high-energy molecule pyruvate into the high-energy molecule NADH (reduced nicotinamide adenine dinucleotide).
[^4]: Mitochondria are organelles within many cells which provide efficient conversion of the products of the TCA cycle into ATP.
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abstract: 'We study the role of the dimer structure of light-harvesting complex II (LH2) in excitation transfer from the LH2 (without a reaction center (RC)) to the LH1 (surrounding the RC), or from the LH2 to another LH2. The excited and un-excited states of a bacteriochlorophyll (BChl) are modeled by a quasi-spin. In the framework of quantum open system theory, we represent the excitation transfer as the total leakage of the LH2 system and then calculate the transfer efficiency and average transfer time. For different initial states with various quantum superposition properties, we study how the dimerization of the B850 BChl ring can enhance the transfer efficiency and shorten the average transfer time.'
author:
- 'S. Yang'
- 'D. Z. Xu'
- 'Z. Song'
- 'C. P. Sun'
title: 'Dimerization-assisted energy transport in light-harvesting complexes'
---
Introduction
============
To face the present and forthcoming global energy crisis, human should search for clean and effective energy source. Recently the investigations on the basic energy science for this purpose has received great attention and experienced impressive progress based on the fundamental physics [Fleming08,arti]{}. In photosynthetic process, the structural elegance and chemical high efficiency of the natural system based on pigment molecules in transferring the energy of sunlight have stimulated a purpose driven investigation [@Venturi08; @HuXiChe971; @Johnson08; @Plenio091; @Nori09; @Guzik09; @Nori08; @Fleming09; @Plenio092; @Guzik08; @Mukamel09], finding artificial analogs of porphyrin-based chromophores. These artificial systems replicate the natural process of photosynthesis [@arti] so that the much higher efficiencies could be gained than that obtained in the conventional solid systems [@arti]. It is because one of the most attractive features of photosynthesis is that the light energy can be captured and transported to the reaction center (RC) within about 100ps and with more than 95% efficiency [@HuXiChe971; @Fleming94].
Actually, in most of the plants and bacterium, the primary processes of photosynthesis are almost in common [@Fleming94; @HuXiChe972; @Venturi08]: Light is harvested by antenna proteins containing many chromophores; then the electronic excitations are transferred to the RC sequentially, where photochemical reactions take place to convert the excitation energy into chemical energy. Most recent experiments have been able to exactly determine the time scales of various transfer processes by the ultra-fast laser technology [@exp1; @exp2; @exp3]. These great progresses obviously offer us a chance to quantitatively make clear the underlying physical mechanism of the photosynthesis, so that people can construct the artificial photosynthesis devices in the future to reach the photon-energy and photon-electricity conversions with higher efficiency. For example, quantum interference effects in energy transfer dynamics [@Guzik08] has been studied for the Fenna-Matthews-Olson (FMO) protein complex, and it was found [@Guzik09] that, for such molecular arrays, the spatial correlations in the phonon bath and its induced decoherence could affect on the efficiency of the primary photosynthetic event. The present paper will similarly study the influences of spatial structure on the primary processes of photosynthesis for the light-harvesting complexes II (LH2).
In the past, by making use of the x-ray crystallographic techniques, the structure of light-harvesting system has been elucidated [HuXiChe96,Venturi08]{}. In the purple photosynthetic bacteria, there exist roughly two types of light-harvesting complexes, referred to as light-harvesting complex I (LH1) and light-harvesting complex II (LH2). In LH1, the RC is surrounded by a B875 bacteriochlorophyll (BChl) ring with maximum absorption peak at 875 nm. The LH2 complex, however, does not contain the RC, but can transfer energy excitation to the RC indirectly through LH1. In the purple bacteria, LH2 is a ring-shaped aggregate built up by $8$ (or $9$) minimal units, where each unit consists of an $\alpha \beta $-heterodimer, three BChls, and one carotenoid. The $\alpha \beta $-heterodimers, i.e., $\alpha $-apoproteins and $\beta $-apoproteins constitute the skeleton of LH2, while the BChls are embedded in the scaffold to form a double-layered ring structure. The top ring including $16$ (or $18$) BChl molecules is named as B850 since it has the lowest-energy absorption maximum at 850 nm. The bottom ring with 8 BChls is called B800 because it mainly absorbs light at 800 nm. In every minimal unit, the carotenoid connects B800 BChl with one of the two B850 BChls. Excitation is transferred from one pigment to the neighbor one through the Föster mechanism [HuXiChe971]{}, while the electron is spatially transferred via the Marcus mechanism [@Leegwater96]. Generally, it is independent of the global geometry configuration of the system.
In the present paper, we will study the energy transfer procedure in LH2 by considering the structure dimerization of the B850 ring. It has been conjectured that the dimerized inter-pigment couplings can cause the energy gap to protect the collective excitations [@HuXiChe972]. Indeed, like the the Su-Schrieffer-Heeger model for the flexible polyacetylene chain [SSH]{}, the dimerization of the spatial configuration with the Peierls distorted ground state will minimize the total energy for the phonon plus electron. As it is well known, this model exhibits a rich variety of nonlinear phenomena and topological excitations including the topological protection of the quantum state transfer [@Song]. Similarly, we will show that, when the B850 ring in LH2 is dimerized the excitation transfer efficiency may be enhanced to some extent.
Based on the open quantum system theory, we simply model the excited and un-excited states of a BChl pigment as a quasispin. The excitation transfer is represented by the total leakage from a LH2. Using the master equation, we calculate the efficiency of excitation transfer and the average transfer time in low temperature for various initial states with different superposition properties. The results explicitly indicate that the dimerization of couplings indeed enhances the quantum transport efficiency and shortens the average transfer time.
![(color online) The model setup of the light-harvesting complex II constructed by 8 unit cells. The couplings between the neighboring quasi-spins in the B850 ring is dimerized as $J_{2}(1+\protect\delta)$ and $%
J_{2}(1-\protect\delta)$. $g_{1}$ denote the nearest couplings between B850 BChls and B800 BChls, while $g_{2}$ denote the next nearest couplings between B850 BChls and B800 BChls. (a) Illustration of the whole system with $g_{2}=0 $. (b) Detailed drawing of three unit cells and their non-local couplings. (c) Legends.[]{data-label="model1"}](fig1.eps){width="8"}
This paper is organized as follows. In Sec. II, a double-ring XY model with $%
N$ unit cells is presented to simulate the LH2 system. In Sec. III, the energy transfer process is described by the quantum master equation. The transfer efficiency $\eta \left( t\right) $ and the average transfer time $%
\tau $ are introduced to characterize the dynamics of the system. In Sec. IV, we represent the master equation in the momentum space and show that only the $\left( k,k\right) $-blocks of the density matrix are relevant to energy transfer. In Sec. V, it is found that the transfer efficiency $\eta
\left( t\right) $ and the average transfer time $\tau $ of an arbitrary initial state can be obtained through the channel decomposition. Some numerical analysis of $\eta ^{\left[ A,k\right] }\left( t_{0}\right) $ and $%
\tau ^{\left[ A,k\right] }$ for all the $k$-channels are presented in Sec. VI. They show that a suitable dimerization of the B850 BChl ring can enhance the transfer efficiency and shorten the average transfer time. Conclusions are summarized at the end of the paper. In Appendix A, we provide an alternative way to deal with the energy leakage problem. In Appendix B, detail derivations of transforming the master equation from the real space to the $k$-space are given. The approximate solution of $\tau ^{\left[ A,k%
\right] }$ for $k=0$ and $k=\pm \pi $ channel is shown in Appendix C.
Model setup
===========
The simplified model of LH2 is shown in Fig. \[model1\]. All the bacteriochlorophylls (big and small green squares) are modeled by the two-level systems with excited state $|e_{j}^{[c]}\rangle $, ground state $%
|g_{j}^{[c]}\rangle $, and energy level spacing $\Omega _{c}$. The raising and lowering quasi-spin operators of the $j$th two-level system on the $[c]$ ring is expressed as $$\sigma _{j}^{+[c]}=|e_{j}^{[c]}\rangle \langle g_{j}^{[c]}|\text{, }\sigma
_{j}^{-\left[ c\right] }=|g_{j}^{[c]}\rangle \langle e_{j}^{[c]}|,$$where $[c]=$ $\left[ a\right] $ ($\left[ b\right] $) denotes the B800 (B850) BChl ring. Approximately, all the couplings are supposed to be of XY type [@Johnson08]. This simplification enjoys the main feature of excitation transfer. The Hamiltonians $$H_{a}=\frac{\Omega _{a}}{2} \sum_{j=1}^{N}\sigma _{j}^{z\left[
a\right] }+J_{1}\sum_{j=1}^{N}\left( \sigma _{j}^{+\left[ a\right]
}\sigma _{j+1}^{-[a]}+\mathrm{H.c.}\right) \label{h1}$$and $$\begin{aligned}
H_{b}& =\frac{\Omega _{b}}{2} \sum_{j=1}^{2N}\sigma _{j}^{z\left[
b\right] }+J_{2}\sum_{j=1}^{N}[\left( 1+\delta \right) \sigma _{2j-1}^{+%
\left[ b\right] }\sigma _{2j}^{-\left[ b\right] } \notag \\
& +\left( 1-\delta \right) \sigma _{2j}^{+\left[ b\right] }\sigma _{2j+1}^{-%
\left[ b\right] }+\mathrm{H.c.}] \label{h2}\end{aligned}$$with $N=8$, describe the excitations of the B800 and B850 BChl rings,respectively. In the B850 BChl ring, the parameter $\delta \neq 0$ characterizes the dimerization due to the spatial deformation of the flexible B850 BChl ring in LH2. The coupling constants of $H_{b}$ are dimerized as $J_{2}\left( 1+\delta \right) $ and $J_{2}\left( 1-\delta
\right) $ since the intra-unit and inter-unit Mg-Mg distance between neighboring B850 BChls may be different. The non-local XY type interaction
$$\begin{aligned}
H_{ab}& =g_{1}\sum_{j=1}^{N}\left[ \sigma _{j}^{+\left[ a\right] }\left(
\sigma _{2j-1}^{-\left[ b\right] }+\sigma _{2j}^{-\left[ b\right] }\right) +%
\mathrm{H.c.}\right] \notag \\
& +g_{2}\sum_{j=1}^{N}\left[ \sigma _{j}^{+\left[ a\right] }\left( \sigma
_{2j-3}^{-\left[ b\right] }+\sigma _{2j-2}^{-\left[ b\right] }+\sigma
_{2j+1}^{-\left[ b\right] }+\sigma _{2j+2}^{-\left[ b\right] }\right) +%
\mathrm{H.c.}\right] \label{Hspin}\end{aligned}$$
is used to describe the interaction between the B800 and B850 BChl rings.
In the single excitation case, the quasi-spin can be represented with a spinless fermion with the mapping$$\sigma _{j}^{+\left[ a\right] }\leftrightarrow A_{j}^{\dag },\sigma
_{2j-1}^{+\left[ b\right] }\leftrightarrow B_{j}^{\dag },\sigma _{2j}^{+%
\left[ b\right] }\leftrightarrow C_{j}^{\dag }$$from the spin space $V_{s}=C_{2}^{\otimes 3N}$ to the subspace $V_{F}$ ofthe Fermion Fock space spanned by $$\left\{ \left\vert O,j\right\rangle =O_{j}^{\dag }\left\vert 0\right\rangle
\text{ }|\text{ }O=A,B,C;j=1,2,\cdots ,N\right\} .$$Hereafter, let us represent the site index as $\left( O,j\right) $, where $j$ refers to a unit cell shown in Fig. \[model1\], and $O=A,B,C$ to a position type inside the unit cell. In the subscripts, the site index $%
\left( O,j\right) $ is written as $Oj$ for simplicity. The vacuum state of the Fermion system $\left\vert 0\right\rangle $ corresponds to the state that all the quasi-spins are in their ground states, $$\left\vert 0\right\rangle \leftrightarrow \prod_{j=1}^{N}\left\vert g_{j}^{
\left[ a\right] }\right\rangle \otimes \prod_{j=1}^{2N}\left\vert g_{j}^{%
\left[ b\right] }\right\rangle .$$Then the total Hamiltonian $H_{S}=H_{a}+H_{b}+H_{ab}$ of LH2 is mapped into $$\begin{aligned}
H_{S}& =\sum_{j=1}^{N}\left[ \Omega _{a}A_{j}^{\dag }A_{j}+\Omega _{b}\left(
B_{j}^{\dag }B_{j}+C_{j}^{\dag }C_{j}\right) \right] \notag \\
& +\sum_{j=1}^{N}\left\{ J_{1}A_{j}^{\dag }A_{j+1}+g_{1}A_{j}^{\dag }\left(
B_{j}+C_{j}\right) \right. \notag \\
& +J_{2}[\left( 1+\delta \right) B_{j}^{\dag }C_{j}+\left( 1-\delta \right)
C_{j}^{\dag }B_{j+1}] \notag \\
& \left. +g_{2}A_{j}^{\dag }\left( B_{j+1}+B_{j-1}+C_{j+1}+C_{j-1}\right) +%
\mathrm{H.c.}\right\} . \label{Hparticle}\end{aligned}$$In the present work, no multi-fermion interactions are considered for simplicity.
On the other hand, we use the Holstein-Primakoff transformation [HPtransform]{} to map the quasi-spin into bosons. The excitations of the BChls can be described by quasi-spins with the total angular momentum $S$. Then $D=A,B,C$ can be regarded as the annihilation operators of bosons for the Fock space spanned by$$\begin{aligned}
\{\left( D_{j}^{\dag }\right) ^{n_{D,j}}\left\vert 0\right\rangle \text{ }|%
\text{ }D &=&A,B,C;j=1,\cdots ,N; \notag \\
n_{D,j} &=&0,1,\cdots ,2S\}.\end{aligned}$$For $S>1/2$, one local bacteriochlorophyll has more than one excited states. In this case, higher order coherence could be included for further generalization.
In the following, we focus on the single excitation case. Then the temperature should be suitable to ensure there is no higher order excited state.
Transfer efficiency and average transfer time via the master equation
=====================================================================
Next we consider the energy transfer from an initial state $$\widehat{\rho }\left( 0\right) =\sum_{j,l}\rho _{Aj,Al}\left( 0\right)
\left\vert A,j\right\rangle \left\langle A,l\right\vert ,$$which is a coherent superposition or a mixture of those local states $%
\left\vert A,j\right\rangle $ on the B800 ring. As time goes by, the initial state will evolves a state distributing around both the B800 and the B850 rings. Since there exists a difference of chemical potential. $\Delta \Omega
=\Omega _{a}-\Omega _{b}$, energy is transferred between the two rings during the time evolution. For an isolated LH2 system, such energy transfer is coherent, namely, the system oscillates between the B800 and the B850 rings . However, when a LH2 is coupled to a heat reservoir with infinite degrees of freedom, irreversible energy transfer occurs. As illustrated in Fig. \[model2\], in the real photosynthetic system, energy is transferred from one LH2 to another LH2 or LH1 through the B850 ring [@HuXiChe972]. Therefore, we regard the first excited LH2 as an open system, and the sum of others as the a heat reservoir. The energy transfer now can be manipulated as the energy leakage from the B850 ring to the environment.
![(color online) The first excited LH2 is treated as an open system while the other LHs are regarded as heat reservoirs. The energy transfer process is equivalent to the excitation leakage from the B850 BChl ring of the LH2 system to the environment.[]{data-label="model2"}](fig2.eps){width="8"}
In order to describe such a procedure that the excitations are finally transferred from the B850 ring to the heat reservoir, the Markovian master equation $$\frac{d\widehat{\rho }}{dt}=-i\left[ H_{S},\widehat{\rho }\right] +{\mathcal{%
L}}\left( \widehat{\rho }\right) \label{mastereq}$$in the Lindblad form is employed for determining the time-evolution of the density matrix. Here two kinds of loss processes, dissipation and dephasing, are considered as Lindblad terms $${\mathcal{L}}\left( \widehat{\rho }\right) =\sum_{j=1}^{N}\left[ {\mathcal{L}%
}_{\mathrm{diss},j}\left( \widehat{\rho }\right) +{\mathcal{L}}_{\mathrm{deph%
},j}\left( \widehat{\rho }\right) \right] .$$ We suppose that each quasi-spin on the B850 ring is coupled to an independent heat reservoir [@Guzik09], which reflects the local modes of phonons and other local fluctuations. Then the dissipation from the $j$th unit cell is described as $${\mathcal{L}}_{\mathrm{diss},j}\left( \widehat{\rho }\right) =\Gamma
_{j}\sum_{O=B,C}(O_{j}\widehat{\rho }O_{j}^{\dag }-\frac{1}{2}\left\{
O_{j}^{\dag }O_{j},\widehat{\rho }\right\} ), \label{dissipation}$$where $\left\{ \cdot ,\cdot \right\} $ denotes the anti-commutator. Here, the sink rate $\Gamma _{j}$ at the $j$th point may be site dependent. For the dynamics constrained on the subsystem described by $O$ operators, the last term of ${\mathcal{L}}_{\mathrm{diss},j}\left( \widehat{\rho }\right) $ gives contribution $-\Gamma _{j}\rho _{Oj,Oj}$ to $d\rho
_{Oj,Oj}/dt$, thus dissipation results in the reduction of the total population. Therefore, the dissipation term Eq. (\[dissipation\]) represents the incoherent transfer of energy into the environment.
On the other hand, the dephasing term reads$${\mathcal{L}}_{\mathrm{deph},j}\left( \widehat{\rho }\right) =\Gamma
_{j}^{\prime }\sum_{O=B,C}(O_{j}^{\dag }O_{j}\widehat{\rho }O_{j}^{\dag
}O_{j}-\frac{1}{2}\left\{ O_{j}^{\dag }O_{j},\widehat{\rho }\right\} ).
\label{dephasing}$$Compared with the dissipation term, the dephasing one ${\mathcal{L}}_{%
\mathrm{deph},j}\left( \widehat{\rho }\right) $ does not contributeto any time local change of the probability distribution, i.e., the derivative of the diagonal elements of the density matrix is irrelevant to this term. Thus the total population $\sum_{O=A,B,C}\sum_{j}\rho _{Oj,Oj}$ would be conserved if only the dephasing term were present. However, the dephasing process is also incoherent since it make the nondiagonal elements of the density matrix tend to zero.
The above two contributions force the LH2 system to ultimately reach a steady state $\widehat{\rho }_{\mathrm{steady}}=\left\vert 0\right\rangle
\left\langle 0\right\vert =\widehat{\rho }_{v,v}$, namely, in the long-time limit, all excitations are sinked away. The same steady state is obtained from Eq. (\[mastereq\]) in the super-operator form$$\frac{d}{dt}[\rho ]=M[\rho ],$$where $[\rho ]$ denotes the column vector defined by all matrix elements in some order, and the super-operator $M$ is determined by
$$M[\rho ]=[-i\left[ H_{S},\widehat{\rho }\right] +{\mathcal{L}}\left(
\widehat{\rho }\right) ].$$
In this sense the steady state is just the non-trivial eigenstate of $M$ with vanishing eigen-energy. Usually, from $\det M=0,$ the steady state can be found.
However, we are interested in the system dynamics on a short timescale, i.e., how soon can the excitations be transferred from one LH2 to the other light-harvesting complexes? To this end, the transfer efficiency $\eta
\left( t\right) $ is defined as the population $\rho _{v,v}\left( t\right) $ of the vacuum state $\left\vert 0\right\rangle $ at time $t$, $$\eta \left( t\right) =\rho _{v,v}\left( t\right) .$$The corresponding master equation (\[mastereq\])$$\begin{aligned}
\frac{d\rho _{v,v}}{dt}& =\sum_{j=1}^{N}\Gamma _{j}\left\langle 0\right\vert
\left( B_{j}\widehat{\rho }B_{j}^{\dag }+C_{j}\widehat{\rho }C_{j}^{\dag
}\right) \left\vert 0\right\rangle \notag \\
& =\sum_{j=1}^{N}\Gamma _{j}\sum_{O=B,C}\rho _{Oj,Oj}\end{aligned}$$means that only the first term of ${\mathcal{L}}_{\mathrm{diss},j}\left(
\rho \right) $ contributes to the time derivatives of $\rho _{v,v}\left(
t\right) $. The transfer efficiency is given by the integral of the above formula [@Johnson08; @Guzik09; @Plenio092],$$\eta \left( t\right) =\int_{0}^{t}\sum_{j=1}^{N}\Gamma _{j}\sum_{O=B,C}\rho
_{Oj,Oj}\left( t^{\prime }\right) dt^{\prime }. \label{Eeta}$$The average transfer time $\tau $ is further defined as [Guzik09,Johnson08]{} $$\begin{aligned}
\tau &=&\lim_{t\rightarrow \infty }\frac{1}{\eta \left( t\right) }%
\int_{0}^{t}t^{\prime }\sum_{j=1}^{N}\Gamma _{j}\sum_{O=B,C}\rho
_{Oj,Oj}\left( t^{\prime }\right) dt^{\prime } \notag \\
&=&\frac{1}{\overline{\eta }}\int_{0}^{\infty }t^{\prime
}\sum_{j=1}^{N}\Gamma _{j}\sum_{O=B,C}\rho _{Oj,Oj}\left( t^{\prime }\right)
dt^{\prime }, \label{Etau}\end{aligned}$$where usually $$\overline{\eta }=\lim_{t\rightarrow \infty }\eta \left( t\right) =1.$$Therefore, an efficient energy transfer requires not only a perfect transmission efficiency $\eta $ but also a short average time $\tau $.
In Appendix A, we present an equivalent non-Hermitian Hamiltonian method, which can also be utilized to study the dynamics of the open system.
$k$-space representation of the master equation
===============================================
In this section we present the $k$-space representation of the above master equation, so that we can reduce the dynamics of time evolution in some invariant subspace. If all the dissipation and dephasing rates are homogeneous on the B850 BChl ring, i.e., $\Gamma _{j}=\Gamma $ and $\Gamma
_{j}^{\prime }=\Gamma ^{\prime }$, the whole system has translational symmetry. For each unit cell containing three BChls shown in Fig. [model1]{}, we introduce the Fourier transformation, $$O_{k}^{\dag }=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}e^{ikj}O_{j}^{\dag }
\label{fourier}$$for $O=A,B,C$. Then in the $k$-space the Hamiltonian (\[Hparticle\]) is represented as $H_{S}=\sum_{k}H_{k}$ with$$\begin{aligned}
H_{k}& =2J_{1}\cos kA_{k}^{\dag }A_{k}+\{\left( g_{1}+2g_{2}\cos k\right)
\left( A_{k}^{\dag }B_{k}+A_{k}^{\dag }C_{k}\right) \notag \\
& +J_{2}\left[ \left( 1+\delta \right) +\left( 1-\delta \right) e^{-ik}%
\right] B_{k}^{\dag }C_{k}+\mathrm{H.c.}\}.\end{aligned}$$Here $k$ are chosen as discrete values $$k=\frac{2\pi l}{N},\text{ for }l=1,2,\cdots ,N.$$
![(color online) Configuration of the density matrix of the $N=8$ system in the subspace expanded by $\{\left\vert 0\right\rangle , \left\vert
O,k\right\rangle \}$ with $O=A,B,C$ and $k=(2\protect\pi/8)
\times1,2,\cdots,8$. An initial state localized in the $(k_{1},k_{2})$-block can be evolved to other $(k,k_{2}+k-k_{1})$-blocks (black hollow dot-dash squares). Only the diagonal $(k,k)$-blocks (green solid squares) are related to the average transfer time.[]{data-label="romatrix"}](fig3.eps){width="8"}
In the subspace of the single excitation plus the vacuum with the basis$$\{\left\vert 0\right\rangle ,\left\vert O,k\right\rangle \equiv O_{k}^{\dag
}\left\vert 0\right\rangle |k=\frac{2\pi l}{N};l=1,2,\cdots N;O=A,B,C\},$$the general density matrix is decomposed into $$\widehat{\rho }=\widehat{\rho }_{v,v}+\sum_{k_{1},k_{2}}\widehat{\rho }%
_{k_{1},k_{2}}+\sum_{k}\left( \widehat{\rho }_{v,k}+\widehat{\rho }%
_{k,v}\right) . \label{rho}$$where $$\widehat{\rho }_{v,v}=\rho _{v,v}\left\vert 0\right\rangle \left\langle
0\right\vert$$is the vacuum block while $$\widehat{\rho }_{k_{1},k_{2}}=\sum_{O,O^{\prime }=A,B,C}\rho
_{Ok_{1},O^{\prime }k_{2}}\left\vert O,k_{1}\right\rangle \left\langle
O^{\prime },k_{2}\right\vert .$$is called the $\left( k_{1},k_{2}\right) $-block. For fixed $k_{1}$ and $%
k_{2}$, $\rho _{Ok_{1},O^{\prime }k_{2}}$ form a matrix $$\left(
\begin{array}{ccc}
\rho _{Ak_{1},Ak_{2}} & \rho _{Ak_{1},Bk_{2}} & \rho _{Ak_{1},Ck_{2}} \\
\rho _{Bk_{1},Ak_{2}} & \rho _{Bk_{1},Bk_{2}} & \rho _{Bk_{1},Ck_{2}} \\
\rho _{Ck_{1},Ak_{2}} & \rho _{Ck_{1},Bk_{2}} & \rho _{Ck_{1},Ck_{2}}%
\end{array}%
\right) .$$The $k$-space representation of the density matrix is illustrated in Fig. \[romatrix\] for the $N=8$ system.
In the $k$-space, the master equation (\[mastereq\]) is reduced to $$\begin{aligned}
& \frac{d\widehat{\rho}_{k_{1},k_{2}}}{dt}=-i\left( H_{k_{1}}\widehat{\rho }%
_{k_{1},k_{2}}-\widehat{\rho}_{k_{1},k_{2}}H_{k_{2}}\right) \notag \\
& +\sum_{O=B,C}\left\{ \frac{\Gamma^{\prime}}{N}\sum_{k}O_{k}^{\dag}O_{k_{1}}%
\widehat{\rho}_{k_{1},k_{2}}O_{k_{2}}^{\dag}O_{k_{2}+k-k_{1}}\right. \notag
\\
& \left. -\frac{1}{2}\left( \Gamma+\Gamma^{\prime}\right) \left(
O_{k_{1}}^{\dag}O_{k_{1}}\widehat{\rho}_{k_{1},k_{2}}+\widehat{\rho}%
_{k_{1},k_{2}}O_{k_{2}}^{\dag}O_{k_{2}}\right) \right\} \label{drok1k2}\end{aligned}$$ for all the $k_{1},k_{2}$, $$\begin{aligned}
\frac{d\widehat{\rho}_{k,v}}{dt} & =-iH_{k}\widehat{\rho}_{k,v}-\frac{1}{2}%
\left( \Gamma+\Gamma^{\prime}\right) \sum_{O=B,C}O_{k}^{\dag}O_{k}\widehat{%
\rho}_{k,v}, \notag \\
\frac{d\widehat{\rho}_{v,k}}{dt} & =i\widehat{\rho}_{v,k}H_{k}-\frac{1}{2}%
\left( \Gamma+\Gamma^{\prime}\right) \sum_{O=B,C}\widehat{\rho}%
_{v,k}O_{k}^{\dag}O_{k}\end{aligned}$$ for all the $k$, and $$\frac{d\widehat{\rho}_{v,v}}{dt}=\sum_{k}\sum_{O=B,C}\Gamma O_{k}\widehat {%
\rho}_{k,k}O_{k}^{\dag}. \label{dro00}$$ The details of the calculation are shown in Appendix B.
We notice that the equations about $\widehat{\rho }_{k,v}$ and $\widehat{%
\rho }_{v,k}$ are completely decoupled from $\widehat{\rho }_{k_{1},k_{2}}$ and $\widehat{\rho }_{v,v}$. It follows from Eq. (\[drok1k2\]) that when no dephasing exists, i.e., $\Gamma ^{\prime }=0$, the $\left(
k_{1},k_{2}\right) $-block $\widehat{\rho }_{k_{1},k_{2}}$ is decoupled with other $\widehat{\rho }_{k_{1}^{\prime },k_{2}^{\prime }}$ for $\left(
k_{1}^{\prime },k_{2}^{\prime }\right) \neq \left( k_{1},k_{2}\right) $. Thus $\widehat{\rho }_{k_{1},k_{2}}$ only evolves in the $\left(
k_{1},k_{2}\right) $-block. However, when the dephasing is present ($\Gamma
^{\prime }\neq 0$), the term $$\sum_{O=B,C}\frac{\Gamma ^{\prime }}{N}\sum_{k}O_{k}^{\dag }O_{k_{1}}%
\widehat{\rho }_{k_{1},k_{2}}O_{k_{2}}^{\dag }O_{k_{2}+k-k_{1}}$$actually induces the coupling between the $\left( k_{1},k_{2}\right) $-block and the $\left( k,k_{2}+k-k_{1}\right) $-block. The initial $\widehat{\rho }%
_{k_{1},k_{2}}$ may evolves to $\widehat{\rho }_{k,k_{2}+k-k_{1}}$ as time goes by. A typical example of $\left( k,k_{2}+k-k_{1}\right) $-blocks are shown by the black hollow dot-dash squares in Fig. \[romatrix\]. The momentum difference $k_{1}-k_{2}$ is conserved during the evolution since$$k_{2}-k_{1}=(k_{2}+k-k_{1})-k.$$In addition, Eq. (\[dro00\]) means that only the $\left( k,k\right) $-blocks of the density matrix result in energy transfer, which are marked by the $8$ green solid squares in Fig. \[romatrix\]. All the other $k_{1}\neq
k_{2}$ blocks do not affect the transfer efficiency $\eta \left( t\right) $ and average transfer time $\tau $ at all. Especially, the initial component $%
\widehat{\rho }_{k_{1},k_{2}}$ with $k_{1}\neq k_{2}$ will not influence $%
\eta \left( t\right) $ or $\tau $ at any time $t$ afterwards since it cannot evolve to the blocks with $k_{1}=k_{2}$. Therefore, only considering the dynamics of the $\left( k,k\right) $-blocks are enough for the present purpose.
Transfer efficiency and average transfer time with channel decomposition
========================================================================
In this section we use the $k$-space representation of master equation to calculate the average transfer time and transfer efficiency by the standard open quantum system method. As a highly organized array of chlorophyll molecules, the LH2 acts cooperatively to shuttle the energy of photons to elsewhere when sunlight shines on it. In this sense, we use the density matrix$$\widehat{\rho }\left( 0\right) =\sum_{k_{1},k_{2}}\sum_{O,O^{\prime
}=A,B,C}\rho _{Ok_{1},O^{\prime }k_{2}}\left( 0\right) \left\vert
O,k_{1}\right\rangle \left\langle O^{\prime },k_{2}\right\vert
\label{initstate1}$$to describe the excitations in the initial state. From the discussions in the last section, only the $k_{1}=k_{2}=k$ blocks relevant to energy transfer. Therefore, there exists an equivalence class of initial states$$\left[ \widehat{\rho }^{\prime }\left( 0\right) \right] =\left\{ \widehat{%
\rho }\text{ }|\text{ }\left\langle O,k\right\vert \widehat{\rho }\left\vert
O^{\prime },k\right\rangle =\rho _{Ok,O^{\prime }k}\left( 0\right) \right\}$$that results in the same transfer efficiency and average transfer time as that for $\widehat{\rho }\left( 0\right) $. For further use, a special density matrix is chosen from the equivalence class $\widehat{\varrho }%
\left( 0\right) \in \left[ \widehat{\rho }^{\prime }\left( 0\right) \right] $,$$\begin{aligned}
\widehat{\varrho }\left( 0\right) &=&\sum_{k}\sum_{O,O^{\prime }=A,B,C}\rho
_{Ok,O^{\prime }k}\left( 0\right) \left\vert O,k\right\rangle \left\langle
O^{\prime },k\right\vert \notag \\
&=&\sum_{k}\widehat{\varrho }^{\left[ k\right] }\left( 0\right) ,\end{aligned}$$which satisfies $\left\langle O,k_{1}\right\vert \widehat{\varrho }\left(
0\right) \left\vert O^{\prime },k_{2}\right\rangle =\rho _{Ok,O^{\prime
}k}\left( 0\right) $ for $k_{1}=k_{2}=k$, and $\left\langle
O,k_{1}\right\vert \widehat{\varrho }\left( 0\right) \left\vert O^{\prime
},k_{2}\right\rangle =0$ for $k_{1}\neq k_{2}$. $\widehat{\varrho }\left(
0\right) $ plays an equivalent role for determining the transfer efficiency and average transfer time. Here, $$\widehat{\varrho }^{\left[ k\right] }\left( 0\right) =\sum_{O,O^{\prime
}=A,B,C}\rho _{Ok,O^{\prime }k}\left( 0\right) \left\vert O,k\right\rangle
\left\langle O^{\prime },k\right\vert$$is called as the $k$-channel component of the density matrix. According to the above observation, we first choose every $\widehat{\varrho }^{\left[ k%
\right] }\left( 0\right) $ as the initial state to obtain the final state $%
\widehat{\varrho }^{\left[ k\right] }\left( t\right) $, which gives the $k$-channel transfer efficiency at time $t$,$$\eta ^{\left[ k\right] }\left( t\right) =\Gamma \int_{0}^{t}\sum_{k^{\prime
}}\sum_{O=B,C}\varrho _{Ok^{\prime },Ok^{\prime }}^{\left[ k\right] }\left(
t^{\prime }\right) dt^{\prime }, \label{etak}$$and the $k$-channel average transfer time$$\tau ^{\left[ k\right] }=\frac{\Gamma }{\overline{\eta }}\int_{0}^{\infty
}t^{\prime }\sum_{k^{\prime }}\sum_{O=B,C}\varrho _{Ok^{\prime },Ok^{\prime
}}^{\left[ k\right] }\left( t^{\prime }\right) dt^{\prime }. \label{tauk}$$Then we prove a general proposition:
*For an arbitrary initial state* $\widehat{\rho }\left( 0\right) $* (Eq. \[initstate1\]) of the LH2 complex, the transfer efficiency at time* $t$* and the average transfer time are the sum of* $\eta ^{\left[ k\right] }\left( t\right) $* and* $\tau ^{\left[ k%
\right] }$* over all* $k$*-channels, respectively.* $$\begin{aligned}
\eta \left( t\right) &=&\sum_{k}\eta ^{\left[ k\right] }\left( t\right)
\notag \\
\tau &=&\sum_{k}\tau ^{\left[ k\right] }. \label{etatau1}\end{aligned}$$
In order to prove the above proposition we notice that the effective initial state $\widehat{\varrho }\left( 0\right) $ evloves to $$\widehat{\varrho }\left( t\right) =\sum_{k}\widehat{\varrho }^{\left[ k%
\right] }\left( t\right) . \label{etatau2}$$Since the corresponding transfer efficiency and average transfer time of $%
\widehat{\varrho }\left( 0\right) $ are$$\begin{aligned}
\eta \left( t\right) &=&\Gamma \int_{0}^{t}\sum_{k^{\prime
}}\sum_{O=B,C}\varrho _{Ok^{\prime },Ok^{\prime }}\left( t^{\prime }\right)
dt^{\prime } \notag \\
\tau &=&\frac{\Gamma }{\overline{\eta }}\int_{0}^{\infty }t^{\prime
}\sum_{k^{\prime }}\sum_{O=B,C}\varrho _{Ok^{\prime },Ok^{\prime }}\left(
t^{\prime }\right) dt^{\prime }, \label{etatau3}\end{aligned}$$Eq. (\[etatau1\]) is obtained from Eqs. (\[etak\]), (\[tauk\]), ([etatau2]{}), and (\[etatau3\]). Namely, $\eta \left( t\right) $ and $\tau $ are the sum of $\eta ^{\left[ k\right] }\left( t\right) $ and $\tau ^{\left[
k\right] }$ for different momentum $k$ channels.
The present experimental observations [@HuXiChe972] have provided some potential pathways for light-harvesting. One of them originates from the excitations on the B800 BChl ring. It shows that the excitations are transferred to the RC through B800 (LH2) $\rightarrow $ B850 (LH2) $%
\rightarrow $ B850 (another LH2) $\rightarrow \cdots \rightarrow $ B875 (LH1) $\rightarrow $ RC. As to our model, the initial state is specialized as$$\widehat{\rho }\left( 0\right) =\sum_{k_{1},k_{2}}\rho
_{Ak_{1},Ak_{2}}\left( 0\right) \left\vert A,k_{1}\right\rangle \left\langle
A,k_{2}\right\vert . \label{initstate2}$$Accordingly, the $k$-channel component of the effective initial state $%
\widehat{\varrho }\left( 0\right) $ becomes$$\widehat{\varrho }^{\left[ k\right] }\left( 0\right) =\rho _{Ak,Ak}\left(
0\right) \left\vert A,k\right\rangle \left\langle A,k\right\vert =\rho
_{Ak,Ak}\left( 0\right) \widehat{\varrho }^{\left[ A,k\right] }\left(
0\right) .$$Taking$$\widehat{\varrho }^{\left[ A,k\right] }\left( 0\right) =\left\vert
A,k\right\rangle \left\langle A,k\right\vert \label{initAk}$$as the initial state, we obtain the transfer efficiency and the average transfer time $$\begin{aligned}
\eta ^{\left[ A,k\right] }\left( t\right) &=&\Gamma
\int_{0}^{t}\sum_{k^{\prime }}\sum_{O=B,C}\varrho _{Ok^{\prime },Ok^{\prime
}}^{\left[ A,k\right] }\left( t^{\prime }\right) dt^{\prime } \notag \\
\tau ^{\left[ A,k\right] } &=&\frac{\Gamma }{\overline{\eta }}%
\int_{0}^{\infty }t^{\prime }\sum_{k^{\prime }}\sum_{O=B,C}\varrho
_{Ok^{\prime },Ok^{\prime }}^{\left[ A,k\right] }\left( t^{\prime }\right)
dt^{\prime }.\end{aligned}$$Hereafter, the superscript $\left[ A,k\right] $ denotes that the initial state is Eq. (\[initAk\]). Similar to the above analysis about the proposition, we present a corollary:
*The transfer efficiency* $\eta \left( t\right) $* and average transfer time* $\tau $* of the initial state in Eq. ([initstate2]{}) are the weighted average of* $\eta ^{\left[ A,.k\right]
}\left( t\right) $* and* $\tau ^{\left[ A,k\right] }$*, respectively. *$$\begin{aligned}
\eta \left( t\right) &=&\sum_{k}\rho _{Ak,Ak}\left( 0\right) \eta ^{\left[
A,.k\right] }\left( t\right) \notag \\
\tau &=&\sum_{k}\rho _{Ak,Ak}\left( 0\right) \tau ^{\left[ A,k\right] }.
\label{corollary}\end{aligned}$$In the following, we will show the analytical and numerical results of $\eta
^{\left[ A,.k\right] }\left( t\right) $ and $\tau ^{\left[ A,k\right] }$.
First we consider the case without dephasing, i.e., $\Gamma ^{\prime }=0$. The time evolution from initial state $\widehat{\varrho }^{\left[ A,k\right]
}\left( 0\right) $ only takes place in the $\left( k,k\right) $-block. According to Eq. (\[drok1k2\]), the master equation of $\widehat{\rho }%
_{k,k}$ $$\begin{aligned}
\frac{d\widehat{\rho }_{k,k}}{dt}& =-i\left( H_{k}\widehat{\rho }_{k,k}-%
\widehat{\rho }_{k,k}H_{k}\right) \notag \\
& -\frac{\Gamma }{2}\sum_{O=B,C}\left( O_{k}^{\dag }O_{k}\widehat{\rho }%
_{k,k}+\widehat{\rho }_{k,k}O_{k}^{\dag }O_{k}\right) . \label{drokk}\end{aligned}$$gives the average transfer time$$\tau ^{\left[ A,k\right] }=\frac{\Gamma }{\overline{\eta }}\int_{0}^{\infty
}t^{\prime }\sum_{O=B,C}\varrho _{Ok,Ok}^{\left[ A,k\right] }\left(
t^{\prime }\right) dt^{\prime }.$$
When $k=0$, Eq. (\[drokk\]) about $\rho _{Ok,O^{\prime }k}$ is rearranged as a system of differential equations about $v_{j}\left( t\right) $ ($%
j=1,\cdots ,4$) and $v_{5}\left( t\right) =\left[ v_{4}\left( t\right) %
\right] ^{\ast }$:
$$\begin{aligned}
v_{1}& =\rho _{Ak,Ak}, \notag \\
v_{2}& =\rho _{Bk,Bk}+\rho _{Ck,Ck},v_{3}=\rho _{Bk,Ck}+\rho _{Ck,Bk},
\notag \\
v_{4}& =\rho _{Ak,Bk}+\rho _{Ak,Ck},v_{5}=\rho _{Bk,Ak}+\rho _{Ck,Ak}.\end{aligned}$$
It is $$\begin{aligned}
\frac{d}{dt}v_{1}\left( t\right) & =ig_{+}\left[ v_{4}\left( t\right)
-v_{5}\left( t\right) \right] , \notag \\
\frac{d}{dt}v_{2}\left( t\right) & =-ig_{+}\left[ v_{4}\left( t\right)
-v_{5}\left( t\right) \right] -\Gamma v_{2}\left( t\right) , \notag \\
\frac{d}{dt}v_{3}\left( t\right) & =-ig_{+}\left[ v_{4}\left( t\right)
-v_{5}\left( t\right) \right] -\Gamma v_{3}\left( t\right) , \notag \\
\frac{d}{dt}v_{4}\left( t\right) & =2ig_{+}v_{1}\left( t\right) -ig_{+}\left[
v_{2}\left( t\right) +v_{3}\left( t\right) \right] \notag \\
& -\left[ 2i\left( J_{1}-J_{2}\right) +i\Delta \Omega +\frac{\Gamma }{2}%
\right] v_{4}\left( t\right) , \label{vvv}\end{aligned}$$with initial conditions$$v_{1}\left( 0\right) =1,v_{2}\left( 0\right) =v_{3}\left( 0\right)
=v_{4}\left( 0\right) =v_{5}\left( 0\right) =0.$$Here $g_{+}=\left( g_{1}+2g_{2}\right) $. Solving the above differential equations, we obtain$$\begin{aligned}
\tau ^{\left[ A,k=0\right] }& =\frac{\Gamma }{\overline{\eta }}%
\int_{0}^{\infty }t^{\prime }v_{2}\left( t^{\prime }\right) dt^{\prime }
\notag \\
& =\frac{g_{+}^{2}+\left( J_{1}-J_{2}+\Delta \Omega /2\right) ^{2}+\Gamma
_{0}^{2}/4}{g_{+}^{2}\Gamma _{0}}, \label{tauk0}\end{aligned}$$with $\Gamma _{0}=\Gamma /2$, which is independent of the dimerization parameter $\delta $. Similarly, when $k=\pm \pi $, the average transfer time of $\widehat{\varrho }^{\left[ A,k\right] }\left( t_{0}\right) $ is$$\tau ^{\left[ A,k=\pm \pi \right] }=\frac{g_{-}^{2}+\left( J_{1}+J_{2}\delta
-\Delta \Omega /2\right) ^{2}+\Gamma _{0}^{2}/4}{g_{-}^{2}\Gamma _{0}},
\label{taukpi}$$where $g_{-}=\left( g_{1}-2g_{2}\right) $. It is a quadratic function with respect to $\delta $. The optimal parameter $\delta $ with the shortest transfer time satisfies $$\delta _{\mathrm{opt}}^{\left[ A,k=\pm \pi \right] }=\frac{\Delta \Omega
/2-J_{1}}{J_{2}}.$$When $g_{1}=2g_{2}$, Eq. (\[taukpi\]) shows that $\tau ^{\left[ A,k=\pm
\pi \right] }=\infty $, corresponds to $\overline{\eta }=0$, the energy transfer is prevented at this time.
![(color online) The average transfer time $\protect\tau ^{[A,k]}$ of $\widehat{\protect\varrho}^{[A,k]}(0) $ (blue scatter lines) and $\protect%
\tau _{\mathrm{mix}}$ of the initial mixed state $\widehat{\protect\rho} _{%
\mathrm{mix}}^{A}(0)$ (red solid lines) with respect to the dimerization degree $\protect\delta $ of the B850 BChl ring. Here $N=8$, $J_{1}/\Gamma=0.3$, $J_{2}/\Gamma=1$, $g_{1}/\Gamma=0.5$, $
\Delta\Omega / \Gamma=0.1$, $\Gamma^{\prime }/\Gamma=1$, $g_{2}/\Gamma=0$ (upper panel) and $g_{2}/\Gamma=0.125$ (lower panel). $\protect\tau $ is in the unit of $(1/\Gamma )$ and $\overline{\protect\eta }=1$. It shows that each $%
\protect\tau ^{[A,k]}$ ($k\neq 0$) curve has a minimum at $\protect\delta _{%
\mathrm{opt}}^{\left[ A,k\right] }\neq 0$. $\protect\tau _{\mathrm{mix}}$ is the equal-weighted average of a complete set of $\left\{ \protect\tau %
^{[A,k]}\right\} $.[]{data-label="figtau"}](fig4.eps){width="8"}
If the dephasing is present, i.e., $\Gamma ^{\prime }\neq 0$, we can provide approximate solutions for $\tau ^{\left[ A,k=0\right] }$ and $\tau ^{\left[
A,k=\pm \pi \right] }$,$$\begin{aligned}
\tau ^{\left[ A,k=0\right] } \!\!&=& \!\! \frac{g_{+}^{2}\!\left( 4\Gamma
_{s} \! -\Gamma ^{\prime }\right) \! \!/\Gamma \! +\!\left( 2J_{1} \!
-2J_{2} \! +\Delta \Omega \right) ^{2}\! \! +\Gamma _{s}^{2}/4}{%
2g_{+}^{2}\Gamma _{s}} \notag \\
\tau ^{\left[ A,k=\pm \pi \right] } \!\! &=& \!\! \frac{g_{-}^{2}\!\left(
4\Gamma _{s} \! -\Gamma ^{\prime }\right) \!\! /\Gamma \! +\!\left( 2J_{1}
\! +2J_{2}\delta \! -\Delta \Omega \right) ^{2} \!\! +\Gamma _{s}^{2}/4}{%
2g_{-}^{2}\Gamma _{s}}, \label{tauappro}\end{aligned}$$ where $\Gamma _{s}=\Gamma +\Gamma ^{\prime }$. They almost exactly agree with the numerical calculation below, and can also be confirmed by Eq. ([tauk0]{}) and (\[taukpi\]) when $\Gamma ^{\prime }=0$. The details are shown in Appendix C.
Energy transfer efficiency and average transfer time in numerical calculation
=============================================================================
For a general $k$, the analytical solution of $\eta ^{\left[
A,k\right] }\left( t\right) $ and $\tau ^{\left[ A,k\right] }$ is not easy to get. Nevertheless, the numerical results of $\tau
^{\left[ A,k\right] }$ as a function of $\delta $ are plotted as blue scatter lines in Fig. \[figtau\]. Here we have chosen $$\begin{aligned}
N &=&8,\frac{J_{1}}{\Gamma }=0.3,\frac{J_{2}}{\Gamma }=1, \notag \\
\frac{g_{1}}{\Gamma } &=&0.5,\frac{\Delta \Omega }{\Gamma }=0.1,\frac{\Gamma
^{\prime }}{\Gamma }=1,\end{aligned}$$$g_{2}/\Gamma =0$ for the upper panel, $g_{2}/\Gamma =0.125$ for the lower panel, and $t$ is in the unit of $\left( 1/\Gamma \right) $ and is long enough to ensure $\overline{\eta }=1$. It shows that when $k\neq 0$ and $\delta $ varies from $-1$ to $1$, there always exist optimum cases $%
\delta _{\mathrm{opt}}^{\left[ A,k\right] }\neq 0$ with and shorter average transfer time. This fact reflects the enhanced effect of dimerization.
We then take the mixed initial density matrix $\widehat{\rho }\left(
0\right) =\widehat{\rho }_{\mathrm{mix}}^{A}\left( 0\right) $ as an example,$$\begin{aligned}
\widehat{\rho }_{\mathrm{mix}}^{A}\left( 0\right) &=&\sum_{j=1}^{N}\rho
_{Aj,Aj}\left( 0\right) \left\vert A,j\right\rangle \left\langle
A,j\right\vert \notag \\
&=&\sum_{k_{1},k_{2}}\rho _{Ak_{1},Ak_{2}}\left( 0\right) \left\vert
A,k_{1}\right\rangle \left\langle A,k_{2}\right\vert .\end{aligned}$$The weight $\rho _{Ak,Ak}$ always satisfies$$\rho _{Ak,Ak}\left( 0\right) =\frac{1}{N}\sum_{j=1}^{N}\rho _{Aj,Aj}\left(
0\right) =\frac{1}{N}.$$From Eq. (\[corollary\]), the transfer efficiency and the average transfer time of $\widehat{\rho }_{\mathrm{mix}}^{A}$ is$$\begin{aligned}
\eta _{\mathrm{mix}}\left( t\right) &=&\frac{1}{N}\sum_{k}\eta
^{\left[
A,k\right] }\left( t\right) \\
\tau _{\mathrm{mix}} &=&\frac{1}{N}\sum_{k}\tau ^{\left[ A,k\right] },\end{aligned}$$$\tau_{\mathrm{mix}}$ is also verified numerically and shown in Fig. \[figtau\] as the red solid lines.
In order to see the dynamics of the transfer process clearly, we plot $\eta _{\mathrm{mix}}$ with respect to the dimerization degree $\delta$ and time $t$ in Fig. \[figeta\](a), i.e., $\eta
_{\mathrm{mix}}=\eta _{\mathrm{mix}}(\delta,t)$. At a certain instant $t_{0}=12$, $\eta _{\mathrm{mix}}(\delta,t_{0})$ as a function of $\delta$ is plotted in Fig. \[figeta\](b), while for a certain dimerization degree $\delta_{0}=-0.5$, $\eta
_{\mathrm{mix}}(\delta_{0},t)$ as a function of $t$ is plotted in Fig. \[figeta\](c). Here the parameters are chosen as same as the ones in Fig. \[figtau\] except that $g_{2}/\Gamma=0.125$. The contour map Fig. \[figeta\](a) and the profiles of $\eta
_{\mathrm{mix}}(\delta,t)$ in Fig. \[figeta\](b) and (c) show that (1) $\eta _{\mathrm{mix}}(\delta,t)$ increases monotonously as time goes by. In the large $t$ limit, $\eta _{\mathrm{mix}}(\delta,t)$ equals to $1$. (2) At any certain short instant, an optimum $\delta$ can enhance the transfer efficiency.
Similar to $\eta _{\mathrm{mix}}$ and $\tau _{\mathrm{mix}}$, in general, there exists an optimal $\delta _{\mathrm{opt}}\neq 0$ for an arbitrary initial $\widehat{%
\rho }\left( 0\right) $, which means that a suitable distortion of the B850 ring is helpful for the excitation transfer. This result agrees with the x-ray observation that the Mg-Mg distance between neighboring B850 BChls is 9.2Å within the $\alpha \beta $-heterodimer and 8.9Å between the heterodimers reported in Ref. [@HuXiChe96]. The B850 ring is indeed dimerized in nature.
![(color online) (a) The contour map of the transfer efficiency of the initial mixed state $\eta
_{\mathrm{mix}}(\delta,t)$ as a function of dimerization degree $\delta$ and time $t$ for the same setup as that in the lower panel of Fig. \[figtau\]. (b) The profile of $\eta _{\mathrm{mix}}$ along $t_{0}=12$ (black dashed line in (a)). (c) The profile of $\eta _{\mathrm{mix}}$ along $\delta_{0}=-0.5$ (red dot line in (a)). It shows that $\eta _{\mathrm{mix}}(\delta,t)$ increases over time, and an optimum $\delta$ can enhance the transfer efficiency.[]{data-label="figeta"}](fig5.eps){width="8"}
As shown in Fig. \[figtau\], $\tau ^{\left[ A,k=0\right] }$ and $\tau ^{%
\left[ A,k=\pm \pi \right] }$ are particular since nearly all the other $%
\tau ^{\left[ A,k\right] }$ are within the range of $\left[ \tau ^{\left[
A,k=0\right] },\tau ^{\left[ A,k=\pm \pi \right] }\right] $, so is the average transfer time $\tau $ of an arbitrary $\widehat{\rho }\left(
0\right) $. Besides, the absolute value of $\delta _{\mathrm{opt}}^{\left[
A,k=\pm \pi \right] }$ for the $k=\pm \pi $ case is larger than the one of other $\rho \left( 0\right) $, i.e., $\left\vert \delta _{\mathrm{opt}%
}\right\vert \leq \left\vert \delta _{\mathrm{opt}}^{\left[ A,k=\pm \pi %
\right] }\right\vert $. Hence, once we have known the properties of $\tau ^{%
\left[ A,k=0\right] }$ and $\tau ^{\left[ A,k=\pm \pi \right] }$, the behavior of a general $\tau $ can be conjectured to some extend. Compared the lower panel of Fig. \[figtau\] with the upper panel, a larger $%
g_{2}/\Gamma $ can increase $\tau ^{\left[ A,k=\pm \pi \right] }$ but decrease $\tau ^{\left[ A,k=0\right] }$. In the $g_{2}/\Gamma =0.125$ case, the homogeneous pure state $\widehat{\varrho }^{\left[ A,k=0\right] }\left(
0\right) $ is better than the mixed state $\widehat{\rho }_{\mathrm{mix}%
}^{A}\left( 0\right) $ for energy transport. However, the upper panel with $%
g_{2}/\Gamma =0$ gives the contrary result.
The minimal $\tau ^{\left[ A,k=\pm \pi \right] }$ is reachable at $\delta _{%
\mathrm{opt}}^{\left[ A,k=\pm \pi \right] }=\left( \Delta \Omega
/2-J_{1}\right) /J_{2}$, $$\tau _{\min }^{\left[ A,k=\pm \pi \right] }=\frac{\left( g_{1}-2g_{2}\right)
^{2}\left( 4\Gamma +3\Gamma ^{\prime }\right) +\Gamma \left( \Gamma +\Gamma
^{\prime }\right) ^{2}/4}{2\left( g_{1}-2g_{2}\right) ^{2}\Gamma \left(
\Gamma +\Gamma ^{\prime }\right) }.$$In the toy model illustrated in Fig. \[figtau\], $J_{2}>J_{1}$ and $%
g_{2}<g_{1}$. When $$g_{2}/g_{1}=\gamma _{g}=\frac{1}{2}+\xi ^{2}-\xi \sqrt{1+\xi ^{2}},$$we have $\tau _{\min }^{\left[ A,k=\pm \pi \right] }=\tau ^{\left[ A,k=0%
\right] }$, where $$\xi =\frac{\left( \Gamma +\Gamma ^{\prime }\right) }{2\left(
2J_{2}-2J_{1}-\Delta \Omega \right) }.$$On the side of $0<g_{2}/g_{1}<\gamma _{g}$, $\tau _{\min }^{\left[ A,k=\pm
\pi \right] }<\tau ^{\left[ A,k=0\right] }$, while on the other side $%
\gamma _{g}<g_{2}/g_{1}<1$, $\tau _{\min }^{\left[ A,k=\pm \pi \right]
}>\tau ^{\left[ A,k=0\right] }$.
In general, the shortest average transfer time of an arbitrary initial $%
\widehat{\rho }\left( 0\right) $ is within the range of $\left[ \tau _{\min
}^{\left[ A,k=\pm \pi \right] },\tau ^{\left[ A,k=0\right] }\right] $. The mean value of $\tau _{\min }^{\left[ A,k=\pm \pi \right] }$ and $\tau ^{%
\left[ A,k=0\right] }$ can roughly reflect the influence of parameters on the transfer process,$$\overline{\tau }=\frac{1}{2}\left( \tau _{\min }^{\left[ A,k=\pm \pi \right]
}+\tau ^{\left[ A,k=0\right] }\right) .$$In Fig. \[tao2\], we plot $\overline{\tau }$ with respect to $g_{1}/\Gamma
$ for different $J_{2}/\Gamma =0,1,2,3$. Here, $J_{1}/\Gamma =0.3$, $\delta
=\left( \Delta \Omega /2-J_{1}\right) /J_{2}$, $g_{2}/g_{1}=0.25$, $\Delta
\Omega /\Gamma =0.1$, $\Gamma ^{\prime }/\Gamma =0.5$, and $\overline{\tau }$ is in the unit of $\left( 1/\Gamma \right) $. It shows that $\overline{\tau }
$ decreases monotonously as $g_{1}/\Gamma $ increases. In the short $%
g_{1}/\Gamma $ limit, $\overline{\tau }$ tends to infinity, which is reasonable since the two BChl rings are decoupled in this case. Moreover, $%
\overline{\tau }$ is larger when $J_{2}/\Gamma $ is larger.
![(color online) Plots of $\overline{\protect\tau}$ as a function of the dissipation ratio $g_{1}/\Gamma$ with $J_{2}/\Gamma=0,1,2,3$, where $%
J_{1}/\Gamma=0.3$, $\protect\delta=(\Delta\Omega/2-J_{1})/J_{2}$, $%
g_{2}/g_{1}=0.25$, $\Delta\Omega / \Gamma=0.1$, $\Gamma^{\prime}/\Gamma=0.5$, and $\overline{\protect\tau}$ is in the unit of $(1/\Gamma) $. It shows that $\overline{\protect\tau}$ decreases as $g_{1}/\Gamma$ increases, but increases with the increasing of $J_{2}/\Gamma$. []{data-label="tao2"}](fig6.eps){width="8"}
Conclusion
==========
In summary, we have studied the craggy transfer in light-harvesting complex with dimerization. We employed the open quantum system approach to show that the dimerization of the B850 BChl ring can enhance the transfer efficiency and shorten the average transfer time for different initial states with various quantum superposition properties. Actually our present investigation only focuses on a crucial stage in photosynthesis – the energy transfer, which is carried by the coherent excitations in the typical light-harvesting complex II (LH2). Here the LH2 is modeled as two coupled bacteriochlorophyll (BChl) rings. With this modeling, the ordinary photosynthesis is roughly described as three basic steps: 1) stimulate an excitation in LH2; 2) transfer it to another LH2 or LH1; 3) the energy causes the chemical reaction that converts carbon dioxide into organic compounds. Namely, the excitations are transferred to the RC through B800 (LH2) $\rightarrow $ B850 (LH2) $\rightarrow $ B850 (another LH2) $\rightarrow \cdots \rightarrow $ B875 (LH1) $\rightarrow $ RC. Obviously, the first two are of physics, thus our present approach can be generalized to investigate these physical processes. Although photosynthesis happens in different fashions for different species, some features are always in common from the point of view of physics. For example, the photosynthetic process always starts from the light absorbing and energy transfer.
Another important issues of the photosynthesis physics concerns about the quantum natures of light [@Glauber1; @Glauber2]. Since the experiments have illustrated the role of the quantum coherence of collective excitations in LH complexes, it is quite natural to believe that the excitation coherence may be induced by the higher coherence of photon. Therefore, in a forthcoming paper we will report our systematical investigation on how the statistical properties of quantum light affects the photosynthesis.
This work is supported by NSFC No. 10474104, 60433050, 10874091 and No. 10704023, NFRPC No. 2006CB921205 and 2005CB724508.
Equivalent non-Hermitian Hamiltonian
====================================
In the case without dephasing, i.e., $\Gamma _{j}^{\prime }=0$, an equivalent non-Hermitian Hamiltonian is introduced to study the dynamics of the open system, $$H=H_{S}-i\sum_{j=1}^{N}\frac{\Gamma _{j}}{2}\left( B_{j}^{\dag
}B_{j}+C_{j}^{\dag }C_{j}\right) . \label{nonHermitian}$$The equivalence between Eq. (\[nonHermitian\]) and (\[mastereq\]) is shown as follows. On the one hand, the Schrödinger equation$$i\frac{d}{dt}\left\vert \psi \right\rangle =\left[ H_{S}-i\sum_{j=1}^{N}%
\frac{\Gamma _{j}}{2}\left( B_{j}^{\dag }B_{j}+C_{j}^{\dag }C_{j}\right) %
\right] \left\vert \psi \right\rangle$$and its Hermitian conjugate$$-i\frac{d}{dt}\left\langle \psi \right\vert =\left\langle \psi \right\vert %
\left[ H_{S}+i\sum_{j=1}^{N}\frac{\Gamma _{j}}{2}\left( B_{j}^{\dag
}B_{j}+C_{j}^{\dag }C_{j}\right) \right] ,$$gives the evolution equation of the density matrix $\widehat{\rho }%
=\left\vert \psi \right\rangle \left\langle \psi \right\vert $,$$\begin{aligned}
\frac{d\widehat{\rho }}{dt}& =\left( \frac{d}{dt}\left\vert \psi
\right\rangle \right) \left\langle \psi \right\vert +\left\vert \psi
\right\rangle \left( \frac{d}{dt}\left\langle \psi \right\vert \right)
\notag \\
& =-i\left[ H_{S},\widehat{\rho }\right] -\sum_{j=1}^{N}\frac{\Gamma _{j}}{2}%
\left\{ B_{j}^{\dag }B_{j}+C_{j}^{\dag }C_{j},\widehat{\rho }\right\} .
\label{ro1}\end{aligned}$$
On the other hand, when the dephasing terms are absent, the master equation Eq. (\[mastereq\]) becomes$$\frac{d\widehat{\rho }}{dt}=-i\left[ H_{S},\widehat{\rho }\right]
+\sum_{j=1}^{N}\sum_{O=B,C}\Gamma _{j}[O_{j}\widehat{\rho }O_{j}^{\dag }-%
\frac{1}{2}\left\{ O_{j}^{\dag }O_{j},\widehat{\rho }\right\} ]. \label{ro2}$$The above equation is written on the expanded Hilbert space with an additive vacuum basis $\left\vert 0\right\rangle $. Compared with Eq. (\[ro1\]), the additive term in Eq. (\[ro2\]) $\sum_{j=1}^{N}\sum_{O=B,C}$ $\Gamma
_{j}O_{j}\widehat{\rho }O_{j}^{\dag }$ has only contribution to $d\widehat{%
\rho }_{v,v}/dt$, which does not change the dynamics of the system. The Eqs. (\[ro1\]) and (\[ro2\]) are equivalent for determining the time evolution of $\widehat{\rho }_{Oj,O^{\prime }j^{\prime }}$.
For the non-Hermitian Hamiltonian, the corresponding transfer efficiency and the average transfer time are$$\begin{aligned}
\eta \left( t\right) &=&\int_{0}^{t}\sum_{j=1}^{N}\Gamma
_{j}\sum_{O=B,C}\left\vert \left\langle 0\right\vert O_{j}\left\vert \psi
\left( t^{\prime }\right) \right\rangle \right\vert ^{2}dt^{\prime } \notag
\\
\tau &=&\frac{1}{\overline{\eta }}\int_{0}^{\infty }t^{\prime
}\sum_{j=1}^{N}\Gamma _{j}\sum_{O=B,C}\left\vert \left\langle 0\right\vert
O_{j}\left\vert \psi \left( t^{\prime }\right) \right\rangle \right\vert
^{2}dt^{\prime } \label{etataunonH}\end{aligned}$$Due to the equivalence of the non-Hermitian Hamiltonian and the dissipative master equation, the results of Eq. (\[etataunonH\]) are as same as the ones calculated by Eq. Eqs. (\[Eeta\]) and (\[Etau\]). The non-Hermitian Hamiltonian method has an advantage over the master equation one for saving computer time. Instead of $N^{2}$ equations, only a system of $N$ equations are needed to be solved in the non-Hermitian Hamiltonian case.
However, when the dephasing terms are present, there is no equivalent non-Hermitian Hamiltonian. In this case, compared with Eq. (\[ro1\]), the additional term $\sum_{O=B,C}\Gamma _{j}^{\prime }$ $O_{j}^{\dag }O_{j}%
\widehat{\rho }O_{j}^{\dag }O_{j}$ cannot be omitted any more. It can also affect the evolution of the density matrix of the LH2 system.
Transform the master equation to the $k$-space
==============================================
In this section, we will transform the master equation from the real space (Eqs. (\[mastereq\])-(\[dephasing\])) to the $k$-space (Eqs. ([drok1k2]{})-(\[dro00\])). Since $H_{S}=\sum_{k}H_{k}$, and $\widehat{\rho }$ is expressed as Eq. (\[rho\]), we have$$\begin{aligned}
-i\left[ H_{S},\widehat{\rho }\right] &=&-i\left[ \sum_{k}H_{k},%
\sum_{k_{1},k_{2}}\widehat{\rho }_{k_{1},k_{2}}+\sum_{k}\left( \widehat{\rho
}_{v,k}+\widehat{\rho }_{k,v}\right) \right] \notag \\
&=&-i\sum_{k_{1},k_{2}}\left( H_{k_{1}}\widehat{\rho }_{k_{1},k_{2}}-%
\widehat{\rho }_{k_{1},k_{2}}H_{k_{2}}\right) \notag \\
&&-i\sum_{k}\left( H_{k}\widehat{\rho }_{k,v}-\widehat{\rho }%
_{v,k}H_{k}\right) . \label{B1}\end{aligned}$$Here $\left[ H_{S},\widehat{\rho }_{v,v}\right] =0$ since they are in the different subspaces.
According to the Fourier transformation Eq. (\[fourier\]), the term $%
\sum_{j}O_{j}\widehat{\rho }O_{j}^{\dag }$ becomes$$\begin{aligned}
&&\sum_{j}O_{j}\widehat{\rho }O_{j}^{\dag }=\sum_{k_{1},k_{2},k_{3},k_{4}}%
\frac{1}{N}\sum_{j}e^{i\left( k_{3}-k_{4}\right) j}O_{k_{3}}\widehat{\rho }%
_{k_{1},k_{2}}O_{k_{4}}^{\dag } \notag \\
&=&\sum_{k_{1},k_{2}}\frac{1}{N}\sum_{j}e^{i\left( k_{1}-k_{2}\right)
j}O_{k_{1}}\widehat{\rho }_{k_{1},k_{2}}O_{k_{2}}^{\dag } \notag \\
&=&\sum_{k_{1},k_{2}}\delta _{k_{1},k_{2}}O_{k_{1}}\widehat{\rho }%
_{k_{1},k_{2}}O_{k_{2}}^{\dag }=\sum_{k}O_{k}\widehat{\rho }%
_{k,k}O_{k}^{\dag }.\end{aligned}$$The term $\sum_{j}O_{j}^{\dag }O_{j}\widehat{\rho }$ is transformed as$$\begin{aligned}
&&\sum_{j}O_{j}^{\dag }O_{j}\widehat{\rho } \notag \\
&=&\sum_{k_{1},k_{3},k_{4}}\frac{1}{N}\sum_{j}e^{-i\left( k_{3}-k_{4}\right)
j}O_{k_{3}}^{\dag }O_{k_{4}}\left( \sum_{k_{2}}\widehat{\rho }_{k_{1},k_{2}}+%
\widehat{\rho }_{k_{1},v}\right) \notag \\
&=&\sum_{k_{1},k_{3}}\frac{1}{N}\sum_{j}e^{-i\left( k_{3}-k_{1}\right)
j}O_{k_{3}}^{\dag }O_{k_{1}}\left( \sum_{k_{2}}\widehat{\rho }_{k_{1},k_{2}}+%
\widehat{\rho }_{k_{1},v}\right) \notag \\
&=&\sum_{k_{1},k_{3}}\delta _{k_{1},k_{3}}O_{k_{3}}^{\dag }O_{k_{1}}\left(
\sum_{k_{2}}\widehat{\rho }_{k_{1},k_{2}}+\widehat{\rho }_{k_{1},v}\right)
\notag \\
&=&\sum_{k_{1},k_{2}}O_{k_{1}}^{\dag }O_{k_{1}}\widehat{\rho }%
_{k_{1},k_{2}}+\sum_{k}O_{k}^{\dag }O_{k}\widehat{\rho }_{k,v}.\end{aligned}$$Similarly,$$\sum_{j}\widehat{\rho }O_{j}^{\dag }O_{j}=\sum_{k_{1},k_{2}}\widehat{\rho }%
_{k_{1},k_{2}}O_{k_{2}}^{\dag }O_{k_{2}}+\sum_{k}\widehat{\rho }%
_{v,k}O_{k}^{\dag }O_{k}.$$Finally, the term $\sum_{j}O_{j}^{\dag }O_{j}\widehat{\rho }O_{j}^{\dag
}O_{j}$ is written as$$\begin{aligned}
&&\sum_{j}O_{j}^{\dag }O_{j}\widehat{\rho }O_{j}^{\dag }O_{j} \notag \\
&=&\sum_{k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}}\frac{1}{N^{2}}%
\sum_{j}e^{-i\left( k_{3}-k_{4}+k_{5}-k_{6}\right) j}O_{k_{3}}^{\dag
}O_{k_{4}}\widehat{\rho }_{k_{1},k_{2}}O_{k_{5}}^{\dag }O_{k_{6}} \notag \\
&=&\sum_{k_{1},k_{2},k_{3},k_{6}}\frac{1}{N^{2}}\sum_{j}e^{-i\left(
k_{3}-k_{1}+k_{2}-k_{6}\right) j}O_{k_{3}}^{\dag }O_{k_{1}}\widehat{\rho }%
_{k_{1},k_{2}}O_{k_{2}}^{\dag }O_{k_{6}} \notag\end{aligned}$$$$\begin{aligned}
&=&\frac{1}{N}\sum_{k_{1},k_{2},k_{3},k_{6}}\delta
_{k_{6},k_{2}+k_{3}-k_{1}}O_{k_{3}}^{\dag }O_{k_{1}}\widehat{\rho }%
_{k_{1},k_{2}}O_{k_{2}}^{\dag }O_{k_{6}} \notag \\
&=&\frac{1}{N}\sum_{k_{1},k_{2},k}O_{k}^{\dag }O_{k_{1}}\widehat{\rho }%
_{k_{1},k_{2}}O_{k_{2}}^{\dag }O_{k_{2}+k-k_{1}}. \label{B5}\end{aligned}$$Therefore, Eqs. (\[drok1k2\])-(\[dro00\]) are obtained by summarizing Eqs. (\[B1\])-(\[B5\]).
Approximative master equations for special cases
================================================
In the cases of $k=0$ and $k=\pm \pi $, we have the approximate master equation,$$\begin{aligned}
\frac{d\widehat{\rho }_{k,k}}{dt} &=&-i\left( H_{k}\widehat{\rho }_{k,k}-%
\widehat{\rho }_{k,k}H_{k}\right) +\sum_{O=B,C}\left\{ \Gamma ^{\prime
}O_{k}^{\dag }O_{k}\widehat{\rho }_{k,k}O_{k}^{\dag }O_{k}\right. \notag \\
&&\left. -\frac{\Gamma +\Gamma ^{\prime }}{2}\left( O_{k}^{\dag }O_{k}%
\widehat{\rho }_{k,k}+\widehat{\rho }_{k,k}O_{k}^{\dag }O_{k}\right)
\right\} . \label{drokkgammaD}\end{aligned}$$It is verified numerically that the term $\Gamma ^{\prime }O_{k}^{\dag }O_{k}%
\widehat{\rho }_{k,k}O_{k}^{\dag }O_{k}$ in Eq. (\[drokkgammaD\]) plays the same role as $(\Gamma ^{\prime }/N)\sum_{k^{\prime }}O_{k^{\prime
}}^{\dag }O_{k}\widehat{\rho }_{k,k}O_{k}^{\dag }O_{k^{\prime }}$ in Eq. (\[drok1k2\]) for $k=0,\pm \pi $. For the $\left( k=0,k=0\right) $-block, Eq. (\[vvv\]) becomes$$\begin{aligned}
\frac{d}{dt}v_{1}\left( t\right) & =ig_{+}\left[ v_{4}\left( t\right)
-v_{5}\left( t\right) \right] , \notag \\
\frac{d}{dt}v_{2}\left( t\right) & =-ig_{+}\left[ v_{4}\left( t\right)
-v_{5}\left( t\right) \right] -\Gamma v_{2}\left( t\right) , \notag \\
\frac{d}{dt}v_{3}\left( t\right) & =-ig_{+}\left[ v_{4}\left( t\right)
-v_{5}\left( t\right) \right] -\left( \Gamma +\Gamma ^{\prime }\right)
v_{3}\left( t\right) , \notag \\
\frac{d}{dt}v_{4}\left( t\right) & =2ig_{+}v_{1}\left( t\right) -ig_{+}\left[
v_{2}\left( t\right) +v_{3}\left( t\right) \right] \notag \\
& -\left[ 2i\left( J_{1}-J_{2}\right) +i\Delta \Omega +\frac{\Gamma +\Gamma
^{\prime }}{2}\right] v_{4}\left( t\right) ,\end{aligned}$$Solving the above differential equation we have $\tau ^{\left[ A,k=0\right]
} $ shown in Eq. (\[tauappro\]). The average transfer time $\tau ^{\left[
A,k=\pm \pi \right] }$ for the $k=\pm \pi $ channel is also obtained similarly.
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title: |
Supplementary Material:\
Learning to Align Images using Weak Geometric Supervision
---
thanks
|
---
abstract: 'In this paper we extend the constructions of Boava and Exel to present the C$^*$-algebra associated with an injective endomorphism of a group with finite cokernel as a partial group algebra and consequently as a partial crossed product. With this representation we present another way to study such C$^*$-algebras, only using tools from partial crossed products.'
author:
- Felipe Vieira
title: 'C$^*$-algebras of endomorphisms of groups with finite cokernel and partial actions'
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Definition]{} \[theorem\][Example]{}
[^1]
Introduction
============
Consider an injective endomorphism $\varphi$ of a discrete countable group $G$ with unit $\{e\}$ with finite cokernel i.e, $$\label{eqintro1}
\left|\dfrac{G}{\varphi(G)} \right|<\infty,$$
as above, for $H$ subgroup of $G$, we use $\frac{G}{H}$ to denote the set of left cosets of $H$ in $G$. Analyzing the natural representation of $G$ and $\varphi$ inside $\mathcal{L}(l^2(G))$ we construct a concrete C$^*$-algebra $C_r^*[\varphi]\subseteq \mathcal{L}(l^2(G))$ and a universal one denoted by $\mathds{U}[\varphi]$. Such constructions were presented by Hirshberg in [@Hirsh], and were later generalized by Cuntz and Vershik in [@CunVer] and also in [@Viei].
Using a semigroup crossed product description of $\mathds{U}[\varphi]$ implies the existence of a (full corner) group crossed product description of it ([@Cuntz2], [@CuntzTopMarkovII] and [@Laca1]), but it is not the only way to represent it as a crossed product: analogously to the work of G. Boava and R. Exel in [@BoEx] one can show that $\mathds{U}[\varphi]$ has a partial group crossed product description, which can also be related to an inverse semigroup crossed product by [@ExVi].
We present in this paper the latter construction cited above and show the simplicity of $\mathds{U}[\varphi]$, which is part of the conclusions in [@Hirsh], using only the partial group crossed product description of that C$^*$-algebra.
Definition
==========
We repeat the constructions of [@Hirsh]. Let $G$ be a discrete countable group with unit $e$ and $\varphi$ an injective endomorphism (monomorphism) of $G$ with finite cokernel (\[eqintro1\]).
Consider the Hilbert space $l^2(G)$ with orthonormal basis $\xi_h$, taking every element of $G$ to 0 apart from the element $h$, which goes to 1. Define the following bounded operators on $l^2(G)$: $$U_g(\xi_h)=\xi_{gh}$$
and $$S(\xi_g)=\xi_{\varphi(g)}.$$
The invertibility property of groups and the injectivity of the endomorphism $\varphi$ imply that the $U_g$’s are unitary operators and $S$ is an isometry respectively. Therefore we define the following C$^*$-algebra.
\[defi1red\]We denote $C_r^*[\varphi]$ the reduced C$^*$-algebra of $\varphi$, to be the C$^*$-subalgebra of $\mathcal{L}(l^2(G))$ generated by the above defined unitaries $\{U_g:\;g\in G\}$ and isometry $S$.
Inspired by the properties of the operators above:
\[defi1\]We call $\mathds{U}[\varphi]$ the universal C$^*$-algebra generated by the unitaries $\{u_g:\;g\in G\}$ and one isometry $s$ such that:
1. $u_gu_h=u_{gh}$;
2. $su_g=u_{\varphi(g)}s$;
3. ${\displaystyle}\sum_{g\in G/\varphi(G)}u_gss^*u_{g^{-1}}=1$;
for all $g$, $h\in G$.
As the universal C$^*$-algebra above is defined using relations satisfied by the generators of the reduced one, obviously there is a canonical surjective $*$-homomorphism from $\mathds{U}[\varphi]$ onto $C_r^*[\varphi]$.
Note that the conditions (i) and (ii) above can be merged into the relation $$u_gs^nu_hs^m=u_{g\varphi^n(h)}s^{n+m}.$$
By (ii) we have, for $g\in G$, the obvious relations $$u_{g}s^*=s^*u_{\varphi(g)}$$
and $$u_{\varphi(g)}ss^*=ss^*u_{\varphi(g)}.$$
Also note that in (iii) there is no ambiguity if we choose different representatives of the cosets: $$\begin{split}
u_{g\varphi(h)}ss^*u_{(g\varphi(h))^{-1}}&=u_gu_{\varphi(h)}ss^*u_{\varphi(h^{-1})}u_{g^{-1}}=u_gss^*u_{\varphi(h)}u_{\varphi(h^{-1})}u_{g^{-1}}\\
&=u_gss^*u_{g^{-1}}.
\end{split}$$
Condition (iii) implies that $u_gss^*u_{g^{-1}}$ and $u_hss^*u_{h^{-1}}$ are orthogonal projections if $g^{-1}h\notin\varphi(G)$, so the multiplication can be described as: $$u_gss^*u_{g^{-1}}u_hss^*u_{h^{-1}}=\left\{
\begin{array}{cl}
u_gss^*u_{g^{-1}}, & \hbox{if }h\in g\varphi(G); \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
This extends to the family of elements of type $u_gs^n{s^*}^nu_{g^{-1}}$ for any $n\in\mathds{N}$ $$u_gs^n{s^*}^nu_{g^{-1}}u_hs^n{s^*}^nu_{h^{-1}}=\left\{
\begin{array}{cl}
u_gs^n{s^*}^nu_{g^{-1}}, & \hbox{if }h\in g\varphi^n(G); \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
And note that, for $g$, $h\in G$ and $n\geq m\in\mathds{N}$: $$\begin{split}
&u_gs^n{s^*}^nu_{g^{-1}}u_hs^m{s^*}^mu_{h^{-1}}\\
&=u_gs^n{s^*}^nu_{g^{-1}}u_hs^m\left({\displaystyle}\sum_{k\in\frac{G}{\varphi^{n-m}(G)}}u_ks^{n-m}{s^*}^{n-m}u_{k^{-1}}\right){s^*}^mu_{h^{-1}}\\
&=u_gs^n{s^*}^nu_{g^{-1}}\left({\displaystyle}\sum_{k\in\frac{G}{\varphi^{n-m}(G)}}u_{h\varphi^m(k)}s^n{s^*}^nu_{(h\varphi^m(k))^{-1}}\right)\\
&=\left\{
\begin{array}{ll}
u_gs^n{s^*}^nu_{g^{-1}},&\hbox{ if }h\varphi^m(k)\in g\varphi^n(G)\hbox{ for some }k\in\frac{G}{\varphi^{n-m}(G)}; \\
0,\hbox{ otherwise}.
\end{array}
\right.
\end{split}$$
Crossed product description of $\mathds{U}[\varphi]$ {#sectioncpdescr}
====================================================
In this section we present a semigroup crossed product description of $\mathds{U}[\varphi]$. The semigroup crossed product definition which we will use is the same as presented in Appendix A of [@Li1], via covariant representations. In our case the semigroup implementing the action will be the semidirect product $$S:=G\rtimes_\varphi\mathds{N}=\{(g,n)\;:\;g\in G, n\in\mathds{N}\}$$
with product $$(g,n)(h,m)=(g\varphi^n(h),n+m).$$
We will also show that the action implemented by $S$ can be split i.e, the semigroup crossed product by $S$ can be seen as a semigroup crossed product by $\mathds{N}$. This crossed product description is a great tool to prove some properties of $\mathds{U}[\varphi]$: we will show that when $G$ is amenable this C$^*$-algebra is nuclear and satisfies UCT. Secondly, that description allows one to use the six-term exact sequence introduced by M. Khoshkam and G. Skandalis in [@Khoska] on $\mathds{U}[\varphi]$.
Moreover, due to M. Laca [@Laca1], sometimes it is possible to see semigroup crossed products as full corners of group ones, which implies that both are Morita equivalent and therefore have the same K-groups. And in case the semigroup action is implemented by $\mathds{N}$, Laca’s dilation turns this $\mathds{N}$-action into a $\mathds{Z}$-action, which fits the requirements to use the classical Pimsner-Voiculescu exact sequence [@Pivo1].
Set $$\overline{G}:={\displaystyle}\lim_{\leftarrow}\left\{\frac{G}{\varphi^m(G)}: p_{m,l+m}\right\}$$
where $$p_{m,l+m}:\dfrac{G}{\varphi^{l+m}(G)}\rightarrow\dfrac {G}{\varphi^m(G)}$$
is the canonical projection. We can see $\overline{G}$ as $$\overline{G}=\left\{(g_m)_m\in{\displaystyle}\prod_{m\in\mathds{N}}\frac{G}{\varphi^m(G)}:\;p_{m,l+m}(g_{l+m})=g_m\right\},$$
with the induced topology on the product ${\displaystyle}\prod_{m\in\mathds{N}}\frac{G}{\varphi^m(G)}$, where each finite set $\dfrac{G}{\varphi^m(G)}$ carries the discrete topology, implying that $\overline{G}$ is a compact space.\
Furthermore, we have the map $$\begin{split}
G&\rightarrow\overline{G}\\
g&\mapsto (g)_m,
\end{split}$$
which is an embedding when $\varphi$ is pure. Also set $$\mathcal{G}:=\lim_{\rightarrow}\{\mathcal{G}_m:\phi_{l+m,m}\}$$
where $\mathcal{G}_m=\overline{G}$ for all $m\in\mathds{N}$ and $\phi_{l+m,m}=\varphi^l$. We can see $\mathcal{G}$ as $$\mathcal{G}={\displaystyle}\bigcup^._{m\in\mathds{N}}\mathcal{G}_m\diagup\thicksim$$
with $x_l\sim y_m$ if and only if $\varphi^m(x_l)=\varphi^l(y_m)$, $x_l\in\mathcal{G}_l$ and $y_m\in\mathcal{G}_m$. Note that $\mathcal{G}$ is a locally compact set.
Denote by $q$ the canonical projection $$q:{\displaystyle}\bigcup_{m\in\mathds{N}}^.\mathcal{G}_m\rightarrow\mathcal{G},$$
and $i_m$ the embedding $$\begin{array}{cccccc}
i_m: & \overline{G} & = & \mathcal{G}_m & \hookrightarrow & \mathcal{G} \\
& x & = & x & \mapsto & q(x).
\end{array}$$
Again we have the identification $$\begin{split}
\overline{G}&\hookrightarrow\mathcal{G}\\
x&\mapsto i_0(x).
\end{split}$$
*\[obs12\] Note that if we suppose that our endomorphism $\varphi$ is *totally normal*, i.e. all the $\varphi^m(G)$ are normal subgroups of $G$, then $\overline{G}$ and $\mathcal{G}$ will be groups; one just has to consider the componentwise multiplication in $\overline{G}$ and $$i_m(x)i_l(y)=i_{l+m}(xy),\; \forall\; x, y\in\overline{G}$$*
on $\mathcal{G}$.
\[prop3\]The map $$\begin{split}
\alpha: C^*(P)&\rightarrow C(\overline{G})\\
u_gs^n{s^*}^nu_{g^{-1}}&\mapsto p_{g\varphi^n(\overline{G})},
\end{split}$$
where the latter denotes the characteristic function on the subset $g\varphi^n(\overline{G})\subseteq\overline{G}$, is an isomorphism.
It is clear that $C^*(P)$ is the inductive limit of $$D_m:=C^*\left(\left\{u_gs^m{s^*}^mu_{g^{-1}}:g\in\frac{G}{\varphi^m(G)}\right\}\right)$$
with the inclusions (using (iii) of Definition \[defi1\]) $$\begin{split}
D_m &\hookrightarrow D_{l+m}\\
u_gs^m{s^*}^mu_{g^{-1}}&\mapsto{\displaystyle}\sum_{h\in\frac{G}{\varphi^l(G)}}u_{g\varphi^m(h)}s^{l+m}{s^*}^{l+m}u_{\varphi^m(h^{-1})g^{-1}}.
\end{split}$$
Furthermore the pairwise orthogonality of the projections $u_gs^m{s^*}^mu_{g^{-1}}$ for fixed $m\in\mathds{N}$ implies that $$spec(D_m)\cong\dfrac{G}{\varphi^m(G)}$$
with $$\begin{split}
spec(D_{l+m})&\rightarrow spec(D_m)\\
\chi &\mapsto\chi|_{D_m}
\end{split}$$
corresponding to $$\begin{split}
p_{l+m,m}:\frac{G}{\varphi^{l+m}(G)}&\rightarrow\frac{G}{\varphi^m(G)}\\
g\varphi^{l+m}(G) &\mapsto g\varphi^m(G).
\end{split}$$
Therefore $$spec(C^*(P))\cong{\displaystyle}\lim_{\leftarrow}\left\{\frac{G}{\varphi^m(G)}:p_{m,l+m}\right\}=\overline{G}.$$
Thus we get the isomorphism $$\begin{split}
\alpha:C^*(P) &\rightarrow C(\overline{G})\\
u_gs^m{s^*}^mu_{g^{-1}}&\mapsto p_{g\varphi^m(\overline{G})}.
\end{split}$$
\[defi2\]The stabilization of $\mathds{U}[\varphi]$, denoted by $\mathds{U}^s[\varphi]$, is the inductive limit of the system $\{\mathds{U}_m^s[\varphi]:\psi_{m,l+m}\}$ where, $\forall\; m\in\mathds{N}$, $\mathds{U}_m^s[\varphi]=\mathds{U}[\varphi]$ and $$\begin{split}
\psi_{m,l+m}:\mathds{U}[\varphi]&\rightarrow\mathds{U}[\varphi]\\
x&\mapsto s^lx{s^*}^l.
\end{split}$$
Furthermore define $C^*(P)^s={\displaystyle}\lim_{\rightarrow}\{C^*(P)_m^s:\psi_{m,l+m}\}$ with $C^*(P)_m^s=C^*(P)$ and $\psi_{m,l+m}$ as above.
\[prop4\]We have $C^*(P)^s\cong C_0(\mathcal{G})$.
The maps $\psi_{m,l+m}$, conjugated by $\alpha$, give maps $$\widetilde{\psi}_{m,l+m}:=\alpha\circ\psi_{m,l+m}\circ\alpha^{-1}: C(\overline{G})\rightarrow C(\overline{G}),$$
where $\widetilde{\psi}_{m,l+m}(f)(x)=f(\varphi^{-l}(x))p_{\varphi^l(\overline{G})}(x):$ $$\begin{split}
\widetilde{\psi}_{m,l+m}(p_{g\varphi^m(\overline{G})})(x)&=\widetilde{\psi}_{m,l+m}\circ\alpha(u_gs^m{s^*}^mu_{g^{-1}})(x)\\
&=\alpha\circ\psi_{m,l+m}(u_gs^m{s^*}^mu_{g^{-1}})(x)\\
&=\alpha(u_{\varphi^l(g)}s^{l+m}{s^*}^{l+m}u_{\varphi^l(g^{-1})})(x)\\
&=p_{\varphi^l(g)\varphi^{l+m}(\overline{G})}(x)\\
&=p_{g\varphi^m(\overline{G})}(\varphi^{-l}(x))p_{\varphi^l(\overline{G})}(x).
\end{split}$$
By the properties of inductive limits, we have an isomorphism $$\overline{\alpha}: C^*(P)^s\rightarrow{\displaystyle}\lim_{\rightarrow}\{C(\overline{G}):\widetilde{\psi}_{m,l+m}\}.$$
Additionally we consider the $*$-homomorphisms $$\begin{split}
\kappa_k:C(\overline{G})&\rightarrow C_0(\mathcal{G})\\
f &\mapsto f\circ i_k^{-1}.p_{i_k(\overline{G})}
\end{split}$$ (where the $i$’s are as defined before Remark \[obs12\]). These $*$-homomorphisms satisfy $\kappa_{l+m}\circ\widetilde{\psi}_{m,l+m}=\kappa_m$, since $$\begin{split}
\kappa_{l+m}\circ\widetilde{\psi}_{m,l+m}(f)(x)&=\widetilde{\psi}_{m,l+m}(f)\circ i_{l+m}^{-1}(x)p_{i_{l+m}(\overline{G})}(x)\\
&=f(i_{l+m}^{-1}(\varphi^{-l}(x)))p_{\varphi^l(\overline{G})}(x)p_{i_{l+m}(\overline{G})}(x)\\
&=f(i_{m}^{-1}(x))p_{i_m(\overline{G})}(x)\\
&=\kappa_m(f)(x).
\end{split}$$
Hence we have a $*$-homomorphism $${\displaystyle}\lim_{\rightarrow}\{C(\overline{G}):\widetilde{\psi}_{m,l+m}\}\rightarrow C_0(\mathcal{G}).$$
This is injective as each $\kappa_k$ is, because of $\kappa_k(f)\circ i_k=f$. It is also surjective as $\mathcal{G}=\overline{{\displaystyle}\cup_{m\in\mathds{N}^*}i_m(\overline{G})}$ and using the Stone-Weierstrass Theorem. So we have $$C^*(P)^s\cong C_0(\mathcal{G}).$$
Now we have all the tools to describe our C$^*$-algebra as a semigroup crossed product using $S=G\rtimes_\varphi\mathds{N}$. Consider the action $$\begin{split}
\alpha: S &\rightarrow \hbox{End}(C^*(P))\\
(g,n)&\mapsto u_gs^n (.) {s^*}^nu_{g^{-1}}.
\end{split}$$
\[teo2\]$\mathds{U}[\varphi]$ is isomorphic to $C^*(P)\rtimes_{\alpha} S$.
By definition, $C^*(P)\rtimes_{\alpha} S$ together with $$\begin{split}
\iota_P: C^*(P)&\rightarrow C^*(P)\rtimes_{\alpha} S\\
x &\mapsto \iota_P(x)
\end{split}$$
and $$\begin{split}
\iota_S: S&\rightarrow \hbox{Isom}(C^*(P)\rtimes_{\alpha} S)\\
(g,n) &\mapsto \iota_S(g,n)
\end{split}$$
satisfying $$\iota_P(u_gs^nx{s^*}^nu_{g^{-1}})=\iota_S(g,n)\iota_P(x)\iota_S(g,n)^*$$
is the crossed product of $(C^*(P),S,\alpha)$. But note that the triple $\mathds{U}[\varphi]$, $$\begin{split}
\pi: C^*(P)&\rightarrow \mathds{U}[\varphi]\\
x &\mapsto x
\end{split}$$
and $$\begin{split}
\rho: S&\rightarrow \hbox{Isom}(\mathds{U}[\varphi])\\
(g,n)&\mapsto u_gs^n
\end{split}$$
is a covariant representation of $(C^*(P),S,\alpha)$ because: $$\rho(g,n)\pi(x)\rho(g,n)^*=u_gs^nx{s^*}^nu_{g^{-1}}=\pi(\alpha_{(g,n)}(x)).$$
Therefore there exists a $*$-homomorphism $$\label{teoiso1}
\Phi: C^*(P)\rtimes_{\alpha} S \rightarrow \mathds{U}[\varphi]$$
such that $\Phi\circ\iota_P=\pi$ and $\Phi\circ\iota_S=\rho$.
In the other hand it is well known [@Laca2] that the crossed product $C^*(P)\rtimes_{\alpha} S$ is generated as a C$^*$-algebra by elements of the form $\iota_S(g,n)$ because we have$$\iota_P(u_gs^n{s^*}^nu_{g^{-1}})=\iota_S(g,n)\iota_S(g,n)^*.$$ But note that $\mathds{U}[\varphi]$ can be viewed as the universal C$^*$-algebra generated by the unitaries $\{u_g:\; g\in G\}$ and the isometry $s$ changing conditions (i) and (ii) in Definition \[defi1\] to the equivalent one $u_gs^nu_hs^m=u_{g\varphi^n(h)}s^{n+m}$.
Therefore we identify $\iota_S(g,n)$ with $u_gs^n$ because the first ones satisfy the condition above, which generate $\mathds{U}[\varphi]$: $$\iota_S(g,n)\iota_S(h,m)=\iota_S(g\varphi^n(h),n+m)$$
and $$\begin{split}
{\displaystyle}\sum_{g\in G/\varphi(G)}\iota_S(g,n)\iota_S(g,n)^*=&{\displaystyle}\sum_{g\in G/\varphi(G)}\iota_P(u_gs^n{s^*}^nu_{g^{-1}})\\
=&\iota_P\left({\displaystyle}\sum_{g\in G/\varphi(G)}u_gs^n{s^*}^nu_{g^{-1}}\right)\\
=&\iota_P(1)=1.
\end{split}$$
Thus we get another $*$-homomorphism $$\label{teoiso2}
\begin{split}
\Delta: \mathds{U}[\varphi] &\rightarrow C^*(P)\rtimes_{\alpha} S\\
u_gs^n&\mapsto \iota_S(g,n).
\end{split}$$
As (\[teoiso1\]) and (\[teoiso2\]) are inverses of each other we can conclude that $\mathds{U}[\varphi]$ and $C^*(P)\rtimes_{\alpha} S$ are isomorphic.
In order to be able to apply the exact sequence presented in [@Khoska] we split the action of $S$ presented above: we show that its semigroup crossed product is isomorphic to a semigroup crossed product implemented by $\mathds{N}$, where $\mathds{N}$ acts on a group crossed product by $G$.
\[Prop1GN\]The C$^*$-algebra $\mathds{U}[\varphi]$ is also isomorphic to the semigroup crossed product $(C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$, where: $$\begin{split}
\omega: G &\rightarrow \hbox{Aut}(C^*(P))\\
g &\mapsto u_g(\cdot)u_{g^{-1}}\\[2\baselineskip]
\tau: \mathds{N} &\rightarrow \hbox{End}(C^*(P)\rtimes_{\omega}G)\\
n &\mapsto s^n(\cdot){s^*}^n
\end{split}$$
such that for $a_g\delta_g\in C^*(P)\rtimes_{\omega}G$, $\tau_n(a_g\delta_g)=s^na_g{s^*}^n\delta_{\varphi^n(g)}$.
We will show that $C^*(P)\rtimes_{\alpha} S$ and $(C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$ are isomorphic, by exploiting the universality of the semigroup crossed products, using two steps analogous to the first part of the proof of theorem above. Consider $C^*(P)\rtimes_{\alpha} S$ together with $$\begin{split}
\iota_P: C^*(P)&\rightarrow C^*(P)\rtimes_{\alpha} S\\
x&\mapsto \iota_P(x)
\end{split}$$
and $$\begin{split}
\iota_S: S&\rightarrow \hbox{Isom}(C^*(P)\rtimes_{\alpha} S)\\
(g,n)&\mapsto \iota_S(g,n)
\end{split}$$
satisfying $$\iota_P(u_gs^nx{s^*}^nu_{g^{-1}})=\iota_S(g,n)\iota_P(x)\iota_S(g,n)^*$$
being the crossed product of $(C^*(P),S,\alpha)$. Analogously take $(C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$ with $$\begin{split}
\iota_G: C^*(P)\rtimes_{\omega}G&\rightarrow (C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}\\
a\delta_g&\mapsto \iota_G(a\delta_g)
\end{split}$$
and $$\begin{split}
\iota_N: \mathds{N}&\rightarrow \hbox{Isom}((C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N})\\
n&\mapsto \iota_N(n)
\end{split}$$
satisfying $$\iota_B(s^na\delta_g{s^*}^n\delta_{\varphi^n(g)})=\iota_N(n)\iota_B(a\delta_g)\iota_N(n)^*$$
as the crossed product of $(C^*(P)\rtimes_{\omega}G,\mathds{N},\tau)$, where $a\delta_g$ represents the generating elements of $C^*(P)\rtimes_{\omega}G$, $g\in G$.
Note that the triple $(C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$, $$\begin{split}
\varrho: C^*(P)&\rightarrow (C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}\\
a&\mapsto \iota_G(a\delta_e)
\end{split}$$
and $$\begin{split}
\sigma: S&\rightarrow \hbox{Isom}((C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N})\\
(g,n)&\mapsto \iota_G(1\delta_g)\iota_N(n)
\end{split}$$
is a covariant representation of $(C^*(P),S,\alpha)$: $$\begin{split}
\sigma(g,n)\varrho(a)\sigma(g,n)^*&=\iota_G(1\delta_g)\iota_N(n)\iota_G(a\delta_e)\iota_N(n)^*\iota_G(1\delta_g)^*\\
&=\iota_G(1\delta_g)\iota_G(s^na{s^*}^n\delta_e)\iota_G(1\delta_g)^*\\
&=\iota_G(u_gs^na{s^*}^nu_{g^{-1}}\delta_g)\iota_G(1\delta_{g^{-1}})\\
&=\iota_G(u_gs^na{s^*}^nu_{g^{-1}}\delta_e)\\
&=\varrho(u_gs^na{s^*}^nu_{g^{-1}}).
\end{split}$$
Therefore we get a $*$-homomorphism $$\label{teo12iso1}
\Phi: C^*(P)\rtimes_{\alpha}S\rightarrow (C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N}$$
such that $\Phi\circ\iota_P=\varrho$ and $\Phi\circ\iota_S=\sigma$.
Let us find an inverse for $\Phi$ using the fact that the triple $C^*(P)\rtimes_{\alpha} S$, $$\begin{split}
\varpi: C^*(P)\rtimes_{\omega}G&\rightarrow C^*(P)\rtimes_{\alpha} S\\
a\delta_g&\mapsto \iota_P(a)\iota_S(g,0)
\end{split}$$
and $$\begin{split}
\vartheta: \mathds{N}&\rightarrow \hbox{Isom}(C^*(P)\rtimes_{\alpha} S)\\
n&\mapsto \iota_S(e,n)
\end{split}$$
is a covariant representation of $(C^*(P)\rtimes_{\omega}G,\mathds{N},\tau)$: $$\begin{split}
\vartheta(n)\varpi(a\delta_g)\vartheta(n)^*&=\iota_S(e,n)\iota_P(a)\iota_S(g,0)\iota_S(e,n)^*\\
&=\iota_S(e,n)\iota_S(g,0)\iota_P(u_{g^{-1}}au_g)\iota_S(e,n)^*\\
&=\iota_S(\varphi^n(g),0)\iota_S(e,n)\iota_P(u_{g^{-1}}au_g)\iota_S(e,n)^*\\
&=\iota_S(\varphi^n(g),0)\iota_P(s^nu_{g^{-1}a{s^*}^n}u_g)\\
&=\iota_S(\varphi^n(g),0)\iota_P(u_{\varphi^n(g^{-1})}s^na{s^*}^nu_{\varphi^n(g)})\\
&=\iota_P(s^na{s^*}^n)\iota_S(\varphi^n(g),0)=\varpi(s^na{s^*}^n\delta_{\varphi^n(g)}).
\end{split}$$
This implies the existence of a $*$-homomorphism $$\label{teo12iso2}
\Delta: (C^*(P)\rtimes_{\omega}G)\rtimes_{\tau}\mathds{N} \rightarrow C^*(P)\rtimes_{\alpha} S$$
satisfying $\Delta\circ\iota_G=\varpi$ and $\Delta\circ\iota_N=\vartheta$.
Straightforward calculations show that the $*$-homomorphisms (\[teo12iso1\]) and (\[teo12iso2\]) are inverses of each other.
*\[ex2\] For any finite group $G$, an injective endomorphism will be surjective and therefore the isometry $s$ defining $\mathds{U}[\varphi]$ will be a unitary (by item (iii) of Definition \[defi1\]). Then as $C^*(P)=\mathds{C}$, $$\mathds{U}[\varphi]\cong C^*(G)\rtimes_\tau\mathds{N}$$*
where $$\tau: \mathds{N}\rightarrow \hbox{End}(C^*(G))$$
with $$\tau_n(\lambda u_g)=\lambda u_{\varphi^n(g)}.$$
If one has the description of the K-theory of $C^*(G)$ it is easy to calculate the K-groups of $\mathds{U}[\varphi]$ by applying the Khoshkam-Skandalis sequence ([@Khoska]).
$\square$
Since more results are known for group crossed products than for semigroup ones it is useful to find such a description of our C$^*$-algebra. We can do this using the minimal automorphic dilation of the semigroup crossed product system above (for more details, see Section 2 in [@Laca1]). One important requirement to use this dilation is that the semigroup must be an Ore semigroup: an Ore semigroup is a cancellative semigroup which is right-reversible i.e, it satisfies $Ss\cap Sr\neq \emptyset$ for all $s,r\in S$.
\[propOre\] The semidirect product $S=G\rtimes_\varphi\mathds{N}$ is an Ore semigroup.
Consider $(g_i,n_i)\in S$ for $i\in\{1,2,3\}$. $S$ is cancellative: $$\begin{split}
&(g_1,n_1)(g_3,n_3)=(g_2,n_2)(g_3,n_3)\\
\Rightarrow\;&(g_1\varphi^{n_1}(g_3),n_1+n_3)=(g_2\varphi^{n_2}(g_3),n_2+n_3)\\
\Rightarrow\; &n_1=n_2 \hbox{ and }g_1\varphi^{n_1}(g_3)=g_2\varphi^{n_1}(g_3)\\
\Rightarrow\; &g_1=g_2
\end{split}$$
$$\begin{split}
&(g_1,n_1)(g_2,n_2)=(g_1,n_1)(g_3,n_3)\\
\Rightarrow\; &(g_1\varphi^{n_1}(g_2),n_1+n_2)=(g_1\varphi^{n_1}(g_3),n_1+n_3)\\
\Rightarrow\; &n_2=n_3 \hbox{ and }\varphi^{n_1}(g_2)=\varphi^{n_1}(g_3)\\
\Rightarrow\; &g_2=g_3\hbox{ as }\varphi\hbox{ is injective}.
\end{split}$$
Also any two principal left ideals of $S$ intersect: $$\begin{split}
(\varphi^{n_2}(g_1^{-1}),n_2)(g_1,n_1)&=(e,n_2+n_1)\\
&=(\varphi^{n_1}(g_2^{-1}),n_1)(g_2,n_2)\in S(g_1,n_1)\cap S(g_2,n_2).
\end{split}$$
It follows that the semigroup $S$ can be embedded in a group, called the enveloping group of $S$, which we will denote as $env(S)$, such that $S^{-1}S=env(S)$ (Theorem 1.1.2 [@Laca1]). It also implies that $S$ is a directed set by the relation defined by $(g,n)< (h,m)$ if $(h,m)\in S(g,n)$. Let us define a candidate for $env(S)$. Consider $$\mathds{G}:={\displaystyle}\lim_{\rightarrow}\{G_n:\varphi^n\}$$
(with $G_n=G$ for all $n\in\mathds{N}$) and with the extended endomorphism $\overline{\varphi}$ construct the group $$\overline{S}:=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}.$$
Then we can define an extended action $\overline{\alpha}$ of $\overline{S}$ over $C^*(P)^s$: $$\begin{split}
\overline{\alpha}:\overline{S}&\rightarrow \hbox{Aut}(C^*(P)^s)\\
(g_j,n)&\mapsto {s^*}^{j}u_gs^{n+j}(\cdot){s^*}^{n+j}u_{g^{-1}}s^{j}
\end{split}$$
(note that we can also find $g_j$ such that $j\geq |\,n|$).
Moreover, consider $i: C^*(P)\rightarrow C^*(P)^s$ the canonical inclusion.
\[proporeG\]The C$^*$-dynamical system $(C^*(P)^s,\overline{S},\overline{\alpha})$ is the minimal automorphic dilation of $(C^*(P),S,\alpha)$.
Since the subset of $S$ containing all elements of the type $(e,n)$ is cofinal in $S$, we need only prove that $\overline{S}=env(S)$ (to use Theorem 2.1.1 in [@Laca1]). For this we need to show that $S$ is a subsemigroup of $\overline{S}$ and $\overline{S}\subset S^{-1}S$ [@CliPre].
First it is obvious that $S$ is a subsemigroup of the group $\overline{S}$ via the inclusion $(g,n)\mapsto (g_0,n)$, where $g_0=g\in G=G_0\hookrightarrow\overline{G}$.
Without loss of generality take $(g_i,j)\in\overline{S}$ with $i>|j|$. Then $$(g_i,j)=(g_i,-i)(e,j+i)=(g_0,i)^{-1}(e,j+i)\in S^{-1}S.$$
We may conclude that the following theorem holds (Theorem 2.2.1 of [@Laca1]).
\[teo22\]The C$^*$-algebra $\mathds{U}[\varphi]$ is also isomorphic to the full corner $$\iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)\footnote{The isomorphism C$^*(P)\cong C(\overline{G})$ implemented in Proposition \ref{prop3} implies that the projection $\iota(1)\in C^*(P)^s$ corresponds to $p_{\overline{G}}\in C(\overline{G})$ viewed inside $C_0(\mathcal{G})$ via $i_0$ (defined before Remark \ref{obs12}).}.$$
$\square$
Let us denote the isomorphism given by the last theorem by $$\beta: \mathds{U}[\varphi]\rightarrow \iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1),$$
and by Theorem 2.2.1 in [@Laca1] we know that $$\beta({s^*}^nu_{h^{-1}}fu_{h'}s^m)=i(1)U^*_{(h,n)}i(f)U_{(h',m)}i(1).$$
Note that the isomorphism above implies that $\mathds{U}[\varphi]$ and $C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ are Morita equivalent and so they have the same K-groups.
To finish our identifications:
\[teo3\] The stabilization (Definition \[defi2\]) $\mathds{U}[\varphi]^s$ is isomorphic to the group crossed product $C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$.
As in Theorem 2.4 in [@Laca1] we know that $\beta(u_g)=V(g_0,0)\iota(1)$ and $\beta(s^n)=V(e_0,n)\iota(1)$, where $V$ represents $\overline{S}$ in the crossed product $C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$.
Define $$\begin{split}
\widetilde{\gamma}_{m,l+m}:\iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)&\rightarrow \iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)\\
x &\mapsto V(e,l)xV(e,l)^*.
\end{split}$$
Remembering from Definition \[defi2\] that $$\begin{split}
\psi_{m,l+m}:\mathds{U}[\varphi]&\rightarrow\mathds{U}[\varphi]\\
x&\mapsto s^lx{s^*}^l,
\end{split}$$
we can conclude that $$\beta\circ\psi_{m,l+m}\circ\beta^{-1}= \widetilde{\gamma}_{m,l+m}$$
which implies the existence of an isomorphism $$\overline{\beta}:\mathds{U}[\varphi]^s\rightarrow{\displaystyle}\lim_{\rightarrow}\{\iota(1)(C^*(P)^s
\rtimes_{\overline{\alpha}}\overline{S})\iota(1),\;\widetilde{\gamma}_{m,l+m}\}.$$
Moreover for $k\geq 0$ set $$\begin{split}
\lambda_k:\iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)&\rightarrow C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}\\
z &\mapsto V^*(e,k)zV(e,k).
\end{split}$$
As $$\begin{split}
\lambda_{l+m}\circ\widetilde{\gamma}_{m,l+m}(z)&=V^*(e,l+m)V(e,l)zV(e,l)^*V(e,l+m)\\
&=V^*(e,m)zV(e,m)=\lambda_m(z),\\
\end{split}$$
we have a $*$-homomorphism $$\lambda:{\displaystyle}\lim_{\rightarrow}\{\iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1):\widetilde{\gamma}_{m,l+m}\} \rightarrow C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}.$$
It is injective because each $\lambda_k$ is. Moreover as $$\lambda_k(\iota(1))=\overline{\alpha}_{(e,-k)}(\iota(1))V_{(e,0)}={s^*}^k\iota(1)s^kV_{(e,0)}$$
is an approximate unit for $C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$, for $z\in C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ we have $$\begin{split}
&{\displaystyle}\lim_k\lambda_k(\iota(1)V(e,k)zV^*(e,k)\iota(1)\iota(1))\\
=&{\displaystyle}\lim_k[\lambda_k(\iota(1)V(e,k)zV^*(e,k)\iota(1))\lambda_k(\iota(1))]\\
=&{\displaystyle}\lim_k[V^*(e,k)\iota(1)V(e,k)zV^*(e,k)\iota(1)V(e,k)][{s^*}^k\iota(1)s^kV_{(e,0)}]\\
=&{\displaystyle}\lim_k{s^*}^k\iota(1)s^kV_{(e,0)}z{s^*}^k\iota(1)s^kV_{(e,0)}{s^*}^k\iota(1)s^kV_{(e,0)}\\
=&z,
\end{split}$$
and so $\lambda$ is surjective.
Consequently $
\mathds{U}[\varphi]^s\cong C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}.
$
*Consider a surjective endomorphism $\varphi$ of a group $G$. The surjectivity of $\varphi$ implies that $s$ is an isometry (by item (iii) of Definition \[defi1\]). Moreover Proposition \[Prop1GN\] together with the fact that $C^*(P)=\mathds{C}$ implies that $$\mathds{U}[\varphi]\cong C^*(G)\rtimes_\tau\mathds{N},$$*
where $$\tau: \mathds{N} \rightarrow \hbox{End}(C^*(G))$$
is defined by $$\tau_n(u_g)=u_{\varphi^n(g)}.$$
Using the six-term exact sequence introduced by Khoshkam and Skandalis in [@Khoska], one can build the sequence $$\begin{array}{ccccc}
K_0(C^*(G)) &\xrightarrow{1-K_0(\tau_1)} &K_0(C^*(G)) &\rightarrow & K_0(\mathds{U}[\varphi]) \\
\uparrow & & & & \downarrow \\
K_1(\mathds{U}[\varphi]) &\leftarrow &K_1(C^*(G)) &\xleftarrow{1-K_1(\tau_1)} & K_1(C^*(G)) \\
\end{array}$$
(note that this example is very similar to Example \[ex2\]).
$\square$
Properties
==========
The crossed product description in last section implies two nice properties of $\mathds{U}[\varphi]$.
\[propl14\]If $G$ is amenable then $\mathds{U}[\varphi]$ is nuclear.
$G$ being amenable implies that $\overline{S}$ is amenable as well (amenability is closed under direct limits by [@vNeu] and also closed under semidirect products). But we know that $C^*(P)^s$ is nuclear because it is commutative, therefore $C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ is nuclear by Proposition 2.1.2 in [@Ror]. Since hereditary C\*-subalgebras of nuclear C\*-algebras are nuclear by Corollary 3.3 (4) in [@ChoiEffros], we conclude that $$\mathds{U}[\varphi]\cong C^*(P)\rtimes_{\alpha}S\cong i(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})i(1)$$
is nuclear.
\[propl15\]If $G$ is amenable then $\mathds{U}[\varphi]$ satisfies the UCT property.
Since $C^*(P)^s$ is commutative, $C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S}$ is isomorphic to a groupoid C$^*$-algebra. When the group $G$ is amenable then $\overline{S}$ also is, and the respective groupoid is also amenable. Therefore using a result by Tu ([@tutu] Proposition 10.7), the crossed product satisfies UCT. By Morita equivalence, $\mathds{U}[\varphi]$ also satisfies it.
We will now prove that our algebra $\mathds{U}[\varphi]$ is purely infinite and simple. We will proceed in the same way as in [@Cuntz2] and in many other papers: we present a particular faithful conditional expectation and a dense $*$-subalgebra of $\mathds{U}[\varphi]$ such that the conditional expectation of any positive element of this $*$-subalgebra can be described using a finite number of pairwise orthogonal projections.
For this purpose we will use the description in Theorem \[teo22\] of $\mathds{U}[\varphi]$ as a corner of a group crossed product. To define the conditional expectation, we require the amenability of the group $G$: therefore the group $\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$ (defined after Proposition \[propOre\]) is also amenable (as mentioned in the proof of Proposition \[propl14\]). This condition is necessary because we want to use the well-known result which says that there exists a canonical faithful conditional expectation on the reduced group crossed product, and the amenability of $\overline{S}$ implies that both the full and the reduced group crossed products (implemented by $\overline{S}$-actions) are isomorphic.
The main tool of this section is the following (proven in Proposition 5.2 of [@Li1]).
\[prop13\]Let $\widetilde{A}$ be a dense $*$-subalgebra of a unital C$^*$-algebra $A$. Assume that $\epsilon$ is a faithful conditional expectation on $A$ such that for every $0\neq x\in\widetilde{A}_{+}$ there exist finitely many projections $f_i\in A$ with
- $f_i\bot f_j$,$\forall\; i\neq j$;
- $f_i\sim_{s_i} 1$, via[^2] isometries $s_i\in A$, $\forall\; i$;
- $\left\|{\displaystyle}\sum_if_i\epsilon(x)f_i\right\|=\|\epsilon(x)\|$;
- $f_ixf_i=f_i\epsilon(x)f_i\in\mathds{C}f_i$,$\forall\; i$.
Then $A$ is purely infinite and simple.
$\square$
Moreover in order to find these projections it is also necessary to require that the injective endomorphism $\varphi$ is pure, i.e: $${\displaystyle}\bigcap_{n\in\mathds{N}}\varphi^n(G)=\{e\}.$$
In order to apply the proposition above the first step is to define a conditional expectation. As mentioned before, we require that the group $G$ is amenable. Remember the isomorphism from Theorem \[teo22\]: $$\beta: \mathds{U}[\varphi]\rightarrow \iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1).$$
\[prop1\]There exists a faithful conditional expectation $$\begin{split}
\epsilon: \mathds{U}[\varphi]&\rightarrow \beta^{-1}(\iota(1)C^*(P)^s\iota(1))\\
{s^n}^*u_{h^{-1}}fu_{h'}s^m&\mapsto\left\{
\begin{array}{ll}
{s^n}^*u_{h^{-1}}fu_hs^n, & \hbox{if }n=m\hbox{ and }h=h'; \\
0, & \hbox{otherwise.}
\end{array}
\right.
\end{split}$$
for all $h$, $h'\in G$ and $n$, $m\in\mathds{N}$.
As $\overline{S}$ is amenable, the isomorphism $\beta$ of Theorem \[teo22\] can be expanded to include also the reduced group crossed product $$\mathds{U}[\varphi]\cong \iota(1)(C^*(P)^s\rtimes_{\overline{\alpha}}\overline{S})\iota(1)\cong
\iota(1)(C^*(P)^s\rtimes_{r,\overline{\alpha}}\overline{S})\iota(1).$$
Let us denote the elements of $\overline{S}$ by $s$ and its identity by $e$. We will also use $\delta_s$ to denote the unitary elements implementing the action of $\overline{S}$ in the crossed product. Consider the well-known faithful conditional expectation on the reduced group crossed product: $$\begin{split}
E: C^*(P)^s\rtimes_{r,\overline{\alpha}}\overline{S}&\rightarrow C^*(P)^s\\
x\delta_s&\mapsto \left\{
\begin{array}{ll}
x, & \hbox{if }s=e; \\
0, & \hbox{otherwise.}
\end{array}
\right.
\end{split}$$
Straightforward calculations show that the following is also a faithful conditional expectation: $$\begin{split}
\overline{E}: \iota(1)(C^*(P)^s\rtimes_{r,\overline{\alpha}}\overline{S})\iota(1)&\rightarrow \iota(1)C^*(P)^s\iota(1)\\
\iota(1)x\delta_s\iota(1)&\mapsto \left\{
\begin{array}{ll}
\iota(1)x\iota(1), & \hbox{if }s=e; \\
0, & \hbox{otherwise.}
\end{array}
\right.
\end{split}$$
Using the isomorphism $\beta$ we can rewrite $\overline{E}$ to conclude that we have the faithful conditional expectation $$\begin{split}
\epsilon: \mathds{U}[\varphi]&\rightarrow \beta^{-1}(\iota(1)C^*(P)^s\iota(1))\\
{s^n}^*u_{h^{-1}}fu_{h'}s^m&\mapsto\left\{
\begin{array}{ll}
{s^n}^*u_{h^{-1}}fu_hs^n, & \hbox{if }n=m\hbox{ and }h=h'; \\
0, & \hbox{otherwise.}
\end{array}
\right.
\end{split}$$
Now, to find projections to describe the image of $y\in span(Q)_{+}$ under the conditional expectation $\epsilon$ presented above, remember that $y$ has the form $$y={\displaystyle}\sum_{m,n,h,h',f}a_{(m,n,h,h',f)}{s^*}^nu_{h^{-1}}fu_{h'}s^m$$
for $m$, $n\in\mathds{N}$, $h$, $h'\in G$, $f\in P$ and $a_{(\ldots)}\neq 0$. As we have finitely many projections of $C^*(P)$ in the description of $y$, write them all as sums of (altogether $N$) mutually orthogonal projections $u_{g_i}s^M{s^*}^Mu_{g_i^{-1}}$, with $g_i\in G/\varphi^M(G)$, for all $1\leq i\leq N$ and $M\in\mathds{N}$ big enough.
\[prop2\]There are $N$ pairwise orthogonal projections $f_1,\ldots f_N\in P$ such that
1. $\Phi$ defined by $$\begin{split}
C^*(\{u_{g_1}s^M{s^*}^Mu_{g_1^{-1}},\ldots,u_{g_N}s^M{s^*}^Mu_{g_N^{-1}}\})&\rightarrow C^*(\{f_1,\ldots,f_N\})\\
z&\mapsto{\displaystyle}\sum_{i=1}^Nf_izf_i
\end{split}$$
is an isomorphism, and
2. $\Phi(\epsilon(y))={\displaystyle}\sum_{i=1}^Nf_iyf_i$, $\forall\; y\in\mathds{U}[\varphi]$.
Define $$f_i:=u_{h_i}s^p{s^*}^pu_{h_i^{-1}}$$
for some $p\in\mathds{N}$ bigger than $M$ (in fact, we may choose $p$ as big as we want), where $g_i^{-1}h_i\in\varphi^M(G)$. This implies that the set of the $f_i$’s is orthogonal and that (i) holds.
For (ii), first note that when $\delta_{m,n}\delta_{h,h'}=1$ it is true that $\epsilon=Id$, and so (ii) is satisfied. So let us take a look on those summands in $y$ with $\delta_{m,n}\delta_{h,h'}=0$ (we will say that such an element has *critical index* $(m,n,h,h',f)$). The conditional expectation $\epsilon$ maps these summands to 0 and in order for (ii) to be satisfied we need that, for all $1\leq i\leq N$, $$f_i{s^*}^nu_{h^{-1}}fu_{h'}s^mf_i=0.$$
We calculate $$\begin{split}
&f_i{s^*}^nu_{h^{-1}}fu_{h'}s^mf_i\\
&={s^*}^nu_{h^{-1}}(u_hs^nf_i{s^*}^nu_{h^{-1}})f(u_{h'}s^mf_i{s^*}^mu_{h'^{-1}})u_{h'}s^m\\
&={s^*}^nu_{h^{-1}}[u_{h\varphi^n(h_i)}s^{n+p}{s^*}^{n+p}u_{\varphi^n(h_i^{-1})h^{-1}}\\ &u_{h'\varphi^m(h_i)}s^{m+p}{s^*}^{n+p}
u_{\varphi^m(h_i^{-1})h'^{-1}}]fu_{h'}s^m.
\end{split}$$
Now, analysing only the expression between the brackets, $$\begin{split}
&[u_{h\varphi^n(h_i)}s^{n+p}{s^*}^{n+p}u_{\varphi^n(h_i^{-1})h^{-1}}u_{h'\varphi^m(h_i)}s^{m+p}{s^*}^{m+p}
u_{\varphi^m(h_i^{-1})h'^{-1}}]\\
&=\left({\displaystyle}\sum_{g\in G/\varphi^m(G)}u_{h\varphi^n(h_i)\varphi^{n+p}(g)}s^{m+n+p}{s^*}^{m+n+p}u_{\varphi^{n+p}(g^{-1})\varphi^n(h_i^{-1})h^{-1}}\right)\\
&\times\left({\displaystyle}\sum_{k\in G/\varphi^n(G)}u_{h'\varphi^m(h_i)\varphi^{m+p}(k)}s^{m+n+p}{s^*}^{m+n+p}u_{\varphi^{m+p}(k^{-1})\varphi^m(h_i^{-1})h^{-1}}\right).
\end{split}$$
This product will be zero if the two sums are mutually orthogonal, which happens if for all $g\in G/\varphi^m(G)$ and $k\in G/\varphi^n(G)$, $$h\varphi^n(h_i)\varphi^{n+p}(g)\varphi^{m+n+p}(x)\neq h'\varphi^m(h_i)\varphi^{m+p}(k)\varphi^{m+n+p}(y),\;\forall\; x,y\in G$$
which is equivalent to $$\varphi^{m+p}(k^{-1})\varphi^m(h_i^{-1})h'^{-1}h\varphi^n(h_i)\varphi^{n+p}(g)\neq\varphi^{m+n+p}(z),\;\forall\; z\in G.$$
A sufficient condition for this to hold is that $\varphi^m(h_i^{-1})h'^{-1}h\varphi^n(h_i)\neq\varphi^p(z)$ $\forall\; z\in G$, for each critical index $(m,n,h,h',f)$. Using the fact that $\varphi$ is pure we may choose some $p_{(m,n,h,h',f)}\in\mathds{N}$ such that $$\varphi^m(h_i^{-1})h'^{-1}h\varphi^n(h_i)\notin\varphi^{p_{(m,n,h,h',f)}}(G).$$
As we have a finite number of critical indices, it is sufficient to take the biggest $p_{(m,n,h,h',f)}$ and call it $p$.
To understand this choice of $p$, consider the following example.
*\[ex1\] Let $G=\mathds{Z}$ and $$\begin{split}
\varphi:\mathds{Z}&\rightarrow\mathds{Z}\\
n &\mapsto 3n.
\end{split}$$ Then we have $\frac{G}{\varphi(G)}=\{\overline{0},\overline{1},\overline{2}\}$, $\frac{G}{\varphi^2(G)}=\{\overline{0},\overline{1},\ldots,\overline{8}\}$ and in general $$\frac{G}{\varphi^n(G)}=\{\overline{0},\ldots,\overline{3^n-1}\}=\mathds{Z}_{3^n}.$$*
Take the following $y\in span(Q)$ $$\begin{split}
y=&2{s^*}^2u_{30}(u_5s^{4}{s^*}^4u_{-5})u_{2187}s^{1}-4{s^*}^7u_0(u_{10}s^{4}{s^*}^4u_{-10})u_{-5}s^9\\ &+{s^*}^8u_{20}s^{4}{s^*}^4u_{-20}s^8
\end{split}$$
and note that in $y$ we have two terms with critical indices (the first ones).
Using the notation of the above proposition, $M=4$ and, for the first term of $y$: $n=2$, $m=1$, $h=-30$, $h'=2187$ and $g_1=5$. Choosing $h_1=86$, it is true that $-g_1+h_1=-5+86=81\in\varphi(G)$. Then: $$\varphi^{1}(-86)-2187-30+\varphi^{2}(86)=-1701=\varphi^5(7)\notin\varphi^6(\mathds{Z}).$$
So $p_1:=p_{(1,2,-30,2187,f)}=6$ (or bigger). For the second term it is not hard to see that $p_2=1$: $$\varphi^{9}(-91)+5-0+\varphi^{7}(91)=-1592131\notin\varphi^1(\mathds{Z}).$$
So one can choose any $p\geq 6$.
$\square$
Using the description above of the faithful conditional expectation $$\epsilon: \mathds{U}[\varphi]\rightarrow \beta^{-1}(\iota(1)C^*(P)^s\iota(1))$$
where $P=\{u_gs^n{s^*}^nu_{g^{-1}}:\;g\in G,\; n\in\mathds{N}\}$, together with the dense $*$-subalgebra $$\hbox{span}(Q)=\hbox{span}(\{{s^*}^nu_{h^{-1}}fu_{h'}s^m:\;f\in P, h, h'\in G, n,m\in\mathds{N}\}),$$
we can prove the main result of this section by applying Propositions \[prop2\] and \[prop13\] (the definition of pure infiniteness comes from [@Cuntz2]).
\[teo1\]Let $G$ be a discrete countable amenable group and $\varphi$ a pure injective endomorphism of $G$ with finite cokernel. Then the C$^*$-algebra $\mathds{U}[\varphi]$ is simple and purely infinite, i.e. for all non zero $x\in\mathds{U}[\varphi]$ there are $a$, $b\in\mathds{U}[\varphi]$ with $axb=1$.
$\square$
When satisfied the conditions of the theorem above, the universal C$^*$-algebra $\mathds{U}[\varphi]$ is isomorphic to $C_r^*[\varphi]$, as defined in Definitions \[defi1\] and \[defi1red\] respectively.
$\square$
If the conditions of the theorem above are satisfied, the universal C$^*$-algebra $\mathds{U}[\varphi]$ is a Kirchberg algebra satisfying the UCT property.
$\square$
It would be interesting to know if the conditions of the theorem above are also necessary: if we construct the C$^*$-algebra associated with some injective endomorphism of an amenable group, is it simple and purely infinite only if $\varphi$ is pure? Unfortunately we don’t answer this question here, but the next trivial example gives some idea about this direction.
*\[ex0\] For some commutative group $G$ (thus amenable), consider $\varphi = id_G$.*
As $u_gs=su_g$ for all $g\in G$ ($\varphi$ is trivial), our C$^*$-algebra will be commutative. Now, as $\frac{G}{\varphi(G)}$ has only the element $\{e\}$, condition (iii) of Definition \[defi1\] implies that the isometry $s$ is a unitary. Then $\mathds{U}[\varphi]$ is the commutative C$^*$-algebra generated by the unitaries $\{u_g, s:g\in G\}$, and this one is the non-simple tensor product $C^*(G)\otimes C^*(\mathds{Z})=C^*(G)\otimes C(\mathcal{S}^1)$.
Moreover using the Künneth Formula [@Schoc] we conclude that $$K_0(\mathds{U}[\varphi])=K_1(\mathds{U}[\varphi])=K_0(C^*(G))\oplus K_1(C^*(G)).$$
$\square$
Description of $\mathds{U}[\varphi]$ via group partial crossed products {#partialcrpr}
=======================================================================
In [@BoEx] Boava and Exel constructed a partial group algebra isomorphic to the C$^*$-algebra $\mathds{U}[R]$ associated with a integral domain $R$ [@Culi1]. Consequently due to Theorem 4.4 of [@ExLaQu] one can define a certain partial crossed product which is isomorphic to $\mathds{U}[R]$. With the latter description it is proven in [@BoEx], using only tools from partial crossed products, that if $R$ is not a field then $\mathds{U}[R]$ is simple (which is part of the conclusion of Li [@Li1], namely, Corollary 5.14).
In this section we will present analogous results adapted to our case, i.e., given a C$^*$-algebra $\mathds{U}[\varphi]$ associated with some injective endomorphism $\varphi$ of a group $G$ with unit $e$, we will show that $\mathds{U}[\varphi]$ can also be viewed as a partial group algebra and, consequently, as a partial crossed product. The ideas follow the ones presented in [@BoEx].
With this description we show that when $G$ is amenable we can rewrite the faithful conditional expectation $\epsilon$ presented in Proposition \[prop1\] in terms of the partial group crossed product. To finish we use a well known result from the theory of group partial crossed products to prove a weaker result than Theorem \[teo1\]: if $G$ is commutative and $\varphi$ is pure then $\mathds{U}[\varphi]$ is simple.
We start with an introduction to partial actions, partial crossed products and partial group algebras, before presenting the right isomorphisms and descriptions of $\mathds{U}[\varphi]$.
\[pcpdefi1\]A partial action $\alpha$ of a group $G$ on a C$^*$-algebra $A$ is a collection of closed two-sided ideals $\{D_g\}_{g\in G}$ of $A$ and $*$-isomorphisms $\alpha_g: D_{g^{-1}}\rightarrow D_g$ satisfying
- $D_e=A$;
- $\alpha_h^{-1}(D_h\cap D_{g^{-1}})\subseteq D_{(gh)^{-1}}$;
- $\alpha_g\circ\alpha_h(x)=\alpha_{gh}(x)$, $\forall\; x\in \alpha_h^{-1}(D_h\cap D_{g^{-1}})$.
Using (PA1) - (PA3) one can show that $\alpha_e=$ id$_A$, $\alpha_{g^{-1}}=\alpha_g^{-1}$ and that$\alpha_h^{-1}(D_h\cap D_{g^-1})=D_{(gh)^{-1}}\cap D_{h^{-1}}$.
Analogously, one can define a partial action of $G$ acting on a locally compact space $X$: just replace the ideals $D_g$ by open sets $X_g\subseteq X$ and the $*$-isomorphisms $\alpha_g$ by homeomorphisms $\theta_g: X_{g^{-1}}\rightarrow X_g$.
We call the triples $(\alpha,G,A)$ or $(\theta,G,X)$ partial dynamical systems, or partial actions when there is no possibility of misunderstanding.
*\[Expa1\] If $\theta$ is a partial action of $G$ on the locally compact space $X$ with $\theta_g: X_{g^{-1}}\rightarrow X_g$, one can easily construct a partial action of $G$ on the C$^*$-algebra $C_0(X)$ considering $D_g=C_0(X_g)$ and $$\begin{split}
\alpha_g:C_0(X_{g^{-1}})&\rightarrow C_0(X_g)\\
f&\mapsto f\circ\theta_{g^{-1}}.
\end{split}$$*
$\square$
Now we want to define partial crossed products. There are three ways to realize them: one using Fell bundles (and we recommend [@ExelFell]), another using enveloping C$^*$-algebras (for details and some interesting examples look at Section 2 of [@Mc]) and the last one as a universal object with respect to covariant pairs (see Section 3 of [@QuRa]). We use the last way in our proofs and therefore we present it.
Let us define first a particular set of representations called partial representations.
\[pcpdefi2\]A partial representation $\pi$ of a group $G$ into a unital C$^*$-algebra $B$ is a map $\pi: G\rightarrow B$ satisfying
- $\pi(e)=1$;
- $\pi(g^{-1})=\pi(g)^*$;
- $\pi(g)\pi(h)\pi(h^{-1})=\pi(gh)\pi(h^{-1})$.
Then the partial group crossed product $A\rtimes_{\alpha}G$ is defined as the universal object with respect to a covariant pair $(\upsilon,\pi)$, which means a $*$-homomorphism ($B$ being a unital C$^*$-algebra) $$\upsilon: A\rightarrow B$$
and a partial representation of $G$ $$\pi: G\rightarrow B$$
satisfying $$\begin{split}
\upsilon(\alpha_g(x))&=\pi(g)\upsilon(x)\pi(g^{-1})\hbox{ for }x\in D_{g^{-1}},\\
\upsilon(x)\pi(g)\pi(g^{-1})&=\pi(g)\pi(g^{-1})\upsilon(x)\hbox{ for }x\in A.
\end{split}$$
To define a partial group algebra, consider the set $[G]:=\{[g]:\;g\in G\}$ (without any operations).
\[defipagr\]The partial group algebra of $G$, denoted $C^*_p(G)$, is the universal C$^*$-algebra generated by $[G]$ with respect to the relations
- $[e]=1$;
- $[g^{-1}]=[g]^*$;
- $[g][h][h^{-1}]=[gh][h^{-1}]$.
The C$^*$-algebra $C_p^*(G)$ is universal with respect to partial representations of $G$ (note the equivalence between relations (R$_p$) and (PR) of Definition \[pcpdefi2\]).
In fact, one can define partial group algebras for more restricted situations, i.e., requiring that $[G]$ satisfies additional relations than the 3 relations above. Let us set $e_g:=[g][g^{-1}]$ and for our constructions consider $\mathcal{R}$ a set of (extra) relations on $[G]$ such that every relation is of the form $$\label{relationspga}
\sum_i\prod_je_{g_{ij}}=0.$$
The partial group algebra of $G$ with relations $\mathcal{R}$, denoted $C_p^*(G,\mathcal{R})$, is defined to be the universal C$^*$-algebra generated by $[G]$ with relations $R_p\cup\,\mathcal{R}$. This C$^*$-algebra is universal with respect to partial representations which satisfy $\mathcal{R}$.
An interesting fact is that the class of partial group algebras without restrictions and of the ones with extra relations of the type (\[relationspga\]) is contained in the class of partial crossed products (Definition 6.4 of [@Exel1] and Theorem 4.4 of [@ExLaQu] respectively). In our case the C$^*$-algebra $\mathds{U}[\varphi]$ will be isomorphic to a partial group algebra with additional relations of the form (\[relationspga\]) above, and we will show how these can be viewed as partial crossed products.
Consider the power set $\mathcal{P}(G)$ (of $G$) with the topology given by identifying it with the compact set $\{0,1\}^G$, and denote $X_G$ the subset of $\mathcal{P}(G)$ of the subsets $\xi$ of $G$ which contain $e\in G$. Note that using the product topology of $\{0,1\}^G$ implies that $X_G$ is compact and Hausdorff.
Denote by $1_g$ the following function in $C(X_G)$: $$1_g(\xi)=\left\{
\begin{array}{ll}
1, & \hbox{if }g\in\xi; \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
Denote $\widehat{\mathcal{R}}$ the subset of $C(X_G)$ given by the functions $\sum_i\prod_j1_{g_{ij}}$ where the relation $\sum_i\prod_je_{g_{ij}}=0$ is in $\mathcal{R}$. The *spectrum* of the relations $\mathcal{R}$ is defined to be the compact (Proposition 4.1 [@ExLaQu]) space $$\Omega_\mathcal{R}:=\{\xi\in X_G:\; f(g^{-1}\xi)=0,\;\forall\; f\in\widehat{\mathcal{R}},\;\forall\; g\in\xi\}.$$
Now for $g\in G$, consider $$\Omega_g:=\{\xi\in\Omega_\mathcal{R}:\; g\in\xi\}$$
and let us define $$\begin{split}
\theta_g: \Omega_{g^{-1}}&\rightarrow\Omega_g\\
\xi&\mapsto g\xi.
\end{split}$$
Then we have defined a partial action $\theta$ of $G$ on $\Omega_\mathcal{R}$. Turning this partial action (as in Example \[Expa1\]) into a partial action $\alpha$ of $G$ on $C(\Omega_\mathcal{R})$, it is well known (by Theorem 4.4 (iii) in [@ExLaQu]) that $$\label{parcpc14}
C_p^*(G,R)\cong C(\Omega_\mathcal{R})\rtimes_{\alpha}G.$$
Now let us find a partial group C$^*$-algebra description of $\mathds{U}[\varphi]$. Therefore recall the set $\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$ whose elements will be denoted by $(g_i,n)$ with $g_i\in G_i\subseteq\mathds{G}$. In case $g\in G=G_0\subseteq\mathds{G}$ we will use the notation $(g,n)$.
Consider the following relations $\mathcal{R}$:
- $[(g,0)][(g,0)^{-1}]=1,\;\forall\; g\in G$;
- $[(e,-n)][(e,-n)^{-1}]=1\;\forall\; n\in\mathds{N}$;
- ${\displaystyle}\sum_{g\in\frac{G}{\varphi^n(G)}}[(g,n)][(g,n)^{-1}]=1,\;\forall\; n\in\mathds{N}$.
Consider also the partial group algebra relations in this case i.e, on the group $\overline{S}$:
- $[(e,0)]=1$;
- $[(g_i,n)^{-1}]=[(g_i,n)]^*,\;\forall\; n\in\mathds{Z},\;\forall\; g_i\in\mathds{G}$;
- $[(g_i,n)][(h_j,m)][(h_j,m)^{-1}]=[(g_i\varphi^n(h_j),n+m)][(h_j,m)^{-1}],{\newline}\;\forall\; m,n\in\mathds{Z},\;\forall\; g_i,h_j\in\mathds{G}$.
Define $$\begin{split}
\pi: \overline{S}&\rightarrow\mathds{U}[\varphi]\\
(g_i,n)&\mapsto {s^*}^iu_gs^{n+i},
\end{split}$$
remembering that we can always suppose $i\geq |\,n|$. Note that when $g\in G$, $\pi(g,n)=u_gs^n$.
The map $\pi$ is a partial representation of $\overline{S}$ which satisfies the relations $\mathcal{R}$.
First we prove that $\pi$ is a partial representation of $\overline{S}$.
(R$_p$1): $\pi((e,0))=u_e=1$;
(R$_p$2): $$\begin{split}
\pi((g_i,n)^{-1})&=\pi((g^{-1}_{i+n},-n))={s^*}^{i+n}u_{g^{-1}}s^i=({s^*}^iu_gs^{i+n})^*\\
&=(\pi((g_i,n)))^*;
\end{split}$$
(R$_p$3): $$\begin{split}
&\pi((\varphi^j(g)\overline{\varphi}^{i+n}(h)_{i+j},n+m))\pi((h_j,m)^{-1})\\
&={s^*}^{i+j}u_{\varphi^j(g)\varphi^{i+n}(h)}s^{i+j+n+m}{s^*}^{j+m}u_{h^{-1}}s^j\\
&={s^*}^iu_g{s^*}^js^{i+n}\underbrace{u_hs^{j+m}{s^*}^{j+m}u_{h^{-1}}}\underbrace{s^j{s^*}^j}s^j\\
&={s^*}^iu_g{s^*}^js^{i+n}s^j{s^*}^ju_hs^{j+m}{s^*}^{j+m}u_{h^{-1}}s^j\\
&={s^*}^iu_gs^{i+n}{s^*}^ju_hs^{j+m}{s^*}^{j+m}u_{h^{-1}}s^j\\
&=\pi((g_i,n))\pi((h_j,m))\pi((h_j,m)^{-1}).
\end{split}$$
Now we show that $\pi$ satisfies the extra relations $\mathcal{R}$.
($\mathcal{R}_1$): $\pi((g,0))\pi((g,0)^{-1})=u_e=1$;
($\mathcal{R}_2$): $\pi((e,-n))\pi((e,-n)^{-1})=\pi((e_n,-n))\pi((e_n,-n)^{-1})={s^*}^ns^n=1$;
($\mathcal{R}_3$): ${\displaystyle}\sum_{g\in\frac{G}{\varphi^n(G)}}\pi((g,n))\pi((g,n)^{-1})={\displaystyle}\sum_{g\in\frac{G}{\varphi^n(G)}}u_gs^n{s^*}^{-n}u_{g^{-1}}=1$.
It follows from the universality of the partial group algebra $C_p^*(\overline{S},\mathcal{R})$ that there exists a $*$-homomorphism $$\label{eqpga1}
\begin{split}
\Phi: C^*_p(\overline{S},\mathcal{R})&\rightarrow\mathds{U}[\varphi]\\
[(g_i,n)]&\mapsto {s^*}^iu_gs^{n+i}.
\end{split}$$
Let us find an inverse for $\Phi$ by using the relations which define $\mathds{U}[\varphi]$.
The (obviously) unitary elements $[(g,0)]$ and isometries $[(e,n)]$ of $C_p^*(\overline{S},\mathcal{R})$ satisfy the relations which define $\mathds{U}[\varphi]$.
Let us show that the elements above satisfy the relations (i) - (iii) of Definition \[defi1\].
(i):$$\begin{split}
[(g,0)][(h,0)]&=[(g,0)][(h,0)][(h^{-1},0)][(h,0)]=[(gh,0)][(h^{-1},0)][(h,0)]\\
&=[(gh,0)];
\end{split}$$
(ii):$$\begin{split}
[(e,1)][(g,0)]&=[(e,1)][(g,0)][(g^{-1},0)][(g,0)]=[(\varphi(g),1)][(g^{-1},0)][(g,0)]\\
&=[(\varphi(g),1)]=[(\varphi(g),1)][(e,-1)][(e,1)]\\
&=[(\varphi(g),0)][(e,1)][(e,-1)][(e,1)]\\
&=[(\varphi(g),0)][(e,1)];
\end{split}$$
(iii): $$\begin{split}
[(g,0)][(e,1)][(e,-1)][(g^{-1},0)]&=[(g,1)][(e,-1)][(g^{-1},0)]\\
&=[(g,1)][(e,-1)][(g^{-1},0)][(g,0)][(g^{-1},0)]\\
&=[(g,1)][(g^{-1}_1,-1)][(g,0)][(g^{-1},0)]\\
&=[(g,1)][(g^{-1}_1,-1)]=[(g,1)][(g,1)]^*,
\end{split}$$
and using $\mathcal{R}_3$ we see that it satisfies condition (iii).
Consequently we have a $*$-homomorphism $$\label{eqpga2}
\begin{split}
\Psi: \mathds{U}[\varphi]&\rightarrow C^*_p(\overline{S},\mathcal{R})\\
u_g&\mapsto [(g,0)]\\
s^n&\mapsto [(e,n)].
\end{split}$$
\[teouppga\] The C$^*$-algebra $\mathds{U}[\varphi]$ is isomorphic to $C^*_p(\overline{S},\mathcal{R})$.
We just have to show that the $*$-homomorphisms (\[eqpga1\]) and (\[eqpga2\]) are inverses of each other on the generators of the respective C$^*$-algebras.
$\,\;\;\;\;\;\;\bullet\;\;\Phi\circ\Psi(u_g)=\Phi([(g,0)])=u_g$;
$\,\;\;\;\;\;\;\bullet\;\;\Phi\circ\Psi(s^n)=\Phi([(e,n)])=s^n$; $$\begin{split}
\bullet\;\;\Psi\circ\Phi([(g_i,n)])&=\Psi({s^*}^iu_gs^{n+i})=[(e,-i)][(g,0)][(e,n+i)]\\
&=[(e,-i)][(g,0)][(e,n+i)][(e,-n-i)][(e,n+i)]\\
&=[(e,-i)][(g,n+i)][(e,-n-i)][(e,n+i)]\\
&=[(e,-i)][(e,i)][(e,-i)][(g,n+i)]\\
&=[(e,-i)][(e,i)][(\overline{\varphi}^{-i}(g),n)]\\
&=[(g_i,n)].
\end{split}$$
In order to define a partial crossed product isomorphic to $C_p^*(\overline{S},\mathcal{R})$ which by the theorem above is isomorphic to $\mathds{U}[\varphi]$, consider $X_{\overline{S}}$ the subset of $\mathcal{P}(\overline{S})$ of the subsets $\xi$ of $\overline{S}$ which contain $(e,0)\in \overline{S}$. Also $1_s\in C(X_{\overline{S}})$ is given by $$1_s(\xi)=\left\{
\begin{array}{ll}
1, & s\in\xi; \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
and the partial group algebra relations $\mathcal{R}$ are
- $e_{(g,\,0)}-1=0,\;\forall\; g\in G$;
- $e_{(e,-n)}-1=0\;\forall\; n\in\mathds{N}$;
- ${\displaystyle}\sum_{g\in\frac{G}{\varphi^n(G)}}e_{(g,n)}-1=0,\;\forall\; n\in\mathds{N}$.
This implies that $\widehat{\mathcal{R}}$ is the subset of $C(X_{\overline{S}})$ consisting of the functions
- $1_{(g,\,0)}-1_{(e,\,0)},\;\forall\; g\in G$;
- $1_{(e,-n)}-1_{(e,\,0)}\;\forall\; n\in\mathds{N}$;
- ${\displaystyle}\sum_{g\in\frac{G}{\varphi^n(G)}}1_{(g,n)}-1_{(e,\,0)},\;\forall\; n\in\mathds{N}$.
The spectrum of the relations $\mathcal{R}$ is defined to be $$\Omega_\mathcal{R}=\{\xi\in X_{\overline{S}}:\; f(g^{-1}\xi)=0,\;\forall\; f\in\widehat{\mathcal{R}},\;\forall\; g\in\xi\}.$$
Consider $$\Omega_s=\{\xi\in\Omega_\mathcal{R}:\; s\in\xi\}$$
and define the partial action $\varpi$ of $\overline{S}$ on $\Omega_\mathcal{R}$ by $$\label{isouapcp}
\begin{split}
\varpi_s: \Omega_{s^{-1}}&\rightarrow\Omega_s\\
\xi&\mapsto s\xi.
\end{split}$$
Then it is well known by Theorem \[teouppga\] and (\[parcpc14\]) respectively that $$\label{isouapcp15}
\mathds{U}[\varphi]\cong C^*_p(\overline{S},\mathcal{R})\cong C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S},$$
where $$\label{isouapcp16}
\begin{split}
\alpha_s: C(\Omega_{s^{-1}})&\rightarrow C(\Omega_s)\\
f&\mapsto f\circ\varpi_{s^{-1}}.
\end{split}$$
The partial crossed product description of $\mathds{U}[\varphi]$ presented above together with the requirement that $G$ is amenable (which implies that $\overline{S}$ is as well) makes it possible to define a certain conditional expectation as done in [@ExelFell] Proposition 2.9 (as in the classical group crossed product construction the amenability of the group implies the isomorphism of both reduced and full constructions by [@Mc], and a faithful conditional expectation exists for the reduced one). We will show that this conditional expectation is the same - modulo the isomorphism already established - as $\epsilon$ as given by Proposition \[prop1\]. The conditional expectation of $C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}$ is given by $$\begin{split}
\overline{E}: C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}&\rightarrow C(\Omega_\mathcal{R})\\
f\delta_s &\mapsto \left\{
\begin{array}{ll}
f, & \hbox{if }s=(e,0); \\
0, & \hbox{otherwise.}
\end{array}
\right.
\end{split}$$ Identifying $C^*_p(\overline{S},\mathcal{R})$ with $C(\Omega_\mathcal{R})\rtimes_{\alpha}\overline{S}$, $\overline{E}$ becomes $$\begin{split}
E: C^*_p(\overline{S},\mathcal{R})&\rightarrow C^*(e_{(g_i,n)})\footnotemark\\
{\displaystyle}\prod_{(g_i,n)\in\overline{S}}^{\scriptscriptstyle{\hbox{finite}}}[(g_i,n)]&\mapsto \left\{
\begin{array}{ll}
{\displaystyle}\prod_{(g_i,n)\in\overline{S}}^{\scriptscriptstyle{\hbox{finite}}}[(g_i,n)], & \hbox{if }{\displaystyle}\prod_{(g_i,n)\in\overline{S}}^{\scriptscriptstyle{\hbox{finite}}}(g_i,n)=(e,0); \\
0, & \hbox{otherwise.}
\end{array}
\right.
\end{split}$$
Using the isomorphism $\Psi$ (from (\[eqpga2\])) and $\epsilon$ (from Proposition \[prop1\]), we shall prove the following.
$E\circ\Psi=\Psi\circ\epsilon$.
Let us prove the equality on the dense $*$-subalgebra of $\mathds{U}[\varphi]$ given by $$\hbox{span}(Q)=\hbox{span}(\{{s^*}^nu_{h^{-1}}fu_{h'}s^m:\;f\in P,\,h, h'\in G,\,n,m\in\mathds{N}\}).$$
Consider $f=u_gs^k{s^*}^ku_{g^{-1}}\in P$, $h, h'\in G$, and $n,m\in\mathds{N}$. $$\begin{split}
&E\circ\Psi({s^*}^nu_{h^{-1}}fu_{h'}s^m)=E\circ\Psi({s^*}^nu_{h^{-1}}u_gs^k{s^*}^ku_{g^{-1}}u_{h'}s^m)\\
&= E([(e,-n)][(h^{-1},0)][(g,0)][(e,k)][(e,-k)][(g^{-1},0)][(h',0)][(e,m)])\\
&=\delta_{n,m}\delta_{h,h'}[(e,-n)][(h^{-1},0)][(g,0)][(e,k)][(e,-k)][(g^{-1},0)][(h,0)][(e,n)]\\
&=\delta_{n,m}\delta_{h,h'}[(e,-n)][(h^{-1},0)]\Psi(f)[(h,0)][(e,n)],
\end{split}$$
while $$\begin{split}
\Psi\circ\epsilon({s^*}^nu_{h^{-1}}fu_{h'}s^m)&=\Psi(\delta_{n,m}\delta_{h,h'}{s^n}^*u_{h^{-1}}fu_hs^n)\\
&=\delta_{n,m}\delta_{h,h'}[(e,-n)][(h^{-1},0)]\Psi(f)[(h,0)][(e,n)].
\end{split}$$
This shows that both conditional expectations $E$ and $\epsilon$ are the same, up to the isomorphism $\Psi$.
Simplicity of $\mathds{U}[\varphi]$
-----------------------------------
To prove that $\mathds{U}[\varphi]$ is simple using partial crossed product theory, we suppose that $G$ is commutative. Therefore our group is amenable and the endomorphism $\varphi$ is totally normal i.e, the images of $\varphi$ are normal subgroups of $G$. This implies that the set $\overline{G}$, defined in the beginning of Section \[sectioncpdescr\], is a group.
We need some definitions (from [@ExLaQu]) concerning partial actions, as they play a role in the proof that $\mathds{U}[\varphi]$ is simple. Consider $(\theta,H,X)$ a partial dynamical system where $X$ is a locally compact space with $X_h$ being the open sets (Definition \[pcpdefi1\]).
We say that a partial action $\theta$ is topologically free if for every $h\in H\backslash\{e\}$ the set $F_h:=\{x\in X_{h^{-1}}:\;\theta_h(x)=x\}$ has empty interior.
In order to define the minimality of $\theta$, we adjust the classical definition of invariance: a subset $V$ of $X$ is invariant under the partial action $(\theta,H,X)$ if $\theta_h(V\cap X_{h^{-1}})\subseteq V$ $\forall\; h\in H$.
The partial action $\theta$ is minimal if there are no invariant open subsets of $X$ other than $\emptyset$ and $X$.
Suited to our setting, there is a result due to Exel, Laca and Quigg (Corollary 2.9 of [@ExLaQu]) which says that the partial action $\varpi$ defined in (\[isouapcp\]) is topologically free and minimal if and only if $C(\Omega_\mathcal{R})\rtimes_{\alpha}(\overline{S})$, as defined in (\[isouapcp15\]) and (\[isouapcp16\]), is simple (in fact their result applies to the reduced crossed product, but as we are assuming $G$ is commutative and thus amenable, we know that $\overline{S}$ is amenable and this implies that both the full and reduced partial crossed products are isomorphic by [@Mc] Proposition 4.2), so it is clear that we have to understand the topology of $\Omega_\mathcal{R}$, which unfortunately is not an easy task.
To avoid difficulties we present a new set which is homeomorphic to $\Omega_\mathcal{R}$, and for which we can easily understand the topology. Consider $\frac{G}{\varphi^k(G)}=\{e\}$ for negative integers $k$ and for $m\leq n$ both integers the canonical projection $$p_{m,n}: \dfrac{G}{\varphi^n(G)}\rightarrow\dfrac{G}{\varphi^m(G)}.$$
Using these, define $$\begin{split}
\widetilde{G}:&=\lim_{\leftarrow \atop n}\left\{\dfrac{G}{\varphi^n(G)}:\;p_{m,n}\right\}\\
&=\left\{(g_n\overline{\varphi}^n(G))_{n\in\mathds{Z}}\in\prod_{n\in\mathds{Z}}\dfrac{G}{\varphi^n(G)}:\;p_{m,n}(g_n)=g_m,\hbox{ if }m\leq n\right\},
\end{split}$$
where $\overline{\varphi}$ is the extension of $\varphi$ defined after Proposition \[propOre\]. Note that when $n\leq0$, $\frac{G}{\varphi^n(G)}=\{e\}$ and therefore for any element in $\widetilde{G}$, the entries indexed by negative integers are $e$. Moreover, when $n>0$, $\overline{\varphi}^n=\varphi^n$. Particularly it makes not necessary to carry the bar over $\varphi$ when denoting the elements of $\widetilde{G}$, and we will also use the notation $(g_m)_{n\in\mathds{Z}}\in\widetilde{G}$. One can see $G$ inside $\widetilde{G}$ through the map $g\mapsto (g\varphi^n(G))_n$, which is injective if $\varphi$ is pure.
Another fact is that the set defined above is isomorphic as a topological group to our previous defined $\overline{G}$ (beginning of Section \[sectioncpdescr\]), because that set is exactly this one except for the negative entries of the vectors in $\widetilde{G}$, which are always $e$. Therefore $\widetilde{G}$ is compact.
Consider $$\begin{split}
\rho:\widetilde{G}&\rightarrow \hbox{P}(\overline{S})\\
(g_n\overline{\varphi}^n(G))_{n\in\mathds{Z}}&\mapsto\{(g_n\overline{\varphi}^n(h),n):\;n\in\mathds{Z},\;h\in G\}.
\end{split}$$
The set $\rho(\widetilde{G})$ is contained in $\Omega_\mathcal{R}$.
Take $(g_m)_m\in\widetilde{G}$ and it is clear from the definition of $\widetilde{G}$ that $$g_m=g_{m-n}\overline{\varphi}^{m-n}(\overline{k}_1)$$ and $$g_{m+n}=g_m\overline{\varphi}^m(\overline{k}_2)$$ for $n\in\mathds{N}$ and $\overline{k}_1,\overline{k}_2\in G$.
Denote $\xi:=\rho((g_m)_m)$. We have to show that $f(g^{-1}\xi)=0$ for all $g\in\xi$ and all $f\in\widehat{\mathcal{R}}=\widehat{\mathcal{R}}_1\cup\widehat{\mathcal{R}}_2\cup\widehat{\mathcal{R}}_3$. Therefore fix $g=(g_m\overline{\varphi}^m(k),m)\in\xi$ for $m\in\mathds{Z}$ and $k\in G$.
$\bullet\; f=1_{(h,0)}-1\in\widehat{\mathcal{R}}_1$: Then $f(g^{-1}\xi)=0\Leftrightarrow g(h,0)\in\xi$, which is true because $g(h,0)=(g_m\overline{\varphi}^m(kh),m)\in\xi$.
$\bullet\; f=1_{(e,-n)}-1\in\widehat{\mathcal{R}}_2$: Similarly $f(g^{-1}\xi)=0\Leftrightarrow g(e,-n)\in\xi$ and the latter holds as $g(e,-n)=(g_m\overline{\varphi}^m(k),m-n)=(g_{m-n}\overline{\varphi}^{m-n}(\overline{k}_1\overline{\varphi}^n(k)),m-n)\in\xi$.
$\bullet\; f={\displaystyle}\sum_{h\in\frac{G}{\varphi^n(G)}}1_{(h,n)}-1\in\widehat{\mathcal{R}}_3$: Here $f(g^{-1}\xi)=0\Leftrightarrow$ there exists only one class $h\varphi^n(G)$ such that $g(h,n)\in\xi$. But $$g(h,n)=(g_m\overline{\varphi}^m(kh),m+n)=(g_{m+n}\overline{\varphi}^m(\overline{k}_2^{-1}kh),m+n)$$
belongs to $\xi$ if and only if $\overline{k}_2^{-1}kh\in\overline{\varphi}^n(G)=\varphi^n(G)$ (as $n\in\mathds{N}$), which is the same as requiring $h\in k^{-1}\overline{k}_2\varphi^n(G)$, and this can be true only for one class in $\frac{G}{\varphi^n{G}}$.
\[homeorho\] $\rho:\widetilde{G}\rightarrow \Omega_\mathcal{R}$ is a homeomorphism.
If $\rho((g_m)_m)=\rho((h_m)_m)$ then $h_m=g_m\varphi^m(k_m)$ for all $m\in\mathds{N}$, with $k_m\in G$. Then $g_m=h_m$ in $\frac{G}{\varphi^m(G)}$ for all $m\in\mathds{N}$ and $(g_m)_m=(h_m)_m$ (note that for $m<0$, $g_m=h_m=e$).
Now let us prove that $\rho$ is surjective. Take $\xi\in\Omega_\mathcal{R}$ and remember that $(e,0)\in\xi$ which, using $f_1^h:=1_{(h,0)}-1\in\widehat{\mathcal{R}}_1$, implies that $(h,0)\in\xi$ $\forall\; h\in G$. Also for each $j\in\mathds{N}$, set $f_3^j:={\displaystyle}\sum_{h\in\frac{G}{\varphi^j(G)}}1_{(h,j)}-1\in\widehat{\mathcal{R}}_3$.
As $f_3^j((e,0)\xi)=0$, for each $j$ there exists only one class $u_j\varphi^j(G)\in\frac{G}{\varphi^j(G)}$ such that $(u_j,j)\in\xi$. Using functions of the type $f_2^n:=1_{(0,-n)}-1\in\widehat{R}_2$, for $n\in\mathds{N}$, one sees that $(u_j\varphi^j(G))_{j\in\mathds{Z}}\in\widetilde{G}$. Now we prove that $\rho((u_j\varphi^j(G))_j)=\xi$.
By construction $(u_j,j)\in\xi$, which implies (using $f_1^h\in\widehat{\mathcal{R}}_1$ defined above) that $(u_j,j)(h,0)=(u_j\varphi^j(h),j)\in\xi$ for all $h\in G$. Doing the same for every $j$ it follows that $\rho((u_j\varphi^j(G))_j)\subseteq\xi$.
Suppose that $h=(k,i)\in\xi\backslash\rho((u_j\varphi^j(G))_j)$ and note that $$(k,i)\notin\rho((u_j\varphi^j(G))_j)\Leftrightarrow(k,i)\notin(u_i\varphi^i(G),i)\Leftrightarrow u^{-1}_ik\notin\varphi^i(G).$$
Now consider the elements $g=(u_i,0)$ and $h'=(u_i,i)$ of $\rho((u_j\varphi^j(G))_j)\subseteq\xi$. Since $u^{-1}_ik\notin\varphi^i(G)$, we have that $g^{-1}h=(u^{-1}_ik,i)$ and $g^{-1}h'=(e,i)$ are different, which implies that $f_3^i(g^{-1}\xi)\neq 0$, and this contradicts the fact that $\xi\in\Omega_\mathcal{R}$.
Last, let us prove that $\rho$ preserves the topology. As the sets are compact and Hausdorff, it is enough to prove that $\rho^{-1}$ is continuous, which we will prove by showing that $\pi_m\circ\rho^{-1}$ is continuous for all $m\in\mathds{Z}$ where $\pi_m:\widetilde{G}\rightarrow\frac{G}{\varphi^m(G)}$ is the canonical projection.
As $\frac{G}{\varphi^m(G)}$ is discrete we just have to show that $\rho\circ\pi_m^{-1}(\{u_m\varphi^m(G)\})$ is open in $\Omega_R$ for all $u_m\varphi^m(G)\in\frac{G}{\varphi^m(G)}$. But note that (by the proof of surjectivity above) $$\rho\circ\pi_m^{-1}(\{u_m\varphi^m(G)\})=\{\xi\in\Omega_R:\;(u_m,m)\in\xi\},$$
which is open in $\Omega_R$ (induced by the product topology in $\{0,1\}^{\overline{S}}$).
Then $\rho:\widetilde{G}\rightarrow \hbox{P}(\overline{S})$ is a homeomorphism.
Using the proposition above, we identify $\Omega_\mathcal{R}$ with $\widetilde{G}$, and thus view $\varpi$ as a partial action of $G$ on $\widetilde{G}$. Remember that $$\Omega_s=\{\xi\in\Omega_\mathcal{R}:\; s\in\xi\}.$$
Set $$\widetilde{G}_s:=\rho^{-1}(\Omega_s)$$
and define $$\varpi_s:\widetilde{G}_{s^{-1}}\rightarrow\widetilde{G}_s.$$
Using $\rho$ we can conclude that for $(g_i,n)\in\overline{S}=\mathds{G}\rtimes_{\overline{\varphi}}\mathds{Z}$ ($g_i\in G_i\hookrightarrow\mathds{G}$) $$\widetilde{G}_{(g_i,n)}=\{(h_m\varphi^m(G))_{m\in\mathds{Z}}\in\widetilde{G}:\;h_n\varphi^n(G)=g_i\varphi^n(G)\}$$
(where $h_n$ is viewed inside $G=G_0\subseteq\mathds{G}$) and $$\varpi_{(g_i,n)}((h_m\varphi^m(G))_m)=(g_i\varphi^n(h_m)\varphi^{n+m}(G))_{n+m}=(g_i\varphi^n(h_{m-n})\varphi^m(G))_m.$$
An easily proven and useful result follows.
\[lema111\]For $(g_i,n)\in\overline{S}$ the following holds:
1. $\widetilde{G}_{(g_i,n)}=\emptyset\Leftrightarrow g_i\notin G\varphi^n(G)$;
2. $\widetilde{G}_{(g_i,n)}=\widetilde{G}\Leftrightarrow G\subseteq g_i\varphi^n(G)$.
$\square$
For $m\in\mathds{Z}$ and a subset $C_m\subseteq\frac{G}{\varphi^m(G)}$ (containing whole cosets) define the open set (it is open because it is the inverse image of a point via a projection) $$V_m^{C_m}=\{(u_n\varphi^n(G))_n\in\widetilde{G}:\;u_m\varphi^m(G)\in C_m\}.$$
Clearly when $m\leq n$ then $V_m^{C_m}=V_n^{C_n}$ where $$C_n=\left\{u\varphi^n(G)\in\dfrac{G}{\varphi^n(G)}:\;u\varphi^m(G)\in C_m\right\}.$$
From the definition of the product topology, we know that finite intersections of open sets $V_m^{C_m}$ form the base for the topology in $\widetilde{G}$. Since $V_{m_1}^{C_{m_1}}\cap V_{m_2}^{C_{m_2}}=V_{m}^{C_{n_1}}\cap V_{m}^{C_{n_2}}=V_m^{C_{n_1}\cap C_{n_2}}$ for $m\geq m_1,m_2$, $\left\{V_m^{C_m}\right\}$ is already a base for the topology.
Also note that if $C_m\neq\emptyset$ then for $k>0$, $C_{m+k}$ has at least 2 elements and therefore we can assume that if $V_m^{C_m}$ is not empty then $C_m$ has at least 2 elements (replacing $V_m^{C_m}$ by $V_n^{C_n}$ for $n>m$ if necessary).
\[prop111\]When $\varphi$ is a pure injective endomorphism of a commutative group $G$, the partial action $\varpi$ from $\widetilde{G}$ defined above is topologically free.
Let us show that $$F_{(g_i,n)}=\{x\in\widetilde{G}_{(g_i,n)^{-1}}:\;\varpi_{(g_i,n)}(x)=x\}$$
has empty interior, for $(g_i,n)\neq (e,0)$.
$\bullet$ Case 1: $n=0$. If $g_i\notin G$ then Lemma \[lema111\] (i) assures that $F_{(g_i,0)}=\emptyset$.
Therefore suppose that $g_i\in G$. If $F_{(g_i,0)}\neq\emptyset$ the equation $\varpi_{(g_i,0)}(x)=x$ implies $g_i\in\varphi^m(G)$ for all $m\in\mathds{Z}$ (using the commutativity of $G$). As $\varphi$ is pure we conclude that $g_i=e$, and then $F_{(g_i,0)}=\emptyset$ for $g_i\neq e$.
$\bullet$ Case 2: Let $(g_i,n)$ with $n\neq 0$. Using again Lemma \[lema111\] (i) we can assume that $g_i\in G\varphi^n(G)$. Take $V$ a non-empty open set of $\widetilde{G}_{(g_i,n)^{-1}}$ and, if needed, shrink $V$ so that $V=V_m^{C_m}$ (and we can assume that $m=ln>0$ for some big $l>0$). Note that we can assume that $C_m$ has at least 2 distinct elements, say $u_1\varphi^m(G)\neq u_2\varphi^m(G)$, which implies that $u_2^{-1}u_1\notin\varphi^m(G)$.
Suppose for a contradiction that $\varpi_{(g_i,n)}(x)=x$, $\forall\; x\in V$. Then, since $(u_j\varphi^k(G))_k\in V$ for $j=1,2$, we have $$\begin{split}
\varpi_{(g_i,n)}((u_j\varphi^k(G))_k)=(u_j\varphi^k(G))_k&\Rightarrow (g_i\varphi^n(u_j)\varphi^k(G))_k=(u_j\varphi^k(G))_k\\
&\Rightarrow u_j^{-1}g_i\varphi^n(u_j)\in\varphi^k(G)\hbox{ for }j=1,2\\
&\Rightarrow\varphi^n(u_2^{-1})u_2u_1^{-1}\varphi^n(u_1)\in\varphi^k(G), \forall\; k\in\mathds{Z}
\end{split}$$
(again we used the commutativity of $G$ to cancel the $g_i$’s). But as $\varphi$ is pure, $$\begin{split}
\varphi^n(u_2^{-1}u_1)=u_2^{-1}u_1&\Rightarrow\varphi^{ln}(u_2^{-1}u_1)=u_2^{-1}u_1\Rightarrow u_2^{-1}u_1\in\varphi^m(G)
\end{split}$$
which contradicts our hypothesis. So no open set can be contained in $F_{(g_i,n)}$, which implies that it has empty interior.
\[prop112\]The partial action $\varpi$ is minimal.
We will show that all $x\in\widetilde{G}$ has dense orbit by showing the following: if $V$ is a non-empty open set then there exists $(g,n)\in\overline{S}$ such that $x\in\widetilde{G}_{(g,n)^{-1}}$ and $\varpi_{(g,n)}(x)\in V$.
Take $x=(u_m\varphi^m(G))_{m\in\mathds{Z}}\in\widetilde{G}$ and $V=V_k^{C_k}\neq\emptyset$. Consider $u\varphi^k(G)\in C_k$ and define $(uu_k^{-1},0)$. By Lemma \[lema111\] (ii), since $uu_k^{-1}G=G$, it follows that $\widetilde{G}_{(uu_k^{-1},0)^{-1}}=\widetilde{G}$ and therefore $x\in\widetilde{G}_{(uu_k^{-1},0)^{-1}}$.
To finish, note that $$\varpi_{(uu_k^{-1},0)}(x)=\varpi_{(uu_k^{-1},0)}((u_m\varphi^m(G))_m)=(uu_k^{-1}u_m\varphi^m(G))_m\in V.$$
We can now conclude (and this result agrees with the previous obtained Theorem \[teo1\]):
If $\varphi$ is a pure injective endomorphism with finite cokernel of some commutative discrete countable group $G$ then the C$^*$-algebra $\mathds{U}[\varphi]$ is simple.
$\square$
In the conditions of theorem above, we have$$C_r^*[\varphi]\cong\mathds{U}[\varphi].$$
$\square$
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DEPARTAMENTO DE MATEMÁTICA - UFSC BLUMENAU - BRAZIL (f.vieira@ufsc.br)
[^1]: Supported by CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
[^2]: I.e.: $\exists\; s_i$ isometries such that $s_is_i^*=f_i$,$\forall\; i$
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---
abstract: 'The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing $XY$ interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the $C^*$-algebraic approach to the phase transition problem. Note that these kinde of models do not have one-dimensional analogues, i.e. the considered model persists only on trees. It turns out that if the Ising part interactions vanish then the model with only competing $XY$-interactions on the Cayley tree of order two does not have a phase transition. By phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model have clustering property which implies that the von Neumann algebras corresponding to the states are factors.'
---
\
\
*Department of Mathematical Sciences,\
College of Science, United Arab Emirates University,\
P.O. Box 15551,Al Ain, Abu Dhabi, UAE*\
E-mail: [far75m@yandex.ru, farrukh.m@uaeu.ac.ae]{}
[Abdessatar Barhoumi]{}\
*Department of Mathematics\
Nabeul Preparatory Engineering Institute\
Campus Universitairy - Mrezgua - 8000 Nabeul,\
Carthage University, Tunisia*\
E-mail: [abdessatar.barhoumi@ipein.rnu.tn]{}\
[Abdessatar Souissi]{}\
*College of Business Administration,\
Qassim university, Buraydah, Saudi Arabia*\
E-mail: [a.souaissi@qu.edu.sa]{}\
[Soueidy EL Gheteb ]{}\
*Department of Mathematics,\
Carthage University, Tunisia*\
E-mail: [elkotobmedsalem@gmail.com]{}\
Introduction {#intr}
=============
The study of magnetic systems with competing interactions in ordering is a fascinating problem of condensed matter physics. One of the most canonical examples of such systems are frustrated Ising models which demonstrate a plethora of critical properties[@Chak; @Lieb; @MS]. The frustrations can be either geometrical or brought about the next-nearest neighbor NNN interactions. Competing interactions frustrations can result new phases, change the Ising universality class, or even destroy the order at all. Another interesting aspect of the criticality in the frustrated Ising models is an appearance of quantum critical points at spacial frustration points of model’s high degeneracy, and related quantum phase transitions [@Sach]. The Ising models with frustrations can be thought as perturbation of the classical Ising model. If the perturbation terms do not commute with the Ising pieces, it outcomes quantum effects. In particular case, if the perturbation is the XY interaction, then the model become more interesting (see [@AAC; @CG; @MS; @Ow] for a systematic study (physical approach) of the Ising model with quantum frustration on 2D lattices). However, a rigorous (mathematical) investigation of the quantum Ising model with competing $XY$ interactions does not exist yet in the literature. We notice that $XY$-interactions are truly quantum, ( i.e. contain pieces not commuting with each other). In the present paper, we propose to investigate the phase transition problem for the mentioned model on the Cayley tree or Bethe lattice [@Ost] within quantum Markov chains (QMC) scheme. Here, the QMC scheme is based on the $C^*$-algebraic approach. We notice that the Ising model with Ising type competing interactions (with commuting interactions) has been recently studied in [@MBS161; @MBS162; @MR1; @MR2] by means of QMC. As we mentioned, in the current paper, the commuting interactions are non-commutative, and this makes big difference between those papers.
On the other hand, our investigation will allow to construct quantum analogous of Markov fields (see [@D; @Geor; @[Pr]; @[Sp75]; @Spa]) which is one of the basic problems in quantum probability. We notice that quantum Markov fields naturally appear in quantum statistical mechanics and quantum fields theories [@DW; @DM][^1].
We point out, even in classical setting, for models over integer lattices, there do not exist analytical solutions (for example, critical temperature) on such lattices. Therefore, it was proposed [@Bax] to consider spin models on regular trees for which one can exactly calculate various physical quantities. One of the simplest tree is a Cayley tree [@Ost]. In [@MBS161; @MBS162] we have established that Gibbs measures of the Ising model with competing (Ising) interactions on a Cayley trees, can be considered as QMC. Note that if the perturbation vanishes then the model reduces to the classical Ising one which was also examined in [@ArE] by means of $C^*$-algebra approach.
In the present paper, we are going to study the Ising model with $XY$-competing interactions on a Cayley trees of order two. We point out that this model has non-commutative interactions, i.e. $XY$ ones, therefore, the investigation of this model is tricky. We notice that, in general, QMC do not have KMS property (see [@AF03; @BR]), therefore, general theory of KMS-states is not applicable for such kind of chains. One of the main questions of this paper is to know whether the considered model exhibits two different QMC associated with the mentioned model on the Cayley trees. Our main result is the following one.
\[Main\] For the Ising model with competing $XY$- interactions , , $J_{0}>0$, $J \in \mathbb{R}\setminus\{-J_0,J_0\}$, $\b>0$ on the Cayley tree of order two, the following statements hold:
- if $\Delta(\theta)\leq 0$, then there is a unique QMC;
- if $\Delta(\theta)>0$, then there occurs a phase transition,
where $$\Delta(\theta)=\frac{\theta^{2J_0}-\theta^{J_0}(\theta^{J}+\theta^{-J})-3}{\theta^{2J_0}-\theta^{J_0}(\theta^{J}+\theta^{-J})+1},~~
\theta=e^{2\beta}.$$
To establish result, we will prove Theorem \[5.3\], from which we conclude that there are three coexisting phases in the region $\Delta(\theta)>0$, and one of it, i.e. $\varphi_{\alpha}$, survives in the region $\Delta(\theta)<0$. This leads us that the state $\varphi_{\alpha}$ describes the disordered phase of the model, which shows a similar behavior with the classical Ising model [@B90; @Roz2]. In comparison with the Ising model, we stress that in the present model, we have a similar kind of phases (translation invariant ones) when $\Delta(\theta)>0$. From Figure 2 (see below), one concludes that the phase transition occurs except for a “triangular region”. This shows how the competing interactions effect to the existence of other phases. Notice that if $J=0$, then we obtain the classical Ising model for which the existence of a disordered phase coexisting with two ordered phases is well-known [@Bax; @Roz2].
On the other hand, we emphasize that both problems, i.e. a construction and phase transitions are non-trivial and, to a large extent, open. In fact, even if several definitions of quantum Markov fields on trees (and more generally on graphs) have been proposed, a really satisfactory, general theory is still missing and physically interesting examples of such fields in dimension $d\geq2$ are very few.
In order to get the existence of the phase transition (see [@MBS161]), one needs to check several conditions, and one of them based on a notion of the quasi-equivalence of quantum Markov chains which essentially uses $C^*$-algebraic approach and techniques. This situation totally differs from the classical (resp. quantum) cases, where it is sufficient to prove the existence of at least two different solutions (resp. KMS states) of associated renormalized equations (see [@Roz2]). Therefore, even for classical models, to check the existence of the phase transition (in the sense of our paper) is not a trivial problem. Here we mention that the quasi-equivalence of product states (which correspond to the classical model without interactions) was a tricky job and investigated in [@Mat0; @PS]. In this paper, we are considering more complicated states (which are QMC associated with the model) than product ones, and for these kind of states we are going to obtain their non-quasi equivalence. We will first show that these states have clustering property, and hence they are factor states. We point out that even this fact presents its own interests since these states associates with non-commutative Hamiltonians having non-trivial interactions.
Let us outline the organization of the paper. After preliminary information (see Section 2), in Section 3 we provide a general construction of backward quantum Markov chains on Cayley tree. Moreover, in this section we give the definition of the phase transition. Using the provided construction, in Section 4 we consider the Ising model with competing $XY$-interactions on the Cayley tree of order two. Section 5 is devoted to the existence of the three translation-invariant QMC $\varphi_\a$, $\varphi_1$ and $\varphi_1$ corresponding to the model. Section 6 is devoted to the proof of Theorem \[Main\]. In this section we will prove that states $\varphi_1$ and $\varphi_2$ do not have overlapping supports. Before, to establish their non-quasi equivalence, we first prove that these states have the clustering property. Section 7 we study a particular case $J_0=0$, which means that we only have $XY$ interactions. In the considered setting, it turns out that the phase transition does not occur.
Preliminaries
=============
Let $\Gamma^k_+ = (L,E)$ be a semi-infinite Cayley tree of order $k\geq 1$ with the root $x^0$ (i.e. each vertex of $\Gamma^k_+$ has exactly $k+1$ edges, except for the root $x^0$, which has $k$ edges). Here $L$ is the set of vertices and $E$ is the set of edges. The vertices $x$ and $y$ are called [*nearest neighbors*]{} and they are denoted by $l=<x,y>$ if there exists an edge connecting them. A collection of the pairs $<x,x_1>,\dots,<x_{d-1},y>$ is called a [*path*]{} from the point $x$ to the point $y$. The distance $d(x,y), x,y\in V$, on the Cayley tree, is the length of the shortest path from $x$ to $y$.
Recall a coordinate structure in $\G^k_+$: every vertex $x$ (except for $x^0$) of $\G^k_+$ has coordinates $(i_1,\dots,i_n)$, here $i_m\in\{1,\dots,k\}$, $1\leq m\leq n$ and for the vertex $x^0$ we put $(0)$. Namely, the symbol $(0)$ constitutes level 0, and the sites $(i_1,\dots,i_n)$ form level $n$ (i.e. $d(x^0,x)=n$) of the lattice (see Fig. 1).
![The first levels of $\G_+^2$[]{data-label="fig1"}](tree.eps){width="10.07cm"}
Let us set $$W_n = \{ x\in L \, : \, d(x,x_0) = n\} , \qquad \Lambda_n =
\bigcup_{k=0}^n W_k, \qquad \L_{[n,m]}=\bigcup_{k=n}^mW_k, \
(n<m)$$ $$E_n = \big\{ <x,y> \in E \, : \, x,y \in \Lambda_n\big\}, \qquad
\Lambda_n^c = \bigcup_{k=n}^\infty W_k$$ For $x\in \G^k_+$, $x=(i_1,\dots,i_n)$ denote $$S(x)=\{(x,i):\ 1\leq
i\leq k\}.$$ Here $(x,i)$ means that $(i_1,\dots,i_n,i)$. This set is called a set of [*direct successors*]{} of $x$.
Two vertices $x,y\in V$ is called [*one level next-nearest-neighbor vertices*]{} if there is a vertex $z\in V$ such that $x,y\in S(z)$, and they are denoted by $>x,y<$. In this case the vertices $x,z,y$ was called [*ternary*]{} and denoted by $<x,z,y>$.
Let us define on $\G^k_+$ a binary operation $\circ:\G^k_+\times\G^k_+\to\G^k_+$ as follows: for any two elements $x=(i_1,\dots,i_n)$ and $y=(j_1,\dots,j_m)$ put $$\label{binar1}
x\circ
y=(i_1,\dots,i_n)\circ(j_1,\dots,j_m)=(i_1,\dots,i_n,j_1,\dots,j_m)$$ and $$\label{binar2}
x\circ x^0=x^0\circ x= (i_1,\dots,i_n)\circ(0)=(i_1,\dots,i_n).$$
By means of the defined operation $\G^k_+$ becomes a noncommutative semigroup with a unit. Using this semigroup structure one defines translations $\tau_g:\G^k_+\to \G^k_+$, $g\in \G^k_+$ by $$\label{trans1}
\tau_g(x)=g\circ x.$$ It is clear that $\tau_{(0)}=id$.
The algebra of observables $\cb_x$ for any single site $x\in L$ will be taken as the algebra $M_d$ of the complex $d\times d$ matrices. The algebra of observables localized in the finite volume $\L\subset L$ is then given by $\cb_\L=\bigotimes\limits_{x\in\L}\cb_x$. As usual if $\L^1\subset\L^2\subset L$, then $\cb_{\L^1}$ is identified as a subalgebra of $\cb_{\L^2}$ by tensoring with unit matrices on the sites $x\in\L^2\setminus\L^1$. Note that, in the sequel, by $\cb_{\L,+}$ we denote the positive part of $\cb_\L$. The full algebra $\cb_L$ of the tree is obtained in the usual manner by an inductive limit $$\cb_L=\overline{\bigcup\limits_{\L_n}\cb_{\L_n}}.$$
In what follows, by ${\cal S}({\cal B}_\L)$ we will denote the set of all states defined on the algebra ${\cal B}_\L$.
Consider a triplet ${\cal C} \subset {\cal B} \subset {\cal A}$ of unital $C^*$-algebras. Recall [@ACe] that a [*quasi-conditional expectation*]{} with respect to the given triplet is a completely positive (CP) linear map $\ce \,:\, {\cal A} \to
{\cal B}$ such that $ \ce(ca) = c \ce(a)$, for all $a\in {\cal
A},\, c \in {\cal C}$.
\[QMCdef\] A state $\varphi$ on ${\cal B}_L$ is called a [*backward quantum Markov chain (QMC)*]{}, associated to $\{\L_n\}$, if for each $\Lambda_n$, there exist a quasi-conditional expectation $\ce_{\Lambda_n}$ with respect to the triplet $$\label{trplt1}
{\cal B}_{{\Lambda}_{n-1}}\subseteq {\cal
B}_{\Lambda_n}\subseteq{\cal B}_{\Lambda_{n+1}}$$ and an initial state $\rho \in S(B_{\Lambda_0})$ such that: $$\label{dfgqmf}
\varphi = \lim_{n\to\infty} \rho_0 \circ
\ce_{\Lambda_0}\circ \ce_{\Lambda_{1}} \circ \cdots \circ
\ce_{\Lambda_n}$$ in the weak-\* topology.
We notice that in [@[AcFiMu07]] a more general definition of backward QMC is given on arbitrary quasi-local algebras.
Construction of Quantum Markov Chains {#dfcayley}
=====================================
In this section we are going to provide a construction of a backward quantum Markov chain which contains competing interactions.
Let us rewrite the elements of $W_n$ in the following lexicographic order (w.r.t. the coordinate system), i.e. $$\begin{aligned}
\overrightarrow{W_n}:=\left(x^{(1)}_{W_n},x^{(2)}_{W_n},\cdots,x^{(|W_n|)}_{W_n}\right).\end{aligned}$$
Note that $|W_n|=k^n$. In this lexicographic order, vertices $x^{(1)}_{W_n},x^{(2)}_{W_n},\cdots,x^{(|W_n|)}_{W_n}$ of $W_n$ are given as follows $$\begin{aligned}
\label{xw}
&&x^{(1)}_{W_n}=(1,1,\cdots,1,1), \quad x^{(2)}_{W_n}=(1,1,\cdots,1,2), \ \ \cdots \quad x^{(k)}_{W_n}=(1,1,\cdots,1,k,),\\
&&x^{(k+1)}_{W_n}=(1,1,\cdots,2,1), \quad
x^{(2)}_{W_n}=(1,1,\cdots,2,2), \ \ \cdots \quad
x^{(2k)}_{W_n}=(1,1,\cdots,2,k),\nonumber\end{aligned}$$ $$\vdots$$ $$\begin{aligned}
&&x^{(|W_n|-k+1)}_{W_n}=(k,k,,\cdots,k,1), \
x^{(|W_n|-k+2)}_{W_n}=(k,k,\cdots,k,2),\ \ \cdots
x^{|W_n|}_{W_n}=(k,k,\cdots,k,k).\end{aligned}$$
Analogously, for a given vertex $x,$ we shall use the following notation for the set of direct successors of $x$: $$\begin{aligned}
\overrightarrow{S(x)}:=\left((x,1),(x,2),\cdots (x,k)\right),\quad
\overleftarrow{S(x)}:=\left((x,k),(x,k-1),\cdots (x,1)\right).\end{aligned}$$
In what follows, by $^\circ\prod$ we denote the lexicographic order, i.e. $$^\circ\prod_{k=1}^n a_k=a_1a_2\cdots a_n,$$ where elements $\{a_k\}\subset {\mathcal{B}}_L$ are multiplied in the indicated order. This means that we are not allowed to change this order.
Note that each vertex $x\in L$ has interacting vertices $\{x, (x,1),\dots,(x,k)\}$. Assume that, to each edges $<x,(x,i)>$ ($i=1,\dots,k$) an operators $K_{<x,(x,i)>}\in\mathcal{B}_{x}\otimes\mathcal{B}_{(x, i)}$ is assigned, respectively. Moreover, for each competing vertices $>(x,i),(x,i+1)<$ and $<x,(x,i),(x,i+1)>$ ($i=1,\dots,k$) the following operators are assigned: $$L_{>(x,i),(x,i+1)<}\in\mathcal{B}_{(x,i)}\otimes\mathcal{B}_{(x,i+1)}, \ \
M_{(x,(x,i),(x,i+1))}\in\mathcal{B}_{x}\otimes\mathcal{B}_{(x,i)}\otimes\mathcal{B}_{(x,i+1)}.$$
We would like to define a state on $\mathcal{B}_{\Lambda_n}$ with boundary conditions $\omega_{0}\in\mathcal{B}_{0,+}$ and $\{h^{x}\in\mathcal{B}_{x,+}:\ x\in L\}$.
For each $n\in \natural$ denote $$\begin{aligned}
\label{K1}
&& A_{x,(x,1),\dots,(x,k)}=\big(^\circ\prod_{i=1}^k
K_{x,(x,i)}\big)\big(^\circ\prod_{i=1}^k
L_{>(x,i),(x,i+1)<}\big)\big(^\circ\prod_{i=1}^k
M_{(x,(x,i),(x,i+1))}\big),\\[2mm]
\label{K11} &&K_{[m,m+1]}:=\prod_{x\in \overrightarrow
W_{m}}A_{x,(x,1),\dots,(x,k)}, \ \ 1\leq m\leq n,\\[2mm]
\label{K2} &&
\bh_{n}^{1/2}:=\prod_{x\in W_n}(h^{x})^{1/2}, \ \ \ \bh_n=\bh_{n}^{1/2}(\bh_{n}^{1/2})^*\\[2mm]
\label{K3}
&&{\mathbf{K}}_n:=\omega_{0}^{1/2} (^\circ\prod_{m=0}^{n-1}K_{[m,m+1]})\bh_{n}^{1/2}\\[2mm]
\label{K4} && \mathcal{W}_{n]}:={\mathbf{K}}_n^{*}{\mathbf{K}}_n\end{aligned}$$ One can see that ${\cw}_{n]}$ is positive.
In what follows, by $\tr_{\L}:\cb_L\to\cb_{\L}$ we mean normalized partial trace (i.e. $\tr_{\L}(\id_{L})=\id_{\L}$, here $\id_{\L}=\bigotimes\limits_{y\in \L}\id$), for any $\Lambda\subseteq_{\text{fin}}L$. For the sake of shortness we put $\tr_{n]} := \tr_{\Lambda_n}$.
Let us define a positive functional $\ffi^{(n)}_{w_0,\bh}$ on $\cb_{\Lambda_n}$ by $$\begin{aligned}
\label{ffi-ff}
\ffi^{(n)}_{w_0,\bh}(a)=\tr(\cw_{n+1]}(a\otimes\id_{W_{n+1}})),\end{aligned}$$ for every $a\in \cb_{\Lambda_n}$. Note that here, $\tr$ is a normalized trace on ${\cal B}_L$ (i.e. $\tr(\id_{L})=1$).
To get an infinite-volume state $\ffi$ on $\cb_L$ such that $\ffi\lceil_{\cb_{\L_n}}=\ffi^{(n)}_{w_0,\bh}$, we need to impose some constrains to the boundary conditions $\big\{w_0,\bh\big\}$ so that the functionals $\{\ffi^{(n)}_{w_0,\bh}\}$ satisfy the compatibility condition, i.e. $$\begin{aligned}
\label{compatibility}
\ffi^{(n+1)}_{w_0,\bh}\lceil_{\cb_{\L_n}}=\ffi^{(n)}_{w_0,\bh}.\end{aligned}$$
\[compa\] Assume that for every $x\in L$ and triple $\{x,(x,i),(x,i+1)\}$ $(i=1,\dots,k-1$) the operators $K_{<x,(x,i)>}$, $L_{>(x,i),(x,i+1)<}$, $M_{(x,(x,i),(x,i+1))}$ are given as above. Let the boundary conditions $w_{0}\in {\cal
B}_{(0),+}$ and ${\bh}=\{h_x\in {\cal B}_{x,+}\}_{x\in L}$ satisfy the following conditions: $$\begin{aligned}
\label{eq1}
&&\Tr(\omega_{0}h_{0})=1, \\
\label{eq2}
&&\Tr_{x]}\big({A_{x,(x,1),\dots,(x,k)}}^\circ\prod_{i=1}^k
h^{(x,i)}A_{x,(x,1),\dots,(x,k)}^{*}\big)=h^x, \ \ \textrm{for
every}\ \ x\in L,\end{aligned}$$ where as before $A_{x,(x,1),\dots,(x,k)}$ is given by . Then the functionals $\{\ffi^{(n)}_{w_0,\bh}\}$ satisfy the compatibility condition . Moreover, there is a unique backward quantum d-Markov chain $\ffi_{w_0,{\bh}}$ on $\cb_L$ such that $\ffi_{w_0,{\bh}}=w-\lim\limits_{n\to\infty}\ffi^{(n)}_{w_0,\bh}$.
Let us check that the states $\ffi^{(n,b)}_{w_0,{\bh}}$ satisfy the compatibility condition. For $a \in B_{\Lambda_n}$, we have:
$\ffi^{(n+1)}_{w_0,{\bh}}(a\otimes \id_{\cw_{n+1}})=\tr(\cw_{n+2]}(a\otimes\id_{\Lambda_{[n+1,n+2]}}))$\
$=\tr(K_{[n,n+1]}^{*}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*} w_0 K_{[0,1]} K_{[1,2]}
\cdots K_{[n+1,n+2]}\bh_{n+2}^{1/2}(a\otimes\id_{\Lambda_{[n+1,n+2]}})\bh_{n+2}^{1/2}K_{[n+1,n+2]}^{*})$\
$=\tr(K_{[n,n+1]}^{*}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*} w_0 K_{[0,1]}K_{[1,2]}
\cdots K_{[n,n+1]}(a\otimes\id_{\Lambda_{[n+1,n+2]}})K_{[n+1,n+2]}\bh_{n+2}K_{[n+1,n+2]}^{*})$\
$=\tr(K_{[n,n+1]}^{*}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*} w_0 K_{[0,1]}K_{[1,2]}
\cdots K_{[n,n+1]}(a\otimes\id_{\Lambda_{[n+1,n+2]}})\Tr_{n+1]}(K_{[n+1,n+2]}\bh_{n+2}K_{[n+1,n+2]}^{*}))$\
$=\tr(K_{[n,n+1]}^{*}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*} w_0 K_{[0,1]}K_{[1,2]}
\cdots K_{[n,n+1]}(a\otimes\id_{\Lambda_{[n+1,n+2]}})\bh_{n+1})$\
$=\tr(\bh_{n+1}^{1/2}K_{[n,n+1]}^{*}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*} w_0 K_{[0,1]}K_{[1,2]}
\cdots K_{[n,n+1]}\bh_{n+1}^{1/2}(a\otimes\id_{\Lambda_{[n+1,n+2]}}))$\
$=\tr(\cw_{n+1]}(a\otimes\id_{\Lambda_{[n+1,n+2]}}))$\
$=\ffi^{(n)}_{w_0,{\bh}}(a)$
To show that $\ffi^{(n)}_{w_0,{\bh}}$ is a backward QMC, we define quasi-conditional expectations $\ce_{n} $ as follows: $$\begin{aligned}
\label{E-n1}
&&\hat\ce_{n}(x_{n+1]})=\tr_{n]}(K_{[n,n+1]}\bh_{n+1}^{1/2}x_{n+1]}\bh_{n+1}^{1/2}K_{[n,n+1]}^*), \ \ x_{n+1]}\in \cb_{\L_{n+1}}\\
&&\label{E-n2}
\ce_{k}(x_{k+1]})=\tr_{k]}(K_{[k,k+1]}x_{k+1]}K_{[k,k+1]}^*),
\ \ x_{k+1]}\in\cb_{\L_{k+1}}, \ \ k=1,2,\dots,n-1,\end{aligned}$$ Then for any monomial $a_{\L_1}\otimes a_{W_2}\otimes\cdots\otimes
a_{W_{n}}\otimes\id_{W_{n+1}}$, where $a_{\L_1}\in\cb_{\L_1},a_{W_k}\in\cb_{W_k}$, ($k=2,\dots,n$), we have:\
$
\ffi^{(n)}_{w_0,\bh}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}})=\tr(\cw_{n+1]}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}}\otimes \id_{W_{n+1}}))$\
$=\tr(w_0 K_{[0,1]}
\cdots K_{[n,n+1]}\bh_{n+1}^{1/2}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}}\otimes \id_{W_{n+1}})\bh_{n+1}^{1/2}K_{[n,n+1]}^{*}\cdots K_{[0,1]}^{*})\\[2mm]
$$=\tr(w_0 K_{[0,1]} \cdots K_{[n-1,n]} \tr_{n]}(K_{[n,n+1]}\bh_{n+1}^{1/2}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}}\otimes \id_{W_{n+1}})\bh_{n+1}^{1/2}K_{[n,n+1]}^{*})K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*})\\[2mm]
$$=\tr(w_0 K_{[0,1]} \cdots K_{[n-2,n-1]} \tr_{n-1]}(K_{[n-1,n]} \hat\ce_{n}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}}) K_{[n-1,n]}^{*})\cdots K_{[0,1]}^{*})\\[2mm]
$$=\tr(w_0 K_{[0,1]} \cdots K_{[n-2,n-1]} \ce_{n-1} \hat\ce_{n}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}})K_{[n-2,n-1]}^{*}\cdots K_{[0,1]}^{*})\\[2mm]
$$=\tr(w_0 \ce_{0}\circ \ce_{1}\cdots \ce_{n-1}\circ \hat\ce_{n}(a_{\L_1}\otimes \cdots\otimes
a_{W_{n}}))$. This means that the limit state $\ffi_{w_0,\bh}$ is a backward QMC. This completes the proof.
We notice that a phase transition phenomena is crucial in higher dimensional quantum models [@BCS],[@Sach; @BR2]. In [@ArE], quantum phase transition for the two-dimensional Ising model using $C^*$-algebra approach. In [@fannes] the VBS-model was considered on the Cayley tree. It was established the existence of the phase transition for the model in term of finitely correlated states which describe ground states of the model. Note that more general structure of finitely correlated states was studied in [@fannes2].
Our goal in this paper is to establish the existence of phase transition for the given family of operators. Heuristically, the phase transition means the existence of two distinct B backward QMC. Let us provide a more exact definition (see [@MBS161]).
We say that there exists a phase transition for a family of operators $\{K_{<x,(x,i)>}\}$, $\{L_{>(x,i),(x,i+1)<}\}$, $\{M_{(x,(x,i),(x,i+1))}\}$ if the following conditions are satisfied:
1. [existence]{}: The equations , have at least two $(u_0,\{h^x\}_{x\in L})$ and $(v_0,\{s^x\}_{x\in
L})$ solutions;
2. [not overlapping supports]{}: there is a projector $P\in B_L$ such that $\ffi_{u_0,\bh}(P)<\ve$ and $\ffi_{v_0,\bs}(P)>1-\ve$, for some $\ve>0$.
3. [not quasi-equivalence]{}: the corresponding quantum Markov chains $\ffi_{u_0,\bh}$ and $\ffi_{v_0,\bs}$ are not quasi equivalent [^2].
Otherwise, we say there is no phase transition.
QMC associated with Ising-XY model with competing interactions {#exam1}
===============================================================
In this section, we define the model and formulate the main results of the paper. In what follows we consider a semi-infinite Cayley tree $\G^2_+=(L,E)$ of order two. Our starting $C^{*}$-algebra is the same $\cb_L$ but with $\cb_{x}=M_{2}(\bc)$ for all $x\in L$. By $\sigma_{x}^{u}$, $\sigma_{y}^{u}$, $\sigma_{z}^{u}$ we denote the Pauli spin operators for a site $u \in L$, i.e.
$$\id^{(u)}=\begin{pmatrix}
0 & 1\\
1& 0
\end{pmatrix},\sigma_{x}^{u}=\begin{pmatrix}
0 & 1\\
1& 0
\end{pmatrix} , \sigma_{y}^{u}=\begin{pmatrix}
0 & -i\\
i& 0
\end{pmatrix}, \sigma_{z}^{u}=\begin{pmatrix}
1 & 0\\
0& -1
\end{pmatrix}.$$
For every vertices $(u,(u,1),(u,2))$ we put $$\begin{aligned}
\label{1Kxy1}
&&K_{<u,(u,i)>}=\exp\{J_0\beta H_{<u,(u,i)>}\}, \ \ i=1,2,\ \ J_0>0, \ \beta>0,\\[2mm] \label{1Lxy1} &&
L_{>(u,1),(u,2)<}=\exp\{J\beta H_{>(u,1),(u,2)<}\}, \ \ J\in\br,\end{aligned}$$ where $$\begin{aligned}
\label{1Hxy1}
&&
H_{<u,(u,1)>}=\frac{1}{2}\big(\id^{(u)}\otimes \id^{(u,1)}+\sigma_{z}^{(u)}\otimes \sigma_{z}^{(u,1)}\big) ,\\
\label{1H>xy<1} &&
H_{>(u,1),(u,2)<}=\frac{1}{2}\big(\sigma_{x}^{(u,1)}\otimes \sigma_{x}^{(u,2)}+\sigma_{y}^{(u,1)}\otimes \sigma_{y}^{(u,2)}\big).\end{aligned}$$
Furthermore, for the sake of simplicity, we assume that $M_{(u,(u,i),(u,i+1))}=\id$ ($i=1,2,\dots,k$) for all $u\in L$.
The defined model is called the [*Ising model with competing $XY$- interactions*]{} per vertices $(u,(u,1),(u,2))$.
For each $m \in \mathbb{N}$, from , it follows that $$\begin{aligned}
\label{1Hxym}
&&H_{<u,v>}^{m}=H_{<u,v>}=\frac{1}{2}\big(\id^{(u)}\otimes\id^{(v)}+\sigma^{(u)}_z\otimes\sigma^{(v)}_z\big),\\[2mm]
\label{1Hxym}
&&H_{>(u,1),(u,2)<}^{2m}=H_{>(u,1),(u,2)<}^2=\frac{1}{2}(\id^{(u,1)}\otimes \id^{(u,2)}-\sigma_{z}^{(u,1)}\otimes \sigma_{z}^{(u,2)})\\[2mm]
\label{1Hxym}
&&H_{>(u,1),(u,2)<}^{2m-1}=H_{>(u,1),(u,2)<}\big.\end{aligned}$$ Therefore, one finds $$K_{<u,(u,i)>}=K_{0}\id^{(u)}\otimes\id^{(u,i)}+K_{3}\sigma_{z}^{(u)}\otimes \sigma_{z}^{(u,i)}$$ $$L_{>(u,1),(u,2)<}= \id^{(u,1)}\otimes \id^{(u,2)} + \sinh(J\beta) H_{>(u,1),(u,2)<}+(\cosh(J\beta)-1)H^{2}_{>(u,1),(u,2)<}$$ where $$\begin{aligned}
&&K_0=\frac{\exp{J_0\beta}+1}{2},\ \ \ K_3=\frac{J_0\exp{\beta}-1}{2},\\[2mm]\end{aligned}$$ Hence, from for each $x\in L$ we obtain $$\begin{aligned}
\label{Ax}
A_{(u,(u,1),(u,2))}&=\gamma_{1}\id^{(u)}\otimes\id^{(u,1)}\otimes\id^{(u,2)}+\gamma_{2}\id^{u}
\otimes\sigma_{x}^{(u,1)}\otimes\sigma_{x}^{(u,2)}\nonumber\\
&+\gamma_{2}\id^{(u)}\otimes\sigma_{y}^{(u,1)}\otimes\sigma_{y}^{(u,2)}
+\gamma_3\id^{(u)}\otimes\sigma_{z}^{(u,1)}\otimes\sigma_{z}^{(u,2)}\nonumber\\
&+\delta_{1}\sigma_{z}^{(u)}\otimes\id^{(u,1)}\otimes\sigma_{z}^{(u,2)}
+\delta_1\sigma_{z}^{(u)}\otimes\sigma_{z}^{(u,1)}\otimes \id^{(u,2)}\end{aligned}$$ where $$\left\{\begin{matrix}
\gamma_1=\frac{1}{4}[\exp(2J_0\beta)+1+2\exp(J_0\beta)\cosh(J\beta)], & \gamma_2=\frac{1}{2}\exp(J_0\beta)\sinh(J\beta). \bigskip \\
\gamma_3=\frac{1}{4}[\exp(2J_0\beta)+1-2\exp(J_0\beta)\cosh(J\beta)], & \delta_1=\frac{1}{4}(\exp(2J_0\beta)-1).
\end{matrix}\right.$$
Recall that a function $\{h^u\}$ is called *translation-invariant* if one has $h^{u}=h^{\tau_g(u)}$, for all $u,g\in \G^2_+$. Clearly, this is equivalent to $h^u=h^v$ for all $u,v\in L$.
In what follows, we restrict ourselves to the description of translation-invariant solutions of ,. Consequently, we assume that: $ h^{u}=h$ for all $u\in L$, here $$h= \left(
\begin{array}{cc}
h_{11} & h_{12} \\
h_{21} & h_{22} \\
\end{array}
\right).$$ Then, equation reduces to $$\begin{aligned}
\label{eqder1}
h&=&\Tr_{u]}A_{(u,(u,1),(u,2))}[\id^{(u)}\otimes h\otimes h]A_{(u,(u,1),(u,2))}^{*}\nonumber\\
&=&[C_1 \tr(h)^2+C_{2}\tr(\sigma_{z}h)^2]\id^{(u)}+ C_3Tr(h)\Tr(\sigma_z h)\sigma_z^{(u)}.\end{aligned}$$ where $$\left\{
\begin{array}{ll}
C_1=\frac{1}{4}(\exp(4 J_0\beta)+1)+\frac{1}{2}\exp(2 J_0\beta)\cosh(2J\beta);\bigskip \\
C_2=\frac{1}{4}(\exp(4 J_0\beta)+1)-\frac{1}{2}\exp(2 J_0\beta)\cosh(2J\beta);\bigskip \\
C_3=\frac{1}{2}(\exp(4 J_0\beta)-1).
\end{array}
\right.$$ Now taking into account $$\Tr(h)=\frac{h_{11}+h_{22}}{2},\ \
\Tr(\sigma_{z} h)=\frac{h_{11}-h_{22}}{2}$$ the equation reduces to the following one $$\label{1EQ1}
\left\{
\begin{array}{lll}
\Tr(h)=C_1\Tr(h)^2+C_2\Tr(\s h)^2,\\
\Tr(\s h)=C_3\Tr(h)\Tr(\s h),\\
h_{21}=0, h_{12}=0.\\
\end{array}
\right.$$ This equation implies that a solution $h$ is diagonal, and through the equation , $\omega_{0}$ could be chosen diagonal as well. In the next sections we are going to examine .
Existence of QMC associated with the model.
===========================================
In this section we are going to solve , which yields the existence of QMC associated with the model. We consider two distinct cases.
Case $h_{11}=h_{22}$ and associate QMC
--------------------------------------
We assume that $h_{11}=h_{22}$ , then is reduced to $$h_{11}=h_{22}=\frac{1}{C_1}.$$ Then putting $\alpha=\frac{1}{C_1}$ we get $$h_{\alpha}=
\left(
\begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}
\right)$$
\[5.1\] The pair $(\omega_{0},\{h^{u}=h_{\alpha}| u\in L\})$ with $\omega_{0}=\frac{1}{\alpha}\id,\ \ h^{u}=h_{\alpha}, \forall
u\in L,$ is a solution of ,. Moreover the associated backward QMC can be written on the local algebra $\mathcal{B}_{L, loc}$ by $$\varphi_{\alpha}(a)=\alpha^{2^{n}-1}\Tr\bigg(\prod_{i=0}^{n-1}K_{[i,i+1]}a\prod_{i=0}^{n-1}K_{[n-i-1,n-i]}^{*}\bigg),
\ \ \forall a\in B_{\Lambda_{n}}.$$
Case $h_{11}\neq h_{22}$ and associate QMC
------------------------------------------
Now we suppose that $h_{11}\neq h_{22}$, and put $\theta=\exp(2\beta)$. Then the equation reduces to $$\begin{aligned}
\label{EQ2}
\left\{
\begin{array}{ll}\
\frac{h_{11}+h_{22}}{2}=\frac{1}{C_3}, \\
\\
(\frac{h_{11}-h_{22}}{2})^{2}=\frac{C_{3}-C_1}{C_2.C_{3}^{2}},
\end{array} \right.\end{aligned}$$
Denote $$\Delta(\theta)=\frac{C_{3}-C_1}{C_2}=\frac{\theta^{2J_0} -\theta^{J_0} (\theta^{J}+\theta^{-J})-3}{\theta^{2J_0}-\theta^{J_0} (\theta^{J}+\theta^{-J})+1}$$
\[hh0\] If $\Delta(\theta) >0$, then the equation has two solutions given by $$\begin{aligned}
\label{hh1}
&& h =\xi_{0}\id+\xi_{3}\sigma,\\
&&\label{hh2}
h'=\xi_{0}\id-\xi_{3}\sigma,\end{aligned}$$ where $$\begin{aligned}
\label{xi01}
\xi_{0}=\frac{1}{C_3}=\frac{2}{\theta^{2J_0}-1}, \ \ \
\xi_{3}=\frac{\sqrt{\Delta(\theta)}}{C_3}=\frac{2\sqrt{\Delta(\theta)}}{\theta^{2J_0}-1}\end{aligned}$$
Assume that $\Delta(\theta)>0$. Then one can conclude that is equivalent to the following system $$\begin{aligned}
\left\{
\begin{array}{ll}
h_{1,1}+h_{2,2}=2\xi_0, \\
h_{1,1}-h_{2,2}=\pm2\xi_3
\end{array}
\right.\end{aligned}$$ It is easy to see that $h_{1,1}=\xi_{0}-\xi_{3}$, $h_{2,2}=\xi_{0}+\xi_3$. Hence, we get ,.
From we find that $\omega_0=\frac{1}{\xi_0}\id\in
\mathcal{B}^{+}$. Therefore, the pairs $\big(\omega_{0},\ \
\{h^{(u)}=h, \ u\in L\}\big)$ and $\big(\omega_0,\{h^{(u)}=h', \
u\in L\}\big)$ define two solutions of ,. Hence, they define two backward QMC $\varphi_1$ and $\varphi_2$, respectively. Namely, for every $ a\in\mathcal{B}_{\Lambda_n}$ one has $$\begin{aligned}
\label{F1}
&&\varphi_1(a)=\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n}^{1/2} a \bh_{n}^{1/2} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\\[2mm] \label{F2}
&&\varphi_2(a)=\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n}^{'1/2} a \bh_{n}^{'1/2} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big).\end{aligned}$$
Hence, we summarize this section in the followigin result.
\[5.3\] The following statements hold:
- if $\Delta(\theta)\leq0$, then there is a unique translation invariant QMC $\varphi_\a$;
- if $\Delta(\theta)>0$, then there are at least three translation invariant QMC $\varphi_\a$, $\varphi_1$ and $\varphi_2$.
From this theorem we conclude that there are three coexisting phases in the region $\Delta(\theta)>0$, and one of it, i.e. $\varphi_{\alpha}$, survives in the region $\Delta(\theta)<0$. This leads us that the state $\varphi_{\alpha}$ describes the disordered phase of the model, which shows a similar behavior with the classical Ising model [@B90; @Roz2]. In comparison with the Ising model, we stress that in the present model, we have a similar kind of phases (translation invariant ones) when $\Delta(\theta)>0$. From Figure 2 (see below), one concludes that the phase transition occurs except for a “triangular region”. This shows how the competing interactions effect to the existence of other phases. Notice that if $J=0$, then we obtain the classical Ising model for which the existence of a disordered phase coexisting with two ordered phases is well-known [@Bax; @Roz2].
Next auxiliary fact gives an equivalent condition for $\Delta(\theta)>0$.
\[DD\] $\Delta(\theta)>0$ iff one of the following statements hold:
1. $J>J_0$ or $J<-J_0$;
2. $-J_0<J<J_0$ and $$J_0>\frac{1}{2\beta}\ln\left(\sinh(2J\beta)+\sqrt{\sinh^2(2J\beta)+3}\right)$$
We know that $\theta=\exp(2\beta)$, $\beta>0$ and $J_0>0$. Then one finds
$$\Delta=\frac{\theta^{2J_0}-\theta^{J_0}\left(\theta^J+\theta^{-J}\right)-3}
{\theta^{2J_0}-\theta^{J_0}\left(\theta^J+\theta^{-J}\right)+1}=\frac{R(J)-4}{R(J)}$$ where $$R(J)=\theta^{2J_0}-\theta^{J_0}\left(\theta^J+\theta^{-J}\right)+1=(\theta^{J_0-J}-1)(\theta^{J_0+J}-1)$$ Thanks to $\theta>1$, we have $$R(J)\left\{
\begin{array}{ll}
>0, & \mbox{if } (J_0-J)(J_0+J)>0\\
<0, & \mbox{if } (J_0-J)(J_0+J)<0
\end{array}\right.$$
[**Case $ (J_0-J)(J_0+J)<0$**]{}. In this setting, one can see that $\Delta(\theta)>0$.\
[**Case $ (J_0-J)(J_0+J)>0$**]{} Note that $\Delta(\theta)>0$ if and only if $R(J)>4$. For convenience, we denote $\theta^{J_0}=a$ and $\theta^J=b$. And consider the following equation $$a^2-\left(b+b^{-1}\right)a-3=0.$$ Then $$a_{1}=\frac{b+b^{-1}+\sqrt{D}}{2}>0$$ $$a_{2}=\frac{b+b^{-1}-\sqrt{D}}{2}<0$$ where $D=\left(b+b^{-1}\right)^2+12$. Due to $a>0$, we may conclude that $$a^2-\left(b+b^{-1}\right)a-3\left\{
\begin{array}{ll}
>0, & \mbox{if } a>a_1\\
<0, & \mbox{if } 0<a<a_1
\end{array}
\right.$$ This means that $\Delta(\theta)>0$ if and only if $$J_0>\frac{1}{2\beta}\ln\left(\sinh(2J\beta)+\sqrt{\sinh^2(2J\beta)+3}\right)$$ which completes the proof.
From this lemma we infer that the phase transition exists in the shaded region shown in the Figure 2 (see $(J,J_0)$ plane).
![Phase diagram[]{data-label="fig1"}](fig1.eps){width="10.07cm"}
Proof of Theorem \[Main\]
=========================
This section is devoted to the proof of Theorem \[Main\]. To realize it, we first show not overlapping supports of the states $\varphi_1$ and $\varphi_2$. Then we show that these states satisfy the clustering property, which yields that they are factor states, and this fact allows us to prove their non- quasi-equivalence.
Not overlapping supports of $\varphi_1$ and $\varphi_2$
-------------------------------------------------------
As usual we put $$e_{11}= \left(
\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}
\right), \ \ \
e_{22}=
\left( \begin{array}{cc}
0 & 0 \\
0 & 1
\end{array}
\right).$$ Now for each $n\in\mathbf{N}$, we denote $$P_n:=\bigg(\bigotimes_{x\in \Lambda_n}e_{11}^{(x)}\bigg)\otimes\id, \ \ Q_n:=\bigg(\bigotimes_{x\in
\Lambda_n}e_{22}^{(x)}\bigg)\otimes\id.$$ Clearly, $P_n$ and $Q_n$ are orthogonal projections in $\mathcal{B}_{\Lambda_n}$.
\[pq\_n\] For every $n\in\bn$, one has
1. $\varphi_{1}(P_n)=\varphi_2(Q_n)=\frac{1}{2\xi_0}\left(\xi_0+\xi_3\right)^{2^{n}}\left(\frac{C_1+C_2+C_3}{4}\right)^{2^{n}-1},$
2. $\varphi_1(Q_n)=\varphi_2(P_n)=\frac{1}{2\xi_0}\left(\xi_0-\xi_3\right)^{2^{n}}\left(\frac{C_1+C_2+C_3}{4}\right)^{2^{n}-1}.$
(i). From we find $$\begin{aligned}
\varphi_1(P_n)&=&\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n}^{1/2} P_n \bh_{n}^{1/2} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\nonumber\\
&=&\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n} P_n K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\end{aligned}$$ Thanks to $he_{11}=(\xi_0+\xi_3)e_{11}$ and one gets $$\begin{aligned}
\Tr_{n-1]}(
K_{[n-1,n]}\bh_nP_nK_{[n-1,n]}^{*})
&=& P_{n-2}\otimes\prod_{u\in \overrightarrow W_{n-1}}\Tr_{u]}(A_{(u,(u,1),(u,2))}e_{11}^{(u)}\otimes h e_{11}^{(u,1)}\otimes h
e_{11}^{(u,2)}A_{(u,(u,1),(u,2))})\nonumber\\
&=& (\xi_0+\xi_3)^{2|W_{n-1]}|}\bigg(\frac{C_1+C_2+C_3}{4}\bigg)^{|W_{n-1}|}P_{n-1}.\nonumber\\\end{aligned}$$ Hence, $$\begin{aligned}
\varphi_1(P_n)&=&(\xi_0+\xi_3)^{|W_{n}|}\left(\frac{C_1+C_2+C_3}{4}\right)^{|W_{n-1}|}\Tr\left[ \omega_0K_{[0,1]}\cdots
K_{[n-2,n-1]}P_{n-1}K_{[n-2,n-1]}^{*}\cdots K_{[0,1]}^{*}\right]\nonumber\\
&&.\nonumber\\
&&.\nonumber\\
&&.\nonumber\\
&=& \left(\xi_0+\xi_3\right)^{|W_{n}|}\left(\frac{C_1+C_2+C_3}{4}\right)^{|W_{n-1}|+ ... +|W_{0}|}\Tr\left[ \omega_0P_{0}\right]\nonumber\\
&=&\left(\xi_0+\xi_3\right)^{|W_{n}|}\left(\frac{C_1+C_2+C_3}{4}\right)^{|\Lambda_{n-1}|}\Tr\left[ \omega_0P_{0}\right]\nonumber\\
&=&\frac{1}{2\xi_0}\left(\xi_0+\xi_3\right)^{2^{n}}\left(\frac{C_1+C_2+C_3}{4}\right)^{2^n-1}.\end{aligned}$$ Analogously, using $h'e_{22}=(\xi_0+\xi_3)e_{22}$ we obtain $$\begin{aligned}
\Tr_{n-1]} (K_{[n-1,n]}\bh'_nQ_nK_{[n-1,n]}^{*})
&=& Q_{n-2}\otimes\prod_{u\in \overrightarrow W_{n-1}}\Tr_{u]}(A_{(u,(u,1),(u,2))}e_{2,2}^{(u)}\otimes h' e_{2,2}^{(u,1)}\otimes h'
e_{22}^{(u,2)}A_{(u,(u,1),(u,2))})\nonumber\\
&=&
(\xi_0+\xi_3)^{2|\mathcal{W}_{n-1]}|}\bigg(\frac{C_1+C_2+C_3}{4}\bigg)^{|\mathcal{W}_{n-1]}|}Q_{n-1}.\end{aligned}$$ which yields $$\varphi_2(Q_n)=\frac{1}{2\xi_0}\left(\xi_0+\xi_3\right)^{2^{n}}\left(\frac{C_1+C_2+C_3}{4}\right)^{2^n-1}.$$
\(ii) Now from $he_{22}=(\xi_0-\xi_3)e_{22}$ and $h'e_{11}=(\xi_0-\xi_3)e_{11}$, we obtain $$\begin{aligned}
&&\Tr_{n-1]} K_{[n-1,n]}\bh_nQ_nK_{[n-1,n]}^{*} =
(\xi_0-\xi_3)^{2|W_{n-1}|}\bigg(\frac{C_1+C_2+C_3}{4}\bigg)^{|W_{n-1}|}Q_{n-1}.\\[2mm]
&&\Tr_{n-1]} K_{[n-1,n]}\bh'_nP_nK_{[n-1,n]}^{*} =
(\xi_0-\xi_3)^{2|W_{n-1}|}\bigg(\frac{C_1+C_2+C_3}{4}\bigg)^{|W_{n-1}|}P_{n-1}.\end{aligned}$$ The same argument as above implies (ii). This completes the proof.
\[6.2\] For fixed $n\in\mathbf{N}$, one has $$\varphi_1(P_n) \rightarrow 1, \ \ \varphi_2(Q_n)\rightarrow 1 \ \ \textrm{as} \ \ \beta\rightarrow +\infty.$$
We know that $\theta=\exp (2\beta)\rightarrow+\infty $ as $\beta\rightarrow+\infty$. Hence, one finds $$\begin{aligned}
&& \frac{1}{2\xi_0}=\frac{\theta^{2J_0}-1}{4}\sim
\frac{\theta^{2J_0}}{4}, \ \ \textrm{as} \ \
\theta\to+\infty\\[2mm]
&&
\left(\xi_0+\xi_3\right)^{2^{n}}=\left(\frac{2}{\theta^{2J_0}-1}(1+\sqrt{\frac{\theta^{2J_0}-\theta^{J_0}(\theta^J+\theta^{-J})+1}{\theta^{2J_0}-\theta^{J_0}(\theta^J+\theta^{-J})-3}})\right)^{2^{n}}
\sim \left(\frac{4}{\theta^{2J_0}}\right)^{2^{n}}, \ \ \textrm{as}
\ \
\theta\to+\infty\\[2mm]
&&\left(\frac{C_1+C_2+C_3}{4}\right)^{2^{n}-1}=\left(\frac{\theta^{2J_0}}{4}\right)^{2^{n}-1}.\end{aligned}$$ Hence, we obtain $$\varphi_{1}(P_n)=\varphi_2(Q_n)
\sim
\frac{\theta^{2J_0}}{4}\left(\frac{4}{\theta^{2J_0}}\right)^{2^{n}}\left(\frac{\theta^{2J_0}}{4}\right)^{2^{n}-1}=1.$$ So, $$\lim_{\theta\to\infty}\varphi_{1}(P_n)=
\lim_{\theta\to\infty} \varphi_2(Q_n)=1.$$ This completes the proof.
We note that from $P_n\leq 1-Q_n$ one gets $$\lim_{\theta\to\infty}\varphi_{1}(Q_n)= \lim_{\theta\to\infty}
\varphi_2(P_n)=0.$$ This implies that the states $\varphi_1$ and $\varphi_2$ have non overlapping supports.
Clustering Property for $\varphi_1$ and $\varphi_2$
----------------------------------------------------
In this subsection, we are going to prove that the states $\ffi_1$, $\ffi_2$ satisfy the clustering property.
Recall that a state $\ffi$ on $\mathcal{B}_L$ satisfies the *clustering property* if for every $a,f\in \mathcal{B}_L$ one has $$\label{clus}
\lim_{|g|\rightarrow \infty}\ffi(a \tau_{g}(f))=\ffi(a)\ffi(f).$$
Thanks to Theorem there are two solutions of , and , these two solutions can be written as follows: $\big(\omega_{0},\ \ \{h^{(u)}=h, \ u\in L\}\big)$ and $\big(\omega_0,\{h^{(u)}=h', \
u\in L\}\big)$, where\
$$\omega_{0}=\frac{1}{\xi_{0}}\id ,~~~h=\xi_{0}\id -\xi_{3}\sigma_{z},$$
$$\omega_{0}=\frac{1}{\xi_{0}}\id ,~~~h'=\xi_{0}\id +\xi_{3}\sigma_{z}.$$
By $\ffi_1$, $\ffi_2$ we denote the corresponding backward quantum Markov chains. To prove the clustering property we need to study the following matrix:
$$A:=\begin{pmatrix}
c_{1}\xi_{0} & -c_{2}\xi_{3}\\
-\frac{c_{3}}{2}\xi_{3} & \frac{c_{3}}{2}\xi_{0}
\end{pmatrix}$$
One can easily prove the following fact.
The above given matrix $A$ is a diagonalizable matrix, and can be written as follows: $$\label{7.2}
A:=P\begin{pmatrix}
1 & 0\\
0 & (c_{1}-\frac{c_{3}}{2})\xi_{0}
\end{pmatrix}P^{-1}$$ where $$P:=\begin{pmatrix}
\frac{c_{2}\xi_{3}}{c_{1}\xi_{0}-1} & 2c_{2}\xi_{3}\\
1 & 1
\end{pmatrix}, \ \ \det(P)=\frac{c_{2}\xi_{3}(3-2c_{1}\xi_{0})}{c_{1}\xi_{0}-1}.$$
\[7.3\] Let $a \in \mathcal{B}_{\L_{N_{0}}}$, for some $N_{0} \in \bn$, and $f_n=\bigotimes\limits_{x\in W_{n}}f^{(x)}=f^{x_{W_{n}}^{(1)}} \otimes\id_{W_{n}\setminus\{x_{W_{n}}^{(1)}\}}\in\mathcal{B}_{W_{n}}$, where $f^{x_{W_{n}}^{(1)}} =f$, then for each backward quantum Markov chains $\ffi_1$, $\ffi_2$ we have $$\lim_{n\rightarrow \infty}\ffi_k(a \otimes \id \cdots\otimes \id \otimes f_n)=\ffi_k(a)\ffi_k(f ), \ \ k=1,2.$$
By symmetry of calculations, it is enough to prove the result for $\big(\omega_{0},\ \ \{h^{(u)}=h, \ u\in L\}\big)$. From and we have:
$$\ffi^{(n)}_{w_{0},\bh}(a_{N_{0}} \otimes \id \cdots\otimes \id \otimes f)
=\tr(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}( a\otimes \ce_{N_{0}+1}( \id \otimes \cdots\otimes \ce_{n-1}( \id \otimes\hat\ce_{n}(\id \otimes f))\cdots))$$ First, let us calculate $\hat\ce_{n}(\id \otimes f)$. From it follows that
$$\begin{aligned}
\hat\ce_{n}(f\otimes \id)&=& \bigotimes\limits_{x\in W_{n}}\tr_{x]}(A_{x \vee S(x)} f^{(x)}\otimes h_{S(x)} A^{*}_{x \vee S(x)})\\
&=& \tr_{x_{W_{n}}^{(1)}]}(A_{x_{W_{n}}^{(1)} \vee S(x_{W_{n}}^{(1)})} f^{(x_{W_{n}}^{(1)})}\otimes h_{S(x_{W_{n}}^{(1)})} A^{*}_{x_{W_{n}}^{(1)} \vee S(x_{W_{n}}^{(1)})}) \otimes\\[2mm]
&&\bigotimes\limits_{x\in W_{n}\setminus\{x_{W_{n}}^{(1)}\}} \tr_{x]}( A_{x \vee S(x)}h_{S(x)}
A^{*}_{x \vee S(x)})\\
&=& \bigg( \alpha_{1} f^{(x_{W_{n}}^{(1)})}+ \alpha_{2} (f^{(x_{W_{n}}^{(1)})}\sigma_{z}^{(x_{W_{n}}^{(1)})}
+ \sigma_{z}^{(x_{W_{n}}^{(1)})}f^{(x_{W_{n}}^{(1)})})\\[2mm]
&&+ \alpha_{3}\sigma_{z}^{(x_{W_{n}}^{(1)})}f^{(x_{W_{n}}^{(1)})}\sigma_{z}^{(x_{W_{n}}^{(1)})}\bigg)\otimes \bigotimes\limits_{x\in W_{n}\setminus\{x_{W_{n}}^{(1)}\}}h_{x}\\
&=& g^{(x_{W_{n}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n}\setminus\{x_{W_{n}}^{(1)}\}}h_{x},
\end{aligned}$$
where $$g^{(x_{W_{n}}^{(1)})}=\alpha_{1} f^{(x_{W_{n}}^{(1)})}+ \alpha_{2} (f_{1}^{(x_{W_{n}}^{(1)})}\sigma_{z}^{(x_{W_{n}}^{(1)})}+ \sigma_{z}^{(x_{W_{n}}^{(1)})}f^{(x_{W_{n}}^{(1)})})+ \alpha_{3}\sigma_{z}^{(x_{W_{n}}^{(1)})}f^{(x_{W_{n}}^{(1)})}\sigma_{z}^{(x_{W_{n}}^{(1)})},$$ and $$\left\{\begin{array}{ll}
\alpha_{1}=(C_{1}-2\delta_{1}^{2})\xi_{0}^{2}+ (C_{2}-2\delta_{1}^{2})\xi_{3}^{2}\\
\alpha_{2}=-\frac{C_{3}}{2}\xi_{0}\xi_{3}\\
\alpha_{3}=2\delta_{1}^{2}(\xi_{0}^{2}+\xi_{3}^{2})
\end{array}\right..$$
Hence, one has $$\begin{aligned}
\ce_{n-1}(\id \otimes \hat\ce_{n}(f\otimes \id))&=& \tr_{x_{W_{n-1}}^{(1)}]}(A_{x_{W_{n-1}}^{(1)} \vee S(x_{W_{n-1}}^{(1)})} \id \otimes g^{(x_{W_{n}}^{(1)})}\otimes h A^{*}_{x_{W_{n-1}}^{(1)} \vee S(x_{W_{n-1}}^{(1)})})
\otimes \\[2mm]
&&\bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}\\
&=&(\alpha_{1,g}\id^{(x_{W_{n-1}}^{(1)})}+\alpha_{2,g}\sigma_{z}^{(x_{W_{n-1}}^{(1)})})\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}\\
&=&\alpha_{1,g} \id^{(x_{W_{n-1}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}\\[2mm]
&&+ \alpha_{2,g} \sigma_{z}^{(x_{W_{n-1}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}
\end{aligned}$$
So, one finds $$\begin{aligned}
\ce_{n-1}(\id \otimes \hat\ce_{n}(f\otimes \id))&=& v_{1}\id^{(x_{W_{n-1}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}+ v^{'}_{1} \sigma_{z}^{(x_{W_{n-1}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}\end{aligned}$$ where $$\left\{\begin{array}{ll}
\alpha_{1,X}=C_{1}\tr(X)\xi_{0}- C_{2}\tr(\sigma_{z}X)\xi_{3}\\
\alpha_{2,X}=\frac{C_{3}}{2}\left(\tr(\sigma_{z}X)\xi_{0}-\tr(X)\xi_{3}\right)\\
v_{1}=\alpha_{1,g}\\
v_{1}^{'}=\alpha_{2,g}\\
\end{array}\right.$$
Then by iteration we obtain $$\begin{aligned}
\ce_{n-k}(\id \otimes... \ce_{n-1}(\id \otimes \hat\ce_{n}(f\otimes \id)))&=& v_{k}\id^{(x_{W_{n-1}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}\\[2mm]
&&+ v^{'}_{k} \sigma_{z}^{(x_{W_{n-1}}^{(1)})}\otimes \bigotimes\limits_{x\in W_{n-1}\setminus\{x_{W_{n-1}}^{(1)}\}} h_{(x)}\end{aligned}$$ where $$\left\{\begin{array}{ll}
v_{k}=v_{k-1}C_{1}\xi_{0}-C_{2}\xi_{3}v_{k-1}^{'}\\
v_{k}^{'}=-\frac{C_{3}}{2}\xi_{3}v_{k-1}+ \frac{C_{3}}{2}\xi_{0}v_{k-1}^{'}\\
\end{array}\right.$$
Now let calculate the explicit form of the sequence $v_{k}$, we can see : $$\begin{aligned}
\begin{pmatrix}
v_{k}\\ v_{k}^{'}
\end{pmatrix}&=& A \begin{pmatrix}
v_{k-1}\\
v_{k-1}^{'}
\end{pmatrix}\\
&\vdots&\\
&=& A^{k-1}\begin{pmatrix}
v_{1}\\
v_{1}^{'}
\end{pmatrix}\end{aligned}$$
Then by we get, $$\begin{aligned}
\begin{pmatrix}
v_{k}\\ v_{k}^{'}
\end{pmatrix}&=&P\begin{pmatrix}
1 & 0\\
0& (c_{1}-\frac{c_{3}}{2})^{k-1}\xi_{0}^{k-1}
\end{pmatrix}P^{-1}\begin{pmatrix}
v_{1}\\
v_{1}^{'}
\end{pmatrix}\\
&=&\begin{pmatrix}
\frac{1}{3-2c_{1}\xi_{0}}+\frac{2\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}(1-c_{1}\xi_{0})}{3-2c_{1}\xi_{0}} & -\frac{2c_{2}\xi_{3}}{3-2c_{1}\xi_{0}}+\frac{2c_{2}\xi_{3}\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}}{3-2c_{1}\xi_{0}}\\[2mm]
\frac{c_{1}\xi_{0}-1}{c_{2}\xi_{3}(3-2c_{1}\xi_{0})}+\frac{\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}(1-c_{1}\xi_{0})}{c_{2}\xi_{3}(3-2c_{1}\xi_{0})}&
\frac{-2(c_{1}\xi_{0}-1)}{3-2c_{1}\xi_{0}}+\frac{\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}}{3-2c_{1}\xi_{0}}
\end{pmatrix}\begin{pmatrix}
v_{1}\\[2mm]
v_{1}^{'}
\end{pmatrix}\\[2mm]
&=&\begin{pmatrix}
\eta_{1}+\widehat{\eta_{1}} \xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1} & \eta_{2}- \eta_{2}\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}\\[2mm]
\widehat{\eta_{2}}-\widehat{\eta_{2}} \xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}&
\widehat{\eta_{1}} + \eta_{1}\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1}
\end{pmatrix}\begin{pmatrix}
v_{1}\\[2mm]
v_{1}^{'}\\
\end{pmatrix}\end{aligned}$$ where $$\eta_{1}=\frac{1}{3-2c_{1}\xi_{0}},\ \
\widehat{\eta_{1}}=\frac{2(1-c_{1}\xi_{0})}{3-2c_{1}\xi_{0}}$$$$\eta_{2}=-\frac{2c_{2}\xi_{3}}{3-2c_{1}\xi_{0}}, \ \
\widehat{\eta_{2}}=\frac{c_{1}\xi_{0}-1}{c_{2}\xi_{3}(3-2c_{1}\xi_{0})}$$
Hence,
$$\left\{\begin{array}{ll}
v_{k}=\left(\eta_{1}+\widehat{\eta_{1}} \xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1} \right)v_{1}+ \left( \eta_{2}- \eta_{2}\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1} \right)v_{1}^{'}\\[2mm]
v_{k}^{'}=\left(\widehat{\eta_{2}}-\widehat{\eta_{2}} \xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1} \right)v_{1}+ \left( \widehat{\eta_{1}} + \eta_{1}\xi_{0}^{k-1}(c_{1}-\frac{c_{3}}{2})^{k-1} \right)v_{1}^{'}\\
\end{array}\right.$$
So, one finds
$$\ffi^{(n)}_{w_{0},\bh}(a\otimes \id \cdots\otimes \id \otimes f)
=v_{n-N_{0}+1}\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}\bigg( a\otimes \id^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} \bigg)\right)$$ $$+ v_{n-N_{0}+1}^{'}\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}\bigg( a\otimes \sigma_{z}^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} \bigg)\right)$$ $$=\left(\eta_{1}+\widehat{\eta_{1}} \xi_{0}^{n-N_{0}}(c_{1}-\frac{c_{3}}{2})^{n-N_{0}} \right)v_{1}\tr\left(\omega_{0}\ce_{0}\circ\cdots\circ \ce_{N_{0}}\bigg( a\otimes \id^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} \bigg)\right)$$ $$+ \left( \eta_{2}- \eta_{2}\xi_{0}^{n-N_{0}}(c_{1}-\frac{c_{3}}{2})^{n-N_{0}} \right)v_{1}^{'}\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}\bigg( a\otimes \id^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} \bigg)\right)$$ $$+ \left(\widehat{\eta_{2}}-\widehat{\eta_{2}} \xi_{0}^{n-N_{0}}(c_{1}-\frac{c_{3}}{2})^{k-1} \right)v_{1}\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}\bigg( a\otimes \sigma_{z}^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x}\bigg )\right)$$ $$+ \left( \widehat{\eta_{1}} + \eta_{1}\xi_{0}^{n-N_{0}}(c_{1}-\frac{c_{3}}{2})^{n-N_{0}} \right)v_{1}^{'}\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}\bigg( a\otimes \sigma_{z}^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} \bigg)\right)$$ One can see that $\xi_{0}^{n-N_{0}}\rightarrow 0$, as $n\rightarrow \infty$, which implies
$$\lim_{n\rightarrow \infty}\ffi_{w_{0},\bh}(a \otimes \id \otimes \cdots\otimes \id \otimes f_n)
=(\eta_{1}v_{1}+ \eta_{2}v_{1}^{'})\tr\bigg(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}( a\otimes \id^{(x_{W_{N_{0}+1}})} \otimes$$ $$\bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} )\bigg)$$ $$+(\widehat{\eta_{2}}v_{1}+\widehat{\eta_{1}}v_{1}^{'})\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}( a\otimes \sigma_{z}^{(x_{W_{N_{0}+1}})} \otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x} )\right)$$ where $$\left\{\begin{array}{ll}
\eta_{1}v_{1}+ \eta_{2}v_{1}^{'}=\frac{\xi_{0}^{2}}{6-4C_{1}\xi_{0}}[(4C_{3}-2C_{1})\tr(f)-\sqrt{\Delta(\theta)}(4C_{2}+C_{3})\tr(\sigma_{z}f)]\\[2mm]
\widehat{\eta_{2}}v_{1}+\widehat{\eta_{1}}v_{1}^{'}=-c_{3}\xi_{3}(\eta_{1}v_{1}+ \eta_{2}v_{1}^{'})\\
\end{array}\right.$$
On other hand we have $$\ffi_{w_{0},\bh}(f)=C_{3}(\eta_{1}v_{1}+ \eta_{2}v_{1}^{'}),$$
Hence, one gets
$$\begin{aligned}
\lim_{n\rightarrow \infty}\ffi_{w_{0},\bh}(a\otimes\cdots\otimes f_n)&=&\ffi_{w_{0},\bh}(f)\tr\bigg(\omega_{0}\ce_{0}\circ \cdots\circ \ce_{N_{0}}( a\otimes h^{(x_{W_{N_{0}+1}})}\otimes \bigotimes\limits_{x\in W_{N_{0}+1}\setminus\{x_{W_{N_{0}+1}}^{(1)}\}} h_{x}\bigg)\\
&=&\ffi_{w_{0},\bh}(f)\tr\left(\omega_{0}\ce_{0}\circ \cdots\circ \hat\ce_{N_{0}}( a)\right)\\
&=&\ffi_{w_{0},\bh}(f)\ffi_{w_{0},\bh}(a).\end{aligned}$$
This completes the prove.
Now we are ready to prove the clustering property.
\[CPr\] The states $\ffi_1$ and $\ffi_2$ satisfy the clustering property.
Thanks to the density argument, without lost of generality, we may assume $a,f \in \mathcal{B}_{loc}$. This means that there are $N_0,m_{0} \in \bn$ such that $a\in \mathcal{B}_{\L_{N_{0}}}$, $f \in \mathcal{B}_{\L_{m_{0}}}$. Moreover, $f$ can be write in the following form $$f=\bigotimes\limits_{x\in\L_{m_{0}}}f^{(x)}.$$ By symmetry of calculations, it is enough to prove the result for $\big(\omega_{0},\ \ \{h^{(u)}=h, \ u\in L\}\big)$.\
In what follows, we assume that $g\in W_n$. Therefore, we put $g_n:=g$. Then one has $$\begin{aligned}
\tau_{g_{n}}(f)&=& \bigotimes\limits_{x\in\L_{m_{0}}}f^{( g_{n} \circ x)}\\
&=& f^{(g_{n})}\otimes f^{(g_{n},W_{1})}\otimes f^{(g_{n},W_{2})}\otimes...\otimes f^{(g_{n},W_{m_{0}})},\end{aligned}$$ where $$f^{(g_{n},W_{k})}= \bigotimes\limits_{x\in W_{k}}f^{(g_{n}\circ x)} \ \ \text{and} \ \ \{g_{n},W_{k}\}=\{(g_{n} \circ x),\ \ x \in W_{k}\}.$$\
We can see $\tau_{g_{n}}(f) $ as an element of $B_{\L_{n+m_{0}}}$, i.e.
$$\begin{aligned}
\tau_{g_{n}}(f)&=& \id_{\L_{n-1}}\otimes( f^{(g_{n})}\otimes \id_{W_{n}\setminus\{g_{n}\}})\otimes(f^{(g_{n},W_{1})}\otimes \id_{W_{n+1}\setminus\{g_{n},W_{1}\}})\otimes\\[2mm]
&&...\otimes(f^{(g_{n},W_{m_{0}})}\otimes \id_{W_{n+m_{0}}\setminus\{g_{n},W_{m_{0}}\}}).\end{aligned}$$
For the sake of simplicity, let us denote $$\begin{aligned}
\tau_{g_{n}}(f)&=& \bigotimes\limits_{x\in\L_{n+m_{0}}}f_{1}^{(x)}.\end{aligned}$$
From and it follows that $$\ffi_{w_{0},\bh}(a\otimes \id \otimes \cdots\otimes \id \otimes \tau_{g_{n}}(f))=\ffi^{(n+m_{0})}_{w_{0},\bh}(a \otimes \id \cdots\otimes \id \otimes \tau_{g_{n}}(f))$$ $$=\tr(\omega_{0}\ce_{0}\circ\cdots\circ \ce_{N_{0}}( a\otimes \ce_{N_{0}+1}( \id \otimes \cdots\otimes \ce_{n+m_{0}-1}( \id \otimes\hat\ce_{n+m_{0}}(\tau_{g_{n}}(f)))...)).$$\
Let us calculate $\hat\ce_{n+m_{0}}(\tau_{g_{n}}(f))$. Indeed, from one gets $$\begin{aligned}
\hat\ce_{n+m_{0}}(\tau_{g_{n}}(f)\otimes h_{W_{n+m_{0}+1}})&=& \tr_{n+m_{0}]}(K_{[n+m_{0},n+m_{0}+1]}\tau_{g_{n}}(f)\otimes h_{W_{n+m_{0}+1}}K_{[n+m_{0},n+m_{0}+1]}^*)\\
&=& \bigotimes\limits_{x\in\L_{n+m_{0}-1}}f_{1}^{(x)}\bigotimes\limits_{x\in W_{n+m_{0}}} \tr_{x]}( A_{x \vee S(x)}
f_{1}^{(x)}\otimes h^{(S(x))} A^{*}_{x \vee S(x)}) \\
&=& \bigotimes\limits_{x\in\L_{n+m_{0}-1}}f_{1}^{(x)}\bigotimes\limits_{x\in W_{m_{0}}} \tr_{g_{n} \circ x]}( A_{g_{n} \circ x \vee S(g_{n} \circ x)}
f^{(g_{n} \circ x)}\otimes h^{(S(g_{n} \circ x))} A^{*}_{g_{n} \circ x \vee S(g_{n} \circ x)}) \\
&&\otimes \bigotimes\limits_{x\in W_{n+m_{0}}\setminus\{g_{n},W_{m_{0}}\}} \tr_{x]}( A_{x \vee S(x)} h^{(x)}
A^{*}_{x \vee S(x)}) \\
&=& \bigotimes\limits_{x\in\L_{n+m_{0}-1}}f_{1}^{(x)}\bigotimes\limits_{x\in W_{m_{0}}} T_{m_{0}}^{(g_{n}\circ x)}\otimes\bigotimes\limits_{x\in W_{n+m_{0}}\setminus\{g_{n},W_{m_{0}}\}} h^{(x)}
\end{aligned}$$ where $$T_{m_{0}}^{(g_{n}\circ x)}=\tr_{g_{n} \circ x]}( A_{(g_{n} \circ x) \vee S(g_{n} \circ x)}
f^{(g_{n} \circ x)}\otimes h^{(S(g_{n} \circ x))} A^{*}_{(g_{n} \circ x )\vee S(g_{n} \circ x)}).$$
Hence,
$$\begin{aligned}
\ce_{n+m_{0}-1}(\id\otimes \hat\ce_{n+m_{0}}(\tau_{g_{n}}(f)))&=& \bigotimes\limits_{x\in\L_{n+m_{0}-2}}f_{1}^{(x)} \tr_{n+m_{0}-1]}(K_{[n+m_{0}-1,n+m_{0}]} \bigotimes\limits_{x\in W_{n+m_{0}-1}}f_{1}^{(x)}\bigotimes\limits_{x\in W_{m_{0}}} T_{m_{0}}^{(g_{n}\circ x)}\\[2mm]&&\otimes\bigotimes\limits_{x\in W_{n+m_{0}}\setminus\{g_{n},W_{m_{0}}\}} h^{(x)}K_{[n+m_{0}-1,n+m_{0}]}^*)\\[2mm]
&=& \bigotimes\limits_{x\in\L_{n+m_{0}-2}}f_{1}^{(x)} \bigotimes\limits_{x\in W_{m_{0}-1}}\tr_{g_{n} \circ x]}\bigg( A_{(g_{n} \circ x )\vee S(g_{n} \circ x)} f^{(g_{n} \circ x)}\\[2mm]
&&\otimes T_{m_{0}}^{(S(g_{n} \circ x))}\otimes A^{*}_{(g_{n} \circ x) \vee S(g_{n} \circ x)}\bigg)\\[2mm]
&&\otimes\bigotimes\limits_{x\in W_{n+m_{0}-1}\setminus\{g_{n},W_{m_{0}-1}\}} \tr_{x]}( A_{x \vee S(x)}h^{(x)} A^{*}_{x \vee S(x)}) \\[2mm]
&=&\bigotimes\limits_{x\in\L_{n+m_{0}-2}}f_{1}^{(x)}\bigotimes\limits_{x\in W_{m_{0}-1}} T_{m_{0}-1}^{(g_{n}\circ x)}\otimes\bigotimes\limits_{x\in W_{n+m_{0}-1}\setminus\{g_{n},W_{m_{0}}\}} h^{(x)}\end{aligned}$$
where $$T_{m_{0}-1}^{(g_{n}\circ x)}=\tr_{g_{n} \circ x]}\left( A_{g_{n} \circ x \vee S(g_{n} \circ x)} f^{(g_{n} \circ x)}\otimes T_{m_{0}}^{(S(g_{n} \circ x))}\otimes A^{*}_{g_{n} \circ x \vee S(g_{n} \circ x)}\right).$$
By iteration, we obtain $$\begin{aligned}
\ce_{n}(\id \otimes\ce_{n+1} (\id \otimes...\otimes \ce_{n+m_{0}-1}( \id \otimes\hat\ce_{n+m_{0}}(\tau_{g_{n}}(f)))...)) &=&\bigotimes\limits_{x\in W_{0}} T_{0}^{(g_{n}\circ x)}\otimes\bigotimes\limits_{x\in W_{n}\setminus\{g_{n},W_{0}\}} h^{(x)}\\
&=&T_{0}^{(g_{n}\circ x_{0})}\otimes\bigotimes\limits_{x\in W_{n}\setminus\{g_{n}\circ x_{0}\}} h^{(x)},
\end{aligned}$$
which yields $$\begin{aligned}
\ffi_{w_{0},\bh}(a_{N_{0}} \otimes \id \otimes \id ...\otimes \id \otimes \tau_{g_{n}}(f))&=&\tr(\omega_{0}\ce_{0}\circ ...\circ \ce_{N_{0}}( a_{N_{0}}\otimes \ce_{N_{0}+1}( \id \\[2mm]
&&\otimes \ce_{N_{0}+2}( \id \otimes...\otimes \ce_{n-1}(T_{0}^{(g_{n}\circ x_{0})}\otimes\bigotimes\limits_{x\in W_{n}\setminus\{g_{n}\circ x_{0}\}} h^{(x)}))).
\end{aligned}$$
Then Lemma \[7.3\] implies
$$\begin{aligned}
\lim_{n \rightarrow\infty}\ffi_{w_{0},\bh}(a_{N_{0}} \otimes \id \otimes \id ...\otimes \id \otimes \tau_{g_{n}}(f)) &=&\ffi_{w_{0},\bh}(a_{N_{0}})\tr(\omega_{0}\ce_{0}\circ ... \circ \ce_{n-1}(T_{0}^{(g_{n}\circ x_{0})}\otimes\bigotimes\limits_{x\in W_{n}\setminus\{g_{n}\circ x_{0}\}} h^{(x)})))\\
&=&\ffi_{w_{0},\bh}(a_{N_{0}})\ffi_{w_{0},\bh}(\tau_{g_{n}}(f)).\end{aligned}$$
This completes the proof.
Non quasi equivalence of $\varphi_1$ and $\varphi_2$
----------------------------------------------------
In this subsection we are going to prove that the states $\varphi_1$ and $\varphi_2$ are not quasi equivalent. To establish the non-quasi equivalence, we are going to use the following result (see [@BR Corollary 2.6.11]).
\[br-q\] Let $\varphi_1,$ $\varphi_2$ be two factor states on a quasi-local algebra $\ga=\cup_{\Lambda}\ga_\Lambda$. The states $\varphi_1,$ $\varphi_2$ are quasi-equivalent if and only if for any given $\varepsilon>0$ there exists a finite volume $\Lambda\subset L$ such that $\|\varphi_1(a)-\varphi_2(a)\|<\varepsilon \|a\|$ for all $a\in B_{\Lambda^{'}}$ with $\Lambda^{'}\cap\Lambda=\emptyset.$
Now due to Theorem \[CPr\] the states $\varphi_1$ and $\varphi_2$ have clustering property, and hence they are factor states. Let us define an element of $\mathcal{B}_{\Lambda_n}$ as follows: $$E_{\Lambda_n}:=e_{11}^{x_{W_n}^{(1)}}\otimes\bigg(\bigotimes_{y\in
\Lambda_n\setminus \{x_{W_n}^{(1)}\}}\id^{y}\bigg),$$ where $x_{W_n}^{(1)}$ is defined in . Now we are going to calculate $\varphi_1(E_{\Lambda_n})$ and $\varphi_2(E_{\Lambda_n})$, respectively. First consider the state $\varphi_1$, then we know that this state is defined by $\omega_0=\frac{1}{\xi_0}\id$ and $h^{x}=h=\xi_0\id+\xi_3\sigma_{z}$. Define two elements of $\mathcal{B}_{W_n}$ by $$\hat{\bh}_n:=\id^{x_{W_n}^{(1)}}\otimes\bigotimes_{x\in W_n\setminus \{x_{W_n}^{(1)}\}}h^{(x)}$$ $$\check{\bh}_{n}:=\sigma_{z}^{x_{W_n}^{(1)}}\otimes\bigotimes_{x\in W_n\setminus \{x_{W_n}^{(1)}\}}h^{(x)}$$
\[f-p-11\] Let $$\hat{\psi}_n:=\Tr_{n-1]}\big[\omega_{0}K_{[0,1]}\cdots K_{[n-1,n]}\hat{\bh}_{n}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big]$$ $$\check{\psi}_n:=\Tr_{n-1]}\big[\omega_{0}K_{[0,1]}\cdots K_{[n-1,n]}\check{\bh}_{n}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big]$$ Then there are two pairs of reals $(\hat{\rho}_{1},\hat{\rho}_{2})$ and $(\check{\rho}_{1},\check{\rho}_{2})$ depending on $\theta$ such that $$\left\{
\begin{array}{ll}
\hat{\psi}_n=\hat{\rho}_{1}+\hat{\rho}_{2}(\frac{C_1}{C_3}-1)^{n}, \\
\\
\check{\psi}_n=\check{\rho}_{1}+\check{\rho}_{2}(\frac{C_1}{C_3}-1)^{n} \\
\end{array}
\right.$$
One can see that $$\begin{aligned}
\left(
\begin{array}{ll}
\hat{\psi}_n \\
\\
\check{\psi}_n
\end{array}
\right) &=& \left(
\begin{array}{c}
\Tr_{n-1]}\big[\omega_{0}K_{[0,1]}\cdots K_{[n-2,n-1]}\Tr_{n-1]}[K_{[n-1,n]}\hat{\bh}_{n}K_{[n-1,n]}^{*}]K_{[n-2,n-1]}^{*}\cdots
K_{[0,1]}^{*}\big], \\
\\
\Tr_{n-1]}\big[\omega_{0}K_{[0,1]}\cdots K_{[n-2,n-1]}Tr_{n-1]}[K_{[n-1,n]}\check{\bh}_{n}K_{[n-1,n]}^{*}]K_{[n-2,n-1]}^{*}\cdots
K_{[0,1]}^{*}\big]
\end{array}
\right).\end{aligned}$$ After small calculations, we find $$\left\{
\begin{array}{ll}
\Tr_{x]}\left[A_{(x,(x,1),(x,2))}\big(\id^{(x)}\otimes\id^{(x,1)}\otimes
h^{(x,2)}\big)A_{(x,(x,1),(x,2))}\right]=C_{1}\xi_0\id^{(x)}+\frac{1}{2}C_{3}\xi_3\sigma_{z}^{(x)} \\
\\
\Tr_{x]}\big[A_{(x,(x,1),(x,2))}\big(\id^{(x)}\otimes\sigma^{(x,1)}\otimes
h_{(\xi_{0},\xi_{3})}^{(x,2)}\big)A_{(x,(x,1),(x,2))}\big]=C_2\xi_3\id^{(x)}+\frac{1}{2}\sigma_{z}^{(x)}
\end{array}
\right.$$ Hence, one gets $$\left\{
\begin{array}{ll}
\Tr_{n-1]}K_{[n-1,n]}\hat{\bh}_{n}K_{[n-1,n]}^{*}=C_1\xi_0\hat{h}_{n-1}+\frac{1}{2}C_3\xi_3\check{h}_{n-1},\\
\\
\Tr_{n-1]}K_{[n-1,n]}\check{\bh}_{n}K_{[n-1,n]}^{*}=C_2\xi_3\hat{h}_{n-1}+\frac{1}{2}\check{h}_{n-1}.
\end{array}
\right.$$ Therefore, $$\begin{aligned}
\left(
\begin{array}{ll}
\hat{\psi}_n \\[2mm]
\check{\psi}_n
\end{array}
\right)
&=&
\left(
\begin{array}{c}
C_1\xi_0\hat{\psi}_{n-1}+\frac{1}{2}C_3\xi_3\check{\psi}_{n-1}
\\[2mm]
C_2\xi_3\hat{\psi}_{n-1}+\frac{1}{2}\check{\psi}_{n-1}
\end{array}
\right)\nonumber\\[2mm]
&=&
\left(
\begin{array}{cc}
C_1\xi_0 & \ \ \frac{1}{2}C_3\xi_3 \\[2mm]
C_2\xi_3 &\ \ \frac{1}{2} \\
\end{array}
\right)
\left(
\begin{array}{c}
\hat{\psi}_{n-1} \\[2mm]
\check{\psi}_{n-1} \\
\end{array}
\right)\nonumber\\
\vdots\nonumber\\
&=&\left(
\begin{array}{cc}
C_1\xi_0 & \ \ \frac{1}{2}C_3\xi_3 \\[2mm]
C_2\xi_3 &\ \ \frac{1}{2} \\
\end{array}
\right)^{n}
\left(
\begin{array}{c}
\hat{\psi}_{0} \\[2mm]
\check{\psi}_{0} \\
\end{array}
\right),\end{aligned}$$ where $$\left\{
\begin{array}{c}
\hat{\psi}_{0}=\Tr (\omega_{0})=\frac{1}{\xi_0}\\[2mm]
\check{\psi}_{0}=\Tr (\omega_{0}\sigma_z) = 0 \\
\end{array}
\right.$$ The matrix $$N := \left(
\begin{array}{cc}
C_1\xi_0 & \ \ \frac{1}{2}C_3\xi_3 \\[2mm]
C_2\xi_3 &\ \ \frac{1}{2} \\
\end{array}
\right)$$ can be written in diagonal form by: $$N=P
\left(
\begin{array}{cc}
1 & \ \ 0 \\
0 &\ \ \frac{C_{1}}{C_3}-\frac{1}{2} \\
\end{array}
\right)
P^{-1}$$ where $$P=\left(
\begin{array}{cc}
\frac{C_3}{2 C_2} & \ \ -\frac{\xi_3}{\xi_0} \\
\frac{\xi_3}{\xi_0} &\ \ 1 \\
\end{array}
\right), \ \ \det(P)=\frac{3 C_{3}-2 C_{1}}{2 C_2}$$ So, $$\begin{aligned}
\left(
\begin{array}{c}
\hat{\psi}_n \\
\check{\psi}_n \\
\end{array}
\right)
&=&P \left(
\begin{array}{cc}
1 & \ \ 0 \\
0 &\ \ (\frac{C_{1}}{C_3}-\frac{1}{2} )^{n}\\
\end{array}
\right)P^{-1}
\left(
\begin{array}{c}
\frac{1}{\xi_0} \\
\\
0
\end{array}
\right)\nonumber\\
&=&
\left(
\begin{array}{ll}
\hat{\rho}_{1}+\hat{\rho}_{2}(\frac{C_1}{C_3}-\frac{1}{2})^{n}\\[2mm]
\check{\rho}_{1}+\check{\rho}_{2}(\frac{C_1}{C_3}-\frac{1}{2})^{n} \\
\end{array}
\right).\end{aligned}$$ where $$\begin{aligned}
\label{r1}
&& \hat{\rho}_{1}=\frac{C_{3}^{2}}{3C_{3}-2C_{1}}, \ \
\hat{\rho}_{2}=\frac{2 C_{3}(C_3-C_1)}{3C_{3}-2C_{1}},\\[2mm]
&& \check{\rho}_{1}=\frac{2C_2 C_{3}^{2}\xi_{3}}{
3C_3-2C_1},\ \ \check{\rho}_{2}=-\frac{2C_2
C_{3}^{2}\xi_{3}}{ 3C_3-2C_1}.\label{r2}\end{aligned}$$ This completes the proof.
\[6.6\] For each $n\in\bn$ one has $$\begin{aligned}
\varphi_1(E_{\Lambda_n})&=&\frac{1}{2}\left[(\xi_0+\xi_3)(C_1\xi_0 +C_2\xi_3
)\hat{\rho}_{1}+\frac{C_3}{2}(\xi_0+\xi_3)^{2}\check{\rho}_{1}\right]\nonumber\\
&&+\frac{1}{2}\left[(\xi_0+\xi_3)(C_1\xi_0 +C_2\xi_3
)\hat{\rho}_{2}+\frac{1}{2}C_3(\xi_0+\xi_3)^{2}\check{\rho}_{2}\right]\bigg(\frac{C_1}{C_3}-\frac{1}{2}\bigg)^{n-1}\end{aligned}$$
From we have $$\begin{aligned}
\varphi_1(E_{\Lambda_n})&=&\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n}^{1/2} E_{\Lambda_n} \bh_{n}^{1/2} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\\
&=&\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n} E_{\Lambda_n} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\end{aligned}$$ One can calculate that $$\begin{aligned}
\Tr_{n-1]} (K_{[n-1,n]}\bh_n
K_{[n-1,n]}^{*}E_{\Lambda_n})&=&\Tr_{x_{W_{n-1}}^{(1)}]}\bigg(A_{(x_{W_{n-1}}^{(1)},
x_{W_n}^{(1)}, x_{W_n}^{(2)})}
\big(\id^{x_{W_{n-1}}^{(1)}}\otimes e_{1,1}h^{x_{W_n}^{(1)}}\otimes h^{x_{W_n}^{(2)}}\big)\nonumber\\
&&A_{(x_{W_{n-1}}^{(1)}, x_{W_n}^{(1)}, x_{W_n}^{(2)})}^{*}\bigg)\otimes\bigotimes_{x\in W_{n-1}\setminus \{x_{W_n}^{(1)}\}}h^{(x)}\nonumber\\
&=&\frac{1}{2}\left[(\xi_0+\xi_3)(C_1\xi_0+C_2\xi_3)\hat{\bh}_{n-1}+\frac{C_3}{2}(\xi_0+\xi_3)^{2}\check{\bh}_{n-1}\right].\end{aligned}$$
Hence $$\begin{aligned}
\varphi_1(E_{\Lambda_n})
&=&\frac{1}{2}(\xi_0+\xi_3)(C_1\xi_0+ C_2\xi_3)\Tr\left[ \omega_0K_{[0,1]}\cdots K_{[n-2,n-1]}\hat{\bh}_{n-1} K_{[n-2,n-1]}^{*}\cdots
K_{[0,1]}^{*}\right]\nonumber\\
&&+C_3\bigg(\frac{\xi_0+\xi_3}{2}\bigg)^{2}\Tr\left[
\omega_0 K_{[0,1]}\cdots K_{[n-2,n-1]}\check{\bh}_{n-1}
K_{[n-2,n-1]}^{*}\cdots
K_{[0,1]}^{*}\right].\nonumber\\
&=&\frac{1}{2}\left[(\xi_0+\xi_3)(C_1\xi_0+C_2\xi_3)\hat{\psi}_{n-1}+\frac{C_3}{2}(\xi_0+\xi_3)^{2}\check{\psi}_{n-1}\right].%\nonumber\\\end{aligned}$$ Now using the values of $\hat{\psi}_{n-1}$ and $\check{\psi}_{n-1}$ given by the previous lemma we obtain the result.
Now we consider the state $\varphi_2$. Recall that this state is defined by $\omega_0=\frac{1}{\xi_0}\id$ and $h^{x}=h'=\xi_0\id-\xi_3\sigma_z$. Define two elements of $\mathcal{B}_{W_n}$ by $$\hat{\bh'}_n:=\id^{x_{W_n}^{(1)}}\otimes\bigotimes_{x\in W_n\setminus \{x_{W_n}^{(1)}\}}h'^{(x)}$$ $$\check{\bh'}_{n}:=\sigma^{x_{W_n}^{(1)}}\otimes\bigotimes_{x\in W_n\setminus \{x_{W_n}^{(1)}\}}h'^{(x)}$$ Using the same argument like in the proof of Lemma \[f-p-11\] we can prove the following auxiliary fact.
\[6.7\] Let $$\hat{\phi}_n:=\Tr_{n-1]}\big[\omega_{0} K_{[0,1]}\cdots K_{[n-1,n]}\hat{\bh'}_{n}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big]$$ $$\check{\phi}_n:=\Tr_{n-1]}\big[\omega_{0} K_{[0,1]}\cdots K_{[n-1,n]}\check{\bh'}_{n}K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big]$$ Then there are two pairs of reals $(\hat{\pi}_{1},\hat{\pi}_{2})$ and $(\check{\pi}_{1},\check{\pi}_{2})$ depending on $\theta$ such that $$\left\{
\begin{array}{ll}
\hat{\phi}_n=\hat{\pi}_{1}+\hat{\pi}_{2}(\frac{C_1}{C_3}-\frac{1}{2})^{n}, \\
\check{\phi}_n=\check{\pi}_{1}+\check{\pi}_{2}(\frac{C_1}{C_3}-\frac{1}{2})^{n} \\
\end{array}
\right.$$ where $$\hat{\pi}_{1}=\frac{C_3^2}{3 C_3-2 C_1}, \ \ \hat{\pi}_{2}=\frac{2 C_{3}(C_3-C_1)}{3C_{3}-2C_1},$$ $$\check{\pi}_{1}=-\frac{2C_{2}C_3^2\xi_{3}}{ 3C_3-2C_1},\ \ \check{\pi}_{2}=\frac{2C_{2}C_3^{2}\xi_{3}}{ 3C_3-2C_1}.$$
\[6.8\] For each $n\in\bn$ one has $$\begin{aligned}
\varphi_2(E_{\Lambda_n})&=&\frac{1}{2}\left[(\xi_0-\xi_3)(C_1\xi_0-C_2\xi_3)\hat{\pi}_{1}+\frac{C_3}{2}(\xi_0-\xi_3)^{2}\check{\pi}_{1}\right]\\[2mm]
&&+\frac{1}{2}\left[(\xi_0-\xi_3)(C_1\xi_0 -C_2\xi_3)\hat{\pi}_{2}+\frac{C_3}{2}(\xi_0-\xi_3)^{2}\check{\pi}_{2}\right]
\bigg(\frac{C_1}{C_3}-\frac{1}{2}\bigg)^{n-1}.\end{aligned}$$
From we find $$\begin{aligned}
\label{6.81}
\varphi_2(E_{\Lambda_n})&=&\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n}^{'1/2} E_{\Lambda_n} \bh_{n}^{'1/2} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\\
&=&\Tr \big(\omega_0 K_{[0,1]}\cdots K_{[n-1,n]}\bh_{n}^{'} E_{\Lambda_n} K_{[n-1,n]}^{*}\cdots K_{[0,1]}^{*}\big)\\\end{aligned}$$ We easily calculate that $$\begin{aligned}
\Tr_{n-1]} (K_{[n-1,n]}\bh'_n E_{\Lambda_n}
K_{[n-1,n]}^{*})=\frac{1}{2}(\xi_0-\xi_3)(C_1\xi_0-C_2\xi_3)\hat{\bh'}_{n-1}+C_3\bigg(\frac{\xi_0-\xi_3}{2}\bigg)^{2}\check{\bh'}_{n-1}.\end{aligned}$$ Hence, from one gets $$\begin{aligned}
\varphi_2(E_{\Lambda_n})
=\frac{1}{2}\left[(\xi_0-\xi_3)(C_1\xi_0-C_2\xi_3)\hat{\phi}_{n-1}+\frac{C_3}{2}(\xi_0-\xi_3)^{2}\check{\phi}_{n-1}\right].%\nonumber\\\end{aligned}$$ Using the values of $\hat{\phi}_{n-1}$ and $\check{\phi}_{n-1}$ given in Lemma \[6.7\], we obtain the desired assertion.
\[6.9\] Assume that $J \in ]-J_0,J_0[$, then the two Backward QMC $\varphi_1$ and $\varphi_2$ are not quasi-equivalent.
For any $\forall n\in\natural$ it is clear that $E_{\Lambda_n}\in\mathcal{B}_{\Lambda_n}\setminus\mathcal{B}_{\Lambda_{n-1}}.$ Therefore, for any finite subset $\Lambda\in L$, there exists $n_0\in\natural$ such that $\Lambda\subset\Lambda_{n_0}$. Then for all $n>n_0$ one has $E_{\Lambda_n}\in\mathcal{B}_{\Lambda_n}\setminus\mathcal{B}_{\Lambda}.$ It is clear that $$\|E_{\Lambda_n}\|=\|e_{1,1}^{x_{W_n}^{(1)}}\bigotimes_{y\in
L\setminus \{x_{W_n}^{(1)}\}}\id^{y}\|=\|e_{1,1}\|=\frac{1}{2}.$$ From Propositions \[6.6\] and \[6.8\] we obtain $$\begin{aligned}
\left|\varphi_1(E_{\Lambda_n})-\varphi_1(E_{\Lambda_n})\right|
&=&\frac{1}{2}\bigg|\big[(\xi_0+\xi_3)(C_1\xi_0 +C_2\xi_3)\hat{\rho}_{1}
+C_3(\xi_0+\xi_3)^{2}\check{\rho}_{1}\big]\\[2mm]
&&-\big[(\xi_0-\xi_3)(C_1\xi_0 -C_2\xi_3 )\hat{\pi}_{1}
+C_3(\xi_0-\xi_3)^{2}\check{\pi}_{1}\big]\nonumber\\
&&+\bigg(\big[(\xi_0+\xi_3)(C_1\xi_0 +C_2\xi_3)\hat{\rho}_{2}+C_3(\xi_0+\xi_3)^{2}\check{\rho}_{2}\big]\\[2mm]
&&-\big[(\xi_0-\xi_3)(C_1\xi_0 - C_2\xi_3 )\hat{\pi}_{2}+C_3(\xi_0-\xi_3)^{2}\check{\pi}_{2}\big]\bigg)
\left(\frac{C_1}{C_3}-\frac{1}{2}\right)^{n-1}\bigg|\nonumber\\
&\geq& I_1-I_2\left|\frac{C_1}{C_3}-\frac{1}{2}\right|^{n-1}\end{aligned}$$ where $$\begin{aligned}
I_1&=&\frac{1}{2}\bigg|\big[(\xi_0+\xi_3)(C_1\xi_0 +C_2\xi_3 )\hat{\rho}_{1}+C_3(\xi_0+\xi_3)^{2}\check{\rho}_{1}\big]\\[2mm]
&&-\big[(\xi_0-\xi_3)(C_1\xi_0 - C_2\xi_3 )\hat{\pi}_{1}+C_3(\xi_0-\xi_3)^{2}\check{\pi}_{1}\big]\bigg|\\[2mm]
I_2&=&\frac{1}{2}\bigg|\big[(\xi_0+\xi_3)(C_1\xi_0 +C_2\xi_3 )\hat{\rho}_{2}+C_3(\xi_0+\xi_3)^{2}\check{\rho}_{2}\big]\\[2mm]
&&-\big[(\xi_0-\xi_3)(C_1\xi_0 - C_2\xi_3 )\hat{\pi}_{2}+C_3(\xi_0-\xi_3)^{2}\check{\pi}_{2}\big]\bigg|.\end{aligned}$$ Due to $\beta> 0, \theta=\exp2\beta>1$, $C_1>0$,$C_3>0,
\xi_0>, \xi_3>0$, one can find that $$\begin{aligned}
I_1=\frac{C_3\xi_3(2 C_2+C_3)}{3 C_3-2 C_1}>0.
\end{aligned}$$ Now we have $\frac{2C_1-C_3}{2C_3} =\frac{\theta^{J_0}(\theta^J+\theta^{-J})+2}{2(\theta^{2J_0}-1)}$, since $J \in ]-J_0,J_0[$ then $$\frac{2C_1-C_3}{2C_3}
=\frac{\theta^{J_0}(\theta^J+\theta^{-J})+2}{2(\theta^{2J_0}-1)}\sim \frac{1}{2\theta^{(J_0-J)}}\leq \frac{1}{2},~~~~\theta \geq \theta_0$$ Then the following equality is hold: $$\begin{aligned}
\bigg|\frac{C_1}{C_3}-\frac{1}{2}\bigg|\leq \frac{1}{2}\end{aligned}$$ which yields $$I_2\left|\frac{C_1}{C_3}-\frac{1}{2}\right|^{n-1}\rightarrow
0 \ \ \textrm{as} \ \ n\rightarrow +\infty.$$ Then there exists $n_1\in \bn$ such that $\forall n\geq n_0$ one has $$I_2\left|\frac{C_1}{C_3}-\frac{1}{2}\right|^{n}\leq
\frac{\varepsilon_1 }{2}.$$ Hence, for all $n\geq n_1$ we obtain $$\left|\varphi_1(E_{\Lambda_n})-\varphi_1(E_{\Lambda_n})\right|\geq
\frac{\varepsilon_1}{2}=\varepsilon_1\|E_{\Lambda_n}\|.$$ This, according to Theorem \[br-q\], means that the states $\varphi_1$ and $\varphi_2$ are not quasi-equivalent. The proof is complete.
Now Theorems \[5.3\],\[6.2\] and \[6.9\] imply Theorem \[Main\].
QMC associated with the XY-interaction model with $J_0=0$
===========================================================
In this section, we consider a model which does not contain the classical Ising part, i.e. $J_0=0$, which means the model has only competing XY-interactions. In this setting, from one gets
$$\begin{aligned}
\label{Axn}
A_{(u,(u,1),(u,2))}&=&L_{>(u,1),(u,2)<}\\
&&=\id^{(u,1)}\otimes \id^{(u,2)} + \sinh(J\beta) H_{>(u,1),(u,2)<}\\
&&+(\cosh(J\beta)-1)H^{2}_{>(u,1),(u,2)<}\\
&&=R_{1}\id^{(u)}\otimes\id^{(u,1)}\otimes\id^{(u,2)}
+R_2\id^{(u)}\otimes\sigma_{x}^{(u,1)}\otimes\sigma_{x}^{(u,2)}\\
&&+R_{2}\id^{(u)}\otimes\sigma_{y}^{(u,1)}\otimes\sigma_{y}^{(u,2)}
+R_3\id^{(u)}\otimes\sigma_{z}^{(u,1)}\otimes \sigma_{z}^{(u,2)}\end{aligned}$$
where $$\left\{
\begin{array}{ll}
R_1=\frac{1}{4}(\cosh(J\beta)+1);\bigskip \\
R_2=\frac{\sinh(J\beta)}{2};\bigskip \\
R_3=\frac{1}{2}(1-\cosh(J\beta)).
\end{array}
\right.$$
Therefore, one finds:
$$\begin{aligned}
\label{eqdern}
h&=&Tr_{x]}A_{(u,(u,1),(u,2))}[\id^{(u)}\otimes h\otimes h]A_{(u,(u,1),(u,2))}^{*}\nonumber\\
&=&[(R_1+2R_1^2+R_3^2) \tr(h)^2]\id^{(u)}.\end{aligned}$$
The equation is reduced to the following one $$\label{EQ1}
\left\{
\begin{array}{lll}
h_{11}=h_{22}=\frac{1}{(R_1+2R_1^2+R_3^2)},\\
h_{21}=0, h_{12}=0.\\
\end{array}
\right.$$
Then putting $\alpha=\frac{1}{(R_1+2R_1^2+R_3^2)}$ we get $$h_{\alpha}=
\left(
\begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}
\right)$$
\[5.1\] The pair $(\omega_{0},\{h^{x}=h_{\alpha}| x\in L\})$ with $\omega_{0}=\frac{1}{\alpha}\id,\ \ h^{x}=h_{\alpha}, \forall
x\in L,$ is solution of ,. Moreover the associated Backward QMC can be written on the local algebra $\mathcal{B}_{L, loc}$ by: $$\varphi_{\alpha}(a)=\alpha^{2^{n}-1}\Tr\bigg(\prod_{i=0}^{n-1}K_{[i,i+1]} a \prod_{i=0}^{n-1} K_{[n-i-1,n-i]}^{*}\bigg),
\ \ \forall a\in B_{\Lambda_{n}}.$$ In this case, there is no phase transition.
We stress that if one takes nearest neighbor XY interactions on the Cayley tree of order two, still there does not occur a phase transition [@AMSa1]. However, if the order of the tree is three or more then for the mentioned model there exists a phase transition [@AMSa2].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to professors L.Accardi for fruitful discussions and useful suggestions on the definition of the phase transition.
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[^1]: The quantum analogues of Markov chains were first constructed in [@[Ac74f]], where the notion of quantum Markov chain (QMC) on infinite tensor product algebras was introduced. Later on, in [@fannes2], finitely correlated states were introduced and studied, which are related to each other. However, satisfactory constructions of such kind of fields were not established, since most of the fields were considered over the integer lattices [@[AcFi01a]; @AF03].
[^2]: Recall that a representation $\pi_1$ of a $C^*$-algebra $\ga$ is *normal* w.r.t. another representation $\pi_2$, if there is a normal $*$- epimorphism $\rho:\pi_2(\ga)''\to \pi_1(\ga)''$ such that $\rho\circ\pi_2=\pi_1$. Two representations $\pi_1$ and $\pi_2$ are called *quasi-equivalent* if $\pi_1$ is normal w.r.t. $\pi_2$, and conversely, $\pi_2$ is normal w.r.t. $\pi_1$. This means that there is an isomorphism $\alpha:\pi_1(\ga)''\to
\pi_2(\ga)''$ such that $\alpha \circ\pi_1=\pi_2$. Two states $\varphi$ and $\psi$ of $\ga$ are said be *quasi-equivalent* if the GNS representations $\pi_\varphi$ and $\pi_\psi$ are quasi-equivalent [@BR].
|
---
author:
- '[**S.Ayadi**]{}[^1]'
title: ' ASYMPTOTIC PROPERTIES OF RANDOM MATRICES OF LONG-RANGE PERCOLATION MODEL '
---
[**Abstract**]{}: We study the spectral properties of matrices of long-range percolation model. These are $N\times N$ random real symmetric matrices $H=\{H(i,j)\}_{i,j}$ whose elements are independent random variables taking zero value with probability $1-\psi\left( (i-j)/b\right)$, $b\in \mathbb{R}^{+}$, where $\psi$ is an even positive function with $\psi(t)\le{1}$ and vanishing at infinity. We study the resolvent $G(z)=(H-z)^{-1}, \ Imz\neq{0}$ in the limit $N,b\rightarrow\infty$, $b=O(N^{\alpha}), \ 1/3<\alpha<1$ and obtain the explicit expression $T(z_{1},z_{2})$ for the leading term of the correlation function of the normalized trace of resolvent $g_{N,b}(z)=N^{-1}Tr G(z)$. We show that in the scaling limit of local correlations, this term leads to the expression $(Nb)^{-1}T(\lambda+r_{1}/N+i0,\lambda+r_{2}/N-i0)=
b^{-1}\sqrt{N}|r_{1}-r_{2}|^{-3/2}(1+o(1))$ found earlier by other authors for band random matrix ensembles. This shows that the ratio $b^{2}/N$ is the correct scale for the eigenvalue density correlation function and that the ensemble we study and that of band random matrices belong to the same class of spectral universality.
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[**AMS Subject Classifications**]{}: 15A52, 45B85, 60F99. 0,2cm
[**Key Words**]{}: [*random matrices, asymptotic properties, percolation model.*]{} 0,2cm
[**running title**]{}: [*Asymptotic properties for percolation model.*]{}
0,5cm
Introduction
============
Random matrices play an important role in various fields of mathemathics and physics. The eigenvalue distribution of large matrices was initially considered by E.Wigner to model the statistical properties of the energy spectrum of heavy nuclei (see e.g. the collection of early papers [@BBB]). Further investigations have led to numerous applications of random matrices of infinite dimensions in such branches of theoretical physics as statistical mechanics of disordered spin systems, solid state physics, quantum chaos theory, quantum field theory and others (see monographs and reviews [@I; @II; @III; @IIII]). In mathematics, the spectral theory of random matrices has revealed deep links with the orthogonal polynomials theory, integrable systems, representation theory, combinatorics, free probability theory, and others [@BleIts; @Deift; @Soshni; @Voi].
The first result of the spectral theory of large random matrices concerns the eigenvalue distribution of the Wigner ensemble $A_{N}$ of $N\times
N$ real symmetric matrices of the form $$\label{b.1ANij}
A_{N}(i,j)=\frac{1}{\sqrt{N}}a(i,j), \quad |i|,|j|\le{n},$$ where $N=2n+1$ and $\{a(i,j); \ -n\le{i}\le{j}\le{n}\}$ are independent and identically distributed random variables defined on the same probability space $(\Omega,\mathfrak{F},{\bf P})$ such that $$\label{b.1Eaij}
{\bf E}\{a(i,j)\}=0, \quad {\bf
E}\{a(i,j)^{2}\}=v^{2}(1+\delta_{ij}),$$ where $$\delta_{ij}=\left\{
\begin{array}{lll}
0 & \textrm{if} & i\neq{j}, \\
1 & \textrm{if} & i=j
\end{array}\right.$$ is the Kronecker symbol and ${\bf E}\{\cdot\}$ is the mathematical expectation with respect to ${\bf P}$.
Denoting by $\lambda^{(n)}_{-n}\le{\ldots}\le{\lambda^{(n)}_{n}}$ the eigenvalues of $A_{N}$, the normalized eigenvalue counting function is defined by $$\label{b.1sigma}
\sigma_{n}(\lambda,A_{N})=N^{-1}\sharp\{\lambda^{(n)}_{j}\le{\lambda}\}.$$ E.Wigner [@BB] proved that if $a(i,j)$ has all order finite moments, the eigenvalue counting measure $d\sigma_{n}(\lambda,A_{N})$ converges weakly in average as $n\rightarrow \infty$ to a distribution $d\sigma_{sc}(\lambda)$, where the nondecreasing function $\sigma_{sc}(\lambda)$ is differentiable and its derivative $\rho_{sc}$ is given by $$\label{b.1rho}
\rho_{sc}{(\lambda)}=\sigma_{sc}^{'}(\lambda)=\frac{1}{2\pi
v^{2}}\left\{
\begin{array}{lll}
\sqrt{4v^{2}-\lambda^{2}} & \textrm{if} &
|\lambda|\le{2\sqrt{v^{2}}}, \\
0 & \textrm{if} & |\lambda|>2v.
\end{array}\right.$$ This limiting distribution (\[b.1rho\]) is known as the Wigner distribution, or the semicircle law. A proof of the Wigner’s result based on the resolvent technique is given in [@EEE; @FFF; @FF].
Important generalizations of the Wigner’s ensemble are given by the band and dilute random matrix ensembles [@CCC]. In the band random matrices model, the matrix elements take zero value outside the band of width $b_{n}$ along the principal diagonal, for some positive sequence $(b_{n})_{n\geq{0}}$ of real numbers. This ensemble can be obtained from $A_{N}$ (\[b.1ANij\]) by multiplying each $a(i,j)$ by $I_{(-1/2,1/2)}\left( (i-j)/b_{n}\right)$, where $$I_{B}(t) = \left\{
\begin{array}{lll}
1 & \textrm{if} & t\in{B}, \\
0 & \textrm{if} & t\in\mathbb{R}\setminus B
\end{array}\right.$$ is the indicator function of the interval $B$. The ensemble of dilute random matrices can be obtained from $A_{N}$ (\[b.1ANij\]) by multiplying $a(i,j)$ by independent Bernoulli random variables of parameter $ p_{n}/N$. Assuming that $b(n)=o(n)$ for large $n$, the semicircle law is observed for both ensembles, in the limit $b_{n}\rightarrow\infty$ (see [@AA]) and $p_{n}\rightarrow\infty$ as $n\rightarrow\infty$ (see [@CCC]).
The crucial observation made numerically [@Casati] and then supported in the theoretical physics (see [@Fyodorv; @syl]) is that the ratio $b^{2}/n$ is the critical one for the corresponding transition in spectral properties of band random matrices. In [@B], it was proved that the ratio $\tilde{\alpha}=\lim_{n\rightarrow\infty}b^{2}/n$ naturally arises when one considers the leading term of this correlation function on the local scale. This can be regarded as the support of the conjecture that the local properties of spectra of band random matrices depend on $\tilde{\alpha}$.
Let us describe our results in more details. We are interested in a generalization of the both ensembles mentioned above. Roughly speaking, we consider the band random matrices with a random width. To proceed, we consider the ensemble $\{H_{n,b}\}$ of random $N\times N$ matrices, $N=2n+1$ whose entries $H_{n,b}$ is obtained as follows: we multiply each matrix element $a(i,j)$ by some Bernoulli random variable $d_{b}(i,j)$ with parameter $\psi\left(
(i-j)/b\right)$.
The family $\{d_{b}(i,j); \ |i|,|j|\le{n}\}$ can be regarded as the adjacency matrix of the family of random graphs $\{\Gamma_{n}\}$ with $N=2n+1$ vertices $(i,j)$ such that the average number of edges attached to one vertex is $b_{n}$. Hence, each edge $e(i,j)$ of the graph is present with probability $\psi\left( (i-j)/b\right)$ and not present with probability $1-\psi\left( (i-j)/b\right)$. Below are some well known examples:
1. In theoretical physics, the ensemble $\{\Gamma_{n}\}$ with $\psi(t)=e^{-|t|^{s}}$ is referred to as the Long-Range Percolation Model (see for example [@K] and references therein). Our ensemble can be regarded as a modification of the adjacency matrices of $\{\Gamma_{n}\}$. To our best knowledge, the spectral properties of this model has not been studied yet.
2. It is easy to see that if one takes $b_{n}=N$ and $\psi\equiv 1$, then one recovers the Wigner ensemble $(1.1)$.
3. If one considers $\psi(t)=I_{(-1/2,1/2)}(t)$, one gets the band random matrix ensemble [@AA].
In present paper, we consider the resolvent $G_{n,b}=\left(H_{n,b}-zI\right)^{-1}$ and study the asymptotic expansion of the correlation function $$C_{n}(z_{1},z_{2})={\bf E}\{g_{n,b}(z_{1})g_{n,b}(z_{2})\}-{\bf
E}\{g_{n,b}(z_{1})\}{\bf E}\{g_{n,b}(z_{2})\},$$ where we denoted $g_{n,b}=N^{-1}{\mathrm{Tr}}G_{n,b}(z)$. Keeping $z_{l}$ far from the real axis, we consider the leading term $T(z_{1},z_{2})$ of this expansion and find explicit expression for it. This term $T(r_{1}+i0,r_{2}-i0)$ regarded on the local scale $r_{1} -r_{2} =
r/N$ exhibits different behavior depending on the rate of decay of the profile function $\psi(t)$.
Our main conclusion is that if $\psi(t)\sim |t|^{-1-\nu}$ as $t\rightarrow\infty$, then the value $\nu=2$ separates two major cases. If $\nu\in(1,2)$, then the limit of $T(r)$ depend on $\nu$. If $\nu\in(2,+\infty)$, then $$\frac{1}{Nb}T(r)=-const \cdot \frac{\sqrt{N}}{b} \cdot
\frac{1}{|r|^{3/2}}(1+o(1)).$$ This asymptotic expression coincides with the result obtained in [@B] for band random matrices. Then one can conclude that the ensemble under consideration and the band random matrix ensemble belong to the same universality class.
The outline of this paper is as follows. In section 2, we define the random matrix ensemble $H_{n,b}$ of long-range percolation model, we state our main results and describe the scheme of their proofs. In section 3, we study the correlation function $C_{n,b}(z_{1},z_{2})$ and obtain the main relation for it. In section 4, we show that ${\bf Var}\{g_{n,b}(z)\}$ is bounded by $(Nb)^{-1}$ and find the leading term $T(z_{1},z_{2})$ of the correlation function under the moment condition that $\sup_{i,j} {\bf E}|a(i,j)|^{14}<\infty$. In section 5, we prove the auxiliary facts used in section 4. Expressions derived in section 4 are analyzed in section 6, where the asymptotic behavior of $T(z_{1},z_{2})$ is studied and the issue of the universal bihaviour is discussed.
The ensemble, main results and technical tools
==============================================
The ensemble and main results
-----------------------------
Let us consider a family of independent real random variables ${\cal
A}_{n}= \{a(i,j); \ |i|, |j|\le{n}\}$ satisfying (\[b.1ANij\]). Let $\psi(t)$, $t\in\mathbf{R}$, be a real continuous even function such that: $$\label{b.2psicondit}
0\le{\psi{(t)}}\le{1}, \quad \int_{\mathbb{R}}\psi{(t)}dt = 1.$$ Given real $b>0$, we introduce a family of independent Bernoulli random variables ${\cal D}_{b}=\{d_{b}(i,j); \ |i|, |j|\le{n}\}$ with the law $$\label{b.2dijb}
d_{b}(i,j) = \left\{
\begin{array}{lll}
1 & \textrm{with probability} & \psi\left( (i-j)/b\right) \\
0 & \textrm{with probability} & 1-\psi\left( (i-j)/b\right).
\end{array}\right.$$ This family is independent of the family ${\cal A}_{n}$. We assume that ${\cal A}_{n}$ and ${\cal D}_{n}$ are defined on the same probability space $(\Omega,\mathfrak{F},{\bf P})$ and we denote by ${\bf E}\{\cdot\}$ the mathematical expectation with respect to ${\bf P}$.
We define a real symmetric $N\times N$ random matrix $H_{n,b}$ by equality: $$\label{b.2Hnbij}
H_{n,b}(i,j)=\frac{1}{\sqrt{b}}a(i,j)d_{b}(i,j), \quad i\le{j},
\quad |i|, |j|\le{n},$$ where $b\le{N}$, $N=2n+1$. Here and below the family $\{H_{n,b}\}$ is referred to as the ensemble of random matrices of long-range percolation model. In what follows, we will need the existence of several absolute moments of $a(i,j)$ that we denote by $$\label{b.2mur}
\mu_{l}=\sup_{|i|, |j|\le{n}}{\bf E}\{|a(i,j)|^{l}\},$$ where the upper bound for $l$ will be specified later.
We consider the resolvent $$G_{n,b}(z)=(H_{n,b}-z)^{-1}, \quad {\mathrm{Im}}z\neq{0}.$$ Its normalized trace $g_{n,b}(z)$ coincides with the Stieltjes transform of the normalized eigenvalue counting function $\sigma_{n,b}(\lambda;H_{n,b})$ (\[b.1sigma\]): $$\label{b.2gnbz}
g_{n,b}(z)=\frac{1}{N}{\mathrm{Tr}}G_{n,b}(z)=\int
(\lambda-z)^{-1}d\sigma_{n,b}(\lambda,H_{n,b}),
\ {\mathrm{Im}}z \neq{0}.$$ In [@A], we have proved that if $\mu_{3}< \infty$ (\[b.2mur\]) and $1\ll b\ll N$, then $$\lim_{n,b\rightarrow\infty}{\bf E}\{g_{n,b}(z)\}=w(z)$$ for $z\in \Lambda_{\eta}$, where $$\label{b.2Lambdaeta}
\Lambda_{\eta}=\{z\in\mathbf{ C}: \ \eta\le{|{\mathrm{Im}}z|}\}, \quad
\eta=2v+1$$ and the limiting function $w(z)$ verifies equation $$\label{b.2wz}
w(z)=\frac{1}{-z-v^{2} w(z)}$$ with $v$ is determined by (\[b.1Eaij\]). Equation (\[b.2wz\]) has a unique solution in the class of functions such that ${\mathrm{Im}}w(z){\mathrm{Im}}z>0$, ${\mathrm{Im}}z\neq{0}$. This solution $w(z)$ is the Stieltjes transform of the semi-circle distribution (\[b.1rho\]). This result shows that the semi-circle law is valid for random matrices of long-range percolation model. As a by-product of proof, we have shown that $$\label{b.2Vargnbz}
{\bf Var}\{g_{n,b}(z)\}=o(1), \ \quad z\in\Lambda_{\eta}, \quad
\hbox{ as } \quad n,b\rightarrow\infty$$ and that the convergence $g_{n,b}(z)\rightarrow{w(z)}$ holds in probability.
In this paper, we improve the result (\[b.2Vargnbz\]) in two stages. On the first one we show that ${\bf
Var}\{g_{n,b}(z)\}=O\left((Nb)^{-1}\right)$ in the limit $n,b\rightarrow\infty$ such that $$\label{b.2bnalpha}
b=O\left(n^{\alpha}\right), \quad 1/3<\alpha<1$$ and this gives the convergence $g_{n,b}(z)\rightarrow{w(z)}$ with probability 1. Next, we find the explicit form of the leading term of the correlation function $$C_{n,b}(z_{1},z_{2})={\bf E}\{g_{n,b}(z_{1})g_{n,b}(z_{2})\}-{\bf
E}\{g_{n,b}(z_{1})\}{\bf E}\{g_{n,b}(z_{2})\}.$$ We now formulate the main result of the paper, where we denote $w_{1}=w(z_{1})$ and $w_{2}=w(z_{2})$ are given by (\[b.2wz\]).
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Let ${\cal A}_{n}$ be such that, in addition to (\[b.1Eaij\]), the following properties are verified : $$\label{b.2Eaij3}
{\bf E}\{a(i,j)^{3}\}={\bf E}\{a(i,j)^{5}\}=0, \quad {\bf
E}\{a(i,j)^{2m}\}=V_{2m}(1+\delta_{ij})^{m}, \quad m=2,3$$ for all $i\le{j}$, $\mu_{14}<\infty$ (\[b.2mur\]) and $$\int_{\R} \sqrt{\psi(t)}dt<\infty.$$ Then in the limit $n,b\rightarrow\infty$ (\[b.2bnalpha\]) and for $z_{l}\in\Lambda_{\eta}$ (\[b.2Lambdaeta\]), $l=1,2$, equality $$\label{b.2Cnbz1z2Th}
C_{n,b}(z_{1},z_{2})
=\frac{1}{Nb}T(z_{1},z_{2})+o\left(\frac{1}{Nb}\right)$$ holds with $T$ is given by the formula $$\label{b.2Tz1z2}
T(z_{1},z_{2})=Q(z_{1},z_{2})+\frac{2\Delta w_{1}^{3}w_{2}^{3}}{(1 -
v^{2}w_{1}^{2})(1-v^{2}w_{2}^{2})}$$ with $$\label{b.2Qz1z2}
Q(z_{1},z_{2})=\frac{v^{2}w_{2}^{2}w_{1}^{2}}{\pi(1-v^{2}w_{1}^{2})(
1- v^{2}w_{2}^{2})} \int_{\R}\frac{\tilde{\psi}(p)}{[ 1-
v^{2}w_{1}w_{2}\tilde{\psi}(p)]^{2}}dp,$$ where $\tilde{\psi}(p)$ is the Fourier transform of $\psi$ $$\tilde{\psi}(p)=\int_{\R}\psi(t)e^{ipt}dt$$ and $$\label{b.2Delta}
\Delta= V_{4}\int_{\R}\psi(t)dt-3v^{4}\int_{\R}\psi^{2}(t)dt.$$
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Under this conditions, Theorem 2.1 and relation (\[b.2Tz1z2\]) remain true with $\Delta$ replaced by $$\lim_{n,b\rightarrow \infty} \sup_{|i|\le{n}}\left(
b\sum_{|j|\le{n}} {\bf E}\{H(i,j)^{4}\} - 3 {\bf
E}\{H(i,j)^{2}\}^{2}\right)$$ $$=\lim_{n,b\rightarrow \infty}
\sup_{|i|\le{n}}\left(\frac{1}{b}\sum_{|j|\le{n}}
(1+\delta_{ij})^{2}\left[V_{4}\psi(\frac{i-j}{b})
-3v^{4}\psi(\frac{i-j}{b})^{2}\right]\right).$$ We would like to note that the form of (\[b.2Tz1z2\]) generalizes the expressions obtained in [@B] and [@C]. Namely, the term $Q(z_{1},z_{2})$ is derived for the case when the entries of random matrices $H$ are gaussian random variables. The ensemble we consider is very similar to the band random matrices, but it represents a different model. The form of the last term is exactly the same as the one obtained in [@C] for the Wigner random matrices. This shows that this term “ forgets ” the band-like structure of our matrices. All our computations and formulas are valid in the case of band random matrices $H_{n,b}(i,j)=b^{-1/2}a(i,j)[\psi\left((i-j)/b\right)]^{1/2}$ with not necessarily gaussian $a(i,j)$. Therefore Theorem 2.1 generalizes the results of paper [@B]. In the case of band random matrices, one obtains the same expressions (\[b.2Cnbz1z2Th\]) and (\[b.2Tz1z2\]) with $\Delta$ (\[b.2Delta\]) replaced by $\Delta_{band}=(V_{4}-3v^{4})\int \psi^{2}(t)dt$, provided $a(i,j)$ are the same as in Theorem 2.1.
The results of Theorem 2.1 are used to study the universality properties of eigenvalue distribution. We do this in Section 6.
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Cumulant expansions and resolvent identities
--------------------------------------------
We prove Theorem 2.1 and Theorem 2.2 by using the method proposed in papers [@C; @CCC] and further developed in a series of works [@A; @B]. The basic tools of this method are given by the resolvent identities combined with the cumulant expansions technique.
### The cumulant expansions formula
Let us consider a family $\{X_{t}: \ t=1,\ldots,m\}$ of independent real random variables defined on the same probability space such that ${\bf
E}\{|X_{t}|^{q+2}\}<\infty $ for some $q\in \N$ and $t=1,\ldots,m$. Then for any complex-valued function $F(u_{1},\ldots,u_{m})$ of the class $\mathcal{C}_{\infty}(\R^{m})$ and for all $j$, one has $$\label{b.2EXjFX1Xm}
{\bf E}\{X_{t} F(X_{1},\ldots,X_{m})\}=\sum_{r=0}^{q}
\frac{K_{r+1}}{r!} {\bf
E}\left\{\frac{\partial^{r}F(X_{1},\ldots,X_{m})}
{(\partial{X_{t}})^{r}}\right\} +\epsilon_{q}(X_{t}),$$ where $K_{r}=Cum_{r}(X_{t})$ is the r-th cumulant of $X_{t}$ and the remainder $\epsilon_{q}(X_{t})$ can be estimated by inequality $$\label{b.2epsilonqXj}
|\epsilon_{q}(X_{t})|\le{C_{q} \sup_{U\in \R^{m}}
\left|\frac{\partial^{q+1}F(U)}{\partial{u_{t}^{q+1}}}\right|{\bf
E}\{|X_{t}|^{q+2}\}},$$ where $C_{q}$ is a constant. Relations (\[b.2EXjFX1Xm\]) and (\[b.2epsilonqXj\]) can be proved by multiple using of the Taylor’s formula (see [@A; @C] for the proofs).
The cumulants $K_{r}$ can be expressed in terms of the moments $\breve{\mu}_{r}={\bf E}(X_{t}^{r})$ of $X_{t}$.
Indeed, let $f_{t}$ be a complex-valued function of one real variable such that $$f_{t}(x)=F(X_{1},\ldots,X_{t-1},x,X_{t+1},\ldots,X_{n})$$ and $f_{t}^{(r)}$ is its r-th derivative.
0,2cm
- If $q=1$ and ${\bf E}\{X_{t}\}=0$, then $$\label{c.K1K2}
K_{1}=\breve{\mu}_{1}=0, \quad K_{2}=\breve{\mu}_{2}$$ and the remainder $\epsilon_{1}(X_{t})$ is given by: $$\label{b.2epsilon1Xj}
\epsilon_{1}(X_{t})=\frac{1}{2}{\bf
E}\left\{X_{t}^{3}f_{t}^{(2)}(x_{0})\right\}-K_{2}{\bf
E}\left\{X_{t}f_{t}^{(2)}(x_{1})\right\}.$$
- If $q=3$ and ${\bf E}\{X_{t}\}= {\bf
E}\{X_{t}^{3}\}=0$, then $$\label{b.2K2K4}
K_{1}=K_{3}=0, \quad K_{2}=\breve{\mu}_{2}, \quad
K_{4}=\breve{\mu}_{4}-3\breve{\mu}^{2}_{2}$$ and the remainder $\epsilon_{3}(X_{t})$ is given by: $$\begin{aligned}
\nonumber \epsilon_{3}(X_{t})=& \frac{1}{4!}{\bf
E}\left\{X_{t}^{5}f_{t}^{(4)}(x_{0})\right\}-\frac{K_{2}}{3!}{\bf
E}\left\{X_{t}^{3}f_{t}^{(4)}(x_{1})\right\}\\
\label{b.2epsilon3Xj} &-\frac{K_{4}}{3!}{\bf
E}\left\{X_{t}f_{t}^{(4)}(x_{2})\right\}.\end{aligned}$$
- If $q=5$ and ${\bf E}\{X_{t}\}= {\bf E}\{X_{t}^{3}\}={\bf
E}\{X_{t}^{5}\}=0$, then the cumulants $K_{r}$, $r=1,\ldots,4$ are given by (\[b.2K2K4\]), $$\label{b.2K5K6}
K_{5}=0, \quad
K_{6}=\breve{\mu}_{6}-15\breve{\mu}_{4}\breve{\mu}_{2}
+30\breve{\mu}^{3}_{2}$$ and the remainder $\epsilon_{5}(X_{t})$ is given by: $$\begin{aligned}
\nonumber \epsilon_{5}(X_{t})= &\frac{1}{6!}{\bf
E}\left\{X_{t}^{7}f_{t}^{(6)}(x_{0})\right\}-\frac{K_{2}}{5!}{\bf
E}\left\{X_{t}^{5}f_{t}^{(6)}(x_{1})\right\}\\
&\label{b.2epsilon5Xj} -\frac{K_{4}}{(3!)^{2}}{\bf
E}\left\{X_{t}^{3}f_{t}^{(6)}(x_{2})\right\}-\frac{K_{6}}{5!}{\bf
E}\left\{X_{t}f_{t}^{(6)}(x_{3})\right\},\end{aligned}$$ where for each $\nu=0,\ldots,3$, $x_{\nu}$ is a real random variable that depends on $X_{t}$ and such that $|x_{\nu}|\le{|X_{t}|}$. In what follows, we denote $f_{t}^{(r)}(x_{\nu})
=[\partial^{r}{F}/\partial{X_{t}^{r}}]^{(\nu)}$.
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### Resolvent identities
For any two real symmetric $n \times n$ matrices $h$ and $\tilde{h}$ and any non-real $z$ the resolvent identity $$\label{b.2h-zI-1}
( h-zI)^{-1}=(\tilde h-zI)^{-1}- ( h-zI)^{-1}( h-\tilde h)(\tilde
h-zI)^{-1}$$ is valid. Regarding (\[b.2h-zI-1\]) with, $\tilde h=0$ and denoting $G=\left(h-zI\right)^{-1}$, we get equality $$\label{b.2Gij}
G(i,j) = \zeta\delta_{ij} - \zeta\sum_{s=1}^{n} G(i,s)h(s,j), \quad
\zeta={-z^{-1}},$$ where $h(i,j), \ i,j=1,\ldots,n$ are the entries of the matrix $h$, $G(i,j)$ are the entries of the resolvent $G$ and $\delta$ denotes the Kronecker symbol.
Using (\[b.2h-zI-1\]) we derive for $G=\left(h-zI\right)^{-1}$, $|{\mathrm{Im}}z|\neq{0}$ equality $$\label{b.2partialG}
\frac{\partial{G(s,t)}}{\partial{h(j,k)}}=
-\frac{1}{1+\delta_{jk}}\left[G(s,j)G(k,t)+ G(s,k)G(j,t)\right].$$ We will also need two more formulas based on (\[b.2partialG\]); these are expressions for $\partial^{2}{G(i,j)}/\partial{h(j,i)^{2}}$ and $\partial^{3}{G(i,j)}/\partial^{3}{h(j,i)}$. We present them later. 0,1cm
### The scheme of the proof of Theorem 2.1
In this subsection we present a schema of computation of the leading terms of $C_{n,b}(z_{1},z_{2})$ (cf. (\[b.2Cnbz1z2Th\])).
Let us denote $g_{l}=g_{n,b}(z_{l})$, $l=1,2$ (everywhere below, we omit the subscripts $n$,$b$ when no confusion can arise). For a given a random variable, we denote $\xi^{0}=\xi-{\bf E}\xi$. Then using identity $$\label{b.2Ef0g0}
{\bf E}\{\xi^{0}g^{0}\}={\bf E}\{\xi^{0}g\},$$ we rewrite $C_{12} =C_{n,b}(z_{1},z_{2})$ as $$C_{12}={\bf E}\{g^{0}_{1}g_{2}\}=
\frac{1}{N}\sum_{|i|\le{n}}R_{12}(i)$$ with $R_{12}(i)={\bf E}\{g^{0}_{1}G_{2}(i,i)\}$. Applying the resolvent identity (\[b.2h-zI-1\]) to $G_{2}(i,i)$, we obtain equality $$\label{b.2R12}
R_{12}(i)=-\zeta_{2}\sum_{|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,p)H(p,i)\}.$$ To compute ${\bf E}\{g^{0}_{1}G_{2}(i,p)H(p,i)\}$, we use the cumulants expansion method (\[b.2EXjFX1Xm\]), and get $$\begin{aligned}
\nonumber {\bf E}\{g^{0}_{1}G_{2}(i,p)H(p,i)\}=& K_{2}{\bf
E}\left\{\frac{\partial\left(g^{0}_{1}G_{2}(i,p)\right)}
{\partial{H(p,i)}}\right\}\\
\label{b.2Eg0G2ipHpi} &+\frac{K_{4}}{6}{\bf
E}\left\{\frac{\partial^{3}\left(g^{0}_{1}G_{2}(i,p)\right)}
{\partial{H(p,i)^{3}}}\right\}+\tau_{ip},\end{aligned}$$ where $K_{r}$ is the r-th cumulant of $H(p,i)$ and $\tau_{ip}$ vanishes. Substituting this equality in (\[b.2R12\]) and using (\[b.2partialG\]), we obtain that $$\begin{aligned}
\nonumber \frac{\partial\{g^{0}_{1}G_{2}(i,p)\}}{\partial{H(p,i)}}=&
g^{0}_{1}\frac{\partial G_{2}(i,p)}
{\partial{H(p,i)}}+G_{2}(i,p)\frac{1}{N}\sum_{|s|\le{n}}\frac{\partial
G_{1}(s,s)}{\partial{H(p,i)}}\\
\nonumber =&
-\frac{1}{1+\delta_{pi}}g^{0}_{1}[G_{2}(i,p)^{2}+G_{2}(i,i)G_{2}(p,p)]\\
\label{b.2partialg10G2ij} &
-\frac{1}{1+\delta_{pi}}\left\{\frac{2}{N}G^{2}_{1}(i,p)G_{2}(i,p)\right\},\end{aligned}$$ where we used (\[b.2partialG\]) in the form $$\frac{\partial\{g^{0}_{1}G_{2}(i,p)\}}{\partial{H(p,i)}}=
\left\{\frac{\partial\left(g^{0}_{1}G_{2}(i,p)\right)}
{\partial{h(p,i)}}\vert_{h=H}\right\}.$$ We get relation $$\begin{aligned}
\nonumber R_{12}(i)= & \zeta_{2}v^{2}{\bf
E}\left\{g^{0}_{1}G_{2}(i,i)\sum_{|p|\le{n}}G_{2}(p,p)
\frac{1}{b}\psi\left(\frac{p-i}{b}\right)\right\}\\
\nonumber &+\frac{\zeta_{2}v^{2}}{b}\sum_{|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,p)^{2}\}\psi\left(\frac{p-i}{b}\right)\\
\nonumber &+ \frac{2\zeta_{2}v^{2}}{Nb}\sum_{|p|\le{n}}{\bf
E}\{G_{1}^{2}(i,p)G_{2}(i,p)\}\psi\left(\frac{p-i}{b}\right)\\
\label{b.2R12form1} &-\frac{\zeta_{2}}{6}\sum_{|p|\le{n}}K_{4}{\bf
E}\left\{\frac{\partial^{3}(g^{0}_{1}G_{2}(i,p)
)}{\partial{H(p,i)^{3}}}\right\}+\Phi_{n,b}(i),\end{aligned}$$ where $\sup_{|i|\leq{n}}|\Phi_{n,b}(i)|$ vanishes as $n,b\rightarrow\infty$ (\[b.2bnalpha\]) (see subsection 3.2 for more details). Also we have taken into account that (cf. (\[b.2K2K4\])) $$K_{2}(p,i)=K_{2}\left(H_{n,b}(p,i)\right)=\frac{1}{b}{\bf
E}\{a(p,i)^{2}d_{n,b}(p,i)^{2}\}=\frac{v^{2}}{b}
\psi\left(\frac{p-i}{b}\right)(1+\delta_{pi}).$$ Let us return to relation (\[b.2R12form1\]). We observe that the first term of the right-hand side (RHS) can be expressed in terms of $R_{12}$. This gives the possibility to obtain an equation of $R_{12}$. The second term vanishes in the limit $n,b\rightarrow\infty$ (we give later the explicit formulation). The third term represents the leading term of the correlations function (which provides the first expression of (\[b.2Tz1z2\])). The fourth term gives the contribution of the order $O((Nb)^{-1})$ to (\[b.2Cnbz1z2Th\]) (which provides the second expression of the leading term (\[b.2Tz1z2\])). The last term $\Phi_{n,b}(i)$ gives the contribution of the order $o((Nb)^{-1})$ to (\[b.2Cnbz1z2Th\]) (see Lemma 3.2).
Correlation function of the resolvent
=====================================
In this section we give the main relation of the correlation function $C_{n,b}(z_{1},z_{2})$. In what follows, we will need two elementary inequalities $$\label{b.3GijleImz}
|G(i,p)|\le{||G||}\le{\frac{1}{|{\mathrm{Im}}z|}},$$ and $$\label{b.3sumGij2leImz2}
\sum_{|p|\le{n}}|G(i,p)|^{2}=||G\vec{e}_{i}||^{2}\le{\frac{1}{|{\mathrm{Im}}z|^{2}}}, \quad |i|\le{n}$$ that hold for the resolvent of any real symmetric matrix. Here and below we consider $||e||_{2}^{2}=\sum_{i}|e(i)|^{2}$ and denote by $||G||=\sup_{||e||_{2}=1}||Ge||_{2}$ the corresponding operator norm.\
Derivation of relations for $R_{12}(i)$
---------------------------------------
Let us consider the average ${\bf E}\{g^{0}_{1}G_{2}(i,p)H(p,i)\}$. For each pair $(i,p)$, $g^{0}_{1}G_{2}(i,p)$ is a smooth function of $H(p,i)$. Its derivatives are bounded because of equation (\[b.2partialG\]) and (\[b.3GijleImz\]). In particular $$|D^{6}_{pi}\{\hat{g}^{0}_{1}\hat{G}_{2}(i,p)\}| \leq{C\left(|{\mathrm{Im}}z_{1}|^{-1} + |{\mathrm{Im}}z_{2}|^{-1}\right)^{8} },$$ where $C$ is an absolute constant. Here and thereafter we use the notation $D_{pi}$ for $\partial/\partial{H(p,i)}$.
According to the definition of $H$ and the condition $\mu_{7}<\infty$ (\[b.2mur\]), the seven absolute moment of $H(p,i)$ is of order $1/(b^{7/2})$. Then we can apply (\[b.2EXjFX1Xm\]) with $q=5$ to ${\bf
E}\{g^{0}_{1}G_{2}(i,p)H(p,i)\}$ and using (\[b.2h-zI-1\]), we get relation
- if $p<i$ $$\begin{aligned}
\nonumber {\bf E}\{g^{0}_{1}G_{2}(i,p)H(p,i)\}= &
K_{2}\left(H(p,i)\right){\bf
E}\left\{D^{1}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\} \\
\nonumber &+\frac{K_{4}\left(H(p,i)\right)}{6}{\bf
E}\left\{D^{3}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}\\
\label{b.3pi} &+\frac{K_{6}\left(H(p,i)\right)}{120}{\bf
E}\left\{D^{5}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}+
\tilde{\epsilon}_{pi}\end{aligned}$$ with $$\begin{aligned}
\nonumber \tilde{\epsilon}_{pi}= &\frac{1}{6!}{\bf
E}\left\{H(p,i)^{7}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(0)}\right\}\\
\nonumber &-\frac{K_{2}\left(H(p,i)\right)}{5!}{\bf
E}\left\{H(p,i)^{5}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(1)}\right\}\\
\nonumber &-\frac{K_{4}\left(H(p,i)\right)}{(3!)^{2}}{\bf
E}\left\{H(p,i)^{3}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(2)}\right\}\\
\label{b.3tildepsilonpi} &-\frac{K_{6}\left(H(p,i)\right)}{5!}{\bf
E}\left\{H(p,i)[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(3)}\right\},\end{aligned}$$ where the cumulants are given by (cf. (\[b.2K2K4\])-(\[b.2K5K6\])) $$\label{b.3K2K4}
K_{2}\left(H(p,i)\right)=
\frac{v^{2}}{b}\psi(\frac{p-i}{b})(1+\delta_{pi}), \quad
K_{4}\left(H(p,i)\right)=\frac{\Delta_{pi}}{b^{2}}(1+\delta_{pi})^{2}$$ with $\Delta_{pi}=
V_{4}\psi\left((p-i)/b\right)-3v^{4}\psi\left((p-i)/b\right)^{2}$ and $$\label{b.3K6}
K_{6}\left(H(p,i)\right)=\frac{\theta_{pi}}{b^{3}}(1+\delta_{pi})^{3},$$ with $\theta_{pi} =
V_{6}\psi\left((p-i)/b\right)-15V_{4}v^{2}\psi\left((p-i)/b\right)^{2}
+30v^{6}\psi\left((p-i)/b\right)^{3}$. In (\[b.3tildepsilonpi\]), we have denoted for each pair $(p,i)$ $$[g^{0}_{1}G_{2}(i,p)]^{(\nu)}=\{g^{(\nu)}\}^{0}_{pi}(z_{1})
G^{(\nu)}_{pi}(i,p;z_{2}), \quad \nu=0,\ldots,3$$ and $G^{(\nu)}_{pi}(z_{l})=(H^{(\nu)}_{pi}-z_{l})^{-1}$, $l=1,2$ with real symmetric $$H^{(\nu)}_{pi}(r,s)= \left\{
\begin{array}{lll}
H(r,s) & \textrm{if} & (r,s)\neq(p,i); \\
H^{(\nu)}(p,i) & \textrm{if} & (r,s)=(p,i),
\end{array}\right.$$ where $|H^{(\nu)}(p,i)|\le{|H(p,i)|}$, $\nu=0,\ldots,3$ (see subsection 2.2.1 for more detail).
- If $i<p$, then using equality $H(p,i)=H(i,p)$, we get $$\begin{aligned}
\nonumber {\bf E}\{g^{0}_{1}G_{2}(i,p)H(i,p)\}=
&K_{2}\left(H(i,p)\right){\bf
E}\left\{D^{1}_{ip}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}\\
\nonumber & + \frac{K_{4}\left(H(i,p)\right)}{6}{\bf
E}\left\{D^{3}_{ip}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}\\
\label{b.3ip} &+\frac{K_{6}\left(H(i,p)\right)}{120}{\bf
E}\left\{D^{5}_{ip}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}+
\tilde{\tilde{\epsilon}}_{ip},\end{aligned}$$ where $\tilde{\tilde{\epsilon}}_{ip}$ is given by (\[b.3tildepsilonpi\]) with replaced $D_{pi}$ by $D_{ip}$ and $K_{r}$ are the cumulants of $H(i,p)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
- If $p=i$, then $$\begin{aligned}
\nonumber {\bf E}\{g^{0}_{1}G_{2}(i,i)H(i,i)\}=
&K_{2}\left(H(i,i)\right){\bf
E}\left\{D^{1}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\}\\
\nonumber &+ \frac{K_{4}\left(H(i,i)\right)}{6}{\bf
E}\left\{D^{3}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\}\\
\label{b.3ii} &+\frac{K_{6}\left(H(i,i)\right)}{120}{\bf
E}\left\{D^{5}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\}+
\tilde{\tilde{\tilde{\epsilon}}}_{ii},\end{aligned}$$ where $\tilde{\tilde{\tilde{\epsilon}}}_{ii}$ is given by (\[b.3tildepsilonpi\]) with replaced $D_{pi}$ by $D_{ii}$ and $K_{r}$ are the cumulants of $H(i,i)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
Substituting (\[b.3pi\]), (\[b.3ip\]) and (\[b.3ii\]) into (\[b.2R12\]) and using (\[b.2partialg10G2ij\]), we obtain equality $$\begin{aligned}
\nonumber R_{12}(i)= &\zeta_{2}v^{2}{\bf
E}\left\{g^{0}_{1}G_{2}(i,i)\sum_{|p|\le{n}}G_{2}(p,p)
\frac{1}{b}\psi\left(\frac{p-i}{b}\right)\right\}\\
\nonumber &+\frac{\zeta_{2}v^{2}}{b}\sum_{|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,p)^{2}\}\psi\left(\frac{p-i}{b}\right)\\
\nonumber & +\frac{2\zeta_{2}v^{2}}{Nb}\sum_{|p|\le{n}}{\bf
E}\{G_{1}^{2}(i,p)G_{2}(i,p)\}\psi\left(\frac{p-i}{b}\right)\\
\nonumber
&-\frac{\zeta_{2}}{6}\sum_{|p|\le{n}}\frac{\Delta_{pi}}{b^{2}}{\bf
E}\left\{D^{3}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}
-\frac{\zeta_{2}\Delta_{ii}}{2b^{2}}{\bf
E}\left\{D^{3}_{ii}\left(g^{0}_{1}G_{2}(i,i)\right)\right\}\\
\label{b.3R12form1}
&-\frac{\zeta_{2}}{120}\sum_{|p|\le{n}}\frac{\theta_{pi}}{b^{3}}(1+\delta_{pi})^{3}{\bf
E}\left\{D^{5}_{pi}\left(g^{0}_{1}G_{2}(i,p)\right)\right\}+\epsilon_{i}\end{aligned}$$ with $$\begin{aligned}
\nonumber \epsilon_{i}= &-\zeta_{2}\sum_{|p|\le{n}}\frac{1}{6!}{\bf
E}\left\{H(p,i)^{7}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(0)}\right\}\\
\nonumber &+\zeta_{2}\sum_{|p|\le{n}} \frac{K_{2}}{5!}{\bf
E}\left\{H(p,i)^{5}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(1)}\right\}\\
\nonumber &+\zeta_{2}\sum_{|p|\le{n}}\frac{K_{4}}{(3!)^{2}}{\bf
E}\left\{H(p,i)^{3}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(2)}\right\}\\
\label{b.3epsiloni} &+\zeta_{2}\sum_{|p|\le{n}}\frac{K_{6}}{5!}{\bf
E}\left\{H(p,i)[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(3)}\right\},\end{aligned}$$ where $K_{r}$ are the cumulants of $H(p,i)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
Main relation for $R_{12}(i)$
-----------------------------
To give the complete description of $R_{12}$, we use the notation $$U(p,i)=\frac{1}{b}\psi\left(\frac{p-i}{b}\right), \quad
U_{G}(i)=\sum_{|p|\le{n}}G(p,p)U(p,i)$$ and introduce the identity $$\label{b.3Efg}
{\bf E}\{\xi g\}={\bf E}\{\xi\}{\bf E}\{g\}+{\bf E}\{\xi g^{0}\}.$$ Then, we rewrite the first term of the RHS of (\[b.3R12form1\]) in the form $$\begin{aligned}
\zeta_{2}v^{2}{\bf
E}& \left\{g^{0}_{1}G_{2}(i,i)\sum_{|p|\le{n}}G_{2}(p,p)U(p,i)\right\}\\
&=\zeta_{2}v^{2}R_{12}(i){\bf E}\{U_{G_{2}}(i)\}+\zeta_{2}v^{2}{\bf
E}\{g^{0}_{1}G_{2}(i,i)U^{0}_{G_{2}}(i)\}.\end{aligned}$$ Now computing the partial derivatives with the help of (\[b.2partialG\]), we obtain the following relation for $R_{12}$ $$\begin{aligned}
\nonumber R_{12}(i)= &\zeta_{2}v^{2}R_{12}(i){\bf
E}\{U_{G_{2}}(i)\}+\zeta_{2}v^{2}{\bf
E}\{g^{0}_{1}G_{2}(i,i)U^{0}_{G_{2}}(i)\} \\
\label{b.3R12form2} &+\frac{2\zeta_{2}v^{2}}{N}
\sum_{|p|\le{n}}F_{12}(i,p)U(p,i)+\frac{1}{Nb}\Upsilon_{12}(i)
+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\end{aligned}$$ with $F_{12}(i,p)={\bf E}\{G_{1}^{2}(i,p)G_{2}(i,p)\}$, $$\label{b.3Upsilon}
\Upsilon_{12}(i)=\frac{\zeta_{2}}{b}\sum_{|p|\le{n}}\Delta_{pi}{\bf
E}\left\{[G_{1}^{2}(i,i)G_{1}(p,p)+G_{1}^{2}(p,p)G_{1}(i,i)]
G_{2}(i,i)G_{2}(p,p)\right\},$$ the terms $Y_{r}(i)$, $r=1,\ldots,7$ are given by relations $$\begin{aligned}
Y_{1}(i)=&\frac{\zeta_{2}}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,i)^{2}G_{2}(p,p)^{2}\}\Delta_{pi},\\
Y_{2}(i)=&\zeta_{2}v^{2}{\bf
E}\left\{g^{0}_{1}\sum_{|p|\le{n}}G_{2}(i,p)^{2}U(p,i)\right\},\\
Y_{3}(i)=&\frac{\zeta_{2}}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{g^{0}_{1}G_{2}(i,p)^{4}
+6g^{0}_{1}G_{2}(i,p)^{2}G_{2}(i,i)G_{2}(p,p)\right\}\Delta_{pi},\\
Y_{4}(i)=&\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}^{2}(i,p)G_{2}(i,p)^{3}
+3G_{1}^{2}(i,p)G_{2}(i,p)G_{2}(i,i)G_{2}(p,p)\right\}\Delta_{pi},\end{aligned}$$
$$\begin{aligned}
Y_{5}(i)= &\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}^{2}(i,p)G_{1}(i,p)G_{2}(i,p)^{2}
+G_{1}^{2}(i,p)G_{1}(i,p)G_{2}(i,i)G_{2}(p,p)\right\}\Delta_{pi}\\
&+\frac{\zeta_{2}}{Nb^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}^{2}(i,i)G_{1}(p,p)G_{2}(i,p)^{2}
+G_{1}^{2}(p,p)G_{1}(i,i)G_{2}(i,p)^{2}\right\}\Delta_{pi}\\
&+\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}^{2}(i,p)G_{1}(i,p)^{2}G_{2}(i,p)
+G_{1}^{2}(i,i)G_{1}(p,p)G_{1}(i,p)G_{2}(i,p)\right\}\Delta_{pi}\\
&+\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}^{2}(i,p)G_{1}(p,p)G_{1}(i,i)G_{2}(i,p)\right\}\Delta_{pi}\\
&+\frac{2\zeta_{2}}{Nb^{2}}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}^{2}(p,p)G_{1}(i,i)G_{1}(i,p)G_{2}(i,p)\right\}\Delta_{pi},\\
Y_{6}(i)= &-\frac{3\zeta_{2}}{b^{2}}\left({\bf
E}\{g_{1}^{0}G_{2}(i,i)^{4}\}+\frac{1}{N}{\bf
E}\{G^{2}_{1}(i,i)G_{2}(i,i)^{3}\}\right)\Delta_{ii}\\
& -\frac{3\zeta_{2}}{Nb^{2}}{\bf
E}\left\{G^{2}_{1}(i,i)G_{1}(i,i)G_{2}(i,i)[G_{1}(i,i)+G_{2}(i,i)]
\right\}\Delta_{ii},\\
Y_{7}(i)=
&-\frac{\zeta_{2}}{120}\sum_{|p|\le{n}}\frac{\theta_{pi}}{b^{3}}
(1+\delta_{pi})^{3}{\bf
E}\left\{D^{5}_{pi}(g^{0}_{1}G_{2}(i,p))\right\}\end{aligned}$$
and $\epsilon_{i}$ given by (\[b.3epsiloni\]). The first and the second terms of the RHS of (\[b.3R12form2\]) is expressed in terms of $R_{12}$ and this finally gives a closed relation for $R_{12}$. The third and forth terms of the RHS of (\[b.3R12form2\]) give a non-zero contribution to $R_{12}$ that provide the expression of the leading term $T(z_{1},z_{2})$ (\[b.2Tz1z2\]). We will compute this contribution later (see subsection 4.3). The two last terms of (\[b.3R12form2\]) contributes with $o((Nb)^{-1})$ to (\[b.2Cnbz1z2Th\]). We formalize this proposition in the following two statements. 0,2cm
Under conditions of Theorem 2.1, the estimate $$\label{b.3Y1Y2lem}
\max_{r=1,2}\left\{\sup_{|i|\le{n}} |Y_{r}(i)|\right\}
=O\left(b^{-2}n^{-1}+b^{-2}[{\bf Var}\{g_{1}\}]^{1/2}\right).$$ is true in the limit $n,b\rightarrow\infty$ (\[b.2bnalpha\]).
0,2cm We postpone the proof of Lemma 3.1 to the next section. 0,2cm
Under conditions of Theorem 2.1, the estimate $$\label{b.3Y3Y6lem}
\max_{r=3,4,5,6,7}\left\{\sup_{|i|\le{n}} |Y_{r}(i)|\right\}
=O\left(b^{-2}n^{-1}+b^{-2}[{\bf Var}\{g_{1}\}]^{1/2}\right)$$ and $$\label{b.3epsilonlem}
\sup_{|i|\le{n}}|\epsilon_{i}|=O\left(b^{-2}n^{-1}+b^{-2}[{\bf
Var}\{g_{1}\}]^{1/2}\right)$$ are true in the limit $n,b\rightarrow\infty$ (\[b.2bnalpha\]).
0,2cm
[*Proof of Lemma 3.2.*]{} We start with (\[b.3Y3Y6lem\]). Inequality (\[b.3GijleImz\]) and (\[b.3sumGij2leImz2\]) implies that if $z_{l}\in \Lambda_{\eta}$, then $$|Y_{3}(i)|\le{\frac{7[V_{4}+3v^{4}]}{\eta^{3}b^{2}}\sum_{|p|\le{n}}{\bf
E}|g_{1}^{0}G_{2}(i,p)^{2}|}=O\left(\frac{1}{b^{2}}\{{\bf
Var}\{g_{1}\}\}^{1/2}\right).$$ To estimate $Y_{r}$, $r=4,5$, we use (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]) and inequality $$\begin{aligned}
\nonumber \sum_{|i|\le{n}}{\bf E}|G^{m}_{1}(i,p)G_{2}(i,p)|&\le{{\bf
E}\left(\sum_{|i|\le{n}}|G^{m}_{1}(i,p)|^{2}\right)^ {1/2}
\left(\sum_{|i|\le{n}}|G_{2}(i,p)|^{2}\right)^{1/2}}\\
\label{b.3sumG1ipG2ip}&\le{\frac{1}{\eta^{m+1}}}\end{aligned}$$ with $m=1,2$. Then we get that $|Y_{4}(i)|\le{8[V_{4}+v^{4}]/(\eta^{6}Nb^{2})}$. Using (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]) and (\[b.3sumG1ipG2ip\]) with $m=1,2$, we obtain that the terms $\sup_{i}|Y_{r}(i)|$, $r=5,6$ are all of the order indicated in (\[b.3Y3Y6lem\]).\
Let us estimate $Y_{7}$. Let us accept for the moment that $$\label{b.3ED5pig10G2ij}
{\bf E}|D^{5}_{pi}\{g^{0}_{1}G_{2}(i,p)\}|=O\left(N^{-1}+[{\bf
Var}\{g_{1}\}]^{1/2}\right), \ \hbox{ as } \quad
n,p\rightarrow\infty$$ holds. Using this estimate and relation (\[b.2psicondit\]), we obtain that $$\sum_{|p|\le{n}}\left|\frac{\theta_{pi}}{b}\right|\le{c\sum_{|p|\le{n}}\frac{1}{b}\psi\left(\frac{p-i}{b}\right)}=O(1)$$ and that $$\sup_{|i|\le{n}} |Y_{7}(i)|=O\left(b^{-2}n^{-1}+b^{-2}\{{\bf
Var}\{g_{1}\}\}^{1/2}\right)$$ where $c$ is a constant.
Now let use prove (\[b.3ED5pig10G2ij\]). Using (\[b.2partialG\]) and (\[b.3GijleImz\]), we get for $z_{1}\in\Lambda_{\eta}$ $$D_{pi}\{g^{0}_{1}\}=\frac{1}{N}\sum_{|t|\le{n}}D_{pi}\{G_{1}(t,t)\}
=-\frac{2}{N}G^{2}_{1}(i,p)=O\left(\frac{1}{N}\right).$$ It is easy to show that $$\label{b.3Drpig10}
D^{r}_{pi}\{g^{0}_{1}\}=O\left(\frac{1}{N}\right), \quad
r=1,2,\ldots, \ z\in\Lambda_{\eta}.$$ Then (\[b.3ED5pig10G2ij\]) follows from (\[b.3Drpig10\]) and (\[b.3GijleImz\]). Estimate (\[b.3Y3Y6lem\]) is proved.0,5cm
To proceed with estimates of $\epsilon_{i}$ (\[b.3epsilonlem\]), we use the following simple statement, proved in the previous work [@A]. 0,2cm
(see [@A]) If $z_{l}\in\Lambda_{\eta},
\ l=1,2$, under conditions of Theorem 2.1, the estimates $$\label{b.3Vargnbnu}
{\bf Var}([g_{n,b}(z_{l})]^{(\nu)})=O\left({\bf
Var}\{g_{n,b}(z_{l})\}+b^{-1}N^{-2}\right), \quad \nu=0,\ldots,3$$ and $$\label{b.3D6pig10G2ip}
D^{6}_{pi}\left\{g^{0}_{1}G_{2}(i,p)\right\}=O\left(N^{-1}+|g_{1}^{0}|\right)$$ are true in the limit $n,b\longrightarrow \infty$ (\[b.2bnalpha\]).
0,2cm
Now regarding the first term of (\[b.3epsiloni\]) and using (\[b.3Vargnbnu\]) and (\[b.3D6pig10G2ip\]), we obtain inequality $$\begin{aligned}
\nonumber &\sum_{|p|\le{n}}{\bf
E}|H(p,i)^{7}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(0)}|\le{c_{1}\sum_{|p|\le{n}}{\bf
E}\left\{\frac{|H(p,i)|^{7}}{N}+|H(p,i)|^{7}|[g^{0}_{1}]^{(0)}|\right\}}\\
\nonumber &\le{c_{1}\sum_{|p|\le{n}}\frac{\hat{\mu}_{7}}{Nb^{7/2}}
\psi\left(\frac{p-i}{b}\right)+c_{1}\sum_{|p|\le{n}}\frac{(\hat{\mu}_{14})^{1/2}}{b^{7/2}}
\left(\psi\left(\frac{p-i}{b}\right)\right)^{1/2}\left({\bf
Var}\{[g_{1}]^{(0)}\}\right)^{1/2}} \\
\label{b.3epsilonestim1} &=O\left(N^{-1}b^{-2}+b^{-2}[{\bf
Var}\{g_{1}\}]^{1/2}\right),\end{aligned}$$ where $c$ is a constant.
Regarding the last term of the right-hand side of (\[b.3epsiloni\]) and using (\[b.3Vargnbnu\]) and (\[b.3D6pig10G2ip\]), we obtain inequality $$\begin{aligned}
\nonumber &\sum_{|p|\le{n}}K_{6}{\bf
E}|H(p,i)[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(3)}|\\
\nonumber
&\le{\frac{c_{2}}{b^{2}}\left(\frac{1}{b}\sum_{|p|\le{n}}\psi\left(\frac{p-i}{b}\right)\right)
\left(\frac{\hat{\mu}_{1}\psi(\frac{p-i}{b})}{Nb^{1/2}}+\frac{\hat{\mu}^{1/2}_{2}
\left(\psi(\frac{p-i}{b})\right)^{1/2}}{b^{1/2}}\left({\bf
Var}\{[g_{1}]^{(3)}\}\right)^{1/2}\right)} \\
\label{b.3epsilonestim2} &=O\left(N^{-1}b^{-2}+b^{-2}[{\bf
Var}\{g_{1}\}]^{1/2}\right),\end{aligned}$$ where $c_{2}$ is a constant.
Repeating previous computations of (\[b.3epsilonestim2\]), we obtain that $$\begin{aligned}
\nonumber &\sum_{|p|\le{n}}K_{4}{\bf
E}|H(p,i)^{3}[D^{6}_{pi}(g^{0}_{1}G_{2}(i,p))]^{(2)}|
+\sum_{|p|\le{n}}K_{2}{\bf
E}|H(p,i)^{5}D^{6}_{pi}[g^{0}_{1}G_{2}(i,p)]^{(1)}|\\
\label{b.3epsilonestim3} &=O\left(N^{-1}b^{-2}+b^{-2}[{\bf
Var}\{g_{1}\}]^{1/2}\right).\end{aligned}$$ Then (\[b.3epsilonlem\]) follows from the estimates given by relations (\[b.3epsilonestim1\]), (\[b.3epsilonestim2\]) and (\[b.3epsilonestim3\]). Lemma 3.2 is proved. $\hfill \blacksquare$
0,2cm Let us come back to relation (\[b.3R12form2\]). Using equality $$\sum_{|p|\le{n}}{\bf E}\{g^{0}_{1}G_{2}(i,i)G^{0}_{2}(p,p)\}U(p,i) =
{\bf E}\{g^{0}_{1}U_{G_{2}}^{0}(i)G_{2}^{0}(i,i)\} +
U_{R_{12}}(i){\bf E}\{G_{2}(i,i)\},$$ we obtain the following relation $$\begin{aligned}
\nonumber R_{12}(i) = &\zeta_{2}v^{2}R_{12}(i)U_{{\bf
E}(G_{2})}(i)+\zeta_{2}v^{2}U_{R_{12}}(i){\bf E}\{G_{2}(i,i)\}\\
\nonumber
&+\frac{2\zeta_{2}v^{2}}{N}\sum_{|p|\le{n}}F_{12}(i,p)U(p,i)+
\frac{1}{Nb}\Upsilon_{12}(i)\\
\label{b.3R12form3} &+\tau(i)+\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i},\end{aligned}$$ where $F_{12}(i,p)$ is the same as in (\[b.3R12form2\]), $\Upsilon_{12}$ is given by (\[b.3Upsilon\]) and $$\label{b.3tau}
\tau(i) = \zeta_{2}v^{2}{\bf
E}\{g^{0}_{1}U_{G_{2}}^{0}(i)G_{2}^{0}(i,i) \}.$$ Relation (\[b.3R12form3\]) is the main equality used for the proof of Theorem 2.1. We use (\[b.3R12form3\]) twice : at the first stage we estimate the variance ${\bf Var}\{g_{n,b}(z)\}$ and at the second one we obtain explicit expressions for the leading term of $C_{n,b}(z_{1},z_{2})$. This will be done this in the next section. 0,5cm
Variance and leading term of $C_{n,b}(z_{1},z_{2})$
===================================================
In this section we give the estimate of the variance and the proof of Theorem 2.1, postponing some technical results to the next section. 0,5cm
Estimate of the variance
------------------------
Let us define an auxiliary variable $$\label{b.4q2i}
q_{2}(i)=\frac{\zeta_{2}}{1-\zeta_{2}v^{2}U_{g_{2}}(i)},$$ where $g_{2}(i)={\bf E}G_{2}(i,i)$. Then we can rewrite (\[b.3R12form3\]) in the form $$\begin{aligned}
\nonumber R_{12}(i)= &
v^{2}q_{2}(i)U_{R_{12}}(i)g_{2}(i)+\frac{1}{Nb}
\left(2v^{2}q_{2}(i)b[F_{12}U](i,i)
+q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)\right) \\
\label{b.4R12form4}
&+q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+\sum_{r=1}^{7}Y_{r}(i)
+\epsilon_{i}\right)\end{aligned}$$ with $$[F_{12}U](i,i)=\sum_{|p|\le{n}}F_{12}(i,p)U(p,i),$$ where $F_{12}(i,p)$ is the same as in (\[b.3R12form2\]) and $\Upsilon_{12}$ is given by (\[b.3Upsilon\]).
Now let us estimate the terms of the RHS of (\[b.4R12form4\]).Taking into account that $U(p,i)\le{b^{-1}}$ and using inequalities (\[b.3sumG1ipG2ip\]) with $m=2$, it is easy to see that if $z_{l}\in\Lambda_{\eta}$, then $$\label{b.4estim1}
\frac{1}{N}|[F_{12}U](i,i)|\le{\frac{1}{\eta^{3}Nb}}
=O(\frac{1}{Nb}).$$ Let us estimate $\Upsilon_{12}$ (\[b.3Upsilon\]). Using (\[b.3GijleImz\]) and inequality $|\Delta_{pi}|\le{[V_{4}+3v^{4}]\psi((p-i)/b)}$, we obtain that $$\label{b.4q2ileImz}
|q_{2}(i)|\le{\frac{1}{|{\mathrm{Im}}z_{2}|}}, \quad z_{2}\in\Lambda_{\eta}$$ and that $$\label{b.4estim2}
|q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)|\le{\frac{2[V_{4}+3v^{4}]}{\eta^{6}}
\sum_{|p|\le{n}}\frac{1}{b}\psi\left(\frac{p-i}{b}\right)}=O(1).$$ To estimate the term $\tau$ (\[b.3tau\]), we use the following statement. 0,2cm
Under the conditions of Theorem 2.1, the estimate $$\label{b.4taulem}
\sup_{|i|,|s|\le{n}}|{\bf E}g^{0}(z)G^{0}(i,i)U^{0}_{G}(s)| =
O\left(n^{-1}b^{-2}+b^{-2}[{\bf Var}\{g(z)\}]^{1/2}\right)$$ is true in the limit $n,b\rightarrow\infty$ (\[b.2bnalpha\]).
0,2cm We prove Lemma 4.1 in section 5. 0,2cm
It follows from results of Lemmas 3.1, 3.2 and relation (\[b.4taulem\]), that if $z_{j}\in \Lambda_{\eta}$, then $$\label{b.4estim3}
\sup_{|i|\le{n}}\left|q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+
\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right)\right|
=O\left(N^{-1}b^{-2}+b^{-2}\{{\bf Var}\{g_{1}\}\}^{1/2}\right).$$ Let us denote $r_{12} = \sup_{i}|R_{12}(i)|$. Regarding estimates (\[b.4estim1\]), (\[b.4estim2\]) and (\[b.4estim3\]), we derive from (\[b.4R12form4\]) inequality $$r_{12}\le{ \frac{v^{2}}{\eta^{2}}r_{12} + \frac{A}{bN} +
\frac{1}{b^{2}}\sqrt{r_{12}}}$$ for some constant A. Since $r_{12}$ is bounded for all $z_{l}\in
\Lambda_{\eta}$, then $r_{12}=O((Nb)^{-1}+b^{-4})$. Using condition (\[b.2bnalpha\]) and taking $z=z_{1}=\overline{z_{2}}$, one obtains that $$\label{b.4Vargnbz}
{\bf Var}\{g_{n,b}(z)\}=O\left(\frac{1}{Nb}\right).$$ Substituting (\[b.4Vargnbz\]) into (\[b.4estim3\]), we obtain that $$\label{b.4estim5}
\sup_{|i|\le{n}}\left|q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+
\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right)\right|
=o\left(\frac{1}{Nb}\right)$$ in the limit $n,b\rightarrow\infty$ (\[b.2bnalpha\]) and for all $z_{l}\in\Lambda_{\eta}$, $l=1,2$. This proves (\[b.2Cnbz1z2Th\]). 0,5cm
Leading term of the correlation function
----------------------------------------
Assuming that (\[b.4estim5\]) is true, we rewrite (\[b.4R12form4\]) in the form $$\label{b.4R12form5}
R_{12}(i)=v^{2}q_{2}(i)g_{2}(i)U_{R_{12}}(i)+\frac{1}{Nb}f_{12}(i)
+\Gamma(i)$$ with $$\label{b.4f12}
f_{12}(i)=
2v^{2}q_{2}(i)b[F_{12}U](i,i)+q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i),$$ where $F_{12}(i,p)$ is the same as in (\[b.3R12form3\]) and $\Upsilon_{12}$ is given by (\[b.3Upsilon\]). We have denoted the vanishing terms by $$\label{b.4Gamma}
\Gamma(i)=q_{2}(i)\zeta^{-1}_{2}\left(\tau(i)+
\sum_{r=1}^{7}Y_{r}(i)+\epsilon_{i}\right).$$ To obtain an explicit expression for the leading term of $C_{n,b}(z_{1},z_{2})$, it is necessary to study in detail the variables $F_{12}$ and $\Upsilon_{12}$. Let us formulate the corresponding statements and the auxiliary relations needed. Given a positive integer $L$, set $$\label{b.4BL}
B_{L}\equiv B_{L}(n,b)=\left\{i\in \Z ; \ |i|\le{n-bL}\right\}.$$
0,2cm
If $z\in\Lambda_{\eta}$, then for arbitrary positive $\epsilon$ and large enough values of $n$ and $b$ (\[b.2bnalpha\]) there exists a positive integer $L=L(\epsilon)$ such that relations $$\label{b.4supbF12U}
\sup_{i\in{B_{L}}}\left|b[F_{12}U](i,i)
-\frac{w_{2}w_{1}^{2}}{2\pi(1- v^{2}w_{1}^{2})}\int_{\R}
\frac{\tilde{\psi}(p)}{[1-
v^{2}w_{1}w_{2}\tilde{\psi}(p)]^{2}}dp\right|\le{\epsilon}$$ and $$\label{b.4supUpsilon}
\sup_{i\in{B_{L}}}\left|q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)-
\frac{2\Delta
w^{3}_{1}w^{3}_{2}}{1-v^{2}w^{2}_{1}}\right|\le{\epsilon}$$ hold for enough $n$ and $b$ satisfying (\[b.2bnalpha\]) with $\Delta$ is given by (\[b.2Delta\]).
0,2cm
The proof of Lemma 4.2 is based on the following statement formulated for the product $G_{1}G_{2}$. 0,2cm
Given positive $\epsilon$, there exists a positive integer $L=L(\epsilon)$ such that relations $$\label{b.4supEG2ii}
\sup_{i\in{B_{L}}}\left|{\bf E}\{G^{2}_{1}(i,i)\} -
\frac{w^{2}_{1}}{1 - v^{2}w^{2}_{1}}\right|\le{\epsilon},$$ $$\label{b.4supbET12U}
\sup_{i\in{B_{L}}}\left|b\sum_{|s|\le{n}}{\bf
E}\{G_{1}(i,s)G_{2}(i,s)\}U^{k}(s,i)- \frac{1}{2\pi}\int_{\R}
\frac{w_{1}w_{2}\tilde{\psi}^{k}(p)}{1-
v^{2}w_{1}w_{2}\tilde{\psi}(p)}dp\right|\le{\epsilon}$$ and $$\label{b.4supET12}
\sup_{i\in{B_{L}}}\left|\sum_{|s|\le{n}}{\bf
E}\{G_{1}(i,s)G_{2}(i,s)\} - \frac{w_{1}w_{2}}{1 -
v^{2}w_{1}w_{2}}\right|\le{\epsilon}$$ hold for enough $n$ and $b$ satisfying (\[b.2bnalpha\]) for all $k\in \N$, all $z_{j}\in \Lambda_{\eta}$, $j=1,2$.
0,2cm We postpone the proof of Lemma 4.3 to the next section. 0,2cm
Proof of Lemma 4.2 and Theorem 2.1
----------------------------------
### Proof of Lemma 4.2
We start with (\[b.4supbF12U\]). Let us consider the average $F_{12}(i,s)={\bf E}\{G^{2}_{1}(i,s)G_{2}(i,s)\}$. Applying to $G_{2}(i,s)$ the resolvent identity (\[b.2h-zI-1\]), we obtain equality $$F_{12}(i,s)=\zeta_{2}\delta_{is}{\bf
E}\{G^{2}_{1}(i,i)\}-\zeta_{2}\sum_{|p|\le{n}}{\bf
E}\{G^{2}_{1}(i,s)G_{2}(i,p)H(p,s)\}.$$ Applying formula (\[b.2EXjFX1Xm\]) to ${\bf
E}\{G^{2}_{1}(i,s)G_{2}(i,p)H(p,s)\}$ with $q=3$ and taking into account (\[b.2partialG\]), we get relation $$\begin{aligned}
\nonumber F_{12}(i,s)= &\zeta_{2}\delta_{is}{\bf
E}\{G^{2}_{1}(i,i)\}+ \zeta_{2}v^{2}[t_{12}U](i,s){\bf
E}\{G^{2}_{1}(s,s)\}\\
\nonumber &+\zeta_{2}v^{2}[F_{12}U](i,s)g_{1}(s)+
\zeta_{2}v^{2}F_{12}(i,s)U_{g_{2}}(s)\\
\label{b.4F12} &+\sum_{r=1}^{5}\beta_{r}(i,s),\end{aligned}$$ where we denoted $g_{l}(s)={\bf E}\{G_{l}(s,s)\}$, $l=1,2$, $t_{12}(i,s)={\bf E}\{G_{1}(i,s)G_{2}(i,s)\}$ and the terms $\beta_{l}$, $l=1,\ldots,5$ are given by: $$\begin{aligned}
\beta_{1}(i,s)=&\zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\{G^{2}_{1}(p,s)G_{1}(i,s)G_{2}(i,p)\}U(p,s)\\
&+\zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\{G^{2}_{1}(i,s)G_{1}(p,s)G_{2}(i,p)\}U(p,s)\\
&+\zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\{G^{2}_{1}(i,s)G_{2}(p,s)G_{2}(i,p)\}U(p,s),\\
\beta_{2}(i,s)=&\zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\{G_{1}(i,p)G_{2}(i,p)(G^{2}_{1}(s,s))^{0}\}U(p,s)\\
&+\zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\{G^{2}_{1}(i,p)G_{2}(i,p)G^{0}_{1}(s,s)\}U(p,s),\\
\beta_{3}(i,s)= &\zeta_{2}v^{2}{\bf
E}\left\{G^{2}_{1}(i,s)G_{2}(i,s)U^{0}_{G_{2}}(s)\right\},\\
\beta_{4}(i,s)= &-\frac{\zeta_{2}}{6}\sum_{|p|\le{n}}K_{4}{\bf
E}\left\{D^{3}_{ps}\left(G^{2}(i,s)G_{2}(i,p)\right)\right\},\end{aligned}$$ and $$\begin{aligned}
\beta_{5}(i,s)=&-\frac{\zeta_{2}}{4!}\sum_{|p|\le{n}}{\bf
E}\left\{H(p,s)^{5}[D^{4}_{ps}(G^{2}(i,s)G_{2}(i,p))]^{(0)}\right\}\\
&+\frac{\zeta_{2}}{3!}\sum_{|p|\le{n}}K_{2}{\bf
E}\left\{H(p,s)^{3}[D^{4}_{ps}(G^{2}(i,s)G_{2}(i,p))]^{(1)}\right\}\\
&+\frac{\zeta_{2}}{3!}\sum_{|p|\le{n}}K_{4}{\bf
E}\left\{H(p,s)[D^{4}_{ps}(G^{2}(i,s)G_{2}(i,p))]^{(2)}\right\}\end{aligned}$$ with $K_{r}$ are the cumulants of $H(p,s)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
Let us accept for the moment that $$\label{b.4betaresti}
\max_{j=1,\ldots,5}\left\{\sup_{|i|,|s|\le{n}}|
\beta_{r}(i,s)|\right\}=O\left(b^{-1}\right), \quad \hbox{ as }
\quad n,b\rightarrow\infty$$ holds for enough $n$ and $b$ satisfying (\[b.2bnalpha\]). Using them and the definition of $q_{2}(s)$ (\[b.4q2i\]), we rewrite (\[b.4F12\]) in the form $$\label{b.4F12form1}
F_{12}(i,s)=
v^{2}g_{1}(s)q_{2}(s)[F_{12}U](i,s)+R_{1}(i,s)+R_{2}(i,s)+\beta(i,s),$$ where we denoted $$\label{b.4R1}
R_{1}(i,s)=q_{2}(i){\bf E}\{G^{2}_{1}(i,i)\}\delta_{is},$$ $$\label{b.4R2}
R_{2}(i,s)=v^{2}q_{2}(s)[t_{12}U](i,s){\bf E}\{G^{2}_{1}(s,s)\}$$ and the vanishing term $$\beta(i,s)=\frac{q_{2}(s)}{\zeta_{2}}\sum_{r=1}^{5}\beta_{r}(i,s).$$ We define the linear operator $W$ that acts on the space of $N\times{N}$ matrices $F$ according to the formula $$[WF](i,s)=v^{2}g_{1}(s)q_{2}(s)\sum_{|p|\le{n}}F(i,p)U(p,s).$$ It is easy to see that if $z_{l}\in\Lambda_{\eta}$, then the estimates (\[b.3GijleImz\]) and (\[b.4q2ileImz\]) imply that $|R_{1}|\le{\eta^{-3}} $ and $|R_{2}|\le{v^{2}\eta^{-5}}$ and that $$\label{b.4Winequ}
||W||_{(1,1)}\le{\frac{v^{2}}{\eta^{2}}}<\frac{1}{2},$$ where the norm of $N\times{N}$ matrix $A$ is determined as $||A||_{(1,1)}=\sup_{i,s}|A(i,s)|$. This estimate verified by the direct computation of the norm $||WA||_{(1,1)}$ with $||A||_{(1,1)}=1$. Then (\[b.4F12form1\]) can be rewritten as $$\label{b.4F12form2}
F_{12}(i,s)=\sum_{m=0}^{\infty}\left[W^{m}\left(R_{1} + R_{2}
+\beta\right)\right](i,s).$$ The next steps of the proof of (\[b.4supbF12U\]) are very elementary. To do this, we start with the following statements, proved in the previous work [@A]. 0,2cm
(see [@A]) Given positive $\epsilon$, there exists a positive integer $L=L(\epsilon)$ such that relations $$\label{b.4supEGii}
\sup_{i\in{B_{L}}}|{\bf E}\{G(i,i;z)\}-w(z)|\le{\epsilon}, \quad
z\in\Lambda_{\eta}$$ and $$\label{b.4supq}
\sup_{i\in{B_{L}}}|q(i;z)-w(z)|\le{2\epsilon} \quad
z\in\Lambda_{\eta}$$ hold for enough $n$ and $b$ satisfying (\[b.2bnalpha\]), where $w$ and $q$ are given by (\[b.2wz\]) and (\[b.4q2i\]).
0,2cm
Now let us return to relation (\[b.4F12form2\]). We consider the first $M$ terms of the infinite series and use the decay of the matrix elements $U(i,s)=U^{(b)}(i,s)$. If one considers (\[b.4R1\]) and (\[b.4R2\]) with $i$ and $s$ taken far enough from the endpoints -$n$, $n$, then the variables $g_{1}(j)$, $q_{2}(k)$ enter into the finite series with $j$ and $k$ also far from the endpoints. Then one can use relations (\[b.4supEGii\]) and (\[b.4supq\]) and replace $g_{1}$ and $q_{2}$ by the constant values $w_{1}$ and $w_{2}$, respectively. This substitution leads to simplified expressions with error terms that vanish as $n,b\rightarrow\infty$. The second step is similar. It is to show that we can use Lemma 4.3 and replace the terms $R_{1}$ and $R_{2}$ of the finite series of (\[b.4R1\])and (\[b.4R2\]) by corresponding expressions given by formulas (\[b.4supEG2ii\]) and (\[b.4supbET12U\]).
Let us start to perform this program. Taking into account the estimate of $\beta$ (\[b.4betaresti\]) and using bounded-ness of the terms $R_{1}$ and $R_{2}$, we can deduce from (\[b.4F12form2\]) equality $$\label{b.4F12form3}
b\sum_{|s|\le{n}}F_{12}(i,s)U(s,i)=b\sum_{m=0}^{M}\left[W^{m}(R_{1}
+ R_{2}).U\right](i,i)+\kappa_{1}(i,i),$$ where $ M>0$ is such that given $\epsilon>0$ and $|\kappa_{1}(i,i)|<\epsilon$ for large enough $b$ and $N$. Now let us find such $h>0$ that the following holds $$\sup_{h\le{|t|}}\psi(t)<\epsilon \ \hbox{ and } \
\int_{h\le{|t|}}\psi(t)dt\le{\epsilon}.$$ We determine the matrix $$\hat{U}(i,p)=\left\{
\begin{array}{lll}
U(i,p) & \textrm{if} & |i-p|\le{bh}; \\
0 & \textrm{if} & |i-p|>bh
\end{array}\right.$$ and denote by $\hat{W}$ the corresponding linear operator $$[\hat{W}F](i,s)=v^{2}g_{1}(s)q_{2}(s)\sum_{|p|\le{n}}F_{12}(i,p)
\hat{U}(p,s).$$ Certainly , $\hat{W}$ admits the same estimate as $W$ (\[b.4Winequ\]). Given $\epsilon>0$ and $L>0$ the large number. Let us denote by $Q$ the first natural greater than $(M+k)h$. Then one can write that $$\label{b.4F12form4}
b\sum_{m=0}^{M}\left[W^{m}(R_{1} +
R_{2}).U\right](i,i)=b\sum_{m=0}^{M}\left[\hat{W}^{m}(R_{1} +
R_{2})\hat{U}\right](i,i) +\kappa_{2}(i,i),$$ where $$\label{b.4supkappa2}
\sup_{i\in{B_{L+Q}}}|\kappa_{2}(i,i)|\le{\epsilon} , \quad \hbox{ as
} \ n,b \longrightarrow{\infty}.$$ The proof of (\[b.4supkappa2\]) uses elementary computations. Indeed, $\kappa_{2}(i,i)$ is represented as the sum of $M+1$ terms of the form $$b\sum_{|s_{r}|\le{n}}^{*}\nu^{2m}g_{1}(s_{1})q_{2}(s_{1})\ldots,g_{1}(s_{m})q_{2}
(s_{m})[R_{1}+R_{2}](i,s_{m+1})$$ $$\times U(s_{m+1},s_{m})\ldots U(s_{1},i),$$ where the sum is taken over the values of $s_{j}$ such that $|s_{j}-s_{j+1}|>bh$ at least for one of the numbers $j\le{m}$.
Now remembering the a priori bounds for $R_{1}$ (\[b.4R1\]) and $R_{2}$ (\[b.4R2\]), one obtains the following estimate of $\kappa_{2}$: $$\begin{aligned}
\nonumber
\sup_{|i|\le{n}}|\kappa_{2}(i,i)|\le&{\sum_{m=0}^{M}\frac{v^{2m}}
{\eta^{2m+3}}\sum_{|s_{r}|\le{n}}^{*}bU(i,s_{1})\ldots
U(s_{m},s_{m+1})} \\
\label{b.4supDelta2estim1}
&+\sum_{m=0}^{M}\frac{v^{2m+2}}{\eta^{2m+5}}
\sum_{|s_{r}|\le{n}}^{*}bU(i,s_{1})\ldots U(s_{m},s_{m+1}).\end{aligned}$$ Assuming that $|s_{j}-s_{j+1}|>bh$ and using inequality $$\begin{aligned}
\sum_{|s_{i}|\le{n}}U(i,s_{1})\ldots U(s_{j-1},s_{j})&\le{
\sum_{s_{i}\in \Z}U(i,s_{1})\ldots U(s_{j-1},s_{j})}\\
\nonumber &\le{\left[\int_{-\infty}^{+\infty}\psi(t)dt
+\frac{\psi(0)}{b}\right]^{j}}\\
\label{b.4sums-r} &\le{(1+1/b)^{j}},\end{aligned}$$ one sees that for large enough $b$ and $n$, $$\sum_{|s_{j}|\le{n}}U^{j}(i,s_{j})\epsilon
U^{m-j}(s_{j+1},s_{m+1})\le{\epsilon}.$$ Let us also mention here that given $\epsilon >0$, one has large enough $n$ and $b$ that $$\label{b.4supUj-1}
\sup_{i\in B_{L+Q}}|\sum_{|s|\le{n}}U^{j}(i,s)-1|\le{\epsilon},$$ where $j\le{M}$. This follows from elementary computations related with the differences $$\label{b.4Pb}
P_{b}=\frac{1}{b}\sum_{t\in
\Z}\psi\left(\frac{t}{b}\right)-\int_{\R}\psi(s)ds$$ and $$\label{b.4Tnb}
T_{n,b}(i)\equiv
T(i)=\frac{1}{b}\sum_{|t|\le{n}}\psi\left(\frac{t-i}{b}\right)-
\frac{1}{b}\sum_{t\in \Z}\psi\left(\frac{t}{b}\right).$$ that vanish in the limit $1\ll b\ll n$ (see previous work [@A] for more details).
This reasoning when slightly modified is used to estimate the second term in the RHS of (\[b.4supDelta2estim1\]). Now one can write that $$\sup_{|i|\le{n}}|\kappa_{2}(i,i)|\le{2\epsilon\sum_{m=0}^{M}m
\left[\frac{v^{2}}{\eta^{2}}\right]^{m}}\le{\epsilon}.$$ Regarding the RHS of (\[b.4F12form4\]) with $i\in B_{L+Q}$, one observes that the summations run over such values of $s_{r}$ that $|i-s_{1}|\le{bh}$, $|s_{r}-s_{r+1}|\le{bh}$, and thus $s_{j}\in
B_{L}$ for all $j\le{k+m-1}$. This means that we can apply relations (\[b.4supEGii\]) and (\[b.4supq\]) to the RHS of (\[b.4F12form4\]) and to replace $g_{1}$ by $w_{1}$, $q_{2}$ by $w_{2}$. From (\[b.4F12form3\]), it follows that $$\begin{aligned}
b[F_{12}U](i,i)=&\sum_{m=0}^{M}[v^{2}w_{1}w_{2}]^{m} \
b\sum_{|s_{m+1}|\le{n}}\left(R_{1}(i,s_{m+1})+R_{2}(i,s_{m+1})\right)
\hat{U}^{m+1}(s_{m+1},i)\\
&+\kappa_{3}(i,i)\end{aligned}$$ with $$\sup_{i\in B_{L+Q}}|\kappa_{3}(i,i)|\le{4\epsilon}.$$ Finally, applying Lemma 4.3 to the expressions involved in $R_{l}$ and taking into account that $$\label{b.4bUm+1}
\sup_{i\in
B_{L+Q}}|bU^{m+1}(i,i)-\frac{1}{2\pi}\int_{\R}\tilde{\psi}^
{m+1}(p)dp|\le{\epsilon},$$ we obtain equality $$\begin{aligned}
\nonumber
b[F_{12}U](i,i)=&\frac{1}{2\pi}\frac{w^{2}_{1}w_{2}}{1-v^{2}w^{2}_{1}}
\sum_{m=0}^{M}[v^{2}w_{1}w_{2}]^{m}\int_{\R}\tilde{\psi}^{m+1}(p)dp\\
\label{b.4F12form5}
&+\frac{1}{2\pi}\frac{w^{2}_{1}w_{2}}{1-v^{2}w^{2}_{1}}
\sum_{m=0}^{M}[v^{2}w_{1}w_{2}]^{m}v^{2}\int_{\R}
\frac{w_{1}w_{2}\tilde{\psi}^{m+1}(p)}{1-v^{2}w_{1}w_{2}
\tilde{\psi}(p)}dp+ \kappa_{4}(i,i)\end{aligned}$$ with $$\sup_{i\in B_{L+Q}}|\kappa_{4}(i,i)|\le{\epsilon}.$$ Passing back in (\[b.4F12form5\]) to the infinite series and simplifying them, we arrive at the expression standing in the RHS of (\[b.4supbF12U\]). Relation (\[b.4supbF12U\]) is proved.
0,5cm
Now let us prove (\[b.4betaresti\]). Inequality $U(p,s)\le{b^{-1}}$, (\[b.3GijleImz\]) and (\[b.3sumG1ipG2ip\]) imply that if $z_{l}\in\Lambda_{\eta}$, the estimate $$\label{b.4beta12}
\max_{r=1,2}\left\{\sup_{|i|,|s|\le{n}}|\beta_{r}(i,s)|\right\}
=O(b^{-1})$$ holds for enough $n$ and $b$ satisfying (\[b.2bnalpha\]). To estimate $\beta_{3}$, we use the following estimate of the diagonal elements of the resolvent $G$, proved in the previous work [@A]. 0,2cm
(see [@A]) If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimate $$\label{b.4supEUG02}
\sup_{|s|\le{n}}{\bf E}\{|U^{0}_{G}(s;z)|^{2}\}=O(b^{-2})$$ holds for enough $n$ and $b$ satisfying (\[b.2bnalpha\]).
0,2cm
Then inequality (\[b.3GijleImz\]) and estimate (\[b.4supEUG02\]), imply that $$\label{b.4beta3}
\sup_{|i|,|s|\le{n}}|\beta_{3}(i,s)|=O(b^{-1}),\quad
z_{1},z_{2}\in\Lambda_{\eta} \quad \hbox { as } \
n,b\rightarrow\infty.$$ Using inequality $$\label{b.4K4estim}
|K_{4}\left(H(p,s)\right)|\le{\frac{4|\Delta_{ps}|}{b^{2}}}
\le{\frac{4[V_{4}+3v^{4}]}{b^{2}}\psi\left(\frac{p-s}{b}\right)}$$ and relations (\[b.3GijleImz\]) and (\[b.2partialG\]), we obtain that $$|{\bf
E}\left\{D^{3}_{ps}\left(G^{2}(i,s)G_{2}(i,p)\right)\right\}|=O(1),
\quad \hbox { as } \ n,b\rightarrow\infty$$ and conclude that $$\label{b.4beta4}
\sup_{|i|,|s|\le{n}}|\beta_{4}(i,s)|=O(b^{-1}),\quad
z_{1},z_{2}\in\Lambda_{\eta} \quad \hbox { as } \
n,b\rightarrow\infty.$$ Regarding the term $\beta_{5}$ and using similar arguments as those to the proof of (\[b.3epsilonlem\]) (see (\[b.3epsilonestim1\])-(\[b.3epsilonestim2\])), we conclude that $$\label{b.4beta5}
\sup_{|i|,|s|\le{n}}|\beta_{5}(i,s)|=O(b^{-1}),\quad
z_{1},z_{2}\in\Lambda_{\eta} \quad \hbox { as } \
n,b\rightarrow\infty.$$ Now (\[b.4betaresti\]) follows from (\[b.4beta12\]), (\[b.4beta3\]), (\[b.4beta4\]) and (\[b.4beta5\]).
0,5cm
To complete the proof of Lemma 4.2, let us prove (\[b.4supUpsilon\]). To do this we use the following simple statement, proved in the previous work [@A]. 0,2cm
(see [@A]) If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimate $$\label{b.4supEGss2}
\sup_{|s|\le{n}}{\bf E}\{|G(s,s;z)^{0}|^{2}\}=O(b^{-1})$$ holds for enough $n$ and $b$ satisfying (\[b.2bnalpha\]).
0,2cm
We introduce the variable $M_{12}(i)=q_{2}(i)\zeta^{-1}_{2}\Upsilon_{12}(i)$ with $\Upsilon_{12}$ is given by (\[b.3Upsilon\]). Using identity (\[b.3Efg\]) and estimate (\[b.4supEGss2\]), we obtain that $$\begin{aligned}
\nonumber M_{12}(i)= &q_{2}(i)g_{2}(i){\bf
E}\{G^{2}_{1}(i,i)\}\sum_{|p|\le{n}}\frac{\Delta_{pi}}{b}g_{1}(p)g_{2}(p)\\
\label{b.4Mesti}
&+q_{2}(i)g_{1}(i)g_{2}(i)\sum_{|p|\le{n}}\frac{\Delta_{pi}}{b}{\bf
E}\{G^{2}_{1}(p,p)\}g_{2}(p)+o(1), \quad \hbox{ as } \
n,b\rightarrow\infty.\end{aligned}$$ If one considers (\[b.4Mesti\]) with $i$ taken far enough from the endpoints $-n$, $n$, then one can use relation (\[b.4supEGii\]) and (\[b.4supq\]) and replace $g_{1}$, $g_{2}$ and $q_{2}$ by the constant values $w_{1}$ and $w_{2}$. This substitution leads to simplified expressions with error terms that vanish as $n,b\rightarrow \infty$. To finish the proof, we use relation (\[b.2psicondit\]) and Lemma 4.3 and replace the terms $\sum_{p}\Delta_{pi}/b$ and $G^{2}_{1}$ of $M_{12}$ by the corresponding expressions given by relations (\[b.2Delta\]) and (\[b.4supEG2ii\]). This proves (\[b.4supUpsilon\]). Lemma 4.2 is proved. $\hfill \blacksquare$0,5cm
### Proof of Theorem 2.1
Let us return to relation (\[b.4R12form5\]). We introduce the linear operator $W^{(g_{2},q_{2})}$ acting on vectors $e\in \C^{N}$ with components $e(i)$ as follows; $$\{W^{(g_{2},q_{2})}(e)\}(i)=
v^{2}g_{2}(i)q_{2}(i)\sum_{|p|\le{n}}e(p)U(p,i).$$ As a matter of fact, we can rewrite (\[b.4R12form5\]) in the following form: $$\label{b.4R12form6}
[I-W^{(g_{2},q_{2})}](R_{12})(i)=\frac{1}{Nb}f_{12}(i)+\Gamma(i),$$ where $f_{12}$ and $\Gamma$ are given by (\[b.4f12\]) and (\[b.4Gamma\]). It is easy to see that if $z\in\Lambda_{\eta}$, then inequalities (\[b.3GijleImz\]) and (\[b.4q2ileImz\]) imply that $$||W^{(g_{2},q_{2})}||_{1}\le{\frac{v^{2}}{(2v+1)^{2}}}<{\frac{1}{2}},$$ where $||W^{(g_{2},q_{2})}||_{1}=\sup_{|V|_{1}=1}|W^{(g_{2},q_{2})}(V)|_{1}$ and $|V|_{1}=\sup_{i}|V(i)|$. Then (\[b.4R12form6\]) can be rewritten in the form $$R_{12}(i)=\frac{1}{Nb}\sum_{m=0}^{\infty}
\left([W^{(g_{2},q_{2})}]^{m}\vec{f}_{12}\right)(i)
+o\left(\frac{1}{Nb}\right).$$ Regarding the trace $$\frac{1}{N}\sum_{|i|\le{n}}R_{12}(i)=\frac{1}{N}\sum_{i\in
B_{L}}R_{12}(i)+\frac{2bL}{N}O\left(\frac{1}{Nb}\right) =
\frac{1}{N}\sum_{i\in B_{L}}R_{12}(i) + o\left(\frac{1}{Nb}\right)$$ and repeating the same arguments of the proof of (\[b.4supbF12U\]) presented above, we can write that $$R_{12}(i)=\frac{1}{Nb}
\sum_{m=0}^{M}\sum_{|t|\le{n}}f_{12}(t)(v^{2}w^{2}_{2}U)^{m}(t,i)
+\frac{1}{Nb} \Delta^{(2)}(i)$$ with $\sup_{i\in B_{L}}|\Delta^{(2)}(i)|=o(1)$. Finally, observing that $f_{12}(t)$ asymptotically does not depend on $t$ (see Lemma 4.2), we arrive with the help of (\[b.4supUj-1\]), at the expression (\[b.2Tz1z2\]). Theorem 2.1 is proved. $\hfill
\blacksquare$
Proof of auxiliary statement
============================
The main goal of this section is to prove Lemmas 3.1, 4.1 and 4.3.
Proof of Lemma 3.1
------------------
### Estimate of the term $Y_{1}$ (\[b.3R12form2\])
Here we have to use the resolvent identity (\[b.2h-zI-1\]) and the cumulants expansion formula (\[b.2EXjFX1Xm\]) twice. However, the computations are based on the same inequalities as those of the proofs of Lemma 3.2. Regarding $Y_{1}(i)=\zeta_{2}b^{-2}\sum_{p}{\bf
E}\{g^{0}_{1}G_{2}(i,i)^{2}G_{2}(p,p)^{2}\}\Delta_{pi}$, we apply to $G_{2}(i,i)$ the resolvent identity (\[b.2h-zI-1\]). Then we get relation $$\begin{aligned}
Y_{1}(i)=&{\zeta^{2}_{2}}{b^{2}}\sum_{|p|\le{n}}{\bf E}\{g_{1}^{0}
G_{2}(i,i)G_{2}(p,p)^{2}\}\Delta_{pi}\\
&-\frac{\zeta^{2}_{2}}{b^{2}}\sum_{|s|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2}H(s,i)\}\Delta_{pi}.\end{aligned}$$ Applying the formula (\[b.2EXjFX1Xm\]) with $q=3$ to ${\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2}H(s,i)\}$, we obtain that $$\begin{aligned}
\nonumber Y_{1}(i)=&\frac{\zeta^{2}_{2}}{b^{2}}\sum_{|p|\le{n}}{\bf
E}(g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2})\Delta_{pi}\\
\label{b.5Y1form1}
&+\frac{\zeta^{2}_{2}v^{2}}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)^{2}G_{2}(p,p)^{2}U_{G_{2}}(i)\}\Delta_{pi}+
\sum_{r=1}^{3}Q_{r}(i)\end{aligned}$$ with $$\begin{aligned}
Q_{1}(i)=&\frac{2\zeta^{2}_{2}v^{2}}{Nb^{2}}\sum_{|s|,|p|\le{n}}{\bf
E}\{G_{1}^{2}(i,s)G_{2}(i,s)G_{2}(i,i)G_{2}(p,p)^{2}\}U(s,i)\Delta_{pi}\\
&+\frac{3\zeta^{2}_{2}v^{2}}{b^{2}}\sum_{|s|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}G_{2}(i,s)^{2}\}U(s,i)\Delta_{pi}\\
&+\frac{4\zeta^{2}_{2}v^{2}}{b^{2}}\sum_{|s|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)G_{2}(i,s)
G_{2}(i,p)G_{2}(p,s)\}U(s,i)\Delta_{pi},\\
Q_{2}(i)=&-\frac{\zeta^{2}_{2}}{b^{2}}\sum_{|s|,|p|\le{n}}\frac{K_{4}}{6}
{\bf
E}\left\{D_{si}^{3}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})
\right\}\Delta_{pi}\end{aligned}$$ and $$\begin{aligned}
Q_{3}(i)=&-\frac{\zeta^{2}_{2}}{b^{2}4!}\sum_{|s|,|p|\le{n}}{\bf
E}\left\{H(s,i)^{5}[D_{si}^{4}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)
G_{2}(p,p)^{2})]^{(0)}\right\}\Delta_{pi}\\
&+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\le{n}}K_{2}{\bf
E}\left\{H(s,i)^{3}[D_{si}^{4}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)
G_{2}(p,p)^{2})]^{(1)}\right\}\Delta_{pi}\\
&+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\le{n}}K_{4}{\bf
E}\left\{H(s,i)[D_{si}^{4}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)
G_{2}(p,p)^{2})]^{(2)}\right\}\Delta_{pi},\end{aligned}$$ where $K_{r}$, $r=2,4$ are the cumulants of $H(s,i)$ as in (\[b.3K2K4\]). Applying to the second term of the RHS of (\[b.5Y1form1\]) identity (\[b.3Efg\]) and using the definition of $q_{2}(i)$ (\[b.4q2i\]), we obtain that $$\begin{aligned}
\nonumber
Y_{1}(i)=&\frac{\zeta_{2}q_{2}(i)}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}\}\Delta_{pi}\\
\nonumber &+\frac{\zeta_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)^{2}G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\}\Delta_{pi}\\
\label{b.5Y1form2}
&+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}Q_{r}(i).\end{aligned}$$ Regarding the first term of the RHS of this equality, we apply the resolvent identity (\[b.2h-zI-1\]) to $G_{2}(i,i)$. Repeating the usual computations based on the formula (\[b.2EXjFX1Xm\]) (with $q=3$) and relation (\[b.2partialG\]), we obtain that $$\frac{\zeta_{2}q_{2}(i)}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}\}\Delta_{pi}
=\frac{\zeta_{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(p,p)^{2}\}\Delta_{pi}$$ $$\label{b.5Y1form3}
+\frac{\zeta_{2}v^{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\}\Delta_{pi}
+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}\breve{Q}_{r}(i)$$ with $$\begin{aligned}
\breve{Q}_{1}(i)=
&\frac{2\zeta^{2}_{2}v^{2}q_{2}(i)}{Nb^{2}}\sum_{|s|,|p|\le{n}} {\bf
E}\{G_{1}^{2}(i,s)G_{2}(i,s)G_{2}(p,p)^{2}\}U(s,i)\Delta_{pi}\\
&+\frac{\zeta^{2}_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|s|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(p,p)^{2}G_{2}(i,s)^{2}\}U(s,i)\Delta_{pi}\\
&+\frac{4\zeta^{2}_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|s|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(p,p)G_{2}(i,s)
G_{2}(i,p)G_{2}(p,s)\}U(s,i)\Delta_{pi},\\
\breve{Q}_{2}(i)=&-\frac{\zeta^{2}_{2}q_{2}(i)}{b^{2}}\sum_{|s|,|p|\le{n}}\frac{K_{4}}{6}
{\bf E}\left\{D_{si}^{3}(g^{0}_{1}G_{2}(i,s)G_{2}(p,p)^{2})
\right\}\Delta_{pi}\end{aligned}$$ and $$\begin{aligned}
\breve{Q}_{3}(i)=
&-\frac{\zeta^{2}_{2}}{b^{2}4!}\sum_{|s|,|p|\le{n}}{\bf
E}\left\{H(s,i)^{5}[D_{si}^{4}(g^{0}_{1}G_{2}(i,s)
G_{2}(p,p)^{2})]^{(0)}\right\}\Delta_{pi}\\
&+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\le{n}}K_{2}{\bf
E}\left\{H(s,i)^{3}[D_{si}^{4}(g^{0}_{1}G_{2}(i,s)
G_{2}(p,p)^{2})]^{(1)}\right\}\Delta_{pi}\\
&+\frac{\zeta^{2}_{2}}{b^{2}3!}\sum_{|s|,|p|\le{n}}K_{4}{\bf
E}\left\{H(s,i)[D_{si}^{4}(g^{0}_{1}G_{2}(i,s)
G_{2}(p,p)^{2})]^{(2)}\right\}\Delta_{pi},\end{aligned}$$ where $K_{r}$, $r=2,4$ are the cumulants of $H(s,i)$ as in (\[b.3K2K4\]).
Substituting (\[b.5Y1form3\]) into (\[b.5Y1form2\]), we obtain that $$\begin{aligned}
\nonumber
Y_{1}(i)=&\frac{\zeta_{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(p,p)^{2}\}\Delta_{pi}\\
\nonumber
&+\frac{\zeta_{2}v^{2}}{b^{2}}q^{2}_{2}(i)\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\}\Delta_{pi}\\
\nonumber &+\frac{\zeta_{2}v^{2}q_{2}(i)}{b^{2}}\sum_{|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,i)^{2}G_{2}(p,p)^{2}U^{0}_{G_{2}}(i)\}\Delta_{pi}\\
\label{b.5Y1form4}
&+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}Q_{r}(i)
+\frac{q_{2}(i)}{\zeta_{2}}\sum_{r=1}^{3}\breve{Q}_{r}(i).\end{aligned}$$ Now let us estimate each term of the RHS of this equality. If one assumes for a while that $$\label{b.5Y1estim1}
\sup_{|p|\le{n}}|{\bf E}\{g_{1}^{0}G_{2}(p,p)^{2}\}
|=O\left(N^{-1}b^{-1}+b^{-1}[{\bf Var}\{g_{1}\}]^{1/2}\right)$$ holds for enough $n$ and $b$ satisfying (\[b.2bnalpha\]). Then this estimate and relations (\[b.4q2ileImz\]), (\[b.4K4estim\]) and (\[b.4supEUG02\]) imply that the fist, the second and the third terms of the RHS of (\[b.5Y1form4\]) are of the order indicated in the RHS of (\[b.3Y1Y2lem\]).
Inequality (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]), (\[b.3sumG1ipG2ip\]) (with $m=1$ and $m=2$) and (\[b.4q2ileImz\]) imply that the term $q_{2}(i)\zeta^{-1}_{2}[Q_{1}(i)+\breve{Q}_{1}(i)]$ is of the order indicated in the RHS of (\[b.3Y1Y2lem\]). Using similar arguments as those of the proof of (\[b.3epsilonlem\]) (see (\[b.3epsilonestim1\])-(\[b.3epsilonestim3\])) and the following estimates (cf. (\[b.3Vargnbnu\])-(\[b.3D6pig10G2ip\])) $$D_{si}^{r}(g^{0}_{1}G_{2}(i,i)G_{2}(i,s)G_{2}(p,p)^{2})=O\left(N^{-1}+|g_{1}^{0}|\right),
\quad r=3,4$$ and $${\bf Var}\{[g_{n,b}(z_{l})]^{(\nu)}\}=O\left({\bf
Var}\{g_{n,b}(z_{l})\}+b^{-1}N^{-2}\right), \quad \nu=0,1,2,$$ we obtain that the terms $Q_{r}$, $r=2,3$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). We conclude that the terms $\breve{Q}_{r}$, $r=2,3$ and $\sup_{i}|Y_{1}(i)|$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). 0,5cm
Now let us prove (\[b.5Y1estim1\]). Let us apply the resolvent identity (\[b.2h-zI-1\]) to $G_{2}(p,p)$. Repeating the usual computations based on the formula (\[b.2EXjFX1Xm\]) (with $q=3$) and relation (\[b.2partialG\]), we obtain that $$\begin{aligned}
\nonumber {\bf E}\{g_{1}^{0}G_{2}(p,p)^{2}\}= &q_{2}(p){\bf
E}\{g_{1}^{0}G_{2}(p,p)\}+q_{2}(p){\bf
E}\{g_{1}^{0}G_{2}(p,p)^{2}U^{0}_{G_{2}}(p)\}\\
\nonumber &+3q_{2}(p)v^{2}\sum_{|s|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(p,s)^{2}G_{2}(p,p)\}U(s,p)\\
\nonumber &+\frac{2v^{2}}{N}q_{2}(p)\sum_{|s|\le{n}}{\bf
E}\{G^{2}_{1}(s,p)G_{2}(s,p)G_{2}(p,p)\}U(s,p) \\
\label{b.5Y1estim2} &-\frac{q_{2}(p)}{6}\sum_{|s|\le{n}}K_{4}{\bf
E}\left\{D_{sp}^{3}(g^{0}_{1}G_{2}(p,p)G_{2}(p,s))\right\}-q_{2}(p)
\tilde{Q}(i)\end{aligned}$$ with $$\begin{aligned}
\tilde{Q}(i)=&-\frac{1}{4!}\sum_{|s|\le{n}}{\bf
E}\left\{H(s,p)^{5}[D_{sp}^{4}(g^{0}_{1}G_{2}(p,p)
G_{2}(p,s))]^{(0)}\right\}\\
&+\frac{1}{3!}\sum_{|s|\le{n}}K_{2}{\bf
E}\left\{H(s,p)^{3}[D_{sp}^{4}(g^{0}_{1}G_{2}(p,p)
G_{2}(p,s))]^{(1)}\right\}\\
&+\frac{1}{3!}\sum_{|s|\le{n}}K_{4}{\bf
E}\left\{H(s,p)[D_{sp}^{4}(g^{0}_{1}G_{2}(p,p)
G_{2}(p,s))]^{(2)}\right\},\end{aligned}$$ where $K_{r}$, $r=2,4$ are the cumulants of $H(s,p)$ as in (\[b.3K2K4\]).
Let us estimate each term of the RHS of (\[b.5Y1estim2\]). It is easy to show that the estimate of the first term of the RHS of (\[b.5Y1estim2\]) follows from the following statement, proved in the previous work [@A]. 0,2cm
(see [@A]) If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimate $$\label{b.5supEg10G2pp}
\sup_{|p|\le{n}}|{\bf
E}\{g_{1}^{0}G_{2}(p,p)\}|=O\left(b^{-1}n^{-1}+b^{-1}[{\bf
Var}\{g_{1}\}]^{1/2}\right)$$ holds in the limit $n,b\rightarrow\infty$ (\[b.2bnalpha\]).
0,2cm
Then (\[b.5Y1estim1\]) follows from this Lemma.and the estimate (\[b.4supEUG02\]) and the similar arguments used in the estimates of the terms $Q_{r}$, $r=1,2,3$ in (\[b.5Y1form4\]). Estimate (\[b.5Y1estim1\]) is proved. 0,2cm
### Estimate of $Y_{2}$ (\[b.3R12form2\])
We rewrite $Y_{2}$ in the form $Y_{2}(i)=\zeta_{2}v^{2}\sum_{s}{\bf
E}\{M(i,s)\}U(s,i)$, where we denoted $${\bf E}\{M(i,s)\}={\bf E}\{g^{0}_{1}G_{2}(i,s)^{2}\}.$$ To proceed with estimate of $Y_{2}$, we use the resolvent identity (\[b.2h-zI-1\]) and the cumulants expansion formula (\[b.2EXjFX1Xm\]) twice. However, the computations are based on the results of Lemma 4.5, 4.6 and 5.1. Therefore we just indicate the main lines of the proof and do not go into the details. Applying to $G_{2}(i,s)$ the resolvent identity (\[b.2h-zI-1\]), we get equality $$\label{b.5Y2form1}
{\bf E}M(i,s)=\zeta_{2}\delta_{is}{\bf E}\{g^{0}_{1}G_{2}(i,i)\}
-\zeta_{2}\sum_{|t|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)H(t,s)\}.$$ Regarding the first term of the RHS of this equality and using relation (\[b.5supEg10G2pp\]), it is easy to see that the term $$\sum_{|s|\le{n}}\zeta_{2}\delta_{is}{\bf
E}\{g^{0}_{1}G_{2}(i,i)\}U(s,i)=\zeta_{2}\frac{\psi(0)}{b}{\bf
E}\{g^{0}_{1}G_{2}(i,i)\}$$ is the value of order indicated in (\[b.3Y1Y2lem\]). Let us consider the second term of (\[b.5Y2form1\]). Applying formula (\[b.2EXjFX1Xm\]) with $q=5$ to ${\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)H(t,s)\}$ and taking account relations (\[b.2partialG\]) and (\[b.3Efg\]), we obtain that $$\label{b.5Y2form2}
-\zeta_{2}\sum_{|t|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)H(t,s)\}
=\sum_{l=1}^{7}\Theta_{l}(i,s),$$ where $$\begin{aligned}
\Theta_{1}(i,s)=&v^{2}\zeta_{2}{\bf
E}\{g^{0}_{1}G_{2}(i,s)^{2}\}{\bf E}U_{G_{2}}(s),\\
\Theta_{2}(i,s)=&v^{2}\zeta_{2}{\bf
E}\{g^{0}_{1}G_{2}(i,s)^{2}U^{0}_{G_{2}}(s)\},\\
\Theta_{3}(i,s)=&\frac{2v^{2}\zeta_{2}}{N}\sum_{|t|\le{n}}{\bf
E}\{G^{2}_{1}(s,t)U(t,s)G_{2}(i,s)G_{2}(i,t)\},\\
\Theta_{4}(i,s)=&v^{2}\zeta_{2}\sum_{|t|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,s)\}U(t,s),\\
\Theta_{5}(i,s)=&2v^{2}\zeta_{2}\sum_{|t|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(t,s)G_{2}(i,t)\}U(t,s),\\
\Theta_{6}(i,s)=&-\zeta_{2}\sum_{|t|\le{n}}\frac{K_{4}\left(H(t,s)\right)}{6}
{\bf E}\{D^{3}_{ts}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))\}\end{aligned}$$ and $$\Theta_{7}(i,s)=-\zeta_{2}\sum_{|t|\le{n}}\frac{K_{6}\left(H(t,s)\right)}{5!}{\bf
E}\{D^{5}_{ts}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))\}+\tilde{\Theta}_{7}(i,s)$$ with $$\begin{aligned}
\tilde{\Theta}_{7}(i,s)=&-\frac{\zeta_{2}}{6!}\sum_{|t|\le{n}}{\bf
E}\left\{H(t,s)^{7}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(0)}
\right\}\\
&+\frac{\zeta_{2}}{5!}\sum_{|t|\le{n}}K_{2}\left(H(t,s)\right){\bf
E}\left\{H(t,s)^{5}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(1)}
\right\}\\
&+\frac{\zeta_{2}}{(3!)^{2}}\sum_{|t|\le{n}}K_{4}\left(H(t,s)\right){\bf
E}\left\{H(t,s)^{3}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(2)}
\right\}\\
&+\frac{\zeta_{2}}{5!}\sum_{|t|\le{n}}K_{6}\left(H(t,s)\right){\bf
E}\left\{H(t,s)[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t))]^{(3)}
\right\},\end{aligned}$$ where $K_{r}\left(H(t,s)\right)$, $r=2,4,6$ are the cumulants of $H(t,s)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
The term $\Theta_{1}$ is of the form $v^{2}\zeta_{2}{\bf
E}\{M(i,s)\}{\bf E}U_{G_{2}}(s)$ and can be put to the left hand side of (\[b.5Y2form1\]). The terms $\Theta_{2}$ and $\Theta_{3}$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). This can be shown with the help of the estimate (\[b.4supEUG02\]) and inequality (eg. [@A]) $$\begin{aligned}
\nonumber &\left|\sum_{|s|,|t|\le{n}}G^{2}_{1}(s,t)G_{2}(i,s)G_{2}(i,t)\right| \\
\label{b.5sumstG1G2}
&\le{||G_{1}^{2}||\left(\sum_{|s|\le{n}}|G_{2}(i,s)|^{2}\right)^{1/2}
\left(\sum_{|t|\le{n}}|G_{2}(i,t)|^{2}\right)^{1/2}}
\le{\frac{1}{\eta^{4}}}.\end{aligned}$$ Regarding $\Theta_{4}$, we apply the resolvent identity (\[b.2h-zI-1\]) to the factor $G_{2}(s,s)$. Repeating the usual computations based on the formula (\[b.2EXjFX1Xm\]) with $q=5$ and taking into account relations (\[b.2partialG\]) and (\[b.3Efg\]), we obtain that $$\label{b.5Theta4}
\Theta_{4}(i,s)=v^{2}\zeta^{2}_{2}\sum_{|t|\le{n}}{\bf
E}\{M(i,t)\}U(t,s) +v^{2}\zeta_{2}\Theta_{4}(i,s){\bf
E}U_{G_{2}}(s)+\sum_{l=1}^{8}\Omega_{l}(i,s),$$ where $$\begin{aligned}
\Omega_{1}(i,s)=&v^{4}\zeta^{2}_{2}\sum_{|t|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,s)U^{0}_{G_{2}}(s)\}U(t,s),\\
\Omega_{2}(i,s)=&v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,t)^{2}G_{2}(s,p)^{2}\}U(p,s)U(t,s),\\
\Omega_{3}(i,s)=&\frac{2v^{4}\zeta^{2}_{2}}{N}\sum_{|t|,|p|\le{n}}{\bf
E}\{G^{2}_{1}(p,s)G_{2}(i,t)^{2}G_{2}(s,p)\}U(p,s)U(t,s),\\
\Omega_{4}(i,s)=&2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,t)G_{2}(i,s)G_{2}(p,t)G_{2}(s,p)\}U(p,s)U(t,s),\\
\Omega_{5}(i,s)=&2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,t)G_{2}(i,p)G_{2}(s,t)G_{2}(s,p)\}U(p,s)U(t,s),\\
\Omega_{6}(i,s)=&-\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\le{n}}\frac{K_{4}\left(H(p,s)\right)}{6}
{\bf E}\{D^{3}_{ps}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))\}U(t,s),\\
\Omega_{7}(i,s)=&-\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\le{n}}\frac{K_{6}\left(H(p,s)\right)}{5!}{\bf
E}\{D^{5}_{ps}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))\}U(t,s)\end{aligned}$$ and $$\begin{aligned}
\Omega_{8}(&i,s)\\
=&-\frac{\zeta^{2}_{2}v^{2}}{6!}\sum_{|t|,|p|\le{n}}{\bf
E}\left\{H(p,s)^{7}[D_{ps}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(0)}
\right\}U(t,s)\\
&+\frac{\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\le{n}}K_{2}\left(H(p,s)\right){\bf
E}\left\{H(p,s)^{5}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(1)}
\right\}U(t,s)\\
&+\frac{\zeta^{2}_{2}v^{2}}{(3!)^{2}}\sum_{|t|,|p|\le{n}}K_{4}\left(H(p,s)\right){\bf
E}\left\{H(p,s)^{3}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(2)}
\right\}U(t,s)\\
&+\frac{\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\le{n}}K_{6}\left(H(p,s)\right){\bf
E}\left\{H(p,s)[D_{ts}^{6}(g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p))]^{(3)}
\right\}U(t,s)\end{aligned}$$ with $K_{r}\left(H(p,s)\right)$, $r=2,4,6$ are the cumulants of $H(p,s)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
The terms $\Omega_{l}$, $l=1,\ldots,5$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). This can be shown with the help of the estimate (\[b.4supEUG02\]) and the inequalities (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]), (\[b.3sumG1ipG2ip\]) and (\[b.5sumstG1G2\]). The term $\Omega_{6}$ contains $272$ terms that are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). This can be checked by direct computations with the use of (\[b.3sumG1ipG2ip\]) and (\[b.5sumstG1G2\]). Using similar argument as those of the proofs of (\[b.3Y3Y6lem\]) and (\[b.3epsilonlem\]) (see (\[b.3epsilonestim1\])-(\[b.3epsilonestim3\])), and the following estimate (cf. (\[b.3ED5pig10G2ij\])) $$\label{b.5ED5}
{\bf
E}|D^{5}_{pi}\{g^{0}_{1}G_{2}(i,t)^{2}G_{2}(s,p)\}|=O\left(N^{-1}+[{\bf
Var}\{g_{1}\}]^{1/2}\right), \ \hbox{ as } \quad
n,p\rightarrow\infty,$$ we conclude that the terms $\Omega_{7}$ and $\Omega_{8}$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). Then, the relation (\[b.5Theta4\]) is of the form that leads to the estimates needed for $\sum_{s}{\bf E}\{M(i,s)\}U(s,i)$.
Regarding $\Theta_{5}(i,s)$, we apply the resolvent identity (\[b.2h-zI-1\]) to the factor $G_{2}(t,s)$. Repeating the usual computations based on the formula (\[b.2EXjFX1Xm\]) with $q=5$ and taking into account relations (\[b.2partialG\]) and (\[b.3Efg\]), we obtain that $$\label{b.5Theta5}
\Theta_{5}(i,s)=2v^{2}\zeta^{2}_{2}\frac{\psi(0)}{b}{\bf E}M(i,s)
+v^{2}\zeta_{2}\Theta_{5}(i,s){\bf
E}U_{G_{2}}(s)+\sum_{l=1}^{9}\Omega^{'}_{l}(i,s),$$ where $$\begin{aligned}
\Omega^{'}_{1}(i,s)=&2v^{4}\zeta^{2}_{2}\sum_{|t|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,s)U^{0}_{G_{2}}(s)\}U(t,s),\\
\Omega^{'}_{2}(i,s)=&2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g_{1}^{0}G_{2}(i,p)G_{2}(s,s)G_{2}(i,t)G_{2}(t,p)\}U(p,s)U(t,s),\\
\Omega^{'}_{3}(i,s)=&\frac{4v^{4}\zeta^{2}_{2}}{N}\sum_{|t|,|p|\le{n}}{\bf
E}\{G^{2}_{1}(p,s)G_{2}(i,s)G_{2}(i,t)G_{2}(t,p)\}U(p,s)U(t,s),\\
\Omega^{'}_{4}(i,s)=&4v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(p,s)G_{2}(i,t)G_{2}(t,p)\}U(p,s)U(t,s),\\
\Omega^{'}_{5}(i,s)=&2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)G_{2}(i,p)G_{2}(t,p)G_{2}(s,t)\}U(p,s)U(t,s),\\
\Omega^{'}_{6}(i,s)=&2v^{4}\zeta^{2}_{2}\sum_{|t|,|p|\le{n}}{\bf
E}\{g^{0}_{1}G_{2}(i,s)^{2}G_{2}(p,t)^{2}\}U(p,s)U(t,s),\\
\Omega^{'}_{7}(i,s)=&-2\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\le{n}}\frac{K_{4}\left(H(p,s)\right)}{6}
{\bf E}\{D^{3}_{ps}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))\}U(t,s),\\
\Omega^{'}_{8}(i,s)=&-2\zeta^{2}_{2}v^{2}\sum_{|t|,|p|\le{n}}\frac{K_{6}\left(H(p,s)\right)}{5!}{\bf
E}\{D^{5}_{ps}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))\}U(t,s)\end{aligned}$$ and $$\begin{aligned}
\Omega^{'}&_{9}(i,s)\\
=&-\frac{2\zeta^{2}_{2}v^{2}}{6!}\sum_{|t|,|p|\le{n}}{\bf
E}\left\{H(p,s)^{7}[D_{ps}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(0)}
\right\}U(t,s)\\
&+\frac{2\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\le{n}}K_{2}\left(H(p,s)\right){\bf
E}\left\{H(p,s)^{5}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(1)}
\right\}U(t,s)\\
&+\frac{2\zeta^{2}_{2}v^{2}}{(3!)^{2}}\sum_{|t|,|p|\le{n}}K_{4}\left(H(p,s)\right){\bf
E}\left\{H(p,s)^{3}[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(2)}
\right\}U(t,s)\\
&+\frac{2\zeta^{2}_{2}v^{2}}{5!}\sum_{|t|,|p|\le{n}}K_{6}\left(H(p,s)\right){\bf
E}\left\{H(p,s)[D_{ts}^{6}(g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p))]^{(3)}
\right\}U(t,s)\end{aligned}$$ with $K_{r}\left(H(p,s)\right)$, $r=2,4,6$ are the cumulants of $H(p,s)$ as in (\[b.3K2K4\])-(\[b.3K6\]).
The terms $\sum_{s}\Omega^{'}_{l}(i,s)U(s,i)$, $l=1,\ldots,6$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). This can be shown with the help of the estimate (\[b.4supEUG02\]) and the inequalities (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]), (\[b.3sumG1ipG2ip\]) and (\[b.5sumstG1G2\]). The term $\Omega^{'}_{7}$ contains $356$ terms that are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). This can be checked by direct computations with the use of (\[b.3sumG1ipG2ip\]) and (\[b.5sumstG1G2\]). Using similar argument as those of the proofs of (\[b.3Y3Y6lem\]) and (\[b.3epsilonlem\]) (see (\[b.3epsilonestim1\])-(\[b.3epsilonestim3\])), and the following estimate (cf. (\[b.3ED5pig10G2ij\])) $$\label{b.5ED5}
{\bf
E}|D^{5}_{pi}\{g^{0}_{1}G_{2}(i,s)G_{2}(i,t)G_{2}(t,p)\}|=O\left(N^{-1}+[{\bf
Var}\{g_{1}\}]^{1/2}\right), \ \hbox{ as } \quad
n,p\rightarrow\infty,$$ we conclude that the terms $\Omega^{'}_{8}$ and $\Omega^{'}_{9}$ are of the order indicated in the RHS of (\[b.3Y1Y2lem\]). Then, the form of (\[b.5Theta5\]) is also such that, being substituted into (\[b.5Y2form2\]) and then into (\[b.5Y2form1\]), it leads to the needed estimates.
The term $\Theta_{6}(i,s)$ contains $67$ terms. These terms can be gathered into three groups. In each group, the terms are estimated by the same values with the help of the same computations.
We give estimates for the typical cases. Using (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]) and (\[b.3sumG1ipG2ip\]) (with $m=1$), we get for the terms of the first group: $$\begin{aligned}
&\left|\frac{\zeta_{2}}{N}\sum_{|t|\le{n}}K_{4}\left(H(t,s)\right){\bf
E}\{G_{1}^{2}(t,t)G_{1}(s,s)G_{1}(t,s)G_{2}(i,s)G_{2}(i,t)\}\cdot
\frac{1}{(1+\delta_{ts})^{3}}\right|\\
&\le{\frac{V_{4}+3v^{4}}{\eta^{5}Nb^{2}}\sum_{|t|\le{n}}{\bf
E}|G_{1}(t,s)G_{2}(i,t)|}\le{\frac{V_{4}+3v^{4}}{\eta^{7}Nb^{2}}}.\end{aligned}$$ For the terms of the second group, we obtain estimates $$\begin{aligned}
&\left|\zeta_{2}\sum_{|t|\le{n}}K_{4}\left(H(t,s)\right){\bf
E}\{g_{1}^{0}G_{2}(s,s)^{2}G_{2}(i,t)^{2}\}\cdot
\frac{1}{(1+\delta_{ts})^{3}}\right|\\
&\le{\frac{V_{4}+3v^{4}}{\eta^{3}b^{2}}{\bf
E}|g_{1}^{0}|\sum_{|t|\le{n}}|G_{2}(i,t)^{2}|}\le{\frac{[V_{4}+3v^{4}]\sqrt{{\bf
Var}\{g_{1}\}}}{\eta^{5}b^{2}}}.\end{aligned}$$ Finally, for the terms of the third group, we get inequalities $$\begin{aligned}
&\left|\sum_{|s|\le{n}}\frac{\zeta_{2}}{N}\sum_{|t|\le{n}}K_{4}\left(H(t,s)\right){\bf
E}\{G_{1}^{2}(s,s)G_{1}(t,t)G_{2}(t,t)G_{2}(i,s)^{2}\}U(s,i)\cdot
\frac{1}{(1+\delta_{ts})^{3}}\right|\\
&\le{\frac{V_{4}+3v^{4}}{\eta^{5}Nb}\sum_{|s|\le{n}}{\bf
E}|G_{2}(i,s)^{2}|\sum_{|t|\le{n}}U(t,s)U(s,i)}=O\left(\frac{1}{Nb^{2}}\right).\end{aligned}$$ Gathering all the estimates of $67$ terms, we obtain that $$\left|\sum_{|s|\le{n}}\Theta_{6}(i,s)U(s,i)\right|=O\left(\frac{1}{Nb^{2}}+\frac{\sqrt{{\bf
Var}\{g_{1}\}}}{b^{2}}\right).$$
Using similar argument as those of the proofs of (\[b.3Y3Y6lem\]) and (\[b.3epsilonlem\]) (see (\[b.3epsilonestim1\])-(\[b.3epsilonestim3\])), we conclude that $\Theta_{7}$ and $\sup_{i}|Y_{2}(i)|$ are of the order indicated in (\[b.3Y1Y2lem\]). Estimate (\[b.3Y1Y2lem\]) is proved and so Lemma 3.1 is proved.$\hfill \blacksquare$
0,5cm
Proof of Lemma 4.1
------------------
Let us consider the variable $$K(i,s)={\bf E}\{RG^{0}(i,i)\}={\bf E}\{R^{0}G(i,i)\},$$ where we denoted $R=g^{0}U^{0}_{G}(s)$. Applying to $G_{2}(i,i)$ the resolvent identity (\[b.2h-zI-1\]) and taking account formula (\[b.2EXjFX1Xm\]) with $q=3$ and relation (\[b.2partialG\]), we obtain that $$\label{b.5RG}
{\bf E}\{R^{0}G(i,i)\}=\zeta v^{2}{\bf E}\{R^{0}G(i,i)U_{G}(i)\} +
\sum_{a=1}^{5}l_{a}(i,s)$$ with $$\begin{aligned}
l_{1}(i,s)=&\zeta v^{2}\sum_{|p|\le{n}}{\bf
E}\{R^{0}G(i,p)^{2}\}U(p,i),\\
l_{2}(i,s)=&2\zeta v^{2}\sum_{|p|,|t|\le{n}}{\bf
E}\{g^{0}G(t,p)G(t,i)G(i,p)\}U(t,s)U(p,i),\\
l_{3}(i,s)=&\frac{2\zeta v^{2}}{N}\sum_{|p|,|t|\le{n}}{\bf
E}\{G(p,t)G(i,t)U^{0}_{G}(s)G(i,p)\}U(p,i),\\
l_{4}(i,s)=&-\frac{\zeta}{6}\sum_{|p|\le{n}}K_{4}{\bf
E}\{D^{3}_{pi}(R^{0}G(i,p))\}\end{aligned}$$ and $$\begin{aligned}
l_{5}(i,s)=&-\frac{\zeta}{4!}\sum_{|p|\le{n}}{\bf
E}\left\{H(p,i)^{5}[D_{pi}^{4}(R^{0}G(i,p))]^{(0)}\right\}\\
&+\frac{\zeta}{3!}\sum_{|p|\le{n}}K_{2}{\bf
E}\left\{H(p,i)^{3}[D_{pi}^{4}(R^{0}G(i,p))]^{(1)}\right\}\\
&+\frac{\zeta}{3!}\sum_{|p|\le{n}}K_{4}{\bf
E}\left\{H(p,i)[D_{pi}^{4}(R^{0}G(i,p))]^{(2)}\right\},\end{aligned}$$ where $K_{r}$, $r=2,4$ are the cumulants of $H(p,i)$ as in (\[b.3K2K4\]). Let us use the identity $${\bf E}R^{0}XY={\bf E}RX^{0}{\bf E}Y+{\bf E}RY^{0}{\bf E}X+{\bf
E}RX^{0}Y^{0}-{\bf E}R{\bf E}X^{0}Y^{0},$$ and rewrite (\[b.5RG\]) in the form $$\label{b.5RGform1}
{\bf
E}\{R^{0}G(i,i)\}=K(i,s)=v^{2}q(i)g(i)\sum_{|t|\le{n}}K(t,s)U(t,i)+
\Pi(i,s)$$ with $$\begin{aligned}
\nonumber \Pi(i,s)=&v^{2}q(i)\left[{\bf
E}\{RU^{0}_{G}(i)G^{0}(i,i)\}-{\bf
E}\{g^{0}U^{0}_{G}(s)\}{\bf E}\{G^{0}(i,i)U^{0}_{G}(i)\}\right]\\
\label{b.5Pi} &+ \frac{q(i)}{\zeta}\sum_{a=1}^{5}l_{a}(i,s),\end{aligned}$$ where $g(i)={\bf E}\{G(i,i)\}$ and $q$ is given by (\[b.4q2i\]). Now we rewrite (\[b.5RGform1\]) in the form of a vector equality $$\vec{K}(.,s)=[I-W^{(q,g)}]^{-1}\vec{\Pi}(.,s),$$ where we denote by $W^{(q,g)}$ the linear operator acting on a vector $e$ with components $e(i)$ as $$[W^{(q,g)}e](i)=v^{2}q(i)g(i)\sum_{|t|\le{n}}e(t)U(t,i)$$ and vectors $[\vec{\Pi}(.,s)](i)=\Pi(i,s)$. It is easy to see that if $z\in \Lambda_{\eta}$, then $||W^{(q,g)}||\le{\frac{1}{2}}$. Thus, to prove relation (\[b.4taulem\]), it is sufficient to show that $$\label{b.5Piestim}
\sup_{|i|,|s|\le{n}}|\Pi(i,s)|=O\left(\frac{1}{Nb^{2}}
+\frac{1}{b^{2}}\left({\bf Var}\{g\}\right)^{1/2}\right).$$ Let us prove (\[b.5Piestim\]). Taking into account inequality (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]), (\[b.5sumstG1G2\]) and estimate (\[b.4supEUG02\]), we obtain that $$|l_{a}(i,s)|\le{\frac{c}{b^{2}}\left({\bf Var}\{g\}\right)^{1/2}}
\quad \ a=1,2$$ and $$|l_{3}(i,s)|\le{\frac{c}{Nb^{2}}},$$ where c is a constant. Using similar arguments as those of the proof of (\[b.3epsilonlem\]) (see (\[b.3epsilonestim1\])-(\[b.3epsilonestim3\])) and the following estimates (cf. (\[b.3Vargnbnu\])-(\[b.3D6pig10G2ip\])) $$D_{pi}^{r}(R^{0}G(i,p))=O\left(N^{-1}+|g_{1}^{0}|\right), \quad
r=3,4$$ and $${\bf Var}\{[g_{n,b}(z)]^{(\nu)}\}=O\left({\bf
Var}\{g_{n,b}(z)\}+b^{-1}N^{-2}\right), \quad \nu=0,1,2,$$ we obtain that the terms $l_{a}$, $a=4,5$ are of the order indicated in the RHS of (\[b.4taulem\]). Finally, we derive inequality $$\begin{aligned}
\nonumber |\Pi(i,s)|\le&{c\left({\bf
Var}\{g\}\right)^{1/2}\left(\left({\bf
E}|U^{0}_{G}(i)|^{4}\right)^{1/2}+\frac{1}{b^{2}}\left({\bf
E}|U^{0}_{G}(i)|^{2}\right)^{1/2}\right)}\\
\label{b.5Piestim2}
&+c\left(\frac{1}{Nb^{2}}+\frac{1}{b^{2}}\left({\bf
Var}\{g\}\right)^{1/2}\right),\end{aligned}$$ where $c$ is a constant. Then (\[b.5Piestim\]), (\[b.5Piestim2\]) and Lemma 4.1 follow from (\[b.4supEUG02\]) and the following estimate. 0,2cm
If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimate $$\label{b.4supEUG04}
\sup_{|s|\le{n}}{\bf E}\{|U^{0}_{G}(s;z)|^{4}\}=O(b^{-4})$$ holds in the limit $n,b\rightarrow\infty$.
0,2cm
[*Proof of Lemma 5.2.*]{} Let us consider variable $${\bf
E}\{U^{0}_{G_{1}}(x_{1})U^{0}_{G_{2}}(x_{2})U^{0}_{G_{3}}(x_{3})U^{0}_
{G_{4}}(x_{4})\}= {\bf
E}[U^{0}_{G_{1}}(x_{1})U^{0}_{G_{2}}(x_{2})U^{0}_{G_{3}}
(x_{3})]^{0}U_{G_{4}}(x_{4}).$$ Set $T=U^{0}_{G_{1}}U^{0}_{G_{2}}U^{0}_{G_{3}}$ and $M(x_{1},x_{2},x_{3},t)={\bf E}T^{0}G_{4}(t,t)$. We apply to $G_{4}(t,t)$ the resolvent identity $(3.2)$ and obtain $${\bf E}T^{0}G_{4}(t,t)=-\zeta_{4}\sum_{|s|\le{n}}{\bf
E}\{T^{0}G_{4}(t,s)H(s,t)\}.$$ Applying (\[b.2EXjFX1Xm\]) to ${\bf E}\{T^{0}G_{4}(t,s)H(s,t)\}$ with $q=3$ and taking into account (\[b.2partialG\]), we get relation $$\begin{aligned}
\nonumber &{\bf E}T^{0}G_{4}(t,t)\\
\nonumber &= \zeta_{4}v^{2}{\bf
E}\{T^{0}G_{4}(t,t)U_{G_{4}}(t)\}+\zeta_{4}v^{2}{\bf
E}\left\{T^{0}\sum_{|s|\le{n}}G_{4}(t,s)^{2}U(s,t)\right\}\\
\nonumber &+ 2\zeta_{4}v^{2}\sum_{(i,j,k)}{\bf
E}\left\{U^{0}_{G_{i}}(x_{i})U^{0}_{G_{j}}(x_{j})\sum_{|y|,|s|\le{n}}
G_{k}(y,s)G_{k}(t,y)G_{4}(t,s)U(y,x_{k})U(s,t)\right\}\\
\label{b.5W4form1} &+\zeta_{4}\Gamma_{1}(t)+\zeta_{4}\Gamma_{2}(t)\end{aligned}$$ with $$\label{b.5daleth1}
\Gamma_{1}(t)=-\sum_{|s|\le{n}}\frac{K_{4}}{3!}{\bf
E}\left\{D^{3}_{st}(T^{0}G_{4}(t,s))\right\}$$ and $$\begin{aligned}
\nonumber \Gamma_{2}(t)=&-\frac{1}{4!}\sum_{|s|\le{n}}{\bf
E}\left\{H(s,t)^{5}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(0)}
\right\}\\
\nonumber &+\sum_{|s|\le{n}}\frac{K_{2}}{3!}{\bf
E}\left\{H(s,t)^{3}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(1)}
\right\} \\
\label{b.5daleth2} &+\sum_{|s|\le{n}}\frac{K_{4}}{3!}{\bf
E}\left\{H(s,t)[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(2)}
\right\},\end{aligned}$$ where $K_{r}$, $r=2,4$ are the cumulants of $H(s,t)$ as in (\[b.3K2K4\]). In (\[b.5W4form1\]), we introduce the notation $$\sum_{(i,j,k)}\xi(x_{i},x_{j},x_{k}) = \xi(x_{1},x_{2},x_{3})
+\xi(x_{1},x_{3},x_{2}) + \xi(x_{2},x_{3},x_{1}).$$ Applying to the first term of the RHS of (\[b.5W4form1\]) relation (\[b.3Efg\]) and using $q_{4}(t)$ (\[b.4q2i\]), we obtain that $$\begin{aligned}
&{\bf E}T^{0}G_{4}(t,t)\\
&= q_{4}(t) v^{2}{\bf E}\{T^{0}G_{4}(t,t)U^{0}_{G_{4}}(t)\}
+q_{4}(t)v^{2}{\bf
E}\left\{T^{0}\sum_{|s|\le{n}}G_{4}(t,s)^{2}U(s,t)\right\}\\
&+2q_{4}(t)v^{2}\sum_{(i,j,k)}{\bf
E}\left\{U^{0}_{G_{i}}(x_{i})U^{0}_{G_{j}}(x_{j})\sum_{|y|,|s|\le{n}}
G_{k}(y,s)G_{k}(t,y)G_{4}(t,s)U(y,x_{k})U(s,t)\right\}\\
&+q_{4}(t)\left(\Gamma_{1}(t)+\Gamma_{2}(t)\right).\end{aligned}$$ Now gathering relation given by (\[b.2psicondit\]), (\[b.3GijleImz\]), (\[b.3sumGij2leImz2\]), (\[b.5sumstG1G2\]), (\[b.4q2ileImz\]) and $$\sup_{|t|\le{n}}{\bf E}|T^{0}U^{0}_{G_{4}}(t)|\le{{\bf
E}|T|\sup_{|t|\le{n}}{\bf E}|U^{0}_{G_{4}}(t)|}
+\sup_{|t|\le{n}}{\bf E}|TU^{0}_{G_{4}}(t)|$$ imply the following inequality $$\begin{aligned}
\nonumber |\sum_{|t|\le{n}}M(x_{1},x_{2},x_{3},t)U(t,x_{4})|\le&{
\frac{v^{2}}{\eta^{2}}\sup_{|t|\le{n}}{\bf
E}|TU^{0}_{G_{4}}(t)|+\frac{v^{2}}{\eta^{2}}{\bf
E}|T|\sup_{|t|\le{n}}{\bf E}|U^{0}_{G_{4}}(t)|}\\
\nonumber &+\frac{2v^{2}}{\eta^{3}b}{\bf E}|T|
+\frac{6v^{2}}{\eta^{4}b^{2}}{\bf
E}|U^{0}_{G_{i}}(x_{i})U^{0}_{G_{j}}(x_{j})|\\
\label{b.5Wineq}
&+\frac{1}{\eta}\sup_{|t|\le{n}}|\Gamma_{1}(t)+\Gamma_{2}(t)|.\end{aligned}$$ Henceforth, for sake of clarity, we consider $G=G_{1}=G_{3}=\bar{G}_{2}=\bar{G}_{4}$ and $x=x_{r}$, $r=1,\ldots,4$, then we get $T=\left(U^{0}_{G}(x)\right)^{2}U^{0}_{\bar{G}}(x)$ and $$\label{b.5ET}
{\bf E}|T|\le{\left({\bf E}|U^{0}_{G}|^{4}\right)^{1/2}\left({\bf
E}|U^{0}_{G}|^{2}\right)^{1/2}}.$$ Let us assume for the moment that $$\label{b.5supdaleth}
\sup_{|t|\le{n}}|\Gamma_{1}(t)+\Gamma_{2}(t)|=O\left(b^{-4}+
b^{-2}\sqrt{W}\right), \quad z\in\Lambda{\eta}$$ with $W=\sup_{x}{\bf E}|U^{0}_{G}(x)|^{4}$. Now returning to (\[b.5Wineq\]) and gathering estimates given by relations (\[b.4supEUG02\]), (\[b.5ET\]) and (\[b.5supdaleth\]) imply the following estimate $$W\le{A_{1}b^{-2}\sqrt{W}+A_{2}b^{-4}},$$ where $A_{1}$, $A_{2}$ are some constants. This proves (\[b.4supEUG04\]).
To complete the proof of Lemma 5.2, let us prove (\[b.5supdaleth\]). To do this, we use the following statement. 0,5cm
If $z\in\Lambda_{\eta}$, then under conditions of Theorem 2.1, the estimates $$\label{b.5DrU0Gx}
D^{r}_{st}\left(U^{0}_{G}(x)\right)=O(b^{-1}), \ \quad
r=1,\ldots,4,$$ $$\label{b.5DrT0Gts}
D^{r}_{st}\left(T^{0}\bar{G}(t,s)\right)=
O\left(b^{-3}+b^{-2}|U^{0}_{G}(x)|+b^{-1}|U^{0}_{G}(x)|^{2}
+|U^{0}_{G}(x)|^{3}\right), \quad r=3,4$$ and $$\label{b.5EU0G2r}
{\bf E}|[U^{0}_{G}(x)]^{(\nu)}|^{2r}=O\left(b^{-3r}+{\bf
E}|U^{0}_{G}(x)|^{2r}\right), \ r=1,2$$ hold for all $\nu=0,1,2$, all $|x|\le{n}$ and large enough $n$ and $b$ satisfying (\[b.2bnalpha\]).
0,2cm We prove this Lemma at the end of this subsection.0,2cm
Let us return to the proof of (\[b.5supdaleth\]). Regarding the variable $\Gamma_{1}$ (\[b.5daleth1\]) and using (\[b.4supEUG02\]), (\[b.4K4estim\]) and (\[b.5DrT0Gts\]), one gets with the help of (\[b.5ET\]) that $$\label{b.5dalethestima1}
\sum_{|s|\le{n}}\frac{2[V_{4}+3v^{4}]}{3b}{\bf
E}|D^{3}_{st}(T^{0}\bar{G}(t,s))|U(s,t)=O\left(b^{-4}+
b^{-2}\sqrt{W}\right).$$ Now let us estimate $\Gamma_{2}$ (\[b.5daleth2\]). Regarding the first term of the RHS of (\[b.5daleth2\]) and using (\[b.4supEUG02\]), (\[b.5DrT0Gts\]) and (\[b.5EU0G2r\]), we obtain inequality $$\begin{aligned}
\nonumber &\sum_{|s|\le{n}}{\bf
E}|H(s,t)^{5}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(0)}|\\
\nonumber &\le{c \sum_{|s|\le{n}}{\bf
E}\left\{\frac{|H(s,t)|^{5}}{b^{3}}+\frac{|H(s,t)|^{5}}{b^{2}}
|[U_{G}^{0}(x)]^{(0)}|+\frac{|H(s,t)|^{5}}{b}
|[U_{G}^{0}(x)]^{(0)}|^{2}\right\}}\\
\nonumber &+c\sum_{|s|\le{n}}{\bf
E}\left\{|H(s,t)|^{5}|[U_{G}^{0}(x)]^{(0)}|^{3} \right\}\\
\nonumber &\le{c
\sum_{|s|\le{n}}\left[\frac{\mu_{5}}{b^{11/2}}\psi\left(\frac{s-t}
{b}\right)+\frac{\mu_{10}^{1/2}}{b^{9/2}} \left({\bf
E}|[U_{G}^{0}(x)]^{(0)}|^{2}\right)^{1/2}\psi\left(\frac{s-t}
{b}\right)^{1/2}\right]}\\
\nonumber &+c \sum_{|s|\le{n}}\left[\frac{\mu_{10}^{1/2}}{b^{7/2}}
\left({\bf
E}|[U_{G}^{0}(x)]^{(0)}|^{4}\right)^{1/2}\psi\left(\frac{s-t}
{b}\right)^{1/2}\right] \\
\label{b.5daleth2estima1}
&=O\left(\frac{1}{b^{9/2}}+\frac{1}{b^{5/2}}\sqrt{W}\right).\end{aligned}$$ Repeating the arguments used to prove (\[b.5daleth2estima1\]), it is easy to show that the term $$\sum_{|s|\le{n}}\frac{K_{2}}{3!}{\bf
E}\left\{H(s,t)^{3}[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(1)}
\right\}+\sum_{|s|\le{n}}\frac{K_{4}}{3!}{\bf
E}\left\{H(s,t)[D^{4}_{st}\left(T^{0}G_{4}(t,s)\right)]^{(2)}
\right\}$$ is of the order indicated in the RHS of (\[b.5supdaleth\]) and that $$\label{b.5dalethestima2}
\sup_{|t|\le{n}}|\Gamma_{2}(t)|=O\left(b^{-4}+
b^{-2}\sqrt{W}\right).$$ Then the estimate (\[b.5supdaleth\]) follows from (\[b.5dalethestima1\]) and (\[b.5dalethestima2\]). Lemma 5.2 is proved. 0,2cm
[*Proof of Lemma 5.3.*]{} We prove Lemma 5.2 with $r=1$ because the general case does not differ from this one. We start with the proof of (\[b.5DrU0Gx\]). Using (\[b.2partialG\]), we obtain that $$D^{1}_{st}\left(U^{0}_{G}(x)\right)=-2\sum_{|k|\le{n}}G(k,s)G(k,t)
U(k,x).$$ Then estimate (\[b.5DrU0Gx\]) (with $r=1$) follows from this relation and inequality $|U(k,x)|\le{b^{-1}}$ and (\[b.3sumG1ipG2ip\]) (with $m=1$). The general case does not differ from this one, so the estimate (\[b.5DrU0Gx\]) is proved.
Let us prove (\[b.5DrT0Gts\]). Remembering that $T=[U^{0}_{G}(x)]^{2}U^{0}_{\bar{G}}(x)$ and using (\[b.2partialG\]) and (\[b.5DrU0Gx\]), we obtain that $$D^{1}_{st}\{T^{0}\}=O(b^{-1}|U^{0}_{G}(x)|^{2}),$$ $$D^{2}_{st}\{T^{0}\}=O\left(b^{-2}|U^{0}_{G}(x)|
+b^{-1}|U^{0}_{G}(x)|^{2}\right),$$ $$D^{3}_{st}\{T^{0}\}=O\left(b^{-3}+b^{-2}|U^{0}_{G}(x)|
+b^{-1}|U^{0}_{G}(x)|^{2}\right).$$ Now it is easy to show that (\[b.5DrT0Gts\]) is true.
0,2cm
Finally, we prove (\[b.5EU0G2r\]) with $r=1$ because the general case does not doffer from this one. To simplify computation, we use the notation: for each pair $(s,t)$ and $\nu=0,1,2$, let $H^{(\nu)}_{st}=H^{(\nu)}=\hat{H}$ be the matrix defined by $$\hat{H}(r,i)= \left\{
\begin{array}{lll}
H(r,i), & \textrm{if} & (r,i)\neq(s,t); \\
\hat{H}(s,t), & \textrm{if} & (r,i)=(s,t)
\end{array}\right.$$ with $|\hat{H}(s,t)|\le{|H(s,t)|}$ and its resolvent by $G^{(\nu)}_{sp}(z)=\hat{G}(z)$. Then the resolvent identity (\[b.2h-zI-1\]) imply that $$\begin{aligned}
U_{\hat{G}}(x)=&U_{G}(x)-\frac{1}{b}\sum_{|k|,|r|,|i|\le{n}}\hat{G}(k,r)
\{\hat{H}-H\}(r,i)G(i,k)\psi\left(\frac{x-k}{b}\right)\\
=&U_{G}(x)-\frac{1}{b}\sum_{|k|\le{n}}B(k,s,t)
\psi\left(\frac{x-k}{b}\right)\end{aligned}$$ with $B(k,s,t)=\hat{G}(k,s) [\hat{H}(s,t)-H(s,t)]G(t,k)$. Then inequality (\[b.3GijleImz\]) implies that $$\begin{aligned}
&{\bf E}|U^{0}_{\hat{G}}(x)|^{2}\\
&\le{2{\bf E}|U^{0}_{G}(x)|^{2}+\frac{2}{b^{2}} {\bf
E}\left|\sum_{|k|\le{n}}B^{0}(k,s,t)\psi\left(\frac{x-k}{b}\right)\right|^{2}}\\
&\le{2{\bf E}|U^{0}_{G}(x)|^{2}+\frac{8}{\eta^{4}b^{2}}{\bf
E}\left(|\hat{H}(s,t)|+{\bf E}|H(s,t)|\right)^{2}}\\
&\le{2{\bf E}|U^{0}_{G}(x)|^{2}+\frac{8}{\eta^{4}b^{3}}\left[{\bf
E}|a(s,p)|^{2}\psi\left(\frac{s-p}{b}\right)+3\left({\bf
E}|a(s,p)|\right)^{2}\psi\left(\frac{s-p}{b}\right)^{2}\right]}.\end{aligned}$$ This proves (\[b.5EU0G2r\]). Lemma 5.3 is proved. $\hfill
\blacksquare$
Proof of Lemma 4.3.
-------------------
We prove relation (\[b.4supbET12U\]) with $k=1$ because the general case does not differ from this one. To derive relations for the average value of the variable $t_{12}(i,s)={\bf E}G_{1}(i,s)G_{2}(i,s)$, we use identity (\[b.2h-zI-1\]) and relation (\[b.2EXjFX1Xm\]) (with $q=3$) and repeat the proof of relation (\[b.4supbF12U\]). Simple computations lead to $$\begin{aligned}
\nonumber t_{12}(i,s) = &\zeta_{2}g_{1}(i)\delta_{is} +
\zeta_{2}v^{2}t_{12}(i,s)U_{g_{2}}(s) \\
\label{b.5t12} &+
\zeta_{2}v^{2}\sum_{|p|\le{n}}t_{12}(i,p)g_{1}(s)U(p,s)+
\sum_{j=1}^{6}\gamma_{j}(i,s),\end{aligned}$$ with $$\begin{aligned}
\gamma_{1}(i,s) =& \zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\{G_{1}(i,s)G_{2}(i,p)G_{1}(p,s)\}U(p,s),\\
\gamma_{2}(i,s)=&\zeta_{2}v^{2}\sum_{|p|\le{n}}{\bf
E}\left\{G_{1}(i,s)G_{2}(i,p)G_{2}(p,s)\right\}U(p,s)\\
\gamma_{3}(i,s)=&\zeta_{2}v^{2}{\bf
E}\{G_{1}(i,s)G_{2}(i,s)U^{0}_{G_{1}}(s)\},\\
\gamma_{4}(i,s)=&\zeta_{2}v^{2}{\bf
E}\left\{G^{0}_{1}(s,s)\sum_{|p|\le{n}}G_{1}(i,p)G_{2}(i,p)U(p,s)
\right\},\\
\gamma_{5}(i,s)=&-\frac{\zeta_{2}}{6}\sum_{|p|\le{n}}K_{4}{\bf
E}\left\{D^{3}_{ps}(G_{1}(i,s)G_{2}(i,p))\right\}\end{aligned}$$ and $$\begin{aligned}
\gamma_{6}(i,s)=&-\frac{\zeta_{2}}{4!}\sum_{|p|\le{n}}{\bf
E}\left\{H(p,s)^{5}[D^{4}_{ps}\left(G_{1}(i,s)G_{2}(i,p)\right)]^{(0)}
\right\}\\
&+\frac{\zeta_{2}}{3!}\sum_{|p|\le{n}}K_{2}{\bf
E}\left\{H(p,s)^{3}[D^{4}_{ps}\left(G_{1}(i,s)G_{2}(i,p)\right)]^{(1)}
\right\}\\
&+\frac{\zeta_{2}}{3!}\sum_{|p|\le{n}}K_{4}{\bf
E}\left\{H(p,s)[D^{4}_{ps}\left(G_{1}(i,s)G_{2}(i,p)\right)]^{(2)}
\right\},\end{aligned}$$ where $K_{r}$, $r=2,4$ are the cumulants of $H(p,s)$ as in (\[b.3K2K4\]). Using (\[b.3sumG1ipG2ip\]), it is easy to show that $$\sup_{|i|,|s|\le{n}}|\gamma_{1}(i,s)|=o(b^{-1}), \quad
\sup_{|i|\le{n}}|\sum_{|s|\le{n}}\gamma_{1}(i,s)|=o(b^{-1}).$$ The same is valid for $\gamma_{2}$. Similar estimates for $\gamma_{3}$, $\gamma_{4}$, $\gamma_{5}$ and $\gamma_{6}$ follow from relations (\[b.4supEUG02\]), (\[b.4supEGss2\]) and simple arguments as those to the proof of (\[b.4betaresti\]) (see (\[b.4K4estim\])-(\[b.4beta5\])). Thus, (\[b.5t12\]) implies that $$\label{b.5t12form1}
t_{12}(i,s) = g_{1}(i)q_{2}(i)\delta_{is} +
v^{2}g_{1}(s)q_{2}(s)\{t_{12}U\}(i,s) + \Delta(i,s),$$ where $$\label{b.5Deltaestim}
\sup_{|i|,|s|\le{n}}|\Delta(i,s)|=o(1) \quad \hbox{ and } \quad
\sup_{|i|\le{n}}|\sum_{|s|\le{n}}\Delta(i,s)| = o(1)$$ in the limit $n,b\rightarrow \infty$. We rewrite relation (\[b.5t12form1\]) in the matrix form (cf. (\[b.4F12form2\])) $$\label{b.5t12form2}
t_{12} = \{ I - W^{(g_{1},q_{2})} \}^{-1}(Diag(g_{1}q_{2}) + \Delta
) = \sum_{m=0}^{+\infty}\{W^{(g_{1},q_{2})}\}^{m}(Diag(g_{1}q_{2}) +
\Delta ).$$ Now we can apply to (\[b.5t12form2\]) the same arguments as in the proof of (\[b.4supbF12U\]). Replacing $g_{1}$ and $q_{2}$ by $w_{1}$ and $w_{2}$, respectively, we derive from (\[b.5Deltaestim\]) that for $i\in B_{L+Q}$, $$\label{b.5t12form3}
t_{12}(i,s) = \sum_{m=0}^{M}v^{2m}(w_{1}w_{2})^{m+1}[U^{m}](i,s) +
o(1),\quad n,b\rightarrow \infty.$$ Multiplying both sides of (\[b.5t12form3\]) by $U(s,i)$ and summing over $s$, we obtain the relation $$\sum_{|s|\le{n}}t_{12}(i,s)U(s,i) =
\sum_{m=0}^{M}v^{2m}(w_{1}w_{2})^{m+1}[U^{m+1}](i,i) + o(1), \
N,b\rightarrow \infty.$$ Now convergence (\[b.4bUm+1\]) implies the relation that leads, with $M$ replaced by $\infty$, to (\[b.4supbET12U\]). 0,2cm
To prove (\[b.4supET12\]), let us sum (\[b.5t12form3\]) over $s$. The second part of (\[b.5Deltaestim\]) tells us that the terms $\Delta$ remain small when summed over $s$. Thus we can write relations $$\label{b.5sumt12is}
\sum_{|s|\le{n}}t_{12}(i,s)=
\sum_{m=0}^{M}(v^{2}w_{1}w_{2})^{m+1}\sum_{|s|\le{n}}[U^{m}](i,s) +
o(1) ,\ N,b\rightarrow \infty.$$ Taking into account estimates for terms (\[b.4Pb\])-(\[b.4Tnb\]) (see previous work [@A] for more details), it is easy to observe that convergence (\[b.4supUj-1\]) together with (\[b.5sumt12is\]) imply (\[b.4supET12\]). Finally, we prove (\[b.4supEG2ii\]). To derive relations for the average value of variable $t_{11}(i,s)={\bf E}G_{1}(i,s)G_{1}(i,s)$, we repeat the proof of (\[b.4supET12\]) and replace $G_{2}$ by $G_{1}$. Then one obtains (\[b.4supEG2ii\]). Lemma 4.3 is proved. $\hfill
\blacksquare$ 0,2cm
Asymptotic properties of $T(z_{1},z_{2})$
=========================================
The asymptotic expression for $T(z_{1},z_{2})$ regarded in the limit $z_{1}=\lambda_{1}+i0$, $z_{2}=\lambda_{2}+i0$ supplies one with the information about the local properties of eigenvalue distribution provided that $\lambda_{1}-\lambda_{2}=O(N^{-1})$. Indeed, according to (\[b.2gnbz\]), the formal definition of the eigenvalue density $\rho_{n,b}(\lambda)=\sigma^{'}_{n,b}(\lambda)$ is $$\rho_{n,b}(\lambda)=\frac{1}{2i}[g_{n,b}(\lambda+i0)-g_{n,b}(\lambda-i0)].$$ We consider the density-density correlation function of $\rho_{n,b}$ $$R_{n,b}(\lambda_{1},\lambda_{2})=-\frac{1}{4}\sum_{\delta_{1},\delta_{2}
=-1,1}\delta_{1}\delta_{2}C_{N,b}(\lambda_{1}+i\delta_{1}0,\lambda_{2}+
i\delta_{2}0).$$ In general, even if $R_{n,b}$ can be rigorously determined, it is difficult to carry out direct study of it. Taking into account relation $(2.13)$, one can simpler-expression $$\label{b.6Xi}
\Xi_{n,b}(\lambda_{1},\lambda_{2})=-\frac{1}{4Nb}
\sum_{\delta_{1},\delta_{2}=-1,+1}\delta_{1}\delta_{2}T(\lambda_{1}+
i\delta_{1}0,\lambda_{1}+i\delta_{1}0)$$ and assume that it corresponds to the leading term to $R_{n,b}(\lambda_{1},\lambda_{2})$ in the limit $n,b\rightarrow\infty$.
It should be noted that for Wigner random matrices this approach is justified by the study of the simultaneous limiting transition $N\rightarrow\infty$, ${\mathrm{Im}}z_{j}\rightarrow0$ in the studies of $C_{N}(z_{1},z_{2})$ [@DD; @AAA; @DDD; @EE].
0,5cm
Let $T(z_{1},z_{2})$ is given by (\[b.2Tz1z2\]). Assume that function $\hat{\psi}(p)$ is such that there exist positive constants $c_{1}$, $\delta$ and $v>1$ that $$\label{b.6hatpsicond}
\hat{\psi}(p)=\hat{\psi}(0)-c_{1}|p|^{\nu}+o(|p|^{\nu})$$ for all $p$ such that $|p|\le{\delta}$, $\delta\rightarrow 0$. Then $$\label{b.6Xith}
\Xi_{n,b}(\lambda_{1},\lambda_{2})=\frac{1}{Nb}\frac{c_{2}}
{|\lambda_{1}-\lambda_{2}|^{2-1/v}}(1+o(1))$$ for $\lambda_{j}$, $j=1,2$ satisfying $$\label{b.6lambdatheor}
\lambda_{1}, \lambda_{2}\rightarrow\lambda \in (-2v,2v).$$
0,2cm
We see from (\[b.2Tz1z2\]) that there are two terms in $T(z_{1},z_{2})$. The first was found in [@B] for band random matrices, the second coincides with that found in [@C] for the ensemble of Wigner random matrices. The proof of (\[b.6Xith\]) consists of two parts already done in [@B] and [@C]. For completeness, we reproduce here these computations. 0,2cm
[*Proof of Theorem 6.1*]{}. Let us start with the term of (\[b.6Xi\]) that correspond to $\delta_{1}\delta_{2}=-1$. It follows from (\[b.2wz\]) that $$\label{b.61-v2w1w2}
\frac{1-v^{2}w_{1}w_{2}}{w_{1}w_{2}}=\frac{z_{1}-z_{2}}{w_{1}-w_{2}}.$$ The above identity yields relations $$\epsilon|w(\lambda+i\epsilon)|^{2}={\mathrm{Im}}w(\lambda+i\epsilon)(1-v^{2}|
w(\lambda+i\epsilon)|^{2}) \quad \hbox{ and } \quad
|w(\lambda+i0)|^{2}=v^{-2}$$ for $\lambda$ such that ${\mathrm{Im}}{w}(\lambda+i0)>0$. Combining these relations with (\[b.1rho\]) for the real and imaginary parts of $w(\lambda+i0)=\tau(\lambda)+i\rho(\lambda)$, we obtain that $$\label{b.6taurho}
v^{2}\tau^{2}=\frac{\lambda^{2}}{4v^{2}} \quad \hbox{ and } \quad
v^{2}\rho^{2}=1-\frac{\lambda^{2}}{4v^{2}}$$ (here and below we omit the variable $\lambda$). This implies the existence of the limits $w(z_{1})=\overline{w(z_{2})}$ for (\[b.6lambdatheor\]). One can easily deduce from (\[b.61-v2w1w2\]) that in the limit (\[b.6lambdatheor\]) $$\label{b.61-v2w1w2estim}
\frac{1-v^{2}w(z_{1})w(z_{2})}{w(z_{1})w(z_{2})}=
\frac{\lambda_{1}-\lambda_{2}}{2i\rho}=o(1).$$ Also we have that $$\label{b.6taurhorelation}
(1-v^{2}w^{2}_{1})(1-v^{2}w^{2}_{2})=2-2v^{2}(\tau^{2}-\rho^{2})=
4v^{2}\rho^{2}.$$ Now let us consider the leading term of the correlation function. Rewrite (\[b.2Tz1z2\]) as $$\begin{aligned}
T(z_{1},z_{2})=&Q(z_{1},z_{2})+Q^{'}(z_{1},z_{2})
+\frac{2v^{2}Q(z_{1},z_{2})}{(1-v^{2}w^{2}_{1})
(1-v^{2}w^{2}_{2})}\\
=&\frac{2v^{2}S(z_{1},z_{2})}{(1-v^{2}w^{2}_{1}) (1-v^{2}w^{2}_{2})}
+Q^{'}(z_{1},z_{2})\end{aligned}$$ with $$\label{b.6S}
S(z_{1},z_{2})=\frac{1}{2\pi}\int_{-\infty}^{+\infty}
\frac{w^{2}_{1}w^{2}_{2}\hat{\psi}(p)}{( 1-
v^{2}w_{1}w_{2}\hat{\psi}(p))^{2}}dp$$ and $$\label{b.6Q'}
Q^{'}(z_{1},z_{2})=\frac{2\Delta v^{4}w_{1}^{3}w_{2}^{3}}{(1 -
v^{2}w_{1}^{2})(1-v^{2}w_{2}^{2})},$$ where $\Delta$ is given by (\[b.2Delta\]). It is easy to observe that relations (\[b.6taurho\]) and $|w(\lambda+i0)|^{2}=|w(\lambda-i0)|^{2}=1/v^{2}$ imply that (cf. [@C]) $$\label{b.6Q'proof}
Q^{'}(\lambda_{1}+i0,\lambda_{2}-i0)+Q^{'}(\lambda_{1}-i0,
\lambda_{2}+i0)=\frac{\Delta}{v^{4}\rho^{2}}.$$ Now let us consider $S(z_{1},z_{2})$ (\[b.6S\]) and let us write $$S(z_{1},z_{2})=\frac{1}{2\pi}\left\{\int_{-\delta}^{\delta}+
\int_{\mathbf{R}\setminus(-\delta,\delta)}\right\}
\frac{w^{2}_{1}w^{2}_{2}\hat{\psi}(p)} {(1-
v^{2}w_{1}w_{2}\hat{\psi}(p))^{2}}dp=I_{1}+I_{2}.$$ Relation (\[b.61-v2w1w2\]) and (\[b.61-v2w1w2estim\]) imply equality $$\label{b.61-v2w1w2hatpsiestim}
[1- v^{2}w_{1}w_{2}\hat{\psi}(p)]^{2}=[\hat{\Psi}(p)-1]^{2}(1+o(1)).$$ Since $\psi(t)$ is monotone, then $$\liminf_{p\in \mathbf{R}\setminus(-\delta,\delta)}[\Psi(p)-1]^{2}>0.$$ This means that $I_{2}<\infty$ in the limit (\[b.6lambdatheor\]). Relations (\[b.6hatpsicond\]), (\[b.61-v2w1w2estim\]) and (\[b.61-v2w1w2hatpsiestim\]) imply in the limit (\[b.6lambdatheor\]) and that if we take $\delta|\lambda_{1}-\lambda_{2}|^{-1/\nu}\rightarrow\infty$, we obtain asymptotically (cf. [@B]) $$\label{b.6I1+I1}
I_{1}(\lambda_{1}+i0,\lambda_{2}-i0)+I_{1}(\lambda_{1}-i0,
\lambda_{2}+i0)=4B_{v}(c_{1})\frac{(2v\rho)^{2-1/\nu}}
{|\lambda_{1}-\lambda_{2}|^{2-1/\nu}},$$ where $$\label{b.6Bvc1}
B_{v}(c_{1})=\frac{1}{2\pi
c^{1/\nu}_{1}}\left[\int_{0}^{\infty}\frac{ds}{1+s^{2\nu}}-2\int_{0}^
{\infty}\frac{ds}{(1+s^{2\nu})^{2}}\right]$$ and $c_{1}$ is as in (\[b.6hatpsicond\]). To prove (\[b.6Xith\]), it remains to consider the sum $$I_{1}(\lambda_{1}+i0,\lambda_{2}-i0)+I_{2}(\lambda_{1}-i0,
\lambda_{2}+i0).$$ It is easy to observe that relations of the form (\[b.6taurhorelation\]) imply the bounded ness of this sum in the limit (\[b.6lambdatheor\]).
Now gathering relations (\[b.6taurhorelation\]), (\[b.6Q’proof\]) and (\[b.6I1+I1\]), we derive that $$\label{b.6Xiproofresult}
\Xi_{n,b}(\lambda_{1},\lambda_{2})=\frac{1}{Nb}\frac{B_{\nu}(c_{1})}
{(2v\rho)^{1/\nu}}\frac{1}{|\lambda_{1}-\lambda_{2}|^{2-1/\nu}}(1+o(1)).$$ This proves (\[b.6Xith\]). $\hfill \blacksquare$ 0,2cm
Let us discuss two consequences of Theorem 6.1.
- If $\nu=2$ and $c_{1}=\int t^{2}\psi(t)<\infty$. Regarding the RHS of (\[b.2Cnbz1z2Th\]) in the limit (\[b.6lambdatheor\]) with $\lambda_{j}=\lambda+\frac{r_{j}}{N}$, $j=1,2$, we obtain the asymptotic relation (see [@B]) $$\begin{aligned}
\nonumber
\Xi(\lambda_{1},\lambda_{2})&=-\frac{B_{2}(c_{1})}{2\sqrt{2}(v^{2}
\rho)^{1/2}}\frac{\sqrt{N}}{b}
\frac{1}{|r_{1}-r_{2}|^{3/2}}(1+o(1))\\
\label{b.6Xiform1}&=-C\frac{\sqrt{N}}{b}
\frac{1}{|r_{1}-r_{2}|^{3/2}}(1+o(1)), \quad C>0.\end{aligned}$$
- If $\Psi(t)=O(|t|^{-1-\nu})$ with $1<\nu<2$, we obtain the asymptotic relation (see [@B]) $$\label{b.6Xiform2}
\Xi(\lambda_{1},\lambda_{2})=\frac{B_{\nu}(c_{1})}{(2v^{2}\rho)^{1/\nu}}
\frac{N^{1-1/\nu}}{b}\frac{1}{|r_{1}-r_{2}|^{2-1/\nu}}(1+o(1))$$ and conclude that the expression for (\[b.6Xi\]) is proportional to $$\frac{N^{1-1/\nu}}{b}\frac{1}{|r_{1}-r_{2}|^{2-1/\nu}}.$$
The form of asymptotic expressions (\[b.6Xiform1\]) and (\[b.6Xiform2\]) coincides with the expressions determined by Khorunzhy and Kirsch (see [@B]) for the spectral correlation function of band random matrices [@B].
The first conclusion is that the leading terms of the ensemble we study (see (\[b.2Hnbij\])) and the ensemble of band random matrices are different but in the local scale, the form (\[b.6Xiform1\]) and (\[b.6Xiform2\]) is the same. More precisely, the tow ensembles mentioned above belong to the same class of spectral universality.
Our main conclusion is that the limiting expression for $\Xi_{n,b}
(\lambda_{1},\lambda_{2})$ exhibits different behavior depending on the rate of decay of $\psi(t)$ at infinity. In both cases (see (\[b.6Xiform1\]) and (\[b.6Xiform2\])) the exponents do not depend on the particular form of the function $\psi(t)$. Moreover, in the first case the exponents do not depend on $\psi$ at all. This can be regarded as a kind of spectral universality for the random matrix ensembles $\{H_{n,b}\}$ (\[b.2Hnbij\]). One can deduce that these characteristics also do not depend on the probability distribution of the random variables $a(i,j)$ (\[b.1ANij\]).
0,5cm
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[^1]: LMV - Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin-en-Yvelines, 78035 Versailles (FRANCE). E-mail: ayadi@math.uvsq.fr
|
---
abstract: |
Motivated by the idea that some characteristics are specific to the relations between individuals and not of the individuals themselves, we study a prototype model for the dynamics of the states of the links in a fixed network of interacting units. Each link in the network can be in one of two equivalent states. A majority link-dynamics rule is implemented, so that in each dynamical step the state of a randomly chosen link is updated to the state of the majority of neighboring links. Nodes can be characterized by a link heterogeneity index, giving a measure of the likelihood of a node to have a link in one of the two states. We consider this link-dynamics model on fully connected networks, square lattices and Erdös-Renyi random networks. In each case we find and characterize a number of nontrivial asymptotic configurations, as well as some of the mechanisms leading to them and the time evolution of the link heterogeneity index distribution. For a fully connected network and random networks there is a broad distribution of possible asymptotic configurations. Most asymptotic configurations that result from link-dynamics have no counterpart under traditional node dynamics in the same topologies.\
author:
- 'J. Fernández-Gracia'
- 'X. Castelló'
- 'V.M. Eguíluz'
- 'M. San Miguel'
title: 'Dynamics of link states in complex networks: the case of a majority rule'
---
INTRODUCTION
============
Collective properties of interacting units have been traditionally studied considering that each of these units has a property or state, and interacts with others in a network of interactions. The result of the interaction depends on the state of the interacting units. For example, in a spin system in a lattice, there is a spin in each node with a given state and it interacts with its neighbors in the lattice, in a way that depends on their relative spin state. The same basic set up has been implemented in individual or agent based models of social collective properties [@castellano_stat_phys]. These models endow individuals with a variable, which can be discrete or continuous, describing for example, an opinion state. The model also prescribes a dynamical rule, which results in changes of the states of the agents that depend on the state of the agents with whom they interact. However, there is a number of possible interactions among individuals in which the state variable is more properly described as a state of the interaction link than a state of the individuals in interaction. This is specially the case for relational interactions such as friendship, trust, communication channel (phone or skype), method of salutation (kiss or handshake), etc. It is also the case in language competition dynamics [@Baronchelli2012]: So far language has been modeled in this context [@castello2006; @patriarca2012] as an individual property, but use of a language, as opposed to knowledge of a language, is more a link than a node property in a social network of interactions. Noteworthy, data on link states associated with trust, friendship or enmity, obtained from on-line games and on-line communities, is now available and has been recently analyzed [@Szell2010; @Leskovec2010].\
Social balance theory [@Heider1946] is a well established precedent in the study on link states and link interactions. This theory states that a relation can be positive or negative and that there is a natural tendency to avoid unbalanced triads of individuals, where an unbalanced triad is defined as one for which the multiplication of the states of the three links is negative. A number of recent studies address social balance in complex networks, implementing stochastic link dynamics that explore when a balanced situation is or is not reached asymptotically [@Antal2005; @ANTAL2006; @Radicchi2007]. Social balance theory has also been confronted with large scale data [@Szell2010; @Leskovec2010], and alternative theories for the interaction of positive/negative relations have also been proposed [@Leskovec2010; @Marvel2010]. Focusing on link properties has also been emphasized in the problem of community detection in complex networks [@Ahn2010; @Evans2009; @Evans2010; @Liu2010; @Traag2009]. This opposes the traditional view of identifying network communities with a set of nodes [@Santo201075], and it makes possible for an individual to be assigned to different communities. Finally, the idea of considering link dynamics is also present in the problem of network dynamics controllability [@Nepusz2012]. Here the aim is to identify the most relevant links to drive the system to a desired global state of the network, instead of focusing on the dynamically most influent nodes [@Klemm2012].\
The aim of this paper is to investigate a prototype model for the dynamics of link states in a fixed network. Links can be in two equivalent states. This departs from the positive/negative interactions, considered for example in social balance, where the two link states play different roles. Equivalent link states can occur in many relational interactions including, for example, salutation or competition of languages of the same prestige. As a first step towards the characterization of such link dynamics we choose to investigate a majority rule dynamics akin to a zero-temperature kinetic Ising model but for the states of the links, instead of the state of the nodes. We show that such link majority rule dynamics on complex networks results in a variety of non-trivial different asymptotic configurations which are generally not found when studying traditional node-dynamics in the same topologies. We also show how a quantity characterizing the node behavior naturally arises for the link states, so that nodes can also be characterized by the state of the links connected to the node.\
The paper is organized as follows: in section II we define our majority rule link dynamics model, as well as some quantities introduced for its characterization. In sections III, IV and V we describe our results on a fully connected network, a square lattice and Erdös-Renyi random networks, respectively. Section VI contains a discussion summary. Besides the material presented in this paper, which is self-explanatory, the supplementary material available in the web will help visualize different aspects of the model on different networks.\
A MODEL FOR LINK DYNAMICS
=========================
We consider a fixed network composed by $N$ nodes and $L$ edges. The state of each link $i-j$ is characterized by a binary variable $s_{i-j}$ which can take two equivalent values $A$ or $B$. We study a majority rule for the dynamics of the state of the links. The basic step of the dynamics is
1. Randomly choose a link $i-j$.
2. Update its state to the one of the majority of links in its first neighborhood (two links are considered first neighbors if they are attached to a common node). In case of a tie, the state of the link is randomly chosen
The time unit is set to $N$ basic steps so that for each node, on the average, the state of one of its links is updated per unit time. This dynamics corresponds to the usual zero temperature Glauber dynamics.
There exist two trivial absorbing ordered configurations, for which all the links in the system have the same state. The dynamics tend to order the system locally. We investigate whether, depending on the topology of the network, the dynamics orders the system globally or if the system reaches asymptotic disordered configurations with coexistence of both states. We will also characterize the transient dynamics towards these asymptotic configurations. For these purposes we will consider the following quantities characterizing the network and its links dynamics\
$k_i$
: [Degree of node $i$.]{}
$k^{A/B}_i$
: [ Number of $A/B$ edges connecting node $i$. Obviously the sum of both types of links is the degree of the node, $k^{A}_i+k^{B}_i=k_i$.]{}
$\rho$
: [ Density of nodal interfaces: It serves as order parameter. It gives a local measure of the level of order in the system. Take a node of degree $k_i$. We can map it to a set of $k_i$ nodes which represent the links attached to that node. Those links are all neighbors of each other through node $i$, therefore they form a fully connected subgraph of size $k_i$. The order parameter is just the density of ties in that graph which connect links of the original network holding different states. Therefore the order parameter is zero only for the fully ordered configurations. In section \[discusion\] we will come back to this mapping of links into nodes. $$\rho=\frac{\sum_{i=1}^Nk_i^Ak_i^B}{\sum_{i=1}^Nk_i(k_i-1)/2},$$]{}
$b_i$
: [Link heterogeneity index of node $i$. It is a node characterization giving a measure of the likelihood of a node to have $A$ or $B$ links ($b_i=+1$ or $b_i=-1$ for all links of the same type, $b_i=0$ for a completely symmetric case). $$b_i=\frac{k_i^A-k_i^B}{k_i}.$$]{}
$P(b,t)$
: [Link heterogeneity index distribution, probability that a randomly chosen node has link heterogeneity index $b$ at time $t$.]{}
$S(t)$
: [Survival probability, probability that a realization of the stochastic link dynamics has not reached a fully ordered configuration at time $t$.]{}
FULLY CONNECTED NETWORK
=======================
We first consider a fully connected network in which every node is connected to every other node so that $L=N(N-1)/2$. Note however that every link is not a first neighbor of every other link.
Time evolution
--------------
Fig. \[average\_FC\] shows the time evolution of the ensemble average of the order parameter $\langle\rho\rangle$ and the survival probability $S(t)$ for random initial conditions. The average order parameter shows a decay towards a plateau, indicating that the absorbing ordered configurations are not always reached. Comparing this result with the survival probability, which also saturates at a certain value after a transient, we conclude that the plateau in the average order parameter is due to realizations which get frozen in a configuration with coexistence of states. The analysis of single realizations of the link dynamics (lower panel of Fig. \[average\_FC\]) shows smooth dynamics to an asymptotic state in which the order parameter is frozen. In the following we investigate the characteristics of these frozen asymptotic configurations.\
Asymptotic configurations
-------------------------
We checked that all asymptotic configurations are frozen, *i.e.* the order parameter $\rho$ and the densities of links in each state reach a value that stays constant over time. The probability of having a certain value of $\rho$ in the asymptotic configurations, $\rho_{\infty}$, is plotted in Fig. \[CG\_final\_rho\]. We observe a very heterogeneous set of possible final configurations in addition of the most probable ordered configuration ($\rho=0$). For classifying the disordered frozen configurations we use the number $n_b$ of elements in the set of link heterogeneity indices present in each configuration. The limiting case $n_b=1$ corresponds to the ordered configuration in which all nodes have the same heterogeneity index $b=1$ or $b=-1$. The case $n_b=2$ corresponds to a family of asymptotic configurations where the number of different link heterogeneity indices of the nodes is two, and therefore the nodes can be divided into two groups. These configurations are depicted in Fig. \[cg\_frozen\].a.
### Simplest frozen configurations
The simplest frozen configurations on a fully connected network are of the type shown in Fig. \[cg\_frozen\]a. They consist of a set of $k$ nodes that have only links in one state (red links), and the rest of nodes, $N-k$, having all their links in the other state (blue links), except for the links with the $k$ nodes of the first set. It is clear that for this kind of configuration there are only two types of nodes in terms of link heterogeneity index. The group of size $k$ having $|b|=1$ and therefore contributing to the asymptotic $p(b,t=\infty)$ in the peaks $b=\pm1$ (see Fig.\[evol\_average\_pdeb\] a), and another group of size $N-k$ with $|b|=(2k-N-1)/(N-1)$. Therefore, for these configurations $n_b=2$.\
![(Color online) a) Simple frozen configurations in a fully connected network. b) Frozen configuration with $n_b=3$ on a fully connected network. The states of the links are encoded by color and line style, so there are red solid lines and blue dashed lines representing both states.\[cg\_frozen\]](FC_frozen_2)
These configurations can be proven to be frozen for a range of sizes $k$. For this purpose one has to find how many of the neighboring links of a given link are in each state and impose that links in state A (B) have more A (B) neighbors than B (A) neighbors. In this way one easily concludes that configurations as the one in Fig. \[cg\_frozen\].a) are frozen whenever $$1<k<N/2-1.$$ These solutions exist for $N>4$. In Fig. \[monofraction\] we show the probability density to reach a configuration with a certain fraction $k/N$ with $|b|=1$, given that the asymptotic configuration is of the type shown in Fig. \[cg\_frozen\]a. All the possible configurations can be reached from random initial conditions.
### Other asymptotic configurations
Figure \[numbil\] shows the probability of reaching an asymptotic configuration with a certain number of different link heterogeneity indices $n_b$ in the system. The ordered configurations $n_b=1$ and the ones with $n_b=2$ described above are the most probable. An example of a configuration with $n_b=3$ is shown in Fig. \[cg\_frozen\]b. These configurations have $k$ nodes with $|b|=1$, $l$ nodes with $b=(2k-N+1)/(N-1)$ and $N-k-l$ nodes with $b=(N-2l-1)/(N-1)$. Following the same argument used for $n_b=2$ configurations, we can conclude that $n_b=3$ configurations are frozen provided that\
$$\begin{aligned}
k&>1\nonumber \\
l&<N/2-1\nonumber \\
k&<N/2-1\nonumber \\
l&>k+1\nonumber\end{aligned}$$ When $n_b$ is increased, the frozen configurations become structurally more complicated, and are much less probable. Empirically we have found that always a group of agents with $|b|=1$ appears. To characterize a frozen solution with $n_b$ we need $n_b-1$ parameters and $n_b(n_b+1)/2$ inequalities, which arise imposing that the state of every link is frozen and give a boundary for the possible architecture of those configurations.\
![(Color online) Probability of reaching a frozen configuration with a certain number of different link heterogeneity indices $n_b$, starting from random initial conditions on a complete graph. Sizes are $N=100$ (black circles), $N=300$ (red squares) and $N=600$ (blue diamonds), and the statistics are over $10^5$ realizations of the system.\[numbil\]](CG_nb_stats_bocetos){width="50.00000%"}
Link heterogeneity index distribution
-------------------------------------
Figure \[evol\_average\_pdeb\].a shows the time evolution of the link heterogeneity index distribution: We observe that it evolves from a distribution peaked around $b=0$ for random initial conditions, to a bimodal distribution peaked around $b=\pm1$, with a quite homogeneous probability of having any link heterogeneity index. This statistics characterized by this distribution includes the most probable realizations that reach ordered states but also others which freeze in configurations with nodes with different possible values of $b$, as discussed in the characterization of the asymptotic configurations. Note that both type of realizations contribute to the peaks at $b=\pm1$ since frozen disordered asymptotic configuration have at least one group of agent with $b=\pm1$.\
SQUARE LATTICE
==============
In order to account for local interaction effects we first consider a square lattice with periodic boundary conditions. We used only square lattices with a square number of sites, *i.e.* $N=l^2$ with $l$ being the length of the side of the network.
Time evolution
--------------
Fig. \[average\_2D\] shows the time evolution the ensemble average of the order parameter $\langle\rho\rangle$ and the survival probability $S(t)$ for random initial conditions. We observe a qualitative behavior very similar to the one in Fig. \[average\_FC\] for a fully connected network, *i.e.* $\langle \rho\rangle$ and $S(t)$ decay smoothly to a plateau value. Together with the plot of single realizations of the stochastic dynamics (lower panel of Fig. \[average\_2D\] ) this indicates that some of the realizations reach an asymptotic ordered state, while others get trapped in a disordered configuration for which the order parameter remains constant for all times. We have found only three different types of realizations characterized by their asymptotic configurations, as we discuss next.\
Asymptotic configurations
-------------------------
The probability of reaching one of the three main possible asymptotic configurations, characterized by their value of the order parameter, is shown in Fig. \[square\_final\_rho\]. These configurations are depicted in Fig. \[square\_asymp\]. Note that these are the asymptotic configurations starting from random initial conditions and using periodic boundary conditions.
![(Color online) Probability of reaching a given asymptotic value of the order parameter on a square lattice with periodic boundary conditions and starting from random initial conditions. There are three different possible configurations, namely ordered state, horizontal/vertical stripes and diagonal stripes. Sizes are $N=2500$ (black circles), $N=3600$ (red squares) and $N=4900$ (blue diamonds). Statistics computed from $10^4$ realizations. \[square\_final\_rho\]](2D_final_rho_bocetos){width="50.00000%"}
- *Ordered configurations:* All links are in the same state and $\rho=0$.
- *Dynamically trapped configurations*, where the order parameter remains constant, $\rho=1/\sqrt{N}$, but the densities of links in each state fluctuate around a certain value. These configurations form vertical/horizontal stripes, as shown in Fig. \[square\_asymp\]a. These configurations are dynamical traps from which the system cannot reach the ordered state: links in the boundaries of the stripes continue to blink without changing the value of the order parameter. Single stripe are the configurations reached from random initial conditions. However configurations with a larger number of stripes ( and thus a value of the order parameter which is a multiple of $1/\sqrt{N}$) are also dynamical traps of the model.
- *Frozen configurations*, where the order parameter and the densities of links in each state remain constant. Configurations reached from random initial conditions are single diagonal stripe as the one shown in Fig. \[square\_asymp\]b, with a value of the order parameter $\rho=\frac{4}{3\sqrt{N}}$. There are other frozen configurations for our dynamical model which we have not observed in our simulations with random initial conditions. These include multiple diagonal stripes and a combination of diagonal front that we call percolating diamond (Fig. \[square\_asymp\]c): It contains a square of links in one state, rotated an angle of 45 degrees, surrounded by links in the opposite state and which percolates through the network. For the percolating diamond configuration $\rho=\frac{4(\sqrt{N}-1)}{3N}$.
Note that given an asymptotic configuration and the size of the network the value of the order parameter $\rho$ can be known and is the same independently of the width of the stripes in the stripe patterns, as the order parameter only contributes in the boundaries of the stripes.
![(Color online) Different asymptotic disordered configurations on a square lattice with periodic boundary conditions. a) Vertical/horizontal single stripe. The gray dash-dotted links keep changing state forever, while all other links are in a frozen state. b) Diagonal single stripe. All links are frozen. c) Percolating diamond. All links are frozen.\[square\_asymp\]](square_asymp_accessible){width="50.00000%"}
Link heterogeneity index distribution
-------------------------------------
For a square lattice the link heterogeneity index takes values $b=\pm1, \pm0.5, 0$. The evolution of the distribution $P(b,t)$ is shown in Fig. \[evol\_average\_pdeb\]b. It evolves from an initial peak at $b=0$ to a distribution with two peaks at $b=\pm1$ and a minimum value at $b=0$. This evolution can be understood from the asymptotic configurations described above: The two peaks at $b=\pm1$ originate in the most probable ordered configurations, but also on the large percentage of nodes with $b=\pm1$ in the two other possible asymptotic configurations. The values $b=\pm0.5$ appear only in the second most probable asymptotic configuration: vertical/horizontal single stripe. For these configurations there are $4\sqrt{N}$ nodes whose heterogeneity index keeps jumping from $b=\pm1$ and $b=\pm0.5$. Last, the probability of having nodes with $b=0$ comes from the third possible asymptotic configuration, diagonal single stripe. In this configuration $2\sqrt{N}-2$ nodes have $b=0$ and the other nodes are divided into two equal groups with $b=1$ and $b=-1$.
RANDOM NETWORKS
===============
In order to account for the role of network heterogeneity we finally consider the link dynamics model on Erdös-Renyi random networks.
Time evolution
--------------
Proceeding as in the previous cases we show in Fig. \[average\_ERm5\] the time evolution of the ensemble average order parameter. The survival probability (not shown) is one at all times, except for small systems or networks of high average degree where it tends to a fully connected like behavior. Our results indicate that all stochastic realizations of the dynamics reach an asymptotic disordered configuration with a constant value of $\rho$.
Asymptotic configurations
-------------------------
We observe a large variety of asymptotic configurations characterized by different values of the order parameter $\rho$, as shown in Fig. \[ER\_final\_rho\]. Increasing the average degree of the network, the distribution of final values of $\rho$ approaches the one for a fully connected network (Fig. \[CG\_final\_rho\]): The distribution develops a peak that moves towards $\rho=0$ and another peak near the maximum asymptotic order parameter value that is accessible starting from random initial conditions.\
For random networks we also find three kinds of asymptotic configurations:
- *Ordered configurations:* All links are in the same state and $\rho=0$. This configuration is only observed in small systems or in systems with high average degree (close to fully connected network).
- *Dynamically trapped configurations:* the order parameter remains constant, but the densities of links in each state vary with time.
- *Frozen configurations*, where the order parameter and the densities of links in each state remain constant.
It is possible to identify some basic mechanisms leading to the observed traps. Among them:
- *The role of hubs:* If a node $i$ is such that $k_i\gg k_j$ for any neighboring node $j$, then the links attached to node $i$ usually end up sharing all the same state, which in most cases is the one of the initial majority state in that set of links. This effect creates *frozen* links, *i.e.* links which do not change state. Initial conditions and the particular topology of the realization will determine how frequent is this effect and whether this leads or not to an ordered configuration.
- *Dynamics conserving the value of $\rho$:* There exist changes of the state of the links which do not cause a change in the value of $\rho$. These changes are those for which the link changing state has a symmetric environment, with the same number of neighbors in each state as shown in Fig. \[isoenergetic2\]. This situation is the one also found in a square lattice Fig. \[square\_asymp\].a. This kind of phenomenon can appear in more complex forms, as shown in Fig. \[animation\]. There one can see that the order parameter is frozen after approximately $10$ time steps, but the configuration of the system keeps changing, as can be seen from the snapshots of the system configuration at different times.
This behavior of the model was already pointed out in Ref. [@redner_ising3d] for the Ising model in three dimensions, where the system wanders ad infinitum on a connected set of equal-energy blinker states. In Ref. [@castello_epl] it was also found how community structure can induce topological traps just as is the case in the present work.
![(Color online) Example of change of state which changes the densities of blue and red links but conserves the value of the order parameter $\rho$. Independently of the state of the grey link this motif will contribute to the order parameter of the whole system with $\rho=1/5$.\[isoenergetic2\]](blinker_random)
![(Color online) One realization on a small random network of size $N=20$. Top left panel shows the evolution of the order parameter, which freezes after approximately $10$ time steps. The other panels show the configuration of the system at different times. The color of the nodes reflects their link heterogeneity index. Red (blue) is for having all links in the red (blue) option, white is for having half of the links in each color. The changes in the configuration do not affect the value of the order parameter. For example the only difference between the configuration at $t=20$ and the one at $t=120$ is the state of a single link. If we count we can see that the link has the same number of neighbors in each state. One can check that all the changes of state are of the type depicted in Fig. \[isoenergetic2\]\[animation\]](rho_seems_frozen){width="49.00000%"}
Link heterogeneity index distribution
-------------------------------------
The evolution of the distribution of link heterogeneity indices in random networks is shown in Fig. \[evol\_average\_pdeb\]c. The initial distribution is broad, but smoothly peaked around $b=0$. This evolves to a bimodal distribution peaked around $b=\pm1$. The hubs of the network are prone to become nodes with $b=\pm1$, which in turn pulls more nodes to this value of $b$. The fact that links can be in frozen states for different parts of the network implies that between patches of ordered *domains*, there are nodes with any value of the link heterogeneity index. This contributes to the almost flat distribution of $b$ values between the two peaks. Blinking links in dynamically trapped configurations also contribute to the broad distribution of intermediate values of $b$.
SUMMARY and DISCUSSION {#discusion}
======================
The study of a majority rule for the dynamics of two equivalent link states in a fixed network uncovers a set of non-trivial asymptotic configurations which are generally not present when studying the classical node-based majority rule dynamics. The characterization given of the asymptotic configurations in fully connected networks, square lattices and Erdös-Renyi random networks provides the basis for the understanding of the evolution of the link heterogeneity index distribution. For a fully connected network and for a square lattice we have fully characterized the asymptotic configurations reached from random initial conditions. In a fully connected network we have found large heterogeneity in the asymptotic configurations. All these configurations, classified by the number $n_b$ of heterogeneity indexes present in the configuration, are frozen. Note that for the corresponding node-dynamics in the same network only an asymptotic ordered configuration is found ($n_b=1$). In a square lattice we have found asymptotic configurations which are ordered, frozen and disordered, or dynamically trapped. The latter do not have an analog in the corresponding node dynamics. In the case of Erdös-Renyi random networks we have described the mechanisms leading to the existence of very heterogeneous asymptotic configurations which are either frozen or dynamical traps.\
It is clear that a link-dynamics model can be mapped into an equivalent node-based problem by changing the network of interaction. The node-equivalent network is the line-graph [@Krawczyk2010; @Manka-Krason2009; @Rooij1965] of the original network. The line-graph is a network where the links of the original network are represented by a node and are connected to those nodes that represent links that were first neighbors in the original network. This mapping of the problem has not been pursued here since it obscures our original motivation and, given the complexity of the line graphs of the networks considered here, it has been found not to be particularly useful for a quantitative description of the dynamics. However, the mapping does provide additional qualitative understanding of our findings: The line graph is a network with higher connectivity since all links that converged originally in a node form a fully connected subgraph in the line-graph, as clearly seen in the line-graph of a fully connected network or a square lattice. This results in an increased cliqueness of the line graph, as compared to the original network. Such cliqueness is behind the topological traps that give rise to the wide range of possible asymptotic configurations that we find for the link-dynamics. In addition, the mapping of a hub of the original network in the corresponding line-graph also helps understanding the different role played by hubs in node or link-based dynamics: as discussed in Section IV, hubs tend to freeze link states in their neighborhood.\
The link heterogeneity index is a useful way of characterizing nodes in a given link configuration. For example in node based models of language competition, a node can be in state $A$ or $B$ corresponding to two competing languages, and bilingualism can only be introduced through a third node bilingual state $AB$ [@castello2006]. In the framework of link dynamics, state $A$ or $B$ characterizes the language used in a given interaction between two individuals, and the link heterogeneity index is a natural measure of the degree of bilingualism of each individual (node). Continuing with this example, a next step is to consider the mixed dynamics of language competence (node dynamics) and language use (link dynamics). In general, consideration of the coevolution of link and node states is a natural framework that emerges in the study of collective behavior of interacting units. In physical terms, the states of the interacting particles are coupled to the state of the field that carries the interaction. Another possible avenue of research is the addition of more realistic features to the model, such as the temporal patterns of human interactions [@juanfpre; @tempnets], which introduces heterogeneity in the activation of different links.\
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We acknowledge financial support from the MINECO (Spain) and FEDER (EU) through projects FISICOS (FIS2007-60327) and MODASS (FIS2011-24785). J.F.-G. acknowledges a predoctoral fellowship from the Government of the Balearic Islands through the Conselleria d’Educació, Cultura i Universitats with funding from the ESF. X.C. acknowledges the Juan de la Cierva programe from the Spanish Government.
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author:
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K. C. Nowack$^{\dagger,1,\ast}$, F. H. L. Koppens$^{\dagger,1}$, Yu. V. Nazarov$^{1}$, L. M. K. Vandersypen$^{1,\ast}$\
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title: Coherent Control of a Single Electron Spin with Electric Fields
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Manipulation of single spins is essential for spin-based quantum information processing. Electrical control instead of magnetic control is particularly appealing for this purpose, since electric fields are easy to generate locally on-chip. We experimentally realize coherent control of a single electron spin in a quantum dot using an oscillating electric field generated by a local gate. The electric field induces coherent transitions (Rabi oscillations) between spin-up and spin-down with $90^\circ$ rotations as fast as $\sim$55ns. Our analysis indicates that the electrically-induced spin transitions are mediated by the spin-orbit interaction. Taken together with the recently demonstrated coherent exchange of two neighboring spins, our results demonstrate the feasibility of fully electrical manipulation of spin qubits.
Spintronics and spin-based quantum information processing provide the possibility to add new functionality to today’s electronic devices by using the electron spin in addition to the electric charge [@awschalom02]. In this context, a key element is the ability to induce transitions between the spin-up and spin-down states of a localized electron spin, and to prepare arbitrary superpositions of these two basis states. This is commonly accomplished by magnetic resonance, whereby bursts of a resonant oscillating magnetic field are applied [@poole]. However, producing strong oscillating magnetic fields in a semiconductor device requires specially designed microwave cavities [@simovic06] or microfabricated striplines [@koppens06], and has proven to be challenging. In comparison, electric fields can be generated much more easily, simply by exciting a local gate electrode. In addition, this allows for greater spatial selectivity, which is important for local addressing of individual spins. It would thus be highly desirable to control the spin by means of electric fields.
Although electric fields do not couple directly to the electron spin, indirect coupling can still be realized by placing the spin in a magnetic field gradient [@tokura06] or in a structure with a spatially varying $g$-tensor, or simply through spin-orbit interaction, present in most semiconductor structures [@bychkov84; @dresselhaus55]. Several of these mechanisms have been employed to electrically manipulate electron spins in two dimensional electron systems [@kato03a; @kato03b; @salis01; @schulte05], but proposals for coherent electrical control at the level of a single spin [@golovach06; @levitov03; @tokura06; @debald05; @walls07] have so far remained unrealized.
We demonstrate coherent single spin rotations induced by an oscillating electric field. The electron is confined in a gate-defined quantum dot (see Fig. 1A) and we use an adjacent quantum dot, containing one electron as well, for read-out. The ac electric field is generated through excitation of one of the gates that forms the dot, thereby periodically displacing the electron wavefunction around its equilibrium position (Fig. 1B).
The experiment consists of four stages (Fig. 1C). The device is initialised in a spin-blockade regime where two excess electrons, one in each dot, are held fixed with parallel spins (spin triplet), either pointing along or opposed to the external magnetic field (the system is never blocked in the triplet state with anti-parallel spins, because of the effect of the nuclear fields in the two dots combined with the small interdot tunnel coupling, see [@koppens07b] for full details). Next, the two spins are isolated by a gate voltage pulse, such that electron tunneling between the dots or to the reservoirs is forbidden. Then, one of the spins is rotated by an ac voltage burst applied to the gate, over an angle that depends on the length of the burst [@SOM] (most likely the spin in the right dot, where the electric field is expected to be strongest). Finally, the read-out stage allows the left electron to tunnel to the right dot if and only if the spins are anti-parallel. Subsequent tunneling of one electron to the right reservoir gives a contribution to the current. This cycle is continuously repeated, and the current flow through the device is thus proportional to the probability of having antiparallel spins after excitation.
To demonstrate that electrical excitation can indeed induce single-electron spin flips, we apply a microwave burst of constant length to the right side gate and monitor the average current flow through the quantum dots as a function of external magnetic field $\mathbf{B_\mathrm{ext}}$ (Fig. 2A). A finite current flow is observed around the single-electron spin resonance condition, i.e. when $|\mathbf{B_\mathrm{ext}}| = h f_\mathrm{ac} / g
\mu_\mathrm{B}$, with $h$ Planck’s constant, $f_\mathrm{ac}$ the excitation frequency, and $\mu_\mathrm{B}$ the Bohr magneton. From the position of the resonant peaks measured over a wide magnetic field range (Fig. 2B) we determine a $g$-factor of $|g|=0.39 \pm
0.01$, which is in agreement with other reported values for electrons in GaAs quantum dots [@hansonrmp06].
In addition to the external magnetic field, the electron spin feels an effective nuclear field $B_\mathrm{N}$ arising from the hyperfine interaction with nuclear spins in the host material and fluctuating in time [@khaetskii02; @merkulov02]. This nuclear field modifies the electron spin resonance condition and is generally different in the left and right dot (by $\Delta B_\mathrm{N}$). The peaks shown in Fig. 2A are averaged over many magnetic field sweeps and have a width of about 10-25 mT. This is much larger than the expected linewidth, which is only 1-2 mT given by the statistical fluctuations of $B_\mathrm{N}$ [@johnson05a; @koppens05]. Looking at individual field sweeps measured at constant excitation frequency, we see that the peaks are indeed a few mT wide (see Fig. 2C), but that the peak positions change in time over a range of $\sim$ 20mT. Judging from the dependence of the position and shape of the averaged peaks on sweep direction, the origin of this large variation in the nuclear field is most likely dynamic nuclear polarization [@baugh07; @rudner06; @koppens06; @klauser06; @laird07].
In order to demonstrate coherent control of the spin, the length of the microwave bursts was varied, and the current level monitored. In Fig. 3A we plot the maximum current per magnetic field sweep as a function of the microwave burst duration, averaged over several sweeps (note that this is a more sensitive method than averaging the traces first and then taking the maximum)[@SOM]. The maximum current exhibits clear oscillations as a function of burst length. Fitting with a cosine function reveals a linear scaling of the oscillation frequency with the driving amplitude (Fig. 3B), a characteristic feature of Rabi oscillations, and proof of coherent control of the electron spin via electric fields.
The highest Rabi frequency we achieved is $\sim 4.7$MHz (measured at $f_\mathrm{ac}=15.2$GHz) corresponding to a $90^\circ$ rotation in $\sim 55$ ns, which is only a factor of two slower than those realized with magnetic driving [@koppens06]. Stronger electrical driving was not possible because of photon-assisted-tunneling. This is a process whereby the electric field provides energy for one of the following transitions: tunneling of an electron to a reservoir or to the triplet with both electrons in the right dot. This lifts spin-blockade, irrespective of whether the spin resonance condition is met.
Small Rabi frequencies could be observed as well. The bottom trace of Fig. 3A shows a Rabi oscillation with a period exceeding $1.5
\mu$s (measured at $f_\mathrm{ac}=2.6$GHz), corresponding to an effective driving field of only about 0.2mT, ten times smaller than the statistical fluctuations of the nuclear field. The reason the oscillations are nevertheless visible is that the dynamics of the nuclear bath is slow compared to the Rabi period, resulting in a slow power law decay of the oscillation amplitude on driving field [@koppens07].
We next turn to the mechanism responsible for resonant transitions between spin states. First, we exclude a magnetic origin as the oscillating magnetic field generated upon excitation of the gate is more than two orders of magnitude too small to produce the observed Rabi oscillations with periods up to $\sim 220$ns, which requires a driving field of about 2mT [@SOM]. Second, we have seen that there is in principle a number of ways in which an ac electric field can cause single spin transitions. What is required is that the oscillating electric field give rise to an effective magnetic field, $\mathbf{B_\mathrm{eff}}(t)$, acting on the spin, oscillating in the plane perpendicular to $\mathbf{B_\mathrm{ext}}$, at frequency $f_\mathrm{ac} = g
\mu_\mathrm{B} |\mathbf{B_\mathrm{ext}}| / h$. The $g$-tensor anisotropy is very small in GaAs so g-tensor modulation can be ruled out as the driving mechanism. Furthermore, in our experiment there is no external magnetic field gradient applied, which could otherwise lead to spin resonance [@tokura06]. We are aware of only two remaining possible coupling mechanisms: spin-orbit interaction and the spatial variation of the nuclear field.
In principle, moving the wavefunction in a nuclear field gradient can drive spin transitions [@erlingsson02; @tokura06] as was recently observed [@laird07]. However, the measurement of each Rabi oscillation took more than one hour, much longer than the time during which the nuclear field gradient is constant ($\sim 100 \mu$s - few s). Because this field gradient and therefore, the corresponding effective driving field slowly fluctuates in time around zero, the oscillations would be strongly damped, regardless of the driving amplitude [@laird07]. Possibly a (nearly) static gradient in the nuclear spin polarization could develop due to electron-nuclear feedback. However, such polarization would be parallel to $\mathbf{B_\mathrm{ext}}$ and can thus not be responsible for the observed coherent oscillations.
In contrast, spin-orbit mediated driving can induce coherent transitions [@golovach06], which can be understood as follows. The spin-orbit interaction in a GaAs heterostructure is given by $H_\mathrm{SO}=\alpha(p_x \sigma_y - p_y \sigma_x) + \beta(-p_x
\sigma_x + p_y \sigma_y)$, where $\alpha$ and $\beta$ are the Rashba and Dresselhaus spin-orbit coefficient respectively, and $p_{x,y}$ and $\sigma_{x,y}$ are the momentum and spin operators in the $x$ and $y$ directions (along the $[100]$ and $[010]$ crystal directions respectively). As suggested in [@levitov03], the spin-orbit interaction can be conveniently accounted for up to the first order in $\alpha, \beta$ by applying a (gauge) transformation, resulting in a position-dependent correction to the external magnetic field. This effective magnetic field, acting on the spin, is proportional and orthogonal to the field applied: $$\mathbf{B}_\mathrm{eff}(x,y) = \mathbf{n}\otimes \mathbf{B}_\mathrm{ext}; \ n_x = \frac{2m^*}{\hbar} \left(-\alpha
y -\beta x\right);\; n_y = \frac{2m^*}{\hbar} \left(\alpha x +\beta y\right);\; n_z=0 \label{eq1}$$
An electric field $\mathbf{E}(t)$ will periodically and adiabatically displace the electron wave function (see Fig. 1B) by $\mathbf{x}(t)=(e l_\mathrm{dot}^2/\Delta) \mathbf{E}(t)$, so the electron spin will feel an oscillating effective field ${\mathbf
B}_\mathrm{eff}(t)\perp {\mathbf B}_\mathrm{ext}$ through the dependence of ${\mathbf B}_\mathrm{eff}$ on the position. The direction of $\mathbf{n}$ can be constructed from the direction of the electric field as shown in Fig. 4C and together with the direction of $\mathbf{B_\mathrm{ext}}$ determines how effectively the electric field couples to the spin. The Rashba contribution always gives $\mathbf{n}\bot \mathbf{E}$, while for the Dresselhaus contribution this depends on the orientation of the electric field with respect to the crystal axis. Given the gate geometry, we expect the dominant electric field to be along the double dot axis (see Fig. 1A) which is here either the $[110]$ or $[1\bar{1}0]$ crystallographic direction. For these orientations, the Dresselhaus contribution is also orthogonal to the electric field (see Fig. 4C). This is why both contributions will give $\mathbf{B}_{\mathrm{eff}}\neq 0$ and lead to coherent oscillations in the present experimental geometry, where $\mathbf{E} \parallel
\mathbf{B}_\mathrm{ext}$. Note that in [@laird07], a very similar gate geometry was used, but the orientation of $\mathbf{B}_\mathrm{ext}$ was different, and it can be expected that $\mathbf{E} \perp \mathbf{B}_\mathrm{ext}$. In that experiment, no coherent oscillations were observed, which is consistent with the considerations here.
An important characteristic of spin-orbit mediated driving is the linear dependence of the effective driving field on the external magnetic field which follows from Eq. 1 and is predicted in [@golovach06; @levitov03; @khaetskii01]. We aim at verifying this dependence by measuring the Rabi frequency as a function of the resonant excitation frequency (Fig. 4A), which is proportional to the external magnetic field. Each point is rescaled by the estimated applied electric field (Fig. 4B). Even at fixed output power of the microwave source, the electric field at the dot depends on the microwave frequency due to various resonances in the line between the microwave source and the gate (caused by reflections at the bonding wires and microwave components). However, we use the photon-assisted-tunneling response as a probe for the ac voltage drop across the interdot tunnelbarrier, which we convert into an electric field amplitude by assuming a typical interdot distance of 100 nm. This allows us to roughly estimate the electric field at the dot for each frequency [@SOM]. Despite the large error bars, which predominantly result from the error made in estimating the electric field, an overall upgoing trend is visible in Fig. 4A.
For a quantitative comparison with theory, we extract the spin-orbit strength in GaAs, via the expression of the effective field $\mathbf{B_\mathrm{eff}}$ perpendicular to $\mathbf{B_\mathrm{ext}}$ for the geometry of this experiment [@golovach06] $$|\mathbf{B}_\mathrm{eff}(t)| = 2|\mathbf{B_\mathrm{ext}}| \frac{l_\mathrm{dot}}{l_\mathrm{SO}}
\frac{e| \mathbf{E}(t)|l_\mathrm{dot}}{\Delta},$$ with $l_\mathrm{SO}$ the spin-orbit length (for the other definitions see Fig. 1B). Here, $l_\mathrm{SO}^{-1}=m^*(\alpha
\mp\beta)/\hbar$ for the case with the gate symmetry axis along $[1\bar{1}0]$ or $[110]$ respectively. Via $f_\mathrm{Rabi}=(g\mu_\mathrm{B} |\mathbf{B_\mathrm{eff}}|)/2h $, the confidence interval of the slope in Fig. 4A gives a spin-orbit length of $28-37 \mu$m (with a level splitting $\Delta$ in the right dot of 0.9 meV extracted from high bias transport measurements). Additional uncertainty in $l_\mathrm{SO}$ is due to the estimate of the interdot distance and the assumption of a homogenous electric field, deformation effects of the dot potential [@walls07] and extra cubic terms in the Hamiltonian [@dresselhaus55]. Still, the extracted spin-orbit length is of the same order of magnitude as other reported values for GaAs quantum dots [@hansonrmp06].
Both the observed trend of $\mathbf{B_\mathrm{eff}}$ with $f_\mathrm{ac}$ and the extracted range for $l_\mathrm{SO}$ are consistent with our supposition (by elimination of other mechanisms) that spin transitions are mediated by spin-orbit interaction. We note that also for relaxation of single electron spins in which electric field fluctuations from phonons couple to the spin, it is by now well established that the spin-orbit interaction is dominant at fields higher than a few 100 mT [@erlingsson02; @khaetskii01; @golovach06; @hansonrmp06]. It can thus be expected to be dominant for coherent driving as well.
The electrically driven single spin resonance reported here, combined with the so-called $\sqrt{{\sc SWAP}}$ gate based on the exchange interaction between two neighbouring spins [@petta05], brings all-electrical universal control of electron spins within reach. While the $\sqrt{{\sc SWAP}}$ gate already operates on sub-nanosecond timescales, single-spin rotations still take about one hundred nanoseconds (the main limitation is photon-assisted-tunneling). Faster operations could be achieved by suppressing photon-assisted-tunneling (e.g. by increasing the tunnel barriers or operating deeper into Coulomb blockade), by working at still higher magnetic fields, by using materials with stronger spin-orbit interaction or through optimized gate designs. Furthermore, the electrical control offers the potential for spatially selective addressing of individual spins in a quantum dot array, since the electric field is produced by a local gate. Finally, we note that the spin rotations were realized at magnetic fields high enough to allow for single-shot read-out of a single spin [@elzerman04], so that both elements can be integrated in a single experiment.
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We thank L. P. Kouwenhoven, C. Barthel, E. Laird, M. Flatté, I. T. Vink and T. Meunier for discussions; R. Schouten, B. van der Enden and R. Roeleveld for technical assistance and J. H. Plantenberg and P. C. de Groot for help with the microwave components. Supported by the Dutch Organization for Fundamental Research on Matter (FOM) and the Netherlands Organization for Scientific Research (NWO).
![(**A**) Scanning electron microscope image of a device with the same gate structure as the one used in this experiment. Metallic TiAu gates are deposited on top of a GaAs heterostructure which hosts a 2DEG 90 nm below the surface. Not shown is a coplanar stripline on top of the metallic gates, separated by a dielectric (not used in this experiment, see also [@koppens06]). In addition to a dc voltage we can apply fast pulses and microwaves to the right side gate (as indicated) through a home made bias-tee. The orientation of the in-plane external magnetic field is as shown. (**B**) The electric field generated upon excitation of the gate displaces the center of the electron wavefunction along the electric field direction and changes the potential depth. Here, $\Delta$ is the orbital energy splitting, $l_{\mathrm{dot}}=\hbar/\sqrt{m^*\Delta}$ the size of the dot, $m^*$ the effective electron mass, $\hbar$ the reduced Planck constant and $\mathbf{E}(t)$ the electric field. (**C**) Schematic of the spin manipulation and detection scheme, controlled by a combination of a voltage pulse and burst, $V(t)$, applied to the right side gate. The diagrams show the double dot, with the thick black lines indicating the energy cost for adding an extra electron to the left or right dot, starting from $(0,1)$, where $(n,m)$ denotes the charge state with n and m electrons in the left and right dot. The energy cost for reaching $(1,1)$ is (nearly) independent of the spin configuration. However, for $(0,2)$, the energy cost for forming a singlet state (indicated by $S(0,2)$) is much lower than that for forming a triplet state (not shown in the diagram). This difference is exploited for initialization and detection, as explained further in the main text.](Fig1.eps){width="4in"}
{width="4in"}
{width="4in"}
![(**A**) Rabi frequency rescaled with the applied electric field for different excitation frequencies. The errorbars are given by $f_{\mathrm{Rabi}}/E\cdot\sqrt{(\delta E/E)^2+(\delta f_{\mathrm{Rabi}}/f_{\mathrm{Rabi}})^2}$ where $\delta f_{\mathrm{Rabi}}$ and $\delta E$ are the error in the Rabi frequency and electric field amplitude respectively. The grey lines are the 95% confidence bounds for a linear fit through the data (weighting the datapoints by the inverse error squared). (**B**) Estimated electric field amplitudes at which the Rabi oscillations of **(**A) were measured at the respective excitation frequencies [@SOM]. (**C**) Construction of the direction of $\mathbf{n}$ resulting from the Rashba and Dresselhaus spin-orbit interaction for an electric field along $[110]$ following equation 1. The coordinate system is set to the crystallographic axis $[100]$ and $[010]$.](Fig4.eps){width="4in"}
****\
A
: Supplementary Materials and Methods
B
: Supplementary Text
- Extraction of Rabi oscillations from magnetic field sweeps
- Estimate of the electric field amplitude at the dot
- Upper bound on the ac magnetic field amplitude at the dot
C
: Supplementary Figures
D
: Supplementary References
[**A Supplementary Materials and Methods**]{}\
The GaAs/AlGaAs heterostructure from which the sample were made was purchased from Sumitomo Electric. The 2DEG has a mobility of $185 \times 10^3 \mathrm{cm}^2/\mathrm{Vs}$ at 77K, and an electron density of $4-5 \times 10^{11} \mathrm{cm}^{-2}$, measured at 30 mK with a different device than used in the experiment. Background charge fluctuations made the quantum dot behaviour excessively irregular. The charge stability of the dot was improved considerably in two ways. First, the gates were biased by +0.5 V relative to the 2DEG during the device cool-down. Next, after the device had reached base temperature, the reference of the voltage sources and IV converter (connected to the gates and the 2DEG) were biased by +2 V. This is equivalent to a -2 V bias on both branches of the coplanar stripline (CPS), which therefore (like a gate) reduces the 2DEG density under the CPS. The sample used is identical to the one in reference [@koppens06].
Based on transport measurements through the double dot, we can be nearly certain that there were only two electrons present in the double dot. Note however that the addition of two extra electrons in one of the two dots does not affect the manipulation and detection scheme.
The microwave bursts were created by sending a microwave signal generated by a Rohde & Schwarz SMR40 source through either a high isolation GaAs RF switch (Minicircuits ZASWA-2-50DR) for frequencies in the range of 10MHz to 4.6GHz or through two mixers in series (Marki Microwave M90540) for frequencies above 5GHz. The switch and the mixers were gated by rectangular pulses from an arbitrary wave form generator (Tektronix AWG520). The microwave bursts and voltage pulses generated by the marker channel of the same waveform generator were combined (splitter Pasternack PE2064) and applied to the right side gate through a home made bias-tee (rise time 150 ps and a RC charging time of $\gg$10ms at 77K).
The measurements were performed in a Oxford Instruments Kelvinox 400 HA dilution refrigerator operating at a base temperature of 38mK.\
[**B Supplementary Text**]{}\
**B.1 Extraction of Rabi oscillations from magnetic field sweeps**\
In Fig. 2C we see that at large external magnetic field, the nuclear field fluctuates over a much larger range than $A/\sqrt{N}$, where $A$ is the nuclear field experienced by the electron spin when the nuclei are fully polarized and $N$ the number of nuclei overlapping with the electron wave function. This made it impossible in the experiment to record a Rabi oscillation at constant $B_{\mathrm{ext}}$. We therefore chose to sweep the external magnetic field through the resonance. We measured a few magnetic field sweeps per microwave burst length and averaged over the max (raw data shown in Fig. S1A)imum current values reached in each sweep.
However, when extracting the Rabi oscillation by looking at the absolute maximum per magnetic field sweep, it is not obvious that the correct Rabi period $T_{\mathrm{Rabi}}=2 h/(g \mu_{\mathrm{B}}B_{\mathrm{eff}})$ is found. For instance, a burst which produces a $2 \pi$ rotation at resonance, gives a tip angle different from $2 \pi$ away from resonance.
In order to illustrate the effect more fully, Fig. S1B shows a map of the probability for flipping a spin, calculated from the Rabi formula [@poole] as a function of the detuning away from resonance and the microwave burst length. When taking for each fixed burst length the maximum probability, a saw tooth like trace is obtained (Fig. S1C). Still the positions of the maxima remain roughly at burst lengths corresponding to odd multiples of $\pi$ and the distance between maxima corresponds to the Rabi period.
In addition, we note that every data pixel in Fig. S1A is integrated for about 50ms, so it presumably represents an average over a number of nuclear configurations. This is additionally taken into account in Fig. S1D by averaging each point over a Gaussian distribution of detunings. The width of the distribution used in Fig. S1D corresponds to statistical fluctuations of the nuclear field along the direction of the external magnetic field of $1.1$mT (at a driving field of $\sim 0.8$mT). This assumes that on top of the large variation of the nuclear field, visible in Fig. 2C, which occurs on a minute time scale, the nuclear field undergoes additional statistical fluctuations on a faster time scale. Taking the maximum in Fig. S1D for each microwave burst length reveals a rather smooth Rabi oscillation (Fig. S1E) with a phase shift [@koppens07], and again with the proper Rabi period.
Presumably neither case, with and without averaging over a distribution of detunings, reflects the actual experimental situation in detail. However in the simulation the Rabi period obtained from the periodicity of the maximum probability as a function of the burst length is *independent* of the width of the gaussian distribution.
Finally, we remark that these conclusions are unchanged when considering the maximum current for each burst length (the current measures parallel spins versus anti-parallel spins) instead of the maximum probability for flipping a single spin. On this basis, we conclude that taking the maximum current value for each burst length gives us a reliable estimate of the Rabi period.\
**B.2 Estimate of the electric field amplitude at the dot**\
The electric field generated at the dot by excitation of a gate is difficult to quantify exactly. While we can estimate the power that arrives at the sample holder from the output power of the microwave source and the measured attenuation in the line, the power that arrives at the gate is generally somewhat less (the coax is connected to the gate via bonding wires). In addition, it is difficult to accurately determine the conversion factor between the voltage modulation of the gate and the electric field modulation of the dot. We here estimate the voltage drop across the interdot tunnel barrier via photon-assisted-tunneling (PAT) measurements, and extract from this voltage drop a rough indication of the electric field at the dot.
The leakage current through the double quantum dot in the spin blockade regime as a function of the detuning $\Delta_{\mathrm{LR}}$ (defined in Fig. S2A) shows at $B_{\mathrm{ext}}=0$T a peak at $\Delta_{\mathrm{LR}}=0$ due to resonant transport and a tail for $\Delta_{\mathrm{LR}}>0$ due to inelastic transport (emission of phonons) [@koppens05] (Fig. S2B). Excitation of the right side gate induces an oscillating voltage drop across the tunnel barrier between the two dots, which leads to side peaks at $\Delta_{\mathrm{LR}}=nhf_{\mathrm{ac}}, n=\pm1,\pm2,...$ away from the resonant peak (Fig. S2C). These side peaks are due to electron tunnelling in combination with absorption or emission of an integer number of photons, a process which is called photon-assisted-tunneling. In the limit where $hf_{\mathrm{ac}}$ is much smaller than the linewidth of the states $h\Gamma$ ($\Gamma$ is the tunnel rate) the individual sidepeaks cannot be resolved, whereas for higher frequencies they are clearly visible (see Fig. S2D).
More quantitatively we describe PAT by following reference [@stoof96]. An ac voltage drop $V(t)=V_{\mathrm{ac}}\cos{2\pi f_{\mathrm{ac}}t}$ across the interdot tunnel barrier modifies the tunnel rate through the barrier as $\tilde{\Gamma}(E)=\sum_{n=-\infty}^{+\infty}J_n^2(\alpha)\Gamma(E+nhf_{\mathrm{ac}})$. Here, $\Gamma(E)$ and $\tilde{\Gamma}(E)$ are the tunnel rates at energy E with and without ac voltage, respectively; $J_n^2(\alpha)$ is the square of the *n*th order Bessel function of the first kind evaluated at $\alpha=(eV_{\mathrm{ac}})/hf_{\mathrm{ac}}$, which describes the probability that an electron absorbs or emits $n$ photons of energy equal to $hf_{\mathrm{ac}}$ (with $-e$ the electron charge). Fig. S1E shows the current calculated from this model including a lorentzian broadening of the current peaks. A characteristic of the $n$-th Bessel function $J_n(\alpha)$, important here, is that it is very small for $\alpha \ll n$ (i.e. when $e V_{ac} \ll n h f_{\mathrm{ac}}$) and starts to increase around $\alpha \approx n$, implying that the number of side peaks is approximately $eV_{\mathrm{ac}}/hf_{\mathrm{ac}}$. This results in a linear envelope visible in Fig. S1E.
We extract $eV_{\mathrm{ac}}$ as the width of the region with non-zero current measured at fixed microwave frequency $f_{\mathrm{ac}}$ and amplitude $V_{\mathrm{mw}}$. Instead of this width, we can take equivalently the number of side peaks times $hf_{ac}$ (this is possible at frequencies high enough such that individual side peaks are resolved). A reasonable estimate of the error made in determining $eV_{\mathrm{ac}}$ is $\pm h f_{\mathrm{ac}}$. Another method to extract $V_{\mathrm{ac}}$ is to determine the slope of the envelope (for which a threshold current needs to be chosen) of the PAT response (see Fig. S2D). Varying the threshold gives a spread in the slope which defines the error of this method. We note that within the error bars both methods give the same result.
In order to estimate from $V_{\mathrm{ac}}$ the amplitude of the oscillating electric field at the dot, $|E|$, we assume that this voltage drops linearly over the distance between the two dot centers (a rough approximation), which is approximately 100 nm. This estimate is used in Fig. 4A in the main text, and in the approximate determination of the spin orbit length. Note that the uncertainty in this estimate of the spin-orbit length only affects the overall scaling in Fig. 4A, but not the fact that there is an up-going trend.\
**B.3 Upper bound on the ac magnetic field amplitude at the dot**\
The oscillating gate voltage produces an oscillating electric field at the dot. Here we determine an upper bound on the oscillating magnetic field that is unavoidably generated as well. Since the distance from the gate to the dot is much smaller than the wavelength (20 GHz corresponds to 1.5 cm), we do this in the near-field approximation, where magnetic fields can only arise from currents (displacement currents or physical currents).
An oscillating current can flow from the right side gate to ground via the 2DEG, the coplanar stripline [@koppens06], or the neighbouring gates (all these elements are capacitively coupled to the right side gate). We first consider the case of the stripline. The right side gate is about 100nm wide and overlaps with the coplanar stripline over a length of about 10 $\mu$m, giving an overlap area of $\approx (1 \mu\mathrm{m})^2$. The gate and stripline are separated by a 100 nm thick dielectric (calixerene [@holleitner03], $\epsilon_r=7.1$), which results in a capacitance of 0.6 fF. For a maximum voltage of 10 mV applied to the right side gate and a microwave frequency of 20 GHz, this gives a maximum displacement current through this capacitor of $\sim 1\mu$A. This is an upper bound as we neglect all other impedances in the path to ground. Even if this entire current flowed at a distance to the dot of no more than 10 nm (whether in the form of displacement currents or physical currents), it would generate a magnetic field $B_{ac}$ of only $\approx 0.02$mT, more than two orders of magnitude too small to explain the observed Rabi oscillations. In reality, the displacement current is distributed along the length of the gate, and most of the current through the gate and stripline flows at a distance very much greater than 10 nm from the dot, so $B_{ac}$ is still much smaller than 0.02 mT. The maximum magnetic field resulting from capacitive coupling to the other gates and to the 2DEG is similarly negligible.
It is also instructive to compare the power that was applied to the gate for electric excitation of the spin with the power that was applied to the microfabricated stripline for magnetic excitation [@koppens06]. For the shortest Rabi periods observed here (220 ns), the power that arrived at the sample holder was less than $\approx -36$dBm (the output power of the microwave source minus the attenuation of the microwave components in between source and sample holder, measured at 6 GHz – at higher frequencies, the attenuation in the coax lines will be still higher). In order to achieve this Rabi frequency through excitation of the stripline, more than 100 times more power ($\approx -14$ dBm) was needed directly at the stripline [@koppens06].
The upper bounds we find for the oscillating magnetic field generated along with the electric field are thus much smaller than the field needed to obtain the measured Rabi frequencies of a few MHz. We therefore exclude magnetic fields as a possible origin for our observations.
[**C Supplementary Figures**]{}\
{width="6in"}
![ (**A**) Schematic of a double dot with $\Delta_{\mathrm{LR}}$ (detuning) the difference in the energy the electron needs to access the left or right dot. (**B,C**) Current through the double dot as a function of detuning with microwaves turned off (**B**) and on (**C**). (**D**)Measured current as a function of detuning $\Delta_{\mathrm{LR}}$ and microwave amplitude $V_{\mathrm{mw}}$ at the gate at $f_{\mathrm{ac}}=15.2$GHz and 2.6GHz (applied in continuous wave). The external magnetic field is zero and therefore spin blockade is lifted due to mixing of the spin states through the fluctuating nuclear field[@koppens05]. At higher microwave amplitude ($V_{\mathrm{mw}}>0.5$mV and $1.5$mV respectively), the transition to the right dot triplet state is also visible (in the upper right corner). $V_{\mathrm{mw}}$ is determined by the estimated attenuation of the coaxial lines and the switching circuit used to create microwave bursts. (**E**) Simulated current as a function of detuning and $\alpha=eV_{\mathrm{ac}}/(hf_{\mathrm{ac}})$ ($h$ Planck’s constant) for $f_{\mathrm{ac}}=15.2$GHz and 2.6GHz respectively. It reproduces the linear envelope of the measured current as well as, qualitatively, a modulation of the current amplitude in detuning. However the asymmetry with respect to detuning visible in (**D**) as well as the observed overall increase of the current with $V_{mw}$ is not captured in this model. \[SOM2\]](SOM2.eps){width="6in"}
[**D Supplementary References**]{}
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C. Poole, [*Electron Spin Resonance, 2nd ed.*]{} (Wiley, New York, 1983).
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F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven and L. M. K. Vandersypen, [*Science*]{} [**309**]{}, 1346 (2005).
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|
---
abstract: 'We present the electronic and structural properties of monolayer [WSe$_{2}$]{} grown by pulsed-laser deposition on monolayer graphene (MLG) on SiC. The spin splitting in the [WSe$_{2}$]{} valence band at $\overline{\mathrm{K}}$ was $\Delta_\mathrm{SO}=0.469\pm0.008$ eV by angle-resolved photoemission spectroscopy (ARPES). Synchrotron-based grazing-incidence in-plane X-ray diffraction (XRD) revealed the in-plane lattice constant of monolayer [WSe$_{2}$]{} to be $a_\mathrm{WSe_2}=3.2757\pm0.0008 \mathrm{\AA}$. This indicates a lattice compression of $-$0.19% from bulk [WSe$_{2}$]{}. By using experimentally determined graphene lattice constant ($a_\mathrm{MLG}=2.4575\pm0.0007 \mathrm{\AA}$), we found that a 3$\times$3 unit cell of the slightly compressed [WSe$_{2}$]{} is perfectly commensurate with a 4$\times$4 graphene lattice with a mismatch below 0.03%, which could explain why the monolayer [WSe$_{2}$]{} is compressed on MLG. From XRD and first-principles calculations, however, we conclude that the observed size of strain is negligibly small to account for a discrepancy in $\Delta_\mathrm{SO}$ found between exfoliated and epitaxial monolayers in earlier ARPES. In addition, angle-resolved, ultraviolet and X-ray photoelectron spectroscopy shed light on the band alignment between [WSe$_{2}$]{} and MLG/SiC and indicate electron transfer from graphene to the [WSe$_{2}$]{} monolayer. As further revealed by atomic force microscopy, the [WSe$_{2}$]{} island size depends on the number of carbon layers on top of the SiC substrate. This suggests that the epitaxy of [WSe$_{2}$]{} favors the weak van der Waals interactions with graphene while it is perturbed by the influence of the SiC substrate and its carbon buffer layer.'
author:
- 'H. Nakamura'
- 'A. Mohammed'
- 'P. Rosenzweig'
- 'K. Matsuda'
- 'K. Nowakowski'
- 'K. K[ü]{}ster'
- 'P. Wochner'
- 'S. Ibrahimkutty'
- 'U. Wedig'
- 'H. Hussain'
- 'J. Rawle'
- 'C. Nicklin'
- 'B. Stuhlhofer'
- 'G. Cristiani'
- 'G. Logvenov'
- 'H. Takagi'
- 'U. Starke'
title: 'Spin splitting and strain in epitaxial monolayer WSe$_2$ on graphene '
---
Introduction\[intro\]
=====================
Two-dimensional (2D) transition metal dichalcogenides (TMDs) MX$_2$ (M = Mo or W, X= S, Se, or Te) possess outstanding electronic, spin, and optical properties at thicknesses of a few layers and hold great promise for future optoelectronic and spintronic applications [@Mak2016; @Rivera2018; @Wang2018; @Mak2018; @Wang2012; @Novoselov2016]. In the monolayer limit, the breaking of structural inversion symmetry gives rise to a large spin splitting in the top valence band located at the $\overline{\mathrm{K}}$ and $\overline{\mathrm{K}}'$ points of the surface Brillouin zone[@Xiao2012; @Zhu2011; @Yuan2013]. Due to time reversal symmetry, the $\overline{\mathrm{K}}$ and $\overline{\mathrm{K}}'$ valleys have opposite out-of-plane spin polarization and each valley is associated with optical selection rules of opposite chirality as well as opposite signs of Berry curvature [@Xiao2012]. This leads to the valley-contrasting physics of monolayer TMDs, such as optical valley polarization and the valley Hall effect [@Mak2016; @Mak2018]. Recent advances in the application of TMDs as a quantum light source is remarkable, especially for [WSe$_{2}$]{} [@Srivastava2015; @He2015; @Koperski2015; @Chakraborty2015; @Luo2018; @Lu2019], where the spin-valley degree of freedom is found to be robust also in local bound carriers [@Lu2019].
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The spin splitting in the valence band at $\overline{\mathrm{K}}$ is revealed directly by angle-resolved photoemission spectroscopy (ARPES) [@Zhang2013; @Chiu2014; @Eknapakul2014; @Riley2014; @Le2015; @Zhang2016; @Aretouli2015; @Diaz2015; @Latzke2015; @Riley2015; @Sugawara2015; @Aretouli2016; @Mo2016; @Aziza2017; @Forti2017; @Henck2017; @Agnoli2018; @Wilson2017]. Spin-resolved ARPES confirmed an out-of-plane spin polarization that disappears for an even number of layers, consistent with the idea that inversion asymmetry is essential for the spin splitting [@Mo2016]. As demonstrated by theory and experiment, [WSe$_{2}$]{} has the largest spin splitting $\Delta_\mathrm{SO}$ amongst all TMDs of 2H-type [@Xiao2012; @Zhu2011; @Le2015; @Zhang2016; @Wilson2017]. Le *et al.* reported $\Delta_\mathrm{SO}=513$ meV in monolayer [WSe$_{2}$]{} exfoliated from a bulk crystal [@Le2015], while very recent work on an exfoliated monolayer [WSe$_{2}$]{} reported $\Delta_\mathrm{SO}=485$ meV [@Nguyen2019]. Zhang *et al.* found $\Delta_\mathrm{SO}=475$ meV in monolayer [WSe$_{2}$]{} grown by molecular beam epitaxy (MBE) on bilayer graphene/SiC [@Zhang2016]. The discrepancy in $\Delta_\mathrm{SO}$ between the MBE-grown and earlier exfoliated monolayer has been attributed to potential strain in the epitaxial TMD layer [@Zhang2016]. However, an evaluation of such strain in monolayer [WSe$_{2}$]{} using a precise structural probe, such as X-ray diffraction, has thus far been missing in any of the ARPES-studied monolayer.
Besides inducing strain, the substrate beneath a TMD could have an effect on its electronic properties by affecting the growth mode or via charge redistribution at the interface [@Sun2017]. TMDs on graphene represent a prototypical van der Waals (vdW) heterostructure where charge transfer could critically influence the physical properties of the TMD [@Froehlicher2018]. In this regard, ARPES of the graphene $\pi$-bands before and after the creation of a vdW heterostructure could provide a direct evidence of charge transfer across the TMD/graphene interface, but no such experiment has yet been reported. Alternatively, the charge transfer can be indirectly assessed from the position of the valence band maximum of the TMD ($E_\mathrm{K}$) with respect to the Fermi level [$E_\mathrm{F}$]{}. In [WS$_{2}$]{} grown by chemical vapor deposition on epitaxial monolayer graphene on SiC, Forti *et al.* found $E_\mathrm{K}=-1.84$ eV [@Forti2017]. Taking into account a band gap $E_\mathrm{G}$ of 2.1 eV for pristine monolayer [WS$_{2}$]{}, where [$E_\mathrm{F}$]{} is assumed to lie mid-gap, this corresponds to a significant downshift of $\sim 0.8$ eV of the [WS$_{2}$]{} bands. This, in turn, indicates electron transfer to [WS$_{2}$]{} across the interface. For MBE-grown [WSe$_{2}$]{} on epitaxial bilayer graphene, ARPES and scanning tunneling spectroscopy (STS) yielded $E_\mathrm{K}\sim -1.1$ eV [@Zhang2016]. Considering a band gap of 1.95 eV as determined by STS, this also suggests a small downshift ($\sim 100$ meV) of the [WSe$_{2}$]{} bands, consistent with an electron transfer to the TMD layer. On the other hand, ARPES of monolayer [WSe$_{2}$]{} transferred to cleaved graphite yielded $E_\mathrm{K}=-0.7$ eV [@Wilson2017]. Assuming the same band gap $E_\mathrm{G}$, this corresponds to [$E_\mathrm{F}$]{} residing closer to the valence band and thus indicates a hole transfer to [WSe$_{2}$]{}. However, we note that the above results are only indirect indications of charge transfer, because the position of the Fermi level can depend on the way the respective heterostructure was prepared. To unambiguously resolve the issue of charge transfer across the TMD/graphene interface, a comparison of ARPES measurements performed both before and after the creation of the vdW heterostructure could be highly useful.
In this paper, we clarify the electronic structure of monolayer [WSe$_{2}$]{} grown by pulsed-laser deposition on epitaxial monolayer graphene on SiC (MLG/SiC). In particular, we address the issue of a potential strain effect on the spin splitting $\Delta_\mathrm{SO}$ by using ARPES and grazing-incidence X-ray diffraction (GIXRD) data, supported by an analysis based on first-principles calculations. The electron transfer from graphene to [WSe$_{2}$]{} is revealed by comparing ARPES of the graphene $\pi$-bands before and after the [WSe$_{2}$]{} deposition. Ultraviolet and X-ray photoelectron spectroscopy (UPS and XPS), which are also conducted before and after the [WSe$_{2}$]{} growth, shed light on the band alignment between monolayer [WSe$_{2}$]{} and graphene. Atomic force microscopy (AFM) further reveals a significant impact of the substrate morphology on the [WSe$_{2}$]{} island size.
Experiment and theory\[exptheory\]
==================================
Monolayer graphene (MLG) on SiC was grown using the well-established recipe of sublimation growth at elevated temperatures in argon atmosphere [@Emtsev2009; @Forti2014]. Note that, on SiC, the graphene monolayer resides on top of a $(6\sqrt{3}\times 6\sqrt{3})\mathrm{R}30^\circ$-reconstructed carbon buffer layer (zerolayer graphene, ZLG) that is covalently bound to the SiC substrate [@Riedl2010]. [WSe$_{2}$]{} films were grown on the thus prepared MLG/SiC substrates via hybrid-pulsed-laser deposition ([*h*-PLD]{}) in ultra-high vacuum (UHV) [@Avaise]. This recently developed, bottom-up technique utilizes a pulsed laser to ablate transition metal targets, supported by chalcogen vapor supplied from an effusion cell, thus combining PLD and MBE. Pure tungsten (99.99 %) was ablated using a pulsed KrF excimer laser (248nm) with a repetition rate of 10Hz, while pure selenium (99.999 %) was evaporated from a Knudsen cell at a flux rate of around 1.5 Å/s as monitored by a quartz crystal microbalance. The deposition was carried out at 450 $^{\circ}$C for three hours, followed by two-step annealing at 640 $^{\circ}$C and 400 $^{\circ}$C for one hour each. Further details on [*h*-PLD]{} can be found elsewhere [@Avaise]. GIXRD measurements were carried out at the I07 beamline of Diamond Light Source [@Nicklin2016], with a photon energy of 12 keV (wavelength 1.0332 Å) and a Pilatus 100K 2D detector (DECTRIS). The incident angle $\alpha \sim 0.2^\circ$ of the X-rays was chosen according to the critical angle of the samples, which were kept in helium atmosphere during the measurements. Topographic AFM images were acquired with a Bruker microscope in peak force tapping mode. For photoelectron spectroscopy and LEED measurements, the freshly prepared samples were capped with a 10 nm-thick selenium layer at room temperature and transported through air into a different UHV facility, where the capping layer was removed by heating to 300 $^{\circ}$C. ARPES and UPS measurements were performed using monochromatized HeI $\alpha$ (21.22 eV) and HeII $\alpha$ (40.81 eV) photons and a 2D hemispherical analyzer equipped with a CCD Detector (SPECS Phoibos 150). The energy resolution of ARPES analyzer was 60 or 90 meV at a pass energy of 20 or 30 eV, respectively, as measured from the Fermi edge of gold at room temperature. XPS was carried out using non-monochromatized Mg K$\alpha$ (1253.6 eV) radiation. All the measurements took place at room temperature. First-principles calculations were performed using density functional theory (DFT) as implemented in WIEN2k [@wien2k] and ADF-BAND [@ADF-BAND; @BAND]. The generalized gradient approximation as parameterized by Perdew-Burke-Ernzerhof [@PBE] was used to describe the exchange-correlation functional. The spin-orbit coupling is included in a second variational procedure (Wien2k) or in the original basis set (ADF-BAND). We used a $k$-point mesh of 16$\times$16$\times$1 and adopted a slab geometry with a 30 Å gap between adjacent layers to suppress the interlayer interaction.
{width="14cm"}
Results and discussion {#res}
======================
Electronic structure and strain {#structstrain}
-------------------------------
The vertical structure of the [WSe$_{2}$]{}/MLG heterostack is schematically shown in Fig. \[arpes\](a). Figures \[arpes\](c) and (d) show the LEED patterns obtained before and after the growth of monolayer [WSe$_{2}$]{} on MLG/SiC with a coverage of approximately 50 %, demonstrating the preferred epitaxial relationship between [WSe$_{2}$]{} and graphene ([WSe$_{2}$]{} \[$10\overline{1}0$\] $\vert \vert$ graphene \[$10\overline{1}0$\]). This epitaxial relationship of the vdW heterostructure results in a reciprocal space alignment as shown in Fig. \[arpes\](b). The APRES intensity map recorded along the $\overline{\Gamma\mathrm{K}}$ direction of [WSe$_{2}$]{} and graphene is shown in Fig. \[arpes\](h). The valence bands of monolayer [WSe$_{2}$]{} are resolved with excellent quality, essentially consistent with the result of the first-principles calculation \[see Fig. \[arpes\](g)\]. As expected from the reciprocal space alignment, the graphene $\pi$-bands with their characteristic linear dispersion in the vicinity of the Fermi level [$E_\mathrm{F}$]{} also appear at higher parallel momenta $k_\parallel$ \[see Figs. \[arpes\](h) and (i)\]. Note that the $\pi$-bands are shifted in energy before and after the [WSe$_{2}$]{} deposition as revealed by the corresponding energy-momentum cuts recorded at the graphene $\overline{\mathrm{K}}$ point perpendicular to the $\overline{\Gamma\mathrm{K}}$ direction \[see Figs. \[arpes\](e) and (f)\]. Before the [WSe$_{2}$]{} deposition, the Dirac point is found 0.41 eV below [$E_\mathrm{F}$]{}, reflecting the $n$-type doping of epitaxial graphene on SiC [@Riedl2010]. After the growth of [WSe$_{2}$]{} on top of graphene, the Dirac point has shifted to 0.27 eV below [$E_\mathrm{F}$]{}. This upshift of 140 meV indicates electron transfer from graphene to the TMD monolayer which will be further discussed in Sec. \[align\].
{width="17cm"}
The large spin splitting $\Delta_\mathrm{SO}$ arising in the topmost [WSe$_{2}$]{} valence band at $\overline{\mathrm{K}}$ due to the breaking of inversion symmetry in monolayer [WSe$_{2}$]{} is clearly resolved in the ARPES data \[see Figs. \[arpes\](h), (i) and \[fpc\](a), (b)\]. To quantify this splitting, an energy distribution curve (EDC) was extracted at the $\overline{\mathrm{K}}$ point of [WSe$_{2}$]{} as indicated by the dashed black line in Fig. \[fpc\](a). By fitting this EDC with two pseudo-Voigt curves as shown in Fig. \[fpc\](b), we obtain $\Delta_\mathrm{SO}=0.469\pm0.008$ eV. The detail of the EDC analysis is shown in Supplemental Material [@SM]. This value is appreciably smaller than the 513 meV observed in monolayer [WSe$_{2}$]{} exfoliated from a bulk crystal [@Le2015], while close to more recent recent result (485 meV) [@Nguyen2019] and MBE-grown [WSe$_{2}$]{} (475 meV) [@Zhang2016]. While it is tempting to relate this difference to strain resulting from the epitaxial TMD growth, we will show in the following that the influence of strain on $\Delta_\mathrm{SO}$ is actually negligible for [WSe$_{2}$]{} on graphene.
We first focus on the results obtained from synchrotron-based GIXRD [@Avaise]. Utilizing an X-ray beam that propagates parallel to the sample surface at a critical angle of incidence $\alpha\sim0.2^{\circ}$ \[see Fig. \[xrd\](a)\], this technique probes the in-plane structure of the [WSe$_{2}$]{} films \[see Fig. \[xrd\](b)\]. The in-plane reciprocal space map shown in Fig. \[xrd\](c) clearly captures diffraction from monolayer [WSe$_{2}$]{}. We find [WSe$_{2}$]{} \[$10\overline{1}0$\] $\parallel$ graphene \[$10\overline{1}0$\], fully consistent with LEED \[see Fig. \[arpes\](d)\]. The weak ring-like elongation of the [WSe$_{2}$]{} diffraction in the reciprocal space map reflects large crystalline mosaic of monolayer [WSe$_{2}$]{} islands with respect to rotation around the surface normal. The wide-angle ($\pm 100^{\circ}$) rocking ($\theta$) scan for the [WSe$_{2}$]{}(110) peak exhibits the expected periodicity of $60^\circ$ as shown in Fig. \[xrd\](e). To evaluate the potential strain in the epitaxial TMD film, the in-plane lattice constant $a$ of [WSe$_{2}$]{} was extracted from the $\delta$-$\theta$ scans shown in Fig. \[xrd\](d). We find $a=3.2757\pm0.0008 \mathrm{\AA}$, which indicates a small compression of $-\,0.19$ % with respect to the bulk reference value ($a=3.282\pm0.001 \mathrm{\AA}$ [@Schutte]). The detail of the extraction of lattice constant and error is shown in the Supplemental Material[@SM]. The lattice constant of MLG directly beneath [WSe$_{2}$]{} is determined to be $a_\mathrm{MLG}=2.4575\pm0.0007 \mathrm{\AA}$ from the same $\delta$-$\theta$ scans. Using these values, we deduce that on MLG/SiC, a 3$\times$3 unit cell of the compressed [WSe$_{2}$]{} is perfectly commensurate with a 4$\times$4 graphene lattice, with an experimentally determined mismatch below 0.03 %. This could explain why monolayer [WSe$_{2}$]{} is compressed on MLG.
We now turn to the result of the first-principles calculation to examine the role of strain. Figures \[fpc\](c-f) show how a compressive or tensile strain modifies the valence band structure of monolayer [WSe$_{2}$]{}. In the respective calculations, the in-plane lattice constant was changed proportionally to include strain while keeping the unit cell volume constant. Qualitatively, our calculations indicate that compressive strain reduces the value of $\Delta_\mathrm{SO}$ \[Fig. \[fpc\](f)\], which is in accordance with previous first-principles results [@Le2015]. We find $\Delta_\mathrm{SO}$ = 452 (475) meV using Wien2k (ADF-BAND) for “zero strain”, i.e., when we fix the lattice constants of monolayer identical to bulk. By introducing strain, inferred change in $\Delta_\mathrm{SO}$ is $+$ ($-$) 18 meV per 1 % of tensile (compressive) strain as shown in Fig. \[fpc\](f). This holds for moderately strained [WSe$_{2}$]{} (as is the case in experiment) while the general dependence of $\Delta_\mathrm{SO}$ on strain is clearly nonlinear. Using the experimentally determined value of the lattice compression of WSe$_2$ on graphene ($-$0.19%), the amount of change in $\Delta_\mathrm{SO}$ that could arise from compressed strain is $-$3.4 meV. This is smaller than the error in experimental $\Delta_\mathrm{SO}$ (469$\pm$8 meV), and much smaller than the difference of 44 meV between our epitaxial monolayer [WSe$_{2}$]{} and the exfoliated one from a bulk crystal [@Le2015]. We thus conclude that a strain effect cannot explain the discrepancy in $\Delta_\mathrm{SO}$.
A subtle issue in the approach we used to estimate strain is that we actually do not know the lattice constant of a freestanding monolayer [WSe$_{2}$]{}. Namely, a monolayer [WSe$_{2}$]{} even without substrate effects may not have an identical lattice constant as that of bulk counterpart. To examine this, we performed additional first-principles calculations for bulk and monolayer [WSe$_{2}$]{} with structural optimization[@SM]. The theoretical results showed that the lattice constant of monolayer [WSe$_{2}$]{} converges to almost identical value as that of bulk (expanded only by $\sim+$0.03% [@SM]). This means that the experimentally observed compression ($-$0.19%) could be attributed to a strain in monolayer [WSe$_{2}$]{} as we have assumed.
Because strain in the epitaxial [WSe$_{2}$]{} is excluded as an origin of discrepancy in $\Delta_\mathrm{SO}$, we point out alternative possibilities. We first examined the possibility that the larger $\Delta_\mathrm{SO}$ observed in exfoliated [WSe$_{2}$]{} came from a tensile strain in the flake when transferred to the substrate. By using the second-order polynomial fit to the $\Delta_\mathrm{SO}$ vs. strain plot \[Fig.\[fpc\](f)\], we found the maximum gain in $\Delta_\mathrm{SO}$ predicted by the theory is $+40$meV for $+4.5$% tensile strain. This is close to the experimentally found difference ($\sim$48meV), which means that $+\sim$4-5% of tensile strain in the exfoliated bulk is needed to reproduce the value observed in the exfoliated bulk by strain. However, this is unlikely because the band dispersion of monolayer [WSe$_{2}$]{} expected for such a large strain \[see Fig.\[fpc\](e)\], which is rather different from the pristine monolayer, is not observed in the ARPES of the exfoliated bulk [@Le2015]. A very recent ARPES on exfoliated [WSe$_{2}$]{} by Nguyen *et al.* [@Nguyen2019] showed that (i) $\Delta_\mathrm{SO}$=0.485$\pm$0.01 eV for monolayer, much closer to the value observed in this study, and (ii) $\Delta_\mathrm{SO}$=0.501$\pm$0.01 eV for bilayer [WSe$_{2}$]{}, which is close to that of earlier exfoliated monolayer result [@Le2015]. Thus, a more plausible origin for discrepancy may be that the larger $\Delta_\mathrm{SO}$ in the previous study was obtained due to some inclusion of bilayer [WSe$_{2}$]{} in an exfoliated monolayer.
Band alignment and charge transfer {#align}
----------------------------------
The sample work function $\phi$ can be measured using UPS. From the secondary cutoffs of the respective spectra as shown in Fig. \[ups\](a), we infer $\phi=4.13$ eV and 4.40 eV ($\pm 0.04$ eV) before and after the growth of [WSe$_{2}$]{}, respectively. In combination with the ARPES results of Sec. \[structstrain\], we derive the band alignment of the [WSe$_{2}$]{}/MLG heterostructure as sketched in Fig. \[ups\](c). In quasi-free standing graphene, the bulk polarization of the SiC substrate induces an upward band bending, which would result in $p$-doping of the surface when terminated by a clean interface [@Ristein2012]. Yet, with the presence of the buffer layer (ZLG) this is overcompensated by donor states at the graphene/SiC interface, resulting in the $n$-type character of epitaxial MLG/SiC [@Mammadov2017] with its Dirac point residing 0.41 eV below [$E_\mathrm{F}$]{} \[see Fig. \[arpes\](e)\]. As discussed in Sec. \[structstrain\], the Dirac point shifts up by 0.14 eV to 0.27 eV below [$E_\mathrm{F}$]{} upon [WSe$_{2}$]{} growth \[see Fig. \[arpes\](f)\]. To our knowledge, such a shift of the graphene $\pi$-bands upon TMD growth on top was not reported previously. There are two possible mechanisms to explain this observation. First, electron transfer from graphene to [WSe$_{2}$]{} could shift the graphene bands upwards. Second, if the donor states at the graphene/SiC interface are partially compensated during the TMD growth (e.g. via chemical reaction with the Se vapor), the amount of $n$-type doping of graphene could change. In the latter case, modified donor states should influence the band bending at the graphene/SiC interface, which can be detected via a shift of the SiC core levels. The fitted XPS core level spectra of C $1s$ and Si $2p$ are shown in Fig. \[ups\](b). The C $1s$ fits consist of four components representing bulk SiC, MLG and the carbon buffer layer with its partial bonding to SiC (S1 and S2) [@Riedl2010]. The Si $2p$ spectra can reasonably well be fitted by one spin-orbit split doublet ($j=3/2$ and $1/2$ with an area ratio of 2:1). We find that the SiC peaks are unshifted in energy after the growth of [WSe$_{2}$]{}, indicating that the band bending at the interface remains unchanged. From this, we can exclude that the reduced $n$-type doping of graphene results from a modification of the interfacial donor states during TMD growth. Upon [WSe$_{2}$]{} growth the C $1s$ MLG component shifts by $0.16\pm0.02$ eV to lower binding energies while S1 and S2 retain their positions. This core level shift of MLG is quantitatively in line with the upshift of the Dirac point observed in ARPES and further supports the scenario of electron transfer from MLG to [WSe$_{2}$]{}. The work function increases by the charge transfer, and we ascribe the remaining increase of $0.27 - 0.14 = 0.13$ eV to an extrinsic upshift $\Delta\phi_\text{ext}$ of the vacuum level due to the change in surface termination from MLG to [WSe$_{2}$]{}\[Fig. \[ups\](c)\]. The valence band maximum $E_\mathrm{K}$ of [WSe$_{2}$]{} is found $\sim 1.1$ eV below [$E_\mathrm{F}$]{} in our ARPES measurements \[see Fig. \[arpes\](i)\], which matches very well with the results obtained from MBE-grown epitaxial [WSe$_{2}$]{} on bilayer graphene [@Zhang2016]. By assuming a band gap of $1.95$ eV as previously determined by STS [@Zhang2016], we estimate that the conduction band minimum $E_\mathrm{C}$ is located $\sim 0.85$ eV above [$E_\mathrm{F}$]{}. As such, the Fermi level in [WSe$_{2}$]{} resides closer to the conduction band minimum than to the valence band maximum. We finally note that a finite density of in-gap states can be expected in our epitaxial [WSe$_{2}$]{} films, stabilizing the position of [$E_\mathrm{F}$]{} inside the band gap after the electron transfer from graphene.
![(a) UPS spectra obtained from pristine MLG (red) and [WSe$_{2}$]{}/MLG (blue). On the final-state-energy axis, the respective sample work function can directly be read off from the secondary cutoff (red and blue arrows). (b) XPS core level spectra of C 1$s$ and Si 2$p$. The shift of the MLG peak (green curve) to lower binding energies upon [WSe$_{2}$]{} growth is consistent with the observed charge transfer from MLG onto [WSe$_{2}$]{}. All other components are found unshifted, indicating that the band bending at the graphene/SiC interface is unperturbed by the [WSe$_{2}$]{} growth. (c) Schematic band alignment of the [WSe$_{2}$]{}/MLG heterostructure as obtained from photoelectron spectroscopy (not drawn to scale). The polarity contribution to the upward band bending at the SiC/ZLG interface (red circles) is partially compensated by donor states (blue circles). Electron transfer from graphene onto [WSe$_{2}$]{} is indicated by the filled arrow. The Fermi energy before (after) the [WSe$_{2}$]{} growth is shown by the green (blue) dashed lines. The additional contribution $\Delta\phi_\text{ext}$ to the work function change results from an upshift of the vacuum level due the change in surface termination from MLG to[WSe$_{2}$]{}. []{data-label="ups"}](XPS_band.pdf){width="0.95\linewidth"}
Morphology of monolayer [WSe$_{2}$]{} {#morph}
-------------------------------------
![Topographic AFM images of epitaxial [WSe$_{2}$]{} on MLG/SiC. Lateral dimensions are (a) $3\times3$ $\mathrm{\mu}$m and (b) $1\times1$ $\mathrm{\mu}$m.[]{data-label="afm"}](AFM.pdf){width="0.9\linewidth"}
The morphology of monolayer [WSe$_{2}$]{} was measured by AFM. The unique feature of the $h$-PLD grown films was a high spatial uniformity with relatively small island sizes. Larger scale AFM images show a high density of nucleation sites distributed uniformly over the surface \[Fig.\[afm\](a)\]. The epitaxial graphene substrate had a minor inhomogeneity on the surface coming from the fabrication process, resulting in only small areas of bilayer graphene (BLG) close to the step edge, and other areas of exposed buffer layer (ZLG, which lacks a Dirac linear dispersion) within the flat MLG terrace region. Notably, [WSe$_{2}$]{} islands were indiscernible on ZLG by AFM \[Fig.\[afm\](b)\]. On the other hand, islands on BLG tended to be larger (frequently approaching $\sim$100 nm) than on MLG \[Fig.\[afm\](b)\]. The different [WSe$_{2}$]{} island sizes throughout the epitaxial graphene substrate are likely related to the distinct chemical nature and morphology of BLG, MLG and ZLG. During the TMD growth process, the migration of species could be severely limited on ZLG due to their covalent bonding to SiC and the resultant buckled surface [@sforzini], in contrast to the weak interaction on a complete vdW layers (MLG and BLG). For the latter, BLG regions may have even smoother surface than that of MLG regions due to the remoteness to the covalent bonds. Thus, our result clearly highlights the advantage of a chemically inert and smooth vdW surface in obtaining larger [WSe$_{2}$]{} domains during the epitaxial growth.
Summary and conclusions\[sum\]
==============================
A spin splitting of $\Delta_\mathrm{SO}=0.469\pm0.008$ eV is found for the topmost [WSe$_{2}$]{} valence band at $\overline{\mathrm{K}}$. The in-plane lattice constant of [WSe$_{2}$]{} was determined by grazing incidence X-ray diffraction, revealing a small compression ($-0.19$ %) of the epitaxial monolayer [WSe$_{2}$]{} film with respect to its bulk counterpart. Supplementing these data with first-principles calculations, we conclude that potential strain effects on $\Delta_\mathrm{SO}$ are negligible in our [WSe$_{2}$]{} film. Furthermore, the overall band alignment between [WSe$_{2}$]{} and graphene was clarified. The electron transfer from graphene to [WSe$_{2}$]{} becomes apparent from an upshift of the Dirac point of graphene with respect to the Fermi level after the growth of the TMD monolayer. The varying [WSe$_{2}$]{} island sizes on substrate areas covered by graphene layers of different thicknesses suggest the importance of atomically smooth, weakly interacting van der Waals surfaces for monolayer TMD epitaxy. Our results provide high-quality data on both electronic and structural properties of monolayer [WSe$_{2}$]{} and shed light on potential substrate influences in bottom-up TMD growth.
We are grateful to D. Huang and D. Weber for discussions and critical reading of the manuscript. We thank S. Prill-Drimmer, K. Pflaum, M. Dueller, and F. Adams for technical support. We acknowledge Diamond Light Source for time on Beamline I07 under Proposal SI18887 and Calipso program for the financial support. This work was supported by the Alexander von Humboldt-Foundation.
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See Supplemental Material for the detail of ARPES EDC fitting to derive spin splitting, the lattice constant determination in GIXRD, as well as additional first-principles calculations with relaxed lattice.
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[**Suppressing Background Radiation Using Poisson Principal Component Analysis**]{}\
[P. Tandon$^{*}$, P. Huggins$^*$, A. Dubrawski$^*$, S. Labov$^{**}$, K. Nelson$^{**}$]{}\
[${}^*$ Auton Lab, Carnegie Mellon University]{} [${}^{**}$ Lawrence Livermore National Laboratory]{}
0.075in [**Introduction.**]{} Performance of nuclear threat detection systems based on gamma-ray spectrometry often strongly depends on the ability to identify the part of measured signal that can be attributed to background radiation. We have successfully applied a method based on Principal Component Analysis (PCA) to obtain a compact null-space model of background spectra using PCA projection residuals to derive a source detection score. We have shown the method’s utility in a threat detection system using mobile spectrometers in urban scenes (Tandon et al 2012). While it is commonly assumed that measured photon counts follow a Poisson process, standard PCA makes a Gaussian assumption about the data distribution, which may be a poor approximation when photon counts are low. This paper studies whether and in what conditions PCA with a Poisson-based loss function (Poisson PCA) can outperform standard Gaussian PCA in modeling background radiation to enable more sensitive and specific nuclear threat detection.
0.05in [**Preliminaries.**]{} Radiation measurements are non-negative integer vectors which are photon counts across subsequent energy bins. In our case there are 128 bins. Figure 1A shows an example measurement where photon counts are low in high energy bins. Figure 1B shows an example of one of these bins. We can see that a Poisson model of the data better matches the true distribution than a Gaussian model. Standard PCA projection is optimal at explaining variance assuming the data is Gaussian. Collins et al. 2002 provides a generalization of PCA to a range of loss functions in the exponential family which they term E-PCA. One variant utilizes a Poisson error model, a formulation which we adopt in our study.
0.05in [**Experiments.**]{} Our radiation data is collected in a city by a vehicle carrying a double 4x16 NaI planar scintillator, with measurements taken over intervals of about 1s each. Three methods for background modeling are compared: standard (Gaussian) PCA, Poisson PCA, and a Gaussian PCA-based spectral anomaly detector currently fielded in our source detection system. All methods were trained on a set of roughly 1,000 background radiation measurements. Twenty testing data sets were created. Each consisted of roughly 1,000 background (negative) data points and also the same number of synthetic positive points created by injecting the negative points with additional counts due to a hypothetical synthesized fissle materials source. There is one testing data set for each distance to source in intervals of 1m, from 1 to 20m. Each of the evaluated methods estimated background models using training data and produced a reconstruction error score for each data point in the test sets.
A successful method will distinguish positive from negative data points. We measured Symmetric Kullback-Leibler Divergence (SKL) between the distributions of scores for negative and positive test data in each testing data set. A histogram estimator was used to compute SKL. Figures 2A-D plot the results for different numbers of principal components used by each method, from 2 to 5. Since the projection obtained by Poisson PCA may vary somewhat depending on the initial starting point of the optimization, 30 experiments were run for each number of principal components, and $[0.20,0.80]$ confidence intervals were drawn.
Figure 2E plots the top SKL performance at each distance (1-20m). For each method and distance to source, the best SKL score is reported by choosing the optimal number of principal components ranging from 1 to 5. When the sensor is near the source, all methods can distinguish background from source-injected data very well. Poisson PCA, however, outperforms other methods at large distances (lower source injection counts), suggesting Poisson PCA may improve source detection times for moving sensors. It may also benefit detection with shorter observation time intervals, potentially improving peak signal-to-noise ratios for measurements taken along a trajectory. We note that our findings match the intuition that at large distances there will be lower measured counts of source-originating photons, so Gaussian approximations of unexplained variance will become less accurate.
\
-0.1in
0.1in [**Discussion and Conclusions.**]{} Detecting faint sources among noisy background is an important practical problem, and our results suggest that Poisson PCA can boost source detection power at large distances, and potentially reduce source detection times for mobile sensors. Interestingly, the more standard PCA methods tend to perform better at close range, suggesting that the optimal model may be an ensemble of different methods.
0.1in [**References.**]{}
Michael Collins, Sanjoy Dasgupta, and Robert Schapire. A generalization of principal components analysis to the exponential family. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems 14 (NIPS), Cambridge, MA, 2002. MIT Press.
Prateek Tandon, Peter Huggins, Artur Dubrawski, Jeff Schneider, Simon Labov and Karl Nelson. Source location via Bayesian aggregation of evidence with mobile sensor data. (under review).
0.1in [This work has been supported by the US Department of Homeland Security, Domestic Nuclear Detection Office, under competitively awarded 2010-DN-077-ARI040-02. This support does not constitute an express or implied endorsement on the part of the Government. Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07NA27344. ]{}
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abstract: |
We present a theory of electronic properties of gated triangular graphene quantum dots with zigzag edges as a function of size and carrier density. We focus on electronic correlations, spin and geometrical effects using a combination of atomistic tight-binding, Hartree-Fock and configuration interaction methods (TB+HF+CI) including long range Coulomb interactions. The single particle energy spectrum of triangular dots with zigzag edges exhibits a degenerate shell at the Fermi level with a degeneracy $N_{edge}$ proportional to the edge size. We determine the effect of the electron-electron interactions on the ground state, the total spin and the excitation spectrum as a function of a shell filling and the degeneracy of the shell using TB+HF+CI for $N_{edge} < 12$ and approximate CI method for $N_{edge}\geq
12$. For a half-filled neutral shell we find spin polarized ground state for structures up to $N=500$ atoms in agreement with previous [*ab initio*]{} and mean-field calculations, and in agreement with Lieb’s theorem for a Hubbard model on a bipartite lattice. Adding a single electron leads to the complete spin depolarization for $N_{edge} \leq 9$. For larger structures, the spin depolarization is shown to occur at different filling factors. Away from half-fillings excess electrons(holes) are shown to form Wigner-like spin polarized triangular molecules corresponding to large gaps in the excitation spectrum. The validity of conclusions is assessed by a comparison of results obtained from different levels of approximations. While for the charge neutral system all methods give qualitatively similar results, away from the charge neutrality an inclusion of all Coulomb scattering terms is necessary to produce results presented here.
author:
- 'P. Potasz'
- 'A. D. Güçlü'
- 'A. W$\rm{\acute{o}}$js'
- 'P. Hawrylak'
title: 'Electronic properties of gated triangular graphene quantum dots: magnetism, correlations and geometrical effects.'
---
I. Introduction
===============
Graphene is an atomically thick layer of carbon atoms arranged in a honeycomb lattice.[@Novoselov+Geim+04; @Novoselov+Geim+05; @Zhang+Tan+05; @Son+PRL+06; @Potemski+deHeer+06; @Geim+Novoselov+07; @Rycerz+Tworzydlo+07; @Xia+Mueller+09; @Mueller+Xia+10; @Neto+Guinea+09] Due to its unique electronic properties and promising potential for applications, there is a growing research interest in graphene based nanostructures.[@Neto+Guinea+09; @Abergel+10; @Rozhkov+11] Attempts at fabricating graphene nanostructures with well defined shape and edge type have been reported starting from the graphene layer and using top-down techniques, [@Li+08; @Ponomarenko+08; @Ci+08; @You+08; @Schnez+09; @Ritter+09; @Jia+09; @Campos+09; @Neubeck+10; @Biro+10; @CruzSilva+10; @Yang+10; @Krauss+10] bottom-up techniques [@Zhi+08; @Treier+10; @Mueller+10; @Morita+11; @Lu+11] starting from carbon based molecules, as well as starting from graphane and removing hydrogen atoms using AFM tips.[@Singh+09; @Tozzini+10; @Xiang+09; @Schmidt+10]
The work on graphene nanostructures is motivated by the expectation that finite size effects significantly modify electronic properties of graphene. As a result of size quantization, an energy gap opens up, making graphene a semiconductor with a gap tunable from THz to UV. The energy gap can be tuned by changing not only the size but also the shape and the type of edge, allowing us to control the material’s optical properties.[@Yamamoto+06; @Zhang+08; @Guclu+10] Two types of edges in graphene are of particular interest due to their stability: armchair and zigzag. For zigzag edges, edge states in the vicinity of the Fermi energy appear. This is related to breaking the sublattice symmetry between the two types of atoms in the unit cell of the graphene honeycomb lattice. The presence of edge states was predicted theoretically [@NFD+96; @Fujita+96; @Son+06; @Son+PRL+06; @Ezawa+06; @Yamamoto+06; @Ezawa+07; @FRP+07; @AHM+08; @Wang+Yazyev+09; @Potasz+10] and confirmed experimentally.[@Niimi+Matsui+05; @Kobayashi+Fukui+05; @Tao+Jiao+11] These edge states form a degenerate band in graphene ribbons [@NFD+96; @Fujita+96; @Son+06; @Son+PRL+06; @Ezawa+06] or can collapse to a degenerate shell in graphene quantum dots.[@Yamamoto+06; @Ezawa+07; @FRP+07; @AHM+08; @Wang+Meng+08; @Wang+Yazyev+09; @Guclu+09; @Potasz+10] It was previously shown that the degeneracy is equal to the difference between the number of atoms corresponding to two sublattices in the bipartite lattice.[@Ezawa+07; @FRP+07; @Wang+Meng+08; @Potasz+10] In particular, the geometry that maximizes the imbalance between the two sublattices is a zigzag edge triangle where the degeneracy of the zero-energy shell is proportional to the number of atoms on the one edge.[@Potasz+10] This presents a unique opportunity to design a quantum system with a macroscopic degeneracy, analogously to the two-dimensional electron gas in a strong magnetic field.
While fabricating and measuring triangular graphene quantum dots with well defined edges [@Campos+09; @Zhi+08; @Lu+11; @Morita+11] remains a challenge, the theory of triangular graphene quantum dots (TGQD) with zigzag edges was developed by several groups.[@Yamamoto+06; @Guclu+10; @Ezawa+07; @FRP+07; @AHM+08; @Wang+Meng+08; @Ezawa+08; @Philpott+08; @HMA+08; @Guclu+09; @Potasz+10; @Kosimov+10; @Ezawa+10; @Sahin+10; @Ezawa+E10; @Voznyy+11; @Morita+11; @Romanovsky+11; @Xi+09; @Kinza+10; @Zarenia+11; @Dai+12] In particular, the macroscopically degenerate zero-energy band and the corresponding wavefunctions were explicitely constructed.[@Potasz+10] For a half-filled shell, TGQDs were studied by Ezawa using the Heisenberg Hamiltonian, [@Ezawa+07] by Fernandez-Rossier and Palacios [@FRP+07] using the mean-field Hubbard model, by Wang, Meng, and Kaxiras [@Wang+Meng+08] using density functional theory (DFT); and Güçlü [*et al.*]{} [@Guclu+09] using exact diagonalization techniques. It was shown that the ground state is fully spin polarized, with a finite magnetic moment proportional to the shell degeneracy. This finding is in agreement with Lieb’s theorem on magnetism of the Hubbard model for bipartite lattice systems.[@Lieb+89]
The effect of defects and disorder was also investigated.[@Potasz+10; @Ezawa+E10; @Voznyy+11] In particular, Voznyy [*et al.*]{} [@Voznyy+11] have shown by using [*ab initio*]{} methods that hydrogen-passivation stabilizes zigzag edges in TGQD over the pentagon-heptagon reconstructed edges.[@Voznyy+11] It was also proved that the zero-energy shell survives when TGQD is deformed to trapezoidal shape.[@Potasz+10] Ezawa studied the stability of magnetization against disorder. He considered three types of randomness: in a hopping integral, a site energy and lattice defects.[@Ezawa+E10] He proved that the magnetism is still governed by Lieb’s theorem but the number of degenerate states changed by the number of lattice defects. Some of us have shown in Ref. by use of methods beyond mean-field approximations, that the magnetization is unstable with respect to additional charge, leading to a complete spin depolarization. The spin depolarization was shown to significantly influence transport properties, blocking current through the graphene quantum dot due to the spin blockade.[@Guclu+09] It was also shown that by changing the population of the degenerate shell using a gate, one can simultaneously control magnetic and optical properties, determined by strong electron-electron and excitonic interactions.[@Guclu+10]
In this work we use improved configuration-interaction (CI) tools to extend our previous results [@Guclu+09] regarding the role of electron-electron interactions, magnetism and correlations in TGQDs to larger structures. We investigate the electronic properties as a function of size and filling factor of the degenerate shell by using a combination of tight-binding (TB), Hartree-Fock (HF) and configuration interaction methods (TB+HF+CI). Our many-body Hamiltonian includes, in addition to the on-site interaction term, all scattering and exchange terms within next-nearest neighbors, and all direct interaction terms in the two-body Coulomb matrix elements. Using full CI combined with the TB+HF method we demonstrate that the ground state for the charge neutral system has maximally polarized edge states for structures consisting of up to 200 atoms with the number of degenerate edge states $N_{edge} \leq 9$. By analyzing a spin-flip excitation spectrum of the spin-polarized ground state, we verify the spin-polarized ground state for up to 500 atoms or $N_{edge}=20$. These results for a system with long ranged Coulomb interaction appear to be consistent with Lieb’s theorem for the Hubbard model. Using TB+HF+full CI method for TGQD charged with an additional electron and a size of up to $N=200$ atoms it is shown that a complete spin depolarization predicted earlier by some of us [@Guclu+09] exists only up to a critical size. The critical size is established by studying the stability of a charged spin-polarized shell to spin-flip excitations. It is shown that for sizes up to the critical size the spin wave and minority spin electron form a bound state, a trion, signaling the tendency to the depolarization. For sizes exceeding the critical size the spin waves are unbound and the spin-polarized state is the ground state up to the sizes studied ($N\approx 500$ atoms). For TGQD structures above the critical size, depolarization effects away from the half-filling are observed. Results of TB+HF+CI calculations allow us to extract the excitation gap as a function of a shell filling. It is found that the largest gaps correspond to the half-filled spin-polarized shell and special filling fractions. At these special filling fractions, we predict a formation of Wigner-like spin polarized molecules, related to long range Coulomb interactions and a triangular geometry of graphene quantum dot. Finally, we compare results obtained at different levels of approximations. We show that, for the charge neutral system, the Hubbard, extended Hubbard, and fully interacting models are in good qualitative agreement. On the other hand, away from the half-filling, only a fully interacting model is able to correctly capture the effect of correlations.
The paper is organized as follows. In Sec. II, we describe our model. Section III contains analysis of the ground state spin and correlations as a function of size and filling factor of the degenerate shell. In Sec. IV, we compare results obtained within different levels of approximations. In Sec. V, we summarize our results.
II. Model of a graphene triangular quantum dot
===============================================
Graphene is a two-dimensional honeycomb crystal formed by carbon atoms on two interpenetrating hexagonal sublattices. The unit cell thus contains two carbon atoms. The distance between nearest-neighbor atoms is $a=1.42$ $\rm{\AA}$. By using vectors ${\bf{R}}=n{\bf{a}}_{1}+m{\bf{a}}_{2}$ with $n,m$ integers and primitive unit vectors defined as ${\bf{a}}_{1,2}=a/2(\pm \sqrt{3},3)$, one can obtain the positions of all the atoms in the structure. By cutting the graphene lattice in three zigzag directions, an equilateral triangle can be obtained, as shown in Fig. \[fig:Fig1\]. Such a system has a broken sublattice symmetry with two properties: (i) all edge atoms (with only two bonds) belong to the same sublattice, (ii) the difference between the number of atoms belonging to each sublattice is proportional to the number of atoms on one of the three edges.
Each carbon atom has four valence electrons. Three of them, on $s$, $p_x$, and $p_y$ orbitals, form $sp^2$ bonds with the three nearest in-plane neighbors. They are strongly bound and responsible for mechanical properties of graphene. The remaining fourth valence electron on each carbon atom $p_z$ orbital, perpendicular to the plane of graphene, is weakly bound and determines electronic properties of the system. Single-particle properties of graphene can be described by using one orbital tight-binding (TB) Hamiltonian.[@Wallace+47] We have previously shown that, within the TB model in the nearest-neighbors approximation, TGQDs with zigzag edges exhibit an energy gap, with a degenerate shell at the Fermi (zero) energy, with a degeneracy proportional to the length of an edge.[@Potasz+10] An example of TB energy levels for a structure consisting of 97 atoms with $N_{edge}=7$ degenerate states is shown in Fig. \[fig:fig2\](a). Our goal is to study the role of electron-electron interactions for electrons occupying this degenerate shell. Solving the full many-body problem even for such a small structure with 97 atoms is not possible at present. However, due to the energy gap separating the valence band and degenerate states, the valence electrons that do not occupy the degenerate band can be treated in a mean-field approximation. The remaining electrons occupying the degenerate shell must, however, be treated using a configuration-interaction method (CI). Therefore, we use a TB+HF+CI approach that allows us to treat the electronic correlations for electrons in the degenerate shell and their interaction with valence electrons at the mean-field level.
We start from the full many-body Hamiltonian for interacting electrons on the $p_z$ orbitals of graphene. It can be written as $$\begin{aligned}
H= \sum_{i,l,\sigma}\tau_{il\sigma}c^\dagger_{i\sigma}c_{l\sigma}
+\frac{1}{2}\sum_{\substack{i,j,k,l,\\\sigma \sigma'}}\langle ij\vert V \vert kl\rangle
c^\dagger_{i\sigma}c^\dagger_{j\sigma'}c_{k\sigma'}c_{l\sigma},
\label{fullH}\end{aligned}$$ where the operator $c^\dagger_{i\sigma}$ creates an electron on the $i$-th $p_z$ orbital with spin $\sigma$, $\tau_{il\sigma}$ is a hopping integral that describes the probability of scattering of electron on the $l$-th $p_z$ orbital $\phi_{l}$ to the $i$-th $p_z$ orbital $\phi_{i}$. The second term describes two-body Coulomb interactions between $p_z$ electrons. Note that at this stage, the unknown hopping terms $\tau_{il\sigma}$ do not include the effect of electron-electron interactions of $p_z$ electrons.
A. Mean-Field approximation for infinite graphene sheet
-------------------------------------------------------
Let us first write the Hamiltonian for graphene, given by Eq. (\[fullH\]), in the mean-field Hartree-Fock (HF) approximation as $$\begin{aligned}
\nonumber
H_{MF}^{o}&=& \sum_{i,l,\sigma}\tau_{il\sigma}c^\dagger_{i\sigma}c_{l\sigma}
+ \sum_{i,l,\sigma} \sum_{j,k,\sigma'}\rho_{jk\sigma'}^{o}
(\langle ij\vert V \vert kl\rangle \\
&-&\langle ij\vert V \vert lk\rangle\delta_{\sigma,\sigma'}) c^\dagger_{i\sigma}c_{l\sigma}
=\sum_{i,l,\sigma}t_{il\sigma}c^\dagger_{i\sigma}c_{l\sigma}.
\label{HMF}\end{aligned}$$ This is effectively a one-body TB Hamiltonian for a graphene layer [@Wallace+47] with density matrix elements $\rho_{jk\sigma'}^{o}=\langle c^\dagger_{j\sigma '}c_{k\sigma '}\rangle$ calculated with respect to the fully occupied valence band. The values of $\rho^{o}_{jk\sigma'}$ are evaluated in Appendix A and their role becomes clear in the next subsection. $t_{il\sigma}$ are experimentally estimated hopping integrals.
B. Mean-Field approximation for graphene quantum dots
-----------------------------------------------------
We now derive a mean-field Hamiltonian for electrons in graphene quantum dots (GQD). First, we apply mean-field approximation to the Hamiltonian given by Eq. (\[fullH\]) for electrons in GQD, with a result written as $$\begin{aligned}
\nonumber
H_{MF}^{GQD}&=& \sum_{i,l,\sigma}\tau_{il\sigma}c^\dagger_{i\sigma}c_{l\sigma}
+ \sum_{i,l,\sigma} \sum_{j,k,\sigma'}\rho_{jk\sigma'}
(\langle ij\vert V \vert kl\rangle \\
&-&\langle ij\vert V \vert lk\rangle\delta_{\sigma,\sigma'}) c^\dagger_{i\sigma}c_{l\sigma},
\label{HDOT}\end{aligned}$$ with density matrix $\rho_{jk\sigma'}$ for GQD. By combining Eq. (\[HMF\]) and Eq. (\[HDOT\]) we get $$\begin{aligned}
\nonumber
&H_{MF}^{GQD}&=H_{MF}^{GQD}-H_{MF}^{o}+H_{MF}^{o} \\
\nonumber
&=& \sum_{i,l,\sigma}t_{il\sigma}c^\dagger_{i\sigma}c_{l\sigma}
+ \sum_{i,l,\sigma} \sum_{j,k,\sigma'}(\rho_{jk\sigma'}-\rho^{o}_{jk\sigma'})
(\langle ij\vert V \vert kl\rangle \\
&-&\langle ij\vert V \vert lk\rangle\delta_{\sigma,\sigma'})c^\dagger_{i\sigma}c_{l\sigma}.
\label{hfhamilton}\end{aligned}$$ Here the subtracted component in the second term corresponds to mean-field interactions included in effective $t_{il\sigma}$ hopping integrals, described by the graphene density matrix $\rho^{o}_{jk\sigma'}$. For the TGQDs, the density matrix elements $\rho_{jk\sigma'}$ are calculated with respect to the many-body ground state of $N_{ref}=N_{site}-N_{edge}$ electrons, where $N_{site}$ is the number of atoms. Since the valence band and the degenerate shell are separated by an energy gap, the closed shell system of $N_{ref}$ interacting electrons is expected to be well described in a mean-field approximation, using a single Slater determinant. This corresponds to a charged system with $N_{edge}$ positive charges, as schematically shown in Fig. \[fig:fig2\](b). The Hamiltonian given by Eq. (\[hfhamilton\]) has to be solved self-consistently to obtain Hartree-Fock quasi particle orbitals. In numerical calculations, in addition to the on-site interaction term, all scattering and exchange terms within next-nearest neighbors and all direct interaction terms are included in the two-body Coulomb matrix elements $\langle ij|V|kl\rangle$ computed using Slater $p_z$ orbitals.[@Ransil+60] The few largest Coulomb matrix elements are given in Ref. . The value of the effective dielectric constant $\kappa$ depends on the substrate and is set to $\kappa=6$ in what follows.[@Reich+02] A method of calculating values of $\rho^{o}_{jk\sigma'}$ for graphene is shown in Appendix A. Values of hopping integrals $t_{il\sigma}$ are taken from the experimental data [@Bostwick+07] or [*ab initio*]{} calculation.[@Reich+02] We use $t=-2.5$ eV for nearest-neighbors and $t'=-0.1$ eV for next-nearest-neighbors [@DCN+07] hopping matrix elements. The HF results were also compared with the results of [*ab initio*]{} calculations.[@Guclu+09; @Voznyy+11]
We now discuss mean-field results for the charge neutral system. In the vicinity of the center of a sufficiently large dot a charge distribution around a given site is identical to that of an infinite system. The density matrices for graphene $\rho^{o}_{jk\sigma'}$ and for GQD $\rho_{jk\sigma'}$ are equal. A second term in Eq. (\[hfhamilton\]) vanishes, leaving only a hopping integral $t_{il\sigma}$. On the other hand, close to the edges, a density matrix for the GQD differs in comparison to its graphene counterpart. After diagonalizing the HF Hamiltonian given by Eq. (\[hfhamilton\]) one obtains eigenvalues and eigenvectors that involve the geometrical properties of the system, shown in Fig. \[fig:fig2\](b). A slight removal of the degeneracy of middle edge states and three corner states with a bit higher energies are observed, with electronic densities shown in Ref. .
C. Configuration Interaction method
-----------------------------------
After the self-consistent procedure we get new orbitals for quasi-particles with a fully occupied valence band and a completely empty degenerate shell. We start filling these degenerate states by adding extra electrons one by one, schematically shown in Fig. \[fig:fig2\](c). Next, we solve the many-body Hamiltonian corresponding to the added electrons, given by $$\begin{aligned}
\nonumber
&H_{MB}&=\sum_{s,\sigma}\epsilon_{s}a^\dagger_{s\sigma}a_{s\sigma}
\\&+&\frac{1}{2}\sum_{\substack{s,p,d,f,\\\sigma,\sigma'}} \langle sp\mid V\mid df\rangle
a^\dagger_{s\sigma}a^\dagger_{p\sigma'}a_{d\sigma'}a_{f\sigma},
\label{MBody}\end{aligned}$$ where the first term describes the energies of the Hartree-Fock orbitals and the second term describes an interaction between quasi particles occupying degenerate HF states denoted by $s,p,d,f$ indices. The two-body quasi particle scattering matrix elements $\langle sp\mid V\mid df\rangle$ are calculated from the two-body localized on-site Coulomb matrix elements $\langle ij\mid V\mid kl\rangle$.
In our calculations, we neglect scattering from/to the states from a fully occupied valence band. Moreover, because of the large energy gap between the shell and the conduction band, we can neglect scatterings to the higher energy states. A validity of these approximations is assessed in Ref. [@Potasz+ring+10]. These approximations allow us to treat the degenerate shell as an independent system that significantly reduces the dimension of the Hilbert space. The basis is constructed from vectors corresponding to all possible many-body configurations of electrons distributed within the degenerate shell. For a given number of electrons $N_{el}$, the Hamiltonian given by Eq. (\[MBody\]) is diagonalized in each subspace with total $S_{z}$.
D. Effect of gate charge
------------------------
In our model, we start from the system with an empty shell that corresponds to the charged system. As in our previous work, Ref. , electrons from the shell are transferred to the metallic gate. The Hamiltonian for $N_{ref}$ electrons in the presence of a gate in the mean-field Hartree-Fock approximation was written as $$\begin{aligned}
\nonumber
&H_{MF}&= \sum_{i,l,\sigma}t_{il\sigma}c^\dagger_{i\sigma}c_{l\sigma}
+ \sum_{i,l,\sigma} \sum_{j,k,\sigma'}(\rho_{jk\sigma'}-\rho^{o}_{jk\sigma'})
(\langle ij\vert V \vert kl\rangle \\
&-&\langle ij\vert V \vert lk\rangle\delta_{\sigma,\sigma'})
c^\dagger_{i\sigma}c_{l\sigma}
+ \sum_{i,\sigma}v^{g}_{ii}(q_{ind})c^\dagger_{i\sigma}c_{i\sigma},
\label{hfgate}\end{aligned}$$ with an electrostatic potential $v_{ii}^{g}$ related to $N_{edge}$ electrons in a gate defined as $$\begin{aligned}
%\nonumber
v_{ii}^{g}(q_{ind})&=& \sum^{N_{site}}_{j=1}\frac{-q_{ind}/N_{site}}
{\kappa\sqrt{(x_i-x_j)^2+(y_i-y_j)^2+d^2_{gate}}}
\label{Hgate}\end{aligned}$$ with $q_{ind}=-N_{edge}$ charge smeared out at positions $(x_i,y_i)$ at a distance $d_{gate}$ from the quantum dot. Next, we derived the many-body Hamiltonian with an inclusion of the effect of gate, written as $$\begin{aligned}
\nonumber
H=\sum_{p,\sigma}\epsilon_{p}a^\dagger_{p\sigma}a_{p\sigma}
+\frac{1}{2}\sum_{\substack{p,q,r,s,\\\sigma,\sigma'}}\langle pq\vert V\vert rs\rangle
a^\dagger_{p\sigma}a^\dagger_{q\sigma'}a_{r\sigma'}a_{s\sigma} \\
+\sum_{p,q,\sigma} \langle p\vert v^{g}(N_{add})\vert q\rangle
a^\dagger_{p\sigma}a_{q\sigma}+ 2\sum_{p'}\langle p'\vert v^{g}(N_{add})\vert p'\rangle,
\label{CIhamilton}\end{aligned}$$ where the indices without the prime sign $(p,q,r,s)$ run over $N_{edge}$ degenerate states, while the index with the prime sign $p'$ runs over $N_{ref}/2$ valence states (below the degenerate shell). A third term in Eq. (\[CIhamilton\]) corresponds to scattering from state $q$ to state $p$ in a degenerate shell as a result of interactions with electrons in a gate. The fourth term is a constant and just shifts the entire spectrum by a constant energy.
III. Magnetism and Correlation effects
======================================
Electronic properties as a function of the filling factor
---------------------------------------------------------
We, first, concentrate on a TGQD consisting of $N=97$ atoms, which is the largest system previously studied in our earlier work in Ref. [@Guclu+09]. It has $N_{edge}=7$ zero-energy degenerate states obtained from TB calculations, shown in Fig. \[fig:fig2\](a). After self-consistent HF calculations neglecting the gate charge (the effect of the gate will be discussed later), we obtain new quasi particle orbitals, shown in Fig. \[fig:fig2\](b). The degeneracy is slightly removed. We fill these degenerate levels by additional electrons and calculate two-body scattering matrix elements. For a given number of quasi particles, the many-body Hamiltonian, Eq. (\[MBody\]), is diagonalized in a basis of configurations of electrons distributed within the shell, as explained in Sec. II. In Fig. \[fig:fig3\], we analyze the dependence of the low energy spectra on the total spin $S$ for \[Fig. \[fig:fig3\](a)\] the charge neutral system, $N_{el}=7$ electrons, and \[Fig. \[fig:fig3\](b)\] one added electron, i.e., $N_{el}=8$ electrons. We see that for the charge neutral TGQD with $N_{el}=7$ electrons the ground state of the system is maximally spin polarized, with $S=3.5$, indicated by a circle. There is only one possible configuration of all electrons with parallel spins that corresponds to exactly one electron per one degenerate state. The energy of this configuration is well separated from other states with lower total spin $S$, which require at least one flipped spin among seven initially spin-polarized electrons. An addition of one extra electron to the system with $N_{el}=7$ spin polarized electrons induces correlations as seen in Fig. \[fig:fig3\](b), where the cost of flipping one spin is very small. Moreover, for $N_{el}=8$, the ground state is completely depolarized with $S=0$, indicated by a circle, but this ground state is almost degenerate with states corresponding to the different total spin.
The calculated many-body energy levels, including all spin states for different numbers of electrons (shell filling), are shown in Fig. \[fig:fig4\]. For each electron number, $N_{el}$, energies are measured from the ground-state energy and scaled by the energy gap of the half-filled shell, corresponding to $N_{el}=7$ electrons in this case. The solid line shows the evolution of the energy gap as a function of shell filling. The energy gaps for a neutral system , $N_{el}=7$ , as well as for $N_{el}=7-3=4$ and $N_{el}=7+3=10$ are found to be significantly larger in comparison to the energy gaps for other electron numbers. In addition, close to the half-filled degenerate shell, the reduction of the energy gap is accompanied by an increase of low energy density of states. This is a signature of correlation effects, showing that they can play an important role at different filling factors.
We now extract the total spin and energy gap for each electron number. Figures \[fig:fig5\](a) and (b) show the phase diagram, the total spin $S$ and an excitation gap as a function of the number of electrons occupying the degenerate shell. The system reveals maximal spin polarization for almost all fillings, with exceptions for $N_{el}=8,9$ electrons. However, the energy gaps are found to strongly oscillate as a function of shell filling as a result of a combined effect of correlations and system’s geometry. We observe a competition between fully spin polarized system that maximizes exchange energy and fully unpolarized system that maximizes the correlation energy. Only close to the charge neutrality, for $N_{el}=8$ and $N_{el}=9$ electrons, are the correlations sufficiently strong to overcome the large cost of the exchange energy related to flipping spin. The excitation gap is significantly reduced and exhibits large density of states at low energies, as shown in Fig. \[fig:fig3\].
Away from half-filling, we observe larger excitation gaps for $N_{el}=4$ and $N_{el}=10$ electrons. These fillings correspond to subtracting/adding three electrons from/to the charge-neutral system with $N_{el}=7$ electrons. In Fig. \[fig:fig6\] we show the corresponding spin densities. Here, long range interactions dominate the physics and three spin polarized \[Fig. \[fig:fig6\](a)\] holes ($N_{el}=7-3$ electrons) and \[Fig. \[fig:fig6\](b)\] electrons ($N_{el}=7+3$ electrons) maximize their relative distance by occupying three consecutive corners. Electron spin density is localized in each corner while holes correspond to missing spin density localized in each corner. We also note that this is not observed for $N_{el}=3$ electrons filling the degenerate shell (not shown here). The energies of HF orbitals of corner states correspond to three higher energy levels \[see Fig. \[fig:fig12\](c)\], with electronic densities shown in Ref. [@Guclu+09]. Thus, $N_{el}=3$ electrons occupy lower-energy degenerate levels corresponding to sides instead of corners. On the other hand, when $N_{el}=7$ electrons are added to the shell, self-energies of extra electrons renormalize the energies of HF orbitals. The degenerate shell is again almost perfectly flat similarly to levels obtained within the TB model. A kinetic energy does not play a role allowing a formation of a spin-polarized Wigner-like molecule, resulting from a long-range interactions and a triangular geometry. We note that Wigner molecules were previously discussed in circular graphene quantum dots with zigzag edges described in the effective mass approximation.[@Wunsch+2008; @Romanovsky+2009] The rotational symmetry of quantum dot allowed for the construction of an approximate correlated ground state corresponding to either a Wigner-crystal or Laughlin-like state.[@Wunsch+2008] Later, a variational rotating-electron-molecule (VREM) wave function was used.[@Romanovsky+2009] Unfortunately, due to a lack of an analytical form of a correlated wave function with a triangular symmetry, it is not possible to do it here.
Figure \[fig:fig5\] also shows the effect of the presence of a gate at a distance $d_{gate}=20a$, where $a=1.42$ is a nearest-neighbor’s inter-atomic distance. Clearly, the effect of a gate is very weak, just slightly changing energy gaps. In Fig. \[fig:fig7\], energy gaps as a function of a gate distance for the charge-neutral $N_{el}=7$ and charged $N_{el}=8$ system for our tested system with $N_{edge}=7$ degenerate states are shown. There are no effects for a gate distance $d_{gate}\geq 20a$. When a gate distance is comparable to graphene-substrate separation, $d_{gate}\sim 5a$, the energy gap for $N_{el}=7$ increases while the energy gap for $N_{el}=8$ decreases. The drop for $N_{el}=8$ is not sufficiently strong to change an observed effect of the spin depolarization. According to the above analysis, we next present results for a Hamiltonian with a gate at infinity.
Electronic properties as a function of the size
-----------------------------------------------
In a previous section, we have analyzed in detail the electronic properties of a particular TGQD with $N=97$ atoms as a function of the filling factor $\nu=N_{el}/N_{edge}$, i.e., the number of electrons per number of degenerate levels. In this section we address the important question of whether one can predict the electronic properties of a TGQD as a function of size.
Figure \[fig:fig8\] shows spin phase diagrams for triangles with odd number of degenerate edge states $N_{edge}$ and increasing size. Clearly, the total spin depends on the filling factor and size of the triangle. However, all charge-neutral systems at $\nu=1$ are always maximally spin polarized and a complete depolarization occurs for $N_{edge}\leq 9$ for structures with one extra electron added (such depolarization also occurs for even $N_{edge}$, not shown). Similar results for small size triangles were obtained in our previous work.[@Guclu+09] However, at $N_{edge}=11$ we do not observe depolarization for $N_{edge}+1$ electrons but for $N_{edge}+3$, where a formation of Wigner-like molecule for a triangle with $N_{edge}=7$ was observed. We will come back to this problem later. We now focus on the properties close to the charge neutrality.
For the charge-neutral case, the ground state corresponds to only one configuration $|GS\rangle=\prod_i a_{i,\downarrow}^\dagger |0\rangle$ with maximum total $S_z$ and occupation of all degenerate shell levels $i$ by electrons with parallel spin. Here $|0\rangle$ is the HF ground state of all valence electrons. Let us consider the stability of the spin polarized state to single spin flips. We construct spin-flip excitations $|kl\rangle=a_{k,\uparrow}^\dagger a_{l,\downarrow} |GS\rangle$ from the spin-polarized degenerate shell. The spin-up electron interacts with a spin-down "hole” in a spin-polarized state and forms a collective excitation, an exciton. An exciton spectrum is obtained by building an exciton Hamiltonian in the space of electron-hole pair excitations and diagonalizing it numerically, as was done, e.g., for quantum dots.[@Hawrylak+Wojs+Brum+96] If the energy of the spin flip excitation turns out to be negative in comparison with the spin-polarized ground state, the exciton is bound and the spin-polarized state is unstable. The binding energy of a spin-flip exciton is a difference between the energy of the lowest state with $S=S_z^{max}-1$ and the energy of the spin-polarized ground state with $S=S_z^{max}$. An advantage of this approach is the ability to test the stability of the spin polarized ground state for much larger TGQD sizes.
Figure \[fig:fig9\] shows the exciton binding energy as a function of the size of TGQD, labeled by a number of the degenerate states $N_{edge}$. The largest system, with $N_{edge}=20$, corresponds to a structure consisting of $N=526$ atoms. The exciton binding energies are always positive, i.e., the exciton does not form a bound state, confirming a stable magnetization of the charge neutral system. The observed ferromagnetic order was also found by other groups based on calculations for small systems with different levels of approximations.[@Ezawa+07; @FRP+07; @Wang+Meng+08; @Guclu+09] The above results confirm predictions based on Lieb’s theorem for a Hubbard model on bipartite lattice relating total spin to the broken sublattice symmetry.[@Lieb+89] Unlike in Lieb’s theorem, in our calculations many-body interacting Hamiltonian contains direct long-range, exchange, and scattering terms. Moreover, we include next-nearest-neighbor hopping integral in HF self-consistent calculations that slightly violates bipartite lattice property of the system, one of cornerstones of Lieb’s arguments.[@Lieb+89] Nevertheless, the main result of the spin-polarized ground state for the charge neutral TGQD seems to be consistent with predictions of Lieb’s theorem and, hence, applicable to much larger systems.
Having established the spin polarization of the charge-neutral TGQD we now discuss the spin of charged TGQD. We start with a spin-polarized ground state $|GS\rangle$ of a charge-neutral TGQD with all electron spins down and add to it a minority spin electron in any of the degenerate shell states $i$ as $|i\rangle=a_{i,\uparrow}^\dagger|GS\rangle$. The total spin of these states is $S_z^{max}-1/2$. We next study stability of such states with one minority spin-up electron to spin-flip excitations by forming three particle states $|lki\rangle=a_{l,\uparrow}^\dagger a_{k,\downarrow}a_{i,\uparrow}^\dagger |GS\rangle$ with total spin $S_z^{max}-1/2-1$. Here there are two spin-up electrons and one hole with spin-down in the spin-polarized ground state. The interaction between the two electrons and a hole leads to the formation of trion states. We form a Hamiltonian matrix in the space of three particle configurations and diagonalize it to obtain trion states. If the energy of the lowest trion state with $S_z^{max}-1/2-1 $ is lower than the energy of any of the charged TGQD states $|i\rangle$ with $S_z^{max}-1/2$, the minority spin electron forms a bound state with the spin-flip exciton, a trion, and the spin-polarized state of a charged TGQD is unstable. The trion binding energy, shown in Fig. \[fig:fig9\], is found to be negative for small systems with $N_{edge}\leq 8$ and positive for all larger systems studied here. The binding of the trion, i.e., the negative binding energy, is consistent with the complete spin depolarization obtained using TB+HF+CI method for TGQD with $N_{edge}\leq
9$ but not observed for $N_{edge}=11$ (and not observed for $N_{edge}=10$, not shown here), as shown in Fig. \[fig:fig8\]. For small systems, a minority spin-up electron triggers spin-flip excitations, which leads to the spin depolarization. With increasing size, the effect of the correlations close to the charge neutrality vanishes. At a critical size, around $N_{edge}=8$, indicated by an arrow in Fig. \[fig:fig9\], a quantum phase transition occurs from minimum to maximum total spin.
However, the spin depolarization does not vanish but moves to different filling factors. In Fig. \[fig:fig8\] we observe that the minimum spin state for the largest structure computed by the TB+HF+CI method with $N_{edge}=11$ occurs for TGQD charged with additional three electrons. We recall that for TGQD with $N_{edge}=7$ charged with three additional electrons a formation of a Wigner-like spin polarized molecule was observed, shown in Fig. \[fig:fig6\]. In the following, the differences in the behavior of these two systems, $N_{edge}=7$ and $11$, will be explained based on the analysis of the many-body spectrum of the $N_{edge}=11$ system.
Figure \[fig:fig10\] shows the many-body energy spectra for different numbers of electrons for $N_{edge}=11$ TGQD to be compared with Fig. \[fig:fig4\] for the $N_{edge}=7$ structure. Energies are renormalized by the energy gap of a half-filled shell, $N_{el}=11$ electrons in this case. In contrast to the $N_{edge}=7$ structure, energy levels corresponding to $N_{el}=N_{edge}+1$ electrons are sparse, whereas increased low-energy densities of states appear for $N_{el}=N_{edge}+2$ and $N_{el}=N_{edge}+3$ electrons. In this structure, electrons are not as strongly confined as for smaller systems. Therefore, for $N_{el}=N_{edge}+3$ electrons, geometrical effects that lead to the formation of a Wigner-like molecule become less important. Here, correlations dominate, which results in a large low-energy density of states.
IV. Different levels of approximations analysis
===============================================
In this section, we study the role of different interaction terms included in our calculations. The computational procedure is identical to that described in Sec. II. We start from the TB model but in self-consistent HF and CI calculations we include only specific Coulomb matrix elements. We compare results obtained with Hubbard model with only the on-site term, the extended Hubbard model with on-site plus long range Coulomb interactions, and a model with all direct and exchange terms calculated for up to next-nearest neighbors using Slater orbitals, and all longer range direct Coulomb interaction terms approximated as $\langle ij|V|ji\rangle=1/(\kappa|r_i-r_j|)$, written in atomic units, 1 a.u.$=27.211$ eV, where $r_i$ and $r_j$ are positions of $i$-th and $j$-th sites, respectively.
The comparison of HF energy levels for the structure with $N_{edge}=7$ is shown in Fig. \[fig:fig11\].
The on-site $U$-term slightly removes degeneracy of the perfectly flat shell \[Fig.\[fig:fig11\](a)\] and unveils the double valley degeneracy. On the other hand, the direct long-range Coulomb interaction separates three corner states from the rest with a higher energy \[Fig.\[fig:fig11\](b)\], forcing the lifting of one of the doubly degenerate subshells. Finally, the inclusion of exchange and scattering terms causes stronger removal of the degeneracy and changes the order of the four lower-lying states. However, the form of the HF orbitals is not affected significantly (not shown here).
In Fig. \[fig:fig12\] we study the influence of different interacting terms on CI results. The phase diagrams obtained within (a) the Hubbard model and (b) the extended Hubbard model show that all electronic phases are almost always fully spin polarized. The ferromagnetic order for the charge-neutral system is properly predicted. For TGQD charged with electrons, only inclusion of all Coulomb matrix elements correctly predicts the effect of the correlations leading to the complete depolarization for $N_{el}=8$ and $9$. We note that the depolarizations at other filling factors are also observed in Hubbard (at $N_{el}=2)$) and extended Hubbard calculation (at $N_{el}=11)$) results.
A more detailed analysis can be done by looking at the energy excitation gaps, which are shown in Fig. \[fig:fig13\]. For the charge-neutral system, all three methods give comparable excitation gaps, in agreement with previous results.[@Ezawa+07; @FRP+07; @Wang+Meng+08; @Guclu+09] In the Hubbard model, the energy gap of the doped system is reduced compared to the charge neutrality but without affecting magnetic properties. The inclusion of a direct long-range interaction in Fig. \[fig:fig13\](b) induces oscillations of the energy gap. For $N_{el}=N_{edge}+1$ electrons the energy gap is significantly reduced but the effect is not sufficiently strong to depolarize the system. Further away from half-filling, a large energy gap for models with long-range interactions for $N_{el}=N_{edge}+3$ appears, corresponding to the formation of a Wigner-like molecule of three spin-polarized electrons in three different corners. The inclusion of exchange and scattering terms slightly reduces the gap but without changing a main effect of Wigner-like molecule formation.
V. Conclusions and remarks
==========================
We have investigated magnetism, correlations, and geometrical effects in TGQDs by use of the TB+HF+CI method. Our many-body Hamiltonian includes all direct long-range terms and exchange and scattering terms up to next-nearest neighbors. We have performed analysis as a function of the filling factor of the degenerate band of edge states for different sizes. Through a full analysis of the many-body energy spectrum of structures consisting of up to 200 atoms, we confirmed the existence of the spin polarized ground state in agreement with Lieb’s theorem. By studying spin exciton binding energies, we also predicted stable magnetization for structures with more than 500 atoms. The complete spin depolarization was observed for one electron added to the charge neutral TGQD up to a critical size. Above a critical size the maximally spin-polarized charged TGQD was predicted using trion binding energy analysis. We have shown that in small systems, three electrons/holes added to the charge neutrality form the spin-polarized Wigner-like molecule. We relate this fact to geometrical effects and direct long-range interaction terms. For larger systems, geometry becomes less important and for the same filling we observe a spin depolarization as a result of correlations. Finally, we compared the fully interacting model with the Hubbard and extended Hubbard models. While qualitative agreement for the charge-neutral system was observed, the effect of correlations can be described only with the inclusion of all direct long-range, exchange, and scattering terms.
[*Acknowledgment*]{}. The authors thank NSERC, NRC-CNRS CRP, the Canadian Institute for Advanced Research, Institute for Microstructural Sciences, and QuantumWorks for support. P. P. acknowledges financial support from fellowship cofinanced by European Union within European Social Fund. A. W. acknowledges support from the EU Marie Curie CIG.
Appendix A
==========
In this appendix, we calculate density matrix elements $\rho_{jl\sigma'}^{o}$ between sites $j$ and $l$ for an infinite graphene sheet. The valence band eigenfunctions of the TB Hamiltonian in the nearest-neighbor approximation given by Eq. (\[HMF\]) are\
$$\begin{aligned}
\nonumber
\Psi_{\bf k}^v\left({\bf r}\right)&=&\frac{1}{\sqrt{2N_c}}(
\sum_{\bf R_A}e^{i{\bf k}{\bf R_A}}\phi_{z}
\left({\bf r}-{\bf R_A}\right)
\\
&+&\sum_{{\bf{R_B}}}e^{i{\bf{k}}{\bf{R_B}}}e^{-i{\bf{k}}{\bf{b}}}
e^{-i\theta_{{\bf{k}}}}\phi_{z}\left({{\bf{r}}-{\bf{R_B}}}\right)),
\label{TBvalen}\end{aligned}$$ where $\phi_{z}\left({{\bf{r}}}\right)$ are $p_{z}$ orbitals. The positions of the sublattice A and B atoms are given by ${\bf R_A}=n{\bf{a}}_{1}+m{\bf{a}}_{2}$ and ${\bf R_B}=n{\bf{a}}_{1}+m{\bf{a}}_{2}+{\bf b}$, described by unit vectors of hexagonal lattice defined as ${\bf{a}}_{1,2}=a/2(\pm \sqrt{3},3)$ and ${\bf{b}}=a(0,1)$, a vector between two nearest-neighbors atoms from the same unit cell with a distance $a=1.42$ $\rm{\AA}$. $N_{c}$ is the number of unit cells, and $\exp({i\theta_{\bf k}})=\frac{f({\bf{k}})}{|f({\bf{k}})|}$ with $f({\bf{k}})=1+e^{i{\bf{k}}{\bf{a}}_{1}}+e^{i{\bf{k}}{\bf{a}}_{2}}$. The density matrix for the graphene layer $\rho^{o}_{jl\sigma}$ for two sites $j$ and $l$ is defined as $$\begin{aligned}
\rho^{o}_{jl\sigma}=\sum_{\bf k}
b_{{\bf R}_j}({\bf k})b_{{\bf R}_l}({\bf k}),\end{aligned}$$ where $b_{\bf R}$’s are the coefficients of the $p_z$ orbitals given in Eq. (\[TBvalen\]). The on-site density matrix element for an arbitrary lattice site [*j*]{} is site and sublattice index independent, $$\begin{aligned}
\rho^{o}_{jj\sigma}=\frac{1}{2N_{c}}\sum_{{\bf{k}}}
e^{-i{\bf k R}_j}e^{i{\bf k R}_j}=
\frac{1}{2N_{c}}\sum_{{\bf{k}}}1=\frac{1}{2},
\label{fk}\end{aligned}$$ where we took into account the fact that the number of occupied states is equal to the number of unit cells in the system. The nearest-neighbors density matrix elements for two atoms from the same unit cell corresponds to ${\bf R}_l-{\bf R}_j={\bf b}$ and can be calculated using $$\begin{aligned}
\nonumber
\rho^{o}_{jl\sigma}&=&
\frac{1}{2N_{c}}\sum_{{\bf{k}}}e^{-i{\bf k R}_j}
e^{i {\bf k R}_l} e^{-i{\bf k b}}e^{-i\theta_{\bf k}}
\\
&=&\frac{1}{2N_{c}}\sum_{{\bf{k}}}e^{-i\theta_{{\bf{k}}}}\simeq 0.262,
\nonumber\end{aligned}$$ where the summation over occupied valence states is carried out numerically. We obtained the same value for two other nearest neighbors. The same results can also be obtained by diagonalizing a sufficiently large hexagonal graphene quantum dot and by computing the density matrix elements for two nearest neighbors in the vicinity of the center of the structure. We have also calculated next-nearest neighbors density matrix elements, obtaining negligibly small values.
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---
abstract: 'This paper considers the use of an entanglement breaking channel in the construction of a secure bit-commitment protocol. It is shown that this can be done via a depolarizing quantum channel.'
author:
- |
S. Arash Sheikholeslam sasheikh@ece.uvic.ca\
Department of Electrical and Computer Engineering, University of Victoria\
PO Box 3055, STN CSC, Victoria, BC Canada V8W 3P6\
- |
T. Aaron Gulliver\
agullive@ece.uvic.ca\
Department of Electrical and Computer Engineering, University of Victoria,\
PO Box 3055, STN CSC, Victoria, BC Canada V8W 3P6
title: 'EPR Secure Non-relativistic Bit Commitment Through Entanglement Breaking Channels'
---
Introduction {#introduction .unnumbered}
============
The most successful cheating strategy against non-relativistic bit commitment schemes is the entanglement attack (also known as the EPR attack) [@Mayers99thetrouble] [@PhysRevLett.78.3414]. In this strategy, one of the parties (Alice) entangles a system with the one she uses for commitment and keeps this second system secret. Then she is able to cheat before the opening phase through local operations on her own system. One approach to counter this cheating strategy is to determine a means of breaking the entanglements. This must be done either through local transforms performed by the other party (Bob), or through local noise applied to Bob’s system (from the transmission channel).
Entanglement breaking channels are a relatively new concept in quantum information that were first introduced in [@15]. The characteristics of two-qubit entanglements are discussed in [@16] and [@17]. In particular, the local two-qubit entanglement-annihilating channel (2-LEA) is examined in [@16]. >From [@15], a local channel $c$ is called entanglement breaking if the output of the channel operating on an entangled state is separable, where separability for a density matrix $\rho$ means $\rho=\sum_{i} p_i \rho_a^i\otimes \rho_b^i$. In the next section, we describe through an example how an entanglement breaking channel can be used to secure the Bennett and Brassard bit commitment scheme [@1984-175-179] against an EPR attack.
Depolarizing Channel Bit Commitment {#depolarizing-channel-bit-commitment .unnumbered}
===================================
As is typical, we assume Alice is working in a noise free environment, i.e., a perfectly shielded and isolated lab. Therefore the joint noise which corrupts the entangled state $\rho_{AB}$ is $I \otimes \varepsilon_{c}[\rho_{AB}]$, where $\varepsilon_{c}$ is the channel noise. This entanglement breaking operation must either be applied by Bob through some apparatus he possesses for adding noise, or by the quantum channel through which Alice sends the qubits to Bob, as shown in Figure 1. Here we use a depolarizing channel to apply the entanglement breaking operation. This channel is defined in [@17] as $$\epsilon(X)=qX+(1-q)tr[X]\frac{1}{2}I$$ The action of the depolarizing channel replaces the qubit with the completely mixed state, $\frac{I}{2}$, with probability $1-q$.
It was shown in [@18][@19] that the evolution of any entangled state in a channel (entanglement breaking channel in this case), is determined by the evolution of a maximally entangled state in the channel. Therefore we only consider the effect of the entanglement breaking channel on the maximally entangled state $\vert \psi^+\rangle=\frac{1}{\sqrt{2}}(\vert0_A0_B\rangle+\vert1_A1_B\rangle)$, where the subscripts $A$ and $B$ denote Alice and Bob, respectively. It was proven in [@19] for a local quantum channel $\mathbb{S}$ (which operates on a qubit), an entangled state $\vert X\rangle$ (which is a bipartite $N \otimes 2$ state), and concurrence $C$ as defined in [@19] (a measure of entanglement), that $$C((I\otimes \mathbb{S})[\vert X\rangle\langle X\vert])=C[\vert X\rangle\langle X\vert]]C[(I\otimes \mathbb{S})[\vert \psi^+\rangle \langle \psi^+ \vert]].$$ Since $C[\vert X\rangle\langle X\vert]]=0$ for $\vert X\rangle$, which is a separable state, if a local channel $\mathbb{S}$ (or $I\otimes \mathbb{S}$ for the entire system) applied on the maximally entangled state $\vert \psi^+\rangle$ disentangles it, then we have $C((I\otimes \mathbb{S})[\vert X\rangle\langle X\vert])=0$, which disentangles any bipartite $N \otimes 2$ state.
Having a maximally entangled state $\vert \psi^+\rangle$ and applying a depolarizing channel on Bob’s state (which means the effect of the channel on the entire system is $(I\otimes\epsilon)[X]$), results in the state $$q \vert \psi^+\rangle \langle \psi^+ \vert+\frac{(1-q)}{4}I_A\otimes I_B$$ which has been shown to be separable for $q \leq \frac{1}{3}$ [@16]. Therefore as discussed above, this channel will disentangle Bob’s qubit from any other system (such as Alice’s secret system). Now assume the two parties use the simple Bennett and Brassard bit commitment scheme in which Alice sends random selections of $\vert \uparrow \rangle$ or $\vert \rightarrow\rangle$ for 0, and $\vert \nearrow \rangle$ or $\vert \searrow \rangle$ for 1. Since $\rho_+=\rho_\times=\rho$, if $\rho$ passes through the channel Bob will expect to receive $q\rho+(1-q)tr[\rho]\frac{1}{2}I$. Thus if Alice attempts to cheat (i.e., entangle her secret system with Bob’s qubit), he will receive a separable state after applying the depolarizing channel ($\rho_{channel}=\sum_{i} p_i \rho_a^i\otimes \rho_b^i$). Therefore the states of Alice will be disentangled from Bob’s system, and more importantly he can determine if Alice has cheated or not. To show this, two cases must be considered, Alice is honest and has not cheated, and Alice attempts to cheat. For the first case, the probability of Bob receiving the same state as Alice sent is $q$. Therefore the probability of Bob measuring the correct state is $\frac{q}{2}$ since the probability of choosing the correct basis is $\frac{1}{2}$. Bob should then expect to measure at least $\frac{q}{2}$ of the states correctly. In the second case, Alice cheats and entangles her secret states with the committed qubits. After Bob performs the depolarizing operation, the state will be disentangled as described previously, and therefore Alice cannot change the state that Bob measures. There is also no guarantee that the probability of Bob measuring the correct state is $\frac{q}{2}$, so Alice may be exposed regarding the entanglements even if she does not change the committed bit.
A simple security analysis regarding our example using the Bennet and Brassard scheme is given below. Alice prepares an entangled state $\vert a_0\rangle_A \vert 0\rangle_B+\vert a_1\rangle_A\vert 1\rangle_B$, where the subscript $A$ means the state is controlled by Alice and $B$ controlled by Bob. Alice then sends Bob’s states to him, and when she wants to change her mind about the committed bit she performs a unitary transform followed by a projective measurement on her own state (we can just assume a projective measurement). The effect of the entanglement breaking channel on the system after Alice’s measurement is $\rho_b = \sum_{i} p_i \langle a_j \vert \rho_a^i \vert a_j\rangle \rho_b^i$, where $a_j$ is the basis for the projective measurement. Thus in order for Alice to determine $\rho_b$ she needs to know the decomposition caused by the channel, i.e., $\rho_{channel}=\sum_{i} p_i \rho_a^i\otimes \rho_b^i$, and therefore she needs to know the value of $q$ (which we assume is controlled by Bob).
Chailloux and Kerenidis [@20] provided lower bounds on an optimal quantum bit commitment (the bounds are tight and the upper bounds are close to the lower bounds), however they assumed that the operations in both the commitment and revealing phases are **unitary transforms** on Alice and Bob’s quantum spaces. Here we take the same approach towards analysing the security of our system. For the Hiding property (i.e. the ability of Bob to guess the committed bit assuming a honest Alice), we know that without considering the channel Bob can guess the bit with probability $\frac{1}{2}+ \frac{\Delta(\sigma_0,\sigma_1)}{2}$ (where $\sigma_b$ is the density matrix assigned to 0 or 1). The effect of the channel, Bob’s ability to cheat is then simply the maximum of his ability to distinguish the states with or without having them passed through the quantum channel, which is given by\
$P_{Bcheat} = \frac{1}{2}+ Max(\frac{\Delta(\sigma_0,\sigma_1)}{2},\frac{\Delta(\mathbb{S}[\sigma_0],\mathbb{S}[\sigma_1])}{2})$\
Where $\mathbb{S}[.]$ is the effect of the channel. For Alice’s cheating probability, consider the following.\
We assume a cheating Alice prepares a state $\rho_{AB}$ and sends it to Bob so that just before Alice opens the bit, the state of that part of the system which Bob possesses is $\sigma_{B}=Tr_A(\alpha[I\otimes \mathbb{S}[\rho_{AB}]])$ where $\alpha[.]$ is Alice’s operation on her own part of the system (i.e. A unitary transform followed by a measurement). Now assuming that $\mathbb{S}[.]$ is an entanglement breaking channel, we have $\sigma_{B}= \sum_{i} p_i Tr_A(A[\rho_a^i])\otimes \rho_b^i$. Assuming Alice wants Bob to measure 0, she should maximize the probability of Bob detecting $\sigma_0$, which is $F^2(\sigma_{Bend},\sigma_0)$, where $F$ is the fidelity. This means Alice must know the $p_i$ and properly choose the value of $Tr_A(A[\rho_a^i])$ (i.e. she also needs to know the $\rho_a^i$). This requires Alice to know the channel characteristics, but these are controlled by Bob and is kept secret by him.
Conclusions {#conclusions .unnumbered}
===========
In this letter we have shown that by using an entanglement breaking channel, the simple Bennett and Brassard bit commitment scheme can be made secure against EPR attacks. We also presented an example of a depolarizing channel which is practically conceivable. Only entanglement attacks were discussed, we leave the unconditional security of these noise based systems as a topic for future research.
[9]{}
Bennett, Charles H and Brassard, Gilles: [Quantum cryptography: Public key distribution and coin tossing]{}, volume 11, Proceedings of IEEE International Conference on Computers Systems and Signal Processing, Bangalore, India, 1984, 175-179
, [Apr]{}, [Phys. Rev. Lett.]{}, [10.1103/PhysRevLett.78.3414]{}, [Mayers, Dominic]{}, [1997]{}, [17]{}, [http://link.aps.org/doi/10.1103/PhysRevLett.78.3414]{}, [American Physical Society]{}, [3414–3417]{}, [78]{}
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, [Nat Phys]{}, [Vol. 4]{}, [No. 4.]{} [(February 2008)]{}, [pp. 99-102.]{}
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, [Andre Chailloux, Lordanis Kerenidis]{}, [Optimal bounds for quantum bit commitment]{}, [arxiv:1102.1678v1]{}, [2011]{}
|
---
abstract: 'We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.'
author:
- Matthew Robert Ballard
title: Wall crossing for derived categories of moduli spaces of sheaves on rational surfaces
---
Introduction {#section: Introduction}
============
A central question in the theory of derived categories is the following: given a smooth, projective variety $X$, how does one find interesting semi-orthogonal decompositions of its derived category, ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }X)}$? Historically, two different parts of algebraic geometry have fed this question: birational geometry and moduli theory. [@Orl92; @BO95; @OrlK3; @Bridgeland-flops; @KawD-K; @Kaw06; @Kuz09b] provides a non-exhaustive highlight reel for this approach.
This paper focuses on the intersection of birational geometry and moduli theory. Namely, given some moduli problem equipped with a notion of stability, variation of the stability condition often leads to birational moduli spaces. As such, it is natural to compare the derived categories in this situation. Let us consider the well-understood situation of torsion-free rank two semi-stable sheaves on rational surfaces [@EG; @FQ; @MW]. The flipping of unstable strata under change of polarization was investigated to understand the change in the Donaldson invariants. It provides the input for the following result, which can be viewed as a categorification of the wall-crossing formula for Donaldson invariants.
\[theorem: intro\] Let $S$ be a smooth rational surface over $\C$ with $L_-$ and $L_+$ ample lines bundles on $S$ separated by a single wall defined by unique divisor $\xi$ satisfying $$\begin{gathered}
L_- \cdot \xi < 0 < L_+ \cdot \xi \\
0 \leq \omega_S^{-1} \cdot \xi.
\end{gathered}$$ Let $\mathcal M_{L_{\pm}}(c_1,c_2)$ be the $\mathbb{G}_m$-rigidified moduli stack of Gieseker $L_{\pm}$-semi-stable torsion-free sheaves of rank $2$ with first Chern class $c_1$ and second Chern class $c_2$.
There is a semi-orthogonal decomposition $$\begin{gathered}
{\operatorname{D}^{\operatorname{b}}(\operatorname{coh } \mathcal M_{L_+}(c_1,c_2) )} = \left\langle \underbrace{{\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^{l_{\xi}})}, \ldots, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^{l_{\xi}})}}_{\mu_{\xi}}, \ldots \right. \\
\left. \underbrace{{\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^0)}, \ldots, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^0)}}_{\mu_{\xi}}, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh } \mathcal M_{L_-}(c_1,c_2))} \right\rangle
\end{gathered}$$ where $$\begin{aligned}
l_{\xi} & := (4c_2 - c_1^2 + \xi^2)/4 \\
H^l & := {\operatorname{Hilb}}^l(S) \times {\operatorname{Hilb}}^{l_{\xi} - l}(S) \\
\mu_{\xi} & := \omega_S^{-1} \cdot \xi
\end{aligned}$$ with the convention that ${\operatorname{Hilb}}^0(S) := {\operatorname{Spec}} \C$.
While Theorem \[theorem: intro\] is interesting in its own right, the method might be moreso. Indeed, Theorem \[theorem: intro\] represents one of multiple possible applications, including to moduli spaces of Bridgeland semi-stable objects on rational surfaces, [@ABCH].
Theorem \[theorem: intro\] follows from the general technology that goes under the heading of windows in derived categories. Windows provide a framework for addressing the central question put forth above; they are a machine for manufacturing interesting semi-orthogonal decompositions of ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }X)}$. They have a rich history with contributions by many mathematicians and physicists [@KawFF; @VdB; @Orl09; @HHP; @Seg2; @HW; @Shipman; @HL12; @BFKGIT; @DS]. However, windows have not yet achieved their final form. Previous work dealt with an Artin stack $\mathcal X$ plus a choice of global quotient presentation $\mathcal X = [X/G]$. Locating an appropriate quotient presentation for a given $\mathcal X$ is not convenient in applications, in particular the above, so one would like a definition of a window more intrinsic to $\mathcal X$. Section \[section: SODs and BB\] provides such a definition using an appropriate type of groupoid in Białynicki-Birula strata and extends the prior results on semi-orthogonal decompositions [@BFKGIT] to this setting, see Theorem \[theorem: elementary wall crossing\]. A similar extension should appear in [@HL14].
**Acknowledgments:** The author has benefited immensely from conversations and correspondence with Arend Bayer, Yujiro Kawamata, Colin Diemer, Gabriel Kerr, Ludmil Katzarkov, and Maxim Kontsevich, and would like to thank them all for their time and insight. The author would especially like to thank David Favero for a careful reading of and useful suggestions on a draft of this manuscript. Finally, the author would also like to thank his parents for fostering a hospitable work environment.
The author was supported by a Simons Collaboration Grant.
Semi-orthogonal decompositons and BB-strata {#section: SODs and BB}
===========================================
For the whole of this section, $k$ will denote an algebraically-closed field. The term, variety, means a separated, reduced scheme of finite-type over $k$. All points of a variety are closed points unless otherwise explicitly stated.
In this section, we extend the results of [@BFKGIT] by removing the global quotient presentation from the input data. We try to keep this section as self-contained as possible.
Truncations of sheaves on BB strata
-----------------------------------
We begin with the following definition. Let $X$ be a smooth quasi-projective variety equipped with a $\mathbb{G}_m$-action. Let $X^{\mathbb{G}_m}$ denote the fixed subscheme of the action and let $X_0$ be a choice of a connected component of the fixed locus. We recall the following well-known result of Białynicki-Birula.
\[theorem: BB\] The fixed locus $X^{\mathbb{G}_m}$ is smooth and is a closed subvariety of $X$. Let $X_0$ be a connected component of $X^{\mathbb{G}_m}$. There exists a unique smooth and locally-closed $\mathbb{G}_m$-invariant subvariety $X^+_0$ of $X$ and a unique morphism $\pi: X^+_0 \to X_0$ such that
1. $X_0$ is a closed subvariety of $X^+_0$.
2. The morphism $\pi: X_0^+ \to X _0$ is an equivariantly locally-trivial fibration of affine spaces over $X_0$.
3. For a point $x \in X_0$, there is an equality $$T_x X_0^+ = \left(T_x X\right)^{\geq 0}$$ where the right hand side is the subspace of non-negative weights of the geometric tangent space.
This is [@BB Theorem 2.1 and Theorem 4.1]
Note that the weights on the affine fibers are all positive and the set of closed points of a $X^+_0$ from Theorem \[theorem: BB\] is $$\{ x \in X \mid \lim_{\alpha \to 0} \sigma(\alpha,x) \in X_0 \}$$ so one may think of $X^+_0$ as the set of points that flow into $X_0$ as $\alpha \to 0$.
\[definition: BB strata\] Let $X$ be a smooth quasi-projective variety equipped with a $\mathbb{G}_m$ action. Let $X_0$ be a choice of a connected component of $X^{\mathbb{G}_m}$. The **BB stratum** associated with $X_0$ is $X^+_0$ appearing in Theorem \[theorem: BB\].
If $X = X^+_0$, then we shall say also say that $X$ is a BB stratum. In particular, we require that $X^{\mathbb{G}_m}=X_0$ is connected in this situation.
Note that $X$ being a BB stratum is equivalent to an extension of the action map $\sigma: \mathbb{G}_m \times X \to X$ to morphism $\mathbb{A}^1 \times X \to X$.
Let $X$ be a BB stratum. Let us consider the local situation first. So $X = {\operatorname{Spec}} R[x_1,\ldots,x_n]$ and we have a coaction map, also denoted by $\sigma$, $$\sigma: R[x_1,\ldots,x_n] \to R[x_1,\ldots,x_n,u,u^{-1}]$$ where the weights the $x_i$ are positive. The fixed locus is then $X_0 = {\operatorname{Spec}} R$. Let $M$ be a $\mathbb{G}_m$ equivariant module over $R$, i.e. a quasi-coherent $\mathbb{G}_m$-equivariant sheaf on $X_0$. Then, we have a map $$\Delta: M \to M[u,u^{-1}]$$ corresponding to the equivariant structure. One sets $$M_i := \{ m \in M \mid \Delta(m) = m \otimes u^i \}.$$ If we try to globalize this construction, then two different $M_i$’s are identified under an automorphism of $M$, which has degree zero with respect to $\mathbb{G}_m$. Thus, for any quasi-coherent $\mathbb{G}_m$-equivariant sheaf, this gives a quasi-coherent $\mathbb{G}_m$-quasi-coherent sheaf on $X_0$, $\mathcal E_i$.
\[lemma: decomposition on the fixed locus\] Let $X$ be a BB stratum and let $\mathcal E$ be a coherent $\mathbb{G}_m$-equivariant sheaf on the fixed locus $X_0$. Then there is a functorial decomposition $$\mathcal E \cong \bigoplus_{i \in \Z} \mathcal E_i.$$ In particular, for each $i \in \Z$, the functor $$\mathcal E \mapsto \mathcal E_i$$ is exact.
This is standard and a proof is suppressed.
\[corollary: decomposition on the fixed locus derived version\] Let $X$ be a BB stratum and let $\mathcal E$ be a bounded complex of coherent $\mathbb{G}_m$-equivariant sheaves on the fixed locus $X_0$. Then there is a functorial decomposition of the complex $$\mathcal E \cong \bigoplus_{i \in \Z} \mathcal E_i.$$ This descends to the derived category, ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X_0/\mathbb{G}_m])}$.
This follows immediately from Lemma \[lemma: decomposition on the fixed locus\].
\[definition: weight decomposition\] For a subset $I \subseteq \Z$, we say that a complex $\mathcal E$ from ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X_0/\mathbb{G}_m])}$ has **weights concentrated in $I$** if $(\mathcal H^p (\mathcal E))_i = 0$ for all $p \in \Z$ and $i \not \in I$.
Now, we turn our attention to $\mathbb{G}_m$-equivariant sheaves on $X$ itself. Let $j: X_0 \to X$ be the inclusion.
Let $\mathcal E$ be an object ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X/\mathbb{G}_m])}$ and let $I \subseteq \Z$. We say that $\mathcal E$ has **weights concentrated in $I$** if $\mathbf{L}j^* \mathcal E$ has weights concentrated in $I$.
Next, we want to give a procedure to truncate the weights. We first again go back to the local case with $X = {\operatorname{Spec}} R[x_1,\ldots,x_n]$ and the $x_i$’s having positive weight. Let $M$ be a $\mathbb{G}_m$-equivariant module over $R[x_1,\ldots,x_n]$. We can still consider $$M_a : = \{ m \in M \mid \sigma(m) = m \otimes u^a \}.$$ However, this is no longer a submodule of $M$ as multiplication by $x_i$ will raise the weight. But, $$M_{\geq a} := \bigoplus_{j \geq a} M_j$$ is a submodule of $M$ and inherits a natural $\mathbb{G}_m$-equivariant structure. The assignment $M \mapsto M_{\geq a}$ is functorial with respect to $\mathbb{G}_m$-equivariant morphisms so gives a exact functor $$\tau_{\geq a} : {\operatorname{coh}} [X/\mathbb{G}_m] \to {\operatorname{coh}} [X/\mathbb{G}_m]$$ which, of course, descends to the derived category $$\tau_{\geq a}: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X/\mathbb{G}_m])} \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X/\mathbb{G}_m])}.$$
Now let us consider the global situation. Since we have a $\mathbb{G}_m$-invariant cover of $X$ of the form ${\operatorname{Spec}} R[x_1,\ldots,x_n]$ with $x_i$ having positive weights and $R$ have zero weights, we can glue this construction to get $$\begin{aligned}
\tau_{\geq a}: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X/\mathbb{G}_m])} & \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X/\mathbb{G}_m])} \\
\mathcal E & \to \mathcal E_{\geq a}.\end{aligned}$$
Let $a \in \Z$. For a bounded complex of coherent $\mathbb{G}_m$-equivariant sheaves on $X$, $\mathcal E$, one calls $\mathcal E_{\geq a}$ a **weight truncation** of $\mathcal E$.
\[lemma: truncation of weights actually truncates weights\] If $\mathcal E$ has weights concentrated in $I$, then $\tau_{\geq a} \mathcal E$ has weights concentrated in $I \cap [a, \infty)$. Moreover, if $\mathcal E$ has weights concentrated in $[a, \infty)$, then there is a natural quasi-isomorphism $\mathcal E \cong \mathcal E_{\geq a}$.
We can check this computation locally and assume that $\mathcal E$ is a bounded complex of locally-free sheaves. Then, one sees that there is a natural isomorphism $$(\mathcal E_{\geq a})|_{X_0} \cong (\mathcal E|_{X_0})_{\geq a}.$$ Looking at the left-hand side we see that the complex has weights in $[a, \infty) \cap I$.
Now we level up and consider an appropriate type of groupoid in BB strata. Let $X^1 \overset{s}{\underset{t}\rightrightarrows} X^0$ be a groupoid scheme with $s$, $t$ smooth and $X^1,X^0$ smooth and quasi-projective. In general, we shall suppress the additional data packaged in a groupoid scheme, including in the next statement. Assume we have a commutative diagram
\(m) \[matrix of math nodes, row sep=2em, column sep=1.5em, text height=1.5ex, text depth=0.25ex\] [ \_m X\^0 & X\^0\
X\^1 & X\^0.\
]{}; (\[yshift=2pt\]m-1-1.east) edge node \[above\] [$\pi$]{} (\[yshift=2pt\]m-1-2.west) (\[yshift=-2pt\]m-1-1.east) edge node \[below\] [$\sigma$]{} (\[yshift=-2pt\]m-1-2.west) (m-1-1) edge node \[left\] [$l$]{} (m-2-1) (\[yshift=2pt\]m-2-1.east) edge node \[above\] [$s$]{} (\[yshift=2pt\]m-2-2.west) (\[yshift=-2pt\]m-2-1.east) edge node \[below\] [$t$]{} (\[yshift=-2pt\]m-2-2.west) (m-1-2) edge node \[right\] [$=$]{} (m-2-2) ;
of groupoid schemes with $l$ a closed embedding and $\sigma$ an action. Then, we can define a morphism $$\begin{aligned}
A: \mathbb{G}_m \times X^1 & \to X^1 \\
(\alpha, x_1) & \mapsto l(\alpha,t(x_1)) \cdot x_1 \cdot l(\alpha^{-1},\sigma(\alpha,s(x_1)))\end{aligned}$$ where the central dot is notation for the multiplication $m: X_1 \times_{s,X^0,t} X^1 \to X^1$.
\[lemma: adjoint action of Gm\] The morphism $A$ defines an action of $\mathbb{G}_m$ on $X^1$ making both $s$ and $t$ equivariant.
This is straightforward to verify so the details are suppressed.
\[definition: BB groupoid\] If we have a commutative diagram of groupoids as above, the action $A$ is the called the **adjoint** of $\mathbb{G}_m$ on $X^1$. We say that a groupoid scheme $X^{\bullet}$ is a **stacky BB stratum** if $A$ extends to a morphism $\mathbb{A}^1 \times X^1 \to X^1$ and $(X^0)^{\mathbb{G}_m}$ is connected. Similarly, we say the associated stack $[X^0/X^1]$ is a stacky BB stratum if $X^{\bullet}$ is such.
We can pass to the fixed loci of the $\mathbb{G}_m$ actions on $X^{\bullet}$ to get another groupoid scheme $(X^1)^{\mathbb{G}_m} \rightrightarrows (X^0)^{\mathbb{G}_m}$ which we call the **fixed substack** and denoted by $[X^0/X^1]_0$. Taking the limit as $\alpha \to 0$ in $\mathbb{G}_m$ we get an induced projection denoted by $\pi: [X^0/X^1] \to [X^0/X^1]_0$.
Note that if $A$ extends to a morphism $\mathbb{A}^1 \times X^1 \to X^1$ then $X^1$ and $X^0$ are unions of BB strata. For simplicity of exposition, we require the connectedness of $(X^0)^{\mathbb{G}_m}$. The disconnected case can be handled using the same arguments with minor modifications.
\[lemma: truncation well-defined operation on stacky BB-strata\] Let $X^{\bullet}$ be a stacky BB stratum and let $\tau_{\geq l}$ be the trunction functor on ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X^0/\mathbb{G}_m])}$ as previously defined. Then, $\tau_{\geq l}$ descends to an endofunctor $$\tau_{\geq l} : {\operatorname{D}^{\operatorname{b}}(\operatorname{coh } X^{\bullet} )} \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh } X^{\bullet} )}.$$ Furthermore, the weight decomposition on ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X^0_0/\mathbb{G}_m])}$ descends to ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }[X^0_0/X^1_0])}$.
Recall that a quasi-coherent sheaf on $X^{\bullet}$ is a pair $(\mathcal E, \theta)$ with $\mathcal E \in {\operatorname{Qcoh}}(X^0)$ and isomorphism $\theta: s^* \mathcal E \overset{\sim}{\to} t^* \mathcal E$ satisfying an appropriate cocycle condition and identity condition. Note that any sheaf on $X^{\bullet}$ carries a $\mathbb{G}_m$-equivariant structure by pulling back along $l$, so truncation is well-defined on $\mathcal E$. For any coherent sheaf $\mathcal E$ on $X^{\bullet}$, we will show that $$\theta(t^* (\tau_{\geq l} \mathcal E)) \subset s^*(\tau_{\geq l} \mathcal E).$$ This suffices to show that $\tau_{\geq l}$ descends as $\theta^{-1} = i^*\theta$ where $i: X^1 \to X^1$ is the inverse over $X^0$.
This question is local so we may assume that $X^1 = {\operatorname{Spec}} S$ and $X^0 = {\operatorname{Spec}} R$. Let $M$ be the module corresponding to $\mathcal E$ so that $\theta : M \otimes_{R,t} S \to M \otimes_{R,s} S$ is an isomorphism. As we will need to use it, we now recall the cocycle condition in this local situation. We have a diagram
\(m) \[matrix of math nodes, row sep=2em, column sep=1.5em, text height=1.5ex, text depth=0.25ex\] [ M \_[R,t]{} S \_[s,R,t]{} S & M \_[R,t]{} S \_[s,R,t]{} S\
S \_[s,R,t]{} S \_[s,R]{} M & S \_[s,R,t]{} S \_[t,R]{} M.\
]{}; (m-1-1) edge node \[above\] [$\theta \otimes_R t$]{} (m-1-2) (m-1-2) edge node \[right\] [$\sim$]{} (m-2-2) (m-2-2) edge node \[below\] [$s \otimes_R \theta$]{} (m-2-1) (m-2-1) edge node \[left\] [$\sim$]{} (m-1-1) ;
which gives us an isomorphism $$M \otimes_{R,t} S \otimes_{s,R,t} S \overset{ (s \otimes_R \theta) \circ (\theta \otimes_R t) }{\to} M \otimes_{R,t} S \otimes_{s,R,t} S.$$ We can also tensor $\theta$ over $S$ with $m : S \to S \otimes_{s,R,t} S$ to get another isomorphism. The cocycle condition is $$(s \otimes_R \theta) \circ (\theta \otimes_R t) = \theta \otimes_S m.$$
Now, we wish to check that $\theta(M_i \otimes_R S) \subset M_{\geq i} \otimes_R S$. We do this as follows. First, since $A: S \to S[u,u^{-1}]$ has image in $S[u]$, we must have that the image of $$(l \otimes 1 \otimes l) \circ (m \otimes 1) \circ m := \tilde{A}: S \to R[u_1,u_1^{-1}] \otimes_{\pi, R, t} S \otimes_{s, R, \sigma} R[u_2,u_2^{-1}]$$ lies in the $S$-subalgebra generated by $u_1^i u_2^j$ with $i - j \geq 0$. Since $\theta$ satisfies the cocyle condition, we can factor $$\theta \otimes_S \tilde{A} = (1 \otimes_R \theta \otimes_R 1) \circ (\overline{\theta} \otimes_R 1 \otimes_R 1) \circ (1 \otimes_R \theta \otimes_R 1) \circ (1 \otimes_R 1 \otimes_R \overline{\theta}).$$ Applying this to $m_j \in M_j$, we have $$(\theta \otimes_S \tilde{A})(m_j) = \sum_i \theta(\theta(m_j)_i)) u_1^iu_2^j$$ where $$\theta(m_j) = \sum_i \theta(m_j)_i u_1^i.$$ As $i-j \geq 0$, we see that the $\theta(m_j) \in M_{\geq j} \otimes_{R,s} S$ and $\theta$ preserves the truncation.
For the final statement, repeating the previous argument and assuming $\mathbb{G}_m$ acts trivially on $R$ and $S$ shows that $\theta$ preserves the whole splitting via weights.
\[example: BFK example\] Let $G$ be a linear algebraic group acting on a variety $X$. Assume we have a one-parameter subgroup, $\lambda: \mathbb{G}_m \to G$, and choice of connected component of the fixed locus, $Z_{\lambda}^0$. Then, we have the BB stratum $Z_{\lambda}$ and its orbit $S_{\lambda} := G \cdot Z_{\lambda}$. Let $$P(\lambda) := \{ g \in G \mid \lim_{\alpha \to 0} \lambda(\alpha) g \lambda(\alpha)^{-1} \text{ exists }\}.$$ There is an induced action of $P(\lambda)$ on $Z_{\lambda}$. In general, $G \overset{P(\lambda)}{\times} Z_{\lambda}$ is a resolution of singularities of $S_{\lambda}$. If we assume this map is an isomorphism, then we get what is called **an elementary stratum** in the language of [@BFKGIT]. Since $$[G \overset{P(\lambda)}{\times} Z_{\lambda} / G ] \cong [Z_{\lambda} / P(\lambda) ]$$ and the groupoid, $$P(\lambda) \times Z_{\lambda} \overset{\pi}{\underset{\sigma}\rightrightarrows} Z_{\lambda}$$ is a stacky BB stratum, we see that $[S_{\lambda}/G]$ is a stacky BB stratum. The simplest case is $G = \mathbb{G}_m$.
Removing stacky BB strata and comparing derived categories
----------------------------------------------------------
Let $\mathcal X$ be a smooth Artin stack of finite type over $k$.
\[definition: weights along stacky fixed locus\] Let $i: \mathcal Z \to \mathcal X$ be a smooth closed substack that is also a stacky BB stratum and let $l: \mathcal Z_0 \to \mathcal X$ be the closed immersion of the fixed substack. Let $\mathcal E$ be a bounded complex of coherent sheaves $\mathcal X$ and let $I \subseteq \Z$. We say that $\mathcal E$ has **weights concentrated in $I$** if $\mathbf{L}l^* \mathcal E \in {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal Z_0)}$ has weights concentrated in $I$.
\[definition: windows\] Let $\mathcal Z$ be a stacky BB stratum in $\mathcal X$ and let $I \subseteq \Z$. The **$I$-window** associated to $\mathcal Z$ is the full subcategory whose objects have weights concentrated in $I$. We denote this subcategory by ${{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I,\mathcal Z_0)$.
\[lemma: fully-faithful\] Assume that the weights of the conormal sheaf of $\mathcal Z$ in $\mathcal X$ are all strictly negative and let $t_{\mathcal Z}$ be the weight of the relative canonical sheaf $\omega_{\mathcal Z \mid \mathcal X}$. Set $\mathcal U := \mathcal X \setminus \mathcal Z$ and $j: \mathcal U \to \mathcal X$ be the inclusion. Then, the functor $$j^*: {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I,\mathcal Z_0) \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U)}$$ is fully-faithful whenever $I$ is contained in a closed interval of length $< -t_{\mathcal Z}$.
The argument here is essentially due to Teleman [@Tel Section 2] which the author first learned of in [@HL12]. It amounts to descending through a few spectral sequences. For the convenience of the reader and to keep the paper self-contained, we recapitulate it in some detail.
For any two objects $\mathcal E,\mathcal F$ of ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal X)}$ there is an exact triangle of graded vector spaces $${\operatorname{Hom}}_{\mathcal X, \mathcal Z}(\mathcal E, \mathcal F) \to {\operatorname{Hom}}_{\mathcal X}(\mathcal E, \mathcal F) \overset{j^*}{\to} {\operatorname{Hom}}_{\mathcal U}(j^* \mathcal E, j^* \mathcal F)$$ coming from applying the exact triangle of derived functors $$\label{equation: local cohomology global sections}
\mathbf{R}\Gamma_{\mathcal Z}(\mathcal X,-) \to \mathbf{R}\Gamma(\mathcal X,-) \overset{j^*}{\to} \mathbf{R}\Gamma(\mathcal U,-)$$ to $\mathbf{R}\mathcal Hom_{\mathcal X}(\mathcal E, \mathcal F)$. Therefore, a necessary and sufficient condition for $j^*$’s fully-faithfulness is the vanishing of ${\operatorname{Hom}}_{\mathcal X, \mathcal Z}(\mathcal E, \mathcal F)$.
The exact triangle in Equation comes from applying $\mathbf{R}\Gamma(\mathcal X,-)$ to the following exact triangle $$\mathcal H_{\mathcal Z} \to {\operatorname{Id}} \to \mathbf{R}j_*j^*$$ of functors. Here $\mathcal H_{\mathcal Z}$ is the derived sheafy local cohomology functor. The complex $\mathcal H_{\mathcal Z}(\mathcal G)$ is not scheme-theoretically supported on $\mathcal Z$ but there is a bounded above filtration by powers of the ideal sheaf $\mathcal I_{\mathcal Z}$. The associated graded sheaves are scheme-theoretically supported on $\mathcal Z$. Furthermore, since $\mathcal Z$ is smooth, the $s$-th associated graded piece is isomorphic to $\mathbf{L}i^*\mathcal G \otimes {\operatorname{Sym}}^s(\mathcal T_{\mathcal Z \mid \mathcal X}) \otimes \omega^{-1}_{\mathcal Z \mid \mathcal X}$.
We may now take global sections on $\mathcal Z$ which we may factor through two pushforwards: one by $\pi: \mathcal Z \to \mathcal Z_0$ and one by the rigidification map $r: \mathcal Z_0 \to \mathcal Z^{\mathbb{G}_m}_0$ [@ACV Theorem 5.15]. The pushforward $r_*: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal Z_0)} \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal Z_0^{\mathbb{G}_m})}$ projects onto the weight $0$ component of the weight decomposition. Thus, to establish vanishing, it suffices to show that there is no weight $0$ component in the splitting of $$\pi_* \left( \mathbf{L}i^*\mathcal G \otimes {\operatorname{Sym}}^s(\mathcal T_{\mathcal Z \mid \mathcal X}) \otimes \omega^{-1}_{\mathcal Z \mid \mathcal X} \right).$$ Since we can resolve anything using pullbacks from $\mathcal Z_0$, we can use the projection formula to get $$\pi_* \left( \mathbf{L}i^*\mathcal G \otimes {\operatorname{Sym}}^s(\mathcal T_{\mathcal Z \mid \mathcal X}) \otimes \omega^{-1}_{\mathcal Z \mid \mathcal X} \right) \cong \mathbf{L}l^* \mathcal G \otimes {\operatorname{Sym}}^s(\mathcal T_{\mathcal Z \mid \mathcal X})|_{\mathcal Z_0} \otimes \omega^{-1}_{\mathcal Z \mid \mathcal X}|_{\mathcal Z_0} \otimes \pi_* \mathcal O_{\mathcal Z}.$$ One notices that the terms $${\operatorname{Sym}}^s(\mathcal T_{\mathcal Z \mid \mathcal X})|_{\mathcal Z_0} \otimes \omega^{-1}_{\mathcal Z \mid \mathcal X}|_{\mathcal Z_0} \otimes \pi_* \mathcal O_{\mathcal Z}$$ have weights $\geq -t_{\mathcal Z}$. When the weights of $\mathcal G$ are concentrated in $(t_{\mathcal Z}, \infty)$, we get a trivial weight zero component and the desired vanishing. Now, setting $\mathcal G = \mathbf{R}\mathcal Hom_{\mathcal X}(\mathcal E, \mathcal F)$ for $\mathcal E, \mathcal F \in {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I,\mathcal Z_0)$, we see that our assumption on $I$ exactly implies this.
\[lemma: essentially surjective\] Assume that the weights of the conormal sheaf of $\mathcal Z$ in $\mathcal X$ are all strictly negative and let $t_{\mathcal Z}$ be the weight of the relative canonical sheaf $\omega_{\mathcal Z \mid \mathcal X}$. Set $\mathcal U := \mathcal X \setminus \mathcal Z$ and $j: \mathcal U \to \mathcal X$ be the inclusion. Then, the functor $$j^*: {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I,\mathcal Z_0) \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U)}$$ is essentially surjective whenever $I$ contains a closed interval of length $\geq -t_{\mathcal Z} - 1$.
We may assume that $\mathcal X \not = \mathcal Z$ and we may reduce to $I$ being an interval of length $-t_{\mathcal Z} -1$. To demonstrate essential surjectivity it suffices to iteratively reduce the weights by forming exact triangles $$\mathcal E' \to \mathcal E \to \mathcal T$$ where $\mathcal T$ is set-theoretically supported on $\mathcal Z$ and the weights of $\mathcal E'$ are concentrated in a strictly smaller interval than $\mathcal E$. There is an exact triangle $$\tau_{\geq b-1} \mathbf{L}i^* \mathcal E \to \mathbf{L}i^*\mathcal E \to \mathcal W$$ where weights of $\mathcal E$ along $\mathcal Z$ are concentrated in $[a,b]$ with $b-a \geq -t_{\mathcal Z}$. By Lemma \[lemma: truncation of weights actually truncates weights\], the weights of $\tau_{\geq b-1} \mathbf{L}i^* \mathcal E$ are concentrated in $[a,b-1]$. Then, the only weight of $\mathcal W$ is $b$. Consider $i_*\mathcal W$ as an object of ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal X)}$. Let us check that the weights of $i_*\mathcal W$ lie in $[b+t_{\mathcal Z},b]$. We must compute $$\mathbf{L}i^* i_* \mathcal W.$$ This has cohomology sheaves isomorphic to $\mathcal W \otimes \bigwedge^*(\Omega_{\mathcal Z \mid \mathcal X})$ which by assumption has weights in $[b+t_{\mathcal Z},b]$. Note that the only contribution to weight $b$ is $\mathcal W$ itself and the map $$\mathbf{L}i^*\mathcal E|_{X_0} \to \mathcal W|_{X_0}$$ induces an isomorphism on the weight $b$ portion of the decomposition. Now we define $\mathcal E'$ to be the cone over the map $\mathcal E \to i_* \mathcal W$. Restricting the exact triangle to $X_0$ and remembering that computing weight spaces is an exact functor, we see that the weights of $\mathcal E'$ along $\mathcal Z$ are concentrated in $[a,b-1]$. We may also raise the weights by conjugating the previous procedure by dualization. Combining the two procedures, we can move the weights of any complex into $I$ up to $\mathcal Z$-torsion.
\[corollary: window equivalence\] Assume that the weights of the conormal sheaf of $\mathcal Z$ in $\mathcal X$ are all strictly negative and let $t_{\mathcal Z}$ be the weight of the relative canonical sheaf $\omega_{\mathcal Z \mid \mathcal X}$. Set $\mathcal U := \mathcal X \setminus \mathcal Z$ and $j: \mathcal U \to \mathcal X$ be the inclusion. Then, the functor $$j^*: {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I,\mathcal Z_0) \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U)}$$ is an equivalence whenever $I$ is interval of length $-t_{\mathcal Z} - 1$.
This is an immediate consequence of Lemmas \[lemma: fully-faithful\] and \[lemma: essentially surjective\].
\[definition: complementary pieces\] Let $s \in \Z$. Denote by $\mathcal C_s(\mathcal Z)$ the full subcategory of ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal Z_0)}$ consisting of objects with weight $s$.
\[lemma: complementary pieces pull back ok\] Assume that the weights of the conormal sheaf of $\mathcal Z$ in $\mathcal X$ are all strictly negative. The functor $$\begin{aligned}
\Upsilon_s : \mathcal C_s(\mathcal Z) & \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal X)} \\
\mathcal E & \mapsto i_* \pi^* \mathcal E
\end{aligned}$$ is fully-faithful.
We use the standard adjunctions. We have $${\operatorname{Hom}}_{\mathcal X}( \Upsilon_s \mathcal E, \Upsilon_s \mathcal F) \cong {\operatorname{Hom}}_{\mathcal Z}( \mathbf{L}i^*i_* \pi^* \mathcal E, \pi^* \mathcal F).$$ The cohomology sheaves of $\mathbf{L}i^*i_* \pi^* \mathcal E$ are isomorphic to $\pi^* \mathcal E \otimes {\operatorname{Sym}}^*(\mathcal T_{\mathcal Z \mid \mathcal X})$. Thus, there is a natural map $$\label{equation: a map in complementary pieces proof}
{\operatorname{Hom}}_{\mathcal Z}( \pi^* \mathcal E, \pi^* \mathcal F) \to {\operatorname{Hom}}_{\mathcal X}( \Upsilon_s \mathcal E, \Upsilon_s \mathcal F).$$ We first check that this is isomorphism. The cone of $\mathbf{L}i^*i_* \pi^* \mathcal E \to \pi^* \mathcal E$ has cohomology sheaves $\pi^* \mathcal E \otimes {\operatorname{Sym}}^{\geq 1}(\Omega_{\mathcal Z \mid \mathcal X})$ which have weights $< s$. Let us show that $${\operatorname{Hom}}_{\mathcal Z}(\pi^* \mathcal E \otimes {\operatorname{Sym}}^{\geq 1}(\Omega_{\mathcal Z \mid \mathcal X}), \pi^* \mathcal F[s]) = 0$$ for any $s$. This vanishing combined with a spectral sequence argument gives that the map in Equation is an isomorphism. We have $$\begin{aligned}
{\operatorname{Hom}}_{\mathcal Z}(\pi^* \mathcal E \otimes {\operatorname{Sym}}^{\geq 1}(\Omega_{\mathcal Z \mid \mathcal X}), \pi^* \mathcal F[s]) & \cong {\operatorname{Hom}}_{\mathcal Z_0}(\mathcal E \otimes {\operatorname{Sym}}^{\geq 1}(\Omega_{\mathcal Z \mid \mathcal X}), \mathcal F \otimes {\operatorname{Sym}}^*(\Omega_{\mathcal Z_0 \mid \mathcal Z})[s]) \\
& \cong {\operatorname{Hom}}_{\mathcal Z_0}(\mathcal E, \mathcal F \otimes {\operatorname{Sym}}^*(\Omega_{\mathcal Z_0 \mid \mathcal Z}) \otimes {\operatorname{Sym}}^{\geq 1}(\mathcal T_{\mathcal Z \mid \mathcal X})[s]).
\end{aligned}$$ We can again factor through pushforward to the rigidification $\mathcal Z_0^{\mathbb{G}_m}$. In this case, the above Hom-space is zero as the weights of the right hand side are concentrated in $(s,\infty)$.
Next we have $$\begin{aligned}
{\operatorname{Hom}}_{\mathcal Z}( \pi^* \mathcal E, \pi^* \mathcal F) & \cong {\operatorname{Hom}}_{\mathcal Z_0}( \mathcal E, \mathcal F \otimes {\operatorname{Sym}}^*(\Omega_{\mathcal Z_0 \mid \mathcal Z})).
\end{aligned}$$ The only piece of $\mathcal F \otimes {\operatorname{Sym}}^*(\Omega_{\mathcal Z_0 \mid \mathcal Z})$ in weight $s$ is $\mathcal F$. Thus, $${\operatorname{Hom}}_{\mathcal Z_0}( \mathcal E, \mathcal F \otimes {\operatorname{Sym}}^*(\Omega_{\mathcal Z_0 \mid \mathcal Z})) \cong {\operatorname{Hom}}_{\mathcal Z_0}( \mathcal E, \mathcal F)$$ which gives fully-faithfulness.
\[lemma: semi-orthogonal decomposition of bigger windows\] Assume that the weights of the conormal sheaf of $\mathcal Z$ in $\mathcal X$ are all strictly negative and let $t_{\mathcal Z}$ be the weight of the relative canonical sheaf $\omega_{\mathcal Z \mid \mathcal X}$. Assume that $v-u \geq -t_{\mathcal Z}$. There is a semi-orthogonal decomposition $${{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}([u,v],\mathcal Z_0) = \langle \Upsilon_{v}, {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}([u,v-1],\mathcal Z_0) \rangle.$$
Take $\mathcal E \in {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}([u,v],\mathcal Z_0)$ and consider the exact triangle from the proof of Lemma \[lemma: essentially surjective\] $$\mathcal E' \to \mathcal E \to i_* \mathcal W.$$ It is clear that from the definition that $i_* \mathcal W$ lies in the image of $\Upsilon_v$ and we saw already that $\mathcal E'$ lies in ${{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}([u,v-1],\mathcal Z_0)$. Thus, the two subcategories generate ${{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}([u,v],\mathcal Z_0)$. It remains to check semi-orthogonality. This follows as in the proof of Lemma \[lemma: fully-faithful\].
\[remark: singular case\] The reader should observe that everything goes through if $\mathcal X$ is singular but smooth along $\mathcal Z$.
\[definition: elementary strata and elementary wall crossing\] Let $\mathcal X$ be a smooth algebraic stack of finite-type over $k$. A stacky BB stratum $\mathcal Z$ in $\mathcal X$ is called an **elementary stratum** if the weights of $\Omega_{\mathcal Z \mid \mathcal X}$ along $\mathcal Z$ are strictly negative.
A pair of elementary strata $\mathcal Z_-$ and $\mathcal Z_+$ is called an **elementary wall crossing** if $\mathcal Z_{-,0} = \mathcal Z_{+,0}$ and the two embeddings of $\mathbb{G}_m$ into the automorphisms of $\mathcal Z_{\pm,0}$ differ by inversion.
\[theorem: elementary wall crossing\] Assume we have elementary wall crossing, $\mathcal Z_-,\mathcal Z_+$. Fix $d \in \Z$.
1. If $t_{\mathcal Z_+} < t_{\mathcal Z_-}$, then there are fully-faithful functors, $$\Phi^+_d: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_-)} \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_+)},$$ and, for $-t_{\mathcal Z_-} + d \leq j \leq -t_{\mathcal Z_+} + d - 1$, $$\widetilde{\Upsilon}_j^-: \mathcal C_j(\mathcal Z_-) \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_+)},$$ and a semi-orthogonal decomposition, $${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_+)} = \langle \widetilde{\Upsilon}^-_{-t_{\mathcal Z_-}+d}, \ldots, \widetilde{\Upsilon}^-_{-t_{\mathcal Z_+}+d-1}, \Phi^+_d \rangle.$$
2. If $t_{\mathcal Z_+} = t_{\mathcal Z_-}$, then there is an exact equivalence, $$\Phi^+_d: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_-)} \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_+)}.$$
3. If $t_{\mathcal Z_+} > t_{\mathcal Z_-}$, then there are fully-faithful functors, $$\Phi^-_d: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_+)} \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_-)},$$ and, for $-t_{\mathcal Z_+} + d \leq j \leq -t_{\mathcal Z_-} + d -1$, $$\widetilde{\Upsilon}_j^+: \mathcal C_j(\mathcal Z_+) \to {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_-)},$$ and a semi-orthogonal decomposition, $${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_-)} = \langle \widetilde{\Upsilon}^+_{-t_{\mathcal Z_+}+d}, \ldots, \widetilde{\Upsilon}^+_{-t_{\mathcal Z_-}+d-1}, \Phi^-_d \rangle.$$
This is the same argument as for the proof of [@BFKGIT Theorem 3.5.2]. Again, we recall it in some detail. Swapping the roles of $\mathcal Z_+$ and $\mathcal Z_-$ we can assume that $t_{\mathcal Z_+} \leq t_{\mathcal Z_-}$. Choose intervals $I_- \subseteq I_+$ with the diameter of $I_{\pm}$ equal to $-t_{\mathcal Z_{\pm}} - 1$ and $d := {\operatorname{min}} I_- = {\operatorname{min}} I_+$. From Lemma \[lemma: semi-orthogonal decomposition of bigger windows\], there is a semi-orthogonal decomposition $${{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I_+,\mathcal Z_0) = \langle \Upsilon^-_{-t_{\mathcal Z_-}+d}, \ldots, \Upsilon^-_{-t_{\mathcal Z_+}+d-1}, {{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I_-,\mathcal Z_0) \rangle$$ Using Corollary \[corollary: window equivalence\], we can pull back to $\mathcal U_+$ to get $${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_+)} = \langle i^*_+ \circ \Upsilon^-_{-t_{\mathcal Z_-}+d}, \ldots, i^*_+ \circ \Upsilon^-_{-t_{\mathcal Z_+}+d-1}, i^*_+{{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I_-,\mathcal Z_0) \rangle.$$ Applying Corollary \[corollary: window equivalence\] again, we know that $i_-^*$ induces an equivalence between ${{{\vphantom{\kern-0.23em\operatorname{W}}}^{=}{\kern-0.23em\operatorname{W}}}^{\kern-0.21em =}}(I_-,\mathcal Z_0)$ and ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal U_-)}$. We set $$\widetilde{\Upsilon}_j^- := i^*_+ \circ \Upsilon^-_j$$ and $$\Phi^+_d := i_+^* \circ (i^*_-)^{-1}$$ to finish.
Stable sheaves on rational surfaces
===================================
In this section, we apply Theorem \[theorem: elementary wall crossing\] using the well-known structure of semi-stable rank two torsion-free sheaves on rational surfaces [@EG; @FQ; @MW].
Let $S$ be a smooth complex projective surface. In this section, we will show how to apply Theorem \[theorem: elementary wall crossing\] to wall-crossing of Gieseker stable sheaves obtained by variation of the polarization on $S$.
Basics
------
Let us recall the main notions of stability. Let $L$ be an ample line bundle on $S$.
\[definition: stability\] Let $E$ be coherent sheaf on $S$. The sheaf, $E$, is **Gieseker $L$-semi-stable** [@Gieseker] if it is torsion-free and for any proper subsheaf $F \subsetneq E$ one has $\overline{p}_L(F) \leq \overline{p}_L(E)$ where $\overline{p}_L$ is the reduced Hilbert polynomial associated to the embedding given by $L$. If the inequality is strict for any proper subsheaf, $E$ is **Gieseker $L$-stable**. If $E$ is not Gieseker $L$-semi-stable, then $E$ is **Gieseker $L$-unstable**.
The sheaf, $E$, is **Mumford $L$-semi-stable** [@Mum62; @Takemoto] if it is torsion-free and for any proper subsheaf $F \subset E$ one has $\mu_L(F) \leq \mu_L(E)$ where $\mu_L$ is the $L$-slope of the sheaf. Again if the inequality is always strict, $E$ is **Mumford $L$-stable** and if $E$ is not Mumford $L$-semi-stable then it is called **Mumford $L$-unstable**.
Fix invariants $c_0,c_1,c_2$ and consider the moduli functors, $\widetilde{\mathcal M}_L(c_0,c_1,c_2)$, $\widetilde{\mathcal Mum}_L(c_0,c_1,c_2)$, given by $$\begin{aligned}
X & \mapsto \{\text{iso. classes of Gieseker $L$-s.s. families with } c_0(\mathcal F_x) = c_0, c_1(\mathcal F_x) = c_1, c_2(\mathcal F_x) = c_2 \} \\
X & \mapsto \{\text{iso. classes of Mumford $L$-s.s. families with } c_0(\mathcal F_x) = c_0, c_1(\mathcal F_x) = c_1, c_2(\mathcal F_x) = c_2 \}.\end{aligned}$$
The following is well-known.
\[lemma: moduli stack is finite-type\] The functor, $\widetilde{\mathcal M}_L(c_0,c_1,c_2)$, is an algebraic stack of finite-type over $k$. The same is true for $\widetilde{\mathcal Mum}_L(c_0,c_1,c_2)$.
First, $\widetilde{\mathcal M}_L(c_0,c_1,c_2)$ is open, [@HL Proposition 2.3.1], in the stack of coherent sheaves on $S$ which is algebraic, [@stacks-project Theorem 75.5.12]. Thus, $\widetilde{\mathcal M}_L(c_0,c_1,c_2)$ is algebaic stack. There is a bounded family of $L$-semi-stable sheaves with fixed numerical invariants, [@HL Theorem 3.3.7]. Base changing the induced map to $\widetilde{\mathcal M}_L(c_0,c_1,c_2)$ gives a smooth, surjective map with source of finite-type. A similar argument shows the statement for $\widetilde{\mathcal Mum}_L(c_0,c_1,c_2)$.
Next, we address smoothness. Recall that $\mathcal E$ is Gieseker (Mumford) polystable if it is the direct sum of Gieseker (Mumford) stable sheaves.
\[lemma: smoothness at wall points\] Assume $K_S < 0$. Let $E$ be Gieseker $L$-polystable or simple. Then, $E$ is a smooth point of $\widetilde{\mathcal M}_L(c_0,c_1,c_2)$. A similar statement holds for $\widetilde{\mathcal Mum}_L(c_0,c_1,c_2)$.
Write $E = \bigoplus_{i \in I} E_i$ with each $E_i$ stable. First, since each $E_i$ is stable, or since $E$ is simple, and $S$ is proper, we have $${\operatorname{Hom}}(E,E) \subset {\operatorname{M}}_m(\Gamma(S,\mathcal O_S)) = M_m(k).$$ where $M_m(k)$ is $m \times m$-matrices in $k$.
To check smoothness, it suffices to show that ${\operatorname{ext}}^1(E, E) - {\operatorname{hom}}(E, E) = \chi({\operatorname{Ext}}^*(E,E))$, which remains constant over $\mathcal M_L(c_0,c_1,c_2)$. It suffices, therefore, to show that ${\operatorname{Ext}}^2(E, E) = 0$. From Serre duality, we have $${\operatorname{ext}}^2(E, E) = {\operatorname{h}}^0( \mathcal Hom(E, E) \otimes \omega_S).$$ Since $K_S < 0$, there is an inclusion $\omega_S \to \mathcal O_S$ and an inclusion $$\mathcal Hom(E, E) \otimes \omega_S \to \mathcal Hom(E,E).$$ Taking global sections, we see that ${\operatorname{h}}^0( \mathcal Hom(E, E) \otimes \omega_S)$ counts the dimension of the subspace of global sections of ${\operatorname{Hom}}(E,E)$ which vanish along $-K_S$. Since all global sections are constant over $S$, we have $${\operatorname{h}}^0( \mathcal Hom(E, E) \otimes \omega_S) = 0.$$ Since Mumford stability implies Gieseker stability, we get the same statement for Mumford semi-stable sheaves.
Rank two stable sheaves
-----------------------
We now restrict ourselves to the case $S$ is a rational surface and $c_0 = 2$. We recall the results of Friedman and Qin [@FQ], see also [@EG; @MW]. Let $L_+$ and $L_-$ be two ample line bundles on $S$. For a divisor $\xi$ satisfying $\xi \equiv c_1 ({\operatorname{mod}} 2)$ and $c_1^2-4c_2 \leq \xi^2 \leq 0$, consider the hyperplane in ${\operatorname{Amp}}(S)_{\mathbb{R}}$ given by $$W^{\xi} := \{ D \mid D \cdot \xi = 0 \}.$$ The hyperplane $W^{\xi}$ is called the wall associated to $\xi$. For simplicity, we shall assume that the line segment joining $L_+$ and $L_-$ intersects only a single $W^{\xi}$, determined by a unique $\xi$. The more general case, where rational multiples of $\xi$ may remain integral and define the same wall, requires minimal modification of the argument [@FQ]. Denote the polarization given by the intersection of the line and the hyperplane by $L_0$. We assume that $L_0$ lies in no other walls.
We have two inclusions $$\widetilde{\mathcal M}_{L_-}(c_1,c_2) \subseteq \widetilde{\mathcal Mum}_{L_0}(c_1,c_2) \supseteq \widetilde{\mathcal M}_{L_+}(c_1,c_2).$$ where we change notation $$\widetilde{\mathcal M}_L(2,c_1,c_2) =: \widetilde{\mathcal M}_L(c_1,c_2)$$ to reflect that the focus of our attention is upon rank two sheaves. Note the switch to Mumford semi-stable in the wall. Friedman and Qin study Mumford $L_0$-semi-stable sheaves of a particular form.
Let $Z^k(F)$ be the set of sheaves $E$ occuring in a short exact sequence $$0 \to I_{Z_1}(F) \to E \to I_{Z_2}(\Delta - F) \to 0$$ where $c_1(\Delta) = c_1$, $\xi = 2F - \Delta$, $Z_1 \in {\operatorname{Hilb}}^k(S)$, and $Z_2 \in {\operatorname{Hilb}}^{l_{\xi}-k}(S)$ with $$l_{\xi} = (4c_2 - c_1^2 + \xi^2)/4.$$
Denote the associated substack of $\widetilde{\mathcal Mum}_{L_0}(c_1,c_2)$ by $\widetilde{\mathcal Z}^k(F)$.
The substack $\widetilde{\mathcal Z}^k(F)$ is closed in $\widetilde{\mathcal Mum}_{L_0}(c_1,c_2)$ and $\widetilde{\mathcal Mum}_{L_0}(c_1,c_2)$ is smooth along $\widetilde{\mathcal Z}^k(F)$.
A Mumford $L_0$-semi-stable sheaf $E$ lies in $Z^{k}(F)$ if and only if there is a surjective map $$E(-F) \to I_{Z_2} \to 0$$ with $\xi = 2F - \Delta$ and $l(Z_2) = l_{\xi} - k$. This is a closed condition as it states $E(-F)$ lies in the Quot scheme for the Hilbert polynomial associated to $I_{Z_2}$.
By [@FQ Lemma 2.2], the non-split extensions in $Z^k(F)$ are simple. The split extensions are polystable. So Lemma \[lemma: smoothness at wall points\] gives the last statement.
Applying this proposition with the switch $F \to \Delta -F$ also gives the corresponding statement for $\widetilde{\mathcal Z}^k(\Delta - F)$.
The stacks $\widetilde{\mathcal Z}^k(F)$ admit particularly simple geometric descriptions. Let $\mathcal E^k(F)$ be the coherent sheaf on $H^k := {\operatorname{Hilb}}^k(S) \times {\operatorname{Hilb}}^{l_{\xi} - k}(S)$ classifying the extensions appearing in $Z^k(F)$. By [@FQ Lemma 2.6], $\mathcal E^k(F)$ is locally-free. Let $$X^k(F) := \underline{{\operatorname{Spec}}}(\mathcal E^k(F))$$ be the associated geometric vector bundle on $H^k$. There is a natural action of $\mathbb{G}_m^2$ on $X^k(F)$ given by endomorphisms of $I_{Z_1}(F) \oplus I_{Z_2}(\Delta - F)$.
There are isomorphisms $$\begin{aligned}
\widetilde{\mathcal Z}^k(F) & \cong [X^k(F)/\mathbb{G}_m^2] \\
\widetilde{\mathcal Z}^k(\Delta-F) & \cong [X^k(\Delta-F)/\mathbb{G}_m^2].
\end{aligned}$$
From [@FQ Lemma 2.2.i], given $E$ in $\widetilde{\mathcal Z}^k(F)$, then $Z_1$ and $Z_2$ are uniquely determined and the map $\mathcal I_{Z_1}(F) \to E$ is unique up to scaling. So the extension class for a given $E$ is determined up to scaling.
To move closer to schemes, we now rigidify our stacks and remove the residual $\mathbb{G}_m$ coming from multiples of the identity. We denote the rigidified stacks by removing the tilde, e.g. the $\mathbb{G}_m$-rigidified moduli stack of Mumford $L_0$-semi-stable sheaves will be denoted by $\mathcal Mum_{L_0}(c_1,c_2)$. We do this now, at the current point in the argument, to guarantee the following result holds.
The substacks $\mathcal Z^k(F)$ and $\mathcal Z^{l_{\xi} - k}(\Delta - F)$ form an elementary wall crossing in the stack $\mathcal Mum_{L_0}(c_1,c_2)$.
We have a presentation of $\mathcal Z^k(F)$ given by $$\mathbb{G}_m \times X^k(F) = \mathbb{G}_m^2/\mathbb{G}_m \times X^k(F) \rightrightarrows X^k(F)$$ Now take the $\mathbb{G}_m$ given by the first summand in $\mathbb{G}_m$, i.e scalar endomorphisms of $I_{Z_1}(F)$. Under this $\mathbb{G}_m$-action, $X^k(F)$ contracts onto the zero locus, $H^k$, so is a BB-stratum. This also gives the morphisms $l: \mathbb{G}_m \times X^k(F) \to \mathbb{G}_m^2 \times X^k(F)$ and $l: \mathbb{G}_m \times X^k(F) \to \mathbb{G}_m^2/\mathbb{G}_m \times X^k(F)$. The adjoint action is trivial on the first factor and is the action on the second. Thus, it extends to $\mathbb{A}^1$ and $\mathcal Z^k(F)$ is a stacky BB stratum. Computations in [@FQ Section 3] identify the conormal sheaf of $\mathcal Z^k(F)$ with $\mathcal E^{l_{\xi}-k}(\Delta - F)$ which has weight $-1$ with respect to this $\mathbb{G}_m$-action. So $\mathcal Z^k(F)$ is an elementary stratum. Similarly, one shows that $\mathcal Z^{l_{\xi} -k}(\Delta - F)$ is an elementary stratum. Since we have rigidified, the two choices of $\mathbb{G}_m$ actions on the fixed substacks differ by inversion.
From here on, we assume that $$L_- \cdot (2F - \Delta) < 0 < L_+ \cdot (2F - \Delta).$$ and $$\omega_S^{-1} \cdot (2F - \Delta) \geq 0.$$
Next, we want to compare the moduli stacks $\mathcal M_{L_+}(c_1,c_2)$ and $\mathcal M_{L_-}(c_1,c_2)$ via a sequence to elementary wall-crossings.
To do so, we consider the following intermediate stacks $$\mathcal M^{\leq l, \geq t} := \mathcal Mum_{L_0}(c_1,c_2) \setminus \left( \bigcup_{k > l} \mathcal Z^k(F) \cup \bigcup_{k < t} \mathcal Z^k(\Delta - F) \right).$$
\[lemma: ends of wall crossings\] We have $$\begin{aligned}
\mathcal M_{L_+}(c_1,c_2) & = \mathcal M^{\leq l_{\xi}, \geq l_{\xi}+1} \\
\mathcal M_{L_-}(c_1,c_2) & = \mathcal M^{\leq -1, \geq 0}.
\end{aligned}$$
This is [@FQ Lemma 3.2.ii].
We get a sequence of elementary wall crossing given by $$\mathcal M^{\leq l, \geq l+1} \subset \mathcal M^{\leq l+1, \geq l+1} \supset \mathcal M^{\leq l+1, \geq l+2}.$$ Applying Theorem \[theorem: elementary wall crossing\], we get the following statement.
\[proposition: one wall crossing\] With the assumptions as above, there is a semi-orthogonal decomposition $${\operatorname{D}^{\operatorname{b}}(\operatorname{coh } \mathcal M^{\leq l+1, \geq l+2} )} = \left\langle \underbrace{{\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^l)}, \ldots, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^l)}}_{\mu_{\xi}}, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh } \mathcal M^{\leq l, \geq l+1})}\right\rangle$$ where $$\begin{aligned}
H^l & := {\operatorname{Hilb}}^l(S) \times {\operatorname{Hilb}}^{l_{\xi} - l}(S) \\
\mu_{\xi} & := \omega_S^{-1} \cdot (2F - \Delta) = \omega_S^{-1} \cdot \xi.
\end{aligned}$$
This is an immediate application of Theorem \[theorem: elementary wall crossing\] using [@FQ Lemma 2.6] to compute the number of copies of ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^l)}$.
\[corollary: SOD for whole wall crossing\] Let $S$ be a smooth rational surface over $\C$ with $L_-$ and $L_+$ ample lines bundles on $S$ separated by a single wall defined by unique divisor $\xi$ satisfying $$\begin{gathered}
L_- \cdot \xi < 0 < L_+ \cdot \xi \\
0 \leq \omega_S^{-1} \cdot \xi.
\end{gathered}$$ Let $\mathcal M_{L_{\pm}}(c_1,c_2)$ be the $\mathbb{G}_m$-rigidified moduli stack of Gieseker $L_{\pm}$-semi-stable torsion-free sheaves of rank $2$ with first Chern class $c_1$ and second Chern class $c_2$.
There is a semi-orthogonal decomposition $$\begin{gathered}
{\operatorname{D}^{\operatorname{b}}(\operatorname{coh } \mathcal M_{L_+}(c_1,c_2) )} = \left\langle \underbrace{{\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^{l_{\xi}})}, \ldots, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^{l_{\xi}})}}_{\mu_{\xi}}, \ldots \right. \\
\left. \underbrace{{\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^0)}, \ldots, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }H^0)}}_{\mu_{\xi}}, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh } \mathcal M_{L_-}(c_1,c_2))} \right\rangle
\end{gathered}$$ where $$\begin{aligned}
l_{\xi} & := (4c_2 - c_1^2 + \xi^2)/4 \\
H^l & := {\operatorname{Hilb}}^l(S) \times {\operatorname{Hilb}}^{l_{\xi} - l}(S) \\
\mu_{\xi} & := \omega_S^{-1} \cdot \xi
\end{aligned}$$ with the convention that ${\operatorname{Hilb}}^0(S) := {\operatorname{Spec}} \C$.
This is an iterated application of Proposition \[proposition: one wall crossing\] using Lemma \[lemma: ends of wall crossings\] to identify first and last moduli spaces.
The results on wall crossing of moduli spaces of semi-stable sheaves in [@EG; @FQ; @MW] were originally obtained to compute the change in the Donaldson invariants under change of the polarization. Thus, Corollary \[corollary: SOD for whole wall crossing\] can be viewed as a categorification of that wall-crossing formula.
An example
----------
Let us work out the consequences of Corollary \[corollary: SOD for whole wall crossing\] in the case $S = \mathbb{P}^1 \times \mathbb{P}^1$ with $c_1 = 5H_1 + 5H_2$ and $c_2 = 14$. We get the following decomposition of the ample cone into walls and chambers
\[scale=1, vertex/.style=[circle,draw=black!100,fill=black!100,thick, inner sep=0.5pt,minimum size=0.5mm]{}, cone/.style=[->,very thick,>=stealth]{}, wall/.style=[->,dashed,very thick,>=stealth]{}\] (0,0) – (6,0) – (6,6) – (0,6) – (0,0); (0,0) – (6,6); (0,0) – (6,2); (0,0) – (2,6); (0,0) – (0,6.5); (0,0) – (6.5,0); at (6.4,6.4) [$W^{\xi_2}$]{}; at (2,6.4) [$W^{\xi_1}$]{}; at (6.6,2) [$W^{\xi_3}$]{}; at (0.5,4) [$\mathcal C_I$]{}; at (2.4,3.6) [$\mathcal C_{II}$]{}; at (4,0.5) [$\mathcal C_{IV}$]{}; at (3.6,2.4) [$\mathcal C_{III}$]{}; in [0,1,...,6]{} in [0,1,...,6]{} [ at (,) ; ]{}
where $$\begin{aligned}
\xi_1 & = 3H_1-H_2 \\
\xi_2 & = H_1 - H_2 \\
\xi_3 & = -H_1 + 3H_2.\end{aligned}$$ We have $$\begin{aligned}
l_{\xi_1} & = 0 , \mu_{\xi_1} = 5 \\
l_{\xi_2} & = 1 , \mu_{\xi_2} = 0 \\
l_{\xi_3} & = 0 , \mu_{\xi_3} = 5.\end{aligned}$$
Let $\mathcal M_J$ denote the (rigidified) moduli stack of Gieseker $L$-semi-stable torsion free sheaves of rank $2$ with $c_1 = 5H_1+5H_2$ and $c_2 = 14$, $L \in \mathcal C_J$ and $J \in \{I,II,III,IV\}$. Applying Corollary \[corollary: SOD for whole wall crossing\] to crossing of $W_1$, we see that there is a semi-orthogonal decomposition $${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_{II})} = \left\langle E_1,\ldots, E_5, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_I)} \right\rangle$$ with $E_i$ exceptional. Applying it to $W_2$, we get an equivalence $$\Phi: {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_{II})} \overset{\sim}{\to} {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_{III})}.$$ Applying it to $W_3$, we get a semi-orthogonal decomposition $${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_{III})} = \left\langle F_1,\ldots, F_5, {\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_{IV})} \right\rangle$$ with $F_i$ exceptional.
The involution, $i$, that exchanges the two factors of $\mathbb{P}^1 \times \mathbb{P}^1$ induces isomorphisms $$\mathcal M_I \cong \mathcal M_{IV}$$ and $$\mathcal M_{II} \cong \mathcal M_{III}$$ so there are really only two moduli spaces here. Note however, that the equivalence $\Phi$ is not the pullback $i^*$ as it leaves unchanged sheaves that semi-stable in both chambers. Combining the two equivalences, we get an interesting autoequivalence of $i^* \circ \Phi$ of ${\operatorname{D}^{\operatorname{b}}(\operatorname{coh }\mathcal M_{II})}$.
[99]{}
D. Abramovich, A. Corti, A Vistoli. [*Twisted bundles and admissible covers*]{}. Special issue in honor of Steven L. Kleiman. Comm. Algebra 31 (2003), no. 8, 3547–3618. Arcara, A. Bertram, I. Coskun, J. Huizenga. [*The minimal model program for the Hilbert scheme of points on $\mathbb{P}\sp 2$ and Bridgeland stability*]{}. Adv. Math. 235 (2013), 580–626. M. Ballard, D. Favero, L. Katzarkov. *Variation of Geometric Invariant Theory quotients and derived categories*. [arXiv:1203.6643](http://arxiv.org/pdf/math/1203.6643.pdf). A. Białynicki-Birula. *Some theorems on actions of algebraic groups*. Ann. of Math. (2) 98 (1973), 480-497. A. Bondal, D. Orlov. [*Semi-orthogonal decompositions for algebraic varieties.*]{} Preprint MPI/95-15. [arXiv:math.AG/9506012](http://arxiv.org/pdf/math.AG/9506012.pdf). T. Bridgeland. [*Flops and derived categories*]{}. Invent. Math. 147 (2002), no. 3, 613-632. W. Donovan, E. Segal. [*Window shifts, flop equivalences, and Grassmannian twists*]{}. [ arXiv:1206.0219](http://arxiv.org/pdf/1206.0219.pdf). G. Ellingsrud, L. Göttsche. [*Variation of moduli spaces and Donaldson invariants under change of polarization*]{}. J. Reine Angew. Math. 467 (1995), 1-49. R. Friedman, Z. Qin. [*Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces*]{}. Comm. Anal. Geom. 3 (1995), no. 1-2, 11–83. D. Gieseker. [*On the moduli of vector bundles on an algebraic surface*]{}. Ann. of Math. (2) 106 (1977), no. 1, 45–60. D. Halpern-Leistner. [*The derived category of a GIT quotient*]{}. [arXiv:1203.0276](http://arxiv.org/pdf/1203.0276.pdf). D. Halpern-Leistner. [*On the derived category of a stack with a $\theta$-stratification*]{}. In preparation. M. Herbst, K. Hori, D. Page. *Phases Of N=2 Theories In 1+1 Dimensions With Boundary*. [arXiv:0803.2045](http://arxiv.org/pdf/0803.2045.pdf). M. Herbst, J. Walcher. *On the unipotence of autoequivalences of toric complete intersection Calabi-Yau categories*. Math. Ann. 353 (2012), no. 3, 783-802. D. Huybrechts, M. Lehn. The geometry of moduli spaces of sheaves. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010. Y. Kawamata. [*$D$-equivalence and $K$-equivalence*]{}. J. Differential Geom. 61 (2002), no. 1, 147-171. Y. Kawamata. [*Francia’s flip and derived categories*]{}. Algebraic geometry, 197-215, de Gruyter, Berlin, 2002. Y. Kawamata. *Derived categories of toric varieties*. Michigan Math. J. 54 (2006), no. 3, 517-535. A. Kuznetsov. *Derived categories of cubic fourfolds*. Cohomological and geometric approaches to rationality problems, 219-243, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010. K. Matsuki, R. Wentworth. [*Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface*]{}. Internat. J. Math. 8 (1997), no. 1, 97–148. D. Mumford. [*Projective invariants of projective structures and applications*]{}. 1963 Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 526–530. D. Orlov. *Projective bundles, monoidal transformations, and derived categories of coherent sheaves*. Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852-862. D. Orlov. [*Equivalences of derived categories and $K3$ surfaces*]{}. Algebraic geometry, 7. J. Math. Sci. (New York) 84 (1997), no. 5, 1361-1381. D. Orlov. *Derived categories of coherent sheaves and triangulated categories of singularities*. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503-531, Progr. Math., 270, Birkhäuser Boston, Inc., Boston, MA, 2009. E. Segal. *Equivalences between GIT quotients of Landau-Ginzburg B-models*. Comm. Math. Phys. 304 (2011), no. 2, 411-432. I. Shipman. [*A geometric approach to Orlov’s theorem*]{}. Compos. Math. 148 (2012), no. 5, 1365-1389. authors. Stacks project. <http://stacks.math.columbia.edu/>. Takemoto. [*Stable vector bundles on algebraic surfaces*]{}. Nagoya Math. J. 47 (1972), 29–48. C. Teleman. [*The quantization conjecture revisited*]{}. Ann. of Math. (2) 152 (2000), no. 1, 1-43. M. Van den Bergh. [*Non-commutative crepant resolutions*]{}. The legacy of Niels Henrik Abel, 749-770, Springer, Berlin, 2004.
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abstract: 'Using density-functional theory in combination with a thermodynamic formalism we calculate the relative stability of various structural models of the polar O–terminated (000$\bar{1}$)–O surface of ZnO. Model surfaces with different concentrations of oxygen vacancies and hydrogen adatoms are considered. Assuming that the surfaces are in thermodynamic equilibrium with an O$_2$ and H$_2$ gas phase we determine a phase diagram of the lowest-energy surface structures. For a wide range of temperatures and pressures we find that hydrogen will be adsorbed at the surface, preferentially with a coverage of 1/2 monolayer. At high temperatures and low pressures the hydrogen can be removed and a structure with 1/4 of the surface oxygen atoms missing becomes the most stable one. The clean, defect-free surface can only exist in an oxygen-rich environment with a very low hydrogen partial pressure. However, since we find that the dissociative adsorption of molecular hydrogen and water (if also the Zn–terminated surface is present) is energetically very preferable, it is very unlikely that a clean, defect-free (000$\bar{1}$)–O surface can be observed in experiment.'
author:
- 'B. Meyer'
title: |
First-principles study of the polar O–terminated ZnO surface\
in thermodynamic equilibrium with oxygen and hydrogen
---
Introduction {#sec:intro}
============
To understand the structure, composition and stability of polar surfaces on a solid theoretical basis is one of the challenging problems in surface science.[@noguera] The most interesting polar surfaces are so called “Tasker-type(3)” surfaces[@tasker] which are formed by alternating layers of oppositely charged ions. Assuming a purely ionic model[@comment1] in which all ions are in their formal bulk oxidation state, such a stacking sequence creates a dipole moment perpendicular to the surfaces which diverges with slab thickness, and with simple electrostatic arguments it can be shown that the surface energy will diverge with sample size.[@tasker] To quench the dipole moment and to make the polar surfaces stable, a redistribution of charges in the surface layers has to take place.[@nosker] For various polar surfaces different mechanisms to accomplish the charge compensation have been observed,[@fks] however, in many cases the underlying stabilization mechanism is very controversially discussed in the literature.
One of the most widely investigated examples of Tasker-type(3) polar surfaces are the two basal planes of ZnO: the O–terminated (000$\bar{1}$)–O and the Zn–terminated (0001)–Zn surface. The two surfaces are the terminating planes of a stacking sequence of hexagonal Zn and O layers along the crystallographic $c$-axis with alternating distances of $R_1$=0.61[Å]{} and $R_2$=1.99[Å]{}. In this case, as can be easily shown[@noguera; @zno34], the polar ZnO surfaces are only stable if the O–terminated face is less negative and the Zn–terminated surface layer less positively charged compared to the formal bulk oxidation state by a factor of $R_1/(R_1+R_2)$$\approx$1/4. In principle, three different scenarios are conceivable to accomplish this charge redistribution: The ionic charge of the surface ions may be reduced from $\pm$2 to $\pm$3/2, which may be regarded as an “electron transfer” from the O– to the Zn–terminated surface (Ia). As a result, partially occupied surface bands will appear with a 3/4 filled O–$2p$ band at the (000$\bar{1}$)–O and a 1/4 filled Zn–$4s$ band at the (0001)–Zn surface. This is often referred to as “intrinsic surface state compensation”[@nosker] or as “metalization of the polar surfaces”.[@wander] However, whether a true metallic state is present will depend on the dispersion of the partially occupied bands. Additionally, in a second step, the surface may reconstruct and undergo a distortion in which, for example for the O–terminated surface, four surface atoms combine in such a way that an unoccupied $2p$–band splits from the other eleven occupied $2p$–bands and the surface becomes insulating again (Ib). Secondly, the charge reduction of the surface layers may take place by removing 1/4 of the surface ions (II). The so created vacancies may be ordered and form a reconstruction or may be randomly distributed. Finally, charged species may be adsorbed to reduce the formal oxidation state of the surface ions (III). For example, water may dissociate and protons (H$^+$) and hydroxyl groups (OH$^-$) adsorb on every second O and Zn surface ion, respectively.[@comment2] All three scenarios represent idealizations of the charge compensation process. In general, any combinations of the three mechanisms are conceivable, like the simultaneous formation of vacancies and partially filled bands, as long as the charge compensation rule is obeyed. The surface structure finally realized for a specific polar surface will then be the one with the lowest surface energy.
For ZnO it was believed for a long time that both polar surfaces exist in an unreconstructed, truncated-bulk-like state. After standard preparation procedures both surfaces show regular (1$\times$1) pattern in low-energy electron diffraction (LEED)[@duke] and other diffraction experiments.[@wander; @jedrecy; @overbury] Some evidence for missing Zn ions on the (0001)–Zn surface was found in gracing incidence X-ray diffraction (GIXD)[@jedrecy], however, for the (000$\bar{1}$)–O surface no evidence for substantial amounts of surface oxygen vacancies was detected in GIXD[@jedrecy] and low-energy alkali-ion scattering (LEIS).[@overbury]
For ideal, truncated-bulk-like surface terminations only mechanism (Ia) can explain the stability of the polar ZnO surfaces. Consequently, in most theoretical first-principles studies of the polar ZnO surfaces ideal surface terminations together with partially filled surface bands were assumed.[@wander; @carlsson; @zno34] Studies exploring the other two stabilizations mechanisms are very scarce. In a pioneering ab-initio study Wander and Harrison[@wander_h] investigated whether the polar surfaces may be stabilized by the dissociation of water and the adsorption of H$^+$ and OH$^-$ groups according to mechanism (III). They found this energetically unfavorable compared to situation (Ia). However, instead of 1/2 monolayers, which would be the ideal configurations for band filling, only full monolayers of H$^+$ and OH$^-$ were considered, thereby overcompensating the needed charge transfer between the polar surfaces. In addition, only one specific adsorption site for the H$^+$ and OH$^-$ groups was probed.
Meanwhile two recent experiments have created considerable doubt that the polar ZnO surfaces really exist in a clean, unreconstructed state. With scanning tunneling microscopy (STM) it was shown[@diebold1; @diebold2] that the Zn–terminated surface is characterized by the presence of nanoscaled, triangular islands with a height of one ZnO double-layer. The shape of the islands is size-dependent and typical pit structures appear for larger islands. Since the step edges are O–terminated, the high step concentration leads to a significant decrease of Zn ions in the surface. A rough analysis of the island and pit size distribution yielded that approximately 1/4 of the Zn ions is missing, in agreement with mechanism (II). With detailed density-functional theory (DFT) calculations[@diebold1; @kresse] it was confirmed that a crystal termination with triangular shaped islands and pits is indeed lower in energy than the perfect bulk-truncated surface for a wide range of oxygen and hydrogen chemical potentials. Under H rich conditions, structures with up to 1/2 monolayer of hydroxyl groups were even more stable, indicating that the actual surface morphology will sensitively depend on the chemical environment.
On the other hand, for the O-terminated polar surface no such island and pit structure was found with STM.[@diebold2] However, with He scattering (HAS) it was discovered[@woell] that at ultrahigh vacuum (UHV) conditions O–terminated surfaces with (1$\times$1) LEED and HAS diffraction pattern are always hydrogen covered, whereas after a careful removal of the hydrogen a (1$\times$3) structure is found. The (1$\times$3) spots are best visible in HAS, but under certain conditions can also be observed in LEED.[@woell] The H–free surface is very reactive and dissociates molecular hydrogen and water and therefore exits only for a limited time even at UHV conditions. In a subsequent study[@zno35; @zno36; @staemmler; @fink] CO was used as a probe molecule to distinguish between clean and hydrogen saturated surfaces. By comparing calculated CO adsorption energies for different surface structures with experimental results it was confirmed that the polar O–terminated surface is usually hydrogen covered whereas a clean (1$\times$1) (000$\bar{1}$)–O surface is very unlikely to exist.
In previous studies the H coverage of the polar O–terminated surface was not observed most likely because LEED and X–rays are not sensitive to hydrogen. However, it is not clear how the structures found in the HAS study, Ref. , lead to a stabilization of the polar O–terminated surface. A full hydrogen monolayer would overcompensate the charge transfer so that again partially occupied bands have to be present, and the nature of the H–free (1$\times$3) structure is still unknown. There is some evidence from X–ray photoelectron spectroscopy (XPS)[@woell] that oxygen vacancies are involved, but 1/3 of the oxygens missing is also not the expected vacancy concentration to stabilize the surface.
Motivated by these experimental findings we explore in the present paper the competition between the three stabilization mechanisms in a very general way. The main focus will be on the O–terminated (000$\bar{1}$)–O surface, and we take an approach very similar in spirit to the investigation of the Zn–terminated surface in Ref. and . For a series of surface models we determine the total energies and the fully relaxed atomic structures using a first-principles DFT approach. Surface structures with various oxygen vacancy concentrations and different amounts of adsorbed hydrogen are considered, including structures corresponding to the three ideal stabilization scenarios and structures compatible with the HAS observations.
Static total-energy DFT calculations only give results for zero temperature, zero pressure and for surfaces in contact with vacuum. However, the actual lowest-energy structure of the (000$\bar{1}$)–O surface will depend on the environment and can change with temperature $T$, pressure $p$ and exposure to O$_2$ and H$_2$ gas phases. Therefore, to determine the equilibrium structure and composition of the surface at finite temperature and oxygen and hydrogen partial pressures, we combine our DFT results with a thermodynamic description of the surfaces. To take deviations in surface composition and the presence of gas phases into account, we introduce appropriate chemical potentials[@chadi] and calculate an approximation of the Gibbs free surface energy.[@finnis1] Depending on the chemical potentials we then determine the surface structure with the lowest free energy which allows us to construct a phase diagram for the surface. If we assume that the surface is in thermodynamic equilibrium with the gas phases, we can relate the chemical potentials to a given temperature $T$ and pressure $p$. In this way we are able to extend our zero temperature and zero pressure DFT results to experimentally relevant environments, thereby bridging the gap between UHV-like conditions and temperatures and gas phase pressures that are typically applied, for example, in catalytic processes like the methanol synthesis.[@catalysis]
Computational Approach {#sec:theorie}
======================
Thermodynamics {#sec:therm}
--------------
In this section we will give a brief description of the thermodynamic formalism which we have used to determine the most stable structures of the polar O–terminated ZnO surface. The formalism has been successfully applied in several previous surface studies[@padilla; @wang; @batyrev; @pojani; @reuter; @diebold1] and is described in more detail in Refs. and .
The general expression for the free energy of a surface in equilibrium with particle reservoirs at the temperature $T$ and pressure $p$ is given by[@cahn] $$\label{def_gsurf}
\gamma(T,p) = \frac{1}{A} \left( G(T,p,\{N_i\}) -
\sum_i N_i\, \mu_i(T,p) \right) \;,$$ where $G(T,p,\{N_i\})$ is the Gibbs free energy of the solid with the surface of interest, $A$ is the surface area, and $\mu_i$, $N_i$ are the chemical potentials and particle numbers of the various species. In contrast to the usual convention in macroscopic thermodynamics we define here the chemical potentials per atom rather than per mole. For the study of the polar O–terminated ZnO surface in contact with an oxygen and a hydrogen gas phase we have to consider the three chemical species $i$ = Zn, O and H.
For simplicity we have assumed two independent reservoirs for O$_2$ and H$_2$ with a common pressure $p$. Experimentally, it is more likely that a mixture of O$_2$ and H$_2$ is present. In this case, the pressure $p$ in Eq. (\[def\_gsurf\]) has to be replaced by appropriate partial pressures $p_{\rm O_2}$ and $p_{\rm H_2}$. However, in the present study we will keep the restriction of separate reservoirs in the sense that we do not allow O$_2$ and H$_2$ to react to H$_2$O, which would be the case in full thermodynamic equilibrium. This is justified by arguing that the reaction barrier for the formation of H$_2$O is high enough that the reaction plays no role on the time scales of interest.
In thermodynamic equilibrium the chemical potentials would be determined uniquely by the temperature $T$, the pressure $p$ and the total particle numbers of the solid and the gas phases. The surface structure, here represented by the particle numbers $N_i$, would then be determined by a unconstrained minimization of the surface free energy, Eq. (\[def\_gsurf\]). However, this is not very practical to do. Therefore we will take a different approach: We calculate the surface free energy of a series of model surfaces with different structures and compositions as a function of the chemical potentials. For given chemical potentials we predict which surface structure is the most stable one by searching for the surface model with the lowest surface free energy. In a second step, the chemical potentials are then related to actual temperature and pressure conditions by assuming that the surface is in thermodynamic equilibrium with the gas phases.
In our calculations all surfaces are represented by periodically repeated slabs so that the Gibbs free energy $G(T,p,\{N_i\})$ refers to the content of one supercell. Since ZnO is not centrosymmetric, slabs representing the polar ZnO surfaces are inevitably O–terminated on one side and Zn–terminated on the other side. It is therefore not possible to assign unique surface energies to the two polar surface terminations. Only the sum of the surface energies and thereby the cleavage energy are well defined quantities. However, in the present study we are only interested in the [*relative*]{} stability of O–terminated surfaces with different structures and compositions. The Zn-face of the slabs is unchanged in all calculations. Therefore we relate all energies relative to a reference state which we have taken to be the ideal, truncated-bulk termination. We define the change of the cleavage energy $\Delta\gamma$ as the difference of Eq. (\[def\_gsurf\]) for a slab with a chosen surface structure and a slab with ideal surface terminations: $$\begin{aligned}
\label{def_dgamma}
\Delta\gamma(T,p)
& = & \frac{1}{A} \Big(
G^{\rm surf}_{\rm slab}(T,p,N_{\rm V},N_{\rm H}) -
G^{\rm ref}_{\rm slab}(T,p) \nonumber \\
& & {}+ N_{\rm V}\,\mu_{\rm O}(T,p) -
N_{\rm H}\,\mu_{\rm H}(T,p) \Big) \;.\end{aligned}$$ Here, $G^{\rm surf}_{\rm slab}$ and $G^{\rm ref}_{\rm slab}$ are the Gibbs free energies of the supercells with the model surface and the reference configuration, respectively, and $A$ is now the area of the surface unit cell. Since we only consider structures of the O–terminated surface with O–vacancies and adsorbed H atoms, only the number of O–vacancies $N_{\rm V}$ (= difference of the number of O–atoms in the slab with the model surface and the reference state) and the number of adsorbed H atoms $N_{\rm H}$ appears in Eq. (\[def\_dgamma\]), i.e. the chemical potential of Zn drops out. The difference $\Delta\gamma$ is negative if a model surface is more stable than the ideal, truncated-bulk-like surface termination and positive otherwise.
In principle we now have to calculate the Gibbs free energy of all slabs representing our surface models, including the contributions coming from changes in volume and in entropy. The entropy term may be calculated, for example, by evaluating the vibrational spectra in a quasiharmonic approximation[@frank], but in practice this is computationally very demanding. As is apparent from Eq. (\[def\_dgamma\]), only the [*difference*]{} of the Gibbs free energy of two surface structures enters the expression for $\Delta\gamma$. In Ref. it was shown that the vibrational contributions to the entropy usually cancel to a large extent and that the influence of volume changes are even smaller. Therefore we will neglect all entropy and volume effects. The Gibbs free energies then reduce to the internal energies of the slabs and we can replace $G^{\rm surf}_{\rm slab}$ and $G^{\rm ref}_{\rm slab}$ in Eq. (\[def\_dgamma\]) by the energies as directly obtained from total–energy (e.g. DFT) calculations.
Finally we have to determine meaningful ranges in which we can vary the chemical potentials. First, there are upper bounds for all three chemical potentials $\mu_{\rm O}$, $\mu_{\rm H}$ and $\mu_{\rm Zn}$, beyond which molecular oxygen and molecular hydrogen would condensate and metallic Zn would crystallize at the surface. These bounds are given by the total energy of the isolated molecules $E_{\rm O_2}$, $E_{\rm H_2}$ and of bulk Zn $E^{\rm bulk}_{\rm Zn}$ (neglecting volume and entropy effects): $$\label{upper_bound}
\mu_{\rm O} \le \frac{1}{2}E_{\rm O_2}\;,\quad
\mu_{\rm H} \le \frac{1}{2}E_{\rm H_2}\;,\quad
\mu_{\rm Zn} \le E^{\rm bulk}_{\rm Zn}\;.$$ In the following we will use these upper bounds as zero point of energy and relate the chemical potentials relative to the total energies of the isolated molecules: $$\label{def_dmu}
\Delta\mu_{\rm O} = \mu_{\rm O} - \frac{1}{2}E_{\rm O_2}\;,\quad
\Delta\mu_{\rm H} = \mu_{\rm H} - \frac{1}{2}E_{\rm H_2}\;.$$ Furthermore we impose that the surface is always in equilibrium with the ZnO bulk phase. Then the sum of $\mu_{\rm O}$ and $\mu_{\rm Zn}$ has to be equal to the total energy $E_{\rm ZnO}$ of bulk ZnO. Thus only one of the two chemical potentials $\mu_{\rm O}$ and $\mu_{\rm Zn}$ is independent, and together with Eq. (\[upper\_bound\]) we introduce lower bounds for the chemical potentials. Using the energy of formation $E_{\rm f}$ of ZnO: $$\label{def_ef}
E_{\rm f} = E_{\rm ZnO} - E_{\rm Zn} - \frac{1}{2}E_{\rm O_2}$$ the allowed range for the chemical potential $\Delta\mu_{\rm O}$ is given by: $$\label{range}
E_{\rm f} \;\le\; \Delta\mu_{\rm O} \;\le\; 0 \;.$$ If we assume that the surfaces are in thermodynamic equilibrium with the gas phases we can relate the chemical potentials $\Delta\mu_{\rm O}$ and $\Delta\mu_{\rm H}$ to a given temperature $T$ and partial pressures $p_{\rm O_2}$ and $p_{\rm H_2}$. For ideal gases we can use the well-known thermodynamic expressions[@reuter] $$\Delta\mu_{\rm O}(T,p_{\rm O_2}) = \frac{1}{2} \Big(
\tilde{\mu}_{\rm O_2}(T,p^0) + k_{\rm B}T \ln(p_{\rm O_2}/p^0) \Big)$$ and $$\Delta\mu_{\rm H}(T,p_{\rm H_2}) = \frac{1}{2} \Big(
\tilde{\mu}_{\rm H_2}(T,p^0) + k_{\rm B}T \ln(p_{\rm H_2}/p^0) \Big)$$ in which $p^0$ is the pressure of a reference state and the temperature dependence of the chemical potentials $\tilde{\mu}_{\rm O_2}(T,p^0)$ and $\tilde{\mu}_{\rm H_2}(T,p^0)$ is tabulated in thermochemical reference tables.[@janaf] However, it should be noted, as pointed out by Finnis[@finnis2], that the equilibrium with the gas phase need not to be perfect. It is sufficient if the surface is in equilibrium with the bulk phase. In this case, the chemical potential is related to defect concentrations of the bulk. For example, if oxygen vacancies are the dominant defects we have[@mayer1; @hagen; @mayer2] $$\Delta\mu_{\rm O}(T,p) = \tilde{\mu}_0 - k_{\rm B}T \ln(c_{\rm V}/c_0)$$ with the vacancy concentration $c_{\rm V}$ and the oxygen occupancy of the O–lattice site of $c_0$.
Ab-Initio Calculations {#sec:method}
----------------------
The total energies of slabs representing different model surfaces as well as the bulk and molecular reference energies were calculated within the framework of density-functional theory (DFT).[@hks] Exchange and correlation effects were included in the generalized–gradient approximation (GGA) using the functional of Perdew, Becke and Ernzerhof (PBE).[@pbe] Normconserving pseudopotentials[@van-pp] were employed together with a mixed-basis set consisting of plane waves and non-overlapping localized orbitals for the O–$2p$ and the Zn–$3d$ electrons.[@mb] A plane-wave cut-off energy of 20Ry was sufficient to get well converged results. Monkhorst-Pack k-point meshes[@mp] with a density of at least (6$\times$6$\times$6) points in the primitive ZnO unit cell were chosen. A dipole correction[@bengtsson; @bm] to the electrostatic potential was included in the calculations to eliminate all artificial interactions between the periodically repeated supercells due to the dipole moment of the slabs. For more details on convergence parameters, the construction of appropriate supercell as well as the calculated bulk and clean surface structures of ZnO we refer to Ref. , where the same computational settings as in the present study were used.
All surfaces were modeled by periodically repeated slabs. Very thick slabs consisting of 8 Zn-O double-layers were used to reduce the residual internal electric field.[@zno34]. To represent different surface structures (1$\times$2), (1$\times$3) and (2$\times$2) surface unit cells with different combinations of O vacancies and H adatoms were considered. All atomic configurations were fully relaxed by minimizing the atomic forces.
In Table \[tab:ref\] we compare the computed binding energies of different bulk and molecular reference structures with experimental heat of formations. While the calculated binding energies for isolated $H_2$ molecules and bulk Zn agree quite well with experiment, there is a noticeable error of 1.2eV in the binding energy of the free O$_2$ molecule. This is a well known deficiency of DFT.[@batyrev; @reuter] The overestimation of the O$_2$ binding energy is also reflected in a formation energy for ZnO which is to low by 0.6eV. Such deviations would seriously influence our analysis of the surface energies, Eq. (\[def\_dgamma\]), and would alter the allowed range for the oxygen chemical potential, Eq. (\[range\]). Therefore, to circumvent errors introduced by a poor description of the O$_2$ molecule, we have applied the following procedure: we take the experimental value for the formation energy $E_{\rm f}$ of ZnO from Table \[tab:ref\] and we use Eq. (\[def\_ef\]) to replace the total energy of O$_2$ by $E_{\rm f}$ and the total energies of Zn and ZnO. $E_{\rm Zn}$ and $E_{\rm ZnO}$ are both bulk quantities which are typically more accurately described in DFT than molecular energies.
H$_2$ O$_2$ Bulk Zn Bulk ZnO
------------------------------ ------- ------- --------- ---------- --
$E_{\rm f}^{\rm PBE}$ \[eV\] 4.50 6.38 1.12 2.84
$H_{\rm f}^{\rm exp}$ \[eV\] 4.52 5.17 1.35 3.50
: \[tab:ref\] Calculated binding energies $E_{\rm f}^{\rm PBE}$ for the isolated H$_2$ and O$_2$ molecules and for the bulk phases of Zn and ZnO. Zero-point vibrations are not included. The experimental heat of formations $H_{\rm f}^{\rm exp}$ are for $T$=298K and $p$=1bar and are taken from Ref. .
Results and Discussion {#sec:results}
======================
Surface Distortions {#sec:dist}
-------------------
First we explore the possibility whether the ideal, truncated-bulk-like (000$\bar{1}$)–O surface may lower its energy by breaking the symmetry of the surface layers, thereby adopting a distorted surface structure according to mechanism (Ib). A tendency for symmetry breaking reconstructions is often indicated by a strong nesting of the Fermi surface. In Fig. \[fig:fermi\] we have plotted the two-dimensional Fermi surface formed by the partially occupied O–$2p$ bands. The plot represents a cut through the Brillouin zone of our supercell including only k-vectors with a zero component perpendicular to the surface. Figure \[fig:fermi\] reveals that actually two surface bands cross the Fermi level. This is well known and in full agreement with band structure plots presented in Refs. and . The corresponding wave functions are strongly localized at the oxygen atoms of the terminating surface layers and are mainly formed by the two O–$2p$ orbitals parallel to the surface. By integrating the occupied and unoccupied areas of the Brillouin-zone we find that indeed roughly 1/2 electron per surface atom is missing to fill the two surface bands. However, both Fermi contours are almost spherical and only a weak nesting is present.
As a second test, we did several calculations in which we randomly displaced the surface atoms in the top atomic layer of our slabs and started an atomic relaxation. Different slabs with (1$\times$2), (1$\times$3) and (2$\times$2) surface unit cells were used but in all cases the surfaces relaxed back toward a symmetric structure with a 3-fold symmetry. The surface energy was always higher than in the fully symmetric state, so that no hint for a tendency toward symmetry breaking reconstructions was found.
Oxygen Vacancies {#sec:ovac}
----------------
In the next step we investigate whether the (000$\bar{1}$)–O surface is stabilized by creating oxygen vacancies. From slabs with (1$\times$1), (1$\times$2), (1$\times$3) and (2$\times$2) surface unit cells we have removed one O–atom, thereby creating vacancy concentrations $c_{\rm V}$ of 1, 1/2, 1/3 and 1/4. In Fig. \[fig:ovac\] we have plotted the change of the surface energy $\Delta\gamma$ of the four defect structures relative to the defect-free surface as a function of the oxygen chemical potential $\Delta\mu_{\rm O}$. As to be expected from mechanism (II) the defective surface with 1/4 of the O atoms missing is the most stable surface structure over a wide range of chemical potentials. Translating the chemical potential into temperature and pressure conditions (assuming that the surface is in equilibrium with an O$_2$ gas phase, see upper $x$–axis in Fig. \[fig:ovac\]) we see that this will be the most stable surface at typical UHV-conditions of surface science experiments. However, at oxygen rich conditions (high pressure and low temperature), exceeding a chemical potential of $-$1.58eV, the ideal, defect-free surface becomes the most stable structure. The other surfaces with 1, 1/2 and 1/3 vacancy concentrations are higher in energy for all chemical potentials and will therefore not be present in thermodynamic equilibrium. In particular, it is very unlikely that the (1$\times$3) structure observed in the HAS experiment[@woell] corresponds to a simple missing-row structure with every third O atom removed from the surface.
At this point we should emphasize that plots like Fig. \[fig:ovac\] strictly only allow to rule out surface structures which are higher in energy than other surface models. Since we can only perform calculations for a limited set of surface models, it is always possible that a not yet considered structure with a lower energy exists. For example, since we use periodically repeated surface unit cells of a specific size in our DFT approach, all our defect structures are perfectly ordered. It is however very well possible, that disordered or even incommensurate structures might lead to a lower energy. Additionally, there are hints that island and pit structures like the ones observed for the Zn–terminated surface[@diebold1] may also for the O–terminated surface be lower in energy than the ideal surface and the surface with 1/4 vacancy concentration considered in our study.[@kresse2]
In Table \[tab:ovac\] we have listed O vacancy formation energies $E_{\rm V}$ which we have defined in the present context as the energy difference between the ideal surface and a vacancy surface structure of concentration $c_{\rm V}$ plus $1/2\,E_{\rm O_2}$, i.e. $E_{\rm V}$ is defined with respect to the total energy $E_{\rm O_2}$ of free oxygen molecules, and not, as usually done, with respect to bulk ZnO. $E_{\rm V}$ depends strongly on the vacancy concentration, indicating a strong interaction between the vacancies. Up to $c_{\rm V}$=1/4 the energy cost to remove O–atoms is modest. This is not surprising since up to a vacancy concentration of 1/4 the removal of O-atoms supports the charge compensation of the O–terminated surface and will result in a better filling of the partially occupied O–$2p$ band. For higher vacancy concentrations, however, we start to overcompensate the charge transfer which stabilizes the surface. The O–$2p$ band is full now, and we have to start to fill Zn–$4s$–states in the conduction band. Therefore the energy cost to remove more O–atoms increases rapidly.
$c_{\rm V}$ 1 1/2 1/3 1/4$^{\rm (a)}$ 1/4$^{\rm (b)}$
-------------------- ------- ------- ------- ----------------- -----------------
$E_{\rm V}$ \[eV\] +3.24 +2.88 +2.51 +1.80 +1.58
: \[tab:ovac\] Calculated vacancy formation energies $E_{\rm V}$ per O atom for removing oxygen from the ideal surface, forming O$_2$ molecules and a surface structure with a O vacancy concentration of $c_{\rm V}$. (a) for a 6–fold (2$\times$2) arrangement of the O vacancies with a separation $2a$, (b) for a rectangular O vacancy pattern with distances of $2a$ and $\sqrt{3}\,a$, $a$ being the ZnO lattice constant.
\(a) Ideal (000$\bar{1}$)–O surface:
\
Site: ‘on–top’ ‘hcp–hollow’ ‘fcc–hollow’
------------------- ---------- -------------- --------------
$\Delta E$ \[eV\] +3.16 0.0 +0.05
: \[tab:h\] Relative stability of the high-symmetry adsorption sites for (a) the surface O atoms of the ideal O–terminated surface and (b) the OH–groups of the H saturated surface for a full monolayer coverage. The $\Delta E$ are the calculated energy differences per O–atom/OH–group for moving the topmost O/OH–surface layer from the regular lattice position of the wurtzite structure (‘hcp–hollow–site’) to the ‘on–top’ and ‘fcc–hollow’ position.
\
(b) Hydrogen covered (000$\bar{1}$)–O surface:
\
Site: ‘on–top’ ‘hcp–hollow’ ‘fcc–hollow’
------------------- ---------- -------------- --------------
$\Delta E$ \[eV\] +1.78 0.0 $-$0.02
: \[tab:h\] Relative stability of the high-symmetry adsorption sites for (a) the surface O atoms of the ideal O–terminated surface and (b) the OH–groups of the H saturated surface for a full monolayer coverage. The $\Delta E$ are the calculated energy differences per O–atom/OH–group for moving the topmost O/OH–surface layer from the regular lattice position of the wurtzite structure (‘hcp–hollow–site’) to the ‘on–top’ and ‘fcc–hollow’ position.
$c_{\rm H}$ 1 3/4 2/3 1/2 1/3 1/4
-------------------- --------- --------- --------- --------- --------- ---------
$E_{\rm b}$ \[eV\] $-$0.71 $-$1.10 $-$1.25 $-$1.90 $-$2.12 $-$2.20
: \[tab:hcov\] Calculated binding energies $E_{\rm b}$ per H atom for dissociating H$_2$ molecules and forming hydrogen layers of coverage $c_{\rm H}$.
Hydrogen Adsorption {#sec:hcov}
-------------------
We turn now to a situation in which the (000$\bar{1}$)–O surface is in equilibrium with a H$_2$ gas phase. H$_2$ molecules can dissociate, and hydrogen atoms may adsorb at the surface thereby forming OH-groups with the surface oxygen ions. Before we start to calculate the surface energy for different surface models with H adatoms, we consider the possibility that the preferred adsorption site of these OH-groups is no longer the regular lattice site of the O ions. Three different high-symmetry adsorption sites are conceivable above the underlying Zn layer: an ‘on-top’ position, a ‘hcp-hollow site’, which is the regular lattice position for the O ions in the wurtzite structure, and a ‘fcc-hollow site’.
First we consider the clean surface without adsorbed hydrogen. Then we see from Table \[tab:h\] that indeed the ‘hcp-hollow’ sites are the most stable positions for the O surface layer. However, moving the whole layer to ‘fcc-hollow’ sites costs only a small amount of energy. Turning to the hydroxylated surface with a full monolayer coverage of hydrogen we find that the OH-groups now prefer the ‘fcc-hollow site’. So by gradually adding hydrogen, the regular lattice site of the surface O ions becomes unstable relative to the ‘fcc-hollow site’. However, the energy difference between ‘fcc-’ and ‘hcp-hollow sites’ is so small that we have neglected this effect in all further calculations and have only considered ‘hcp-hollow sites’ for surface oxygen atoms and OH–groups.
In the next step we construct different surface models of hydrogen covered (000$\bar{1}$)–O surfaces using a similar procedure as in Sec. \[sec:ovac\]. We take slabs with (1$\times$1), (1$\times$2), (1$\times$3) and (2$\times$2) surface unit cells, and we add different amounts of hydrogen to create hydrogen coverages of $c_{\rm H}$ of 1/4, 1/3, 1/2, 2/3, 3/4 and 1 monolayer. The calculated surface energy changes $\Delta\gamma$ relative to the clean (000$\bar{1}$)–O surface are plotted in Fig. \[fig:hcov\] as a function of the hydrogen chemical potential $\Delta\mu_{\rm H}$. A very similar behavior as in Section \[sec:ovac\] arises: At H–rich conditions the structure with a 1/2 monolayer hydrogen coverage is the most stable surface, in agreement with mechanism (III). On the other hand, at H–poor conditions the clean surface without hydrogen becomes the most stable structure. As a new feature we find that between the two limiting cases the surfaces with 1/3 and 1/4 monolayer coverage are slightly more stable for small intervals of the chemical potentials.[@comment3] Thus, by lowering the chemical potential $\Delta\mu_{\rm H}$ from H–rich to H–poor conditions it is possible to gradually reduce the hydrogen coverage from 1/2 monolayer to 1/3, 1/4 and zero coverage. Translating the chemical potential into temperature and pressure conditions we find that we will start to remove H at UHV-pressures at a temperature of roughly 750K, which is in reasonable agreement with the experimental observation.[@woell] The other surface structures with hydrogen coverages larger than 1/2 are always higher in energy and will be unstable for all temperatures and hydrogen partial pressures. In particular a surface with a full monolayer of hydrogen as postulated from the results of the HAS experiment[@woell] is not likely to exist in thermodynamic equilibrium. However, from the intensity of the He–specular peak it was deduced that only about 0.1% of the (000$\bar{1}$)–O surface consists of flat terraces with diameters exceeding 50[Å]{} which contribute to the (1$\times$1) HAS signal.[@woell] Therefore, the H covered surface with a (1$\times$1) HAS diffraction pattern may well be a minority phase which is formed under suitable kinetic conditions.
In Table \[tab:hcov\] we have summarized the H binding energies $E_{\rm b}$ per atom which we have calculated as energy difference between the ideal (000$\bar{1}$)–O surface plus $1/2\,E_{\rm H_2}$ and surface structures with H coverages of $c_{\rm H}$. We see that it becomes rapidly unfavorable to adsorb more hydrogen as soon as the concentration for ideal charge compensation of the polar surface of 1/2 monolayer is reached. The reason for this behavior is the same as in the case of the oxygen vacancies: Up to 1/2 monolayer of hydrogen we fill the partially occupied O–$2p$–band, beyond 1/2 monolayer the O–$2p$–band is completely filled and we have to populate the conduction band. The decrease in $E_{\rm b}$ from 1/2 to 1/4 monolayer coverage indicates that also at low coverages a weak repulsive interaction between the hydrogen atoms remains. This is the reason why also the low-coverage structures appear in the surface phase diagram. If we extrapolate $E_{\rm b}$ toward zero coverage of the surface we can estimate an initial binding energy of about 2.3eV per H atom for the dissociative adsorption of H$_2$.
Water Dissociation {#sec:water}
------------------
As evident from Fig. \[fig:hcov\] hydrogen adsorption plays a major role for the stabilization of the polar (000$\bar{1}$)–O surface and for almost all conceivable experimental conditions hydrogen will be present on the surface. But until now we have only considered molecular hydrogen as the reservoir for the surface hydrogen. However, in many chemical reactions and catalytic processes also water is present. Therefore we will briefly explore if also water can act as a source for surface hydrogen in the presence of the (0001)–Zn face.
In a DFT study using the hybrid B3LYP functional Wander and Harrison[@wander_h] found that dissociating water and forming a full H and OH monolayer on the O– and Zn–terminated surface, respectively, is energetically unfavorable by 0.1eV. As adsorption sites for the OH–groups they assumed the Zn ‘on-top’ positions which would be the next lattice sites for O ions if the crystal is extended. However, on the polar surfaces two more high-symmetry adsorption sites exist: the ‘hcp-hollow site’ position above atoms in the second surface layer and a ‘fcc-hollow’ site with no atoms beneath.
Using the same adsorption geometry as Wander and Harrison we also find that dissociating water is energetically unfavorable with a slightly larger energy cost of 0.3eV. However, as shown in Table \[tab:oh\], the configuration with the OH groups adsorbed at the ‘fcc-hollow sites’ instead of the ‘on-top’ positions is much lower in energy. Considering the correct adsorption positions for the OH groups we now find that even for the thermodynamically unstable monolayer coverages the dissociation of water is energetically preferred by about 0.4eV. Taking only 1/2 monolayer coverages into account, this energy gain will be significantly larger.
Site: ‘on–top’ ‘hcp–hollow’ ‘fcc–hollow’
------------------- ---------- -------------- --------------
$\Delta E$ \[eV\] 0.0 $-$0.04 $-$0.72
: \[tab:oh\] Relative stability of the different OH adsorption sites on the polar Zn–terminated surface, calculated for a monolayer coverage.
Surface Phase Diagram {#sec:phase}
---------------------
Finally we combine the results of the previous subsections and assume that the polar (000$\bar{1}$)–O surface is now simultaneously in equilibrium with an O$_2$ and a H$_2$ gas phase. In addition to the surface models described in Sec. \[sec:ovac\] and Sec. \[sec:hcov\] we have furthermore considered various mixed structures of O vacancies and adsorbed H atoms in the (1$\times$2), (1$\times$3) and (2$\times$2) surface unit cells.
The surface free energy now depends on two chemical potentials $\Delta\mu_{\rm O}$ and $\Delta\mu_{\rm H}$. The graphs of Fig. \[fig:ovac\] and \[fig:hcov\] therefore have to be extended to a 3-dimensional plot. Such a diagram would be rather complex and hard to follow. The most important information contained in the plot of the surface free energies is which of the surface models has the lowest surface energy for a given combination of chemical potentials $\Delta\mu_{\rm O}$ and $\Delta\mu_{\rm H}$. This information is better visualized if we project the 3-dimensional diagram onto the ($\Delta\mu_{\rm O}$,$\Delta\mu_{\rm H}$) plane and only mark the regions for which a certain surface structure is the most stable one. The result is a phase diagram of the (000$\bar{1}$)–O surface which is shown in Fig. \[fig:phase\].
The surface phase diagram in Fig. \[fig:phase\] summarizes in a condensed fashion the essential results of our study. This phase diagram is dominated by two structures: a surface with 1/2 monolayer of adsorbed H and a hydrogen-free surface with 1/4 of the oxygen atoms removed. These are the two scenarios denoted as mechanism (II) and (III) in the Introduction, indicating that filling the O–$2p$–bands is a very important mechanism to stabilize the (000$\bar{1}$)–O surface. Next to these two phases we find two structures in which H is gradually removed from the surface and only at very H–poor conditions and when plenty of oxygen is available, the ideal O–terminated surface stabilized by mechanism (I) becomes the most stable structure.
None of the additionally considered mixed structures which simultaneously contain O vacancies and H adatoms appears in the phase diagram. This is not very surprising since for the given sizes of the surface unit cells no combination of O vacancies and H adatoms exists that leads to fully occupied O–$2p$–bands. However, it is very well conceivable that for larger surface unit cells mixed structures with, for example, an O vacancy concentration of 1/8 and a H coverage of 1/4, become more stable which would then appear as new phases between the H covered and the O vacancy structures.
Relating the chemical potentials to temperature conditions and partial pressures of the gas phase shows, see Fig. \[fig:phase\], that for almost all realistic experimental conditions hydrogen will be present at the (000$\bar{1}$)–O surface. Even at UHV-conditions with a low hydrogen partial pressure one has to go to rather high temperatures to fully remove all hydrogen. In this case, a surface structure with O vacancies will become the most stable one. In order to stabilize the ideal O–terminated surface an oxygen atmosphere with an extremely low content of hydrogen (and also water vapor) is necessary, which is basically not achievable in experiment.
In Table \[tab:dij\] we have summarized the surface relaxations for the three most important surface structures appearing in the surface phase diagram, Fig. \[fig:phase\]. For the extended surface structures with H adatoms and O vacancies we have averaged in each atomic plane parallel to the surface the atomic displacements, and we define the interlayer distances $d$ as the separation of the averaged atomic positions. Depending on the surface structure and the charge compensation process, very different surface relaxations occur. The largest relaxations are found for the clean, defect-free surface termination with a compression of the first double-layer distance of almost 50% and also a significant contraction of the second and subsequent double-layer spacings. This is in agreement with other previous ab-initio studies.[@noguera; @zno34; @wander; @carlsson] After filling the partially occupied surface bands by adsorbing 1/2 monolayer of H or by removing 1/4 of the O ions, however, the surface layers relax back to almost truncated-bulk-like positions. Thus, for surfaces with a lower H adatom or O vacancy concentration, surface relaxations between the two extremes are conceivable. This may explain why experimentally very different results for the surface relaxations were found. GIXD measurements[@wander; @jedrecy] have predicted an inward relaxation of the upper O–layer with a contraction of the first double-layer distance of 40% and 20%, respectively, whereas from LEED[@duke] and LEIS[@overbury] experiments it was concluded that the first double-layer spacing is close to its bulk value.
[llll]{}
$d$ & ideal surface & H covered & O vacancies\
H$-$O$_1$ & & 0.1825\
O$_1$$-$Zn$_2$ & 0.0628 ($-$48%) & 0.1207 (0.0%) & 0.1151 ($-$4.6%)\
Zn$_2$$-$O$_3$ & 0.3985 (+5.1%) & 0.3779 ($-$0.4%) & 0.3767 ($-$0.7%)\
O$_3$$-$Zn$_4$ & 0.1077 ($-$11%) & 0.1246 (+3.2%) & 0.1238 (+2.6%)\
Zn$_4$$-$O$_5$ & 0.3813 (+0.5%) & 0.3773 ($-$0.5%) & 0.3772 ($-$0.6%)
Summary and Conclusions {#sec:summary}
=======================
By combining first-principles density-functional calculations with a thermodynamic formalism we have determined lowest-energy structures of the polar O–terminated (000$\bar{1}$)–O surface of ZnO in thermal equilibrium with an O$_2$ and H$_2$ gas phase. This scheme allows us to extend our zero-temperature and zero-pressure DFT results to more realistic temperature and pressure conditions which are usually applied in surface science experiments or in catalytic processes like the methanol synthesis, and thus to bridge computationally the ‘pressure gap’.
The essential result of this approach is a phase diagram of the (000$\bar{1}$)–O surface which is plotted in Fig. \[fig:phase\]. From this surface phase diagram we predict that hydrogen is adsorbed at the (000$\bar{1}$)–O surface for a wide range of temperatures and H$_2$ partial pressures, including UHV–conditions. This is in agreement with the recent observations of a HAS experiment[@woell] and was also confirmed in a study of the CO adsorption on the polar ZnO surfaces.[@zno35]
We find a H binding energy of roughly 2.3eV per atom if molecular hydrogen dissociates and adsorbs at the clean O–terminated surface. Furthermore we predict that in situations where both polar surface terminations are present (for example for powder samples) also the dissociative adsorption of water with H and OH–groups being adsorbed at the O– and Zn–terminated surface, respectively, is energetically preferable. However, as soon as a coverage of 1/2 monolayer of hydrogen is reached, the energy gain of adsorbing more hydrogen on the (000$\bar{1}$)–O surface drops very rapidly with increasing hydrogen coverage. Therefore no stable phases with more than 1/2 monolayer H coverage appear in the surface phase diagram, Fig. \[fig:phase\]. In particular, a structure with a full monolayer of H as predicted in Ref. is not very likely to exist globally in thermodynamic equilibrium (which does not exclude a kinetic or local stabilization).
Going to low hydrogen partial pressures and higher temperatures it is possible to gradually remove the hydrogen from the surface and to form stable phases with less than 1/2 monolayer coverage of hydrogen. If all hydrogen is removed, oxygen vacancies will be created as was speculated in Ref. . However, we find that a surface with a vacancy concentration of 1/4 is much more stable than a missing-row structure where 1/3 of the oxygens has been removed. Therefore, based on the limited set of surface structures taken into consideration in our study, we currently do not understand the (1$\times$3) structure observed in Ref. .
Finally, at higher oxygen partial pressures the O vacancies will be filled and the clean, defect-free O–terminated surface becomes the most stable structure. However, the hydrogen partial pressure has to be very low so that we consider it very unlikely that a clean, defect-free (000$\bar{1}$)–O surface can be observed in experiment.
Acknowledgments
===============
The author would like to thank Dominik Marx, Christof Wöll, Georg Kresse, and Ulrike Diebold for fruitful discussions. The work was supported by SFB 558 and FCI.
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|
**A systematic method for constructing discrete Painlevé equations**
**in the degeneration cascade of the E$_8$ group**
and [B. Grammaticos]{}
We present a systematic and quite elementary method for constructing discrete Painlevé equations in the degeneration cascade for E$_8^{(1)}$. Starting from the invariant for the autonomous limit of the E$_8^{(1)}$ equation one wishes to study, the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms. These new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping. We explain the method on three examples, two symmetric mappings and an asymmetric one, and we discuss the link between our results and the known geometric structure of these mappings.
PACS numbers: 02.30.Ik, 05.45.Yv
Keywords: discrete Painlevé equations, singularity confinement, deautonomisation, degeneration cascade, affine Weyl groups, singular fibres
1\. [Introduction]{}
Singularities play an essential role in integrability. For continuous systems the study of their singularities gave birth to a most powerful integrability criterion, the famous Painlevé property \[\]. The situation for discrete systems is at once similar to that for continuous systems and also more complex. The idea of a singularity in the discrete case is that, starting from generic values, at some point during the iteration of the mapping one suddenly loses a degree of freedom. More specifically, in the case of second-order nonlinear mappings which we shall focus on in this paper, such a situation arises at a value of the variable $x_n$ which is such that its iterate $x_{n+1}$ is independent of $x_{n-1}$. That the precise iteration step at which this phenomenon occurs actually depends on the initial conditions is what makes the singularity ‘movable’, in analogy to the well-known notion of movable singularities in the continuous case.
One particularly crucial observation led to the formulation of a discrete integrability property which came to be known as [*singularity confinement*]{} \[\]: by examining a host of discrete systems known to be integrable with spectral methods it was observed that a singularity that appeared at some iteration step, in fact again disappeared after a few more steps. We consider this property to be the discrete analogue of the Painlevé property for continuous systems. Though systems that are integrable through spectral methods possess the singularity confinement property, the latter is not a sufficient criterion for integrability. Whereas a general nonintegrable mapping has unconfined singularities (in the sense that a singularity that appears at some iteration never disappears) there exist many examples of systems that have only confined singularities but that are not integrable \[\]. To address this difficulty we introduced in \[\] a method we dubbed [*full deautonomisation*]{}, which allows to detect the integrable or nonintegrable character of a mapping with confined singularities.
The absence of unconfined singularities in a given mapping signals the possibility of its regularisation through a succession of blow-ups \[,\]. In \[\] we have used this algebro-geometric method to establish the reliability of our ‘deautonomisation through singularity confinement’ approach. We remind here that the deautonomisation method consists in extending an autonomous mapping to one where the various, previously constant, coefficients are (appropriately chosen) functions of the independent variable. In a recent paper \[\], Carstea and co-workers have put this notion of deautonomisation on a rigorous algebro-geometric footing in the case of so-called Quispel-Roberts-Thompson (QRT) \[\] mappings. In particular, they show how to deautonomise a given QRT mapping starting from a particular fibre in the elliptic fibration that is left invariant by the mapping and, as a result, explain how the choice of fibre determines the type of surface the deautonomised mapping will be associated with in Sakai’s classification of discrete Painlevé equations \[\].
As a matter of fact, deautonomisation of integrable autonomous mappings, using singularity confinement, is the method through which the vast majority of discrete Painlevé equations have been derived. It is, in practice, much faster and easier to apply to a given mapping than a full-blown algebro-geometric analysis. However, both approaches are closely related and often one can be used to shed light on the other. As was shown on a collection of examples in \[\], not only does the deautonomisation of a mapping through blow-up yield exactly the same conditions on the parameters in the mapping as the singularity confinement criterion does, these conditions actually turn out to be intimately related to a linear transformation on a particular part of the Picard lattice for the surface obtained from the blow-ups. The part in question is characterised by the fact that its complement is invariant under the automorphism induced on the Picard lattice by the mapping; the automorphism essentially acts as a permutation of the generators for that invariant sublattice. Quite often, this permutation actually shows up in the singularity analysis of the mapping, as a typical singularity pattern which we call [*cyclic*]{}. In such a cyclic singularity pattern, a specific pattern of fixed length keeps repeating for all iterations. Note however that such cyclic patterns are not necessarily periodic as some finite values appearing in the repeating pattern may change from one pattern to the next. Strictly speaking, the singularities that are contained in a cyclic pattern are not confined, at least not in the spirit of the original definition as conceived in \[\], which demands that once one exits a singularity one should be genuinely home free and not re-enter the same singularity after some further iterations. However, they are most definitely not unconfined singularities either but they should rather be thought of as fixed ones, in the sense that if the initial conditions are not special (i.e. already part of the cycle) then it is impossible to enter such a singularity. Therefore, from an integrability point of view, they are completely innocuous, which is why they are often simply neglected when performing the singularity analysis for a mapping. In \[\] Halburd showed how, using the knowledge of confined and cyclic singularities, one can obtain in a simple way the exact degree growth for a given mapping. However, as we showed in \[\], when one is interested in a yes or no answer concerning the integrable character of a mapping with both cyclic patterns and confined singularities, the study of the latter does suffice and there is no need to take into account the cyclic singularities.
Be that as it may, we shall argue here that such cyclic singularity patterns can provide valuable information in the context of the deautonomisation method, as they will often betray the precise identity of the deautonomised mapping in Sakai’s classification, even before actually carrying out the deautonomisation. In this paper we are going to study discrete Painlevé equations that belong to the degeneration cascade starting from E$_8^{(1)}$ in that classification. Our starting point will be three different equations associated to E$_8^{(1)}$, already obtained in \[\]. For each of these equations we start by taking the autonomous limit, thereby obtaining, for each case, a mapping of QRT-type that will play the role of a master mapping. Our guide will be the classification of canonical forms for the QRT mapping which we presented in \[\]. We remind that the QRT invariant, in the symmetric case, has the form $$K={\alpha_0x_{n-1}^2x_n^2+\beta_0x_{n-1}x_n(x_{n-1}+x_n)+\gamma_0(x_{n-1}^2+x_n^2)+\epsilon_0x_{n-1}x_n+\zeta_0(x_{n-1}+x_n)+\mu_0\over \alpha_1x_{n-1}^2x_n^2+\beta_1x_{n-1}x_n(x_{n-1}+x_n)+\gamma_1(x_{n-1}^2+x_n^2)+\epsilon_1x_{n-1}x_n+\zeta_1(x_{n-1}+x_n)+\mu_1}.\eqdef\qrtinv$$ Eight canonical forms have been obtained in \[\]. For completeness we give below the list of the corresponding denominators, $D$, of the QRT invariant () together with the canonical form of the mapping: $$\leqno {\rm(I)} \qquad D=1\qquad\qquad x_{n+1}+x_{n-1}=f(x_n),$$ $$\leqno {\rm(II)} \qquad D=x_{n-1}x_n\qquad\qquad x_{n+1}x_{n-1}=f(x_n),$$ $$\leqno {\rm(III)} \qquad D=x_{n-1}+x_n\qquad\qquad (x_{n+1}+x_{n})(x_{n}+x_{n-1})=f(x_n),$$ $$\leqno {\rm(IV)} \qquad D=x_{n-1}x_n-1\qquad\qquad (x_{n+1}x_{n}-1)(x_{n}x_{n-1}-1)=f(x_n),$$ $$\leqno{\rm(V)}\qquad D=(x_{n-1}+x_n)(x_{n-1}+x_n+2z)\qquad\qquad {(x_{n+1}+x_{n}+2z)(x_{n}+x_{n-1}+2z)\over(x_{n+1}+x_{n})(x_{n}+x_{n-1})}=f(x_n),$$ $$\leqno {\rm(VI)} \qquad D=(x_{n-1}x_n-z^2)(x_{n-1}x_n-1)\qquad\qquad {(x_{n+1}x_{n}-z^2)(x_{n}x_{n-1}-z^2)\over(x_{n+1}x_{n}-1)(x_{n}x_{n-1}-1)}=f(x_n),$$ $$\leqno {\rm(VII)} \qquad D=(x_{n-1}+x_n-z^2)^2-4x_{n-1}x_n\qquad\qquad {(x_{n+1}-x_{n}-z^2)(x_{n-1}-x_{n}-z^2)+x_nz^2\over x_{n+1}-2x_{n}+x_{n-1}-2z^2}=f(x_n),$$ $$\leqno {\rm(VIII)} \qquad D=x_{n-1}^2+x_n^2-(z^2+1/z^2)x_{n-1}x_n+(z^2-1/z^2)^2\quad
{(x_{n+1}z^2-x_{n})(x_{n-1}z^2-x_{n})-(z^4-1)^2\over(x_{n+1}z^{-2}-x_{n})(x_{n-1}z^{-2}-x_{n})-(z^{-4}-1)^2}=f(x_n).$$ It turns out that in the cases V and VI there exists an alternative way to write the invariant which we have dubbed V$'$ and VI$'$. In this case we have: $$\leqno{\rm(V')}\qquad D=(x_{n-1}+x_n)^2-z^2\qquad\qquad {(x_{n+1}+x_{n}+z)(x_{n}+x_{n-1}+z)\over(x_{n+1}+x_{n}-z)(x_{n}+x_{n-1}-z)}=f(x_n),$$ $$\leqno {\rm(VI')} \qquad D=x_{n-1}^2+(z+1/z)x_{n-1}x_n+x_n^2\qquad\qquad {(zx_{n+1}+x_{n})(x_{n}+zx_{n-1})\over(x_{n+1}+zx_{n})(zx_{n}+x_{n-1})}=f(x_n).$$ Implementing homographic transformations of the dependent variable in order to cast the invariant of each master mapping into a judiciously chosen canonical form, we obtain all the possible canonical forms of mappings related to the master mapping. The deautonomisation of these mappings then leads to the discrete Painlevé equations in the corresponding degeneration cascade of the original E$_8^{(1)}$ mapping. We study the structure of the singularities for all these mappings and show that apart from the confined singularities which are always there, for certain canonical forms of the invariant and given the right choice of dependent variable, cyclic patterns do also exist. The relation of these patterns to reducible (singular) fibres for the associated invariant and, ultimately, with the method presented in \[\] will also be explained.
The equations we shall work with stem from \[\] where we presented a list of discrete Painlevé equations associated to the affine Weyl group E$_8^{(1)}$, written in what we call [*trihomographic form*]{}. As shown in \[\] the trihomographic form is a very convenient way to represent the various discrete Painlevé equations, not limited to those related to E$_8^{(1)}$. The general symmetric form (in the QRT \[\] terminology) of the additive trihomographic equation is $${x_{n+1}-(z_n+z_{n-1}+k_n)^2\over x_{n+1}-(z_n+z_{n-1}-k_n)^2}{x_{n-1}-(z_n+z_{n+1}+k_n)^2\over x_{n-1}-(z_n+z_{n+1}-k_n)^2}{x_{n}-(2z_n+z_{n-1}+z_{n+1}-k_n)^2\over x_{n}-(2z_n+z_{n-1}+z_{n+1}+k_n)^2}=1,\eqdef\zunu$$ and the asymmetric one $${x_{n+1}-(\zeta_n+z_n+k_n)^2\over x_{n+1}-(\zeta_n+z_n-k_n)^2}{x_{n}-(\zeta_n+z_{n+1}+k_n)^2\over x_{n}-(\zeta_n+z_{n+1}-k_n)^2}{y_{n}-(z_n+2\zeta_n+z_{n+1}-k_n)^2\over y_{n}-(z_n+2\zeta_n+z_{n+1}+k_n)^2}=1\eqdaf\zduo$$ $${y_{n}-(\zeta_{n-1}+z_n+\kappa_n)^2\over y_{n}-(\zeta_{n-1}+z_n-\kappa_n)^2}{y_{n-1}-(\zeta_n+z_{n}+\kappa_n)^2\over y_{n-1}-(\zeta_n+z_{n}-\kappa_n)^2}{x_{n}-(\zeta_n+2z_n+\zeta_{n-1}-\kappa_n)^2\over x_{n}-(\zeta_n+2z_n+\zeta_{n-1}+\kappa_n)^2}=1,\eqno(\zduo b)$$ where $z_n, \zeta_n, k_n, \kappa_n$ are functions of the independent variables, the precise form of which is obtained by the application of the singularity confinement criterion. The multiplicative form of the trihomographic equation is obtained from () or () by replacing the terms $(z_n+z_{n-1}+k_n)^2, $ etc. , by $\sinh^2(z_n+z_{n-1}+k_n), \dots$ \[\]. For the elliptic case we have squares of elliptic sines in the left-hand side, i.e. $(z_n+z_{n-1}+k_n)^2, \dots$ should be replaced by ${\rm sn}^2(z_n+z_{n-1}+k_n)$ etc., while in the right-hand side we have a ratio of a product of squares of theta functions, instead of 1.
Hereafter we shall focus on three particular equations among those obtained in \[\]. The parameters in these equations are most conveniently expressed using two auxiliary variables. The first one, $t_n$, is related to the independent variable $n$ in a purely secular way by $t_n\equiv\alpha n+\beta$. The second one is a periodic function $\phi_m(n)$ with period $m$, i.e. $\phi_m(n+m)=\phi_m(n)$, given by $$\phi_m(n)=\sum_{l=1}^{m-1} \epsilon_l^{(m)} \exp\left({2i\pi ln\over m}\right).\eqdef\ztre$$ Note that the summation in () starts at 1 and not at 0 and thus $\phi_m(n)$ introduces $m-1$ parameters. Two of the equations we are going to study have the symmetric form () and parameter dependence $$z_n=u_{n+1}-u_{n}+u_{n-1},\ k_n=u_{n+2}-u_{n}+u_{n-2} \quad {\rm where}\quad u_n=t_n+\phi_2(n)+\phi_7(n),\eqdef\parsi$$ $$z_n=u_{n},\ k_n=u_{n+1}+u_{n}+u_{n-1} \quad {\rm where}\quad u_n=t_n+\phi_2(n)+\phi_3(n)+\phi_5(n),\eqdef\parsii$$ while the third one has the asymmetric form () and parameter dependence: $$z_n=u_n-u_{n+1}+u_{n-1}-u_{n-2},\ \zeta_n=u_{n+2}-u_n+u_{n-2},\ k_n=u_n,\ \kappa_n=u_{n+2}+u_{n-3}, \quad u_n=t_n+\phi_8(n).\eqdef\parsiii$$
2\. [The symmetric equation with periods 2 & 7]{}
Before proceeding to the autonomous limit of the additive equation with periods 2 and 7, i.e. () with parameter dependence (), we first study the structure of its singularities already in the nonautonomous form. For this purpose, in fact, it suffices to consider just the secular dependence in $t_n$, neglecting the periodic terms. We have thus the trihomographic expression $${x_{n+1}-(3t_n-\alpha)^2\over x_{n+1}-(t_n-\alpha)^2}{x_{n-1}-(3t_n+\alpha)^2\over x_{n-1}-(t_n+\alpha)^2}{x_{n}-9t_n^2\over x_{n}-25t_n^2}=1,\eqdef\zcua$$ for which two singularity patterns exist, one of length six $\{x_{n-2}=25t_{n-2}^2, x_{n-1}=(3t_{n-2}-\alpha)^2, x_{n}=(t_{n-2}-4\alpha)^2, x_{n+1}=(t_{n+3}+4\alpha)^2, x_{n+2}=(3t_{n+3}+\alpha)^2, x_{n+3}=25t_{n+3}^2\}$ and one of length four $\{x_{n-1}=9t_{n-1}^2, x_{n}=(t_{n-1}-\alpha)^2, x_{n+1}=(t_{n+2}+\alpha)^2, x_{n+2}=9t_{n+2}^2\}$.
Next we go to the autonomous limit by taking $\alpha=0$ (and neglecting the periodic terms). Equation () now becomes $${x_{n+1}-9\beta^2\over x_{n+1}-\beta^2}{x_{n-1}-9\beta^2\over x_{n-1}-\beta^2}{x_{n}-9\beta^2\over x_{n}-25\beta^2}=1,\eqdef\zqui$$ and we can put $\beta$ to 1 by a simple rescaling of $x_n$. Regardless, as shown in \[\] it suffices to put $${2\over3}X_n={x_{n}-9\beta^2 \over x_{n}-\beta^2},\eqdef\zsex$$ for () to become $$X_{n+1}X_{n-1}=A{X_n-1\over X_n},\eqdef\zhep$$ for $A=27/4$. (Note that, had we worked with the multiplicative form of the equation, we would have found for $A$ an expression involving $\lambda=e^{\beta}$). The mapping () being autonomous and of QRT type it is straightforward to obtain its conserved quantity. We find readily $$K={(X_nX_{n-1}+A)(X_n+X_{n-1}-1)\over X_nX_{n-1}},\eqdef\znov$$ where we shall now leave the parameter $A$ essentially free, albeit non-zero.
Next we study the singularity patterns of (). The singularity of length six for () which starts with $x_n=25$ corresponds to a singularity for () starting with $X_n=1$. Iterating from this value we find the confined pattern $\{1,0,\infty,\infty,0,1\}$, which fits exactly with the first pattern for (). Next we turn to the length-four pattern for () starting from $x_n=9$, which corresponds to $X_n=0$. Starting with a finite generic value for $X_{n-1}=f$ and from $X_n=0$, we find the following repeating pattern $\{f,0,\infty,\infty,0,f',\infty,f'',0,\infty,\infty,0,\dots\}$ where $f,f',f'',\dots$ are finite values depending on $X_{n-1}$. We remark readily that the pattern $\{f,0,\infty,\infty,0,f',\infty\}$, of length seven, is simply repeated cyclically. Moreover, the confined pattern of length four one would have expected from that of (), i.e. a pattern $\{0,\infty,\infty,0\}$ in the variable $X_n$, is now embedded in the cyclic one, bracketed by two finite values.
Deautonomising () is straightforward. We assume that $A$ is a function of the independent variable $n$ and require that the confined pattern be still confined in the nonautonomous case. We find the confinement constraint $$A_{n+2}A_{n-1}=A_{n+1}A_n,\eqdef\zoct$$ which is integrated to $\log A_n=\alpha n+\beta+\gamma(-1)^n$ \[\]. This means that the corresponding discrete Painlevé equation is associated to the affine Weyl group (A$_1$+A$_1)^{(1)}$. Compared to () we lost the period 7 in the coefficients but, on the other hand, it can be checked that the cyclic pattern of length seven persists in the nonautonomous case of (). Note that in Sakai’s classification, a discrete Painlevé equation of multiplicative type with (A$_1$+A$_1)^{(1)}$ symmetry is usually associated with a (generalized) Halphen surface of type A$_6^{(1)}$. In the next section we shall explain that this is exactly the intersection type of the reducible singular fibre in the elliptic fibration defined by () that is responsible for the cyclic singularity pattern of length seven we detected through our singularity analysis.
Next we look for the various ways in which the invariant () can be made to coincide with one of the 8 canonical forms presented in the introduction. For this we introduce a homographic transformation of the dependent variable (and we add a constant $\kappa$ to the invariant since the latter is at best defined up to an additive constant). We shall not enter into the details of this calculation but give directly the results. Four canonical cases could be identified. For the first two there is no need to add a constant to the invariant. We find thus that introducing the homography $$X_n={1\over2}{y_n+z\over y_n-z},\eqdef\zdec$$ (with $z\neq 0$) and giving to $A$ the value $A=-1/4$, we find the invariant $$K={(y_n^2-z^2)(y_{n-1}^2-z^2)\over (y_n+y_{n-1})(y_n+y_{n-1}-2z)},\eqdef\dunu$$ leading to the mapping $$\left({y_n+y_{n+1}-2z\over y_n+y_{n+1}}\right)\left({y_n+y_{n-1}-2z\over y_n+y_{n-1}}\right)={y_n-3z\over y_n+z}.\eqdef\dduo$$ On the other hand, the homography $$X_n={1\over z+1/z}{1-zy_n\over z-y_n},\eqdef\dtre$$ with $A=-1/(z+1/z)^2\neq -1/4$ leads to the invariant $$K={(y_n-z)(y_{n-1}-z)(1-zy_n)(1-zy_{n-1})\over (y_ny_{n-1}-z^2)(y_ny_{n-1}-1)},\eqdef\dcua$$ and the mapping $$\left({y_ny_{n+1}-z^2\over y_ny_{n+1}-1}\right)\left({y_ny_{n-1}-z^2\over y_ny_{n-1}-1}\right)={y_n-z^3\over y_n-1/z}.\eqdef\dqui$$ Deautonomising the mappings () and () is straightforward and the results have already been presented in \[\]. In short, we find the additive mapping $${(y_n+y_{n+1}-z_n-z_{n+1})(y_n+y_{n-1}-z_n-z_{n-1})\over (y_n+y_{n+1})(y_n+y_{n-1})}=
{y_n-z_{n+3}-z_n-z_{n-3} \over y_n-z_{n+3}+z_{n+1}+z_n+z_{n-1}-z_{n-3}},\eqdef\dsex$$ with $z_n=\alpha n+\beta+\phi_7(n)$, from (), and the multiplicative mapping $${(y_ny_{n+1}-z_nz_{n+1})(y_ny_{n-1}-z_nz_{n-1})\over (y_ny_{n+1}-1)(y_ny_{n-1}-1)}=
{y_n-z_{n+3}z_nz_{n-3} \over y_n-z_{n+3}z_{n-3}/(z_{n+1}z_nz_{n-1})},\eqdef\dhep$$ with $\log z_n=\alpha n+\beta+\phi_7(n)$, from (). Both have E$_7^{(1)}$ symmetry and, in particular, it should be noted that the period 2 which was present in the parameters of the original mapping we started from, has been lost. A study of their singularities shows that both equations have two confined patterns of lengths 6 and 4, just like (), and this even when one neglects the periodic term in $z_n$, keeping only the secular dependence. Thus, in the nonautonomous case the length-7 cyclic pattern has disappeared but a periodic term of period 7 has made its appearance in the coefficients of the equation.
The interesting thing is the appearance of the length-four singularity. The way singularity confinement was always implemented in the deautonomisation procedure was to look for the confined singularity patterns in the autonomous case and preserve them upon deautonomisation. The existence of cyclic patterns where a shorter pattern–in this case of length four–is bracketed by two finite values shows that there exist situations where one may decide that a pattern is confined where, in reality, it is just a part of a longer, cyclic, one. However it turns out that such a misidentification is innocuous in practice. First, as explained in the introduction, the existence of cyclic singularity patterns is compatible with integrability, and second, as seen here, upon deautonomisation the pattern embedded between two finite values may indeed become confined.
There are still two other possible homographies we can implement. Obviously, when $A=27/4$, the homography () is among the possible ones, leading back to the mapping we started with. Which leaves one final homography which is of the form $$X_n={z^2+1+1/z^2\over z^2+1/z^2}{y_n+z^3+1/z^3\over y_n+z+1/z}.\eqdef\doct$$ Together with the values $A=(z^2+1+1/z^2)^3/(z^2+1/z^2)^2$ and $\kappa=-(z+1/z)^4/(z^2+1/z^2)$ this yields the invariant $$K={(y_n+z+1/z)(y_{n-1}+z+1/z)(y_n+z^3+1/z^3)(y_{n-1}+z^3+1/z^3)\over y_n^2+y_{n-1}^2-(z^2+1/z^2)y_ny_{n-1}+(z^2-1/z^2)^2}.\eqdef\dnov$$ The corresponding mapping is $${(y_n-z^2y_{n+1})(y_n-z^2y_{n-1})-(z^4-1)^2\over (z^2y_n-y_{n+1})(z^2y_n-y_{n-1})-(z^4-1)^2/z^4}={y_n+z^4(z+1/z)\over z^4y_n+z+1/z}.\eqdef\ddec$$ Note that this is the canonical form of the autonomous limit of a multiplicative discrete Painlevé equation associated to the affine Weyl group E$_8^{(1)}$. Its deautonomisation has been presented in \[\].
It is interesting to point out here that elliptic discrete Painlevé equations can also be obtained, be it at the price of slightly generalising our starting point to the autonomous, trihomographic, form $${\delta x_{n+1}-\eta\over \mu x_{n+1}-\nu}{\delta x_{n-1}-\eta\over \mu x_{n-1}-\nu}{\delta x_{n}-\eta\over \rho x_{n}-\sigma}=1,\eqdef\wunu$$ where the parameters are expressed in terms of theta functions as follows: $\delta=\theta_0^2(3z)$, $\eta=\theta_1^2(3z)$, $\mu=\theta_0^2(z)$, $\nu=\theta_1^2(z)$, $\rho=\theta_0^2(5z)$, $\sigma=\theta_1^2(5z)$. (Here we are using the conventions of the monograph of Byrd and Friedman \[\] for the naming of the theta functions). Introducing the homography $${\delta\sigma-\eta\rho\over\mu\sigma-\nu\rho}X_n={\delta x_n-\eta \over \mu x_n-\nu},\eqdef\wduo$$ we obtain precisely equation () with $A=(\mu\sigma-\nu\rho)^3/((\delta\nu-\eta\mu)(\delta\sigma-\eta\rho)^2)$. Thus the analysis presented above also leads to elliptic equations associated to the E$_8^{(1)}$ group. For example, starting from the parameter dependence () and introducing $$\displaylines{a_n=z_n+z_{n-1}+k_n,\ b_n=z_n+z_{n-1}-k_n,\ c_n=z_n+z_{n+1}+k_n,\ d_n=z_n+z_{n+1}-k_n, \hfill\cr\hfill e_n=2z_n+z_{n-1}+z_{n+1}-k_n,\ f_n=2z_n+z_{n-1}+z_{n+1}+k_n,\quad\eqdisp\fduo\cr}$$ we obtain the full, nonautonomous, form of this discrete Painlevé equation: $${\theta_0^2(a_n)x_{n+1}-\theta_1^2(a_n)\over \theta_0^2(b_n)x_{n+1}-\theta_1^2(b_n)}\ {\theta_0^2(c_n)x_{n-1}-\theta_1^2(c_n)\over \theta_0^2(d_n)x_{n-1}-\theta_1^2(d_n)}\ {\theta_0^2(e_n)x_{n}-\theta_1^2(e_n)\over \theta_0^2(f_n)x_{n}-\theta_1^2(f_n)}= 1.\eqdef\ftre$$
The equations presented above are not the only ones we can obtain starting from () or, equivalently, (). Given the form of () it is clear that we can also consider an evolution where we skip one out of two indices and thus establish a mapping relating $x_{n+2}, x_n$ and $x_{n-2}$. The simplest way is to solve () for $X_n$ and obtain an invariant involving $X_{n+1}$ and $X_{n-1}$. Downshifting the indices we find $$K={(X_nX_{n-2}-X_n-A)(X_nX_{n-2}-X_{n-2}-A)\over X_nX_{n-2}-A}.\eqdef\vunu$$ Putting $A=a^2$ and rescaling $X$, so as to absorb one factor of $a$, we obtain from () the mapping $$(X_nX_{n+2}-1)(X_nX_{n-2}-1)={X_n\over a^2X_n-a}.\eqdef\vduo$$ The singularity patterns of () can be directly read off from those of (). We have the confined pattern $\{1/a,\infty,0\}$, its mirror image $\{0,\infty,1/a\}$ and a cyclic one $\{\infty,0,\infty,0,\infty,f,f'\}$, of length 7, as expected. In order to deautonomise () we write its right-hand side as $X_n/(b_nX_n-a_n)$. Applying the confinement criterion we then find $\log a_n=\alpha n+\beta$ and $b_n=da_n^2$ where $d$ is a free constant \[\]. Note that in this case the parameters in the mapping only have secular dependence on $n$.
It is interesting at this point to introduce the invariant associated to the autonomous limit of () for the right-hand side $X_n/(d a^2 X_n-a)$. We find $$K={a^2dX_n^2X_{n-2}^2-a X_nX_{n-2}(X_n+X_{n-2})+a(X_n+X_{n-2})-a^2d+1\over X_nX_{n-2}-1}\eqdef\aunu$$ In order to obtain double-step equations associated to E$_7^{(1)}$ or E$_8^{(1)}$ we can now either start from () and (), or use the invariant (). Both approaches are not equivalent however, since working with () and () is tantamount to setting $d=1$ (and thus to working with equations obtained from ()), which turns out to be a necessary constraint to obtain E$_7^{(1)}$ related equations, but not for those related to E$_8^{(1)}$.
Working with (), along the lines set out above, we find the invariant $$K={(y_{n-2}-z)(y_n-z)\big((z+1/z)(y_{n-2}y_n+1)-2(y_{n-2}+y_n)\big)\over(z^2y_n-y_{n-2})(z^2y_{n-2}-y_n)},\eqdef\vtre$$ and the corresponding mapping $$\left({z^2y_{n+2}-y_n\over y_{n+2}-z^2y_n}\right)\left({z^2y_{n-2}-y_n\over y_{n-2}-z^2y_n}\right)={(y_n-z)^2(y_n-z^3)\over z(zy_n-1)^3}.\eqdef\vcua$$ We remark that the mapping is given in the form we dubbed (VI$'$) in \[\]. Its deautonomisation, albeit in the canonical form (VI), was given in \[\], where we have derived E$_7^{(1)}$-associated equations with period-7 periodicity in their parameters. The non-autonomous form of () corresponds to equation (20) of \[\]. An analogous derivation centred around () leads to the mapping $$\left({y_{n+2}-y_n+2z\over y_{n+2}-y_n-2z}\right)\left({y_{n-2}-y_n+2z\over y_{n-2}-y_n-2z}\right)={(y_n-z)^2(y_n-3z)\over (y_n+z)^3}.\eqdef\vqui$$ This is a mapping of canonical form (V$'$). Clearly it can be brought to the canonical (V) form through some elementary transformations: changing the sign of two consecutive $y_n$ out of four we can bring the left hand side to one where the dependent variables have the same sign whereupon a simple translation can absorb the terms proportional to $z$ in the denominator. Again the deautonomisation results can be found in \[\], equation (11).
Where the explicit presence of the parameter $d$ becomes essential is in the derivation of the double-step equations associated to the E$_8^{(1)}$ group. For this, we start from the invariant () and look for the proper homographic transformation bringing the invariant to a canonical E$_8^{(1)}$ form. We focus on the additive case, for which a detailed analysis was presented in \[\]. Without going into the lengthy details we find that the autonomous mapping we obtain is the autonomous limit of the equation 5.2.7 of \[\] where the parameter $d$ above and the constant $c$ figuring in 5.2.7 are related through $d=(c-3)(c+1)^3/((c+3)(c-1)^3)$. The deautonomisation of this mappings is obviously the one given in \[\]. Thus equation (), which corresponds to case 3.3 in \[\], is related, once we consider evolution with double step, to case 5.2.7 of that paper. It goes without saying that similar results could be obtained for multiplicative and elliptic systems.
3\. [Invariants, singular fibres and deautonomisations]{}
As is well-known since the seminal works of Veselov \[\] and Tsuda \[\], a QRT map with invariant () can be described by a birational automorphism on a rational elliptic surface. This surface, say $X$, is obtained from blowing up ${\Bbb P}^1\times{\Bbb P}^1$ at eight points $(u_i,v_i)$. These are the eight points (taking into account multiplicities) that do not lie on a unique member of the pencil of quadratic curves $$\displaylines{ (\alpha_0+K\alpha_1)u^2v^2+(\beta_0+K\beta_1)uv(u+v)+(\gamma_0+K\gamma_1)(u^2+v^2)\hfill\cr\hfill+
(\epsilon_0+K\epsilon_1)uv+(\zeta_0+K\zeta_1)(u+v)+(\mu_0+K\mu_1)=0,\quad \eqdisp\antonii\cr}$$ obtained from the invariant () by taking $x_{n-1}=u$, $x_n=v$. Each member of this family of quadratic curves (), parametrised by $K\in{\Bbb P}^1$, is invariant under the QRT map at hand. What is important here, is that after blow-up, the QRT map is so to speak ‘lifted’ to an automorphism on $X$ that preserves the elliptic fibration $\pi: X \to {\Bbb P}^1$ described by (). For a generic value of $K$, the fibre $\pi^{-1}(K)\in X$ is a genuine elliptic curve (i.e. an irreducible smooth curve of genus one). There exist however a finite number of special values for $K$ for which the corresponding curve is either not smooth simply because it contains a node or a cusp (the so-called multiplicative and additive reductions, respectively) or because it is in fact reducible and therefore contains singular points. The fibres $\pi^{-1}(K)$ that correspond to these special values of $K$ are called [*singular*]{} and are either reducible or irreducible, in the sense explained above. Reducible fibres can be represented graphically by their intersection diagrams, which are actually Dynkin diagrams of ADE-type. For surfaces with a section, which is the case for the QRT-maps, all possible configurations of reducible fibres have been studied in \[\], resulting in a relatively short list of 74 different surface types, which makes the list rather easy to consult. A full classification, including all possible configurations that involve irreducible singular fibres, and which also distinguishes between multiplicative and additive A$_1^{(1)}$ or A$_2^{(1)}$ fibres (which is not the case in \[\]), can be found in \[\] and \[\] but the result is a much longer list (almost four times as long!) which is much less accessible than the one in \[\]. In the following we shall mainly concern ourselves with reducible singular fibres for the invariants we wish to treat, as these are easily obtained without resorting to special algorithms (see e.g. Appendix C in \[\]) and, whenever necessary, we shall use the lists in \[\] or \[\] to supplement our results with information on the irreducible fibres.
Let us consider for example the three point mapping () where we take $X_{n-1}$ to be $u_n$ and $X_n$ to be $v_n$ and where $A$ is a non-zero (complex) constant. Note that, rewriting the invariant () in these new variables, we see readily that $K$ becomes infinite for the four lines $u=0$, $v=0$, $u=\infty$, $v=\infty$ on ${\Bbb P}^1\times{\Bbb P}^1$. Moreover, the mapping () has two indefinite points, $(u,v)=(0,1)$ and $(u,v)=(\infty,0)$, and there are two curves on ${\Bbb P}^1\times{\Bbb P}^1$ that contract to a point under its action: $\{v=1\}$ and $\{v=0\}$. Under the action of the mapping, the first of these curves gives rise to the confined singularity pattern of length six described in section 2, $$\{v=1\}=\pmatrix{f \cr 1}\to\pmatrix{1\cr 0}\to\pmatrix{0\cr\infty}\to\pmatrix{\infty\cr\infty}\to\pmatrix{\infty\cr 0}\to\pmatrix{0\cr 1}\to\{u=1\},\eqdef\antonv$$ which can be read off from the $v$-coordinates in the bottom row of the vectors in the middle of the pattern. The second curve is part of a cyclic pattern of length seven $$\{v=0\}\to\pmatrix{0\cr\infty}\to\pmatrix{\infty\cr\infty}\to\pmatrix{\infty\cr 0}\to\{u=0\}\to\{v=\infty\}\to\{u=\infty\}\to\{v=0\}\to\cdots,\eqdef\antonvi$$ which, after blow-up, gives rise to a cycle of curves on a rational elliptic surface $X$, $$\cdots\to\{v=0\}\to C_{0,\infty}\to C_{\infty,\infty}\to C_{\infty, 0}\to\{u=0\}\to\{v=\infty\}\to\{u=\infty\}\to\{v=0\}\to\cdots\eqdef\antonvii$$ (where $C_{0,\infty}, C_{\infty,\infty}$ and $C_{\infty, 0}$ are curves on $X$ obtained from the blow-ups at the points indicated, as shown in Figure 1). This cycle corresponds of course to the length-seven cyclic pattern we discovered by singularity analysis, for mapping (). More importantly however, it is clear that the union of all seven curves in this cycle, in fact, constitutes an (invariant) reducible singular fibre on $X$ for the value $K=\infty$.
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The intersection diagram for this reducible fibre–obtained by associating a vertex with each component of the fibre and with each intersection of the components an edge between the corresponding vertices–is easily seen to form a heptagon, i.e. the Dynkin diagram for A$_6^{(1)}$. The mapping induced on $X$ acts as a rotation of this diagram, as shown in Figure 2.
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Oguiso and Shioda in \[\] list only two surfaces that contain such a singular fibre: one where it is the sole reducible fibre to exist on the surface (number 25 in their list) and one where it is accompanied by an A$_1^{(1)}$ fibre (number 47). It is easy to check directly on the invariant () itself that there is (exactly) one other reducible fibre: for $K=0$ we find that the invariant decomposes into two curves (on ${\Bbb P}^1\times{\Bbb P}^1$) $$C_1 :\quad u v = -A\qquad\quad{\rm and}\quad\qquad C_2 :\quad u+v = 1,\eqdef\antonviii$$ which are interchanged by the mapping: the image under the mapping of $C_1$ is $C_2$ and vice versa. After blow-up, these two curves are the components of a singular fibre of $A_1^{(1)}$ type for the surface $X$, but we need to distinguish two cases. When $A\neq-1/4$, $C_1$ and $C_2$ intersect transversally at two distinct points, whereas when $A=-1/4$ they are tangential at a single point. The former type of fibre is usually denoted as A$_1^{(1)}$ and the latter, its so-called additive reduction, as A$_1^{(1)*}$.
In order to better understand the deautonomisation of the mapping () that led to (), i.e. to an equation associated with the affine Weyl group (A$_1$+A$_1)^{(1)}$, it is important to notice first that the set of coordinates (in fact, the homography ()) we have chosen is such that the curves and singular points in the cyclic pattern () are all independent of the parameter in the equation. Hence, after blow-up, the resulting A$_6^{(1)}$ fibre $\{v=0\} \cup C_{0,\infty}\cup C_{\infty,\infty}\cup C_{\infty, 0}\cup \{u=0\}\cup \{v=\infty\}\cup \{u=\infty\}$ will be, in this sense, completely ‘fixed’ on $X$. (As we shall see in the following this situation might be too restrictive; it suffices to require that the intersections of the components of the fibre are independent of the parameters). This is important since, when deautonomising the mapping, we wish to keep the intersection pattern of the components of the fibre independent of $n$. Now, when deautonomising, the base points (for the blow-ups that lead to $X$) which depend on the parameters of the mapping will also be $n$ dependent, in which case, strictly speaking, we do not obtain a single algebraic surface $X$, but rather a family $\{X_n\}$ of such surfaces. A deautonomisation of an integrable mapping is only meaningful \[\] when the map induced between the Picard lattices ${\rm Pic}X_n$ and ${\rm Pic}X_{n+1}$ by its deautonomisation is an isomorphism and, moreover, when this induced action actually coincides with that induced by the original map on ${\rm Pic}X$. In \[\] it is explained how, under these circumstances, any given invariant fibre for a QRT-map (smooth or singular) can be used as an anti-canonical divisor that will define the generalized Halphen surface (in the sense of \[\]) that is associated with the discrete Painlevé equation obtained from that deautonomisation.
In a nutshell, in the present case, what happens is that since the isomorphism between ${\rm Pic}X_n$ and ${\rm Pic}X_{n+1}$ is the same as that induced by the original autonomous map () on ${\rm Pic}X$, the unitary action on the part of ${\rm Pic}X$ that contains the components of the A$_6^{(1)}$ fibre – embodied by the cyclic pattern () – is still present in the nonautonomous setting. Choosing this particular singular fibre as the basic geometric structure to be preserved under deautonomisation, forces the base points for the blow-ups to follow the same permutation pattern as the components of the singular fibre to which they belong. In general, the movement of the base points under the action of the map imposes conditions on the parameters in the map that appear in the coordinates of the base points. However, it turns out that these conditions–equivalent to the confinement conditions–only pick up information about the movement of curves relative to this global periodic movement of the base points. Hence, in this particular deautonomisation of (), the period 7 parameter dependence that was present in the original map (), subject to (), can no longer be present, but the period 2 dependence in the parameters that is associated with the movement of the components $C_1$ and $C_2$ of the A$_1^{(1)}$ fibre () does persist in ().
Let us turn our attention now to the deautonomisations associated with the A$_1^{(1)}$ and A$_1^{(1)*}$ singular fibres. Under the homography (), $u=(1-z U)/((z+1/z)(z-U))$ and a similar transformation for $v$ with $A=-1/(z+1/z)^2$ $(z^2\neq 1)$, the curves $C_1$ and $C_2$ given by () are transformed into $$\tilde C_1 :\quad U V = 1\qquad\quad{\rm and}\quad\qquad \tilde C_2 :\quad U V = z^2,\eqdef\antonix$$ which intersect transversally at $(U,V) = (0,\infty)$ and $(\infty,0)$. Identifying $y_{n-1}$ with $U$ and $y_n$ with $V$ in the invariant () for the mapping (), it is obvious that these two curves are the two components of the A$_1^{(1)}$ type singular fibre associated with this particular canonical form of the invariant for the value $K=\infty$. (It should be noted that the invariant ($\dcua$) is in fact proportional to the reciprocal of the original invariant (), which explains why this singular fibre is obtained for the value $K=\infty$ and not at $K=0$ as was originally the case). The generalized Halphen surface obtained by regularising the deautonomisation () of mapping (), by blow-up, is therefore of type A$_1^{(1)}$, which is perfectly consistent with the fact that () is a multiplicative Painlevé equation associated with the E$_7^{(1)}$ Weyl group.
Similarly, the homography (), $u=(U+z)/(2(U-z)), v=(V+z)/(2(V-z))$ ($z\neq0$) for $A=-1/4$, transforms the curves $C_1$ and $C_2$ to two curves which resemble the ultra-discrete limits of (): $$C_1^* :\quad U + V = 0\qquad\quad{\rm and}\quad\qquad C_2^* :\quad U + V = 2 z.\eqdef\antonx$$ The curves $C_1^*$ and $C_2^*$ intersect tangentially at the point $(\infty,\infty)$ and are, obviously, the two components of the singular fibre associated with the invariant () for the value $K=\infty$. Hence, the deautonomisation () of mapping () is associated with a generalized Halphen surface of type A$_1^{(1)*}$, as indicated by the fact that it is an additive discrete Painlevé equation with E$_7^{(1)}$ Weyl group symmetry.
As expected, in both cases the period 2 parameter dependence that was originally present in () has now disappeared, but the period 7 behaviour in the parameters related to the A$_{6}^{(1)}$ fibre has resurfaced. Note also that in () as well as in (), one of the components of the singular fibre now does depend on the parameter $z$ of the mapping, but that the intersection points do not. Similarly, in the new coordinates we chose for these two mappings, the components of the A$_{6}^{(1)}$ singular fibre will now depend on the parameter $z$, which makes that the values of the parameters will not match up perfectly and no cyclic pattern of length 7 will show up in the singularity analysis of the nonautonomous mappings.
As we saw in section 2, these are however not the only possible deautonomisations for the mapping (). A generic, smooth (so-called A$_0^{(1)}$ type) fibre for the elliptic fibration obtained from the invariant will result in an elliptic discrete Painlevé equation such as (). Furthermore, closer inspection of the detailed lists in \[\] (or \[\]) shows that there are in fact three types of rational elliptic surfaces that contain an A$_6^{(1)}$, A$_1^{(1)}$ configuration (allowing for possible additive cases). The first surface, with a configuration of A$_6^{(1)}$ A$_1^{(1)}$ singular fibres and three nodal curves (denoted as 3A$_0^{(1)*}$), corresponds in our case to a generic choice of the parameter $A$ in the QRT-mapping (), i.e. $A\neq -1/4$ or $27/4$. For each such value of $A$ for the homography () we find indeed three different values of $\kappa$ (due to the symmetry of the latter under $z\to -z$ and $z\to1/z$), that is three different singular fibres, but the form of the mapping does not change. For a given generic $A$, by deautonomising (), we have one $q$-Painlevé equation in E$_8^{(1)}$, one in E$_7^{(1)}$ and one in (A$_1$+A$_1)^{(1)}$ (equations (), () and () with () respectively, obtained in section 2), which are indeed associated to A$_0^{(1)*}$, A$_1^{(1)}$ and A$_6^{(1)}$ type generalized Halphen surfaces in Sakai’s classification.
A second surface has a configuration A$_6^{(1)}$ A$_1^{(1)*}$ 2A$_0^{(1)*}$, i.e. with one additive type A$_1^{(1)}$ type fibre, which in our case corresponds to the choice $A=-1/4$ in (). For this value of $A$ we have now two different values of $\kappa$ (again taking into account the invariances of $\kappa$) which are compatible with the homography (). This choice of the value of $A$ does not affect the equations with E$_8^{(1)}$ or (A$_1$+A$_1)^{(1)}$ symmetry, but the equation with E$_7^{(1)}$ symmetry, obtained after deautonomisation, is now of additive type (equation ()).
The final type of surface has the configuration A$_6^{(1)}$ A$_1^{(1)}$ A$_0^{(1)**}$ A$_0^{(1)*}$, where A$_0^{(1)**}$ stands for a cuspal curve (additive A$_0^{(1)}$ type). This is our case $A=27/4$, for which we find now one value of $\kappa$ that still gives rise (after deautonomisation) to a multiplicative equation of E$_8^{(1)}$. An additive Painlevé equation with E$_8^{(1)}$ symmetry does of course exist: it was our starting equation (), subject to ().
Finally, it is interesting to point out that as the mapping () was obtained by doubling the step in (), it possesses two invariant curves, $$a u v -u - a = 0\qquad{\rm and}\qquad a u v- v - a = 0,\eqdef\antonxi$$ where $u=X_{n-2}, v=X_n$. These are obtained from the factors in the numerator of the invariant (), i.e. from the fibre $K=0$, and are obviously relics of the curves $C_1$ and $C_2$ that were permuted under the original mapping. As was pointed out in section 2, mapping () still possesses a cyclic pattern of length 7 but unsurprisingly, its deautonomisation has lost the period 2 dependence in the parameters.
4\. [The symmetric equation with periods 2, 3 & 5]{}
We shall now investigate the degeneration cascade of the symmetric equation () with the parameter dependence (), along the lines set out in the previous sections. In order to analyse the singularity properties of this map it is in fact sufficient to consider the trihomographic expression $${x_{n+1}-(5t_n-\alpha)^2\over x_{n+1}-(t_n+\alpha)^2}{x_{n-1}-(5t_n+\alpha)^2\over x_{n-1}-(t_n-\alpha)^2}{x_{n}-t_n^2\over x_{n}-49t_n^2}=1,\eqdef\vsex$$ with only secular behaviour in the parameters.
Two singularity patterns exist, one of length eight, $\{x_{n-3}=49t_{n-3}^2, x_{n-2}=(5t_{n-3}-\alpha)^2,x_{n-1}=(3t_{n-3}-4\alpha)^2,x_{n}=(t_{n-3}-9\alpha)^2,x_{n+1}=(t_{n+4}+9\alpha)^2, x_{n+2}=(3t_{n+4}+4\alpha)^2,x_{n+3}=(5t_{n+4}+\alpha)^2 ,x_{n+4}=49t_{n+4}^2\}$ and one of length two, $\{x_{n}=t_{n}^2,x_{n+1}=t_{n+1}^2\}$.
Going to the autonomous limit, taking $\alpha=0$ and scaling $\beta$ to 1, we find the mapping $${x_{n+1}-25\over x_{n+1}-1}{x_{n-1}-25\over x_{n-1}-1}{x_{n}-1\over x_{n}-49}=1.\eqdef\vhep$$ Introducing a new dependent variable $X$ by $${1\over2}X_n={x_n-25\over x_n-1},\eqdef\voct$$ we find the mapping $$X_{n+1}X_{n-1}=A(X_n-1),\eqdef\vnov$$ where $A=4$. The singularity pattern starting with $x=49$ corresponds to a pattern starting with $X=1$ found to be $\{1,0,-A,\infty, \infty,-A,0,1\}$. The length-two pattern, starting with $x=1$, corresponds to one starting with $X=\infty$. More precisely, starting from a generic (finite) value $f$ for $X_{n-1}$ and with $X_n=\infty$, we find the cyclic pattern $\{f,\infty,\infty,f',0,f'',\infty,\infty,\dots\}$ (where $f,f',f'',\dots$ are finite values depending on $X_{n-1}$), consisting of a repetition of the length-five pattern $\{f,\infty,\infty,f',0\}$. Again, the length-two confined pattern expected from that of (), i.e. $\{\infty,\infty\}$ in the variable $X_n$, is embedded in the cyclic one, bracketed by two finite values. However, it is worth pointing out that if, with hindsight, one would have continued the iteration for the mapping () starting from $x_{n-1}=w, x_n=t_{n}^2$ for a few steps more, i.e. beyond the point where the singularity actually confines, one would have found the succession of values $$x_{n-1}=w,~ x_{n}=t_{n}^2, ~x_{n+1}=t_{n+1}^2, ~x_{n+2}= g(w), ~x_{n+3} = ~(5 t_{n+2}-\alpha)^2, x_{n+4} =~g'(w), ~\dots,\eqdef\remxx$$ (where $g(w)$ and $g'(w)$ are complicated functions of the initial value $w$) which corresponds to the succession $f,\infty,\infty,f',0,f'', \dots$ in the autonomous case (). This shows that the nonautonomous mapping () in fact picks up the slightly longer subpattern $\{f,\infty,\infty,f',0,f''\}$ from the cyclic pattern than is apparent from a simple check of the singularity confinement property.
Assuming now $A$ to be a function of $n$ we can try to deautonomise (). The singularity confinement constraint yields $$A_{n+3}A_{n-2}=A_{n+1}A_n,\eqdef\tdec$$ which can be integrated to $A_n=\alpha n+\beta +\phi_2(n)+\phi_3(n)$. The corresponding discrete Painlevé equation is thus associated to the affine Weyl group A$_4^{(1)}$ \[\] . Although the period 5 behaviour in the parameters has disappeared compared to those of () with (), the singularity analysis shows that the length-5 cyclic pattern persists even when () is deautonomised.
The mapping (), being of QRT-type, has an invariant of the form $$K={(X_nX_{n-1}+A(X_n+X_{n-1})-A)(X_n+X_{n-1}+A)\over X_nX_{n-1}},\eqdef\tunu$$ from which it is obvious that the cycle $\{v=\infty\} \to (\infty,\infty) \to \{u=\infty\} \to \{v=0\} \to \{u=0\} \to \{v=\infty\} \to \cdots$ (where $u=X_{n-1}$ and $v=X_n$) is nothing but the footprint on ${\Bbb P}^1\times{\Bbb P}^1$ of an A$_4^{(1)}$ type singular fibre associated with () at $K=\infty$. As explained in section 3, it is this fibre which underlies the geometric structure of the deautonomisation, which is therefore associated with a surface of A$_4^{(1)}$ type, resulting in a discrete Painlevé equation with A$_4^{(1)}$ Weyl group symmetry. Here again, as the curves in this singular fibre do not depend on the parameter $A$ of the mapping, the cyclic pattern resulting from the invariance of the fibre still persists in the nonautonomous equation.
As in section 2, we can now use the invariant () as a starting point to look for the possible canonical forms that can be obtained from (), through a homographic change of the dependent variable and a shift of the value of the invariant by adding a constant $\kappa$. First we must discard the trivial case $A=-1$ which corresponds to a mapping with a periodic, period-5, solution. Then, in total, six canonical cases are left. First, for $A\neq-1/2$, introducing the homography $$X_n=-{Ay_n+A+1\over y_n-1},\eqdef\tduo$$ we obtain the invariant (with $\kappa=-1-A$) $$K={(y_n-1)(y_{n-1}-1)(Ay_n+A+1)(Ay_{n-1}+A+1)\over y_ny_{n-1}-1},\eqdef\ttre$$ corresponding to the mapping $$(y_ny_{n+1}-1)(y_ny_{n-1}-1)=(1-y_n)(1+y_n(1+1/A)).\eqdef\tcua$$ Rewriting the right-hand side of () as $(1-y_n)(1-z_ny_n)$ we find that, for the singularities of the mapping to confine, $z_n$ must satisfy the constraint $$z_{n+5}z_{n-2}=z_{n+3}z_n,\eqdef\tqui$$ the integration of which leads to $\log z_n=\alpha n+\beta+\phi_2(n)+\phi_5(n)$ \[\]. The corresponding $q$-discrete Painlevé equation is associated to the affine Weyl group E$_6^{(1)}$ (as the period 3 dependence that was originally present in () has now disappeared). It is easily checked on the invariant () that there is a singular fibre for $K=\infty$ that decomposes into three curves (on ${\Bbb P}^1\times{\Bbb P}^1$) which form the 3-cycle $~\cdots \to \{u=\infty\} \to \{u v=1\} \to \{v=\infty\} \to \cdots~$ (where $u=y_{n-1}$ and $v=y_n$). Since the intersection of these curves in non-degenerate, this singular fibre is of type A$_2^{(1)}$, which indeed gives the correct surface in Sakai’s classification. Note also that this 3-cycle does not give rise to a cyclic singularity pattern for mapping (), since it does not contain any singularities (none of the curves in it contract to a point under the action of the mapping).
The situation changes when $A=-1/2$. In this case we use the homographic transformation $$X_n={y_n+z\over 2y_n},\eqdef\tsex$$ which, with $\kappa=-1/2$, yields an invariant of the form $$K={y_ny_{n-1}(y_n+z)(y_{n-1}+z)\over y_n+y_{n-1}}.\eqdef\thep$$ The corresponding mapping is now of additive type: $$(y_n+y_{n+1})(y_n+y_{n-1})=y_n(y_n-z).\eqdef\toct$$ Deautonomising () by applying the confinement criterion we find for $z$ the constraint $$z_{n+5}+z_{n-2}=z_{n+3}+z_n,\eqdef\tnov$$ and $z_n=\alpha n+\beta+\phi_2(n)+\phi_5(n)$ \[\]. Hence, the discrete Painlevé equation we obtain is again associated to the affine Weyl group E$_6^{(1)}$, but is now of additive type. Indeed, the singular fibre obtained from () for $K=\infty$, is this time given by three lines that intersect at a single point, $(\infty,\infty)\in{\Bbb P}^1\times{\Bbb P}^1$, and that form the 3-cycle $~\cdots \to \{u=\infty\} \to \{u+v=0\} \to \{v=\infty\} \to \cdots~$ and this fibre is therefore of A$_2^{(1)*}$ type, i.e. of additive type. We would like to point out that both discrete Painlevé equations associated to E$_6^{(1)}$ have been first identified in \[\] and then in \[\] and \[\] respectively.
The next homographic transformation is $$X_n={ay_n-b\over y_n-1},\eqdef\tdec$$ where the constants $a,b ~(a\neq b)$ are related by the constraint $a^2+ab+b^2-a-b=0$. The corresponding invariant, obtained for $A=-(a+b)$ and $\kappa=0$, is $$K={(ay_n-b)(ay_{n-1}-b)(y_n-1)(y_{n-1}-1)\over(by_ny_{n-1}-a)(y_ny_{n-1}-1)},\eqdef\qunu$$ leading to the mapping $$\left(y_ny_{n+1}-a/b\over y_ny_{n+1}-1\right)\left(y_ny_{n-1}-a/b\over y_ny_{n-1}-1\right)={y_n-a^2/b^2\over y_n-1}.\eqdef\qduo$$ An additive mapping can be similarly obtained by taking $$X_n={2\over3}{y_n+z\over y_n},\eqdef\qcua$$ $(z\neq0)$ with $A=-4/3$ and $\kappa=0$, which leads to the invariant $$K={y_ny_{n-1}(y_n+z)(y_{n-1}+z)\over(y_n+y_{n-1}-z)(y_n+y_{n-1})},\eqdef\qqui$$ and the mapping $$\left(y_n+y_{n+1}-z\over y_n+y_{n+1}\right)\left(y_n+y_{n-1}-z\over y_n+y_{n-1}\right)={y_n-2z\over y_n}.\eqdef\qsex$$ Note that a different choice for (), $X_n=(2y_n-3z)/(3y_n-3z/2)$, leads to a mapping dual to () where the right-hand side is $(y_n-z/2)/(y_n+3z/2)$.
The deautonomisation of () and () has been performed in \[\] where we have shown that the resulting discrete Painlevé equations are related to the affine Weyl group E$_7^{(1)}$, the periodicities of their coefficients being 3 and 5. The period 2 behaviour that (compared to the original mapping) has disappeared from the parameters, is related to the singular fibre $ \{uv=1\}\cup\{b\;u v = a\}$ for $K=\infty$ that is apparent on the canonical form of the invariant () (the components of which are obviously interchanged by the mapping ()) or to the singular fibre $\{u+v=0\} \cup \{u+v=z\}$ for the invariant () of (), again for $K=\infty$. Note that, as before, although the components of these fibres depend on the parameter $z$, their intersections do not. The singular fibre for equation () intersects in two different points and is therefore of type A$_1^{(1)}$, whereas that for () is of type A$_1^{(1)*}$, as was to be expected.
Further choices for the homographic transformations do exist. Obviously when $A=4$ the homography () takes us back to the additive E$_8^{(1)}$ mapping we started with. A multiplicative mapping can be obtained by introducing the homography $$X_n={z^2+1/z^2\over z^2-1+1/z^2}{y_n+z^5+1/z^5\over y_n+z+1/z},\eqdef\qhep$$ and the values $A=(z^2+1/z^2)^2/(z^2-1+1/z^2)$, $\kappa=-(z^2+1/z^2)(z+1/z)^4/(z^2-1+1/z^2)$. The corresponding invariant is $$K={(y_n+z+1/z)(y_{n-1}+z+1/z)(y_n+z^5+1/z^5)(y_{n-1}+z^5+1/z^5)\over y_n^2+y_{n-1}^2-(z^2+1/z^2)y_ny_{n-1}+(z^2-1/z^2)^2},\eqdef\qoct$$ and the mapping has the form $${(y_n-z^2y_{n+1})(y_n-z^2y_{n-1})-(z^4-1)^2\over (z^2y_n-y_{n+1})(z^2y_n-y_{n-1})-(z^4-1)^2/z^4}={y_n+z^4(z^3+1/z^3)\over z^4y_n+z^3+1/z^3}.\eqdef\qnov$$ By deautonomising (as we did in \[\]) we obtain a multiplicative discrete Painlevé equation with dependence () in the coefficients, associated to the affine Weyl group E$_8^{(1)}$.
At this point it is worth summarising the results we obtained so far. In total, discarding the case $A=-1$ which leads to a periodic mapping, we have found four different cases for the value of the parameter $A$ in mapping (). When $A$ is generic, i.e. $A\neq 4, -1/2$ or $-4/3$, we found that () can be deautonomised to $q$-Painlevé equations with E$_8^{(1)}$, E$_7^{(1)}$, E$_6^{(1)}$ and A$_4^{(1)}$ symmetries (the deautonomisations of equations (), (), () and () respectively). For such generic values of $A$, the QRT-map () is therefore associated with a rational elliptic surface with an A$_4^{(1)}$ A$_2^{(1)}$ A$_1^{(1)}$ 2A$_0^{(1)*}$ configuration of singular fibres. When $A=4$, the equation with E$_8^{(1)}$ symmetry is of additive type (equation () with ()) and the corresponding configuration of singular fibres on the elliptic surface for () becomes A$_4^{(1)}$ A$_2^{(1)}$ A$_1^{(1)}$ A$_0^{(1)**}$. On the other hand, when $A=-4/3$, the multiplicative equation with E$_8^{(1)}$ symmetry still exists but the E$_7^{(1)}$ one is now additive (equation ()), which corresponds to a A$_4^{(1)}$ A$_2^{(1)}$ A$_1^{(1)*}$ A$_0^{(1)*}$ configuration on the elliptic surface. The final configuration is A$_4^{(1)}$ A$_2^{(1)*}$ A$_1^{(1)}$ A$_0^{(1)*}$, obtained for $A=-1/2$, for which the equation with E$_6^{(1)}$ symmetry becomes additive (equation ()). These four types of configurations exhaust all possibilities for such a surface, as can be verified in Persson’s list \[\] and as has been explained in great detail in the monograph of Duistermaat, chapter 11.4 \[\].
Obviously an elliptic discrete Painlevé equations can also be constructed, following the same procedure we presented in section 2. We shall not go here into these details. The calculations are elementary and the interested readers can repeat them for themselves.
Just as in the case of the periods 2 and 7 mapping we can consider double-step evolutions. We start from (), solve for $X_n$ and, after a downshift of the indices, we obtain the invariant $$K={(X_nX_{n-2}-aX_n-1+a^2)(X_nX_{n-2}-aX_{n-2}-1+a^2)\over X_nX_{n-2}-1},\eqdef\qdec$$ where we have taken $A=-a^2$ and have rescaled $X$ so as to absorb a factor of $a$. The resulting mapping is $$(X_nX_{n+2}-1)(X_nX_{n-2}-1)= a^2(1-aX_n).\eqdef\punu$$ Its singularity patterns are $\{1/a,a,\infty,0\}$, its mirror image $\{0,\infty,a,1/a\}$ and the cyclic pattern of length five $\{f,\infty,0, \infty, f',1/f', \infty, 0, \cdots\}$.
In order to deautonomise () we rewrite it as $(X_nX_{n+2}-1)(X_nX_{n-2}-1)=b_n(1-a_nX_n)$. Using the singularity confinement criterion we find that $b_n=da_{n+1}a_{n-1}$, where $d$ is a constant, while $a_n$ obeys the relation $a_{n+4}a_n=a_{n+3}a_{n+1}$ the solution of which is $\log a_n=\alpha n+\beta+\phi_3(n)$. This result was first obtained in \[\], equation 124. The period 2 that was present in the parameter dependence in the original mapping (), subject to (), has disappeared here because of the doubling of the step. Similarly the period 5 has disappeared because this deautonomisation involves the A$_4^{(1)}$ singular fibre that underlies the cyclic pattern we found for ().
Introducing the invariant associated to the autonomous limit of () keeping the constant $d$ and putting $h=a^2d$ we find $$K={X_n^2X_{n-2}^2-a X_nX_{n-2}(X_n+X_{n-2})+a(1-h)(X_n+X_{n-2})+a^2h+h-1\over X_nX_{n-2}-1}.\eqdef\aduo$$ Using this invariant we can now derive the double-step equivalent of () and (). Starting from the additive equation we introduce the homography $$X_n=a{y_n-z\over y_n+z},\eqdef\bunu$$ and, under the constraint $h=1-a^2$, we obtain, for $\kappa=0$, the invariant $$K={(y_ny_{n-2}-mz(y_n+y_{n-2})+z^2)(y_n+z)(y_{n-2}+z)\over y_n+y_{n-2}},\eqdef\bduo$$ where $m=(a^2+1)/(a^2-1)$. The corresponding mapping is $$(y_n+y_{n+2})(y_n+y_{n-2})={(y_n+z)(y_n-z)^2\over y_n-mz}.\eqdef\btre$$ The deautonomisation of this mapping was presented in \[\], equation (35), which has coefficients with period five as well as one free parameter.
Next we turn to the multiplicative equation. We introduce the homography $$X_n=a{y_n+1/c\over y_n+c},\eqdef\bcua$$ and provided we set $h= 1-a^2/c^2$ we find the invariant (with $\kappa=a^2 (1-1/c^2)$) $$K={(c^2 (a^2-1) y_n y_{n-2} + c (a^2-c^2) (y_n + y_{n-2}) + a^2 -c^4)(y_n+c)(y_{n-2}+c)\over y_ny_{n-2}-1}.\eqdef\bqui$$ The multiplicative mapping is now $$(y_ny_{n+2}-1)(y_ny_{n-2}-1)={(1+y_n/c)(1+cy_n)^2\over 1+c \mu y_n},\eqdef\bsex$$ where we have introduced the auxiliary quantity $\mu=(a^2-1)/(a^2-c^2)$. We have obtained the deautonomisation of this mapping in \[\], equation (51), which again has coefficients with period five as well as one free parameter.
In order to obtain double step equation starting for the E$_7^{(1)}$-associated ones we start from the multiplicative equation (). (Using the invariant () would not have led to a different result since a constraint for the obtention of the E$_7^{(1)}$-associated equations is $d=1$). Solving () for $y_n$ we find from () the invariant $$K={(y_{n-2}-1)(y_n-1)\big((z+1)y_{n-2}y_n-(1+z^2)(y_{n-2}+y_n)+z\big)\over(zy_n-y_{n-2})(zy_{n-2}-y_n)},\eqdef\pduo$$ where we have taken $a/b=z$. The corresponding mapping is $$\left({zy_{n+2}-y_n\over y_{n+2}-zy_n}\right)\left({zy_{n-2}-y_n\over y_{n-2}-zy_n}\right)={y_n^2-(1+z^2)y_n+z^2\over (zy_n-1)^2},\eqdef\ptre$$ again given in (VI$'$) form. Its deautonomisation, in the canonical form (VI), was given in \[\]. As expected, the same procedure applied to the additive-type mapping () leads to a (V$'$) form $$\left({y_{n+2}-y_n+z\over y_{n+2}-y_n-z}\right)\left({y_{n-2}-y_n+z\over y_{n-2}-y_n-z}\right)={y_n(y_n-2z)\over (y_n+z)^2}.\eqdef\pcua$$ In order to obtain the double-step equations related to E$_8^{(1)}$ group we start from the invariant (), obtain the double step mapping and deautonomise it. We find that equation (), which corresponds to case 3.1 in \[\], is related, once we consider the double step evolution, to case 4.3.4 of the same paper. By performing a similar analysis we can obtain a multiplicative double step equation and, in fact, also an elliptic one.
It turns out that for the present system it is also possible to consider a triple step evolution. Starting from the invariant () we obtain $$K={(AX_nX_{n-3}-A-1)(AX_nX_{n-3}-X_n-X_{n-3}-A-2)\over X_nX_{n-3}-1},\eqdef\triena$$ where we have absorbed one factor of $A$ in $X$ so as to bring the denominator to a canonical form. The corresponding mapping is $$(X_nX_{n-3}-1)(X_nX_{n+3}-1)={1\over A^2}{(X_n+1)^2\over X_n-1/A}.\eqdef\tridyo$$ The deautonomisation of this mapping leads to a right-hand side of the form $az_nz_{n+1}(X_n^2-(b+1/b)X_n+1)/(X_n-z_n)$ where $\log z_n=\alpha n+\beta+\gamma(-1)^n$ and $a,b$ are constant. The form of its left-hand side notwithstanding, this mapping is indeed in A$_4^{(1)}$, a result obtained in \[\]. The autonomous form we should therefore work with is $$(X_nX_{n-3}-1)(X_nX_{n+3}-1)=a{(X_n-b)(X_n-1/b)\over X-c},\eqdef\tridyo$$ corresponding to the invariant $$K={X_n^2X_{n-3}^2-cX_nX_{n-3}(X_n+X_{n-3})+(a+c)(X_n+X_{n-3})-a(b+1/b)-1\over X_nX_{n-3}-1}.\eqdef\tritri$$ By implementing the proper homographic transformation we can now obtain from () a mapping which, when deautonomised, leads to an equation associated to the affine Weyl group E$_7^{(1)}$. We shall not go into all the details but just give the corresponding multiplicative discrete Painlevé equation. It has the form $$\displaylines{\left({y_ny_{n+3}- z_{n-1} z_{n} z_{n+1} z_{n+2} z_{n+3} z_{n+4}\over y_ny_{n+3}-1}\right)\left({y_ny_{n-3}- z_{n-4} z_{n-3} z_{n-2} z_{n-1} z_{n} z_{n+1}\over y_ny_{n-3}-1}\right)
\hfill\cr\hfill= {(y_n- z_{n} z_{n+1} z_{n+4})(y_n- z_{n-4} z_{n-1} z_{n})(y_n- z_{n-2} z_{n-1} z_{n} z_{n+1} z_{n+2})\over (y_n-z_n ) (y_n-1/z_{n+3}) (y_n-1/z_{n-3}) }, \quad\eqdisp\trites \cr}$$ where $\log z_n=\alpha n+\beta+\phi_3(n)+\phi_5(n)$. Note that since the evolution takes place with a triple step the $\phi_3(n)$ term could have been neglected in $z_n$, to be replaced by two constants in the appropriate places in the equation. Similarly we can find the corresponding equation of additive type $$\displaylines{\left({y_n+y_{n+3}\!-\!z_{n-1}\!-\!z_{n}\!-\!z_{n+1}\!-\!z_{n+2}\!-\!z_{n+3}\!-\!z_{n+4} \over y_n+y_{n+3}} \right)\left({y_n+y_{n-3}\!-\!z_{n-4}\!-\!z_{n-3}\!-\!z_{n-2}\!-\!z_{n-1}\!-\!z_{n}\!-\!z_{n+1}\over y_n+y_{n-3}}\right)\hfill\cr\hfill
= {(y_n-z_{n}-z_{n+1}-z_{n+4})(y_n-z_{n-4}-z_{n-1}-z_{n})(y_n-z_{n-2}-z_{n-1}-z_{n}-z_{n+1}-z_{n+2})\over (y_n-z_n ) (y_n+z_{n+3}) (y_n+z_{n-3}) }, \quad\eqdisp\tripen \cr}$$ where $z_n=\alpha n+\beta+\phi_3(n)+\phi_5(n)$, and the same remark on $\phi_3(n)$ applies here. Curiously these equations do not seem to have been derived before.
Triple-step equations associated to E$_6^{(1)}$ can also be obtained. In this case the extension of the invariant to the form () is not mandatory, the form () being sufficient. We give directly the final result for the multiplicative equation $$\left({x_{n+3}-z_{n+1}x_n\over x_{n+3}z_{n+2}-x_n}\right)\left({x_{n-3}-z_{n-1}x_n\over x_{n-3}z_{n-2}-x_n}\right)={(x_nz_{n+1}-1)(x_nz_{n-1}-1)\over (x_n-1)(x_n-z_n)},\eqdef\trihex$$ where $\log z_n=\alpha n+\beta+\phi_2(n)+\phi_5(n)$ and the additive one $$\left({x_{n+3}-z_{n+1}-x_n\over x_{n+3}-z_{n+2}-x_n}\right)\left({x_{n-3}-z_{n-1}-x_n\over x_{n-3}-z_{n-2}-x_n}\right) ={(x_n+z_{n+1})(x_n+z_{n-1})\over x_n(x_n-z_n)},\eqdef\trihep$$ with $z_n=\alpha n+\beta+\phi_2(n)+\phi_5(n)$. Again, these two equations do not seem to have been previously derived. Obtaining triple-step equations associated to E$_8^{(1)}$ is also possible but it turns out that the resulting additive equation is the case 5.2.5 of () and thus need not be presented explicitly here. (Obviously, once the result for the additive equation are obtained, transcribing it to the multiplicative and elliptic cases is elementary, as explained in the introduction).
5\. [The asymmetric equation with period 8]{}
The discrete Painlevé equation () with period 8 (as given by ()) was first derived in \[\]. In \[\] it was again obtained during the complete classification of trihomographic equations associated to the group E$_8^{(1)}$. It can only be expressed in asymmetric (in the QRT parlance) form: $${x_{n+1}-4t_n^2\over x_{n+1}}{x_{n}-4t_n^2\over x_{n}}{y_{n}-t_n^2\over y_{n}-9t_n^2}=1\eqdaf\pqui$$ $${y_{n}-(3t_n-2\alpha)^2\over y_{n}-t_n^2}{y_{n-1}-(3t_n-\alpha)^2\over y_{n-1}-(t_n-\alpha)^2}{x_{n}\over x_{n}-(4t_n-2\alpha)^2}=1.\eqno(\pqui b)$$ Four singularity patterns do exist: $\{x_n=0,y_n=t_n^2\}$, $\{y_n=t_n^2,x_{n+1}=0\}$, $\{y_{n-1}=9t_{n-1}^2, x_n=4t_{n-1}^2, y_n=(t_{n-1}-\alpha)^2, x_{n+1}=4\alpha^2, y_{n+1}=(t_{n+2}+\alpha)^2, x_{n+2}=4t_{n+2}^2, y_{n+2}=9t_{n+2}^2\}$ and $\{x_{n-2}=(4t_{n-2}-2\alpha)^2, y_{n-2}=(3t_{n-2}-2\alpha)^2, x_{n-1}=(2t_{n-2}-2\alpha)^2, y_{n-1}=(t_{n-2}-3\alpha)^2, x_n=16\alpha^2, y_n=(t_{n+2}+2\alpha)^2, x_{n+1}=4t_{n+2}^2, y_{n+1}=(3t_{n+2}-\alpha)^2, x_{n+2}=(4t_{n+2}-2\alpha)^2\}$. Next we autonomise () obtaining $${x_{n+1}-4\over x_{n+1}}{x_{n}-4\over x_{n}}{y_{n}-1\over y_{n}-9}=1,\eqdaf\psex$$ $${y_{n}-9\over y_{n}-1}{y_{n-1}-9\over y_{n-1}-1}{x_{n}\over x_{n}-16}=1.\eqno(\psex b).$$ First let us point out that since $y$ and $x$ appear homographically in (a) and (b) we can eliminate either of them and obtain an equation for the other variable. We find thus $${(x_{n+1}-x_n-4)(x_{n-1}-x_n-4)+16x_n\over x_{n+1}-2x_n+x_{n-1}-8}=-{x_n(x_n+20)\over2(x_n+2)}\eqdef\atre$$ and $${(y_{n+1}-y_n-4)(y_{n-1}-y_n-4)+16y_n\over y_{n+1}-2y_n+y_{n-1}-8}=-{y_n^2+18y_n-27\over 2 y_n}.\eqdef\acua$$ They are the autonomous limits of equations 4.2.3 and 4.3.2 of E$_8^{(1)}$ type derived in \[\]. Their deautonomisation was presented in full detail in that paper.
Going back to () we can introduce two new variables $X$ and $Y$ as $$X_n={x_n-4\over x_n}, \quad Y_n={y_n-9\over y_n-1},\eqdef\phep$$ and rewrite () as $$X_nX_{n+1}=Y_n,\qquad Y_nY_{n-1}=AX_n+B,\eqdef\miu$$ where $A=4$ and $B=-3$, to obtain the invariant $$K={X_nY_n^2+AX_n^2+BX_n+AY_n\over X_nY_n}.\eqdef\eunu$$ Equation () can be easily deautonomised, leading to $\log A_n=\alpha n+\beta$ and $B$ a constant and an equation in A$_1^{(1)}$. When we view equation () as a mapping on ${\Bbb P}^1\times{\Bbb P}^1$, $(u,v)\to(v/u,\;A/u + B/v)$ with $u=X_n$ and $v=Y_n$, it is easily verified that there are only 3 curves that contract to a point and give rise to a singularity: $\{A v + B u =0\} ,~\{u=0\}$ and $\{v=0\}$. The first curve is the only one that gives rise to a genuinely confined singularity pattern $$\{A v + B u =0\}\to (-{B\over A}, 0)\to (0, \infty) \to (\infty^2, \infty) \to (0,0) \to \{u=-{B\over A}\},\eqdef\uvpatti$$ which corresponds exactly to the length-nine pattern we found for the mapping (), $$\{X_n= -{B\over A}, Y_n= X_{n+1}=0, Y_{n+1}=\infty, X_{n+2}=\infty^2, Y_{n+2}=\infty, X_{n+3}=Y_{n+3}=0, X_{n+4}=-{B\over A}\}.\eqdef\uvpattii$$ (The notation $\infty^2$ is shorthand for the following: had we introduced a small quantity $\epsilon$, assuming that $X_{n+1}=\epsilon$, we would have found that $X_{n+2}$ is of the order of $1/\epsilon^2$). The two other curves turn out to be part of a cyclic singularity pattern of length 8: $$\cdots\to\{u=0\}\to(\infty,\infty)\to\{v=0\}\to(0,\infty)\to(\infty^2,\infty)\to(0,0)\to\{v=\infty\}\to\{u=\infty\}\to\cdots,\eqdef\uvpattiii$$ which is nothing but the footprint on ${\Bbb P}^1\times{\Bbb P}^1$ of the A$_7^{(1)}$ singular fibre that exists for () at $K=\infty$. It is easily verified that the three remaining patterns we found for () are contained in this cyclic pattern.
Next we look for canonical forms starting from the invariant () by introducing two different homographies for $X$ and $Y$. Obviously the homography () would bring us back to our starting point. However another possibility exists, leading to a multiplicative mapping. Introducing the homography $$X_n=a{x_n+z^2+1/z^2\over x_n+2}, \quad Y_n=a^2{y_n+z^3+1/z^3\over y_n+z+1/z},\eqdef\eduo$$ and parametrising $A=a^3(z+1/z)^2$, $B=-a^4(z^2+1+1/z^2)$, where the parameter $a$ satisfies the equation $a^4-Aa-B=0$. We then obtain, from () with $\kappa=-a^2(z^2+4+1/z^2)$, the invariant $$K={(x_n+z^2+1/z^2)(x_n+2)(y_n+z^3+1/z^3)(y_n+z+1/z)\over (x^2+y^2)z^2-xyz(z^2+1)+(z^2-1)^2}.\eqdef\etre$$ From the conservation of $K$ we find the system of equations $${x_{n+1}+z^2+1/z^2\over x_{n+1}+2}{x_{n}+z^2+1/z^2\over x_{n}+2}{y_n+z+1/z\over y_n+z^3+1/z^3}=1,\eqdaf\ecua$$ $${y_n+z+1/z\over y_n+z^3+1/z^3}{y_{n-1}+z+1/z\over y_{n-1}+z^3+1/z^3}{x_{n}+z^4+1/z^4\over x_{n}+2}=1.\eqno(\ecua b).$$ Since these equations have a trihomographic form we can eliminate either $x$ or $y$ and obtain an equation for a single variable. Eliminating $y$ from (a) and (b) we find $${(x_n-z^2x_{n+1})(x_n-z^2x_{n-1})-(z^4-1)^2\over (z^2x_n-x_{n+1})(z^2x_n-x_{n-1})-(z^4-1)^2/z^4}=z^4{x_n^2+x_n(z^2+1)^2-z^8+2z^6+2z^4+2z^2-1\over x_n^2z^8+x_nz^{8}(z^2+1)^2-z^8+2z^6+2z^4+2z^2-1}.\eqdef\equi$$ Similarly, eliminating $x$ we obtain $${(y_n-z^2y_{n+1})(y_n-z^2y_{n-1})-(z^4-1)^2\over (z^2y_n-y_{n+1})(z^2y_n-y_{n-1})-(z^4-1)^2/z^4}=z^4{y_n^2+y_n(3z^3+1/z)-z^8+3z^6+3z^2-1\over y_n^2z^8+y_nz^5(z^4+3)-z^8+3z^6+3z^2-1}.\eqdef\esex$$ Equations of the form () and () where first derived in \[\] where they have been deautonomised albeit only as far as the secular dependence is concerned. Under a complete deautonomisation we expect these two equations to be the multiplicative analogues of the E$_8^{(1)}$ associated equations 4.2.3 and 4.3.2 of \[\].
In Persson’s classification \[\] there are two types of configurations that only have an A$_7^{(1)}$ fibre as a reducible fibre, A$_7^{(1)}$ 4A$_0^{(1)*}$ and A$_7^{(1)}$ A$_0^{(1)**}$ 2A$_0^{(1)*}$. The first case corresponds to the generic situation where the values of $A$ and $B$ are such that $A^4/B^3\neq-256/27$. When this is true, the equation $a^4-Aa-B=0$ has generically four distinct roots giving rise to four different values of $\kappa$. In this case we have two multiplicative Painlevé equations, one of A$_1^{(1)}$ and one of E$_8^{(1)}$ type. The second case corresponds precisely to $A^4/B^3=-256/27$. (Notice that Duistermaat, who has also studied this mapping, in chapter 11.7.1 of \[\], gives a constraint in the form $\delta^5=-8/27$. Unfortunately this value is wrong and our analysis shows that the correct value is $\delta^5=256/27$). Under this constraint we can still have a meaningful homography, like (), to a multiplicative equation (in E$_8^{(1)}$ when deautonomised) for values of $z$ that satisfy $27z^8+68z^6+98z^4+68z^2+27=0$, which leads to two distinct values for $\kappa$ (given the invariance of the latter under $z\to-z$ and $z\to1/z$). On the other hand, the scaling freedom which exists at the level of the constraint $A^4/B^3=-256/27$ translates itself to a mere scaling freedom in equation () an thus, up to a rescaling of $X$ and $Y$, the only homography leading to an additive equation of E$_8^{(1)}$ type is ().
A different approach is also possible, but it leads to the same conclusions. Going back to the mapping () for $X$ and $Y$ we can eliminate $Y$ and find for $X$ the mapping $$X_{n+1}X_{n-1}={A\over X_n}+{B\over X_n^2}.\eqdef\poct$$ By rescaling the variable $X$ we can always put $B=-1$ which corresponds to the customary form of (). The singularity patterns of () are $\{-B/A,0,\infty^2,0,-B/A\}$ and the cyclic one $\{0,\infty^2,0,f,\infty,0,\infty, f'\}$ of length 8, which are easily read off from the patterns () and () respectively. The deautonomisation of () leads to $\log A_n=\alpha n+\beta$, (with $B=-1$), which means that the corresponding discrete Painlevé equation has just one degree of freedom and is associated to A$_1^{(1)}$.
The invariant of () is $$K={X_{n}^2X_{n-1}^2+A(X_{n}+X_{n-1})+B\over X_{n}X_{n-1}}.\eqdef\pnov$$ This can be the starting point for the derivation of other canonical forms to be obtained through homographic transformations of the dependent variable. We shall not go into all the details but no such possibility exists apart from forms associated to the group E$_8^{(1)}$ and in fact we find precisely the result obtained above working with the asymmetric form, i.e. equation ().
Similarly, eliminating $X$ from the mapping for () we obtain $$(Y_nY_{n+1}-B)(Y_nY_{n-1}-B)=A^2Y_n,\eqdef\stre$$ where we can again scale $B$ to 1 by redefining the variable $Y$. The singularity patterns of () are $\{0,\infty,\infty,0\}$ and the length-8 cyclic one $\{\infty,0,\infty,\infty,0,\infty,f,f'\}$, which can again be read off directly from () and (), while its deautonomisation is of course $\log A_n=\alpha n+\beta$, (with $B=1$) \[\]. The invariant of () is $$K={(Y_nY_{n-1}-B)(Y_n+Y_{n-1})+A^2\over Y_nY_{n-1}-B}\eqdef\scua$$ As expected, starting from () we find again the result obtained above, namely equation (). Let us point out here that the asymmetric equation () is just the Miura relation between () and (), a result first obtained in \[\].
6\. [Conclusions]{}
In this paper we addressed the question of the systematic construction of discrete Painlevé equations in the degeneration cascade of the affine Weyl group E$_8^{(1)}$. Our starting point was the list of discrete Painlevé equations in E$_8^{(1)}$ derived in trihomographic form in previous works of ours. In order to illustrate our method, while keeping the paper at a manageable length, we chose to work with three systems we selected among a dozen candidates. The first step in our method was to consider the autonomous limit of a given equation and, using the results of \[\], to write the simplest possible mapping in the cascade as well as its invariant. Next, introducing homographic transformations, we obtained all possible canonical forms of the said invariant, the corresponding mapping ensuing automatically. To check that our approach does populate the degeneration cascade without omissions, we compared our results to the classification of singular fibres in \[\], obtaining a perfect agreement.
Our method is an alternative to that of Carstea, Dzhamay and Takenawa, who explicitly calculate all singular fibres for a given QRT mapping and then proceed to its degeneration (without however taking into account any possible additive reductions that might arise for special parameter values in the mapping). Another advantage of our method is that once a mapping is obtained in canonical form, its deautonomisation, leading to a discrete Painlevé equation, is quite straightforward. As it turned out, in most cases presented here this last step was not even necessary since we could find a relation to a previously derived result of ours. Still some new discrete Painlevé equations did make their appearance.
This is a first exploratory work, intended to showcase our method and the extreme usefulness of the classification of the QRT canonical forms. We intend to return to our study of the degeneration cascade of E$_8^{(1)}$ in a future work of ours.
RW would like to acknowledge support from the Japan Society for the Promotion of Science (JSPS), through the the JSPS grant: KAKENHI grant number 15K04893.
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|
---
author:
- Shihao Zou
- Xinxin Zuo
- Yiming Qian
- Sen Wang
- Chi Xu
- Minglun Gong
- Li Cheng
bibliography:
- 'egbib.bib'
title: Polarization Human Shape and Pose Dataset
---
20SubNumber[0]{}
Introduction
============
Polarization images are known to be able to capture polarized reflected lights that preserve rich geometric cues of an object, which has motivated its recent applications in reconstructing detailed surface normal of the objects of interest. Meanwhile, inspired by the recent breakthroughs in human shape estimation from a single color image, we attempt to investigate the new question of whether the geometric cues from polarization camera could be leveraged in estimating detailed human body shapes. This has led to the curation of Polarization Human Shape and Pose Dataset (PHSPD)[^1], our home-grown polarization image dataset of various human shapes and poses.
Our PHSPD dataset synchronizes four cameras, one polarization camera and three Kinects v2 in three different views (each Kinect v2 has a depth and a color camera). The depth and color images from three-view Kinects v2 are used to get more accurate annotations of shape and pose in 3D space. We propose an economic yet effective approach to annotating shape and pose in 3D space. Compared with Human3.6M [@human36m] that uses expensive Motion Capture system to annotate human poses, we do not require subjects to wear special tight clothes and a lot of sensors, which makes the acquired images restrictive and impractical.
We show some of our annotated shapes and poses in Fig. \[fig:multiview-demo\], where the shapes are rendered on the image plane and the poses are shown in 3D space. We can see that our annotated shapes and poses align well with the subjects in the image plane from four camera views.
![The figure shows our annotated shapes and poses. The first column is the polarization image for reference. The second to the fifth columns show the annotated shape rendered on the polarization image and three-view color images. The sixth column shows the annotated pose in 3D space.[]{data-label="fig:multiview-demo"}](fig/supplement/fig2_1.pdf "fig:"){width="\columnwidth"} ![The figure shows our annotated shapes and poses. The first column is the polarization image for reference. The second to the fifth columns show the annotated shape rendered on the polarization image and three-view color images. The sixth column shows the annotated pose in 3D space.[]{data-label="fig:multiview-demo"}](fig/supplement/fig2_2.pdf "fig:"){width="\columnwidth"} ![The figure shows our annotated shapes and poses. The first column is the polarization image for reference. The second to the fifth columns show the annotated shape rendered on the polarization image and three-view color images. The sixth column shows the annotated pose in 3D space.[]{data-label="fig:multiview-demo"}](fig/supplement/fig2_3.pdf "fig:"){width="\columnwidth"} ![The figure shows our annotated shapes and poses. The first column is the polarization image for reference. The second to the fifth columns show the annotated shape rendered on the polarization image and three-view color images. The sixth column shows the annotated pose in 3D space.[]{data-label="fig:multiview-demo"}](fig/supplement/fig2_4.pdf "fig:"){width="\columnwidth"} ![The figure shows our annotated shapes and poses. The first column is the polarization image for reference. The second to the fifth columns show the annotated shape rendered on the polarization image and three-view color images. The sixth column shows the annotated pose in 3D space.[]{data-label="fig:multiview-demo"}](fig/supplement/fig2_5.pdf "fig:"){width="\columnwidth"}
Data Acquisition
================
Our acquisition system synchronizes four cameras, one polarization camera and three Kinects V2 in three different views (each Kinect v2 has a depth and a color camera). The layout is shown in Fig. \[fig:camera-layout\]. The main task in data acquisition is multi-camera synchronization. As one desktop can only control one Kinect v2, we develop a soft synchronization method. Specifically, each camera was connected with a desktop (the desktop with the polarization camera is the master and the other three ones with three Kinects are clients). We use socket to send message to each desktop. After receiving certain message, each client will capture the most recent frame from the Kinect into the desktop memory. At the same time, the master desktop sends a software trigger to the polarization camera to capture one frame into the buffer. Practically, our synchronization system can be as fast as 15 frames per second (fps). Fig. \[fig:camera-layout\] shows the synchronization performance of the system that we develop. We let a bag fall down and compare the position of the bag in the same frame from four views. We can find that the positions of the bag captured by four cameras are almost the same in terms of its distance to the ground.
![Left figure: the layout of our multi-camera system. Three Kinects are placed around a circle of motion area with one polarization camera. Right figure: the synchronization result of our multi-camera system. The same frame of the three-view color images and one-view polarization image are displayed. Note that the layout of our multi-camera system has been changed to the left figure, but other settings are the same.[]{data-label="fig:camera-layout"}](fig/supplement/camera_config.png "fig:"){width="0.39\columnwidth"} ![Left figure: the layout of our multi-camera system. Three Kinects are placed around a circle of motion area with one polarization camera. Right figure: the synchronization result of our multi-camera system. The same frame of the three-view color images and one-view polarization image are displayed. Note that the layout of our multi-camera system has been changed to the left figure, but other settings are the same.[]{data-label="fig:camera-layout"}](fig/supplement/synchronization.png "fig:"){width="0.59\columnwidth"}
group \# actions
---------- ------------------------------------------------------------------------
1 warming-up, walking, running, jumping, drinking, lifting dumbbells
2 sitting, eating, driving, reading, phoning, waiting
3 presenting, boxing, posing, throwing, greeting, hugging, shaking hands
: The table displays the actions in each group. Subjects are required to do each group of actions for four times, but the order of the actions each time is random.[]{data-label="tab:dataset-action"}
gender
------- ---------- -------- -------- -------------
1 female 22561 22241 320 (1.4%)
2 male 24325 24186 139 (0.5%)
3 male 23918 23470 448 (1.8%)
4 male 24242 23906 336 (1.4%)
5 male 24823 23430 1393 (5.6%)
6 male 24032 23523 509 (2.1%)
7 female 22598 22362 236 (1.0%)
8 male 23965 23459 506 (2.1%)
9 male 24712 24556 156 (0.6%)
10 female 24040 23581 459 (1.9%)
11 male 24303 23795 508 (2.1%)
12 male 24355 23603 752 (3.1%)
total - 287874 282112 5762 (2.0%)
: The table shows the detail number of frames for each subject and also the number of frames that have SMPL shape and 3D joint annotations.[]{data-label="tab:dataset-details"}
Our dataset has 12 subjects, 9 males and 3 females. Each subject is required to do 3 different groups of actions (18 different actions in total) for 4 times plus one free-style group. Details are shown in Tab. \[tab:dataset-action\]. So each subject has 13 short videos and the total number of frames for each subject is around 22K. Overall, our dataset has 287K frames with each frame including one polarization image, three color and three depth images. Quantitative details of our dataset are shown in Tab. \[tab:dataset-details\].
Annotation Process
==================
Shape and Pose Representation
-----------------------------
We represent the 3D human body shape (mesh) using SMPL model [@loper2015smpl], which is a differentiable function $\mathcal{M}(\boldsymbol{\beta}, \boldsymbol{\theta})\in\mathbb{R}^{6890\times3}$ that outputs a triangular mesh with 6890 vertices given 82 parameters $[\boldsymbol{\beta}, \boldsymbol{\theta}]$. The shape parameter $\boldsymbol{\beta}\in\mathbb{R}^{10}$ is the linear coefficients of a PCA shape space that mainly determines individual body features such height, weight and body proportions. The shape space is learned from a large dataset of body scans [@loper2015smpl]. The pose parameter $\boldsymbol{\theta}\in\mathbb{R}^{72}$ mainly describes the articulated pose, which consists of one global rotation of the body and the relative rotations of 23 joints in axis-angle representation. The final body mesh is produced by first applying shape-dependent and pose-dependent deformations to the template body, then using forward-kinematics to articulate the body and finally deforming the surface with linear blend skinning. The 3D joint positions $\mathbf{J}\in\mathbb{R}^{24\times3}$ can be obtained by linear regression from the output mesh vertices.
Annotation of Shape and Pose
----------------------------
The reason that we use multi-camera system to acquire image data is that multi-camera system provides much more information than a single-camera system. So the annotation of human shape and pose in 3D is more reliable.
After camera calibration and plane segmentation of human body in depth images, we have a point cloud of human surface fused from three-view depth image, and also noisy 3D pose by Kinect SDK at hand. The annotation of SMPL human shape and 3D joint position has three main steps as follows.
### Initial guess of 3D pose
As the 3D pose given by Kinect SDK is noisy, we use the predicted 2D pose by OpenPose [@openpose] as the criterion to decide which joint position given by Kinect SDK is correct. We select 14 aligned joints that both Openpose and Kinect have. For joint $i$ in view $j=\{1,2,3\}$, 2D joint position by OpenPose is denoted by $(v_{ij}^o, u_{ij}^o)$, and 3D joint position by Kinect by $(x_{ij}, y_{ij}, z_{ij})$ and its projected 2D joint position by $(v_{ij}^k, u_{ij}^k)$. Since we cannot figure out which joint is detected correctly by Kinect, we use the joint position by OpenPose as the criterion to decide whether this joint is correctly estimated by Kinect, that is $$w_{ij} =
\begin{cases}
1 & \text{if}\ \sqrt{(v_{ij}^k - v_{ij}^o)^2 + (u_{ij}^k - u_{ij}^o)^2} < 50,\\
0 & \text{otherwise},
\end{cases}$$ where $50$ means the pixel distance. Then, we get the initial guess of the 3D joint position $(\hat x_i, \hat y_i, \hat z_i)$ of joint $i$ by averaging the valid positions given by three-view Kinects as $$\hat x_i = \frac{\sum_{j=1}^3 w_{ij}x_{ij}}{\sum_{j=1}^3 w_{ij}}, \hat y_i = \frac{\sum_{j=1}^3 w_{ij}y_{ij}}{\sum_{j=1}^3 w_{ij}}, \hat z_i = \frac{\sum_{j=1}^3 w_{ij}z_{ij}}{\sum_{j=1}^3 w_{ij}}.$$ If none of the three-view joint positions by Kinect is correct, we consider it as a missing joint. We discard the frame with more than 2 joints missing (14 in total).
### Fitting shape to pose
The next step is similar to SMPLify [@bogo2016keep], but instead of fitting to the 2D joints which have inherent depth ambiguity, we fit SMPL model to the initial guess of 3D pose.
### Fine-tuning shape to the point cloud
The final step is fine-tuning the shape to the point cloud of human surface so that the annotated SMPL shape parameters are more accurate. We iteratively optimize SMPL parameters by minimizing the distance between vertices of SMPL shape to their nearest point. Finally, we have the annotated SMPL shape parameters and 3D pose.
Besides, we render the boundary of SMPL shape on the image to get the mask of background, and calculate the target normal using three depth images based on [@qi2018geonet]. Although the target normal is noisy, our experiment result shows our model can still learn to predict good and smooth normal maps.
The annotation process is shown in Fig. \[fig:annoataion-demo\]. Starting from the initial guess of 3D pose, we fit SMPL shape to the initial 3D pose and further fine-tune to the point cloud of human surface. Finally, we get the annotated human shape and pose for each frame. We can find from Fig. \[fig:annoataion-demo\] that the third step is critical to make the annotated shape align better to the subject in the image in that the pint cloud of human surface gives much more information than a skeleton-based pose. So the third step can adjust the shape to improve the alignment of body parts. Besides, we also show our annotated shape on multi-view images (one polarization image and three-view color image) and the human pose in 3D coordinate space in Fig \[fig:multiview-demo\].
![The figure shows our three-step annotation process. The first column shows the initial guess of 3D pose, which is projected on the polarization image. After fitting the SMPL shape to the initial pose, we show the initial fitted shape with the point cloud of human surface (black points) in the second column and the rendered shape on the image in the third column. The fourth and fifth columns show the annotated shape after fine-tuning the shape to the point cloud of human surface. The sixth column shows the corresponding annotated 3D pose projected on the polarization image.[]{data-label="fig:annoataion-demo"}](fig/supplement/fig1_1.pdf "fig:"){width="0.9\columnwidth"} ![The figure shows our three-step annotation process. The first column shows the initial guess of 3D pose, which is projected on the polarization image. After fitting the SMPL shape to the initial pose, we show the initial fitted shape with the point cloud of human surface (black points) in the second column and the rendered shape on the image in the third column. The fourth and fifth columns show the annotated shape after fine-tuning the shape to the point cloud of human surface. The sixth column shows the corresponding annotated 3D pose projected on the polarization image.[]{data-label="fig:annoataion-demo"}](fig/supplement/fig1_2.pdf "fig:"){width="0.9\columnwidth"} ![The figure shows our three-step annotation process. The first column shows the initial guess of 3D pose, which is projected on the polarization image. After fitting the SMPL shape to the initial pose, we show the initial fitted shape with the point cloud of human surface (black points) in the second column and the rendered shape on the image in the third column. The fourth and fifth columns show the annotated shape after fine-tuning the shape to the point cloud of human surface. The sixth column shows the corresponding annotated 3D pose projected on the polarization image.[]{data-label="fig:annoataion-demo"}](fig/supplement/fig1_3.pdf "fig:"){width="0.9\columnwidth"} ![The figure shows our three-step annotation process. The first column shows the initial guess of 3D pose, which is projected on the polarization image. After fitting the SMPL shape to the initial pose, we show the initial fitted shape with the point cloud of human surface (black points) in the second column and the rendered shape on the image in the third column. The fourth and fifth columns show the annotated shape after fine-tuning the shape to the point cloud of human surface. The sixth column shows the corresponding annotated 3D pose projected on the polarization image.[]{data-label="fig:annoataion-demo"}](fig/supplement/fig1_4.pdf "fig:"){width="0.9\columnwidth"} ![The figure shows our three-step annotation process. The first column shows the initial guess of 3D pose, which is projected on the polarization image. After fitting the SMPL shape to the initial pose, we show the initial fitted shape with the point cloud of human surface (black points) in the second column and the rendered shape on the image in the third column. The fourth and fifth columns show the annotated shape after fine-tuning the shape to the point cloud of human surface. The sixth column shows the corresponding annotated 3D pose projected on the polarization image.[]{data-label="fig:annoataion-demo"}](fig/supplement/fig1_5.pdf "fig:"){width="0.9\columnwidth"}
[^1]: Our PHSPD dataset will be released soon for academic purpose only.
|
---
abstract: 'We propose new multi-dimensional atom optics that can create coherent superpositions of atomic wavepackets along three spatial directions. These tools can be used to generate light-pulse atom interferometers that are simultaneously sensitive to the three components of acceleration and rotation, and we discuss how to isolate these inertial components in a single experimental shot. We also present a new type of atomic gyroscope that is insensitive to parasitic accelerations and initial velocities. The ability to measure the full acceleration and rotation vectors with a compact, high-precision, low-bias inertial sensor could strongly impact the fields of inertial navigation, gravity gradiometry, and gyroscopy.'
author:
- 'B. Barrett'
- 'P. Cheiney'
- 'B. Battelier'
- 'F. Napolitano'
- 'P. Bouyer'
bibliography:
- 'References.bib'
title: 'Multi-Dimensional Atom Optics and Interferometry'
---
Inertial sensors based on cold atoms and light-pulse interferometry [@Borde1989; @Kasevich1991; @Cronin2009] exhibit exquisite sensitivity that could potentially revolutionize a variety of fields including geophysics and geodesy [@Carraz2014; @Menoret2018], gravitational wave detection [@Canuel2018], tests of fundamental laws and inertial navigation [@Hogan2009; @Barrett2014a]. Their state-of-the-art sensitivity and ultra-low measurement bias are particularly appropriate for long-term integration as in precision measurements [@Bouchendira2011; @Rosi2014] or space experiments [@Becker2018]. They also offer great potential for autonomous inertial navigation systems [@Jekeli2005; @Battelier2016; @Fang2016; @Cheiney2018], where the attitude and position of a moving body is determined by integrating the equations of motion.
The measurement principle of light-pulse atom interferometers (AIs) is linked to a retro-reflected laser beam that is referenced to an atomic transition. This defines a phase ruler to which the free-falling atom’s trajectory is compared [@Borde2001], in analogy to classical falling-corner-cube gravimeters [@Niebauer1995]. In general, the direction of the retro-reflected beam defines the inertially-sensitive axis of these quantum sensors. They can be sensitive to accelerations [@Peters1999; @Gillot2014; @Freier2016; @Hardman2016; @Bidel2018] and acceleration gradients [@Snadden1998; @Rosi2015; @Asenbaum2017] parallel to the effective optical wavevector $\bm{k}$, and to rotations perpendicular to the plane defined by $\bm{k} \times \bm{v}_0$ [@Gustavson1997; @Stockton2011; @Barrett2014b; @Tackmann2014; @Rakholia2014; @Dutta2016; @Yao2018], where $\bm{v}_0$ is the initial velocity of the atomic source. So far, the challenge of realizing multi-axis inertial measurements has been addressed only in a sequential manner [@Canuel2006; @Wu2017], where the direction of $\bm{k}$ was changed between measurement cycles. In this work, we propose new multi-dimensional AI geometries that are *simultaneously* sensitive to accelerations and rotations in 3D, and can discern their vector components within a single shot.
In what follows, we define a multi-dimensional AI as one where the light interaction exchanges momentum with the atomic sample along more than one spatial direction at a time. This momentum exchange is accompanied by an independent phase shift along each axis, which is imprinted on the corresponding diffracted wavepacket [@Borde2004]. This mechanism creates a unique type of atom-optical element that satisfies all the requirements of a multi-dimensional AI—enabling one to split, reflect and recombine matter waves along two or more axes simultaneously.
![(a,b) Multi-dimensional atom-optical beamsplitters realized with mutually-orthogonal pairs of independent, counter-propagating Raman beams. (c) Energy-momentum diagram showing velocity-sensitive Raman transitions associated with each pair of beams shown in (a,b). To avoid excitation of parasitic resonances, different detunings $\Delta_{\mu}$ are used for each beam pair. This leads to a small difference in the wavevectors $k_{\mu}$, which has been exaggerated for clarity.[]{data-label="fig:2D+3DAtomOptics"}](Fig1-2D+3DAtomOptics.pdf){width="48.00000%"}
We have developed a semi-classical model for 3D atom optics involving Raman transitions [@SupMat]. Figure \[fig:2D+3DAtomOptics\] shows examples of 2D and 3D atom-optical beamsplitters, where mutually-orthogonal pairs of counter-propagating Raman beams couple an atom with initial momentum $\bm{p}_0$ to two or three diffracted states moving perpendicular to one another. Along each axis $\mu = x,y,z$, two beams with wavevectors $\bm{\kappa}_{1\mu}$ and $\bm{\kappa}_{2\mu}$, and corresponding frequencies $\omega_{1\mu}$ and $\omega_{2\mu}$, excite two-photon Raman transitions [^1] between two ground states ${\left| 1 \right\rangle}$ and ${\left| 2 \right\rangle}$ separated by frequency $\omega_{21}^{\rm A}$. During this process, a momentum $\hbar \bm{k}_{\mu} = \hbar (\bm{\kappa}_{1\mu} - \bm{\kappa}_{2\mu})$ is transferred to the atom, where $\hbar$ is the reduced Planck’s constant and $\bm{k}_{\mu} \simeq 2\bm{\kappa}_{1\mu}$ is the effective Raman wavevector along axis $\mu$ [@Aspect1989; @Moler1992]. The laser frequencies $\omega_{n\mu}$ are detuned by $\Delta_\mu$ from an intermediate excited state ${\left| 3,\bm{p}_0 \right\rangle}$ as shown in \[fig:2D+3DAtomOptics\](c), such that $|\Delta_\mu|$ is large compared to the natural linewidth of the atomic transition and, for $\mu \neq \nu$, $|\Delta_{\mu} - \Delta_{\nu}|$ is much larger than the effective Rabi frequency. This second condition strongly inhibits scattering processes involving absorption along one axis and re-emission along an orthogonal one. In the region of beam overlap, an atom initially in ${\left| 1,\bm{p}_0 \right\rangle}$ undergoes 3D diffraction—splitting the wavepacket into a superposition of this undiffracted state and three orthogonal diffracted states: ${\left| 2, \bm{p}_0 + \hbar \bm{k}_{\mu} \right\rangle}$. The dynamics of this coherent 3D diffraction process are described by Rabi oscillations in an effective 4-level system: ${\left| \Psi \right\rangle} = \mathcal{C}_0 {\left| 1,0,0,0 \right\rangle} + \mathcal{C}_x {\left| 2,\hbar k_x,0,0 \right\rangle} + \mathcal{C}_y {\left| 2,0,\hbar k_y,0 \right\rangle} + \mathcal{C}_z {\left| 2,0,0,\hbar k_z \right\rangle}$, where the states are labelled by their internal energy and the photon momentum transfer along each direction. This system exhibits Rabi oscillations in the population between states, where the vector of state amplitudes $\bm{\mathcal{C}}^{\rm T} = (\mathcal{C}_0,\mathcal{C}_x, \mathcal{C}_y, \mathcal{C}_z)$ evolves according to \[C(t)\] (t) = (0). Here, $\chi_{\mu}$ is the Rabi frequency and $\delta_{\mu} \simeq (\omega_{1\mu} - \omega_{2\mu}) - \omega_{21}^{\rm A} - \delta_{\mu}^{\rm D} - \delta_{\mu}^{\rm R}$ is the two-photon detuning of each beam pair, where $\delta_{\mu}^{\rm D} = \bm{k}_{\mu} \cdot \bm{v}_0$ is the Doppler shift for initial velocity $\bm{v}_0 = \bm{p}_0/m$, $\delta_{\mu}^{\rm R} = \hbar k_{\mu}^2/2m$ is a photon recoil shift, and $m$ is the atomic mass. For the special case when $\delta_x = \delta_y = \delta_z \equiv \delta$, the effective Rabi frequency for this system can be written analytically as $\Omega_{\rm Rabi} = \frac{1}{2} \sqrt{\delta^2 + 4(|\chi_x|^2 + |\chi_y|^2 + |\chi_z|^2)}$.
We now discuss the specific case of 2D atom optics. Figure \[fig:2DAtomOptics-Rabi\](a) displays Rabi oscillations corresponding to a 2D beamsplitter, where population is transferred between atoms initially in ${\left| 1,0,0,0 \right\rangle}$ and the two diffracted states ${\left| 2,\hbar k_x,0,0 \right\rangle}$ and ${\left| 2,0,\hbar k_y,0 \right\rangle}$. A beamsplitter is achieved at an interaction time $\tau$ corresponding to a pulse area $\Omega_{\rm Rabi}\tau = \pi/2$, where 50% of the population is accumulated in the two target states. Contrary to a 1D beamsplitter, where one usually desires a 50/50 superposition of the initial and final states, here the population in the initial state is fully depleted—closely resembling 1D double-diffraction beamsplitters [@Leveque2009; @Malossi2010; @Giese2013; @Kuber2016; @Ahlers2016]. Similarly, \[fig:2DAtomOptics-Rabi\](b) shows the Rabi oscillation corresponding to a 2D mirror, where a $\pi$-pulse of duration $2\tau$ achieves 100% population transfer between ${\left| 2,\hbar k_x,0,0 \right\rangle}$ and ${\left| 2,0,\hbar k_y,0 \right\rangle}$. The resonance frequency for this transition is identical to the population-reversed case (${\left| 2,0,\hbar k_y,0 \right\rangle} \to {\left| 2,\hbar k_x,0,0 \right\rangle}$), which is ideal for reflecting the two arms of an interferometer. We emphasize that this population transfer is possible only through the coupling with the undiffracted state ${\left| 1,0,0,0 \right\rangle}$.
![Rabi oscillations for a 2D beamsplitter (a) and mirror (b) realized using pulse areas of $\Omega_{\rm Rabi} t = \pi/2$ and $\pi$, respectively. The insets show the corresponding physical picture.[]{data-label="fig:2DAtomOptics-Rabi"}](Fig2-2DAtomOptics-Rabi.pdf){width="48.00000%"}
A key aspect of any matter-wave optical element is the transfer of a “classical” phase to the atoms [@Borde1989; @Borde2001; @CohenTannoudji1992]. In the case of light-pulse atom optics, this is the optical phase difference between excitation beams at the position of the atoms [@Kasevich1991]. To illustrate how these optical phases play a role for 2D atom optics, we consider the specific case of resonant fields ($\delta_x = \delta_y = 0$) and Rabi frequencies of identical magnitude ($\chi_{\mu} = |\chi| e^{i \phi_{\mu}}$). Here, $\phi_{\mu} = \bm{k}_\mu \cdot \bm{r} + \varphi_{\mu}$ is the total phase difference between Raman beams along the $\mu$-axis, with the atomic position denoted by $\bm{r}$ and the laser phase difference by $\varphi_{\mu} = \varphi_{1\mu} - \varphi_{2\mu}$. In a truncated basis with $\bm{\mathcal{C}}^{\rm T} = (\mathcal{C}_0,\mathcal{C}_x, \mathcal{C}_y)$, the 2D beamsplitter and mirror pulses can then be summarized by the following matrices
\[M2D\] $$\begin{aligned}
& \mathbb{M}_{\rm 2D}(\tau) = -\begin{pmatrix}
0 & \frac{i}{\sqrt{2}} e^{-i\phi_x} & \frac{i}{\sqrt{2}} e^{-i\phi_y} \\
\frac{i}{\sqrt{2}} e^{i\phi_x} & -\frac{1}{2} & \frac{1}{2} e^{i(\phi_x - \phi_y)} \\
\frac{i}{\sqrt{2}} e^{i\phi_y} & \frac{1}{2} e^{i(\phi_y - \phi_x)} & -\frac{1}{2}
\end{pmatrix}, \\
& \mathbb{M}_{\rm 2D}(2\tau) = -\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & e^{i(\phi_x - \phi_y)} \\
0 & e^{i(\phi_y - \phi_x)} & 0
\end{pmatrix}.\end{aligned}$$
Here, the role of the optical phases becomes immediately clear. For atoms undergoing a two-photon transition from ${\left| 1,0,0,0 \right\rangle}$ to the diffracted state along axis $\mu$, the phase $\phi_{\mu}$ is imprinted on the wavepacket. This is a result of absorbing a photon from the field propagating along $\bm{\kappa}_{1\mu}$, followed by stimulated emission into the field along $\bm{\kappa}_{2\mu}$. Similarly, the phase $-\phi_{\mu}$ is imprinted when making the transition from the same diffracted state back to ${\left| 1,0,0,0 \right\rangle}$. Finally, atoms transferred between diffracted states acquire the phase $\pm(\phi_x - \phi_y)$. This arises because there is no direct coupling between ${\left| 2,\hbar k_x,0,0 \right\rangle}$ and ${\left| 2,0,\hbar k_y,0 \right\rangle}$—atoms must make a four-photon transition through the intermediate state ${\left| 1,0,0,0 \right\rangle}$ in a similar manner to double diffraction [@Leveque2009; @Giese2013].
A 2D Mach-Zehnder interferometer can be formed by combining a sequence of three 2D atom-optical pulses of duration $\tau - 2\tau - \tau$, each separated by an interrogation time $T$. Figure \[fig:2D-MZ-Diagrams\](a) shows the atomic trajectories associated with this new AI geometry, where atoms are split, reflected, and recombined along two spatial directions. A simple matrix representation of this process is obtained from the following product \_[MZ]{} = \_[2D]{}() \_[free]{}(T) \_[2D]{}(2) \_[free]{}(T) \_[2D]{}(), where $\mathbb{U}_{\rm free}(T)$ is a unitary matrix describing the free evolution between laser interactions [@Cheinet2008]. For an atom initially in ${\left| 1,0,0,0 \right\rangle}$, and allowing for different optical phases $\phi_{\mu,i}$ during the $i^{\rm th}$ pulse, one can show that the two internal state populations—corresponding to the two complimentary output ports of the AI—are given by $$\begin{aligned}
\label{2DFringes}
|\!{\left\langle 1 \right|} \mathbb{M}_{\rm MZ} {\left| 1,0,0,0 \right\rangle}\!|^2 & = \tfrac{1}{2} \left(1 \!+\! \cos\Delta\Phi\right) = |\mathcal{C}_0|^2, \\
|\!{\left\langle 2 \right|} \mathbb{M}_{\rm MZ} {\left| 1,0,0,0 \right\rangle}\!|^2 & = \tfrac{1}{2} \left(1 \!-\! \cos\Delta\Phi\right) = |\mathcal{C}_x|^2 \!+\! |\mathcal{C}_y|^2,\end{aligned}$$ where $\Delta\Phi = \Delta\phi_1 - 2\Delta\phi_2 + \Delta\phi_3$ is the total AI phase shift, with $\Delta\phi_i \equiv \phi_{x,i} - \phi_{y,i}$. We point out that the populations of the two diffracted states, $|\mathcal{C}_x|^2$ and $|\mathcal{C}_y|^2$, are identical and hence carry the same information.
The state-labelled architecture of this AI enables one to readout the two AI ports by spatial-integration using resonant fluorescence or absorption imaging [@Rocco2014]. Although the output ports are spatially separated, the 2D AI shown in \[fig:2D-MZ-Diagrams\](a) does not require a spatially-resolved detection system [@Dickerson2013; @Sugarbaker2013; @Hoth2016]. An interference fringe can be obtained from either port by scanning the optical phases—allowing one to probe for inertial effects.
The atomic trajectories shown in \[fig:2D-MZ-Diagrams\](a) give this 2D AI a unique sensitivity to inertial effects. Intuitively, since the two pathways enclose a rectangular spatial area in the $xy$-plane, the inertial phase is sensitive to the rotation component perpendicular to this plane, $\Omega_z$. This sensitivity is proportional to the area enclosed by the two pathways and does not require an initial velocity. In addition, when projected onto the $xt$- and $yt$-planes, these pathways enclose the same space-time area as a 1D Mach-Zehnder geometry—yielding sensitivity to the two acceleration components $a_x$ and $a_y$.
![(a) A sequence of 2D atom-optical pulses constituting a 2D Mach-Zehnder interferometer. (b) A retro-reflected beam geometry enabling 2D double-diffraction atom optics, which symmetrically transfer $\pm\hbar k_{\mu}$ of momentum along each axis $\mu = x,y$. (c) Four simultaneous 2D Mach-Zehnder AIs derived from the same atomic source. Linear combinations of the four phase shifts allow one to isolate the three inertial components $a_x$, $a_y$ and $\Omega_z$ with increased sensitivity.[]{data-label="fig:2D-MZ-Diagrams"}](Fig3-2DMachZehnder-Diagrams-v3.pdf){width="48.00000%"}
The full inertial dynamics resulting from the interference between any two atomic trajectories are encoded in the phase shift $\Delta\Phi$. We compute this phase shift for the 2D Mach-Zehnder geometry using the ABCD$\xi$ formalism developed by Bordé and Antoine [@Borde2001; @Antoine2003a; @Antoine2003b; @Borde2004]. Briefly, $\Delta\Phi$ can be written as \[DeltaPhi\] = \_[i=1]{}\^N \_i \_i + \_i, where $\Delta\bm{K}_i \equiv \bm{k}_{{\rm A},i} - \bm{k}_{{\rm B},i}$ is the difference between the effective wavevectors $\bm{k}_{{\rm A},i}$ and $\bm{k}_{{\rm B},i}$ associated with the momentum transfer from the $i^{\rm th}$ light pulse along paths “A” and “B”, respectively. Similarly, $\bm{Q}_i \equiv \half\big(\bm{q}_{\rm A}(t_i) + \bm{q}_{\rm B}(t_i)\big)$ is the position on the mid-point trajectory, and $\Delta\varphi_i = \varphi_{{\rm A},i} - \varphi_{{\rm B},i}$ is a control parameter arising from the relative laser phases. The atomic position $\bm{q}$ and momentum $\bm{p}$ trajectories are computed from the solution to the classical equations of motion [@Antoine2003a]. Due to the symmetry of the Mach-Zehnder geometry ($\bm{k}_{\rm A,2} + \bm{k}_{\rm B,2} = 0$), the phase shift is entirely determined by the choice of initial wavevectors $\bm{k}_{\rm A,1}$ and $\bm{k}_{\rm B,1}$ [^2]. In what follows, we label the AI phase shift with the subscript “A,B” which specifies both its geometry and initial wavevectors. To leading order in $T$, the generalized Mach-Zehnder phase shift is [@SupMat] \[DeltaPhiAB\] \_[A,B]{} = \_1 T\^2. Here, $\bm{a} = (a_x,a_y,a_z)$ is the acceleration vector due to external motion and gravity, $\bm{\Omega} = (\Omega_x,\Omega_y,\Omega_z)$ is the rotation vector, $\bm{v}_1$ is the atomic velocity at the time of the first light pulse, $\bm{K}_1 \equiv \half \big(\bm{k}_{\rm A,1} + \bm{k}_{\rm B,1} \big)$ corresponds to the momentum transferred to the atom’s center of mass by the first pulse, and we have omitted the control phases $\Delta\varphi_i$ for clarity. The first two terms in correspond to the well-known first-order phase shift $\Delta\bm{K}_1\cdot(\bm{a} + 2\bm{v}_1\times\bm{\Omega})T^2$ which exhibits sensitivity to the components of $\bm{a}$ and the Coriolis acceleration $2\bm{v}_1\times\bm{\Omega}$ that are parallel to $\Delta\bm{K}_1$. The third term is a purely rotational phase which can be written as $2\frac{\hbar}{m} \big(\Delta\bm{K}_1 \times \bm{K}_1\big) \cdot \bm{\Omega} \, T^2$. We emphasize that this phase is not present in 1D light-pulse AIs where $\Delta\bm{K}_1 \times \bm{K}_1 = 0$. This key point leads to additional rotation and gravity gradient sensitivity [@SupMat] with multi-dimensional geometries that has not yet been exploited experimentally. In contrast to previous atomic gyroscopes [@Gustavson1997; @Stockton2011; @Barrett2014b; @Tackmann2014; @Rakholia2014; @Dutta2016; @Yao2018], here an initial launch velocity is not required to achieve rotation sensitivity—instead this velocity is provided by the first 2D beamsplitter. This is advantageous for two reasons: (*i*) the magnitude of this velocity kick can be as precise as the value of $k$ (typically better than one part in $10^9$), and (*ii*) the direction of the kick can be changed by simply reversing the sign of $\bm{k}_{\rm A,1}$ and $\bm{k}_{\rm B,1}$. With a single atomic source, these features can then be exploited to suppress contributions from pure accelerations and initial velocities—which are the main sources of error in atomic gyroscopes [@Barrett2014b].
$\Delta\Phi_{x,y}$ $\Delta\Phi_{-x,y}$ $\Delta\Phi_{-x,-y}$ $\Delta\Phi_{x,-y}$ Sum phase
-------------------- --------------------- ---------------------- --------------------- ------------------------------------------ --
$+$ $-$ $-$ $+$ $4 k_x a_x^{\rm tot} T^2$
$-$ $-$ $+$ $+$ $4 k_y a_y^{\rm tot} T^2$
$+$ $-$ $+$ $-$ $8 \frac{\hbar}{m} k_x k_y \Omega_z T^2$
: Linear combinations of 2D Mach-Zehnder phases obtained from the four symmetric geometries shown in \[fig:2D-MZ-Diagrams\](c). Here, the laser phase contribution $\Delta\varphi_1 - 2\Delta\varphi_2 + \Delta\varphi_3$ cancels in the sum phase since it is common to all geometries.[]{data-label="tab:DeltaPhiComb"}
For the 2D Mach-Zehnder geometry shown in \[fig:2D-MZ-Diagrams\](a), with initial wavevectors in the $xy$-plane ($\bm{k}_{\rm A,1} = k_x \hat{\bm{x}}$, $\bm{k}_{\rm B,1} = k_y \hat{\bm{y}}$, $\Delta\bm{K}_1 \times \bm{K}_1 = k_x k_y \hat{\bm{z}}$), gives \[DeltaPhixy\] \_[x,y]{} = k\_x a\_x\^[tot]{} T\^2 - k\_y a\_y\^[tot]{} T\^2 + k\_x k\_y \_z T\^2, where $\bm{a}^{\rm tot} \equiv \bm{a} + 2\bm{v}_1\times\bm{\Omega}$. Although this phase contains a mixture of different inertial effects, one can isolate each of them by using linear combinations of phases obtained from area-reversed geometries, as shown in Table \[tab:DeltaPhiComb\]. Each phase $\Delta\Phi_{\rm A,B}$ can be obtained from a single measurement by employing double-diffraction [@Leveque2009; @Malossi2010; @Giese2013; @Kuber2016; @Ahlers2016] or double-single-diffraction [@Barrett2016a] atom optics in two dimensions [@SupMat]. Figures \[fig:2D-MZ-Diagrams\](b) and (c) display a scheme in which four simultaneous interferometers are generated from the same atomic source via 2D double diffraction pulses—enabling one to isolate $a_x$, $a_y$, and $\Omega_z$ in a single shot. Here, the phase readout requires spatial resolution of the adjacent interferometer ports, therefore the cloud diameter at the final beamsplitter must be less than the separation between adjacent clouds. This implies a sub-recoil-cooled source with an initial cloud size $\sigma_0 \ll 2 \hbar k_{\mu} T/m$ ($\sigma_0 \ll 1$ mm for $T = 10$ ms). This scheme is well-suited to inertial navigation applications, where strong variations of rotations and accelerations between measurement cycles would compromise the common-mode rejection of a sequential measurement protocol. Additionally, with strongly-correlated measurements, one can reject both the phase noise between orthogonal Raman beams and common systematic effects.
![A sequence of 3D atom optical pulses generating three 2D Mach-Zehnder interferometers in mutually-orthogonal planes. Parasitic trajectories excited by the beams are not shown. Three sets of spatially-separated atomic clouds arrive at opposite corners of a cube where they can be read out simultaneously—yielding sensitivity to the full acceleration and rotation vectors.[]{data-label="fig:3D-MZ-Diagrams"}](Fig4-3DMachZehnder-Diagrams-v4.pdf){width="49.00000%"}
These principles can be extended to a 3D geometry, where three mutually-perpendicular pairs of Raman beams intersect to generate three separate 2D interferometers in orthogonal planes, as shown in \[fig:3D-MZ-Diagrams\]. At $t = 0$, a 3D beamsplitter diffracts an atom initially in the undiffracted state ${\left| 1,0,0,0 \right\rangle}$ into three equal proportions traveling along $\hat{\bm{x}}$, $\hat{\bm{y}}$, and $\hat{\bm{z}}$. These three diffracted states continue along their respective axes ($\hat{\bm{\mu}}$) until $t = T$, when a 3D atom-optical mirror freezes the motion along $\hat{\bm{\mu}}$ and diffracts each wavepacket equally along the two directions orthogonal to $\hat{\bm{\mu}}$ [^3]. Finally, at $t = 2T$ the atoms intersect at three opposite corners of a cube, as shown in \[fig:3D-MZ-Diagrams\](c), where a recombination pulse transfers population from the diffracted states in each plane to an undiffracted one. Detection of the resulting 9 spatially-separated clouds yields sensitivity to the full acceleration and rotation vectors in a single shot. Individual inertial components can then be isolated in the same manner previously described for a 2D geometry—that is, by exciting symmetrically with double-diffraction pulses and imaging the clouds separately in each plane.
This 3D geometry allows one to easily construct a three-axis gyroscope. For instance, the combination of phases: $\Theta \equiv \Delta\Phi_{x,y} + \Delta\Phi_{y,z} + \Delta\Phi_{z,x}$, obtained from three corners of the cube, and $\Upsilon \equiv \Delta\Phi_{-x,y} + \Delta\Phi_{y,z} + \Delta\Phi_{z,-x}$, acquired by reversing the Raman wavevector on the $x$-axis, yields
\[Phi3D\] $$\begin{aligned}
\label{Phi3D-A}
\Theta & = \frac{2\hbar}{m} (k_y k_z \Omega_x + k_x k_z \Omega_y + k_x k_y \Omega_z) T^2, \\
\label{Phi3D-B}
\Upsilon & = \frac{2\hbar}{m} (k_y k_z \Omega_x - k_x k_z \Omega_y - k_x k_y \Omega_z) T^2.\end{aligned}$$
These phase combinations allow one to access $\Omega_x$ through the sum $\Theta + \Upsilon = 4\frac{\hbar}{m} k_y k_z \Omega_x T^2$. Hence, can be used as a building block to isolate each rotation component. A key point here is that both $\Theta$ and $\Upsilon$ arise from a single measurement of three simultaneous 2D interferometers in orthogonal planes. Yet they are each immune to spurious velocities, accelerations and laser phase noise, and hence can be combined to isolate a given rotation component. Since all quantities appearing in these rotation phases are precisely known, future gyroscopes based on this architecture could benefit from the same relative accuracy as cold-atom-based accelerometers [@Menoret2018].
We have presented a novel approach for manipulating atomic wavepackets in multiple spatial dimensions. These new atom-optical tools can be utilized to generate simple 2D interferometers sensitive to inertial effects in 3D. More complex planar geometries involving 2D double diffraction pulses enable one to isolate two components of acceleration and one rotation in a single shot, while also rejecting laser phase noise and common systematic effects. Finally, we discussed an extension to a 3D geometry, where the full acceleration and rotation vectors can be retrieved. These concepts can easily be extended to other AI configurations involving four or more pulses [@Dubetsky2006; @Cadoret2016], which could be advantageous for applications such as multi-axis gravity gradiometry, gyroscopy, or gravitational wave detection. The sensitivity of these AIs could also benefit from multi-photon momentum transfer pulses [@Malinovsky2003; @Clade2009; @Kovachy2012; @McDonald2013; @Kotru2015; @Jaffe2018], which would aid the realization of a 3D inertial sensor in a compact volume. We anticipate that this work will influence future generations of quantum accelerometers and gyroscopes, and will offer new perspectives for inertial navigation systems.
This work is supported by the French national agencies ANR (l’Agence Nationale pour la Recherche), DGA (Délégation Générale de l’Armement) under the ANR-17-ASTR-0025-01 grant, IFRAF (Institut Francilien de Recherche sur les Atomes Froids), and action spécifique GRAM (Gravitation, Relativité, Astronomie et Métrologie). P. Bouyer thanks Conseil Régional d’Aquitaine for the Excellence Chair.
[^1]: These multi-dimensional atom optics can also be achieved using Bragg transitions.
[^2]: The final wavevectors are predetermined by the initial wavevectors in order to close the two interferometer arms, $\bm{k}_{\rm A,1} + \bm{k}_{\rm B,3} = 0$ and $\bm{k}_{\rm A,3} + \bm{k}_{\rm B,1} = 0$.
[^3]: After the 3D mirror pulse, 1/9 of the population on each arm remains in the undiffracted state (not shown in \[fig:3D-MZ-Diagrams\]), which leads to a loss of interference contrast.
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abstract: 'While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then discussing 4-term homology for Boolean algebras. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.'
author:
- 'Józef H. Przytycki'
title: Distributivity versus associativity in the homology theory of algebraic structures
---
Gdansk, July 2010 – Bethesda, September 2011
Introduction
============
This paper is a summary of numerous talks I gave last year[^1]. My goal was to understand homology theory related to distributivity (and motivated by knot theory), but along the way I discovered various elementary structures, probably new, or at least not studied before. Thus I will devote the second section to the monoid of binary operations and its elementary properties. This, in addition to being a basis for my multi-term distributive homology, may be of interest to people working on universal algebras.
Because I hope for a broad audience I do not assume any specific knowledge of homological algebra or algebraic topology and will survey the basic concepts like chain complex chain homotopy and abstract simplicial complex in Sections 3 and 4. In the fifth section I recall two classical approaches to homology of a semigroup: group homology and Hochschild homology. In the sixth section we build a one-term homology of distributive structures and recall the definition of the rack homology of Fenn, Rourke, and Sanderson [@FRS]. In further sections we deepen our study of distributive homology, define multi-term distributive homology and show a few examples of computations. In the tenth section we relate distributivity to the third Reidemeister move (or braid relation) and discuss motivation coming from knot theory. In a concluding remark we speculate on relations with the Yang-Baxter operator and a potential path to Khovanov homology.
Monoid of binary operations
===========================
Let $X$ be a set and $*:X\times X \to X$ a binary operation. We call $(X;*)$ a [*magma*]{}. For any $b\in X$ the adjoint maps $*_b: X\to X$, is defined by $*_b(a)=a*b$. Let $Bin(X)$ be the set of all binary operations on $X$.
\[Proposition 2.1\] $Bin(X)$ has a monoidal (i.e. semigroup with identity) structure with composition $*_1*_2$ given by $a*_1*_2b= (a*_1b)*_2b$, and the identity $*_0$ being the right trivial operation, that is, $a*_0b=a$ for any $a,b\in X$.
Associativity follows from the fact that adjoint maps $*_b$ compose in an associative way, $(*_3)_b((*_2)_b(*_1)_b) = ((*_3)_b(*_2)_b)(*_1)_b$; we can write directly: $a(*_1*_2)*_3b= ((a*_1b)*_2b)*_3b = (a*_1b)(*_2*_3)b= a*_1(*_2*_3)b$.
The submonoid of $Bin(X)$ of all invertible elements in $Bin(X)$ is a group denoted by $Bin_{inv}(X)$. If $* \in Bin_{inv}(X)$ then $*^{-1}$ is usually denoted by $\bar *$.
It is worth mentioning here that the composition of operations in the monoid $Bin(X)$ may be thought as taking first the diagonal coproduct $\Delta: X\to X\times X$ (i.e., $\Delta(b) = (b,b)$) and applying the result on $a\in X$; Berfriend Fauser suggested after my March talk in San Antonio to try other comultiplications (he did some unpublished work on it).
One should also remark that $*_0$ is distributive with respect to any other operation, that is, $(a*b)*_0c= a*b= (a*_0c)*(b*_0c)$, and $(a*_0b)*c= a*c= (a*c)*_0(b*c)$. This distributivity later plays an important role[^2].
While the associative magma has been called a semigroup for a long time, the right self-distributive magma didn’t have an established name, even though C.S. Peirce considered it in 1880. Alissa Crans, in her PhD thesis of 2004, suggested the name [*right shelf*]{} (or simply [*shelf*]{}) [@Cr]. Below we write the formal definition of a shelf and the related notions of [*spindle*]{}, [*rack*]{}, and [*quandle*]{}.
\[Definition 2.2\] Let $(X;*)$ be a magma, then:
1. If $*$ is right self-distributive, that is, $(a*b)*c=(a*c)*(b*c)$, then $(X;*)$ is called a shelf.
2. If a shelf $(X;*)$ satisfies the idempotency condition, $a*a=a$ for any $a\in X$, then it is called a [*right spindle*]{}, or just a spindle (again the term coined by Crans).
3. If a shelf $(X;*)$ has $*$ invertible in $Bin(X)$ (equivalently $*_b$ is a bijection for any $b\in X$), then it is called a [*rack*]{} (the term wrack, like in “wrack and ruin", of J.H.Conway from 1959 [@C-W], was modified to rack in [@F-R]).
4. If a rack $(X;*)$ satisfies the idempotency condition, then it is called a [*quandle*]{} (the term coined in Joyce’s PhD thesis of 1979; see [@Joy]). Axioms of a quandle were motivated by three Reidemeister moves (idempotency by the first move, invertibility by the second, and right self-distributivity by the third move); see Section 10 and Figures 10.2-10.4.
5. If a quandle $(X;*)$ satisfies $**=*_0$ (i.e. $(a*b)*b=a$) then it is called [*kei*]{} or an involutive quandle. The term kei () was coined in a pioneering paper by Mituhisa Takasaki in 1942 [@Tak]
The main early example of a rack (and a quandle) was a group $G$ with a $*$ operation given by conjugation, that is, $a*b=b^{-1}ab$; Conway jokingly thought about it as a wrack of a group. The premiere example given by Takasaki was to take an abelian group and define $a*b=2b-a$. We will give many more examples later (mostly interested in the possibility of having shelves which are not racks; e.g. Definition \[Definition 2.13\]).
Definition \[Definition 2.2\] describes properties of an individual magma $(X;*)$. It is also useful to consider subsets or submonoids of $Bin(X)$ satisfying the related conditions described in Definition \[Definition 2.3\].
\[Definition 2.3\]
1. We say that a subset $S \subset Bin(X)$ is a distributive set if all pairs of elements $*_{\alpha},*_{\beta} \in S$ are right distributive, that is, $ (a*_{\alpha}b)*_{\beta}c= (a*_{\beta}c)*_{\alpha}(b*_{\beta}c)$ (we allow $*_{\alpha}=*_{\beta}$).
1. The pair $(X;S)$ is called a multi-shelf if $S$ is a distributive set. If $S$ is additionally a submonoid (resp. subgroup) of $Bin(X)$, we say that it is a distributive monoid (resp. group).
2. If $S \subset Bin(X)$ is a distributive set such that each $*$ in $S$ satisfies the idempotency condition, we call $(X;S)$ a multi-spindle.
3. We say that $(X;S)$ is a multi-rack if $S$ is a distributive set, and all elements of $S$ are invertible.
4. We say that $(X;S)$ is a multi-quandle if $S$ is a distributive set, and elements of $S$ are invertible and satisfy the idempotency condition.
5. We say that $(X;S)$ is a multi-kei if it is a multi-quandle with $**=*_0$ for any $*\in S$. Notice that if $*_1^2=*_0$ and $*_2^2=*_0$ then $(*_1*_2)^2=*_0$; more generally if $*_1^n=*_0$ and $*_2^n=*_0$ then $(*_1*_2)^n=*_0$. This follows from Proposition \[Proposition 2.8\].
2. We say that a subset $S \subset Bin(X)$ is an associative set if all pairs of elements $*_{\alpha},*_{\beta} \in S$ are associative with respect to each another, that is, $ (a*_{\alpha}b)*_{\beta}c= a*_{\alpha}(b*_{\beta}c)$.
\[Proposition 2.4\]
1. If $S$ is a distributive set and $*\in S$ is invertible, then $S\cup \{\bar *\}$ is also a distributive set.
2. If $S$ is a distributive set and $M(S)$ is the monoid generated by $S$ then $M(S)$ is a distributive monoid.
3. If $S$ is a distributive set of invertible operations and $G(S)$ is the group generated by $S$, then $G(S)$ is a distributive group.
We divide our proof into three elementary but important lemmas.
\[Lemma 2.5\] Let $(X;*)$ be a magma and $f:X\to X$ a magma homomorphism (i.e. $f(x*y)=f(x)*f(y)$). If $f$ is invertible (we denote $f^{-1}$ by $\bar f$) then $\bar f$ is also a magma homomorphism.
Our goal is to show that $\bar f(x*y)=\bar f(x)*\bar f(y)$. For this, let $\bar x = \bar f(x)$ and $\bar y = \bar f(y)$ (equivalently $f(\bar x)=x$ and $f(\bar y)=y$). Then, from $f(\bar x*\bar y)=f(\bar x)*f(\bar y)$ follows $f(\bar x*\bar y)= x*y$. Therefore, $\bar x*\bar y = \bar f(x*y)$ which gives $\bar f(x)*\bar f(y)=\bar f(x*y)$.
\[Corollary 2.6\]
1. If $*,*' \in Bin(X)$ and $*$ is invertible and (right) distributive with respect to $*'$, then $\bar *$ is (right) distributive with respect to $*'$.
2. If $*,*' \in Bin(X)$, $*$ is invertible, and $*'$ is (right) distributive with respect to $*$, then $*'$ is (right) distributive with respect to $\bar *$.
3. If $(X;*)$ is a rack, then $(X;\bar *)$ is a rack.
4. If $\{*',*\}$ is a distributive set and $*$ is invertible, then $\{*',*,\bar *\}$ is a distributive set.
\(i) Because $(a*'b)*c= (a*c)*'(b*c)$, the map $*_c : X\to X $ is a $*'$-shelf homomorphism; thus by Lemma \[Lemma 2.5\], $\bar *_c: X\to X $ is a $*'$-shelf homomorphism. The last property can be written as $(a*'b)\bar *c= (a \bar *c)*'(b \bar *c)$, that is, $\bar *$ is (right) distributive with respect to $*'$.\
(ii) To prove the distributivity of $*'$ with respect to $\bar *$ we consider the formula that follows from the distributivity of $*'$ with respect to $*$: $ ((a\bar * b)*b)*'c = ((a\bar * b)*'c)*(b*'c)$. This is equivalent to $ a*'c = ((a\bar * b)*'c)*(b*'c)$ and thus: $$(a*'c)\bar * (b*'c)= ((a\bar * b)*'c).$$ (iii) To see the (right) self-distributivity of $\bar *$ we notice that the (right) self-distributivity of $*$ gives, by (ii), the distributivity of $*$ with respect to $\bar *$. Thus, $*_c$ is a $\bar *$-shelf homomorphism so, by Lemma \[Lemma 2.5\], $\bar *_c$ is a $\bar *$-shelf homomorphism which gives the (right) self-distributivity of $\bar *$.\
(iv) follows from (i), (ii), and (iii).
Proposition \[Proposition 2.4\](i) follows from Corollary \[Corollary 2.6\]. Part (ii) of Proposition \[Proposition 2.4\] follows from the following elementary lemma, and (iii) is a combination of (i) and (ii).
\[Lemma 2.7\]
1. Let $*,*_1, *_2\in Bin(X)$ and let $*$ be (right) distributive with respect to $*_1$ and $*_2$. Then $*$ is (right) distributive with respect to $*_1*_2$.
2. Let $*,*_1, *_2\in Bin(X)$ and let $*_1$ and $*_2$ be (right) distributive with respect to $*$. Then $*_1*_2$ is (right) distributive with respect to $*$.
3. If $\{S,*_1,*_2\}$ is a distributive set, then $\{S,*_1,*_2,*_1*_2\}$ is also a distributive set.
\(i) We have $(a*_1*_2b)*c= ((a*_1b)*_2b)*c=((a*c)*_1(b*c))*_2(b*c)=(a*c)*_1*_2((b*c)$, as needed.\
(ii) We have $(a*b)*_1*_2c= ((a*b)*_1c)*_2c= ((a*_1c)*_2c)*((b*_1c)*_2c)= (a*_1*_2c)*(b*_1*_2c)$, as needed.\
(iii) Because of (i) and (ii) we have to only prove the (right) self-distributivity of $*_1*_2$. We have $$(a*_1*_2b)*_1*_2c=(((a*_1b)*_2b)*_1c)*_2c= (((a*_1b)*_1c)*_2(b*_1c))*_2c=$$ $$(((a*_1c)*_1(b*_1c))*_2c)*_2((b*_1c)*_2c)=$$ $$((a*_1c)*_2c)*_1((b*_1c)*_2c))*_2((b*_1c)*_2c)) = (a*_1*_2c)*_1*_2(b*_1*_2c).$$ This proves the (right) self-distributivity of $*_1*_2$.
Our monoidal structure of $Bin(X)$ behaves well with respect to (right) distributivity, as demonstrated by Proposition \[Proposition 2.4\]. It is interesting to notice that the analogue of Proposition \[Proposition 2.4\] does not hold for associative sets. For example, if $(X;*)$ is a group, then, $\bar *$ is seldom associative. We would need $(a\bar * b)\bar *c = (a\bar *c)\bar * (b\bar *c)$, that is, $a*b^{-1}*c^{-1}= a*c^{-1}*b*c^{-1}$ or equivalently $b^{-1}*c*b= c^{-1}$ for any $b$ and $c$. But for $b=c$ we get $c^2=1$, and thus $X$ should be an abelian group of the form ${{\mathbb Z}}_2Y$ for some set $Y$ CHECK(we would need $(a\bar * b)\bar *c=a\bar * (b\bar *c)$. Similarly, it very seldom happens that if $\{*_1,*_2\}$ is an associative set then the operation $*_1*_2$ is associative.\
\
When is a distributive monoid commutative?
------------------------------------------
Soon after I gave the definition of a distributive submonoid of $Bin(X)$ Michal Jablonowski, a graduate student at Gdańsk University, noticed that any distributive monoid whose elements are idempotent operations is commutative. We have:
\[Proposition 2.8\]
1. Consider $*_{\alpha},*_{\beta}\in Bin(X)$ such that $*_{\beta}$ is idempotent ($a*_{\beta}a=a$) and distributive with respect to $*_{\alpha}$, then $*_{\alpha}$ and $*_{\beta}$ commute. In particular:
2. If $M$ is a distributive monoid and $*_{\beta}\in M$ is an idempotent operation, then $*_{\beta}$ is in the center of $M$.
3. A distributive monoid whose elements are idempotent operations is commutative.
We have: $(a*_{\alpha}b)*_{\beta}b \stackrel{distrib}{=} (a*_{\beta}b)*_{\alpha}(b*_{\beta}b)
\stackrel{idemp}{=}
(a*_{\beta}b)*_{\alpha}b$.
A few months later Agata Jastrz[ȩ]{}bska (also a graduate student at Gdańsk University), checked that any distributive group in $Bin_{inv}(X)$ for $|X|\leq 5$ is commutative. Finally, in July of 2011 Maciej Mroczkowski (attending my series of talks at Gdańsk University) constructed noncommutative distributive submonoids of $Bin(X)$, the smallest for $|X|=3$. Here is Mroczkowski’s construction.
\[Construction 2.9\] Consider a pair of sets $X\supset A$ and the set of all retractions from $X$ to $A$ (denoted by $R(X,A)$. Then the set of all shelfs $(X;*_r)$ with $r\in R(X,A)$ and $a*_rb=r(b)$ forms a distributive subsemigroup of $Bin(X)$ which is non-abelian for $|X|> |A|>1$. This semigroup, denoted $SR(X,A)$, has a presentation: $\{R\ | \ *_{r_{\alpha}}*_{r_{\beta}}= *_{ r_{\beta}}\}$ and is clearly not commutative. Notice that it is a semigroup with a left trivial operation.
The simplest example is given by $X=\{b,a_1,a_2\}$ and $A=\{a_1,a_2\}$; then $SR(X,A)$ has 2 elements $*_{r_1}$ and $*_{r_2}$ with $r_1(b)=a_1$ and $r_2(b)=a_2$.
The choices above are related to the following:\
(i) $(X;*_g)$ with $a*_gb=g(b)$ is a shelf if and only if $g^2=g$.\
(ii) Two operations $*_{g_1}$ and $*_{g_2}$ are distributive with respect to each other iff $g_1g_2=g_2$ and $g_2g_1=g_1$, since:\
$(a*_{g_1}b)*_{g_2}c=g_2(c)$ and $(a*_{g_2}c)*_{g_1}(b*_{g_2}c)=g_1(b*_{g_2}c)=g_1(g_2(c))$.\
(iii) $g_1$ and $g_2$ form a distributive set if $g_1(X)=g_2(X)$ and $g_1$ and $g_2$ are retractions.
$SR(X,A)$ is a distributive semigroup. If we add $*_0$ to it we obtain a distributive monoid $MR(X,A)$.
It still remains an open problem whether an invertible operation is in the center of a distributive submonoid of $Bin(X)$, or whether a distributive subgroup of $Bin(X)$ is abelian. With relation to these questions, we propose a few problems for a computer savvy student, possibly for her/his senior thesis or master degree:
1. For small $X$, say $|X| \leq 6$, find all distributive submonoids of $Bin(X)$. In fact, such monoids form a poset with respect to inclusion, so it is sufficient to find all maximal distributive monoids.
2. Consider only distributive subgroups of $Bin(X)$. As in (i) find all maximal subgroups. Are they all abelian?
3. Now assume that we have a distributive monoid of idempotent operations (not necessarily invertible). Again find maximal distributive monoids in this category. It is interesting that, for $|X| =2$, we have four-spindle structures and they form a unique maximal distributive submonoid of 4 elements (related to the two element Boolean algebra).
4. Consider now submonoids of $Bin(X)$ such that their elements satisfy all quandle conditions. Find all maximal distributive subgroups of $Bin(X)$ in this category. This is stronger than classifying small quandles since we build posets of them.
For $|X| = 6$ the problems above may test the strength of a computer and the quality of the algorithm. For $|X| = 5$ it is feasible and for $|X| = 4$ even a small computer and not that efficient program should work and a solution will still be of great interest.
Every abelian group is a distributive subgroup of $Bin(X)$ for some $X$
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In the previous subsection we stressed that the question of whether every distributive subgroup of $Bin(X)$ is abelian is open; it is easy, however, to construct any abelian group as a distributive subsemigroup of some $Bin(X)$. The following proposition describes an elementary generalization of this:
\[Proposition 2.11\] Let $X$ be a semigroup. Consider a map $\tau : X \to Bin(X)$ given by $x \tau(a)y = xa$. Then:
1. $\tau$ is a homomorphism of semigroups.
2. If $1_r$ is a right unit of $X$ (i.e. $x1_r=x$) then $\tau (1_r) = *_0$.
3. If $X$ is a group, or more generally a semigroup with the property[^3] that if $xa = xb$ for every $x\in X$ then $a=b$, then $\tau$ is a monomorphism.
4. For any function $f:X\to X$ we define a shelf $(X;*_f)$ by $a*_fb = f(a)$ (this is a rack if $f$ is invertible and a spindle if $f=Id_X$). Then $\{*_{f_1},*_{f_2}\}$ forms a distributive set iff $f_1$ and $f_2$ commute.
5. If $X$ is a commutative semigroup such that if $xa = xb$ for any $x$ then $a=b$, then $X$ embeds as a distributive subsemigroup in $Bin(X)$.
\(i) We have $x\tau(ab)y= xab$ and $x\tau(a)\tau(b)y=(x\tau(a)y)\tau(b)y= xa\tau(b)y=xab$.\
(ii) $x\tau(1_r)y=x1_r=x$ thus $\tau(1_r)=*_0$.\
(iii) If $\tau(a)= \tau(b)$, then for all $x$ we have $xa=xb$. Thus, by our property, $a=b$ and $\tau$ is a monomorphism.\
(iv) We have: $(a*_{f_1}b)*_{f_2}c= f_2(a*_{f_1}b)=f_2f_1(a)$,\
$(a*_{f_2}c)*_{f_1}(b*_{f_2}c)= f_1((a*_{f_2}c)=f_1f_2(a)$. Thus, right distributivity holds iff $f_1$ and $f_2$ commute.\
(v) With our assumption $\tau$ is a monomorphism, and by (iv) its image is a distributive semigroup (compare Proposition \[Proposition 7.2\] where we show that commutativity of $X$ is not needed if we replace distributivity by chronological-distributivity).
Multi-shelf homomorphism
------------------------
Homomorphism of multi-shelves is a special case of a homomorphism of universal algebras (heterogeneous two-sorted algebras). Concretely, consider two multi-shelves $(X_1;S_1)$ and $(X_2;S_2)$ and a map $h: S_1 \to S_2$. We say that $f: X_1 \to X_2$ is a multi-shelves homomorphism if for any $* \in S_1$ we have $f(a*b)= f(a)h(*)f(b)$.
\[Proposition 2.12\] Let $(X;S)$ be a multi-shelf and $* \in S$. Then for any $c\in X$ the adjoint map $*_c: X \to X$ is a multi-shelf endomorphism of $X$ (with $h=Id: S \to S$).
The map is a homomorphism because, for any $*_{\alpha}\in S$, from right distributivity we have:\
$*_c(a*_{\alpha}b)=(a*_{\alpha}b)*c= (a*c)*_{\alpha}(b*c)= *_c(a)*_{\alpha}*_c(b).$
Examples of shelves and multi-shelves from a group
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Consider the three classical classes of quandles: the trivial quandles, the conjugate quandles, and the core quandles. They have (also classical) generalizations (e.g. [@Joy; @A-G]), or we can say deformations, important for us because they also produce interesting shelves which are often not quandles or racks, and lead to interesting families of multi-shelves.
\[Definition 2.13\] Let $G$ be a group and $h: G \to G$ a group homomorphism. Then we define three classes of spindles with $(G,*_h)$ as follows:\
(i) $a*_hb = h(ab^{-1})b$;\
(ii) $a*_hb = h(b^{-1}a)b$;\
(iii) $a*_hb = h(ba^{-1})b$, here we assume that $h^2 = h$.
We comment on each class below:\
(i) $(G,*_h)$ is a quandle iff $h$ is invertible, and for $h = Id$ it is a trivial quandle. If $G$ is an abelian group we obtain an Alexander spindle (Alexander quandle for $h$ invertible); in an additive convention we write $a *_h b = h(a)-h(b) + b = (1- h)(b) + h(a)$.\
(ii) $(G,*_h)$ is a quandle iff $h$ is invertible, and for $h = Id$ we obtain the conjugacy quandle ($a*b=b^{-1}ab$). If $G$ is an abelian group we obtain an Alexander spindle, the same as in case (i).\
(iii) We need $h^2 = h$ for right self-distributivity, as the following calculation demonstrates: $$(a*_hb)*_hc=(h(ba^{-1})b)*_hc = h(c(h(ba^{-1})b)^{-1})c= h(cb^{-1})h^2(ab^{-1})c \stackrel{h^2=h}{=}$$ $$h(cb^{-1} ab^{-1} )c$$ $$(a*_h c)*_h (b*_h c) = h((b*_h c)(a*_h c)^{-1})(b*_h c)=h(h(cb^{-1})c(h(ca^{-1} )c)^{-1}))h(cb^{-1})c {=}$$ $$h^2(cb^{-1} )h(c)h(c^{-1} )h^2 (ac^{-1} )h(cb^{-1} )c = h^2 (cb^{-1} ac^{-1} )h(cb^{-1} )c \stackrel{h^2=h}{=}$$ $$h(cb^{-1} ab^{-1} )c.$$ Because of the condition $h^2 = h$, our spindle is a quandle only if $h = Id$, in which case we obtain a core quandle ($a*b = ba^{-1} b$).\
It is interesting to compose $*_h*_h$ in (iii), as we obtain example (i). We can interpret this by saying that $*_h$ from (i), for $h^2 = h$ has a square root. One can also check that for (iii) $*_h^3 = *_h$, thus the monoid in $Bin(X)$ generated by $*_h$ is the three element cyclic monoid $\{*_h |\ *_h^3 = *_h \}$. We have: $a*_h^3b= ((a*_hb)*_hb)*_hb= ((h(ba^{-1})b)*_hb)*_hb=(h(bb^{-1}h(ab^{-1})b))*_hb =
(h^2(ab^{-1})b)*_hb \stackrel{h^2=h}{=} (h(ab^{-1})b)*_hb= h(bb^{-1}h(ba^{-1})b=h^2(ba^{-1})b
\stackrel{h^2=h}{=} h(ba^{-1})b= a*_hb.$\
Let us go back to case (ii):\
We check below that $*_h$ given by $a *_h b = h(b^{-1} a)b$ is right self-distributive. Thus by Proposition \[Proposition 2.4\](ii) the monoid generated by $*_h$ is a distributive monoid; however $*_{h_1}$ and $*_{h_2}$ are seldom right distributive as the calculation below shows (proving also distributivity for $h_1=h_2$): $$(a *_{h_1} b) *_{h_2} c = (h_1 (b^{-1} a)b) *_{h_2} c = h_2 (c^{-1} (h_1 (b^{-1} a)b)c =$$ $$h_2 (c^{-1} )h_2 h_1 (b^{-1} a)h_2 (b)c$$ $$(a *_{h_2} c) *_{h_1} (b *_{h_2} c) = (h_2 (c^{-1} a)c) *_{h_1} (h_2 (c^{-1} b)c) =$$ $$h_1 (h_2 (c^{-1} b)c)^{-1} h_2 (c^{-1} a)c)h_2 (c^{-1} b)c =
h_1 (c^{-1} )h_1 h_2 (b^{-1} c)h_{1} h_2 (c^{-1} a)h_1 (c)h_2 (c^{-1} b)c =$$ $$h_1 (c^{-1} )h_1 h_2 (b^{-1} a)h_1 (c)h_2 (c^{-1})h_2( b)c.$$ Again back in case (i) ($a *_h b = h(ab^{-1} )b$) we get: $$(a*_{h_1}b)*_{h_2}c = (a *_{h_2} c) *_{h_2 h_1 h_2^{-1}} (b *_{h_2} c).$$ In particular, $*_{h_1}$ and $*_{h_2}$ are right distributive if the functions commute ($h_1 h_2 = h_2 h_1$). The last equation can be interpreted as twisted distributivity, for $G$-families of quandles, the concept developed by Ishii, Iwakiri, Jang, and Oshiro [@Is-Iw; @Ca-Sa; @IIJO].
In the next few sections we compare associativity and distributivity in developing homology theory. In Section 3 we recall the basic notions of a chain complex, homology, and a chain homotopy, in order to make this paper accessible to non-topologists. We also recall the notion of a presimplicial and simplicial module, the basic concepts that are not familiar to nonspecialists.
Chain complex, homology, and chain homotopy
===========================================
Let $\{C_n\}_{n\in Z}$ be a graded abelian group (or an $R$-module[^4]). A chain complex ${\mathcal C}= \{C_n,\partial_n\}$ is a sequence of homomorphisms $\partial_n: C_n \to C_{n-1}$ such that $\partial_{n-1}\partial_{n}=0$ for any $n$. So $\mathrm{Im}(\partial_{n+1}) \subset \mathrm{Ker}(\partial_{n})$, and the quotient group $\frac{\mathrm{Ker}(\partial_{n})}{\mathrm{Im}(\partial_{n+1})}$ is called the $n$th homology of a chain complex ${\mathcal C}$, and denoted by $H_n({\mathcal C})$. Elements of $\mathrm{Ker}(\partial_{n})$ are called n-cycles, and we write $Z_n=\mathrm{Ker}(\partial_{n})$, and elements of $\mathrm{Im}(\partial_{n+1})$ are called n-boundaries and we write $B_n=\mathrm{Im}(\partial_{n+1})$ .
A map of chain complexes $f: {\mathcal C}' \to {\mathcal C}$ is a collection of group homomorphisms $f_n: C'_n \to C_n$ such that all squares in the diagram commute, that is, $ f_{n-1}\partial'_n= \partial_{n}f_n$. A chain map induces a map on homology $f_*: H_n( {\mathcal C}') \to H_n( {\mathcal C})$.
One important and elementary tool we use in the paper is a [*chain homotopy*]{}, so we recall the notion:
\[Definition 3.1\] Two chain maps $f,g : {\mathcal C}' \to {\mathcal C}$ are chain homotopic if there is a degree 1 map $h: C' \to C$ (that is $h_i: C'_i \to C_{i+1}$) such that $$f-g = {\partial_{i+1}}h_i + h_{i-1}\partial'_i.$$
The importance of chain homotopy is given by the following classical result:
If two chain maps $f$ and $g$ are chain homotopic, then they induce the same homomorphism of homology $f_* =g_*: H({\mathcal C}') \to H({\mathcal C})$. In particular, if ${\mathcal C}'= {\mathcal C}$, $f=Id$, and $g$ is the zero map, then the chain complex ${\mathcal C}$ is acyclic, that is $H_n({\mathcal C})=0$ for any $n$.
Presimplicial module and Simplicial module
------------------------------------------
It is convenient to have the following terminology, whose usefulness is visible in the next sections and which takes into account the fact that, in most homology theories, the boundary operation $\partial_n:C_n \to C_{n-1}$ can be decomposed as an alternating sum of [*face maps*]{} $d_i:C_n \to C_{n-1}$. Often we also have [*degeneracy*]{} maps $s_i:C_n \to C_{n+1}$. Formal definitions mostly follow [@Lod].
\[Definition 3.3\]
1. A simplicial module $(C_n,d_i,s_i)$, over a ring $R$, is a collection of $R$-modules $C_n$, $n\geq 0$, together with face maps $d_i:C_n\to C_{n-1}$ and degenerate maps $s_i: C_n\to C_{n+1}$, $0\leq i \leq n$, which satisfy the following properties: $$(1) \ \ \ d_id_j = d_{j-1}d_i\ for\ i<j.$$ $$(2)\ \ \ s_is_j=s_{j+1}s_i,\ \ 0\leq i \leq j \leq n,$$ $$(3) \ \ \ d_is_j= \left\{ \begin{array}{rl}
s_{j-1}d_i &\mbox{ if $i<j$} \\
s_{j}d_{i-1} &\mbox{ if $i>j+1$}
\end{array} \right.$$ $$(4) \ \ \ d_is_i=d_{i+1}s_i= Id_{C_n}.$$
2. $(C_n,d_i)$ satisfying (1) is called a [*presimplicial module*]{} and leads to the chain complex $(C_n,\partial_n)$ with $\partial_n = \sum_{i=0}^n(-1)^id_i$.
3. A [*weak simplicial module*]{} $(M_n,d_i,s_i)$ satisfies conditions (1)-(3) and a weaker condition in place of condition (4):\
(4’) $d_is_i=d_{i+1}s_i$.
4. A [*very weak simplicial module*]{} $(M_n,d_i,s_i)$ satisfies conditions (1)-(3).
We defined weak and very weak simplicial modules motivated by homology of distributive structures (as it will be clear later, Proposition \[Proposition 6.4\]). We use the terms weak and very weak simplicial modules as the terms pseudo and almost simplicial modules are already in use[^5].
Subcomplex of degenerate elements
---------------------------------
Consider a graded module $(C_n,s_i)$ where $s_i: C_n\to C_{n+1}$ for $0\leq i \leq n$. We define a graded module of [*degenerated submodules*]{} $C_n^D$ as follows: $$C_n^D= span\{s_0(C_{n-1}),...,s_{n-1}(C_{n-1}) \}.$$ If $(C_n,d_i,s_i)$ is a presimplicial module with degeneracy maps, then $C_n^D$ forms a subchain complex of $(C_n,\partial_n)$ with $\partial_n=\sum_{i=1}^n(-1)^id_i$ provided that conditions (3) and (4’) of Definition \[Definition 3.3\] hold (in particular, if $(C_n,d_i,s_i)$ is a weak simplicial module). We compute: $$\partial_ns_p= (\sum_{i=0}^n(-1)^id_i)s_p= \sum_{i=0}^n(-1)^i(d_is_p)=$$ $$\sum_{i=0}^{p-1}(-1)^i(d_is_p) + (-1)^pd_ps_p + (-1)^{p+1}d_{p+1}s_p + \sum_{i=p+2}^{p-1}(-1)^i(d_is_p)=$$ $$\sum_{i=0}^{p-1}(-1)^i(s_{p-1}d_i) + \sum_{i=p+2}^{p-1}(-1)^i(s_pd_{i-1}) \in C^D_{n-1}.$$
It is a classical result that if $(C_n,d_i,s_i)$ is a simplicial module, then $C_n^D$ is an acyclic subchain complex. The result does not hold, however, for a weak simplicial module, and we can have nontrivial degenerate homology $H_n^D=H_n(C^D)$ and normalized homology $H^{Norm}_n=H_n(C/C^D)$ different from $H_n(C)$. These play an important role in the theory of distributive homology.
\[Remark 3.4\] Even if $(C_n,d_i,s_i)$ is only a very weak simplicial module, that is $d_is_i$ is not necessarily equal to $d_{i+1}s_i$, we can construct the analogue of a degenerate subcomplex. We define $t_i: C_n \to C_n$ by $t_i=d_is_i-d_{i+1}s_i$, and define subgroups $C_n^{(t)}\subset C_n$ as $span (t_0(C_n),...,t_{n-1}(C_n),t_n(C_n))$. Then we define the subgroups $C_n^{(tD)}$ as $span(C_n^{(t)},C_n^D)$. We check directly that $C_n^{(t)}$ and $C_n^{(tD)}$ are subchain complexes of $(C_n,\partial_n)$ and they play an important role in distributive homology. In Theorem \[Theorem 6.6\] we show how to use the triplet of chain complexes $C_n^{(t)}\subset C_n^{(tD)} \subset C_n$ to find the homology of a shelf $(X;*_g)$ with $a*_gb= g(b)$, $g:X \to X$, and $g^2=g$. The generalization of this is given in [@P-S].
Homology for a simplicial complex
=================================
The homology theories that we introduce are modelled on the classical homology of simplicial complexes. We review this for completeness below.
Let $K=(X,S)$ be an abstract simplicial complex with vertices $X$ (which we order) and simplexes $S\subset 2^X$. That is, we assume elements of $S$ are finite, include all one-element subsets[^6], and that if $s'\subset s \in S$, then also $s'\in S$. The associated chain complex has a chain group $C_n$ that is a subgroup of ${{\mathbb Z}}X^{n+1}$ (i.e. a free abelian group with basis $X^{n+1}$) generated by $n$-dimensional simplexes $(x_0,x_1,...,x_n)$: we assume that $x_0<x_1<...<x_n$ in our ordering. The boundary operation is defined by: $$\partial (x_0,x_1,...,x_n) = \sum_{i=0}^n (-1)^i (x_0,...,x_{i-1},x_{i+1},...,x_n).$$ Notice that we can put $d_i(x_0,x_1,...,x_n)= (x_0,...,x_{i-1},x_{i+1},...,x_n)$ with $\partial_n=\sum_{i=0}^n (-1)^id_i$, and that $(C_n,d_i)$ is a simplicial module (i.e. $d_id_j= d_{j-1}d_i$ for $0\leq i <j \leq n$).
We do not require any structure on $X$, but as we will see later we can think of $X$ as a (trivial) semigroup or a shelf, $(X,*_0)$, with $a*_0b=a$ for any $a,b\in X$.
One proves classically that homology does not depend on the ordering of $X$. Alternatively, one can consider a chain complex with bigger chain groups $\bar C_n\subset {{\mathbb Z}}X^{n+1}$ generated by sequences $(x_0,x_1,...,x_n)$ such that the set $\{x_0,x_1,...,x_n\}$ is a simplex in $S$; as before we put $\partial (x_0,x_1,...,x_n) = \sum_{i=0}^n (-1)^i (x_0,...,x_{i-1},x_{i+1},...,x_n).$ In this approach, our definition is ordering independent and allows degenerated simplexes. The homology is the same as we can consider the acyclic subcomplex of $\bar C_n$ generated by degenerate elements $(x_0,x_1,...,x_n)$, that is, elements with $x_i=x_{i+1}$ for some $i$, and “transposition" elements $(x_0,...x_{i-1},x_i,x_{i+1},x_{i+2},...,x_n)+ (x_0,...x_{i-1},x_{i+1},x_i,x_{i+2},...,x_n)$.
In this second approach we have a simplicial module $(C_n,d_i,s_i)$ with $s_i(x_0,...,x_n)=
(x_0,...,x_{i-1},x_i,x_i,x_{i+1},...,x_n)$.
[**The motivation for the boundary operation**]{} comes from the geometrical realization of an abstract simplicial complex as illustrated below: $$\partial(x_0,x_1,x_2)= \partial({\parbox{2.7cm}{\psfig{figure=SC-triangle.eps,height=2.5cm}}}) =
{\parbox{2.7cm}{\psfig{figure=SC-bound-triangle.eps,height=2.5cm}}} =$$ $$(x_1,x_2)-(x_0,x_2)+(x_0,x_1).$$
Homology of an associative structure: group homology and Hochschild homology
============================================================================
We describe below two classical homology theories for semigroups. Our homology of distributive structures is related to these theories.
Group homology of a semigroup
-----------------------------
Let $(X,*)$ be a semigroup. We define a chain complex $\{C_n,\partial_n\}$ as follows: $C_n(X)= {{\mathbb Z}}X^n$ and $\partial_n : {{\mathbb Z}}X^n \to {{\mathbb Z}}X^{n-1}$ is defined by $$\partial(x_1,...,x_n)= (x_2,...,x_n) +$$ $$\sum_{i=1}^{n-1}(-1)^i(x_1,...,x_{i-1},x_i*x_{i+1},x_{i+2},...,x_n) +$$ $$(-1)^n(x_1,...x_{n-1})$$ We also assume that $H_0(X)={{\mathbb Z}}$ and $\partial_1(x)=1$. We can check that $ \partial^2=0$ if and only if $*$ is associative.
\[Example 5.1\] Checking this is quite illuminative, so we perform it for $n=3$: $$\partial_2(\partial_3(x_1,x_2,x_3))=\partial_2((x_1,x_2) -(x_0*x_1,x_2)+ (x_0,x_1*x_2)-(x_0,x_1))=$$ $$x_2 - x_1*x_2 +x_1$$ $$- x_2 + (x_0*x_1)*x_2 -x_0*x_1$$ $$+ x_1*x_2 - x_0*(x_1*x_2) + x_0$$ $$-x_1 + x_0*x_1 - x_0 =$$ $$(x_0*x_1)*x_2 -x_0*(x_1*x_2),$$ which is 0 iff $*$ is associative.
Let $\partial^{(\ell)}$ be a boundary map obtained from the group homology boundary operation by dropping the first term from the sum. Analogously, let $\partial^{(r)}$ be a boundary map obtained from the group homology boundary operation by dropping the last term from the sum. It is a classical observation that $(C_n,\partial^{(\ell})$ and $(C_n,\partial^{(r)})$ are acyclic for a group (or a monoid). We show this below in a slightly more general context (used later in the distributive case).
\[Example 5.2\]
1. Assume that a semigroup $(X,*)$ has a left identity $1_{\ell}$ (i.e. $1_{\ell}x=x$), then the chain homotopy $$H_{\ell}(x_1,...,x_n)= (1_{\ell},x_1,...,x_n)$$ satisfies: $$(\partial^{(\ell)} H_{\ell} + H_{\ell}\partial^{(\ell)})(x_1,...,x_n)= Id_X.$$ Thus the identity map is chain homotopic to the zero map, and the related homology groups are trivial.
2. Assume that a semigroup $(X,*)$ has a right identity $1_{r}$ (i.e. $x1_r=x$), then the chain homotopy $$H_{r}(x_1,...,x_n)= (-1)^{n+1}(x_1,...,x_n,1_r)$$ and we get: $$(\partial^{(r)} H_{r} + H_{r}\partial^{(r)})(x_1,...,x_n)= Id_X.$$ Thus the identity map is chain homotopic to the zero map, and the related homology groups are trivial.
One of the classical observations in group homology is that if $(X,*)$ is a finite group, then the cardinality of $X$, $|X|$, annihilates homology groups. We demonstrate this below in a slightly more general context; we use the observation later for distributive homology.
Assume that $(X,*)$ is a semigroup which contains a finite right orbit $A$, that is, $A$ is a finite subset of $X$ such that for each $b\in X$, we have $*_b(A)=A$ (i.e. $*_b:A \to A$ is a bijection). Then $|A|$ annihilates $H_n(X)$. In particular, if $(X,*)$ is a finite group we can take $A=X$. If $(X,*)$ has a left zero[^7] $p_{\ell }$ (i.e. $p_{\ell}*x=p_{\ell}$), then we can take $A=\{p_{\ell}\}$ and the homology groups are trivial.
Let $\Sigma=\sum_{a\in A}a$, in ${{\mathbb Z}}X$. We have $\Sigma *b=\Sigma$. We consider the chain homotopy $h_n(x_1,...,x_n)=(\Sigma,x_1,...,x_n)$ (with the convention that $h_{-1}(1)=\Sigma$). This is a chain homotopy between $|A|Id$ and the zero map, i.e. we have $\partial_{n+1}h+ h\partial_{n} = |A|Id$. Thus we conclude that $|A|$ is an annihilator of homology ($|A|H_n(X)=0$).
\[Remark 5.4\]
1. If we define $d_i: C_n\to C_{n-1}$ by: $$d_0(x_1,...,x_n)= (x_2,...,x_n)$$ $$d_i(x_1,...,x_n)= (x_1,...,x_{i-1},x_i*x_{i+1},x_{i+2},...,x_n) \ \ for\ \ 0<i<n$$ $$and \ \ d_n(x_1,...,x_n)=(x_1,...x_{n-1})$$ then $(C_n,d_i)$ is a presimplicial module.
2. If $(X;*)$ is a monoid, we define degeneracy maps $s_0(x_1,...,x_n)= (1,x_1,...,x_n)$, and for $i>0$, $s_i(x_1,...,x_n)= (x_1,...,x_i,1,x_{i+1},...,x_n)$. Then $(C_n,d_i,s_i)$ is a simplicial module.
Hochschild homology of a semigroup
----------------------------------
Let $(X;*)$ be a semigroup. We define a Hochschild chain complex $\{C_n,\partial_n\}$ as follows [@Hoch; @Lod]: $C_n(X)= {{\mathbb Z}}X^{n+1}$ and the Hochschild boundary $\partial_n : {{\mathbb Z}}X^n \to {{\mathbb Z}}X^{n-1}$ is defined by: $$\partial(x_0,x_1,...x_n)=$$ $$\sum_{i=0}^{n-1}(-1)^i(x_0,...,x_{i-1},x_i*x_{i+1},x_{i+2},...,x_n) +$$ $$(-1)^n(x_n*x_{0},x_1,...x_{n-1})$$
The resulting homology is called the Hochschild homology of a semigroup $(X,*)$ and denoted by $H\!H_n(X)$ (introduced by Hochschild in 1945 [@Hoch]). It is useful to define $C_{-1}={{\mathbb Z}}$ and define $\partial_0(x)=1$ to obtain the augmented Hochschild chain complex and augmented Hochschild homology.
Again if $(X,*)$ is a monoid then dropping the last term gives an acyclic chain complex.
More generally (and similarly to group homology), we check that if $(X,*)$ has a left unit $1_{\ell}$, then the chain homotopy $H_{\ell}(x_0,...,x_n)= (1_{\ell},x_0,...,x_n)$ satisfies $(\partial H_{\ell} + H_{\ell}\partial)(x_0,...,x_n)= (x_0,x_1,...,x_n)$, so the identity map is chain homotopic to the zero map. For $(X,*)$ with a right unit $1_r$ we use the chain homotopy $H_r((x_0,...,x_n)= (-1)^{n+1}(x_0,...,x_n,1_r)$ to get a chain homotopy between the identity and the zero map.
Notice that dropping the last term in the definition of the boundary operation in Hochschild homology is like dropping the first and the last terms in $\partial$ for the group homology of a semigroup (up to a grading shift).
1. If we define $d_i: C_n\to C_{n-1}$ by: $$d_i(x_0,...,x_n)= (x_0,...,x_{i-1},x_i*x_{i+1},x_{i+2},...,x_n) \ \ for\ \ 0\leq i<n$$ $$and \ \ d_n(x_0,...,x_n)=(x_n*x_0,x_1,...x_{n-1}),$$ then $(C_n,d_i)$ is a presimplicial module.
2. If $(X,*)$ is a monoid, we define degeneracy maps for $0\leq i\leq n$ by the formula $s_i(x_0,...,x_n)= (x_0,...,x_i,1,x_{i+1},...,x_n)$. Then $(C_n,d_i,s_i)$ is a simplicial module.
\[Remark 5.6\] To build a Hochschild chain complex we do not have to restrict ourselves to the case of a semigroup $X$ or a semigroup ring $RX$. We can consider a general (associative) ring $A$ and our definitions still work due to the homogeneity of the boundary operation. Thus we put $C_n(A)= A^{\otimes n+1}$, $d_i(a_0,...,a_n)= (a_0,..,a_i*a_{i+1},...a_n)$ for $0\leq i <n$, and $d_n(a_0,...,a_n)= (a_n*a_0,a_1,...,a_{n-1})$. Notice that $d_id_{i+1}=d_id_i$ iff $a_i*(a_{i+1}*a_{i+2})= (a_i*a_{i+1})*a_{i+2}$, that is, iff $*$ is associative.
Homology of distributive structures {#Section 6}
===================================
Recall that a shelf $(X,*)$ is a set $X$ with a right self-distributive binary operation $*:X \times X \to X$ (i.e. $(a*b)*c= (a*c)*(b*c)$).
\[Definition 6.1\] We define a (one-term) distributive chain complex ${\mathcal C}^{(*)}$ as follows: $C_n={{\mathbb Z}}X^{n+1}$ and the boundary operation $\partial^{(*)}_n: C_n \to C_{n-1}$ is given by: $$\partial^{(*)}_n(x_0,...,x_n)= (x_1,...,x_n) +$$ $$\sum_{i=1}^{n}(-1)^i(x_0*x_i,...,x_{i-1}*x_i,x_{i+1},...,x_n).$$ The homology of this chain complex is called a one-term distributive homology of $(X,*)$ (denoted by $H_n^{(*)}(X)$).
We directly check that $\partial^{(*)}\partial^{(*)}=0$ (see Example 6.3 and Proposition \[Proposition 6.4\]).
We can put $C_{-1}={{\mathbb Z}}$ and $\partial_0(x)=1$. We have $\partial_0\partial_1^{(*)}=0$, so we obtain an augmented distributive chain complex and an augmented (one-term) distributive homology, $\tilde H^{(*)}_n$. As in the classical case we get:
\[Proposition 6.4\] $
H_n^{(*)}(X)=
\begin{cases}
{{\mathbb Z}}\oplus \tilde H^{(*)}_n(X) & n = 0 \\
\tilde H^{(*)}_n(X) & \text{otherwise}
\end{cases}
$
\[Example 6.3\] We check here that $\partial^{(*)}_1(\partial^{(*)}_2(x_0,x_1.x_2))=0$ is equivalent to $*$ being right self-distributive: $$\partial^{(*)}_1(\partial^{(*)}_2(x_0,x_1,x_2))= \partial^{(*)}_1((x_1,x_2) - (x_0*x_1,x_2) + (x_0*x_2,x_1*x_2))=$$ $$x_2 - x_1*x_2 +$$ $$-x_2 + (x_0*x_1)*x_2 +$$ $$x_1*x_2 - (x_0*x_2)*(x_1*x_2)=$$ $$(x_0*x_1)*x_2 - (x_0*x_2)*(x_1*x_2)\stackrel{distrib}{=}0$$
\[Proposition 6.4\]
1. Let $d_0(x_0,...x_n)= (x_1,...,x_n)$ and $d_i(x_0,...x_n)= (x_0*x_i,...,x_{i-1}*x_i,x_{i+1},...,x_n)$, for $0<i\leq n$. Then $(C_n,d_i)$ is a presimplicial module. In fact, $d_id_{i+1}=d_id_i$ for $i>0$ is equivalent to right self-distributivity.
2. Let $s_i(x_0,...x_n)= (x_0,...,x_{i-1},x_i,x_i,x_{i+1},...,x_n)$, then $(C_n,d_i,s_i)$ is a very weak simplicial module.
3. If $(X,*)$ is a spindle, then $(C_n,d_i,s_i)$ is a weak simplicial module.
\(i) This is a direct calculation and in the cases of $0=i\leq j$ and $i\leq j-1$ the equality $d_id_j=d_{j-1}d_i$ holds without any assumption on $*$. The equality $d_id_{i+1}-d_id_i=0$ for $0<i=j-1$ is equivalent to right self-distributivity. We have: $$(d_id_{i+1}-d_id_i)(x_0,...,x_n)=$$ $$d_i((x_0*x_{i+1},...,x_{i-1}*x_{i+1},x_{i}*x_{i+1},x_{i+2},...,x_n) -
(x_0*x_i,...,x_{i-1}*x_i,x_{i+1},x_{i+1},x_{i+2},...,x_n))=$$ $$((x_0*x_{i+1})*(x_{i}*x_{i+1}),...,(x_{i-1}*x_{i+1})*(x_{i}*x_{i+1}), x_{i+2},...,x_n) -$$ $$((x_0*x_i)*x_{i+1},...,(x_{i-1}*x_i)*x_{i+1},x_{i+2},...,x_n) \stackrel{distr}{=} 0.$$ (ii) A short calculation shows that conditions (2) and (3) of a very weak simplicial module hold without any assumption on $*$.\
(iii) We check Condition (4’) of Definition \[Definition 3.3\]: $(d_is_i - d_{i+1}s_i)(x_0,...,x_n)=$ $$(d_i-d_{i+1})((x_0,...,x_{i-1},x_i,x_i,x_{i+1},...,x_n))=$$ $$(x_0*x_i,...,x_{i-1}*x_i,{\bf x_i-x_i*x_i},x_{i+1},...,x_n) \stackrel{idemp}{=}0.$$ We notice that distributivity was not needed here, only the idempotency property of $*$.
Proposition \[Proposition 6.4\] is generalized in Lemma \[Lemma 7.1\].
Computation of one-term distributive homology {#Subsection 6.1}
---------------------------------------------
If $(X;*)$ is a rack, then the one-term (augmented) distributive chain complex is acyclic. This may be the reason that this homology was not studied before. The first systematic calculations are given in [@P-S]. We observe there, in particular, that if there is given $b\in X$ in a shelf $(X;*)$ such that $*_b$ is invertible, then $\tilde H_n^{(*)}(X)=0$. To this effect, consider a chain homotopy $(-1)^{n+1}h_b$, where $h_b(x_0,...,x_n)=(x_0,...,x_n,b)$ to get $(\partial_{n+1}^{(*)}(-1)^{n+1}h_b + (-1)^{n}h_b\partial_{n}^{(*)})(x_0,...,x_n)= (x_0,...,x_n)*b$. Thus the map $(x_0,...,x_n) \to (x_0,...,x_n)*b$ is chain homotopic to zero and if $*_b$ is invertible, $\tilde H_n^{(*)}(X)=0$; compare Proposition \[Proposition 8.5\](v).
Below we show another result in this direction, motivated by an analogous observation from group homology (Proposition 5.3).
\[Proposition 6.5\] Assume that $(X;*)$ is a shelf which contains a finite right orbit $A$, that is, $A$ is a finite subset of $X$ such that for each $b\in X$, we have $*_b(A)= A*b=A$ (i.e. $*_b:A \to A$ is a bijection). Then $|A|$ annihilates $H_n(X)$. In particular, if $(X;*)$ has a left zero $p_{\ell }$ (i.e. $p_{\ell}*x=p_{\ell}$ for any $x\in X$), then we can take $A=\{p_{\ell}\}$ and the (augmented) homology groups are trivial.
The element $\sum_{a\in A}a \in {{\mathbb Z}}X$ is invariant under the right action, that is, $(\sum_{a\in A}a) *b=\sum_{a\in A}a$. We consider the chain homotopy $$h(x_1,...,x_n)=((\sum_{a\in A}a),x_1,...,x_n) \text{ with the convention that } h(1)=\sum_{a\in A}a.$$ This is a chain homotopy between $|A|Id$ and the zero map, i.e. we have $\partial_{n+1}h+ h\partial_{n} = |A|Id$. Thus we conclude that $|A|$ is an annihilator of homology ($|A|H_n(X)=0$).
In Section 7, we introduce a multi-term distributive homology and Proposition \[Proposition 6.5\] can also be generalized to this, case, that is, for $\partial^{(a_1,...,a_k)}= \sum_{i=1}^k a_i\partial^{(*_i)}$ with $\sum_{i=1}^ka_i\neq 0$ and $A$ right invariant for any operation $*_i$.
In general, we conjecture in [@P-S] that one-term distributive homology is always torsion free. Thus in the case of Proposition \[Proposition 6.5\] homology groups are conjectured to be trivial. In the special case of invertible $*$ (so $A=X$), we proved already at the beginning of this Subsection that the (augmented) homology groups are trivial (see also [@P-S] and Corollary \[Corollary 8.2\](ii)).
Computation for a shelf with $a*_gb=g(b)$
-----------------------------------------
In [@P-S] we compute the one-term distributive homology for a family of shelves with a premiere example of a left trivial shelf $(X;*_g)$, where $a*_gb=g(b)$ with $g^2=g$.
[@P-S]\[Theorem 6.6\] $$\tilde H^{(*_g)}_n(X) \simeq {{\mathbb Z}}((g(X)-\{x_0\})\times X^n)$$ where $x_0$ is any fixed element of $g(X)$. In other words, $\tilde H^{(*_g)}_n(X)$ is isomorphic to a free abelian group with basis $(g(X)-\{x_0\})\times X^n$.\
For a finite $X$, we can write it as $\tilde H^{(*_g)}_n(X)= {{\mathbb Z}}^{(|g(X)|-1)|X|^n}$.
We give a relatively short “ideological" computation of $H_n^{(*_g)}(X)$ based on the short exact sequence of chain complexes introduced in Section 3 (compare Remark \[Remark 3.4\]). More precisely, let $F_0^{(t)}=F_0^{(t)}(C_n)= t_0(C_n)$, and $F_0^{(tD)}=F_0^{(tD)}(C_n)= span(t_0(C_n),s_0(C_{n-1}))$. We consider three nested chain complexes $F_0^{(t)} \subset F_0^{(tD)} \subset C_n$. The idea of our proof is to observe that $F_0^{(t)}$ has trivial boundary operations, $F_0^{(tD)}/F_0^{(t)}$ is acyclic, and $C_n/F_0^{(tD)}$ has trivial boundary operations. Finally, we have to study carefully the long exact sequence corresponding to the short exact sequence of chain complexes $0\to F_0^{(tD)} \to C_n \to C_n/F_0^{(tD)} \to 0$ to get the conclusion of the theorem. In more detail, we are mostly interested in the case of $*_g$ from the theorem, but much of what follows applies in more general setting.We have $t_0(x_0,x_1,...,x_n)=(x_0,x_1,...,x_n) - (x_0*x_0,x_1,...,x_n)=(x_0-x_0*x_0,x_1,...,x_n)$; we use a “bilinear notation". We have $\partial t_0= 0$ as long as the equality $x*a = (x*x)*a$ holds[^8] in $(X,*)$. Thus we have:\
(I) $H_n(F_0^t)= F_0^t=t_0C_n= {{\mathbb Z}}^{(|X|-|X/\sim|)|X|^{n}}$, where $\sim$ is an equivalence relation on $X$ generated by $x \sim x*x$. For $*=*_g$, we can take as a basis of $H_n(F_0^t)= F_0^t$ elements $(x_0-g(x_0),x_1,...,x_n)$, and for a finite $X$, $H_n(F_0^t)={{\mathbb Z}}^{(|X|-|g(X)|)|X|^{n}}$.\
(II) For any shelf, $F_0^{(tD)}/F_0^{(t)}$ is acyclic. Namely, $s_0$ is a chain homotopy between the identity and the zero map on $F_0^{(tD)}/F_0^{t}$. We have: $$\partial s_0 + s_0\partial = t_0 + s_0d_0 \equiv s_0d_0 \equiv Id\ on \ F_0^{(tD)}/F_0^{t}$$ (here we use the fact that in a shelf $d_0s_0=Id$, so $s_0d_0s_0=s_0$, thus $s_0d_0$ is the identity on $F_0^{(tD)}/F_0^{t}$).\
As a corollary, we have that the embedding $F^t_0 \to F_0^{(tD)}$ induces an isomorphism on homology.\
(III) Consider now the chain complex $C_n/F_0^{(tD)}$. Here, for $a*_gb=g(b)$, $g^2=g$, the boundary operation is trivial as $$\partial (x_0,x_1,...,x_n)= t_0(x_1,...,x_n)+\sum_{i=2}^n(-1)^i(g(x_i),...,g(x_i),x_{i+1},...,x_n)\in F_0^{(tD)}.$$ From this we conclude that $H_n(C_n/F_0^{(tD)})=C_n/F_0^{(tD)}$. As a basis of the group we can take elements $(x_0,x_1,...,x_n)$ with $x_0=g(x_0)$ and $x_1\neq x_0$. Thus the group is isomorphic to ${{\mathbb Z}}(g(X)\times (X-\{x_0\}|)\times X^{n-1})$ and for a finite $X$, the group is isomorphic to ${{\mathbb Z}}^{|g(X)|(|X|-1)|X|^{n-1}}$.\
(IV) We consider the long exact sequence of homology corresponding to\
$0\to F_0^{(tD)}\to C_n \to C_n/F^{(tD)}\to 0$: $$...\stackrel{b_*}{\rightarrow} H_n(F_0^{(tD)}) \to H_n(C) \to H_n(C/F^{(tD)}) \stackrel{b_*}{\rightarrow}
H_{n-1}(F_0^{(tD)})\to...$$ We now show that the connecting homomorphism $b_*:H_n(C/F_0^{(tD)}) \to H_{n-1}(F_0^{(tD)})$ is an epimorphism. In fact, the element $(x_0,x_1-x_1*x_1,x_2,...,x_n)$ is a chain in $C_n$ but it is a cycle in $C_n/F_0^{(tD)}$. Thus its boundary $\partial(x_0,x_1-x_1*x_1,x_2,...,x_n)= (x_1-x_1*x_1,x_2,...,x_n)\in F_0^{(tD)}$, and this yields our connecting homomorphism $b: C_n/F_0^{(tD)} \to F_0^{(tD)}$, defined on the level of chains, with the image equal to $C_n^{(t)}$. However, because of (II), $b$ yields an epimorphism $b_*:H_n(C/F_0^{(tD)}) \to H_{n-1}(F_0^{(tD)})$. Thus the long exact sequence of homology gives the short exact sequence: $$0 \to H_n(C) \to H_n(C/F^{(tD)}) \stackrel{b_*}{\rightarrow} H_{n-1}(F_0^{(tD)})\to 0.$$ We now compute $H_n(C)$ as the kernel of $b_*$ to get the free abelian group with a basis obtained from the basis of $ H_n(C/F^{(tD)})$ by deleting elements of the form $(x_0,x_1-x_1*x_1,x_2,...,x_n)$ for fixed $x_0$. Thus, $H_n(X)$ is isomorphic to $ {{\mathbb Z}}((g(X))-\{x_0\})\times X^n)$ for $n>0$ and $H_0(X)= {{\mathbb Z}}(g(X))$. If $X$ is finite we get $rank\ \tilde H_n = |g(X)|(|X|-1)|X|^{n-1}-(|X|-|g(X)|)|X|^{n-1}=
(|g(X)|-1)|X|^{n}$.
We can make a small but useful generalization of Theorem \[Theorem 6.6\] by considering a new chain complex $C^{(d)}_n(X)$ obtained from $C_n^{(*_g)}(X)$ by taking, for any number $d$, the boundary operation $\partial^{(d)} = d\partial^{(*_g)}$. See Theorem \[Theorem 9.1\] and [@Pr-Pu] for further generalizations of the case $g=Id$.
\[Corollary 6.7\] Assume that $X$ is a finite set.
1. If $d\neq 0$ then $$\tilde H_n^{(d)}(X)= {{\mathbb Z}}^{(|g(X)|-1)|X|^n} \oplus {{\mathbb Z}}_d^{|X|^{n+1}-|g(X)|u_n},$$ where $u_n=u_n(|X|)=|X|^n - |X|^{n-1} +...+ (-1)^{n}|X|+(-1)^{n+1}$. In particular, if $g=Id$ we get $$\tilde H_n^{(d)}(X)= {{\mathbb Z}}^{(|X|-1)|X|^n} \oplus {{\mathbb Z}}_d^{u_n-(-1)^n}.$$
2. If $d=0$, then $H^{(d)}_n(X)= C^{(d)}_n(X)= {{\mathbb Z}}^{|X|^{n+1}}$.
\(i) As long as $d\neq 0$ the free part of the homology does not depend on $d$, so we know the free part from Theorem \[Theorem 6.6\]. We see that the torsion part is $(\partial_{n+1}(C_{n+1}))\otimes {{\mathbb Z}}_d$, so for a finite $X$ we have to compute the rank of $\partial_{n+1}(C_{n+1})$. We do this by observing that $$rk \partial_{n+1}(C_{n+1})+ rk \tilde H^{(*_g)}_n +rk \partial_{n}(C_{n})=
rk C_n(X)= |X|^{n+1}.$$ For example, for $n=0$ we get $(|X|- |g(X)|) + (|g(X)|-1) +1 = |X|$ (we work with the reduced homology $\tilde H^{(*_g)}_n$). Knowing initial data, the rank of homology, and the ranks of the chain groups we compute that $rk \partial_{n+1}(C_{n+1}) = |X|^{n+1}- |g(X)||X|^{n}+ |g(X)||X|^{n-1}+...+ (-1)^{n+1}|g(X)| =
|X|^{n+1}-|g(X)|u_n$, and the formula for homology is proven.\
(ii) Boundary operations are trivial, so the formula follows.
Multi-term distributive homology {#Section 7}
================================
The first homology theory related to a self-distributive structure was constructed in early 1990s by Fenn, Rourke, and Sanderson [@FRS] and motivated by (higher dimensional) knot theory[^9]. For a rack $(X,*)$, they defined rack homology $H_n^R(X)$ by taking $C^R_n={{\mathbb Z}}X^n$ and $\partial_n^R: C_n \to C_{n-1}$ is given by $\partial_n^R = \partial_{n-1}^{(*)}-\partial_{n-1}^{(*_0)}$. Our notation has grading shifted by 1, that is, $C_n(X)= C^R_{n+1}= {{\mathbb Z}}X^{n+1}$. It is routine to check that $\partial^R_{n-1}\partial_n^R=0$. However, it is an interesting question what properties of $*_0$ and $*$ are really used. With relation to the paper [@N-P-4] we noticed that it is distributivity again which makes $(C^R(X),\partial_n^R)$ a chain complex. More generally we observed that if $*_1$ and $*_2$ are right self-distributive and distributive with respect to each other, then $\partial^{(a_1,a_2)}= a_1\partial^{(*_1)}+a_2\partial^{(*_2)}$ leads to a chain complex (i.e. $\partial^{(a_1,a_2)}\partial^{(a_1,a_2)}=0$). Below I answer a more general question: for a finite set $\{*_1,...,*_k\}\subset Bin(X)$ and integers $a_1,...,a_k \in {{\mathbb Z}}$, when is $(C_n,\partial^{(a_1,...,a_k)})$ with $\partial^{(a_1,...,a_k)}=a_1\partial^{(*_1)}+...+a_k\partial^{(*_k)}$ a chain complex? When is $(C_n,d_i^{(a_1,...,a_k)})$ a presimplicial set? We answer these questions in Lemma \[Lemma 7.1\]. In particular, for a distributive set $\{*_1,...,*_k\}$ the answer is affirmative.
\[Lemma 7.1\]
1. If $*_1$ and $*_2$ are right self-distributive operations, then $(C_n,\partial^{(a_1,a_2)})$ is a chain complex if and only if the operations $*_1$ and $*_2$ satisfy: $$(a*_1b)*_2c + (a*_2b)*_1c = (a*_2c)*_1(b*_2c) + (a*_1c)*_2(b*_1c).$$ We call this condition [*weak distributivity*]{}.
2. We say that a set $\{*_1,...,*_k\}\subset Bin(X)$ is weakly distributive if each operation is right self-distributive and each pair of operations is weakly distributive (with two main cases: distributivity $(a*_1b)*_2c = (a*_2c)*_1(b*_2c)$ and chronological distributivity[^10] $(a*_1b)*_2c=(a*_1c)*_2(b*_1c)$). We have: $(C_n,d_i^{(a_1,...,a_k)})$ is a presimplicial set if and only if the set $\{*_1,...,*_k\}\subset Bin(X)$ is weakly distributive.
3. $(C_n,\partial_n^{(a_1,...,a_k)})$ is a chain complex if and only if the set $\{*_1,...,*_k\}\subset Bin(X)$ is weakly distributive.
We have $\partial_{n-1}^{(a_1,a_2)} \partial_n^{(a_1,a_2)}=$ $$(a_1\partial_{n-1}^{(*_1)}+a_2\partial_{n-1}^{(*_2)}) (a_1\partial_{n}^{(*_1)}+a_2\partial_{n}^{(*_2)}) =$$ $$a_1^2\partial_{n-1}^{(*_1)}\partial_{n}^{(*_1)} + a_2^2\partial_{n-1}^{(*_2)}\partial_{n}^{(*_2)}
+ a_1a_2(\partial_{n-1}^{(*_1)}\partial_{n}^{(*_2)} + \partial_{n-1}^{(*_2)}\partial_{n}^{(*_1)})
=$$ $$a_1a_2(\partial_{n-1}^{(*_1)}\partial_{n}^{(*_2)} + \partial_{n-1}^{(*_2)}\partial_{n}^{(*_1)})$$ To see that the condition $(a*_1b)*_2c + (a*_2b)*_1c = (a*_2c)*_1(b*_2c) + (a*_1c)*_2(b*_1c)$ is necessary, let us consider the case $n=2$. We have $$(\partial_{1}^{(*_1)}\partial_{2}^{(*_2)}) + \partial_{1}^{(*_2)}\partial_{2}^{(*_1)})(x_0,x_1,x_2)=$$ $$\partial_{1}^{(*_1)}((x_1,x_2) - (x_0*_2x_1,x_2)+ (x_0*_2x_2,x_1*_2x_2)) +$$ $$\partial_{1}^{(*_2)}((x_1,x_2) - (x_0*_1x_1,x_2)+ (x_0*_1x_2,x_1*_1x_2)) =$$ $$x_2 - x_1*_1x_2 - x_2 +(x_0*_2x_1)*_1x_2 + x_1*_2x_2 - (x_0*_2x_2)*_1(x_1*_2x_2) +$$ $$x_2 - x_1*_2x_2 - x_2 +(x_0*_1x_1)*_2x_2 + x_1*_1x_2 - (x_0*_1x_2)*_2(x_1*_1x_2) =$$ $$(x_0*_2x_1)*_1x_2 - (x_0*_2x_2)*_1(x_1*_2x_2) + (x_0*_1x_1)*_2x_2 - (x_0*_1x_2)*_2(x_1*_1x_2),$$ which is equal to zero iff weak distributivity holds.\
On the other hand, we show below that weak distributivity is sufficient to have $d_i^{(a_1,a_2)}d_{i+1}^{(a_1,a_2)}=d_i^{(a_1,a_2)}d_i^{(a_1,a_2)}$ for $0<i<n$ and is sufficient for $(C_n,d_i^{(a_1,a_2)})$ being a presimplicial module (the other needed equalities $d_id_j=d_{j-1}d_i$ for $i<j$ follow without using any special conditions). Namely, we have: $$(d_i^{(a_1,a_2)}d_{i+1}^{(a_1,a_2)}-d_i^{(a_1,a_2)}d_i^{(a_1,a_2)})(x_0,...,x_n) =$$ $$d_i^{(a_1,a_2)}(a_1((x_0,...,x_i)*_1x_{i+1},x_{i+2},...,x_n) + a_2((x_0,...,x_i)*_2x_{i+1},x_{i+2},...,x_n) -$$ $$(a_1((x_0,...,x_{i-1})*_1x_i,x_{i+1},...,x_n) + a_2((x_0,...,x_{i-1})*_2x_i,x_{i+1},...,x_n))) =$$ $$a_1^2(((x_0,...,x_{i-1})*_1x_{i+1})*_1(x_i*_1x_{i+1}),x_{i+2},...,x_n) -$$ $$(((x_0,...,x_{i-1})*_1x_i)*_1x_{i+1},x_{i+2},...,x_n)) +$$ $$a_2^2(((x_0,...,x_{i-1})*_2x_{i+1})*_2(x_i*_2x_{i+1}),x_{i+2},...,x_n) -$$ $$(((x_0,...,x_{i-1})*_2x_i)*_2x_{i+1},x_{i+2},...,x_n)) +$$ $$a_1a_2(((x_0,...,x_{i-1})*_1x_{i+1})*_2(x_i*_1x_{i+1}),x_{i+2},...,x_n) +$$ $$a_1a_2(((x_0,...,x_{i-1})*_2x_{i+1})*_1(x_i*_2x_{i+1}),x_{i+2},...,x_n) -$$ $$a_1a_2(((x_0,...,x_{i-1})*_1x_i)*_2x_{i+1},x_{i+2},...,x_n))-$$ $$a_1a_2(((x_0,...,x_{i-1})*_2x_i)*_1x_{i+1},x_{i+2},...,x_n))$$ which is equal to zero by the weak distributivity property. This completes our proof of (i); (ii) and (iii) follow from this directly.
There is some justification for studying the concept of chronological-distributivity or weak distributivity, as every semigroup $A$ (with the property: $xa=xb$, for every $x$, implies $a=b$) can be embedded as a chronological-distributive semigroup in $Bin(A)$ (compare Proposition \[Proposition 2.11\]):
\[Proposition 7.2\]
1. For $f:X \to X$ we define $*_f$ by $a*_fb=f(a)$. We have $*_f*_g= *_{gf}$ as $a*_f*_gb= (a*_fb)*_gb=g(f(a))= a*_{gf}b$.
2. For any pair of functions $f,g:X\to X$ the pair $(*_f,*_g)$ is chronologically distributive; namely we have: $$(a*_fb)*_gc= g(f(a)),$$ $$(a*_fc)*_g((a*_fc)= g(a*_fc)=g(f(a)).$$
3. Any semigroup $A$ with the property: $xa=xb$, for every $x$, implies $a=b$, is a chronologically distributive subsemigroup of $Bin(A)$.
4. Any commutative semigroup $A$ with the property: $xa=xb$, for every $x$, implies $a=b$, is a distributive subsemigroup of $Bin(A)$.
The proof is a simple application of ideas from Proposition \[Proposition 2.11\]. but we will comment on (iii) and (iv) as it is still an open problem whether any distributive subgroup of $Bin(X)$ is abelian, Section 2.\
The embedding $\tau: A \to Bin(A)$ is given by $\tau(a) = *_{f_a}$ where $f_a(x)=xa$ and thus $x*_{f_a}y= xa$, It is a homomorphism as $x\tau(ab)y= x*_{f_{ab}}y= xab = (x*_{f_a}y)b=(x*_{f_a}y)*_{f_b}y=
x*_{f_a}*_{f_b}y=x\tau(a)\tau(b)y$. $\tau(A)$ is chronological-distributive by (ii). $\tau$ is a monomorphism because of quasigroup property. Finally, in our case, distributivity and chronological distributivity are the same if $f$ and $g$ commute.
From distributivity to associativity
------------------------------------
We observed that to linearly combine two self-distributive operations into a new operation we need weak distributivity. We can ask the similar question for associative operations, say $*_{\alpha}$ and $*_{\beta}$ on $X$. For $\partial^{(a,b)}= a\partial^{\alpha}+ b\partial^{\beta},$ is it a boundary operation? We consider group or Hochschild homology. A sufficient condition is that $\partial^{\alpha}\partial^{\beta} = -\partial^{\beta}\partial^{\alpha}$. This allows us not only to create linear combinations of boundary operations but also to create a chain bicomplex using $\partial^{\alpha}$ horizontally and $\partial^{\beta}$ vertically. The condition $\partial^{\alpha}\partial^{\beta} = -\partial^{\beta}\partial^{\alpha}$ follows from: $$(a*_{\alpha}b)*_{\beta}c + (a*_{\beta}b)*_{\alpha}c =$$ $$a*_{\beta}(b*_{\alpha} c)+ a*_{\alpha}(b*_{\beta}c)$$
I do not know a good name for this so I will call it weak associativity (following the terminology from the distributive case), as it is a combination of associativity $(a*_{\alpha}b)*_{\beta}c= a*_{\alpha}(b*_{\beta}c)$, and chronological associativity (that is: $(a*_{\alpha}b)*_{\beta}c= a*_{\beta}(b*_{\alpha}c)$).
Of course weak associativity follows from each, associativity and chronological-associativity, separately.\
\
\
[**Checking for $n=3$ and group homology**]{}:\
Let $*_{\alpha}$ and $*_{\beta}$ be two associative operations on a set $X$. We have: $$(\partial^{\beta}\partial^{\alpha}+ \partial^{\alpha}\partial^{\beta})(x_1,x_2,x_3)=$$ $$\partial^{\beta}((x_2,x_3)-(x_1*_{\alpha}x_2,x_3)+ (x_1,x_2*_{\alpha}x_3)- (x_1,x_2))+$$ $$\partial^{\alpha}((x_2,x_3)-(x_1*_{\beta}x_2,x_3)+ (x_1,x_2*_{\beta}x_3)- (x_1,x_2))=$$ $$x_3-x_2*_{\beta}x_3+x_2 -x_3 + (x_1*_{\alpha}x_2)*_{\beta}x_3 -x_1*_{\alpha}x_2 +$$ $$x_2*_{\alpha}x_3 - x_1*_{\beta}(x_2*_{\alpha}x_3)+ x_1 -x_2 +x_1*_{\beta}x_2 -x_1 +$$ $$x_3-x_2*_{\alpha}x_3+x_2 -x_3 + (x_1*_{\beta}x_2)*_{\alpha}x_3 -x_1*_{\beta}x_2 +$$ $$x_2*_{\beta}x_3 - x_1*_{\alpha}(x_2*_{\beta}x_3)+ x_1 -x_2 +x_1*_{\alpha}x_2 -x_1 =$$ $$(x_1*_{\alpha}x_2)*_{\beta}x_3 - x_1*_{\beta}(x_2*_{\alpha}x_3)+$$ $$(x_1*_{\beta}x_2)*_{\alpha}x_3 - x_1*_{\alpha}(x_2*_{\beta}x_3)$$ which is equal to zero iff weak associativity holds.
Techniques to study multi-term distributive homology
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The remarkable map $f:X^{n+1}\to X^{n+1}$;\
$f(x_0,x_1,...,x_{n-1},x_n) = (x_0*x_1*...*x_n, x_1*...*x_n,...,x_{n-1}*x_n,x_n)$
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\
I noticed this very interesting map only in September 2010, but it looks similar to the well known change of coordinates in homology of groups.
Let $*_0$ denote the trivial right action on $X$ (i.e. $a*_0b=a$), and let operations $*,*_1, *_2,...*_k$ be elements of a distributive submonoid of $Bin(X)$, that is, they are right self-distributive operations on a set $X$ which are distributive with respect to another.[^11]\
Let $f=f^{(*)}: RX^n \to RX^{n} $ be given by[^12]: $$f(x_0,x_1,...,x_{n-1},x_n) = (x_0*x_1*...*x_n, x_1*...*x_n,...,x_{n-1}*x_n,x_n)$$ and $\partial^{(*)}(x_0,x_1,...,x_{n-1},x_n) = \sum_{i=0}^n (-1)^i(x_0*x_i,...,x_{i-1}*x_i, x_{i+1},...,x_n)$then $ f^{(*)}\partial^{(*_2)} = \partial^{(*_1)} f^{(*)} $ where $*_2= **_1$ (recall that the composition of operations is: $a (**_1) b = (a*b)*_1 b$), as the following calculation demonstrates:\
$f^{(*)}\partial^{(*_2)}(x_0,...,x_n)= $ $$\sum_{i=0}^n (-1)^i(x_0*...*x_{i-1}*_2x_i*x_{i+1}*...*x_n,
x_{i-1}*_2x_i*x_{i+1}*...*x_n,x_{i+1}*...*x_n,...,x_n)$$ and $\partial^{(*_1)}f^{(*)}(x_1,...,x_n)= $ $$\sum_{i=1}^n (-1)^i (x_0*...*x_{i-1}*x_i*_1x_i*x_{i+1}*...*x_n,
x_{i-1}*x_i*_1x_i*x_{i+1}*...*x_n,x_{i+1}*...*x_n,...,x_n).$$ Here are interesting applications/special cases:\
\[Corollary 8.2\]
1. Consider the multi-term boundary operation $\partial^{(a_1,...,a_n)}= \sum_{i=1}^k a_i\partial^{(*_i)}$, then $f^{(*)}$ is a chain map from the chain complex on ${{\mathbb Z}}X^{n+1}$ with a composite boundary operation $*\circ\partial^{(a_1,...,a_n)} \stackrel{def}{=}
\sum_{i=1}^k a_i\partial^{(**_i)}$ to $({{\mathbb Z}}X^{n+1},\partial^{(a_1,...,a_n)})$.\
Thus if $*$ is invertible (like in a rack) then this chain map is invertible and induces an isomorphism of homology. In particular:
2. If $\partial^{(*)}$ is a one-term operation with invertible $*$ then it has the same homology as $\partial^{(*_0)}$ which is acyclic. Here let us stress that we proved acyclicity for one-term homology for racks (for one-term homology we can prove acyclicity in a more general case: it suffices to assume that there is $b$ such that $*_b$ is a bijection (as usually $*_b(a)=a*b$)). (See Theorem \[Theorem 6.6\] for examples of shelves that are not racks and with a chain complex that is not acyclic).
3. In classical (two-term) rack homology ($\partial = \partial^{(*_0)} -\partial^{(*)}$) the above result gives an isomorphism with the chain complex $\bar\partial = \partial^{(\bar *)} -\partial^{(*_0)}$ which describes the classical homology of the dual complex ($\bar *$ in place of $*$)[^13].
4. More generally, we can consider any two-term complex with a boundary operation $a_1\partial^{(*_1)} + a_2\partial^{(*_2)}$ and for an invertible $*_1$ we get an isomorphic complex with $a_1\partial^{(*_0)} +a_2\partial^{(\bar*_1*_2)}$. This can be interpreted as saying that any 2-term homology of racks is equivalent to the twisted homology [@CES-1] $\partial^T= t \partial^{0} -\partial^{1}$ (noninvertible $a_2$ gives slightly more possibilities).\
Notice that, on the way from the twisted homology of $(X;*)$ and its dual $(X;\bar *)$, we also invert $t$.
Splitting multi-term distributive homology into degenerate and normalized parts {#Subsection 8.2}
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For a quandle $(X;*)$ and its chain complex $(C_n,\partial^R)$, Carter, Kamada, and Saito at al. [@Car; @CJKLS; @CKS] considered the degenerate subcomplex and its quotient which they call [*quandle chain complex*]{}. Litherland and Nelson [@L-N] proved that this complex splits. Their result extends to multi-spindle $(X;*_1,...,*_k)$ (that is, a multi-shelf with every operation idempotent). Our proof follows that given in [@N-P-2].
Consider a multi-spindle $(X;*_1,...,*_k)$ and its chain complex $C_n(X)={{\mathbb Z}}X^{n+1}$, $\partial^{(a_1,...,a_k)}=
\sum_{i=1}^ka_i\partial^{(*_i)}$. Recall that we deal with a weak simplicial module $(C_n,d_i,s_i)$ with $d_i=d_i^{(a_1,...,a_k)}= \sum_{i=1}^ka_id^{(*_i)}_i$ and\
$s_i(x_0,...x_n)= (x_0,...,x_{i-1},x_i,x_i,x_{i+1},...,x_n)$. Thus, we know, in general, that $C_n^D=span(s_0C_{n-1},s_1C_{n-1},...,s_{n-1}C_{n-1})$ is a subchain complex of $(C_n,\partial^{(a_1,...,a_k)})$. This complex is usually not acyclic but it always splits. Let $C^{Norm}=C/C^D$ be the quotient complex, called the [*normalized*]{} complex of a multi-spindle.
\[Theorem 8.3\]
1. Consider the short exact sequence of chain complexes: $$0 \to C_n^D(X) \to C_n(X) \to C_n^{Norm}(X) \to 0$$ Then this complex splits with a split map $\alpha : C^{Norm}(X) \to C_n(X)$ given by the formula: $$\alpha(x_0,x_1,x_2,...,x_n)= (x_0,x_1-x_0,x_2-x_1,...,x_n-x_{n-1}).$$ We will the use multilinear convention as in [@N-P-2], e.g. $\alpha(x_0,x_1,x_2)=$ $$(x_0,x_1-x_0,x_2-x_1)= (x_0,x_1,x_2)-(x_0,x_0,x_2) - (x_0,x_1,x_1)+ (x_0,x_0,x_1).$$
2. $H_n(X) = H_n^D(X) \oplus H_n^{Norm}(X)$.
\(i) First observe that $\alpha$ is well defined since $\alpha(s_i(x_0,...,x_{n-1}))=
(x_0,...,x_i-x_i,...,x_{n-1}) = 0$, so $\alpha (C_n^D)=0$. We also have $\beta\alpha = Id_{C^{Norm}}$, because $(\alpha - Id)(C_n) \subset C^D_n$ and $\beta (C_n^D)=0$. This shows that $\alpha$ splits $\{C_n\}$ as a graded group. To show this split of a chain complex we should show that $\alpha$ is a chain map, that is, $\partial^{(a_1,...,a_k)}\alpha = \alpha\partial^{(a_1,...,a_k)}$. Of course it suffices to prove the relation $\partial^{(*_i)}\alpha = \alpha\partial^{(*_i)}$ for any $i$. This follows from Lemma \[Lemma 8.4\] below. Part (ii) follows directly from (i).
\[Lemma 8.4\]
1. For any spindle $(X,*)$ and its related presimplicial module $(C_n,d_i)$ we have $$d_i\alpha - \alpha d_i = r_{i-1}+ r_i, \ for \ 0\leq i \leq n, \ where$$ $$r_{−1} = r_0 = 0 \text{ and for } 0 < i < n:$$ $$r_i = −((x_0 , x_1 - x_0 , ..., x_i - x_{i-1} )* x_i , x_{i+2} - x_{i+1} , ..., x_n - x_{n-1})).$$ In particular, $r_0 = -(x_0 , x_2 - x_1 , ..., x_n - x_{n-1}))$ and\
$r_{n-1} = -(x_0 , x_1 - x_0 , ..., x_{n-1} - x_{n-2} ) * x_{n-1}$.
2. $\partial^{(*)}_n \alpha - \alpha\partial^{(*)}_n = 0$.
We check immediately that $d_0 \alpha - \alpha d_0 = r_0$ and that $d_n \alpha - \alpha d_n = r_{n-1}.$ Then, for $0 < i < n$ we compute: $$(d_i\alpha - \alpha d_i)(x_0,x_1,...,x_n)=$$ $$d_i((x_0 , x_1 -x_0 ,...,x_n -x_{n-1} )- \alpha((x_0 , ..., x_{i-1})*x_i , x_{i+1} , ..., x_n ) =$$ $$((x_0 , x_1 -x_0 , ..., x_{i-1} -x_{i-2} )*(x_{i} -x_{i-1}), x_{i+1} -x_i , x_{i+2} -x_{i+1} , ..., x_n - x_{n-1})-$$ $$((x_0 , x_1 -x_0, ..., x_{i-1} -x_{i-2})*x_i , x_{i+1} - x_{i-1}*x_i , x_{i+2}-x_{i+1},...,x_{n}-x_{n−1}) =$$ $$-((x_0 , x_1 - x_0 , ..., x_{i-1} - x_{i-2} ) * x_{i-1} , x_{i+1} - x_i , x_{i+2} - x_{i+1} , ..., x_n - x_{n-1})+$$ $$((x_0,x_1 -x_0,...,x_{i-1} -x_{i-2} )*x_i , x_{i+1} -x_i -x_{i+1} +x_{i-1} *x_i,x_{i+2}-x_{i+1},...,x_n -x_{n-1})=$$ $$r_{i-1} +((x_0 , x_1 -x_0 , ..., x_{i-1} -x_{i-2} )*x_i , (x_{i-1} -x_i )*x_i , x_{i+2} -x_{i+1},...,x_n -x_{n−1}) =$$ $r_{i-1} + r_i$, as needed.
\(ii) follows from (i) as $\partial^{(*)}_n = \sum_{i=0}^n(-1)^id_i$ and $\sum_{i=0}^n (-1)^i(r_{i-1}+ r_i) = 0.$
Basic properties of multi-term distributive homology {#Subsection 8.3}
----------------------------------------------------
Let $(X;*_1,...,*_k)$ be a multi-shelf. We say that $A\subset X$ is a submulti-shelf if it is closed under all operations $*_i$. In particular, for an element $t\in X$, the set $\{t\}$ is a submulti-shelf iff it satisfies the idempotency condition for any operation ($t*_it=t$). For a submulti-shelf $A$ we have the short exact sequence of chain complexes (recall that $\partial^{(a_1,...,a_k)}= \sum_{i=1}^ka_i\partial^{(*_i)}$ and to shorten notation we often write $\Sigma=\sum_{i=1}^ka_i$): $$0 \to C_n(A) \to C_n(X) \to C_n(X,A) \to 0, \text{\ \ where } C_n(X,A)= C_n(X)/C_n(A).$$
\[Proposition 8.5\]
1. Assume that for a submulti-shelf $A\subset X$ there is an operations-preserving retraction $r: X \to A$. Then $r$ extends to a (chain complex) split of the above short exact sequence $\tilde{r}:{{\mathbb Z}}X^{n+1} \to {{\mathbb Z}}A^{n+1}$. In particular, $H_n(X)=H_n(A) \oplus H_n(X,A)$.
2. If $\{t\}\subset X$ is a one element submulti-shelf of $X$, then $X\to \{t\}$ is a multi-shelf retraction, thus, by (i) $C_n(X,\{t\})$ splits and $H_n(X)=H_n(\{t\}) \oplus H_n(X,\{t\})$. We think about the homology of $\{t\}$ as a multi-shelf homology of a point, and call $H_n(X,\{t\})$ a [*reduced homology*]{}.
3. Let $$\Sigma = \Sigma_{i=1}^k a_i\neq 0, \text{\ \ then } H_n^{(a_1,...,a_n)}(\{t\})=
\begin{cases}
{{\mathbb Z}}& n = 0 \\
0 & n>0 \text{ even}\\
{{\mathbb Z}}_{\Sigma} & \text{ $n$ is odd}
\end{cases}$$ and for $\Sigma = 0$, $H_n^{(a_1,...,a_n)}(\{t\})={{\mathbb Z}}$ for any $n$.
4. Let $(X;*)$ be a shelf, $(x*t)*t=x*t$ for every $x\in X$, and $X*t$ be the orbit of the left action of $X$ on $t$ that is, $X*t=\{y\in X\ | \ y=x*t, \text{\ for some } x\in X\}$. Then $r_t=*_t: X \to X*t$ is a retraction; thus, by (i), $H_n^{(*)}(X)=H_n^{(*)}(X*t) \oplus H_n^{(*)}(X,X*t)$.
5. Let $(X;*_1,...,*_k)$ be a multi-shelf. Consider the map $h_t: C_n \to C_{n+1}$ given by $h_t(x_0,...,x_n)= (x_0,...,x_n,t)$, and the map $f_t=\sum_{i=1}^ka_i (*_i)_t$, given by $f_t(x_0,...,x_n)= \sum_{i=1}^ka_i((x_0,...,x_n)*_it)$, then $(-1)^{n+1}h_t: C_n \to C_{n+1}$ is a chain homotopy between the map $f_t$ and the zero map.
6. Let $(X;*_1,...,*_k)$ be a multi-shelf and $*_0$ the identity operation of $Bin(X)$. Let $\partial^{(a_0,a_1,...,a_k)}=\sum_{i=0}^ka_i\partial^{(*_i)}$. Then $a_0Id_X$ is chain homotopic to $-f_t= -\sum_{i=1}^ka_i (*_it)$.
7. Let $(X,*)$ be a shelf, and consider a rack boundary operation $\partial^R= \partial^{(*_0)}- \partial^{(*)}$, then for any $t\in X$, we have $f_t=*_t$ is chain homotopic to $-Id_X$ and it is a (chain complex) retraction, thus $H^R_n(X,X*t)=0$ and $H^R_n(X)= H^R_n(X*t)$ for any $t\in X$ such that $(x*t)*t=x*t$ for every $x\in X$. A generalization of this observation plays an important role in the computation of the 4-term homology of distributive lattices in [@Pr-Pu].
\(i) If $i: A \to X$ is an embedding and $\tilde{i}: C_n(A) \to C_n(X)$ its linear extension to chain complexes, then $\tilde{r}\tilde{i}=Id_A$ and $\partial \tilde{r} = \tilde{r} \partial$, so $\tilde{r}$ is a map that splits chain complex $C_n(X)$ and (i) of Proposition \[Proposition 8.5\] follows.\
(ii) It follows from (i) and idempotency $t*_it=t$.\
(iii) $C_n(\{t\}) = {{\mathbb Z}}$ with basic element $(t,t,...,t)$. The chain complex reduces to: $$... \stackrel{0}{\rightarrow} {{\mathbb Z}}\stackrel{\times \Sigma}{\rightarrow} {{\mathbb Z}}\stackrel{0}{\rightarrow} {{\mathbb Z}}\stackrel{\times \Sigma}{\rightarrow}
{{\mathbb Z}}\stackrel{0}{\rightarrow} {{\mathbb Z}}\stackrel{\times \Sigma}{\rightarrow} {{\mathbb Z}}\stackrel{0}{\rightarrow} {{\mathbb Z}}\to 0$$ and the homology follows immediately.\
(iv) This follows from (i).\
(v) We have $$\partial_{n+1}^{(a_1,...,a_k)}h_t - h_t\partial_{n}^{(a_1,...,a_k)} =
(-1)^{n+1}\sum_{i=1}^ka_i((x_0,...,x_0)*_it)$$ and (v) follows.\
(vi) This follows immediately from (v).\
(vii) This is a consequence of (vi) but it should be stressed that it is a tautology for a rack (as then $X*t=X$ for any $t$). If $(X,*)$ is not a rack, that is, there is $t$ with $*_t$ not invertible, then we have a reduction in the computation of rack homology ($\partial^R= \partial^{(*_0)}- \partial^{(*)}$) from $X$ to $X*t$.
We refer to [@Pr-Pu] for some useful generalizations of Proposition \[Proposition 8.5\].
We end this section by showing that the [*reduced early degenerate complex*]{} $(F_0,\{t\}) =s_0(C_{n-1})/C_n(\{t\})$ splits from the reduced chain complex $C(X,\{t\})$ of a multi-spindle $(X;*_1,...,*_n)$. The second factor $C(X,\{t\})/(F_0,\{t\})$ is called the [*reduced early normalized chain complex*]{} and denoted by $C^{eN}(X,\{t\})$. We also show how $F_0=\{F_n^0\}$ and $\{C_n\}$ are related.
\[Proposition 8.6\]
1. The short exact sequence of multi-spindle chain complexes: $$0 \to (F_n^0,\{t\}) \to C_n(X,\{t\}) \to C_n^{eN}(X,\{t\}) \to 0$$ splits with a split map $s_0p_0: C(X,\{t\}) \to (F_0,\{t\})$. where $p_0(x_0,x_1,...,x_n)=(x_1,...,x_n)$.
2. $s_0: C_{n-1}(X)\otimes {{\mathbb Z}}_{\sum_{i=1}^ka_k} \to s_0{C_{n-1}}\otimes {{\mathbb Z}}_{\sum_{i=1}^ka_k}$ yields an isomorphism on $mod (\sum_{i=1}^ka_k)$ homology.
Proposition \[Proposition 8.6\] follows from Lemma \[Lemma 8.7\] (see also [@Pr-Pu] for further developments of these ideas).
\[Lemma 8.7\]
1. The map $s_0: C_n \to C_{n+1} $ is a chain homotopy between $(\sum_{i=1}^k a_i) s_0p_0$ and the zero map. In particular $(\sum_{i=1}^k a_i)$ annihilates $H_n(F^0(X))$. Furthermore, $s_0p_0$ is a chain that splits the chain complex of Proposition \[Proposition 8.6\](i).
2. The map $p_0: C_n \to C_{n-1} $ is a chain homotopy between $(\sum_{i=1}^k a_i)p_0p_0$ and the zero map. Furthermore, $p_0p_0$ is a chain map.
3. If $(\sum_{i=1}^k a_i)=0$, then $(-1)^ns_0$ and $(-1)^np_0$ are chain maps (we write $\sigma$ for $(-1)^ns_0$). Furthermore, $p_0s_0= Id_{C_n}$ and $s_0p_0=Id_{F^0}$. In particular, $\sigma: C_n \to F^0_{n+1}$ is an isomorphism of chain complexes.
4. More generally, $\sigma \otimes Id$ is a chain complex isomorphism $C_n(X)\otimes {{\mathbb Z}}_{\Sigma} \to F^0_{n+1}\otimes {{\mathbb Z}}_{\Sigma}$. In particular, $H_n(X,{{\mathbb Z}}_{\Sigma})$ is isomorphic to $H_{n+1}(F_0,{{\mathbb Z}}_{\Sigma})$.
\(i) We use the fact that $d_0s_0=d_1s_0 = (\sum_{i=1}^k a_i)Id_{C_n}$ and that $(C_n,d_i,s_i)$ is a weak simplicial module and, in particular, $d_is_0=s_0d_{i-1}$ for $i>1$. Thus we have: $$\partial^{(a_1,...,a_k)}s_0 + s_0\partial^{(a_1,...,a_k)}= \sum_{i=0}^{n+1}(-1)^{i}d_is_0 +
\sum_{i=0}^n(-1)^{i}s_0d_i=$$ $$(d_0s_0- d_1s0) + \sum_{i=2}^{n+1}(-1)^{i}d_is_0 + \sum_{i=0}^n(-1)^{i}s_0d_i=$$ $$\sum_{i=2}^{n+1}(-1)^{i}s_0d_{i-1} + \sum_{i=0}^n(-1)^{i}s_0d_i=$$ $$\sum_{i=1}^{n+1}(-1)^{i+1}s_0d_{i} + \sum_{i=0}^n(-1)^{i}s_0d_i= s_0d_0 =
(\sum_{i=1}^k a_i) s_0p_0.$$ $(\sum_{i=1}^k a_i) s_0p_0$ is a chain map, and because $C_n$ is a complex of free groups, the map $s_0p_0$ is a chain map.\
We also can check directly that $s_0p_0: C_n \to C_n$ is a chain map that is a (chain) retraction to $F^0_n=s_0C_{n-1}$. First, $s_0p_0$ is the identity on $F^0_n$; further we have: $$(d^{(*)}_0s_0p_0 -s_0p_0d^{(*)}_0)(x_0,x_1,x_2,...,x_n)= (x_0,x_1,x_2,...,x_n)- (x_1,x_1,x_2...,x_n),$$ $$(d^{(*)}_1s_0p_0 -s_0p_0d^{(*)}_1)(x_0,x_1,x_2,...,x_n)= (x_0,x_1,x_2,...,x_n)- (x_1,x_1,x_2...,x_n),$$ $$(d^{(*)}_is_0p_0 -s_0p_0d^{(*)}_i)(x_0,x_1,x_2,...,x_n)=0 \text{ for $i>1$ }.$$ Thus $\partial^{(*)}s_0p_0 = s_0p_0\partial^{(*)}$ and finally $\partial^{(a_1,...,a_k)}s_0p_0 = s_0p_0\partial^{(a_1,...,a_k)}$.
\(ii) We notice that $d_0p_0= (\sum_{i=1}^k a_i)p_0p_0$ and $d_ip_0=p_0d_{i+1}$. Thus: $\partial^{(a_1,...,a_k)}p_0 + p_0\partial^{(a_1,...,a_k)}= (\sum_{i=1}^k a_i)p_0p_0$ and (ii) follows.\
(iii) For $\sum_{i=1}^k a_i =0$ we directly see that $(-1)^ns_0$ and $(-1)^np_0$ are chain maps. (iv) We see immediately that $s_0p_0=Id_{F_0}\mod \Sigma$ and $p_0s_0=Id_{C_n} \text{ mod } \Sigma$.
Examples {#Examples 9}
========
In this section we illustrate our theory by various calculations of homology of multi-spindles. With the exception of racks (e.g. [@N-P-2; @Nos; @Cla]) no calculations were done before. We offer calculations of varying difficulties, starting from two-term homology. In Subsection 9.4 we make a detailed calculation using the following idea: in the homology of a point, the chain groups $C_n(\{t\})$ are one-dimensional which makes the computation easy (see Proposition \[Proposition 8.5\] (iv)). For $|X|>1$ the chain groups grow exponentially, but there is one case when the computation is not difficult, but still illuminating: the case of $|X|=2$ and normalized homology, in which $C_n(X)$ is two-dimensional. For example, it works nicely for the group homology of ${{\mathbb Z}}_2$ and for the Hochschild homology of ${{\mathbb Z}}({{\mathbb Z}}_2)={{\mathbb Z}}[x]/(x^2-1)$, or ${{\mathbb Z}}[x]/(x^2)$ (the underlying ring of Khovanov homology). Here we show the calculation for a $4$-term distributive homology of a 4-spindle (in fact, the maximal multi-spindle for $|X|=2$; see Subsection 9.3 and the 2-element Boolean algebra $B_1$).
The case of 2-term homology with $\partial^{(a,d)}=a\partial^{(*_0)}+ d\partial^{(*_{\sim})}$
---------------------------------------------------------------------------------------------
Define $*_{\sim}: X\times X \to X$ as the left trivial operation, that is $a*_{\sim}b=b$ (we will explain our notation in the section on Boolean algebras).
Below we consider the homology of the chain complex $(C(X);\partial^{(a,d)})$ where $\partial^{(a,d)}=a\partial^{(*_0)}+ d\partial^{(*_{\sim})}$. This generalizes Theorem \[Theorem 6.6\] for $g=Id$ and is further generalized in [@Pr-Pu].
\[Theorem 9.1\]
1. The chain complex $(C_n(X),\partial^{(a,d)})$ splits into three pieces:\
(i) $C_n(\{t\})$, the chain complex of a point (we fix a point $t\in X$),\
(ii) $F_0(X,\{t\})=\{F^0_n(X,\{t\})\}=\{F^0_n/C_n(\{t\})\}=\{s_0C_{n-1}/C_n(\{t\})\}$, the reduced early degenerate chain complex, and\
(iii) $C_n^{eN}(X,\{t\})= C_n(X,\{t\})/F^0_n$, the reduced early normalized chain complex.\
2. If $a+d\neq 0$, then $H_n(\{t\})=
\begin{cases}
{{\mathbb Z}}& \text{if $n = 0$} \\
0 & \text{if $n$ is even and } n>0 \\
{{\mathbb Z}}_{a+d} & \text{if $n$ is odd}
\end{cases}
$\
If $a+d=0$ then $H_n(\{t\})={{\mathbb Z}}$.
3. For a finite $X$, and $a$ or $d$ different from $0$ we have:\
$H_n(F_0(X,\{t\}))=
\begin{cases}
{{\mathbb Z}}_{gcd(a,d)}^{u_n-1} & \text{ if $n$ is even} \\
{{\mathbb Z}}_{gcd(a,d)}^{u_n} & \text{ if $n$ is odd}
\end{cases} $\
where $u_n(|X|)=u_n$ is defined by: $u_0=1$, $u_1=|X|-1$, and $u_n+u_{n-1} = |X|^n$, that is $u_n= |X|^n-u_{n-1}= |X|^n-|X|^{n-1}+...+(-1)^n=
\frac{|X|^{n+1}+(-1)^n}{|X|+1}$.
4. For a finite $X$, and $a\neq 0$, we have $$H_n(X,\{t\})/F_0)=
{{\mathbb Z}}_a^{u_{n+1}-u_n+(-1)^n}.$$
5. If $a\neq 0$, and $a+d\neq 0$ then\
$H_n(X)=
\begin{cases}
{{\mathbb Z}}\oplus {{\mathbb Z}}_a^{|X|-1}& \text{if $n = 0$} \\
{{\mathbb Z}}_a^{u_{n+1}-u_n+1}\oplus {{\mathbb Z}}_{gcd(a,d)}^{u_{n}-1} & \text{if $n$ is even and } n>0 \\
{{\mathbb Z}}_{a+d} \oplus {{\mathbb Z}}_a^{u_{n+1}-u_n -1}\oplus {{\mathbb Z}}_{gcd(a,d)}^{u_{n}} & \text{if $n$ is odd}
\end{cases}
$\
The case of $a=0$ was already considered in Corollary \[Corollary 6.7\]. The case of $a+d=0$ differs only from the general case in the factor $H_n(\{t\})$ so can be easily derived from (2)-(4).
\(1) This follows from Propositions \[Proposition 8.5\] and \[Proposition 8.6\].\
(2) This is a special case of Proposition \[Proposition 8.5\](iii).\
(3) This follows from (4) and Lemma \[Lemma 8.7\](iii).\
(4) First we notice that $\partial^{(a,d)}= a\partial^{(*_0)}$ in our chain group. The result follows from the fact that for $a=1$ we get an acyclic chain complex and from a careful analysis of the rank of $\partial_n(C^{eN})$.\
(5) This is the summary of (2)-(4).
Example: 3-term distributive homology of a spindle with $1_r$ and $0_r$
-----------------------------------------------------------------------
For any spindle $(X,*)$ we have the 3-element distributive set $\{*_0,*,*_{\sim}\}$; we check directly: $$(x*y)*_{\sim}z=z \text{ and } (x*_{\sim}z)*(y*_{\sim}z)= z*z=z,$$ and $$(x*_{\sim}y)*z= y*z \text{ and } (x*z)*_{\sim}(y*z)= y*z.$$ Thus we can consider 3-term distributive homology of a multi-spindle $(X;*_0,*,*_{\sim})$ with the boundary operation $\partial^{(a,c,d)}= a\partial^{(*_0)} + c\partial^{(*)} + d \partial^{(*_{\sim})}$. Computation of this homology, in general, is a difficult problem as it contains quandle homology as a special case. However, for $*$ with a right unit $1_r$ (i.e. $x*1_r=x$), and a right projector $0_r$ (i.e. $x*0_r=0_r$), the solution can be obtained in a manner similar to that of Theorem \[Theorem 9.1\]. Namely, we have:
\[Theorem 9.2\]
1. The chain complex $(C_n(X),\partial^{(a,c,d)})$ splits into three pieces:\
(i) $C_n(\{1_r\})$, the chain complex of a point (we fix a point $1_r$),\
(ii) $F_0(X,\{1_r\})=\{F^0_n/C_n(\{1_r\})\}=\{s_0C_{n-1}/C_n(\{1_r\})\}$, the reduced chain complex of early degenerate elements, and\
(iii) $C_n^{eN}(X,\{t\}) =C_n(X,\{1_r\})/F^0_n$, the reduced early normalized chain complex.\
2. If $a+c+d\neq 0$, then $H_n(\{t\})=
\begin{cases}
{{\mathbb Z}}& \text{if $n = 0$} \\
0 & \text{if $n$ is even and } n>0 \\
{{\mathbb Z}}_{a+c+d} & \text{if $n$ is odd}
\end{cases}
$\
If $a+c+d=0$ then $H_n(\{1_r\})={{\mathbb Z}}$.
3. For a finite $X$, and $a$, $c$, or $d$ different from $0$, we have:\
$H_n(s_0C_{n-1}/C_n(\{1_r\})=
\begin{cases}
{{\mathbb Z}}_{gcd(a,c,d)}^{u_n-1} & \text{ if $n$ is even} \\
{{\mathbb Z}}_{gcd(a,c,d)}^{u_n} & \text{ if $n$ is odd}
\end{cases} $\
where $u_n=u_n(|X|)= |X|^n-|X|^{n-1}+...+(-1)^n$, as in Theorem \[Theorem 9.1\].
4. For a finite $X$, and $a$ or $c$ different from $0$, we have $$H_n((X,\{1_r\})/F_0)=
{{\mathbb Z}}_{gcd(a,c)}^{u_{n+1}-u_n+(-1)^n}.$$
5. If $a$ or $c$ $\neq 0$, and $a+c+ d\neq 0$ then\
$H_n(X)=
\begin{cases}
{{\mathbb Z}}\oplus {{\mathbb Z}}_{gcd(a,c)}^{|X|-1}& \text{if $n = 0$} \\
{{\mathbb Z}}_{gcd(a,c)}^{u_{n+1}-u_n+1}\oplus {{\mathbb Z}}_{gcd(a,c,d)}^{u_{n}-1} & \text{if $n$ is even and } n>0 \\
{{\mathbb Z}}_{a+c+d} \oplus {{\mathbb Z}}_{gcd(a,c)}^{u_{n+1}-u_n -1}\oplus {{\mathbb Z}}_{gcd(a,c,d)}^{u_{n}} & \text{if $n$ is odd}
\end{cases}
$\
The case $a=c=0$ was already described in Corollary \[Corollary 6.7\]. The case $a+c+d=0$ differs from other cases only at $H_n(\{1_r\})$.
The proof is a refinement of the proof of Theorem \[Theorem 9.1\]. We first consider the chain homotopy $h_{1_r}$ to get: $$(-1)^{n+1}(\partial^{(a,c,d)}h_{1_r} - h_{1_r}\partial^{(a,c,d)})(x_0,...,x_n)= a(x_0,...,x_n) +
(c+d)(1_r,...,.1_r),$$ and the chain homotopy $h_{0_r}$ to get: $$(-1)^{n+1}(\partial^{(a,c,d)}h_{0_r} - h_{0_r}\partial^{(a,c,d)})(x_0,...,x_n)= (a+c)(x_0,...,x_n) +
d(0_r,...,.0_r),$$ From this we conclude that $H_n(C,\{1_r\})$ is annihilated by $gcd(a,c)$. Further we proceed like in the proof of Theorem \[Theorem 9.1\]; see [@Pr-Pu] for details.
Example: 4-term normalized distributive homology of the 2-element Boolean algebra {#Subsection 9.3}
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Our first interesting example of a distributive monoid is given by a distributive lattice (e.g. Boolean algebra) $({\mathcal L},\cup,\cap)$ because lattice operations $\cup$ and $\cap$ form a distributive set. (We refer to [@B-D; @Gra; @Si; @Tra] for an extensive coverage of distributive lattices[^14] and Boolean algebras). In this paper, we denote these binary operations by $*_{\cup}$ and $*_{\cap}$. The distributive monoid spanned by these operations is a commutative monoid of 4 idempotent elements: $*_0$ - identity element, $*_{\cup}$, $*_{\cap}$, and the composition $*_{\sim}= *_{\cap} *_{\cup}$. One can present the monoid as: $$\{*_{\cup},*_{\cap}\ | \ *_{\cup}*_{\cap}=*_{\cap}*_{\cup}, *_{\cup}*_{\cup}=*_{\cup},
*_{\cap}*_{\cap}=*_{\cap}\}.$$ Notice that $*_{\sim}$ is the left trivial operation, $a*_{\sim}b=b$.
Using our 4-element distributive monoid we can consider the 4-term boundary operation: $\partial^{(a,b,c,d)}: C_n({\mathcal L}) \to C_{n-1}({\mathcal L})$, where $C_n({\mathcal L})= {{\mathbb Z}}{\mathcal L}^{n+1}$ and $\partial=\partial^{(a,b,c,d)} =
a\partial^{(*_0)}+ b\partial^{(*_{\cup})} + c\partial^{(*_{\cap})} + d\partial^{(*_{\sim})}$.
The computation of the four-term distributive homology of ${\mathcal L}$ is generally difficult, but it is done fully in [@Pr-Pu]; see Theorem \[Theorem 9.5\]. For normalized homology we are able to make a very elementary (and illuminating, in my opinion) calculation in the simplest nontrivial case of $B_1=\{0,1\}$, the two element Boolean algebra of subsets of the one element set. This case is approachable because $C^{Norm}_n(B_1)$ is 2-dimensional for any $n$. Choose the basis $e_n= (0,1,0,1,0,...)$, $e'_n=(1,0,1,0,1,...)$ of $C^{Norm}_n(B_1)=C_n(B_1)/C_n^D(B_1)$. To be able to deduce homology, it is enough to write $\partial$ in this basis. We have to consider the case of $n$ even and odd separately.\
Detailed calculation of the quandle homology of $\bf{X=B_1}$
------------------------------------------------------------
$$\partial^{(*_0)}(e_n) = (-1)^{n}e_{n-1}+ e'_{n-1}= (-1)^n\partial^{(*_0)}(e_n').$$
For $n$ even we have $e_n= (0,1,...,1,0)$ and $e_n'= (1,0,...,0,1)$; then $$\partial^{(*_{\cup})}(0,1,...,1,0)= (-1)^{n}e_{n-1}$$ $$\partial^{(*_{\cup})}(1,0,...,0,1)= e_{n-1}$$
For $n$ odd we have $e_n= (0,1,...,0,1)$ and $e'_n= (1,0,...,1,0)$; then $$\partial^{(*_{\cap})}(0,1,...,1,0,1)= (-1)^{n} e_{n-1} +e_{n-1}'$$ $$\partial^{(*_{\cap})}(1,0,...,0,1,0)= 0.$$ $$\partial^{(*_{\sim})}(e_n)= \partial^{(*_{\sim})}(e'_n)=0.$$ For $\partial^{(a,b,c,d)}= a\partial^{*_0} + b\partial^{*_{\cup}} + c\partial^{*_{\cap}} +d\partial^{*_{\sim}},$ and for $n$ even $$\partial^{(a,b,c,d)}(e_n)= (-1)^n(a+b)e_{n-1} + (a+c)e'_{n-1} =(-1)^n\partial^{(a,b,c,d)}(e'_n).$$ For $n$ odd: $$\partial^{(a,b,c,d)}(e_n)= (-1)^n(a+c)e_{n-1} +(a+c)e'_{n-1} ,$$ $$\partial^{(a,b,c,d)}(e'_n)= (a+b)(e_{n-1}+ (-1)^ne'_{n-1}) .$$ Therefore, we have the following matrices of relations in $C^{Norm}_n/\partial(C^{Norm}_{n+1})$. For $n$ even: $$\left(
\begin{array}{cc}
(-1)^n(a+b) & a+c \\
a+b & (-1)^n(a+c) \end{array}
\right)$$
For $n$ odd: $$\begin{pmatrix}
(-1)^n(a+c) & a+c \\
a+b & (-1)^n(a+b)
\end{pmatrix}$$
From this we get:
\[Proposition 9.3\]
1. $C^{Norm}_n/\partial(C^{Norm}_{n+1}) = {{\mathbb Z}}\oplus {{\mathbb Z}}_{gcd(a+b,a+c)}$.
2. For $n>0$, $\partial (C^{Norm}_n) = {{\mathbb Z}}$, unless $a+b=a+c=0$ in which case $\partial (C^{Norm}_n) =0$.
3. For $n>0$, $H^{Norm}_n(B_1)= {{\mathbb Z}}_{gcd(a+b,a+c)}$, unless $a+b=a+c=0$ in which case $H^{Norm}_n(B_1)= {{\mathbb Z}}\oplus {{\mathbb Z}}$.\
$H^{Norm}_0(B_1)= C^{Norm}_0/\partial(C^{Norm}_{1}) = {{\mathbb Z}}\oplus {{\mathbb Z}}_{gcd(a+b,a+c)}$.
In the proof we use the standard but important observations that\
(i) $rank(H_n) + rank (\mathrm{Im} \partial_{n+1})+ rank (\mathrm{Im} \partial_{n}) = X^n$, and\
(ii) $tor H_n(X) = tor ({{\mathbb Z}}X^{n+1}/\mathrm{Im} (\partial_{n+1})).$\
The degenerate part of the homology is much more difficult. We started with computer experiments (with the help of Michal Jablonowski and Krzysztof Putyra) and eventually proved the following:
[@Pr-Pu]\[Theorem 9.4\] Assume that $a+b+c+d \neq 0$ and $a+b\neq 0$ or $a+c\neq 0$ then $rank H_n^D(B_1)=0$ and $$H_n^D(B_1)=
\left\{ \begin{array}{rl}
{{\mathbb Z}}_{gcd(a+b,a+c)}^{a_n-1} \oplus {{\mathbb Z}}_{gcd(a+b,a+c,c+d)}^{a_n-1} &\mbox{ if $n$ is even} \\
{{\mathbb Z}}_{a+b+c+d} \oplus {{\mathbb Z}}_{gcd(a+b,a+c)}^{a_n-1} \oplus {{\mathbb Z}}_{gcd(a+b,a+c,c+d)}^{a_n} & \mbox{ if $n$ is odd}
\end{array} \right.$$ In the formula above $a_n=u_n(2) $ (see Theorem \[Theorem 9.1\]), that is, $a_0=a_1=1$, $a_n+a_{n-1}= 2^n$, and thus $a_n= 2a_{n-1}+(-1)^n=2^n-2^{n-1}+...+ (-1)^n = \frac{2^{n+1}+(-1)^n}{3}$.
More about homology for $\bf{ \partial^{(a,b,c,d)}= a\partial^{*_0} + b\partial^{*_{\cup}} + c\partial^{*_{\cap}} +d\partial^{*_{\sim}}}$
-----------------------------------------------------------------------------------------------------------------------------------------
\
Two months after a June seminar talk I gave at Warsaw Technical University, we found a general formula for the four-term distributive homology of any finite distributive lattice. For $b=c=0$ it gives Theorem \[Theorem 9.1\]. To formulate Theorem \[Theorem 9.5\] we need some basic terminology: let ${\mathcal L}$ be a distributive lattice; we say that an element a of ${\mathcal L}$ is join-irreducible if for any decomposition $a = b \cup c$, we have $a = b$ or $a = c$. Let $J({\mathcal L})$ be the set of non-minimal (different from $\emptyset$), join-irreducible elements in ${\mathcal L}$ and $J$ its cardinality. In what follows $L$ denotes the cardinality of ${\mathcal L}$. If ${\mathcal L}$ is finite, then $J$ is equal to the length of every maximal chain in ${\mathcal L}$ (see Corollary 14 in [@Gra]).
[@Pr-Pu]\[Theorem 9.5\]. Let ${\mathcal L}$ be a finite distributive lattice. Assume for simplicity that $a+b+c+d\neq 0$, $a+b$ or $a+c$ is not equal to $0$, and one of $a$, $b$ and $c$ is not equal to $0$. Then $ H^{(a,b,c,d)}_n({\mathcal L})=$ $${{\mathbb Z}}\oplus {{\mathbb Z}}_{gcd(a+b,a+c)}^{J}\oplus {{\mathbb Z}}_{gcd(a,b,c)}^{L-J-1} \text{ if $n=0$ },$$ $${{\mathbb Z}}_{gcd(a+b,a+c)}^{Ju_n(2)} \oplus {{\mathbb Z}}_{gcd(a,b,c)}^{u_{n+1}(L)-u_n(L)+1-Ju_n(2)}
\oplus {{\mathbb Z}}_{gcd(a+b,a+c,c+d)}^{Ju_n(2)-J}
\oplus {{\mathbb Z}}_{gcd(a,b,c,d)}^{u_n(L)-1-Ju_n(2)+J}$$ if $n$ is even, and $${{\mathbb Z}}_{a+b+c+d} \oplus
{{\mathbb Z}}_{gcd(a+b,a+c)}^{Ju_n(2)} \oplus {{\mathbb Z}}_{gcd(a,b,c)}^{u_{n+1}(L)-u_n(L)-1-Ju_n(2)} \oplus {{\mathbb Z}}_{gcd(a+b,a+c,c+d)}^{Ju_n(2)}
\oplus {{\mathbb Z}}_{gcd(a,b,c,d)}^{u_n(L)-Ju_n(2)}$$ if $n$ is odd.
Generalized lattices
--------------------
Our computation in [@Pr-Pu] of the four-term homology of a distributive lattice can be partially generalized and this justifies an introduction of the following multi-spindle, in which commutativity or associativity of operations are not assumed.
A [*generalized lattice*]{} $(X;*_1,*_2)$ is a set with two binary operations which satisfy the following three conditions:\
(1) Each operation is right self-distributive.\
(2) Absorption conditions hold: $(a*_1b)*_2b=b=(a*_2b)*_1b$ (in particular each action satisfies the idempotency condition).\
(3) $(a*_1b)*_1b= a*_1b$ and $(a*_2b)*_2b= a*_2b$\
If additionally our operations are right distributive with respect to each other:\
(4) $(a*_1b)*_2c= (a*_2c)*_1(b*_2c)$ and $(a*_2b)*_1c= (a*_1c)*_2(b*_1c)$, we call $(X;*_1,*_2)$ a [*generalized distributive lattice*]{}.
We should comment here that absorption implies that $*_1*_2=*_2*_1=*_{\sim}$ and idempotency of each operation $a*_1a=a=a*_2a$ (we have: $((a *_1 a) *_2 a) *_1 a = a*_1 a$ (absorption for $b=a$, i.e. $(a *_1 a) *_2 a = a $). We also have $((a *_1 a) *_2 a) *_1 a = a$ (absorption for $b=a *_1 a)$; thus $a *_1 a = a$. The monoid in $Bin(X)$ generated by $(*_1,*_2)$ is isomorphic to the four element monoid from classical (distributive) lattices (Subsection 9.3).
Motivation from Knot Theory
===========================
The fundamental result in combinatorial knot theory, envisioned by Maxwell and proved by Reidemeister and Alexander and Briggs around 1927, is that links in $R^3$ are equivalent (isotopic) if and only if their diagrams are related by a finite number of local moves (now called Reidemeister moves). Three Reidemeister moves are illustrated in Figures 10.2- 10.4; see [@Prz-1; @Prz-2] for an early history of knot theory. Thus, one can think about classical knot theory as analyzing knot diagrams modulo Reidemeister moves. One can, naively but successfully, construct knot invariants as follows: choose a set $X$ with a binary operation $*: X \times X \to X$, and consider “colorings" of arcs of an oriented diagram $D$ (arcs are from undercrossing to undercrossing) by elements of $X$ so that, at every crossing, the coloring satisfies the condition from Fig. 10.1. This gives a different condition for a positive and negative crossing, which can be put together as in Fig. 10.1 (iii) (here only the overcrossing has to be oriented and, of course, we need an orientation of the plane of the projection). We interpret the use of the operation $*$ as saying that an overcrossing is acting on an undercrossing. We define a diagram invariant $col_X(D)$ as a cardinality of a set of all allowed colorings of $D$, that is, $col_X(D)= |\{f:arcs(D) \to X\ | \ f \text{ satisfies the rules of Fig. 10.1} \}|$.\
\
Figure 10.1; local coloring by $(X,*)$
In order to be a link invariant, $col_X(D)$ should be invariant under the Reidemeister moves, which provides motivation for the axioms of a quandle.
1. The first Reidemeister move of Figure 10.2 requires the idempotency $a*a=a$ (left part), and in the case of the right part we need a unique solution $x=a$ for the equation $x*a=a$, this follows from the idempotency and invertibility of $*$.
\
\
Figure 10.2; $col_X(R_1(D))= col_X(D)$
1. The second Reidemeister move requires invertibility of $*$. In fact, the move from Fig. 10.3(i) requires $*_b$ to be injective ($a*b=a'*b \Rightarrow a=a'$) and that of Fig. 10.3(ii) requires $*_b$ to be bijective (for any $a,b$ there is the unique $x$ such that $a=x*b$).
\
\
Figure 10.3; $col_X(R_2(D))= col_X(D)$
1. We illustrate the need for right self-distributivity of $*$ in Figure 10.4, where we choose all crossings to be positive. If $*$ is also invertible, then all other choices of orientation follow as well (Proposition \[Proposition 2.4\] can be used then).\
\
\
\
Figure 10.4; $col_X(R_3(D))= col_X(D)$
Motivation for degenerate chains and quandle homology {#Subsection 10.1}
------------------------------------------------------
Carter, Kamada and Saito at al. noticed in 1998 [@CKS; @CJKLS] that if one colors a link diagram $D$ by elements of a given quandle $(Q,*)$ as described above, and then considers a sum over all crossings of $D$ of pairs in $Q^2$, $\pm (a,b)$, according to the following convention:\
\
\
Figure 10.5; building the 1-chain for an oriented link diagram and its coloring
\
then the sum $c(D)=\sum_{v \in \{Crossings\}} sgn(v)(a(v),b(v))$ is not only a 1-chain, but is a 1-cycle in $C_1(Q)$, and its class in the first homology $H^Q_1(Q)$ is invariant under Reidemeister moves. We show this carefully, and in particular, stress the difference (shift) in grading. The history of discovering quandle homology is surveyed in [@Car].
1. Carter, Kamada, and Saito have considered cocycle invariants, and in their convention, an element $Q^2 \to {{\mathbb Z}}$ is a 2-cocycle. For us, however, the sum constructed above is a 1-cycle, an element of $C_1(Q)={{\mathbb Z}}Q^2$. The (rack or quandle) boundary operation they consider is $\partial^R= \partial^{(*)}- \partial^{(*_0)}$, and for ${{\mathbb Z}}Q^2$ we get $\partial^R(x_0,x_1)= x_0 - x_0*x_1$. In our case, if a contribution of the crossing $v$ is $sgn(v) (a,b)$, then its contribution to $\partial^R(c(D)$ is $sgn(v) (x_0 - x_0*x_1)$. Figure 10.5 informs us that the contribution is exactly the difference between the label of an undercrossing at the entrance minus the label at the exit (when moving according to the orientation). Thus, obviously, each component contributes zero to $\partial^R(c(D))$, thus $\partial^R(c(D))$ is a 1-cycle.
2. The first Reidemeister move introduces $(a,a)$ into the sum, so we have to declare it to be zero; here the need to consider normalized or quandle homology arises. Now $\partial^Q: C/C^D \to C/C^D$. The second Reidemeister move always works as the crossings involved in it have opposite signs, so the new contribution to $w(c)$ cancels.
3. With the third Reidemeister move, we consider the move from Figure 10.4 (it requires some topological manipulation, but it is well known that it is sufficient). Thus the contributions to $c(D)$ of three crossings from the top diagram is $(b,c)+(a,c)+(a*c,b*c)$ and of the bottom diagram is $(a,b)+ (a*b,c) + (b,c)$. Now $\partial^R(a,b,c)= (a,c)-(a,b)-(a*b,c)+(a*c,b*c)$, which is exactly $c(D)-c(R_3(D))$. Thus $c(D)$ and $c(R_3(D))$ are homologous in $H_1^R(D)$ and $H^Q_1(D)$.
4. We showed that if $(Q,*)$ is any quandle and we choose a quandle coloring of $D$, then $c(D)$ yields an element of $H_1^Q(Q)$ preserved by all Reidemeister moves. However, if $(Q,*)$ is only a rack, then $c(D)$ is an invariant of $R_2$ and $R_3$ (a so called invariant of regular isotopy), thus $c(D)$ yields an element of $H_1^Q(Q)$ invariant up to regular isotopy.
5. One can improve (iv) slightly and make our cycle invariant $c(D)$ more useful by noting that $c(D)$ yields an invariant of framed isotopy. Here we observe that we can move a “kink" of the first Reidemeister move under another arc using $R_2$ and $R_3$ only, and cancel contributions from “kinks" of the opposite sign, as long as they are in the same component.
The above considerations have been generalized to surfaces in 4-space, or more generally, to codimension two embeddings; in fact, it was an initial motivation for Fenn, Rourke, and Sanderson to introduce rack homology around 1990. There is another remarkable cocycle invariant developed in [@R-S; @CKS] for codimension 2 embeddings, coming from shadow colorings by elements of $(X;*)$. It is a 3-cocycle invariant in classical knot theory (we formulate it below in a homology language and with a dimension shift; thus we construct a 2-cycle in $C_n(X)$).
[@R-S; @CKS]\[Definition 10.1\] Let $(X,*)$ be a rack and $D$ an oriented link diagram. We decorate arcs of $D$ by elements of $X$ as in the previous definition (Figure 10.1). Additionally, we color regions of $R^2 -D$ by elements of $X$ according to the convention: (the small arrow is added to record a positive orientation of the projection surface). For a given shadow coloring we define a 2-cycle $c_2(D)\in C_2^R(X)$ as the sum over all crossings of $D$ of terms $\pm (x,a,b)$ according to the convention of Figure 10.6:\
\
\
Figure 10.6; building a 2-chain for an oriented link diagram using a shadow coloring
\
One can check that $c_2(D)$ is a 2-cycle in $C_2(X)$. Further, $c_2(D)$ is preserved by the second Reidemeister move (to see the cancellation of contributions from two new crossings after $R_2$, we should just put together crossings of Figure 10.6). With a little more effort one shows that $c_2(R_3(D)) - c_2(D)$ is a boundary (e.g. if we shade regions of Figure 10.4, with the bottom region labelled by $x$, then $c_2(R_3(D)) - c_2(D) = \partial (x,a,b,c)$). Thus $c_2(D)$ and $c_2(R_3(D)$ are homologous in $H_2^R(D)$. To summarize, the homology class of $c_2(D)$ is a regular isotopy invariant.\
If $(X,*)$ is a quandle, we can work with quandle homology $H^Q_2(X)$, and because the contribution of the new crossing in a first Reidemeister move is a degenerate element, the class of $c_2(D)$ in $H^Q_2(X)$ is preserved by all Reidemeister moves.
If we only care about the third Reidemeister move of Figure 10.4, we can work with any shelf $(X,*)$. The usefulness of working only with some Reidemeister moves may be debated, but there is already a considerable body of literature on the topic [@CESS].
\[Remark 10.2\] Recall that the map $p_0: C_n(X) \to C_{n-1}(X)$ is given by\
$p_0(x_0,x_1,...,x_n) =
(x_1,...,x_n)$ and that, as noted in Lemma \[Lemma 8.7\], $(-1)^{n+1}p_0$ is a chain map on $(C_n\otimes {{\mathbb Z}}_{\Sigma}, \partial_n^{(a_1,...,a_k)})$. If $\Sigma =\sum_{i=1}^ka_i =0$, as is the case for rack homology, then $(-1)^{n+1}p_0$ is a chain map. Our observation is that $p_0(c_2(D))= c(D)$, which follows from the construction, but should have some interesting consequences. It is true, in general, that for a given $n$-dimensional “diagram" $D$ of an $n$-dimensional manifold in $R^{n+1}$, the $n$-chain corresponding to a shadow coloring of $D$ is sent by $p_0$ to a coloring of $D$. We plan to address the significance of this in [@P-R].
Yang-Baxter Homology?
=====================
From self-distributivity to Yang Baxter equation
------------------------------------------------
Let $(X;*)$ be a shelf and $kX$ a free module over a commutative ring $k$ with basis $X$ (we can call $kX$ a [*linear shelf*]{}). Let $V=kX$, then $V\otimes V = k(V^2)$ and the operation $*$ yields a linear map $R=R_{(X;*)}: V\otimes V \to V\otimes V$ given by $R(a,b)=(b,a*b)$. Right self-distributivity of $*$ gives the equation of linear maps $V\otimes V \otimes V \to V\otimes V\otimes V$: $$(R\otimes Id)(Id \otimes R)(R\otimes Id) = (Id \otimes R)(R\otimes Id)(Id \otimes R).$$ In general, the equation of type (1) is called a Yang-Baxter equation and the map $R$ a Yang-Baxter operator. We also often require that $R$ is invertible. With relation to this, we notice that if $*$ is invertible then $R_{(X;*)}$ is invertible with $R^{-1}_{(X;*)}(a,b)= (b\bar * a, a)$.
In our case $R_{(X;*)}$ permutes the base $X\times X$ of $V\otimes V$, so it is called a permutation or a set theoretical Yang-Baxter operator. Our distributive homology, in particular our rack homology $(C_n,\partial^R=\partial^{(*)}-\partial^{(*_0)})$, can be thought of as the homology of $R$. It was generalized from the Yang-Baxter operator coming from a self-distributive $*$ to any permutational Yang-Baxter operator (coming from biracks or biquandles), [@CES-2]. For a general Yang-Baxter operator, there is no general homology theory (compare [@Eis-1; @Eis-2]). The goal/hope is to define homology for any Yang-Baxter operator, so that the Yang-Baxter operator defining the Jones polynomial leads to a version of Khovanov homology.
Acknowledgements
================
I was partially supported by the NSA-AMS 091111 grant, by the Polish Scientific Grant: Nr. N-N201387034, and by the GWU REF grant.
I would like to thank participants of my lectures, readers of early versions of this paper, and the referee for many useful comments and suggestions.
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\
Address:\
Department of Mathematics,\
George Washington University,\
przytyck@gwu.edu\
and Gdańsk University
[^1]: From my Summer 2010 talk at Knots in Poland III to a seminar at Warsaw Technical University in June 2011 (e.g. Knots in Chicago conference, AMS meeting in Chile, Conference in San Antonio, Colloquium at U. Louisiana, seminars at GWU, Columbia University, George Mason U., Universidad de Valparaiso, SUNY at Buffalo, University of Maryland, Warsaw University and Gdansk University, and Knots in Washington XXXI and XXXII). I am grateful for the opportunity given and stress that I gained a lot from interaction with the audience.
[^2]: Notice that $*_0$ and $*$ are seldom associative, as $(a*_0b)*c=a*c$ but $a*_0(b*c)=a$.
[^3]: Functions satisfying this property are called functionally equal. The property holds, for example, for for a semigroup with the left cancellation property $xa=xb \Rightarrow a=b$, or an abelian semigroup whose elements are all idempotent (if $xa=xb$ for every $x$, then $a=aa=ab=ba=bb=b$).
[^4]: For simplicity we work mostly with abelian groups, i.e. ${{\mathbb Z}}$-modules, but we could also assume that we work with $R$-modules, where $R$ is a ring with identity.
[^5]: According to [@Fra], a pseudo-simplicial module $(M_n,d_i,s_i)$ satisfies only conditions (1),(3),(4) of Definition \[Definition 3.3\] [@Ti-Vo; @In]. An almost-simplicial module satisfies conditions (1)-(4) of Definition \[Definition 3.3\] except $s_is_i= s_{i+1}s_i$. A pseudo-simplicial module satisfies the Eilenberg-Zilber Theorem described in [@Fra1] and proved in [@In].
[^6]: We find it convenient to also allow an empty simplex, say of dimension $-1$; it will lead to augmented chain complexes.
[^7]: Sometimes called a left projector.
[^8]: It holds if a subshelf $A=*(X\times X)=\{z\in X\ | \ z=x*y \text{ for some $x,y\in X$} \}$ is a spindle. For example, if there is a retraction $p: X \to A$ of $X$ to a spindle $A$ with $x_1*x_2=p(x_1)*p(x_2)$. Two basic examples are: $a*_gb=g(b)$, $g^2=g=p$ and $a*_fb=f(a)$, $f^2=f=p$; this idea is considered in [@P-S].
[^9]: The recent paper by Roger Fenn, [@Fenn] states: “Unusually in the history of mathematics, the discovery of the homology and classifying space of a rack can be precisely dated to 2 April 1990.”
[^10]: I did not see this concept considered in literature, but it seems to be important in K.Putyra’s work on odd Khovanov homology [@Put]; see also Proposition \[Proposition 7.2\].
[^11]: Historical note: The concept of a monoid of operations on a set $X$, $Bin(X)$, can be found in a classical literature, e.g. [@R-S], however a multi-term distributive homology which followed, while motivated by rack and quandle homology, was only conceived in July 2010 at the end of my visit to Gdansk and before Knots in Poland III. Seeds of the concepts were in the paper [@N-P-4] and the following:
1. If $*: X\times X$ is a right self-distributive binary operation on $X$ then $$*^k=
\underbrace{**...*}_{\mbox{$k$-times}}: X\times X \to X,$$ is also self-distributive.
2. If $*_1$ and $*_2$ are right self-distributive operations that are also right distributive with respect to each other then the composition $*_1*_2$ is right self-distributive.
3. $*_0$ defined by $a*_0b=a$ is right distributive with respect to any other operation $(a*b)*_0c= (a*_0c)*(b*_0c)$ and $(a*_0b)*c= (a*c)*_0(b*c)$.
4. If two binary operations $*_1$ and $*_2$ are distributive with respect to each other then $$\partial^{(*_2)}\partial^{(*_1)}=-\partial^{(*_1)}\partial^{(*_2)}.$$
5. If $*_1$ and $*_2$ are self-distributive and distributive with respect to each other then $\partial^{(a_1,a_2)}= a_1\partial^{(*_1)}+a_2\partial^{(*_2)}$ leads to a chain complex (i.e. $\partial^{(a_1,a_2)}\partial^{(a_1,a_2)}=0$).
[^12]: We use a standard convention for products in non-associative algebras, called the left normed convention, that is, whenever parentheses are omitted in a product of elements $a_1$, $a_2,\ldots,$ $a_n$ of $X$ then $a_1*a_2*\ldots *a_n=((\ldots ((a_1*a_2)*a_3)*\ldots)*a_{n-1})*a_n$ (left association), for example, $a*b*c=(a*b)*c$).
[^13]: The observation that rack or quandle homology are the same for $(X;*)$ for $(X;\bar *)$ was proven first by S.Kamada and was known to the authors of [@FRS].
[^14]: In our language a distributive lattice is a multi-spindle $({\mathcal L},\cup,\cap)$ with commutative and associative operations, satisfying absorption axioms: $(a\cup b)\cap b = b = (a\cap b)\cup b$.
|
---
abstract: 'We first review the derivation of the exact expression for the average distance $\left \langle r_n \right \rangle$ of the $n$-th neighbour of a reference point among a set of $N$ random points distributed uniformly in a unit volume of a $D$-dimensional geometric space. Next we propose a mean-field theory of $\left \langle r_n \right \rangle$ and compare it with the exact result. The result of the mean-field theory is found to agree with the exact expression only in the limit $D \to \infty$ and $n \to \infty$. Thus the mean-field approximation is useless in this context.'
author:
- |
P. Bhattacharyya, B. K. Chakrabarti and A. Chakraborti\
\
[Saha Institute of Nuclear Physics,]{}\
[Sector - I, Block - AF, Bidhannagar, Kolkata 700 064, India]{}
title: 'The average distance of the $n$-th neighbour in a uniform distribution of random points'
---
Introduction to the average $n$-th neighbour distance
=====================================================
Consider $N$ (a large number) points distributed randomly and uniformly in a unit volume of a $D$-dimensional geometric space. A point is said to be the $n$-th neighbour of another (the reference point) if there are exactly $n - 1$ other points that are closer to the latter than the former. The average distance to the first neighbour is exactly known [@Chandrasekhar1943]; though originally calculated in three dimensions the method can be used for any finite dimension $D$ : The probability distribution $P(r_1) {\mathrm d}r_1$ of the first neighbour distance is defined by the probability of finding the first neighbour of a given reference point at a distance between $r_1$ and $r_1 + {\mathrm d}r_1$ :
$$P(r_1) \: {\mathrm d}r_1 = \left [ 1 - V(r_1) \right ]^{N - 1} \:
(N - 1) \: {\mathrm d}V(r_1) ,
\label{eq:prob-1}$$
where $V(r_1) = \pi^{D/2} \cdot (r_1)^D / \Gamma(D/2 + 1)$ is the volume of a $D$-dimensional hypersphere of radius $r_1$ centered at the reference point. The average first neighbour distance is defined as :
$$\left \langle r_1 \right \rangle = \int_0^R \: r_1 \: P(r_1) \:
{\mathrm d}r_1 , \label{eq:defn-av1}$$
where $R$ is the radius of a $D$-dimensional hypersphere of unit volume :
$$R = {1 \over \pi^{1/2}} \: \left [\Gamma \left ({D \over 2}
+ 1\right )\right ] ^{1/D} . \label{eq:unitvol-rad}$$
With the probability distribution of equation \[eq:prob-1\] we get
$$\begin{aligned}
\left\langle r_1\right\rangle & = & \int_0^1 r_1 \: [1 - V(r_1)]^{N - 1} \:
(N - 1) \: {\mathrm d}V(r_1)
\nonumber \\
~ & = & {1 \over \pi^{1/2}} \:
\left [ \Gamma\left({D \over 2} + 1\right ) \right ]^{1/D}
\Gamma \left ( 1 + {1 \over D} \right ) \: \left ( 1 \over N \right )^{1/D} .
\label{eq:av-1}\end{aligned}$$
Now we address the general problem : What is the form of the average $n$-th neighbour distance, for any finite $n$? Though this is a problem of purely geometric nature, the quantity $\left\langle r^{(D)}_N(n)\right\rangle$ is relevant in physical and computational contexts; for example, in astrophysics we need to know the average distance between neighbouring stars distributed independently in a homogeneous universe [@Chandrasekhar1957], and in the traveling salesman problem we need the average distance of the neighbours of each site for estimating the optimal path-length [@Beardwood1959].
We proceed by extending the line of argument used in the case of the first neighbour [@Chandrasekhar1943] to the $n$-th neighbour. The probability distribution of the $n$-th neighbour distance $r_n$ is defined as the probablity $P(r_n) {\mathrm d}r_n$ of finding the $n$-th neighbour of a given reference point at a distance between $r_n$ and $r_n + {\mathrm d}r_n$. This is a [*conditional probability*]{} because we look for the $n$-th neighbour of a point when its first $(n - 1)$ neighbours have already been located :
$$P\left (r_n\right ) \: {\mathrm d}r_n= \left [ 1 - {V(r_n) -
V(r_{n - 1}) \over
1- V(r_{n - 1})}\right ]^{N - n} \:
{(N - n) \: {\mathrm d}V(r_n) \over 1 - V(r_{n - 1})} .
\label{eq:prob-n}$$
The quantity $V(r_n)$ is the volume of a $D$-dimensional hypersphere of radius $r_n$ centered at the reference point. For a given reference point and its first $n - 1$ neighbours the average $n$-th neighbour distance is obtained as :
$$\left \langle r_n\right\rangle_{\mathrm (particular)}
= \int_{r_{n-1}}^R \: r_n \: P\left ( r_n\right) \: {\mathrm d}r_n
\label{eq:particular}$$
where, as before, $R$ is the radius of a $D$-dimensional hypersphere of unit volume. The quantity $\left \langle r_n\right\rangle_{\mathrm (particular)}$ is a function of a particular $r_{n-1}$, $r_{n-2}$, $\ldots$, $r_1$ which are the distances of the first $n-1$ neighbours of the given reference point. To calculate the ensemble average of $r_n$ the quantity $\left \langle r_n\right\rangle_{\mathrm (particular)}$ must be averaged successively over the probability distributions of each of the first $n-1$ neighbours :
$$\begin{aligned}
\left \langle r_n\right\rangle &
= & \int_0^R \: {\mathrm d}r_1 \: P(r_1) \:
\int_{r_1}^R \: {\mathrm d}r_2 \: P(r_2) \: \cdots
\int_{r_{n-3}}^R \: {\mathrm d}r_{n-2} \: P(r_{n-2})
\nonumber \\
~ & ~ & \times
\int_{r_{n-2}}^R \: {\mathrm d}r_{n-1} \: P(r_{n-1}) \:
\int_{r_{n-1}}^R \: {\mathrm d}r_n \: r_n \: P( r_n)
\label{eq:general}\end{aligned}$$
where the probability distribution of the $i$-th neighbour is given by equation \[eq:prob-n\] with $i$ replacing $n$. After a change in the order of the integrals in equation \[eq:general\] :
$$\begin{aligned}
\left \langle r_n\right\rangle &
= & (N - 1) (N - 2) \cdots (N - n) {\left [\Gamma\left ({D \over 2}
+ 1\right )\right ]^{1/D} \over \pi^{1/2}}
\nonumber \\
~ & ~ & \times \int_0^1 \: {\mathrm d}V(r_n) \:
\left [V(r_n)\right ]^{1/D} \:
\left [1 - V(r_n)\right ]^{N - n}
\int_0^{V(r_n)} \: {\mathrm d}r_1 \nonumber \\
~ & ~ & \times
\int_{V(r_1)}^{V(r_n)} \: {\mathrm d}r_2 \:
\cdots
\int_{V(r_{n-3})}^{V(r_n)} \:
{\mathrm d}r_{n-2} \:
\int_{V(r_{n-2})}^{V(r_n)} \:
{\mathrm d}r_{n-1}
\label{eq:changed-general}\end{aligned}$$
which gives the final form of the average $n$-th neighbour distance :
$$\begin{aligned}
\left \langle r_n\right \rangle & = &
\int_0^1 \: \left ( \begin{array}{c}
N - 1\\
n - 1
\end{array}
\right ) \: \left [V(r_n)\right ]^{n + (1/D) -1} \:
\left [1 - V(r_n)\right ]^{N - n} (N - n) \:
{\mathrm d}V(r_n) \nonumber \\
~ & = & {1 \over \pi^{1/2}} \:
\left [\Gamma\left ({D \over 2} + 1\right )\right ]^{1/D} \:
{\Gamma\left (n + {1 \over D}\right ) \over \Gamma(n)} \:
\left ({1 \over N}\right )^{1/D} .
\label{eq:percus-av}\end{aligned}$$
This result was reported in [@Percus1996].
Next we consider fluctuations $\delta r_n$ occuring in $r_n$. This can be calculated exactly for any neighbour $n$ ; the mean square deviation in $r_n$ from its average value is given by :
$$\begin{aligned}
\left (\delta r_n\right )^2 & = & \left\langle r_n^2\right\rangle
- \left\langle r_n\right\rangle^2
\nonumber \\
~ & = & {1 \over \pi} \left [\Gamma\left ({D \over 2} + 1\right )\right ]
^{2/D}
\left [{\Gamma\left (n + {2 \over D}\right ) \over \Gamma(n)}
- {\Gamma^2\left (n + {1 \over D}\right ) \over \Gamma^2(n)}
\right ] \left (1 \over N\right )^{2/D}
\label{eq:fluct}\end{aligned}$$
which vanishes as $D \to \infty$. This suggests that the form of $\left\langle r_n\right\rangle$ for large $D$ can be arrived at by neglecting fluctuations, an approach which corresponds to mean-field theories in statistical mechanics.
A mean-field theory
===================
By the following mean-field argument we derive an expression for the average $n$-th neighbour distance in large dimensions $D$. Since the average first neighbour distance can be found easily, we derive $\left\langle r_n\right\rangle$ in terms of $\left\langle r_1\right\rangle$. As before we consider $N$ (a large number of) random points distributed uniformly within a unit volume of a $D$-dimensional geometric space. We choose any one of them as the reference point and locate its $n$-th neighbour. Neglecting fluctuations, which we can do for large $D$, the distance between them is $r_n(N) \approx
\left\langle r_n(N)\right\rangle$. Keeping these two points fixed we change the number of points in the unit volume to $N \alpha$ by adding or removing points at random; the factor $\alpha$ is arbitrary to the extent that $N \alpha$ and $n \alpha$ are natural numbers. Since the distribution of points is uniform, the hypersphere that had originally enclosed just $n$ points will now contain $n \alpha$ points. Therefore, what was originally the $n$-th neighbour of the reference point now becomes the $n \alpha$-th neighbour. Since the two points under consideration are fixed, so is the distance between them. Consequently,
$$\left\langle r_n(N)\right\rangle
\approx \left\langle r_{n \alpha}(N \alpha)\right\rangle .$$
Now we take $\alpha = 1/n$, so that
$$\left\langle r_n(N)\right\rangle
\approx \left\langle r_1(N/n)\right\rangle ,$$
which shows that the average $n$-th neighbour distance for a set of $N$ random points distributed uniformly is approximately given by the average distance for a depleted set of $N/n$ random points in the same volume. Using the expression for $\left\langle r_1(N)\right\rangle$ from equation \[eq:av-1\] we get
$$\left\langle r_n\right\rangle \approx {1 \over \pi^{1/2}} \:
\left [ \Gamma\left ({D \over 2} + 1\right) \right ]^{1/D}
\Gamma \left (1 + {1 \over D}\right ) \: \left ( n \over N \right )^{1/D} .
\label{eq:mf-av-n}$$
Since the above argument neglects fluctuations the result of equation \[eq:mf-av-n\] ought to be exact in the limit $D \to \infty$.
The exact expression of $\left\langle r_n\right\rangle$ for a finite dimension $D$ is expected to reduce to the form of equation \[eq:mf-av-n\] as $D \to \infty$ where fluctuations do not affect. For large $D$ equation \[eq:percus-av\] takes the following form :
$$\left\langle r_n\right\rangle \approx {1 \over \pi^{1/2}} \:
\left [ \Gamma\left ({D \over 2} + 1\right) \right ]^{1/D}
\Gamma \left (1 + {1 \over D}\right ) \:
\left (1 + {1 \over D} \sum_{k = 1}^{n - 1} {1 \over k} \right )
\left ( 1 \over N \right )^{1/D} .
\label{eq:percus-largeD}$$
For the above expression to reduce to the form of equation \[eq:mf-av-n\] the sum $\sum_{k = 1}^{n - 1} {1 \over k}$ must be equal to $\log_e n$ which happens only in the limit $n \to \infty$. Thus for any finite $n$ the exact result of equation \[eq:percus-av\] fails to produce the fluctuation-free form of equation \[eq:mf-av-n\] in large dimensions $D$. This shows that the mean-field approximation is useless in the present context. However the mean-field approach for $\left\langle r_n\right\rangle$ may be used as a crude approximation in other distributions (non-uniform) of random points where an exact calculation is not possible beyond the first neighbour.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank D. Dhar, S. S. Manna and A. Percus for their comments.
[99]{}
S. Chandrasekhar, Rev. Mod. Phys. [**15**]{} (1943) 1.
S. Chandrasekhar, An introduction to the theory of stellar structure (University of Chicago Press, Chicago, 1957).
J. Beardwood, J. H. Halton and J. M. Hammersley, Proc. Camb. Phil. Soc. [**55**]{} (1959) 299.
A. Percus and O. Martin, Phys. Rev. Lett. [**76**]{} (1996) 1188.
|
---
abstract: 'We consider the problem of identifying people on the basis of their walk (gait) pattern. Classical approaches to tackle this problem are based on, *e.g.* video recordings or piezoelectric sensors embedded in the floor. In this work, we rely on the acoustic and vibration measurements, obtained from a microphone and a geophone sensor, respectively. The contribution of this work is twofold. First, we propose a feature extraction method based on an (untrained) shallow scattering network, specially tailored for the gait signals. Second, we demonstrate that fusing the two modalities improves identification in the practically relevant open set scenario.'
address: 'Technicolor, Cesson-Sévigné, France'
bibliography:
- 'main.bib'
title: SCATTERING FEATURES FOR MULTIMODAL GAIT RECOGNITION
---
identification, walk, acoustic, vibration, scattering transform
Introduction {#sec:intro}
============
Identification lies at the heart of many user-defined services, ranging from movie recommendations to online banking. Due to its practical relevance, the problem of identifying people using various biometrics has triggered a significant amount of research in the signal processing and machine learning communities. Traditional means of identification, such as face [@jain2011handbook] or speaker [@hansen2015speaker; @reynolds2008text] recognition, often require active participation in the recognition process, which may be intrusive in many applications. Therefore, a method that can reliably and *passively* identify people is advantageous in such a context.
In this work, we consider human gait as biometrics for identifying people present in a room. A number of approaches to gait-based identification have been proposed in the past, exploiting different signal modalities influenced by walk pattern, *e.g.* based on video [@phillips2002gait; @hofmann2014tum], depth [@hofmann2014tum] or underfloor accelerometer measurements [@bales2016gender]. An appealing modality is structural (*e.g.* floor) vibration induced by walking, and acquired through *geophones* [@pan2015indoor], since it offers several practical advantages over other commonly used types of signals. One of them is increased security - it stems from the fact that there is no simple method (to the authors’ knowledge) that can accurately reproduce one’s gait in terms of the vibration signal. Another is preservation of privacy as vibration data is usually not considered a confidential, or sensitive information. Finally, the proposed setup is simple and cheap – typically one geophone is sufficient to monitor a medium-sized room. Unfortunately, geophone measurements are not very rich in content, due to their very limited bandwidth. Currently, geophones are reliably measuring ground vibrations only in the very low frequency range [@hons2006transfer], while the human footstep energy spans up to ultrasonic frequencies [@ekimov2007ultrasonic]. Hence, the loss of information is substantial.
In addition to vibrations (wave propagation in solids), a walking human also produces audible signals, which can be registered by standard microphones and used for identification [@geiger2014acoustic; @hofmann2014tum]. These have a much wider bandwidth, and, in addition to footsteps, they are also generated due to friction of the upper body (*i.e.* due to leg and arm movements). However, modestly-priced microphones suffer from poor frequency response at very low frequencies, and the measured signals are susceptible to environmental noise, such as speech or music. Therefore, it seems that the vibration and acoustic modality somehow complement each other: while the former is secure, robust and “senses” the low-frequency range, the latter carries more information, particularly at high frequencies. The goal of this work is to demonstrate that gait-based recognition using each of the modalities is a viable means of human identification, and that the two can be successfully coupled together in order to boost identification performance.
In the following section, we discuss the physical origin of acoustic and vibration gait measurements. Then, we introduce a feature extraction technique based on the *scattering transform* [@anden2014deep] and the specificities of the gait signal. In addition, we propose a simple feature fusion technique to enhance performance when bimodal measurements are available. Finally, we provide open set identification results, obtained from exhaustive experiments on a home-brewed dataset.
Gait signals {#sec:signals}
============
A microphone and a geophone, placed (fixed) at the same location in a room, simultaneously acquire signals of a walking person. Their example outputs are shown in Fig. \[figSignals\]: while the two time series are markedly different, the envelope peaks (corresponding to footfalls) are obviously correlated. In fact, the two modalities are linked through latent physical quantity – (vertical) vibration particle velocity at the impact point – as described in the remainder of the section. Hereafter, $\vec{r}$ denotes the coordinates of the impact (footfall) point relative to the position of the sensors, $\var{t}$ denotes time and $\omega$ denotes the angular frequency. The hat notation $\hat{\cdot}$ is used to denote the Fourier representation $\mathcal{F}(\cdot)$ of a signal.
![Vibration (top) and acoustic (bottom) recording of a person walking in silence.[]{data-label="figSignals"}](Signals.png){width=".4\textwidth"}
Acoustic pressure signal $\hat{x}_{\aud}(\omega,\vec{r}) = \mathcal{F}\left(x_{\aud}(\var{t},\vec{r}) \right)$ can be related to the particle velocity $\hat{v}(\omega)$, as follows [@ekimov2006vibration]: $$\label{eqMicTransfer}
\hat{x}_{\aud}(\omega,\vec{r}) = \hat{h}_{\aud}(\omega,\vec{r}) \hat{v}(\omega) + \hat{e}_{\aud}(\omega) = \hat{g}_{\aud}(\omega,\vec{r})\frac{\hat{v}(\omega)}{\hat{z}(\omega)} + \hat{e}_{\aud}(\omega),$$ where $\hat{e}_{\aud}(\omega)$ is the additive noise of the microphone, and $\hat{h}_{\aud}(\omega,\vec{r})$ denotes the microphone transfer function. The latter comprises *specific acoustic impedance* $\hat{z}(\omega)$ (which is a material-related quantity of a medium [@fahy2000foundations]) at the impact point, and the (air) impulse response $\hat{g}_{\aud}(\omega,\vec{r})$, relating the impact point and the microphone location. While we may assume that the floor is an isotropic solid – thus $z(\omega)$ does not change significantly with regards to $\vec{r}$ – the impulse response $\hat{g}_{\aud}(\omega,\vec{r})$ is influenced to a larger extent by the change in position (this has been empirically verified in [@bard2008human]).
Geophone measures the voltage corresponding to the velocity of the proof mass relative to the device case. In the prescribed frequency range, the velocity of the proof mass can be related to the impact point velocity $\hat{v}(\omega)$ [@ekimov2006vibration] as $$\label{eqGeoTransfer}
\hat{x}_{\geo}(\omega,\vec{r}) = \hat{h}_{\geo}(\omega,\vec{r}) \hat{v}(\omega) + \hat{e}_{\geo}(\omega) = S_{\geo} \hat{g}_{\geo}(\omega,\vec{r}) \hat{v}(\omega) + \hat{e}_{\geo}(\omega),$$ where $\hat{e}_{\geo}(\omega)$ is the additive noise of the geophone, and $\hat{h}_{\geo}(\omega,\vec{r})$ is the geophone transfer function. Furthermore, $S_{\geo}$ denotes the sensitivity constant, while $\hat{g}_{\geo}(\omega,\vec{r})$ is the impulse response *within the floor* (hence different from $\hat{g}_{\aud}(\omega,\vec{r})$).
Transfer functions $\hat{h}_{\aud}(\omega,\vec{r})$ and $\hat{h}_{\geo}(\omega,\vec{r})$ (analogously, signals $\hat{x}_{\aud}(\omega,\vec{r})$ and $\hat{x}_{\geo}(\omega,\vec{r})$) are therefore, dependent on $\vec{r}$ - the parameter we cannot control. This is the relative position between the walking person and the immobile sensors, which thus depends on time $\var{t}$, *i.e.*, $\vec{r} := \vec{r}(\var{t})$. We assume that $\vec{r}(\var{t})$ is a slowly varying function, *i.e.*, the impulse responses are (locally) stationary with respect to $\vec{r}$ within a relatively short temporal window, and one can write $$\begin{aligned}
x_{\aud}(\var{t},\vec{r}) \approx x_{\aud}(\var{t}) & = h_{\aud}(\var{t})*v(\var{t}) + e_{\aud}(\var{t}) \; \text{and}\\
x_{\geo}(\var{t},\vec{r}) \approx x_{\geo}(\var{t}) & = h_{\geo}(\var{t})*v(\var{t}) + e_{\geo}(\var{t}), \end{aligned}$$ where $h_{\aud}(\var{t})$ and $h_{\geo}(\var{t}) $ are time-domain representations of $\hat{h}_{\aud}(\omega) \approx \hat{h}_{\aud}(\omega,\vec{r})$ and $\hat{h}_{\geo}(\omega) \approx \hat{h}_{\geo}(\omega,\vec{r})$, respectively.
Feature extraction {#sec:features}
==================
The hypothesis is that the impact velocity $v(\var{t})$ is sufficiently informative to discriminate people. The sensors, however, measure only the bandlimited convolution of $v(\var{t})$ with the corresponding transfer functions. Fortunately, the local stationarity assumption enables us to exploit cancellation property of the so-called *normalized scattering* representation.
Scattering transform
--------------------
Scattering tranform is a novel feature extraction method, based on a cascade of wavelet transforms and modulus nonlinearities, bearing some resemblence to convolutional neural networks [@bruna2013invariant; @mallat2016understanding]. An appealing property of scattering networks is that their filters are pre-defined, hence they require no training. Yet, classifiers using scattering features exhibit almost state-of-the-art performance on several problems, *e.g.* [@anden2014deep; @bruna2013invariant]. In the following, we briefly describe how the scattering transform is computed on the audio signal $x_{\aud}(\var{t})$. The features from $x_{\geo}(\var{t})$ are extracted in the same manner.
For a scattering of *order* $\ing{p}$, the features are computed as $S_{\lambda_1 \hdots \lambda_{\ing{p}}}(x_{\aud}(\var{t}),\var{t}) = \phi_T(\var{t}) * U_{\lambda_1 \hdots \lambda_{\ing{p}}} (x_{\aud}(\var{t}),\var{t})$. Here, $\phi_T$ denotes the real-valued lowpass filter of bandwidth $2\pi/T$ (where $T$ is the targeted extent of time-invariance), and $U_{\lambda_1 \hdots \lambda_{\ing{p}}}(\cdot)$ is the so-called the *wavelet propagator*[^1]: $$U_{\lambda_1 \hdots \lambda_{\ing{p}}} (x_{\aud}) = | \psi_{\lambda_{\ing{p}}} * | \psi_{\lambda_{\ing{p-1}}} * | \hdots | \psi_{\lambda_1} * x_{\aud} | \hdots |,$$ where $\psi_{\lambda_{\ing{i}}} := \psi_{\lambda_{\ing{i}}}(\var{t})$ is a complex analytic wavelet filterbank at $0 < \ing{i} \leq \ing{p}$. The set of scales $\lambda_{\ing{i}} \in \Lambda_{\ing{i}}$ is chosen such that the filterbank covers the frequency range $[\pi/T, \omega_{\aud}/2]$ ($\omega_{\aud}$ is the sampling frequency), possibly with certain redundancy. The expression above defines the recursion $U_{\lambda_1 \hdots \lambda_{\ing{p}}} (x_{\aud}) = | \psi_{\lambda_{\ing{p}}} * U_{\lambda_1 \hdots \lambda_{\ing{p-1}}} (x_{\aud}) |$, with $U_{\varnothing} (x_{\aud}) = x_{\aud}$ at $\ing{i} = 0$.
In [@anden2014deep], the authors further refine scattering features by making them nearly invariant to convolution by a filter $h$, when $\hat{h}$ is almost constant on the support of $\psi_{\lambda_{\ing{i}}}$, which we assume to hold in our application. These normalized scattering coefficients are computed as component-wise division $$\begin{aligned}
& \tilde{S}_{\lambda_1}(x_{\aud}) = \frac{S_{\lambda_1}(x_{\aud})}{\phi_T * |x_{\aud}| + \varepsilon}, \, \varepsilon > 0, \; \text{for} \; \ing{i}=1,\\
\text{and} \; & \tilde{S}_{\lambda_1 \hdots \lambda_{\ing{i}}} (x_{\aud}) = \frac{S_{\lambda_1 \hdots \lambda_{\ing{i}}} (x_{\aud})}{S_{\lambda_1 \hdots \lambda_{\ing{i}-1}} (x_{\aud})}, \; \text{for} \; \ing{i}>1.\end{aligned}$$ The zero-order coefficients $\tilde{S}_{\varnothing}(x_{\aud}) := S_{\varnothing}(x_{\aud}) = \phi_T * x_{\aud}$ remain unchanged.
Hence, if we independently consider signal segments of duration $\tau$ for which our local stationarity assumption holds, the normalized scattering features should be invariant to filtering by $h_{\aud}$ (accordingly, filtering by $h_{\geo}$ for the geophone signal), and would mostly reflect the behavior of the fingerprint function $v$ in a given bandwidth.
Feature fusion
--------------
When bimodal (microphone and geophone) measurements are available, one can exploit the fact that their effective bandwidths – frequency ranges for which *SNRs (Signal-to-Noise-Ratio)* is high – are somewhat complementary. Excluding $\tilde{S}_{\varnothing}(\cdot)$, their respective normalized scattering representations should be complementary as well: the most informative coefficients of each modality appear at scales that do not overlap with one another, except perhaps within a narrow band. Indeed, while the vibration signal $x_{\geo}$ has a very low and narrow frequency range, the audio $x_{\aud}$ is a wideband signal.
This intuition can be verified in Fig. \[figScattering\], where dark color indicates low magnitude coefficients, and vice-versa. For simplicity, the geophone signal $x_{\geo}$ is upsampled to match the length of the audio signal $x_{\aud}$, thus the feature matrices have the same size. This suggests a simple fusion technique: since the coefficients are nonnegative, one can simply compute a weighted average of the two modalities to obtain a more informative representation, whose (implicit) effective bandwidth is extended. We remark that this is not a pure heuristics, as normalized scattering approach described before places the two representations in the same “impact velocity feature space”.
![Normalized ($\ing{p}=1$) scattering features for the vibration (top) and audio (bottom) modality, representing the same person, at different time instances (left/right).[]{data-label="figScattering"}](Scatt_geo2.png "fig:"){width=".23\textwidth"} ![Normalized ($\ing{p}=1$) scattering features for the vibration (top) and audio (bottom) modality, representing the same person, at different time instances (left/right).[]{data-label="figScattering"}](Scatt_geo1.png "fig:"){width=".23\textwidth"}\
![Normalized ($\ing{p}=1$) scattering features for the vibration (top) and audio (bottom) modality, representing the same person, at different time instances (left/right).[]{data-label="figScattering"}](Scatt_aud2.png "fig:"){width=".23\textwidth"} ![Normalized ($\ing{p}=1$) scattering features for the vibration (top) and audio (bottom) modality, representing the same person, at different time instances (left/right).[]{data-label="figScattering"}](Scatt_aud1.png "fig:"){width=".23\textwidth"}
The fused scattering $\iter{S_{\cdot}}{f}$ at orders $\ing{i} \geq 1$ is given as $$\iter{S_{\lambda_1,\hdots \lambda_{\ing{i}}}}{f} = \alpha(x_{\aud}) \tilde{S}_{\lambda_1,\hdots \lambda_{\ing{i}}}(x_{\aud}) + \alpha(x_{\geo}) \tilde{S}_{\lambda_1,\hdots \lambda_{\ing{i}}}(x_{\geo}),$$ with a weight $\alpha(\cdot)$ defined as $$\alpha(\cdot) = \left( \max_{ \{ \lambda_{\ing{i}} \in \Lambda_{\ing{i}} \}_{1\leq \ing{i} \leq \ing{p}} } \tilde{S}_{\lambda_1,\hdots \lambda_{\ing{i}}}(\cdot) \right)^{-1},$$ to account for the magnitude disparity among the modalities.
The rows corresponding to zero-order coefficients $\tilde{S}_{\varnothing}(x_{\aud})$ and $\tilde{S}_{\varnothing}(x_{\geo})$ are simply concatenated with the fused ones.
Feature postprocessing
----------------------
The lowpass filtering by $\phi_T$ makes the output invariant to translations smaller than $T$. It was shown [@mallat2016understanding] that the information loss introduced by lowpass filtering is compensated by the higher-order scattering coefficients, with the scattering order $\ing{p}$ predominatelly driven by the signal content [@bruna2013invariant; @anden2014deep]. The rule of thumb is that the larger $T$ is, the higher order the scattering transform should be. Unfortunately, this significantly increases computational complexity: scattering transform yields a tree-like representation (*cf.* Fig. 2 in [@bruna2013invariant]), where each “path” $\{\lambda_1,\lambda_2,\hdots \lambda_{\ing{p}} \}_{\lambda_1 \in \Lambda_1,\lambda_2 \in \Lambda_2,\hdots \lambda_{\ing{p}} \in \Lambda_{\ing{p}}}$ needs to be traversed (*i.e.* a full sequence of convolutions needs to be performed) to reach a leaf node. As applications enabled by person identification often require real-time processing, our aim is to reduce the computational burden and compute normalized scattering features only up to $\ing{p}=1$ order (“shallow” scattering network), which implies that $T$ cannot be large. However, features computed from very short signal segments cannot capture temporal dynamics of the gait, which we deem useful for identification. Indeed, the average period of normal walk is about $~1.22$s (two footfalls with the same leg) [@ekimov2011rhythm], and computing sufficiently informative scattering features with $T$ that large is computationally prohibitive. While sophisticated classifiers, such as those based on Hidden Markov Models [@geiger2014acoustic], could be used to model the temporal dynamics between successive feature vectors, we opted for a simpler alternative. By inspecting two scattering feature matrices, with the same label but computed at different time instances (Fig. \[figScattering\] left and right), one may notice that the main source of variability is due to global temporal offset. This can be easily suppressed by computing the Fourier transform of the scattering matrix across temporal direction, and applying the modulus operator, *i.e.* by discarding the phase. Thus, we extract a segment of duration $\tau > 1.22\text{s} \gg T$, and then postprocess the obtained feature matrix by applying the Fourier modulus row-wise. Since very long segments violate the local stationarity assumption, we set $\tau \approx 1.5$s. Hence, the postprocessing phase introduces additional layer to the first order scattering network.
As suggested in [@anden2014deep], to separate multiplicative signal components and reduce dimensionality, we apply logarithm and *PCA (Principal Component Analysis)* – or its approximation through *DCT (Discrete Cosine Transform)* – to postprocessed the feature matrix. The features are standardized (centered and scaled to unit variance) before PCA (DCT).
Results {#sec:results}
=======
While gait recognition attracted considerable amount of research, vibration- and audio-based bimodal identification has not been investigated so far, to the best knowledge of the authors. This led us to build our own dataset, by simultaneously recording signals using one ION^^ SM-24 geophone (sampling rate $\approx 1$kHz), and one Samson Meteor^^ microphone ($44.1$kHz). The recordings involved $8$ male and $4$ female participants, each recorded during three days, and asked to wear the same type of shoes on (at least) two different days. All recordings were taken in the same room with carpet floor covering. The participants walked the same route $10$ times per day: starting $\sim6$m away, they approached the sensors, and returned to the initial point.
30 50 100 150
------- ------------ -------- -------- ------------
0.046 23.18% 20.00% 16.13% 15.08%
0.093 16.56% 15.48% 15.96% **13.35%**
0.186 16.67% 15.96% 15.08% 16.13%
0.371 20.00% 19.34% 19.35% 21.64%
0.046 26.67% 25.81% 27.92% 26.80%
0.093 25.49% 23.33% 23.47% 31.94%
0.186 **20.78%** 24.44% 29.51% 30.03%
0.371 23.87% 29.22% 28.89% 29.03%
0.046 19.05% 16.67% 12.86% 12.09%
0.093 16.13% 12.79% 12.38% **10.00%**
0.186 15.96% 14.74% 13.33% 13.33%
0.371 19.15% 16.67% 18.21% 19.68%
: EER performance of the audio (top), vibration (middle) and fused (bottom) features (lower is better).[]{data-label="tabResults"}
Open set identification refers to the case when classes not seen during training may appear in the test phase, and the recognition system needs to label them as “unknowns”. This type of problem is common in speaker recognition, which shares many traits with gait-based identification (interestingly, in the referenced literature, we found no connection between the two). The gist of current state-of-technology in speaker recognition are variants of *GMM-UBM (Gaussian Mixture Model - Universal Background Model)* framework – an interested reader may consult *e.g.* [@hansen2015speaker; @reynolds2008text] – which we here apply to gait identification. The gait dataset is divided into the “training” and “test” sets, such that the “training” set contains recordings taken on those two days when the participants wore different type of shoes. In this way, we ensure that the training data is sufficiently diverse. The data recorded on the third day constitutes the test set.
We split the training dataset such that the recordings of $6$ randomly chosen individuals are used for training the UBM, and the training data of $3$ among the remaining $6$ (also randomly chosen), is used for the enrollement *cf.* [@reynolds2008text]. The test data of these $6$ participants is used in the evaluation phase (thus, there are $3$ unknown persons). This random partitioning is repeated $100$ times, to verify that the results are consistent.
Normalized scattering, with a redundant Morlet wavelet filterbank, is computed on overlapping signal segments of duration $\tau$ (stepsize $=0.25$s), using the Scatnet toolbox [@sifre2013scatnet]. The GMM-UBM system [@sadjadi2013msr], with $64$ Gaussian components, is then fed with the postprocessed scattering features.
Series of experiments is performed by varying the hyperparameters $T$ and $\ing{N}$ (the number of retained DCT coefficients), for each random partition. The median results, in terms of *EER (Equal Error Rate)* [@hansen2015speaker], are presented in Table \[tabResults\]. Overall, as expected, with the geophone-only features the recognition is somewhat poor. The audio modality performs better, while the fused features perform best, regardless of parameterization. Boxplots for the best-performing parameterizations (boldface values in the table), in Fig. \[figBoxplot\], show that the EERs of the fused representation have the smallest variance. Concerning the choice of time-invariance parameter $T$, the optimal value is between $0.093$s and $0.186$s, which is consistent with average duration of the footfall impact event [@ekimov2011rhythm]. The preferred number of features seems to be modality-dependent (*e.g.* richer representations favor larger $\ing{N}$), and may be related to the preset number of GMM components.
![Best performance for each feature type.[]{data-label="figBoxplot"}](boxplot.png){width="35.00000%"}
Conclusion {#sec:conclusion}
==========
We have presented a novel feature extraction approach for person identification based on audio and vibration gait measurements. In a low ambient noise environment, using the audio modality increases recognition accuracy, as demonstrated by the exhaustive experimentation on our bimodal signal dataset. Additionally, we have shown that the two modalities can be fused together to further improve recognition performance. Future work will focus on recognition in adverse conditions, *e.g.* in the presence of auditory noise, and/or several people walking. For the latter, we feel that a body of work on speaker diarization [@anguera2012speaker] could be exploited to target such problems. Moreover, bimodal data may offer distinct advantages, both in terms of “walker diarization”, but also in terms of robustness to ambient noise, since the two modalities are usually not simultaneously affected by the same noise source. Finally, in this work we opted for (deterministic) scattering feature extraction, due to the size of our training dataset. If this is not a limiting factor, recent trends in machine learning suggest that a deep neural network may achieve superior performance.
[^1]: For notational convenience, the variables $\var{t}$ and $\omega$ are dropped when the dependence is obvious.
|
---
abstract: 'We observed two-photon emission signal from the first excited state of parahydrogen gas coherently excited by counter-propagating laser pulses. A single narrow-linewidth laser source has roles in the excitation of the parahydrogen molecules and the induction of the two-photon emission process. We measured dependences of the signal energy on the detuning, target gas pressure, and input pulse energies. These results are qualitatively consistent with those obtained by numerical simulations based on Maxwell-Bloch equations with one spatial dimension and one temporal dimension. This study of the two-photon emission process in the counter-propagating injection scheme is an important step toward neutrino mass spectroscopy.'
address:
- '$^1$ Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan'
- '$^2$ PRESTO, Japan Science and Technology, Okayama 700-8530, Japan'
author:
- 'Takahiro Hiraki$^1$, Hideaki Hara$^1$, Yuki Miyamoto$^1$, Kei Imamura$^1$, Takahiko Masuda$^1$, Noboru Sasao$^1$, Satoshi Uetake$^{1,2}$, Akihiro Yoshimi$^1$, Koji Yoshimura$^1$, and Motohiko Yoshimura$^1$'
title: 'Coherent two-photon emission from hydrogen molecules excited by counter-propagating laser pulses'
---
May 2018
[*Keywords*]{}: coherence, parahydrogen, two-photon emission, counter-propagating laser injection, Maxwell-Bloch equations\
Introduction
============
Emission processes of atoms or molecules could be modified when coherent phenomena them. A famous example is superradiance, which was first predicted by Dicke [@bib:superradiance], and has been observed in various systems [@bib:SR1; @bib:SR2; @bib:SR3; @bib:SR4; @bib:SR5; @bib:SR6; @bib:SR7; @bib:SR8]. In superradiant emission processes, an ensemble of atoms or molecules behaves cooperatively. Consequently, the emission rate of the ensemble is significantly enhanced compared usual cases where each atom or molecule behaves independently [@bib:SRreview]. By taking advantage of this enhancement property of superradiance, very weak processes of atomic transitions could be observed [@bib:SRamplification].
One of the authors of the current paper proposed another coherent amplification scheme [@bib:SPAN2008]. The principle of this scheme is reviewed in [@bib:ptepreview] and is described here. Let us consider a system where atoms or molecules are excited by lasers to a metastable excited state and then they de-excite with emitting particles. In the excitation process, coherence among laser photons is imprinted into the ensemble. We denote $\bm{k}_{\rm in}$ $(\bm{k}_{\rm out})$ by the sum of the wave of excitation (de-excitation) particles. The emission rate $R$ of this process is proportional to $$\left|\int d\bm{r}\sum_{a=1}^{N} \exp({\rm i}(\bm{k}_{\rm in}-\bm{k}_{\rm out})\cdot(\bm{r}-\bm{r}_a))\mathcal{M}(\bm{r}_a)\right|^2,$$ where $a$ denotes an atom or a molecule, $\bm{r}_a$ denotes spatial position, and $\mathcal{M}$ represents the transition amplitude of the . If each particle behaves independently, the emission rate is proportional to $N$, which indicates the total number of atoms or molecules involved. However, in the case that $\bm{k}_{\rm in}=\bm{k}_{\rm out}$ holds and decoherence time is sufficiently long, the emission rate becomes $$R\propto\left|\sum_{a=1}^{N} \mathcal{M}(\bm{r}_a)\right|^2\propto N^2$$ and thus huge rate amplification can be achieved. Here rate amplification is realized only when momentum conservation among excitation and de-excitation particles holds.
This emission rate amplification method can be applied to resolve fundamental questions in a different context, neutrino physics [@bib:SPAN1; @bib:ptepreview]. In recent neutrino physics, the squared-mass differences and the mixing angles of neutrinos have been measured in various neutrino oscillation experiments [@bib:SKsolar; @bib:KamLAND; @bib:DayaBay; @bib:T2K17; @bib:Nova; @bib:nuosc]. However, the absolute scale of neutrino masses, mass ordering, CP phases, and mass type (Dirac or Majorana) still remain [@bib:directmass; @bib:Planck; @bib:nufit; @bib:DBD]. Uncovering these properties is important in terms of particle physics beyond the standard model and cosmology [@bib:nucosmo]. We use atomic or molecular de-excitation processes emitting a single photon and a neutrino pair, which we call RENP (Radiative Emission of Neutrino Pair). In the RENP process, emission rate spectra of the photon have abundant information about these neutrino properties [@bib:observable; @bib:boosted]. A serious issue when we use RENP processes in experiments is their extremely small rates [@bib:SPAN1; @bib:observable; @bib:calculation]. Thus to amplify the rate by using this amplification method is a key to the observation of RENP processes.
As of this moment, it is quite difficult to observe RENP processes. One of the reasons of this is our understandings of the rate amplification mechanism is insufficient. Although multi-photon emission processes has been extensively studied, most of them did not focus on rate amplification of the processes. Moreover, since multi-photon emission processes are possible dominant background sources in RENP observation experiments, it is necessary to understand in detail the rate amplification in multi-photon emission processes. Thus, we have been studying the coherent amplification mechanism using two-photon emission (TPE) processes. TPE processes are the simplest multi-photon emission processes and observation of TPE processes is easier than those of the RENP or higher-order multi-photon emission processes.
In the previous experiments, we observed the TPE signal from vibrational states of parahydrogen (p-H$_2$) molecules [@bib:TPE1; @bib:TPE2; @bib:TPE3; @bib:TPE4], which have long decoherence time . In these experiments two lasers of different colors were injected coaxially in the same direction for the Raman excitation of p-H$_2$. In study, in contrast, p-H$_2$ are excited by monochromatic counter-propagating lasers another requirement for the amplification of the RENP processes can be satisfied as described below.
We focus on invariant masses of excitation and de-excitation particles in the RENP processes. The invariant mass should be conserved in order to amplify the emission rate. This is because in addition to the momentum conservation condition, energy conservation among excitation and de-excitation particles should also hold. The RENP processes occur only when the invariant mass is larger than the sum of the masses of the emitted neutrinos.
Invariant mass of the two free photons for the excitation is given by $$s=2c^2|\bm{p}_1|^2|\bm{p}_2|^2(1-\cos\theta),$$ where $|\bm{p}_1|$ and $|\bm{p}_2|$ indicate momentum of each photon and $\theta$ indicates the crossing angle between the photons. In the case of one-side laser injection scheme ($\theta=0$), $s$ is equal to 0 and the amplification conditions of the RENP process cannot be satisfied. In the counter-propagating laser excitation scheme ($\theta=\pi$), in contrast, it is possible to satisfy the amplification conditions of the RENP process. Another merit of the counter-propagating excitation scheme is related to soliton formulation, which is described in [@bib:soliton; @bib:dynamics]. For these reasons, to observe and understand the TPE process in the counter-propagating injection scheme is an important step toward observation of RENP processes.
As in the case of the previous experiments [@bib:TPE1; @bib:TPE2; @bib:TPE3], it is helpful for understandings of properties of the coherent amplification to compare actual data to results of the numerical simulation. These simulations are based on Maxwell-Bloch equations, which represent developments of the laser fields and the coherence among the ensemble. In this paper, we investigate properties of the TPE process in detail through comparisons with simulation results.
The TPE process observed in our previous experiments is a four-wave mixing (FWM) process, which has been studied in nonlinear optics and quantum electronics [@bib:FWMreview]. In the current experiment setup, frequencies of all the four waves are identical. This process is called degenerate four-wave mixing process (DFWM). DFWM generation experiments in the counter-propagating scheme [@bib:boyd_DFWM1; @bib:boyd_DFWM3] or those using H$_2$ [@bib:ph2_DFWMFG] have been conducted, though motivations of the previous experiments are different substantially from that in this study.
The rest of this paper is structured as follows: In section \[sec:setup\] we note properties of the p-H$_2$ molecule and describe the experimental setup. In section \[sec:results\] experimental results are described. In section \[sec:discussion\] these results are compared to the results of the numerical simulations. Detailed information about the derivation of the Maxwell-Bloch equations and the numerical simulation is described in Appendix.
Experimental setup {#sec:setup}
==================
Para-hydrogen (p-H$_2$) is hydrogen molecule whose spins of the two nuclei are aligned antiparallel and the total nuclear spin angular momentum $I_N$ is zero. Rotational quantum numbers of p-H$_2$ in the electronic ground state are even ($J=0,2,\dots$) because of the Pauli exclusion principle. At low temperature, most of p-H$_2$ molecules lie in the $J=0$ state.
In order to achieve huge rate amplification, it is favorable to use a target with small decoherence. The intermolecular interaction is one of the causes of decoherence. The intermolecular interactions of $J=0$ p-H$_2$ molecules are weaker than those of ortho-hydrogen molecules ($I_N=1$, odd-$J$) because of the spherical symmetry of rotational wavefunction of them. For this reason we use the p-H$_2$ target rather than normal hydrogen gas target.
Figure \[fig:diagram\] shows a schematic energy-level diagram of the p-H$_2$ molecule. We generate coherence between the ground state $|g\rangle$ ($v=0$, $J=0$) and the first vibrationally excited state $|e\rangle$ ($v=1$, $J=0$). The energy difference between $|g\rangle$ and $|e\rangle$ is $\hbar\omega_{eg} =$ 0.5159 eV. Single-photon electric dipole ($E1$) transitions between these states are forbidden and two-photon $E1\times E1$ transitions are allowed. For this reason $|e\rangle$ is metastable and its spontaneous emission rate is $\mathcal{O}(10^{-11})$ Hz, which is dominated by the two-photon $E1\times E1$ transition process [@bib:TPE1]. Effect of the decoherence due to the spontaneous de-excitation is negligibly small in this setup. Intermediate states $|j\rangle$ of the two-photon transitions are electronically excited states. The energy level differences between $|g\rangle$ or $|e\rangle$ and $|j\rangle$, $\hbar\omega_{jg}$ and $\hbar\omega_{je}$ are much larger than $\hbar\omega_{eg}$ (far-off-resonance condition). In the previous coherent two-photon emission (TPE) signal generation experiments via one-side excitation [@bib:TPE1; @bib:TPE2; @bib:TPE3], coherence between $|g\rangle$ and $|e\rangle$ was prepared using the Raman process, where were necessary. In this experiment, in contrast, we generate coherence using identical-frequency photons which originate from a single laser. This laser frequency is denoted by $\omega_l$ and the two-photon detuning $\delta$ is defined as $\delta=2\omega_l-\omega_{eg}$.
![Schematic energy-level diagram of the parahydrogen molecule. The energy difference between the $v=0$ ground state $|g\rangle$ and the $v=1$ state $|e\rangle$ corresponds to a wavelength of 2403 nm. Intermediate states $|j\rangle$ of the two-photon transitions are electronically excited states. The two-photon detuning $\delta$ is defined as $\delta=2\omega_l-\omega_{eg}$.[]{data-label="fig:diagram"}](fig1.pdf){width="40.00000%"}
The laser system is basically the same as the previous experiment where third-harmonic generation from the p-H$_2$ gas target was observed [@bib:THG]. Here we describe the laser system briefly. Figure \[fig:setup\] (a) shows the schematic view of the mid-infrared (MIR) laser system. A continuous-wave (cw) laser ($\lambda=871.4$ nm) was prepared using the external cavity diode laser (ECDL). The $\delta$ was precisely controlled by adjusting the frequency of the ECDL. The cw near infrared laser intensity was amplified by the tapered amplifier (TA). The cw laser was injected into the triangle cavity, where pulse near-infrared laser was generated in lithium triborate (LBO) crystals with the second harmonic of an injection-seeded Nd:YAG laser (Continuum, Powerlite DLS 9010, $\lambda=532.2$ nm) through optical parametric generation (OPG). Next, through optical parametric amplification (OPA) in LBO crystals, 1367 nm pulses were generated from the near-infrared laser and the second harmonic of the injection-seeded Nd:YAG pulse laser. The MIR pulses ($\lambda=4806$ nm) were then generated through difference-frequency generation (DFG) in potassium titanyl arsenate (KTA) crystals between the fundamental pulses of the injection-seeded Nd:YAG laser ($\lambda=1064$ nm) and the 1367 nm pulses. The repetition rate and duration of the MIR pulses were 10 Hz and roughly 5 ns (FWHM), respectively. Laser linewidth of the MIR pulses was measured with the absorption spectroscopy using rovibrational transitions of the carbonyl sulfide gas. Estimated FWHM linewidth is 145 (16) MHz, which is roughly 1.6 times larger than the Fourier-transform-limited linewidth if Gaussian-shaped pulses are assumed.
Figure \[fig:setup\] (b) shows the schematic view of the experimental setup. High-purity ($>99.9\%$) p-H$_2$ gas was prepared from normal hydrogen gas by using a magnetic catalyst (Fe(OH)O) cooled to approximately 14 K. The prepared p-H$_2$ gas was enclosed in a copper cell, which was located inside a cryostat at approximately 78 K by liquid nitrogen. At this temperature, a conversion rate from p-H$_2$ molecules to orthohydrogen molecules during the experiment is slow and it is enough to replace the p-H$_2$ gas target roughly once a week. The length of the cell was 15 cm, which was much shorter than the MIR pulse length. Anti-reflection coated BaF$_2$ (Thorlabs, WG01050-E) substrates were used for the optical windows of the cryostat and the cell.
The MIR beam was divided into three beams by using beam splitters. For excitation of p-H$_2$ gas, counter-propagating pump MIR pulses (pump1, pump2) were injected into the cell. beam, the trigger laser, was injected into the cell simultaneously with the pump beams to induce the TPE process. The timing jitter among the beams was negligibly small. The trigger beam was roughly 1$^{\circ}$ tilted from the pump beam axis in the horizontal direction so that the signal light was separated from the pump beams while keeping overlapped region between the pump and trigger beams. The trigger beam was almost overlapped with the pump beams around the center of the target cell. Diameter (D4$\sigma$) of the input beams were $2-3$ mm and were loosely focused around the center of the cell. The input pulse energies of the beams were measured by using an energy detector (Gentec-EO, QE12). Measured energies for the pump beams $\mathcal{E}_{\rm p1}$ and $\mathcal{E}_{\rm p2}$ were roughly 1 mJ/pulse and that for the trigger beam $\mathcal{E}_{\rm trig}$ was 0.6 mJ/pulse. There existed roughly a 10% pulse-by-pulse energy fluctuations.
The pump beams and the trigger beam were circularly polarized by using quarter-wave plates before they were injected into the target. Both of the pump lasers were right-handed circularly polarized (RH) from the point of view of the source. The $z$ axis is defined in figure \[fig:setup\] (b). We chose the $z$ axis as the quantization axis. The polarization of the pump1 (pump2) beam was $\sigma^+$ ($\sigma^-$) and the excitation process by the pump1 (pump2) beam was a $\Delta m_J=+1$ ($\Delta m_J=-1$) process. The two-photon excitation with only the pump1 beam or the pump2 beam was forbidden because of the selection rule of the two-photon transition. In contrast, the two-photon excitation by the counter-propagating photon pair was allowed. The signal light was generated by the trigger pulse from excited p-H$_2$ molecules and went back along with the trigger beam line because of the amplification condition (momentum conservation). The trigger light was left-handed circularly polarized (LH), and because of the selection rule the signal light was also left-handed circularly polarized. It was actually allowed that p-H$_2$ molecules were excited by the pump1 laser and the trigger laser. However, in this case direction of the de-excited photons was the same as that of the pump1 or the trigger laser because of the amplification condition and these photons were not observed by the signal detector. Excitation by the pump2 and the signal photons were also allowed but its effect on signal energy was negligibly small.
The signal light was horizontally polarized after it passed the quarter-wave plate. In this experiment it was difficult to reduce background component, which comprised scattering lights of the pump and the trigger beams. This is because the wavelength of these lights were the same as that of the signal light and wavelength filters could not be used for the reduction of the background stray light. A portion of the signal light was reflected by using a beam splitter (Thorlabs, BSW511) and was then reflected by a polarizing beam splitter (Research Electro-Optics, Product 15625) which was used for reduction of the background RH scattering lights. Furthermore, the cryostat was placed on a rotation stage and was rotated so that the reflected lights of the pump and the trigger beams by the optical windows the minimum. The MIR signal pulses were detected by using a mercury-cadmium-telluride detector (Vigo system, PC-3TE-9).
![(a) Schematic view of the laser system with the external cavity diode laser (ECDL), the tapered amplifier (TA), the lithium triborate crystals (LBO), and the potassium titanyl arsenate crystals (KTA). (b) Schematic view of the experimental setup with the quarter-wave plates (QWP), the beam splitter (BS), the polarizing beam splitter (PBS), and the mercury cadmium telluride (MCT) detector. The pump1 (pump2) beam was injected from the right (left) side of the figure. The trigger beam was injected from the right side of the figure.[]{data-label="fig:setup"}](fig2.pdf){width="80.00000%"}
Results {#sec:results}
=======
Detuning curve
--------------
First, we confirmed the two-photon transition signal was generated from the p-H$_2$ target. The signal energy varied by changing the detuning $\delta$ and could be separated from the background component. Figure \[fig:detuning\] shows observed spectra as a function of $\delta$, where target gas pressure was set to 280 kPa. At each detuning, the signal pulse was measured 200 or 300 times. The error bars in the plot indicate the standard errors, where only the statistical uncertainty is considered. The signal energy was fluctuated because of the fluctuation of position and energies of the input beams during the experiment. The systematic uncertainty which includes these fluctuations is much larger than the statistical uncertainty, but its estimation is difficult and we do not discuss further in this paper. For comparison, a signal energy distribution when the polarization of one of the pump lasers was changed to LH is shown in blue square points. In this case, the two-photon excitation process was suppressed and only background scattering light was observed. This confirms the signal peak stems from the TPE process.
The solid line in figure \[fig:detuning\] is a fit to data points by a Lorentzian function and a constant. The constant term of the fit represents the background components. As mentioned in the previous section, the origin of the $\delta$ is defined as the point $2\omega_l=\omega_{eg}$. The MIR laser frequency was estimated during the experiment by monitoring the laser frequencies of the ECDL and the seed laser of the Nd:YAG laser with a high-precision wavemeter (HighFinesse WS-7). Before this experiment was conducted we measured the $\omega_{eg}$ through the Raman scattering process between $\ket{g}$ and $\ket{e}$. The previously measured $\omega_{eg}$ was $124748.7-10.9 \times p$ GHz at 78 K, where $p$ indicates the p-H$_2$ pressure in MPa and the second term represents the pressure shift. This value is consistent with those measured by external experiments [@bib:linewidth; @bib:energylevel; @bib:pressureshift] within the systematic uncertainty of our measurement. The center value of the Lorentzian spectrum by the fit was -27 MHz. The uncertainty of the $\delta$ is roughly 170 MHz, which is determined from the absolute accuracy of the wavemeter, and this center value is close to the origin. The width of the fitted Lorentzian profile is described in subsection \[sec:pdep\].
By considering the detector responsivity and transmittance of optics, the signal energy $\mathcal{E}_{\rm sig}$ around $\delta=0$ is estimated to be roughly 20 nJ. This value is $\mathcal{O}(10^{-5})$ times smaller than those of pump and trigger pulses. We have defined the “enhancement factor” as a ratio of the observed photon number to that expected due to spontaneous two-photon emissions with experimental acceptance [@bib:TPE1; @bib:TPE4]. The enhancement factor in the current experiment is roughly comparable to that in the previous measurement [@bib:TPE2].
Signal energy dependence on input pulse energies {#sec:edep}
------------------------------------------------
Next, we present a dependence of signal energy on the energies of the pump and the trigger beams.
Figure \[fig:edep\] shows the dependence of signal energy at 288 kPa and $\delta\approx0$ on input beam energy. At each data point measurement was conducted approximately 3000 times and background components were subtracted by using signal energy data measured at off-resonant points. Blue solid line indicates experimental data fitted with $\mathcal{E}_{\rm sig}=A\times I^{B}$, where $I$ indicates the input pulse intensity before division and $A$ and $B$ are the fit parameters. We obtained $B=2.99\pm0.03$, though actual systematic uncertainty is considered to be much larger than the obtained error.
Dependence of detuning curves on target pressure {#sec:pdep}
------------------------------------------------
Finally, we present dependences of detuning curve width and signal energy on p-H$_2$ gas pressure. Target p-H$_2$ gas pressure was varied from 10 to 340 kPa. Signal energies especially in the low-pressure region were weak and were largely affected by the fluctuation of background lights and detector noises. For confirmation of reproducibility, measurements were conducted several times in 10-60 kPa region. Peak energies and detuning curve widths were obtained from fitted detuning dependence curves. The were calculated .
Dependence of the FWHM detuning curve width on target pressure is shown in figure \[fig:pres\_width\]. In the low-pressure region the detuning curve width is considered to be dominated by the laser linewidth. The detuning curve width is increasing as target pressure becomes higher, which is due to the pressure broadening effect. The size of the pressure broadening effect is approximately proportional to the gas pressure [@bib:linewidth].
Dependence of the signal peak energy on target pressure is shown in figure \[fig:pres\_height\]. The data points are normalized by that at the highest pressure. Signal energy increases as the target density becomes higher because more p-H$_2$ molecules are excited. On the other hand, the increasing rate of the signal energy becomes lower because effect of decoherence by the pressure broadening is larger in the high-pressure region.
![Signal energy distributions as a function of detuning $\delta$ for RH+RH polarization pump beams (black circle points) and RH+LH polarization pump beams (blue square points). At each point, signal pulse was measured 200 or 300 times. The error bars indicate standard errors. Black solid line indicates a fit to black points by a Lorentzian function and a constant. The constant term of the fit represents the background components. Red dashed line indicates the normalized detuning curve of the simulation result with the same constant background. []{data-label="fig:detuning"}](fig3.pdf){width="70.00000%"}
![Dependence of signal energy $\mathcal{E}_{\rm sig}$ on input beam intensity $I$. Black square points indicate experimental data. At each data point measurement was conducted approximately 3000 times and background components were subtracted by using data taken at off-resonant point. Blue solid line indicates experimental data fitted with $\mathcal{E}_{\rm sig}=A\times I^{B}$, where $A$ and $B$ are the fit parameters. []{data-label="fig:edep"}](fig4.pdf){width="70.00000%"}
![Target pressure dependence of spectral widths of detuning curves. Black circle points indicate experimental data. In the low-pressure region, measurements were conducted several times for confirmation of reproducibility. Blue solid line indicates the FWHM width of the simulation result. Red dotted line and green dashed line indicate the Lorentzian component and the Gaussian component of the Voigt width, respectively. []{data-label="fig:pres_width"}](fig5.pdf){width="70.00000%"}
![Target pressure dependence of peak signal energy normalized by that of the measured data at the highest pressure. Black circle points indicate experimental data. In the low-pressure region, measurements were conducted several times for confirmation of reproducibility. Blue line indicates the simulation result.[]{data-label="fig:pres_height"}](fig6.pdf){width="70.00000%"}
Discussion {#sec:discussion}
==========
Maxwell-Bloch equations {#sec:discussion1}
-----------------------
In order to understand the experimental results , we constructed a numerical simulation which reproduces the experimental situation. The simulation is based on Maxwell-Bloch equations with one spatial and one temporal (1+1) dimension. In this simulation, intensity and duration of the pump beams and the trigger beam, and p-H$_2$ gas target pressure were set to the same values as those of the experiment. The temporal shape of the input pulses are assumed to be Gaussian. Temporal electric field distribution of each input beam at the target edge is written as $$|E_{\rm X}(t)|^2=\frac{2c}{\varepsilon_0\sqrt{2\pi}\sigma_t}I_{\rm X}\exp\left(-\frac{(t-t_0+L/2c)^2}{2\sigma_t^2}\right),$$ where $I_{\rm X}$ (X=p1, p2, trig), $\sigma_t$, and $L$ indicate pulse intensity of each input beam, duration of each pulse, and the target length, respectively. The center of these beams passes through the target center at $t=t_0$. We describe the whole equations in Appendix and here we describe an essential part of the equations: $$\begin{aligned}
\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial z}\right) E_{\rm sig} \approx \frac{{\rm i}\omega_lN_t}{2}\left(\alpha_{gg}E_{\rm sig}+2\alpha_{eg}\rho_{ge}^{0*}E_{\rm trig}^{*}\right), \label{eq:Maxwells4_approx} \\
\frac{\partial\rho_{ge}^{0}}{\partial t} = -({\rm i}\delta+\gamma_{2})\rho_{ge}^{0} -{\rm i}\Omega_{eg}^{0*}, \label{eq:Blochs3_0} \\
\Omega_{eg}^{0}\approx \frac{\varepsilon_0\alpha_{eg}}{2\hbar}E_{\rm p1}E_{\rm p2}. \label{eq:Rabiapprox}\end{aligned}$$ Equation (\[eq:Maxwells4\_approx\]) represents the development of the envelope of the signal electric field $E_{\rm sig}$. $N_t$ indicates the number density of the p-H$_2$ and $\alpha_{gg}$ and $\alpha_{eg}$ indicate polarizabilities of the p-H$_2$. Equation (\[eq:Blochs3\_0\]) represents the temporal development of the coherence generated by the pump lasers, where $\gamma_2$ indicates the transverse relaxation rate. The $\Omega_{eg}^{0}$ represents the two-photon Rabi frequency.
In the previous one-side laser injection scheme, the Maxwellian part of the partial differential equations can be simplified to ordinary differential equations by introducing the co-moving coordinates [@bib:TPE1]. we directly treat the partial differential equations by using the method of lines [@bib:methodoflines] After that treatment we numerically solve ordinary differential equations where the space derivative ($\partial$/$\partial z$) is discretized. For the discretization of the space derivative, we used the weighted essentially non-oscillatory schemes [@bib:WENO].
In this simulation the $\gamma_2$ is an input parameter. In previous one-side excitation experiments [@bib:ptepreview] we used the Raman transition linewidth $\Delta_{\mathrm{Raman}}$ between $|g\rangle$ and $|e\rangle$ as the value of the $2\gamma_2$. The pressure dependence of $\Delta_{\mathrm{Raman}}$ is approximately written [@bib:linewidth] as $$\Delta_{\mathrm{Raman}}=C_p p + \frac{C_{\rm D}}{p}, \label{eq:Ramanwidth}$$ where $p$ indicates the p-H$_2$ pressure and $C_p$ and $C_{\rm D}$ are constant. The first term represents the pressure broadening effect and the second term represents the Doppler broadening effect with Dicke narrowing [@bib:Dicke]. In the current experimental scheme, in contrast, the signal photons were mostly generated from $|e\rangle$ excited by the same-frequency counter-propagating pump lasers. Thus the system is almost Doppler-free [@bib:Dopplerfree1; @bib:Dopplerfree2] and the size of the residual Doppler effect is negligibly small compared to the other broadening effect. For this reason we take into account only the pressure broadening term as the $\gamma_2$.
From equations (\[eq:Maxwells4\_approx\])-(\[eq:Rabiapprox\]), it is found that signal energy depends on input pulse energies, the $\delta$, $N_t$, and the $\gamma_2$. We can confirm consistency of signal energy dependences of these parameters between the measured data and the simulation results from the detuning curves and the signal energy dependences on input pulse energy and target pressure.
Analytical estimation of the signal energy dependence on input pulse energies {#sec:discussion2}
-----------------------------------------------------------------------------
We can predict how the input energy dependence behaves from the Maxwell-Bloch equations in a simplified case where we assume the energies of the pump and the trigger beams are constant. They actually decrease due to the two-photon absorption, but their decrease rates after the beams pass through the target are expected to be $\mathcal{O}$(1)%. Temporal development of the coherence (equation (\[eq:Blochs3\_0\])) is $$\frac{\partial\rho_{ge}^{0}}{\partial t} = -({\rm i}\delta+\gamma_{2})\rho_{ge}^{0}-{\rm i}\frac{c\alpha_{eg}}{\hbar\sigma_t\sqrt{2\pi}}\sqrt{I_{\rm p1}I_{\rm p2}}\exp\left(-\frac{(t-t_0)^2}{2\sigma_t^2}-\frac{z^2}{2\sigma_t^2c^2}\right),
\label{eq:approx_coh_diff}$$ where $\sigma_t$ indicates duration of each pulse. By considering the initial condition $\rho_{ge}^{0}=0$ at $t=-\infty$, $$\begin{aligned}
\rho_{ge} &=& -{\rm i}\frac{c\alpha_{eg}}{2\hbar}\sqrt{I_{\rm p1}I_{\rm p2}}\exp\left(-\Gamma(t-t_0)+\frac{\sigma_t^2\Gamma^2}{2}-\frac{z^2}{2\sigma_t^2c^2}\right) \nonumber \\
&\times& {\rm erfc}\left(\frac{-(t-t_0)+\sigma_t^2\Gamma}{\sqrt{2}\sigma_t}\right),
\label{eq:approx_coh}\end{aligned}$$ where $\Gamma=\gamma_{2}+\rm{i}\delta$ and erfc indicates the complementary error function. From equation (\[eq:approx\_coh\]) $|\rho_{ge}^{0}|$ is proportional to $\sqrt{I_{\rm p1}I_{\rm p2}}$. In the development of the signal electric field (equation (\[eq:Maxwells4\_approx\])), the last term of the right-hand side indicates generation of the signal photons induced by the trigger field. Since $I_{\rm sig}\propto|E_{\rm sig}|^2$ is proportional to $$N_t^2|\rho_{ge}^0|^2I_{\rm trig}\propto N_t^2I_{\rm p1}I_{\rm p2}I_{\rm trig}, \label{eq:approx}$$ the signal energy $\mathcal{E}_{\rm sig}$ has the cubic dependence on $I$. We also obtained $\mathcal{E}_{\rm sig}=AI^{x}$, $x\approx 3.0$ from the simulation, where this approximation is not applied. The measured data are consistent with the theoretical predictions and the simulation result.
Comparison of the detuning curve width dependence on target pressure {#sec:discussion3}
--------------------------------------------------------------------
The detuning curve shape of the simulation is well-approximated by a Voigt profile, which is the convolution of a Lorentzian profile and a Gaussian profile. Red dashed line in figure \[fig:detuning\] shows the signal energy dependence on the detuning obtained by the simulations. Here the signal energy is normalized so that the peak height of the detuning curve of the simulation is the same as that of the fitted data. The detuning curve shape of the simulation result is consistent with that of the fitted data.
Blue solid line in figure \[fig:pres\_width\] indicates detuning curve widths calculated from the simulations. The detuning curve width of the experimental data is slightly wider with the simulation results. A width of a Voigt profile is approximately written as $w_{\rm V}\approx0.5346w_{\rm L} + \sqrt{0.2166w^2_{\rm L} + w^2_{\rm G}}$, where $w_{\rm L}$ (red dotted line), and $w_{\rm G}$ (green dashed line) indicate the FWHM width of Lorentzian and Gaussian function, respectively [@bib:Voigt]. In this case $w_{\rm L}$ is almost linear to the p-H$_2$ pressure and $w_{\rm G}$ is almost constant. The Gaussian term arises from the Fourier-transform-limited laser linewidth. As mentioned in section \[sec:setup\], actual laser linewidth is wider than the Fourier-transform-limited one. This is considered to be the reason of the difference of the detuning curve width between the data and the simulation.
Blue line in figure \[fig:pres\_height\] indicates the pressure dependence of the signal peak energy obtained by the simulation. Signal energy of the simulation is also normalized by that of the data point at the highest pressure. Pressure dependences of normalized signal energy of the simulation are also basically consistent with the data. These results indicate decoherence of the system is dominated by the pressure broadening effect. There exists a dip near the pressure of 200 kPa in the data. This is considered to be due to unexpected variation of some of beam properties during taking those data points.
Comparison of absolute signal energy {#sec:discussion4}
------------------------------------
Finally, we mention a comparison of the absolute signal energy between the simulation result and the measured data. In the calculation of the signal energy of the simulation we assume the spatial profile of the input beams are assumed to be flat-top and have a diameter of 2 mm. We calculate the absolute signal energy by integrating the temporal profile of the signal pulse obtained by the numerical simulation. The signal energy at $\delta=0$ and at 280 kPa of the simulation is roughly 2.2$\times10^2$ nJ, which is roughly 10 times larger than that of the data.
Signal energies largely depend on the size of the relaxation rate. However, as described in the previous subsection, normalized signal energy dependence on target pressure of the simulations is generally consistent with those of the measured data. Thus it is unlikely that wrong estimation of the relaxation rate is the cause of the discrepancy. reason for this discrepancy is that our numerical simulation is too simplified. In particular, in this simulation we assume that the trigger and the pump beams are completely overlapped throughout the target, which is not correct in the actual experiment because the trigger beam was tilted from the pump beamline. By considering the overlapped region of the beams inside the target, the generated signal energy with this effect is estimated to be roughly half compared with the case where the beams completely overlap. However, the absolute signal energy of the simulation is still roughly five times larger than that of the data. Another simplification is that we numerically solved $1+1$-dimensional Maxwell-Bloch equations, where transverse terms of the laser fields were not taken into account. If we execute more realistic simulations, this discrepancy might be resolved, though this is beyond the scope of the current paper.
Conclusion {#sec:conclusion}
==========
In order to deepen understandings of the rate amplification mechanism using atomic or molecular coherence, we conducted an experiment where coherence was prepared by counter-propagating excitation laser pulses. We successfully observed the two-photon emission (TPE) signal from parahydrogen (p-H$_2$) molecules. This observation is an important step for future neutrino spectroscopy experiments because the counter-propagating excitation scheme will be adopted in those experiments, though further studies are necessary for observation of RENP processes.
It is also an important advance to improve understandings of the rate amplification mechanism through numerical simulations. We numerically solved Maxwell-Bloch equations with one spatial and one temporal dimension. We compared the signal energies of the measured data and those dependences on p-H$_2$ gas pressure and input beam energies with those calculated by using the numerical simulations. We found dependences of the data are qualitatively consistent with the simulation.
This work was supported by JSPS KAKENHI Grant No. JP15H02093, No. JP15H03660, No. JP15K13486, No. JP15K17651, No. JP16J10938, No. JP17K14292, No. JP17K14363, No. JP17K18779, , and . This work was also supported by JST, PRESTO and the Matsuo Foundation.
Construction of simulation based on Maxwell-Bloch equations {#sec:simulation}
===========================================================
In this Appendix a derivation of Maxwell-Bloch equations and description of the numerical simulation are shown.
Laser field and the parahydrogen states of the experimental system
------------------------------------------------------------------
The electric fields of the pump, trigger, signal lasers are represented as $\tilde{\bm{E}}_{\rm p1}$, $\tilde{\bm{E}}_{\rm p2}$, $\tilde{\bm{E}}_{\rm trig}$, and $\tilde{\bm{E}}_{\rm sig}$, respectively. We assume the pump beams are right-handed circularly polarized (RH) and the trigger and the signal beams are left-handed circularly polarized (LH). In order to make the simulation simple and the numerical simulation fast, we conducted $1+1$-dimensional simulations. We also ignore third or higher harmonic generation processes, which are mainly generated through the two-photon excitation between the pump1 and the trigger beam. Our interest lies in the slowly varying envelopes of the electromagnetic fields. Because all the fields oscillate with the frequency around the laser frequency $\omega_l$ thanks to the simplification, they are expressed with the electromagnetic field envelopes ($E_{\rm p1}$, $E_{\rm p2}$, $E_{\rm trig}$, and $E_{\rm sig}$) by $$\begin{aligned}
\tilde{\bm{E}}_{\rm p1}(z,t)&=&\frac{1}{2}\left(E_{\rm p1}(z,t)\hat{\bm{\epsilon}}_{R}\exp\Bigl({-{\rm i}\omega_l(t-\frac{z}{c})\Bigr)}+({\rm c.c.})\right), \\
\tilde{\bm{E}}_{\rm p2}(z,t)&=&\frac{1}{2}\left(E_{\rm p2}(z,t)\hat{\bm{\epsilon}}_{R}\exp\Bigl({-{\rm i}\omega_l(t+\frac{z}{c})\Bigr)}+({\rm c.c.})\right), \\
\tilde{\bm{E}}_{\rm trig}(z,t)&=&\frac{1}{2}\left(E_{\rm trig}(z,t)\hat{\bm{\epsilon}}_{L}\exp\Bigl({-{\rm i}\omega_l(t-\frac{z}{c})\Bigr)}+({\rm c.c.})\right), \\
\tilde{\bm{E}}_{\rm sig}(z,t)&=&\frac{1}{2}\left(E_{\rm sig}(z,t)\hat{\bm{\epsilon}}_{L}\exp\Bigl({-{\rm i}\omega_l(t+\frac{z}{c})\Bigr)}+({\rm c.c.})\right),\end{aligned}$$ where $\hat{\bm{\epsilon}}_{R}$ and $\hat{\bm{\epsilon}}_{L}(=\hat{\bm{\epsilon}}_{R}^{*})$ represent circular polarization unit vectors and $\tilde{\bm{E}}$ or $\tilde{E}$ represent that electric fields include fast oscillating phase terms. Next, we denote the wave function of the p-H$_2$ system by $$\ket{\psi}=c_g e^{-{\rm i} \omega_g t}\ket{g}+c_e e^{-{\rm i} (\omega_e+\delta) t}\ket{e}+c_{j+} e^{-{\rm i} \omega_{j} t}\ket{j_+} + c_{j-} e^{-{\rm i} \omega_{j} t}\ket{j_-},$$ where $\ket{j_+}$ ($\ket{j_-}$) represents $m_J=+1$ ($m_J=-1$) intermediate states of the p-H$_2$. The Schr[ö]{}dinger equation of the system is $${\rm i}\hbar\frac{\partial}{\partial t}\ket{\psi}=(H_0+H_I)\ket{\psi},
\label{eq:schrodinger}$$ where $H_0$ indicates the free term of the p-H$_2$ states and $H_I$ indicates the interaction Hamiltonian: $$\begin{aligned}
H_0\ket{g}=\hbar\omega_g\ket{g},\; H_0\ket{e}=\hbar\omega_e\ket{e}, \;H_0\ket{j_\pm}=\hbar\omega_j\ket{j_\pm}, \\
H_I=-{\bm{d}}\cdot\tilde{\bm{E}}=-{\bm{d}}\cdot(\tilde{\bm{E}}_{\rm p1}+\tilde{\bm{E}}_{\rm p2}+\tilde{\bm{E}}_{\rm trig}+\tilde{\bm{E}}_{\rm sig}).\end{aligned}$$ The $H_I$ consists of electric dipole interactions between $\ket{j_\pm}$ and $\ket{g}$ or $\ket{e}$. We also introduce transition electric dipole moments $d_{jg}=\bra{j_{+(-)}}-\bm{d}\cdot\bm{\hat{\epsilon}}_{R(L)}\ket{g}$ and $d_{je}=\bra{j_{+(-)}}-\bm{d}\cdot\bm{\hat{\epsilon}}_{R(L)}\ket{e}$. The other transition electric dipole moments are zero since they are $E1$-forbidden transitions.
Optical Bloch equations
-----------------------
In this subsection we derive Optical Bloch equations for the two-level-reduced system and their approximation for the numerical simulation. First, we focus on $\ket{j_\pm}$ of the Schr[ö]{}dinger equation (\[eq:schrodinger\]): $$\begin{aligned}
{\rm i}\hbar\frac{\partial c_{j+}}{\partial t}&=&\frac{1}{2}\left(d_{jg}\exp({\rm i}\omega_{jg}t)c_g+d_{je}\exp({\rm i}\omega_{je'}t)c_e\right)\left(\tilde{E}_{\rm p1}+\tilde{E}_{\rm p2}^{*}+\tilde{E}_{\rm trig}^{*}+\tilde{E}_{\rm sig}\right), \nonumber \\
{\rm i}\hbar\frac{\partial c_{j-}}{\partial t}&=&\frac{1}{2}\left(d_{jg}\exp({\rm i}\omega_{jg}t)c_g+d_{je}\exp({\rm i}\omega_{je'}t)c_e\right)\left(\tilde{E}_{\rm p1}^{*}+\tilde{E}_{\rm p2}+\tilde{E}_{\rm trig}+\tilde{E}_{\rm sig}^{*}\right), \nonumber\end{aligned}$$ where we introduce $\omega_{e'}=\omega_e+\delta$ and $\omega_{je'}=\omega_j-\omega_e-\delta$. By using the Markovian approximation and an initial condition ($c_{j\pm}(t=0)=0$), we obtain $$\begin{aligned}
c_{j+}=-\frac{1}{2\hbar}\sum_m^{g, e'}d_{jm}c_m&&\Big(\frac{\exp({\rm i}(\omega_{jm}-\omega_l)t)-1}{\omega_{jm}-\omega_l}(\bar{E}_{\rm p1}+\bar{E}_{\rm sig}) \nonumber \\
&&+\frac{\exp({\rm i}(\omega_{jm}+\omega_l)t)-1}{\omega_{jm}+\omega_l}(\bar{E}_{\rm p2}^{*}+\bar{E}_{\rm trig}^{*})\Big), \label{eq:cjplus} \\
c_{j-}=-\frac{1}{2\hbar}\sum_m^{g, e'}d_{jm}c_m&&\Big(\frac{\exp({\rm i}(\omega_{jm}-\omega_l)t)-1}{\omega_{jm}-\omega_l}(\bar{E}_{\rm p2}+\bar{E}_{\rm trig}) \nonumber \\
&&+\frac{\exp({\rm i}(\omega_{jm}+\omega_l)t)-1}{\omega_{jm}+\omega_l}(\bar{E}_{\rm p1}^{*}+\bar{E}_{\rm sig}^{*})\Big), \label{eq:cjminus}\end{aligned}$$ where $d_{je'}=d_{je}$ and $c_{e'}=c_{e}$ and $$\begin{aligned}
\bar{E}_{\rm p1}=E_{\rm p1}\exp\Bigl({\rm i}\omega_l\frac{z}{c}\Bigr), \ \ \bar{E}_{\rm p2}=E_{\rm p2}\exp\Bigl(-{\rm i}\omega_l\frac{z}{c}\Bigr), \\
\bar{E}_{\rm trig}=E_{\rm trig}\exp\Bigl({\rm i}\omega_l\frac{z}{c}\Bigr), \ \ \bar{E}_{\rm sig}=E_{\rm sig}\exp\Bigl(-{\rm i}\omega_l\frac{z}{c}\Bigr). \end{aligned}$$ By using equations (\[eq:cjplus\]) and (\[eq:cjminus\]) and the slowly varying envelope approximation, the original Schr[ö]{}dinger equation (\[eq:schrodinger\]) is reduced to a Schr[ö]{}dinger equation of the two-level system, $\ket{g}$ and $\ket{e}$: $$\begin{aligned}
{\rm i}\hbar\frac{\partial c_{g}}{\partial t}&=&-\frac{\varepsilon_0\alpha_{gg}}{4}c_g \left(|\bar{E}_{\rm p1}+\bar{E}_{\rm sig}|^2+|\bar{E}_{\rm p2}+\bar{E}_{\rm trig}|^2\right) \nonumber \\ &&-\frac{\varepsilon_0\alpha_{eg}}{2}c_e(\bar{E}_{\rm p1}^{*}+\bar{E}_{\rm sig}^{*})(\bar{E}_{\rm p2}^{*}+\bar{E}_{\rm trig}^{*}), \label{eq:cg}\\
{\rm i}\hbar\frac{\partial c_{e}}{\partial t}&=&-\hbar\delta c_e-\frac{\varepsilon_0\alpha_{ee}}{4}c_g \left(|\bar{E}_{\rm p1}+\bar{E}_{\rm sig}|^2+|\bar{E}_{\rm p2}+\bar{E}_{\rm trig}|^2\right) \nonumber \\ &&-\frac{\varepsilon_0\alpha_{eg}^{*}}{2}c_e(\bar{E}_{\rm p1}+\bar{E}_{\rm sig})(\bar{E}_{\rm p2}+\bar{E}_{\rm trig}), \label{eq:ce}\end{aligned}$$ where $\alpha_{gg}$, $\alpha_{ee}$, and $\alpha_{eg}$ represent polarizabilities of hydrogen. They are given by $$\begin{aligned}
\alpha_{gg}(\omega)=\sum_{j}\frac{|d_{gj}|^2}{\varepsilon_0\hbar}\left(\frac{1}{\omega_{jg}-\omega}+\frac{1}{\omega_{jg}+\omega}\right), \\
\alpha_{ee}(\omega)=\sum_{j}\frac{|d_{je}|^2}{\varepsilon_0\hbar}\left(\frac{1}{\omega_{je'}-\omega}+\frac{1}{\omega_{je'}+\omega}\right), \\
\alpha_{eg}(\omega)=\sum_{j}\frac{d_{gj}d_{je}}{\varepsilon_0\hbar}\frac{1}{\omega_{je'}+\omega}. \end{aligned}$$ Note that addition of angular momenta should be considered in the products of the electric dipole moments. We use an abbreviated notation $\alpha_{gg, ee, eg}(\omega_l)=\alpha_{gg, ee, eg}$ for the case of $\omega=\omega_l$. In this simulation we took into account most part of the transitions via intermediate states, that is, the $0-36$th vibrational transitions of the Lyman band and $0-13$th transitions of the Werner band [@bib:sqrt3].
The effective Hamiltonian of the two-level system is summarized by $${\rm i}\hbar\frac{\partial}{\partial t}\left(
\begin{array}{c}
c_g \\
c_e
\end{array}
\right)=H_{\rm eff}\left(
\begin{array}{c}
c_g \\
c_e
\end{array}
\right), \ \ H_{\rm eff}=-\hbar\left(
\begin{array}{cc}
\Omega_{gg} & \Omega_{ge} \\
\Omega_{eg} & \Omega_{ee}+\delta
\end{array}
\right),$$ where $\Omega_{ee}$, $\Omega_{gg}$ represent the ac Stark shifts and $\Omega_{eg}$ represents the complex two-photon Rabi frequencies, respectively. The $\Omega_{ee}$, $\Omega_{gg}$, and $\Omega_{eg}$ are given by $$\begin{aligned}
\Omega_{gg}&=&\frac{\varepsilon_0\alpha_{gg}}{4\hbar}\left(|\bar{E}_{\rm p1}+\bar{E}_{\rm sig}|^2+|\bar{E}_{\rm p2}+\bar{E}_{\rm trig}|^2\right), \label{eq:acstark} \\
\Omega_{ee}&=&\frac{\varepsilon_0\alpha_{ee}}{4\hbar}\left(|\bar{E}_{\rm p1}+\bar{E}_{\rm sig}|^2+|\bar{E}_{\rm p2}+\bar{E}_{\rm trig}|^2\right), \label{eq:acstark2} \\
\Omega_{eg}&=&\Omega_{ge}^{*}= \frac{\varepsilon_0\alpha_{eg}}{2\hbar}(\bar{E}_{\rm p1}+\bar{E}_{\rm sig})(\bar{E}_{\rm p2}+\bar{E}_{\rm trig}). \label{eq:Omega1} \end{aligned}$$ In order to take into account relaxation effects, we introduce the two-level density matrix by $$\rho(z,t) = \left(
\begin{array}{cc}
|c_g|^2 & c_gc_e^* \\
c_ec_g^* & |c_e|^2
\end{array}
\right) = \left(
\begin{array}{cc}
\rho_{gg} & \rho_{ge} \\
\rho_{eg} & \rho_{ee}
\end{array}
\right),$$ where $|\rho_{ge}|$ represents the size of the coherence. Time development of the density matrix (von Neumann equation or optical Bloch equations) which includes the relaxation effect is given by $$\begin{aligned}
\frac{\partial\rho_{gg}}{\partial t} &=& {\rm i}(\Omega_{ge}\rho_{eg}-\Omega_{eg}\rho_{ge}) + \gamma_{1}\rho_{ee},
\label{eq:Bloch1} \\
\frac{\partial\rho_{ee}}{\partial t} &=& {\rm i}(\Omega_{eg}\rho_{ge}-\Omega_{ge}\rho_{eg}) - \gamma_{1}\rho_{ee}
=-\frac{\partial\rho_{gg}}{\partial t},
\label{eq:Bloch2} \\
\frac{\partial\rho_{ge}}{\partial t} &=& {\rm i}(\Omega_{gg}-\Omega_{ee}-\delta)\rho_{ge} +{\rm i}\Omega_{ge}(\rho_{ee}-\rho_{gg}) - \gamma_{2}\rho_{ge}.
\label{eq:Bloch3}\end{aligned}$$ Parameters $\gamma_{1}$ and $\gamma_{2}$ indicate longitudinal and transverse relaxation rates respectively and represent the sizes of decoherence effect. In the case of the current setup, the longitudinal relaxation comprises the natural lifetime of the excited state and the transit-time broadening, but the $\gamma_{1}$ is negligibly small compared to the laser duration. In contrast, the $\gamma_{2}$ largely affects the signal energy.
In the Bloch equations, $\Omega_{gg}$, $\Omega_{ee}$, and $\Omega_{eg}$ include fast oscillating phases: $$\begin{aligned}
|\bar{E}_{\rm p1}+\bar{E}_{\rm sig}|^2+|\bar{E}_{\rm p2}+\bar{E}_{\rm trig}|^2 = |E_{\rm p1}|^2 + |E_{\rm p2}|^2 + |E_{\rm trig}|^2 + |E_{\rm sig}|^2 \nonumber \\
+ \left(E_{\rm p1}E_{\rm sig}^{*}\exp\Bigl(2{\rm i}\omega_l\frac{z}{c}\Bigr)
+ E_{\rm p2}E_{\rm trig}^{*}\exp\Bigl(-2{\rm i}\omega_l\frac{z}{c}\Bigr)+({\rm c.c.})\right), \\
(\bar{E}_{\rm p1}+\bar{E}_{\rm sig})(\bar{E}_{\rm p2}+\bar{E}_{\rm trig}) = E_{\rm p1}E_{\rm p2}+E_{\rm trig}E_{\rm sig} \nonumber \\ +E_{\rm p1}E_{\rm trig}\exp\Bigl(2{\rm i}\omega_l\frac{z}{c}\Bigr)+E_{\rm p2}E_{\rm sig}\exp\Bigl(-2{\rm i}\omega_l\frac{z}{c}\Bigr).\end{aligned}$$ Since it is difficult to treat such fast oscillating phase terms in our numerical simulations, we introduce further approximations. In this experimental setup, the ac Stark shift term in equation (\[eq:Bloch3\]) is small ($\max(|\Omega_{gg}-\Omega_{ee}|)=\mathcal{O}(1)$ MHz) compared to the laser linewidth and we ignore that term. Furthermore, $\rho_{ee}$ is at most $\mathcal{O}(10^{-5})$ and equation (\[eq:Bloch3\]) is approximated to $$\frac{\partial\rho_{ge}}{\partial t} = -({\rm i}\delta+\gamma_{2})\rho_{ge} -{\rm i}\Omega_{ge}.\label{eq:Blochtmp}$$ Finally, we separately consider fast oscillating terms in $\rho_{ge}$ and $\Omega_{eg}$ as $$\begin{aligned}
\rho_{ge}=\rho_{ge}^{0}+\rho_{ge}^{+}\exp\Bigl(2{\rm i}\omega_l\frac{z}{c}\Bigr)+\rho_{ge}^{-}\exp\Bigl(-2{\rm i}\omega_l\frac{z}{c}\Bigr), \\
\Omega_{eg}=\Omega_{eg}^{0}+\Omega_{eg}^{+}\exp\Bigl(2{\rm i}\omega_l\frac{z}{c}\Bigr)+\Omega_{eg}^{-}\exp\Bigl(-2{\rm i}\omega_l\frac{z}{c}\Bigr), \\
\Omega_{eg}^{0}= \frac{\varepsilon_0\alpha_{eg}}{2\hbar}(E_{\rm p1}E_{\rm p2}+E_{\rm trig}E_{\rm sig}), \\
\Omega_{eg}^{+}=\frac{\varepsilon_0\alpha_{eg}}{2\hbar}E_{\rm p1}E_{\rm trig}, \ \ \ \Omega_{eg}^{-}=\frac{\varepsilon_0\alpha_{eg}}{2\hbar}E_{\rm p2}E_{\rm sig},\end{aligned}$$ and each fast frequency component of equation (\[eq:Blochtmp\]) becomes $$\begin{aligned}
\frac{\partial\rho_{ge}^{0}}{\partial t} &=& -({\rm i}\delta+\gamma_{2})\rho_{ge}^{0} -{\rm i}\Omega_{eg}^{0*}, \label{eq:Blochs3} \\
\frac{\partial\rho_{ge}^{+}}{\partial t} &=& -({\rm i}\delta+\gamma_{2}')\rho_{ge}^{+} -{\rm i}\Omega_{eg}^{-*}, \label{eq:Blochs4} \\
\frac{\partial\rho_{ge}^{-}}{\partial t} &=& -({\rm i}\delta+\gamma_{2}')\rho_{ge}^{-} -{\rm i}\Omega_{eg}^{+*}. \label{eq:Blochs5} \end{aligned}$$ Each coherence component $\rho_{ge}^{0}$, $\rho_{ge}^{+}$, and $\rho_{ge}^{-}$ respectively represent coherence generated by the pump1 + pump2 and the trigger + signal fields, the pump2 + signal fields, and the pump1 + trigger fields. The development of the population (equations (\[eq:Bloch1\], \[eq:Bloch2\])) still include fast oscillating terms, but as shown in the next subsection the signal intensity does not depend on the population by adapting the approximation $\rho_{ee}\ll\rho_{gg}\simeq1$.
In the simulations used in our previous experiments [@bib:TPE3], values of the $\gamma_2$ were calculated from Raman linewidths (equation (\[eq:Ramanwidth\])) of an external experiment [@bib:linewidth] as approximation. In this simulation we use the same value for the $\gamma_2'$ in equation (\[eq:Blochs4\]) and (\[eq:Blochs5\]). However, as described in section \[sec:discussion1\], the Doppler broadening effect is negligible in the case of the excitation by the counter-propagating lasers. Thus, the value of the $\gamma_2$ in equation (\[eq:Blochs3\]) is calculated from only the pressure broadening term of the Raman linewidth.
Maxwell equations
-----------------
We also consider developments of the laser fields and the macroscopic polarization of the p-H$_2$ molecules. Maxwell’s equations in a homogeneous dielectric gas medium without sources and magnetization are given by $$\begin{aligned}
\nabla\cdot\tilde{\bm{D}}=0, \ \ \ &&\nabla\cdot\tilde{\bm{B}}=0, \nonumber \\
\nabla\times\tilde{\bm{E}}=-\frac{\partial}{\partial t}\tilde{\bm{B}}, \ \ \
&&\nabla\times\tilde{\bm{B}}=\mu_0 \frac{\partial}{\partial t}\tilde{\bm{D}}, \nonumber \\
\tilde{\bm{D}}=\varepsilon_0\tilde{\bm{E}}+N_t\tilde{\bm{P}}, \ \ \ &&\nabla\cdot\tilde{\bm{E}}=0, \nonumber\end{aligned}$$ where $N_t$ denotes the number density of the p-H$_2$ gas and $\tilde{\bm{E}}$ and $\tilde{\bm{P}}$ represent the laser fields and the macroscopic polarization of the p-H$_2$ molecules, respectively. From these equations, developments of the laser fields propagating in the $z$ direction are given by $$\frac{\partial^2\tilde{\bm{E}}}{\partial t^2}-c^2\frac{\partial^2\tilde{\bm{E}}}{\partial z^2}=-\frac{N_t}{\varepsilon_0}\frac{\partial^2\tilde{\bm{P}}}{\partial t^2}. \label{eq:Maxwell0}$$ In the current experimental setup, the $\tilde{\bm{E}}$ and the $\tilde{\bm{P}}$ are written by $$\begin{aligned}
\tilde{\bm{E}}=\tilde{\bm{E}}_{\rm p1}+\tilde{\bm{E}}_{\rm p2}+\tilde{\bm{E}}_{\rm trig}+\tilde{\bm{E}}_{\rm sig}, \\
-\tilde{\bm{P}}=-\bra{\psi}\bm{d}\ket{\psi} \label{eq:polarization} \\
=(c_{j-}^{*}c_g e^{{\rm i} \omega_{jg}t}d_{jg}+c_{g}^{*}c_{j+} e^{{\rm i} \omega_{gj}t}d_{gj} +c_{e}^{*}c_{j+} e^{{\rm i} \omega_{e'j}t}d_{ej}+c_{j-}^{*}c_e e^{{\rm i} \omega_{je'}t}d_{je})\hat{\bm{\epsilon}}_{R} + ({\rm c.c.}). \nonumber \end{aligned}$$ By using equations (\[eq:cjplus\]) and (\[eq:cjminus\]) and ignoring all the frequency components other than $\exp(\pm{\rm i}\omega_l t)$, equation (\[eq:polarization\]) becomes $$\begin{aligned}
\frac{2}{\varepsilon_0}\tilde{\bm{P}}&=&\Bigl[((\alpha_{gg}\rho_{gg}+\alpha_{ee}\rho_{ee})E_{\rm p1}+2\alpha_{eg}\rho_{ge}^{*}E_{\rm p2}^{*})\exp\left(-{\rm i}\omega_l\left(t-\frac{z}{c}\right)\right) \nonumber \\
&+& ((\alpha_{gg}\rho_{gg}+\alpha_{ee}\rho_{ee})E_{\rm sig}+2\alpha_{eg}\rho_{ge}^{*}E_{\rm trig}^{*})\exp\left(-{\rm i}\omega_l\left(t+\frac{z}{c}\right)\right) \nonumber \\
&+& ((\alpha_{gg}\rho_{gg}+\alpha_{ee}\rho_{ee})E_{\rm p2}^{*}+2\alpha_{eg}^{*}\rho_{ge}E_{\rm p1})\exp\left({\rm i}\omega_l\left(t+\frac{z}{c}\right)\right) \nonumber \\
&+& ((\alpha_{gg}\rho_{gg}+\alpha_{ee}\rho_{ee})E_{\rm trig}^{*}+2\alpha_{eg}^{*}\rho_{ge}E_{\rm sig})\exp\left({\rm i}\omega_l\left(t-\frac{z}{c}\right)\right)\Bigr]\bm{\hat{\epsilon}}_{R} \nonumber \\
&+&({\rm c.c.}).\end{aligned}$$ By using following approximations $$\begin{aligned}
\alpha_{gg}\rho_{gg}+\alpha_{ee}\rho_{ee}\simeq\alpha_{gg}, \\
\left(\frac{\partial}{\partial t}\pm{\rm i} \omega_l \right) (\rm{slowly\ varying\ term}) \simeq \pm{\rm i} \omega_l (\rm{slowly\ varying\ term}) , \\
\left(c\frac{\partial}{\partial z}\pm{\rm i} \omega_l \right) (\rm{slowly\ varying\ term}) \simeq \pm{\rm i} \omega_l (\rm{slowly\ varying\ term}),\end{aligned}$$ we obtain the developments of the envelopes of the electric fields: $$\begin{aligned}
\left(\frac{\partial}{\partial t}-c\frac{\partial}{\partial z}\right) E_{\rm p1} &=& \frac{{\rm i}\omega_lN_t}{2}\left(\alpha_{gg}E_{\rm p1}+2\alpha_{eg}(\rho_{ge}^{0*}E_{\rm p2}^{*}+\rho_{ge}^{-*}E_{\rm trig}^{*})\right),
\label{eq:Maxwells1} \\
\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial z}\right) E_{\rm p2} &=& \frac{{\rm i}\omega_lN_t}{2}\left(\alpha_{gg}E_{\rm p2}+2\alpha_{eg}(\rho_{ge}^{0*}E_{\rm p1}^{*}+\rho_{ge}^{+*}E_{\rm sig}^{*})\right),
\label{eq:Maxwells2} \\
\left(\frac{\partial}{\partial t}-c\frac{\partial}{\partial z}\right) E_{\rm trig} &=& \frac{{\rm i}\omega_lN_t}{2}\left(\alpha_{gg}E_{\rm trig}+2\alpha_{eg}(\rho_{ge}^{0*}E_{\rm sig}^{*}+\rho_{ge}^{-*}E_{\rm p1}^{*})\right),
\label{eq:Maxwells3} \\
\left(\frac{\partial}{\partial t}+c\frac{\partial}{\partial z}\right) E_{\rm sig} &=& \frac{{\rm i}\omega_lN_t}{2}\left(\alpha_{gg}E_{\rm sig}+2\alpha_{eg}(\rho_{ge}^{0*}E_{\rm trig}^{*}+\rho_{ge}^{+*}E_{\rm p2}^{*})\right).
\label{eq:Maxwells4} \end{aligned}$$ As evident from these equations, signal photons are generated by the trigger field with the $\rho_{ge}^0$ and the pump2 field with the $\rho_{ge}^+$. Since the $\rho_{ge}^+$ is developed by the pump2 + signal fields and $|\rho_{ge}^+|$ is small, most part of the signal light is generated by the trigger field with the coherence generated by the pump beams.
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|
---
abstract: 'Three-dimensional tracking of animal systems is the key to the comprehension of collective behavior. Experimental data collected via a stereo camera system allow the reconstruction of the 3d trajectories of each individual in the group. Trajectories can then be used to compute some quantities of interest to better understand collective motion, such as velocities, distances between individuals and correlation functions. The reliability of the retrieved trajectories is strictly related to the accuracy of the 3d reconstruction. In this paper, we perform a careful analysis of the most significant errors affecting 3d reconstruction, showing how the accuracy depends on the camera system set-up and on the precision of the calibration parameters.'
author:
- 'Andrea Cavagna$^{*}$, Chiara Creato$^{*,\ddagger}$, Lorenzo Del Castello$^{*,\ddagger}$, Irene Giardina$^{*,\ddagger}$, Stefania Melillo$^{*,\ddagger}$, Leonardo Parisi$^{*,\S}$, Massimiliano Viale$^{*,\ddagger}$'
title: ' Error control in the set-up of stereo camera systems for 3d animal tracking'
---
Introduction {#introduction .unnumbered}
============
Technological improvements in the field of digital cameras are strongly simplifying the study of collective behavior in animal groups. The use of single or multi-camera systems to record the time evolution of a group is by far the most common tool to understand collective motion. The emergence of collective behavior in human crowds [@gallup2012; @moussaid2009], fish schools [@katz2011; @butail20103d], bird flocks [@attanasi2014information; @attanasi2014emergence] and insect swarms [@attanasi2014prl; @attanasi2014collective; @butail20113d; @butail2012reconstructing] have been investigated. Events of interest are recorded from one or more cameras and images are then processed to reconstruct the trajectory of each individual in the group. Positions of single individuals at each instant of time are used to characterize the system. Density, mean velocity, mean acceleration, size of the group, as well as single velocities and accelerations are computed to understand how collective behavior arises and following which rules. The reliability of these results is strictly connected to the accuracy of the reconstructed trajectories.
In a previous work Cavagna et al. in [@cavagnaAnimal] suggested how to reduce the error on the retrieved trajectories choosing the proper set up for the system. More recently in [@theriault2014], a tool to check the accuracy of a multicamera system on the reconstructed trajectories is provided, together with a software for the calibration of the intrinsic and extrinsic parameters. In [@towne2012], Towne et al. give a way to measure the reconstruction error and to quantify it when a DLT technique is used to calibrate the extrinsic parameters of the system. But a theoretical discussion on the propagation of the errors from experimental measures to the reconstructed positions and trajectories is still missing.
From a collective behavior perspective, distances between targets are much more interesting than absolute positions. Indeed, quantities like density, velocity, acceleration are not referred to the position of single animals, but they involve a measure of a distance. For this reason, in our discussion we will show how to estimate the error on the reconstructed $3d$ position of a single target, but we will give more emphasis to the error on mutual distances between two targets. We will focus the analysis on how experimental measurements and calibration uncertainty affect mutual distances between targets and we will give some suggestions on how to choose a suitable set up in order to achieve the desired and acceptable error.
In the first section of the paper we give a description of the pinhole model, which is by far the simplest but effective approximation of a digital camera. We introduce the nomenclature used in the entire paper. Moreover we describe the mathematical relations holding between the position of a target in the three dimensional real world and the position of its image on the camera sensor. In the same section, we describe the general principles of the $3d$ reconstruction making use of systems of two pinhole cameras. In the second section we will show the error formulation for both absolute position of a single target and mutual distance between pair of targets considering at first one camera only, and then generalizing the results to the case of a camera system. In the third section we give an interpretation of the error formulation to suggest how the reconstruction error can be reduced by properly choosing the suitable intrinsic and extrinsic parameters of the system. In the fourth and last section we consider the set up we use in the field to record starling flocks and midge swarms. We give a description on the tests we perform to check the accuracy of the retrieved trajectories. Moreover we practically show how the results of these tests are affected by experimental measurements inaccuracy.
Three dimensional reconstruction: general principles {#section::general}
====================================================
The reconstruction of the position of a target imaged by one or more camera can be intuitively address with a geometric perspective. In some special situations, the images of only one camera gives enough information to determine where the target is located in the real world. However in the general case at least two cameras are needed.
In this section we address the geometric formulation of the $3d$ reconstruction problem for one camera and for a system of two cameras, knowing intrinsic and extrinsic parameters of the system. Intrinsic parameters fix the geometry of the camera and its lens: the position of the image center, the focal length and the distortion coefficients. We calibrate the intrinsic parameters taking into account the radial distortion up to the first order coefficient only, while we do not consider tangential distortion. Tests presented in Section \[section::test\] show that this is sufficient to obtain the desired accuracy in the $3d$ reconstruction. Extrinsic parameters, instead, describe the position and orientation of the camera system with respect to a world reference frame. They generally include a set of angles, fixing the orientation of the cameras and a set of length measures expressed in meters defining the cameras positions. Depending on the experiment they can be practically measured with high precision instruments or calibrated through software tools. In the paper, we consider cameras in the pinhole approximation, which is the easiest but effective camera model. Note that the pinhole approximation does not take into account lens distortion. In the following when talking about the position of a target on the image plane, we will always refer to its coordinates already undistorted.
![ [**Pinhole model.**]{} [**a:**]{} Similarity between the dashed and the filled orange triangles shows the proportionality between the $x$ coordinate of a $3d$ target $P$ and the $u$ coordinate of its image $Q$ by a factor $z/\Omega$. Similarity between the two dashed and filled green triangles shows instead the proportionality between the $y$ coordinate of the point $P$ and the $v$ coordinate of its image. [**b:**]{} The two targets $P_1$ and $P_2$ lying on the same $z$-plane are imaged in $Q_1$ and $Q_2$. Similarity between the two dashed and filled orange triangles shows that the distance $ R=|\protect\overrightarrow{P_1P}_2|$ is proportional to the $2d$ distance $r=|\protect\overrightarrow{Q_1Q}_2|$ through the coefficient $z/\Omega$. []{data-label="fig:pinhole_model"}](Fig1.pdf){width="1.0\columnwidth"}
A camera maps the three dimensional real world into the two dimensional space of its image. In the pinhole approximation the correspondence between the $3d$ space and the $2d$ image plane is defined as a central projection with center in the focal point $O$, see Fig.\[fig:pinhole\_model\]a. The image plane is located at a distance $f$ from the focal point $O$, with $f$ being the focal length of the lens coupled with the camera. Under the pinhole camera model, a point $P$ in the three dimensional world is mapped to the point $Q$ on the image plane belonging to the line connecting $O$ and $P$. Note that in this approximation the image plane is located in front of the focal point. Thus the resulting image on the sensor does not reproduce the scene rotated of $180^\circ$ as it would be in a real camera. We choose this representation for its simplicity and because it does not affect in any way the discussion of the present paper.
The natural $3d$ reference frame for a camera in the pinhole approximation is the one having the origin in the focal point $O$, $z$-axis coincident with the optical axis of the camera and the $xy$-plane parallel to the sensor, as shown in Fig.\[fig:pinhole\_model\]a. Coordinates in this reference frame are expressed in meters, while coordinates of a point belonging to the sensor plane are usually expressed in pixels. The size of a single pixel, $w_p\times h_p$, defines the correspondence between the two units of measurements. Thus the point $Q$ belonging to the image plane and such that $Q^\prime\equiv(x^\prime,y^\prime,f)$ in the $3d$ pinhole reference frame corresponds to $Q\equiv(u,v)$ in the sensor reference frame, with $u=x^\prime/w_p$ and $v=y^\prime/h_p$. For the sake of simplicity, we will assume to deal with sensor made of squared pixels, i.e. $w_p=h_p$ so that the conversion factor from meters to pixel is the same in both the direction $x$ and $y$. From now on, we denote the coordinates expressed in meters by $x$, $y$ and $z$, and the coordinate expressed in pixels by $u$, $v$ while $\Omega$ represents the focal length $f$ expressed in pixels.
Pseudo-$\mathbf{3d}$ reconstruction: single camera case
-------------------------------------------------------
As shown in Fig.\[fig:pinhole\_model\]b, target $P=(x,y,z)$ in the real world is projected into a point $Q=(u,v)$ on the image plane. Using similarity between triangles it can be shown that $u=\Omega x/z$ and $v=\Omega y/z$. The correspondence between the $3d$ real space and the $2d$ image plane is then defined as: $
(x,y,z)\rightarrow(u,v)=(\Omega x/z,\Omega y/z),
$ or equivalently: $$\label{eq::pinhole}
u=\Omega x/z \mbox{ and } v=\Omega y/z$$ Thus, knowing the $3d$ position of a target we can determine the $2d$ coordinates of its image. But in the general case we would like to do the opposite: knowing the position of a target on the sensor plane, we would like to retrieve the corresponding $3d$ coordinates in the pinhole reference frame. If no other informations are available, eqs.(\[eq::pinhole\]) do not have a unique solution in the unknown $(x,y,z)$ so that $3d$ reconstruction is not feasible making use of one camera only, and at least two cameras are needed.
In the special case of targets lying on a plane, extra information about the mutual position between the camera and the plane where the motion occurs, i.e. extrinsic parameters, can be used to define an homography. Eqs.(\[eq::pinhole\]) can then be inverted and the $3d$ positions of the targets can be retrieved making use of one camera only. For the sake of simplicity, in this section we will not address this general case, but only the easier particular situation when the motion happens on a plane parallel to the camera sensor. The interested reader can retrieve the exact formulation for the general planar motion putting together the information written in this section and in the following one, where the general $3d$ case is discussed.
In the special case of targets lying on a plane parallel to the sensor, i.e. fixed $z$, eqs.(\[eq::pinhole\]) can be inverted and the $3d$ position of a target projected in $Q=(u,v)$ can be computed as: $$\label{eq::pinhole_XYZ}
x=u\displaystyle\frac{z}{\Omega} \mbox{ and } y=v\displaystyle\frac{z}{\Omega}$$
{width="100.00000%"}
Consider now two targets $P_1\equiv(x_1,y_1,z)$ and $P_2\equiv(x_2,y_2,z)$ on the same $z$-plane, and their images $Q_1\equiv(u_1,v_1)$ and $Q_2\equiv(u_2,v_2)$ as in Fig.\[fig:pinhole\_model\]b. Thanks again to similarity between triangles, the distance between the two targets $ R$ and the distance between their projections $r$ satisfy the following equation: $$\label{eq::2d_deltaR}
R= r\displaystyle\frac{z}{\Omega}$$ so that $z/\Omega$ fixes the ratio between distances in the $3d$ and $2d$ space.
With the same argument it can be shown that: $$\label{eq::2d_deltaX}
\Delta x=\Delta u\displaystyle\frac{z}{\Omega} \mbox{ , }\Delta y=\Delta v\displaystyle\frac{z}{\Omega}$$ where $\Delta x=x_1-x_2$, $\Delta y=y_1-y_2$, $\Delta u=u_1-u_2$, $\Delta v=v_1-v_2$.
In the field of collective behavior when we deal with cells or bacteria moving on a glass slide or fish in swallow water, the motion happens on a preferential plane. Setting up the camera in such a way that the sensor is parallel to this special plane, the displacement in the $z$ direction is negligible; $z$ can be considered constant and the equations above hold. Thus, measuring with an high precision instrument the distance, $z$, between the sensor and the plane where the experiment takes place, it is possible to reconstruct the position of each target in the pinhole reference frame. In the following we will refer to these particular cases as $2d$ experiments.
$\mathbf{3d}$ reconstruction: stereometric camera system
--------------------------------------------------------
The ambiguity of the $3d$ position of a target from its image on one camera only, can be easily solved making use of two or more cameras. For the sake of simplicity, from now on we take into account only system of two synchronized cameras, having the same focal length and sensors of the same size.
Consider, as in Fig\[fig:pinhole\_system\]a, a three dimensional target $P$ in the $3d$ real world and its projections $Q_L$ and $Q_R$ on the sensor plane of two cameras, denoted by left and right. $P$ belongs to the line $l_L$ passing by $O_L$ and $Q_L$ as well as to the line $l_R$ passing by $O_R$ and $Q_R$. $P$ is the crossing point between the two lines $l_L$ and $l_R$ as shown in Fig.\[fig:pinhole\_system\]a.
In each of the two cameras, eq.(\[eq::pinhole\]) holds, and the two lines $l_L$ and $l_R$ can be defined as:
- in the reference frame of the left camera a parametric equation, with parameter $a$ for the line $l_L$ is: $l_L=a(u_L,v_L,\Omega)$;
- in the reference frame of the right camera a parametric equation, with parameter $b$ for the line $l_R$ is: $l_R=b(u_R,v_R,\Omega)$.
In order to find the crossing point, $P$, between $l_L$ and $l_R$ we need to express both the lines in the same reference frame: we choose the reference frame of the left camera to be the world reference frame.
In the world reference frame $l_R=\vec{D}^T+bM(u_R,v_R,\Omega)^T$ where $\vec{D}=\overrightarrow{O_LO}_R$, $M$ is the rotation matrix which brings the world reference frame parallel to the reference frame of the right camera, and the superscript $T$ denotes that the correspondent vector is transposed. The crossing point between the two lines is then obtained making use of the solution the system defined by $l_L=l_R$ in the unknown $a$ and $b$: $$\label{eq::lLlR}
a(u_L,v_L,\Omega)^T=\vec{D}^T+bM(u_R,v_R,\Omega)^T.$$ The solution $(a^\star, b^\star)$ identifies the position $P\equiv(x,y,z)=(a^\star u_L, a^\star v_L, a^\star\Omega)=(u_Lz/\Omega, v_Lz/\Omega, a^\star\Omega)$. Eqs. (\[eq::lLlR\]) are well defined when $\vec{D}$ and $M$, i.e. the extrinsic parameters of the system, are known.
$\vec{D}$ represents the vector distance between the two cameras. Its modulus $|\vec{D}|=d$ is the distance expressed in meters between the two focal points $O_L$ and $O_R$. Instead, the orientation of $\vec{D}$ can be expressed in spherical coordinates through two angles, $\delta$ and $\epsilon$: $\vec{D}=d(\cos\delta\cos\epsilon, \sin\delta\cos\epsilon, \sin\epsilon)^T$, see Fig.\[fig:pinhole\_system\]a. Denote by $\vec{D}_{xy}$ the projection of $\vec{D}$ on the $xy$-plane. $\delta$ is defined as the angle between the $x$-axis and $\vec{D}_{xy}$, while $\epsilon$ is the angle between $\vec{D}$ and $\vec{D}_{xy}$. The rotation matrix $M$ can be parametrized by the three angles of yaw, pitch and roll denoted respectively by $\alpha$, $\beta$ and $\gamma$: $M=M_{\alpha}M_{\beta}M_{\gamma}$. The mutual position and orientation of the cameras is then defined through the distance $d$ and the $5$ angles $\alpha$, $\beta$, $\gamma$, $\delta$ and $\epsilon$. These parameters are directly measured or calibrated when performing the experiment. We will show in the next section that inaccuracy on these quantities can strongly affect the retrieved position of the target $P$.
Consider, as in Fig.\[fig:pinhole\_system\]b, two targets $P_1\equiv(x_1,y_1,z_1)$ and $P_2\equiv(x_2,y_2,z_2)$ and their images $Q_{1L}=(u_{1L},v_{1L})$ and $Q_{2L}=(u_{2L},v_{2L})$ in the left camera and $Q_{1R}=(u_{1R},v_{1R})$ and $Q_{2R}=(u_{2R},v_{2R})$ in the right camera. The expression for $ R$ becomes more complicated passing from $2d$ to $3d$ experiments, since eq.(\[eq::2d\_deltaX\]) does not hold anymore. $\Delta x$ depends on both $\Delta u$ and $\Delta z$, as well as $\Delta y$ depends on $\Delta v$ and $\Delta z$. From eq. (\[eq::pinhole\]), $x_1=u_{1L}z_1/\Omega$ and $x_2=u_{2L}z_2/\Omega$. This implies that $$\label{eq::3d_deltaX}
\Delta x=x_1-x_2=(\Delta u\bar{z}+\Delta z\bar{u})/\Omega$$ where $\bar{z}=(z_1+z_2)/2$, and $\bar{u}=(u_{1L}+u_{2L})/2$. With the same argument it can be proved that $\Delta y=(\Delta v\bar{z}+\Delta z\bar{v})/\Omega$, with $\bar{v}=(v_{1L}+v_{2L})/2$. As a consequence:
$$\label{eq::3d_deltar}
R=\left(\displaystyle\frac{(\Delta u\bar{z}+\Delta z\bar{u})^2}{\Omega^2}+\displaystyle\frac{(\Delta v\bar{z}+\Delta z\bar{v})^2}{\Omega^2}+\Delta z^2\right)^{1/2}$$
For short $\Delta z$, eq.(\[eq::3d\_deltar\]) becomes $ R= r\bar{z}/\Omega$ giving back eq.(\[eq::2d\_deltaR\]) for the $2d$ experiments.
The introduction of a non constant third coordinate $z$, makes the expression of the reconstructed position not transparent. For this reason we will not discuss the general case, for further information see [@hartley2003book], but in the following we will retrieve the exact solution of eq.(\[eq::lLlR\]) for the two special cases described in Fig.\[fig:common\_fov\] and highlighted respectively in black and red.
### **[Pure translation along x axis.]{}**
In this special case, the two cameras have the same orientation and the two focal points both lie on the $x$ axis with a mutual distance equal to $d$, see Fig.\[fig:common\_fov\] where this set up is highlighted in black together with its field of view. $\vec{D}=d(1,0,0)^T$ and the rotation matrix $M$ is equal to the identity matrix. Eqs.(\[eq::lLlR\]) become: $$a(u_L,v_L,\Omega)^T=(d,0,0)^T+b(u_R,v_R,\Omega)^T$$ In order to retrieve the position of the target $P$ imaged in $Q_L\equiv(u_L,v_L)$ and $Q_R\equiv(u_R,v_R)$ the above system has to be solved in the unknown $a$ and $b$. We find the solution $a^\star=b^\star=d/(u_L-u_R)=d/s$. $a^\star$ represents the ratio between the metric distance between the two focal points and the disparity $s=u_L-u_R$ expressed in pixels.
From equations (\[eq::lLlR\]) we obtain: $$\left\{
\begin{array}{rl}
x &= u_Lz/\Omega\\
y &= v_Lz/\Omega\\
z &= \Omega d/s
\end{array}
\right.$$
![[**Parallel vs rotated cameras set up.**]{} In black a set up with the two cameras with the same orientation but with a displacement in the direction of $x$, while in red a symmetric set up with a mutual rotation about the $y$ axis only. The two different system field of views are highlighted respectively in black and red. Note that increasing the focal length of the two cameras makes the common field of view narrower. Increasing $d$ moves the working distance further in the $z$ direction and also reduces the portion of $3d$ space imaged by both cameras. Moreover the parallel set up has an optimal field of view at large $z$ while the rotated set up is optimal for short $z$, indicating that $\alpha$ affects the optimal working distance. []{data-label="fig:common_fov"}](Fig5.pdf){width="1.0\columnwidth"}
### **[Translation along the x axis plus symmetric rotation about the y axis.]{}**
This is the special case obtained applying a translation along the $x$ axis and then rotating the left camera of an angle $\alpha/2$ in the clockwise direction and the right camera of an angle $\alpha/2$ in the counterclockwise direction about the $y$ axis, as shown in Fig.\[fig:common\_fov\] where this set up is highlighted in red.
The mutual angle of rotation about the $y$ axis between the cameras is equal to $\alpha$, so that the rotation matrix is: $$M=
\begin{pmatrix}
\cos\alpha & 0 & -\sin\alpha\\
0 & 1 & 0\\
\sin\alpha & 0 & \cos\alpha
\end{pmatrix}$$
Eqs. (\[eq::lLlR\]) are then:
$$\begin{array}{lcl}
au_L &=& d\cos\epsilon+b[u_R\cos\alpha -\Omega\sin\alpha]\\
av_L &=& bv_R\\
a\Omega &=& d\sin\epsilon+b[u_R\sin\alpha +\Omega\cos\alpha]
\end{array}$$
The solution of the above system is not trivial and different approximations can be made to simplify the problem. In our case we can assume that the angle of rotation $\alpha$ is small and since the set up is symmetric $\epsilon=\alpha/2$.
{width="100.00000%"}
For small angles $\alpha$, $\sin\alpha\sim\alpha$ and $\cos\alpha\sim 1$, and the previous equations become:
$$\begin{array}{lcl}
au_L &=& d+b[u_R-\alpha\Omega]\\
av_L &=& bv_R\\
a\Omega &=& d\alpha/2+b[\alpha u_R+\Omega]
\end{array}$$
Solving the above system we obtain: $$a^\star=\displaystyle\frac{1}{\Omega}\left(\Omega d - u_Ld\displaystyle\frac{\alpha}{2}\right)\displaystyle\frac{\alpha u_R+\Omega}{\alpha(u_Lu_R+\Omega^2)+\Omega s}$$ and with the additional assumption that $u_L$, $u_R\ll\Omega$, $a^\star=d/(s+\alpha\Omega)$. So that, eqs.(\[eq::lLlR\]) become: $$\label{eq::3d_xyz}
\left\{
\begin{array}{rl}
x &= u_Lz/\Omega\\
y &= v_Lz/\Omega\\
z &= a^\star\Omega=\Omega d/(s+\alpha\Omega)
\end{array}
\right.$$ Note that for $\alpha=0rad$ the solution is exactly what we obtained in the case of pure translation.
The approximation $\sin\alpha\sim\alpha$ and $\cos\alpha\sim 1$ holds for angles approaching $0~rad$. For angles smaller than $0.2~rad$, the error in the approximation is of the third order for the sine and of the second order for the cosine, so that if $\alpha=0.2~rad$, $\sin\alpha-\alpha\sim 10^{-3}~rad$ and $\cos\alpha-1\sim 10^{-2}~rad$. When $\alpha$ is not small, eq.(\[eq::lLlR\]) can not be simplified and the solution is not trivial anymore. For the sake of simplicity, we do not give here the formulation of the solution for the general case.
Error control: theoretical relations
====================================
In the two previous sections we described systems of one or more cameras in the pinhole approximation. We showed how to retrieve the three dimensional position of a target and the mutual distance between two targets knowing only the parameters of the system. Through the error analysis we want to quantify how errors in the experimental measures and calibration of the intrinsic and extrinsic parameters affect the reconstruction process. Moreover we want to investigate the possibility to reduce the error choosing the proper experimental set up. We will focus our analysis on the reconstruction of the three dimensional position of a target, but we will give more emphasis to the propagation of the error in the retrieved mutual distance between two targets.
We will first address the error theory in the case of $2d$ experiments showing how to quantify the error making use of geometry only, then we will approach the error theory from a more formal and mathematical point of view. Finally we will consider the more general case of $3d$ experiments only in the formal way, since the geometric interpretation is not very intuitive.
2d experiments
--------------
This is the special case where objects move on a plane parallel to the sensor at a distance $z$ from the focal point. The position of target $P$ projected in $Q\equiv(u,v)$ is: $x=uz/\Omega$ and $y=vz/\Omega$. Instead the distance $ R$ between two targets $P_1$ and $P_2$ is computed making use of eq.(\[eq::2d\_deltaR\]); so that $R= rz/\Omega$ where $r$ is the distance in pixels between the projections of the two targets $P_1$ and $P_2$. The quantities involved are then $u,v,z,\Omega$ and we want to investigate how each of these parameters affect $x$, $y$ and $ R$.
### **Error on absolute position.**
Consider the case when $z$ is directly measured with an error $\delta z$. Given a target $P\equiv(x,y,z)$, it is reconstructed in $P^\prime$ on the $(z+\delta z)$-plane on the line passing by $O$, $P$ and $Q$, as shown in Fig.\[fig:error\_camera\_system\]a. Making use of similarity between triangles the following relation between the error on the $x$-coordinate of $P$, $\delta x$ and $\delta z$ can be shown: $$\delta x/x=\delta z/z$$ $\delta z/z$ is constant for all the targets in the field of view. The equation above implies that the position of each target projected on the sensor is affected by the same relative error $\delta x/x=\delta z/z$. In other words, the error $\delta x$ depends on the position $x$ of the target and the larger $x$, the larger is the error $\delta x$, while the ratio $\delta x/x$ is constant and equal to $\delta z/z$. In $2d$ experiments, the error in $z$ is the accuracy of the instrument used to measure it. If an instrument with an accuracy of $1mm$ is used on $z=10cm$, the relative error on $z$, and as a consequence of $x$, is equal to $0.01$. Note that the relative error is dimensionless. If instead the same instrument is used to take the measure of $z=10m$ the relative error becomes negligible being equal to $0.0001$, and producing a negligible relative error on $x$. While designing the set up an acceptable threshold for the relative error on $x$ has to be defined, then the working distance $z$ and the measure instrument can be chosen accordingly.
Fig.\[fig:error\_camera\_system\]b represents a system where the focal length is calibrated with an error $\delta\Omega$. This makes the sensor of the camera to be at a distance $\Omega+\delta\Omega$ from the focal point, instead of at a distance $\Omega$. $P$ is projected in $Q^\prime$ with the same $u$ coordinate of $Q$, but on the $(\Omega+\delta\Omega)$-plane, while the retrieved $P^\prime$ lies on the same $z$ plane of $P$ but on the line passing by $O$ and $Q^\prime$ and not on the correct one. So that its position on the $xz$-plane is $P^\prime\equiv(x+\delta x, z)$. Note that $x+\delta x<x$, meaning that the error $\delta x$ is negative, as shown in Fig.\[fig:error\_camera\_system\]b. Making use of similarity between triangles, it can be shown that: $x/z=u/\Omega$ and $(x+\delta x)/z=u/(\Omega+\delta\Omega)$. Putting together these two equations we obtain: $$\delta x/x =-\delta\Omega/(\Omega+\delta\Omega)$$ The negative sign in this equation indicates that a positive error on $\Omega$, produces a negative relative error $\delta x/x$, i.e. if the incorrect focal length is bigger than the correct one than the retrieved $x^\prime=x+\delta x$ is smaller than the correct $x$, as shown in Fig.[\[fig:error\_camera\_system\]b]{}.
As for the error on $z$, the error on $\Omega$ fixes the relative error on $x$. An error $\delta\Omega=30px$ with $\Omega=3000px$ produces a relative error on $x$ equal to $0.01$. The error on $\delta\Omega$ is generally completely due to the calibration procedure used for $\Omega$, so that it can be easily reduce using a precise calibration software.
Fig.\[fig:error\_camera\_system\]c represents a system where an error $\delta u$ on the determination of the position of the projection of the target occurs. The point $P$ is then considered to be projected in $Q^\prime$ instead of $Q$. The retrieved position $P^\prime$ lies on the same $z$-plane of $P$, but on the line passing by $O$ and $Q^\prime$. Similarity between triangles shows that: $$\delta x=\delta uz/\Omega$$ The error $\delta u$ affects $x$ in a different way than the other two parameters. Unlike the error on $z$ and $\Omega$, it does not produce a constant relative error. Moreover it does not depend on the $x$ coordinate of the target. In a set up with $z=100m$ and $\Omega=3000px$, a target in the image segmented with an error $\delta u=3px$ produces an error $\delta x=0.1m$. If the position of a target is $x=1m$, an error of $0.1m$ corresponds to a relative error of $0.1$. If the error occurs on a target at $x=10m$, its retrieved position is $10.1m$ corresponding to a relative error of $0.01$. Note that, since the camera pinhole model does not include distortion effect, the error $\delta u$ includes the segmentation error due to noise on the picture and the error in the position of the point of interest when the distortion coefficient are not properly calibrated.
The error $\delta x$ can be kept under control choosing the proper parameters of the set up, in particular the ratio $z/\Omega$. If the maximum acceptable error on $\delta x$ is defined as $c$ and $\delta u\sim 1px$, $z/\Omega$ has to be chosen in order to verify: $z\delta u/\Omega=z/\Omega<c$.
In the general case the error on $x$ is the sum of the three contributes due to $\delta z$, $\delta\Omega$ and $\delta u$, so that: $$\label{eq::2d_percentage_deltax}
\delta x=x\left(\displaystyle\frac{\delta z}{z}-\displaystyle\frac{\delta\Omega}{\Omega+\delta\Omega}\right)+\delta u\displaystyle\frac{z}{\Omega}.$$
For large $x$, targets at the edge of the field of view, the dominant term of the error is the relative part due to $\delta z$ and $\delta\Omega$, while for small $x$, targets in the center of the field of view, the dominant part is the one due to the error on $\delta u$.
Note that the entire discussion of this section could have been addressed in a more formal way simply computing the derivative of $x$ respect to the parameters $u$, $z$ and $\Omega$: $\delta x=x\left(\delta z/z-\delta\Omega/\Omega\right)+\delta uz/\Omega.$ The difference between this last equation and eq.(\[eq::2d\_percentage\_deltax\]) is only in the term depending on $\delta\Omega$ but in the general case we can assume that $\delta\Omega\ll\Omega$ so that the two terms can be considered equal.
The same arguments used to retrieve the error on $x$ can be used to write the formulation of the error on the $y$ coordinate only referring the schemes in Fig.\[fig:error\_camera\_system\] to the $yz$-plane: $$\label{eq::2d_percentage_deltay}
\delta y=y\left(\displaystyle\frac{\delta z}{z}-\displaystyle\frac{\delta\Omega}{\Omega}\right)+\delta v\displaystyle\frac{z}{\Omega}.$$
### **Error on mutual distances between targets.**
The error $\delta R$ on the distance $ R$ can be obtained deriving eq.(\[eq::2d\_deltaR\]) with respect to $z$, $\Omega$ and $ R$: $$\label{eq::2d_deltaDeltaR}
\delta R= R\left(\displaystyle\frac{\delta z}{z}-\displaystyle\frac{\delta\Omega}{\Omega}\right)+\displaystyle\frac{z}{\Omega}\delta r.$$ On the $xz$-plane the previous equation is: $$\label{eq::3d_deltaDeltax}
\delta\Delta x=\Delta x\left(\displaystyle\frac{\delta z}{z}-\displaystyle\frac{\delta\Omega}{\Omega}\right)+\displaystyle\frac{z}{\Omega}\delta\Delta u.$$ The error on $z$ and $\Omega$ produce the same effect on the distance $\Delta x$ then on the absolute position of a target. They both induce a constant relative error on the distances between targets. The error $\delta\Delta x$ on large $\Delta x$ is higher than on small $\Delta x$.
The third term, instead does not depend on $\Delta x$. As for $\delta x$, the first two terms of the error on $\Delta x$ can be reduced choosing a proper instrument to measure $d$ and a precise calibration software to calibrate $\Omega$, while the third term can be kept under a certain threshold choosing a set up with the proper ratio $z/\Omega$.
Referring the same arguments to the $yz$-plane: $$\delta\Delta y=\Delta y\left(\displaystyle\frac{\delta z}{z}-\displaystyle\frac{\delta\Omega}{\Omega}\right)+\displaystyle\frac{z}{\Omega}\delta\Delta v.$$ Putting together the equation for $\delta\Delta x$ and $\delta\Delta y$ we find eq.(\[eq::2d\_deltaDeltaR\]).
The discussion made on $\delta\Delta x$ can be referred to $\delta R$. For large $ R$ the dominant term of the error is the constant relative error $\delta z/z-\delta\Omega/\Omega$, while for short $ R$ the dominant term is $\delta rz/\Omega$ which can be kept small choosing a set up with the proper ratio $z/\Omega$, as shown in the next section.
3d experiments
--------------
The error analysis is not trivial when dealing with real $3d$ experiments, i.e. targets are free to move in the entire $3d$ space without any preferential plane. The graphical interpretation of the errors is not as intuitive as in the $2d$ experiments. For this reason we find a formulation of the error on the position of a target and on distances between pairs of targets making use of derivatives. Moreover we analyze in detail only the special case introduced in the previous section: a set up with the two cameras translated on the $x$ axis and symmetrically rotated of an angle $\alpha/2$ about the $y$ axis, as shown in red in Fig.\[fig:common\_fov\]. The expression of the error in the case of a set up with parallel cameras can then be obtained imposing $\alpha=0rad$.
### **Error on absolute position.**
Under the additional hypotheses that $\alpha$ is a small angle, and $u_L$, $u_R\ll\Omega$ eq.(\[eq::3d\_xyz\]) holds and the position of a target $P$ projected in $Q_L\equiv(u_L,v_L)$ and $Q_R\equiv(u_R,v_R)$ in the left and in the right camera is defined by: $P\equiv(x,y,z)=(u_Lz/\Omega,v_Lz/\Omega,\Omega d/(s+\alpha\Omega))$.
$x$ and $y$ strictly depend on $z$, as well as $\delta x$ and $\delta y$ are affected by $\delta z$. For this reason in the analysis of the error on the absolute position of the targets we will focus first on the error on $z$ and then we will write the expression for $\delta x$ and $\delta y$ too. Computing the derivative of $z$ defined in eq.(\[eq::3d\_xyz\]), we find: $$\delta z=z\displaystyle\frac{\delta d}{d}+z\displaystyle\frac{\delta\Omega}{\Omega}\left(1-\displaystyle\frac{z}{d}\alpha\right)-\displaystyle\frac{z^2}{d}\left(\displaystyle\frac{\delta s}{\Omega}+\delta\alpha\right)$$ Note that negative signs in the previous equations indicate that a positive error $s$ and $\alpha$ produce a negative error on $z$.
The relative error on $z$ is then: $$\label{eq::3d_percentage_deltaz}
\displaystyle\frac{\delta z}{z}=\displaystyle\frac{\delta d}{d}+\displaystyle\frac{\delta \Omega}{\Omega}\left(1-\displaystyle\frac{z}{d}\alpha\right)-\displaystyle\frac{z}{d}\left(\displaystyle\frac{\delta s}{\Omega}+\delta\alpha\right)$$ where: $\delta d/d$ is the relative error on the measured baseline, $\delta\Omega/\Omega$ is the relative error on the calibrated focal length, $\delta\alpha$ is the error on the measure of the angle $\alpha$ and $\delta s$ is the error on the disparity $s=u_L-u_R$. $\delta s$ represents the difference between the error in the determination of $u_L$ and $u_R$. As for the $2d$ case, an error on $s$ can be due to noise in the image but also to an error in the calibration of the distortion coefficients.
The relative error on $z$ is then made by one constant term, $\delta d/d$, and by three terms which grow linearly in $z$. The constant term due to the error on the measure of the baseline can be reduced choosing the proper instrument, as already discussed in the previous section about the error on $z$.
The other three terms, instead, can be reduced choosing the system parameters, $z$, $\Omega$ and $d$ in the proper way. A typical working distance $z$ is generally chosen and typical $\delta\alpha$ and $\delta s$ are estimated. The three linear terms of the equation above can then be kept smaller than a certain threshold, $c$, imposing the following inequalities: $z\delta s/\Omega d<c$ and $z\delta\alpha /d<c$. These two relations fix a lower bound for $\Omega$ and for $d$.
Concerning $x$, substituting eq.(\[eq::3d\_percentage\_deltaz\]) in eq.(\[eq::2d\_percentage\_deltax\]) we find that: $$\delta x=x\left[\displaystyle\frac{\delta d}{d}-\displaystyle\frac{z}{d}\left(\alpha\displaystyle\frac{\delta \Omega}{\Omega}+\displaystyle\frac{\delta s}{\Omega}+\delta\alpha\right)\right]+\delta u\displaystyle\frac{z}{\Omega}$$ and substituting eq.(\[eq::3d\_percentage\_deltaz\]) in eq.(\[eq::2d\_percentage\_deltay\]): $$\delta y=y\left[\displaystyle\frac{\delta d}{d}-\displaystyle\frac{z}{d}\left(\alpha\displaystyle\frac{\delta \Omega}{\Omega}+\displaystyle\frac{\delta s}{\Omega}+\delta\alpha\right)\right]+\delta v\displaystyle\frac{z}{\Omega}$$ Note that an error on $\Omega$ does not affect the three components $x$, $y$ and $z$ in the same way. In the expression for $\delta z/z$, the coefficient of the term due to $\delta\Omega/\Omega$ is equal to $(1-\alpha z/d)$ , while for $\delta x/x$ and $\delta y/y$ it is equal to $-\alpha z/d$. A positive error on $\Omega$ produces a negative error on $x$ and $y$, while the error on $z$ can be positive or negative, depending on the position of the target.
When the angle $\alpha$ is not small the approximation $\sin\alpha\sim\alpha$ does not hold anymore and the original system of equations has to be used. In the general case the complete expression of $z$ has to be derived with respect to all the variables and the expression of the error gets much more complicated, including extra terms. The sources of errors are always the same, i.e. measure or calibration errors for intrinsic and extrinsic parameters and segmentation inaccuracy, but their contributions are different. For the sake of simplicity we do not discuss here the general case. The reader interested in error formulation for the general problem has only to compute partial derivatives of the complete solution with respect to each parameter included in the expression for $z$.
### **Error on mutual distances between targets.**
This is by far the more interesting issue. All the analysis we will do on trajectories is not based on the absolute position of the targets, but on their mutual position. In order to guarantee the accuracy we want on our analysis, we need to have accurate measure of the distances between targets. Consider two targets in the three dimensional space, $P_1$ and $P_2$, and their distance $ R=|\overrightarrow{P_1P_2}|=(\Delta x^2+\Delta y^2+\Delta z^2)^{1/2}$, as shown in Fig.\[fig:pinhole\_system\]. In the special case when $\Delta z\sim0$ the $3d$ error on the mutual distance between the two targets is essentially the case of the $2d$ experiment and the error on the mutual distances is $ R\sim rz/\Omega$ as shown in the previous section. Instead when $ R\sim\Delta z$ the error analysis is much more complicated. In the following we will retrieve a formulation of $\delta\Delta z$ for the special set up with a translation along the $x$ axis and a symmetric rotation about the $y$ axis of an angle $\alpha/2$.
From eq.(\[eq::3d\_xyz\]) $z=\Omega d/(s+\alpha\Omega)$. $$\Delta z=\Omega d\left[\displaystyle\frac{1}{s_1+\alpha\Omega}-\displaystyle\frac{1}{s_2+\alpha\Omega}\right]$$ where $s_1$, $s_2$ represent the disparity of the projection of $P_1$ and $P_2$. Deriving the above equation we find: $$\begin{aligned}
\delta\Delta z= \Delta z&\left[\displaystyle\frac{\delta d}{d}+ \displaystyle\frac{\delta\Omega}{\Omega}\left(1-2\alpha\displaystyle\frac{\bar{z}}{d}\right)-2\displaystyle\frac{\bar{z}}{d}\left(\displaystyle\frac{\delta\bar{s}}{\Omega}+\delta\alpha\right)\right]+\\
-& 2\displaystyle\frac{\bar{z}^2}{\Omega d}\delta\Delta s\end{aligned}$$ where $\bar{z}=(z_1+z_2)/2$, $\delta\bar{s}=(\delta s_1+\delta s_2)/2$ and $\delta\Delta s=\delta s_1-\delta s_2$ is the difference between the error on the disparity of the two targets.
For large $\Delta z$ the first term of the previous equation is the dominant part of the error and: $$\label{eq::large_Deltaz}
\displaystyle\frac{\delta\Delta z}{\Delta z}\sim \displaystyle\frac{\delta d}{d}+\displaystyle\frac{\delta\Omega}{\Omega}\left(1-2\alpha\displaystyle\frac{\bar{z}}{d}\right)-2\displaystyle\frac{\bar{z}}{d}\left(\displaystyle\frac{\delta \bar{s}}{\Omega}+\delta\alpha\right)$$
Passing from $2d$ to $3d$ experiments the relative error on the mutual distances between two targets is not constant anymore. The only constant term is $\delta d/d$, while all the others depend linearly on the position of the two targets $P_1$ and $P_2$. The ratio $\bar{z}/d$ controls how much $\Delta z$ is affected by $\delta s$, $\delta\alpha$ and $\delta\Omega$. So that the error can be kept low choosing $z/d$ smaller than a desired value.
On the other side, for small $\Delta z$ the dominant part of the error is: $$\label{eq::short_Deltaz}
\delta\Delta z\sim-2\displaystyle\frac{\bar{z}^2}{\Omega d}\delta\Delta s$$
This term is not relative, but absolute. Each pair of targets segmented with an error $\delta\Delta s$ is affected by the same error on $\Delta z$, independently on the size of $\Delta z$. Thus, this error has a bigger effect on short distances then on large ones. Moreover the dependence on $z^2$ makes the error growing very fast when the targets get farther from the cameras. When designing the experiment it is very important to estimate this error, and to choose the set up in order to keep it small, because it will affect all the small distances.
In the general case the two expressions in eq.(\[eq::large\_Deltaz\]) and eq.(\[eq::short\_Deltaz\]) contribute together at the error on $\Delta z$. Note that the segmentation error appears in two different terms, one linearly depending on $z$, $\Delta z \bar{z}\delta s/\Omega d$, which is due to the error in the segmentation of the single pair of targets mostly affecting large distances. The other one grows with $z^2$ and depends on the difference between the errors on the segmentation of the two pairs of targets and mostly affecting short distances.
With similar arguments it can be shown that: $$\begin{aligned}
\delta\Delta x= \Delta x&\left[\displaystyle\frac{\delta d}{d}-2\displaystyle\frac{\bar{z}}{d}\left(\alpha\displaystyle\frac{\delta\Omega}{\Omega}+\displaystyle\frac{\delta\bar{s}}{\Omega}+\delta\alpha\right)\right]+2\displaystyle\frac{\bar{z}^2}{\Omega d}\delta\Delta u\end{aligned}$$ and $$\begin{aligned}
\delta\Delta y= \Delta y&\left[\displaystyle\frac{\delta d}{d}-2\displaystyle\frac{\bar{z}}{d}\left(\alpha\displaystyle\frac{\delta\Omega}{\Omega}+\displaystyle\frac{\delta\bar{s}}{\Omega}+\delta\alpha\right)\right]+2\displaystyle\frac{\bar{z}^2}{\Omega d}\delta\Delta v\end{aligned}$$ The error on $ R=(\Delta x^2+\Delta y^2+\Delta z^2)^{1/2}$ is then: $\delta R=(\Delta x\delta\Delta x+\Delta y\delta\Delta y+\Delta z\delta\Delta z)/ R$ So that, for large $R$ the error, $\delta R$, is dominated by: $$\label{eq::3d_rel_dR}
\displaystyle\frac{\delta R}{ R}\sim \displaystyle\frac{\delta d}{d}-2\displaystyle\frac{\bar{z}}{d}\left(\alpha\displaystyle\frac{\delta\Omega}{\Omega}+\displaystyle\frac{\delta\bar{s}}{\Omega}+\delta\alpha\right)+\displaystyle\frac{\Delta z^2}{ R^2}\displaystyle\frac{\delta\Omega}{\Omega}$$ While for short $R$ the dominant part of the error is the absolute term: $$\label{eq::3d_abs_dR}
\delta R\sim -2\displaystyle\frac{\bar{z}^2}{\Omega d}\delta\Delta s$$
Error control: setting up the system
====================================
![[**Field of view of a single camera.**]{} Similarity between the dashed and the filled orange triangles show that the width of the field of view, $W$, grows linearly with the sensor width, $w$, with the coefficient of proportionality equal to $z/\Omega$. While through the similarity between the dashed and the filled green triangles it can be shown that $H=hz/\Omega$. []{data-label="fig:single_fov"}](Fig4.pdf){width="1.0\columnwidth"}
Designing the set up of a $3d$ experiments, intrinsic and extrinsic parameters of the system have to be chosen taking into account the volume of the $3d$ space to be imaged by the cameras, and the accuracy of the $3d$ reconstruction. In this section we give some suggestions on how to choose the properly set up when performing $2d$ and $3d$ experiments making use of the theoretical relations for the error described in the previous sections.
Single camera.
--------------
When dealing with one camera only, the magnification ratio, $z/\Omega$ plays a crucial role in the choice of the set up. As shown in Section \[section::general\], the magnification ratio fixes the correspondence between distances expressed in meters in the real world and distances expressed in pixels units on the sensor plane. The magnification ratio has to be chosen very carefully taking into account some properties of the objects to be tracked, but also taking care of the desired accuracy of the $3d$ reconstruction.
First of all, an object of size $l$ in the real world would be imaged in an object of size $w_l$ on the sensor, such that $w_l=l/(z/\Omega)$. The smaller the magnification ratio, the bigger the imaged object on the screen. The previous relation can be seen as a way to fix a lower bound for $z/\Omega$. As an example, in our experience we want the image of the objects of interest to be at least as large as four pixels. Thus, when recording birds with body size $l\sim 0.4~m$, in order to have their image as large as $4$ pixels we need $z/\Omega$ to be larger than $0.1~m/px$, while when recording midges with body size of about $2~mm$, the magnification ratio should be larger than $0.0005~m/px$.
A second issue is related to the minimum appreciable distance. With the same argument used above, it can be shown that the minimum reconstructable metric distance in the $3d$ real world corresponds to one pixel on the sensor and it is defined by $r_{min}=1*z/\Omega$ and expressed in meters. This means that two objects at a mutual distance shorter than $r_{min}$ can not be distinguished in the picture. It is generally very useful to have an estimate of the interparticle distance of the group of interest and choose $z/\Omega$ in such a way that on average the distance between imaged objects is larger than $3$ or $4$ pixels. Otherwise objects would be too close to each other and optical occlusions would occur frequently. As an example if the interparticle distance is about $10cm$, we want $z/\Omega$ to be greater than $0.025m/px$.
The third aspect related to $z/\Omega$ is the choice of the size of the field of view. Denoting by $W$ and $H$ the width and the height of the field of view, it is easy to show that $W=wz/\Omega$ and $H=hz/\Omega$ where $w\times h$ represents the size of the sensor, see Fig.\[fig:single\_fov\]. The larger the ratio $z/\Omega$ the larger the field of view. Denoting by $W^\star$ and $H^\star$ the minimum size of the field of view, we would like to choose $z$ and $\Omega$ such that: $$\label{eq::2d_setup_zO_wh}
z/\Omega\geq W^\star/w \mbox{ and } z/\Omega\geq H^\star/h.$$
The fourth and last issue is related to the error control. Eq(\[eq::2d\_deltaDeltaR\]) tells that the error on the distance $ R$ is: $\delta R= R(\delta z/z-\delta\Omega/\Omega)+\delta rz/\Omega$. The first two terms of this equation are constant and depend only on the precision in the measure of $z$ and in the calibration of $\Omega$, so that they can be kept as small as we want only choosing the measurement instrument with the proper accuracy. Instead the last term depends on the chosen set up and the larger the magnification ratio, the larger the absolute error on short distances. Denote by $c$ the maximum acceptable error. Given an estimate of $\delta\Delta u$ we would like to choose $z$ and $\Omega$ such that: $$\label{eq::2d_setup_zO_cu}
z/\Omega\leq c/\delta\Delta u.$$
The first three issues above give lower bound for the magnification ratio, while the last one gives an upper bound. In principle one would like to have a large field of view and a small error, but they are both controlled by $z/\Omega$, so that a compromise between the two issues has to be found. Note that the pixel size is crucial for the two problems related to the object size and the interparticle distance, while the size of the sensor plays a crucial role in the two inequalities for $W$ and $H$. In practice, the ratio $z/\Omega$ is chosen to guarantee the desired accuracy through eq.(\[eq::2d\_setup\_zO\_cu\]) and then the size of the sensor needed is determined by eq.(\[eq::2d\_setup\_zO\_wh\]).
Two cameras system
------------------
In the case of real $3d$ experiments the choice of the parameters is a bit more complicated. For the sake of simplicity, we refer only to a symmetric set up with a rotation about the $y$-axis. This is the set up we use when performing our experiment on bird flocks. It has the big advantage that the angle $\epsilon$ can be derived from the measure of the angle $\alpha$, reducing the number of experimental parameters.
The considerations made above about the lower bound for the magnification ratio in order to guarantee the desired size of the imaged objects and the desired interparticle distance on the sensor plane, are still valid when designing a multicamera set up, but unlike $2d$ experiment, the volume of interest is not determined anymore by the field of view of one camera only. What matters now is the common field of view of the two cameras. The size of the common field of view does not depend only on $z$ and $\Omega$ but also on $d$ and $\alpha$, see Fig.\[fig:common\_fov\]. $\Omega$ influences the angle of view of each camera, so that the larger $\Omega$ the narrower each field of view and as a consequence the narrower the common field of view. $d$ affects the portion of $3d$ space in the common field of view. The larger $d$ the smaller the portion of $3d$ space imaged by the cameras. $\alpha$ affects the distance from the cameras of the common field of view. An angle $\alpha=0rad$, see Fig.\[fig:common\_fov\] where this set up is highlighted in black, makes the common field of view optimal for very large $z$. While $\alpha\neq 0 rad$ makes the common field of you optimal for short distances. In particular, the larger $\alpha$ the shorter the working distance.
The same parameter, $z$, $\Omega$, $d$ and $\alpha$ , control also the accuracy of the reconstructed distances. In fact, from eq.(\[eq::3d\_rel\_dR\]) and eq.(\[eq::3d\_abs\_dR\]), the error on $ R$ is $$\begin{aligned}
\delta R=R&\left[\displaystyle\frac{\delta d}{d}-2\displaystyle\frac{\bar{z}}{d}\left(\alpha\displaystyle\frac{\delta\Omega}{\Omega}+\displaystyle\frac{\delta\bar{s}}{\Omega}+\delta\alpha\right)+\displaystyle\frac{\Delta z^2}{ R^2}\displaystyle\frac{\delta\Omega}{\Omega}\right]+\\
&-2\displaystyle\frac{\bar{z}^2}{\Omega d}\delta\Delta s\end{aligned}$$
The constant term $\delta d/d$ depends only on the instrument used to take its measure and it can be strongly reduced choosing an instrument with the proper accuracy. The three terms linear in $z$ are controlled by the ratio $z/d$, while the last term by the ratio $z^2/\Omega d$. In principle the larger $d$ and $\Omega$ the lower the error, while $z$ should be as short as possible. In practice many environmental constraints are involved in the choice of the parameters and a trade off between the biological characteristic of the group of interest and the accuracy has to be found.
As for the $2d$ experiment, if we denote by $c$ the acceptable threshold for the absolute error on short $R$ and by $c^\prime$ the acceptable relative error for the large distances, we can define the set up the system imposing the two following inequalities: $$2\displaystyle\frac{z^2}{\Omega d}\delta\Delta s<c \mbox{ and } 2\displaystyle\frac{z}{d}\left(\displaystyle\frac{\delta\Omega}{\Omega}+\delta\alpha+\displaystyle\frac{\delta s}{\Omega}\right)< c^\prime$$ The above inequalities can be used to define an upper bound for both the ratios $z/d$ and $z/\Omega$ and find a set of suitable parameters which allow accuracy in the $3d$ reconstruction in the desired common field of view, respecting also the constraint due to the objects size and interparticle distances.
In many cases some of the parameters are fixed by the location where the experiments is performed.
Indeed, when designing our experiment on bird flocks, we could not choose $z$. The experiment is performed on the roof of a building and birds are almost at $125m$ from the cameras. We can not go closer. Moreover, the baseline can not be larger than $25~m$. We put the cameras the furthest we can, so that the ratio $z/d$ is defined by the environmental constraint and it is equal to $5$. We estimated $\delta\Omega/\Omega=0.001$, $\delta\alpha=0.001~rad$ (when directly measured through the method described in [@cavagnaAnimal]) and $\delta s=1~px$. As a consequence $c^\prime\sim 0.01$, telling that the relative error on large distances is smaller than $0.01$. Instead, for short distances we choose $c=0.4~m$, which is a typical bird to bird distance and we estimate $\delta\Delta s\sim 0.5~px$. The previous inequality gives, than, the following lower bound for $\Omega>2z^2\delta\Delta s/c d\sim 1500$.
Instead, we perform the experiment on midge swarms in a park and we can go as close as we want to the swarm. The working distance is than not fixed by environmental constraints. But we can not choose $d$ as large as we want. In fact, we take pictures of midges using the scattering of the sun light, so that they appear as white dots on a black background. For very large $d$ it is difficult to have a good scatter effects for both the cameras. For this reason when performing the experiments with swarms we first fix the maximum $d$ and then we choose $z$ and $\Omega$ accordingly. In practice we put the cameras the furthest possible and we set the working distance choosing $z$ in such a way to guarantee a small error on the short distances. For this experiment we choose $\Omega=7000px$, because we do not need a wide field of view since swarms are generally very stable. We define $c\sim 0.002~m$ which is the body length of a midge. The inequality above, implies that $z$ should verify: $z^2<c\Omega d/2\delta\Delta s$. If the baseline is $6~m$, and we estimate $\delta\Delta s\sim 0.5~px$ than $z^2<84~m^2$ and we find that $z<9~m$.
Error control: reconstruction tests {#section::test}
===================================
Every time the experiment is performed, an estimate of the reconstructed error should be taken, in order to check the experimental accuracy in measuring and calibrating that specific set up. The idea is to put some targets in the common field of view of the cameras, to measure their distance with a precise instrument, and to reconstruct their $3d$ positions. The comparison between the measured distances between pairs of targets and the reconstructed distances tells how accurate the reconstruction is. Moreover, a careful analysis of the results can reveal the source of inaccuracy and can be used when trying to fix problems.
Note that the theoretical formulation of the reconstruction problem described in this paper is meant to give an estimate of the errors when designing the experiment. In practice, eqs.(\[eq::lLlR\]) in general does not have an exact solution. This happens because of the error in the segmented objects due to image noise and to all the errors in the measure and calibration of intrinsic and extrinsic parameters. An approximation of eqs.(\[eq::lLlR\]) is then found, generally making use of a least squares method.
We perform experiments in the field with starling flocks and midge swarms. The camera system set up is similar in both cases. We use two synchronized cameras shooting at $170$ fps. For flocking events we choose a baseline of $25$m and a working distance of $125$m, while for swarming events the baseline is about $6$m with a working distance of $8$m. The main difference between the two systems is the way we measure and calibrate the extrinsic parameters.
Postcalibration: swarms
-----------------------
![[**Midges swarms. Relative error on mutual distances between targets.**]{} Orange circles represent relative errors on the targets used in the extrinsic parameters postcalibration process, while green circles represent the relative errors on targets not used in the calibration. For both the sets of measures the relative error is lower than $0.01$. []{data-label="fig:midges_postcal"}](Fig6.pdf){width="0.8\columnwidth"}
For swarming events we decide the orientation of each camera independently; we find the interesting swarm, we fix the baseline and then we rotate each camera in order to center the swarm in the image. We measure the baseline but we do not directly measure the mutual orientation of the stereometric cameras. Instead we retrieve the $5$ angles $\alpha$, $\beta$, $\gamma$, $\delta$ and $\epsilon$ making use of a post calibration procedure. Two targets, $2\times 2$ checkerboard, are mounted on a bar and their distance is accurately measured. $25$ pictures of the targets are taken in different positions, moving the bar in the $3d$ volume where the event of interest take place. A montecarlo algorithm is then used to find the $5$ angles minimizing the error in the reconstruction of the distances between the postcalibration targets. In addition we take some pictures of the targets on the bar, which are not used for the calibration procedure but only to check the reconstruction error. Typical reconstruction errors for the targets used during the calibration process are shown in Fig.\[fig:midges\_postcal\], orange circles, and compared with the reconstruction error on the control targets not used in the calibration process, green circles. The errors on the two sets of targets are comparable and in both cases the relative errors are lower than $0.01$. This guarantees the reliability of our retrieved trajectories.
Precalibration: flocks
----------------------
![[**Birds flocks. Errors on mutual distances between targets in the reconstruction test.**]{} [**a:**]{} Relative errors in the reconstruction of the distance between targets at with large $ R$, $ R$ between $5m$ and $40m$. Relative errors are lower than $0.01$. [**b:**]{} Absolute reconstruction errors on short distances of about $0.2m$. For all the targets the absolute error is lower than $1cm$. []{data-label="fig:birds_precal"}](Fig7.pdf){width="1.0\columnwidth"}
In the set up for the experiment on birds, we can not use a post calibration procedure, because we would need to take pictures of targets in the sky at at least $100$m from the cameras, nor we can take pictures of known targets to check the quality of the $3d$ reconstruction. For this reason we fix the mutual orientation of the cameras a priori as described in [@cavagnaAnimal], and we record only those events happening in the common field of view. But we still need to check the accuracy. For this aim we perform reconstruction tests in a different location, setting up the cameras in a smaller set up. We want to check errors especially on the reconstruction of large distances $R$, which are the ones affected by errors in the measure of intrinsic and extrinsic parameters. For this reason we perform reconstruction tests, keeping the ratio $z/d$ as in the field. Thus we choose a baseline of $10$m and we put targets at a distance in $z$ between $20$m and $60$m.
We accurately measure the distances between all pairs of targets. We take a picture of those targets and then we use the measured extrinsic parameters to reconstruct the distances between pairs of targets. The difference between the measured distances and the reconstructed ones gives the error on the $3d$ distances. Fig.\[fig:birds\_precal\]a shows typical relative errors for our $3d$ reconstruction test on targets at a large mutual distance $ R$. As shown in the plot, our reconstruction error is smaller than $0.01$. In Fig.\[fig:birds\_precal\]b absolute reconstruction errors on the distances of targets at short $ R\sim 0.2m$ are shown. The short distance of $0.2m$ is chosen to simulate the distance between birds in a quite dense flock. The results in Fig.\[fig:birds\_precal\]b show that we have errors of the order of $1cm$, showing the high quality of the reconstructed distances.
![[**Effects of errors on the measures of intrinsic and extrinsic parameters on the reconstruction of large $\mathbf{R}$.**]{} [**a: error on the measure of the baseline.**]{} Relative reconstruction error when $\delta d/d=0.1$ on the baseline $d$, green circles, compared with the error using the correct measure of $d$, orange circles. As expected relative errors are constant and equal to $0.1$. [**b: error on the measure of the mutual angle about the y axis.**]{} Relative reconstruction error when $\delta\alpha=-0.015rad$, green circles, compared with the error using the correct measure of $\alpha$, orange circles. The slope of the linear fit is equal to $0.0039$ corresponding to $2\delta\alpha/d=0.003$. The error is quite big and it reaches the value $0.15$ for $z\sim 60m$. [**c: error on the focal length.**]{} Relative reconstruction error when $\delta\Omega/\Omega=0.1$, green circles, compared with the error using the correct measure of $\Omega$, orange circles. The slope of the linear fit is equal to $0.0025$ corresponding to $2\alpha\delta\Omega/(\Omega d)=0.003$. Relative error reaches the value $0.15$ for $z\sim 60m$. Note that the term $\Delta z^2\delta\Omega/(R^2\Omega)$ is added to $\delta R/R$, in order to not affect the fit with a quantity not constant for all the pairs of target. [**d: segmentation error.**]{} Relative reconstruction error when $\delta s=-30px$, green circles, compared with the error using the correct segmentation, orange circles. The slope of the linear fit is equal to $0.0021$ corresponding to $2\delta s/(\Omega d)$. The relative error at $z\sim 60m$ is quite close to $0.1$. []{data-label="fig:plots"}](Fig8.pdf){width="1.0\columnwidth"}
The average of the errors on the reconstructed distances is by far the first measure to look in the results of a reconstruction test. But it is also interesting and more useful to analyze the results looking for sources of errors. The big span of $z$ for targets used in the test, allows a more detailed analysis. Fig.\[fig:plots\] shows the results on the same reconstruction test of Fig.\[fig:birds\_precal\], but where we manually added errors on the intrinsic and extrinsic parameters of the system. These results perfectly match the theory described in the paper.
The constant relative error due to a wrong measure of $d$ is shown in Fig.\[fig:birds\_precal\]a. Fig.\[fig:birds\_precal\]b,\[fig:birds\_precal\]c and \[fig:birds\_precal\]d show the linear trend of the three terms of the relative error on $R$ depending respectively on $\alpha$, $\Omega$ and $s$. Instead in Fig.\[fig:short\_distances\] the effect of a wrong segmentation of targets at short distances of about $0.2m$ are shown. As expected this term is quadratic in $z$ and it reaches $1m$ for $z\sim 50m$.
![[**Effect of segmentation error on the reconstruction of short $\mathbf{R}$.**]{} Relative reconstruction error when $\delta\Delta s=-10px$, green circles, compared with the error using the correct segmented points, orange circles. The coefficient of the quadratic fit is equal to $0.00047$ and it is compatible with $2\delta\Delta s/(\Omega d)$. []{data-label="fig:short_distances"}](Fig9.pdf){width="0.8\columnwidth"}
Note that we forced the system to have errors on intrinsic and extrinsic parameters to be much bigger than the typical experimental errors. A relative error of $0.1$ on $d$ would correspond, in our birds experimental set up, to an absolute error of about $2.5m$, which is not realistic at all. The only reasonable error is the one on $\alpha$, and as shown in Fig.\[fig:plots\], it is the one mostly affecting the $3d$ reconstruction accuracy.
Whenever we run a test on the reconstruction quality we plot the relative error on large $\Delta z$ vs $z$; we first look at the average value of the errors. If we obtain high and almost constant errors the most probable cause is a bad measure of $d$ and we check it taking again the distance, or measuring the baseline more carefully. Then we look if there is a linear trend relating the relative error on $\Delta z$ to $z$. If we find a clear linear trend we try to understand if the error is coming from a bad measure of $\alpha$, $\Omega$ or $\delta s$ performing again the test and in the worst case calibrating a new time the intrinsic parameters of the system.
In the nasty case when we find high reconstruction errors due to a miscalibration of the intrinsic parameters or due to a bad measure of the extrinsic parameters, we throw away the correspondent collected data, so that we are sure that our analysis is based only on reliable trajectories.
Conclusions {#conclusions .unnumbered}
===========
In the design of a $3d$ experiment the choice of intrinsic and extrinsic parameters is very delicate. A trade off between biological necessity, environmental constraints and accuracy of the reconstruction of the $3d$ position of the imaged targets has to be found.
In the paper we showed how errors in the measurement of the system parameters affect the reconstruction of the mutual distance between targets. As a consequence they affect the analysis of quantities like velocity, acceleration and correlation functions. Moreover errors on different parameters influence the reconstructed distances depending on their size and on their positions. In particular, large distances are mostly affected by errors on the orientation of the cameras, while short distances by segmentation errors. In the example of Fig.\[fig:plots\]b, a small error on $\alpha$ of $0.01rad$ produces relative errors up to $0.16$, while in the example of Fig.\[fig:short\_distances\] a segmentation error of $10px$ produces errors up to $1m$ at $z\sim40m$ over mutual distances of about $0.2m$. In our experiment we manage to keep relative errors on large distances smaller than $0.01$ and absolute errors on short distances below $1cm$ (over distances of about $0.2m$).
Independently on the intrinsic and extrinsic parameters calibration procedures and on the segmentation software used, the best way to reduce the reconstruction error is to design the proper set up. The strategy is to choose large $\Omega$ and $d$ trying to be as close as possible to the group of interest. But at the end of the day, the only way to guarantee the reliability of the retrieved trajectories is to take care of the error while planning the experiment and then test the accuracy.
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---
abstract: 'An exciton theory for quasi-one dimensional organic materials is developed in the framework of the Su-Schrieffer-Heeger Hamiltonian augmented by short range extended Hubbard interactions. Within a strong electron-electron correlation approximation, the exciton properties are extensively studied. Using scattering theory, we analytically obtain the exciton energy and wavefunction and derive a criterion for the existence of a $B_u$ exciton. We also systematically investigate the effect of impurities on the coherent motion of an exciton. The coherence is measured by a suitably defined electron-hole correlation function. It is shown that, for impurities with an on-site potential, a crossover behavior will occur if the impurity strength is comparable to the bandwidth of the exciton, corresponding to exciton localization. For a charged impurity with a spatially extended potential, in addition to localization the exciton will dissociate into an uncorrelated electron-hole pair when the impurity is sufficiently strong to overcome the Coulomb interaction which binds the electron-hole pair. Interchain coupling effects are also discussed by considering two polymer chains coupled through nearest-neighbor interchain hopping $t_{\perp}$ and interchain Coulomb interaction $V_{\perp}$. Within the $t$ matrix scattering formalism, for every center-of-mass momentum, we find two poles determined only by $V_{\perp}$, which correspond to the interchain excitons, and four poles only involving intrachain Coulomb $V$, which are intrachain excitons. The interchain exciton wavefunction is analyzed in terms of inter- and intra-chain character. Finally, the exciton state is used to study the charge transfer from a polymer chain to an adjacent dopant molecule. From a variational wave function for the total system, we explore the dependence of the probability of charge transfer on the acceptor level, the hopping, and the wavefunction of the exciton.'
address: 'Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545'
author:
- 'Z. G. Yu, A. Saxena, and A. R. Bishop'
title: 'Excitons in quasi-one dimensional organics: Strong correlation approximation'
---
introduction
============
In recent years, the exciton concept of electron-hole bound states has gained popularity in conjugated organic polymers. Experimentally, it has been discovered that poly(phenylene vinylene) (PPV) and its derivatives, e.g., poly\[2-methoxy, 5-(2’ ethyl-hexoxy)-1,4 phenylene vinylene\] (MEH-PPV) can be used as the active luminescent layer in electroluminescent light-emitting diode devices.[@Bu90] By using different conjugated polymers, polymer light-emitting diodes (PLED) have been fabricated which emit throughout the visible region of the spectrum.[@Bu90; @BH91; @Bu92; @Be94; @Br92; @Gu92] It is believed that radiative recombination of excitons gives rise to luminescence, so a comprehensive understanding of exciton properties in polymer chains is very important to guide improvements in quantum efficiency of these PLED devices. Also, conjugated polymers have shown potential application in photonics for their large optical nonlinearity and ultrafast response time.[@HKSS; @CZ87] In view of the significant role excitons play in optical properties of a system, a detailed study of the exciton is also an important issue for polymer photonic device design. Theoretically, excitons in conjugated polymers are both attractive and challenging because of the coexistence of low-dimensional confinement and strong electron-electron ([*e-e*]{}) correlation in these systems. These ingredients have in fact led to many controversies during the study of excitons in polymers.[@Ch94; @CC93; @SBS94; @Yu95]
Discussion of excitons in solids can be traced back more than half a century to the pioneering work by Frenkel[@Fr31] which even preceded than the band theory in solids. After numerous studies on excitons over several decades, the exciton, as an elementary excitation, has been well established in bulk insulators. This is the reason why in the existing exciton theories for polymers,[@Su84; @HN85; @TIH87; @AYS92] the standard exciton theory in semiconductors[@NTA80] was usually borrowed with limited justification. In these theories, the polymer is regarded as a Peierls insulator which means the gap between the conduction and valence bands arises from the Peierls dimerization, a well-known nesting effect in one-dimensional metals.[@Pe55] Then the exciton states are solved considering the [*e-e*]{} interaction as a perturbation as in traditional exciton theory. But the polymer is significantly different from the conventional semiconductor. In a semiconductor, the electron correlation effects can largely be neglected and it is plausible to treat the [*e-e*]{} interaction as a perturbation. However, the polymer is typically a strongly correlated system with a moderate to large on-site Hubbard repulsion, and much of the band gap is due to the electron correlation rather than the dimerization.[@OUK72; @Uk79; @BM85; @sun] Thus the foundation of existing exciton theories in conjugated polymers is not so firm, and these theories have already led to some qualitatively incorrect results. For example, from these theories, the 1$B_u$ exciton is lower than the 2$A_g$, but typically the order should be reversed. Again, the threshold of the conduction band is independent of $U$, the on-site Hubbard repulsion.[@AYS92] This is unreasonable since, when an electron is excited to the conduction band, double occupation must occur and cause an additional Hubbard repulsion.
Having appreciated the central importance of electron correlation in polymers, several efforts have been undertaken to take account of the strong correlation effect on exciton states.[@Shuai; @Gu93; @SEGR; @Ma96; @GCM95] In these works, numerical exact diagonalization and density matrix renormalization group (DMRG) approaches are employed to handle small systems. From those numerical results, some useful information on the electron structure of the polymer can be captured by extrapolating the results to a long chain. However, these works cannot be regarded as constituting an exciton theory, since only some specific states are focused on in the calculations and the exciton is not treated as a quasi-particle. We must seek a more complete exciton theory of conjugated polymers, in which the correlation is stressed and the exciton is an elementary excitation. This kind of theory will then be useful to study the optical and transport properties related to the exciton. In this paper, we will develop a new exciton theory in the limit in which the Hubbard $U$ is the main origin of the band gap (i.e., the polymer is regarded as a Mott insulator). In this regime, the spin has little effect on the exciton states and the energy difference between the singlet and triplet states is negligibly small compared with their binding energies. Using scattering theory, we will analytically calculate the exciton states and find a critical strength of the [*e-e*]{} interaction for the existence of bound exciton states.
Currently, chemically synthesized polymers cannot be free from impurities and the “pristine” samples of polymers contain a non-negligible density of impurities and defects from the cross-linking, complex morphological effects, conjugation length effects, and some extrinsic defects. Such impurities sometimes critically influence the properties of the system, e.g., the transport. For example, the conductivity in [*trans*]{}-polyacetylene can be dramatically enhanced by 13 orders of magnitude by doping.[@HKSS] In an impurity-free system, the exciton states with different center-of-mass momenta form exciton bands, and within the band the exciton moves coherently as a composite particle. The disorder tends to produce localized states, and in one-dimensional systems, any nonvanishing impurity potential will lead to a localized electronic state.[@AALR] Thus it is interesting to examine the localization of an exciton, a composite particle, due to the impurities, and furthermore to determine if the exciton ceases to be a composite particle of electron and hole when the impurity is strong enough. In this paper, we will address the interplay of coherent motion of the exciton and different types of impurities in our conjugated polymer model.
Strictly speaking, the polymer is only a quasi-one dimensional system, in which interchain couplings always exist, and sometimes their effects are striking. Many calculations have shown that nonlinear excitations like solitons and polarons may be unstable by taking the interchain coupling into account.[@BM83; @MC93] Recently, great attention has been paid to the interchain effects in luminescent polymers, since many experiments demonstrated that a large fraction of primary photoexcitations are interchain excitons or polaron pairs.[@HYJR; @Yan; @YRKM] Also some theoretical calculations have been carried out to explore the interchain coupling effects on exciton properties.[@MC94; @Yu96] Although the terminology of interchain exciton and intrachain exciton are widely used in current literature, these concepts are not so clearly delineated. From the principles of quantum mechanics, the wavefunction of every eigenstate in the coupled system must be distributed over the whole system, so it is difficult to distinguish from the wavefunction which one corresponds to the interchain and which one to the intrachain exciton. We will clarify what the interchain and intrachain exciton states are, and calculate their energies and wavefunctions.
Photoconductivity in conjugated polymers can sometimes be greatly enhanced by intercalating or doping the polymer with a particular species of molecule. Interesting examples of this phenomenon occur when MEH-PPV is doped by fullerene C$_{60}$ molecules.[@Sa92; @KC93] This is because the exciton, the bound electron and hole state in the polymer, will decay when the dopant molecule is introduced. The electron (or hole) in the exciton will transfer from the polymer chain to the doped molecule, giving rise to a free carrier. Rice and Gartstein recently proposed a theory to explain the ultrafast time scale for this charge transfer.[@RG96] From a quantum mechanics perspective, assuming we have an exciton state in the polymer chain due to photoexcitation, when the coupling between the chain and the molecule is switched on, the electron will move in the whole system (including the chain and the molecule), and this state must have a lower energy than the initial state. So another point of view from which to study the charge transfer is to ask what percentage of the electron (hole) has transferred from the polymer chain to the dopant. This percentage should depend on the acceptor level and the coupling between the chain and the molecule, as well as the initial exciton wavefunction in the polymer chain. We will discuss this issue here.
The paper is organized as follows. First, we develop an exciton theory for conjugated polymers in the strong correlation (large Hubbard $U$) approximation in Sec. II. In Sec. II.A we simplify a Peierls-extended Hubbard model to a model represented by spinless fermions with short-range [*e-e*]{} interactions in real space. Then we use $t$ matrix scattering theory to determine the wavefunction and binding energy of exciton states analytically and derive a criterion for the existence of the $B_u$ exciton in Sec. II.B. This criterion is further proved according to the Levinson’s theorem in scattering theory in Sec. II.C. In Sec. II.D a more formal and compact formalism for optical absorption in conjugated polymers is presented based on our exciton theory in the large-$U$ limit. Sec. III is devoted to the impurity effects on the coherent motion of the exciton. Using a suitably defined electron-hole correlation function, we study different types of impurity. In Sec. IV, we investigate interchain coupling effects by considering a two-chain system supplemented by nearest-neighbor interchain hopping $t_{\perp}$ and interchain [*e-e*]{} interaction $V_{\perp}$. Using $t$ matrix formalism, we analytically determine the poles corresponding to intrachain and interchain excitons, respectively. We also show the wavefunction of the interchain exciton. In Sec. V, the static $A_g$ and $B_u$ excitons are used to study the charge transfer in a molecularly-doped polymer. By constructing a variational wavefunction for the whole system, the energy of this variational state, and accordingly the probability of charge transfer, can be obtained. Finally, we summarize our results in Sec. VI.
A new exciton theory in conjugated polymers
===========================================
In existing theories, the polymer is regarded as a Peierls insulator, and then the exciton state is determined by treating the [*e-e*]{} interaction (including the on-site Hubbard interaction) as a perturbation. In this picture, the single particles (electron and hole) are defined based on a non-interacting Su-Schrieffer-Heeger model, so the band gap, from these theories, is independent of the [*e-e*]{} interaction. In a strongly correlated system, the electronic states are quite different from the non-interacting model. Since the ground state is half-filled, an electron excited to the conduction band must cost the additional Hubbard repulsion energy caused by the double occupation. In conjugated polymers and related organic conductors, it is now accepted that the origin of the band gap comes typically from the Hubbard repulsion rather than the Peierls dimerization. So the Hubbard term should be given priority when one develops an exciton theory. In this section, we will regard the polymer as a Mott insulator and develop the exciton theory in large-$U$ limit. In this energy regime, double and higher order electron-hole excitations can be neglected because of their high energies ($\ge 2U$), and a single configuration interaction approximation is reasonable in determining exciton states. Before carrying out the calculation, let us recall how large the Hubbard $U$ (in units of electronic hopping energy $t$) is in real systems: $U/t\sim 3-4$ in conjugated polymers and $U/t\sim 8-10$ in segregated stack charge transfer salts.[@BCM92; @Soos] Strictly, this strong correlation limit is applicable only when $U\gg t$, thus real conjugated polymers only marginally satisfy this approximation.
Hamiltonians
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The theoretical model we consider is the Peierls-extended Hubbard model, i.e., the Su-Schrieffer-Heeger[@HKSS] model augmented by an extended Hubbard interaction. For a one-dimensional chain, this model Hamiltonian is $$H=-t\sum_{l\sigma}[1-(-
1)^l\delta](c^{\dagger}_{l\sigma}c_{l+1\sigma}+{\rm
H.c.})+U\sum_l
n_{l\uparrow}n_{l\downarrow}+V\sum_l(\rho_l-1)(\rho_{l+1}-1)~.$$ Here $c^{\dagger}_{l\sigma}$ creates an electron of spin $\sigma$ on site $l$, $t$ is the one-electron hopping integral, $\delta$ is a bond-alternation parameter, $U$ and $V$ are respectively the on-site and nearest-neighbor Coulomb interaction, $n_{l\sigma}=c^{\dagger}_{l\sigma}c_{l\sigma}$ is the number operator, and $\rho_l=n_{l\uparrow}+n_{l\downarrow}$. Since we will be concerned only with electronic excitations in this work, we consider a rigid dimerized ground state as a starting point, and do not specify its explicit origin (e.g., [*e-e*]{} interactions, electron-phonon couplings, or crystal structure). Strictly speaking, this Peierls-extended Hubbard model is directly applicable only to [*trans*]{}-polyacetylene. However, recent calculations have shown that the primary excitation in luminescent polymers like PPV can also be described within an effective linear chain model.[@SEGR] In these luminescent polymers, the lowest excitonic wave function extends over several repeat units. The properties of exciton are therefore not very sensitive to the detailed structure within the unit cell. From the viewpoint of renormalization, we can map the complex structure of a luminescent polymer into an effective Peierls-extended Hubbard system with the same significant physical properties by integrating out the superfluous degrees of freedom caused by the complicated unit cell structure. We have also neglected lattice relaxation, since many experiments and theories have demonstrated that [*e-e*]{} interactions dominate electron-lattice interactions in many luminescent polymers.[@Le94; @Co90; @GB91; @RBBH] This simplification enables us to handle [*e-e*]{} interactions in long chains and arrive at an understanding of [*electronic states*]{} in conjugated polymers without loss of essential physics, although the quantitative explanation of some [*lattice*]{} property like vibronic structure or bond length should, indeed, take into account lattice relaxation effects.[@BSFB]
To emphasize the electron correlation, we begin with the Hubbard model $$H_0=-t\sum_{l\sigma}(c^{\dagger}_{l\sigma}c_{l+1\sigma}+{\rm
H.c.})+U\sum_l n_{l\uparrow}n_{l\downarrow}~.$$ Although the exact wave function and the ground-state energy of this Hamiltonian have been obtained by Lieb and Wu,[@LW68] the Green’s function and correlation functions are difficult to calculate directly by using the exact wave function, and it is also difficult to use their solution to study the exciton. As a practical alternative, here we make the strong correlation (large-$U$) approximation. In this approximation, as we will see later, the band gap is essentially $U$, which is not the same as the exact solution for the Hubbard model by Lieb and Wu,[@LW68] where the charge excitation gap is $U-4t+\sum_{n=1}^{\infty}(-1)^n[\frac{1}{2}
nU-(t^2+\frac{1}{4}n^2U^2)^{1/2}]$. However, for the strong correlation limit $U\gg 4t$, this difference is not important, and does not affect the exciton trends we wish to establish.
The density product $n_{l\uparrow}n_{l\downarrow}$ can be expressed by the on-site electron number and spin operator, and the Hubbard model is rewritten as $$H_0=-t\sum_{l\sigma}(c^{\dagger}_{l\sigma}c_{l+1\sigma}+{\rm
H.c.})+U\sum_l [\frac{1}{2}\rho_l-(-1)^l{\rm\bf S}_l\cdot{\rm\bf
n}_l]~,$$ where ${\bf S}_l$ is the electron spin operator at site $l$, $${\bf S}_l=\frac{1}{2}\sum_{\sigma \sigma'}c^{\dag}_{l\sigma}
{\boldmath\sigma}_{\sigma \sigma'}c_{l\sigma'}~,$$ and ${\bf n}_l$ is a unit vector along the spin polarization axis of the electron.[@WSTS]
When $U\gg t$, the ground state is expected to have Néel order. We make an approximation by assuming that [**n**]{}$_l$ always coincides with the $z$ axis, which implies that the spin excitations have been ignored. Thus, in this approximation, the singlet and triplet excitons have the same energy and are not distinguishable. This is reasonable since for a one-dimensional Hubbard model, the spin and charge excitations are separated when $U\to \infty$ and by order $U/t$ for $U\gg t$, and the exciton is a charge excitation. This point is also directly justified by comparing energies of the singlet and triplet states obtained from a finite chain exact diagonalization calculation.[@note]
$$H_0\simeq
-t\sum_{l\sigma}(c^{\dagger}_{l\sigma}c_{l+1\sigma}+{\rm
H.c.})+\frac{U}{2}\sum_{l\sigma}[1-(-
1)^l\sigma]c^{\dagger}_{l\sigma}c_{l\sigma}~.$$
This Hamiltonian is readily diagonalized by introducing $$c_{l\sigma}=\frac{1}{\sqrt{N}}{\sum_k}' e^{ikl}[(u_k+\sigma
p_lv_k)\alpha_{k\sigma}+p_l(u_k-\sigma p_lv_k)\beta_k]~.$$ Here the prime means that the summation runs over the reduced Brillouin zone $|k|<\pi/2$ and $p_l=(-1)^l$. Then $$H_0=-{\sum_{k\sigma}}'\Bigl[(E_k-
\frac{U}{2})\alpha^{\dagger}_{k\sigma}
\alpha_{k\sigma}+(E_k+\frac{U}{2})\beta^{\dagger}_{k\sigma}
\beta_{k\sigma}\Bigr]~,$$ with $$E_k=\sqrt{\frac{U^2}{4}+\varepsilon^2_k}~,$$ $$\varepsilon_k=-2t\cos k~.$$ The functions $u_k$ and $v_k$ are $$u_k=\frac{1}{\sqrt{2}}\sqrt{1+\frac{|\varepsilon_k|}{E_k}}
\simeq\frac{1}{\sqrt{2}}\Bigl(1+\frac{|\varepsilon_k|}{U}\Bigr)~,$$ $$v_k=\frac{1}{\sqrt{2}}\sqrt{1-\frac{|\varepsilon_k|}{E_k}}
\simeq\frac{1}{\sqrt{2}}\Bigl(1-\frac{|\varepsilon_k|}{U}\Bigr)~.$$
In the case of $U\gg t$, a localized picture is more convenient. Two spinless fermions can be defined in the lattice representation as follows[@WSTS] $$\begin{aligned}
\alpha_l&=&\sum_{\sigma}\theta(p_l\sigma)\sqrt{2/N}{\sum_k}'
e^{ikl}\alpha_{k\sigma}~,\\
\beta_l&=&\sum_{\sigma}\theta(p_l\sigma)\sqrt{2/N}{\sum_k}'
e^{ikl}\beta_{k\sigma}~,\end{aligned}$$ where $\theta(x)$ is the step function. Expanded in powers of $t/U$, $c_{l\sigma}$ can be expressed by $\alpha_l$ and $\beta_l$: $$c_{l\sigma}=\theta(p_l\sigma)\alpha_l+\theta(-p_l\sigma)p_l\beta_l
+\theta(-p_l\sigma)\frac{t}{U}(\alpha_{l+1}+\alpha_{l-1})
+\theta(p_l\sigma)p_l\frac{t}{U}(\beta_{l+1}+\beta_{l-
1})+O(\frac{t^2}{U^2})~.$$
If we include the bond alternation part in our unperturbed Hamiltonian, then $$\begin{aligned}
H'_0&=&H_0+\sum_{l\sigma}(-1)^l\delta
t(c^{\dagger}_{l\sigma}c_{l+1\sigma}+{\rm H.c.})\nonumber\\
&=&J\sum_l(h^{\dagger}_lh_l+\beta^{\dagger}_l\beta_l)+
U\sum_l\beta^{\dagger}_l\beta_l+\frac{J}{2}\sum_l(h^{\dagger}_{l+2}h_l
+\beta^{\dagger}_l\beta_{l+2}+{\rm H.c.})\nonumber\\
&+&\delta t\sum_l(-h_l\beta_{l+1}+h_{l+1}\beta_l
-\beta^{\dagger}_{l+1}h^{\dagger}_l+\beta^{\dagger}_lh^{\dagger}_{l+1})~.\end{aligned}$$ Here we have introduced the hole operator $h^{\dagger}_i=\alpha_i$, and $J=2t^2/U$.
By introducing the Fourier transformations $$\begin{aligned}
h_l&=&\frac{1}{\sqrt{N}}\sum_k e^{ikl}h_k~,\\
\beta_l&=&\frac{1}{\sqrt{N}}\sum_k e^{-ikl}\beta_{-k}~,\end{aligned}$$ we can rewrite $H'_0$ in momentum space as $$H'_0=\sum_k[(J+J\cos 2k)h^{\dagger}_kh_k+(U+J+J\cos
2k)\beta^{\dagger}_k\beta_k+2i\delta t\sin
k(h_k\beta_{-k}-\beta^{\dagger}_{-k}h^{\dagger}_k)]~.
\label{h0'}$$
Making the Bogoliubov transformation $$\begin{aligned}
\eta_k&=&\cos\theta_kh_k+i\sin\theta_k\beta^{\dagger}_{-k}~,\\
\gamma^{\dagger}_{-k}&=&-
i\sin\theta_kh_k+\cos\theta_k\beta^{\dagger}_{-k}~,\end{aligned}$$ Hamiltonian (\[h0’\]) can be diagonalized if the relation $$\sin2\theta_k=\frac{-4\delta t\sin k}{U}$$ is satisfied, yielding $$H'_0=\sum_k(\epsilon_k\eta^{\dagger}_k\eta_k+\tilde{\epsilon}_k
\gamma^{\dagger}_k\gamma_k)~,$$ with $$\begin{aligned}
\epsilon_k&=&J(1+\delta^2)+J(1-\delta^2)\cos 2k~,\\
\tilde{\epsilon}_k&=&U+J(1-3\delta^2)+J(1+3\delta^2)\cos 2k~.\end{aligned}$$ Operators $\eta^{\dagger}_k$ and $\gamma^{\dagger}_k$ create the hole and electron in the new valence and conduction band, respectively. Their lattice representations are $$\begin{aligned}
\eta_l&=&\frac{1}{\sqrt{N}}\sum_k e^{ikl}\eta_k~,\\
\gamma_l&=&\frac{1}{\sqrt{N}}\sum_k e^{-ikl}\gamma_{-k}~,\end{aligned}$$ which can be expressed by $h_l$ and $\beta_l$ to order $1/U$: $$\begin{aligned}
\eta_l\simeq h_l-\frac{\delta
t}{U}(\beta^{\dagger}_{l+1}-\beta^{\dagger}_{l-1})~,\\
\gamma^{\dagger}_l\simeq \beta^{\dagger}_l+\frac{\delta
t}{U}(h_{l+1}-h_{l-1})~.\end{aligned}$$
The inter-site Coulomb interaction is necessary to bind the electron and hole. We consider the $V$-term in the Peierls-extended Hubbard model as a scattering potential, which has the local representation $$\begin{aligned}
H_{\rm int}&=&V\sum_{l}(\rho_l-1)(\rho_{l+1}-1)\nonumber\\
&=&V\sum_l(h^{\dagger}_{l+1}h^{\dagger}_lh_lh_{l+1}
+\beta^{\dagger}_{l+1}\beta^{\dagger}_{l}\beta_l\beta_{l+1}
-h^{\dagger}_{l+1}\beta^{\dagger}_l\beta_lh_{l+1}
-\beta^{\dagger}_{l+1}h^{\dagger}_lh_l\beta_{l+1})~.\end{aligned}$$
Since the main interest here is an exciton, only the interaction between the electron and hole is relevant. To order $1/U$, we have $$H^{\rm e-h}_{\rm int}=-V\sum_l(\eta^{\dag}_{l+1}\gamma^{\dag}_l\gamma_l
\eta_{l+1}
+\gamma^{\dag}_{l+1}\eta^{\dag}_l\eta_l\gamma_{l+1})~.$$
Exciton states: $t$ matrix theory
---------------------------------
Since the Hamiltonian is invariant with respect to translation, the exciton states can be classified according to the total quasimomentum $K$. We can write the exciton wave function as $$|\Psi_K \rangle=\sum_sB_{s,K}|\psi_{s,K}\rangle~,$$ where $K$ is the center-of-mass momentum. The basis is chosen as $$|\psi_{s,K}\rangle=\frac{1}{\sqrt{N}}\sum_le^{iKl}\gamma^{\dagger}_{l+s}
\eta^{\dagger}_l|g\rangle~,
\label{bas}$$ representing a created electron-hole pair from the ground state $|g\rangle$ with a separation $s$ in real space. We will determine the exciton state by using $t$ matrix scattering theory. According to $t$ matrix theory[@Ca91] $${\cal T}(z)={\cal U}+{\cal UG}(z){\cal T}(z)~,
\label{t=}$$ where ${\cal T}(z)$ is the $t$ matrix, ${\cal G}(z)$ the notation for resolvent $1/(z-H'_0)$, and ${\cal U}$ the potential operator. Equation (\[t=\]) has the formal solution $${\cal T}(z)={\cal U}/[1-{\cal G}(z){\cal U}]~.
\label{t=1}$$ Using the basis of Eq. (\[bas\]), we obtain the Green’s function $$\begin{aligned}
(r|{\cal G}(z)|s)&\equiv&G(r-s;z)\nonumber\\
&=&\langle\psi_{r,K}|(z-H'_0)^{-1}|\psi_{s,K}\rangle
=\frac{1}{N}\sum_k
\frac{e^{ik(r-s)}}{z-(\tilde{\epsilon}_k+\epsilon_{-k+K})}~.
\label{gre}\end{aligned}$$ Here $z=E_K+i0^+$ and the potential matrix is $$\begin{aligned}
(s|{\cal U}|s')&\equiv&\langle \psi_{s,K}|H^{\rm e-h}_{\rm
int}|\psi_{s',K}\rangle\nonumber\\
&=&-V\delta_{ss'}(\delta_{s,-1}+\delta_{s,1})~.\end{aligned}$$
The utility of Eq. (\[t=1\]) rests on the possibility of actually constructing the inverse operator $1/(1-{\cal GU})$. This can be achieved exactly in our case since, conveniently, the potential is of short range in the local representation. Actually, the portion of the potential ${\cal U}$ containing nonzero elements forms a $2\times 2$ submatrix under the basis Eq. (\[bas\]),
$$\left( \begin{array}{cc}
{\cal U}_{-1-1} & {\cal U}_{-11}\\
{\cal U}_{1-1} & {\cal U}_{11}
\end{array} \right)
=-V\left( \begin{array}{cc}
1 & 0\\
0 & 1
\end{array} \right)~.$$
Denoting $D$ as the determinant of $1-{\cal GU}$, we have $$D(E_K)=\left| \begin{array}{cc}
1+G(0;E_K)V & G(-2;E_K)V\\
G(2;E_K)V & 1+G(0;E_K)V
\end{array} \right|~.$$ The determinant will vanish for some specific values of the energy, which are the energies of the localized states. Consequently, to find the energy $E_K$ of the bound exciton state we look for the root of $$D(E_K)=0~.$$ Subsequently, the wave function is calculated by solving the equation $$B_{r,K}=\sum_{st}(r|G(E_K)|s)(s|{\cal U}|t)B_{t,K}~.$$
First, let us focus on the static exciton, i.e., $K=0$. In this case, the system is symmetric with respect to spatial inversion. Introducing the transformation $$\begin{aligned}
B^+_l&=&\frac{1}{\sqrt{2}}(B_l+B_{-l})~,\\
B^-_l&=&\frac{1}{\sqrt{2}}(B_l-B_{-l})~,\end{aligned}$$ where $B_l \equiv B_{l,K=0}$, and noticing $$G(s-t;E_0)=G(t-s;E_0)~,$$ we can write the determinant $D$ as the product of two parts: $$D(E_0)=D_-(E_0)D_+(E_0)~.$$ Here $$D_-(E_0)=1+[G(0;E_0)-G(2;E_0)]V$$ is for the $A_g$ state with the wave function $$B^-_l=-V[G(l-1;E_0)-G(l+1;E_0)]B^-_1~;$$ and $$D_+(E_0)=1+[G(0;E_0)+G(2;E_0)]V$$ is for the $B_u$ state with the wave function $$B^+_l=-V[G(l-1;E_0)+G(l+1;E_0)]B^+_1~.$$
We denote $x$ as the exciton binding energy: $$x=E_G-E_0~,$$ where $E_G$ is the band gap: $$E_G=(\tilde{\epsilon}_k+\epsilon_k)|_{k=\pi/2}=U-4\delta^2J~.
\label{gap}$$ When $\delta=0$, $E_G$ does not equal the exact result by Lieb and Wu [@LW68]. However, this is not a problem for estimating correct binding energies of the exciton states, since we will directly calculate the binding energy rather than first calculating the energy of the exciton. The Green’s functions are readily calculated; for $x>0$ $$G(2l+1;E_0)\equiv 0~,$$ $$G(2l;E_0)=\frac{(-1)^{l+1}}{\sqrt{x^2+4J(1+\delta^2)x}}
\Bigl(\frac{1-\sqrt{1-u^2}}{u}\Bigr)^{2l}~,$$ with $$u^2=\frac{4J(1+\delta^2)}{x+4J(1+\delta^2)}<1~.$$
Then, by requiring $D_-(E_0)=0$, we obtain the binding energy of the $A_g$ exciton as $$x_-=\frac{V^2}{V+J(1+\delta^2)}~,$$ with corresponding wave function $$\begin{aligned}
B^-_{2l-1}&=&-\frac{G(2l;E_0)-G(2l-2;E_0)}{\sqrt{-2[G'(0;E_0)-
G'(2;E_0)]}}~,\\
B^-_{2l}&=&0~.\end{aligned}$$ Here $G'(n;E)\equiv \frac{d}{dE}G(n;E)$.
For the $B_u$ state, when $V>2J(1+\delta^2)$, the binding energy of the exciton is $$x_+=\frac{[V-2J(1+\delta^2)]^2}{V-J(1+\delta^2)}~,$$ and the wave function is $$\begin{aligned}
B^+_{2l-1}&=&-\frac{G(2l;E_0)+G(2l-2;E_0)}{\sqrt{-
2[G'(0;E_0)+G'(2;E_0)]}}~,\\
B^+_{2l}&=&0~.\end{aligned}$$ When $V<2J(1+\delta^2)$, a solution $E_0<E_G$ satisfying equation $D_+(E_0)=0$ cannot be found, i.e., there is no bound state. This result gives a criterion for the existence of a $B_u$ exciton. Finite chain numerical DMRG and exact diagonalization calculations show that the binding energy of the 1$B_u$ state is near zero when $V$ is not large, but are not conclusive as to whether the state is strictly bound or free.[@shuai2] Our analytical results clearly indicate that the stable 1$B_u$ exciton does not exist when $V$ is less than $2J(1+\delta^2)$. Since this critical value is half of the width of the continuum band, $4J(1+\delta^2)$, and the bandwidth describes the kinetic energy of a free particle, this criterion is a reflection of the competition between the kinetic energy and the attraction of the electron and hole.
Proof of the criterion: Levinson’s theorem
------------------------------------------
The criterion derived above can be proven by Levinson’s theorem.[@Ne66] Namely, the number of bound states in representation $s$ lying either above or below the continuum band can be determined by using $$\delta_s(E_i)-\delta_s(E_f)=\pi n_s~.
\label{lev}$$ Here $E_i$ is the lowest energy in the band and $E_f$ the highest. $n_s$ is number of states in any row of representation $s$ separated from the band. $\delta_s$ is the phase shift that appears in the usual partial wave expansion for the scattering amplitude,[@Ca91] which can also be extracted from the subdeterminants of $\det(1-{\cal GU})$, $$\tan \delta_s=-\Im D_s/\Re D_s~.$$ In the exciton case, $$\begin{aligned}
E_i&=&E_G~,\\
E_f&=&(\epsilon_k+\tilde{\epsilon}_k)_{k=0}=E_G+4J(1+\delta^2)~.\end{aligned}$$ It should be noted that in a one-dimensional system the density of states at the edge of the continuum band will diverge, so the form of Eq. (\[lev\]) in our case is $$\delta_s(E_i-0^+)-\delta_s(E_f+0^+)=\pi n_s~.$$
The Green’s functions in different energy regions can be calculated according to the definition Eq. (\[gre\]). We give the explicit expressions in Appendix A. Then the subdeterminants $D_-$ and $D_+$ are obtained by straightforward calculations.
For $E_0=E_i-0^+$, i.e., the energy is just below the onset of the continuum band, $$\begin{aligned}
D_-&=&-\infty-i0^+~,\\
D_+&=&1-\frac{V}{2J(1+\delta^2)}-i0^+~,\end{aligned}$$ and the phase shifts are $$\delta_-(E_i-0^+)=\pi~,$$ $$\delta_+(E_i-0^+)=\left\{\begin{array}{lc}
0~~~V<2J(1+\delta^2)\\
\pi~~~V>2J(1+\delta^2)~.
\end{array}\right.$$
For $E_0=E_f+0^+$, i.e., the energy is just above the top of the continuum band, $$\begin{aligned}
D_-&=&1+\frac{V}{2J(1+\delta^2)}-i0^+~,\\
D_+&=&+\infty-i0^+~,\end{aligned}$$ and the corresponding phase shifts are $$\delta_-(E_f+0^+)=0~,$$ $$\delta_+(E_f+0^+)=0~.$$
Thus we have $$\delta_-(E_i-0^+)-\delta_-(E_f+0^+)=\pi~,$$ showing that there is always an $A_g$ bound exciton for a nonvanishing $V$. However, when $V<2J(1+\delta^2)$, $$\delta_+(E_i-0^+)-\delta_+(E_f+0^+)=0~,$$ indicating that there is no $B_u$ bound exciton. The bound exciton will appear only when $V>2J(1+\delta^2)$, since then $$\delta_+(E_i-0^+)-\delta_+(E_f+0^+)=\pi~.$$
To get an overall picture of the phase shift in this one-dimensional system, we also examine the phase shifts at $E_i+0^+$ and $E_f-0^+$. For $E_0=E_i+0^+$, i.e., the energy is just above the onset of the continuum band, $$\begin{aligned}
D_-&=&1+\frac{V}{2J(1+\delta^2)}-i\infty~,\\
D_+&=&1-\frac{V}{2J(1+\delta^2)}-i0^+~,\end{aligned}$$ and the phase shifts are $$\delta_-(E_i+0^+)=\pi/2~,$$ $$\delta_+(E_i+0^+)=\left\{\begin{array}{lc}
0~~~V<2J(1+\delta^2)\\
\pi~~~V>2J(1+\delta^2)~.
\end{array}\right.$$ For $E_0=E_f-0^+$, i.e., the energy is just below the top of the continuum band, $$\begin{aligned}
D_-&=&1+\frac{V}{2J(1+\delta^2)}-i0^+~,\\
D_+&=&1-\frac{V}{2J(1+\delta^2)}-i\infty~,\end{aligned}$$ and the phase shifts are $$\delta_-(E_f-0^+)=0~.$$ $$\delta_+(E_f-0^+)=\left\{\begin{array}{lc}
\pi/2~~~V<2J(1+\delta^2)\\
\pi/2~~~V>2J(1+\delta^2)~.
\end{array}\right.$$
Now let us study the behavior of the phase shift as a function of energy. For the $A_g$ state, when the energy passes through the edge of the band the phase shift falls discontinuously from $\pi$ to $\pi/2$, then gradually approaches zero at the top of the band. So there is always a bound state. For the $B_u$ exciton, when $V<2J(1+\delta^2)$ the phase shift increases from zero as the energy increases from the bottom of the band, and approaches $\pi/2$ just below the top of the band, and then drops to zero again when we pass through the top edge of the band. Thus no bound state exists. When $V>2J(1+\delta^2)$, the phase shift decreases from $\pi$ to $\pi/2$ as the energy increases from the bottom to the top of band, and abruptly falls to zero when we cross the edge of the band. Thus a bound state appears. The discontinuity of the phase shift at the band edges is due to the infinite density of states at the bottom and top of the band.
Figure 1 shows the wave functions of the $A_g$ and $B_u$ states with $U=10t$, $V=0.5t$, and $\delta=0.2$, corresponding to binding energies $x_+=0.024t$ and $x_-=0.353t$, respectively. We can see that the wave function of the $A_g$ exciton decays more rapidly than that of the $B_u$ one. For $x>0$, we introduce the parameter $z$, $$z=-\ln\Bigl(\frac{1-\sqrt{1-u^2}}{u}\Bigr)>0~.
\label{width}$$ Since for large $l$, $$G(2l;E_0)\sim e^{-z|2l|}~,$$ we can define the width $R$ of the $K=0$ exciton by $R=2/z$. From Eq. (\[width\]), we estimate the width is about 3 lattice constants for the $A_g$ exciton and about 12 lattice constants for the $B_u$, as shown in Fig. 1.
We calculate the energy of the exciton for $K\ne 0$ from $D(E_K)=0$. A straightforward computation of Eq. (\[gre\]) gives $$\begin{aligned}
G(0;E_K)&=&-\frac{1}{\sqrt{ac-b^2}}~,\\
G(2;E_K)&=&G^*(-2;E_K)\nonumber~,\\
&=&-\frac{1}{4b^2+(a-c)^2}\Bigl\{\bigl[2(a-c)
+\frac{c^2-a^2}{\sqrt{ac-b^2}}\bigr]
+i\bigl[4b-\frac{2b(a+c)}{\sqrt{ac-b^2}}\bigr]\Bigr\}~,\end{aligned}$$ where $$a=x+3J+5\delta^2 J+J(1-\delta^2)\cos 2K~,$$ $$b=J(1-\delta^2)\sin 2K~,$$ $$c=x+J(1-\delta^2)-J(1-\delta^2)\cos 2K~.$$ By solving the equations $$\begin{aligned}
1+G(0;E_K)V+|G(2;E_K)|V&=&0~,\\
1+G(0;E_K)V-|G(2;E_K)|V&=&0~,\end{aligned}$$ we can compute the energy of the moving excitons. In Fig. 2, we have described the energy of the excitons as a function of the center-of-mass momentum $K$. There are two branches in the energy spectra. The energy difference between these two branches becomes smaller when $K$ increases, and reaches a minimum at $K=\pi/2$. The bandwidths of these two branches are approximately $J$.
When $V\gg J$, our results show that both $A_g$ and $B_u$ excitons have binding energy $V$, which is consistent with physical intuition.[@GCM95] In the strong coupling limit $U\gg V\gg t$, Guo [*et al.*]{} used a local (zero hopping limit) picture and argued, since the ground state has all sites singly occupied and the exciton states are linear combinations of configuration ...11120111..., where the numbers denote site occupancies, that the exciton energy is $U-V$. The electron-hole continuum consists of all states in which the double occupancy (electron) and the empty site (hole) are separated by more than one site (e.g., ...11211..1011...), which has energy $U$. Thus the binding energy is $V$. Another prediction from our theory is that the $2A_g$ state has a lower energy than the $1B_u$, which is the observed ordering in many non-luminescent conjugated polymers.[@BCM92; @KSW88; @HKS82] The strong Coulomb interaction regime we consider here is the reason for this ordering in our model.
Optical absorption
------------------
We will calculate the optical absorption from Fermi’s golden rule $$\alpha(\omega)\propto \frac{1}{\omega}\sum_n|\langle n|{\bf J}|g
\rangle |^2\delta(\omega-E_n)~,$$ where ${\bf J}$ is the current opertor and $|n\rangle$ is the excited state with one electron-hole pair. This expression can be written in a more general form if we denote $|\nu \rangle={\bf J}|g \rangle$, $$\alpha(\omega)\propto -\frac{1}{\pi\omega}\lim_{\varepsilon\to 0^+}
\Im \langle \nu|\frac{1}{\omega+i\varepsilon-H}|\nu\rangle~.$$ Hamiltonian $H$ referred to here is that determining the energy of the electron-hole pair with $K=0$. The current operator in the polymers reads $${\bf J}=it\sum_{l\sigma}[1-(-1)^l\delta](c^{\dag}_{l+1\sigma}c_{l\sigma}-
c^{\dag}_{l\sigma}c_{l+1\sigma})~.
\label{curr}$$ Using the spinless fermions $\eta_l$ and $\gamma_l$ defined in Sec. II.A, we rewrite Eq. (\[curr\]) as $${\bf J}=-it\delta\sum_{l}(\eta_{l+1}\gamma_l-\gamma^{\dag}_l\eta^{\dag}_{l+1}
+\eta_{l}\gamma_{l+1}-\gamma^{\dag}_{l+1}\eta^{\dag}_{l})~.$$ Thus the optical absorption can be expressed by the electron-hole Green’s function $$\alpha(\omega)\propto -\frac{t^2\delta^2}{\pi\omega}\Im[\tilde{G}(0;\omega)
+\tilde{G}(2;\omega)]~,$$ where $\tilde{G}(l;\omega)$ is the Green’s function of $H$, satisfying $$\tilde{\cal G}(z)={\cal G}(z)+{\cal G}(z){\cal T}(z){\cal G}(z)~.$$ If we denote $G(n;\omega)=G_n$, then $$\alpha(\omega)\propto -\frac{t^2\delta^2}{\pi\omega}\Im
\Bigl\{(G_0+G_{-2})
-\frac{V}{D(\omega)}[
(G_0+G_2)^2+V(G^3_0-G_0G^2_2+G^2_0G_2-G^3_2)]\Bigr \}$$ and $D(\omega)=[1+(G_0-G_2)V][1+(G_0+G_2)V]$.
From Fig. 3, the $B_u$ exciton has acquired 52% oscillator strength when $U=5t$ and $V=t$. If we increase $V$ and thus have an exciton with a larger binding energy, the $B_u$ exciton will gain more oscillator strength. For $U=5t$ and $V=2t$, the $B_u$ exciton has achieved 95% strength, and the strength of the transition to the continuum is correspondingly diminished, as shown in Fig. 4. The large transition strength for the exciton state is a characteristic feature of one-dimension.
impurities and the coherent motion of exciton
=============================================
As stated in the introduction, “pristine” samples of a polymer cannot eliminate all impurities and defects. Moreover, the fluctuations (both quantum and thermal) of the lattice are also a kind of intrinsic disorder for the electronic states in polymers.[@MW92] Excitons represent a coherent composite particle motion of correlated electrons and holes, whereas impurities tend to produce localized wavefunction. We know that an impurity has a strong effect on transport properties in many systems, especially low-dimensional materials, so a natural question arises: how do the impurities affect the coherent motion of the exciton?
In polymers, two kinds of impurity are often referred to in the literature.[@PBBL] A site impurity is represented by a local potential at site 0: $$H_1=V_0\sum_{\sigma}c^{\dag}_{0\sigma}c_{0\sigma}~,$$ and a bond impurity which acts on the bond between sites 0 and 1: $$H_2=-W_0\sum_{\sigma}(c^{\dag}_{0\sigma}c_{1\sigma}+{\rm H.c.})~.$$ Both of these impurities have very localized (on-site) potentials. However, for a charged impurity, its potential, in principle, may be of long-range. Thus we have two length scales here: one is the range of the impurity potential ($l_i$) and the other the range of the (screened) Coulomb interaction ($l_V$). The latter is equal to one lattice constant in our model. Since the impurity competes with the Coulomb interaction (in exciton states) differently in the two regimes ($l_V>l_i$ or $l_V<l_i$), impurity effects are expected to be different.
On-site impurity potentials
---------------------------
For the site and bond impurities described above, we can rewrite them in the spinless fermion representation, giving $$H_1=V_0(-\eta^{\dag}_0\eta_0+\gamma^{\dag}_0\gamma_0)~,$$ $$H_2=-W_0(-\eta_0\gamma_1-\gamma^{\dag}_1\eta^{\dag}_0+\gamma^{\dag}_0
\eta^{\dag}_1
+\eta_1\gamma_0)~.$$ Since $H_2$ involves the creation and annihilation of an electron-hole pair, it must be less important than $H_1$ by order $1/U$. This can be seen more clearly by using the unitary transformation $H^S=e^{-S}H e^S$, $$H^S_2=\frac{W^2_0}{U+2J(1-\delta^2)}(-\eta^{\dag}_0\eta_0-\eta^{\dag}_1\eta_1
-\gamma^{\dag}_0\gamma_0-\gamma^{\dag}_1\gamma_1)~.$$ Although the site impurity seems more realistic from the above analysis, we will study three kinds of impurity to arrive at a unified picture of impurity effects: $$\begin{aligned}
H^{\rm imp}_1&=&V_0(-\eta^{\dag}_0\eta_0+\gamma^{\dag}_0\gamma_0),\\
H^{\rm imp}_2&=&V_0(-\eta^{\dag}_0\eta_0-\gamma^{\dag}_0\gamma_0),\\
H^{\rm imp}_3&=&V_0(\eta^{\dag}_0\eta_0+\gamma^{\dag}_0\gamma_0).\end{aligned}$$ Hamiltonian $H^{\rm imp}_1$, in which the impurity attracts the hole and repels the electron, or vice versa, imitates a local charged impurity. Hamiltonian $H^{\rm imp}_2$, in which the impurity attracts both the electron and hole, acts as a trap for particles. In Hamiltonian $H^{\rm imp}_3$, the impurity potentials are repulsive for both the electron and hole, describing a barrier effect. The last two types of impurity can be viewed as simulating the cross-linking and conjugation breaking effects in conjugated polymers.
There is no translation invariance once the impurity is included, so we will work in real space and the Hamiltonian we must study reads $$\begin{aligned}
H_i&=&\sum_l\bigl\{J(1+\delta^2)\eta^{\dag}_l\eta_l+\frac{1}{2}J(1-\delta^2)
(\eta^{\dag}_l\eta_{l+2}
+\eta^{\dag}_{l+2}\eta_l)+[U+J(1-3\delta^2)]\gamma^{\dag}_l\gamma_l\nonumber\\
&+&\frac{1}{2}J(1-\delta^2)(\gamma^{\dag}_l\gamma_{l+2}
+\gamma^{\dag}_{l+2}\gamma_l)\bigr\}
+V\sum_l(\eta^{\dag}_{l+1}\eta^{\dag}_l\eta_l\eta_{l+1}+
\gamma^{\dag}_{l+1}\gamma^{\dag}_l\gamma_l\gamma_{l+1}\nonumber\\
&-&\eta^{\dag}_{l+1}\gamma^{\dag}_l\gamma_l\eta_{l+1}
-\gamma^{\dag}_{l+1}\eta^{\dag}_l\eta_l\gamma_{l+1})
+H^{\rm imp}_i~.\end{aligned}$$
The key issue here is how to measure the coherence in the excitonic composite particle. We can do this by defining the correlation function between the electron and hole in the lowest state in the one electron and one hole subspace: $${\cal R}(l,l')=\frac{\langle\delta\rho_h(l)\delta\rho_e(l')\rangle}
{\sqrt{\langle(\delta\rho_h(l))^2\rangle \langle(\delta\rho_e(l'))^2\rangle}}~,
\label{cor}$$ where the deviations are $$\delta A=A-\langle A \rangle~,$$ and the density operators of electron and hole are $$\begin{aligned}
\rho_h(l)&=&\eta^{\dag}_l\eta_l~,\\
\rho_e(l)&=&\gamma^{\dag}_l\gamma_l~.\end{aligned}$$
In the impurity-free system, this correlation function (\[cor\]) is connected with the relative wavefunction of the lowest exciton state $|\Psi_0 \rangle=\frac{1}{\sqrt{N}}\sum_sB_s\gamma^{\dag}_{l+s}\eta^{\dag}_l|
g\rangle$ by $${\cal R}^{\rm free}(l,l')=|B_{l-l'}|^2~.$$
When we add an impurity, the correlation is expected to decrease. The closer ${\cal R}$ is to ${\cal R}^{\rm free}$, the more correlated are the electron and hole in the lowest excitonic states, while ${\cal R}$ approaching zero means that there is no correlation between the electron and hole; in other words, this excitonic state has lost all its coherence.
The effects of the first kind of impurity are illustrated in Fig. 5, which shows the electron-hole correlation functions for different sites in a finite system of size $N=10$. We emphasize that the parameters we use ($U=10t$ and $V=t$) ensure that the exciton has a very localized [*relative*]{} wavefunction, so that finite system corrections and the boundary condition effects are not important. In Fig. 5, all the correlation functions exhibit a crossover behavior around $V_0\sim 0.1t$. This can be understood in term of a relevant energy scale of the exciton, namely, the width of the exciton band, which equals $J=2t^2/U$. This crossover behavior, which occurs at $V_0\sim J$, describes the localization of the exciton, i.e., the free exciton becomes trapped. We can also calculate the charge density at the impurity site as the impurity strength increases. Since in an impurity-free system the hole (electron) is uniformly distributed, and from our exciton theory the electron and hole do not tend to occupy the same site, the hole density at the impurity site is $2/N$. We see a crossover again in Fig. 6 when $V_0$ is comparable to the bandwidth $J$. After this crossover, the hole density at the impurity site approaches 1, clearly showing that the exciton is trapped by the impurity, and the correlation between the electron and hole gradually vanishes, although, as indicated in Fig. 7, they are bound together near the impurity.
For the second type of impurity, from Fig. 8, a crossover is also observed if the impurity strength is similar to the exciton bandwidth, again indicating that the exciton is trapped. But when $V_0$ is larger than $V$, the correlation function abruptly falls to zero, which implies the total breakdown of the exciton as a composite particle. This is because, when $V_0$ is large enough, it is a lower energy for the impurity to trap the electron and hole separately rather than the impurity trapping the hole and then the electron being trapped near the hole due to the Coulomb interaction (as for the first kind of impurity). Thus the electron and the hole occupy the same site and they have no Coulomb interaction. This is not an exciton.
Now we consider the third type of impurity. The correlation function behaviors in Fig. 9 seem more complicated than for the other two types of impurity. The correlation function in which the hole is at the impurity site shows an analogous crossover behavior when $V_0$ is near the exciton bandwidth $J$ to the first and second types of impurity. However, if both the electron and the hole are left (or right) of the impurity, from the correlation function we see that they have not felt the impurity. On the other hand, if the electron and the hole sit on different sides of the impurity, there is a crossover at $V_0\sim J$, but part of correlation between the electron and hole survives.
The different effects of these three kinds of impurity can be further understood if we project the lowest excitonic state to the free exciton states with momentum $K$. We depict in Figs. 10, 11 and 12 the distribution $|Z_K|^2$, where $Z_K=\langle \Psi_K|\Psi \rangle$, and $|\Psi\rangle$ is the lowest excitonic state in these three disordered systems. In the impurity-free system, the lowest state is the linear combination of exciton states with $K=0$ and $K=\pi$ (they are degenerate). In the presence of an impurity, the exciton state will be scattered to other exciton states with different $K$ and the distribution of momenta will broaden from the $\delta$ function. A free exciton state with a specific $K$ can be defined only when the width of the distribution of momentum is narrow enough (i.e., the lifetime of this state is long enough). This is analogous to the quasi-particle in Landau’s Fermi Liquid theory.[@AGD] From these figures, we see that for the first and second types of impurities, after the crossover at $V_0\sim J$ the distribution in momentum space becomes so broad that we can hardly identify the original exciton state with momentum $K=0$ or $K=\pi$. For the second type of impurity, when $V_0$ is larger than $V$, the final state has no distribution at all on any free exciton state, also indicating that the final state is no longer excitonic. However, for the third kind of impurity, after the crossover at $V_0\sim J$, the distribution in momentum space still has two sharp peaks at $K=0$ and $K=\pi$. This is the reason why the exciton is still coherent, as shown in the correlation functions, and in this sense the exciton can be regarded as a quasi-particle in this disordered system.
Extended impurity potentials
----------------------------
For a charged impurity, the range of (screened Coulomb) potential can be extended over several lattice constants. As an illustration, here we consider a specific impurity potential $$H^{\rm imp}_4=V_0(-\eta^{\dag}_0\eta_0+\gamma^{\dag}_0\gamma_0)+\frac{V_0}{2}
(-\eta^{\dag}_1\eta_1+\gamma^{\dag}_1\gamma_1
-\eta^{\dag}_{-1}\eta_{-1}+\gamma^{\dag}_{-1}\gamma_{-1})~.$$ Its range ($l_i$) is three lattice constants which is longer than that of the Coulomb interaction ($l_V$). The correlation functions are illustrated in Fig. 13, from which we observe again a crossover around $V_0\sim J$, indicating the free exciton becomes trapped. Interestingly, when $V_0$ is sufficiently large compared to $V$, the correlation functions falls further abruptly to zero, indicating the dissociation of the exciton into an uncorrelated electron-hole pair. Note that this does not occur for the charged impurity $H^{\rm imp}_1$ with the on-site potential, since the impurity range is then less than the trapped exciton size. From charge densities shown in Fig. 14, we find that for $V_0=0.5t$ (just after the first crossover), both the electron and the hole (thus the exciton) are trapped around the impurity. For $V_0=5t$ (after the correlation goes to zero), the hole is trapped by the impurity while the electron is repelled from the impurity. The dissociation of excitons here is easily understood. Because the impurity attracts the hole and repels the electron, when the impurity strength becomes sufficiently strong, the Coulomb attraction cannot overcome the impurity repulsion to bind the electron and hole together. This impurity-induced exciton dissociation may be invoked to interpret impurity-enhanced photoconductivity observed in certain experiments. We can project the lowest excitonic state in the system with the impurity to free exciton states with different momenta. In Fig. 15, the distribution in momentum space changes from a sharply localized one ($V_0=0.1t$, before the crossover) to a very broad Gaussian distribution ($V_0=0.5t$, after the crossover) and finally goes to zero ($V_0=5t$). This is consistent with the picture that the free exciton becomes trapped, then dissociates into an uncorrelated electron-hole pair with increasing impurity strength.
interchain coupling and interchain excitons
===========================================
The interchain coupling can strongly influence the energy and stability of the nonlinear excitations such as solitons and polarons. Current interest in interchain coupling and the intrachain and interchain exciton crossover in polymers stems from the experimentally observed large amount of interchain excitations in luminescent polymers like PPV. However, the concept of an interchain exciton and how to distinguish the interchain and intrachain excitons, are not very clear. The wavefunction is not so useful to specify whether a state is an interchain or intrachain exciton, because the wavefunction of any state, in principle, will spread over the whole system if interchain coupling is present.
To demonstrate interchain exciton states in our approach, we study a two-chain system coupled by the nearest-neighbor hopping, $$H_{\rm hop}=-t_{\perp}\sum_{l\sigma}(c^{\dag}_{1l\sigma}c_{2l\sigma}+{\rm H.c.}
)~,$$ and nearest-neighbor interchain Coulomb interaction, $$H_{\rm Cou}=V_{\perp}\sum_l(\rho_{1l}-1)(\rho_{2l}-1)~.$$ Here $1,2$ are chain indices. Now the unperturbed Hamiltonian in the large-$U$ limit becomes $$\begin{aligned}
\tilde{H}_0&=&H^{\prime}_0+H_{\rm hop}\nonumber\\
&=&\sum_{i,k}(\epsilon_k\eta^{\dag}_{ik}\eta_{ik}
+\tilde{\epsilon}_k\gamma^{\dag}_{ik}\gamma_{ik})+t_{\perp}\sum_l
(\eta^{\dag}_{2l}\eta_{1l}-\gamma^{\dag}_{1l}\gamma_{2l}+{\rm H.c.})~.\end{aligned}$$ In momentum space, this is $$\tilde{H}_0=\sum_{i,k}(\epsilon_k\eta^{\dag}_{ik}\eta_{ik}
+\tilde{\epsilon}_k\gamma^{\dag}_{ik}\gamma_{ik})+t_{\perp}\sum_k
(\eta^{\dag}_{2k}\eta_{1k}-\gamma^{\dag}_{1k}\gamma_{2k}+{\rm H.c.})~,
\label{inter}
\label{INTER}$$ which can be readily diagonalized, yielding $$\tilde{H}_0=\sum_{I,k}(E^I_{k}\tilde{\eta}^{\dag}_{Ik}\tilde{\eta}_{Ik}
+\tilde{E}^I_k\tilde{\gamma}^{\dag}_{Ik}\tilde{\gamma}_{Ik})~.$$ Here $I=1,2$ is the band index (hereafter we use small $i,j$ for the chain indices, and capital $I,J$ for the band indices). The details of the diagonalization are given in Appendix B. Now we have two conduction and two valence bands with the dispersion relations $$\begin{aligned}
E^{1,2}_k&=&\epsilon_k\mp t_{\perp}~,\\
\tilde{E}^{1,2}_k&=&\tilde{\epsilon}_k\mp t_{\perp}~.\end{aligned}$$ The interaction Hamiltonian now contains two parts $$H_V=H_{\rm int}+H_{\rm Cou}~.$$ The electron-hole interaction part of $H_V$, which is relevant to the exciton state, is $$H^{\rm e-h}_V=-V\sum_{i,l}(\eta^{\dag}_{il+1}\gamma^{\dag}_{il}\gamma_{il}
\eta_{il+1}
+\gamma^{\dag}_{il+1}\eta^{\dag}_{il}\eta_{il}\gamma_{il+1})
-V_{\perp}\sum_l(\eta^{\dag}_{1l}\gamma^{\dag}_{2l}\gamma_{2l}\eta_{1l}
+\gamma^{\dag}_{1l}\eta^{\dag}_{2l}\eta_{2l}\gamma_{1l})~.
\label{hveh}$$ If we define local fermion operators $$\begin{aligned}
\tilde{\eta}_{Il}&=&\frac{1}{\sqrt{N}}\sum_k e^{ikl}\tilde{\eta}_{Ik}~,\\
\tilde{\gamma}_{Il}&=&\frac{1}{\sqrt{N}}\sum_k e^{ikl}\tilde{\gamma}_{Ik}~,\end{aligned}$$ we can rewrite Eq. (\[hveh\]) as $$\begin{aligned}
H^{\rm e-h}_V&=&-\frac{V}{2}\sum_{IJ}\sum_l(\tilde{\eta}^{\dag}_{Il+1}
\tilde{\gamma}^{\dag}
_{Jl}\tilde{\gamma}_{Jl}\tilde{\eta}_{Il+1}
+\tilde{\gamma}^{\dag}_{Il+1}\tilde{\eta}^{\dag}
_{Jl}\tilde{\eta}_{Jl}\tilde{\gamma}_{Il+1})\nonumber\\
&+&\frac{V}{2}\sum_l(\tilde{\eta}^{\dag}_{1l+1}\tilde{\gamma}^{\dag}
_{1l}\tilde{\gamma}_{2l}\tilde{\eta}_{2l+1}
+\tilde{\eta}^{\dag}_{1l+1}\tilde{\gamma}^{\dag}
_{2l}\tilde{\gamma}_{1l}\tilde{\eta}_{2l+1}+
\tilde{\gamma}^{\dag}_{1l+1}\tilde{\eta}^{\dag}
_{1l}\tilde{\eta}_{2l}\tilde{\gamma}_{2l+1}
+\tilde{\gamma}^{\dag}_{1l+1}\tilde{\eta}^{\dag}
_{2l}\tilde{\eta}_{1l}\tilde{\gamma}_{2l+1}+1\Longleftrightarrow 2)\nonumber\\
&-&\frac{V_{\perp}}{2}\sum_{IJ}\sum_l\tilde{\gamma}^{\dag}_{Il}
\tilde{\eta}^{\dag}
_{Jl}\tilde{\eta}_{Jl}\tilde{\gamma}_{Il}
-\frac{V_{\perp}}{2}\sum_l(\tilde{\eta}^{\dag}_{1l}\tilde{\gamma}^{\dag}
_{1l}\tilde{\gamma}_{2l}\tilde{\eta}_{2l}
+\tilde{\eta}^{\dag}_{1l}\tilde{\gamma}^{\dag}
_{2l}\tilde{\gamma}_{1l}\tilde{\eta}_{2l}+1\Longleftrightarrow 2)~.\end{aligned}$$ Thus we can construct the exciton wavefunction in this two-chain system as $$|\Psi_K\rangle=\sum_{IJ}\sum_s B^{IJ}_{s,K}|\psi^{IJ}_{s,K}\rangle~.$$ Here the center-of-mass momentum $K$ is still a good quantum number, and the basis is $$|\psi^{IJ}_{s,K}\rangle=\frac{1}{\sqrt{N}}\sum_le^{iKl}
\tilde{\gamma}^{\dag}_{Il+s}
\tilde{\eta}^{\dag}_{Jl}|g\rangle~,
\label{bas2}$$ which means the exciton state is a combination of every possible electron-hole excitation in different conduction and valence bands. The free electron-hole pair Green’s function under the basis Eq. (\[bas2\]) is $$\langle\psi^{I'J'}_{r,K}|\frac{1}{z_K-\tilde{H}_0}|\psi^{IJ}_{s,K}\rangle
=\delta_{II'}\delta_{JJ'}G^{IJ}(r-s;z_K)$$ $$G^{IJ}(l;z_K)=\frac{1}{N}\sum_k\frac{e^{ikl}}{z_K-(\tilde{E}^I_k+E^J_{-k+K})}~,$$ and the scattering potential is written as $$\langle\psi^{I'J'}_{s,K}|H^{\rm e-h}_V|\psi^{IJ}_{s',K}\rangle
=\delta_{ss'}\Bigl(-\frac{V}{2}\delta_{s1}-\frac{V}{2}\delta_{s-1}-
\frac{V_{\perp}}{2}
\delta_{s0}\Bigr)F(I',J';I,J)$$ with $F(I,J;I,J)=1$ and $F(1,1;2,2)=F(2,2;1,1)=F(1,2;2,1)=F(2,1;1,2)=1$, and otherwise $F=0$. We can solve for the exciton states by locating the roots of the determinant $\det (1-{\cal GU})$ according to $t$ matrix theory. The whole determinant can be decomposed into blocks by appropriate transformations and we achieve two subdeterminants, $$D_1=\left| \begin{array}{cc}
1+\displaystyle\frac{V_{\perp}}{2}G^{11}(0;z_K)
& \displaystyle\frac{V_{\perp}}{2}G^{11}(0;z_K)\\
\displaystyle\frac{V_{\perp}}{2}G^{22}(0;z_K) &
1+\displaystyle\frac{V_{\perp}}{2}G^{22}(0;z_K)
\end{array} \right|$$ and $$D_2=\left| \begin{array}{cc}
1+\displaystyle\frac{V_{\perp}}{2}G^{12}(0;z_K)
& \displaystyle\frac{V_{\perp}}{2}G^{12}(0;z_K)\\
\displaystyle\frac{V_{\perp}}{2}G^{21}(0;z_K)
& 1+\displaystyle\frac{V_{\perp}}{2}G^{21}(0;z_K)
\end{array} \right|~.$$ $D_1$ and $D_2$ are determined only by interchain Coulomb interaction $V_{\perp}$ corresponding to [*interchain*]{} excitons, and two subdeterminants, $$D_3=\left| \begin{array}{cc}
1+\displaystyle\frac{V}{2}[G^{11}(0;z_K)+G^{22}(0;z_K)]
& \displaystyle\frac{V}{2}[G^{11}(-2;z_K)+G^{22}(-2;z_K)]\\
\displaystyle\frac{V}{2}[G^{11}(2;z_K)+G^{22}(2;z_K)] &
1+\displaystyle\frac{V}{2}[G^{11}(0;z_K)+G^{22}(0;z_K)]
\end{array} \right|$$ and $$D_4=\left| \begin{array}{cc}
1+\displaystyle\frac{V}{2}[G^{12}(0;z_K)+G^{21}(0;z_K)]
& \displaystyle\frac{V}{2}[G^{12}(-2;z_K)+G^{21}(-2;z_K)]\\
\displaystyle\frac{V}{2}[G^{12}(2;z_K)+G^{21}(2;z_K)] &
1+\displaystyle\frac{V}{2}[G^{12}(0;z_K)+G^{21}(0;z_K)]
\end{array} \right|~,$$ which are determined only by the intrachain Coulomb interaction $V$ corresponding to [*intrachain*]{} excitons. Equations $D_1=0$ and $D_2=0$ have a single root, respectively, whereas both $D_3=0$ and $D_4=0$ have two roots. Thus there are a total of six exciton bands: two interchain exciton bands and four intrachain exciton bands. Figures 16 and 17 show these intrachain and interchain exciton bands. The relative energy ordering of the interchain and intrachain excitons depends on the ratio $V_{\perp}/V$.
It is interesting to study the wave function of the interchain excitons. The static interchain exciton can be represented in real space as $$|\Psi_{K=0}\rangle=\sum_{ij}\sum_s A^{ij}_s\frac{1}{\sqrt{N}}\sum_l
\gamma^{\dag}_{il+s}\eta^{\dag}_{jl}|g\rangle~.$$ For the lower exciton state determined by $D_1(z_0)=0$, we obtain $$A^{11}_s=-A^{22}_s=-\frac{G^{11}(s;z_0)-G^{22}(s;z_0)}{2\sqrt{-[G^{11}(0;z_0)
+G^{22}(0;z_0)]'}}~,$$ which represents the amplitude for the electron and hole being on the same chain; while $$A^{21}_s=-A^{12}_s=-\frac{G^{11}(s;z_0)+G^{22}(s;z_0)}{2\sqrt{-[G^{11}(0;z_0)
+G^{22}(0;z_0)]'}}$$ is the amplitude that the electron and hole are on different chains. Here $$G^{11}(0;z)=-\frac{1}{\sqrt{(E_G-z-2t_{\perp})(E_G-z-2t_{\perp}+W)}}$$ $$G^{22}(0;z)=-\frac{1}{\sqrt{(E_G-z+2t_{\perp})(E_G-z+2t_{\perp}+W)}}~,$$ where $E_G$ is defined as Eq. (\[gap\]) and $W=4J(1+\delta^2)$. Figure 18 illustrates the intrachain wavefunction $A^{11}_s$ and interchain wavefunction $A^{21}_s$ for this interchain exciton state. We can see that although the state is for an interchain exciton, there is still some probability for the electron and hole to be on the same chain.
For the higher interchain exciton determined by $D_2(z_0)=0$, we have $$A^{11}_s=A^{22}_s=0~,$$ $$A^{21}_s=A^{12}_s=-\frac{G^{12}(s;z_0)}{\sqrt{-[G^{12}(0;z_0)]'}}~,$$ $$G^{12}(0;z)=-\frac{1}{\sqrt{(E_G-z)(E_G-z+W)}}~.$$ In this interchain exciton, there is no relative amplitude between the electron and hole if they are on the same chain. In Fig. 19, we plot the wavefunction $A^{21}_s$ for this interchain exciton.
For more complicated Coulomb interactions within and between the chains, the exact interchain and intrachain exciton poles are difficult to obtain analytically. Instead, we can measure the correlation between the electron and hole using Eq. (\[cor\]). In Fig. 20, we depict the intrachain and interchain electron-hole correlation functions for our simple interchain coupling situation for two $N=12$ chains. Here we choose a fixed interchain hopping $t_{\perp}/t=0.2$. The transition at $V_{\perp}/V=1.18$ shows that the lowest exciton state changes from an intrachain exciton to an interchain one.
charge transfer in a molecularly-doped polymer
==============================================
Photoinduced charge transfer from a polymer chain to an adjacent dopant molecule, such as in PPV/C$_{60}$ blends has attracted much recent attention, because it can greatly increase the photoconductivity in polymers. In recent theoretical work on this phenomenon, Rice and Gartstein[@RG96] proposed a mechanism to explain the observed ultrafast time scale of this process. In this section, instead of discussing the time scale, we attempt to calculate the final state wavefunction of the whole system comprising the polymer chain and dopant molecule. This can tell us what part of the electron in the exciton has transferred from the chain to the dopant. To make our idea more transparent, let us briefly describe this photoinduced charge transfer process. In the ground state, there is no overlap between the chain and the dopant. The photoexcitation produces an exciton state in the polymer chain. Then, due to the coupling between the polymer and the dopant molecule, the electron (or hole) will transfer from the chain to the adjacent molecule. As a simplified Hamiltonian, we consider $$H=H_{\rm chain}+\Delta_e\sum_mc^{\dag}_mc_m+H_{\rm tran}~.$$ Here we are modeling the dopant molecule by assuming it has an acceptor level with energy $\Delta_e$, which couples to the polymer chain only by nearest-neighbor hopping $$H_{\rm tran}=-v\sum_m(c^{\dag}_m\gamma_0+{\rm H.c.})
=-\frac{v}{\sqrt{N}}\sum_{m,k}(c^{\dag}_m\gamma_k+{\rm H.c.})~.$$ A schematic diagram of our model is shown in Fig. 21. The whole system consists of a polymer chain and $N_d$ dilute noninteracting dopants. Before the coupling between the polymer chain and the dopant is switched on, the system has an exciton state on the polymer chain. When we turn on the coupling, the electron in the exciton will transfer between the chain and the molecule. Thus we can construct a variational wavefunction $$|\Psi\rangle=a|\Psi_0\rangle +\frac{1}{\sqrt{N_d}}\sum_m\sum_k
a_kc^{\dag}_m\gamma_k|\Psi_0\rangle~,$$ with the condition $\langle \Psi|\Psi \rangle=1$. The first term describes the electron remaining on the chain as a component of the exciton and the second term describes that the electron with different momentum has different probability to transfer to the dopant molecule. Here $|\Psi_0 \rangle$ is the assumed static exciton state within the polymer chain, $$|\Psi_0 \rangle=\sum_s B_s\frac{1}{\sqrt{N}}\sum_l
\gamma^{\dag}_{l+s}\eta^{\dag}_l|g\rangle~,$$ which can be represented by the relative momentum between the electron and hole $$|\Psi_0 \rangle=\sum_kB_k\frac{1}{\sqrt{N}}\gamma^{\dag}_k\eta^{\dag}_{-k}|
g\rangle~.$$ Its energy is $$\langle \Psi_0|H_{\rm chain}|\Psi_0\rangle=E_0~.$$ The variational state $|\Psi\rangle$ must have a lower energy than $E_0$ , $$\epsilon=\langle\Psi|H|\Psi \rangle-E_0<0~.$$ From $\displaystyle\frac{\partial\epsilon}{\partial a}=0$ and $\displaystyle\frac{\partial\epsilon}{\partial a_k}=0$, we obtain two coupled equations: $$\begin{aligned}
-\sqrt{\frac{N_d}{N}}a v&=&a_k[\epsilon+E_0-(\epsilon_k+\Delta_e)]~,
\label{eq1}\\
a\epsilon&=&\frac{1}{N}\sum_k a_k|B_k|^2\sqrt{\frac{N_d}{N}} v~.
\label{eq2}\end{aligned}$$ Using Eq. (\[eq1\]), $a_k$ can be eliminated from Eq. (\[eq2\]) and we have the eigenvalue equation for $\epsilon$ $$\epsilon=c\frac{1}{N}\sum_k \frac{v^2|B_k|^2}{\epsilon+E_0-(\epsilon_k+
\Delta_e)}\equiv c F(\epsilon)~.$$ Here $c\equiv N_d/N$ is the dopant concentration. Once we have found the negative solution of $\epsilon$, the probability that the exciton remains on the chain is $$a^2=\frac{1}{1-c F'(\epsilon)}~,$$ where $F'(\epsilon)=\frac{dF(\epsilon)}
{d\epsilon}$. So the probability of charge transfer is $$P=1-a^2=\frac{-cF'(\epsilon)}{1-cF'(\epsilon)}~.$$
For the $B_u$ exciton in the polymer chain $$B_k=\sum_lB^+_le^{-ikl}~.$$ If we assume for demonstration purposes that the exciton is highly localized (this is not necessary in our theory), i.e., $B^+_1=B^+_{-1}=\frac{1}{\sqrt{2}}$ and $B^+_l=0$ for other $l$, then $$B_k=\sqrt{2}\cos k~,$$ and we can write $F(\epsilon)$ in a very compact form: $$F(\epsilon)=\frac{v^2}{J(1-\delta^2)}\Bigl(\sqrt{\frac{E+2J(1-\delta^2)}{E}}
-1\Bigr)~,$$ with $$E=-(\epsilon+E_0-\Delta_e-2J\delta^2)~.$$
Figure 22 illustrates the probability of charge transfer to the dopant. The dopant concentration is set to be 0.2, a typical value for a molecularly-doped polymer. We see that when the acceptor level is near the exciton energy $E_0$, a crossover will occur. When the acceptor level is below this crossover value, the electron is mainly on the dopant. Otherwise, the electron is mainly on the polymer chain. The coupling strength $v$ affects the charge transfer by controlling the width of the crossover. The smaller $v$ is, the more rapid is the crossover.
For the $A_g$ exciton, if we make the assumption $B^-_1=-B^-_{-1}=\frac{1}
{\sqrt{2}}$, then $$|B_k|=|\sum_lB^-_le^{ikl}|=\sqrt{2}\sin k~,$$ and we have $$F(\epsilon)=\frac{v^2}{J(1-\delta^2)}\Bigl(1-\sqrt{\frac{E}{E+2J(1-\delta^2)}}
\Bigr)~.$$ We plot the charge transfer probability for the $A_g$ state in Fig. 23. We can see that there is a threshold for $\Delta_e$. Below this value, the electron will thoroughly transfer to the dopant molecule. However for the $B_u$ state there is a long tail below the critical value, indicating some fraction of the electron can still be found in the polymer. Having gained the knowledge of how $\Delta_e$, $v$, and the exciton wavefunction influence the probability of charge transfer, one will be able to control this transfer process in the conjugated polymer.
concluding remarks
==================
In this paper, we have extensively studied the exciton states in conjugated polymers by emphasizing the dominant role of [*e-e*]{} correlations. The model we studied here is the widely-used Peierls-extended Hubbard model with frozen bond dimerization. First, in the large-$U$ approximation, we mapped this model to a spinless fermion model with only nearest-neighbor Coulomb interaction in real space. The short range interaction enabled us to apply $t$ matrix theory to analytically calculate the energy spectrum and wavefunction of bound (exciton) states. We have found that there always exists a stable $A_g$ exciton as long as the nearest-neighbor Coulomb $V$ is nonzero; for the $B_u$ state, however, a stable exciton state can exist only when $V$ is larger than the half width of the continuum band. This criterion has been proven based on Levinson’s theorem. In our results, we have a correct ordering for $A_g$ and $B_u$ states, i.e., $2A_g<1B_u$, as observed in most conjugated polymers. The impurity effects on the coherent motion of the excitons were also investigated in this large-$U$ approximation. The coherence of the exciton can be measured by an appropriately defined electron-hole correlation function. We have studied impurities with on-site potentials as well as a charged impurity with a more extended potential. There are three kinds of impurity with the on-site potential: the first is like a local charge, attracting holes but repelling electrons (or vice versa); the second acts as a well, attracting both electron and hole; the third is like a barrier which repels both electron and hole. We have found that for the first and second type of impurities, the electron-hole correlations exhibit a crossover when the impurity strength $V_0$ is comparable to the exciton bandwidth $J$, which describes the exciton being trapped by the impurity. For the second type of impurity, if the impurity strength is larger than the Coulomb interaction $V$, the deep well will trap the electron and hole separately, leading to the total de-correlation of the exciton as a particle. For the third type of impurity, the exciton coherence can survive the impurity and the distribution in momentum space has a sharp peak which means the exciton still moves freely. For the charged impurity with an extended potential of range greater than $l_V$, the range of the Coulomb interaction, the free exciton becomes trapped at $V_0\sim J$, analogous to the situation for on-site impurity potentials. However, unlike the charged impurity with the on-site potential, the exciton dissociates into an uncorrelated electron-hole pair when $V_0$ is sufficiently large compared to the Coulomb strength $V$.
We have also investigated the effects of interchain coupling and the resulting interchain exciton states within the strong correlation approximation by considering a two-chain system with nearest-neighbor interchain hopping $t_{\perp}$ and Coulomb interaction $V_{\perp}$. In this coupled system, we have two conduction bands and two valence bands. Within the $t$ matrix formalism, we have found six poles for every center-of-mass momentum $K$, in which two poles are determined solely by $V_{\perp}$, corresponding to interchain excitons, while the other four poles are determined solely by the intrachain Coulomb interaction $V$, corresponding to intrachain excitons. We have also illustrated the wavefunctions of the static interchain exciton. There is still some amplitude for the electron and hole being on the same chain for the interchain exciton state. For more complicated Coulomb potentials, we propose a way to distinguish the interchain and intrachain excitons, namely by comparing the interchain electron-hole correlation function with the intrachain one.
The charge transfer in a molecularly-doped conjugated polymer has been studied by constructing a variational wavefunction for the whole system including the polymer chain and dopant molecule. We modeled this coupled system by simply regarding the molecule has an acceptor level which interacts with the polymer chain by nearest-neighbor hopping $v$. Minimizing the energy of the state, we have obtained the energy of the variational state and, accordingly, the probability of charge transfer. We have shown that a crossover behavior will occur when the acceptor level is near to the exciton energy. When the acceptor level is higher than this crossover value, the electron mainly remains on the polymer chain. Otherwise, most of the electron density will transfer to the dopant molecule. The hopping $v$ controls the width of this crossover, the larger $v$ is, the more gentle is the crossover. The wavefunction is also an important influence on the charge transfer. For the $A_g$ state, there is a threshold for the acceptor level. If $\Delta_e$ is less than this value, the charge transfers to the molecule thoroughly and the percentage of electron density in the polymer chain is zero.
Our calculations in this paper presented a comprehensive picture of the exciton in conjugated polymers, in a limit in which the electron correlation effects have been taken seriously and consistently. Our exciton theory can be readily extended to a system with a relatively long range Coulomb interaction. Also using our spinless fermion representation for the Peierls-extended Hubbard model, biexciton states can be obtained either by the Heitler-London method or diagonalization of the Hamiltonian in two electron-hole pair space. Although for real conjugated polymers, the Hubbard $U$ is not so strong and our results cannot quantitatively match the energy levels in luminescent polymers, our theory is useful for understanding several puzzles which have arisen from correlation effects in conjugated polymers. Finally, we note that recent experimental evidence has demonstrated that there is an excitonic contribution to the pairing mechanism in YBa$_2$Cu$_3$O$_{7-\delta}$.[@Ho94] We expect that our exciton theory can give some guidance for exciton effects in high-T$_c$ superconductors by extending the formalism to two dimensions.
We are grateful to X. Sun, Z. Shuai, W. Z. Wang, J. T. Gammel, S. A. Brazovskii, N. N. Kirova, and D. Schmeltzer for many helpful discussions and criticisms. This work was supported by the U. S. Department of Energy.
Explicit expressions of Green’s functions
=========================================
In this Appendix, we will give explicit expressions for $G(0;E_0)$ and $G(2;E_0)$ in different energy regions, calculated according to the definition Eq. (\[gre\]). Here we use the same notation as in the text; $x=E_G-E_0$. When $E_0>E_f$, i.e., $x<-4J(1+\delta^2)$, the energy is above the top of the continuum band, and the Green’s functions are $$G(0;E_0)=\frac{1}{\sqrt{x^2-4J(1+\delta^2)|x|}}~,$$ $$G(2;E_0)=-\frac{1}{2J(1+\delta^2)}+\Bigl[\frac{|x|}{2J(1+\delta^2)}-
1 \Bigr]
\frac{1}{\sqrt{x^2-4J(1+\delta^2)|x|}}~.$$ When $E_i<E_0<E_f$, i.e., $-4J(1+\delta^2)<x<0$, the energy is within the continuum band, $$G(0;E_0)=-i\frac{1}{\sqrt{-x^2-4J(1+\delta^2)x}}~,$$ $$G(2;E_0)=-\frac{1}{2J(1+\delta^2)}
+i\Bigl[\frac{x}{2J(1+\delta^2)}+1 \Bigr]
\frac{1}{\sqrt{-x^2-4J(1+\delta^2)x}}~.$$ When $E_0<E_i$, i.e., $x>0$, the energy is in the gap, $$G(0;E_0)=-\frac{1}{\sqrt{x^2+4J(1+\delta^2)x}}~,$$ $$G(2;E_0)=-\frac{1}{2J(1+\delta^2)}
+\Bigl[\frac{x}{2J(1+\delta^2)}+1 \Bigr]
\frac{1}{\sqrt{x^2+4J(1+\delta^2)x}}~.$$
Diagonalization of Hamiltonian (\[inter\])
==========================================
Hamiltonian (\[inter\]) can be written as $$\tilde{H}_0=\sum_k (\eta^{\dag}_{1k}~~\eta^{\dag}_{2k})
\left( \begin{array}{cc}
\epsilon_{k} & t_{\perp} \\
t_{\perp} & \epsilon_k \\
\end{array} \right)
\left( \begin{array}{cc}
\eta_{1k} \\
\eta_{2k} \\
\end{array} \right)+
\sum_k (\gamma^{\dag}_{1k}~~\gamma^{\dag}_{2k})
\left( \begin{array}{cc}
\tilde{\epsilon}_{k} & -t_{\perp} \\
-t_{\perp} & \tilde{\epsilon}_k \\
\end{array} \right)
\left( \begin{array}{cc}
\gamma_{1k} \\
\gamma_{2k} \\
\end{array} \right)~.$$
Making the transformations $$\left( \begin{array}{c}
\eta_{1k}\\
\eta_{2k}\\
\end{array} \right)=\frac{1}{\sqrt{2}}\left( \begin{array}{cc}
1 & 1 \\
-1 & 1 \\
\end{array} \right)
\left( \begin{array}{c}
\tilde{\eta}_{1k} \\
\tilde{\eta}_{2k} \\
\end{array} \right)$$ and $$\left( \begin{array}{c}
\gamma_{1k}\\
\gamma_{2k}\\
\end{array} \right)=\frac{1}{\sqrt{2}}\left( \begin{array}{cc}
1 & -1 \\
1 & 1 \\
\end{array} \right)
\left( \begin{array}{c}
\tilde{\gamma}_{1k} \\
\tilde{\gamma}_{2k} \\
\end{array} \right),$$ we have $$\tilde{H}_0=\sum_{Ik}(E^I_{k}\tilde{\eta}^{\dag}_{Ik}\tilde{\eta}_{Ik}
+\tilde{E}^I_k\tilde{\gamma}^{\dag}_{Ik}\tilde{\gamma}_{Ik})$$ and the relation between the two types of local operators $\eta_{il}$ ($\gamma_{il}$) and $\tilde{\eta}_{Il}$ ($\tilde{\gamma}_{Il}$): $$\begin{aligned}
\eta_{1l}&=&\frac{1}{\sqrt{2}}(\tilde{\eta}_{1l}+\tilde{\eta}_{2l})~,\\
\eta_{2l}&=&\frac{1}{\sqrt{2}}(-\tilde{\eta}_{1l}+\tilde{\eta}_{2l})~,\\
\gamma_{1l}&=&\frac{1}{\sqrt{2}}(\tilde{\gamma}_{1l}-\tilde{\gamma}_{2l})~,\\
\gamma_{2l}&=&\frac{1}{\sqrt{2}}(\tilde{\gamma}_{1l}+\tilde{\gamma}_{2l})~.\end{aligned}$$
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|
---
abstract: 'We consider the bias arising from time discretization when estimating the threshold crossing probability $w(b) := {\mathbb{P}}(\sup_{t\in[0,1]} B_t > b)$, with $(B_t)_{t\in[0,1]}$ a standard Brownian Motion. We prove that if the discretization is equidistant, then to reach a given target value of the relative bias, the number of grid points has to grow quadratically in $b$, as $b$ grows. [When considering non-equidistant discretizations (with threshold-dependent grid points), we can substantially improve on this]{}: we show that for such grids the required number of grid points is independent of $b$, and in addition we point out how they can be used to construct a strongly efficient algorithm for the estimation of $w(b)$. [Finally, we show how to apply the resulting algorithm for a broad class of stochastic processes; it is empirically shown that the threshold-dependent grid significantly outperforms its equidistant counterpart.]{}'
author:
- 'Krzysztof Bisewski[^1]'
- Daan Crommelin
- Michel Mandjes
bibliography:
- 'bibliografia\_optimal\_grids.bib'
date: 'September 13, 2017 '
title:
- |
Controlling the time discretization bias\
for the supremum of Brownian Motion
- |
Controlling the time discretization bias\
for the supremum of Brownian Motion
---
[0.8]{}\[0.5,0.5\](0.535,0.97)
Introduction
============
Extreme values of random processes play a prominent role in a broad range of practical problems. It is often of interest to find the tail of the distribution of the supremum of a continuous-time [stochastic]{} process $(X_t)_{t\ge 0}$ over a finite time interval. [In this paper the focus is on the level crossing probability $$w(b) := {\mathbb{P}}\left(\sup_{t\in[0,1]}X_t > b\right).$$ For many classes of processes, such as the Gaussian processes [@adler1990introduction], typically no explicit expressions for $w(b)$ are available, with Brownian Motion and the Ornstein-Uhlenbeck process being notable exceptions. When an explicit expression for $w(b)$ is unavailable one usually resorts to using high-dimensional numerical integration and simulation-based methods, see e.g. [@genz2009computation] for further reading.]{}\
For most of the available numerical methods, the underlying [continuous-time]{} process needs to be discretized in time. One chooses a certain *finite grid* $T \subset [0,1]$ and then approximates $w(b)$ with $w_T(b) := {\mathbb{P}}\big(\sup_{t\in T}X_t > b\big)$. We note that this always leads to an underestimation, i.e., $w_T(b) \leq w(b)$. We quantify this underestimation by $\beta_T(b) := (w(b)-w_T(b))/w(b)$, the relative *discretization bias*[^2]. Typically $T$ is chosen to be an *equidistant grid* $T = \{\frac{1}{n},\frac{2}{n},\ldots, 1\}$ and in that case, $\beta_T(b)$ can be reduced only by changing the *grid size* $n$. The finer the grid, the smaller the bias, but also, the larger the computational effort to estimate $w_T(b)$. The main drawback of using equidistant grids is that typically, to reach a given target value of the discretization bias, the grid size $n$ has to grow with the threshold $b$. In that case, for large $b$, the appropriate grid size can become so large that the computation is not feasible. Two central questions arise from these observations: How fast does $n$ have to grow in $b$? Furthermore, can we identify a more efficient family of grids?
In this paper we address these issues for standard Brownian Motion. Although in this case $w(b)$ can be computed explicitly, there are no available expressions for $\beta_T(b)$. We conduct a thorough study of the influence of the choice of the grid on the corresponding relative bias. Furthermore, we argue that exploring the [case]{} of standard Brownian Motion is a first step towards finding efficient grids for [a more general class of processes. We demonstrate numerically how our analysis of efficient grids for Brownian Motion leads to a useful procedure to determine efficient grids for a broad range of other processes.]{}
The contributions of this paper are the following. [(i) The first finding can be seen as a negative result]{}: we show that to *uniformly control*[^3] the relative bias, the size $n$ of the equidistant grid must grow at least quadratically in $b$; see Theorem \[thm\_eq\] in Section \[s:BM\_equi\]. [(ii) The second finding is that we can do much better by using a *[threshold-dependent]{}* family of grids]{}, meaning that grid points change their location with $b$ (but the number of points does not increase). The discretization bias induced by this particular family of grids is uniformly controlled without having to increase the number of grid points; see Theorem \[THEorem\] in Section \[s:BM\_bb\]. According to the best of the authors’ knowledge, this is the first result which shows that a careful choice of the grid can drastically increase the accuracy of the discrete estimator of $w(b)$. Using [threshold-dependent]{} grids makes it feasible to estimate $w(b)$ with moderate grid sizes even for very high thresholds $b$, which would be impossible to estimate using equidistant grids. In particular, in Section \[s:algorithm\] we present a strongly efficient algorithm for the estimation of $w(b)$ that relies on [threshold-dependent]{} grids. [(iii) In the third place, we point out how the ideas underlying our threshold-dependent grid can be used for a broad class of stochastic processes (including Gaussian processes, such as fractional Brownian Motion, and Lévy processes); it is empirically shown that the threshold-dependent grid significantly outperforms its equidistant counterpart.]{}
An efficient grid (both small in size and inducing a small discretization bias) is particularly relevant for situations with large $b$. In this respect, the work presented here connects to the rare event simulation literature. As $b$ approaches infinity, $w(b)$ decays exponentially to $0$ and standard simulation-based methods like Crude Monte Carlo to estimate $w(b)$ become extremely time consuming. We emphasize that rare event simulation methods commonly aim to control the sampling error, not the bias due to the discretization. [@adler2012efficient] develop an algorithm that is strongly efficient (with bounded relative sampling error) for estimation of $w_T(b)$ (rather than $w(b)$). We will show that combining their algorithm with the use of [threshold-dependent]{} grids provides a strongly efficient algorithm for estimation of $w(b)$.
A topic closely related to ours concerns the quantification of the difference between the supremum of the stochastic process taken over $[0,1]$ and the supremum taken over a finite grid $T\subset [0,1]$, i.e. $$\Delta(T) = \sup_{t\in[0,1]} X_t - \sup_{t\in T} X_t.$$ There are several results in the literature that study the behavior of $\Delta(T)$ for standard Brownian Motion. [@asmussen1995discretization] shown that for the equidistant grids $T^\text{eq}_n = \{\frac{1}{n},\ldots,\frac{n}{n}\}$, $\sqrt{n}\,\Delta(T^\text{eq}_n)$ has a tight, non-degenerate weak limit, as $n\to\infty$ and [@janssen2009equidistant] derived an expansion for ${\mathbb{E}}\Delta(T^\text{eq}_n)$. For *random grids* $T^\text{rnd}_n = \{U_1,\ldots,U_n\}$, where $U_1,\ldots,U_n$ are i.i.d. uniform samples on $(0,1)$, independent of the Brownian Motion $(X_t)_{t\in[0,1]}$, [@calvin1997aaverage] establish the weak limit of $\sqrt{n}\,\Delta(T^\text{rnd}_n)$. Finally, [@calvin1997baverage] proposed a class of *adaptive* grids, meaning that the consecutive grid-points $t_{k+1}$ are chosen based on $((t_1,B_{t_1}),\ldots,(t_{k},B_{t_k}))$; given any $\delta>0$, an adaptive grid $T^\delta_n = \{t_1^\delta,\ldots,t_n^\delta\}$ is provided such that $n^{1-\delta/2}\Delta(T^\delta_n)$ has a weak limit.
In our study we do not focus on the [difference $\Delta(T)$ between the values]{} of the maxima of the discrete and continuous-time Brownian Motion, but rather on the $\beta_T(b)$, i.e., the relative [difference between the probabilities]{} that these maxima lie above a certain fixed threshold.
There are several approaches to tackle the discretization bias available in the literature. Arguably, the most widely applicable method is *Multilevel Monte Carlo* (MLMC) [@giles2008multilevel]. It can be applied together with any numerical method that relies on discretization. The idea is to use several different *levels of discretization* and spend less computational effort (draw less samples) at the finest levels of discretization. MLMC effectively reduces the computational effort, and the time saved can be used to produce even finer levels of discretization. It could be interesting to explore the combination of MLMC method together with the idea of [threshold-dependent]{} grids but further exploiting this procedure lies beyond the scope of this article.
One of the methods that aims to directly decrease the bias [induced by equidistant grids]{} is *continuity correction*. Since the discrete-time approximation $w_T(b)$ is always smaller than $w(b)$, one could *slightly* lower the threshold $b$ to compensate for the underestimation. [[@broadie1997continuity], using the machinery developed in [@siegmund1985sequential], proposed a way of lowering the threshold which improves the rate of convergence of the relative bias from $O(n^{-1/2})$, cf. Proposition \[prop:eq\], to $O(n^{-1})$, as the number of grid points $n$ grows large. However, in the non-Brownian case, it remains a non-trivial problem how much $b$ should be decreased. In fact, there is no direct way of making sure whether lowering $b$ decreases the absolute relative bias, as lowering $b$ by *too much* leads to overcompensation and thus to an estimate that is [*larger*]{} than $w(b)$.]{} By contrast, it is straightforward to compare the bias induced by two different grids — the larger the discrete estimator $w_T(b)$, the smaller the relative bias.
There are also several simulation-based algorithms that do not rely on pre-discretization. [@li2015rare] propose a strongly efficient algorithm for estimation of $w(b)$ for a large class of Gaussian processes (most prominently, processes with constant variance function). However, when the underlying process has a unique point of maximal variance (such as Brownian Motion), the algorithm requires the simulation of a random time $\tau\in[0,1]$ from a density $f(t) \propto {\mathbb{P}}(X_t>b)$, which becomes a rare event simulation problem when $b$ is large. While for an arbitrary process, the random discretization proposed in the algorithm requires a computational effort cubic in the number of grid points (in order to simulate a discrete Gaussian path), pre-discretization requires only quadratic effort; see the discussion in Section \[s:algorithm\].
This paper is organized as follows. Section \[s:pre\_res\] provides definitions, preliminaries, and develops a general intuition. In Section \[s:BM\_equi\] we introduce useful upper and lower bounds for the discretization bias (see Lemma \[lem:BM\_LB\]) and show that the number of points on the equidistant grid has to grow quadratically in the threshold $b$ in order to uniformly control the discretization bias. In Section \[s:BM\_bb\], as an alternative to equidistant grids, we study [threshold-dependent]{} grids, which control the relative bias with a constant grid size, independently of $b$. The proofs of all lemmas and a proposition are postponed to Section \[s:proofs\]. In Section \[s:algorithm\] we present an algorithm by [@adler2012efficient], that we use throughout the paper for producing the numerical results; combining this algorithm with the use of [threshold-dependent]{} grids yields a strongly efficient algorithm for estimation of $w(b)$, see Corollary \[cor:strongly\_efficient\]. [In Section \[s:application\] we apply threshold-dependent grids developed in previous section to stochastic processes other than Brownian Motion: Brownian Motion with jumps, Ornstein-Uhlenbeck process and fractional Brownian Motion.]{} Lastly, in Section \[s:discussion\] we present concluding remarks and discuss some ideas for future research of *optimal grids*. In the appendices we collect various technical results used throughout the paper.
Preliminary results {#s:pre_res}
===================
Let $(B_t)_{t\in [0,1]}$ be a standard Brownian Motion on the time interval $[0,1]$ with $B_0 = 0$. We consider the probability of crossing a positive threshold $b$, that is $$\begin{aligned}
\label{defi_wb}
w(b) := {\mathbb{P}}\bigg(\sup_{t\in [0,1]} B_t > b\bigg).\end{aligned}$$ For a standard Brownian Motion, an explicit formula for the threshold-crossing probability (\[defi\_wb\]) is known, namely $w(b) = 2\,{\mathbb{P}}(B_1>b)$, which follows directly using the *reflection principle* (see e.g. [@morters2010brownian]). Given a *finite grid* $T$ we define a discrete-time approximation of $w(b)$: $$\begin{aligned}
\label{defi_wbT}
w_T(b) := {\mathbb{P}}\bigg(\sup_{t\in T} B_t > b\bigg),\end{aligned}$$ where $T = \{t_1, \ldots, t_n\}$ is a finite subset of the interval $[0,1]$, ordered such that $t_1<\ldots<t_n$. As we are mostly interested in choosing the grid $T$ efficiently, we define the following performance measure.
\[def:rel\_bias\] Let $T$ be a finite grid on $[0,1]$, then $$\begin{aligned}
\beta_T(b) := \frac{w(b) - w_T(b)}{w(b)} = {\mathbb{P}}\Big(\sup_{t\in T} B_t < b \ \big| \sup_{t\in [0,1]} B_t > b \Big)\end{aligned}$$ is called the relative bias induced by the grid $T$.
The second representation of relative bias in Definition \[def:rel\_bias\] is especially intuitive. It means that the relative bias is *the probability that $B_t$ stays below $b$ on the grid $T$, given that its supremum over $[0,1]$ is greater than $b$*. Notice that any grid which includes the endpoint $t=1$ will induce a relative bias no greater than $\frac{1}{2}$. Indeed, if $1\in T$, then $w_T(b) = {\mathbb{P}}(\sup_{t\in T} B_t > b) \geq {\mathbb{P}}(B_1 > b)$ and thus $$\begin{aligned}
\beta_T(b) = 1 - \frac{w_T(b)}{w(b)} \leq 1 - \frac{{\mathbb{P}}\big(B_1 > b\big)}{2\,{\mathbb{P}}\big(B_1 > b\big)} = \frac{1}{2}.\end{aligned}$$ Our objective is to accurately estimate $w(b)$ using discrete approximations $w_T(b)$, in a computationally efficient manner. Brownian Motion has continuous paths and thus it is always possible for a given $b$ to find a fine enough grid to bound the bias up to a desired accuracy. However, the computational cost of estimating $w_T(b)$ grows in the grid size and thus it might be infeasible to numerically compute $w_T(b)$ for large grids.\
At this point, we emphasize that we are not as much interested in the behaviour of $\beta_T(b)$ for a fixed $b$ or a fixed $n$ but rather in asymptotic regimes in which $b$ and/or $n$ approach infinity. For every $b$ we allow to use a different grid so it seems natural to treat the grid as a function of threshold. For every $b$ we define a collection of grids of all possible sizes $\{T_1(b), T_2(b), \ldots \}$, where $T_n(b)$ has $n$ elements, and we denote $\beta_n(b) := \beta_{T_n(b)}(b)$. For a given *family of grids* we are interested in behavior of $\beta_n(b)$ as $n$ or $b$ tend to infinity. The most straightforward choice for the family of grids is the following.
\[def:equi\_grid\] The family $\{T_n\}_{n\in{\mathbb{N}}}$, where $T_n := \{t^n_1,\ldots,t^n_n\}$ with $t^n_k:=\frac{k}{n}$ is called the equidistant family of grids.
Notice that the location of grid points on the equidistant grid is independent of $b$. Since the distance between neighboring points is equal to $\frac{1}{n}$, and since Brownian paths are continuous, it follows that $\beta_n(b) \to 0$, as $n \to \infty$ for any fixed $b$. It has been established in [@asmussen1995discretization] that for $T_n$, equidistant grid, the difference between the continuous-time and discrete-time supremum $\varepsilon_n = \sup_{t\in[0,1]}B_t-\sup_{t\in T_n} B_t$ is of order $n^{-1/2}$. More precisely, the sequence $(\sqrt{n}\varepsilon_n)_{n\in{\mathbb{N}}}$ has a tight and non-degenerate weak limit.
\[prop:eq\] Let $(B_t)_{t\in[0,1]}$ denote standard Brownian Motion and $\{T_n\}_{n\in{\mathbb{N}}}$ be the equidistant family of grids from Definition $\ref{def:equi_grid}$ with $\beta_n(b) := \beta_{T_n}(b)$. For any threshold $b>0$ there exist positive constants $C_1,C_2$ such that $$\begin{aligned}
C_1\,n^{-1/2} \leq \beta_n(b) \leq C_2\,n^{-1/2}.\end{aligned}$$
The proof of the Proposition \[prop:eq\] is given in Section \[s:proofs\]. The proof we give strongly resembles the proof of Theorem \[thm\_eq\] below in Section \[s:BM\_equi\], but we remark that it is also possible to derive it using the tools developed in [@broadie1997continuity].
Proposition \[prop:eq\] states that $\beta_n(b)$ decays like $n^{-1/2}$, when $n$ grows large *for a fixed $b$* but it does not describe the behavior of the relative bias when $b$ varies. In Theorem \[thm\_eq\] in the following section, we derive an upper bound for $\beta_n(b)$ for $n$ and $b$ simultaneously.
Figure \[fig:relative\_bias\_equi\] shows the evolution of the relative bias for four different thresholds $b=5, 6, 7, 8$ against the size of the equidistant grid. Even though all four graphs show the $n^{-1/2}$ decay, the graphs rise up with growing threshold. In particular, for thresholds $b=5$ and $8$ respectively $n=700$ and $1700$ points are needed to arrive at around $10\%$ relative bias. It indicates that, as $b$ grows *increasingly many grid-points are needed to arrive at the target relative bias*. Using the [threshold-dependent]{} grid that we develop in Section \[s:BM\_bb\] one can arrive at $10\%$ relative bias using approximately $n=100$ grid-points, independently of the value of the threshold. This amounts to a substantial improvement of the computational efficiency.
[0cm]{} ![Plots of the relative bias $\beta_n(b)$ against the grid size $n$ for the equidistant family of grids for four different thresholds. The numerical results are computed using an algorithm described in Section \[s:algorithm\]. []{data-label="fig:relative_bias_equi"}](equi_levels5-8_points50-2000.eps "fig:")
In some cases, the equidistant family of grids is the best possible choice, in the sense that other grid families require at least equally fast asymptotic growth of $n$ as $b$ increases, in order to control the relative bias. [@adler2012efficient] prove that for *centered, homogeneous and twice continuously differentiable (in a mean squared sense) Gaussian processes*, $n$ has to grow linearly in $b$ to uniformly control the relative bias. Moreover, if $n$ grows sublinearly in $b$, then the relative bias of any family of grids (not necessarily equidistant) tends to its maximal value, as $b$ approaches infinity. It is noted, however, that Brownian Motion does not belong to the family of Gaussian processes for which the result of [@adler2012efficient] applies.\
In the following two sections we analyze the asymptotic behavior of the relative bias $\beta_n(b)$ for two families of grids. We prove that the equidistant grid requires quadratic growth of $n$ in $b$ (see Theorem \[thm\_eq\] in Section \[s:BM\_equi\]). As an alternative, we develop the [threshold-dependent]{} family of grids, for which we prove that the relative bias can be made arbitrarily small, uniformly in $b$ for fixed $n$ (see Theorem \[THEorem\] in Section \[s:BM\_bb\]). We obtain a uniform rate of convergence in $n$ and also provide a closed-form expression for the [threshold-dependent]{} family of grids (see Definition (\[THE\_grid1\]) in Section \[s:BM\_bb\]).
Equidistant family of grids for Brownian Motion {#s:BM_equi}
===============================================
This section is devoted to analyzing the asymptotic behavior of the relative bias for the equidistant family of grids. The methodology developed in this section will be used later to prove Theorem \[THEorem\]; in particular, the crucial part of the proof concerns bounds for the relative bias induced by an arbitrary finite grid, developed in Lemma \[lem:BM\_LB\].\
The following theorem describes the asymptotic behaviour of the relative bias, under the equidistant family of grids.
\[thm\_eq\] Let $(B_t)_{t\in[0,1]}$ denote standard Brownian Motion and $\{T_n\}_{n\in{\mathbb{N}}}$ be the equidistant family of grids from Definition \[def:equi\_grid\] with $\beta_n(b) := \beta_{T_n}(b)$.
(a) Let $b_0$ be any positive, real number. There exist positive constants $C_0,C_1$, independent of $b$ and $n$ such that $$\begin{aligned}
\beta_n(b) \leq C_0\cdot bn^{-1/2},\end{aligned}$$ for all $b \geq b_0$, and $$\begin{aligned}
\beta_n(b) \leq C_1 \cdot n^{-1/2},\end{aligned}$$ for all $b\in(0,b_0]$.
(b) Let $m : (0,\infty) \to (0,\infty)$ be such that ${\lim_{b\to \infty}}{m(b)}/{b^2} = 0$. Then, as $b\to\infty$, $$\begin{aligned}
\inf_{n \leq m(b)} \beta_n(b) \longrightarrow \frac{1}{2}.\end{aligned}$$
Part (a) of Theorem \[thm\_eq\] states that $\beta_n(b) \leq C_0\, bn^{-1/2}$, so that in order to bound $\beta_n(b)$ uniformly in $b$ it suffices to take $n = O(b^2)$. The second part of the Theorem \[thm\_eq\] states that if $n=o(b^2)$ then $\beta_n(b)\to 1/2$, meaning that the relative bias cannot be bounded by an arbitrarily small number. Together, the two parts entail that the growth $n=O(b^2)$ is sufficient and there is no better (slower) growth which would guarantee a uniformly bounded relative bias.
The crucial part of the proof of Theorem \[thm\_eq\] is the method of bounding the relative bias. Since no explicit expressions for $w_T(b)$ or $\beta_T(b)$ are known (even if $T$ is an equidistant grid) we develop a general upper bound for $\beta_T(b)$ in the following lemma, in which we use the quantities [$$\begin{aligned}
a_j(b) &:= {\mathbb{P}}\big(B_{t_j(b)-t_{j-1}(b)} < 0, \ldots, B_{t_n(b)-t_{j-1}(b)} < 0\big),\:\:\:\:
a_{n+1}(b):=\sfrac{1}{2}, \\
w_j(b)& := {\mathbb{P}}\big(\tau_b\in (t_{j-1}(b),t_j(b)] \ \big| \ \tau_b\in(0,1]\big),\\
\tau_b &:= \inf\{t\geq 0 : B_t > b \}. \end{aligned}$$Notice that in this definition of $a_j(b)$ and $w_j(b)$ we allow grid points $t_1,\ldots,t_n$ to change their location with $b$. In the present section, which is on equidistant grids, the grid points obviously do not depend on $b$, but in later sections they do.]{}
\[lem:BM\_LB\] Let $T(b) = \{t_1(b), \ldots, t_n(b)\}\subset[0,1]$, where $0<t_1(b)<\ldots < t_n(b)\leq 1$, and let $t_0(b) = 0$. The following lower and upper bounds for $\beta_T(b)$ apply: $$\begin{aligned}
\underbar{$\beta$}_T(b) \leq \beta_T(b) \leq \bar\beta_T(b)\end{aligned}$$ with $$\begin{aligned}
\underbar{$\beta$}_T(b) := \frac{1}{2}\sum_{j=1}^{n} a_{j+1}(b) \, w_j(b), \ \ \ \ \ \bar\beta_T(b) := \sum_{j=1}^n a_j(b) \, w_j(b).\end{aligned}$$
A short proof of Lemma \[lem:BM\_LB\] is included in Section \[s:proofs\]. The bounds consist of elements of two types: $a_j(b)$, the probability that $B_t$ stays negative at times $t_j-t_{j-1}, \ldots, t_n-t_{j-1}$, and $w_j(b)$, the probability that $B_t$ hits $b$ for the first time in the interval $[t_{j-1},t_j]$ given that its supremum over $[0,1]$ is greater than $b$.\
For a general grid [$T(b)$]{}, the [probabilities $a_j(b)$]{} are difficult to control. However, when $T(b)$ is equidistant [(thus independent of $b$)]{}, then also the probabilities $a_j$ are independent of $b$; [we emphasize this independence by writing $a_j$ instead of $a_j(b)$ throughout this section.]{} As a result, there exists a tight asymptotic bound for them (see Lemma \[lem\_feller\] below); we were inspired to look into such quantities while reading [@morters2010brownian Section 5]. [The probabilities $w_j(b)$]{} are controlled using a mean value theorem, see Appendix \[appendix:results\].\[mvt\_taub\].
\[lem\_feller\] There exist constants $C^*_1, C^*_2 > 0$ such that: $$\begin{aligned}
C^*_1 n^{-1/2} \leq {\mathbb{P}}\Big(B_1 > 0, \ldots, B_n > 0\Big) \leq C^*_2 n^{-1/2}\end{aligned}$$ for all $n\in{\mathbb{N}}$.
In fact, the assertion in Lemma \[lem\_feller\] is true for any symmetric random walk; [see [@feller1971introduction Theorem 4 in Section XII.7, and Lemma 1 in Section XII.8]]{}. Before proving Theorem \[thm\_eq\] we present one more lemma.
\[equi\_grid\_increasing\_b\] Let $T = \{t_1, \ldots, t_n\}$ be such that $t_k = \frac{k}{n}$ and let $t_0 = 0$. Then the upper bound $\bar\beta_T(b)$ developed in Lemma \[lem:BM\_LB\] is an increasing function of $b$.
An important implication of Lemma \[equi\_grid\_increasing\_b\] is that for any $b_0>0$ we have that $\beta_T(b) \leq \bar\beta_T(b) \leq \bar\beta_T(b_0)$ uniformly for all $b\leq b_0$, which completely covers the statement on the situation that $b\leq b_0$ in part (a) of Theorem \[thm\_eq\]. The proof of Lemma \[equi\_grid\_increasing\_b\] is given in Section \[s:proofs\].
Thanks to Lemma \[equi\_grid\_increasing\_b\] it suffices to prove the first part of Theorem \[thm\_eq\](a), i.e. we assume that $b\geq b_0$. Without loss of generality we put $b_0 = 1$. Exploiting the upper bound developed in Lemma \[lem:BM\_LB\] we decompose the sum $\sum_{j=1}^n a_j\cdot w_j(b)$ into three parts, which we treat separately: $$\begin{aligned}
\label{to_bound_thm_eq}
\beta_{n}(b) & \leq a_1\cdot w_1(b) + \sum_{j=2}^{n-1} a_j\cdot w_j(b) + a_n\cdot w_n(b),\end{aligned}$$ Using the definition of the equidistant grid and the scaling property of Brownian motion we can see that $a_j = {\mathbb{P}}\big( B_{t_j-t_{j-1}} < 0, \ldots, B_{t_n-t_{j-1}} < 0 \big) = {\mathbb{P}}\big( B_1 < 0, \ldots, B_{n-j+1} < 0 \big)$ and the bound in Lemma \[lem\_feller\] yields $a_j \leq C^*_2 (n-j+1)^{-1/2}$ for all $j\in\{1,\ldots,n\}$. Since all $w_j(b)\leq 1$, we thus have a straightforward bound for the first term in (\[to\_bound\_thm\_eq\]): $$\begin{aligned}
a_1\cdot w_1(b) \leq C^*_2 \,n^{-1/2}\end{aligned}$$ The second term we bound in the following fashion, relying on the [upper bound]{} that we have for $w_j(b)$ (stated in Result B.V), $$\begin{aligned}
\nonumber \sum_{j=2}^{n-1} a_j\cdot w_j(b) & \leq \sum_{j=2}^{n-1} C_2^*\,(n-j+1)^{-1/2} \cdot \frac{b(b+\sqrt{b^2+4})}{4} \frac{\sqrt{n}}{(j-1)^{3/2}} e^{-\frac{b^2}{2} \cdot \left( \frac{n}{j} - 1\right)} \\
\label{rieman_summ_nice} & \leq C_1 \cdot bn^{-1/2} \cdot \sum_{j=2}^{n-1} \frac{1}{n} \cdot \left( \frac{b}{\sqrt{1-\frac{j}{n}}} \cdot \left(\frac{j}{n}\right)^{-3/2} e^{-\frac{b^2}{2} \cdot \left( \frac{n}{j} - 1\right)} \right) \\
\label{rieman_b0small} & \leq C_1 \cdot bn^{-1/2} \cdot \int_0^1 \frac{b}{\sqrt{1-x}} \cdot x^{-3/2} \cdot e^{-\frac{b^2}{2}\, (1/x - 1)}\,dx \\
\nonumber & \leq C_1 \cdot bn^{-1/2},\end{aligned}$$ where $C_2^*$ comes from Lemma \[lem\_feller\] and $C_1$ is a positive constant, independent of $b$ and $n$. To arrive at (\[rieman\_summ\_nice\]) we use that $2(j-1) \geq j$ for all relevant $j$. In the transition from (\[rieman\_summ\_nice\]) to (\[rieman\_b0small\]) we use the definition of the Riemann sum for the function $$f(b,x) := \frac{b}{\sqrt{1-x}}x^{-3/2}e^{-\frac{b^2}{2}(1/x-1)};$$ note that, since $f(b,x)$ is an increasing function of $x$ when $b\geq 1$ (see Result \[appendix:results\].\[res\_increasing\] in the Appendix), the Riemann sum in (\[rieman\_summ\_nice\]) underestimates the integral, i.e., $\sum_{j=2}^{n} \frac{1}{n}f(b,\frac{j-1}{n}) \leq \int_0^1 f(b,x)\,dx = \sqrt{2\pi}$.\
Lastly, since $a_n = {\mathbb{P}}(B_{t_n}<0) = \frac{1}{2}$ we have $$\begin{aligned}
a_n\cdot w_n(b) \leq \frac{1}{2} \cdot \frac{b(b+\sqrt{b^2+4})}{4} \frac{\sqrt{n}}{(n-1)^{3/2}} \leq C_2 \,\frac{b^2}{n},\end{aligned}$$ where $C_2$ is a positive constant independent of $n$ and $b$. Since $w_n(b) \leq 1$ this results in $$a_n\cdot w_n(b) \leq \min \left\{ C_2 \,\frac{b^2}{n},\frac{1}{2}\right\} \leq \sqrt{\min \left\{ C_2 \,\frac{b^2}{n},\frac{1}{2}\right\}} \leq \sqrt{C_2}\,bn^{-1/2}.$$ Combining the above bounds, $$\begin{aligned}
\beta_{n}(b) & \leq a_1\cdot w_1(b) + \sum_{j=2}^{n-1} a_j\cdot w_j(b) + a_n\cdot w_n(b) \leq C_2^* n^{-1/2} + C_1 \,bn^{-1/2} + \sqrt{C_2} \,bn^{-1/2}\\
& \leq C_0 \, bn^{-1/2},\end{aligned}$$ where $C_0$ is a positive constant independent of $b$ and $n$. This concludes the proof.
Without loss of generality we can assume $m(b)\to\infty$ as $b\to\infty$. Similar to the proof of Lemma \[lem:BM\_LB\] in Section \[s:proofs\] we obtain: $$\begin{aligned}
\nonumber w(b)\,\beta_T(b) & = {\mathbb{P}}\Big(\sup_{t\in T} B_t < b, \sup_{t\in [0,1]} B_t > b \Big) = {\mathbb{P}}\Big(\sup_{t\in T} B_t < b, \tau_b\in[0,1] \Big) \\
\nonumber & = \sum_{j=1}^n {\mathbb{P}}\Big(\sup_{t\in \{t_j, \ldots, t_{n}\}} B_t < b, \tau_b\in(t_{j-1},t_j] \Big) \geq {\mathbb{P}}\Big( B_{t_n} < b, \tau_b\in(t_{n-1},t_n] \Big)\\
\nonumber & = \int_{t_{n-1}}^{t_n} {\mathbb{P}}(B_{t_n} < b | B_s = b) {\mathbb{P}}(\tau_b\in ds) = \frac{1}{2}\,{\mathbb{P}}(\tau_b\in(t_{n-1},t_n))\end{aligned}$$ Dividing both sides of the inequality by $w(b)$ yields an elementary lower bound on $\beta_T(b)$: $$\begin{aligned}
\label{very_simple_lower_bound}
\beta_T(b) & \geq \frac{1}{2}\,{\mathbb{P}}\big(\tau_b\in (t_{n-1},t_n] \ \big| \ \tau_b\in(0,1]\big) = \frac{1}{2} \cdot \frac{\Phi(-b/\sqrt{t_n}) - \Phi(-b/\sqrt{t_{n-1}})}{\Phi(-b)}\end{aligned}$$ [where $\Phi(\cdot)$ denotes the standard normal cdf, and we use the fact that ${\mathbb{P}}(\tau_b\leq t) = 2\,{\mathbb{P}}(B_t>b)$.]{} In our case $t_n = 1$ and $t_{n-1} = \frac{n-1}{n} \leq \frac{m-1}{m}$, so that [due to the monotonicity of $\Phi(\cdot)$]{} $$\begin{aligned}
\label{ineq_funna}
\inf_{n\leq m(b)}\beta_n(b) \geq \frac{1}{2} - \frac{1}{2} \frac{\Phi(-b/\sqrt{(m-1)/m})}{\Phi(-b)} \, .\end{aligned}$$ Taking the limit $b\to\infty$ on both sides of inequality (\[ineq\_funna\]) yields: $$\begin{aligned}
{\lim_{b\to \infty}}\nonumber \inf_{n\leq m(b)}\beta_n(b) & \geq \frac{1}{2} - \frac{1}{2}{\lim_{b\to \infty}}\frac{\Phi(-b/\sqrt{(m-1)/m})}{\Phi(-b)} \\
\label{using_some_limit_result} & = \frac{1}{2} - \frac{1}{2}{\lim_{b\to \infty}}\frac{\frac{\sqrt{(m-1)/m}}{b}\phi(b/\sqrt{(m-1)/m})}{\frac{1}{b}\phi(b)} \\
\nonumber & = \frac{1}{2} - \frac{1}{2}{\lim_{b\to \infty}}e^{-b^2/(2(m-1))} = \frac{1}{2}\end{aligned}$$ where [$\phi(\cdot)$ denotes the standard normal pdf]{}, we use result \[appendix:results\].\[limit1\] in (\[using\_some\_limit\_result\]), and the last equality is a consequence of the assumption that ${\lim_{b\to \infty}}{m(b)}/{b^2} = 0$.
In this section we have proven that in order to uniformly control the relative bias, the size of the equidistant grid must grow at least quadratically in $b$, as $b$ approaches infinity. In the next section we present a [threshold-dependent]{} grid, which yields a uniform bound on the relative bias using a grid of given size. In other words, in order to control the relative bias with increasing $b$, instead of adding more and more points to the grid, it suffices to suitably shift their location.
[Threshold-dependent]{} grids for Brownian Motion {#s:BM_bb}
=================================================
In this section we prove the main result of the paper. We explicitly present a [threshold-dependent]{} family of grids which uniformly (in $b$) bounds the relative bias.\
Before we introduce the result, we give some intuition why it is possible to control the relative bias as $b$ grows, without increasing $n$. [Firstly, for any given $\varepsilon>0$, we have that $${\mathbb{P}}(\sup_{t\in[0,1-\varepsilon]}B_t>b) = 2\,{\mathbb{P}}(B_{1-\varepsilon}>b) = o(w(b)),$$as $b\to\infty$. Therefore,]{} $${{\mathbb{P}}\Big(\sup_{t\in[1-\varepsilon,1]} B_t > b \,\Big| \sup_{t\in[0,1]} B_t > b\Big) \longrightarrow 1, \ \text{ as } b \to \infty.}$$ It means that with growing $b$, the ‘hitting of the threshold’ occurs closer and closer to time $t=1$. It indicates that the grid points should be gradually shifted towards the point $t=1$, as $b$ is increasing. Moreover, the result in Theorem \[thm\_eq\] indicates how fast the points should be shifted. It states that for the family of equidistant grids, the uniform bound on the bias is achieved if the number of grid points grows quadratically in $b$. Equivalently, the distances between neighboring points are decreasing proportionally to $b^{-2}$. It turns out that this is indeed the pace at which the points should be shifted towards $t=1$.
In the following result, $\Phi(\cdot)$ and $\Phi^{-1}(\cdot)$ denote the standard normal cdf and its inverse, respectively.
\[THEorem\] Let $(B_t)_{t\in[0,1]}$ be a standard Brownian Motion. Fix $b_0 > 0$ and let $\{T_n(b)\}_{n\in{\mathbb{N}},b>0}$ be a family of grids such that $T_n(b) = \{t^n_1(b), \ldots, t^n_n(b)\}$; here $t^n_k(b) := \frac{k}{n}$ for $b\leq b_0$, and $$\begin{aligned}
\label{THE_grid1}
t^n_k(b) := \Bigg(\frac{b}{\Phi^{-1}\left(\frac{k}{n}\,\Phi(-b)\right)} \Bigg)^{2},\end{aligned}$$ for $b>b_0$. Denote $\beta_n(b) := \beta_{T_n(b)}(b)$. There exists a positive $C$, independent of $b$ and $n$, such that $$\begin{aligned}
\beta_n(b) \leq C\,n^{-1/4}\end{aligned}$$ for all $b>0$.
We emphasize that the bound for the relative bias $\beta_n(b)$ developed above does not depend on the threshold $b$ and thus holds *uniformly*, for all $b$. Figure \[fig:relative\_bias\_equi\_vs\_adapt\] shows the comparison between the relative bias of the equidistant and the [threshold-dependent]{} grid, both of size $n=100$. The bias induced by the [threshold-dependent]{} grid remains uniformly bounded (by circa $0.1$), while the former tends to $0.5$, the worst possible relative bias, cf. Theorem \[thm\_eq\], part (b).\
[0cm]{} ![A plot of the relative bias of the equidistant grid $\beta_{100}^{\text{eq}}(b)$ and the [threshold-dependent]{} grid [$\beta_{100}^{\text{td}}(b)$]{}, both with fixed grid size $n=100$, as a function of $b$. Notice that $\beta_{100}^\text{eq}(b)$ tends to 0.5 with growing $b$ (the largest possible bias), while [$\beta_{100}^\text{td}(b)$]{} remains bounded by about $0.1$. The numerical results are computed with the algorithm discussed in Section \[s:algorithm\]. The relative error due to finite sample size is negligible (smaller than 0.006).[]{data-label="fig:relative_bias_equi_vs_adapt"}](table_ad_vs_eq_2-20_n=100_v2.eps "fig:")
Notice that for small $b$, $\{T_n(b)\}_{n\in{\mathbb{N}},b\in(0,b_0]}$ in Theorem \[THEorem\] is identical to the equidistant family of grids. In fact this is exactly the setting of [the second part of the Theorem \[thm\_eq\](a)]{}. The real contribution of Theorem \[THEorem\] is the regime when $b>b_0$. The grid defined in (\[THE\_grid1\]) is the unique solution to the set of equations $$\begin{aligned}
\label{THE_grid2}
{\mathbb{P}}\Big(\tau_b \in(t^n_{k-1}(b),t^n_k(b)] \ \Big| \ \tau_b \in (0,1]\Big) = \frac{1}{n}\end{aligned}$$ for all $k \in \{1, \ldots, n\}$ and $t_0 := 0$. To see this, we sum up the first $k$ equations in (\[THE\_grid2\]) and obtain an explicit equation for $t_k^n(b)$: $$\begin{aligned}
\label{THE_grid3}
{\mathbb{P}}\Big(\tau_b \in(0,t^n_k(b)] \ \Big| \ \tau_b \in (0,1]\Big) = \frac{k}{n}.\end{aligned}$$ [Since for Brownian Motion it holds that $${\mathbb{P}}(\tau_b \in(0,t^n_k(b)]) = 2\,{\mathbb{P}}(B_{t^n_k(b)}>b) = 2\,\Phi(-b/\sqrt{t^n_k(b)}),$$ and in particular ${\mathbb{P}}(\tau_b \in (0,1]) = 2\,{\mathbb{P}}(B_1>b) = 2\,\Phi(-b)$, Eqn. (\[THE\_grid3\]) can be equivalently expressed as $$\begin{aligned}
\label{THE_grid4}
\frac{{\mathbb{P}}(B_{t^n_k(b)}>b)}{{\mathbb{P}}(B_1>b)} = \frac{k}{n}\end{aligned}$$ or, in terms of the cdf $\Phi(\cdot)$, $$\begin{aligned}
\Phi\left(-b/\sqrt{t^n_k(b)}\right) = \frac{k}{n}\Phi(-b).\end{aligned}$$ ]{}
Finally, after taking the inverse $\Phi^{-1}(\cdot)$ from both sides of the equation above we see that $t_k^n(b)$ satisfies (\[THE\_grid1\]). Figure \[fig:grid\_evolution\] shows the placement of the grid-points on the grid $T_{5}(b)$, as defined in (\[THE\_grid1\]), for increasing $b$. In fact, one can prove that $$\begin{aligned}
\label{limit_adaptive_grid}
b^2\big(1-t^n_k(b)\big) \xrightarrow{b\to\infty} -2\log(k/n)\end{aligned}$$ and thus $$t^n_k(b)\approx 1 - \frac{2\log(n/k)}{b^2}$$ for large $b$. It means that the points of the grid (\[THE\_grid1\]) are clustered around $t=1$, with distances between the points proportional to $b^{-2}$. Here we see [an important connection with Theorem \[thm\_eq\](a)]{}, where the distances between grid-points decrease at the same pace, as already mentioned in the opening paragraph of this section.
[0cm]{} ![Location of the grid-points $t^{5}_1(b), \ldots, t^{5}_{5}(b)$ defined in (\[THE\_grid1\]) with increasing threshold $b$. Note that with growing $b$ all the points are gradually shifted towards the end-point $t=1$.[]{data-label="fig:grid_evolution"}](times_against_levels.eps "fig:")
For $b>b_0$, the points $t^n_1(b), \ldots, t^n_n(b)$ of the [threshold-dependent]{} grid (\[THE\_grid1\]) do not coincide with the equidistant grid, entailing that we can not directly use Lemma \[lem\_feller\] to control the terms of type $a_j(b)$ in the upper bound developed in Lemma \[lem:BM\_LB\] in Section \[s:BM\_equi\]. The following lemma resolves this issue.
\[lemma2\_me\] Let $t_0 = 0 < t_1 < t_2 < \ldots < t_n < \infty$. Then $$\begin{aligned}
{\mathbb{P}}\Big(B_{t_1} > 0, \ldots, B_{t_n} > 0\Big) \leq {\mathbb{P}}\Big(B_1 > 0, \ldots, B_N > 0\Big),\end{aligned}$$ for any $N \leq N_n$, where $$N_n:=\left(\frac{t_n}{\max_{k=1,\ldots,n}(t_k - t_{k-1})}\right)^{1/2}.$$
A proof of this lemma is provided in Section \[s:proofs\]. Lemma \[lem\_feller\] applied to the upper bound in Lemma \[lemma2\_me\] yields a simple upper bound for ${\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_n} > 0\big)$ for any choice of $t_0 = 0 < t_1 < t_2 < \ldots < t_n < \infty$. In our case, after applying Lemma \[lem:BM\_LB\] we have to control probabilities of the type ${\mathbb{P}}\big( B_{t_j-t_{j-1}} < 0,\ldots, B_{t_n-t_{j-1}} < 0 \big)$, and thus we need a lower bound on $$\frac{t_n-t_{j-1}}{\max_{k=j,\ldots,n}(t_k-t_{k-1})},$$ which we give in the following lemma.
\[lemming\] For the grid in (\[THE\_grid1\]), for $k>j$, $b>0$ and $n\in{\mathbb{N}}$ we have: $$\begin{aligned}
\textnormal{(a)} \ \ \ \ \frac{t_n^n(b) - t_j^n(b)}{t_k^n(b) - t_j^n(b)} \geq \frac{\log n - \log j}{\log k - \log j}.\end{aligned}$$ and when additionally $b\geq\sqrt{3}$ we have $$\begin{aligned}
\textnormal{(b)} \ \ \ \ \max_{k=j,\ldots,n} (t^n_{k}(b)-t^n_{k-1}(b)) = t^n_{j}(b)-t^n_{j-1}(b)\end{aligned}$$
Lemma \[lemming\] is proven in Section \[s:proofs\]. The lower bound in part (a) of Lemma \[lemming\] is in fact $$\lim_{b\to\infty}\frac{1 - t_j^n(b)}{t_k^n(b) - t_j^n(b)}.$$ With these lemmas we can prove Theorem \[THEorem\].
Part (a) of Theorem \[thm\_eq\] states that for any choice of $b_0$ there exists positive $C_1$ such that $\beta_n(b) \leq C_1n^{-1/2}$ for $b\leq b_0$ and thus also $\beta_n(b) \leq C_1n^{-1/4}$. Without the loss of generality, from now on we assume that $b>b_0 = \sqrt{3}$. Fix $n\in{\mathbb{N}}$ and denote $t_k := t^n_k(b)$ for notational simplicity. After combining the general upper bound from Lemma \[lem:BM\_LB\] with the equivalent definition (\[THE\_grid2\]) of the [threshold-dependent]{} grid (\[THE\_grid1\]) we obtain $$\begin{aligned}
\beta_n(b) & \leq \sum_{j=1}^n a_j(b)w_j(b) = \frac{1}{n} \sum_{j=1}^n a_j(b);\end{aligned}$$ observe that in our setting $w_n(b)=\frac{1}{n}.$ Moreover, Lemma \[lemma2\_me\] yields (recalling the definition of $a_n(b)$) $$\begin{aligned}
\beta_n(b) \leq \frac{1}{2n} + \frac{1}{n} \sum_{j=2}^{n-1} {\mathbb{P}}\big( B_1 > 0, \ldots, B_{N_n(j)} > 0 \big) + \frac{1}{2n},\end{aligned}$$ where $$N_n(j) := \left[ \left(\frac{t_n^n(b) - t_{j-1}^n(b)}{\max_{k\geq j}|t^n_k(b) - t^n_{k-1}(b)|}\right)^{1/2}\right].$$ Combining Lemma \[lem\_feller\] with Lemma \[lemming\] gives $$\begin{aligned}
\beta_n(b) \leq \frac{1}{n} + C\frac{1}{n} \sum_{j=2}^{n-1} \widetilde{N}_n(j)^{-1/4}, \ \ \text{ where } \ \widetilde{N}_n(j) := \frac{\log n - \log (j-1)}{\log j - \log (j-1)}\end{aligned}$$ with a constant $C>0$ that is independent of $n$ and $b$. Notice that $\widetilde{N}_n(j)$ does not depend on $b$. For $b>b_0$ we thus obtain $$\begin{aligned}
\nonumber \beta_n(b) & \leq \frac{1}{n} + C\,n^{-1} \sum_{j=2}^{n-1} \left(\frac{\log j-\log(j-1)}{\log n-\log(j-1)}\right)^{1/4} \\
\nonumber & = \frac{1}{n} + C\,n^{-1} \sum_{j=2}^{n-1} \left(\frac{\log\big(1 + \frac{1}{j-1}\big)}{\log\big(\frac{n}{j-1}\big)}\right)^{1/4} \\
\label{log_nierownosc} & \leq \frac{1}{n} + C\,\,n^{-1/4} \sum_{j=2}^{n-1}\frac{1}{n}\left(\frac{\frac{n}{j-1}}{\log\big(\frac{n}{j-1}\big)}\right)^{1/4} \\
\label{rieman_b0big} & \leq \frac{1}{n} + C\,\,n^{-1/4} \int_0^1 \left( \frac{1}{-x\log x} \right)^{1/4}\,dx \\
\nonumber & \leq C\,n^{-1/4}\end{aligned}$$ where $C$ is a constant, independent from $n$ and $b$, that might differ from line to line. In (\[log\_nierownosc\]) we use the inequality $\log(1+x)\leq x$ and in (\[rieman\_b0big\]) we use the convergence of the Riemann sum to the integral. This concludes the proof of Theorem \[THEorem\].
[For the purpose of showing that for any confidence level $\alpha$ and bias $\varepsilon$, [see also (\[confidence\]),]{} the ‘equiprobable’ grid (as defined through (\[THE\_grid1\])) requires a computational effort that is bounded in $b$, it suffices that the decay of the upper bound for $\beta_n(b)$ in Theorem \[THEorem\] is of order $n^{-1/4}$; [see Corollary \[cor:strongly\_efficient\] in Section \[s:algorithm\]]{}. As an aside we remark that we hypothesize that this decay is actually of order $n^{-1/2}$. This is supported by numerical experiments; see Figure $\ref{fig:relative_bias_M=1}$ where plots of $\beta_n(b)$ versus $n$ are shown for the [threshold-dependent]{} grid (\[THE\_grid1\]). The step we expect to be ‘loose’, [in obtaining the bound of Theorem \[THEorem\],]{} is the one corresponding to Lemma \[lemma2\_me\]. We conjecture that Lemma \[lemma2\_me\] is valid with $$N_n:=\frac{t_n}{\max_{k=1,\ldots,n}(t_k - t_{k-1})}.$$ (i.e., without the square root), which suffices to yield the $n^{-1/2}$ decay of $\beta_n(b)$. ]{} \[remark1\]
[0cm]{} ![Relative bias $\beta_n(b)$ versus grid size $n$ for the [threshold-dependent]{} grid (\[THE\_grid1\]). The threshold is fixed at $b=3$. The right panel shows a loglog plot, left panel a linear plot. The results suggest that $\beta_n(b)$ decays proportionally to $n^{-1/2}$ rather than $n^{-1/4}$ [(see also Remark \[remark1\])]{}. []{data-label="fig:relative_bias_M=1"}](b3_plot_10-200.eps "fig:") ![Relative bias $\beta_n(b)$ versus grid size $n$ for the [threshold-dependent]{} grid (\[THE\_grid1\]). The threshold is fixed at $b=3$. The right panel shows a loglog plot, left panel a linear plot. The results suggest that $\beta_n(b)$ decays proportionally to $n^{-1/2}$ rather than $n^{-1/4}$ [(see also Remark \[remark1\])]{}. []{data-label="fig:relative_bias_M=1"}](b3_loglog_10-200.eps "fig:")
Numerical algorithm for estimation of $w(b)$ {#s:algorithm}
============================================
As mentioned in the introduction, the family of [threshold-dependent]{} grids (\[THE\_grid1\]) can be used to construct a strongly efficient algorithm for estimation of $w(b)$, see Corollary \[cor:strongly\_efficient\] below. In this paper, by ‘strongly efficient’ we mean that for any given accuracy $\varepsilon>0$ and confidence level $\alpha>0$ the computational time of an estimator $\widehat{w}(b)$ for $w(b)$ that satisfies $$\begin{aligned}
{\mathbb{P}}\left( \left| \frac{\widehat{w}(b)}{w(b) }-1 \right| > \varepsilon \right) < \alpha
\label{confidence}\end{aligned}$$ is bounded independently of the threshold $b$.\
In all numerical experiments throughout this paper we used an algorithm developed by [@adler2012efficient], see Algorithm \[alg:blanchet\] below. Although it is applicable for estimation of quantities such as ${\mathbb{P}}(\max_{i\in\{1,\ldots,n\}} X_i > b)$, where $X\in{\mathbb{R}}^n$ is normally distributed with an arbitrary positive-definite covariance matrix, we present their algorithm for the specific case of Brownian Motion, as considered in this paper.
\[alg:blanchet\] Choose a threshold $b$ and a finite grid $T = \{t_1, \ldots, t_n\} \in [0,1]$. The estimator $\widehat{w}_T(b)$, computed according to the following algorithm, is an unbiased estimator of $w_T(b)$.
1. Generate a random time $\tau$ on the grid, i.e. $\tau \in T$, according to the law $${\mathbb{P}}(\tau = t_k) = \frac{{\mathbb{P}}(B_{t_k}>b)}{\sum_{j=1}^n {\mathbb{P}}(B_{t_j} > b)}.$$
2. Generate $B_\tau$ under the condition $B_\tau > b$.
3. Generate a discrete path of the Brownian Motion $(B_{t_1}, \ldots, B_{t_n})$ conditioned on the pair $(\tau, B_\tau)$ generated in the previous steps.
4. Compute $$\widehat{w}_T(b) := \frac{\sum_{j=1}^n {\mathbb{P}}(B_{t_j} > b)} {\sum_{j=1}^n {\mathds{1}}(B_{t_j} > b)}.$$
[@adler2012efficient] prove that the Algorithm \[alg:blanchet\] gives an *unbiased* estimator of $w_T(b)$ (not of $w(b)$) and that for a fixed $T$ (independent of $b$), the relative variance ${\mathbb{V}\textnormal{\textrm{ar}}}(\widehat{w}_T(b))/w_T^2(b) \to 0$, as $b\to\infty$. The authors also propose an estimator for $w(b)$, which relies on a *random discretization*. However, with growing $b$, one needs increasingly many random grid-points in order to control the relative bias, therefore the continuous-time algorithm *is not* strongly efficient. In order to reduce the sampling error one generates multiple replicas of the estimator and takes their average. Since every replica is based on a different grid, one must repeatedly calculate the Cholesky decomposition (whose computational time is cubic in the number of grid-points) in order to sample discrete Gaussian paths in Step 3 of Algorithm \[alg:blanchet\]. Choosing a predefined grid speeds up this computation, as in that case the Cholesky decomposition has to be performed *only once*, making its computational cost negligible.\
Combining the [threshold-dependent]{} grids as proposed in Section \[s:BM\_bb\] with Algorithm \[alg:blanchet\] yields a strongly efficient estimator for $w(b)$ which is given in the corollary below.
\[cor:strongly\_efficient\] Fix an accuracy $\varepsilon>0$ and a confidence level $\alpha>0$. Choose a grid $T := T_n(b)$ from the family of grids defined in $(\ref{THE_grid1})$ such that $\beta_T(b) := \beta_n(b) < \varepsilon$ [for all $b>0$]{} (this is possible due to the result in Theorem \[THEorem\]). Let $\widehat{w}^{(1)}_{T}(b), \ldots \widehat{w}^{(N)}_{T}(b)$ be i.i.d copies of the estimator from Algorithm \[alg:blanchet\], with $$N \geq \frac{n^2}{\alpha (\varepsilon-\beta_T(b))^2}.$$ Then $$\widehat{w}(b) := \frac{1}{N}\sum_{i=1}^N \widehat{w}^{(i)}_T(b)$$ satisfies $$\begin{aligned}
\label{to_prove:strong_eff}
{\mathbb{P}}\left( \left| \frac{\widehat{w}(b)}{w(b) }-1 \right| > \varepsilon \right) < \alpha,\end{aligned}$$ and the computational effort to simulate $\widehat{w}(b)$ is bounded independently of $b$.
First notice that since $\beta_T(b)$ is uniformly bounded in $b$ (see Theorem \[THEorem\]), so that $N$ is fixed independently of $b$, it follows that $\widehat{w}(b)$ can be computed in bounded time, independently of $b$. It remains to prove that $\widehat{w}(b)$ satisfies the strong efficiency property (\[to\_prove:strong\_eff\]). Note that $\widehat{w}(b)$ is an unbiased estimator of $w_T(b)$, not of $w(b)$. The relative variance of $\widehat{w}(b)$ with respect to $w_T(b)$ can be bounded independently of $b$ for an arbitrary choice of the grid in terms of the grid size $n$, $$\begin{aligned}
\frac{{\mathbb{V}\textnormal{\textrm{ar}}}(\widehat{w}_T(b))}{(w_T(b))^2} \leq \frac{{\mathbb{E}}(\widehat{w}_T(b))^2}{(w_T(b))^2} \leq \left( \frac{\sum_{j=1}^n {\mathbb{P}}(B_{t_j} > b)}{\max_{j\in\{1,\ldots,n\}} {\mathbb{P}}(B_{t_j} > b)}\right)^2 \leq n^2.\end{aligned}$$ Due to Chebyshev’s inequality, $$\begin{aligned}
{\mathbb{P}}\left( \left| \frac{\widehat{w}(b)}{w(b)} - 1 \right| > \varepsilon \right) & = {\mathbb{P}}\left( \left| \frac{\widehat{w}(b) - w_T(b)}{w(b)} + \frac{w_T(b) - w(b)}{w(b)} \right| > \varepsilon \right) \leq {\mathbb{P}}\left( \left| \frac{\widehat{w}(b) - w_T(b)}{w(b)} \right| > \varepsilon - \beta_T(b) \right) \\
& \leq \frac{{\mathbb{V}\textnormal{\textrm{ar}}}(\widehat{w}(b))}{(\varepsilon-\beta_T(b))^2 (w(b))^2} = \frac{1}{N} \cdot \frac{{\mathbb{V}\textnormal{\textrm{ar}}}(\widehat{w}_T(b))}{(\varepsilon-\beta_T(b))^2 (w(b))^2} \\
& \leq \frac{1}{N} \cdot \frac{n^2}{(\varepsilon-\beta_T(b))^2} \leq \alpha.\end{aligned}$$ This concludes the proof.
We conclude this section by a remark on the simulation of the conditioned Brownian Motion in Step 3 of Algorithm \[alg:blanchet\]. The naïve method would be to construct the covariance matrix of the conditioned process, calculate the Cholesky decomposition of that matrix (cubic in the number of grid points) and then simulate the process in a standard manner. Notice that this step must be repeated for every replica $\widehat{w}^{(i)}_T(b)$ and thus its computational cost scales with the number of samples. The following algorithm, which can be found e.g. in [@doucet2010note], requires only a single calculation of the Cholesky decomposition for all replicas.
\[alg:doucet\] Let $X = (X_1, X_2)^T \in {\mathbb{R}}^n$, where $X_1 \in {\mathbb{R}}^{n-1}$ and $X_2 \in {\mathbb{R}}$, be normally distributed with mean $\mu$ and covariance matrix $\Sigma$, $$\begin{aligned}
& \mu = \begin{pmatrix}
\mu_1\\
\mu_2
\end{pmatrix}, \text{ where } \mu_1\in{\mathbb{R}}^{n-1} \text{ and } \mu_2\in{\mathbb{R}}\, , \\
& \Sigma = \begin{pmatrix}
\Sigma_{11} & \Sigma_{12} \\
\Sigma_{12}^T & \Sigma_{22}
\end{pmatrix}, \text{ where } \Sigma_{11}\in{\mathbb{R}}^{(n-1)\times (n-1)}, \Sigma_{12}\in{\mathbb{R}}^{n-1} \text{ and } \Sigma_{22}\in{\mathbb{R}}.\end{aligned}$$ The following algorithm generates a sample $\overline{X} \sim (X_1 | X_2 = x_2)$:
1. Sample $Z = (X_1, X_2)^T \sim N(\mu,\Sigma)$
2. Compute $\overline{X} = X_1 + \Sigma_{12}\Sigma_{22}^{-1}(x_2 - X_2)$.
Note that the computational effort to produce the conditioned Gaussian random variable $\overline{X}$ in Step 2 of Algorithm \[alg:doucet\] is linear in the dimension $n$. Thus, this algorithm significantly reduces the computation time of Step 3 of Algorithm \[alg:blanchet\] when that step is repeated for each replica.
Efficient grids for a broad class of stochastic processes {#s:application}
=========================================================
In this section we discuss how the idea of threshold-dependent grids can be applied to stochastic processes other than Brownian Motion. We let $(X_t)_{t\in[0,1]}$ be a real-valued stochastic process and $ t^*(b) := \operatorname*{arg\,max}_{t\in[0,1]} {\mathbb{P}}(X_t > b)$. For simplicity we here assume that $t\mapsto{\mathbb{P}}(X_t>b)$ is [continuous and strictly increasing]{} so that $t^*(b)=1$ (but situations in which $t^*(b)\in(0,1)$ can be dealt [with similarly, see also the discussion in Section \[s:discussion\]]{}).
As argued in the previous sections, it is efficient to let the position of the grid points depend on $b$. We constructed for Brownian Motion a grid by finding $T(b)=\{t_1(b),\ldots,t_n(b)\}$ such that $$\label{G1}{\mathbb P}\big( \tau_b\in({0},t_k(b)]\,|\, \tau_b\in(0,1]\big) =\frac{k}{n};$$ [cf. (\[THE\_grid3\]).]{} An inherent problem is that the class of processes for which the distribution of $\tau_b$ is known is very limited, so that the approach does not seem to be useful for relevant stochastic processes other than Brownian motion. We saw, however, that for Brownian Motion the $t_k(b)$ satisfying (\[G1\]) also solve $$\label{G2}\frac{{\mathbb P}(X_{t_k(b)}>b)}{{\mathbb P}(X_1>b)}=\frac{k}{n};$$ [cf. (\[THE\_grid4\])]{}. The idea now is to use the level-dependent (or: ‘equiprobable’) grid (\[G2\]) for general real-valued processes. The [major]{} advantage of the grid (\[G2\]) is that to calculate the position of the grid points $t_k$ the sole prerequisite is that the process’ [*marginals*]{} are known (rather than the distribution of $\tau_b$). In addition, even if the marginal distributions of $X_t$ are not available, but the [*asymptotics*]{} of $\mathbb{P}(X_t>b)$ (as $b\to\infty$) are, then a good approximation of this grid can be found. (In the sequel we write, for brevity, $T=\{t_1,\ldots,t_n\}$ instead of $T(b)=\{t_1(b),\ldots,t_n(b)\}$) and $t^*$ instead of $t^*(b).$)
We now provide the rationale behind the grid (\[G2\]). Let $T$ be a grid such that $t^*\in T$. Evidently, by the union bound, $${\mathbb{P}}(X_{t^*} > b) \leq w_T(b) \leq \sum_{t\in T} {\mathbb{P}}(X_t>b)$$ Now notice that if the grid $T$ is such that for $t\in T\setminus \{t^*\}$ $$\begin{aligned}
\label{def:as_no_contr}
{\mathbb{P}}(X_t>b) = o\big({\mathbb{P}}(X_{t^*}>b)\big), \ \ \text{ as } b\to\infty\end{aligned}$$ then it does not make sense to include the point $t$ [for large $b$]{}. Property (\[def:as\_no\_contr\]) clearly compromises the performance of equidistant grids as $b\to\infty.$ Considering however the grid points $t_k$ of the [threshold]{}-dependent grid, as defined by (\[G2\]), these will by design not experience (\[def:as\_no\_contr\]).
To assess the performance of the above threshold-dependent grid (\[G2\]), we introduce a measure of performance closely related to the relative bias. Note that when no formulas for $w(b)$ are available, nor it is known how to reliably approximate $w(b)$, we cannot determine the exact value of the relative bias. We now make the following two observations. (1) As $w_T(b)<w(b)$ for any choice of $T$, the larger $w_T(b)$ is, the better; if $w_{T_1}(b)>w_{T_2}(b)$ for grids $T_1,T_2$, then also $\beta_{T_1}(b)<\beta_{T_2}(b)$. (2) The crude lower bound $w(b)\geq{\mathbb{P}}(X_{t^*}>b)$ provides us with a useful benchmark. Combining these two thoughts motivates the following performance measure of a grid $T$: $$\begin{aligned}
\gamma_T(b) := \frac{w_T(b)}{{\mathbb{P}}(X_{t^*}>b)}\end{aligned}$$ Notice that for any $T$ such that $t^*\in T$ we have $$\gamma_T(b) \in\bigg[1,\frac{w(b)}{{\mathbb{P}}(X_{t^*}>b)}\bigg].$$ What is more, for any two grids $T_1,T_2$ we have $\gamma_{T_1}(b) \geq \gamma_{T_2}(b)$ if and only if $\beta_{T_1}(b) \leq \beta_{T_2}(b)$; this means that the bigger the $\gamma_T(b)$ is, the better. [As our main aim is to efficiently approximate $w(b)$ using discrete-time approximations $w_T(b)$, we see that if $\gamma_T(b)\approx 1$ then there is little gain from using $w_T(b)$ over a deterministic estimator ${\mathbb{P}}(X_{t^*}>b)$.]{}
In a series of examples we compare $\gamma_T(b)$ induced by (i) the threshold-dependent (equiprobable) grid and (ii) the equidistant grid of the same size; we consistently use $n=100$ grid points. In all cases $t\mapsto{\mathbb{P}}(X_t>b)$ is a continuous, strictly increasing function (so that $t^*=1$). The most important conclusion is that the experiments below uniformly indicate that the equiprobable grid outperforms the equidistant one, not only in the asymptotic regime, as threshold $b$ grows large, but already for moderate values of $b$. This shows how the ideas the we developed earlier this paper, that have provable optimality properties for Brownian motion, lead to an efficient estimation procedure for a much broader class of stochastic processes. In all examples, we observe that $\gamma_T(b)$ induced by the equidistant grid converges to $1$, [thus the corresponding $w_T(b)$ is asymptotically equivalent to ${\mathbb{P}}(X_{t^*}>b)$, as $b\to\infty$.]{}
*Let $(X_t)_{t\in[0,1]}$ be a Brownian Motion with jumps, i.e. $$\begin{aligned}
\label{def:bm_with_jumps}
X_t := B_t + N_t,\end{aligned}$$ where $B_t$ is a standard Brownian Motion and $N_t$ is a standard Poisson process with intensity $\lambda=1$.*
Even though there are no closed-form expressions for $w(b)$, it is still possible to generate exact samples from $\sup_{t\in[0,1]} X_t$ (see [@dkebicki2015queues Section 10.1]). We can use this to construct an unbiased estimator of $w(b)$ and thus can estimate the relative bias of the tested grids. The results in Figure \[fig:bm\_poiss\_bias\] show the substantial gain achieved by the level-dependent grid. The graphs look similar to those of Brownian Motion, which is indicative of the threshold-dependent grid having a uniformly bounded relative bias.
[.5]{} ![Brownian Motion with jumps. Plots of $\beta_n(b)$ (left) and $\gamma_n(b)$ (right) as a function of the threshold $b$ for threshold-dependent and equidistant grids of size $n=100$. []{data-label="fig:bm_poiss_bias"}](BMP_beta.eps "fig:"){width="1.00\linewidth"}
[.5]{} ![Brownian Motion with jumps. Plots of $\beta_n(b)$ (left) and $\gamma_n(b)$ (right) as a function of the threshold $b$ for threshold-dependent and equidistant grids of size $n=100$. []{data-label="fig:bm_poiss_bias"}](BMP_gamma.eps "fig:"){width="1.00\linewidth"}
\[ex:num\_exp2\][ *Let $(X_t)_{t\in[0,1]}$ be an Ornstein-Uhlenbeck process, i.e., a strong solution to the following SDE: with $X_0=0$, $$\begin{aligned}
dX_t &\,= -X_t\,dt + dW_t.\end{aligned}$$ Then $(X_t)_{t\in[0,1]}$ is a zero-mean Markovian Gaussian process with covariance function $$c(s,t) := {\mathbb C}{\rm ov}(X_s,X_t) = \frac{1}{2} \left( e^{-|t-s|} - e^{-(t+s)} \right).$$ [The exact value of $w(b)$ is known only in terms of special functions, see [@alili2005representations] and it is not straightforwardly evaluated. However, the exact asymptotics of $w(b)$, as $b$ grows large, [*are*]{} known:]{} $$\begin{aligned}
w(b) = C\, {\mathbb{P}}(X_1>b) (1+o(1)), \text{ as } b\to\infty\end{aligned}$$ where $C$ is a positive constant independent of $b$, [see e.g. [@dkebicki2003exact Theorem 5.1] or the original theorem by [@piterbarg1978asymptotic]]{}; this explains why for the level-dependent grid $\gamma_n(b)$ goes to a constant in Figure \[fig:ou\_bias\]. Again the equidistant grid is significantly outperformed by the threshold-dependent grid.*]{}
[.5]{} ![Ornstein-Uhlenbeck process. Plot of $\gamma_n(b)$ as a function of the threshold $b$ for threshold-dependent and equidistant grids of size $n=100$. Notice that with growing $b$ the equidistant estimator converges to ${\mathbb{P}}(X_1>b)$. []{data-label="fig:ou_bias"}](OU_beta.eps "fig:"){width="1\linewidth"}
[.75]{} ![Ornstein-Uhlenbeck process. Plot of $\gamma_n(b)$ as a function of the threshold $b$ for threshold-dependent and equidistant grids of size $n=100$. Notice that with growing $b$ the equidistant estimator converges to ${\mathbb{P}}(X_1>b)$. []{data-label="fig:ou_bias"}](OU_gamma.eps "fig:"){height="0.7\linewidth" width="1\linewidth"}
\[ex:num\_exp3\][ *Let $(X_t)_{t\in[0,1]}$ be a fractional Brownian Motion (fBM) with a Hurst parameter $H\in(0,1)$, that is a zero-mean Gaussian process with the covariance function $$\begin{aligned}
C_H(s,t) := {\mathbb C}{\rm ov}(X_s,X_t) = \frac{1}{2}\left(s^{2H} + t^{2H} - |t-s|^{2H}\right).\end{aligned}$$ Observe that fBM with Hurst parameter $H=1/2$ is a standard Brownian Motion. For any $H$ we have $C_H(t,t) = t^{2H}$ (strictly increasing variance in time) and thus $t^* = 1$.\
[The exact value of the probability $w(b)$ for $H\neq 1/2$ remains unknown. However, like in Example \[ex:num\_exp2\], the exact asymptotics of $w(b)$ are known:]{} $$\begin{aligned}
w(b) = \begin{cases}
C_H b^{1/H-2}{\mathbb{P}}(X_1>b)(1+o(1)), & \text{ for } H\in(0,\frac{1}{2}) \\
{\mathbb{P}}(X_1>b)(1+o(1)), & \text{ for } H\in(\frac{1}{2},1)
\end{cases}\end{aligned}$$ where $C_H$ is a constant only depending on $H$; [we again refer to [@dkebicki2003exact Theorem 5.1] or the original theorem by [@piterbarg1978asymptotic]]{}. We apply threshold-dependent grids in these two different asymptotic regimes for $H=0.4$ and $H=0.6$, see the results in Figure \[fig:fbm\_bias\]. Again the threshold-dependent grid performs considerably better. In case $H=0.4$ the above asymptotic result explains why for the level-dependent grid $\gamma_n(b)$ keeps increasing ($w(b)/{\mathbb P}(X_1>b)$ behaves as the increasing function $b^{1/H-2}$). In case $H=0.6$, again using the asymptotic result, [$\gamma_n(b) \to 1$ as $b$ grows large, both for the equidistant grid and for the threshold-dependent grid (equivalently, the relative bias vanishes for both as $b \to \infty$). Note however that with the threshold-dependent grid, $\gamma_n(b)$ tends to 1 slower than with the equidistant grid, as can be seen in Figure \[fig:fbm\_bias\] [(right panel)]{}, showing the more favorable performance of the threshold-dependent grid.]{}* ]{}
[.5]{} ![fBm with Hurst parameter $H=0.4$ (left) and $H=0.6$ (right). Plot of $\gamma_n(b)$ as a function of the threshold $b$ for threshold-dependent and equidistant grids of size $n=100$. []{data-label="fig:fbm_bias"}](fBM04_gamma.eps "fig:"){width="1.00\linewidth"}
[.5]{} ![fBm with Hurst parameter $H=0.4$ (left) and $H=0.6$ (right). Plot of $\gamma_n(b)$ as a function of the threshold $b$ for threshold-dependent and equidistant grids of size $n=100$. []{data-label="fig:fbm_bias"}](fBM06_gamma.eps "fig:"){width="1.00\linewidth"}
Concluding remarks and discussion {#s:discussion}
=================================
In this paper we have demonstrated that the errors due to time discretization when estimating threshold-crossing probabilities $w(b)$ can be significantly reduced by using other grids than the commonly used equidistant grid. We have analyzed this in considerable detail for the case of standard Brownian Motion. In particular, we have shown that in order to control the error as $b$ grows large, it suffices to properly *shift* the grid points instead of refining the grid with more and more points. At the same time, controlling the error using equidistant grids requires *quadratic* growth of the number of grid points, as $b$ grows large.
Numerical estimation is evidently not needed for Brownian Motion due to the availability of analytical results.
Our paper however indicates that the underlying ideas can be used to construct efficient grids for a broad class of stochastic processes (notably, [Lévy processes]{} and Gaussian processes, such as fractional Brownian Motion). The results presented in this paper are intended to develop valuable insight and useful heuristics for tackling the estimation of tail probabilities of these more general classes of processes. [We have demonstrated such heuristics for several processes in Section \[s:application\]. There,]{} we presented a procedure, that is empirically shown to work well for stochastic process $(X_t)_{t\in[0,1]}$ of which the marginal distributions are known:
- Identify $$t^*(b) := \operatorname*{arg\,max}_{t\in[0,1]} {\mathbb{P}}(X_t>b);$$ in case $(X_t)_{t\in[0,1]}$ is a zero-mean Gaussian process, $t^*$ is a point of maximal variance, i.e., $\operatorname*{arg\,max}_{t\in[0,1]} {\mathbb{V}\textnormal{\textrm{ar}}}\, X_t$. As argued, for many key models we have that $t^*=1.$
- Construct a grid $T = \{t_1,\ldots,t_n\}$ clustered around it, such that $t_k$ solves (\[G2\]), for $k\in\{1,\ldots,n\}$.
As we pointed out, even if the marginal distribution of $X_t$ is not available but only the corresponding asymptotics, as $b\to\infty$, this procedure can be applied.
It is also noted that it is straightforward to compare two different grids: the larger the value of $w_T(b)$, the closer it is to the target quantity $w(b)$.
A natural question that arises in relation to Theorem \[THEorem\] is whether we can find a grid that is even better than the one defined in (\[THE\_grid1\]). Constructing an *optimal n-grid* $T^*_n(b)$, i.e. a grid of size $n$ that minimizes the relative bias for a given $b$, remains elusive. However we have been able to find an explicit formula for an optimal 2-grid, namely $T^*_2(b) = \{t^*_1(b), t^*_2(b)\}$, with $$\begin{aligned}
t^*_1(b) = \frac{\pi b^2}{4} \left( \sqrt{1+\frac{8}{\pi b^2}} - 1\right), \ \ \ \ \text{and} \ \ \ \ t^*_2(b) = 1\end{aligned}$$ where ${\lim_{b\to \infty}}\beta_{T^*_2(b)}(b) = 1 - \frac{1}{2} \Phi(\sqrt{2/\pi}) - \frac{1}{4}e^{-1/\pi} \approx 0.4244$. For comparison, the [threshold-dependent]{} grid defined in (\[THE\_grid1\]) yields ${\lim_{b\to \infty}}\beta_2(b) = \frac{3}{8} + \frac{1}{2} \Phi(-\sqrt{2\log2}) \approx 0.4348$, hence the grid (\[THE\_grid1\]) is not minimizing the bias (although the difference with the optimal 2-grid is small). Additionally, we were able to prove that for an optimal n-grid, $T^*_n(b) = \{t_1^*(b),\ldots,t_n^*(b)\}$, the limits ${\lim_{b\to \infty}}b^2(1-t^*_k(b))$ must exist, and are all finite and pairwise distinct. As a result we were able to numerically calculate the limit ${\lim_{b\to \infty}}\beta_{T^*_3(b)} \approx 0.3796$. Finding optimal grids for larger $n$ remains an open problem. We note, however, that with the [threshold-dependent]{} grid we can bound the relative bias uniformly in $b$ (see Theorem \[THEorem\]) and in this sense the grid (\[THE\_grid1\]) is already (asymptotically) optimal.
Proofs of Lemmas \[lem:BM\_LB\], \[equi\_grid\_increasing\_b\], \[lemma2\_me\], \[lemming\] and Proposition \[prop:eq\] {#s:proofs}
=======================================================================================================================
In part (a) of Theorem \[thm\_eq\] it has been proven already that $\beta_n(b) \leq C_0 bn^{-1/2}$. Thus, when $b$ is fixed it is straightforward that the upper bound in the assertion of the theorem holds.\
The lower bound developed in Lemma \[lem:BM\_LB\] reads $\beta_n(b) \geq \frac{1}{2} \sum_{j=1}^{n-1} a_{j+1}\cdot w_j(b) + \frac{1}{2}w_n(b)$. Since we have $a_j < a_{j+1}$ for the equidistant grid and all $a_j$ and $w_j$ are non-negative, we may use the weaker inequality $$\beta_n(b) \geq \frac{1}{2}\sum_{j=2}^n a_j\cdot w_j(b).$$ In the following we use Lemma \[lem\_feller\] for a lower bound on terms $a_j$ and Result \[appendix:results\].\[mvt\_taub\] for a lower bound on $w_j$. $$\begin{aligned}
\nonumber \sum_{j=2}^{n} a_{j}\cdot w_j(b) & \geq \frac{b\,(3b + \sqrt{b^2 +8})}{8} \, \sum_{j=2}^{n} C_1^*(n-j+1)^{-1/2}\frac{\sqrt{n}}{j^{3/2}} \, e^{-\frac{b^2}{2} \, \left(\frac{n}{j-1}-1\right)} \\
\nonumber & \geq C \, n^{-1/2} \, \sum_{j=2}^{n} \frac{1}{n} \, \left( \frac{b}{\sqrt{1-\frac{j-1}{n}}} \, \left(\frac{j-1}{n}\right)^{-3/2} e^{-\frac{b^2}{2} \, \left( \frac{n}{j-1} - 1\right)} \right) \\
\label{riemann_sum_prop} & \geq C \, n^{-1/2} \, \int_0^1 \frac{b}{\sqrt{1-x}} \, x^{-3/2} \, e^{-\frac{b^2}{2}\, (1/x - 1)}\,dx \\
\nonumber & \geq C \, n^{-1/2},\end{aligned}$$ where $C$ is a positive constant independent of $n$ (but dependent on $b$) that may vary from line to line. To arrive at (\[riemann\_sum\_prop\]) we use the convergence of the Riemann sum, noting that $b$ is fixed and that the function $$f(x) := \frac{b}{\sqrt{1-x}} \, x^{-3/2} \, e^{-\frac{b^2}{2}\, (1/x - 1)}$$ is integrable on $(0,1)$. This concludes the proof.
Notice that the events $\{\sup_{t\in [0,1]} B_t > b\}$ and $\{\tau_b\in(0,1]\}$ are equivalent. We thus find $$\begin{aligned}
w(b)\,\beta_T(b) & = {\mathbb{P}}\Big(\sup_{t\in T} B_t < b, \sup_{t\in [0,1]} B_t > b \Big) = {\mathbb{P}}\Big(\sup_{t\in T} B_t < b, \tau_b\in[0,1] \Big) \\
& = \sum_{j=1}^n {\mathbb{P}}\Big(\sup_{t\in \{t_j, \ldots, t_{n}\}} B_t < b, \tau_b\in(t_{j-1},t_j] \Big) \\
& = \sum_{j=1}^n \int_{t_{j-1}}^{t_j} {\mathbb{P}}\Big( \sup_{t\in \{t_j, \ldots, t_{n}\}} B_t < b \mid B_s = b \Big)\,{\mathbb{P}}(\tau_b\in ds) \\
& = \sum_{j=1}^n \int_{t_{j-1}}^{t_j} {\mathbb{P}}\Big( B_{t_j-s} < 0, \ldots, B_{t_n-s} < 0 \Big)\,{\mathbb{P}}(\tau_b\in ds)\end{aligned}$$ To prove the upper bound we use the fact that ${\mathbb{P}}( B_{t_j-s} < 0, \ldots, B_{t_n-s} < 0)$ is a non-increasing function of $s \in [t_{j-1},t_j]$ (see Appendix \[appendix:grid\_transformations\], Transformation T\[pb\]), so that $$\begin{aligned}
w(b)\,\beta_T(b) & \leq \sum_{j=1}^n \int_{t_{j-1}}^{t_j} {\mathbb{P}}\Big( B_{t_j-t_{j-1}} < 0, \ldots, B_{t_n-t_{j-1}} < 0 \Big)\,{\mathbb{P}}(\tau_b\in ds) \\
& = \sum_{j=1}^n {\mathbb{P}}\Big( B_{t_j-t_{j-1}} < 0, \ldots, B_{t_n-t_{j-1}} < 0 \Big) \cdot {\mathbb{P}}\big(\tau_b\in (t_{j-1},t_j]\big).\end{aligned}$$ Dividing both sides of the inequality by $w(b) = {\mathbb{P}}(\tau_b\in(0,1])$ gives $\beta_T(b) \leq \bar{\beta}_T(b)$. To prove the lower bound we use Result \[appendix:results\].\[grid\_ineq\] from the Appendix, so as to obtain $$\begin{aligned}
\nonumber w(b)\,\beta_T(b) & = \sum_{j=1}^n \int_{t_{j-1}}^{t_j} {\mathbb{P}}\Big( B_{t_j-s} < 0, \ldots, B_{t_n-s} < 0 \Big)\,{\mathbb{P}}(\tau_b\in ds) \\
\label{przejscieLB} & \geq \sum_{j=1}^{n-1} \int_{t_{j-1}}^{t_j} \frac{1}{2}\,{\mathbb{P}}\Big( B_{t_{j+1}-t_{j}} < 0, \ldots, B_{t_n-t_{j}} < 0 \Big)\,{\mathbb{P}}(\tau_b\in ds) + \frac{1}{2}\,{\mathbb{P}}\big(\tau_b\in (t_{n-1},t_n]\big) \\
\nonumber & \geq \sum_{j=1}^{n-1} \frac{1}{2}\,{\mathbb{P}}\Big( B_{t_{j+1}-t_j} < 0, \ldots, B_{t_n-t_{j}} < 0 \Big) \cdot {\mathbb{P}}\big(\tau_b\in (t_{j-1},t_j]\big) + \frac{1}{2}\,{\mathbb{P}}\big(\tau_b\in (t_{n-1},t_n]\big).\end{aligned}$$ Dividing both sides of the inequality by $w(b)$ leads to $\beta_T(b) \geq \underbar{$\beta$}_T(b)$ and concludes the proof.
Recall the definitions of $a_j(b)$ and $w_j(b)$, and $\bar\beta_T(b) := \sum_{j=1}^n a_j(b) w_j(b)$. Notice that if we put $t_k = \frac{k}{n}$, then by the scaling property of Brownian Motion $$a_j(b) = {\mathbb{P}}\big( B_{1} < 0, \ldots, B_{1 + n-j} < 0 \big)$$ and thus $a_1 < a_2 < \ldots < a_n$ (since the $a_j(b)$s are independent of $b$, we abbreviate $a_j := a_j(b)$).\
\
Assume that for any $0 < b_1 < b_2$ there exists $k \in \{1,\ldots,n-1\}$ such that $$\begin{aligned}
\label{to_prove_funnylemma2}
w_j(b_1) \geq w_j(b_2), \text{ for } j\leq k \ \ \ \text{ and } \ \ \ w_j(b_1) \leq w_j(b_2), \text{ for } j> k.\end{aligned}$$ Since the weights $w_j(b)$ must satisfy $\sum_{j=1}^n w_j(b) = 1$ we have $\sum_{j=1}^n \big(w_j(b_2) - w_j(b_1)\big) = 0$ and thus $$\sum_{j=k+1}^{n} \big(w_j(b_2) - w_j(b_1)\big) = \sum_{j=1}^{k} \big(w_j(b_1) - w_j(b_2)\big).$$ Finally, $$\begin{aligned}
\bar\beta_T(b_2) - \bar\beta_T(b_1) & = \sum_{j=1}^n a_j \big(w_j(b_2) - w_j(b_1)\big) = \sum_{j=k+1}^n a_j \big(w_j(b_2) - w_j(b_1)\big) - \sum_{j=1}^{k} a_j \big(w_j(b_1) - w_j(b_2)\big) \\
& \geq a_{k+1} \sum_{j=k+1}^n \big(w_j(b_2) - w_j(b_1)\big) - a_{k} \sum_{j=1}^{k} \big(w_j(b_2) - w_j(b_1)\big) \\
& = \big(a_{k+1} - a_k\big) \sum_{j=k+1}^n \big(w_j(b_2) - w_j(b_1)\big) > 0.\end{aligned}$$ For the remainder of the proof we prove the existence of $k\in\{1,\ldots,n-1\}$ satisfying (\[to\_prove\_funnylemma2\]). Let $\tau_b := \inf\{t\geq 0 : B_t \geq b\}$ be the first hitting time of level $b$ and let $f(b,t)$ be the density of $\tau_b$ given that $\tau_b \leq 1$, i.e., $$\begin{aligned}
f(b,t) := \frac{b}{2\Phi(-b)}t^{-3/2} \phi\left(-\frac{b}{\sqrt{t}}\right),\end{aligned}$$ where $b>0$, $t\in(0,1)$, and $\phi(\cdot)$ denotes the density of a standard normal random variable. We will prove that for any $0 < b_1 < b_2$ there exists $t^*$ such that: $$\begin{aligned}
\label{funny_property}
f(b_1,t) > f(b_2,t), \text{ for } t\in(0,t^*) \ \ \ \text{ and } \ \ \ f(b_1,t) < f(b_2,t), \text{ for } t\in(t^*,1].\end{aligned}$$ Then the weights $$w_j(b) := \int_{t_{j-1}}^{t_j} f(b,t) \,dt$$ are decreasing for all $\frac{j}{n}\leq t^*$, and increasing for all $\frac{j}{n}\geq \frac{1}{n}+t^*$. If $nt^*$ is not an integer, it is not known whether $w_{[t^*n] +1}(b)$ increases or not, but for sure there exists $k \in \{1,\ldots,n-1\}$ satisfying (\[to\_prove\_funnylemma2\]). For the remainder we prove the existence of $t^*$ satisfying (\[funny\_property\]). For $t\in(0,1)$: $$\begin{aligned}
f(b_1,t) - f(b_2,t) & = \frac{1}{2}\,t^{-3/2} \, \left( \frac{b_1\,\phi\left(-\frac{b_1}{\sqrt{t}}\right)}{\Phi(-b_1)} - \frac{b_2\,\phi\left(-\frac{b_2}{\sqrt{t}}\right)}{\Phi(-b_2)} \right) \\
& = \frac{1}{2}\,t^{-3/2} \frac{b_2\,\phi\left(-\frac{b_2}{\sqrt{t}}\right)}{\Phi(-b_2)} \, \left( \frac{b_1\,\phi\left(-\frac{b_1}{\sqrt{t}}\right)\Phi(-b_2)}{b_2\,\phi\left(-\frac{b_2}{\sqrt{t}}\right)\Phi(-b_1)} - 1 \right) \\
& = \underbrace{\frac{1}{2}\,t^{-3/2} \frac{b_2\,\phi\left(-\frac{b_2}{\sqrt{t}}\right)}{\Phi(-b_2)}}_{> 0} \, \underbrace{\left( e^{\frac{b_2^2-b_1^2}{2t}}\,\frac{b_1\,\Phi(-b_2)}{b_2\,\Phi(-b_1)} - 1 \right)}_{=:g(t)}\end{aligned}$$ Note that $\lim_{t\to0^+}g(t) = + \infty$ and $g(1) < 0$ (for example due to the Result \[appendix:results\].\[res1\] in the Appendix) and that $g(\cdot)$ is strictly decreasing, hence $g(\cdot)$ has exactly one zero $t^*$ and $g(t) > 0$ for $t<t^*$ and $g(t) < 0$ for $t>t^*$. The observation that $\text{sign}(f(b_1,t) - f(b_2,t)) = \text{sign}(g(t))$ concludes the proof.
Let $h := \max_{k=1,\ldots,n}(t_k - t_{k-1})$. We transform the grid $T = \{t_1, \ldots, t_n\}$ with Transformations T\[pa\]–T\[pc\], see Appendix \[appendix:grid\_transformations\], in such a way that after all transformations we end up with $\{h, \ldots, Nh\}$.
1. Using Transformation T\[pb\], translate the grid to the right by $h-t_1$, i.e., put $$\begin{aligned}
t_j := t_j + h-t_1 \text { \ \ for all } j\in\{1,\ldots,n\}\end{aligned}$$
2. Put $\sigma_1 := 1$, $c_1 = 1$ and $k := 2$. While $k \leq N$ do:
- Put $\sigma_k := \inf\{j : t_j \geq kh\}$.
- Using Transformation T\[pc\], contract the grid after time $t_{\sigma_{k-1}}$ by a factor $c_k$, where $c_k$ is defined by ${h}/({t_{\sigma_k}-t_{\sigma_{k-1}}})$. Formally, we put $$\begin{aligned}
t_j := \begin{cases}
t_j, & j \in \{1,\ldots,\sigma_{k-1}\} \\
t_{\sigma_{k-1}} + c_k(t_j - t_{\sigma_{k-1}}), & j \in \{\sigma_{k-1}+1, \ldots, n\}
\end{cases}\end{aligned}$$ Notice that after this operation $t_{\sigma_k} = kh$.
- Put $k:=k+1$.
3. Using Transformation T\[pa\], delete all the points $t_k$ such that $t_k \not\in\{h, \ldots, hN\}$.
Now we prove that the algorithm is well-defined, more precisely, we confirm that all $\sigma_k$’s exist. First, see that $\sigma_1$ is well-defined. By induction, assume that $\sigma_k$ is well-defined and prove that $\sigma_{k+1}$ is well-defined as well. Notice that after the $k$th loop in Step 2 of the algorithm, the distances between the points shrunk at most by a factor $p_k = \prod_{j=1}^k c_j$ compared with the initial maximal distance $h$. Moreover, we observe that $$\begin{aligned}
\label{ckplus1}
c_{k} = \frac{h}{t_{\sigma_{k}}-t_{\sigma_{k-1}}} \geq \frac{h}{h + (t_{\sigma_{k}} - t_{\sigma_{k}-1})} \geq \frac{h}{h+\max_{j>\sigma_{k-1}} | t_{j+1} - t_{j}|} \geq \frac{1}{1 + \prod_{j=1}^{k-1} c_j}\end{aligned}$$ We prove by induction that $p_k = \prod_{j=1}^k c_j \geq \frac{1}{k}$ for all $k\in\{1,\ldots,N\}$. Obviously $p_2 = c_2 \geq \frac{1}{2}$. Assume that $p_{k-1} \geq \frac{1}{k-1}$. After multiplying inequality (\[ckplus1\]) by $p_{k-1}$ we obtain $$\begin{aligned}
p_{k} \geq \frac{p_{k-1}}{1 + p_{k-1}} = 1 - \frac{1}{1+p_{k-1}} \geq 1 - \frac{1}{1+\frac{1}{k-1}} = \frac{1}{k},\end{aligned}$$ which ends the inductive proof. Next, in order to show that $\sigma_{k+1}$ is well defined for $k\in\{1, \ldots N-1\}$ it suffices to prove that the endpoint $t_n$, after the $k$th loop of Step 2, is greater than $h(k+1)$. We prove a stronger statement, namely that the endpoint $t_n$ after being shrunk by a factor $p_k$ is still greater than $h(k+1)$, i.e. $h(k+1) \leq t_np_k$. By the definition of $N$, $h$ satisfies the inequality $h \leq {t_n}/{N^2}$, thus $$\begin{aligned}
h(k+1) \leq \frac{t_n(k+1)}{N^2} = \frac{t_n(k+1)}{N^2p_k}p_k = \frac{k(k+1)}{N^2} t_np_k \leq t_np_k,\end{aligned}$$ which concludes the proof that $\sigma_{k+1}$ is well-defined. As all transformations used in steps 1-3 satisfy (\[non-descreasity\_of\_n-grid\_transformations\]) we have $$\begin{aligned}
{\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_n} > 0\big) \leq {\mathbb{P}}\big(B_h > 0, \ldots, B_{Nh} > 0\big)\end{aligned}$$ We finish the proof by observing that ${\mathbb{P}}\big(B_h > 0, \ldots, B_{Nh} > 0\big) = {\mathbb{P}}\big(B_1 > 0, \ldots, B_N > 0\big)$, due to the scaling property of Brownian Motion.
Notice that the grid points $t^n_k(b)$ defined in (\[THE\_grid1\]) depend only on the threshold $b$ and the *ratio* $\frac{k}{n} \in [0,1]$. We are able to extend the definition of $t^n_k(b)$ to $t:(0,1]\times(0,\infty)\to[0,1]$, $$\begin{aligned}
t(s,b) := \Bigg(\frac{b}{\Phi^{-1}\big(s\,\Phi(-b)\big)} \Bigg)^{2}\end{aligned}$$ such that $t^n_k(b) = t(\frac{k}{n},b)$. Equivalently, $t(s,b)$ can be defined as the unique solution to $$\begin{aligned}
\label{general_tsb}
\Phi\left(-\frac{b}{\sqrt{t(s,b)}}\right) = s\,\Phi\left(-b\right)\end{aligned}$$ This extension makes it possible to inspect the derivative of $t^n_k(b)$ with respect to the ratio $\frac{k}{n}$. Using the extension function of $t^n_k(b)$, we aim to prove the more general statement that for $0<s_1<s_2<1$, $$\begin{aligned}
\label{tobeequiv}
\frac{1 - t(s_1,b)}{t(s_2,b) - t(s_1,b)} \geq \frac{-\log s_1}{\log s_2 - \log s_1} \ \iff \ \ \frac{1 - t(s_1,b)}{-\log s_1} \leq \frac{1 - t(s_2,b)}{-\log s_2}.\end{aligned}$$ Moreover, using the definition (\[general\_tsb\]) we may substitute $$s = \Phi\Big(-\frac{b}{\sqrt{t(s,b)}}\Big)/\Phi\left(-b\right)$$ and arrive at another equivalent form of inequality (\[tobeequiv\]): $$\begin{aligned}
\label{lhsfuntolemming}
\frac{1 - t(s_1,b)}{\log\big( \Phi(-b) \big) - \log\big( \Phi(-b/\sqrt{t(s_1,b)})\big)} \leq \frac{1 - t(s_2,b)}{\log\big( \Phi(-b) \big) - \log\big( \Phi(-b/\sqrt{t(s_2,b)})\big)} \, ,\end{aligned}$$ which is Result \[appendix:results\].\[superresult\] in the Appendix. For part (b) see that the density of the first hitting time, $$\begin{aligned}
{\mathbb{P}}(\tau_b\in ds) = \frac{b}{\sqrt{2\pi}}s^{-3/2}e^{-b^2/(2s)}\,ds, \text{ \ for } s>0\end{aligned}$$ is an increasing function on the interval $s\in[0,\frac{b^2}{3}]$ and thus part (b) follows from the second definition of the grid points $t^n_k(b)$ in (\[THE\_grid2\]).
Grid transformations {#appendix:grid_transformations}
====================
Let $T = \{t_1, \ldots, t_n\}$ with $0 < t_1 < \ldots < t_n < \infty$. We introduce three *grid transformations*, i.e. operations $T \mapsto \widetilde{T}$ satisfying $$\begin{aligned}
\label{non-descreasity_of_n-grid_transformations}
{\mathbb{P}}\big(B_t > 0 \text{ for all } t\in T\big) \leq {\mathbb{P}}\big(B_t > 0 \text{ for all } t\in \widetilde{T}\big).\end{aligned}$$
1. \[pa\] **Deleting**. For any $k\in\{1,\ldots,n\}$ $$\begin{aligned}
{\mathbb{P}}\Big(B_{t_1} > 0, \ldots, B_{t_n} > 0\Big) \leq {\mathbb{P}}\Big(B_{t_1} > 0,\ldots,B_{t_{k-1}} > 0,B_{t_{k+1}} > 0,\ldots,B_{t_n} > 0\Big)\end{aligned}$$
2. \[pb\] **Translation to the right of the whole sequence**. For any $s>0$ $$\begin{aligned}
{\mathbb{P}}\Big(B_{t_1} > 0, \ldots, B_{t_n} > 0\Big) \leq {\mathbb{P}}\Big(B_{t_1+s} > 0, \ldots, B_{t_n+s} > 0\Big)\end{aligned}$$
3. \[pc\] **Contraction of time after some point**. For any $k\in\{1,\ldots,n-1\}$ and $c\in(0,1)$: $$\begin{aligned}
{\mathbb{P}}\Big(B_{t_1} > 0, \ldots, B_{t_n} > 0\Big) \leq {\mathbb{P}}\Big(B_{t_1} > 0, \ldots,B_{t_k} > 0, B_{t_k + c(t_{k+1}-t_k)} > 0,\ldots,B_{t_k + c(t_n-t_k)} > 0\Big)\end{aligned}$$
Assertion T\[pa\] is straightforward to verify. Observe for T\[pb\] that $$\begin{aligned}
{\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_n} > 0\big) & = \int_0^\infty {\mathbb{P}}\big(B_{t_2} > 0, \ldots, B_{t_n} > 0 \mid B_{t_1} = x\big)\frac{1}{\sqrt{2\pi t_1}} e^{-x^2/(2t_1)}\,dx \\
& = \int_0^\infty {\mathbb{P}}\big(B_{t_2-t_1} < x, \ldots, B_{t_n-t_1} < x\big)\frac{1}{\sqrt{2\pi t_1}} e^{-x^2/(2t_1)}\,dx \\
& = \int_0^\infty {\mathbb{P}}\big(B_{t_2-t_1} < y\sqrt{t_1}, \ldots, B_{t_n-t_1} < y\sqrt{t_1} \big)\frac{1}{\sqrt{2\pi}} e^{-y^2/2}\,dy \\
& \leq \int_0^\infty {\mathbb{P}}\big(B_{t_2-t_1} < y\sqrt{t_1+s}, \ldots, B_{t_n-t_1} < y\sqrt{t_1+s} \big)\frac{1}{\sqrt{2\pi}} e^{-y^2/2}\,dy \\
& = \int_0^\infty {\mathbb{P}}\big(B_{t_2-t_1} < x, \ldots, B_{t_n-t_1} < x\big)\frac{1}{\sqrt{2\pi (t_1+s)}} e^{-x^2/(2(t_1+s))}\,dx \\
& = {\mathbb{P}}\big(B_{t_1+s} > 0, \ldots, B_{t_n+s} > 0\big)\end{aligned}$$ and for T\[pc\] that $$\begin{aligned}
& {\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_n} > 0\big) \\
& = \int_0^\infty {\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_{k-1}} > 0 \mid B_{t_k} = x\big)\, {\mathbb{P}}\big(B_{t_{k+1}} > 0, \ldots, B_{t_n} > 0 \mid B_{t_k} = x\big)\frac{1}{\sqrt{2\pi t_k}} e^{-x^2/(2t_k)}\,dx \\
& = \int_0^\infty {\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_{k-1}} > 0 \mid B_{t_k} = x\big)\, {\mathbb{P}}\big(B_{t_{k+1}-t_k} < x, \ldots, B_{t_n-t_k} < x \big)\frac{1}{\sqrt{2\pi t_k}} e^{-x^2/(2t_k)}\,dx \\
& \leq \int_0^\infty {\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_{k-1}} > 0 \mid B_{t_k} = x\big)\, {\mathbb{P}}\left(B_{t_{k+1}-t_k} < \frac{x}{\sqrt{c}}, \ldots, B_{t_n-t_k} < \frac{x}{\sqrt{c}} \right)\frac{1}{\sqrt{2\pi t_k}} e^{-x^2/(2t_k)}\,dx \\
& = \int_0^\infty {\mathbb{P}}\big(B_{t_1} > 0, \ldots, B_{t_{k-1}} > 0 \mid B_{t_k} = x\big)\, {\mathbb{P}}\big(B_{c(t_{k+1}-t_k)} < x, \ldots, B_{c(t_n-t_k)} < x \big)\frac{1}{\sqrt{2\pi t_k}} e^{-x^2/(2t_k)}\,dx \\
& = {\mathbb{P}}\big(B_{t_1} > 0, \ldots,B_{t_k} > 0, B_{t_k + c(t_{k+1}-t_k)} > 0,\ldots,B_{t_k + c(t_n-t_k)} > 0\big)\end{aligned}$$
Miscellaneous results {#appendix:results}
=====================
Let $\Phi(\cdot)$ denote the standard normal cumulative distribution function and $\phi(\cdot)$ the standard normal density function. Below we list various results that we use. Results \[appendix:results\].\[ineq1\]–\[appendix:results\].\[ineq2\] are standard, and not proven here.
1. \[ineq1\] For $x>0$: $$\begin{aligned}
\frac{x}{1+x^2} \leq \frac{\Phi(-x)}{\phi(x)} \leq \frac{1}{x}\end{aligned}$$
2. \[limit1\] As $x\to\infty$, $$\lim_{x\to\infty} \frac{\Phi(-x)}{\frac{1}{x}\phi(x)} \to 1.$$
3. \[ineq2\] [@szarek1999nonsymmetric]. For $x>-1$: $$\begin{aligned}
\frac{2}{x + (x^2+4)^{1/2}} \leq \frac{\Phi(-x)}{\phi(x)} \leq \frac{4}{3x + (x^2+8)^{1/2}}\end{aligned}$$
4. \[grid\_ineq\] Let $0 < t_1 < \ldots < t_n < \infty$, then: $$\begin{aligned}
{\mathbb{P}}\Big(B_{t_1} > 0, \ldots, B_{t_n} > 0\Big) \geq \frac{1}{2}\,{\mathbb{P}}\Big(B_{t_2-t_1} > 0, \ldots, B_{t_n-t_1} > 0\Big)\end{aligned}$$
5. \[mvt\_taub\] Let $T = \{t_1, \ldots, t_n\}$, where $t_j := \frac{j}{n}$, $\tau_b := \inf\{t\geq 0 : B_t \geq b\}$ and $b > 0$, then: $$\begin{aligned}
{\mathbb{P}}\Big(\tau_b\in (t_{j-1},t_j] \ \big| \ \tau_b\in(0,1]\Big) \leq \frac{b\,(b + \sqrt{b^2 +4})}{4} \cdot \frac{\sqrt{n}}{(j-1)^{3/2}} e^{-\frac{b^2}{2} \cdot \left(\frac{n}{j}-1\right)}\end{aligned}$$ and $$\begin{aligned}
{\mathbb{P}}\Big(\tau_b\in (t_{j-1},t_j] \ \big| \ \tau_b\in(0,1]\Big) \geq \frac{b\,(3b + \sqrt{b^2 +8})}{8} \cdot \frac{\sqrt{n}}{j^{3/2}} e^{-\frac{b^2}{2} \cdot \left(\frac{n}{j-1}-1\right)}\end{aligned}$$ for $j \in \{2, \ldots n\}$.
6. \[res\_increasing\] Let $f:(0,\infty)\times(0,1)\to(0,\infty)$ such that $$\begin{aligned}
f(b,x) := \frac{b}{\sqrt{1-x}}x^{-3/2}e^{-\frac{b^2}{2}(1/x-1)}\end{aligned}$$ Then $f(b,x)$ is an increasing function of $x$, when $b\geq 1$.
7. \[res1\] Let $f:(0,\infty)\to(0,\infty)$ such that $$\begin{aligned}
f(x) := \frac{\Phi(-x)}{\phi(x)}\end{aligned}$$ Then $f$ is a strictly decreasing function.
8. \[res2\] Let $f:(0,\infty)\to(0,\infty)$ such that $$\begin{aligned}
f(x) := \frac{\Phi(-x)}{\frac{1}{x}\phi(x)}\end{aligned}$$ Then $f$ is a strictly increasing function.
9. \[superresult\] Let $f:[0,1]\to[0,\infty)$ be such that $$\begin{aligned}
f(t):= \begin{cases}
0, & t=0; \\
\vspace{-4mm}\\
\frac{\displaystyle 1-t}{\displaystyle \log\big( \Phi(-b) \big) - \log\big( \Phi(-b/\sqrt{t})\big)}, & t\in(0,1); \\
\vspace{-4mm}\\
\frac{\displaystyle 2\,\Phi(-b)}{\displaystyle b\,\phi(b)}, & t = 1.
\end{cases}\end{aligned}$$ Then $f$ is continuous and increasing.
Proofs of results \[appendix:results\].\[grid\_ineq\]–\[appendix:results\].\[superresult\]
------------------------------------------------------------------------------------------
The proof is very similar to the proofs from Appendix \[appendix:grid\_transformations\]. Note that $$\begin{aligned}
{\mathbb{P}}\Big(B_{t_1} > 0, \ldots, B_{t_n} > 0\Big) & = \int_0^\infty {\mathbb{P}}\Big(B_{t_2} > 0, \ldots, B_{t_n} > 0 \mid B_{t_1} = x \Big)\frac{1}{\sqrt{2\pi t_1}} e^{-x^2/(2t_1)}\,dx \\
& = \int_0^\infty {\mathbb{P}}\Big(B_{t_2-t_1} < x, \ldots, B_{t_n-t_1} < x \Big)\frac{1}{\sqrt{2\pi t_1}} e^{-x^2/(2t_1)}\,dx \\
& \geq \int_0^\infty {\mathbb{P}}\Big(B_{t_2-t_1} < 0, \ldots, B_{t_n-t_1} < 0 \Big)\frac{1}{\sqrt{2\pi t_1}} e^{-x^2/(2t_1)}\,dx \\
& = \frac{1}{2}\,{\mathbb{P}}\Big(B_{t_2-t_1} > 0, \ldots, B_{t_n-t_1} > 0\Big),\end{aligned}$$ which concludes the proof.
Using the mean value theorem and monotonicity of $\phi(\cdot)$ on the negative half-line, we have $|\Phi(-x) - \Phi(-y)| \leq |x-y| \cdot \phi(-y)$ for $0<y<x$. Furthermore, $$\begin{aligned}
{\mathbb{P}}\Big(\tau_b\in (t_{j-1},t_j] \Big) = 2\Phi(-b/\sqrt{t_j}) - 2\Phi(-b/\sqrt{t_{j-1}}) \leq 2\,\left(\frac{b}{\sqrt{t_{j}}} - \frac{b}{\sqrt{t_{j}}}\right) \, \phi\left(\frac{b}{\sqrt{t_{j}}}\right)\end{aligned}$$ Thus, for $b > 0$ and $j\in\{2,\ldots,n\}$, after substituting $t_j = \frac{j}{k}$, the above combined with the inequality \[appendix:results\].\[ineq2\] yield: $$\begin{aligned}
{\mathbb{P}}\Big(\tau_b\in (t_{j-1},t_j] \ \big| \ \tau_b\in(0,1]\Big) & \leq \frac{1}{\Phi(-b)}\,\left(\frac{b}{\sqrt{t_{j-1}}} - \frac{b}{\sqrt{t_{j}}}\right) \, \phi\left(\frac{b}{\sqrt{t_{j}}}\right) \\
& = \frac{b \sqrt{n}}{\sqrt{2\pi} \Phi(-b)} \, \frac{\sqrt{j}-\sqrt{j-1}}{\sqrt{(j-1)j}} e^{-b^2n/(2j)} \\
& \leq \frac{b \sqrt{n}}{2\sqrt{2\pi} \Phi(-b)} \, \frac{1}{(j-1)^{3/2}} e^{-b^2n/(2j)} \\
& \leq \frac{b\,(b + \sqrt{b^2 +4})}{4} \, \frac{\sqrt{n}}{(j-1)^{3/2}} e^{-\frac{b^2}{2} \, \left(\frac{n}{j}-1\right)}\end{aligned}$$ The proof of the second inequality is analogous.
It suffices to prove that $\frac{d}{dx}f(b,x) \geq 0$ for $b\geq 1$. See that $$\begin{aligned}
\frac{d}{dx}f(b,x) & = be^{b^2/2} \, \frac{d}{dx}\, \frac{e^{-b^2/(2x)}}{\sqrt{1-x}\, x^{3/2}} \\
& = \frac{be^{b^2/2}}{(1-x)x^3} \, \left( \frac{b^2}{2x^2}e^{-b^2/(2x)}\sqrt{1-x}\, x^{3/2} - e^{-b^2/(2x)} \, \left( -\frac{x^{3/2}}{2\sqrt{1-x}} + \frac{3}{2}\sqrt{x(1-x)} \right) \right) \\
& = \frac{be^{b^2/2\,(1-1/x)}}{2(1-x)^{3/2}x^{7/2}} \, \left( b^2(1-x) + x^2 - 3x(1-x) \right) \\
& = \underbrace{\frac{be^{b^2/2\,(1-1/x)}}{2(1-x)^{3/2}x^{7/2}}}_{>0} \, \big( \underbrace{4x^2-(b^2+3)x+b^2}_{=:g(x)} \big)\end{aligned}$$ Note that $g(x)$ has at most one root when $b\in[1,3]$ thus $g(x) \geq 0$ for $b\in[1,3]$. Moreover, when $b>3$, then $g'(x) = 8x - (b^2+3) < -1$ (for $x\in[0,1]$) thus $g(x)$ is strictly decreasing for $x\in[0,1]$. From the observation that $g(0) = b^2>0$ and $g(1) = 1 >0$ we conclude that $g(x)$ is nonnegative on the interval $[0,1]$ for $b\geq1$ and thus $\frac{d}{dx} f(b,x) \geq 0$, when $b\geq 1$.
We have that $$\begin{aligned}
f'(x) = \frac{-\phi(x) + \Phi(-x)\, x}{\phi(x)},\end{aligned}$$ thus $f'(x) \leq 0$ iff $-\phi(x) + \Phi(-x)\, x \leq 0$, which is equivalent to Result \[appendix:results\].\[ineq1\].
See that $$\begin{aligned}
f'(x) = \frac{\Phi(-x) - x\phi(x) + x^2\Phi(-x)}{\phi(x)},\end{aligned}$$ thus $f'(x) \geq 0$ iff $\frac{\Phi(-x)}{\phi(x)} \geq \frac{x}{1+x^2}$, which is an implication of the lower bound from result \[appendix:results\].\[ineq2\].
It is easy to see that $\lim_{t\to0^+} f(t) = 0$. To see that $\lim_{t\to 1^-} f(t) = \frac{2\,\Phi(-b)}{b\,\phi(b)}$ we expand $\log\big(\Phi(-b/\sqrt{t})\big)$ in a series around $t_0 = 1$ and obtain $$\begin{aligned}
\log\big(\Phi(-b/\sqrt{t})\big) = \log\big(\Phi(-b)\big) + \frac{b\,\phi(b)}{2\Phi(-b)}(t-1) + o(t-1)\end{aligned}$$ Thus $$\begin{aligned}
\lim_{t\to1^-} f(t) = \lim_{t\to1^-} \frac{1-t}{\frac{b\,\phi(b)}{2\Phi(-b)}(1-t) + o(t-1)} = \frac{2\,\Phi(-b)}{b\,\phi(b)}.\end{aligned}$$ To prove that $f$ is increasing we study the first derivative. For $t\in(0,1)$: $$\begin{aligned}
\nonumber\frac{d}{dt}\,f(t) & = \frac{- \log\left( \frac{\Phi(-b)}{\Phi(-b/\sqrt{t})}\right) + \frac{(1-t)\Phi(-b/\sqrt{t})\Phi(-b)}{\Phi(-b)} \cdot \frac{bt^{-3/2}}{2\Phi(-b/\sqrt{t})^2}\phi(-b/\sqrt{t}) }{\Big(\log\big( \Phi(-b) \big) - \log\big( \Phi(-b/\sqrt{t})\big)\Big)^2} \\
\label{nominator_line}& = \frac{ \log\left( \frac{\Phi(-b/\sqrt{t})}{\Phi(-b)}\right) + \frac{b(1-t)}{2t^{3/2}} \cdot \frac{\phi(-b/\sqrt{t})}{\Phi(-b/\sqrt{t})} }{\Big(\log\big( \Phi(-b) \big) - \log\big( \Phi(-b/\sqrt{t})\big)\Big)^2}\end{aligned}$$ Due to Result \[appendix:results\].\[res1\] we have the lower bound $$\frac{\phi(-b/\sqrt{t})}{\Phi(-b/\sqrt{t})} \geq \frac{\phi(-b)}{\Phi(-b)}$$ and thus the numerator of the fraction in (\[nominator\_line\]) can be bounded from below by the function $g:(0,1)\to{\mathbb{R}}$ defined as below: $$\begin{aligned}
g(t) := \log\left( \frac{\Phi(-b/\sqrt{t})}{\Phi(-b)} \right) + \frac{b(1-t)}{2\,t^{3/2}} \, \frac{\phi(b)}{\Phi(-b)}
$$ Notice that $g(t)\geq 0$ implies $\frac{d}{dt}f(t)\geq 0$ which is exactly what we want to establish. For the remainder of the proof we show that $g(t)$ is non-negative. Since $\lim_{t\to0^+}g(t) = +\infty$ and $g(1) = 0$, it suffices to show that $g'(t)$ is monotone (non-increasing). We study the first derivative $$\begin{aligned}
g'(t) & = \frac{b}{2t^{3/2}} \frac{\phi(b/\sqrt{t})}{\Phi(-b/\sqrt{t})} + \frac{b}{4t^{3/2}} \frac{\phi(b)}{\Phi(-b)} - \frac{3b}{4t^{5/2}} \frac{\phi(b)}{\Phi(-b)} \\
& = \frac{b^2}{4t^2} \left( 2\, \frac{\frac{\sqrt{t}}{b}\phi(b/\sqrt{t})}{\Phi(-b/\sqrt{t})} + \left(t^{1/2} - 3t^{-1/2} \right) \, \frac{\frac{1}{b}\phi(b)}{\Phi(-b)}\right) \\
& \leq \frac{b^2}{4t^2} \left( 2\, \frac{\frac{\sqrt{t}}{b}\phi(b/\sqrt{t})}{\Phi(-b/\sqrt{t})} -2 \, \frac{\frac{1}{b}\phi(b)}{\Phi(-b)}\right) \leq 0,\end{aligned}$$ where the last inequality is a consequence of the application of Result \[appendix:results\].\[res2\], that is $$t \longmapsto \frac{\frac{\sqrt{t}}{b}\phi(b/\sqrt{t})}{\Phi(-b/\sqrt{t})}$$ is an increasing function of $t$.
**Acknowledgments**. [The authors would like to thank Ankush Agarwal and Johan van Leeuwaarden for useful discussions and suggestions. In addition, the reviewers’ reports helped improving the quality of our work considerably.]{} This work is part of the research programme ‘Rare Event Simulation for Climate Extremes’ with grant number 657.014.033, which is (partly) funded by the Netherlands Organisation for Scientific Research (NWO). Michel Mandjes’ research is partly funded by the NWO Gravitation Programme NETWORKS, grant number 024.002.003.
[^1]: Email: bisewski@cwi.nl
[^2]: [As $b\to\infty$, both $w_T(b)$ and $w(b)$ tend to $0$, so that the *absolute* bias is not a meaningful accuracy measure.]{}
[^3]: In this context *uniform control* means that for a fixed $\varepsilon>0$, we have that $\beta_T(b) < \varepsilon$ for all $b>0$; the grid $T$ can change in $b$.
|
---
abstract: 'The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring on even-dimensional generators. In this paper we review the cases in which the conjecture is known to hold and we highlight the properties that permit one to prove the conjecture in these cases.'
address: |
Departamento de Matemáticas y Estadística\
Universidad del Norte\
Km. 5 via Puerto Colombia, Barranquilla, Colombia
author:
- Bernardo Uribe
bibliography:
- 'Evenness-bibliography.bib'
title: The evenness conjecture in equivariant unitary bordism
---
Introduction {#introduction .unnumbered}
============
The $G$-equivariant unitary bordism groups for $G$ a compact Lie group are the bordism groups of $G$-equivariant tangentially stable almost complex manifolds, also known as $G$-equivariant unitary manifolds. These are closed $G$-manifolds $M$ for which a stable tangent bundle $TM \oplus \underline{{\mathbb{R}}}^k$, where $\underline{{\mathbb{R}}}^k$ denotes the trivial bundle ${\mathbb{R}}^k \times M$ with trivial $G$-action, can be endowed with the structure of a $G$-equivariant complex bundle. Two tangentially stable almost complex $G$-structures are identified if after stabilization with further $G$-trivial $\underline{{\mathbb{C}}}$ summands the structures become $G$-homotopic through complex $G$-structures. Being unitary is inherited by the fixed points sets. Whenever $H$ is a closed subgroup of $G$ the fixed points $M^H$ are also tangentially stable almost complex, and moreover a $N_{H}$-tubular neighbourhood around $M^H$ in $M$ possesses a complex $N_{H}$ structure [@May-book §XVIII, Proposition 3.2].
For a cofibration of $G$-spaces $Y \to X$, the geometric $G$-equivariant unitary bordism groups $\Omega^G_n(X,Y)$ are the $G$-bordism classes of $G$-equivariant $n$-dimensional manifolds with map $(M^n, \partial M^n) \to (X,Y)$.
The $G$-equivariant unitary bordism groups of a point $\Omega_*^G$ become a ring under the cartesian product of manifolds with the diagonal $G$-action, and therefore a module over the unitary bordism ring $\Omega_*$ where we consider a unitary manifold as a trivial $G$-manifold. Milnor and Novikov [@Milnor; @Novikov], making use of the Adams spectral sequence, showed that the unitary bordism ring is a polynomial ring $\Omega_*= {\mathbb{Z}}[x_{2i} \colon i \geq 1]$ with one generator in each even degree. In this work we will be interested in the $\Omega_*$-module structure of the equivariant unitary bordism groups $\Omega_*^G$.
Explicit calculations carried out by Landweber [@Landweber-cyclic] in the cyclic case and by Stong [@Stong-complex] in the abelian $p$-group case permitted them to conclude that in these cases the equivariant unitary bordism group $\Omega_*^G$ is a free $\Omega_*$-module on even-dimensional generators. Ossa in [@Ossa] generalized this result to any finite abelian group and Löffler in [@Loffler] and Comezaña in [@May-book §XXVIII, Theorem 5.1] showed that this also holds whenever $G$ is a compact abelian Lie group. Explicit calculations done for the dihedral groups $D_{2p}$ with $p$ prime by Ángel, Gómez and the author [@AngelGomezUribe], for groups of order $pq$ with $p$ and $q$ different primes by Lazarov [@Lazarov] and for groups on which all its Sylow subgroups are cyclic by Rowlett [@Rowlett-metacyclic] show that for these groups this phenomenon also occurs. We believe that this property should hold in the $G$-equivariant unitary bordism groups for any compact Lie group $G$, in the same way that the coefficients for $G$-equivariant K-theory are trivial in odd degrees and a free module over the integers on even degrees.
The theme of this work is the
[**Evenness conjecture for the equivariant unitary bordism groups**]{}
which states that the $G$-equivariant unitary bordism group is a free $\Omega_*$-module on even-dimensional generators whenever $G$ is a compact Lie group. Rowlett explicitly mentions this conjecture in his work of 1980 [@Rowlett-metacyclic] and later Comezaña in his work of 1996 [@May-book §XXVIII.5]. We also believe that this conjecture holds in general and we do hope that this paper will help to spread it to the mathematical community for its eventual proof.
In this work we survey the original proofs of the known cases of the evenness conjecture for finite groups. We start in section \[section bordism\] with the definition of the equivariant unitary bordism groups for pairs of families and the long exact sequence associated to them. In section \[section bundles\] we review the decomposition of equivariant complex vector bundles restricted to fixed point sets done by Ángel, Gómez and the author [@AngelGomezUribe] and how this decomposition allows one to write the equivariant unitary bordism groups of adjacent pair of families as the bordism groups of equivariant classifying spaces. In section \[section abelian\] we review the proofs of the evenness conjecture done by Landweber [@Landweber-cyclic] for cyclic groups, by Stong [@Stong-complex] for abelian $p$-groups and by Ossa [@Ossa] for general finite abelian groups. In section \[section non abelian\] we review the proof of the evenness conjecture for groups for which all its Sylow subgroups are cyclic done by Rowlett [@Rowlett-metacyclic] and we finish in section \[section conclusion\] with some conclusions.
We would like to thank Prof. Peter Landweber for reading this manuscript and for providing valuable comments and suggestions that improved the presentation of this survey.
Equivariant unitary bordism for families of subgroups {#section bordism}
=====================================================
To study the equivariant bordism groups Conner and Floyd introduced the study of bordism groups of manifolds with prescribed isotropy groups [@ConnerFloyd-Odd §5]. A family of subgroups ${\mathcal{F}}$ of $G$ is a set of subgroups of $G$ which is closed under taking subgroups and under conjugation. The classifying space for the family $E{\mathcal{F}}$ is a $G$-space which is terminal in the category of ${\mathcal{F}}$-numerable $G$-spaces [@tomDieck-transformation §1, Theorem 6.6] and characterized by the following properties on fixed point sets: $E{\mathcal{F}}^H\simeq*$ if $H \in {\mathcal{F}}$ and $E{\mathcal{F}}^H = \emptyset$ if $H \notin {\mathcal{F}}$. This classifying space may be constructed in such a way that whenever ${\mathcal{F}}' \subset {\mathcal{F}}$, the induced map $E {\mathcal{F}}' \to E{\mathcal{F}}$ is a $G$-cofibration.
The equivariant unitary bordism groups for pairs of families may be defined as follows $$\Omega_*^G[{\mathcal{F}},{\mathcal{F}}'](X,A):= \Omega_*^G(X \times E{\mathcal{F}},X \times E{\mathcal{F}}' \cup A \times E{\mathcal{F}}'),$$ see [@tomDieck-Orbit-I page 310], or alternatively they may be defined in a geometric way as in [@Stong-complex §2].
A $({\mathcal{F}}, {\mathcal{F}}')$ [*free geometric unitary bordism element*]{} of $(X,A)$ is an equivalence class of 4-tuples $(M,M_0,M_1,f)$, where:
- $M$ is an $n$-dimensional $G$-manifold endowed with tangentially stable almost $G$ structure which is moreover ${\mathcal{F}}$-free, i.e. such that all isotropy groups $G_m=\{ g \in G \ | \ gm=m \}$ for $m \in M$ belong to ${\mathcal{F}}$, and such that $f:M \to X$ is $G$-equivariant; and
- $M_0, M_1$ are compact submanifolds of the boundary of $M$, with $\partial M = M_0 \cup M_1$, $M_0 \cap M_1 = \partial M_0=\partial M_1$ having tangentially stable almost complex structures induced from $M$, both $G$-invariant, such that $f(M_1) \subset A$ and $M_0$ if ${\mathcal{F}}'$-free, i.e. all isotropy groups of $M_0$ belong to ${\mathcal{F}}'$.
Two four-tuples $(M,M_0,M_1,f)$ and $(M',M'_0,M'_1,f)$ are equivalent if there is a 5-tuple $(V,V^+,V_0,V_1,F)$ where
- $V$ is a ${\mathcal{F}}$-free manifold and $F : V \to X$ is a $G$-equivariant map;
- The boundary of $V$ is the union of $M$, $M'$ and $V^+$ with $M \cap V^+=\partial M$, $M' \cap V^+= \partial M'$, $M \cap M' = \emptyset$, $V^+ \cap(M \cup M') = \partial V^+$, with $V$ inducing the tangentially stable almost complex $G$-structure on $M$ and the opposite one on $M'$; $V^+$ is $G$-invariant and $F$ restricts to $f$ in $M$ and to $f'$ on $M'$; and
- $V^+$ is the union of the $G$-invariant submanifolds $V_0$, $V_1$ with intersection a submanifold $V^-$ in their boundaries, such that $\partial V_i=M_i \cup V^-\cup M_i'$, $M_i \cap V^-= \partial M_i$, $M'_i \cap V^-= \partial M'_i$ with $V_0$ is ${\mathcal{F}}'$-free and $F(V_1) \subset A$.
The set of equivalence classes of $n$-dimensional $({\mathcal{F}}, {\mathcal{F}}')$-free geometric unitary bordism elements of $(X,A)$, consisting of classes $(M,M_0,M_1,f)$ where the dimension of $M$ is $n$, and under the operation of disjoint union, forms an abelian group denoted by $$\Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X,A).$$ Call these groups the geometric $G$-equivariant unitary bordism groups of the pair $(X,A)$ restricted to the pair of families ${\mathcal{F}}' \subset {\mathcal{F}}$.
Note that if $N$ is a tangentially stable almost complex closed manifold, we can define $N \cdot (M,M_0,M_1,f) = (N \times M,N \times M_0,N \times M_1,f \circ \pi_M)$ thus making $\Omega^G_* \{{\mathcal{F}},{\mathcal{F}}' \}(X,A)$ a graded module over the unitary bordism ring $\Omega_*$.
The covariant functor $\Omega^G_* \{{\mathcal{F}},{\mathcal{F}}' \}$ defines a $G$-equivariant homology theory [@Stong-complex Proposition 2.1], the boundary map on $A$ $$\begin{aligned}
\delta : \Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X,A) & \to \Omega^G_{n-1} \{{\mathcal{F}},{\mathcal{F}}' \}(A,\emptyset) \\
(M,M_0,M_1,f) & \mapsto (M_1,\partial M_1,\emptyset,f|_{M_1})\end{aligned}$$ defines the long exact sequence in homology for pairs $$\cdots \Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X,A) \stackrel{\delta}{\to} \Omega^G_{n-1} \{{\mathcal{F}},{\mathcal{F}}' \}(A,\emptyset) \to \Omega^G_{n-1} \{{\mathcal{F}},{\mathcal{F}}' \}(X,\emptyset) \to \cdots,$$ and for families ${\mathcal{F}}'' \subset {\mathcal{F}}' \subset {\mathcal{F}}$, choosing the boundary which is ${\mathcal{F}}'$-free $$\begin{aligned}
\partial : \Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X,A) & \to \Omega^G_{n-1} \{{\mathcal{F}}',{\mathcal{F}}'' \}(X,A) \\
(M,M_0,M_1,f) & \mapsto (M_0, \emptyset, \partial M_0,f|_{M_0})\end{aligned}$$ one obtains by [@Stong-complex Proposition 2.2] the long exact sequence in homology for families $$\cdots \Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X,A) \stackrel{\partial}{\to} \Omega^G_{n-1} \{{\mathcal{F}}' ,{\mathcal{F}}'' \}(X,A) \to \Omega^G_{n-1} \{{\mathcal{F}},{\mathcal{F}}'' \}(X,A) \to \cdots.$$
The bordism condition restricted to the non-relative case $\Omega^G_* \{{\mathcal{F}},{\mathcal{F}}' \}(X)$ can be read as the set of bordism classes of maps $f:M \to X$ such that $M$ is ${\mathcal{F}}$-free and $\partial M$ is ${\mathcal{F}}'$-free with $M$ endowed with a tangentially stable almost complex $G$-structure. Two become equivalent if there exists a $G$-manifold $F:V \to X$ which is ${\mathcal{F}}$-free such that $\partial V=M \cup M' \cup V^+$ and $M \cap V^+=\partial M$, $M' \cap V^+= \partial M'$, $M \cap M' = \emptyset$, $V^+ \cap(M \cup M') = \partial V^+$, with the property that $F$ restricts to $f$ on $M$ and to $f'$ on $M'$ and with $V^+$ ${\mathcal{F}}'$-free.
In [@tomDieck-Orbit-I Satz 3] it is shown that the canonical map that one can define $$\begin{aligned}
\label{iso geometric families EF}
\mu : \Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X,A) \to \Omega^G_n [{\mathcal{F}},{\mathcal{F}}' ](X,A)\end{aligned}$$ becomes a natural isomorphism of homology theories.
A key fact about the $({\mathcal{F}}, {\mathcal{F}}')$-free geometric unitary bordism elements of $X$ is the following result proven in [@ConnerFloyd-Odd Lemma 5.2]. Whenever $(M^n,\partial M^n ,f)$ is a $({\mathcal{F}}, {\mathcal{F}}')$-free geometric unitary bordism element of $X$ and $W^n$ a compact manifold with boundary regularly embedded in the interior of $M^n$ and invariant under the $G$-action, such that $G_m \in {\mathcal{F}}'$ for all $m \in M^n \backslash W^n$, then $[M^n,\partial M^n ,f]=[W^n, \partial W_n, f|_{W^n}]$ in $\Omega^G_n \{{\mathcal{F}},{\mathcal{F}}' \}(X)$.
Whenever the pair of families ${\mathcal{F}}' \subset {\mathcal{F}}$ differ by a fixed group $A$, i.e. ${\mathcal{F}}\backslash {\mathcal{F}}' = (A)$ with $(A)$ the set of subgroups of $G$ conjugate to $A$, then the pair $({\mathcal{F}}, {\mathcal{F}}')$ is called an [*adjacent pair of families of groups*]{}. In the case that $A$ is normal in $G$ a $({\mathcal{F}}, {\mathcal{F}}')$-free geometric unitary bordism class $[M, \partial M, f]$ of $X$ is equivalent to $\sum_{j=1}^l [U_j, \partial U_j, f|_{U_j}]$ where the $U_j$’s are disjoint $G$-equivariant tubular neighborhoods of the $M^A_j$’s and these sets are the connected components of the $A$-fixed point set $M^A$. Since the normal bundle of the fixed point set $M^A_j$ may be classified by a map to an appropriate classifying space, the groups $\Omega^G_* \{{\mathcal{F}}, {\mathcal{F}}'\}(X)$ become isomorphic to the direct sum of $G/A$-free equivariant unitary bordism groups of the product of $X^A$ with an appropriate classifying space (see [@AngelGomezUribe Theorem 4.5]). To introduce this result we need to understand how the fixed points of universal equivariant bundles behave. This is the subject of the next section.
Equivariant vector bundles and fixed points {#section bundles}
===========================================
Complex representations
-----------------------
Let $G$ be a compact Lie group and $A$ a closed and normal subgroup of $G$ fitting into the exact sequence $$1 \to A \to G \to Q \to 1.$$ Let $\rho: A \to U(V_\rho)$ be an irreducible unitary representation of $A$, denote by $\operatorname{\textup{Irr}}(A)$ the set of isomorphism classes of irreducible representations of $A$ and let $W$ be a finite dimensional complex $G$-representation. Then we have an isomorphism of $A$-representations $$\bigoplus_{\rho \in \operatorname{\textup{Irr}}(A)} V_\rho \otimes \operatorname{\textup{Hom}}_A(V_\rho,W) \stackrel{\cong}{\to} W.$$
The group $G$ acts on the set of $A$-representations $$(g\cdot \rho) (a) := \rho(g^{-1}ag)$$ and therefore it acts on $\operatorname{\textup{Irr}}(A)$. Denote by $G_\rho := \{g \in G \ | \ g \cdot \rho \cong \rho \}$ the isotropy group of the isomorphism class of $\rho$ and denote $Q_\rho:= G_\rho/A$.
If $g \cdot \rho \cong \rho$ then there exists $ M\in U(V_\rho)$ such that $g \cdot \rho (a) = M^{-1}\rho(a) M$. Since this matrix $M$ is unique up to a central element, we obtain a homomorphism $f: G_\rho \to PU(V_{\rho})$ which fits into the following diagram $$\xymatrixrowsep{0.5cm}
\xymatrix{ A \ar[d]_\rho \ar@{^{(}->}[r]^{\iota} & G_\rho \ar[d]^f \\
U(V_\rho) \ar[r]^p & PU(V_{\rho}),
}$$ thus making $V_\rho$ into a projective $G_\rho$-representation.
Define the ${\mathbb{S}}^1$-central extension $ \widetilde{G}_{\rho} : = f^*U(V_\rho)$ of $G_\rho$ which fits into the following diagram $$\xymatrixrowsep{0.4cm}
\xymatrix{
& {\mathbb{S}}^1 \ar[d] & {\mathbb{S}}^1 \ar[d] \\
A \ar[r]^{\widetilde{\iota}} \ar[d]^= & \widetilde{G}_{\rho}
\ar[r]^{\widetilde{f}} \ar[d] & U(V_\rho) \ar[d] \\
A \ar[r]^\iota & G_\rho \ar[r]^f & PU(V_\rho),
}$$ endowing $V_\rho$ with the structure of a $\widetilde{G}_{\rho}$-representation where ${\mathbb{S}}^1$ acts by multiplication with scalars.
The vector space $\operatorname{\textup{Hom}}_A(V_\rho, W)$ is also a $\widetilde{G}_{\rho}$-representation where for $\phi \in \operatorname{\textup{Hom}}_A(V_\rho, W)$ and $\widetilde{g} \in \widetilde{G}_{\rho}$ we set $$( \widetilde{g} \bullet \phi)(v) := g \phi(\widetilde{f}(\widetilde{g})^{-1} v).$$ It follows that $A$ acts trivially on $\operatorname{\textup{Hom}}_A(V_\rho, W)$ and moreover the elements of ${\mathbb{S}}^1$ act by multiplication with their inverse.
Hence $V_\rho \otimes \operatorname{\textup{Hom}}_A(V_\rho, W)$ is a $G_\rho$ representation, where $\operatorname{\textup{Hom}}_A(V_\rho, W)$ is a $\widetilde{Q}_{\rho}:= \widetilde{G}_{\rho}/A$ representation where ${\mathbb{S}}^1$ acts by multiplication of the inverse. Here $\widetilde{Q}_{\rho}$ is an ${\mathbb{S}}^1$-central extension of $Q_\rho$.
Since the isotropy group $G_\rho$ contains the connected component of the identity in $G$, the index $[G \colon G_\rho]$ is finite and we may induce the $G_\rho$-representation $V_\rho \otimes \operatorname{\textup{Hom}}_A(V_\rho, W)$ to $G$ thus obtaining the following result.
There is a canonical isomorphism of $G$-representations $$\bigoplus_{\rho \in G \backslash \operatorname{\textup{Irr}}(A)} \operatorname{Ind}_{G_\rho}^G \left( V_\rho \otimes \operatorname{\textup{Hom}}_A(V_\rho, W) \right) \cong W$$ where $\rho$ runs over representatives of the orbits of the action of $G$ on $\operatorname{\textup{Irr}}(A)$.
Equivariant complex bundles
---------------------------
The previous result generalizes to equivariant complex vector bundles, but prior to showing this generalization we need to recall the multiplicative induction map introduced in [@Bix-tomDieck §4]. Let $H$ be a closed subgroup of the compact Lie group $G$. The right adjoint to the restriction functor $r_H^G$ from $G$-spaces to $H$-spaces is called the multiplicative induction functor and takes an $H$-space $Y$ and returns the $G$-space $$m_H^G(Y):=\operatorname{map}(G,Y)^H$$ of $H$-equivariant maps from $G$ to $Y$, with $G$ considered as an $H$-space by left multiplication. The $G$-action on $m_H^G(Y)$ is given by $(g \cdot f)(k) := f(kg)$, $m_H^G(Y)$ is homeomorphic to the space of sections of the projection map $G \times_H Y \to G/H$ and, in the case that $G/H$ is finite, it is homeomorphic to $[G:H]$ copies of $Y$.
There is a homeomorphism $$\operatorname{map}(X,m_H^G(Y))^G \stackrel{\cong}{\to} \operatorname{map}(r_H^G(X), Y)^H, \ \ F \mapsto (x \mapsto F(x)(1_G))$$ whose inverse maps $f$ to $m_H^G(f) \circ p_H^G$ where $p_H^G:X \to m_H^G(r_H^G(X))$, $p_H^G(x)(g)=gx$, is the unit of the adjunction.
Now consider a $G$-space $X$ on which the closed and normal subgroup $A$ acts trivially. Take a $G$-equivariant complex vector bundle $p: E \to X$ and assume that $E$ has an hermitian metric in such a way that $G$ acts through unitary matrices on the complex fibers. For a complex $A$-representation $\rho : A \to U(V_\rho)$ denote by ${\mathbb{V}}_{\tau}$ the trivial $A$-vector bundle $\pi_2 : V_\rho \times X \to X$.
The complex vector bundle $\operatorname{\textup{Hom}}_{A}({\mathbb{V}}_{\rho},E)$ is a $\widetilde{Q}_{\rho}$-equivariant complex vector bundle where ${\mathbb{S}}^1$ acts on the fibers by multiplication of the inverse, ${\mathbb{V}}_{\rho} \otimes\operatorname{\textup{Hom}}_{A}({\mathbb{V}}_{\rho},E)$ is a $G_\rho$-equivariant complex vector bundle and $$(p_{G_\rho}^G)^* \left(m_{G_\rho}^G({\mathbb{V}}_{\rho} \otimes\operatorname{\textup{Hom}}_{A}({\mathbb{V}}_{\rho},E)) \right) \to X$$ is a $G$-equivariant complex vector bundle over $X$.
[@AngelGomezUribe Theorem 2.7] \[thm decomposition\] Let $G$ be a compact Lie group, $A$ a closed and normal subgroup, $X$ a $G$-space on which $A$ acts trivially and $E \to X$ a $G$-equivariant complex vector bundle. Then there is an isomorphism of $G$-equivariant complex vector bundles $$\bigoplus_{\rho \in G \backslash \operatorname{\textup{Irr}}(A)} (p_{G_\rho}^G)^* \left(m_{G_\rho}^G({\mathbb{V}}_{\rho}
\otimes\operatorname{\textup{Hom}}_{A}({\mathbb{V}}_{\rho},E)) \right)\stackrel{\cong}{\to} E$$ where $\rho$ runs over representatives of the orbits of the $G$-action on the set of isomorphism classes of $A$-irreducible representations.
With the same hypothesis as in the previous theorem, there is an induced decomposition in equivariant K-theory $$K^*_G(X) \cong \bigoplus_{\rho \in G \backslash \operatorname{\textup{Irr}}(A)}
{}^{\widetilde{Q}_{\rho}} K^*_{Q_\rho}(X), \ \ E \mapsto \bigoplus_{\rho \in G \backslash \operatorname{\textup{Irr}}(A)}
\operatorname{\textup{Hom}}_{A}({\mathbb{V}}_{\rho},E)$$ where ${}^{\widetilde{Q}_{\rho}} K^*_{Q_\rho}(X)$ is the $\widetilde{Q}_{\rho}$-twisted $Q_\rho$- equivariant K-theory of $X$ which is built out of the Grothendieck group of $\widetilde{Q}_{\rho}$-equivariant complex vector bundles over $X$ on which the central ${\mathbb{S}}^1$ acts on the fibers by multiplication.
Classifying spaces
------------------
The decomposition described above can also be written at the level of classifying spaces; let us set up the notation first. Let $G$ be a compact Lie group and $\widetilde{G}$ a ${\mathbb{S}}^1$-central group extension of $G$. Let $\widetilde{\bf{C}}^\infty$ be a countable direct sum of all complex irreducible $\widetilde{G}$ representations on which elements of ${\mathbb{S}}^1$ act by multiplication with their inverse. Denote by ${}^{\widetilde{G}}B_GU(n)$ the Grassmannian of $n$-planes of $\widetilde{\bf{C}}^\infty$ and denote by ${}^{\widetilde{G}}\gamma_GU(n)$ the canonical $n$-plane bundle over ${}^{\widetilde{G}}B_GU(n)$. The complex vector bundle $${\mathbb{C}}^{n} \to
{}^{\widetilde{G}}\gamma_GU(n) \to
{}^{\widetilde{G}}B_GU(n)$$ is a universal $\widetilde{G}$-twisted $G$-equivariant complex vector bundle of rank $n$. Denote by $\gamma_GU(n) \to B_GU(n)$ the universal $G$-equivariant complex vector bundle of rank $n$.
For a closed subgroup $A$ of $G$, let $N_A$ denote the normalizer of $A$ in $G$ and $W_A:= N_A/A$. Consider the fixed point set $B_GU(n)^A$ and the restriction $\gamma_GU(n)|_{B_GU(n)^A}$ of the universal bundle to this fixed point set. For $\rho \in \operatorname{\textup{Irr}}(A)$, by the arguments above we have that $$\operatorname{\textup{Hom}}_{A}({\mathbb{V}}_{\rho},\gamma_GU(n)|_{B_GU(n)^A})$$ is a $(\widetilde{W_A})_\rho$-twisted $(W_A)_\rho$-equivariant complex bundle, but since the space $B_GU(n)^A$ is not necessarily connected, it may not have constant rank. Therefore Theorem \[thm decomposition\] implies the following equivariant homotopy equivalence.
[@AngelGomezUribe Theorems 3.3 & 3.5] \[thm homotopy decomposition\] There are $W_A$-equivariant homotopy equivalences $$\bigsqcup_{n=0}^{\infty}\gamma_GU(n)^A \simeq \left(
\bigsqcup_{n=0}^{\infty}\gamma_{W_A}U(n_1) \right)\times
\prod_{\rho \in W_A \backslash \operatorname{\textup{Irr}}(A) \atop \rho \ne1} m^{W_A}_{(W_A)_\rho}
\left( \bigsqcup_{n_\rho=0}^{\infty}{}^{(\widetilde{W_A})_\rho}B_{(W_A)_\rho}U(n_\rho) \right),$$ $$\bigsqcup_{n=0}^{\infty}B_GU(n)^A \simeq
\prod_{\rho \in W_A \backslash \operatorname{\textup{Irr}}(A)} m^{W_A}_{(W_A)_\rho}
\left( \bigsqcup_{n_\rho=0}^{\infty}{}^{(\widetilde{W_A})_\rho}B_{(W_A)_\rho}U(n_\rho) \right).$$
If $G$ is abelian then $A$ is normal, $G$ acts trivially on $\operatorname{\textup{Irr}}(A)$ and all the irreducible representations are 1-dimensional. Therefore we get a $G/A$-homotopy equivalence $$\gamma_GU(n)^A \simeq \bigsqcup_{(n_\rho)_{\rho \in \operatorname{\textup{Irr}}(A)} \atop \sum_\rho n_\rho = n} \left(
\gamma_{G/A}U(n_1) \times \prod_{\rho \in \operatorname{\textup{Irr}}(A) \atop \rho \neq 1}
B_{G/A}U(n_\rho)\right).$$ In order to get a similar formula for the case in which $G$ is not abelian we need to introduce further notation and make some choices. Let ${\mathcal{P}}(n,A)$ be the set of arrangements of non-negative integers $(n_\rho)_{\rho \in \operatorname{\textup{Irr}}(A)}$ such that $$\sum_{\rho \in \operatorname{\textup{Irr}}(A)} n_\rho |\rho|=n,$$ then non-equivariantly there is a homotopy equivalence $$B_GU(n)^A \simeq \bigsqcup_{{(n_\rho) \in {\mathcal{P}}(n,A) }} \prod_{\rho \in \operatorname{\textup{Irr}}(A)}
\left({}^{(\widetilde{W_A})_\rho}B_{(W_A)_\rho}U(n_\rho) \right).$$ The group $W_A$ acts on ${\mathcal{P}}(n,A)$ on the right by permuting the arrangements, i.e. the action of $g \in W_A$ on the arrangement $(n_\rho)$ is the arrangement $(n_\rho) \cdot g:= (n_{g\cdot \rho})$ meaning that it has the number $n_{g \cdot \rho}$ in the coordinate $\rho$. Denote by $(W_A)_{(n_\rho)}$ the isotropy group of the arrangement $(n_\rho)$. Rearranging the terms we obtain the following $W_A$-equivariant homotopy equivalence $$\begin{aligned}
&B_GU(n)^A \simeq \label{homotopy type BGU(n)A} \\ \nonumber
\bigsqcup_{{(n_\rho) \in {\mathcal{P}}(n,A) / W_A }} &
W_A \underset{(W_A)_{(n_\rho)}}{\times} \left(
\prod_{\rho \in (W_A)_{(n_\rho)} \backslash \operatorname{\textup{Irr}}(A)} m_{(W_A)_\rho \cap (W_A)_{(n_\rho)}}^{(W_A)_{(n_\rho)}}
\left({}^{(\widetilde{W_A})_\rho}B_{(W_A)_\rho}U(n_\rho) \right) \right)
\end{aligned}$$ where $(n_\rho)$ runs over representatives of the orbits of the action of $W_A$ on ${\mathcal{P}}(n,A)$, and $\rho$ runs over representatives of the orbits of the action of $(W_A)_{(n_\rho)}$ on $\operatorname{\textup{Irr}}(A)$.
For the calculation of the equivariant unitary bordism of adjacent families of groups we need to consider only the arrangements of non-negative integers $(n_\rho)$ such that the number associated to the trivial representation is zero, i.e. $n_1=0$. Denote by $\overline{{\mathcal{P}}}(n,A)$ the set of arrangements $(n_\rho)$ such that $n_1=0$ and define the $W_A$-space: $$\begin{aligned}
&C_{N_A,A}(k) := \label{space CGA(k)} \\ \nonumber
\bigsqcup_{{(n_\rho) \in \overline{{\mathcal{P}}}(k,A) / W_A }}&
W_A \underset{(W_A)_{(n_\rho)}}{\times} \left(
\prod_{\rho \in (W_A)_{(n_\rho)} \backslash \operatorname{\textup{Irr}}(A) \atop \rho \neq 1} m_{(W_A)_\rho \cap (W_A)_{(n_\rho)}}^{(W_A)_{(n_\rho)}}
\left({}^{(\widetilde{W_A})_\rho}B_{(W_A)_\rho}U(n_\rho) \right) \right)
\end{aligned}$$
Therefore we have the following $W_A$-homotopy equivalence $$\begin{aligned}
\label{formula with C}
\gamma_GU(n)^A \simeq
\bigsqcup_{k=0}^n \gamma_{W_A}U(n-k) \times C_{N_A,A}(k)\end{aligned}$$ such that in the case that $G$ is abelian we have the simple formula $$\begin{aligned}
C_{G,A}(k) = \bigsqcup_{{(n_\rho) \in \overline{{\mathcal{P}}}(k,A) }}
\prod_{\rho \in \operatorname{\textup{Irr}}(A) \atop \rho \neq 1}
B_{G/A}U(n_\rho).
\end{aligned}$$
Now we are ready to state the relation between the $G$-equivariant unitary bordism groups of adjacent pair of families of groups and the classifying spaces defined above.
[@AngelGomezUribe Corollary 4.6] \[theorem adjacent\] Let $G$ be a finite group, $X$ a $G$-space and $({\mathcal{F}}, {\mathcal{F}}')$ an adjacent pair of families differing by the conjugacy class of the subgroup $A$, then there is an isomorphism $$\Omega^G_n\{{\mathcal{F}}, {\mathcal{F}}'\}(X) \cong
\bigoplus_{0 \leq 2k \leq n} \Omega^{W_A}_{n-2k}\{\{1\}\}(X^A \times C_{N_A,A}(k) )$$ where $\{1\}$ is the family of subgroups of $W_A$ which only contains the trivial group.
Take a bordism class $[M, \partial M , f:M \to X]$ in $\Omega^G_n\{{\mathcal{F}}, {\mathcal{F}}'\}(X)$ and note that $M^A \cap M^{gAg^{-1}} = \emptyset$ whenever $g$ does not belong to $N_A$. Then choose an $N_A$-equivariant tubular neighbourhood $U$ of $M^A$ such that its $G$-orbit $G \cdot U$ is a $G$-equivariant tubular neighbourhood of $G \cdot M^A$ and such that $$G \underset{N_A}{\times} U \stackrel{\cong}{\to} G \cdot U, \ \ [(g,u)] \to gu$$ is a $G$-equivariant diffeomorphism. The assignment $[M, \partial M , f:M \to X] \mapsto
[U, \partial U , f|_U:U \to X]$ induces an isomorphism $$\Omega^G_n\{{\mathcal{F}}, {\mathcal{F}}'\}(X) \stackrel{\cong}{\to} \Omega^{N_A}_n\{{\mathcal{F}}|_{N_A}, {\mathcal{F}}'|_{N_A}\}(X).$$
Let $M^A_{n-2k}$ denote the component of $M^A$ which is a $(n-2k)$-dimensional $W_A$-free manifold and such that $M^A= \bigcup_{0 \leq 2k \leq n} M^A_{n-2k}$. The tubular neighbourhood $U$ is $N_A$-equivariantly diffeomorphic to $ \bigcup_{0 \leq 2k \leq n} D(\nu_{n-2k})$ where $\nu_{n-2k} \to M^A_{n-2k}$ is the normal bundle of the inclusion $M^A_{n-2k} \to M$. Since the trivial $A$-representation does not appear on the fibers of the normal bundles, by Theorem \[thm homotopy decomposition\] and formula we know that the bundle $\nu_{n-2k}$ is classified by a $W_A$-equivariant map $h_{n-2k}:M^A_{n-2k} \to C_{N_A,A}(k)$. The bordism class $[ M^A_{n-2k}, f|_{M^A_{n-2k}} \times h_{n-2k} : M^A_{n-2k} \to X^A \times C_{N_A,A}(k)]$ belongs to $\Omega^{W_A}_{n-2k}\{\{1\}\}(X^A \times C_{N_A,A}(k) )$ and the assignment $$[M, \partial M , f:M \to X] \mapsto \bigoplus_{0 \leq 2k \leq n} [ M^A_{n-2k}, f|_{M^A_{n-2k}}\times h_{n-2k} : M^A_{n-2k} \to X^A \times C_{N_A,A}(k)]$$ induces the desired isomorphism.
The evenness conjecture for finite abelian groups {#section abelian}
=================================================
In this section we will outline the main ingredients used by Landweber [@Landweber-cyclic] in the cyclic group case, Stong [@Stong-complex] in the $p$-group case and Ossa [@Ossa] in the general case to show that the evenness conjecture holds for finite abelian groups. The conjecture also holds for compact abelian groups, Löffler [@Loffler] showed it for the homotopic $G$-equivariant unitary bordism groups in the case that $G$ is a unitary torus, and Comezaña [@May-book §XXVIII] generalized it to any compact abelian group. Comezaña furthermore showed that the map from the $G$-equivariant unitary bordism groups to the homotopic ones is injective whenever $G$ is compact abelian thus proving the evenness conjecture for any compact abelian group. In this work we will address the finite group case.
Prior to addressing the study of the $G$-equivariant unitary bordism groups for finite abelian groups we need to recall some results on the unitary bordism groups.
Thom’s remarkable Theorem [@Thom] shows that the unitary bordism groups $\Omega_*$ can be calculated as the stable homotopy groups $\lim_k\pi_{n+k}(MU(k))$ of the Thom spaces $MU(k)$ of the canonical complex vector bundles over $BU(k)$. Milnor in [@Milnor Theorem 3] showed that these stable homotopy groups are zero if $n$ is odd and free abelian if $n$ is even with a number of generators equal to the number of partitions of $n/2$. Independently Novikov [@Novikov1 Theorem 1] showed that as a ring the unitary bordism groups are isomorphic to the ring of polynomials over the integers with generators $x_{2i}$ of degree $2i$ for $i\geq 1$. The spectrum $MU$ that the Thom spaces $MU(k)$ define permitted Atiyah [@Atiyah-bordism] to define the homotopy unitary bordism groups $MU_*(X)$ and the homotopy unitary cobordism groups $MU^*(X)$ of a space $X$ as a generalized homology and cohomology theory respectively. Thom’s theorem implies that for $X$ a CW-complex the unitary bordism groups over $X$ are equivalent to the homotopic ones: $\Omega_*(X) \cong MU_*(X)$ via the Thom-Pontrjagin map.
The Atiyah-Hirzebruch spectral sequence [@AtiyahHirzebruch] (cf. [@Kochman §4.2]) applied to the unitary bordism groups of a CW-complex $X$ produces a spectral sequence which converges to $\Omega_*(X)$ and whose second page is $E^2_{p,q}\cong H_p(X; \Omega_q)$; let us call this spectral sequence the [*bordism spectral sequence*]{}. The Thom homomorphism $$\mu: \Omega_*(X) \to H_*(X; {\mathbb{Z}}), \ \ [M, f:M \to X] \mapsto f_*[M],$$ which takes a unitary bordism element in $X$ and maps it to the image under $f$ of the fundamental class $[M] \in H_*(M; {\mathbb{Z}})$, is a natural transformation of homology theories and is also the edge homomorphism $\Omega_*(X) \to E^2_{*,0} \cong H_*(X; {\mathbb{Z}})$ of the spectral sequence.
Whenever $X$ is a CW-complex whose homology $H_*(X; {\mathbb{Z}})$ is free abelian then by [@ConnerSmith Lemma 3.1] the bordism spectral sequence collapses, the unitary bordism group $\Omega_*(X)$ is a free $\Omega_*$-module and the homomorphism induced by the Thom map $$\widetilde{\mu}: {\mathbb{Z}}{\otimes}_{\, \Omega_*} \Omega_*(X) \to H_*(X; {\mathbb{Z}})$$ is an isomorphism.
Applying the bordism spectral sequence to the unitary bordism groups of $BU(n)$ it is shown in [@Kochman Proposition 4.3.3] that $\Omega_*(BU(n))$ is a free $\Omega_*$-module with basis $$\Omega_*(BU(n)) \cong \Omega_* \{ \alpha_{k_1}\alpha_{k_2} \dots \alpha_{k_n} \colon k_1 \leq \cdots \leq k_n \}$$ where $\alpha_{k_1}\alpha_{k_2} \dots \alpha_{k_n}$ is the unitary bordism class of the bordism element $$({\mathbb{C}}P^{k_1} \times \cdots \times {\mathbb{C}}P^{k_n} ,F: {\mathbb{C}}P^{k_1} \times \cdots \times {\mathbb{C}}P^{k_n} \to BU(n))$$ where the map $F$ classifies the canonical rank $n$ complex vector bundle over the product of projective spaces.
In [@ConnerSmith Proposition 3.6] it is shown that if $X$ is a finite CW-complex such that the Thom homomorphism is surjective then the bordism spectral sequence collapses. Whenever $BG$ is the classifying space of a finite group $G$ Landweber showed in [@Landweber-complex Theorem 3] that the following conditions are equivalent:
- The Thom homomorphism $\mu : \Omega_*(BG) \to H_*(BG; {\mathbb{Z}})$ is surjective.
- The bordism spectral sequence collapses.
- $G$ has periodic cohomology, i.e. every abelian subgroup of $G$ is cyclic.
- $H^n(BG; {\mathbb{Z}})=0$ for all odd $n$.
- The projective dimension of $\Omega_*(BG)$ as a $\Omega_*$-module is $1$ or $0$.
The previous result implies that whenever we consider the cyclic group $G= {\mathbb{Z}}/k$ of order $k$, the bordism classes $[L^{2n+1}(k), \iota: L^{2n+1}(k) \to B {\mathbb{Z}}/k]$ of the lens spaces $L^{2n+1}(k):= S^{2n+1}_k/ ({\mathbb{Z}}/k)$ , where $S^{2n+1}_k$ denotes the sphere of unit vectors in ${\mathbb{C}}^{n+1}$ with the ${\mathbb{Z}}/k$-action given by multiplication of the root of unity $e^{\frac{2 \pi i}{k}}$, generate $\Omega_*(B {\mathbb{Z}}/k)$ as a $\Omega_*$-module.
One property of finite abelian groups that will be used is the following. If $A$ is a subgroup of a finite abelian group $G$ and $\Gamma$ is a product of classifying spaces of the form $B_GU(k)$, then by Theorem \[thm homotopy decomposition\] the fixed point set $\Gamma^A$ is a product of classifying spaces of the form $B_{G/A}U(l)$. This fact allows one to use an induction hypothesis when calculating the equivariant unitary bordism groups of products of spaces of the form $B_GU(k)$.
Now we can start the proof of the evenness conjecture for finite abelian groups. First we will handle the case of cyclic $p$-groups following [@Landweber-cyclic], then we will review the case of general abelian $p$-groups following [@Stong-complex] and we will prove the general case using a simple argument on localization shown in [@Ossa].
Cyclic $p$-groups {#subsection cyclic}
-----------------
Let $G$ be a cyclic group of order $p^s$ a power of the prime $p$. Let $\Gamma:= \prod_{i=1}^lB_GU(k_i)$ be a product of spaces of the form $B_GU(k)$ and ${\mathcal{F}}_t= \{ H \subset G \colon |H| \leq p^t\}$ the family of of subgroups or order bounded by $p^t$; the family ${\mathcal{F}}_s$ is the family of all subgroups of $G$ and therefore $\Omega_*^G( \ ) = \Omega_*^G\{ {\mathcal{F}}_s \}( \ )$. Let us split $\Omega_*^G( \ ) = \Omega_+^G( \ ) \oplus \Omega_-^G( \ )$ where $ \Omega_+^G( \ )$ denotes the even degree bordism groups and $\Omega_-^G( \ )$ the odd degree ones. We will prove by induction on the size of the group that for any $0 \leq t < s$ the following properties hold:
- $\Omega^G_*\{{\mathcal{F}}_s,{\mathcal{F}}_t\}(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators.
- $\Omega^G_+\{{\mathcal{F}}_t,{\mathcal{F}}_{t-1}\}(\Gamma)$ is a free $\Omega_*$-module.
- The boundary homomorphism is surjective $$\Omega^G_+\{{\mathcal{F}}_s,{\mathcal{F}}_t\}(\Gamma) \stackrel{\partial
}{\to} \Omega^G_-\{{\mathcal{F}}_t,{\mathcal{F}}_{t-1}\}(\Gamma).$$
Let us see that these properties imply that $\Omega^G_*(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators. Since $\Omega^G_*\{{\mathcal{F}}_s,{\mathcal{F}}_0\}(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators, the long exact sequence associated to the families of groups ${\mathcal{F}}_{0} \subset {\mathcal{F}}_s$ induce the exact sequence $$0 \to \Omega^G_+\{{\mathcal{F}}_0\}(\Gamma) \to \Omega^G_+(\Gamma) \to \Omega^G_+\{{\mathcal{F}}_s,{\mathcal{F}}_{0}\}(\Gamma) \stackrel{\partial}{\to} \Omega^G_-\{{\mathcal{F}}_0\}(\Gamma) \to \Omega_-^G( \Gamma) \to 0.$$
The unitary bordism group of free actions $\Omega^G_*\{{\mathcal{F}}_0\}(\Gamma)$ is isomorphic to $\Omega_*(BG \times \prod_{i=1}^l BU(k_i))$ since both $EG \times B_GU(k_i)$ and $EG \times BU(k_i)$ classify $G$-equivariant complex vector bundles of rank $k_i$ over free $G$-spaces. The unitary bordism groups of $BU(k_i)$ are free $\Omega_*$-modules on even-dimensional generators, and therefore by the Künneth theorem we have that $$\Omega^G_*\{{\mathcal{F}}_0\}(\Gamma) \cong \Omega_*(BG) \underset{\Omega_*}{\otimes} \Omega_*\left(\prod_{i=1}^l BU(k_i)\right).$$
Hence we have that $\Omega^G_+\{{\mathcal{F}}_0\}(\Gamma)$ is a free $\Omega_*$-module in even degrees and that $\Omega^G_-\{{\mathcal{F}}_0\}(\Gamma)$ is all $p$-torsion. Consider a unitary bordism class defined by the map $h:M \to \prod_{i=1}^l BU(k_i)$ and denote by $E:= E_1 \oplus \cdots \oplus E_l$ with $E_j$ the complex vector bundle that the map $\pi_j \circ h : M \to BU(k_j)$ defines. Take the ball $B^{2n+2}_{p^s}$ of vectors in ${\mathbb{C}}^{n+1}$ with norm less than 1 endowed with the action of $G$ given by multiplication by $e^{\frac{2 \pi i}{p^s}}$ and consider the $G$-equivariant $\prod_{i=1}^l U(k_i)$ complex bundle that the product $B^{2n+2}_{p^s} \times E \to B^{2n+2}_{p^s}\times M$ defines.
This $G$-equivariant $\prod_{i=1}^l U(k_i)$ complex bundle is classified by a $G$-equivariant map $$f : B^{2n+2}_{p^s}\times M \to \Gamma$$ and its $G$-equivariant unitary bordism class $[ B^{2n+2}_{p^s}\times M, f]$ belongs to $\Omega_+^G\{{\mathcal{F}}_s, {\mathcal{F}}_0\}(\Gamma)$. Its boundary is $[ S^{2n+1}_{p^s}\times M, f|_{S^{2n+1}_{p^s}\times M}]$ and it belongs to $\Omega_-^G\{ {\mathcal{F}}_0\}(\Gamma)$. By the Künneth isomorphism described above we know that the unitary bordism classes $[ S^{2n+1}_{p^s}\times M, f|_{S^{2n+1}_{p^s}\times M}]$ generate $\Omega_-^G\{ {\mathcal{F}}_0\}(\Gamma)$ and therefore the boundary homomorphism $\Omega^G_+\{{\mathcal{F}}_s,{\mathcal{F}}_{0}\}(\Gamma) \stackrel{\partial}{\to} \Omega^G_-\{{\mathcal{F}}_0\}(\Gamma)$ is surjective. This implies that $\Omega_-^G(\Gamma)$ is trivial.
Since $\Omega^G_+\{{\mathcal{F}}_0\}(\Gamma) \cong \Omega_+(\prod_{i=1}^l BU(k_i))$ is a free $\Omega_*$-module, and by hypothesis $\Omega^G_+\{{\mathcal{F}}_s,{\mathcal{F}}_{0}\}$ also, then it implies that $\Omega^G_+(\Gamma)$ is a free $\Omega_*$-module.
In particular we have that the $G$-equivariant unitary bordism group $\Omega_*^G$ is a free $\Omega_*$-module on even-dimensional generators.
Now let us sketch the proof of the properties cited above. Let us assume that the properties hold for cyclic groups of order less than $p^s$ and let us proceed by induction on the families of subgroups of $G$. For the adjacent pair of families $({\mathcal{F}}_s, {\mathcal{F}}_{s-1})$ differing by the group $G$, we know by Theorem \[theorem adjacent\] that $\Omega_*^G \{{\mathcal{F}}_s,{\mathcal{F}}_{s-1}\}(\Gamma)$ is a direct sum of groups $\Omega_*(\Gamma^G \times \Gamma')$ where both $\Gamma^G$ and $\Gamma'$ are products of classifying spaces of unitary groups. Therefore $\Omega_*^G \{{\mathcal{F}}_s,{\mathcal{F}}_{s-1}\}(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators and we have started our induction.
Now let us assume that the properties hold for the pair of families $({\mathcal{F}}_s, {\mathcal{F}}_j)$ for $s >j \geq t$. Therefore we get the following exact sequence of groups $$\begin{aligned}
\nonumber
0 \to \Omega^G_+\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma) \to \Omega^G_+&\{{\mathcal{F}}_s, {\mathcal{F}}_{t-1}\}(\Gamma) \to \Omega^G_+\{{\mathcal{F}}_s,{\mathcal{F}}_t\}(\Gamma) \stackrel{\partial}{\to} \\ &\Omega^G_-\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma) \to \Omega_-^G\{{\mathcal{F}}_s, {\mathcal{F}}_{t-1}\}( \Gamma) \to 0. \label{LES t t-1}
\end{aligned}$$
Since the pair of families $({\mathcal{F}}_t, {\mathcal{F}}_{t-1})$ differ by the cyclic group $H$ of order $p^t$, $G/H$ is a cyclic group of order $p^{s-t}$, and $\Gamma^{H}$ is a product of classifying spaces of the form $B_{G/H}U(k)$, then by Theorem \[theorem adjacent\] there is an isomorphism $$\begin{aligned}
\label{iso t t-1}
\Omega^G_*\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma) \cong \bigoplus_{k \geq 0}\Omega_{*-2k}^{G/H}\{{\mathcal{F}}_0\} (\Gamma^{H} \times C_{G,H}(k))
\end{aligned}$$ where both $\Gamma^{H}$ and $C_{G,H}(k)$ are disjoint unions of products of classifying spaces of the form $B_{G/H}U(k)$.
Therefore we know that $\Omega^G_+\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma)$ is a free $\Omega_*$-module and by the induction hypothesis we know that the boundary map $$\begin{aligned}
\label{induction sujectivity}
\Omega_+^{G/H}\{{\mathcal{F}}_{s-t},{\mathcal{F}}_0\} (\Gamma^{H} \times C_{G,H}(k)) \stackrel{\partial}{\to} \Omega_-^{G/H}\{{\mathcal{F}}_0\} (\Gamma^{H} \times C_{G,H}(k))
\end{aligned}$$ is surjective. A bordism class in $\Omega^G_-\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma)$ can be represented by a class $[D(E), f: D(E) \to \Gamma]$ where $D(E)$ is the disk bundle of a $G$-equivariant vector bundle $E \to M$ over a manifold $M$ on which $H$ acts trivially and $G/H$ acts freely, and such that the trivial representation of $H$ does not appear on the fibers of $E$. This bundle is classified by a $G/H$-equivariant map $h: M \to C_{G,H}(k)$ for some $k$, and the bordism class $[M, f|_M \times h : M \to \Gamma^{H}\times C_{G,H}(k)]$ lives in $\Omega_-^{G/H}\{{\mathcal{F}}_0\} (\Gamma^{H} \times C_{G,H}(k))$. By the surjectivity of there is a bordism class $[Z, F \times \tilde{h}: Z \to \Gamma^{H}\times C_{G,H}(k)]$ in $ \Omega_+^{G/H}\{{\mathcal{F}}_{s-t},{\mathcal{F}}_0\} (\Gamma^{H} \times C_{G,H}(k))$ such that $\partial Z = M$, $F|_{M}=f|_M$ and $\tilde{h}|_M=h$. Let $p: V \to Z$ denote the $G$-equivariant vector bundle over $Z$ that the map $\tilde{h}$ defines and note that the bordism class $[D(V), F \circ p : D(V) \to \Gamma]$ defines an element in $\Omega_*^G\{{\mathcal{F}}_s, {\mathcal{F}}_{t}\}(\Gamma)$ since the trivial $H$-representation does not appear on the fibers of $V$ and the action of $G/H$ over $M$ is free. The boundary of $D(V)$ is the union of the sphere bundle $S(V)$ and $D(V)|_{M}= D(E)$, but since $S(V)$ is ${\mathcal{F}}_{t-1}$-free we have that $$\partial [D(V), F \circ p : D(V) \to \Gamma] = [D(E), p|_{E} \circ f|_{M}: D(E) \to \Gamma]=[D(E), f: D(E) \to \Gamma]$$ and therefore the boundary map $$\Omega_+^{G}\{{\mathcal{F}}_s, {\mathcal{F}}_t\}(\Gamma) \stackrel{\partial}{\to} \Omega_-^{G}\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma)$$ is surjective.
Now, the group $\Omega^G_*\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma)$ has projective dimension 1 as a $\Omega_*$-module. This follows from the following two facts, first that $\Omega_*(B(G/H))$ has projective dimension 1 over $\Omega_*$, and second that equation induces the isomorphism $$\Omega^G_*\{{\mathcal{F}}_t, {\mathcal{F}}_{t-1}\}(\Gamma) \cong \bigoplus_{k \geq 0}\Omega_{*-2k}(B(G/H) \times Z)$$ where $Z$ is a product of copies of $BU(k)$’s. By Schanuel’s lemma we know that the kernel of the boundary map $\partial$ is projective and therefore free [@ConnerSmith Proposition 3.2]. By the long exact sequence described in we deduce that $\Omega_*^{G}\{{\mathcal{F}}_s, {\mathcal{F}}_{t-1} \}(\Gamma)$ is a free $\Omega_{*}$-module on even-dimensional generators and we conclude that the properties also hold for the pair of families $({\mathcal{F}}_s, {\mathcal{F}}_{t-1})$.
Therefore the evenness conjecture holds for cyclic $p$-groups [@Landweber-cyclic Theorem 1’].
General abelian $p$-groups
--------------------------
The argument to show the evenness conjecture for general abelian $p$-groups is more elaborate than the one done above for cyclic $p$-groups. We will follow the original proof of Stong done in [@Stong-complex] in which the author uses very cleverly the Thom isomorphism and the long exact sequence for pairs of spaces in order to understand the long exact sequence for a pair of families once restricted to a special kind of actions on manifolds. Here we shorten the original proof and we highlight its main ingredients.
Let $G = H \times {\mathbb{Z}}/q$ with $q=p^s$ such that all elements in $H$ have order less or equal than $p^s$ and let $$\Gamma:= \prod_{i=1}^lB_GU(k_i)$$ be a product of spaces of the form $B_GU(k)$. We will show by induction on the order of the group $G$ that the bordism group $\Omega_*^G(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators. Therefore let us assume that $\Omega_*^K(\Gamma')$ is a free $\Omega_*$-module on even-dimensional generators for all $p$-groups of order less than the order of $G$ and $\Gamma'$ any product of classifying spaces of the form $B_{K}U(l)$.
Following the notation of Stong in [@Stong-complex] let us consider the following families of subgroups of $G$:
- ${\mathcal{F}}_a$ is the family of all subgroups of $G$,
- ${\mathcal{F}}_s$ is the family of subgroups whose intersection with ${\mathbb{Z}}/q$ is a proper subgroup of $W$, i.e. ${\mathcal{F}}_s: = \{ W \subset H \times {\mathbb{Z}}/q \, | \, \{1\} \times {\mathbb{Z}}/q \not\subset W \}$,
- ${\mathcal{F}}_f$ is the family of subgroups whose intersection with ${\mathbb{Z}}/q$ is the unit subgroup, i.e. ${\mathcal{F}}_f: = \left\{ W \subset H \times {\mathbb{Z}}/q \, | \, \left( \{1\} \times {\mathbb{Z}}/q \right) \cap W = \{(1,1)\} \right\}$.
A manifold $M$ is ${\mathcal{F}}_s$-free if for every $x \in M$ the isotropy group $({\mathbb{Z}}/q)_x \neq {\mathbb{Z}}/q$, and it is ${\mathcal{F}}_f$-free if the restriction of the action to ${\mathbb{Z}}/q$ is free.
The classifying space $E {\mathcal{F}}_f$ has a free ${\mathbb{Z}}/q$-action and can be understood as the universal $H$-equivariant ${\mathbb{Z}}/q$-principal bundle $E_H {\mathbb{Z}}/q$ [@LueckUribe Theorem 11.4] . Hence $E {\mathcal{F}}_f = E_H {\mathbb{Z}}/q$ and its quotient $E {\mathcal{F}}_f / ({\mathbb{Z}}/q) = B_H {\mathbb{Z}}/q$ is the classifying space of $H$-equivariant ${\mathbb{Z}}/q$-principal bundles. By the isomorphism of , and since the action of ${\mathbb{Z}}/q$ is free, we get the following isomorphisms [@Stong-complex Proposition 3.1]: $$\begin{aligned}
\label{iso gf}
\Omega_*^G\{ {\mathcal{F}}_f\}(X) \cong \Omega_*^G(X \times E_H {\mathbb{Z}}/q) \cong \Omega_*^H(X \times_{{\mathbb{Z}}/q} E_H {\mathbb{Z}}/q).
\end{aligned}$$
Since both spaces $E_H {\mathbb{Z}}/q \times B_GU(k_i)$ and $E_H {\mathbb{Z}}/q \times B_H U(k_i)$ classify $H \times {\mathbb{Z}}/q$-equivariant $U(k_i)$-principal bundles over spaces with free ${\mathbb{Z}}/q$-action, we may take the maps $$B_H U(k_i) \to B_G U(k_i) \to B_H U(k_i),$$ where the left hand side map classifies the $G$-equivariant complex bundles such that the action of ${\mathbb{Z}}/q$ is trivial over the total space of the bundle, and the right hand side is the one that forgets the ${\mathbb{Z}}/q$-action, thus producing $G$-equivariant homotopy equivalences $$E_H {\mathbb{Z}}/q \times E_H U(k_i) \stackrel{\simeq}{\to} E_H {\mathbb{Z}}/q \times E_G U(k_i) \stackrel{\simeq}{\to} E_H {\mathbb{Z}}/q \times E_H U(k_i).$$
If we denote by $$\Gamma': = \prod_{i=1}^lB_HU(k_i)$$ and the map $\iota: \Gamma' \to \Gamma$ is the one that classifies trivial ${\mathbb{Z}}/q$-bundles over $H$-spaces, then the argument above implies that the isomorphism [@Stong-complex Proposition 3.2] holds: $$\begin{aligned}
\label{iso gf for classifying space}
\Omega_*^H\left( B_H {\mathbb{Z}}/q \times \Gamma' \right) \cong \Omega_*^G\left( E_H {\mathbb{Z}}/q \times \Gamma' \right) \underset{\cong}{\stackrel{\iota_*}{\to}}\Omega_*^G\{ {\mathcal{F}}_f\}\left(\Gamma \right).
\end{aligned}$$
Let $T$ be the generator of the group ${\mathbb{Z}}/q$ and denote by ${\mathbb{Z}}/p^t$ the subgroup generated by $T^{p^{s-t}}$. A manifold is ${\mathcal{F}}_f$-free if and only if $T^{p^{s-1}}$ acts freely and therefore a $({\mathcal{F}}_a, {\mathcal{F}}_f)$-manifold $M$ can be localized to the normal bundle of the fixed point set $M^{{\mathbb{Z}}/p}$ of the subgroup ${\mathbb{Z}}/p$. The normal bundle is a $G$-equivariant complex bundle over the trivial ${\mathbb{Z}}/p$ space and once it is classified to the appropriate spaces $C_{G, {\mathbb{Z}}/p}(k)$ of we obtain the isomorphism [@Stong-complex Proposition 3.4]: $$\begin{aligned}
\label{iso g,gf}
\Omega_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(X) \cong \bigoplus_{0\leq 2k \leq *} \Omega_{*-2k}^{G/ ({\mathbb{Z}}/p)}\left( X^{{\mathbb{Z}}/p} \times C_{G, {\mathbb{Z}}/p}(k) \right).
\end{aligned}$$
Applying the previous isomorphism to $\Gamma = \prod_{i=1}^lB_GU(k_i)$ we obtain that $$\Omega_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(\Gamma)$$ is a free $\Omega_*$-module on even-dimensional generators since both $\Gamma^{{\mathbb{Z}}/p}$ and $C_{G, {\mathbb{Z}}/p}(k)$ are products of spaces of the form $B_{G/ ( {\mathbb{Z}}/p)} U(l)$ and by induction we assumed that the evenness conjecture was true for groups of order less than the one of $G$ and spaces of this type. Therefore the long exact sequence for the pair of families $({\mathcal{F}}_a, {\mathcal{F}}_f)$ becomes: $$\begin{aligned}
\label{exact sequence g,gf}
0 \to \Omega_+^G\{{\mathcal{F}}_f\}(\Gamma) \to \Omega_+^G(\Gamma){\to} \Omega_+^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma)
\stackrel{\partial}{\to} \Omega_-^G\{{\mathcal{F}}_f\}(\Gamma) \to \Omega_-^G(\Gamma) \to 0.
\end{aligned}$$
A $({\mathcal{F}}_a, {\mathcal{F}}_s)$-free manifold $M$ once restricted to the action of ${\mathbb{Z}}/q$ becomes a ${\mathbb{Z}}/q$-manifold on which the boundary has no fixed points of the whole group. Therefore the manifold can be localized on the normal bundle of the fixed point set $M^{{\mathbb{Z}}/q}$ and the information of the normal bundle can be recorded with appropriate maps to the classifying spaces $C_{G, {\mathbb{Z}}/q}(k)$ of . Following the same proof as in Theorem \[theorem adjacent\] one obtains the following isomorphism [@Stong-complex Proposition 3.3]: $$\begin{aligned}
\label{iso g,gs}
\Omega_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(X) \cong \bigoplus_{0\leq 2k \leq *} \Omega_{*-2k}^{H}\left( X^{{\mathbb{Z}}/q} \times C_{G, {\mathbb{Z}}/q}(k) \right).
\end{aligned}$$ Since both $\Gamma^{{\mathbb{Z}}/q}$ and $C_{G, {\mathbb{Z}}/q}(k)$ are products of spaces of the form $B_HU(l)$, by the induction hypothesis we obtain that $ \Omega_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators.
In order to understand the image of the boundary map of Stong restricted the equivariant bordism groups to manifolds with a special type of $G$ action. Stong noticed that the image of the boundary map could be determined by restricting to manifolds on which the ${\mathbb{Z}}/q$-fixed points are of codimension 2 and therefore he studied the class of [*[special $G$ actions]{}*]{}.
Let $G= H \times {\mathbb{Z}}/q$ be a finite abelian group. The class of [*[special $G$ actions]{}*]{} is the collection of $G$-equivariant unitary manifolds $M$ satisfying:
- The restriction to a ${\mathbb{Z}}/q$-action is semi-free, i.e. for each $x \in M$ the isotropy group $({\mathbb{Z}}/q)_x$ is either ${\mathbb{Z}}/q$ or $\{1\}$.
- The set $M^{{\mathbb{Z}}/q}$ of fixed point sets has codimension 2 in $M$ and ${\mathbb{Z}}/q$ acts on the normal bundle of $M^{{\mathbb{Z}}/q}$ so that the generator $T$ of ${\mathbb{Z}}/q$ acts by multiplication by $e^{\frac{2 \pi i }{q}}$, or the fixed point set $M^{{\mathbb{Z}}/q}$ is empty.
The class of special $G$ actions is sufficiently large to permit all constructions done in section \[section bordism\], and for a pair of families $({\mathcal{F}}, {\mathcal{F}}')$ in $G$ we denote by $\overline{\Omega}_*^G\{{\mathcal{F}},{\mathcal{F}}'\}$ the equivariant homology theory defined by using only special $G$ actions. The inclusion of special $G$ actions in the class of all $G$ actions defines natural transformations of homology theories $$I_* \colon \overline{\Omega}_*^G\{{\mathcal{F}},{\mathcal{F}}'\} \to {\Omega}_*^G\{{\mathcal{F}},{\mathcal{F}}'\}$$ preserving the relations between these functors. The $G$-equivariant unitary bordism groups of special $G$ actions satisfy the following properties:
- The natural transformation $$I_* \colon \overline{\Omega}_*^G\{{\mathcal{F}}_f\} \stackrel{\cong}{\to} {\Omega}_*^G\{{\mathcal{F}}_f\}$$is an equivalence since every ${\mathcal{F}}_f$ action is a special $G$ action.
- The inclusion $({\mathcal{F}}_a, {\mathcal{F}}_f) \subset ({\mathcal{F}}_a, {\mathcal{F}}_s)$ induces an isomorphism $$\overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(X) \stackrel{\cong}{\to} \overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(X)$$ since ${\mathcal{F}}_s$-free special $G$ actions are ${\mathcal{F}}_f$-free.
- From the equation we get the isomorphism $$\overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(X) \cong \Omega_{*-2}^H(X^{{\mathbb{Z}}/q} \times B_HU(1)),$$ thus implying that $\overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(X)$ maps isomorphically to a direct summand in ${\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(X)$.
- For $\Gamma:=\prod_{i=1}^lB_GU(k_i)$ the induction hypothesis implies that $\overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators. Therefore the canonical maps $$\overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(\Gamma) \to {\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(\Gamma) \to {\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_s\}(\Gamma)$$ imply that $\overline{\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(\Gamma)$ also maps isomorphically to a direct summand in ${\Omega}_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_f\}(\Gamma)$.
Let us now concentrate on understanding the five-term exact sequence restricted to special $G$ actions $$\begin{aligned}
\label{exact sequence special g,gf}
0 \to \overline{\Omega}_+^G\{{\mathcal{F}}_f\}(\Gamma) \to \overline{\Omega}_+^G(\Gamma){\to} \overline{\Omega}_+^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma)
\stackrel{\partial}{\to} \overline{\Omega}_-^G\{{\mathcal{F}}_f\}(\Gamma) \to \overline{\Omega}_-^G(\Gamma) \to 0.
\end{aligned}$$ Note that the map $\iota_*:\Gamma' \to \Gamma$ induces the commutative diagram $$\xymatrixrowsep{0.5cm}
\xymatrix{
\Omega_{*-2}^H(\Gamma^{{\mathbb{Z}}/q} \times B_HU(1)) \ar[r]^{\cong} & \overline{\Omega}_*^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma) \ar[r]^\partial & \ \overline{\Omega}_{*-1}^G\{{\mathcal{F}}_f\}(\Gamma) \\
\Omega_{*-2}^H(\Gamma' \times B_HU(1)) \ar[r]^{\cong} \ar[u]& \overline{\Omega}_*^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma') \ar[u]^{\iota_*} \ar[r]^\partial & \ \overline{\Omega}_{*-1}^G\{{\mathcal{F}}_f\}(\Gamma') \ar[u]^{\iota_*}_{\cong}
}$$ where the middle homomorphism $\iota_*$ maps isomorphically $\overline{\Omega}_*^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma')$ into a direct summand in $\overline{\Omega}_*^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma)$ since $\Gamma'$ is mapped to one connected component of the fixed point set $\Gamma^{{\mathbb{Z}}/q}$. Therefore the image of the boundary homomorphism $\partial$ is the same in both cases.
In what follows we will study the induced boundary homomorphism $$\begin{aligned}
\label{boundary homomorpshim gamma'}
\Omega_{*-2}^H(\Gamma' \times B_HU(1)) \to \overline{\Omega}_{*-1}^G\{{\mathcal{F}}_f\}(\Gamma') \cong
{\Omega}_{*-1}^H(\Gamma' \times B_H {\mathbb{Z}}/q)\end{aligned}$$ using the Thom isomorphism, the long exact sequence for pairs and a particular model for $E_H{\mathbb{Z}}/q$.
Let ${\bf{C}}^\infty_H$ be a countable direct sum of all complex irreducible $H$-representations and consider the ${\mathbb{Z}}/q$ action on ${\bf{C}}^\infty_H$ such that the generator $T$ of ${\mathbb{Z}}/q$ acts by mutliplication with $e^{\frac{2 \pi i}{q}}$. The sphere $S({\bf{C}}^\infty_H)$ of vectors of norm 1 is an $G={\mathbb{Z}}/q \times H$ space on which ${\mathbb{Z}}/q$ acts freely and moreover is a ${\mathcal{F}}_f$-free space. Since the non-empty fixed point sets are infinite dimensional spheres we know that this sphere $S({\bf{C}}^\infty_H)$ is a model for $E_H{\mathbb{Z}}/q$. The Grassmannian $Gr_1 {\bf{C}}^\infty_H$ is a model for $B_HU(1)$ since ${\mathbb{Z}}/q$ acts trivially on the one dimensional vector spaces, and ${\mathbb{Z}}/q$ acts on the fibers of the canonical line bundle $\gamma_HU(1) \to B_HU(1)$ by multiplication by $e^{\frac{2 \pi i}{q}}$. To simplify the notation denote by $$\gamma_1 : = \gamma_HU(1)$$ and note that $S({\bf{C}}^\infty_H) \cong S(\gamma_1)$ where $S(\gamma_1)$ denotes the sphere bundle of $\gamma_1$.
Consider now the line bundle $\gamma_1^{\otimes q}$ over $B_HU(1)$ which is the $q$-fold tensor product of $\gamma_1$. The diagonal map $$\Delta: \gamma_1 \to \gamma_1^{\otimes q}, \ \ v \mapsto v \otimes \cdots \otimes v$$ is a $q$ to 1 map on the fibers of the line bundles and therefore it induces an $H$-equivariant homeomorphism $$S(\gamma_1)/( {\mathbb{Z}}/q) \cong S(\gamma_1^{\otimes q}).$$ Therefore we may take either $S(\gamma_1)/( {\mathbb{Z}}/q)$ or $S(\gamma_1^{\otimes q})$ as a model for $B_H{\mathbb{Z}}/q$. The Thom isomorphism $$\Omega_*^H(( D(\gamma_1^{\otimes q}), S(\gamma_1^{\otimes q})) \times \Gamma') \cong
\Omega_{*-2}^H(B_HU(1) \times \Gamma')$$ together with the long exact sequence for the pair $( D(\gamma_1^{\otimes q}), S(\gamma_1^{\otimes q}))$ and the induction hypothesis provides a four-term exact sequence $$\begin{aligned}
0 \to \Omega_+^H( \Gamma' \times S(\gamma_1^{\otimes q})) \to \Omega_+^H&( \Gamma' \times B_HU(1)) \to \\
& \Omega_{+}^H( \Gamma' \times B_HU(1)) \to \Omega_{-}^H( \Gamma' \times S(\gamma_1^{\otimes q})) \to 0,\end{aligned}$$ where the right hand side homomorphism is precisely the one of . Therefore we obtain that the boundary homomorphism of is surjective, and since by the induction hypothesis $ \Omega_+^H( \Gamma' \times B_HU(1))$ is a free $\Omega_*$-module, we conclude that $ \Omega_+^H( \Gamma' \times S(\gamma_1^{\otimes q}))$ is also a free $\Omega_*$-module.
Therefore we obtain the following commutative diagram with exact rows $$\xymatrixrowsep{0.4cm}
\xymatrixcolsep{0.4cm}
\xymatrix{
0 \ar[r] & {\Omega}_{+}^G\{{\mathcal{F}}_f\}(\Gamma) \ar[r] & {\Omega}_+^G(\Gamma) \ar[r] & {\Omega}_+^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma) \ar[r]^\partial & {\Omega}_{-}^G\{{\mathcal{F}}_f\}(\Gamma) \ar[r] &0 \\
0 \ar[r] &\overline{\Omega}_{+}^G\{{\mathcal{F}}_f\}(\Gamma) \ar[r] \ar[u]_{\cong}&\overline{\Omega}_+^G(\Gamma) \ar[r] \ar[u]& \overline{\Omega}_+^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma) \ar[r]^\partial \ar@{^{(}->}[u]& \overline{\Omega}_{-}^G\{{\mathcal{F}}_f\}(\Gamma) \ar[u] _\cong \ar[r] &0\\
0 \ar[r] & \overline{\Omega}_{+}^G\{{\mathcal{F}}_f\}(\Gamma') \ar[r] \ar[u]^{\iota_*}_{\cong} &\overline{\Omega}_+^G(\Gamma') \ar[r] \ar[u]& \overline{\Omega}_+^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma') \ar@{^{(}->}[u] \ar[r]^\partial & \ \overline{\Omega}_{-}^G\{{\mathcal{F}}_f\}(\Gamma') \ar[u]^{\iota_*}_{\cong} \ar[r] &0,
}$$ thus implying that $\Omega_-^G(\Gamma)=0$ and that $\Omega_+^G(\Gamma)$ is a free $\Omega_*$-module since both ${\Omega}_{+}^G\{{\mathcal{F}}_f\}(\Gamma)$ and ${\Omega}_+^G\{{\mathcal{F}}_a,{\mathcal{F}}_f\}(\Gamma)$ are free $\Omega_*$-modules.
Therefore the evenness conjecture holds for finite abelian $p$-groups [@Stong-complex].
The general case
----------------
The proof of the evenness conjecture for general finite abelian groups was done by Ossa [@Ossa] and is based on the proof of Stong for $p$-groups and appropriate localizations at different primes. For a finite abelian group $K$ denote by $Z_K:= {\mathbb{Z}}[1/|K|]$ the localization of the integers at the ideal generated by the order of $K$.
Let $G = K \times L$ with $K$ and $L$ finite abelian with $|K|$ and $|L|$ relatively prime and consider the homomorphism $\Omega_*^{K \times L}\{{\mathcal{F}}\} \to \Omega_*^{ L}\{{\mathcal{F}}\}$ which forgets the $K$ action and ${\mathcal{F}}$ is any family of subgroups of $L$ . Let us show that the localized homomorphism $$\Omega_*^{K \times L}\{{\mathcal{F}}\}(\Gamma)\otimes Z_K \to \Omega_*^{ L}\{{\mathcal{F}}\}(\Gamma) \otimes Z_K$$ is an isomorphism whenever $\Gamma:= \prod_{i=1}^l B_GU(k_i)$. Let us proceed by induction over $L$ and over the family $\{{\mathcal{F}}\}$.
For the trivial family ${\mathcal{F}}=\{\{1\}\}$ we obtain the isomorphism $$\Omega_*(BK \times BL \times \prod_i BU(k_i)) \otimes Z_K \stackrel{\cong}{\to}
\Omega_*( BL \times \prod_i BU(k_i))\otimes Z_K$$ since $\Omega_*(BK) \otimes Z_K \cong \Omega_* \otimes Z_K$.
Whenever the adjacent pair of families $( {\mathcal{F}}, {\mathcal{F}}')$ differ by $H \subset L$ we obtain the homomorphism of long exact sequences $$\xymatrixrowsep{0.4cm}
\xymatrixcolsep{0.4cm}
\xymatrix{
{} \ar[r] & \Omega_*^{K \times L}\{{\mathcal{F}}'\}(\Gamma) \ar[r] \ar[d] & \Omega_*^{K \times L}\{{\mathcal{F}}\}(\Gamma) \ar[r] \ar[d] &
\Omega_*^{K \times L/H}\{\{1\}\}(\Gamma^H \times \Gamma') \ar[r]\ar[d] & \\
{} \ar[r] & \Omega_*^{ L}\{{\mathcal{F}}'\}(\Gamma) \ar[r] & \Omega_*^{ L}\{{\mathcal{F}}\}(\Gamma) \ar[r] &
\Omega_*^{ L/H}\{\{1\}\}(\Gamma^H\times \Gamma') \ar[r] &
}$$ with $\Gamma'$ a disjoint union of products of spaces of the form $B_{K \times L/H}U(l)$. Tensoring with $Z_K$ induces an isomorphism on the left vertical arrow by the induction hypothesis on the families and an isomorphism on the right vertical arrow by the induction hypothesis on the group $L/H$. The 5-lemma implies the desired isomorphism.
Now let ${\mathcal{F}}$ be any family of subgroups of $K$ and denote by ${\mathcal{F}}\times \Phi$ the family of subgroups of $G$ whose elements are groups $J\times H$ with $J \in {\mathcal{F}}$ and $H$ any subgroup of $L$. Let us show by induction on ${\mathcal{F}}$ and on the group $K$ that the localized module $$\Omega_*^{K \times L}\{{\mathcal{F}}\times \Phi\}(\Gamma)\otimes Z_K$$ is a free $\Omega_* \otimes Z_K$-module on even-dimensional generators. Whenever ${\mathcal{F}}$ is the trivial family we have shown above that $$\Omega_*^{K \times L}\{\{1\} \times \Phi\}(\Gamma)\otimes Z_K \stackrel{\cong}{\to} \Omega^L(\Gamma) \otimes Z_K$$ is an isomorphism, and by the induction hypothesis we know that $\Omega^L(\Gamma)$ is a free $\Omega_*$-module on even-dimensional generators.
If the adjacent pair of families $({\mathcal{F}}, {\mathcal{F}}')$ differ by the subgroup $J$, then we obtain the long exact sequence $$\cdots \to \Omega_*^{K \times L}\{{\mathcal{F}}'\times \Phi\}(\Gamma) \to \Omega_*^{K \times L}\{{\mathcal{F}}\times \Phi\}(\Gamma) \to
\Omega_*^{K/J}\{\{1\}\}(\Gamma^{J\times L} \times \Gamma'') \to \cdots$$ where $\Gamma''$ is a disjoint union of spaces of the form $B_{K/J}U(l)$. Tensoring with $Z_K$ gives us free $\Omega_* \otimes Z_K$-modules on even-dimensional generators on the left hand side by the induction on families and free $\Omega_* \otimes Z_K$-modules on even-dimensional generators on the right hand side by the induction on the group $K$. By [@ConnerSmith Proposition 3.2] projective $\Omega_* \otimes Z_K$-modules are free, hence the middle term is also a free $\Omega_* \otimes Z_K$-module on even-dimensional generators.
Therefore we have proved that if $\Omega^L(\Gamma)$ is a free $\Omega_*$-module then $\Omega^{K \times L}(\Gamma)
\otimes Z_K$ is a free $\Omega_* \otimes Z_K$-module. Let us now write $G = P_1 \times \cdots \times P_k$ with $P_i$ its Sylow $p_i$-subgroup. Since the evenness conjecture holds for $p$-groups, we have that $\Omega_*^{P_i}(\Gamma)$ is a free $\Omega_*$-module and therefore $\Omega_*^G(\Gamma) \otimes {\mathbb{Z}}[1/[G :P_i]]$ is a free $\Omega_* \otimes {\mathbb{Z}}[1/[G :P_i]]$-module. Since the numbers $[G :P_i]$ are relatively prime it follows that $\Omega_*^G(\Gamma)$ is a free $\Omega_*$-module.
Therefore the evenness conjecture holds for finite abelian groups [@Ossa].
The equivariant unitary bordism groups for non-abelian groups. {#section non abelian}
==============================================================
The evenness conjecture has been shown to be true for the dihedral groups $D_{2p}$ with $p$ prime by Ángel, Gómez and the author [@AngelGomezUribe], for groups of order $pq$ where $p$ and $q$ are different primes by Lazarov [@Lazarov] and for the more general case of groups for which all its Sylow subgroups are cyclic by Rowlett [@Rowlett-metacyclic]. In these cases the group $G$ is a semidirect product ${\mathbb{Z}}/r \rtimes {\mathbb{Z}}/s$ of cyclic groups with $r$ and $s$ relatively prime [@Hall Theorem 9.4.3], and the study of the equivariant unitary bordism groups is also carried out by calculating the equivariant unitary bordism groups of adjacent pairs of families of subgroups as is done in the cyclic group case of section \[subsection cyclic\].
The main tool used by Rowlett to study the case on which all Sylow subgroups are cyclic is the equivariant unitary spectral sequence constructed by himself in [@Rowlett Prop.osition 2.1]. Suppose that $A$ is a normal subgroup of $G$ and that $Q=G/A$. A family ${\mathcal{F}}$ of subgroups of $A$ is called $G$-invariant if it is closed under conjugation by elements of $G$. Consider a pair $({\mathcal{F}}, {\mathcal{F}}')$ of $G$-invariant families of $A$ and note that $\Omega_*^A\{{\mathcal{F}}, {\mathcal{F}}'\}$ becomes a $Q$-module in the following way. Consider an $A$-manifold $M$ with action $\theta: A \times M \to M$ and take an element $g \in G$. Define a new action on $M$ by the map $g_* \theta: A \times M \to M$, $ g_*(a,m) := \theta(g^{-1}ag,m)$ and denote the action of $g$ on the bordism class $[M, \theta]$ by $\overline{g}[M,\theta]: =[M, g_* \theta]$. This action is trivial on elements of $A$ and therefore it boils down to an action of $Q$. Then there is a first quadrant spectral sequence $E^r$ converging to $\Omega_*^G\{{\mathcal{F}}, {\mathcal{F}}'\}$ whose second page is $$E^2_{p,q}\cong H_p(Q,\Omega_q^A\{{\mathcal{F}}, {\mathcal{F}}'\}).$$
In the case that both groups $A$ and $Q$ are cyclic of relatively prime order, the action of $Q$ on $\Omega_+^A\{{\mathcal{F}}, {\mathcal{F}}'\}$ factors through a permutation action on the free generators and therefore the second page is not difficult to calculate. If we take the family ${\mathcal{F}}_A$ of all subgroups of $A$, the second page of the spectral sequence becomes $H_q(Q, \Omega_q^A)$, and since $\Omega_*^A$ is a free $\Omega$-module on even-dimensional generators, then we obtain that $\Omega_+^G\{{\mathcal{F}}_A\}$ is a free $\Omega_*$-module. Moreover, the same explicit construction carried out in section \[subsection cyclic\] can be adapted in this case to show that the long exact sequence associated to the pair of families $\{{\mathcal{F}}_a, {\mathcal{F}}_A\}$, with ${\mathcal{F}}_a$ the family of all subgroups, becomes $$0 \to \Omega_+^G\{{\mathcal{F}}_A \} \to \Omega_+^G \to \Omega_+^G\{{\mathcal{F}}_a, {\mathcal{F}}_A\} \stackrel{\partial}{\to} \Omega_-^G\{{\mathcal{F}}_A\} \to 0.$$
The same argument as in shows that $\Omega_*^G\{{\mathcal{F}}_a, {\mathcal{F}}_A\}$ is a free $\Omega_*$-module on even-dimensional generators and therefore we conclude that $\Omega_-^G$ is zero and $\Omega_+^G$ is a free $\Omega_*$-module.
The spectral sequence defined above can also be used in order to understand the torsion-free part of the $G$-equivariant unitary bordism groups for any abelian group. Take any subgroup $A$ of $G$ and let $({\mathcal{F}}_A, {\mathcal{F}}_A')$ be the adjacent pair of families of $G$ which differ by the conjugacy class of $A$. Tensoring with the rationals we obtain an isomorphism $$\Omega_*^G \{{\mathcal{F}}_A, {\mathcal{F}}_A'\} \otimes {\mathbb{Q}}\cong \Omega_*^A \{{\mathcal{F}}_A, {\mathcal{F}}_A'\}^{W_A} \otimes {\mathbb{Q}}$$ where the right hand side consists of the $W_A$-invariant part and $W_A:= N_A/A$. Since $\Omega_*^A \{{\mathcal{F}}_A, {\mathcal{F}}_A'\}$ is a free $\Omega_*$-module on even-dimensional generators we obtain the isomorphism $$\Omega_*^G \otimes {\mathbb{Q}}\cong \bigoplus_{(A)}\Omega_*^A \{{\mathcal{F}}_A, {\mathcal{F}}_A'\}^{W_A} \otimes {\mathbb{Q}}$$ where $(A)$ runs over the conjugacy classes of subgroups of $G$ [@Rowlett Theorem 1.1], cf. [@tomDieck-Mackey Theorem 1]. In particular the torsion-free component of $\Omega_*^G$ is all of even degree.
Apart from the non-abelian groups in which all their Sylow subgroups are cyclic, there is no other finite non-abelian group for which the evenness conjecture has been shown to hold.
The main difficulty lies in the understanding of the equivariant bordism groups $\Omega_*^G\{{\mathcal{F}}\}( {}^{\widetilde{G}}B_GU(n))$ of the classifying spaces ${}^{\widetilde{G}}B_GU(n)$ associated to ${\mathbb{S}}^1$-central extensions $\widetilde{G}$ of $G$ for different families ${\mathcal{F}}$ of subgroups. These bordism groups are the ones appearing once we try to calculate the equivariant unitary bordism groups for adjacent pair of families. Any development in the understanding of these equivariant unitary bordism groups will shed light on the proof of the evenness conjecture for a bigger class of groups.
Conclusion {#section conclusion}
==========
The evenness conjecture for equivariant unitary bordism has been an important question in algebraic topology for more than forty years. The conjecture has been proved to hold only for compact abelian Lie groups and finite groups for which all their Sylow subgroups are cyclic, for all other groups the conjecture remains open. We do hope that the present summary of known results will help settle the conjecture in the near future.
|
---
abstract: 'For an evaporating black hole which is a radiation-black hole combined system, we express the entanglement entropy and the Page time in terms of the conformal time in the RST model. The entropy change of the black hole is nicely written in terms of Hawking flux. Integrating the first law of thermodynamics, we can obtain the decreasing black hole entropy and the increasing radiation entropy, and the entanglement entropy for this system based on the Page argument. We also obtain analytically the critical temperature to release black hole information, which corresponds to the Page time, and discuss the relation between the conserved total entropy and information recovering of the black hole in this model.'
author:
- Myungseok Eune
- Yongwan Gim
- Wontae Kim
bibliography:
- 'references.bib'
title: Note on explicit form of entanglement entropy in the RST model
---
Bekenstein has suggested that the entropy of a black hole is proportional to the area of the horizon [@Bekenstein:1972tm; @Bekenstein:1973ur; @Bekenstein:1974ax], and subsequently Hawking’s discovery has led to the result that the black hole has thermal radiation with the temperature $T_{\rm H} = \kappa_{\rm H} / 2\pi
$ [@Hawking:1974sw], where $\kappa_{\rm H}$ is the surface gravity at the event horizon. It has also been claimed that the black hole would eventually disappear completely through thermal radiation, which gives rise to information loss problem [@Hawking:1976ra]. However, if Hawking radiation plays a role of carrier of information, information will come out so slowly until the Page time [@Page:1993wv] when the entanglement entropy becomes maximum such that the dimension of radiation equals to that of the black hole in the Hilbert space. When the dimension of radiation is larger than that of the black hole, information is naturally contained in radiation. Moreover, it has been shown that in Ref. [@Page:1993wv] the above statistical analysis can be realized in the Callan-Giddings-Harvey-Strominger (CGHS) model [@Callan:1992rs] by taking into account the classical metric along with the corresponding constant temperature which is independent of black hole mass so that radiation does not reflect the back reaction of the geometry. Actually, the black hole entropy of a two-dimensional black hole with the back reaction was studied for the static case in Refs. [@Solodukhin:1994yz; @Solodukhin:1995te].
In this work, we are going to study the entanglement entropy based on the Page formulation using the RST model [@Russo:1992ax; @Russo:1992yh] to take into account back reaction of the geometry, which yields naturally the time-dependent geometry. The essential difficulty is to identify the time-dependent temperature which is quite awkward in standard thermodynamics. So we would like to assume that a radiation-black hole combined system is in equilibrium at each time such that the radiation temperature measured by the fixed observer at the future null infinity is identified with the black hole temperature. Then, the thermodynamic first law is also read off from the differential form of the energy conservation law [@Kim:1995wr], so that the entropy change of the black hole is neatly written in terms of Hawking flux. Integrating the first law of thermodynamics, we can obtain the decreasing black hole entropy and the increasing radiation entropy, and the entanglement entropy based on the Page argument [@Page:1993wv]. So, the total entropy is always constant while the total information is not conserved locally because of the time-dependent entanglement entropy; however, it is expected that the total information is recovered after complete evaporation of the black hole. Now, let us start with the RST model given by the action [@Russo:1992ax] $$\begin{aligned}
I &= I_{\rm DG} + I_f + I_{\rm PL} + I_{\rm
corr}, \label{action:total}\end{aligned}$$ with $$\begin{aligned}
I_{\rm DG} &= \frac{1}{2\pi} \int d^2 x \sqrt{-g}\, e^{-2\phi}
\left[R + 4(\nabla\phi)^2 + 4\lambda^2
\right], \label{action:DG} \\
I_f &= \frac{1}{2\pi} \int d^2 x \sqrt{-g} \left[- \frac12
\sum_{i=1}^{N} (\nabla f_i)^2 \right], \label{action:f} \\
I_{\rm PL} &= \frac{\kappa}{2\pi} \int d^2 x \sqrt{-g}
\left[-\frac14 R \frac{1}{\Box} R\right], \label{action:PL} \\
I_{\rm corr} &= \frac{\kappa}{2\pi} \int d^2 x \sqrt{-g}
\left[-\frac12 \phi R \right], \label{action:corr}\end{aligned}$$ where $\kappa = (N-24)/12$ which can be positive by taking into account the ghost decoupling term [@Strominger:1992th] and $\lambda$ is a cosmological constant. Eq. is added to obtain an exact black hole solution and it is reduced to the CGHS model without this term [@Callan:1992rs]. From the action , the equations of motion are given by $\Box f_i =0$, $e^{-2\phi} \left[R - 4 (\nabla\phi)^2 + 4\Box\phi +
4\lambda^2\right] + \frac{\kappa}{4} R =0$, and $ G_{\mu\nu} =
T_{\mu\nu}^f + T_{\mu\nu}^{\rm qt} $, where $ G_{\mu\nu} \equiv
\frac{2\pi}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu\nu}} \left(I_{\rm
DG} + I_{\rm corr}\right) = 2e^{-2\phi} \left[\nabla_\mu
\nabla_\nu \phi + g_{\mu\nu} \left((\nabla \phi)^2 - \Box\phi -
\lambda^2\right) \right] + \frac{\kappa}{2} \left(\nabla_\mu
\nabla_\nu \phi - g_{\mu\nu} \Box\phi \right)$. The energy-momentum tensors for matter are defined as $$\begin{aligned}
T_{\mu\nu}^f &\equiv - \frac{2\pi}{\sqrt{-g}} \frac{\delta
I_f}{\delta g^{\mu\nu}} \notag \\
&= \sum_{i=1}^N \left[\frac12 \nabla_\mu f_i \nabla_\nu f_i -
\frac14 g_{\mu\nu} (\nabla f_i)^2 \right], \label{T:cov:cl} \\
T_{\mu\nu}^{\rm qt} &\equiv - \frac{2\pi}{\sqrt{-g}} \frac{\delta
I_{\rm PL}}{\delta g^{\mu\nu}}. \label{T:cov:qt}\end{aligned}$$ It can be checked that the dilaton improved Bianchi identity for the RST model is satisfied, *i.e.,* $\nabla^\mu G_{\mu\nu} = 0$, which yields covariant conservation relations for classical matter and quantum matter as $\nabla^\mu T_{\mu\nu}^f = 0$ = $\nabla^\mu
T_{\mu\nu}^{\rm qt} = 0$. Thus, the definitions of the classical and the quantum energy-momentum tensors given by Eqs. and in the RST model are compatible with those of the CGHS model as long as covariant conservation relations are concerned. In the conformal gauge given by $ds^2 = e^{2\rho} dx^+ dx^-$, the classical energy-momentum tensor is written as $
T_{\pm\pm}^f = \frac12 \sum_{i=1}^N (\partial_\pm f_i)^2$ and $
T_{\pm\mp}^f =0$ and the quantum energy-momentum tensor is given by $T_{\pm\pm}^{\rm qt} = \kappa
\left[\partial_\pm^2 \rho - (\partial_\pm \rho)^2 - t_\pm (x^\pm)
\right]$ and $T_{\pm\mp}^{\rm qt} = -\kappa \partial_+ \partial_-
\rho$, which agrees with the quantum energy-momentum tensor introduced in the CGHS model [@Callan:1992rs]. The unknown functions $t_\pm(x^\pm)$ reflect the nonlocal property of the effective action. Of course, one may define the energy-momentum tensor for matter in a different way by including the contribution from Eq. because it is actually not unique. However, it can be well-defined at asymptotic future null infinity since it can be expressed by only the boundary function as $T_{\pm\pm}^{\rm qt} =
-\kappa t_{\pm} $ when we consider Hawking radiation in that region.
By introducing new variables given by $\chi =
\sqrt\kappa \rho - (\sqrt\kappa/2) \phi + e^{-2\phi}/\sqrt\kappa $ and $\Omega= (\sqrt\kappa/2) \phi + e^{-2\phi}/\sqrt\kappa$ for simplicity, the action can be written as [@Russo:1992ax; @Russo:1992yh], $$\begin{aligned}
I = \frac{1}{\pi} \int d^2 x \left[- \partial_+
\chi \partial_- \chi + \partial_+
\Omega \partial_- \Omega + \lambda^2 e^{(2/\sqrt\kappa)(\chi - \Omega)}
+ \frac12 \sum_{i=1}^N \partial_+ f_i \partial_- f_i
\right], \label{S:+-}\end{aligned}$$ and the two constraints are given by $\kappa t_\pm = -
\partial_\pm \chi \partial_\pm \chi + \sqrt\kappa \partial_\pm^2 \chi +
\partial_\pm \Omega \partial_\pm \Omega + \frac12
\sum_{i=1}^N \partial_\pm f_i \partial_\pm f_i$. The equations of motion derived from the action can be exactly solved. In the Kruskal coordinates where $\chi = \Omega$, the evaporating black hole formed by an incoming shock wave of $T^f_{++} = [M/(\lambda
x^+_0)] \delta(x^+ - x^+_0)$ is described by the solution of $\Omega
(x^+, x^-)= - \lambda^2 x^+ x^- / \sqrt\kappa - \frac{\sqrt\kappa}{4}
\ln(-\lambda^2 x^+ x^-) - \frac{M}{\lambda \sqrt\kappa x^+_0} (x^+ -
x^+_0) \Theta (x^+ - x^+_0)$, where the linear dilaton vacuum is chosen for $x^+< x^+_0$. An asymptotically static coordinate can be obtained from the coordinate transformations defined by $x^+ =
(1/\lambda) e^{\lambda \sigma^+} $ and $x^- = - (1/\lambda)
e^{-\lambda \sigma^-} - ( M/\lambda^2) e^{-\lambda \sigma^+_0}
\Theta(\sigma^+ - \sigma^+_0)$, where $\sigma^+_0 = \lambda^{-1}
\ln(\lambda x^+_0) $.
Note that the RST model is known to be quantum-mechanically inconsistent after appearance of the naked singularity [@Strominger:1994tn]. The curvature singularity and apparent horizon collide in a finite proper time and the singularity is naked after the two have merged [@Russo:1992ax]. In order to avoid the naked singularity, a vacuum state can be patched at the intersection point ($\sigma^+_s, \sigma^-_s$) of the singularity curve and the apparent horizon as shown in the Penrose diagram of Fig. \[fig:penrose\], where the intersection point is given by $\sigma^-_s = \sigma^+_0 + \lambda^{-1} \ln \left[(\lambda/M)
\left(\exp(4M/(\kappa\lambda)) - 1 \right) \right]$ and $\sigma^+_s = \sigma^-_s + \lambda^{-1} \ln (\kappa/4)$ [@Russo:1992ax; @Russo:1992yh; @Kim:1995wr; @Strominger:1994tn]. However, this patching procedure requires the thunderpop energy which is the negative classical energy emanated from the black hole. So we are going to mainly discuss the entanglement entropy of the RST model before the negative Bondi mass appears.
![It shows the Penrose diagram of the black hole formed by a shock wave at $\sigma^+ = \sigma^+_0$.[]{data-label="fig:penrose"}](penrose){width="48.00000%"}
On the other hand, from the covariant conservation law, one can get the ordinary conserved quantity by expanding the metric and dilaton fields around the linear dilaton vacuum. Then, the linearized equation of motion becomes $G^{(1)}_{ \mu \nu} = T^f_{\mu \nu} - G^{(2)}_{\mu
\nu}$ [@Weinberg:1972book], where $T^f_{\mu \nu}$ is a classical energy-momentum tensor, $G^{(1)}_{\mu \nu}$ is the linear perturbed part of $G_{\mu \nu}$, and $G^{(2)}_{\mu \nu}$ is the rest. Then, one can choose the time and space coordinate so that it is easy to show that the linearized equation of motion identically satisfies the ordinary conservation law, $\partial_\mu G^{(1) \mu 0} = 0,$ by the use of the linearized Bianchi identity [@Kim:1995jta]. It implies that the current defined as $J^\mu = T_f^{\mu 0} - G^{(2) \mu
0}$ satisfies the ordinary conservation law $\partial_\mu J^{\mu}=
0$. Thus we can define the Bondi mass $B(\sigma^-)$ which is the energy evaluated along the null line [@Bondi:1962px], $B(\sigma^-)
= (1/2)\int_{-\infty}^\infty d\sigma^+ G^{(1) - 0} (\sigma^+, \sigma^-
)$ while the ADM mass is calculated at the spatial infinity as $E_{\rm
ADM}(t) = \int_{-\infty}^\infty dq G^{(1)00} (t,q)$ [@Arnowitt:1962hi]. Using the integrated form of the linearized equation of motion, after some calculations the difference between the ADM mass and the Bondi mass can be obtained as [@Kim:1995wr] $$\label{bt}
E_{\rm ADM}(t)-B(\sigma^-) = \int_{-\infty}^{\sigma^-} d\sigma^-
\left. \left( T_{--}^f +T_{--}^{\rm qt} \right) \right|_{\sigma^+ \to
\infty}.$$ Note that the classical infalling energy-momentum tensor does not exist since it cannot appear in the asymptotic future null infinity. However, in the RST model there may exist the classical out-going negative energy density called the thunderpop energy at $\sigma^{-}_s$. From now on, we will consider radiation-hole combined system before the thunderpop appears, and then the conformal time is restricted to $-\infty < \sigma <\sigma^{-}_s$. It means that we can take vanishing out-going classical energy-momentum tensor in this analysis.
Next, the integrated Hawking flux is given by $ H(\sigma^-) =
\int_{-\infty}^{\sigma^-} d\sigma^- h(\sigma^-)$, where the Hawking flux is $h(\sigma^-) = \left. T_{--}^{\rm qt} \right|_{\sigma^+ \to
\infty}$. The Hawking flux is simply reduced to the boundary function as $h(\sigma^-) = - t_- (\sigma^-)$ since $\sigma^\pm$ is a quasi-static coordinate system at infinity, and so the fields approach the linear dilaton vacuum at $\sigma^+ \to \infty$. In this black hole, the Hawking radiation is written as $ h(\sigma^-) =
(\kappa\lambda^2/4) [1 - (1 + (M/\lambda) e^{\lambda (\sigma^- -
\sigma^+_0)} )^{-2} ]$ [@Callan:1992rs]. Note that the Bondi mass is the remaining energy after quantum-mechanical Hawking radiation has been emitted from the system. So it is plausible to regard the Bondi mass as a black hole mass in the quantum back-reacted model. From Eq. , we can get the conservation law as [@Kim:1995wr] $$\begin{aligned}
B(\sigma^-) + H(\sigma^-)=M, \label{E:BH}\end{aligned}$$ where $M$ is the ADM mass. The energy can be conserved in this evaporating black hole system so that the Bondi energy plus the Hawking radiation should be equal to the initial infalling energy by the scalar fields.
Now, we will assume the radiation-black hole combined system as a thermal equilibrium system for each conformal time $\sigma^-$ in order to apply the thermodynamic first law. Let us first relate the Hawking flux to the Hawking temperature in analogy with the static case [@KeskiVakkuri:1993xi], then one can read off the black hole temperature $T(\sigma^-)$ from the Hawking radiation by identifying $$h(\sigma^-) = \kappa \pi^2 T^2(\sigma^-),$$ which yields $$\begin{aligned}
T(\sigma) &= \frac{\lambda}{2\pi} \left[1 - \frac{1}{\left(1 +
\frac{M}{\lambda} e^{\lambda (\sigma^- - \sigma^+_0)}
\right)^2} \right]^{1/2}. \label{T:def}\end{aligned}$$ Note that it vanishes at $\sigma^- \to -\infty$ since the black hole did not radiate yet and the well-known Hawking temperature is recovered as $T_{\rm H} = \lambda/2\pi$ at $\sigma^- \to \infty$ which is compatible with the previous static results [@Page:1993wv; @KeskiVakkuri:1993xi]. Using the differential form of the energy conservation law , the change of the black hole entropy can be written as $$\begin{aligned}
\Delta S_h & = S_h(\sigma^-) - S_h^0 = \int \frac{dB}{T} \notag \\
&= - \pi \sqrt{\kappa} \int_{-\infty}^{\sigma^-} d\sigma^-
\sqrt{h(\sigma^-)}, \label{entropy:change}\end{aligned}$$ where $S_h^0$ denotes the entropy of the black hole at $\sigma^-
\to -\infty$. The entropy change is essentially due to Hawking radiation such that the entropy of the black hole is decreasing. From Eq. the entropy is calculated as $$\begin{aligned}
S_h (\sigma^-) &=\frac{2\pi M}{\lambda} - \frac{\pi\kappa}{2}
\left[\sec^{-1} \gamma(\sigma^-) + \ln\left(\gamma(\sigma^-) +
\sqrt{\gamma^2(\sigma^-) -1}\right) \right], \label{S:bh}\end{aligned}$$ where $\gamma(\sigma^-) = 1 + (M/\lambda) e^{\lambda(\sigma^- -
\sigma^+_0)} $ and we employed the fact that the entropy of the black hole is given by $S_h^0 = 2\pi M/\lambda$ at the initial time of $\sigma^- \to -\infty$ since the entropy of the black hole starts with the maximum thermal entropy of the area law, and at the same time the Hawking temperature is zero. As time goes on, the black hole entropy is decreasing according to the increasing Hawking temperature which amounts to $T_{\rm H} = \lambda/2\pi$ at $\sigma^- \to \infty$. Note that in the conventional thermodynamic analysis, the black hole entropy and the temperature are given as $S =
2\pi M/\lambda$, and $T_{\rm H} = \lambda/2\pi $.
On the other hand, for a system consisting of the black hole subsystem and the radiation subsystem, the entanglement entropy for $S_h, S_r \gg 1$ is given by the Page argument as [@Page:1993wv] $$\begin{aligned}
S_{\rm ent} &\simeq \left\{
\begin{array}{ll}
%\displaystyle
S_r - \frac12 e^{S_r-S_h} &\qquad
\mathrm{for}\ S_r\le S_h \\
\vspace{2mm}
%\displaystyle
S_h - \frac12 e^{S_h-S_r} & \qquad
\mathrm{for}\ S_r\ge S_h
\end{array}
\right., \label{S:ent}\end{aligned}$$ where $S_h $ and $S_r$ are the black hole entropy and the radiation entropy, respectively. Note that the total entropy of the system is preserved such that it is given as $S_r +
S_h = 2\pi M/\lambda$. The entanglement entropy has a maximum value at the Page time when the black hole emits a half of its initial Bekenstein-Hawking entropy, $i.e.$, $S_r =\pi M/\lambda$. Using Eq. , we can write the entanglement entropy explicitly in terms of $\sigma^-$, and it becomes $$\begin{aligned}
S_{\rm ent} (\sigma^-)&\simeq \frac{\pi\kappa}{2} \Bigg[\sec^{-1}
\gamma(\sigma^-) + \ln\left(\gamma(\sigma^-) +
\sqrt{\gamma^2(\sigma^-) -1}\right) \Bigg] \notag \\
&\quad - \frac12 \left[ \left(\gamma(\sigma^-) +
\sqrt{\gamma^2(\sigma^-) -1}\right) \exp \left(\sec^{-1}
\gamma(\sigma^-) - \frac{2 M}{\kappa \lambda}\right)
\right]^{\pi \kappa}\end{aligned}$$ for $\sigma^- \le \sigma^-_c$ and $$\begin{aligned}
S_{\rm ent} (\sigma^-)&\simeq \frac{2\pi M}{\lambda} - \frac{\pi\kappa}{2}
\left[\sec^{-1}
\gamma(\sigma^-) + \ln\left(\gamma(\sigma^-) +
\sqrt{\gamma^2(\sigma^-) -1}\right) \right] \notag \\
&\quad - \frac12 \left[ \left(\gamma(\sigma^-) +
\sqrt{\gamma^2(\sigma^-) -1}\right) \exp \left(\sec^{-1}
\gamma(\sigma^-) - \frac{2 M}{\kappa \lambda}\right)
\right]^{-\pi \kappa}\end{aligned}$$ for $\sigma^- \ge \sigma^-_c$. Note that the entanglement entropy becomes maximum at the conformal time of $\sigma^{-}_c$ which comes from the maximization of the entanglement entropy formally given in the closed form of $\gamma(\sigma^-_c) \cos [\ln(\gamma(\sigma^-_c) +
\sqrt{\gamma^2(\sigma^-_c) - 1}) - 2M/(\kappa\lambda) ] =1$.
![The solid, the dashed, and the thick-dotted lines show the behaviors of the entanglement entropy $S_{\rm {ent}}$, the black hole entropy $S_h$, and the thermodynamic radiation entropy $S_r$, respectively. The entanglement entropy has a maximum at $\sigma^-_c$ and it vanishes at $\sigma^-_p$.[]{data-label="fig:I:S.ent:S.r::sigma"}](ISeSr){width="50.00000%"}
That point is just the Page time expressed by the conformal time since the radiation entropy is the same with the black hole entropy as shown in Fig. \[fig:I:S.ent:S.r::sigma\]. The radiation entropy is monotonically increasing while the black hole entropy is monotonically decreasing, and their sum is constant.
By the way, there is a deficiency in this calculation that the black hole entropy is negative for $\sigma^- > \sigma^-_p$ because the Bondi mass in the RST model is negative due to the surplus Hawking radiation after $\sigma^{-}_p$ [@Strominger:1994tn] so that the present calculations are meaningful before $\sigma^-_p$. Moreover, the expression for the entanglement entropy based on the Page argument is valid only for many degrees of freedom as was noticed below such as $S_h, S_r \gg 1$. So, it seems to be inappropriate to discuss beyond the end point of the entropy in our formulation, and the calculation of the entanglement entropy becomes a good approximation around $\sigma^-_c$. One more thing to be mentioned is that we distinguished the definition of the entropy depending on the subsystem: the black hole entropy is defined by employing the Bondi mass, which is plausible in that the entropy change of the black hole should be negative because the black hole radiates while the entropy change of radiation should be positive because Hawking radiation is increased monotonically.
As for the naked singularity of the black hole, the black hole can generally form a singularity. However, as seen from the original work [@Page:1993wv], the black hole system was assumed to have many degrees of freedom such as $m, n \gg 1$ in order to formulate the system and derive the explicit form of the average value of the entanglement entropy of Eq. (\[S:ent\]). It means that even in spite of the small black hole, it should have many degrees of freedom in this formulation so that the black hole does not lose its mass completely and then the naked singularity is no more concerned. Based on this argument, we have employed the same entropy formula in the present RST model so that our result is also valid only for the many degrees of freedom just at the conformal time $\sigma^- \ll \sigma_p^-$ for which $S_h \gg 1$. Therefore, the entanglement entropy turns out to be well-defined only around $\sigma_c^-$ in Fig. \[fig:I:S.ent:S.r::sigma\] except the extreme limits of very small degrees of freedom. Furthermore, the advantage of the RST model in Ref. [@Russo:1992ax] is that it has been designed to be free from the naked singularity because the flat metric can be patched with the black hole metric when the singularity forms at $\sigma_s^-$ as seen from Fig. \[fig:penrose\].
As a result, we have obtained the decreasing black hole entropy, the increasing radiation entropy, the entanglement entropy, and the Page time in terms of the conformal time in the exactly soluble RST model. Moreover, we can find a Page temperature at the Page time since $\sigma^{-}_c$ was identified so that it becomes formally $T(\sigma^-_c)$ from Eq. . In other words, information is significantly leased above the critical temperature of $T({\sigma^{-}_c})$.
In Refs. [@Myers:1994sg; @Hayward:1994dw], the black hole entropy and increase theorem related to the second law of black hole thermodynamics have been studied for the RST model, and we would like to mention some differences between our work and them. First, the system in our work was divided into two subsystems so that the black hole has the black hole entropy and radiation has the radiation entropy, respectively, while there appears only a single system and the single entropy to define the black hole system in the previous works. Additionally, the entanglement entropy in this work has been defined throughout correlation between the two subsystems, so that the entropy in Refs. [@Myers:1994sg; @Hayward:1994dw] behaves like not the entanglement entropy but the radiation entropy in our work in the sense that it is always increasing as time goes on and the entropy change is always positive, which guarantees the second law of black hole thermodynamics.
In the original work done by Page in Ref. [@Page:1993wv], the system was divided into two subsystems; one is for the black hole with dimension $m$ and the other is for radiation with dimension $n$. The most important assumption is that these subsystems form a total system in a pure state in a Hilbert space of fixed dimension $mn$. It means that the total entropy is constant, and consequently $\Delta S=0$. Following this assumption in Ref. [@Page:1993wv], we also assumed that the total entropy should be constant in order to realize the Page argument in the RST model. In particular, for the case of $\Delta S>0$, the information will be eventually lost like ordinary thermal systems. Therefore, the requirement of the fixed total entropy is a sort of constraint based on the hypothesis that no information is lost in black hole formation and evaporation as was claimed in Ref. [@Page:1993wv].
W. Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MOE) (2010-0008359).
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[****]{}
Gauge principle: received and passive view {#passive}
==========================================
The received view of gauge theories and their underlying gauge principle is that we are able to derive the coupling of a formerly free field to an interaction field from the requirement of local gauge covariance. The usual derivative $\partial_\mu$ must be replaced by a covariant derivative \[cov-der1\] D\_= \_- iq A\_(x), where $A_\mu$ is a Lie algebra-valued 1-form—mathematically a connection in a principal fiber bundle, physically the gauge potential of an interaction field. However, this gauge argument would be close to a miracle, if the connection, which arises from the local symmetry requirement, were non-flat (i.e. with non-vanishing curvature).
But this, of course, is not—and can hardly be—the case. What rather happens is that we explicitly allow for a freedom in the choice of the position representation of the wave function (figuring as a free Schrödinger or Dirac field in the matter field equation from which the gauge principle starts). To see this immediately, consider a system of vectors $\Big\{\ket{\phi}\Big\}$ spanning an abstract Hilbert space $\Hil$, such that a wave function in the position representation $\ket{x}$ simply reads $\Psi(x) = \braket{x}{\phi}$. Accordingly, local gauge transformations are expressed as $\ket{x'} = e^{i \chi(x)} \ket{x} = \hat U \ket{x}$. Such a change of representation must apply as well to all operators $\hat O$ acting on $\Hil$, viz. $\hat O' = \hat U \hat O \hat U^+$. In particular, for the derivative operator we get \[cov-der2\] D\_= \_- i \_(x). Identifying the gradient of the phase with the gauge potential $A_\mu$ (multiplied by some charge $q$ in order to get the units[^1] straight) \[chi-alpha\] \_(x) = q A\_(x) leads to (\[cov-der1\]). Obviously, $A_\mu$ is a flat connection.
Thus, the celebrated gauge principle is not sufficient to ‘derive’ the coupling to a new interaction-field, but rather makes the built-in covariance under local gauge transformations explicit. We are, from the mere logic of the argument, not enforced to consider the connection term non-flat. Moreover, the Noether current connected to the covariance of the Dirac Lagrangian under global $U(1)$ transformations is just the probability density current $S^\mu=\bar\psi \gamma^\mu \psi$ and not the charge current $j^\mu=q \ S^\mu$, since there simply is no charge occurring in the Dirac equation. Again, in the standard textbook presentation the charge is put in by hand by means of (\[chi-alpha\]). Call all this the *passive view* on the gauge principle, since local gauge transformations are treated in full analogy to coordinates—coordinates, however, in the fiber bundle rather than in the space-time base space. The analogy is complete if we draw a comparison to the Levi-Civita connection in General Relativity. Here, Christoffel symbols already occur in the geodesic equation simply because of curvilinear coordinates in flat Minkowski space—without entailing a real gravitational field, i.e. non-vanishing Riemann curvature.
The non-observability of local phase transformations
====================================================
Let us now consider the gauge principle’s local phase transformations (a.k.a. gauge transformations of the first kind) in more detail. Under such transformations the wave function yields (x) ’(x) = (x) e\^[i (x)]{}. Thus, we obtain $\partial_\mu \psi(x) \to \partial_\mu \psi'(x) =
e^{i \chi(x)} \Big( \partial_\mu + i \partial_\mu \chi(x) \Big) \psi(x)$, which confirms the covariant derivative (\[cov-der2\]), i.e. Dirac equations $(i \partial_\mu \gamma^\mu - m) \psi(x) = 0$ and $(i D_\mu \gamma^\mu - m) \psi'(x) = 0$ are equivalent. We also get with $\hat{p}_\mu = - i \partial_\mu$ and $e^{i \chi}=e^{i px}$ the phase transformation behavior $\chi \to \chi - \partial_\mu \chi(x) \cdot x^\mu$, which leads to the holonomy \[holonomy\] = \_(x) d x\^. Of course, written as such and provided that spacetime is simply connected, expression (\[holonomy\]) is trivial and $\Delta \chi=0$. We will come back to this point in a moment.
As an intermediate step, let us ask for the possibility to observe local phase transformations. A widespread argument says that—in contrast to the passive view—the phase transformed wave function leads to new expectation values. We get, for instance, for the momentum operator $\hat{p}_\mu \psi' = (p + \partial_\mu \chi ) \psi'$ as opposed to $\hat{p}_\mu \psi = p \psi$. But this is of course misleading since we must use the transformed momentum operator $\hat{P}_\mu = \hat{p}_\mu - \partial_\mu \chi$ corresponding to (\[cov-der2\]) and, thus, $\hat{P}_\mu \psi' = \hat{p} \psi'$. Indeed, just like their global counterparts local phase transformations do not change any expectation values at all.
Another argument can be found in ’t Hooft (1980), who considers an ordinary double-split experiment showing an interference pattern. Inserting a phase shifter behind the slit in one of the two paths results in a corresponding shift of the interference pattern—which can be calculated from (\[holonomy\]). ’t Hooft now argues that the phase shifter can be seen as a realization of a local phase transformation. But this is impossible since local phase transformations would then change the holonomy, which is, however, invariant under (global and local) $U(1)$. What is rather observed in this case is the relative phase shift. Let $\psi = \psi_I + \psi_{II}$, where $\psi_I$ and $\psi_{II}$ are the two partial wave functions on the two paths $I$ and $II$, then, say, a $\lambda/4$-phase shifter in path $I$ corresponds to $\psi_I \to \psi_I e^{i \lambda/4}$ and hence $\psi \to \tilde{\psi} = \psi_I e^{i \lambda/4} + \psi_{II}$, where $\psi$ and $\tilde{\psi}$ obviously do have different expectation values in general (cf. Brading and Brown 2003). We must therefore very well distinguish between relative phase shifts and local phase transformations. The former are observable, the latter clearly are not.
“Phase charges” and Aharonov-Bohm effect
========================================
As already mentioned, only non-trivial holonomies are of interest. In this case the loop cannot be contracted to a point and the underlying fiber bundle is non-trivial, too. The double-slit experiment or the Mach-Zehnder interferometer are cases at hand. Another example is provided by the Aharonov-Bohm (AB) effect. Here we have an observable effect despite the fact that the connection is flat. At first glance, this seems to contradict the passive view statement from Sect. \[passive\], where we made the claim that from flat connections alone the physical situation can hardly be changed. However, the ultimate cause of the AB effect is of course the magnetic field confined to the region of the solenoid in the experimental setting. Nevertheless, there is no magnetic field in the region of the electron (the configuration space of the electron). Here the connection is flat, but the non-trivial holonomy of this connection causes observable effects. Thus, the lesson of the AB effect decidedly is that we must consider the holonomy outside the solenoid as a real entity! This conclusion is inevitable if we want to avoid an interpretation that either considers gauge-dependent quantities as physically real—such as the gauge potential—or does not conform to the idea of local action—which happens in the case where we allow for a non-local interaction between the confined magnetic field and the electron wave function (Eynck, Lyre, Rummell 2001). As a real entity, the holonomy couples to some property of the electron, and this, in the usual picture, is just the charge $q$. We shall, however, rather write $q^{(p)}$ (the superscript will become clear soon). In the AB case there is, indeed, more to (\[chi-alpha\]) than a mere rewriting of the phase function, since from (\[holonomy\]) together with (\[chi-alpha\]) we get the observed phase shift as a function of $q^{(p)}$ and the magnetic flux \[q\_phase\] q\^[(p)]{} A\_d x\^= q\^[(p)]{} F\_ d x\^ = q\^[(p)]{} \_[mag]{} .
But what really is the origin of the charge $q^{(p)}$ in (\[q\_phase\])? Obviously, it is a certain property of the electron, but it is not the property of a ‘usual’ charge being the source and drain of the electromagnetic field, since there is no field in the configuration space of the electron (or, at least, we may abstract from it). The AB effect itself has its origin in the topological nature of the non-trivial holonomy, since mappings $\setS^1 \to \setS^1$ from the electron’s configuration space to the gauge group are non-trivial and constitute the fundamental group $\pi_1(U(1)) = \setZ$. This holonomy now, as a physical entity, couples in some way to the electron, so we may very well interpret $q^{(p)}$ as the *coupling strength* between the electron wave function and the holonomy.
Let us call the ‘usual’ charge—the source and drain of the field—the active or passive *field charge* $q^{(f)}$—in full analogy to the active or passive gravitational mass (the charge of the gravitational field). By way of contrast, $q^{(p)}$ can be called the *phase charge*, since it originates from the phase factor of the wave function only. A first argument for this conceptual maneuver of distinguishing two different kinds of charges is that we have in fact no *a priori* reason to identify them. A more compelling physical argument is that $q^{(p)}$ is in principle measurable in a isolated experiment. Whereas $q^{(f)}$ is tested whenever we perform measurements where charges figure as sources or drains of the electromagnetic field (e.g. in measuring the Coulomb force between two electrons), $q^{(p)}$ only becomes visible, if we consider (\[holonomy\]) and its observable consequences. And this is exactly what happens in the AB effect.
All this leaves us with a remarkable conclusion: *It might very well be the case that particles with one and the same field charge do have different phase charges and, therefore, show different AB effects.* This can clearly be seen from (\[q\_phase\]). We have (on the l.h.s.) the observable shift of the interference pattern, which is proportional to the phase charge $q^{(p)}$ and the magnetic flux $\Phi_{mag}$ (on the r.h.s.). The latter can be measured independently by testing, for instance, the Lorentz force of the magnetic field on electrons and muons (which is obviously the same, because of their common field charge). However, from the way (\[q\_phase\]) was derived, we have no reason to expect the same interference shift in an AB experiment for electrons as compared to muons.
Note that the claim is not that in an actual experiment electrons and muons will show different AB effects. One would, in fact, expect the same $q^{(p)}$ for both. The above arguments are rather intended to show that there is no theoretical principle in our known physics which precludes the possibility of differing phase charges. Insofar as $q^{(p)}$ and $q^{(f)}$ are conceptually different, their equivalence \[gep\] q\^[(p)]{} = q\^[(f)]{} must be tested experimentally. Therefore the real claim here is that some of our Standard Model’s experiments—involving topological effects from flat connections—are in fact “null experiments” on the *equivalence principle of field and phase charge*.
A gauge theoretic equivalence principle
=======================================
In Lyre (2000) the attempt was made to propose a gauge theoretic generalization of the equivalence principle. The analogy is indeed striking: Field charges and gravitational mass appear in the field equations of a field theory (e.g. Maxwell or Einstein equations), whereas phase charges and inertial mass appear in the corresponding equations of motion (e.g. Dirac or geodesic equations).[^2] The generalized equivalence principle is then intended to fill the explanatory gap in the architecture of a gauge theory arising from the mere passive view of the gauge principle. This gap is simply the following: Let ${\cal L}_D = \bar\psi (i \partial_\mu \gamma^\mu - m) \psi$ be the Dirac Lagrangian and ${\cal L}_{coup}=j_\mu A^\mu$ the inhomogeneous ‘coupling’ part of the Lagrangian \[LagD\] [L]{}\_D’ = [L]{}\_D + [L]{}\_[coup]{}, which arises due to the replacement of the usual derivative by the covariant derivative (based on the requirement of local gauge covariance). As we have seen, however, the covariant derivative (\[cov-der2\]) only entails a flat connection in ${\cal L}_{coup}$. There is, on the other hand, the Maxwell theory with the Lagrangian \[LagM\] [L]{}\_M’ = [L]{}\_M + [L]{}\_[coup]{}. Here ${\cal L}_M = \frac{1}{4} F_{\mu\nu} F^{\mu\nu}$ is the kinetic term of the free Maxwell field and ${\cal L}_{coup}$ the inhomogeneous term including field charges. In this case the connection in ${\cal L}_{coup}$ is non-flat.
In order to arrive at the full Lagrangian of the Dirac-Maxwell gauge theory (or QED, analogously), we have to combine ${\cal L}_D'$ and ${\cal L}_M'$ in order to get \[LagDM\] [L]{}\_[DM]{} = [L]{}\_D + [L]{}\_[coup]{} + [L]{}\_M. But, obviously, ${\cal L}_{coup}$ figures in two different meanings here. Because of its mere passive nature, the gauge ‘principle’ does not allow to generalize from a flat connection in (\[LagD\]) to a non-flat connection in (\[LagM\]). Thus, the two ${\cal L}_{coup}$-terms cannot simply be identified.
There is, however, the possibility of non-trivial holonomies with a phase charge $q^{(p)}$ in the inhomogeneous term in (\[LagD\]), which we therefore indicate as ${\cal L}_{coup}^{(p)}$ as opposed to ${\cal L}_{coup}^{(f)}$ in (\[LagM\]). From the equivalence of field and phase charges (\[gep\]) we then get \_[coup]{}\^[(p)]{} = [L]{}\_[coup]{}\^[(f)]{} \_[coup]{} and, thus, the desired Lagrangian (\[LagDM\]). The $U(1)$ gauge theory is therefore not based on the physically vacuous gauge ‘principle’, a mere passive symmetry requirement, but on the gauge theoretic equivalence principle, which in the form of (\[gep\]) must be verified empirically.
Discussion and conclusion
=========================
The concept of a phase charge and, hence, the gauge theoretic equivalence principle is based on the existence of a non-trivial holonomy. Unfortunately, there are no AB effects for higher $SU(n)$ groups, since $\pi_1(SU(n)) = 1$. As far as other topological effects in gauge theories are concerned (e.g. instantons or $\theta$-vacua), it is not so clear whether they do perhaps only arise because of some clever approximations (e.g. the assumption of *vanishing fields* in the infinity of *Euclidean spacetime* for $SU(2)$-instantons). In these cases the holonomy should not be considered a really existing entity. This then means that our considerations only apply for $U(1)$ and that no simple extension of the gauge theoretic equivalence principle to the general Yang-Mills case will be possible.
One should also mention that the analogy between the gauge theoretic and the usual gravitational equivalence principle, striking as it may be, is of course only a heuristic one. As already mentioned, it certainly breaks down if we compare the concepts of phase charge and inertial mass. The reason for this might very well be seen in the classical nature of the latter and may perhaps be overcome in a future theory of quantum gravity.
In this respect the similarity of the above proposal of phase charges to Anandan’s conception of gauge fields as “interference fields” should perhaps be emphasized. For the particular case of general relativity, Anandan (1979) has shown that the “gravitational phase” is $\Delta \chi = \frac{mc}{\hbar} \int g_{\mu\nu} dx^{\mu\nu}$, that is the spacetime distance measured in Compton wavelengths and, hence, $m$ being the inertial mass! The same is true for neutron interferometry, the famous COW experiments. Here the phase shift is actually proportional to the product of inertial and gravitational mass, since the gravitational potential includes the latter (cf. Greenberger 1983). A possible objection against the phase charge proposal is that, in order to experimentally realize non-trivial bundles with flat connections, a region with a non-flat connection (i.e.a real field and, hence, field charges) must exist simultaneously. This seems to show that we cannot observe $q^{(p)}$ independently from $q^{(f)}$. But again: the two charges are of different origin in the sense that we take, for instance, the electron’s field charge to realize the electric current in the solenoid, but may perform the AB experiment with electrons, muons or tauons to measure their phase charge. Moreover, strictly speaking we cannot make an independent measurement of the inertial mass either. We always have to make the idealization of neglecting the gravitational masses of the measuring devices.
The reader may finally ask what the world were looking like if the proposed equivalence between phase and field charge would empirically be violated. The astonishing answer is that this would not change so much the phenomenology of our elementary particles world, since the violation only becomes visible in experiments where non-trivial holonomies with flat connections are involved. Conceptually, however, such a violation would leave us with a serious puzzle, since then—again—the explanatory gap in the logical structure of the empirically so eminently successful gauge theories persists. This gap is certainly not filled by the gauge ‘principle’, but may perhaps be construed as an equivalence principle between phase and field charges.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Harvey Brown and Alfred Pflug for helpful comments.
References {#references .unnumbered}
==========
: Anandan, J. (1979). Interference, gravity and gauge fields. , 53 A(2):221–249.
: Brading, K. and Brown, H. R. (2003). Observing gauge symmetry transformations? , fortcoming.
: Eynck, T. O., Lyre, H., and Rummell, N. v. (2001). A versus [B]{}! [T]{}opological nonseparability and the [A]{}haronov-[B]{}ohm effect. (PITT-PHIL-SCI00000404).
: Greenberger, D. M. (1983). The neutron interferometer as a device for illustrating the strange behaviour of quantum systems. , 55(4):875–905.
: Lyre, H. (2000). A generalized equivalence principle. , 9(6):633–647. (gr-qc/0004054).
: ’t Hooft, G. (1980). Gauge theories of the forces between elementary particles. , 242(6):104–138.
[^1]: If not otherwise stated we set $c=\hbar=1$ throughout this paper.
[^2]: The presentation in Lyre (2000) was insufficient, since no isolated experiment for $q^{(p)}$ was proposed. This is the main task of the present paper. A rather rhetoric move is to rename the misleading term ‘inertial charge’ of the former paper into ‘phase charge’ here.
|
---
abstract: 'Our analysis on the two magnitude-limited samples of LINERs suggests a correlation between $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$ or f(25)/f(60), and Hubble-type index at $>$99.99% significance level. As $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$ and f(25)/f(60) are considered as the indicators of star-formation activity and AGN activity, respectively, our result suggests that LINERs with higher AGN activity may have a lower star-formation contribution. The ones with highest AGN activity and lowest star-formation contributions are ellipticals. All well-studied LINER 1s belong to this group. On the other hand, LINERs with higher star-formation activity present lower AGN contributions. We find all well-studied LINER 2s in this parameter space. Most of LINERs having inner ring structures belong to this group. Statistics with other indicators of star-formation or AGN activity (nulear mass-to-light ratio at H band, and the ratio of X-ray-to-UV power) provide further evidence for such a trend. We have seen that along with the evolution of galaxies from late-type spirals to early-type ones, and up to ellipticals, the intensity of AGN activity increases with decreasing star-formation contributions, The above analyses may suggest a possible connection between the host galaxies and nuclear activities, and it might also indicate a possible evolutionary connection between AGN and starburst in LINERs.'
author:
- 'S.J. Lei, J.H. Huang, W. Zheng, L. Ji, Q.S. Gu'
title: A Possible Evolutionary Connection between AGN and Starburst in LINERs
---
Introduction
============
It was the study of Simkin, Su, and Schwarz (1980) that revealed a more frequent occurrence of outer and inner rings in the host galaxies of Seyferts than those of non-Seyfert galaxies. This pioneering investigation has been confirmed recently (Hunt & Malkan 1999) of the possible connection between the host galaxies and nuclear activities.
The galaxy morphology can be modified by environmental effects, such as mergers or interactions. It could be formed, on the other hand, as an evolutionary sequence (Martinet 1995), without galaxy interactions.
The role of minor mergers in the formation of AGN has been tested by Corbin (2000) . A very intriguing result obtained from this study is that the nuclear spectral type of galaxies is strongly dependent on their Hubble type instead of environmental effects, indicating the existence of a connection between the host galaxy and nuclear activity.
When Condon et al. suggested (1982), and Rush et al. (1996) conducted, to distinguish starbursts from AGN by using of FIR-radio correlation, the possible connection between the host galaxy and nuclear activity is involved. The nuclear spectral types of sources can be designated by their global properties. The FIR-radio correlation has proven a useful tool successfully classifying the LINERs’ type, LINER 1s and LINER 2s, in recent investigation (Ji et al.2000). In their study on a small sample of LINERs, a higher frequency of inner rings in LINERs found by Hunt & Malkan (1999) turned out to be related to LINER 2s.
It is worth exploring the astrophysical implication of this result, especially for understanding a possible connection between the host galaxy and nuclear activity, or even further a possible connection between starbursts and AGN in terms of the evolution of galaxies.
LINERs’ sample
==============
To faciliatate the study of active nuclei and their hosts, it is desirable to utilize a LINER sample on the basis of host galaxy flux (Krolik 1999). One of the best sample in the literature is the spectroscopic survey by Ho, Filippenko and Sargent (1997, HFS hereafter), from which we have a magnitude-limited LINER sample, with a statistically meaningful size of 94.
A second, and deeper, magnitude-limited LINER sample has been constructed from a catalog of LINER compiled by Carrillo et al. (1999, CMD hereafter), which is based on criteria of m$_{\rm B} \leq$ 14.5 and $\delta \geq
0^{\circ}$. The CMD sample contains 223 sources.
Statistical Results
===================
Star Formation and AGN Activity
-------------------------------
In order to test a possible connection between the host galaxies and nuclear activities, we need to study the star-formation activity among LINER samples, along with their AGN activity. As an indicator of star-formation activity, the luminosity ratio of $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$ has been widely used (Keel 1993, Huang et al. 1996, Hunt & Malkan 1999). On the other hand, the mid-infrared flux ratio f(25)/f(60) has been known a good indicator of AGN activity (Hunt & Malkan 1999).
Following Hunt & Malkan (1999), we make plots of f(25)/f(60) vs Hubble-type index for the HFS sample and the CMD sample in Fig 1a and 1b respectively. Significant correlations between them, $>$99.99% for the HFS sample and $>$99.99% for the CMD sample, imply a possible relation of nuclear activity to the evolutionary status of galaxy, similar to the results obtained by Corbin (2000) and Hunt & Malkan (1999) for AGN.
The different distributions shown in Fig 1a and 1b lie mostly in the lack of LINERs in late-types for the HFS sample, as compared with those for the CMD sample. It is something related to the selection effect in the HFS sample, which makes it harder to find AGN in late-type galaxies. Though the CMD sample is not complete for sources with 12.5 $< m_{\rm B} \leq$ 14.5, the situation is much improved. This selection effect may cause some problems on the study of evolutionary status of AGN activity along the Hubble sequence. However, it is not crucial for our investigation as the statistics given above indicated.
In Fig 2 we have shown the star-formation indicator of $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$ vs Hubble-type index for the CMD sample only. The correlation is significant at the level of $>$99.99%. What the trend in Fig 2 shows is certainly consistent with the fact that Hubble-type sequence is also a sequence of star formation rate (Kennicutt 1992). The most active star formation occurs in late-type spirals. In fact, the majority of LINERs with inner rings are located in these types, see Fig 1 and 2.
Further Test
------------
There are other significant indicators for star-formation or AGN activity, such as the nuclear mass-to-light ratio at H band, $M/L_{\rm H}$, and the ratio of X-ray-to-UV power, [$\nu f_{\nu}$]{}(2-10kev)/[$\nu f_{\nu}$]{}(1300Å) (Oliva et al. 1999; Maoz et al. 1998).
A diagram of $M/L_{\rm H}$ vs f(25)/f(60) for Seyferts is shown in Fig 3a with the $M/L_{\rm H}$ data obtained by Oliva et al. (1999), which again demonstrates the f(25)/f(60) ratio as a good indicator of AGN activity. The sources in the lower-left part of the diagram are Seyfert 2 with circum-nuclear starbursts, while those in the upper-right part are Seyfert 1 without circum-nuclear starbursts. The trend shown in Fig 3a might provide evidence for the evolutionary hypothesis for Seyferts (Hunt & Malkan 1999).
Following this approach, we made a similar plot for LINERs in Fig 3b, with the $M/L_{\rm H}$ data obtained by Devereux et al. (1987). The correlation between $M/L_{\rm H}$ and f(25)/f(60) is significant at 97% level. Two sources shown with symbol of star, NGC1052 and NGC4486, are those with the $M/L_{\rm H}$ data obtained by Oliva et al. (1999). The systematic difference between the two observations is obvious, as it can be seen from different positions of the same object, NGC1052.
Further evidence for this evolutionary trend might be provided using another AGN activity, i.e. the ratio of X-ray-to-UV power. Due to the limited LINERs with both X-ray detections and the UV observations, we use the Einstein data (Carrillo et al. 1999) for the X-ray fluxes instead of the hard X-ray fluxes used by Maoz et al. (1998). The UV fluxes are retrieved from the IUE archive data.[^1] The correlation between $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$ and [$\nu f_{\nu}$]{}(0.2-4.0kev)/[$\nu f_{\nu}$]{}(1300Å) is shown in Fig 4, significant at 99% level.
Discussions
===========
A key point in AGN unification hypotheses is to claim that Seyfert 1s and 2s are intrinsically similar, and the different types of nuclear activity are caused by different view angles. Increasing evidence (Malkan et al. 1998; Hunt et al. 1999) suggests that Seyfert nuclei may be same objects seen at different evolutionary sequence. Seyfert 2s tend to reside in later morphological types than Seyfert 1s. Starbursts and massive stars play an important energetic role in a significant fraction of Seyfert 2s (Heckman 1999), while few Seyfert 1s have such circum-nuclear starbursts (Gonzalez Delgado et al. 1997; Hunt et al. 1997).
The data points in the lower right part of Fig. 4 of the diagram have higher ratio of [$\nu f_{\nu}$]{}(0.2-4.0kev)/[$\nu f_{\nu}$]{}(1300Å) and lower ratio of $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$, in other words, have higher AGN activity and lower star-formation contributions. It is the region the well studied LINERs with broad emission or with active AGN are located at, e.g. NGC3998, NGC4486, NGC4594(Larkin et al. 1998; Ho 1998; Nicholson et al. 1998; Maoz et al. 1998). We have found few LINERs with inner rings in this region. Sources with the highest AGN activity and the lowest star-formation contributions are absolutely ellipticals. It has been known that most of X-rays detected in ellipticals come from the extended halo where elliptical galaxies reside. Removing these ellipticals from the sample used in Fig 4, the correlation between $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$ and [$\nu f_{\nu}$]{}(0.2-4.0kev)/[$\nu f_{\nu}$]{}(1300Å) becomes significant at 90% level.
On the contrary, the sources shown in the upper left part of the figure have lower ratio of [$\nu f_{\nu}$]{}(0.2-4.0kev)/[$\nu f_{\nu}$]{}(1300Å) and higher ratio of $L_{{\,{\rm FIR}}}/L_{{\,{\rm B}}}$, i.e. lower AGN contribution and higher star-formation activity. It is the region some well studied LINERs supported by massive stars are found, e.g. NGC4569, NGC4736, NGC4826, NGC5194 (Maoz et al. 1998; Larkin et al. 1998; Alonso-Herrero et al. 1999; Barth & Shields 2000). The above argument holds for Fig. 3b, too. The AGN-supported LINERs, NGC1052, NGC4579, NGC4486, are distributed in the region with higher ratio of f(25)/f(60) (strong AGN activity) and higher nuclear $M/L_{\rm H}$ (small starburst contribution). While NGC7217, a well studied starburst(SB)-supported LINER, is in the region having higher star-formation activity and lower AGN contribution.
If adopting the Hubble sequence as an evolutionary sequence (Pfenniger, Combes & Martinet 1994, Martinet 1995), the correlations in Fig 1 and 2 show that the AGN and the star-formation activity are anticorrelated to the evolutionary status of galaxies. Along with the evolution from late-type spirals to early-type ones, and up to ellipticals, the intensity of AGN activity in LINERs increases with decrease of star-formation contributions, consistent with what the statistics performed for Fig 3 and 4 implied. Due to the large sample size used for Fig 1 and 2, more well studied LINERs, AGN- or SB-supported, can be found in their corresponding regions, e.g. SB-supported LINERs NGC404, NGC3504, NGC5055, NGC7743 (Maoz et al. 1998; Alonso-Herrero et al. 1999; Larkin et al 1998), and AGN-supported ones NGC2639, NGC3718, NGC4203, NGC4278, NGC6500 (Alonso-Herrero et al. 1999; Ho 1998; Iyomoto et al. 1998; Shields et al. 2000; Falcke et al. 1999; Maoz et al. 1998).
Probably, this evolutionary trend might indicate processes in which massive black holes in LINERs are competing with the starbursts for the inflowing gas, a suggestion proposed by de Carvalho & Coziol (1999).
The LINERs residing in late-type spirals, with strong star-formation and lower AGN activity, might be at an early evolutionary stage. They are LINER 2s. The growing bulges of galaxies are bound to be followed by the evolution of galaxies from late-type to early-type spirals. The formation of a massive black hole may turn out to be a natural evolution of the massive bulges of galaxies (Norman & Scoville 1988). Ellipticals, S0 galaxies and very early-type spirals containing LINERs with broad emission or active AGN have massive bulges for the formation of massive or supermassive black holes (Wandel 1999; Margorrian et al. 1998), being active enough to suppress the circum-nuclear starbursts in LINERs. These LINERs might be at late evolutionary stages of LINERs, they are LINER 1s.
In conclusion, the above analyses may suggest a possible connection between the host galaxies and nuclear activities, and it might also indicate a possible evolutionary connection between AGN and starburst in LINERs.
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[^1]: Based on INES data from the IUE satellite
|
---
abstract: |
For $q\in(0,1),$ let $B_q$ denote the limit $q$-Bernstein operator. In this paper, the distance between $B_q$ and $B_r$ for distinct $q$ and $r$ in the operator norm on $C[0,1]$ is estimated, and it is proved that $1\leqslant \|B_q-B_r\|\leqslant 2,$ where both of the equalities can be attained. To elaborate more, the distance depends on whether or not $r$ and $q$ are rational powers of each other. For example, if $r^j\neq q^m$ for all $j,m\in \mathbb{N},$ then $\|B_q-B_r\|=2,$ and if $r=q^m, m\in \mathbb{N},$ then $\|B_q-B_r\|=2(m-1)/m.$
[**Keywords:**]{} Limit $q$-Bernstein operator, Peano kernel, positive linear operators
[**2010 MSC:**]{} 47A30, 47B30, 41A36
author:
- Sofiya Ostrovska and Mehmet Turan
title: '**The distance between two limit $q$-Bernstein operators**'
---
\[section\] \[theorem\][Definition]{} \[theorem\][Lemma]{} \[theorem\][Observation]{} \[theorem\][Example]{} \[theorem\][Corollary]{} \[theorem\][Remark]{}
[*Atilim University, Department of Mathematics, Incek 06836, Ankara, Turkey*]{}\
[*e-mail: sofia.ostrovska@atilim.edu.tr, mehmet.turan@atilim.edu.tr*]{}\
Introduction and statement of results
=====================================
The limit $q$-Bernstein operator $B_q$ can be viewed as an analogue of the Szász-Mirakyan operator pertinent the Euler distribution, also known as the $q$-deformed Poisson distribution, see [@charalam Ch. 3, Sec. 3.4] and [@india]. The latter is used in the $q$-boson theory, which delivers a $q$-deformation of the quantum harmonic oscillator formalism [@bieden]. Going into details, the $q$-deformed Poisson distribution defines the distribution of energy in a $q$-analogue of the coherent state [@bieden; @jing]. The $q$-analogue of the boson operator calculus is recognized as an indispensable area within theoretical physics. It brings out explicit expressions for the representations of the quantum group $SU_q(2),$ which plays an important role in various problems such as exactly solvable lattice models of statistical mechanics, integrable model field theories, conformal field theory, only to mention a few. For additional information the reader is referred to [@qgroup]. Therefore, linear operators related to the $q$-deformed Poisson distribution, in particular the limit $q$-Bernstein operator, are of significant interest for applications.
The operator $B_q$ also emerges as a limit for a sequence of $q$-Bernstein polynomials in the case $0<q<1.$ Over the past years, the limit $q$-Bernstein operator has been studied widely from different perspectives. Its approximation, spectral, and functional-analytic properties, probabilistic interpretation, the behavior of iterates, and the impact on the analytic characteristics of functions have been examined. See, for example, [@nazimhigher; @parametric; @wangmeyer]. The review of obtained results along with extensive bibliography can be found in [@JAM].
Let $q>0.$ For any $a\in{\mathbb C},$ as given in [@askey Ch. 10], we denote: $$(a;q)_0:=1, \quad
(a;q)_k:=\prod_{s=0}^{k-1}(1-aq^s), \quad
(a;q)_\infty:=\prod_{s=0}^{\infty}(1-aq^s).$$
$\cite{jat}$ For each $q\in(0,1),$ and $f\in C[0,1],$ the limit $q$-Bernstein operator is defined by $f\mapsto B_qf$ where $$\begin{aligned}
B_q (f;x) =
\begin{cases}
\displaystyle \sum_{k=0}^{\infty} f(1-q^k)p_k(q;x) & \textrm{if } \: x\in[0,1) \\
f(1) & \textrm{if } \: x = 1
\end{cases} \label{bq}\end{aligned}$$ in which $$\begin{aligned}
p_k(q;x)=\frac{x^k (x;q)_\infty}{(q;q)_k}. \label{pk}\end{aligned}$$
As it can be readily seen that $B_q(f;x)$ is defined by the values of $f$ on the set $$\begin{aligned}
{\mathbb J}_q:=\{1-q^k: k\in{\mathbb N}_0\}.\end{aligned}$$
Further, Euler’s identity [@askey page 490, Corollary 10.2.2] $$\begin{aligned}
\label{euler}
\frac{1}{(x;q)_\infty} = \sum_{k=0}^{\infty} \frac{x^k}{(q;q)_k}, \quad |x|<1 , \quad |q|<1,\end{aligned}$$ implies that $$\begin{aligned}
\sum_{k=0}^\infty p_k(q;x)=
\begin{cases}
1, & x\in[0, 1) \\
0, & x=1.
\end{cases} \label{sumpk}\end{aligned}$$
Formulae and show that $B_q$ is a positive linear operator on $C[0,1]$ with $\|B_q\|=1.$ Recently, the continuity of the operator $B_q$ with respect to the parameter has been examined in [@manal] where the outcome below has been presented.
For every $f\in C[0,1],$ one has: $$\lim_{q\to a}B_q(f;x)= B_a(f;x)$$ and the convergence is uniform on $[0,1].$
This demonstrates the continuity of $B_q$ in strong operator norm. The aim of the current paper is to investigate whether the continuity persists with respect to the topology produced by the uniform operator norm. It turns out that in this topology, $\{B_q\}_{q\in(0,1)}$ forms a discrete set of operators where each $B_q$ is an isolated point so that $\|B_q-B_r\| \geqslant 1$ whenever $q\neq r.$
The reasoning of the present paper is based essentially on the next theorem, which in itself can be of interest. The idea of the proof is attributed to a statement made by a Mathoverflow user under the nickname ‘fedja’, see [@fedja].
\[thmfed\] Let $q\in (0,1)$ and $m\geqslant 2$ be an integer. Then, for all $x\in[0,1]$ and all $k\in{\mathbb N}_0,$ the following inequality holds: $$\begin{aligned}
p_{mk}(q;x) \leqslant p_k(q^m;x). \label{eqfed}\end{aligned}$$
Obviously, by the triangle inequality, $\|B_q-B_r\| \leqslant 2.$ In some cases, the equality is attained, as claimed by the below-mentioned result.
\[thmqr2\] Let $q,r\in[0,1].$ If ${\mathbb J}_q \cap {\mathbb J}_r=\{0\},$ then $\|B_q - B_r \| = 2.$
Now, comes the case ${\mathbb J}_q\cap{\mathbb J}_r\neq\{0\}.$ This situation occurs when $r^j = q^m$ for some positive integers $j$ and $m$ and reveals:
\[thmBqrmj\] Let $0<r<q<1$ and $r^j=q^m$ where $j$ and $m$ are mutually prime positive integers. Then $\|B_q - B_r \| \geqslant \frac{2(m-1)}{m}.$
In the case when $j=1,$ i.e., ${\mathbb J}_{q^m}={\mathbb J}_r\subset {\mathbb J}_q,$ the exact value of $\|B_q-B_r\|$ has been obtained.
\[thmBqr1\] Let $q\in (0,1)$ and $r=q^{m}$ for some integer $m\geqslant 2.$ Then $\|B_q - B_r \| = \frac{2(m-1)}{m}.$
\[corBqr\] For any $q\neq r,$ one has $1 \leqslant \|B_q-B_r\| \leqslant 2.$ The inequalities are sharp in the sense that both equalities are attained.
Some auxiliary results
======================
In this section, some results which will later contribute in proving our theorems, are presented. To begin with, we point out the following:
\[obs\] For any positive integer $m,$ one has: $$\begin{aligned}
\label{limpmk}
\lim_{x\to 1^-} \sum_{k=0}^{\infty} p_{mk}(q;x)=\frac{1}{m}.\end{aligned}$$
Clearly, $$\begin{aligned}
\sum_{k=0}^\infty p_{mk}(q;x)=(x;q)_\infty \sum_{k=0}^{\infty} \frac{x^{mk}}{(q;q)_{mk}} \leqslant
(x;q)_\infty \sum_{k=0}^{\infty} \frac{x^{mk}}{(q;q)_{k}} = \frac{(x;q)_\infty}{(x^m;q)_\infty}\end{aligned}$$ where the identity is used. Therefore, $$\begin{aligned}
\sum_{k=0}^\infty p_{mk}(q;x)\leqslant \frac{(x;q)}{(x^m;q)_\infty}=\frac{1-x}{1-x^m}\, \frac{(qx;q)_\infty}{(qx^m;q)_\infty} \to \frac{1}{m}\quad \text{as} \quad x\to 1^-.\end{aligned}$$ On the other hand, $$\begin{aligned}
\sum_{k=0}^\infty p_{mk}(q;x)&=(x;q)_\infty \sum_{k=0}^{\infty} \frac{x^{mk}}{(q;q)_{mk}} \geqslant
(x;q)_\infty \sum_{k=0}^{\infty} \frac{x^{mk}}{(q;q)_{\infty}}\\
&= \frac{(x;q)_\infty}{(q;q)_\infty (1-x^m)} =
\frac{1-x}{1-x^m}\, \frac{(qx;q)_\infty}{(q;q)_\infty} \to \frac{1}{m}\quad \text{as} \quad x\to 1^-.\end{aligned}$$
\[lemint\] If, for every $k\in{\mathbb N},$ inequality holds for $x\in[0, 1-q^{mk+m/2}],$ then it holds for all $x\in[0,1].$
Clearly, $[0,1]=\{1\}\bigcup_{k=1}^\infty [0, 1-q^{mk+m/2}].$ For $x=1,$ the inequality is obvious. Now, by , inequality can be expressed as $$\frac{x^{mk}(x;q)_\infty}{(q;q)_{mk}} \leqslant \frac{x^k(x;q)_\infty}{(q^m;q^m)_k}$$ or, equivalently, $$\begin{aligned}
u_k(x):=x^{(m-1)k}\prod_{j=0}^{k-1}\prod_{\ell=1}^{m-1} \frac{1}{1-q^{\ell+mj}} \leqslant
\prod_{j=0}^{\infty}\prod_{\ell=1}^{m-1} \frac{1}{1-q^{\ell+mj}x}. \label{umk}\end{aligned}$$ Clearly, $$\max_{j\in{\mathbb N}} u_j(x)=u_k(x)$$ on $[1-\alpha_{k-1}, 1-\alpha_k]$ where $\alpha_k$ can be found from the equation $u_{k+1}(x)=u_k(x).$ Therefore, if for every $k\in{\mathbb N},$ holds on $[0, 1-\alpha_k],$ then it holds on $[1-\alpha_{k-1}, 1-\alpha_k],$ and, as a result, on $[0,1].$ That is why, is going to be proved on $[0,1-q^{mk+m/2}] \supseteq [0, 1-\alpha_k]$ for every $k\in{\mathbb N}.$ To justify this inclusion, one has to show that $\alpha_k \geqslant q^{mk+m/2}.$ Indeed, $$\begin{aligned}
(1-\alpha_k)^{m-1}&=\prod_{\ell=1}^{m-1}(1-q^{mk+\ell}) \\
&= \sqrt{\prod_{\ell=1}^{m-1}\left[(1-q^{mk+\ell})(1-q^{mk+m-\ell})\right]} \\
&= \prod_{\ell=1}^{m-1} \sqrt{1-q^{mk+\ell}-q^{mk+m-\ell}+q^{2mk+m}} \\
& \leqslant \prod_{\ell=1}^{m-1} \sqrt{(1-q^{mk+m/2})^2} = (1-q^{mk+m/2})^{m-1}\end{aligned}$$ by virtue of the arithmetic-geometric mean inequality. This completes the proof.
\[lemrho\] Let $\rho(t)=1/(e^t+e^{-t}-2),$ $t>0.$ Then, for all $s, t >0,$ the following inequality is valid: $$\begin{aligned}
\rho(s+t) \leqslant e^{-s}\rho(t). \label{rho}\end{aligned}$$
Equivalently, one may prove that $1/\rho(s+t) \geqslant e^{s}/\rho(t),$ that is, $e^{-t}-2 \leqslant e^{-2s-t}-2e^{-s}.$ If $s=0,$ then both sides are equal, while for $s>0,$ the derivative of the right hand side with respect to $s$ is positive, which yields the statement.
For the sequel, a special quadrature formula to approximate $\int_a^b f(t)dt$ is needed. More precisely, we set, for $m\geqslant 2,$ $$\begin{aligned}
\label{qm}
Q_m(f;a,b):=\frac{b-a}{m-1}\sum_{j=1}^{m-1} f\left(a+\frac{b-a}{m}\,j\right).\end{aligned}$$ It is not difficult to see that the quadrature formula gives the exact value of the integral for polynomials of degree at most 1. Denote by $R_{a,b}(f)$ the error in this approximation, i.e., $$\begin{aligned}
\label{err}
R_{a,b}(f)=\int_a^b f(t)dt - Q_m(f;a,b).\end{aligned}$$
The error is given by $$\begin{aligned}
\label{peano}
R_{a,b}(f)=\int_a^b K_{a,b}(t) f''(t)dt\end{aligned}$$ where $$\begin{aligned}
\label{kab}
K_{a,b}(t)&=
\begin{cases}
\frac{1}{2}(t-a)^2 & {\rm if } \:\: t\in[a, a+h_1] \\
\frac{1}{2}\left(t-a-\frac{mh_1k}{m-1}\right)^2+\frac{mh_1^2k(m-k-1)}{2(m-1)^2} & {\rm if }\:\: t\in[a+kh_1, a+(k+1)h_1], 1\leqslant k \leqslant m-2\\
\frac{1}{2}(b-t)^2 & {\rm if } \:\: t\in[b-h_1,b]
\end{cases}\end{aligned}$$ and $h_1=\frac{b-a}{m}.$
By Peano’s Theorem (see, for example [@stoer Theorem 3.2.3, page 123]), the error is expressed by where $K_{a,b}(t)=R_{a,b}((x-t)_+)$ and $$\begin{aligned}
(x-t)_+=
\begin{cases}
x-t & {\rm if } \:\: x \geqslant t, \\
0 & {\rm if } \:\: x < t.
\end{cases}\end{aligned}$$ Here, $(x-t)_+$ is considered as a function of $x.$ Plain calculation of $R_{a,b}((x-t)_+)$ using results in .
In what follows, given $h>0,$ denote by $K(t)$ the $h$-periodic function on ${\mathbb R}$ such that $K(t)=K_{0,h}(t)$ for $t\in[0,h]$ where $K_{a,b}(t)$ is given by . In other words, $K(t+h)=K(t)$ for all $t\in{\mathbb R}$ and $$\begin{aligned}
\label{kern}
K(t)=
\frac{1}{2}\, t^2-\frac{hk}{m-1} \, t+\frac{h^2k(k+1)}{2m(m-1)} \quad {\rm for }\:\: t\in[kh_1, (k+1)h_1],\:\: 0\leqslant k \leqslant m-1\end{aligned}$$ where $h_1=h/m.$
For all $m\geqslant 2,$ the following inequality holds: $$\begin{aligned}
\int_0^h K(t) \frac{dt}{t^2} \geqslant 8 \int_h^{3h} K(t) \frac{dt}{t^2}. \label{intk}\end{aligned}$$
For $j=0,1,2,$ set $$I_j:=\int_0^h \frac{K(t)}{(t+jh)^2}dt.$$ The inequality is equivalent to $I_0\geqslant 8(I_1+I_2).$ Now, $$\begin{aligned}
I_0 &=\int_{0}^{h} \frac{K(t)}{t^2} \, dt = \int_{0}^{h_1} \frac{K(t)}{t^2}\,dt+\sum_{k=1}^{m-1} \int_{kh_1}^{(k+1)h_1} \frac{K(t)}{t^2}\,dt \\
&=\frac{h}{2m}+\sum_{k=1}^{m-1}\left[\frac{h}{2m}-\frac{hk}{m-1}\ln\left(1+\frac{1}{k}\right)+\frac{h^2k(k+1)}{2m(m-1)}\left(\frac{1}{kh_1}-\frac{1}{(k+1)h_1}\right)\right]\\
&=h-\frac{h}{m-1}\sum_{k=1}^{m-1} k\ln\left(1+\frac{1}{k}\right)\end{aligned}$$ which gives $$\begin{aligned}
\frac{(m-1)I_0}{h} = m-1-m\ln m + \ln(m!).\end{aligned}$$ On the other hand, for $j\geqslant 1,$ using and the substitution $x=t+jh,$ one obtains $$\begin{aligned}
I_j &=\int_{0}^{h} \frac{K(t)}{(t+jh)^2} \, dt = \sum_{k=0}^{m-1} \int_{kh_1}^{(k+1)h_1} \frac{K(t)}{(t+jh)^2} \, dt.\\
&=\sum_{k=0}^{m-1} \int_{jh+kh_1}^{jh+(k+1)h_1} \left[\frac{1}{2}-h\left(j+\frac{k}{m-1}\right)\, \frac{1}{x}+h^2\left(\frac{j^2}{2}+\frac{jk}{m-1}+ \frac{k(k+1)}{2m(m-1)}\right)\frac{1}{x^2}\right]\,dx\\
&=\frac{h}{2}-h\sum_{k=0}^{m-1}\left(j+\frac{k}{m-1}\right)\ln\left(1+\frac{1}{jm+k}\right)+S_j\end{aligned}$$ where $$\begin{aligned}
S_j &= \frac{h^2}{h_1}\sum_{k=0}^{m-1}\left(\frac{j^2}{2}+\frac{jk}{m-1}+ \frac{k(k+1)}{2m(m-1)}\right)\left(\frac{1}{jm+k}-\frac{1}{jm+k+1}\right)\\
&=\frac{h}{2}\left[j^2\left(\frac{1}{j}-\frac{1}{j+1}\right)+\frac{1}{m-1}\sum_{k=0}^{m-1}[2mjk+k(k+1)]\left(\frac{1}{jm+k}-\frac{1}{jm+k+1}\right)\right]\\
&=\frac{h}{2}\left\{\frac{j}{j+1}+\frac{1}{m-1}\sum_{k=0}^{m-1}\left[1-jm(jm+1)\left(\frac{1}{jm+k}-\frac{1}{jm+k+1}\right)\right]\right\}\\
&=\frac{h}{2}\left[\frac{j}{j+1}+\frac{m}{m-1}-\frac{jm(jm+1)}{m-1}\left(\frac{1}{jm}-\frac{1}{jm+m}\right)\right]
=\frac{h}{2}.\end{aligned}$$ Therefore, $$\begin{aligned}
I_j =h-h\sum_{k=0}^{m-1}\left(j+\frac{k}{m-1}\right)\ln\left(1+\frac{1}{jm+k}\right)\end{aligned}$$ or $$\begin{aligned}
\frac{I_j}{h} =1-j\ln\left(1+\frac{1}{j}\right)-\frac{m}{m-1}\ln(jm+m)+\frac{\ln[(jm+m)!]-\ln[(jm)!]}{m-1}.\end{aligned}$$ With the help of Stirling’s formula $$\begin{aligned}
\sqrt{2\pi} n^{n+1/2} e^{-n+1/(12n+1)} < n! < \sqrt{2\pi} n^{n+1/2} e^{-n+1/(12n)},\end{aligned}$$ one gets $$\begin{aligned}
\frac{I_1+I_2}{h} &= 2+\ln 2 -2\ln 3 - \frac{m[\ln(2m)+\ln(3m)]}{m-1}+
\frac{\ln[(3m)!]-\ln(m!)]}{m-1} \\
& \leqslant 2+\ln 2 -2\ln 3 - \frac{m[\ln(2m)+\ln(3m)]}{m-1}\\
& \qquad + \frac{1}{m-1}\left[\left(3m+\frac{1}{2}\right)\ln(3m)-\left(m+\frac{1}{2}\right)\ln(m)-2m+\frac{1}{36m}-\frac{1}{12m+1}\right]\end{aligned}$$ and as a result $$\begin{aligned}
\frac{(m-1)(I_1+I_2)}{h} \leqslant -2+\frac{5}{2} \ln 3 - \ln 2 + \frac{1}{36m}-\frac{1}{12m+1},\end{aligned}$$ while $$\begin{aligned}
\frac{(m-1)I_0}{h} \geqslant -1 + \ln{\sqrt{2\pi m}} + \frac{1}{12m+1}.\end{aligned}$$ The needed inequality $I_0\geqslant 8(I_1+I_2)$ for all $m\geqslant 2$ follows from the fact that $$\begin{aligned}
-1+ \ln\sqrt{2\pi m} +\frac{1}{12m+1} \geqslant 8\left(-2 + \frac{5}{2} \ln 3 - \ln 2 + \frac{1}{36m} - \frac{1}{12m+1}\right),\end{aligned}$$ or equivalently, $$\begin{aligned}
\ln\sqrt{2\pi m} +\frac{9}{12m+1} - \frac{2}{9m} + 15 - 20 \ln 3 + 8\ln 2 \geqslant 0.\end{aligned}$$ To see this, let $$\begin{aligned}
\theta(x)= \ln\sqrt{2\pi x} +\frac{9}{12x+1} - \frac{2}{9x} + 15 - 20 \ln 3 + 8\ln 2.\end{aligned}$$ Then, $$\begin{aligned}
\theta'(x)= \frac{1}{2x} - \frac{108}{(12x+1)^2} + \frac{2}{9x^2} > \frac{1}{2x} - \frac{108}{(12x)^2} + \frac{2}{9x^2}
= \frac{18x-19}{36x^2} > 0 \quad \text{for} \quad x\geqslant 2.\end{aligned}$$ Hence, for all $m\geqslant 2,$ one has: $$\theta(m)\geqslant \theta(2) \approx 0.0073,$$ which completes the proof.
In the case $m=2,$ an alternative proof is presented in [@fedja].
Let $\rho(t)$ be the function from Lemma \[lemrho\]. Then $$\int_0^h K(t) \rho(t) dt \geqslant 8 \int_h^{3h} K(t) \rho(t) dt.$$
The statement is a consequence of the fact that $t\mapsto \frac{1}{t^2 \rho(t)}=\frac{e^{t}+e^{-t}-2}{t^2}$ is an increasing function for $t>0.$
\[corh0\] For $h \leqslant h_0:=\ln 4,$ $$\int_0^h K(t) \rho(t) dt \geqslant e^{3h/2} \int_h^{3h} K(t) \rho(t) dt.$$
Next, for a given $f:[a,b]\to {\mathbb R},$ denote by $E_{a,b}$ the error in the composite quadrature formula to approximate $\int_a^b f(t) dt$ when the interval $[a,b]$ is divided into $n$ subintervals of equal length $h$ and the rule is applied on each subinterval. That is, $$\begin{aligned}
E_{a,b}=\int_a^b f(t)dt- \sum_{j=1}^{n} Q_m(f;a+(j-1)h, a+jh), \label{errab}\end{aligned}$$ where $h=(b-a)/n.$ If $b=\infty,$ we take $n=\infty.$
\[lemeras\] Let $f(t)=-\ln(1-e^{-t}),$ $t>0.$ Then, for all $a>0$ and any step size, one obtains: $$\begin{aligned}
E_{s+a,\infty} \leqslant e^{-s} E_{a,\infty}.\end{aligned}$$
By Peano’s Theorem on the integral representation of the error term, $$E_{s,\infty}=\int_s^{\infty} f''(t) K(t-s)dt,$$ where $K(t)$ is defined by . Since $f''(t)=\rho(t)$, from Lemma \[lemrho\], application of yields: $$\begin{aligned}
E_{s+a,\infty}=\int_{s+a}^{\infty} \rho(t) K(t-s-a) dt =\int_{a}^{\infty} \rho(t+s) K(t-a) dt \leqslant e^{-s} \int_a^\infty \rho(t)K(t-a) dt =e^{-s}E_{a,\infty}.\end{aligned}$$
\[intst\] If $f(t)=-\ln(1-e^{-t}),$ $t>0,$ then for all $S,T>0,$ there holds: $$\int_S^\infty f(t) dt \geqslant -ST + \int_0^T f(t)dt.$$
It can be observed geometrically since $f(t)$ is a decreasing continuous function symmetric about the line $y=x.$
Proofs of the main results
==========================
As it was done in the proof of Lemma \[lemint\], inequality is equivalent to $$\begin{aligned}
x^{(m-1)k}\prod_{j=0}^{k-1}\prod_{\ell=1}^{m-1} \frac{1}{1-q^{\ell+mj}} \leqslant
\prod_{j=0}^{\infty}\prod_{\ell=1}^{m-1} \frac{1}{1-q^{\ell+mj}x}.\end{aligned}$$ Taking the logarithm of both sides leads to $$\begin{aligned}
-(m-1)k\ln\left(\frac{1}{x}\right)+\sum_{j=0}^{k-1}\sum_{\ell=1}^{m-1} \ln\left(\frac{1}{1-q^{\ell+mj}}\right) \leqslant
\sum_{j=0}^{\infty}\sum_{\ell=1}^{m-1} \ln\left(\frac{1}{1-q^{\ell+mj}x}\right).\end{aligned}$$ Set $h=\ln(1/q^m),$ $S=\ln(1/x),$ $T=kh$ and $f(t)=-\ln(1-e^{-t}).$ Then the inequality becomes $$\begin{aligned}
-ST+\sum_{j=0}^{k-1} \frac{h}{m-1}\sum_{\ell=1}^{m-1}f(jh+\frac{h}{m}\ell)
\leqslant \sum_{j=0}^{\infty} \frac{h}{m-1}\sum_{\ell=1}^{m-1}f(S+jh+\frac{h}{m}\ell)\end{aligned}$$ which can be written as $$\begin{aligned}
-ST+\sum_{j=0}^{k-1} Q_m(f;jh,(j+1)h)
\leqslant \sum_{j=0}^{\infty} Q_m(f;S+jh,S+(j+1)h).\end{aligned}$$ The sums in the last inequality can be viewed as the composite quadrature formulas for the integrals $\int_0^T f(t)dt$ and $\int_S^\infty f(t)dt,$ respectively. Therefore, by , one has $$\begin{aligned}
-ST+\int_0^T f(t)dt - E_{0,T}
\leqslant \int_S^\infty f(t)dt - E_{S,\infty}.\end{aligned}$$ Using Lemma \[intst\], if one can show that $$\begin{aligned}
E_{0,T} \geqslant E_{S,\infty} \label{ineq}\end{aligned}$$ for $h\leqslant h_0,$ the proof will be complete for $q\geqslant 1/2.$ Due to Lemma \[lemint\], we need only to deal with the case $x\in[0,1-q^{mk+m/2}]$ or $$\begin{aligned}
e^{-S} \leqslant 1-e^{-T-h/2}. \label{est}\end{aligned}$$ By Lemma \[lemeras\], one derives that $E_{S,\infty} \leqslant e^{-S}E_{0,\infty}$ and also $E_{T,\infty} \leqslant e^{-T+h} E_{h,\infty}$ As $f''(t)=\rho(t),$ using Corollary \[corh0\] along with Peano’s Theorem implies that $E_{h,3h}\leqslant e^{-3h/2}E_{0,h}$ whenever $h\leqslant h_0.$ Thence, $$\begin{aligned}
E_{h,\infty} = E_{h,3h}+E_{3h,\infty} \leqslant e^{-3h/2} E_{0,h}+e^{-2h}E_{h,\infty} \leqslant e^{-3h/2}E_{0,\infty},\end{aligned}$$ whence $$\begin{aligned}
E_{0,T}=E_{0,\infty}-E_{T,\infty} \geqslant \left(1-e^{-T-h/2}\right)E_{0,\infty} \geqslant e^{-S}E_{0,\infty} \geqslant E_{S,\infty}\end{aligned}$$ due to . This is the desired inequality . To finish the proof, we observe that for all $q\in(0, \frac{1}{2}),$ $x\in[0,1]$ and $i\in{\mathbb N},$ it is true that $$\frac{x}{1-q^i} \leqslant \frac{1}{1-q^ix}$$ and thence, for $q\in(0, \frac{1}{2})$ and $x\in[0,1]$ $$\prod_{j=0}^{k-1}\prod_{\ell=1}^{m-1} \frac{x}{1-q^{\ell+mj}}
\leqslant
\prod_{j=0}^{k-1}\prod_{\ell=1}^{m-1} \frac{1}{1-q^{\ell+mj}x}
\leqslant
\prod_{j=0}^{\infty}\prod_{\ell=1}^{m-1} \frac{1}{1-q^{\ell+mj}x}.$$ The proof is complete.
For given $\varepsilon>0,$ one can find $\delta>0$ such that $p_0(q;x)<\varepsilon/4$ for $x\in[1-2\delta, 1-\delta].$ Because of , for $N\in{\mathbb N},$ one may write: $$\begin{aligned}
\sum_{k=1}^N p_k(q;x) = 1 - p_0(q;x)-\sum_{k=N+1}^{\infty} p_k(q;x), \quad x\in[0,1).\end{aligned}$$ Notice that the series in converges uniformly on any closed subinterval of $[0,1),$ in particular, on $[0,1-\delta].$ Therefore, there exists $N_0\in{\mathbb N},$ such that $$\begin{aligned}
\sum_{k=N+1}^\infty p_k(q;x) < \frac{\varepsilon}{4}, \quad \text{for all} \quad x\in[0,1-\delta], \quad N>N_0.\end{aligned}$$ Hence, on $[1-2\delta, 1-\delta],$ there holds: $$\begin{aligned}
\sum_{k=1}^N p_k(q;x) > 1-\frac{\varepsilon}{2}. \label{1Npk}\end{aligned}$$ Apply this procedure to find $N_1, N_2$ and $\delta$ satisfying both $$\begin{aligned}
\sum_{k=1}^{N_1} p_k(q;x) > 1-\frac{\varepsilon}{2} \quad \text{and} \quad
\sum_{k=1}^{N_2} p_k(r;x) > 1-\frac{\varepsilon}{2}\end{aligned}$$ for $x\in[1-2\delta, 1-\delta].$ Setting $N=\max\{N_1, N_2\},$ one obtains $$\begin{aligned}
\sum_{k=1}^{N} [p_k(q;x)+p_k(r;x)] > 2-\varepsilon \quad \text{for} \quad x\in[1-2\delta, 1-\delta].\end{aligned}$$ At this point, for every $N\in \mathbb{N},$ consider a function $f_N\in C[0,1]$ such that $\|f_N\|=1$ and $$\begin{array}{ll}
f_N(1-q^k) = 1 & \mathrm{when} \;\; \:\: k = 1,2,\dots , N,\\
f_N(1-r^k) = -1 & \mathrm{when} \;\; \:\: k =1,2,\dots , N,\\
f_N(1-q^k) = f_N(1-r^k)= 0 & \mathrm{when}\;\; k \neq 1,2,\dots , N,.
\end{array}$$ Then, $$\left(B_q-B_r\right)f(x)=\sum_{k=1}^{N} [p_k(q;x)+p_k(r;x)] > 2-\varepsilon \quad \text{for} \quad x\in[1-2\delta, 1-\delta].$$ Since, $$\|B_q-B_r\| \geqslant \|\left(B_q-B_r\right)f(x)\| \geqslant 2-\varepsilon$$ and as $\varepsilon >0$ has been chosen arbitrarily, one concludes that $\|B_q-B_r\| \geqslant 2.$ On the other hand, by the triangle inequality, one has $\|B_q-B_r\| \leqslant 2,$ and the statement follows.
Let $f\in C[0,1].$ Then, for $x\in [0,1),$ $$\begin{aligned}
\left(B_q-B_r\right)f(x)
&=\sum_{k=0}^\infty f(1-q^k) p_k(q;x)-\sum_{k=0}^\infty f(1-r^k) p_k(r;x) \\
&=\sum_{m|k} f(1-q^k) p_k(q;x)+\sum_{m\nmid k} f(1-q^k) p_k(q;x) \\
& \qquad -\sum_{j|k} f(1-r^k) p_k(r;x)-\sum_{j\nmid k} f(1-r^k) p_k(r;x) \\
&=\sum_{k=0}^\infty f(1-q^{mk}) \left[p_{mk}(q;x)-p_{jk}(r;x)\right] \\
&\qquad +\sum_{m\nmid k} f(1-q^k) p_k(q;x)-\sum_{j\nmid k} f(1-r^k) p_k(r;x)\end{aligned}$$ For each $N\in \mathbb{N}$, choose $f_N\in C[0,1]$ with $\|f_N\|=1$ in such a way that $$\begin{array}{ll}
f_N(1-r^k) = -1 & \mathrm{for} \;\;j\nmid k, \:\: k \leqslant N,\\
f_N(1-q^k) = -1 & \mathrm{for} \;\;m | k, \:\: k \leqslant N,\\
f_N(1-q^k) = 1 & \mathrm{for}\;\;m\nmid k,\:\: k \leqslant N,\\
f_N(1-q^k) = f_N(1-r^k)= 0 & \mathrm{for}\;\; k > N.
\end{array}$$ Then, $$\begin{aligned}
\left(B_q-B_r\right)f_N(x)
&=\sum_{k=0}^N p_k(q;x)+\sum_{k=0}^N p_k(r;x)- 2\sum_{k=0}^{\lfloor N/m \rfloor} p_{mk}(q;x).\end{aligned}$$ Following the line of reasoning in the preceding proof and bearing in mind , one may opt for $\delta >0$ and $N\in \mathbb{N}$ such that, for every $\varepsilon >0,$ $$\sum_{k=0}^N p_k(q;x)+\sum_{k=0}^N p_k(r;x)- 2\sum_{k=0}^{\lfloor N/m \rfloor} p_{mk}(q;x) \geqslant 2-\frac{2}{m}-\varepsilon\quad \text{when} \quad x\in[1-2\delta, 1-\delta].$$ The statement is now immediate.
Let $f\in C[0,1].$ Then $$\begin{aligned}
\left(B_q-B_r\right)f(x)
&=\sum_{k=0}^\infty f(1-q^k) p_k(q;x)-\sum_{k=0}^\infty f(1-r^k) p_k(r;x) \\
&=\sum_{m|k} f(1-q^k) p_k(q;x)+\sum_{m\nmid k} f(1-q^k) p_k(q;x)-\sum_{k=0}^\infty f(1-q^{mk}) p_k(q^m;x) \\
&=\sum_{k=0}^\infty f(1-q^{mk}) \left[p_{mk}(q;x)-p_{k}(q^m;x)\right]+\sum_{m\nmid k} f(1-q^k) p_k(q^m;x)\end{aligned}$$ Using Theorem \[thmfed\] and , the last inequality becomes $$\begin{aligned}
\left|\left(B_q-B_r\right)f(x)\right|
&\leqslant 2\|f\|\left(1-\sum_{k=0}^\infty p_{mk}(q;x)\right), \quad \text{for} \quad x\in[0, 1).\end{aligned}$$ Now, by , for all $\varepsilon >0$ there exists $\delta>0$ such that $$\sum_{k=0}^\infty p_{mk}(q;x) > \frac{1}{m}-\frac{\varepsilon}{2}, \quad \text{for} \quad x\in[1-2\delta, 1-\delta].$$ Therefore, $$\begin{aligned}
\left|\left(B_q-B_r\right)f(x)\right|
&\leqslant 2\|f\|\left(1-\frac{1}{m}+\frac{\varepsilon}{2}\right), \quad \text{for} \quad x\in[1-2\delta, 1-\delta].\end{aligned}$$ This implies that $\|B_{q}-B_{r}\| \leqslant 2-2/m.$ Together with Theorem \[thmBqrmj\], this yields the statement.
Acknowledgments {#acknowledgments .unnumbered}
===============
During the work on this paper, the authors were lucky to see the discussion on Mathoverflow [@fedja] concerning inequality in the case $m=2.$ We would like to express our sincere gratitude to all MO users, who participated in this fruitful discussion, especially to the user whose nickname is ‘fedja’ and whose grasp of the subject was of a significant inspiration. We are also pleased to thank Prof. Alexandre Eremenko (Purdue University, USA) for his encouragement and valuable help throughout the entire process of our work.
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https://mathoverflow.net/questions/269740/inequality-for-functions-on-0-1
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---
abstract: 'We prove a lower bound and an upper bound for the total variation distance between two high-dimensional Gaussians, which are within a constant factor of one another.'
author:
- |
Luc Devroye[^1]\
McGill University
- |
Abbas Mehrabian[^2]\
McGill University
- |
Tommy Reddad[^3]\
McGill University
title: |
The total variation distance between\
high-dimensional Gaussians
---
Introduction
============
The Gaussian (or normal) distribution is perhaps the most important distribution in probability theory due to the central limit theorem. For a positive integer $d$, a vector $\mu \in {\mathbb{R}}^d$, and a positive definite matrix $\Sigma$, the Gaussian distribution with mean $\mu$ and covariance matrix $\Sigma$ is a probability distribution over ${\mathbb{R}}^d$ denoted by ${\mathcal{N}}(\mu,\Sigma)$ with density $$\det(2\pi \Sigma)^{-1/2} \exp (- (x-\mu){^{\mathsf{T}}}\Sigma^{-1} (x-\mu)) \qquad \forall x\in{\mathbb{R}}^d.$$ We denote by $N(\mu,\Sigma)$ a random variable with this distribution. Note that if $X\sim {\mathcal{N}}(\mu,\Sigma)$ then ${\mathbf{E}}X = \mu$ and ${\mathbf{E}}X X{^{\mathsf{T}}}= \Sigma$.
If the covariance matrix is positive semi-definite but not positive definite, the Gaussian distribution is singular on ${\mathbb{R}}^d$, but has a density with respect to a Lebesgue measure on an affine subspace: let $r$ be the rank of $\Sigma$, and let ${\operatorname{range}}(\Sigma)$ denote the range (also known as the image or the column space) of $\Sigma$. Let $\Pi$ be a $d\times r$ matrix whose columns form an orthonormal basis for ${\operatorname{range}}(\Sigma)$. Then the matrix $\Sigma'\coloneqq \Pi{^{\mathsf{T}}}\Sigma \Pi$ has full rank $r$, and ${\mathcal{N}}(\mu,\Sigma)$ has density given by $$\det(2\pi \Sigma')^{-1/2} \exp (- (x-\mu){^{\mathsf{T}}}\Pi \Sigma'^{-1} \Pi{^{\mathsf{T}}}(x-\mu))$$ with respect to the $r$-dimensional Lebesgue measure on $\mu +{\operatorname{range}}(\Sigma)$. The density is zero outside this affine subspace. For general background on high-dimensional Gaussian distributions (also called multivariate normal distributions), see [@Tong; @Vershynin].
Given two Gaussian distributions, our goal is to understand how different they are. Our measure of similarity is the [*total variation distance (t.v.d.)*]{}, which for any two distributions $P$ and $Q$ over ${\mathbb{R}}^d$ is defined as $${\operatorname{TV}\left(P, Q\right)} \coloneqq \sup_{A\subseteq {\mathbb{R}}^d} |P(A)-Q(A)|.$$ If $P$ and $Q$ have densities $p$ and $q$, then it is easy to verify that the set $A\coloneqq \{x : p(x)>q(x)\}$ attains the supremum here, and this observation leads to the following identity: $$\label{tvl1}
{\operatorname{TV}\left(P, Q\right)} = \frac12 \int_{{\mathbb{R}}^d} |p(x) - q(x)| \, dx,$$ that is, the t.v.d. is half of the $L^1$ distance. In the following, we will sometimes write ${\operatorname{TV}\left(X, Y\right)}$ for ${\operatorname{TV}\left(P, Q\right)}$, where $X$ and $Y$ are random variables distributed as $P$ and $Q$, respectively. Observe that ${\operatorname{TV}\left(P, Q\right)}$ is a metric and is always between 0 and 1. For a survey on measures of distance between distributions and the inequalities between them, see [@Verdu].
We have seen that the t.v.d. can be written as an integral or as a supremum, but in general there is no known closed form for it. In this note we give lower and upper bounds in a closed form for the t.v.d. between two Gaussians, which are within a constant factor of one another.
Note that if $\mu_1 + {\operatorname{range}}(\Sigma_1)
\neq \mu_2 + {\operatorname{range}}(\Sigma_2)$, in particular if ${\operatorname{rank}}(\Sigma_1)\neq{\operatorname{rank}}(\Sigma_2)$, then it is easy to see that $
{{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\Sigma_1), {\mathcal{N}}(\mu_2,\Sigma_2)\right)}}=1$, since the intersection of the supports have zero Lebesgue measure; so we will not explicitly treat this case.
To state our results we need some matrix definitions. The $d$-dimensional identity matrix is denoted $I_d$. The *Frobenius norm* (also called the Hilbert–Schmidt norm or the Schur norm) of a matrix $A$ is denoted by $\|A\|_F \coloneqq \sqrt{{\operatorname{tr}}(AA{^{\mathsf{T}}})}$. Note that $\|A\|_F^2$ equals the sum of squares of entries of $A$. If $A$ is symmetric, $\|A\|_F^2$ equals the sum of squares of eigenvalues of $A$. For general background on matrix norms, see [@Horn Chapter 5].
Our first main result concerns the same-mean case. Note that we have not tried to optimize the constants in our results.
\[thm:meanzero\] If $\mu\in {\mathbb{R}}^d$ and $\Sigma_1$ and $\Sigma_2$ are positive definite $d\times d$ matrices, then $$\frac{1}{100}
\leq
\frac{{\operatorname{TV}\left({\mathcal{N}}(\mu,\Sigma_1), {\mathcal{N}}(\mu,\Sigma_2)\right)}}{\min \{1, \|\Sigma_1^{-1}\Sigma_2-I_d\|_F\}}
\leq
\frac32 .$$ If $\Sigma_1$ and $\Sigma_2$ are positive semi-definite and ${\operatorname{range}}(\Sigma_1)={\operatorname{range}}(\Sigma_2)$ and $r = {\operatorname{rank}}(\Sigma_1) = {\operatorname{rank}}(\Sigma_2)$, then let $\Pi$ be a $d\times r$ matrix which has the same range as $\Sigma_1$ and $\Sigma_2$. Then we have $$\frac{1}{100}
\leq
\frac{{\operatorname{TV}\left({\mathcal{N}}(\mu,\Sigma_1), {\mathcal{N}}(\mu,\Sigma_2)\right)}}{\min \{1, \|(\Pi{^{\mathsf{T}}}\Sigma_1 \Pi)^{-1}(\Pi{^{\mathsf{T}}}\Sigma_2 \Pi)-I_{r}\|_F\}}
\leq
\frac32 .$$
This theorem generalizes two previously known bounds: in [@Klartag Lemma 4.8] the upper bound of this theorem is proved in the case when $\Sigma_1=\alpha I_d$ and $\Sigma_2=\beta I_d$, while in [@lower_bound_improved Lemma 3.8] we have proved the lower bound of this theorem in the special case when the diagonal entries of $\Sigma_1^{-1}$ and $\Sigma_2^{-1}$ are all ones.
The paper [@ulyanov] proves a bound similar to Theorem \[thm:meanzero\] for Gaussian distributions in a general Hilbert space. Their Corollary 2 states the following for ${\mathbb{R}}^d$: If $\Sigma_1$ and $\Sigma_2$ are positive definite $d\times d$ matrices and $\|\Sigma_1^{-1}\Sigma_2-I_d\|_F \leq 1/50$, then $$\frac{1}{100}
\leq
\frac{{\operatorname{TV}\left({\mathcal{N}}(\mu,\Sigma_1), {\mathcal{N}}(\mu,\Sigma_2)\right)}}{ \|\Sigma_1^{-1}\Sigma_2-I_d\|_F}
\leq
2 .$$ This result has the advantage that it covers infinite-dimensional spaces as well, but it holds only when $\|\Sigma_1^{-1}\Sigma_2-I_d\|_F$ is smaller than a threshold. For the case where the means are different, we prove the following theorem.
\[thm:main\] Suppose $d>1$, let $\mu_1\neq\mu_2\in {\mathbb{R}}^d$ and let $\Sigma_1,\Sigma_2$ be positive definite $d\times d$ matrices. Let $v\coloneqq \mu_1-\mu_2$ and let $\Pi$ be a $d\times d-1$ matrix whose columns form a basis for the subspace orthogonal to $v$. Define the function $$tv(\mu_1,\Sigma_1,\mu_2,\Sigma_2) \coloneqq
\max\left\{ \frac{|v{^{\mathsf{T}}}(\Sigma_1-\Sigma_2)v|}{v{^{\mathsf{T}}}\Sigma_1 v},\red{\frac{{v{^{\mathsf{T}}}v}}{\sqrt{v{^{\mathsf{T}}}\Sigma_1 v}}},
\| (\Pi{^{\mathsf{T}}}\Sigma_1 \Pi)^{-1}
\Pi{^{\mathsf{T}}}\Sigma_2 \Pi - I_{d-1} \|_F\right\}
.$$ Then, we have $$\frac{1}{200} \leq
\frac{{{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\Sigma_1), {\mathcal{N}}(\mu_2,\Sigma_2)\right)}}}{\min\{1,tv(\mu_1,\Sigma_1,\mu_2,\Sigma_2)\}} \leq
\frac{9}{2} .$$
Note that the positive definiteness of the covariance matrices can be assumed without loss of generality; if $\mu_1+{\operatorname{range}}(\Sigma_1)=\mu_2+{\operatorname{range}}(\Sigma_1)\neq {\mathbb{R}}^d$, then one can work in this affine subspace instead.
Along the way of proving this theorem, we also give bounds for the one-dimensional case.
\[thm:onedimensional\] In the one dimensional case, $d=1$, we have $$\frac{1}{200} \min \left\{1, \max \left\{ \frac{|\sigma_1^2-\sigma_2^2|}{\sigma_1^2} , \frac{40 |\mu_1-\mu_2|}{\sigma_1} \right\} \right\}
\leq
{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)}
\leq \frac{3|\sigma_1^2-\sigma_2^2|}{2\sigma_1^2}+ \frac{|\mu_1-\mu_2|}{2\sigma_1} .$$
Observe that while the t.v.d. is symmetric, our lower and upper bounds are not symmetric, so they can be automatically strengthened; for instance the following symmetric version of Theorem \[thm:onedimensional\] holds: $$\begin{aligned}
\frac{1}{200} \min \left\{1, \max \left\{ \frac{|\sigma_1^2-\sigma_2^2|}{\min\{\sigma_1,\sigma_2\}^2} , \frac{40 |\mu_1-\mu_2|}{\min\{\sigma_1,\sigma_2\}} \right\} \right\}
& \leq
{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)}
\\& \leq \frac{3|\sigma_1^2-\sigma_2^2|}{2\max\{\sigma_1,\sigma_2\}^2}+ \frac{|\mu_1-\mu_2|}{2\max\{\sigma_1,\sigma_2\}} .\end{aligned}$$
Some preliminaries and other known bounds for the t.v.d. between Gaussians appear in Section \[sec:prelim\]. We start by proving Theorem \[thm:meanzero\] in Section \[sec:thm:meanzero\], then we prove Theorem \[thm:onedimensional\] in Section \[sec:thn:onedimensional\], and finally we prove Theorem \[thm:main\] in Section \[sec:thm:main\].
Preliminaries {#sec:prelim}
=============
#### The coupling characterization of the t.v.d.
For two distributions $P$ and $Q$, a pair $(X,Y)$ of random variables defined on the same probability space is called a *coupling* for $P$ and $Q$ if $X \sim P$ and $Y\sim Q$. An extremely useful property of the t.v.d. is [*the coupling characterization*]{}: for any two distributions $P$ and $Q$, we have ${\operatorname{TV}\left(P, Q\right)} \leq t$ if and only if there exists a coupling $(X,Y)$ for them such that ${{\mathbf{P}}\left\{{X\neq Y}\right\}} \leq t$ (see, e.g., [@Levin Proposition 4.7]). This implies in particular that there exists a coupling $(X,Y)$ such that ${{\mathbf{P}}\left\{{X\neq Y}\right\}} = {\operatorname{TV}\left(P, Q\right)}$.
This characterization implies that for any function $f$ we have ${\operatorname{TV}\left(f(X), f(Y)\right)} \leq {\operatorname{TV}\left(X, Y\right)}$. If $f$ is invertible (for instance if $f(v)=Av+b$ where $A$ is full-rank) this also implies ${\operatorname{TV}\left(f(X), f(Y)\right)} = {\operatorname{TV}\left(X, Y\right)}$.
An important property of the Gaussian distribution is that any linear transformation of a Gaussian random variable is also Gaussian. In particular, if $X\sim {\mathcal{N}}(\mu,\Sigma)$ then $$AX + b \sim {\mathcal{N}}(A\mu +b, A\Sigma A + A\mu b{^{\mathsf{T}}}+ b\mu{^{\mathsf{T}}}A{^{\mathsf{T}}}+ b b{^{\mathsf{T}}}).$$
For a positive semi-definite matrix $\Sigma$ with eigendecomposition $\Sigma = \sum_{i = 1}^d \lambda_i v_i v_i{^{\mathsf{T}}}$ where the $v_i$ are orthonormal, we define $\Sigma^{1/2} \coloneqq \sum_{i = 1}^d \sqrt{\lambda_i} v_i v_i{^{\mathsf{T}}}$ and $\Sigma^{-1/2} \coloneqq \sum_{i = 1}^d v_i v_i{^{\mathsf{T}}}/\sqrt{\lambda_i}$. It is easy to observe that if $g\sim {\mathcal{N}}(0,I)$ then $\Sigma^{1/2} g \sim {\mathcal{N}}(0,\Sigma)$.
We will use the inequality $$0 \leq x-\log(1+x) \leq x^2
\qquad \forall x \geq -2/3$$ throughout, which implies that for any $x\geq -2/3$ there exists a $b\in[0,1]$ such that $x-\log(1+x)=bx^2$.
We next state some known bounds for the t.v.d. between two Gaussians, which may be more convenient than the above bounds for some applications.
For the case when the two Gaussians have the same covariance matrices, [@ulyanov Theorem 1] gives $${{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\Sigma), {\mathcal{N}}(\mu_2,\Sigma)\right)}}
= {{\mathbf{P}}\left\{{ N(0,1) \in \left[
-\frac{ \sqrt{(\mu_1-\mu_2){^{\mathsf{T}}}\Sigma^{-1} (\mu_1-\mu_2)} }{2},
\frac{\sqrt{(\mu_1-\mu_2){^{\mathsf{T}}}\Sigma^{-1} (\mu_1-\mu_2)}}{2}
\right]}\right\}}.$$
The following bounds follow from known relations between statistical distances.
#### An upper bound for the t.v.d. using the KL-divergence.
For distributions $P$ and $Q$ over ${\mathbb{R}}^d$ with densities $p$ and $q$, their Kullback–Leibler divergence (KL-divergence) is defined as $$\begin{aligned}
{\operatorname{KL}\left(P \parallel Q\right)} \coloneqq \int_{{\mathbb{R}}^d} p(x) \log\left( \frac{p(x)}{q(x)} \right) dx,
\end{aligned}$$ and Pinsker’s inequality [@Tsybakov Lemma 2.5] states that ${\operatorname{TV}\left(P, Q\right)}\leq \sqrt{{\operatorname{KL}\left(P \parallel Q\right)}/2}$ for any pair of distributions. The KL-divergence between two Gaussians has a closed form (e.g., [@Rasmussen Formula (A.23)]): $$\operatorname{KL}({\mathcal{N}}(\mu_1,\Sigma_1)\parallel{{\mathcal{N}}(\mu_2,\Sigma_2)})
~=~
\frac{1}{2} \left( {\operatorname{tr}}(\Sigma_1^{-1}\Sigma_2 - I)
+ (\mu_1-\mu_2) {^{\mathsf{T}}}\Sigma_1^{-1} (\mu_1-\mu_2)
- \log \det (\Sigma_2 \Sigma_1^{-1}) \right).$$ Combining these gives the following proposition.
\[upperkl\] If $\Sigma_1$ and $\Sigma_2$ are positive definite, then $${\operatorname{TV}\left({\mathcal{N}}(\mu_1,\Sigma_1), {\mathcal{N}}(\mu_2,\Sigma_2)\right)} \leq \frac 1 2 \sqrt { {\operatorname{tr}}(\Sigma_1^{-1}\Sigma_2 - I)
+ (\mu_1-\mu_2) {^{\mathsf{T}}}\Sigma_1^{-1} (\mu_1-\mu_2)
- \log \det (\Sigma_2 \Sigma_1^{-1}) }.$$
#### Bounds for the t.v.d. using the Hellinger distance.
For distributions $P$ and $Q$ over ${\mathbb{R}}^d$ with densities $p$ and $q$, their Hellinger distance is defined as $$\begin{aligned}
\operatorname{H}({P},{Q}) \coloneqq \frac{1}{\sqrt2} \sqrt{ \int_{{\mathbb{R}}^d} \left(\sqrt {p(x)}-\sqrt{q(x)}\right)^2 \, dx } ,
\end{aligned}$$ and it is known that $$\operatorname{H}({P},{Q})^2 \leq {\operatorname{TV}\left(P, Q\right)} \leq
\operatorname{H}({P},{Q}) \sqrt{2-\operatorname{H}({P},{Q})^2}
\leq\sqrt2 \operatorname{H}({P},{Q}),$$ see [@LeCam page 25]. The Hellinger distance between two Gaussians has a closed form (e.g., [@Pardo page 51]): $$\operatorname{H}({\mathcal{N}}(\mu_1,\Sigma_1), {{\mathcal{N}}(\mu_2,\Sigma_2)})^2
~=~
1 - \frac{\det(\Sigma_1)^{1/4}\det(\Sigma_2)^{1/4}}{\det\left(\frac{\Sigma_1+\Sigma_2}{2}\right)^{1/2}} \exp \left\{ -\frac18 (\mu_1-\mu_2){^{\mathsf{T}}}\left(\frac{\Sigma_1+\Sigma_2}{2}\right)^{-1}(\mu_1-\mu_2) \right\}.$$ Combining these gives the following proposition.
\[upperhellinger\] Assume that $\Sigma_1,\Sigma_2$ are positive definite, and let $$h=h(\mu_1,\Sigma_1,\mu_2,\Sigma_2) \coloneqq \left(1 - \frac{\det(\Sigma_1)^{1/4}\det(\Sigma_2)^{1/4}}{\det\left(\frac{\Sigma_1+\Sigma_2}{2}\right)^{1/2}} \exp \left\{ -\frac18 (\mu_1-\mu_2){^{\mathsf{T}}}\left(\frac{\Sigma_1+\Sigma_2}{2}\right)^{-1}(\mu_1-\mu_2) \right\}\right)^{1/2}.$$ Then, we have $$h^2 \leq {\operatorname{TV}\left({\mathcal{N}}(\mu_1,\Sigma_1), {\mathcal{N}}(\mu_2,\Sigma_2)\right)} \leq h \sqrt{2-h^2} \leq h\sqrt2.$$
Same-mean case: proof of Theorem \[thm:meanzero\] {#sec:thm:meanzero}
=================================================
In this section we consider the case when both Gaussians have the same mean. For proving the theorem we will need two lemmas.
\[lem:zeromeancore\] Suppose $\lambda_1,\dots,\lambda_d \geq -2/3$ and let $\rho \coloneqq \sqrt{\sum_{i=1}^{d} \lambda_i^2}$. If $C$ is a diagonal matrix with diagonal entries $1+\lambda_1,\dots,1+\lambda_d$, then $
{{\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)}}
\geq
\rho/6 - \rho^2/8 - (e^{\rho^2}-1)/2.
$
From we have $$\begin{aligned}
2 {\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I)\right)} &= (2\pi)^{-d/2}\int_{{\mathbb{R}}^d} \left| e^{-x{^{\mathsf{T}}}x/2} - \sqrt{\det(C)} e^{-x{^{\mathsf{T}}}C x/2} \right| \, dx \\
& = (2\pi)^{-d/2}\int_{{\mathbb{R}}^d} e^{-x{^{\mathsf{T}}}x/2} \left| 1 - \sqrt{\det(C)} e^{-x{^{\mathsf{T}}}(C-I_d) x/2} \right| \, dx \\
& = {\mathbf{E}}\left| 1 - \sqrt{\det(C)} e^{-g{^{\mathsf{T}}}(C-I_d) g/2} \right| \\
& = {\mathbf{E}}\left| 1 - \exp \left(
\sum_{i=1}^{d} \log(1+\lambda_i)/2 - \lambda_i g_i^2/2
\right) \right|,
\end{aligned}$$ where $g=(g_1,\dots,g_d)\sim{\mathcal{N}}(0,I_d)$. Since $\lambda_i\geq -2/3$ for all $i$, we have $\log (1+\lambda_i)/2
= \lambda_i/2 - b_i \lambda_i^2/2$ for some $b_i\in[0,1]$, and summing these up we find $\sum_{i = 1}^d \log (1+\lambda_i)/2
= \sum_{i = 1}^d \lambda_i/2 - b \rho^2$ for some $b\in[0,1]$. Also let $h_i = 1 - g_i^2$ and $X = \sum_{i = 1}^d \lambda_i h_i/2$, whence $$\begin{aligned}
2 {\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)} &={\mathbf{E}}\left| 1 - e^{-b\rho^2} e^{X} \right|
\geq {\mathbf{E}}\left| 1 - e^X \right| - {\mathbf{E}}\left| e^X - e^{-b\rho^2} e^{X} \right| \notag\\
& \geq
{\mathbf{E}}\left| X \right| - {\mathbf{E}}X^2/2 - (1-e^{-b\rho^2}){\mathbf{E}}e^{X} \notag\\ &
\geq
\frac {({\mathbf{E}}X^2)^{3/2}}{({\mathbf{E}}X^4)^{1/2}} - {\mathbf{E}}X^2/2 - (1-e^{-b\rho^2}){\mathbf{E}}e^{X}
\label{3piece} \end{aligned}$$ where the first inequality is the triangle inequality, the second one follows from $$|1 - e^x| \geq |x| - x^2/2 \qquad \forall x \in {\mathbb{R}},$$ and the third one follows from Hölder’s inequality. We control each term on the right-hand-side of . First, we observe that since $h_i$ is mean-zero, we have ${\mathbf{E}}h_i h_j=0$ for all $i\neq j$, and so $${\mathbf{E}}X^2 = {\mathbf{E}}\left(\sum_{i = 1}^d \lambda_i h_i/2\right)^2
= \sum_{i = 1}^d (\lambda_i/2)^2 {\mathbf{E}}h_i^2
= \sum_{i = 1}^d \lambda_i^2/2 = \rho^2/2,$$ since ${\mathbf{E}}h_i^2 = 2$. Since ${\mathbf{E}}g_i^2 = 1,{\mathbf{E}}g_i^4 = 3,{\mathbf{E}}g_i^6 =15,{\mathbf{E}}g_i^8 =105,$ one can compute ${\mathbf{E}}h_i^4 = 60$, and so $$\begin{aligned}
{\mathbf{E}}X^4 & = {\mathbf{E}}\left(\sum_{i = 1}^d \lambda_i h_i/2\right)^4 \\
&= \sum_{i = 1}^d (\lambda_i/2)^4 {\mathbf{E}}h_i^4
+ 3 \sum_{i\neq j} (\lambda_i/2)^2 (\lambda_j/2)^2 {\mathbf{E}}h_i^2{\mathbf{E}}h_j^2
\\& = 60 \sum_{i = 1}^d (\lambda_i/2)^4
+ 12 \sum_{i\neq j} (\lambda_i/2)^2 (\lambda_j/2)^2 \\
& \leq 60 \left(\sum_{i = 1}^d (\lambda_i/2)^2 \right)^2 = 15 \rho^4/4.
\end{aligned}$$ Finally, for the exponential moment, we note that ${\mathbf{E}}\exp(tg_i^2) = (1-2t)^{-1/2}$ for any $t<1/2$, hence $$\begin{aligned}
{\mathbf{E}}e^{X} =
\prod_{i = 1}^d \left(e^{\lambda_i/2}
{\mathbf{E}}e^{-\lambda_i g_i^2 /2}\right)
= \prod_{i = 1}^d \left(e^{\lambda_i/2}
e^{\frac{-1}{2} \log (1+\lambda_i) }\right)
=\exp\left( \sum_{i = 1}^d
\lambda_i/2
-\log (1+\lambda_i)/2
\right) = e^{b\rho^2},
\end{aligned}$$ consequently, $$\begin{aligned}
2 {\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)}
\geq
\frac{(\rho^2/2)^{3/2}}{(15\rho^4/4)^{1/2}} - \rho^2/4
- e^{b\rho^2}+1 \geq \rho/3 - \rho^2/4 - (e^{\rho^2}-1),
\end{aligned}$$ completing the proof.
\[lem:onedimconstant\] If $\lambda^2\geq 0.01$ then ${{\operatorname{TV}\left({\mathcal{N}}(0,1), {\mathcal{N}}(0,1+\lambda)\right)}}>0.01$.
If $\lambda>0$ then $1+\lambda\geq1.1$, so we have $$\begin{aligned}
{{\operatorname{TV}\left({\mathcal{N}}(0,1), {\mathcal{N}}(0,1+\lambda)\right)}} &\geq
{{\mathbf{P}}\left\{{N(0,1)\in[-1,1]}\right\}}-{{\mathbf{P}}\left\{{N(0,1+\lambda)\in[-1,1]}\right\}} \\ &\geq
{{\mathbf{P}}\left\{{N(0,1)\in[-1,1]}\right\}}-{{\mathbf{P}}\left\{{N(0,1.1)\in[-1,1]}\right\}} \\ &> 0.68-0.66 > 0.01,
\end{aligned}$$ while if $\lambda<0$ then $1+\lambda\leq0.9$ so we have $$\begin{aligned}
{{\operatorname{TV}\left({\mathcal{N}}(0,1), {\mathcal{N}}(0,1+\lambda)\right)}} &\geq
{{\mathbf{P}}\left\{{N(0,1+\lambda)\in[-1,1]}\right\}}-{{\mathbf{P}}\left\{{N(0,1)\in[-1,1]}\right\}} \\ &\geq
{{\mathbf{P}}\left\{{N(0,0.9)\in[-1,1]}\right\}}-{{\mathbf{P}}\left\{{N(0,1)\in[-1,1]}\right\}} \\ &> 0.70-0.69 = 0.01. \qedhere
\end{aligned}$$
We can now prove Theorem \[thm:meanzero\].
For both parts of the theorem, we may assume that $\mu=0$. We start with the case that $\Sigma_1$ and $\Sigma_2$ are positive definite, i.e., they have full rank. Let $\Sigma_1^{-1}\Sigma_2$ have eigenvalues $1+\lambda_1,\dots,1+\lambda_d$, and let $\rho \coloneqq \|\Sigma_1^{-1}\Sigma_2-I\|_F = \sqrt{\sum_{i = 1}^d {\lambda_i^2}}$.
We first prove the upper bound. If some $\lambda_i<-2/3$ then trivially $${\operatorname{TV}\left({\mathcal{N}}(0,\Sigma_1), {\mathcal{N}}(0,\Sigma_2)\right)} \leq 1 \leq \frac32 |\lambda_i| \leq \frac32 \sqrt{\sum_{i=1}^{d} \lambda_i^2}=3\rho/2.$$ Otherwise, by Proposition \[upperkl\], $$4{\operatorname{TV}\left({\mathcal{N}}(0,\Sigma_1), {\mathcal{N}}(0,\Sigma_2)\right)}^2
\leq \sum_{i=1}^{d} (\lambda_i - \log(1+\lambda_i)) \leq \sum_{i=1}^{d} \lambda_i^2=\rho^2,$$ and the upper bound in the theorem is proved.
For proving the lower bound, we first claim that if $C$ is a diagonal matrix with diagonal entries $1+\lambda_1,\dots,1+\lambda_d$, then $${{\operatorname{TV}\left({\mathcal{N}}(0,\Sigma_1), {\mathcal{N}}(0,\Sigma_2)\right)}}
=
{{\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)}}.
\label{diagonalize}$$ To prove this, let $g\sim {\mathcal{N}}(0,I_d)$. We first claim if $E$ and $F$ are positive definite matrices with the same spectrum, then ${\operatorname{TV}}(Eg, g) = {\operatorname{TV}}(Fg, g)$. To see this, let $s_1, \dots, s_d$ be the eigenvalues of $E$ and $F$, and let $g_1, \dots, g_d$ be the components of $g$. Then by rotation-invariance of $g$, both ${\operatorname{TV}}(Eg, g)$ and ${\operatorname{TV}}(Fg, g)$ are equal to ${\operatorname{TV}}( (s_1 g_1, s_2 g_2, \dots, s_d g_d), (g_1, g_2, \dots, g_d) )$, and the claim is proved. This also implies $${{\operatorname{TV}\left({\mathcal{N}}(0,I_d), {\mathcal{N}}(0,E)\right)}}={{\operatorname{TV}\left({\mathcal{N}}(0,I_d), {\mathcal{N}}(0,F)\right)}},$$ for any two positive definite matrices $E$ and $F$ with the same spectrum.
Next, we have $$\begin{aligned}
{\operatorname{TV}}( {\mathcal{N}}(0, \Sigma_1), {\mathcal{N}}(0, \Sigma_2)) &=
{\operatorname{TV}}(\Sigma_1^{1/2} g, \Sigma_2^{1/2} g) =
{\operatorname{TV}}(\Sigma_2^{-1/2} \Sigma_1^{1/2} g,g) \\
&=
{\operatorname{TV}}( {\mathcal{N}}(0, \Sigma_2^{-1/2} \Sigma_1 \Sigma_2^{-1/2}),{\mathcal{N}}(0,I_d)).
\end{aligned}$$ Now $\Sigma_2^{-1/2} \Sigma_1 \Sigma_2^{-1/2}$ has the same spectrum as $\Sigma_2^{-1}\Sigma_1$, which has the same spectrum as $C^{-1}$, whence is proved.
For proving the lower bound in the theorem we consider three cases.
**Case 1: there exists some $i$ with $|\lambda_i|\geq0.1$.** Observe that if we project a random variable distributed as ${\mathcal{N}}(0,C^{-1})$ onto the $i$-th component, we obtain a ${\mathcal{N}}(0,(1+\lambda_i)^{-1})$ random variable . Since projection can only decrease the t.v.d., using Lemma \[lem:onedimconstant\] we obtain $${{\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)}}
\geq
{{\operatorname{TV}\left({\mathcal{N}}(0,(1+\lambda_i)^{-1}), {\mathcal{N}}(0,1)\right)}}
=
{{\operatorname{TV}\left({\mathcal{N}}(0,1), {\mathcal{N}}(0,1+\lambda_i)\right)}}
\geq 0.01,$$ as required. The equality above follows since the t.v.d. is invariant under any linear transformation.
**Case 2: $|\lambda_i|<0.1$ for all $i$ and $\rho \leq 0.17$.** In this case Lemma \[lem:zeromeancore\] gives $${{\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)}}
\geq
\rho/6 - \rho^2/8 - (e^{\rho^2}-1)/2
\geq \rho / 100,$$ as required.
**Case 3: $|\lambda_i|<0.1$ for all $i$ and $\rho > 0.17$.** Define $$f(\rho)\coloneqq \rho/6 - \rho^2/8 - (e^{\rho^2}-1)/2,$$ and observe that $f(x)\geq0.01$ for $0.1\leq x \leq 0.17$. Let $1\leq j < d$ be the largest index such that $\sum_{i=1}^{j} \lambda_i^2 \leq 0.17^2$, and observe that since $|\lambda_i|<0.1$ for all $i$, we have $\rho'^2\coloneqq \sum_{i=1}^{j} \lambda_i^2 \geq 0.17^2 - 0.1^2 >0.01$ and so $f(\rho') \geq 0.01$. Let $C'$ be the diagonal $j\times j$ matrix with diagonal entries $1+\lambda_1,\dots,1+\lambda_j$. Observe that if we project a random variable distributed as ${\mathcal{N}}(0,C^{-1})$ onto the first $j$ coordinates, we would obtain a $ {\mathcal{N}}(0,C'^{-1})$ random variable. Since projection can only decrease the t.v.d., using Lemma \[lem:zeromeancore\] we obtain $${{\operatorname{TV}\left({\mathcal{N}}(0,C^{-1}), {\mathcal{N}}(0,I_d)\right)}}
\geq
{{\operatorname{TV}\left({\mathcal{N}}(0,C'^{-1}), {\mathcal{N}}(0,I_j)\right)}}
\geq f(\rho') \geq 0.01,$$ as required.
We finally consider the case that $\Sigma_1$ and $\Sigma_2$ are not positive definite, but they are positive semi-definite, and ${\operatorname{range}}(\Sigma_1)={\operatorname{range}}(\Sigma_2)$. Recall that $\Pi$ is a $d\times r$ matrix whose columns form a basis for ${\operatorname{range}}(\Sigma_1)$. Then observe that $v \mapsto \Pi{^{\mathsf{T}}}v$ is an invertible map from ${\operatorname{range}}(\Sigma_1)$ to ${\mathbb{R}}^{r}$, with the inverse given by $w \mapsto \Pi (\Pi{^{\mathsf{T}}}\Pi)^{-1} w$. This implies $${\operatorname{TV}\left(N(0,\Sigma_1), N(0,\Sigma_2)\right)}
=
{\operatorname{TV}\left(\Pi{^{\mathsf{T}}}N(0,\Sigma_1), \Pi{^{\mathsf{T}}}N(0,\Sigma_2)\right)}
=
{\operatorname{TV}\left({\mathcal{N}}(0,\Pi{^{\mathsf{T}}}\Sigma_1 \Pi), {\mathcal{N}}(0,\Pi{^{\mathsf{T}}}\Sigma_2 \Pi)\right)},$$ and $\Pi{^{\mathsf{T}}}\Sigma_1 \Pi$ and $\Pi{^{\mathsf{T}}}\Sigma_2 \Pi$ are now positive definite $r \times r$ matrices, hence the second part of the theorem follows from the first part.
One-dimensional case: proof of Theorem \[thm:onedimensional\] {#sec:thn:onedimensional}
=============================================================
We start with the upper bound. If $\frac{|\sigma_1^2-\sigma_2^2|}{\sigma_1^2}\geq 2/3$, then the right-hand-side is at least 1, and the bound holds because the t.v.d. is at most 1. Otherwise, since ${\sigma_2^2}/{\sigma_1^2}-1\geq -2/3$, we have ${\sigma_2^2}/{\sigma_1^2}-1-\log({\sigma_2^2}/{\sigma_1^2})\leq ({\sigma_2^2}/{\sigma_1^2}-1)^2$, so from Proposition \[upperkl\] we have $$\begin{aligned}
{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)} & \leq \frac12 \sqrt{{\sigma_2^2}/{\sigma_1^2}-1-\log({\sigma_2^2}/{\sigma_1^2}) + (\mu_1-\mu_2)^2/\sigma_1^2} \\&\leq \frac12
\sqrt{{\sigma_2^2}/{\sigma_1^2}-1-\log({\sigma_2^2}/{\sigma_1^2})} + \frac12\sqrt{(\mu_1-\mu_2)^2/\sigma_1^2} \\&\leq \frac12 |{\sigma_2^2}/{\sigma_1^2}-1| + \frac 12 |(\mu_1-\mu_2)/\sigma_1|,
\end{aligned}$$ completing the proof of the upper bound.
The lower bound follows from the following two lower bounds: $$\begin{aligned}
\frac{1}{200} \min \left\{1, \frac{|\sigma_1^2-\sigma_2^2|}{\sigma_1^2}\right\}
& \leq
{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)}, \label{lb1} \\
\frac15 \min \left\{1, \frac{|\mu_1-\mu_2|}{\sigma_1} \right\}
& \leq
{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)}. \label{lb2}
\end{aligned}$$ We start with proving . We show $$\label{changemean}
\frac12
{\operatorname{TV}\left({\mathcal{N}}(0,\sigma_1^2), {\mathcal{N}}(0,\sigma_2^2)\right)} \leq
{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)},$$ and then follows from Theorem \[thm:meanzero\]. Assume without loss of generality that $\sigma_1\leq\sigma_2$ and $\mu_1\leq \mu_2$. By the form of the density of the normal distribution, this implies there exists some $c=c(\sigma_1,\sigma_2)$ such that $${\operatorname{TV}\left({\mathcal{N}}(0,\sigma_1^2), {\mathcal{N}}(0,\sigma_2^2)\right)}
= {{\mathbf{P}}\left\{{N(0,\sigma_2^2) \notin [-c,c]}\right\}}- {{\mathbf{P}}\left\{{N(0,\sigma_1^2) \notin [-c,c]}\right\}},$$ and thus $${{\mathbf{P}}\left\{{N(0,\sigma_2^2) >c}\right\}}
= {{\mathbf{P}}\left\{{N(0,\sigma_1^2) >c}\right\}} + {\operatorname{TV}\left({\mathcal{N}}(0,\sigma_1^2), {\mathcal{N}}(0,\sigma_2^2)\right)}/2.$$ Therefore, $$\begin{aligned}
{{\mathbf{P}}\left\{{N(\mu_2,\sigma_2^2) >c}\right\}}
& = {{\mathbf{P}}\left\{{N(\mu_2,\sigma_1^2) >c}\right\}} + {\operatorname{TV}\left({\mathcal{N}}(0,\sigma_1^2), {\mathcal{N}}(0,\sigma_2^2)\right)}/2
\\ & \geq
{{\mathbf{P}}\left\{{N(\mu_1,\sigma_1^2) >c}\right\}} + {\operatorname{TV}\left({\mathcal{N}}(0,\sigma_1^2), {\mathcal{N}}(0,\sigma_2^2)\right)}/2,
\end{aligned}$$ and is proved.
To complete the proof we need only prove . By symmetry we may assume $\mu_1\leq \mu_2$. Let $X \sim {\mathcal{N}}(\mu_1,\sigma_1^2)$. Then $$\begin{aligned}
{{\operatorname{TV}\left({\mathcal{N}}(\mu_1,\sigma_1^2), {\mathcal{N}}(\mu_2,\sigma_2^2)\right)}} &\geq {{\mathbf{P}}\left\{{N(\mu_2,\sigma_2^2)\geq \mu_2}\right\}}
- {{\mathbf{P}}\left\{{X\geq \mu_2}\right\}}\\
& = 1/2 - (1/2-{{\mathbf{P}}\left\{{X\in[\mu_1,\mu_2]}\right\}})
\\& ={{\mathbf{P}}\left\{{X\in[\mu_1,\mu_2]}\right\}}.
\end{aligned}$$ If $\mu_2-\mu_1 \geq \sigma_1$, then $${{\mathbf{P}}\left\{{X\in[\mu_1,\mu_2]}\right\}}
\geq
{{\mathbf{P}}\left\{{X\in[\mu_1,\mu_1+\sigma_1]}\right\}}
=
{{\mathbf{P}}\left\{{N(0,1)\in[0,1]}\right\}} > \frac15,$$ while if $\mu_2-\mu_1 < \sigma_1$ then $$\begin{aligned}
{{\mathbf{P}}\left\{{X\in[\mu_1,\mu_2]}\right\}}
= \int_{\mu_1}^{\mu_2} \frac{e^{-(x-\mu_1)^2/2\sigma_1^2}}{\sqrt{2\pi}\sigma_1} dx \geq (\mu_2-\mu_1) \frac{e^{-(\mu_2-\mu_1)^2/2\sigma_1^2}}{\sqrt{2\pi}\sigma_1} > \frac{e^{-1/2}}{\sqrt{2\pi}} \frac{|\mu_1-\mu_2|}{\sigma_1} > \frac{ |\mu_1-\mu_2|}{5\sigma_1},
\end{aligned}$$ which proves and completes the proof of the theorem.
Proof of Theorem \[thm:main\] {#sec:thm:main}
=============================
Let $u\coloneqq (\mu_1+\mu_2)/2$. Any vector in ${\mathbb{R}}^d$ has a component in the direction of $v$ and a component orthogonal to $v$. In particular, any $w$ can be written uniquely as $$w = u + f_1(w) v + f_2(w),\qquad f_2(w){^{\mathsf{T}}}v = 0,$$ with $f_1$ and $f_2$ given by $$f_1(w) = \frac{(w-u){^{\mathsf{T}}}v}{v{^{\mathsf{T}}}v}\in {\mathbb{R}}, \qquad f_2(w) = w-u-f_1(w)v = P(w-u),$$ with $P\coloneqq I_d - vv{^{\mathsf{T}}}/v{^{\mathsf{T}}}v$.
Let $X\sim {\mathcal{N}}(\mu_1,\Sigma_1)$ and $Y\sim {\mathcal{N}}(\mu_2,\Sigma_2)$. Then we have $$\begin{aligned}
\max \{ {\operatorname{TV}\left(f_1(X), f_1(Y)\right)}, {\operatorname{TV}\left(f_2(X), f_2(Y)\right)} \}
& \leq {\operatorname{TV}\left(X, Y\right)} \\& \leq {{\operatorname{TV}\left(f_1(X), f_1(Y)\right)} + {\operatorname{TV}\left(f_2(X), f_2(Y)\right)}},
\end{aligned}$$ where the last inequality is by the coupling characterization of the t.v.d.: indeed, let $(X_1,Y_1)$ be a coupling with $X_1\sim f_1(X)$ and $Y_1\sim f_1(Y)$ and ${{\mathbf{P}}\left\{{X_1\neq Y_1}\right\}} = {\operatorname{TV}\left(f_1(X), f_1(Y)\right)}$, and define $(X_2,Y_2)$ similarly. Then $(u+X_1 v + X_2, u+Y_1 v + Y_2)$ is a coupling for $(X,Y)$ and it satisfies $${{\mathbf{P}}\left\{{u+X_1 v + X_2\neq u+Y_1 v + Y_2}\right\}}
\leq {{\mathbf{P}}\left\{{X_1\neq Y_1}\right\}}+{{\mathbf{P}}\left\{{X_2\neq Y_2}\right\}} ={\operatorname{TV}\left(f_1(X), f_1(Y)\right)} + {\operatorname{TV}\left(f_2(X), f_2(Y)\right)},$$ where the first inequality is simply the union bound.
We next claim that $\displaystyle f_1(X)\sim {\mathcal{N}}\left(\frac 1 2 ,\frac{v{^{\mathsf{T}}}\Sigma_1 v}{\red{(v{^{\mathsf{T}}}v)^2}} \right)$. To see this, observe that $f_1(X) = (X-u){^{\mathsf{T}}}v / v{^{\mathsf{T}}}v$ is a linear map of a Gaussian, so it is Gaussian. Its mean and covariance can be computed from those of $X$. Similarly, one can compute $\displaystyle f_1(Y)\sim {\mathcal{N}}\left(-\frac 1 2,\frac{v{^{\mathsf{T}}}\Sigma_2 v}{\red{(v{^{\mathsf{T}}}v)^2}}\right)$. So, Theorem \[thm:onedimensional\] gives $$\frac{1}{200} \min \left\{1, \max \left\{ \frac{|v{^{\mathsf{T}}}\Sigma_1 v-v{^{\mathsf{T}}}\Sigma_2 v|}{v{^{\mathsf{T}}}\Sigma_1 v} , {\frac{{40 v{^{\mathsf{T}}}v} }{{\sqrt{v{^{\mathsf{T}}}\Sigma_1 v}}}} \right\} \right\}
\leq
{\operatorname{TV}\left(f_1(X), f_1(Y)\right)}
\leq
\frac{3|v{^{\mathsf{T}}}\Sigma_1 v-v{^{\mathsf{T}}}\Sigma_2 v|}{2v{^{\mathsf{T}}}\Sigma_1 v}+ \red{\frac{{v{^{\mathsf{T}}}v}}{2{\sqrt{v{^{\mathsf{T}}}\Sigma_1 v}}}} .$$
On the other hand, since $f_2(w)=P(w-u)$ with $P = I_d - vv{^{\mathsf{T}}}/v{^{\mathsf{T}}}v$, $f_2(X)$ and $f_2(Y)$ are also Gaussians, with $f_2(X)\sim{\mathcal{N}}(0,P\Sigma_1P)$ and $f_2(Y)\sim{\mathcal{N}}(0,P\Sigma_2P)$. Note that ${\operatorname{range}}(P\Sigma_1P)={\operatorname{range}}(P\Sigma_2P)={\operatorname{range}}(\Pi)$. Also observe that since each column of $\Pi$ is orthogonal to $v$, we have $\Pi{^{\mathsf{T}}}P = \Pi$ and $P \Pi = \Pi$. Hence Theorem \[thm:meanzero\] gives $$\frac{1}{100} \min \{1, \| (\Pi{^{\mathsf{T}}}\Sigma_1 \Pi)^{-1}
\Pi{^{\mathsf{T}}}\Sigma_2 \Pi - I_{d-1} \|_F\}
\leq
{\operatorname{TV}\left(f_2(X), f_2(Y)\right)}
\leq
\frac{3}{2} \| (\Pi{^{\mathsf{T}}}\Sigma_1 \Pi)^{-1}
\Pi{^{\mathsf{T}}}\Sigma_2 \Pi - I_{d-1} \|_F,$$ completing the proof of the theorem.
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[^1]: Supported by NSERC Grant A3456.
[^2]: Supported by an IVADO-Apogée-CFREF Postdoctoral Fellowship.
[^3]: Supported by NSERC PGS D Scholarship 396164433.
|
---
abstract: 'A static self-gravitating electrically charged spherical thin shell embedded in a (3+1)-dimensional spacetime is used to study the thermodynamic and entropic properties of the corresponding spacetime. Inside the shell, the spacetime is flat, whereas outside it is a Reissner-Nordström spacetime, and this is enough to establish the energy density, the pressure, and the electric charge in the shell. Imposing that the shell is at a given local temperature and that the first law of thermodynamics holds on the shell one can find the integrability conditions for the temperature and for the thermodynamic electric potential, the thermodynamic equilibrium states, and the thermodynamic stability conditions. Through the integrability conditions and the first law of thermodynamics an expression for the shell’s entropy can be calculated. It is found that the shell’s entropy is generically a function of the shell’s gravitational and Cauchy radii alone. A plethora of sets of temperature and electric potential equations of state can be given. One set of equations of state is related to the Hawking temperature and a precisely given electric potential. Then, as one pushes the shell to its own gravitational radius and the temperature is set precisely equal to the Hawking temperature, so that there is a finite quantum backreaction that does not destroy the shell, one finds that the entropy of the shell equals the Bekenstein-Hawking entropy for a black hole. The other set of equations of state is such that the temperature is essentially a power law in the inverse Arnowitt-Deser-Misner (ADM) mass and the electric potential is a power law in the electric charge and in the inverse ADM mass. In this case, the equations of thermodynamic stability are analyzed, resulting in certain allowed regions for the parameters entering the problem. Other sets of equations of state can be proposed. Whatever the initial equation of state for the temperature, as the shell radius approaches its own gravitational radius, the quantum backreaction imposes the Hawking temperature for the shell in this limit. Thus, when the shell’s radius is sent to the shell’s own gravitational radius the formalism developed allows one to find the precise form of the Bekenstein-Hawking entropy of the correlated black hole.'
author:
- 'José P. S. Lemos'
- 'Gonçalo M. Quinta'
- 'Oleg B. Zaslavskii'
title: 'Entropy of a self-gravitating electrically charged thin shell and the black hole limit'
---
Introduction
============
In general relativity, in 3+1 dimensions, a black hole spacetime is characterized by its conserved charges and the fundamental constants. The conserved charges are for example the Arnowitt-Deser-Misner (ADM) mass $m$ and the electric charge $Q$. The fundamental constants are the two constants of the theory, namely, the gravitational constant $G$, and the velocity of light (which is set to one). In an analysis of quantum aspects of a black hole, such as the black hole entropy and its inherent degrees of freedom, the other fundamental constant in physics, Planck’s constant $\hbar$, also appears naturally. With these three constants one makes the Planck length $l_p=\sqrt{G\hbar}$ and the Planck area $A_p=l_p^2$. Also, $m$, $Q$, $G$, and the velocity of light give the horizon radius $r_+$ and so the horizon area $A_+=4\pi r_+^2$. Then, the Bekenstein-Hawking entropy of a black hole, given by $S_{\rm
bh}=\frac14\,\frac{A_+}{A_p}$ [@beken; @hawk1; @hawk2], where the Boltzmann constant is set to one, is a measure of how many Planck areas there are in the horizon area. It also shows that black hole quantum mechanics, and consequently black hole entropy, is in its essence and generality a process of pure quantum gravity, as no other constants besides the gravitational constant $G$, the velocity of light, and Planck’s constant $\hbar$ enter, through the Planck area ${A_p}$, in the final process. In addition, it suggests that the ultimate degrees of freedom that inhabit the realm of quantum gravity are in the area of the enclosing region, rather than within the volume as it is the case for ordinary matter [@thooft; @suss] (for a review see, e.g., [@lemos]). However, since there is no quantum gravity theory at hand, black hole entropy is still an enigma although there has been progress in its understanding, especially through the resort to gravitational low-energy quantum theories.
Since black holes are vacuum solutions, and our primitive concepts of entropy are based on the quantum properties of matter, it would be useful to have a spacetime with matter and study its thermodynamic and entropic properties. One then can look for a limit where a black hole might emerge. In this way, one can have hints to how a black hole’s entropy develops. We are thus interested in a system which contains both gravitational and material degrees of freedom but which does not introduce too many complexities due to the matter constitution.
The next simplest solution to a black hole solution, is a vacuum solution except for an infinitesimally thin region of spacetime where there is matter, i.e., a self-gravitating thin shell. As a thin shell is the nearest to a vacuum solution one can have, it is a very useful system that allows one to probe almost pure spacetime properties. A thin shell is defined as an infinitesimally thin surface which partitions spacetime into an interior region and an exterior region. Since it corresponds to some sort of matter and the spacetime properties must reflect it, the thin shell should satisfy some conditions in order for the entire spacetime to be a valid solution of the Einstein equations. Such conditions relate the stress-energy tensor of the shell to the extrinsic curvature of the spacetime. The stress-energy tensor yields the density and pressure, and in general the matter properties are also the equations of state, such as the temperature and possibly others, and the entropy. A particularly simple thin shell is one that is static and is spherically symmetric.
Suppose then a self-gravitating static spherical thin shell. Assume the simplest case, the inner spacetime is Minkowski and the outer spacetime is Schwarzschild. One can then work out its dynamics and thermodynamic properties, such as the temperature and entropy. In an elegant work, by finding the surface energy density and pressure, and imposing that the shell is at a given local temperature $T$, and so using a canonical ensemble, Martinez [@Mart] found those thermodynamic properties for the simplest shell, characterized by its rest mass $M$ and radius $R$. In [@Mart] only shells whose matter obeyed the dominant energy condition, and so the radius $R$ greater than a given value, were considered. Martinez’z approach [@Mart] draws in many respects from York’s work [@york1] where the thermodynamic properties of a pure Schwarzschild black hole is treated using a canonical ensemble, i.e., imposing a fixed temperature on some fictitious massless shell at a definite radius outside the event horizon. Another reason that motivates the use of thin shells is the fact that they can be taken with some ease to their own gravitational radius, i.e., to the black hole limit. If one does that, as was done in [@lemoszaslavskii], one recovers the black hole entropy, i.e., the entropy $S$ of the shell at its own gravitational radius is $S=S_{\rm
bh}=\frac14\,\frac{A_+}{A_p}$ for such a matter configuration at the black hole limit. Such a configuration is called a quasiblack hole. Thus, the black hole thermodynamic properties can be studied by a direct computation if thin shells are used.
It is important to generalize Martinez’s work [@Mart] for electrically charged shells, which we will do here. We consider that the shell has an electric charge $Q$. In this case, the inner spacetime is Minkowski and the outer spacetime is Reissner-Nordström. One can then work out the shell’s dynamics and thermodynamic properties, such as the energy density, the pressure, the electric potential temperature, and the entropy. Due to the introduction of a new state variable in the thermodynamic system, namely, the additional thermodynamic electric potential, the calculations become considerably more complex. At the same time the richness of the physical results increases as well. We take the shell to its own gravitational radius, the black hole end point, which is meaningful in the calculation of the shell’s entropy $S$, and find that the entropy is equal to the Bekenstein-Hawking entropy. The extremal $\sqrt{G}m= Q$ limit can then be taken which gives the same expression for the entropy, i.e., $S=S_{\rm
bh}=\frac14\,\frac{A_+}{A_p}$, see, however, [@pretvolisr; @lemoszaslaextremal] for a discussion of the entropy of extremal black holes taken from extremal shells. Electrically charged black holes were studied in [@yorketal; @pecalemos], where the thermodynamic properties of a pure Reissner-Nordström black hole are treated using a grand canonical ensemble, i.e., imposing a fixed temperature and electric potential at some definite radius outside the event horizon.
There are other works that used thin shells to understand the thermodynamics and the evolution of the entropy in certain spacetimes. In [@lemosquinta1; @lemosquinta2] the formalism of Martinez [@Mart] was used to study three-dimensional thin shells including the thermodynamics of a thin shell with a static Bañados-Teitelboim-Zanelli (BTZ) outer spacetime. In [@daviesfordpage; @hiscock] thin shells with a black hole inside were used to understand how the entropy of the spacetime evolves as the shell approaches its own event horizon.
We analyze static thin shells using the junction condition formalism established in [@Israel] with the complement to electrically charged shells developed in [@Kuchar]. Our thermodynamic approach, follows the general approach for thermodynamic systems given in [@callen], as does the approach of [@Mart].
We will adopt the following line of work. In Sec. \[thinsh\] we study a static spherical symmetric thin shell whose interior is Minkowski and exterior is Reissner-Nordström. We find the main properties of the global spacetime as well as the rest energy density, and thus the rest mass, the pressure in the shell, and the shell’s electric charge. In Sec. \[thermo\] we exhibit the first law of thermodynamics, find the generic integrability conditions and the stability conditions. In Sec. \[eqsos\] we use the spherical shell whose dynamics is displayed in Sec. \[thinsh\]. We present the three independent thermodynamic variables $(M,R,Q)$ and then through the integrability conditions find the functional dependence for the temperature $T$ and the thermodynamic electric potential $\Phi$ on those variables. Then in Sec. \[entro\] the differential for the entropy $S$ of the shell is obtained as a differential on the gravitational radius $r_+$ and the Cauchy horizon radius $r_-$ and up to two functions which depend on $r_+$ and $r_-$. Those functions are essentially the inverse of the temperature and the electric potential of the shell if it were located at infinity. Moreover, it is shown that the two functions are related by a specific differential equation, and that the entropy of the shell is a function of $r_+$ and $r_-$ alone, which themselves are functions of $(M,R,Q)$. In Sec. \[bhlimit\], to advance further, one needs to specify the form of the equations of state. We give a particular set of equations of state that will lead us with some ease to the black hole entropy when the shell is taken to its own gravitational radius. Indeed, by choosing the Hawking temperature due to quantum-mechanical arguments and a precise electric potential, the entropy of a charged black hole will naturally emerge. We then compare our approach with the usual thermodynamic approach for black holes. In Sec. \[eqstate\] we give another simple set of phenomenological equations of state for the temperature and the electric potential, where free parameters encoding the details of the matter fields will naturally appear. This set of equations of state allows us to also find the entropy and study analytically the stability conditions. In Sec. \[other\] we briefly discuss other interesting equations of state. Finally, in Sec. \[conc\] we conclude. We leave for Appendix A a study of the dominant energy condition of the matter fields in the shell which is not important in the thermodynamic study, but which is interesting to have. In Appendix B we derive the equations of thermodynamic stability for a system with three independent variables.
The thin-shell spacetime {#thinsh}
========================
The Einstein-Maxwell equations {#emeqs}
------------------------------
We start with the Einstein-Maxwell equations in 3+1 dimensions $$G_{{\alpha}{\beta}}=8\pi G\,
T_{{\alpha}{\beta}}\,,
\label{ein}$$ $$\nabla_{\beta}F^{\alpha\beta}=4\pi J^{\alpha}\,.
\label{max}$$ $G_{{\alpha}{\beta}}$ is the Einstein tensor, built from the spacetime metric $g_{{\alpha}{\beta}}$ and its first and second derivatives, $8\pi G$ is the coupling, with $G$ being the gravitational constant in 3+1 dimensions and we are using units in which the velocity of light is one, and $T_{{\alpha}{\beta}}$ is the energy-momentum tensor. $F_{\alpha\beta}$ is the Faraday-Maxwell tensor, $J_\alpha$ is the electromagnetic four-current and $\nabla_\beta$ denotes covariant derivative. The other Maxwell equation $\nabla_{[\gamma}F_{\alpha\beta]}=0$, where $[...]$ means antisymmetrization, is automatically satisfied for a properly defined $F_{\alpha\beta}$. Greek indices will be used for spacetime indices and run as $\alpha,\beta=0,1,2,3$, with 0 being the time index.
The thin-shell gravitational junction conditions {#gravj}
------------------------------------------------
We consider now a two-dimensional timelike massive electrically charged shell with radius $R$, which we will call $\Sigma$. The shell partitions spacetime into two parts, an inner region $\mathcal{V}_i$ and an outer region $\mathcal{V}_o$. In order to find a global spacetime solution for the Einstein equation, Eq. (\[ein\]), we will use the thin-shell formalism developed in [@Israel].
First, we specify the metrics on each side of the shell. In the inner region $\mathcal{V}_i$ ($r< R$) we assume the spacetime is flat, i.e. $$\begin{aligned}
\label{LEI}
ds_i^2 & = g_{{\alpha}{\beta}}^i dx^\alpha dx^\beta =\nonumber\\&
-dt_i^2 + dr^2 + r^2\, d\Omega^2\,,\quad r< R\,,\end{aligned}$$ where $t_i$ is the inner time coordinate, polar coordinates $(r,\theta,\phi)$ are used, and $d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2$. In the outer region $\mathcal{V}_o$ ($r> R$), the spacetime is described by the Reissner-Nordström line element $$\begin{aligned}
\label{LEO}
ds_o^2 & = g_{{\alpha}{\beta}}^o dx^\alpha dx^\beta = \nonumber\\
& \hspace{-3mm} -\left(1 - \frac{2Gm}{r} + \frac{G Q^2}{r^2}\right)
dt_o^2 + \frac{dr^2}{1 - \dfrac{2Gm}{r} + \dfrac{G Q^2}{r^2}} \nonumber \\
& \hspace{7mm} + r^2 d\Omega^2 \,, \quad r> R\,,\end{aligned}$$ where $t_o$ is the outer time coordinate, and again $(r,\theta,\phi)$ are polar coordinates, and $d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2$. The constant $m$ is to be interpreted as the ADM mass, or energy, and $Q$ as the electric charge. Finally, on the hypersurface itself, $r=R$, the metric $h_{ab}$ is that of a 2-sphere with an additional time dimension, such that, $$ds_{\Sigma}^2 = h_{ab} dy^a dy^b =
-d\tau^2 + R^2(\tau) d\Omega^2\,, \quad r= R\,,
\label{intrinsmetr}$$ where we have chosen $y^a=(\tau,\theta,\phi)$ as the time and spatial coordinates on the shell. We have adopted the convention to use latin indices for the components on the hypersurface. The time coordinate $\tau$ is the proper time for an observer located at the shell. The shell radius is given by the parametric equation $R= R(\tau)$ for an observer on the shell. On each side of the hypersurface, the parametric equations for the time and radial coordinates are denoted by $t_{i}=T_{i}(\tau)$, $r_{i}=R_{i}(\tau)$, and $t_{o}=T_{o}(\tau)$, $r_{o}=R_{o}(\tau)$. The metric $h_{ab}$ is also called the induced metric and can be written in terms of the 3+1-dimensional spacetime metric $g_{{\alpha}{\beta}}$. In particular, viewed from each side of the shell, the induced metric is given by $$h^{i}_{ab} = g^{i}_{{\alpha}{\beta}} \, e^{{\alpha}}_{i}{}_a \,
e^{{\beta}}_{i}{}_b\,,
\quad
h^{o}_{ab} = g^{o}_{{\alpha}{\beta}} \, e^{{\alpha}}_{o}{}_a \,
e^{{\beta}}_{o}{}_b\,,$$ where $e^{{\alpha}}_{i}{}_a$ and $e^{{\alpha}}_{o}{}_a$ are tangent vectors to the hypersurface viewed from the inner and outer regions, respectively. With these last expressions, we have all the necessary information to employ the formalism developed in [@Israel]. We will also apply this formalism to electrically charged systems which was displayed first in [@Kuchar].
The thin-shell formalism states that two junction conditions are needed in order to have a smooth change across the hypersurface. The first junction condition is expressed by the relation $$[h_{ab}]=0\,,$$ where the parentheses symbolize the jump in the quantity across the hypersurface, which in this case is the induced metric. This condition immediately implies that $h^{i}_{ab}=h^{o}_{ab}=h_{ab}$, or explicitly $$\begin{aligned}
\label{J1}
& -\left(1 - \frac{2Gm}{r} + \frac{G Q^2}{r^2}\right) \dot{T}_o^2 +
\nonumber \\ & \hspace{8mm} \frac{\dot{R}_o^2}{\left(1 -
\dfrac{2Gm}{r} + \dfrac{G Q^2}{r^2}\right)} = -\dot{T}_i^2 +
\dot{R}_i^2 = -1\,,\end{aligned}$$ where a dot denotes differentiation with respect to $\tau$. The second junction condition is related to the inner and outer extrinsic curvature $K^a_{i}{}_{b}$ and $K^a_{o}{}_{b}$, respectively, defined as $$K^a_{i}{}_{b} = \left(\nabla_{{\beta}}\, n^{i}_{{\alpha}}\right) \, e^{{\alpha}}_{i}{}_c
\, e^{{\beta}}_{i}{}_b \, h^{ca}_{i}\,,\,
K^a_{o}{}_{b} = \left(\nabla_{{\beta}}\, n^{o}_{{\alpha}}\right) \, e^{{\alpha}}_{o}{}_c
\, e^{{\beta}}_{o}{}_b \, h^{ca}_{o}
\,,
\label{extr1}$$ where $n^{i}_\alpha$ and $n^{o}_\alpha$, are the inner and outer normals to the shell, respectively. The second junction condition then says $\left[K^{a}{}_{b}\right]=0$ if the metric is to be smooth across the hypersurface. However, this condition can be violated, in which case it can be physically interpreted as the existence of a thin matter shell where the hypersurface is located. In addition, the shell’s stress-energy tensor $S^{a}{}_{b}$ is related to the jump in the extrinsic curvature through the Lanczos equation, namely, $$S^{a}{}_{b}=-\frac{1}{8 \pi G}
\left([K^{a}{}_{b}]-[K]h^{a}{}_{b}\right)\,,
\label{extr2}$$ where $K = h^{b}{}_{a} K^{a}{}_{b}$. Proceeding then to the calculation of the extrinsic curvature components, one can show that they are given by the general expressions $$\begin{aligned}
K^{\tau}_i{}_{\tau} &= \frac{\ddot{R}}{\sqrt{1+\dot{R}^2}}\,,
\label{a1}\\
K^{\tau}_o{}_{\tau} &= \frac{-\frac{G \dot{m}}{R\,\dot{R}}-\frac{G
Q^2}{R^3}+\frac{G m}{R^2}+\ddot{R}}{\sqrt{1-\frac{2Gm}{R} + \frac{G
Q^2}{R^2}+\dot{R}^2}}\,, \label{a2}\\
K^{\phi}_i{}_{\phi} = K^{\theta}_i{}_{\theta} &=
\frac{1}{R}\sqrt{1+\dot{R}^2}\,,\label{a3}\\
K^{\phi}_o{}_{\phi} = K^{\theta}_o{}_{\theta} &=
\frac{1}{R}\sqrt{1-\frac{2Gm}{R} + \frac{G Q^2}{R^2}+\dot{R}^2}\,.
\label{a4}\end{aligned}$$ Using Eqs. (\[a1\])-(\[a4\]) in Eq. (\[extr2\]), one can calculate the non-null components of the stress-energy tensor $S_{ab}$ of the shell. In particular, we will assume a static shell, such that $\dot{R}=0$, $\ddot{R}=0$, and $\dot{m}=0$. In that case, we are led to $$\begin{aligned}
S^{\tau}{}_{\tau} & = \frac{\sqrt{1-\frac{2Gm}{R} + \frac{G Q^2}{R^2}}
- 1}{4 \pi G R}\,, \label{S1} \\
S^{\phi}{}_{\phi} = S^{\theta}{}_{\theta} & =
\frac{\sqrt{1-\frac{2Gm}{R}+\frac{G Q^2}{R^2}}-1}{8 \pi G R} +
\nonumber\\
&
\frac{\frac{m G}{R}-\frac{G Q^2}{R^2}}{8 \pi G R
\sqrt{1-\frac{2Gm}{R}+\frac{G Q^2}{R^2}}} \label{S2}\,.\end{aligned}$$ To further advance, one needs to specify what kind of matter the shell is made of, which we will consider to be a perfect fluid with surface energy density $\sigma$ and pressure $p$. This implies that the stress-energy tensor will be of the form $$S^{a}{}_{b} = (\sigma +p) u^a u_b + p
h^{a}{}_{b}\,,
\label{perffluid}$$ where $u^a$ is the three-velocity of a shell element. We thus find that $$S^{\tau}{}_{\tau} = -\sigma\,,
\label{lam}$$ $$S^{\theta}{}_{\theta} = S^{\phi}{}_{\phi} = p
\label{press}\,.$$ Combining Eqs. (\[lam\])-(\[press\]) with Eqs. (\[S1\])-(\[S2\]) results in the equations $$\begin{aligned}
\sigma = & \frac{1-\sqrt{1-\frac{2Gm}{R} + \frac{G Q^2}{R^2}}}{4 \pi G
R}\,, \label{SS1} \\
p = & \frac{\sqrt{1-\frac{2Gm}{R}+\frac{G
Q^2}{R^2}}-1}{8 \pi G R} + \nonumber \\
&
\frac{\frac{m G}{R}-\frac{G Q^2}{R^2}}{8
\pi G R \sqrt{1-\frac{2Gm}{R}+\frac{G Q^2}{R^2}}}
\label{SS2}\,.\end{aligned}$$ Note that Eq. (\[SS2\]) is purely a consequence of the Einstein equation which is encoded in the junction conditions. Thus, although no information about the matter fields of the shell has been given, we know that they must have a pressure equation of the form (\[SS2\]), otherwise no mechanical equilibrium can be achieved.
It is useful to define the shell’s redshift function $k$ as $$k=\sqrt{
1 - \frac{2Gm}{R} + \frac{G Q^2}{R^2}}
\label{red0}\,.$$ Equation (\[red0\]) allows Eqs. (\[SS1\])-(\[SS2\]) to be written as $$\begin{aligned}
\sigma = & \frac{1-k}{4\pi G R} \label{Sig1}\,, \\
p = & \frac{R^2(1-k)^2 - GQ^2}{16 \pi G R^3 k}\,.
\label{pQk1}\end{aligned}$$ From the energy density $\sigma$ of the shell we can define the rest mass $M$ through the equation $$\sigma = \frac{M}{4\pi R^2} \,.
\label{sigmarestmasssigma}$$ Note that from Eqs. (\[Sig1\]) and (\[sigmarestmasssigma\]) one has $$\label{M1}
M = \frac{R}{G}(1-k).$$ Using Eqs. (\[red0\]) and (\[M1\]), we are led to an equation for the ADM mass $m$, $$\label{m0}
m = M - \frac{G M^2}{2R} + \frac{Q^2}{2R}\,.$$ This equation is intuitive on physical grounds as it states that the total energy $m$ of the shell is given by its mass $M$ minus the energy required to built it against the action of gravitational and electrostatic forces, i.e., $- \frac{G M^2}{2R} + \frac{Q^2}{2R}$. For $Q=0$, we recover the result derived in [@Mart]. Note that Eq. (\[m0\]) is also purely a consequence of the Einstein equation encoded in the junction conditions, i.e., although no information about the matter fields of the shell has been given, we know that they must have an ADM mass given by Eq. (\[m0\]).
The gravitational radius $r_+$ and the Cauchy horizon $r_-$ of the shell spacetime are given by the zeros of the $g_{00}^o$ in Eq. (\[LEO\]). They are then $$r_{+} = G\,m +\sqrt{G^2m^2 - G Q^2}\,,
\label{horradi}$$ $$r_{-} = G\,m - \sqrt{G^2m^2 - G Q^2}\,,
\label{horradicauch}$$ respectively. The gravitational radius $r_+$ is also the horizon radius when the shell radius $R$ is inside $r_+$, i.e., the spacetime contains a black hole. Although they have the same expression, conceptually, the gravitational and horizon radii are distinct. Indeed, the gravitational radius is a property of the spacetime and matter, independently of whether there is a black hole or not. On the other hand, the horizon radius exists only when there is a black hole.
The gravitational radius $r_+$ and the Cauchy horizon $r_-$ in Eqs. (\[horradi\])-(\[horradicauch\]) can be inverted to give $$m=\frac{1}{2G}\left( r_{+} + r_{-}\right)\,,
\label{invhorradi}$$ $$Q=\sqrt{
\frac{
r_{+}r_{-}}{G}
}\,.
\label{invhorradicauch}$$ From Eq. (\[horradi\]) one can define the gravitational area $A_+$ as $$\label{arear+}
A_+=4\pi\,r_+^2\,.$$ This is also the event horizon area when there is a black hole. Using Eqs. (\[horradi\])-(\[horradicauch\]) implies that $k$ in Eq. (\[red0\]) can be written as $$\label{red}
k=\sqrt{\Big(1-\frac{r_+}{R}\Big)\Big(1-\frac{r_-}{R}\Big)}\,.$$
The area $A$ of the shell, an important quantity, is from Eq. (\[intrinsmetr\]) given by $$A=4 \pi R^2\,.
\label{area1}$$
The thin-shell electromagnetic junction conditions {#emj}
--------------------------------------------------
Now we have to deal with Eq. (\[max\]). The Faraday-Maxwell tensor $F_{\alpha\beta}$ is usually defined in terms of an electromagnetic four-potential $A_\alpha$ by $$F_{{\alpha}{\beta}} = \partial_{{\alpha}}A_{{\beta}}-\partial_{{\beta}}A_{{\alpha}}\,,
\label{empot}$$ where $\partial_{\beta}$ denotes partial derivative.
To use the thin-shell formalism related to the electric part we need to specify the vector potential $A_\alpha$ on each side of the shell. We assume an electric ansatz for the electromagnetic four-potential $A_\alpha$, i.e., $$A_{\alpha} = (-\phi,
0,0,0)\,,
\label{empotspec}$$ where $\phi$ is thus the electric potential. In the inner region $\mathcal{V}_i$ ($r < R$) the spacetime is flat. So the Maxwell equation $\nabla_{\beta}F^{\alpha\beta}=\frac{1}{\sqrt{-g}}
\partial_{\beta}\left(\sqrt{-g}F^{\alpha\beta}\right)
=0$ has as a constant solution for the inner electric potential $\phi_i$ which, for convenience, can be written as $$\label{phiin}
\phi_i= \frac{Q}{R}+{\rm constant}\,,\quad r< R\,,$$ where $Q$ is a constant, to be interpreted as the conserved electric charge. In the outer region $\mathcal{V}_o$ ($r> R$), the spacetime is Reissner-Nordström and the Maxwell equation $\nabla_{\beta}F^{\alpha\beta}=\frac{1}{\sqrt{-g}}
\partial_{\beta}\left(\sqrt{-g}F^{\alpha\beta}\right)
=0$ now yields $$\label{phiout}
\phi_o= \frac{Q}{r}+{\rm constant}\,,\quad r>R\,.$$
Due to the existence of electricity in the shell, another important set of restrictions must also be considered. These restrictions are related to the discontinuity present in the electric field across the charged shell. We are interested in the projection $$A_a = A_{{\alpha}} \,e^{{\alpha}}_a$$ of the four-potential in the shell’s hypersurface, since it will contain quantities which are intrinsic to the shell. Indeed, following [@Kuchar], $$\label{potjunct1}
\left[A_{a}\right]=0\,,$$ with $A_{i\,a} = (-\phi_{i}, 0,0)$, and $A_{o\,a} = (-\phi_{o}, 0,0)$ being the vector potential at $R$, on the shell, seen from each side of it. Thus, the constants in Eqs. (\[phiin\]) and (\[phiout\]) are indeed the same and so at $R$ $$\phi_o =
\phi_i \,,\quad r=R\,.$$ Following [@Kuchar] further, the tangential components $F_{ab}$ of the electromagnetic tensor $F_{{\alpha}{\beta}}$ must change smoothly across $\Sigma$, i.e. $$\label{JC1}
\left[F_{ab}\right]=0\,,$$ with $$F^{i}_{ab} = F^{i}_{{\alpha}{\beta}} e^{{\alpha}}_{i}{}_a \,
e^{{\beta}}_{i}{}_b\,,\quad
F^{o}_{ab} = F^{o}_{{\alpha}{\beta}} e^{{\alpha}}_{o}{}_a \,
e^{{\beta}}_{o}{}_b\,,$$ while the normal components $F_{a\perp}$ must change by a jump as, $$\label{JC2}
\left[F_{a\perp}\right] = 4\pi \sigma_e u_{a}\,,$$ where $$F^{i}_{a\perp} = F^{i}_{{\alpha}{\beta}} e^{{\alpha}}_{i}{}_a \,
n^{{\beta}}_{i}\,,\quad
F^{o}_{a\perp} = F^{o}_{{\alpha}{\beta}} e^{{\alpha}}_{o}{}_a \,
n^{{\beta}}_{o}\,,$$ and $\sigma_eu_{a}$ is the surface electric current, with $\sigma_e$ being the density of charge and $u_{a}$ its 3-velocity, defined on the shell. One can then show that Eq. (\[JC1\]) is trivially satisfied, while Eq. (\[JC2\]) leads to the single nontrivial equation at $R$, on the shell, $$\label{JCn1}
\frac{\partial\phi_o}{\partial r}
-\frac{\partial\phi_i}{\partial r}
= - 4\pi \sigma_e\,,\quad r=R\,.$$ Then, from Eqs. (\[phiin\]), (\[phiout\]), and (\[JCn1\]) one obtains $$\label{PhiJC}
\frac{Q}{R^2}=4\pi \sigma_e \,,$$ relating the total charge $Q$, the charge density $\sigma_e$, and the shell’s radius $R$ in the expected manner. This section with its equations forms the dynamical side of the electric thin shell solution.
Restrictions on the thin-shell radius {#restric}
-------------------------------------
A natural inequality that the shell should obey is to consider the shell to be outside its gravitational radius in all instances, so $$R \geq r_+.
\label{notrapped}$$ It is then clear that the physical allowed values for $k$ in Eq. (\[red\]) are in the interval $[0,1]$. It is also interesting to consider the restrictions imposed by the dominant energy condition. However, since it will not take part in our analysis we leave this discussion for Appendix \[apa\].
Thermodynamics and stability conditions for the thin shell: Generics {#thermo}
====================================================================
Thermodynamics and integrability conditions for the thin shell {#thermo2}
--------------------------------------------------------------
We now turn to the thermodynamic side and to the calculation of the entropy of the shell. We use units in which the Boltzmann constant is one. We start with the assumption that the shell in static equilibrium possesses a well-defined temperature $T$ and an entropy $S$ which is a function of three variables, call them $M$, $A$, $Q$, i.e., $$\label{entropy0}
S=S(M,A,Q)\,.$$ $(M,A,Q)$ can be considered as three generic parameters. In our connection they are the shell’s rest mass $M$, area $A$, and charge $Q$. The first law of thermodynamics can thus be written as $$\label{TQ}
T dS = dM + pdA - \Phi dQ$$ where $dS$ is the differential of the entropy of the shell, $dM$ is the differential of the rest mass, $dA$ is the differential of the area of the shell, $dQ$ is the differential of the charge, and $T$, $p$ and $\Phi$ are the temperature, the pressure, and the thermodynamic electric potential of the shell, respectively. In order to find the entropy $S$, one thus needs three equations of state, namely, $$\label{press0}
p=p(M,A,Q)\,,$$ $$\label{temper}
\beta=\beta(M,A,Q)\,,$$ $$\label{electr}
\Phi=\Phi(M,A,Q)\,,$$ where $$\label{beta}
{\beta}\equiv \frac1T$$ is the inverse temperature.
It is important to note that the temperature and the thermodynamic electric potential play the role of integration factors, which implies that there will be integrability conditions that must be specified in order to guarantee the existence of an expression for the entropy, i.e. that the differential $dS$ is exact. These integrability conditions are $$\begin{aligned}
\left(\frac{{\partial}{\beta}}{{\partial}A}\right)_{M,Q} & = \hspace{3mm}
\left(\frac{{\partial}{\beta}p}{{\partial}M}\right)_{A,Q}\,, \label{Ione} \\
\left(\frac{{\partial}{\beta}}{{\partial}Q}\right)_{M,A} & = - \left(\frac{{\partial}{\beta}\Phi}{{\partial}M}\right)_{A,Q}\,, \label{Itwo} \\
\left(\frac{{\partial}{\beta}p}{{\partial}Q}\right)_{M,A} & = - \left(\frac{{\partial}{\beta}\Phi}{{\partial}A}\right)_{M,Q} \,.
\label{Ithree}\end{aligned}$$ These equations enable one to determine the relations between the three equations of state of the system.
Stability conditions for the thin shell {#thermo3}
---------------------------------------
With the first law of thermodynamics given in Eq. (\[TQ\]), one is able to perform a thermodynamic study of the local intrinsic stability of the shell. To have thermodynamic stability the following inequalities should hold $$\label{B1}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right)_{A,Q} \leq 0\,,$$ $$\label{B2}
\left(\frac{{\partial}^2 S}{{\partial}A^2}\right)_{M,Q} \leq 0\,,$$ $$\label{B3}
\left(\frac{{\partial}^2 S}{{\partial}Q^2}\right)_{M,A} \leq 0\,,$$ $$\label{B4}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right)\left(\frac{{\partial}^2 S}{{\partial}A^2}\right) - \left(\frac{{\partial}^2 S}{{\partial}M {\partial}A}\right)^2 \geq 0\,,$$ $$\label{B5}
\left(\frac{{\partial}^2 S}{{\partial}A^2}\right)\left(\frac{{\partial}^2 S}{{\partial}Q^2}\right) - \left(\frac{{\partial}^2 S}{{\partial}A {\partial}Q}\right)^2 \geq 0\,,$$ $$\label{B6}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right)\left(\frac{{\partial}^2 S}{{\partial}Q^2}\right) - \left(\frac{{\partial}^2 S}{{\partial}M {\partial}Q}\right)^2 \geq 0\,,$$ $$\label{B7}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right) \left(\frac{{\partial}^2 S}{{\partial}Q {\partial}A}\right) - \left(\frac{{\partial}^2 S}{{\partial}M {\partial}A}\right) \left(\frac{{\partial}^2
S}{{\partial}M {\partial}Q}\right) \geq 0\,.$$ The derivation of these expressions follows the rationale presented in [@callen], see Appendix \[apb\].
The thermodynamic independent variables and the three equations of state: equations for the pressure, temperature and electric potential {#eqsos}
========================================================================================================================================
The three independent thermodynamic variables $(M,R,Q)$
-------------------------------------------------------
We will work from now onwards with the three independent variables $(M,R,Q)$ instead of $(M,A,Q)$. The rest mass $M$ of the shell is from Eq. (\[sigmarestmasssigma\]) given by $$M = 4\pi R^2 \, \sigma \,,
\label{restmasssigma}$$ where $\sigma$ is given by Eq. (\[Sig1\]) and $R$ is the radius of the shell. The first law of thermodynamics written in generic terms is simpler when expressed using the area $A$ of the shell, but here it is handier to use the radius $R$ in this specific study. The radius $R$ is related to the area $A$ through Eq. (\[intrinsmetr\]), i.e., $$R= \sqrt
{
\dfrac{A}{4\,\pi}}.
\label{radiusarea}$$ As for the charge $Q$, using Eq. (\[PhiJC\]), it is given by $$Q = 4\pi R^2 \, \sigma_e .$$ The three independent thermodynamic variables are thus $(M,R,Q)$.
We should now envisage Eq. (\[m0\]) and Eqs. (\[horradi\])-(\[horradicauch\]) as functions of $(M,R,Q)$, i.e. $$\label{m}
m(M,R,Q) = M - \frac{G M^2}{2R} + \frac{Q^2}{2R}\,,$$ and $$r_{+}(M,R,Q) = G\,m(M,R,Q) +\sqrt{G^2m(M,R,Q)^2 - G Q^2}\,,
\label{horradi2}$$ $$r_{-}(M,R,Q) = G\,m(M,R,Q) - \sqrt{G^2m(M,R,Q)^2 - G Q^2}\,,
\label{horradicauch2}$$ respectively. The function $k$ in Eq. (\[red\]) is also a function of $(M,R,Q)$, $$\begin{aligned}
\label{red2}
&k(r_{+}(M,R,Q),r_{-}(M,R,Q),R)=&\nonumber\\
&\sqrt{\Big(1-\frac{r_+(M,R,Q)}{R}\Big)\Big(1-
\frac{r_-(M,R,Q)}{R}\Big)}\,.\end{aligned}$$
The pressure equation of state
------------------------------
Expressing the pressure equation of state in the form of Eq. (\[press0\]), we obtain from Eqs. (\[SS2\]) and (\[m0\]) \[or Eq. (\[m\])\], $$\label{pinMRQ}
p(M,R,Q) = \frac{GM^2-Q^2}{16 \pi R^2 (R-GM)}\,,$$ or changing from the variables $(M,R,Q)$ to $(r_+,r_-,R)$ which is more useful, we find \[see Eqs. (\[pQk1\]) and (\[invhorradicauch\])\], $$\begin{aligned}
\label{pQk}
&p(r_+,r_-,R) = \nonumber \\
&\frac{R^2(1-k)^2
- r_+ r_-}{16 \pi G R^3 \,k}\,, \end{aligned}$$ where $k$ can be envisaged as $k=k(r_+,r_-,R)$ as given in Eq. (\[red2\]) and $r_+$ and $r_-$ are functions of $\,(M,R,Q)$, see Eqs. (\[horradi2\])-(\[horradicauch2\]). This reduces to the expression obtained in [@Mart] in the limit $Q=0$ or $r_-=0$. This equation, Eq. (\[pQk\]), is a pure consequence of the Einstein equation, encoded in the junction conditions.
The temperature equation of state
---------------------------------
Turning now to the temperature equation of state (\[temper\]), we will need to focus on the integrability condition (\[Ione\]). Changing from the variables $(M,R,Q)$ to $(r_+,r_-,R)$, Eq. (\[Ione\]) becomes $$\left(\frac{{\partial}{\beta}}{{\partial}R}\right)_{r_+,r_-} =
{\beta}\frac{R(r_+ + r_-)-2r_+ r_-}{2 R^3 k^2}$$ which has the analytic solution $$\label{BS5}
{\beta}(r_+,r_-,R) = b(r_+,r_-) \,k$$ where $k$ is a function of $r_+$, $r_-$, and $R$, as given in Eq. (\[red2\]), and $b(r_+,r_-)\equiv {\beta}(r_+,r_-,\infty)$ is an arbitrary function, representing the inverse of the temperature of the shell if its radius were infinite. Hence, in a sense, from Eq. (\[BS5\]), we recover Tolman’s formula for the temperature of a body in curved spacetime. The arbitrariness of this function is due to the fact that the matter fields of the shell are not specified. Note that $b$ and $k$ are still functions of $(M,R,Q)$ through the variables $r_+$ and $r_-$, see Eqs. (\[horradi2\])-(\[horradicauch2\]) and Eq. (\[red2\]).
The electric potential equation of state
----------------------------------------
The remaining equation of state to be studied is the electric potential. Using Eqs. (\[M1\]) and (\[red2\]), one can deduce $\left(\frac{\partial M}{\partial A}\right)_{r_{+},r_{-}}=-p$, i.e., $$\left(\frac{\partial M}{\partial R}\right)_{r_{+},r_{-}}=-8\pi\,R\,p\,.
\label{mp}$$ Then, it follows from Eqs. (\[Ione\])-(\[Ithree\]) and Eq. (\[mp\]) that the differential equation $$\left(\frac{{\partial}p}{{\partial}Q}\right)_{M,R} + \frac{1}{8 \pi
R}\left(\frac{{\partial}\Phi}{{\partial}R}\right)_{r_+,r_-} + \Phi \left(\frac{{\partial}p}{{\partial}M}\right)_{R,Q} = 0\,,
\label{phieqdiff}$$ holds, where the second term has been expressed in the variables $(r_+,r_-,R)$ and the other terms in the variables $(M,R,Q)$ for the sake of computational simplicity. Then, after using Eq. (\[pinMRQ\]) in Eq. (\[phieqdiff\]), we obtain that Eq. (\[phieqdiff\]) takes the form $$R^{2}\,
\left(\frac{\partial \Phi k}{\partial R}\right)_{r_+,r_-}
- \frac{\sqrt{r_+ r_-}}
{\sqrt{G}}=0\,,
\label{above}$$ where $k$ can be envisaged as $k=k(r_+,r_-,R)$ as given in Eq. (\[red2\]). The solution of Eq. (\[above\]) is then $$\label{Phi0}
\Phi(r_+,r_-,R) = \frac{\phi(r_+,r_-) - \frac{\sqrt{r_+ r_-}}
{\sqrt{G}R}}{k}$$ where $\phi(r_+,r_-)\equiv \Phi(r_+,r_-,\infty)$ is an arbitrary function that corresponds physically to the electric potential of the shell if it were located at infinity. This thermodynamic electric potential $\Phi$ is the difference in the electric potential $\phi$ between infinity and $R$, blueshifted from infinity to $R$ (see a similar result in [@yorketal; @pecalemos] for an electrically charged black hole in a grand canonical ensemble). We also see that, once again, by changing to the variables $(r_+,r_-,R)$ we are able somehow to reduce the number of arguments of the arbitrary function from three to two.
It is convenient to define a function $c(r_+,r_-)$ through $c(r_+,r_-) \equiv \frac{\phi(r_+,r_-)}{Q}$, i.e., $$\label{cr+r-}
c(r_+,r_-) \equiv \sqrt{G} \frac{\phi(r_+,r_-)}{\sqrt{r_+ r_-}}\,,$$ where we have used $Q=\sqrt{r_+r_-/G}$ as given in Eq. (\[invhorradicauch\]). Then, Eq. (\[Phi0\]) is written as $$\label{Phi}
\Phi(r_+,r_-,R) = \dfrac{ {c(r_+,r_-)}
- \dfrac{1}{R} }
{k}\,\sqrt{\frac{r_+r_-}{G}}\,,$$ where $k$ can be envisaged as $k=k(r_+,r_-,R)$ as given in Eq. (\[red2\]).
Entropy of the thin shell {#entro}
=========================
At this point we have all the necessary information to calculate the entropy $S$. By inserting the equations of state for the pressure, Eq. (\[pQk\]), for the temperature, Eq. (\[BS5\]), and for the electric potential, Eq. (\[Phi\]), as well as the differential of $M$ given in Eq. (\[M1\]) and the differential of the area $A$ or of the radius $R$, see Eq. (\[radiusarea\]), into the first law, Eq. (\[TQ\]), we arrive at the entropy differential $$\begin{aligned}
\label{dSQ}
dS = b(r_+,r_-) & \frac{1-c(r_+,r_-) r_-}{2G} dr_+ \nonumber \\
& + b(r_+,r_-)\frac{1-c(r_+,r_-) r_+}{2G} dr_-\,,\end{aligned}$$ Now, Eq. (\[dSQ\]) has its own integrability condition if $dS$ is to be an exact differential. Indeed, it must satisfy the equation $$\label{bc5}
\frac{{\partial}b}{{\partial}r_-} (1-r_- c) - \frac{{\partial}b}{{\partial}r_+}(1-r_+ c) =
\frac{{\partial}c}{{\partial}r_-} b r_- - \frac{{\partial}c}{{\partial}r_+} b r_+.$$ This shows that in order to obtain a specific expression for the entropy one can choose either $b$ or $c$, and the other remaining function can be obtained by solving the differential equation (\[bc5\]) with respect to that function. Since Eq. (\[bc5\]) is a differential equation there is still some freedom in choosing the other remaining function. In the first examples we will choose to specify the function $b$ first and from it obtain an expression for $c$. We also give examples where the function $c$ is specified first.
From Eq. (\[dSQ\]) we obtain $$\label{entr5}
S=S(r_+,r_-)\,,$$ so that the entropy is a function of $r_+$ and $r_-$ alone. In fact $S$ is a function of $(M,R,Q)$, $S(M,R,Q)$, but the functional dependence has to be through $r_+(M,R,Q)$ and $r_-(M,R,Q)$, i.e., in full form $$\label{entr6}
S(M,R,Q)=S(r_+(M,R,Q),r_-(M,R,Q))\,.$$ This result shows that the entropy of the thin charged shell depends on the $(M,R,Q)$ through $r_+$ and $r_-$ which themselves are specific functions of $(M,R,Q)$.
It is also worth noting the following feature. From Eq. (\[entr6\]) we see that shells with the same $r_+$ and $r_-$, i.e., the same ADM mass $m$ and charge $Q$, but different radii $R$, have the same entropy. Let then an observer sit at infinity and measure $m$ and $Q$ (and thus $r_+$ and $r_-$). Then, the observer cannot distinguish the entropy of shells with different radii. This is a kind of thermodynamic mimicker, as a shell near its own gravitational radius and another one far from it have the same entropy.
The thin shell and the black hole limit {#bhlimit}
=======================================
The temperature equation of state and the entropy {#peos}
-------------------------------------------------
Let us consider a charged thin shell, for which the differential of the entropy has been deduced to be Eq. (\[dSQ\]). We are free to choose an equation of state for the inverse temperature. Let us pick for convenience the following inverse temperature dependence, $$\label{bBH}
b(r_+,r_-) = \gamma\,
\frac{r_+^2}{r_+-r_-}\,,$$ where $\gamma$ is some constant with units of inverse mass times inverse radius, i.e., units of angular momentum.
For a charged shell we must also specify the function $c(r_+,r_-)$, whose form can be taken from the differential equation (\[bc5\]) upon substitution of the function (\[bBH\]). There is a family of solutions for $c(r_+,r_-)$ but for our purposes here we choose the following specific solution, $$\label{cBH}
c(r_+,r_-) = \frac{1}{r_+}\,.$$ The rationale for the choices above becomes clear when we discuss the shell’s gravitational radius, i.e., black hole, limit. Inserting the choice for $b(r_+,r_-)$, Eq. (\[bBH\]), along with the choice for the function $c(r_+,r_-)$, Eq. (\[cBH\]), in the differential (\[dSQ\]) and integrating, we obtain the entropy differential for the shell $$\label{diffSS}
dS = \frac{\gamma}{2\, G} \,r_+\, dr_+\,.$$ Thus, the entropy of the shell is $S=\frac{\gamma}{4\, G} \,r_+^2+ S_0$, where $S_0$ is an integration constant. Imposing that when the shell vanishes (i.e., $M=0$ and $Q=0$, and so $r_+=0$) the entropy vanishes we have that $S_0$ is zero, and so $S=\frac{\gamma}{4\, G} \,r_+^2$. Thus, we can write the entropy $S(M,R,Q)$ as $$\label{SSa}
S = \frac{\gamma}{16\pi G}\, A_+\,,$$ where $A_+$ is the gravitational area of the shell, as given in Eq. (\[arear+\]). This result shows that the entropy of this thin charged shell depends on $(M,R,Q)$ through $r_+^2$ only, which itself is a specific function of $(M,R,Q)$.
Now, what is the constant $\gamma$? It should be determined by the properties of the matter in the shell, and cannot be decided a priori.
The stability conditions for the specific temperature ansatz {#stabil1}
------------------------------------------------------------
The thermodynamic stability of the uncharged case ($Q=0$, i.e., $r_-=0$) can be worked out [@Mart] and elucidates the issue. In the uncharged case the nontrivial stability conditions are given by Eqs. (\[B1\]) and (\[B4\]). Equation (\[B1\]) gives immediately $R\leq\frac32 r_+$, i.e., $R\leq3Gm$. On the other hand, Eq. (\[B4\]) yields $R\geq r_+$, i.e., $R\geq2Gm$. Thus, the stability conditions yield the following range for $R$, $r_+\leq
R\leq\frac32 r_+$, or in terms of $m$, $2Gm\leq R\leq 3Gm$. This is precisely the range for stability found by York [@york1] for a black hole in a canonical ensemble in which a spherical massless thin wall at radius $R$ is maintained at fixed temperature $T$. In [@york1] the criterion used for stability is that the heat capacity of the system should be positive, and physically such a tight range for $R$ means that only when the shell, at a given temperature $T$, is sufficiently close to the horizon can it smother the black hole enough to make it thermodynamically stable. The positivity of the heat capacity is equivalent to our stability conditions, Eqs. (\[B1\]) and (\[B4\]) in the uncharged case.
The stability conditions, Eqs. (\[B1\])-(\[B7\]), for the general charged case cannot be solved analytically in this instance, they require numerical work, which will shadow what we want to determine. Nevertheless, the approach followed in [@yorketal; @pecalemos] for the heat capacity of a charged black hole in a grand canonical ensemble gives a hint of the procedure that should be followed.
The black hole limit {#bhl}
--------------------
### The black hole limit properly stated {#bhlps}
Although $\gamma$ should be determined by the properties of the matter in the shell, there is a case in which the properties of the shell have to adjust to the environmental properties of the spacetime. This is the case when $R
\to r_+$. In this case, one must note that, as the shell approaches its own gravitational radius, quantum fields are inevitably present and their backreaction will diverge unless we choose the black hole Hawking temperature $T_{\rm bh}$ for the temperature of the shell. In this case, $R\to r_+$, we must take the temperature of the shell as $T_{\rm bh} = \frac{\hbar}{4 \pi} \frac{r_+-r_-}{r_+^2}$, where $\hbar$ is Planck’s constant. So we must choose $$\gamma= \frac{4\pi}{\hbar} \,,
\label{ga}$$ i.e., $\gamma$ depends on fundamental constants. Then, $$\label{thawk}
b(r_+,r_-) = \frac{1}{T_{\rm bh}}
=\frac{4 \pi} {\hbar}
\frac{r_+^2}{r_+-r_-}\,.$$ In this case the entropy of the shell is $S= \frac14\frac{A_+}{G\hbar}$, i.e., $$\label{SS}
S=
\frac{1}{4}\frac{A_+}{A_p}\,,$$ where $l_p=\sqrt{G\hbar}$ is the Planck length, and $A_p=l_p^2$ the Planck area. Note now that the entropy given in Eq. (\[SS\]) is the black hole Bekenstein-Hawking entropy $S_{\rm bh}$ of a charged black hole since $$\label{SSBH}
S_{\rm bh}= \frac{1}{4}\frac{A_+}{A_p}\,,$$ where $A_+$ is here the horizon area. Thus, when we take the shell to its own gravitational radius the entropy is the Bekenstein-Hawking entropy. The limit also implies that the pressure and the thermodynamic electric potential go to infinity as $1/k$, according to Eqs. (\[pQk\]) and (\[Phi\]), respectively. Note, however, that the local inverse temperature goes to zero as $k$, see Eq. (\[BS5\]), and so the local temperature of the shell also goes to infinity as $1/k$. These well-controlled infinities cancel out precisely to give the Bekenstein-Hawking entropy (\[SS\]).
Note that, since $A=A_+$ when the shell is at its own gravitational radius, at this point the entropy of the shell is proportional to its own area $A$, indicating that all the shell’s fundamental degrees of freedom have been excited.
Note also that the shell at its own gravitational radius, at least in the uncharged case, is thermodynamically stable, since in this case stability requires $r_+\leq R\leq\frac32 r_+$, as mentioned above.
Note yet that our approach and the approach followed in [@pretvolisr] to find the black hole entropy have some similarities. The two approaches use matter fields, i.e., shells, to find the black hole entropy. Here we use a static shell that decreases its own radius $R$ by steps, maintaining its staticity at each step. In [@pretvolisr] a reversible contraction of a thin spherical shell down to its own gravitational radius was examined, and it was found that the black hole entropy can be defined as the thermodynamic entropy stored in the matter in the situation that the matter is compressed into a thin layer at its own gravitational radius.
Finally we note that the extremal limit $\sqrt{G}m=Q$ or $r_+=r_-$ is well defined from above. Indeed, when one takes the limit $r_+\to
r_-$ one finds that $1/b(r_+,r_-)=0$ (i.e., the Hawking temperature is zero) and the entropy of the extremal black hole is still given by $S_{\rm extremal\,bh}= \frac{1}{4}\frac{A_+}{A_p}$. It is well known that extremal black holes and in particular their entropy have to be dealt with care. If, ab initio, one starts with the analysis for an extremal black hole one finds that the entropy of the extremal black hole has a more general expression than simply being equal to one quarter of the area [@pretvolisr; @lemoszaslaextremal]. This extremal shell is an example of a Majumdar-Papapetrou matter system. Its pressure $p$ is zero, and it remains zero, and thus finite, even when $R\to r_+$. This limit of $R\to r_+$ is called a quasiblack hole, which in the extremal case is a well-behaved one.
### The rationale for the choice of $b(r_{+},r_{-})$ and $c(r_{+},r_{-})$ {#rational}
We have started with a thin shell and imposed a temperature equation of state of the Hawking type, see Eq. (\[bBH\]) \[see also Eqs. (\[ga\]), and (\[thawk\])\], and a specific thermodynamic electric potential, see Eq. (\[cBH\]). This set of equations of state gives an entropy for the shell proportional to its gravitational radius area $A_+$. One can moreover set the temperature of the shell at any $R$ precisely equal to the Hawking temperature $T_{\mathrm{bh}}$, see Eq. (\[thawk\]). Remarkably, we have then shown that a self-gravitating electric thin shell at the Hawking temperature and with a specific electric potential has a Bekenstein-Hawking entropy.
A priori, Hawking-type choices for the temperature \[Eqs. (\[bBH\]), (\[ga\]), and (\[thawk\])\], and black hole-type choices for the electric potential \[Eq. (\[cBH\])\], are simply choices, many other choices for the set of equations of state can be taken. However, this set is really imposed on the shell when it approaches its gravitational radius, where it takes the precise forms given in Eqs. (\[bBH\]) and (\[ga\]) \[or, Eq. (\[thawk\])\], and (\[cBH\]) as the spacetime quantum effects get a hold on the shell.
We would like to stress that the requirement $b=T_{\mathrm{bh}}^{-1}$ \[see Eqs. (\[bBH\]), (\[ga\]), and (\[thawk\])\] is compulsory only for shells that approach their own gravitational radius. Otherwise, if we consider the radius of the shell within some constrained region outside the gravitational radius, the shell temperature can be arbitrary since away from the horizon, quantum backreaction remains modest and does not destroy the thermodynamic state. One can discuss whole classes of functions $b(r_{+},r_{-})\neq
T_{\mathrm{bh}}^{-1}$.
In addition we stress that the choice for $c$, Eq. (\[cBH\]), is necessary only for shells at the gravitational radius limit. According to Eq. (\[cr+r-\]), this gives us $\phi =\frac{ \sqrt{r_-} }{ \sqrt{G\, r_{+} }}$, i.e., $$\phi =\frac{Q}{r_{+}} \label{rn}$$ that coincides with the standard expression for the electric potential for the Reissner-Nordström black hole. In addition, Eq. (\[Phi\]) acquires the form $$\Phi =\frac{Q}{k}\left(\frac{1}{r_{+}}-\frac{1}{R}\right)
\label{frn}$$ that coincides entirely with the corresponding formula for the Reissner-Nordström black hole in a grand canonical ensemble [@yorketal]. Meanwhile, in our case there is no black hole. Moreover, if we go to the uncharged case, $Q\rightarrow0$ or $r_{-}\rightarrow 0$, and thus the outer space is described by the Schwarzschild metric, then it is seen from Eq. (\[dSQ\]) that the quantity $c$ drops out from the entropy, so the choice of $c$ is relevant for the charged case only, of course.
### Similarities between the thin-shell approach and the black hole mechanics approach {#sim}
There are similarities between the thin-shell approach and the black hole mechanics approach [@hawk1]. These are evident if we express the differential of the entropy of the charged shell (\[dSQ\]) in terms of the black hole ADM mass $m$ and charge $Q$, given in terms of the variables $(r_+,r_-)$ by Eqs. (\[invhorradi\])-(\[invhorradicauch\]). The differential for the entropy of the shell reads in these variables $$T_0 dS = dm - c\, Q\, dQ$$ where we have defined $T_0 \equiv 1/b(r_+,r_-)$ which is the temperature the shell would possess if located at infinity. Here, $T_0 =1/b(r_+,r_-)$ and $c=c(r_+,r_-)$ should be seen as $T_0 (m,Q)=1/b(m,Q)$ and $c(m,Q)$, respectively, since $r_+$ and $r_-$ are functions of $m$ and $Q$. As we have seen, if we take the shell to its gravitational radius, we must fix $T_0 = T_{\rm bh}$ and $c = 1/r_+$. This suggests that $Q/r_+$ should play the role of the black hole electric potential $\Phi_{\rm bh}$, which in fact is true, as shown in Eq. (\[rn\]) (see [@hawk1], see also [@yorketal; @pecalemos]). So the conservation of energy of the shell is expressed as $$\label{1T7}
T_{\rm bh} dS_{\rm bh} = dm - \Phi_{\rm bh}\, dQ\,.$$ We thus see that the first law of thermodynamics for the shell at its own gravitational radius is equal to the energy conservation for the Reissner-Nordström black hole.
The thin shell with another specific equation of state for the temperature {#eqstate}
==========================================================================
The temperature equation of state and the entropy {#the-temperature-equation-of-state-and-the-entropy}
-------------------------------------------------
The previous equation of state is not prone to a simple stability analysis. Here we give another equation of state that permits finding both an expression for the shell’s entropy and performing a simple stability analysis.
We must first specify an adequate thermal equation of state for $b(r_+,r_-)$. A possible simple choice is a power law in the ADM mass $m$, i.e., $b(r_+,r_-)$ has the form $$\label{bF6}
b(r_+,r_-) = 2G \, a(r_++r_-)^{{\alpha}}$$ where $a$ and ${\alpha}$ are free coefficients related to the properties of the shell. Power laws occur frequently in thermodynamic systems, and so this is a natural choice as well. The simple choice above allows one to find the form of the function $c$. Indeed, the integrability equation (\[bc5\]) gives that the function $c$ can be put in the form $c(r_+,r_-) = 2G \frac{f(r_+ r_-)}{(r_+ + r_-)^{{\alpha}}}$, where $f(r_+ r_-)$ is an arbitrary function of the product $r_+ r_-$ and supposedly also depends on the intrinsic constants of the matter that makes up the shell. For convenience we choose $f(r_+ r_-)=d\,(r_+ r_-)^{\delta}$, where $d$ and $\delta$ are parameters that reflect the shell’s properties, so that $$\label{cpower}
c(r_+,r_-) = 2G \,d\, \frac{(r_+ r_-)^{\delta}}
{(r_+ + r_-)^{{\alpha}}}\,.$$ The gravitational constant $G$ was introduced in Eqs. (\[bF6\]) and (\[cpower\]) for convenience. Inserting Eqs. (\[bF6\])-(\[cpower\]) into Eq. (\[dSQ\]) and integrating, gives the entropy $$\label{SF6}
S(r_+,r_-) = a \, \left[\frac{(r_++r_-)^{{\alpha}+1}}{{\alpha}+1} - d \,
\frac{(r_+ r_-)^{{\delta}+1}}{{\delta}+1}\right] \,,$$ where the constant of integration $S_0$ has been put to zero, as expected in the limit $r_+
\to 0$ and $r_- \to 0$. Again, the entropy of this thin charged shell depends on $(M,R,Q)$ through $r_+$ and $r_-$ only, which in turn are specific functions of $(M,R,Q)$.
We consider positive temperatures and positive electric potentials, so $$\label{adgeq0}
a>0\,,\quad d>0\,.$$ We consider only $$\label{ageq0}
{\alpha}>0 \,,$$ for the simplicity of the upcoming stability analysis. Although this choice somewhat narrows down the range of cases to which the analysis is applicable, it only rules out the cases where $-1<{\alpha}<0$, since for values ${\alpha}\leq-1$ it would give a diverging entropy in the limit $r_+
\to 0$ and $r_- \to 0$, something which is not physically acceptable. Indeed, in such a limit we would expect the entropy to be zero which requires ${\alpha}> -1$.
The stability conditions for the specific temperature ansatz {#stabil2}
------------------------------------------------------------
Proceeding to the thermodynamic stability treatment, we start with Eq. (\[B1\]), which can be shown to be equivalent to $$\label{B1s}
r_+ r_- - 2 R^2 k^2 {\alpha}+ (1-k^2)R^2 \geq 0.$$ Solving for $k$, this leads to the restriction $$\label{k1s}
k \leq \sqrt{\frac{1}{2{\alpha}+1}\left(1+\frac{r_+ r_-}{R^2}\right)}.$$
Going now to Eq. (\[B2\]), it gives $$\begin{aligned}
\label{B2s}
[r_+ r_- - (1-k)^2 R^2]&[{\alpha}(r_+ r_- - (1-k)^2 R^2) \nonumber \\
& + 3(r_+ r_- + (1-k^2)R^2)] \leq 0.\end{aligned}$$ Since the second multiplicative term on the left must be positive, one can solve for $k$ and obtain the set of values which satisfy the inequality, $$\begin{aligned}
\label{k2s}
\frac{{\alpha}}{{\alpha}+3} - \sqrt{\frac{9}{({\alpha}+3)^2} + \frac{r_+ r_-}{R^2}}
\leq k \nonumber \\
\leq\frac{{\alpha}}{{\alpha}+3} + \sqrt{\frac{9}{({\alpha}+3)^2} +
\frac{r_+ r_-}{R^2}}\,.\end{aligned}$$ As for Eq. (\[B3\]), it reduces to $$\frac{d R(2{\delta}+1)(r_+ r_-)^{{\delta}}}{\left(\frac{r_+ r_-}{R} +
(1-k^2)R\right)^{{\alpha}}} \geq \frac{R^2(1-k^2)+(2 {\alpha}+ 1)r_
+ r_-}{R^2(1-k^2)+r_+ r_-}.$$ Although one cannot conclude anything directly from the above inequality, it is nonetheless worth noting that the right-hand side is greater than zero, and so ${\delta}$ must obey the condition $$\label{intDe}
{\delta}\geq -\frac{1}{2}.$$
Regarding Eq. (\[B4\]), it is possible to show that it implies the condition $$\begin{aligned}
\label{B4s}
r_+^2 r_-^2 & ({\alpha}+3) - 2r_+ r_-R^2(2k^2{\alpha}+ 2k^2 -
k + {\alpha}-1) \nonumber \\
& + (1-k)^2 R^4 (3k^2 {\alpha}+ k^2 + 2{\alpha}k + {\alpha}-1) \leq 0,\end{aligned}$$ which does not provide any information on its own since it is a polynomial of order four in the variable $k$. Nonetheless, it does need to be satisfied once a region of allowed values for $k$ is known, which will be ascertained in the following.
Concerning Eq. (\[B5\]), we are led to
$$\frac{d R(2{\delta}+1)(r_+ r_-)^{{\delta}}}{\left(\frac{r_+ r_-}{R} +
(1-k^2)R\right)^{{\alpha}}} \leq
\frac{r_+^2 r_-^2(3{\alpha}+ 1) + 2(1-k)r_+ r_-R^2(2{\alpha}(k-1) +
2k -1)-(1-k)^3 R^4(k({\alpha}+3) - {\alpha}+ 3)}{\left[(1-k)^2 R^2 -
r_+ r_-\right]\left[(k-1) R^2 (k({\alpha}+3) - {\alpha}+ 3)\right]},$$
which does not contain any new information.
On the other hand, when Eq. (\[B6\]) is simplified to $$\begin{aligned}
\frac{d R(2{\delta}+1)(r_+ r_-)^{{\delta}}}{\left(\frac{r_+ r_-}{R} +
(1-k^2)R\right)^{{\alpha}}} & \nonumber \\
& \hspace{-15mm} \geq \frac{R^2(1-k^2)+(2 {\alpha}+ 1)r_+ r_-
- 2R^2 k^2 {\alpha}}{R^2(1-k^2)+r_+ r_- -2R^2k^2{\alpha}}\,,\end{aligned}$$ and one notices that the numerator on the right side must be positive, another constraint on $k$ naturally appears, namely $$\label{k6s}
k \leq \sqrt{\frac{1}{2{\alpha}+1} + \frac{r_+ r_-}{R^2}}.$$
Finally, the last condition (\[B7\]) gives the inequality $$r_+ r_-({\alpha}+1) - R^2 \left[({\alpha}+1)k^2 + {\alpha}-1\right] \geq 0$$ which constricts the values of $k$ to be within the interval $$\label{k7s}
k \leq \sqrt{-\frac{{\alpha}-1}{{\alpha}+1} + \frac{r_+ r_-}{R^2}}.$$
The definitive region of permitted values for $k$ is the intersection of the conditions (\[k1s\]), (\[k2s\]), (\[k6s\]) and (\[k7s\]). It is possible to show that such an intersection gives the range $$\label{intK}
\frac{{\alpha}}{{\alpha}+3} - \sqrt{\frac{9}{({\alpha}+3)^2} + \frac{r_+ r_-}{R^2}}
\leq k \leq \sqrt{-\frac{{\alpha}-1}{{\alpha}+1} + \frac{r_+ r_-}{R^2}}$$ where ${\alpha}$ must be restricted to $$\label{intAl}
{\alpha}\geq \frac{1+\frac{r_+ r_-}{R^2}}{1-\frac{r_+ r_-}{R^2}}.$$ Returning to Eq. (\[B4s\]), it is now possible to verify if the interval (\[intK\]) satisfies said condition, which indeed it does.
The black hole limit {#bhl2}
--------------------
If one takes the shell to its own gravitational radius, the chosen temperature equation of state (\[bF6\]) is wiped out, and a new equation of state sets in to adapt to the quantum spacetime properties. The new equation of state is then given by Eq. (\[thawk\]) and the black hole entropy (\[SS\]) follows.
Other equations of state {#other}
========================
Naturally, other equations of state can be sough. We give four examples, one fixing $b(r_+,r_-)$ and three others fixing $c(r_+,r_-)$.
If we fix the inverse temperature $$b(r_+,r_-) = \gamma\,
\frac{r_+^2}{r_+-r_-}\,,$$ for some $\gamma$, as we did before, then generically, from Eq. (\[bc5\]), we find $$c(r_+,r_-) = \frac{a(r_+r_-)(r_+-r_-)+r_-}{r_+^2} \,,$$ where $a(r_+r_-)$ is an arbitrary function of integration of the product $r_+r_-$ and presumably also depends on the intrinsic constants of the matter that makes up the shell. Then, from Eq. (\[dSQ\]), the entropy is $$S(r_+,r_-)=\frac{\gamma}{4G}\left(r_+^2 +
\int_{0}^{r_{+}r_{-}}
\left(1-a(x)\right)\,dx\right)\,,$$ where again we are assuming zero entropy when $r_+=0$. In the example we gave previously we have put $a(r_+r_-)=1$, so that $c(r_+,r_-) = \frac{1}{r_+}$. This case $a(r_+r_-)=1$ gives precisely that the entropy of the shell is proportional to the area of its gravitational radius and for $\gamma= \frac{4\pi}{\hbar}$ gives that the entropy of the shell is equal to the corresponding black hole entropy as we have discussed previously. Of course, many other choices can be given for $a(r_+r_-)$ and quite generally the entropy will be a function of $r_+$ and $r_-$, $S=S(r_+,r_-)$.
Inversely, instead of $b(r_+,r_-)$ one can give $c(r_+,r_-)$. One equation for $c(r_+,r_-)$ could be $$c(r_+,r_-) =\frac{1}{r_{+}}\,,$$ as for the black hole case. The integrability condition, Eq. (\[bc5\]), for the temperature then gives $$b(r_+,r_-)=\frac{h(r_{+})}{r_{+}-r_{-}}\,,
\label{neweos1}$$ where $h(r_+)$ is a function that can be fixed in accord with the matter properties of the shell. Then, from Eq. (\[dSQ\]), the entropy is $$S(r_+)=\frac{1}{2G}\int_{0}^{r_{+}}\,
\frac{h(x)}{x}\,dx\,,$$ where it is implied that the function $h(x)$ vanishes at $x=0$ rapidly enough so that the entropy goes to zero when $r_+=0$. If we choose $h(r_+)=\frac{4\pi}{\hbar}
r_+^2$, then one recovers the black hole temperature and the black hole entropy for the shell.
Another equation of state one can choose for $c(r_+,r_-)$ is $$c(r_+,r_-) =\frac{1}{r_{-}}\,.$$ The integrability condition, Eq. (\[bc5\]), similarly gives $$b(r_+,r_-) =\frac{h(r_{-})}{r_{+}-r_{-}}\,.$$ where $h(r_-)$ is a function that can be fixed in accord with the matter properties of the shell. In this case, from Eq. (\[dSQ\]), the entropy of the shell depends on $r_{-}$ only, and is given by $$S(r_-)=\frac{1}{2G}\int_{0}^{r_{-}}\frac{h(x)}{x}\,dx\,,$$ where we are assuming zero entropy when $r_-=0$.
Yet another example can be obtained if one puts $$c(r_+,r_-) =c(r_{+}r_{-})\,,$$ i.e., $c$ is a function of the product $r_{+}r_{-}$ and may also depend on the intrinsic constants of the matter that makes up the shell. The integrability condition then gives $$b=b_{0}\,,$$ where $b_0$ is a constant, and so in this case, the temperature measured at infinity does not depend on $r_{+}$ or $r_{-}$. The entropy is then $$S(r_+,r_-)
=\frac{b_{0}}{2G}
\left(r_{+}+r_{-}-\int_{0}^{r_{+}r_{-}} c(x)\,dx\right)\,,$$ where we are assuming zero entropy when $r_+=0$ and $r_-=0$.
One could study in detail these four cases for the thermodynamics of a shell performing in addition a stability analysis for each one. We refrain here to do so. Certainly other interesting cases can be thought of.
Conclusions {#conc}
===========
We have considered the thermodynamics of a self-gravitating electrically charged thin shell thus generalizing previous works on the thermodynamics of self-gravitating thin-shell systems. Relatively to the simplest shell where there are two independent thermodynamic state variables, namely, the rest mass $M$ and the size $R$ of the shell, we have now a new independent state variable in the thermodynamic system, the electric charge $Q$, out of which, using the first law of thermodynamics and the equations of state one can construct the entropy of the shell $S(M,R,Q)$. Due to the additional variable, the charge $Q$, the calculations are somewhat more complex. Concomitantly, the richness in physical results increases in the same proportion.
The equations of state one has to give are the pressure $p(M,R,Q)$, the temperature $T(M,R,Q)$, and the electric potential $\Phi(M,R,Q)$. The pressure can be obtained from dynamics alone, using the thin-shell formalism and the junction conditions for a flat interior and a Reissner-Nordström exterior. The form of the temperature and of the thermodynamic electric potential are obtained using the integrability conditions that follow from the first law of thermodynamics.
The differential for the entropy in its final form shows remarkably that the entropy must be a function of $r_+$ and $r_-$ alone, i.e., a function of the intrinsic properties of the shell spacetime. Thus, shells with the same $r_+$ and $r_-$ (i.e., the same ADM mass $m$ and charge $Q$) but different radii $R$, have the same entropy. From the thermodynamics properties alone of the shell one cannot distinguish a shell near its own gravitational radius from a shell far from it. In a sense, the shell can mimic a black hole.
The differential for the entropy in its final form gives that $T$ and $\Phi$ are related through an integrability condition. One has then to specify either $T$ or $\Phi$ and the form of the other function is somewhat constrained. We gave two example cases and mentioned other possibilities.
First, we gave the equations of state where the temperature has the form of the Hawking temperature, apart from a constant factor, and the electric potential has a simple precise form $Q/r_+$, and found the entropy. When the factor is the Hawking factor it was shown that the resulting entropy was equal to the Bekenstein-Hawking entropy of a nonextremal charged black hole. The need to set the temperature of the shell equal to the Hawking temperature is justified when the shell is taken to its own gravitational radius. At this radius the backreaction of the nearby quantum fields diverges unless the shell has precisely the Hawking temperature. Conversely, one should note that if instead, the function for the electric potential $Q/r_+$ was given, the integrability equation would then fix the function $T$ apart from an arbitrary function. A simple choice for this arbitrary function is the Hawking temperature.
Second, the other set of equations of state were given as a simple ansatz. For the thermal equation of state, we set the temperature as proportional to some power in the ADM mass $m$, and the thermodynamic electric potential was set to be a power in the electric charge and an inverse power in $m$. This choice also allows one to find an expression for the entropy of the shell and, furthermore, allows for an analytic stability analysis. Indeed, despite the increase in complexity in the thermodynamic stability analysis due to the existence of four new stability equations, it was possible to obtain a unique range for the redshift parameter $k$, as well as the regions of allowed values for the parameters ${\alpha}$ and ${\delta}$.
Many other interesting equations of state can be chosen and some of them were indeed given. However at the gravitational radius all turn into the Hawking equation of state, i.e., the Hawking temperature. Since the area of the shell $A$ is equal to the gravitational radius area $A_+$, $A=A_+$, when the shell is at its own gravitational radius, and $S=\frac{1}{4}\frac{A_+}{A_p}$ in this limit, we conclude that the entropy of the shell is proportional to its own area $A$. This indicates that all its fundamental degrees of freedom have been excited. Matter systems at their own gravitational radius are called quasiblack holes and have thermodynamic properties similar to black holes.
We thank FCT-Portugal for financial support through Project No. PEst-OE/FIS/UI0099/2013. GQ also acknowledges the grant No. SFRH/BD/92583/2013 from FCT. OBZ has been partially supported by the Kazan Federal University through a state grant for scientific activities.
The dominant energy condition {#apa}
=============================
With the expressions for the mass density and pressure, Eqs. (\[Sig1\])-(\[pQk1\]), we can consider some mechanical constraints which the shell should naturally obey. One can impose that the shell satisfies the weak energy condition. It requires that $\sigma$ and $p$ be positive, which is always verified. One can also insist that the shell satisfies the dominant energy condition, i.e., $$p\leq \sigma\,.$$ It is then possible to show that the dominant energy condition imposes the constraint $k \in [k_1,k_2]$, where $$k_{1} = \frac{3}{5}\left(1 - \sqrt{1-
\frac{5}{9}\left(1-\frac{r_+ \, r_-}{R^2}\right)}\,\right)\,,$$ and $k_{2} = \frac{3}{5}\left(1 + \sqrt{1-
\frac{5}{9}\left(1-\frac{r_+ \, r-}{R^2}\right)}\,\right)$. Since $k_2>1$, and $k$ trivially obeys $k\leq1$, we conclude that the dominant energy condition restricts the values of $k$ to obey $$\label{d1}
k_1\leq k\,.$$ In the case where there is no charge, i.e., $Q=0$ or $r_-=0$, one gets $k_1=1/5$, thus regaining the result obtained in [@Mart]. When expressed in terms of the variables $R/m$ and $R/Q$, the relation (\[d1\]) can be written as $$\frac{R}{m}
\geq \frac{25}{6+10 \frac{G Q^2}{R^2} +3\sqrt{4+5\frac{G Q^2}{R^2}}}$$ or in terms of $R/r_+$ and $r_-/R$, $$\frac{R}{r_+} \geq
\frac{
12\sqrt{1- \frac{r_-}{R} - \left(\frac{r_-}{R}\right)^2 +
\left(\frac{r_-}{R}\right)^3}
+31 \frac{r_-}{R} -20 \left(\frac{r_-}{R}\right)^2 -12}
{24 \frac{r_-}{R} - 25
\left(\frac{r_-}{R}\right)^2}$$ This is a mechanical constraint. A fundamental constraint, the no-trapped-surface condition for the shell, is $R\geq r_+$, as was given in Eq. (\[notrapped\]).
Derivation of the equations of thermodynamic stability for a system with three independent variables {#apb}
====================================================================================================
In this appendix we shall show the derivation of the equations of thermodynamic stability for an electrically charged system, i.e., Eqs. (\[B1\])-(\[B7\]). Thus the approach used for two independent variables in [@callen] is extended here by us to three independent variables. We name these independent variables $M$, $A$, and $Q$.
We start by considering two identical subsystems, each with an entropy $S = S(M,A,Q)$, where $M$ is the internal energy of the system (equivalent to the rest mass), $A$ is its area and $Q$ its electric charge. The usual state variables of a thermodynamic system are the internal energy $U$, volume $V$ and the number of particles, $N$, say. However, the system we wish to study is an electrically charged thin shell, and thus it is natural to use the variables $(M,A,Q)$. Thermodynamic stability is guaranteed if $dS = 0$ and $d^2 S < 0$ are both satisfied, or in other words, if the entropy is an extremum and a maximum respectively.
Now suppose we keep $A$ and $Q$ constant and remove a positive amount of internal energy $\Delta M$ from one subsystem to the other. The total entropy of the two subsystems goes from the value $2 S(M,A,Q)$ to $S(M+\Delta M, A, Q) + S(M-\Delta M,A,Q)$. If the initial entropy $S(M,A,Q)$ is a maximum, then the sum of initial entropies must be greater or equal to the sum of final entropies, i.e. $$\label{Beq1}
S(M+\Delta M, A, Q) + S(M-\Delta M,A,Q) \leq 2 S(M,A,Q).$$ Expanding $S(M+\Delta M, A, Q)$ and $S(M-\Delta M, A, Q)$ in a Taylor series to second order in $\Delta M$, we see that Eq. (\[Beq1\]) becomes $$\label{Beq2}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right)_{A,Q} \leq 0$$ in the limit $\Delta M \to 0$. The same reasoning applies if we fix $M$ and $Q$ instead and apply a positive change of area $\Delta A$, so we must have $$\label{Beq3}
S(M, A+\Delta A, Q) + S(M,A-\Delta A,Q) \leq 2 S(M,A,Q).$$ which in the limit $\Delta A \to 0$ gives $$\label{Beq4}
\left(\frac{{\partial}^2 S}{{\partial}A^2}\right)_{M,Q} \leq 0.$$ If we fix $M$ and $A$ and make a positive change $\Delta Q$ on the charge, we have $$\label{Beq5}
S(M, A, Q+\Delta Q) + S(M,A,Q-\Delta Q) \leq 2 S(M,A,Q).$$ and so it follows that $$\label{Beq6}
\left(\frac{{\partial}^2 S}{{\partial}Q^2}\right)_{M,A} \leq 0.$$ However, if we keep only one quantity fixed, like $Q$ for example, we must also have a final sum of entropies smaller than the initial sum if we apply a simultaneous change of area and internal energy rather than separately, i.e. $$\begin{aligned}
\label{Beq7}
& S(M+\Delta M, A+\Delta A, Q) \nonumber \\
& \hspace{4mm} + S(M-\Delta M,A-\Delta A,Q) \leq 2 S(M,A,Q).\end{aligned}$$ This inequality is satisfied by Eq. (\[Beq2\]) and Eq. (\[Beq4\]), but it also implies a new requirement. If we expand the left side in a Taylor series to second order in $\Delta M$ and $\Delta A$, and use the abbreviated notation $S_{ij} = {\partial}^2 S/{\partial}x_i {\partial}x_j$, we get $$S_{MM} (\Delta M)^2 + 2 S_{MA} \Delta M \Delta A +
S_{AA} (\Delta A)^2 \leq 0.
\label{smmsmasaa}$$ Multiplying Eq. (\[smmsmasaa\]) by $S_{MM}$ and adding and subtracting $S_{MA}^2 (\Delta A)^2$ to and from the left side, allows the last inequality to be written in the form $$(S_{MM} \Delta M + S_{MA} \Delta A)^2 + (S_{MM}S_{AA} -
S_{MA}^2)(\Delta A)^2 \geq 0.$$ Since the first term on the left side is always greater than zero, we see that it is sufficient to have $$\label{Beq8}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right) \left(\frac{{\partial}^2 S}{{\partial}A^2}\right) - \left(\frac{{\partial}^2 S}{{\partial}M {\partial}A}\right)^2 \geq 0.$$ This concludes the derivation of Eqs. (\[B1\]), (\[B2\]) and (\[B4\]).
To derive the other stability equations, namely, Eqs. (\[B3\]), (\[B5\]), (\[B6\]), and (\[B7\]), we note that we can repeat the same calculations but now we fix $M$ and $A$ in turns. It is now straightforward to see that, when fixing $M$, we must have $$S_{AA} (\Delta A)^2 + 2 S_{AQ} \Delta A \Delta Q + S_{QQ} (\Delta Q)^2 \leq 0,$$ which is satisfied by $$\label{Beq9}
\left(\frac{{\partial}^2 S}{{\partial}A^2}\right) \left(\frac{{\partial}^2 S}{{\partial}Q^2}\right) - \left(\frac{{\partial}^2 S}{{\partial}A {\partial}Q}\right)^2 \geq 0.$$ Finally, by fixing $A$ we get the inequality $$S_{MM} (\Delta M)^2 + 2 S_{MQ} \Delta M \Delta Q + S_{QQ} (\Delta Q)^2 \leq 0$$ which implies the sufficient condition $$\label{Beq10}
\left(\frac{{\partial}^2 S}{{\partial}M^2}\right) \left(\frac{{\partial}^2 S}{{\partial}Q^2}\right) - \left(\frac{{\partial}^2 S}{{\partial}M {\partial}Q}\right)^2 \geq 0.$$ The last case left consists of doing a simultaneous change in all the state variables of the system, i.e., $$\begin{aligned}
\label{Beq11}
& S(M+\Delta M, A+\Delta A, Q+\Delta Q) \nonumber \\
& \hspace{4mm} + S(M-\Delta M,A-\Delta A,Q-\Delta Q) \leq 2 S(M,A,Q).\end{aligned}$$ To investigate the sufficient differential condition that this inequality implies, one must first expand $S(M+\Delta M, A+\Delta A,
Q+\Delta Q)$ and $S(M-\Delta M,A-\Delta A,Q-\Delta Q)$ in a Taylor series to second order in $\Delta M$, $\Delta A$ and $\Delta Q$, which can be shown to lead to $$\begin{aligned}
\label{Beq12}
& S_{MM} (\Delta M)^2 + S_{AA} (\Delta A)^2 + S_{QQ} (\Delta Q)^2
\nonumber \\ & + 2 S_{MA} \Delta M \Delta A + 2 S_{MQ} \Delta M \Delta
Q + 2 S_{QA} \Delta A \Delta Q \leq 0.\end{aligned}$$ Multiplying the above relation by $S_{MM}$, noting that $$\begin{aligned}
& (S_{MM} \Delta M + S_{MA} \Delta A + S_{MQ} \Delta Q)^2 = \nonumber
\\ & S_{MM}^2 (\Delta M)^2 + S_{MA}^2 (\Delta A)^2 + S_{MQ}^2 (\Delta
Q)^2 + \nonumber \\
& + 2 S_{MM} S_{MA} \Delta M \Delta A + 2 S_{MM} S_{MQ} \Delta M
\Delta Q + \nonumber \\
& +
2 S_{MA} S_{MQ} \Delta A \Delta Q \,,\end{aligned}$$ and inserting this into Eq. (\[Beq12\]), gives $$\begin{aligned}
& (S_{MM} \Delta M + S_{MA} \Delta A + S_{MQ} \Delta Q)^2 \nonumber \\
& \hspace{1mm} + (S_{MM}S_{AA} - S_{MA}^2)(\Delta A)^2 +
\nonumber \\
& \hspace{1mm} +
(S_{MM}S_{QQ}
- S_{MQ}^2)(\Delta Q)^2 +\nonumber \\
& + 2 (S_{MM}S_{QA}-S_{MA}S_{MQ}) \Delta A \Delta Q \geq 0.\end{aligned}$$ Recalling Eq. (\[Beq8\]) and Eq. (\[Beq10\]), and noting that the first term in the above inequality is always positive, we conclude that the condition $$\left(\frac{{\partial}^2 S}{{\partial}M^2}\right) \left(\frac{{\partial}^2 S}{{\partial}Q {\partial}A}\right) - \left(\frac{{\partial}^2 S}{{\partial}M {\partial}A}\right) \left(\frac{{\partial}^2
S}{{\partial}M {\partial}Q}\right) \geq 0$$ is sufficient to satisfy Eq. (\[Beq11\]). This concludes the derivation of Eqs. (\[B3\]), (\[B5\]), (\[B6\]) and (\[B7\]). Thus all stability equations, Eqs. (\[B1\])-(\[B7\]), have been derived.
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|
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abstract: |
The results of LO [*Fixed point*]{} QCD (FP-QCD) analysis of the CCFR data for the nucleon structure function $~xF_3(x,Q^2)~$ are presented. The predictions of FP-QCD, in which $~\alpha_{s}(Q^2)~$ tends to a nonzero coupling constant $\alpha_{0}$ as $~Q^2\to \infty~,$ are in good agreement with the data. Constraints for the possible values of the $~\beta~$ function parameter $~b~$ regulating how fast $~\alpha_ s(Q^2)~$ tends to its asymptotic value $~\alpha_{0}~$ are found from the data. The corresponding values of $~\alpha_{0}~$ are also determined. Having in mind the recent QCD fits to the same data we conclude that in spite of the high precision and the large $~(x,Q^2)~$ kinematic range of the CCFR data they cannot discriminate between QCD and FP-QCD predictions for $~xF_3(x,Q^2)~$.\
---
=-1.5 cm =-12 truemm 23.6cm
Ł
[**Constraints on “Fixed Point” QCD\
from the CCFR Data on Deep Inelastic\
Neutrino-Nucleon Scattering**]{}
[**Aleksander V. Sidorov** ]{}\
[*Bogoliubov Theoretical Laboratory\
Joint Institute for Nuclear Research\
141980 Dubna, Russia\
E-mail: sidorov@thsun1.jinr.dubna.su*]{} 0.5cm [^1]\
[*International Centre for Theoretical Physics, Trieste, Italy*]{}\
0.3cm
0.5 cm
[**1. Introduction.**]{} The success of perturbative Quantum Chromodynamics (QCD) in the description of the high energy physics of strong interactions is considerable. The QCD predictions are in good quantitative agreement with a great number of data on lepton-hadron and hadron-hadron processes in a large kinematic region (e.g. see reviews [@Altarelli] and the references therein). Despite of this success of QCD,we consider that it is useful and reasonable to put the question: Do the present data fully exclude the so-called [*fixed point*]{} (FP) theory models [@Pol] ?\
We remind that these models are not asymptotically free. The effective coupling constant $~\alpha_{s}(Q^{2})~$ approaches a constant value $~\alpha_{0}\ne 0~$ as $~Q^{2}\to \infty~$ (the so-called fixed point at which the Callan- Symanzik $\beta$-function $~\beta(\alpha_{0}) = 0~$). Using the assumption that $~\alpha_{0}~$ is [*small*]{} one can make predictions for the physical quantities in the high energy region, as like in QCD, and confront them to the experimental data. Such a test of FP theory models has been made [@GR; @BS] by using the data of deep inelastic lepton-nucleon experiments started by the SLAC-MIT group [@SLAC] at the end of the sixties and performed in the seventies [@data70]. It was shown that
$~i$) the predictions of the FP theory models with [*scalar*]{} and [*non- colored (Abelian) vector*]{} gluons [*do not agree*]{} with the data
$ii$) the data [*cannot distinguish*]{} between different forms of scaling violation predicted by QCD and the so-called [*Fixed point*]{} QCD (FP-QCD), a theory with [*colored vector*]{} gluons, in which the effective coupling constant $~\alpha_{s}(Q^{2})~$ does not vanish when $Q^{2}$ tends to infinity.\
We think there are two reasons to discuss again the predictions of FP-QCD. First of all, there is evidence from the non-perturbative lattice calculations [@Pat] that the $\beta$- function in QCD vanishes at a nonzero coupling $~\alpha_{0}~$ that is small. (Note that the structure of the $\beta$-function can be studied only by non-perturbative methods.) Secondly, in the last years the accuracy and the kinematic region of deep inelastic scattering data became large enough, which makes us hope that discrimination between QCD and FP-QCD could be performed.\
Recently we have analyzed the CCFR deep inelastic neutrino-nucleon scattering data [@prep1] in the framework of the [*Fixed point*]{} QCD. It was demonstrated [@SiSt] that the data for the nucleon structure function $~xF_3(x,Q^2)~$ are in good agreement with the LO predictions of this theory model using the assumption that the [*fixed point*]{} coupling $~\alpha_{0}~$ is small. In contrast to the results of the fits to the previous generations of deep inelastic lepton-nucleon experiments, the value of this constant was determined with a good accuracy: $$\alpha_{0} = 0.198\pm0.009~.
\label{a0res}$$
However, this value of $~\alpha_0~$ is not consistent with $~\alpha_s(M^2_z)~$ measurements at LEP. For instance, from the scaling violation in the fragmentation functions in $~e^{+}e^{-}~$ annihilation $~\alpha_s(M^2_z)~$ has been determined [@fitdel] as: $$\alpha_s(M^2_z) = 0.118\pm0.005~.
\label{amz}$$
This discrepancy follows from the fact that in our analysis pure asymptotic formula for the structure function $~xF_3~$ has been used , i.e. the effective coupling constant $~\alpha_s(Q^2)~$ has been approximated with its asymptotic value $~\alpha_0~$. Results (1) and (2) have shown that in the $~Q^2~$ range studied up to now $~\alpha_0~$ is not yet reached and therefore, to determine $~\alpha_0~$ properly from the data the preasymptotic behaviour of $~\alpha_s(Q^2)~$ has to be taken into account.\
In this paper we present a leading order [*Fixed point*]{} QCD analysis of the CCFR data [@prep1], in which the next corrections to the pure asymptotic expression for $~xF_3(x,Q^2)~$ are taken into account. We remind that the structure function $~xF_3~$ is a pure non-singlet and the results of the analysis are independent of the assumption on the shape of gluons. As in a previous analysis the method [@Kriv] of reconstruction of the structure functions from their Mellin moments is used. This method is based on the Jacobi - polynomial expansion [@Jacobi] of the structure functions. In [@KaSi] this method has been already applied to the QCD analysis of the CCFR data.\
[**2. Method and Results of Analysis.**]{} Let us start with the basic formulas needed for our analysis.
The Mellin moments of the structure function $~xF_3(x,Q^2)~$ are defined as: $$M_n^{NS}(Q^2)=\int_{0}^{1}dxx^{n-2}xF_{3}(x,Q^2)~,
\label{mom}$$ where $~n=2,3,4,...~.$
In the case of FP-QCD the effective coupling constant $~\alpha_s(Q^2)~$ at large $~Q^2~$ takes the form: $$\alpha_s(Q^2) = \alpha_0 + f(Q^2)~,
\label{apre}$$ where $~f(Q^2)\rightarrow 0~$ when $~Q^2\rightarrow\infty~.$
Let us assume that $~\alpha_0~$ is a [*first order*]{} ultraviolet fixed point for the $\beta$-function, i.e. $$\beta(\alpha) = -b(\alpha - \alpha_0)~,~~~~~~b~>~0~,
\label{beta}$$ Then $$\alpha_s(Q^2) = \alpha_0 +
[\alpha_s(Q^2_0)-\alpha_0]({Q_0^2\over Q^2})^b~.
\label{aq2}$$ and instead of Eq.(5) in [@SiSt] we obtain now for the moments of $~xF_3~$ the following LO expression: $$M_{n}^{NS}(Q^2)
=M_{n}^{NS}(Q_{0}^2) \left [ \frac{Q_0^{2}}
{Q^{2}} \right ]^{\frac{1}{2}d^{NS}_{n}}F_n(Q^2)~,
\label{mfp}$$ where $$F_n(Q^2) = exp\{{(\alpha_s(Q^2_0)- \alpha_0)\over 2b\alpha_0}
d_n^{NS}[({Q^2_0\over Q^2})^b-1]\}~.
\label{cf}$$
In (\[mfp\]) and (\[cf\]) $$d_n^{NS} = \frac{\alpha_0}{4\pi}\gamma^{(0)NS}_n
\label{dn}$$ and $$\gamma^{(0)NS}_{n} ={8\over 3}[1 - {2\over n(n+1)} + 4\sum_{j=2}^{n}
{1\over j}]~.
\label{goa0}$$
The $n$ dependence of $~\gamma^{(0)NS}_{n}~$ is exactly the same as in QCD. However, the $~Q^2~$ behaviour of the moments is different. In contrast to QCD, the Bjorken scaling for the moments of the structure functions is broken by powers in $~Q^2~$. In (\[cf\]) and (\[dn\]) $~\alpha_0~$ and $b$ are parameters, to be determined from the data.\
Having in hand the moments (\[mfp\]) and following the method [@Kriv; @Jacobi], we can write the structure function $~xF_3~$ in the form: $$xF_{3}^{N_{max}}(x,Q^2)=x^{\a}(1-x)^{\beta}\sum_{n=0}^{N_{max}}\Theta_n ^{\a ,
\beta}
(x)\sum_{j=0}^{n}c_{j}^{(n)}{(\a ,\beta )}
M_{j+2}^{NS} \left ( Q^{2}\right ), \\
\label{e7}$$ where $~\Theta^{\alpha \beta}_{n}(x)~$ is a set of Jacobi polynomials and $~c^{n}_{j}(\alpha,\beta)~$ are coefficients of the series of $~\Theta^{\alpha,\beta}_{n}(x)~$ in powers in x: $$\Theta_{n} ^{\a , \beta}(x)=
\sum_{j=0}^{n}c_{j}^{(n)}{(\a ,\beta )}x^j .
\label{e9}$$
$N_{max},~ \alpha~$ and $~\beta~$ have to be chosen so as to achieve the fastest convergence of the series in the R.H.S. of Eq.(\[e7\]) and to reconstruct $~xF_3~$ with the accuracy required. Following the results of [@Kriv] we use $~\alpha =
0.12~,
{}~\beta = 2.0~$ and $~N_{max} = 12~$. These numbers guarantee accuracy better than $~10^{-3}~$.\
Finally we have to parametrize the structure function $~xF_3~$ at some fixed value of $~Q^2 = Q^2_{0}~$. We choose $~xF_3(x,Q^2)~$ in the form: $$xF_{3}(x,Q_0^2)=Ax^{B}(1-x)^{C}~.
\label{e10}$$
The parameters A, B and C in Eq. (\[e10\]) and the FP-QCD parameters $~\alpha_{0}~$ and $b$ are free parameters which are determined by the fit to the data.
In our analysis the target mass corrections [@tmc] are taken into account. To avoid the influence of higher–twist effects we have used only the experimental points in the plane $~(x,Q^2)~$ with $~10 < Q^2\leq 501~(GeV/c)^2~$. This cut corresponds to the following $~x~$ range:$~0.015\leq x \leq 0.65~$.\
The results of the fit are presented in Table 1. In all fits only statistical errors are taken into account.\
0.2 cm
$b$ $\chi^2_{d.f.}$ $\alpha_0$ A B C $\alpha_s(M_z^2)$
------ ----------------- --------------- --------------- --------------- -------------- -------------------
0.15 82.7/61 .057$\pm$.026 6.96$\pm$.20 .799$\pm$.013 3.44$\pm$.03 .121$\pm$.034
0.20 82.3/61 .097$\pm$.021 6.95$\pm$..20 .799$\pm$.013 3.44$\pm$.03 .132$\pm$.025
0.25 82.0/61 .122$\pm$.018 6.94$\pm$.19 .798$\pm$.013 3.45$\pm$.03 .142$\pm$.020
------------------ ---------------------------------------------------------------------------------------
[**Table 1.**]{} The results of the LO FP-QCD fit to the CCFR $~xF_3~$ data.
$\chi^2_{d.f.}$ is the $\chi^2$-parameter normalized to the degree of freedom $d.f.$.
------------------ ---------------------------------------------------------------------------------------
0.8 cm
Summarizing the results in the Table one can conclude:
1\. The values of $~\chi^2_{d.f.}~$ are slightly smaller than those obtained in the LO QCD analysis [@KaSi] of the CCFR data and indicate a [*good description*]{} of the data.
2\. The values of $~b$, for which the asymptotic coupling $~\alpha_0~$ is smaller than $~\alpha_ s( M^2_z)~$, are found to range in the following interval: $$0 < b < 0.25~.
\label{ib}$$
For the values of $b$ smaller than 0.15 $~\alpha_0~$ can not be determined from CCFR data. The errors in $~\alpha_0~$ exceed the mean values of this parameter. For the values of $~b \geq 0.25~$ the mean value of $~\alpha_0~$ is equal or bigger than $~\alpha_ s(M^2_z)~$.
3\. The accuracy of determination of $~\alpha_0~$ is not good enough. The accuracy increases with increasing $b$.
4\. $\alpha_0 = 0.057~$ corresponding to $~b = 0.15~$ is preferred to the other values of $~\alpha_0~$ determined from the data.
5\. The values of $~\alpha_ s(M^2_z)~$ corresponding to the values of $~b~$ from the range (\[ib\]) are in agreement within one standard deviation with $~\alpha_ s(M^2_z)~$ determined from the LEP experiments.
6\. The values of the parameters A, B and C are in agreement with the results of [@KaSi]. They are found to be independent of $~b~$ and $~\alpha_0~$. We have found also that multiplying the R.H.S. of (\[e10\]) by term $~(1+ \gamma x)~$ one can not improve the fit.\
[**Summary.**]{} The CCFR deep inelastic neutrino-nucleon scattering data have been analyzed in the framework of the [*Fixed point*]{} QCD. It has been demonstrated that the data for the nucleon structure function $~xF_3(x,Q^2)~$ are in good agreement with the LO predictions of this quantum field theory model using the assumption that $~\alpha_{0}~$ is a [*first order ultraviolet fixed point*]{} of the $~\beta~$ function and $~\alpha_{0}~$ is [*small*]{}. Some constraints on the behaviour of the $~\beta~$ function near $~\alpha_{0}~$ have been found from the data. The value $~\alpha_{0} = 0.057~$ corresponding to the $~\beta~$ function parameter $~b=0.15~$ we have obtained is preferred to the other ones determined from the data.\
In conclusion, we find that the CCFR data, the most precise data on deep inelastic scattering at present, [*do not eliminate*]{} the FP-QCD and therefore other tests have to be made in order to distinguish between QCD and FP-QCD.\
We are grateful to Profs. A.L. Kataev, D.I. Kazakov, N.N. Nikolaev, E.A. Paschos, E. Seiler, D.V. Shirkov and N.G. Stefanis for useful discussions and remarks. One of us (D.S) would like to thank also the International Atomic Energy Agency and UNESCO for the hospitality at the International Centre for Theoretical Physics in Trieste where this work was completed.\
This research was partly supported by INTAS (International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union) under Contract nb 93-1180, by the Russian Fond for Fundamental Research Grant N 94-02-03463-a and by the Bulgarian Science Foundation under Contract\
-1cm
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[^1]: Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences\
Boul. Tsarigradsko chaussee 72, Sofia 1784, Bulgaria. E-mail:stamenov@bgearn.bitnet
|
---
author:
- |
[^1]\
II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany\
Joint Institute for Nuclear Research,$141980$ Dubna (Moscow Region), Russia\
E-mail:
- |
Vladimir V. Bytev[^2]\
Joint Institute for Nuclear Research,$141980$ Dubna (Moscow Region), Russia\
II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany\
E-mail:
- |
Bernd A. Kniehl\
II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany\
E-mail:
- |
B.F.L. Ward\
Department of Physics, Baylor University, One Bear Place, Waco, TX 76798, USA\
E-mail:
- |
Scott A. Yost\
Department of Physics, The Citadel, 171 Moultrie St., Charleston, SC 29409, USA\
E-mail:
title: 'Feynman Diagrams, Differential Reduction, and Hypergeometric Functions'
---
Introduction
============
Any dimensionally-regularized [@dimreg] multiloop Feynman diagram with propagators $1/(p^2-m^2)$ can be written in the form of a finite sum of multiple Mellin-Barnes integrals [@calan; @MB] obtained via a Feynman-parameter or “$\alpha$” representation: [@QFT] $$\begin{aligned}
F\left(a_{js},b_{km},c_i,d_j,\vec{x} \right) =
\int_{\gamma+i\mathbb{R}} dz_1 \ldots dz_r
\frac{\Pi_{j=1}^{p} \Gamma\left(\sum_{s=1}^r a_{js} z_s+c_j\right)}
{\Pi_{k=1}^{q} \Gamma\left(\sum_{m=1}^r b_{km} z_m+d_k\right)}
x_1^{-z_1} \ldots x_r^{-z_r} \; ,
\label{MB}\end{aligned}$$ where $
a_{js}, b_{km} \in \mathbb{Q}, c_i, d_j \in \mathbb{C},\
$ and $\gamma$ is chosen such that the integral exists. In this integral, the $x_a$ are rational functions of kinematic invariants (masses and momenta) and the matrices $a_{js}, b_{kr}$ and vectors $c_i, d_j$ depend linearly on the dimension $n$ of space-time.[^3]. In dimensional regularization, $n$ is treated as an arbitrary parameter, so that $n=I-2{\varepsilon}$ for a positive integer $I$.
Formally,[^4] this integral can be expressed in terms of a sum of residues of the integrated expression $$\begin{aligned}
F\left(a_{js},b_{km},\vec{c},\vec{d},\vec{\alpha}, \vec{x} \right) =
\sum_{\vec{\alpha}} B_{\vec{\alpha}} \vec{x}^{\;\vec{\alpha}}
\Phi(\vec{\gamma};\vec{\sigma};\vec{x}) \;,
\label{FI}\end{aligned}$$ where the components of the vector $\vec{\alpha}$ are defined in terms of the matrix components $a_{js},b_{km}$ and vectors $\vec{c},\vec{d}$, the coefficients $
B_{\vec{\alpha}}
$ are ratios of $\Gamma$-functions with arguments depending on $\vec{\alpha}$, and the functions $\Phi(\vec{\gamma};\vec{\sigma};\vec{x})$ have the form $$\begin{aligned}
\Phi(\vec{\gamma};\vec{\sigma};\vec{x})
=
\sum_{m_1,m_2,\cdots, m_r=0}^\infty
\Biggl(
\frac{
\Pi_{j=1}^K
\Gamma\left( \sum_{a=1}^r \mu_{ja}m_a+\gamma_j \right)
}
{
\Pi_{k=1}^L
\Gamma\left( \sum_{b=1}^r \nu_{kb}m_b+\sigma_k \right)
}
\Biggr)
x_1^{m_1} \cdots x_r^{m_r} \;,
\label{Phi}\end{aligned}$$ with $
\mu_{ab}, \nu_{ab} \in \mathbb{Q},\
\gamma_j,\sigma_k \in \mathbb{C}.
$
Let $
\vec{e}_j = (0,\cdots,0,1,0,\cdots,0)
$ denote the unit vector with unity in its $\j^{\rm th}$ entry, and define $
\vec{x}^{\vec{m}} = x_1^{m_1} \cdots x_r^{m_r}
$ for any integer multi-index $\vec{m} = (m_1, \cdots, m_r)$. In accordance with the definition of Refs. [@bateman; @Gelfand], the (formal) multiple series $
\sum_{\vec{m}=0}^\infty C(\vec{m}) \vec{x}^{\vec{m}},
$ is called [*hypergeometric*]{} if, for each $i=1, \cdots, r$, the ratio $
C(\vec{m}+\vec{e}_i)/C(\vec{m})$ is a rational function in the index of summation $(m_1, \cdots, m_r)$. Ore and Sato [@Ore:Sato] found (see also Ref. [@Gelfand]) that the coefficients of such a series have the general form $$\begin{aligned}
C(\vec{m})
= \Pi_{i=1}^r \lambda_i^{m_i}R(\vec{m})
\Biggl(
\frac{
\Pi_{j=1}^N \Gamma(\mu_j(\vec{m})+\gamma_j)
}
{
\Pi_{k=1}^M \Gamma(\nu_k(\vec{m})+\delta_k)
}
\Biggr) \;,
\label{ore}\end{aligned}$$ where $
N, M \geq 0,
$ $
\lambda_j,\delta_j, \gamma_j \in \mathbb{C}
$ are arbitrary complex numbers, $\mu_j, \nu_k: \mathbb{Z}^r \to \mathbb{Z}$ are arbitrary integer-valued linear maps, and $R$ is an arbitrary rational function. A series of this form is called a [*Horn-type*]{} hypergeometric series.
We deduce from Eqs. (\[Phi\]) and (\[ore\]) that, assuming there is a region of variables where each multiple series of Eq. (\[FI\]) is convergent, any Feynman diagram can be understood as a special case of a Horn-type hypergeometric function with the functions $R(\vec{m})$ equal to unity.[^5] One interesting property of Horn-type hypergeometric functions is the existence of a set of differential contiguous relations between functions with shifted arguments. We consider such relations in the following section.
Contiguous Relations via Linear Differential Operators
======================================================
Let us consider a formal series of type (\[Phi\]). The sequences $\vec{\gamma}=(\gamma_1,\cdots, \gamma_K)$ and $\vec{\sigma}=(\sigma_1,\cdots, \sigma_L)$ are called [*upper* ]{} and [*lower*]{} parameters of the hypergeometric function, respectively. Two functions of type (\[Phi\]) with sets of parameters shifted by a unit, $\Phi(\vec{\gamma}+\vec{e_c};\vec{\sigma};\vec{x})$ and $\Phi(\vec{\gamma};\vec{\sigma};\vec{x})$, are related by a linear differential operator: $$\begin{aligned}
\Phi(\vec{\gamma}+\vec{e_c};\vec{\sigma};\vec{x})
& = &
\sum_{m_1,m_2,\cdots, m_r=0}^\infty
\Biggl(
\frac{
(\sum_{a=1}^r \mu_{ca}m_a+\gamma_c)
\Pi_{j=1}^K
\Gamma\left(
\sum_{c=1}^r \mu_{jc}m_c+\gamma_j \right)
}
{
\Pi_{k=1}^L
\Gamma\left( \sum_{b=1}^r \nu_{kb}m_b+\sigma_k \right)
}
\Biggr)
x_1^{m_1} \cdots x_r^{m_r}
\nonumber \\
& = &
\sum_{m_1,m_2,\cdots, m_r=0}^\infty
\Biggl(
\frac{
\left ( \sum_{a=1}^r
\mu_{ca} x_a \frac{\partial}{\partial x_a}+\gamma_c \right)
\Pi_{j=1}^K
\Gamma\left ( \sum_{c=1}^r \mu_{jc}m_c+\gamma_j \right)
}
{
\Pi_{k=1}^L
\Gamma\left( \sum_{b=1}^r \nu_{kb}m_b+\sigma_k \right)
}
\Biggr)
x_1^{m_1} \cdots x_r^{m_r}
\nonumber \\
& = &
\left ( \sum_{a=1}^r \mu_{ca} x_a \frac{\partial}{\partial x_a}+\gamma_c \right)
\Phi(\vec{\gamma};\vec{\sigma};\vec{x})
\equiv
U_{[\gamma_c \to \gamma_c+1]}^+
\Phi(\vec{\gamma},\vec{\sigma}, \vec{x})
\;.
\label{do1}\end{aligned}$$ Similar relations also exist for the lower parameters: $$\begin{aligned}
\Phi(\vec{\gamma};\vec{\sigma}-\vec{e}_c;\vec{x})
& = &
\sum_{m_1,m_2,\cdots, m_r=0}^\infty
\sum_{b=1}^r (\nu_{cb}m_b+\sigma_c)
\Biggl(
\frac{
\Pi_{j=1}^K
\Gamma\left(\sum_{a=1}^N \mu_{ja}m_a+\gamma_j \right)
}
{
\Pi_{k=1}^L
\Gamma\left(\sum_{b=1}^r \nu_{kb}m_b+\sigma_k \right)
}
\Biggr)
x_1^{m_1} \cdots x_r^{m_r}
\nonumber \\
& = &
\left(
\sum_{b=1}^r
\nu_{cb} x_b \frac{\partial}{\partial x_b} + \sigma_c
\right)
\Phi(\vec{\gamma};\vec{\sigma};\vec{x})
\equiv
L_{[\sigma_c \to \sigma_c-1]}^-
\Phi(\vec{\gamma};\vec{\sigma};\vec{x})
\;.
\label{do2}\end{aligned}$$ The linear differential operators $U_{\gamma_c \to \gamma_c+1}^+$, $L_{\sigma_c \to \sigma_c-1}^-$ are called [*step-up*]{} and [*step-down*]{} operators for the upper and lower index, respectively. If additional step-down and step-up operators $U_{\gamma_c}^-$, $L_{\sigma_c}^+$ satisfying $$U_{[\gamma_{c}+1 \to \gamma_c ]}^- U_{[\gamma_c \to \gamma_c+1]}^+ \Phi(\vec{\gamma},\vec{\sigma},\vec{x}) =
L_{[\sigma_{c}-1 \to \sigma_{c}]}^+ L_{[\sigma_c \to \sigma_c-1]}^- \Phi(\vec{\gamma},\vec{\sigma},\vec{x}) =
\Phi (\vec{\gamma},\vec{\sigma},\vec{x})
\;,$$ ([*i.e.*]{}, the inverses of $U_{\gamma_c}^+$, $L_{\sigma_c}^-$), are constructed, we can combine these operators to shift the parameters of the hypergeometric function by any integer. This process of applying $U_{\gamma_c}^\pm, L_{\sigma_c}^\pm$ to shift the parameters by an integer is called [**differential reduction**]{} of a hypergeometric function (\[Phi\]).
Algebraic relations between functions $\Phi(\vec{\gamma},\vec{\sigma};\vec{x})$ with parameters shifted by integers are called [**contiguous relations**]{}. The development of systematic techniques for obtaining a complete set of contiguous relations has a long story. It was started by Gauss, who found the differential reduction for the $_2F_1$ hypergeometric function in 1823. [@Gauss] Numerous papers have since been published [@contiguous] on this problem. An algorithmic solution was found by Takayama in Ref. [@theorem], and those methods have been extended in a later [^6] series of publications [@Japan].
Let us recall that any hypergeometric function can be considered to be the solution of a proper system of partial differential equations (PDEs). In particular, for a Horn-type hypergeometric function, the system of PDEs can be derived from the coefficients of the series. In this case, the ratio of two coefficients can be presented as a ratio of two polynomials, $$\frac{C(\vec{m}+e_j)}{C(\vec{m})} = \frac{P_j(\vec{m})}{Q_j(\vec{m})} \;,
\label{pre-diff}$$ so that the Horn-type hypergeometric function satisfies the following system of equations: $$0 =
D_j (\vec{\gamma},\vec{\sigma},\vec{x})
\Phi(\vec{\gamma},\vec{\sigma}, \vec{x})
=
\left[
Q_j\left(
\sum_{k=1}^r x_k\frac{\partial}{\partial x_k}
\right)
\frac{1}{x_j}
-
P_j\left(
\sum_{k=1}^r x_k\frac{\partial}{\partial x_k}
\right)
\right]
\Phi(\vec{\gamma},\vec{\sigma}, \vec{x}) \;, \quad j=1, \cdots, r.
\label{diff}$$
Let $\mathfrak{R}$ be the left ideal of the ring $\mathfrak{D}$ of differential operators generated by the system of differential equations for a hypergeometric function (\[diff\]), $D_j (\vec{\gamma},\vec{\sigma},\vec{x}),j=1,.\cdots,r$. The first step in Takayama’s algorithm is the construction of a Gröbner basis $\mathfrak{G}=\left\{G_i(\vec{\gamma}, \vec{\sigma},\vec{x}), i=1,\cdots,q \right\}$ of $\mathfrak{R}$. Then the step-up operator corresponding to $
U_{\gamma_c}^+
$ and step-down operator corresponding to $
L_{\sigma_c}^-
$ are solutions to the linear equations $$\begin{aligned}
&&
\sum_{i=1}^q C_i G_i(\vec{\gamma}, \vec{\sigma},\vec{x})
+
U_{[\gamma_{c}+1 \to \gamma_c ]}^- U_{[\gamma_c \to \gamma_c+1]}^+ = 1 \;,
\\ &&
\sum_{i=1}^q E_i G_i(\vec{\gamma}, \vec{\sigma},\vec{x})
+
L_{[\sigma_{c}-1 \to \sigma_{c}]}^+ L_{[\sigma_c \to \sigma_c-1]}^- = 1\;, \end{aligned}$$ where $
C_i, E_i
$ are arbitrary functions. This system has a solution if the left ideal generated by $\mathfrak{G} \cup \left\{ U_{\gamma_c}^+ \right\}$ is equal to $\mathfrak{D}$ (see details in Ref. [@theorem]).
In this way, the Horn-type structure provides an opportunity to reduce hypergeometric functions to a set of basis functions with parameters differing from the original values by integer shifts: $$P_0(\vec{x})
\Phi(\vec{\gamma}+\vec{k};\vec{\sigma}+\vec{l};\vec{x})
=
\sum_{m_1, \cdots, m_p=0}^{\sum{|k_i|+\sum|l_i|}} P_{m_1, \cdots, m_r} (\vec{x})
\left( \frac{\partial}{\partial x_1} \right)^{m_1} \cdots
\left( \frac{\partial}{\partial x_r} \right)^{m_r}
\Phi(\vec{\gamma};\vec{\sigma};\vec{x}) \;,
\label{reduction}$$ where $P_0(\vec{x})$ and $P_{m_1, \cdots, m_p}(\vec{x})$ are polynomials with respect to $\vec{\gamma},\vec{\sigma}$ and $\vec{x}$ and $\vec{k},\vec{l}$ are lists of integers.
Differential Reduction in Practice
==================================
In real physical problems, the variables are generally not linearly independent: some of them can be equal to one another $x_i=x_j, i \neq j$. In this case, the Horn-type hypergeometric function generally is not expressible in terms of Horn hypergeometric functions with fewer variables. Another important physical case is when variables belong to the surfaces where coefficient $P_0(\vec{x})$ of Eq. (\[reduction\]) vanishes (in the one-variable case it corresponds to $z=1$). In all of these cases, the differential reduction cannot be directly applied. But if the l.h.s. of Eq. (\[reduction\]) is defined for that limit, the smooth limit of the r.h.s. of Eq. (\[reduction\]) will exist too.
The problem is then to find this smooth limit. Here, “physics” plays a role. For the evaluation of physical processes, exact results in terms of hypergeometric function are not necessary, but only the coefficients of a Laurent expansion around an integer value of the space-time dimension. From that point of view, to guarantee that smooth limit exists, it is enough to prove that a smooth limit exists for all coefficients of the all-order ${\varepsilon}$ expansion.
Recently, physicists have proven several theorems on the all-order ${\varepsilon}$ expansion of hypergeometric functions about integer and/or rational values of parameters [@DK; @nested; @Kalmykov; @KK08]. A remarkable property of this construction is that for special values of parameters, the coefficients are expressible in terms of multiple polylogarithms [@mp]. For functions of this type, the limiting procedure is well understood. Unfortunately, existing theorems are not adequate to cover all values of parameters for hypergeometric functions generated by Feynman diagrams [@KK08; @sunset].
Discussion and Conclusions
==========================
We have presented formal arguments that any Feynman diagram can be treated as a finite sum of Horn-type hypergeometric functions. This applies, in particular, for off-shell diagrams with different values of masses in internal lines. In the physically interesting cases when some of the arguments are equal to one another, or belong to some singularity surface, a limiting procedure should be constructed. To find this limit, and strongly prove that the proper limit exists, the all-order ${\varepsilon}$ expansion of hypergeometric function around rational values of parameters can be used.
The Horn-type hypergeometric functions possess useful properties: the system of differential equation they satisfy is enough (i) for reduction of original function to a restricted set of basis functions (the number of basis functions follows directly from the system of equations); (ii) for the construction of the all-order ${\varepsilon}$ expansion for the basis hypergeometric functions in form of the Lappo-Danilevky solution. The first part of this algorithm has been discussed in [@MKL06; @LCWS08]; the second part in [@Kalmykov; @KK08]. The validity of our approach has been confirmed (in particular cases) by full agreement with the evaluation of the first coefficients of the ${\varepsilon}$ expansion constructed in [@kalmykov:expansion], by theorems about the all-order ${\varepsilon}$ expansion of hypergeometric functions proven in [@nested] with the help of another technique, and by comparisons of the results of the differential reduction of some Feynman diagrams with the results of reductions obtained via computer programs [@kalmykov:programs].
The above-mentioned properties of the hypergeometric representation, in particular Takayama’s reduction algorithm, demonstrate that the hypergeometric representation is a universal tool for the evaluation of Feynman diagrams. Obtaining a decomposition of the integration region into regions where each individual term is well-defined is one of the main problems in the hypergeometric approach (beyond the one-loop and one-fold cases).
Series representations (in four dimensions) have been used in Ref. [@Kershaw] to obtain a system of partial differential equations for $N$-point one-loop diagrams. In Ref. [@Davydychev], the hypergeometric representation has been derived for an $N$-point one-loop diagram with arbitrary powers of the propagators. The possibility of using a hypergeometric representation for the reduction of Feynman diagrams was considered in Ref. [@MKL06]. The idea of using the Gröbner basis technique for the reduction of Feynman diagrams has been proposed by Tarasov [@Tarasov:Grobner] and received further extension in Ref. [@Grobner:difference]. For practical applications of the “limiting” procedure, the first few coefficients of the ${\varepsilon}$ expansion are necessary. The ${\varepsilon}$ expansion is implemented in several packages [@packages] for a restricted class of hypergeometric functions. Some results for the finite harmonic sums are available in [@finite].
Another important class of developments includes techniques such as integration-by-parts [@ibp], generalized recurrence relations [@numerator:massive], and the differential equation approach [@DE], which make it possible to work directly with parameters of the Feynman diagram (the l.h.s. of Eq. (\[FI\])) without splitting the diagram into a linear combination of Horn-type hypergeometric functions. Such direct analyses of Feynman diagrams as hypergeometric functions are based on the properties of dimensional regularization [@dimreg] , which treats all types of singularities (IR and UV) simultaneously.
Some partial results of our research have been used in Ref. [@application]. A more detailed discussion will be presented in a forthcoming publication [@BKK].
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[^1]: Fellow of the Conference Organizing Committee.
[^2]: Research supported by MK 1607.2008.2
[^3]: Starting from the $\alpha$-representation for Feynman diagrams with (irreducible) numerators [@QFT], the Mellin-Barnes representation can be written explicitly in terms of Symanzik polynomials.[@calan] In Refs. [@numerator:massless; @numerator:massive], it was shown that this $\alpha$-representation can be understood as a linear combination of scalar Feynman diagrams of the original type with shifted powers of propagators and space-time dimension multiplying a tensor factor depending on external momenta. Through fifty years of the evaluation of Feynman diagrams, many auxiliary programs have been created for the generation of Symanzik polynomials. In particular, the matrix representation for these polynomials derived by Nakanishi (see Eqs. (3.23)$-$(3.34) in Ref. [@Nakanishi]) has been realized on FORM by Oleg Tarasov in Ref. [@Symanzik1], by Oleg Veretin in [@Symanzik2], and recently in Ref. [@Symanzik3].
[^4]: This is true under the condition that there is a common domain of convergence. For particular values of the kinematic variables and powers of the propagators, the Mellin-Barnes representation may contain terms like $\hbox{$\Gamma(a-s)$} \times \hbox{$\Gamma(a+s)$}$ or $\Gamma^2(a-s) \times \Gamma^2(b+s)$, where $a$ and $b$ are parameters and $s$ is an integration variable. Two algorithms for the practical construction of the ${\varepsilon}$-expansion of Mellin-Barnes integrals with such singularities have been described in Refs. [@Smirnov-Tausk], These are now implemented in several packages [@Gzakon]. It will be quite interesting to apply modern mathematical algorithms, as was done in Ref. [@hironaka], to the problem of singularities in the Mellin-Barnes representation as well. In the framework of our approach, such singularities can be regularized by introducing an additional analytical regularization for each propagator (see Ref. [@extra]) or by introducing masses to regulate IR singularities. General theorems on the properties of Feynman diagrams in dimensional regularization guarantee that a smooth limit of the auxiliary regularization should exist.
[^5]: Using the technique presented in Ref. [@Melnikov], this statement can also be shown to be valid for the phase-space integral.
[^6]: The problem also can be solved via an Ore algebra [@Ore] approach to the relevant system of linear differential and difference (shift) operators.
|
---
abstract: 'We show that the information gained in spectroscopic experiments regarding the number and distribution of atomic environments can be used as a valuable constraint in the refinement of the atomic-scale structures of nanostructured or amorphous materials from pair distribution function (PDF) data. We illustrate the effectiveness of this approach for three paradigmatic disordered systems: molecular C$_{60}$, a-Si, and a-SiO$_2$. Much improved atomistic models are attained in each case without any *a-priori* assumptions regarding coordination number or local geometry. We propose that this approach may form the basis for a generalised methodology for structure “solution” from PDF data applicable to network, nanostructured and molecular systems alike.'
author:
- 'Matthew J. Cliffe'
- 'Martin T. Dove'
- 'D. A. Drabold'
- 'Andrew L. Goodwin'
title: Structure determination of disordered materials from diffraction data
---
Many materials of fundamental importance possess structures that do not exhibit long-range periodicity: examples include metallic and covalent glasses [@Byrne_2008], amorphous biominerals [@Weiner_2005], the so-called “phase-change” chalcogenides of DVD-RAM technology [@Sun_2006], and amorphous semiconductors such as a-Si and a-Ge [@Armatas_2006]. The absence of Bragg reflections in the diffraction patterns of these materials precludes the use of traditional crystallographic techniques as a means of determining their atomic-scale structures. Yet it is clear that these materials do possess well-defined local structure on the nanometre scale [@Billinge_2007]; moreover it is often this local structure that is implicated in the particular physical properties of interest [@Billinge_2004]. For this reason, the development of systematic information-based methodologies for the determination of local structure in disordered materials remains one of the key challenges in modern structural science [@Juhas_2006].
Historically, local structure has been studied experimentally using two principal approaches: (i) the diffraction techniques of neutron and x-ray total scattering, from which the distribution of interatomic separations can be measured via the pair distribution function (PDF) [@Egami_2003], and (ii) resonance and spectroscopic methods (NMR, EXAFS, IR, Raman) that yield information concerning the number and population of distinct atomic environments, together with (in favourable cases) metal-coordination/molecular geometries [@Brodsky_1978; @Mullerwarmuth_1982]. These techniques afford a rich body of information, and over the past 5–10 years a number of sophisticated methods of analysis have emerged that aim to derive structural models via fitting to these experimental data. The Reverse Monte Carlo (RMC) [@McGreevy_2001] and Empirical Potential Structure Refinement (EPSR) [@Soper_2005] methods have been used widely in the glass and amorphous materials community, while the PDFfit [@Proffen_1999] and “Liga” [@Juhas_2008] methods have been applied more recently to nanostructured solids such as C$_{60}$ [@Juhas_2006] and ferrihydrite [@Michel_2007]—systems that present similar crystallographic challenges.
There is, however, a fundamental problem: markedly different structural models can be equally consistent with the same PDF data [@Gereben_1994]. Moreover, the task of fitting simultaneously to PDF and spectroscopic data is almost always either too computationally demanding or in fact not quantitatively possible. Taken together, these factors have meant that it is often difficult to determine the atomic-level structure of these materials, and that there is no “routine” information-based approach analogous to those for crystalline materials.
In this Letter, we show that this problem can largely be solved by using information gained via spectroscopy—the number and population of distinct atomic environments—to guide refinement of experimental PDF data. Structural refinement based on reproducing the experimental PDF alone is, in general, not sufficiently well-constrained to produce models that reflect the “true” local structure in a material; however, if refinement is forced also to reflect the correct number and distribution of atom environments then convergence on the correct local structure usually follows. This approach is easily implemented and generic. Moreover, we show that successful refinement can be initiated using entirely random atomistic models and, in being driven wholly by experimental data, one avoids any other *a-priori* assumptions concerning *e.g.* coordination numbers or geometries. While our focus lies on proof-of-principle at this stage, our results show that routine information-based structure determination of disordered materials is now a viable prospect.
Our paper is arranged as follows. We begin by describing the particular implementation of our methodology through a “variance”-based term in the cost function used to drive PDF refinement. Three principal case studies follow: nanoparticulate C$_{60}$ (single cluster; one atom environment), amorphous silicon (continuous network solid; one atom environment) and amorphous silica (continuous network solid; two atom environments). In all three instances we show that a conventional RMC approach fails to obtain the correct structure solution—often spectacularly—but that inclusion of the variance term is sufficient to recover almost-perfect models of material structure in each case. We conclude by discussing a number of different possible implementations of our underlying methodology.
In outlining our methodology, it is useful to consider first the simplest type of disordered material: namely a phase that contains a single atom type and for which spectroscopy indicates a single atom environment. The existence of a single atomic environment demands that structural correlation functions calculated for different individual atoms within the material should take similar forms. In order to recast this statement with specific reference to the PDF, we first define atomic PDFs $p_j(r)$ for an atomistic model such that the “bulk” (measurable) PDF $G(r)$ corresponds to the average $\langle p(r)\rangle$ taken over all atoms $j$. Then the existence of a single atom environment dictates a similarity $p_j(r)\sim p_{j^\prime}(r)\sim G(r)$ for all atoms $j,j^\prime$. Whereas a standard PDF-based structure refinement would involve minimising a function of the form
$$\label{oldchi2}
\chi^2=\sum_r[\langle p(r)\rangle-G_{\rm expt}(r)]^2,$$
what we would propose is the alternative cost function
$$\label{newchi2}
\chi^2=\frac{1}{N}\sum_j\sum_r[p_j(r)-G_{\rm expt}(r)]^2.$$
Note that in this reformulation one obtains $\chi^2=0$ if and only if the model PDF matches $G_{\rm expt}(r)$ *and* each individual $p_j(r)$ has the same form. It is straightforward to show that the new penalty function $\chi^2$ of is in fact equal to that of plus a variance term $\chi^2_{\rm Var}=\langle p(r)^2\rangle-\langle p(r)\rangle^2$. What the spectroscopic result suggests is to add to a conventional PDF refinement a term that penalises variance in local coordination environments; for this reason we are terming our approach an INVariant Environment Refinement Technique (INVERT).
In practice, the individual $p_j(r)$ for a static atomistic model consist of a series of delta functions, and in order to obtain a well-behaved variance term, it is necessary to adopt a modified formulation such as:
$$\label{varchi2}
\chi^2_{\textrm{Var}}=\frac{1}{N}\sum_j\sum_i\frac{\left[d_j(i)-\langle d(i)\rangle\right]^2}{\langle d(i)\rangle^2},$$
where $d_j(i)$ measures the distance from atom $j$ to its $i$-th neighbour, and $\langle d(i)\rangle$ is the average such distance over all atoms $j$. The term in the denominator of Eq. appears in order to account for the fact that the number of neighbours at a distance $d$ scales with $d^2$ [@othermodifications]. The extension to multiple atom types and/or atom environments is straightforward. A separate variance term is included for each different pair of atom types (A and B, say); the form of each individual term is the same as in Eq. except that the $d_j(i)$ will refer to $i$-th neighbour of type B around the $j$-th atom of type A, and so on.
At this point we emphasise that no assumption has been made regarding the actual distribution of neighbours around each atom—only that this distribution should be similar for equivalent atoms. Moreover, we are able to constrain the partial PDF functions for multi-component systems despite the experimental PDF data representing a sum over these separate contributions.
![\[fig1\]RMC refinement of the experimental PDF of C$_{60}$: (a) the neutron PDF itself, corrected to remove inter-molecular correlations as described in Ref. ; (b) a random starting configuration of 60 carbon atoms; (c) a typical configuration produced by conventional RMC refinement of either idealised or experimental PDF data; and those produced by INVERT+RMC using (d) idealised, and (e) experimental PDF data. In panels (b)–(e), atoms with three nearest neighbours are coloured blue and others are coloured red.](fig1.png)
We have chosen C$_{60}$ as a simple first case study, not least because the task of determining its well-known icosahedral structure from the experimental PDF \[Fig. \[fig1\](a)\] has recently been highlighted as a benchmark challenge in nanostructure determination [@Juhas_2008]. As straightforward as the task might seem, conventional RMC refinement from a random starting configuration \[Fig. \[fig1\](b)\] fails entirely, giving a set of small clusters that contains only a few of the real set of interatomic separations \[Fig. \[fig1\](c)\]. The same result is obtained even if idealised PDF data are used.
The INVERT modification exploits the experimental NMR result that C$_{60}$ contains a single C environment [@Johnson_1990]. Clearly the RMC configuration in Fig. \[fig1\](c) violates this property, and so would now give rise to a large $\chi^2_{\rm Var}$ term that will help drive refinement forward. Indeed, INVERT+RMC refinement from the same random starting configuration gives the correct solution for idealised data \[Fig. \[fig1\](d)\] and a near-perfect solution for the experimental neutron PDF data of Ref. \[Fig. \[fig1\](e)\]. We note that such a result has only ever been achieved previously using the highly-sophisticated cluster optimisation methods of the “Liga” algorithm or using genetic algorithms based on the principle of mating or crossover (the latter only giving correct solutions in 56% of attempts) [@Juhas_2006; @Hartke_1999; @Deaven_1995]. Here, INVERT+RMC consistently obtains a topologically-identical solution from random starting coordinates in approximately 2000–4000 accepted moves.
We find the extension to a cluster with two atom environments—namely, S$_{12}$ [@Steidel_1981]—enjoys similar success [@s12note]. Videos that illustrate the refinement process for C$_{60}$ and S$_{12}$ are provided as supporting information [@Movies].
The paradigmatic “stumbling block” for RMC, however, has always been amorphous Si, whose structure is believed to consist of a continuous random network (CRN) of tetrahedral Si centres [@Drabold_2009]. Rather than generating a network of four-fold-coordinated Si atoms, RMC refinements of a-Si PDF data yield configurations with unphysically-broad distributions of Si coordination numbers [@Gereben_1994]. This is allowed because, crudely speaking, a pair of atoms of coordination numbers three and five will contribute to the average PDF indistinguishably from two four-fold coordinated atoms, and yet the former state is statistically more likely during a sequence of random moves. Various work-arounds have been proposed and implemented (*e.g.* constraining coordination numbers to equal four), and in the most favourable of cases these yield CRNs comparable to those obtained from bond-switching (Wooten-Weaire-Winer, “WWW” [@Wooten_1985]) methods, molecular dynamics and *ab-initio* calculations [@Drabold_2009].
![\[fig2\]Comparison of a-Si configurations obtained using (left) “Native RMC” and (centre) “INVERT+RMC”’ refinement with (right) the trusted “WWW”model of Ref. . (a) Slices of the configurations themselves, with four-coordinate Si coloured blue and others coloured red. (b) The PDFs calculated from each configuration, which are essentially identical. (c) Corresponding PDF variances calculated using Eq. .](fig2.png)
However, there is a sense in which one recovers from these approaches only the very information already used to generate the constraints: if the coordination number is constrained to equal four during refinement, then four-fold coordination cannot be considered an independent result. Consequently, our motivation for considering a-Si as a second case study was primarily to determine whether, by using the evidence for a single Si environment from NMR studies [@Shao_1990], INVERT+RMC refinement could yield reasonable structural models without recourse to explicit coordination number constraints.
First, a conventional RMC refinement was performed using $G(r)$ “data” generated from the trusted WWW model of Ref. . The starting configuration was a random collection of 512 Si atoms in a cubic box of side 21.7Å. Refinement gave a highly-disordered configuration that displayed all the hallmarks of previously-described problematic RMC studies [@Gereben_1994]: only 27% of Si atoms are four-fold coordinated, there are substantial density variations, and large numbers of unphysical Si$_3$ “triangles” \[left-hand panel of Fig. \[fig2\](a)\]. In contrast, a parallel INVERT+RMC refinement achieved an almost perfect coordination distribution (95% four-fold). The improvement extended even to the higher-order correlations (discussed in more detail below): in particular, the number of Si$_3$ triangles is halved, and the density distribution is much more even. Inspection of the configuration itself \[centre panel of Fig. \[fig2\](a)\] now reveals obvious similarities to the trusted WWW model \[right-hand panel of Fig. \[fig2\](a)\]. The PDF itself is relatively insensitive to this fundamental improvement in local structure modelling \[Fig. \[fig2\](b)\], while the variance term of Eq. clearly acts a much better figure-of-merit \[Fig. \[fig2\](c)\].
Similar results are obtained for amorphous SiO$_2$, which is a conceptual extension in that it contains two distinct atom environments: that of the Si atoms and that of the O atoms. Experimental neutron PDF data were taken from Ref. , and starting configurations generated from a random distribution of 64 Si atoms and 128 O atoms in a periodic cubic box of side 14.37Å. RMC refinement both with and without the INVERT modification gave excellent fits to the PDF, but the INVERT+RMC model had a much higher percentage of fourfold Si coordination (97% *vs*. 59% for the RMC-only configuration). Indeed, we believe the INVERT+RMC configuration to be the first information-based CRN model of a-SiO$_2$ \[Fig. \[fig3\]\].
![\[fig3\]A slice of the information-driven CRN model of a-SiO$_2$ obtained using INVERT+RMC refinement as described in the text. Si and O atoms shown in dark and light colours, respectively; atoms in shades of blue have the expected coordination numbers of 4 (Si) and 2 (O), while the few in shades of red have incorrect coordination numbers.](fig3.png){width="4cm"}
The INVERT methodology is by no means applicable only to RMC refinement. Our focus on RMC in this Letter arises from a desire to demonstrate the effectiveness of the INVERT approach using a refinement method that is known to favour *disorder*. The incorporation of variance-based cost functions in any refinement approach is straightforward, and such a modification to more sophisticated PDF fitting approaches than RMC, *e.g.* as suggested in Ref. , might reasonably be expected to produce even more realistic configurations.
Speaking more generally, we would note that the concept of local invariance encompasses more than minimising the PDF variance alone. One can imagine, for example, that minimising the variance in higher-order correlation functions, such as angle distributions, coordination geometry, and CRN ring statistics might also improve refinement further. Importantly, these constraints can be implemented despite the functions not being measurable experimentally. In practice, however, we have found that the calculation of higher-order correlation functions is too computationally-demanding for speedy refinement at this stage; the extension to constraining geometric invariance using spherical harmonics and/or the triplet distribution function is an approach we hope to pursue further in the near future.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We gratefully acknowledge financial support from Trinity College, Cambridge to A.L.G., from the EPSRC (UK) to A.L.G. and M.J.C., and from the US NSF to D.A.D. under grant DMR 09-03225. We thank D. A. Keen (Rutherford Appleton Laboratory) for useful discussions, and acknowledge the University of Cambridge’s CamGrid infrastructure for computational resources.
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---
abstract: 'Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $\phi:\mathbb{D}\rightarrow\mathbb{C}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.'
address:
- 'P. Universidad Católica de Chile'
- University of Michigan
- Stanford University
author:
- 'M. Chuaqui'
- 'P. Duren'
- 'B. Osgood'
bibliography:
- 'hol-lift.bib'
title: 'Injectivity Criteria for Holomorphic Curves in $\mathbb{C}^n$'
---
Introduction {#section:intro}
============
Let $f:\mathbb{D}\rightarrow\mathbb{C}$ be a locally injective holomorphic mapping defined in the unit disk, and let $${\mathcal{S}}f=(f''/f')'-(1/2)(f''/f')^2$$ be its Schwarzian derivative. A classical univalence criterion of Nehari [@nehari:nehari-old-p] relates the size of $|Sf|$ to the univalence of $f$. Nehari stated the result in the form: $$\label{eq:p-criterion}
|{\mathcal{S}}f(z)| \leq 2p(|z|)$$ implies that $f$ is injective in $\mathbb {D}$, if $p(x)$ is a positive, even, and continuous function defined for $x\in(-1,1)$, with the properties
1. $(1-x^2)^2p(x)$ is non-increasing for $x\in[0,1)$;
2. the differential equation $u''+pu=0$ has no nontrivial solutions with more than one zero in $(-1,1)$.
Condition includes the criteria $|{\mathcal{S}}f(z)|\leq \pi^2/2$ and $|{\mathcal{S}}f(z)|\leq 2(1-|z|^2)^{-2}$ from [@nehari:schlicht], but also many others. A function $p$ satisfying the hypotheses above will be referred to as a *Nehari function*. Later, in Section \[section:conf-sch\], we will also introduce the notion of an *extremal Nehari function*.
Before its connection with univalence was understood, the attributes of the Schwarzian that made it interesting are that it vanishes identically precisely for Möbius transformations, $${\mathcal{S}}f(z) = 0 \quad \text{if and only if} \quad f(z) = \frac{az+b}{cz+d}\,,
\quad ad-bc \ne 0\,,$$ and that it is invariant under post-composition with a Möbius transformation, $$S(f \circ g) = Sg$$ if $f$ is Möbius. More generally, one has the chain rule $${\mathcal{S}}(f\circ g) = (({\mathcal{S}}f)\circ g)(g')^2 + {\mathcal{S}}g\,.
\label{eq:S-chain-rule}$$
Consider now a locally injective holomorphic curve $\phi:\mathbb{D}\rightarrow\mathbb{C}^n$, $n \ge 1$. Write $\phi =(f_1,\dots,f_n)$, with each $f_k$ holomorphic in $\mathbb{D}$, and define the smooth real-valued function $\sigma$ on $\mathbb{D}$ by $$\sigma = \frac{1}{2}\log(|f_1'|^2+\cdots |f_n'|^2)\,.$$ We define the Schwarzian derivative of $\phi$ to be $$\label{eq:hol-schwarzian}
{\mathcal{S}}\phi = 2(\sigma_{zz}-\sigma_z^2)\,,$$ where $$\sigma_z = \frac{1}{2}\left(\frac{\partial\sigma}{\partial x} - i
\frac{\partial\sigma}{\partial y}\right)\,.$$ This reduces to the classical Schwarzian when $n=1$, so there is no ambiguity in using the same name and symbol. Further background on this definition is in Section \[section:conf-sch\]; it derives from a generalization of the Schwarzian to conformal mappings of Riemannian manifolds.
A straightforward calculation based only on the definition together with ${\mathcal{S}}$ vanishing on Möbius transformations shows that $${\mathcal{S}}(\phi \circ T) = (({\mathcal{S}}\phi)\circ T)(T')^2 \label{eq:S-circ-T}$$ when $T$ is a Möbius transformation of $\mathbb{D}$. We will need this in an number of places. We do not consider $M\circ
\phi$ when $M$ is a Möbius transformation of $\mathbb{R}^{2n}$, nor do we have ${\mathcal{S}}(M\circ \phi)={\mathcal{S}}\phi$, since, in general, $M\circ \phi$ is not holomorphic and so its Schwarzian is not defined (at least not so simply). However, there is a substitute for Möbius invariance that we will also need. It involves a version of the Schwarzian introduced by Ahlfors, discussed in Section \[section:S1\]. A very general version of is in Section \[section:conf-sch\].
Let $\Sigma = \phi(\mathbb{D})$, which we can regard as a (real) 2-dimensional surface in $\mathbb{R}^{2n}$. The Gaussian curvature, $K(\phi(z))$, of $\Sigma$ at $\phi(z)$ is given by $$\label{eq:gauss-curvature}
K(\phi(z)) = -e^{-2\sigma(z)}\Delta\sigma(z)\,,$$ and so is nonpositive. We shall prove:
\[theorem:hol-p-criterion\] Let $p$ be a Nehari function and let $\phi: \mathbb{D} \rightarrow \mathbb{C}^n$ be holomorphic with $\phi' \ne 0$. If $$\label{eq:hol-p-criterion}
|{\mathcal{S}}\phi(z)| + \frac{3}{4}|\phi'(z)|^2|K(\phi(z)| \le 2p(|z|)\,,\quad z
\in \mathbb{D}\,,$$ then $\phi$ is injective.
We recover Nehari’s theorem if $n=1$, since then $\phi(\mathbb{D}) \subset \mathbb{C}$ and $K(\phi(z)) =0$.
We are also able to describe just how injectivity fails on $\partial\mathbb{D}$, and it does so in a rather special way. To have such a statement make sense it is first necessary to know that a mapping satisfying Theorem \[theorem:hol-p-criterion\] extends continuously to the boundary. In Section \[section:boundary\] we will give a precise analysis of the situation, but as a preliminary result we now state:
\[theorem:boundary-extension-1\] A holomorphic curve $\phi:\mathbb{D} \rightarrow \mathbb{C}^n$ satisfying Theorem \[theorem:hol-p-criterion\] has an extension to $\overline{\mathbb{D}}$ that is uniformly continuous in the spherical metric.
Thus a function $\phi$ satisfying the condition maps the unit circle to a continuous closed curve $\Gamma\subset{\mathbb{C}}^n\cup\{\infty\}$. We say that $\phi$ is an *extremal function* for the criterion if $\Gamma$ is *not* a [simple]{} closed curve. In this case there is a pair of points $\zeta_1, \zeta_2 \in \partial\mathbb{D}$ with $f(\zeta_1)=f(\zeta_2)=P$, and one says that $P$ is a *cut point* of $\Gamma$. We now have the following characteristic property of extremal mappings.
\[theorem:extremal\] Under the hypotheses of Theorem 1, suppose the closed curve $\Gamma = \phi(\partial\Bbb D)$ is not simple and let $P$ be a cut point. Then there exists a Euclidean circle or line $C$ such that $C\setminus\{P\}
\subset
\Sigma$. Furthermore, equality holds in along $\phi^{-1}
(C \setminus\{P\})$.
In addition to these results, in Section \[section:covering\] we will derive a covering theorem for holomorphic curves satisfying that generalizes some one-dimensional results. In Section \[section:example\] we will construct examples showing that the criterion is sharp.
When we began our work on generalizing Nehari’s theorem it was in a rather different context, namely a harmonic mapping $f$ of $\mathbb{D}$ and its Weierstrass-Enneper lift $\widetilde{f}$ mapping $\mathbb{D}$ to a minimal surfaces in $\mathbb{R}^3$, see [@cdo:harmonic-lift]. The lift $\widetilde{f}$ is a conformal mapping of $\mathbb{D}$, say with conformal factor $e^\sigma$, and again there is a generalization of the Schwarzian derivative, ${\mathcal{S}}f =
2(\sigma_{zz}-\sigma_z^2)$, just as in . One then has: If $p$ is a Nehari function and $$\label{eq:harmonic-p-criterion}
|{\mathcal{S}}f(z)| + e^{2\sigma(z)}|K(\widetilde{f}(z))| \le 2p(|z|)\,, \quad
z\in\mathbb{D}\,,$$ then $\widetilde{f}$ is injective in $\mathbb{D}$. Here $K(\widetilde{f}(z))$ is the Gaussian curvature of the minimal surface $\widetilde{f}(\mathbb{D})$ at $\widetilde{f}(z)$. There are also results analogous to Theorems 2 and 3.
It was very surprising, to us at least, that such similar statements hold in these two different settings. Comparing and the most visible difference is in the multiple of the curvature terms, $3/4$ in the former and $1$ in the latter. This is a reflection of the shared nature of the proofs, with small changes in the constants in some of the preliminary results. Things did not have to be this way, one might have thought. The cause of this commonality comes from the differential geometry of the holomorphic curve as a surface in $\mathbb{R}^{2n}$, to be explained in Section \[section:S1\].
We have certainly borrowed from the exposition in [@cdo:harmonic-lift], but while the two papers run a parallel course in many – but not all – ways, we have tried to make this paper reasonably self-contained. There are a few instances where we refer to [@cdo:harmonic-lift] for details that would have been reproduced verbatim here.
Ahlfors’ Schwarzian and the Second Fundamental Form {#section:S1}
===================================================
We begin the same way as in [@cdo:harmonic-lift], with Ahlfors’ Schwarzian for curves in $\mathbb{R}^m$ and its relationship to curvature, but here we find an important difference with what was done in the case of harmonic maps. Here to make use of the properties of Ahlfors’ operator to study injectivity we must relate the second fundamental form of the holomorphic curve as a surface in $\mathbb{R}^{2n}$ to its Gaussian curvature.
Ahlfors [@ahlfors:schwarzian-rn] defined a Schwarzian derivative for mappings $\varphi : (a,b)\rightarrow {\Bbb R}^m$ of class $C^3$ with $\varphi'(x)\neq0$ by generalizing separately the real and imaginary parts of the Schwarzian for analytic functions. We only need the operator corresponding to the real part, which is $$S_1\varphi = \frac{\langle
\varphi',\varphi'''\rangle}{|\varphi'|^2} - 3\frac{\langle
\varphi',\varphi''\rangle^2}{|\varphi'|^4} +
\frac32\frac{|\varphi''|^2}{|\varphi'|^2}\,,$$ where $\langle\cdot\, ,\cdot\, \rangle$ denotes the Euclidean inner product. If $T$ is a Möbius transformation of $\mathbb{R}^m$ then, as Ahlfors showed, $$S_1(T \circ \varphi)(t) = S_1{\varphi}(t)\,. \label{eq:S1-invariance}$$ We also record the fact that if $\gamma(t)$ is a smooth function with ${\gamma}'(t) \ne 0$ then $$S_1({\varphi}\circ \gamma)
= ((S_1{\varphi}) \circ \gamma)(\gamma')^2 + {\mathcal{S}}\gamma\,,\label{eq:S_1-chain-rule}$$ analogous to the chain rule for the analytic Schwarzian.
As in the introduction, let $\phi :\mathbb{D} \rightarrow
\mathbb{C}^n$ be a holomorphic curve, and let $\Sigma=\phi(\mathbb{D})$. Ahlfors’ Schwarzian enters the proof of Theorem \[theorem:hol-p-criterion\] for two reasons. First, it is related to ${\mathcal{S}}\phi$ via the geometry of $\Sigma$ as a surface in $\mathbb{R}^{2n}$. Second, bounds on $S_1{\varphi}$ imply injectivity along curves, and this will be sufficient to prove injectivity in $\mathbb{D}$. We take up the first point in this section and the second point in Section \[section:injectivity\].
We recall the notion of the second fundamental form of a submanifold. Let $M$ be a submanifold of $\mathbb{R}^m$ with the metric induced by the Euclidean metric on $\mathbb{R}^m$. Let $D$ be the covariant derivative on $\mathbb{R}^m$ and let $D'$ be the covariant derivative on $M$. If $X$ and $Y$ are vector fields tangent to $M$ then $D_XY$ need not be tangent to $M$ but rather has components tangent and normal to $M$: $$D_XY = D'_XY+I\!I(X,Y)\,.$$ The normal component is the second fundamental form, $I\!I(X,Y)$. It is a tensor.
For holomorphic curves the second fundamental form is related to the Gaussian curvature of $\Sigma$ in the following way.
\[lemma:II-gaussian\] Let $\phi:\mathbb{D}\rightarrow\mathbb{C}^n$ be holomorphic with $\phi'\neq 0$, and let $V(x)=\phi'(x)/|\phi'(x)|$, $x\in(-1,1)$. Then along $\phi$ the second fundamental form of $\Sigma=\phi(\mathbb{D})$ satisfies $$\label{eq:II-gaussian}
|I\!I(V,V)|^2=\frac12|K(\phi)|\,.$$
We need to find the components of $D_VV$ tangent and normal to $\Sigma$. First, using $V(x) =
\phi'(x)/|\phi'(x)|=e^{-\sigma(x)}\phi'(x)$ we have, along $(-1,1)$, $$\label{eq:DVV}
D_VV=e^{-\sigma}(e^{-\sigma}\phi')'=
e^{-\sigma}(e^{-\sigma}\phi''-e^{-\sigma}\sigma_x\phi')
=e^{-2\sigma}(\phi''-\sigma_x\phi') = e^{-2\sigma}(\phi_{xx}-\sigma_x\phi_x)\, .$$ Next let $Y$ be the vector field $e^{-\sigma}\phi_y$ along $(-1,1)$. Since $\phi$ is conformal the pair $\{V,Y\}$ is an orthonormal frame for $\Sigma$ along the curve $\phi((-1,1))$. In order to determine the component of $D_VV$ normal to $\Sigma$ we will compute $\langle D_VV,V\rangle$ and $\langle
D_VV,Y\rangle$.
>From $\langle V,V\rangle= 1$ it follows that $\langle
D_VV,V\rangle=0$. Then, $$\langle D_VV,Y\rangle = e^{-3\sigma}\langle \phi_{xx}-\sigma_x\phi_x, \phi_y\rangle
=e^{-3\sigma}\langle \phi_{xx},\phi_y\rangle \, ,$$ because $\langle \phi_x,\phi_y\rangle =0$. But $\phi$ is also harmonic, hence $$\begin{aligned}
\langle D_VV,Y\rangle &= - e^{-3\sigma}\langle \phi_{yy},\phi_y\rangle
=-\frac12\, e^{-3\sigma}\frac{\partial}{\partial y}\langle \phi_y,\phi_y\rangle\\
&=-\frac12\,e^{-3\sigma}\frac{\partial}{\partial
y}(e^{2\sigma})-e^{-\sigma}\sigma_y \, .
\end{aligned}$$ It follows that $$D_VV = -e^{-\sigma}\sigma_yY+I\!I(V,V) \, ,$$ that is, from , $$I\!I(V,V) =
e^{-2\sigma}(\phi_{xx}-\sigma_x\phi_x+\sigma_y\phi_y) \, .$$ Therefore $$\begin{aligned}
e^{4\sigma}|\Pi(V,V)|^2
&=|\phi_{xx}|^2+e^{2\sigma}\sigma_x^2+e^{2\sigma}\sigma_y^2
-2\sigma_x\langle\phi_{xx},\phi_x\rangle+2\sigma_y\langle\phi_{xx},\phi_y\rangle\\
&=|\phi_{xx}|^2+e^{2\sigma}(\sigma_x^2+\sigma_y^2)-2\sigma_x\langle
\phi_{xx},\phi_x\rangle-2\sigma_y\langle\phi_{yy},\phi_y\rangle\\
&=
|\phi_{xx}|^2+e^{2\sigma}(\sigma_x^2+\sigma_y^2)-\sigma_x
\frac{\partial}{\partial x}(e^{2\sigma})
-\sigma_y\frac{\partial}{\partial y}(e^{2\sigma})\\
&= |\phi_{xx}|^2 - e^{2\sigma}(\sigma_x^2+\sigma_y^2)\,.
\end{aligned}$$ More compactly, $$\label{eq:|II|^2}
e^{4\sigma}|I\!I(V,V)|^2=|\phi_{xx}|^2-e^{2\sigma}|\nabla\sigma|^2 \, .$$
On the other hand, using $\phi=(f_1,\ldots,f_n)$ and $\sigma =
(1/2\log\left(\,|f_1'|^2+\cdots+|f_n'|^2\right)$, together with $\phi_x=\phi'=(f_1',\ldots,f_n')$ and $\phi_{xx}=(f_1'',\ldots,f_n'')$, we obtain $$\label{eq:sigma_z}
2\sigma_{z}=\frac{\overline{f_1'}f_1''+\cdots\overline{f_n'}f_n''}
{|f_1'|^2+\cdots+|f_n'|^2} \, ,$$ and after some algebra, $$|\nabla\sigma|^2=|2\sigma_{z}|^2=e^{-4\sigma}\left|\sum_i
\overline{f_i'}f_i''\right|^2
\, .$$ This inserted in gives $$\label{eq:|II|^2-alternate}
e^{6\sigma}|\Pi(V,V)|^2=\sum_i\left|f_i''\right|^2\sum_j\left|f_j'\right|^2
-\sum_i\overline{f_i'}f_i''\sum_jf_j'\overline{f_j''}
=\sum_{i<j}\left|f_i'f_j''-f_j'f_i''\right|^2 \, . $$
Finally we compute $K(\phi(x))=-e^{-2\sigma(x)}\Delta\sigma(x)$. From it follows that $$\frac12\Delta\sigma=2\sigma_{z\bar{{z}}}=\frac{|f_1''|^2+\cdots+|f_n''|^2}
{|f_1'|^2+\cdots+|f_n'|^2}-
\frac{\left|\overline{f_1'}f_1''+\cdots\overline{f_n'}f_n''\right|^2}
{\left(\,|f_1'|^2+\cdots+|f_n'|^2\right)^2}\,,$$ and after some manipulation one obtains $$\label{eq:Laplacian-sigma}
\Delta\sigma=2e^{-4\sigma}\sum_{i<j}\left|f_i'f_j''-f_j'f_i''\right|^2
\, . $$ Comparing with we deduce that $$|\Pi(V,V)|^2=\frac12|K(\phi)| \, ,$$ as desired.
Ahlfors’ Schwarzian $S_1$ and the holomorphic Schwarzian ${\mathcal{S}}$ from appear together in the following relationship.
\[lemma:ahlfors-and-hol-schwarzian\] Let $\phi :\mathbb{D} \rightarrow \mathbb{C}^n$ be holomorphic with $\phi' \ne 0$. Let $\gamma(t)$ be a Euclidean arc-length parametrized curve in $\mathbb{D}$ with curvature $\kappa(t)$, and let ${\varphi}(t) = \phi(\gamma(t))$ be the corresponding parametrization of $\Gamma = \phi(\gamma)$ on $\Sigma=\phi(\mathbb{D})$. Let $V(t)$ be the Euclidean unit tangent vector field along ${\varphi}(t)$, given by $$V(t) = \frac{{\varphi}'(t)}{|{\varphi}'(t)|} = \frac{\phi'({\gamma}(t)){\gamma}'(t)}
{|\phi'({\gamma}(t))|}\,.$$ Then $$\label{eq:S1-curvature-general}
S_1{\varphi}(t) = \text{Re}\{{\mathcal{S}}\phi({\gamma}(t))({\gamma}'(t))^2\}
+\frac{3}{4}|\phi'({\gamma}(t))|^2 K({\varphi}(t)) +
\frac{1}{2}\kappa^2(t)\,.$$
This is the most general form of the relationship between $S_1$ and ${\mathcal{S}}$. Compare this formula to the one in Lemma 1 in [@cdo:harmonic-lift]. We will also need in the special case when ${\gamma}(t)$ is a diameter of $\mathbb{D}$, say from $-1$ to $1$. In this case $\kappa = 0$ and we can write the equation as $$\label{eq:S1-curvature-special}
S_1 \phi(x) = \text{Re} \{{\mathcal{S}}\phi(x)\} + \frac{3}{4}|\phi'(x)|^2|K(\phi(x))|\,.$$ Here, by $S_1 \phi(x)$ we mean $S_1$ applied to $\phi$ restricted to the interval $(-1,1)$.
Let $v(t) = |{\varphi}'(t)|$. The proof is based on a formula of Chuaqui and Gevirtz in [@chuaqui-gevirtz:S1], according to which $$S_1{\varphi}= \left(\frac{v'}{v}\right)'-\frac{1}{2}\left(\frac{v'}{v}\right)^2
+\frac{1}{2}v^2 k^2\,, \label{eq:S1-Ch-G}$$ where $k$ is the Euclidean curvature of the curve ${\varphi}(t)$ in $\mathbb{R}^{2n}$. We compute the terms on the right-hand side.
First, by definition, $$v(t) = |{\varphi}'(t)|= |\phi'({\gamma}(t))| = e^{\sigma({\gamma}(t))}\,,$$ and hence $$\frac{v'}{v} = \langle \nabla\sigma\,,\,{\gamma}'\rangle\,,\quad
\left(\frac{v'}{v}\right)' = {\operatorname{Hess}}\sigma({\gamma}',{\gamma}') +
\langle\nabla\sigma\,,\,{\gamma}''\rangle\,,$$ where $${\operatorname{Hess}}\sigma =
\begin{pmatrix}
\sigma_{xx} & \sigma_{xy}\\
\sigma_{xy} & \sigma_{yy}
\end{pmatrix}$$ is the Hessian matrix regarded as a bilinear form and ${\gamma}'$ is identified with unit tangent vector $(x'(t),y'(t))$. Moreover, with a similar identification, ${\gamma}'' = \kappa \mathbf{n}$, where $\mathbf{n}$ is the unit normal to ${\gamma}(t)$. Thus $$\left(\frac{v'}{v}\right)'-\frac{1}{2}\left(\frac{v'}{v}\right)^2
= {\operatorname{Hess}}\sigma({\gamma}',{\gamma}') + \kappa\langle\nabla\sigma\,,\,
\mathbf{n}\rangle -\frac{1}{2}\langle\nabla\sigma\,,\,{\gamma}'\rangle^2\,.
\label{eq:Ch-G-first-term}$$
Next we work with the curvature term $k^2v$. We can write $$k^2 = k_i^2 + |I\!I(V,V)|^2\,, \label{eq:k-and-ki}$$ where $k_i$ is the intrinsic (geodesic) curvature of ${\varphi}({\gamma}(t))$ on the surface ${\varphi}(\mathbb{D}) = \Sigma
\subset \mathbb{R}^{2n}$. Furthermore, ${\varphi}\colon(\mathbb{D},e^{2\sigma}\mathbf{g}_0) \rightarrow
(\Sigma,\mathbf{g}_0)$ is a local isometry between $\Sigma$ with the Euclidean metric, denoted here by $\mathbf{g}_0$, and $\mathbb{D}$ with the conformal metric $e^{2\sigma}\mathbf{g}_0$, and so $k_i = \hat{\kappa}$, the curvature of ${\gamma}(t)$ in the metric $e^{2\sigma}\mathbf{g}_0$. In turn, a classical formula in conformal geometry states that $$e^\sigma \hat{\kappa} = \kappa - \langle \nabla \sigma\,,\,
\mathbf{n}\rangle;$$ see, for example, Section 3 in [@os:sch]. Combining these formulas gives us $$v^2k^2 = k^2e^{2\sigma} = \kappa^2 - 2\kappa\langle\nabla \sigma\,,\,
\mathbf{n}\rangle + \langle\nabla\sigma\,,\,\mathbf{n}\rangle^2 +
e^{2\sigma}|I\!I(V,V)|^2\,. \label{eq:Ch-G-second-term}$$
For $S_1{\varphi}$ in we combine and and manipulate some terms to write $$\begin{aligned}
S_1{\varphi}&= {\operatorname{Hess}}(\sigma)({\gamma}',{\gamma}') -\frac{1}{2}\langle
\nabla\sigma\,,\,{\gamma}'\rangle^2+ \frac{1}{2}\langle\nabla\sigma\,,\,
\mathbf{n}\rangle^2 + \frac{1}{2}\kappa^2 + \frac{1}{2}e^{2\sigma}|I\!I(V,V)|^2\\
&= {\operatorname{Hess}}({\varphi})({\gamma}',{\gamma}')+ \frac{1}{2}|\nabla\sigma|^2 -
\langle\nabla\sigma\,,\,{\gamma}'\rangle^2 + \frac{1}{2}\kappa^2 +
\frac{1}{2}e^{2\sigma}|I\!I(V,V)|^2\\
&= {\operatorname{Hess}}(\sigma)({\gamma}',{\gamma}') -
\langle\nabla\sigma\,,\,{\gamma}'\rangle^2 -
\frac{1}{2}(\Delta\sigma - |\nabla\sigma|^2)
+ \frac{1}{2}\Delta \sigma + \frac{1}{2}\kappa^2
+ \frac{1}{2}e^{2\sigma}|I\!I(V,V)|^2
\end{aligned}$$ Next, one finds by straight calculation (see also Section \[section:conf-sch\]) that $${\operatorname{Hess}}(\sigma)({\gamma}',{\gamma}') -\langle\nabla\sigma\,,\,
{\gamma}'\rangle^2 -\frac{1}{2}(\Delta\sigma - |\nabla\sigma|^2) =
\text{Re}\{{\mathcal{S}}\phi({\gamma})({\gamma}')^2\}$$ while from , $$\frac{1}{2}\Delta \sigma = -\frac{1}{2}e^{2\sigma}K(\phi) =
\frac{1}{2}e^{2\sigma}|K(\phi)|\,.$$ Substituting these and $e^{2\sigma} = |\phi'|^2$ gives $$S_1{\varphi}(t) = \text{Re}\{{\mathcal{S}}\phi({\gamma}(t))({\gamma}'(t))^2\} +
\frac{1}{2}|\phi'({\gamma}(t))|^2(K({\varphi}(t)) +
|I\!I(V(t),V(t))|^2) + \frac{1}{2}\kappa^2(t)\,.$$ Now appealing to Lemma \[lemma:II-gaussian\] brings this into final form, $$S_1{\varphi}(t) = \text{Re}\{{\mathcal{S}}\phi({\gamma}(t))({\gamma}'(t))^2\} +
\frac{3}{4}|\phi'({\gamma}(t))|^2 K({\varphi}(t)) + \frac{1}{2}\kappa^2(t)\,.$$
Injectivity, Extremal Functions, and the Proofs of Theorems \[theorem:hol-p-criterion\] and \[theorem:extremal\] {#section:injectivity}
================================================================================================================
We recall the hypothesis of Theorem \[theorem:hol-p-criterion\], that $\phi$ satisfies , $$|{\mathcal{S}}\phi(z)| + \frac{3}{4}|\phi'(z)|^2|K(\phi(z))| \le 2p(|z|)\,,
\quad z \in \mathbb{D}\,.
\label{eq:hol-p-criterion-2}$$ The proof of injectivity rests on a result of Chuaqui and Gevirtz [@chuaqui-gevirtz:S1] giving a criterion for univalence on (real) curves in terms of $S_1$.
Let $p(x)$ be a continuous function such that the differential equation $u''(x)+p(x)u(x)=0$ admits no nontrivial solution $u(x)$ with more than one zero in $(-1,1)$. Let $\varphi : (-1,1)\rightarrow {\Bbb R}^m$ be a curve of class $C^3$ with tangent vector $\varphi'(x)\neq 0$. If $S_1\varphi(x)\leq2p(x)$, then $\varphi$ is univalent.
We pass immediately to
If $\phi$ satisfies then, from , along the diameter $(-1,1)$, $$S_1\phi(x) \le 2p(x)\,,$$ and so $\phi$ is injective there by Theorem A. The same holds for any rotation $\phi(e^{i\theta} z)$ of $\phi$ and hence $\phi$ is injective along any diameter of $\mathbb{D}$.
Suppose now that $z_1$ and $z_2$ are distinct points not on a diameter. Let $\gamma$ be the hyperbolic geodesic through $z_1$ and $z_2$. By a rotation of $\mathbb{D}$ we may assume that $\gamma$ meets the imaginary axis orthogonally at a point $i\rho$. The Möbius transformation $$T(z) = \frac{z-i\rho}{1+i\rho z}$$ maps $\mathbb{D}$ onto itself, preserves the imaginary axis, and carries $\gamma$ to the diameter $(-1,1)$. The function $$\psi(z) = \phi(T(z))$$ is a holomorphic reparametrization of $\Sigma = \phi(\mathbb{D})$ with $\psi'\ne 0$ and we claim that $$|{\mathcal{S}}\psi(x)| + \frac{3}{4}|\psi'(x)|^2|K(\psi(x))| \le 2p(x)\,,\quad -1 < x <1\,.$$ If so, then $S_1\psi(x) \le 2p(x)$ as above, whence $\psi$ is injective along $(-1,1)$ and $\phi(z_1) \ne \phi(z_2)$.
For this, first note that $$|\psi'(x)| =|\phi'(T(x))|\,|T'(x)|$$ while also $${\mathcal{S}}\psi(x) = {\mathcal{S}}\phi(T(x))T'(x)^2\,,$$ from . Next, by hypothesis, $$|{\mathcal{S}}\phi(T(x))|\,|T'(x)|^2 + \frac{3}{4}|\phi'(T(x))|^2|T'(x)|^2
|K(\phi(T(x)))|\le 2p(|T(x)|)|T'(x)|^2\,,$$ and so the claim will be established if we show $$p(|T(x)|T'(x)|^2) \le p(|x|)\,,\quad -1 <x <1\,. \label{eq:injectivity-punchline}$$ But now a simple calculation gives that $$|x| \le |T(x)| \label{eq:nehari-trick}$$ for this particular Möbius transformation, whence $$(1-|T(x)|^2)^2p(|T(x))| \le (1-x^2)^2p(|x|)$$ by the assumption that $(1-x^2)^2p(x)$ is non-increasing for $x\in[0,1)$. Furthermore $$\frac{|T'(x)|}{1-|T(x)|^2} = \frac{1}{1-x^2}$$ for any Möbius transformation of $\mathbb{D}$ onto itself. Thus $$|T'(x)|^2p(|T(x)|) = \frac{(1-|T(x)|^2)^2}{(1-x^2)^2}p(|T(x)|) \le p(|x|)\,,$$ finishing the proof.
The trick of passing from injectivity along diameters to the general case via this special Möbius transformation, using to obtain , goes back to Nehari. We used the same argument in [@cdo:harmonic-lift].
Theorem \[theorem:extremal\] states that if $\phi$ satisfies the univalence criterion and fails to be injective on $\partial \mathbb{D}$ (assuming continuous extension) then it must do so in a particular way, that the surface $\Sigma = \phi(\mathbb{D})$ contains a Euclidean circle minus the cut point where injectivity fails. Moreover, equality holds in along the preimage of the circle. These properties of such extremal functions depend upon another result of Chuaqui and Gevirtz in the same paper [@chuaqui-gevirtz:S1]. To state it we need one additional construction, which will be used again in later sections.
If the function $p(x)$ of Theorem A is even, as will be the case for a Nehari function, then the solution $u_0$ of the differential equation $u''+pu=0$ with initial conditions $u_0(0)=1$ and $u_0'(0)=0$ is also even, and therefore $u_0(x)\neq0$ on $(-1,1)$, since otherwise it would have at least two zeros. Thus the function $$\Phi(x) = \int_0^x u_0(t)^{-2}\,dt\,, \qquad -1 < x < 1\,,
\label{eq:Phi}$$ is well defined and has the properties $\Phi(0)=0$, $\Phi'(0)=1$, $\Phi''(0)=0$, $\Phi(-x)=-\Phi(x)$, and $\mathcal{S}\Phi=2p$. Furthermore, $S_1\Phi=\mathcal{S}\Phi$ since $\Phi$ is real-valued, and so $S_1\Phi=2p$ as well.
In terms of $\Phi$, the second Chuaqui-Gevirtz theorem is as follows.
Let $p(x)$ be an even function with the properties assumed in Theorem A, and let $\Phi$ be defined as above. Let $\varphi: (-1,1)\rightarrow {\Bbb R}^m$ satisfy $S_1\varphi(x)\leq2p(x)$ and have the normalization $\varphi(0)=0$, $|\varphi'(0)|=1$, and $\varphi''(0)=0$. Then $|\varphi'(x)|\leq\Phi'(|x|)$ for $x\in(-1,1)$, and $\varphi$ has an extension to the closed interval $[-1,1]$ that is continuous with respect to the spherical metric. Furthermore, there are two possibilities:
$(i)$ If $\Phi(1)<\infty$, then $\varphi$ is univalent in $[-1,1]$ and $\varphi([-1,1])$ has finite length.
$(ii)$ If $\Phi(1)=\infty$, then either $\varphi$ is univalent in $[-1,1]$ or $\varphi=R\circ\Phi$ for some rotation $R$ of ${\Bbb R}^m$.
We can now proceed with
Suppose that $\phi$ satisfies and fails to be injective on $\partial\mathbb{D}$, and let $\zeta_1,\zeta_2\in\partial\mathbb{D}$ be points for which $\phi(\zeta_1)=\phi(\zeta_2)=P$. We first show that we can form $\psi = \phi \circ T$ for a suitable Möbius transformation of $\mathbb{D}$ onto itself, with $\psi$ still satisfying and with $\psi(1)= \psi(-1) = P$. We refer to the calculations in the proof of Theorem \[theorem:hol-p-criterion\], and we distinguish two cases.
Suppose first that $(1-x^2)^2p(x)$ is constant. Then the condition is fully invariant under the Möbius transformations of $\mathbb{D}$, and for a Möbius modification $\psi = \phi \circ T$ with $T(1)=\zeta_1$, $T(-1)=\zeta_2$ we obtain $\psi(1) = \psi(-1)=P$.
Now suppose that $(1-x^2)^2p(x)$ is not constant. Suppose also, by way of contradiction, that $\zeta_1$ and $\zeta_2$ are not on a diameter. Since is, in any case, invariant under rotations of $\mathbb{D}$, we may assume that $\zeta_1$ and $\zeta_2$ are both in the upper half plane and are symmetric in the imaginary axis. Just as in the proof of Theorem \[theorem:hol-p-criterion\], let $T(z) = ({z-i\rho})/({1+i\rho z})$ with $T(-1)=\zeta_1$, $T(1)=\zeta_2$, and let $\psi=\phi\circ T$. Then holds for $\psi$, together with $\psi(1)=\psi(-1)=P$ and $S_1\psi(x)\leq 2p(x)$. Moreover we must have a strict inequality $S_1\psi(x)<2p(x)$ on some interval in $(-1,1)$ because $(1-x^2)^2p(x)$ is not constant. However, this last statement stands in contradiction with Theorem B. To be precise, there is a Möbius transformation $M$ of $\Bbb{R}^{2n}$ such that $M\circ \psi$ satisfies the hypotheses of Theorem B with $(M\circ \psi)(-1) = (M\circ \psi)(1)$, and because $\psi$ is not injective on $[-1,1]$ we have, using , $S_1(M\circ \psi) = S_1 \psi = 2p$. The contradiction shows that $\zeta_1$ and $\zeta_2$ must lie on a diameter.
In all cases, by a suitable modification we can now assume that the injectivity of $\phi$ on $\partial\mathbb{D}$ fails by $\phi$ mapping the interval $[-1,1]$ to a closed curve on $\phi(\overline{\mathbb{D}})$ with $\phi(-1)=\phi(1)=P$, and by a post composition with a Möbius transformation of $\mathbb{R}^{2n}$ we may further assume that $\phi$ is normalized as in Theorem B. Then again by Theorem B, $\phi = V \circ \Phi$ for some Möbius transformation of $\mathbb{R}^{2n}$. Hence $S_1 \phi = S_1\Phi = 2p$ and $\phi$ maps $[-1,1]$ to a Euclidean circle or a line, since $\Phi$ maps $[-1,1]]$ onto $\mathbb{R}\cup\{\infty\}$ and $V$ preserves circles. Finally, since $S_1\phi(x) = 2p(x)$ and $\phi$ satisfies , it follows from that $$|{\mathcal{S}}\phi(x)| + \frac{3}{4}|\phi'(x)|^2|K(\phi(x))| = 2p(x),\quad x \in [-1,1]\,.$$ This concludes the proof.
A Covering Theorem {#section:covering}
==================
We continue to assume that $\phi$ satisfies the injectivity criterion . In this section we will derive a lower bound for the radius of a metric disk centered at $\phi(0)$ on $\Sigma$. We will assume an additional normalization of $|\phi'(0)|=1$, and then the lower bound will depend on $|\phi''(0)|$ and on a second extremal function associated with the Nehari function $p$. We also need to assume that $p$ is nondecreasing on $[0,1)$, which is the case for many examples. The result we obtain is very much in line with those from classical geometric function theory, see, for example, [@essen-keogh:schwarzian].
Let $U$ be the solution of $$U''-pU=0\,,\qquad U(0)=1\,,\,\, U'(0)=0\,$$ (note the minus sign in the differential equation) and define $$\Psi(x) = \int_0^x U(t)^{-2}\,dt\,.$$
\[theorem:covering\] Let $\phi$ be a holomorphic curve satisfying with $|\phi'(0)|=1$, and suppose that $p(x)$ is nondecreasing on $[0,1)$. Then $$\min_{|z|=r}d_{\Sigma}(\phi(z), \phi(0)) \geq\,
\frac{2\Psi(r)}{2+|\phi''(0)|\Psi(r)} \,, \label{eq:min-d}$$ where $d_{\Sigma}$ denotes distance on $\Sigma= \phi(\mathbb{D})$. In particular, $\Sigma$ contains a metric disk of radius $$\frac{2\Psi(1)}{2+|\phi''(0)|\Psi(1)} $$ centered at $\phi(0)$.
The proof relies ultimately on comparing solutions of two differential equations, one involving the extremal $\Psi$, where the relevant inequalities come to us by way of the formulas from Lemma \[lemma:ahlfors-and-hol-schwarzian\] and its proof. This requires some preparation.
Let $z_r$ be a point on $|z|=r$ for which the minimum on the left hand side of is attained. Since $\phi$ is injective the minimum is positive and $z_r \ne 0$. The geodesic, $\Gamma$, on $\Sigma$ that joins $\phi(0)$ with $\phi(z_r)$ is contained in $\phi(\{|z|\leq r\})$ and we let $\gamma=\phi^{-1}(\Gamma)$. Write $\gamma(t)$ for the parametrization of $\gamma$ by Euclidean arc-length, and let $\varphi(t)=\phi(\gamma(t))$ be the corresponding parametrization of $\Gamma$. Further, let $$v(t) = |\phi'(\gamma(t))|\,.$$ Now compare the expressions for $S_1\varphi(t)$ in and , using relating the Euclidean and intrinsic curvatures of $\Gamma$ and the second fundamental form of $\Sigma$: $$\begin{split}
(v'/v)'-&\frac12(v'/)^2+\frac12\,v^2
\left(\, k_i^2+|I\!I(V,V)|^2\right) =\\
&{\rm Re}\{S\phi(\gamma)(\gamma')^2\}+\frac12\,v^2\left(\,
|K(\phi(\gamma))|+|I\!I(V,V)|^2\right)+\frac12\kappa^2 \, ,
\end{split}$$ where $\kappa$ is the Euclidean curvature of $\gamma$ and $V$ is the Euclidean unit tangent vector field along $\Gamma$; all expressions are to be evaluated at $t$. Since $\Gamma$ is a geodesic $k_i=0$ and this becomes $$(v'/v)'-\frac{1}{2}(v'/v)^2 = {\rm
Re}\{{\mathcal{S}}\phi(\gamma)(\gamma')^2\}+
\frac{1}{2}v^2|K(\phi(\gamma))|+\frac{1}{2}\kappa^2\, .$$ But the univalence criterion implies that $${\rm Re}\{S\phi(\gamma)(\gamma')^2\} \geq -|S\phi(\gamma)| \geq
-2p(\gamma)+\frac34v^2|K(\phi(\gamma))| \, ,$$ hence $$\begin{aligned}
(v'/v)'-\frac12(v'/v)^2 &\geq
-2p(|\gamma|)+\frac54v(t)^2|K(\phi(\gamma))|+\frac12\kappa^2\\
&\geq
-2p(|\gamma|) \, . \label{eq:covering-comparison}
\end{aligned}$$ Now let $$h(t)=\int_0^tv(\tau)\,d\tau \, .$$ Then ${\mathcal{S}}h = (v'/v)'-\frac{1}{2}(v'/v)^2$, and from the preceding estimate, $${\mathcal{S}}h(t) \geq -2p(|\gamma(t)|) \geq
-2p(t)\, , $$ the final inequality holding because $|\gamma(t)|\leq t$ and we have assumed that $p(x)$ is nondecreasing on $[0,1)$.
The result ${\mathcal{S}}h(t) \ge -2p(t)$ is the main inequality we need in order to apply the Sturm comparison theorem. For this, first note that the function $w=v^{-1/2}$ is the solution of $$w''+\displaystyle{\frac12}(Sh)w=0\,,\quad w(0)=1\,,\,w'(0) =
-\frac{1}{2} v'(0)\,.$$ Next, consider also the solution $y(t)$ of $$y''-py=0\,,\quad y(0)=1\,,y'(0) = \frac{1}{2}|\phi''(0)| \, .$$ Since $
-\frac{1}{2}{\mathcal{S}}h(t) \le p(t)
$, and also $$w'(0)=-\displaystyle{\frac12} v'(0) \leq
\displaystyle{\frac12}|\phi''(0)|\,,$$ it follows by comparison that $$w(t) \le y(t)\,.$$ Finally, observe that, explicitly, $ y= (H')^{-1/2}$, where $$H=\frac{2\Psi}{2+|\phi''(0)|\Psi} \, .$$ Consequently, $$h(t)=\int_0^tw^{-2}(\tau)\,d\tau \geq \int_0^ty^{-2}(\tau)\,d\tau =
\int_0^tH'(\tau)\,d\tau = H(t) \, ,$$ and hence $$d_{\Sigma}(\phi(z_r),\phi(0))= \int_{\gamma}v \geq \int_0^r
v(t)\,dt = h(r) \geq H(r) \, .$$ This completes the proof.
Conformal Schwarzians, Extremal Nehari Functions, and Convexity {#section:conf-sch}
===============================================================
To advance further in the analysis of mappings satisfying , in particular to study continuous extension to the boundary, we need estimates based on convexity. This requires a notion of the Schwarzian for conformal metrics and an associated differential equation. This section generally follows the treatment of these ideas in [@cdo:harmonic-lift], abbreviated somewhat and modified to serve the case of holomorphic maps rather than lifts of harmonic maps. We refer to that paper and also to [@os:sch] for (many) more details.
Let $\mathbf{g}$ be a Riemannian metric on the disk $\Bbb D$. We may assume that $\mathbf{g}$ is conformal to the Euclidean metric, $\mathbf{g}_0=dx\otimes dx +dy \otimes dy= |dz|^2$. Let $\sigma$ be a smooth function on $\Bbb D$ and form the symmetric 2-tensor $${\operatorname{Hess}}_{\mathbf{g}}(\sigma) - d\sigma \otimes d\sigma.
\label{eq:Hessian}$$ Here ${\operatorname{Hess}}$ denotes the Hessian operator. If $\gamma(s)$ is an arc-length parametrized geodesic for $\mathbf{g}$, then $${\operatorname{Hess}}_{\mathbf{g}}(\sigma)(\gamma',\gamma') =
\frac{d^2}{ds^2}(\sigma\circ \gamma)\,.$$ The Hessian depends on the metric, and since we will be changing metrics we indicate this dependence by the subscript $\mathbf{g}$.
We now form $$B_\mathbf{g}(\sigma)={{\operatorname{Hess}}}_\mathbf{g}(\sigma)-d\sigma\otimes
d\sigma-\frac{1}{2}(\Delta_\mathbf{g} \sigma-||{\operatorname{grad}}_\mathbf{g}
\sigma||^2)\mathbf{g}\,.
\label{eq:B-sigma}$$ The final term has been subtracted to make the trace zero.
This is the Schwarzian tensor of $\sigma$. Before explaining its connection to conformal maps, metrics and the Schwarzian derivative, first note that in standard Cartesian coordinates one can represent $B_{\mathbf{g}}(\sigma)$ as a symmetric, traceless $2\times 2$ matrix, say of the form $$\begin{pmatrix}
a & -b \\
-b & -a
\end{pmatrix}\,.$$ Further identifying such a matrix with the complex number $a+bi$ then allows us to associate the tensor $B_{\mathbf{g}}(\sigma)$ with $a+bi$, and then $$||B_\mathbf{g_0}(\sigma)(z)||_\mathbf{g_0} = |a+bi|\,.$$
A locally injective holomorphic curve $\phi:\mathbb{D}\rightarrow\mathbb{C}^n$ is a conformal mapping of $\mathbb{D}$ with the Euclidean metric into $\mathbb{R}^{2n}$ with the Euclidean metric and if, as before, we write $\phi =(f_1,\dots,f_n)$ then the conformal factor is $$\phi^*(\mathbf{g_0}) = e^{2\sigma} \mathbf{g_0}\,,\quad \sigma =
\frac{1}{2}\log(|f_1'|^2+\cdots |f_n'|^2)\,.$$ The Schwarzian derivative of $\phi$ is defined to be $$\mathcal{S}_\mathbf{g} \phi = B_\mathbf{g}(\sigma)\,.$$ When $n=1$ and $\sigma = \log|\phi'|$ we find that $$B_{\mathbf{g}_0}(\log|\phi'|)=\left( \begin{array}{rr} {\rm Re}\,
\mathcal{S}\phi & -{\rm Im}\,\mathcal{S}\phi \\ -{\rm Im}\,\mathcal{S}\phi
& -{\rm Re}\,\mathcal{S}\phi
\end{array}\right) \, ,$$ writing the tensor in matrix form as above, where $\mathcal{S}\phi$ is the classical Schwarzian derivative of $\phi$. When $n \ge 1$, identifying the tensor with a complex number leads to $${\mathcal{S}}\phi = 2(\sigma_{zz}-\sigma_z^2)\,,$$ the definition that we gave in Section \[section:intro\].
The tensor $B_\mathbf{g} \sigma$ changes in a simple way if there is a conformal change in the background metric $\mathbf{g}$. Specifically, if $\widehat{\mathbf{g}} = e^{2\rho}\mathbf{g}$ then $$B_\mathbf{g}(\rho+\sigma) = B_\mathbf{g}(\rho)+B_{\widehat{\mathbf{g}}}(\sigma).$$ This is actually a generalization of the chain rule for the classical Schwarzian. An equivalent formulation is $$B_{\widehat{\mathbf{g}}}(\sigma -\rho) = B_\mathbf{g}(\sigma) -
B_\mathbf{g}(\rho)\,,
\label{eq:subtraction-formula}$$ which is what we will need in later calculations.
Next, just as the linear differential equation $w''+(1/2)p w=0$ is associated with $Sf=p$, there is also a linear differential equation associated with the Schwarzian tensor. If $$B_\mathbf{g}(\sigma) = p\,,$$ where $p$ is a symmetric, traceless 2-tensor, then $\eta=e^{-\sigma}$ satisfies $${\operatorname{Hess}}_\mathbf{g}(\eta) + \eta p = \frac{1}{2}(\Delta_\mathbf{g}\eta)\mathbf{g}\,.
\label{eq:Hess-eq}$$
We now turn to convexity. In this setting, a function $\eta$ is convex relative to the metric $\mathbf{g}$ if $${\operatorname{Hess}}_\mathbf{g} \eta \ge \alpha\mathbf{g}\,,$$ where $\alpha$ is a nonnegative function. This is equivalent to $$\frac{d^2}{ds^2}(\eta \circ \gamma) \ge \alpha \ge 0$$ for any arc-length parametrized geodesic $\gamma$.
Convexity is an important notion for us because we will find that an upper bound for $\mathcal{S}_\mathbf{g}\phi$ coming from the injectivity criterion leads via and to just such a positive lower bound for the Hessian of an associated function, and this is what we need to study boundary behavior. This fact obtains, however, not relative to the Euclidean metric but when the background metric $\mathbf{g}$ is a complete, radial metric coming from an *extremal* Nehari function. We explain this now.
It follows from the Sturm comparison theorem that if $p$ is a Nehari function then so is a multiple $kp$ for any $k$ with $0<k<1$. This need not be so if $k >1$ and we say that $p$ is an *extremal Nehari function* if $kp$ is *not* a Nehari function for *any* $k>1$. For example, $p(x) = 1/(1-x^2)^2$ and $p(x)=\pi^2/4$ are both extremal Nehari functions. In [@co:noncomplete] it was shown that some constant multiple of each Nehari function is an extremal Nehari function. Observe that since a holomorphic curve $\phi$ satisfying a condition of the type $|{\mathcal{S}}\phi| +\cdots \le 2p$, as in , also then satisfies $|{\mathcal{S}}\phi| +\cdots \le 2kp$ for any $k>1$ we may always assume when is in force that $p$ is an extremal Nehari function.
There is another way to describe this situation in terms of the extremal *function* associated with a given $p$. Recall the definition from , $$\Phi(x) = \int_0^x u_0(t)^{-2}\,dt\,, \qquad -1 < x < 1\,,$$ where $u_0$ is the solution of $u''+pu=0$ with initial conditions $u_0(0)=1$ and $u_0'(0)=0$. We use $\Phi$ to form the radial conformal metric $$\mathbf{g}_\Phi = \Phi'(|z|)^2|dz|^2 \label{eq:Phi-metric}$$ on $\mathbb{D}$. It is implicit in [@co:noncomplete], without the terminology, that the following conditions are equivalent:
1. $p$ is an extremal Nehari function.
2. $\Phi(1) = \infty$.
3. The metric $\Phi'(|z|)^2|dz|^2$ is complete.
We recall that for a complete metric any two points can be joined by a geodesic and that a geodesic can be extended indefinitely. We let $d_{\mathbf{g}_\Phi}$ be the distance in the ${\mathbf{g}_{\Phi}}$ metric and note that since ${\mathbf{g}_{\Phi}}$ is radial $$d_{\mathbf{g}_\Phi}(0,z) = \Phi(|z|)\,.$$
The curvature of a radial metric of the form , complete or not, can be expressed as $$K_{\mathbf{g}_{\Phi}}(z) = -2 \Phi'(|z|)^{-2}(A(|z|) + p(|z|))\,,\quad r=|z|\,,
\label{eq:K-and-A}$$ where $$A(r) = \frac{1}{4}\left(\frac{\Phi''(r)}{\Phi'(r)}\right)^2+\frac{1}{2r}
\frac{\Phi''(r)}{\Phi'(r)}\,, \quad r\ge 0\,.
\label{eq:A}$$ >From the properties of $\Phi$ it follows that $A(r)$ is continuous at $0$ with $A(0)= p(0)$ and that the curvature is negative. Thus, as we will need, $$|K_{\mathbf{g}_{\Phi}}(z)| = 2 \Phi'(|z|)^{-2}(A(|z|) + p(|z|)) \,
.\label{eq:|K|}$$ If the metric is complete, or equivalently comes from an extremal Nehari function, then as was shown in [@co:noncomplete] $$p(r) \le A(r)\,. \label{eq:p-and-A}$$
All of these comments go into the proof of the following theorem.
\[theorem:convexity\] Let $\phi$ satisfy the injectivity criterion for an extremal Nehari function $p$. Then $$w(z)=\sqrt{\frac{\Phi'(|z|)}{|\phi'(z)|}} \label{eq:associated-fnc}$$ satisfies $${\rm Hess}_{\mathbf{g}_{\Phi}}(w) \geq \,
\frac18\,w^{-3}|K_{\mathbf{g}_{\Phi}}|{\mathbf{g}_{\Phi}} \, , \label{eq:hess-w}$$ In particular, $w$ is a convex function relative to the metric ${\mathbf{g}_{\Phi}}$.
Compare this result to the corresponding result, Theorem 4 in [@cdo:harmonic-lift], for lifts of harmonic mappings (in [@cdo:harmonic-lift] we did not use the term “extremal Nehari function") where the constant in the inequality bounding the Hessian from below is $1/4$ rather than $1/8$.
As in [@cdo:harmonic-lift] we formulate a separate lemma.
\[lemma:convexity\] Let $\phi$ satisfy the injectivity criterion for an extremal Nehari function $p$ and let $\rho=\log|\Phi'|$. Then $$\|B_{\mathbf{g}_{\Phi}}(\sigma - \rho)\|_{\mathbf{g}_{\Phi}}
+\frac{3}{4}e^{2(\sigma - \rho)}|K(\phi)| \le
\frac{1}{2}|K_{\mathbf{g}_{\Phi}}|\,. \label{eq:B(sigma-rho)}$$ Here recall that $K(\phi(z))$ is the Gaussian curvature of the surface $\Sigma = \phi(\mathbb{D})$ at $\phi(z)$.
The proof is very much like the proof of Lemma 2 in [@cdo:harmonic-lift], but to show how the assumptions enter we will present the argument.
First note from that, in terms of $\rho$, the absolute value of the curvature of ${\mathbf{g}_{\Phi}}=
\phi'(|z|)^2|dz|^2=e^{2\rho(z)}|dz|^2$ is $$|K_{\mathbf{g}_{\Phi}}(z)| = e^{-2\rho(z)}(A(|z|) + p(|z|)) \,
.\label{eq:|K|-2}$$ Also, from we have $$B_{\mathbf{g}_{\Phi}}(\sigma-\rho) = B_{\mathbf{g}_0}(\sigma)-
B_{\mathbf{g}_0}(\rho) \, .$$ Since ${\mathbf{g}_{\Phi}} = e^{2\rho}\mathbf{g}_0$, the norm scales to give $$\|B_g(\sigma-\rho)\|_{\mathbf{g}_\Phi}=
e^{-2\rho}\|B_{\mathbf{g}_0}(\sigma)-B_{\mathbf{g}_0}
(\rho)\|_{\mathbf{g}_0}=e^{-2\rho}|B_{\mathbf{g}_0}
(\sigma)-B_{\mathbf{g}_0}(\rho)| \, ,$$ where in the last equation we have identified the Euclidean norm of the tensor with the magnitude of the corresponding complex number.
Next, a calculation (see also [@co:noncomplete]) produces $$B_{\mathbf{g}_0}(\sigma)-B_{\mathbf{g}_0}(\rho)=
\zeta^2S\phi(z)+A(|z|)-p(|z|) \; ,
\quad \zeta=\frac{z}{|z|} \,.$$ In light of these statements, establishing is equivalent to $$\left|\zeta^2S\phi(z)+A(|z|)-p(|z|)\right|+\frac34\,e^{2\sigma(z)}
|K(\phi(z))| \leq A(|z|)+p(|z|) | \,.$$ This in turn follows from the assumption that $\phi$ satisfies the injectivity criterion and, crucially, from the inequality : $$\begin{aligned}
\left|\zeta^2S\phi(z)+A(|z|)-p(|z|)\right|+\frac34\,e^{2\sigma}|K|
&\leq |\zeta^2S\phi(z)|+|A(|z|)-p(|z|)|+\frac34\,e^{2\sigma}|K|\\
&= |S\phi(z)|+\frac34\,e^{2\sigma}|K|+A(|z|)-p(|z|)\\
& \leq A(|z|)+p(|z|) | \, .
\end{aligned}$$
The deduction of Theorem \[theorem:convexity\] from Lemma \[lemma:convexity\], relies on . Write $w=e^{(\rho-\sigma)/2}$, $v = w^2 = e^{\rho-\sigma}$, and then, according to , $${\operatorname{Hess}}_{\mathbf{g}_{\Phi}} v +vB_{\mathbf{g}_{\Phi}}(\sigma-\rho)=
\frac{1}{2}(\Delta_{\mathbf{g}_{\Phi}} v)\mathbf{g}\,.$$ With this, the proof is almost word-for-word the same as the corresponding proof in [@cdo:harmonic-lift], and we omit the details. Instead, let us present one consequence of Theorem \[theorem:convexity\] here, with more to come in the next section.
\[lemma:unique-critical-point\] Under the assumptions of Theorem \[theorem:convexity\], if $w$ has at least two critical points then the range of $\phi$ lies in a plane.
Suppose $z_1$ and $z_2$ are critical points of $w$. Then, because $w$ is convex, $w(z_1)$ and $w(z_2)$ are absolute minima, and so is every point on the geodesic segment $\gamma$ (for the metric $\mathbf{g}_\Phi$) joining $z_1$ and $z_2$ in $\mathbb{D}$. Hence ${\rm Hess}_g(w)(\gamma', \gamma')=0$, which from implies that $|K|\equiv 0$ along $\Gamma=\phi(\gamma)$. From in Section \[section:S1\] we have that $$|K| = 2e^{-6\sigma}\sum_{i<j}\left|f_i'f_j''-f_j'f_i''\right|^2\, ,$$ hence $f_i'f_j''-f_j'f_i''=0$ along $\gamma$, for all $i<j$. By analytic continuation, $f_i'f_j''-f_j'f_i''=0$ everywhere, and using that $\phi'\neq 0$, it follows that for some $i$ and all $j$ there are constants $a_j, b_j$ such that $f_j=a_jf_i+b_j$. This proves the lemma.
A corresponding result for harmonic maps is Lemma 3 in [@cdo:harmonic-lift].
Boundary Behavior and the Proof of Theorem \[theorem:boundary-extension-1\] {#section:boundary}
===========================================================================
We now study the boundary behavior for functions $\phi$ satisfying . As in the results just above we suppose that $p$ is an extremal Nehari function and we set $$w(z)=\sqrt{\frac{\Phi'(|z|)}{|\phi'(z)|}}$$ where $\Phi$ is the extremal function associated with $p$. We just saw that if $w$ has at least two critical points then the range of $\phi$ lies in a plane, and so the setting is effectively that of an analytic function satisfying Nehari’s criterion . The boundary behavior in this case has been thoroughly studied; see [@gehring:gehring-pommerenke], [@co:gp] and also the summary of the classical results in [@cdo:harmonic-lift].
We next consider the situation when $w$ has a unique critical point, and here the basic estimate is as follows.
\[lemma:distortion\] If $w$ has a unique critical point then there are positive constants $a$ and $b$ and a number $r_0$, $0<r_0<1$, such that $$|\phi'(z)| \le \frac{\Phi'(|z|)}{(a\Phi(|z|)+b)^2}\,,\quad r_0<|z|<1\,.
\label{eq:distortion}$$
Let $z_0$ be the unique critical point of $w$. Let $\gamma(s)$ be an arc-length parametrized geodesic in the metric $\mathbf{g}$ starting at $z_0$ in a given direction. Let $\widetilde{w}(s)=w(\gamma(s))$. Now the critical point is unique, and therefore $\widetilde{w}'(s)>0$ for all $s>0$. Thus there is an $s_0>0$ and an $a>0$ such that $\widetilde{w}'(s) > a$ for all $s>s_0$. This implies that $\widetilde{w}(s) > as+b$ for some positive constant $b$ and $s>s_0$. It is easy to see from compactness that the constants $s_0$, $a$, and $b$ in this estimate can be made uniform, independent of the direction of the geodesic starting at $z_0$. In other words, $$w(z) \geq ad_{\mathbf{g}_\Phi}(z,z_0)+b$$ for all $z$ with $d_{\mathbf{g}_\Phi}(z,z_0)>s_0$. By renaming the constant $b$ and for suitable $r_0$ we will then have $$w(z) \geq ad_{\mathbf{g}_\Phi}(z,0)+b$$ for all $z$ with $1>|z| > r_0$. The theorem follows from the definition of $w$ because $d_{\mathbf{g}_\Phi}(z,0)=\Phi(|z|)$.
The estimate in this lemma allows one to deduce that $\phi$ has a continuous extension to $\overline{\mathbb{D}}$, and the argument is just as in [@cdo:harmonic-lift]. We will give only a few details here, enough for a more precise accounting of the regularity of the extension (also as in [@cdo:harmonic-lift]).
Since the function $(1-x^2)^2p(x)$ is positive and decreasing on $[0,1)$, we can form $$\lambda= \lim_{x\rightarrow 1}(1-x^2)^2p(x)\,.$$ It was shown in [@co:noncomplete] that $\lambda\leq 1$, and that $\lambda=1$ if and only if $p(x)=(1-x^2)^{-2}$. In this case, the function $\Phi$ is given by $$\Phi(z)=\frac12 \log\frac{1+z}{1-z}\,.$$ Thus amounts to $$|\phi'(z)| \leq
\frac{1}{(1-|z|^2)\left(\displaystyle{\frac{a}{2}\log\frac{1+|z|}{1-|z|}}+b\right)^2}
\; , \quad |z|> r_0 \, .$$ >From this, the technique of integrating along hyperbolic segments in $\mathbb{D}$, see also [@gehring:gehring-pommerenke], leads to $$|\phi(z_1)-\phi(z_2)| \leq \, C\left(\log\frac{1}{|z_1-z_2|}\right)^{-1} \; ,
\label{eq:cont-ext-1}$$ for some constant $C$ and points $z_1$, $z_2$ for which the hyperbolic geodesic segment joining them is contained in the annulus $r_0<|z|<1$. This implies that $\phi$ is uniformly continuous in the closed disk, and its continuous extension also satisfies . Thus when $\lambda =1$ the extension has a logarithmic modulus of continuity.
Suppose now that $\lambda<1$. We appeal to a result from [@co:noncomplete], according to which $$\lim_{x\rightarrow 1}(1-x^2)\frac{\Phi''}{\Phi'}(x) =
2(1+\sqrt{1-\lambda})=2\mu \, .$$ Note that $1<\mu\leq 2$. It follows that for any $\epsilon>0$ there exists $0<x_0<1$ such that $$\frac{\mu-\epsilon}{1-x} \leq \frac{\Phi''}{\Phi'}(x) \leq
\frac{\mu+\epsilon}{1-x}\; , \quad x> x_0 \,,$$ which implies that $$\frac{1}{(1-x)^{\mu-\epsilon}} \leq \Phi'(x) \leq
\frac{1}{(1-x)^{\mu+\epsilon}}\; , \quad x> x_0 \, ,$$ Then $$\frac{\Phi'(x)}{(a\Phi(x)+b)^2} \leq \frac{C}{(1-x)^{\alpha+3\epsilon}} \, ,$$ where $\alpha=2-\mu=1-\sqrt{1-\lambda}$ and $C$ depends on $a$, $b$ and the values of $\Phi$ at $x_0$. This estimate, together with the technique of integration along hyperbolic segments, implies that $$|\phi(z_1)-\phi(z_2)| \leq C|z_1-z_2|^{1-\alpha-3\epsilon}
= C|z_1-z_2|^{\sqrt{1-\lambda}-3\epsilon} \, ,$$ for all points $z_1$, $z_2$ for which the hyperbolic geodesic segment joining them is contained in the annulus $\max\{r_0,x_0\}<|z|<1$. This shows that $\phi$ admits a continuous extension to the closed disk, with at least a Hölder modulus of continuity.
We also point out that if one has the additional information that $x=1$ is a regular singular point of the differential equation $u''+pu=0$, then from an analysis of the Frobenius solutions at $x=1$ one can deduce that $$\Phi'(x) \sim \frac{1}{(1-x)^{\mu}} \; , \: x\rightarrow 1 \, .$$ This then provides exact Hölder continuous extension when $\lambda >0$ and a Lipschitz continuous extension when $\lambda=0$.
All of this discussion has been under the assumption that $w$ has a unique critical point. The argument in the case where $w$ has no critical points, though still based on convexity, requires additional work. This, too, is very close to what was done in [@cdo:harmonic-lift], so we only sketch the key points.
As will be explained momentarily, it is necessary to consider shifts $T\circ\phi$ of the holomorphic curve $\phi$ by Möbius transformations $T$ of $\mathbb{R}^{2n}$. The composition $T\circ\phi$ will not, in general, be holomorphic, though it is still conformal as a mapping of $\mathbb{D}$ into $\mathbb{R}^{2n}$. Write the corresponding conformal metric on the disk as $e^{2\tau}\mathbf{g}_0$ and, restricting $\tau$ to the radial segment $re^{i\theta}$, $0 \le r <1$, let $$\Upsilon_\theta(r) = e^{\tau(re^{i\theta})}$$ Also, let $s=\Phi(r)$ be the arc-length parameter of $[0,1)$ in the metric $\mathbf{g}_\Phi$, so that $r=\Phi^{-1}(s)$. Replacing Theorem \[theorem:convexity\], along radial segments, we find that the function $$\omega_\theta(s) = \left\{\frac{\Phi'(\Phi^{-1}(s))}
{\Upsilon_\theta(\Phi^{-1}(s))}\right\}^{1/2}$$ is convex, meaning in this case simply that $\omega_\theta''(s) \ge 0$.
This is Lemma 5 in [@cdo:harmonic-lift], in a slightly different notation, and we will not give the (identical) proof. The reason, however, why one can compose with a Möbius transformation of the range and still get a convexity result on radial segments is that $S_1(T\circ \phi) = S_1\phi$ on radial segments, from . Then via Lemma \[lemma:ahlfors-and-hol-schwarzian\], bounds on ${\mathcal{S}}\phi$ entail bounds on $S_1(T\circ \phi)$ and the convexity of $\omega(s)$ can be deduced from such bounds for $S_1$.
Now one shows, as in [@cdo:harmonic-lift], that for any $\theta_0$ it is possible to choose a Möbius transformation $T$ so that $\omega_{\theta_0}'(0) >0$. Therefore by convexity $\omega_{\theta_0}(s)
\ge as+b$, $a$, $b >0$, and then $$\Upsilon_{\theta_0}(r) \le \frac{\Phi'(r)}{(a\Phi(r)+b)^2}\,,$$ which provides a substitute for . By continuity this estimate holds for $\theta$ near $\theta_0$, so in a small angular sector about the radius $re^{i\theta_0}$. This in turn implies that $T\circ \phi$ has a continuous extension to the part of $\overline{\mathbb{D}}$ in the sector. Since $\theta_0$ was arbitrary and since we allow for Möbius transformations of the range, we obtain an extension of $\phi$ to $\overline{\mathbb{D}}$ that is continuous in the spherical metric.
Examples {#section:example}
========
In this section we present some examples to show that the injectivity criterion is sharp. In the setting of lifts of harmonic maps the corresponding examples were provided by mappings into a catenoid in $\mathbb{R}^3$. Surprisingly, the formulas are similar here, though some of the analytical details are different.
[**Example 1.**]{} Let $p(x)=\pi^2/4$. Then the criterion becomes $$|S\phi|+\frac34|\phi'|^2|K|\leq \frac{\pi^2}{2} \,
.\label{eq:example-1}$$ Define $\phi:\mathbb{D}\rightarrow\mathbb{C}^2$ to be $$\phi(z)=( c\, e^{\pi z},\, e^{-\pi z} ) \, ,$$ where the constant $c$ is to be chosen later. Then $$e^{2\sigma} = \pi^2( c^2\,e^{2\pi x} + e^{-2\pi x}) \, .$$ Straightforward calculations produce $$\sigma_x = \pi\,\frac{c^2e^{4\pi x}-1}{c^2e^{4\pi x}+1} \, ,$$ and $$\sigma_{xx}= \frac{8\pi^2c^2e^{4\pi x}}{(c^2e^{4\pi x}+1)^2} \, .$$ Hence for the Schwarzian, $${\mathcal{S}}\phi = 2(\sigma_{zz}-\sigma^2_z) =
\frac12(\sigma_{xx}-\sigma_x^2) = \frac{4\pi^2c^2e^{4\pi
x}}{(c^2e^{4\pi x}+1)^2}-\frac{\pi^2}{2} \left( \frac{c^2e^{4\pi
x}-1}{c^2e^{4\pi x}+1}\right)^2 \, .$$
If $c$ is chosen large enough, then $$|{\mathcal{S}}\phi| = \frac{\pi^2}{2} \left( \frac{c^2e^{4\pi
x}-1}{c^2e^{4\pi x}+1}\right)^2-\frac{4\pi^2c^2e^{4\pi
x}}{(c^2e^{4\pi x}+1)^2} \, .$$ Thus $$|S\phi|+\frac34|\phi'|^2|K| = |S\phi|+\frac34\,\sigma_{xx} =
\frac{\pi^2}{2} \left( \frac{c^2e^{4\pi x}-1}{c^2e^{4\pi
x}+1}\right)^2+\frac{2\pi^2c^2e^{4\pi x}}{(c^2e^{4\pi x}+1)^2} =
\frac{\pi^2}{2} \, .$$ Therefore, equality holds in everywhere, and $\phi$ is injective, but barely, since $\phi(1)=\phi(-1)$.
[**Example 2.**]{} The previous example can be extrapolated to give a general construction. Let $p$ be a Nehari function with the additional property that it is the restriction to $(-1,1)$ of an analytic function in the disk $p(z)$ that satisfies $|p(z)|\leq p(|z|)$. Typical examples are $p(z)=(1-z^2)^{-2}$ and $p(z)=2(1-z^2)^{-1}$. The extremal map $\Phi$ is then analytic and univalent in the disk, and satisfies ${\mathcal{S}}\Phi(z)=2p(z)$ there. Moreover, the image $\Phi(\mathbb{D})$ is a parallel strip like domain, symmetric with respect to the real and imaginary axes, and containing the entire real line. Let $$f(z)=\frac{c\Phi(z)+i}{c\Phi(z)-i} \, ,$$ where $c>0$ is to be chosen later and sufficiently small so that $i/c \notin \Phi(\mathbb{D})$ (it can be shown that the map $\Phi$ is always bounded along the imaginary axis, see [@co:gp]). The function $f$ maps $\mathbb{D}$ onto a simply-connected domain containing the unit circle minus the point 1. The smaller the value of $c$ the thinner the image of $f$.
Define $\phi:\mathbb{D}\rightarrow\mathbb{C}^2$ by $$\phi(z)=(f(z),\frac{1}{f(z)}\,) \, .$$ Then $$e^{2\sigma}=|f'(z)|^2\left(1+\frac{1}{|f(z)|^4}\right) \,,$$ and a lengthy calculation results in $${\mathcal{S}}\phi=2(\sigma_{zz}-\sigma_z)={\mathcal{S}}\Phi+6\frac{(\overline{f}f')^2}{(1+|f|^4)^2}
\, , \label{eq:Sf-example-2}$$ and $$e^{2\sigma}|K|=8\frac{|ff'|^2}{(1+|f|^4)^2} \, . \label{eq:sigma-example-2}$$ Condition then reads $$\left|{\mathcal{S}}\Phi+6\frac{(\overline{f}f')^2}{(1+|f|^4)^2} \,\right|
+6\frac{|ff'|^2}{(1+|f|^4)^2} \leq {\mathcal{S}}\Phi(|z|) \, .\label{eq:criterion-example-2}$$ Suppose, for example, we let $p(z)=(1-z^2)^{-2}$, for which the extremal function is $$\Phi(z)=\frac12\log\frac{1+z}{1-z}\,.$$ Then $\Phi'(z)=(1-z^2)^{-1}$ and $f'=-2ic\Phi'/(c\Phi-i)^2$, and after some simplifications becomes $$\left|\frac{2}{(1-z^2)^2}-\frac{24c^2(c\overline{\Phi}-i)^2}
{(1-z^2)^2(c\overline{\Phi}+i)^2(c\Phi-i)^4(1+|f|^4)}\,\right|
+\frac{24c^2|c\Phi+i|^2}{|1-z^2|^2|c\Phi-i|^6(1+|f|^4)} \leq\,
\frac{2}{(1-|z|^2)^2} \, ,$$ which further reduces to $$\left|1-\frac{12c^2(1+c^2\Phi^2)^2}{(|c\Phi-i|^4+|c\Phi+i|^4)^2}\,\right|
+\frac{12c^2|1+c^2\Phi^2|^2}{(|c\Phi-i|^4+|c\Phi+i|^4)^2} \leq \,
\frac{|1-z^2|^2}{(1-|z|^2)^2} \, .\label{eq:reduced-criterion}$$ Let $$\zeta=\displaystyle{\frac{12c^2(1+c^2\Phi^2)^2}{(|c\Phi-i|^4+|c\Phi+i|^4)^2}}\,.$$ In order to guarantee , we need the following estimates for $|1-{\rm Re}\{\zeta\}|$ and $|{\rm Im}\{\zeta\}|$.
\[lemma:example-estimates\] If $c$ is small then there exist absolute constants $A$, $B$, $C$ such that $$|1-{\rm Re}\{\zeta\}| \leq 1-|\zeta| +Ac^4|{\rm Im}\{{\Phi}\}|^2 \, ,
\label{eq:hard-estimate-1}$$ $$|{\rm Im}\{\zeta\}| \leq Bc^3|{\rm Im}\{{\Phi}\}| \, ,\label{eq:hard-estimate-2}$$ and $$|1-\zeta| \leq 1-|\zeta| +Cc^4|{\rm Im}\{{\Phi}\}|^2 \, .\label{eq:hard-estimate-3}$$
It is clear that $|\zeta| < 1$ if $c$ is small, hence $|1-{\rm
Re}\{\zeta\}| = 1-{\rm Re}\{\zeta\}$. Thus amounts to $$|\zeta|- {\rm Re}\{\zeta\} \leq Ac^4|{\rm
Im}\{{\Phi}\}|^2 \, . \label{eq:hard-estimate-1a}$$ We have $$\begin{aligned}
|\zeta|-{\rm Re}\{\zeta\} &=
\frac{12c^2}{(|c{\Phi}-i|^4+|c{\Phi}+i|^4)^2}\left[\, |1+c^2{\Phi}^2|^2-{\rm
Re}\{(1+c^2{\Phi}^2)^2\}\right]\\
& = \frac{12c^6( |{\Phi}|^4-{\rm
Re}\{{\Phi}^4\})}{(|c{\Phi}-i|^4+|c{\Phi}+i|^4)^2}\\
& =
-\frac{6c^6({\Phi}^2-\overline{{\Phi}}^{\,2})^2}{(|c{\Phi}-i|^4+|c{\Phi}+i|^4)^2}\\
&= \frac{24c^6({\Phi}+\overline{{\Phi}})^2|{\rm
Im}\{{\Phi}\}|^2 }{(|c{\Phi}-i|^4+|c{\Phi}+i|^4)^2} \,
\end{aligned}$$ which shows and thus because $c^2(\Phi+\overline{\Phi})^2/(|c\Phi-i|^4+|c\Phi+i|^4)^2$ is bounded for small $c$.
To establish observe that $$2i\,{\rm Im}\{(1+c^2{\Phi}^2)^2\} =
(1+c^2{\Phi}^2)^2-(1+c^2\overline{{\Phi}}^2)^2=
c^4({\Phi}^4-\overline{{\Phi}}^4)+2c^2({\Phi}^2-\overline{{\Phi}^2}) \, ,$$ from which follows directly. Finally, is a consequence of and because for $\zeta=x+iy$ small then $|1-\zeta|
\leq |1-x|+2y^2$.
|
---
abstract: 'In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the realization of some sparse semigroups as Weierstrass semigroups. The smoothness property of monomial curves associated to (hyper)ordinary semigroups presented by Pinkham and Rim-Vitulli, and the results on double covering of curves by Torres are crucial in this.'
address:
- 'Instituto de Matemática, Universidade Federal de Alagoas (UFAL). Avenida Lourival de Melo Mota, 57072-970, Maceió – AL, Brazil'
- 'Instituto de Matemática Pura e Aplicada (IMPA). Estrada Dona Castorina 110, 22460-320, Rio de Janeiro – RJ, Brazil'
- 'Instituto de Matemática e Estatítica, Universidade Federal Fluminense (UFF). Rua Mário Santos Braga S/N, Campus do Valonguinho, 24020-140 Niterói – RJ, Brazil'
author:
- André Contiero
- 'Carlos Gustavo T. A. Moreira'
- 'Paula M. Veloso'
title: On the structure of numerical sparse semigroups and applications to Weierstrass points
---
numerical semigroups ,genus ,sparse semigroups ,Weierstrass points ,Weierstrass semigroups 20M13 ,14H55
Introduction
============
Let ${\ensuremath{\mathcal{H}}}$ be a numerical semigroup of genus $g>1$. We say that ${\ensuremath{\mathcal{H}}}$ is a *sparse semigroup* if every two subsequent gaps of ${\ensuremath{\mathcal{H}}}$ are spaced by at most $2$. The concept of a sparse semigroup was introduced by Munuera–Torres–Villanueva in [@MunueraTorresVillanueva] and emerged as a generalisation of *Arf semigroups*. The latter appear naturally in the study of one-dimensional analytically unramified domains by analysing their valuation semigroups (see [@Arf],n[@BarucciDobbsFontana] and [@Lipman] for further details on Arf semigroups). Furthermore, one of the main subjects related to numerical semigroups are Weierstrass points on algebraic curves (points whose gap sequence of the numerical semigroup associated to a smooth projective pointed curve is the sequence of orders of vanishing of the holomorphic differentials of the curve at the base point.)
In this work, we are concerned with studying the structure of sparse semigroups and some applications to Weierstrass points. Looking for a classification and for an upper bound for genus of sparse semigroups, we introduce a *leap* count assertion (Theorem \[leapthm\]), which involves an interplay between single and double leaps. Besides, it plays a fundamental role in the main results of this work.
Section \[SparseSec\] presents several consequences of Theorem \[leapthm\]. We prove that, if the genus of a sparse semigroup is large enough, then the last few gaps are spaced by $2$. This results is proved regardless the parity of the Frobenius number ${\ell}_{g}$ (Proposition \[finalDblLeapsProp\]). Additionally, we classify some sparse semigroups with few single leaps or with large Frobenius number. At this point, *(hyper)ordinary semigroups* (see [@Rim-Vitulli]) and $\gamma$-*hyperelliptic semigroups* show up (see [@Torres]).
Looking for an upper bound for the genus of a sparse semigroup we introduce, in Section \[limitsparse\], the concept of a *limit sparse semigroup*: sparse semigroups with as many single as double leaps. Considering the parity of the Frobenius number, we classify limit sparse semigroups with even Frobenius number (Theorem \[charactSparseLimTheo\]), which are all hyperordinary with multiplicity $3$. As a consequence, we get an upper bound for the genus of any sparse semigroup with even Frobenius number, namely $g<4r$ where ${\ell}_{g}=2g-2r$ (Corollary \[genusQuotaSparseTheo\]).
We also classify limit sparse semigroups with odd Frobenius number. In this case, the *multiplicity* of the semigroup plays an import role. If the multiplicity of the limit sparse semigroup ${\ensuremath{\mathcal{H}}}$ is even, then ${\ensuremath{\mathcal{H}}}$ is an $r$-hyperelliptic semigroup, where ${\ell}_{g}=2g-2r-1$, (Theorem \[oddFrobEvenMultTheo\]). On the other hand, if the multiplicity is odd, then either ${\ensuremath{\mathcal{H}}}$ is hyperordinary of multiplicity $3$, or a semigroup of multiplicity $2r+1$, where ${\ell}_{g}=2g-2r-1$ is the Frobenius number of ${\ensuremath{\mathcal{H}}}$ (Theorem \[oddFrobOddMul\]). With the classification of limit sparse semigroups with odd Frobenius number in mind, we find an upper bound for the genus of these sparse semigroups, namely $g\leq4r+1$, except when all nongaps smaller than the Frobenius number are even (Corollary \[genusQuotaSparseOddTheo\]).
Finally, in the last section, we study the realization of (limit) sparse semigroups as Weierstrass semigroups. At this point the smoothness property of monomial curves associated to (hyper)ordinary semigroups presented by Pinkham [@Pinkham] and Rim-Vitulli [@Rim-Vitulli] is crucial. Furthermore, regarding $\gamma$-hyperelliptic sparse semigroups, the results by Torres [@Torres] on double covering of curves are applied.
Sparse semigroups {#SparseSec}
=================
Let ${\ensuremath{\mathbb{N}}}$ be the set of natural numbers. A *numerical semigroup* ${\ensuremath{\mathcal{H}}}= \{0 = n_{0} < n_{1} < \ldots \} \subseteq {\ensuremath{\mathbb{N}}}$ of finite genus $g\geq 1$ is an additive subset of ${\ensuremath{\mathbb{N}}}$ containing $0$, closed under addition and such that there are only $g$ elements in the set ${\ensuremath{\mathbb{N}}}\setminus{\ensuremath{\mathcal{H}}}=\{1={\ell}_1<{\ell}_2<\dots<{\ell}_{g}\}$. The elements in ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}$ are called *gaps* and the largest gap ${\ell}_{g}$ is called the *Frobenius number* of ${\ensuremath{\mathcal{H}}}$. The elements of ${\ensuremath{\mathcal{H}}}$ are referred to as *nongaps*, and the smallest positive nongap is said to be the *multiplicity* of ${\ensuremath{\mathcal{H}}}$.
A *sparse* (numerical) *semigroup* ${\ensuremath{\mathcal{H}}}$ is a numerical semigroup where two subsequent gaps of ${\ensuremath{\mathcal{H}}}$ with $1\leq{\ell}_{i-1},{\ell}_{i}\in\{{\ell}_1 < \ldots < {\ell}_g\}$ are spaced by at most $2$, $${\ell}_{i}-{\ell}_{i-1}\leq 2, \, i=2,\ldots,g \ \ , \ {\ell}_i\in{\ensuremath{\mathbb{N}}}\setminus{\ensuremath{\mathcal{H}}}.$$ Equivalently, ${\ensuremath{\mathcal{H}}}$ is sparse if its first $c-g$ nongaps satisfy $$n_{i + 1} -
n_ i \geq 2, \, i = 1,\ldots , c - g\ ,$$ where $c:={\ell}_g+1$ is the least integer such that $c+h\in {\ensuremath{\mathcal{H}}}$ for every $h\in {\ensuremath{\mathbb{N}}}$. The integer $c$ is said to be the *conductor* of ${\ensuremath{\mathcal{H}}}$ (clearly, $c=n_{c-g}$).
Two particular classes of sparse semigroups will appear frequently in this work: *ordinary sparse semigroups* (${\ensuremath{\mathcal{H}}}_g=\{0,g+1,g+2,\dots\}$) and *hyperordinary sparse semigroups* (${\ensuremath{\mathcal{H}}}=m{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{g}$, $0<m<g$).
Another class of sparse semigroups are Arf semigroups [@MunueraTorresVillanueva Corollary 2.2]. We recall that a numerical semigroup ${\ensuremath{\mathcal{H}}}$ is an *Arf semigroup* if $n_{i} + n_{j} - n_{k} \in {\ensuremath{\mathcal{H}}}$, for $i \geq j \geq k$ (see [@BarucciDobbsFontana Theorem I.3.4] for fifteen alternative characterizations of Arf semigroups, among which we call attention to the following: ${\ensuremath{\mathcal{H}}}$ is an Arf semigroup if and only if $2n_i-n_j\in {\ensuremath{\mathcal{H}}}$, for all $i\geq j\geq 1$). There are, however, sparse semigroups that are not Arf see Remark \[nonArf\] or [@MunueraTorresVillanueva Example 2.3]).
It is well known that for any numerical semigroup the Frobenius number ${\ell}_{g}$ satisfies ${\ell}_{g} \leq 2g - 1$ (see, for instance, [@Oliveira Theorem 1.1]). We may, thus, define the parameter $${\kappa}: = 2g - {\ell}_{g}>1\, ,$$ and we notice that ${\kappa}\leq g$.
Since sparse semigroups are the ones where subsequent gaps are either consecutive or spaced by $2$, it is only natural to count how many pairs of subsequent gaps are in either situation. Given a sparse semigroup ${\ensuremath{\mathcal{H}}}$, consider the sets: $${\ensuremath{\mathcal{D}}}:= \{ i \, ; \, {\ell}_{i + 1} - {\ell}_{i} = 2 \} \ \text{ (``double leaps'')},$$ $${\ensuremath{\mathcal{S}}}:= \{ i \, ; \, {\ell}_{i + 1} - {\ell}_{i} = 1 \} \ \text{ (``single leaps'')},$$ and their cardinalities: $$D :=\# {\ensuremath{\mathcal{D}}}\ \ \text{and} \ \ S :=\# {\ensuremath{\mathcal{S}}}.$$
\[leapthm\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup of genus $g$. Then:
1. $D + S = g - 1$.
2. $D = g - {\kappa}$.
3. $S = {\kappa}- 1$.
(1): Every gap, except the last one, ${\ell}_{g}$, is the starting point of a leap. So the total number of leaps, either single or double, is $g - 1$. Thus, $D + S = g - 1$.
\(2) and (3): Between $1$ and ${\ell}_{g} = 2g - {\kappa}$, there are $S$ single leaps and $D$ double leaps, regardless of their order. So, ${\ell}_{g} = 2g - {\kappa}= 1 + S +2D$. This equation together with the previous one yield the desired results.
We shall denote leaps by an ordered pair of subsequent gaps $({\ell}_{i}, {\ell}_{i+1})$, where ${\ell}_{i} < {\ell}_{i+1}$. Clearly, a leap is single if ${\ell}_{i} + 1 = {\ell}_{i+1}$, and double if ${\ell}_{i} + 2 = {\ell}_{i+1}$.
Next proposition gives us a little bit more information on the structure of sparse semigroups. It tells us that, if $g \geq 2{\kappa}- 1$, then the last few gaps occur every two integers.
\[finalDblLeapsProp\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup of genus $g$. If $g\geq 2{\kappa}-1$, then ${\ell}_{i+1} - {\ell}_{i} = 2$, for every $i=2{\kappa}-2,\ldots,g-1$.
If $g=2{\kappa}-1$, then ${\ensuremath{\mathcal{H}}}$ has ${\kappa}-1$ single leaps. Let us assume that ${\ell}_g-{\ell}_{g-1}=1$. Thus ${\ell}_{g-1}-n_i,{\ell}_{g}-n_i$ are consecutive gaps for $i=1,\ldots,{\kappa}-1$. Since ${\ell}_{g-1},{\ell}_g$ are consecutive gaps, the total number of single leaps for ${\ensuremath{\mathcal{H}}}$ is bigger than ${\kappa}-1$, which is a contradiction. Now, if ${\ensuremath{\mathcal{H}}}$ is a sparse semigroup of genus $g=2{\kappa}+j$, with $j\in{\ensuremath{\mathbb{N}}}$. Then $\widetilde{{\ensuremath{\mathcal{H}}}}={\ensuremath{\mathcal{H}}}\cup\{{\ell}_g\}$ is a sparse semigroup of genus $\widetilde{g}=2{\kappa}+j-1$. Thus the gaps of ${\ensuremath{\mathcal{H}}}$ satisfy ${\ell}_{i+1}-{\ell}_{i}=2$, for $i=2{\kappa}-2,\ldots,g-2$. Hence we just have to analyze ${\ell}_{g}-{\ell}_{g-1}$, which is analogous to the case where $g=2{\kappa}-1$.
For even values of ${\kappa}$,the previous result had been stated and proved by Munuera, Villanueva and Torres by means of a completely diverse approach.
Upon researching sparse semigroups, it became clear to us that those having genus $g = 2{\kappa}-1$ and Frobenius number ${\ell}_{g} = 2g - {\kappa}= 3{\kappa}-2$ are quite special. In fact, the lemma below suggests that they are “limit” in some sense; this notion will become clearer in the next section.
\[reductionLemma\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup of genus $g = 2{\kappa}+ j$, $j \geq 0$, with Frobenius number ${\ell}_g=2g - {\kappa}$. Then there is a sparse semigroup $\widetilde{{\ensuremath{\mathcal{H}}}}$ of genus $\widetilde{g} = 2{\kappa}-1$ and Frobenius number ${\ell}_{\widetilde{g}} = 2\widetilde{g} - {\kappa}= 3{\kappa}- 2$ such ${\ensuremath{\mathcal{H}}}$ is a subsemigroup of $\widetilde{{\ensuremath{\mathcal{H}}}}$.
Since $g \geq 2{\kappa}-1$, Proposition \[finalDblLeapsProp\] tells us that the last $j + 2$ gaps of ${\ensuremath{\mathcal{H}}}$ are spaced by $2$. Consider the set $\widetilde{{\ensuremath{\mathcal{H}}}} = {\ensuremath{\mathcal{H}}}\cup \{{\ell}_{g-j-2}, {\ell}_{g-j-1},\dots,{\ell}_{g}\}$. Clearly, $\widetilde{{\ensuremath{\mathcal{H}}}}$ contains $0$ and is additively closed. From the fact that ${\ensuremath{\mathcal{H}}}$ is sparse, we see that so is $\widetilde{{\ensuremath{\mathcal{H}}}}$. So $\widetilde{{\ensuremath{\mathcal{H}}}}$ is a sparse semigroup and, by construction, it has $\widetilde{g} = 2{\kappa}- 1$ gaps, ${\ensuremath{\mathcal{H}}}$ is a subsemigroup of $\widetilde{{\ensuremath{\mathcal{H}}}}$, and its Frobenius number is ${\ell}_{\widetilde{g}} = 2\widetilde{g} - {\kappa}= 3{\kappa}- 2$.
We present now several consequences and applications of our leap-count result (Theorem \[leapthm\]), which illustrate the techniques used in the theory of sparse semigroups. It will be clear, trough the next results, that there are only few sparse semigroups with large Frobenius number (or, equivalently, with few single leaps). We will make this statement more precise in the next section.
We recall that a numerical semigroup is said to be *symmetric* (resp. *quasi-symmetric*) if ${\ell}_g=2g-1$ (resp. ${\ell}_g=2g-2)$.
\[symCor\] If ${\ensuremath{\mathcal{H}}}$ is a symmetric sparse semigroup, then ${\ensuremath{\mathcal{H}}}$ is the hyperelliptic semigroup ${\ensuremath{\mathcal{H}}}=\langle 2,2g+1 \rangle $.
Since ${\ell}_{g}=2g-1$, the sparse semigroup ${\ensuremath{\mathcal{H}}}$ does not have single leaps i.e. ${\kappa}=0$. Then $2\in{\ensuremath{\mathcal{H}}}$ and all the odd numbers between $1$ and ${\ell}_{g}$ are gaps.
\[quasiSymCor\] If ${\ensuremath{\mathcal{H}}}$ is a quasi-symmetric sparse semigroup, then, either ${\ensuremath{\mathcal{H}}}= \langle 3,4,5 \rangle $, or ${\ensuremath{\mathcal{H}}}= \langle 3,5,7 \rangle $.
Since ${\ell}_g=2g-2$, we have that ${\kappa}= 2$ and so $S=1$ and $D=g-2$. We must have $1,2 \not\in {\ensuremath{\mathcal{H}}}$, which already accounts for the only single leap, so $3 \in {\ensuremath{\mathcal{H}}}$. Since all subsequent leaps must be double, the remaining gaps must all be even numbers. Then $g\leq 3$. Now, notice that there are no numerical semigroups of genus $g = 1$, otherwise ${\ell}_g={\ell}_1=2g-2=0$, a contradiction. If $g=2$, we have ${\ell}_g= {\ell}_2=2g-2=2$, and so ${\ensuremath{\mathcal{H}}}= \langle 3,4,5 \rangle $. Finally, for $g=3$, we have ${\ell}_g=2g-2=4$, so $1$ and $2$ are also gaps, for they divide $4$, and thus ${\ensuremath{\mathcal{H}}}= \langle 3,5,7 \rangle $.
We say that a numerical semigroup is *$\gamma$-hyperelliptic* if it has exactly $\gamma$ even gaps. For the sake of clarity, we note that a $\gamma$-hyperelliptic semigroup may have odd gaps and the integer $\gamma$ is not necessarily its genus. Such semigroups are closely related with double covering of curves [@BallicoCentina; @Torres; @Torres2]. Additionaly, they arise when we deal with the characterization of sparse semigroups having as many single as double leaps (see next section).
\[charactOdd3\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup having genus $g \geq 3$ and ${\ell}_{g} = 2g - 3$. Then ${\ensuremath{\mathcal{H}}}$ is one of the following:
1. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{5}$, ${\ensuremath{\mathcal{H}}}$ is $2$-hyperelliptic;
2. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{7}$, ${\ensuremath{\mathcal{H}}}$ is $2$-hyperelliptic;
3. ${\ensuremath{\mathcal{H}}}= 2({\ensuremath{\mathbb{N}}}\setminus \{1\}) \cup {\ensuremath{\mathcal{H}}}_{2g - 2}$, ${\ensuremath{\mathcal{H}}}$ is $1$-hyperelliptic.
Theorem \[leapthm\] (3) tells us that $S=2$. Clearly, ${\ell}_{1} = 1$ and ${\ell}_{2} = 2$, which accounts for one single leap. If $3 \in {\ensuremath{\mathcal{H}}}$, by the sparse property, we must have that $1, 2, 4, 5 \not\in {\ensuremath{\mathcal{H}}}$, and this accounts for all $2$ single leaps. Thus, either ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 4, 5\}$ and $g = 4$ (1), or ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 4, 5, 7\}$ and $g = 5$ (2). Otherwise, ${\ell}_{3} = 3$, and all leaps from this point on must be double. So ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 5, 7, \ldots 2g - 3\}$ and ${\ensuremath{\mathcal{H}}}= \{4, 6, \ldots , 2g - 4\} \cup \{n \in {\ensuremath{\mathbb{N}}}\, ; \, n \geq 2g - 2 \}$ (3).
Let ${\ensuremath{\mathcal{H}}}$ be a numerical semigroup having genus $g \geq 6$ and ${\ell}_{g} = 2g - 3$. Then the following are equivalent:
- ${\ensuremath{\mathcal{H}}}$ is sparse;
- ${\ensuremath{\mathcal{H}}}$ is $1$-hyperelliptic.
The implication $a. \implies b.$ follows immediately from the previous theorem. The other implication can be proved as follows: since ${\ensuremath{\mathcal{H}}}$ is $1$-hyperelliptic, the only even gap of ${\ensuremath{\mathcal{H}}}$ must be $2$. Indeed, if ${\ell}>2$ were an even gap, then, from ${\ell}=2+({\ell}-2)$, $2$ or ${\ell}-2$ would be a smaller even gap, a contradiction. So, every even number larger than $2$ is a nongap, and thus ${\ell}_r \ge 2r-3, \forall r \ge 3$, and, if the equality holds for $r=g$, it must hold for every $r$ with $3 \le r \le
g$, which implies that ${\ensuremath{\mathcal{H}}}$ is sparse.
\[charactEven4\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup having genus $g \geq 4$ and ${\ell}_{g} = 2g - 4$. Then ${\ensuremath{\mathcal{H}}}$ is one of the following:
1. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{8}$, and ${\ensuremath{\mathcal{H}}}$ is $3$-hyperelliptic;
2. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{10}$ , and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
3. ${\ensuremath{\mathcal{H}}}= 4{\ensuremath{\mathbb{N}}}+ {\ensuremath{\mathcal{H}}}_{6}$, and ${\ensuremath{\mathcal{H}}}$ is $2$-hyperelliptic;
4. ${\ensuremath{\mathcal{H}}}= {\ensuremath{\mathcal{H}}}_{4}$, and ${\ensuremath{\mathcal{H}}}$ is $2$-hyperelliptic;
5. ${\ensuremath{\mathcal{H}}}= 5{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6}$, and ${\ensuremath{\mathcal{H}}}$ is $3$-hyperelliptic;
6. ${\ensuremath{\mathcal{H}}}= \{0, 5, 7\} \cup {\ensuremath{\mathcal{H}}}_8$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
Theorem \[leapthm\] (3) tells us that $S = 3$. Again, ${\ell}_{1} = 1$ and ${\ell}_{2} = 2$, which account for one single leap.
If $3 \in {\ensuremath{\mathcal{H}}}$, since ${\ensuremath{\mathcal{H}}}$ is sparse and $S=3$, we must have that $1, 2, 4, 5, 7, 8 \not\in {\ensuremath{\mathcal{H}}}$ accounting for all $3$ single leaps. Thus, either ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 4, 5, 7, 8\}$ and $g = 6$ (1), or ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 4, 5, 7, 8, 10\}$ and $g = 7$ (2).
Otherwise, ${\ell}_{3} = 3$, which accounts for a second single leap. If $4 \in {\ensuremath{\mathcal{H}}}$, since ${\ensuremath{\mathcal{H}}}$ is sparse, $5 \not\in {\ensuremath{\mathcal{H}}}$. Notice that $6 \not\in
{\ensuremath{\mathcal{H}}}$ (otherwise, every even number $n \geq 4$ is in ${\ensuremath{\mathcal{H}}}$, and we would only have $2$ single leaps, a contradiction). So we have the remaining single leap $(5, 6)$, and all leaps from his point on must be double. On the other hand, we should have $8=4+4 \in {\ensuremath{\mathcal{H}}}$. Thus, ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 5, 6\}$ (3).
Finally, if $4 \not\in {\ensuremath{\mathcal{H}}}$, then all $3$ single leaps occur on gaps $1, 2, 3,
4$ and all other leaps are double. Thus, $5 \in {\ensuremath{\mathcal{H}}}$ and the only possibilities are ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 4\}$ (4), ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 4, 6\}$ (5) and ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 4, 6, 8\}$ (6).
Notice that the previous result implies, in particular, that, if ${\ensuremath{\mathcal{H}}}$ is a sparse semigroup having genus $g \geq 4$ and ${\ell}_{g} = 2g - 4$, then $g \le 7$. We will see in the next section (Theorem \[genusQuotaSparseTheo\]) that it is possible to generalize this fact (and the result about quasi-symmetric sparse semigroups) in the sense that for any fixed $r$, there are only a finite number of (sorts of) sparse semigroups, which can be explicitly listed.
\[charactOdd5\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup having genus $g \geq 5$ and ${\ell}_{g} = 2g - 5$. Then ${\ensuremath{\mathcal{H}}}$ is one of the following:
1. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{11}$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
2. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+ {\ensuremath{\mathcal{H}}}_{13}$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
3. ${\ensuremath{\mathcal{H}}}= 2({\ensuremath{\mathbb{N}}}\setminus \{1, 3\}) \cup {\ensuremath{\mathcal{H}}}_{2g -4}$, with $g \ge 6$, and ${\ensuremath{\mathcal{H}}}$ is $2$-hyperelliptic;
4. ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 7\} \cup {\ensuremath{\mathcal{H}}}_{9}$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
5. ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 7, 10\} \cup {\ensuremath{\mathcal{H}}}_{11}$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
6. ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 7, 10, 12\} \cup {\ensuremath{\mathcal{H}}}_{13}$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
7. ${\ensuremath{\mathcal{H}}}= \{ 0, 5\} \cup {\ensuremath{\mathcal{H}}}_{7}$, and ${\ensuremath{\mathcal{H}}}$ is $3$-hyperelliptic;
8. ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 8\} \cup {\ensuremath{\mathcal{H}}}_{9}$, and ${\ensuremath{\mathcal{H}}}$ is $3$-hyperelliptic;
9. ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 8, 10\} \cup {\ensuremath{\mathcal{H}}}_{11}$, and ${\ensuremath{\mathcal{H}}}$ is $4$-hyperelliptic;
10. ${\ensuremath{\mathcal{H}}}= 2({\ensuremath{\mathbb{N}}}\setminus \{1, 2\}) \cup {\ensuremath{\mathcal{H}}}_{2g -4}$, with $g \ge 5$, and ${\ensuremath{\mathcal{H}}}$ is $2$-hyperelliptic;
The proof technique is very similar to the one in the previous theorem.
Theorem \[leapthm\] (3) tells us that there are $S=4$. Again, ${\ell}_{1} = 1$ and ${\ell}_{2} = 2$, which account for one single leap. If $3 \in {\ensuremath{\mathcal{H}}}$, since ${\ensuremath{\mathcal{H}}}$ is sparse, we must have that $1, 2, 4, 5, 7, 8, 10, 11 \not\in {\ensuremath{\mathcal{H}}}$, accounting for all $4$ single leaps. Thus, either ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 4, 5, 7, 8, 10, 11\}$ and $g = 8$ (1), or ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 4, 5, 7, 8, 10, 11, 13\}$ and $g = 9$ (2).
Otherwise, ${\ell}_{3} = 3$, which accounts for a second single leap. If $4 \in {\ensuremath{\mathcal{H}}}$, since ${\ensuremath{\mathcal{H}}}$ is sparse, we must have that, for every $n \in 4{\ensuremath{\mathbb{N}}}$, $n < {\ell}_{g}$, both $n - 1$ and $n + 1$ are gaps. In particular, $5, 7 \not\in {\ensuremath{\mathcal{H}}}$. Notice that $6 \not\in {\ensuremath{\mathcal{H}}}$ (otherwise, every even number $n \geq 4$ is in ${\ensuremath{\mathcal{H}}}$, and we would only have $2$ single leaps, a contradiction). So we have other $2$ single leaps (as $5, 6, 7 \not\in {\ensuremath{\mathcal{H}}}$), and all leaps from this point on must be double. Thus, ${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 5, 6, 7, 9, 11, \ldots, 2g - 5\}$ (and $2g-5 \ge 7$, so $g \ge 6$) (3).
Otherwise, ${\ell}_{4} = 4$, which accounts for a third single leap. If $5 \in {\ensuremath{\mathcal{H}}}$, since ${\ensuremath{\mathcal{H}}}$ is sparse and $5 < {\ell}_{g}$, $6 \not\in {\ensuremath{\mathcal{H}}}$ and exactly one of the following possibilities holds: $7 \not\in {\ensuremath{\mathcal{H}}}$ or $8 \not\in {\ensuremath{\mathcal{H}}}$ (the two cannot happen simultaneously, as we would then have $6$ single leaps).
If $7 \in {\ensuremath{\mathcal{H}}}$, then $8 \not\in {\ensuremath{\mathcal{H}}}$ and, since $10=5+5 \in {\ensuremath{\mathcal{H}}}$ and $9 \le {\ell}_{g}$, $9 \not\in {\ensuremath{\mathcal{H}}}$. This accounts for the fourth and last single leap of ${\ensuremath{\mathcal{H}}}$. So all leaps from his point on must be double, which means that all even numbers greater than $9$ must be in ${\ensuremath{\mathcal{H}}}$. Since $14, 15 \in {\ensuremath{\mathcal{H}}}$ and ${\ensuremath{\mathcal{H}}}$ is sparse, $\{n \geq 14\} \subset {\ensuremath{\mathcal{H}}}$. Thus, in this case, we have the following possibilities for ${\ensuremath{\mathcal{H}}}$: ${\ensuremath{\mathcal{H}}}= \{0, 5, 7\} \cup \{n \geq 10\}$ (4), ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 7, 10\} \cup \{n \geq 12\}$ (2) and ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 7, 10, 12\} \cup \{n \geq 14\}$ (6).
If $8 \in {\ensuremath{\mathcal{H}}}$, then $7 \not\in {\ensuremath{\mathcal{H}}}$, and all single leaps occur on gaps $1, 2, 3, 4, 6, 7$. Thus, all leaps from this point on must be double, which means that all even numbers greater than $7$ must be in ${\ensuremath{\mathcal{H}}}$. So $13=5+8 \in {\ensuremath{\mathcal{H}}}$, and thus, in this case, we have the following possibilities for ${\ensuremath{\mathcal{H}}}$: ${\ensuremath{\mathcal{H}}}= \{ 0, 5\} \cup \{n \geq 8\}$ (7), ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 8\} \cup \{n \geq 10\}$ (8) and ${\ensuremath{\mathcal{H}}}= \{ 0, 5, 8, 10\} \cup \{n \geq 12\}$ (9).
On the other hand, if $5 \not\in {\ensuremath{\mathcal{H}}}$, then all $4$ single leaps occur on gaps $1, 2, 3, 4, 5$ and all other leaps are double. Thus, $${\ensuremath{\mathbb{N}}}\setminus {\ensuremath{\mathcal{H}}}= \{1, 2, 3, 4, 5, 7, 9, \ldots, 2g - 5\}$$ (here $2g-5 \ge 5$, and so $g \ge 5$) (10).
An important feature of Theorems \[charactOdd3\] and \[charactOdd5\] will be generalized in the next section (Corollary \[genusQuotaSparseOddTheo\]): if ${\ensuremath{\mathcal{H}}}$ is a sparse semigroup having genus $g$ for which ${\ell}_g=2g-(2r+1)$, where $r \in {\ensuremath{\mathbb{N}}}$, then, if $g>4r+1$, all nongaps of ${\ensuremath{\mathcal{H}}}$ smaller than ${\ell}_g$ are necessarily even. In this case all the $({\ell}_g+1)/2=g-r$ odd positive integers smaller than ${\ell}_g+1$ are gaps, so there are $r$ even gaps (i.e., ${\ensuremath{\mathcal{H}}}$ is $r$-hyperelliptic), and the set $\{m \in {\ensuremath{\mathbb{N}}}\, ; \, 2m \in {\ensuremath{\mathcal{H}}}\}$ is a semigroup of genus $r$. Theoretically, one could use the proof technique in Theorems \[charactOdd3\] and \[charactOdd5\] (and/or the above remark) to characterize all sparse semigroups having ${\ell}_{g} = 2g - (2r +1)$ for any fixed $r \in {\ensuremath{\mathbb{N}}}$. However, as $r$ grows, the number of cases to be analyzed quickly add up, and the arguments in the proof become more intricate.
Limit sparse semigroups {#limitsparse}
=======================
Having in mind the results of the previous section, it is only natural for one to ask about the existence of sparse semigroups of genus $g \geq 4r-1$ and Frobenius number ${\ell}_g=2g-2r$ (where $\kappa=2r$). A preliminary analysis of examples suggests that such semigroups do not exist if $g > 4r - 1$. This fact together with Lemma \[reductionLemma\] reinforces the idea that sparse semigroups of genus $g=4r-1$ with even Frobenius number ${\ell}_{g} = 2g - 2r$ are special.
We note that Theorem \[leapthm\] assures us that a sparse semigroup has genus $g=2{\kappa}-1$ if and only if $S = D$.
We call a sparse semigroup with as many single as double leaps (equivalently, of genus $g=2{\kappa}-1$) a *limit sparse semigroup*.
Starting from searching for an upper bound to the genus of limit sparse semigroups with even Frobenius number, our aim is to analyze more closely the structure of such semigroups regardless of its Frobenius number’s parity. When needed, we shall denote ${\ell}_{g} = 2g - 2r$ (${\kappa}=2r$) for even Frobenius numbers, or ${\ell}_{g} = 2g - 2r -1$ (${\kappa}=2r+1$) for odd Frobenius numbers.
\[bijSDHLemma\] If ${\ensuremath{\mathcal{H}}}$ is a limit sparse semigroup, then $\# \{ {\ensuremath{\mathcal{H}}}\cap \{1, 2, 3, \ldots, {\ell}_{g}\} \} = S = D$.
Clearly, there are ${\ell}_{g} = 2g - {\kappa}$ natural numbers in the set $\{1, \ldots, {\ell}_{g}\}$, $g$ of which are gaps. So, the $g - {\kappa}$ remaining ones are all in ${\ensuremath{\mathcal{H}}}\cap \{1, \ldots, {\ell}_{g}\}$ and, thus, $\# \{ {\ensuremath{\mathcal{H}}}\cap \{1, 2, 3, \ldots, {\ell}_{g}\} \} = g - {\kappa}= D = S$.
\[4isGapLemma\] If ${\ensuremath{\mathcal{H}}}$ is a limit sparse semigroup with even Frobenius number ${\ell}_g=2g - 2r$. Then $4 \not\in {\ensuremath{\mathcal{H}}}$.
We know that ${\ell}_g=6r-2$, ${\ell}_{g-1}=6r-4$, so $3r-2$, $3r-1$ are also gaps. Assume, by contradiction, that $4 \in {\ensuremath{\mathcal{H}}}$, then, analyzing the residual class of $r$ modulo $4$, one of the integers $6r-2$, $3r-2$, $3r-2$ is a nongap, a contradiction.
Let us now see that for each $r$ there is precisely one limit sparse semigroup with even Frobenius number. First, we state and prove another technical lemma:
\[3isNongapLemma\] Let ${\ensuremath{\mathcal{H}}}$ be a limit sparse semigroup with even Frobenius number ${\ell}_{g}= 2g-2r = 6r-2$. Then $3 \in {\ensuremath{\mathcal{H}}}$ if and only if $6r-5 \notin {\ensuremath{\mathcal{H}}}$.
It is clear that if $3 \in {\ensuremath{\mathcal{H}}}$ then ${\ensuremath{\mathcal{H}}}=3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6r-2}$, so $6r-5\notin{\ensuremath{\mathcal{H}}}$.
Now, let us assume that $6r-5 \notin {\ensuremath{\mathcal{H}}}$. Since $S = D$ and because of Lemma \[bijSDHLemma\], there are $2r-2$ nongaps in the interval $[1,6r-5]$. Then for each such nongap $n \in {\ensuremath{\mathcal{H}}}$, the consecutive numbers $6r-5-n$ and $6r-4-n$ are gaps, which produces $2r-2$ single leaps. Notice that the single leaps $(6r-5-n, 6r-4-n)$ are disjoint, for every $n \in {\ensuremath{\mathcal{H}}}$, because of the sparse property.
Suppose, by contradiction, that $3 \notin {\ensuremath{\mathcal{H}}}$. Thus $(1,2)$ and $(2,3)$ are single leaps, and $(6r-5-n,6r-4-n)$ are $2r-2$ disjoint single leaps, for every $n \in {\ensuremath{\mathcal{H}}}\cap [1,6r-5]$, and so at least one of the two single leaps $(1,2)$ and $(2,3)$ are not of this form. Hence, since $(6r-5,6r-4)$ is also a single leap, the number of single leaps is bigger than $2r-1$, which is a contradiction.
\[charactSparseLimTheo\] Let ${\ensuremath{\mathcal{H}}}$ be a limit sparse semigroup with even Frobenius number ${\ell}_g=2g-2r$. Then ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+ {\ensuremath{\mathcal{H}}}_{6r-2}$.
In view of Lemma \[3isNongapLemma\], all we have to do is to show that $6r-5$ is a gap. Suppose otherwise. So, there is an integer $x \geq 3$ such that the last single leap is $(6r-2x-1,6r-2x)$. So, in the interval $[6r-2x,6r-2]$, all even numbers are gaps and all odd numbers are nongaps.
Let $n_1$ be the multiplicity of ${\ensuremath{\mathcal{H}}}$. We first notice that it cannot be that $n_1$ is an odd number and $n_1 < 2x-1$. Otherwise, $6r-2-n_1$ would be an odd gap and $6r-2-n_1 > 6r-2x-1$, a contradiction. Moreover, it cannot be that $n_1$ is an even number and $n_1 \leq 2x$. Otherwise, there would be an even nongap, a multiple of $n_1$, between $6r-2x$ and $6r-2$, which is also a contradiction. Then there are only two possibilities for $n_1$:
1. $n_1 > 2x$;
2. $n_1 = 2x-1$.
Since the nongaps in the interval $[6r-2x,6r-2]$ are the odd numbers, there are $2r-x$ positive nongaps smaller than $6r-2x-1$. For each one of them, say $n \in {\ensuremath{\mathcal{H}}}\cap [1,6r-2x-2]$, $6r-2x-1-n$ and $6r-2x-n$ are consecutive gaps, producing $2r-x$ single leaps.
Let us consider each case separately:
Consider (A). In this case, $1,2,3,\ldots,n_1-1$ are consecutive gaps. The single leaps $(6r-2x-1-n,6r-2x-n)$ are all disjoint for every $n \in {\ensuremath{\mathcal{H}}}$, $n < 6r-2x-1$. Then, among the single leaps in the interval $[1,n_1-1]$, there are at least $x-1$ different from the ones in $\{ (6r-2x-1-n,6r-2x-n) \, ; \, n \in {\ensuremath{\mathcal{H}}}, \, n < 6r -2x-1 \}$. Since $(6r-2x-1,6r-2x)$ is also a single leap, we have, so far, $(x-1)+(2r-x)+1=2r$ single leaps, which is a contradiction, because $S=2r-1$.
Now, let us suppose (B). Since $S = 2r -1$, there are exactly $x-2$ single leaps in the interval $[1,n_1-1]$ that are not in $\{ (6r-2x-1-n,6r-2x-n) \, ; \, n \in {\ensuremath{\mathcal{H}}}, \, n < 6r -2x-1 \}$. Thus, there are $x-1$ single leaps in the interval $[1,2x-2]$ of the type $(6r-2x-1-n,6r-2x-n)$, where $n\in{\ensuremath{\mathcal{H}}}\cap[1,6r-2x-2]$, which are necessarily $(1,2), (3,4), \ldots, (2x-3,2x-2)$. Hence $6r-4x+2, \ldots, 6r-2x-4, 6r-2x-2 \in {\ensuremath{\mathcal{H}}}$. In particular, since $x \ge 3$, $6r-2x-4 \in {\ensuremath{\mathcal{H}}}$.
Now, consider the interval $[2x-1,6r-4x+1]$ which has $3(2r-2x+1)\ne 0$ integers and contains $2r-x-(x-1)=2r-2x+1$ nongaps. For each nongap $n \in {\ensuremath{\mathcal{H}}}\cap [2x-1,6r-4x+1]$, we consider the consecutive gaps $6r-2x-1-n$ and $6r-2x-n$, which both belong to $[2x-1,6r-4x+1]$ (and thus are $2r-2x-1$ single leaps contained in this interval). Then, since ${\ensuremath{\mathcal{H}}}$ is sparse, the nongaps in the interval $[2x-1,6r-4x+1]$ are necessarily the integers congruent to $2x-1 \pmod 3$. Hence, $2x+2\in {\ensuremath{\mathcal{H}}}$ (this is clear when the interval $[2x-1,6r-4x+1]$ has more than $3$ elements – it has at least $3$ elements, since $3(2r-2x+1)$ is a positive multiple of $3$; if $3(2r-2x+1)=3$, then $2x+2=6r-4x+2 \in {\ensuremath{\mathcal{H}}}$). Thus, in this case, $(2x+2)+(6r-2x-4)=6r-2={\ell}_g$ would be a nongap, a contradiction.
So, in any case, we arrive at a contradiction, and conclude that $6r-5$ is a gap, which implies, by Lemma \[3isNongapLemma\], that $3 \in {\ensuremath{\mathcal{H}}}$ and it is straightforward to see that ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+ {\ensuremath{\mathcal{H}}}_{6r-2}$.
If ${\ensuremath{\mathcal{H}}}$ is a limit sparse semigroup with even Frobenius number ${\ell}_g=2g-2r$, then ${\ensuremath{\mathcal{H}}}$ is an Arf semigroup.
We now have a tight bound for the genus of a sparse semigroup with even Frobenius number, greatly improving that by Munuera, Torres and Villanueva ($g\leq 6r-n_1$ if $g \geq 4r-1$ [@MunueraTorresVillanueva Theorem 3.1]):
\[genusQuotaSparseTheo\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup of genus $g$ with even Frobenius number ${\ell}_g=2g-2r$. Then $g \leq 4r - 1$.
Suppose ${\ensuremath{\mathcal{H}}}$ is a sparse semigroup of genus $g = 4r+ j$, $j \geq 0$, with ${\ell}_g=2g-2r$. Then, by Lemma \[reductionLemma\], there exists a sparse semigroup $\widetilde{{\ensuremath{\mathcal{H}}}}$ of genus $\widetilde{g} = 4r - 1$ and ${\ell}_{\widetilde{g}} = 6r - 2$ such ${\ensuremath{\mathcal{H}}}$ is a subsemigroup of $\widetilde{{\ensuremath{\mathcal{H}}}}$. Theorem \[charactSparseLimTheo\] tells us that $\widetilde{{\ensuremath{\mathcal{H}}}} =3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6r-2}$. By the construction of $\widetilde{{\ensuremath{\mathcal{H}}}}$ in the proof of Lemma \[reductionLemma\], we see that $\{n \in {\ensuremath{\mathbb{N}}}\, ; \, n < 6r \} \cap {\ensuremath{\mathcal{H}}}=
\{n \in {\ensuremath{\mathbb{N}}}\, ; \, n < 6r \} \cap \widetilde{{\ensuremath{\mathcal{H}}}}$; in particular, $3 \in {\ensuremath{\mathcal{H}}}$ and thus $6r \in {\ensuremath{\mathcal{H}}}$, which contradicts the fact that $6r = {\ell}_{4r} = {\ell}_{g -
j}$ is a gap of ${\ensuremath{\mathcal{H}}}$.
Let us now analyze some constraints on limit sparse semigroups having odd Frobenius number.
\[oddFrobEvenMultTheo\] Let ${\ensuremath{\mathcal{H}}}$ be a limit sparse semigroup with odd Frobenius number ${\ell}_{g}= 2g - (2r+1) = 6r+1$. If the multiplicity $n_1$ of ${\ensuremath{\mathcal{H}}}$ is even, then every nongap smaller then ${\ell}_g$ is even, and so ${\ensuremath{\mathcal{H}}}$ is $r$-hyperelliptic.
Assume there is an odd nongap $x \in {\ensuremath{\mathcal{H}}}$, $x < {\ell}_g$. Let $\tilde n$ be the largest odd nongap smaller than ${\ell}_g$. The interval $({\ell}_g-n_1,{\ell}_{g}]$ contains a complete residue system of the integers module $n_1$, so $\tilde n$ satisfies ${\ell}_g-n_1<\tilde n<{\ell}_g$. By construction, $\tilde n+1$ and $\tilde n+2$ are consecutive gaps. So, since ${\ensuremath{\mathcal{H}}}$ is sparse, there are less than $n_1/2$ nongaps in the interval $({\ell}_{g}-n_1,{\ell}_{g}]$, thus there are at least $2r - n_1/2 +1$ positive nongaps smaller than ${\ell}_{g}-n_1$. Define $\Gamma:=\{n\in{\ensuremath{\mathcal{H}}}\ ; 0 < n < {\ell}_g-n_1\}$. Being $\tilde n+1$ and $\tilde n+2$ consecutive gaps, for each $n\in\Gamma$, $\tilde n+1-n$ and $\tilde n+2-n$ are also consecutive gaps. But in the interval $[1,n_1-1]$ there are at least $n_1/2-1$ single leaps distinct from each $(\tilde n+1-n,\tilde n+2-n)$. By considering also the single leap $(\tilde n+1,\tilde n+2)$ we already count a number of $(2r - n_1/2 + 1) + (n_1/2-1) + 1 = 2r + 1$ single leaps, a contradiction. Now, the odd gaps are all the odd numbers between $1$ and ${\ell}_g=6r+1$, and so there are $3r + 1$ odd gaps and $r$ even gaps, then ${\ensuremath{\mathcal{H}}}$ is $r-$hyperelliptic.
\[oddFrobOddMul\] Let ${\ensuremath{\mathcal{H}}}$ be a limit sparse semigroup with odd Frobenius number ${\ell}_{g}= 2g - (2r+1) = 6r+1$. If the multiplicity $n_1$ of ${\ensuremath{\mathcal{H}}}$ is odd, then ${\ensuremath{\mathcal{H}}}$ is one of the following:
1. ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6r+1}$;
2. ${\ensuremath{\mathcal{H}}}= \langle 2j+1; j \in {\ensuremath{\mathbb{N}}}, r \le j \le 2r-1\rangle \cup {\ensuremath{\mathcal{H}}}_{6r+1}$, with $r>1$.
Let us first show that, if $6r-2 \notin {\ensuremath{\mathcal{H}}}$, then $3 \in {\ensuremath{\mathcal{H}}}$ (and so ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6r+1}$). If $6r-2 \notin {\ensuremath{\mathcal{H}}}$, then, since $S = D$ and because of Lemma \[bijSDHLemma\], there are $2r-1$ nongaps in the interval $[1,6r-2]$. Then for each such nongap $n \in {\ensuremath{\mathcal{H}}}$, the consecutive numbers $6r-2-n$ and $6r-1-n$ are gaps, which produces $2r-1$ single leaps. Notice that the single leaps $(6r-2-n, 6r-1-n)$ are all disjoint, for every $n \in {\ensuremath{\mathcal{H}}}$, because of the sparse property.
Suppose, by contradiction, that $3 \notin {\ensuremath{\mathcal{H}}}$. Thus $(1,2)$ and $(2,3)$ are single leaps, and $(6r-2-n,6r-1-n)$ are $2r-1$ disjoint single leaps, for every $n \in {\ensuremath{\mathcal{H}}}\cap [1,6r-2]$, and so at least one of the two single leaps $(1,2)$ and $(2,3)$ are not of this form. Hence, since $(6r-2,6r-1)$ is also a single leap, the number of single leaps is bigger than $2r$, which is a contradiction.
Suppose now that $6r-2$ is a nongap. So, there is an integer $x \geq 2$ such that the last single leap is $(6r-2x,6r-2x+1)$. So, in the interval $[6r-2x+1,6r+1]$, all odd numbers are gaps and all even numbers are nongaps.
First notice that it cannot be that $n_1 < 2x+1$. Otherwise, $6r+1-n_1$ would be an even gap and $6r+1-n_1 > 6r-2x$, a contradiction. Then there are only two possibilities for $n_1$:
1. $n_1 > 2x+1$;
2. $n_1 = 2x+1$.
Since the nongaps in the interval $[6r-2x+1,6r+1]$ are the even numbers, there are $2r-x$ positive nongaps smaller than $6r-2x$. For each one of them, say $n \in {\ensuremath{\mathcal{H}}}\cap [1,6r-2x-1]$, $6r-2x-n$ and $6r-2x+1-n$ are consecutive gaps, producing $2r-x$ single leaps.
Let us consider each case separately:
If we assume (A), then $1,2,3,\ldots,n_1-1$ are consecutive gaps. The single leaps $(6r-2x-n,6r-2x+1-n)$ are all disjoint for every $n \in {\ensuremath{\mathcal{H}}}$, $n < 6r-2x$. Then, among the single leaps in the interval $[1,n_1-1]$, there are at least $x$ different from the ones in $\{ (6r-2x-n,6r-2x+1-n) \, ; \, n \in {\ensuremath{\mathcal{H}}}, \, n < 6r - 2x \}$. Since $(6r-2x,6r-2x+1)$ is also a single leap, we have, so far, $x+(2r-x)+1=2r+1$ single leaps, which is a contradiction, because $S=2r$.
Let us assume (B). Since $S = 2r$, there are exactly $x-1$ single leaps in the interval $[1,n_1-1]$ that are not in $\{ (6r-2x-n,6r-2x+1-n) \, ; \, n \in {\ensuremath{\mathcal{H}}}, \, n < 6r -2x \}$. Thus, there are $x$ single leaps in the interval $[1,2x]$ of the type $(6r-2x-n,6r-2x+1-n)$, where $n\in{\ensuremath{\mathcal{H}}}\cap[1,6r-2x-1]$, which are necessarily $(1,2), (3,4), \ldots, (2x-1,2x)$. Hence $6r-4x+1, \ldots, 6r-2x-3, 6r-2x-1 \in {\ensuremath{\mathcal{H}}}$. In particular, since $x \ge 2$, $6r-2x-3 \in {\ensuremath{\mathcal{H}}}$, and, by the sparse property, $6r-4x$ is a gap.
Now, consider the interval $[2x+1,6r-4x]$ which has $6(r-x)$ integers and contains $2r-x-x=2r-2x$ nongaps. For each nongap $n \in {\ensuremath{\mathcal{H}}}\cap [2x+1,6r-4x]$, we consider the consecutive gaps $6r-2x-n$ and $6r-2x+1-n$, which both belong to $[2x-1,6r-4x+1]$ (and thus are $2r-2x$ single leaps contained in this interval). Then, since ${\ensuremath{\mathcal{H}}}$ is sparse, the nongaps in the interval $[2x+1,6r-4x]$ are necessarily the integers congruous to $2x+1 \pmod 3$. Hence, provided $r-x \neq 0$ (in which case the interval $[2x+1,6r-4x]$ has $6(r-x) \ge 6$ integer elements), we have $2x+4\in {\ensuremath{\mathcal{H}}}$. Thus, in this case, $(2x+4)+(6r-2x-3)=6r+1={\ell}_g$ would be a nongap, a contradiction. Thus we must have $r-x=0$, i.e., $x=r$, and so $6r-4x+1=2x+1$ and $6r-2x+2=4x+2$. In this case, ${\ensuremath{\mathcal{H}}}= \{0\}\cup \{2j+1; j \in {\ensuremath{\mathbb{N}}}, r \le j \le 2r-1\} \cup \{2j; j \in {\ensuremath{\mathbb{N}}}, 2r+1 \le j \le 3r\}\cup \{n \in {\ensuremath{\mathbb{N}}}\, ; \, n \geq 6r+2\}$, which is clearly the semigroup ${\ensuremath{\mathcal{H}}}= \langle 2j+1; j \in {\ensuremath{\mathbb{N}}}, r \le j \le 2r-1\rangle \cup {\ensuremath{\mathcal{H}}}_{6r+1}$.
\[nonArf\] The semigroups of the form ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+ {\ensuremath{\mathcal{H}}}_{6r+1}$ are clearly Arf, but the semigroups of the form ${\ensuremath{\mathcal{H}}}= \langle 2j+1; j \in {\ensuremath{\mathbb{N}}}, r \le j \le 2r-1\rangle + {\ensuremath{\mathcal{H}}}_{6r+1}$ with $r>1$ are not Arf, since $4r-3, 4r-1 \in {\ensuremath{\mathcal{H}}}$, but $2.(4r-1)-(4r-3)=4r+1 \notin {\ensuremath{\mathcal{H}}}$.
\[genusQuotaSparseOddTheo\] Let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup of genus $g$ with ${\ell}_g=2g-(2r+1)$. Then either $g \leq 4r + 1$, or all nongaps smaller than ${\ell}_g$ are even.
Suppose ${\ensuremath{\mathcal{H}}}$ is a sparse semigroup of genus $g = 4r + j=2(2r+1)+(j-2)$, $j \geq 2$, with ${\ell}_g=2g-(2r+1)$. Then, by Lemma \[reductionLemma\], there exists a sparse semigroup $\widetilde{{\ensuremath{\mathcal{H}}}}$ of genus $\widetilde{g} = 4r + 1$ and ${\ell}_{\widetilde{g}} = 6r + 1$ such ${\ensuremath{\mathcal{H}}}$ is a subsemigroup of $\widetilde{{\ensuremath{\mathcal{H}}}}$. The previous theorems tell us that we have the following possibilities:
1. All nongaps of $\widetilde{{\ensuremath{\mathcal{H}}}}$ (and so of ${\ensuremath{\mathcal{H}}}$) smaller than ${\ell}_{\widetilde{g}} = 6r + 1$ are even
In this case, by the construction of $\widetilde{{\ensuremath{\mathcal{H}}}}$ in the proof of Lemma \[reductionLemma\], the gaps of ${\ensuremath{\mathcal{H}}}$ larger than $6r$ are given by a sequence of double leaps starting at $6r+1$, and so all the remaining nongaps of ${\ensuremath{\mathcal{H}}}$ smaller than ${\ell}_g$ are even.
2. $\widetilde{{\ensuremath{\mathcal{H}}}} =3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6r+1}$ Here, by the construction of $\widetilde{{\ensuremath{\mathcal{H}}}}$ in the proof of Lemma \[reductionLemma\], we see that $\{n \in {\ensuremath{\mathbb{N}}}\, ; \, n < 6r+2 \} \cap {\ensuremath{\mathcal{H}}}= \{n \in {\ensuremath{\mathbb{N}}}\, ; \, n < 6r+2 \} \cap \widetilde{{\ensuremath{\mathcal{H}}}}$; in particular, we have $6r+3 \in {\ensuremath{\mathcal{H}}}$, which contradicts the fact that $6r+3 = {\ell}_{4r+2} = {\ell}_{g - j+2}$ is a gap of ${\ensuremath{\mathcal{H}}}$.
3. $\widetilde{{\ensuremath{\mathcal{H}}}} = \langle 2j+1; r \le j \le 2r-1\rangle \cup {\ensuremath{\mathcal{H}}}_{6r+1}$. Since $2r+1 \in {\ensuremath{\mathcal{H}}}$ we have $6r+3=3(2r+1) \in {\ensuremath{\mathcal{H}}}$ – which also contradicts the fact that $6r+3 = {\ell}_{4r+2} = {\ell}_{g - j+2}$ is a gap of ${\ensuremath{\mathcal{H}}}$.
On sparse Weierstrass semigroups
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One of the main applications of numerical semigroups is the study of Weierstrass points on curves. Having dealt (and classified limit) sparse semigroups, it is natural to ask whether those are realized as a Weierstrass semigroup.
By a *curve*, we mean a smooth projective curve defined over an algebraically closed field of characteristic zero. If $\cX$ is a curve and $P\in\cX$ is a point of $\cX$, then the Weierstrass semigroup $H(P)$ of the pair $(\cX,P)$ consists of the integers $n$ for which there does exists a meromorphic function on $\cX$ with pole divisor $nP$, i.e. $$H(P):=\{n\in{\ensuremath{\mathbb{N}}}\, ; \,dim H^0(\cX\, ,{\mathcal{O}}_{\cX}((n-1)P))< \dim H^0(\cX\, ,{\mathcal{O}}_{\cX}(nP))\}$$ It is clear that $H(P)$ is a numerical semigroup. Now, it follows from Riemann-Roch Theorem that the gap sequence of $H(P)$ is the sequence of orders of vanishing of the holomorphic differentials of $\cX$ at $P$. In particular, the genus of $H(P)$ is equal to the genus of the curve $\cX$. A numerical semigroup is a *Weierstrass semigroup* if it is realized as Weierstrass semigroup of some pair $(\cX,P)$. A point on a curve is a *Weierstrass point* if its associated Weierstrass semigroup is different from ${\ensuremath{\mathcal{H}}}_{g}$, where $g$ is the genus of the curve.
It is known, from Rim–Vitulli [@Rim-Vitulli], that the negatively graded semigroups are only the ordinary, hyperordinary and those semigroups of multiplicity $m>1$ having precisely one gap between $m$ and $2m$. By a theorem of Pinkham [@Pinkham], we know that a monomial curve associated to a negatively graded semigroup can be smoothed. In particular, a (hyper)ordinary semigroup is a Weierstrass semigroup. Thus, from those works and Theorem \[charactSparseLimTheo\], we get:
Let ${\ensuremath{\mathcal{H}}}$ be a limit sparse semigroup of genus $g$. If the Frobenius number ${\ell}_g$ is even, then ${\ensuremath{\mathcal{H}}}$ is a Weierstrass semigroup;
It is clear that if ${\ensuremath{\mathcal{H}}}$ is a limit sparse semigroup with odd Frobenius number of the type ${\ensuremath{\mathcal{H}}}= 3{\ensuremath{\mathbb{N}}}+{\ensuremath{\mathcal{H}}}_{6r+1}$ (see Theorem \[oddFrobOddMul\]), then ${\ensuremath{\mathcal{H}}}$ is also a Weierstrass semigroup.
Let ${\ensuremath{\mathcal{H}}}'$ be any numerical semigroup of genus $r$. Consider ${\ensuremath{\mathcal{H}}}:=2{\ensuremath{\mathcal{H}}}' \cup {\ensuremath{\mathcal{H}}}_{6r+1}$. Then ${\ensuremath{\mathcal{H}}}$ is a limit sparse semigroup with odd Frobenius number with even multiplicity (see Theorem \[oddFrobEvenMultTheo\]). Reciprocally, every semigroup satisfying the hypothesis of Theorem \[oddFrobEvenMultTheo\] arises in this way. Moreover, ${\ensuremath{\mathcal{H}}}$ is Arf if and only if ${\ensuremath{\mathcal{H}}}'$ is Arf.
It is clear that if ${\ensuremath{\mathcal{H}}}'$ is a semigroup of genus $r$, then the Frobenius number of ${\ensuremath{\mathcal{H}}}:=2{\ensuremath{\mathcal{H}}}' \cup {\ensuremath{\mathcal{H}}}_{6r+1}$ is $6r+1$. The gaps of ${\ensuremath{\mathcal{H}}}$ are all the odd integers in $[1,6r+1]$ and the even numbers $2m$ where $m$ is a gap of ${\ensuremath{\mathcal{H}}}'$. Thus the genus of ${\ensuremath{\mathcal{H}}}$ is $4r+1$. The sparse condition follows from the construction of ${\ensuremath{\mathcal{H}}}$. On the other hand, let ${\ensuremath{\mathcal{H}}}$ be a sparse semigroup of genus $g=4r+1$, with Frobenius number ${\ell}_g=2g-(2r+1)$, and even multiplicity $n_1$. Thus Theorem \[oddFrobEvenMultTheo\] ensures that all nongaps smaller than ${\ell}_g=6r-1$ are even. The odd gaps are all the odd numbers between $1$ and ${\ell}_g=6r+1$. So there are $3r+1$ odd gaps and $r$ even gaps. Thus ${\ensuremath{\mathcal{H}}}\cap [1,\dots,{\ell}_{g}]=\{2\,n_1,2\,n_2,\dots,2\,n_{2r}\}$. We consider the set ${\ensuremath{\mathcal{H}}}':=\{0,n_1,\dots,n_{2r},n_{2r}+1,n_{2r}+2,\dots\}$. Since ${\ensuremath{\mathcal{H}}}$ is additively closed, ${\ensuremath{\mathcal{H}}}'$ a semigroup. An integer is a gap of ${\ensuremath{\mathcal{H}}}'$ if and only if $2n$ is a gap of ${\ensuremath{\mathcal{H}}}$. Thus the genus of ${\ensuremath{\mathcal{H}}}'$ is $r$. The Arf condition follows from the construction of ${\ensuremath{\mathcal{H}}}'$. We notice that the last gap in ${\ensuremath{\mathcal{H}}}'$ is at most $2r-1$, and so the last even gap of ${\ensuremath{\mathcal{H}}}$ is at most $4r-2$.
Following the same steps of the above Corollary, we get:
\[rhiperg4r1\] Let ${\ensuremath{\mathcal{H}}}$ be a semigroup as in Corollary \[genusQuotaSparseOddTheo\], with $g>4r+1$. Then ${\ensuremath{\mathcal{H}}}$ is obtained in the following way: take any numerical semigroup ${\ensuremath{\mathcal{H}}}' \subset {\ensuremath{\mathbb{N}}}$ of genus $r$, and set ${\ensuremath{\mathcal{H}}}:=2{\ensuremath{\mathcal{H}}}' \cup {\ensuremath{\mathcal{H}}}_{2g-2r-1}$.
If ${\ensuremath{\mathcal{H}}}$ is a numerical semigroup as in Corollary \[genusQuotaSparseOddTheo\], with $g>4r+1$, then its odd gaps are all the odd numbers between $1$ and ${\ell}_g=2g-(2r+1)$. So, there are $g-r$ odd gaps and $r$ even gaps. Thus ${\ensuremath{\mathcal{H}}}:=2{\ensuremath{\mathcal{H}}}' \cup {\ensuremath{\mathcal{H}}}_{2g-2r-1}$ where ${\ensuremath{\mathcal{H}}}'$ is a numerical semigroup of genus $r$. Note that the last gap of ${\ensuremath{\mathcal{H}}}'$ is at most $2r-1$, and so the last even gap of ${\ensuremath{\mathcal{H}}}$ is at most $4r-2$.
As can be noted of from the last two corollaries above, and the Corollary \[genusQuotaSparseOddTheo\], we may expect that some sparse semigroups with odd Frobenius number arise as a double covering of a genus $r$ curve. We recall that Torres [@Torres] characterized $r$-hyperelliptic curves of genus $g$ which arise as a double covering of a genus $r$-curves under the assumption $g\ge 6r+4$. Gathering the Corollary \[rhiperg4r1\] and the comment after the proof of Theorem A of [@Torres] we get:
Let ${\ensuremath{\mathcal{H}}}$ a numerical sparse semigroup of genus $g\ge 6r+4$ and Frobenius number ${\ell}_{g}=2g-(2r+1)$. If ${\ensuremath{\mathcal{H}}}$ is a Weierstrass semigroup, then it arises as a double covering of a genus $r$-curve. In this case, we have ${\ensuremath{\mathcal{H}}}:=2{\ensuremath{\mathcal{H}}}' \cup {\ensuremath{\mathcal{H}}}_{2g-2r-1}$, where ${\ensuremath{\mathcal{H}}}'=\{n/2\mid\, n\in {\ensuremath{\mathcal{H}}}\text{ is even}\}$ is a Weierstrass semigroup of genus $r$.
Let ${\ensuremath{\mathcal{H}}}' \subset {\ensuremath{\mathbb{N}}}$ any Weierstrass semigroup of genus $r$, and $g\ge 4r+1$. Is the semigroup ${\ensuremath{\mathcal{H}}}:=2{\ensuremath{\mathcal{H}}}' \cup {\ensuremath{\mathcal{H}}}_{2g-2r-1}$ always Weierstrass?
Let ${\ensuremath{\mathcal{H}}}$ be a sparse Weierstrass semigroup with odd Frobenius number ${\ell}_{g}= 2g - (2r+1)$ and $g\ge 4r+1$ such that the multiplicity $n_1$ of ${\ensuremath{\mathcal{H}}}$ is even. Is the semigroup ${\ensuremath{\mathcal{H}}}'=\{n/2\mid\, n\in {\ensuremath{\mathcal{H}}}\text{ is even}\}$ always Weierstrass?
If the answer to Problem A of [@Komeda] is affirmative then the answer to this last question is also affirmative (indeed it would be enough that there were no numerical semigroup belonging to the box numbered by viii) in [@Komeda]).
Is the limit sparse semigroup ${\ensuremath{\mathcal{H}}}= \langle 2j+1; j \in {\ensuremath{\mathbb{N}}}, r \le j
\le 2r-1\rangle \cup {\ensuremath{\mathcal{H}}}_{6r+1}$ Weierstrass for every $r>1$?
Acknowledgements {#acknowledgements .unnumbered}
================
The authors warmly thank Prof. Luciane Quoos (Universidade Federal do Rio de Janeiro) for introducing sparse semigroups to them.
References {#references .unnumbered}
==========
[99]{}
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abstract: 'A triangulation of a surface is *irreducible* if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus $g\ge0$ with $b\ge0$ boundaries is $O(g+b)$. So far, the result was known only for surfaces without boundary ($b=0$). While our technique yields a worse constant in the $O(.)$ notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.'
author:
- 'Alexandre Boulch [^1]'
- 'Éric Colin de Verdière [^2]'
- 'Atsuhiro Nakamoto [^3]'
bibliography:
- './biblio.bib'
title: 'Irreducible Triangulations of Surfaces with Boundary [^4]'
---
**Keywords:** Topological graph theory, surface, triangulation, irreducible triangulation, homotopy.
**MSC Classification:** 05C10, 57M15, 57N05.
Introduction
============
Let ${\mathcal{S}}$ be a surface, possibly with boundary. A *triangulation* is a simplicial complex whose underlying space is ${\mathcal{S}}$. Contracting an edge of the triangulation (identifying two adjacent vertices in the simplicial complex) is allowed if this results in another triangulation of the same surface. An *irreducible* triangulation, sometimes called *minimal* triangulation, is a triangulation on which no edge is contractible. Every triangulation can be reduced to an irreducible triangulation by iteratively contracting some of its edges.
Irreducible triangulations have been much studied in the context of surfaces without boundary. In this paper, we initiate the study of irreducible triangulations for surfaces that may have boundary. We prove that the number of vertices of an irreducible triangulation is linear in the genus and the number of boundary components of the surface. Compared to previous works, our theorem and its proof have two interesting features: the result is more general, since it applies to surfaces with boundary, and the arguments of the proof are simpler.
Previous Works for Surfaces Without Boundary
--------------------------------------------
We first describe previous related works, on surfaces without boundary. Barnette and Edelson [@be-ao2mh-88; @be-a2mhf-89] proved that the number of irreducible triangulations of a given surface is finite. Nakamoto and Ota [@no-nits-95] were the first to show that the number of vertices in an irreducible triangulation admits an upper bound that is linear in the genus of the surface. The best upper bound known to date was developed by Joret and Wood [@jw-its-10], who proved that this number is at most $\max{\{ 13g-4,4 \}}$. (Here and in the sequel, $g$ is the *Euler genus*, which equals twice the usual genus for orientable surfaces and equals the usual genus for non-orientable surfaces.) This bound is asymptotically tight, as there are irreducible triangulations with $\Omega(g)$ vertices; however, the minimal number of vertices in a triangulation is $\Theta(\sqrt{g})$ [@jr-mtos-80].
Some low genus cases were studied. Steinitz [@sr-vtp-34] proved that the unique irreducible triangulation of the sphere is the boundary of the tetrahedron. The two irreducible triangulations of the projective plane were found by Barnette [@b-gtpp-82], followed by the 21 triangulations of the torus by Lawrencenko [@l-itt-87] and the 25 triangulations of the Klein bottle by Lawrencenko and Negami [@ln-itkb-97]. More recently, Sulanke [@s-gits-06; @s-itlgs-06] developed a method to generate all the irreducible triangulations of surfaces without boundary. His algorithm rediscovered the irreducible triangulations for the projective plane, the Klein bottle, and the torus; it also built the irreducible triangulations of the double-torus ([396,784]{} triangulations) and the non-orientable surfaces of genus three ([9,708]{}) and four ([6,297,982]{}). Irreducible triangulations can also be used to generate all the triangulations [@s-gt2m-91].
To solve problems on triangulations, it sometimes suffices to solve them on irreducible triangulations. For example, on a triangulation of an orientable surface with Euler genus $g\geq 4$ (at least two handles), Barnette [@mt-gs-01 Conjecture 5.9.3] conjectured that there always exists a cycle without repeated vertices that is non-null-homotopic and separating. More generally, Mohar and Thomassen [@mt-gs-01 Conjecture 5.9.5] conjectured that for every even $h$, $0<h<g$, there exists a cycle without repeated vertices that splits the surface into two surfaces of genus $h$ and $g-h$, respectively. To prove these conjectures, it suffices to prove them for irreducible triangulations. (See also the discussion by Sulanke [@s-itlgs-06 Sect. 5].)
Irreducible triangulations are also a good tool to study diagonal flips on triangulations. Negami [@n-dftss-99; @n-dfts-94] used the fact that there are finitely many irreducible triangulations to prove that two triangulations of a surface with the same number of vertices are equivalent under diagonal flips, provided the number of vertices is greater than an integer depending only on the surface.
For a more detailed survey on results on irreducible triangulations, see Mohar and Thomassen [@mt-gs-01 Sect. 5.4]; for further applications, see the recent paper by Joret and Wood [@jw-its-10] and references therein. Generalizations of the notion of irreducible triangulations, such as *$k$-irreducible triangulations* ($k\ge3$), have also been studied [@mn-kts-95; @grs-its-96].
Our Result
----------
It turns out that the notion of irreducible triangulations extends directly to the case of surfaces with boundary. In this paper, we prove that the number of vertices of such an irreducible triangulation admits an upper bound that is linear in the genus $g$ and the number of boundaries $b$ of the surface. In more details:
\[Th:main\] Let ${\mathcal{S}}$ be a (possibly non-orientable) surface with Euler genus $g$ and $b$ boundaries. Assume $g\geq1$ or $b\geq2$. Then every irreducible triangulation of ${\mathcal{S}}$ has at most $570g+385b-573$ vertices, except for the case of the projective plane, in which the bound is $186$.
![For any even $g\ge0$ and any $b\ge1$ (one of these two inequalities being strict), there exists an irreducible triangulation of an orientable surface with Euler genus $g$ with $b$ boundary components, and with $5g/2+4b-2$ vertices. The figure illustrates the case $g=4$ and $b=3$. Starting with a set of triangles glued together, all meeting at a vertex (bottom part), attach a set of $g/2$ pairs of interlaced rectangular strips (top left) and a set of $b-1$ non-interlaced rectangular strips (top right), and triangulate every strip by adding an arbitrary diagonal (not shown in the picture). That the resulting triangulation is irreducible follows from the fact that every edge belongs to a non-null-homotopic 3-cycle or is a linking edge (a non-boundary edge whose endpoints are both boundary vertices). Also, note that all vertices are on the boundary. In particular, taking $b=1$, we obtain an irreducible triangulation whose single boundary component contains $5g/2+2$ vertices.[]{data-label="F:example"}](./figures/example.eps){width=".7\linewidth"}
This bound is asymptotically tight; see Figure \[F:example\]. Compared to the case of surfaces without boundary, the main difficulty we encountered was to prove that the number of boundary vertices is $O(g+b)$ (there are indeed irreducible triangulations of surfaces whose single boundary contains $\Theta(g)$ vertices, as Figure \[F:example\] also illustrates); the known methods for surfaces without boundary do not seem to extend easily to surfaces with boundary. Our strategy is roughly as follows. Let $T$ be an irreducible triangulation. First, we show that every matching of (the vertex-edge graph of) $T$ has $O(g+b)$ vertices. Then, we show that every inclusionwise maximal matching contains a constant fraction of the vertices of $T$. For technical reasons, in the case of surfaces with boundary, we actually need to restrict ourselves to a matching satisfying some additional mild conditions.
In particular, we reprove that, on a surface without boundary of genus $g$, the number of vertices of an irreducible triangulation is $O(g)$. Our method does not improve over the current best bound of $\max{\{ 13g-4,4 \}}$ by Joret and Wood [@jw-its-10]. However, it is substantially different and simpler than the other known proofs of this result. These former proofs, by Nakamoto and Ota [@no-nits-95] and Joret and Wood [@jw-its-10] (see also Gao et al. [@grt-itteo-91]), rely on a deep theorem by Miller [@m-atgg-87] (see also Archdeacon [@a-nga-86]) stating that the genus of a graph (the minimum Euler genus of a surface on which a graph can be embedded) is additive over 2-vertex amalgams (identification of two vertices of disjoint graphs). While the method yields the current best bounds on the number of vertices, it seems a bit unnatural to use the genus of a graph to derive a result on graphs embedded on a fixed surface. Another paper by Cheng et al. [@cdp-hsmit-04] also claims a linear bound without using Miller’s theorem, but this part of their paper has a flaw (personal communication with the authors).[^5] In contrast, our proof is short and uses only elementary topological lemmas.
Finally, we refine the above technique in the case of surfaces without boundary, and obtain a bound that is better than that of Theorem \[Th:main\], but no better than the current best result by Joret and Wood [@jw-its-10].
We shall introduce some definitions of topology and preliminary lemmas in Section \[S:prelim\]. We then prove our main theorem (Section \[S:main\]). Finally, in Section \[S:improv\], we describe the improvement of the technique for surfaces without boundary.
Preliminaries {#S:prelim}
=============
We present a few notions of combinatorial topology; for further details, see also Stillwell [@s-ctcgt-93], Armstrong [@a-bt-83], or Henle [@h-cit-94].
Topological Background
----------------------
#### Surfaces, Cycles, and Homotopy.
A *surface* ($2$-manifold with boundary) is a topological Hausdorff space where each point has an open neighborhood homeomorphic to the plane or the closed half-plane; the points in the latter case are called *boundary points*. Henceforth, ${\mathcal{S}}$ denotes a compact, connected surface.
By the classification theorem, ${\mathcal{S}}$ is homeomorphic to a surface obtained from a sphere by removing finitely many open disks and attaching handles (*orientable case*) or Möbius bands (*non-orientable case*) along some of the resulting boundaries. In the orientable case, the *Euler genus* of ${\mathcal{S}}$, denoted by $g$, equals *twice* the number of handles; in the non-orientable case, it equals the number of Möbius bands. The number of remaining *boundary components* is denoted by $b$.
In this paper, a *cycle* on ${\mathcal{S}}$ is the image of a one-to-one continuous map $S^1\to{\mathcal{S}}$, where $S^1$ is the standard circle. In particular, we emphasize that cycles are undirected and simple. Two cycles are *homotopic* if one can be deformed continuously to the other; more formally, two cycles $C_0$ and $C_1$ are homotopic if there exists a continuous map $h:[0,1]\times S^1\to{\mathcal{S}}$ such that $h(0,\cdot)$ is one-to-one and has image $C_0$, and similarly $h(1,\cdot)$ is one-to-one and has image $C_1$. A cycle is *null-homotopic* if and only if it bounds a disk on ${\mathcal{S}}$. We emphasize that only homotopy of cycles is considered in this paper; for example, we say that two loops are homotopic if and only if the corresponding cycles are homotopic (without fixing any point of the loops).
A cycle is *two-sided* if cutting along it results in a (possibly disconnected) surface with two boundaries, and *one-sided* otherwise. Equivalently, a cycle is two-sided if it has a neighborhood homeomorphic to an annulus, and one-sided if it has a neighborhood homeomorphic to a Möbius band (which implies that the surface is non-orientable). Two homotopic cycles in general position cross an even number of times if they are two-sided, and an odd number of times if they are one-sided.
#### Graph Embeddings, Triangulations, and Edge Contractions.
Let $G$ be a graph, possibly with loops and multiple edges. An *embedding* of $G$ on ${\mathcal{S}}$ is a “crossing-free” drawing of $G$ on ${\mathcal{S}}$. More precisely, the vertices of $G$ are mapped to distinct points of ${\mathcal{S}}$; each edge is mapped to a path in ${\mathcal{S}}$, meeting the image of the vertex set only at its endpoints, and such that the endpoints of the path agree with the points assigned to the vertices of that edge. Moreover, all the paths must be without intersection or self-intersection except, of course, at common endpoints. We sometimes identify $G$ with its embedding on ${\mathcal{S}}$. The *faces* of $G$ are the connected components of the complement of the image of $G$ in ${\mathcal{S}}$. The graph embedding $G$ is *cellular* if each of its faces is an open disk. If it is the case, *Euler’s formula* states that $|V|-|E|+|F|=2-g-b$, where $V$, $E$, and $F$ are the sets of vertices, edges, and faces of $G$, respectively.
Let $e$ be an edge of a graph $G$ embedded in the interior of ${\mathcal{S}}$. Assume that $e$ is not a loop. The *contraction* of $e$ shrinks $e$ to a single vertex; the resulting graph is in the interior of ${\mathcal{S}}$. Loops and multiple edges may appear during this process (Figure \[contractionGraph\]).
![Edge contraction on an embedded graph.[]{data-label="contractionGraph"}](./figures/contraction2.eps){width="8cm"}
A *triangulation* $T$ on ${\mathcal{S}}$ is a graph without loops or multiple edges embedded on ${\mathcal{S}}$ such that each face is an open disk with three distinct vertices, and such that two such triangles intersect on a single edge (and its two incident vertices), a single vertex, or not at all. In other words, the vertices, edges, and faces of $G$ form a simplicial complex whose underlying space is ${\mathcal{S}}$.
The definition of edge contraction on a triangulation is slightly different from an edge contraction on a graph embedding. Let $uv$ be an edge of $T$; contracting edge $uv$ identifies both vertices $u$ and $v$ in the simplicial complex $T$; the dimension of some simplices decreases by one. We say that $uv$ is *contractible* if the new simplicial complex is still homeomorphic to ${\mathcal{S}}$ (Figure \[contractionTriangulation\]).
![Edge contraction on a triangulation.[]{data-label="contractionTriangulation"}](./figures/contraction.eps){width="8cm"}
In more details, assume for now that $e=uv$ is an interior edge, incident with triangles $uvx$ and $uvy$. Contracting $e$ shrinks $e$, identifying its two vertices $u$ and $v$, and identifies the two pairs of parallel edges ${\{ ux,vx \}}$ and ${\{ uy,vy \}}$. The definition is similar if $e$ is a boundary edge, except that it has a single incident triangle $uvx$. If $uv$ is not a boundary edge but exactly one vertex (say $u$) is incident to a boundary, then the edge $uv$ is contracted to $u$, on the boundary. If this operation results in a triangulation of ${\mathcal{S}}$, we say that $e$ is *contractible*. In particular, a *linking edge* of $T$ is a non-boundary edge whose both vertices are on the boundary; a linking edge is never contractible.
A triangulation of a surface is *irreducible* if it contains no contractible edge. For example, it is known that the only irreducible triangulation of the sphere is the boundary of a tetrahedron [@sr-vtp-34]. Using a similar argument, it is not hard to show that the only irreducible triangulation of the disk is made of a single triangle.
Preliminary Lemmas
------------------
We list here a series of basic facts that will be used in our proof.
\[L:noncontr\] Assume ${\mathcal{S}}$ is not the sphere or the disk, and let $T$ be an irreducible triangulation of ${\mathcal{S}}$. Then every non-linking edge of $T$ belongs to a non-null-homotopic 3-cycle.
This was proved by Barnette and Edelson [@be-ao2mh-88 Lemma 1] for surfaces without boundary: In this case, every edge of $T$ belongs to a 3-cycle that is not the boundary of a triangle; if that 3-cycle is null-homotopic, then it bounds a disk, and an edge inside that disk must be contractible. The argument immediately extends to non-linking edges of surfaces with boundary. (For boundary edges, we need to distinguish whether the boundary has length at least four, in which case the previous argument applies, or exactly three, in which case the result is obvious.)
\[degreeVertex\] The degree of a non-boundary vertex of an irreducible triangulation of ${\mathcal{S}}$ is at least four.
This is a direct consequence of a result by Sulanke [@s-gits-06 Theorem 1]. Specifically, he uses Lemma \[L:noncontr\] to show that every vertex of an irreducible triangulation belongs to two non-separating 3-cycles crossing at that vertex. Again, the argument extends to non-boundary vertices of surfaces with boundary.
\[nonHomotopicCycles\] Let $G$ be a one-vertex graph with $\ell$ loop edges, embedded in the interior of ${\mathcal{S}}$. Assume that no face of $G$ is a disk bounded by one or two edges. Then $\ell\leq 3g+2b-3$, except if ${\mathcal{S}}$ is a sphere or a disk (in which cases $\ell=0$).
Barnette and Edelson [@be-a2mhf-89 Corollary 1] prove a similar result; see also Chambers et al. [@ccelw-scsh-08 Lemma 2.1]. Here is a sketch of proof. Without loss of generality, we can assume that $G$ is inclusionwise maximal; namely, no edge can be added to $G$ without violating the hypotheses of the lemma. Then it follows from the classification of surfaces that, unless the surface is the sphere, the disk, or the projective plane, every face of the graph is a disk bounded by three edges, or an annulus bounded by a single edge and a single boundary component of ${\mathcal{S}}$. A standard double-counting argument combined with Euler’s formula concludes.
\[nonHomotopicCycles2\] Let $G$ be a one-vertex graph with $\ell$ loop edges, embedded in the interior of ${\mathcal{S}}$. Assume that no loop of $G$ is null-homotopic and that no two loops of $G$ are homotopic. Then $\ell$ is at zero if ${\mathcal{S}}$ is a sphere or a disk, at most one if ${\mathcal{S}}$ is the projective plane, and at most $3g+2b-3$ otherwise.
The hypotheses imply that no face of $G$ is a disk bounded by one or two edges (and thus Lemma \[nonHomotopicCycles\] concludes), unless that disk is bounded by twice the same edge (in which case ${\mathcal{S}}$ is the projective plane).
\[sizeHomotopyClass\] Let $C$ be a non-null-homotopic 3-cycle in an irreducible triangulation $T$ of ${\mathcal{S}}$. No more than nine pairwise edge-disjoint 3-cycles of $T$ are homotopic to $C$.
*First case: $C$ is two-sided.* This case is a small variation on a lemma by Barnette and Edelson [@be-ao2mh-88 Lemma 9]. Any two distinct 3-cycles homotopic to $C$ must cross an even number of times, hence cannot cross at all; thus two such 3-cycles bound an annulus, possibly “pinched” on a vertex or an edge. So the set of 3-cycles homotopic to a given 3-cycle can be ordered linearly. Assume there are at least ten pairwise edge-disjoint 3-cycles of $T$ homotopic to $C$; let us consider ten such consecutive cycles in this ordering, $C_1,\ldots,C_{10}$. See Figure \[homotopyClass\].
For every $i$, the annulus between $C_i$ and $C_{i+3}$ cannot be pinched along a vertex: otherwise, it is easy to see that an edge between $C_{i+1}$ and $C_{i+2}$ would be contractible [@be-ao2mh-88 Lemma 7]. This annulus cannot be pinched along an edge, since the cycles are edge-disjoint. So any two consecutive cycles in the sequence $C_1,C_4,C_7,C_{10}$ bound a non-pinched annulus. Now, similarly, some edge between $C_4$ and $C_7$ is contractible [@be-ao2mh-88 Lemma 9], which is a contradiction.
![Illustration of the proof of Lemma \[sizeHomotopyClass\] (in the two-sided case): if there are ten homotopic edge-disjoint 3-cycles, there must be four pairwise disjoint homotopic 3-cycles, so there is at least one contractible edge.[]{data-label="homotopyClass"}](./figures/homotopy.eps){width="8cm"}
*Second case: $C$ is one-sided.* In this case, any 3-cycle homotopic to $C$ crosses $C$ exactly once, and must therefore share exactly one vertex with $C$. Let $v$ be any vertex of $C$; we prove below that at most two 3-cycles different from $C$ and homotopic to $C$ pass through $v$. This proves that there are at most seven 3-cycles homotopic to $C$ (including $C$ itself), which concludes.
So assume that (at least) four 3-cycles (including $C$) homotopic to $C$ share together a vertex $v$. These cycles lie in a Möbius band “pinched” at $v$, and we can order them linearly; let $C_1,\ldots,C_4$ be consecutive cycles in this ordering. As in the first case [@be-ao2mh-88 Lemma 7], an edge between $C_2$ and $C_3$ would be contractible, a contradiction.
Proof of Theorem \[Th:main\] {#S:main}
============================
A *matching* $M$ of a graph $G$ is a set of edges of $G$ such that every vertex of $G$ belongs to at most one edge of $M$.
The Size of a Matching
----------------------
Our first task is to prove that a matching of an irreducible triangulation has size $O(g+b)$.
\[sizeMatching\] Let $T$ be an irreducible triangulation of ${\mathcal{S}}$, where $g\geq 1$ or $b\geq 2$. Let $M$ be a matching of $T$ containing no linking edge. Then the number of edges of $M$ is at most $27$ if ${\mathcal{S}}$ is the projective plane and $81g+54b-81$ otherwise.
The structure of the proof is as follows. In three steps, we remove edges to $M$, obtaining successively matchings $M_1$, $M_2$, and $M_3$, each of them satisfying additional properties. We show that the edge set of $M_3$ is in bijection with the edge set of a one-vertex graph on ${\mathcal{S}}$ where no edge is null-homotopic and no two edges are homotopic. By Corollary \[nonHomotopicCycles2\], this implies that $M_3$ has $O(g+b)$ edges. Furthermore, we show that $M_3$ contains some constant fraction of the edges of $M$, so that also $M$ has $O(g+b)$ edges.
Recall that every edge $e$ of $M$ belongs to a non-null-homotopic 3-cycle (Lemma \[L:noncontr\]); let $C_e$ be such a cycle.
*Construction of $M_1$.* Assume that an edge $e$ belongs to two cycles $C_{e_1}$ and $C_{e_2}$. Then $e$ cannot be in $M$. Moreover, each of the four vertices of $C_{e_1}\cup C_{e_2}$ is an endpoint of $e_1$ or $e_2$; so no edge of $C_{e_1}\cup C_{e_2}$ belongs to a 3-cycle $C_{e_3}$ for $e_3\not\in{\{ e_1,e_2 \}}$. Iteratively, for every such edge $e$ belonging to two 3-cycles $C_{e_1}$ and $C_{e_2}$, we remove one of $e_1$ and $e_2$ from $M$. The previous discussion implies that we remove at most half of the edges in $M$. Let $M_1$ be the resulting set of edges; we have $|M|\leq2|M_1|$. The set $M_1$ satisfies the hypotheses of the lemma, but now the cycles $C_e$, $e\in M_1$, are edge-disjoint. Now, we forget $M$ and focus on bounding the size of $M_1$.
*Construction of $M_2$.* We partition the edges $e$ of $M_1$ according to the homotopy class of the corresponding 3-cycle $C_e$. Let $M_2$ be obtained by choosing one arbitrary representative edge per class; the cycles $C_e$, $e\in M_2$, are in distinct homotopy classes. We have $|M_1|\leq 9|M_2|$ by Lemma \[sizeHomotopyClass\] and since the cycles $C_e$, $e\in M_1$, are edge-disjoint. Now, the cycles $C_e$, $e\in M_2$, are in distinct non-trivial homotopy classes and are edge-disjoint.
*Construction of $M_3$.* For every $e\in M_2$, let $\pi_e$ be the path of length two obtained from $C_e$ by removing $e$. We orient the two edges of $\pi_e$ towards the extremities of $\pi_e$ (which are also the endpoints of $e\in M_2$). Since $M_2$ is a matching, every vertex of the triangulation $T$ is the target of at most one oriented edge.
![In light lines, the matching $M_2$; in bold lines, the graph $\Pi_2$, here forming a tree plus an edge.[]{data-label="graph"}](./figures/orientedGraph.eps){width="8cm"}
Let $\Pi_2$ be the union of the graphs $\pi_e$, $e\in M_2$. We claim that $\Pi_2$ is a *pseudoforest*: every connected component $\Pi'_2$ of $\Pi_2$ contains at most one cycle (see Figure \[graph\]). Indeed, every vertex of $\Pi'_2$ is the target of at most one oriented edge, so the number of edges of $\Pi'_2$ is at most the number of vertices of $\Pi'_2$, and $\Pi'_2$ is connected; so a spanning tree of $\Pi'_2$ contains all but at most one edge of $\Pi'_2$.
If $\Pi'_2$ is not a tree, let $e'$ be an edge such that $\Pi'_2-e'$ is a tree. The edge $e'$ belongs to some $\pi_e$. We remove $e$ from $M_2$ (and consequently $\pi_e$ from $\Pi'_2$); the graph $\Pi'_2$ becomes one or two trees, and the other connected components of $\Pi_2$ are unaffected. We do this iteratively for every connected component $\Pi'_2$ of $\Pi_2$. Let $M_3$ be obtained from $M_2$ after removing these edges.
Before the removal of any edge of $M_2$, if a connected component $\Pi'_2$ of $\Pi_2$ is not a tree, it contains a cycle $C$ of length at least three (Figure \[graph\]); since each vertex of $C$ has at most one incoming edge, the edges of $C$ belong to distinct $C_e$. Therefore, when removing edges of $M_2$ to form $M_3$, we remove at most one third of the edges of $M_2$. So $|M_2|\leq\frac32|M_3|$.
*End of the proof.* Let $\Pi_3$ be the union of the graphs $\pi_e$, for $e\in M_3$; by construction, $\Pi_3$ is a forest. We now view $T$ as a graph embedded on ${\mathcal{S}}$ (slightly moving it towards the interior of ${\mathcal{S}}$, if ${\mathcal{S}}$ has a boundary), and contract all the edges of $\Pi_3$ in this graph; this is legal since this set contains no cycle. Each edge $e$ of $M_3$ is transformed into a loop $\ell_e$ homotopic to $C_e$. The loops $\ell_e$ form a graph $\Gamma$ embedded on ${\mathcal{S}}$; that graph has a single vertex per connected component. There exists a tree $U$ embedded on ${\mathcal{S}}$ meeting $\Gamma$ exactly at its vertex set. We may contract $U$ on the surface; now $\Gamma$ is transformed into a set of simple, pairwise disjoint loops $\Gamma'$ with the same vertex. Furthermore, the loops are non-null-homotopic and pairwise non-homotopic, so Corollary \[nonHomotopicCycles2\] implies that $|M_3|=|\Gamma'|\leq3g+2b-3$ (unless ${\mathcal{S}}$ is the projective plane, in which case the upper bound is one). By construction, we also have $|M|\leq 2|M_1|\leq 2*9|M_2|\leq 2*9*\frac32|M_3|=27|M_3|$, which concludes the proof.
An Inclusionwise Maximal Matching Covers Many Vertices
------------------------------------------------------
Now, we prove that an inclusionwise maximal matching of $T$ must cover a constant fraction of the edges of $T$.
\[verticesBound\] Let $T$ be an irreducible triangulation of ${\mathcal{S}}$. Let $W$ be a set of vertices of $T$. Let $M$ be an inclusionwise maximal matching of $T$ among those that avoid $W$. Assume further that every boundary vertex of $T$ is either in $W$ or incident to an edge of $M$. Then the number of vertices of $T$ is at most $7|M|+4|W|+3g+3b-6$.
Let us denote by $V$, $E$, and $F$ the vertices, edges, and faces of $T$, respectively. Let ${V_M}$ be the vertices reached by $M$ and $X$ be the vertices neither in ${V_M}$ nor in $W$. Let $\bar{M}$ be the set of the edges of $T$ that are not in $M$. Thus ${\{ W,{V_M},X \}}$ is a partition of $V$, and ${\{ M,{\bar M}\}}$ is a partition of $E$.
Let $v\in X$. Recall that $v$ is a non-boundary vertex by hypothesis. According to Lemma \[degreeVertex\], $v$ has degree at least four, so it is incident to at least four edges in ${\bar M}$. By maximality of the matching $M$, the other vertex of each of these four edges is not in $X$. So, charging each vertex $v$ of $X$ with these four edges, we obtain that $4|X|\leq |{\bar M}|$.
The rest of the proof is standard machinery. Since $T$ is a triangulation, by double-counting we obtain $|F|\leq\frac23|E|$ (this is not an equality in general since ${\mathcal{S}}$ may have boundary). Plugging this relation into Euler’s formula $|V|-|E|+|F|=2-g-b$, we obtain: $$(|W|+|{V_M}|+|X|)-\frac13(|M|+|{\bar M}|)\geq 2-g-b.$$ $|V_M|=2|M|$ gives, after some rearranging: $$|{\bar M}|-3|X|\leq 5|M|+3|W|+3g+3b-6.$$ As shown above, $4|X|\leq |{\bar M}|$, implying $|X|\leq|{\bar M}|-3|X|$ and so $$|X|\leq5|M|+3|W|+3g+3b-6.$$ This bound on $|X|$ allows to bound $|V|=2|M|+|W|+|X|$ in terms of $|M|$, $|W|$, $g$, and $b$, implying the result.
End of Proof
------------
The proof of Theorem \[Th:main\] combines Propositions \[sizeMatching\] and \[verticesBound\]:
Let $W$ be a set of vertices, one on each boundary component of ${\mathcal{S}}$ having an odd number of vertices. Build a matching $M$ made of edges on the boundary of ${\mathcal{S}}$ and covering the vertices on the boundary of ${\mathcal{S}}$ that are not in $W$. Extend $M$ to an inclusionwise maximal matching of $T$ that avoids $W$; we still denote it by $M$.
$M$ contains no linking edge by construction so, by Proposition \[sizeMatching\], $M$ has less than $81g+54b-81$ edges ($27$ if ${\mathcal{S}}$ is the projective plane). By Proposition \[verticesBound\], and since $|W|\leq b$, the number of vertices of $T$ is at most $7|M|+3g+7b-6$.
Combining these equations proves that $T$ has at most $570g+385b-573$ vertices ($186$ if ${\mathcal{S}}$ is the projective plane).
Improvement for Surfaces Without Boundary {#S:improv}
=========================================
The purpose of this section is to improve the previous bound when ${\mathcal{S}}$ has no boundary ($b=0$). The strategy is to improve the bound of Proposition \[verticesBound\] using a more careful analysis.
\[Th:main2\] Let ${\mathcal{S}}$ be a (possibly non-orientable) surface with Euler genus $g\ge1$ and without boundary. Then every irreducible triangulation of ${\mathcal{S}}$ has at most $f(g)$ vertices, where $f(1)=55$, $f(2)=194$, $f(3)=333$, and $f(g)=163g-164$ if $g\ge4$.
The following lemma and its proof appear in a manuscript by Fujisawa et al. [@fno-hcbtg-10 p. 4]; we reproduce the proof in slightly more details here for convenience.
\[lem:conncomp\] Let ${\mathcal{S}}$ be a surface of Euler genus $g\ge1$ without boundary, and let $G=(V,E)$ be a 4-connected graph embedded on ${\mathcal{S}}$. Then for every $U\subseteq V$, the number of components of $G-U$ is at most $\max{\{ 1,|U|+g-2 \}}$.
We can assume that $U\neq\emptyset$ and that $G-U$ has at least two connected components; otherwise, the result is clear. Let $K$ be the graph obtained from $G$ by the following steps:
1. Contract the edges of a spanning forest of $G-U$. Now the current graph has vertex set $U\cup W$, where $W$ has one element for each component of $G-U$. The following steps will only add and remove edges of this graph.
2. Delete each edge with both endpoints in $U$. Similarly, delete each edge with both endpoints in $G-U$ (such edges are actually loops, by the first step). Now the current graph is bipartite.
3. On each face of the resulting graph that is not a disk, add edges to cut that face into a disk. This can be done without violating bipartiteness, because every face has a boundary component with at least one vertex in $U$ and at least one vertex in $W$ (since $U$ and $W$ are non-empty).
4. If there exists a face with two incident edges, remove one of these two edges. (The two edges incident to the face are distinct, because ${\mathcal{S}}$ is not the sphere and the edge is not a loop.) Repeat this step as much as possible.
We now have: $$\begin{aligned}
4|W| & \le & |E(K)|\\
& \le & 2(|E(K)|-|F(K)|)\\
& = & 2(|W|+|U|+g-2).\\
\end{aligned}$$ Indeed, the first inequality holds by 4-connectivity of $G$: since $|W|\ge2$, every component of $G-U$ is adjacent to at least four different vertices of $U$; therefore, in $K$, every vertex of $W$ is adjacent to at least four different vertices of $U$. The second line follows from the fact that each face is incident to at least four edges (by bipartiteness of $K$ and using Step 4). The third line holds by virtue of Euler’s formula, since $K$ is cellularly embedded on ${\mathcal{S}}$.
\[verticesBound2\] Let ${\mathcal{S}}$ be a surface of Euler genus $g\geq1$ without boundary, and let $G=(V,E)$ be a 4-connected graph embedded on ${\mathcal{S}}$. Let $M$ be a maximum-size matching of $G$. Then the number of vertices of $G$ is at most $2|M|+\max{\{ 1,g-2 \}}$.
The Tutte-Berge formula [@b-cmg-58][@s-cope-03 Sect. 24.1] asserts that the number of vertices of $G$ not covered by a maximum-size matching of $G$ is the maximum, over all $U\subseteq V$, of $o(G-U)-|U|$, where $o(G-U)$ denotes the the number of components of the graph $G-U$ with an odd number of vertices. By Lemma \[lem:conncomp\], for every $U\subseteq V$, we have $o(G-U)-|U|\le\max{\{ 1,g-2 \}}$. The result follows.
If $T$ is 4-connected, by Proposition \[verticesBound2\], $T$ has at most $2|M|+\max{\{ 1,g-2 \}}$ vertices where $M$ is a maximum-size matching of $T$. Using the bound on the size of a maximal matching $M$ (Proposition \[sizeMatching\]), we deduce that $T$ has at most $h(g)$ vertices, where $h(1)=55$, $h(2)=163$, and $h(g)=163g-164$ if $g\ge3$.
If $T$ is not 4-connected, this means that a vertex set $U$ of size at most three separates $T$. Actually, $|U|=3$, and $U$ forms a 3-cycle $C$ in $T$. This cycle $C$ must be separating, but also non-null-homotopic, for otherwise some edge of $T$ would be contractible (as in the proof of Lemma \[L:noncontr\]). Let ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ be the surfaces obtained by cutting ${\mathcal{S}}$ along $C$ and attaching a triangle to each copy of $C$. The Euler genera of ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ add up to $g$. Furthermore, $C$ is two-sided (since it is separating), so the number of 3-cycles homotopic to $C$ in ${\mathcal{S}}$ is at most $27$ [@be-ao2mh-88 Lemma 9]. Any edge that is contractible in ${\mathcal{S}}_1$ or ${\mathcal{S}}_2$ belongs to such a cycle. So the total number of edges in ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ that are contractible is at most $3\times27+3=84$ (the “$+3$” term comes from the fact that the three edges of $C$ may be contractible in both ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$.) A similar reasoning is used by Barnette and Edelson [@be-ao2mh-88 Proof of Theorem 2].
It follows that the number of vertices of an irreducible triangulation of a surface without boundary with Euler genus $g$ is bounded from above by $f(g)$, where $f$ satisfies the induction formula: $$f(g)=\max\left\{h(g),\max_{\substack{g_1+g_2=g\\g_1,g_2\ge1}}\big\{f(g_1)+f(g_2)+84\big\}\right\}.$$ Thus, we have $f(1)=55$, $f(2)=194$, and for $g\ge3$: $$f(g)=\max\left\{163g-164,\max_{\substack{g_1+g_2=g\\g_1,g_2\ge1}}\big\{f(g_1)+f(g_2)+84\big\}\right\}.$$ It is easily checked by induction that $f(3)=333$ and $f(g)=163g-164$ for $g\ge4$.
[^1]: École Polytechnique (member of ParisTech), Palaiseau, France. E-mail: [alexandre.boulch@polytechnique.edu](alexandre.boulch@polytechnique.edu).
[^2]: Laboratoire d’informatique, École normale supérieure, CNRS, Paris, France. E-mail: [Eric.Colin.de.Verdiere@ens.fr](Eric.Colin.de.Verdiere@ens.fr).
[^3]: Department of Mathematics, Yokohama National University, Yokohama, Japan. Email: [nakamoto@ynu.ac.jp](nakamoto@ynu.ac.jp).
[^4]: This work was done during the first author’s internship at École normale supérieure. The internship was funded by the *Agence Nationale de la Recherche* under the *Triangles* project of the *Programme blanc* ANR-07-BLAN-0319.
[^5]: Specifically, in the proof of their Lemma 3, the authors incorrectly claim that there are at most $g$ pairwise non-homologous cycles on an orientable surface of Euler genus $g$.
|
---
abstract: |
The exponential kernel $$E{g}(\lambda,w) = \exp -\frac{1}{\pi}\int_{\mathbb{C} }
\frac{g(u)}{\overline{u-w} (u-\lambda) } da(u ),$$ where the compactly supported bounded measurable function $g$ satisfies $0 \leq g\leq 1,$ and suitably defined for all complex $\lambda, w,$ plays a role in the theory of Hilbert space operators with one-dimensional self-commutators and in the theory of quadrature domains. This article studies continuity and integral representation properties of $E_{g}$ with further applications of this exponential kernel to operators with one-dimensional self-commutator.
author:
- 'Kevin F. Clancey'
bibliography:
- 'Ework.bib'
date: August 2018
title: 'An exponential kernel associated with operators that have one-dimensional self-commutators'
---
Introduction
============
For $g$ a compactly supported bounded measurable function defined on the complex plane $\mathbb{C}$ that satisfies $0\leq g\leq1,$ let $E_{g}(\lambda,w)=E(\lambda,w)$ be defined by $$\begin{gathered}\label{key} E(\lambda,w) = \exp -\frac{1}{\pi}\int_{\mathbb{C} }
\frac{u-w}{u-\lambda }
\frac{g(u)}{\vert u-w\vert^2} da(u )= \\ \exp -\frac{1}{\pi}\int_{\mathbb{C} }
\frac{g(u)}{\overline{u-w} (u-\lambda) } da(u )\end{gathered}$$ for $\lambda\neq w,$ with $E(w,w)$ defined to be $0$ if $\frac{1}{\pi}\int_{\mathbb{C}}g(u)\vert u-w\vert^{-2} = \infty$ and equal to $$\exp-\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )$$ when $$\label{finite}\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )<\infty .$$ Here $a$ denotes area measure.
The function $E_g$ first appeared in the study of bounded linear operators on a Hilbert space with one-dimensional self-commutator. To show this connection, let $T$ be a bounded linear operator on a Hilbert space $\mathcal H$ satisfying $T^{*}T-TT^{*}= \varphi\otimes\varphi.$ Throughout the following it will be assumed that $T$ is irreducible, which in this case is equivalent to the statement that there are no non-zero subspaces of $\mathcal H$ reducing $T$ where $T$ restricts to a normal operators. Pincus [@PNAS] established that there is a one-to-one correspondence between the unitary equivalence classes of the collection of such operators with the collection of equivalence classes of compactly supported Lebesgue measurable functions $g$ satisfying $0\leq g\leq 1.$ The (equivalence class of the) function $g_T$ associated with $T$ is called the principal function of $T.$ The principal function $g$ (the subscript $T$ will usually not be included on the principal function) first appeared in [@pincus1] in the study of spectral theory of self-adjoint singular integral operators on the real line. To continue the story of the connection of $E_{g_{_{T}}}$ with $T,$ we introduce the local resolvent. It develops that for $\lambda$ in $\mathbb{C}$ there is a unique solution of the equation $T_{\lambda}^{*}x=\varphi$ orthogonal to the kernel of $T_{\lambda}^{*} = (T-\lambda)^{*}.$ This solution is denoted $T_{\lambda}^{*-1}\varphi.$ The $\mathcal H$-valued function $T_{\lambda}^{*-1}\varphi$ defined for $\lambda \in \mathbb{C}$ was first investigated by Putnam [@Putnam] and Radjabalipour [@Radj] and will be called the global-local resolvent associated with the operator $T.$ The following result from [@Clancey84] shows the connection between the function $E_{g_{T}}$ and the $\mathcal H$-valued function $T_{\lambda}^{*-1}\varphi$ .
Let $T$ be an irreducible operator with one-dimensional self-commutator $T^{*}T-TT^{*}= \varphi\otimes\varphi.$ Let $g=g_{T}$ be the associated principal function and $T_{\lambda}^{*-1}\varphi, \lambda\in \mathbb{C}$ the associated global-local-resolvent. Then for $\lambda$ and $w$ in $\mathbb{C}$ $$1- (T_{w}^{*-1}\varphi,T_{\lambda}^{*-1}\varphi )= E_{g}(\lambda, w ) = \exp-\frac{1}{\pi}{\int_{\mathbb{C}} \frac{u-w}{u-\lambda }\frac{g(u)}{\vert u-w\vert^2}} da(u ).\label{KFC}$$
For $\vert\lambda\vert$ and $\vert w\vert$ larger than $\Vert T\Vert,$ the result in the above theorem follows from early work on the principal function as presented in [@pincus]. The identity (\[KFC\]) has consequences for both the operator $T$ and the function $E.$ For example, it follows easily from the weak continuity of the global-local resolvent and (\[KFC\]) that the function $E$ is separately continuous. However, the fact that $E$ is separately continuous without assuming the identity is far from obvious (see, [@putinarmartin p. 260]). In descending chronological order, expository accounts of the relevant operator theory can be found in [@putinarmartin], [@peller], [@xia], [@semiKFC], and [@putnam1]. We also refer to these sources for historical accounts and many of the references to the area.
In an unexpected direction, when the function $g$ is the characteristic function of a planar domain $\Omega,$ Putinar [@Putinar] made connections between the exponential kernel $E_{g},$ quadrature domains, and operators with one dimensional self-commutator. A recent account of these connections, including a discussion of properties of $E,$ and citations of earlier work can be found in the book [@bookgusput].
The main focus here is on the continuity and integral representation properties of $E_{g}.$ This is the content of the first part of this paper. We will study the continuity properties function $E_{g}$ without any reference to the associated operator $T.$ In the second part, we will offer some comments about the operator $T$ that can be gleaned from the identity (\[KFC\]) and results in the first part.
Continuity properties of E
==========================
In this section we will establish the sectional continuity of the function $E_{g}$ and show this function is locally Lipschitz at points of positive density of the measure $gda.$ A study of Cauchy transform representations of $E_{g}$ will also be presented. It should be remarked that the sectional continuity of $\vert E_{g}\vert$ was established in [@Clancey84] using methods similar to but less exact than those employed here. It will be assumed that $w$ is fixed and $E_g(\lambda, w)$ will be considered as a function of $\lambda.$ Unless stated otherwise, the function $g$ will be assumed to satisfy $0\leq g\leq1.$
Sectional continuity
--------------------
Fix the point $w.$ For $\lambda\neq w,$ the continuity of the function $$f_{w}(\lambda) = -\frac{1}{\pi}\int_{\mathbb{C}}\frac{g(u)}{(\overline{u-w})(u-\lambda)}da(u) \label{logE}$$ can be established by elementary means. In particular, this follows from basic properties of the Cauchy transform that will be introduced below. Thus for $w$ fixed, a study of the sectional continuity of $E_{g}(\lambda, w)$ reduces to investigating continuity at $w.$ This will accomplished by first studying the continuity of (\[logE\]) in the case $\frac{1}{\pi}\int_{\mathbb{C}}\frac{g(u)}{\vert u-w\vert^{2}}da(u) < \infty.$
We first derive some integral formulas for the case where the function $g$ in (\[logE\]) is the characteristic function of well-chosen discs.
The linear fractional mapping $$T_{w,\lambda}(u)=\frac{u-w}{u-\lambda}$$ has invariant properties relative to the measure $\frac{1}{\vert u-w\vert ^2}da(u)$. This property can be used to compute the real and imaginary parts of the integral $$\frac{-1}{\pi}\int_{D} \frac{u-w}{u - \lambda}\vert u-w\vert^{-2} da(u)$$ over specific discs.
For $\alpha \in\bf R$ let $D_{\lambda,\alpha}$ be the disc with center $c_{\alpha}=\frac{w+\lambda +\alpha (\lambda - w)}{2}$ of radius $r_{\alpha}=\frac{\vert (\lambda-w) (1-\alpha)\vert}{2}$. For $\alpha < 1$, $$D_{\lambda, \alpha}=\{u:Re\left[\frac{u-w}{u-\lambda} \right] < \frac{\alpha}{\alpha -1}\}$$ and for $\alpha >1$ $$D_{\lambda, \alpha}=\{u:Re\left[\frac{u-w}{u-\lambda} \right] > \frac{\alpha}{\alpha -1}\}.$$ We note that for $N>1$ $$D_{\lambda, \frac{N}{N+1}}=\{u:Re\left[\frac{u-w}{u-\lambda} \right] < -N\}$$ and $$D_{\lambda, \frac{N}{N-1}}=\{u:Re\left[\frac{u-w}{u-\lambda} \right] > N\}.$$
For $0\leq\alpha$, a direct computation using a change of variables and polar coordinates shows $$-\frac{1}{\pi}\int_{D_{\lambda,\alpha}} Re\frac{u-w}{u-\lambda }\frac{1}{\vert u-w\vert^2} da(u ) = \frac{-1}{\pi}\int_{0}^{\frac{\pi}{2}} \log(\alpha^2\cos^2\theta + \sin^2\theta)d\theta =\ln\frac{2}{1+\alpha},$$ and for $\alpha <0$
$$\label{real}-\frac{1}{\pi}\int_{D_{\lambda,\alpha}} Re\frac{u-w}{u-\lambda }\frac{1}{\vert u-w\vert^2} da(u ) =\ln\frac{2}{1+\vert\alpha\vert}.$$
In a similar manner, for $\beta\neq 0$ in $\bf R,$ let $\Delta_{\lambda,\beta}$ be the disc of radius $r_{\beta} = \vert\frac{\beta (\lambda - w)}{2}\vert$ centered at $c_{\beta} =\lambda +\frac{i(\lambda-w)\beta}{2}.$
For $\beta >0,$$$\Delta_{\lambda,\beta}=\{u:Im\frac{u-w}{u-\lambda} < -\frac{1}{\beta}\},$$ and for $\beta <0$
$$\Delta_{\lambda,\beta}=\{u:Im\frac{u-w}{u-\lambda} > -\frac{1}{\beta}\}.$$
Another polar-coordinates computation shows that for $\beta\neq 0$
$$\label{Imaginary}\begin{gathered}-\frac{1}{\pi}\int_{\Delta_{\lambda,\beta}} Im\frac{u-w}{u-\lambda }\frac{1}{\vert u-w\vert^2} da(u ) =\\ \frac{1}{\pi}\int_{0}^{\pi} \arctan (\beta +\cot\theta) d\theta =\arctan\{\frac{\beta}{2}\}.\end{gathered}$$
At the end of this paper it will be shown that the identities (\[real\]) and (\[Imaginary\]) are closely aligned and derivable from (\[KFC\]) in the case where the operator $T$ is the unilateral shift.
One consequence of the identities (\[real\]) and (\[Imaginary\]) is the following. Given $\varepsilon >0,$ there is an $M=M(\varepsilon)$ independent of $\lambda$ and $w$ such that for $N>M$ $$\label{RE} \frac{1}{\pi}\int_{\left| Re\frac{u-w}{u-\lambda }\right| >N} \left| Re\frac{u-w}{u-\lambda }\right|\frac{1}{\vert u-w\vert^2} da(u ) < \varepsilon$$
and $$\label{IM}\frac{1}{\pi}\int_{\vert Im\frac{u-w}{u-\lambda }\vert>N} \left| Im\frac{u-w}{u-\lambda }\right|\frac{1}{\vert u-w\vert^2} da(u ) < \varepsilon .$$ For technical reasons, it is required that $M>1.$
Using the above, one can directly establish the sectional continuity of the function $E.$
\[continuity\] For $w$ fixed in $\mathbb{C}$ the function $E_{g}(\cdot, w)$ is continuous on $\mathbb{C}.$
Case 1. $\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )<\infty .$
Let $\varepsilon >0.$ Let $N >M$ be fixed so that the inequalities (\[RE\]) and (\[IM\]) hold for all $\lambda\neq w.$ Choose $\delta_{0} >0$ such that for $0<\delta\leq\delta_{0}$ one has
$$\label{bound}\frac{1}{\pi}\int_{\vert u-w\vert <\delta} \frac{g(u)}{\vert u-w\vert^2} da(u )<\frac{\varepsilon}{N} .$$
There exists a $\delta_1< \delta_0$ such that for $\vert \lambda -w\vert <\delta_1,$ one has $$\bigl\lvert\frac{u-w}{u-\lambda} -1\bigl\rvert < \varepsilon\ \text{for}\ \vert u-w\vert \geq \delta_0.$$
Note for $\lambda$ sufficiently close to $w,$ say for $\vert \lambda -w \vert <\delta_{2},$ the set $$U_N^{\lambda}=\{u:\vert Re\frac{u-w}{u-\lambda }\vert >N\} = D_{\lambda ,\frac{N}{N-1}}\cup D_{\lambda ,\frac{N}{N+1}}$$ and the set $$V_N^{\lambda}= \{u:\vert Im\frac{u-w}{u-\lambda }\vert >N\} =\Delta_{\lambda, 1/ N} \cup\Delta_{\lambda, -1/ N}$$ will be in the disc $\{ u: \vert u-w\vert <\delta_{1}\}.$ For $\vert \lambda -w\vert <\delta_2,$ we estimate $$D(\lambda ):=\left|\frac{1}{\pi}\int_{\mathbb{C}} \frac{u-w}{u-\lambda}\frac{g(u)}{\vert u-w\vert ^2} da(u) - \frac{1}{\pi}\int_{\mathbb{C}}\frac{g(u)}{\vert u-w\vert ^2} da(u)\right|$$ separately over the sets $$A = \{ u:\vert u-w\vert \geq \delta_1\}, B = \{u: \vert u-w\vert < \delta_1\}\backslash (U_N^{\lambda}\cup V_{N}^{\lambda}), \text{and}\ C= U_N^{\lambda}\cup V_{N}^{\lambda}.$$
On $A$ we have the estimate
$$\begin{aligned} \left|\frac{1}{\pi}\int_{A} \frac{u-w}{u-\lambda}\frac{g(u)}{\vert u-w\vert ^2} da(u) - \frac{1}{\pi}\int_{A}\frac{g(u)}{\vert u-w\vert ^2} da(u)\right| \leq \\ \frac{1}{\pi}\int_{A} \left|\frac{u-w}{u-\lambda}-1 \right|\frac{g(u)}{\vert u-w\vert ^2} da(u) \leq \frac{\varepsilon}{\pi} \int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2}da(u) . \end{aligned}$$
On $B$ $$\begin{gathered} \left|\frac{1}{\pi}\int_{B} \frac{u-w}{u-\lambda}\frac{g(u)}{\vert u-w\vert ^2} da(u) - \frac{1}{\pi}\int_{B}\frac{g(u)}{\vert u-w\vert ^2} da(u)\right| \leq \\ \frac{1}{\pi}\int_{B}\left|\frac{u-w}{u-\lambda}\right|\frac{g(u)}{\vert u-w\vert^2}da(u) + \frac{1}{\pi}\int_{B}\frac{g(u)}{\vert u-w\vert^2}da(u) \leq\\ \frac{1}{\pi}\int_{B}\left| Re \left[\frac{u-w}{u-\lambda}\right]\right|\frac{g(u)} {\vert u-w\vert^2}da(u) + \frac{1}{\pi}\int_{B}\left| Im \left[\frac{u-w}{u-\lambda}\right]\right|
\frac{g(u)}{\vert u-w\vert^2} da(u) +\frac{\varepsilon}{N} \leq \\ 2N\frac{1}{\pi}\int_{B}\frac{g(u)}{\vert u-w\vert ^2} da(u) +\frac{\varepsilon}{N} \leq 3\varepsilon. \end{gathered}$$
On $C$
$$\begin{gathered} \left|\frac{1}{\pi}\int_{C} \frac{u-w}{u-\lambda}\frac{g(u)}{\vert u-w\vert ^2} da(u) - \frac{1}{\pi}\int_{C}\frac{g(u)}{\vert u-w\vert ^2} da(u)\right| \leq \\ \frac{1}{\pi}\int_{C}\left| Re\left[\frac{u-w}{u-\lambda} \right] \right| \frac{g(u)}{\vert u-w\vert ^2} da(u) + \frac{1}{\pi}\int_{C}\left| Im\left[\frac{u-w}{u-\lambda}\right]\right| \frac{g(u)}{\vert u-w\vert ^2} da(u) + \frac{1}{\pi}\int_{C}\frac{g(u)}{\vert u-w\vert ^2} da(u).\end{gathered}$$ The last integral is less than the corresponding integral over $\{u:\vert u -w\vert < \delta_1\}$ and consequently less than $\varepsilon.$ The first of the two integrals on the right side of this last inequality can be estimated as follows:
$$\begin{gathered} \frac{1}{\pi}\int_{C}\left| Re\left[\frac{u-w}{u-\lambda} \right] \right| \frac{g(u)}{\vert u-w\vert ^2} da(u) = \\ \frac{1}{\pi}\int_{U_N^{\lambda}}\left| Re\left[\frac{u-w}{u-\lambda} \right] \right|\frac{g(u)}{\vert u-w\vert ^2} da(u) + \frac{1}{\pi}\int_{V_N^{\lambda}\backslash U_N^{\lambda}}\left| Re\left[\frac{u-w}{u-\lambda} \right] \right| \frac{g(u)}{\vert u-w\vert ^2} da(u).\end{gathered}$$ The first of these last two integrals is less than $\varepsilon$ and using the fact that $\left|Re\left[\frac{u-w}{u-\lambda} \right] \right|$ is less than $N$ off $U_N^{\lambda},$ it follows from equation (\[bound\]) that the second integral is also less than $\varepsilon.$ A similar argument shows $$\frac{1}{\pi}\int_{C}\left| Im\left[\frac{u-w}{u-\lambda}\right]\right| \frac{g(u)}{\vert u-w\vert ^2} da(u) < 2\varepsilon.$$ It follows from the above discussion that for $\vert \lambda -w\vert <\delta_{2}$ $$D(\lambda) < \left( 7 + \frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u)\right)\varepsilon .$$ This completes the proof in Case 1.
Case 2. $\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )=\infty .$ This case is easier then the first case and was established in [@Clancey84]. The result follows once it is shown that $\lim_{\lambda\to w} \vert E(\lambda, w)\vert = 0.$ For completeness, we include the details. The notation $D(w,r)$ will be used for the disc in $\mathbb{C}$ centered at $w$ of radius $r.$ By the monotone convergence theorem $$\lim_{\lambda\to w}\frac{1}{\pi}\int_{D(w, \vert\lambda - w\vert )} \frac{g(u)}{\vert u-w\vert^2} da(u) = \infty.$$ In the notation introduced above, the disc $D(w, \vert\lambda - w\vert )$ coincides with $D_{\lambda, -1},$ which can be written as the disjoint union $(D_{\lambda, -1}\backslash D_{\lambda, 0})\cup D_{\lambda, 0}.$ Then $$\begin{gathered}\vert E(\lambda, w)\vert = \exp -\frac{1}{\pi}\int_{\mathbb{C}} Re\left[\frac{u-w}{u-\lambda }\right]\frac{g(u)}{\vert u-w\vert^2} da(u )\leq \\
\left(\exp -\frac{1}{\pi}\int_{\mathbb{C}\backslash D_{\lambda, -1}} Re\left[\frac{u-w}{u-\lambda}\right]\frac{g(u)}{\vert u-w\vert^2} da(u )\right)\left(\exp -\frac{1}{\pi}\int_{D_{\lambda, 0}} Re\left[\frac{u-w}{u-\lambda}\right]\frac{g(u)}{\vert u-w\vert^2} da(u )\right)\leq\\ 2 \exp -\frac{1}{2\pi}\int_{\mathbb{C}\backslash D( w,\vert \lambda -w\vert)} \frac{g(u)}{\vert u-w\vert^2} da(u ).\end{gathered}$$ Here we used the facts that $Re\left[\frac{u-w}{u-\lambda}\right]$ is greater than $\frac{1}{2}$ on $\mathbb{C}\backslash D( w,\vert \lambda -w\vert)$ and non-negative on $D_{\lambda, -1}\backslash D_{\lambda, 0}$ as well as the result that $$-\frac{1}{\pi}\int_{D_{\lambda, 0}} Re\left[\frac{u-w}{u-\lambda}\right]\frac{1}{\vert u-w\vert^2} da(u) =\ln 2.$$ As noted above, the desired result now follows from the monotone convergence theorem.
Assuming, as is the case here, that $0\leq g\leq 1,$ one consequence of the last integral inequality is the inequality $$\vert E_{g}(\lambda, w) \vert\leq 2, \ \text{for all}\ \lambda, w$$ with equality holding if and only if $g$ is the characteristic function of a disc and where $\lambda, w$ are antipodal boundary points.
\[forrep\] With $w$ fixed, with minor modifications, the proof of Case 1 in Theorem \[continuity\] establishes the continuity of the integral in (\[logE\]) as a function of $\lambda$ for any compactly supported bounded measurable function $g$ under the assumption $$\frac{1}{\pi}\int_{\mathbb{C}} \frac{|g(u)|}{\vert u-w\vert^2} da(u )<\infty .$$
Local Lipschitz continuity
--------------------------
The function $E_{g}$ is Lipschitz at almost every point in the support of $g.$ To establish this we will use the following elementary lemmas.
Let $h=h(x)$ be continuous on the interval $[0,R]$ with $h(0)=0$ and $R>0$ is fixed. For $t\in (0,R]$ define $$H(t) = \int_{t}^{R}\frac{h(x)}{x}dx.$$ Given $0< \varepsilon $ there exists a $\delta >0$ and constant $K=K(\varepsilon)$ such that for $0<t<\delta$ one has the estimate $$\vert H(t)\vert\leq K -\varepsilon \ln t$$
Let $\varepsilon >0$ and find $\delta >0$ such that $0<t<\delta$ implies $\vert h(t)\vert <\varepsilon.$ Then for $0<t<\delta$ $$\begin{aligned}\vert H(t)\vert \leq \int_{t}^{\delta}\vert h(x)\vert\frac{dx}{x} + \int_{\delta}^{R}\vert h(x)\vert\frac{dx}{x}\leq\varepsilon\int_{t}^{\delta}\frac{dx}{x}+M\ln R -M\ln\delta =\\M\ln R +(\varepsilon-M)\ln \delta - \varepsilon\ln t ,\end{aligned}$$ where $M$ is the maximum of $\vert h\vert$ on $[0,R].$ The result follows with $K(\varepsilon ) = M\ln R +(\varepsilon - M)\ln\delta.$
We continue to assume that $g$ is a measurable function with compact support satisfying $0\leq g\leq 1$ and introduce the notation $\bold{L}_{g}$ for the set of points of positive Lebesgue density of $g.$ Thus $\bold{L}_{g}$ is the set of points $w$ that satisfy $$\lim_{R\to 0}\frac{1}{\pi r^2} \int_{D(w, r)} g(u) da(u) = \gamma >0.$$
\[Lip\] Let $g=g(u)$ be a bounded non-negative measurable function defined in a neighborhood of $D(0,R)$ of where $0\in\bold{L}_{g},$ that is, $$\lim_{r\to 0}\frac{1}{\pi r^2}\int_{D(0,r)}g(u)da(u) =\gamma >0.$$ Given $\varepsilon >0$ there is a $\delta =\delta (\varepsilon )$ and a constant $K = K(\varepsilon )$ such that with $0<t<\delta$ one has the estimate $$\frac{1}{2\pi}\int_{D(0,R)\backslash D(0,t)}\frac{g(u)}{\vert u\vert^2}da(u) < K - (\gamma -\varepsilon )\ln t.$$
For $0\leq s\leq R,$ let $G(s) = \frac{1}{2\pi}\int_{0}^{2\pi} g(se^{i\theta})d\theta$ and for $0<t\leq R$ set $$f(t)=\int_{t}^{R}\frac{G(s)}{s^2}sds.$$ Note that $$f(t) =\frac{1}{2\pi}\int_{D(0,R)\backslash D(0,t)}\frac{g(u)}{\vert u\vert^2}da(u).$$Applying integration by parts for Lebesgue-Stieltjes integration, see, for example, [@Integration], to this first integral above for $f$ one obtains the identity $$\begin{gathered}f(t) = \frac{\int_{0}^{x}G(s)sds}{x^2}\bigg\rvert_{t}^{R} + \int_{t}^{R}\{\frac{2}{x^2}\int_{0}^{x}G(s)sds\}\frac{dx}{x}=\\\frac{1}{R^2}\int_{0}^{R}G(s)sds-\frac{1}{t^2}\int_{0}^{t}G(s)sds + \int_{t}^{R}\left[\frac{2}{x^2}\int_{0}^{x}G(s)sds-\gamma\right]\frac{dx}{x}+\gamma\ln R-\gamma\ln t\end{gathered}.$$ We note that as $t\to 0$ the second term in this last expression approaches $\frac{1}{2}\gamma$ and as a consequence is bounded on $[0,R].$ If one applies the preceding lemma to the third integral in the right side of this last identity with$$h(x)=\frac{2}{x^2}\int_{0}^{x}G(s)sds-\gamma$$ the result follows.
\[main\] Let $w$ be in $\bold{L}_{g},$ that is, assume $$\lim_{\lambda\to w}\frac{1}{\pi\vert\lambda - w\vert ^2}\int_{D(w,\vert\lambda -w\vert)}g(u)da(u) =\gamma >0,$$ so that, $E(w,w)=0.$Then the function $E_{g}(\lambda , w)$ is Lipschitz at $w.$ More specifically, given $\varepsilon <\gamma$ there is a disc $D(w,\delta)$ with $$|E(\lambda, w)|\leq K\vert\lambda -w\vert^{\gamma-\varepsilon}, \ \ \lambda\in D(w, \delta).\label{lipord}$$
First note that the disc $D(w,\vert\lambda -w\vert)$ is precisely $D_{\lambda, -1}$ and the complement of this disc is precisely the set where $Re\left[\frac{u-w}{u-\lambda}\right]\geq\frac{1}{2}.$ As a consequence $$\begin{gathered}\label{prelim}\vert E(\lambda, w)\vert = \exp -\frac{1}{\pi}\int_{\mathbb{C}} Re\frac{u-w}{u-\lambda }\frac{g(u)}{\vert u-w\vert^2} da(u )\leq \\ 2\exp -\frac{1}{2\pi}\int_{\mathbb{C}\backslash D(w,\vert\lambda - w\vert )} \frac{g(u)}{\vert u-w\vert^2} da(u ). \end{gathered}$$ This last identity is obtained by writing the integral $$-\frac{1}{\pi}\int_{D(w,\vert\lambda -w\vert)} Re\frac{u-w}{u-\lambda }\frac{g(u)}{\vert u-w\vert^2} da(u)$$ over $D(w,\vert\lambda -w\vert) = D_{\lambda, -1}$ as the sum of the integrals over $D_{\lambda, -1}\backslash D_{\lambda, 0}$ and $D_{\lambda, 0}.$ The first of these integral being negative contributes nothing to the last inequality. In the second integral one can replace $g$ by $1$ and use the identity $$-\frac{1}{\pi}\int_{D_{\lambda, 0}} Re\frac{u-w}{u-\lambda }\frac{1}{\vert u-w\vert^2} da(u ) =\ln 2$$ to obtain the above bound. The result then follows from (\[prelim\]) and Proposition \[Lip\].
A simple example to keep in mind is the case where $g$ is the characteristic function $\mathbbm{1}_{\bf{D}} $ of the unit disc $\bf{D}.$ In this case, we denote the function $E_{g}$ by $E_{\bf{D}}.$ One computes $$\label{unitdisc}E_{\bf{D}}(\lambda, w) = \begin{cases} \frac{\vert\lambda - w\vert^2}{1-\overline{w}\lambda} & \text{if}\ \lambda, w\in\bf{D} \\ \\
\overline{\left[\frac{w-\lambda}{w}\right]} & \text{if}\ \lambda\in\bf{D}\ \text{and}\ w\notin\bf{D}\\ \\1-\frac{1}{\overline{w}\lambda} &\text{if}\ \lambda, w\notin\bf{D} .\end{cases}$$
We remark that this example suggests that, in general, the local Lipschitz order of $E_{g}(w,\cdot)$ at $w$ in (\[lipord\]) should be $2(\gamma - \varepsilon).$ This is the case if $g$ is smooth in a neighborhood of $w.$
Cauchy Transform Representations of $E_{g}$
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Given a compactly supported measure $\mu$ on $\mathbb{C},$ the Cauchy Transform $\hat{\mu}$ is the locally integrable function defined for $a.e.$ $\lambda$ in $\mathbb{C}$ by $$\label{ct} \hat{\mu} (\lambda) = \frac{1}{\pi}\int_{\mathbb{C}}\frac{d\mu (u)}{u-\lambda} .$$ For a measure of the form $d\mu = fda,$ where $f$ is a compactly supported integrable function, the Cauchy transform will be denote $\hat{f}.$ A good place to read about the Cauchy Transform is the monograph of Garnett [@Garnett]. In the sense of distributions $$-\bar{\partial}\hat{\mu} = \mu.$$ Formally, for $w$ fixed and $\lambda\neq w,$ one expects in the sense of distributions $$\label{dbarE} \bar{\partial_{\lambda}} E_{g}(\lambda, w) =\frac{E_{g}(\lambda, w)}{\overline{\lambda - w}}g(\lambda)$$ and, consequently, $$\label{distr} E_{g}(\lambda, w) =1-\left(\frac{E_{g}(\lambda, w)}{\overline{\lambda - w}}g(\lambda )\right)^{\widehat{ }}.$$ Here, $hat$ denotes the distributional Cauchy transform, given for a distribution $S$ with compact support on the test function $\phi$ by $$<\phi, \hat{S} > = -<\hat{\phi},S>.$$ Taking into account behavior at infinity and Weyl’s Lemma, for $w$ fixed and $\lambda\neq w,$ formally, one further expects $$\label{keyidentity} E(\lambda, w) = 1-\frac{1}{\pi}\int_{\mathbb{C}}\frac{E(u,w)}{\overline{u-w}} g(u)\frac{da(u)}{u-\lambda}.$$ All of the above distributional identities are subtle. Although, for $w$ fixed, the function $\frac{E_{g}(\lambda, w)}{\overline{\lambda - w}}g(\lambda)$ is integrable, it is unclear whether the identity (\[keyidentity\]) holds for all $\lambda.$ The goal of this subsection is to establish circumstances where, for $w$ fixed, (\[keyidentity\]) holds for all $\lambda.$ More specifically, it will be shown that this is the case when $w\in\bold{L}_{g}$ or when $\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )<\infty .$ We will take a somewhat ad hoc path to the results.
We will first consider an easy case. Let $\bold{G}$ denote the essential support of the function $g.$ Thus $z$ is in $\bold{G}$ if and only if every neighborhood of $z$ intersects the set $\{u : g(u)\neq 0\}$ in a set of positive measure. For the case $w\notin \bold{G}$ one can give a direct proof that (\[keyidentity\]) holds for all $\lambda.$ To this end, recall that for $h$ and $k$ bounded measurable functions with compact support in $\mathbb{C}$ one has $$\hat{h}\hat{k} = \widehat{\hat{h}k}+\widehat{h\hat{k}}$$ see, for example, [@Garnett p.107]. We remark that fact that $h$ and $k$ are bounded with compact support implies that $\hat{h}$ and $\hat{k}$ are continuous. As a consequence, for $h$ bounded with compact support and $N\geq 1$ one has $$\label{power}\hat{h}^{N} = N\widehat{\hat{h}^{N-1} h}.$$.
Let $w\notin\bold{G}.$ Then (\[keyidentity\]) holds for all $\lambda$ in $\mathbb{C}.$
Applying the identity (\[power\]) with $h(u)=\frac{g(u)}{\overline{u-w}}$ one sees $$\begin{gathered}1- E(\lambda, w) = \sum_{N=1}^{\infty}\frac{ (-1)^{N-1} }{N!}
\left( \frac{1}{\pi}\int_{\mathbb{C}}\frac{g(u)}{\overline{u-w}} \frac{da(u)}{u-\lambda}\right)^N = \\ \sum_{N=1}^{\infty}\frac{ (-1)^{N-1} }{(N-1)!} \frac{1}{\pi}\int_{\mathbb{C}}
\left( \frac{1}{\pi}\int_{\mathbb{C}}\frac{g(v)}{\overline{v-w}} \frac{da(v)}{v-u}\right)^{N-1} \frac{g(u)}{\overline{u-w}}\frac{da(u)}{u-\lambda} =\\ \frac{1}{\pi}\int_{\mathbb{C}}\frac{E(u,w)}{\overline{u-w}}g(u)\frac{da(u)}{u-\lambda},\end{gathered}$$ where, for $\lambda$ fixed, the interchange of summation and integration to produce the last expression follows from the uniform convergence of the series for $E_{g} (u, w)$ in the variable $u$ on $\bold{G},$ which is the support of the finite measure $\frac{g(u)}{\overline{u-w}}\frac{da(u)}{u-\lambda}.$
The extent of the validity of (\[power\]) for arbitrary planar integrable $h$ is unclear; however, when $N=2$ it does hold $a.e.$ when $\hat{h}$ is integrable with respect to $\vert h\vert da$ (see, [@Volberg]).
We continue our study of the validity of (\[dbar\]) and (\[distr\]) by first deriving and analogue of (\[power\]) for functions of the form $$\label{singh} h_{w}(u)=\frac{g(u)}{\overline{u-w}}.$$
Without loss of generality, it is sufficient to consider this last identity when $w=0.$ We begin by noting that for $\lambda\neq 0$ $$\label{power1}\int_{\mathbb{C}}\frac{g(u)}{\overline{u}(u-\lambda)} da(u)= \frac{1}{\lambda}\left[\int_{\mathbb{C}}\frac{ug(u)}{\overline{u}(u-\lambda)}da(u) - \int_{\mathbb{C}}\frac{g(u)}{\overline{u}}da(u)\right].$$ We introduce the notations$$h_{0}(u)=\frac{g(u)}{\overline{u}}\ \ k_{0}(u)=\frac{ug(u)}{\overline{u}}\ \ C=-\frac{1}{\pi}\int_{\mathbb{C}}\frac{g(u)}{\overline{u}}da(u).$$ Thus equation (\[power1\]) can be written in the compact form $$\widehat{h_{0}}(\lambda) =\frac{1}{\lambda}\left[ \widehat{k_{0}}(\lambda) + C\right]\ \lambda\neq 0.$$ Note that $$C=-\widehat{k_{0}}(0).$$Since the function $k_{0}$ is bounded with compact support, we can apply (\[power\]) to this function. Using a binomial expansion we see for $\lambda\neq 0,$
$$\label{binomial} \begin{gathered} \widehat{h_{0}}^{N}(\lambda) = \frac{1}{\lambda^N}
\sum_{j=0}^{N}\frac{N!}{j!(N-j)!}\widehat{k_{0}}^{j}(\lambda)C^{N-j} =\\
\frac{C^{N}}{\lambda^{N}} + \frac{N}{\lambda^N}\sum_{j=1}^{N}\frac{(N-1)!}{(j-1)!(N-j)!} \widehat{(\widehat{k_{0}})^{j-1}k_{0}}(\lambda)C^{N-j}= \\
\frac{C^{N}}{\lambda^N} + \frac{N}{\lambda^N}\left( \left(\widehat{k_{0}+ C}\right)^{N-1}k_{0}\right) ^\bold{\widehat{ }}(\lambda).\end{gathered}$$
Therefore we have the following:
Let $h_{0}$ be the function defined by (\[singh\]) with $w=0,$ $C=-\frac{1}{\pi}\int_{\mathbb{C}}\frac{g(u)}{\overline{u}}da(u),$ and $\lambda\neq 0.$ For $N\geq 1$ $$\widehat{h_{0}}^{N}(\lambda) = \frac{C^{N}}{\lambda^{N}} + \frac{N}{\lambda^{N}}\frac{1}{\pi}\int_{\mathbb{C}}u^{N}\left(\frac{1}{\pi}\int_{\mathbb{C}}\frac{h_{0}(v)}{v-u}da(v)\right)^{N-1}\frac{h_{0}(u)}{u-\lambda}da(u).$$ As a consequence, in the sense of distributions, on $\mathbb{C}\backslash \lbrace 0\rbrace,$ $$\label{power2} -\overline{\partial}\left( \widehat{h_{0}}^{N}\right) = N\widehat{h_{0}}^{N-1}h_{0}.$$
We remark that the function $ f_{N}(u)=u^{N}\left(\frac{1}{\pi}\int_{\mathbb{C}}\frac{h_{0}(v)}{v-u}da(v)\right)^{N-1}$ appearing in this last integral extends to be continuous on $\mathbb{C}.$
On $\mathbb{C}\backslash \lbrace 0\rbrace,$ the series $$E(\lambda,0) = \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\widehat{h_0}^{n}(\lambda)$$ converges in the sense of distributions. Consequently, using (\[power2\]), on $\mathbb{C}\backslash \lbrace 0\rbrace,$ we have the distributional identity $$\begin{gathered}-\overline{\partial}E(\lambda, 0) = \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}(-\overline{\partial})(\widehat{h_0}^{n})(\lambda) =\\ -\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(n-1)!}(\widehat{h_0}^{n-1})h_{0}(\lambda)=\\ \frac{E(\lambda, 0)}{\overline{\lambda}}g(\lambda ).\end{gathered}$$
Thus, for fixed $w,$ we have established the distributional identity (\[dbarE\]) which we emphasize is in the sense of distributions on $\mathbb{C}\backslash \lbrace w\rbrace.$ The distribution $$\frac{E(u, w)}{\overline{u-w}}g(u)$$ now considered as a locally integrable distribution on $\mathbb{C}$ differs from the distribution $$-\overline{\partial}[1-E(u,w)],$$ on $\mathbb{C},$ by a first-order distribution supported on $\lbrace w\rbrace.$ Consequently in the sense of distributions on $\mathbb{C}$ $$\label{firstorder} -\overline{\partial}[1-E(u,w)] - \frac{E(u, w)}{\overline{u-w}}g(u) =\alpha\delta_{w} + \beta\partial\delta_{w}+ \gamma\bar{\partial}\delta_{w},$$ for some constants $\alpha,\beta,\gamma.$ Let $S_{w}$ be the locally integrable distribution $$\label{rightside} S_{w}(u) =1- E(u,w) - \frac{1}{\pi}\int_{\mathbb{C}}\frac{E(u,w)}{\overline{u-w}}\frac{g(u)}{u-\lambda} da(u)$$ on $\mathbb{C}.$ Then equation (\[firstorder\]) can be written in the form $$-\overline{\partial} S_{w} = \alpha\delta_{w} + \beta\partial\delta_{w}+ \gamma\bar{\partial}\delta_{w}.$$ We remark that when $w$ is a Lebesgue point for the function $g,$ then the estimate (\[lipord\]) of Theorem \[main\] implies that $\frac{E(\lambda, w)}{\overline{\lambda -w}}g(\lambda)$ is in $L^{2+\sigma}(\mathbb{C})$ for some $\sigma >0.$ As indicated in Remark \[continuity\] with $w$ fixed and$\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )<\infty,$ the integral on the right side in \[rightside\] is continuous. Consequently, in these cases, the distribution $S_{w}$ is a continuous function. It is an exercise in distribution theory to show that if $S$ is continuous and $-\overline{\partial} S = \alpha\delta_{w} + \beta\partial\delta_{w}+ \gamma\bar{\partial}\delta_{w},$ then $\alpha=\beta=\gamma=0$ and, therefore, $S$ is an entire function. Since $S_{w}$ vanishes at infinity, S must be zero. We have therefore established the following:
\[rep\] Let $w$ be in $\bold{L}_{g}$ or satisfy $\frac{1}{\pi}\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )<\infty .$ Then (\[keyidentity\]) holds for all $\lambda$ in $\mathbb{C}.$
It would be interesting to know the extent to which (\[keyidentity\]) holds and, in particular, whether the integral in this equation always converges at $w.$
We also remark that when $f$ is a compactly supported function that belongs to $L^{q}$ for $q >2,$ then $\hat{f}$ satisfies the Lipschitz condition $|\hat{f} (\lambda) - \hat{f} (w)| \leq K|\lambda - w|^{1-\frac{2}{q}}.$ For a proof of this last statement, see [@Brennan]. The results on the function $E$ in Theorem \[rep\] and Theorem \[main\] fall inline with this result.
Operators with one-dimensional self-commutator
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As described in the introduction, there is a close connection between the function $E_{g}$ and operators with one dimensional self-commutator. In this section, we will describe some of these connections and derive a few consequences of the discussion of $E_{g}$ from the preceding section. Let $T$ be a bounded operator on the Hilbert space $\mathcal H$ satisfying $T^*T-TT^* = \varphi\otimes\varphi,$ where $\varphi$ is an element of $\mathcal H.$ It will always be assumed that the operator $T$ is irreducible, equivalently, there no non-trivial subspaces of $\mathcal H$ reducing $T$ where the restriction is a normal operator. Note that we have elected to assume $T$ is hyponormal, that is, the self-commutator $[T^*,T]=T^*T-TT^*$ is non-negative. The spatial behaviors of the hyponormal operator $T$ and its cohyponormal adjoint $T^*$ are quite distinct. As noted in the introduction, for $\lambda$ in $\mathbb{C},$ there is a unique solution of the equation $T_{\lambda}^{*}x=\varphi$ orthogonal to the kernel of $T_{\lambda}^{*} = (T-\lambda)^{*},$ which will be denoted $T_{\lambda}^{*-1}\varphi.$ This follows easily from the range inclusion theorem of Douglas [@Douglas] when one notes that for all $\lambda\in\mathbb{C}$ one has $T_{\lambda}^*T_{\lambda} - T_{\lambda}T_{\lambda}^* =\varphi\otimes\varphi$ and, consequently, $T_{\lambda}^*T_{\lambda}\geq\varphi\otimes\varphi.$ The $\mathcal H$-valued function $T_{\lambda}^{*-1}\varphi$ defined for all $\lambda \in \mathbb{C}$ is called the global-local resolvent associated with the operator $T^*.$ Using the result of Douglas mentioned above one can also see that for all $\lambda\in\mathbb{C}$ there is a contraction operator $K(\lambda)$ satisfying $T_{\lambda}^* = K(\lambda) T_{\lambda},$ where $T_{\lambda} = T-\lambda.$ The contraction operator $K(\lambda)$ is unique if one requires it to be zero on the orthogonal complement of the range of $T_{\lambda}.$ The following identity, specialized here to the case of rank-one self-commutators, was first established for general hyponormal operators in [@Wadhwa] $$I=T_{\lambda}^{*-1}\varphi\otimes T_{\lambda}^{*-1}\varphi + K(\lambda) K^{*}(\lambda) + P_{\lambda},\ \lambda\in\mathbb{C},$$ where $P_{\lambda}$ denotes the orthogonal projection onto the (at most one-dimensional) kernel of $T^*_{\lambda}.$ There is no equivalent of the global-local resolvent for the operator $T.$ To see this, we recall the following:
\[Kinvert\] Let $T$ be an irreducible operator on the Hilbert space $\mathcal H$ with one-dimensional self-commutator $T^*T-TT^* = \varphi\otimes\varphi.$ Fix $\lambda\in\mathbb{C}.$ Then the following are equivalent:
1. There is a solution of the equation $(T-\lambda)x=\varphi$
2. The operator $K(\lambda)$ is invertible
3. $\| T_{\lambda}^{*-1}\varphi \| < 1$
4. $\int_{\mathbb{C}} \frac{g(u)}{\vert u-w\vert^2} da(u )<\infty,$ where $g$ is the principal function of $T.$
Let $\rho_{T}(\varphi)$ be the set of $\lambda\in\mathbb{C}$ such that there is a solution of the equation $(T-\lambda)x=\varphi.$ The last condition in the above proposition shows that condition ($I$) cannot hold at Lebesgue points of $g.$ This implies the result of Putnam [@Putnam] which establishes that the interior of $\rho_{T}(\varphi)\cap\sigma (T),$ where $\sigma (T)$ denotes the spectrum of $T,$ is empty and points out the significant difference between the local resolvents $T_{\lambda}^{*-1}\varphi$ and $T_{\lambda}^{-1}\varphi. $
We will continue to use the notation $\bold{L}_g$ for the set of points of positive Lebesgue density of a bounded measurable function $g.$ It develops that the local resolvent function is locally Lipschitz on $\bold{L}_g.$ This is the content of the following:
Let $g$ be the the principal function associated with the operator $T$ having one-dimensional self-commutator $T^*T-TT^* = \varphi\otimes\varphi.$ Let $w$ be in the set $\bold{L}_g$ with $0<\gamma =\lim_{R\to 0}\frac{1}{\pi R^{2}} \int_{D(w,\delta)}g(u)da(u).$ Then there is a disc $D(w, \delta)$ such that $$\| T_{\lambda}^{*-1}\varphi - T_{w}^{*-1}\varphi\|\leq K|\lambda - w|^{\gamma -\varepsilon},$$ for $\lambda\in D(w,\delta)\cap\bold{L}_{g},$ where $K$ is a constant, and $\epsilon < \gamma.$
It follows from Theorem \[main\] that there is a disc $D(w, \delta)$ so that for $\lambda\in D(w,\delta)$ one has $$\label{est} |E_{g}(\lambda,w)| \leq K|\lambda - w|^{\gamma -\varepsilon}.$$ Since, $\| T_{\lambda}^{*-1}\varphi\| = 1$ for $\lambda\in\bold{L}_{g}$ $$\begin{gathered} \| T_{\lambda}^{*-1}\varphi - T_{w}^{*-1}\varphi\|^2=\\\|T_{\lambda}^{*-1}\varphi \|^2 -<T_{\lambda}^{*-1}\varphi, T_{w}^{*-1}\varphi > - <T_{w}^{*-1}\varphi, T_{\lambda}^{*-1}\varphi >
+ \|T_{w}^{*-1}\varphi\|^2= \\1 - <T_{\lambda}^{*-1}\varphi, T_{w}^{*-1}\varphi > + 1 - <T_{w}^{*-1}\varphi, T_{\lambda}^{*-1}\varphi > =\\ \overline{E_{g}(\lambda,w)} +E_{g} (\lambda,w) \leq 2|E_{g}(\lambda, w)|,
\end{gathered}$$ where we have made use of Theorem \[KFC\]. The result follows from (\[est\]).
The result in this last proposition will not be true at points $w$ in the spectrum $\sigma (T)$ where any of the conditions in Proposition \[Kinvert\] hold. It is easy to construct an example of an operator $T$ with this property, so that, $w\in\sigma (T)$ and $\|T_{w}^{*-1}\varphi\| <1.$ By Putnam’s result mentioned above, in every neighborhood of $w,$ there will be points with $\| T_{\lambda}^{*-1}\varphi\| =1.$ Thus the conclusion of this last proposition cannot hold at the point $w.$
Integral representations using the global-local resolvent
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The connection between the global local resolvent and the principal function can be seen in the following result from [@ClanceyIU]
\[CL\] Let $T$ be an operator with one-dimensional self-commutator $T^*T-TT^* = \varphi\otimes\varphi$ and $g$ the associated principal function. For $r=r(u)$ a rational function with poles off the spectrum of $T$ and $\lambda\in\mathbb{C}$ $$\label{Cauchy} (r(T)\varphi, T_{\lambda}^{*-1}\varphi) = \frac{1}{\pi}\int_{\mathbb{C}} r(u) \frac{g(u)}{u-\lambda }da(u).$$ As a consequence, in the sense of distributions, $$\label{dbar} -\bar{\partial} (\varphi, T_{\lambda}^{*-1}\varphi) = g.$$
We record the following analogue of this last result for the operator $T^{*}.$
\[CLstar\] Let $T$ be an operator with one-dimensional self-commutator $T^*T-TT^* = \varphi\otimes\varphi$ and $g$ the associated principal function. Assume $\lambda\in\bold{L}_{g}.$ Let $p=p(\bar{u})$ be a polynomial in the variable $\bar{u},$ then $$\label{Cauchystar} (p(T^{*})\varphi, T_{\lambda}^{*-1}\varphi) = \frac{1}{\pi}\int_{\mathbb{C}} p(\bar{u}) \frac{ E_{g} (\lambda, u) }{u-\lambda } g(u)da(u).$$
Since $\lambda\in\bold{L}_{g},$ then by Theorems \[KFC\] and \[rep\] for all $z$ $$\label{twovariable}< T_{\lambda}^{*-1}\varphi, T_{z}^{*-1}\varphi> = 1 - E(z,\lambda) = \frac{1}{\pi}\int_{\mathbb{C}}\frac{E(u,\lambda)}{\overline{u-\lambda}} g(u)\frac{da(u)}{u-z}.$$ Equating powers of $z$ at infinity one obtains $$< T_{\lambda}^{*-1}\varphi, T^{*k}\varphi> =\frac{1}{\pi}\int_{\mathbb{C}}u^{k}\frac{E(u,\lambda)}{\overline{u-\lambda}} g(u)da(u),$$ for $k=0, 1,\cdots .$ The result follows by taking complex conjugates in this last identity.
The integral kernel $$\mathcal{T} (\lambda, u) =\frac{E_{g}(\lambda, u)}{u-\lambda}g(u),$$ appearing in Theorem \[CLstar\] $a.e.\ gda$ satisfies $\vert \mathcal{T} (\lambda, u)\vert \leq \vert u-\lambda\vert^{-\sigma}$ near $\lambda,$ where $0<\sigma <1.$ In particular, has the advantage that $\mathcal{T} (\lambda, \cdot)$ is in $L^2(\mathbb{C}).$
It is known that the closed span of $T_{\lambda}^{*-1}\varphi,\ \lambda\in\sigma(T),$ is $\mathcal{H}.$ Since the closure of $\bold{L}_{g}$ equals $\sigma (T),$ it follows that the closed span of $T_{\lambda}^{*-1}\varphi,\ \lambda\in\bold{L}_{g},$ also is $\mathcal{H}.$
We conclude this subsection with a few examples of applications of Theorem \[CLstar\].. A test-function model for the operator $T^{*}$ was constructed in [@putdist] (see also [@putinarmartin p. 151, p.261] ) using the map $$\eta\in\mathcal{D}(\mathbb{C})\ \rightarrow\ U(\eta) = \frac{1}{\pi}\int\partial\eta (\lambda) T_{\lambda}^{*-1}\varphi da(\lambda ),$$ so that $U(\bar{z}\eta) = T^{*}U(\eta).$ This test function model is dual to the distributional model described in [@ClanceyJOT], where the map $V:\mathcal{H} \rightarrow \mathcal{E}^{'}(\mathbb{C})$ defined by $V(f)=-\bar{\partial} <f, T_{\lambda}^{*-1}\varphi > $ was studied. It is easily verified that for $\eta$ a test function and $f\in\mathcal{H}$ $$<U(\eta), f> = \frac{1}{\pi}<\eta, \overline{V(f)}>.$$
Note that using (\[twovariable\]) for $w\in\bold{L}_{g}$ $$\begin{gathered} < T_{w}^{*-1}\varphi, U(\eta ) > = \frac{1}{\pi}\int_{\mathbb{C}}\overline{\partial}\overline{\eta} < T_{w}^{*-1}\varphi, T_{\lambda}^{*-1}\varphi > da(\lambda) = \\ \frac{1}{\pi}\int_{\mathbb{C}}\bar{\partial} \bar{\eta}(\lambda)\left(\frac{1}{\pi}\int_{\mathbb{C}}\frac{E(u,w)}{\overline{u-w}} \frac{g(u)}{u-\lambda} da(u)\right) da(\lambda)=\\\frac{1}{\pi}\int_{\mathbb{C}}\ \bar{\eta}(\lambda)\frac{E(\lambda,w)}{\overline{\lambda-w}} g(\lambda)da(\lambda).
\end{gathered}$$ Equivalently, $$<U(\eta ), T_{w}^{*-1}\varphi > = \frac{1}{\pi}\int_{\mathbb{C}}\eta(\lambda)\frac{E(w,\lambda )}{\lambda-w} g(\lambda)da(\lambda).$$
In some cases, when combined with Theorem \[CLstar\], this last equation allows one to to identify the vector $U(\eta) .$ For example, if $\eta (u) = \bar{u}^{k}, \ k= 0, 1, 2, \cdots$ on the set where $g$ is non-zero, then $U(\eta ) = T^{*k}\varphi.$ It is noted that this result can also be obtained directly from the definition of $U(\eta).$
For simplicity, suppose the essential support $\bold{G}$ of the principal function $g$ is contained in the open unit disc $\mathbb{D}.$ Let $$\Phi_{1} = \frac{1}{\pi}\int_{\mathbb{D}}T_{\lambda}^{*-1}\varphi da(\lambda ) \ \text{and}\ \Phi_{g} = \frac{1}{\pi}\int_{\mathbb{C}}T_{\lambda}^{*-1}\varphi g(\lambda) da(\lambda ).$$ For $w\in \bold{L}_{g},$ using (\[twovariable\]) one computes $$<T_{w}^{*-1}\varphi, \Phi_{1} +\Phi{g} > = <T_{w}^{*-1}\varphi, T\varphi>.$$ This yields $$T\varphi = \Phi_{1} + \Phi_{g} = \frac{1}{\pi}\int_{\mathbb{D}} (1+g) T_{\lambda}^{*-1}\varphi da(\lambda),$$ which gives a concrete representation of $T\varphi$ in terms of the dense family $\lbrace T_{\lambda}^{*-1}\varphi : \lambda\in\sigma (T)\rbrace.$ As is often the case, the unilateral shift provides an illuminating version of this last identity.This last identity can be viewed as a realization of the formula for $T\varphi$ given in the test function model in [@putinarmartin p. 261]. That is $$T\varphi = \frac{1}{\pi}\int\partial\left(\eta -\overline{\widehat{g\bar{\eta}}}\right) T_{\lambda}^{*-1}\varphi,$$ where the test function $\eta$ satisfies $n(\lambda) = \lambda$ on the support of $g.$
Non-cyclic vectors
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Let $T$ be an operator with one-dimensional self-commutator $T^*T-TT^* = \varphi\otimes\varphi$ and $T_{\lambda}^{*-1}\varphi\ \lambda\in\mathbb{C}$ the corresponding global-local resolvent. Given a compactly supported planar measure $\mu,$ one can define the vector $$\label{vector}\phi_{\mu} =\int_{\mathbb{C}}T_{\lambda}^{*-1}\varphi d\mu$$ as a weak integral, that is, for $f\in\mathcal {H}$ $$<\phi_{\mu}, f > = \int_{\mathbb{C}}<T_{\lambda}^{*-1}\varphi, f > d\mu (\lambda).$$ In an extremely formal sense $\phi_{\mu} = -(\widehat{\overline{\mu}}(T))^{*}\varphi.$ For example, if $\mu = \delta_{w},$ with $\mu\notin\sigma (T),$ one has $-\widehat{\overline{\mu}}(\lambda) = \frac{1}{\pi (\lambda -w)} $ and $\phi_{\mu} = T_{w}^{*-1}\varphi.$
It follows from Theorem \[CL\] that for $r$ a rational function with poles off the spectrum of $T$ we have $$<r(T)\varphi, \phi_{\mu} > = -\int_{\mathbb{C}} r(u)\widehat{\overline{\mu}} (u) g(u) da(u).$$ We remark on the connection between this last identity and results concerning rational approximation in [@Thomson] and more recently [@yang]. For $X$ a compact set in the plane, let $P(X)$ respectively, $R(X)$ be the closure in the space $C(X)$ of continuous function on $X$ of the polynomials, respectively, the rational functions with poles off $X.$ It was shown in [@Thomson] when $X$ is nowhere dense, then the closure of the module $\bar{z} P(X) + R(X)$ is $C(X)$ if and only if $R(X)=C(X).$ Here $z$ is the function $z(u)=u.$ This is in contrast to the result in [@trent] that establishes when X is a compact nowhere dense set, then the closure of $\bar{z}R(X)+R(X)$ is $C(X).$
We are interested here in the case where the characteristic function $\mathbbm{1}_{X}$ “is" the principal function of an operator with one-dimensional self-commutator. Since the principal function of an operator with one-dimensional self commutator is only determined up to sets of measure zero, this has to be properly interpreted. If $g$ is the principal function associated with $T,$ then $\sigma (T)$ is the essential closure of the set $\lbrace u:g(u)\neq 0 \rbrace.$ Consequently, we only consider the class of closed nowhere dense sets of positive measure that are essentially closed, i.e., equal their essential closure. For such a set $X$ there is a unique associated irreducible operator $T_{X}$ with one-dimensional self-commutator having principal function $\mathbbm{1}_{X}.$ Moreover, different such sets $X$ correspond to different operators $T_{X}$ and $\sigma (T_{X}) =X.$ It should also be noted that for a compact set $X$ the essential closure of $X$ is a closed subset of $X$ that differs from $X$ by a set of planar measure zero. Moreover, if $R(X)\neq C(X),$ the same is true for its essential closure [@rubel].
Based on the result of Thomson [@Thomson], as noted in [@ClanceyIU], one can establish the following:
Suppose $T$ is an operator with one-dimensional self-commutator $T^*T-TT^* = \varphi\otimes\varphi$ associated with the principal function $\mathbbm{1}_{X},$ where $X$ is a compact essentially closed nowhere dense set of positive measure. If the closure of $\bar{z} P(X) + R(X)$ is not equal to $ C(X), $ equivalently, the closure of $\bar{z} P(\sigma (T)) + R(\sigma (T))$ is not equal to $C(\sigma (T)),$ then the vector $\varphi$ is not cyclic for the operator $T.$
Let $\mu$ be a non-zero measure on $X$ that annihilates $\bar{z} P(X)+R(X).$ Let $\phi_{\bar{\mu}}$ be given by (\[vector\]) with $\bar{\mu}$ replacing $\mu.$ Then for $r$ a rational function with poles off $X$ the last equation results in the identity $$<r(T)\varphi, \phi_{\bar{\mu}} > = -\int_{X} r(u)\hat{\mu} (u) da(u)= \int_{X}\bar{u}r(u)d\mu (u).$$ By the result of [@trent] the closure of $\bar{z}R(X) + R(X)$ is $C(X)$ and therefore for some $r\neq 0$ the right side of this last equation is non-zero. This implies $\phi_{\bar{\mu}}$ is non-zero. Since $\int_{X}\bar{z}pd\mu = 0$ for all polynomials $p$ it follows that $\varphi$ is not (polynomially) cyclic for the operator $T.$
In the context of the last result, a natural example to consider is that of a Swiss cheese $X,$ that is, where $X$ is a closed nowhere dense set of positive planar measure obtained by removing a collection of open discs $D(w_{n}, r_{n}),\ n=1,2\cdots $ with disjoint closures from the closed unit disc. If one assumes $\Sigma_{1}^{\infty} r_{n} <\infty,$ then $R(X)\neq C(X).$ Let $T$ be the irreducible operator with one-dimensional self-commutator associated with the principal function $\mathbbm{1}_{X},$ where $X$ is a Swiss cheese with $\Sigma_{1}^{\infty} r_{n} <\infty.$ It is known that the $\varphi$ is rationally cyclic for both $T$ and $T^{*},$ see [@ClanceyJOT]. The result above shows $\varphi$ is not (polynomially) cyclic for $T.$
In a fundamental paper, Brown [@brown] established the existence of invariant subspaces for a hyponormal operators $H$ whenever there is a closed disc $D$ such that $R(D\cap\sigma (H))\neq C(D\cap\sigma (H)).$ Thus the result in the last proposition does not advance the theory of invariant subspaces. However, the fact that $\varphi$ is not cyclic for $T$ is new, albeit depending on the deep result of Thomson [@Thomson].
The result in the last proposition can be extended to the case, again assuming $X$ is nowhere dense, where for some closed disc $D$ the closure of $\bar{z} P(X\cap D) + R(X\cap D)$ is not equal to $ C(X\cap D).$ To see this, note if $\mu$ is a non-zero measure on $X\cap D$ annihilating $\bar{z} P(X\cap D) + R(X\cap D),$ then one has $$<r(T)\varphi, \phi_{\bar{\mu}} > = \int_{X\cap D}\bar{u}r(u)d\mu (u)$$ for $r$ a rational function with poles off $X$ and the last integral will be zero for $r = p$ a polynomial. In order to see that $\phi_{\bar{\mu}}$ is not zero, note that the last integral will be non-zero for some rational function $r$ with poles off $X\cap D.$ If one does a partial fractions decomposition of $r$ the part of this decomposition with poles in $X\backslash(X\cap D)$ can be approximated by a polynomial on $X\cap D.$ In this way, $r$ can be replaced by a rational function $r_{0}$ with poles off $X$ where $\int_{X\cap D}\bar{u}r_{0}(u)d\mu (u)\neq 0.$
It would be interesting to see if the results in [@trent1], [@Thomson] and [@yang] appropriately extend so that the above techniques can be used to establish that the vector $\varphi$ is not cyclic under the conditions that $\sigma (T)$ is nowhere dense and there is a closed disc $D$ such that $R(D\cap\sigma (H))\neq C(D\cap\sigma (H)).$
Some definite integral values computed using $E$ and the unilateral shift
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In the case, where the operator $T$ is the unilateral shift $Uf(z)=zf(z)$ acting on the Hardy space $H^{2}$ consisting of analytic functions $f$ on the open unit disc $\bold{D}$ with norm $$\| f\| = \left(\lim_{r\to 1}\frac{1}{2\pi}\int_{0}^{2\pi} |f(re^{i\theta}|^2d\theta\right)^{\frac{1}{2}},$$ the principal function $g_{T}$ is the characteristic function of the disc $\bold{D}.$ It is easy to verify that for $|\lambda |<1, $ $$\label{inside}U_{\lambda}^{*-1} \bold{1} (z) =\frac{z-\lambda}{1-\bar{\lambda}z}$$ and for $|\lambda |\geq 1,$ $$\label{outside}U_{\lambda}^{*-1} \bold{1} (z) = -\frac{\bold{1}}{\bar{\lambda}},$$ where we are using the notation $\bold{1}$ for the constant function $\bold{1}(z) = 1$ that appears in the self-commutator $U^{*}U - UU^{*} = \bold{1}\otimes\bold{1}.$ A straightforward computation can be used to verify $$\label{shift}1- (U_{w}^{*-1}\bold{1},U_{\lambda}^{*-1}\bold{1})= E_{\bold{D}}(\lambda, w ) = \exp-\frac{1}{\pi}{\int_{\bold{D}} \frac{u-w}{u-\lambda }\frac{1}{\vert u-w\vert^2}} da(u ),$$ where $E_{\bold{D}}$ is given by (\[unitdisc\]).
Using this last formula one can directly verify the integral formulas (\[real\]) and (\[Imaginary\]). For example, consider the formula (\[real\]) where $\alpha <1.$ The map $u = (\lambda -c_{\alpha})z +c_{\alpha}$ maps $\bold{D}$ onto $D_{\lambda, \alpha}$ sending $-1$ to $\lambda_{\alpha} = w +\alpha(\lambda - w)$ and $1$ to $\lambda.$ This change of variables results in the equality $$\label{change}-\frac{1}{\pi}\int_{D_{\lambda,\alpha}} \frac{u-w}{u-\lambda }\frac{1}{\vert u-w\vert^2} da(u ) =-\frac{1}{\pi}\int_{\bold{D}}\frac{z - s_{\alpha}}{z-1}\frac{1}{|z-s_{\alpha}|^{2}} da(z),$$ where $s_{\alpha} = \frac{1+\alpha}{\alpha -1}$ is the image of $w$ under the inverse map $z=z(u).$ The right side of this last equation is recognized as the exponent in the following special case of equation (\[shift\]): $$1- (U_{s_{\alpha}}^{*-1}\bold{1},U_{1}^{*-1}\bold{1})= E_{\bold{D}}(s_{\alpha}, 1 ) = \exp-\frac{1}{\pi}{\int_{\bold{D}} \frac{u-s_{\alpha}}{u-1 }\frac{1}{\vert u-s_{\alpha}\vert^2}} da(u ).$$ Depending on whether $s_{\alpha}$ is inside the open unit disc ($\alpha <0$) or outside the open unit disc ($0\leq \alpha <1$) one uses (\[inside\]) or (\[outside\]) to compute the left side of this last equality. For example, in the case where $s_{\alpha}$ is inside the open unit disc $$1- (U_{s_{\alpha}}^{*-1}\bold{1},U_{1}^{*-1}\bold{1}) = \frac{2}{1+|\alpha |}.$$ Combining this last identity with (\[change\]) equation (\[real\]) follows. Similar arguments using the unilateral shift can be used to obtain the other instances of (\[real\]) and (\[Imaginary\]).
Department of Mathematics
University of Georgia
Athens, GA
email: kclancey@uga.edu
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abstract: 'We study the localization properties of weakly interacting Bose gas in a quasiperiodic potential commonly known as Aubry-André model. Effect of interaction on localization is investigated by computing the ‘superfluid fraction’ and ‘inverse participation ratio’. For interacting Bosons the inverse participation ratio increases very slowly after the localization transition due to ‘multisite localization’ of the wave function. We also study the localization in Aubry-André model using an alternative approach of classical dynamical map, where the localization is manifested by chaotic classical dynamics. For weakly interacting Bose gas, Bogoliubov quasiparticle spectrum and condensate fraction are calculated in order to study the loss of coherence with increasing disorder strength. Finally we discuss the effect of trapping potential on localization of matter wave.'
author:
- 'Sayak Ray, Mohit Pandey, Anandamohan Ghosh and S. Sinha'
title: Localization of weakly interacting Bose gas in quasiperiodic potential
---
Introduction
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In recent years ‘Anderson localization’ of particles and waves has regained interest in quantum many particle physics. ‘Anderson localization’ is a remarkable quantum phenomenon for which the propagating wave becomes exponentially localized in the presence of disorder[@anderson; @rmp] and is a well studied subject in the context of electronic systems in presence of disorder. Like particles, waves can also localize in disordered medium. Localization of ‘matter wave’ has recently been observed in experiments on ultracold Bose gases in presence of speckle potential[@aspect] and in bichromatic optical lattices[@inguscio1]. For certain parameters, Bosons in bichromatic lattice can be mapped on to Aubry-André (AA) model[@aa] with ‘quasi-periodic’ potential. Also localization of light in AA model has been observed experimentally[@lahini]. Unlike Anderson model with random disorder, the AA model exhibits localization transition in one dimension at a critical strength of the potential. Apart from that the quasi-periodicity of AA model gives rise to various interesting spectral properties[@pichard]. In recent experiment [@inguscio2] effect of interaction on AA-model has been studied. Repulsive interaction plays an important role in the formation of correlated phases like ‘Bose-glass’ phase[@BHM; @lugan; @inguscio3; @pasienski]. Also the dynamics and diffusion of interacting Bose gas in presence of disorder have been investigated both experimentally[@inguscio4; @clement] and theoretically[@larcher; @flach]. In recent years ‘many body localization’[@huse1] and localization at finite temperature[@huse2; @alt; @shlyapnikov] have generated an impetus to study disordered Bose gas.
In this work we investigate localization of weakly interacting Bose gas in AA potential. Apart from calculating the ground state properties, we also study localization of wavefunction by using a classical dynamical map. The paper is organized as follows: In section II, we discuss non-interacting Bose gas in quasiperiodic potential. In subsection A, we review the AA model and discuss the single particle localization properties. Localization in non-interacting system using dynamical map approach is presented in subsection B. In section III, various physical quantities like inverse participation ratio and superfluid fraction are computed to investigate localization of weakly interacting Bose gas within mean-field theory. Effect of non-linearity due to interactions on the dynamical map is studied. Further, we compute the Bogoliubov quasiparticle energies and amplitudes to investigate quantum fluctuations. We study the localization of both Bogoliubov amplitudes and non-condensate densities.
Effect of trapping potential on localization and effect of disorder on the center of mass motion are presented in section IV. Finally we summarize our results in section V.
Non-interacting Bosons in quasiperiodic potential
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In the original experiment[@inguscio1] localization of ultracold Bosons has been studied using bichromatic optical lattice which can be mapped on to Aubry-André model within tight binding approximation and for certain parameter regime[@modugno2]. The Aubry-André model is defined by the Hamiltonian, $$H= - J\sum_n (|n\rangle \langle n + 1|+ |n+1\rangle \langle n|) + \lambda \sum_n \cos(2\pi \beta n) |n\rangle \langle n|
\label{aah}$$ where $|n\rangle$ is a Wannier state at lattice site $n$, $J$ is nearest neighbour hopping strength, $\lambda$ is the strength of onsite potential and period of potential is determined by $\beta$. In the rest of the paper, we would be working in the units in which $J=1$.
For $\lambda=2$, the Hamiltonian is equivalent to the Harper model[@harper] describing the motion of an electron in a square lattice in the presence of a perpendicular magnetic field, where the flux $\Phi$ through each plaquette in units of flux quantum $\Phi_{0}= h/e$ is given by the parameter $\beta = \Phi/\Phi_{0}$. Energy spectrum of this problem gives rise to the well known ‘Hofstadter butterfly’[@hofstadter]. The Hamiltonian given in Eq., poses very interesting properties for irrational values of $\beta$. When $\beta$ is chosen to be a ‘Diophantine number’, AA model undergoes a localization transition at a critical value of the potential strength $\lambda = 2$[@aa; @jit1]. On the contrary, localization transition is absent in one dimensional Anderson model with random disorder. The quasiperiodic potential generates a correlated disorder in the AA model.
Single particle localization
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In order to study localization transition in AA model, we choose $\beta= (\sqrt{5}-1)/2$ which is the inverse of ‘golden mean’ and a Diophantine number . This is particularly helpful since rational approximation of $\beta$ can be done by Fibonacci series. Although incommensurability of the potential with the underlying lattice plays an crucial role in AA model, for numerical studies one can approximate $\beta = F_{n-1}/F_{n}$, where $F_{n}$ is $n$th Fibonacci number for sufficiently large value $n$. This rational approximation of $\beta$ fixes the lattice size $N_{s}= F_{n}$ to impose periodic boundary condition[@hanggi].
Duality in AA model can be shown by introducing new basis states in momentum space, $$|k\rangle = N_{s}^{-1/2} \sum_n \exp(i 2\pi k\beta n)| n \rangle.
\label{dual_basis}$$ The dual model is obtained by substituting Eq.\[dual\_basis\] in Eq.\[aah\], $$H= \frac{\lambda}{2} \left[ \sum_k |k\rangle \langle k + 1|+ \mbox{h.c} + \frac{4}{\lambda} \sum_k \cos(2\pi \beta k) |k\rangle \langle k| \right].
\label{dual_aa}$$ It is important to note that AA model becomes self-dual at a critical coupling $\lambda = 2$, where the localization transition occurs. To obtain the eigenvalues and eigenfunctions of the Hamiltonian we expand the state vector in terms of Wannier states, $|\psi\rangle = \sum_{n} \psi_{n}|n\rangle$, where $\psi_{n}$ is the wavefunction at $n$th lattice site. The eigenvalue equation of the Hamiltonian Eq.\[aah\] is reduced to a discrete Schrödinger equation, $$-(\psi_{n+1} + \psi_{n-1}) + \lambda \cos(2\pi \beta n) \psi_{n} = \epsilon \psi_{n},
\label{dse}$$ where $\epsilon$ is the energy eigenvalue. The degree of localization of a normalized state $|\psi\rangle$ can be quantified by ‘inverse participation ratio’ (IPR) $I$, $$I = \sum_{n}|\psi_{n}|^{4}
\label{ipr_eq}$$ The wavefunction of an extremely localized particle at a site $n_{0}$ is given by $\psi_{n} = \delta_{n,n_{0}}$, for which the IPR becomes unity. On the other hand, for the completely delocalized wavefunction $\psi_{n} = 1/\sqrt{N_{s}}$, IPR is $1/N_{s}$ which vanishes in the thermodynamic limit. For AA model all energy eigenfunctions in real space are exponentially localized and IPR sharply increases to unity above the critical coupling $\lambda = 2$. Due to the duality of AA model, the localization of wavefunction in real space and in dual momentum space shows opposite behavior. In real space the wavefunctions are localized above $\lambda =2$, wheras localization in dual momentum space occurs for $\lambda \leq 2$. The IPR of the ground state wavefunction in real space and in dual monentum space as a function of $\lambda$ are shown in Fig.\[fig1\]. It is interesting to note that IPR in real and momentum space intersects at the self dual point $\lambda = 2$.
![IPR of ground state wavefunction as a function of strength of the potential $\lambda$. IPR in real space $I_{r}$ and in dual momentum space $I_{k}$ are shown by solid(black) line and dashed (blue) line respectively.[]{data-label="fig1"}](Fig1.pdf){width="8cm"}
The spectrum of AA model also shows various interesting properties. For $\lambda =0$ it has usual single band energy spectrum of periodic lattice. Addition of the potential generates quasiperiodic structure and destroys the band like dispersion. Successive rational approximation of $\beta$ generates more periodicity over the underlying lattice, which in turn breaks the original energy band into many subbands and it leads to opening of band gaps. The energy spectrum of AA model is obtained by numerical diagonalization and is depicted in Fig.\[fig2\]a for increasing values of $\lambda$. The variation of energy gaps with the coupling strength can be noticed in this figure. The spectral statistics and distribution of energy gaps are analyzed in [@pichard]. Mathematically it can be shown that the energy spectrum of AA model forms a Cantor set[@jit]. Self-similarity in energy levels can be understood from the integrated level density, $$N(\epsilon) = \sum_{i}\theta(\epsilon - \epsilon_{i}),
\label{int_ds}$$ where $\epsilon_{i}$ is the i’th eigenvalue and $\theta(x)$ is heaviside step function. The integrated density of states $N(\epsilon)$ shows ‘Devil’s staircase’ like fractal structure, which is evident from Fig.\[fig2\]b where we plotted the normalized integrated density of states $N(\epsilon)$ as a function of scaled energy within the interval of zero to one. For different values of $\lambda$, the Devil’s staircase structures of $N(\epsilon)$ do not overlap, which indicates that the fractal dimension changes with the strength of the potential $\lambda$.
Transport properties also show significant changes in localization transition. Due to spatial localization of single particle states transport coefficients vanish in the thermodynamic limit. In electronic systems the conductivity vanishes exponentially with the system size due to ‘Anderson localization’. For neutral superfluid corresponding physical quantity is ‘superfluid fraction’(SFF), which is measured by generating a superflow by applying a phase twist[@fisher]. In presence of the phase twist the original Hamiltonian in Eq.\[aah\] becomes, $$H_{\Theta} = -\sum_{n=1}^{N_s} (e^{- i \Theta/ N_s} |n\rangle \langle n + 1|+ \mbox{h.c}) + \lambda \cos(2\pi \beta n) |n\rangle \langle n|,
\label{aa_twist}$$ where $\Theta$ is arbitrarily small phase difference across the boundary. For one dimensional system with periodic boundary condition this is exactly similar to a quantum ring in presence of a flux which generates supercurrent through the ring. The superfluid fraction $f_{s}$ is defined as[@burnett], $$f_s = N_{s}^{2} \frac{E_{0}(\Theta)-E_{0}(0)}{\Theta^2},
\label{sf}$$ where $E_{0}(\Theta)$ is the ground state energy of the Hamiltonian $H_{\Theta}$ with arbitrary small value of $\Theta$, and $E_{0}(0)$ is the ground state energy of the original Hamiltonian given in Eq.\[aah\]. To obtain the ground state energy upto $\Theta^2$ order, the Hamiltonian $H_{\Theta}$ is expanded as, $$H_{\Theta} \approx H + \frac{\Theta}{N_s} \hat{J} - \frac{\Theta^2}{2 N_s^2}\hat{T}
\label{pert}$$ where we define a current operator $\hat{J} = -i \sum_n (|n\rangle \langle n + 1| - \mbox{h.c}) $ and the usual kinetic energy $\hat{T} =-\sum_n (|n\rangle \langle n + 1|+ \mbox{h.c})$. Using second order perturbation, the SFF in Eqn.\[sf\] can be written as, $$f_{s} = -\frac{1}{2} \langle{\psi_0}|{\hat{T}}|{\psi_0}\rangle + \sum_{i \neq 0} \frac{{|\langle \psi_{i}|}{\hat{J}}{|\psi_0}\rangle|^2}{\epsilon_{i}-\epsilon_{0}},
\label{sf_pert}$$ where $\epsilon_{i}$, $|\psi_{i}\rangle$ are eigenvalues and eigenfunctions of Hamiltonian given in Eq.\[aah\] and $|\psi_{0}\rangle$ is ground state wavefunction. Variation of SFF with the coupling strength of the potential is shown in Fig.\[fig3\]. For increasing values of $\lambda$ the SFF decreases from unity and vanishes at the critical point $\lambda = 2 $.
![Superfluid fraction $f_{s}$ (solid line) and IPR of ground state in real space $I_{r}$ (dashed line) as a function of $\lambda$.[]{data-label="fig3"}](Fig3.pdf){width="8cm"}
Hamiltonian map
---------------
Localization phenomena can also be studied by an alternative method of classical Hamiltonian map(CHM), which is very useful to obtain analytical estimate of localization length[@izrailev_rev; @izrailev]. The discrete Schrödinger equation given in Eq.\[dse\] can be written as following dynamical map, $$\begin{aligned}
p_{n+1} & = & p_{n} + (\lambda \cos(2\pi \beta n) -\epsilon -2)x_{n},\\
x_{n+1} & = & x_{n} + p_{n+1}
\label{class_eqn}\end{aligned}$$ where the classical dynamical variables are given by, $x_{n} = \psi_{n}$ and $p_{n} = \psi_{n} - \psi_{n-1}$. Here the wavefunction $\psi_{n}$ plays the role of position in CHM and the number of lattice site becomes the number of iteration or time axis of the dynamics. We can choose initial real wavefunctions $\psi_{0}$ and $\psi_{1}$ (or equivalently $x_{1}$ and $p_{1}$) and evolve the dynamical variables by the transfer matrix, $$T_{n} = \left[\begin{array}{c c} (\lambda \cos(2\pi \beta n) -\epsilon -1) & 1\\
(\lambda \cos(2\pi \beta n) -\epsilon -2) & 1
\end{array} \right].
\label{trans_mat}$$
By iteration of this map we obtain the asymptotic behavior of the dynamical variables for a fixed energy eigenvalue $\epsilon$. Mathematically it can be shown that the dynamical instability of the above Hamiltonian map corresponds to the localization phenomena. Exponentially localized wavefunctions asymptotically fall off as $\psi_{n} \sim e^{-n/\xi}$, where $\xi$ is the localization length. But for arbitrary choice of the initial wavefunction, the solution of Eq.\[dse\] also has an exponentially growing component $\sim e^{n/\xi}$. Hence the localization is manifested by the exponential growth of the dynamical variables of the Hamiltonian map which gives rise to chaos in the classical phase space. Whereas the extended states represents periodic motion in the phase space. To understand the localization transition in AA model, we choose an eigenvalue close to the band center and calculate the phase space trajectories of the Hamiltonian map for an ensemble of random initial values of $x$ and $p$. The phase portraits of Eq.\[class\_eqn\] are shown in Fig.\[fig3\] for increasing values of $\lambda$. In the delocalized regime with small disorder strength $\lambda$, the phase portrait consists of closed elliptic orbitals as seen from Fig.3a. Increasing the strength of quasiperiodic potential $\lambda$ leads to diffusive behavior at the outer part of the phase space and regular portion of phase space with periodic orbits is reduced (see Fig.3b and Fig.3c). Finally for $\lambda$ close to the critical value, an instability sets in the CHM and phase space trajectories in Fig.3d show chaotic behavior which indicates localization transition.
In dynamical systems ‘Lyapunov exponent’ (LE) is a measure to quantify chaos. Since the number of lattice site plays the role of time in the classical map, the LE corresponds to the inverse localization length of the wavefunction. For periodic motion LE vanishes and in the chaotic regime the non-vanishing LE gives finite localization length $\xi$. To calculate the LE we construct the matrix $U_{N_s}= T_{N_s}T_{N_s-1}.....T_{2}T_{1}$ by multiplying the transfer matrices $T_{n}$ sequentially for $N_s$ iteration. The LE $l$ can be obtained by using the formula, $$l = \lim_{N_s\rightarrow \infty} \frac{\log(\lambda_{+})}{N_s},
\label{le}$$ where $\lambda_{+}$ is the largest eigenvalue of the matrix $U_{N_s}$ and LE is obtained from $\xi = 1/l$. The localization length can also be calculated by using Thouless formula [@thouless]. Using the duality of AA model the analytical expression of LE is given by $\xi = \log(\lambda/2)$[@aa], which is independent of energy. Within CHM approach we numerically compute the localization length $\xi$ from Eq.\[le\] using QR decomposition method [@lauter]. Numerically obtained localization length as a function of $\lambda$ for different energy eigenvalues are compared with the analytical result in Fig.\[fig5\].
![Variation of Lyapunov exponent or inverse localization length $1/\xi$ with disorder strength $\lambda$. Dotted and dot-dashed lines represent numerically obtained Lyapunov exponents from Eq.\[le\] and for comparison the analytical result is shown by solid (red) line.[]{data-label="fig5"}](Fig5.pdf)
For the localization transition in AA model with quasiperiodic potential, the choice of $\beta$ as irrational number (particularly a Diophantine number) plays a crucial role [@jit1]. To understand this mathematical condition, we analyze the phase space of CHM for successive rational approximation of $\beta$, which is shown in Fig.\[fig6\]. Using Fibonacci series, the rational approximation of $\beta$ can be written as $\beta_{n} = F_{n-1}/F_{n}$ and the potential has periodicity $F_{n}$ for successive integer $n$. For sufficiently large value of $n$, $\beta_{n}$ approaches to the inverse of ‘golden mean’ and the potential becomes quasiperiodic. In the localized regime, for $\lambda=2.2$ the phase space trajectories of CHM with increasing order of rational approximation of $\beta$ are presented in Fig.\[fig6\] (a) to (c) for fixed initial condition. Even in the localized regime phase space contains periodic orbits for $\beta_{5}$. As shown in Fig.6b to Fig.6d, the periodic orbits break as $\beta$ approaches to the inverse of ‘golden mean’ by successive rational approximation and finally dynamics become chaotic which is consistent with the localization phenomena.
Localization of weakly interacting Bose gas
===========================================
Interacting Bosons in presence of quasiperiodic potential can be described by Bose-Hubbard model [@BHM], $$H = -\sum_{\langle ij\rangle}(a^{\dagger}_{i}a_{j} + \mbox{h.c}) + \lambda \sum_{i}\cos(2\pi \beta i) \hat{n}_{i} + \frac{U}{2}\sum_{i} \hat{n}_{i}(\hat{n}_{i}-1)
\label{Bose_hubbard}$$ where $a^{\dagger}_{i}$($a_{i}$) are creation (annihilation) operators for Bosons at site $i$, $\hat{n}_{i} = a^{\dagger}_{i}a_{i}$, and $U$ is the strength of onsite repulsive interaction. Above quantum many-body Hamiltonian can capture various correlated phases of strongly interacting Bosons which undergoes quantum phase transition. It is known that Bose-Hubbard model in the presence of random disorder can give rise to ‘Bose glass’ phase [@BHM]. For sufficiently weak interaction strength $U$ and for high average density of Bosons a ‘quasi-condensate’ may form in one dimensional systems [@1d_shlyap]. In this regime, one can replace the bosonic operators $a_{i}$ by a classical field $\psi_{i}$ which represents the macroscopic wavefunction of the ‘quasi-condensate’. Minimization of the classical energy corresponding to Eq.\[Bose\_hubbard\] leads to the ‘discrete nonlinear Schrödinger equation’ (DNLS), $$-(\psi_{i+1} + \psi_{i-1}) +\lambda \cos(2\pi \beta i)\psi_{i}+ U|\psi_{i}|^{2} \psi_{i} = \mu \psi_{i},
\label{dnlse}$$ where $\mu$ is the chemical potential, and the normalization of the wavefunction of $N_{b}$ number of Bosons gives $\sum_{i} |\psi_{i}|^{2} = N_{b}$.
We numerically find out the ground state wavefunction of Eq.\[dnlse\] by imaginary time propagation method. For weak disorder the wavefunction is extended and numerical convergence is fast, whereas close to the localization transition more care is needed to obtain the ground state since many metastable states appear in this region. To obtain the degree of localization of the ground state wavefunction in presence of repulsion we calculate the IPR in real space using Eq.\[ipr\_eq\]. The variation of IPR of condensate wavefunction with increasing strength of disorder $\lambda$ for different values of effective interaction $UN_{b}$ is shown in Fig.\[fig7\](a). The IPR becomes nonzero for $\lambda\ge 2$ and increases with a slower rate compared to the non-interacting system, indicating spreading of the wavefunction due to the repulsive interaction. The degree of localization decreases with increasing strength of repulsive interaction $UN_{b}$. In Fig.7(b) the spatial variation of the wavefunction with disorder strength $\lambda$ is represented by color scale plot. It is evident from this figure that single site localization is not favorable energetically and the wavefunction is localized at almost degenerate but spatially separated sites. This particular feature of the fragmented condensate has also been studied in [@Samuel]. Due to the multisite localization the IPR is much less than unity and increases very slowly with $\lambda$. A change in the slope of IPR with increasing $\lambda$ occurs when the number of localized sites decreases and we have checked that finally at very large value of $\lambda$ the wavefunction becomes localized at a single site.
To study the interplay between disorder and interaction in the transport properties of dilute Bose gas, we calculate the superfluid fraction $f_{s}$. To generate a superflow we introduce a small amount of phase twist ($\Theta \sim 0.1$) in the hopping term of DNLS (in Eq.\[dnlse\]) similar to Eq.\[aa\_twist\] and then the superfluid fraction is computed using the formula, $$f_s = N_s ^2 \frac{E_{cl}(\Theta)-E_{cl}(0)}{\Theta^2},
\label{fs_int}$$ where $E_{cl}(\Theta) $ is the classical energy corresponding to the Hamiltonian in Eq.\[Bose\_hubbard\], $$\begin{aligned}
E_{cl}(\Theta) & = & -\sum_{\langle ij\rangle}\left[e^{i\Theta/N_s}\phi^{\ast}_{i}\phi_{j} +\mbox{h.c}\right] + \lambda \sum_{i}\cos(2\pi \beta i) |\phi_{i}|^{2}\nonumber\\
& & + \frac{UN_{b}}{2}\sum_{i} |\phi_{i}|^{4}.
\label{energy_gp}\end{aligned}$$ The wavefunction $\phi_{i}$ minimize $E_{cl}(\Theta)$ and is normalized to unity. The superfluid fraction $f_{s}$ as a function of $\lambda$ for different values of repulsive interactions $UN_{b}$ is shown in Fig.\[fig8\]. The SFF of weakly interacting Bose gas obtained from ‘density matrix renormalization group’ also shows similar behavior [@minguzzi]. Due to the repulsive interaction, $f_{s}$ vanishes at larger strength of quasiperiodic potential $\lambda > 2$, however the IPR rises from zero at $\lambda \approx 2$. This behavior is different from the noninteracting AA model.
![Variation of ‘superfluid fraction’ $f_s$ with $\lambda$ for different interaction strength $UN_b$.[]{data-label="fig8"}](Fig8.pdf)
We also investigated the localization properties of DNLS by Hamiltonian map approach. Eq.\[dnlse\] can be written in the form of nonlinear classical map, $$\begin{aligned}
p_{i+1} & = & p_{i} + (\lambda \cos(2\pi \beta i) -\mu -2)x_{i} + UN_{b}x_{i}^{3},\\
x_{i+1} & = & x_{i} + p_{i+1}
\label{class_dnls}\end{aligned}$$
where, $x_{i} = \phi_{i}$ and $p_{i} = \phi_{i} - \phi_{i-1}$. The repulsive nonlinearity gives rise to an unstable classical potential $ \sim - \frac{UN_{b}}{4}x_{i}^{4}$ due to which the classical trajectories become unstable at large values of phase space variables. To avoid this problem we choose relatively small region of phase space within which the potential is metastable, and study the effect of disorder in the phase space dynamics. In Fig.\[fig9\], for a fixed value of nonlinearity $UN_{b}$ we show the phase portrait with an ensemble of initial configurations for increasing values of disorder strength $\lambda$. Similar qualitative features like non interacting system are also seen in the phase space dynamics of DNLS, however the classical periodic orbits are modified due to the nonlinearity.
Fluctuations within Bogoliubov approximation
--------------------------------------------
So far we studied the weakly interacting Bosons within mean-field approximation using macroscopic wavefunction for the condensate. It is also important to analyze the quantum fluctuations induced by the quasiperiodic disorder potential. Within Bogoliubov approximation, the quantum field operators can be approximated by, $$a_{i} = e^{-i\mu t}\left[\psi_{i} + \sum_{\nu}\left(u_{i}^{\nu}b_{\nu}e^{-i\omega_{\nu}t} + v_{i}^{\ast \nu}b_{\nu}^{\dagger}e^{i\omega_{\nu}t}\right)\right],
\label{fluctuation_op}$$ where $\psi_{i}$ is the macroscopic wavefunction of the condensate satisfies Eq.\[dnlse\], $u_{i}^{\nu}$,$v_{i}^{\nu}$ are amplitudes corresponding to $\nu$th eigenmode with bosonic operators $b_{\nu}$, $b_{\nu}^{\dagger}$. Bogoliubov quasiparticle energies $\omega_{\nu}$ can be obtained from, $$\begin{aligned}
& & -(u_{i+1} + u_{i-1}) +[\lambda \cos(2\pi \beta i) + 2 U|\psi_{i}|^{2} -\mu]u_{i}\nonumber\\
& & + U \psi_{i}^{2}v_{i} = \omega u_{i},\\
& & -(v_{i+1} + v_{i-1}) +[\lambda \cos(2\pi \beta i) + 2 U|\psi_{i}|^{2} -\mu]v_{i} \nonumber\\
& & + U \psi_{i}^{\ast 2}u_{i} = -\omega v_{i},
\label{bog_eqns}\end{aligned}$$ and normalization condition gives $ \sum_{i} (u_{i}^{\nu}u_{i}^{\ast \nu'} - v_{i}^{\nu}v_{i}^{\ast \nu'}) = \delta_{\nu \nu'}$.
For a given interaction strength $U$ and average density of Bosons $n_{0} = N_{b}/N_{s}$, we first numerically obtain the ground state macroscopic wavefunction $\psi_{i}$, then diagonalize Eq.\[bog\_eqns\] to calculate Bogoliubov quasiparticle energies and amplitudes $u_{i}^{\nu}$, $v_{i}^{\nu}$. The Bogoliubov energy spectrum for increasing strength of disorder $\lambda$ is depicted in Fig.\[fig10\](a) for a fixed value of interaction $UN_{b}=1$ and $N_{s}=144$. The normalized integrated density of states $N(\omega)$ also shows Devil’s staircase like structure as shown in Fig.\[fig10\](b).
Due to the quantum fluctuations, depletion of the condensate occurs and the non-condensate density $\rho_{nc}$ at zero temperature can be obtained from the Bogoliubov theory, $$\rho_{nc}(i) = \sum_{\nu} |v_{i}^{\nu}|^{2}.
\label{den_nc}$$ The condensate fraction is given by $N_{c}/N_{b} = 1 - \sum_{i,\nu}|v_{i}^{\nu}|^{2}/N_{b}$. In one dimensional system the noncondensate fraction diverges as $ \log(N_s)$, which prohibits the formation of condensate in the thermodynamic limit. However for sufficiently weak interaction a quasi condensate can form in quasi one dimensional and finite system [@1d_shlyap]. To investigate the disorder induced quantum fluctuation in a quasi-condensate in a finite lattice with $N_{s}= 144$, we calculate the condensate fraction with increasing disorder strength $\lambda$, which is shown in Fig.\[fig11\](a). In absence of disorder the quantum depletion is small in weakly interacting gas of Bosons and increases with the interaction strength. With the increase in disorder strength $\lambda$ the condensate fraction remains close to unity in the delocalized regime and then rapidly decreases around the critical point $\lambda \approx 2$. This qualitative feature (as shown in Fig.\[fig11\](a)) indicates that enhanced quantum fluctuations near the localization transition can destroy the quasi condensate above $\lambda \approx 2$ and strongly correlated phases can appear. The variation of non-condensate density $\rho_{nc}$ with disorder strength $\lambda$ is depicted in Fig.\[fig11\](c) by color scale plot. It is interesting to note that the distinct feature of multisite localization for $\lambda > 2$ is observed even for non-condensate density. Also the IPR of normalized non-condensate density shows similar behavior as that of the condensate wavefunction and increases from $\lambda \approx 2$ (as seen from Fig.\[fig11\](c)). Localization of Bogoliubov quasiparticles and enhancement of quantum fluctuations in presence of quasiperiodic potential particularly for $\lambda \ge 2$ are clearly evident from this analysis.
Localization in the presence of a trap
======================================
Although the localization transition in AA model occurs in thermodynamic limit, in real experimental setup a weak trapping potential is always present in order to confine the ultracold atoms. Main features of the localization can also be observed in trapped system provided the length scale of the trapping potential is larger compared to the localization length. Additionally some interesting effects due to the trap can also be seen. A harmonic trap is introduced by adding a potential $V_{i} = \frac{1}{2}\omega_{HO}^{2}(i-i_{c})^{2}$ in the Hamiltonian (Eq.\[Bose\_hubbard\]), where $\omega_{HO}$ is related to the trapping frequency, $i$ is the site index, and $i_{c}$ is the center of the trap.
Although in a trap the wavefunction is always localized, but in a weak trapping potential the width of the wavepacket is sharply reduced due to disorder induced localization. The density profile of the condensate in presence of a harmonic trap for different disorder strength $\lambda$ is shown in Fig.\[fig12\]b. We calculate the IPR of the ground state wavefunction with increasing disorder strength $\lambda$ and an increase of IPR around $\lambda \approx 2$ is observed as expected. However the IPR does not vanish in the delocalized regime $\lambda <2$ and takes a small value due to the tapping potential. We have noticed earlier that in the absence of a trap the IPR increases very slowly after the localization transition due to the repulsive interaction and the wavefunction is localized at spatially separated sites with quasi-degenerate energies. These quasi-degeneracy of onsite energies can be lifted by introducing a harmonic trap which leads to enhancement of the degree of localization which is elucidated in Fig.\[fig12\]a where a rapid increase of IPR to unity is shown by increasing the trap frequency by a small amount.
Collective oscillation of a Bose-Einstein condensate in a trap can also show its superfluid properties. A small displacement of the condensate from the center of the trap generates center of mass (COM) oscillation which is a well studied collective mode of Bose-Einstein condensate. Here we numerically study the motion of COM of trapped condensate in presence of quasi periodic disorder. Since the superfluidity is affected by the disorder, we calculate the coherence factor $\Psi$ of the oscillating condensate which is given by [@Smerzi] $$\Psi = \sum_{i} \psi_{i}^{\ast} \psi_{i+1},
\label{coherence_fac}$$ We can notice that the above expression contains information of relative phase difference between neighboring sites and $|\Psi|^{2}$ gives a quantitative measure of overall phase coherence of the oscillating condensate.
In Fig.\[fig13\] (a) and (b), we have shown the COM motion of the trapped condensate and coherence factor of corresponding macroscopic wavefunction for increasing disorder strength $\lambda$. It is clear that the coherence of the time dependent wavefunction decreases with increasing disorder. It is also interesting to study the variation of condensate fraction with disorder. In one dimensional trapped condensate the divergence of non-condensate fraction is less severe due to the finiteness of the trap. Like the homogeneous system, we expect enhancement of quantum fluctuation due to disorder which may destroy the condensate around a critical disorder strength $\lambda \approx 2$.
Conclusions
===========
In conclusion, we have investigated localization of both non-interacting Bosons as well as weakly interacting quasi-condensate in presence of a quasiperiodic potential. Apart from calculating various physical properties, understanding localization transition through the approach of classical Hamiltonian map is one of the main result of this work.
In the non-interacting Aubry-André model the localization transition can be identified by the vanishing ‘superfluid fraction’ and the rise of IPR at the critical strength of quasiperiodic potential $\lambda=2$. A classical Hamiltonian map is constructed from the Schrödinger equation. The phase space trajectories of the corresponding classical map shows periodic orbits for small disorder strength. With the increasing disorder strength $\lambda$ chaotic behavior is observed at the outer region of the phase space and finally near the critical value $\lambda = 2$ all periodic orbits are destroyed due to the onset of chaos in the localized regime. The parameter $\beta=(\sqrt{5}-1)/2$ is chosen to be an irrational Diophantine number(inverse of golden mean) which is an essential requirement for localization transition. This has been elucidated by means of Hamiltonian map for $\lambda > 2$ where the destruction of periodic orbits by successive rational approximation of $\beta$ indicates onset of localization.
We have investigated the localization of weakly interacting Bose gas in quasiperiodic potential by mean-field approach. Unlike non-interacting case, the vanishing of SFF and rise of IPR does not take place at same strength of disorder $\lambda$. Due to the repulsive interactions, the macroscopic wavefunction is localized at many sites with quasi-degenerate energies for $\lambda >2$. The multisite localization of the interacting system is manifested by a much slower increase of IPR starting from $\lambda =2$. With increasing the strength of quasiperiodic potential, the number of sites over which the condensate is localized decreases and finally the wavefunction becomes localized at a single site for very large value of $\lambda$ and IPR approaches to unity. The SFF decreases with increasing disorder strength $\lambda$ and vanishes at $\lambda>2$ due to the repulsive interaction. The repulsive interaction gives rise to an unstable nonlinear potential in the CHM approach, due to which the stable region of phase space decreases. In the phase portrait the stable region containing the periodic orbits decreases with increasing $\lambda$ and finally the onset of chaos at $\lambda \approx 2$ signifies localization of the wavefunction. Further, the Bogoliubov quasiparticle spectrum has been calculated numerically. We notice that disorder enhances the quantum fluctuations due to which the condensate fraction of a finite system decreases rapidly around $\lambda \approx 2$. This indicates the possible formation of glassy phase and multisite localized insulators.
Finally we also considered the effect of trapping potential on the localization transition. In the localized regime, the number of sites over which the wavefunction is localized reduces due to the presence of a trap which has been shown from the rapid increase in the IPR by tuning the trap frequency. The center of mass motion of the condensate in a harmonic trap also shows the signature of localization. The center of mass oscillations become incoherent with increasing disorder. To summarize, the present study provides a clear picture of localization of non-interacting and weakly interacting Bose gas in the presence of a quasiperiodic disorder and it reveals various interesting features which are interesting for both academic point of view, as well for future experiments.
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abstract: 'Parton Distribution Functions (PDFs) are essential non-perturbative inputs for calculation of any observable with hadronic initial states. These PDFs are released by individual groups as discrete grids as a function of the Bjorken-$x$ and energy scale $Q$. The LHAPDF project at HepForge maintains a repository of PDFs from various groups in a new standardized [[[LHAPDF6]{}]{}]{} format, as well as older formats such as the [CTEQ]{} PDS grid format. [[[ManeParse]{}]{}]{} is a package that provides PDFs within the [[[Mathematica]{}]{}]{}framework to facilitate calculating and plotting. The program is self-contained so there are no external links to any [[[FORTRAN]{}]{}]{}, [C]{} or [[[C$++$]{}]{}]{} programs. The package includes the option to use the built-in [[[Mathematica]{}]{}]{} interpolation or a custom cubic Lagrange interpolation routine which allows for flexibility in the extrapolation (particularly at small $x$ values). [[[ManeParse]{}]{}]{} is fast enough to enable simple calculations (involving even one or two integrations) to be easily computed in the [[[Mathematica]{}]{}]{}framework.'
author:
- 'D. B. Clark'
- '[E. Godat]{}'
- '[F. I. Olness]{}'
bibliography:
- 'mp.bib'
date: 'Received: date / Accepted: date'
title: '[[[ManeParse]{}]{}]{}: a [[[Mathematica]{}]{}]{} reader for Parton Distribution Functions '
---
author title
@pre@post
What is [[[ManeParse]{}]{}]{}? \[sec:intro\]
============================================
![ a) We display $x\,f_{u}^A(x,Q)$ for the up-quark at $Q = 2~\mathrm{GeV}$ as a function of $x$ for the 10 PDFs listed in Table \[tab:MomSum\]. the ratio of the PDFs in a) compared to CT10 proton PDF ($A = 1$) as a function of x. While we don’t identify them individually, the one curve (yellow) that distinctly deviates from the others is the nuclear PDF for lead $ A = 208 $. \[fig:manyPDFs\]](images/abs "fig:"){width="45.00000%"} ![ a) We display $x\,f_{u}^A(x,Q)$ for the up-quark at $Q = 2~\mathrm{GeV}$ as a function of $x$ for the 10 PDFs listed in Table \[tab:MomSum\]. the ratio of the PDFs in a) compared to CT10 proton PDF ($A = 1$) as a function of x. While we don’t identify them individually, the one curve (yellow) that distinctly deviates from the others is the nuclear PDF for lead $ A = 208 $. \[fig:manyPDFs\]](images/ratio "fig:"){width="45.00000%"}
Parton Distribution Functions (PDFs) are essential elements for making predictions involving hadrons (protons and nuclei) in the initial state. For example, at the LHC, we can compute the Higgs production cross section ($\sigma$) using the formula $\sigma=f_{a/P}\otimes f_{b/P}\otimes\omega_{ab\to H}$ where PDFs $f_{a/P}$ and $f_{b/P}$ give the probability density for finding partons “$a$” and “b” in the two proton beams, and the hard cross section, $\omega_{ab\to H}$, gives the probability density for partons $a$ and $b$ producing the Higgs, $H$. The PDFs cannot be computed from first principles at this time[^1], so they must be extracted using fits to experimental data. This analysis is performed by a number of collaborations, and the PDFs are generally distributed as a grid of values in $x$ and $Q$ which must be interpolated to generate the PDF $f_{a}(x,Q)$ for flavor “$a$” at momentum fraction $x$ and energy scale $Q.$
[[[ManeParse]{}]{}]{}[^2] is a flexible, modular, lightweight, stand-alone package used to provide access to a wide variety of PDFs within [[[Mathematica]{}]{}]{}. To illustrate the flexibility, in Fig. \[fig:manyPDFs\] we show how [[[ManeParse]{}]{}]{} can work simultaneously with different PDF sets from a variety of groups. This figure[^3]displays the selected PDF sets listed in Table \[tab:MomSum\]. Some of the sets are in the [[[LHAPDF6]{}]{}]{} grid format[@Buckley:2014ana], and others are in the older [CTEQ]{} PDS grid format.[@Nadolsky:2008zw] These sets also have different numbers of active flavors, $N_{F}$, different values for the initial evolution scale, $Q_{0}$, different values for the heavy quark masses, $\{m_{c},m_{b},m_{t}\}$, and they can represent either protons or nuclei. Nevertheless, [[[ManeParse]{}]{}]{} is able to easily compare and contrast sets from different groups in a common framework.
We describe the key features of [[[ManeParse]{}]{}]{} available to the user. In Section \[sec:example\], we sketch a minimal example of how the program is used. In Section \[sec:Inside-the-ManeParse\], we provide some details of how the PDFs are parsed, stored and interpolated. In Section \[sec:Sample-Plots\], we display some example plots that are easily constructed using [[[ManeParse]{}]{}]{}. In Section \[sec:Error-PDFs:\], we provide examples of the functions in the [[pdfError]{}]{} module. In the Appendix, we discuss files provided by [[[ManeParse]{}]{}]{} and how to obtain the external PDF files.
A simple example\[sec:example\]
===============================
We begin by outlining a simple example of how [[[ManeParse]{}]{}]{} may be used. After loading the [[[ManeParse]{}]{}]{} packages into [[[Mathematica]{}]{}]{}, the user can enter the following commands:
[[Get\[pdfParseLHA.m\]]{}]{}
[[iSet1=pdfParseLHA[\[]{}LHA\_file.info,LHA\_file.dat[\]]{}]{}]{}
[[pdfFunction[\[]{}iSet1,iParton,x,Q[\]]{}]{}]{}
[[Get\[pdfParseCTEQ.m\]]{}]{}
[[iSet2=pdfParseCTEQ[\[]{}PDS\_file.pds[\]]{}]{}]{}
[[pdfFunction[\[]{}iSet2,iParton,x,Q[\]]{}]{}]{}
The first and fourth line load the parsing subpackages included in [[[ManeParse]{}]{}]{}. Loading either of these, causes the [[pdfCalc]{}]{} package to be loaded as well. The second line reads an [[[LHAPDF6]{}]{}]{} formatted external data file (LHA\_File.dat) and its associated information file (LHA\_File.info), and generates an internal PDF set that is referenced by the integer [[iSet1]{}]{}. The fifth line reads a PDS formatted external data file[^4] (PDS\_File.pds) and generates an internal PDF set that is referenced by the integer [[iSet2]{}]{}.
After reading these data files, the user is provided with the core function for computing the PDFs: [[pdfFunction[\[]{}iSet,iParton,x,Q[\]]{}.]{}]{} Here, [[iSet]{}]{} selects the individual PDF set, [[iParton]{}]{} selects the parton flavor as shown in Table \[tab:flavTable\], and {[[x,Q]{}]{}} specify the momentum fraction, $x$, and the energy scale, $Q$, in GeV.
[[pdfFunction]{}]{} performs the bulk of the work for the [[[ManeParse]{}]{}]{}program, so the package has been optimized for speed to make it practical to perform single or double integrals in a reasonable amount of time; specifically, the [[pdfFunction]{}]{} call generally takes less than 1ms per core on a standard laptop or desktop.
Additionally, [[[ManeParse]{}]{}]{} can handle an arbitrary number of PDF sets and can switch between sets without delay. When the external PDF file is parsed, the data is stored internally (about 1Mb per PDF set) and the [[iSet]{}]{} variable essentially functions as a pointer to the set; thus, it is trivial to loop over many PDF sets as was done in Fig. \[fig:manyPDFs\]. This feature contrasts to some of the older [[[FORTRAN]{}]{}]{} programs, which could only store a fixed number of sets in memory and often had to re-read the data files.
These are the key elements of the package, however, we also provide many auxiliary functions described below. Consistent with the [[[Mathematica]{}]{}]{} convention, all our public functions begin with the prefix “[[pdf]{}]{}”. One can obtain a complete list with the command [[?pdf[\*]{}]{}]{}. The usage message for individual functions is displayed in a similar manner to:
[[?pdfFunction]{}]{}\
pdfFunction\[setNumber, flavor, x, Q\]
- This function returns the interpolated value of the PDF for the [[[.pds]{}]{}]{}/[[[.dat]{}]{}]{} file specified by $setNumber$, for the given $flavor$ and value of Bjorken $x$ and scale $Q$.
- *Warning*: The results of this function are only reliable between the maximum and minimum values of $x$ and $Q$ in the [[[.pds]{}]{}]{}/[[[.dat]{}]{}]{} file[^5].
Inside the [[[ManeParse]{}]{}]{} Package\[sec:Inside-the-ManeParse\]
====================================================================
{width="80.00000%"}
flavor \# $0$ or $21$ $\pm 1$ $\pm 2$ $\pm 3$ $\pm 4$ $\pm 5$ $\pm 6$
----------- ------------- ----------- --------- -------------- ------------ ------------- ----------
parton gluon down/dbar up/ubar strange/sbar charm/cbar bottom/bbar top/tbar
Overview of package
-------------------
[[[ManeParse]{}]{}]{} internally consists of four modules (or sub-packages) as illustrated in Fig. \[fig:Flow-chart:\]. The modular structure of [[[ManeParse]{}]{}]{} allows for separate parsers for the [[[LHAPDF6]{}]{}]{} ([[pdfParseLHA]{}]{}) and PDS ([[pdfParseCTEQ]{}]{}) grids which read the individual file types and pass the information on to a common calculation ([[pdfCalc]{}]{}) module.
The new [[[LHAPDF6]{}]{}]{} format is intended as a standard that all groups can use to release their results; additionally, many older PDF sets have been converted into this format.
The [[[ManeParse]{}]{}]{} modular structure provides flexibility, as the user can use both [[[LHAPDF6]{}]{}]{} and PDS format, or even write a custom parser to read a set that is not in one of these formats.
The error PDFs module ([[pdfError]{}]{}) uses [[pdfCalc]{}]{} to construct PDF uncertainties, luminosities, and correlations as illustrated in Sec. \[sec:Error-PDFs:\].
The key elements of each PDF set include the 3-dimensional $\{x,Q,N_{F}\}$ grid and the associated information, which is stored as a set of [[[Mathematica]{}]{}]{}rules. We now describe the features and some details of these structures.
The PDF {x,Q,NF} grid
---------------------
The parsing routines [[pdfParseLHA]{}]{} and [[pdfParseCTEQ]{}]{} read the external files and assemble the PDF sets into a common data structure that is used by the [[pdfCalc]{}]{} module. The central structure is a 3-dimensional grid of PDF values in {x,Q,$N_{F}$} space, which uses vectors $\{x_{vec},Q_{vec}\}$ to specify the grid points. The spacing of $\{x_{vec},Q_{vec}\}$ need not be uniform; typically, $Q_{vec}$ uses logarithmic spacing, and $x_{vec}$ is commonly logarithmic at small $x$ and linear at large $x$. Different spacings in $x_{vec}$ and $Q_{vec}$ do not pose a problem for the [[pdfCalc]{}]{} package, as the grid points are simply interpolated to provide the PDF at a particular point in {x,Q,$N_F$}. The user is agnostic to the specific grid spacing chosen in a PDF release.
### NF Convention
{width="45.00000%"} {width="45.00000%"}
{width="45.00000%"} {width="45.00000%"}
The $N_{F}$ flavor dimension is determined by the [[iSet]{}]{} value passed to [[pdfFunction]{}]{}. The association between the grid slice in $N_{F}$ and [[iSet]{}]{} is specified in the [[[LHAPDF6]{}]{}]{} info file using the “[[key:data]{}]{}” format such as “[[Flavors: $[-5,-4,-3,-2,-1,1,2,3,4,5,21]$]{}]{}”. This tells us which partons are in the grid, and their proper order.[^6] Note: we use the standard Monte Carlo (MC) convention[^7] throughout [[[ManeParse]{}]{}]{} where $d=1$ and $u=2$ rather than the mass-ordered convention (see Table \[tab:flavTable\]).[^8] The standard MC convention also labels the gluon as [[iParton]{}]{} $=21$; for compatibility, the gluon in [[[ManeParse]{}]{}]{} can be identified with either [[iParton]{}]{} $=21$ or [[iParton]{}]{} $=0$.
[[[ManeParse]{}]{}]{} is able to work with PDF sets with different numbers of flavors. For example, in Fig. \[fig:manyPDFs\], the [NNPDF]{} set includes $N_{F}= 6$ where [[iParton]{}]{} $ = \{\bar{t},\ldots ,t\}$, while most of the other sets have $N_{F}=5$. If a flavor, [[iParton]{}]{}, is not defined, [[pdfFunction]{}]{} will return zero. This feature allows the user to write a sum over all quarks $\sum\,f_{i}(x,Q)$ for $i=\{-6, \ldots ,6\}$ without worrying whether some PDF sets might have less than 6 active flavors.
Additionally, the [[[ManeParse]{}]{}]{} framework has the flexibility to handle new particles such as a 4^th^ generation of quarks with [[iParton]{}]{} $=\{b',t'\}=\{7,8\}$ or a light gluino with [[iParton]{}]{} $=\tilde{g}=1000021$ PDF by identifying the flavor index, [[iParton]{}]{}, with the appropriate grid position in the [[[LHAPDF6]{}]{}]{} info file.
### Q Sub-Grids
At NNLO and beyond, the PDFs can become discontinuous across the mass flavor thresholds. This is illustrated using the NNLO MSTW set in Fig. \[fig:mstw-disc \] where we observe a discontinuity of both the gluon and b-quark PDF across the b-quark threshold at $m_{b}=4.75$ GeV. [[[ManeParse]{}]{}]{} accommodates this by using sub-grids in $Q$ as illustrated in Fig. \[fig:qGrids\]-a); for example, we use separate grids below and above the threshold at $Q=m_{b}=4.75\ \mathrm{GeV}$. When we call the PDF at a specific $Q$ value, [[[ManeParse]{}]{}]{} looks up the relevant heavy quark thresholds, $\{m_{c},m_{b},m_{t}\}$, to determine which sub-grid to use for the interpolation. For $Q < m_{b}$, sub-grid \#2 ($N_{F}=4$) is used, and for $Q \geq m_{b}$, sub-grid \#3 ($N_{F}=5$) is chosen.
Note that for the $x$ value ($10^{-4}$) displayed in Fig. \[fig:mstw-disc \], the b-quark PDF is negative for $Q$ just above $m_{b}$; this is the correct higher-order result and justifies (in part) why we do not force the PDFs to be positive definite. This behavior also makes sense in terms of the momentum sum rule, which we will discuss in Sec. \[sub:Momentum-Sum-Rules\].
YAML [[[Mathematica]{}]{}]{}
-------------------------------------------------- -------------------------------------------------------------
key: “data” “key” $\rightarrow$ “data”
SetDesc: “nCTEQ15 ...” “SetDesc” $\rightarrow$ “nCTEQ15 ...”
NumFlavors: 5 “NumFlavors” $\rightarrow$ $5$
Flavors: [\[]{}-5,-4,-3,-2,-1,1,2,3,4,5,21[\]]{} “Flavors”$\rightarrow${-5,-4,-3,-2,-1,1,2,3,4,5,21}
AlphaS\_Qs: [\[]{}1.299999e+00, ...[\]]{} “AlphaS\_Qs”$\rightarrow \{ 1.299999\times10^{+00}, ... \}$
UnknownKey: data “UnknownKey”$\rightarrow$”data”
### An NF-dependent PDF: f(x,Q,NF)
Note, the use of sub-grids in $Q$ also enables the use of overlapping $N_{F}$ ranges as in a hybrid scheme as described in Ref. [@Kusina:2013slm]; in this case, we generalize the PDF so that it also becomes a function of the number of flavors: $f(x,Q,{\color{red}N_{F}})$. This feature is useful if, for example, we are performing a fit to data in the region $Q\sim m_{b}$; we can perform a consistent $N_{F}=4$ flavor fit even if some of the data are above the $N_{f}=5$ threshold ($Q>m_{b}$) by selecting $f(x,Q,N_{F}=4)$; thus, we avoid encountering any discontinuities in the region of the data.[^9] We illustrate this generalized case for $f(x,Q,N_{F})$ in Fig. \[fig:qGrids\]-b). Here, the user has the freedom to choose the active number of flavors, $N_{F}$, rather than being forced to transition at the quark mass values as in Fig. \[fig:qGrids\]-a).
The [[[LHAPDF6]{}]{}]{} Info File
---------------------------------
In addition to the 3-dimensional $\{x,Q,N_{F}\}$ grid, there is auxiliary material associated with each PDF set. In the LHA format, each PDF collection has an associated “info” file which contains the additional data in a YAML format,[^10] whereas in the CTEQ PDS format files, the auxiliary information is contained at the top of each PDS data file. Each parser interprets this information and builds a list of [[[Mathematica]{}]{}]{} rules.
The basic syntax of YAML is [[key: “data”[\]]{}]{}]{}, and the LHA parser converts this into a [[[Mathematica]{}]{}]{} rule as [[{“key”$\rightarrow$“data”}]{}]{}. This can be viewed within [[[ManeParse]{}]{}]{} using the function [[pdfGetInfo[\[]{}iSet[\]]{}]{}]{}, and Table \[tab:yamlTable\] demonstrates the some sample mappings between the two.
If “[[key]{}]{}” is known to be a number, “[[data]{}]{}” is converted from a string into a number. This behavior applies to values such as {NumFlavors, QMin, MTop, ...}. If “[[key]{}]{}” is known to be a list such as {Flavors, AlphaS\_Qs}, “[[data]{}]{}” is converted from a string into a [[[Mathematica]{}]{}]{} list. If “” is unknown, “[[data]{}]{}” is left as a string. This means that [[[ManeParse]{}]{}]{} can handle any unknown “[[key]{}]{}”, and the user can modify these rules after the fact, or introduce a custom modification by identifying “[[key]{}]{}” to the parser.
Interpolation
-------------
Once the 3-dimensional {x,Q,$N_{F}$} grid and auxiliary rules are given to the [[pdfCalc]{}]{} module, we are ready to interact with the PDFs. When the user calls for $f_{i}(x,Q)$, the [[pdfCalc]{}]{} module will determine the appropriate $N_{F}$ index and $Q$ grid and do a 4-point interpolation in the 2-dimensional $\{x,Q\}$ space. Generally, [[pdfCalc]{}]{} will interpolate $\{x,Q\}$ values with 2 grid points on each side, but at the edges of the grid, it will use a 3-1 split. It also will extrapolate beyond the limits of the grid and will return a number, even if it is unphysical. Except for setting $f_{i}(x,Q)=0$ for $x>1$, we do not check bounds, as this would slow the computation; in the sample files, we do provide examples of how the user can implement particular boundaries if desired.
Additionally, we allow the interpolated PDF to be negative. At very large $x$ this can happen due to numerical uncertainty, but there are also instances where a negative PDF is the physical result, such as at NNLO (illustrated in Fig. \[fig:mstw-disc \]). Within [[[Mathematica]{}]{}]{}, it is easy for the user to impose particular limits (*i.e.* positivity) if desired. The interpolation can be performed either with the [[[Mathematica]{}]{}]{} [[Interpolate]{}]{} function (default) or a custom 4-point Lagrange interpolator and is set with the [[pdfSetInterpolator]{}]{} function. We set the [[[Mathematica]{}]{}]{} [[Interpolate]{}]{} function as the default, as it is slightly faster, but the custom 4-point Lagrange interpolator often will provide better extrapolation of the PDFs beyond the grid boundaries and has some adjustable parameters which are useful in the low $x$ region.
![The ratio of PDFs can sometimes lead to interpolation problems; we display the ratio of two gluon PDFs at $Q = 100\ \mathrm{GeV}$. Fig.-a) on the left was generated with the default [[[Mathematica]{}]{}]{} interpolator, and Fig.-b) on the right was generated with the custom 4-point Lagrange interpolation with the default scaling of $a = 1$. \[fig:badInterp\]](images/badInterp "fig:"){width="45.00000%"} ![The ratio of PDFs can sometimes lead to interpolation problems; we display the ratio of two gluon PDFs at $Q = 100\ \mathrm{GeV}$. Fig.-a) on the left was generated with the default [[[Mathematica]{}]{}]{} interpolator, and Fig.-b) on the right was generated with the custom 4-point Lagrange interpolation with the default scaling of $a = 1$. \[fig:badInterp\]](images/badInterp2 "fig:"){width="45.00000%"}
Because the PDFs typically scale as $1/x$ for small $x$, the grids actually store $x\:f_{i}(x,Q)$, as this generally improves the interpolation. To return $f_{i}(x,Q)$ we divide by $x$, but to avoid dividing by zero we internally impose a default minimum $x$ value of $x_{min}=10^{-30}$.
In principle, the interpolation of $f_{i}(x,Q)$ can be generalized using a scaling factor $1/x^{a}$ with a variable $a$. This is one of the adjustable parameters in the custom 4-point Lagrange interpolation, and can be set with the [[pdfSetXpower[\[]{}a[\]]{}]{}]{} function.
We have bench-marked many of the PDF sets to ensure our interpolations are accurate across the defined grid in $\{x,Q\}$ space. For the PDS files, our interpolation (with [[xPower]{}]{} $=1.0$) essentially uses the same algorithm so our results match better than one part in $10^{3}$. When bench-marking with [[[LHAPDF6]{}]{}]{} interpolation, our interpolation is generally matches to 2 parts in $10^{3}$, and this may increase at very high or low $x$ values. We also compared our interpolator while varying [[xPower]{}]{} from 0.5 to 1.5 as well as the [[[Mathematica]{}]{}]{} interpolator to [[[LHAPDF6]{}]{}]{} interpolation; these agree better than 1 part in $10^{3}$ for $x>3\times10^{-3}$, but can increase for small $x$. If necessary, the user can supply a custom interpolation routine.
We find that ratios of PDFs are more sensitive to the interpolation than the PDFs themselves. For illustrative purposes, in Fig. \[fig:badInterp\], we show an example of a poor interpolation generated with the [[[Mathematica]{}]{}]{} interpolator compared to a good interpolation by the custom 4-point Lagrange interpolation with the default $a = 1$ scaling; in general, we find the custom 4-point Lagrange interpolation computes smoother ratios and provides better extrapolation beyond the grid limits.
alpha-S Function
----------------
![$\alpha_{S}(Q)$ vs. $Q$ in GeV from [NNPDF]{}. Note the discontinuity across the $m_{b} = 4.18\ \mathrm{GeV}$ threshold which is enlarged in Fig. b). \[fig:alphas\]](images/as1 "fig:"){width="45.00000%"} ![$\alpha_{S}(Q)$ vs. $Q$ in GeV from [NNPDF]{}. Note the discontinuity across the $m_{b} = 4.18\ \mathrm{GeV}$ threshold which is enlarged in Fig. b). \[fig:alphas\]](images/as2 "fig:"){width="45.00000%"}
For some of the PDF sets, the value of $\alpha_{S}(Q)$ is provided as a list of points associated with $Q_{vec}$. For these sets, we interpolate[^11] $\alpha_{S}(Q)$ to provide a matched function called [[pdfAlphaS[\[]{}iSet,Q[\]]{}]{}]{}; this is displayed in Fig. \[fig:alphas\] for a sample PDF set. The [[pdfGetInfo[\[]{}iSet[\]]{}]{}]{} function will display the information associated with the corresponding PDF set (including any $\alpha_{S}$ values). If the PDF set does not have any $\alpha_{S}$ information, the [[pdfAlpha]{}]{} function will return [[Null]{}]{}. In Fig. \[fig:alphas\]-a) we display $\alpha_{S}(Q)$ for the [NNPDF]{} set, and in Fig. \[fig:alphas\]-b) we enlarge the region near $m_{b}=4.18\ \mathrm{GeV}$ to display the discontinuity. In general, $\alpha_{S}(Q)$ will be discontinuous at NNLO and higher and at all mass thresholds, $\{m_{c},m_{b},m_{t}\}$.
Sample Plots & Calculations\[sec:Sample-Plots\]
===============================================
The advantage of importing the PDF sets into Mathematica is that we have the complete set of built-in tools that we can use for calculating and graphing. We illustrate some of these features here.
Graphical Examples
------------------
![Sample linear and log ratio plots of the gluon PDFs from Table \[tab:MomSum\] compared to CT14 as a function of $x$ at $Q = 2.0\ \mathrm{GeV}$. \[fig:LogLinear\]](images/linear "fig:"){width="45.00000%"} ![Sample linear and log ratio plots of the gluon PDFs from Table \[tab:MomSum\] compared to CT14 as a function of $x$ at $Q = 2.0\ \mathrm{GeV}$. \[fig:LogLinear\]](images/log "fig:"){width="45.00000%"}
To highlight the graphical capabilities, in Figure \[fig:LogLinear\] we display a selection of PDFs using both linear (top) and log (bottom) scale. Using the flexible graphics capabilities of [[[Mathematica]{}]{}]{}it is easy to automatically generate such plots for different PDF sets.
Small [*x*]{} Extrapolation
---------------------------
![Small $x$ extrapolation of the gluon PDF from the nCTEQ15 proton at $Q = 100~\mathrm{GeV}$ using [[pdfLowFunction]{}]{}. Here, $x_{min}=5\times10^{-6}$, and the extrapolation exponent $1/x^{a}$ is set to $a=\{0.4,\,0.6,\,0.8,\,1.0,\,1.2,\,1.4,\,1.6\}$. \[fig:smallX\]](images/pdfLow){width="45.00000%"}
Sometimes it is useful to extrapolate to low $x$ values beyond the limits of the PDF grid; for example, the study of high energy cosmic ray experiments that use very small $x$ extrapolations. @Arguelles:2015wba [@Bhattacharya:2015jpa] We provide [[pdfLowFunction[\[]{}iSet,iParton,x,Q,power[\]]{}]{}]{} which allows the user to choose the extrapolation power in the small $x$ region.[^12] An example is displayed in Fig. \[fig:smallX\] for the nCTEQ15 proton PDF. The minimum $x$ value for this set for the grid is $x_{min}=5\times10^{-6}$; beyond this limit [[pdfLowFunction]{}]{} will extrapolate using the form $1/x^{a}$. In this example, we vary the power from $0.4$ to $1.6$; using the [[[Mathematica]{}]{}]{} integration routines it is easy to find that this range of variation in the small $x$ behavior will only change the momentum fraction of the gluon by $1/2\%$.
Momentum Sum Rules\[sub:Momentum-Sum-Rules\]
--------------------------------------------
![The integrated momentum fraction Eq. \[eq:sum\] of the PDF flavors vs. $Q$ in GeV for the NNPDF set. At large $Q$ the curves are (in descending order) $\{g,u,d,\bar{u},\bar{d},s,c,b,t\}$. \[fig:IntMom\]](images/momfracs){width="45.00000%"}
The PDFs satisfy a number of momentum and number sum rules, and this provides a useful cross check on the results. The momentum sum rule: $$\sum_{i}\int_{0}^{1}dx\,x\,f_{i}(x,Q)=1,
\label{eq:sum}\\$$ says that the total momentum fraction of the partons must sum to 100%. If any single parton flavor were not imported correctly, this cross-check would be violated; hence, this provides a powerful “sanity check” on our implementation. In Table \[tab:MomSum\] we display the partonic momentum fractions (in percent) and the total; for each PDF set the momentum sum rule checks within numerical accuracy.[^13]
While Table \[tab:MomSum\] presented the momentum fraction for a single $Q$ value ($3\ \mathrm{GeV}$), it is interesting to see how these values change with the energy scale. In Fig. \[fig:IntMom\] we show the momentum carried by each PDF flavor (in percent) as a function of $Q$ in GeV. We can see the heavy quarks, $\{c,b,t\}$ enter as we cross the flavor mass thresholds. In the limit of large $Q$, the $\{\bar{u},\bar{d},
\bar{s}\}$ PDFs approach each other asymptotically.
Nuclear Correction Factors
--------------------------
![Nuclear correction ratios $F_{2}^{A}/F_{2}^{N}$ vs. $x$ for $Q=10~\mathrm{GeV}$ for the nCTEQ15 PDFs over an iso-scalar target. The left plot is on a linear scale, and the right plot is a log scale. This figure is comparable to Fig. 1 of Ref. [@Schienbein:2009kk]. \[fig:NucRatios\] ](images/structRatios "fig:"){width="45.00000%"} ![Nuclear correction ratios $F_{2}^{A}/F_{2}^{N}$ vs. $x$ for $Q=10~\mathrm{GeV}$ for the nCTEQ15 PDFs over an iso-scalar target. The left plot is on a linear scale, and the right plot is a log scale. This figure is comparable to Fig. 1 of Ref. [@Schienbein:2009kk]. \[fig:NucRatios\] ](images/structRatiosLog "fig:"){width="45.00000%"}
Given the PDFs, it is then trivial to build up simple calculations. In Fig. \[fig:NucRatios\] we display the nuclear correction factors $F_{2}^{A}/F_{2}^{N}$ for a variety of nuclei. Here, the $F_{2}$ structure functions are related to the PDFs via $F_{2}^{A}(x,Q)=x\sum_{q}\,
e_{q}^{2}\,f_{q/A}(x,Q)$ at leading order where $F_{2}^{N}$ is an isoscalar, and $F_{2}^{A}$ is the scaled structure function[^14] for nuclei $A$. We have also superimposed the uncertainty bands; we will discuss this more in the following Section.
Luminosity
----------
![The differential parton-parton luminosity $ d L_{a \bar{a}}/d M_{X}^{2} $ vs. $ M_{x} $ in GeV at $ \sqrt{s} = 14\,\mathrm{TeV}$ for (in descending order) $a = \{g,u,d,s,c,b\} $. \[fig:lumi\]](images/lum){width="45.00000%"}
Using the integration capabilities of [[[Mathematica]{}]{}]{} it is easy to compute the differential parton-parton luminosity @Campbell:2006wx for partons $a$ and $b$: $$\frac{d {\cal L}_{ab}}{d \hat{s}}
= \frac{1}{s}\frac{1}{1+
\delta_{ab}}\int_{\tau}^{1}\frac{dx}{x}\,f_{a}(x, \sqrt{\hat{s}})\,f_{b}(
\frac{\tau}{x}, \sqrt{\hat{s}})\,+(a\leftrightarrow b),
\label{eq:lum}\\$$ where $\tau = \hat{s} / s$, and the cross section is $$\sigma = \sum_{a,b} \int \left( \frac{d \hat{s}}{\hat{s}} \right)
\left( \frac{d {\cal L}_{ab}}{d \hat{s}} \right)
(\hat{s}\ \hat{\sigma}_{ab}).
\label{eq:lum2}\\$$ Note, the luminosity definition[^15] of Eq.\[eq:lum\] has dimensions of a cross section ($1/\hat{s}$), and in Eq.\[eq:lum2\] we multiply by a scaled (dimensionless) cross section ($\hat{s} \, \hat{\sigma}_{ab}$).
We define the [[pdfLuminosity]{}]{} function to compute Eq. \[eq:lum\]. The hadron-hadron production cross section for producing particle of mass $\sqrt{\hat{s}} = M_{X}$ is proportional to the luminosity times the scaled partonic cross section $\hat{s} \sigma$ as in Eq. \[eq:lum2\]. In Fig. \[fig:lumi\] we display the differential luminosity $ d L_{a\bar{a}}/ d M_{X}^{2}$ for parton– anti-parton ($a\bar{a}$) combinations; this luminosity would be appropriate if we were interested in estimating the size of the cross section for the process of quark–anti-quark annihilation into a Higgs boson, $b\bar{b}\to H$ for example [^16].
W Boson Production
------------------
![Leading-Order $W^{+}$ production cross section, $d\sigma/dy$ at the Tevatron ($p\bar{p},$ $ 1.96\ \mathrm{TeV} $) and the LHC ($pp$, $ 8\
\mathrm{TeV}$). We display the total cross section and the individual partonic contributions. \[fig:wprod\]](images/wPlusXsecWerrTeva "fig:"){width="45.00000%"} ![Leading-Order $W^{+}$ production cross section, $d\sigma/dy$ at the Tevatron ($p\bar{p},$ $ 1.96\ \mathrm{TeV} $) and the LHC ($pp$, $ 8\
\mathrm{TeV}$). We display the total cross section and the individual partonic contributions. \[fig:wprod\]](images/wPlusXsecWerr "fig:"){width="45.00000%"}
Next, we compute a simple leading-order (LO) cross section for $W^{+}$ boson production at the Tevatron proton–anti-proton collider ($ 1.96\
\mathrm{TeV} $) and the LHC proton-proton collider ($ 8\ \mathrm{TeV} $). Schematically, the cross section is $\sigma(W^{+})=f_{a}\otimes f_{b}
\otimes\omega_{ab\to W^{+}}$. There are two convolution integrals, but the constraint that the partonic energies sum to the boson mass $W^{+}$ eliminates one.[@Kusina:2012vh; @Kovarik:2012te] Hence, this can easily be performed inside of [[[Mathematica]{}]{}]{}, and the results are displayed in Fig. \[fig:wprod\]. It is interesting to note the much larger width of the rapidity distribution at the LHC as well as the increased relative contribution of the heavier quark channels (such as $c\bar{s}$ and $u\bar{s}$).
Error PDFs & Correlations\[sec:Error-PDFs:\]
============================================
PDF Uncertainties
-----------------
![The fractional PDF uncertainty vs. $x$ at $Q=10\ \mathrm{GeV}$. a) The upper (red) curve is CT14 using the [[pdfHessianError]{}]{} function, and the lower (blue) curve is the [NNPDF]{} using the [[pdfMCError]{}]{} function for the gluon. (Note, these curves do not necessarily represent the same confidence level.) b) The down quark PDF uncertainty band for the CTEQ6.6 PDFs (inner, red) and the [[nCTEQ15]{}]{} lead 208 (outer, blue); \[fig:errorPDF\]](images/error1 "fig:"){width="45.00000%"} ![The fractional PDF uncertainty vs. $x$ at $Q=10\ \mathrm{GeV}$. a) The upper (red) curve is CT14 using the [[pdfHessianError]{}]{} function, and the lower (blue) curve is the [NNPDF]{} using the [[pdfMCError]{}]{} function for the gluon. (Note, these curves do not necessarily represent the same confidence level.) b) The down quark PDF uncertainty band for the CTEQ6.6 PDFs (inner, red) and the [[nCTEQ15]{}]{} lead 208 (outer, blue); \[fig:errorPDF\]](images/error2 "fig:"){width="45.00000%"}
We now examine some of the added features provided by the [[pdfError]{}]{} module. To accommodate the PDF errors, it is common for the PDF groups to release a set of grids to characterize the uncertainties; the number of PDFs in each error is typically in the range 40 to 100, but can in principle be as many as 1000.
As [[[Mathematica]{}]{}]{} handles lists naturally, we can exploit this feature to manipulate the error PDFs. The [[pdfFamilyParseLHA]{}]{} and [[pdfFamilyParseCTEQ]{}]{} functions will read an entire directory of PDFs and return the associated set numbers as a list; this list can then be used to manipulate the entire group of error PDFs.
For example, we can use this feature to read the 100 PDFs of the [NNPDF]{}set displayed in Fig. \[fig:errorPDF\], capture the returned list of [[iSet]{}]{} values, and pass this to the plotting function; we’ll describe this more in the following.
When working with the error PDFs, the first step is to take the list of [[iSet]{}]{} values and obtain a list of the PDF values. Constructing the PDF error depends on whether the set is based on the Hessian or the Monte Carlo method.
The Hessian PDF error sets can be organized as follows $\{X_{0},\,X_{1}^{+},\,
X_{1}^{-},\,X_{2}^{+},\,X_{2}^{-},...,\,X_{N}^{+},\,X_{N}^{-}\}$ where $X_{0}$ represents the central set, $\{X_{1}^{+},\,X_{1}^{-}\}$ represent the plus and minus directions along eigenvector \#1, and so on up to eigenvector $N$. For the Hessian PDF sets, there should be an odd number equal to $2N+1$ where N is the number of eigenvector directions. The PDF errors can then be constructed using symmetric, plus, or minus definitions:[@Pumplin:2002vw; @Campbell:2006wx] $$\begin{aligned}
\Delta X^{Hess}_{sym} & = & \frac{1}{2}\sqrt{\sum_{i=1}^{N}\left[X_{i}^{+}-X_{i}^{-}
\right]^{2}}\label{eq:sym}\\
\Delta X^{Hess}_{plus} & = & \sqrt{\sum_{i=1}^{N}\left[\mbox{\ensuremath{\max}}\left
\{ X_{i}^{+}-X_{0},\,X_{i}^{-}-X_{0},\,0\right\} \right]^{2}}\label{eq:plus}\\
\Delta X^{Hess}_{minus} & = & \sqrt{\sum_{i=1}^{N}\left[\mbox{\ensuremath{\max}}
\left\{ X_{0}-X_{i}^{+},\,X_{0}-X_{i}^{-},\,0\right\} \right]^{2}}\label{eq:minus}\end{aligned}$$ These can be computed using the function [[pdfHessianError]{}]{}, and can take an optional “[[method]{}]{}” argument, {[[“sym”,“plus”,“minus”]{}]{}}, to specify which formula is used to compute the error; the default being [[“sym”]{}]{}.
We next turn to the Monte Carlo sets. For example, the [NNPDF]{} set (\#3 in Table \[tab:MomSum\]) has 101 elements; the “zeroth” set is the central set, and the remaining 100 replica sets span the PDF uncertainty space. The central set is the average of all the sets, and the PDF error is the standard deviation of the 100 replica sets. For these sets, [[pdfMCCentral]{}]{} will return the central PDF value. [[pdfMCError]{}]{} will return the associated error. This function can also take an optional “[[method]{}]{}” argument, {[[“sym”, “plus”,“minus”]{}]{}}, defined by Eqs. \[eq:MCsym\], \[eq:MCplus\], \[eq:MCminus\].@Alekhin:2011sk@Gao:2013bia
The modification from the Hessian case is due to the MC error PDFs using replica sets, not eigenvector pairs.[^17] The formula for $\Delta X^{MC}_{sym}$ is a straightforward extension of the Hessian case: $$\begin{aligned}
\Delta X^{MC}_{sym} & = & \sqrt{ \frac{1}{N_{rep}}\sum_{i=1}^{N}
\left[X_{i}-X_{0}
\right]^{2}} \quad .
\label{eq:MCsym}\end{aligned}$$ where $N_{rep}$ counts the 100 replica sets not including the “zeroth” central set. This quantity is simply the standard deviation of the values. The $1/\sqrt{N_{rep}}$ factor compensates for the fact that Monte Carlo sets can have an arbitrary number of replicas, in contrast to the Hessian sets which have a fixed number of eigenvector sets.
It is possible to define extensions for Monte Carlo “plus” and “minus” uncertainties as:[@Nadolsky:2001yg] $$\begin{aligned}
\Delta X^{MC}_{plus} & = & \sqrt{\frac{1}{N^{+}_{rep}}\sum_{i=1}^{N}
\left[\mbox{\ensuremath{\max}}
\left
\{ X_{i}-X_{0},\,0
\right\}
\right]^{2}}\label{eq:MCplus}
\\
\Delta X^{MC}_{minus} & = & \sqrt{\frac{1}{N^{-}_{rep}}\sum_{i=1}^{N}
\left[\mbox{\ensuremath{\max}}
\left\{ X_{0} - X_{i} ,\,0
\right\}
\right]^{2}} \quad ,
\label{eq:MCminus}\end{aligned}$$ where $N^{\pm}_{rep}$ are the number of replicas above/below the mean.
In Fig. \[fig:errorPDF\]-a), we compute the fractional PDF error for the CT14 PDF gluon using the [[pdfHessianError]{}]{} function with the [[“sym”]{}]{} formula of Eq. \[eq:sym\]. The same is done for the [NNPDF]{} set [[pdfMCError]{}]{} function, using Eq. \[eq:MCsym\]. As expected, we see the uncertainty increase both as $x \to 1$ and at very small $x$ values.
In Fig. \[fig:errorPDF\]-b), we compute the error bands for the down quark in the CTEQ6.6 proton PDF and also the [[nCTEQ15]{}]{} lead-208 PDF; as expected, we see the uncertainties on the nuclear PDF are larger than the proton PDF uncertainties.
Correlation Angle
-----------------
{width="45.00000%"}{width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
Finally, we can compute the correlation cosines via the relation: [@Nadolsky:2008zw] $$\begin{aligned}
\cos\varphi&=&
\frac{\overrightarrow{\nabla}X\cdot\overrightarrow{\nabla}Y}{\triangle X\,\triangle Y}
\nonumber \\
&=&\frac{1}{4\triangle X\,\triangle Y}\;
\sum_{i=1}^{N}
\left(X_{i}^{+}-X_{i}^{-}\right)\left(Y_{i}^{+}-Y_{i}^{-}\right)\;.
\label{eq:cosphi}\end{aligned}$$ We have implemented separate functions [[pdfHessianCorrelation]{}]{} and [[pdfMCCorrelation]{}]{} as the computation of the uncertainty in the denominator $\Delta X\,\Delta Y$ could depend on Eqs. \[eq:sym\], \[eq:plus\], \[eq:minus\] or \[eq:MCsym\] \[eq:MCplus\] \[eq:MCminus\].
In Fig. \[fig:correlation\] we display an example where we show the correlation cosine between the $W^+$ cross section and the partonic flavors for both the Tevatron and LHC. We observe the behavior of the flavors is quite similar except for the $u$ and $d$ quarks which stand out at large $x$.
The cosine of the correlation angle indicates the degree to which the error on a particular parton’s PDF contributes to the uncertainty on some function of the PDFs, usually a physical observable. A value close to one for some parton indicates that the PDF error on the observable is being driven by the error on that parton’s PDF. Similarly, a value close to zero indicates that the error on the parton’s PDF does not contribute significantly to the error on the observable. More details can be found in Ref. [@Nadolsky:2008zw].
Conclusions\[sec:Conclusions\]
==============================
We have presented the [[[ManeParse]{}]{}]{} package which provides PDFs within the [[[Mathematica]{}]{}]{} framework. This is designed to work with any of the [[[LHAPDF6]{}]{}]{} format PDFs, and is extensible to other formats such as the CTEQ PDS format. [[[ManeParse]{}]{}]{} can also work with nuclear PDFs such as the nCTEQ15 sets.
The [[[ManeParse]{}]{}]{} package implements a number of novel features. It adapts YAML relations into [[[Mathematica]{}]{}]{} rules including unknown keys, and can handle discontinuities in both the PDFs and $\alpha_s(Q^2)$. We have implemented a flexible interpolation with a tuneable parameter, and it can extrapolate to small $x$ with a variable power. Additionally, we have implemented functions to facilitate the calculation of PDF uncertainties for both Hessian and Monte Carlo PDF sets.
[[[ManeParse]{}]{}]{} provides many tools to simplify calculations involving PDFs, and is fast enough such that even one or two convolutions can easily be computed within the [[[Mathematica]{}]{}]{} framework. We illustrated these features with examples of $W$ production, luminosity calculations, nuclear correction factors, and $N_F$-dependent PDFs.
In summary, the [[[ManeParse]{}]{}]{} package is a versatile, flexible, user-extensible tool that can be used by beginning users to make simple PDF plots, as well as by advanced users investigating subtle features of higher-order discontinuities and PDF uncertainty calculations.
Acknowledgments\[sec:Acknowledgments\] {#acknowledgmentssecacknowledgments .unnumbered}
======================================
The authors would like to thank V. Bertone, M. Botje, A. Buckley, P. Nadolsky, and V. Radescu for help and suggestions. We are also grateful to our nCTEQ colleagues K. Kovarik, A. Kusina, T. Jezo, C. Keppel, F. Lyonnet, J.G. Morfin, J.F. Owens, I. Schienbein, & J.Y. Yu for valuable discussions, testing the program, and providing useful feedback.
We acknowledge the hospitality of CERN, DESY, Fermilab, and KITP where a portion of this work was performed. We thank HepForge for hosting the project files. This work was also partially supported by the U.S. Department of Energy under Grant No. DE-SC0010129 and by the National Science Foundation under Grant No. NSF PHY11-25915.
Appendices
==========
[[[ManeParse]{}]{}]{} Distribution Files\[sec:ManeParse-Distribution-Files\]
----------------------------------------------------------------------------
The [[[ManeParse]{}]{}]{} package is distributed as a gzipped tar file (about 2.6Mb), and this is available at [[cteq.org]{}]{} or [[ncteq.HepForge.org]{}]{}.
When this is unpacked, the [[ManeParse]{}]{} modules {[[pdfCalc]{}]{}, [[pdfErrors]{}]{}, [[pdfParseCTEQ]{}]{}, [[pdfParseLHA]{}]{}} will be in the [[./MP\_Packages/]{}]{}
There is a [[Demo.nb]{}]{} [[[Mathematica]{}]{}]{} notebook which will illustrate the basic functionality of the program; we also include a [[Demo.pdf]{}]{} file so the user can see examples of the correct output.
We do not distribute any PDF files, so these must be obtained from the [[[LHAPDF6]{}]{}]{} website[^18] or the CTEQ website.[^19] The [[README]{}]{} file will explain how to run the [[MakeDemo.py]{}]{} python script to download and set up the necessary directories for the PDF files.[^20]
The [[MakeDemo.py]{}]{} script will also run the [Perl]{} script [[noe2.perl]{}]{} on the CT10 data files. Older versions of these files use a two digit exponent (e.g. ), but occasionally three digits are required in which case the value is written as instead of . While the GNU compiler writes and reads this properly, other programs (including [[[Mathematica]{}]{}]{}) do not, so the [[noe2.perl]{}]{} script fixes this. This script can also be run interactively, in which case it will print out any lines that are modified.
There is a manual in both [[[Mathematica]{}]{}]{} format (manual\_v1.nb) and PDF format (manual\_v1.pdf); this allows the user to execute the notebook directly, but also see how the output should look. The manual provides examples of all the functions of [[[ManeParse]{}]{}]{}.
There is also a glossary file [[User.pdf]{}]{} which provides a list and usage of all the commands.
A Simple Example
----------------
First we define some directory paths. You should adjust for your particular machine. Note, for LHAPDF6, the individual “dat” and “info” files are stored in subdirectories.
\
\
\
\
\
Next, we load the [[[ManeParse]{}]{}]{} packages. The [[pdfCalc]{}]{} package is automatically loaded by both [[pdfParseLHA]{}]{} and [[pdfParseCTEQ]{}]{}, so we do not need to do this separately.
\
\
\
[[pdfParseLHA]{}]{} will read the PDF set and assign an “iSet” number, which in this case is $1$.
\
\
\
\
Out\[...\]:=$1$\
The “iSet” numbers are assigned sequentially, and are returned by [[pdfParseLHA]{}]{} which we use to define the variable [[iSetMSTW]{}]{} (=1 in this example). We can then evaluate the PDF values.
\
\
\
\
Out\[...\]:=$11.714$\
Next, we can read in an NNPDF PDF set.
\
\
\[1.0\][, ]{}\
\[1.0\][\] ]{}\
Out\[...\]:=$2$\
We can then evaluate this PDF. We find it is similar (but not identical) to the value above.
\
Out\[...\]:=$11.8288$\
Finally, we load a [[ctq66]{}]{} PDF file in the older “pds” format using the [[pdfParseCTEQ]{}]{} function; note this only takes a single file as the “info” details are contained in the “pds” file header.
\
\
Out\[...\]:= 3\
\
Out\[...\]:= 11.0883\
Now that we have these functions defined inside of Mathematica, we can make use of all the numerical and graphical functions. Detailed working examples are provided in the auxiliary files.
NF-Dependent PDF Example
------------------------
![ We display the gluon PDF $f_g (x,Q)$ at $x=0.03$ vs. $Q$ for $N_F=\{3,4,5,6\}$; $N_F=3$ is the largest, and $N_F=6$ is the smallest curve. \[fig:nfPDF\]](images/nfPDF.pdf){width="45.00000%"}
We provide an example of implementing the $N_F$-dependent PDFs within the [[[ManeParse]{}]{}]{} framework using the matched set of PDFs[^21] with $N_F=\{3,4,5,6\}$ from Ref. [@Kusina:2013slm]. We load the [[[ManeParse]{}]{}]{} packages as above, and then read in the grid files which are in “pds” format.\
\
\
\
\
\
[[pdfParseCTEQ]{}]{} returns the “iSet” number and we store these in {[[iSetNF3, ...]{}]{}}. The below function [[pdfNF]{}]{} allows the user to choose $N_F$, and then returns the appropriate PDF.\
\
\
\
\
\
\
\
\
\
The the [[pdfNF]{}]{} function, the “iSet” variable is local to the [[Module]{}]{}. We now compute some sample values.\
\
\
\
\
\
Out\[...\]:={123.288, 117.694, 115.331, 115.341}\
As we have taken $Q=10~$GeV, we are above the charm and bottom transition, but below the top transition; hence the $N_F=\{5,6\}$ results are the same, but the $N_F=\{3,4\}$ values differ.
In Fig. \[fig:nfPDF\] we display the gluon PDF vs. $Q$ for $N_F=\{3,4,5,6\}$. We observe as we activate more flavors in the PDF evolution the gluon is reduced as a function of $N_F$. This decrease in the gluon PDF will be (partially) compensated by the new $N_F$ channels.
[^1]: Lattice QCD has made great strides in computing PDFs in recent years. [@Ma:2014jga; @Alexandrou:2016tjj]
[^2]: The [[[ManeParse]{}]{}]{} program was originally developed to run on the SMU computing cluster “ManeFrame” which is a play on words inspired by the school mascot, Peruna the pony.
[^3]: All plots presented here have been generated in [[[Mathematica]{}]{}]{}.
[^4]: Note that the [[[LHAPDF6]{}]{}]{} files have both a data file and an info file whereas the older [CTEQ]{} PDS files have only a data file.
[^5]: If interpolation outside the given grid is requested by the user, [[[ManeParse]{}]{}]{} is equipped to handle this. The [[[Mathematica]{}]{}]{} interpolater will throw a warning message and proceed to use built-in extrapolation techniques. The [[[ManeParse]{}]{}]{} interpolator will extrapolate using the behavior defined with [[pdfSetXpower]{}]{}.
[^6]: For the PDS files, this information is contained in the header of the data file so there is not a separate info file; [[pdfParseCTEQ]{}]{} extracts the proper association.
[^7]: See Ref. [@Agashe:2014kda] “Review of Particle Physics,” Chapter 34 entitled “Monte Carlo particle numbering scheme.”
[^8]: Caution is required here as many of the older [CTEQ]{} releases use the mass-ordered convention with $u=1$ and $d=2$. [[[ManeParse]{}]{}]{} converts these mass-ordered sets into the MC ordering.
[^9]: Note that the APFEL PDF evolution library[@Bertone:2013vaa] is in the process of implementing these features.
[^10]: “YAML Ain’t Markup Language” <http://yaml.org/>
[^11]: Since at Leading Order (LO), $\alpha_{S}(Q)=1/[\beta_{0}\ln(Q^{2}/\Lambda^{2})]$, we obtained improved results by interpolating in $1/\alpha_{S}(Q)$.
[^12]: The “[[a]{}]{}” argument is optional; the default power is $1.0$. We use a separate function [[pdfLowFunction]{}]{} so as not to slow the computation of [[pdfFunction]{}]{}.
[^13]: Numerical uncertainties arise from the extrapolation down to $x\to0$, the interpolation, and the integration precision.
[^14]: More specifically, $F_{2}^{N}$ is the average of the proton and neutron $(p+n)/2$ and $F_{2}^{A}$ is composed of Z protons, (A-Z) neutrons, and scaled by A to a make it “per nucleon:” $\left[Z\,p+(A-Z)\,n\right]/A$.
[^15]: There are other definitions of the luminosity in the literature which are dimensionless such as ${\cal L}=f_a \otimes f_b$.
[^16]: [[[ManeParse]{}]{}]{} also has the capability to handle custom PDFs. This allows the user to explore a wide variety of phenomena, such as intrinsic heavy quarks, as long as the custom PDFs are written in either [[[LHAPDF6]{}]{}]{} or [CTEQ]{} format
[^17]: See the [[[LHAPDF6]{}]{}]{} reference@Buckley:2014ana for a more complete description of the error definitions and calculation.
[^18]: <http://lhapdf.hepforge.org/>
[^19]: <http://cteq.org/>
[^20]: Python is not essential to [[[ManeParse]{}]{}]{} as the files can be setup manually.
[^21]: These PDF sets are available at [[http://ncteq.hepforge.org/]{}]{}.
|
---
abstract: 'This proceeding briefly summarizes our recent work on calculating the correlated fluctuations of net protons on the hydrodynamic freeze-out surface near the QCD critical point. For both Poisson and Binomial baselines, our calculations could roughly reproduce the energy dependent cumulant $C_4$ and $\kappa \sigma^2$ of net protons, but always over-predict $C_2$ and $C_3$ due to the positive contributions from the static critical fluctuations.'
address:
- 'Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China'
- 'Collaborative Innovation Center of Quantum Matter, Beijing 100871, China'
- 'Center for High Energy Physics, Peking University, Beijing 100871, China'
author:
- Lijia Jiang
- Pengfei Li
- Huichao Song
title: 'Multiplicity fluctuations of net protons on the hydrodynamic freeze-out surface'
---
Relativistic heavy-ion collisions, QCD critical point, correlations and fluctuations
Introduction
============
One of the fundamental goals of Relativistic Heavy Ion Collisions (RHIC) is to locate the critical point of the QCD phase diagram [@Aggarwal:2010cw]. To search the critical point in experiment, the STAR collaboration has systematically measured the higher cumulants of net-protons in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27, 39, 62.4 and 200 GeV [@Aggarwal:2010wy; @Adamczyk:2013dal; @Luo:2015ewa]. With the maximum transverse momentum increased from $0.8$ to $2$ GeV, the measured cumulant ratios $\kappa \sigma ^{2}$ of net protons show large deviations from the Poisson baselines and present an obvious non-monotonic behavior in the most central Au+Au collisions [@Luo:2015ewa], which hints the signal of the QCD critical point. To quantitatively study these experimental data, we need to develop dynamical models near the QCD critical point. Currently, most of the dynamical models near the QCD critical point, e.g. chiral fluid dynamics, focus on the dynamical evolution of the bulk matter [@Paech:2003fe]. In a recent paper [@Jiang], we introduced a freeze-out scheme for the dynamical models near the QCD critical point through coupling the decoupled classical particles with the order parameter field. With a modified distribution function, we calculated the correlated fluctuations of net protons on the hydrodynamic freeze-out surface in Au+Au collisions at various collision energies. In this proceeding, we will briefly summarize the main results of that paper.
The formalism and set-ups
=========================
In traditional hydrodynamics, the particles emitted from the freeze-out surface can be calculated through the Cooper-Frye formula with a classical distribution function $f(x,p)$ [@Song:2007fn]. In the vicinity of the critical point, we assume the effective mass of the classical particles strongly fluctuates through interacting with the order parameter field: $\delta m =g\sigma(x)$, which introduces a modified distribution function that also correlated fluctuates in position space [@Jiang]. To the linear order of $\sigma(x)$, the modified distribution function can be expanded as: $$f=f_{0}+\delta f=f_{0}\left( 1-g\sigma /\left( \gamma T\right) \right) ,$$where $f_{0}$ is the traditional equilibrium distribution function, $\delta f$ is the fluctuation deviated from the equilibrium part, $\gamma =\frac{p^{\mu }u_{\mu }}{m}$ is the covariant Lorentz factor and the coupling constant $g=\frac{dm}{d\sigma }$. With such expansion, the connected 2-point 3-point and 4-point correlators $\left\langle \delta f_{1}\delta f_{2}\right\rangle _{c}$, $\left\langle \delta f_{1}\delta f_{2} \delta f_{3}\right\rangle _{c}$ and $\left\langle \delta f_{1}\delta f_{2} \delta f_{3} \delta f_{4}\right\rangle _{c}$ are proportional to $\left\langle \sigma _{1}\sigma _{2}\right\rangle _{c}$, $\left\langle \sigma _{1}\sigma _{2}\sigma_{3}\right\rangle _{c}$ and $\left\langle \sigma _{1}\sigma _{2}\sigma_{3}\sigma_{4}\right\rangle _{c}$, respectively. Integrating over the whole freezeout surface for theses correlators gives the explicit forms of the critical fluctuations for produced hadrons [@Jiang]: $$\begin{aligned}
\left\langle \left( \delta N\right) ^{2}\right\rangle _{c}=& \left( \frac{%
1}{\left( 2\pi \right) ^{3}}\right) ^{2}\prod_{i=1,2}\left( \int \frac{1%
}{E_{i}}d^{3}p_{i}\int_{\Sigma _{i}}p_{i\mu }d\sigma _{i}^{\mu }\right)
\frac{f_{01}f_{02}}{\gamma _{1}\gamma _{2}}\frac{g^{2}}{T^{2}}%
\left\langle \sigma _{1}\sigma _{2}\right\rangle _{c}, \\
\left\langle \left( \delta N\right) ^{3}\right\rangle _{c}=& \left( \frac{%
1}{\left( 2\pi \right) ^{3}}\right) ^{3}\prod\limits_{i=1,2,3}\left(
\int \frac{1}{E_{i}}d^{3}p_{i}\int_{\Sigma _{i}}p_{i\mu }d\sigma _{i}^{\mu
}\right)
\frac{f_{01}f_{02}f_{03}}{\gamma _{1}\gamma _{2}\gamma _{3}}\left(
-1\right) \frac{g^{3}}{T^{3}}\left\langle \sigma _{1}\sigma _{2}\sigma
_{3}\right\rangle _{c}, \\
\left\langle \left( \delta N\right) ^{4}\right\rangle _{c}=& \left( \frac{%
1}{\left( 2\pi \right) ^{3}}\right) ^{4}\prod\limits_{i=1,2,3,4}\left(
\int \frac{1}{E_{i}}d^{3}p_{i}\int_{\Sigma _{i}}p_{i\mu }d\sigma _{i}^{\mu
}\right)
\frac{f_{01}f_{02}f_{03}f_{04}}{\gamma _{1}\gamma _{2}\gamma
_{3}\gamma _{4}}\frac{g^{4}}{T^{4}}\left\langle \sigma _{1}\sigma _{2}\sigma
_{3}\sigma _{4}\right\rangle _{c}.\end{aligned}$$ where the correlators of the sigma field can be derived from the probability distribution function with cubic and quartic terms $P[\sigma ]=\exp \left\{ -\Omega \left[ \sigma \right] /T\right\}
= \exp \left\{-\int d^{3}x\left[ \frac{1}{2}\left( \nabla
\sigma \right) ^{2}+\frac{1}{2}m_{\sigma }^{2}\sigma ^{2}+\frac{\lambda _{3}%
}{3}\sigma ^{3}+\frac{\lambda _{4}}{4}\sigma ^{4}\right]/T\right\}$ [@Stephanov:2008qz; @Stephanov:2011pb]. With such equilibrium distribution $P[\sigma]$, the deduced $\left\langle \left( \delta N\right) ^{2}\right\rangle _{c}$, $\left\langle \left( \delta N\right) ^{3}\right\rangle _{c}$ and $\left\langle \left( \delta N\right) ^{4}\right\rangle _{c}$ belong to the category of static critical fluctuations. If replacing the related integrations on the freeze-out surface by integrations over the whole position space, the standard formula for a static and infinite medium given by Stephanov in 2009 [@Stephanov:2008qz] can be reproduced [@Jiang].
To obtain the needed freezeout surface $\Sigma$, we implement the viscous hydrodynamic code VISH2+1[@Song:2007fn] and extend its 2+1-d freezeout surface to the longitudinal direction with the momentum and space rapidity correlations. For simplicity, we neglect succeeding hadronic scatterings and resonance decays below $T_c$. We assume the critical and noncritical fluctuations are independent, and use the Poission and Binomial distributions as the non-critical fluctuations baselines. Correspondingly, the total cumulants are expressed as: $C_n=(C_n)^{non-critical} + (C_n)^{critical}$, $n = 2, 3, 4$ (with $(C_n)^{critical}=\left\langle \left( \delta N\right) ^{n}\right\rangle _{c}$). To roughly fit the trends of $C_2$, $C_3$ and $C_4$ of net protons, we tune the couplings $g$, $\tilde{\lambda}_{3}$ $\left( \lambda _{3}=\tilde{\lambda}_{3}T\left( T\xi \right)
^{-3/2}\right)$ and $\tilde{\lambda}_{4}$ $\left( \lambda _{4}=\tilde{\lambda}
_{4}\left( T\xi \right) ^{-1}\right) $, and the correlation length $\xi$ within the allowed parameter ranges for each collision energy (please refer to [@Jiang] for details).
[**Au+Au 0-5%, Thermal+Critical fluctuations (Poisson baselines )** ]{}
{width="2.9"} {width="2.9"}
[**Au+Au 0-5%, Thermal+Critical fluctuations (Binomial baselines )** ]{}
{width="2.9"} {width="2.9"}
{width="2.9"} {width="2.9"}
Numerical results
=================
Fig. 1 and Fig. 2 show the energy dependence of cumulants $C_{1}-C_{4}$ for net protons in the most central Au+Au collisions with either Poisson or Binomial baselines. After tuning $g$, $\xi$, $\tilde{\lambda}_{3}$ and $\tilde{\lambda}_{4}$ within the allowed parameter ranges, we roughly describe the decreasing trend of $C_2$ and $C_3$ and the non-monotonic behavior of $C_4$ with the increase of collision energy. However, $C_{2}$ and $C_{3}$ from our model calculations are always above the Poisson/Binomial baselines due to the positive contributions from the critical fluctuations. For the Binomial baselines, our model calculations can nicely fit the energy dependent $C_{4}$ within two different $p_{T}$ ranges. However, if using the Poisson baselines, our calculations can not simultaneously describe the $C_{4}$ data at lower collision energies. For Au+Au collisions below 11.5 GeV, the measured $C_{4}$ are higher than the Poisson expectation values for $0.4<p_{T}<2 \ \mathrm{GeV}$, but lower than the Poisson expectation values for $0.4<p_{T}<0.8 \ \mathrm{GeV}$. For Eqs.(2-4), the change of the $p_{T}$ ranges only affects the magnitude of the $C_{n}^{critial}$ from the critical fluctuations, rather than their signs, which thus can not explain the $C_{4}$ data at lower collision energies with the Poisson baseline.
Fig. 1 and Fig. 2 also show that, with the maximum $p_{T}$ increased from 0.8 GeV to 2 GeV, the higher cumulants from both experiment and model dramatically increase, showing large deviations from the Poisson/Binomial baselines. In fact, the $n_{th}$ order critical fluctuations from Eqs.(2-4) are closely related to the $n_{th}$ power of the total net-proton numbers within the defined $p_T$ and rapidity range. With the maximum $p_T$ increased from $0.8 \ \mathrm{GeV}$ to $2 \ \mathrm{GeV}$, the averaged numbers of the net protons almost increase by a factor of two, leading to large increase of higher cumulants in our calculations.
Fig. 3 shows the energy dependence of cumulant ratios $S\sigma
=C_{3}/C_{2}$ and $\kappa \sigma ^{2}=C_{4}/C_{2}$ in 0-5% Au+Au collisions with Poisson/Binomial baselines. Although our model calculations over-predict $C_{2}$ and $C_{3}$, the cumulants ratios $S\sigma $ and $%
\kappa \sigma ^{2}$ show better agreement with the experimental measurements in the most central collisions, except for $S\sigma $ with $0.4<p_{T}<2 \ \mathrm{GeV}$. Ref [@Jiang] also showed that the critical fluctuations dramatically decreased from most central to semi-peripheral Au+Au collisions. For 30-40% centrality, our model calculations (in [@Jiang]) can nicely fit $S\sigma$ and $\kappa \sigma ^{2}$ with the Binomial baselines, but fail to fit $\kappa \sigma ^{2}$ with the Poisson baselines since the Poisson expectations largely deviate from the experimental data there.
summary
=======
Based on Ref. [@Jiang], this proceeding briefly introduced the freezeout scheme near the QCD critical point and outlined the formulism to calculate the correlated fluctuations of net protons on the hydrodynamic freeze-out surface with the presence of an external order parameter field. Our model calculations could roughly describe the decrease trend of $C_2$ and $C_3$ and the non-monotonic behavior of $C_4$ and $\kappa \sigma^2$ through tuning the related parameters, but always over-predict the values of $C_{2}$ and $C_{3}$ for both Poisson and Binomial baselines due to the positive contributions from the static critical fluctuations. To solve this sign problem of $C_{2}^{critcal}$ and $C_{3}^{critcal}$, the dynamical evolution of the sigma field and more realistic thermal fluctuation baselines should be investigated in the near future.\
$Acknowledgement:$ This work is supported by the NSFC and the MOST under grant Nos. 11435001 and 2015CB856900.
References {#references .unnumbered}
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abstract: 'Tunneling of electrons of definite chirality into a quantum wire creates counterpropagating excitations, carrying both charge and energy. We find that the partitioning of energy is qualitatively different from that of charge. The partition ratio of energy depends on the excess energy of the tunneling electrons (controlled by the applied bias) and on the interaction strength within the wire (characterized by the Luttinger liquid parameter $\kappa$), while the partitioning of charge is fully determined by $\kappa$. Moreover, unlike for charge currents, the partitioning of energy current should manifest itself in $dc$ experiments on wires contacted by conventional (Fermi-liquid) leads.'
author:
- Torsten Karzig
- Gil Refael
- 'Leonid I. Glazman'
- Felix von Oppen
title: Energy Partitioning of Tunneling Currents into Luttinger Liquids
---
*Introduction.—*Recent experiments try to elucidate the out-of-equilibrium physics of one-dimensional (1D) electron systems [@review10], with experimental systems including quantum wires [@barak10], carbon nanotubes [@chen09], as well as quantum Hall edge channels [@granger09; @altimiras10]. At low energies, the electron kinetics is dominated by processes within the electron liquid, and the kinetics in 1D is quite distinct from that in higher dimensions [@lunde07; @gutman09; @bagrets09; @micklitz10; @karzig10; @lunde10; @micklitz11]. The differences appear already in the most elementary process, namely the accommodation of an additional electron with well-defined energy and momentum which is injected into the liquid. In higher dimensions, the energy and momentum are transferred to a quasiparticle of the Fermi liquid, while the injected charge spreads away from the injection point isotropically in space and on a short time scale governed by the (collective) plasmon excitations. In 1D, such a momentum-conserving tunneling process creates an excited state of the liquid, involving correlated multiple electron-hole excitations. The description of such a state is quite complex [@gutman09] even within the Tomonaga-Luttinger model. That raises the question of finding measurable characteristics which quantify the state of the liquid perturbed by electron injection.
Perhaps the simplest characteristic is the partition ratio $Q_-/Q_+$ of the injected charge $e$. The latter creates two pulses which carry unequal charges, $Q_+$ and $Q_-$, propagating, respectively, in and against the direction of motion of the injected charge [@pham00; @steinberg08]. In the absence of interactions, the entire injected charge moves in the direction of motion of the injected electron, i.e., $Q_-=0$. In the interacting (Luttinger) liquid, $Q_-/Q_+$ is simply related to the ratio of compressibilities of the liquid with and without interactions, and can be readily obtained from the conservation laws of particle number and momentum which yields $Q_\pm = (1 \pm \kappa)/2$ in units of $e$ [@review10]. \[Here, the Luttinger liquid parameter $\kappa$ measures the interaction strength, with $\kappa =1$ ($\kappa<1$) for non-interacting (repulsively interacting) particles.\] The two pulses propagate freely unless they encounter an inhomogeneity of the interaction constant [@safi95; @maslov95]. Unfortunately, such inhomogeneities are inevitable in experiment which probe the Luttinger liquid by attaching Fermi liquid leads. Because of multiple scattering at the two interfaces, the net charges $ Q_L$ and $Q_R$ flowing into left and right leads differ from the intrinsic values $Q_-$ and $Q_+$. Indeed, $Q_L=0$ in the case of Fermi liquid leads, rendering interaction effects in the Luttinger liquid irrelevant for the charge partitioning measured in $dc$ experiments [@steinberg08; @berg09].
The energy of the injected electron is another conserved quantity in the tunneling process which plays a crucial role in the non-equilibrium physics of the electron liquid. In this paper, we show that the energy is also partitioned between left- and right-moving excitations, in a way which is quite distinct from the partitioning of the injected charge and which sensitively probes the interaction strength. When momentum is conserved in the injection process, the initial splitting of the excess energy (measured from the Fermi energy) depends on both energy and momentum of the injected electron as well as the interaction strength $\kappa$. The actual amounts of energy deposited into the two Fermi-liquid leads depend in general on the nature of the interface between Luttinger liquid and leads. The interface is transparent to the flow of energy at high energies, and has finite transparency in the opposite limit. In both limits, the partition of energy deposited in the two leads becomes independent of the properties of the interface but remains a function of $\kappa$ and excess energy. We suggest relatively simple $dc$ experiments to detect energy partitioning and also extend our considerations to include energy partitioning in tunneling into quantum Hall edge states.
*Energy currents in Luttinger liquids.—*We consider a Luttinger liquid of spinless fermions at zero temperature. Decomposing the Luttinger-liquid displacement and phase fields $\phi$ and $\theta$ into right- and left-moving excitations $\theta_\pm(x)=\theta(x) \pm \phi(x)/\kappa$, the relevant Hamiltonian takes the form [@pham00] $$H=\frac{v_F}{4\pi}\int {\rm d}x \sum_{\alpha=\pm}(\nabla \theta_\alpha)^2$$ with commutation relations $[\theta_\alpha(x),\theta_{\alpha^\prime}(x^\prime)] = \delta_{\alpha\alpha^\prime} (i\pi\alpha/\kappa)\, {\rm sgn}(x-x^\prime)$.
![Illustration of proposed experimental setups. (a) Nonlocal injection by momentum-conserving tunneling between parallel quantum wires. The quantum dots to the left and right of the injection region serve to probe the energy partitioning. (b) Local injection into one of two closeby quantum Hall edge channels. The figure indicates both the initial splitting of charge and energy at injection and the resulting splitting in the Fermi-liquid leads. While the charge partitioning is identical for both setups, the energy partitioning is different and distinct from the charge partitioning.[]{data-label="fig1"}](Fig1.pdf)
We first consider tunneling from a parallel source wire of length $L_S$ \[nonlocal injection, cp. Fig. \[fig1\](a)\]. In this case, the dispersions of quantum wire and source can be shifted relative to each other in momentum by applying a magnetic field and in energy by applying a bias voltage $V$ [@barak10]. Following recent experiments [@barak10], we assume that these shifts are such that tunneling is only allowed for left movers from the source \[field operator $\psi_S(x)$\] which tunnel into right-moving free-electron states in the quantum wire \[field operator $\psi_R^\dagger(x)$\]. This is described by the tunneling Hamiltonian $H_{TR}=t\int_{S}{\rm d}x [\psi_{R}^{\dagger}(x)\psi_{S}(x)+\psi_{S}^{\dagger}(x)\psi_{R}(x)]$ where the nature of the chirality of the states is included through the dispersions, cf. Fig. \[fig2\](a).
The ensuing right- and left-moving energy currents $I^E_\pm$ in the Luttinger liquid are now described by the operators $$\begin{aligned}
I_\pm^{E}\! & = & \!\mathrm{i}[H_{TR},\frac{v_{F}}{4\pi}\int\mathrm{d}x\left(\nabla\theta_\pm\right)^{2}]
\nonumber \\
&\! = & \!\frac{\pm c t}{2 {\rm i}}Q_\pm \! \int_S \! {\rm d} x (\!\{\psi_R^\dagger(x),\nabla \theta_\pm(x) \}\psi_S(x)-\rm{h.c.}\! )
\label{energycurrentop} \end{aligned}$$ To leading order in the tunneling, the expectation value of $I_\pm^E$ becomes $$\left\langle I_{\pm}^{E}(\tau)\right\rangle=-\mathrm{i}\int_{-\infty}^{\tau}\mathrm{d}t' \left\langle\left[ I_{\pm}^{E}(\tau),H_{TR}(t')\right]\right\rangle\,.
\label{avIE}$$ The resulting correlators can be efficiently computed by writing $\psi_{R}^{\dagger}\sim\mathrm{e}^{-\mathrm{i} \left(Q_{+}A_{+}\nabla \theta_{+}+Q_{-}A_{-}\nabla\theta_{-}\right)}$, expressing them in terms of formal derivatives with respect to the auxiliary operators $A_{\pm}=\nabla^{-1}$, and tracing the modifications due to $A_\pm$ in the standard calculation [@giamarchi] of the Luttinger liquid Green function. We then find
$$\begin{aligned}
\left\langle I_{\pm}^{E}\right\rangle = \frac{Q_{\pm}^{2}t^{2} L_{S}}{\kappa}\int \frac{{\mathrm d}\epsilon_S} {2\pi}\int \frac{\mathrm{d}k}{2\pi} \int_{0}^{\infty}{\mathrm{d} \omega_q} \Big\{ G^>_{R,k\mp q}(\epsilon_S - \omega_{q}) G_{S,k}^<(\epsilon_S-eV)
+G^<_{R,k \pm q}(\epsilon_S + \omega_q ) G^>_{S,k}(\epsilon_S-eV) \Big\}\, .
\label{parallelwire}\end{aligned}$$
Here, $G^{<,>}_{R,k}(\epsilon)$ denotes the lesser ($<$) or larger ($>$) Green function of the right-moving electrons (with chemical potential $\mu=0$), $G^{<,>}_{S,k}(\epsilon)$ the corresponding Green functions of the left-movers in the source (with chemical potential $\mu_S=eV$), and $\omega_q=cq$ is the plasmon dispersion. The two terms in Eq. (\[parallelwire\]) describe spontaneous plasmon emission in the course of tunneling from source to wire and vice versa, yielding a zero-temperature energy current which is strictly positive.
A complementary experimental setup would consist of two quantum Hall edge channels spaced such that there is appreciable Coulomb interaction but negligible interedge tunneling. This system shares the same interaction physics with the quantum wire [@berg09], but allows for locally injecting electrons of fixed chirality and fixed energy $\epsilon_{\rm in}$ by selective tunneling into one of the edge channels from a nearby single-level quantum dot \[local injection, cp. Fig. \[fig1\](b)\]. For tunneling into right-moving states, the tunneling Hamiltonian takes the form $H_{TR} = t_{\rm loc}[\psi_R^\dagger(x=0)\psi_S + {\rm h.c.}]$. Focusing on tunneling from the quantum dot into the quantum wire, i.e., on voltages for which the quantum dot is occupied and described by the Green function $G_S^<(k,\epsilon)=2\pi i \delta(\epsilon + eV - \epsilon_{\rm in})$, we obtain $$\left\langle I_{\pm}^{E}\right\rangle=\frac{i Q_{\pm}^{2} t_{\rm loc}^{2}}{\kappa}\int_{0}^{\infty}\mathrm{d}\omega
\int \frac{{\rm d}k}{2\pi} G^>_{R,k}(\epsilon_{\rm in}-\omega).
\label{IElocal}$$ for the left- and right-moving energy currents.
It is instructive to compare these results for the energy current to the charge current. Charge partitioning is already evident from the [*operator*]{} relation $I_\pm=Q_{\pm} I$ between the right- and left-moving charge currents $I_\pm=\frac{d}{dt}\{\pm(\kappa/2\pi) \int dx \nabla \theta_\pm\}$ and the total current operator $I=I_++I_-$ [@pham00]. Thus, charge partitioning depends only on the interaction parameter and is independent of the particular tunneling process. In contrast, energy partitioning generally depends on the energy and momentum of the tunneling electron which requires one to go beyond the operator level and which makes it sensitive to details of the tunneling process, as we will now discuss in detail.
*Energy partitioning.—*Focus first on the case of nonlocal injection \[Fig. \[fig1\](a)\]. The energetics of the tunneling process from a noninteracting source wire is illustrated in Fig. \[fig2\](a). While electrons with a distribution of energies and momenta can tunnel into the lower wire, it is easy to separate out the contribution of electrons of well-defined energy and momentum by measuring the differential energy currents ${\rm d}\langle I^E_\pm\rangle/{\rm d}V$. Indeed, Eq. (\[parallelwire\]) yields $$\frac{{\rm d} \langle I_{\pm}^E \rangle/{\rm d} V}{{\rm d} \langle I \rangle/{\rm d} V}=\frac{1}{2} (eV \pm c k_V)\,,
\label{diffInonlocal}$$ where $V$ is the applied bias and $k_V$ the momentum of the highest-energy electron in the source, cp. Fig. \[fig2\](a). For the double-wire geometry [@barak10], the change in $V$ must be accompanied by an adjustment in the magnetic field such that the crossing of source dispersion and Luttinger liquid mass shell (injection point when both source and wire are noninteracting) remains fixed.
To obtain Eq. (\[diffInonlocal\]), we assume the bias to be such that electrons are tunneling from the source into the lower wire, i.e., we can restrict attention to the first term in Eq. (\[parallelwire\]). Then, Eq. (\[diffInonlocal\]) follows by using the Luttinger-liquid spectral function [@giamarchi] (for right- and left-movers) $A_{R/L} = (2\pi/\phi\Gamma^2(\phi))(\Lambda/2c)^{2\phi}|\omega\mp ck|^{\phi-1} |\omega \pm ck|^\phi \theta(|\omega| - c |k|)$ as well as the relation $G^>_{k}(\epsilon) = - i A(k,\epsilon)[1- n_F(\epsilon)]$. Here, $\phi = (\kappa -2 + \kappa^{-1})/4$, $\Lambda$ denotes a large-momentum cutoff, and $n_{F}(\epsilon)$ is the Fermi function.
In essence, this result for energy partitioning can be understood from energy and momentum conservation. A right-moving electron with wavevector $k_F=m v_F$ injected into the Luttinger liquid causes right- and left-moving charge excitations of charges $Q_\pm$ moving with velocity $\pm c$. Since charge transport is accompanied by mass transport, momentum and charge conservation imply $Q_+ m c-Q_- mc=mv_F$ and $Q_+ + Q_- = 1$. This immediately fixes [@review10] the charge partitioning $Q_\pm$. Now, consider injection of an electron above the Fermi energy, with energy $\epsilon_F+\epsilon_{\rm in}$ and momentum $k_F+k_{\rm in}$. While the argument for the charge currents remains untouched, conservation of the excess energies and momenta requires $$\begin{aligned}
\epsilon_{\rm in} = c |k_+| + c |k_-| \,\,\, ; \,\,\, k_{\rm in} = k_+ + k_- \label{con2}\,.\end{aligned}$$ Here, $k_\pm$ denotes the excess momenta of the left- and right-moving excitations \[see Fig. \[fig2\](b)\]. In this way, we find the corresponding excess energies$$\epsilon_\pm=c |k_\pm| = (\epsilon_{\rm in} \pm c k_{\rm in})/2\,,
\label{energycon}$$ which explains Eq. (\[diffInonlocal\]). This result implies that the energy partitioning is entirely independent of the charge partitioning and can be tuned to arbitrary values by varying experimental parameters. In fact, when the momentum of the injected right-moving electron is smaller than the Fermi momentum \[$k_{\rm in}<0$, cf. Fig \[fig2\](b)\] , and its energy close to $c|k|$, Eq. (\[energycon\]) implies that essentially all its excess energy is propagating to the left, while most of the charge moves to the right. A crucial ingredient in this result is the interaction-induced broadening of the Luttinger-liquid spectral function which allows for injection of particles away from the mass shell.
![Illustration of nonlocal injection process. (a) Overlap of occupied states in the (non-interacting) source wire (thick blue line) and the Luttinger liquid, as described by the spectral function. The difference between the Fermi energies of source and Luttinger liquid is controlled by the voltage $V$. The Luttinger-liquid spectral function is indicated as a gray-scale background. (b) Illustration of the energy-conservation argument for energy partitioning. (c) For an interacting source, the tunneling current is determined by the overlap of the spectral functions of source and wire.[]{data-label="fig2"}](Fig2.pdf)
For local injection, Eq. (\[IElocal\]) implies $$\left\langle I_{\pm}^{E}\right\rangle = \frac{Q_\pm^2} {Q_+^2+Q_-^2} \left\langle I^E\right\rangle
\label{local}$$ in terms of the total energy current $\left\langle I^E\right\rangle = \epsilon_{\rm in} \langle I \rangle$. Unlike for nonlocal injection, this energy partitioning depends only on the interaction constant, but it is still distinctly different from the charge splitting. This difference can be traced to the fact that the charge density is linear in the Luttinger liquid fields, while the energy is quadratic. *Experimental consequences.—*We now turn to experimental signatures of energy partitioning, emphasizing that unlike charge partitioning, it is not masked by the presence of Fermi-liquid leads. For nonlocal injection, the right- and left-moving charge excitations have different maximal energies, given by $\epsilon_{\rm max}^R=(1/2)(eV+ck_V)$ and $\epsilon_{\rm max}^L=(1/2)(eV-ck_V)$ when injecting right-movers. Here, we assume for definiteness that the source wire has a larger charge velocity. Note that these maximal energies remain valid even for an interacting source, cf. Fig. \[fig2\](c). These results can be tested experimentally in some detail in the setup sketched in Fig. \[fig1\](a), in which the Luttinger liquid is probed by single-level quantum dots both to the left and to the right of the injection region (cp. [@takei10]). First consider a long Luttinger liquid in the absence of Fermi-liquid leads. In this case, the maximal energies of right- and left-moving excitations are directly observable as thresholds in the current flowing into the quantum dots. Indeed, current can flow into the quantum dots with gate-tunable dot level $\epsilon_{\rm out}^{R/L}$ only as long as $\epsilon_{\rm out}^{R/L} < \epsilon_{\rm max}^{R/L}$.
In the vicinity of the threshold, the charge currents into the quantum dots will exhibit a power-law dependence on $\epsilon_{\rm max}^{R/L} - \epsilon_{\rm out}^{R/L}$. Extending the approach of Ref. [@takei10] to the nonlocal injection of electrons of definite chirality, we find for the injection of right movers that $$\begin{aligned}
dI_R/dV &\propto& (\epsilon_{\rm max}^{R} - \epsilon_{\rm out}^{R})^{\phi-1}
\\
dI_L/dV &\propto& (\epsilon_{\rm max}^{L} - \epsilon_{\rm out}^{L})^{3\phi-\sqrt{\phi(\phi-1)}},\end{aligned}$$ where $I_{R/L}$ denote the charge currents into right and left quantum dot. These results are valid for a noninteracting source. The expressions for the current in the case of an interacting source are more involved [@unpublished].
In the presence of Fermi-liquid leads, their interface with the Luttinger liquid causes reflection of the energy currents which depends sensitively on the energy $\epsilon$ of the excitations. One may model the interface by $\kappa$ which varies spatially (over a length $d$) from its nominal value in the Luttinger liquid to $\kappa=1$ in the lead. For low energies, $\epsilon \ll c/d$, the interface can be viewed as abrupt, and the reflection of the energy current is, in close analogy with the Fresnel equations of optics, given by $R_E=1-T_E=(c-v_F)^2/(c+v_F)^2$ [@gutman09; @footnote]. For larger energies, $\epsilon \gg c/d$, the interface becomes smooth and reflection of the energy current is exponentially suppressed. Thus, when $\epsilon_{\rm max}^{R/L} \ll c/d$, there will be multiple reflection of energy currents. In this case, only the larger of the two thresholds can be directly probed experimentally. However, when the threshold energies are sufficiently large, $\epsilon_{\rm max}^{R/L} \gg c/d$, energy reflection at the interfaces becomes negligible and both thresholds are directly accessible.
While we considered the spin-polarized case above, the presence of thresholds carries over to the case of a spin-degenerate system supporting spinon excitations. For a linear spectrum with SU(2) symmetry, the injected right mover can excite both left and right-moving charge (with velocity $c_\rho$) modes but only right-moving spin modes (with velocity $c_s$). Consider first the region with $eV > c_\rho |k_V|$. Then, the right threshold $\epsilon^R_{\rm max}$ remains the same as in the spinless case (with $c \to c_\rho$) while the left threshold becomes $\epsilon_{\rm max}^{L} = c_\rho (eV - c_s k_V)/(c_\rho+c_s)$. At lower voltages, $c_\rho |k_V| > eV > c_s |k_V|$, there is no tunneling for $k_V < 0$, while we find $\epsilon_{\rm max}^R = \max\{c_s(eV+c_\rho k_V)/(c_\rho+c_s) ; c_\rho(eV-c_s k_V)/(c_\rho-c_s)\}$ and $\epsilon_{\rm max}^L = c_\rho (eV - c_s k_V)/(c_\rho+c_s)$ for $k_V>0$.
For local injection into a quantum-Hall edge channel, the thresholds for electron extraction are equal on both sides of the injection point, but the overall right- and left-moving energy currents are different. This remains true after multiple reflections from the Luttinger-liquid-lead interfaces although these reflections affect the overall energy current flowing into the left and right leads. Assuming that the injection energies are sufficiently small such that the Luttinger-liquid-lead interfaces can be treated as abrupt, the energy currents flowing into the right and left leads would be ${\cal T} I_+^E + {\cal R} I_-^E$ and ${\cal R} I_+^E + {\cal T} I_-^E$, respectively. Here, we define ${\cal T} = 1/(1+R_E)$ and ${\cal R} = R_E/(1+R_E)$. These energy currents can in principle be measured directly by probing the electron distribution functions in the outgoing edge channels of the leads (cp., Refs. [@chen09]).
*Conclusions.—*While energy and charge of an injected electron travel together in a non-interacting system, this is no longer the case in the presence of interactions. The decoupling caused by interactions is peculiar in one dimension, where it is impossible to separate the excitations into plasmons and Fermi-liquid quasiparticles. In the Luttinger liquid picture, interactions leave the [*dc*]{} conductance unchanged [@safi95; @maslov95], while significantly affecting, e.g., the thermal conductance [@kane96; @fazio98; @gutman09; @footnote]. The decoupling leads to particularly striking consequences when injecting electrons with fixed chirality into a 1d electron system where one may reach conditions such that charge and energy of an injected particle propagate in directions opposite to each other. Finally, energy partitioning is accessible experimentally with existing abilities and unlike charge partitioning, is detectable in $dc$ setups which include Fermi-liquid leads.
We acknowledge discussions with G. Barak, A. Levchenko, T. Micklitz, and A. Yacoby, financial support through DFG SPP 1538 (FUB), DOE Contract No. DE-FG02-08ER46482 (Yale), and the Packard Foundation (Caltech), as well as the hospitality of the Aspen Center for Physics during part of this work.
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This implies that the thermal conductance $G_T$ of a Luttinger liquid depends on the interaction strength $\kappa$. Multiple reflections from the lead-Luttinger-liquid interface give rise to an overall energy transmission of $T_E^{\rm ( tot)} = 2\kappa/(1+\kappa^2)$ and thus $G_T=(\pi T/6)T_E^{\rm (tot)}$, cp. Ref. [@gutman09].
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abstract: 'We experimentally demonstrated strong adiabatic mixed-field orientation $\Nuptot=0.882$ of carbonyl sulfide molecules (OCS) in their absolute ground state. OCS was oriented in combined non-resonant laser and static electric fields inside a two-plate velocity map imaging spectrometer. The transition from non-adiabatic to adiabatic orientation for the rotational ground state was studied by varying the applied laser and static electric field. Above static electric field strengths of 10 kV/cm and laser intensities of $10^{11}~\SI{}{\wsqcm}$ the observed degree of orientation reached a plateau. These results are in good agreement with computational solutions of the time-dependent Schrödinger equation.'
author:
- 'Jens S. Kienitz'
- Sebastian Trippel
- Terry Mullins
- Karol Długołęcki
- 'Rosario Gonz[á]{}lez-F[é]{}rez'
- Jochen Küpper
bibliography:
- 'string.bib'
- 'cmi.bib'
title: 'Adiabatic mixed-field orientation of ground-state-selected carbonyl sulfide molecules'
---
=1
Introduction
============
Molecular samples with directional order, , oriented molecules, enable the extraction of information directly in the molecular frame, for instance, from photoelectron angular distributions [@Meckel:Science320:1478; @Bisgaard:Science323:1464; @Holmegaard:NatPhys6:428; @Kelkensberg:PRA84:051404; @Boll:PRA88:061402], high-order harmonic generation [@Itatani:Nature432:867; @Vozzi:NatPhys7:822; @Kraus:PRL113:023001], electron and x-ray diffractive imaging [@Hensley:PRL109:133202; @Kuepper:PRL112:083002; @Yang:NatComm7:11232], and stereochemistry experiments [@Brooks:Science193:11; @Loesch:JCP93:4779; @Rakitzis:Science303:1852]. For diffraction experiments, data has often to be recorded and averaged over many shots. If the molecules in an ensemble have directional order, a molecular-frame diffraction pattern corresponding to the single-molecule signal above noise can be obtained [@Stern:FD171:393; @Filsinger:PCCP13:2076; @Spence:PRL92:198102].
Various approaches have been developed to generate oriented molecules, including brute-force orientation using strong dc electric [@Loesch:JCP93:4779; @Friedrich:Nature353:412; @Block:PRL68:1303] and magnetic [@Slenczka:PRL72:1806] fields, shaped [@Ghafur:NatPhys5:289; @Goban:PRL101:013001] and two-color [@Vrakking:CPL271:209; @De:PRL103:153002; @Kraus:PRL113:023001] near-infrared laser pulses, terahertz pulses [@Harde:PRL66:1834; @Machholm:PRL87:193001; @Fleischer:PRL107:163603; @Egodapitiya:PRL112:103002], multi-pulse schemes [@Kraus:PRL113:023001; @Egodapitiya:PRL112:103002], and mixed laser and dc electric field orientation [@Friedrich:JCP111:6157; @Holmegaard:PRL102:023001; @Ghafur:NatPhys5:289; @Filsinger:JCP131:064309; @Trippel:MP111:1738]. In mixed-field orientation, the non-resonant laser field creates near degenerate doublets that are efficiently oriented by the dc electric field [@Friedrich:JCP111:6157; @Holmegaard:PRL102:023001]. However, for the two components of the doublet the dipole moments, and thus the molecules, point in opposite directions, resulting in a reduced or vanishing macroscopic orientation depending on the populations of the two components [@Filsinger:JCP131:064309]. Quantum-state selection allows for the preparation of ensembles of molecules all in a single rovibronic state [@Stern:ZP39:751; @Reuss:StateSelection; @Nielsen:PCCP13:18971; @Putzke:PCCP13:18962; @Chang:IRPC34:557]. Strong orientation can be achieved if these populations can be adiabatically transferred to the oriented field-dressed states. According to the adiabaticity theorem [@Born:ZP51:165], the field-dressed dynamics are adiabatic if a molecule remains in its eigenstate as the field strength, , the laser intensity, is changed. This condition can be fulfilled for most quantum mechanical systems, including non-resonant adiabatic alignment [@Ortigoso:PRA86:032121], when the Hamiltonian evolves sufficiently slowly in time. However, in contrast to adiabatic alignment, adiabatic mixed-field orientation tends to be more challenging, because the energy-spacing within the doublets becomes very small inside the laser field [@Nielsen:PRL108:193001; @Trippel:PRL114:103003].
Here, we present a combined experimental and computational investigation of the degree of orientation of rovibronic-ground-state-selected OCS molecules for various laser intensities and dc electric field strengths. The presented experimental setup allows for the use of comparably strong dc electric fields of 20 kV/cm. The experimental findings are compared to the theoretical description obtained by solving the time-dependent Schrödinger equation for the mixed-field orientation of the populated states within the state-selected molecular beam.
Methods
=======
Experiment
----------
![(Color online) (a) Experimental setup and definition of the axis system and relevant directions and angles. $\beta$ is the angle between the static electric field and the polarization vector of the control laser pulse; probe and control laser polarization are perpendicular to each other. Colored wireframes mark the positions of the cross sections depicted below. (b) Cross section of the deflector and the corresponding electric field strength. The position of the molecular beam is marked as a black dashed circle. (c) Cross section of the VMI and the corresponding electric potential. The dashed circle marks the interaction region of the molecular beam with the laser pulses and the white frames the electrodes. (d) Typical velocity map of S$^+$ ions from ionization of oriented OCS. is defined by the angle between the laboratory fixed $Y$-axis and the molecule fixed $z$-axis projected onto the detector surface. Velocity cuts used in the analysis are illustrated by red circles in the detector image.[]{data-label="fig:experimental_setup"}](setup){width="\linewidth"}
The experimental setup is depicted in . A supersonic molecular beam was generated by expanding a mixture of 500 ppm OCS seeded in of helium into vacuum through an Even-Lavie-valve [@Even:JCP112:8068] at a repetition rate of . The beam was collimated by two skimmers, and downstream from the nozzle. An electrostatic deflector [@Filsinger:ACIE48:6900; @Chang:IRPC34:557], placed downstream from the nozzle, dispersed the molecular beam according to its quantum states [@Filsinger:JCP131:064309; @Chang:IRPC34:557]. A cross section of the deflector and its electric field are shown in b. A third skimmer was positioned behind the deflector for further differential pumping.
The molecules are oriented and probed inside a velocity map imaging spectrometer (VMI) consisting of two electrodes [@Papadakis:RSI77:3101]. The control and probe laser pulses with a central wavelength of were provided by an amplified femtosecond laser system [@Trippel:MP111:1738]. The temporal profile of the control laser pulse had a sawtooth shape with a slow rising edge ($\SI{600}{\pico\second}$, 2.5-97.5%) and a fast falling edge ($\SI{250}{\pico\second}$). We measured the spatial beam profile, in intensity, with a beam profiler (SpiriCon SP620U), which yields $\sigma=17~\um$ and $\sigma'=18~\um$ along the two principal axes of the profile; the first (short) axis is rotated away from the laboratory $Y$ axis. This resulted in a peak intensity up to $\Icontrol\approx\SI{6E11}{\wsqcm}$. The intensity was controlled by a half-wave plate mounted on a rotation stage in combination with a polarization filter. The degree of orientation was probed through Coulomb explosion imaging following multiple ionization of OCS by a 30 fs laser pulse focused down to $\sigma_1=\SI{13}{\micro\meter}, \sigma_2=\SI{17}{\micro\meter}$, which resulted in a peak intensity of $\Iprobe\approx\SI{1E14}{\wsqcm}$. The first principal axis is rotated by towards the laboratory $Y$ axis. The relative timing between the control and probe laser pulses was varied with a delay stage positioned in the laser beam path of the probe laser. Both laser pulses were linearly polarized with polarization vectors perpendicular to each other. The orientation of both polarization vectors around the laboratory fixed $X$-axis, and, therefore, the angle $\beta$ between the ac and the dc electric fields were simultaneously controlled by a half wave plate. At $\beta=\ang{35}$ we achieved the best compromise between the strongest orientation, with its maximum at $\beta=\ang{0}$, and the ideal detection efficiency, with its maximum at $\beta=\ang{90}$. For $\beta=\degree{35}$, the component of the dc electric field parallel to the polarization axis of the control laser is reduced to $\cos{(\ang{35})\,\Estat\approx0.82\,\Estat}$.
The temporal intensity profile of the chirped control laser pulse has been determined as described in appendix \[sec:laser-pulse\] and is shown in c. For use in computations we applied a Fourier-transform-based low-pass filter to remove frequencies above 81 GHz in the temporal profile of the pulse, which correspond to the limit of the pixel size of the CCD within the spectrometer; the resulting pulse is depicted by the blue area in a.
The deflection of the molecular beam was characterized by vertically scanning the $Y$ position of the focus of the probe laser pulse across the molecular beam using a corresponding translation of the focusing lens. The integrated ion signal at each position is proportional to the column density at the corresponding $Y$ position in the molecular beam.
![(Color online) Measured profiles along the laboratory $Y$-axis for the deflected (rhombuses) and undeflected (squares) molecular beams. The solid lines represent simulations of the density profile for the deflected , , states in red, brown, and purple, respectively and for the sum of these states in green. The blue line is the sum of the undeflected states. The vertical yellow line depicts the position of the laser focus used for the orientation experiments in the molecular beam.[]{data-label="fig:deflection"}](deflection){width="\linewidth"}
shows the measured normalized density profiles of the deflected (rhombuses) and the undeflected (squares) molecular beams. The molecules are deflected to positive $Y$ values by the interaction of their quantum-state-specific dipole moment with the inhomogeneous electric field. The solid lines represent numerical simulations [@Chang:CPC185:339; @Chang:IRPC34:557] of the density profiles for the undeflected (blue) and deflected (green) molecular beams as well as the individual contributions to the deflected beam by the $\ket{J,M}=\ket{0,0}$, , states in red, brown, and purple, respectively. From these simulations we obtain a rotational temperature of 2 K for the original molecular beam; this fairly high temperature [@Chang:IRPC34:557] is ascribed to the low stagnation pressure and not fully optimized operation conditions of the valve, but is not critical for the investigation performed here. The yellow bar at in indicates the position where the orientation experiments were performed. The estimated ground state population at this position was $0.95\pm0.05$, with the remaining population in the state.
It was predicted that dc field strengths on the order of $\Estat=\SI{10}{\kvcm}$ are required to achieve adiabatic orientation with 500 ps pulses [@Nielsen:PRL108:193001]. In order to achieve such strong fields a two plate velocity map imaging spectrometer [@Papadakis:RSI77:3101] was set up. A sectional view of the electrodes and typical potentials are shown in c. The two-plate design allows for a much stronger field for a given repeller-electrode potential than for the classical three-plate VMI [@Eppink:RSI68:3477], , $\Estat=\SI{20.7}{\kvcm}$ at $U_{r}=$. In addition, as the magnification of the velocity map scales with $1/\sqrt{\text{U}_{\text{r}}}$, the measured detector images are larger compared to a classical VMI operated at the same electric field strength. Velocity-focusing conditions were obtained by positioning the laser focus at a specific $Z$ position between the two electrodes. This position is independent of the applied repeller voltage, which allows for continuous tuning of without a change of the VMI focusing conditions.
The velocity maps are detected by a position sensitive detector, a combination of two multi-channel plates (MCPs) in Chevron configuration and a phosphor screen. A CMOS camera (Optronis CL600X2) with a repetition rate of was used to film the screen. The positions of individual ions were determined via a centroiding algorithm [@Wester1998]. Gating the detector by a fast high-voltage switch (Behlke HTS 31-03-GSM) allowed to distinguish ionic fragments by their time of flight and to record VMIs for individual fragments. d shows a typical S$^+$-position histogram. Red circles indicate the area between $v_{\parallel}=\sqrt{v^2_x + v^2_y}=\SI[per-mode=symbol]{2500}{\meter\per\second}$ and $v_{\parallel}=\SI[per-mode=symbol]{4700}{\meter\per\second}$; this range was used to determine the degree of orientation in all measurements. Ions recorded in this area originate from the Coulomb fragmentation channel $\text{OCS}+n\gamma\rightarrow\text{OC}^{+}+\text{S}^+$, which is a directional fragmentation along the C-S bond of the molecule. Slower fragmentation channels with velocities $v_{\parallel}<\SI[per-mode=symbol]{2500}{\meter\per\second}$, resulting from singly-ionized molecules, are much more intense and not shown [@Trippel:PRL114:103003]. The degree of orientation is characterized by the ratio $\Nuptot$ of S$^+$ ions hitting the detector on the upper half $\text{N}_{\text{up}}$ divided by the total number of S$^+$ ions $\text{N}_{\text{tot}}$.
Theory {#ssec:methods:theoretical}
------
To obtain physical insight into the experimental orientation, a theoretical description of the rotational dynamics of OCS in the experimental field configuration was performed. The dc electric field is always turned on adiabatically, which was computationally checked to be valid for the current experimental parameters, and the adiabatic pendular states of the dc-field configuration were taken as the initial states in these calculations. Then, the time-dependent Schrödinger equation was solved for a constant dc field using the temporal profile of the experimental control laser pulses. To compare with the experimental observations, the theoretical orientation ratio was computed including the volume effect, which took into account the spatial intensity profiles of the control and the probe laser pulses (*vide supra*) and the experimental velocity distribution of the ions after the Coulomb explosion in the range $2500\text{~m/s}\le{}v_{\parallel}\le4700\text{~m/s}$ [@Omiste:PCCP13:18815]. If the mixed-field dynamics was adiabatic, the molecule remained in the same pendular eigenstate as the laser pulse is turned on and the Hamiltonian evolves with time [@Born:ZP51:165]. The rotational dynamics were analyzed by projecting the time-dependent wave function on the basis formed by the adiabatic pendular states, which was obtained by solving the time-independent Schrödinger equation for the instantaneous Hamiltonian at time $t$, including both, the interactions with the ac and dc fields. These projections onto the adiabatic pendular basis allowed us to disentangle the field-dressed dynamics for each state of the molecular beam as well as to identify the sources of non-adiabatic effects.
Experimental Results {#sec:results}
====================
![(Color online) Experimental (points) and theoretical (solid and dashed lines) degree of orientation as a function of the peak control laser intensity for $\beta=\degree{35}$ and static electric field strengths of 5.2 kV/cm, 10.4 kV/cm, 15.6 kV/cm, and 20.7 kV/cm. In the inset the experimental (points) and theoretical (solid line) mean degree of orientation, averaged over control laser intensities between $\Icontrol=\SI{3e11}{\wsqcm}$ and (blue area of the main graph), is shown as a function of the dc electric field . For the theoretical degree of orientation, depicted by the solid red line, we assumed the populations obtained from the simulated deflection profiles shown in . The dashed line represents calculations with an adjusted population distribution as described in the text.[]{data-label="fig:intensity-35"}](intensity-35){width="\linewidth"}
shows the experimental degree of orientation $\Nuptot$ as a function of the peak control laser intensity for experimental dc electric field strengths $\Estat=5.2$ kV/cm, 10.4 kV/cm, 15.6 kV/cm, and 20.7 kV/cm. A small degree of anti-orientation[^1] at zero and weak control laser intensities was observed. This is due to the combined effect of “brute-force” orientation, generated by the static electric field of the VMI spectrometer, and geometric alignment, , selective ionization of OCS by the probe laser, which results in preferred ionization of anti-oriented molecules. For all four electric field strengths, the degree of orientation increased with increasing peak control laser intensity up to $\Icontrol\approx\SI{1E11}{\wsqcm}$. The slope of the experimental degree of orientation was the same for $\Estat=10.4$, 15.6, and , while it was slightly lower for . For $\Icontrol>\SI{2E11}{\wsqcm}$, the degree of orientation was nearly constant with further increasing laser intensities. In this plateau region the degree of orientation was, within error estimates, independent of the dc field strengths for $\Estat\ge10$ kV/cm, while it was reduced for . The inset of shows the experimental and theoretical mean degree of orientation in the plateau region as a function of the dc field strength. All data points between $\Icontrol=\SI{3e11}{\wsqcm}$ and , indicated by the blue area in the main figure, were taken into account. The experimentally obtained orientation was $\Nuptot=0.882\pm0.004$ for dc field strengths of and above, indicating nearly adiabatic orientation for the ground state. For , we had a 7% smaller degree of orientation of $\Nuptot=0.820$.
Discussion
==========
To understand the saturation of the degree of orientation as a function of the dc field strength the eigenenergies of the adiabatic pendular states of OCS were examined.
![(Color online) (a) Energy of the [$\left|0,0,e\right>_\textup{p}$]{}, [$\left|1,1,e\right>_\textup{p}$]{}, [$\left|1,0,e\right>_\textup{p}$]{}, [$\left|2,2,e\right>_\textup{p}$]{}, [$\left|2,1,e\right>_\textup{p}$]{}, and [$\left|2,0,e\right>_\textup{p}$]{} states as a function of the control laser intensity for a static electric field strengths with $\Estat=\SI{5.2}{\kvcm}$ (dashed lines), and $\Estat=\SI{20.7}{\kvcm}$ (solid lines). (b) Sketch of the square of the [$\left|0,0,e\right>_\textup{p}$]{} and [$\left|1,1,e\right>_\textup{p}$]{} wave functions at $\Estat=\SI{20.7}{\kvcm}$.[]{data-label="fig:theory"}](theory){width="\linewidth"}
a shows the eigenenergies for $\Estat=\SI{5.2}{\kvcm}$ (dashed lines) and (solid lines) as a function of the control laser intensity.[^2] As the laser intensity increases, the two pendular states [$\left|0,0,e\right>_\textup{p}$]{} and [$\left|1,1,e\right>_\textup{p}$]{} form a near-degenerate doublet; their energy spacing in the strong-ac-field limit is given by $\Delta{E}\approx2\mu_\text{p}\Estat\cos{\beta}$ with the permanent dipole moment $\mu_\text{p}$. In order to ensure adiabatic orientation, any time scale contained in the temporal envelope of the control laser pulse has to be longer than the time scale that corresponds to the instantaneous energy difference in the laser field. Therefore, , the rise time of the control laser pulse has to be longer for a dc field of $\Estat = \SI{5.2}{\kvcm}$ than for a dc field of . If this requirement is not fulfilled, the dynamics becomes non-adiabatic and population is transferred between the two pendular states forming the doublet. In this case the resulting orientation for a system starting in the ground state is reduced since the two pendular states orient in opposite directions, see b. For rotationally excited states, the field-dressed dynamics is more complicated. In addition to the pendular doublet formation, the field-free-degenerate manifold splits into distinct $M$ components. This results in narrow avoided crossings between energetically neighboring pendular states in the combined field. The avoided crossing between the [$\left|1,1,e\right>_\textup{p}$]{} and [$\left|1,0,e\right>_\textup{p}$]{} states is shown in c for $\Estat=\SI{5.2}{\kvcm}$ (dashed lines) and (solid lines). A larger splitting is observed for stronger dc fields and, therefore, the corresponding dynamics is more adiabatic. Our calculations have shown that a field strength on the order of is required to provide adiabatic orientation for the $J=1$ states with a control pulse similar to the experimental one, but without roughness.
In the slopes of all experimentally determined degrees of orientation were steeper than for the calculated ones. A possible reason for this discrepancy are errors in the experimentally determined intensity of the control laser, which relies on a determination of the spatial profile of the laser focus. For these beam-profile measurements, the laser beam had to be attenuated by seven orders of magnitudes, which was achieved by using reflections of optical flats and neutral density filters and which might have affected the beam profile. Furthermore, the profile of the laser focus might change slightly with pulse energy, for instance, due to self focusing at high pulse energies. The slopes of the experiment and the calculations match nicely if we assume a 1.43 times more intense control laser beam, well within our error estimates.
The theoretical degree of orientation taking into account our experimental conditions are shown as solid lines in . The initial rotational state distribution is given by $w({\ensuremath{\left|0,0,e\right>_\textup{p}}\xspace})=0.95$, $w({\ensuremath{\left|1,1,e\right>_\textup{p}}\xspace})=0.025$, and $w({\ensuremath{\left|1,1,o\right>_\textup{p}}\xspace})=0.025$, corresponding to the state distribution obtained from the deflection profile in assuming that the individual states are adiabatically transferred from the deflector to the interaction region inside the velocity map imaging spectrometer. The experimentally determined temporal laser intensity profile, shown in a, was taken into account. In comparison to the experimental results we observe a higher degree of orientation for the theoretical curves. We attribute this discrepancy to the following effects: First, simulations have shown that molecules in excited rotational states are not adiabatically transferred from the deflector to the velocity map imaging spectrometer. The non adiabatic transfer is mostly caused by rotating electric field vectors [@Wall:PRA81:033414] in the fringe field regions of the deflector and the VMI. Moreover, Majorana transitions could occur in field free regions. Second, a smoother temporal profile of the laser (*vide supra*) pulse would result in a larger (smaller) weight of the [$\left|1,1,e\right>_\textup{p}$]{} ([$\left|1,0,e\right>_\textup{p}$]{}) state due to a more-adiabatic passage at the corresponding avoided crossing. Because [$\left|1,1,e\right>_\textup{p}$]{} is anti-oriented while [$\left|1,0,e\right>_\textup{p}$]{} is oriented this would lead to a decreased degree of orientation.
Adjusting the input parameters for the calculations, a better agreement between experiment and theory was obtained. The dashed lines in correspond to calculations where the peak intensity was increased by a factor of 1.43 and an initial state population of $w({\ensuremath{\left|0,0,e\right>_\textup{p}}\xspace})=0.8$, $w({\ensuremath{\left|1,1,e\right>_\textup{p}}\xspace})=0.066$, $w({\ensuremath{\left|1,1,o\right>_\textup{p}}\xspace})=0.066$, and $w({\ensuremath{\left|1,0,e\right>_\textup{p}}\xspace})=0.066$ was assumed. The results of these calculations show a better agreement with the experimental measurements.
The theoretical curve for shows a decrease of the degree of orientation for increasing laser intensities above $3\times10^{11}$ W/cm$^2$. This can be attributed to increased non-adiabatic coupling and population transfer between the [$\left|0,0,e\right>_\textup{p}$]{} and [$\left|1,1,e\right>_\textup{p}$]{} states, which gets more important for higher intensities as the slope of the intensity-change and, therefore, the temporal variation of the Hamiltonian gets faster. The theoretical decrease is within the error of the experimental data and is not resolved in the experimental results.
The non adiabatic dynamics can be further investigated by studying the time dependent weights of the decomposition of the time-dependent wave function ${\ensuremath{\left|J,M,s\right>_{\textup{t}}}\xspace}$ in the basis of the adiabatic pendular states ${\ensuremath{\left|J,M,s\right>_\textup{p}}\xspace}$.
![(Color online) (a) Degree of orientation and alignment as a function of the delay between the control and the probe laser with a peak intensity of the control laser $\Icontrol=\SI{6E11}{\wsqcm}$, a static electric field of $\Estat=\SI{20.7}{\kvcm}$, and a laser polarization direction of $\beta=\ang{35}$. The purple area depicts the temporal profile of the control laser. (b,c,d) weights of the basis functions ${\ensuremath{\left|J,M,s\right>_\textup{p}}\xspace}$ of the time-dependent wave functions [$\left|0,0,e\right>_{\textup{t}}$]{}, [$\left|1,1,o\right>_{\textup{t}}$]{} and [$\left|1,1,e\right>_{\textup{t}}$]{}, respectively. The dashed line marks the delay that we used for the measurements shown in []{data-label="fig:delay"}](delay){width="\linewidth"}
The purple area in a depicts the temporal laser beam profile with its slow rising and a fast falling edge obtained from the spectrum of the chirped control laser pulse. In addition, the experimental degree of orientation and alignment as a function of the relative timing between the control laser pulse and the probe pulse is shown. The dashed line mark the delay we used for the intensity measurements. Both, the degree of orientation and alignment peak at this intensity. The intensity used for the calculation of the time dependent weights is given by $\SI{6e11}{\wsqcm}$. The dc field strengths is $\SI{20.7}{\kvcm}$. In b–d the time dependent weights $w$ of systems initially in the ${\ensuremath{\left|0,0,e\right>_{\textup{t}=0}}\xspace}={\ensuremath{\left|0,0,e\right>_\textup{p}}\xspace}$, ${\ensuremath{\left|1,1,o\right>_{\textup{t}=0}}\xspace}={\ensuremath{\left|1,1,o\right>_\textup{p}}\xspace}$, and ${\ensuremath{\left|1,1,e\right>_{\textup{t}=0}}\xspace}={\ensuremath{\left|1,1,e\right>_\textup{p}}\xspace}$ states are presented. A deviation from a weight $w=1$ indicates population transfer and, therefore, non-adiabatic dynamics.
The field-dressed dynamics of the ground state [$\left|0,0,e\right>_{\textup{t}}$]{} in b is characterized by the formation of the pendular pair at a delay of approximately 50 ps, when population is transferred to the first rotational excited state resulting in a weight $w({\ensuremath{\left|1,1,e\right>_\textup{p}}\xspace})=5.5\cdot10^{-3}$. In addition, population transfer to states that correlate adiabatically to the $J=2$ manifold is observed in the calculation. This is attributed to the roughness of the time-profile and the sudden changes in intensity of the experimental pulse, and not due to the slow $\SI{600}{\pico\second}$ rise time of the laser pulse, which is long compared to the rotational period of 82 ps. Initially, population is transferred to [$\left|2,2,e\right>_\textup{p}$]{}. At stronger ac fields, this state encounters an avoided crossing at which practically all population is diabatically transferred to the [$\left|2,1,e\right>_\textup{p}$]{} state. Furthermore, for stronger fields further population transfer from the ground-state pendular doublet proceeds to this [$\left|2,1,e\right>_\textup{p}$]{} state, which reaches a population of $w({\ensuremath{\left|2,1,e\right>_\textup{p}}\xspace})\approx4.5\cdot10^{-2}$. Although the field-dressed dynamics of [$\left|0,0,e\right>_{\textup{t}}$]{} including the experimental laser profile is non-adiabatic, the pendular states contributing to the dynamics, [$\left|0,0,e\right>_\textup{p}$]{}, [$\left|1,0,e\right>_\textup{p}$]{} and [$\left|2,1,e\right>_\textup{p}$]{}, are all strongly oriented and a computed degree of orientation of $\Nuptot=0.976$ is obtained for [$\left|0,0,e\right>_{\textup{t}}$]{} at the peak laser intensity. We have also performed calculations with an ideal pulse without roughness, which was generated by fitting error functions to the experimental pulse. For this completely smooth theoretical pulse the field-dressed dynamics of the ground-state state is only affected by the formation of the pendular doublet and the dynamics would be completely adiabatic, confirming that the remaining non-adiabaticity is due to the, albeit small, experimental noise in the laser intensity. For the ground state of the odd irreducible representation, [$\left|1,1,o\right>_{\textup{t}}$]{}, we encounter an equivalent field-dressed dynamics, shown in c; it is slower because the pendular doublet formation occurs at stronger ac fields. At approximately $100$ ps we observe an ac-field-induced population transfer between the initially populated [$\left|1,1,o\right>_\textup{p}$]{} state and the [$\left|3,3,o\right>_\textup{p}$]{} state. At $200$ ps later this population transfers diabatically to the [$\left|3,2,o\right>_\textup{p}$]{} state. Almost simultaneously, and due to the formation of the first pendular doublet in this irreducible representation, there is some population transferred from the [$\left|1,1,o\right>_\textup{p}$]{} state to [$\left|2,2,o\right>_\textup{p}$]{}. At the peak intensity, the pendular states significantly contributing to the time-dependent wave function [$\left|1,1,o\right>_{\textup{t}}$]{} are $w({\ensuremath{\left|1,1,o\right>_\textup{p}}\xspace})=0.993$, $w({\ensuremath{\left|2,2,o\right>_\textup{p}}\xspace})=2.4\cdot10^{-3}$, and $w({\ensuremath{\left|3,2,o\right>_\textup{p}}\xspace})=5.2\cdot10^{-3}$. Analogously to the absolute ground-state, the dynamics of this state [$\left|1,1,o\right>_{\textup{t}}$]{} in an ideal pulse would only be affected by the formation of the pendular doublet between the states [$\left|1,1,o\right>_\textup{p}$]{} and [$\left|2,2,o\right>_\textup{p}$]{}.
In d the weights of the expansion coefficients of the time-dependent [$\left|1,1,e\right>_{\textup{t}}$]{} wave function for the experimental pulse with peak intensity $\SI{6e11}{\wsqcm}$ and a dc field with strength $\SI{20.7}{\kvcm}$ is presented. The mixed-field dynamics of this state is more complicated. In the presence of a tilted static electric field, the pendular states [$\left|1,1,e\right>_\textup{p}$]{} and [$\left|1,0,e\right>_\textup{p}$]{} are energetically very close, and even a weak ac field provokes a strong coupling between them. At weak laser intensities, when the splitting of the [$\left|1,M,e\right>_\textup{p}$]{} manifold due to the strong dc field takes place, a significant amount of population is transferred between them. Thereafter, the state [$\left|1,0,e\right>_\textup{p}$]{} possesses the dominant contribution to the [$\left|1,1,e\right>_{\textup{t}}$]{} state. This non-adiabatic behavior takes place before the pendular doublet formation, , at low intensities and short times. By further increasing the ac field strength, population is transferred to [$\left|0,0,e\right>_\textup{p}$]{} and [$\left|2,2,e\right>_\textup{p}$]{} due to the formation of the pendular doublets with [$\left|1,1,e\right>_\textup{p}$]{} and [$\left|1,0,e\right>_\textup{p}$]{}, respectively. Due to ac-field-induced couplings with other pendular states and narrow avoided crossings, we observed that many adiabatic pendular states contribute to the dynamics. At the peak intensity, the pendular states that significantly contribute to the time-dependent wave function [$\left|1,1,e\right>_{\textup{t}}$]{} are $w({\ensuremath{\left|0,0,e\right>_\textup{p}}\xspace})=3.9\cdot10^{-3}$, $w({\ensuremath{\left|1,1,e\right>_\textup{p}}\xspace})=0.435$, $w({\ensuremath{\left|1,0,e\right>_\textup{p}}\xspace})=0.553$, $w({\ensuremath{\left|2,2,e\right>_\textup{p}}\xspace})=1.1\cdot10^{-3}$, $w({\ensuremath{\left|2,1\right>_\textup{p}}\xspace})=6\cdot10^{-4}$, $w({\ensuremath{\left|2,0,e\right>_\textup{p}}\xspace})=3\cdot10^{-3}$, $w({\ensuremath{\left|2,0,e\right>_\textup{p}}\xspace})=3\cdot10^{-3}$, and $w({\ensuremath{\left|3,1,e\right>_\textup{p}}\xspace})=2.8\cdot10^{-3}$. Since [$\left|1,1,e\right>_\textup{p}$]{} is anti-oriented and cancels the contributions form other states, [$\left|1,1,e\right>_{\textup{t}}$]{} shows no orientation at the peak intensity, , $\Nuptot=0.50$. In contrast, the dynamics of this [$\left|1,1,e\right>_{\textup{t}}$]{} state in an ideal pulse without roughness is mainly affected by the splitting of the [$\left|1,M,e\right>_\textup{p}$]{} manifold, and weakly by the subsequent formation of the pendular pairs. For this ideal pulse, only two adiabatic pendular states contribute significantly, with $w({\ensuremath{\left|1,1,e\right>_\textup{p}}\xspace})=0.652$ and $w({\ensuremath{\left|1,0,e\right>_\textup{p}}\xspace})=0.346$, resulting in anti-orientation. For the ideal pulse with peak intensity $\SI{6e11}{\wsqcm}$ and a dc field of , the non-adiabaticity during the splitting of the [$\left|J,M,e\right>_\textup{p}$]{} manifold would be significantly reduced, and the population-transfer from [$\left|1,1,e\right>_\textup{p}$]{} to [$\left|1,0,e\right>_\textup{p}$]{} would be $<0.011$.
Conclusion
==========
Adiabatic mixed field orientation of ground-state-selected OCS molecules has been demonstrated using strong dc electric fields of $10$–$20$ kV/cm. The experiments demonstrate strong orientation with $\Nuptot=0.882$ in agreement with our theoretical description. For dc electric fields of or stronger, the observed degree of orientation was independent of the dc electric field strength, which indicated that the molecules in their ground state are oriented adiabatically. Comparison with calculations showed that only a very small fraction of the population was transferred to excited states. The deviation of the degree of orientation from its maximal possible value of $\Nuptot=0.976$ was attributed to contributions of excited rotational states present in the deflected part of the molecular beam that was used in the orientation experiments. Preparing the molecules in the absolute ground state would result in full adiabatic orientation dynamics and, therefore, in an even higher degree of orientation. The adiabatic orientation of an excited rotational state is more challenging. Avoided crossing and the degeneracy at low laser intensity make non-adiabatic behavior more likely.
Compared to other techniques [@Ghafur:NatPhys5:289; @Goban:PRL101:013001; @Vrakking:CPL271:209; @De:PRL103:153002; @Kraus:PRL113:023001] our approach ensures a strong degree of orientation while employing only moderately strong laser intensities on the order of , which are far below the onset of ionization, even for larger molecules [@Strohaber:PRA84:063414]. Moreover, these moderate fields strengths allow for the investigation of chemical dynamics, using molecular-frame-imaging approaches, without significant distortions of the dynamics.
Our findings hold for polar molecules in general, as the Hamiltonian can be rescaled accordingly. Due to the complexity of the rotational level structure of asymmetric tops, non-adiabatic effects will have a larger impact on the orientation dynamics of these more complex molecules [@Omiste:PRA88:033416; @Omiste:JCP135:064310]. Nevertheless, our finding hold for any molecule prepared in the rotational ground state; the experimental realization of such a sample was experimentally demonstrated for C$_7$H$_5$N using the alternating-gradient $m/\mu$ selector [@Filsinger:PRL100:133003; @Filsinger:PRA82:052513; @Putzke:PCCP13:18962].
The improved adiabaticity in mixed field orientation and the resulting increase in the degree of orientation will improve, especially for complex molecules, imaging experiments with fixed-in-space molecules such as the investigation of molecular-frame photoelectron angular distributions [@Holmegaard:NatPhys6:428; @Pullen:NatComm6:7262; @Zeidler:PRL95:203003] or the recording of molecular movies by x-ray [@Kuepper:PRL112:083002] and electron diffraction [@Hensley:PRL109:133202; @Yang:NatComm7:11232] or photoelectron holography [@Boll:PRA88:061402; @Boll:FD17171:57].
Acknowledgements
================
Besides DESY, this work has been supported by the *Deutsche Forschungsgemeinschaft* (DFG) through the excellence cluster “The Hamburg Center for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale” (CUI, EXC1074) and the Helmholtz Association “Initiative and Networking Fund”. R.G.F. gratefully acknowledges financial support by the Spanish project FIS2014-54497-P (MINECO) and the Andalusian research group FQM-207.
Measurement of the temporal control laser profile {#sec:laser-pulse}
=================================================
The temporal profiles of the control-laser pulses were deduced from a measurement of their spectrum and a calibrated wavelength-to-time conversion. All spectra were recorded with a commercial spectrometer (Photon Control SPM-002-X).
![(Color online) (a) Spectrum of the control laser beam. (b) Interference spectrum of the control and probe laser beam. The red ellipse marks the position of the spectral interference pattern. The green line indicates the amplitude of the interference. (c) Measurement (rhombuses) and quadratic fit (line) of the wavelength of the interference pattern as a function of the delay stage position. (d) Temporal evolution of the control laser beam.[]{data-label="fig:hv-pico-time"}](hv-pico-time){width="\linewidth"}
The conversion function between the measured spectrum, shown in a, and the corresponding temporal intensity profile of the control laser pulse, shown in d, was determined from the spectral interference of the control laser and the time-delayed $30$ fs probe laser pulses. Both laser beams were linearly polarized, parallel to each other, colinearly overlapped, and their simultaneous spectrum recorded. The resulting spectral interference between the two laser pulses manifests itself as a localized large amplitude fluctuation in the spectrum. These fluctuations mark the position “in time” of the short probe pulse within the spectrum of the temporarily stretched – chirped – control laser pulse. As an example, the combined spectrum of both lasers is shown for one relative timing in b. The position of the spectral interference is highlighted by the red ellipse and the amplitude of its fluctuation is indicated by the green curve. The wavelength $\lambda$ of the spectral interference pattern in the spectrum as a function of the delay-stage position $d$ is shown in c. The blue line is a quadratic fit according to $\lambda(d)=\lambda_0 + a\,d + b\,d^2$ from which we obtained $\lambda_0=\SI[per-mode=symbol]{829.99}{\nano\meter}$, $a=\SI[per-mode=symbol]{-0.57}{\nano\meter\per\milli\meter}$ and $b=\SI[per-mode=symbol]{-8.65e-4}{\nano\meter\per\milli\meter\squared}$. The conversion from delay stage position $d$ to time $t$ is given by $t=2d/c$, where $c$ is the speed of light. The factor of two is taking into account that the light was traveling back and forth in the translation stage. The resulting temporal profile of the control laser pulse, calculated according to $$t(\lambda) = \SI{860}{\pico\second} + \frac{2}{c} \left( \SI{698.2}{\milli\meter} -
\SI[per-mode=fraction]{3.41}{\milli\meter\per\nano\meter} \lambda +
\SI[per-mode=fraction]{3.098}{\milli\meter\per\nano\meter\squared} \lambda^2 \right)
\notag$$ is shown in d.
[^1]: For an anti-oriented state, it holds that $-1\le \oricost < 0$ and $0 \le \Nuptot < 0.5$.
[^2]: The adiabatic pendular states [$\left|J,M,s\right>_\textup{p}$]{} are labeled by the quantum numbers $J$ and $M=|m_J|$ of the adiabatically correlated field-free rotor states and the symmetry $s$ with respect to reflection at the plane defined by the dc and ac electric field vectors, denoted $e$ for even and $o$ for odd. For the adiabatic following, we take into account that the dc field is turned on first. Once the maximum dc electric field strength is achieved, the ac laser pulse is turned on. These adiabatic-pendular-state labels are amended by a subscript $p$.
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[**Tubular initial conditions and ridge formation**]{} M.S. Borysova$^{\ddag}$, O.D. Borysov$^{\flat}$, Iu.A. Karpenko$^{\dag,2}$, V.M. Shapoval$^\dag$ and Yu.M. Sinyukov$^\dag$,\
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**Abstract**
The 2D azimuth & rapidity structure of the two-particle correlations in relativistic A+A collisions is altered significantly by the presence of sharp inhomogeneities in superdense matter formed in such processes. The causality constraints enforce one to associate the long-range longitudinal correlations observed in a narrow angular interval, the so-called (soft) ridge, with peculiarities of the initial conditions of collision process. This study’s objective is to analyze whether multiform initial tubular structures, undergoing the subsequent hydrodynamic evolution and gradual decoupling, can form the soft ridges. Motivated by the flux-tube scenarios, the initial energy density distribution contains the different numbers of high density tube-like boost-invariant inclusions that form a bumpy structure in the transverse plane. The influence of various structures of such initial conditions in the most central A+A events on the collective evolution of matter, resulting spectra, angular particle correlations and $v_n$-coefficients is studied in the framework of the HydroKinetic Model (HKM).
PACS number(s): 25.75.Gz, 24.10.Nz
Introduction
============
The correlation and fluctuation study is a valuable tool for probing the dynamics of heavy-ion collisions, the initial conditions, and for exploring the thermodynamic properties of strongly interacting matter at extremely high temperatures and/or densities. The data from the experiments at RHIC revealed interesting features in the two-particle correlation landscape [@Horner; @Alver; @Alver2; @Abelev; @Adare; @Wosiek]. Specifically, an excess of correlated particles in a wide pseudorapidity interval and a narrow azimuthal angle range $\Delta\varphi$ was firstly measured in the relativistic heavy-ion collisions by the STAR collaboration [@Horner]. The pair correlation on the near side of the trigger particle (with $p_T= 2.5-15$ GeV) was found to be extended across the entire detector pseudorapidity acceptance region $\Delta\eta\sim 3-4$ units. Such a correlation structure was called “the ridge”. The ridges are observed both in the correlations of particles with a jet triggering (the “hard” ridge) and in the correlations without a high $p_T$ trigger particle (the “soft” ridge). The analyses of measurements by the PHENIX [@Adare] and PHOBOS [@Wosiek] RHIC collaborations confirmed the STAR results. In ALICE LHC experiment it was found that the soft ridge is consistent with expectations from collective response to anisotropic initial conditions [@ALICE]. Moreover, the recent measurements by CMS collaboration [@CMS1; @CMS2] detected unexpected similar effect in proton-proton collisions at LHC. The clear and significant ridge structure emerges at $\Delta\varphi\approx 0$, extending to $\left|\Delta\eta\right|$ of at least 4 units. This novel feature of the data has never been seen in two-particle correlation functions in $pp$ or $p\bar p$ collisions before.
The discovery of the ridge structures has aggravated quantitative theoretical analyses of nucleus-nucleus and proton-proton collisions but brought the new physical ideas. Early models of the ridge formation were based on the opposite physical mechanisms. Some authors treated the ridge as an initial-state effect [@Dumitru]; the others explored final-state effects, soft interactions and jet propagation in anisotropic plasma, as the origin of the ridge [@Schenke]. In the former case the authors argued that due to causality constraints the long-range correlations within 4 units of pseudorapidity could be explained only if they originated from the very initial collision stage. They could be a consequence of the correlations in the classical color fields responsible for multi-particle production in relativistic heavy-ion collisions. Then, due to fluctuations of color charges in colliding nuclei, the longitudinally boost-invariant and transversally bumpy structure of the matter can be formed. As for the hard ridges, these correlations are associated with jet propagation, that gives relatively narrow correlations in pseudorapidity. The interesting attempt to combine soft and hard ridge structures is done in [@Werner]. It is based on interaction of jets with pieces of expanding bulk matter boosted by the transverse flow.
The role of fluctuations in formation of the soft observables was first considered in [@Hama0; @Hama], and their ability to produce ridges in [@Hama2]. Alver and Roland [@Roland] observed that the correlations arising from geometrical fluctuations in the initial conditions result in odd flow harmonics such as a triangular flow $v_{3}$. Then one can suppose that $v_{3}$, together with flow harmonics of all higher orders could explain the excess in the long-ranged two-particle correlations. It has been recently shown in measurements [@CMS1; @CMS2; @Lacey; @Grosse; @Jia] and demonstrated in various modelings [@Ollitrault; @Luzum; @Bass] that there are fairly strong fluctuations in the initial matter profile from event to event. Such fluctuations contain various higher order harmonics in azimuthal angle $\sim cos (n\phi)$. After the collective expansion of the bulk matter these fluctuations lead to observed harmonic flows up to about $n = 6$. There is a hope that such high harmonics can explain the soft di-hadron azimuthal correlations – the soft ridge. It means, in fact, that the soft ridges originate from the fluctuating initial conditions altogether with the specific set of Fourier $n$-harmonics. It also seems, that the fluctuations of the initial conditions, associated with ridges, are of specific type. This issue is studied in the present article.
If the soft ridge is caused by the peculiarities of the initial space structure of the bulk matter, then this effect should be analyzed within hydrodynamics models of A+A collisions, which are now the standard approach to the description of such processes. During more than 50 years the hydrodynamic models were based on the smooth initial energy density profiles (see, e.g. [@Gavin; @Hirano] for the ridge problem). However, it has been shown, that fluctuations in the positions of nucleons within the colliding nuclei from event to event may lead to significant deviations from the smooth regular profiles in one event [@Manly]. The similar effect can be caused by the fluctuations of the local color charge in Color Glass Condensate effective field theory [@Gavin1]. This irregular structure of the initial conditions for hydrodynamic evolution was explored in [@Hama2; @Petersen; @Karpenko; @Andrade] for analysis of the ridge phenomenon. It becomes clear, that the most important factor for ridge formation is not just the variation of the geometry form of initial system from event to event, but strongly inhomogeneous bumpy structure of the initial energy density profile in the transverse plane. Such initial structure is subjected to further non-trivial evolution during the system expansion. The different mechanisms can be responsible for the formation of the bumps in the initial energy density profile, e.g. it can be longitudinal strings as in Nexus [@Hama2], color flux tubes, arising between large locally fluctuating color charges in colliding nuclei. These fluctuations have the form of very narrow and dense, approximately boost-invariant longitudinal tubes, shifted differently from the center in transverse plane, on the top of smoothly distributed energy density. The number of the tubes also can be different. The analysis of the ridge formation in the case of one peripheral tube has been done in [@Hama2; @Andrade] with the Cooper-Frye prescription for freeze-out. The analogous analysis for multi-tube systems was provided in [@Borysova; @multi].
In this work we continue to analyze possibilities of ridge formation at sharp disturbances of the initial energy density. The evolution and spectra formation for the system with bumpy initial energy density profile is analyzed within the HydroKinetic Model (HKM)[@Sinyukov; @Akkelin; @Sinyukov2; @Sinyukov3] which incorporates description of all the stages of the system evolution, including the afterburn one, responsible for continuous particle liberation from expanding decoupling system. Previously it was found that the effect of the initial bump-like fluctuations is not washed out during the system expansion and leads to the specific final energy density distributions [@Borysova; @Borysova2; @Borysova3]. In this paper we examine the influence of tube-like fluctuations of different type on the observed particle spectra, azimuthal correlations and magnitudes of the Fourier harmonics. To highlight the role of the bumpy structure, we limit our present study to the most central collisions only, say $c=0-2\%$. In contrast to the non-central collisions, with strong initial eccentricity already in average (in the framework of the variable geometry analysis) and large $2^{nd}$ flow harmonic produced, in the perfectly central collisions $(b = 0)$ the background geometry is isotropic, and the anisotropy and $v_n$-coefficients, caused by the fluctuating bumpy structure only, will be best manifested.
Model description
=================
The hydrokinetic approach, which we use as the basic model of the matter evolution, is described in detail in [@Akkelin; @Sinyukov2]. It contains the perfect hydrodynamic component, related to expanding quark-gluon matter, and the kinetic one, describing the system decay and the spectra formation due to gradual particle liberation from expanding hadron-resonance matter. We consider the transversally bumpy, tube-like IC aiming to study how the initial fluctuations in the energy density distribution affect the spectra, azimuthal correlations and flows. For the sake of simplicity, these results are presented only for one kind of particles – negative pions $\pi^-$.
The HKM in its original version [@Akkelin; @Sinyukov2] is (2+1)D model based on the boost-invariant Bjorken-type IC, where the longitudinal flow has quasi-inertial form and is related to the initial proper time $\tau_i=\sqrt{t^2-z^2}$, when (partial) thermalization is established and further evolution can be described by hydrodynamics. The transverse dynamics of the prethermal stage starts at very early time $\tau_0 =0.1-0.2$ fm/c just after the c.m.s. energy in the overlapping region of colliding nuclei exceeds their binding energy. The thermalization can hardly happen before 1 fm/c (1–1.5 fm/c is the lowest known estimate of the thermalization time). However, if one starts hydrodynamics at that time, neither radial nor elliptic flow can develop well enough to explain experimental transverse spectra and their anisotropy ($v_2$-coefficients). The solution of this problem was proposed in the papers [@Sinyukov3; @Sinyukov4], where it was demonstrated how efficiently the transverse collective flow and its anisotropy can develop at prethermal stage (even without pressure) in spatially finite systems typical for A+A collisions. Now the description of this pre-thermal stage, based on the Ref. [@Akkelin2] is in progress. Meanwhile, in [@KarpSinWern] a rough approach was proposed, which allows one to calculate the developing of the transverse flow and its anisotropy at the pre-thermal stage by means of hydro evolution that starts at very early time (0.1 fm/c) just to account for the energy-momentum conservation law. It brings a good agreement with all bulk observables. It does not mean that thermalisation actually happens at the proper time 0.1–0.2 fm/c [^1], but only that hydrodynamic approach introduces no big errors being applied out of its applicability region, at the prethermal stage, to describe collective flow and its anisotropy at the thermalization time $\sim 1$ fm/c. With this in mind the hydrodynamic evolution in our approach begins from the starting time of the collision process, $\tau_i=\tau_0$. Here we use the starting proper time $\tau_0 = 0.2$ fm/c. At this very early moment there is no collective transverse flow, and our analysis is related just to this case.
We assume the initial energy density distribution in the corresponding flux-tubes (produced by the fluctuating local nucleons distribution or color charge fluctuations in colliding nuclei) to be fairly homogeneous in a long rapidity interval because of the boost-invariance and rather thin transversally with the transverse (Gaussian) radii $a_i = 1$ fm. The initial energy-density distribution $\epsilon(x,y) $ at $\tau_0$ is supposed to be decomposed in general case as
$$\begin{aligned}
\label{edd}
\epsilon(x,y) = \epsilon_{bkg}(r)+\sum\limits_{i=0}^{N_t} \epsilon_i e^{-\frac{(x-x_i)^2+(y-y_i)^2}{a^2}},\end{aligned}$$
where $r^2=x^2+y^2$, ${\bf r}_i=(x_i,y_{i})$ are the positions of the tubes’ centers, $N_t$ is the number of tubes, $\epsilon_i$ are the values of maximal energy density in the tube-like fluctuations. In our analysis we use the two different types of the background $\epsilon_{bkg}$, on which tubes are placed: it takes either the Gaussian form $$\begin{aligned}
\label{edd1}
\epsilon_{bkg}^G(r)=\epsilon_{b}e^{-\frac{r^2}{R^2}},\end{aligned}$$ or the Woods-Saxon form with the surface thickness parameter $\delta$ $$\begin{aligned}
\label{edd2}
\epsilon_{bkg}^{WS}(r)=\frac{\epsilon_{b}}{e^{\frac{\sqrt{r^2} -R_a}{\delta}}+1},\end{aligned}$$ where $\epsilon_b$ is the maximum value of the background energy-density distribution.
The results are demonstrated for various kinds of the bumpy IC structure with different number $N_t$ of tubes – two odd ones ($N_t$ = 1 and 3) and two even ones ($N_t$ = 4 and 10) having variable radial distances from the center. Also the two previously mentioned types of the background are utilized. Specifically, the analysis is carried out for the following tube-like initial configurations:
The configuration with a smooth Gaussian profile without fluctuations, as it was considered in [@Sinyukov3], where the parameters obtained from the fit to the Color Glass Condensate model result are as follows: $R = 5.4$ fm and maximum energy density at $r = 0$ is $\epsilon_{b} = 90 $ GeV/fm$^3$.
The configuration with one tube in the center, where the tube energy density profile is the Gaussian one with $a = 1.0$ fm and $\epsilon_i=270$ GeV/fm$^3$. The background corresponds to the case (i).
The configuration with one tube shifted from the center, where $\epsilon_i = 270$ GeV/fm$^3$, $r_1 = 3$ fm or $r_1 = 5.6$ fm, and $a = 1$ fm. The background corresponds to item (i). The initial configuration with $r_1 = 3$ fm is presented in Fig. 1 (left).
The configuration with three tubes: $\epsilon_i= 250$ GeV/fm$^3$; $\textbf{r}_i = (0, 5.6)$, $(-1, 3.6)$, $(-1, -3.6)$ fm or $\textbf{r}_i =(0, 0)$, $(-1, 3.6)$, $(-1, -3.6)$ fm; $a_i = 1$ fm. The background is Gaussian with $\epsilon_{b} = 85$ GeV/fm$^3$, $R = 5.4$ fm.
The configuration with four symmetrically located tubes: $\epsilon_i=250$ GeV/fm$^3$, $r_i = 5.6$ fm and $a_i = 1$ fm. The background is the Gaussian one with $\epsilon_b=85$ GeV/fm$^3$. The corresponding initial energy density profile is presented in Fig. 1 (center).
The configuration with ten tubes, $\epsilon_{b} = 25$ GeV/fm$^3$, $R = 5.4$ fm, $r_1 = 0$ fm, $r_{2,3,4} = 2.8$ fm, $r_{i> 4} = 4.7$ fm, $a = 1$ fm, and $\epsilon_i=4\epsilon_{b}$ (see Fig. 1 right).
Besides these configurations we consider also the case of IC with background profile $\epsilon_{bkg}(r)$ in the form (3) ($\epsilon_{b} = 90$ GeV/fm$^3$, $R = 6.37$ fm, and $\delta = 0.54$ fm) and one tube shifted from the center and placed at $r_i = 0.5R$ or $1.1R$. The maximum energy densities are $\epsilon_i = 270$ GeV/fm$^3$, $a = 1$ fm. This type of IC with $r_i = 0.5R$ is presented in Fig. 2.
![3D plots of the initial energy density profiles with tube-like IC on the background in the form (\[edd1\]) for $\tau_0 = 0.2$ fm/c. Left – 1 tube (iii) displaced at $r_1 = 3$ fm. Center – 4 tubes (v). Right – 10 tubes (vi).](fig1a.png "fig:"){width="\linewidth"} ![3D plots of the initial energy density profiles with tube-like IC on the background in the form (\[edd1\]) for $\tau_0 = 0.2$ fm/c. Left – 1 tube (iii) displaced at $r_1 = 3$ fm. Center – 4 tubes (v). Right – 10 tubes (vi).](fig1b.png "fig:"){width="\linewidth"} ![3D plots of the initial energy density profiles with tube-like IC on the background in the form (\[edd1\]) for $\tau_0 = 0.2$ fm/c. Left – 1 tube (iii) displaced at $r_1 = 3$ fm. Center – 4 tubes (v). Right – 10 tubes (vi).](fig1c.png "fig:"){width="\linewidth"}
{width="60.0mm"}
The configurations described above serve as the initial conditions for hydrodynamic evolution of the superdense system. The quark-gluon plasma and hadron gas are supposed to be in complete local equilibrium above the chemical freeze-out temperature $T_{ch}$ with EoS described below. With the given IC, the evolution of thermally and chemically equilibrated matter is described with the help of the ideal hydrodynamics approximation. The latter is based on (2+1)D numerical hydrodynamic code, described in [@Sinyukov2]. For this stage of evolution we use the lattice QCD-inspired equation of state of quark-gluon phase [@laine] together with corrections for small but non-zero baryon chemical potentials [@Sinyukov2], matched with chemically equilibrated hadron-resonance gas via crossover-type transition. The hadron-resonance gas consists of all ($N=329$) well-established hadron states made of u,d,s-quarks, including $\sigma$-meson, $f_0$(600). The chemical freeze-out hypersurface at $T = 165$ MeV, which corresponds to the end of chemically and thermally equilibrated evolution and start of the hydrokinetic stage of A+A collision process, is presented in Fig. 3 for the initial configuration (iii).
At the temperatures below $T_{ch}$ the system loses chemical and thermal equilibrium and gradually decays. In the hydrokinetic approach, HKM, the dynamical decoupling is described by the particle escape probabilities from the inhomogeneous hydrodynamically expanding system (with resonance decays taken into account) in a way consistent with the kinetic equations in the relaxation-time approximation for emission functions [@Akkelin; @Sinyukov2]. The HKM gives the possibility to calculate the momentum spectra formation during continuous process of the particle liberation. Within HKM one can consider the single event (= fixed IC) basing on numerical calculations of the analytical formulas for spectra using the temperatures and particle concentrations from numerical hydrodynamic solution. An extensive description of the approach can be found in [@Akkelin; @Sinyukov2]. In the employed version of HKM there is no Monte Carlo cascade algorithm for the hadron stage, that typically brings the new ensemble of events with energy-momentum conservation fulfilled only in average for the fixed initial conditions and fixed hydrodynamic evolution (in this aspect the results of some studies do not correspond, strictly speaking, to the true event-by-event analysis). We just calculate the analytical structure describing gradual decay of expanding fluid into the particles in the single event (related to the fixed IC) in the sense that solutions of the Boltzmann equation describe continuously the [*mean*]{} distribution functions at given IC in correspondence with the energy-momentum conservation law.
In this paper the HKM is applied to the systems with the bumpy ICs which are described above.
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Results and discussion
======================
In this Section we present the results obtained within HKM model with the tube-like IC. It is obvious, that the angular dependence of transverse pion spectra in the cases (i) and (ii), described in the previous Section, is flat, and the effective temperature of the $p_T$-spectrum (its inverse slope) is higher for configuration (ii). A non-trivial angular dependence appears for the other ICs, where at least one tube is shifted from the center and so brings an azimuthal asymmetry into the system. In Fig. 3, which fully corresponds to the actual calculations, one can see that the fluctuation in the initial distribution leads to the appearance of concavity on freeze-out hypersurface in the azimuthal direction corresponding to the initial high-energy fluctuation. It means, in particular, that the freeze-out temperature is reached earlier in this domain.
*3.1. Integrated Pion Spectra.* The analysis of the spectra at various $p_T$ shows their angular dependence to be different. In this subsection we demonstrate the results for integrated over $p_T$ spectra, aiming to see a possibility for the ridge formation in such a tubular picture. In Fig. 4 we present $|p_T|$-integrated pion spectra $dN/d\phi$ for different initial configurations. The black line corresponds to the integrated spectra at $p_T > 0.9$ GeV and the red one corresponds to the momentum values $p_T > 2.5$ GeV [^2]. One can see that behavior of the curves in the case of the Woods-Saxon background distribution with the single very peripheral tube (Fig. 4, $2^{nd}$ row, right) is similar to the results obtained in [@Hama1] for similar configuration; in particular, both local minima coincide at $\phi$=0, but in contrast to [@Hama1], for $p_T> 2.5$ GeV the minimum is not absolute but local. This probably points out that Woods-Saxon background is not similar to the one used in [@Hama1], for the Nexus-inspired IC. For the Gaussian-like background or not so peripheral single-tube disposition (Fig. 4, top left) the angular behavior of the spectrum is different from the above result. Nevertheless, the azimuthal positions of the maximal values of the spectra are correlated for high momentum “trigger” component and “associated” soft one. In the $3^{rd}$ row of Fig. 4 for three initial tubes there are no positive correlations in a front of the initial tube at $\phi=0$. Nevertheless, there are such correlations at the left figure at the points $-\pi$ and $\pi$ and at the right figure at $-1$ and 1 rad. The angular positions of the maximal values are synchronized for 4 and, partially, for 10 tubes (Fig. 4, bottom).
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![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_1tube3_G.png "fig:"){width="57mm"} ![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_1tube56_G.png "fig:"){width="57mm"}
![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_1tube06_WS.png "fig:"){width="57mm"} ![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_1tube11_WS.png "fig:"){width="57mm"}
![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_3tubes0_56_WS.png "fig:"){width="57mm"} ![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](Fig5_nf23_tubes_vs_ALICE.png "fig:"){width="57mm"}
![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_4tubes_G.png "fig:"){width="57mm"} ![The correlation function $C(\Delta\phi)$ for various ICs. The top row demonstrates the case of 1 displaced tube with the Gaussian background (iii): left – at $r_1 =0.55R$, right – at $r_1 =1.04R$. The $2^{nd}$ row corresponds to the case of 1 displaced tube with the Woods-Saxon background (vii): left – at $r_i =0.5 R$, right – at $r_i =1.1 R$. The $3^{rd}$ row describes 3 tubes with the Gaussian background (iv): left – $\textbf{r}_i = (0, 0)$,$(-1, 3.6)$,$(-1,-3.6)$ fm, right – $\textbf{r}_i = (0, 5.6)$,$(-1, 3.6)$,$(-1, -3.6)$ fm. The $4^{th}$ row is related to the Gaussian background: left – 4 tubes (v), right – 10 tubes (vi). The ALICE LHC data are taken from [@ALICE].](sptr_dphi_pt0925_10tubes_G.png "fig:"){width="57mm"}
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The results demonstrate that if one particle with relatively large $p_T$ is triggered, then, most probably, its azimuthal direction will correspond to the one of the peaks in the distributions $dN/d\phi$. Then, as it follows from the $dN/d\phi$ distributions for $p_T> 0.9$ GeV and $p_T> 2.5$ GeV, the probability to find the second “associated” particle with the same or smaller transverse momentum will be maximal in a narrow range $\Delta\phi$ near this peak. Such effects are typically expressed through the ratio $C(\Delta\phi)$ of the dihadron distribution in $\Delta \phi$ for the [*same*]{} events to the one extracted from the [*mixed*]{} events:
$$C(\Delta\phi)-1=\left(\frac{dN^{mixed}}{d\Delta\phi}\right)^{-1}\left(\frac{dN^{same}}{d\Delta\phi}-\frac{dN^{mixed}}{d\Delta\phi}\right),$$
where in our model for each identical (in the sense of variable geometry analysis) initial tube-like configuration we have
$$\begin{aligned}
\label{edd4}
&&\frac{dN^{same}}{d\Delta\phi}=\int\limits_{-\pi}^{\pi} f\left(\phi+\frac{\Delta\phi}{2}\right)\cdot g\left(\phi-\frac{\Delta\phi}{2}\right) d\phi, \nonumber \\
&&\frac{dN^{mixed}}{d\Delta\phi}=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\int\limits_{-\pi}^{\pi} d\phi_1 d\phi_2 f\left(\phi_1+\frac{\Delta\phi}{2}\right)\cdot g\left(\phi_2-\frac{\Delta\phi}{2}\right)=\frac{1}{2\pi}N_{as}N_{tr}.\end{aligned}$$
Here $f(\phi) =\int\limits_{0.9}^{3.0} s(p,\phi) dp$ and $g(\phi) =\int\limits_{2.5}^{3.0} s(p,\phi) dp$ are $dN/d\phi$ distributions for $p_T> 0.9$ GeV and $p_T> 2.5$ GeV respectively, $s(p,\phi)$ are the pion spectra, and $N_{tr}=\int\limits_{-\pi}^{\pi} g(\phi)d\phi$ and $N_{as}=\int\limits_{-\pi}^{\pi} f(\phi)d\phi$ are the normalization constants for the “trigger” and “associated” components respectively.
The correlation functions $C(\Delta\phi)$ are presented in Fig. 5 for the different initial configurations. One can see that for many of considered initial configurations (but not for all of them!) there are narrow azimuthal near-side correlations which are typical for the ridge. The inclusive correlations between the triggered particle and the particle corresponding to the other peaks will be relatively weak, since these peaks will change their angular location from event to event with respect to the “triggered”, or near-side, peak, and this will wash out the two-particle correlations outside the near-side peak. It is not unlikely, however, that the doubly-peaked away-side structure, which is seen in Fig. 5 ($2^{nd}$ row right and $3^{rd}$ right) can be supported by the multi-peak configurations similar to those presented in Fig. 5 (bottom), and so the doubly-peaked away-side structure may survive, if the weights of such triangular initial configurations are fairly big. One can also see that if the tube is not in the IC peripheral region, just the near-side ridge structure appears, and the away-side structure is washed out. Thus, not only the number of the tubes is important, but also their positions. The comparison of the ridge and doubly-peaked away-side structure with the data of ALICE LHC [@ALICE] supports such a hypothesis. This is the possible mechanism of formation of the so-called soft ridge structure [^3]. The comparison between the results for the Gaussian background (Fig. 5, $1^{st}$ row, right) and the Woods-Saxon one (Fig. 5, $2^{nd}$ row, right) demonstrates sensitivity of ridge formation to the background initial profile: in the case of one tube the ridges prefer Woods-Saxon type of the profile.
As for the formation of the hard ridges with triggered particle momenta $p_T> 5-6$ GeV, it is very likely that they are the result of superposition of the bulk spectrum structure, caused by the tubular bumpy IC, and the jet spectrum one, conditioned by the jet formation mechanism accounting for the interaction with the bulk matter. Even a 10% coincidence between the azimuth jet direction and the angular position of one of the peaks in soft spectra, similar to those in Fig. 4, can be enough to form the observed hard ridge [@Werner].
*3.2. The Flow Harmonics and $p_T$ Dependence of Their Magnitudes.* The anisotropy of the initial tubular conditions that leads to non-trivial ridge structure, results also in a non-trivial transverse momentum spectrum angular structure, even in the most central collisions. This structure can be investigated quantitatively using a discrete Fourier decomposition. Then the azimuthal momentum distribution of the emitted particles is commonly expressed as $$\label{edd6}
\frac{dN}{p_T dp_T d\phi dy} = \frac{1}{2\pi} \frac{dN}{p_T dp_T dy} \left(1+ \sum\limits_{n=1}^{\infty} 2 v_n(p_T) cos(n(\phi-\Psi_n(p_T)))\right),$$ where $v_n$ is the magnitude of the $n^{th}$ order harmonic term, relative to the angle of the initial-state spatial plane of symmetry $\Psi_n$. In Fig. 6 we show our results for the $p_T$ dependence of $v_n$ for different initial configurations. Here, since our purpose is to show clearly the effects of bumpy IC, we plot the results obtained for $v_n$ up to $n=6$. One can note that $v_1$ coefficients take negative and positive values, and so the integral $\int_0^{3.5 \mathrm{GeV}}dp_T p_T^2 v_1(dN/p_Tdp_T)$ is close to zero. In fact, this integral is less than $10\%$ of the differences between its values in the regions where $v_1>0$ (large $p_T$) and $v_1<0$ (small $p_T$). This smallness of the integral (for different configurations it is $0.004-0.029$ GeV) reflects the transverse momentum (it is zero initially) conservation. Since we consider $v_n$ coefficients for the pion subsystem only, one cannot expect exact zero value for considered integral.
Note that for configurations (ii)–(vii) for odd number of the initial tubes the odd harmonics are dominating, and the analogous tendency takes place for the even harmonics at even number of the tubes. However, to make general conclusion, more configurations have to be considered.
The experimentally valuable results have to be based on some model of the tube-like initial conditions and the correspondingly built event generator for IC, and also include an event by event evolution procedure. It will define the weight of the single events, such as those presented in Fig. 6, and finally give the resulting event-averaged $v_n$-coefficients and the quantitative structure of the ridge. We plan to realize such a quantitative experimental analysis in subsequent studies.
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Conclusions
===========
The tube-like fluctuating structures in the initial energy density distribution are considered with the aim to study the influence of their presence on the pion spectra, flow harmonics and ridge formation. These very dense color-field flux tubes are formed at the very initial stage of the nucleus-nucleus collision and lead to the long-range longitudinal correlations in pseudorapidity. It is found that the presence of the corresponding bumpy structures in transverse direction in IC strongly affects the hydrodynamic evolution and leads to emergence of conspicuous structures in azimuthal distributions of the pion transverse momentum spectra. The hydrokinetic evolution for different initial configurations with different numbers of tubes is calculated. As the result, one can see that most configurations can bring the ridge structure accompanied by the specific set of $v_n(p_T)$-coefficients. Not only the triangular structures of the initial conditions are responsible for the soft ridge formation, but also the odd number of the initial tubes can support this job. It means that the hydrodynamic mechanism of the near-side “soft ridges” formation is sufficiently plausible. Also one can note that the doubly-peaked away-side structure appears when there is one outer/peripheral tube in the initial conditions. To constrain event by event fluctuating IC in A+A collisions with subsequent hydrodynamic expansion within the viscous HKM and provide the detailed quantitative analysis of the ridge structure a further systematic analysis is planned.
Acknowledgments
===============
Yu. M. Sinyukov thanks Y. Hama for the fruitful discussions and FAPESP (Contract 2013/10395-2) for financial support. The research was carried out within the scope of the European Ultra Relativistic Energies Agreement (EUREA) of the European Research Group GDRE: Heavy ions at ultrarelativistic energies and is supported by the State Fund for Fundamental Researches of Ukraine, Agreement F33/24-2013 and the National Academy of Sciences of Ukraine, Agreement F4-2013. Iu. A. Karpenko acknowledges the financial support by the ExtreMe Matter Institute EMMI and Hessian LOEWE initiative.
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[^1]: the same concerns the hydrodynamic models where the starting time is 0.4–0.6 fm/c
[^2]: It is demonstrated in different hydrodynamic models, see e.g. [@KarpSinWern], that the hydrodynamics works in $p_T$-region until 3 GeV for the central events; here we consider very central collisions with $b\approx 0$. Note that we analyze the contribution of only hydro-component to the soft ridge formation.
[^3]: Note, that the dihadron correlations are sensitive not only to the tube configurations, but also to the tube energy, as it is analyzed for one-tube case in [@Hama2; @Hama1].
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---
abstract: 'Practical implementations of quantum computing are always done in the presence of decoherence. Geometric phase is useful in the context of quantum computing as a tool to achieve fault tolerance. Recent experimental progresses on coherent control of single electron have suggested that electron in quantum dot systems is promising candidate of qubit in future quantum information processing devices. In this paper, by considering a feasible quantum dot model, we calculate the geometric phase of the quantum dot system in nonuitary evolution and investigate the effect of environment parameters on the phase value.'
author:
- 'Sun Yin[^1] and D. M. Tong[^2]'
title: Geometric Phase of a Quantum Dot System in Nonunitary Evolution
---
The quantal geometric phase was first discovered by Berry [@Berry] in 1984 in considering the quantum systems under cyclic adiabatic evolution. It has aroused much attention of researchers due to its importance. Since then the original notion of Berry phase has been extended to a general concept of geometric phase for pure states as well as for mixed states. The extension to pure states in nonadiabatic cyclic evolution was developed by Aharonov and Anandan [@Aharonov] in 1987, and that to pure states in nonadiabatic and noncyclic evolution was done by Samuel and Bhandari [@Samuel] in 1988. Further generalizations and refinements, by relaxing the constrains of adiabaticity, unitarity, and cyclicity of the evolution, have since been carried out [@Mukunda; @pati95]. While all these extensions are of quantum systems in pure states, Uhlmann [@Uhlmann] was the first to address the geometric phases of mixed states within the mathematical context of purification. A physical definition of geometric phases for mixed states in unitary evolution was put forward by Sjöqvist [*et al.*]{} [@Sjoqvist2000] in 2000 based on quantum interferometry, and it was recast in a kinematic description by Singh [*et al.*]{} [@Singh2003]. The generalization of mixed geometric phases to quantum systems in nonunitary evolution was given by Tong [*et al.*]{} [@Tong2004] in 2004. More works on geometric phases related to states for open systems may be seen in Refs. [@Carollo03].
The geometric property of the geometric phase has stimulated many applications. It has been found that the geometric phase plays important roles in quantum phase transition, quantum information processing, etc. [@Bohm]. The geometric phase shift can be fault tolerant with respect to certain types of errors, thus several proposals using NMR, laser trapped ions, etc. have been given to use geometric phase to construct fault-tolerant quantum information processer [@Falci2000], and the fault-tolerant geometric quantum computation gate has been demonstrated in experiments using NMR [@Jones2000].
Geometric phase is useful in the context of quantum computing as a tool to achieve fault tolerance. Practical implementations of quantum computing are always done in the presence of decoherence. Fortunately, recent experimental progresses on coherent control of single electron have suggested that electron in quantum dot systems is a promising candidate of qubit in future quantum information processing devices [@Loss1998], because it has long spin coherence time. This start us to investigate the geometric phase of quantum dot systems in nonuitary evolution. In this paper, we calculate the geometric phase of a feasible quantum dot model and investigate the effects of the environment parameters to the phase value.
![Illustration of the model. \[model\]](GPfig1.eps){width="4.5cm" height="2.5cm"}
The model is illustrated as Fig.\[model\]. Two quantum dots, QD$_1$ and QD$_2$ , are coupled to each other with strength $s_1$. An electron is trapped in the quantum dots and it tunnels between the two quantum dots. Only one energy level is considered in each quantum dot, and hence the electron and the two quantum dots construct a two-level quantum system, a qubit. The environment of the system consists of another quantum dot, QD$_0$, and two leads connecting to QD$_0$. The left lead has higher chemical potential than the right lead. Electrons can tunnel from the left lead to QD$_0$ and then tunnel out to the right lead. For simplicity, we assume that only one electronic state with the energy level $E_0$ in QD$_0$ is correlated and $\mu_L>E_0>\mu_R$, where $\mu_L$ and $\mu_R$ are the chemical potentials of the left lead and the right lead respectively. Once there is an electron in QD$_0$, it will affect the coupling between QD$_1$ and QD$_2$ by changing the coupling strengthes from $s_1$ to $s_2$. This is an interesting model, of which the relaxation and decoherence and quantum measurement have been well studied [@Stace2004; @Gurvitz2008]. The model is easily performed in experiment, and it may play a potential selection for geometric quantum computation of using quantum dot systems.
Noting that the qubit system, comprising the trapped electron and the two dots, is an open system being in mixed state, we use the formula of geometric phases for mixed states in nonunitary evolution given in Ref. [@Tong2004]. For an open quantum system, described by the reduced density operator, $\rho(t)=\sum_{k=1}^2\omega_k(t)|\phi_k(t)\rangle\langle\phi_k(t)|,~t\in[0,\tau]$ the geometric phase is given by the formula, $$\begin{aligned}
\gamma(\tau)=\textrm{Arg}\bigg(\sum_{k=1}^2
\sqrt{\omega_k(0)\omega_k(\tau)}
\langle\phi_k(0)|\phi_k(\tau)\rangle e^{ -\int_0^\tau \langle
\phi_k(t) | \dot{\phi}_k(t) \rangle dt}\bigg), \label{GP}\end{aligned}$$ where $\omega_k(t)$ is the $k-$th eigenvalue of the reduced density matrix, $|\phi_k(t)\rangle$ is the corresponding eigenvector, and $\tau$ is the total evolutional time.
In order to calculate the geometric phase of the qubit system, we need to obtain the reduced density operator. The Hamiltonian of the large system can be expressed as $$\begin{aligned}
H=&H_s+H_e+H_i,\nonumber\\
H_s=&E_1a_1^{\dagger}a_1+E_2a_2^{\dagger}a_2+
s_1(a_1^{\dagger}a_2+a_2^{\dagger}a_1),\nonumber\\
H_e=&E_0c_0^{\dagger}c_0+\sum_lE_lc_l^{\dagger}c_l+\sum_r
E_rc_r^{\dagger}c_r +\sum_{l,r}(\Omega_lc_l^{\dagger}c_0
+\Omega_rc_0^{\dagger}c_r+\textrm{H.c.}),\nonumber\\
H_i=&(s_2-s_1) c_0^{\dagger}c_0(a_1^{\dagger}a_2+a_2^{\dagger}a_1).
\label{hamiltonian}\end{aligned}$$ Here, $H_s,~H_e,~H_i$ are the Hamiltonians corresponding to the system itself, the environment and the interaction between the system and its environment, respectively; $a_1^{\dagger}$ and $a_2^{\dagger}$ $(a_1$ and $a_2)$ are the electron creation (annihilation) operators in the two quantum dots; $c_l^{\dagger}$ and $c_r^{\dagger}$ ($c_l$ and $c_r$) are the electron creation (annihilation) operators in the environment corresponding to the left lead and the right lead respectively; $E_1$ and $E_2$ are the energy level of QD$_1$ and QD$_2$; $\Omega_l $ $(\Omega_r)$ is the coupling parameter of left (right) lead with the quantum dot QD$_0$. For simplicity, we have considered electrons as spinless fermions, and we have used $E_l$, $E_r$, $\Omega_l $, and $\Omega_r$ to represent $E_{Ll}$, $E_{Rr}$, $\Omega_{Ll} $, and $\Omega_{Rr}$ respectively.
The wave function of the large system, $|\Psi(t)\rangle$, satisfies the Schrödinger equation, $i\frac{d|\Psi(t)\rangle}{dt}=H(t)|\Psi(t)\rangle$. The reduced density operator $\rho(t)$ may be expressed as the partial traces of $|\Psi(t)\rangle\langle \Psi(t)|$ with respect to the environment consisting of the quantum dot QD$_0$ and the two leads, $\rho(t)=\text{tr}_{D_0}\varrho(t)$, where $\varrho(t)=\text{tr}_{Ls} |\Psi(t)\rangle\langle \Psi(t)|$. Following the method used in Refs. [@Gurvitz2008], we may get the equations of motion for the elements of density matrix $\varrho(t)$. The bases of $\varrho(t)$ consists of four discrete states, $|1\rangle\equiv|1,0,0\rangle$, $|2\rangle\equiv|1,0,1\rangle$, $|3\rangle\equiv|0,1,0\rangle$, $|4\rangle\equiv|0,1,1\rangle$, where $|n_1,n_2,n_3\rangle$ means that there are $n_1,~n_2,~n_3$ electrons in QD$_1$, QD$_2$, QD$_0$ respectively. In the approximation of constant density of states, let $\Gamma_L=2\pi |\Omega_{L} |^2\rho_{L}$ and $\Gamma_{R}=2\pi
|\Omega_{R}|^2\rho_{R}$, where $\rho_{L}$ ($\rho_{R}$) is the density of states for the left (right) lead, and $\Omega_L$ ($\Omega_R$) denotes the constant coupling parameter $\Omega_l$ $(\Omega_r)$. $\Gamma_L$ ($\Gamma_R$) depicts the tunneling rate between the left (right) lead and QD$_0$. In this case, the elements $\varrho_{ij}$ of the density matrix $\varrho(t)$ satisfy[@note], $$\begin{aligned}
\dot{\varrho}_{11}&=-\Gamma_L\varrho_{11}+\Gamma_R\varrho_{22}
-is_1(\varrho_{13}-\varrho_{31}),\nonumber\\
\dot{\varrho}_{22}&=-\Gamma_R\varrho_{22}+\Gamma_L\varrho_{11}
-is_2(\varrho_{24}-\varrho_{42}),\nonumber\\
\dot{\varrho}_{33}&=-\Gamma_L\varrho_{33}+\Gamma_R\varrho_{44}
-is_1(\varrho_{31}-\varrho_{13}),\nonumber\\
\dot{\varrho}_{44}&=-\Gamma_R\varrho_{44}+\Gamma_L\varrho_{33}
-is_2(\varrho_{42}-\varrho_{24}),\nonumber\\
\dot{\varrho}_{13}&=-i\epsilon_0\varrho_{13}-is_1(\varrho_{11}
-\varrho_{33})-\Gamma_L\varrho_{13}+\Gamma_R\varrho_{24},\nonumber\\
\dot{\varrho}_{24}&=-i\epsilon_0\varrho_{24}-is_2(\varrho_{22}
-\varrho_{44})-\Gamma_R\varrho_{24}+\Gamma_L\varrho_{13}.
\label{density}\end{aligned}$$ The initial condition is taken as $\varrho_{ij}|_{t=0}=1$, for $i=j=1$, or $0$, for all other $i,~j$, corresponding to the case that the electron is in QD$_1$ and no electron is in QD$_0$. Here $\epsilon_0=E_1-E_2$, is the energy difference of the energy levels of QD$_1$ and QD$_2$. The elements $\rho_{ij}$ of the reduced density matrix $\rho(t)$ of qubit can be then expressed as $$\begin{aligned}
\rho_{11}=1-\rho_{22}=\varrho_{11}+\varrho_{22},~\rho_{12}=\rho_{21}^\ast=\varrho_{13}+\varrho_{24}.
\label{density-reduce}\end{aligned}$$ Once the reduced density matrix is obtained, we can calculate its eigenvalues $\omega_k(t)$ and eigenvectors $|\phi_k(t)\rangle$, and we have $$\begin{aligned}
&\omega_{1,2}(t)=\frac{1\pm\sqrt{(\rho_{11}
-\rho_{22})^2+4|\rho_{12}|^2}}{2},\nonumber\\
&|\phi_1(t)\rangle=\frac{1}{\sqrt{1+\frac{|\rho_{12}|^2}{(\omega_1
-\rho_{22})^2}}}\left[\begin{array}{c}1 \\
\frac{\rho_{21}}{\omega_1-\rho_{22}}\end{array}\right],\nonumber\\
&|\phi_2(t)\rangle=\frac{1}{\sqrt{1+\frac{|\rho_{12}|^2}{(\omega_2
-\rho_{11})^2}}}\left[\begin{array}{c} \frac{\rho_{12}}{\omega_2
-\rho_{11}}\\ 1
\end{array}\right].\label{omega1}\end{aligned}$$ The initial condition taken above implies $\omega_1(0)=1,~\omega_2(0)=0$, and $|\phi_1(0)\rangle=\Big[\begin{array}{c} 1\\
0\end{array}\Big], ~~|\phi_2(0)\rangle=\Big[ \begin{array} {c} 0\\
1\end{array}\Big].$ The evolution of the system can be illustrated by the path traced in Bloch sphere. The three-dimensional coordinates in the Bloch sphere are $x=\rho_{12}+\rho_{21}$, $y=i(\rho_{12}-\rho_{21})$, and $z=\rho_{11}-\rho_{22}$, respectively. By using the Four-order Runge-Kutta method, we may numerically resolve the differential equations in (\[density\]) and obtain the value of the density operator. Fig. \[BlochSphere\] shows the path traced by the state of the system, where the parameters are chosen as $\Gamma_L=1.0,~
\Gamma_R=2.0,~s_1=1.0,~s_2=0.5, ~\epsilon_0=-2.0$. Hereafter, we take the parameter $s_1$ as the base unit. All the other parameters with energy dimension, such as $\Gamma_L,~ \Gamma_R,~s_2$, are measured by the unit $s_1$, and the time is measured by $1/s_1$. As the time goes on, the path starts from $(0,0,1)$, which corresponds to the state that the trapped electron is in QD$_1$, and moves spirally to $(0,0,0)$, which corresponds to the state that the electron has half probability in QD$_1$ and half in QD$_2$.
![The Bloch sphere of the density matrix. \[BlochSphere\]](GPfig2.eps){width="6.cm" height="4.5cm"}
Substituting Eq. (\[omega1\]) into Eq. , we can calculate the geometric phase of the system. It may be simply expressed as $\gamma(\tau)=i\int_0^\tau \langle \phi_1(t) |
\dot{\phi}_1(t)\rangle dt.$ To sketch out the changing tendency of the geometric phase, we numerically calculate the geometric phase. The parameters are again chosen as $\Gamma_L=1.0,~ \Gamma_R=2.0,~s_1=1.0,~s_2=0.5,
~\epsilon_0=-2.0$. The result is shown as Fig. \[GP-t\].
![The geometric phase as a function of time. \[GP-t\]](GPfig3.eps){width="4.5cm" height="2.8cm"}
The geometric phase is usually put in region $[0,2\pi)$ (mod $2\pi$). In order to show entirely the changing tendency of the phase and express clearly the path dependence of the geometric phase, here we give the schematic by using the calculated values without making a $2\pi$-modulus. The recast of the results in $[0,2\pi)$ is trivial.
From Fig. \[GP-t\], we find that the geometric phase is changing as the time is going on, and it finally saturates to a constant value. The saturation value is a characteristic value for a given configuration of parameters, which may be simply called as the characteristic geometric phase (CGP). This is consistent with the ‘geometricity’ of the geometric phase, that is, the geometric phase is only dependent on the path traced by the state of the system, but not on the dynamics. When evolutional time is small, the spiral path has large spiral radius and the changing of the path is notable, and thus the changing of the geometric phase is obvious too. With evolutional time going on, the spiral radius of the path becomes small and the changing rate of the spiral path are reduced, and therefore the changing of the geometric phase will be reduced too. The system will finally evolves to the point $(0,0,0)$, and from then on the path will be little changing, and so does the geometric phase.
In the model, there are three environment parameters $s_2$, $\Gamma_L$, $\Gamma_R$. We now investigate the effects of these parameters on the phase values. For this, we will consider two kinds of geometric phase values, the geometric phase corresponding to the whole evolutional time, i.e., the CGP, and the geometric phase corresponding to a special time interval $T$.
Firstly, we observe the effect of the parameters on the CGP.
![The geometric phases, CGP and $\gamma(T)$, as functions of the parameters $s_2$, $\Gamma_L$, and $\Gamma_R$. The parameters except for the one taken as variable are chosen as $\Gamma_L=1.0,~
\Gamma_R=2.0,~s_1=1.0,~s_2=0.5, ~\epsilon_0=-2.0$. \[GP-parameter\]](GPfig4.eps){width="7.5cm" height="7.0cm"}
Fig. \[GP-parameter\](a) shows the effect of $s_2$ on CGP. From the figure, we see that CGP is strongly dependent on the parameter $s_2$. Specially, CGP is infinitely large at $s_2=s_1$. This is a reasonable result, because $s_2=s_1$ means that the environment does not affect the qubit system. In the case, the qubit is in the pure state, which is evolving repeatedly along a closed circle in the Bloch sphere, and CGP will accumulate infinitely as the time is going on. However, as the parameter $s_2-s_1$ is becoming large from zero, the value of the phase will reduce. The phase values will approach to zero when $s_2-s_1$ is large enough. This may be explained by the following argument. The larger $s_2-s_1$ means the larger correlation between the environment and the qubit system, which leads to the smaller spiral radius of the path traced by the state of the system. When the environments’ effect is stronger enough, the path may approaches to a line directly from $(0,0,1)$ to $(0,0,0)$ and the corresponding geometric phase will be near to zero.
Figs. \[GP-parameter\] (b) and (c) show the effect of parameters $\Gamma_{L}$ and $\Gamma_{R}$ on CGP. From the figures, we find that the two curves in the figures are similar. CGP becomes infinitely large at $\Gamma_{L}=0$ or $\Gamma_{R}=0$, and it is also approaching to infinity as $\Gamma_{L}$ or $\Gamma_{R}$ is going to large values. These observations are consistent with the physical construction in the model, as we have taken $\Gamma_{R}=2$ in Fig. \[GP-parameter\] (b) and $\Gamma_{L}=1$ in \[GP-parameter\] (c). Roughly speaking, when $\Gamma_L$ is small and $\Gamma_R$ is large, electrons are hard to tunnel into QD$_0$ from the left lead but easy to tunnel out of QD$_0$. There is nearly no electron staying in QD$_0$ in all the time, i.e., the coupling between QD$_1$ and QD$_2$ is mainly $s_1$. The effect of the environment on the qubit is negligible, and the qubit may be taken as a closed two-level system with coupling strength $s_1$. The picture of CGP corresponding to the case is the left part of Fig. \[GP-parameter\](b) or the right part of \[GP-parameter\](c). When $\Gamma_{L}$ is large and $\Gamma_{R}$ is small, electrons are easy to tunnel into QD$_0$ from the left lead but hard to tunnel out of QD$_0$. There is an electron staying in QD$_0$ almost in all the time, i.e., the coupling between QD$_1$ and QD$_2$ is dominated by $s_2$. The effect of the environment on the qubit is only to change the coupling strength between QD$_1$ and QD$_2$ from $s_1$ to $s_2$, and the qubit system may be taken as a closed system but with coupling $s_2$. The picture corresponding to this case is the right part of the curve in Fig. \[GP-parameter\](b) or left part of the curve in Fig. \[GP-parameter\](c). When $\Gamma_{L}$ and $\Gamma_{R}$ are in the same order, the qubit is an open system in mixed state. The path traced by the mixed state is a spiral curve and so corresponds to finite values of CGP.
Secondly, we observe the effect of the parameters on the geometric phase for the special time interval $T$. If there is no coupling between the qubit and the environment, or $s_2=s_1$, the qubit system will be in a pure state and it will evolve from the initial state $(0,0,1)$ back to itself after a time interval $T$, making up a closed circle in the Bloch sphere. In the case where $\epsilon_0=-2$ and $s_1=1$, we have $T=\pi/\sqrt{2}$, and the geometric phase corresponding to the closed cycle is $\gamma(T)=\pi-\pi/\sqrt{2}$. However, if $s_2\ne s_1$, the path traced by the state in the Bloch sphere will become an unclosed curve and the geometric phase $\gamma(T)$ will be changed under the effect of the environment. Therefore, $\gamma(T)$ may be used to describe the effect of the environment on geometric phase in an finite time, during which the pure state evolves one circle. Figs. \[GP-parameter\](d), \[GP-parameter\](e) and \[GP-parameter\](f) show the effect of parameters $s_2$, $\Gamma_L$ and $\Gamma_R$ on $\gamma(T)$, respectively. The curves in the figures may be explained by applying a similar discussion as above.
In conclusion, we have calculated the geometric phase of a feasible quantum dot model and investigate the effects of the environment parameters to the phase value. Here, we not only presented the parameters’s effect on the characteristic geometric phase, which corresponding to the whole evolutional time, but also studied their effect on the geometric phase in a finite time interval $T$, defined by using pure state without the effect of environment. The approach of calculating the geometric phase in the paper is reliable. While the other approaches of defining the geometric phase of open systems have met criticisms [@Ericsson2003], the kinematic approach used in the paper has been widely applied to investigate the open systems in various environments [@Hamonic]. Our investigation on geometric phase is helpful to completely understand the properties of the quantum dot system.
This work is supported by NSF of China under Grant Nos.10675076, 10875072 and 10804062.
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[^1]: yinsun@sdu.edu.cn
[^2]: tdm@sdu.edu.cn
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\#1[\#1]{}
Understanding of long-time dynamics of interacting quantum many-body systems or quantized fields is a long standing open problem. In particular, one would like to understand the conditions for the emergence of quantum mixing (implying ergodicity), i.e. general decay of time autocorrelation functions. Inspired by rich quantum behaviour of non-integrable few body systems having (partially) chaotic classical limit[@QChaos] few papers appeared recently concerning ‘quantum chaos’ in non-integrable many-body systems [@JLP; @MBQC; @KtV]. In [@KtV] [*dynamical phase transition*]{} from non-ergodic dynamics exhibiting ideal transport to mixing dynamics exhibiting normal transport has been demonstrated numerically in a non-integrable kicked t-V fermion model on 1d lattice.
Below we construct an exact [*linear mapping*]{} from dynamics of a certain large class of interacting infinite spin$-{{\textstyle{\frac{1}{2}}}}$ chains in Heisenberg picture to a class of non-linear [*One-Body Image dynamical systems*]{} (OBI) in Schr" odinger picture which are realized either on configurational 2d torus or on 2d lattice (tight-binding (TB) formulation). Further we will show by working out two examples: (i) how integrable behavior of the infinite [*XX spin chain in spatially modulated transversal magnetic field*]{} is connected to the integrability of OBI and to the Harper equation [@Harper], and (ii) demonstrate the phase transition of the related [*non-integrable kicked XX chain*]{} from non-ergodic dynamics to ergodic and mixing dynamics corresponding to the stochastic transition from regular to chaotic motion of OBI in the classical limit.
Let $\sigma^s_j,j\in\Z,s\in\{x,y,z\}$ denote a chain of independent Pauli spin$-{{\textstyle{\frac{1}{2}}}}$ variables. We start by generalizing the result of [@KI], namely we find that the operator space spanned by the following set of spatially modulated observables $\{U_n(\vartheta),V_n(\vartheta); n\in\Z,\vartheta\in[-\pi,\pi)\}$ $$\begin{aligned}
U_n(\vartheta) &=&\sum_{j=-\infty}^{\infty}e^{i(j+{{\scriptstyle{\frac{1}{2}}}}|n|)\vartheta}
\cases{
\sigma^x_j\,(\sigma^z_j)_{n-1}\,\sigma^x_{j+n} & $n\ge 1$,\cr
-\sigma^z_j & $n = 0$,\cr
\sigma^y_j\,(\sigma^z_j)_{-n-1}\,\sigma^y_{j-n} & $n\le-1$,\cr}\\
V_n(\vartheta) &=&\sum_{j=-\infty}^{\infty}e^{i(j+{{\scriptstyle{\frac{1}{2}}}}|n|)\vartheta}
\cases{
\sigma^x_j\,(\sigma^z_j)_{n-1}\,\sigma^y_{j+n} & $n\ge 1$, \cr
1 & $n = 0$,\cr
-\sigma^y_j\,(\sigma^z_j)_{-n-1}\,\sigma^x_{j-n} & $n\le-1$. \cr}\end{aligned}$$ where $(\sigma^z_j)_k := \prod_{l=1}^k \sigma^z_{j+l}$ for $k\ge 1$ and $(\sigma^z_j)_0:=1$, is closed under the Lie bracket $[A,B]=AB-BA$ and forms an infinitely dimensional Dynamical Lie Algebra (DLA) $$\begin{aligned}
\lbrack U_{n}(\vartheta),U_{l}(\varphi) \rbrack
&=& 2i \exp\left({{\textstyle{\frac{i}{2}}}}(l\vartheta+n\varphi)s_{l-n}\right)V_{n-l}(\vartheta+\varphi)
\nonumber \\
&-& 2i \exp\left({{\textstyle{\frac{i}{2}}}}(l\vartheta+n\varphi)s_{n-l}\right)V_{l-n}(\vartheta+\varphi),
\nonumber \\
\lbrack U_{n}(\vartheta),V_{l}(\varphi) \rbrack
&=& 2i \exp\left({{\textstyle{\frac{i}{2}}}}(-l\vartheta+n\varphi)s_l\right)U_{n+l}(\vartheta+\varphi)
\nonumber\\
&-& 2i \exp\left({{\textstyle{\frac{i}{2}}}}(l\vartheta+n\varphi)s_l\right)U_{n-l}(\vartheta+\varphi),
\nonumber\\
\lbrack V_n(\vartheta),V_l(\varphi) \rbrack &=& (s_n+s_l)\Bigl\{
\sin\left({{\textstyle{\frac{1}{2}}}}(l\vartheta+n\varphi)\right)\times \nonumber \\
\bigl(
(s_{n-l}s_l+1)&V_{n-l}&(\vartheta+\varphi) - (s_{l-n}s_l+1)V_{l-n}(\vartheta+\varphi)
\bigr) \nonumber \\
+\;\; 2\sin\bigl({{\textstyle{\frac{1}{2}}}}(l\vartheta\!\!&-&\!\!n\varphi)\bigr)V_{n+l}(\vartheta+\varphi)
\Bigr\}. \label{eq:DLA}\end{aligned}$$ where $s_n:=-1,0,1$ for $n<,=,>0$, resp., is a sign of integer $n$. Few notable members of DLA are: Ising or XX hamiltonian $H_I=J U_1(0)$,$H_{XX} = J (U_1(0)+U_{-1}(0))$, spin interaction with modulated transversal magnetic field $h_z=h\cos(\epsilon j)$ with period $2\pi/\epsilon$ lattice spacings $H_{mh} = {{\textstyle{\frac{1}{2}}}}h (U_0(\epsilon) + U_0(-\epsilon))$, spin current $j_s = V_{1}(0) + V_{-1}(0)$, etc. Let us fix the [*fundamental field modulation*]{} $\epsilon$ and introduce the following notation: $$\begin{aligned}
(n,k)\in\Z^2,\qquad\quad
U^\pm_{n,k} &=& {{\textstyle{\frac{1}{2}}}}\left( U_n(k\epsilon) \pm U_{-n}(-k\epsilon)\right),\\
V^+_{n,k} &=& {{\textstyle{\frac{1}{2}}}}\left( V_n(k\epsilon) + V_{-n}(-k\epsilon)\right)s_n,\\
V^-_{n,k} &=& {{\textstyle{\frac{1}{2}}}}\left( V_{n}(k\epsilon) - V_{-n}(-k\epsilon)\right),\\
W^\pm_{n,k} &=& U^\pm_{n,k} + i V^\pm_{n,k},\\
(y,x)\in\T^2,\quad
W^\pm(y,x) &=& \frac{1}{2\pi}\sum_{n,k=-\infty}^\infty
e^{i(ny+kx)} W^\pm_{n,k}.\end{aligned}$$ We may also consider $\epsilon$ as a [*lattice spacing*]{} and treat $W^\pm_{n,k}$ as a set of spatially $2\pi$-periodic fields. We will assume that the modulation is [*incommensurable*]{} with the lattice spacing, i.e. that $\epsilon/2\pi$ is [*irrational*]{}, otherwise obsrvables $W^\pm_{n,k}$ are periodic w.r.t. index $k$. DLA becomes a Hilbert space when we introduce an infinite temperature (grand) canonical scalar product [@KI] $(A|B) :=
\lim_{L\rightarrow\infty}{{\textstyle{\frac{1}{L}}}}2^{-L} {{\rm tr\,}}A^\dagger B$, where $L$ is a diverging length of the spin-chain. Let the two linear subspaces spanned by $W^\sigma_{n,k}$ (or $W^\sigma(y,x)$) for $\sigma\in\{+,-\}$ be denoted by ${\frak M}_\sigma$. The spaces ${\frak M}_+$ and ${\frak M}_-$ are [*orthogonal*]{} and observables $W^+_{n,k}$ ($n,k\in\Z$) and $W^-_{n,k}$ (for $n\ge 1$) form orthonormal bases in each of them, since one can show $$\begin{aligned}
(W^+_{n,k}|W^+_{m,l}) &=& \delta_{n,m} \delta_{k,l},\quad
(W^+_{n,k}|W^-_{m,l}) = 0,\nonumber\\
(W^-_{n,k}|W^-_{m,l}) &=& (\delta_{n,m}-\delta_{n,-m})\delta_{k,l}.
\label{eq:scal}\end{aligned}$$ The full set $\{W^-_{n,k}\}$ is over-complete, since $W^-_{-n,k}=-W^-_{n,k}$, while the subspace ${\frak M}_+={\frak M}^\dagger_+$ is self-adjoint, since $W^{+\dagger}_{n,k}=W^+_{-n,-k}$. One can write analogous relations in terms of continuous variables $(y,x)$. We have $\text{DLA} =
{\frak M}^\dagger_- \oplus {\frak M}_+ \oplus {\frak M}_-$. Note that the [*adjoint map*]{} $({{\rm ad\,}}A)B=[A,B]$ generates the Heisenberg motion on DLA, $\exp(it{{\rm ad\,}}A)B=e^{i t A} B e^{-i t A}$. In particular, the motion generated by $U^+_{n,k}$ has a beautiful structure. Let us write the self-adjoint Hamiltonian in a general form as $$H = \sum_{n,k} {{\textstyle{\frac{1}{4}}}}g_{n,k} \left(
U^{+}_{n,k} e^{i\gamma_{n,k}} + U^{+}_{n,-k} e^{-i\gamma_{n,-k}}
\right)
\label{eq:H}$$ using two sets of possibly time dependent real coefficients $g_{n,k}=g_{n,k}(t),\gamma_{n,k}=\gamma_{n,k}(t)$. Tedious but straightforward calculation, using algebra (\[eq:DLA\]), gives the action of ${{\rm ad\,}}H$ on two continuous sets of observables $W^+(y,x),W^-(y,x)$, $(y,x)\in \T^2$ which can be written in terms of two non-local ‘Schr" odinger operators’ $\H^\pm$ $$\begin{aligned}
({{\rm ad\,}}H)&&W^\pm(y,x) =
- {\textstyle\frac{1}{\hbar}}\H^\pm W^\pm(y,x), \label{eq:adH} \\
\H^+ = \sum_{n,k}\hbar g_{n,k}&\bigl(&
\sin(n\px\!-\!k\py)\sin(kx\!+\!ny\!-\!\gamma_{n,k})
\nonumber\\ \noalign{\vskip -0.18in}
-&& \sin(n\px\!+\!k\py)\sin(kx\!-\!ny\!-\!\gamma_{n,k})\bigr),\nonumber\\
\H^- = \sum_{n,k}\hbar g_{n,k}&\bigl(&
\cos(n\px\!-\!k\py)\cos(kx\!+\!ny\!-\!\gamma_{n,k})
\nonumber\\ \noalign{\vskip -0.18in}
+&& \cos(n\px\!+\!k\py)\cos(kx\!-\!ny\!-\!\gamma_{n,k})\bigr),\label{eq:H1}\end{aligned}$$ where $\hat{p}_{x,y} = -i\hbar\partial/\partial_{x,y}$ are momentum operators conjugate to $x,y$ with an ‘effective Planck constant’[@hb] $$\hbar = {{\textstyle{\frac{1}{2}}}}\epsilon. \label{eq:hbar}$$ Since Heisenberg dynamics generated by $H$ is closed on ${\frak M}_\sigma$, $({{\rm ad\,}}H){\frak M}_\sigma \subseteq {\frak M}_\sigma$, one may write a general time-evolving operator $A(t)\in{\frak M}_\sigma$ in terms of a complex-valued ‘Schr" odinger wave function’, in either ‘momentum’ $\Psi^A_{n,k}(t)$ or ‘position’ $\Psi^A(y,x;t)$ representation $$A(t) = \sum_{n,k} \Psi^A_{n,k}(t)^* W^\sigma_{n,k} =
\int_{\T^2}\!\!dy dx \Psi^A(y,x;t)^* W^\sigma(y,x).$$ By means of eq. (\[eq:adH\]) and the fact that $\H^\sigma$ is Hermitian on $L^2(\T^2)$ (which can be checked directly using the expressions (\[eq:H1\])) one can easily show that the Heisenberg evolution of the observable $A(t)$, $(d/dt)A(t) = i({{\rm ad\,}}H)A$, is [*fully equivalent*]{} to the Scr" odinger equation $$i\hbar\frac{d}{dt}\Psi^A(y,x;t) = \H^\sigma \Psi^A(y,x;t).
\label{eq:Sch}$$ governing time evolution of one particle on a torus $\T^2$ (OBI). The [*bilinear*]{} map $(H,A(t))\leftrightarrow (\H^\sigma,\Psi^A(y,x;t))$ is a central result of this Letter. To conclude a general exposition we make few remarks: (i) A non-trivial ‘classical limit’ $\hbar\rightarrow 0$ of OBI exists, being [*equivalent*]{} (\[eq:hbar\]) to the continuum field limit of the quantum spin chain model $\epsilon\rightarrow 0$, if $\hbar g_{n,l}$ (and not $g_{n,l}$ alone) are kept constant and finite. (ii) The operators $\H^+$ and $\H^-$ (\[eq:H1\]) commute $$[\H^+,\H^-] \equiv 0,
\label{eq:com}$$ and the Poisson bracket of the corresponding classical counterparts vanishes. (iii) As a consequence of the previous remark we find that OBI (\[eq:Sch\]) (and its classical limit) is [*integrable*]{}, $\H^{-\sigma}$ being the second integral of motion, provided the original spin-field Hamiltonian $H$ (\[eq:H\]) or OBI Hamiltonian $\H^\sigma$ is autonomous, i.e. $(\partial/\partial t) H\equiv 0$. However, one has a possibility of chaotic motion in classical limit and emergence of ‘quantum chaos’ when the problem is explicitly time-dependent, say that coefficients are periodic functions, $g_{n,k}(t+1)=g_{n,k}(t)$. In such case one integrates the evolution over one period of time and defines the unitary Floquet maps $U=\hat{\cal T}\exp(-i\int_0^1 dt H(t))$, $\U^\sigma=\hat{\cal T}\exp(-i\int_0^1 dt \H^\sigma(t))$. (iv) Temporal correlation functions of the quantum field problem are mapped (using eqs. (\[eq:scal\])) onto transition amplitudes of OBI $$(A(t)|B(t')) =
\cases{{\langle \Psi^A(t)|\Psi^B(t')\rangle} & $\sigma=+$,\cr
{\langle \Psi^A(t)|\hat{\cal P}_y\Psi^B(t')\rangle} &
$\sigma=-$,\cr}
\label{eq:CF}$$ where $\hat{\cal P}_y\Psi(y,x)=\Psi(y,x)-\Psi(-y,x)$. Therefore, ergodic properties of many-body dynamics on DLA are determined by the spectral properties of OBI: (a) Spin chain is [*quantum mixing*]{} in ${\frak M}_\sigma$, $\lim_{t\rightarrow\infty}(A(t)|B) = 0$, $A,B\in {\frak M}_\sigma$, iff the spectrum of OBI Hamiltonian $\H^\sigma$ (or of OBI Floquet map $\U^\sigma$) does not have (non-trivial) [*point*]{} component. (b) Spin chain is [*quantum ergodic*]{} in ${\frak M}_\sigma$, $\lim_{T\rightarrow\infty}T^{-1}\int_0^T dt (A(t)|B) = 0$, iff $0$ (or $1$) is [*not*]{} in the non-trivial point spectrum of $\H^\sigma$ (or $U^\sigma$). In autonomous case, $\partial H/\partial t\equiv 0$, the Hamiltonian $H$ and the [*trivial*]{} zero-frequency eigenstate, $\H^+\Psi^H=-\hbar \Psi^{[H,H]}=0$, $\Psi^H_{n,k}={{\textstyle{\frac{1}{8}}}}(g_{n,k}e^{-i\gamma_{n,k}}\!+\!g_{n,-k}e^{i\gamma_{n,-k}}
\!+\!g_{-n,k}e^{-i\gamma_{-n,k}}\!+\!g_{-n,-k}e^{i\gamma_{-n,-k}})$ should be excluded from ${\frak M}_+$ and $L^2(\T^2)$, respectively, i.e. $(A|H)=(B|H)=0$.
We apply the above results to work out two interesting examples. [*Example I:*]{} [*XX spin chain in spatially modulated quasi-periodic transversal magnetic field*]{} $\vec{h}_j=(0,0,h\cos(\epsilon j))$ (XXmh) $$H = H_{XX} + H_{mh} = J U^+_{1,0} +
{{\textstyle{\frac{1}{2}}}}h (U^+_{0,1} + U^+_{0,-1}).$$ Here the Heisenberg dynamics on DLA is governed by the following commuting one-body problems $$\begin{aligned}
\H^+ &=& \alpha \sin\px\sin y - \beta\sin\py\sin x,\label{eq:Hex}\\
\H^- &=& \alpha \cos\px\cos y + \beta\cos\py\cos x,\nonumber\end{aligned}$$ where $\alpha = 2\epsilon J=4\hbar J,\beta = 2\epsilon h=4\hbar h$. This models are directly related to the electron motion on 2d rectangular $a\times b$ lattice in a uniform perpendicular magnetic field $h'$ within the TB approximation [@Harper; @Sokoloff]. In the symmetric gauge $\vec{A} = {{\textstyle{\frac{1}{2}}}}h'(-y,x,0)$ the TB problem with the band energy ${\cal E}(\vec{K})=\frac{\alpha}{2}\cos(a K_1)
+\frac{\beta}{2}\cos(b K_2)$ reads $$\begin{aligned}
\H \Psi_{n,k} &=& {\textstyle\frac{\alpha}{2}}\left(
e^{i{{\scriptstyle{\frac{1}{2}}}}\epsilon k}\Psi_{n+1,k} + e^{-i{{\scriptstyle{\frac{1}{2}}}}\epsilon k}\Psi_{n-1,k}\right)
\nonumber\\
&+& {\textstyle\frac{\beta}{2}}\left(
e^{-i{{\scriptstyle{\frac{1}{2}}}}\epsilon n}\Psi_{n,k+1} + e^{i{{\scriptstyle{\frac{1}{2}}}}\epsilon n}\Psi_{n,k-1}\right)
\label{eq:TB}\end{aligned}$$ where $\epsilon=e_o a b h'/c_o\hbar_{\text{phys}}$ [@hb] is here the dimensionless magnetic flux thru lattice cell. We note that discrete indices $(n,k)\in\Z^2$ now label the position lattice $(na,kb)$ while continuous indices $(y,x)\in\T^2$ are the conjugate quasi-momenta. OBI Hamiltonians $\H^\pm$ can be written in terms of $\H$ and its [*time-reversal*]{} $\H^* = \H|_{h'\rightarrow -h'}$, namely $\H^\pm = \H \mp \H^*$, and hence $[\H,\H^*]=0$. Using a different, Landau gauge $\vec{A}=h'(0,x,0)$ the TB problem (\[eq:TB\]) can be re-written in terms of 1d Harper equation[@Harper] $${{\textstyle{\frac{1}{2}}}}\alpha (u_{n+1}+u_{n-1}) +
\beta \cos(n\epsilon-\vartheta)u_n = E u_n.
\label{eq:Harper}$$ Let us assume for the moment that $\alpha < \beta$. Then $u_n(\vartheta;E)=u_n$ is a unique exponentially localized eigenfunction (EF) of eq. (\[eq:Harper\]) which has a dense pure point spectrum, and $\Psi_{n,k}(\vartheta;E) =
\exp\left(i(\vartheta-{{\textstyle{\frac{1}{2}}}}\epsilon n)k\right) u_n(\vartheta;E)$ is a [*degenerate dense*]{} set of EFs of TB problem (\[eq:TB\]), $\H \Psi_{n,k}(\vartheta;E) = E\Psi_{n,k}(\vartheta;E)$, for a [*dense*]{} set of parameters $\vartheta$ [@Sokoloff]. Though $\H$ and $\H^*$ should have a common set of EFs, $\Psi_{n,k}(\vartheta;E)$ is not an EF of $\H^*$, neither it is in $L^2$ since it is [*extended*]{} in variable $k$. We search for such EF with an ansatz $\Phi_{n,k}(\vartheta;E,E')=\sum_j v_j \Psi_{n,k}(\vartheta+\epsilon j;E)$ and require $\H^* \Phi_{n,k}(\vartheta;E,E')=E'\Phi_{n,k}(\vartheta;E,E')$ yielding the Harper equation (\[eq:Harper\]) for coefficients $v_n=u_n(\vartheta;E')$. Thus we obtain a common set of EFs of $\H$ and $\H^*$ in terms of a ‘convolution’ of two 1d Harper functions $$\!\!\!\!\!\Phi_{n,k}(\vartheta;E,E')\!=\!
\sum_l u_{l+n}(\vartheta;E) u_l(\vartheta;E')
e^{i(\vartheta - \epsilon l - {{\scriptstyle{\frac{1}{2}}}}\epsilon n)k}
\label{eq:EF}$$ which is also a common set of EFs of $\H^\pm$ (\[eq:Hex\]) $$\H^\pm \Phi_{n,k}(\vartheta;E,E') = (E\mp E')\Phi_{n,k}(\vartheta;E,E').$$ The property $\Phi_{n,k}(\vartheta+\epsilon;E,E')=\Phi_{n,k}(\vartheta;E,E')$ suggests independence of EF on parameter $\vartheta$ provided $\epsilon/2\pi$ is irrational. If $\alpha > \beta$ localized EFs can be constructed analoglously by ‘duality transformation’ $n\leftrightarrow k,y\leftrightarrow x$. Thus, we found that OBI $\H^\pm$ have a dense pure point spectrum for $\alpha\ne\beta$, hence XXmh is non-mixing, non-ergodic and even [*completely integrable*]{}, namely ‘zero-energy’ eigenstates of $\H^\sigma$ are the images of a complete set of [*conserved charges*]{}, $Q_\sigma(E) = \sum_{n,k} \Phi^*_{n,k}(E,\sigma E) W^\sigma_{n,k}$, $[H,Q_\sigma(E)]\equiv 0$. However, time autocorrelation function $(A(t)|A)$ of a certain observable $A$ may still decay to zero provided the image function $\Psi^A$ is orthogonal to all (localized) EFs (\[eq:EF\]) of $\H^\pm$. Interestingly, this happens with the spin-current $j_s=W^+_{1,0}-W^+_{-1,0}$ if $\alpha < \beta$ (i.e. $J < h$), since EFs (\[eq:EF\]) have the following properties $\Phi_{n,k}(E,E')^* = \Phi_{n,k}(E',E) = \Phi_{-n,-k}(E,E')$ and $\Phi_{n,0}(E,E')^* = \Phi_{n,0}(E,E')$ (we put $\vartheta:=0$) implying $\Phi_{n,0}(E,E') = \Phi_{-n,0}(E,E')$. So we have ${\langle \Psi^{j_s}|\Phi(E,E')\rangle} \equiv 0$. This proves $(j_s(t\rightarrow\infty)|j_s)\rightarrow 0$ and non-ballistic spin-transport (vanishing [*spin stiffness*]{} $D_s:=\lim_{T\rightarrow\infty}(1/T)\int_0^T dt(j_s(t)|j_s)$) for $J < h$, while for $J > h$ we find in general ballistic transport ($D_s > 0$) since no similar symmetry exists for the other index $k$.
[*Example II:*]{} [*kicked*]{} XXmh model (kXXmh) with time-dependent Hamiltonian $$H(t) = J U^+_{1,0} + {{\textstyle{\frac{1}{2}}}}h (U^+_{0,1} + U^+_{0,-1})
\sum_m \delta(t-m).$$ One-period propagator from just after the kick $$U=\exp\left(-i{{\textstyle{\frac{1}{2}}}}h (U^+_{0,1} + U^+_{0,-1})\right)
\exp\left(-i J U^+_{1,0}\right)$$ is equivalent to Floquet quantum maps of two kicked OBI $$\begin{aligned}
\U^+ &=& \exp\bigl({\textstyle\frac{i\beta}{\hbar}}\sin\py\sin x\bigr)
\exp\left({\textstyle\frac{-i\alpha}{\hbar}}\sin\px\sin y\right),\label{eq:U1}\\
\U^- &=& \exp\bigl({\textstyle\frac{-i\beta}{\hbar}}\cos\py\cos x\bigr)
\exp\left({\textstyle\frac{-i\alpha}{\hbar}}\cos\px\cos y\right). \nonumber\end{aligned}$$ In the following we will consider only the map $\U^+$ since the space ${\frak M}_+$ contain physically more interesting observables. The Floquet evolution $\Psi^A(m) = \U^{+ m}\Psi^A(0)$ yielding Heisenberg evolution of observables $A(m)\in{\frak M}_+$ is in the ‘classical’ limit equivalent to a volume-preserving $(2\times2)$d map on $\T^2\times\R^2$ $$\begin{aligned}
x' &=& x + \alpha\cos p_x\sin y, \quad p'_y = p_y - \alpha\sin p_x\cos y,
\label{eq:clmap}\\
y' &=& y - \beta\cos p'_y\sin x',\quad p'_x = p_x + \beta\sin p'_y\cos x',
\nonumber\end{aligned}$$ which is non-integrable and (almost) fully chaotic for sufficiently large kick parameters, $\alpha,\beta\gg 1$. Interesting question is now if and when the dynamics of kXXmh is quantum mixing and how it corresponds to dynamics of the ‘classical map’ (\[eq:clmap\]) as $\hbar={{\textstyle{\frac{1}{2}}}}\epsilon\rightarrow 0$. This problem has been approached numerically by iterating the one-body Floquet map $\U^+$ on a finite (truncated) momentum space $(n,k)\in\{-N/2\ldots N/2\}^2$. The position states are then discretized as $x_j = s j,y_j = s j, s=2\pi/N$. The truncated Floquet map $\U^+$ can be efficiently implemented by means of Fast Fourier Transformation (FFT), namely if $F$ is 1d FFT on $N$ sites then $N^2\times N^2$ Floquet matrix is de-composed as $(F^{-1}\otimes 1)(\text{diag\,}C_{n,k})(F\otimes F^{-1})
(\text{diag\,}D_{n,k})(1\otimes F)$, with diagonal matrices $C_{n,k}=\exp\left(i(\beta/\hbar)\sin s n \sin \hbar k\right)$ and $D_{n,k}=\exp\left(-i(\alpha/\hbar)\sin \hbar n\sin s k\right)$, requiring $\sim 4 N^2 \log_2 N$ computer operations per time step. In order to avoid recurrences of quantum probability due to finiteness of momentum space we use an absorbing boundary in momentum space, namely after each iteration of the truncated Floquet map we multiply the wave-function by a box-window, $\Psi_{n,k}(m)\rightarrow
\theta(N/2-\alpha/\hbar-|n|)\theta(N/2-\beta/\hbar-|k|)\Psi_{n,k}(m)$, Convergence to true dynamics on a torus has been checked by comparing results for different truncations, say $N$ and $N/2$ (we went up to $N=2^{14}$).
In Fig.1 we show numerical results for the auto-correlation function of the spin current $C(m) = (j_s(m)|j_s(0))$ while similar, compatible results have been obtained for the time-correlations of other observables. (i) For sufficiently large kick parameters $\alpha,\beta$ the classical map (\[eq:clmap\]) is strongly chaotic and mixing exhibiting normal diffusion in momentum plane $(p_x,p_y)$. However, kXXmh is not exactly mixing for any finite $\hbar$: $|C(m)|$ is rapidly (possibly [*exponentially*]{}) decreasing down to some value $C^* = \overline{|C(m)|}$ where it saturates. When we decrease $\hbar$, $C^*$ decreases proportionally, $C^* \propto \hbar$, and so in the ‘quasi-classical’/continuum limit $\hbar={{\textstyle{\frac{1}{2}}}}\epsilon\rightarrow 0$ the [*point*]{} spectrum of $\U^+$ vanishes and kXXmh approaches mixing behaviour in accordance with the map (\[eq:clmap\]). (ii) For smaller but still finite values of $\alpha,\beta$ the classical map enters into the regime of KAM quasi-integrability with invariant tori suppressing the diffusion of momenta $(p_x,p_y)$. Correspondingly, kXXmh is non-mixing and $C^* \sim 1$ for any value of $\hbar$. In this regime, $C(m)$ is very weakly $\hbar-$dependent. In both regimes, (i) and (ii), the square widths of the ‘wavepackets’ ${\langle \Psi^{j_s}(m)|}\hat{p}^2_{x,y}{|\Psi^{j_s}(m)\rangle}$ have been found to be [*uniformly*]{} increasing in time and limited only by the size of the truncated momentum space $N$. This rules out the possibility of [*quantum localization*]{} and existence of [*pure point*]{} spectrum, and indicates coexistence of [*point*]{} and [*continuous*]{} spectrum for any finite $\hbar$ (and finite $\alpha,\beta$), a situation similar to (possibly related) 1d kicked Harper model [@KH]. In the limit $\alpha,\beta\rightarrow 0$, the continuous spectral component vanishes and we recover integrable XXmh model with [*pure point*]{} spectrum as discussed above. The quantum correlation function $C(m)$ seem to follow the quasi-classical propagator only up to logarithmically short time, namely we found empirically that deviation (when it is small) increases exponentially $|C(m)-C_{\hbar\rightarrow 0}(m)|\approx 0.022 \hbar^2 e^{\lambda m}$ with $\lambda\approx 0.59$ for $\alpha=3,\beta=0.75$ and $\lambda\approx 1.1$ for $\alpha=6,\beta=1.5$.
[*Conclusions.*]{} In a specific $\infty$d class of (Pauli spin, or spinless fermion) quantum field models in 1d, the Heisenberg time evolution in two disjoint $\infty$d linear subspaces of essential field observables has been shown to be formally equivalent to the Schr" odinger dynamics of a class of one-body image problems on a 2d torus (or 2d lattice). Autonomous models of this class were found to be completely integrable, pointing out a novel class of integrable one-body problems (\[eq:H1\],\[eq:com\]). For example, dynamics of XX chain in a static quasiperiodic transversal field has been solved in terms of Harper equation[@Satija]. However, time-dependent (e.g. periodically kicked) models of our class behave in a non-integrable fashion being mapped onto one-body problems with chaotic classical limit. It seems that spatial modulation is crucial to break integrability since spin chain kicked with [*homogeneous*]{} transversal field remains completely integrable as found in [@KI]. In the [*contunuum field limit*]{} our kicked spin chain model (kXXmh) has been demonstrated to undergo a (phase) transition from mixing to non-mixing dynamics (similar to a transition found in [@KtV]), as its one-body counterpart in the [*classical limit*]{} undergoes a stochastic transition from chaotic to quasi-regular motion. This is an interesting link between quantum field theory and chaotic dynamics and should inspire future research in this direction. Such approach to long-time dynamics of certain (non-integrable) quantum many-body systems, since it makes time evolution formally equivalent to ‘quantum chaos’ in few degrees of freedom, overcomes the traditional problems due to huge Fock space in thermodynamic limit. Financial support by the Ministry of Science and Technology of R. Slovenia is acknowledged.
G. Casati and B.V. Chirikov, eds., ‘Quantum Chaos: Between Order and Disorder’, (Cambridge U.P. 1994).
G.J.-Lasinio, C.Presilla, Phys.Rev.Lett.[**77**]{}, 4322 (1996).
G. Montambaux, D. Poilblanc, J. Bellisard, C. Sire, Phys.Rev.Lett.[**70**]{}, 497 (1993); D. Poilblanc, T. Ziman, J. Bellisard, F. Mila, G. Montanbaux, EPL [**22**]{}, 537 (1993).
T. Prosen, Phys.Rev.Lett. [**80**]{} (1998) 1808; J.Phys.A: Math.Gen.[**31**]{} (1998) L645; e-print cond-mat/9808150.
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abstract: 'Experimental data on thin films of cylinder-forming block copolymers (BC) – free-standing BC membranes as well as supported BC films – strongly suggest that the local orientation of the BC patterns is coupled to the geometry in which the patterns are embedded. We analyze this phenomenon using general symmetry considerations and numerical self-consistent field studies of curved BC films in cylindrical geometry. The stability of the films against curvature-induced dewetting is also analyzed. In good agreement with experiments, we find that the BC cylinders tend to align along the direction of curvature at high curvatures. At low curvatures, we identify a transition from perpendicular to parallel alignment in supported films, which is absent in free-standing membranes. Hence both experiments and theory show that curvature can be used to manipulate and align BC patterns.'
author:
- Giang Thi Vu
- 'Anabella A. Abate'
- 'Leopoldo R. Gómez'
- 'Aldo D. Pezzutti'
- 'Richard A. Register'
- 'Daniel A. Vega'
- Friederike Schmid
title: Curvature as a guiding field for patterns in thin block copolymer films
---
Because of their ability to self-assemble into well-defined periodic nanostructures, block copolymers (BC) are attracting great interest as potential template materials for cost-effective nanofabrication techniques [@Harrison2000; @Segalman2005; @Bita2008; @Ruiz2008; @Singh2012; @Marencic2010b; @Vega2013; @Garcia2015; @Abate2016]. With BC systems, one can produce high-resolution patterns with tunable wavelength using traditional processing techniques. This offers promising perspectives for applications in scalable nanoscale devices. However, one frequent problem with the self-assembly approach is lack of long-range order due to pattern undulations and defects, e.g., dislocations, disclinations, or grain boundaries [@Harrison2000; @Harrison2002; @Vega2005; @Nagpal2012; @Hur2015]. Numerous methods to produce patterns with well-defined orientational and positional order have been proposed, such as shear alignment [@Angelescu2004; @Kim2014b; @Davis2015], alignment in electric fields [@Amundson1993; @Morkved1996; @Mansky1998], zone annealing [@Berry2007; @Yager2010], or grapho- and chemo-epitaxy, where surface interactions and confinement effects are exploited to order patterns [@Kim2003; @Gomez2009; @Garcia2014; @Garcia2015; @Luo2015; @Sundrani2004; @Yager2010; @Marencic2010b; @Hamley2009; @Angelescu2004; @Vega2013] or to control defect positions [@Nelson2002].
Here, we analyze another possible source of alignment, the [*geometry*]{} in which the system is embedded. Experiments and simulations on curved systems have indicated that the pattern configurations are affected by both intrinsic and extrinsic geometry. Even in Euclidean systems, a strong coupling between patterns and curvature seems to drive the equilibrium configurations and the coarsening process [@Gomez2009; @Vega2013; @Matsumoto2015; @Pezzutti2015].
As a first step towards a more quantitative understanding of the nature of the coupling between BC thin films or membranes and curvature, in the present paper, we study curved monolayers of cylinder-forming BC systems by complementary experiments, symmetry considerations, and self-consistent field theory (SCFT) calculations [@Abate2016; @Mueller2005]. We consider two types of model systems: a) free-standing BC membranes and b) BC thin films deposited onto a curved substrate.
The geometric features of a 2D curved surface can be characterized in terms of a shape operator ${\bf S}$, which has two Eigenvalues $k_{1,2}=1/R_{1,2}$ corresponding to the inverse maximal and minimal radii of curvature $R_i$ \[see Supplemental Material (SM) for more details\]. The determinant and the trace of ${\bf S}$ define the Gaussian curvature $K=k_1
k_2$ and twice the mean curvature $2 H = k_1+k_2$, respectively [@Deserno]. The experimental systems studied here have a non-Euclidean metric ($K \neq 0$, free-standing membrane) or a Euclidean metric with zero Gaussian curvature ($K
= 0$, curved substrate).
In both systems we employ the same BC system, a cylinder-forming polystyrene-block-poly(ethylene-alt-propylene) diblock (PS-PEP 4/13) [@Marencic2010]. The number-average block molecular weights for the BC are 4.3 kg/mol for PS and 13.2 kg/mol for PEP. In bulk, the PS blocks arrange in hexagonally packed cylinders embedded in the PEP matrix. In thin films the PS cylinders adopt a configuration parallel to the film surface. The center-to-center spacing of the cylinders is $d_{sm}=21$ nm. Thin films of thickness $\sim 30$ nm are prepared by spin-coating from a 1 wt. $\%$ solution in toluene, a good solvent for both blocks. Order is induced by annealing at a temperature $T$, above the glass transition of the PS block ($T_g\sim 330$K) and below the order-disorder transition temperature $T_{ODT}=417$K of the BC. Details of the preparation of the experimental systems are given in SM. To obtain free-standing membranes, the films are first annealed on a flat substrate, then further cooled down below the $T_g$ of the PS block and finally lifted off and redeposited on a transmission electron microscopy (TEM) grid. During this process, the system retains the symmetry, average inter-cylinder distance, and structure of defects established during annealing. To obtain supported films, the BCs are directly spin-coated onto curved substrates and the thermal annealing process is monitored by atomic force microscopy (AFM) at selected time points.
Fig. \[fig:fig1n\] shows an AFM image of a free-standing film where height and cylinder locations are measured simultaneously. The light and dark regions correspond to PS-rich and PEP-rich regions, respectively. After releasing the membrane from the confining substrate, it develops a non-Euclidean shape to relieve the elastic energy of topological defects that have survived the thermal annealing. The shape results from a competition between the strain field of the defects, the bending energy associated with the curvature of the membrane, and the membrane tension [@Matsumoto2015]. Fig.\[fig:fig1n\] (d) shows the correlation between the orientation of the underlying pattern and the local orientation of membrane wrinkles. Although the different defects impose competing out-of-plane deformations, one clearly notices that wrinkles have a tendency to be oriented either perpendicular ($\theta = 0$) or parallel ($\theta = \pm \pi/2$) to the underlying cylinders, suggesting that the bending energy is anisotropic and coupled to the liquid crystalline order of the BC \[see also Figs. \[fig:figS1\], \[fig:figS2\] in SM\].
![\[fig:fig1n\] a) Phase-height AFM image of a free-standing thin film (image size: 2.6 $\mu$m $\times$ 2.6 $\mu$m). b) Local mean curvature for the membrane shape. Here the vectors $u_{1,2}$ indicate the directions of the principal curvatures and $\textbf{N}$ is the normal vector to the membrane surface (image size: 1.8 $\mu$m $\times$ 1.8 $\mu$m, region indicated by dashed lines in panel a; $H_{max}=-H_{min}=3.84 \times 10^{-3}$ nm${}^{-1}$). c) Local orientation of the director field $\alpha$ of the pattern with regard to the x-axis. d) Histogram showing the local distribution of angles $\theta=\alpha-\beta$ between $\alpha$ and the local orientation of the membrane wrinkles $\beta$ \[see also Fig. \[fig:figSB\] in SM\].](Fig1New.jpg){width="8.5cm"}
![\[fig:fig2n\] Top panels: 3D AFM phase-height images of the BC thin film on a curved substrate after annealing at T=373K. Panels a) and b) show the pattern configuration after 90 min (image size: 2.0 $\mu$m $\times$ 1.5 $\mu$m) and 3.5 h of thermal annealing (image size: 1.0 $\mu$m $\times$ 1.25 $\mu$m), respectively. Height scale: 80 nm from crest to valley, $H_{max}^2 = 6.25$ $\mu$m${}^{-2}$). The presence of a dislocation and $+1/2$ disclinations has been emphasized with a rectangle and circles, respectively. Bottom panels: c) Local orientation of the smectic pattern (color map indicated at the bottom). d) Histograms showing the distribution of angles $\theta$ between the local cylinder orientation and the direction of curvature at two different annealing times. ](Fig2New.jpg){width="8.cm"}
A similar observation is made for the thin films on curved substrates. Fig. \[fig:fig2n\] shows AFM phase and height-phase images of the BC thin film deposited onto a curved substrate. Right after the spin coating, the pattern is characterized by a very small orientational correlation length ($\xi_2 \sim 20 $ nm) and a high density of defects. During annealing at T=373 K, the system orders via annihilation of dislocations and disclinations. Already at an early stage of annealing, the thin film becomes unstable and dewets at the regions with the highest curvature (Fig. \[fig:fig2n\]). Upon further annealing, the order in the system increases and the pattern develops a clear preferential orientation with regard to the substrate. Fig.\[fig:fig2n\] shows that the PS cylinders tend to align perpendicular to the crest of the substrate. Thus the topography of the substrate seems to act as an external field that breaks the azimuthal symmetry \[see also Fig.\[fig:figS3\] in SM\]. Note that the equilibrium configuration obtained here is opposite to that predicted in previous theories for curved columnar phases [@Santangelo2007; @Kamien2009], where it was assumed that bending along the cylinder direction is energetically more costly than bending in the perpendicular direction [@Kamien2009].
The phenomena described above can be analyzed using general symmetry considerations. The curvature free energy per area of [*isotropic*]{} fluid-like membranes can be expanded in the invariants of the shape operator ${\bf S}$ as $F_{HC}=\frac{\kappa_b}{2} (2H-c_0)^2 + \kappa_g K$, where $\kappa_b$ and $\kappa_g$ are the bending and Gaussian rigidity, respectively, and $c_0$ is the spontaneous curvature [@Helfrich1973; @Canham1970]. Here we consider anisotropic [*nematic*]{} membranes with in-plane order characterized by a director ${\bf n}$ (the orientation of the cylinders), thus additional terms become possible. Including all terms up to second order in ${\bf S}$ that are compatible with the in-plane nematic symmetry, i.e., $({\bf
n} \cdot {\bf S} \cdot {\bf n})$, $({\bf n} \cdot {\bf S} \cdot {\bf n})^2$ [@powers1995; @biscari2006], and $({\bf S} \cdot {\bf n})^2$ [@oda1999; @chen1999], we can derive the following expression for the anisotropic part of the curvature free energy per area \[see SM\]: $$\label{eq:ani}
%
F_{\mbox{\tiny ani}}
= -\frac{\kappa'}{2} (k_1 - k_2) (2 H -c_0') \cos(2 \theta)
- \frac{\kappa''}{2} (H^2 - K) \cos(4 \theta) .
%$$ Here $\theta \in [0:\pi/2]$ denotes the angle between the director and the direction of largest curvature $k_1$ ($|k_1| > |k_2|$), and $\kappa',
\kappa''$, $c_0'$ are anisotropic elastic parameters. In symmetric membranes, $c_0'$ vanishes ($c_0'=0$). We emphasize that Eq.(\[eq:ani\]) gives the generic form of the free energy of curved nematic films up to second order in the curvatures, which should be generally valid regardless of molecular details. For $\kappa''>0$, the second term describes a quadrupolar coupling between the curvature tensor and the director that favors [*two*]{} directions of preferential alignment of the director ${\bf n}$ along the two principal directions of curvature. Such a competition between two stable/metastable aligned states was also predicted in other continuum models for nematic shells [@Napoli2012; @Napoli2012b]. The first term selects between the two directions.
The results of the symmetry analysis are compatible with the experiments: As discussed above, in the BC membranes, wrinkles form preferentially parallel or perpendicular to the director \[Fig. \[fig:fig1n\] (d)\]. Similarly, in the thin films, the distribution of local cylinder orientations $\theta$ is bimodal at early annealing time (30 min), with two characteristic peaks separated by $\sim \pi/2$ \[Fig. \[fig:fig2n\](d)\]. During the first stage of coarsening, the parallel and perpendicular configurations compete. After long annealing times, $C_{\perp}$ dominates, and the histogram becomes sharply peaked at the orientation $\theta=0$, suggesting $\kappa'>0$ in Eq. (\[eq:ani\]). We note, however, that supported films are asymmetric and hence the spontaneous curvature parameter $c_0'$ will very likely not vanish, in which case Eq. (\[eq:ani\]) predicts that the preferred orientation switches from $\theta=0$ to $\theta=\pi/2$ in a region of very small curvatures $k_1 \in [0:c_0']$. We will discuss this further below.
In order to obtain a more quantitative theoretical description, we use SCFT [@Matsen2002; @Mueller2005] to study the two systems considered in the experiment, the free-standing membrane and the curved supported thin film. We consider a melt of asymmetric $AB$ diblock copolymer molecules with degree of polymerizaton $N$ and statistical segment length $b$ at temperature $T$ confined to a curved film of thickness $\epsilon$ by two coaxial cylindrical surfaces (see schematics in Fig.\[fig:fig3n\](a), \[fig:fig4n\](a), and \[fig:fig4n\](b). Periodic or tilted periodic boundary conditions \[see SM\] are applied in the two in-plane directions.
In the following, lengths and energies are given in units of $R_g^2 =
\frac{1}{6} N b^2$ and $G k_B T$, respectively, where $k_B$ is the Boltzmann constant and $G =\rho_c \: R_g^3$ is the rescaled dimensionless copolymer density in the bulk. The incompatibility between the blocks is specified by the product $\chi N$, where $\chi$ is the Flory-Huggins parameter. Here we use $\chi N=20$ and $f=0.7$ to match the experimental values ($f$ is the volume fraction of the $A$-block). Our calculations are done in the grand canonical ensemble with the chemical potential $\mu=(2.55 + \ln G) k_BT$ [@Abate2016; @fn1] and inverse isothermal compressibility $\kappa N=25$ [@Helfand1975; @Pike2009; @Detcheverry2010]. Monomers $\alpha = A,B$ close to a surface experience a surface field, which we characterize in terms of the surface energy per area $\gamma_\alpha$ of a fluid of $\alpha$-monomers in contact with the same surface \[see SM\], given in units $\hat{\gamma} = G k_B T/R_g^2$. To account for the experimental fact that cylinders align parallel to the film, the interaction parameters are chosen such that majority A-blocks preferentially adsorb to the surface, i.e., $\gamma_A < \gamma_B$. In planar films, the copolymers then self-assemble into aligned cylinders with a spacing $\lambda= 3.6 R_g$. Matching this with the value $d_{sm} = 21$ nm observed experimentally, we can identify $R_g \approx 5.8$ nm [@fn3] and hence $G = 5.77$ for our experimental systems (assuming an average copolymer density of 0.861 g/cm${}^3$ at 363K).
\[t\] ![\[fig:fig3n\] (a) Schematic representation of curvature radius $R_m$ in free-standing membranes. (b) Density profiles from SCFT for the parallel (left) and perpendicular (right) configurations $C_\parallel$ and $C_\perp$ at $R_m=9Rg$. (c) Free energy per area as a function of inverse curvature radius $R_g/R_m$ for the $C_{||}$ and $C_{\perp}$ configurations. Inset shows the free energy shift per area as a function of angle $\theta$ between the cylinders and the direction of curvature relative to the $C_{\perp}$ configuration ($\theta = 0$) at $R_m = 50 R_g$. ](Figure3.png "fig:"){width="8.2cm"}
We first consider free-standing membranes, which we model as a symmetric film with surface interaction energies $\gamma_A N = - 24 \hat{\gamma}$ and $\gamma_B N = - 23 \hat{\gamma}$. We calculate the free energy per area as a function of the curvature radius $1/R_m$ of the mid-surface of the film (see Fig. \[fig:fig3n\](a) for the two cases where cylinders are aligned parallel or perpendicular to the curvature ($C_\parallel$, $C_\perp$, see Fig. \[fig:fig3n\](b). In each case, the film thickness $\epsilon$ and the wavelength of the characteristic pattern are optimized to obtain the lowest free energy state. Fig. \[fig:fig3n\](b) shows the resulting density profiles for the parallel and perpendicular configurations in a system with a relatively large curvature ($R_m=9Rg$). The differences are small, indicating that curvature affects neither the position of the cylinder with regard to the plane of symmetry, nor the segregation strength. The optimum inter-cylinder spacing is $\lambda \sim 3.6 R_g$, which is slightly smaller than the bulk value, $\lambda_{bulk} \sim 3.7 R_g$. The ratio, $\lambda /\lambda _{bulk} \sim 0.97$, is in good agreement with SCFT calculations and experiments on flat substrates, where it was found that in thin films the unit cell is stretched perpendicular to the plane of the film resulting in lateral distances smaller than those in bulk [@Knoll2007; @Abate2016]. The optimal thickness is $\epsilon \sim
3.5R_g$ for both the parallel and perpendicular configurations \[see Fig.\[fig:figS7\] in SM\]. None of these features appears to be severely affected by the curvature within the range of curvatures explored here.
The behavior of the free energy per area for the two configurations is shown in Fig. \[fig:fig3n\](c). The perpendicular orientation is clearly favored. Furthermore, in agreement with Eq. (\[eq:ani\]), both $C_{\perp}$ and $C_{||}$ represent local free energy minima with respect to variations of the angle $\theta$ between cylinders and the direction of curvature, (see inset of Fig. \[fig:fig3n\]c). Since the free energy grows almost quadratically with the curvature, ${F}/{A}={\kappa}/{2R_m^2}$, we can calculate bending stiffness parameters for the $C_{||}$ and $C_{\perp}$ configurations. By fitting the free energy per area up to a quadratic order of the mean curvature, we obtain $\kappa_{||}=(1.056 \pm 0.002)G k_BT$ and $\kappa_{\perp}=(0.376\pm 0.002)G k_BT$ for the parallel and perpendicular configurations, respectively. Comparing this with Eq. (\[eq:ani\]) and using $k_1 = \pm 1/R_m$, $k_2 = 0$, we can deduce $\kappa'=(\kappa_{||} -
\kappa_\perp) = 0.68 G k_B T \approx 4 k_B T$, which corresponds to $\kappa'
\approx 1.6 \times \: 10^{-13}$ erg at room temperature. In contrast, the total bending energy of the membranes has been estimated to be of order $\kappa_b
\sim 10^{-9}$ erg, which is much higher due to the large contribution of the glassy PS block [@Matsumoto2015]. Hence, the influence of $\kappa'$ on the membrane shapes is presumably negligible.
\[t\] ![\[fig:fig4n\] (a,b) Schematic representation of supported thin films (green) on curved substrates (gray). (c,d) Free energy difference per area $\Delta F = (F_{||} - F_{\perp})/A$ of supported thin films with parallel ($C_{||}$) vs. perpendicular ($C_{\perp}$) orientation, versus curvature, for two sets of surface interaction energies I (top) and II (bottom) as described in the main text. Blue shaded regions highlight curvature regimes where the parallel orientation is more favorable ($\Delta F < 0$). (e,f) Corresponding curves for the free energy per area (red) and thickness (green) in the perpendicular configuration $C_\perp$.](Figure4.png "fig:"){width="7.8cm"}
The situation is different when looking at copolymer ordering in supported films, where the curvatures are kept fixed and energy differences of order $k_B
T$ do significantly influence the selection of the pattern orientations.
Thin films differ from membranes in two respects. First, the reference surface which is kept fixed during the free energy minimization is the interface between the substrate and the film (not the mid-plane of the film as in membranes), and second, the interaction energies may be different at the substrate and air interfaces. Fig. \[fig:fig4n\](c),\[fig:fig4n\](d) shows results for $\gamma_B^{a,s}= -6 \hat{\gamma}$ and two representative parameter sets for $\gamma_A^{a,s}$: (I) $\gamma_A^s N = -10 \hat{\gamma},
\gamma_A^a N = -20 \hat{\gamma}$, and (II) $\gamma_A^s N = -24 \hat{\gamma},
\gamma_A^a N = -10 \hat{\gamma}$, where superscripts $s$ and $a$ denote “substrate” and “air”, respectively. In both cases, the perpendicular configuration is more favorable at large curvatures. At small curvatures, however, there exists a small regime where parallel configurations have lower free energy. This is in agreement with our symmetry considerations (see above, Eq. (\[eq:ani\])) and also with the experimental observations. Indeed, Fig. \[fig:fig2n\](b) and Fig. \[fig:figS1\] in SM suggest that the locally preferred orientation switches from perpendicular to parallel in a region around the inflection point of the surface profile (green shaded areas in Fig. \[fig:fig2n\]), and this induces defects in that region. Hence curvature can be used not only to orient patterns, but also to generate defects at specific regions in space.
Fig. \[fig:fig4n\] also shows the behavior of the free energy and the minimum-energy thickness as a function of mean curvature for the $C_{\perp}$ configuration. For $H\ge 0$, the free energy increases as the curvature increases, indicating that the thin film is likely to become unstable and dewet from the substrate, also in good agreement with the experimental data shown in Fig. \[fig:fig2n\]. Conversely, for $H<0$ the film remains stable, since the free energy decreases as the curvature increases. In the experiments (Fig. \[fig:fig2n\]), it can be observed that the thin film dewets at the region with the highest curvature, where $H_{max}R_g \sim 0.015$. These results are in good agreement with recent experiments on curved substrates by Park and Tsarkova [@Park2017], who also found dewetting for $H>0$ and thin film thickening for $H<0$ in agreement with Fig.\[fig:fig4n\](e),\[fig:fig4n\](f) (green curves).
In conclusion, we have shown through experiments, symmetry considerations, and SCFT calculations that curvature can be employed as a guiding field to produce well-ordered patterns. The SCFT calculations provide a rough estimate of the equilibrium configuration for curved systems and predict dewetting in regions with high local positive curvature $H>0$. From a technological perspective, our results indicate that through appropriate control over the surface interactions, it should be possible to prevent dewetting while keeping a geometric field with sufficient strength to guide order.
Acknowledgements
================
We gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft (Grant Schm 985/19 and SFB TRR 146), the National Science Foundation MRSEC Program through the Princeton Center for Complex Materials (DMR-1420541), the Universidad Nacional del Sur, and the National Research Council of Argentina (CONICET). The SCFT calculations were done on the high performance computing center MOGON in Mainz. PS-PEP 4/13 was synthesized by Dr. Douglas Adamson.
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Supplementary Information on:\
Curvature as a guiding field for patterns in thin block copolymer films
=======================================================================
This supplementary material provides additional information about the experimental systems and details on the theory and the SCFT calculations.
Experimental Systems
--------------------
### System Preparation
The polystyrene-block-poly(ethylene-alt-propylene) diblock (PS-PEP 4/13) copolymer was synthesized through sequential living anionic polymerization of styrene and isoprene followed by selective saturation of the isoprene block (see ref. [@Marencic2010] for details). Thin films of thickness $\sim 30$ nm are prepared by spin-coating from a 1 wt. $\%$ solution in toluene.
To obtain a free-standing membrane, we first spin coat a monolayer of BC cylinders onto a 50 nm thick flat layer of sucrose deposited onto a silicon wafer, then thermally anneal it at $T$=363K until a prescribed orientational correlation length of $\xi_2 \sim 200$ nm is obtained [@Matsumoto2015], and finally cool it to room temperature. The sucrose layer is then used as a sacrificial layer to float the thin film off the substrate and onto the surface of water, and subsequently redeposit it as a free-standing membrane on a transmission electron microscopy (TEM) grid (grid spacing $25 \, \mu$m $> \xi_2 >
d_{sm}$). As the film is lifted off at a temperature well below the glass transition temperature of the PS block, the system retains the symmetry, average inter-cylinder distance, and structure of defects established during annealing. The pattern order is mainly disrupted by $\pm \frac{1}{2}$ disclination multipoles.
To prepare the substrate, we employ a solvent-annealing technique on a photoresist array of trench patterns deposited onto a silicon nitride wafer. Details about the method of sustrate preparation can be found elsewhere [@Vega2013]. It yields Gaussian-like smooth substrates with a pitch of 2.2 $\mu$m and crests with maximum height of 80 nm. The largest mean curvature of the substrate is found at the crests, where $H=H_{max}\sim 2.5 \,
\mu$m${}^{-1}$.
The thin films are imaged using a Veeco Dimension 3000 atomic force microscope (AFM) in tapping mode. The spring constant of the tip (uncoated Si) is $\sim$ 40 N/m and its resonant frequency is 300 kHz.
### Membrane topography
In order to determine the local curvatures, the metric tensor for the membrane was obtained at room temperature from the AFM height profiles. Through AFM we parametrize the surface in the Monge gauge. In this representation, the coordinates of each point $\textbf{r}$ are expressed as $\textbf{r}=(x,y,h(x,y))$, where $x$ and $y$ are planar coordinates and $h(x,y)$ the out-of-plane displacement. The metric tensor can then be calculated as $g_{ij} = \delta_{ij} + h_i h_j$ and the shape tensor as
$$\begin{aligned}
{\bf S} & = & \frac{1}{(1+h_x^2 + h_y^2)^{3/2}}
\\ \nonumber &&
\left( \begin{array}{cc}
(1+h_y^2) h_{xx} - h_x h_y h_{xy} & (1+h_x^2) h_{xy} - h_{xx} h_x h_y \\
(1+h_y^2) h_{xy} - h_x h_y h_{yy} & (1+h_x^2) h_{xy} - h_{xy} h_x h_y
\end{array} \right).\end{aligned}$$
The Eigenvalues of ${\bf S}$ give the principal curvatures. Standard methods are employed to determine the metric tensor and the principal curvatures at each point on the membrane surface [@Matsumoto2015].
Figure \[fig:figSA\] shows the direction of the principal minimum and maximum curvatures. Once these directions and the principal radii of curvature $R_{1,2}$ have been determined, the geometric properties of the membrane shape can be obtained. To determine the correlation between the pattern orientation $\alpha$ and membrane distortions, we determine the local orientation of the membrane wrinkles $\beta$ relative to the x-axis from the distribution of the maximum main curvature $k_1$. The wrinkle orientation $\beta$ was obtained by measuring the local gradient of $k_1$ (see Fig. \[fig:figSB\]), i.e. $\tan{(\beta)}=\nabla_y K_1 / \nabla_x K_1$.
![\[fig:figSA\] [Unit vectors indicating the directions of the principal maximum (red lines) and minimum (light blue lines) curvatures.]{}](FigureSA.jpg){width="8.5cm"}
{width="15cm"}
Figs. \[fig:figS1\] and \[fig:figS2\] show the mean and Gaussian curvatures for a free-standing cylinder-forming BC thin film membrane. Here a semitransparent mask with the maps for $H$ and $K$ was applied to also show the block copolymer texture (70$\%$ transparency).
Note the wrinkled topography of the membrane and the coupling with the smectic-like texture of the BC system.
![\[fig:figS1\] Height-phase AFM image of a freestanding thin film overlapped with mean curvature. ($H_{max}=-H_{min}=3.84 \times 10^{-3}$ nm${}^{-1}$; image size: 2.6 $\mu$m $\times$ 2.6 $\mu$m).](FigureS1.jpg){width="8.5cm"}
![\[fig:figS2\] Height-phase AFM image of a freestanding thin film overlapped with Gaussian curvature. ($K_{max}=-K_{min}=1.2 \times 10^{-5}$ nm${}^{-2}$; image size: 2.6 $\mu$m $\times$ 2.6 $\mu$m).](FigureS2.jpg){width="8.5cm"}
### Thin films on curved substrates
Fig. \[fig:figS3\] emphasizes the coupling between the pattern orientation and the mean curvature of the substrate. As here the Gaussian curvature is zero, the substrate is topologically equivalent to flat space. Thus, while for a strictly 2D system no coupling can be expected, the finite thickness of the film leads to an interaction that penalizes those configurations that involve an inter-cylinder elastic distortion.
![\[fig:figS3\] (a) 3D AFM phase-height image of the BC thin film lying on a curved substrate after 3.5 h of thermal annealing at $T$=373K. (image size: 2.5 $\mu$m $\times$ 1.0 $\mu$m).](FigureS3.jpg){width="8cm"}
Theory: Symmetry considerations.
--------------------------------
In its own Eigensystem, the shape tensor ${\bf S}$ can be written as ${\bf S} = \sum_{i=1}^2 k_i {\bf u}_i \otimes {\bf u}_i$, where $k_i$ are the principal curvatures and ${\bf u}_i$ the corresponding Eigenvectors, where we can choose $|k_1|>|k_2|$ without loss of generality. If a membrane or thin film has in-plane order characterized by a director field ${\bf n}$, the curvature free energy per area no longer has to be rotationally symmetric, and it may contain additional terms of the form $({\bf n} \cdot {\bf S} \cdot {\bf n})$, $({\bf n} \cdot {\bf S} \cdot {\bf n})^2$, and $({\bf n} \cdot {\bf S})^2$. We write the contribution of these terms to the curvature free energy per area in the general form $$F_{\bf n} = A {\bf n} \cdot {\bf S} \cdot {\bf n}
- B ({\bf n} \cdot {\bf S} \cdot {\bf n})^2
- C ({\bf n} \cdot {\bf S})^2.$$ The product $({\bf n} \cdot {\bf u}_1)$ defines the angle $\theta$ between the director and the direction of largest curvature [*via*]{} $({\bf n} \cdot {\bf u}_1)^2 = \cos^2 \theta$. Inserting this and using ${\bf n} = ({\bf n} \cdot {\bf u}_1) {\bf u}_1 + ({\bf n} \cdot {\bf u}_2) {\bf u}_2$ and $({\bf n} \cdot {\bf u}_2)^2 = 1 - ({\bf n} \cdot {\bf u}_1)^2$, we obtain $$\begin{aligned}
F_{\bf n} &=& A H - (\frac{3}{2} B+2 C) H^2 + (\frac{1}{2} B+C) K
\\ \nonumber
&& + \Big( \frac{A}{2} (k_1 - k_2)
- (B+C) (k_1 - k_2) \: H \Big)\: \cos(2 \theta)
\\ \nonumber
&& - \frac{B}{2} (H^2 -K) \cos(4 \theta).\end{aligned}$$ The first term can be absorbed in the spontaneous curvature $c_0$, and the second two terms in the bending and Gaussian modulus $\kappa_b$ and $\kappa_g$, respectively. The last two terms give the expression for the anisotropic curvature free energy per area $F_{\mbox{\tiny ani}}$ in the main text (Eq.(1)), with $\kappa'= (B+C)$ and $\kappa''= B$, and $c_0'=A/(B+C)$.
Theory: SCFT calculations.
--------------------------
### Basic equations
We consider a melt of asymmetric $AB$ diblock copolymer molecules confined in a volume $V$ between two coaxial cylindrical surfaces of radius $R_1$ and $R_2=R_1+\epsilon$, where $\epsilon$ is the thickness of the confined film. The two surfaces preferentially attract $A$-monomers. Dirichlet boundary conditions are applied in the radial direction and periodic boundary conditions are applied in the in-plane directions. Each diblock copolymer molecule consists of $N$ segments of which a fraction $f$ forms the majority block $A$. We assume that $A$ and $B$ segments have the same statistical segment length $b$. The microscopic concentration operators of $A$ and $B$ segments at a given point $\mathbf{r}(r,\varphi,z)$ are defined as $$\begin{aligned}
\hat{\phi}_A(\mathbf{r}) = \frac{1}{\rho_c} \sum_{j=1}^{n} \int_0^f ds
\: \delta (\mathbf{r}-\mathbf{r}_{j}(s)) \\
\hat{\phi}_B(\mathbf{r}) = \frac{1}{\rho_c} \sum_{j=1}^{n} \int_f^1 ds
\: \delta (\mathbf{r}-\mathbf{r}_{j}(s))\end{aligned}$$ respectively. These concentrations are made dimensionless by dividing by the average copolymer density $\rho_c$. The interaction potential of the melt is $$\begin{aligned}
\frac{\mathcal{H_I}}{k_BT} & = & \rho_c \int d\mathbf{r}
\left[ \chi N \hat{\phi}_A(\mathbf{r})\hat{\phi}_B(\mathbf{r}) \right.
\\ &&
\left. + \frac{1}{2} \kappa N
\left( \hat{\phi}_A(\mathbf{r}) + \hat{\phi}_B(\mathbf{r}) -1 \right)^2 \right]
\nonumber \\ \nonumber
&& + \rho_c \int d\mathbf{r} \: H(\mathbf{r})
\left[ \Lambda_A^{s,a} N \hat{\phi}_A(\mathbf{r})
+ \Lambda_B^{s,a} N \hat{\phi}_B(\mathbf{r}) \right]\end{aligned}$$ where the Flory-Huggins parameter $\chi$ specifies the repulsion of $A$ and $B$ segments. The second term describes a finite compressibility of the melt [@Helfand1975], which is fixed to $\kappa N = 25$, similar to previous work on similar systems [@Pike2009; @Detcheverry2010]. The terms $\Lambda_{A,B}^{a,s} \: H(\mathbf{r})$ are surface fields. We choose a form
$$H(\mathbf{r}) = \left\{
\begin{array}{l l}
(1 + \cos(\pi (r-R_1)/\delta)) & \quad \text{ $R_1 \leq r \leq R_1 + \delta$}\\
0 & \quad \text{ $R_1 + \delta < r < R_2-\delta$} \\
(1+\cos(\pi(R_2-r)/\delta)) & \quad \text{ $R_2-\delta \leq r \leq R_2$}
\end{array} \right.$$
with $\delta = 0.2 R_g$. The value of $\delta$ must be chosen small enough relative to the domain size so that its finite size does not affect the phase behavior of the thin films significantly. $\Lambda_{A,B}^{s,a}$ gives the strength of the interaction between block $A$ or $B$, respectively, and the substrate ($s$) and air ($a$) interface. The “surface interaction energies per area” of component $A$ or $B$ are defined as the integrated surface energy per area of a hypothetical film of $A$ or $B$ monomers with density $\hat{\phi}_{A,B} \equiv 1$, i.e., $\gamma_{A,B}^{s,a} =
\rho_c \int d\mathbf{r} \: H(\mathbf{r}) \Lambda_{A,B}^{s,a}$. They will be given in units of $\hat{\gamma} = \rho_c R_g k_BT$ (which is a unit of energy per area). In the following, we shall set $k_BT=1$, for notational simplicity.
In the membrane study we assume that the two surfaces are symmetric for each block, $\Lambda_A^a N = -120$ and $\Lambda_B^a N = -115$ corresponding to $\gamma_A N = -24 \hat{\gamma}$ and $\gamma_B N = -23 \hat{\gamma}$. In the curved supported thin films, we choose symmetric surface interactions for the $B$-block $\Lambda_B^s N = \Lambda_B^a N =
-30$ corresponding to $\gamma_B N = -6 \hat{\gamma}$, and asymmetric conditions for the $A$-block. Specifically, we study two cases:
- The substrate attracts the $A$-block more strongly than the free (air) surface ($\Lambda_A^s N = -120, \Lambda_A^a N = -50$) with corresponding surface energy per area $\gamma_A^s N = -24 \hat{\gamma}$, $\gamma_A^a N= - 10 \hat{\gamma}$.
- The free surface attracts the $A$-block more strongly than the substrate ($\Lambda_A^s = -50, \Lambda_A^a N = -100$) with corresponding surface energy per area $\gamma_A^s N= -10 \hat{\gamma}$ and $\gamma_A^a N= -20 \hat{\gamma}$.
Our calculations are done in the grand canonical ensemble with the free energy $$\begin{aligned}
F_{GC} & = &
-e^{\mu}Q + \rho_c \int d\mathbf{r}
\left[\chi N\phi_A(\mathbf{r})\phi_B(\mathbf{r}) \right.
\nonumber \\ &&
\quad + \left. \frac{\kappa}{2}\left(\phi_A(\mathbf{r})
+\phi_B(\mathbf{r})-1 \right)^2\right] \nonumber \\
& & - \rho_c \int d\mathbf{r}
\left[ \omega_A(\mathbf{r})\phi_A(\mathbf{r})
+\omega_B(\mathbf{r})\phi_B(\mathbf{r}) \right] \nonumber \\
& & + \rho_c \int d\mathbf{r} H(\mathbf{r}) N
\left[ \Lambda_A^{s,a} \phi_A(\mathbf{r}) + \Lambda_B^{s,a} \phi_B(\mathbf{r}) \right]
\label{eq:fgc}\end{aligned}$$ where $\mu$ is chemical potential, $Q$ the partition function of a single non-interacting polymer chain, $$Q = \int d\mathbf{r} q(\mathbf{r},s) q^{\dag}(\mathbf{r},1-s)$$ and $q(\mathbf{r},s)$ and $q^{\dag}(\mathbf{r},1-s)$ satisfy the modified diffusion equation $$\begin{aligned}
\frac{\partial q(\mathbf{r},s)}{\partial s}
= \Delta q(\mathbf{r},s) -\omega_{\alpha}(\mathbf{r},s) q(\mathbf{r},s) \end{aligned}$$ with $$\omega_{\alpha}(\mathbf{r},s) = \left\{
\begin{array}{l l}
\omega_A(\mathbf{r}) & \quad \text{for $0<s<f$}\\
\omega_B(\mathbf{r}) & \quad \text{for $f<s<1$}
\end{array} \right.$$ and the initial condition $q(\mathbf{r},0) = 1$. The diffusion equation for $q^{\dag}(\mathbf{r},1-s)$ is similar with $\omega_\alpha (\mathbf{r},s)$ replaced by $\omega_\alpha (\mathbf{r},1-s)$ and the same initial condition, $q^{\dag}(\mathbf{r},0)=1$. By finding the extremum of the free energy, Eq. (\[eq:fgc\]) with respect to $\omega_{A,B}(\mathbf{r})$ and $\phi_{A,B}(\mathbf{r})$, we get the self-consistent equations, $$\begin{aligned}
\frac{\omega_A(\mathbf{r})}{N} & = & \chi \phi_B(\mathbf{r})
+ \kappa \left[\phi_A(\mathbf{r})+\phi_B(\mathbf{r})-1\right]
+ \Lambda_A H(\mathbf{r}) \nonumber \\
\frac{\omega_B(\mathbf{r})}{N} & = & \chi \phi_A(\mathbf{r})
+ \kappa \left[\phi_A(\mathbf{r})+\phi_B(\mathbf{r})-1 \right]
+ \Lambda_B H(\mathbf{r}) \nonumber \\
\phi_A(\mathbf{r}) & = &
\frac{1}{\rho_c} \: {\rm e}^\mu
\int_0^f ds \: q(\mathbf{r},s) \: q^{\dag}(\mathbf{r},1-s) \nonumber \\
\phi_B(\mathbf{r}) & = &
\frac{1}{\rho_c} \: {\rm e}^\mu
\int_f^1 ds \: q(\mathbf{r},s)\: q^{\dag}(\mathbf{r},1-s)\end{aligned}$$
### Boundary conditions
The SCFT calculations are done in cylindrical coordinates $(r,\phi,z)$, where $r$ is the direction normal to the film or membrane surface and $z$ is the direction of zero curvature. Configurations $C_{\parallel}$, $C_\perp$ with cylinder orientations parallel or perpendicular to the direction of curvature can be obtained with periodic boundary conditions in the ($\phi,z$) directions. In order to impose a given tilted orientation with tilt angle $\theta$ (as in Fig. \[fig:fig3n\]c, inset), we must apply tilted boundary conditions, either in the $z$ or in the $\phi$ direction. We do this by using affine coordinates $(r,u,v)$ with periodic boundary conditions in $(u,v)$ and (1) $u =
\varphi, v = z-\varphi a$, $a=R \tan \theta$ or (2) $(r,u,v)$, $u=\varphi - b
z, v=z$, $b = 1/(R\tan \theta$). In case (1), $a=0$ corresponds to the perpendicular configuration $C_\perp$ ($\theta = 0$), and in case (2) $b=0$ corresponds to the parallel configuration $C_\parallel$ ($\theta = \pi/2$).
We solve the modified diffusion equations with periodic boundary conditions in effectively two dimensions: $(r,v)$, independent of $u$ in case (1), and $(r,u)$, independent of $v$ in case (2). This enforces tilted orientations of cylinders. In general, the Laplace-Beltrami operator has the following form $$\begin{aligned}
\Delta_{LB} = \frac{1}{\sqrt{|\det g|}}
\sum_{ij} \frac{\partial}{\partial x_i} \left( g^{ij} \sqrt{|\det g|} \frac{\partial}{\partial x_j}\right)\end{aligned}$$ where $g_{ij}$ is the metric tensor and $g^{ij}$ its inverse. The Laplace-Beltrami operator in cases (1) and (2) is thus given by $$\begin{aligned}
\Delta_{LB}^{(1)}
& = & \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^2}{\partial r^2}
+ \frac{1}{r^2} \: \frac{\partial^2}{\partial u^2}
- \frac{2a}{r^2} \frac{\partial^2}{\partial u \: \partial v} \nonumber \\
& & + \left( 1 + \frac{a^2}{r^2} \right) \frac{\partial^2}{\partial v^2}
\qquad \mbox{case (1)}\\
\Delta_{LB}^{(2)} & = & \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^2}{\partial r^2}
+ \left( \frac{1}{r^2} + b^2 \right) \frac{\partial^2}{\partial u^2} \nonumber \\
& & -2b \frac{\partial^2}{\partial u \: \partial v} + \frac{\partial^2}{\partial v^2}
\qquad \mbox{case (2)}.\end{aligned}$$
The modified diffusion equations were solved using the the Crank-Nicolson method. We used the setup (1) for small angles $0 < \theta <
\pi/4$ and the setup (2) for large angles $ \pi/4 < \theta < \pi/2$, and compared the results from both setups at the angle $\theta = \pi/4$, to verify that both setups give the same result.
### Discretization errors and correction
The discretizations in the azimuthal and thin film directions ($z$ and $r$, respectively) were chosen as $\Delta z=0.05R_g$ and $\Delta r =0.01R_g$, and the parameter $s$ was discretized in steps of $\Delta s = 0.0001$. Whereas most of the choices are not critical, we found that the discretization in the $r$ direction has a significant influence on the resulting free energies, and discretization errors could not be neglected. On the other hand, we also found that they lead to an energy shift $\Delta F$ which depends only on $\Delta r$ and not on the film thickness or curvature. We therefore studied the dependence of $\Delta F$ on $\Delta r$ systematically for different values of the film thickness and curvature. Then we fitted the result to a third order polynomial, resulting in the estimate $\Delta F (\Delta r) = -11.75 \Delta r - 270 \Delta
r^2 + 5035 \Delta r^3 $ ($\Delta F (\Delta r) = -11.95 \Delta r - 167 \Delta
r^2 + 3737 \Delta r^3 $) and $\Delta F (\Delta r) = -12.6 \Delta r - 277 \Delta
r^2 - 2367 \Delta r^3 $ with asymmetric and homogeneous surface interactions, respectively. Figs. \[fig:figS4\], \[fig:figS5\] and \[fig:figS6\] present the fitting results. These corrections were then applied to the results of the SCFT calculations.
![\[fig:figS4\] Shift of free energy $\Delta F$ as a function of discretization $\Delta r$ for symmetric films with surface interactions $\Lambda_A^a N = -120, \Lambda_B^a N = -115$, for different curvatures and film thicknesses $\epsilon$ as indicated. Solid line: fit function $f(x)= -12.6x-277x^2-2367x^3$.](FigureS4.png){width="7.5cm"}
![\[fig:figS5\] Shift of free energy $\Delta F$ as a function of discretization $\Delta r$ for films with asymmetric surface interactions $\Lambda_A^s N =
-120, \Lambda_B^s N = -30$ at the fixed substrate and $\Lambda_A^a N = -50,
\Lambda_B^a N = -30$ at the free surface for different curvatures and film thicknesses $\epsilon$ as indicated. Solid line: fit function $f(x)=-11.75x-270x^2+5035x^3$.](FigureS5.png){width="7.5cm"}
![\[fig:figS6\] Shift of free energy $\Delta F$ as a function of discretization $\Delta r$ for films with asymmetric surface interactions $\Lambda_A^s N =
-50, \Lambda_B^s N = -30$ at the fixed substrate and $\Lambda_A^a N = -100,
\Lambda_B^a N = -30$ at the free surface for different curvatures and film thicknesses $\epsilon$ as indicated. Solid line: fit function $f(x)=-11.95x-167x^2+3737x^3$.](FigureS6.png){width="7.5cm"}
### Optimum thickness of free-standing membranes
Fig. \[fig:figS7\] shows the behavior of the optimum film thickness of free-standing membranes as a function of curvature both parallel and perpendicular configurations. Interestingly, the optimal thickness is not affected strongly by curvature in the range of curvatures considered here.
![\[fig:figS7\] The optimal film thickness of free-standing membranes as a function of curvature. ](FigureS7.png){width="8cm"}
|
---
author:
- 'Geoffrey T. Bodwin and Andrea Petrelli'
date: 'March 11, 2013'
title: |
Erratum: Order-$\bm{v^4}$ corrections to $\bm{S}$-wave quarkonium decay\
\[0pt\]\[Phys. Rev. D [**66**]{}, 094011 (2002)\]
---
In this erratum, we correct several errors in Ref. [@Bodwin:2002hg].
In Eqs. (6.25), (6.26), (6.27), (6.28b), and (6.30) of Ref. [@Bodwin:2002hg], the denominator factor $81$ should be written as $27(D-1)$, where $D$ is the space-time dimension. The factor $D-1$ arises from the projection of bilinears of $p$ onto $S$-wave states. Here $p$ is one-half the relative momentum of the quark and antiquark in the quark-antiquark rest frame. Specifically, the $S$-wave projections yield $$p_ip_jp_i'p_j'\to
\frac{\bm{p}^2}{D-1}\delta_{ij}\frac{\bm{p}^2}{D-1}\delta_{ij}
=\frac{\bm{p}^4}{D-1},$$ where $\bm{p}^2={\bm{p}'}^2$, but we distinguish the orientations of the momentum $\bm{p}$ in the incoming state and the momentum $\bm{p}'$ in the outgoing state in order to project each onto an $S$-wave angular-momentum state. The denominator factors $27(D-1)$ affect the $\overline{\rm MS}$ renormalization of the NRQCD matrix element. In $\overline{\rm MS}$ renormalization, one constructs the counterterm for a UV-divergent subdiagram by subtracting the poles in $\epsilon=(4-D)/2$ that appear in that subdiagram, plus some associated constants. In order to ensure that the counterterm contribution removes the contribution that is proportional to the UV divergence in the divergent subdiagram, one must compute in $D$ dimensions all factors that are external to the divergent subdiagram. These external factors include factors that derive from the external momenta, such as the projections onto specific orbital-angular-momentum states. A failure to follow this procedure at the one-loop level leads to a shift of the result by a constant term. However, at the level of two or more loops, it leads to a breakdown in the consistency of the renormalization program, with the consequence that uncanceled poles appear in short-distance coefficients [@Bodwin:2012xc]. It follows from these corrections that the term $-833/972$ in Eq. (6.31c) should be replaced with $-805/972$ and that the term $7.62$ in the last line of Table IV should be replaced with $8.22$.
Equation (6.3) of Ref. [@Bodwin:2002hg] should read $$f(p)=\left[\frac{E(p)}{m}\right]^{3(D-2)-D}
=\left[\frac{E(p)}{m}\right]^2\left[1-4\epsilon\log\frac{E(p)}{m}
\right]+O(\epsilon^2).$$ This correction does not affect the term of order $\epsilon^0$ in Eq. (6.3). Hence, as is explained just after Eq. (6.3), it does not affect the result of the calculation.
Equation (6.21) of Ref. [@Bodwin:2002hg] should read $$\begin{aligned}
\tilde{\cal M}_{\rm IR}&=&\frac{\bm{p}^4}{m^8}
\frac{(N_c^2-1)(N_c^2-4)}{N_c^2}
\frac{128\pi^3\alpha_s^3}{(3-2\epsilon)^3}\mu^{6\epsilon}
\left\{
\frac{(3-2\epsilon)(1-\epsilon)}{x^2}
-\frac{2(2-\epsilon)y(1-y)}{x^2}\right.
+\frac{(3-2\epsilon)(1-\epsilon)}{(1-xy)^2}\nonumber\\
&&-\frac{2(2-\epsilon)x(1-x)(1-y)}{(1-xy)^4}
\left. +\frac{(3-2\epsilon)(1-\epsilon)}{[1-x(1-y)]^2}
-\frac{2(2-\epsilon)x(1-x)y}{[1-x(1-y)]^4}\right\}.\end{aligned}$$ The terms that contain factors of $x^2$ in the denominator, the terms that contain factors of $(1-xy)^2$ in the denominator, and the terms that contain factors of $[1-x(1-y)]^2$ in the denominator are related to each other by cyclic permutations of the energy fractions $x_1$, $x_2$, and $x_3$ and give equal contributions to the integral over the final-state phase space. This error in Eq. (6.21) of Ref. [@Bodwin:2002hg] was not propagated into the remainder of the calculation.
In Eq. (2.16) of Ref. [@Bodwin:2002hg], $0.82n_f$ should be $0.81n_f$. An exact expression for $F_{ee}({}^3S_1)$ through order $\alpha^2\alpha_s^2$ is [@Beneke:1997jm] $$\begin{aligned}
F_{ee}({}^3S_1)&=&\frac{2\pi Q^2\alpha^2}{3}\bigg\{1
-4C_F\frac{\alpha_s(m)}{\pi}+\bigg[\frac{103}{27}-\frac{511\pi^2}{162}
-\frac{224\pi^2\ln(2)}{27}
-\frac{250\,\zeta(3)}{9}+\frac{22}{27}n_f
+\frac{140\pi^2}{27}\ln\biggl(\frac{2m}{\mu_\Lambda}\biggr)
\biggl(\frac{\alpha_s}{\pi}\biggr)^2\biggr]\biggr\},\nonumber\\\end{aligned}$$ where $\zeta(z)$ is the Riemann zeta function.
There are several additional errors of a minor nature in Ref. [@Bodwin:2002hg]. Just after Eq. (2.10), $m^2 F_1({}^3S_1)$ should be $F_1({}^3S_1)$. Just after Eq. (2.14), $m^2
F_{\gamma\gamma}({}^1S_0)$ should be $F_{\gamma\gamma}({}^1S_0)$. In Eq. (5.5c), $58/54$ should be $29/27$. In Eq. (6.13), $k_1+k_2+k_3$ should be $k_1^0+k_2^0+k_3^0$.
We thank Chaehyun Yu for pointing out the error in Eq. (6.3) of Ref. [@Bodwin:2002hg] and Wen-Long Sang for pointing out the error in Eq. (6.27) of Ref. [@Bodwin:2002hg]. We also thank Jungil Lee for bringing to our attention some errors in an earlier version of this erratum. The work of G.T.B. in the High Energy Physics Division at Argonne National Laboratory was supported by the U. S. Department of Energy, Division of High Energy Physics, under Contract No.DE-AC02-06CH11357.
G. T. Bodwin and A. Petrelli, Phys. Rev. D [**66**]{}, 094011 (2002) \[hep-ph/0205210\]. G. T. Bodwin, U-R. Kim, and J. Lee, J. High Energy Physics 11 (2012) 020 \[arXiv:1208.5301 \[hep-ph\]\]. M. Beneke, A. Signer, and V. A. Smirnov, Phys. Rev. Lett. [**80**]{}, 2535 (1998) \[hep-ph/9712302\].
|
psfig.sty &=& [**v**]{} [**H**]{}
Measurements of the linear term in the temperature dependence of the electromagnetic penetration depth [@Hardy] $\delta \lambda(T)\equiv
\lambda(T)-\lambda(0)\sim T$ played a pivotal role in the identification of the $d$-wave symmetry of the cuprate superconductors. It was always clear, however, that they were sensitive only to the depletion of the superfluid fraction by the thermal excitations of quasiparticles with momenta near the order parameter nodes, and hence provided information only on the energy gap on these regions of the Fermi surface. Furthermore, in a $d$-wave superconductor with tetragonal crystal symmetry, the penetration depth is independent of the angle $\theta$ the supercurrent makes with the crystal axes since the linear response tensor is isotropic; the gap [*shape*]{} is thus not directly reflected in the angle dependence of the effect. Yip and Sauls[@YipSauls] argued, however, that information about the gap shape in an unconventional superconductor with line nodes could be obtained by measuring the nonlinear (magnetic field-dependent) penetration depth $\delta \lambda(H)=\lambda(H)-\lambda(0)$, and predicted an unusual field dependence, $$\begin{aligned}
{\delta \lambda (H)\over \lambda (0)}\simeq {\zeta_\theta\over \sqrt{2}}
{H\over H_0}, \label{ys}\end{aligned}$$ where $\zeta_\theta$ is a (weak) angle-dependent function, $H$ is the applied magnetic field, and $H_0=3\Phi_0/(\pi^2 \xi_0\lambda_0)$ is of the order the thermodynamic critical field of the system ($\Phi_0$ is the flux quantum, $\xi_0$ the coherence length and $\lambda_0$ the penetration depth). Later, it was shown[@Xhuetal] that impurities alter this prediction, leading to an asymptotic $\delta \lambda \sim H^2$ dependence, but only at concentrations such that the linear term in the $H\rightarrow 0$ temperature dependence $\delta\lambda(T)\sim T$ is replaced by $\delta\lambda(T)\sim
T^2$ [@Grossetal; @Prohammer; @felds]. From an experimental standpoint, the search for confirmation of this fundamental manifestation of $d$-wave symmetry has been controversial. In 1995, Maeda et al. reported the first observation of a linear $H$ term on $Bi_2Sr_2CaCu_2O_8$ $(BSCCO-2212)$ films [@Maedaetal], but the results were questionable since they were obtained at relatively high temperatures, where the Yip-Sauls theory predicts a $\delta\lambda\sim H^2$. The clean $d$-wave nonlinear Meissner effect should be obtained, according to the simple theory, only for temperatures below a crossover $T\ll E_{nonlin}$, with $E_{nonlin}=v_sk_F\simeq (H/H_0)\Delta_0$ and $\Delta_0$ the gap maximum, giving typically $E_{nonlin}\simeq 1K$ for the cuprates, for fields close to $H_{c1}$, the lower critical field. Measurement of a linear-$H$ term was also claimed at much higher temperatures for $YBa_2Cu_3O_{7-\delta}$ ($YBCO$) samples [@Maeda2]. It is important to recall that several extrinsic effects can give rise to spurious field dependences, most importantly contributions from trapped vortices which enter near sample edges. An extrinsic contribution $\delta\lambda\sim\sqrt{H}$ to the effective penetration depth can arise from coupling of the rf field used in resonant coil experiments to the pinned vortices [@Campbellpendepth]. Such dependences were also found in measurements at much lower temperatures and on higher quality single crystals of $YBCO$ by Carrington and Giannetta [@CarringtonGiannetta]. Finally, a measurement to detect higher-harmonic nonlinear modes in the ac response predicted on the basis of Yip-Sauls theory by $\breve{\rm Z}$utić and Valls [@ZuticValls], apparently less sensitive to the effects of trapped flux, failed to observe the predicted signal [@Goldmanprivate]. We therefore take the point of view that the available experimental data do not support the existence of a nonlinear Meissner effect in the best cuprate samples, and ask why. Our answer is that the effects of nonlocal electrodynamics, neglected in the work of Yip and Sauls, serve to cut off the (local) nonlinear Meissner effect as the field is lowered, replacing the linear-$H$ response with a quadratic one below a crossover field $H^*$. In most experimental geometries, this crossover is of the order of the lower critical field $H_{c1}$, meaning the $\delta\lambda(H)\sim H$ behavior cannot generically be observed in the Meissner state. However, we show that nonlocal effects are negligible in geometries with currents parallel to a nodal direction. [*Nonlocal electrodynamics.*]{} The Yip-Sauls description of the nonlinear Meissner effect is performed in the local limit, i.e. the $d$-wave Cooper pairs, although extended objects, are assumed to respond to the electromagnetic field at a single point at the center of the pair. Such an approximation is normally justified in classic type-II superconductors, where $\lambda_0$ is much larger than $\xi_0$. Although the cuprates are strongly type-II superconductors, with small in-plane $\xi_0\simeq{\cal O}(15A)$ and large in-plane $\lambda_0\simeq {\cal O}(1500A)$, effects of nonlocal electrodynamics are to be seen in several special situations, as a result of the gap nodes, along which the “effective coherence length" $v_F/\pi\Delta_k$ diverges [@KL]. Such effects were shown to significantly suppress the shielding supercurrents in measurements of the ac Meissner effect [@PHW] in unconventional superconductors with line nodes, with application to heavy fermion materials. In the context of the cuprates, it was shown by Zucarro et al.[@Scharnberg] that significant variations in the electromagnetic response of clean $d$-wave superconductors were to be expected when nonlocal effects were included. But it was Kosztin and Leggett[@KL] who first pointed out that the very signature of a superconductor with line nodes, the linear-$T$ term in the dc penetration depth, was modified at the lowest temperatures [@SD]. The crossover temperature below which this occurs is $E_{nonloc}\simeq \kappa^{-1}\Delta_0$, where $\kappa\equiv
\lambda_0/\xi_0$ is the Ginzburg-Landau parameter. The characteristic crossover field defined by $E_{nonloc}=E_{nonlin}$ is given by $H^*\simeq
\kappa^{-1} H_0\simeq H_{c1}$. One might therefore expect that nonlocal effects “cut off" the nonlinear Meissner effect. This observation needs, however, to be supported by a full calculation, which we now sketch.
[*General current response.*]{} The current response for a $d$-wave superconductor may be derived from a full calculation of the BCS free energy for fixed external field $\H$ corresponding to vector potential $\A$. We have performed such a calculation (including both nonlocal and nonlinear effects analytically) under the assumptions of i) negligible spatial variations of the amplitude of the order parameter, $\Delta_\k(\r)$; and ii) slow spatial variations of the superfluid velocity $\vs$. The details of the derivation will be given elsewhere [@LHWPRB], but the final result for the Fourier component of the current in an infinite system at [*constant*]{} ${\vs}$ is ${\bf j}(\q)=-\K(\q,\vs,T) \A(\q)$, where $$\begin{aligned}
{\K(\q,\vs,T)\over c/(4\pi \lambda_0^2)}=1+{2T\over nm}
\sum_{\omega_n,{\bf k}}{\bf k}^2_{\parallel}\frac{z_\k^2+\xi_+\xi_- +
\Delta_+\Delta_-}{[z_\k^2-E^2_+][z_\k^2-E^2_-]} \, ,
\label{bubble}\end{aligned}$$ with $\lambda_0=\sqrt{ mc^2/4\pi ne^2}$ the [*local*]{} penetration depth at $T=0$ in [*linear*]{} response theory, $n$ the electron density, $m$ the electron mass, $c$ the speed of light, $\k_\parallel$ the projection of $\k$ onto the surface, $z_\k\equiv i\omega_n+\vs\cdot\k_F$, $E_\pm^2\equiv \xi_\pm^2+\Delta_\pm^2$, $\xi_\pm\equiv\xi_{\k\pm\q/2}$, and $\xi_\k={k^2\over 2m}-\mu$ ($\mu$ the chemical potential). This is the standard linear response result with all quasiparticle Matsubara energies $\omega_n$ modified by the semiclassical Doppler shift $\vs\cdot\k_F$ [@MakiTsuneto]. For the real physical system in the Meissner state, a surface boundary is present and $v_s$ decays with the distance from the surface $y$. In the linear, local approximation, $v_s(y)=(e\lambda_0 H/mc) \exp(-y/\lambda_0)$ decays exponentially and $v_s(y=0)$ is proportional to the external field, neither of which hold generally. But we may imagine the system to be subdivided into many layers, in each of which $v_s$ is roughly constant. Furthermore, we assume [*specular*]{} scattering surface boundary condition on the quasiparticle wavefunctions. In this case, the response is given by Eq. (\[bubble\]). The geometry which we study first is such that the magnetic field is along the $c$ axis and thus $\vs$ lies in the $ab$ plane; the normal of the boundary plane $\hat q$ is also in the $ab$ plane, forming an angle $\theta$ with the $b$ axis. We will discuss the situation where [**$H$**]{} is perpendicular to the $c$ axis later. To a first approximation, we neglect the space dependence of $v_s$ and replace it by its value at the surface $v_s\simeq v_s(y=0)=e\lambda_0 H/mc$ as given by the solution to the linear, local electrodynamics problem. Now we separate out the $T=0$ local, linear response as $\K(\q,\vs,T)=
c/(4\pi\lambda_0^2)+\delta\K(\q,\vs,T)$, and then [*define*]{} the nonlinear, nonlocal penetration depth to be $$\begin{aligned}
\lambda_{\rm spec}=\int^\infty_0{H(y)\over H(0)}dy\simeq\lambda_0
-{8\over c}\int^\infty_0 dq\frac{\delta{\cal K}(\q,{\bf v}_s,T)}
{(\frac{1}{\lambda^2_0}+q^2)^2}.
\label{pene1}\end{aligned}$$ This expression has the virtue of reducing exactly to the nonlocal expression of Kosztin and Leggett[@KL] if the $\vs$-dependence is neglected, and (qualitatively) to the nonlinear expression of Yip and Sauls [@YipSauls] if the $q$-dependence is neglected. It is worthwhile reviewing the latter case. When $T\ll v_sk_F$ we obtain $$\begin{aligned}
{\delta{\cal K}(\q\rightarrow 0,{\bf v}_s,T)\over c/(4\pi\lambda_0^2)}=
-\frac{\zeta_\theta}{\sqrt{2}}\frac{v_sk_F}{\Delta_0} \, ,
\label{response9}\end{aligned}$$ where $\zeta_\theta={1\over 2}\sum_{l=\pm 1} |\cos\theta+l\sin\theta|^3$. Thus, $$\begin{aligned}
\frac{\delta\lambda^{({\rm loc})}_{\rm spec}}{\lambda_0}
=\frac{\lambda^{({\rm loc})}_{\rm spec}-\lambda_0}{\lambda_0}
\simeq\frac{\zeta_\theta}{2\sqrt{2}}\frac{v_sk_F}{\Delta_0}
\simeq\frac{3}{2}\frac{\zeta_\theta}{\sqrt{2}}\frac{H}{H_0}\; .
\label{penetra2}\end{aligned}$$ We recall that Yip and Sauls [@YipSauls] and Xu et al. [@Xhuetal] defined penetration depth from its initial decay: $\lambda=-H(0)/[dH(y)/dy]|_{y=0}$ which is equivalent to the definition in Eq. (\[pene1\]) only in the linear limit. Using their definition we find that the prefactor 3/2 in Eq. (\[penetra2\]) is modified to 3. At this stage, we may improve our approximation by restoring the spatially varying nature of $v_s$. We follow Yip et al. to solve directly a nonlinear London equation $$\begin{aligned}
\nabla^2 \v_s={4\pi\over c}\K(0^+, \v_s,T)\v_s=
{\v_s\over \lambda^2_0}(1-\frac{\zeta_\theta}{\sqrt{2}}
\frac{k_F}{\Delta_0}v_s) \, ,
\label{london1}\end{aligned}$$ under the boundary condition $dv_s/dy|_{y=0}=(e/mc)H$. Using the relation $dv_s/dy=(e/mc)H(y)$ and their definition for the penetration depth again, we see that $\delta\lambda/\lambda_0=2\frac{\zeta_\theta}{\sqrt{2}}
\frac{H}{H_0} $. So in the local limit we get qualitatively the same nonlinear correction to the penetration depth as Yip and Sauls [@YipSauls], comparing to Eq. (\[ys\]), but with a larger prefactor. This deviation arises from the perturbative treatment of the response of the superconductor to the $\A(\q\neq 0)$ modes in the present theory. The nonlinear effects obtained at $T\ll v_sk_F$, due to the Doppler shift coming from the coupling of the $\A(\q=0)$ mode to the quasiparticles, are nonperturbative results. A discussion of this point can be found in Ref. [@LHWPRB].
[*Low-energy scaling properties.*]{} Explicit analytical results for the full response in the approximation of constant $v_s$ may now be obtained by expanding $\delta\K$ for low energies, $v_s k_F, q v_F, T\ll\Delta_0$. A 2-parameter scaling relation may be derived, $$\begin{aligned}
\delta{\cal K}(q,v_s,T)=-\frac{c}{8\pi\lambda^2_0}\frac{T}{\Delta_0}
\sum_{l=\pm 1} u^2_{\theta l}
F_\lambda({\alpha w_{\theta l}\over T},{\varepsilon u_{\theta l}\over T}),
\nonumber\\
F_\lambda(z_1,z_2)=\frac{\pi}{4}z_1+[\ln (e^{z_2}+1)+\ln (e^{-z_2}+1)]\nonumber\\
-\int^{z_1}_{0}dx[f(x-z_2)+f(x+z_2)][1-(\frac{x}{z_1})^2]^{1/2},
\label{scaling}\end{aligned}$$ with $\alpha=v_Fq/(2\sqrt{2})$, $\varepsilon=v_sk_F/\sqrt{2}$, $u_{\theta l}=|\cos\theta+l \sin\theta|$, $w_{\theta l}=|\sin\theta-l \cos\theta|$ and $f(x)=1/(1+e^x)$. We consider primarily the limit $T\ll \alpha,\varepsilon$. It is straightforward to see that for $z_2\gg 1$ $$\begin{aligned}
F_\lambda(z_1,z_2)\simeq
\left \{
\begin{array}{ll}
z_2\;\;\; & z_1\ll z_2 \\
{\pi z_1\over 4}+z_2(z^2_2+\pi^2)/(6z^2_1) \;\;\; & z_2\ll z_1\, .
\end{array}
\right.
\label{scaling1}\end{aligned}$$ The scaling form (\[scaling\]) may now be inserted into (\[pene1\]) to find the penetration depth. Its field dependence turns out to be determined by the ratio $\alpha w_{\theta l}/\varepsilon u_{\theta l}$, with a typical $q$ set to $1/\lambda_0$ in (\[pene1\]) by the denominator. If we define two new dimensionless parameters $h_{\theta l}=(u_{\theta l}
/w_{\theta l}) h$, $l=\pm 1$, with $h\equiv mv_s\lambda_0\rightarrow
(3/\pi)\kappa H/H_0$, we get $\delta\lambda_{\rm spec}=\sum_{l=\pm 1}
\delta\lambda^{(l)}_{\rm spec}$, where $$\begin{aligned}
\frac{\delta\lambda^{(l)}_{\rm spec}}{\lambda_0}\simeq
{\pi\over 4\sqrt{2}}u^3_{\theta l}\kappa^{-1} h+c_{\theta l1}
&\kappa^{-1}&\frac{1}{h^2}
+c_{\theta l2}\kappa{1\over h^4}{T^2\over \Delta_0^2} \nonumber \\
&&{\rm for}\;\;\; h_{\theta l} \gg 1, \label{lamnonlin}\end{aligned}$$ with $c_{\theta l1}=0.008 w^3_{\theta l}$ and $c_{\theta l2}=0.006
w^3_{\theta l}/u^2_{\theta l}$, and $$\begin{aligned}
\frac{\delta\lambda^{(l)}_{\rm spec}}{\lambda_0}\simeq
{\pi\over 8\sqrt{2}}d_{\theta l1}\kappa^{-1}
+d_{\theta l2}&\kappa^{-1}& h^2 +
d_{\theta l3} \kappa {T^2\over\Delta_0^2} \nonumber \\
&&{\rm for}\;\;\; h_{\theta l} \ll 1, \label{lamnonloc}\end{aligned}$$ with $d_{\theta l1}=0.5 u^2_{\theta l}w_{\theta l}$, $d_{\theta l2}=1.1
u^4_{\theta l}/w_{\theta l}$ and $d_{\theta l3}=0.47 u^2_{\theta l}/w_{\theta l}$. The singularity which leads to the linear term in Eq. (\[lamnonlin\]) if nonlocal effects are simply neglected is seen to be cut off by the constant term in Eq. (\[lamnonloc\]). This happens at $h_{\theta l}\sim 1$. For $\theta$ not too close to a nodal value, $w_{\theta l}$ and $u_{\theta l}$ are of order unity for both $l=\pm 1$, and the crossover field may be simply determined from $h=1$, $H^*\sim\kappa^{-1}H_0$. The thermodynamical critical field $H_c\simeq H_0$ is the geometric mean $\sqrt{H_{c1}H_{c2}}$ of the upper critical field $H_{c2}=\Phi_0/(2\pi\xi_0^2)$ and the lower critical field $H_{c1}=\Phi_0/(2\pi\lambda_0^2)$. Thus the crossover field [*for such geometries*]{} is of the same order as the lower critical field, $H^*\simeq H_{c1}$. Since the Meissner state is unstable to the Abrikosov vortex state for fields above $H_{c1}$, the linear-$H$ term in $\delta\lambda$ is asymptotically unobservable in these geometries. In the Meissner state it is, in fact, replaced by a quadratic variation, as seen in (\[lamnonloc\]). In Fig. 1, we have plotted the penetration depth calculated in Kosztin and Leggett’s geometry $\theta=0$, according to the exact scaling function given in (\[scaling\]), to explicitly exhibit this effect.
When $\theta$ is near any node, however, $u_{\theta l}$ can be much larger than $w_{\theta l}$ and the nonlocal energy scale much smaller than the nonlinear one, leading to a very small crossover field $H^*\simeq H_{c1}w_{\theta l}
/u_{\theta l}\ll H_{c1}$. We conclude that the nonlinear Meissner effect may be observable in such special geometries, e.g. a sample with a dominant (110) surface and $H\parallel c$.
(200,180)
There is another special situation in the quasi-2D cuprates in which the characteristic energy of nonlocal effects becomes very small and the previous discussion might not be expected to apply. If we assume that the quasiparticles are strictly confined to the $ab$ plane and if the experimental configuration is dominated by a (001) surface with $\H\parallel ab$, then $E_{nonloc}^{(ab)}=
\q\cdot\v_F=0.$ In this case, however, the magnetic field is not screened ($\lambda\rightarrow\infty$) [@GK]. Whenever the quasiparticles are allowed to move along the $c$-direction we expect nonzero $E_{nonloc}^{(ab)}$ [@schopohlcomm]. This can be demonstrated directly for any model of coherent transport along the $c$-axis. The materials of greatest current interest are the cuprate materials $YBCO$ and $BSCCO-2212$. In the $YBCO$ case, it is reasonable to treat the system as weakly-3D. The characteristic nonlocal energy is then $v_{Fc}q\simeq
\xi_{0c}\Delta_0/\lambda_0$ much smaller than in the (010) case since the $c$-axis coherence length is $\xi_{0c}\simeq 3A$ as opposed to the in-plane coherence length of $\xi_{0}\simeq 15A$. On the other hand, the lower critical field is also much smaller for this geometry, $H_{c1}^{(ab)}\simeq \Phi_0/(2\pi
\lambda_0\lambda_{0c})$. Using $\lambda_{0c}~\simeq ~0.5-0.8\times10^4 A$, we again find a large crossover field $H^{*(ab)}\simeq (\xi_{0c}/\xi_0) H^*
\simeq 0.5-1 H_{c1}^{(ab)}$, making it still impossible from a practical point of view to extract a linear-$H$ term [@Cooper].
The BSCCO system is so anisotropic that it is questionable whether the above argument applies. A full treatment of this problem awaits a generally accepted theory of the (incoherent) $\hat c$-axis transport in the normal state.
In conclusion, with the exception of the claim by Maeda et al. [@Maedaetal; @Maeda2], all attempts to measure the nonlinear Meissner effect, a fundamental consequence of $d$-wave symmetry, have failed. As the order parameter symmetry is by now well-established in these materials, an explanation is required. To this end, we have considered the interplay of nonlinear and nonlocal effects in the electromagnetic response of a $d$-wave superconductor in the Meissner regime. This situation differs from the similar problem considered by Amin et al. [@Aminetal] in the vortex phase due to the fact that the nonlinear and nonlocal energies $E_{nonlin}$ and $E_{nonloc}$ are not independent in their case, as a consequence of the fluxoid quantization condition. By contrast, in the Meissner state, while the nonlocal scale is set by intrinsic normal-state parameters, i.e. $T_c$ and $\kappa$, the nonlinear energy $E_{nonlin}\simeq
v_sk_F$ is effectively tunable by an external field. We have shown that the nonlocal terms in the response generically cut off the nodal singularity in the nonlinear response, eliminating the linear-$H$ term in the penetration depth.
Neither the current theory nor the Yip-Sauls result, both based on a consideration of the shielding currents in the Meissner phase, can explain the results of Maeda et al. Maeda’s results were obtained on samples which did not clearly exhibit a $\delta\lambda(T;H\rightarrow 0)\sim T$ behavior in any range, and therefore should not show nonlinear effects even according to the Yip - Sauls theory. Furthermore, they were performed at temperatures of 9K and above, and a linear-$H$ behavior was reported up to 50K. As we show in Fig. 1, even for temperatures as low as $0.1T_c$, no linear term [*should*]{} be visible even if nonlocal effects are neglected. Therefore we must conclude that Maeda et al. observed an extrinsic effect, probably related to trapped vortices. Very recently, new high-resolution resonant coil measurements[@CarringtonGiannetta; @BHBL] on very clean $YBCO$ samples observed (in some geometries) linear $H$ terms with magnitudes close to the Yip-Sauls prediction [@CarringtonGiannetta; @BHBL], but simultaneously measured qualitatively inconsistent temperature dependence. These authors also observed effects attributed to trapped vortices [@CarringtonGiannetta], and we therefore conclude that the linear term must be of extrinsic origin. Clearly, a test of the nonlinear effects which is independent of trapped flux is desirable. The harmonics of the nonlinear response predicted by $\breve{\rm Z}$utić and Valls in principle provide such a test, but to date, a high resolution nonlinear transverse magnetization experiment designed to observe these resonances has failed to do so [@Goldmanprivate]. This null result is consistent with the current theory, but by itself cannot confirm it due to uncertainties regarding the identification of $H_{c1}$ [@Goldmanprivate].
Finally, in all experiments up to now the orientation of the supercurrent was along the crystal axes. Our analysis suggests the best possibility to observe the effect is in geometries with $v_s$ along the nodal directions.
The authors are grateful to M. Franz, W. Hardy, and N. Schopohl for helpful communications, and particularly to R. Giannetta and A. Goldman for discussion of their unpublished results. Partial support was provided by NSF-DMR-96–00105 and the A. v. Humboldt Foundation.
-0.8cm
W. N. Hardy et al., Phys. Rev. Lett. 70, 3999 (1993). S. K. Yip and J. A. Sauls, Phys. Rev. Lett. 69, 2264 (1992); D. Xu, S. K. Yip and J. A. Sauls, Phys. Rev. B 51, 16233 (1995). F. Gross et al., Z. Phys. B 64, 175 (1986). M. Prohammer and J. P. Carbotte, Phys. Rev. B 43, 5370 (1991). P. J. Hirschfeld and N. Goldenfeld, Phys. Rev. B 48, 4219 (1993). A. Maeda et al., Phys. Rev. Lett. 74, 1202 (1995). A. Maeda et al., J. Phys. Soc. Jpn. 65, 3638 (1996). M. W. Coffey and J. R. Clem, Phys. Rev. Lett. 67, 386 (1991); E. H. Brandt, Phys. Rev. Lett. 67, 2219 (1991). A. Carrington and R. Giannetta, private communication. I. $\breve{\rm Z}$utić and O. T. Valls, cond/mat/9806288. A. Goldman, private communication. I. Kosztin and A. J. Leggett, Phys. Rev. Lett. 79, 135 (1997). Recently it was suggested that the rigorous linear $T$ dependence of the magnetic penetration depth would violate the third law of thermodynamics \[N. Schopohl and O. V. Dolgov, Phys. Rev. Lett. 80, 4761 (1998)\]. However, the nonlocal and nonlinear effects serve to cut off this linear $T$ dependence at low temperatures (Ref.[@schopohlcomm]). W. O. Putikka, P. J. Hirschfeld, and P. Wölfle, Phys. Rev. B 41, 7285 (1990). C. Zuccaro, C. T. Rieck and K. Scharnberg, Physica C 235-240, 1807 (1994). M. -R. Li, P. J. Hirschfeld, and P. Wölfle, unpublished. K. Maki and T. Tsuneto, Prog. Theor. Phys. 27, 228 (1962). D. I. Glazman and A. E. Koshelev, Sov. Phys. JETP 70, 774(1990). P. J. Hirschfeld, M. -R. Li, and P. Wölfle, Phys. Rev. Lett. 81, 4024 (1998). For representative values of $\xi_0$ and $\lambda_0$, see S. L. Cooper and K. E. Gray, in [*Physical Properties of High Temperature Superconductors*]{}, IV, D. M. Ginsberg, ed., Singapore: World Scientific, 1994. M. H. S. Amin, I. Affleck, and M. Franz, Phys. Rev. B 58, 5848 (1998). C. P. Bidinosti et al., cond/mat/9808231.
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---
abstract: 'This paper is a continuation of [@Ishitani-Kato_COSA1], in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we show that the function is continuous and has the semigroup property, which is strongly related to the Hamilton–Jacobi–Bellman quasi-variational inequality. Moreover, we show that noise in market impact causes risk-neutral assessment to underestimate the impact cost. We also study typical examples under a log-linear/quadratic market impact function with Gamma-distributed noise.'
address:
- 'Kensuke Ishitani: Department of Mathematics, Faculty of Science and Technology, Meijo University, Tempaku, Nagoya 468-8502, Japan'
- 'Takashi Kato: Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan'
author:
- Kensuke Ishitani
- 'Takashi Kato\*'
title: Theoretical and Numerical Analysis of an Optimal Execution Problem with Uncertain Market Impact
---
[^1]
Introduction and the Model {#sec_intro}
==========================
In [@Ishitani-Kato_COSA1], we derive a continuous-time value function corresponding to an optimal execution problem with uncertain market impact (MI) as a limit of a discrete-time value function. In this paper, we study some mathematical properties of the value function, and give an interpretation from the point of view of mathematical finance.
First, we recall the continuous-time value function derived in [@Ishitani-Kato_COSA1]. Denote by $\mathcal {C}$ the set of non-decreasing, non-negative, and continuous functions $u$ on $D := \Bbb {R}\times [0, \Phi_0] \times [0,\infty )$, with $\Phi _0 > 0$ fixed, such that $$\begin{aligned}
\label{growth_C}
u(w,\varphi ,s)\leq C_u(1+|w|^{m_u}+s^{m_u}), \ \ (w,\varphi ,s)\in D\end{aligned}$$ for some constants $C_u, m_u>0$. For $t\in [0, 1]$, $(w,\varphi ,s)\in D$ and $u\in \mathcal {C}$, define $$\begin{aligned}
\label{def_conti_v}
V_t(w,\varphi ,s ; u) = \sup _{(\zeta _r)_{r}\in \mathcal {A}_t(\varphi )}
{\mathop {\rm E}}[u(W_t,\varphi _t, S_t)] \end{aligned}$$ subject to $$\begin{aligned}
\nonumber
dW_r &= \zeta _rS_rdr, \\\nonumber
d\varphi _r &= -\zeta _rdr, \\
dX_r &= \sigma (X_r)dB_r+b(X_r)dr-g(\zeta _r)dL_r,
\label{SDE_X_g}\\\nonumber
S_r &= \exp (X_r) \end{aligned}$$ and $(W_0,\varphi _0,S_0) = (w,\varphi ,s)$, where $(B_r)_{0\leq r\leq 1}$ is a standard one-dimensional Brownian motion defined on a complete probability space $(\Omega , \mathcal {F}, P)$ and $(L_r)_{0\leq r\leq 1}$ is a one-dimensional non-decreasing Lévy process (subordinator) defined on the same probability space. (Note that $V_0(w,\varphi ,s ; u) = u(w,\varphi ,s)$.) Assume that $(B_r)_r$ and $(L_r)_r$ are independent. Further assume that $\sigma , b : \Bbb {R}\longrightarrow \Bbb {R}$ are Lipschitz continuous bounded functions satisfying $$\begin{aligned}
\label{Bdd_Lipschitz_Constant}
|\sigma (x) - \sigma (y)| + |b(x) - b(y)| \leq K|x - y|, \ \
|\sigma (x)| + |b(x)| \leq K, \ \ x, y\in \Bbb {R} \end{aligned}$$ for some $K > 0$, and $g : [0, \infty )\longrightarrow [0, \infty )$ is a function defined by $$\begin{aligned}
g(\zeta)=\int_0^{\zeta} h(\zeta ')d\zeta ' ,\end{aligned}$$ where $h:[0, \infty) \to [0, \infty)$ is a non-decreasing continuous function. $\mathcal {A}_t(\varphi )$ is the set of $(\mathcal {F}_r)_{0\leq r\leq t}$-adapted and caglad processes (i.e., those that are left-continuous with finite right-limit for arbitrary values of $r$) $\zeta = (\zeta _r)_{0\leq r\leq t}$ such that $\zeta _r\geq 0$ for each $r\in [0,t]$, $\int ^t_0\zeta _r dr\leq \varphi $ almost surely, and $$\begin{aligned}
\label{def_sup}
||\zeta ||_\infty := \sup _{(r,\omega )\in [0,t]\times \Omega }\zeta _r(\omega ) < \infty , \end{aligned}$$ where $\mathcal {F}_r=\sigma \{ B_v, L_v;v\leq r\}\vee \{\mbox{Null sets}\} $. Here, the supremum in (\[def\_sup\]) is taken over all values in $[0, t]\times \Omega $. As noted in [@Ishitani-Kato_COSA1], we may use the essential supremum in (\[def\_sup\]) in place of the supremum.
We assume that the Lévy measure $\nu $ of $(L_r)_r$ satisfies $$\begin{aligned}
\label{assumption_C}
||\nu ||_1 + ||\nu ||_2 < \infty , \end{aligned}$$ where $||\nu ||_p = \left( \int _{(0, \infty )}z^p\nu (dz)\right)^{1/p} $. Note that the Lévy decomposition of $(L_r)_r$ is given by $$\begin{aligned}
L_r = \gamma r + \int ^r_0\int _{(0, \infty )}zN(dv, dz), \end{aligned}$$ where $\gamma \geq 0$ and $N(\cdot, \cdot)$ is a Poisson random measure (see, for example, [@Papapantoleon; @Sato]).
Here, we introduce the financial interpretation of these notations. We consider a simple market model in which only two financial assets are traded: cash and a security. Assume that a single trader is to sell (liquidate) the owned shares of the security by time $t$. Also assume that the price of the cash is always $1$ (in other words, the risk-free rate is $0$) and that the security price fluctuates due to market noise and in response to the trader’s sales. The function $u$ in $\mathcal {C}$ is regarded as the trader’s utility function. With this, $V_t(w, \varphi , s; u)$ is the supremum of the expected utility of the trader with initial cash amount $w$, initial shares $\varphi \in [0, \Phi_0]$, and initial security price $s$. Here, $\Phi _0 > 0$ denotes an upper bound of $\varphi $ and can be arbitrarily chosen; $(\zeta _r)_{0\leq r\leq t}$ denotes the trader’s execution strategy; and $\zeta _r$ denotes the execution speed at time $r$. The trader chooses an admissible execution strategy from $\mathcal {A}_t(\varphi )$ to optimize the expected utility of the triplet $(W_t, \varphi _t, S_t)$, where $S_r$ describes the security price at time $r$ and $X_r$ is its log-price; $W_r$ denotes the cash amount at time $r$; and $\varphi_r$ denotes the shares of the security at time $r$. The fluctuation of the triplet $(W_r, \varphi _r, S_r)_{0\leq r\leq t}$ is characterized by the differential equations in (\[SDE\_X\_g\]). $(B_r)_r$ represents the component of the market noise reflected in fluctuation of the security price. The term $$\begin{aligned}
\label{MI_decomp}
g(\zeta _r)dL_r = \gamma g(\zeta _r)dr + g(\zeta _r)\int _{(0, \infty )}zN(dr, dz) \end{aligned}$$ describes the (infinitesimal) MI of the trader’s selling with speed $\zeta _r$. $\gamma $ (resp., $g$) denotes the magnitude (resp., shape) of the MI. Because $g$ is non-decreasing and convex, the MI becomes huge when $\zeta _r$ is large. The last term in the right-hand side of (\[MI\_decomp\]) indicates the effect of noise in the MI, which is mathematically described by the jump of $(L_r)_r$.
In this paper, we study some properties of the continuous-time value function $V_t(w,\varphi ,s ; u)$. We find that the value function is continuous in $(w, \varphi , s)\in D$ and $t > 0$. In addition, right-continuity at $t = 0$ depends on the state of $h(\infty ) := \lim _{\zeta \rightarrow \infty }h(\zeta )$. In particular, noise in the MI does not affect the continuity of the value function. We also show that the Bellman principle (the semi-group property) holds and perform a comparison with the result in the case of a deterministic MI, which was studied in [@Kato], and show that noise in the MI causes risk-neutral assessment to underestimate the MI cost. This means that a trader who attempts to minimize the expected liquidation cost is not sensitive enough to uncertainty in the MI. Last, we present generalizations of the examples from [@Kato] and investigate the effects of noise in the MI on the optimal strategy of a trader, by numerical experiments. We consider a risk-neutral trader’s execution problem with a log-linear/quadratic MI function with Gamma-distributed noise.
The rest of this paper is organized as follows. In Section \[section\_Results\], we present our results on the properties of the value function. In Section \[sec\_SO\], we consider the case where the trader must sell all shares of the security, which is referred to as the “sell-off condition.” We also study the optimization problem under the sell-off condition and show that the results in [@Kato Sect. 4] also hold in our model. Section \[section\_comparison\] compares deterministic MIs with random (stochastic) MIs in a risk-neutral framework. In Section \[section\_examples\], we present some examples based on the proposed model. We conclude this paper in Section \[section\_conclusion\]. All proofs are in Section \[sec\_proof\].
Properties of Value Functions {#section_Results}
=============================
Regarding the continuity of the continuous-time value function, we have the following theorem:
\[conti\_random\] Let $u\in \mathcal {C}$.\
$\mathrm {(i)}$ If $h(\infty )=\infty $, then $V_t(w,\varphi ,s ; u)$ is continuous in $(t,w,\varphi ,s)\in [0,1]\times D$.\
$\mathrm {(ii)}$ If $h(\infty )<\infty $, then $V_t(w,\varphi ,s ; u)$ is continuous in $(t,w,\varphi ,s)\in (0,1]\times D$ and $V_t(w,\varphi ,s ; u)$ converges to $Ju(w,\varphi ,s)$ uniformly on any compact subset of $D$ as $t\downarrow 0$, where $Ju(w,\varphi ,s)$ is given as $$\begin{aligned}
\left\{
\begin{array}{l}
\sup _{\psi \in [0,\varphi ]}
u\Big (w+\frac{1-e^{-\gamma h(\infty )\psi }}{\gamma h(\infty )}s,\varphi -\psi , s e^{-\gamma h(\infty )\psi} \Big )
\hspace{5mm} (\gamma h(\infty )>0), \\
\sup _{\psi \in [0,\varphi ]}
u(w+\psi s,\varphi -\psi ,s) \hspace{35.0mm} (\gamma h(\infty )=0).
\end{array}
\right. \end{aligned}$$
- The assertions of Theorem \[conti\_random\] are also quite similar to the result in [@Kato], which showed that continuities in $w$, $\varphi $, and $s$ of the value function are always guaranteed, but continuity in $t$ at the origin depends on the state of the function $h$ at infinity. When $h(\infty ) = \infty $, MI for large sales is sufficiently strong ($g(\zeta )$ diverges rapidly with $\zeta \rightarrow \infty $) to prevent the trader from performing instant liquidation: an optimal policy is “no trading” in infinitesimal time, and thus $V_t$ converges to $u$ as $t\downarrow 0$. When $h(\infty ) < \infty $, the value function is not always continuous at $t = 0$ and has the right limit $Ju(w, \varphi ,s)$. In this case, MI for large sales is not particularly strong ($g(\zeta )$ still diverges, although with low divergence speed) and there is room for liquidation within infinitesimal time. The function $Ju(w,\varphi ,s)$ corresponds to the utility of liquidation by the trader, who sells part of the shares of a security $\psi $ by dividing it infinitely within an infinitely short time (sufficiently short that the fluctuation in the price of the security can be ignored) and obtains an amount $\varphi -\psi $; that is, $$\begin{aligned}
\label{almost_block}
\zeta ^\delta _r = \frac{\psi }{\delta }1_{[0, \delta ]}(r), \ \ r\in [0, t] \ \ (\delta \downarrow 0). \end{aligned}$$
Note that, similarly to the argument in Remark 2.6 in [@Ishitani-Kato_COSA1], we obtain significant improvement in the strength of the proofs over that given in [@Kato], and this is one of the main mathematical contributions of this paper. See Section \[sec\_proof\] for details.
- Note that the jump part $$\begin{aligned}
\label{jump_term}
g(\zeta _r)\int _{(0, \infty )}z N(dr, dz) \end{aligned}$$ does not change the result. Also note that if $\gamma = 0$ and $h(\infty ) < \infty $, then the effect of MI disappears in $Ju(w, \varphi , s)$. This situation may occur even if ${\mathop {\rm E}}[c^n_k] \geq \varepsilon _0$ (or ${\mathop {\rm E}}[L_1]\geq \varepsilon _0$) for some $\varepsilon _0 > 0$.
Here, we present the Bellman principle (dynamic programming principle or “semi-group” property). Let us define $Q_t:\mathcal {C}\longrightarrow \mathcal {C}$ by $Q_t u(w,\varphi,s)=V_t(w,\varphi,s;u)$. Then we can easily show that $Q_t$ is well defined as a nonlinear operator. The same proof as that for Theorem 3.2 in [@Kato] gives the following proposition:
\[th\_semi\] For each $r, t\in [0, 1]$ with $t + r\leq 1$, $(w, \varphi , s)\in D$ and $u\in \mathcal {C}$, it holds that $Q_{t+r}u(w,\varphi ,s)=Q_t Q_r u(w,\varphi ,s)$.
By using the above proposition, we can formally derive the Hamilton–Jacobi–Bellman (HJB) equation corresponding to our value function on the generalized domain of the utility function $\hat{D} = \Bbb {R}\times [0, \infty ) \times [0, \infty )$: $$\begin{aligned}
\label{HJB_V}
\frac{\partial }{\partial t}V_t(w,\varphi ,s ; u) - \sup _{\zeta \geq 0}
\mathscr {L}^\zeta V_t(w,\varphi ,s ; u) = 0 \end{aligned}$$ with the same boundary conditions as (3.5) in [@Kato], where $$\begin{aligned}
\mathscr {L}^\zeta v(t,w,\varphi ,s)
&=
\overline{\mathscr {L}}^{\zeta }v(t,w,\varphi , s) - \tilde{\mathscr {L}}^{\zeta }v(t,w,\varphi , s), \\
\overline{\mathscr {L}}^{\zeta }v(t,w,\varphi , s)
&=
\frac{1}{2}\hat{\sigma }(s)^2\frac{\partial ^2}{\partial s^2}v(t,w,\varphi ,s) +
\hat{b}(s)\frac{\partial }{\partial s}v(t,w,\varphi ,s)\\
& +
\zeta \Big (s\frac{\partial }{\partial w}v(t,w,\varphi ,s) -
\frac{\partial }{\partial \varphi }v(t,w,\varphi ,s)\Big ) -
\gamma g(\zeta )s\frac{\partial }{\partial s}v(t,w,\varphi ,s), \\
\tilde{\mathscr {L}}^{\zeta }v(t,w,\varphi , s)
&= \int _{(0, \infty )}
\left\{ v(w, \varphi , s) - v(w, \varphi , s e^{-g(\zeta )z})\right\} \nu (dz). \end{aligned}$$ (\[HJB\_V\]) is a partial integro-differential equation (PIDE). When $\tilde {\mathscr {L}}^{\zeta }\equiv 0$, that is, when there is no jump, characterization of our value function as the unique viscosity solution of (\[HJB\_V\]) is studied by [@Kato] under some additional technical conditions. Showing these properties in the general case is a more challenging task. Here we introduce some related literature in place of presenting a detailed argument on the solvability of (\[HJB\_V\]): in [@Holden], the existence (i.e., characterization of a value function as a viscosity solution) and uniqueness of the solution of the HJB equation corresponding to the optimal investment/consumption problem with durability and local substitution in the Lévy version of the Black–Scholes-type market model is studied. Reference [@Seydel] shows existence and uniqueness of a solution to the Hamilton–Jacobi–Bellman quasi-variational inequalities (HJBQVIs) appearing in combined impulse and (regular) stochastic control problems with jump diffusions (existence in this case is also introduced in [@Oksendal-Sulem] without detailed technical arguments). In [@Bouchard-Touzi], by means of the weak dynamic programming principle, the characterization of a value function of stochastic control problems under Lévy processes with finite Lévy measure, which arises as a discontinuous viscosity solution of the corresponding HJB equation, is studied. The strong comparison principle (which is closely related to the uniqueness of viscosity solutions) for second-order non-linear PIDEs on a bounded domain is studied in [@Ciomaga].
Sell-Off Condition {#sec_SO}
==================
In this section, we consider the optimal execution problem under the “sell-off condition" introduced in [@Kato]. A trader has a certain quantity of shares of a security at the initial time, and must liquidate all of them by the time horizon. Then, the space of admissible strategies is reduced to $$\begin{aligned}
\mathcal {A}^{\mathrm {SO}}_t(\varphi ) =
\left \{ (\zeta _r)_r\in \mathcal {A}_t(\varphi )\ ; \
\int ^t_0\zeta _r dr = \varphi \right \} . \end{aligned}$$ We define a value function with the sell-off condition by $$\begin{aligned}
V^{\mathrm {SO}}_t(w,\varphi ,s ; U) &=& \sup _{(\zeta _r)_r\in \mathcal {A}^{\mathrm {SO}}_t(\varphi )}{\mathop {\rm E}}[U(W_t)] \end{aligned}$$ for a continuous, non-decreasing and polynomial growth function $U : \Bbb {R}\longrightarrow \Bbb {R}$.
The following theorem is analogous to Theorem 4.1 in [@Kato] (we omit the proof because it is nearly identical):
\[thesame\]$V^{\mathrm {SO}}_t(w,\varphi ,s ; U) = V_t(w,\varphi ,s ; u)$, where $u(w, \varphi , s) = U(w)$.
By Theorem \[thesame\], we see that the sell-off condition does not introduce changes in the value of the value function in a continuous-time model.
Analogously to Theorem 4.2 in [@Kato], a similar result to Theorem 3 in [@Lions-Lasry] holds when $g(\zeta )$ is linear:
\[th\_LL\]Assume $g(\zeta ) = \alpha _0\zeta $ for $\alpha _0> 0$.\
$\mathrm {(i)}$ $V^\mathrm {SO}_t(w, \varphi , s ; U) = \overline{V}^\varphi _t
\left( w + \frac{1 - e^{-\gamma \alpha _0 \varphi }}{\gamma \alpha _0}s, e^{-\gamma \alpha _0 \varphi }s ; U\right)$, where $$\begin{aligned}
\overline{V}^\varphi _t(\bar{w}, \bar{s} ; U) &=& \sup _{(\overline{\varphi }_r)_r\in \overline {\mathcal {A}}_t(\varphi )}
{\mathop {\rm E}}[U(\overline{W}_t)]\\
&&\hspace{1mm}\mathrm {s.t.}\hspace{3.3mm}d\overline{S}_r =
e^{-\gamma \alpha _0\overline{\varphi }_r}\hat{b}(\overline{S}_re^{\gamma \alpha _0\overline{\varphi }_r})dr +
e^{-\gamma \alpha _0 \overline{\varphi }_r}\hat{\sigma }(\overline{S}_re^{\gamma \alpha _0 \overline{\varphi }_r})dB_r\\
&&\hspace{19mm} -\overline{S}_{r-}dG_r, \\
&&\hspace{8mm}d\overline{W}_r = \frac{e^{\gamma \alpha _0\overline{\varphi }_r} - 1}{\gamma \alpha _0}d\overline{S}_r, \\
&&\hspace{11mm}\overline{S}_0 = \bar{s}, \ \ \overline{W}_0 = \bar{w} \end{aligned}$$ and $$\begin{aligned}
\overline {\mathcal {A}}_t(\varphi ) &=& \left\{ \left( \varphi - \int ^r_0\zeta _vdv\right) _{0\leq r\leq t}\ ; \
(\zeta _r)_{0\leq r\leq t} \in \mathcal {A}^\mathrm {SO}_t(\varphi )\right\} , \\
G_r &=& \int_0^r \int_{(0, \infty)}(1-e^{-\alpha _0\zeta _sz})N(ds, dz). \end{aligned}$$ $\mathrm {(ii)}$ If $U$ is concave and $\hat{b} (s)\leq 0$ for $s\geq 0$, then $$\begin{aligned}
\label{eq_LL}
V^\mathrm {SO}_t(w, \varphi , s ; U) =
U\left( w + \frac{1 - e^{-\gamma \alpha _0 \varphi }}{\gamma \alpha _0 }s\right) . \end{aligned}$$
The proof is in Section \[sec\_proof\_th\_LL\]. Note that the assertion (ii) is the same as Theorem 3 in [@Lions-Lasry], and in this case we can also obtain the explicit form of the value function. The right side of (\[eq\_LL\]) is equal to $Ju(w,\varphi ,s)$ for $u(w, \varphi , s) = U(w)$ and the nearly optimal strategy for $V^{\mathrm {SO}}_t(w,\varphi ,s ; U) = V_t(w,\varphi ,s ; u)$ is given by (\[almost\_block\]). This implies that when considering a linear MI function, a risk-averse (or risk-neutral) trader’s optimal liquidation strategy with negative risk-adjusted drift is nearly the same as block liquidation (i.e., selling all shares at once) at the initial time.
Effect of Uncertainty in MI in the Risk-neutral Framework {#section_comparison}
=========================================================
The purpose of this section is to investigate how noise in the MI function affects the trader. Particularly, we focus on the case where the trader is risk-neutral, that is, $u(w, \varphi , s) = u_{\mathrm {RN}}(w, \varphi , s) = w$. Note that such a risk-neutral setting is a typical and standard assumption in the study of the execution problem (see e.g. [@Alfonsi-Fruth-Schied; @Cheng-Wang; @Kato2; @Kato_VWAP; @Konishi-Makimoto; @Makimoto-Sugihara; @Schied-Zhang]).
First, we prepare a value function of the execution problem with a deterministic MI function to compare with the case of random MI. Let $\bar{V}_t(w, \varphi , s ; u)$ be the same as in (\[def\_conti\_v\]) by replacing $g(\zeta )$ and $L_t$ with $\tilde{\gamma }g(\zeta )$ and $t$, that is, the SDE for $(X_r)_r$ is given as $$\begin{aligned}
dX_r = \sigma (X_r)dB_r + b(X_r)dr - \tilde{\gamma }g(\zeta _r)dr, \end{aligned}$$ where $$\begin{aligned}
\label{def_tilde_gamma}
\tilde{\gamma } = {\mathop {\rm E}}[L_1] = \gamma + \int _{(0, \infty )}z\nu (dz). \end{aligned}$$ The following proposition is proved in Section \[sec\_proof\_th\_comp\_noise\]:
\[th\_comp\_noise\]We have $$\begin{aligned}
\label{comp_noise}
V_t(w, \varphi , s ; u_\mathrm {RN}) \geq \bar{V}_t(w, \varphi , s ; u_\mathrm {RN}). \end{aligned}$$
This proposition shows that noise in MI is welcome because it decreases the liquidation cost for a risk-neutral trader.
For instance, we consider a situation where the trader estimates the MI function from historical data and tries to minimize the expected liquidation cost. Then, a higher sensitivity of the trader to the volatility risk of MI results in a lower estimate for the expected proceeds of the liquidation. This implies that accommodating the uncertainty in MI makes the trader prone to underestimating the liquidation cost. Thus, as long as the trader’s target is the expected cost, the uncertainty in MI is not an incentive for being conservative with respect to the unpredictable liquidity risk. In Section \[section\_examples\], we present the results of numerical experiments conducted to simulate the above phenomenon.
Examples {#section_examples}
========
In this section, we show two examples of our model, which are both generalizations of the ones in [@Kato].
Motivated by the Black–Scholes-type market model, we assume that $b(x) \equiv -\mu $ and $\sigma (x) \equiv \sigma $ for some constants $\mu , \sigma \geq 0$ and assume that $\tilde{\mu } := \mu - \sigma ^2/2$ is positive. We also assume a risk-neutral trader with utility function $u(w, \varphi ,s) = u_\mathrm {RN}(w) = w$. In this case, if there is no MI, then a risk-neutral trader will fear a decrease in the expected stock price, and thus will liquidate all the shares immediately at the initial time.
We consider MI functions that are log-linear and log-quadratic with respect to liquidation speed, and assume Gamma-distributed noise; that is, $g(\zeta ) = \alpha _0\zeta ^p$ for $\alpha _0 > 0$ and $p = 1, 2$, and $L_t$ satisfies $$\begin{aligned}
P(L_t - \gamma t\in dx) &=& \mathrm {Gamma}(\alpha _1t, \beta _1)(dx) \\
&:=&
\frac{1}{\Gamma (\alpha _1t)(\beta _1)^{\alpha _1t}}x^{\alpha _1t - 1}e^{-x/\beta _1} 1_{(0,\infty )}(x)\,dx, \end{aligned}$$ where $\Gamma (x)$ is the Gamma function. Here, $\alpha _1, \beta _1$, and $\gamma > 0$ are constants. The corresponding Lévy measure is $$\begin{aligned}
\nu (dz) = \frac{\alpha _1}{z}e^{-z/\beta _1}1_{(0, \infty )}(z)\,dz. \end{aligned}$$ Note that for the discrete-time model studied in [@Ishitani-Kato_COSA1], we can define the corresponding discrete-time MI function as $g^n_k(\psi ) = c^n_kg_n(\psi )$, where $g_n(\psi ) = n^{p-1}\alpha _0\psi ^p$ and $(c^n_k)_k$ is a sequence of i.i.d. random variables with distribution $$\begin{aligned}
P(c^n_k - \gamma \in dx) = \mathrm {Gamma}(\alpha _1/n, n\beta _1)(dx). \end{aligned}$$ In each case, assumptions \[A\], \[B1\]–\[B3\], and \[C\] of [@Ishitani-Kato_COSA1] are satisfied.
Log-Linear Impact & Gamma Distribution {#sec_linear_eg}
--------------------------------------
In this subsection, we set $g(\zeta ) = \alpha _0\zeta $ ($p=1$). Theorem \[th\_LL\] directly implies the following:
\[th\_eg\_random\] We have $$\begin{aligned}
\label{eg_Ju}
V_t(w,\varphi ,s ; u_\mathrm {RN}) = w+\frac{1-e^{-\gamma \alpha _0\varphi }}{\gamma \alpha _0}s \end{aligned}$$ for each $t\in (0,1]$ and $(w,\varphi ,s)\in D$.
The implication of this result is the same as in [@Kato]: the right side of (\[eg\_Ju\]) is equal to $Ju(w,\varphi ,s)$ and converges to $w+\varphi s$ as $\alpha _0\downarrow 0$ or $\gamma \downarrow 0$, which is the profit gained by choosing the execution strategy of block liquidation at $t = 0$. Therefore, the optimal strategy in this case is to liquidate all shares by dividing infinitely within an infinitely short time at $t = 0$ (we refer to such a strategy as a nearly block liquidation at the initial time). Note that the jump part of MI (\[jump\_term\]) does not influence the value of $V_t(w,\varphi ,s ; u_\mathrm {RN})$.
Log-Quadratic Impact & Gamma Distribution {#sec_quad_eg}
-----------------------------------------
Next we study the case of $g(\zeta ) = \alpha _0\zeta ^2$ ($p=2$). In [@Kato], we obtained a partial analytical solution to the problem: when $\varphi $ is sufficiently small or large, we obtain the explicit form of optimal strategies. However, the noise in MI complicates the problem, and deriving the explicit solution is more difficult. Thus, we rely on numerical simulations. Under the assumption that the trader is risk-neutral, we can assume that an optimal strategy is deterministic. Here, we introduce the following additional condition:\
$[D]$ $\gamma \geq \alpha _1\beta _1 / 8$.\
In fact, we can replace our optimization problem with the deterministic control problem $$\begin{aligned}
f(t, \varphi ) = \sup _{(\zeta _r)_r}\int ^t_0
\exp \left( -\int ^r_0q(\zeta _v)dv \right) \zeta _rdr \end{aligned}$$ for a deterministic process $(\zeta _r)_r$ under the above assumption, where $$\begin{aligned}
q(\zeta ) &= \tilde {\mu } + \hat{g}(\zeta ), \\
\hat{g}(\zeta ) &= \gamma \alpha _0\zeta ^2 + \alpha _1\log (\alpha _0\beta _1 \zeta ^2 + 1). \end{aligned}$$ This gives the following theorem:
\[th\_f\] $V_t(w, \varphi , s ; u_\mathrm {RN}) = w + sf(t, \varphi )$ under $[D]$.
This theorem is obtained by a similar proof to Proposition 5.1 in [@Kato] by using the following Laplace transform of the Gamma distribution: $$\begin{aligned}
{\mathop {\rm E}}[e^{- \lambda c^n_k}] =
\exp \left( -\gamma \lambda - \frac{ \alpha_{1}}{n}\log ( n\beta_{1}\lambda + 1) \right). \end{aligned}$$
From Theorem \[th\_f\] and (\[HJB\_V\]), we derive the HJB equation for the function $f$ as $$\begin{aligned}
\label{HJB_f_eg}
\frac{\partial }{\partial t}f + \tilde{\mu }f - \sup _{\zeta \geq 0}
\left\{ \zeta \left( 1 - \frac{\partial }{\partial \varphi }f\right) -
\hat{g}(\zeta )f\right\} = 0 \end{aligned}$$ with the boundary condition $$\begin{aligned}
\label{boundary_f}
f(0, \varphi ) = f(t, 0) = 0. \end{aligned}$$ When $\gamma \geq \alpha _1 / 2$, the function $\hat{g}$ becomes convex, so we can apply Theorems 3.3 and 3.6 in [@Kato] to show the following proposition:
\[prop\_eg\_HJB\] Assume $\gamma \geq \alpha _1/2$. Then $f(t, \varphi )$ is the viscosity solution of $(\ref {HJB_f_eg})$. Moreover, if $\tilde{f}$ is a viscosity solution of $(\ref {HJB_f_eg})$ and $(\ref {boundary_f})$ and has a polynomial growth rate, then $f = \tilde{f}$.
It is difficult to obtain an explicit form of the solution of (\[HJB\_f\_eg\]) and (\[boundary\_f\]). Instead, we solve this problem numerically by considering the deterministic control problem $f^n_{[nt]}(\varphi )$ in the discrete-time model for a sufficiently large $n$: $$\begin{aligned}
f^n_k(\varphi ) &=
\sup _{
\substack{(\psi ^n_l)^{k-1}_{l = 0}\subset [0, \varphi ]^k,\\
\sum _l\psi ^n_l \leq \varphi }
}
\sum ^{k-1}_{l = 0}\psi ^n_l\exp \left( -\tilde{\mu }\times \frac{l}{n} - \sum ^l_{m = 0}
I_m\right), \\
I_m &= n\gamma \alpha _0(\psi ^n_m)^2
+ \frac{\alpha _1}{n}\log (n^2\alpha _0\beta _1 (\psi ^n_m)^2 + 1). \end{aligned}$$ Note that the convergence $\lim_{n\to \infty} f^n_{[nt]}(\varphi ) = f(t,\varphi )$ is guaranteed by Theorem 2.3 of [@Ishitani-Kato_COSA1]. We set each parameter as follows: $\alpha _0 = 0.01, t = 1,
\tilde{\mu } = 0.05, w = 0, s = 1$, and $n = 500$. We examine three patterns for $\varphi$, $\varphi = 1, 10$, and $100$.
### The case of fixed $\gamma $ {#sec_fixed_gamma}
In this subsection, we set $\gamma = 1$ to examine the effects of the shape parameter $\alpha _1$ of the noise in MI. Here, we also set $\beta _1 = 2$. As seen in the numerical experiment in [@Kato], the forms of optimal strategies vary according to the value of $\varphi$. Therefore, we summarize our results separately for each $\varphi$.
Figure \[graph\_phi1\_0\] shows graphs of the optimal strategy $(\zeta _r)_r$ and its corresponding process $(\varphi _r)_r$ of the security holdings in the case of $\varphi=1$, that is, the number of initial shares of the security is small. As found in [@Kato], if there is no noise in the MI function (i.e., if $\alpha _1 = 0$), then the optimal strategy is to sell the entire amount at the same speed (note that the roundness at the corner in the left graph of Figure \[graph\_phi1\_0\] represents the discretization error and is not essential). The same tendency is found in the case of $\alpha _1 = 1$, but in this case the execution time is longer than in the case of $\alpha _1 = 0$. When we take $\alpha _1 = 3$, the situation is completely different. In this case, the optimal strategy is to increase the execution speed as the time horizon approaches.
When the amount of the security holdings is $10$, which is larger than in the case of $\varphi=1$, the optimal strategy and the corresponding process of the security holdings are as shown in Figure \[graph\_phi10\_0\]. In this case, a trader’s optimal strategy is to increase the execution speed as the end of the trading time approaches, which is the same as in the case of $\varphi=1$ with $\alpha _1 = 3$. Clearly, a larger value of $\alpha _1$ corresponds to a higher speed of execution closer to the time horizon. We should add that a trader cannot complete the liquidation when $\alpha _1 = 3$. However, as mentioned in Section \[sec\_SO\], we can choose a nearly optimal strategy from $\mathcal {A}^\mathrm {SO}_1(\varphi )$ without changing the value of the expected proceeds of liquidation by combining the execution strategy in Figure \[graph\_phi10\_0\] (with $\alpha _1 = 3$) and the terminal (nearly) block liquidation. See Section 5.2 of [@Kato] for details.
When the amount of the security holdings is too large, as in the case of $\varphi = 100$, a trader cannot complete the liquidation regardless of the value of $\alpha _1$, as Figure \[graph\_phi100\_0\] shows. This is similar to the case of $\varphi = 10$ with $\alpha _1 = 3$. The remaining amount of shares of the security at the time horizon is larger for larger noise in MI. Note that the trader can also sell all the shares of the security without decreasing the profit by combining the strategy with the terminal (nearly) block liquidation.
![Result for $\varphi = 100$ in the case of fixed $\gamma $. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100_0"}](zeta_1.eps "fig:") ![Result for $\varphi = 100$ in the case of fixed $\gamma $. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100_0"}](phi_1.eps "fig:")
![Result for $\varphi = 100$ in the case of fixed $\gamma $. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100_0"}](zeta_10.eps "fig:") ![Result for $\varphi = 100$ in the case of fixed $\gamma $. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100_0"}](phi_10.eps "fig:")
![Result for $\varphi = 100$ in the case of fixed $\gamma $. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100_0"}](zeta_100.eps "fig:") ![Result for $\varphi = 100$ in the case of fixed $\gamma $. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100_0"}](phi_100.eps "fig:")
### The case of fixed $\tilde{\gamma }$
In the above subsection, we presented a numerical experiment performed to compare the effects of the parameter $\alpha _1$ by fixing $\gamma $. Here, we perform numerical comparison from a different viewpoint.
The results in Section \[section\_comparison\] imply that accounting for the uncertainty in MI will cause a risk-neutral trader to be optimistic about the estimation of liquidity risks. To obtain a deeper insight, we investigate the structure of the MI function in more detail. In Theorems \[conti\_random\](ii) and \[th\_eg\_random\], the important parameter is $\gamma $, which is the infimum of $L_1$ and is smaller than or equal to ${\mathop {\rm E}}[L_1]$. We can interpret this as a characteristic feature whereby the (nearly) block liquidation eliminates the effect of positive jumps of $(L_t)_t$. However, there is another decomposition of $L_t$ such that $$\begin{aligned}
L_t = \tilde{\gamma }t + \int ^t_0\int _{(0, \infty )}z\tilde{N}(dr, dz), \end{aligned}$$ where $\tilde{\gamma }$ is given by (\[def\_tilde\_gamma\]) and $$\tilde{N}(dr, dz) = N(dr, dz) - \nu (dz)dr.$$ This representation is essential from the viewpoint of martingale theory. Here, $\tilde{N}(\cdot, \cdot)$ is the compensator of $N(\cdot, \cdot)$ and $\tilde{\gamma }$ can be regarded as the “expectation” of the noise in MI. Just for a risk-neutral world (in which a trader is risk-neutral), as studied in Section \[section\_comparison\], we can compare our model with the case of deterministic MI functions as in [@Kato] by setting $\tilde{\gamma } = 1$. Based on this, we conduct another numerical experiment with a constant value of $\tilde{\gamma }$.
Note that in our example $$\begin{aligned}
\label{temp_mean}
\tilde{\gamma } = \gamma + \alpha _1\beta _1\end{aligned}$$ and $$\begin{aligned}
\label{temp_var}
\frac{1}{t}\mathrm {Var}\left( \int ^t_0\int _{(0, \infty )}z\tilde{N}(dr, dz)\right) = \alpha _1\beta ^2_1 \end{aligned}$$ hold. Here, (\[temp\_mean\]) (respectively, (\[temp\_var\])) corresponds to the mean (respectively, the variance) of the noise in the MI function at unit time. Comparisons in this subsection are performed with the following assumptions: We set the parameters $\beta _1$ and $\gamma $ to satisfy $$\begin{aligned}
\gamma + \alpha _1\beta _1= 1, \ \ \alpha _1\beta _1^2= 0.5. \end{aligned}$$ We examine the cases of $\alpha _1 = 0.5$ and $1$, and compare them with the case of $\gamma = 1$ and $\alpha _1 = 0$.
Figure \[graph\_phi1\] shows the case of $\varphi = 1$, where the trader has a small amount of security holdings. Compared with the case in Section \[sec\_fixed\_gamma\], the forms of all optimal strategies are the same; that is, the trader should sell the entire amount at the same speed. The execution times for $\alpha _1 > 0$ are somewhat shorter than for $\alpha _1 = 0$.
Figure \[graph\_phi10\] corresponds to the case of $\varphi = 10$. The forms of the optimal strategies are similar to the case of $\varphi = 10$, $\alpha _1 = 0, 1$ in Section \[sec\_fixed\_gamma\]. Clearly, the speed of execution near the time horizon increases with increasing $\alpha _1$.
The results for $\varphi = 100$ are shown in Figure \[graph\_phi100\]. The forms of the optimal strategies are similar to the case of $\varphi = 100$ in Section \[sec\_fixed\_gamma\]. However, in contrast to the results in the previous subsection, the remaining amount of shares of the security at the time horizon is smaller for larger $\alpha _1$.
Finally, we investigate the total MI cost introduced in [@Kato2] (which is essentially equivalent to an implementation shortfall (IS) cost [@Almgren-Chriss; @Perold]): $$\begin{aligned}
\mathrm {TC}(\varphi) = -\log \frac{V_T(0, \varphi, s)}{\varphi s}. \end{aligned}$$ As noted at the beginning of this section, when the market is fully liquid and there is no MI, then the total proceeds of liquidating $\varphi$ shares of the security at $t=0$ are equal to $\varphi s$. In the presence of MI, however, the optimal total proceeds decrease to $V_T(0, \varphi, s) = \varphi s\times \exp (-\mathrm {TC}(\varphi))$. Thus, the total MI cost $\mathrm {TC}(\varphi)$ denotes the loss rate caused by MI in a risk-neutral world.
Figure \[graph\_MIcost\] shows the total MI costs in the cases of $\varphi = 1$ and $10$. Here, we omit the case of $\varphi = 100$ because the amount of shares of the security is too large to complete the liquidation unless otherwise combining terminal block liquidations (which may crash the market). In both cases of $\varphi= 1$ and $10$, we find that the total MI cost decreases by increasing $\alpha _1$. Since the expected value $\tilde{\gamma }$ of the noise in MI is fixed, an increase in $\alpha _1$ implies a decrease in $\gamma $ and $\beta _1$. Risk-neutral traders seem to be more sensitive to the parameter $\gamma $ than to $\alpha _1$, and thus the trader can liquidate the security without concern about the volatility of the noise in MI. Therefore, the total MI cost for $\alpha _1 > 0$ is lower than that for $\alpha _1 = 0$.
![Result for $\varphi = 100$ in the case of fixed $\tilde{\gamma }$. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100"}](fig_r_zeta_phi1.eps "fig:") ![Result for $\varphi = 100$ in the case of fixed $\tilde{\gamma }$. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100"}](fig_r_phi_phi1.eps "fig:")
![Result for $\varphi = 100$ in the case of fixed $\tilde{\gamma }$. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100"}](fig_r_zeta_phi10.eps "fig:") ![Result for $\varphi = 100$ in the case of fixed $\tilde{\gamma }$. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100"}](fig_r_phi_phi10.eps "fig:")
![Result for $\varphi = 100$ in the case of fixed $\tilde{\gamma }$. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100"}](fig_r_zeta_phi100.eps "fig:") ![Result for $\varphi = 100$ in the case of fixed $\tilde{\gamma }$. Left : The optimal strategy $\zeta _r$. Right : The amount of security holdings $\varphi _r$. []{data-label="graph_phi100"}](fig_r_phi_phi100.eps "fig:")
![Total MI cost $\mathrm {TC}(\varphi)$ for a risk-neutral trader. Left: the case of $\varphi = 1$. Right: the case of $\varphi = 10$. The horizontal axes denote the shape parameter $\alpha _1$ of the Gamma distribution.[]{data-label="graph_MIcost"}](fig_r_cost1_2_new.eps "fig:") ![Total MI cost $\mathrm {TC}(\varphi)$ for a risk-neutral trader. Left: the case of $\varphi = 1$. Right: the case of $\varphi = 10$. The horizontal axes denote the shape parameter $\alpha _1$ of the Gamma distribution.[]{data-label="graph_MIcost"}](fig_r_cost10_2_new.eps "fig:")
Concluding Remarks {#section_conclusion}
==================
In this paper, we studied an optimal execution problem with uncertain MI by using the model derived in [@Ishitani-Kato_COSA1]. Our main results discussed in Sections \[section\_Results\] and \[sec\_SO\] are almost the same as in [@Kato].
When considering uncertainty in MI, there are two typical barometers of the “level” of MI: $\gamma $ and $\tilde{\gamma }$. By using the parameter $\gamma $, we can decompose MI into a deterministic part $\gamma g(\zeta _t)dt$ and a pure jump part $g(\zeta _t)\int _{(0, \infty )}zN(dt, dz)$. Then, the pure jump part can be regarded as the difference from the deterministic MI case studied in [@Kato]. On the other hand, as mentioned in Sections \[section\_comparison\] and \[section\_examples\], the parameter $\tilde{\gamma }$ is important not only in martingale theory but also in a risk-neutral world. Studying $\tilde{\gamma }$ also provides some hints about actual trading practices. Regardless of whether we accommodate uncertainty into MI, it may result in an underestimate of MI for a risk-neutral trader.
Studying the effects of uncertainty in MI in a risk-averse world is also meaningful. As mentioned in Section \[sec\_SO\], when the deterministic part of the MI function is linear, the uncertainty in MI does not significantly influence the trader’s behavior, even when the trader is risk-averse. In future work, we will investigate the case of nonlinear MI.
Explicitly introducing trading volume processes is another important generalization. In some studies of the optimization problem of volume-weighted average price (VWAP) slippage, the trading volume processes are introduced as stochastic processes. For instance, [@Frei-Westray] studies a minimization problem of the tracking error of VWAP execution strategies (see [@Kato_VWAP] for a definition of VWAP execution strategies). In [@Frei-Westray], a cumulative trading volume process is defined as a Gamma process. Moreover, [@Kato_VWAP] treats a generalized Almgren–Chriss model such that a temporary MI function depends on instantaneous trading volume processes, and shows that an optimal execution strategy of a risk-neutral trader is actually the VWAP execution strategy. Since a trading volume process is unobservable, we can regard it as a source of the uncertainty of MI functions. Therefore, studying the case where MI functions are affected by trading volumes is within our focus.
Finally, in our settings the MI function is stationary in time, but in the real market the characteristics of MI change according to the time zone. Therefore, it is meaningful to study the case where the MI function is not time-homogeneous. This is another topic for future work.
Proofs {#sec_proof}
======
We first recall some lemmas from [@Ishitani-Kato_COSA1].
\[lemm\_conti\_u\] Let $\Gamma _k$ $(k\in \Bbb {N})$ be sets, $u\in {\mathcal {C}}$, and let $(W^i(k,\gamma ), \varphi ^i(k,\gamma ), S^i(k,\gamma ))\in D$ $(\gamma \in \Gamma _k$, $k\in \Bbb {N}$, $i=1,2)$ be random variables. Assume that $$\begin{aligned}
&&\lim _{k\rightarrow \infty }\sup _{\gamma \in \Gamma _k}
{\mathop {\rm E}}[|W^1(k,\gamma )-W^2(k,\gamma )|^{m_1} + |\varphi ^1(k,\gamma )-\varphi ^2(k,\gamma )|^{m_2}\\
&&\hspace{57mm} + |S^1(k,\gamma )-S^2(k,\gamma )|^{m_3}] = 0\end{aligned}$$ and $$\begin{aligned}
\sum ^2_{i=1}\sup _{k\in \Bbb {N}}\sup _{\gamma \in \Gamma _k}
{\mathop {\rm E}}[|W^i(k,\gamma )|^{m_4}+(S^i(k,\gamma ))^{m_4}] < \infty \end{aligned}$$ for some $m_1, m_2, m_3 > 0$ and $m_4 > m_u$, where $m_u$ is as appeared in $(\ref {growth_C})$. Then we have $$\begin{aligned}
&&\lim _{k\rightarrow \infty }\sup _{\gamma \in \Gamma _k}
\big| {\mathop {\rm E}}[u(W^1(k,\gamma ), \varphi ^1(k,\gamma ), S^1(k,\gamma ))] \\
&&\qquad \qquad \quad -
{\mathop {\rm E}}[u(W^2(k,\gamma ), \varphi ^2(k,\gamma ), S^2(k,\gamma ))]\big| = 0 .\end{aligned}$$
\[Lemma\_Moment\_Estimate\] Let $Z(t; r, s) = \exp (Y(t; r, \log s))$ and $\hat{Z}(s) = \sup_{0\leq r\leq 1}Z(r; 0, s)$. Then, for each $m>0$, there is a constant $C_{m, K}>0$ depending only on $K$ and $m$ such that $E [\hat{Z}(s)^m]\leq C_{m, K} s^m$, where $K>0$ is a constant appearing in (\[Bdd\_Lipschitz\_Constant\]).
\[Ishi\_Lem\] Let $(X^{k, i}_r)_{r\in [0,1]}$, $i = 1, 2$, $ k \in \Bbb {N}$, be $\mathbb{R}$-valued $(\mathcal {F}_r)_r$-progressive processes satisfying $$\begin{aligned}
X^{k, i}_r = x^{k, i} + \int ^r_0b(X^{k, i}_v)dv + \int ^r_0\sigma (X^{k, i}_v)dB_v + F^{k, i}_r ,
\ \ r\in [0,1],\end{aligned}$$ with $x^{k, i} \in \Bbb{R}$ for $i = 1, 2$ and $k \in \Bbb {N}$, where $(F^{k, i}_r)_r$ are $(\mathcal {F}_r)_r$-adapted processes of bounded variation, and let $\Pi_k \subset [0,1]$, $k \in \Bbb {N}$, be Borel sets. Moreover, assume that
(i)
: $x^{k, 1} - x^{k, 2}\longrightarrow 0, \ \ k\rightarrow \infty $,
(ii)
: $\lim_{k\to \infty} \left\{ D^k_1+\int_0^1 D^k_r dr \right\}= 0$, where $$\begin{aligned}
D^k_r = {\mathop {\rm E}}\left [\sup_{v\in \Pi_k(r) }|F^{k, 1}_v - F^{k, 2}_v|\right ], \ \
\Pi_k(r)=([0,r]\cap \Pi_k)\cup \{r\}. \end{aligned}$$
Then it holds that $$\begin{aligned}
{\mathop {\rm E}}\left [\sup_{v \in \Pi_k } \left| X^{k, 1}_v - X^{k, 2}_v \right| \right ]
\longrightarrow 0, \ \ k \rightarrow \infty . \end{aligned}$$
\[lem\_comparison\] Let $t\in [0,1]$, $\varphi \geq 0$, $x\in \Bbb {R}$, $(\zeta _r)_{0\leq r\leq t}, (\zeta '_r)_{0\leq r\leq t}
\in \mathcal {A}_t(\varphi )$ and suppose $(X_r)_{0\leq r\leq t}$ $($resp., $(X'_r)_{0\leq r\leq t}$$)$ is given by $(\ref {SDE_X_g})$ with $(\zeta _r)_r$ $($resp., $(\zeta '_r)_r$$)$ and $X_0 = x \leq X'_0$. Suppose $\zeta _r\leq \zeta '_r$ for any $r\in [0,t]$ almost surely. Then $X_r\geq X'_r$ for any $r\in [0,t]$ almost surely.
Proof of Theorem \[conti\_random\] {#subsec_conti}
-----------------------------------
Continuity in $(w, \varphi , s)$ can be easily proved in the same manner as in the previous study [@Kato], so we focus on the continuity in $t$ (uniformly on any compact subset of $D$).
First of all, we prove the following lemma:
\[eval\_0\] Assume $h(\infty ) = \infty $. Then, for any $t\in [0,1]$, $\varphi \in [0, \Phi_0]$, and $(\zeta _r)_{0\leq r\leq t}\in \mathcal {A}_t(\varphi )$, $$\begin{aligned}
\label{temp_lemma_1}
{\mathop {\rm E}}\Big[ \int ^r_0\exp \Big( -\int ^v_0g(\zeta _{v'})dL_{v'}\Big) \zeta _v dv \Big]
\leq \phi (r), \ \ r\in [0,t], \end{aligned}$$ where $\phi (r),\ r\in (0,1]$ is a continuous function depending only on function $h(\zeta )$ and $\Phi _0$, such that $\lim _{r\rightarrow 0}\phi (r) = 0$.
\[Proof of Lemma \[eval\_0\]\] We may assume that $\tilde{\gamma } > 0$. Let $\pi _r = \int ^r_0g(\zeta _v)dL_v$ and $\tau _R = \inf \{ v\in [0,t]\ ; \ \pi _v>R \} \wedge r$ for $r\in (0,t]$ and $R>0$. Since $(\pi _v)_v$ is nondecreasing and $(\exp (-\pi _{v-}) \zeta _v)_v$ is left-continuous, we have that $$\begin{aligned}
{\mathop {\rm E}}\Big[ \int ^r_0\exp ( -\pi _v) \zeta _v dv \Big] &\leq
{\mathop {\rm E}}\Big[ \int ^r_0\exp ( -\pi _{v-}) \zeta _v dv \Big] \nonumber
=\frac{1}{\tilde{\gamma}}{\mathop {\rm E}}\Big[ \int ^r_0\exp ( -\pi _{v-}) \zeta _v dL_v \Big] \\
&\leq \frac{1}{\tilde{\gamma}}{\mathop {\rm E}}\Big[ \int ^{(\tau _R+\varepsilon)\wedge r}_0 \zeta _v dL_v \Big]
+\frac{e^{-R}}{\tilde{\gamma}} {\mathop {\rm E}}\Big[ \int ^r_{(\tau _R+\varepsilon)\wedge r}\zeta _v dL_v \Big]
\label{Ineq_E_int_0_to_r_exp_mPIv_zeta_v_dv}\end{aligned}$$ holds for $r\in (0,t]$, $R>0$ and $\varepsilon >0$. Using the left-continuity of $(\zeta _v)_v$, we obtain $$\begin{aligned}
\frac{e^{-R}}{\tilde{\gamma}} {\mathop {\rm E}}\Big[ \int ^r_{(\tau _R+\varepsilon)\wedge r}\zeta _v dL_v \Big]
\leq \frac{e^{-R}}{\tilde{\gamma}} {\mathop {\rm E}}\Big[ \int ^r_{0}\zeta _v dL_v \Big]
=e^{-R}\int ^r_{0}{\mathop {\rm E}}[ \zeta _v ] dv \leq \Phi _0e^{-R}. \end{aligned}$$ The first term on the right side of (\[Ineq\_E\_int\_0\_to\_r\_exp\_mPIv\_zeta\_v\_dv\]) is rewritten as $$\begin{aligned}
\label{Ineq_E_int_0_to_tauR_zeta_v_dv}
\frac{1}{\tilde{\gamma}}{\mathop {\rm E}}\Big[ \int ^{(\tau _R+\varepsilon)\wedge r}_0 \zeta _v dL_v \Big]
=r {\mathop {\rm E}}\Big[ \int^r_0 \zeta _v 1_{[0, \tau _R+\varepsilon]}(v) \frac{dL_v}{\tilde{\gamma}r}\Big]. \end{aligned}$$ Since $g(\zeta )$ is convex and $(\tilde{\gamma} r)^{-1}dL_v(\omega)P(d\omega)$ is a probability measure on\
$([0, r]\times \Omega, \mathcal{B}([0,r])\otimes \mathcal{F})$, we apply the Jensen inequality to obtain $$\begin{aligned}
& g\Big( {\mathop {\rm E}}\Big[ \int^r_0 \zeta _v 1_{[0, \tau _R+\varepsilon]}(v) \frac{dL_v}{\tilde{\gamma}r}\Big] \Big)
\leq {\mathop {\rm E}}\Big[ \int^r_0 g(\zeta _v 1_{[0, \tau _R+\varepsilon]}(v)) \frac{dL_v}{\tilde{\gamma}r}\Big]
=\frac{{\mathop {\rm E}}[ \pi_{(\tau _R+\varepsilon)\wedge r}]}{\tilde{\gamma}r} .\end{aligned}$$ Combining this with (\[Ineq\_E\_int\_0\_to\_tauR\_zeta\_v\_dv\]) we get $$\begin{aligned}
&\frac{1}{\tilde{\gamma}}{\mathop {\rm E}}\Big[ \int ^{(\tau _R+\varepsilon)\wedge r}_0 \zeta _v dL_v \Big]
\leq r g^{-1}\left( \frac{{\mathop {\rm E}}[ \pi_{(\tau _R+\varepsilon)\wedge r}]}{\tilde{\gamma}r} \right) ,\end{aligned}$$ where $g^{-1}(y) := \sup \{ \zeta \in [0,\infty ) \ ; \ g(\zeta ) = y \} $, $y\geq 0$. Since $\big( \int^v_0 \zeta_{v'}dL_{v'} \big)_v$ and $(\pi_v)_v$ are right-continuous, and $g^{-1}(y)$ is a continuous function on $y\in [0, \infty)$, we have that $$\begin{aligned}
&\frac{1}{\tilde{\gamma}}{\mathop {\rm E}}\Big[ \int ^{\tau _R}_0 \zeta _v dL_v \Big]
\leq
\lim_{\varepsilon \to 0}
r g^{-1}\left( \frac{{\mathop {\rm E}}[ \pi_{(\tau _R+\varepsilon)\wedge r}]}{\tilde{\gamma}r} \right)
=r g^{-1}\left( \frac{{\mathop {\rm E}}[ \pi_{\tau _R}]}{\tilde{\gamma}r} \right)
\leq r g^{-1}\left( \frac{R}{\tilde{\gamma}r} \right) . \end{aligned}$$ Summarizing the above arguments, we arrive at $$\begin{aligned}
{\mathop {\rm E}}\left[ \int ^r_0\exp ( -\pi _v) \zeta _v dv\right] \leq
r g^{-1}\left( \frac{R}{\tilde{\gamma}r}\right) + \Phi _0e^{-R}. \end{aligned}$$ Therefore, if we can find a positive function $R(r)$ that satisfies $$\begin{aligned}
\label{cond_Rr}
R(r) \longrightarrow \infty \ \ \mathrm {and} \ \
r g^{-1}\left (\frac{R(r)}{\tilde{\gamma}r}\right ) \longrightarrow 0\ \ \mathrm {as} \ \ r\rightarrow 0, \end{aligned}$$ we complete the proof of (\[temp\_lemma\_1\]). To construct such an $R(r)$, mimicking the proof of Lemma B.12 in [@Kato], we define $$\begin{aligned}
R(r) = \tilde{\gamma} r g(M(r)), \ \
M(r) = f^{-1}\left( \frac{1}{r}\right), \ \
f(\zeta ) = \zeta \sqrt{h\left( \frac{\zeta }{2}\right) }, \ \ r > 0, \end{aligned}$$ where the inverse function $f^{-1}(y)$ is defined in the same manner as $g^{-1}(y)$. We can easily verify (\[cond\_Rr\]) by the same arguments as in [@Kato].
The following proposition can be proved by the same proof as Theorem 3.1(ii) in [@Kato] in combination with Lemma \[eval\_0\] and Proposition \[th\_semi\].
Assume $h(\infty)=\infty$. Then for any compact set $E\subset D$, $$\begin{aligned}
\lim _{t\downarrow 0} \sup _{(w,\varphi ,s)\in E}
\vert V_{t}(w,\varphi ,s;u)-u(w, \varphi ,s)\vert = 0.\end{aligned}$$
Next we consider the case where $h(\infty)<\infty$. Hereinafter, for each $(w, \varphi, s) \in D$ and $(\zeta _r)_r\in \mathcal {A}_t(\varphi )$, we denote by $\Xi _t(w,\varphi ,s ; (\zeta _r)_r)$ the ordered triplet of processes $(W_r, \varphi _r, S_r)_{0\leq r\leq t}$ given by the differential equations in (\[SDE\_X\_g\]).
\[thm2\_pro1\] Assume $h(\infty ) < \infty $. Then for any compact set $E\subset D$ we have $$\begin{aligned}
\limsup _{t\downarrow 0 } \sup _{(w,\varphi ,s)\in E}
(Ju(w,\varphi ,s)-V_t(w,\varphi ,s;u))\leq 0. \end{aligned}$$
Take any $t\in (0,1)$, $(w,\varphi, s)\in E$, and $\psi \in [0, \varphi]$. Set $(\zeta _r)_r \in \mathcal{A}_t(\varphi )$ by $\zeta _r = \frac{\psi }{t}\ (0\leq r\leq t)$, and let $(W_r,\varphi _r, S_r)_{0\leq r\leq t}=\Xi _t(w, \varphi, s; (\zeta _r)_r)$ and $X_r=\log S_r$. A standard argument leads us to $$\begin{aligned}
{\mathop {\rm E}}\Big[ \sup_{r\in [0, t]}\vert \exp (X_r) - s\exp \left(-g({\psi}/{t})L_r \right) \vert \Big]
&\leq C_K s \sqrt{t}, \\
{\mathop {\rm E}}\left[ \big\vert W_t - w - \psi s \int_0^1 \exp \left(-g({\psi}/{t})L_{tv} \right) dv \big\vert \right]
&\leq C_K \psi s \sqrt{t}\end{aligned}$$ for some $C_K > 0$. Thus, using Lemma \[lemm\_conti\_u\], we get $$\begin{aligned}
\nonumber
&\sup_{\substack{(w, \varphi ,s)\in E \\ \psi \in [0,\varphi ]}}
\left\{
I_1((\zeta _r)_r) -
V_t(w,\varphi , s;u) \right\} \\
&\quad \leq \sup_{\substack{(w, \varphi ,s)\in E \\ \psi \in [0,\varphi ]}}
\left\{
I_1((\zeta _r)_r)
- {\mathop {\rm E}}[u(W_t,\varphi_t , \exp(X_t)) ]
\right\} \longrightarrow 0 \ \ \,\,t \downarrow 0,
\label{prop6_conv1} \end{aligned}$$ where $$\begin{aligned}
I_1((\zeta _r)_r) = {\mathop {\rm E}}[u(w+\psi s\int_0^1 \exp (-g(\psi /t)L_{tv})dv,\varphi - \psi ,
s\exp \left(-g(\psi /t)L_t \right)) ]. \end{aligned}$$
Next we will show $$\begin{aligned}
\label{LOLN}
\sup_{\substack{(w, \varphi ,s)\in E \\ \psi \in [0,\varphi ]}}
\left| I_1((\zeta _r)_r) - I_2((\zeta _r)_r)\right |
\longrightarrow 0, \ \ t\downarrow 0, \end{aligned}$$ where $$\begin{aligned}
I_2((\zeta _r)_r) = {\mathop {\rm E}}\Big[ u\Big( w+\psi s\int_0^1 \exp (-g(\psi /t)\gamma tv)dv,\varphi - \psi ,
s\exp \left(-g(\psi /t)\gamma t \right) \Big) \Big]. \end{aligned}$$ Theorem 9.43.20 in [@Sato] implies $$\begin{aligned}
\label{conv_L}
\lim _{t\downarrow 0 }\frac{L_t}{t} = \gamma \ \ \mathrm {a.s.} \end{aligned}$$ Hence, we obtain $$\begin{aligned}
&\sup_{\substack{(w, \varphi ,s)\in E \\ \psi \in [0,\varphi ]}}
{\mathop {\rm E}}\left[ \vert \exp(-g(\psi /t)\gamma t) - \exp(-g(\psi /t)L_t) \vert \right] \\
&\quad \leq {\mathop {\rm E}}\left[ 1 - \exp\left( t g(\varphi ^* /t)
\left\{ \gamma - \frac{L_t}{ t} \right\}\right) \right]
\longrightarrow 0, \ \ t\downarrow 0, \end{aligned}$$ where we denote $\varphi ^* := \sup _{(w, \varphi , s)\in E} \varphi$. Similarly, we obtain $$\lim_{t\downarrow 0 }\sup_{\substack{(w, \varphi ,s)\in E \\ \psi \in [0,\varphi ]}}
{\mathop {\rm E}}\left[ \left\vert \psi s \int_0^1
\left\{ \exp(-g(\psi /t)\gamma tv) -\exp(-g(\psi /t) L_{tv}) \right\} dv
\right\vert \right] = 0 .$$ Thus we get (\[LOLN\]) by using Lemma $\ref {lemm_conti_u}$.
We now complete the proof of Proposition \[thm2\_pro1\]. By the monotonicity of $u(w,\varphi ,s)$ (especially in $w$ and $s$) and the inequality $(0\leq ) t g(\psi /t)\leq \psi h(\infty)$, we see that $$\begin{aligned}
I_2((\zeta _r)_r) \geq u(w+F(\psi)s,\varphi - \psi , s e^{-\gamma h(\infty )\psi }) , \end{aligned}$$ where $$\begin{aligned}
F(\psi) = \int_0^{\psi}e^{-\gamma h(\infty )p}dp
= \psi \int^1_0 \exp (-\gamma h(\infty) \psi v) dv. \end{aligned}$$ Therefore, $$\begin{aligned}
\label{temp_F0}
\sup _{(w,\varphi ,s)\in E}(Ju(w,\varphi ,s)-V_t(w,\varphi ,s;u))
\leq
\sup_{\substack{(w, \varphi ,s)\in E \\ \psi \in [0,\varphi ]}}\left( I_2((\zeta _r)_r) - {\mathop {\rm E}}[u(W_t, \varphi _t, S_t)] \right) . \end{aligned}$$ Now our assertion is shown immediately from (\[prop6\_conv1\]), (\[LOLN\]), and (\[temp\_F0\]).
\[thm2\_pro2\] Assume $h(\infty ) < \infty $. Then for any compact set $E\subset D$ , $$\begin{aligned}
\limsup _{t\downarrow 0 } \sup _{(w,\varphi ,s)\in E}
(V_t(w,\varphi ,s;u)-Ju(w,\varphi ,s))\leq 0. \end{aligned}$$
Take any $t \in (0, 1)$, $(w, \varphi , s) \in E$, and $(\zeta_r)_r \in \mathcal{A}_t(\varphi)$. Denote\
$(W_r,\varphi _r, \allowbreak S_r)_{0\leq r\leq t}=\Xi _t(w, \varphi , s ; (\zeta _r)_r)$ and $X_r=\log S_r$. Since $g$ is convex, the Jensen inequality implies $$\begin{aligned}
\int_0^r g(\zeta _v)dL_v \geq
\gamma \int_0^r g(\zeta _v)dv
\geq \gamma r g\left( \frac{1}{r} \int_0^r \zeta_v dv\right)
= \gamma \int_0^{\eta_r}h(\zeta / r)d\zeta ,\quad r\in [0, t],\end{aligned}$$ where $\eta _r = \int_0^r \zeta_v dv $. Then we have $$\begin{aligned}
\label{Th2_Uniform_Estimate1}
&u\Big( w + s \int_0^t \zeta_r \exp \big(-\int_0^r g(\zeta _v)dL_v \big) dr,
\varphi - \eta_t, s e^{-\int_0^t g(\zeta _v)dL_v } \Big) \nonumber \\
&\quad \leq
u\Big( w + s \int_0^t \zeta_r \exp \big( -\gamma \int_0^{\eta_r}h(\zeta/r)d\zeta \big)dr ,
\varphi - \eta_t, s e^ {-\gamma \int_0^{\eta_t}h(\zeta /t)d\zeta } \Big) . \end{aligned}$$
As in the proof of Proposition \[thm2\_pro1\], we get $$\begin{aligned}
{\mathop {\rm E}}\Big[ \sup_{r\in [0, t]}\Big\vert \exp (X_r) - s\exp \big(-\int_0^r g(\zeta _v)dL_v \big) \Big\vert \Big]
&\leq C_K s \sqrt{t}, \label{Th2_Uniform_Estimate2}\\
{\mathop {\rm E}}\left[ \Big\vert W_t - w - s \int_0^t \zeta_r \exp \big(-\int_0^r g(\zeta _v)dL_v \big) dr \Big\vert \right]
&\leq C_K \Phi_0 s \sqrt{t} \label{Th2_Uniform_Estimate3}\end{aligned}$$ for some $C_K > 0$. Then we can apply Lemma \[lemm\_conti\_u\] with (\[Th2\_Uniform\_Estimate2\]) and (\[Th2\_Uniform\_Estimate3\]) to obtain $$\begin{aligned}
&\sup_{\substack{(w, \varphi ,s)\in E \\ (\zeta_r)_r \in \mathcal{A}_t(\varphi)}}
\Big\vert
{\mathop {\rm E}}\Big[ u\Big( w + s \int_0^t \zeta_r \exp \big(-\int_0^r g(\zeta _v)dL_v \big) dr,
\varphi - \eta_t, s e^{-\int_0^t g(\zeta _v)dL_v } \Big)\Big]
\nonumber \\
&\qquad \qquad \quad -
{\mathop {\rm E}}[u(W_t, \varphi_t, S_t)]
\Big\vert
\longrightarrow 0\ \ \mathrm {as} \ \ t \downarrow 0. \label{Th2_Uniform_Conv1}\end{aligned}$$
We can also see that $$\begin{aligned}
\sup_{r\in [0, t]}\Big\vert \exp \left( - \gamma \int_0^{\eta_r} h(\zeta / r) d\zeta \right)
- e^{- \gamma h(\infty ) \eta_r} \Big\vert
&\leq 2\gamma \widetilde{\varepsilon}_t , \label{Th2_Uniform_Estimate4}\\
\Big\vert {\mathop {\rm E}}\Big[ \int_0^t \zeta_r \Big\{
\exp \Big( -\gamma \int_0^{\eta_r}h(\zeta / r)d\zeta \Big)
-e^{- \gamma h(\infty ) \eta_r} \Big\} dr \Big] \Big\vert
&\leq 2\gamma \Phi_0 \widetilde{\varepsilon}_t , \label{Th2_Uniform_Estimate5}\end{aligned}$$ where $\widetilde{\varepsilon}_t = \int_0^{\Phi_0} \big( h(\infty) - h(\zeta /t) \big)d\zeta
(\longrightarrow 0, \ \ t\downarrow 0)$. Applying Lemma \[lemm\_conti\_u\] again with (\[Th2\_Uniform\_Estimate4\]) and (\[Th2\_Uniform\_Estimate5\]), we have that $$\begin{aligned}
&\sup_{\substack{(w, \varphi ,s)\in E \\ (\zeta_r)_r \in \mathcal{A}_t(\varphi)}}
\Big\vert
{\mathop {\rm E}}\Big[ u\Big( w + s \int_0^t \zeta_r \exp \Big( -\gamma \int_0^{\eta_r}h(\zeta / r)d\zeta \Big)dr ,
\varphi - \eta_t, s e^{-\gamma \int_0^{\eta_t}h(\zeta / t)d\zeta } \Big) \Big]\nonumber \\
&\qquad \qquad \quad - {\mathop {\rm E}}\Big[
u\Big(w + s \int_0^t \zeta_r e^{- \gamma h(\infty ) \eta_r} dr,
\varphi - \eta_t, s e^{- \gamma h(\infty ) \eta_t} \Big) \Big]
\Big\vert \longrightarrow 0\ \ \mathrm {as} \ \ t \downarrow 0. \label{Th2_Uniform_Conv2}\end{aligned}$$ Moreover, from the definition of $Ju(w, \varphi , s)$, we see that $$\begin{aligned}
\nonumber
&\sup_{\substack{(w, \varphi ,s)\in E \\ (\zeta_r)_r \in \mathcal{A}_t(\varphi)}}
\Big\{
{\mathop {\rm E}}\Big[ u\Big(w + s \int_0^t \zeta_r e^{- \gamma h(\infty ) \eta_r} dr,
\varphi - \eta_t, s e^{- \gamma h(\infty ) \eta_t}\Big) \Big]
-Ju(w, \varphi , s) \Big\} \\
&\quad = \sup_{\substack{(w, \varphi ,s)\in E \\ (\zeta_r)_r \in \mathcal{A}_t(\varphi)}}
\Big\{
{\mathop {\rm E}}\Big[ u\Big(w + s F(\eta _t), \varphi - \eta_t, s e^{- \gamma h(\infty ) \eta_t}\Big) \Big] - Ju(w, \varphi , s)
\Big\} \leq 0.
\label{Th2_Uniform_Ineqality3}\end{aligned}$$ Combining (\[Th2\_Uniform\_Estimate1\]), (\[Th2\_Uniform\_Conv1\]), (\[Th2\_Uniform\_Conv2\]), and (\[Th2\_Uniform\_Ineqality3\]), we obtain our assertion.
Finally, we consider the continuity with respect to $t\in (0, 1]$.
\[thm2\_pro3\] Let $E\subset D$ be a compact set. Then we have the following:\
$\mathrm {(i)}$ $\lim _{t'\uparrow t }
\sup _{(w,\varphi ,s)\in E}
\vert V_{t'}(w,\varphi ,s;u) - V_{t}(w,\varphi ,s;u)\vert = 0$, $t\in (0,1]$.\
$\mathrm {(ii)}$ $\lim _{t'\downarrow t }
\sup _{(w,\varphi ,s)\in E}
\vert V_{t'}(w,\varphi ,s;u) - V_{t}(w,\varphi ,s;u)\vert = 0$, $t\in (0,1)$.
All we have to do is to show that $$\begin{aligned}
JV_t(w,\varphi,s;u)\leq V_t(w,\varphi,s;u) ,
\quad (w, \varphi , s) \in D , \quad t \in (0, 1)
\label{obj_t_ineq}\end{aligned}$$ under $h(\infty ) < \infty $, because all the other assertions are obtained in the same way as in the proof of Proposition B.17 in [@Kato] combined with Proposition \[th\_semi\] and (\[obj\_t\_ineq\]).
Take any $t \in (0, 1)$, $(w, \varphi , s) \in D$, $\psi \in [0,\varphi ]$, and $(\zeta_r)_{0\leq r\leq t} \in \mathcal{A}_t(\varphi - \psi )$. Define $(W_r,\varphi _r, S_r)_{0\leq r\leq t}=\Xi _t(w+F(\psi)s, \varphi - \psi, s e^{-\gamma h(\infty) \psi} ; (\zeta _r)_r)$ and $X_r=\log S_r$. For any $\delta \in (0,t)$, we define $(\tilde{\zeta} _r)_{0\leq r\leq t}\in \mathcal{A}_t(\varphi)$ by $\tilde{ \zeta }_r = (\psi / \delta )1_{[0,\gamma \delta ]} (L_{r-}) + \zeta_r$. Note that the admissibility of $(\tilde {\zeta _r})_r$ comes from $L_r \geq \gamma r$. Furthermore, we denote $(\tilde{W}_r,\tilde{\varphi }_r,\tilde{S}_r)_{0\leq r\leq t} $ $=$ $\Xi _t(w, \varphi , s ; (\tilde{\zeta} _r)_r)$ and $\tilde{X}_r = \log \tilde{S}_r$.
From the definition, we have that $$\begin{aligned}
&X_r = \log s +
\int _0^r \sigma (X_v)dB_v + \int _0^r b(X_v)dv + F^{(\delta), 1}_r, \\
&\tilde{X}_r = \log s +
\int _0^r \sigma (\tilde{X}_v)dB_v +
\int _0^r b(\tilde{X}_v)dv + F^{(\delta), 2}_r, \quad \mbox{for}\ r\in [0 ,t], \end{aligned}$$ where $$\begin{aligned}
F^{(\delta), 1}_r = -\gamma h(\infty ) \psi - \int _0^r g(\zeta_v)dL_v,
\quad F^{(\delta), 2}_r =- \int _0^r g(\tilde{ \zeta }_v)dL_v . \end{aligned}$$ We will apply Lemma \[Ishi\_Lem\] with $F^{(\delta), 1}_r$, $F^{(\delta), 2}_r$, and $\Pi^{(\delta)} = [\delta , t]$ to show $$\begin{aligned}
\label{Prop8_IshiLem_result}
{\mathop {\rm E}}\Big[ \sup_{r\in [\delta ,t]} |\tilde{X}_r - X_r| \Big] \longrightarrow 0 , \ \
\delta \downarrow 0.\end{aligned}$$ Set $D^{(\delta)}_r = {\mathop {\rm E}}\Big[ \sup_{v\in \Pi^{(\delta)}(r)}
\vert F^{(\delta), 1}_v - F^{(\delta), 2}_v \vert \Big]$. Obviously it holds that $\Pi^{(\delta)}(r)=[\delta , r]$ ($r\geq \delta$), $\{r \}$ ($r< \delta$) and $$\begin{aligned}
D^{(\delta)}_t + \int _0^t D^{(\delta)}_r dr
\leq (2-\delta) {\mathop {\rm E}}\Big[ \sup_{v\in [\delta, t]}\vert F^{(\delta), 1}_v - F^{(\delta), 2}_v \vert \Big]
+ \int _0^{\delta} {\mathop {\rm E}}[\vert F^{(\delta), 1}_r - F^{(\delta), 2}_r \vert] dr . \end{aligned}$$ Since $(L_v)_v$ is nondecreasing, we see that $$\begin{aligned}
\tilde{u}(\delta):=\sup \{v\in [0, t]; L_{v-}\leq \gamma \delta \}
= \sup \{v\in [0, t]; L_{v}\leq \gamma \delta \} . \end{aligned}$$ Moreover, $\tilde{u}(\delta)\leq \delta $ holds from the definition of $(L_r)_r$. Then we have $$\begin{aligned}
&F^{(\delta), 2}_r-F^{(\delta), 1}_r = \gamma h(\infty ) \psi
- \frac{1}{\delta} \int^{r\wedge \tilde{u}(\delta)}_0
\Big\{ \int^{\psi}_0 h\left (\frac{1}{\delta}\zeta ' + \zeta_v\right ) d\zeta ' \Big\} dL_v
\label{Diff_F_delta_eq_1}\\
&\quad =h(\infty ) \psi \Big\{ \gamma - \frac{L_{r\wedge \tilde{u}(\delta)}}{\delta}\Big\}
+ \frac{1}{\delta} \int^{r\wedge \tilde{u}(\delta)}_0 \Big\{ \int^{\psi}_0
\Big( h(\infty) - h\left (\frac{1}{\delta}\zeta ' + \zeta_v\right ) \Big) d\zeta ' \Big\} dL_v
\nonumber \end{aligned}$$ for $0\leq r \leq t$. From (\[Diff\_F\_delta\_eq\_1\]), we have $$\begin{aligned}
&{\mathop {\rm E}}\Big[ \sup_{v\in [\delta, t]}\vert F^{(\delta), 1}_v - F^{(\delta), 2}_v \vert \Big] \nonumber \\
&\quad \leq h(\infty)\psi{\mathop {\rm E}}\left [ \gamma - \frac{ L_{\tilde{u}(\delta)}}{\delta }\right ] +
\gamma \int^{\psi}_0 \left( h(\infty)-h\left (\frac{1}{\delta}\zeta '\right )\right ) d\zeta ' ,
\label{temp_last_ineq_1}\\
&\int _0^{\delta} {\mathop {\rm E}}[\vert F^{(\delta), 1}_r - F^{(\delta), 2}_r \vert] dr
\leq \delta h(\infty ) \psi \gamma
+ \delta \tilde{\gamma} \int^{\psi}_0 \left ( h(\infty)-h\left (\frac{1}{\delta}\zeta '\right )\right ) d\zeta '.
\label{temp_last_ineq_2}\end{aligned}$$ The second terms of the right sides of both (\[temp\_last\_ineq\_1\]) and (\[temp\_last\_ineq\_2\]) converge to $0$ as $\delta \downarrow 0$. Moreover we can show the following lemma:
\[temp\_tilde\_u\] $\frac{\tilde{u}(\delta )}{\delta } \longrightarrow 1, \ \ \delta \downarrow 0$ a.s.
By the above lemma and (\[conv\_L\]), we have $$\begin{aligned}
\label{conv_Lu}
\frac{L_{\tilde{u}(\delta )}}{\delta}
\longrightarrow \gamma, \ \ \delta \downarrow 0
\ \ \mathrm {a.s.} \end{aligned}$$ Then the dominated convergence theorem implies that the first term of the right side of (\[temp\_last\_ineq\_1\]) also converges to $0$ as $\delta \downarrow 0$. Now we arrive at $$\begin{aligned}
D^{(\delta)}_t +
\int _0^t D^{(\delta)}_r dr \longrightarrow 0 , \ \
\delta \downarrow 0, \end{aligned}$$ which immediately implies (\[Prop8\_IshiLem\_result\]) together with Lemma \[Ishi\_Lem\].
A standard argument with (\[Prop8\_IshiLem\_result\]) gives $$\begin{aligned}
&{\mathop {\rm E}}\Big[\sup _{r\in [\delta ,t]} |\exp(\tilde{X}_r)-\exp(X_r)|^{1/2}\Big]
\nonumber \\
&\quad \leq
(2sC_{1, K})^{1/2}
{\mathop {\rm E}}\Big[\sup _{r\in [\delta ,t]} |\tilde{X}_r - X_r|\Big]^{1/2}
\longrightarrow 0, \ \ \delta \downarrow 0 . \label{Prop8_Conv_E_Diff_exp_X}\end{aligned}$$ On the other hand, we see that $$\begin{aligned}
{\mathop {\rm E}}[ \vert W_t - \tilde{W}_t \vert ^{1/2} ]
& \leq
J_1 + J_2 + J_3, \end{aligned}$$ where $$\begin{aligned}
J_1 &= {\mathop {\rm E}}\Big[\Big\vert
\frac{\psi}{\delta} \int^{\tilde{u}(\delta )}_0 \exp(\tilde{X}_r)dr
-s\int^{\psi}_0 e^{-\gamma h(\infty) p} dp
\Big\vert ^{1/2} \Big], \\
J_2 &= E\Big[ \Big\{ \int^t_{\delta} \zeta_r \vert \exp (\tilde{X}_r)- \exp (X_r) \vert dr \Big\}^{1/2} \Big] , \\
J_3 &= {\mathop {\rm E}}\Big[ \Big\{ \int^{\delta}_0 \zeta_r \vert \exp (\tilde{X}_r)- \exp (X_r) \vert dr \Big\}^{1/2} \Big] .\end{aligned}$$ Easily we get $$\begin{aligned}
&J_2
\leq \sqrt{\varphi - \psi} {\mathop {\rm E}}\Big[ \sup _{r\in [\delta ,t]} \big|e^{\tilde{X}_r}-e^{X_r}\big|^{1/2}\Big]
\longrightarrow 0, \ \ \delta \downarrow 0,
\nonumber \\
&J_3
\leq (\delta \|\zeta \|_{\infty})^{1/2}
{\mathop {\rm E}}\Big[ \sup _{r\in [0, \delta]} \{ e^{\tilde{X}_r}+e^{X_r} \}^{1/2}\Big]
\longrightarrow 0, \ \ \delta \downarrow 0
\nonumber\end{aligned}$$ by virtue of (\[Prop8\_Conv\_E\_Diff\_exp\_X\]) and Lemma \[Lemma\_Moment\_Estimate\]. As for $J_1$, a similar calculation to (\[Diff\_F\_delta\_eq\_1\]) gives $$\begin{aligned}
\nonumber
J_1
&\quad \leq \sqrt{sC_{1, K}\psi }{\mathop {\rm E}}\Big[1-\frac{\tilde{u}(\delta )}{\delta }\Big]^{1/2}\nonumber \\
&\qquad + \sqrt{\psi } {\mathop {\rm E}}\left[ \left( \frac{1}{\delta}\int^{\delta}_0
\left\vert \exp (\tilde{X}_r) - s \exp \left(-\frac{\gamma h(\infty) \psi r}{\delta}\right )\right\vert dr
\right)^{1/2} \right] \nonumber \\
&\quad \leq
\sqrt{sC_{1, K}\psi } {\mathop {\rm E}}\Big[1-\frac{\tilde{u}(\delta )}{\delta }\Big]^{1/2} +
\sqrt{ s(1+C_{1, K})\psi } \left\{ A^{1/2}_1 + A^{1/2}_2\right\} ,
\label{Prop8_decomp2_Diff_W} \end{aligned}$$ where $$\begin{aligned}
A_1 &= \frac{1}{\delta}{\mathop {\rm E}}\left [ \int^{\delta}_0 \left \{
\Big\vert \int^r_0 \sigma (\tilde{X}_v)dB_v \Big\vert
+\Big\vert \int^r_0 b(\tilde{X}_v)dv \Big\vert
+\Big\vert \int^r_0 g(\zeta_v)dL_v \Big\vert \right \} dr\right] , \\
A_2 &= \frac{1}{\delta}{\mathop {\rm E}}\left [ \int^{\delta}_0
\Big\vert \int^r_0 (g(\tilde{ \zeta }_v)-g(\zeta_v)) dL_v - \frac{\gamma h(\infty)\psi r}{\delta}\Big\vert
dr \right ] . \end{aligned}$$ Straightforward calculations lead us to $$\begin{aligned}
\label{Prop8_decomp4_Diff_W}
A_1 \leq
\frac{2K}{3}\sqrt{\delta } + \frac{(K + \tilde{\gamma }g(||\zeta ||_\infty ))\delta }{2} .\end{aligned}$$ Moreover, by Lemma \[temp\_tilde\_u\] and (\[conv\_Lu\]), we see that $$\begin{aligned}
A_2
&\quad \leq \gamma \int^{\psi}_0 (h(\infty)-h(\zeta'/\delta))d\zeta'
+\frac{ \psi h(\infty)}{\delta} {\mathop {\rm E}}\Big[ \frac{1}{\delta } \int^{\delta }_0
\vert \gamma r -L_{r\wedge \tilde{u}(\delta )}\vert dr \Big]
\nonumber \\
&\quad \leq \gamma \int^{\psi}_0 (h(\infty)-h(\zeta'/\delta))d\zeta'
+\psi h(\infty) {\mathop {\rm E}}\Big[ \frac{1}{\delta} \int^{\tilde{u}(\delta )}_0 \Big\{ \frac{L_r}{r}-\gamma \Big\} dr \Big]
\nonumber \\
&\qquad + \psi h(\infty) {\mathop {\rm E}}\Big[ \Big( 1-\frac{\tilde{u}(\delta )}{\delta }\Big)
\Big\{ \gamma \Big( 1-\frac{\tilde{u}(\delta )}{\delta }\Big)
+ \Big(\gamma - \frac{L_{\tilde{u}(\delta )}}{\delta}\Big)\Big\}\Big]
\longrightarrow 0, \ \ \delta \downarrow 0 . \label{Prop8_decomp6_Diff_W} \end{aligned}$$ Combining Lemma \[Lemma\_Moment\_Estimate\], Lemma \[lem\_comparison\], (\[Prop8\_decomp2\_Diff\_W\]), (\[Prop8\_decomp4\_Diff\_W\]), and (\[Prop8\_decomp6\_Diff\_W\]), we get $J_1\longrightarrow 0$ as $\delta \downarrow 0$, hence we arrive at $\lim_{\delta \downarrow 0}{\mathop {\rm E}}[ \vert W_t - \tilde{W}_t \vert ^{1/2} ] = 0$. Therefore, by Lemma \[lemm\_conti\_u\] we obtain $$\begin{aligned}
&{\mathop {\rm E}}[u(W_t,\varphi_t,\exp(X_t))] -V_t(w,\varphi,s;u)\\
&\quad \leq \lim_{\delta \downarrow 0}|{\mathop {\rm E}}[u(W_t,\varphi_t,\exp(X_t))] -
{\mathop {\rm E}}[u(\tilde{X}_t,\tilde{\varphi }_t,\exp(\tilde{X}_t))]| = 0. \end{aligned}$$ Since $(\zeta_r)_{0\leq r\leq t} \in \mathcal{A}_t(\varphi - \psi )$ is arbitrary, we get $$\begin{aligned}
V_t(w+F(\psi)s, \varphi - \psi,
s e^{-\gamma h(\infty) \psi};u)\leq V_t(w,\varphi,s;u). \end{aligned}$$ for an arbitrary $\psi \in [0, \varphi ]$. Now we complete the proof of (\[obj\_t\_ineq\]).
We may assume $\gamma > 0$. Fix any $\varepsilon \in (0, 1)$ and set $\varepsilon ' =\gamma \varepsilon / (1 - \varepsilon ) $. By (\[conv\_L\]), we see that for almost all $\omega $, there exists a $\delta _0 = \delta _0(\omega ) > 0$ such that $L_\delta / \delta < \gamma + \varepsilon '$ for each $\delta \in (0, \delta _0)$. Let $\delta _1 = \delta _1(\omega ) = (1 + \varepsilon ' / \gamma )^{-1}\delta _0$ and take any $\delta \in (0, \delta _1)$. Moreover, let $\delta ' = (1 + \varepsilon ' / \gamma )^{-1}\delta $. Then we see that $\delta ' < \delta _0$ and thus $L_{\delta '} < (\gamma + \varepsilon ')\delta ' = \gamma \delta $. By this inequality and the definition of $\tilde{u}(\delta )$, we get $1 \geq \tilde{u}(\delta ) / \delta \geq \delta ' / \delta = 1 - \varepsilon $, which implies the assertion.
Proof of Theorem \[th\_LL\] {#sec_proof_th_LL}
---------------------------
We can confirm assertion (i) by applying Itô’s formula to $\overline{S}_r$ and $\overline {W}_r$. By a similar argument to that in Section 7.9 in [@Kato], we obtain $$\begin{aligned}
{\mathop {\rm E}}[U(\overline {W}_t)]
&\leq &
U\Bigg( \bar{w} +
\int ^t_0{\mathop {\rm E}}\Bigg [ \frac{1 - e^{-\gamma \alpha _0\overline {\varphi }_r}}{\gamma \alpha _0}
\hat{b}(\overline {S}_re^{\gamma \alpha _0\overline {\varphi }_r})\\
&&\hspace{23mm} -
\int _{(0, \infty )}
\frac{e^{\gamma \alpha _0\overline {\varphi }_r} - 1}{\gamma \alpha _0}
\overline {S}_r(1-e^{-\alpha _0\zeta _rz})\nu (dz) \Bigg]dr\Bigg) \end{aligned}$$ for any $(\overline {\varphi }_r)_r\in \overline {\mathcal {A}}_t(\varphi )$ by virtue of the Jensen inequality. Since $\hat{b}$ is non-positive, the function $U$ is non-decreasing, and the terms $$\begin{aligned}
1 - e^{-\gamma \alpha _0\overline {\varphi }_r}, \
e^{\gamma \alpha _0\overline {\varphi }_r} - 1, \
1-e^{-\alpha _0\zeta _rz}\end{aligned}$$ are all non-negative, we see that ${\mathop {\rm E}}[U(\overline {W}_t)]\leq U(\overline {w})$ for any $(\overline {\varphi }_r)_r\in \overline {\mathcal {A}}_t(\varphi )$, which implies $\overline {V}^\varphi _t(\bar{w}, \bar{s})\leq U(\bar{w})$. The opposite inequality $\overline {V}^\varphi _t(\bar{w}, \bar{s})\geq U(\bar{w})$ is obtained similarly to the result in Section 7.9 in [@Kato]. This completes the proof.
Proof of Proposition \[th\_comp\_noise\] {#sec_proof_th_comp_noise}
----------------------------------------
The following proposition immediately leads us to (\[comp\_noise\]).
$V^n_k(w, \varphi , s ; u_\mathrm {RN}) \geq \bar{V}^n_k(w, \varphi , s ; u_\mathrm {RN})$, where $V^n_k$ is defined as in [@Ishitani-Kato_COSA1] and $\bar{V}^n_k$ is obtained from $V^n_k$ by replacing $c^n_k$ with $\tilde{\gamma }$.
We use the notation of [@Ishitani-Kato_COSA1]. Take any $(\psi ^n_l)_l\in \mathcal {A}^n_k(\varphi )$ and let $(W^n_l, \varphi ^n_l, S^n_l)_l = \Xi ^n_k(w, \varphi , s ; (\psi ^n_l)_l)$ be the triplet for $\bar{V}^n_k(w, \varphi , s ; u_{\mathrm {RN}})$. Since $c^n_l$ is independent of $\mathcal {F}^n_l$, the Jensen inequality implies $$\begin{aligned}
{\mathop {\rm E}}[W^n_k]
&=
w + \sum ^{k-1}_{l = 0}{\mathop {\rm E}}[\psi ^n_lS^n_l\exp (-{\mathop {\rm E}}[c^n_l | \mathcal {F}^n_l]g_n(\psi ^n_l))]\\
&\leq
w + \sum ^{k-1}_{l = 0}{\mathop {\rm E}}[\psi ^n_lS^n_l{\mathop {\rm E}}[\exp (-c^n_lg_n(\psi ^n_l)) | \mathcal {F}^n_l]] \leq
V^n_k(w, \varphi , s ; u_\mathrm {RN}). \end{aligned}$$ Since $(\psi ^n_l)_l$ is arbitrary, we obtain the assertion.
[**Acknowledgment.**]{} The authors are grateful to Prof. Tai-Ho Wang (Baruch College, The City University of New York) for helpful comments and discussions on the subject matter. In addition, the authors thank the reviewer for various comments and constructive suggestions to improve the quality of the paper.
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[^1]: \* This work was supported by a grant-in-aid from the Zengin Foundation for Studies on Economics and Finance.
|
[^1], [^2], [^3] [^4]
> [***Keywords***]{}: Backward stochastic equation, nonlinear § equation, optimal control, Wiener process.\
> [**2000 Mathematics Subject Classification:**]{} 60H15, 35Q40, 49K20, 35J10.
Introduction
============
We consider the controlled stochastic system governed by the nonlinear § equation $$\label{equa-x}
\barr{rcll}
idX(t,\xi)&\!\!=\!\!&\D X(t,\xi)dt+\lbb|X(t,\xi)|^{\a-1}X(t,\xi)dt -i\mu(\xi)X(t,\xi)dt \\
&& +V_0(\xi)X(t,\xi)dt+\dd\sum\limits^m_{j=1} u_j(t)V_j(\xi)X(t,\xi)dt \\
&&\dd+iX(t,\xi)dW(t,\xi),\ \ \ t\in(0,T), \ \xi\in\rr^d,\vsp
X(0)&\!\!=\!\!&x\ \mbox{ in }\rr^d. \earr$$ Here $\lbb=\pm1,\ \a>1,\ V_j\in W^{1,\9}(\rr^d),\ 0\leq j\leq m,$ are real valued functions, $W$ is the Wiener process, $$\label{e2}
W(t,\xi)=\sum^N_{j=1}\mu_j e_j(\xi)\b_j(t),\ t\ge0,\ \xi\in\rr^d,$$ and $$\barr{rcl}
u(t)&=&(u_1(t),...,u_m(t)) \in \bbr^m,\ t\in(0,T),\vsp
\mu(\xi)&=&\dd\frac12\sum^N_{j=1}|\mu_j|^2e^2_j(\xi), \xi\in\rr^d,\ d\ge1,\earr$$ with $\mu_j$ purely imaginary numbers (i.e. $Re \mu_j =0$), $e_j(\xi)$ real-valued functions and $\b_j$ independent real Brownian motions on a probability space $(\Omega,\calf,\mathbb{P})$ with natural filtration $(\calf_t)_{t\ge0}$, $1\le j\le N.$ For simplicity, we assume $N<\9$, but the arguments in this paper easily extend to the case where $N= \9$.
The physical significance of is well known. $X=X(t,\xi,\oo),$ $\xi\in\rr^d$, $t\ge0,$ $\oo\in\ooo$, represents the quantum state at time $t$, while the stochastic perturbation $iX\,dW$ represents a stochastic continuous measurement via the pointwise quantum observables $R_j(X)=\mu_je_jX$. The functin $V_0$ describes an external potential.
In the conservative case considered in this paper (i.e. $Re \mu_j=0$, $1\leq j\leq N$), $-i\mu X dt + i X dW$ is indeed the Stratonovitch differential. It follows by Itô’s formula that $|X(t)|^2_{L^2}=|x|^2_{L^2}$, $\ff t\ge0$. Hence, normalizing the initial state $|x|_{L^2}=1$, we have $|X(t)|_{L^2}=1$, $\ff t\in[0,T],$ and so, the quantum system evolves on the unit ball of $L^2$ and verifies the conservation of probability. See e.g. [@4; @5].
We also mention that, for the general case when $\mu_j$ are complex numbers, one of the main feature is that the mean norm square $|X(t)|_2^2$, $t\in [0,T]$, is a continuous martingale. This fact enables one to define the “physical” probability law and implies the conservation of $\bbe |X(t)|_2^2$, $t\in[0,T]$, which plays an important role in the application to open quantum systems. See, e.g., [@bg] for more details. See also [@1; @2] for global well-posedness with exponents of the nonlinearity in the optimal subcritical range.
As regards the real valued input control $u$, in most situations it represents an external applied force due to the interaction of the quantum system with an electric field or a laser pulse applied to a quantum system.
Here we shall study an optimal control problem associated with the control system which, in a few words, can be described as follows (see Problem (P) below): find an input control $u$ that steers in time $T$ the state $X$ as close as possible of a target state $\bbx_T$ and a given trajectory $\bbx_1$, and with a reasonable minimum energy. Roughly speaking, this means to find the quantum mechanical potential $u$ from observation of the quantum state $X(T)$ at the end of time interval $[0,T]$.
It should be mentioned that, there is an extensive literature on the deterministic bilinear control equation mainly concerned with exact controllability in time $T$ of § equations or with the optimal control problem (see, for instance, [@5b], [@5a], [@11a], [@11], [@12], [@13]). However, there are very few results on optimal control problems governed by nonlinear § equations and, to the best of our knowledge, none for stochastic control systems with linear multiplicative noise. In the latter case, the existence of an optimal control is largely an open problem, since the cost functional is not simultaneously lower semicontinuous and coercive in the basic control space.
The approach we used here is based on Skorohod’s representation theorem and Ekeland’s variational principle, and this is one of the main novelties of this work. The approach is also based on an existence result of the linearized backward dual stochastic equation, which is also new in the literature and uses sharp stochastic estimates for linear Schrödinger equations with time dependent coefficients (see [@1; @2]). As a matter of fact, a great effort of this work is dedicated to this issue.
Formulation of problem and the main results {#PROBLEM-RESULT}
===========================================
To begin with, we recall the definition of a strong solution to equation (see [@1], [@2]).
\[d2.1\]Let $x\in L^2$ (resp. $H^1$), $0<T<\9$. Let $\a$ satisfy $1<\a<1+\frac 4 d$ (resp. $1<\a<1+\frac{4}{(d-2)_+}$), $d\geq 1$. A strong $L^2$-(resp. $H^1$-)solution to on $[0,T]$ is an $L^2$-(resp. $H^1$-)valued continuous $(\calf_t)_{t\ge0}$-adapted process $X=X(t)$ such that $|X|^{\a-1}X\in L^1(0,T;H^{-1}),$ and $\pas$, $$\begin{aligned}
\label{e2.1}
X(t) =& x-\int^t_0 \bigg( i\Delta X(s)+\mu X(s)+\lbb i|X(s)|^{\a-1} X(s,\xi) +iV_0(\xi)X(s) \nonumber \\
& \qquad\qquad +i\dd\sum^m_{j=1}u_j(s)V_j(\xi)X(s) \bigg)ds
+ \int^t_0X(s)dW(s),\ t\in[0,T],\end{aligned}$$ as an Itô equation in $H^{-2}$ (resp. $H^{-1}$).
It is easy to check that $\int^t_0X(s)dW(s)$ in Definition \[d2.1\] is an $L^2$-(resp. $H^1$-)valued continuous stochastic integral. (We refer, e.g., to [@6; @LR15] for the general theory of infinite dimensional stochastic integrals.)
Following [@1; @2], we introduce the hypotheses below.
1. $1<\a<1+ \frac 4 d$. For each $1\leq j\leq N$, $e_j\in C^\9_b(\rr^d)$ satisfies $$\begin{aligned}
\label{decay}
\lim_{|\xi|\to\9}\zeta(\xi)|\pp^\g e_j(\xi)|=0,\end{aligned}$$ where $\g$ is a multi-index such that $1\leq |\g|\le 2$, and $$\zeta(\xi)=\left\{\barr{ll}
1+|\xi|^2,&\mbox{if }d\ne2,\vsp
(1+|\xi|^2(\ln(3+|\xi|^2))^2,\ &\mbox{if }d=2.\earr\right.$$
2. In the defocusing case $\lbb=-1$, $1 < \a < 1+ \frac{4}{(d-2)_+}$, and in the focusing case $\lbb=1$, $1< \a < 1+ \frac{4}{d}$. For each $1\leq j\leq N$, $e_j\in C^\9_b(\rr^d)$ satisfies for any multi-index $1\leq |\g| \leq 3$.
The global existence, uniqueness and uniform estimates of the solution to used in this paper are summarized in Proposition \[Pro-Equ-X\] below.
\[Pro-Equ-X\] Assume $(H0)$ (resp. $(H1)$). For each $x\in L^2$ (resp. $H^1$), $u\in \calu_{ad}$ and $0<T<\9$, there exists a unique strong $L^2$-(resp. $H^1$-)solution $X^u$ to , satisfying $|X(t)|_2 = |x|_2$, $t\in [0,T]$ (resp. for any $\rho \geq 1$, $$\begin{aligned}
\label{bdd-Xn-H1-Wpq}
\sup\limits_{u\in \calu_{ad}}\ \bbe \|X^u\|^\rho_{C([0,T]; H^1)} < \9).\end{aligned}$$
Moreover, assuming that the exponent $\a$ is in the range specified in $(H1)$ and that $e_k$ are constants, $1\leq k\leq N$, we have for any $\rho \geq 1$, $$\begin{aligned}
\label{Cons-bdd-Xn-Lpq}
\sup\limits_{u\in \calu_{ad}} ( \|X^u\|_{L^\9(\Omega; L^q(0,T; L^{p}))} + \|X^u\|_{L^\rho(\Omega; L^q(0,T; W^{1,p}))}) < \9,\end{aligned}$$ where $(p,q)$ is any Strichartz pair, i.e., $(p,q) \in [2,\9] \times [2,\9], \frac{2}{q} = \frac{d}{2} - \frac{d}{p}$, if $d\not = 2$, or $(p,q) \in [2,\9) \times (2,\9], \frac{2}{q} = \frac{d}{2} - \frac{d}{p}$, if $d = 2$.
The global existence and uniqueness can be proved similarly as in [@1; @2] by the rescaling approach and the Strichartz estimates for lower order perturbations of the Laplacian. We refer to [@1 Lemma 4.1] and [@2 Lemma 2.7] for explicit formulations of Strichartz estimates in the $L^p$ and Sobolev spaces respectively. The technical proof of the estimates and is postponed to the Appendix for simplicity of the exposition.
In the following, let $L^2_{ad}(0,T;\rr^m)$ denote the space of all $(\calf_t)_{t\ge0}$-adapted $\rr^m$-valued processes $u:[0,T]\to\rr^m$ such that $u\in L^2((0,T)\times\ooo;\rr^m)$. Similarly, $L^2_{ad}(0,T;L^2(\ooo;L^2))$ denotes the space of $L^2$-valued $(\calf_t)_{t\ge0}$-adapted processes $u$ such that $\E\int^T_0|u(t)|^2_2dt<\9.$
The optimal control problem we study in the following is
[*Minimize $$\E\(\!|X(T)-\mathbb{X}_T|^2_2+\g_1\!\dd\int^T_0\!\!|X(t)-\bbx_1(t)|^2_2dt
+\!\dd\int^T_0\!\! (\g_2|u(t)|^2_m + \g_3|u'(t)|^2_m) dt\!\)$$\
on all $(X,u)\in L^2_{ad}(0,T; L^2(\Omega; L^2)) \times \calu_{ad}$ subject to .*]{}Here, $\g_j\ge0$, $1\leq j\leq 3$, $\bbx_T\in L^2(\ooo,\calf_T, \mathbb{P} ;L^2)$ and $\bbx_1\in L^2_{ad}(0,T;L^2(\ooo;L^2))$ are given. In most situations, $\bbx_1$ is a given trajectory of the uncontrolled system or, in particular, a steady state solution. The admissible set $\calu_{ad}$ is defined by $$\begin{aligned}
\label{e2.4}
\calu_{ad} =&\bigg\{u\in L^2_{ad}(0,T;\rr^m);\ u\in U,\ \ a.e.\ on\ (0,T)\times\ooo. \bigg\},\end{aligned}$$ where $U$ is a compact convex subset of $\rr^m$. Let $D_U$ denote the diameter of $U$. Then, $\sup_{u\in \calu_{ad}}\|u\|_{L^\9(0,T; \bbr^m)} \leq D_U <\9$.
As seen earlier, due to the conservation of $|X(t)|_2^2$, by normalizing the initial state we have $ |X(t)|_{2}=1$, and so Problem (P) reduces to $$\begin{aligned}
&\st{(u,X)}{\rm Min} \bigg\{ - 2 \bbe Re \<X(T),\bbx_T\>_2
-2 \g_1 \int_0^T Re \<X(t), \bbx_1(t)\>_2 dt \\
&\qquad \qquad \ + \int_0^T(\g_2 |u(t)|_m^2 + \g_3 |u'(t)|_m^2 ) dt .
\bigg\}\end{aligned}$$
It should be said that in the quantum model $V$ is a given potential which describes the spatial profile of an external field, while the control input $u =\{u_j \}^m_{j=1}$ is its intensity. The objective of the control process is to steer the quantum system from an initial state $x$ to a target state $\bbx_T$ and also in the neighborhood of a given trajectory $\bbx_1$. The last term in the cost functional is the energy cost to obtain the desired objective.
Taking into account that in quantum mechanics the wave function $X$ is not a physical observable, a more realistic situation is where in the cost functional $|X(T)-\bbx_T|^2_2$ is replaced by $\<Q(X(T))-\bbx_T,X(T)-\bbx_T\>_2$, where $Q$ is a self-adjoint operator in $L^2$. However, its treatment is essentially the same.
By $\Phi:L^2_{ad}(0,T;\rr^m)\to\rr$ we denote the objective functional $$\begin{aligned}
\label{def-Phi}
\Phi(u)
=& \bbe |X^u(T)-\bbx_T|_2^2
+ \g_1 \bbe \int_0^T |X^u(t)-\bbx_1(t)|_2^2 dt + \g_2 \bbe \int_0^T |u(t)|_m^2dt \nonumber \\
& +\g_3 \bbe \int_0^T |u'(t)|_m^2 dt,\end{aligned}$$ we may reformulate Problem (P) as $$\label{e2.5}
{\rm(P)}\ \ \ {\rm Min}\{\Phi(u);\ u\in\calu_{ad},\ X^u\ satisfies\ \eqref{equa-x}\}.$$
It should be said that, since Problem (P) is a nonconvex minimization problem, in general it is not well posed. However, if $\g_2, \g_3 =0$, we have the following generic existence result.
\[p2.3\] Assume Hypothesis $(H0)$. Then, there is a residual set $$\mathcal{G}\subset L^2(\ooo,\calf_T,\mathbb{P},L^2)\times L^2_{ad}(0,T;L^2(\ooo;L^2))$$ such that, for every $(\bbx_T,\bbx_1)\in \mathcal{G}$, problem [(P)]{} has at least one solution $u\in \calu_{ad}$.
This is an immediate consequence of a well-known result of Edelstein [@7] on existence of nearest points of closed sets in uniformly convex Banach spaces. Indeed, if we set $\mathcal{Y}=\{Y=(X^u(T),X^u);\ u\in\calu_{ad}\}$, it follows that $\mathcal{Y}$ is a closed subset of $L^2(\ooo;\calf_T,\mathbb{P},L^2)\times L^2_{ad}(0,T;L^2(\ooo;L^2))$ (see e.g. the proof of Lemma \[Lem-Xn\*\] and \[Lem-conv\]) and so, rewriting Problem [(P)]{} as $${\rm Min}\{ \|(\bbx_T,\bbx_1)-Y\|^2_*;\ Y\in \mathcal{Y}\},$$where $\|\cdot\|_*$ is the norm of $L^2(\ooo;\calf_T,\mathbb{P},L^2)\times L^2_{ad}(0,T;L^2(\ooo;L^2)),$ we arrive at the desired conclusion.
However, for the general cases $\g_2, \g_3 \not =0$, the existence of a solution in Problem $(P)$ does not follow by standard compactness techniques used in deterministic optimization problems. The main reason is that, even if a space $\caly$ is compactly imbedded into another space $\mathcal{Z}$, one generally does not have the compact imbedding from $L^p(\Omega; \caly)$ to $L^p(\Omega; \mathcal{Z})$, $1\leq p\leq \9$. Here, we consider the existence for relaxed versions of Problem (P) to be defined below.
\[d2.4\] Let $\caly:= L^2(\bbr^d) \times L^2((0,T)\times \bbr^d) \times C([0,T]; \bbr^N) \times L^2(0,T; \bbr^m) \times L^2( (0,T) \times \bbr^d)$ and $(\Omega^*, \mathcal{F}^*, (\mathcal{F}_t^*)_{t\geq 0})$ be a new filtered probability space, carrying $(\bbx^*_T, \bbx_1^*, \beta^*, u^*, X^*)$ in $\mathcal{Y}$. Define $L^2_{ad^*}(0,T; L^2(\Omega; L^2))$, $\calu_{ad^*}$ and $\Phi^*(u^*)$ similarly as above on this new filtered probability space.
The system $(\ooo^*,\calf^*,\mathbb{P}^*,(\calf^*_t)_{t\ge0},\beta^*, u^*,X^*)$ is said to be [*admissible*]{}, if $\bbx^*_T \in L^2(\Omega, \mathcal{F}^*_T, \bbp^*; L^2)$, $\bbx^*_1 \in L^2_{ad^*}(0,T; L^2(\Omega; L^2))$, $\beta^* = (\beta^*_1, \ldots, \beta^*_N)$ is an $(\calf^*_t)_{t\ge0}$-adapted $\bbr^N$-valued Wiener process, the joint distributions of $(\bbx^*_T, \bbx^*_1, \beta^*)$ and $(\bbx_T, \bbx_1, \beta)$ coincide, $u^*\in \mathcal{U}_{ad^*}$, and $X^*$ is an $(\calf^*_t)_{t\ge0}$-adapted $L^2$-valued process that satisfies equation corresponding to $(\beta^*, u^*)$.
The admissible system $(\ooo^*,\calf^*,\mathbb{P}^*,(\calf^*_t)_{t\ge0},\beta^*, u^*,X^*)$ is said to be a [*relaxed solution*]{} to the optimal control problem (P), if $$\label{e2.6}
\Phi^*(u^*)\le\inf\{\Phi(u);\ u\in\calu_{ad},\ X^u\ satisfies\ \eqref{equa-x}\}.$$
We first prove that, under the regular condition of controls (i.e., $\g_3>0$), there exists a relaxed solutions for the exponents of the nonlinearity in exactly the mass-subcritical range. A similar problem was studied in [@11a] in the deterministic case. We have
\[t2.5\] Consider $\Phi$ with $\g_3>0$. Assume $(H0)$. Then, for each $x\in L^2$, $0<T<\9$, there exists at least one relaxed solution in the sense of Definition \[d2.4\] to the optimal problem $(P)$.
The proof is mainly based on the Skorohod representation theorem and pathwise analysis of solutions by the rescaling approach devoloped in [@1]. We would also like to mention that the rescaling approach allows to obtain pathwise continuous dependence of solutions on controls.\
In order to construct a relaxed solution with equality in in the more difficult irregular case (i.e., $\g_3=0$), we will employ the Ekeland principle and work with the dual backward stochastic equation below $$\begin{aligned}
\label{back-equa}
&d Y= -i\Delta Y\,dt - \lbb i h_1(X^u)Y dt +\lbb i h_2(X^u) \ol{Y} dt + \mu Y dt - iV_0Y dt - i u\cdot V Y dt\nonumber \\
&\qquad \quad + \g_1 (X^u - \bbx_1) dt - \sum\limits_{k=1}^N \ol{\mu_k} e_k Z_k dt + \sum\limits_{k=1}^N Z_k d\beta_k(t), \\
& Y(T) = -(X^u(T)-\bbx_T), \nonumber\end{aligned}$$ where Im denotes the imaginary part, and $$\label{e4.4}
h_1(X^u):= \frac{\a+1}{2} |X^u|^{\a-1},\ \ h_2(X^u):= \frac{\a-1}{2} |X^u|^{\a-3} (X^u)^2 .$$ $h_j$, $j=1,2$, are the complex derivatives of the complex function $z\to |z|^{\a-1}z$, i.e. $h_1(z) = \partial_z (|z|^{\a-1}z)$ and $h_2(z)= \partial_{\ol{z}} (|z|^{\a-1}z)$, $z\in \mathbb{C}$.
However, the singular coefficient $h_{2}(X^u) $ in and the weak regularity effect of the Schrödinger group make it quite difficult to obtain the existence and integrability of the backward solution. The standard method to derive a global estimate for $\bbe \|Y\|^2_{C([0,T]; L^2)}$ from the Itô formula applied to $|Y(t)|_2^2$ are not applicable in the nonlinear case.
The idea here is to apply duality analysis to reduce the analysis of the backward stochastic equation to that of the dual equation below (see also the equation of variation below). By virtue of the forward character of the dual equation, we will apply the rescaling approach and the Strichartz estimates, instead of the Itô formula for $|Y(t)|_2^2$, to control the singular coefficient $h_2(X^u)$ and to obtain pathwise estimates of solutions on small intervals, which then by iteration yield the global pathwise estimates , below. To this aim, we consider in this case the following basic hypothesis.
1. $2\leq \a < 1+ \frac{4}{d}$, $1\leq d\leq 3$, and $e_k$ are constants, $1\leq k\leq N$.
(In the case where $e_k$ are not constant, which is ruled out here, there arise some delicate problems related to the nonintegrability of $(B_j)^{c(B_j)^{2/\theta}}$, where $B_k:= \sup_{t\in [0,T]} |\beta_k(t)|$, $\theta\in (0,1)$ and $c>0$, $1\leq k\leq N$.)
It is easily seen that $(H2)$ implies $(H0)$ and $(H1)$ and also that $(H2)$ is fulfilled in some important physical models, for instance the Gross-Pitaevskii model when $d=1,2$ ([@11a]). As a matter of fact, under Hypothesis $(H2)$, one has not only with equality, but also that the optimal pair $(X,u)$ satisfies the stochastic maximum principle. The main result is formulated below.
\[t2.6\] Consider $\Phi$ with $\g_3=0$. Assume Hypothesis $(H2)$, and $\bbx_T\in L^{2+\nu}(\Omega;H^1) $, $ \bbx_1 \in L^{2+ \nu}(\Omega; L^2(0,T; H^1)) $ for some small $\nu \in (0,1)$.
Then, for each $x\in H^1$, $0<T<\9$, there exists a relaxed solution $(\Omega^*, \mathscr{F}^*, \bbp^*, (\mathscr{F}^*_t)_{t\geq 0}, \beta^*, u^*, X^*)$ in the sense of Definition \[def-Phi\] to Problem (P), such that $$\label{e2.10}
\Phi^*(u^*)=\inf\{\Phi(u);\ u\in\calu_{ad},\ X^u\ satisfies\ \eqref{equa-x} \}.$$ Moreover, we have (the stochastic maximum principle) $$\label{e5.1}
u^* (t)=P_U\(\frac 1{\g_2}\,{\rm Im}\int_{\rr^d}V(\xi) X^*(t,\xi) \ol{Y^*}(t,\xi)d\xi\),\ \ff t\in[0,T],\ \bbp^*-a.s.$$ where $P_U$ is the projection on $U$, and $(Y^*,Z^*)$ is the solution to the dual backward stochastic equation with $\bbx_T, \bbx_1, \beta, u, X^u$ replaced by $\bbx^*_T, \bbx^*_1, \beta^*, u^*, X^*$ respectively.
In the deterministic case (i.e. $\mu_k=0$, $1\leq k\leq N$), for the initial datum $x\in H^1$, the optimal control indeed exists for the exponent $\a\geq 2$ and in the energy-subcritical case $(H1)$, which is also new in the literature.
\[Thm-deter\] In the deterministic case (i.e. $\mu_k=0$, $1\leq k\leq N$), consider $\Phi$ in with $\g_3=0$ and the exponent $\a\geq 2$ in the range specified in Hypothesis $(H1)$. Assume that $\bbx_T\in H^1 $ and $ \bbx_1 \in L^2(0,T; H^1) $.
Then, for each $x\in H^1$, $0<T<\9$, there exists an optimal control $u$ to Problem (P) such that $$\label{e2.10-deter}
\Phi(u)=\inf\{\Phi(v);\ v\in\calu_{ad},\ X^v\ satisfies\ \eqref{e2.1}\}.$$ Moreover, $$\label{e5.1-deter}
u(t)=P_U\(\frac 1{\g_2}\,{\rm Im}\int_{\rr^d}V(\xi) X(t,\xi) \ol{Y}(t,\xi)d\xi\),\ \ff t\in[0,T],$$ where $P_U$ is the projection on $U$, and $Y$ is the solution to the backward equation with $Z=0$.
Optimal bilinear control is also studied in [@11] and [@11a] for linear and nonlinear deterministic Schrödinger equations respectively. In both papers, some compactness conditions of initial data or controls are needed for the existence of the optimal control. More precisely, in [@11] the initial data belong to a compact subspace of $L^2$, while in [@11a] the minimizing controls are bounded in $H^1[0,T]$, hence compact in $L^2[0,T]$. In contrast to this, in Theorem \[Thm-deter\], the existence of the optimal control is obtained without these conditions, and the proof is quite different and applies as well to the stochastic case. Moreover, unlike in [@11a], less regularity of the initial data is required in Theorem \[Thm-deter\] for the maximum principle .
The proof of Theorem \[Thm-deter\] follows the lines of that of Theorem \[t2.6\], the main part of which are the analysis of equation of variation as well as of the backward stochastic equation , and the tightness of controls.
The key idea to obtain the tightness of controls in this irregular case is to employ the Ekeland principle, as well as the directional derivative of $\Phi$, to obtain the representation formula of the minimizing controls (see below). Then, by virtue of the integrability of the forward and backward solutions to and respectively, one is able to obtain the tightness of controls in the space $L^1(0,T; \bbr^d)$, which consequently yields equality in by analogous arguments as in the proof of Theorem \[t2.5\].
As mentioned above, the proof of integrability of the stochastic backward solution relies on duality analysis, which is also of independent interest.
The remaining part of this paper is organized as follows. Section \[PROOF-THM1\] includes the proof of Theorem \[t2.5\]. Section \[GD-PHI\] and Section \[PROOF-THM2\] are mainly devoted to the proof of Theorem \[t2.6\]. Section \[GD-PHI\] is concerned with the directional derivative of $\Phi$, which requires the analysis of the equation of variation and of the backward stochastic equation . Section \[PROOF-THM2\] mainly contains the proof for the tightness of controls. The proof of Theorem \[Thm-deter\] is also included there. For simplicity of the exposition, some auxiliary lemmas and technical proofs are postponed to the Appendix, i.e. Section \[APPDIX\].
#### Notations.
For $1\le p\le\9$, we denote by $L^p(\rr^d)=L^p$ the space of all Lebesgue $p$-integrable (complex-valued) functions on the real Euclidean space $\rr^d$. The norm of $L^p$ is denoted by $|\cdot|_{L^p}$, and $p'\in[1,\9]$ denotes the unique number such that $\frac1p+\frac1{p'}=1$. In particular, the Hilbert space $L^2(\rr^d)$ is endowed with the scalar product $$\<y,z\>_2=\int_{\rr^d} y(\xi)\bar z(\xi)d\xi;\ \ y,z\in L^2(\rr^d),$$where $\bar z$ is the complex conjugate of $z\in\mathbb{C}$. We also use $|\cdot|_2=|\cdot|_{L^2}$. $W^{1,p}=W^{1,p}(\rr^d)$ is the classical Sobolev space $\{v\in L^p;\ \nabla v\in L^p\}$ with the norm $\|v\|_{W^{1,p}} = |v|_2 + |\na v|_2$, $H^1=W^{1,2}$ and $H^{-1}$ is the dual space of $H^1$.
By $L^q(0,T;L^p)$ we denote the space of all integrable $L^p$-valued functions $u:(0,T)\to L^p$ with the norm $$\|u\|_{L^q(0,T;L^p)}=\left(\int^T_0\left(\int_{\rr^d}|u(t,\xi)|^pd\xi\right)^{\frac qp}dt\right)^{\frac1q}.$$ By $C([0,T];L^p)$ we denote the standard space of all $L^p$-valued continuous functions on $[0,T]$ with the sup norm in $t$. $L^q(0,T; W^{1,p})$ and $C([0,T]; H^1)$ are defined similarly. $\mathcal{D}(0,T; \bbr^m)$ is the set of all $\bbr^m$-valued smooth and compactly supported functions, and $\mathcal{D}'(0,T; \bbr^m)$ is its dual space.
We denote by $|\cdot|_m$ the Euclidean norm in $\rr^m$ and by $u\cdot v$ the scalar product of vectors $u,v\in\bbr^m$. We shall use the standard notations to represent spaces of infinite dimensional stochastic processes (see, e.g., [@6; @LR15]).
Throughout this paper, we use $C$ for various constants that may change from line to line.
Proof of Theorem \[t2.5\]. {#PROOF-THM1}
===========================
We set $$\begin{aligned}
I:= \inf \{\Phi(u); u\in \mathcal{U}_{ad}, \ X^u\ satisfies\ \eqref{equa-x}\}> 0\end{aligned}$$ and consider a sequence $\{u_n\} \subset \mathcal{U}_{ad}$ such that $$\begin{aligned}
\label{bbd-Psi}
I \leq \Phi(u_n) \leq I + n^{-1}, \ \ \forall n\in \mathbb{N}.\end{aligned}$$ Since $\g_3 >0$, this yields $$\begin{aligned}
\label{bdd-un}
\sup\limits_{n\geq 1} \bbe \int_0^T (|u_n(t)|_m^2 + |u'_n(t)|_m^2) dt <\9.\end{aligned}$$
\[Lem-tight\] Let $\bbp \circ u_n^{-1}$ be the probability measures induced by $u_n$, $n\geq 1$. Then, $\{\bbp \circ u_n^{-1}\}$ is tight in $C([0,T]; \bbr^m)$.
[***Proof.***]{} By the Arzelà theorem, it suffices to show that $$\begin{aligned}
\label{AA-bdd}
\lim\limits_{R\to \9} \sup\limits_{n\geq 1}
\bbp \circ u_n^{-1}\bigg\{v\in C([0,T];\bbr^m): \sup\limits_{t\in[0,T]} |v(t)|_m > R \bigg\} = 0,\end{aligned}$$ and for any $\ve >0$, $$\begin{aligned}
\label{AA-equicon}
\lim\limits_{\delta\to 0} \sup\limits_{n\geq 1}
\bbp \circ u_n^{-1} \bigg\{v\in C([0,T]; \bbr^m): \sup\limits_{|t-s|\leq \delta} |v(t)-v(s)|_m >\ve \bigg\} =0.\end{aligned}$$
In fact, follows immediately form the uniform boundedness of $\{u_n\}$, while follows by and $$\begin{aligned}
&\sup\limits_{n\geq 1} \bbp \circ u_n^{-1} \bigg\{v\in C([0,T]; \bbr^m): \sup\limits_{|t-s|\leq \delta} |v(t)-v(s)|_m >\ve \bigg\} \nonumber \\
\leq& \frac 1 \ve \sup\limits_{n\geq 1} \bbe \sup\limits_{|t-s|\leq\delta} |u_n(t)-u_n(s)|_m \nonumber \\
\leq& \frac {\delta^{\frac 12}} {\ve} \sup\limits_{n\geq 1} \bbe \|u'_n\|_{L^2(0,T;\bbr^m)} \to 0, \ \ as\ \delta\to 0.\end{aligned}$$ $\square $
Now, consider the sequence $\calx_n:=(\bbx_T, \bbx_1, \beta, u_n)$ with $\beta=(\beta_1,\cdots,\beta_N)$ in the space $\mathcal{Y}:= L^2( \bbr^d) \times L^2( (0,T)\times \bbr^d) \times C([0,T];\bbr^N) \times C([0,T]; \bbr^m)$. Lemma \[Lem-tight\] implies that the induced probability measures of $\calx_n$, $n\in\mathbb{N}$, are tight in the space $\mathcal{Y}$. Then, by Prohorov’s theorem, they are weakly compact and so, by the Skorohod representation theorem, there exist a probability space $(\Omega^*, \mathcal{F}^*, \bbp^*)$ and $\calx^*_n := ((\bbx^*_{T})_n, \bbx^*_{1,n},\beta^*_n, u^*_n)$, $\calx^* :=(\bbx^*_T,\bbx^*_{1}, \beta^*, u^*)$ in $\mathcal{Y}$, $n\in \mathbb{N}$, such that the joint distribution of $\calx_n^*$ and $\calx_n$ coincide, and $\bbp^*$-a.s., as $n\to \9$, $$\begin{aligned}
&\beta^*_{n} \to \beta^*\ \ in\ C([0,T]; \bbr^N), \label{conv-bn*}\\
&(\bbx^*_{T})_n \to \bbx^*_T,\ \ in\ L^2(\bbr^d),\ \ \bbx^*_{1,n} \to \bbx^*_{1},\ \ in\ L^2( (0,T)\times \bbr^d), \label{conv-XT*-X1*}\end{aligned}$$ and $$\begin{aligned}
&u^*_n \to u^*\ \ in\ C([0,T]; \bbr^m). \label{conv-un*}\end{aligned}$$ Note that, $((\bbx_T^*)_n, \bbx^*_{1,n}, \beta^*_n)$ has the same distribution as $(\bbx_T, \bbx_1, \beta)$, and so does the limit $(\bbx^*_T, \bbx^*_1, \beta^*)$.
For each $n\geq 1$, define $\calf^*_{t,n}:=\sigma(\calx^*_n(s), s\leq t)$. Then, $\bbx^*_n(T) \in L^2(\Omega, \mathcal{F}^*_T, \bbp^*; L^2)$, $\bbx^*_{1,n} \in L^2_{ad^*}(0,T; L^2(\Omega; L^2))$, $u^*_n \in \calu_{ad^*}$, and $(\beta^*_n(t), \calf^*_{t,n})$, $t\in [0,T]$, is a Wiener process. It follows from Proposition $2.2$ that, under the hypothesis $(H0)$, for each $(\beta^*_n, u^*_n)$ there exists a unique strong $L^2$-solution $X^*_n$ to . Hence, $(\Omega^*, \mathcal{F}^*, \bbp^*, (\mathcal{F}^*_t)_{t\geq 0}, \beta_n^*, u_n^*, X_n^*)$ is an admissible system.
Moreover, since the solution to is a measurable map of Brownian motions and controls, we also have that the distribution of $((\bbx^*_T)_n, \bbx^*_{1,n}, \beta_n^*, u_n^*, X_n^*)$ are the same to that of $(\bbx_T, \bbx_{1}, \beta, u_n, X_n)$, where $X_n$ is the solution to corresponding to $(\beta, u_n)$. In particular, $\Phi^*(u^*_n) = \Phi(u_n)$.
Similarly, set $\calf^*_t:= \sigma(\calx^*(s), s\leq t)$ and let $X^*$ be the unique strong $L^2$-solution to corresponding to $(\beta^*, u^*)$. Then, $(\Omega^*, \mathcal{F}^*, \bbp^*, (\mathcal{F}^*_t)_{t\geq 0}, \beta^*, u^*, X^*)$ is an admissible system.
Below, we consider the derivatives of $u^*_n$ and $u^*$. For each $n\geq 1$, define $(u_n^*)' \in \mathcal{D}'(0,T; \bbr^m)$ in the distribution sense, i.e., $((u_n^*)' , v)= - (u_n^*, v'), \forall v\in \mathcal{D}(0,T; \bbr^m)$, where $(\ ,\ )$ denotes the pairing between $\mathcal{D}'(0,T; \bbr^m)$ and $\mathcal{D}(0,T; \bbr^m)$. We claim that $(u_n^*)'$ has the same distribution as $u_n'$ in $\mathcal{D}'(0,T; \bbr^m)$, and $\bbe^*\|(u^*_n)'\|^2_{L^2(0,T; \bbr^m)} = \bbe\|u_n'\|^2_{L^2(0,T; \bbr^m)}$. It follows from that there exists $v^* \in L^2(\Omega^*; L^2(0,T;\bbr^m))$, such that $$\begin{aligned}
\label{conv-v*}
(u_n^*)' \to v^*,\ \ weakly\ in \ L^2(\Omega^*; L^2(0,T; \bbr^m)),\ n\to \9.\end{aligned}$$ Indeed, for any $l\geq 1$, $v_j \in \mathcal{D}(0,T; \bbr^m)$, $1\leq j\leq l$, and $c \in \bbr^l$, $$\begin{aligned}
&\bbe^* exp(i\sum\limits_{j=1}^l c_j ((u^*_n)', v_j))
= \bbe^* exp(-i(u^*_n, \sum\limits_{j=1}^l c_j v'_j)) \\
=& \bbe exp(-i(u_n, \sum\limits_{j=1}^l c_j v'_j))
= \bbe exp(i\sum\limits_{j=1}^l c_j (u'_n, v_j)),\end{aligned}$$ which implies that the distributions of $(u^*_n)'$ and $u'_n$ coincide. Moreover, if $\mathscr{D}:= \{v_n\}$ is a dense subset in $\mathcal{D}(0,T; \bbr^m)$ (hence also dense in $L^2(0,T; \bbr^m)$), we have $$\begin{aligned}
\bbe^* \sup\limits_{n\geq 1} \frac{|((u^*_n)',v_n)|^2}{ \|v_n\|^2_{L^2(0,T; \bbr^m)}}
= \bbe \sup\limits_{n\geq 1} \frac{|(u'_n,v_n)|^2}{ \|v_n\|^2_{L^2(0,T; \bbr^m)}}
= \bbe \|u'_n\|^2_{L^2(0,T; \bbr^m)} < \9,\end{aligned}$$ which implies that $\bbe^*\|(u^*_n)'\|^2_{L^2(0,T; \bbr^m)} = \bbe\|u_n'\|^2_{L^2(0,T; \bbr^m)}$, as claimed.
Similarly, define $(u^*)' \in \mathcal{D}'(0,T; \bbr^m)$ in the distribution sense. We have $$\begin{aligned}
\label{v*-u'*}
(u^*)' = v^*,\ \ in\ L^2(0,T; \bbr^m),\ \bbp^*-a.s.\end{aligned}$$ Indeed, let $\mathcal{E}$ be a countable dense set in $L^\9(\Omega^*)$. For any $v\in \mathscr{D}$ and $\psi \in \mathcal{E}$, by , the dominated convergence theorem and the weak convergence , it follows that $$\begin{aligned}
&\bbe^* ((u^*)',v) \psi
= - \bbe^* \psi (\int_0^T u^*(t) \cdot v'(t)dt)
= - \lim\limits_{n\to \9} \bbe^* \psi(\int_0^T u_n^*(t) \cdot v'(t)dt) \\
=& \lim\limits_{n\to \9} \bbe^* \int_0^T (u_n^*)'(t) \cdot v(t) \psi dt
= \bbe^* \int_0^T v^*(t) \cdot v(t)\psi dt
= \bbe^* (v^*,v) \psi.\end{aligned}$$ Hence, $ ((u^*)',v) = (v^*,v)$, $\bbp^*$-a.s., $v \in \mathscr{D}$. Since $\mathscr{D}$ is countable and dense in $L^2(0,T; \bbr^m)$, follows.\
Next, we show that the solutions to depend pathwisely continuous with respect to controllers, by using the rescaling approach developed in [@1; @2].
\[Lem-Xn\*\] Let $X^*_n$ (resp. $X^*$) be the solution to corresponding to $(\beta^*_n, u^*_n)$ (resp. $(\beta^*, u^*)$) as above, $n\geq 1$. Assume the conditions in Theorem $2.5$ to hold. Then, for each $x\in L^2$, $0<T<\9$ and Strichartz pair $(p,q)$, we have $\bbp^*$-a.s., as $n\to \9$, $$\begin{aligned}
\label{conv-Xn*}
\|X^*_n - X^*\|_{C([0,T];L^2)} + \|X^*_n - X^*\|_{L^q(0,T; L^p)} \to 0.\end{aligned}$$
[***Proof.*** ]{} Set $W^*_n(t,\xi) = \sum_{j=1}^N \mu_j e_j(\xi) \beta^*_{j,n}(t)$, $W^*(t,\xi) = \sum_{j=1}^N \mu_j e_j(\xi) \beta^*_{j}(t)$, $t\geq 0$, $\xi\in \bbr^d$. We may assume $T \geq 1$ without loss of generality.
Note that, in the conservative case, $$\begin{aligned}
\label{bdd-Xn*}
|X^*_n(t)|_2 = |x|_2 < \9, \ \ t\in[0,T],\ n\geq 1,\ \bbp^*-a.s.\end{aligned}$$
Using the rescaling transformation, $$\begin{aligned}
\label{rescal}
y^*_n = e^{-W^*_n} X^*_n,\end{aligned}$$ we deduce from with $X$, $u$, $\beta_j$ replaced by $X^*_n$, $u^*_n$ and $\beta_{j,n}^*$ respectively that $$\begin{aligned}
\label{equa-yn*}
dy^*_n
=& A^*_n(t)y^*_n dt - \lbb i |y^*_n|^{\a-1}y^*_n dt,
+ f(u_n^*) y^*_n dt \\
y^*_n(0)=&x, \nonumber\end{aligned}$$ where $A^*_n(t) = -i (\Delta + b^*_n(t)\cdot \na + c^*_n(t))$, $b^*_n(t)= 2 \na W^*_n(t)$, $c^*_n(t)= \sum_{j=1}^d (\partial_j W^*_n(t))^2 + \Delta W^*_n(t)$, $\wt{\mu}=2^{-1}\sum_{j=1}^N \mu_j^2 e^2_j$, and $f(u_n^*) = - i ( V_0 + u^*_n \cdot V)$.
It suffices to prove that for each Strichartz pair $(p,q)$, $\bbp^*$-a.s., $$\begin{aligned}
\label{conv-y-Xn*}
\|y^*_n - y^*\|_{L^\9(0,T;L^2)} + \|y^*_n - y^*\|_{L^q(0,T; L^p)} \to 0,\ \ as\ n\to\9.\end{aligned}$$ Now, we will prove for the Strichartz pair $(p,q)=(\a+1,\frac{4(\a+1)}{d(\a-1)})$. The general case will follow immediately from the Strichartz estimates (see, e.g., [@1 Lemma 4.1]).
To this end, we prove first that $$\begin{aligned}
\label{bdd-y-un*}
\sup\limits_{n\geq 1} \|y^*_n\|_{L^q(0,T; L^p)} < \9,\ \ \bbp^*-a.s.\end{aligned}$$
Applying the Strichartz estimates to yields $$\begin{aligned}
\|y^*_n\|_{L^q(0,t; L^p)}
\leq C_T \bigg[|x|_2
+ \|\lbb i |y^*_n|^{\a-1}y^*_n\|_{L^{q'}(0,t; L^{p'})}
+ \|f(u^*_n) y^*_n\|_{L^1(0,t;L^2)} \bigg].\end{aligned}$$ (Note that, the Strichartz coefficient $C_T$ is independent of $n$, since by $\sup_{n\geq 1}\|W^*_n\|_{C([0,T];L^\9)} <\9$, $\bbp^*$-a.s.)
By Hölder’s inequality, $$\begin{aligned}
\label{esti-lp}
\|\lbb i |y^*_n|^{\a-1}y^*_n\|_{L^{q'}(0,t; L^{p'})}
\leq t^\theta \|y^*_n\|^\a_{L^q(0,T;L^p)},\end{aligned}$$ where $\theta= 1- \frac{d(\a-1)}{4} >0$, and by the conservation , $$\begin{aligned}
\|f(u^*_n)y^*_n\|_{L^1(0,t;L^2)}
\leq T( |V_0|_{L^\9} + D_U\|V\|_{L^\9(\bbr^d;\bbr^m)})|x|_2.\end{aligned}$$ Thus, $$\begin{aligned}
\label{esti-yn*}
\|y^*_n\|_{L^q(0,t; L^p)} \leq C_T (D(T)|x|_2 + t^\theta \|y^*_n\|^\a_{L^q(0,T; L^p)}),\end{aligned}$$ where $ D(T):= 1+T ( | V_0|_{\9}
+ D_U \|V\|_{L^\9(0,T; \bbr^m)} ). $
Choose $t \in [0,T]$ such that $ C_T D(T) (|x|_2+1) = (1-\frac 1 \a) (\a C_T t^\theta)^{-\frac{1}{\a-1}}$, i.e., $$\begin{aligned}
t = \a^{-\frac{\a}{\theta}} (\a-1)^{\frac{\a-1}{\theta}} (|x|_2+1)^{-\frac{\a-1}{\theta}} C_T^{-\frac{\a}{\theta}} D(T)^{-\frac{\a-1}{\theta}} (\leq T).\end{aligned}$$ Then, by Lemma \[Lem-Bound\], we get $$\begin{aligned}
\|y^*_n\|_{L^q(0,t; L^{p})} \leq \frac{\a}{\a-1} C_T D(T) |x|_2 .\end{aligned}$$ Iterating similar estimates on $[jt,(j+1)t\wedge T]$, $1\leq j\leq [\frac{T}{t}]$, we obtain $$\begin{aligned}
\label{esti-yn*-lpq}
\|y^*_n\|_{L^q(0,T; L^p)}
\leq& \([\frac{T}{t}]+1\)^{\frac 1 q} \frac{\a}{\a-1} C_T D(T) |x|_2 ,\end{aligned}$$ which yields .\
It remains to prove . Applying the Strichartz estimates to the equations of $y^*_n$ and $y^*$, we have for any $t\in(0,T)$, $$\begin{aligned}
& \|y^*_n - y^*\|_{L^\9(0,t;L^2)} + \|y^*_n - y^*\|_{L^q(0,t; L^p)} \\
\leq& C_T \bigg[\|f(u_n^*) y^*_n - f(u^*) y^*\|_{L^1(0,t;L^2)}
+ \| |y^*_n|^{\a-1}y^*_n - |y^*|^{\a-1}y^* \|_{L^{q'}(0,t; L^{p'})} \bigg].\end{aligned}$$ Proceeding as in [@1 (4.12)], we have $$\begin{aligned}
&\| |y^*_n|^{\a-1}y^*_n - |y^*|^{\a-1}y^* \|_{L^{q'}(0,t; L^{p'})} \\
\leq& \a t^\theta (\|y^*_n\|^{\a-1}_{L^q(0,T;L^p) } + \|y^*\|^{\a-1}_{L^q(0,T;L^p)}) \| y^*_n - y^*\|_{L^q(0,t;L^p) }.\end{aligned}$$ Moreover, for $t\leq 1$, $$\begin{aligned}
& \|f(u^*_n)y^*_n- f(u^*)y^*\|_{L^1(0,t;L^2)}
\leq t^\frac 12 D \( \|u^*_{n} - u^* \|_{L^2(0,T;\bbr^m)}
+ \|y^*_n-y^*\|_{L^\9(0,t; L^2)} \),\end{aligned}$$ where $D = 2(|V_0|_{L^\9} + \|V\|_{L^\9(\bbr^d; \bbr^m)} (|x|_2+D_U))$.
Hence, $$\begin{aligned}
\label{diff-yn*}
& \|y^*_n - y^*\|_{L^\9(0,t;L^2)} + \|y^*_n - y^*\|_{L^q(0,t; L^p)} \nonumber \\
\leq& C_T\bigg[ \a t^\theta (\|y^*_n\|^{\a-1}_{L^q(0,T;L^{p})} + \|y^*\|^{\a-1}_{L^q(0,T;L^{p})} ) \|y^*_n - y^*\|_{L^{q}(0,t; L^p)} \nonumber \\
& \qquad + t^{\frac 12} D \|u^*_{n} - u^*\|_{L^2 (0,T;\bbr^m)}
+ t^{\frac 12} D \|y^*_n - y^*\| _{L^\9(0,t; L^2)} \bigg] \nonumber \\
\leq& (t^\theta + t^{\frac 12})C_T \wt{D}(T) (\|y^*_n- y^*\|_{L^\9(0,t;L^{2})}+\|y^*_n- y^*\|_{L^q(0,t;L^{p})} +\|u^*_{n} - u^*\|_{L^2 (0,T;\bbr^m)}),\end{aligned}$$ where $ \wt{D}(T):= \a (\sup _{n\geq 1}\|y^*_n\|^{\a-1}_{L^q(0,T;L^{p})} + \|y^*\|^{\a-1}_{L^q(0,T;L^{p})} ) +D <\9$, $ \bbp^*$-a.s. Choosing $t$ small enough and independent of $n$, such that $t^\theta + t^{\frac{1}{2}} \leq (2\wt{D}(T)C_T)^{-1}$, we get that $\bbp^*$-a.s. as $n\to \9$, $$\begin{aligned}
\label{esti-yn*-diff}
\|y^*_n - y^*\|_{L^\9(0,t;L^2)} + \|y^*_n - y^*\|_{L^q(0,t; L^p)}
\leq 2 \|u^*_{n} - u^*\|_{L^2 (0,T;\bbr^m)} \to 0.\end{aligned}$$ Since $t$ is independent of the initial data, iterating this procedure finite times we obtain , thereby completing the proof. $\square $
\[Lem-conv\] Let $X^*_n$, $(\bbx^*_T)_n$, $\bbx^*_{1,n}$, $X^*$, $\bbx^*_T$ and $\bbx^*_1$ be as above, $n\geq 1$. We have $\bbp^*$-a.s., as $n\to \9$, $$\begin{aligned}
\label{App-conv-Xtau}
\bbe^* Re\<X^*_n(T), (\bbx^*_T)_n\>_2
\to \bbe^* Re \<X^*(T), \bbx^*_T\>_2,\end{aligned}$$ and $$\begin{aligned}
\label{App-conv-intX}
\bbe^* \int_0^{T} Re\<X^*_n(t), \bbx^*_{1,n}(t)\>_2 dt
\to \bbe^* \int_0^{T} Re\<X^*(t), \bbx^*_{1}(t)\>_2 dt.\end{aligned}$$
[***Proof.***]{} By and , $\bbp^*$-a.s., as $n\to \9$, $$\begin{aligned}
\label{App-conv-X.1}
Re \<X^*_n(T), (\bbx^*_T)_n\>_2
\to Re \<X^*(T), \bbx^*_T\>_2,\end{aligned}$$ $$\begin{aligned}
\label{App-conv-X.2}
\int_0^{T} Re \<X^*_n(t),\bbx^*_{1,n}(t)\>_2 dt
\to \int_0^{T} Re \<X^*(t), \bbx^*_{1}(t)\>_2 dt.\end{aligned}$$
Then, for any $\ve \in (0,1)$ fixed, by the Young inequality $ab\leq \frac{1-\ve}{2}a^{\frac{2}{1-\ve}} + \frac{1+\ve}{2} b^{\frac{2}{1+\ve}}$, we get $$\begin{aligned}
&\sup\limits_{n\geq 1} \bbe^* |\<X^*_n(T), (\bbx^*_T)_n\>_2|^{1+\ve}
\leq \sup\limits_{n\geq1} \bbe^* |X^*_n(T)|_2^{1+\ve} |(\bbx^*_T)_n|_2^{1+\ve} \\
\leq& \frac{1-\ve}{2} \sup\limits_{n\geq 1}\bbe^* |X^*_n(T)|_2^{\frac{2(1+\ve)}{1-\ve}}
+ \frac{1+\ve}{2} \sup\limits_{n\geq 1} \bbe^* |(\bbx^*_T)_n|_2^2 \\
=& \frac{1-\ve}{2} |x|_2^{\frac{2(1+\ve)}{1-\ve}}
+ \frac{1+\ve}{2}\bbe | \bbx_T |_2^2,\end{aligned}$$ which implies the uniform integrability of $\{Re \<X^*_n(T), (\bbx^*_T)_n\>_2\}_{n\geq 1}$, thereby yielding by .
Similarly, for $\ve \in (0,1)$ fixed, we have $$\begin{aligned}
&\sup\limits_{n\geq 1}\bbe^* \bigg| \int_0^{T} Re \<X^*_n(t), \bbx^*_{1,n}(t)\>_2 dt \bigg|^{1+\ve} \\
\leq& \frac{1-\ve}{2} \sup\limits_{n\geq 1} \bbe^* \|X^*_n\|_{L^2(0,T; L^2)}^{\frac{2(1+\ve)}{1-\ve}}
+ \frac{1+\ve}{2} \sup\limits_{n\geq 1} \bbe^* \|\bbx^*_{1,n}\|_{L^2(0,T; L^2)}^2 \\
= & \frac{1-\ve}{2} T^{\frac{1+\ve}{1-\ve}} |x|_2^{\frac{2(1+\ve)}{1-\ve}}
+\frac{1+\ve}{2} \bbe \int_0^{T} |\bbx_1(t)|_2^2 dt <\9,\end{aligned}$$ which in view of implies , as claimed. $\square$\
[***Proof of Theorem \[t2.5\].***]{} By the conservation identity we have $$\begin{aligned}
\Phi^*(u^*_n)
=& (1+\g_1)|x|_2^2 + \bbe^*|(\bbx^*_T)_n|_2^2 + \g_1 \bbe^* \int_0^{T} |\bbx^*_{1,n}(t)|_2^2 dt\\
& -2 \bbe^* Re \<X^*_n(T), (\bbx^*_T)_n\>_2 - 2 \g_1 \bbe^* \int_0^{T} Re \<X^*_n(t), \bbx^*_{1,n}(t)\>_2 dt \\
& + \g_2 \bbe^* \int_0^{T} |u^*_n(t)|_m^2 dt + \g_3 \bbe^* \int_0^{T} |(u^*_n)'(t)|_m^2 dt,\end{aligned}$$
Note that, since the distributions of $(\bbx^*_T, \bbx^*_1)$ and $((\bbx^*_T)_n, \bbx^*_{1,n})$ coincide for $n\geq 1$, we have $$\begin{aligned}
\label{X-X1}
\bbe^*| \bbx^*_T |_2^2 + \g_1 \bbe^* \int_0^{T} |\bbx^*_{1}(t)|_2^2 dt
= \lim \limits_{n\to \9} \(\bbe^*|(\bbx^*_T)_n|_2^2 + \g_1 \bbe^* \int_0^{T} |\bbx^*_{1,n}(t)|_2^2 dt\).\end{aligned}$$ Moreover, by and the bounded dominated convergence theorem, it follows that $$\begin{aligned}
\g_2 \bbe^* \int_0^{T} |u^*_n(t)|_m^2 dt \to \g_2 \bbe^* \int_0^{T} |u^* (t)|_m^2 dt,\ as\ n\to \9,\end{aligned}$$ and by and , $$\begin{aligned}
\bbe^* \int_0^T |(u^*)'(t)|^2_m dt
\leq& \liminf\limits_{n\to \9} \bbe^* \int_0^T |(u^*_n)'(t)|^2_m dt.\end{aligned}$$ Thus, taking into account Lemma \[Lem-conv\], we obtain $$\begin{aligned}
\Phi^*(u^*)
\leq \liminf\limits_{n\to\9} \Phi^*(u^*_n)
= \liminf\limits_{n\to \9} \Phi(u_n) = I.\end{aligned}$$ which completes the proof. $\square$
The proofs above show also that, in the case $\g_3=0$, the objective functional $\Phi$ depends continuously on controls.
The directional derivative of function $\Phi$ {#GD-PHI}
=============================================
This section is devoted to the calculation of the directional derivative of function $ \Phi $ on the convex set $\calu_{ad}$. Namely, one has
\[l4.1\] Assume that $\g_3=0$ and that the conditions of Theorem \[t2.6\] to hold. Then, for each $x\in L^2$ and all $u, v\in \calu_{ad}$, we have $$\label{e4.1}
\lim_{\vp\to0}\frac1\vp\,(\Phi(u+\vp\wt u)-\Phi(u))=\E\int^{T}_0\eta(u)(t)\cdot{\wt u}(t)dt,$$ where $\wt{u} = v-u$, and $$\label{e4.2}
\eta(u)=2\(\g_2 u-{\rm Im}\int_{\rr^d}V(\xi) X^u(\xi) \ol{Y^u}(\xi)d\xi\).$$ Here $(Y^u, Z^u)$ is the solution to the dual backward stochastic equation .
To prove Proposition \[l4.1\], we first study the equation of variation associated with Problem (P), namely,
$$\begin{aligned}
\label{forward-equa}
&id \vf = \Delta \vf dt + \lbb h_1(X^u) \vf dt + \lbb h_2(X^u) \ol{\vf} dt -i \mu \vf dt \nonumber \\
&\qquad \quad + V_0 \vf dt + u\cdot V \vf dt + \wt{u}\cdot V X^u dt +i \vf d W(t), \nonumber \\
&\vf(0)= 0,\end{aligned}$$
where $\wt{u}= v-u$, $u,v\in \calu_{ad}$, $X^u$ is the solution to , and $h_j(X^u)$, $j=1,2$, are defined as in . The strong $H^1$-(and $L^2$-)solution to can be defined similarly as in Definition \[d2.1\].
\[WP-Forward\] $(i)$ Under Hypothesis $(H0)$, for $u,v \in \calu_{ad}$, $\wt{u}:=v-u$, there exists a unique strong $L^2$-solution $\vf^{u,\wt{u}}$ to on $[0,T]$.
$(ii)$ Under Hypothesis $(H2)$, for any Strichartz pair $(p,q)$, $$\begin{aligned}
\label{bdd-forward-y}
\sup\limits_{u,v\in \calu_{ad}} \(\|\vf^{u,\wt{u}}\|_{L^\9(\Omega; C([0,T]; L^2))} + \|\vf^{u,\wt{u}}\|_{L^\9(\Omega;L^q(0,T; L^p))} \) < \9.\end{aligned}$$
Moreover, set $u_\ve:= u+\ve \wt{u}$ and let $X^u$ and $X^{u_\ve}$ be the corresponding solutions to with the initial datum $x\in H^1$. Then, $$\begin{aligned}
\label{asy-X-y}
\lim\limits_{\ve \to 0} \bbe \sup\limits_{t\in [0,T]} |\ve^{-1}(X^{u_\ve}(t) - X^u(t)) -\vf^{u,\wt{u}}(t)|^2_2 =0.\end{aligned}$$
\[Rem-Diff-h2\]
In comparison with $(ii)$, the weaker Hypothesis $(H0)$ is sufficient for the pathwise existence and uniqueness of the solution to , thanks to the linear structure of . However, as mentioned in Section \[PROBLEM-RESULT\], Hypothesis $(H2)$ is needed in order that the estimate holds. The arguments presented below, particularly in the proof of the estimate , will also be used in the analysis of the dual backward stochastic equation in the proof of Proposition \[WP-Backward\] below.
[***Proof of Lemma \[WP-Forward\].***]{} $(i)$ We set $z^{u,\wt{u}}:= e^{-W} \vf^{u,\wt{u}}$, $\wt{u}:=v-u$, and for simplicity, we omit the dependence of $u,\wt{u}$ in $z^{u,\wt{u}}$ below. It follows from that $$\begin{aligned}
\label{equa-z}
&dz = A(t) z dt - \lbb i h(X^u, z) dt + f(u) z dt - i \wt{u} \cdot V e^{-W}X^u dt, \ t\in(0,T), \\
&z(0)=0, \nonumber\end{aligned}$$ where $A(t)$ is similar as $A^*_n(t)$ in , i.e., $A(t) = -i (\Delta + b(t)\cdot \na + c(t))$, $b(t)= 2 \na W(t)$, $c(t)= \sum_{j=1}^d (\partial_j W(t))^2 + \Delta W(t)$, $h(X^u,z):= h_1(X^u) z + h_2(X^u) e^{-2 i Im W} \ol{z}$, and $f(u):= -i( V_0 + u\cdot V )$.
It is equivalent to prove the existence and uniqueness of the solution to (see the proof of [@1 Lemma 6.1]).
To this purpose, we reformulate in the mild form as $$\begin{aligned}
\label{equa-mild-z}
z(t) = \int_0^t U(t,s) \big[-\lbb i h(X^u, z)(s) & + f(u(s)) z(s) - i\wt{u}(s)\cdot V e^{-W(s)}X^u(s) \big] ds,\end{aligned}$$ where $0\leq t\leq T$, and $U(t,s)$, $0\leq s,t\leq T$, are the evolution operators corresponding to the operator $A(t)$ (see [@1 Lemma 3..3]). Choose the Strichartz pair $(p,q) = (\a+1, \frac{4(\a+1)}{d(\a-1)})$. Define the operator $F$ on $C([0,T]; L^2) \cap L^q(0,T;L^p)$ by $$\begin{aligned}
F(\phi)(t) :=& \int_0^t U(t,s) \big[-\lbb i h(X^u,\phi)(s) +f(u(s)) \phi(s) - i\wt{u}\cdot V e^{-W(s)}X^u(s) \big] ds,\end{aligned}$$ where $0\leq t\leq T$, $\phi \in C([0,T]; L^2) \cap L^q(0,T;L^p)$. Set $\calz^{\tau_1}_{M_1} :=\{ \phi \in C([0,\tau_1]; L^2) \cap L^q(0,\tau_1;L^p): \|\phi\|_{C([0,\tau_1]; L^2)} + \|\phi\|_{L^q(0,\tau_1;L^p)} \leq M_1\} $, where $\tau_1$ and $M_1$ are two random variables to be determined later.
Note that, by Hölder’s inequality, for any $\phi_j \in C([0,T]; L^2) \cap L^q(0,T;L^p)$, $j=1,2$, $$\begin{aligned}
\|h(X^u, \phi_1) - h(X^u, \phi_2)\|_{L^{q'}(0,t; L^{p'})}
\leq \a t^\theta \|X^u\|^{\a-1}_{L^q(0,t; L^{p})} \|\phi_1-\phi_2\|_{L^q(0,t; L^p)},\end{aligned}$$ where $\theta = 1- d(\a-1)/4 \in (0,1)$, and $$\begin{aligned}
\|f(u)(\phi_1-\phi_2)\|_{L^{1}(0,t; L^{2})}
\leq t \|f(u)\|_{L^\9(0,t; L^\9)} \|\phi_1-\phi_2\|_{C([0,t]; L^2)}.\end{aligned}$$ Then, let $R_1(t) =\a t^\theta \|X^u\|^{\a-1}_{L^q(0,t; L^{p})} + t \|f(u)\|_{L^\9(0,t; L^\9)} $, $t\in [0,T]$. By the Strichartz estimates and the above estimates we get for any $t\in [0,T]$, $$\begin{aligned}
\label{esti-diff}
\|F(\phi_1)-F(\phi_2)\|_{C([0,t]; L^2) \cap L^q(0,t;L^p) }
\leq C_t R_1(t) \|\phi_1-\phi_2\|_{C([0,t]; L^2)\cap L^q(0,t; L^p)}.\end{aligned}$$ Similarly, for $\phi \in C([0,T]; L^2) \cap L^q(0,T;L^p)$, $t\in [0,T]$, $$\begin{aligned}
\|F(\phi)\|_{C([0,t]; L^2) \cap L^q(0,t;L^p) }
\leq C_t R_1(t) \|\phi\|_{C([0,t]; L^2)\cap L^q(0,t; L^p)} + C_t \|\wt{u}\cdot V X^u\|_{L^1(0,t; L^2)}.\end{aligned}$$
Setting $\tau_1 = \inf\{t\in[0,T]:C_t R_1(t) \geq \frac 12\}\wedge T$, $M_1 = 2 C_{\tau_1} \|\wt{u}\cdot V X^u\|_{L^1(0,\tau_1; L^2)}$, it follows by that $F$ is a contraction map in $\calz_{M_1}^{\tau_1}$, implying that there exists $\wt{z}_1 \in \calz_{M_1}^{\tau_1}$ such that $F(\wt{z}_1)=\wt{z}_1$. Setting $z_1(\cdot):= \wt{z}_1(\cdot \wedge \tau_1)$ and using similar arguments as in [@1], we deduce that $z_1$ is $(\mathcal{F}_t)$-adapted, continuous in $L^2$, and solves on $[0,\tau_1]$, and $\|z_1\|_{C([0,\tau_1]; L^2) \cap L^q(0,\tau_1;L^p) } \leq M_1 $. We also note that $\tau_1 \geq \sigma_*$, where $$\begin{aligned}
\label{def-sig}
\sigma_* := \inf\{t\in [0,T]; Z(t) \geq \frac 12 \} \wedge T\end{aligned}$$ with $ Z(t):= t^\theta \a C_T \|X^u\|^{\a-1}_{L^q(0,T; L^{p})} + t C_T \|f(u)\|_{L^\9(0,T; L^\9)}$, $t\in [0,T].$
Suppose that at the $n^{th}$-step ($n\geq 1$) we have an increasing sequence of stopping times $\{\tau_j\}_{j=0}^n$ and an $L^2$-valued continuous $(\mathcal{F}_t)$-adapted process $z_n$, which satisfy that $\tau_0 = 0$, $\tau_j - \tau_{j-1} \geq \sigma_*$, $1\leq j\leq n$, $z_n$ solves on $[0,\tau_n]$, $z_n (\cdot )= z_n( \cdot \wedge \tau_n)$, and $$\|z_n\|_{C([0,\tau_n]; L^2) \cap L^q(0,\tau_n;L^p) }
\leq \sum\limits_{j=1}^n (2 C_{\tau_n})^{n+1-j}\ \|\wt{u}\cdot V X^u\|_{L^1(\tau_{j-1}, \tau_j; L^2)}.$$
Set $\calz^{\sigma_n}_{M_{n+1}} =\{\phi\in C([0,\sigma_n]; L^2) \cap L^q(0,\sigma_n;L^p): \|\phi\|_{C([0,\sigma_n]; L^2)} + \|\phi\|_{L^q(0,\sigma_n;L^p)} \leq M_{n+1}\} $, where $\sigma_n$ and $M_{n+1}$ are random variables to be determined later. Define the operator $F_n$ on $C([0,T]; L^2) \cap L^q(0,T;L^p)$ by $$\begin{aligned}
F_n(\phi)(t) :=& U(\tau_n+t, \tau_n) z_n(\tau_n) + \int_0^t U(\tau_n+t,\tau_n+s) \big[-\lbb i h(X^u(\tau_n+s),\phi(s)) \\
& \qquad \quad +f(u(\tau_n+s))\phi(s) - i\wt{u}(\tau_n+s)\cdot V e^{-W(\tau_n+s)}X^u(\tau_n+s) \big] ds,\end{aligned}$$ where $0\leq t\leq T$, $\phi\in C([0,T]; L^2) \cap L^q(0,T;L^p)$.
Similarly, for any $\phi_j \in \calz^{\sigma_n}_{M_{n+1}}$, $j=1,2$, $$\begin{aligned}
&\|F_n(\phi_1)-F_n(\phi_2)\|_{C([0,\sigma_n]; L^2) \cap L^q(0,\sigma_n;L^p) } \\
\leq& C_{\tau_n+\sigma_n} R_{n+1}( \sigma_n) \|\phi_1-\phi_2\|_{C([0,\sigma_n]; L^2)\cap L^q(0,\sigma_n; L^p)},\end{aligned}$$ where $R_{n+1}(t) = \a t^\theta\|X^u\|^{\a-1}_{L^q(\tau_n,\tau_n+t; L^{p})} + t \|f(u)\|_{L^\9(\tau_n,\tau_n+t; L^\9)} $, $t\in [0,T-\tau_n]$, while for $\phi \in \calz^{\sigma_n}_{M_{n+1}}$, we have $$\begin{aligned}
&\|F_n(\phi)\|_{C([0,\sigma_n]; L^2) \cap L^q(0,\sigma_n;L^p) } \\
\leq& C_{\tau_n+\sigma_n} |z_n(\tau_n)|_2
+ C_{\tau_n+\sigma_n} \|\wt{u}\cdot V X^u\|_{L^1(\tau_n,\tau_n+\sigma_n; L^2)} \\
&+ C_{\tau_n+\sigma_n} R_{n+1}( \sigma_n) \|\phi \|_{C([0,\sigma_n]; L^2)\cap L^q(0,\sigma_n; L^p)} \\
\leq& \frac 12 \sum\limits_{j=1}^{n+1} (2 C_{\tau_n+\sigma_n})^{n+2-j} \|\wt{u}\cdot V X^u\|_{L^1(\tau_{j-1},\tau_j; L^2)} \\
& + C_{\tau_n+\sigma_n} R_{n+1}( \sigma_n) \|\phi\|_{C([0,\sigma_n]; L^2)\cap L^q(0,\sigma_n; L^p)}.\end{aligned}$$ Then, let $\sigma_n(t) := \inf\{t\in[0,T -\tau_n]: C_{\tau_n+t} R_{n+1}( t) \geq \frac 12\} \wedge (T-\tau_n)$, $\tau_{n+1} := \tau_n + \sigma_n$, and $M_{n+1} := \sum_{j=1}^{n+1} (2 C_{\tau_{n+1}})^{n+2-j}\ \|\wt{u}\cdot V X^u\|_{L^1(\tau_{j-1}, \tau_j; L^2)} $. It follows that $\tau_{n+1} - \tau_n = \sigma_n \geq \sigma_*$, $F_n$ is a contraction map in $\calz^{\sigma_n}_{M_{n+1}}$, and so there exists $\wt{z}_{n+1} \in \calz^{\sigma_n}_{M_{n+1}}$ satisfying $F_n(\wt{z}_{n+1}) = \wt{z}_{n+1}$. As in [@1], letting $$\begin{aligned}
z_{n+1}(t) = \left\{
\begin{array}{ll}
z_n(t), & \hbox{$t\in [0,\tau_n]$;} \\
\wt{z}_{n+1}((t-\tau_n)\wedge \sigma_n), & \hbox{$t\in (\tau_n, T]$,}
\end{array}
\right.\end{aligned}$$ it follows that $z_{n+1}$ is continuous $(\mathcal{F}_t)$-adapted, satisfies on $[0,\tau_{n+1}]$, $z_n (\cdot )= z_n( \cdot \wedge \tau_{n+1})$, and $$\|z_n\|_{C([0,\tau_{n+1}]; L^2) \cap L^q(0,\tau_{n+1};L^p) }
\leq \sum\limits_{j=1}^{n+1} (2 C_{\tau_{n+1}})^{n+2-j}\ \|\wt{u}\cdot V X^u\|_{L^1(\tau_{j-1}, \tau_j; L^2)}.$$
Iterating this procedure, since $\sigma_n \geq \sigma_*$, we see that after at most $[T/\sigma_*]+1$ steps the stopping time $\tau_n$ reaches $T$. Hence, $\bbp$-a.s. there exists a global solution (denoted by $z$) on $[0,T]$ which satisfies $$\begin{aligned}
\label{esti*}
\|z\|_{C([0,T]; L^2) \cap L^q(0,T;L^p) }
\leq \sum\limits_{j=1}^{[T/\sigma_*]+1} (2 C_T)^{[T/\sigma_*]+2-j}\ \|\wt{u}\cdot V X^u\|_{L^1(\tau_{j-1}, \tau_j; L^2)}.\end{aligned}$$
As regards the uniqueness, given any two solutions $\vf_j$, we set $z_j = e^{-W} \vf_j$, $j=1,2$. Then, similarly to , for any $s,t \in (0,T)$, $s+t \leq T$, we have $$\begin{aligned}
&\|z_1 - z_2\|_{C([s,s+t]; L^2) \cap L^q(s,s+t;L^p)} \\
\leq& C_T (\a t^\theta \|X^u\|^{\a-1}_{L^q(0,T;L^p)} + t \|f(u)\|_{L^\9(0,T; L^\9)})
\|z_1 - z_2\|_{C([s,s+t]; L^2) \cap L^q(s,s+t;L^p)} ,\end{aligned}$$ which implies that $z_1=z_2$ on $[s,s+t]$, $\bbp$-a.s., for $t$ sufficiently small and independent of $s$, thereby yielding the uniqueness by the arbitrariness of $s$.\
$(ii)$ Under Hypothesis $(H2)$, the Strichartz coefficient $C_T$ is now identically a deterministic constant. Moreover, and imply that $\sigma_*$ has a deterministic lower bound, namely, $$\begin{aligned}
\sigma_* \geq t_* := \inf \{t\in[0,T]: Z^*(t) \geq T\} \wedge T, \ \ \bbp-a.s.,\end{aligned}$$ where $Z^*(t) := \a C (t^\theta+t) \sup_{u\in \calu_{ad}} (\|X^u\|^{\a-1}_{L^\9(\Omega; L^q(0,T; L^p))} + \|f(u)\|_{L^\9(\Omega;L^{\9}(0,T; L^\9))})$. Thus, taking into account and the uniform boundedness of $u,v\in \calu_{ad}$, we obtain .
Now, set $\wt{X}^u_\ve : = \ve^{-1}(X^{u_\ve} - X^u) - \vf$ and $\wt{y}^u_\ve:= e^{-W} \wt{X}^u_\ve $. We need to prove that $$\begin{aligned}
\label{asy-X-eta}
\lim\limits_{\ve \to 0} \bbe \|\wt{y}^u_\ve\|^2_{C([0,T]; L^2)} =0.\end{aligned}$$
To this purpose, note that $$\begin{aligned}
\ve^{-1} (u_\ve \cdot V X^{u_\ve} - u \cdot V X^u) = \wt{u} \cdot V X^u + u_\ve \cdot V (\wt{X}^u_\ve + \vf),\end{aligned}$$ and $$\begin{aligned}
&\ve^{-1} ( |X^{u_\ve}|^{\a-1}X^{u_\ve} - |X^u|^{\a-1}X^u) - (h_1(X^u) \vf + h_2(X^u)\ol{\vf}) \\
=& \(\int_0^1 h_1(X_{u,r,\ve}) dr\) \wt{X}^u_\ve + \(\int_0^1 h_2(X_{u,r,\ve}) dr\) \ol{\wt{X}^u_\ve} \\
& + \vf \int_0^1 (h_1(X_{u,r,\ve}) dr - h_1 (X^u)) dr + \ol{\vf} \int_0^1 (h_2(X_{u,r,\ve}) dr - h_2(X^u)) dr.\end{aligned}$$ where $X_{u,r,\ve}= X^u + r(X^{u_\ve}-X^u)$, $r\in [0,1]$. For simplicity, set $R_j(\ve):= \int_0^1 (h_j(X_{u,r,\ve}) - h_j (X^u)) dr$, $j=1,2$, and $R(\ve,\vf):= -i(\lbb R_1(\ve)\vf + \lbb R_2(\ve)\ol{\vf} + \ve\wt{u}\cdot V\vf)$.
Then, by and , $\wt{X}^u_\ve$ satisfies the equation $$\begin{aligned}
\label{equa-Xue}
d \wt{X}^u_\ve =& -i \Delta \wt{X}^u_\ve dt
- \lbb i \int_0^1 h_1(X_{u,r,\ve}) dr \wt{X}^u_\ve dt
- \lbb i \int_0^1 h_2(X_{u,r,\ve}) dr \ol{\wt{X}^u_\ve} dt
\nonumber \\
& -(\mu+iV_0 + iu_\ve \cdot V) \wt{X}^u_\ve dt + R (\ve, \vf) dt
+ \wt{X}^u_\ve d W(t).\end{aligned}$$ This yields $$\begin{aligned}
\label{equa-yue}
d \wt{y}^u_\ve
= -i\Delta \wt{y}^u_\ve dt -\lbb i \int_0^1 h(X_{u,r,\ve}, \wt{y}_\ve^u)dr dt
+ f(u_\ve) \wt{y}^u_\ve dt
+ e^{-W} R (\ve, \vf) dt,\end{aligned}$$ where $h(X_{u,r,\ve}, \wt{y}^u_\ve)$ and $f(u^\ve)$ are similar to those arising in , with $X^u$, $z$, $u$ replaced by $X_{u,r,\ve}$, $\wt{y}^u_\ve$ and $u^\ve$ respectively.
Choose the Strichartz pair $(p,q)=(\a+1, \frac{4(\a+1)}{d(\a-1)})$. Then, by Strichartz’s estimates, Hölder’s inequality and Minkowski’s inequality, we have $$\begin{aligned}
&\|\wt{y}^u_\ve\|_{C([0,t];L^2)} + \|\wt{y}^u_\ve\|_{L^q(0,t; L^p)} \\
\leq& C \int_0^1 \|h(X_{u,r,\ve}, \wt{y}^u_\ve)\|_{L^{q'}(0,t,L^{p'})} dr
+ C t \|f(u_\ve) \|_{L^\9(0,T;L^\9)} \|\wt{y}^u_\ve\|_{C([0,t];L^2)} \\
&+ C \|R (\ve, \vf)\|_{L^1(0,t; L^2) + L^{q'}(0,t; L^{p'})}.\end{aligned}$$ Note that, $$\begin{aligned}
\label{esti-hi-lpq}
&\|h(X_{u,r,\ve}, \wt{y}^u_\ve)\|_{L^{q'}(0,t,L^{p'})} \nonumber \\
\leq& \a t^\theta \|\wt{y}^u_\ve\|_{L^q(0,t; L^p)} \|X_{u,r,\ve}\|^{\a-1}_{ L^{q}(0,t; L^{p})} \nonumber \\
\leq& \a (1\vee 2^{\a-1}) t^\theta \|\wt{y}^u_\ve\|_{L^q(0,t; L^p)} (\|X^{u_\ve} \|^{\a-1}_{L^q(0,T; L^p)} + \|X^u \|^{\a-1}_{L^q(0,T; L^p)} ),\end{aligned}$$ where $\theta = 1 - d(\a-1)/4 >0$. Then, $$\begin{aligned}
\|\wt{y}^u_\ve\|_{C([0,t];L^2)} + \|\wt{y}^u_\ve\|_{L^q(0,t; L^p)}
\leq& CD_3(T) (t^\theta + t) (\|\wt{y}^u_\ve\|_{C([0,t]; L^2)} + \|\wt{y}^u_\ve\|_{L^q(0,t; L^p)}) \\
& +C \|R (\ve, \vf)\|_{L^1(0,T; L^2) + L^{q'}(0,T; L^{p'})}.\end{aligned}$$ where $D_3(T) = \a 2^{\a+1} \sup_{\ve \in[0,1]} (\|X^{u_\ve}\|^{\a-1}_{L^\9(\Omega; L^q(0,T; L^p))}
+ \|f(u_\ve)\|_{L^\9(\Omega; L^\9(0,T; L^\9))})$. Using similar iterating arguments as in the proof of , we obtain $$\begin{aligned}
\sup\limits_{u\in \calu_{ad}}(\|\wt{y}^u_\ve\|_{C([0,T];L^2)} + \|\wt{y}^u_\ve\|_{L^q(0,T; L^p)})
\leq C(T)\| R (\ve, \vf)\|_{L^1(0,T; L^2) + L^{q'}(0,T; L^{p'})}\end{aligned}$$ with $C(T) \in L^\9(\Omega)$.
Thus, by Hölder’s inequality, $$\begin{aligned}
& \bbe \|\wt{y}^u_\ve\|^2_{C([0,T];L^2)} + \bbe \|\wt{y}^u_\ve\|^2_{L^q(0,T; L^p)} \\
\leq& C(T) \bbe \|R (\ve, \vf)\|^2_{L^1(0,T; L^2) + L^{q'}(0,T; L^{p'})} \\
\leq& C(T) \big( \ve^2 D_U^2 T^2 \|V\|^2_{L^\9(0,T;\bbr^m)} \bbe \|\vf\|^2_{C([0,T]; L^2)}
+\sum\limits_{j=1}^2 \bbe \|R_j(\ve)\vf\|^2_{L^{q'}(0,T; L^{p'})} \big).\end{aligned}$$
Therefore, in order to prove , we only need to show that $$\begin{aligned}
\label{conv-R1}
\bbe \|R_j(\ve) \vf\|^2_{L^{q'}(0,T;L^{p'})} \to 0,\ \ as\ \ve\to 0,\ j=1,2.\end{aligned}$$
Below, we prove only for $R_1(\ve)$, but the argument applies as well to $R_2(\ve)$. As in the proof of we get $$\begin{aligned}
\label{conv-Xue-Xu}
\|X^{u_\ve} - X^u\|_{C([0,T]; L^2)} + \|X^{u_\ve} - X^u\|_{L^q(0,T; L^{p})}
\to 0,\ \ as \ \ve\to 0,\ \ \bbp-a.s.\end{aligned}$$
Note that $$\begin{aligned}
\label{diff-h1}
h_1(X_{u,r,\ve}) - h_1(X^u)
=& \int_0^1 \partial_z h_1(X^u + r' (X_{u,r,\ve}-X^u))dr' (X_{u,r,\ve} - X^u) \nonumber \\
& + \int_0^1 \partial_{\ol{z}} h_1(X^u + r' (X_{u,r,\ve}-X^u))dr' (\ol{X_{u,r,\ve}} - \ol{X^u}).\end{aligned}$$ Since $|\partial_z h_1(z)| + |\partial_{\ol{z}} h_1(z)| \leq C |z|^{\a-2}$ for $z\in \mathbb{C}$, using the Minkowski inequality and the Hölder inequality we get that $\bbp$-a.s. for each $r\in [0,1]$, $$\begin{aligned}
\label{esti-Xue-Xu}
&\| h_1(X_{u,r,\ve}) - h_1(X^u)\|_{L^{\frac{q}{\a-1}}(0,T; L^{\frac{p}{\a-1}})} \nonumber \\
\leq& C \int_0^1 \|X^u + r' (X_{u,r,\ve}-X^u)\|^{\a-2}_{L^q(0,T; L^p)} dr' \|X_{u,r,\ve} - X^u\|_{L^q(0,T; L^p)} \nonumber \\
\leq& C \sup\limits_{\ve\in[0,1]} \|X^{u_\ve}\|^{\a-2}_{L^q(0,T; L^p)} \|X^{u_\ve} - X^u\| _{L^q(0,T; L^p)}
\to 0,\ \ as\ \ve \to 0,\end{aligned}$$ where we also used $\a\geq 2$ and the last step is due to .
Thus, using the Hölder inequality combined with the Minkowski inequality and the bounded dominated convergence theorem we obtain that $$\begin{aligned}
\label{conv-h1-lpq}
\|R_1(\ve)\vf\|_{L^{q'}(0,T; L^{p'})}
\leq& T^\theta \|\vf\|_{L^q(0,T; L^p)} \int_0^1 \| h_1(X_{u,r,\ve}) - h_1(X^u)\|_{L^{\frac{q}{\a-1}}(0,T; L^{\frac{p}{\a-1}})} dr \nonumber \\
\to& 0,\ \ as\ \ve \to 0, \ \bbp-a.s.\end{aligned}$$
Moreover, taking into account , and , we have $$\begin{aligned}
\|R_1(\ve) \vf \|_{L^{q'}(0,T;L^{p'})}
\leq C T^{ \theta}\sup\limits_{\ve\in[0,1]}\|X^{u_\ve}\|^{ \a-1 }_{L^q(0,T; L^p)} \|\vf\| _{L^q(0,T; L^{p})} \in L^\9(\Omega),\end{aligned}$$ which along with and the bounded dominated convergence theorem yields for $j=1$. Therefore, the proof is complete. $\square $\
We shall prove now the following result.
\[WP-Backward\] $(i)$ Assume Hypothesis $(H2)$ and that $\bbx_T \in L^{2+\nu}(\Omega; L^2)$, $\bbx_1\in L^{2+\nu}(\Omega; L^2(0,T; L^2))$ for some small $\nu\in(0,1)$.
Then, there exists a unique $(\mathcal{F}_t)$-adapted solution $(Y^u,Z^u)$ to corresponding to $u\in \calu_{ad}$, satisfying for any Stirchartz pair $(p,q)$, $$\begin{aligned}
\label{bdd-Yn}
\sup\limits_{u\in \calu_{ad}} (\|Y^u\|_{L^{2+\nu}(\Omega; C([0,T]; L^2))} + \|Y^u\|_{L^{2+\nu}(\Omega; L^q(0,T; L^p))}) < \9,\end{aligned}$$ and $$\begin{aligned}
\label{bdd-Zn}
\sup\limits_{u\in \calu_{ad}} \|Z_k^u\|_{L^{2+\nu}(\Omega; L^2(0,T; L^2))} < \9,\ \ 1\leq k\leq N.\end{aligned}$$
$(ii)$ Assume in addition that $\bbx_T \in L^{2+\nu}(\Omega; H^1)$, $\bbx_1\in L^{2+\nu}(\Omega; L^2(0,T; H^1))$ for some small $\nu \in (0,1)$. Then, for any $\rho\in [2,2+\nu)$ and any Strichartz pair $(p,q)$, we have $$\begin{aligned}
\label{bdd-Yn-Wpq}
\sup\limits_{u\in \calu_{ad}} (\|Y^u\|_{L^{\rho}(\Omega; C([0,T]; H^1))} + \|Y^u\|_{L^{\rho}(\Omega; L^q(0,T; W^{1,p}))}) < \9,\end{aligned}$$ and $$\begin{aligned}
\label{bdd-Zn-Wpq}
\sup\limits_{u\in \calu_{ad}} \|Z_k^u\|_{L^{\rho}(\Omega; L^2(0,T; H^1))} < \9,\ \ 1\leq k\leq N.\end{aligned}$$
As mentioned in Section \[PROBLEM-RESULT\], the main difficulty in the analysis of backward stochastic equation mainly comes from the singular term $\lbb i h_2(X^u)\ol{Y}$. The proof of Proposition \[WP-Backward\] follows the following steps.
First, we will consider the truncated approximating equation and introduce the dual equation below, which are related together by the formula . Then, the uniform estimate of dual solutions imply, via duality arguments, those of the approximating solutions $\{Y_n\}$ (see below), which in turn imply the uniform estimates of $\{Z_n\}$ (see below). Consequently, in view of the linear structure of , one can pass to the limit and obtain the well-posedness of problem as well as the estimates and .
Analogous arguments are applicable to prove estimates and in Sobolev spaces, which requires the condition $\a\geq 2$ and the integrability conditions on $\bbx_T$ and $\bbx_1$ in Sobolev spaces.\
[***Proof.*** ]{} $(i)$. Let $g$ be a radial smooth cut-off function such that $g=1$ on $B_1(\bbr)$, and $g=0$ on $B^c_2(\bbr)$. For $j=1,2$, set $h_{j,n}(X^u):= g(\frac{|X^u|}{n}) h_j(X^u)$. Note that $ |h_{1,n}(X^u)| + |h_{2,n}(X^u)| \leq \a 2^{\a-1} |g|_{L^\9} n^{\a-1}$.
Consider the approximating backward stochastic equation $$\begin{aligned}
\label{appro-back-equa}
&d Y_n= -i\Delta Y_n\,dt - \lbb i h_{1,n}(X^u)Y_n dt +\lbb i h_{2,n}(X^u) \ol{Y_n} dt + \mu Y_n dt - iV_0Y_n dt \nonumber \\
&\qquad \quad - i u\cdot V Y_n dt + \g_1 (X^u - \bbx_1) dt - \sum\limits_{k=1}^N \ol{\mu_k} e_k Z_{k,n} dt + \sum\limits_{k=1}^N Z_{k,n} d\beta_k(t),\nonumber \\
& Y_n(T) = -(X^u(T)-\bbx_T).\end{aligned}$$ By standard theory for stochastic backward infinite dimensional equations (see, e.g., [@9a], [@HP91]), it follows that there exists a unique $(\mathcal{F}_t)$-adapted solution $(Y_n,Z_n) \in L^2(\Omega; C([0,T]; L^2)) \times (L^2_{ad}(0,T; L^2(\Omega; L^2)))^N$ to .
In order to pass to the limit $n\to \9$, we are going to obtain uniform estimates of $Y_n$ in the space $\caly_{\rho_\nu}:= L^{\rho_\nu}(\Omega; L^\9(0,T; L^2)) \cap L^{\rho_\nu} (\Omega; L^q(0,T; L^p))$ with $\rho_\nu := 2+\nu$ and $(p,q)=(\a+1, \frac{4(\a+1)}{d(\a-1)})$.
To this purpose, for each $n\geq 1$, define the functional $\Lambda_n$ on the space $L^\9(\Omega\times (0,T) \times \bbr^d)$, $$\begin{aligned}
\label{Oper-Phi}
\Lambda_n(\Psi) := \bbe& Re \<X^u(T)- \bbx_T, \psi_n(T)\>_2
+\g_1 \bbe \int_0^{T} Re\<X^u(t)-\bbx_1(t), \psi_n(t)\>_2dt,\end{aligned}$$ where $\Psi \in L^\9(\Omega\times (0,T) \times \bbr^d)$, $X^u$ is the solution to , and $\psi_n$ satisfies $$\begin{aligned}
\label{forward-equa-Psi}
&d \psi_n = -i \Delta \psi_n dt - \lbb i h_{1,n}(X^u) \psi_n dt - \lbb i h_{2,n}(X^u) \ol{\psi_n} dt -\mu \psi_n dt \nonumber \\
&\qquad \quad - i V_0 \psi_n dt -i u\cdot V \psi_n dt - \Psi dt + \psi_n d W(t), \nonumber \\
&\psi_n(0)= 0.\end{aligned}$$ (Note that, is similar to but with $\Psi$ in place of $i\wt{u}\cdot V X^u$.)
Since $|h_{j,n}(X^u)|\leq \a |g|_{L^\9} |X^u|^{\a-1}$, $j=1,2$, arguing as in the proof of Lemma \[WP-Forward\] we infer that there exists a unique strong $L^2$-solution $\psi_n$ to on $[0,T]$, satisfying $$\begin{aligned}
\label{psi-Psi*}
\sup\limits_{n} (\|\psi_n\|_{ C([0,T]; L^2)} + \|\psi_n\|_{ L^q(0,T; L^p )})
\leq C(T) \|\Psi\|_{L^1(0,T; L^2) + L^{q'}(0,T;L^{p'})},\end{aligned}$$ where $C(T) \in L^\9(\Omega)$ is independent of $n$ and $\Psi$. It follows also that for $\rho >1$, $$\begin{aligned}
\label{psi-Psi}
\sup\limits_{n\geq 1} \|\psi_n\|_{\caly_{\rho'}}
\leq C(\rho,T) \|\Psi\|_{\mathcal{Y}'_{\rho'}},\end{aligned}$$ where $C(\rho, T)$ is independent of $n$ and $\Psi$ and $\caly'_{\rho'}:= L^{\rho'}(\Omega; L^1(0,T; L^2)) + L^{\rho'}(\Omega; L^{q'}(0,T;L^{p'}))$.
Moreover, by Itô’s formula, for every $n\geq 1$ and $\Psi\in L^\9(\Omega\times (0,T) \times \bbr^d)$, we have $$\begin{aligned}
\label{dual-Psi-Y}
\Lambda_n(\Psi)= \bbe \int_0^{T} Re \<\Psi, Y_n\>_2 dt.\end{aligned}$$
Thus, by the conservation of $|X^u(t)|_2$ and by estimates , , we have $$\begin{aligned}
\label{dual-esti}
|\Lambda_n(\Psi)|
\leq& \g_1 \|X^u-\bbx_1\|_{L^{\rho_\nu}(\Omega; L^2(0,T; L^2))} \|\psi_n\|_{L^{\rho_\nu'}(\Omega; L^2(0,T; L^2))} \nonumber \\
& + \|X^u(T) - \bbx_T\|_{L^{\rho_\nu}(\Omega; L^2)} \|\psi_n(T)\|_{L^{\rho_\nu'}(\Omega; L^2)} \nonumber \\
\leq& C \|\Psi\|_{\mathcal{Y}'_{\rho_\nu'}},\end{aligned}$$ where $C$ is independent of $n$. Since $L^\9( \Omega\times (0,T) \times \bbr^d )$ is dense in $\caly'_{\rho_\nu'}$ and $\mathcal{Y}_{\rho_{\nu}}$ is the dual space of $\caly'_{\rho_\nu'}$, it follows by , that $$\begin{aligned}
\label{esti-Yn-Lp}
\sup\limits_{n\geq 1}\| Y_n\|_{\mathcal{Y}_{\rho_\nu}} <\9.\end{aligned}$$
Hence, there exists $\wt{Y} \in \mathcal{Y}_{\rho_\nu}$, such that along a subsequence of $\{n\}\to \9$ (still denoted by $\{n\}$), $$\begin{aligned}
\label{conv-Yn-Lp}
Y_n \to \wt{Y},\ \ weak-star\ in\ \mathcal{Y}_{\rho_\nu}.\end{aligned}$$
Note that, for each $j=1,2$, $h_{j,n}(X^u) \to h_j(X^u)$, $d\bbp \otimes dt \otimes d\xi-a.e.$, and $\sup_{n\geq 1}|h_{j,n}(X^u)| \leq \a |g|_{L^\9} |X^u|^{\a-1} \in L^{2\rho_\nu'}(\Omega; L^{\frac{q}{\a-1}}(0,T;L^{\frac{p}{\a-1}}))$, by the dominated convergence theorem, $$\begin{aligned}
\label{conv-hin}
h_{j,n}(X^u) \to h_j(X^u), \ in\ L^{2\rho_\nu'}(\Omega; L^{\frac{q}{\a-1}}(0,T;L^{\frac{p}{\a-1}})), \ \ as\ n\to \9.\end{aligned}$$ Since $\frac{1}{(2\rho'_\nu)'} = \frac{1}{\rho_\nu} + \frac{1}{2\rho_\nu}$, from Hölder’s inequality, and it follows that $$\begin{aligned}
\label{conv-hin-Yn}
h_{1,n} (X^u) Y_n \to h_1(X^u) \wt{Y},\ h_{2,n} (X^u) \ol{Y_n} \to h_2(X^u) \ol{\wt{Y}},\end{aligned}$$ weakly in $L^{(2\rho_\nu')'}(\Omega; L^{\frac{q}{\a}}(0,T;L^{p'}))$.
Moreover, we claim that for $1\leq k\leq N$, $$\begin{aligned}
\label{esti-Zn-L2}
\sup\limits_{n\geq 1} \|Z_{k,n}\|_{L^{\rho_\nu}(\Omega; L^2(0,T; L^2))} \leq C <\9.\end{aligned}$$ In particular, for each $1\leq k\leq N$, there exists $Z^u_k\in L^{\rho_\nu}(\Omega; L^2(0,T; L^2))$ such that (selecting a further subsequence if necessary) $$\begin{aligned}
\label{conv-Zn-L2}
Z_{k,n} \to Z^u_k,\ \ weakly\ in\ L^2(\Omega; L^2(0,T; L^2)).\end{aligned}$$ Since $v \mapsto \int_\cdot^T v d\beta_k(s)$ is a bounded linear operator in $L^2(\Omega; L^2(0,T; L^2))$, it follows that for each $1\leq k\leq N$, $$\begin{aligned}
\label{conv-Zn-L2*}
\int_\cdot^T Z_{k,n} d\beta_k(s) \to \int_\cdot^T Z^u_{k} d\beta_k(s),\ \ weakly\ in\ L^2(\Omega; L^2(0,T; L^2)).\end{aligned}$$
To prove , we apply Itô’s formula to to get that for $\eta>0$, $$\begin{aligned}
\label{ito-pn}
&e^{\eta t}|Y_n(t)|_2^2 \nonumber \\
=& e^{\eta T} |X^{u}(T) - \bbx_T|_2^2
- \eta \int_t^T e^{\eta s} |Y_n|_2^2 ds \nonumber \\
& + 2\lbb \int_{t}^{T} Im \int e^{\eta s} h_{2,n}(X^u) \ol{Y_n^2} d\xi ds
-2 \int_{t }^{T} \int e^{\eta s} \mu |Y_n|^2 d\xi ds \nonumber \\
& -2 \g_1 \int_{t }^{T} Re \int e^{\eta s} Y_n (\ol{X^{u}} - \ol{\bbx_1}) d\xi ds
+2 \sum\limits_{k=1}^N \int_{t }^{T} Re \int e^{\eta s} \mu_k e_k Y_n\ol{Z_{k,n}} d\xi ds \nonumber \\
& - \sum\limits_{k=1}^N \int_{t }^{T} e^{\eta s} |Z_{k,n}|_2^2 ds
-2 \sum\limits_{k=1}^N\int_{t }^{T} Re \int e^{\eta s} Y_n \ol{Z_{k,n}} d\xi d\beta_k(s) \nonumber \\
=:& e^{\eta T} |X^{u}(T) - \bbx_T|_2^2 - \eta \int_t^T e^{\eta s} |Y_n|_2^2 ds + \sum\limits_{j=1}^6 K_j(t ).\end{aligned}$$
Note that, since $|h_{2,n}(X^u)| \leq \a |g|_{L^\9} |X^u|^{\a-1}$, by Hölder’s inequality, $$\begin{aligned}
\label{esti-K1}
|K_1(t)|
\leq \a |g|_{L^\9} e^{\eta T} T^{\theta} \|X^{u}\|^{\a-1}_{L^q(0,T; L^{p})} \|Y_n\|^2_{L^q(0,T; L^p)},\end{aligned}$$ where $\theta = 1 - d(\a-1) / 4 \in (0,1)$.
Moreover, using $ab\leq ca^2 + c^{-1}b^2$, $c>0$, we get $$\begin{aligned}
\label{esti-K24}
\sum\limits_{j=2}^4 |K_j(t)|
\leq& (2|\mu|_\9 + 2 \g_1 + 8\sum\limits_{k=1}^N|\mu_ke_k|^2_{L^\9}) \int_{t }^{T} e^{\eta s} |Y_n|_2^2 ds \nonumber \\
& + 2 \g_1 \int_{t }^{T} e^{\eta s} |X^{u} - \bbx_1|_2^2 ds
+ \frac 12 \int_{t}^{T} e^{\eta s} |Z_{k,n}|_2^2 ds.\end{aligned}$$
Thus, choosing $\eta > 2|\mu|_\9 + 2 \g_1 + 8\sum_{k=1}^N|\mu_ke_k|^2_{L^\9}$, it follows that for any $t\in[0,T]$, $$\begin{aligned}
\label{esti-YZ}
e^{\eta t}|Y_n(t)|_2^2 + \frac 12 \sum\limits_{k=1}^N\int_{t}^{T} e^{\eta s} |Z_{k,n}|_2^2 ds
\leq V_{T,n} -2 \sum\limits_{k=1}^N\int_{t }^{T} Re \int e^{\eta s} Y_n \ol{Z_{k,n}} d\xi d\beta_k(s),\end{aligned}$$ where $$\begin{aligned}
\label{VT}
V_{T,n} : =& e^{\eta T} |X^{u}(T) - \bbx_T|_2^2
+ 2\g_1 \int_{0}^{T} e^{\eta s} |X^{u} - \bbx_1|_2^2 ds \nonumber \\
& + \a |g|_{L^\9} e^{\eta T} T^{\theta} \|X^{u}\|^{\a-1}_{L^q(0,T; L^{p})} \|Y_n\|^2_{L^q(0,T; L^p)}.\end{aligned}$$ This yields $$\begin{aligned}
\label{esti-Z}
& \sum\limits_{k=1}^N \bbe \(\int_{0}^{T} e^{\eta s} |Z_{k,n}|_2^2 ds\)^{\frac{\rho_\nu}{2}} \nonumber \\
\leq& C (\rho_\nu) \( \bbe V^{\frac{\rho_\nu}{2}}_{T,n} + \bbe \sup\limits_{t\in[0,T]}
\bigg|\sum\limits_{k=1}^N\int_{0 }^{t} Re \int e^{\eta s} Y_n \ol{Z_{k,n}} d\xi d\beta_k(s)\bigg|^{\frac{\rho_\nu}{2}} \).\end{aligned}$$
Note that, $\bbe V^{\rho_\nu/2}_{T,n} <\9$, due to , and to the integrability conditions on $\bbx_T$ and $\bbx_1$. Moreover, by the Burkholder-Davis-Gundy inequality, the second term in the right hand side of is bounded by $$\begin{aligned}
& C (\rho_\nu) \bbe \(\int_{0 }^{T} \sum\limits_{k=1}^N \bigg|\int e^{\eta s} Y_n \ol{Z_{k,n}} d\xi \bigg|^2 ds\)^{\frac{\rho_\nu}{4}} \\
\leq& C(\rho_\nu) \sum\limits_{k=1}^N \bbe \sup\limits_{t\in[0,T]} e^{\frac 14 \rho_\nu \eta t} |Y_n(t)|_2^{\frac{\rho_\nu}{2}}
\( \int_0^T e^{\eta s}|Z_{k,n}|_2^2 ds \)^{\frac {\rho_\nu}{4}} \\
\leq& C(\rho_\nu, N) e^{\frac 12 \rho_\nu\eta T} \bbe |Y_n|_{C([0,T]; L^2)}^{ \rho_\nu}
+ \frac 12 \sum\limits_{k=1}^N \bbe \(\int_{0}^{T} e^{\eta s} |Z_{k,n}|_2^2 ds\)^{\frac{\rho_\nu}{2}}.\end{aligned}$$ Plugging this into we get $$\begin{aligned}
\label{esti-Z-Y}
& \sup\limits_{n\geq 1}\frac 12 \sum\limits_{k=1}^N \bbe \(\int_{0}^{T} e^{\eta s} |Z_{k,n}|_2^2 ds\)^{\frac{\rho_\nu}{2}} \nonumber \\
\leq& C(\rho_\nu) \sup\limits_{n\geq 1}\bbe V^{\frac{\rho_\nu}{2}}_{T,n}
+ C(\rho_\nu, N) e^{\frac 12 \rho_\nu \eta T} \sup\limits_{n\geq 1}\bbe|Y_n|_{C([0,T]; L^2)}^{\rho_\nu},\end{aligned}$$ which by implies , as claimed.
Now, set $$\begin{aligned}
Y^u(\cdot):=& -(X^u(T) - \bbx_T) - \int_{\cdot }^{T} \bigg(-i \Delta \wt{Y} - \lbb i h_1(X^u) \wt{Y}
+\lbb i h_2(X^u) \ol{\wt{Y}} \\
&\qquad + \mu \wt{Y}
- iV_0 \wt{Y} - i u\cdot V \wt{Y} + \g_1 (X^u - \bbx_1) - \sum\limits_{k=1}^N \ol{\mu_k} e_k Z^u_{k} \bigg) ds \\
&- \sum\limits_{k=1}^N \int_{\cdot }^{T} Z^u_{k} d\beta_k(s) .\end{aligned}$$ By virtue of , , and , we may pass to the limit in and obtain that for any $v\in H^2$, $f \in L^\9(\Omega\times(0,T))$, $$\begin{aligned}
\label{back-equa-n.1}
\bbe \int_0^T {}_{H^{-2}}\<\wt{Y}(t), f(t)v\>_{H^2} dt
= \bbe \int_0^T {}_{H^{-2}}\<Y^u(t), f(t)v\>_{H^2} dt,\end{aligned}$$ which implies that $Y^u=\wt{Y}$, in $H^{-2}$, $d\bbp \otimes dt$-a.e. Since $Y^u$ is continuous in $H^{-2}$, $\bbp$-a.s., we can find a null set $N'$, such that for any $\omega\not \in N'$, $(Y^u(\omega), Z^u(\omega))$ solves in $H^{-2}$ for all $t\in [0,T]$, which proves the existence of solution to . Estimates and follow immediately by and respectively. Moreover, as in the proof of [@1 Lemma 4.3], we have for $|Y^u(t)-Y^u(s)|_2^2$ an Itô formula similar to , which implies $t\to Y^u(t)$ is $L^2$ continuous.
The uniqueness can also be proved by the duality arguments. Indeed, let $(Y^u_j, Z_j^u)$, $j=1,2$, be any two solutions to . For any $\Psi\in L^\9(\Omega \times (0,T) \times \bbr^d)$, let $\psi$ be the unique solution to but with $h_{j,n}(X^u)$ replaced by $h_j(X^u)$, $j=1,2$. Define $\Lambda(\Psi)$ similarly as in with $\psi_n$ replaced by $\psi$. Then, similarly to , $\Lambda(\Psi) = \bbe\int_0^T Re \<\Psi, Y^u_j\>_2dt$, $j=1,2$. It follows that $Y^u_1 = Y^u_2$ by the arbitrariness of $\Psi $, and so $Z^u_1 = Z^u_2$ by the estimate similar to .\
$(ii)$. Fix $1\leq j \leq N$. Consider the approximating equation of $(\partial_j Y^u, \partial_j Z^u)$ below ($\partial_j := \frac{\partial}{\partial \xi_j}$), $$\begin{aligned}
\label{appro-back-equa-h1}
& d Y'_n= -i\Delta Y'_n dt + G_n (Y'_n) dt
- \sum\limits_{k=1}^N \ol{\mu}_k e_k Z'_{k,n} dt + \g_1 \partial_j (X-\bbx_1) dt \nonumber \\
& \qquad \qquad \ + F_n(X^u, Y^u, Z^u) dt + \sum\limits_{k=1}^N Z'_{k,n} d\beta_k(t), \nonumber \\
& Y'_n= - (\partial_j X^u(T) - \partial_j \bbx_T),\end{aligned}$$ where $X^u$ and $(Y^u, Z^u)$ are the solutions to and respectively, $$\begin{aligned}
G_n (Y'_n )
:= - \lbb i h_{1,n}(X^u)Y'_n + \lbb i h_{2,n}(X^u) \ol{Y'_n }
+( \mu - i V_0 - iu\cdot V) Y'_n ,\end{aligned}$$ $$\begin{aligned}
F_n(X^u, Y^u, Z^u) :=& -\lbb i h'_{1,n}(X^u)Y^u + \lbb i h'_{2,n}(X^u) \ol{Y^u}
+ \partial_j \mu Y^u - i \partial_j V_0Y^u \\
& - i u\cdot \partial_j V Y^u
- \sum\limits_{k=1}^N \partial_j (\ol{\mu_k} e_k) Z^u_{k},\end{aligned}$$ $h_{k,n}(X^u)$ is as in and $h'_{k,n}(X^u) = g(\frac{|X^u|+|\na X^u|}{n}) \partial_j(h_{k}(X^u))$, $k=1,2$. By truncation, $|h_{k,n}(X^u)| +| h'_{k,n}(X^u)| \leq C n^{\a-1}$, $k=1,2$, it follows that there exists a unique $(\mathcal{F}_t)$-adapted solution $(Y'_n, Z'_n) \in L^2(\Omega; C([0,T]; L^2)) \times (L^2_{ad}(0,T; L^2(\Omega; L^2)))^N$ to .
For each $\Psi\in L^\9(\Omega \times (0,T) \times \bbr^d)$, let $\psi_n$ be the solution to . Similarly to , set $$\begin{aligned}
\wt{\Lambda}_{j,n}(\Psi) :=& \bbe Re \<\partial_j X^u(T)- \partial_j \bbx_T, \psi_n(T)\>_2 \\
& + \g_1 \bbe \int_0^{T} Re\< \partial_j X^u(t)- \partial_j \bbx_1(t), \psi_n(t))\>_2dt .\end{aligned}$$ By Itô’s formula, we have $$\begin{aligned}
\label{def-Lamj}
\wt{\Lambda}_{j,n}(\Psi) = \bbe \int_0^T Re\<\Psi, Y'_n\>_2 dt - \bbe \int_0^T Re \int F_n(X^u,Y^u, Z^u)\ol{\psi_n} d\xi dt.\end{aligned}$$
Note that, since $|h'_{1,n}(X^u)| \leq C |X^u|^{\a-2} |\partial_j X^u|$ and $2\leq \a < 1+ 4/d$, by Hölder’s inequality, we have for $(p,q)=(\a+1, \frac{4(\a+1)}{d(\a-1)})$, $$\begin{aligned}
\label{restrict}
& \bigg| \int_0^T Re \int (-\lbb i) h'_{1,n}(X^u) Y^u \ol{\psi_n} d\xi dt \bigg| \nonumber \\
\leq& \|h'_{1,n}(X^u)Y^u \|_{L^{q'}(0,T; L^{p'})} \|\psi_n\|_{L^q(0,T;L^p)} \nonumber \\
\leq& C_\a T^\theta \|X^u\|^{\a-2}_{L^q(0,T;L^p)} \|\partial_j X^u\|_{L^q(0,T;L^p)}\|Y^u \|_{L^q(0,T;L^p)} \|\psi_n\|_{L^q(0,T;L^p)} \nonumber \\
\leq& C_\a T^\theta \|X^u\|^{\a-1}_{L^q(0,T;W^{1,p})} \|Y^u\|_{L^q(0,T;L^p)} \|\psi_n\|_{L^q(0,T;L^p)},\end{aligned}$$ where $\theta = 1- d(\a-1)/4\in (0,1)$. Hence, for any $\rho \in [2,\rho_\nu)$, $$\begin{aligned}
&\bigg| \bbe \int_0^T Re \int (-\lbb i) h'_{1,n}(X^u) Y^u \ol{\psi_n} d\xi dt \bigg|\nonumber \\
\leq& C(T) \bbe (\|X^u\|^{\a-1}_{L^q(0,T;W^{1,p})} \|Y^u\|_{L^q(0,T;L^p)} \|\psi_n\|_{L^q(0,T;L^p)}) \nonumber \\
\leq& C(T) \|X^u\|^{\a-1}_{L^\eta(\Omega; L^q(0,T;W^{1,p}))} \|Y^u\|_{L^{\rho_\nu}(\Omega; L^q(0,T;L^p))} \|\psi_n\|_{L^{\rho'}(\Omega;L^q(0,T;L^p))}),\end{aligned}$$ where $\eta$ satisfies $\frac{1}{(\a-1)\eta} = \frac{1}{\rho'_\nu} - \frac{1}{\rho'}>0$. Similar arguments apply to the term involving $\lbb i h'_{2,n}(X^u) \ol{Y^u}$. Moreover, the other terms in the integration $\bbe \int_0^T Re \int F_n(X^u,Y^u,Z^u )\ol{\psi_n} d\xi dt$ are bounded by $$\begin{aligned}
C \(\|Y^u \|_{L^\rho(\Omega; L^\9(0,T;L^2))} + \sum\limits_{k=1}^N\|Z^u_k\|_{L^\rho(\Omega; L^2(0,T;L^2))}\) \|\psi_n\|_{L^{\rho'}(\Omega; L^\9(0,T;L^2))}.\end{aligned}$$ Plugging the estimates above into and using , , and we obtain for any $\rho \in [2,\rho_\nu)$, $$\begin{aligned}
\bigg|\bbe \int_0^T Re\<\Psi,Y'_n\>_2 dt\bigg|
\leq& |\wt{\Lambda}_{j,n}(\Psi)| + \bigg|\bbe \int_0^T Re \int F_n(X^u,Y^u,Z^u)\ol{\psi_n} d\xi dt \bigg| \\
\leq& C \|\Psi\|_{\mathcal{Y}'_{\rho'}}\end{aligned}$$ with $C$ independent of $n$ and $\Psi$, which implies that for any $\rho \in [2,\rho_\nu)$, $$\begin{aligned}
\label{esti-Yn'}
\sup\limits_{n\geq 1} \| Y'_n \|_{\mathcal{Y}_{\rho}} \leq C.\end{aligned}$$ Once we obtain , using similar arguments as those below , we can prove the assertion $(ii)$. The details are omitted. $\square$\
[***Proof of Proposition \[l4.1\].***]{} Using in Lemma \[WP-Forward\] we have $$\begin{aligned}
\label{e4.9}
& \lim_{\vp\to0}\frac1\vp\,(\Phi(u+\vp\wt u)-\Phi(u)) \nonumber \\
=& 2\E \bigg(\!{\rm Re}\<X^u(T)-\bbx_T,\vf^{u,\wt{u}}(T)\>_2 \nonumber \\
&\qquad +\g_1\E\int^{T}_0 {\rm Re}\<X^u(t)-\bbx_1(t),\vf^{u,\wt{u}}(t)\>_2dt
+\g_2\int^{T}_0 u\cdot{\wt u}\,dt \bigg).\end{aligned}$$ Then, similarly to , by and , we obtain via Itô’s formula, $$\begin{aligned}
\E\,{\rm Re}\<X^u(T)-\bbx_T,\vf^{u,\wt{u}}(T)\>_2
+&\g_1\E\dd\int^{T}_0{\rm Re}\<X^u(t)-\bbx_1(t),\vf^{u,\wt{u}}(t)\>_2 dt\nonumber \\
&= -\E\,{\rm Im}\int^{T}_0\int_{\rr^d}\wt u\cdot V\,X^u \ol{Y^u} d\xi dt.\end{aligned}$$ Combining these formulas we get as claimed. $\square$
Proof of Theorem \[t2.6\]. {#PROOF-THM2}
==========================
As in the proof of Lemma \[Lem-conv\], we note that $\Phi$ is continuous on the metric space $\calu_{ad}$ endowed with the distance $d(u,v)= \|u-v\|=(\bbe \int_0^T |u(t)-v(t)|_m^2 dt)^{1/2}$. Applying Ekeland’s variational principle in $\calu_{ad}$ (see [@9 Theorem 1], or [@8]), for every $n\in\nn$ we get $u_n\in\calu_{ad}$ such that $$\begin{aligned}
&\Phi(u_n)\le\Phi(u)+\dd\frac{1}{n}\,d(u_n,u),\ \ \forall u\in\calu_{ad}. \label{inf-phi-n}\end{aligned}$$ In particular, it follows that $$\label{e4.10a}
u_n={\rm arg\,min}\left\{\Phi(u)+\frac{1}{n}\,\|u_n-u\|;\ u\in\calu_{ad}\right\}.$$
We define the function $\wt{\Phi}: L^2_{ad}(0,T; \bbr^m) \to \ol{\bbr} = (-\9,+\9]$ by $$\begin{aligned}
\wt{\Phi} = \Phi(u) + I_{\calu_{ad}}(u), \ \ \forall u\in L^2_{ad}(0,T; \bbr^m),\end{aligned}$$ where $$\begin{aligned}
I_{\calu_{ad}} (u) = \left\{
\begin{array}{ll}
0, & \hbox{if $u\in \calu_{ad}$;} \\
+\9, & \hbox{otherwise.}
\end{array}
\right.\end{aligned}$$ The subdifferential $\partial \wt{\Phi} (u) \subset L^2_{ad}(0,T; \bbr^m)$ of $\wt{\Phi}$ at $u$ in the sense of R.T. Rockafellar [@R79] is defined as the set of all $z\in L^2_{ad}(0,T; \bbr^m)$ such that the function $v \to \wt{\Phi}(v)- \bbe \int_0^T v(t) z(t) dt$ has $u$ as a substationary point in the sense of [@R79].
We have $$\begin{aligned}
\label{phi-eta}
\partial \wt{\Phi} (u) \subset \eta(u) + \mathcal{N}_{\calu_{ad}} (u), \ \ \forall u \in \calu_{ad},\end{aligned}$$ where $\eta (u)$ is defined by , $\mathcal{N}_{\calu_{ad}}(u)$ is the normal cone to $\calu_{ad} $ at $u_n$, i.e., $$\mathcal{N}_{\calu_{ad}}(u_n)=\{v\in L^2_{ad}(0,T; \bbr^m) ;\ \<v, u_n-{\wt v}\> \ge0,\ \ \ff\wt v\in \calu_{ad}\},$$ and $\<\ ,\ \>$ denotes the inner product of $ L^2_{ad}(0,T; \bbr^m)$.
To prove , as mentioned in [@R79 (2.4)], for each $u\in \calu_{ad}$, one has $$\begin{aligned}
\partial \wt{\Phi}(u)
= \{ z\in L^2_{ad}(0,T; \bbr^m): \wt{\Phi}^\uparrow (u, y) \geq \<y, z\>, \forall y \in L^2_{ad}(0,T; \bbr^m) \},\end{aligned}$$ where $\wt{\Phi}^\uparrow (u,y)$ is the subderivative at $u$ with respect to $y$ $$\begin{aligned}
\wt{\Phi}^\uparrow (u, y)
= \sup\limits_{V \subset \mathcal{N}(y)}
\left[ \limsup\limits_{\substack{u'\to u,\a'\to \wt{\Phi}(u) \\ \a'\geq \wt{\Phi}(u'), t\to 0}}
\inf\limits_{y'\in V}
\left( \frac{\Phi(u'+ty')-\a'}{t}
+ \frac{I_{\calu_{ad}}(u'+ty')}{t} \right) \right],\end{aligned}$$ and $\mathcal{N}(y)$ is the set of all neighborhoods of $y$. This yields $$\begin{aligned}
\wt{\Phi}^\uparrow (u, y)
= \lim\limits_{t\to 0} \frac{\Phi(u+ty)-\Phi(u)}{t} + I'_{\calu_{ad}}(u,y),\end{aligned}$$ where $I'_{\calu_{ad}}(u,y) =0$ if $y\in T_{\calu_{ad}}(u)$, and $I'_{\calu_{ad}}(u,y) = \9$ if $y\not \in T_{\calu_{ad}}(u)$, $T_{\calu_{ad}}(u)$ is the (Clarke) tangent cone to $\mathcal{U}_{ad}$ at $u$ defined in [@R79]. Then, by Proposition \[l4.1\], for any $z\in \partial \wt{\Phi}(u)$, we have that $\<\eta(u), y\> \geq \<y,z\>$, $\forall y=v-u$, $v \in \calu_{ad}$. Thus, $z\in \eta(u) + \mathcal{N}_{\calu_{ad}}(u)$. This implies .
On the other hand, by Theorem $2$ in [@R79] we have $$\begin{aligned}
\label{subdiff-subset}
\partial ( \wt{\Phi}(u) + \frac 1n \|u_n-u\| ) \subset \partial \wt{\Phi}(u) + \frac 1n \partial \|u_n-u\|.\end{aligned}$$
Thus, by - we get $$\begin{aligned}
0 \in& \partial (\wt{\Phi}(u)+ \frac 1n \|u_n-u\|) (u=u_n) \\
\subset& \eta(u_n) + \frac 1n (\partial \|u_n-u\|) (u=u_n) + \mathcal{N}_{\calu_{ad}}(u_n),\end{aligned}$$ which implies that there exist $\zeta_n \in \mathcal{N}_{\calu_{ad}}(u_n)$ and $\eta_n \in (\partial \|u_n-u\|) (u=u_n)$, such that $$\begin{aligned}
\label{e4.11}
\eta(u_n) + \zeta_n + \frac 1n \eta_n = 0.\end{aligned}$$
We claim that, $$\begin{aligned}
\label{equa-cone}
\mathcal{N}_{\calu_{ad}}(u_n)=\{v\in L^2_{ad}(0,T;\rr^m): v\in N_U(u_n),\ \ a.e.\ on\ (0,T) \times \Omega.\},\end{aligned}$$ where $N_U(u_n)$ is the normal cone to $U\subset \rr^m$ at $u_n\in U,$ that is, $N_U(u_n)=\{v\in\rr^m;\ v\cdot( u_n-{\wt v}) \ge0,\ \ \ff\wt v\in U\}.$
Indeed, for any $\eta \in \mathcal{N}_{\calu_{ad}}(u_n)$, we have $$\begin{aligned}
\label{cone-eta}
\bbe \int_0^T \eta \cdot (u_n - v) dt \geq 0, \ \ \forall\ v\in \calu_{ad}.\end{aligned}$$ Since for each closed convex set $U$, $\forall \nu>0$, $(I+\nu N_U)^{-1}=P_U$, where $P_U$ is the projection on $U$, there exists a unique $v \in \calu_{ad}$, such that $$\begin{aligned}
\label{equa-v}
v + N_U(v) \ni u_n + \eta,\ \ a.e.\ on\ (0,T) \times \Omega,\end{aligned}$$ i.e. $v = P_U (u_n + \eta)$, a.e. on $(0,T) \times \Omega$. Hence, there exists $\zeta_v \in N_U(v)$, such that $ v + \zeta_v = u_n + \eta$, $d \bbp \otimes dt $-a.e. Then, by , $$\begin{aligned}
0 \leq \bbe \int_0^T (v-u_n + \zeta_v ) \cdot (u_n - v) dt
= - \|u_n - v\|^2 + \bbe \int_0^T \zeta_v (u_n - v) dt.\end{aligned}$$ Since $d \bbp \otimes dt $-a.e., $\zeta_v \in N_U(v)$, $\zeta_v \cdot (v-u_n) \geq 0$, we get $$\begin{aligned}
\|u_n - v\|^2 \leq \bbe \int_0^T \zeta_v\cdot (u_n -v) dt \leq 0.\end{aligned}$$ It follows that $u_n = v$, $d \bbp \otimes dt$-a.e., which yields by that $\eta \in N_U(u_n)$, $d \bbp \otimes dt$-a.e., thereby implying $$\begin{aligned}
\mathcal{N}_{\calu_{ad}}(u_n) \subset \{v\in L^2_{ad}((0,T);\rr^m): v\in N_U(u_n),\ \ a.e.\ on\ (0,T) \times \Omega.\}.\end{aligned}$$ The inverse inclusion is obvious. Thus we obtain , as claimed.\
Now, by virtue of , we may rewrite as $$\label{e4.12}
\barr{r}
\dd u_n(t)+\frac1{2\gamma_2}\,\zeta^0_n(t)= \frac 1{\gamma_2}\,{\rm Im}\int_{\rr^d} V(\xi) X_n(t,\xi) \ol{Y_n}(t,\xi)d\xi-\frac1{2\g_2 n}\,\eta_n(t)\vsp\mbox{ a.e. on }(0,T)\times\ooo,\earr$$ where $\zeta^0_n(t)\in N_U(u_n(t)),$ $X_n :=X^{u_n}$, and $(Y_n,Z_n)$ is the solution to corresponding to $u_n$. We get by that $$\label{e4.13}
u_n(t)=P_U\(\dd\frac 1{\g_2}\,{\rm Im}\int V(\xi) X_n(t,\xi) \ol{Y_n}(t,\xi)d\xi-\frac1{2\g_2 n}\,\eta_n(t)\),$$ with $$\label{e4.14}
\E\int^{T}_0|\eta_n(t)|^2_mdt=1.$$
We claim that there exists a probability space $(\Omega^*, \mathcal{F}^*, \bbp^*)$, $u^*_n, u^*\in \calu_{ad^*}$, $n\geq 1$, such that the distributions of $u^*_n$ and $u_n$ coincide on $L^1(0,T; \bbr^m)$, and as $n\to \9$, $$\begin{aligned}
\label{un-u-L1}
u^*_n \to u^*, \ \ in\ L^1(0,T; \bbr^m), \ \ \bbp^*-a.s.\end{aligned}$$ Then, it follows from the boundedness of $\{u^*_n\}$ that $$\begin{aligned}
u^*_n \to u^* , \ \ in\ L^2(0,T; \bbr^m), \ \ \bbp^*-a.s.\end{aligned}$$ Hence, similar arguments as in the proof of Theorem \[t2.5\] imply that $$\begin{aligned}
\Phi^*(u^*) = \lim\limits_{n\to \9} \Phi^*(u^*_n) = \lim\limits_{n\to \9} \Phi(u_n) = I,\end{aligned}$$ thereby yielding the sharp equality in .
It remains to prove . By virtue of Skorohod’s representation theorem, we only need to show the tightness of the distributions of $u_n$ in $L^1(0,T; \bbr^m)$, $n\geq 1$. For this purpose, in view of Lemma \[Lem-tight-L1\], it suffices to prove that $\mu_n:=\bbp \circ u_n^{-1}$, $n\geq 1$, satisfy and .
Indeed, follows immediately from the uniform boundedness of $\{u_n\}$. Regarding , by Markov’s inequality, it suffices to show that there exists a positive exponent $b >0$ such that for any $\delta\in(0,1)$, $$\begin{aligned}
\label{L1-h-un}
\limsup\limits_{n\to \9} \bbe \sup\limits_{0<h\leq \delta} \int_0^{T-h} | u_n(t+h) - u_n(t)|_m dt \leq C \delta^b.\end{aligned}$$
To this end, since $P_U$ is Lipschitz, using , the Chauchy inequality, and we get $$\begin{aligned}
\label{L1-h-un-esti}
&\bbe \sup\limits_{0<h\leq \delta} \int_0^{T-h} | u_n(t+h) - u_n(t) |_m dt \nonumber \\
\leq& \frac{1}{\g_2 n} T^{\frac 12} + \frac{1}{\g_2} |V|_{L^{\9}} \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \bigg( |X_n(t+h)-X_n(t)|_{H^{-1}} |Y_n(t+h)|_{H^1} \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \ \ \ + |Y_n(t+h)-Y_n(t)|_{H^{-1}} |X_n(t)|_{H^1} \bigg) dt \nonumber \\
\leq& \frac{1}{\g_2 n} T^{\frac 12} + C \(\bbe \int_0^{T} \sup\limits_{0<h\leq \delta} |X_n(t+h)- X_n(t)|_{H^{-1}}^2 dt\)^{\frac 12} \nonumber \\
& + C \(\bbe \int_0^{T} \sup\limits_{0<h\leq \delta} |Y_n(t+h)- Y_n(t)|_{H^{-1}}^2 dt\)^{\frac 12}.\end{aligned}$$
Let us estimate $\bbe \int_0^{T}\sup_{0<h\leq \delta} |Y_n(t+h)- Y_n(t)|_{H^{-1}}^2 dt$ in the right hand side of . Similar arguments apply to $\bbe \int_0^{T}\sup_{0<h\leq \delta} |X_n(t+h)- X_n(t)|_{H^{-1}}^2 dt$.
By the backward equation , $$\begin{aligned}
\label{K1-K4}
& \bbe \int_0^{T} \sup\limits_{0<h\leq \delta}|Y_n(t+h)- Y_n(t)|_{H^{-1}}^2 dt \nonumber \\
\leq & \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \bigg| \int_t^{t+h} i\Delta Y(s) ds \bigg|_{H^{-1}}^2 dt \nonumber \\
& + \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \bigg| \int_t^{t+h} ( -\lbb i h_1(X_n(s) )Y_n(s) + \lbb i h_2(X_n(s) )\ol{Y_n}(s) ) ds \bigg|_{H^{-1}}^2 dt \nonumber \\
& + \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \bigg| \int_t^{t+h} (\mu - i V_0 - i u_n(s)\cdot V )Y_n(s)ds \bigg|_{H^{-1}}^2 dt \nonumber \\
& + \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \bigg| \int_t^{t+h} ( \g_1(X_n(s) -\bbx_1(s) ) - \sum\limits_{k=1}^N \ol{\mu_k} e_kZ_{k,n}(s) ) ds \bigg|_{H^{-1}}^2 dt \nonumber \\
& + \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \bigg| \int_t^{t+h} \sum\limits_{k=1}^N Z_{k,n}(s) d\beta_k(s) \bigg|_{H^{-1}}^2 dt
=: \sum\limits_{j=1}^5 K_j.\end{aligned}$$
For $K_1$, by , $$\begin{aligned}
\label{K1}
K_1 \leq \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \( \int_t^{t+h} |Y_n(s)|_{H^1} ds\)^2dt
\leq \delta^2 T\ \bbe \sup\limits_{t\in[0,T+1]} |Y_n(t)|^2_{H^1}
\leq C \delta^2,\end{aligned}$$ where $C$ is independent of $n$.
Similarly, by Cauchy’s inequality and by , $$\begin{aligned}
\label{K2}
K_3+ K_4 \leq& \delta^2 T (|\mu|_{\9} + |V_0|_{\9} + D_U\|V\|_{L^\9(0,T+1; L^\9)}) \bbe \sup\limits_{t\in[0,T+1]}|Y_n(t)|_2^2 \nonumber \\
& + \delta \g_1 \bbe \int_0^{T} \int_t^{t+\delta} |X_n(s)-\bbx_1(s)|_2^2 ds dt \nonumber \\
& + \delta \sum\limits_{k=1}^N |\mu_k||e_k|_{\9} \bbe \int_0^{T} \int_t^{t+\delta} |Z_{k,n}(s)|_2^2 ds dt \nonumber \\
\leq& C (\delta+\delta^2),\end{aligned}$$ where $C$ is independent of $n$.
Regarding $K_2$, choose the Strichartz pair $(p,q)=(\a+1, \frac{4(\a+1)}{d(\a-1)})$. Since $p\in(2,\frac{2d}{d-2})$, $L^{p'}(\bbr^d)\hookrightarrow H^{-1}(\bbr^d) $, we have $$\begin{aligned}
K_2
\leq& \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \( \int_t^{t+h} \bigg| -\lbb i h_1(X_n(s))Y_n(s) + \lbb i h_2(X_n(s))\ol{Y_n}(s) \bigg|_{L^{p'}} ds \)^2 dt \nonumber \\
\leq& \a^2 \bbe \int_0^{T} \sup\limits_{0<h\leq \delta} \( \int_t^{t+h} \bigg| X_n^{\a-1}(s) Y_n(s) \bigg|_{L^{p'}} ds \)^2 dt \nonumber \\
\leq& \delta^{2/q} \a^2 \bbe \int_0^{T} \|X_n^{\a-1}Y_n\|^2_{L^{q'}(t,t+\delta; L^{p'})} dt \nonumber \\
\leq& \delta^{2/q} \a^2 T\ \bbe \| X_n^{\a-1}Y_n\|^2_{L^{q'}(0,T+1; L^{p'})}.\end{aligned}$$ Note that, by Hölder’s inequality, $$\begin{aligned}
\|X_n^{\a-1} Y_n\|_{L^{q'}(0,T; L^{p'})}
\leq& T^\theta \|X_n\|^{\a-1}_{L^q(0,T; L^p)} \|Y_n\|_{L^q(0,T; L^p)},\end{aligned}$$ where $\theta = 1- d(\a-1) /4 \in (0,1)$. Hence $$\begin{aligned}
K_2 \leq& \delta^{2/q} \a^2 T^{2\theta +1}\ \bbe \| X_n\|^{2(\a-1)}_{L^q(0,T; L^p)} \|Y_n\|^2_{L^{q}(0,T+1; L^{p})} \nonumber \\
\leq& \delta^{2/q} \a^2 T^{2\theta +1} \|X_n\|^{2(\a-1)}_{L^\9(\Omega; L^q(0,T; L^p))} \|Y_n\|^2_{L^2(\Omega; L^q(0,T+1; L^p))}\end{aligned}$$ Then, by and we obtain $$\begin{aligned}
K_4\leq C \delta^{2/q},\end{aligned}$$ where $C$ is independent of $n$.
For $K_5$, using the Burkholder-Davis-Gundy inequality we get $$\begin{aligned}
K_5
\leq C \int_0^{T} \bbe \int_t^{t+\delta} \sum\limits_{k=1}^N |Z_{k,n}(s)|_2^2 ds dt.\end{aligned}$$ Then, using Fubini’s theorem to interchange the sum and integrals, by we have, $$\begin{aligned}
\label{K4}
K_5
\leq& C \sum\limits_{k=1}^N \(\int_0^\delta\int_0^s + \int_\delta^T \int_{s-\delta}^s + \int_T^{T+\delta}\int_{s-\delta}^T \) |Z_{k,n}(s)|_2^2\ dt ds\nonumber \\
\leq& 3 \delta C\ \sum\limits_{k=1}^N \bbe \int_{0}^{T+1} |Z_{k,n}(s)|_2^2 ds \leq C \delta,\end{aligned}$$ where $C$ is independent of $n$.
Plugging - into , since $2/q<1$ and $\delta <1$, we obtain $$\begin{aligned}
\bbe \int_0^{T} \sup\limits_{0<h\leq \delta} |Y_n(t+h)- Y_n(t)|_{H^{-1}}^2 dt
\leq C (\delta+ \delta^2 +\delta^{\frac 2 q}) \leq C \delta^{\frac 2q},\end{aligned}$$ where $C$ is independent of $n$.
The term $\bbe \int_0^{T} \sup_{0<h\leq \delta} |X_n(t+h)- X_n(t)|_{H^{-1}}^2 dt$ in the right hand side of can be estimated similarly.
Therefore, in view of we obtain with $b = 1/q$, thereby proving the tightness of $\{\mu_n\}$ and yielding the equality in .
Finally, the stochastic maximal principle follows from Proposition \[l4.1\], taking into account that (see ) for the optimal $u^*$, $\eta(u^*) + \zeta^* = 0$, where $\zeta^* \in \mathcal{N}_{\calu_{ad^*}}(u^*)$. The proof is complete. $\square$
#### An example.
We consider the case $m=1$ and $U=[0,\ell]$, where $\ell>0.$ Then, equation reduces to
$$u^*(t)=\left\{\barr{lll}
0&\mbox{\ \ if}&{\rm Im}\int_{\rr}V(\xi)X^*(t,\xi)\ol{Y^*}(t,\xi)d\xi\le0,\vsp
\frac\ell\g_2&\mbox{\ \ if}&{\rm Im}\int_{\rr}V(\xi)X^*(t,\xi) \ol{Y^*}(t,\xi)d\xi\ge\ell,\vsp
\multicolumn{3}{l}{\frac1{\g_2}\, {\rm Im}\int_{\rr}V(\xi)X^*(t,\xi)\ol{Y^*} (t,\xi)d\xi\ \ \ \hfill\mbox{otherwise.}} \earr\right.$$ For the numerical computation of the optimal controller $u^*$, one can use the standard gradient descent algorithm suggested by . Namely, $$\label{e5.2}
u_{n+1} = P_U ( \frac{1}{1+ 2\g_2 \rho_n} u_n + \frac{2\rho_n}{1+ 2\g_2 \rho_n}{\rm Im} \int_{\rr^d}V(\xi)X_n(t,\xi) \ol{Y}_n(t,\xi)d\xi),$$ where $\rho_n>0$ are suitable chosen and $X_n$, $Y_n$ are solutions to the forward–backward system , with $u=u_n$.\
[***Proof of Theorem \[Thm-deter\].*** ]{} The proof follows the lines as that of Theorem \[t2.6\]. As a matter of fact, in the deterministic case, the analysis of the equation of variation and of the backward equation is much easier.
Similarly to Proposition \[l4.1\], we have $$\begin{aligned}
\sup\limits_{u, v\in \calu_{ad}} (\|\vf^{u,\wt{u}}\|_{C([0,T]; H^1)} + \|\vf^{u,\wt{u}}\|_{L^q(0,T; W^{1,p})}) <\9,\end{aligned}$$ where $\wt{u}=v-u$, $u,v \in \calu_{ad}$, and $\vf^{u,\wt{u}}$ is the solution to the deterministic equation of variation (i.e. without $W$). Moreover, $$\begin{aligned}
\label{deter-asy-X-y}
\lim\limits_{\ve\to 0} \sup\limits_{t\in [0,T]}
|\ve^{-1} (X^{u_\ve}(t)-X^u(t)) - \vf^{u,\wt{u}}(t)|_2^2 \to 0,\end{aligned}$$ where $X^{u_\ve}$ and $X^u$ are the solutions to corresponding to $u_\ve:= u+\ve \wt{u}$ and $u$ respectively.
Using strichartiz estimates we have that (see e.g. [@2]) $$\begin{aligned}
\sup\limits_{u, v\in \calu_{ad}} (\|\vf^{u,\wt{u}}\|_{C([0,T]; H^1)} + \|\vf^{u,\wt{u}}\|_{L^q(0,T; W^{1,p})}) <\9,\end{aligned}$$ where $\wt{u}=v-u$, and $\vf^{u,\wt{u}}$ is the solution to the equation of variation but without $W$.
Moreover, let $X^{u_\ve}$ and $X_{u,r,\ve}$ be as in the proof of Proposition \[l4.1\]. Similarly to , we have that, without the restriction $\a\geq 2$, $$\begin{aligned}
\label{deter-asy-X-y}
\lim\limits_{\ve\to 0} \sup\limits_{t\in [0,T]}
|\ve^{-1} (X^{u_\ve}(t)-X^u(t)) - \vf^{u,\wt{u}}(t)|_2^2 \to 0.\end{aligned}$$
Moreover, let $X^{u_\ve}$ and $X_{u,r,\ve}$ be as in the proof of Proposition \[l4.1\]. Similarly to , we have that, without the restriction $\a\geq 2$, $$\begin{aligned}
\label{deter-asy-X-y}
\lim\limits_{\ve\to 0} \sup\limits_{t\in [0,T]}
|\ve^{-1} (X^{u_\ve}(t)-X^u(t)) - \vf^{u,\wt{u}}(t)|_2^2 \to 0.\end{aligned}$$ To this end, instead of , we have $$\begin{aligned}
\|X^{u_\ve} - X^u\|_{C([0,T]; H^1)} + \|X^{u_\ve} - X^u\|_{L^q(0,T; W^{1,p})} \to 0, \ as\ \ve \to 0.\end{aligned}$$ It follows that $h_1(X_{u,r,\ve}) \to h_1(X^u)$, $dt\otimes d\xi$-a.e., and along with the Sobolev imbedding $L^p\hookrightarrow H^1 $, $$\begin{aligned}
|h_1(X_{u,r,\ve})|_{L^{\frac{p}{\a-1}}}
= \frac{\a+1}{2} |X_{u,r,\ve}|^{\a-1}_{L^p}
\to \frac{\a+1}{2} |X^u|^{\a-1}_{L^p}
= |h_1(X^u)|_{L^{\frac{p}{\a-1}}}.\end{aligned}$$ Thus, for each $r\in (0,1)$, $|h_1(X_{u,r,\ve}) - h_1(X^u)|_{L^{\frac{p}{\a-1}}} \to 0$, $dt$-a.e. Taking into account $ |h_1(X_{u,r,\ve})|_{L^{\frac{p}{\a-1}}} \leq C(\a) \sup_{u\in \calu_{ad}}\|X^u\|^{\a-1}_{C([0,T]; H^1)} \in L^{\frac{q}{\a-1}}(0,T)$, we obtain the convergence without the restriction $\a\geq 2$, thereby yielding , as claimed.
Regarding the backward deterministic equation, we can now use the reversing time arguments and the Strichartz estimates to obtain directly the estimate below, $$\begin{aligned}
\sup\limits_{u\in\calu_{ad}} ( \|Y^u\|_{L^\9(0,T; H^1)} + \|Y^u\|_{L^q(0,T; W^{1,p})} ) < \9.\end{aligned}$$ Based on these, one has also the directional derivative of $\Phi$ as in Proposition \[l4.1\], and similarly to , the estimate below for the minimizing sequence of controls $\{u_n\}$ from Ekeland’s principle, $$\begin{aligned}
\limsup\limits_{n\to \9} \sup\limits_{0<h\leq \delta}
\int_0^{T-h} |u_n(t+h) - u_n(t)|_m dt \leq C \delta^{1/q},\end{aligned}$$ which by the Riesz-Kolmogorov theorem implies that $\{u_n\}$ is relative compact in $L^1(0,T; \bbr^m)$, thereby yielding the result. $\square$
Appendix {#APPDIX}
========
([@S15 Lemma 2.17]) \[Lem-Bound\] Let $T>0$ and $f\in C([0,T]; \bbr_+)$, such that $$\begin{aligned}
f\leq a + b f^\a,\ \ on\ [0,T],\end{aligned}$$ where $a,b>0$, $\a>1$, $a<(1-\frac 1 \a) (\a b)^{-\frac{1}{\a-1}}$, and $f(0) \leq (\a b)^{-\frac{1}{\a-1}}$. Then, $$\begin{aligned}
f\leq \frac{\a}{\a-1} a,\ \ on\ [0,T].\end{aligned}$$
[***Proof of and .***]{} For simplicity, we omit the dependence of $u$ in $X^u$. We may assume $T\geq 1$ without loss of generality. Set $$\begin{aligned}
H(X(t)) : = \frac 12 |\na X(t)|_2^2 - \frac{\lbb}{\a +1} |X(t)|_{L^{\a+1}}^{\a+1}\end{aligned}$$ As in the proof of [@2 Theorem 3.1] we have for $t\in [0,T]$, $$\begin{aligned}
\label{Ito-Hami}
&H(X(t)) - H(x) \nonumber \\
=& - \int_0^t \( Im \<(\na V_0 + u(s) \cdot \na V) X(s), \na X(s)\>\) ds \nonumber \\
& +\int_0^t \( Re \<-\nabla(\mu X(s)),\nabla X(s) \>_2ds
+ \frac{1}{2} \sum\limits_{j=1}^N |\nabla (X(s)\phi_j)|_2^2 \) ds \nonumber \\
& -\frac{1}{2}\lambda (\alpha-1)\sum\limits_{j=1}^N \int_0^t \int (Re\phi_j)^2 |X(s)|^{\alpha+1} d\xi ds
+ M(t),\end{aligned}$$ where $M(t):= \sum^N_{j=1} \int_0^t (Re \<\nabla(\phi_j
X(s)),\nabla X(s) \>_2
- \lbb \int Re\phi_j |X(s)|^{\alpha+1} d\xi)d\beta_j(s)$.
Below we shall treat the focusing and defocusing cases respectively.
$(i)$ (The focusing case $\lbb=1$.) Note that, by [@2 Lemma 3.5], $$\begin{aligned}
\label{spli-lp}
|X(t)|_{L^{\a+1}}^{\a+1}
\leq C_\ve |X(t)|_2^p + \ve |\na X(t)|_2^2,\end{aligned}$$ where $p = 2 \frac{2(\a+1)-d(\a-1)}{4-d(\a-1)} >2$. As in the proof of [@2 Theorem 3.7], the first three terms on the right hand side of are bounded by $ C\int_0^t (|X(s)|_2^p + |X(s)|_2^2 + |\na X(s)|_2^2) ds$, where $C$ is independent of $u$. Thus, taking $\ve < \frac{\a+1}{4}$ yields $$\begin{aligned}
\label{esti-foc-hami}
|\na X(t)|_2^2
\leq 4H(x) + C (|X(t)|_2^p + D(t) ) + 4 M(t),\end{aligned}$$ where $D(t) := \int_0^t ( |X(s)|_2^p + |X(s)|_2^2 +|\na X(s)|_2^2) ds$. It follows that for any $\rho \geq 4$, $$\begin{aligned}
\label{naX-D-M}
|\na X(t)|_2^{2\rho}
\leq& C + C( |X(t)|_2^{\rho p} + D^\rho(t) + |M(t)|^\rho)\end{aligned}$$ with $C$ independent of $u$.
Note that, by Jensen’s inequality and the conservation $|X(t)|_2=|x|_2$, $t\in [0,T]$, $$\begin{aligned}
\label{D}
\bbe \sup\limits_{s\in[0,t]} D^\rho(s)
\leq& \bbe \sup\limits_{s\in[0,t]} s^{\rho -1} \int_0^s ( |X(r)|_2^{p\rho} + |X(r)|_2^{2\rho} +|\na X(r)|_2^{2\rho}) dr \nonumber \\
\leq& C(\rho, T)\(1 + \int_0^t \bbe \sup\limits_{r\in[0,s]} |\na X(r)|_2^{2\rho}ds \).\end{aligned}$$ Moreover, by the BDG inequality we get $$\begin{aligned}
\label{esti-M}
& \bbe \sup\limits_{s\in[0,t]} |M(s)|^\rho \nonumber \\
\leq& C(\rho) \bbe \(\int_0^t \sum^N_{j=1} \(|Re \<\nabla(\phi_j
X(s)),\nabla X(s) \>_2 |^2
+ \bigg|\int Re\phi_j |X(s)|^{\alpha+1} d\xi \bigg|^2\) ds \)^{\frac \rho 2} \nonumber \\
\leq& C(\rho) \bbe \(\int_0^t |\na X(s)|_2^4 + |X(s)|_2^4 + |X(s)|_{L^{\a+1}}^{2(\a+1)} ds \)^{\frac \rho 2} \nonumber \\
\leq& C(\rho, T) \bbe \int_0^t |\na X(s)|_2^{2\rho} + |X(s)|_2^{2\rho } + |X(s)|_{L^{\a+1}}^{(\a+1)\rho} ds,\end{aligned}$$ Then, using the conservation and one obtains the estimate $$\begin{aligned}
\label{m-lp}
\bbe \sup\limits_{s\in[0,t]} |M(s)|^\rho \leq C(\rho,T) \(1 + \int_0^t \bbe \sup\limits_{r\in[0,s]} |\na X(r)|_2^{2\rho}ds\).\end{aligned}$$
Thus, plugging and into and using the conservation yields $$\begin{aligned}
\bbe \sup\limits_{s\in[0,t]} |\na X(s)|_2^{2\rho}
\leq C+ C \int_0^t \sup\limits_{r\in[0,s]} |\na X(r)|_2^{2\rho}ds,\end{aligned}$$ which implies by Gronwall’s inequality.\
$(ii)$ (The defocusing case $\lbb=-1$.) Similarly to , we have by , $$\begin{aligned}
& \frac 12 |\na X(t)|_2^2 + \frac {1}{\a+1} |X(t)|_{L^{\a+1}}^{\a+1} \\
\leq& H(x)
+ C \int_0^t (|X(s)|_2^2 + |\na X(s)|_2^2 + |X(s)|^{\a+1}_{L^{\a+1}}) ds
+ M(t).\end{aligned}$$ Using the conservation and we get for $\rho \geq 4$, $$\begin{aligned}
&\bbe \sup\limits_{s\in[0,t]} (|\na X(t)|_2^{2\rho} + |X(t)|^{(\a+1)\rho}_{L^{\a+1}}) \\
\leq& C + C\int_0^t \bbe \sup\limits_{r\in[0,s]} (|\na X(r)|_2^{2\rho} + |X(r)|^{(\a+1)\rho}_{L^{\a+1}}) ds,\end{aligned}$$ and so follows.\
It remains to prove . Indeed, in the case that $e_k$ are constants, $1\leq k\leq N$, by the rescaling transformation $y= e^{-W}X$, we have $$\begin{aligned}
\label{equa-y}
\partial_t y = -i\Delta y - \lbb i |y|^{\a-1} y + f(u)y,\end{aligned}$$ where $f(u):= -i(V_0 + u\cdot V)$. Note that, the Strichartz coefficient $C_T$ is now identically a deterministic constant. Then, arguing as in we obtain that $\sup_{u\in\calu_{ad}}\|y^u\|_{L^\9(\Omega; L^q(0,T; L^p))}<\9$ for any Strichartz pair $(p,q)$.
As regards the estimate for $\|X^u\|_{L^\rho(\Omega; L^q(0,T; W^{1,p}))}$, it suffices to prove that for any $\rho \geq 1$, $$\begin{aligned}
\label{esti-y-Wpq}
\sup\limits_{u\in \calu_{ad}} \bbe \|y^u\|^\rho _{L^q(0,T; W^{1,p})} < \9,\end{aligned}$$ where $(p,q)=(\a+1, \frac{4(\a+1)}{d(\a-1)})$.
Since $|\na (|y|^{\a-1}y)| \leq \a |y|^{\a-1} |\na y|$, the Hölder inequality implies that $$\begin{aligned}
\label{h1-stri-nonl}
\| |y|^{\a-1}y\|_{L^{q'}(0,t; W^{1,p'})}
\leq 2\a t^{\theta} \|y\|^\a_{L^q(0,t; W^{1,p})},\end{aligned}$$ where $\theta=1- \frac{d(\a-1)}{4}>0$. Moreover, $$\begin{aligned}
\|f(u)y\|_{L^1(0,t; H^1)}
\leq T\|f(u)\|_{L^\9(0,T; W^{1,\9})} \|y\|_{C([0,t]; H^1)},\end{aligned}$$
Thus, applying Strichartz estimates to and using the estimates above, we get $$\begin{aligned}
\label{y-h1-t}
&\|y\|_{L^q(0,t; W^{1,p})} \nonumber \\
\leq& C (|x|_{H^1} + 2\a t^\theta \|y\|^\a_{L^q(0,t; W^{1,p})}
+ T \|f(u)\|_{L^\9(0,T; W^{1,\9})} \|y\|_{C([0,t];H^1)}) \nonumber \\
\leq& D(T) (\|y\|_{C([0,T]; H^1)} + t^\theta \|y\|^\a_{L^q(0,T; W^{1,p})} ),\end{aligned}$$ where $D(T) = C(1+ 2\a + T \sup_{u\in \calu_{ad}}\|f(u)\|_{L^\9(\Omega; L^\9(0,T; W^{1,\9}))})$.
Then, similarly to we have $$\begin{aligned}
\|y^u\|_{L^q(0,T; W^{1,p})}
\leq& \([\frac{T}{t}]+1\)^{\frac 1 q} \frac{ \a}{\a-1} D(T) \|y^u\|_{C([0,T];H^1)},\end{aligned}$$ where $t= \a^{-\frac{\a}{\theta}} (\a-1)^{\frac{\a-1}{\theta}} ( \|y^u\|_{C([0,T]; H^1)} +1)^{-\frac{\a-1}{\theta}} D(T)^{-\frac{\a}{\theta}} (\leq T)$.
Therefore, taking into account we obtain , thereby completing the proof. $\square$
\[Lem-tight-L1\] Let $\mu_n$, $n\geq 1$, be a family of probability measures on $L^1(0,T; \bbr^m)$. Assume that $$\begin{aligned}
\label{tight-space}
\lim\limits_{R\to \9} \limsup\limits_{n \to \9} \mu_n\bigg\{ v\in L^1(0,T; \bbr^m): \int_0^T |v(t) |_m dt > R \bigg\} =0,\end{aligned}$$ and for any $\ve >0$, $$\begin{aligned}
\label{tight-time}
\lim\limits_{\delta \to 0} \limsup\limits_{n\to \9} \mu_n \bigg\{v\in L^1(0,T; \bbr^m): \sup\limits_{0<h\leq \delta}
\int_0^{T-h} |v(t+h)-v(t)|_m dt >\ve \bigg\} =0.\end{aligned}$$ Then, $\{\mu_n\}_{n\geq 1}$ is tight in $L^1(0,T; \bbr^m)$.
[***Proof.***]{} Set $K_1(R) = \{ v\in L^1(0,T; \bbr^m): \int_0^T |v(t) |_m dt \leq R \}$ and $K_2(\delta, \ve) = \{v\in L^1(0,T; \bbr^m): \sup_{0<h\leq \delta} \int_0^{T-h} |v(t+h)-v(t)|_m dt \leq \ve \}$, where $R, \delta, \ve >0$.
Fix $\ve >0$. By there exists $N(=N(\ve))$, $R_1(=R_1(\ve))\geq 1$, such that $\sup_{n\geq N} \mu_n ( K^c_1(R_1) ) \leq \frac{\ve}{2}$. Since for each $n\geq 1$, $ \lim_{R\to \9} \mu_n ( K^c_1(R) ) =0, $ we can choose $R_2(=R_2 (\ve)) $ sufficiently large, such that $\sup_{1\leq n\leq N} \mu_n ( K^c_1(R_2) ) \leq \frac{\ve}{2N}$. Thus, letting $R= R_1 \vee R_2$ we get $\sup _{n\geq 1} \mu_n ( K^c_1(R) ) \leq \ve.$
Similarly, since for each $k, n\geq 1$, $\lim _{\delta \to 0} \mu_n ( K^c_2(\delta, \frac 1k) ) =0$, by and similar arguments as above, we can choose $\delta_k >0$ sufficiently small such that $\sup _{n\geq 1} \mu_n ( K^c_2(\delta_k, \frac 1k) ) \leq \frac{\ve}{2^{k}}$.
Then, set $K:= K_1(R) \cap (\bigcap_{k\geq 1} K_2(\delta_k, \frac 1k))$. It follows from [@S87 Theorem 1] that $K$ is relatively compact in $L^1(0,T; \bbr^m)$, and by the estimates above we have $\sup_{n\geq 1} \mu_n(K^c) \leq 2 \ve$, which implies the tightness of $\{\mu_n\}_{n\geq 1}$ in $L^1(0,T; \bbr^m)$. $\square$\
[***Acknowledgements.***]{} Financial support through SFB701 at Bielefeld University is gratefully acknowledged. V. Barbu was partially supported by a CNCS UEFISCDI (Romania) grant, project DN-II-ID-DCE-2012-4-0156. D. Zhang is partially supported by NSFC (No. 11501362), China Postdoctoral Science Foundation funded project (2015M581598).
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[^1]: Octav Mayer Institute of Mathematics (Romanian Academy) and Al.I. Cuza University, 700506, Iaşi, Romania.
[^2]: Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.
[^3]: Department of Mathematics, Shanghai Jiao Tong University, 200240 Shanghai, China. Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.
[^4]: E-mail address: vb41@uaic.ro (V. Barbu), roeckner@math.uni-bielefeld.de (M. Röckner), zhangdeng@amss.ac.cn (D. Zhang)
|
---
abstract: 'We present timing solutions for ten pulsars discovered in 350 MHz searches with the Green Bank Telescope. Nine of these were discovered in the Green Bank Northern Celestial Cap survey and one was discovered by students in the Pulsar Search Collaboratory program in analysis of drift-scan data. Following discovery and confirmation with the Green Bank Telescope, timing has yielded phase-connected solutions with high precision measurements of rotational and astrometric parameters. Eight of the pulsars are slow and isolated, including PSR J0930$-$2301, a pulsar with nulling fraction lower limit of $\sim$30% and nulling timescale of seconds to minutes. This pulsar also shows evidence of mode changing. The remaining two pulsars have undergone recycling, accreting material from binary companions, resulting in higher spin frequencies. PSR J0557$-$2948 is an isolated, 44 [ms]{} pulsar that has been partially recycled and is likely a former member of a binary system which was disrupted by a second supernova. The paucity of such so-called ‘disrupted binary pulsars’ (DRPs) compared to double neutron star (DNS) binaries can be used to test current evolutionary scenarios, especially the kicks imparted on the neutron stars in the second supernova. There is some evidence that DRPs have larger space velocities, which could explain their small numbers. PSR J1806+2819 is a 15 [ms]{} pulsar in a 44 day orbit with a low mass white dwarf companion. We did not detect the companion in archival optical data, indicating that it must be older than 1200 Myr.'
author:
- 'A. M. Kawash, M. A. McLaughlin, D. L. Kaplan, M. E. DeCesar, L. Levin, D. R. Lorimer R. S. Lynch, K. Stovall, J. K. Swiggum, E. Fonseca, A. M. Archibald, S. Banaszak, C. M. Biwer, J. Boyles, B. Cui, L. P. Dartez, D. Day, S. Ernst, A. J. Ford, J. Flanigan, S. A. Heatherly J. W. T. Hessels, J. Hinojosa, F. A. Jenet, C. Karako-Argaman, V. M. Kaspi, V. I. Kondratiev, S. Leake, G. Lunsford, J. G. Martinez, A. Mata, T. D. Matheny, A. E. Mcewen, M. G. Mingyar, A. L. Orsini, S. M. Ransom, M. S. E. Roberts, M. D. Rohr, X. Siemens, R. Spiewak, I. H. Stairs, J. van Leeuwen, A. N. Walker, B. L. Wells,'
bibliography:
- 'modrefs.bib'
- 'psrrefs.bib'
title: 'The Green Bank Northern Celestial Cap Pulsar Survey II: The Discovery and Timing of Ten Pulsars'
---
Introduction {#intro}
============
The Green Bank Northern Celestial Cap (GBNCC) survey is a 350 MHz all-sky pulsar survey being conducted with the Green Bank Telescope (GBT; @slr14). The primary goals of the survey include the discovery of millisecond pulsars (MSPs) suitable for pulsar timing arrays (PTAs) for direct detection of gravitational wave (GW; @abb18 [@ltm15; @srm13]), exotic pulsar systems such as ‘black widows’ [@fst88] and ‘redbacks’ [@r11; @asr09], rotating radio transients (RRATs; [@mll06; @dsm16]), and characterization of the Galactic pulsar population.
This survey is 75% complete and will ultimately cover the entire sky visible to the GBT ($\delta>-40^\circ$) with $\sim$125000 pointings, 120 [s]{} each. To date, there have been 156 pulsars, including 20 MSPs, discovered in this survey. Collection of data is expected to be complete in 2020. Lynch et al. 2018 (submitted) report on the discovery of 45 pulsars, including five millisecond pulsars (MSPs), a new relativistic double neutron star system, an intermediate mass binary pulsar, a mode-changing pulsar, a non-recycled pulsar with a very low magnetic field, and several nulling pulsars. Here, we report on the discovery of nine pulsars in the GBNCC survey, including a millisecond pulsar, a partially recycled pulsar, and a nulling pulsar.
The Pulsar Search Collaboratory (PSC) is a joint outreach program between West Virginia University (WVU) and the Green Bank Observatory (GBO). The primary goals of the PSC are to stimulate high-school students’ interest in science and to discover unknown pulsars. Since its start in 2008, the PSC has integrated high-school students into the pulsar searching process resulting in the discovery of seven pulsars [@rsm13; @srm15]. PSR J1954$+$1021 is the seventh pulsar discovered by the PSC, and its timing solution is presented here, in addition to those of the nine newly discovered GBNCC survey pulsars.
To date, 9% of the $\sim$2600 known pulsars are in binary systems [@mhth05]. These include pulsars orbiting white dwarf (WD), neutron star (NS), and main sequence (MS) companions. Pulsars with WD and NS companions are likely recycled through the following process. From a binary star system, the more massive star will undergo a supernova explosion and form a pulsar. Estimating using the virial theorem, the supernova explosion will cause the binary star system to be disrupted if more than half the pre-supernova mass is ejected from the system [@h83; @bh91]. The presence of a kick could change this requirement significantly. If the system remains bound, and the companion is massive enough to evolve into a giant and overflow its Roche lobe, material is accreted onto the pulsar. This process, commonly referred to as ‘recycling,’ transfers angular momentum to the pulsar and spins it up [@acrs82].
The amount and timescale of recycling depends on the binary companion type. MSPs, which undergo the greatest amount of recycling, usually have WD companions [@tlm12]. However, a surprising number are isolated, presumably due to ablation of their companion stars [@fst88] or three-body interactions [@phl11]. Partially-recycled pulsars, with spin periods $P$ and period derivatives $\dot{P}$ in the 30 [ms]{} <$P$ <100 [ms]{} and $\dot{P}$ <10$^{-17}$ range, typically have more massive WD or NS companions [@l08]. Some are also isolated due to the binary system being disrupted during the supernova explosion of the secondary [@cnt93].
Two of the pulsars presented in this paper have undergone a phase of recycling, presumably in a binary evolution scenario described above. PSR J0557$-$2948 is a partially-recycled pulsar that has no evidence of a companion star and is discussed in Section \[0557\]. PSR J1806+2819 is a recycled pulsar in a binary system that is discussed in further detail in Section \[1806\].
Methodology and Results {#results}
=======================
A description of the survey and the searching pipeline are given in [@slr14]. The search processing responsible for these discoveries took place on the Guillimin cluster operated by CLUMEQ and Compute Canada and the candidate plots were analyzed using the CyberSKA interface. Timing observations of the pulsars presented in this paper were conducted with the GBT using a combination of competitively awarded GBNCC timing time and time purchased by WVU. The observations were typically 15 to 20 minutes in duration.
Using the Green Bank Ultimate Pulsar Processing Instrument (GUPPI; @rdf09), data were recorded over 100 MHz of bandwidth divided in to 4096 frequency channels at a central frequency of 350 MHz every 81.92 $\mu$s. In off-line processing, the filterbank data were dedispersed at the dispersion measure (DM) of the pulsar to account for frequency-dependent time delays due to the interaction of the pulse and the interstellar medium.
To calculate the pulse times of arrival (TOAs) at the telescope, integrated profiles were formed by summing many thousands of pulses modulo the pulse period at each epoch. The pulse profile for each of the pulsars can be found in Figure \[fig:profiles\]. These pulse profiles were cross-correlated by the least squares method to template profiles, created by summing together profiles at multiple epochs, to achieve a high precision TOA measurement of the pulse at the telescope [@t93]. TOAs were calculated using `get_TOAs.py` in the PRESTO software package [@rs11].
{width="100.00000%"}
A timing model incorporating $P$, $\dot{P}$, right ascension (RA), and declination (Dec.) was then fit to the TOAs using the TEMPO software package. This yields a timing solution that accounts for every rotation of the pulsar over the entire data span. The results of these fits are shown in Tables 1 and 2. Each of our timing solutions uses the DE430 Solar System ephemeris [@fwb14] and the Terrestrial Time scale TT(BIPM). The uncertainties in TOAs were scaled using EFACs to assure that the chi-squared value for each fit was equal to one. A full phase-connected timing solution reveals how well the timing model can predict when the next pulse will arrive at the telescope. The timing residuals are the measured difference between the observed and predicted arrival times. The RMS values of the timing residuals are listed in Table \[tb:prd\].
The DMs listed in Table \[tb:posn\] were calculated for eight of the pulsars by creating TOAs in four subbands of the 100 MHz bandpass at one epoch. The frequency-dependent time delay of the pulse was then modeled as pulse dispersion using the TEMPO software package. This yielded a measurement of the DM of the pulsar at that epoch. PSRs J0930$-$2301 and J1954$+$1021 were observed at a central frequency of 820 MHz and bandwidth of 200 MHz in addition to the 350 MHz central frequency observations. This allowed the DM to be measured between epochs observed at different frequencies. Figure 2 shows these pulsars on the $P$–$\dot{P}$ diagram with all of the known Galactic pulsars (i.e. excluding pulsars found in globular clusters). As seen in this Figure, the timing-derived parameters for the pulsars presented in this paper lie within the range of parameters for known Galactic pulsars.
\[tb:posn\]
-------------- ------------------ -------------------- --------- ---------- ---------------- -------------- --------------
Name R.A. (J2000) Dec. (J2000) [$l$]{} [$b$]{} DM Distance$^a$ Distance$^b$
(h m s) ( ) [()]{} [()]{} (pc cm$^{-3}$) (kpc) (kpc)
J0557$-$2948 05:57:32.9995(7) $-$29:48:16.804(7) 235.5 $-$23.99 49.05(1) 4.3 2.9
J0930$-$2301 09:30:02.82(8) $-$23:01:45(1) 253.9 20.14 78.3(5) $>$25$^c$ $>$50$^c$
J1116$-$2444 11:16:23.26(3) $-$24:44:56.1(5) 277.0 33.29 29.8(3) 0.9 1.2
J1234$-$3630 12 34 12.00(2) $-$36 30 41.1(1) 299.1 26.23 58.8(4) 5.7 2.5
J1336$-$2522 13:36:20.84(2) $-$25:22:01.6(5) 315.6 36.40 37.5(2) 5.5 1.7
J1806$+$2819 18:06:25.0658(5) 28:19:01.115(5) 54.6 21.67 18.6802(4) 1.3 1.3
J1929$+$3817 19:29:07.014(5) 38:17:57.5(1) 71.2 9.69 93.4(2) 9.3 5.1
J1954$+$3852 19:54:01.083(4) 38:52:15.88(5) 74.0 5.70 65.4(1) 4.7 3.7
J1954$+$1021 19:54:36.80(1) 10:21:10.5(9) 49.5 $-$8.99 80.87(4) 4.3 3.6
J2154$-$2812 21:54:17.37(7) $-$28:12:41(1) 82.0 $-$20.16 32.1(9) 2.7 2.0
-------------- ------------------ -------------------- --------- ---------- ---------------- -------------- --------------
Timing-derived positions, Galactic longitudes and latitudes, DMs, and distances. The numbers in parentheses after position and DM are the 1-$\sigma$ errors in the last digit reported by TEMPO.
$^a$ Distances derived from the YMW16 [@ymw17] electron density model.
$^b$ Distances derived from the NE2001 [@cl02] electron density model.
$^c$ The DM value for this pulsar is larger than the expected highest DM given by both the YMW16 and the NE2001 models, and, therefore, these models do not provide a reliable distance estimate for this source.
\[tb:prd\]
-------------- --------------------- -------------- ------- ---------- ---------- ------------- ------- ------------- -------------------------- -- -- --
Name $P$ $\dot{P}$ Epoch $w_{50}$ RMS Data Span Age $B$ $\dot{E}$
(s) (10$^{-15}$) (MJD) (ms) ($\mu$s) (MJD) (Myr) (10$^{9}$G) (10$^{30}$ erg s$^{-1}$)
J0557$-$2948 0.0436426389000(3) 0.000073(5) 57381 1.1 23.3 57062–57700 9400 1.8 35
J0930$-$2301 1.80706867799(8) 3.362(7) 57254 62.1 3586.1 56663–57846 8.5 2500 22
J1116$-$2444 0.86794888009(5) 0.985(5) 57513 24.3 736.0 57213–57813 14 950 61
J1234$-$3630 0.569242225079(8) 0.866(3) 57602 11.2 259.1 57385–57819 10. 710 190
J1336$-$2522 0.478145482800(6) 0.3306(5) 57514 19.0 237.0 57215–57813 24 400 120
J1806$+$2819 0.01508366732422(2) 0.0000375(5) 57004 2.0 16.2 56254–57753 6400 0.76 430
J1929$+$3817 0.81421524225(1) 0.6097(9) 57545 29.7 206.7 57216–57875 22 700 43
J1954$+$3852 0.352933478726(1) 6.5998(1) 57485 11.0 166.5 57094–57875 0.85 1500 5900
J1954$+$1021 2.09944017034(5) 1.735(3) 57393 65.2 934.1 56911–57875 19 1900 7.4
J2152$-$2812 1.3433614881(1) 0.613(8) 57297 23.1 1615.5 56838–57756 35 920 1.0
-------------- --------------------- -------------- ------- ---------- ---------- ------------- ------- ------------- -------------------------- -- -- --
[ Periods, period derivatives, MJDs of the epoch used for the period determination, the average pulse widths at 50% of the peak, the RMS values of the post-fit timing residual, the MJD ranges covered, the spin-down age, surface magnetic field strength, and spin-down luminosity. No attempt was made to correct for bias in the observed spin period derivative, $\dot{P}$, by accounting for Galactic acceleration [@nt95] or proper motion [@s70; @dt92]. The numbers in parentheses after $P$ and $\dot{P}$ are the 1-$\sigma$ errors in the last digit reported by TEMPO. All timing solutions use the DE430 Solar System Ephemeris.]{}
Discussion {#conclusions}
==========
Here, we discuss in further detail the most interesting pulsars presented in this paper. In Section \[0557\] we discuss PSR J0557$-$2948 and the discrepancy in the number of predicted and known disrupted recycled pulsars to double neutron star systems. Section \[1806\] discusses the measurement of the binary parameters of PSR J1806+2819, the search for an optical counterpart, and the possibility of including this MSP in PTAs. In Section \[0930\] we explain the nulling analysis performed on PSR J0930$-$2301. We also discuss the significant differences between the distances derived by two different electron density models for all ten pulsars in Section \[models\].
{width="50.00000%"} \[fig:ppdot\]
PSR J0557$-$2948 {#0557}
----------------
PSR J0557$-$2944 is a partially recycled pulsar with a spin period of 43.6 [ms]{} and a DM of 49 pc cm$^{-3}$. The timing solution indicates that this pulsar has undergone a phase of recycling but is now isolated. More specifically, this pulsar has a low period derivative ($\dot{P} = 7.3 \times10^{-20}$) and the spin period shows no evidence of a periodic Doppler shift. This pulsar is likely the end result of a disrupted neutron star (DNS) system. This class of neutron stars began to be identified in the 1990s [@cnt93] and have since been dubbed disrupted recycled pulsars [DRPs; @lma04; @h83; @bh91; @tt98].
An interesting diagnostic of binary neutron star evolution scenarios is the relative numbers and distribution of DRPs and DNSs in the Galaxy. As discussed by previous authors [see, e.g., @lma04], the number of DRPs relative to DNSs will depend on the survival probability of the second supernova explosion that formed these systems: higher survival probabilities (for example, due to relatively runaway velocities) will increase the numbers of DNS binaries relative to DRPs. It is also possible that DRPs actually have larger runaway velocities, and, hence, are fainter, and/or they escape the galaxy more frequently than bound systems. To investigate the latter hypothesis, we examined the height ($z$) from the Galactic plane, flux and luminosity distributions of DRPs and DNSs. Given the small numbers of both types of pulsars, statistical estimates such as the Kolmogorov-Smirnov test have limited ability to determine whether there are any significant differences between the two populations. From a simple comparison based largely on DM-derived distances, we note that median and mean $z$-heights are 200 pc and 300$\pm$100 pc for DNS binaries versus 385 pc and 580$\pm$160 pc for DRPs. While this appears to support the idea that DRPs have higher space velocities, we find no significant difference in the luminosity or flux distributions for the two classes. The lack of any difference in the fluxes is surprising, since for a common luminosity distribution to both sources, one would naively expect a lower average flux from the DRPs if their scale heights were truly larger. A more detailed observational and simulation study, which considers the selection biases in both DRPs and DNS binaries, should be undertaken to further investigate these issues.
PSR J1806+2819 {#1806}
--------------
Initial timing observations of PSR J1806+2819 indicated that the barycentric spin period was varying between epochs, presumably due to the Doppler effect as a result of its motion in a binary orbit with a companion star. Initial measurements of the binary system’s projected semi-major axis, orbital period, and time of periastron passage were determined by using `fit_circular_orbit.py` from the PRESTO software package, which models the varying spin period as an edge-on, circular orbit. These parameters, along with the orbital eccentricity and longitude of periastron, were then incorporated into the timing model. The results of the fits of these five Keplerian parameters, derived using the ELL1 binary model [@lcw01], are shown in Table 3.
\[tb:bin\]
Measured Parameters
---------------------------------------------------------- -----------------
Projected semi-major axis, $A_{1}$$\sin{i}$ (lt-s) 21.608784(8)
Orbital eccentricity times, $\sin{\omega}$ $EPS1$ $-$0.0000852(5)
Orbital eccentricity times, $\cos{\omega}$ $EPS2$ $-$0.0000192(7)
Orbital period, $P_{b}$ (days) 43.866963(2)
Epoch of ascending node passage, $T_{asc}$ (MJD) 57040.76929(2)
Derived Parameters
Orbital eccentricity, $e$ 0.0000874(5)
Longitude of periastron argument, $\omega$ ($\deg$) 257.3(4)
Time of periastron argument, $T_{0}$ (MJD) 57072.13(5)
Mass function, $f_{M}$ ($\mathrm{M}_\odot$) 0.005629873(6)
Minimum companion mass, $M_{c,min}$ ($\mathrm{M}_\odot$) 0.25
: Timing-Derived Binary Parameters for PSR J1806$+$2819
[ Projected pulsar semi-major axis of the orbit, orbital eccentricity multiplied by sine and cosine of the longitude of periastron argument, orbital period, and longitude of ascending node passage of PSR J1806$+$2819 measured using the ELL1 binary model. The orbital eccentricity, longitude of periastron argument, and time of periastron argument were derived from the measured quantities. The numbers in parentheses are the 1-$\sigma$ errors in the last digit reported by TEMPO.]{}
This pulsar is in a highly circular orbit of 44 days with a projected semi-major axis of 21.6 lt-s. Assuming a pulsar mass of 1.4 $\mathrm{M}_\odot$, we find that the minimum (where inclination, $i$ = 90) and median (where inclination, $i$ = $60\degr$) companion masses are 0.25 $\mathrm{M}_\odot$ and 0.29 $\mathrm{M}_\odot$, respectively. The binary period companion mass relation [@ts99; @imt16] gives a companion mass of 0.30 $\mathrm{M}_\odot$. This suggests a WD companion.
We searched for an optical counterpart to PSR J1806+2819 using data release 1 from the Panoramic Survey Telescope and Rapid Response System (PanSTARRS) 3$\pi$ survey [@cmm16]. No source was present in the catalog at the position of the pulsar. We searched the stacked images for each of the five bands (`grizy`) manually, and detect no source at the pulsar’s position (see Figure 3). We use the average 5-$\sigma$ magnitude lower limits for the stacked survey data from [@cmm16] for this source: 23.3, 23.2, 23.1, 22.3, 21.4 for `grizy`, respectively. The estimated reddening of the source was determined using 3-D map of interstellar dust reddening by [@gsf15] at the DM-derived distance of 1.3 kpc. This was converted to an extinction in all five bands using Table 6 from [@sf11].
![Three-color [*gri*]{} composite image from the PanSTARRS image cutout server at the position of PSR J1806+2819. This image is 2$\arcmin$ $\times$ 2$\arcmin$ in size where east is to the left and north is up. A 2$\arcsec$ error at the position of the pulsar is indicated by the circle. No optical counterpart was found for this pulsar. ](f3.pdf "fig:"){width="50.00000%"} \[fig:gri\]
We translated these limits into limits on the effective temperature and age of the presumed WD companion for an assumed companion mass of 0.30 $\mathrm{M}_\odot$. We first determined colors as a function of effective temperature for a 0.30 $\mathrm{M}_\odot$ WD using the Bergeron model [@hb06; @ks06; @tbg11; @bwd11]. Note that the colors do not change significantly as a function of mass. We used absolute normalizations from the models of [@imt16] appropriate for this mass value. We find an upper limit to the effective temperature of 7100 K, with the most constraining limits being determined from the $r$ band. We then used these limits to determine a lower limit on the cooling age of 1200 Myr. For the minimum companion mass of 0.25 $\mathrm{M}_\odot$ (at an $i$ = $90\degr$), the effective temperature and cooling age limits are essentially unchanged.
PSR J1806+2819 is the only MSP presented in this paper. For MSPs to be considered for use in PTAs, the RMS timing residuals must be less than $\sim$1 $\mu$s. PSR J1806+2819 has a RMS value of 16.2 $\mu$s, which is too high to be considered for use in PTAs. There are numerous factors that could contribute to the high RMS value of PSR J1806+2819. The spin period for this pulsar (15 [ms]{}) is larger than most MSPs in PTAs. The spin periods of MSPs used in the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) nine-year data set range from 1.65 [ms]{} to 16.05 [ms]{} [@abb16]. If this pulsar was included in this PTA, it would have the second longest spin period. The large spin period of PSR J1806+2819 is not enough on its own to exclude it from PTA use; however, this pulsar also has a relatively wide pulse profile, shown in Figure \[fig:profiles\]. The full width at half maximum (FWHM), listed in Table \[tb:prd\], is roughly 2 ms. This is much larger than the FWHM values for any of the MSPs in the NANOGrav PTA.
One caveat to this discussion is that our timing measurements for PSR J1806$+$2819 were performed at 350 MHz, whereas most PTA timing is done around 1 GHz. The profile of PSR J1806$+$2819 could be narrower at 1 GHz and the RMS residual could decrease, but it is unlikely that these improvements would be sufficient to make it a suitable PTA addition. However, this pulsar is in the declination range visible to the Arecibo Observatory and should be tested with the very high sensitivity of Arecibo to see if it is suitable for inclusion in NANOGrav.
PSR J0930$-$2301 {#0930}
----------------
The discovery observation of PSR J0930$-$2301 revealed that the pulse intensity was modulated, presumably due to pulse nulling. Pulse nulling is a phenomenon where the pulsed emission suddenly appears to drop to zero and then returns to its normal state [@b70]. Possible explanations for pulse nulling include: the pulsar undergoing complete cessation of emission [@klo06; @gjk12], the pulsar transitioning to weaker emission modes [@elg05; @yws14], the acceleration zone of the pulsar not being completely filled with electron-positron pairs [@dr01; @jvl04], and the pulsar beam of emission moving out of the line of sight from Earth [@dzg05; @zgd07]. Mode changing is a related effect where the average pulse profile suddenly changes between two or more stable states.
Pulse nulling has been observed in about 100 pulsars [@gjk12]. The nulling fraction (NF), or fraction of the pulses with no detectable emission, can range from less than a percent to nearly 100% [@r76; @wmj07; @gjw14]. The NF does not necessarily describe the duration of an individual null or the time between nulls. The null length is the timescale that the pulsar spends in the null state. This can range from a few pulse periods to years [@wmj07; @llm12].
This pulsar was not detectable at many observed epochs possibly due to its nulling behavior, relative faintness, and spin frequency similar to harmonics of common sources of radio frequency interference (RFI). The high DM of PSR J0930$-$2301 makes it unlikely that scintillation played a role in these non-detections. After RFI excision was performed, using `rfifind` from the PRESTO software package, this pulsar was detected at enough epochs to achieve a full phase-connected timing solution.
To study the nulling behavior of PSR J0930$-$2301, a single pulse analysis was performed on a 56 minute timing observation. In offline processing, the dedispersed data were folded to 512 phase bins across the spin period to form subintegrations. Emission detected from PSR J0930$-$2301 was weak, so a subintegration length of 12 pulse periods was used to average enough pulses together to ensure a sufficient signal-to-noise ratio ($ \mathrm{S/N} \geq 5$). This analysis is insensitive to nulls less than or similar to the subintegration length. Therefore, the NF will be an underestimate and only the lower limit on this value can be determined.
We estimated the baseline using the off-pulse bins, and subtracted it from each subintegration to estimate the NF. Pulse energies of each subintegration were determined after subtraction for both the on-pulse and off-pulse regions. The pulse energies were then normalized by the mean energy of the on-pulse region. The summed energy of the on- and off-pulse regions was calculated using the same number of phase bins.
Figure 4 shows the histograms of the on- and off-pulse energy distributions, where the total number of bins for each energy distribution is equal to the number of subintegrations $N$. The excess at zero energy for the on-pulse energy distribution gives the fraction of nulled pulses for the pulsar [@r76]. The NF was calculated by first scaling the off-pulse energy distribution so that the number of pulses with energies less than zero was equal to that in the on-pulse energy distribution. Then we subtracted the on-pulse distribution from this scaled off-pulse distribution. The NF is simply this scale factor.
{width="50.00000%"} \[fig:hist\]
The null lengths were defined as the number of subsequent pulses all with pulse energies below a conservative threshold of five times the off-pulse RMS variation. Since the used subintegration length was 12 pulse periods, our analysis was insensitive to null lengths shorter than this duration. The overlap in the on-pulse and off-pulse histograms (Fig. 4) and the necessary integration of multiple pulses to obtain the required S/N mean we operate in different regime from high-precision nulling studies in which individual pulses are unambiguously identified (e.g. Figure 1 from @lkr02). But these do allow us to place lower limits on the NF, and a qualitative estimate of the null lengths.
The lower limit of the NF for PSR J0930$-$2301 was estimated to be 30%. The null lengths were found to be 36 pulse periods on average and ranged from 12 to 156 pulse periods. Given that the spin period of this pulsar is 1.8 [s]{}, it can be concluded that it usually nulls for seconds up to minutes, with the longest null observed being roughly five minutes. This conclusion was supported via visual inspection of the phase$-$time diagram (Figure 5). Also, this figure shows that the pulse profile has multiple components that turn on and off independently of each other. We conclude that this is evidence of mode changing.
{width="50.00000%"} \[fig:phasetime\]
Electron Density Models {#models}
-----------------------
As seen in the last two columns of Table 1, the distances estimated to the pulsars presented in this paper by the two most recent electron density models are very different for some pulsars. Since the YMW16 model is relatively new, these pulsars provide important data points on the relationship between the two models.
The NE2001 electron density model [@cl02] has been the standard tool to estimate distances to pulsars since 2001. This model incorporates 112 independent pulsar distances and 269 scattering measures to define an electron density model for the Galaxy. The components of the model include: thin and thick axisymmetric disks, spiral arms, a local arm, a local hot bubble around the Sun, super-bubbles in the first and third Galactic quadrants, and over-dense regions surrounding the Gum Nebula, the Vela supernova remnant, Galactic Loop I, and a small region around the Galactic Center. This model also incorporates clumps and voids in directions of known pulsars with DMs higher and lower, respectively, than predicted by the quasi-smooth component of the model.
The YMW16 [@ymw17] model has the same basic structure as the NE2001 model with some important differences. The YMW16 model utilizes 189 independent pulsar distances to define its electron density model. It incorporates many of the same Galactic components as the NE2001 model; however, it does not make use of interstellar scattering in building the model. Also, YMW16 does not attempt to correct individual pulsars with discrepant distances by adding clumps or voids in their respective directions. This type of feature is only added when a number of pulsars in a region have discrepant distances and/or there is independent evidence for this feature.
A majority of the distances estimated to the pulsars presented in this paper derived from the two models are significantly different. Neither model gives a reliable distance to PSR J0930$-$2301. The maximum DM predicted by the YMW16 model in the direction of PSR J0930$-$2301 is $\sim$60.8 pc cm$^{-3}$, which is lower than the measured DM of the pulsar (78.3 pc cm$^{-3}$). It is likely that this electron density model will significantly over-estimate the distances to pulsars in the direction of PSR J0930$-$2301. The maximum DM predicted by the NE2001 model in the direction of PSR J0930$-$2301 is 77.12 pc cm$^{-3}$. This is closer to the timing-derived DM of the pulsar.
The two models are in agreement for only one pulsar, assuming 25% errors, PSR J1806+2819. This is the pulsar with the lowest DM of the ten presented in this paper, suggesting that distance discrepancies become greater with increasing DM. None of the pulsars presented in this paper are at high latitude ($ |b| > 40\degr $), where the NE2001 model has been found to systematically under-estimate distances [@lfl+06]. The distances derived using the YMW16 model are larger than the distances from the NE2001 for seven of the pulsars presented in this paper. The NE2001 model estimates a larger distance for only the closest pulsar, PSR J1119$-$2444.
Conclusions and Future Work {#future}
===========================
In this paper, we reported on the discovery and timing solutions of pulsars discovered through the Green Bank Northern Celestial Cap survey and the Pulsar Search Collaboratory. For each pulsar, a full phase-connected timing solution was achieved by measuring rotational and astrometric parameters. PSR J0557$-$2948 is a partially recycled pulsar with no evidence of a binary companion. We conclude that this pulsar is likely the end result of a disrupted double neutron star system. PSR J1806$+$2819 is a MSP in a 44 day orbit with a WD companion. No optical counterpart was found, so lower limits were placed on the magnitude and age for the WD companion. This pulsar will not likely be considered for PTA use. We placed lower limits on the nulling fraction of PSR J0930$-$2301, a nulling pulsar, and also saw evidence of mode changing.
The GBNCC survey is currently 75% complete, with an additional $\sim$ 50 pulsars expected to be discovered. Future observations of the pulsars discussed in Section \[conclusions\], could lead to additional results. Further timing observations and analysis of PSR J0557$-$2948 could result in a measurement of its proper motion. This would provide insight to the runaway velocity which neutron stars receive and the kick involved in the supernova explosions of their companions, and ultimately help better our understanding about how DNS systems and DRPs form. Higher frequency observations of PSR J1806$+$2819 could support including this pulsar in a PTA, depending on how significantly the RMS timing residuals is reduced. Future discoveries of pulsars with timing-derived DMs, and in particular, independent distances estimates will help to build electron density models of the Galaxy and lead to more accurate distance estimates to pulsars.
Acknowledgements {#ack .unnumbered}
================
The Green Bank Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
We thank West Virginia University for its financial support of GBT operations, which enabled some of the observations for this project.
MAM and AMK were supported by NSF awards AST-1327526 and OIA-1458952.
We thank Compute Canada, the McGill Center for High Performance Computing, and Calcul Quebec for provision and maintenance of the Guillimin supercomputer and related resources.
VMK receives support from an NSERC Discovery Grant, a Gerhard Herzberg Award, an R. Howard Webster Foundation Fellowship from the Canadian Institute for Advanced Research, the Canada Research Chairs Program, and the Lorne Trottier Chair in Astrophysics and Cosmology.
Pulsar work at UBC is supported by an NSERC Discovery Grant and by the Canadian Institute for Advanced Research.
MAM, DRL, JKS, FAJ, MED, DLK, KS, ALO, TDM, RS, ALM, SMR, and XS were supported by NSF Physics Frontiers Center award PtHY-1430284
JWTH and VIK acknowledge support from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement nr. 337062.
|
---
abstract: 'Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how this treatment can be used to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety (see [@Mum]). The same construction generalizes the classical theory of theta functions to vector bundles of higher rank on Abelian varieties and K3 surfaces. We also discuss the geometry behind these constructions.'
author:
- Andrei Tyurin
date: Apr 1999
title: 'Quantization and “theta functions”'
---
Introduction
============
It is a fruitful question to ask for some special basis of the complete linear systems ${\mathbb P}H^0(X, L^k)$, where $X$ is a smooth complete algebraic variety and $L$ a polarization. After this, following Mumford, we can ask for special equations defining $X$ under its embedding in ${\mathbb P}H^0(X,L^k)^*$. Of course, this is a priori impossible (for example, for ${\mathbb P}H^0({\mathbb P}^n,{\mathcal O}_{{\mathbb P}^n}(k))$), but it can be done after “rigidification” – that is, fixing some discrete structure on $X$. This is the subject of Invariant Theory in its pre-Hilbert form; however, any proposed “geometric” rigidification depends on the level $k$, and there is no universal way of doing it. The amazing fact is we can do it in many cases using the “classical” [*Geometric Quantization Procedure*]{} (GQP); but for this, we must leave algebraic geometry and go over to symplectic geometry instead. I would like to call this method the [*general theory of theta functions*]{}.
The starting point is that, together with a complex structure $I$ on $X$, a polarization $L$ gives us a quadruple $(X,{\omega},L,a_L)$, where ${\omega}$ is the Kähler form and $a_L$ a Hermitian connection on $L$ with curvature form $F_a=2\pi i\cdot{\omega}$ of Hodge type $(1,1)$, giving the holomorphic structure on $L$. The pair $(X,{\omega})$ is a symplectic manifold; we can thus view it as the [*phase space*]{} of some classical mechanical system, and the pair $(L,a_L)$ as [*prequantization data*]{} of this system.
We should start by recalling the construction of spaces of wave functions for a pair $(S,{\omega})$, where $S$ is a smooth symplectic manifold of dimension $2n$ with a given symplectic form ${\omega}$. To switch on any quantization procedure, we suppose that the cohomology class $[{\omega}]$ of the symplectic form is integral, that is, there exists a complex line bundle $L$ with $c_1(L)=[{\omega}]$. Moreover, suppose that $L$ has a Hermitian connection $a$ with curvature form $F_a=2\pi i\cdot{\omega}$. Any quadruple of this type $$(S,{\omega},L,a)
\label{eq1.1}$$ is called a [*prequantization*]{} of the classical mechanical system with phase space $(S,{\omega})$.
There are two approaches to the geometric quantization of $(S,{\omega},L,a)$ (\[eq1.1\]) (see [@A], [@S1] or [@W]). We discuss here the simplest version of these constructions, avoiding questions such as the choice of metaplectic structures, densities and half densities specifying geometric conditions on the manifold $S$. (Roughly speaking, $S$ should be a [*Calabi–Yau manifold*]{}). The usual slogan is that we have to choose “one half” of the set of all functions on $S$ using some “polarization” conditions. The first approach is as follows:
### Complex polarizations {#complex-polarizations .unnumbered}
To define a complex polarization, we give $S$ a complex structure $I$ such that $S_I=X$ is a Kähler manifold with Kähler form ${\omega}$. Then the curvature form of the Hermitian connection $a$ is of type $(1,1)$, hence for any [*level*]{} $k\in{\mathbb Z}^+$, the line bundle $L^k$ is a holomorphic line bundle on $S_I$. Complex quantization provides the space of [*wave functions of level $k$*]{}: $${\mathcal H}_{L^k}=H^0(S_I,L^k),
\label{eq1.2}$$ – that is, the space of [*holomorphic*]{} sections of $L^k$. Thus a [*complex polarization*]{} of $(S,{\omega},L,a)$ (\[eq1.1\]) returns to the algebraic geometry $S=X$ we started from.
In particular, the spaces of wave functions (\[eq1.2\]) obtained in this way is the collection of [*complete linear systems*]{} in the usual sense. We will suppose $L$ to be an [*ample*]{} holomorphic line bundle, and in particular, $$H^i(S_I,L)=0 \quad\text{for all $i>0$.}$$
The second approach is the choice of a real polarization:
### Real polarizations {#real-polarizations .unnumbered}
A real polarization of $(S,{\omega},L,a)$ is a Lagrangian fibration $$\pi\colon S\to B,
\label{eq1.3}$$ such that $${\omega}{{}_{{\textstyle{|}}\pi^{-1}(b)}}=0 \quad\text{for every point $b\in B$,}$$ and the fibre $\pi^{-1}(b)$ is a smooth Lagrangian submanifold for generic $b$.
Thus a mechanical system admits a real polarization if and only if it is [*complete integrable*]{}.
Actually, for the ordinary technical tricks of the theory of geometric quantization to work, we should require that the fibration has regular geometric behavior (see, for example, [@S2]). But beginning with Guillemin and Sternberg’s paper [@GS2], it is reasonable to consider more general fibrations, namely, [*real polarizations with singularities*]{}.
Then restricting $L$ to a Lagrangian fibre gives a flat connection $a{{}_{{\textstyle{|}}\text{fibre}}}$ or equivalently, a character of the fundamental group $$\chi\colon\pi_1(\text{fibre})\to{\operatorname U}(1).$$
Let ${\mathcal L}_{\pi}$ be the sheaf of sections of $L$ that are covariant constant along fibres. Then we get the space $${\mathcal H}_{\pi}=\bigoplus_{i} H^i(S,{\mathcal L}_{\pi}).$$ In the regular case, Śniatycki proved that $$H^i(S,{\mathcal L}_{\pi})=0 \quad\text{for $i \ne n$.}$$
1. A fibre of $\pi$ is a [*Bohr–Sommerfeld*]{} cycle of $(S,{\omega},L,a)$ if $\chi=1$.
2. ${\mathrm{BS}}\subset B$ is the subset of Bohr–Sommerfeld fibres.
3. $k$-BS${}\subset B$ is the subset of Bohr–Sommerfeld fibres for $(S,{\omega},L^k, ka)$.
According to the general theory of real quantizations, we expect to get a finite number of Bohr–Sommerfeld fibres, and in the regular case, $$H^n(S,{\mathcal L}_{\pi})=\bigoplus_{{\mathrm{BS}}}{\mathbb C}\cdot s_{i},$$ where $s_{i}$ is a nonzero covariant constant section of the restriction of $(L,a)$ to a Bohr–Sommerfeld fibre of the real polarization $\pi$.
In the general case, we can use this to [*define*]{} the new collection of spaces of wave functions (of level $k$): $${\mathcal H}_{\pi}^k=\bigoplus_{\text{$k$-BS}}{\mathbb C}\cdot s_{i},
\label{eq1.4}$$ and use special tricks to compare (1.4) with (\[eq1.2\]).
There is a canonical way of describing the Bohr–Sommerfeld subset. For this, we must choose special coordinates on $B$, the so-called [*action coordinates*]{}, which are part of the [*action angle*]{} coordinates (see [@A], [@GS1], [@GS2]). Locally around a point $b\in B$, the action coordinates $c_i$ are given as periods along 1-cycles of the fibre $\pi^{-1}(b)$ of a 1-form ${\alpha}$ such that $${\mathrm{d}}{\alpha}={\omega}.
\label{eq1.5}$$ This system of coordinates $\{c_i\}$ is defined up to additive constants and an [*integral*]{} linear transformations. Thus, if $B$ is simply connected, the action coordinates map $B$ locally diffeomorphically to some open subset $$B_c\subset {\mathbb R}^n_{(c_1, \dots, c_n)}
\label{eq1.6}$$ with coordinates $\{c_i\}$. If $(0,\dots, 0)$ is a Bohr–Sommerfeld point, then $${\mathrm{BS}}=B_c \cap {\mathbb Z}^n
\label{eq1.7}$$ is the [*set of integral points*]{} in $B_c$.
Let us return to our collections of spaces of wave functions.
An important observation, proved mathematically in a number of cases, is that the projectivization of the spaces (\[eq1.2\]) are given purely by the symplectic prequantization data and do not depend on the choice of complex structure on $S$. The same is true for the projectivization of the spaces (\[eq1.4\]). Moreover, these spaces do not depend on the real polarization $\pi$ (\[eq1.3\]), provided that we extend our prequantization data $(S,{\omega},L,a,{\mathcal F})$ by adding some “half density” ${\mathcal F}$ (see [@GS1]).
Our [*main problem*]{} is to compare the spaces $${\mathcal H}_{L^k} \quad \text{and} \quad {\mathcal H}_{\pi}^k.$$ If we are lucky enough to be able to construct a canonical isomorphism between these spaces, we get a special basis in the space of wave functions of a complex polarization, and in particular in any ample complete linear system. To distinguish this basis from others, we call it the [*system of theta functions*]{} of level $k$, with “characteristics” which are Bohr–Sommerfeld fibres.
Actually, this generalization of the theory of theta functions requires the final ingredient of the quantization procedure – the algebra of [*observables*]{} represented as an algebra of operators on spaces of wave functions (like the Heisenberg algebra on spaces of classical theta functions). We avoid using such algebras in this article, but they underlie our constructions, so it is reasonable to recall briefly the general shape of this ingredient.
### Algebra of observables and its space of states {#algebra-of-observables-and-its-space-of-states .unnumbered}
As a result of any quantization procedure, we get a ${\mathbb C}^*$-algebra of observables represented as some algebra $A$ of operators on the spaces of wave functions (\[eq1.2\]) or (\[eq1.4\]). As usual, this algebra is a noncommutative extension of some commutative ${\mathbb C}^*$-algebra $A_0\subset
A$. For example, if $S=T^*M$ for some manifold $M$ then $A_0$ is the algebra of continuous complex valued functions, so that $M$ is the [*space of maximal ideals*]{} of $A_0$.
A pair $A_0\subset A$ gives us a space $ {\mathcal H}$ of wave functions (\[eq1.2\]) or (\[eq1.4\]) as the subset of the [*space of states*]{}. Recall that a [*state*]{} is a map: $$\psi\colon A\to{\mathbb C}\quad\text{such that} \quad \psi(a^*a)\ge 0 \quad \text{and} \quad \Vert
\psi \Vert=1.
\label{eq1.8}$$ The set ${\mathcal S}(A)$ of all states of $A$ is a convex space and its [*boundary elements*]{} are called [*pure states*]{} (for example, in the previous example, delta functions of points are pure states). If our ${\mathbb C}^*$-algebra is represented on ${\mathcal H}$ by bounded operators then every vector $\left|\psi\right>$ defines the state as the [*expectation value*]{}.
The known strategy to identify spaces (\[eq1.2\]) and (\[eq1.4\]) is to represent both as irreducible representation spaces of some algebra admitting a [*unique irreducible representation*]{}.
The constructions of Berezin, Toeplitz and Rawnsley (see for example [@R]) are extremely useful for our geometric investigations, and we consider them in §6.
Model for our theory: the classical theory of theta functions
=============================================================
Let $A$ be a principally polarized Abelian variety of complex dimension $g$ with flat metric $g$. Then the tangent bundle $TA$ has the standard constant Hermitian structure (that is, the Euclidean metric, symplectic form and complex structure $I$). The Kähler form $2\pi i{\omega}$ gives a polarization of degree 1. We fix a [*smooth*]{} Lagrangian decomposition of $A$ $$A=T^g_+\times T^g_-,
\label{eq2.1}$$ such that both tori are Lagrangian with respect to ${\omega}$. (In the smooth category, $A$ is the standard torus ${\mathbb R}^{2g}/{\mathbb Z}^{2g}$ with the standard constant integral form ${\omega}$, and the decomposition (\[eq2.1\]) just consists of putting ${\omega}$ in normal form.) Let $L$ be a holomorphic line bundle with holomorphic structure given by a Hermitian connection $a$ with curvature form $F_a=2\pi i\cdot{\omega}$, and $L={\mathcal O}_A(\Theta)$, where $\Theta$ is the classical [*symmetric*]{} theta divisor. The decomposition (\[eq2.1\]) induces a decomposition $$H^1(A,{\mathbb Z})={\mathbb Z}^g_+\times{\mathbb Z}^g_-,
\label{eq2.2}$$ and a Lagrangian decomposition $$A_k=(T^g_+)_k \times (T^g_-)_k
\label{eq2.3}$$ of the group of points of order $k$. Any smooth “irreducible” $g$-cycle in $A$ is the image ${\varphi}(T^g)$ of a smooth linear embedding ${\varphi}\colon
T^g\to A$.
### Complex quantization {#complex-quantization .unnumbered}
This is nothing other than the [*classical theory of theta functions*]{}. Indeed, the decomposition (\[eq2.2\]) defines the collection of compatible [*theta structures*]{} of every level $k$: the decomposition (\[eq2.3\]) defines a Lagrangian decomposition $A_k=({\mathbb Z}^g)_k^+\times({\mathbb Z}^g)_k^-$, and a decomposition of the spaces of wave functions $${\mathcal H}_{L^k}=H^0(A, L^k)=\bigoplus_{w\in ({\mathbb Z}^g)_k^-}{\mathbb C}\cdot\theta_w,
\quad\text{with}\quad {\operatorname{rank}}{\mathcal H}_{L^k}=k^g,
\label{eq2.4}$$ where $\theta_c$ is the theta function with [*characteristic*]{} $c$ (see [@Mum]).
The decomposition (\[eq2.4\]) is given by the following recipe: we identify the torus $T^g_-$ with the dual torus, and consider vectors $w\in(T^g_-)_k$ as (periodic) linear differential forms on $T^g_-$. Applying the symplectic form ${\omega}$ gives a collections of linear vector fields $\xi_w$ on $A$ parallel to the fibration by the tori $T^g_+$. Finally, the translations $t_w$ on $A$ obtained as the exponentials of these vector fields give a finite subgroup of the translations group of $A$.
Now by choosing $\theta_0\in H^0(A, L^k)$ to be a [*very symmetric*]{} section (actually, the section with divisor the sum of all the translates of the theta divisor $\Theta$ by points of $(T^g_+)_k)$), we get a basis of $H^0(A, L^k)$: $$\{\theta_w=t_w^*(\theta_0)\}.
\label{eq2.5}$$
### Real polarization {#real-polarization .unnumbered}
The projection of the direct product (\[eq2.1\]) gives us a real polarization $$\pi\colon A\to T^g_-=B.
\label{eq2.6}$$ Remark that in this case the action coordinates (\[eq1.6\]) are just [*flat*]{} coordinates on $T^g_-=B$, and under this identification $${k\mathrm{\text{-}BS}}=(T^g_-)_k
\label{eq2.7}$$ is the subgroup of points of order $k$.
Now we can consider the dual fibration $$\pi'\colon A'={\operatorname{Pic}}(A/T^g_-)\to T^g_-=B,
\label{eq2.8}$$ with fibres $$(\pi')^{-1} (p)={\operatorname{Hom}}(\pi_1(\pi^{-1}(p),{\operatorname U}(1)).$$ This fibration admits the section $$s_0\in A \quad\text{with}\quad s_0 \cap (\pi')^{-1}
(p)={\operatorname{id}}\in{\operatorname{Hom}}(\pi_1(\pi^{-1}(p),{\operatorname U}(1)),
\label{eq2.9}$$ so that we have a decomposition $$A'=(T^g)'\times T^g_-=B.
\label{eq2.10}$$
An amazing fact recently proved by Golyshev, Lunts and Orlov [@GLO] is that the $2g$-torus $A'$ is canonically equipped with
1. a symplectic form ${\omega}'$;
2. a complex structure $I'$.
Now we can apply geometric quantization to the real polarization (\[eq2.6\]) of the phase space $(A,{\omega}, L^k,a_k)$, where $a_k$ is the Hermitian connection defining the holomorphic structure on $L^k$. Sending the line bundle $L^k$ to the character of the fundamental group of a fibre gives a section $$s_{L^k}\subset A'={\operatorname{Pic}}(A/T^g_-);
\label{eq2.11}$$ and the Bohr–Sommerfeld subset of $B=T^g_-$ is $$s_0 \cap s_{L^k} \,\subset\, s_0=B=T^g_-\,.$$
Under the identification $s_0=T^g_-={\operatorname U}(1)^g$, the intersection points $$s_0 \cap s_{L^k}=({\operatorname U}(1)^g)_k$$ are elements of order $k$ in $T^g={\operatorname U}(1)^g$. We thus get a decomposition $${\mathcal H}_{\pi}^k=\bigoplus_{\rho\in{\operatorname U}(1)^g_k}{\mathbb C}\cdot s_{\rho}.
\label{eq2.12}$$
1. ${\operatorname{rank}}{\mathcal H}_{L^k}={\operatorname{rank}}{\mathcal H}_{\pi}^k$.
2. Moreover, there exists a canonical isomorphism $${\mathcal H}_{L^k}={\mathcal H}_{\pi}^k,$$ up to a scaling factor.
We get already this isomorphism up to the action of the $k^g$-torus $({\mathbb C}^*)^{k^g}$ (compare decompositions (\[eq2.5\]) and (\[eq2.12\])). But the canonical isomorphism is defined by the action of the Heisenberg group $H_k$ on holomorphic sections of the line bundle $L^k$ (= the theory of theta functions, see [@Mum]) and the natural extension of the action of $H_k$ on the collection of Bohr–Sommerfeld orbits. Each of these representations is irreducible; thus the uniqueness of the irreducible representation of $H_k$ gives a canonical identification of these spaces up to scaling.
The functions making up the special bases of these spaces are called [*classical theta functions with characteristics of level $k$*]{}.
A real polarization without degenerate fibres such as $\pi$ in (\[eq2.6\]) is called [*regular*]{}. Using more sophisticated techniques (as in [@GS2]) we get a basis of the same type for real polarizations with degenerate fibres (see Remark after (\[eq1.3\])). But if we start with [*any*]{} polarized Kähler manifold $X$, the main question is the following:
> how to find a real polarization like (\[eq1.3\]) on $X$ (possibly with degenerate fibres)?
The amazing fact is that we can do it in many absolutely unpredictable cases. For example, we now show how to find a real polarization of complex projective space ${\mathbb P}^3$. [**Warning:**]{} We construct some real polarization of ${\mathbb P}^3$, but not a special theta basis in ${\mathbb P}H^0({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(k))$!
For this, we consider a special presentation of the complex threefold ${\mathbb P}^3$ as a real 6-manifold: let ${\Sigma}_2$ be a Riemann surface of genus 2. Then as a 6-manifold, $${\mathbb P}^3={\operatorname{Hom}}(\pi_1({\Sigma}_2),{\operatorname{SU}}(2)) /{\operatorname{PU}}(2)=R_2
\label{eq2.13}$$ is the space of classes of ${\operatorname{SU}}(2)$-representations of the fundamental group of a Riemann surface of genus 2.
Thus ${\mathbb P}^3$ is the first manifold of the collection of manifolds $R_g$. If we solve the problem of real polarizations of these, we get in particular a real polarization of ${\mathbb P}^3$. We do this in the following section, but we first extend the direct approach by giving a description in terms of general theories giving rise to these constructions.
Chern–Simons quantizations of $R_g$
===================================
According to the general procedure, we must present $R_g$ as the classical phase space of some mechanical system. We begin by recalling the full steps of this procedure.
A classical field theory on a manifold $M$ has three ingredients:
1. a collection ${\mathcal A}$ of [*fields*]{} on $M$, which are geometric objects such as sections of vector bundles, connections on vector bundles, maps from $M$ to some auxiliary manifold (the target space) and so on;
2. an [*action functional*]{} $$S \colon {\mathcal A}\to{\mathbb C}$$ which is an integral of a function $L$ (the Lagrangian) of fields;
3. a collection of observable functionals on the space of fields, $${\mathcal W}\colon {\mathcal A}\to{\mathbb C}.$$
Our case is the following.
### Example: Chern–Simons functional {#example-chernsimons-functional .unnumbered}
Here $M$ is a 3-manifold, $${\mathcal A}={\Omega}^1(M) {\otimes}{\operatorname{\mathfrak{su}}}(2)$$ and $$S(a)=\frac{1}{8} \pi^2\int_M {\operatorname{tr}}(a{\mathrm{d}}a + \frac{2}{3} a^3).
\label{eq3.1}$$ As observable, we can consider a [*Wilson loop*]{}, given by some knot $K\subset M$: $${\mathcal W}_K(a)={\operatorname{tr}}({\operatorname{Hol}}_K(a))$$ – the trace of the holonomy of a connection $a$ around the knot $K$.
Now let $$R_g={\operatorname{Hom}}(\pi_1({\Sigma}_g), {\operatorname{SU}}(2))/ {\operatorname{PU}}(2)
\label{eq3.2}$$ be the space of classes of ${\operatorname{SU}}(2)$-representations of the fundamental group of a Riemann surface of genus $g$. This space is stratified by the subspace of reducible representations $$R_g{^{\mathrm{red}}}\subset R_g, \quad R_g{^{\mathrm{irr}}}=R_g-R_g{^{\mathrm{red}}}.
\label{eq3.3}$$ To get this space as the phase space of some mechanical system, consider a compact smooth Riemann surface ${\Sigma}$ of genus $g > 1$ and the trivial Hermitian vector bundle $E_h$ of rank 2 on it. As usual, let ${\mathcal A}_h$ be the affine space (over the vector space ${\Omega}^1({\operatorname{End}}E_h)$) of Hermitian connections and ${\mathcal G}_h$ the Hermitian gauge group. This space admits a stratification: $${\mathcal A}_h{^{\mathrm{red}}}\subset {\mathcal A}_h$$ where the left-hand side is the subset of reducible connections. As usual, let $${\mathcal A}_h{^{\mathrm{irr}}}={\mathcal A}_h-{\mathcal A}_h{^{\mathrm{red}}}.$$
Sending a connection to its curvature tensor defines a ${\mathcal G}_h$-equivariant map $$F \colon {\mathcal A}(E_h)\to{\Omega}^2({\operatorname{End}}E_h)={\operatorname{Lie}}({\mathcal G}_h)^*
\label{eq3.4}$$ to the coalgebra Lie of the gauge group.
We can consider this map as the moment map with respect to the action of ${\mathcal G}_h$. The subset $$F^{-1}(0)={\mathcal A}_F
\label{eq3.5}$$ is the subset of flat connections and $${\mathcal A}_F{^{\mathrm{irr}}}={\mathcal A}_F \cap {\mathcal A}_h{^{\mathrm{irr}}}$$ the subspace of irreducible flat connections.
For a connection $a\in {\mathcal A}_F$ and a tangent vector to ${\mathcal A}_h$ at $a$ $${\omega}\in{\Omega}^1({\operatorname{End}}E_h)=T {\mathcal A}_h,$$ we have $${\omega}\in T {\mathcal A}_F \iff {\nabla}_a ({\omega})=0.
\label{eq3.6}$$ The trivial vector bundle $E_h$ admits the trivial connection $\theta$, which is interesting and important from many points of view, and it provides in particular the possibility of identifying ${\mathcal A}_h$ with ${\Omega}^1({\operatorname{End}}E_h)$ by sending a connection $a$ to the form $a-\theta$. We will identify forms and connections in this way.
The space ${\mathcal A}_h={\Omega}^1({\operatorname{End}}E_h)$ is the [*collection of fields*]{} of YM-QFT with the [*Yang–Mills functional*]{} $$S(a)=\int_{{\Sigma}} |F_a|^2.$$ Thus ${\mathcal A}_F$ is a [*classical phase space*]{}, that is, the space of solutions of an [*Euler–Lagrange equation*]{} ${\delta}S(a)=0$.
There exists a symplectic structure on the affine space ${\mathcal A}_h$, induced by the canonical 2-form given on the tangent space ${\Omega}^1({\operatorname{End}}E_h)$ at a connection $a$ by the formula $${\Omega}_0 ({\omega}_1,{\omega}_2)=\int_{{\Sigma}} {\operatorname{tr}}({\omega}_1 \wedge{\omega}_2).
\label{eq3.7}$$ This form is ${\mathcal G}_h$-invariant, and its restriction to ${\mathcal A}_F{^{\mathrm{irr}}}$ is degenerate along ${\mathcal G}_h$-orbits: at a connection $a$, for a tangent vector ${\omega}\in{\Omega}^1({\operatorname{End}}E_h)$, we have $${\omega}\in T{\mathcal G}_h \iff {\omega}={\nabla}_a {\varphi}\quad\text{for $
{\varphi}\in{\Omega}^0({\operatorname{End}}E_h)={\operatorname{Lie}}({\mathcal G}_h)^*$,}$$ and $$\int_{{\Sigma}} {\operatorname{tr}}({\nabla}_a {\varphi}\wedge{\omega})=\int_{{\Sigma}}{\operatorname{tr}}({\varphi}\wedge{\nabla}_a{\omega})=0.$$ Hence $${\omega}\in T {\mathcal A}_F \iff {\Omega}_0 ({\nabla}_a {\varphi},{\omega})=0.
\label{eq3.8}$$
Interpreting (\[eq3.4\]) as a moment map and using symplectic reduction arguments, we get a nondegenerate closed symplectic form ${\Omega}$ on the space $${\mathcal A}_F /{\mathcal G}_h=R_g$$ of classes of ${\operatorname{SU}}(2)$-representations of the fundamental group of the Riemann surface, and a stratification of this space. The form ${\Omega}$ defines a symplectic structure on $R_g{^{\mathrm{irr}}}$ and a symplectic orbifold structure on $R_g$.
On the other hand, the form ${\Omega}_0$ on ${\mathcal A}_h$ is the differential of the 1-form $D$ given by the formula $$D({\omega})=\int_{{\Sigma}} {\operatorname{tr}}((a) \wedge{\omega}).
\label{eq3.9}$$ We consider this form as a unitary connection $A_0$ on the trivial principal $L_0$ on ${\mathcal A}_h$.
To descend this Hermitian bundle and its connection to the orbit space, one defines the $\Theta$-cocycle (or $\Theta$-torsor) on the trivial line bundle (see [@RSW]). This cocycle is the ${\operatorname U}(1)$-valued function $\Theta$ on ${\mathcal A}_h\times{\mathcal G}_h$ defined as follow: for any triple $({\Sigma},a,g)$ where $(a,g)\in{\mathcal A}_h\times{\mathcal G}_h$, we can find a triple $(Y,A,G)$ where $Y$ is a smooth compact 3-manifold, $A$ a ${\operatorname{SU}}(2)$-connection on the trivial vector bundle ${\mathcal E}$ on $Y$ and $G$ a gauge transformation of it, such that $${\partial}Y={\Sigma}, \quad a=A {{}_{{\textstyle{|}}{\Sigma}}} \quad \text{and} \quad g=G{{}_{{\textstyle{|}}{\Sigma}}}.$$ Then $$\Theta (a, g)=e^{i({\operatorname{CS}}(A^G)-{\operatorname{CS}}(A))}.
\label{eq3.10}$$
Recall that the Chern–Simons functional on the space ${\mathcal A}({\mathcal E}_h)$ of unitary connections on the trivial vector bundle is given by the formula $${\operatorname{CS}}_Y (a_0 +{\omega})=\int_Y {\operatorname{tr}}\left({\omega}\wedge F_{a_0}-\frac{2}{3}
{\omega}\wedge{\omega}\wedge{\omega}\right).
\label{eq3.11}$$ It can be checked that the function (\[eq3.10\]) does not depend on the choice of the triple $(Y,A,G)$ (see [@RSW], §2).
The differential of $\Theta$ at $(a, g)$ is given by the formula $$\begin{gathered}
{\mathrm{d}}\Theta({\omega},{\varphi})= \\
\frac{\pi i}{4}\,\Theta\int_{{\Sigma}}\Bigl({\operatorname{tr}}(g^{-1}{\mathrm{d}}g\wedge g^{-1}{\omega}g)-{\operatorname{tr}}(a\wedge{\nabla}_{a^g}{\varphi})+2{\operatorname{tr}}(F_{a^g} \wedge {\varphi}) \Bigr),
\label{eq3.12}
\end{gathered}$$ where ${\omega}\in{\Omega}^1({\operatorname{End}}E_h)$ and ${\varphi}\in{\Omega}^0({\operatorname{End}}E_h)={\operatorname{Lie}}({\mathcal G}_h)$.
But the restriction of this differential to the subspace of flat connections is much simpler: $${\mathrm{d}}\Theta ({\omega}, {\varphi})=\frac{\pi i}{4}\,\Theta\int_{{\Sigma}}{\operatorname{tr}}(g^{-1}{\mathrm{d}}g
\wedge g^{-1}{\omega}g),
\label{eq3.13}$$ and is independent of the second coordinate.
That this function is in fact a cocycle results from the functional equation $$\Theta (a, g_1 g_2)=\Theta (a, g_1) \Theta (a^{g_1}, g_2).
\label{eq3.14}$$ Using this function as a torsor ${\mathcal A}_h \times_{\Theta}{\operatorname U}(1)$ we get a principal ${\operatorname U}(1)$-bundle $S^1(L)$ on the orbit space ${\mathcal A}_h/{\mathcal G}_h$: $$S^1(L)=({\mathcal A}_h\times S^1)/{\mathcal G}_h,
\label{eq3.15}$$ where the gauge group ${\mathcal G}_h$ acts by $$g(a, z)=(a^g, \Theta(a,g) z),$$ or the line bundle $L$ with a Hermitian structure.
Following [@RSW], let us restrict this bundle to the subspace of flat connections ${\mathcal A}_F$. Then one can check that the restriction of the form $D$ (\[eq3.9\]) to ${\mathcal A}_F$ defines a ${\operatorname U}(1)$-connection $A_{{\operatorname{CS}}}$ on the line bundle $L$.
By definition, the curvature form of this connection is $$F_{A_{{\operatorname{CS}}}}=i\cdot{\Omega}.
\label{eq3.16}$$
Thus the quadruple $$(R_g,{\Omega}, L, A_{{\operatorname{CS}}})
\label{eq3.17}$$ is a [*prequantum system*]{} and we are ready to switch on the Geometric Quantization Procedure.
Complex polarization of $R_g$
=============================
The standard way of getting a complex polarization is to give a Riemann surface ${\Sigma}$ of genus $g$ a conformal structure $I$. We get a complex structure on the space of classes of representations $R_g$ as follows: let $E$ be our complex vector bundle and ${\mathcal A}$ the space of all connections on it. Every connection $a\in{\mathcal A}$ is given by a covariant derivative ${\nabla}_a\colon{\Gamma}(E)\to{\Gamma}(E{\otimes}T^*X)$, a first order differential operator with the ordinary derivative ${\mathrm{d}}$ as the principal symbol and a complex structure gives the decomposition ${\mathrm{d}}={\partial}+{\overline{\partial}}$, so any covariant derivative can be decomposed as ${\nabla}_a={\partial}_a+{\overline{\partial}}_a$, where ${\partial}_a\colon{\Gamma}(E)\to{\Gamma}(E{\otimes}{\Omega}^{1,0})$ and ${\overline{\partial}}_a\colon{\Gamma}(E)\to{\Gamma}(E{\otimes}{\Omega}^{0,1})$. Thus the space of connections admits a decomposition $${\mathcal A}={\mathcal A}'\times {\mathcal A}'',
\label{eq4.1}$$ where ${\mathcal A}'$ is an affine space over ${\Omega}^{1,0}({\operatorname{End}}E)$ and ${\mathcal A}''$ an affine space over ${\Omega}^{0,1}({\operatorname{End}}E)$.
The group ${\mathcal G}$ of all automorphisms of $E$ acts as the group of gauge transformations, and the projection ${\operatorname{pr}}\colon {\mathcal A}\to{\mathcal A}''$ to the space ${\mathcal A}''$ of ${\overline{\partial}}$-operators on $E$ is equivariant with respect to the ${\mathcal G}$-action.
Giving $E$ a Hermitian structure $h$, we get the subspace ${\mathcal A}_h\subset
{\mathcal A}$ of Hermitian connections, and the restriction of the projection ${\operatorname{pr}}$ to ${\mathcal A}_h$ is one-to-one. Under this Hermitian metric $h$, every element $g\in{\mathcal G}$ gives an element ${\overline{g}}=(g^*)^{-1}$ such that $${\overline{g}}=g \iff g\in{\mathcal G}_h.$$ Now for $g\in{\mathcal G}$, the action of ${\mathcal G}$ on the component ${\mathcal A}''$ is standard: $${\overline{\partial}}_{g(a)}=g\cdot {\overline{\partial}}_a\cdot g^{-1}={\overline{\partial}}_a-({\overline{\partial}}_a g)\cdot g^{-1};$$ and the action on the first component ${\mathcal A}'$ of ${\partial}$-operators is $${\partial}_{g(a)}={\overline{g}}\cdot{\partial}_a\cdot{\overline{g}}^{-1}={\partial}_a-(({\overline{\partial}}_a g)\cdot g^{-1})^*.$$ It is easy to see directly that the action just described preserves unitary connections: $${\mathcal G}({\mathcal A}_h)={\mathcal A}_h,
\label{eq4.2}$$ and that the identification ${\mathcal A}_h={\mathcal A}$ is equivariant with respect to this action.
It is easy to see that ${\overline{\partial}}_a^2\in{\Omega}^{0,2}({\operatorname{End}}E)=0$. Thus the orbit space $${\mathcal A}'' /{\mathcal G}=\bigcup {\mathcal M}_i
\label{eq4.3}$$ is the union of all components of the moduli space of topologically trivial bundles on ${\Sigma}_I$. (This union doesn’t admit any good structure, as it contains all unstable vector bundles). Finally, the image of ${\mathcal A}_F\in {\mathcal A}_h$ is the component ${\mathcal M}{^{\mathrm{ss}}}$ of maximal dimension ($3g-3$) of s-classes of semistable vector bundles. Thus by classical technique of GIT of Kempf–Ness type we get:
[*(Narasimhan–Seshadri)*]{} $$R_{{\Sigma}}=R_g={\mathcal M}{^{\mathrm{ss}}}.$$
The form $F_{A_{{\operatorname{CS}}}}$ (\[eq3.16\]) is a $(1,1)$-form and the line bundle $L$ admits a unique holomorphic structure compatible with the Hermitian connection $A_{{\operatorname{CS}}}$.
On the other hand, a complex structure $I$ on ${\Sigma}$ defines a Kähler metric on ${\mathcal M}{^{\mathrm{ss}}}$ (the so-called Weyl–Petersson metric) with Kähler form $${\omega}_{\mathrm{WP}}=i F_{A_{{\operatorname{CS}}}}=i\cdot{\Omega}.
\label{eq4.4}$$ This metric defines the Levi-Civita connection on the complex tangent bundle $T {\mathcal M}{^{\mathrm{ss}}}$, and hence a Hermitian connection $A_{{\mathrm{LC}}}$ on the line bundle $$\det T {\mathcal M}{^{\mathrm{ss}}}=L^{{\otimes}4},
\label{eq4.5}$$ and a Hermitian connection $\frac{1}{4}A_{{\mathrm{LC}}}$ on $L$ compatible with the holomorphic structure on $L$. Thus we have
\[prop4.3\] $$\frac{1}{4} A_{{\mathrm{LC}}}=A_{{\operatorname{CS}}}.$$
Finally, considering ${\mathcal M}{^{\mathrm{ss}}}$ as a family of ${\overline{\partial}}$-operators, we get the Quillen determinant line bundle $L$ having a Hermitian connection $A_Q$ with curvature form $$F_{A_Q}=i\cdot{\Omega}.
\label{eq4.6}$$ Hence we can extend the equality of Proposition \[prop4.3\]:
$$\frac{1}{4} A_{{\mathrm{LC}}}=A_{{\operatorname{CS}}}=A_Q.$$
Summarizing, the result of the complex quantization procedure of the prequantum system (\[eq3.17\]) can be considered to be the spaces of wave functions of level $k$, that is, the spaces of $I$-holomorphic sections $${\mathcal H}_{L^k}=H^0 (L^k)
\label{eq4.7}$$
One knows that this system of spaces and monomorphisms is related to the system of representations of ${\operatorname{\mathfrak{su}}}(2,{\mathbb C})$ in the Weiss–Zumino–Novikov–Witten model of CQFT. Namely, for a half integer $i$, consider the irreducible representation $V_i$ of dimension $2i+1$ of ${\operatorname{\mathfrak{su}}}(2,{\mathbb C})$. The tensor product of two such representations is given by the Clebsch–Gordan rule $$V_i{\otimes}V_j=V_{i+j}\oplus V_{i+j-1}\oplus\dots\oplus V_{i-j}\quad
\text{for $i\ge j$,}
\label{eq4.8}$$ and the level of $V_i$ is $2i$.
Then the [*fusion ring*]{}$R_k({\operatorname{\mathfrak{su}}}(2,{\mathbb C}))$ of level $k$ is the quotient $$R_k({\operatorname{\mathfrak{su}}}(2,{\mathbb C}))=R({\operatorname{\mathfrak{su}}}(2,{\mathbb C}))/{\left< V_{(k+1)/2} \right>}
\label{eq4.9}$$ of the [*representation ring*]{} $R({\operatorname{\mathfrak{su}}}(2,{\mathbb C}))$ by the ideal generated by $V_{(k+1)/2}$.
Moreover, every character of the ring $R({\operatorname{\mathfrak{su}}}(2,{\mathbb C}))$ is given by a complex number $z\in{\mathbb C}$ which we can consider as a diagonal $2\times2$ matrix ${\operatorname{diag}}(iz,-iz)$. This matrix acts on ${\operatorname{\mathfrak{su}}}(2,{\mathbb C})$ and $V_i$ and $$\chi_z(V_i)={\operatorname{tr}}(\exp({\operatorname{diag}}(iz,-iz)))=\frac{\sin((2i+1)z)}{\sin z}.
\label{eq4.10}$$ Thus $$\chi_z(V_i)=0 \iff z=\frac{n \pi}{k+2}\quad \text{for $1 \le n \le 2i+1$.}
\label{eq4.11}$$ In these terms we get: $${\mathcal H}_{L^k}=\frac{(k+2)^{g-1}}{2^{g-1}} \sum_{n=1}^{k+1}
\frac{1}{(\sin(\frac{n\pi}{k+2}))^{2g-2}}\,.
\label{eq4.12}$$
See [@B] for a mathematical derivation of this formula.
Real polarization of $R_g$
==========================
The collection of real polarizations of the prequantum system $$(R_g,{\Omega}, L, A_{{\operatorname{CS}}})$$ is given in a very geometric way in the set-up of perturbation theory of Chern–Simons theory. The crucial point is a [*trinion decomposition*]{} of a Riemann surfaces, given by a choice of a maximal collection of disjoint, noncontractible, pairwise nonisotopic smooth circles on ${\Sigma}$. An isotopy class of such a collection of circles is called a [*marking*]{} of the Riemann surface. It is easy to see ([@HT]) that any such system contains $3g-3$ simple closed circles $$C_1, \dots, C_{3g-3}\subset {\Sigma}_g,
\label{eq5.1}$$ and the complement is the union $${\Sigma}_g-\{C_1, \dots, C_{3g-3}\}=\bigcup_{i=1}^{2g-2} P_i
\label{eq5.2}$$ of $2g-2$ trinions $P_i$, where every trinion is a 2-sphere with 3 disjoint discs deleted: $$P_i=S^2 \setminus \bigl(D_1\cup D_2\cup D_3\bigr) \quad \text{with}
\quad{\overline{D}}_i\cap{\overline{D}}_j=\emptyset \quad \text{for} \quad i \ne j.$$ A collection $\{C_i\}$ with these conditions is called a [*trinion decomposition*]{} of ${\Sigma}$. The invariant of such a decomposition by marking class is given by its [*$3$-valent dual graph*]{} ${\Gamma}(\{C_i\})$, associating a vertex to each trinion $P_i$, and an edge linking $P_i$ and $P_j$ to a circle $C_l$ (\[eq5.1\]) such that $$C_l\,\subset\,{\partial}P_i \cap{\partial}P_j.$$
If we fix the isotopy class of a trinion decomposition $\{C_i\}$, we get a map $$\pi_{\{C_i\}} \colon R_g\to {\mathbb R}^{3g-3}
\label{eq5.3}$$ with fixed coordinates $(c_1, \dots, c_{3g-3})$ such that $$c_i (\pi_{\{C_i\}} (\rho))=\frac{1}{\pi}\,\cos^{-1}\bigl(\frac{1}{2}{\operatorname{tr}}\rho([C_i])\bigr)\in [0, 1].$$
We see that
1. The map $\pi_{\{C_i\}}$ is a real polarization of the system $(R_g,k\cdot{\omega},L^k,k\cdot A_{{\operatorname{CS}}})$.
2. The coordinates $c_i$ are [*action coordinates*]{} for this Hamiltonian system (see (\[eq1.6\]) and [@D]).
These functions $c_i$ are continuous on all $R_g$ and smooth over $(0,1)$. Moreover, Goldman [@G] constructed ${\operatorname U}(1)$-actions on the open dense set $$U_i=c_i^{-1}(0,1)$$ for which the function $c_i$ is the [*moment map*]{}, and all these ${\operatorname U}(1)$-actions commute with each other. Hence we get:
1. The restriction $$\pi_{\{C_i\}}{{}_{{\textstyle{|}}\bigcap_i U_i}}\colon\bigcap_i U_i\to (0, 1)^{3g-3}$$ is the moment map for the ${\operatorname U}(1)^{3g-3}$-action on $\bigcap_i U_i$.
2. The image of $R_g$ under $\pi_{\{C_i\}}$ is a convex polyhedron $$I_{\{C_i\}}\subset [0, 1]^{3g-3}.
\label{eq5.4}$$
3. The symplectic volume of $R_g$ equals the Euclidean volume of $I_{\{C_i\}} $: $$\int_{R_g}{\omega}^{3g-3}={\operatorname{Vol}}I_{\{C_i\}}.
\label{eq5.5}$$
4. The expected number of Bohr–Sommerfeld orbits of the real polarization $\{C_i\}$ $$N_{{\mathrm{BS}}}(\pi_{\{C_i\}},R_g,{\omega},L,A_{{\operatorname{CS}}})
\label{eq5.6}$$ equals the number of half integer points in the polyhedron $I_{\{C_i\}}$, and $$\lim_{k\to\infty} k^{3-3g}\cdot N_{{k\mathrm{\text{-}BS}}}=\int_{R_g}{\omega}^{3g-3}
={\operatorname{Vol}}I_{\{C_i\}}.
\label{eq5.7}$$
[From]{} the combinatorial point of view, the number $N_{{\mathrm{BS}}}$, or more generally the numbers $N_{{k\mathrm{\text{-}BS}}}$ of $k$-BS fibres, is determined as follows: consider functions $$w\colon\{C_i\}\to\frac{1}{2k}\{0,1,2,\dots,k\}
\label{eq5.8}$$ on the collection of edges of the 3-valent graph ${\Gamma}(\{C_i\})$ to the collection of $\frac{1}{2k}$ integers, such that, for any three edges $C_l,
C_m, C_n$ meeting at a vertex $P_i$, the following 3 conditions hold:
1. $w(C_l) + w(C_m) + w(C_n)\in \frac{1}{k}\cdot {\mathbb Z}$;
2. $w(C_l) + w(C_m) + w(C_n) \le 1$;
3. for any ordering of the triple $C_l, C_m, C_n$, $$|w(C_l)-w(C_m)|\le w (C_n) \le w(C_l) + w(C_m).
\label{eq5.9}$$
Such a function $w$ is called an [*admissible integer weight of level*]{} $k$ on the graph ${\Gamma}(\{C_i\})$.
1. The number $|W_g^k|$ of admissible weights of level $k$ is independent of the graph ${\Gamma}(\{C_i\})$;
2. $$|W_g^k|=N_{{k\mathrm{\text{-}BS}}}.
\label{eq5.10}$$
The conditions (\[eq5.9\]) are called [*Clebsch–Gordan conditions*]{} for ${\operatorname{\mathfrak{su}}}(2,{\mathbb C})$, for obvious reasons. We can view the space of all real functions with values in $[0,1]$ subject to these conditions to get a complex $I_{\{C_i\}}$.
Following this combinatorial approach, we can construct a two dimensional complex $Y_g$: the set of vertices is the set of all dual graphs associated with all types of markings of ${\Sigma}$. Two vertices are joined by an edge if and only if the two graphs are related by an [*elementary fusion operation*]{}. The $2$-cells correspond to [*pentagons*]{}, and so on (see [@MS]). The topology of this complex reflects the combinatorial properties of real polarizations of this type.
The geometric meaning of this combinatorial description is as follows: consider the space ${\mathbb R}^{3g-3}$ with action coordinates $c_i$ (\[eq5.3\]). This space contains the integer sublattice ${\mathbb Z}^{3g-3}\subset {\mathbb R}^{3g-3}$, and we can consider the [*action torus*]{}: $$T^A={\mathbb R}^{3g-3} / {\mathbb Z}^{3g-3}.
\label{eq5.11}$$
In particular, we get a map $$\pi_A \colon R_g\to T^A
\label{eq5.12}$$ which glues at most points of the boundary of $I_{\{C_i\}}$.
Now every integer weight $w$ (\[eq5.8\]) satisfying (1) and (2), but a priori without the Clebsch–Gordan conditions (\[eq5.9\]), defines a point of order $2k$ on the action torus $$w\in T^A_{2k}.$$ In particular, the collection $W_g^k$ of admissible integer weights (subject to (\[eq5.9\])) can be considered as a subset of points of order $2k$ on the action torus: $$W_g^k\subset T^A_{2k}.
\label{eq5.13}$$ On the other hand, every vector $w\in {\mathbb R}^{3g-3}$ can be interpreted as a [*differential $1$-form*]{} on ${\mathbb R}^{3g-3}$, and by the usual construction using the symplectic form ${\Omega}$, this defines a vector field $\xi_w$ tangent to the fibres of $\pi$. Integrating such vector fields defines the collection of transformations $$\{t_w\}=e^{\xi_w}\subset{\operatorname{Diff}}^+ (R_g).
\label{eq5.14}$$
These transformations preserve the curvature form $A_{{\operatorname{CS}}}$ of the connection. Thus (because $R_g$ is simple connected), there exists a collection of gauge transformations ${\alpha}_w\in{\mathcal G}_L$ of $L$ such that $$(t_w)^*(A_{{\operatorname{CS}}})=A_{{\operatorname{CS}}}^{{\alpha}_w}.
\label{eq5.15}$$ We can view such gauge transformations as ${\operatorname U}(1)$-[*torsors*]{}, just as in describing the formulas for classical theta functions for Abelian varieties in §2.
Moreover, if $R_{{\Sigma}}$ is given the Kähler structure induced from ${\Sigma}$ and $s\in H^0(R_{{\Sigma}}, L^k)$ is a holomorphic section, then we have the following.
$$(t_w)^*(s)\in H^0(R_{{\Sigma}}, L^k)
\label{eq5.16}$$
is also a holomorphic section.
If $s_0$ is a [*sufficiently symmetric*]{} holomorphic section of $L^k$, then the system $$\{s_w=t_w^*(s_0)\}\subset H^0(R_{{\Sigma}}, L^k)
\label{eq5.17}$$ is a special theta basis of [*some subspace*]{} of $H^0(L^k)$.
Comparing (\[eq5.17\]) and (\[eq2.6\]), we see that the recipe to construct the theta basis is the same as for Abelian varieties with the action space $T^A$ (see §2) but instead of the full collection $T^A_{2k}$ of points of order $2k$, we only use the subset $W_g^k\subset T^A_{2k}$.
In our realistic situation, the prequantum system $(R_g,{\Omega},L,A_{{\operatorname{CS}}})$ is far from the regular “theoretical” case. But in the fundamental papers [@JW1] and [@JW2] there is a well-defined correction to the “theoretical” situation. Here we only explain what we must do at a maximally degenerate Bohr–Sommerfeld fibre. We get proofs of the central statements of Proposition 5.4 and Corollary 5.1 by a quite fruitful method: we give new definitions making the statements almost obvious. We do this in the following special section.
### Unitary Schottky representations {#unitary-schottky-representations .unnumbered}
Every oriented trinion $P_i$ defines a [*handle*]{} ${\mathrm{HP}}_i$, and all these handles glue together to give a [*handlebody*]{} $H_{\{C_i\}}$, a compact 3-manifold such that $${\partial}H_{\{C_i\}}={\Sigma}.
\label{eq5.18}$$ We get a surjection $${\varphi}\colon \pi_1({\Sigma})\to \pi_1(H_{\{C_i\}}),
\label{eq5.19}$$ which defines the subspace $$B_{\{C_i\}}=\bigl\{\rho\in R_g \bigm| \rho {{}_{{\textstyle{|}}\ker {\varphi}}}=1\bigr\}.
\label{eq5.20}$$
1. $B_{\{C_i\}}$ is a Lagrangian subspace of $R_g$.
2. More precisely, it is a fibre of the real polarization $\pi_{\{C_i\}}$ (\[eq5.3\]): $$B_{\{C_i\}}=\pi_{\{C_i\}}^{-1} (1,\dots,1).
\label{eq5.21}$$
3. Moreover, $B_{\{C_i\}}$ is a Bohr–Sommerfeld orbit of $\pi_{\{C_i\}}$.
This Lagrangian subspace $B_{\{C_i\}}$ is singular: $$B_{\{C_i\}}{^{\mathrm{red}}}=B_{\{C_i\}} \cap R_g{^{\mathrm{red}}}={\operatorname{Sing}}B_{\{C_i\}};$$ and $B_{\{C_i\}}{^{\mathrm{irr}}}=B_{\{C_i\}} \cap R_g{^{\mathrm{irr}}}$ is a nonsingular Lagrangian subvariety.
Under the identification of $R_g$ with the moduli space of s-classes of semistable vector bundles on the algebraic curve ${\Sigma}$, the subspace $B_{\{C_i\}}$ is called the subset of [*unitary Schottky subbundles*]{}.
Obviously for this Bohr–Sommerfeld fibre $w_{{\mathrm{US}}}\in T^A_k$, we must use a special description of the symplectomorphism $t_{w_{{\mathrm{US}}}}$. This was done in the papers [@JW1] and [@JW2].
Returning to the general geometric quantization procedure and summarizing these results, we get two spaces of wave functions: complex quantization gives the spaces $${\mathcal H}_{{\Sigma}}^k=H^0(L^k)$$ of $I$-holomorphic sections of $L^k$, and real quantization gives the direct sum $${\mathcal H}_{\pi}^k\,=\sum_{k\text{-BS fibres}}\!{\mathbb C}\cdot s_{{k\mathrm{\text{-}BS}}}
\label{eq5.22}$$ of lines generated by covariant constant sections of restrictions of our prequantum line bundle $(L^k,k\cdot A_{{\operatorname{CS}}})$ to the Bohr–Sommerfeld fibres of $\pi$ of level $k$.
The amazing fact is the following:
For any level $k$, any complex Riemann surface ${\Sigma}$, and any trinion decomposition $\{C_i\}$ with the real polarization $\pi$ of $R_g$ we have $${\operatorname{rank}}{\mathcal H}_{{\Sigma}}=H^0(L^k)={\operatorname{rank}}{\mathcal H}_{\pi},
\label{eq5.23}$$ and these ranks can be computed by the Verlinde calculus.
Our construction gives a distinguished theta basis of the first space $H^0(L^k)$.
This isomorphism between spaces of wave functions underlies all the “modular” behavior of gauge theory invariants in dimensions 2, 3 and 4.
The final “classical” question concerns the Fourier decomposition of our non-Abelian theta functions $s_w$ (5.17). It can be done using the Fourier decomposition along coordinates $\{c_i\}$ of the action torus $T^A$ (\[eq5.11\]) twisting by the system of torsors $\{{\alpha}_w\}$ (5.15). Roughly speaking, the theta functions $s_w$ (5.17) are [*truncated*]{} theta functions on the $(6g-6)$-dimensional “Fourier torus” $$T_F={\operatorname U}(1)^{3g-3} \times T^A.$$ Namely all coefficients of Fourier decompositions not satisfying the Clebsch–Gordan conditions (\[eq5.9\]) must go to zero. Can this condition be interpreted in terms of the heat equation?
Other definition of a theta basis
=================================
We must first recall the main constructions of GQP. Let $h$ be the Hermitian form on $L$, and $$\mu=\frac{1}{(3g-3)!}{\omega}^{3g-3}
\label{eq6.1}$$ the volume form on $R_g$. Then we have a scalar product and norm on the space ${\Gamma}(L^k)$ of global differentiable sections of $L^k$: $${\left< s_1,s_2 \right>}=\int_{R_g} h(s_1, s_2)\cdot \mu \quad \text{and} \quad
\Vert s \Vert=\sqrt{{\left< s,s \right>}}\,.
\label{eq6.2}$$ Let $L^2(L^k)$ be the $L^2$-completion of ${\Gamma}(L^k)$ and $$P_k \colon L^2(L^k)\to H^0(L^k)
\label{eq6.3}$$ the orthogonal projection to the finite dimensional subspace of [*holomorphic*]{} sections $H^0(L^k)\subset L^2(L^k)$.
The ring $C^{\infty}(R_g)$ of smooth functions on $R_g$ acts on $L^2(L^k)$ by multiplication $s\to f\cdot s$, and acts on the space $H^0(L)$ as a [*Toeplitz operator*]{}: $$T_f=P \odot f\in{\operatorname{End}}(H^0(L^k));
\label{eq6.4}$$ the map $C^{\infty}(R_g)\to{\operatorname{End}}(H^0(L^k))$ is called the [*Berezin–Toeplitz map*]{}.
Now, let $$p \colon L^*\to R_g
\label{eq6.5}$$ be the principal ${\mathbb C}^*$-bundle of $L$. Every point $x\in L^k$ defines a linear form $$l_x \colon H^0(L^k)\to{\mathbb C}, \quad\text{given by}\quad s(p(x))=l_x (s)\cdot x,
\label{eq6.6}$$ and the [*coherent state*]{} vector $s_x\in H^0(L^k)$ associated to $x$, which is uniquely determined by the equation $${\left< s_x,s \right>}=l_x(s).
\label{eq6.7}$$ Thus we get a map $${\varphi}_k \colon R_g\to {\mathbb P}H^0(L^k),
\label{eq6.8}$$ which is nothing other than the Hermitian conjugate of the standard algebraic geometric map by a complete linear system to the dual space, because of the equality $$s_{{\alpha}\cdot x}={\overline{{\alpha}}}^{-1}\cdot s_x \quad \text{for ${\alpha}\in{\mathbb C}^*$.}$$ Now, following John Rawnsley, we can define [*coherent projectors*]{} $P_{p(x)}$ and the [*Rawnsley epsilon function*]{} ${\varepsilon}\colon R_g\to{\mathbb R}^+$ in such a way that: $${\varepsilon}(p(x))=|x|^2\cdot {\left< s_x,s_x \right>} \quad \text{and} \quad
h(s_1, s_2)_{p(x)}={\varepsilon}(p(x))\cdot {\left< s_1, P_{p(x)} s_2 \right>}.
\label{eq6.9}$$ Since ${\varepsilon}> 0$ we can modify the old measure on $R_g$: $$\mu_{{\varepsilon}}={\varepsilon}\cdot \mu,
\label{eq6.10}$$ where $\mu$ is (6.1). This measure gives an integral representation of Toeplitz operators: $$T_f(s)=\int_{R_g} f(p(x))\cdot P_{p(x)}(s)\cdot \mu_{{\varepsilon}}.
\label{eq6.11}$$
Up to now, we have been working with a [*complex polarization*]{}. Let us return to the real polarization $\pi$ (\[eq5.3\]). We can identify the target real space ${\mathbb R}^{3g-3}$ of $\pi$ with the dual space $${\mathbb R}^{3g-3}=({\mathbb R}^{3g-3})^*$$ and we can consider our vectors $w\in W^k_g\subset T^A_{2k}$ as [*linear functions*]{} on the target space ${\mathbb R}^{3g-3}$. Thus we get a collection of functions $$\pi^* w \colon R_g\to {\mathbb R}\label{eq6.12}$$ and a collection of Toeplitz operators $$T_{\pi^* w}\in{\operatorname{End}}(H^0(L^k)).
\label{eq6.13}$$ Let us choose one (very symmetric) section $s_0$ in the following way: for $k=1$, the space $H^0(L)$ is the space of ordinary theta functions (see, for example, [@BL]) and every semistable bundle $E$ defines a theta divisor $$\Theta_E=\bigl\{L\in{\operatorname{Pic}}_{g-1}({\Sigma})\bigm| h^0(E{\otimes}L)>0\bigr\}.$$ Let $s_E$ be the section with this divisor as its zero set. Then one has the section $$s_0=s_{{\mathcal O}_{{\Sigma}}\oplus{\mathcal O}_{{\Sigma}}}^k\in H^0(R_{{\Sigma}},L^k),
\label{eq6.14}$$ and the collection of sections $$\{T_{\pi^* w}(s_0)=s_w\}\subset H^0(R_{{\Sigma}},L^k).
\label{eq6.15}$$
Using the integral representation (\[eq6.11\]), Rawnsley’s localization, and Proposition 5.6, we get immediately
The sections $\{T_{\pi^* w}(s_0)=s_w\}$ form a basis of $H^0(R_{\Sigma}, L^k)$.
The reader not wishing to check the following statement may take (\[eq6.15\]) as the [*definition*]{} of the theta basis:
The basis (5.17) coincides with the basis (\[eq6.15\]).
Thus, we do indeed get a generalization of theta functions.
What next?
==========
The theory of theta functions outlined above is just a small sample of the applications of the Geometric Quantization Procedure to algebraic geometry. Here we extend the list, mentioning applications which are natural generalizations of the above constructions.
### Generalization to vector bundles of higher rank {#generalization-to-vector-bundles-of-higher-rank .unnumbered}
This construction is new even in the classical set-up. Let us return to a real polarization of an Abelian variety $\pi\colon A\to T^g_-=B$, and its dual fibration $$\pi'\colon A'={\operatorname{Pic}}(A/T^g_-)\to T^g_-=B,$$ with fibres $$(\pi')^{-1} (p)={\operatorname{Hom}}(\pi_1(\pi^{-1}(p)),{\operatorname U}(1))$$ and section $$s_0\in A' \quad\text{with}\quad s_0 \cap (\pi')^{-1}
(p)={\operatorname{id}}\in{\operatorname{Hom}}(\pi_1(\pi^{-1}(p)),{\operatorname U}(1)).$$ Every stable holomorphic vector bundle $E$ on a [*generic*]{} principally polarized Abelian variety $A$ carries a Hermitian–Einstein connection $a_E$ that defines a holomorphic structure on $E$ with curvature $$F_{a_E}={\Lambda}i\cdot{\omega},
\label{eq7.1}$$ where ${\Lambda}$ is any constant element of ${\operatorname U}({\operatorname{rank}}(E))$, for example, ${\operatorname{id}}$. Hence the restriction of $a_E$ to every fibre of $\pi$ is a flat Hermitian connection on a $g$-torus, and thus $$\begin{gathered}
(a_E){{}_{{\textstyle{|}}\pi^{-1}(b)}}=\chi_1\oplus\dots\oplus \chi_{{\operatorname{rank}}E},
\quad\text{where}\\
\chi_i\in{\operatorname{Hom}}(\pi_1(\pi^{-1}(b)),{\operatorname U}(1)))={\operatorname{Pic}}(\pi^{-1}(b))=(\pi')^{-1}(b).
\end{gathered}
\label{eq7.2}$$
For vector bundles of higher rank, an [*$E$-[BS]{} fibre*]{} is a fibre $\pi^{-1}(b)$ such that the restriction $(E,a_E){{}_{{\textstyle{|}}\pi^{-1}(b)}}$ admits a covariant constant section.
Suppose that $E$ is [*ample*]{}, and in particular that $$H^i(A,E)=0\quad\text{for}\quad i>0.$$ Then, alongside the complex “space of wave functions” $H^0(A,E)$, we get a new space of wave functions $${\mathcal H}_{\pi}^E=\bigoplus_{\text{$E$-BS}}{\mathbb C}\cdot s_{i},
\label{eq7.3}$$ where $s_i$ is a covariant constant section of the restriction of $E$ to a $E$-BS fibre.
We again have the problem of comparing the spaces $${\mathcal H}_{E}=H^0(A, E) \quad \text{and} \quad {\mathcal H}_{\pi}^E.
\label{eq7.4}$$
This problem can be solved by analogous (but more sophisticated) methods from GQP. In particular
The space $H^0(A,E)$ admits a canonical theta basis.
Of course, if $X$ is any Kähler manifold with some real polarization (\[eq1.3\]) and stable holomorphic vector bundle $E$ admitting an Hermitian connection with curvature of the form (\[eq7.1\]), we get two spaces $${\mathcal H}_{E}=H^0(X, E) \quad \text{and} \quad {\mathcal H}_{\pi}^E
\label{eq7.5}$$ to compare.
In particular if $X=R_{{\Sigma}_g}$ one has
For a stable vector bundle of higher rank $E$ on $R_{{\Sigma}}$, $$H^0(R_{{\Sigma}_g}, E)=\bigoplus_{E\text{-}{\mathrm{BS}}}{\mathbb C}\cdot s_{i}.$$
This holds in particular for all the symmetric powers of the Poincaré bundles.
K3 surfaces {#k3-surfaces .unnumbered}
-----------
If the transcendental lattice $T_S$ of a K3 surface $S$ contains an even unimodular sublattice $H$ of rank 2, then $S$ admits a real polarization (see for example [@G1], [@G2] or [@T]). Then every ample complete linear system on $S$ admits a special theta basis.
Geometry behind these constructions: mirror reflection of holomorphic geometry {#geometry-behind-these-constructions-mirror-reflection-of-holomorphic-geometry .unnumbered}
------------------------------------------------------------------------------
If our real polarization (\[eq1.3\]) is regular, that is, the differential of the map $\pi\colon X\to B$ is surjective then all fibres are $n$-tori, and the second fibration $\pi'\colon X'\to B$ can be defined fibrewise in the usual way: $$(\pi')^{-1}(b)={\operatorname{Hom}}(\pi_1(\pi^{-1}(b)),{\operatorname U}(1))
\quad\text{for any point $b\in B$;}$$ that is, the fibre of $\pi'$ is the space of classes of flat connections on the trivial line bundle on $\pi^{-1}(b)$. This is a fibration in groups, and we want to consider its zero section $s\colon B\to X'$ as a submanifold $s_0\subset X'$.
The restriction of a pair $(E, a_E)$ to any fibre $\pi^{-1}(b)$ defines a finite set of points $(\pi')^{-1}(b)$, and hence a multisection $${\operatorname{GFT}}(E)\subset X',
\label{eq8.1}$$ which we again consider as a middle dimensional submanifold of $X'$. This cycle is called the [*Geometric Fourier Transformation*]{} of $E$.
Under the identification $s_0=B$, the set of $E$-BS fibres is defined now as the set of intersection points $$E\text{-BS}=s_0 \cap {\operatorname{GFT}}(E),
\label{eq8.2}$$ and under some geometric conditions, we expect that the number $$\# E\text{-BS}=[s_0] \cap [{\operatorname{GFT}}(E)]
\label{eq8.3}$$ where $[\ ]$ is the cohomology class of a submanifold.
In the general case of a polarization with degenerate fibres, this construction can be performed over the open subset $B_0\subset B$ of smooth tori and a number of questions arise:
1. to construct a smooth compactification $S'$;
2. to construct a symplectic form ${\omega}'$ and extend it to $S'$, in such a way that $\pi'$ is a new real polarization;
3. to construct a complex polarization of $S'$ such that the fibration $\pi$ is given by construction we have described, starting from $(S',{\omega}',L',a')$.
In full generality these problems are very hard (see for example [@G1], [@G2]).
The ideal picture is described by the mirror diagram $$\renewcommand{\arraystretch}{1.5}
\begin{matrix}
&&S &\longleftarrow & E \\
&& \kern3mm \big\downarrow \pi &&\\
& & B&& \\
&& \kern4.5mm \big\uparrow \pi'\\
{\operatorname{GFT}}(E)&\longrightarrow &S' &\longleftarrow & s_0 \\
\end{matrix}
\label{eq8.4}$$ with holomorphic objects (vector bundles) corresponding to the top of (\[eq8.4\]) and special Lagrangian cycles to the bottom.
The inverse problem {#the-inverse-problem .unnumbered}
-------------------
Every stable holomorphic vector bundle $E$ on an $S_I$ (top of (\[eq8.4\])) is a point in the moduli space of stable holomorphic vector bundles $$E\in {\mathcal M}_{[E]}^\mathrm{s}
\label{eq8.5}$$ of topological type $[E]$.
But the cycle ${\operatorname{GFT}}(E)$ (bottom of (\[eq8.4\])) is a point in the moduli space of special Lagrangian cycles (see [@HL]) $${\operatorname{GFT}}(E)\in {\mathcal M}^{[{\operatorname{GFT}}(E)]}
\label{eq8.6}$$ of topological type $[{\operatorname{GFT}}(E)]$. Thus we get a map $${\operatorname{GFT}}\colon {\mathcal M}_{[E]}^\mathrm{s}\to {\mathcal M}^{[{\operatorname{GFT}}(E)]}
\label{eq8.7}$$ sending $E$ to ${\operatorname{GFT}}(E)$.
However, we have not used all the information contained in $E$. Namely, ${\operatorname{GFT}}(E)$ can be defined as a [*supercycle*]{} (or [*brane*]{}). It’s easy to see that any cycle ${\operatorname{GFT}}(E)$ admits a tautological topologically trivial line subbundle $L$ with Hermitian connection $s_{\tau}$. A pair $$({\operatorname{GFT}}(E),a_{\tau})={\operatorname{sGFT}}(E)
\label{eq8.8}$$ of this type is called a [*supercycle*]{} (or brane).
The attempt to reconstruct the vector bundle $E$ (top of (\[eq8.4\])) from the supercycle ${\operatorname{sGFT}}(E)$ (bottom of (\[eq8.4\])) is called the [*inverse problem*]{}. More formally, let $S{\mathcal M}^{[{\operatorname{GFT}}(E)]}$ be the moduli space of supercycles of topological type $[{\operatorname{GFT}}(E)]$. Then in many special cases, one can prove that the map $${\operatorname{sGFT}}\colon{\mathcal M}^\mathrm{s}_{[E]}\to S{\mathcal M}^{([{\operatorname{GFT}}(E)])}$$ is an embedding at the general point. That is, a general stable vector bundle $E$ (top of (\[eq8.4\])) is uniquely determined by the supercycle ${\operatorname{sGFT}}(E)$ on $S'$ (bottom of (\[eq8.4\])).
For example, if the fibration $\pi \colon X\to B$ is the family of all deformations (with degenerations) of the general fibre $\pi^{-1}(b)=T^n$ as a torus with special structure inside $S$, then $$X'=S{\mathcal M}^{[T^n]}$$ is the family of all deformations (with degenerations) of the pair $(T^n,
\tau_0)$, where $\tau_0$ is the trivial connection.
At present this program is only realized in part (see, for example, [@T]). We must first use the experience of the geometric quantization procedure, and apply it in the Calabi–Yau realm of [*simply connected Kähler manifolds with canonical class zero*]{}. But in this paper, we want to emphasize that there exists the collection of singular Fano varieties $R_g$ for which these constructions are very important, although this is an extremely irregular case.
### Acknowledgments {#acknowledgments .unnumbered}
I would like to express my gratitude to the Institut de Mathématiques de Jussieu and the Ecole Normale Supérieure, and personally to Joseph Le Potier and Arnaud Beauville for support and hospitality. I wish to thank Yves Laslo and Christoph Sorger for many helpful discussions. Thanks are again due to Miles Reid for tidying up the English.
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[*or*]{} Tyurin@mpim-bonn.mpg.de
|
---
abstract: |
For a stationary set $S\subseteq \omega_1$ and a ladder system $C$ over $S$, a new type of gaps called $C$-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set $S$, for every ladder $C$ over $S$, every gap contains a subgap that is $C$-Hausdorff. But for every ladder $E$ over $\omega_1\setminus S$ there exists a gap with no subgap that is $E$-Hausdorff.
A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c posets is again a polarized c.c.c poset.
author:
- |
Uri Abraham\
Departments of Mathematics and Computer Science,\
Ben-Gurion University, Beér-Sheva, Israel\
Saharon Shelah [^1]\
Institute of Mathematics\
The Hebrew University, Jerusalem, Israel
title: Ladder gaps over stationary sets
---
Introduction
============
We first review some notations and definitions related to Hausdorff gaps. In fact we follow here the terminology given by M. Scheepers in his monograph [@Scheepers] on Hausdorff gaps, but since we restrict ourselves to $(\omega_1,\omega^*_1)$ gaps our nomenclature is somewhat simpler. The collection of all infinite subsets of $\omega$ is denoted $[\omega ]^\omega$, and for $a,b \in [\omega ]^\omega$, $a
\subseteq^* b$ means that $a\setminus b$ is finite. In this case $X(a,b)$ (the “excess” number) is defined to be the least $k$ such that $a\setminus b \subseteq k$. Thus $a \setminus X(a,b) \subseteq b$, but if $X(a,b)>0$ then $X(a,b)-1 \in a
\setminus b$.
A pre-gap is a pair of sequences $g=\{ (a_i\mid i \in I), (b_j \mid j \in J)\}$ where $I,\; J\subseteq
\omega_1$ are uncountable and $a_i,b_j\in [\omega ]^\omega$ are such that $$a_{i_0} \subseteq^* a_{i_1} \subseteq^* b_{j_1}\subseteq^* b_{j_0}$$ whenever $i_0 <i_1$ are in $I$ and $j_0 < j_1$ in $J$. In most cases $I=J=\omega_1$. Given a pre-gap as above, and uncountable subsets $I'\subseteq I$ and $J'\subseteq J$, the restriction $g\restriction(I',J')$ of $g$ is the pre-gap $\{ (a_i\mid i \in I'), (b_j \mid j \in J')\}$. We write $g\restriction I$ for $g\restriction(I,I)$.
An interpolation for a pre-gap $g$ is a set $x\subseteq \omega$ such that $$a_i \subseteq^* x \subseteq^* a_j$$ for every $i$ and $j$. A pre-gap with no interpolation is called a gap. A famous construction of Hausdorff produces gaps in ZFC (which are now called Hausdorff gaps). Specifically, a Hausdorff gap is a pre-gap $g =\{ (a_i\mid i \in \omega_1),
(b_j \mid j \in \omega_1)\}$ such that for every $\alpha \in \omega_1$ and $n\in \omega$ the set $$\{ \beta \in \alpha \mid a_\beta \setminus n \subseteq b_\alpha \}$$ is finite. A [*special*]{} (or Kunen) gap is a pre-gap $g =\{ (a_i\mid i \in \omega_1), (b_j \mid j \in \omega_1)\}$ such that for some $n_0\in \omega$:
1. $a_\alpha \setminus n_0 \subseteq b_\alpha$ for every $\alpha \in
\omega_1$, and
2. for all $\alpha < \beta < \omega_1$, $$(a_\alpha \cup a_\beta)\setminus n_0 \not \subseteq b_\alpha \cap
b_\beta$$ (equivalently, $a_\alpha\setminus n_0 \not \subseteq b_\beta$ or $a_\beta\setminus n_0 \not \subseteq b_\alpha$).
The interest in these definitions arises from the fact (not too difficult to prove) that these Hausdorff and Kunen pre-gaps are gaps and remain gaps as long as $\omega_1$ is not collapsed: they have no interpolation in any extension in which $\omega_1$ remains uncountable.
In this paper we define two additional types of “special” gaps: $S$-Hausdorff gaps where $S\subseteq \omega_1$ is a stationary set, and $C$-Hausdorff gaps, where $C$ is a ladder system over $S$.
The motivation for this work is the desire to find an example with gaps of the phenomenon in which $\omega_1$ is “split” in a certain behavior on a stationary set $S\subset \omega_1$ and an opposite behavior on its complement $\omega_1 \setminus S$.
Let $S\subseteq \omega_1$ be a stationary set. A pre-gap $g =\{ (a_i\mid i \in \omega_1), (b_j \mid j \in \omega_1)\}$ is $S$-Hausdorff iff for some closed unbounded (club) set $D \subseteq \omega_1$, for every $\delta \in S \cap D$ and for every sequence of ordinals $(i_n \in \delta \mid n \in \omega )$ increasing and cofinal in $\delta$ $$\label{Lim}
\lim_{n\to \infty} X(a_{i_n},b_\delta ) = \infty.$$ That is, for every $k$, there is only a finite number of $n\in \omega$ for which $a_{i_n} \setminus k \subseteq b_\delta$. Since, for $\delta < \delta'$, $b_{\delta'} \subseteq^* b_\delta$, it follows that eventually $X(a_{i_n},b_\delta) \leq X(a_{i_n}, b_{\delta'})$, and hence (\[Lim\]) holds for every $\delta' \geq \delta$ in $\omega_1$. If the pre-gap $g =\{ (a_i\mid i \in I), (b_j \mid j \in J)\}$ is defined only on uncountable sets $I,J \subseteq \omega_1$, we can still define it to be $S$-Hausdorff if for some closed unbounded set $D\subseteq \omega_1$, for every $\delta\in S \cap D$, for every $j\in J \setminus \delta$, and for every increasing sequence of ordinals $ (i_n\in \delta \cap I \mid n \in \omega )$ cofinal in $\delta$, $$\label{Lim2}
\lim_{n \to \infty} X(a_{i_n}, b_j)= \infty$$
Clearly, every Hausdorff gap is an $\omega_1$-Hausdorff gap, and the closed unbounded set $D$ can be taken to be $\omega_1$. The converse of this also holds, in the sense that every $\omega_1$-Hausdorff gap contains a Hausdorff gap. For suppose that $g= \{ (a_i\mid i \in I), (b_j \mid j \in J)\}$ is some $\omega_1$-Hausdorff gap, and let $D\subseteq \omega_1$ be the closed unbounded set given by the definition of $g$ as an $\omega_1$-Hausdorff gap. Define $I' \subseteq I$ such that every two members of $I'$ contain a point from $D$ in between. We claim that $g' = \{ (a_i\mid i \in I'), (b_j \mid j \in J)\}$, is a Hausdorff gap. Indeed, if $\alpha \in J$ then for every $n\in
\omega$ the set $E= \{ \beta\in \alpha\cap I' \mid a_\beta \setminus n
\subseteq b_\alpha \}$ is necessarily finite. For if not, then let $\delta$ be an accumulation point of $E$, and let $\beta_i\in E$, for $i\in \omega$, be increasing and converging to $\delta$. Necessarily $\delta \in D$, and $a_{\beta_i} \setminus n \subseteq b_\alpha$ shows that (\[Lim2\]) does not hold.
\[StatGap\] If $S\subseteq \omega_1$ is stationary, then any $S$-Hausdorff pre-gap is a gap. (So that any pre-gap containing an $S$-Hausdorff pre-gap is a gap.)
[**Proof.**]{} Assume that this not so and let $x$ be an interpolation of an $S$-Hausdorff pre-gap $g= \{ (a_i\mid i \in I), (b_j \mid j \in J)\}$. Then there is a fixed $n_0\in \omega$ such that for unbounded sets of indices $I'\subseteq I$ $J'\subseteq J$, for every $\alpha\in I',\; \beta \in J'$ $$a_\alpha \setminus n_0 \subseteq x \setminus n_0 \subseteq b_\beta.$$ And as a consequence $$\label{E1}
a_\alpha \setminus n_0 \subseteq b_\beta$$ holds. Since $g$ is assumed to be $S$-Hausdorff there exists a closed unbounded set $D\subseteq \omega_1$ as in the definition. We may assume that every $\delta\in D$ is an accumulation point of $I'$. Take now a limit $\delta \in S\cap D$ and any sequence $i_n\in I'\cap \delta$ increasing to $\delta$. Take any $j\in J' \setminus \delta$. Then equation (\[E1\]) implies that $X(a_{i_n},b_\delta)\leq n_0$, which is a contradiction.[[$\dashv$]{}]{}
A stronger notion than that of being $S$-Hausdorff can be defined if the rate at which the sequences in (\[Lim\]) tend to infinity is uniform. For this we must recall the definition of a scale or ladder system on a stationary set.
If $S\subseteq \omega_1$ is a stationary set, then a ladder (system) over $S$ is a sequence $C = ( c_\alpha \mid \alpha \in S\ \mbox{\it is a limit ordinal} )$ such that every $c_\alpha=( c_\alpha(n)\mid n \in \omega )$ is an increasing, cofinal in $\alpha$ $\omega$-sequence.
\[DHG\] For a ladder system $C$ over $S$, we say that a pre-gap $g = {\{ (a_i\mid i \in I), (b_j \mid j \in J)\} }$ is $C$-Hausdorff iff for some closed-unbounded set $D\subseteq \omega_1$ for all $\delta \in S\cap D$ and $j\in J
\setminus \delta$ there is $k\in \omega$ such that for every $n\geq k$ in $\omega$, if $i \in I\cap (\delta \setminus c_\delta(n))$, then $X(a_i, b_j)> n$.
Every $C$-Hausdorff gap (where $C$ is a ladder system over a stationary set $S$) is $S$-Hausdorff. Our aim in this paper is to prove the following consistency result.
Assume G.C.H for simplicity. Suppose that $\kappa$ is a cardinal such that ${\mbox{\rm cf}}(\kappa)> \aleph_1$. Let $S$ be a stationary co-stationary subset of $\omega_1$. Then there is a c.c.c poset of size $\kappa$ such that in every generic extension made via $P$ $2^{\aleph_0}=\kappa$ and the following hold.
1. For every ladder system $C$ over $S$, every gap contains a subgap that is $C$-Hausdorff.
2. For every ladder system $E$ over $\omega_1\setminus S$ there is a gap $g$ with no subgap that is $E$-Hausdorff.
Gaps introduced by forcing {#GF}
==========================
Gaps can be created by forcing with finite conditions (a method due to Hechler [@Hechler]). These gaps are not $S$-Hausdorff for any stationary set, as we are going to see.
If $f\in 2^n$ ($f$ is a function defined on $n$ with range included in $\{ 0,1 \}$) then $f$ is a characteristic function and we let $[ f ] = \{ k \mid f(k) =1 \}$ be the subset of $n$ represented by $f$.
Let $(I,<_I)$ be any ordering isomorphic to $\omega_1 + \omega_1^\ast$. For example take $I = (\omega_1 \times \{ 0 \}) \cup (\omega_1 \times\{ 1 \})$ with $\langle \alpha,0\rangle <_I \langle \beta,0\rangle
<_I \langle \beta,1 \rangle <_I \langle \alpha, 1 \rangle$ whenever $\alpha < \beta < \omega_1$.
Define the poset $P$ by $p \in P$ iff $p$ is a finite function defined on $I$ and such that:
1. For some $n$ (called the “height” of $p$) $p(i) \in 2^n$ for every $i\in
{\mbox{\rm dom}}(p)$. (The height of the empty function is defined to be $0$.)
2. For every $\alpha \in \omega_1$, $\langle \alpha, 0 \rangle \in{\mbox{\rm dom}}(p)$ iff $\langle \alpha, 1 \rangle \in{\mbox{\rm dom}}(p)$, and in this case $[
p(\langle \alpha, 0 \rangle) ] \subseteq [ p(\langle \alpha, 1
\rangle) ]$.
The intuition behind this definition is that for $\alpha \in \omega_1$, $p(\langle \alpha,0\rangle)$ will “grow” to become $a_\alpha$, and $p(\langle \alpha,
1\rangle)$ will finally become $b_\alpha$, as $p$ runs over the generic filter. So that $(\langle a_\alpha \mid
\alpha \in \omega_1\rangle, \langle b_\alpha\mid \alpha\in \omega_1\rangle)$ will be the generic gap with the additional property that $a_\alpha
\subseteq b_\alpha$ for every $\alpha$. The ordering of $P$ reflects this intuition as follows.
For $p_1,p_2\in P$ define $p_1 \leq p_2$ ($p_2$ extends $p_1$) iff
1. $d_1 = {\mbox{\rm dom}}(p_1) \subseteq d_2={\mbox{\rm dom}}(p_2)$, and for every $i\in d_1$, $p_1(i) \subseteq p_2(i)$ (so ${\mbox{\rm height}}(p_1)\leq {\mbox{\rm height}}(p_2)$).
2. For every $i,j\in {\mbox{\rm dom}}(p_1)$, if $i <_I j$ then $$[p_2(i)] \setminus [ p_1(i)] \subseteq [ p_2(j)].$$
It is easy to see that any condition in $P$ has extensions with arbitrarily large height and with domains that extend arbitrarily over $I$. In fact, given $i\in {\mbox{\rm dom}}(p)$ and $k\in \omega$ above height $p$, we can require that the extension $p'$ puts $k$ in $[p'(i)]$.
If $\alpha \in \omega_1$, we can write $\alpha \in {\mbox{\rm dom}}(p)$ instead of $\langle \alpha, 0\rangle\in {\mbox{\rm dom}}(p)$ (which is equivalent to $\langle \alpha, 1\rangle\in {\mbox{\rm dom}}(p)$). So ${\mbox{\rm dom}}(p)$ has two meanings, and the context decides if it means a set of ordinals or a set of pairs.
Suppose that $A \subseteq I$ is such that $\langle \alpha, 0 \rangle\in A$ iff $\langle \alpha, 1 \rangle\in A$. Let $P_A$ be the subposet of $P$ consisting of all conditions $p$ such that ${\mbox{\rm dom}}(p) \subseteq A$. If $p\in P$ then $p\restriction A \in P_A$ and $p\restriction A \leq p$. We prove some additional properties of this restriction map taking $p$ to $p\restriction A$.
In the definition of $p\leq q$ what really counts is the restriction of $q$ to the domain of $p$. That is, $p \leq q $ iff $p \leq q \restriction {\mbox{\rm dom}}(p)$. It follows that $p \leq q$ implies that $p \restriction A \leq q
\restriction A$. It also follows that if $p$ and $q$ are conditions such that for $C ={\mbox{\rm dom}}(p) \cap {\mbox{\rm dom}}(q)$, $p\restriction C = q \restriction C$, then $p$ and $q$ are compatible. In fact, in this case, $p\cup q$ is the minimal extension of $p$ and $q$.
Suppose that ${\mbox{\rm dom}}(p) = {\mbox{\rm dom}}(q)$. Then $p$ and $q$ are compatible in $P$ iff $p \leq q$ or $q \leq p$.
For compatible conditions $p$ and $q$, we define a canonical extension $p\vee q$ of both $p$ and $q$. However, $P$ is not a lattice and $p\vee q$ is not the minimum of all extensions of $p$ and $q$. To define it, we first make an observation. Consider $C = {\mbox{\rm dom}}(p) \cap {\mbox{\rm dom}}(q)$. Then $p\restriction C$ and $q\restriction C$ are comparable in $P_C$ (since they are compatible and have the same domain), and hence we can assume without loss of generality that $q\geq p\restriction A$ and $n={\mbox{\rm height}}(q)\geq m = {\mbox{\rm height}}(p)$ where $A = {\mbox{\rm dom}}(q)$ (the restriction on the heights is needed only in case $p\restriction A = \emptyset$ since it follows from $q\geq p\restriction A$ otherwise). Then $r=p \vee q$ is defined as follows on ${\mbox{\rm dom}}(p) \cup {\mbox{\rm dom}}(q)$, and it will be evident that $p \vee q$ is an extension of $p$ and $q$.
For $i\in {\mbox{\rm dom}}(q)$ define $r(i) = q(i)$. For $i\in {\mbox{\rm dom}}(p) \setminus A$ define $r(i)\in 2^n$ by the following two conditions: $$p(i) \subseteq r(i).$$ $$\label{cc}
[ r(i)] \setminus [ p(i) ] = \bigcup \{ [q(k)]\setminus m \mid k <_I i\ \mbox{and } k
\in A \cap {\mbox{\rm dom}}(p) \}.$$ This definition makes sense since $A\cap {\mbox{\rm dom}}(p) \subseteq {\mbox{\rm dom}}(q)$.
It is clear that $r\in P$, ${\mbox{\rm dom}}(r)={\mbox{\rm dom}}(p) \cup {\mbox{\rm dom}}(q)$ and $r\restriction A = q$. We prove that $r \geq p$. Clause 1 in the definition of extension is obvious, and we have to check clause 2. Suppose that $i,j \in
{\mbox{\rm dom}}(p)$ and $i <_I j$. We have to show that $$\label{aim}
[r(i)] \setminus [p(i)] \subseteq [r(j)].$$ So consider any $a \in [r(i)] \setminus [p(i)]$.
[**Case 1:**]{} $i\in A$. Then $r(i) = q(i)$. If $j\in A$ as well, then (\[aim\]) follows from our assumption that $q\geq p\restriction A$, and since $r(i) = q(i)$, $r(j)
= q(j)$ in this case. If, on the other hand, $j \not \in A$, then $$[r(j) ] \setminus [p(j)] = \bigcup \{ [q(k)]\setminus m \mid k <_I j\ \mbox{ and } k
\in A \cap {\mbox{\rm dom}}(p)\}$$ by the definition of $r$. Since $i\in A \cap {\mbox{\rm dom}}(p)$, $i<_I j$, and $a\in q(i) \setminus m$, $a \in [ r(j) ] \setminus [ p(j) ]$ as required.
[**Case 2:**]{} $i \not \in A$. Then $i \in {\mbox{\rm dom}}(p) \setminus A$ and (\[cc\]) implies that for some $k\in A\cap {\mbox{\rm dom}}(p)$ such that $k <_I i$, $a \in
[ q(k) ]\setminus m$. Then $k <_I j$, both indices are in ${\mbox{\rm dom}}(p)$, and $k \in A$, which brings us back to Case 1. [[$\dashv$]{}]{}
This argument has the following corollary.
Suppose that $p_1,p_2\in P$ and $C={\mbox{\rm dom}}(p_1) \cap {\mbox{\rm dom}}(p_2)$ are such that $p_1\restriction C \geq p_2\restriction C$ and ${\mbox{\rm height}}(p_1)\geq {\mbox{\rm height}}(p_2)$. Then $p_1
\vee p_2$ can be formed (an extension of $p_1$ and $p_2$).
[**Proof.**]{} Define $A = {\mbox{\rm dom}}(p_1)$. Then $p_1 \geq p_2\restriction A$ (because $p_1 \geq p_1\restriction C \geq p_2\restriction C = p_2\restriction A$). So $r = p_1 \vee p_2$ can be formed.[[$\dashv$]{}]{}
$P$ satisfies the c.c.c. In fact if $\{ p_\alpha \mid \alpha\in S\} \subseteq
P$ where $S\subseteq \omega_1$ is stationary, then for some stationary set $S'\subseteq S$, every finite set of conditions in $\{ p_\alpha \mid \alpha
\in S'\}$ is compatible. (This is Talayaco’s condition [@Talayaco].)
[**Proof.**]{} If $p,q\in P$ have the same height and for $C={\mbox{\rm dom}}(p) \cap {\mbox{\rm dom}}(q)$ it happens that $p\restriction C= q\restriction C$, then $p\cup q$ is an extension of $p$ and $q$. Hence a $\Delta$-system argument works here.[[$\dashv$]{}]{}
If $G\subset P$ is some generic filter over $P$, define for every $\alpha \in \omega_1$ $a_\alpha = \bigcup \{ [ p(\langle\alpha,0
\rangle)] \mid p \in G \}$, and $b_\alpha = \bigcup \{ [
p(\langle\alpha,1\rangle)] \mid p \in G \}$. A standard density argument shows that $g$ is a pre-gap, and we denote it as $g$.
\[Gap\] The generic pre-gap $g$ is a gap.
[**Proof.**]{} Suppose that $x\in V^P$ is a name, forced to be an interpolation for the generic pre-gap $g$. For every $\alpha \in \omega_1$ find a condition $p_\alpha\in P$ and a number $n_\alpha\in \omega$ such that $$\label{PN}
p_\alpha {{\: \Vdash_P\:}}a_\alpha \setminus n_\alpha \subseteq x\setminus n_\alpha
\subseteq b_\alpha.$$ Then for some stationary set $S\subseteq \omega_1$, and some fixed $n\in
\omega$, $n=n_\alpha$ for every $\alpha \in S$, and the sets ${\mbox{\rm dom}}(p_\alpha)$ form a $\Delta$-system with core $C\subset I$ (a finite set). We also assume that $p_\alpha \restriction C$ is fixed for $\alpha \in
S$. For $\alpha < \beta$, both in $S$ and above the ordinals involved in $C$, consider $p_\alpha$ and $ p_\beta$. Pick any $k\geq n$ such that $k\geq {\mbox{\rm height}}(p_\alpha)$ as well. Let $i = \langle \alpha , 0 \rangle$, and $j=\langle \beta, 1
\rangle$. We shall find an extension $r$ of $p_\alpha$ and $p_\beta$ such that $r(i)(k) = 1$ and $r(j)(k) = 0$. Then $r{{\: \Vdash\:}}k \in
a_\alpha \wedge k \not \in b_\beta$. But this contradicts (\[PN\]).
To define $r$, define first an extension $p'_\alpha \geq p_\alpha$ by requiring that $p'_\alpha(i)(k) = 1$ and $[p'_\alpha(\langle \gamma,0\rangle)] =
[p_\alpha(\langle \gamma,0\rangle)]$ for every $\langle \gamma,0\rangle \in C$. This is possible since $i$ is never $<_I$ below $\langle \gamma, 0 \rangle\in C$. Now $p'_\alpha$ extends $p_\beta\restriction C$ and hence $r = p'_\alpha \vee
p_\beta$ can be formed. Since the only members of $C$ below $j$ (in $<_I$) are of the form $\langle \gamma,0\rangle$, it follows that $[r(j)]=[p_\beta(j)]$. Thus $r(j)(k)=0$. [[$\dashv$]{}]{}
The following lemma implies that if $G$ is a $(V,P)$-generic filter, $g$ the generic gap, and $U\in V[G]$ is any stationary subset of $\omega_1$ in the extension, then no uncountable restriction of $g$ is $U$-Hausdorff.
\[Prop\] The following holds in $V^P$ for the generic gap $g=
\{ (a_i \mid i \in \omega_1), (b_j\mid j \in \omega_1) \}$. If $J, K \subseteq \omega_1$ are unbounded, then there is a club set $D_0\subseteq \omega_1$ such that for every $\delta \in D_0 $ and $k\in K\setminus \delta$ there are $m\in \omega$ and a sequence $j(n)\in \delta \cap J$ increasing and cofinal in $\delta$ such that $a_{j(n)}\setminus m \subset b_k$ for all $n\in \omega$.
[**Prof.**]{} Let $J, K \in V^P$ be names forced by every condition in $P$ to be unbounded subsets of $\omega_1$. Define in $V^P$ the following set $D_0\subseteq \omega_1$: $\delta \in D_0$ if and only if $\delta\in \omega_1$ is a limit ordinal such that:
$$\begin{minipage}[t]{110mm}
\it for all $k \in K \setminus \delta$
{ there is some $m\in \omega$ and an increasing, cofinal in }
$\delta$ { sequence }
$j(n)\in \delta \cap J$ \mbox{ with } $a_{j(n)}\setminus m \subseteq b_k$.
\end{minipage}$$
We want to prove that $D_0$ contains a closed unbounded subset of $\omega_1$, and assume that it does not. So $R=\omega_1 \setminus D_0$ is (forced by some condition to be) stationary in $V^P$, and hence the set, defined in $V$, of ordinals that are potentially in $R$ is stationary in $V$. Namely, the set $R_0\subset \omega_1$ of ordinals forced by some condition to be in $R$ is stationary. For every $\delta\in
R_0$ pick a condition $p_\delta$ that forces $\delta \not \in D_0$. By extending $p_\delta$ we can find some $k_\delta\geq \delta$ such that $$p_\delta{{\: \Vdash_P\:}}k_\delta \in K\ \mbox{shows that } \delta \not
\in D_0.$$ By extending $p_\delta$ again, we can find some $j_\delta\in
\omega_1 \setminus \delta$ forced by $p_\delta$ to be in $J$ (which is possible since $J$ is supposed to be unbounded in $\omega_1$). If necessary, a further extension ensures that both $j_\delta$ and $k_\delta$ are in the domain of $p_\delta$. Now there exists some $m = m_\delta \in \omega$ such that $p_\delta {{\: \Vdash\:}}\; a_{j_\delta} \setminus m \subseteq b_{k_\delta}$ (the height of $p_\delta$ will do). We can extend $p_\delta$ once again and find $f(\delta) < \delta$ such that $$\label{FEq}
p_\delta {{\: \Vdash_P\:}}\mbox{ there is no } j \in J,\ f(\delta)< j <
\delta,\ \mbox{for which } a_j \setminus m \subseteq b_{k_\delta}.$$
We may assume that, for a stationary set $T\subseteq R_0$, the domains of $p_\alpha$, for $\alpha \in T$, form a $\Delta$ system, that they all have the same height, say $n$, and the same restriction to the core. We also assume that the functions $p_\alpha(\langle j_\alpha, 0 \rangle): n \to \{ 0 , 1 \}$ do not depend on $\alpha$, and that $f(\alpha)$ and $m= m_\alpha$ are fixed on $T$ ($m\leq n$). Now by Talayaco’s chain condition for $P$, there is a stationary $T'\subseteq T$ such that for every $\alpha,\beta\in T'$, $p_\alpha \vee p_\beta$ is a common extension. Pick some $\alpha \in T'$ that is an accumulation point of $T'$ (and such that for every $\beta < \alpha$, $j_\beta < \alpha$). Then find $\beta<\alpha$, $\beta\in T'$ such that $f(\alpha) < \beta$. Then (as we shall see) $$p_\beta\vee p_\alpha {{\: \Vdash\:}}a_{j_\beta} \subseteq a_{j_\alpha},\
j_\beta\in J,\
\mbox{\it and } \alpha > j_\beta > f(\alpha).$$ Yet $$p_\alpha {{\: \Vdash\:}}\; a_{j_\alpha} \setminus m \subseteq b_{k_\alpha},$$ and this is a contradiction to (\[FEq\]). Why does $p_\beta \vee p_\alpha$ force $a_{j_\beta} \subset a_{j_\alpha}$? Because the functions $p_\alpha(\langle j_\alpha,0 \rangle)$ and $p_\beta(\langle j_\beta,0 \rangle)$ are the same, they describe $a_{j_\alpha}\cap n$ and $a_{j_\beta}\cap n$, so $p_\beta\vee p_\alpha$ forces $a_{j_\beta}\subset a_{j_\alpha}$. [[$\dashv$]{}]{}
Specializing pre-gaps on a ladder system
========================================
\[Sp\] For every ladder system $C$ over a stationary set $S\subseteq \omega_1$, and gap $g$, there is a c.c.c forcing notion $Q = Q_{g,C}$ such that in $V^Q$ a restriction of $g$ to some uncountable set is $C$-Hausdorff (and hence $S$-Hausdorff). In fact $Q$ satisfies a stronger property than c.c.c, the polarized chain condition, which we shall define later.
[**Proof.**]{} Fix for the proof a ladder system $C = \langle c_\delta \mid \delta \in S \rangle$ over a stationary set $S
\subseteq \omega_1$ consisting of limit ordinals, and a pre-gap $g={\{ (a_i\mid i \in \omega_1), (b_j \mid j \in \omega_1)\} }$. The forcing poset $Q =Q_{g,C}$ defined below is designed to make an uncountable restriction of $g$ into a $C$-Hausdorff gap.
Define $p\in Q$ iff $p=(w,s)$ where
1. $w\in [ \omega_1 ] ^{<\aleph_0}$ (i.e. a finite subset of $\omega_1$), and
2. $s\in [ S ] ^{<\aleph_0}$.
If $p\in Q$ then we write $p=(w^p,s^p)$ for the two components of $p$.
The ordering $p\leq q$ ($q$ extends $p$) is defined by
a.
: $w^p \subseteq w^q$, $s^p\subseteq s^q$, and
b.
: If $\delta \in s^p$ and $i\in w^p$ are such that $\delta \leq i$, then for every $j\in (w^q\setminus w^p) \cap \delta$,
$$a_j \setminus \mid c_\delta \cap j\mid\; \not \subseteq b_i.$$
Or, equivalently, $X(a_j,b_i) > |c_\delta \cap j |$. It is easy to check that this is indeed an ordering defined on $Q$.
If $G$ is generic over $Q$, define $W = \bigcup \{ w \mid \exists s (w,s)\in G \}$. We will prove that if $g$ is a gap then $Q$ satisfy the c.c.c. So $\omega_1$ is preserved. Clearly, if $p=(w,s)$ is a condition, then for any $\sigma\in S$, $(w,s\cup\{ \sigma\})$ extends $p$, and if $j\in \omega_1$ and $j>\max(w)$, then $(w\cup\{ j \},s)$ extends $p$. (If, however, $j < \max(w)$, then $(w\cup \{ j \}, s)$ may be incompatible with $(w,s)$.) It follows that $W$ is unbounded in $\omega_1$ and $\{ ( a_i \mid i \in W), (b_j \mid j \in
W)\}$ is $C$-Hausdorff.
So the generic filter over $Q$ selects an unbounded in $\omega_1$ restriction of $g$ that is $C$-Hausdorff.
If $p=(w,s)$ is a condition then for every $\alpha\in \omega_1$ the restriction $p\restriction \alpha = ( w\cap \alpha,
s\cap \alpha)$ is defined. Clearly $p\restriction \alpha \leq p$.
If $p=(w,s)$ and $q=(v,r)$ are conditions in $Q$ then define $p\cup q=(w\cup v, s\cup r)$. If $p$ and $q$ are compatible in $Q$, then $p\cup q\in Q$ is the least upper bound of $p$ and $q$.
The following lemma describes a situation in which the compatibility of $p_1$ and $p_2$ can be deduced. This is the situation resulting when $p_1$ and $p_2$ come from a $\Delta$-system, with core fixed below $\gamma$, and such that $p_1$ is bounded by some $\alpha$ such that the domain of $p_2$ has empty intersection with the ordinal interval $[\gamma,\alpha]$. The proof is straightforwards.
\[l32\] Suppose that
1. $p_1=(w_1,s_1)$ and $p_2=(w_2,s_2)$ are in $P$.
2. $\gamma < \alpha < \omega_1$ are such that
1. $ w_1 \subseteq \alpha,$ and $p_1\restriction \gamma$ is compatible with $p_2$.
2. $ w_2 \cap \alpha \subset \gamma$, and $s_2 \cap (\alpha + 1) \subset \gamma$. $p_2\restriction \alpha = p_2 \restriction \gamma$ is compatible with $p_1$.
3. Define $$A = \bigcap \{ a_i \mid i \in w_1 \setminus \gamma \}$$ $$B = \bigcup \{ b_j \mid j \in w_2 \setminus \gamma \}$$ and suppose that there is $n \in A \setminus B$ such that, for every $\delta \in s_2 \setminus \alpha$, $n > | c_\delta \cap
\alpha|$.
Then $p_1$ and $p_2$ are compatible.
[**Proof.**]{} Form $p = p_1 \cup p_2$ and prove that $p_1, p_2 \leq p$. $p_1\leq
p$ is immediate. As for $p_2 \leq p$, observe that $X(a_i,b_j) > | c_\delta
\cap \alpha |$ for every $i \in w _1 \setminus \gamma$, $j \in w_2\setminus
\gamma$, and $\delta \in s_2 \setminus \gamma$. [[$\dashv$]{}]{}
The following simple lemma is used in proving that $Q$ is a c.c.c poset.
\[simp\] Suppose $g=\{ (A_i\mid i \in I), ( B_j\mid j \in J) \}$ is a pre-gap such that for every $i\in I$ and $j\in J$, $i<j$ implies that $A_i\subseteq
B_j$. Then $g$ is not a gap.
[**Proof.**]{} By throwing away a countable set of indices from $J$ we can assume for every $n\in \omega$ that if $n\not\in B_j$ for some $j$, then $n\not \in
B_j$ for uncountably many $j$’s. Define then $x=\bigcup_{i\in I} A_i$. Then $x\subseteq B_j$ for every $j$, because otherwise there are some $i\in I$, $j\in J$, and $n\in \omega$ such that $n\in A_i\setminus B_j$. But then we may find uncountably many indices $j'$ such that $n\not\in B_{j'}$ and in particular there is such $j' > i$. Thence $A_i\not\subseteq B_{j'}$, contradicting our assumption. [[$\dashv$]{}]{}
\[PCC\] Suppose that the domain of our ladder system $C$, namely $S$, is co-stationary.
1. If $g$ is a gap then $Q=Q_{g,C}$ satisfies the c.c.c.
2. Suppose that $T_1$, $T_2 \subseteq \omega_1 \setminus S$ are stationary sets and ${{\overline}p}= ( p^\ell_\delta \mid \delta \in T_\ell)$, for $\ell =
1,2$ , are two sequences of conditions in $Q$ such that, for some fixed $p^*\in
Q$, $p^* \geq p_\delta^1\restriction \delta ,\ p^2_\mu\restriction \mu$, for every $\delta \in T_1$ and $\mu \in T_2$, is such that $p^*$ is compatible with every $p_\delta^1$ and with every $p^2_\mu\restriction \mu$. Then there are stationary subsets $T_1'\subseteq T_1$ and $T_2'\subseteq T_2$ such that, for every $\alpha_1\in T'_1$ and $\alpha_2\in T'_2$, if $\alpha_1 < \alpha_2$ then $p^1_{\alpha_1}$ and $p^2_{\alpha_2}$ are compatible in $Q$.
[**Proof.**]{} We prove [*2*]{} since the proof of [*1*]{} is similar. For any condition $p=(w,s)$ define ${\mbox{\rm dom}}(p)=w\cup s$. Suppose that ${\mbox{\rm dom}}(p^*)\subseteq \gamma$. Then ${\mbox{\rm dom}}(p_\delta^1) \cap \delta \subseteq \gamma$, and ${\mbox{\rm dom}}(p_\mu^2)\cap \mu
\subseteq \gamma$, for every $\delta \in T_1$ and $\mu \in T_2$. We may assume that if $i<j$ then ${\mbox{\rm dom}}(p^\ell_i) \subset \cap\;
({\mbox{\rm dom}}(p^m_j)\setminus \gamma)$ for $\ell,m\in\{1,2\}$.
Since $\delta \in T_\ell$ implies that $\delta \not\in S$, it follows for $p_\delta^\ell = (w,s)$ that the ladder sequence $c_i$ for any $i \in s$ is bounded below $\delta$. So the finite union $$\delta\cap \bigcup\{ c_i\mid i \in s^{p^\ell_\delta} \setminus \delta \}$$ is bounded below $\delta$. Using Fodor’s lemma we may even assume that this intersection is bounded below $\gamma$ (extend $\gamma$ if necessary) and has a fixed finite cardinality.
For every $\delta \in T_1$ define $$A_\delta = \bigcap \{ a_i \mid i\in w^{ p^1_\delta}\setminus
\gamma\}$$ Similarly, for $\delta \in T_2$ define $$B_\delta = \bigcup\{ b_i \mid i\in w^{ p^2_\delta}\setminus
\gamma\}.$$ Clearly, any interpolation for $G=\{ (A_\delta\mid \delta\in T_1\},\{ B_\delta\mid \delta\in
T_2\})$ is also an interpolation for $g$, and hence $G$ is a gap.
Let $k\in \omega$ be such that for every $\delta\in T_\ell$, if $p^\ell_\delta=(w,s)$ and $\alpha\in s\setminus \gamma$, then $\mid
c_\alpha\cap \delta\mid < k$.
Now we find a stationary set $T'_1\subset T_1$ such that for every $n\in
\omega$ if $n\in A_\delta$ for some $\delta \in T'_1$ then $n\in
A_\delta$ for a stationary set of $\delta$’s in $T'_1$. Simply throw away countably many non-stationary sets from $T_1$. Similarly, find a stationary $T'_2\subseteq T_2$ such that if $n\not \in B_\delta$ for some $\delta\in T'_2$ then $n\not \in B_\delta$ for a stationary set of $\delta\in T'_2$.
Now lemma \[simp\] gives $\alpha_1\in T'_1$ and $\alpha_2\in T'_2$ with $\alpha_1<\alpha_2$ such that $A_{\alpha_1}\setminus k \not \subseteq
B_{\alpha_2}$. If we pick $n\in A_{\alpha_1}\setminus B_{\alpha_2}$ such that $n\geq k$ then there are stationary sets $T^{''}_1\subseteq T'_1$ and $T^{''}_2\subseteq T'_2$ such that $n\in A_{\alpha_1}\setminus B_{\alpha_2}$ [*for every*]{} $\alpha_1\in T^{''}_1$ and $\alpha_2\in T^{''}_2$. Hence if $\alpha_1\in T^{''}_1$, $\alpha_2\in T^{''}_2$, and $\alpha_1<\alpha_2$, then $p^1_{\alpha_1}$ and $p^2_{\alpha_2}$ are compatible in $Q$ by lemma \[l32\].[[$\dashv$]{}]{}
Polarized chain condition
-------------------------
Theorem \[PCC\] shows that the poset $Q_{g,C}$ for a gap $g$ and ladder $C$ over a stationary co-stationary set $S$ satisfies some kind of a chain condition, suitable for two sequences indexed by stationary subsets of $\omega_1\setminus S$. We formulate this condition in general and later prove that the iteration with finite support preserves this condition.
Let $T\subseteq \omega_1$ be a stationary set. A c.c.c poset $P$ satisfies the polarized chain condition ([*p.c.c*]{}) for $T$ if it satisfies the following requirement. Suppose that
1. $${{\overline}p}^\ell=(p^\ell_\delta\mid \delta\in T_\ell)\ \mbox{for } \ell
= 1,2$$ are two sequences of conditions in $P$, where $T_\ell\subseteq T$ are stationary for $\ell = 1,2$.
2. $p^*\in P$ is such that for each $\ell = 1,2$ $$p^*{{\: \Vdash_P\:}}\{ \delta\in T_\ell\mid p^\ell_\delta\in G_P\}\ \mbox{is
stationary in} \ \omega_1,$$ where $G_P$ is the name of the generic filter over $P$.
Then there are stationary sets $T'_\ell\subseteq T_\ell$ for $\ell=1,2$ such that $p^1_{\alpha_1}$ and $p^2_{\alpha_2}$ are compatible in $P$ whenever $\alpha_1< \alpha_2$ are in $T'_1$ and $T'_2$ respectively.
We want to prove that if $g$ is a gap and $C$ a ladder over a stationary set $S$ such that $T=\omega_1 \setminus S$ is also stationary, then $Q=Q_{g,C}$ satisfies the p.c.c. for $T$. The problem is that if $p^*$ is as in the p.c.c. definition then it is not necessarily of the form to which theorem \[PCC\] is applicable, and so we need some argument to deduce that $Q$ is p.c.c.
Recall that every club (closed unbounded) subset of $\omega_1$ in a generic extension of $V$ made via a c.c.c poset contains a club subset in $V$. The following property of c.c.c posets is also needed.
\[CCC\] Let $P$ be a c.c.c poset. Suppose that $T\subseteq \omega_1$ is stationary, and $\langle p_\alpha \mid \alpha \in T
\rangle$ is a sequence of conditions in $P$ indexed along $T$. Then there exists some $p_\alpha$ such that $$p_\alpha {{\: \Vdash\:}}\{ \beta \in T \mid p_\beta \in G \}\ \mbox{is
stationary}.$$ In fact, the set of these $\alpha$’s is stationary in $\omega_1$.
[**Prof.**]{} Assume that this is not the case and, for some club $D\subseteq \omega_1$, for every $\alpha \in T\cap D$ there is a club set $C_\alpha$ (necessarily in $V$) and an extension $p_\alpha'$ of $p_\alpha$ such that $$\label{Cl}
p_\alpha' {{\: \Vdash\:}}\beta \in C_\alpha \cap T \rightarrow p_\beta \not
\in G.$$
Let $C=\{ \beta\in \omega_1\mid (\forall \alpha < \beta)
\beta \in C_\alpha\}$ be the diagonal intersection of these club sets. Then $C$ is closed unbounded in $\omega_1$. Take a maximal antichain (surely countable) from the set of extensions $\{ p_\alpha' \mid \alpha\in T\cap D \}$, and let $\alpha_0$ be an index in $T\cap C \cap D$ higher than all indexes of this countable antichain. Then $p'_{\alpha_0}$ is compatible with some $p'_\alpha$ with $\alpha < \alpha_0$. But $\alpha_0\in C_\alpha$ which leads to a contradiction since $p'_{\alpha_0}$ forces that $p_{\alpha_0}\in G$, and $p'_\alpha$ forces that $p_{\alpha_0}\not \in G$ (by \[Cl\]).[[$\dashv$]{}]{}
Now we prove that $Q$ is p.c.c. for $T = \omega_1 \setminus S$.
\[LCC\] If $C$ is a ladder system over $S$, and $T=\omega_1 \setminus S$ is stationary, then, for any gap $g$, $Q_{g,c}$ is p.c.c. over $T$.
[**Proof.**]{} Suppose that $T_1,T_2\subseteq T$ are stationary, and ${{\overline}p}^1$, ${{\overline}p}^2$ are two sequences of conditions indexed along $T_1$ and $T_2$. Let $q^*\in Q$ be such that for $\ell = 1 ,2$ $$\label{EQ}
q^* {{\: \Vdash_Q\:}}\{ \delta \in T_\ell \mid p_\delta^\ell \in G \}\ \mbox{is
stationary in }\ \omega_1.$$
We claim first that we may assume that $q^* \leq p_\delta^1$ for every $\delta\in T_1$. This can be achieved as follows. First, observe that the set of $\delta\in T_1$ for which $p_\delta^1$ and $q^*$ are compatible is stationary. Then use Fodor’s theorem on this stationary set to fix $p_\delta^1\restriction \delta$. Rename $T_1$ to the the resulting stationary set. Redefine $p_\delta^1$ as $p_\delta^1\cup q^*$, and finally apply lemma \[CCC\] to obtain $\delta_0$ such that $$p_{\delta_0}^1 {{\: \Vdash_Q\:}}\{ \delta \in T_1 \mid p_\delta^1\in G \} \
\mbox{is stationary in }\ \omega_1.$$
Now $q^*$ is as required. Since $q^*$ extends the original $q^*$, it still satisfies (\[EQ\]) with respect to $T_2$. Repeat this procedure for $T_2$, and obtain two sequences to which Theorem \[PCC\] is applicable. [[$\dashv$]{}]{}
We note here for a possible future use a stronger form of polarized chain condition (strong-p.c.c) which is not used in this paper.
Let $T\subseteq \omega_1$ be stationary. A poset $P$ is said to satisfy the strong-p.c.c over $T$ if whenever two sequences are given $${{\overline}p}^\ell=(p^\ell_\delta\mid \delta\in T_\ell)\ \mbox{for } \ell
= 1,2$$ of conditions in $P$, where $T_\ell\subseteq T$ are stationary for $\ell = 1,2$, and for some $p^*\in P$, for every $\ell = 1,2$, $$p^*{{\: \Vdash_P\:}}\{ \delta\in T_\ell\mid p^\ell_\delta\in G_P\}\ \mbox{is
stationary in} \ \omega_1,$$ then there are stationary subsets $T_\ell'\subseteq T_\ell$ for $\ell = 1,2$, and conditions $q_\delta \geq p^2_\delta$ for $\delta \in T'_2$ such that:
For every $\delta\in T_2'$ and $q\in P$ such that $q_\delta\leq q$ there exists $\alpha <
\delta$ such that for every $\beta$ that satisfies $\alpha < \beta\in T_1'\cap \delta$ $$q \ \mbox{and}\ p_\beta^1\ \mbox{are compatible in}\ P.$$
Iteration of p.c.c posets
=========================
Our aim in this section is to prove that the iteration with finite support of p.c.c posets is again p.c.c. It is well known (by Martin and Solovay [@MartinSolovay]) that since each of the iterands satisfies the countable chain condition the iteration is again c.c.c, but we have to prove the preservation of the polarized property.
Our posets are separative (and if not, they can be made separative). A poset is separative iff $p \not \leq q$ implies that some extension of $q$ is incompatible with $p$.
\[Pres\] Suppose that $P$ is a p.c.c. poset, and that $Q\in V^P$ is (forced by every condition in $P$ to be) a p.c.c. poset. Then the iteration $P\ast Q$ satisfies the polarized chain condition too.
[**Proof.**]{} Suppose that $(p^\ell_\delta, q^\ell_\delta)\in
P\ast Q$ are given for $\delta \in T_\ell \subseteq T$ and for $\ell = 1,2$, such that for some condition $(p,q)\in P\ast Q$ $$\label{St}
(p,q) {{\: \Vdash\:}}\{ \delta \in T_\ell \mid (p^\ell_\delta, q^\ell_\delta) \in
G_{P\ast Q} \} \ \mbox{is stationary}$$ for $\ell = 1,2$. Then $$p {{\: \Vdash_P\:}}\{ \delta \in T_\ell \mid p^\ell_\delta \in G_P\}\ \mbox{is stationary}.$$ Let $G\subset P$ be $V$-generic, with $p \in G$. In $V[G]$ form the interpretations $q[G] $ (interpretation of $q$) and $Q[G]$ (interpretation of $Q$). Then $q[G] \in Q[G]$. Define the sets $$T'_\ell = \{ \delta \in T_\ell \mid p_\delta^\ell \in G
\},\ \ell = 1,2$$ (which are stationary) and define the sequences $$\langle q^\ell_\delta[G] \mid \delta \in T'_\ell \rangle,\
\mbox{ for } \ell = 1,2 .$$ Then in $V[G]$ $$q[G] {{\: \Vdash\:}}_{Q[G]}\ \{ \delta \in T_\ell ' \mid
q^\ell_\delta [G] \in H \}\ \mbox{ is stationary}$$ where $H$ is the name for the $V[G]$ generic filter over $Q[G]$. (This follows from (\[St\]) and since forcing with $P\ast Q$ is equivalent to the iteration of forcing with $P$ and then with $Q[G]$.)
Since $Q[G]$ satisfies the polarized chain condition for $T$, there are stationary sets $T_\delta^{''}
\subseteq T'_\delta$ such that:
> if $\delta_1\in
> T^{''}_1$, $\delta_2\in T_2^{''}$, and $\delta_1 <
> \delta_2$, then $q^1_{\delta_1}[G]$ and $q^2_{\delta_2}[G]$ are compatible in $Q[G]$.
Back in $V$, let $S_1$ and $S_2$ be $V^P$ names of $T^{''}_1$ and $ T_2^{''}$ respectively, forced to have these properties. The following short lemma will be applied to $S_1$ and to $S_2$.
Suppose that $\langle p_\delta \mid \delta \in T
\rangle$ is a sequence in $P$, $S$ is a name of a subset of $\omega_1$ and $p\in P$ a condition such that $$p {{\: \Vdash\:}}_P\ S \subseteq \{ \alpha \in T \mid
p_\delta \in G \}\ \mbox{and } S\ \mbox{ is stationary in }
\omega_1.$$ Then there is a stationary subset $T^\ast \subseteq
T$, and conditions $p^\ast_\delta$ extending both $p_\delta$ and $ p$ for each $\delta \in T^\ast$ such that $p_\delta^\ast {{\: \Vdash\:}}\delta \in S$.
[**Proof.**]{} Define $T^\ast$ by the condition that $\delta \in T^\ast$ iff $\delta \in T$ and there is a common extension of $p$ and $p_\delta$ that forces $\delta \in S$. We must prove that $T^\ast$ is stationary. If $C\subseteq \omega_1$ is any closed unbounded set, find $p'\geq p$ and $\delta \in C$ such that $p'{{\: \Vdash\:}}\delta \in S$. Then $\delta \in T$ and $p'{{\: \Vdash\:}}\ p_\delta \in G$. Hence $p_\delta \leq p'$ (because $P$ is separative). So $\delta \in T^\ast$.[[$\dashv$]{}]{}
Apply the lemma to $S_1$ and find a stationary set $T^\ast _1\subseteq T_1$ and conditions $p_\delta^{\ast 1 }\geq p_\delta^1, p$, for $\delta \in
T^\ast_1 $ such that $$p_\delta^{\ast 1} {{\: \Vdash\:}}\delta \in S_1.$$ Then (lemma \[CCC\]) find an extension of $p$, denoted $p^\ast$, such that $$p^{\ast} {{\: \Vdash\:}}\{ \delta \in T^\ast_1\mid
p_\delta^{\ast 1} \in G \} \ \mbox{is stationary}.$$ Apply the same argument to $S_2$, and find a stationary set $T^\ast_2\subseteq T_2$ and conditions $p^{\ast 2}_\delta \geq p_\delta^2, p^\ast$ for $\delta \in T_2^\ast$ such that $p^{\ast
2}_\delta {{\: \Vdash\:}}\delta \in S_2$. Now $p^{\ast\ast}\geq
p^\ast$ can be found such that $$p^{\ast\ast}{{\: \Vdash\:}}\{ \delta \in T^\ast_2 \mid p_\delta
^{\ast 2} \in G \} \ \mbox{is stationary}.$$ Since $P$ satisfies the p.c.c., there are stationary sets $T^{\ast\ast}_1 \subseteq T^\ast_1$ and $T^{\ast\ast}_2
\subseteq T^\ast_2$ such that for every $\delta_1 <
\delta_2$ in $T^{\ast\ast}_1$ and $T^{\ast\ast}_2$ (respectively) $p^{\ast 1}_{\delta_1}$ and $p^{\ast
2}_{\delta_2}$ are compatible in $P$, say by some condition $p'$ extending both. But then $p' {{\: \Vdash\:}}\delta_1\in S_1\
\mbox{and } \delta_2 \in S_2$. It follows that $(p^1_{\delta_1},q^1_{\delta_1})$ and $(p^2_{\delta_2},q^2_{\delta_2})$ are compatible in $P\ast Q$ showing that $P\ast Q$ satisfies the p.c.c. The point is that $$p' {{\: \Vdash_P\:}}q^1_{\delta_1}\ \mbox{and}\ q^2_{\delta_2}\ \mbox{are
compatible in}\ Q$$ and hence for some $q'\in V^P$, $p'{{\: \Vdash_P\:}}\ q' \geq
q^1_{\delta_1}, q^2_{\delta_2}$. That is, $(p'q') \geq
(p^1_{\delta_1}, q^1_{\delta_1}),\ (p^2_{\delta_2},
q^2_{\delta_2})$. [[$\dashv$]{}]{}
Let $T$ be a stationary subset of $\omega_1$. An iteration with finite support of p.c.c. for $T$ posets is again p.c.c. for $T$.
[**Proof.**]{} The theorem is proved by induction on the length, $\delta$, of the iteration. For $\delta$ a successor ordinal, this is essentially Lemma \[Pres\]. So we assume that $\delta$ is a limit ordinal, and $\langle P_\alpha \mid \alpha \leq \delta \rangle$ is a finite support iteration, where $P_{\alpha+1}= P_\alpha \ast Q_\alpha$ is obtained with $Q_\alpha \in V^{P_\alpha}$ a p.c.c poset for $T$. Thus conditions in $P_\delta$ are finite functions $p$ defined on a finite subset ${\mbox{\rm dom}}(p)\subset \delta$, and are such that for every $\alpha \in {\mbox{\rm dom}}(p)$, $p\restriction \alpha {{\: \Vdash\:}}_{P_\alpha}\
p(\alpha) \in Q_\alpha$. It is well-known that $P_\delta$ satisfies the c.c.c, and we must prove the polarized property.
Suppose that $\overline{p}^\ell = \langle p_i ^\ell \mid i \in
T_\ell \rangle$ for $\ell = 1,2$ are two sequences of conditions in $P_\delta$, where $T_\ell \subseteq T$ are stationary, and suppose also that $p^\ast\in P_\delta$ is such that $$\label{Stat}
p^\ast {{\: \Vdash\:}}_{P_\delta}\ \{ i \in T_\ell \mid p^\ell_i \in G
\}\ \mbox{is stationary for } \ell = 1,2.$$ We may assume that $p^\ast$ is compatible with every $p^\ell_i$ (just throw away those conditions that are not). The case ${\mbox{\rm cf}}(\delta)>\omega_1$ is trivial, because the support of all conditions is bounded by some $\delta' < \delta$ to which induction is applied). So there are two cases to consider.
[**Case 1:**]{} ${\mbox{\rm cf}}(\delta)=\omega$. Let $\langle \delta_n \mid n
\in \omega \rangle$ be an increasing $\omega$-sequence converging to $\delta$. For every $p_i^\ell$ there is $n\in \omega$ such that ${\mbox{\rm dom}}(p_i^\ell)\subseteq \delta_n$. It follows from (\[Stat\]) that for some specific $n\in \omega$, for some extension $p^{\ast\ast}\geq p^\ast$ $$p^{\ast\ast}{{\: \Vdash\:}}\{ i \in T_\ell \mid p_i^\ell \in P_{\delta_n}\cap G \}\ \mbox{is
stationary for } \ell = 1,2.$$ Now we can apply the inductive assumption to $P_{\delta_n}$.
[**Case 2:**]{} ${\mbox{\rm cf}}(\delta)=\omega_1$. Let $\langle \delta_\alpha
\mid \alpha \in \omega_1\rangle$ be an increasing, continuous, and cofinal in $\delta$ sequence. Intersecting $T_1$ and $T_2$ with a suitable closed unbounded set, we may assume that for every $\alpha< \beta$ $\alpha\in T_1$ and $\beta\in T_2$, ${\mbox{\rm dom}}(p_\alpha^1)\subset \beta$.
We claim that we may without loss of generality assume that, for some $\gamma < \delta$, ${\mbox{\rm dom}}(p_\alpha^\ell)\cap \delta_\alpha$ for all $\alpha \in T_\ell$. We get this in two steps.
In the first step, find a stationary $T_1'\subseteq T_1$ such that the sets ${\mbox{\rm dom}}(p^1_\alpha) \cap \delta_\alpha$, for $\alpha \in T_1'$, are bounded by some $\gamma < \delta$. For each $\alpha\in T'_1$ let $p^{1\ast}_\alpha$ be a common extension of $p^1_\alpha$ and $p^\ast$. Then (use lemma \[CCC\]) find an extension $p^{\ast\ast}\geq p^\ast$ such that $$p^{\ast\ast}{{\: \Vdash\:}}\{ \alpha \in T_1'\mid p_\alpha^1\in G \} \
\mbox{is stationary}.$$ Since $p^{\ast\ast}$ extends $p^\ast$, $p^{\ast\ast}{{\: \Vdash\:}}\{ i \in T_2\mid p_i^2\in G \}\
\mbox{\em is stationary}$. We can again assume that each $p_i^2$ is compatible with $p^{\ast\ast}$ and get $T_2'\subseteq T_2$ stationary such that ${\mbox{\rm dom}}(p^2_\alpha)\cap \delta_\alpha$ is bounded by some $\gamma' < \delta$ (we rename $\gamma$ to be the maximum of $\gamma$ and $\gamma'$). Rename the stationary sets as $T_1$, $T_2$ and we have our assumption.
Apply induction to $P_\gamma$ and to the conditions $p^\ell_\alpha\restriction \gamma$. This yields two stationary subsets which are as required. [[$\dashv$]{}]{}
The model
=========
Assuming the consistency of ZFC, the following property is consistent with ZFC. There is a stationary co-stationary set $S\subseteq \omega_1$ such that
1. For every ladder system $C$ over $S$, every gap contains a $C$-Hausdorff subgap.
2. For every ladder system $H$ over $T = \omega_1\setminus S$ there is a gap $g$ with no subgap that is $H$-Hausdorff.
To obtain the required generic extension we assume that $\kappa$ is a cardinal in $V$ (the ground model) such that ${\mbox{\rm cf}}(\kappa) > \omega_1$ and even $\kappa^{\aleph_1} = \kappa$. We shall obtain a generic extension $V[G]$ in which $2^{\aleph_0}=\kappa$ and the two required properties of the theorem hold. For this we define a finite support iteration of length $\kappa$, iterating posets $P$ as in section \[GF\], which introduce generic gaps, and posets of the form $Q_{g,C}$, as in section \[Sp\], which are designed to introduce a $C$-Hausdorff subgap to $g$.
We denote this iteration $\langle P_\alpha \mid \alpha < \kappa \rangle$. So $P_{\alpha + 1} \simeq P_\alpha \ast R(\alpha)$, where the $\alpha$-th iterand $R(\alpha)$ is either some $P$ or some $Q_{g,C}$. The rules to determine $R(\alpha)$ are specified below. For any limit ordinal $\delta \leq \kappa$, $P_\delta$ is the finite support iteration of the posets $\langle P_\alpha \mid \alpha < \delta \rangle$. We define $P_\kappa$ as our final poset, and we shall prove that in $V^{P_\kappa}$ the two properties of the theorem hold.
Recall that $P$ satisfies Talayaco’s condition and is hence a p.c.c. poset, and each $Q_{g,C}$ is p.c.c. over $T=\omega_1 \setminus S$ (by lemma \[LCC\]).
Since the iterand posets satisfy the p.c.c over $T$, each $P_\alpha$ is a p.c.c. poset over $T$ (and in particular a c.c.c poset). It follows that every ladder system and every gap in $V^{P_\kappa}$ are already in some $V^{P_\alpha}$ for $\alpha < \kappa$. It is obvious that if $g\in V^{P_\alpha}$ is forced by $p\in P_\kappa$ to be a gap in $V^{P_\kappa}$, then $p\restriction \alpha$ forces it to be a gap already in $V^{P_\alpha}$.
To determine the iterands, we assume a standard bookkeeping scheme which ensures two things:
1. For every ladder system $C$ over $S$ and gap $g$ in $V^{P_\kappa}$, there exists a stage $\alpha < \kappa$ so that $C,g \in V^{P_\alpha}$, and $R(\alpha)$ is $Q_{g,C}$.
2. For some unbounded set of ordinals $\alpha \in \kappa$ the iterand $R(\alpha)$ is $P$, producing a generic gap $g$, and the subsequent iterand $R(\alpha+1)$ is $Q_{g,C}$ for some ladder sequence $C$ over $S$.
The first item ensures that, in $V^{P_\kappa}$, for every ladder system $C$ over $S$, every gap contains a $C$-Hausdorff subgap. (A $C$-Hausdorff subgap in $V_{\alpha+1}$ remains $C$-Hausdorff at every later stage and in the final model).
Suppose now that $H$ is a ladder over $T = \omega_1\setminus S$. Then $H$ appears in some $V^{P_\alpha}$ such that $R(\alpha)$ is the poset $P$, and $R(\alpha+1)$ is the poset $Q_{g,C}$ where $g$ is the generic gap introduced by $R(\alpha)$, and $C$ is some ladder sequence over $S$. We want to prove that $g$ is a gap that has no $H$-Hausdorff subgap in $V^{P_\kappa}$. We first prove that $g$ remains a gap in $V^{P_\kappa}$. It is clearly a gap in $V^{P_{\alpha+1} }$ by Lemma \[Gap\]. Since $g$ is $C$-Hausdorff in $V^{P_{\alpha+2} }$, it remains a gap in $V^{P_\kappa}$ (by Lemma \[StatGap\]).
This generic gap $g$ satisfies the conclusion of lemma \[Prop\] in $V^{P_{\alpha+1} }$:
> If $J, K \subseteq \omega_1$ are unbounded, then there is a club set $D_0\subseteq \omega_1$ such that for every $\delta \in D_0 $ and $k\in K\setminus \delta$ there are $m\in \omega$ and a sequence $j(n)\in \delta \cap J$ increasing and cofinal in $\delta$ such that $a_{j(n)}\setminus m \subset b_k$ for all $n\in \omega$.
Since $P_\kappa \simeq P_{\alpha+1} \ast R$, where the remainder $R \simeq
P_\kappa/P_{\alpha+1}$ is interpreted in $V^{P_{\alpha+1}}$ as a finite support iteration of p.c.c. posets over $T$, we can view $P_\kappa$ as a two-stage iteration in which the second stage is a p.c.c. poset over $T$. Thus, for simplicity of expression, we can assume that $V^{P_{\alpha+1}}$ is the ground model. The following lemma then ends the proof.
Suppose in the ground model $V$ a ladder system $H$ over a stationary set $T\subseteq \omega_1$, and a gap $g$ that has the property quoted above (the conclusion of lemma \[Prop\]). Suppose also a poset $R$ that is p.c.c. over $T$. Then in $V^R$ the gap $g$ contains no $H$-Hausdorff subgap.
[**Proof.**]{} Let $g = \{ (a_i \mid i \in \omega_1), (b_j \mid j \in
\omega_1)\}$ and assume (for the sake of a contradiction) some condition $q'$ in $R$ forces that $g' = \{ (a_\alpha \mid \alpha \in A), (b_\beta \mid \beta \in B)\}$ is a $H$-Hausdorff subgap, where $A$ and $B$ are names forced by $q$ to be unbounded in $\omega_1$. Since every club subset of $\omega_1$ in a c.c.c. generic extension contains a club subset in the ground model, we may assume that the club, $D$, which appears in definition \[DHG\] (of $g'$ being $H$-Hausdorff) is in $V$.
For every $\delta\in T \cap D$ define two conditions in $R$ (extending the given condition $q$):
1. $p_\delta \in R$ is such that for some $\alpha(\delta) \in \omega_1\setminus \delta$, $p_\delta{{\: \Vdash_R\:}}\alpha(\delta) \in A$. (This is possible since $A$ is forced to be unbounded.)
2. $q_\delta\in R$ extending $p_\delta$ is such that, for some $\beta(\delta) \in \omega_1 \setminus \delta$, $ q_\delta {{\: \Vdash_R\:}}\ \beta(\delta) \in B$. Moreover, as $g'$ is forced to be $H$-Hausdorff, we can assume that for some $m_\delta \in \omega$, $$q_\delta {{\: \Vdash_R\:}}\ \mbox{for every}\ n \geq m_\delta,\ \mbox{if}\ i \in A
\cap (\delta \setminus c_\delta(n)),\ \mbox{then}\ X(a_i, b_{\beta(\delta}))> n.$$
By Lemma \[CCC\] some condition forces that $q_\delta\in G$ (and hence $p_\delta\in G$) for a stationary set of indices $\delta\in T\cap D$. Since $R$ is p.c.c. for $T$, there are stationary subsets $T_1, T_2 \subseteq
T$ such that any $p_{\delta_1}$ is compatible with $q_{\delta_2}$ if $\delta_1
\in T_1$, $\delta_2\in T_2$ and $\delta_1 < \delta_2$.
Consider now the two unbounded sets $J = \{ \alpha(\delta) \mid \delta
\in T_1 \}$, and $K = \{ \beta(\delta) \mid \delta \in T_2\}$. Apply the conclusion of lemma \[Prop\] quoted above to $J$ and $K$, and let $D_0$ be the club set that appears there. Pick any $\delta \in D \cap T_2 \cap D_0$. Consider $k = \beta(\delta)$. Then $k \in K\setminus \delta$, and so there are $m\in \omega$ and a sequence $j(n)\in \delta \cap J$ cofinal in $\delta$ such that $$\label{Contra}
a_{j(n)} \setminus m \subset b_k\ \mbox{ for all } n\in \omega.$$ Yet every $j(n)$ is of the form $\alpha(\delta_n)$ for some $\delta_n \in T_1 \cap \delta$, and the $\delta_n$’s tend to $\delta$. So (\[Contra\]) can be written as $$\label{Contra1}
X(a_{\alpha(\delta_n)}, b_k) \leq m.$$ It follows from the definition of $T_1$ and $T_2$ that $p_{\delta_n}$ and $q_\delta$ are compatible in $R$. If $q'$ is a common extension, then $q'$ forces that for every $n\geq m_\delta$ if $i=\alpha(\delta_n) \geq c_\delta(n)$ then $X(a_i, b_k) > n$. It suffices now to take $n\geq \max \{ m,m_\delta \}$ to get the contradiction to (\[Contra1\]).
[30]{}
S. H. Hechler, Short nested sequences in $\beta N \setminus N$ and small maximal almost disjoint families, [*General Topology and its Applications,*]{} 2, pp. 139–149, 1972.
D. A. Martin and R. Solovay, Internal Cohen extensions, [*Ann. Math. Logic*]{} [**2**]{}, 143,178, (1970).
M. Scheepers, Gaps in $\omega^\omega$, in [*Set Theory of the Reals*]{} (Ramat Gan, 1991), Israel Mathematical Conference Proceedings, Vol. 6, Bar-Ilan Univ., Ramat Gan (1993), 439–561.
D. Talayco, Applications of cohomology to set theory. I. Hausdorff gaps, [*Ann. Pure Appl. Logic, vol. 71*]{}, 1995, no. 1, 69–106.
[^1]: The author would like to thank the Israel Science Foundation, founded by the Israel Academy of Science and Humanities. Publication \# 598.
|
---
abstract: 'Given a $(k+1)$-tuple $A, B_1,\ldots, B_k$ of $(m\times n)$-matrices with $m\le n$ we call the set of all $k$-tuples of complex numbers $\{{\lambda}_1,\ldots,{\lambda}_k\}$ such that the linear combination $A+{\lambda}_1B_1+{\lambda}_2B_2+\ldots+{\lambda}_kB_k$ has rank smaller than $m$ the [*eigenvalue locus*]{} of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see [@He] and [@Vol], we study a number of properties of the eigenvalue locus in the most important case $k=n-m+1$.'
address:
- 'Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden'
- 'Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden'
- 'Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA'
author:
- Julius Borcea
- Boris Shapiro
- Michael Shapiro
title: On eigenvalues of rectangular matrices
---
Introduction and Main Results {#introduction-and-main-results .unnumbered}
=============================
In recent years there appeared a number of publications discussing the eigenvalues of pencils of non-square matrices and their approximations, see, e.g., [@BEGM], [@CG], [@TW] and references therein. But to the best of our knowledge the following natural problem either has been overlooked by specialists in linear algebra or is deeply buried in the (enormous) literature on this topic.
Given a $(k+1)$-tuple of $(m\times n)$-matrices $A, B_1,\ldots, B_k$, $m\le n$, describe the set of all values of the parameters ${\lambda}_1,\ldots{\lambda}_k$ for which the rank of the linear combination $A+{\lambda}_1B_1+\ldots+{\lambda}_kB_k$ is less than $m$ or, in other words, when the linear system $v*(A+{\lambda}_1B_1+\ldots+{\lambda}_kB_k)=0$¾ has a nontrivial left solution $0\neq v\in {\mathbb {C}}^m$ which we call an [*eigenvector*]{}, where the symbol “$*$” denotes the usual matrix/vector multiplication.
Let ${\mathcal {M}}(m,n)$, $m\le n$, be the linear space of all $(m\times n)$-matrices with complex entries. In what follows we will consider $k$-tuples of $(m\times n)$-matrices $B_1,\ldots, B_k$ which are linearly independent in ${\mathcal {M}}(m,n)$ and denote their linear span by ${\mathcal L}={\mathcal L}(B_1,\ldots,B_k)$. Given a matrix pencil ${\mathcal P}=A+{\mathcal L}$, where $A\in {\mathcal {M}}(m,n)$, let ${\mathcal E}_{\mathcal P}\subset {\mathcal P}$ be its [*eigenvalue locus*]{}, i.e., the set of matrices in ${\mathcal P}$ whose rank is less than $m$. Elements of ${\mathcal E}_{\mathcal P}$ will be called [*(generalized) eigenvalues.*]{} Denote by ${\mathcal {M}}^1\subset {\mathcal {M}}(m,n)$ the set of all $(m\times n)$-matrices with positive corank, i.e., whose rank is non-maximal. Its codimension equals $n-m+1$ and its degree as an algebraic variety equals $\binom {n}{m-1}$, see [@BV Proposition 2.15]. Consider the natural left-right action of the group $GL_m\times GL_n$ on ${\mathcal {M}}(m,n)$, i.e., $GL_m$ (respectively, $GL_n$) acts on $(m\times n)$-matrices by left (respectively, right) multiplication. This action on ${\mathcal {M}}(m,n)$ has finitely many orbits, each orbit being the set of all matrices of a given (co)rank, see, e.g., [@AVG Chap. I §2]. Note that by the well-known product formula for coranks the codimension of the set of matrices of rank $r$ equals $(m-r)(n-r)$. Obviously, for any pencil ${\mathcal P}$ one has that the eigenvalue locus coincides with ${\mathcal E}_{\mathcal P}={\mathcal {M}}^1\cap {\mathcal P}$. Thus for a generic pencil ${\mathcal P}$ of dimension $k$ the eigenvalue locus ${\mathcal E}_{\mathcal P}$ is a subvariety of ${\mathcal P}$ of codimension $n-m+1$ if $k\ge n-m+1$ and it is empty otherwise. The most interesting situation for applications occurs when $k=n-m+1$, in which case ${\mathcal E}_{\mathcal P}$ is generically a finite set. From now on we assume that $k=n-m+1$. Denoting as above by ${\mathcal L}$ the linear span of $B_1,\ldots,B_{n-m+1}$ we say that ${\mathcal L}$ is [*transversal to ${\mathcal {M}}^1$*]{} if the intersection ${\mathcal L}\cap {\mathcal {M}}^1$ is finite and [*non-transversal to ${\mathcal {M}}^1$*]{}¾ otherwise. Notice that due to the homogeneity of ${\mathcal {M}}^1$ any $(n-m+1)$-dimensional linear subspace ${\mathcal L}$ transversal to it intersects ${\mathcal {M}}^1$ only at $0$ and that the multiplicity of this intersection at $0$ equals $\binom{n}{m-1}$. An important and most natural example of such a subspace ${\mathcal L}$ is motivated by the Heine-Stieltjes theory [@He] and its higher order generalizations [@BBS]. Denote by $J_s$, $s=1,\ldots,n-m+1$, the $(m\times n)$-matrix whose entries are given by $a_{i,j}=0$ if $i-j\neq s$ and $1$ otherwise. We call $J_s$ the [*$s$-th unit matrix*]{} or the [*the $s$-th diagonal matrix*]{}. Let us denote the linear span of $J_1,\ldots,J_{n-m+1}$ by ${\mathfrak L}$ and call ${\mathfrak L}$ the [*standard diagonal subspace*]{}. Note that ${\mathfrak L}$ is transversal to ${\mathcal {M}}^1$ since any matrix in ${\mathfrak L}$ different from $0$ has full rank, as one can easily check.
We start with the following simple statement.
\[lm:int\] If ${\mathcal L}\subset {\mathcal {M}}(m,n)$ has dimension $(n-m+1)$ and is tranversal to ${\mathcal {M}}^1$ then for any matrix $A\in {\mathcal {M}}(m,n)$ the eigenvalue locus ${\mathcal E}_{\mathcal P}$ of the pencil ${\mathcal P}=A+{\mathcal L}$ consists of exactly $\binom {n}{m-1}$ points counted with multiplicities.
Notice that since ${\mathcal {M}}^1$ is an incomplete intersection the same holds for the eigenvalue locus ${\mathcal E}_{\mathcal P}$ of a generic pencil ${\mathcal P}=A+{\mathcal L}$, i.e., in order to find ${\mathcal E}_{\mathcal P}$ for a given generic matrix $A$ and a given generic subspace ${\mathcal L}$ one has to solve an overdetermined system of determinantal equations.
However, as was essentially discovered by Heine [@He], the situation is different if one considers the standard diagonal subspace ${\mathfrak L}$ and any $A=(a_{i,j})\in {\mathcal {M}}(m,n)$ which is upper-triangular – that is, such that $a_{i,j}=0$ whenever $i>j$ – and has additionally distinct elements on the first main diagonal.
\[th:Heine\] For any upper-triangular matrix $A=(a_{i,j})\in {\mathcal {M}}(m,n)$ with all distinct entries $a_{i,i}$ on the first main diagonal the eigenvalue locus ${\mathcal E}_{\mathcal P}$ of the pencil ${\mathcal P}=A+{\mathfrak L}$, where ${\mathfrak L}$ is the standard diagonal subspace, is the union of $m$ complete intersections enumerated by the first component of the eigenvalue.
An explicit defining system of $(n-m)$ algebraic equations in $(n-m)$ variables for each such complete intersection is presented in the proof of Theorem \[th:Heine\], see §\[s1\] below.
Given ${\mathcal L}$ as above consider the natural projection map $\pi_{\mathcal L}:{\mathcal {M}}(m,n)\to {\mathcal L}^\perp$ along ${\mathcal L}$, where ${\mathcal L}^\perp={\mathcal {M}}(m,n)/{\mathcal L}$. Noticing that $\dim {\mathcal {M}}^1=\dim {\mathcal L}^\perp$ we define the [*set of critical values*]{} of $\pi_{\mathcal L}$ to be the set ${\mathcal C}_{\mathcal L}$ of all points in ${\mathcal {M}}^1$ where $\pi_{\mathcal L}$ is not a local diffeomorphism of ${\mathcal {M}}^1$ on its image $\pi_{\mathcal L}({\mathcal {M}}^1)$. In other words, ${\mathcal C}_{\mathcal L}$ is the set of all points $p\in {\mathcal {M}}^1$ such that the sum of ${\mathcal L}$ and the tangent space to ${\mathcal {M}}^1$ at $p$ does not coincide with the whole ${\mathcal {M}}(m,n)$. In particular, independently of ${\mathcal L}$ the critical value set ${\mathcal C}_{\mathcal L}$ always includes the set ${\mathcal {M}}^2$ of all $(m\times n)$-matrices with corank at least $2$.
Recall that ${\mathcal {M}}^1\subset {\mathcal {M}}(m,n)$ has the classical small resolution of singularities $\widetilde{{\mathcal {M}}^1}\subset {\mathcal {M}}(m,n)\times {\mathbb {CP}}^{m-1}$. Here $\widetilde{{\mathcal {M}}^1}$ consists of all pairs $\left(A,pker(A)\right)$, where $A\in {\mathcal {M}}^1$ and $pker(A)$ is the projectivization of the left kernel of $A$. Using this construction one can parameterize a Zariski open subset of ${\mathcal {M}}^1$ as follows. Consider the product $P(m,n)= {\mathcal {M}}(m-1,n)\times {\mathbb {C}}^{m-1}$. Take the map $\nu: P(m,n)\to {\mathcal {M}}^1\subset {\mathcal {M}}(m,n)$ sending a pair $({\mathcal A}; k_1,\ldots,k_{m-1})$ to the matrix $A\in {\mathcal {M}}(m,n)$ obtained by appending to ${\mathcal A}$ the last row such that its sum with the linear combination with the coefficients $(k_1,\ldots,k_{m-1})$ of the respective rows of ${\mathcal A}$ vanishes.
The main result of this paper is a simple determinantal representation of ${\mathcal C}_{\mathcal L}$ in the above coordinates.
\[th:determ\] Let ${\mathcal L}$ be any $(n-m+1)$-dimensional linear subspace in ${\mathcal {M}}(m,n)$ transversal to ${\mathcal {M}}^1$ and denote by $L_1,\ldots,L_{n-m+1}$ some basis of ${\mathcal L}$. Then in the coordinates of $P(m,n)$ the critical value set ${\mathcal C}_{\mathcal L}$¾ is given the determinantal equation $$\label{eq:1}
\det\begin{pmatrix} {\mathcal A}\\
V_1\\
\vdots\\
V_{n-m+1}
\end{pmatrix}=0.$$ Here ${\mathcal A}$ is a $(m-1,n)$-matrix with undetermined entries and $V_j$, $j=1,\ldots,n-m+1$, are row vectors given by $V_j=\mathbf{\kappa}* L_j$, where $\mathbf{\kappa}=(k_1,\ldots,k_{m})$.
If one expands equation (\[eq:1\]) in the variables $(k_1,\ldots,k_{n-m+1})$ then the coefficient of each monomial in these variables is a linear combination of the maximal minors of ${\mathcal A}$ (i.e., the Plücker coordinates) with complex coefficients depending only on the choice of ${\mathcal L}$. Moreover, the above equation contains a lot of information of geometric nature.
Our next result shows that for the standard diagonal subspace ${\mathfrak L}$ the determinantal equation in Theorem \[th:determ\] can be made quite a bit more explicit, which is particularly convenient from a computational viewpoint. We need first some additional notation. If $s\ge 1$ is an integer and $1\le r\le s$ let $Q_{r,s}$ be the set of all strictly increasing sequences of $r$ integers chosen from $1,\ldots,s$. Note in particular that $Q_{s,s}$ consists of a single sequence, namely $\{1,\ldots,s\}$. For ${\alpha}=({\alpha}_1,\ldots,{\alpha}_r)\in Q_{r,s}$ set $\rho({\alpha})=\sum_{j=1}^{r}{\alpha}_j$. Given $A\in{\mathcal {M}}(m,n)$, $1\le k\le m$, $1\le l\le n$, ${\alpha}\in Q_{k,m}$ and ${\beta}\in Q_{l,n}$ denote by $A[{\alpha}|{\beta}]\in{\mathcal {M}}(k,l)$ the submatrix of $A$ lying in rows ${\alpha}$ and columns ${\beta}$. Let ${\mathcal{HP}}(i,d)$ denote the complex space of all homogeneous polynomials in $i$ variables of degree $d$ and define the $d\times (i+d-1)$ matrix $$T_{i,d}=T_{i,d}(k_1,\ldots,k_i)=k_1J_1+\ldots+k_iJ_i,$$ where $J_j$, $1\le j\le i$, is as before the $j$-th diagonal $d\times (i+d-1)$ matrix and $k_1,\ldots,k_i$ are indeterminates. We will also need a result that may be of independent interest, namely the following lemma.
\[l:top\] In the above notation, the $\binom{i+d-1}{d}$ polynomials in $k_1,\ldots,k_i$ given by the determinants $$\Big|T_{i,d}[{\alpha}|{\beta}]\Big|,\quad {\alpha}\in Q_{d,d},\,{\beta}\in Q_{d,i+d-1},$$ build a basis of ${\mathcal{HP}}(i,d)$.
The usual determinant expansion formula provides an explicit expression (albeit tedious and not really needed for the present purposes) for the $\binom{i+d-1}{d}\times\binom{i+d-1}{d}$ matrix relating the standard monomial basis of ${\mathcal{HP}}(i,d)$ to the one constructed in Lemma \[l:top\].
\[th:sds\] Let ${\mathcal A}\in{\mathcal {M}}(m-1,n)$ be as in Theorem \[th:determ\]. The homogeneous defining polynomial of ${\mathcal C}_{\mathfrak L}$ with respect to the standard diagonal subspace ${\mathfrak L}$ is given by $$\sum_{{\beta}\in Q_{m-1,n}}(-1)^{\rho({\beta})}\Big|{\mathcal A}\big[\{1,\ldots,m-1\}|{\beta}\big]\Big|\cdot
\Big|T_{m,n-m+1}\big[\{1,\ldots,n-m+1\}|\{1,\ldots,n\}\setminus{\beta}\big]\Big|.$$
For $m=2$ the homogeneous defining polynomial of ${\mathcal C}_{\mathfrak L}$ with respect to the standard diagonal subspace ${\mathfrak L}$ is given by $$a_nk_1^n+a_{n-1}k_1^{n-1}k_2+a_{n-2}k_1^{n-2}k_2^2+\ldots+a_1k_1k_2^{n-1},$$ where $a_j=a_{1,j}$, $j=1,\ldots,n$.
For the standard diagonal subspace ${\mathfrak L}$ in the case of ${\mathcal {M}}(3,4)$ the homogeneous defining polynomial of ${\mathcal C}_{\mathfrak L}$ may be written as $$\begin{gathered}
\begin{vmatrix} a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}\\
a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}\\
k_1&k_2&k_3&0\\
0&k_1&k_2&k_3
\end{vmatrix}=\Delta_{3,4}k_1^2+\Delta_{1,4}k_2^2+\Delta_{1,2}k_3^2-\Delta_{2,4}k_1k_2\\
+(\Delta_{2,3}-\Delta_{1,4})k_1k_3-\Delta_{1,3}k_2k_3,\end{gathered}$$ where $\Delta_{i,j}$ is the $(2\times 2)$-determinant of the upper part ${\mathcal A}$ including the $i$-th and $j$-th columns.
The multiplicity of an eigenvalue $A\in {\mathcal E}_{\mathcal P}$ can be expressed in terms of the dimension of the corresponding local algebra. More exactly, for an $(m\times n)$-matrix $A$ we define the ideal $I_A$ in the algebra ${\mathbb {C}}[[t_1,\dots,t_k]]$ of formal power series as the ideal generated by all Plücker polynomials $\Delta_{i_1,\dots,i_m}(A+\sum_{l=1}^m
t_lB_l)$, where $\Delta_{i_1,\dots,i_m}(X_{m\times n})$ is the determinant of the $(m\times m)$ submatrix of X formed by the columns with the indices $i_1, i_2,\dots,i_m$. Now define the local algebra ${\mathcal A}_{loc}$ as the quotient algebra $A_{loc}={\mathbb {C}}[[t_1,\dots,t_k]]/I_A$. Then the multiplicity of the eigenvalue $A$ in the pencil $A+{\mathcal L}$ equals $\dim_{\mathbb {C}}A_{loc}$.
The main result of this note (Theorem \[th:determ\]) gives a simple explicit determinantal formula for the critical value set ${\mathcal C}_{\mathcal L}$ (in coordinates on the resolution of singularities $P(m,n)$). Its inverse image $\pi_{\mathcal L}^{-1}({\mathcal C}_{\mathcal L})$ is an important hypersurface consisting of all matrices in ${\mathcal {M}}(m,n)$ having a multiple eigenvalue. However, the problem of obtaining explicitly its defining polynomial in matrix entries seems to be quite delicate in general. As an illustration, let us show how this can be done in the simplest case of $(2\times 3)$-matrices.
For $m=2, n=3$ we will write the defining equation for the hypersurface $\pi_{\mathcal L}^{-1}({\mathcal C}_{\mathcal L})$ of matrices with multiple eigenvalues in the space ${\mathcal {M}}(2,3)$ itself.
For any pair of positive integers $m<n$ consider the extended matrix space ${\mathcal {M}}(m,n)\times {\mathbb {C}}P^{m-1}\times {\mathbb {C}}^{n-m+1}$, where the $m$-tuple of homogeneous coordinates in ${\mathbb {C}}^m$ is denoted by $\kappa=(\kappa_1:\dots:\kappa_m)$ and the coordinates in ${\mathbb {C}}^{n-m+1}$ are denoted by $\lambda=(\lambda_1,\dots,\lambda_{n-m+1})$.
Given a matrix $M\in {\mathcal {M}}(m,n)$ we will write a system of polynomial equations for $A+\sum_{i=1}^{n-m+1}\lambda_i J_i$, $$\sum_{s=1}^m \kappa_s [A+\sum_{i=1}^{n-m+1}\lambda_i
J_i]_{s*}=0,$$ expressing the fact that $\lambda$ is an eigenvalue of $M$ while the $\kappa_i$’s are the corresponding coefficients of a linear dependence between the rows of the matrix.
Using resultants we can get rid of the additional variables $\lambda$ and $\kappa$. This elimination leads to the defining equation for the hypersurface in question.
Namely, consider a $(2\times 3)$-matrix $A=\left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
\end{array}\right)$ and let as before $J_1=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array}\right)$ and $J_2=\left(\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}\right)$. A generic element of the pencil ${\mathcal P}$ is thus given by $$A(\lambda_1,\lambda_2):=A-\lambda_1 J_1-\lambda_2 J_2=\left(\begin{array}{ccc}
a_{11}-\lambda_1 & a_{12}-\lambda_2 & a_{13} \\
a_{21} & a_{22}-\lambda_1 & a_{23}-\lambda_2 \\
\end{array}\right).$$
For a generic matrix $A$ the condition that the rank of $A(\lambda_1,\lambda_2)$ is less than $2$ translates into two equations: the minor consisting of the second and third columns vanishes, and the minor consisting of the first and third columns vanishes. These equations have the form $$\begin{aligned}
\label{eq:minors}
(a_{12}-\lambda_2)(a_{23}-\lambda_2)-a_{13} (a_{22}-\lambda_1) &=& 0 \label{eq:minor23}\\
(a_{11}-\lambda_1)(a_{23}-\lambda_2)-a_{13} a_{21} &=&
0 \label{eq:minor13}\end{aligned}$$ Note that $$\label{eq:kappa}
\kappa:=\kappa_1=\frac{a_{23}-\lambda_2}{a_{13}}.$$ Moreover, from the determinantal equation of Theorem \[th:determ\] we obtain a third equation. Substituting expression into the latter gives the equation $$\label{eq:critical}
a_{13}^2 a_{11}-a_{13}^2\lambda_1+a_{23} a_{13}
a_{12}- 3 a_{23} a_{13} \lambda_2+ a_{13}
a_{23}^2-\lambda_2 a_{13} a_{12}+2
a_{13}\lambda_2^2=0.$$
Now equation (\[eq:minor23\]) has bidegree $(1,2)$ with respect to $\lambda_1, \lambda_2$. Analogously, (\[eq:minor13\]) has bidegree $(1,1)$ and (\[eq:critical\]) has bidegree $(1,2)$ with respect to the same variables. Clearly, any solution $s$ of the system of equations consisting of (\[eq:minor23\]), (\[eq:minor13\]) and (\[eq:critical\]) annihilates any polynomial in the ideal generated by these three equations. In particular, the following eight equations have $s$ as a common solution: (\[eq:minor23\]), (\[eq:minor23\]) multiplied by $\lambda_2$, (\[eq:minor23\]) multiplied by $\lambda_2^2$, (\[eq:minor13\]), (\[eq:minor13\]) multiplied by $\lambda_2$, (\[eq:critical\]), (\[eq:critical\]) multiplied by $\lambda_2$, (\[eq:critical\]) multiplied by $\lambda_2^2$. Therefore, the $(8\times 8)$ determinant $$\begin{split}
&D=\\
&\det\left(\begin{array}{cccccccc}
\Delta_{23} & a_{13} & -a_{12}-a_{23} & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & \Delta_{23} & a_{13} & -a_{12}-a_{23} & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & \Delta_{23} & a_{13} & -a_{12}-a_{23} & 1 \\
\Delta_{13} & -a_{23} & -a_{11} & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & \Delta_{13} & -a_{23} & -a_{11} & 1 & 0 & 0 \\
\delta & -a_{13}^2 & \sigma & 0 & 2 a_{13} & 0 & 0 & 0 \\
0 & 0 & \delta & -a_{13}^2 & \sigma & 0 & 2 a_{13} & 0\\
0 & 0 & 0 & 0 & \delta & -a_{13}^2 & \sigma & 2 a_{13} \\
\end{array}\right)
\end{split}$$ vanishes when (\[eq:minor23\]), (\[eq:minor13\]) and (\[eq:critical\]) have a common root. Here we use the following notation: $\delta=a_{13}^2 a_{11}+a_{23} a_{13} a_{12}+a_{13} a_{23}^2$, $\Delta_{23}=a_{12} a_{23}-a_{13} a_{22}$, $\Delta_{13}=a_{11}
a_{23}-a_{21} a_{13}$ and $\sigma=-a_{13} a_{12}-3 a_{13} a_{23}$.
This observation implies that the required defining polynomial for $\pi_{\mathcal L}^{-1}({\mathcal C}_{\mathcal L})$ is the product of some (but not necessarily all) irreducible factors of the polynomial $D$. Factorizing $D$ we obtain $D=a_{11}^6 D_0$, where $$\begin{split}
D_0=& -12a_{13} a_{22}^2 a_{11}+a_{22}^2 a_{12}^2+12 a_{13} a_{22}
a_{11}^2+ a_{11}^2 a_{23}^2+4 a_{21}a_{12}^3\\
&-4 a_{21}a_{23}^3+a_{11}^2a_{12}^2
+12 a_{12}a_{23}^2a_{21} -12 a_{12}^2 a_{23} a_{21}-2 a_{12} a_{23}
a_{22}^2\\
&- 2 a_{12} a_{23} a_{11}^2-2
a_{22} a_{11} a_{23}^2-18 a_{13} a_{22} a_{23} a_{21}
- 2 a_{22}a_{11} a_{12}^2\\
&+18 a_{13} a_{22} a_{21} a_{12}
+18 a_{11}a_{23} a_{13} a_{21}- 18 a_{21} a_{13} a_{12} a_{11}\\
&+4 a_{13}
a_{22}^3-27 a_{21}^2 a_{13}^2-4 a_{13} a_{11}^3+ 4 a_{12} a_{23}
a_{22} a_{11}+a_{22}^2 a_{23}^2=0
\end{split}$$
Note that $D_0$ is of second degree in the variable $a_{13}$ and its discriminant (with respect to this variable) $W=
16(3a_{12}a_{21}-3a_{21}a_{23}-2a_{11}a_{22}+a_{11}^2+a_{22}^2)^3$ is not a complete square. Thus, we conclude that $D_0$ is irreducible. Hence the variety given by $\{D=0\}$ is the union of the variety given by $\{D_0=0\}$ and the hyperplane $\{a_{11}=0\}$ taken with multiplicity $6$.
Since the hyperplane $\{a_{11}=0\}$ is obviously not contained in $\pi_{\mathcal L}^{-1}({\mathcal C}_{\mathcal L}) \subset {\mathcal {M}}(2,3)$ we obtain that $\pi_{\mathcal L}^{-1}({\mathcal C}_{\mathcal L})$ is given by $\{D_0=0\}$.
The authors are grateful to J. M. Landsberg and T. Ekedahl for relevant discussions and to R. Fröberg for help with some of the calculations.
Proofs {#s1}
======
This follows almost directly from homogeneity of ${\mathcal {M}}^1$. Indeed, take any matrix $0\neq A\in {\mathcal {M}}(m,n)$. Let $\tilde l \in {\mathcal L}$ be its eigenvalue, that is a matrix from ${\mathcal L}$ such that $A+\tilde l$ belongs to ${\mathcal {M}}^1$. Notice that for any ${\epsilon}\in (0,1]$ the matrix ${\epsilon}\tilde l$ is the eigenvalue of the matrix ${\epsilon}A$. Considering the family of matrices ${\epsilon}A$ with ${\epsilon}\in [0,1]$ we conclude that the total multiplicity of eigenvalues of the pencil $A+{\mathcal L}$ coincides with that of the linear pencil ${\mathcal L}$ if the latter multiplicity is finite, which gives the required statement.
To get the defining system of algebraic equations for ${\mathcal E}_{\mathcal P}$ under the assumptions of Theorem \[th:Heine\] we proceed exactly as in [@He]. For a given upper-triangular matrix $A\in {\mathcal {M}}(m,n)$ with distinct entries on the main diagonal we want to find all $(n-m+1)$-tuples $({\lambda}_1,\ldots,{\lambda}_{n-m+1})$ such that the matrix $A+{\lambda}_1J_1+{\lambda}_2J_2+\ldots+{\lambda}_{n-m+1}J_{n-m+1}$ has positive corank. Since $A$ is upper-triangular with distinct $a_{i,i}$ then in order to get a positive corank it is necessary to require ${\lambda}_1+a_{i,i}=0$ for some $i=1,\ldots,m$. The next observation is that under the above assumptions on $A$ for any given $i=1,\ldots,m$ the total number of eigenvalues with ${\lambda}_1+a_{i,i}=0$ equals $\binom {n-i}{m-i}$ which gives the following count of the eigenvalues of $A$ noticed already by Heine: $\binom{n}{m-1}=\binom{n-1}{m-1}+\binom{n-2}{m-2}+\ldots+\binom{n-m}{0}.$ Indeed, if ${\lambda}_1+a_{i,i}=0$ then ${\lambda}_1+a_{j,j}\neq 0$ for all $j\neq i$ and, in particular due to the assumptions on $A$ the first $i-1$ rows of $A-a_{i,i}J_1+{\lambda}_2J_2+\ldots+{\lambda}_{n-m+1}J_{n-m+1}$ are linearly independent for all values of ${\lambda}_2,\ldots,{\lambda}_{n-m+1}$. On the other hand, the remaining rows $i$, $i+1,\ldots,m$ can become linearly dependent under an appropriate choice of ${\lambda}_2,\ldots,{\lambda}_{n-m+1}$. Since the matrix $A-a_{1,1}J_1$ is upper-triangular with the $(i,i)$-th entry vanishing the condition that $A-a_{i,i}J_1+{\lambda}_2J_2+\ldots+{\lambda}_{n-m+1}J_{n-m+1}$ has positive corank is equivalent to the condition that the matrix obtained by removing its first $i$ rows and $i-1$ columns has positive corank. By Lemma \[lm:int\] the total number of eigenvalues of the matrix of the size $(m-i+1)\times (n-i)$ equals $\binom {n-i}{m-i}$. Let us now for any given $i=1,\ldots,m$¾ derive a system of algebraic equations in the variables ${\lambda}_2,\ldots,{\lambda}_{n-m+1}$ whose solutions are exactly all the eigenvalues of $A$ with ${\lambda}_1+a_{i,i}=0$. We will concentrate on the case $i=1$ since all other cases are covered in exactly the same way by working with a smaller matrix obtained from $A$ by removing the first $(i-1)$ rows and $(i-1)$ columns. Using $(k_1,\ldots,k_m)$ for the coordinates of the left kernel and ${\lambda}_1,{\lambda}_2,\ldots,{\lambda}_{n-m+1}$ for the eigenvalues we get the following system of equations $$\begin{cases}
0=k_1(a_{1,1}+{\lambda}_1)\\
0=k_1(a_{1,2}{\lambda}_2)+k_2(a_{2,2}+{\lambda}_1) \\
...................................................\\
0=k_1(a_{1,m}+{\lambda}_m)+k_2(a_{2,m}+{\lambda}_{m-1})+...+k_m(a_{m,m}+{\lambda}_1)\\
0=k_1(a_{1,m+1}+{\lambda}_{m+1})+k_2(a_{2,m+1}+{\lambda}_{m})+...+k_m(a_{m,m+1}+{\lambda}_2)\\
......................................................................................................\\
0=k_1(a_{1,m+1}+{\lambda}_{m+1})+k_2(a_{2,m+1}+{\lambda}_{m})+...+k_m(a_{m,m+1}+{\lambda}_2)\\
0=k_1(a_{1,n}+{\lambda}_{n})+k_2(a_{2,n}+{\lambda}_{n-1})+...+k_m(a_{n,n}+{\lambda}_{n-m+1})
\end{cases}$$ expressing the existence of a nontrivial left kernel of $A+{\lambda}_1 J_1+{\lambda}_2J_2+\ldots+{\lambda}_{n-m+1}J_{n-m+1}$. (To simplify notations we assume here that ${\lambda}_j=0$ for $j>n-m+1$.) In order to get the required system of equations in ${\lambda}_1,\ldots,{\lambda}_{n-m+1}$ we have to eliminate from the above system the variables $k_1,\ldots,k_m$. Notice that under our assumptions on $A$ the possible corank of $A+{\lambda}_1 J_1+{\lambda}_2J_2+\ldots+{\lambda}_{n-m+1}J_{n-m+1}$ can be at most $1$ and in the case of corank $1$ the linear dependence must necessarily include the first row, i.e., $k_1=1$. Note also that the first $m$ equations are triangular with respect to $k_1,\ldots,k_m$, which together with our assumptions on $A$ allows us to successfully eliminate them. Namely, from the first equation we get ${\lambda}_1=-a_{1,1}$ and $k_1=1$. Then for any $i=2,\ldots,m$ we solve the $i$-th equation with respect to $k_i$ and get $$k_i=\frac{1}{a_{1,1}-a_{i,i}}\left(k_1(a_{1,i}+{\lambda}_i)+k_2(a_{2,i}+{\lambda}_{i-1})+\ldots+k_{i-1}(a_{i-1,i}+{\lambda}_2)\right).$$ With the initial value $k_1=1$ and taking into account that the only possible denominators occurring in the above expressions for $k_i$ are $a_{1,1}-a_{i,i}$ we recurrently find all $k_i,\; i=1,\ldots,m$ as the functions of the matrix entries and ${\lambda}$’s. Substituting these found expressions in the remaining $n-m$ equations we get the required system of algebraic equations to determine ${\lambda}_2,\ldots,{\lambda}_{n-m+1}$. (Notice that ${\lambda}_1=-a_{1,1}$ was already obtained from the first equation.)
Any matrix $A\in {\mathcal {M}}(2,4)$ has four eigenvalues (counted with multiplicities) with respect to the standard diagonal subspace ${\mathfrak L}$. If $A$ is upper-triangular with distinct elements on the first main diagonal then these eigenvalues split into two groups depending on the value of ${\lambda}_1$. Namely, there are $3$ eigenvalues for which ${\lambda}_1=-a_{1,1}$ and $1$ eigenvalue for ${\lambda}_1=-a_{2,2}$. For ${\lambda}_1=-a_{1,1}$ the above system (before elimination) has the form: $$\begin{cases} 0=k_1(a_{1,1}+{\lambda}_1)\\
0=k_1(a_{1,2}+{\lambda}_2)+k_2(a_{2,2}+{\lambda}_1)\\
0=k_1(a_{1,3}+{\lambda}_3)+k_2(a_{2,3}+{\lambda}_2)\\
0=k_1a_{1,4}+k_2(a_{2,4}+{\lambda}_3).
\end{cases}$$ From the first equation we get $k_1=1$ and ${\lambda}_1=-a_{1,1}$. From the second equation we get $k_2=\frac{a_{1,2}+{\lambda}_2}{a_{1,1}-a_{2,2}}$. Substituting in the remaining two equations we get the next system to determine ${\lambda}_2$ and ${\lambda}_3$: $$\begin{cases}
({\lambda}_2+a_{1,2})({\lambda}_2+a_{2,3})+(a_{1,1}-a_{2,2})({\lambda}_3+a_{1,3})=0\\
({\lambda}_2+a_{1,2})({\lambda}_3+a_{2,4})+(a_{1,1}-a_{2,2})a_{1,4}=0.
\end{cases}$$ In the case ${\lambda}_1+a_{2,2}=0$ one gets a very simple linear system: $$k_2(a_{2,2}+{\lambda}_1)=k_2(a_{2,3}+{\lambda}_2)=k_2(a_{2,4}+{\lambda}_3)=0$$ which gives $k_2=1,\;{\lambda}_1=-a_{2,2},\;{\lambda}_2=-a_{2,3},\;{\lambda}_3=-a_{2,4}.$
As we already mentioned in the introduction the set ${\mathcal C}_{\mathcal L}$ can be determined as the set of all matrices $M \in {\mathcal {M}}^1$ such that the sum of the tangent space to ${\mathcal {M}}^1$ at $M$ and the linear space ${\mathcal L}$ does not coincide with the whole ${\mathcal {M}}(m,n)$. Let us describe a basis of the tangent space to ${\mathcal {M}}^1$ at a sufficiently generic matrix $M$. Since $GL_m\times GL_n$ acts on ${\mathcal {M}}(m,n)$ with finitely many orbits the tangent space to the $GL_m\times GL_n$-orbit of $M$ under this action coincides with the tangent space to ${\mathcal {M}}^1$ at $M$. Note that $GL_m\times GL_n$ acts on ${\mathcal {M}}(m,n)$ by elementary row and column operations. Thus, if we take for example the affine chart in which the determinant formed by the first $(m-1)$ rows and columns is non-vanishing then the tangent space to ${\mathcal {M}}^1$ at any matrix $M$ belonging to this chart is generated by the following two groups of operations: (i) add to each column of $M$ one of its first $m-1$ columns and (ii) add to the last row of $M$ one of its other rows. One has therefore a total of $n(m-1)+(m-1)=(n+1)(m-1)=\dim {\mathcal {M}}^1$ generators. Taking the wedge of these generators with the chosen basis of ${\mathcal L}$ and representing an $(m\times n)$-matrix as a $mn$-vector by patching together its rows we obtain the following $(mn\times mn)$-matrix that has a block structure of an $(m\times m)$-matrix with $(n\times n)$-blocks of the form given below:
$$\mathfrak D=\begin{pmatrix}
a_{1,1}I_n& a_{2,1}I_n& \cdots&a_{m-1,1}I_n& -\sum_{j=1}^{m-1}k_ja_{j,1}\cdot I_n\\
a_{1,2}I_n& a_{2,2}I_n& \cdots&a_{m-1,2}I_n& -\sum_{j=1}^{m-1}k_ja_{j,2}\cdot I_n\\
\vdots & \vdots & \ddots & \vdots & \vdots \\
a_{1,m-1}I_n& a_{2,m-1}I_n& \cdots&a_{m-1,m-1}I_n& -\sum_{j=1}^{m-1}k_ja_{j,m-1}\cdot I_n\\
0_{m-1,n}&0_{m-1,n}&\cdots &0_{m-1,n}& {\mathcal A}_{m-1,n}\\
L_{1,1} &L_{2,1} & \cdots & L_{m-1,1} & L_{m,1}\\
L_{1,2} &L_{2,2} & \cdots & L_{m-1,2} & L_{m,2}\\
\vdots & \vdots & \ddots & \vdots & \vdots \\
L_{1,n-m+1} &L_{2,n-m+1} & \cdots & L_{m-1,n-m+1} & L_{m,n-m+1}\\
\end{pmatrix}.$$ Here ${\mathcal A}={\mathcal A}_{m-1,n}=(a_{i,j})$, $i=1,\ldots,m-1$, $j=1,\ldots,n$, $I_n$ is the identity $(n\times n)$-matrix, $0_{m-1,n}$ is the $((m-1) \times n)$-matrix with all vanishing entries, and, finally, $L_{i,j}$ is the $i$-th row of the matrix $L_j$, see Theorem \[th:determ\]. Notice that the determinant $\det(a_{i,j}I_n)$, $i=1,\ldots,m-1$, $j=1,\ldots,m-1$, of the upper-left block of $\mathfrak D$ equals $\Delta^{m-1}$, where $\Delta=\det (a_{i,j})$, $i=1,\ldots,m-1$, $j=1,\ldots,m-1$, is the leftmost principal minor of $A_{m-1,n}$. By the above assumption the matrix ${\mathcal A}$ lies in the affine chart where $\Delta\neq 0$. Finally, we clear the low-left block $(L_{i.j})$, $i=1,\ldots,m-1$, $j=1,\ldots,n-m+1$, of $\mathfrak D$ by “killing” all its elements through row operations using the above upper-left block (which is a square and non-degenerate $((m-1)n\times (m-1)n)$-matrix) to obtain the low-right block coinciding exactly with the matrix in formula (\[eq:1\]). Thus the determinant of the whole matrix $\mathfrak D$ equals the product between $\Delta^{m-1}$ and the determinant from Theorem \[th:determ\]. Since in the considered chart one has $\Delta\neq 0$ the result follows.
Set $t=i+d$, so that $t\ge 2$. Since $$\dim {\mathcal{HP}}(i,d)=\binom{i+d-1}{d}$$ we have to show that the polynomials constructed in the lemma are linearly independent, which we prove this by induction on $t$. Note that this is trivially true for $t=2$. Assume that it holds for some $t\ge 2$ and let $i,d$ be such that $i+d=t+1$. Suppose that $c_{{\alpha}{\beta}}\in{\mathbb {C}}$ are such that $$\sum_{{\alpha}\in Q_{d,d}\atop {\beta}\in Q_{d,i+d-1}}c_{{\alpha}{\beta}}\Big|T_{i,d}(k_1,\ldots,k_i)[{\alpha}|{\beta}]\Big|=0.$$ Clearly, this may be rewritten as $$\begin{gathered}
\label{eq-lin}
\sum_{{\alpha}\in Q_{d,d}\atop {\beta}\in Q_{d,i+d-2}}c_{{\alpha}{\beta}}\Big|T_{i,d}(k_1,\ldots,k_i)[{\alpha}|{\beta}]\Big|\\
+k_i\cdot\!\!\!\sum_{{\alpha}\in Q_{d-1,d-1}\atop {\beta}\in Q_{d-1,i+d-2}}c_{{\alpha}{\beta}}\Big|T_{i,d-1}(k_1,\ldots,k_i)[{\alpha}|{\beta}]\Big|=0.\end{gathered}$$ In particular, setting $k_i=0$ we get $$\sum_{{\alpha}\in Q_{d,d}\atop {\beta}\in Q_{d,i+d-2}}c_{{\alpha}{\beta}}\Big|T_{i-1,d}(k_1,\ldots,k_{i-1})[{\alpha}|{\beta}]\Big|=0$$ hence $c_{{\alpha}{\beta}}=0$, ${\alpha}\in Q_{d,d}$, ${\beta}\in Q_{d,i+d-2}$, by the induction assumption since $(i-1)+d=t$. Together with this implies that $$\sum_{{\alpha}\in Q_{d-1,d-1}\atop {\beta}\in Q_{d-1,i+d-2}}c_{{\alpha}{\beta}}\Big|T_{i,d-1}(k_1,\ldots,k_i)[{\alpha}|{\beta}]\Big|=0,$$ which in turn yields $c_{{\alpha}{\beta}}=0$, ${\alpha}\in Q_{d-1,d-1}$, ${\beta}\in Q_{d-1,i+d-2}$, again by the induction hypothesis since $i+(d-1)=t$. We conclude that $c_{{\alpha}{\beta}}=0$ for all ${\alpha}\in Q_{d,d}$ and ${\beta}\in Q_{d,i+d-1}$, which proves the desired statement hence also the lemma.
We will use the setting and notation of Lemma \[l:top\] with $i=m$ and $d=n-m+1$. Fix the sequence ${\alpha}=\{1,\ldots,m-1\}\in Q_{m-1,n}$. Now consider the left-hand side of the determinantal equation in Theorem \[th:determ\] in the case when ${\mathfrak L}$ is the standard diagonal subspace and $L_j=J_j$, $1\le j\le n-m+1$. In view of the generalized Laplace expansion theorem, see, e.g., [@MM §2.4.11], when expanding it by the rows ${\alpha}$ this left-hand side becomes $$\begin{gathered}
(-1)^{\frac{m(m-1)}{2}}\sum_{{\beta}\in Q_{m-1,n}}(-1)^{\rho({\beta})}\Big|{\mathcal A}\big[\{1,\ldots,m-1\}|{\beta}\big]\Big|\times\\
\Big|T_{m,n-m+1}(k_1,\ldots,k_m)\big[\{1,\ldots,n-m+1\}|\{1,\ldots,n\}\setminus{\beta}\big]\Big|,\end{gathered}$$ which proves the theorem.
Remarks and open questions
==========================
A {#a .unnumbered}
-
By analogy with the above case, for a given triple $n,m,r$ one can also consider $(m-r)(n-r)$-dimensional pencils of matrices in ${\mathcal {M}}(m,n)$ and study their intersections with the subvariety ${\mathcal {M}}^r$ of all matrices of corank at least $r$. In particular, a natural question is to find an analog of Theorem \[th:determ\] in this situation.
B {#b .unnumbered}
-
It would be interesting to determine the equation for $\pi^{-1}_{\mathcal L}({\mathcal C}_{\mathcal L})$ in general, see Example 3 in the Introduction. Another important direction is to determine the local multiplicity of a given eigenvalue in terms of the defining polynomial for ${\mathcal C}_{\mathcal L}$. Is there any analog of the Jordan normal form allowing to determine the multiplicity of a given eigenvalue?
C {#c .unnumbered}
-
Notice that the left-right action of $GL_m\times GL_n$ extends from the space ${\mathcal {M}}(m,n)$ to every space of (in)complete flags in ${\mathcal {M}}(m,n)$. For simple dimensional reasons, in most cases this action cannot have finitely many orbits.
On which spaces of (in)complete flags the above left-right action of $GL_m\times GL_n$ has finitely many orbits?
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abstract: 'Turbulent flows preferentially concentrate inertial particles depending on their stopping time or Stokes number, which can lead to significant spatial variations in the particle concentration. Cascade models are one way to describe this process in statistical terms. Here, we use a direct numerical simulation (DNS) dataset of homogeneous, isotropic turbulence to determine probability distribution functions (PDFs) for cascade *multipliers*, which determine the ratio by which a property is partitioned into sub-volumes as an eddy is envisioned to decay into smaller eddies. We present a technique for correcting effects of small particle numbers in the statistics. We determine multiplier PDFs for particle number, flow dissipation, and enstrophy, all of which are shown to be scale dependent. However, the particle multiplier PDFs collapse when scaled with an appropriately defined *local* Stokes number. As anticipated from earlier works, dissipation and enstrophy multiplier PDFs reach an asymptote for sufficiently small spatial scales. From the DNS measurements, we derive a cascade model that is used it to make predictions for the radial distribution function (RDF) for arbitrarily high Reynolds numbers, $Re$, finding good agreement with the asymptotic, infinite $Re$ inertial range theory of Zaichik & Alipchenkov \[New Journal of Physics 11, 103018 (2009)\]. We discuss implications of these results for the statistical modeling of the turbulent clustering process in the inertial range for high Reynolds numbers inaccessible to numerical simulations.'
author:
- Thomas Hartlep
- 'Jeffrey N. Cuzzi'
- Brian Weston
bibliography:
- 'bibliography.bib'
date: '12 December 2016 (Submitted to [*Physical Review E*]{}), 28 February 2017 (Accepted)'
title: 'Scale Dependence of Multiplier Distributions for Particle Concentration, Enstrophy and Dissipation in the Inertial Range of Homogeneous Turbulence'
---
Background and Introduction {#Section:Introduction}
===========================
Clustering of inertial (finite-stopping-time) particles into dense zones in fluid turbulence has applications in many fields [for a general review see @2009AnRFM..41..375T]. A number of recent papers have focussed on understanding the basic mechanisms responsible for this effect; several of these [@ZaichikAlipchenkov2009; @BraggCollins2014I; @BraggCollins2014II; @GustavssonMehlig2014] provide thorough reviews and comparisons of previous studies dating back to the early work of @Maxey1987 and @SquiresEaton1991 which we will only sketch briefly. The early work emphasized the role of centrifugation of finite-inertia particles out of vortical structures in turbulence. More recent evidence that clustering arises even in random, irrotational flows suggests that, while vorticity still plays a role, the dominant role is played by so-called “history effects", in which inertial particle velocity dispersions at any location carry a memory of particle encounters with more remote flow regimes which have larger characteristic velocity differences [@PanPadoan2010; @2013ApJ...776...12P; @Braggetal2015JFM]. These history effects lead to spatial gradients in particle random relative velocities, and these gradients in turn generate systematic flows or currents which can outweigh dispersive effects and produce zones of highly variable particle concentration [@ZaichikAlipchenkov2003; @ZaichikAlipchenkov2009; @BraggCollins2014I; @BraggCollins2014II].
To date, by far the most attention regarding particle clustering in turbulence has been devoted to very small spatial scales $r < \eta$ or even $r \ll \eta$, where $\eta$ is the Kolmogorov scale, partly because it is on these scales that particle collisions occur and partly because numerical simulations to date have produced only very limited inertial ranges, at best [see however @2010JFM...646..527B; @Irelandetal2015]. Theories by @ZaichikAlipchenkov2003 [*et seq.*]{}, and @PanPadoan2010 [*et seq.*]{} have been shown to be promising in explaining the cause of particle clustering in terms of history effects, with helpful contributions from the traditional local centrifugation mechanism [@ZaichikAlipchenkov2009; @BraggCollins2014II; @Braggetal2015JFM; @Irelandetal2015]. A thorough review of the effects of clustering and relative velocity effects on particle collisions, emphasizing the astronomical literature, can be found in @PanPadoan2014 [@PanPadoan2015].
Our focus is on clustering at larger scales in the [*inertial range*]{} $\eta < r < L$, where $L$ is the integral scale. Inertial range clustering has important applications for remote sensing of terrestrial clouds [@Shawetal2002], the formation of primitive planetesimals (asteroids and comets) in the early solar nebula [@2008ApJ...687.1432C; @2010Icar..208..518C; @Chambers2010; @0004-637X-740-1-6; @2014ApJ...797...59H; @Johansenetal2015], and even the structure of the interstellar medium [@2015arXiv151002477H]. While little studied in the context of particle clustering, inertial range scaling is known to have different properties than seen in the dissipation range $r < \eta$ [@2007PhRvL..98h4502B; @Braggetal2015PRE]. Only limited predictions have been made of its scaling properties at very high Reynolds number $Re$ [@ZaichikAlipchenkov2003; @ZaichikAlipchenkov2009].
In the inertial range, so-called [*cascade models*]{} which reproduce the statistics of fluid behavior, even if not realistic flow structures, may be valuable for modeling high Reynolds number ($Re$) regimes too demanding for direct numerical simulations. Their application is quite general [@1987PhRvL..59.1424M; @1991JFM...224..429M; @1989PhRvA..40.5284C; @1992PhRvL..68.2762C; @1994ActaMechSup4..113S; @1995JSP....78..311S see [@2007PhRvE..75e6305H] for more references]. We and others have used cascades to model particle clustering in turbulence in the astronomical applications mentioned above.
Cascade models operate by simply applying a partition function or *multiplier* $0 \leq m \leq 1$ to any property $\cal P$ in some given volume of the flow, thus determining the ratio by which the property (dissipation, particle density, etc.) is partitioned into sub-volumes as an eddy is envisioned to decay into smaller eddies. The most common treatment is a binary cascade, in which $\cal P$ is partitioned into two equal subvolumes; however the approach can be applied to arbitrary numbers of subvolumes [@1994ActaMechSup4..113S]. The binary cascade operates on each volume of space, partitioning $\cal P$ into two equal subvolumes by multipliers $m$ and $1-m$, with the multiplier $m$ at each bifurcation drawn from a probability distribution function (PDF) of multipliers $P(m)$. If $P(m)=\delta(m-0.5)$, where $\delta$ is the delta function, the cascade has no effect because the property $\cal P$ is evenly divided, and remains constant per unit volume. On the other hand, broad $P(m)$ functions generate highly *intermittent* spatial distributions in which $\cal P$ has a wide range of values, fluctuating dramatically on small scales such as seen in dissipation [@1991JFM...224..429M; @1995JSP....78..311S] (figure \[fig:inertialrange\]b).
The *dissipation range*, a range of small scales approaching the Kolmogorov scale $\eta$, is found where $r < 20-30\eta$ [@1972fct..book.....T; @1995tlan.book.....F]; in this range, where viscosity is important, the equations of motion are no longer completely scale-free, and fluid scaling properties differ from those in the inertial range. The properties of particle clustering [*do*]{} seem to be scale independent in this region, however [@2007PhRvL..98h4502B; @Braggetal2015PRE], and one expects this regime to be flow-independent for high $Re$. There is also a range of *large* scales near the integral scale $L$, over which deviation from scale invariance surely occurs, but this range has not been well studied and is surely flow-dependent. The application to planetesimal formation has become focussed on particle concentration at scales much larger than the Kolmogorov scale [@2008ApJ...687.1432C; @2010Icar..208..518C] because large clumps are needed for sufficiently rapid gravitational collapse. In previous particle clustering cascade models, @2007PhRvE..75e6305H determined the multiplier PDFs for particle concentration and fluid enstrophy at *small* spatial scales, not too far from $\eta$ (to obtain better statistics), and applied them across all scales ranging up to the integral scale (see section \[Subsection:Discussion:Particles\] for more discussion). Realizing the risks in this, they performed tests which seemed to validate the approach. However, discrepancies between @2007PhRvE..75e6305H and @0004-637X-740-1-6 at the low probabilities of interest for the planetesimal problem [@2010Icar..208..518C] have led us to explore the scale dependence of $P(m)$ in more detail, in order to improve the fidelity of the cascade models.
It is worth noting at this point that it is not a requirement of cascade models that the PDFs be scale-independent; it is merely the first and most obvious assumption. In this paper we present evidence that the multiplier PDFs for particle concentration are scale-*dependent* and present simple guidelines for how this scale-dependence can be included in cascades. Multiplier PDFs can also be *conditioned* on local properties [@1995JSP....78..311S], and indeed were treated this way in our previous work to allow for particle mass-loading on the process [@2007PhRvE..75e6305H]. Scale-dependence *per se* is, however, a different effect than local conditioning, and in this paper we do not address local conditioning.
{width="0.9\linewidth"}
Before describing our own work, we review some experimental results on high-Reynolds number atmospheric boundary layer turbulence, which provide a useful background in scale invariance and complement the more typical, but lower-$Re$, numerical simulations. Studies of the properties of turbulence in atmospheric boundary layer flows have been conducted by @Kholmyanskiy:1973tg [@1975JFM....71..417V] and @1987NuPhS...2...49M; further analysis of the @1987NuPhS...2...49M data was done by @1989PhRvA..40.5284C.[^1] The best reference for the basic experimental data is @1987NuPhS...2...49M [see their Table 1], who conducted an experiment on boundary layer turbulence using a sensor mounted 2 m above the flat roof of a four story building. The Reynolds number for the flow is calculated using the free stream velocity $U$ = 6 m/s and the height $h$ = 2 m of the sensor above the roof: $Re = Uh/\nu = 8 \times 10^5$ where the kinematic viscosity is $1.5\times10^{-5}~$m$^2$/sec, consistent with tabulated values in @1987NuPhS...2...49M of the Taylor scale Reynolds number $Re_{\lambda}$, and its characteristic lengthscale $\lambda$ and velocity $u'$. @1987NuPhS...2...49M give the Kolmogorov scale as $\eta = 7 \times 10^{-4}$ m (we retain their preferred units). Analyses of flow structures by @1989PhRvA..40.5284C (their figure 5 and our figure \[fig:chhabra\]) show fairly well-behaved power law scaling of dissipation for weightings which suppress regions that are strongly anomalous (panels g and h), i.e. strongly differing from the mean, to almost $r \sim 1.8\times 10^4~\eta$ = 12.6 m $\gg h$, suggesting an extensive inertial range. The large or integral scale $L$, which contains the energy in this flow, is thus apparently much larger than the vertical distance of the sensor from the boundary ($h = 2$ m) and plausibly the same as the *longitudinal* integral scale given as $L$ = 180 m [@1987NuPhS...2...49M], see also @Hunt:2000kb, and thus $L/\eta \sim 2 \times 10^5$. However, when the role of strongly anomalous regions is emphasized (panels i through l of figure \[fig:chhabra\]) the scalable inertial range contracts.
![Plot of a family of normalized $q$-th moments of the dissipation $E_r$ in an atmospheric boundary layer, as averaged over binning lengthscale $r$, plotted against $r/\eta$ (taken from figure 5b of @1989PhRvA..40.5284C). The quantity $\mu_i(q,l) \equiv (E_r)_i^q/\Sigma_j(E_r)_j^q$, where $(E_r)_i$ is the dissipation in the $i$th bin of size $r$. Smaller $|q|$ values suppress the effect of strongly anomalous regions, while large negative values of $q$ select for regions of anomalously low turbulent dissipation. Abrupt changes in the slope of these plots (most obvious for large $|q|$) might indicate departure from the true scale-free inertial range, both in the dissipation range at $< 20-30\eta$, and at very large scales where vortex stretching has yet to become effective. Vertical arrows on horizontal axis are the authors [@1989PhRvA..40.5284C] estimate of the inertial scaling range, but the scaling range is narrower for larger $|q|$. \[fig:chhabra\]](fig2.pdf){width="1.00\linewidth"}
In cascade applications, it may be more meaningful to assess the scale dependence of $P(m)$ at large scales not in terms of multiples of $\eta$ as in @1995JSP....78..311S and most other work [e.g., @2007PhRvL..98h4502B], but in terms of fractions of $L$ which more closely connects to causality and energy flow. We will also express scale fractions $r/L$ in terms of cascade bifurcation *levels* $N$ needed to achieve cubes $r$ on a side: $$r/L = 2^{-N/3}.
\label{Eqn:r_over_L}$$
For example, @1995JSP....78..311S compared multiplier PDFs for dissipation in the atmospheric boundary layer over a range of scales (see figure \[fig:inertialrange\]b) and showed that $P(m)$ is highly scale independent over a wide range of scales: $372\eta-3072\eta$, or $372\eta$ to $L/86$ [see also @1992PhRvL..68.2762C]. Dissipation depends on higher-order moments of the velocity gradients, so we are drawn for guidance to the behavior seen in the higher-order moments (larger $|q|$) in figure \[fig:chhabra\] [@1989PhRvA..40.5284C]. The results of @1995JSP....78..311S are consistent with the *generally* power law behavior seen for $25\eta - 10000\eta$ (roughly $25\eta$ to $L/26$) in figure \[fig:chhabra\]. That is, one might infer from where the plots in @1989PhRvA..40.5284C deviate from power law behavior, that the scale-free behavior demonstrated by @1995JSP....78..311S (figure \[fig:inertialrange\]b) might carry on to larger sizes than they actually presented, possibly until $r \sim L/26$ or $10000\eta$, but deviate noticeably for scales larger than $r \sim L/15$ (and at the smaller end below $25\eta$). Moreover, we can conclude from these comparisons that the scale-free behavior seen by @1995JSP....78..311S was safely out of the viscous range, and continued through the inertial range at scales up to $L/86 < L/10$.
The goal of this paper is to use DNS data to derive probability distribution functions for cascade multipliers and construct a cascade model that can be used for modeling higher $Re$-number flows not accessible to direct numerical simulations. The paper is organized as follows: section \[Section:Dataset\] describes the DNS dataset used in this study; section \[Section:Analysis\] describes the data analysis including a novel technique for correcting the effects of small particle number statistics, and presents results for the multiplier PDFs for particle concentration, dissipation and enstrophy; section \[Section:NewCascade\] presents predictions of the new cascade model and comparison with DNS data at two different $Re$; and section \[Section:Discussion\] discusses the results and their implications. A summary and conclusions are given in section \[Section:Conclusions\].
Dataset {#Section:Dataset}
=======
In this paper, we use data from the direct numerical simulations of @2010JFM...646..527B; see also @2008PhRvL.100y4504A and [@2009PhRvE..80f6318B]. The simulation computes forced, homogenous and isotropic turbulence in an incompressible fluid, and the dynamics of inertial particles suspended in the flow. The fluid flow is solved on a $2048^3$ Cartesian grid with a grid spacing that is approximately the Kolmogorov length scale $\eta \approx \Delta x = \Delta y = \Delta z$. Tracer and inertial particles are introduced into the flow and their trajectories are tracked. Particles are considered point particles and are dragged with the flow by viscous forces only; there is no back-reaction on the flow. Particles of different Stokes numbers $St \equiv \tau_s / \tau_\eta$ are considered, where $\tau_s$ is the aerodynamic stopping time of the particle ($\tau_s = 0$ for tracers) and $\tau_\eta$ is the Kolmogorov time.
Figure \[fig:inertialrange\]a shows the second order structure function for particle velocity for this numerical flow [taken along trajectories of different $St$ particles, from figure 2 of @2010JFM...645..497B], as a function of normalized scale $r/\eta$. While the structure function seems to show an inertial range to several thousand $\eta$, in reality the integral eddy scale for this simulation seems to be about half the computational box size, $L \sim 1024\eta$, and for this flow $Re_{\lambda} \approx 4(L/\eta)^{2/3} \sim$ 400 [table 1 of @2010JFM...645..497B]. In blue-green below the horizontal axis we indicate the corresponding values for $r/L$, and the corresponding cascade level $N$. The blue dashed line indicates the expected inertial range for the atmospheric flow of @1987NuPhS...2...49M, with corresponding values of $r/L$ and $N$ also indicated in blue above the figure. Note that the range where $P(m)$ for dissipation was observed to be scale independent by @1995JSP....78..311S corresponds roughly to the scale range $L/860 - L/86$, well below the expected integral scale for that flow and well above the viscous subrange.
Data from this simulation are available publicly online [@RM-2007-GRAD-2048], and we have downloaded and analyzed all of the publicly available data in the present work. This data consists of the entire flow field sampled at 13 instances in time, and particle trajectories sampled at 4,720 equidistant times, both covering about 6 large-eddy time scales $\tau_L$. All flow components and their first derivatives are available at the particle locations. In total there are $3 \times 64$ files of particle trajectories each containing 3,184 particles (a total of $N_p \approx 600\textrm{k}$ particles) for each $St = 0, 0.16, 0.6$ and $1.0$, and 64 files containing 3,184 particles each (i.e., a total of $N_p \approx 200\textrm{k}$ particles) for each $St = 2, 3, 5, 10, 20, 30, 40, 50,$ and $70$.
Analysis {#Section:Analysis}
========
Determining concentration multipliers amounts to counting particles in cubic sub-volumes of size $r^3$ and calculating the fraction of particles falling in each half of the sampling box. We bisect each cube in all three orthogonal directions $x$, $y$, and $z$ each yielding 2 multiplier values totaling 6 multiplier values for $r^3$ cube. The available trajectory data is highly resolved in time (4,720 instances of time over approximately 6 large eddy times $\tau_L$), much more than what is needed for this analysis. The number of snapshots required for good statistics depends on the scale of interest since structures at large scale evolve more slowly than structures at small scale (and contain more particles), and therefore can be sampled less often. We choose to sample the particle data with a temporal spacing of $\tau_\textrm{sample} \approx 0.55~\tau_r$ where $\tau_r$ is the characteristic eddy life time at spatial scale $r$ estimated using Kolomogorov 1941 arguments as $\tau_r = \tau_L (r/L)^{2/3}$. For the box sizes considered, $512\eta$, $256\eta$, $128\eta$, $64\eta$, $45\eta$, $32\eta$, $24\eta$, $16\eta$ and $12\eta$, this results in sampling intervals of $0.34\tau_L$, $0.22\tau_L$, $0.14\tau_L$, $0.086\tau_L$, $0.067\tau_L$, $0.055\tau_L$, $0.044\tau_L$, $0.034\tau_L$ and $0.028\tau_L$, respectively. We populate the sample volume using the positions of all particles from the high-resolution trajectory files at these various discrete times.
Tracer particles {#Subsection:Analysis:Tracers}
----------------
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First, we will look at the *tracer* particles (Stokes number $St=0$) which follow the flow exactly. Particles are initially homogeneously distributed, and since the flow is incompressible, will stay that way on the average, with particle multiplier distributions given by a delta function at $m=0.5$. However, the observed distributions will only approach this behavior for very large number of particles; otherwise, effects of small number statistics complicate matters. The simulation dataset at hand has $N_p \approx 6\times10^5$ particles with $St=0$, and this is small enough to cause significant deviations from ideal behavior at small scales.
For $St=0$ particles, we can quantify these small-number effects analytically. First, the probability of finding $n$ particles in a given box of size $r$ is governed by a Poisson distribution $$\label{Eqn:Poisson}
P_P(n;\bar{n}) = \frac{\bar{n}^n e^{-\bar{n}}}{n!}$$ with the expectation value $E(n) = \bar{n}(r)$, which is here the average number of particles in a box of size $r$, that is $\bar{n}(r) = N_p r^3 / {\cal L}^3$, and where $\cal L$ is the size of the simulation domain. This probability is also the probability of finding a concentration $C=n/{\bar n}$. A comparison of the observed probability distribution functions with equation (\[Eqn:Poisson\]) is shown in figure \[Fig:TracersConcentrationAndMultiplierPDF\]a.
Then, for a given box with exactly $n$ particles, consider the particles one at a time and ask if they fall into one half of the box, say the left half, or the other, right half. For tracer particles, the probability to fall into the left side is the same as falling into the right side of the box, i.e. $p_\textrm{left}=p_\textrm{right}=p=0.5$, and the probability of having exactly $k$ particles fall into one side of the box is then given by a binomial distribution with the probability $$\label{Eqn:Binomial}
P_B(k;n,p) = {n \choose k} p^n (1-p)^{n-k}.$$ This is then also the probability of finding a multiplier $m=k/n$ in a box of $n$ particles.
By combining the probabilities (\[Eqn:Poisson\]) and (\[Eqn:Binomial\]), we see that the probability of finding a multiplier $m$ in the entire simulation domain is $$\label{Eqn:PoissonBinomial}
P(m;\bar{n},p) = \sum_n P_P(n;\bar{n}) P_B(k=mn;n,p).$$ A comparison of this analytical relation with the multiplier PDFs computed from the tracer particle trajectories in the simulation is shown in figure \[Fig:TracersConcentrationAndMultiplierPDF\]b. It shows that equation (\[Eqn:PoissonBinomial\]) models the observed distributions very accurately, and also that the distributions become rather wide at small scales even though the underlying probability distributions are delta functions at $m=0.5$. This effect is a kind of “false intermittency” due to small-number statistics alone.
Particle multipliers for non-zero Stokes numbers {#Subsection:Analysis:NonZeroStokesNumbers}
------------------------------------------------
### Correcting for finite particle numbers {#Subsubsection:Correcting}
As we have seen in the previous section, the number of particles in the dataset is small enough to significantly affect the observed multiplier distributions. In the following we will describe how we can account for these effects and estimate what the underlying PDFs would be, given infinite particle numbers. The goal is to separate the finite-particle number effects, which may be important in many applications, from the effects of the turbulent concentration process. The finite-particle-number effects in a specific application can always be added back into our model later (see, e.g., section \[Subsection:ComparisonCascadeWithDNS\]).
For the following analysis, we will assume a shape for these PDFs. It has been suggested that, at least in the atmospheric context [@1995JSP....78..311S], symmetric beta-distributions provide a good approximation for multiplier distributions of dissipation. Such distributions have also been used in previous studies of particle concentrations [e.g., @2007PhRvE..75e6305H] and are defined by $$\label{Eqn:BetaPDF}
f(m;\beta) = \left( m - m^2 \right)^{\beta-1} \frac{\Gamma(2\beta)}{2\Gamma(\beta)}$$ with $\Gamma$ being the Gamma function. The parameter $\beta$ determines the width of the distribution with small values of $\beta$ corresponding to wide, i.e. more intermittent, distributions. The width of the $\beta$ distribution (its standard deviation) is given by $$\label{Eqn:BetaWidth}
\sigma(\beta) = \sqrt{\frac{1}{4\left( 2\beta + 1 \right)}}.$$ However, similarly to the tracers, the *observed* distribution width, $\sigma_0(\beta)$, will not only depend on the underlying $\beta$ value but also on the number of particles in a given sample, $n$, and the number of samples, $N_s$, used to compute the distribution. In order to characterize this dependence, we have conducted Monte-Carlo experiments. They mimic the finite-particle-number effects in the DNS under the assumption that the underlying probability distributions are $\beta$ distributions. The procedure works in the following way: First, for a given value of $\beta$, we draw a random multiplier $m$ from the $\beta$ distribution. Given the number of particles $n$ in a sample volume (a given box), this corresponds to a partition into $n_\textrm{left} = mn$ and $n_\textrm{right} = (1-m)n$ particles for the two halves of the sample volume, where $n_\textrm{left}$ and $n_\textrm{right}$ are non-integers in general. Then, we draw a random particle number $k_\textrm{left}$ from the corresponding Poisson distribution with expectation value $E(k_\textrm{left}) = n_\textrm{left}$. This value (and $k_\textrm{right} = n - k_\textrm{left}$) represent one random sample of the number of particles found in two halves of a box, and correspond to “observed” multiplier values $m_\textrm{left}=k_\textrm{left}/n$ and $m_\textrm{right}=k_\textrm{right}/n$. We repeat the procedure many times and compute the standard deviation $\sigma$ from all random samples $m_\textrm{left}$ and $m_\textrm{right}$ combined ($N_s$ total number of samples). The result is a random sample of the “observed” distribution width given $n$, $N_s$, and the true underlying value of $\beta$. Figure \[ModelingSamples\] shows the results from such experiments for selected parameters $n$, $N_s$. As one would expect, the scatter is large for a small number of samples, $N_s$, and becomes smaller with increasing $N_s$. Also, one can see that a small number of particles per box, $n$, causes the observed distribution width to be systematically larger than the value from equation (\[Eqn:BetaWidth\]) (red and green symbols), but approaches the exact value as the number of particles gets large (blue and orange symbols).
![*Symbols:* Randomly sampled distribution widths for 4 sets of parameters $N_s, n$ as a function of $\beta$. For each parameter combination, 50 random samples are plotted. *Smooth curve:* $\beta(\sigma)$ given by equation (\[Eqn:BetaWidth\]). \[ModelingSamples\]](fig4.pdf){width="0.95\linewidth"}
Now that we understand how we can model the small-number statistics effects, we can proceed to correct the observations. Given values of $n$ and $N_s$, we conduct the above described Monte-Carlo experiments for many random values of $\beta$, and find the value of $\beta$ that results in a distribution width closest to an observed width $\sigma_0$. This gives us an estimate of the true underlying $\beta$ value. In practice, we consider random values of $\beta$ between $1$ and $\infty$ by drawing samples with equal probability with respect to their true width $\sigma(\beta)$ (equation \[Eqn:BetaWidth\]). Once we have accumulated at least $100$ samples resulting in distribution widths falling in a tolerance range between $\sigma_0 (1+\Delta)^{-1}$ and $\sigma_0 (1+\Delta)$, we compute the appropriate mean and standard deviation of those $\beta$ values. Initially, a large value of $\Delta$ is chosen, but as more and more samples are accumulated, $\Delta$ is reduced step by step by factors of $2$ until the standard deviation converges, i.e. does not change significantly if $\Delta$ is reduced further. We also require that the final $\Delta$ is no more than $0.01$ (the width is matched to within at least $1\%$ of the measured width). To speed up the process, once $\Delta \le 0.1$, we restrict the range of $\beta$ samples drawn to values between $\bar{\beta} e^{ -2 \sqrt{\text{VAR}(\ln\bar\beta)}}$ and $\bar{\beta} e^{2 \sqrt{\text{VAR}(\ln\bar\beta)}}$ where $\bar\beta$ and $\sqrt{\text{VAR}(\ln\beta)}$ are the mean value and standard deviation of $\beta$ in the tolerance range. Specifically, we define $\bar{\beta}$ as the $\beta$ value of the distribution whose variance (width squared) is the same as the arithmetic mean of the individual variances, i.e. $$\label{Eqn:MeanBeta}
\bar{\beta} = \beta(\bar{\sigma})$$ with $$\label{Eqn:MeanSigma}
\bar{\sigma}^2 = \frac{1}{M} \sum_{i=1}^{M} \sigma(\beta_i)^2$$ using equation (\[Eqn:BetaWidth\]) and its inverse $\beta(\sigma^2) = (8\sigma^2)^{-1}-\frac{1}{2}$, and where $\beta_i$ are the $\beta$ values of all the samples having widths within the tolerance range around $\sigma_0$. The uncertainty in $\bar\beta$ is measured by its variance: $$\begin{aligned}
\text{VAR}(\ln\bar\beta) = \frac{\text{VAR}(\bar\beta)}{\bar\beta} \approx \nonumber \\
\left( \frac{\partial \beta}{\partial \sigma^2}(\bar\sigma^2) \right)^2 \text{VAR}(\bar\sigma^2) = \frac{\text{VAR}(\bar\sigma^2)}{64 \bar\sigma^4}, \end{aligned}$$ following non-linear uncertainty propagation truncating the series after the first order, and where the variance of $\bar{\sigma}^2$ is computed by: $$\text{VAR}(\bar{\sigma}^2) = \frac{1}{M} \sum_{i=1}^{M} \left(\bar{\sigma}^2 - \sigma(\beta_i)^2 \right)^2.$$ Once converged, the final values of $\bar\beta$ and $\text{VAR}(\ln\bar\beta)$ are estimates of the true $\beta$ value corrected for finite-particle-number effects, and an estimate of its uncertainty.
### Volume-averaged $\beta$-distributions {#Subsubsection:MainResult}
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Now that we know how to remove the effects that a finite number of particles has on the multiplier distributions for a given box size or scale, we compute corrected volume averages of the $\beta$ values for any given spatial scale and Stokes number. Specifically, at any given scale $r$, we first compute the multiplier values for all the sampling boxes. We then compute the distribution widths $\sigma_{0,j}$ from each subset of boxes having the same number of particles $n_j$. This width is then used to derive a corrected $\beta$ value, which corresponds to a corrected width $\sigma_j$ (through equation \[Eqn:BetaWidth\]). We then combine all results by summing the square of these corrected $\sigma_j$ values weighted with the fractional volume that the boxes with $n_j$ particles occupy. The final, combined $\beta$ value is then given by this average width through the inverse of (\[Eqn:BetaWidth\]).
Figure \[Fig:MultiplierBetas\]a shows the combined concentration multiplier $\beta$ values for all Stokes numbers and all spatial scales considered. We did not consider scales smaller than $12\eta$ since the number of particles in such small boxes is too small to make reliable statistical inferences, even with our correction procedure. From the results, it is very apparent that the concentration multiplier distributions are not only functions of $St$, as has been known, but are also very much scale dependent. It seems intuitive, however, that at any given scale, the multipliers may depend only on the ratio of the particle stopping time and the dynamical time at that spatial scale. An effect of this sort was seen in particle concentration PDFs by @2007PhRvL..98h4502B. Along these lines, we construct a *local* Stokes number $$St_r \equiv \frac{\tau_s}{\tau_r} = St \left(\frac{r}{\eta}\right)^{-2/3},
\label{Eqn:LocalStokesNumber}$$ where we have assumed that the dynamical time at scale $r$ is given by $$\tau_r = \tau_\eta \left(\frac{r}{\eta}\right)^{2/3}
\label{Eqn:TauR}$$ following @1962JFM....13...82K. By plotting the multiplier $\beta$ results against the rescaled Stokes number, the curves approximately collapse into one (figure \[Fig:MultiplierBetas\]b), at least for scales not too close to the integral scale. This means that when local scale and stopping time are accounted for, the *scaled* multiplier $\beta$ curves are cascade level *independent* for scales $r < L/10$ or so, and are thus highly amenable to cascade models at, in principle, arbitrarily large $Re$ at least within the inertial range. For some caveats about $Re$-dependence however, see section \[Section:Discussion\].
Also note that $St_r$ can be written in terms of an *integral-scale* Stokes number $St_L$: $$St_r = 2^{2N/9} St_L,
\label{Eqn:Local_Stokes_Rewritten}$$ where $$St_L \equiv \frac{\tau_s}{\tau_L} = St \left(\frac{L}{\eta}\right)^{-2/3}
\label{Eqn:IntegralScaleStokesNumber}$$ using the same Kolmogorov scaling as in equation (\[Eqn:TauR\]). Equation (\[Eqn:Local\_Stokes\_Rewritten\]) separates terms that depend only on cascade level (*first term*), and particle properties (*second term*), and indicates that particles in different flows behave the same (have the same statistical cascade) if they have the same integral-scale Stokes number $St_L$. We will use this fact later in section \[Subsection:ComparisonCascadeWithDNS\] when we compare the cascade with DNS results from two different simulations at different $Re$.
It should be noted our observed $r^{-2/3}$-scaling seems to contradict the scaling found by @2007PhRvL..98h4502B for “quasi-Lagrangian” probability distribution functions of the mass density. Following an idea by @Maxey1987, they approximated the dynamics of inertial particle by those of tracers in an appropriate synthetic compressible velocity field and derived a scaling for the rate at which an $r$-sized “blob” of particles contracts. They argued that the scaling of the contraction rate relates to the scaling of the pressure field, give the contraction rate as being proportional to $r^{-5/3}$ for the $Re$ of their simulation [^2], and find that their density PDFs collapse when scaled with this contraction rate.
### Composite PDFs {#Subsubsection:CompositePDFs}
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Combining the statistics from boxes with different particle number to form a single $\beta$-distribution that represents the concentration multipliers assumes that multiplier distributions are the same for all concentrations. Since the particles are independent, that is, they do not feel the presence of each other, one would assume that there is no such concentration dependence. However, particles concentrate in particular regions of the flow, and therefore flow properties differ in regions of different particle density and so the multiplier distributions may also be different. We can relax the assumption of equal multiplier distributions and compute a *composite* multiplier distribution by first computing $\beta$-distributions for each particle concentration separately, and then summing these distributions weighted with the fractional volume that the boxes with the particular density occupy. These composite distributions are shown in figures \[PDFs512and256\], \[PDFs128and64\] and \[PDFs45and32\].
The following observations can be made: First, composite and mean $\beta$-distributions may differ in shape, although by construction they have identical widths (second moments). That is, in general, the composite $\beta$-distribution is itself *not* a $\beta$-distribution. At large spatial scales, *flat* or exponential tails are apparent in the composite PDFs; these do not have the same shape as *any* $\beta$-distribution. At small scales, however, differences disappear within the margin of accuracy. Also, at the largest scale ($512\eta$) there are enough particles such that finite-particle-number effects are small and the raw PDFs (shown in green) are essentially identical to the composite PDFs (shown in black). However, the importance of correcting for sampling effects becomes apparent as we look at smaller scales where fewer particles cause spurious widening of the raw distributions or “false intermittency.” Finally, we should mention that at the smallest scales, the number of particles is so small that we are only measuring multipliers in high-concentration regions, which causes a sampling bias since particles are known to avoid vorticity, and such regions may produce more intermittency or broader multiplier PDFs. For instance, the average number of particles in a $32\eta$ box, for the Stokes numbers for which we have a total of $6\times10^5$ particles (see section \[Section:Dataset\]) is only $\left( 32\eta/2048\eta \right)^3 N_p \approx 0.44$. The situation is even worse for, say, $r=12\eta$ and a case with only $2\times10^5$ total particles. The average is then only 0.04 particles per sampling box. Multipliers are measured only in regions with a particle concentration that is at least 100 times larger than the mean concentrations since we can only reasonably measure multipliers if we have at least several particles in a sampling box.
Dissipation and enstrophy multiplier distributions {#Subsection:Analysis:DissipationAndEnstrophy}
--------------------------------------------------
![ Dissipation and enstrophy multiplier distribution $\beta$-values as a function of spatial scale computed from the flow data and from tracer particle trajectories ($St=0$). Results from tracer particles are only shown for large spatial scales since at smaller scales not enough particles are available for a reasonable estimation of the multiplier distributions. \[Fig:DissipationEnstrophyBeta\]](fig9.pdf){width="1.0\linewidth"}
We also calculated the multipliers for fluid dissipation and enstrophy. The rate of turbulent dissipation is given by $$\epsilon = 2 \nu S_{ij} S_{ij} = \nu \left[ (\partial_i u_j) (\partial_i u_j) + (\partial_i u_j) (\partial_j u_i) \right],$$ where $S_{ij}$ and $u_i$ are the strain rate tensor and the components of the velocity field, respectively, and where we use the Einstein summation convention. The enstrophy on the other hand is defined as the square of the vorticity: $${\cal E} = |\nabla \times \vec{u} |^2 = (\partial_i u_j) (\partial_i u_j) - (\partial_i u_j) (\partial_j u_i).$$ All of these flow velocity derivatives are available in the dataset, both in the particle trajectory data files and in the flow snapshots. We have computed multiplier distribution $\beta$ values for $\epsilon$ and $\cal E$ ($\beta_\epsilon$ and $\beta_{\cal E}$) from the trajectory of tracer particles (as they sample the flow more homogeneously than non-zero Stokes number particles), and from the full resolution flow snapshots. The results are shown in figure \[Fig:DissipationEnstrophyBeta\], although the tracer data is only shown for the largest spatial scales since it also suffers from finite-particle effects (not corrected here).
Note the presence of asymptotes for $r \lesssim 20\eta$ (perhaps better thought of as $r \lesssim L/50$, see figure \[fig:hypothesis\]) for both, as anticipated [@1995JSP....78..311S]. Enstrophy is shown to have wider multiplier distributions (smaller $\beta$ values) than dissipation, and is therefore more intermittent. This is consistent with the findings of @1990PhRvA..41..894M in several flows including atmospheric flow, and in numerical simulations [e.g., @1996PhRvL..77.3799C; @1997PhRvL..79.1253C]. For a review, see @1997AnRFM..29..435S. Also, we note that even for the smallest spatial scales considered, still well within the inertial range, the dissipation rate multiplier $\beta$ does not reach the atmospheric flow values of $\beta_\epsilon \sim 3$ [@1995JSP....78..311S]. See section \[Section:Discussion\] for more discussion.
New cascade model with level-dependent multipliers {#Section:NewCascade}
==================================================
Cascade simulations {#Subsection:CascadeSimulation}
-------------------
From the collapsed $\beta(St_r)$ curves (figure \[Fig:MultiplierBetas\]b), we can build an empirical model for the particle multiplier distributions. A sum of two power laws approximates the curves for fixed scale $r$ well: $$\beta(St_r) \approx \beta_\textrm{min} \left( \left( \frac{St_r}{a_1} \right)^{b_1} + \left( \frac{St_r}{a_2} \right)^{b_2} \right),
\label{Eqn:BetaModel}$$ with parameters $a_1$, $a_2$, $b_1$, $b_2$ determining the slopes and positions of the exponentials, respectively, and $\beta_\textrm{min}$ setting the minimum $\beta$ value. From the figure it is evident that there is some residual scale dependence – the curves for different spatial scales don’t overlap exactly. The following parameterization approximates this residual dependence: $$\begin{aligned}
& a_1 = 0.15, & \nonumber \\
& a_2 = 0.45 - 0.25 \exp\left( - \frac{2}{3} \ln \left( 20.5~\frac{r}{L} \right)^2 \right), & \nonumber \\
& b_1 = -1.2, & \nonumber \\
& b_2 = 0.85 + 0.35 \left[1+\operatorname{erf}\left( - 1.8 \ln\left( 29.3~\frac{r}{L} \right) \right) \right], & \nonumber \\
& \beta_\textrm{min} = 4 + 4 \left[ 1+\operatorname{erf}\left( \ln\left( 4~\frac{r}{L} \right) \right) \right], &
\label{Eqn:Parameters}\end{aligned}$$ where $\operatorname{erf}$ and $\ln$ are the error function and the natural logarithm. The parameters asymptote for small $r/L \lesssim 3\times10^{-3}$ (or large cascade level $N \gtrsim 25$) to: $$\begin{aligned}
& a_1 = 0.15, \,\,\, a_2 = 0.45, \nonumber \\
& b_1 = -1.2, \,\,\, b_2 = 1.55, \,\,\, \beta_\textrm{min} = 4. &
\label{Eqn:AsymptoticParameters}\end{aligned}$$ The model is shown in figure \[Fig:NominalModel\] compared to the DNS results. An even simpler model could probably be constructed using a single average curve of all $\frac{r}{L} \le \frac{1}{8}$.
![Beta values of the cascade model following equations (\[Eqn:BetaModel\]) and (\[Eqn:Parameters\]) (*dashed lines*). Results from the present DNS data are shown for comparison (*solid lines*), they are the same curves as in figure \[Fig:MultiplierBetas\] except that the symbols have been suppressed here for legibility. Spatial scales, $r$, are differentiated by color (see figure legend). \[Fig:NominalModel\]](fig10.pdf){width="0.9\linewidth"}
With this model for the particle multiplier distributions, we can perform statistical cascade simulations to predict the probability distribution function for the particle concentration. We start at cascade level 0 with a single concentration value of $C^{(0)}=1.0$. At every cascade level $N$, we then draw a random multiplier value $m^{(N)}$ from the corresponding distribution with a $\beta$ value given by the model (equation \[Eqn:Parameters\]), and split the concentration value from the previous level into two values $C^{(N)}_\textrm{left} = 2 C^{(N-1)} m^{(N)}$ and $C^{(N)}_\textrm{right} = 2 C^{(N-1)} (1-m^{(N)})$. The factor 2 here comes from the fact that the concentration is the ratio of the particle number in a half box and the mean number in a half box (which itself is one half of the mean number of particles in full box). Such a cascade produces $2^N$ random samples of concentrations values at each cascade level $N$ which we use to compute concentration PDFs. For good statistics, however, we need many more samples, and for the predictions shown in the following section we computed 50,000 such cascade simulations.
Level-dependent cascade predictions compared to DNS at two different $Re$ {#Subsection:ComparisonCascadeWithDNS}
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In order to demonstrate and assess the cascade predictions, we compare the probability distribution functions of the concentration factor generated by the cascade model with those measured directly from DNS datasets. In order to do so, we need to account for the small-number effects present in the DNS results which, as we have seen before, can cause observed distributions to be significantly widened relative to *ideal* ones that the cascade produces. Instead of *correcting* the DNS PDFs as we have done before for the measured multiplier distributions, we will *degrade* the cascade PDFs for this purpose by introducing finite-particle-number effects into them.
For motivating the procedure, let us imagine a hypothetical simulation $\cal H$ with a number of particles so large that finite-particle-number effects are negligible, and let $\bar{n}_\infty(r)$ be the average number of particles in a sampling box at length scale $r$ in that simulation. Our present dataset, in comparison, has on average only $\bar{n}(r) = N_p r^3 / {\cal L}^3$ particles in a box of scale $r$, where as before $N_p$ is the total number of particles in the dataset with a given Stokes number, and $\cal{L}$ is the linear extent of the simulation domain, respectively. One can think of our current dataset as a randomly selected subset of the hypothetical simulation $\cal H$, generated by retaining particles from $\cal H$ with a probability of $p(r)=\bar{n}(r)/\bar{n}_\infty(r)$. For brevity, we will suppress the $r$ below. Specifically, let’s say some sampling box in $\cal H$ has $n_\infty$ particles in it (and therefore a concentration factor $C_\infty$ = $n_\infty / \bar{n}_\infty$). From these, we select particles with a probability of $p$, retaining in total $n$ particles, where $n$ is an integer random number with an expectation value of $E(n) = C_\infty \bar{n}$. For $\bar{n}_\infty \rightarrow \infty$, this is a Poisson process and $n$ is a random number with a probability mass function $$P_P(n;C_\infty \bar{n}) = \frac{(C_\infty \bar{n})^n e^{-C_\infty \bar{n}}}{n!}.$$
Using this idea, the recipe for introducing finite-particle-number effects into the cascade PDFs is as follows: First, we draw a random sample $C_\infty$ from a cascade-derived concentration PDF. Second, we draw a random sample $n$ from a Poisson distribution with the corresponding expectation value $E(n) = C_\infty \bar{n}$, where $\bar{n}$ is again the average number of particles at the spatial scale of interest in the DNS data we want to compare. The (integer) particle number $n$ corresponds to a discrete concentration factor $C = n / \bar{n}$. By repeating the procedure $N_s$ times, we can build from the samples a discrete probability distribution function of $C$. It accounts for the finite-particle-number effects and can be directly compared to PDFs measured from the DNS dataset.
Figure \[Fig:ComparisonCascadeWithOurDNS\] shows, for different Stokes numbers, a comparison between the PDFs predicted by the cascade model, and degraded in this way using $N_s = 10^7$, with those calculated directly from the DNS dataset we analyzed in this paper [@RM-2007-GRAD-2048].
![\[Fig:ComparisonCascadeWithPan\] Cumulative probability distribution functions for the concentration factor $C$, for different cascade levels comparing DNS results from figure 8 of @0004-637X-740-1-6 with an estimated Stokes number $St \approx 1.2$ or $St_L \approx 3.5 \times 10^{-2}$ (*solid curves*) with the prediction of our cascade model degraded to account for the finite-particle-number effects in Pan’s DNS dataset (*dotted curves*). For our cascade, we have used our $St = 5$ model with a $St_L \approx 5 \times 10^{-2}$ close to the value of Pan.](fig12.pdf){width="0.95\linewidth"}
Under the assumption that the collapsed multiplier $\beta$ curve is universal and does not depend on Reynolds number, we can use cascade simulation to model conditions at different Reynolds numbers. For caveats to this and a suggestion regarding plausible $Re$ dependence, see the discussion in section \[Section:Discussion\]. It follows from equation (\[Eqn:Local\_Stokes\_Rewritten\]) that particles of different $Re$ flows behave the same and have the same cascade statistics, if they have identical integral-scale Stokes numbers $St_L$.
Here, we compare our cascade with @0004-637X-740-1-6 who performed direct numerical simulation of a compressible flow with suspended initial particles. Their simulation is on a $512^3$ Cartesian grid with an estimated $L/\eta \sim 200$ (compared to $L/\eta \sim 1024$ in the dataset we used here), and contains $8.6\times 10^6$ particles per Stokes number. Initial comparison of their uncorrected multiplier PDFs with ours showed a clear disagreement but most of that disagreement disappears once we take into account finite-particle number effects. Figure \[Fig:ComparisonCascadeWithPan\] shows a comparison of cumulative PDFs of the concentration between @0004-637X-740-1-6 and our cascade model showing reasonable agreement, bearing in mind that the inviscid simulations of @0004-637X-740-1-6 leave a little uncertainty about the value of $St$.
Model prediction for the radial distribution function {#Subsection:RDF}
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In many applications, e.g. in terrestrial clouds, particle collisions play an important role, and it is therefore of great interest to model this process. The rate of collisions depends on two statistical quantities: the radial relative velocity between particles, and the radial distribution function (RDF), $g(r)$, defined as the probability of finding two particles at a given separation normalized with respect to homogeneously distributed particles [@0004-637X-740-1-6; @BraggCollins2014I]. Relative velocities are beyond the scope of the present model, but the cascade model can be used to make predictions for the RDF. For this purpose, we performed statistical simulations similar to section \[Subsection:CascadeSimulation\] but with an important difference: we are here interested in the spatial distribution of the concentration, which then can be used to compute the RDF.
Our cascade model, however, only describes the particle multiplier PDF as a function of cascade level, and does not explicitly contain information about *spatial correlations*. There is therefore some ambiguity in how to compute concentration fields from the cascade model. We have explored three different methods to assess the range of possible solutions. Starting at the largest scale, we divide a cube of space in half along each spatial direction. This results in eight sub-cubes with half the linear size. In order to solve for the concentrations in these sub-cubes uniquely, we need eight equations. The first method – *method A* – makes the following choice: one constraint it given by the fact that the average of the concentrations over all eight sub-cubes is equal to the mean concentration, and seven additional constraints are given by relating the concentration in seven sets of neighboring sub-cubes through multipliers chosen randomly from the cascade model. The specific choice of equations is given in appendix \[Appendix\]. Solving for concentrations to ever smaller cubes until some small cutoff length scale $(r/L)_\textrm{min}$ then yields a statistical realization of the concentration field that can be used to compute the RDF. The probability of finding two particles at a given distance is simply the product of the concentrations at the two points in space a given distance apart, averaged over the whole domain. If we start with a unit concentration at the largest scale, the normalization, that is the probability for homogeneously distributed particles, is simply 1. In order to reduce the computational cost and storage requirements to trackable amounts, we do not follow all sub-cubes to ever smaller scale but only a random selection of them. One half of sub-cubes are followed at each cascade level. Two more methods are obtained by relating the concentrations in the two half-cubes, for each direction separately, through a random multiplier. This set of equations is underdetermined. For *method B* we pick one particular solution, while for *method C* we use a least-square solver to determine the minimum-norm solution. For specifics, again, we refer to appendix \[Appendix\]. Conceptually, it is clear that the three methods allow for different amounts of spatial randomness. *Method A* clearly maximizes intermittency while *method C* leads to the least spatially intermittent solution.
Figure \[Fig:CascadeRDFs\] shows RDFs predicted by our cascade model simulations down to cascade level 72 (spatial scales of $(r/L)_\textrm{min} \approx 6 \times 10^{-8}$). All methods give qualitatively the same results, all in good agreement with the @ZaichikAlipchenkov2009 asymptotic $Re=\infty$ analytical solution, especially regarding the shape and the active range of scales. In fact, the magnitudes are even close enough, within a factor of order unity. RDFs for different $St_L$ computed using method A are plotted in figure \[Fig:CascadeRDFs\]a as a function of the scaled distance $St_L^{-3/2} r/L$, and are shown to collapse for small $St_L \lesssim 10^{-2}$. Effectively, the scaling behavior of the multiplier PDFs (figure \[Fig:MultiplierBetas\]) is carried over to the RDF. At large scales, $St_L^{-3/2} r/L \gtrsim 10^2$, the RDF has a value of 1, that is, particles are homogeneously distributed, and over an *active* range of scales, $10^{-2} \gtrsim St_L^{-3/2} r/L \gtrsim 10^2$, it rises and reaches an asymptote, $g_0$. For larger $St_L$ in our sample, however, the active range is shortened by being too close to the integral scale $L$, and $g(r)$ asymptotes at smaller values that are $St_L$-dependent. The theory of @ZaichikAlipchenkov2009 predicts a very similar behavior and their curve for infinite Reynolds number, effectively for infinitely small $St_L$, is also shown in figure \[Fig:CascadeRDFs\].
In figure \[Fig:CascadeRDFs\]b, we compare RDFs from the different methods and the @ZaichikAlipchenkov2009 model by scaling them with their respective $g_0$. The curves are nearly identical. Method A produces the highest RDF values, i.e. the most intermittent concentration distributions, and method C, as it is biased towards lower intermittency, results in the smallest asymptotic value (see table \[Tab:AsymptoticValues\]) for values of $g_0$. Interestingly, $g_0$ for method C is the same as for the @ZaichikAlipchenkov2009 theory. @ZaichikAlipchenkov2009 use various approximations in their derivation. Among them, they model the turbulence by a Gaussian process which would underestimate the tails of their probability distributions and therefore underestimate intermittency.
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Simulation / Model $g_0$
\[3pt\] Cascade simulation – Method A 13
Cascade simulation – Method B 5.9
Cascade simulation – Method C 3.9
@ZaichikAlipchenkov2009, figure 3 3.9
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: RDF asymptotic values, $g_0$, for $St_L^{-3/2} r/L \ll 1$ and $St_L \ll 1$ for the different cascade simulation methods, and the @ZaichikAlipchenkov2009 model.[]{data-label="Tab:AsymptoticValues"}
Discussion {#Section:Discussion}
==========
The nature of scale-dependent particle concentration {#Subsection:Discussion:Particles}
----------------------------------------------------
As described in section \[Section:Introduction\], the traditional explanation of clustering in terms of centrifugation of particles from eddies has been replaced with a somewhat more complicated and nuanced combination of physical processes [@BraggCollins2014I; @BraggCollins2014II]. We believe that our results (figures \[Fig:MultiplierBetas\] (right) and \[Fig:NominalModel\]), stripped of their minor variations, represent a kind of universal curve (“U-curve") for $\beta(St_r)$ that can be interpreted in terms of these different processes acting on a particle of some $St$ over a range of scales $r$.
In the $St_r < 0.1$ regime, the effect is dominated by centrifugation, which weakens as $St_r$ decreases [@Chunetal2005 and others]; thus $\beta$ increases (the multiplier PDF narrows) with decreasing $St_r$ in this regime. As $St_r$ increases beyond 1, the concentration effect is weakened by the decreasing sensitivity of particles to perturbations of any kind by eddies with timescales much shorter than their stopping times, so $\beta$ again increases. This effect, which could be thought of as an inertia impedance mismatch, has been described in terms of response functions , but see @2010JFM...645..497B [@2013ApJ...776...12P; @2012MNRAS.426..784H] and @2014ApJ...797...59H for other more recent and more sophisticated analyses.
The strongest clustering effect is produced (the multiplier PDF has the lowest $\beta$) across perhaps one or two decades of eddy scale $r$ for a given $St$, centered on the combined parameter $St_r \sim 0.3$, presumably the regime where history effects in particle velocities play the dominant role. While our results support the idea that concentration is generically due to “eddies on the scale of $\eta St_{\eta}^{3/2}$.." [@Irelandetal2015; @2007PhRvL..98h4502B], we think a more refined description one could infer from the U-curve is that clustering is the [*cumulative result of a history of interactions*]{} with the flow of energy as it cascades over eddies ranging over two decades in size, driving particles ever deeper into a concentration “attractor" even in the inertial range [@2009AnRFM..41..375T; @BraggCollins2014I; @BraggCollins2014II]. The other “universal curve" of @ZaichikAlipchenkov2003 (their figure 1) and their improved model [@ZaichikAlipchenkov2009 their figure 3] reproduced in figure \[Fig:CascadeRDFs\] also has this sense. That is, we see a parallel based on causality, between time-asymmetrical “history effects" on particles of some $St$ as they are affected by energy flowing down the cascade through eddies of different scale, and a trajectory down one side and then back up the other side of our U-curve. Such a picture would lead the particle concentration as a function of spatial binning scale to increase sharply over some particular range of scales $r/L$ or $r/\eta$ related only to $St$ (the active range), and then remain constant towards smaller scales where eddy perturbations are felt only weakly because of, essentially, the poor impedance match with the particle stopping time.
A natural prediction of this model is thus that, at infinite $Re$ where energy is available on all timescales, far from the dissipation range, and in the absence of complications such as gravitational settling or fragmentation limits, the maximum particle concentration should not only arise over a similar range of scales near $r \sim \eta St_{\eta}^{3/2}$ [@ZaichikAlipchenkov2003; @ZaichikAlipchenkov2009; @2007PhRvL..98h4502B; @Irelandetal2015], but also should have a “saturation" amplitude that is $St$-independent. Indeed this would seem to be the prediction of @ZaichikAlipchenkov2009 [their figure 3]. In current simulations [@Irelandetal2015; @ZaichikAlipchenkov2009], as well in simulations we have conducted using the cascade, the clustering of larger St particles ($St_L \gtrsim 0.03$) asymptotes at smaller values of the RDF than seen for smaller particles (figure \[Fig:CascadeRDFs\]). We expect this is because the scale at which the larger particles reach $St_r \sim 0.3$ is too close to the integral scale, so their potentially two-decade-wide range of interest, that our U-curve indicates is needed to reach a true asymptote, is truncated at large scales.
Within the dissipation range at $r < 20-30 \eta$, the energy spectrum of the flow changes as a result of the now-fixed eddy timescale $t_r =t_{\eta}$ [@Braggetal2015PRE]. Particles of $St_{\eta} \sim 1$ are now unique in that they do not experience the usual impedance mismatch with faster eddy forcing going to smaller spatial scales, so can continue to increase in concentration going to smaller scales. As noted by @Braggetal2015PRE the question remains as to whether there is any sort of rollover at $r \ll \eta$ in the RDF of $St_{\eta}$=1 particles, or whether an actual singularity would exist for point particles. In terrestrial applications, finite particle sizes comparable to $\eta$ preclude unlimited singular behavior; however, in protoplanetary nebula applications [@PanPadoan2010; @2013ApJ...776...12P; @Johansenetal2015], particle sizes of interest (submillimeter to dm) are orders of magnitude smaller than the Kolmogorov scale (km) so this is a question of significant interest.
@2007PhRvL..98h4502B explicitly described the dissipation regime as characterized by an “attractor" having fractal properties, and indeed multifractal properties were demonstrated by Hogan et al (1999) for clustering in this regime. It is known that cascades lead to fractal and multifractal spatial distributions [@1990PhRvA..41..894M; @1995JSP....78..311S and references therein]. We now suspect that the cascade of @2007PhRvE..75e6305H, in which the multiplier distributions do seem to obey level-[*independent*]{} scaling, were effectively [*dissipation range cascades*]{}. Tests by @2007PhRvE..75e6305H showed good agreement between their level-independent cascade model and DNS. However, the multiplier PDFs were determined at $3\eta$ and all their DNS results were for low-to-moderate $Re_{\lambda} < 140$ such that the integral scales were 14$\eta$, 24$\eta$, 45$\eta$, and 86$\eta$. At least the first three of these runs lie mostly within the dissipation range, where scaling does support level independence [@2007PhRvL..98h4502B; @Braggetal2015PRE]. It might be worth exploring the use of dissipation range cascades further from the standpoint of modeling fractal structure or to study higher moments of the particle density PDF. Indeed @BecChetrite2007 present what is, essentially, a cascade model that reproduces aspects of the particle concentration PDF.
Moreover, to our knowledge, while fractal/multifractal behavior has been shown for particle clustering within the dissipation regime, either at scales of a few to tens of $\eta$ [@Hoganetal1999], or scales smaller than $\eta$ [@Bec2003; @Becetal2006], no explicit study of this property in the inertial range has been done. It would be of interest to find whether the inertial range cascade as described by the U-function (figure \[Fig:MultiplierBetas\]), which is level-[*dependent*]{} but in a predictable way that is level-[*independent*]{}, would also produce such a distribution, when suitably scaled for St. This could be of use in modeling radiative transfer properties [@Shawetal2002].
Dissipation, enstrophy, and $Re$-dependence {#Subsection:Discussion:DissipationAndEnstrophy}
-------------------------------------------
As mentioned earlier in section \[Subsection:Analysis:DissipationAndEnstrophy\], our dissipation multiplier PDFs have larger $\beta$ (are narrower) than the expected $\beta_\epsilon \sim 3$ for all binning sizes we could usefully study, reaching an asymptote of $\beta_\epsilon \sim 8$ at $r \lesssim L/86$. Based on the very extensive inertial range manifested in figure \[fig:inertialrange\] [@2010JFM...645..497B], apparently extending up to $>2000\eta$, we had expected to find scale-free behavior in the dissipation multiplier PDF over most of this range. However, recalling figure \[fig:chhabra\], especially as selected by larger $|q|$, which more strongly weight the structures where most dissipation occurs, the properties of dissipation are not invariant over as wide a range as is the second order velocity structure function that defines the inertial range in @2010JFM...645..497B. Figure \[fig:hypothesis\] summarizes the overall scale variation of $\beta$ for dissipation and enstrophy, showing the spatial scale both in terms of $\eta$ and $L$ following figure \[fig:inertialrange\]. At large scales, the PDF is narrow for both (large $\beta$) but widens with decreasing scale. At scales of $\sim 12\eta$ ($\sim L/86$) it asymptotes at a value which seems to remain scale-free to smaller scales. Our new asymptotic values do not agree with values ($\beta_{\cal E} \sim 10$ for enstrophy) found in @2007PhRvE..75e6305H, or ($\beta_\epsilon \sim 3$ for dissipation) in @1995JSP....78..311S. It may be that the @2007PhRvE..75e6305H $\beta_{\cal E}$ is more properly associated with the dissipation range, but the discrepancy in $\beta_\epsilon$ alone merits some discussion.
We hypothesize that at much higher $Re$ than we can study here, the scale-dependence of $\beta_{\epsilon}$ morphs in a fashion so as to be consistent with @1989PhRvA..40.5284C and @1995JSP....78..311S; that is, has a scale-free $\beta_{\epsilon} \sim 3$ for all scales less than at least 3000$\eta$ (based on @1989PhRvA..40.5284C) and probably less than $L/16$ (based on @1989PhRvA..40.5284C and @1989PhRvL..62.1327C). At larger scales we expect $\beta$ must increase in some smooth fashion similar to ours, with an overall behavior schematically shown by the red dotted line in figure \[fig:hypothesis\]. Meanwhile, by the logic that enstrophy $\cal E$ is always more intermittent (has smaller $\beta$) than dissipation, we then also hypothesize that $\beta_{\cal E}$ varies as suggested by the blue dotted line.
We suspect that our observed $\beta_\epsilon$ and $\beta_{\cal E}$ asymptote (for $r \lesssim L/86$) at larger values than would be true for much higher $Re$, because the viscous or dissipation range, which bounds the inertial range on its small-scale end and extends to 20-30$\eta$ in general, here impinges on the small-scale end of the nascent inertial range, and may prevent the dissipation and enstrophy from ever fully realizing their high-$Re$ intermittency. In contrast, at high atmospheric $Re$, the large-scale onset of the inertial range, at $3000-10^4 \eta$ based on @1989PhRvA..40.5284C and @1995JSP....78..311S, is completely isolated from the viscous range, as indicated in figure \[fig:inertialrange\] by the blue axis labels on the upper horizontal axis.
The scale-dependence and asymptotic value of $\beta_{\cal E}$ is important, because the process of particle concentration may track the properties of $\cal E$ rather than those of $\epsilon$ based on the physics involved (section \[Subsection:Discussion:Particles\]). While it may be coincidental, in our DNS results, the $\beta$ for inertial particles minimizes at a value *very close to our value for $\beta_{\cal E}$*, and considerably smaller (more intermittent) than our value of $\beta_\epsilon$. A secondary, related hypothesis is that the particle concentration multipliers may track the behavior of enstrophy (if velocities and accelerations are dominated by vorticity), and the minima seen in the collapsed curves of figure \[Fig:MultiplierBetas\], which now never fall below 3.0, might drop to significantly lower values, making the particle concentration field more intermittent. For this reason, cascades developed using our *current* collapsed $\beta(St_r)$ curves may underestimate the abundance of zones of high concentration at high $Re$ to some degree. A better understanding of this $Re$-dependence will be needed to put cascade modeling of particle concentration on quantitatively solid ground. The original dataset of @1995JSP....78..311S probably contains enough information to assess the validity of these hypotheses regarding dissipation, but not for enstrophy.
![Hypothetical scaling behavior of dissipation and enstrophy: The solid red and blue symbols, as in figure \[Fig:DissipationEnstrophyBeta\], are those we calculate from the numerical flow at $Re_{\lambda}=400$. Dotted lines are hypothetical values at very high $Re$. High-$Re$ atmospheric dissipation [@1995JSP....78..311S] is scale-free at $\beta \approx 3$ at least between $L/86-L/860$; it is interesting that whatever changes affect the weighted quantities in figure \[fig:chhabra\] and the structure function in figure \[fig:inertialrange\], the $\beta$ values for dissipation and enstropy do not seem to vary through the viscous (dissipation) subrange $r/\eta < 20$. It is plausible and expected that enstrophy will always be more intermittent than dissipation (have smaller $\beta$ values). The atmospheric (high-$Re$) value for the scale-free asymptote for dissipation is roughly $\beta$=3 (black dotted line). We hypothesize that $\beta$ values at high $Re$ follow trajectories similar to the red and blue dotted lines for dissipation and enstrophy respectively. That is, the observed behaviors (symbols) is the effect of an incompletely developed inertial range. \[fig:hypothesis\]](fig14.pdf){width="\linewidth"}
Speculations regarding the effect of higher $Re$-numbers
--------------------------------------------------------
As described in section \[Section:NewCascade\], a level-dependent cascade can be described which captures the inertial range behavior for arbitrary $Re$ (at least to the point where mass loading starts to affect the physics, e.g., see @2007PhRvE..75e6305H). This might be of use in modeling particle concentration in rain clouds, in the protoplanetary nebula, or in other applications at very high $Re$ where numerical simulations are impractical (such as spatial variations in microwave opacity). There are several reasons to expect different behavior in the dissipation range.
Recalling, however, our numerical discrepancy with @1995JSP....78..311S and others regarding the asymptotic value of $\beta$ for dissipation, we think the possibility of $Re$-dependence of even our collapsed $\beta(St_r)$ might imply that our results (and other inertial range results at low $Re$) underestimate effects of concentration, in the sense that at higher $Re$, the minimum in the universal curve would move to lower $\beta$ (more intermittency and higher concentrations). One might speculate that the minimum $\beta$ for particles should track the $\beta$ for enstrophy instead of for dissipation. In the application to planetesimal formation proposed by @2010Icar..208..518C, it is necessary to create a joint PDF of particle concentration *and* enstrophy. This motivates a better understanding of high $Re$ behavior of $\beta$ for both particles and enstrophy.
Conclusions {#Section:Conclusions}
===========
Our results indicate that the multiplier PDFs for particle concentration (section \[Subsection:Analysis:NonZeroStokesNumbers\]), dissipation, and enstrophy (section \[Subsection:Analysis:DissipationAndEnstrophy\]) vary with scale, at least over the largest decade of spatial scales. We also find that the multiplier PDFs for particle concentration have two components: a traditional “$\beta$-function" component, and an exponential-tail component (section \[Subsubsection:CompositePDFs\]). We find that the concentration multiplier $\beta$ values collapse to a *scale*-independent universal curve when plotted against an appropriately scaled *local Stokes number* $St_r = St (r/\eta)^{-2/3}$ (section \[Subsubsection:MainResult\], in particular equation (\[Eqn:LocalStokesNumber\]) and figure \[Fig:MultiplierBetas\]), allowing the cascade model to be used for modeling higher $Re$ conditions not accessible to numerical simulation.
For dissipation, the “$\beta \sim 3$" asymptotic behavior of @1995JSP....78..311S in high $Re$ atmospheric flows probably appears at around $r \sim L/30$ or $L/40$, and remains constant to smaller scales, at least to $r \sim 20-30\eta$ where the dissipation range begins. In the present simulation, the integral scale and the dissipation range are not separated far enough for the dissipation $\beta_\epsilon$ to reach such low values, and instead it asymptotes for scales below $r \sim 20\eta$ to a value of $\beta_\epsilon \sim 8$. Enstrophy, believed to be always more intermittent (smaller $\beta$) than dissipation, asymptotes in the DNS to a value of $\beta_{\cal E} \sim 3$. In light of the connection between vorticity and the acceleration, centrifuging and concentrating of particles, it may not be surprising that this value coincides with the minimum $\beta$ value for particle concentration multipliers at the optimal $St_r$. Given that dissipation, and presumably enstrophy, have not reached their scale-independent, asymptotic values seen in very high $Re$ atmospheric flows, it could be expected that our collapsed particle multiplier $\beta(St_r)$ curve is also $Re$ dependent. Analyses of this sort for DNS of particle-laden flows at significantly higher Reynolds number are therefore highly desirable, as are measurements of enstrophy multipliers in very high Reynolds number flows such as atmospheric flows.
We have also found that the cascade model can be used to construct a spatial distribution of particle concentration, that can be carried to arbitrarily high Reynolds numbers, and has a very good resemblance to the analytical theory of @ZaichikAlipchenkov2009. More work is needed to assess the asymptotic level of maximum concentration for particles of any size (which we find, as did @ZaichikAlipchenkov2009, is size invariant, in the infinite $Re$ limit).
**Acknowledgements:** The authors would like to thank Nic Brummell, Karim Shariff, Katepalli Sreenivasan, Federico Toschi and Alan Wray for insightful discussions on the topic of this paper, and Federico Toschi and Enrico Calzavarini for help accessing and using the DNS data.
We would like to dedicate this paper to Bob Hogan, who passed away in February 2012. Bob was responsible for all the previous computational work done on turbulent concentration by our group from 1992-2010, including multifractal behavior and our first cascades. He would be very interested in these new results, which explain a discrepancy that arose too late for him to resolve.
Three methods for computing the spatial distribution of concentrations {#Appendix}
======================================================================
We here provide more details about the three methods for computing spatial concentration fields from multiplier distributions that we used in section \[Subsection:RDF\] for computing radial distribution functions.
Let us consider a cube of size $r/L=2^{-N/3}$ (cascade level $N$) having a concentration $C^{(N)}$, initially we start with a cascade level 0 cube having a unit concentration $C^{(0)}=1$, and divide it in half along each spatial direction. This results in eight sub-cubes with half the linear size which correspond to a cascade level of $N+3$ (equation \[Eqn:r\_over\_L\]). We denote the concentrations in these sub-cubes as $C_{ijk}^{(N+3)}$, where $i,j,k \in [1,2]$ are indices denoting the left (1) or right (2) sub-cube in the three spatial directions. Since there are eight unknowns, we need eight equations to uniquely determine the concentrations.
The first method – *method A* – makes the following choice: One constraint is given by the fact that the average of the concentrations over all eight sub-cubes is equal to the mean concentration, that is $$\sum_{i,j,k} \frac{1}{8} C^{(N+3)}_{ijk} = C^{(N)}.
\label{Eqn:MethodA1}$$ We get seven more constraints by relating the concentration of neighboring sub-cubes to multipliers that are chosen randomly from the cascade model. A possible choice are the combinations $$\begin{aligned}
(C^{(N+3)}_{111} + C^{(N+3)}_{211}) m_1^{(N+3)} & = & C^{(N+3)}_{111}, \nonumber \\
(C^{(N+3)}_{112} + C^{(N+3)}_{212}) m_3^{(N+3)} & = & C^{(N+3)}_{112}, \nonumber \\
(C^{(N+3)}_{111} + C^{(N+3)}_{121}) m_5^{(N+3)} & = & C^{(N+3)}_{111}, \nonumber \\
(C^{(N+3)}_{112} + C^{(N+3)}_{122}) m_7^{(N+3)} & = & C^{(N+3)}_{112}, \nonumber \\
(C^{(N+3)}_{121} + C^{(N+3)}_{221}) m_2^{(N+3)} & = & C^{(N+3)}_{221}, \nonumber \\
(C^{(N+3)}_{122} + C^{(N+3)}_{222}) m_4^{(N+3)} & = & C^{(N+3)}_{122}, \nonumber \\
C^{(N+3)}_{121} + C^{(N+3)}_{112}) m_6^{(N+3)} & = & C^{(N+3)}_{121},
\label{Eqn:MethodA2}\end{aligned}$$ where $m^{(N+3)}_i$ with $i \in [1,...,7]$ are seven random multiplier values at cascade level $N+3$. Equations (\[Eqn:MethodA1\]) and (\[Eqn:MethodA2\]) are linearly independent and can be solved directly. Applying this procedure recursively to ever smaller cubes until some small cutoff length scale $(r/L)_\textrm{min}$ yields one statistical realization of the concentration field.
The two other methods are obtained by relating the concentrations in the two half-cubes, for each direction separately, through a random multiplier, that is $$\begin{aligned}
\sum_{j,k} C^{(N+3)}_{1ik} = 2 C^{N} m_1^{(N+1)},& \,\,\,\,\, & \sum_{i,k} C^{(N+3)}_{i1k} = 2 C^{N} m_2^{(N+1)},\nonumber \\
\sum_{i,j} C^{(N+3)}_{ij1} = 2 C^{N} m_3^{(N+1)}, & \,\,\,\,\, &
\label{Eqn:MethodBC}\end{aligned}$$ where $m^{(N+1)}_i$ with $i \in [1,...,3]$ are multipliers randomly drawn from the level $N+1$ cascade model. Since these are only three equations, the linear system is underdetermined.
For *method B* we choose one particular solution to equations (\[Eqn:MethodBC\]), namely $$\begin{aligned}
\frac{C^{(N+3)}_{111}}{C^{N}} & = & \left(\;\;\;\;\;\; m_1^{(N+1)}\right) \left(\;\;\;\;\;\; m_2^{(N+1)}\right) \left(\;\;\;\;\;\; m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{211}}{C^{N}} & = & \left(1 - m_1^{(N+1)}\right) \left(\;\;\;\;\;\; m_2^{(N+1)}\right) \left(\;\;\;\;\;\; m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{121}}{C^{N}} & = & \left(\;\;\;\;\;\; m_1^{(N+1)}\right) \left(1 - m_2^{(N+1)}\right) \left(\;\;\;\;\;\; m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{221}}{C^{N}} & = & \left(1 - m_1^{(N+1)}\right) \left(1 - m_2^{(N+1)}\right) \left(\;\;\;\;\;\; m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{112}}{C^{N}} & = & \left(\;\;\;\;\;\; m_1^{(N+1)}\right) \left(\;\;\;\;\;\; m_2^{(N+1)}\right) \left(1 - m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{212}}{C^{N}} & = & \left(1 - m_1^{(N+1)}\right) \left(\;\;\;\;\;\; m_2^{(N+1)}\right) \left(1 - m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{122}}{C^{N}} & = & \left(\;\;\;\;\;\; m_1^{(N+1)}\right) \left(1 - m_2^{(N+1)}\right) \left(1 - m_3^{(N+1)}\right), \nonumber \\
\frac{C^{(N+3)}_{222}}{C^{N}} & = & \left(1 - m_1^{(N+1)}\right) \left(1 - m_2^{(N+1)}\right) \left(1 - m_3^{(N+1)}\right). \nonumber \\
& \, &
\label{Eqn:MethodB}\end{aligned}$$ There is an ambiguity, of course, as to whether multiplier $m^{(N+1)}_i$ is applied to the “left” or “right” half of each box, but since the multipliers $m^{(N+1)}_i$ and $(1-m^{(N+1)}_i)$ have equal probability, either choice would be paired with the opposite given enough random samples (and we do average many random samples to construct each cascade). The eight sub-cubes from each large box have the identical concentrations, just differently distributed, for each set of $m^{(N+1)}_i, i \in [1,...,3]$ whether $m^{(N+1)}_i$ goes to the left or right box in each case.
For the third and final method, *method C*, we use a least-square solver to determine the minimum-norm solution of (\[Eqn:MethodBC\]), that is the solution that is closest to equally distributed concentrations. Clearly, this causes a bias towards the least intermittent spatial distribution. For strongly intermittent multipliers, this method can even lead to negative concentrations in one of the sub-cubes. Such solutions have to be discarded, and this further biases method C towards minimal intermittency.
[^1]: These results were cited and reanalyzed by @1995JSP....78..311S, however the reference to the basic data given by @1995JSP....78..311S is confused with an interpretive article by @1989PhRvL..62.1327C, who themselves cite @1987PhRvL..59.1424M and @1989PhRvA..40.5284C [at the time unpublished] for discussions and analysis of the basic data.
[^2]: They argue that the scaling should change to $r^{-4/3}$ for very high Reynolds numbers $Re_\lambda \ge 600$
|
---
abstract: 'In this paper it is shown how the Heisenberg group of order 27 can be used to construct quotients of degenerate Sklyanin algebras. These quotients have properties similar to the classical Sklyanin case in the sense that they have the same Hilbert series, the same character series and a central element of degree 3. Regarding the central element of a 3-dimensional Sklyanin algebra, a better way to view this using Heisenberg-invariants is shown.'
address: |
Department of Mathematics, University of Antwerp\
Middelheimlaan 1, B-2020 Antwerp (Belgium)\
[kevin.delaet2@uantwerpen.be]{}
author:
- Kevin De Laet
title: Quotients of degenerate Sklyanin algebras
---
Introduction
============
The 3-dimensional Sklyanin algebras form an important class of noncommutative graded algebras, as they correspond to the notion of a noncommutative ${\mathbb{P}}^2$ following Artin, Tate, Van den Bergh and others (see for example [@ATV1] and [@ATV2]). These algebras are parametrized by an elliptic curve $\xymatrix{E \ar@{^{(}->}[r] & {\mathbb{P}}^2}$ and a point $\tau \in E$. They form the largest class of examples of quadratic 3-dimensional AS-regular algebras, that is, graded algebras of global dimension 3 with relations in degree 2 with excellent homological properties. These AS-regular algebras can be described as the quotient of $ {\mathbb{C}}\langle x,y,z\rangle$ by the relations $$\begin{cases}
a yz + b zy + c x^2,\\
a zx + b xz + c y^2,\\
a xy + b yx + c z^2,
\end{cases}
\label{eq:Sklyanin}$$ with $[a:b:c]\in {\mathbb{P}}^2$ not one of the 12 points of $$\begin{aligned}
\{[0:0:1],[0:1:0],[1:0:0]\}\cup \{[a:b:c]\in {\mathbb{P}}^2 | a^3=b^3=c^3=1\}.
\label{eq:nonreg}\end{aligned}$$ It was remarked in many early papers (see for example [@ASregular]) about these algebras that there was a central element of degree 3, which was somewhat mysterious. In [@DeLaetLeBruyn], a intrinsic presentation of this central element was found. It turns out that the central element gave a connection between the Sklyanin algebra $\mathcal{A}_\tau(E)$ and the algebra $\mathcal{A}_{-2\tau}(E)$ if $E$ is an elliptic curve and $\tau \in E$. This was proved using the concept of superpotentials, as explained in for example [@Walton].
The first purpose of this paper is to give a better statement of this theorem in terms of Heisenberg invariants and the degree 3 part of the Koszul dual of a Sklyanin algebra.
The second purpose is the study of quotients of the 12 nonregular algebras. In particular, we show that there is a 1-dimensional family of quotients of each of these 12 algebras parametrized by ${\mathbb{C}}^*$ such that the quotients have Hilbert series $\frac{1}{(1-t)^3}$. In addition, these algebras also have a central element of degree 3, fixed by the Heisenberg group. We also show that for the constructed quotients of the algebra ${\mathbb{C}}\langle x,y,z\rangle/(x^2,y^2,z^2)$, the $n$th roots of unity give quotients that are finite modules over their center of PI-degree $2n$.
Notation
--------
In this article, we use the following notations:
- $\mathbf{V}(I)$ for $I \subset {\mathbb{C}}[a_1,\ldots,a_n]$ an ideal is the Zariski-closed subset of ${\mathbb{A}}^n$ or ${\mathbb{P}}^{n-1}$ determined by $I$, it will be clear from the context if the projective or affine variety is used.
- $D(I)$ for $I$ an ideal $I \subset {\mathbb{C}}[a_1,\ldots,a_n]$ is the open subset ${\mathbb{A}}^n \setminus \mathbf{V}(I)$ or ${\mathbb{P}}^{n-1} \setminus \mathbf{V}(I)$, it will be clear from the context if it is an open subset of affine space or of projective space. If $I = (a)$, then we write $D(a)$ for $D(I)$.
- ${\mathbb{Z}}_n = {\mathbb{Z}}/n{\mathbb{Z}}$ for $n \in \mathbb{N}$.
- $\operatorname{Grass}(m,n)$ will be the projective variety parametrizing $m$-dimensional vector spaces in ${\mathbb{C}}^n$.
- For an algebra $A$ and elements $x,y \in A$, $\{x,y\} = xy + yx$ and $[x,y] = xy-yx$.
- For $V$ a $n$-dimensional vector space, we set $T(V) = \oplus_{k=0}^\infty V^{\otimes k}$, the tensor algebra over $V$.
- Every graded algebra $A$ will be positively graded, finitely generated over ${\mathbb{C}}$ and connected, that is $A_0 = {\mathbb{C}}$.
- The group ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_m(p)$ (respectively ${{\text{\em \usefont{OT1}{cmtt}{m}{n} PSL}}}_m(p)$) is the special linear group (respectively projective special linear group) of degree $m$ over the finite field with $p$ elements.
- For any vector space $V$, ${\mathbb{C}}[V] = T(V)/(wv-vw|w,v \in V)$.
- If $A$ is a connected, finitely generated, positively graded algebra, then the Hilbert series is defined as $H_A(t) = \sum_{k=0}^\infty \dim A_k t^k$.
- Take a reductive group $G$ and 2 finite dimensional representations $V,W$ of $G$. Then $\operatorname{Emb}_G(V,W)$ is the set of injective linear $G$-maps from $V$ to $W$ up to $G$-isomorphisms of $V$.
- Modules will always be left modules unless otherwise mentioned.
$G$-algebras {#sec:Galg}
============
This section is a summary of the general theory developed in [@DeLaet2].
Let $G$ be a reductive group. We call a positively graded connected algebra $A$, finitely generated in degree 1, a *$G$-algebra* if $G$ acts on it by gradation preserving automorphisms.
This definition implies that there exists a finite dimensional representation $V$ of $G$ such that $T(V)/I \cong A$ with $I$ a graded ideal of $T(V)$, which is itself a $G$-subrepresentation of $T(V)$.
One can make quadratic $G$-algebras as follows. Let $V$ be a $G$-representation. Then $V \otimes V$ is also a $G$-representation which decomposes as a summation of simple representations, say $V \otimes V \cong \oplus_{i=1}^m S_i^{a_i}$ where the $S_i$ are distinct simple representations of $G$ and $a_i \geq 0$. A $G$-algebra $A$ is then constructed by taking embeddings of the $S_i$ in $V \otimes V$ as relations of $A$.
One can of course do the same for other degrees by taking relations in $T(V)_i = V^{\otimes i} \cong \oplus_{j=1}^m S_j^{a_j}$ and take different embeddings of the simple representations of $G$ in $V^{\otimes i}$ as relations.
Let $A$ be a $G$-algebra with corresponding ideal $I$ of $T(V)$. We call $B$ a *$G$-deformation* of $A$ up to degree $k$ if $B$ is also a quotient of $T(V)$ such that $\forall 1 \leq i \leq k: A_i \cong B_i$ as $G$-representations. We will call $B$ a $G$-deformation if $\forall i \in {\mathbb{N}}: A_i \cong B_i$ as $G$-representations.
If the relations for $A$ are all of the same degree $k$, then all $G$-deformations up to degree $k$ of $A$ depend on a product of Grassmannians. For example, let $A$ be a quadratic algebra of which we want to find all $G$-deformations up to degree 2. Let $I_2 = \oplus_{i=1}^m S_i^{e_i} \subset V \otimes V = \oplus_{i=1}^m S_i^{a_i}$ with $S_i$ distinct simple representations and $0 \leq e_i \leq a_i$ natural numbers. Then the $G$-deformations up to degree 2 are parametrized by $\operatorname{Emb}_G(\oplus_{i=1}^m S_i^{e_i},\oplus_{i=1}^m S_i^{a_i})=\prod_{i=1}^m \operatorname{Grass}(e_i,a_i)$.
In general, the total set of $G$-deformations up to degree $k$ of a $G$-algebra $A = T(V)/I$ are determined by a Zariski closed subset of $$Z_k=\prod_{j=1}^k \prod_{S_i \text{ simple}}\operatorname{Grass}(e_{i,j},a_{i,j})$$ with $I_j = \oplus_{S_i \text{ simple}} S_i^{e_{i,j}} \subset T(V)_i = \oplus_{S_i \text{ simple}} S_i^{a_{i,j}}$
We say that a variety $Z$ *parametrizes $G$-deformations up to degree $k$* of a $G$-algebra $A$ if $\xymatrix{Z \ar@{^{(}->}[r]^-\phi & Z_k}$ can be embedded in $Z_k$ and the point corresponding to $A$ in $Z_k$ belongs to the image of $\phi$. We say that $Z$ *parametrizes $G$-deformations* of $A$ if $Z$ parametrizes $G$-deformations up to degree $k$ for some $k$ and for each point $x \in Z$ with corresponding algebra $A_x$, we have $$\forall i \in {\mathbb{N}}: (A_x)_i \cong A_i \text{ as $G$-representations}.$$
We will show in the next section that the 3-dimensional Sklyanin algebras are $H_3$-deformations of the polynomial ring ${\mathbb{C}}[V]$. We first show a computational way to decode how a $G$-algebra decomposes as a $G$-module.
Character series {#sec:kos}
----------------
Given a $G$-algebra $A$, it is a natural question to ask how $A$ behaves as a $G$-module. As $G$ acts as gradation preserving automorphisms, we have a decomposition $$A = \bigoplus_{k=0}^{\infty} \bigoplus_{S \text{ simple}} S^{e_{k,S}}$$ with almost all $e_{k,S}$ equal to 0. We will only consider the case that $G$ is finite.
Let $G$ be a finite group. The *character series* for an element $g \in G$ and for a $G$-algebra $A$ is a formal sum $$Ch_A(g,t) = \sum_{n \in \mathbb{Z}} \chi_{A_n}(g) t^n.$$
For example, if $g = 1$, then we have $Ch_A(1,t) = H_A(t)$, the Hilbert series of $A$. As a character of a representation is constant on conjugacy classes, we can represent the decomposition of $A$ in simple $G$-representations as a vector of length equal to the number of conjugacy classes and on the $i$th place the character series $Ch_A(g,t)$ with $g \in C_i$, the $i$th conjugacy class.
Let $V$ be a simple representation of $G$ and let $A$ be a $G$-algebra constructed from $T(V)$. For every element $z$ of the center, we have that $Ch_A(z,t) = H_A(\lambda t)$, where $z$ acts on $V$ by multiplication with $\lambda$. \[lem:cent\]
It follows that in degree $k$ the action of $z$ on $A_k$ is given by multiplication with $\lambda^k$, so the character series for the element $z$ in this case is given by $$Ch_A(z,t)=\sum_{k=0}^\infty \lambda^k \dim A_k t^k = H_A(\lambda t).$$
The finite Heisenberg group of order $27$
=========================================
While in previous papers (see [@DeLaet] and [@DeLaet2]) we needed the finite Heisenberg group of order $p^3$ for any odd prime $p$, we will only consider here the special case $p=3$.
The *Heisenberg group of order 27* is the finite group given by generators and relations $$H_3=\langle e_1, e_2 |~[e_1,e_2]\text{ central},~e_1^3=e_2^3=1 \rangle.$$
$H_3$ is a central extension of the group ${\mathbb{Z}}_3 \times {\mathbb{Z}}_3$ with ${\mathbb{Z}}_3$, $$\begin{aligned}
\xymatrix{1 \ar[r]& \mathbb{Z}_3 \ar[r] & H_3 \ar[r]&\mathbb{Z}_3 \times \mathbb{Z}_3 \ar[r]& 1}.
\label{eqn:centralext}\end{aligned}$$ $H_3$ has 9 1-dimensional representations coming from the quotient $H_3/([e_1,e_2]) = {\mathbb{Z}}_3 \times {\mathbb{Z}}_3$ and 2 3-dimensional simple representations, corresponding to the primitive 3rd roots of unity. These 2 representations are defined in the following way: let $\omega $ be a primitive 3rd root of unity and let $V_1={\mathbb{C}}^3 = {\mathbb{C}}x_0 + {\mathbb{C}}x_1 + {\mathbb{C}}x_2$. Then the action is defined by $$\begin{aligned}
e_1 \cdot x_i = x_{i-1},& e_2 \cdot x_i = \omega^i x_i.\end{aligned}$$ $V^*$ is the representation corresponding to $\omega^2$ and will be denoted by $V_2$. We will use $\chi_{a,b}$ for the character $\xymatrix{H_3 \ar[r]^-{\chi_{a,b}}& {\mathbb{C}}}$ defined by $$\begin{aligned}
\chi_{a,b}(e_1) = \omega^a, & \chi_{a,b}(e_2) = \omega^b.\end{aligned}$$ There is an action of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ on $H_3$ as group automorphisms by the rule $$\begin{bmatrix}
A & B \\ C & D
\end{bmatrix}\cdot e_1 = e_1^A e_2^C,
\begin{bmatrix}
A & B \\ C & D
\end{bmatrix}\cdot e_2 = e_1^B e_2^D.$$ The central element $[e_1,e_2]$ is fixed by this action. From this it follows that the induced action of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ on the simple representations fixes $V_1$ and $V_2$.
3-dimensional Sklyanin algebras are $H_3$-deformations
------------------------------------------------------
Using the construction of Section \[sec:Galg\], we will now show that the 3-dimensional Sklyanin algebras are $H_3$-deformations of ${\mathbb{C}}[V]$ with $V= V_1$ as defined above.
Write $ \mathcal{A}={\mathbb{C}}[V] = T(V)/(V\wedge V)$. In order to find all $H_3$-deformations up to degree 2 of $\mathcal{A}$, we need to decompose $V \otimes V$ in $H_3$-representations. A quick calculation shows that $V \otimes V = (V^*)^3$ and that the $H_3$-generators of $V \otimes V$ are given by $$\begin{aligned}
x_1 x_2 - x_2 x_1, & x_1 x_2 + x_2 x_1, & x_0^2.\end{aligned}$$ The first generator corresponds to the wedge product $V \wedge V$. Taking another copy of $V^*$ in $V\otimes V$ corresponds to an element of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Grass}}}(1,3)$. Given $p=[A:B:C] \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} Grass}}}(1,3) = {\mathbb{P}}^2$, then $p$ determines the quotient ${\mathbb{C}}\langle x_0,x_1,x_2 \rangle/(I)$ with $I$ generated by the relations $$\begin{cases}
A(x_1 x_2 - x_2 x_1) + B(x_1 x_2 + x_2 x_1) + C(x_0^2),\\
A(x_2 x_0 - x_0 x_2) + B(x_2 x_0 + x_0 x_2) + C(x_1^2),\\
A(x_0 x_1 - x_1 x_0) + B(x_0 x_1 + x_1 x_0) + C(x_2^2).
\end{cases}$$ Putting $a = A+B$, $b = B-A$ and $c=C$, one gets the familiar relations of the 3-dimensional Sklyanin algebras as in equation \[eq:Sklyanin\].
In particular, the 3-dimensional Sklyanin algebras have the same Hilbert series as the polynomial ring in 3 variables. Using the results of [@DeLaet2], we see that the character series with respect to $H_3$ of any Sklyanin algebra $\mathcal{B}$ is the same as the character series of $\mathcal{A}$. In fact, this is true for all Artin-Schelter regular algebras parametrized by points of $\operatorname{Emb}_{H_3}(V^*,(V^*)^3)$.
In addition, the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ (left) action on $H_3$ induces a (right) action on $\operatorname{Emb}_{H_3}(V^*,(V^*)^3) \cong {\mathbb{P}}^2$. This projective representation of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ has the property that points lying in the same orbit determine isomorphic algebras. In particular, the only non-regular algebras are those lying in the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$-orbit of either the point $[1:0:0]$ or $[0:0:1]$. For more information, see amongst others [@abdelgadir2014compact], [@DeLaet].
Central elements and Heisenberg invariants {#sec:central}
==========================================
In [@DeLaetLeBruyn], it was proved that there was a connection between the superpotential defining the Sklyanin algebra $\mathcal{A}_{-2\tau}(E)$ and the central element $c_3$ of degree 3 in $\mathcal{A}_{\tau}(E)$. This connection however can better be explained using Heisenberg invariant elements of $V \otimes V \otimes V$.
Let $p=[a:b:c]$ and define $S_p = {\mathbb{C}}\langle x,y,z \rangle /(R_p)$ to be the algebra with relations $$R_p=\begin{cases}
a yz + b zy + c x^2,\\
a zx + b xz + c y^2,\\
a xy + b yx + c z^2.
\end{cases}$$ Let $W_p \subset V \otimes V$ be the vector space spanned by these relations, with $V = {\mathbb{C}}x + {\mathbb{C}}y +{\mathbb{C}}z$. Then $W_p \otimes V \cap V \otimes W_p$ is generically a 1-dimensional vector space, generated by $$a(zxy+xyz+yzx) + b(yxz+zyx+xzy) + c (x^3+y^3+z^3).$$ This element is easily seen to be fixed by $H_3$. In turn, any element $g \in {\mathbb{P}}((V \otimes V \otimes V)^{H_3})$ determines quadratic relations by taking the cyclic derivatives $\delta_x,\delta_y,\delta_z$ as in [@DeLaetLeBruyn]. For AS-regular algebras, $W_p \otimes V \cap V \otimes W_p$ is 1-dimensional, so on the open subset ${\mathbb{P}}^2 \setminus D$ with $D = \bigcup_{g \in {{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)} g \cdot\{[1:0:0],[0:0:1]\}$, we have an injective morphism $$\xymatrix{{\mathbb{P}}^2\setminus D \ar[r]^-\phi& {\mathbb{P}}((V \otimes V \otimes V)^{H_3}) = {\mathbb{P}}^2}$$ In order for this to extend to $\operatorname{Emb}_{H_3}(V^*,(V^*)^3) = {\mathbb{P}}^2$ and to get an isomorphism, one should need that $(W_p \otimes V \cap V \otimes W_p)^{H_3}$ is always 1-dimensional. This is indeed the case. Let $\chi_{a,b}$ be the 1-dimensional representation of $H_3$ defined by $\chi_{a,b}(e_1) = \omega^a, \chi_{a,b}(e_2) = \omega^b$.
Let $\mathbb{V}=\mathbf{V}(abc)\subset {\mathbb{P}}^2_{[a:b:c]}$. Then for each vertex $p$ of $\mathbb{V}$, the decomposition of $W_p\otimes V \cap V \otimes W_p$ in $H_3$-representations is given by $\chi_{0,0}\oplus \chi_{1,0}\oplus\chi_{2,0}$. In particular, $(W_p\otimes V \cap V \otimes W_p)^{H_3}$ is 1-dimensional.
As these algebras are monomial algebras, it is easy to find a basis of $W_p\otimes V \cap V \otimes W_p$. We have
- for $[0:0:1]$, we find ${\mathbb{C}}x^3+{\mathbb{C}}y^3+{\mathbb{C}}z^3$,
- for $[1:0:0]$, we find ${\mathbb{C}}zxy+{\mathbb{C}}xyz+{\mathbb{C}}yzx$,
- for $[0:1:0]$, we find ${\mathbb{C}}yxz+{\mathbb{C}}zyx+{\mathbb{C}}xzy$.
In these 3 cases, it is clear that $e_2$ works trivially on these elements and $e_1$ works by cyclic permutation, leading to the claimed decomposition.
Now, the other points that correspond to nonregular algebras lie in the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$-orbit of these 3 points. In order to prove that $(R_p \otimes V \cap V \otimes R_p)^{H_3}$ is indeed 1-dimensional, we need to work out what the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$-orbits are in the set of simple representations of $H_3$. From the natural action of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ on ${\mathbb{Z}}_3 \times {\mathbb{Z}}_3$ we find that
- $\chi_{00}$ is fixed,
- $V$ and $V^*$ are fixed because the center is fixed,
- the action is transitive on the set $\chi_{a,b}$, $(a,b) \neq (0,0)$.
Now, the vector spaces of the theorem have one thing in common: the action of $e_2$ is fixed. If one takes the $H_3$-representations $W_{a,b} = \chi_{a,b} \oplus \chi_{-a,-b}, a,b \in \{0,1\} $, then the center of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ works trivially on the set $\{W_{a,b}|a,b \in \{0,1\} \}$, so the action is really a ${{\text{\em \usefont{OT1}{cmtt}{m}{n} PSL}}}_2(3)$-action.
From this, we see that the induced action of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$ on the decomposition of $W_{[1:0:0]} \otimes V \cap V \otimes W_{[1:0:0]}$ or $W_{[0:0:1]} \otimes V \cap V \otimes W_{[0:0:1]}$ sends $\chi_{0,0}\oplus W_{1,0}$ to $\chi_{0,0} \oplus W_{a,b}$ for some $a,b \in \{0,1\}$. We have proved
$(W_p\otimes V \cap V \otimes W_p)^{H_3}$ is 1-dimensional for every $p \in {\mathbb{P}}^2$.
So $\phi$ indeed extends to an isomorphism of ${\mathbb{P}}^2$.
Now, theorem 1 of [@DeLaetLeBruyn] can be described as
Let $A = T(V)/R_p$, $p=[a:b:c]$ be a Sklyanin algebra, $c_3$ be the central element of degree 3 in $A$ and let $$\xymatrix{(V \otimes V \otimes V)^{H_3} \ar[r]^-\pi & A_3^{H_3} }$$ be the natural projection map. Then $\pi^{-1}({\mathbb{C}}c_3)$ is a 2-dimensional vector space of $(V \otimes V \otimes V)^{H_3}$, which corresponds in ${\mathbb{P}}((V \otimes V \otimes V)^{H_3})$ to the tangent line of the elliptic curve $E$ at the point $p$.
According to [@DeLaetLeBruyn], the vector space corresponding to the point $-2p \in {\mathbb{P}}((V \otimes V \otimes V)^{H_3})$ is indeed mapped to ${\mathbb{C}}c_3$. As the vector space $${\mathbb{C}}(a(zxy+xyz+yzx) + b(yxz+zyx+xzy) + c (x^3+y^3+z^3))$$ is the kernel of $\pi$, it follows that the vector space generated by $p$ and $-2p$ is indeed $\pi^{-1}({\mathbb{C}}c_3)$. The fact that this is the tangent line to $p$ at $E$ follows as the third point of intersection of the line through $p$ and $-2p$ is $p$ itself.
Quotients of non-regular quadratic algebras
===========================================
In the projective plane ${\mathbb{P}}^2_{[a:b:c]}$, there are 12 points where the corresponding algebra is not AS-regular: the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$-orbit of $[1:0:0]$ (containing 8 elements) and the orbit of $[0:0:1]$ (containing 4 elements). All these algebras have as Hilbert series $\frac{1+t}{1-2t}$ and are clearly not domains, for a detailed description of these algebras, see [@smithdegenerate] and [@DegenerateWalton]. However, it seems that these algebras have a 1-dimensional family of quotients that ‘behave’ like the 3-dimensional AS-regular algebras in the following sense:
- the Hilbert series is the same,
- the character series is the same for each element of $H_3$ and
- there exists a central element of degree 3, fixed by the $H_3$-action.
Let us consider the following example: take the Clifford algebra $C$ over ${\mathbb{C}}[u_0,u_1,u_2]$ with associated quadratic form $$\begin{bmatrix}
0 & u_2 & u_1 \\ u_2 & 0 & u_0 \\ u_1 & u_0 & 0
\end{bmatrix}.$$ In terms of generators and relations of $C$, we have 3 generators $x_0,x_1,x_2$ with relations $$\begin{cases}
x_0^2 = x_1^2 = x_2^2 = 0,\\
[\{x_i,x_{i+1}\},x_{i+2}] = 0, 0 \leq i \leq 2.
\end{cases}$$ This algebra is a quotient of the algebra $S_{[0:0:1]}$ by 2 elements of degree 3 (adding 2 commutation relations of degree 3 automatically implies the third relation).
The character series of $C$ is the same as the character series of the polynomial ring in 3 variables.
Define on $C$ an action of $H_3$ by $$\begin{aligned}
e_1 \cdot x_i = x_{i-1} & e_1 \cdot u_i = u_{i-1},\\
e_2 \cdot x_i = \omega^i x_{i} & e_2 \cdot u_i = \omega^{2i}u_{i}.\end{aligned}$$ $C$ is a free module of rank $8$ over ${\mathbb{C}}[u_0,u_1,u_2]$ with basis $$\{1,x_0,x_1,x_2,x_1x_2,x_2x_0,x_0x_1,x_0x_1x_2\}.$$ It is then easy to compute that under these conditions, $C$ is a graded algebra with character series equal to the polynomial ring in 3 variables with the standard action of $H_3$ (see [@DeLaet] for the AS-regular case, this one is similar).
In particular, this means that the natural epimorphism $$\xymatrix{S_{[0:0:1]} \ar[r]^-\pi & C}$$ has as kernel an ideal generated by 2 elements of degree 3. The ${\mathbb{C}}$-vector space generated by these 2 elements decomposes as $H_3$-representations into $\chi_{1,0} \oplus \chi_{-1,0}$, which is the ‘$H_3$-surplus’ of $S_{[0:0:1]}$ in degree 3.
Considering the representations of $C$, we have
$C$ is Azumaya over every point of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(C))$ except over the trivial ideal $(u_0,u_1,u_2)$.
$C$ is not Azumaya over a point if and only if the associated quadratic form after specialization is of rank $\leq 1$. Taking the $2 \times 2$-minors of the quadratic form, this only happens if $u_i^2$ is mapped to $0$ for all $0 \leq i \leq 2$.
This means that, considering representations, $C$ has more in common with the Sklyanin algebras associated to points of order 2 than the quantum algebra ${\mathbb{C}}_{-1}[x,y,z]$, although the last one is AS-regular.
In the next section, we will find a 1-dimensional family of quotients of $S_{[0:0:1]}$ by degree 3 elements that have the correct character series up to degree 4. we will show that there is an open subset in this quotient that gives algebras with the correct character series.
The Hilbert series
------------------
We work for now in the algebra $S=S_{[0:0:1]}$. $V$ will be the vector space ${\mathbb{C}}x + {\mathbb{C}}y + {\mathbb{C}}z$ and the action of $H_3$ is defined by $$\begin{aligned}
e_1 \cdot x = z, & e_1 \cdot y = x, & e_1 \cdot z = y,\\
e_2 \cdot x = x, & e_2 \cdot y = \omega y, & e_2 \cdot z = \omega^2 z.\end{aligned}$$
Decomposing the degree 3 part $S_3$ in $H_3$-modules, we find $$S_3 \cong \sum_{i,j \in {\mathbb{Z}}_3} \chi_{i,j} \oplus \chi_{0,0} \oplus \chi_{1,0} \oplus \chi_{2,0}.$$ In particular, the multiplicity of $\chi_{1,0}$ and $\chi_{2,0}$ is 2 in $S_3$. This means that the variety parametrizing quotients of $S$ with the right character series up to degree 3 is given by ${\mathbb{P}}^1 \times {\mathbb{P}}^1$.
Let $I_p$ be the ideal of $S$ generated by $$\begin{aligned}
v_1 = A_1 (zxy+\omega xyz+ \omega^2 yzx) + B_1 (yxz+\omega zyx+\omega^2 xzy),\\
v_2 = A_2 (zxy+\omega^2 xyz+ \omega yzx) + B_2 (yxz+\omega^2 zyx+\omega xzy),\end{aligned}$$ with $p=([A_1:B_1],[A_2:B_2]) \in {\mathbb{P}}^1 \times {\mathbb{P}}^1$. Let $W_p = {\mathbb{C}}v_1 + {\mathbb{C}}v_2$. One of the similarities we want to investigate is whether we can get quotients such that the Hilbert series is correct, in particular, correct in degree 4. As $\dim S_4 = 24$ and $\dim {\mathbb{C}}[x,y,z]_4 = 15$, we need to have that $\dim (I_p)_4 = 9$. We have $$\dim (I_p)_4 = \dim W_p \otimes V + \dim V \otimes W_p - \dim V \otimes W_p \cap W_p \otimes V.$$ We don’t need to worry about character series up to degree 4 as $S_4 \cong V^8$. We need to find $W_p$ such that $\dim V \otimes W_p \cap W_p \otimes V$ is 3-dimensional. However, as this vector space is an $H_3$-representation and isomorphic to $V^e$ for some $e \in {\mathbb{N}}$, it is enough to find $W_p$ such that $(V \otimes W_p \cap W_p \otimes V)^{e_2}$ is 1-dimensional.
$S/I_p$ has the correct Hilbert series up to degree 4 iff $p$ lies on the line $\mathbf{V}(A_1B_2-A_2B_1) = \Delta \subset {\mathbb{P}}^1 \times {\mathbb{P}}^1$. \[lem:Hil\]
The elements fixed by $e_2$ in $W_p \otimes V+ V \otimes W_p$ lie in the vector space generated by $$\begin{aligned}
xv_1 &= A_1 xzxy + A_1 \omega^2 xyzx + B_1 xyxz + B_1 \omega xzyx,\\
xv_2 &= A_2 xzxy + A_2 \omega xyzx + B_2 xyxz + B_2 \omega^2 xzyx,\\
v_1x &= A_1 zxyx + A_1 \omega xyzx + B_1 yxzx + B_1 \omega^2 xzyx,\\
v_2x &= A_2 zxyx + A_2 \omega^2 xyzx + B_2 yxzx + B_2 \omega xzyx.\end{aligned}$$ Then $V \otimes W_p \cap W_p \otimes V \neq 0$ iff the following matrix has rank $\leq 3$: $$\begin{bmatrix}
A_1 & 0 & A_1 \omega^2 & B_1 & 0 & B_1 \omega \\
A_2 & 0 & A_2 \omega & B_2 & 0 & B_2 \omega^2 \\
0 & A_1 & A_1 \omega & 0 & B_1 & B_1 \omega^2 \\
0 & A_2 & A_2 \omega^2 & 0 & B_2 & B_2 \omega
\end{bmatrix}.$$ The first $4\times 4$-minor is equal to $$A_1B_2A_1A_2(\omega^2-\omega)-A_2B_1 A_1A_2(\omega^2-\omega) = (\omega^2-\omega)A_1A_2(A_1B_2-A_2B_1).$$ If $A_1 = 0$, then we can take $B_1=1$ and the $4 \times 4$-minor given by the columns $(2,3,4,5)$ becomes $$\begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & A_2 \omega & B_2 & 0\\
0 & 0 & 0 & 1 \\
A_2 & A_2 \omega^2 & 0 & B_2
\end{bmatrix}.$$ Taking the determinant of this matrix, one gets $A_2^2 \omega$. So this also implies $A_1B_2-A_2B_1=0$. A similar result is true if we set $A_2=0$.
If $A_1B_2-A_2B_1=0$, then one immediately checks that $$\begin{bmatrix}
-B_2 & B_1 & -B_2 & B_1 \\
-A_2 & A_1 & -A_2 & A_1
\end{bmatrix}
\begin{bmatrix}
A_1 & 0 & A_1 \omega^2 & B_1 & 0 & B_1 \omega \\
A_2 & 0 & A_2 \omega & B_2 & 0 & B_2 \omega^2 \\
0 & A_1 & A_1 \omega & 0 & B_1 & B_1 \omega^2 \\
0 & A_2 & A_2 \omega^2 & 0 & B_2 & B_2 \omega
\end{bmatrix}=0.$$ As either one of the rows of the first matrix is not 0, we have that $\dim V \otimes W_p \cap W_p \otimes V \geq 3$. The only points where this inequality is possibly strict is when either both $A_1$ and $B_2$ are 0 or both $A_2$ and $B_1$ are 0, this follows from taking the determinants of $$\begin{bmatrix}
A_2 & 0 & A_2 \omega \\
0 & A_1 & A_1 \omega\\
0 & A_2 & A_2 \omega^2
\end{bmatrix} \text{ and }
\begin{bmatrix}
B_2 & 0 & B_2 \omega^2 \\
0 & B_1 & B_1 \omega^2 \\
0 & B_2 & B_2 \omega
\end{bmatrix}.$$ But if for example $A_1 = 0$, then necessarily $A_2 = 0$, but $B_2$ and $A_2$ can not be 0 at the same time. Similar results hold for the other cases, so we are done.
This means that the only points we have to consider lie on the diagonal $\Delta \subset {\mathbb{P}}_{[A_1:B_1]}^1 \times {\mathbb{P}}_{[A_2:B_2]}^1$. From now on, we write $t = \frac{B}{A}$ for $[A:B] \in {\mathbb{P}}^1$ and let $I_t$ be the ideal in $S$ generated by the elements $$\begin{aligned}
(v_1)_t = A (zxy+\omega xyz+ \omega^2 yzx) + B (yxz+\omega zyx+\omega^2 xzy),\\
(v_2)_t = A (zxy+\omega^2 xyz+ \omega yzx) + B (yxz+\omega^2 zyx+\omega xzy),\end{aligned}$$
The next obvious question is whether these algebras parametrized by ${\mathbb{P}}^1$ have the correct Hilbert series. For ${\mathbb{C}}^* = {\mathbb{P}}^1\setminus \{0,\infty \}$ this is true.
The Clifford algebra $C$ corresponds to taking the quotient for the value $t=-1$.
It is enough to prove that the relation $[\{x,y\},z]$ belongs to the vector space ${\mathbb{C}}(v_1)_{-1}+{\mathbb{C}}(v_2)_{-1}$. This means that the following matrix should have rank 2 $$\begin{bmatrix}
1 & \omega & \omega^2 & -1 & -\omega & -\omega^2 \\
1 & \omega^2 & \omega & -1 & -\omega^2 & -\omega \\
-1 & 1 & 0 & 1 & -1 & 0
\end{bmatrix},$$ which is indeed true.
We will later see that all these algebras can be embedded in a smashed product $C \# {\mathbb{Z}}$ if $t \neq 0,\infty$, from which it will follow that
Each algebra $S/I_t$ has the correct Hilbert series if $t \neq 0,\infty$. \[th:Hilbert\]
Point modules of $T_t$
----------------------
In [@smithdegenerate] the point modules of $S$ were classified. We can classify the point modules of $T_t=S/I_t$ using these results. Recall that point modules of $S$ were determined by the following way: let $\mathbb{V}=\mathbf{V}(XYZ) \subset {\mathbb{P}}^2$ be the union of 3 lines and let $q_0,q_1,q_2$ be the intersection points. Then a point module $P$ of $S$ depends on a point sequence $p_0 p_1 p_2 \ldots$ fulfilling the following requirements
- for every $i \in {\mathbb{N}}$, $p_i \in \mathbb{V}$,
- if $p_i \neq q_j$ for any $j$, then $p_{i+1}$ is the intersection point not lying on the same line as $p_i$,
- if $p_i = q_j$ for some $j$, then $p_{i+1}$ is any point on the line opposite of $q_j$.
If we write $P = \oplus_{n\in {\mathbb{N}}} {\mathbb{C}}e_n$, then the point $p_i= [a_i:b_i:c_i]$ corresponds defining an action of $S$ on $P$ by $$\begin{aligned}
x \cdot e_i = a_i e_{i+1}, & y \cdot e_i = b_i e_{i+1}, &z \cdot e_i = c_i e_{i+1}.\end{aligned}$$
The point modules of $T_t$ for $t \neq 0,\infty$ are parametrized by 6 lines, call this set $\mathbb{W}$. The isomorphism $\phi$ induced by sending a point module $P$ to $P[1]$ on $\mathbb{W}$ is such that $\phi^2$ fixes the intersection points and sends each line of $\mathbb{W}$ to itself.
Let $P$ be a point module of $T_t$ with associated point sequence $p_0 p_1 p_2 \ldots$. Let $p_i p_{i+1} p_{i+2}$ be a subtriple of this sequence.
- Assume that $p_i$ is one of the intersection points. Using the Heisenberg action, we may assume that $p_i = [1:0:0]$. Then $p_{i+1} = [0:\alpha:\beta]$.
- Assume that $[0:\alpha:\beta] \neq [0:1:0]$ and $[0:\alpha:\beta] \neq [0:0:1]$. Then $p_{i+2} = p_i$ and the relations of $S/(I_t)$ are trivially fulfilled.
- Assume that $[0:\alpha:\beta] = [0:1:0]$. Then $p_{i+2} = [a:0:b]$. But from the degree 3 relations it follows that $b = 0$ and so $p_{i+2} = p_i$.
- The case $[0:\alpha:\beta] = [0:0:1]$ is similar to the previous case.
- Assume that $p_i$ is not one of the intersection points. Again using the Heisenberg-action, we may assume that $p_i = [0:\alpha:\beta]$, $p_{i+1}=[1:0:0]$ and $p_{i+2} = [0:\gamma:\delta]$. But then it follows from the degree 3 relations that $ \delta \alpha = -t \gamma \beta$ or differently put, $\phi^2$ is an isomorphism of $\mathbf{V}(XYZ)$ fixing the intersection points.
From the proof of this theorem we also notice that if $-t$ is primitive $n$th root of unity, then $\phi^{2n}$ is the identity. This implies that a point module $P$ of $T_t$ in this case parametrizes a ${\mathbb{C}}^*$-family of $2n$-dimensional simple representations of $T_t$.
The point module corresponding to $q_0 q_1 q_0 q_1 \ldots$ corresponds to the ${\mathbb{C}}^*$-family of 2-dimensional simple representations coming from the quotient $T_t/(z) = {\mathbb{C}}\langle x,y \rangle /(x^2,y^2)$. Similar results hold for the Heisenberg-orbit of this point module.
The central element
-------------------
One of the many similarities we want for these new algebras is that there exists a central element of degree 3, fixed by the action of the Heisenberg group.
For every element $t \in \Delta \subset {\mathbb{P}}^1 \times {\mathbb{P}}^1$ there exists a degree 3 central element in $(T_t)_3$ fixed by $H_3$.
Using for example MAGMA, one checks that $$g_t = (zxy + xyz + yzx)+t(yxz+zyx+xzy)$$ is central in $T_t$.
One also checks that $g_t$ acts trivially on each point module. These observations show that the constructed quotients have indeed much in common with the AS-regular algebras, as each point module of the generic AS-regular algebra is also annihilated by the unique central element in degree 3.
An analogue of the twisted coordinate ring
------------------------------------------
We can now calculate the Hilbert series of $M_t = T_t/(g_t)$.
The Hilbert series of $M_t$ is $\frac{1-t^3}{(1-t)^3}$ except if $t=0,\infty$.
Adding the relation $(zxy + xyz + yzx)+t(yxz+zyx+xzy)$, we can find an easier basis for the degree 3 relations of $M_t$. Taking linear combinations, we find $$\begin{cases}
zxy = -t yxz,\\
xyz = -t zyx, \\
yzx = -t xzy,
\end{cases}$$ for $t \neq 0$. For $t=-1$, the corresponding algebra is a quotient of the Clifford algebra, so in this particular case the Hilbert series is correct. But it then follows that the Hilbert series is correct for all these cases, as all these algebras are monomial algebras with the same basis as the quotient of the Clifford algebra.
$g_t$ is not a zerodivisor of $T_t$.
This follows directly from Hilbert series considerations and from Theorem \[th:Hilbert\].
We can find an easy basis for $M_t=T_t/(g_t)$ if $t \neq 0,\infty$.
A monomial $w \in M_t$ is 0 iff any letter is both on an even and an odd place in $w$.
If $w$ is a monomial with not one variable on both an even and an odd place, then $w$ can never be 0, as the 3 relations preserve the even and odd places of the variables. So the combinations $x^2,y^2,z^2$ can never occur in $w$.
Suppose now that $x$ occurs on both an even and an odd place. There is then a submonomial of $w$ of the form $xzyzy\ldots yx$ or $xyzyz\ldots zx$. But then we can bring the last $x$ to place 2 in the submonomial, so we get $x^2$ and $w$ becomes 0. Similar results are true for $y$ and $z$.
If we now take a monomial which is not 0 in $M_t$, then it follows that either all odd or all even places have the same variable, say $x$. But then we can ‘jump’ over this variable to get 1 variable to the left and the other to the right. Fixing a lexicographical order $x>y>z$, we can therefore find a nice basis in the following way: take 2 variables, say for example $x,y$, take the basis of ${\mathbb{C}}[x,y]$ an put a $z$ between each variable. Put $z$ first on the odd places and then the even places. Apply this procedure 3 times (one for each variable), but remember to discard 6 elements (for $xzxz\ldots$, $yzyzyzy\ldots$ and others have been counted twice). For example, a basis for $(M_t)_4$ is given by $$\begin{array}{ccc}
xyxy & xyzy & zyzy, \\
xzxz & xzyz & yzyz, \\
yxyx & yxzx & zxzx, \\
yxyz & zxzy & xyxz.
\end{array}$$
In the even cases, we then find a linearly independent set consisting of $3(\frac{n}{2}+1) + 3(\frac{n}{2}+1)-6 = 3n$ elements, so this is a basis in even degree. In odd degree, we find $3(\frac{n+1}{2}+1) + 3 (\frac{n-1}{2}+1) - 6 = 3n$, so also in this case the constructed linearly independent set is a basis.
The next question is to determine the center of $M_t$.
$M_t$ has no central elements in odd degree.
$M_t$ maps surjectively to the algebras ${\mathbb{C}}\langle y,z \rangle/(y^2,z^2)$, ${\mathbb{C}}\langle x,z \rangle/(x^2,z^2)$ and ${\mathbb{C}}\langle x,y \rangle/(x^2,y^2)$. This means that any central element of odd degree, say $w$, has to have all variables in its monomials, as each of these quotients doesn’t have central elements in odd degree, so $w$ has to belong to the kernel of each of these algebra morphisms. So say that $w$ has as one of its monomials $xyxy\ldots x z x$, with all $x$ on the odd places. For $w$ to belong to the center, we need to get the first $y$ in $y xyxy\ldots x z x$ to the last place. This is however impossible, as the first $y$ is in a odd place and the last place is even. Similar results hold for monomials of the form $yxyx \ldots zy$ and $zxzx \ldots yz$. For monomials of the form $yx\ldots xz$ with $x$ on every even place, we have to get the first $x$ in $xyxyx\ldots xz$ to the last place, but again this is impossible as the first place is odd and the last place is even. Similar results hold for $xy \ldots yz$ ($y$ at even place) and $yz\ldots zx$ ($z$ at even place). This means that $w$ does not contain any monomials, so $w=0$.
If $-t$ is a primitive $n$th root of unity, then the elements $(x+y)^{2n}, (x+z)^{2n}$ and $(y+z)^{2n}$ generate the center.
It is clear that these 3 elements belong to the center of $M_t$, as we have that $$(x+y)^2x = x(x+y)^2,(x+y)^2y = y(x+y)^2,(x+y)^{2n}z = z(x+y)^{2n}.$$ The other claimed elements are central by the fact that the center is stable under the action of the Heisenberg group.
Consider the second Veronese subalgebra $P_t=M_t^{(2)}$ with generators $xy,yx,xz,zx,yz,zy$. We then have $$\begin{cases}
(xy)(yx) = 0 ,\\
(xy)(xz) = (-t)^{-1} (xz)(xy),\\
(xy)(zx) = 0, \\
(xy)(yz) = 0,\\
(xy)(zy) = (-t) (zy)(xy),
\end{cases}$$ together with Heisenberg orbits (in particular, any word consisting of 3 different couples is necessarily 0). It then follows that the center of $P_t$ is generated by these 6 monomials of degree 2 to the $n$th power. Take a degree $2n$ central element $w$ of $M_t$ and suppose it contains the monomial $(xy)^{na}(xz)^{nb}$. We then have $$y(xy)^{na}(xz)^{nb} = (yx)^{na} y (xz)^{nb} = (yx)^{na} (zx)^{nb} y.$$ We find that this monomial can only be in $w$ if and only if the monomial $(yx)^{na} (zx)^{nb}$ also occurs in $w$, with the same coefficient. But this sum of 2 monomials is equal to $$(xy+yx)^{na}(xz+zx)^{nb}.$$ Similar results hold for the other monomials, so we are done.
From now on, assume that $-t$ is a primitive $n$th root of unity. We have $(xy+yx)^{n}(yz+zy)^{n}(zx+xz)^{n}=0$, so the dimension of the center is $\leq 2$.
$M_t$ is a finite module over its center.
$M_t$ is a finite module over $P_t$, which in turn is a finite module over its center. The center of $P_t$ is generated by $u=(xy)^n,u'=(yx)^n,v'=(xz)^n,v=(zx)^n,w=(yz)^n,w'=(zy)^n$. The center of $M_t$ is then equal to $u+u',v+v',w+w'$. Now, $Z(P_t)$ is generated as a module over $Z(M_t)$ by the elements $1,u,u',v,v',w,w'$, as we have for example $$u^av'^b = (u+u')(v+v')(u^{a-1}v'^{b-1})$$ so by induction we conclude that $M_t$ is a finite module over its center.
The dimension of the center is 2.
We know that the dimension of the center is $\leq 2$. If it was $< 2$, then $M_t$ would not be a finite module over its center.
We can now give a description of the simple representations of $M_t$.
The PI-degree of $M_t$ is $2n$.
As the elements of the center of smallest degree are of degree $2n$, it follows from the Cayley-Hamilton polynomial that the PI-degree of $M_t$ is at least $2n$. If we can now find an open subset of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(M_t))$ with $2n$-dimensional simple representations, we are done. However, by determining the point modules of $T_t$ and the observation that each point module is annihilated by $g_t$, we have indeed found a 2-dimensional family of $2n$-dimensional simple representations of $M_t$, as the induced automorphism $\phi$ on the point variety of $M_t$ coming from the shift functor of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Proj}}}(M_t)$ has the property $\phi^{2n}= Id$.
Connection with the Clifford algebra
------------------------------------
Let $\lambda \in {\mathbb{C}}^*$, for the moment not a root of unity. Define an action of the infinite cyclic group ${\mathbb{Z}}= C_\infty = \langle \sigma \rangle $ on the Clifford algebra $C$ by $$\sigma(x) = \lambda^{-1}x, \sigma(y) = y,\sigma(z) = \lambda z$$ and take the smash product $Q=C \# {\mathbb{Z}}$, which is graded by the classic grading on $C$ and $\deg(\sigma) = 0$. Consider the elements $x'=x \# 1, y'=y \# \sigma, z'= z\# \sigma^{-1}$. We then have $$\begin{gathered}
z'x'y' = (z\# \sigma^{-1})(x\# 1)(y\# \sigma) = \lambda (zxy \# 1), \\
x'y'z' = (x\# 1)(y\# \sigma)(z\# \sigma^{-1}) = \lambda (xyz \# 1), \\
y'z'x' = (y\# \sigma)(z\# \sigma^{-1})(x\# 1) = \lambda (yzx \# 1), \\
y'x'z' = (y\# \sigma)(x\# 1)(z\# \sigma^{-1}) = (yzx \# 1),\\
z'y'x' = (z\# \sigma^{-1})(y\# \sigma)(x\# 1) = (zyx \# 1), \\
x'z'y' = (x\# 1)(z\# \sigma^{-1})(y\# \sigma) = (xzy \# 1).\end{gathered}$$ This means that $x',y',z'$ are solutions for the relations of $T_{-\lambda}$.
The Hilbert series of the algebra $Q$ generated by $x',y',z'$ is equal to $\frac{1}{(1-t)^3}$ and all the relations holding in this algebra come from degree 3 relations that hold in $T_{-\lambda}$.
The algebra $\mathcal{C}=C \# {\mathbb{Z}}$ is defined by generators and relations by $$\mathcal{C}={\mathbb{C}}\langle x,y,z ,t ,t^{-1} \rangle/(x^2,y^2,z^2,[\{x,y\},z],[\{y,z\},x],tx - \lambda^{-1}xt,ty - yt,tz - \lambda zt).$$ We give $\mathcal{C}$ the gradation determined by $\deg(x)=\deg(y)=\deg(z)=1,\deg(t)=0$. Then a basis in degree $k$ is determined by fixing a monomial basis $W_k$ for $C$ in degree $k$ and taking all powers of $t$ $$\{f \# t^m | f \in W_k, m \in {\mathbb{Z}}\}.$$ Now, write an element $f \in W_k$ as $f(x,y,z)$. Then the elements $f(x',y',z')$ are also linearly independent: let $a$ reps. $b,c$ be the number of times $x$, resp. $y,z$ is in $f(x,y,z)$. Then $f(x',y',z')$ is equal to $\lambda^{\mu} f(x,y,z) \# t^{b-c}$ for some $\mu \in {\mathbb{Z}}$.
The set $\{f(x',y',z') | f \in W_k \}$ forms a basis of $Q_k$: let $g(x,y,z)\#t^{r}$ be an element of $Q_k$ with $g(x,y,z)$ a monomial of degree $k$, $g\neq 0$. Then necessarily $r=b-c$ where $b$ is the number of times $y$ occurs in $g$ and $c$ is the number of times $z$ occurs in $g$. $g$ can be written as a unique linear combination of elements in $W_k$, say using the terms $f_1,\ldots,f_i$. However, due to the fact that in the defining relations of $C$ the number of times a variable occurs does not change (unless the monomial becomes $0$), this linear combination has the following property: for each $f_i$, the number of occurrences of $x,y,z$ stays the same. But then $g(x,y,z)\#t^{b-c}$ can be written as a linear combination of $f_i(x,y,z) \# t^{b-c} = \lambda^{-\mu}f(x',y',z')$. So the Hilbert series indeed stays the same.
The fact that the only relations are of degree $3$ follows as the only relations between monomials can occur if all variables occur and the relations come from $[\{x,y\},z]$ and $[\{y,z\},x]$.
The algebra $T_{-\lambda}$ can be embedded in a smash product $C \# {\mathbb{Z}}$.
From this, it follows that Theorem \[th:Hilbert\] is proved.
When $\lambda$ is a primitive root of unity of order $n$, we can take the same action of ${\mathbb{Z}}_n = \langle \sigma \rangle$ on $C$, take the smash product $C \# {\mathbb{Z}}_n$ and take the same elements $x',y',z'$ as generators of a subalgebra of $C \# {\mathbb{Z}}_n$. These 3 elements then fulfil the relations of $T_{-\lambda}$ and again we have found a subalgebra of $P$ isomorphic to $T_{-\lambda}$.
In the case that $\lambda$ is a primitive $n$th root of unity, we can lift the $3$ linearly independent central elements of degree $n$ of $T_{-\lambda}/(g_{-\lambda})$.
The element $(x'z'+z'x')^n$ belongs to the center of $C \# {\mathbb{Z}}_n$.
We have $$x'z'+z'x' = (x \# 1)(z \# \sigma^{-1})+(z \# \sigma^{-1})(x \# 1) = (xz+\lambda zx)\# \sigma^{-1}.$$ $xz$ and $zx$ are fixed by $\sigma$, so we get $$((xz+\lambda zx)\# \sigma^{-1})^n=(xz)^n+\lambda^n(zx)^n \# 1= (xz+zx)^n \# 1$$ which is indeed central.
This implies that $(xz+zx)^n$ belongs to the center of $T_{-\lambda}$. Then we can use the Heisenberg action in $T_{-\lambda}$ to see that $(yz+zy)^n$ and $(xy+yx)^n$ are also central in $T_{-\lambda}$. From [@smith1994center Lemma 3.6], we now deduce
The center of $T_{-\lambda}$ with $\lambda$ a primitive $n$th root of unity is generated by 1 element of degree 3 $g_{-\lambda}$ and 3 linear independent elements of degree $2n$, say $u,v,w$, with one relation of the form $uvw = \alpha g^{2n}$ for some $\alpha \in {\mathbb{C}}^*$. $T_{-\lambda}$ is also a finite module over its center.
All the conditions of the mentioned lemma are satisfied: $g_{-\lambda}$ is regular and the image of ${\mathbb{C}}[u,v,w]$ generates the center of $M_{-\lambda}$. Moreover, as we have in $M_{-\lambda}$ that the relation in the center is $\overline{u}\overline{v}\overline{w} = 0$, it follows that the only relation is of the form $uvw = \alpha g^{2n}$. $\alpha$ is not 0 as $(x'z'+z'x')^n(x'y'+y'x')^n(y'z'+z'y')^n$ is not 0 in $C \# {\mathbb{Z}}_n$.
Representation theory
---------------------
Fix $\lambda$ a primitive root of unity of order $n$, $n \neq 1$. Let $P=P_\lambda$ be the smash product of $C$ with ${\mathbb{Z}}_n$ defined in last section and let $Q = Q_\lambda = T_{-\lambda}$ be the subalgebra of $P$ coming from a quotient of $S$.
The center of $P$ is isomorphic to the ring ${\mathbb{C}}[a,b,c,d,e]/(ae - d^2,bc-e^n)$.
The center of $C$ is generated by $(x+y)^2,(y+z)^2,(x+z)^2,g=zxy-yxz$ with one relation of the form $(x+y)^2(y+z)^2(x+z)^2 = \alpha g^2$ for some $\alpha \in {\mathbb{C}}^*$. The elements in $Z(C)$ fixed by $\sigma$ are $xz+zx$, $(yz+zy)^n$, $(yx+xy)^n$, $g$ and $(yz+zy)(yx+xy)$. But then the relations become $$\begin{gathered}
(yz+zy)((xz+zx)(yx+xy)) = \alpha g^2,\\
(xz+zx)^n(yx+xy)^n = ((xz+zx)(yx+xy))^n.\end{gathered}$$ Up to a scalar, these are the relations of the claimed ring.
The PI-degree of $P$ is $2n$.
$P$ is a free module of rank $n$ over $C$. $C$ is a module of rank $4$ over its center $Z(C)$, which in turn is a module of rank $n$ over its ring of invariants $Z(C)^{{\mathbb{Z}}_n}$. The claim follows.
The PI-degree of $Q$ is $2n$.
We know that the PI-degree is at least $2n$, as we have found simple $2n$-dimensional representations coming from the quotient of $Q$ by the degree 3 central element $g_{-\lambda}$. As $Z(P)$ is a finite $Z(Q)$-module of rank $n$, every simple $2n$-dimensional representation of $P$ induces a representation of $Q$ and the induced map $\xymatrix{{{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(P)) \ar[r] & {{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(Q))}$ is surjective. But then there is an open subset of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(Q))$ containing simple $2n$-dimensional representations. This proves the claim.
In fact, by using the $({\mathbb{C}}^*)^3$-action as gradation preserving algebra automorphisms, we find
The Azumaya locus of ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(Q))$ is equal to ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(Q))$ minus 3 lines, the 2 by 2 intersections of the 3 planes $\mathbf{V}(u),\mathbf{V}(v),\mathbf{V}(w)$.
The $({\mathbb{C}}^*)^3$-action divides ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Spec}}}(Z(Q))$ into 8 orbits: 1 of dimension 3 (the largest), 3 of dimension 2, 3 of dimension 1 and 1 of dimension 0. The first orbit is open and therefore intersects ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Azu}}}_{2n}(Q)$, but then this orbit is contained in ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Azu}}}_{2n}(Q)$. The 3 orbits of dimension 2 are the ones coming from simple representations of $M_t$ of dimension $2n$, so these also belong to ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Azu}}}_{2n}(Q)$. The orbits corresponding to lines can not belong to ${{\text{\em \usefont{OT1}{cmtt}{m}{n} Azu}}}_{2n}(Q)$, as there are simple 2-dimensional representations coming from the quotient $Q/(z)={\mathbb{C}}\langle x,y \rangle/(x^2,y^2)$.
The algebra $S_{[1:0:0]}$
=========================
In this case we can be brief: $S_{[1:0:0]}$ is a Zhang-twist of $S_{[0:0:1]}$ by the automorphism $x \mapsto z \mapsto y \mapsto x$. Recall that if $\mathcal{A}$ is a graded algebra with a graded automorphism $\phi$, then the Zhang twist is defined as the algebra with the same generators but with multiplication rule $a * b = a \phi^i(b)$ if $b \in \mathcal{A}_i$. A Zhang twist of a graded algebra preserves the Hilbert series.
$S_{[1:0:0]}$ has a 1-dimensional family of quotients parametrized by ${\mathbb{C}}^*$ such that these quotients have the right Hilbert series. The relations of degree 3 are defined by $$\begin{aligned}
(v_1)_t = (zyx+\omega xzy+ \omega^2 yxz) + t (y^3+ \omega z^3+\omega^2 x^3),\\
(v_2)_t = (zyx+\omega^2 xzy+ \omega yxz) + t (y^3+ \omega^2 z^3+\omega x^3).\end{aligned}$$ The central element becomes $g_t=(zyx+xzy+ yxz) + t (y^3+ z^3+ x^3)$
We find that $$\begin{aligned}
x*x*x = xzy,y*y*y = yxz,z*z*z=zyx,\\
z*y*x = zxy,y*x*z = yzx,x*z*y=xyz\end{aligned}$$ and the other 3 monomials become 0. It is then clear that the relations from the proposition are really the Zhang twists of the relations in $S_{[0:0:1]}$.
If we calculate $(zyx+xzy+ yxz) + t (y^3+ z^3+ x^3)$ in $S_{[1:0:0]}$, we find $$zxy+xyz+yzx+t(yxz+zyx+xzy),$$ which is central in the original quotient of $S_{[0:0:1]}$. As this element is also fixed under $\phi$, we are done.
The controlling variety
=======================
In order to get algebras with the correct Hilbert series up to degree 4, we replaced the point $[0:0:1]$ with a ${\mathbb{P}}^1$. One of course hopes that the corresponding variety parametrizing these algebras is just the blow-up of ${\mathbb{P}}^2$ in this point.
The variety parametrizing $H_3$-deformations up to degree 3 of the polynomial ring ${\mathbb{C}}[x,y,z]$ with the correct Hilbert series up to degree 4 is the blow-up of ${\mathbb{P}}^2$ in 12 points.
We will show that the variety $$Z\subset \operatorname{Emb}_{H_3}(V^*,(V^*)^3) \times \operatorname{Emb}_{H_3}(\chi_{1,0}^2,\chi_{1,0}^3) \cong² {\mathbb{P}}^2 \times {\mathbb{P}}^2$$ parametrizing nice quotients with the right multiplicity of $\chi_{1,0}$ in degree 3 is the blow-up of ${\mathbb{P}}^2$ in the $H_3$-orbit of $[0:0:1]$.
Let $R$ correspond to the relations $$\begin{cases}
a yz + b zy + c x^2,\\
a zx + b xz + c y^2,\\
a xy + b yx + c z^2.
\end{cases}$$ Let $\phi$ be the subspace of $V\otimes V \otimes V$ generated by $$\begin{cases}
A(x^3+\omega y^3+\omega^2 z^3)+B(zxy + \omega xyz + \omega^2 yzx) + C (yxz + \omega zyx + \omega^2 xzy),\\
D(x^3+\omega y^3+\omega^2 z^3)+E(zxy + \omega xyz + \omega^2 yzx) + F (yxz + \omega zyx + \omega^2 xzy)
\end{cases}$$ such that the matrix $$\begin{bmatrix}
A & B & C \\ D & E & F
\end{bmatrix}$$ has rank 2. Decomposing $V \otimes R + R \otimes V$ in $H_3$-representations, we find that the $(R,\phi)$ belongs to $Z$ if and only if the following matrix has rank 2 $$M=
\begin{bmatrix}
c & a \omega^2 & b \omega \\
c & a \omega & b \omega^2 \\
A & B & C \\
D & E & F
\end{bmatrix}.$$ We may assume that $c = 1$, putting $a=1$ or $b=1$ gives similar results. Put $a_{01} = AE-BD, a_{20} = CD-AF, a_{12} = BF-CE$. $M$ has rank 2 if and only if $$\begin{aligned}
a_{12}+a\omega a_{20} + b \omega^2 a_{01} = 0,\\
a_{12}+a\omega^2 a_{20} + b \omega a_{01} = 0.\end{aligned}$$ From this it follows that $a a_{20} = b a_{01}$, which is indeed the equation for the blow-up of ${\mathbb{P}}^2$ in $[a:b:c]=[0:0:1]$. The same result holds for the points $[1:0:0]$ and $[0:1:0]$, call $Z_{1,0}$ the blow-up of ${\mathbb{P}}^2$ in these 3 points.
We could now do the same for the representation $\chi_{2,0}$ to get $Z_{2,0}$, which is also isomorphic to ${\mathbb{P}}^2$ blown-up in 3 points. However, in order to get the right Hilbert series up to degree 4, we have seen that we need to take the ‘diagonal’ $\Delta \subset Z_{1,0} \times_{{\mathbb{P}}^2} Z_{2,0}$, which is of course just the blow-up of ${\mathbb{P}}^2$ in 3 points.
For the other $9$ points, we can use the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$-action.
The bad case $t=0,\infty$
=========================
In the blow-up of ${\mathbb{P}}^2$ in 12 points, there are still 24 points where the Hilbert series of the corresponding algebras explodes. Still, it would be useful to know what the Hilbert series of these algebras are. As all these algebras are isomorphic to each other (or isomorphic to a Zhang twist) by courtesy of the ${{\text{\em \usefont{OT1}{cmtt}{m}{n} SL}}}_2(3)$-action, it is enough to calculate the Hilbert series of the algebra $$\mathcal{C} = {\mathbb{C}}\langle x,y,z \rangle /(x^2,y^2,z^2, zxy + \omega xyz + \omega^2 yzx,zxy + \omega^2 xyz + \omega yzx).$$
The element $g_0 = zxy + xyz + yzx$ fulfils the conditions $$g_0x = x g_0 = g_0y = y g_0=g_0z = z g_0=0.$$
Using the Heisenberg action, it is enough to prove this for $x$. Then it is an easy computer calculation with for example MAGMA.
It is therefore enough to compute the Hilbert series of $$\mathcal{A}=\mathcal{C}/(g_0) = {\mathbb{C}}\langle x,y,z \rangle /(x^2,y^2,z^2,xyz,yzx,zxy).$$ This algebra has the advantage that 2 words $w$ and $w'$ can only be equal to each other if and only if $w = w' = 0$. From this it follows that each $\mathcal{A}_n$ has a unique monomial basis $B_n$ on which $e_1$ acts by a permutation of order 3, without fixed points. This means that $$\begin{aligned}
\# \{ w \in B_n | w \text{ begins with } x \} &= \# \{w \in B_n | w \text{ begins with } y \} \\ &= \# \{w \in B_n | w \text{ begins with } z \}.\end{aligned}$$
We know from Lemma \[lem:Hil\] that the Hilbert series starts with the terms $$H_\mathcal{A}(t)= 1 + 3t + 6t^2 + 9t^3 + 15t^4 + \ldots$$
Let $f(t)=H_\mathcal{A}(t) = \sum_{n=0}^\infty a_n t^n$ be the Hilbert series of $H_\mathcal{A}(t)$ with $a_n = \dim \mathcal{A}_n$. Then the coefficients of $f(t)$ fulfil the recurrence relation $a_n = 2 a_{n-1} - a_{n-3}$ for $n \geq 4$.
For $n=4$, we are done by Lemma \[lem:Hil\]. Let $n \geq 5$. Let $$\xymatrix{\mathcal{A}_{n-1} \ar[r]^-{f_x} & \mathcal{A}_n}$$ be the linear map defined by $w \mapsto xw$. It then follows that $a_n = 3 \dim \operatorname{Im}(f_x)$. So we need to calculate $\ker(f_x)$. Due to the relations, $\ker(f_x)$ is the subspace of $\mathcal{A}_{n-1}$ spanned by the words beginning with $x$ or $yz$. The space spanned by the words beginning with $x$ has dimension $\frac{a_{n-1}}{3}$. The space spanned by the words beginning with $yz$ is the image of the map $$\xymatrix{\mathcal{A}_{n-3} \ar[r]^-{f_{yz}} & \mathcal{A}_{n-1}}.$$ The kernel of $f_{yz}$ is the subspace spanned by the monomials in $\mathcal{A}_{n-3}$ starting with $x$ or $z$. So $\dim \operatorname{Im}(f_{yz}) = a_{n-3} - \frac{2a_{n-3}}{3} = \frac{a_{n-3}}{3}$. This gives $$\begin{aligned}
a_n &= 3 \dim \operatorname{Im}(f_x) \\
&=3 (a_{n-1} - \dim \ker(f_x))\\
&=3 (a_{n-1} - \left(\frac{a_{n-1}}{3} +\frac{a_{n-3}}{3}\right)\\
&=2 a_{n-1}- a_{n-3}.\end{aligned}$$
For $n\geq 3$, we have the recurrence relation $a_n = a_{n-1} + a_{n-2}$
$n=3$ is trivial. By induction, we may assume that $a_{n-1} = a_{n-2} + a_{n-3}$. But then we get $$a_{n} = 2 a_{n-1}- a_{n-3} = a_{n-1} + (a_{n-2} + a_{n-3})- a_{n-3} = a_{n-1} + a_{n-2}.$$
$$H_\mathcal{A}(t) = \frac{3}{1-(t+t^2)} - 2$$
Unfortunately, this is not exactly the Fibonacci sequence. However, if we take $\mathcal{B}= \mathcal{A}^{e_1}$, then we get $$b_n = \dim \mathcal{B}_n = F_n$$ with $F_n$ the $n$th Fibonacci number starting from $F_0 = 1, F_1 = 1$.
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psfig
Low-dimensional quantum spin systems like spin chains, ladders or plaquettes received considerable attention from both theoretical and experimental points of view due to their manifold of unconventional spin excitation spectra. Of particular interest is the so-called spin-Peierls transition. The degeneracy of the ground state of an isotropic one-dimensional (1d) spin-1/2 system is lifted due to the coupling to the lattice, leading to a dimerized singlet ground state and the opening of a singlet-triplet gap.
The magnetic excitation spectrum of a dimerized spin chain consists of a triplet branch at $\Delta_{01}$, a corresponding two-particle continuum of triplet excitations starting at 2$\Delta_{01}$, and well defined magnetic bound states [@uhrig; @bouz; @Affleck]. The latter consist of strongly interacting triplet excitations with a high energy cutoff at 2$\Delta_{01}$. The existence of a singlet bound state at $\sqrt{3}\Delta_{01}$ is predicted by a 1d model with frustrated next-nearest neighbor interaction $J_2= 0.24 J_1$, acting as a binding potential. A further triplet bound state should appear for higher values of $J_2$ [@bouz]. A study of magnetic bound states in quantum spin systems therefore gives valuable insight into the low energy spin excitations which govern the physics of these systems.
Half-filled spin ladder systems attracted enormous interest recently due to the surprising changes of the ground state and excitation spectrum interpolating between one- and two-dimensional quantum spin systems [@dago96]. For even-leg ladders the doping by holes is found to lead to superconductivity. On the other hand, for the quarter-filled ladder system as the corresponding diluted system the ground state and spin excitation spectrum are not well understood until now. However it is expected that they show a similar rich excitation scheme as the half-filled system [@normand].
The inorganic compound $\alpha^\prime$-NaV$_{2}$O$_{5}$ with double chains of edge-sharing distorted tetragonal VO$_{5}$ and a spin-Peierls like transition at T$_{SP}$=35K [@isobe; @weiden] has been interpreted as a quarter-filled spin ladder system [@smolinski]. Recent X-ray diffraction data at room temperature are in favor of a centrosymmetric (D$_{2h}^{13}$) structure with only one type of V site of average valence +4.5. The spins are therefore attached to a V-O-V molecular orbital on the rungs [@smolinski; @roth; @horsch]. The exchange interaction across the rung is by a factor of five larger compared to that along the legs [@smolinski]. Hence the ladder can be described by an effective spin chain.
The formation of a singlet ground state in $\alpha^\prime$-NaV$_{2}$O$_{5}$ below T$_{SP}$ is observed as a drop in the magnetic susceptibility [@isobe; @weiden]. The observation of superlattice peaks in x-ray scattering proves a lattice dimerization with a doubling of the unit cell along the a- and b-axis and a quadrupling along the c-axis [@fuji97]. The singlet-triplet gap was estimated using magnetic susceptibility: $\Delta_{01}$=85 K [@weiden], NMR: 98 K [@Ohama], ESR: 92 K [@vasilev] and neutron scattering: 115 K [@fuji97]. Anomalous behavior of the spin-Peierls like transition is evidenced by drastic deviations from the weak coupling regime, e.g. a large value of the reduced gap $2\Delta_{01}/k_BT_{SP}$=4.8-6.6 K [@weiden; @fuji97; @Ohama; @vasilev]. Moreover, hints for dynamic two-dimensional spin correlations are observed in neutron scattering [@fuji97].
In this paper we will present the first experimental investigation of the spin excitation spectrum of dimerized spin ladders in $\alpha^\prime$-NaV$_{2}$O$_{5}$ yielding evidence for magnetic bound states. Raman scattering as a powerful experimental technique allows to probe dimerization-induced new modes and the relevant exchange paths by applying polarization selection rules. Three new modes appearing for T$<$ T$_{\rm SP}$ are identified by the selection rules and the temperature dependence of their intensity and linewidth as magnetic bound states. The ground state, however, is assumed to be composed by a dynamic superposition of energetically degenerate dimer configurations along the rungs, legs and diagonals of the ladders.
The scattering geometries available for our phonon Raman experiments on $\alpha\prime$-NaV$_{2}$O$_{5}$ with our crystals, i.e. with the polarization of the incident and scattered phonons parallel to the directions (aa), (bb), (cc), and (ab) one expects 15 A$_{1}$ modes and 7 B$_{1}$ modes for the previously assumed non-centrosymmetric structure (C$_{2v}^7$) [@galy66; @golu], whereas experimentally we observed only 8 and 3 modes [@fisch], respectively, in agreement with the factor group analysis for the centrosymmetric structure (D$_{2h}^{13}$). In addition, infrared absorption data [@smirnov; @dama] did not show any phonons coinciding with our Raman-active phonon modes. Therefore the centrosymmetric D$_{2h}^{13}$ structure should be favored. Hence there are no two inequivalent V sites which would give rise to the weakly coupled pairs of V$^{4+}$ and V$^{5+}$ chains as proposed by Isobe et al. [@isobe].
In Fig. 1 we compare the Raman scattering intensity with incident and scattered light polarization parallel to the legs of the ladders (b-axis) for temperatures above (100 K) and below (T=5 K) T$_{\rm SP}$. At high temperature several phonons are observed ranging from 90 to 1000 cm$^{-1}$ assigned as A$_{1g}$ modes. In contrast to CuGeO$_{3}$ no spinon continuum is observed with polarization parallel to the chains. This is in agreement with previous results that a considerable frustration is required to observe a spinon continuum in light scattering experiments [@mutu]. In $\alpha\prime$-NaV$_{2}$O$_{5}$ the frustration is supposed to be weak [@smolinski; @horsch]. A broad phonon at 422 cm$^{-1}$ ($\approx$ J [@isobe; @weiden]) hardens in energy unusually strong by 2.5 % upon lowering temperature below T=T$_{SP}$. This demonstrates a non-negligible spin-phonon coupling in this system.
Several new modes are detected in the dimerized phase in bb-polarization. The frequencies of these additional excitations are: 67, 107, 134, 151, 165, 202, 246, 294 (shoulder), 396, 527 (shoulder), 652 (remnant from aa-polarization), and 948 cm$^{-1}$. In addition, for frequencies below 120 cm$^{-1}$ an overall drop of the background intensity is observed. This is a typical phenomenon for the occurrence of an energy gap. The value of this gap can be determined experimentally to be about 120-125 cm$^{-1}$. If we attribute this gap to be the onset of the two-particle continuum of triplet excitations 2$\Delta_{01}$, we obtain $\Delta_{01}\approx$ 60-62 cm$^{-1}$, in good accordance with the value determined from susceptibility data [@weiden].
For the discrimination between phonon modes arising because of the existence of a superstructure and possible magnetic bound states in Raman spectroscopy some criteria have been developed in the case of CuGeO$_3$ [@lemmens]. In this frustrated spin chain compound one singlet bound state is observed only for incident and scattered light parallel to the chain direction. These polarization selection rules are consistent with the spin conserving nature of the exchange light scattering mechanism ($\Delta$s=0) with a Hamiltonian which is identical to the Heisenberg exchange Hamiltonian. This process allows no off-diagonal scattering matrix element (in crossed polarizations) as observed for one magnon excitations in 2d or 3d antiferromagnets. As the singlet bound state responds very sensitively by defects and thermal fluctuations its scattering intensity as function of temperature reaches its maximum intensity at lower temperatures compared with the dimerization induced phonon modes [@lemmens]. On the other hand, phonon related bound state phenomena observed, e.g., in rare earth chalcogenides showed different properties concerning their selection rules and frequencies [@vitins].
The temperature dependence of scattering intensity, frequency shift and linewidth for various modes observed in the quarter-filled ladder system $\alpha\prime$-NaV$_{2}$O$_{5}$ leads us to an unambiguous distinction between the three modes with energies close to 2$\Delta_{01}$ (67, 107 and 134 cm$^{-1}$) and the other dimerization induced modes at higher energies. Before pointing this out we have to exclude that the 67-cm$^{-1}$ mode is a singlet-triplet excitation (one magnon or spin-flip scattering). This would only be allowed if magnetic light scattering were dominated by spin-orbit coupling. In light scattering experiments in a static magnetic field of up to 7 T neither a shift nor a broadening of this mode was observed. Therefore a one-magnon scattering process can be excluded and hence spin-orbit coupling plays a negligible role in the present context.
Fig. 2a shows the intensity of several modes at low temperatures (T$<$T$_{SP}$) normalized to the adjacent most intense ones. Clearly one can distinguish between the modes at 67, 107 and 134 cm$^{-1}$ on the one hand, and the modes at, e.g., 202, 246, and 948 cm$^{-1}$ on the other hand. The second group follows the behavior expected for a second order type phase transition, i.e. the intensity increases quite sharply upon cooling and then saturates towards low temperatures. The first group of modes obviously increases in intensity upon cooling much more gradually and shows no saturation towards lower temperatures. A similar temperature dependence of the intensity was found for the singlet bound state at 30 cm$^{-1}$ in CuGeO$_{3}$ [@lemmens].
Fig. 2b gives further support for this interpretation. The change of linewidth normalized to the strongest change (67-cm$^{-1}$ mode) is shown as function of temperature. While the three modes at 202, 246, and 948 cm$^{-1}$ do not show any broadening upon approaching the phase transition temperature from below, a behavior typical for simple zone boundary folded phonons, the two modes at 67 and 107 cm$^{-1}$ become remarkably broader. This effect is also accompanied by a small softening of their frequency. The mode at 134 cm$^{-1}$ has been omitted in this analysis due to a strongly changing background. Nevertheless, the tendency is the same.
Further insight can be gained from the comparison with a formerly investigated single crystal of $\alpha^\prime$-NaV$_{2}$O$_{5}$ in Ref. [@fisch] with a slightly broader transition width. The temperature dependence of the intensity of the additional modes at 64, 103, and 130 cm$^{-1}$ shows a similar distinction from the other dimerization-induced modes. However, their frequencies are lower by about 4 cm$^{-1}$ and their absolute intensities are reduced compared to the single crystals used in the present investigation. The energy of all other modes is independent of sample quality. As shown in substitution experiments in CuGeO$_3$ [@lemmens] magnetic bound states in quantum spin systems are extremely sensitive to any defect or thermal fluctuation. Hence from an experimental point of view we have presented reliable evidence that the three modes at 67, 107 and 134 cm$^{-1}$ are related to the singlet-triplet gap and therefore indeed magnetic bound states.
Spin chain models, which neglect magnetic or magnetoelastic inter-chain coupling successfully explain the excitation spectrum of CuGeO$_{3}$ with one singlet bound state [@uhrig; @bouz]. Affleck [@Affleck] proposed a different physical picture which allows for more than one bound state. Due to a linear confining potential mediated via interchain coupling magnetic bound states of soliton-antisoliton pairs arise. If the slope of this confining potential is not too strong, there would be a chance for more than one bound state below the two particle continuum. However, this would imply a more pronounced one-dimensionality in $\alpha^\prime$-NaV$_2$O$_5$ compared to CuGeO$_3$.
The experimental data in the dimerized phase do not confirm this. Fig. 3 shows the polarization dependence of the low energy excitation spectrum. It can be clearly stated that the intensity of the 67-cm$^{-1}$ mode is not restricted to bb-polarization parallel to the legs. It appears in the aa- (i.e. perpendicular to the ladder) and ab-polarizations, as well. Also the 134-cm$^{-1}$ mode and the onset of a continuum at 120 cm$^{-1}$ are observed in all three polarization configurations. So the one-dimensionality concerning the polarization selection rules is obviously not pronounced. On the other hand, the mode at 107 cm$^{-1}$ fulfills the selection rules expected for a singlet bound state in a dimerized chain. It only appears in bb-polarization along the ladder direction. Also its energy, as pointed out above, corresponds to the energy of the singlet bound state mode in CuGeO$_{3}$, i.e., $\sqrt{3}$$\Delta_{01}$= $\sqrt{3}$(60-62 cm$^{-1}$).
This leads us to the following interpretation of our experiments based on $\alpha^\prime$-NaV$_{2}$O$_{5}$ as a quarter-filled ladder: It is not surprising that this compound can have a spin-Peierls transition into a dimerized ground state assuming an effective spin chain model. To understand the full excitation spectrum this point of view obviously is not enough. As more bound states exist than allowed in models based on dimerized chains we consider the possible exchange paths of the quarter-filled ladder to construct further possible dimer states. These might occur along the rung (a-axis), and the ladder legs (b-axis) and its diagonals. To obtain the observed superlattice structure and allowing only weak interaction between ladders the possible dimers are restricted to a square of adjacent sites. Fig. 4 shows a representation of these possible states. The energies of a- and b-axis dimers should be nearly degenerate leading to a system of several competing ground states. Hopping processes between these states will form a quantum superposition which splits the dimer energy levels by $\Delta_{el}$ and leads to a lowering of the lowest eigenstate. This dynamic configuration is allowed as long as the energy of phonons $\hbar\omega_{ph}$ with a dominant spin-phonon coupling that induce the dimerization, fulfill $\hbar\omega_{ph}\gg\Delta_{el}$. This picture is comparable with the dynamic Jahn-Teller effect or the RVB model.
In this way the breakdown of the selection rules observed in light scattering as well as the unexpected low energy of the bound state at 67 cm$^{-1}$ is described qualitatively. A quantitative discussion will have to explain the exact energies of the singlet modes as well as of the modes in the triplet channel. Therefore neutron scattering data are of crucial importance. In recent experiments of Yosihama [@fuji97] a triplet mode with a steep dispersion along the b\*-axis (ladder direction) splits into two branches along the a\*-axis. Its maximum splitting of 34 K (=23.5 cm$^{-1}$) gives a new low energy scale and may be understood as a bonding-antibonding band of the dimers on adjacent plaquettes. This additional scale is much smaller than the charge transfer gap $\Delta_{CT}$=0.7 eV of the V-O-V orbital, the lowest excited state in the spin chain models of Refs. [@smolinski; @horsch]. As a rough estimate the energy separation between the first and the second singlet bound state (40 cm$^{-1}$) observed in Raman scattering experiments should be comparable with twice the maximum splitting of the triplet dispersion along the a\*-axis. The factor of two results from the binding of two triplet states to a singlet bound state.
Using light scattering in $\alpha^\prime$-NaV$_{2}$O$_{5}$, magnetic singlet bound states were identified. These states consist of triplet excitations that are bound with respect to the “free” two-particle continuum of states above 2$\Delta_{01}$. While the mode at 107 cm$^{-1}$ fits both in energy and selection rules to a singlet bound state in a dimerized spin-1/2 Heisenberg chain, the other two bound states do not fit to such a simple picture. Instead, we propose the origin of the singlet modes at 67 and 134 cm$^{-1}$ as due to a superposition of several nearly degenerate dimer configurations on the quarter-filled ladder. We therefore give evidence that, in contrast to the homogeneous high temperature phase, the dimerized phase of $\alpha^\prime$-NaV$_{2}$O$_{5}$ cannot be understood on the basis of a dimerized spin chain. Instead, it is necessary to consider the topology and enlarged exchange degrees of freedom of a ladder.
Acknowledgment: We acknowledge valuable discussions with C. Gros, G.S. Uhrig, G. Bouzerar, W. Brenig, P.H.M. v. Loosdrecht, and G. Roth. Financial support by DFG through SFB 341 and SFB 252, BMBF/Fkz 13N6586/8 and INTAS 96-410 is gratefully acknowledged.
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0.3cm
CERN-TH/96-261\
hep-ph/9610362
\
[**D. de Florian[^1]**]{}\
[*Theoretical Physics Division, CERN, CH 1211 Geneva 23, Switzerland*]{}\
e-mail: Daniel.de.Florian@cern.ch\
[**R. Sassot**]{}\
[*Departamento de Física, Universidad de Buenos Aires\
Ciudad Universitaria, Pab.1 (1428) Bs.As., Argentina*]{}\
[ **Abstract**]{}\
CERN-TH/96-261\
October 1996
[**1. Introduction**]{}\
In recent years, a considerable degree of attention has been paid to the semi-inclusive production of $\Lambda$ hyperons and their polarization. The approaches include experimental programmes to measure polarized fragmentation functions in $e^+e^-\rightarrow \Lambda+X$ [@burjaf] and semi-inclusive lepton-proton deep inelastic scattering (DIS) [@comp; @jaf], both in the current fragmentation region, as well as models for the production of these hyperons in existent $p\bar{p}$ and DIS experiments, but in the target fragmentation region [@elko]. Many of these studies are closely related to the interpretation of the proton spin structure [@elliot; @elka], due to the possibility of reconstructing it from the observed $\Lambda$ polarization.
Notwithstanding this increasing experimental interest in spin-dependent semi-inclusive $\Lambda$ physics, and also the important phenomenological insights related to the production of these hyperons in different processes, QCD corrections to such cross sections have received little attention. These corrections, however, have been shown to be crucial for the interpretation of totally inclusive polarized DIS experiments related to the spin structure of the proton [@elliot; @elka], and they certainly would have a non-negligible role in polarized $\Lambda$ production. On the other hand, one would naturally expect any parton model-inspired phenomenological description of the processes to mould into the more formal QCD-improved description, at least in some adequate limit. A description including fracture functions [@ven; @graudenz] allows not only a consistent factorization of the collinear divergences characteristic of semi-inclusive processes, but also a perturbative picture of events where the identified final-state hadron comes from the target fragmentation region.
In addition to their phenomenological interest, QCD corrections to polarized semi-inclusive processes involve some theoretical subtleties, such as factorization procedures [@vogel], which in this case have to be extended to polarized fragmentation functions, and final-state polarized fracture functions, and which deserve a very careful treatment. In a recent paper [@npb] we have computed the $\cal{O}$$(\alpha_s)$ corrections to the one-particle inclusive polarized DIS cross sections, using dimensional regularization [@bollini] and the HVBM [@HVBM] prescription for the $\gamma_5$ matrix, discussing different factorization schemes. The analysis includes also target fragmentation effects by means of polarized fracture functions, but the polarization of the identified hadron is not taken into account, given that the observables under study are those related to pions and kaons in the final state.
In this paper we extend the approach of ref. [@npb] in order to include the possibility of polarized hadrons, such as the $\Lambda$, in the final state. This inclusion requires the computation of $\cal{O}$$(\alpha_s)$ corrections to polarized fragmentation functions, which dominate $\Lambda$ production in the current fragmentation region and can be measured in $e^+e^- \rightarrow \Lambda + X$, and also to final-state polarized fracture functions, which, as it has been said, account for target fragmentation effects and make possible a consistent factorization of collinear singularities in semi-inclusive DIS.
Next to leading order (NLO) corrections to polarized fragmentation functions have been addressed recently in refs. [@rav], using massive gluons to regulate infrared singularities. In this paper, however, we keep dimensional regularization and we propose a factorization prescription similar to the ones of our previous work [@npb; @prd]. The factorization prescriptions for final-state polarized fracture functions can be generalized directly from those for polarized parton distributions, which guarantee the usual current conservation properties [@gordon; @prd]. The use of a common regularization procedure and natural factorization prescription allows a consistent and more direct analysis of cross sections for different processes.
In the following section we introduce the formal framework for obtaining the QCD corrections to polarized fragmentation functions in $e^+ e^-$ annihilation processes, and we specify our choice for factorization. In the third section we do the same for fracture functions, discussing there the structure of the different cross sections resulting from the combinations of polarization in the initial and final states. Finally, we summarize our results and present conclusions.\
[**2. $e^+ e^-$ annihilation**]{}\
In this section we obtain the NLO corrections for the semi-inclusive production of a longitudinally polarized hadron ($h$) in $e^+ e^-$ annihilation. This process, besides offering the clearest way to define at leading order, and eventually measure, the polarized fragmentation functions, allows us to show our choice for the factorization prescription, which defines the fragmentation functions beyond the leading order.
These corrections have recently been obtained in ref. [@rav], using off-shell gluons as regularization prescription. Since we want to make contact with previous calculations for other semi-inclusive DIS processes, which were performed using dimensional regularization, we first obtain the above-mentioned corrections using the latter prescription.
The usual kinematical variables used for the description of processes like the one under consideration, i.e. $e^+(l) e^-(l') \rightarrow h(p)+ X$, are $$z=\frac{2\, p.q}{Q^2} \,\,\,\,\, {\rm and} \,\,\,\,\, Q^2=q^2 ,$$ where the variable $q$ denotes the four-momentum of the virtual photon, and is defined by the momentum of the incoming leptons, $q=l+l'$. In terms of these variables it is possible to write down the photon fragmentation tensor $W_{\mu\nu}$, whose antisymmetric part $W_{\mu\nu}^A$ is $$W_{\mu\nu}^{A} = 2 \frac{i N_c}{z^2 Q^2} \epsilon_{\mu\nu\rho\sigma} q^\rho s^\sigma g^h_1 .$$ We do not include here the piece related to the additional semi-inclusive structure function $g^h_2$, which only contributes in the case of transverse polarization.
Taking the difference between the cross sections for the processes where the incoming electron and the outgoing hadron are polarized parallel and antiparallel to each other, we obtain $$\frac{d\sigma (\uparrow\uparrow- \downarrow\uparrow)}{dz\, d(\cos\theta)} =
\alpha^2 \frac{N_c \pi}{Q^2} g^h_1 \cos\theta ;$$ here $\theta$ is the angle between the hadron and the beam directions.
In the naive parton model, the polarized structure function $g^h_1 (z)$ is given by the convolution between the partonic cross section corresponding to the Feynman diagram of fig. 1a and the polarized semi-inclusive fragmentation functions $$\Delta D_{h/q}(z)= D^{h\uparrow}_{q\uparrow} (z) - D^{h\uparrow}_{q\downarrow} (z) ,$$ i.e. $$\label{naive}
g^h_1 (z)= \sum^{N_f}_q e_q^2 [\Delta D_{h/q}(z)+\Delta D_{h/{\bar{q}}}(z)] .$$
In NLO of QCD, the interference between the diagrams of figs. 1a and 1b, that between the first one and those in 1c and 1d, as well as the whole contributions from the diagrams of fig. 2 are included, in both cases taking into account that either quarks or gluons undergo hadronization. Then, eq. (\[naive\]) develops the less trivial convoluted expression: $$\begin{aligned}
\label{1loop}
g^h_1 (z,Q^2) = \sum^{N_f}_q e_q^2 \int_z^1 \frac{dy}{y} \left\{\left[ \delta(1-y)+
\Delta C_q \left(y, \frac{Q^2}{\mu^2}\right)\right] \times \right. \nonumber \\
\left. \left[\Delta D_{h/q}\left(\frac{z}{y}\right) +
\Delta D_{h/{\bar{q}}}\left(\frac{z}{y}\right) \right]
+\Delta C_g\left(y, \frac{Q^2}{\mu^2}\right) \Delta D_{h/g}\left(\frac{z}{y}\right) \right\} .\end{aligned}$$
Since in some intermediate steps of the calculation, both infrared and soft collinear divergences appear, (the UV divergences can be avoided by working in the Landau gauge), it is necessary to define a prescription to isolate them. For this purpose, and throughout this paper, we use dimensional regularization and the HVBM scheme, which allows us to deal with typically four-dimensional objects such as $\epsilon^{\mu\nu\rho\sigma}$ and $\gamma^5$ in a fully consistent way. As is well known [@vogel; @webber], spurious terms appear in this scheme because of the breaking of the chiral symmetry. These terms can be absorbed in the factorization procedure in order to keep the naive interpretation of the first moment of the polarized fragmentation function, i.e. as the fraction of the spin of the parton carried by the outgoing hadron.
In order to calculate the NLO corrections, it is useful to define the variables $x_i=2\, k_i . q/ Q^2 = 2\, E_i/Q$, in terms of the momentum carried by the three-final state partons. For the sake of simplicity, we choose the $z$ direction as that of the parton labelled by the index 1, and $\phi$ as the angle between the parton labelled 2 and the $z$ direction. Momentum conservation implies that $$x_2 = \frac{1-x_1}{1-x_1(1-y)},$$ where $$y=\frac{1+\cos\phi}{2},$$ and integrating over the particles labelled 2 and 3 (particle 1, which undergoes hadronisation, is not integrated), the three ($2+1$) particle phase space reduces to $$PS^{(2+1)} = \frac{(4\pi)^\epsilon}{8\pi}\int_0^1 dy \,
\frac{x_1^{1-2\epsilon}}{1-x_1}\, x_2^2\, \int_0^{Q^2 x_2^2 y(1-y)}
d|\hat{k}|^2 \, \frac{|\hat{k}|^{-2(1+\epsilon)}}{\Gamma (-\epsilon)}.$$ Here $|\hat{k}|^2$ is the square of the $d-4=-2 \,\epsilon$ dimensional component of the momentum of the outgoing (integrated) particles 2 and 3.
Projecting conveniently the matrix elements corresponding to the diagrams in fig. 2, and integrating over the phase space, we obtain the real emission contributions to $\Delta C_q$ (we label the outgoing quark as number 1): $$\begin{aligned}
\label{realq}
\Delta C_q \left(y,\frac{Q^2}{\mu^2}\right) &=&
\left( \frac{4\pi
\mu^2}{Q^2} \right)^\epsilon \frac{\Gamma(1-\epsilon)}{\Gamma(1-2 \epsilon)}
\frac{\alpha_s}{2\pi} C_F \left\{ -\frac{1}{\epsilon} \left[ \frac{3}{2}
\delta(1-y) + \frac{1+y^2}{(1-y)_+} \right] \right.\nonumber \\
&+& \left. \, \delta(1-y) \left[ \frac{2}{\epsilon^2} + \frac{3}{\epsilon}
+ 8 -\frac{2 }{3} \pi^2 \right] + \Delta f_q^D(y) \frac{}{} \right\}, \,\,\,\,\,\,\,\,\end{aligned}$$ with $$\begin{aligned}
\Delta f_q^D (y) &=& \delta(1-y) \left[ \frac{2}{3} \pi^2 -\frac{9}{2}
\right] -\frac{3}{2} \left(\frac{1}{1-y}\right)_+ + (1+y^2)
\left(\frac{\ln (1-y)}{1-y}\right)_+ \nonumber \\
&+& 2\, \frac{1+y^2}{1-y} \ln y -\frac{3}{2} (1-y) -2 \widehat {(1-y)} . \end{aligned}$$ In the last equation, we have kept track of terms originated in the $4-2\, \epsilon$ dimensional momentum integration, writing them under hats just for factorization purposes. The second term in the r.h.s. of eq. (\[realq\]), which contains the infrared divergences, cancels identically when adding the virtual contributions coming from the interference of diagrams in fig. 1, as is generally expected.
The same procedure can be applied in order to obtain the gluonic coefficient (now, labelling the gluon as parton 1) obtaining $$\begin{aligned}
\label{realg}
\Delta C_g \left(y,\frac{Q^2}{\mu^2}\right) =
\left(\frac{4\pi\mu^2}{Q^2} \right)^\epsilon \frac{ \Gamma(1-\epsilon)}
{\Gamma(1-2 \epsilon)} \frac{\alpha_s}{2\pi} C_F \left\{ -\frac{1}{\epsilon}
\left[ 2 \, (2-y) \right] + \Delta f_g^D(y) \right\} , \end{aligned}$$ with $$\begin{aligned}
\Delta f_g^D (y) = 2\, \left\{ (2-y) \ln [(1-y)y^2]- (2-y) - 2 \widehat{(1-y)}
\right\} .\end{aligned}$$
Having computed the whole cross section up to next to leading order, we are now able to factorize the divergences by means of the definition of scale-dependent polarized fragmentation functions ($\Delta D_{h/q}(z,Q^2)$ and $\Delta D_{h/g}(z,Q^2)$). Fixing the factorization scale $\mu^2$ equal to $Q^2$, in the $\overline{MS}$ scheme the prescription amounts to absorbing only the $1/\hat\epsilon$ terms. It has been shown that within the HVBM method this prescription leaves unsubtracted some soft finite contributions directly related to the hat terms. It is also frequent to define a different scheme, called $\overline{MS_p}$ [@gordon], where these contributions are subtracted. The general expression for the distributions, whithout specifying any scheme, is then given by $$\begin{aligned}
\label{dd}
\Delta D_{h/q}(z) = \int^1_z \frac{dy}{y} \left\{ \left[ \delta(1-y) +
\frac{\alpha_s}{2\pi} \left( {{1\over \hat{\epsilon}}}\Delta P_{q \leftarrow q} (y) - C_F \Delta
\tilde{f}_q^D (y) \right) \right] \times \right. \,\,\,\, \nonumber \\
\left. \Delta D_{h/q}\left(\frac{z}{y},Q^2\right) + \left( {{1\over \hat{\epsilon}}}\Delta P_{g \leftarrow q}
(y) - C_F \Delta \tilde{f}_g^D (y) \right) \Delta D_{h/g}\left(\frac{z}{y},Q^2\right)
\right\} , \,\,\,\,\,\,\,\end{aligned}$$ where $${{1\over \hat{\epsilon}}}\equiv {1\over \epsilon}
{{\Gamma[1-\epsilon]}\over{\Gamma[1-2
\epsilon]}}\left({{4\pi\mu^2}\over{Q^2}}\right)^\epsilon
=\frac{1}{\epsilon}-\gamma_E+\log 4\pi+\log \frac{\mu^2}{Q^2}+\cal{O}(\epsilon)$$
In the $\overline{MS_p}$ scheme, the finite subtraction terms $\Delta
\tilde{f}_q^D (y) = \Delta \tilde{f}_g^D (y) = 4\, (y-1)$ are designed to absorb the soft contributions coming from the real gluon emission diagrams in fig. 2 and, therefore, they are the same as the one obtained for the definition of NLO polarized quark distributions in DIS. In fact, the hat terms obtained for the real gluon emission diagrams in DIS are equal to the ones obtained in this analysis.
Using the former definition for NLO distributions, the final expression for the semi-inclusive structure function reads $$\begin{aligned}
\label{1loopfact}
g^h_1 (z,Q^2) = \sum^{N_f}_q e_q^2 \int_z^1 \frac{dy}{y} \left\{\left[
\delta (1-y) + \frac{\alpha_s}{2\pi} C_F \left(\Delta f_q^D(y) -
\Delta \tilde{f}_q^D (y)\right) \right] \times \right. \nonumber \\
\left.
\left[\Delta D_{h/q}\left(\frac{z}{y}\right) +
\Delta D_{h/{\bar{q}}}\left(\frac{z}{y}\right) \right]
+ \frac{\alpha_s}{2\pi} C_F \left[ \Delta f_g^D(y) - 2 \Delta \tilde{f}_g^D(y)
\right]\Delta D_{h/g}\left(\frac{z}{y}\right) \right\} \,\,\,\,\,\,\,\,\,\end{aligned}$$ where, as usual, in the $\overline{MS}$ scheme the finite subtracted terms are chosen to be zero.
Notice that although the full expression for the cross sections, and the definition of scale-dependent densities, differs from those of refs.[@rav] in the finite terms, the evolution kernels, which are not scheme-dependent, are identical, as expected.\
[**3. Semi-Inclusive Deep Inelastic Scattering**]{}\
With the definition for polarized fragmentation functions given in the previous section, eq. (14), we are now able to compute the NLO corrections for semi-inclusive DIS in the case in which the final-state hadron is polarized. NLO contributions for processes with unpolarized final-state hadrons (with either polarized or unpolarized initial states) have been computed in refs. [@npb] and [@graudenz], respectively, so we refer the reader to these for most of the definitions and conventions.
Using the usual kinematical DIS variables for the interaction between a lepton of momentum $l$ and helicity $\lambda_l$ and a nucleon $A$ of momentum $P$ and helicity $\lambda_A$ $$x=\frac{Q^2}{2\, P\cdot q} ,\,\,\,\,\,\,\, y=\frac{P\cdot q}{P\cdot l},\,\,\,\,\,\,\, Q^2=-q^2\,\, {\rm and}\,\,\,\, S_H=(P+l)^2 ,$$ the differential cross section for the production of a hadron $h$ with energy $E_h= z\, E_A (1-x)$ and helicity $\lambda_h$ (with $n$ partons in the final state) can be written as $$\begin{aligned}
\label{sec}
\frac{d \sigma^{\lambda_l \lambda_A \lambda_h}}{dx\,dy\,dz} &=& \int \frac{du}{u}\sum_n\sum_{j=q,\bar q , g} \int dPS^{(n)}
\, \frac{\alpha^2}{S_{H}x}\, \frac{1}{e^2 (2\pi)^{2d}}\, \nonumber \\
&\times& \left[
Y_M(-g^{\mu\nu})+Y_L \frac{4x^2}{Q^2}P_\mu P_\nu + \lambda_l Y_{P}\,
\frac{x}{Q^2}\, i \epsilon^{\mu \nu q P } \right] \nonumber \\
&\times& \sum_{\lambda_1,\lambda_2 =\pm 1}H_{\mu\nu}(\lambda_1,\lambda_2)
\left\{ M_{j,h/A} \left( \frac{x}{u},\frac{E_h}{E_A},\frac{\lambda_1}{\lambda_A},\frac{\lambda_h}{\lambda_A}\right)(1-x) \right. \nonumber \\
&+& \left. f_{j/A}\left(\frac{x}{u},\frac{\lambda_1}{\lambda_A}\right)
\sum_{i_\alpha =q,\bar q , g } D_{h/i_{\alpha}}\left(\frac{z}{\rho},
\frac{\lambda_h}{\lambda_2}\right)
\frac{1}{\rho} \right\}\end{aligned}$$ where $\rho= E_\alpha/E_A$. The helicity-dependent partonic tensor is defined by $$H_{\mu\nu} (\lambda_1,\lambda_2) = M_{\mu}(\lambda_1,\lambda_2)
M_{\nu}^\dagger(\lambda_1,\lambda_2) ,$$ where $M_\mu$ is the parton-photon matrix element with the photon polarization vector subtracted. Here $\lambda_1$ and $\lambda_2$ are the helicities of the initial- and final-state partons respectively. Both $f_{j/A}$ and $D_{h/i_{\alpha}}$ are the usual parton distribution and fragmentation functions, respectively, which represent the probabilities of finding a parton $j$ or a hadron $h$ with a given momentum fraction and helicity with respect from the parent particle (the target $A$ or the parton $i_\alpha$, respectively).
As was shown in ref. [@ven], it is necessary to introduce a new distribution for semi-inclusive processes, the fracture function, which gives the probability of finding a polarized parton $j$ and a polarized hadron $h$ in the nucleon, denoted by $M_{j,h/A} \left( \frac{x}{u},\frac{E_h}{E_A},
\frac{\lambda_1}{\lambda_A},\frac{\lambda_h}{\lambda_A}\right) $. Including them, it will be possible to factorize all the collinear divergences that remain in the NLO contributions after the redefinition of the scale-dependent parton distribution and fragmentation functions.
Taking different linear combinations of cross sections for targets and final-state hadrons with equal or opposite helicities, it is possible to isolate parton distributions, fragmentation functions and fracture functions of different kind and their QCD corrections. For example, taking the sum over all the helicity states $$\sigma=\sigma^{\lambda_l ++} +\sigma^{\lambda_l+-} +\sigma^{\lambda_l-+}
+\sigma^{\lambda_l--},$$ the result is proportional to a convolution of unpolarized parton distributions, fragmentation and fracture functions [@graudenz] [^2]. Taking the difference between cross sections with opposite target helicities, as in ref. [@npb]: $$\Delta \sigma=(\sigma^{\lambda_l ++} +\sigma^{\lambda_l+-}) -(\sigma^{\lambda_l-+}
+\sigma^{\lambda_l--}),$$ the result contains polarized parton distributions, single polarized fracture functions (single initial-state polarization), and unpolarized fragmentation functions. In the following two subsections, we consider the two remaining possibilities, which represent the case where only the final-state hadron polarization is put in evidence (single final-state polarization) at cross section level: $$\sigma\Delta=(\sigma^{\lambda_l ++} -\sigma^{\lambda_l+-}) +(\sigma^{\lambda_l-+}
-\sigma^{\lambda_l--}),$$ and that where both polarizations are relevant (double polarization): $$\Delta \sigma\Delta=(\sigma^{\lambda_l ++} -\sigma^{\lambda_l+-}) -(\sigma^{\lambda_l-+}
-\sigma^{\lambda_l--}).$$
[**3a. Single (final-state) polarization**]{}\
It is straightforward to show that the single (final-state) polarization cross section $\sigma \Delta$ is proportional to the convolutions $$\begin{aligned}
\label{conv}
\sigma \Delta \, \propto \, \, f_{j/A}(x/u)\otimes H\Delta(u,\rho) \otimes
\Delta D_{h/i_{\alpha}}(z/\rho) \nonumber \\
+(1-x) \, M\Delta_{j,h/A} (x/u, (1-x) z) \otimes
\Delta H'(u)\end{aligned}$$ where the first term corresponds to current fragmentation processes and the second to fragmentation of the target. In the former, $f_{j/A}(x/u)$ are just the unpolarized parton distributions, $\Delta D_{h/i_{\alpha}}(z/\rho)$ are the polarized fragmentation functions defined in section 2, and $H\Delta = \sum_{\lambda_1,\lambda_2} \lambda_2 \, H(\lambda_1,\lambda_2)$ is the partonic tensor for polarized final states. The convolutions are those defined by eq. (\[sec\]). In the second term of eq. (\[conv\]) we have defined the final state polarized fracture function as $$M\Delta = M_{j,h/A} \left( +,+\right)+
M_{j,h/A} \left( -,-\right) - M_{j,h/A} \left( +,-\right) - M_{j,h/A} \left( -,+\right),$$ where we have omitted the first two kinematical arguments in the fracture functions, and the single polarized tensor $\Delta H'= \sum_{\lambda_1,\lambda_2} \lambda_1 \,H(\lambda_1,\lambda_2)$ (integrated in $\rho$).
Computing the matrix elements of the diagrams in figs. 4, 5 and 6, contracting with the adequate projector $$\begin{aligned}
P_{pol}^{\mu\nu}\equiv \frac{\alpha^2}{S_{H}x}\,\, \frac{1}{e^2 (2\pi)^{2d}}\,\, \frac{x}{Q^2} \,i \epsilon^{\mu \nu q P}, \end{aligned}$$ and integrating them over the phase space as it done in ref. [@npb], we finally find: $$\begin{aligned}
\label{cucusa}
&&\frac{d \sigma \Delta }{dx\,dy\,dz} =
Y^p \lambda_l \sum_{i=q,\bar q} c_i \left\{ \int \int_{A} \frac{du}{u} \frac{d\rho}{\rho}
\left\{ q_i\left(\frac{x}{u}\right) \, \Delta D_{q_i}\left(\frac{z}{\rho}\right)\,
\delta(1-u)\delta(1-\rho)\frac{}{} \right.\right. \nonumber \\
&+& \left.\left. q_i\left(\frac{x}{u}\right)\, \Delta D_{q_i}\left(\frac{z}{\rho}\right)\,
\right. \right. \nonumber \\
&&\hspace*{-4mm}\times \left. \left.
\frac{\alpha_s}{2\pi} \left[ {-{1\over \hat{\epsilon}}}\left(\Delta P_{q\leftarrow q}(\rho)\delta(1-u)+
P_{q\leftarrow q}(u)\delta(1-\rho)\right) +C_f
\Phi\Delta_{qq}(u,\rho)\right]
\right.\right. \nonumber\\
&+& \left. \left. q_i\left(\frac{x}{u}\right)\,\Delta D_g\left(\frac{z}{\rho}\right)\,
\right. \right. \nonumber \\
&&\hspace*{-4mm}\times \left. \left.
\frac{\alpha_s}{2\pi} \left[ {-{1\over \hat{\epsilon}}}\left(\Delta P_{g\leftarrow q}(\rho)\delta(1-u)+
\hat P\Delta_{gq\leftarrow q}(u)\delta(\rho-a)\right) +C_f
\Phi\Delta_{qg}(u,\rho)\right]
\right. \right.\nonumber\\
&+ & \left. \left. g\left(\frac{x}{u}\right)\,\Delta D_{q_i}\left(\frac{z}{\rho}\right)\,
\right. \right. \nonumber \\
&&\hspace*{-4mm}\times \left. \left.
\frac{\alpha_s}{2\pi} \left[ {-{1\over \hat{\epsilon}}}\left( P_{q\leftarrow g}(u)\delta(1-\rho)+
\hat P\Delta_{q\bar{q}\leftarrow g}(u)\delta(\rho-a)\right) + T_f
\Phi\Delta_{gq}(u,\rho)\right] \right\}\right. \nonumber \\
& & \hspace*{40mm}\left. + \, \int_{B} \frac{du}{u} (1-x)
\left\{ M\Delta_{q_i} \left(\frac{x}{u},(1-x) z\right) \delta(1-u) \right. \right. \nonumber \\
&+&\left.\left. M\Delta_{q_i} \left(\frac{x}{u},(1-x) z\right) \, \frac{\alpha_s}{2\pi} \left[ {-{1\over \hat{\epsilon}}}\Delta P_{q\leftarrow
q}(u)+C_f \Phi\Delta_{q}(u) \right] \right. \right.\nonumber \\
&+& \left.\left. M\Delta_g \left(\frac{x}{u},(1-x) z\right) \,
\frac{\alpha_s}{2\pi} \left[ {-{1\over \hat{\epsilon}}}\Delta P_{q\leftarrow g}(u)+ T_f
\Phi\Delta_g (u)\right] \right\}\right\} \end{aligned}$$ where the coefficients $\Phi\Delta_{ij}$ are given in the Appendix, $$c_j= 4\pi Q^2_{q_j} \, {{\alpha^2}\over {S_H x}} \, ,$$ and the integration limits can be found in ref. [@npb].
The poles proportional to $\delta (1-u)$ and to the polarized Altarelli-Parisi kernels $\Delta P_{i\leftarrow j}$ [@ap] correspond to final-state singularities and are subtracted with the definition of the NLO polarized fragmentation functions given in section 2. The coefficients $\Phi\Delta_{ij}$ contain finite soft terms (proportional to the terms marked with hats) related to polarized final-states which are also subtracted by means of eq. (\[dd\]) in the $\overline{MS_p}$ scheme.
Those poles proportional to $\delta (1-\rho)$ and to the unpolarized kernels are related to the initial-state singularities and are therefore subtracted by the NLO definition of unpolarized parton distributions. In this case both the $\overline{MS}$ and the $\overline{MS_p}$ schemes coincide, since we are dealing here with unpolarized initial-state processes, so there are no finite soft terms. Notice that there are no hat terms proportional to $\delta (1-\rho)$ in the coefficients.
Finally, the poles proportional to $\delta (\rho -a)$, where $a=x(1-u)/u(1-x)$, represent configurations where the hadrons are produced by the current but in the direction of the incoming nucleon. These are subtracted along with the target-initiated singularities (homogeneous) by the NLO definition of final-state polarized fracture functions $$\begin{aligned}
\label{1p}
M\Delta_{q_i} (\xi,\zeta) &=& \nonumber \\
&&\hspace*{-25mm}\int_{\frac{\xi}{1-\zeta}}^1 \frac{du}{u} \left\{
\left[ \delta(1-u) + \frac{\alpha_s}{2\pi} \left(
{{1\over \hat{\epsilon}}}\Delta P_{q\leftarrow q}(u) - C_{f} \Delta
\tilde{f}_q^{MH}(u)\right) \right]
M\Delta_{q_i}\left(\frac{\xi}{u},\zeta,Q^2\right) \right. \nonumber \\
& & \left. +\,\frac{\alpha_s}{2\pi} \left[
{{1\over \hat{\epsilon}}}\Delta P_{q\leftarrow g}(u) - T_{f} \Delta
\tilde{f}_g^{MH}(u) \right] M\Delta_g\left(\frac{\xi}{u},\zeta,Q^2\right)
\right\}\nonumber \\
& &\hspace*{-15mm} +\,\int_{\xi}^{\frac{\xi}{\xi+\zeta}} \frac{du}{u} \frac{u}{x(1-u)}
\left\{ \frac{\alpha_s}{2\pi} \left[
{{1\over \hat{\epsilon}}}\hat P\Delta_{gq\leftarrow q}(u) - C_{f} \Delta
\tilde{f}_q^{MI}(u) \right]
\right. \nonumber \\&& \left. \hspace*{3mm} \times
q_i\left(\frac{\xi}{u},Q^2\right)
\Delta D_g\left(\frac{\zeta u}{\xi (1-u)},Q^2\right) \right. \nonumber \\
&&\left. + \, \frac{\alpha_s}{2\pi} \left[
{{1\over \hat{\epsilon}}}\hat P\Delta_{q\bar{q}\leftarrow g}(u) - \frac{\alpha_s}{2\pi} T_{f} \Delta
\tilde{f}_g^{MI}(u) \right]
\right. \nonumber \\ &&\left. \hspace*{3mm} \times \,
g\left(\frac{\xi}{u},Q^2\right) \,
\Delta D_{q_i}\left(\frac{\zeta u}{\xi (1-u)},Q^2\right) \right\} \hspace*{10mm}\end{aligned}$$ where the homogeneous $(MH)$ and inhomogeneous $(MI)$ counterterms in the $\overline{MS_p}$ are given by $$\begin{aligned}
\Delta \tilde{f}_q^{MI}(u) &=& \Delta\tilde{f}_q^{MH}(u) = 4\,(u-1) \\ \nonumber
\Delta \tilde{f}_g^{MI}(u) &=& \Delta\tilde{f}_g^{MH}(u) = 2\,(1-u).
\end{aligned}$$ These are identical to those in the final-state polarized case, since the structure of the corrections is the same (actually, the homogeneous coefficients are exactly the same).
One interesting feature of the divergences is that the poles corresponding to $\delta (1-\rho)$ and $\delta (\rho-a)$ in eq. (\[cucusa\]) differ only in a global sign $(-\hat P\Delta_{q\bar q \leftarrow g} (x) = P_{q\leftarrow g} (x))$. This sign accounts for the opposite polarizations of the quark in the $\gamma g \rightarrow q\bar q$ process according to whether the quark is emitted in the photon direction ($\rho=1$) or in the opposite configuration ($\rho=a$).
Applying the factorization procedure, we obtain the following cross section $$\begin{aligned}
\frac{d\sigma\Delta}{dx\,dy\,dz} & =& \nonumber \\
&& \hspace*{-8mm} Y^p \lambda_l \sum_{i=q,\bar q} c_i\left\{\int \int_{A} \frac{du}{u}
\frac{d\rho}{\rho}
\left\{
q_i\left(\frac{x}{u},Q^2\right)\, \Delta D_{q_i}\left(\frac{z}{\rho},Q^2\right) \,
\delta(1-u)\delta(1-\rho)\frac{}{}
\right. \right. \nonumber \\
&+& \hspace*{-3mm} \left.\left.
q_i\left(\frac{x}{u},Q^2\right)\, \Delta D_{q_i}\left(\frac{z}{\rho},Q^2\right) \,
\frac{\alpha_s}{2\pi} C_f \left[
\Phi\Delta_{qq}(u,\rho) - \Delta \tilde{f}_q^D(\rho) \delta(1-u) \right] \right.\right.
\nonumber \\
&+& \hspace*{-3mm} \left.\left. q_i\left(\frac{x}{u},Q^2\right)\, \Delta D_g\left(\frac{z}{\rho},Q^2\right)\,
\right.\right. \nonumber \\
&& \hspace*{-3mm}\left. \left. \times \,\frac{\alpha_s}{2\pi} C_f \left[
\Phi\Delta_{qg}(u,\rho)- \Delta \tilde{f}_g^{D}(\rho) \delta(1-u) - \Delta \tilde{f}_q^{MI}(u) \delta(\rho-a)\right]
\right. \right.\nonumber\\
& +& \hspace*{-3mm} \left. \left. g\left(\frac{x}{u},Q^2\right)\, \Delta D_{q_i}\left(\frac{z}{\rho},Q^2\right)\,
\frac{\alpha_s}{2\pi} T_f\left[
\Phi\Delta_{gq}(u,\rho) - \Delta
\tilde{f}_g^{MI}(u) \delta (\rho-a) \right]
\right\} \right. \nonumber\\
& & \hspace*{12mm} \left. + \, \int_{B} \frac{du}{u} (1-x)
\left\{ M\Delta_{q_i}\left(\frac{x}{u},(1-x) z,Q^2\right) \delta(1-u) \right.\right. \nonumber \\
& + & \hspace*{-3mm}\left. \left. M\Delta_{q_i}\left(\frac{x}{u},(1-x) z,Q^2\right) \, \frac{\alpha_s}{2\pi} C_f \left[
\Phi\Delta_{q}(u) - \Delta\tilde{f}_q^{MH}(u) \right]
\right.\right. \nonumber \\
&+&\hspace*{-3mm} \left.\left. M\Delta_g \left(\frac{x}{u},(1-x) z,Q^2\right) \,
\frac{\alpha_s}{2\pi} T_f \left[ \Phi\Delta_g(u)
-\Delta\tilde{f}_g^{MH}(u) \right]
\right\}\right\}, \end{aligned}$$ which is free of collinear divergences.\
[**3b. Double (initial- and final-state) Polarization**]{}\
An analysis similar to that of subsection 3.a shields for $\Delta \sigma \Delta$ the following convolutions $$\begin{aligned}
\label{conv2}
\Delta \sigma \Delta \, \propto \, \, \Delta f_{j/A}(x/u)\otimes \Delta H\Delta(u,\rho) \otimes
\Delta D_{h/i_{\alpha}}(z/\rho) \nonumber \\
+(1-x) \, \, \Delta M\Delta_{j,h/A} (x/u, (1-x) z) \otimes
H' (u),\end{aligned}$$ where again the first term corresponds to current fragmentation processes and the second to fragmentation of the target. In the former, $\Delta f_{j/A}(x/u)$ are now polarized parton distributions, $\Delta D_{h/i_{\alpha}}(z/\rho)$ are polarized fragmentation functions, and $\Delta H\Delta = \sum_{\lambda_1,\lambda_2} \lambda_1 \lambda_2 \, H(\lambda_1,\lambda_2)$ is the partonic tensor for polarized initial and final states. In the second term of eq. (\[conv2\]) we define the doubly polarized fracture functions as $$\Delta M\Delta = M_{j,h/A} \left( +,+\right)-
M_{j,h/A} \left( -,-\right) - M_{j,h/A} \left( +,-\right) + M_{j,h/A} \left( -,+\right),$$ and tensor $ H'= \sum_{\lambda_1,\lambda_2} H(\lambda_1,\lambda_2)$, also integrated in $\rho$.
Using now the metric and longitudinal projectors $$\begin{aligned}
P^{\mu\nu}_M = - g^{\mu\nu} \,\,\,\,\, {\rm and} \,\,\,\,\,\, P^{\mu\nu}_L = \frac{4 x^2}{Q^2} P^\mu P^\nu\end{aligned}$$ and defining the metric and longitudinal components of the total cross section $$\Delta\sigma\Delta=\Delta\sigma^M\Delta+\Delta\sigma^L\Delta ,$$ we obtain $$\begin{aligned}
& & \hspace*{-5mm}\frac{d \Delta\sigma^M \Delta }{dx\,dy\,dz} =
Y^M \sum_{i=q,\bar q} 2\, c_i \left\{ \int \int_{A} \frac{du}{u} \frac{d\rho}{\rho}
\left\{ \Delta q_i\left(\frac{x}{u}\right) \, \Delta D_{q_i}\left(\frac{z}{\rho}\right)\,
\delta(1-u)\delta(1-\rho)\frac{}{}
\right.\right. \nonumber \\
&+& \left.\left.
\Delta q_i\left(\frac{x}{u}\right)\, \Delta D_{q_i}\left(\frac{z}{\rho}\right) \frac{\alpha_s}{2\pi}\,
\right.\right. \nonumber \\
\times&& \hspace*{-12mm}\left.\left.
\left[ {-{{1+\epsilon}\over \hat{\epsilon}}}\left(\Delta P_{q\leftarrow q}(\rho)\delta(1-u)+
\Delta P_{q\leftarrow q}(u)\delta(1-\rho)\right) +C_f
\Delta\Phi^M\Delta_{qq}(u,\rho) \right] \right.\right. \nonumber\\
&+& \left. \left. \Delta q_i\left(\frac{x}{u}\right)\,\Delta D_g\left(\frac{z}{\rho}\right) \frac{\alpha_s}{2\pi}\,
\right.\right. \nonumber \\
\times& &\hspace*{-12mm} \left.\left.
\left[ {-{{1+\epsilon}\over \hat{\epsilon}}}\left(\Delta P_{g\leftarrow q}(\rho)\delta(1-u)+
\Delta \hat P\Delta_{gq\leftarrow q}(u)\delta(\rho-a)\right) +C_f
\Delta \Phi^M\Delta_{qg}(u,\rho)\right]
\right. \right.\nonumber\\
&+& \left. \left. \Delta g\left(\frac{x}{u}\right)\,\Delta D_{q_i}\left(\frac{z}{\rho}\right) \frac{\alpha_s}{2\pi}\,
\right.\right. \nonumber \\
\times && \hspace*{-12mm}\left.\left.
\left[ {-{{1+\epsilon}\over \hat{\epsilon}}}\left( \Delta P_{q\leftarrow g}(u)\delta(1-\rho)+
\Delta \hat P\Delta_{q\bar{q}\leftarrow g}(u)\delta(\rho-a)\right) + T_f
\Delta \Phi^M\Delta_{gq}(u,\rho)\right] \right\}\right. \nonumber\\
&& \hspace*{35mm}\left. +\, \int_{B} \frac{du}{u} (1-x)
\left\{ \Delta M\Delta_{q_i}\left(\frac{x}{u},(1-x) z\right) \delta(1-u) \right. \right.\nonumber \\
&+& \left.\left. \Delta M\Delta_{q_i}\left(\frac{x}{u},(1-x) z\right) \frac{\alpha_s}{2\pi} \left[ {-{{1-\epsilon}\over \hat{\epsilon}}}P_{q\leftarrow
q}(u)+C_f
\right] \right. \right.\nonumber \\
&+& \left.\left. \Delta M\Delta_g \left(\frac{x}{u},(1-x) z\right) \,
\frac{\alpha_s}{2\pi} \left[ {-{{1-\epsilon}\over \hat{\epsilon}}}P_{q\leftarrow g}(u)+ T_f
\Delta\Phi^M\Delta_g (u)\right] \right\}\right\} \hspace*{6mm}\mbox{(36)}\nonumber\end{aligned}$$
and $$\begin{aligned}
\frac{d \Delta\sigma^L \Delta }{dx\,dy\,dz} &=&
Y^L \sum_{i=q,\bar q} 2\, c_i \left\{ \int \int_{A} \frac{du}{u} \frac{d\rho}{\rho}
\left\{
\Delta q_i\left(\frac{x}{u}\right)\, \Delta D_{q_i}\left(\frac{z}{\rho}\right)\,
\frac{\alpha_s}{2\pi} C_f
\Delta\Phi^L\Delta_{qq}(u,\rho) \right.\right. \nonumber\\
&+& \left. \left. \Delta q_i\left(\frac{x}{u}\right)\,\Delta D_g\left(\frac{z}{\rho}\right)\,
\frac{\alpha_s}{2\pi} C_f
\Delta \Phi^L\Delta_{qg}^A(u,\rho)
\right. \right.\nonumber\\
&+& \left. \left. \Delta g\left(\frac{x}{u}\right)\,\Delta D_{q_i}\left(\frac{z}{\rho}\right)\,
\frac{\alpha_s}{2\pi} T_f
\Delta \Phi^L\Delta_{gq}(u,\rho) \right\}\right. \nonumber\\
&+& \left. \int_{B} \frac{du}{u} (1-x)
\left\{ \Delta M\Delta_{q_i}\left(\frac{x}{u},(1-x) z\right) \frac{\alpha_s}{2\pi} C_f \Delta\Phi^L\Delta_{q}(u) \right. \right.\nonumber \\
&+& \left.\left. \Delta M\Delta_g \left(\frac{x}{u},(1-x) z\right) \,
\frac{\alpha_s}{2\pi} T_f
\Delta\Phi^L\Delta_g (u) \right\}\right\}\hspace*{32mm} \mbox{(37)}\nonumber \end{aligned}$$
While the longitudinal component, which is zero at leading order, is finite, the collinearly divergent metric component has yet to be subtracted, using the method indicated in the last subsection. We then use the NLO definition of polarized parton distributions [@gordon; @prd] and polarized fragmentation functions in the $\overline{MS_p}$ scheme, and introduce the NLO doubly polarized fracture functions by means of $$\begin{aligned}
\label{2p}
\Delta M\Delta_{q_i} (\xi,\zeta) =
\int_{\frac{\xi}{1-\zeta}}^1 \frac{du}{u} \left\{
\left[ \delta(1-u) + \frac{\alpha_s}{2\pi}
{{{1+\epsilon}\over \hat{\epsilon}}}P_{q\leftarrow q}(u) \right] \Delta M\Delta_{q_i}\left(\frac{\xi}{u},\zeta,Q^2\right) \right. \nonumber \\
\left. + \frac{\alpha_s}{2\pi}
{{{1+\epsilon}\over \hat{\epsilon}}}P_{q\leftarrow g}(u) \Delta M\Delta_g\left(\frac{\xi}{u},\zeta,Q^2\right) \right\}
\hspace*{38mm}
\nonumber \\
\hspace*{5mm} +\int_{\xi}^{\frac{\xi}{\xi+\zeta}} \frac{du}{u} \frac{u}{x(1-u)}
\left\{ \frac{\alpha_s}{2\pi}
{{{1-\epsilon}\over \hat{\epsilon}}}\Delta \hat P\Delta_{gq\leftarrow q}(u) \Delta q_i\left(\frac{\xi}{u},Q^2\right) \,
\Delta D_g\left(\frac{\zeta u}{\xi (1-u)},Q^2\right) \right. \nonumber \\
\left. + \frac{\alpha_s}{2\pi}
{{{1-\epsilon}\over \hat{\epsilon}}}\Delta \hat P\Delta_{q\bar{q}\leftarrow g}(u) \Delta g\left(\frac{\xi}{u},Q^2\right) \,
\Delta D_{q_i}\left(\frac{\zeta u}{\xi (1-u)},Q^2\right) \right\}. \hspace*{4mm} \mbox{(38)}\nonumber \end{aligned}$$
In this case no, finite terms are subtracted since the corrections for the fracture functions have exactly the same structure as those in the unpolarized case. In fact, as can be seen in the contribution from the box diagram, finite soft terms (which for the box diagram are just the hat terms) coming from the effects of the initial- and final-state polarization in the HVBM prescription cancel leaving no soft contributions proportional to $\delta(\rho-a)$.
We then obtain the following finite metric component of the cross section $$\begin{aligned}
\frac{d\Delta \sigma^M\Delta}{dx\,dy\,dz} & = & \nonumber \\
&& \hspace*{-9mm} Y^M \sum_{i=q,\bar q} 2\, c_i\left\{\int \int_{A} \frac{du}{u}
\frac{d\rho}{\rho}
\left\{
\Delta q_i\left(\frac{x}{u},Q^2\right)\, \Delta D_{q_i}\left(\frac{z}{\rho},Q^2\right) \,
\delta(1-u)\delta(1-\rho)\frac{}{}
\right. \right. \nonumber \\
&+& \left.\left.
\Delta q_i\left(\frac{x}{u},Q^2\right)\, \Delta D_{q_i}\left(\frac{z}{\rho},Q^2\right) \,
\right. \right. \nonumber \\
&& \hspace*{-2mm}\left.\left. \times \,
\frac{\alpha_s}{2\pi} C_f \left[
\Delta\Phi^M\Delta_{qq}(u,\rho) - \Delta \tilde{f}_q^D(\rho) \delta(1-u) - \Delta \tilde{f}_q^F(u) \delta(1-\rho) \right] \right.\right.
\nonumber \\
&+& \left.\left. \Delta q_i\left(\frac{x}{u},Q^2\right)\, \Delta D_g\left(\frac{z}{\rho},Q^2\right)\,
\right. \right. \nonumber \\
&& \hspace*{-2mm}\left.\left. \times \,
\frac{\alpha_s}{2\pi} C_f \left[
\Delta\Phi^M\Delta_{qg}(u,\rho)- \Delta \tilde{f}_g^{D}(\rho) \delta(1-u) \right]
\right. \right.\nonumber\\
& +& \left. \left. \Delta g\left(\frac{x}{u},Q^2\right)\, \Delta D_{q_i}\left(\frac{z}{\rho},Q^2\right)\,
\right. \right. \nonumber \\
&& \hspace*{-2mm}\left.\left. \times \,
\frac{\alpha_s}{2\pi} T_f\left[
\Delta\Phi^M\Delta_{gq}(u,\rho) - \Delta
\tilde{f}_g^{F}(u) \delta(1-\rho) \right]
\right\} \right. \nonumber\\
& &\hspace*{20mm}\left. + \, \int_{B} \frac{du}{u} (1-x)
\left\{ \Delta M\Delta_{q_i}\left(\frac{x}{u},(1-x) z,Q^2\right) \delta(1-u) \right. \right. \nonumber \\
& +& \left. \left. \Delta M\Delta_{q_i}\left(\frac{x}{u},(1-x) z,Q^2\right)\frac{\alpha_s}{2\pi} C_f
\Delta \Phi^M\Delta_{q}(u) \right.\right. \nonumber \\
&+ & \left.\left. \Delta M\Delta_g \left(\frac{x}{u},(1-x) z,Q^2\right) \,
\frac{\alpha_s}{2\pi} T_f \Delta \Phi^M\Delta_g(u)
\right\}\right\} \hspace*{23mm} \mbox{(39)} \nonumber\end{aligned}$$
where the finite terms subtracted from the polarized parton distributions are $$\begin{aligned}
\Delta\tilde{f}_q^{F}(u) = 4\,(u-1) \,\,\,\, {\rm and} \,\,\,\,
\Delta\tilde{f}_g^{F}(u) = 2\,(1-u)
\end{aligned}$$ and the coefficients can be found in the Appendix.
Notice that both the single polarized (final-state) fracture function $M \Delta$ and the doubly polarized one $\Delta M \Delta$ are not mere artefacts motivated by factorization, but they are physically motivated parton densities related to target fragmentation. In the latter case they parametrize unpolarized parton densities of a nucleon that fragments into a hadron with a definite polarization state. In the former case, they involve polarized parton distributions of fragmented nucleons[^3].
At this stage, it is worth noticing that any factorization scheme can be chosen as long as the distributions used had been obtained by means of a consistent NLO analysis (with the corresponding splitting functions and coefficients in the same scheme). For instance, in the case of polarized parton distributions, NLO parametrizations can be obtained from ref. [@forteball] in the $\overline{MS_p}$ scheme, or from refs. [@gs; @grsv], in a mixed factorization scheme where only the $\Delta \tilde{f}^F_q(u)=4 (u-1)$ has been substracted ($\Delta \tilde{f}^F_g(u)=0$ in eq. (40)). In the case of polarized fragmentation functions, no parametrizations are available yet, but the same concepts apply to them, both in the case of coputations using dimensional regularization or off-shell gluons, as in ref. [@rav], to regularize the divergencies. The corresponding NLO polarized splitting functions have been obtained very recently [@vogstrat] in the mixed factorization scheme ($\Delta \tilde{f}^D_q(y)=4 (y-1)$ and $\Delta \tilde{f}^D_g(u)=0$ in eq. (14)) and will allow for a complete NLO analysis of polarized fragmentation functions.
[**4. Summary and Conclusions**]{}\
Summarizing our results, we have presented here the full $\cal{O}$$(\alpha_s)$ QCD corrections to the semi-inclusive production cross section of polarized hadrons in $e^+e^-$ annihilation processes and polarized deep inelastic lepton-hadron scattering using dimensional regularization and the HVBM scheme.
Generalizing the factorization prescription that excludes soft contributions in polarized deep inelastic ($\overline{MS_p}$) to polarized fragmentation functions and to two new types of fracture functions, we have shown that this prescription smoothly subtracts all the collinear divergences and soft contributions present in the semi-inclusive production of polarized hadrons in polarized DIS without introducing kinematical cuts or other process-dependent procedures. We present for completeness also our results in the more familiar $\overline{MS}$ scheme, that offer the same advantages with respect to factorization, but with a less trivial interpretation for parton distributions.
In the case of polarized fragmentation functions, we find our results compatible with the ones presented in ref. [@rav], computed using a different regularization prescription. Of course, both results can be consistently used as long as the scheme dependence is cancelled by the use of the corresponding NLO splitting functions in the chosen scheme.
The evolution kernels obtained here for fracture functions also allow to study the pertubative component of the scale dependence, at least to leading order, of these densities, unveiling new aspects of target fragmentation processes induced by DIS. In this way we produce a fully consistent, physically motivated and complete set of QCD-corrected cross sections, of interest in future phenomenological approaches and forthcoming experiments.\
[ **Acknowledgements**]{}
We gratefully acknowledge C. García Canal, G. Veneziano and D. Graudenz for enlightening comments and discussions.\
[**Appendix** ]{}\
We append here the coefficients, some definitions and useful relations.
a\. $$\begin{aligned}
&& \Phi\Delta_{qq}(u,\rho) = \nonumber \\
&-& 8\, \delta (1-u) \delta (1-\rho)
- \frac{1}{(1-u)_+} (1+\rho)
- \frac{1}{(1-\rho)_+} (1+u) \nonumber \\
&+& \delta (1-u) \left[ \rho-1-(1+\rho) \ln (1-\rho) + \frac{1+\rho^2}{1-\rho} \ln\rho + 2 \left( \frac{\ln (1-\rho)}{1-\rho}\right)_+ \right] \nonumber \\
&+& \delta (1-\rho) \left[ 1 - u -(1+u) \ln (1-u) + \frac{1+u^2}{1-u}\ln\frac{1-x}{u-x} + 2 \left( \frac{\ln (1-u)}{1-u}\right)_+ \right] \nonumber \\
&+& 2 \frac{1}{(1-\rho)_+} \frac{1}{(1-u)_+} + 2\,\left( 2 + u \right) \,\left( 1 - x \right) -
2\,\left( 2 - \rho \right) \,{{\left( 1 - x \right) }^2}\nonumber \\
&+&
{{x\,\left( 6\, x\, (1-\rho) (1-x) +x\, (1-\rho)-2+2\,\rho\, (1-x^2)\right) } \over {u - x}} \nonumber \\
&+& {{\left( 1 - \rho \right) \,\left( 1 - x \right) \,{x^2}\,
\left( 2\,x -1 \right) }\over {{{\left( u - x \right) }^2}}} \end{aligned}$$ $$\begin{aligned}
\Phi\Delta_{qg}(u,\rho) &=& \delta (1-u) (2-\rho)\ln[(1-\rho)\,\rho] + \left(\frac{1}{1-u}\right)_+ ( 2 -\rho)
\nonumber \\
&+& \delta (\rho-a) (1+u) \ln\frac{1-u}{u} + \left(\frac{1}{\rho-a}\right)_+ (1+ u) \nonumber \\
&-& 2 (1-\rho)^2 (1+\rho^2)\, \widehat{\delta (1-u)} -1 +
{{2\,\left( 1 -\rho \right) \,\left( 2\,x - 1 \right) }\over {1 - x}} \nonumber \\
&+& {{u -2 \, u^2 (1-x) }\over {u - x}} - {{ 2 \left( 1 - \rho \right) \,\left( 1 - u \right) \, \, x \, u \, (1-x) }\over { \,{{\left( u - x \right) }^2}}}
\nonumber \\
&+& {{ 2\left( 1 - \rho \right) \,\left( 1 - u \right) \,x^2\,
}\over {\left( 1 - x \right) \,{{\left( u - x \right) }}}} + {{\left( 1 - \rho \right) \,\left( 1 - u \right) \,x^2\,
}\over { \,{{\left( u - x \right) }^2}}} - 2 \widehat{\delta (\rho-a)} \nonumber \\\end{aligned}$$ $$\begin{aligned}
\Phi\Delta_{gq}(u,\rho) &=&
\delta(1-\rho)\left[1+(1-2u+2u^2)\left(-1+\ln\frac{(1-u)(1-x)}{u-x}\right)
\right] \nonumber \\
&+& \left(\frac{1}{1-\rho}\right)_+ (1-2 u+2u^2 )
+ 2(1-u) \widehat{\delta (\rho-a)} \nonumber \\
&+& \delta (\rho-a)
\left[1-(1-2u+2u^2)\left(-1+\ln\frac{1-u}{u}\right)\right]
\nonumber \\
&-& \left(\frac{1}{\rho-a}\right)_+ (1-2u+2u^2) \end{aligned}$$
$$\begin{aligned}
\Phi\Delta_{q}(u) & = &
-\frac{1+u^2}{1-u}\ln(u)-(1+u)\ln(1-u)
+2\left(\frac{\ln(1-u)}{(1-u)} \right)_{+} \nonumber \\
&-& \frac{3}{2} \left(\frac{1}{1-u}\right)_+ + 3 u + (-\frac{9}{2}-\frac{\pi^2}{3})\delta(1-u) -2\widehat{(1-u)}\end{aligned}$$
$$\begin{aligned}
\Phi\Delta_{g} (u) &=& 2\widehat{(1-u)} +(2 u-1)
\left[ \ln \left( {{1-u}\over u} \right) -1 \right] \end{aligned}$$
b.
i\. $$\begin{aligned}
&& \Delta\Phi^M\Delta_{qq}(u,\rho)= \nonumber \\
&-& 8\, \delta (1-u) \delta (1-\rho) - \frac{1}{(1-u)_+} (1+\rho)
- \frac{1}{(1-\rho)_+} (1+u) \nonumber \\
&+& \delta (1-u) \left[ \rho-1-(1+\rho) \ln (1-\rho) + \frac{1+\rho^2}{1-\rho} \ln\rho + 2 \left( \frac{\ln (1-\rho)}{1-\rho}\right)_+ \right] \nonumber \\
&+& \delta (1-\rho) \left[ u - 1 -(1+u) \ln (1-u) + \frac{1+u^2}{1-u}\ln\frac{1-x}{u-x} + 2 \left( \frac{\ln (1-u)}{1-u}\right)_+ \right] \nonumber \\
&+& 2 \frac{1}{(1-\rho)_+} \frac{1}{(1-u)_+} + {{\left( 1 - \rho \right) \,x\,\left( 2\,u - x - u\,x \right) }\over {{{\left( u - x \right) }^2}}}\nonumber \\
&+& {{2\,\left( u - x - u\,x \right) }\over {u - x}}
\end{aligned}$$ $$\begin{aligned}
\Delta \Phi^M\Delta_{qg}(u,\rho) &=& \delta (1-u) (2-\rho)\ln[(1-\rho)\, \rho]
+ \left(\frac{1}{1-u}\right)_+ ( 2 -\rho)
\nonumber \\
&+& \delta (\rho-a) (1+u)\left[2+\ln\frac{1-u}{u}\right] + \left(\frac{1}{\rho-a}\right)_+ (1+ u) \nonumber \\
&-& 2\, (1-\rho^2)\, \widehat{\delta (1-u)}
-2\,\left( 1 - x \right) + {{\left( 1 - \rho \right) \,x}\over {u - x}} - {{\rho\,\left( 1 - x \right) \,x}\over {u - x}} \nonumber \\
&+& {{\left( 1 - \rho \right) \,\left( 1 - x \right) \,{x^2}}\over
{{{\left( u - x \right) }^2}}} + {{{x^2}}\over {u - x}} - 2\, u \, \widehat{\delta (\rho-a)} \end{aligned}$$ $$\begin{aligned}
\Delta \Phi^M\Delta_{gq}(u,\rho) &=& \delta (1-\rho) (2u-1)\left(\ln\frac{(1-u)(1-x)}{u-x}\right) \nonumber \\
&+& \left(\frac{1}{1-\rho}\right)_+ (2u-1 )- \left(\frac{1}{\rho-a}\right)_+ (2u-1)
\nonumber \\
&+& \delta (\rho-a)
(2u-1)\left(-2-\ln\frac{1-u}{u}\right) \nonumber \\
&+& 2(1-u) \widehat{\delta (1-\rho)}
\end{aligned}$$ $$\begin{aligned}
\Delta\Phi^M\Delta_{q}(u) & = &
-\frac{1+u^2}{1-u}\ln(u)-(1+u)\ln(1-u)
+2\left(\frac{\ln(1-u)}{1-u} \right)_{+} \nonumber \\
&-&\frac{3}{2}
\left(\frac{1}{1-u}\right)_+ + 3 -u + (-\frac{9}{2}-\frac{\pi^2}{3})\delta(1-u) \end{aligned}$$ $$\begin{aligned}
\Delta\Phi^M\Delta_g (u)= (1-2 u+2u^2) \left(\ln\frac{1-u}{u} -1 \right)\end{aligned}$$
ii\. $$\begin{aligned}
\Delta\Phi^L\Delta_{qq}(u,\rho)=\frac{2 (\rho-a) u^3 (1-x)^3}{(u-x)^2}\end{aligned}$$ $$\begin{aligned}
\Delta \Phi^L\Delta_{qg}(u,\rho)=-2(1-\rho) u^3 \frac{(1-x)^2 }{(u-x)^2} \end{aligned}$$ $$\begin{aligned}
\Delta \Phi^L\Delta_{gq}(u,\rho)= 0\end{aligned}$$ $$\begin{aligned}
\Delta\Phi^L\Delta_{q}(u)= u\end{aligned}$$ $$\begin{aligned}
\Delta\Phi^L\Delta_g (u)= 2u(1-u) \end{aligned}$$
c.
We include here the expressions for the virtual [@ap] and real [@uemven] splitting functions: $$\begin{aligned}
\Delta P_{q\leftarrow q}(u)&=& P_{q\leftarrow q}(u)=
C_f \left[ 2\left(\frac{1}{1-u}\right)_+ + \frac{3}{2}
\delta(1-u)-1-u\right]
\nonumber \\
\Delta P_{q\leftarrow g}(u)&=& \Delta \hat P_{q\bar{q}\leftarrow g}(u) = - \Delta \hat P\Delta_{q\bar{q}\leftarrow g}(u) = T_f \left[2u-1\right]\nonumber \\
P_{q\leftarrow g}(u)&=& - \hat P\Delta_{q\bar{q}\leftarrow g}(u) = T_f \left[2u^2 -2 u+1\right] \\
\Delta P_{g\leftarrow q}&=& \hat P\Delta_{gq\leftarrow q}(1-u) = \Delta \hat P\Delta_{gq\leftarrow q}(1-u) =
C_f \left[ 2-u\right]\nonumber \\
P_{g\leftarrow q}(u)&=& \Delta\hat P_{gq \leftarrow q}(1-u) = C_f \left[ 2\frac{1}{u} -2+u \right] \nonumber
\end{aligned}$$
d.
We include here the coefficients from ref. [@npb] adapted to the definition of the $()_+$ prescriptions used in ref. [@graudenz]: $$\begin{aligned}
& & \Delta\Phi_{qq}(u,\rho) = \nonumber \\
& - & 8\, \delta(1-\rho) \delta(1-u) -
\left(\frac{1}{1-\rho}\right)_+ (1+u)- \left(\frac{1}{1-u}\right)_+(1+\rho)
\nonumber \\
&+& \left[(1-\rho)+\frac{1+\rho^2}{1-\rho}
\ln\rho-(1+\rho)\ln(1-\rho)+2\left(\frac{\ln(1-\rho)}{1-\rho} \right)_{+}\right]
\delta(1-u)\nonumber\\
&+& \left[-(1-u)+\frac{1+u^2}{1-u}
\ln(\frac{1-x}{u-x})-(1+u)\ln(1-u)+2\left(\frac{\ln(1-u)}{1-u} \right)_{+}
\right]
\delta(1-\rho) \nonumber\\
&-&{{\left( 1 - \rho \right) \,\left( 1 - u +
\left( 1 - x \right) \,\left( -1 +
u\,\left( 1 + 2\,u \right) \,\left( 1 - x \right) - x \right)
\right) }\over {{{\left( u - x \right) }^2}}} +
{{4\,u\,\left( 1 - x \right) }\over {u - x}} \nonumber \\
&-& {{2\,\left( 1 - u \right) \,u\,\left( 1 - x \right) }\over {u - x}}- {{2\,x}\over {u - x}} + 2 \left(\frac{1}{1-\rho}\right)_+ \left(\frac{1}{1-u}\right)_+ -2 (1-u) \widehat{\delta(1-\rho)}\nonumber \\\end{aligned}$$ $$\begin{aligned}
\Delta \Phi_{qg}(u,\rho) & = & \delta (1-u) \left[\rho + \left(\rho+{2\over \rho}-2 \right)
\ln \left(\rho(1-\rho) \right) \right] \nonumber
\\
&+& \delta (\rho-a) \left[-2\widehat{(1-u)}-(1-u) +
{{1+u^2}\over{1-u}} \ln \frac{1-u}{u} \right]
- {{2\,{{\left( 1 - \rho \right) }^2}}\over {\rho\,\left( 1 - u \right) }}
\nonumber \\
&+&
{{ -2\, u}\over { (\rho-a)(1-u)}} - {{2\,\left( 1 - u \right) \,u\,
\left( 1 - x \right) }\over {u - x}} +
{{2\,{{\left( 1 - \rho \right) }^2}\,{u^3}\,{{\left( 1 - x \right) }^2}}\over
{\left( \rho-a \right) \,\left( 1 - u \right) \,
{{\left( u - x \right) }^2}}} \nonumber\\
& +& {{\left( 2 - \rho \right) \,x}\over {u - x}}
+\left({1\over{\rho-a}}\right)_+ {{1+u^2}\over{1-u}} +\left({1\over{1-u}}\right)_+ \left(\rho+{2\over \rho}-2 \right) \nonumber \\
&+& {{\left( 1 - \rho \right) \,u\,\left( 1 - x \right) \,x}\over
{{{\left( u - x \right) }^2}}} \end{aligned}$$ $$\begin{aligned}
\Delta\Phi_{gq}(u) &=& \delta (1-\rho) \left[2\widehat{(1-u)} + (2 u-1) \ln \left(
{{(1-x)(1-u)} \over {u-x }} \right) \right] \nonumber \\
&+& \delta (\rho - a) \left[2\widehat{(1-u)} + (2 u-1) \ln\frac{1-u}{u} \right] \nonumber \\
& + & (2 u-1) \left[ \left({1\over{1-\rho}}\right)_+
+\left({1\over{\rho-a}}\right)_+ \right] -2 u{{1-x}\over{u-x}} \end{aligned}$$ $$\begin{aligned}
\Delta \Phi_{q}(u) & = &
-\frac{1+u^2}{1-u}\ln(u)-(1+u)\ln(1-u)
+2\left(\frac{\ln(1-u)}{1-u} \right)_{+} \nonumber \\
&-&\frac{3}{2}
\left(\frac{1}{1-u}\right)_+
+ 3 u + \left(-\frac{9}{2}-\frac{\pi^2}{3}\right)\delta(1-u) -2\widehat{(1-u)}\end{aligned}$$ $$\begin{aligned}
\Delta \Phi_{g} (u,\rho)& = & 2\widehat{(1-u)} +(2 u-1)
\left[ \ln \left( {{1-u}\over u} \right) -1 \right] \end{aligned}$$
[99]{} M. Burkardt and R.L. Jaffe, Phys. Rev. Lett. 70, 2537 (1993). Compass Proposal, CERN/SPSLC 96-14, SPSC/P 297 (1996). R.L. Jaffe, MIT-CTP-2534 (1996) hep-ph/9605456. J. Ellis, D. Kharzeev and A. Kotzinian, Z. Phys. C69, 467 (1996). M. Anselmino, A. Efremov and E. Leader, Phys. Rep. 261 (1995). J. Ellis and M. Karliner, CERN-TH/95-334 (1995) hep-ph/9601280. L. Trentadue and G. Veneziano, Phys. Lett. B323, 201 (1994). D. Graudenz, Nucl. Phys. B432, 351 (1994). W. Vogelsang, Z. Phys. C50, 275 (1991). D. de Florian, C.A. García Canal and R. Sassot, Nucl. Phys. B470, 195 (1996). C.G. Bollini and J.J. Giambiaggi, Nuovo Cimento 12B, 20 (1972). G. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972);\
P. Breitenlohner and D. Maison, Commun. Math. Phys. 52, 11 (1977). V. Ravindran, hep-ph/9607384, hep-ph/9606273, hep-ph/9606272. D. de Florian and R. Sassot, Phys. Rev. D51, 6052 (1995). L.E. Gordon and W. Vogelsang, Phys. Rev D48, 3136 (1993). A. Weber, Nucl. Phys. B382, 63 (1992). G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977). E. Konishi, A. Ukawa and G. Veneziano, Nucl. Phys. B157, 45 (1979). R.D. Ball, S. Forte and G. Ridolfi, Phys. Lett. B378, 255 (1996). T. Gehrmann and W.J. Stirling, Phys. Rev. D53, 6100 (1996). M. Glück, E. Reya, M. Stratmann and W. Vogelsang, Phys. Rev. D53, 4775 (1996). M. Stratmann and W. Vogelsang, hep-ph/9612250 (1996).
[^1]: On leave of absence from Departamento de Física, Universidad de Buenos Aires, Ciudad Universitaria Pab.1 (1428) Bs.As., Argentina.
[^2]: Notice the slight difference in the normalization used here (a factor of 2) with respect to ref. [@graudenz], where an average over the polarizations of the target was taken.
[^3]: The notation used may be, up to a point, misleading: “double” and “single” refer to a given combination of cross sections that will be proportional, in the end, to one or the other fracture function. The partonic pictures for fracture functions are given by eqs. (25) and (33), also confirmed by the kernels that drive their evolution (polarized or unpolarized) in eqs. (27) and (36).
|
---
abstract: 'In this article we investigate the relationships between the classical notions of weakest precondition and weakest liberal precondition, and provide several results, namely that in general, weakest liberal precondition is neither stronger nor weaker than weakest precondition, however, given a deterministic and terminating sequential while program and a postcondition, they are equivalent. Hence, in such situation, it does not matter which definition is used.'
address: 'School of Computing, National University of Singapore'
author:
- 'Andrew E. Santosa'
title: Comparing Weakest Precondition and Weakest Liberal Precondition
---
programming languages ,program verification ,program analysis ,symbolic execution
Introduction {#sec:intro}
============
Recent years have seen the prevalent usage of *symbolic execution* [@king76symbolic] for program analysis. Typical symbolic execution system builds *path conditions* corresponding to execution paths. A path condition is a constraint that represents logical relation between the input and output of an execution path. Its components are constraints modeling the input and output relations of each program statement along the execution path. A path condition can be used to determine the set of inputs that causes a program to reach an error state. For example, given an array index $i$ and array size bound $s,$ the path condition represents the conditions on the input variables that makes array bounds violation $i\geq s$ holding after executing a path. A constraint solver can be used to compute an actual program inputs that causes the violation. There are two well-known notions of the set of inputs represented by a path and violation conditions: *weakest precondition* and *weakest liberal precondition*. These notions are elements of the general notion of *predicate transformation* introduced in [@dijkstra75gcl]. Whereas weakest and weakest liberal preconditions computes input conditions in a “backward” manner, in the literature, the notion of predicate transformation also includes a “forward” transformation called *strongest postcondition*.
In this article, we explain how weakest and weakest liberal preconditions are different. We also explain how that under a very common condition of deterministic and terminating programs they are equivalent. In Section \[sec:nonequivalence\] we provide some preliminary definitions together with our first result that in general, weakest and weakest liberal preconditions are not equivalent. We also present their relationships when the program is deterministic, and when the program induces a satisfiable transition relation. In Section \[sec:equivalence\] we show that given a deterministic and terminating while program, weakest and weakest liberal preconditions are the same, and in Section \[sec:discussion\] we show how to define weakest liberal precondition in terms of weakest precondition, and in Section \[sec:conclusion\] we make some concluding remarks.
Weakest and Weakest Liberal are Not Equivalent {#sec:nonequivalence}
==============================================
Here we clarify some terminologies. In this article, we adopt the more common definition of weakest liberal precondition as in [@dijkstra76lang]. However, in some literature [@bjornerinv97], weakest liberal precondition is instead termed weakest precondition. Our definition of weakest liberal precondition is equivalent to the weakest precondition of [@bjornerinv97]. On the other hand, weakest precondition that we mean in this article is that of [@dijkstra75gcl] or [@dijkstra76lang] which is also sometimes also termed *pre-image* in the literature (cf. the backward CTL decision procedure in [@Huth:ModelChecking00]). Compared to weakest liberal precondition, the notion of weakest precondition as in [@dijkstra76lang] and [@dijkstra75gcl] adds the requirement that the precondition should guarantee the termination of the execution.
We now start with some formal definitions. We denote by $\tilde{x}$ a sequence $x_0, \ldots, x_n$ of (program) variables with some unspecified $n.$ We abuse the notion of program to also mean any of its fragments such as, e.g., a statement is also a program. Now, any program $P$ induces a *transition relation* $\rho_P(\tilde{x},
\tilde{x}')$ on free variables $\tilde{x}$ and $\tilde{x}'$, where $\tilde{x}$ represents the program variables before the transition and $\tilde{x}'$ represents the program variables after the transition. For example, an assignment statement $\tilde{x} ~{\mbox{:=}}~
f(\tilde{x})$ induces the transition relation $\tilde{x}' =
f(\tilde{x}).$ In general, for any condition $\varphi,$ we write $\varphi(\tilde{x})$ to clarify that $\tilde{x}$ and no other are the free variables in $\varphi.$ Given a program $P$ and a postcondition $\varphi(\tilde{x})$, the weakest liberal precondition of $\varphi(\tilde{x})$ wrt. $P,$ written ${\mbox{\textsc{wlp}}}(P, \varphi(\tilde{x})),$ is the formula $$\forall \tilde{x}' : \rho_P(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')$$ where $\varphi(\tilde{x}')$ is $\varphi(\tilde{x})$ with all its free variables renamed to their primed versions. On the other hand, the weakest precondition of $\varphi(\tilde{x})$ wrt. $P,$ written ${\mbox{\textsc{wp}}}(P, \varphi(\tilde{x})),$ is the formula $$\exists \tilde{x}' : \rho_P(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}')$$ We remove the subscript $P$ from the transition relation symbol whenever it is clear from the context.
Weakest liberal precondition and weakest precondition are not equivalent in general, as stated in the following theorem.
\[theorem:neither\] In general, weakest liberal precondition is neither stronger nor weaker than weakest precondition.
If weakest precondition was stronger than weakest liberal precondition, then the following would be unsatisfiable: $$(\exists \tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}')) \wedge \neg(\forall \tilde{x}' :
\rho(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')).$$ This is equivalent to: $$(\exists \tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}')) \wedge (\exists \tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \neg\varphi(\tilde{x}')).$$ There exists some $\rho$ such that this formula is satisfiable, that is, in case $\rho$ comes from nondeterministic statement. For example, when $\rho(\tilde{x},\tilde{x}')$ is just a satisfiable constraint $\varphi(\tilde{x})$, which says nothing about $\tilde{x}'$. More concrete example is when $\rho(\tilde{x},\tilde{x}')$ comes from the statements `c=read();` or `c=rand(seed);` assuming the `read()` and `rand(seed)` can return any value.
On the other hand, if weakest liberal precondition was stronger than weakest precondition, then the following would be unsatisfiable: $$(\forall \tilde{x}' : \rho(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')) \wedge \neg(\exists
\tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}')).$$ This is equivalent to: $$(\forall \tilde{x}' : \rho(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')) \wedge (\forall \tilde{x}' : \rho(\tilde{x},\tilde{x}') \rightarrow \neg\varphi(\tilde{x}')).$$ However, also in this case there is some $\rho$ such that the formula is satisfiable, that is, when $\rho$ is ${\mbox{\textbf{false}}}.$ A concrete example of such $\rho$ is an `exit(0);` statement in C, or any other statement that aborts the program.
Equivalence of Weakest and Weakest Liberal for Deterministic and Terminating While Programs {#sec:equivalence}
===========================================================================================
\[theorem:wpstronger\] When the transition relation is deterministic, weakest precondition is stronger than weakest liberal precondition.
We can infer this from the proof of Theorem \[theorem:neither\] above. More formally, we show this by proving that the following is unsatisfiable when $\rho(\tilde{x},\tilde{x}')$ is $\tilde{x}' = f(\tilde{x})$ for some deterministic function $f$: $$(\exists \tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}')) \wedge \neg(\forall \tilde{x}' :
\rho(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')).$$ This is equivalent to: $$(\exists \tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}')) \wedge (\exists \tilde{x}' :
\rho(\tilde{x},\tilde{x}') \wedge \neg\varphi(\tilde{x}')).$$ Substituting $\rho(\tilde{x},\tilde{x}')$ with $\tilde{x}'=f(\tilde{x})$ we have: $\varphi(f(\tilde{x}))
\wedge \neg\varphi(f(\tilde{x}))$, which is unsatisfiable if f is deterministic.
\[theorem:wlpstronger\] When the transition relation is satisfiable, weakest liberal precondition is stronger than weakest precondition.
We can infer this from the proof of Theorem \[theorem:neither\] above. More formally, we proceed by showing a contradiction that $$\label{eqn:contradict2}
{\mbox{\textsc{wlp}}}(P, \varphi(\tilde{x})) \not\rightarrow
{\mbox{\textsc{wp}}}(P, \varphi(\tilde{x}))$$ is unsatisfiable in case $\rho(\tilde{x},\tilde{x}')$ is satisfiable. It is easy to see that (\[eqn:contradict2\]) is equivalent to: $$\forall \tilde{x}' : \rho(\tilde{x},\tilde{x}') \rightarrow (\varphi(\tilde{x}') \wedge \neg\varphi(\tilde{x}'))$$ which is absurd as $\varphi(\tilde{x}') \wedge
\neg\varphi(\tilde{x}')$ is ${\mbox{\textbf{false}}}$ and $\rho(\tilde{x},\tilde{x}') \not\rightarrow {\mbox{\textbf{false}}}.$
In the special case of sequential programs, since the weakest liberal precondition is actually equivalent to weakest precondition. Following is the proof why, for sequential programs, weakest liberal precondition is equivalent to weakest precondition.
A deterministic sequential while program may contain assignments, if conditionals, and while loops, and their sequential compositions in the usual manner. In addition, for any assignment $\tilde{x} ~{\mbox{:=}}~ f(\tilde{x})$, $f$ is a deterministic function.
Let us now examine the transition relation induced by each of the statement of a deterministic sequential while program:
1. For an assignment $\tilde{x} ~{\mbox{:=}}~ f(\tilde{x})$, the transition relation $\rho(\tilde{x},\tilde{x}')$ is $\tilde{x}'=f(\tilde{x})$.
2. For an if conditional $${\mbox{\textbf{if}}}~ \varphi(\tilde{x}) ~{\mbox{\textbf{then}}}~P~ {\mbox{\textbf{else}}}~ P'$$ when the transition relation for $P$ is $\rho_P(\tilde{x},\tilde{x}')$ and the transition relation for $P'$ is $\rho_{P'}(\tilde{x},\tilde{x}')$, the transition relation $\rho(\tilde{x},\tilde{x}')$ induced by the if conditional is $$(\varphi(\tilde{x}) \wedge \rho_{P}(\tilde{x},\tilde{x}')) \vee (\neg \varphi(\tilde{x}) \wedge \rho_{P'}(\tilde{x},\tilde{x}'))$$
3. For a while loop $${\mbox{\textbf{while}}}~ \varphi(\tilde{x})~ {\mbox{\textbf{do}}}~P$$ when the transition relation for $P$ is $\rho_P(\tilde{x},\tilde{x}')$, then the transition relation for the while loop is the infinite formula $$\begin{array}{c}
(\neg \varphi(\tilde{x}) \wedge \tilde{x}'=\tilde{x}) \vee\\
(\varphi(\tilde{x}) \wedge \rho_P(\tilde{x},\tilde{x}_1) \wedge \neg \varphi(\tilde{x}_1) \wedge \tilde{x}'=\tilde{x}_1) \vee\\
(\varphi(\tilde{x}) \wedge \rho_P(\tilde{x},\tilde{x}_1) \wedge \varphi(\tilde{x}_1) \wedge \rho_P(\tilde{x}_1,\tilde{x}_2) \wedge\neg
\varphi(\tilde{x}_2)\wedge \tilde{x}'=\tilde{x}_2) \vee\\
\ldots
(\varphi(\tilde{x}) \wedge \rho_P(\tilde{x},\tilde{x}_1) \wedge (\bigwedge_{i=2}^n : \varphi(\tilde{x}_{i-1}) \wedge
\rho_P(\tilde{x}_{i-1},\tilde{x}_i)) \wedge \neg \varphi(\tilde{x}_n) \wedge \tilde{x}'=\tilde{x}_n) \vee\\
\ldots
\end{array}$$ or, $$\bigvee_{i=0}^{\infty} (\exists \tilde{x}_0,\ldots ,\tilde{x}_i :
(\bigwedge_{j=0}^{i-1} (\varphi(\tilde{x}_j) \wedge \rho_P(\tilde{x}_j,\tilde{x}_{j+1})) \wedge \neg \varphi(\tilde{x}_i) \wedge \tilde{x}'=\tilde{x}_i \wedge \tilde{x}=\tilde{x}_0))$$ It is important to note here that for any nonterminating program $P$, $\neg
\varphi(\tilde{x}_i)$ for all $i$ is unsatisfiable, hence $\rho_P(\tilde{x},\tilde{x}')$ is ${\mbox{\textbf{false}}}.$
Note that a deterministic while program induces a transition relation that is always satisfiable, since if and while conditionals construct two guarded program paths which guards are opposite of each other. Hence, given a program execution state, both guards cannot be unsatisfiable. Since a deterministic while program is both deterministic and has transition relation that is always satisfiable, Theorems \[theorem:wpstronger\] and \[theorem:wlpstronger\] seem to have already suggested that a deterministic while program would have equivalent weakest liberal precondition and weakest precondition, however, here we will proceed more formally and carefully.
\[lemma:equivalent\] The weakest liberal precondition of an assignment is equivalent to its weakest precondition.
Given an assignment $\tilde{x} ~{\mbox{:=}}~ f(\tilde{x})$ and a postcondition $\varphi(\tilde{x})$, the weakest liberal precondition is $$\forall \tilde{x}' : \tilde{x}'=f(\tilde{x}) \rightarrow \varphi(\tilde{x}')$$ and the weakest precondition is $$\exists \tilde{x}' : \tilde{x}'=f(\tilde{x}) \wedge \varphi(\tilde{x}'),$$ each one is equivalent to $\varphi(f(\tilde{x})),$ given $f$ deterministic function.
\[lemma:sequence\] When for each program $P$ and $P',$ the weakest liberal precondition is equivalent to the weakest precondition given any postcondition, then given a postcondition $\varphi(\tilde{x}),$ the sequence $P P'$ has equivalent weakest liberal precondition and weakest precondition.
Given the postcondition $\varphi(\tilde{x}),$ the weakest liberal precondition of of $\varphi({\mbox{\textit{x}}})$ wrt. $P'$ is ${\mbox{\textit{Pre}}}_{P'}$, which is necessarily equivalent to the weakest precondition of $\varphi({\mbox{\textit{x}}})$ wrt. $P'$. Now, given ${\mbox{\textit{Pre}}}_{P'}$ as postcondition, the weakest liberal precondition and weakest precondition of ${\mbox{\textit{Pre}}}_{P'}$ wrt. $P$ are necessarily equivalent from our assumption that for any postcondition $\varphi$ and program $P,$ ${\mbox{\textsc{wlp}}}(P,\varphi)
\equiv {\mbox{\textsc{wp}}}(P', \varphi).$
\[theorem:deterministic\] Given a deterministic and terminating sequential while program $P$ and a postcondition, the weakest liberal precondition of the program wrt. the postcondition is equivalent to the weakest precondition of the program wrt. the postcondition.
We prove inductively. When $P$ is just a sequence of assignments, from Lemma \[lemma:equivalent\] and Lemma \[lemma:sequence\] we obtain the desired result.
Now let us assume $P$ to be an if conditional, say of the form $${\mbox{\textbf{if}}}~\varphi(\tilde{x})~ {\mbox{\textbf{then}}}~P~ {\mbox{\textbf{else}}}~P'$$ As our induction hypothesis, we also assume that both $P$ and $P'$ have equivalent weakest liberal precondition and weakest precondition given any postcondition. Now suppose that the postcondition of the statement is $\varphi.$ Recall that the transition relation of an if conditional is $$(\varphi(\tilde{x}) \wedge \rho_P(\tilde{x},\tilde{x}')) \vee (\neg \varphi(\tilde{x}) \wedge \rho_{P'}(\tilde{x},\tilde{x}'))$$ The weakest liberal precondition of the if condition, given $\varphi$ as postcondition is therefore $$(\forall \tilde{x}' : ((\varphi(\tilde{x}) \wedge \rho_{P}(\tilde{x},\tilde{x}')) \vee (\neg \varphi(\tilde{x}) \wedge
\rho_{P'}(\tilde{x},\tilde{x}'))) \rightarrow \varphi(\tilde{x}'))$$ which is equivalent to $$(\varphi(\tilde{x}) \rightarrow (\forall \tilde{x}' : \rho_P(\tilde{x},\tilde{x}') \rightarrow
\varphi(\tilde{x}'))) \wedge (\neg \varphi(\tilde{x}) \rightarrow (\forall \tilde{x}' :
\rho_{P'}(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')))$$ Note that in the above, $$\forall \tilde{x}' : \rho_P(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')$$ and $$\forall \tilde{x}' : \rho_{P'}(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')$$ are the weakest liberal preconditions of $\varphi(\tilde{x})$ wrt. respectively $P$ and $P'$. We name them ${\mbox{\textit{Pre}}}_{P}(\tilde{x})$ and ${\mbox{\textit{Pre}}}_{P'}(\tilde{x})$, respectively, obtaining (\[eqn:one\]) below: $$\label{eqn:one}
(\varphi(\tilde{x}) \rightarrow {\mbox{\textit{Pre}}}_P(\tilde{x})) \wedge (\neg \varphi(\tilde{x}) \rightarrow {\mbox{\textit{Pre}}}_{P'}(\tilde{x}))$$ Now the weakest precondition of $\varphi$ wrt. the ${\mbox{\textbf{if}}}$ condition, is: $$(\exists \tilde{x}': (\varphi(\tilde{x}) \wedge \rho_P(\tilde{x},\tilde{x}')) \vee (\neg \varphi(\tilde{x}) \wedge
\rho_{P'}(\tilde{x},\tilde{x}')) \wedge \varphi(\tilde{x}'))$$ which is equivalent to $$(\varphi(\tilde{x}) \wedge (\exists \tilde{x}': \rho_P(\tilde{x},\tilde{x}') \wedge \varphi(\tilde{x}'))) \vee (\neg \varphi(\tilde{x}) \wedge (\exists \tilde{x}': \rho_{P'}(\tilde{x},\tilde{x}’) \wedge \varphi(\tilde{x}')))$$ Since the weakest precondition and weakest liberal preconditions of $P$ and $P'$ are equivalent, we get: $$(\varphi(\tilde{x}) \wedge {\mbox{\textit{Pre}}}_P(\tilde{x})) \vee (\neg \varphi(\tilde{x}) \wedge {\mbox{\textit{Pre}}}_{P'}(\tilde{x}))$$ This is equivalent to (\[eqn:one\]).
While loop of the syntax $${\mbox{\textbf{while}}}~\varphi(\tilde{x})~ {\mbox{\textbf{do}}}~P$$ has the same semantics as the following infinite program consisting of if conditionals. $$\begin{array}{l}
{\mbox{\textbf{if}}}~\varphi(\tilde{x})~ {\mbox{\textbf{then}}}\\
{~~~~}P\\
{~~~~}{\mbox{\textbf{if}}}~\varphi(\tilde{x})~ {\mbox{\textbf{then}}}\\
{~~~~}{~~~~}P\\
{~~~~}{~~~~}\ldots
\end{array}$$ The infinite programs exactly induces the same transition relation as the while loop presented above. Due to termination assumption, the same while loop can be written using a finite number of if conditionals (from the first if conditional up to the last (innermost) if conditional where $\varphi(\tilde{x})$ is ${\mbox{\textbf{false}}}$). More importantly, the while loop induces a transition relation that is satisfiable (not ${\mbox{\textbf{false}}}$), that is, there is a possible execution from the point before the loop to the point right after the loop. Since one if conditional preserves the equivalence of weakest liberal precondition and weakest precondition, as above, so does terminating while loops (which are representable as finite number of ifs).
Discussion {#sec:discussion}
==========
It is easy to see that the following relationship holds between weakest liberal precondition and weakest precondition, where the weakest liberal precondition $$\forall \tilde{x}' : \rho(\tilde{x},\tilde{x}') \rightarrow \varphi(\tilde{x}')$$ is actually equivalent to the negation of the weakest precondition of the negated postcondition. $$\neg(\exists \tilde{x}' : \rho(\tilde{x},\tilde{x}') \wedge \neg\varphi(\tilde{x}')).$$ This fact has been mentioned by Bourdoncle in his abstract debugging approach [@bourdoncle93debug], where he introduced two kinds of assertions to be guaranteed by a correctly running programs: *always* assertions and *eventually* assertions. The proofs of both require program state-space exploration using backward fixpoint computations. The state-space exploration of the always assertions employ weakest liberal precondition while the state-space exploration of the eventually assertions employ weakest precondition. The intuitive relations between both assertions is that, if suppose that we had an always assertion of some program correctness condition, and if the assertion holds, then in no circumstance that a program state where that assertion is violated can be eventually reached. That is, it is *not* the case that a *negation* of the correctness condition eventually holds.
Weakest precondition guarantees the total correctness of a Hoare’s triples $\{{\mbox{\textit{Pre}}}\} ~ S ~ \{\varphi\}$, where ${\mbox{\textit{Pre}}}$ is a precondition, $\varphi$ a postcondition, and $S$ a statement. The notion of weakest liberal precondition, on the other hand, guarantees only partial correctness of the triples, where the postcondition is guaranteed to hold only when the statement was executed successfully.
As a note, we can define weakest liberal precondition using weakest precondition. This does not mean, however, that we cannot implement weakest liberal precondition propagation indirectly using weakest precondition computation. Note that in a sequence $P P'$ the weakest liberal precondition of a condition $\varphi(\tilde{x})$ wrt. the program $P'$ is ${\mbox{\textsc{wlp}}}(\varphi(\tilde{x}), P')$, which is equivalent to $\forall \tilde{x}'' :
(\rho_{P'}(\tilde{x},\tilde{x}'') \rightarrow \varphi(\tilde{x}''))$, where $\rho_{P'}$ is the state transition relation defined by the program $P'.$ Now, the weakest liberal precondition of the sequence is $$\forall \tilde{x}' : \rho_P(\tilde{x},\tilde{x}') \rightarrow (\forall \tilde{x}'' : (\rho_{P'}(\tilde{x}',\tilde{x}'') \rightarrow \varphi(\tilde{x}'')))$$ which is equivalent to $$\forall \tilde{x}', \tilde{x}'' : (\rho_P(\tilde{x},\tilde{x}') \wedge \rho_{P'}(\tilde{x}',\tilde{x}'')) \rightarrow \varphi(\tilde{x}'').$$ Notice that $\rho_P(\tilde{x},\tilde{x}') \wedge \rho_{P'}(\tilde{x}',\tilde{x}'')$ is ${\mbox{\textsc{wp}}}(P P', {\mbox{\textbf{true}}}).$
Concluding Remarks {#sec:conclusion}
==================
The semantics of the guarded commands language introduced in [@dijkstra75gcl] embeds the notion of termination. In [@dijkstra75gcl], weakest precondition has to satisfy an additional condition $Q$ (satisfiability of at least one guard in case of guarded ifs, and a measure for the termination of a guarded loop), which ensures the termination of the statement. However, $Q$ does not exclude nondeterminism, and therefore from Theorems \[theorem:neither\], \[theorem:wpstronger\], and \[theorem:wlpstronger\], we infer that the notion of weakest precondition and $Q$ in [@dijkstra76lang] is stronger than the notion of weakest precondition used in this article.
We note that in this article, we have considered *value* nondeterminism of functions, while [@dijkstra75gcl] consider *control* nondeterminism where multiple guards can be true at the same time and the semantics does not specify which branch is taken. However, control nondeterminism can always be modeled using value nondeterminism by having some guards which depend on random value.
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|
---
abstract: 'It is well established that linear dispersive modes in a flowing quantum fluid behave as though they are coupled to an Einstein-Hilbert metric and exhibit a host of phenomena coming from quantum field theory in curved space, including Hawking radiation. We extend this analogy to any nonrelativistic Goldstone mode in a flowing spinor Bose-Einstein condensate. In addition to showing the linear dispersive result for all such modes, we show that the quadratically dispersive modes couple to a special nonrelativistic spacetime called a Newton-Cartan geometry. The kind of spacetime (Einstein-Hilbert or Newton-Cartan) is intimately linked to the mean-field phase of the condensate. To illustrate the general result, we further provide the specific theory in the context of a pseudo-spin-1/2 condensate where we can tune between relativistic and nonrelativistic geometries. We uncover the fate of Hawking radiation upon such a transition: it vanishes and remains absent in the Newton-Cartan geometry despite the fact that any fluid flow creates a horizon for certain wave numbers. Finally, we use the coupling to different spacetimes to compute and relate various energy and momentum currents in these analogue systems. While this result is general, present day experiments can realize these different spacetimes including the magnon modes for spin-1 condensates such as $^{87}$Rb, $^{7}$Li, $^{14}$K (Newton-Cartan), and $^{23}$Na (Einstein-Hilbert).'
author:
- 'Justin H. Wilson'
- 'Jonathan B. Curtis'
- 'Victor M. Galitski'
bibliography:
- 'spinorBEC-curvedspace.bib'
- 'arxivpapers.bib'
title: Analogue spacetimes from nonrelativistic Goldstone modes in spinor condensates
---
\[sec:intro\]Introduction
=========================
The marriage of quantum mechanics and general relativity is one of the greatest outstanding problems in modern physics. This is in part due to the fact that this theory would only become truly necessary under the most extreme conditionsthe singularity of a black-hole or the initial moments after the big bang. As such, it is extremely difficult to theoretically describe, let alone physically probe.
Despite the seeming intractability, some headway may be made in the understanding of such extreme theories by way of analogy. This idea traces back to Unruh, who in 1981[@unruhExperimentalBlackHoleEvaporation1981] suggested that a flowing quantum fluid could realize a laboratory scale analogue of a quantum field theory in a curved spacetime. Access to even the most rudimentary quantum simulator for such a curved spacetime could provide valuable insights into this otherwise inaccessible regime.
Since Unruh’s initial proposal, many systems have been advanced as candidates for realizing analogue spacetimes [@barceloAnalogueGravity2011], including liquid helium [@jacobsonEventHorizonsErgoregions1998; @volovikUniverseHeliumDroplet2009; @volovikTopologyQuantumVacuum2013], Bose-Einstein condensates [@garaySonicAnalogGravitational2000; @*garaySonicBlackHoles2001; @eckelRapidlyExpandingBoseEinstein2018; @keserAnalogueStochasticGravity2018; @macherBlackholeRadiationBoseEinstein2009; @fischerQuantumSimulationCosmic2004; @fedichevGibbonsHawkingEffectSonic2003; @schutzholdSweepingSuperfluidMott2006; @chaProbingScaleInvariance2017; @steinhauerObservationSelfamplifyingHawking2014; @*steinhauerObservationQuantumHawking2016], nonlinear optical media [@leonhardtRelativisticEffectsLight2000], electromagnetic waveguides [@schutzholdHawkingRadiationElectromagnetic2005], magnons in spintronic devices [@roldan-molinaMagnonicBlackHoles2017], semi-conductor microcavity polaritons [@nguyenAcousticBlackHole2015], Weyl semi-metals [@volovikBlackHoleHawking2016], and even in classical water waves [@euveObservationNoiseCorrelated2016]. Analogue gravity systems are no longer a theoretical endeavor; recent experiments have realized the stimulated Hawking effect [@droriObservationStimulatedHawking2019], and in the case of a Bose-Einstein condensate a spontaneous Hawking effect [@steinhauerObservationSelfamplifyingHawking2014; @*steinhauerObservationQuantumHawking2016].
----------- ------------------- ------------------ ------------
Goldstone Dispersion Analogue Lagrangian
mode spacetime
Type-I $\omega \sim k$ Einstein-Hilbert Eq.
Type-II $\omega \sim k^2$ Newton-Cartan Eq.
----------- ------------------- ------------------ ------------
: Analogue spacetimes which appear for the different Goldstone modes in the presence of a background condensate flow. These spacetimes emerge as effective field theories governing the long-wavelength behavior. As we demonstrate in this work, the emergent geometry is determined by the flow profile of the background condensate. This is explicitly demonstrated in Sec. \[sec:relativistic\] for the Type-I modes and Sec. \[sec:nonrelativistic\] for the Type-II modes, where we also provide an overview of the Newton-Cartan formalism. []{data-label="tab:Key_results"}
In this paper we introduce a new kind of analogue gravity system, one which exhibits Newton-Cartan geometry [@cartanVarietesConnexionAffine1923; @*cartanVarietesConnexionAffine1924; @son2013newtoncartan]. This geometry naturally arises from a full analysis of all Goldstone modes in a flowing spinor (or multicomponent) condensate. Spinor condensates [@stamper-kurnSpinorBoseGases2013] have been studied in the context of analogue curved space before [@watanabeUnifiedDescriptionNambuGoldstone2012; @hidakaCountingRuleNambuGoldstone2013]; however a full accounting of all gapless modes has not been done to the best of our knowledge. The Goldstone modes which realize the Newton-Cartan geometry exhibit a quadratic $\omega\sim \mathbf{k}^2$ dispersion, known as “Type-II” Goldstone modes [@watanabeUnifiedDescriptionNambuGoldstone2012; @hidakaCountingRuleNambuGoldstone2013]. For example, the spin wave excitations about an SU(2) symmetry breaking ferromagnetic mean-field are such a mode. Distinct from the linearly dispersing case (called “Type-I” modes), Newton-Cartan spacetimes implement local Galilean invariance, as opposed to local Lorentz invariance. These results are general and summarized in Table \[tab:Key\_results\], where we give a general prescription for separating out all Goldstone modes into either Type-I (linearly dispersing) or Type-II (quadratically dispersing) modes and assigning them either an Einstein-Hilbert or Newton-Cartan spacetime geometry.
Newton-Cartan geometry was developed by Cartan [@cartanVarietesConnexionAffine1923; @*cartanVarietesConnexionAffine1924] and refined by others [@kunzleGalileiLorentzStructures1972] as a geometric formulation and extension of Newtonian gravity. It has since found application across different areas of physics, including in quantum Hall systems [@son2013newtoncartan; @gromovThermalHallEffect2015; @bradlynLowenergyEffectiveTheory2015] and effective theories near Lifshitz points [@christensenBoundaryStressenergyTensor2014; @christensenTorsionalNewtonCartanGeometry2014] with interest to the high-energy community with implications for quantum gravity [@hartongHoravaLifshitzGravityDynamical2015; @taylorLifshitzHolography2016]. We extend these applications here to flowing condensates for the case of Type-II Goldstone modes.
Heuristically, one may view the quadratic dispersion relation $\omega \sim |\mathbf{k}|^2 + ...$ as the limit of a linear dispersion relation $\omega \sim v|\mathbf{k}| + ...$ with vanishing group velocity $v\rightarrow 0$. In terms of the analogue spacetime, this corresponds to an apparent vanishing of the speed of light. As such, the formation of event horizons and their corresponding Hawking radiation ought to be ubiquitous in such spacetimes; however our results contradict this intuition. Specifically, we find that fields propagating in Newton-Cartan geometries exhibit an additional conservation law which precludes the emission of Hawking radiation.
The immediate implication of this is that any Type-I mode can have an effective event horizon and therefore a Hawking effect (similar things have been noticed for specific other Type-I modes), and further, no Hawking effect can occur for Type-II modes, at least not without introducing quasiparticle interactions (which corresponds to going being a quadratic treatment of fluctuations).
Finally, we discuss the relationship between transport phenomena and gravitational metrics in our theory [@luttingerTheoryThermalTransport1964; @gromovThermalHallEffect2015; @geracieSpacetimeSymmetriesQuantum2015; @son2013newtoncartan]. Specifically, we obtain the stress-tensor, energy flux, and momentum density for theories both with the Einstein-Hilbert and Newton-Cartan geometries. In particular, we relate the energy-momentum tensor calculated in an analogue Einstein-Hilbert geometry to its nonrelativistic counterparts through the use of Newton-Cartan geometry. This helps identify how the analogue Hawking effect results in nontrivial energy and momentum currents in the underlying nonrelativistic system.
The outline of the paper is as follows. Section \[sec:spacetimes\] shows that in the presence of a flowing background condensate Type-I and -II Goldstone modes couple to Einstein-Hilbert (Section \[sec:relativistic\]) and Newton-Cartan (Section \[sec:nonrelativistic\]) geometries respectively. In Section \[sec:model\], we present a minimal model for these space-times and the phase transition that connects them. In Sec. \[subsec:bdg\] we develop the Bogoliubov-de Gennes framework which we then use to analyze this system. In Sec. \[sec:step\] we apply this to a specific step-like flow geometry and show the effect of the geometry on the emitted Hawking radiation. We then discuss transport of energy and momentum in these different analogue spacetimes systems in Sec. \[sec:transport\]. We conclude the paper in Section \[sec:conclusion\]. Our two appendices include Appendix \[app:fluctuations\] where we put the full fluctuation calculation of the Lagrangian and Appendix \[app:BogoliubovHawking\] where we review the Hawking calculation for the phonon problem. Throughout, we take $\hbar=k_B=1$ and our relativistic metrics have signature ($+$ $-$ $-$ $-$). We also indicate spatial vector with a boldface (e.g. $\mathbf{r}$), while spacetime vectors are indicated without boldface (e.g. $x = (t,\mathbf{r})$).
Relationship between spacetime and Goldstone’s theorem {#sec:spacetimes}
======================================================
In this work we consider models of ultra-cold bosonic spinor quantum gases described by an $N$-component field variable $\Psi(\mathbf{r},t) = [ \Psi_1, \Psi_2, \ldots, \Psi_N ]^T$ residing in $d$ spatial dimensions (we do not make the distinction between “spinor" and higher multiplet fields in this work). The Lagrangian describing this system is taken to be of the general form $$\mathcal L = \tfrac{i}{2}(\Psi^\dagger \overrightarrow{\partial_t} \Psi - \Psi^\dagger \overleftarrow{\partial_t} \Psi) - \tfrac1{2m} \nabla \Psi^\dagger \cdot \nabla \Psi - V(\Psi^\dagger, \Psi), \label{eq:general-Lagrangian}$$ where $m$ is the mass of the atoms in the gas and $V(\Psi^\dagger, \Psi)$ is a general potential energy function that includes interactions with an external potential as well as local inter-particle interactions. Such a system may be realized by cold-atoms, where in addition to the inter-particle interactions external potentials such as a harmonic trap, optical lattice, or magnetic field may be present. For a comprehensive review regarding the theory and experimental realization of spinor condensates see Ref. [@stamper-kurnSpinorBoseGases2013].
We consider the case where the Lagrangian exhibits invariance under an internal symmetry described by a Lie group $G$, according to which $\Psi$ transforms via a linear unitary representation $\mathcal{R}(G)$ such that the action $\mathcal{S} = \int \mathcal{L}\ d^{d+1} x$ remains invariant. That is, $$\Psi(x) \rightarrow U \Psi(x)\Rightarrow \mathcal{S} \rightarrow \mathcal{S} \quad \forall U \in \mathcal{R}(G).$$ Recall that a Lie group $G$ is generated by its corresponding Lie algebra $\mathfrak{g}$, and this has a representation of $\mathcal{R}(\mathfrak{g})$ when acting on the field $\Psi$. For ease of calculations, we use the mathematical convention that Lie algebras consist of anti-Hermitian elements. Hence, if $A$ is an element of $\mathcal{R}(\mathfrak{g})$, then $A = -A^\dagger$ and the corresponding group element is $e^{A} = ((e^{A})^{-1})^\dagger$.
We pursue a semi-classical analysis of our system by first obtaining the classical equations of motion (i.e. the saddle-point of the action). Then we linearize the action around the saddle-point, obtaining a description of the symmetry-broken phases in terms of their Goldstone modes. The primary point of our work is that this linearized action admits a simple description in terms of different emergent analogue spacetimes and depending on the nature of the saddle-point, this analogue spacetime may develop non-trivial curved geometry.
The rest of this section is organized as follows. We perform a quadratic fluctuation analysis in Section \[sec:fluctuations\]. In Section \[sec:nonrel-goldstone-proof\] we review the proof of the Goldstone theorem in non-relativistic settings [@watanabeUnifiedDescriptionNambuGoldstone2012; @hidakaCountingRuleNambuGoldstone2013] and show how this allows us to classify Goldstone modes into Type-I and Type-II. Section \[sec:full-lagrangian\] then presents the full Lagrangian for the Goldstone modes while Sections \[sec:relativistic\] and \[sec:nonrelativistic\] make explicit the connection to curved space geometry.
Saddle-Point Expansion {#sec:fluctuations}
----------------------
We begin by looking for saddle-points of the Lagrangian Eq. , the spinor Gross-Pitaevskii equation $$i\partial_t \Psi = -\frac1{2m}\nabla^2 \Psi + \frac{\partial V}{\partial \Psi^\dagger}. \label{eq:EulerLagrange}$$ Suppose that we have found a mean-field solution to this equation $\Psi_0(\mathbf{r},t) \equiv \braket{\Psi(\mathbf{r},t)}$ which describes the dynamics of a mean-field condensate (neglecting fluctuation back-reaction); for a general out-of-equilibrium system, the space-time dependence of $\Psi_0(\mathbf{r},t)$ may be non-trivial [@barnettGeometricalApproachHydrodynamics2009; @keserAnalogueStochasticGravity2018; @stamper-kurnSpinorBoseGases2013].
The presence of a non-zero mean-field solution $\Psi_0$ spontaneously breaks the internal symmetry group $G$ down to a subgroup $H \subset G$. Let $\mathfrak{h}$ be the Lie algebra that generates the subgroup $H$. This is defined by the set of generators $$\mathfrak{h} = \{ \bm \tau\in \mathfrak{g} \;|\; \bm \tau \Psi_0 = 0 \}.$$ We can form a complete basis for $\mathfrak{h} = \operatorname{span}\{\tau_k\}$. The original Lie algebra then separates into two sub-spaces; $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{h}^c$, where $\mathfrak{h}^c$ is simply the complement of $\mathfrak{h}$. It is useful to form an explicit basis for $\mathfrak{h}^c \equiv \operatorname{span}\{\sigma_l\}$ so that $\mathfrak{g} = \operatorname{span}\{\tau_k\}\cup\{ \sigma_l\} = \operatorname{span}\{\sigma_l, \tau_k\}$. Formally, $\mathfrak{h}^c$ is isomorphic to the quotient algebra $\mathfrak{g}/\mathfrak{h}$, and the basis elements $\sigma_l$ are isomorphic to coset spaces.
It is important to emphasize that, although in general the mean-field $\Psi_0(x)$ may break the symmetry group $G$ down to different subgroups $H=H(x)$ at each spacetime point, we do not consider this in full generality since it leads to a very complicated (but interesting) structure involving a non-Abelian connection on the spacetime. However, we later consider [*flowing*]{} condensates which inhomogeneously break the $U(1)$ subgroup of $G$.
We now examine the quadratic fluctuations of the field $\Psi$ about the mean-field by expanding the Lagrangian in powers of $\delta\Psi(x) = \Psi(x) - \Psi_0(x)$. This separates into two distinct contributions; the massless Goldstone modes $\theta_l(x)$ which correspond to spontaneously broken symmetries, and massive fields $\beta_n(x)$ which describe all the remaining modes. Each Goldstone mode corresponds to a broken generator $\sigma_l \in \bar{\mathfrak{h}}$ acting on the mean-field condensate $\Psi_0(x)$. These contribute to the fluctuation action as $$\left(\delta \Psi(x) \right)_{\textrm{Goldstone}} = \sum_{l} \theta_l(x) \sigma_l \Psi_0(x)\equiv \bm\sigma(x) \Psi_0(x) ,\label{eq:goldstone-modes-define}$$ which serves to define the Goldstone matrix field $\bm\sigma(x)$. The remaining degrees of freedom are generically massive and are not amenable to a description in terms of the Lie algebra’s generators. It is advantageous to parameterize the fluctuations $\delta \Psi$ in terms of real fields with massive terms orthogonal to the massless terms in the sense described below. Within the quadratic theory, this implies the fluctuations reside within a real vector space $\mathbb{R}^{2N}\sim \mathbb{C}^N$. The Goldstone modes $\sigma_l \Psi_0(x)$ form a subspace of this manifold while the remaining basis elements are generically massive and are written as $\xi_n(x)$. We note that in general the basis elements are spacetime dependent simply because the mean-field is also spacetime dependent.
In order to make the notion of orthogonality precise we lift the standard complex ($\mathbb{C}^N$) inner product onto our real vector space $\mathbb{R}^{2N}$ to obtain the *real* inner product $g$ defined by $$g(\xi,\chi) \equiv \tfrac12 (\xi^\dagger \chi + \chi^\dagger \xi) \label{eq:real-inner-product}.$$ In terms of the Goldstone manifold and its complement, the variation $\delta \Psi(x) $ takes the compact form $$\delta \Psi(x) = \bm\sigma(x)\Psi_0(x) + \xi(x) ,\label{eq:fluctuation-expanded}$$ where we have defined the massive modes by $$\xi(x) = \sum_n \beta_n(x) \xi_n(x) .\label{eq:massive-modes-define}$$
We proceed to the expansion of the Lagrangian in terms of the variation $\delta \Psi$. First, we consider the potential. It is locally invariant under under $G$, so we can write $$V(\Psi^\dagger, \Psi) = V(\Psi^\dagger e^{\bm \sigma(x)}, e^{-\bm\sigma(x)} \Psi).$$ Furthermore, we can use our expansion of $\Psi(x)$ to obtain $$\begin{split}
e^{-\bm\sigma} \Psi & \approx e^{-\bm\sigma}[\Psi_0 + \bm\sigma\Psi_0 + \xi] \\
& \approx (1 - \bm\sigma +\tfrac12 \bm\sigma^2)[\Psi_0 + \bm\sigma\Psi_0 + \xi] \\
& \approx \Psi_0 + \xi - \bm \sigma \xi - \tfrac12 \bm\sigma^2 \Psi_0,
\end{split}$$ keeping terms up to quadratic order in fluctuations. This allows us to expand the potential energy up to quadratic order (dropping the terms constant and linear in the variation) $$\begin{gathered}
V(\Psi^\dagger, \Psi) = - \left[ \frac{\partial V}{\partial \Psi}\cdot \left(\tfrac12 \bm \sigma^2 \Psi_0 + \bm \sigma \xi \right) + \mathrm{c.c.} \right]\\
+\frac12 \xi^* \xi^*\cdot \frac{\partial^2 V}{\partial \Psi^\dagger \partial \Psi^\dagger} + \xi^*\cdot \frac{\partial^2 V}{\partial \Psi^\dagger \partial \Psi} \cdot \xi + \frac12 \frac{\partial^2 V}{\partial \Psi \partial \Psi}\cdot \xi \xi , \label{eq:potential-expand}\end{gathered}$$ where all derivatives of the potential are understood as being evaluated at the mean-field. The terms quadratic in $\xi,\xi^*$ represent massive terms, and the first line of Eq. drops out when combined on-shell with similar terms from the kinetic part of the Lagrangian. Deriving the full fluctuation Lagrangian is not instructive, and has been relegated to Appendix \[app:fluctuations\]; the final result is given below.
Focusing on the Goldstone modes, written in terms of the “angle fields" $\theta_l(x)$, the resulting Lagrangian for fluctuations is given by $$\begin{gathered}
\mathcal L_{\mathrm{fluc}} = \theta_m P^{\mu}_{mn} (\partial_\mu \theta_n)
+ \beta_m Q^{\mu}_{mn} (\partial_\mu \theta_n) \\ + (\partial_j\theta_n)T^{jk}_{mn}(\partial_k \theta_n) + \mathcal L_{\mathrm{mass}}(\beta_m, \partial_\mu \beta_m), \label{eq:partial-fluctuation-Lagrangian}\end{gathered}$$ where we have instituted the Einstein summation convention. In this and the following, Roman indices $i,j,k,\ldots$ run over spatial dimensions while Greek indices $\mu,\nu,\ldots$ run over both temporal and spatial dimensions (with $\mu = 0 = t$ the temporal index). The Roman indices $n,m,\ldots$ enumerate the different Goldstone modes or massive modes and are similarly summed. The terms $P^{\mu}_{mn}$, $Q^{\mu}_{mn}$, and $T^{jk}_{mn}$ depend on both space and time, and are given by $$\begin{split}
P^{t}_{mn} & = \tfrac{i}{2} \Psi_0^\dagger [\sigma_n, \sigma_m] \Psi_0, \\
P^{j}_{mn} & = \tfrac1{4m} ( \partial_j \Psi_0^\dagger [\sigma_m, \sigma_n] \Psi_0 - \Psi_0^\dagger [\sigma_m, \sigma_n] \partial_j \Psi_0), \\
Q^{t}_{mn} & = i (\Psi_0^\dagger \sigma_n \xi_m + \xi_m^\dagger \sigma_n \Psi_0), \\
Q^{j}_{mn} & = \tfrac1{2m} ( \xi_m^\dagger \sigma_n \partial_j \Psi_0 - \partial_j \Psi_0^\dagger \sigma_n \xi_m \\ & \phantom{= = \quad \quad \quad\quad } + \Psi_0^\dagger \sigma_n \partial_j \xi_m - \partial_j \xi_m^\dagger \sigma_n \Psi_0), \\
T^{jk}_{mn} & = \tfrac{1}{2m} \delta^{jk} \Psi_0^\dagger \sigma_n \sigma_m \Psi_0.
\end{split}$$ As mentioned previously, it is also important to keep track of the massive modes in the full Lagrangian and we offer that full analysis in Appendix \[app:fluctuations\].
Proof of the nonrelativistic Goldstone theorem {#sec:nonrel-goldstone-proof}
----------------------------------------------
Before proceeding to simplify the Lagrangian and derive the curved space analogues, we need to understand and make use of the nonrelativistic Goldstone theorem [@hidakaCountingRuleNambuGoldstone2013; @watanabeUnifiedDescriptionNambuGoldstone2012], providing a complementary proof in the process.
We consider the following ansatz for the mean-field $$\Psi_0(x) = \sqrt{\rho(x)} e^{i\vartheta(x)} \chi, \quad \chi^\dagger \chi = 1, \quad \partial_\mu \chi = 0. \label{eq:mean-field_ansatz}$$ Importantly the spinor structure given by $\chi$ is independent of space and time. The global $U(1)$ symmetry implies the phase and density obey a continuity relation which can be conveniently written as $$\partial_\mu J^\mu = 0 ,$$ with the condensate four-current given by $J^\mu = \rho v_s^\mu$, where the superfluid four-velocity field is $v_s^\mu = (1, \frac{1}{m}\nabla \vartheta )$. This simplifies the term $$P^\mu_{mn} = -\tfrac{i}2 J^\mu \chi^\dagger [\sigma_n,\sigma_m] \chi,$$ which dictates which real fields $\theta_n$ are canonically conjugate to each other. In non-relativistic systems, the relationship between broken symmetry generators and Goldstone modes is not one-to-one. Instead, we must separate out our modes into Type-I and Type-II Goldstone modes, which is done by going to the preferred basis of the matrix $P^\mu_{mn}$.
To understand this, we return to the *real* vector space defined by the Goldstone mode manifold, which we label $\mathcal{A}_{\mathbb{R}}$. That is, $$\mathcal{A}_{\mathbb{R}} = \operatorname{span}_{\mathbb{R}}\{\sigma_l \Psi_0(x) \}.\label{eq:real-Goldstone}$$ The real dimension $D_\mathbb{R}$ of this subspace is simply equal to the number of broken generators. We can complexify this vector space by allowing for complex-valued coefficients $$\mathcal{A}_{\mathbb{C}} \equiv \operatorname{span}_{\mathbb C}\{ \sigma_n \Psi_0 \}.$$ It may be the case that two generators which are linearly independent under real coefficients are linearly dependent when multiplied by complex coefficients. For this reason, this vector space has an associated *complex* dimension $D_{\mathbb{C}} \leq D_{\mathbb{R}}$. The essence of the Goldstone mode theorem is that $D_{\mathbb{R}}$ is the number of broken generators and $D_{\mathbb{C}}$ is the number of modes, and these two quantities can be formally related by classifying each basis element $\sigma_l \Psi_0(x) \in \mathcal{A}_{\mathbb{R}}$ due to whether $i\sigma_n \Psi_0 \in \mathcal{A}_{\mathbb{R}}$ or not.
To establish this we need to return to our real inner product $g(\cdot,\cdot)$. We can use the operation of multiplication by $i$ to define a symplectic bilinear form $\omega(\cdot,\cdot)$ by $$\omega(\eta, \xi) \equiv g(i\eta, \xi) = \tfrac{i}2(\xi^\dagger \eta - \eta^\dagger \xi).$$ The multiplication by $i$ (acting on the basis vectors $\sigma_l \Psi_0(x)$) can be restricted to the real vector space $\mathcal{A}_{\mathbb{R}}$, which we define by the notation $$i|_{\mathcal{A}_{\mathbb{R}}} \equiv I : \mathcal{A}_{\mathbb{R}} \rightarrow \mathcal{A}_{\mathbb{R}}.$$ Similarly, we define $\operatorname{range}I \equiv \mathcal{A}_{\mathrm{II}} \subset \mathcal{A}_{\mathbb{R}}$ as the range of $I$. The null space of $I$ is then defined to be $\mathcal A_{\mathrm{I}}$ and represents states $\eta \in \mathcal{A}_{\mathbb{R}}$ which leave the real vector space upon multiplication by $i$. As a simple example, consider unit vectors $\hat{e}_1 = (1,0)^T$ and $\hat{e}_2 = (i,0)^T$. As elements of a real vector space these are linearly independent, however $i \hat{e}_1 = \hat{e}_2$ and so these are not linearly independent in a complex vector space. In this case, we have $D_{\mathbb{R}} = 2,\ D_{\mathbb{C}} = 1$ and $\operatorname{range}I = \mathcal{A}_{\mathbb{R}},\ \operatorname{null}I = 0$. However, if $\hat{e}_1 = (1,0)^T$ and $\hat{e}_2 = (0,1)^T$ then $D_{\mathbb{R}} = 2 = D_{\mathbb{C}}$ and $\operatorname{range}I = 0,\ \operatorname{null}I = \mathcal{A}_{\mathbb{R}}$.
The classification of basis elements may be accomplished by taking the real inner product of $i\eta$ with the other elements of $\mathcal A$if this vanishes, then $\eta$ is in the kernel of $I$. But this is exactly given by the symplectic bilinear form defined above so that $$\mathcal{A}_{\mathrm{I}} \equiv \operatorname{null}I = \{ \eta\in \mathcal{A}_{\mathbb{R}} \;|\;\omega(\eta, \chi) = 0, \forall \chi \in \mathcal A\}.$$ This condition can be simplified into a matrix condition if we note that we can let $\eta = \sum_n a_n \sigma_n \Psi_0$ and $\chi = \sum_m b_m \sigma_m \Psi_0$, so that $$0 = \omega(\eta,\chi) = -\tfrac{i}2 a_n \Psi_0^\dagger [\sigma_n,\sigma_m] \Psi_0 b_m.$$ This relates the null-space of $I$ to the null-space of the matrix $\Psi_0^\dagger [\sigma_n,\sigma_m] \Psi_0 \propto P^\mu_{mn}$, the term appearing in our Lagrangian which determines the canonically conjugate pairs of modes. Using the rank-nullity theorem, we have $$\mathcal{A}_{\mathbb{R}} = \mathcal{A}_{\mathrm I} \oplus \mathcal{A}_{\mathrm{II}}.$$ Since the matrix given by elements $-\tfrac{i}2\Psi_0^\dagger [\sigma_n,\sigma_m] \Psi_0$ is real and antisymmetric, we can block-diagonalize the matrix with a special orthogonal transformation. Going to this basis and using our ansatz for the flowing mean-field $\Psi_0 = \sqrt{\rho} e^{i\vartheta} \chi$, the result is $$\begin{split}
-\tfrac{i}2\Psi_0^\dagger& [\sigma_n,\sigma_m] \Psi_0 = -\tfrac{i}2 \rho \chi^\dagger [\sigma_n,\sigma_m] \chi \\ \phantom{\Big(} \\
& = \rho
\begin{pmatrix}[ccccc|cc]
{\tikz[overlay,remember picture,baseline=(x1.base)]
\node (x1) {\strut};} 0 & \lambda_1 & 0 & 0 & \phantom{0}{\tikz[overlay,remember picture,baseline=(x2.base)]
\node (x2) {\strut};} & {\tikz[overlay,remember picture,baseline=(y1.base)]
\node (y1) {\strut};}\phantom{0} & \phantom{0}{\tikz[overlay,remember picture,baseline=(y2.base)]
\node (y2) {\strut};} \\
-\lambda_1 & 0 & 0 & 0 & \cdots & 0 & \cdots \\
0 & 0 & 0 & \lambda_2 & & \\
0 & 0 & -\lambda_2 & 0 & & \\
& \vdots & & & \ddots & \\
\hline
& 0 & & & & 0 & \\
& \vdots & & & & & \ddots
\end{pmatrix},
\end{split}\label{eq:typeiandtypeiimatrix}\begin{tikzpicture}[overlay, remember picture,decoration={brace,amplitude=2pt}]
\draw[decorate,thick] (x1.north) -- (x2.north)
node [midway,above=5pt] {$\mathcal A_{\mathrm{II}}$};
\draw[decorate,thick] (y1.north) -- (y2.north)
node [midway,above=5pt] {$\mathcal A_{\mathrm{I}}$};
\end{tikzpicture}$$with $\lambda_j>0$. This defines a preferred basis for the broken generators $\{\sigma_l\}$ which we henceforth assume is the basis we are in. Note that in this basis $\mathcal{A}_{\mathrm{II}}$ takes the form of a direct sum of decoupled symplectic forms.
This matrix provides a natural way to break up the generators. First, we can define $\sigma_n^{\mathrm{II}}$ and its conjugate generator $\overline{\sigma_n^{\mathrm{II}}}$ via $-\tfrac{i}2 \Psi_0^\dagger[\sigma_n^{\mathrm{II}},\overline{\sigma_n^{\mathrm{II}}}]\Psi_0 = \rho \lambda_n$. This implies that $\overline{\sigma_n^{\mathrm{II}}}\Psi_0 = i\sigma_n^{\mathrm{II}}\Psi_0$ (however $\overline{\sigma_n^{\mathrm{II}}} \neq i\sigma_n^{\mathrm{II}}$). Let $n_{\mathrm{II}}$ be the number of $\lambda_j$’s, so that $\dim(\mathcal A_{\mathrm{II}}) = 2n_{\mathrm{II}}$. As the coefficient of the temporal derivative term in the Lagrangian, this matrix tells us that the two Goldstone fields described by $\sigma_n^{\mathrm{II}} \Psi_0(x)$ and $\overline{\sigma_n^{\mathrm{II}} }\Psi_0(x)$ are canonically conjugate to each other and therefore describe the [**same mode**]{}, a Type-II Goldstone mode. Finally, let $\dim(\mathcal A_{\mathrm{I}}) = n_{\mathrm{I}}$ be dimension of the null-space of $I$. This is the number of Type-I Goldstone modes; they represent modes which are canonically conjugate to a massive mode. It is evident by the rank-nullity result that $$2n_{\mathrm{II}} + n_{\mathrm{I}} = D_{\mathbb{R}}$$ is the number of broken generators, while $$n_{\mathrm{II}} + n_{\mathrm{I}} = D_{\mathbb{C}}$$ is the number of Goldstone modes in the system.
With this particular grating into $n_{\mathrm{II}}$ basis elements $\sigma_n^{\mathrm{II}}\Psi_0$ and $n_{\mathrm{I}}$ basis elements $\sigma_n^{\mathrm{I}}\Psi_0$, we can rewrite our real vector space $$\mathcal{A}_{\mathbb{R}} = \operatorname{span}\{ \sigma_n^{\mathrm{II}}\Psi_0, \overline{\sigma_n^{\mathrm{II}}}\Psi_0, \sigma_n^{\mathrm{I}}\Psi_0 \},$$ and similarly, we can write the complexified vector space in two equivalent ways $$\begin{split}
\mathcal{A}_{\mathbb{C}} & = \operatorname{span}_{\mathbb{C}} \{ \sigma_n^{\mathrm{II}}\Psi_0, \sigma_n^{\mathrm{I}}\Psi_0 \}, \\
\mathcal{A}_{\mathbb{C}} & = \operatorname{span}\{ \sigma_n^{\mathrm{II}}\Psi_0, \overline{\sigma_n^{\mathrm{II}}}\Psi_0, \sigma_n^{\mathrm{I}}\Psi_0, i \sigma_n^{\mathrm{I}}\Psi_0 \}.
\end{split}$$ The modes represented by $i \sigma_n^{\mathrm{I}}\Psi_0$ are exactly the massive modes conjugate to $\sigma_n^{\mathrm{I}}\Psi_0$ (by definition, they are not in $\mathcal A$ and are thus not associated with a broken generator).
At low energies (below the relevant mass gaps), massive modes that are not conjugate to Goldstone modes can be trivially integrated out and do not contribute in the IR. This then leaves the Goldstone modes, which are gapless, and a few massive modes which are canonically conjugate to the Type-I Goldstone modes. These massive modes cannot be trivially integrated out and they are to be included in the low-energy theory. Doing so amounts to adding the basis elements $i \sigma_n^{\mathrm{I}} \Psi_0$ to our fluctuation manifold.
Lagrangian for Goldstone Modes {#sec:full-lagrangian}
------------------------------
We now employ this classification into Type-I and -II modes to our benefit by using it to simplify the fluctuation Lagrangian. Recall that in this work we restrict ourselves to flowing condensates which have a spatial texture to the phase mode (and thus inhomogeneously break the global $U(1)$ part of the symmetry group), but have a homogeneous and static spinor texture. For instance, one may consider a condensate of pseudo-spin-$\frac12$ atoms in its ferromagnetic phase which has a definite homogeneous magnetization $\braket{S_z} = \chi^\dagger S_z \chi = \frac12$ but a non-zero density and phase profile. As remarked earlier, this flow produces a non-zero spatial component for the Noether current $J_\mu(x)$. Going to the preferred basis of $P^\mu_{mn}$, obtained in Sec. \[sec:nonrel-goldstone-proof\] then yields the partitioning into the Goldstone modes given by $\{ \sigma_n^{\mathrm{II}}\Psi_0, \overline{\sigma_n^{\mathrm{II}}}\Psi_0, \sigma_n^{\mathrm{I}}\Psi_0 \}$. Let us remind the reader that Type-I modes are those for which $i\sigma_n \Psi_0$ cannot be written as a broken generator $\sigma'_n\Psi_0$ and therefore, the associated real field comes with a massive term in the Lagrangian.
The basis elements $\{ \sigma_n^{\mathrm{II}}\Psi_0, \sigma_n^{\mathrm{I}}\Psi_0 \}$ have the property that they are orthogonal in the conventional sense (e.g. $\eta^\dagger \chi = 0$). As a result of this, $$\begin{split}
\Psi_0^\dagger \sigma_n^{\mathrm{I}} \sigma_m^{\mathrm{II}} \Psi_0 & = 0, \\
-\Psi_0^\dagger \sigma_n^{\mathrm{II}} \sigma_m^{\mathrm{II}} \Psi_0 & = \lambda_n \delta_{nm} \rho(x) , \\
-\Psi_0^\dagger \sigma_n^{\mathrm{I}} \sigma_m^{\mathrm{I}} \Psi_0 & = \mu_n\delta_{nm} \rho(x) ,
\end{split}$$ where we have defined $\mu_n \equiv -\chi^\dagger (\sigma_n^{\mathrm{I}})^2 \chi>0$ and used the fact that $\lambda_n = -\chi^\dagger (\sigma_n^{\mathrm{II}})^2 \chi>0$.
In this basis, the field variation $\delta\Psi(x)$ may be described by three real Goldstone fields $\theta_n$, $\bar \theta_n$, and $\phi_n$ along with the real massive field $\beta_n$ via $$\begin{split}
\bm\sigma & = \sum_{n=1}^{n_{\mathrm{II}}}\left( \theta_n \sigma_n^{\mathrm{II}} + \bar \theta_n \overline{\sigma_n^{\mathrm{II}}} \right) + \sum_{n=1}^{n_\mathrm{I}} \phi_n \sigma_n^{\mathrm{I}}, \\
\xi & = \sum_{n=1}^{n_\mathrm{I}} \beta_n i \sigma_n^{\mathrm{I}} \Psi_0 + \cdots,
\end{split}$$ where “$\cdots$” represents other massive modes that can be trivially integrated out. In this basis, the coefficient $P^\mu_{mn}$ simplifies to $$P^{\mu}_{mn} = \delta_{n\bar{m}} \lambda_n \rho(x) v_s^\mu,$$ where $\bar{m}$ is defined as the index of the conjugate field to the field labeled by $m$. Similarly, we may simplify $Q^\mu_{mn}$ which connects Type-I Goldstone modes to their conjugate massive fields. We indeed find $$Q^\mu_{mn} = 2 \delta_{nm} \mu_n \rho(x) v_s^\mu,$$ where the massive field with index $m$ is indicated by the basis element $i\sigma_m^{\mathrm{I}}\Psi_0$. Lastly, we have the kinetic energy term which we can separate out into its contribution to Type-I and Type-II fields $$\begin{split}
T^{jk}_{mn}|_{\mathrm{I}} & = -\tfrac1{2m} \delta^{jk}\rho(x) \mu_n \delta_{mn} \\
T^{jk}_{mn}|_{\mathrm{II}} & = -\tfrac1{2m} \delta^{jk}\rho(x) \lambda_n \delta_{mn}
\end{split}$$
Notice that $\lambda_n$ or $\mu_n$ multiplies all elements in the Lagrangian where that field (or its conjugate) appears, so we can simply absorb this constant into a re-definition of $\theta_n$, $\bar\theta_n$, $\phi_n$, and $\beta_n$. Then, substituting the form of our fluctuations, the Lagrangian is
$$\begin{gathered}
\mathcal L_{\mathrm{fluc}} = \sum_{n=1}^{n_{\mathrm{I}}} \rho(x) \left[- 2\beta_n v_s^\mu(x) \partial_\mu \phi_n - \tfrac1{2m}[(\nabla \phi_n)^2 + (\nabla \beta_n)^2] - 2 m c_n^2(x) \beta_n^2 \right] \\ + \sum_{n=1}^{n_{\mathrm{II}}} \rho(x) \left\{ -v_s^\mu(x) (\bar{\theta}_n \overrightarrow{\partial_\mu} \theta_n - \bar{\theta}_n \overleftarrow{\partial_\mu} \theta_n) - \tfrac1{2m}[(\nabla \theta_n)^2 + (\nabla \bar\theta_n)^2]\right\}. \label{eq:full-fluc-Lagrangian}\end{gathered}$$
Since the basis for Type-I modes is not uniquely fixed by the canonical conjugate structure of Eq. , this leaves us free to diagonalize the mass tensor produced by the variation of the potential in Eq. . Doing so produces the effective chemical potential terms, $m c_n^2(x)$.
We end this section with a note about the validity of this fluctuation Lagrangian: it can be seen that the overall size of this action is set by the condensate density $\rho(x)$, which uniformly multiplies all terms. Thus, the condensate density $\rho(x)$ acts to enforce the saddle-point in the sense that if it is large, the fluctuation contribution from $\mathcal{L}_{\textrm{fluc}}$ is suppressed. This tells us that our approach ought not be valid if either the condensate density is strongly fluctuating or vanishing all-together, as might happen at finite temperatures or near e.g. the core of a vortex. Additionally, there may be breakdowns in smaller dimensional systems, where long-range order is prohibited by Mermin-Wagner [@merminAbsenceFerromagnetismAntiferromagnetism1966; @hohenbergExistenceLongRangeOrder1967; @colemanThereAreNo1973]. Barring these considerations, we proceed on to study the properties of the effective field theory described in Eq. . We first consider the case where the Goldstone mode is Type-I, and then we study the case of a Type-II mode.
Type-I Goldstones: Relativistic Spacetime {#sec:relativistic}
-----------------------------------------
Consider an isolated Type-I Goldstone mode, with Lagrangian $$\begin{gathered}
\mathcal L_{\mathrm{I}} = \rho(x) [ - 2 \beta v_s^\mu(x) \partial_\mu \phi - \tfrac1{2m}[(\nabla \phi)^2 + (\nabla \beta)^2]\\ - 2 m c^2(x) \beta^2 ],\end{gathered}$$ we assume that $m c^2(x)$ is large enough to dominate over the kinetic energy for $\beta$, so that $\beta$ can be easily integrated out via $ m c^2(x)\beta = -2v_s^\mu \partial_\mu \phi$. We get the resulting Lagrangian, valid at long wavelengths and times $$\label{eqn:type-i-fluc}
\mathcal L_{\mathrm{I}}^{\mathrm{eff}} = \frac{\rho(x)}{2m} \left[ \left( \frac{v_s^\mu(x)\partial_\mu \phi}{c(x)}\right)^2 - (\nabla \phi)^2\right].$$ This describes a scalar field propagating along geodesics of an emergent space-time metric $\mathcal{G}_{\mu\nu}$ with $$\mathcal L_{\mathrm{I}}^{\mathrm{eff}} = \tfrac12\sqrt{-\mathcal G} \mathcal G^{\mu \nu} \partial_\mu \phi \partial_\nu \phi, \label{eq:relativistic-scalar}$$ and $\mathcal G^{\mu \nu}$ given by the line-element $$ds^2 = \frac{\rho}{c} [ c^2 dt^2 - (d\mathbf x - \mathbf v dt)^2] = \mathcal{G}_{\mu \nu} dx^\mu dx^\nu.$$ This was first observed by Unruh in Ref. [@unruhExperimentalBlackHoleEvaporation1981] where he showed that metrics of the form given above can possess non-trivial features including event-horizons. Indeed, the metric for a Schwarschild black hole can take a very similar form in certain coordinate systems. One of the central results of this paper is the extension of this analogue to include the Type-II modes, which do not have emergent Lorentz invariance. This is shown below.
Type-II Goldstones: Non-relativistic Spacetime {#sec:nonrelativistic}
----------------------------------------------
We focus on a single Type-II Goldstone mode, for which there is no massive field to integrate out. We are left with the fluctuation Lagrangian $$\begin{gathered}
\mathcal L_{\mathrm{II}} = \rho(x) \{ -v_s^\mu(x) (\bar{\theta} \overrightarrow{\partial_\mu} \theta - \bar{\theta} \overleftarrow{\partial_\mu} \theta)\\ - \tfrac1{2m}[(\nabla \theta)^2 + (\nabla \bar\theta)^2]\}.\end{gathered}$$ To simplify things, we group the two real fields into one complex field $$\psi = \theta + i \bar{\theta},$$ so that we have $$\mathcal L_{\mathrm{II}} = \rho [\tfrac{i}2 v_s^\mu (\psi^* \overrightarrow{\partial_\mu} \psi - \psi^* \overleftarrow{\partial_\mu} \psi) - \tfrac{1}{2m} | \nabla \psi|^2 ]. \label{eq:single-typeii}$$
It turns out this too has a simple geometric description in terms of an emergent curved space-time. However, instead of being an “Einsteinian" geometry, the resulting description is in terms of a Newton-Cartan geometry [@cartanVarietesConnexionAffine1923; @*cartanVarietesConnexionAffine1924; @son2013newtoncartan; @gromovThermalHallEffect2015; @geracieSpacetimeSymmetriesQuantum2015; @bradlynLowenergyEffectiveTheory2015].
Newton-Cartan geometry consists of three key objects: $(n_\mu, v^\mu, h^{\mu \nu})$. These are not all independent, but rather must satisfy the constraints $$n_\mu v^\mu = 1, \quad n_\mu h^{\mu\nu} = 0.$$ Also note that the indices on these objects are given as covariant and contravariant specifically and cannot be freely raised/lowered without the definition of a metric tensor (which we describe how to construct in Sec. \[sec:transport\]).
To understand the geometry these objects encode, we begin with the fundamental object that enforces time’s special status within a nonrelativistic theory: $n_\mu$. As a one-form, $n_\mu$ (colloquially, we call it the “clock” one-form) can be imagined as a series of surfaces (foliations), and when a spacetime displacement vector is contracted with it, it gives the elapsed time in a covariant manner. In conjunction with the clock one-form, we have the velocity field $v^\mu$, which must go forward a unit of time (hence the constraint $n_\mu v^\mu = 1$) as a four-velocity; flow along $v^\mu$ causally connects spatial surfaces. Lastly, the spatial metric $h^{\mu\nu}$ is degenerate ($n_\mu h^{\mu\nu}=0$) since it solely describes the geometry confined to the $d$-dimensional spatial foliations. While in what follows we describe $h^{\mu\nu}$ emerging from intrinsic properties of the fluid flow, it can also inherit extrinsic contributions (i.e. if the fluid is flowing on an actual curved manifold).
In the presence of this curved Newton-Cartan geometry, the Lagrangian for a massless scalar field takes the form $$\mathcal L = n_0 \sqrt{h} [\tfrac{i}2 v^\mu (\psi^* \overrightarrow{\partial_\mu} \psi - \psi^* \overleftarrow{\partial_\mu} \psi) - \tfrac{h^{\mu\nu}}{2m} \partial_\mu \psi^* \partial_\nu \psi]
\label{eq:NC-lagrangian-II}$$ where $h = (|\det h^{ij}|)^{-1}$.
The Lagrangian of a Type-II Goldstone mode may be brought into this form. Relating Eq. to Eq. , we can extract the geometric objects $n_\mu$, $v^\mu$, and $h^{\mu\nu}$. We see that in our systems $h^{00} = 0 = h^{0i}$, and that $h^{ij} = h^{-1/d} \delta^{ij}$ in $d$ spatial dimensions. Therefore, we know $n_i = 0$; hence, $n_0 v^0 = 1$. Relating terms, we have $$\begin{split}
\sqrt{h} & = \rho, \\
n_0 \sqrt{h} v^i & = \rho v_{\mathrm{s}}^i, \\
n_0 h^{(d-2)/(2d)} & = \rho.
\end{split}$$ This gives us the geometric quantities $$h = \rho^2 , \quad n_0 = \rho^{2/d},$$ and hence $$\label{eqn:geometry-result}
\begin{split}
n_\mu & = [\rho^{2/d}, \bm{0}], \\
v^\mu & = \rho^{-2/d} v_s^\mu, \\
h^{ij} & = \rho^{-2/d} \delta^{ij}.
\end{split}$$
One important aspect of Newton-Cartan geometry is the notion of “torsion" [@bergshoeffNewtonCartanGravityTorsion2017]. Regarded as a differential form, the clock one-form $n= n_\mu dx^\mu$ is in general not an exact differential. This is seen by taking the exterior derivative, which defines the “torsion tensor" $\omega = dn $. Explicitly, $$\omega_{\mu\nu} = \partial_\mu n_\nu - \partial_\nu n_\mu.$$ It is straightforward to see that in general, the torsion tensor in our geometry is non-zero; $$\omega_{0 j} = \partial_j n_0 = \partial_j \rho^{2/d}.$$ Were the torsion zero, we could define an absolute time coordinate $T$, from which we would get the clock one-form as $n = dT$. While the non-zero torsion implies there is no such absolute time, we may confirm that the more general condition $$n \wedge dn = 0$$ is satisfied. This is a necessary and sufficient condition for the foliation of spacetime into “space-like" sheets which are orthogonal to the flow of time [@bergshoeffNewtonCartanGravityTorsion2017]. As such, there is still a notion of causality in this geometry.
We conclude by commenting that the Newton-Cartan geometry we find here is in fact intimately related to the gravitational field first considered by Luttinger in the context of calculating heat transport [@luttingerTheoryThermalTransport1964]. In that limit $n_\mu \propto [e^\Phi, \bm 0]$, and so the gravitational potential (up to scale factor in the logarithm) would be $$\Phi = \frac{2}d \log(\rho).$$ Using this connection, quantities like energy current and the stress-momentum tensor can be calculated as we discuss in Sec. \[sec:transport\]. First, we explore a minimal realization of these geometries and the associated quantum phases in Sec. \[sec:model\] as well as the fate of the Hawking effect across such a transition in Sec. \[sec:step\].
Minimal Theoretical Model {#sec:model}
=========================
In this section, we introduce a minimal model which exhibits a transition between an Einstein-Hilbert and Newton-Cartan spacetime. We begin by analyzing the ground state within mean-field theory. Once this is understood, we study the behavior of fluctuations about the mean-field by employing a Bogoliubov-de Gennes (BdG) description.
The model is that of a pseudo-spin-$\frac12$ bosonic field $\Psi(x) = \left(\Psi_{\uparrow}(x), \Psi_{\downarrow}(x)\right)^T$ with the following Lagrangian density $$\begin{gathered}
\label{eqn:lagrangian}
\mathcal{L} = \Psi^\dagger\left(i\partial_t + \frac{1}{2m}\nabla^2 + \mu \right)\Psi -\frac12g_0\left(\Psi^\dagger \Psi\right)^2 \\ -\frac12g_3\left(\Psi^\dagger \sigma_3 \Psi\right)^2\end{gathered}$$ where $\sigma_j$ are the Pauli matrices for the pseudo-spin and $\mu$ is the chemical potential, which controls the conserved density of the bosons, $\rho = \Psi^\dagger \Psi$. The coupling $g_0>0$ describes a $U(2) = U(1)\times SU(2)$ invariant repulsive density-density contact interaction, as may be expected in a typical spinor BEC, while the $g_3$ parameter introduces anisotropy into the spin exchange interaction. The $g_3$ coupling explicitly breaks the $SU(2)$ symmetry down to $ U(1)\otimes\mathbb{Z}_2$ comprised of rotations of the Bloch vector by any angle about the $z$ axis and reflections of the Bloch vector through the $xy$ mirror plane. Note that stability requires that $g_3 > - g_0$.
Let us briefly comment that, while Lagrangian is a perfectly valid model, a more natural set-up may be realized by the more experimentally available spin-1 systems such as condensed $^7$Li, $^{23}$Na, or $^{87}$Rb. All of these atoms are bosons which have a total hyperfine spin $F=1$ manifold [@stamper-kurnSpinorBoseGases2013]. In this case, the phase transition is between two phases which both respect the full $SU(2)$ spin-rotation symmetrythe ferromagnetic phase and polar (nematic) phase [@hoSpinorBoseCondensates1998; @barnettGeometricalApproachHydrodynamics2009]. In this case, rather than being driven by anisotropy, the transition is driven by the overall sign of the spin-exchange interaction. It turns out that the different ground-state phases have different types of Goldstone modes and therefore exhibit different analogue spacetimes for the spin waves once condensate flow is introduced. The relevant coupling constant is the spin-exchange coupling $c_2$, which is given in terms of the scattering lengths by $$c_2 = \frac{4\pi}{m} \frac{a_2 - a_0}{3}.$$ For $^7$Li and $^{87}$Rb,$c_2< 0 $ while for $^{23}$Na $c_2 >0$ [@stamper-kurnSpinorBoseGases2013]. Thus, all else equal we can realize both the polar (nematic) phase (which occurs for $c_2>0$) as well as the ferromagnetic phase ($c_2<0$) by using two different species of trapped atom. All this is to say that, while Eq. is not as easily realized experimentally, there may be more experimentally feasible models which realize the same physics. We now move on to the analysis of the technically simpler model proposed above.
The mean-field ground state of Eq. is identified as the homogeneous minimum of the energy density $$V = \frac12 g_0 \left(\Psi^\dagger \Psi\right)^2 +\frac12g_3 \left(\Psi^\dagger \sigma_3 \Psi\right)^2 - \mu \Psi^\dagger \Psi.$$ For $\mu <0$ the ground state is trivial and there is no condensate. For $\mu >0$ there is Bose-Einstein condensation and the ground state is a BEC with a uniform condensate density which obeys the equation of state $$\rho = \Psi^\dagger \Psi =\begin{cases}
\frac{\mu}{g_0}, & g_3>0, \\
\frac{\mu}{g_0-|g_3|}, & -g_0<g_3<0.
\end{cases}$$ A non-zero condensate density always spontaneously break the overall $U(1)$ phase symmetry. The corresponding Goldstone mode corresponds to the broken generator $i\sigma_0 = i\mathds{1}$ where $\mathds{1}$ is the $2\times 2$ identity matrix.
![Illustration of the different ground-state Bloch-vector manifolds as the parameter $g_3$ is tuned. For $g_3 < 0$ the ground state manifold consists of the north and south poles and thus the system realizes an Ising ferromagnet, spontaneously breaking the $\mathbb{Z}_2$ symmetry while maintaining the $U(1)$ symmetry. For $g_3 =0 $ the full $SU(2)$ symmetry is realized and the ground-state manifold consists of the entire Bloch sphere. Thus, the system is a Heisenberg ferromagnet which spontaneously breaks the full $SU(2)$ down to $U(1) \subset SU(2)$. Finally, for $g_3 > 0$ the ground state manifold consists of the equatorial plane, rendering the system an XY (easy-plane) ferromagnet. Thus, the initial symmetry is $U(1)$ which is spontaneously broken to the trivial group. []{data-label="fig:phasediagram"}](pseudospinhalf_phasediagram.pdf)
Phase Sound waves Spin waves
------------------------ ----------------- -------------------
Ising Ferromagnet $\omega \sim k$ Gapped
SU(2) Ferromagnet $\omega \sim k$ $\omega \sim k^2$
Easy-plane Ferromagnet $\omega \sim k$ $\omega \sim k$
: Goldstone modes associated to each phase shown in Fig. \[fig:phasediagram\]. All phases have a Type-I Goldstone mode associated to the spontaneous breaking of the global $U(1)$ phase, corresponding to the conventional sound mode. Additionally, there may also be Goldstone modes associated with spontaneous breaking of spin symmetries, leading to spin waves. In the Ising phase, the broken symmetry is discrete and there are no Goldstone modes. In the SU(2) invariant Heisenberg phase there is a Type-II Goldstone mode describing transverse fluctuations of the magnetization, while in the XY easy-plane phase there is a Type-I Goldstone describing equatorial fluctuations of the magnetization.[]{data-label="tab:mode-properties"}
Depending on the value of $g_3$, additional symmetries may be broken, resulting in the phase diagram illustrated in Fig. \[fig:phasediagram\]. We write the condensed $\Psi$ in the density-phase-spinor representation as $$\label{eqn:spinor-form}
\Psi = \sqrt{\rho}e^{i\Theta} \chi, \quad \chi^\dagger \chi = 1$$ where $\chi$ yields the local magnetization density. It may be parameterized in terms of one complex parameter $\zeta$ via $$\chi = \frac1{\sqrt{2(1+|\zeta|^2)}}\begin{pmatrix}
1 + \zeta \\
1 - \zeta
\end{pmatrix}, \quad \zeta \in \mathbb C.$$ Alternatively, it may be represented in the more canonical Euler angle representation as $$\chi = \begin{pmatrix}
\cos\tfrac{\theta}{2} \\
\sin\tfrac{\theta}{2}e^{i\varphi}
\end{pmatrix}, \quad \varphi \in [0,2\pi)\quad \theta \in [0,\pi).$$ We use both of these representations throughout. In terms of $\zeta$ and $\theta,\varphi$ the anisotropic interaction is $$V = \frac12 g_3 \rho^2 \frac{(\zeta + \zeta^*)^2}{(1 + |\zeta|^2)^2} = \frac12 g_3 \rho^2 \cos^2 \theta.$$ We now proceed to study the mean-field phase diagram of the ground state.
[*Ising phase.*]{}We begin by considering the case of $g_3 <0$, i.e. the “Ising ferromagnet" phase. The interaction has a $U(1)\times \mathbb{Z}_2$ symmetry generated by $\tfrac{i}{2} \sigma_3$ composed with inversion of the $z$ component of the magnetization. In this case it is energetically favorable for the Bloch vector to align with the $z$ axis. This breaks the $\mathbb{Z}_2$ symmetry and preserves $U(1)$ so the ground state manifold is the symmetric space $U(1)\times \mathbb{Z}_2 / U(1) \sim \mathbb{Z}_2$. This is depicted in the left-most panel of Fig. \[fig:phasediagram\], which shows the ground-state manifold for the spinor $\chi$ for various couplings. The Goldstone modes associated with the broken-symmetry ground-state, along with their dispersions are shown in Table \[tab:mode-properties\]. As the ground-state manifold is discrete there is no additional Goldstone mode in this phase and we no longer consider this portion of the phase diagram in this work.
[*Heisenberg phase.*]{}When $g_3 =0$ the interaction term is isotropic and the model has the full $SU(2)$ invariance. The ground state then spontaneously break the $SU(2)$ symmetry down to $U(1)$ so that the ground state manifold is the symmetric space $SU(2)/U(1)\sim S^2$the full Bloch sphere. This is illustrated in the middle panel of Fig. \[fig:phasediagram\]. Without loss of generality, we take the ground state magnetization to point along the positive $x$ direction. Thus, $\zeta = 0$ and $\chi = \frac{1}{\sqrt{2}}(1,1)^T$. Then the unbroken generators are $\{\tfrac{i}{2}( \sigma_1 - \mathds{1} )\}$ and the broken generators are $\{\tfrac{i}{2}(\sigma_1 + \mathds{1}), \tfrac{i}{2}\sigma_2 , \tfrac{i}{2}\sigma_3, \}$. Using the formalism from Sec. \[sec:spacetimes\], we find that the $P$ matrix appearing in the Goldstone mode Lagrangian is $$P^t = \rho \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & \tfrac14 \\
0 & -\tfrac14 & 0
\end{pmatrix},$$ where the columns refer, in order, to the generators $\{\tfrac12 i \sigma_0 + \tfrac12 i \sigma_1, \tfrac12 i \sigma_2, \tfrac12 i\sigma_3\}$. In this case, we have one Type-II Goldstone mode associated with the two generators $\{\tfrac12 i \sigma_2, \tfrac12 i\sigma_3\}$ which exhibits a quadratic dispersion relation and hence realize the Newton-Cartan geometry in the presence of inhomogeneous condensate flow. This is summarized in Table \[tab:mode-properties\].
[*XY phase.*]{}We now move on to the case where $g_3>0$. In this case there is an energy penalty associated with a non-zero $z$ component of the magnetization and thus the ground state lies in the manifold defined by $\cos \theta = 0 \Rightarrow \theta = \pi/2$. Thus, the ground state breaks the $U(1)$ symmetry but remains invariant under reflections through the $z=0$ plane. As such, the ground state resides in the symmetric space $U(1)\times \mathbb{Z}_2 / \mathbb{Z}_2 = U(1) \sim S^1$, as depicted in the right panel of Fig. \[fig:phasediagram\]. Without loss of generality we again take the Bloch vector to lie along the $+x$ direction. Thus, only two generators remain unbroken in the Lagrangian $\{i \mathds{1}, \tfrac12 i\sigma_3\}$ and the mean-field breaks both of them. We again refer to Eq. to obtain $$P^t_{mn} = 0.$$ Thus, there are no Type-II Goldstone modes in this system, but instead two Type-I modes which are linearly dispersing and therefore exhibit an analogue Einstein-Hilbert spacetime, summarized in Table \[tab:mode-properties\].
Bogoliubov-de Gennes Analysis {#subsec:bdg}
-----------------------------
We now proceed to examine the fluctuations about the mean-field by obtaining and diagonalizing the Bogoliubov-de Gennes equations of motion. To see how the analogue spacetime emerges we consider a mean-field condensate $\psi_0$ which is inhomogeneous, but has a constant magnetization density. Taking the spin to point in the $+x$ direction, we obtain $$\label{eqn:gs-mf}
\psi_0 = \sqrt{\rho(x)}e^{i\Theta(x)}\chi_0 = \sqrt{\rho(x)}e^{i\Theta(x)} \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}\\
\end{pmatrix} .$$ In this case, the mean-field describes a flowing condensate with superfluid density $\rho(x) = \psi_0^\dagger(x) \psi_0(x)$ and superfluid velocity $\mathbf{v}_s = \frac{1}{m} \nabla \Theta(x)$. Fluctuations about this mean-field can be fully parameterized in terms of the two complex fields $\phi$ and $\zeta$ as $$\label{eqn:fluc-form}
\delta \Psi = \left(\phi \sigma_0 + i\zeta \sigma_2\right)\psi_0.$$ To quadratic order, the Lagrangian from Eq. decouples into two quadratic BdG Lagrangians $$\begin{aligned}
\mathcal L_\phi & = \rho \left[ \tfrac i2 (\phi^* D_t \phi - \phi D_t \phi^*) - \tfrac{|\nabla \phi|^2}{2m} + \tfrac12 g_0 \rho (\phi + \phi^*)^2 \right],\nonumber \\
\mathcal L_\zeta & = \rho \left[ \tfrac i2 (\zeta^* D_t \zeta - \zeta D_t \zeta^*) - \tfrac{|\nabla \zeta|^2}{2m} + \tfrac12 g_3 \rho (\zeta + \zeta^*)^2 \right],\label{eqn:Bdg-Lagrangians}\end{aligned}$$ with $D_t = \partial_t + \mathbf{v}_s \cdot \nabla$ the material derivative in the frame co-moving with the superfluid flow. These two Lagrangians are specific examples of the more general Eq. . In particular, for $g_3 > 0$ at long wavelengths we can apply the analysis of Sec. \[sec:relativistic\] to obtain the relativistic analogue spacetime. If on the other hand, $g_3 = 0$, then at long wavelengths we can apply the analysis of Sec. \[sec:nonrelativistic\] to obtain the nonrelativistic Newton-Cartan analogue spacetime. Nevertheless, it is instructive to instead follow Ref. [@curtisEvanescentModesSteplike2019; @recatiBogoliubovTheoryAcoustic2009], and directly employ the BdG equations when determining the consequences of the changing spacetime structure. This is because the BdG equations provide us with a single unified description with which we may capture both phases, as well as the transition between them.
The BdG equations are obtained as the Euler-Lagrange equations of Lagrangians $\mathcal{L}_{\phi}, \mathcal{L}_{\zeta}$ and are most transparently expressed in terms of the Nambu spinors $$\label{eqn:Nambu}
\Phi_0 = \left(\begin{array}{c}
\phi \\
\phi^*\\
\end{array}\right), \ \Phi_3 = \left(\begin{array}{c}
\zeta\\
\zeta^*\\
\end{array}\right)$$ for condensate and spin wave fluctuations, respectively. We then find the BdG equations $\hat{K}_0 \Phi_0 = 0$, and $\hat{K}_3 \Phi_3 = 0$, with the BdG differential operators $$\label{eqn:BdG-EOM}
\begin{aligned}
& \hat{K}_{0} = \tau_3 \left( i \partial_t +i\mathbf{v}_s\cdot\nabla\right) + \frac{1}{2m\rho}\nabla\cdot\rho\nabla\tau_0 - g_0\rho\left(\tau_0 + \tau_1\right) \\
& \hat{K}_{3} = \tau_3 \left( i \partial_t +i\mathbf{v}_s\cdot\nabla\right) + \frac{1}{2m\rho}\nabla\cdot\rho\nabla\tau_0 - g_3\rho\left(\tau_0 + \tau_1\right), \\
\end{aligned}$$ written in terms of the Nambu particle-hole Pauli matrices $\tau_a$. Let us emphasize that the only difference between $\hat{K}_0$ and $\hat{K}_3$ is the coupling constant appearing in front of the $\tau_0+\tau_1$ term. For sound waves it is $g_0$, while for the spin waves it is $g_3$. Thus, both Goldstone modes end up coupling to the same background condensate density and velocity, albeit with different speeds of sound. Sound waves end up propagating with the local group velocity $$c_0(x) = \sqrt{\frac{g_0 \rho(x)}{m}}$$ while the spin waves have the local group velocity $$c_3(x) = \sqrt{\frac{g_3\rho(x)}{m}}.$$ Thus, we see that the coupling $g_3$ allows us to independently tune the two speeds of sound relative to each other.
For generic values of $g_3>0$ and arbitrary condensate flows we cannot find quantum numbers with which we can diagonalize $\hat{K}_3$. However, at the $SU(2)$ symmetric point $g_3 =0$ we observe that the BdG kernel for spin waves obeys $$\hat{K}_{3} = \tau_3 \left( i \partial_t +i\mathbf{v}\cdot\nabla\right) + \frac{1}{2m\rho}\nabla\cdot\rho\nabla\tau_0 \Rightarrow \left[ \tau_3, \hat{K}_3 \right] = 0.$$ Since $\tau_3$ now commutes with the kernel, the two components of the BdG spinor decouple and each independently obeys a Galilean-invariant dispersion relation. This also results in an additional $U(1)$ symmetry generated by $\tau_3$ which imposes a selection rule for the allowed Bogoliubov transformations. In particular, there is no matrix element which scatters a “particle-like" Bogoliubov quasiparticle into a “hole-like" particle. this process is the one responsible for Hawking radiation and as such we find, counter-intuitively, that it is impossible to generate Hawking radiation in the Newton-Cartan spacetime despite the fact that all flow velocities $\mathbf{v}_s$ are now supersonic. This is explicitly demonstrated for the case of a step-like horizon, which we analyze in the following section.
Step-Like Horizon {#sec:step}
=================
In order to get a more quantitative understanding of how the changing spacetimes affect observable physics, we imagine a specific flow profile and use the BdG equations to solve for the spin-wave scattering matrix. We imagine a quasi-one-dimensional stationary condensate flow with a superfluid density and velocity which obeys $\partial_t \rho = \partial_t v_s = 0$. The continuity equation for the condensate then implies $$\label{eqn:continuity}
\partial_x(\rho v_s) = 0 \Rightarrow \rho(x) v_s(x) = \textrm{const}.$$ The local speed of sound for the spin-waves (henceforth simply written as $c$) is therefore $c(x) = \sqrt{g_3 \rho(x)/m}$.
To further simplify calculations, we consider the case of a step-like profile for $\rho(x),v(x)$ of the form $$\label{eqn:step-flow}
\begin{aligned}
& \rho(x) = \left\{\begin{aligned}
&\rho_l & x < 0 \\
&\rho_r & x \geq 0 \\
\end{aligned}\right. \\
& v(x) = \left\{\begin{aligned}
& - |v_l| & x < 0 \\
& - |v_r| & x \geq 0 .\\
\end{aligned}\right. \\
\end{aligned}$$ Note that continuity requires $v_l \rho_l = v_r \rho_r \Leftrightarrow v_lc_l^2 = v_r c_r^2$. In this work we adopt the convention that $v$ is negative, so that the condensate flows from the right to the left. With this set-up, we can employ the BdG techniques usually used for phonon modes to these spin waves [@recatiBogoliubovTheoryAcoustic2009; @curtisEvanescentModesSteplike2019].
This step-like potential has the advantage that away from the jump, momentum eigenstates solve the BdG equations, and the scattering matrix reduces to a simple plane-wave matching condition at the boundary. The details of this procedure may be found, e.g. in Appendix \[app:BogoliubovHawking\]. Here we simply discuss the results of the calculation. We start by considering $g_3 >0$ to be large and then decrease down to zero. As we do so, while keeping the flow profile fixed, we pass through three regimes.
The first regime occurs for large $g_3$ so that $c_l>|v_l|$ and $c_r > |v_r|$. Thus, there is no sonic horizon and no Hawking radiation.
Eventually as we continue decreasing $g_3$ we enter the regime where $|v_r| < c_r$ but $c_l<|v_l|$. This exhibits a sonic horizon at $x=0$ and is thus accompanied by Hawking radiation.
Finally, we reach the regime where $|v_l| > c_l $ and $|v_r|> c_r$. This is a novel regime wherein both the interior and exterior of the jump are supersonic. However, due to the non-linear Bogoliubov dispersion, there are still some short-wavelength modes for which one or both sides of the flow are not supersonic (this is due to the convex dependence of the group-velocity on momentum). Thus there is still Hawking radiation, however we find that as we decrease $g_3$ further, the total “flux" of modes which are emitted decreases until we recover the result that at $g_3=0$ there is no radiation at all.
To see this, we define the “total number of Hawking modes" at a given frequency to be $N(\omega)$ (see Eqs. and ). This is obtained by calculating the “Hawking" element of the scattering matrix for the BdG equations. From $N(\omega)$ we can then define the total “luminosity” [@corleyHawkingSpectrumHigh1996] leaving the horizon by $$L_{\mathrm{H}} = \int_0^\infty d\omega \ \frac{\omega}{2\pi} N(\omega).$$ Note that in the conventional black hole case, $N(\omega)$ is the number of photons at frequency $\omega$ seen at asymptotic infinity and thus this is simply the number flux per unit frequency of the radiation.
![The total luminosity due to the Hawking radiation for a fixed density profile $\rho(x)$ and velocity profile $v(x)$. We see that there is no Hawking radiation when $c_r$ is sufficiently large so that $c_l>v_l$ (recall these are constrained by the continuity equation). When $c_l<v_l$ but $c_r> v_r$ we get a region of subsonic flow that flows into a supersonic region and we begin seeing traditional Hawking radiation. As we further tune $g_3$, $c_r$ drops below $v_r$ and both regions become supersonic at low frequencies. Evidently, there is still a channel for Hawking radiation emission as seen by the non-zero integrated flux. However, as $c_r$ drops to zero this channel closes, vanishing precisely at the quantum phase transition into the Newton-Cartan geometry ($c_r=0=g_3$). In this plot, $v_l = 0.9$, $v_r=1.3$, $m=10$, and $\rho(x)v(x)=1$.[]{data-label="fig:totalflux"}](totalflux.pdf)
The upshot is given by Fig. \[fig:totalflux\] which plots $L_\mathrm{H}$ as a function of $(c_r/v_r)^2 = g_3 \rho_r /m v_r^2$. Thus, for fixed flow density and velocity, this is essentially plotting as a function of the control parameter $g_3$. We see the three distinct regions and importantly at $g_3=0$ we see the Hawking effect vanish.
To understand this effect, we consider the dispersion relation of the waves away from the horizon, for which momentum is a good quantum number. In the right and left half-spaces we have the relations $$(\omega - v_{\alpha} k)^2 = c_{\alpha}^2 k^2 + \frac{k^4}{4 m^2},$$ where $\alpha = l,r$ for the left and right regions respectively. This relates the lab-frame frequency of a wave $\omega$ to the lab-frame momentum $k$. This dispersion relation is plotted in Figs. \[fig:SuperToSubHawking\] and \[fig:SuperToSuperHawking\]. Due to the presence of a discontinuity at $x = 0$ modes with different momenta mix and only $\omega$ can be fixed globally. Thus, the dispersion relation is to be solved by finding the allowed momenta at each fixed lab-frame frequency. This amounts to finding the roots of a quartic polynomial with real coefficients, and as such there are always four solutions (which are either real or complex conjugate pairs). The real momenta represent propagating modes while we later find that the complex roots describe evanescent modes localized around the horizon.
Subsonic-Supersonic Jump
------------------------
First, we consider the case of a jump between a subsonic and supersonic flow, depicted graphically in Fig. \[fig:SuperToSubHawking\]. In this case, we recover the well-known result that there is Hawking radiation emitted. The dispersion relation in each half-plane is plotted and intercepts with a constant $\omega>0$ are found. These intercepts yield the momenta of the propagating modes in each region for the given frequency. Each curve is depicted with a color indicating the sign of the group velocity in the [**co-moving**]{} frame, which is what is used to distinguish between “particle-like" (red) and “hole-like" (blue), in accordance with the BdG norm (see Appendix \[app:BogoliubovHawking\] and in particular Eq. for definition). We see that the outgoing Hawking mode (combined with an evanescent piece at the horizon) is connected to three incoming waves, one of which is a negative norm state originating from the interior of the horizon. This particle-hole conversion processes is the origin of the Hawking effect, as this induces a Bogoliubov transformation which connects the vacuum of the asymptotic past to a one-particle state in the asymptotic future (and vice-versa).
![The Hawking effect for $g_3$ such that $c_r> v_r$ and $c_l < v_l$ (sub-sonic to super-sonic). In this situation, one side (left) flows faster than the speed of some excitations, and the other side (right) flows slower than the speed of any excitation. The dashed line represents the constant lab frame energy $\omega$. The mode that carries away energy from the horizon is the “Hawking mode,” shown by the star marker. Tracing this mode back in time (bottom of figure), we find that it comes from a scattering process that includes positive (red) and negative (blue) norm states. It is the negative norm state to the left of the horizon that is responsible for particle creation in the Hawking channel. Notice that for frequencies larger than those in the labeled “Hawking region,” there is no Hawking effect due to lack of negative energy modes to have scattered from at earlier times. []{data-label="fig:SuperToSubHawking"}](SubToSuper-HawkingProcess.pdf)
We see that due to the convex non-linear Bogoliubov dispersion relation, there is a maximum frequency of the emitted Hawking radiation obtained by finding the local maximum of the negative norm dispersion relation. Above this frequency, the flow is no longer supersonic since the group velocity of modes depends non-trivially on the frequency.
Supersonic-Supersonic Jump
--------------------------
As we decrease $g_3$ beyond a critical value the system enters a parameter regime where both sides of the jump are supersonic flows. In this case, the dispersion relation still exhibits a Hawking-like region, as we see in Fig. \[fig:SuperToSuperHawking\]. However, we also see a new region emerge at low energies (labeled “super-Hawking” in the figure) in which now both a positive and negative norm mode can be scattered into. This opens a new channel in the scattering matrix which leads to a reduction in the amplitude for scattering into the Hawking channel, as per generalized unitarity constraints. This is seen in Fig. \[fig:HawkingFlux\], which compares $N(\omega)$ for the case of a subsonic-supersonic (red) and supersonic-supersonic jump (blue). Both curves are qualitatively similar at high frequencies, corresponding to the “Hawking" region of frequencies in Figure \[fig:SuperToSuperHawking\]. On the other hand, we see that at low $\omega$, when we have subsonic-to-supersonic flow, $N(\omega)$ diverges in the universal thermal manner, while in the supersonic-to-supersonic regime, there is a noticeable change in behavior between the Hawking and super-Hawking regimes, cutting off this low $\omega$ divergence.
![The Hawking effect for $g_3$ such that $c_r< v_r$ and $c_l < v_l$ (super-sonic to super-sonic). With both regions flowing faster than the speed of excitations (relative to the horizon), we still have a Hawking region, but now we also have a “Super-Hawking” region where the positive and negative normalization modes from both regions can scatter between one another.[]{data-label="fig:SuperToSuperHawking"}](SuperToSuper-HawkingProcess.pdf)
![Hawking flux $N(\omega)$ as a function of frequency for the subsonic-to-supersonic case (red) and the supersonic-to-supersonic case (blue). As we approach the Heisenberg symmetric point $g_3=0$, we find the Hawking flux disappears both in its overall magnitude and singular behavior. The black arrow indicates the onset of the “super-Hawking region” responsible for the absence of the singular distribution. []{data-label="fig:HawkingFlux"}](NumbervsOmega.pdf){width="\columnwidth"}
There are two effects occurring which are responsible for decreasing the Hawking luminosity $L_\mathrm{H}$. First, in the Hawking region the incoming negative norm states now begin to more strongly backscatter into their corresponding negative norm state, occupying the evanescent mode on the right side of the horizon. Second, in this super-Hawking region, the appearance of an outgoing negative-norm mode provides an opportunity for the ingoing negative norm channel to avoid scattering into the positive norm channel. We indeed find that the two channels begin to decouple from each other, diminishing the amount of Hawking radiation that can be produced.
Absence of Hawking Radiation for Type-II modes\[subsec:noHawking\]
------------------------------------------------------------------
This takes us directly into the point where $g_3 = 0$, which exhibits the new Newton-Cartan spacetime geometry. One might expect that there should be something akin to a Hawking effect since some modes “see” a horizon for any difference in $|v_l|$ and $|v_r|$. However, this horizon does not translate into a Hawking effect. As explained earlier, at this point the BdG kernel $\hat{K}_3$ commutes with $\tau_3$. In terms of the BdG Lagrangian of Eq. , we find that there is now a new global $U(1)$ symmetry $\zeta \rightarrow e^{i\vartheta} \zeta$. We can see explicitly from the BdG analysis that this conserved charge density is given by $$Q_{\mathrm{BdG}} = \int d^3 x \, \rho |\zeta|^2.$$ On the other hand, by applying Noether’s theorem directly on the general Newton-Cartan action of Eq. , in the limit where $n_0$ is the only nonzero component of $n_\mu$ and the Lagrangian is independent of the $x^0$, we find $$Q_{\mathrm{BdG}} = \int d^3 r \sqrt{h}|\psi|^2 . \label{eq:NCconservedQ}$$ If we identify $\psi = \zeta$ and use the results of Eq. we find that these two indeed match each other. In particular, Eq. describes a conserved charge for the field $\psi$ on a curved manifold given by $h^{\mu\nu}$.
Since, unlike the charge in Eq. , this density is positive definite it can be genuinely interpreted as the number of BdG quasiparticles. This symmetry then imposes a selection rule on the scattering matrix which prohibits the scattering processes responsible for the Hawking process, which leads to a creation of BdG quasiparticles. This is evident if we see that when $g_3 = 0$, $$\left[ i\left(\partial_t + \mathbf{v}\cdot\nabla\right)+ \frac{1}{2m\rho}\nabla\cdot\rho\nabla \right]\zeta = 0,$$ and hence $\zeta$ and $\zeta^*$ do not mix. Indeed, as Fig. \[fig:ksqr\] illustrates, though Hawking radiation is permissible by conservation of energy and momentum, as seen by the dispersion relation in Fig. \[fig:ksqr\], there is no permissible matrix element for any scattering process which mixes positive and negative norm modes. Thus, at low frequencies (below the cutoff frequency on the right), negative norm modes may be transmitted across the horizon but only as outgoing negative norm modes. This is analogous to the “super-Hawking" regime earlier, but since there is no conversion between positive and negative norm modes, there is no Hawking radiation effect.
Above the cutoff frequency on the right (in what we refer to as the “regular Hawking regime"), all negative norm modes incident from the interior of the horizon must be reflected back. Even in this case, there is still a finite penetration of the negative norm state across the event horizon in the form of an evanescent mode which is decaying away from the horizon, as originally predicted in Ref. [@curtisEvanescentModesSteplike2019]. In fact, this evanescent tail is also present when $g_3 > 0$, but now it is not accompanied by any other outgoing mode. Again, let us emphasize that this evanescent mode is associated with a negative norm mode and therefore does not couple to positive norm modes. Thus, it cannot be spontaneously excited from the ingoing vacuum. Ultimately, as the negative norm mode must be reflected, all the amplitude which initially went into the outgoing positive norm states when $g_3 >0$ is now transferred into the reflected negative norm state and the evanescent tail.
![For $g_3 = 0$ in the Newton-Cartan geometry there is an excitation number conservation that protects negative norm states from scattering into positive norm states and as a result, if we scatter a negative norm state in what used to be the “Hawking region,” we find it fully back scatters into a negative norm state and leaks past the horizon only with an evanescent tail characteristic to a “classically forbidden” region.[]{data-label="fig:ksqr"}](noHawkingProcess.pdf){width="\columnwidth"}
Transport in Newton-Cartan Geometry\[sec:transport\]
====================================================
In this section we take up the issue of energy transport in systems exhibiting Newton-Cartan geometry. Building on Luttinger’s work on computing heat transport via coupling to a gravitational field [@luttingerTheoryThermalTransport1964], there has been a well-established method of coupling systems to Newton-Cartan geometry in order to extract their heat transport properties [@gromovThermalHallEffect2015; @son2013newtoncartan; @bradlynLowenergyEffectiveTheory2015; @geracieSpacetimeSymmetriesQuantum2015]. With these methods, we can begin with the results in Sec. \[sec:nonrelativistic\] and find the stress tensor $T^{\mu\nu}$, energy current $\epsilon^\mu$, and momentum density $p_\mu$. However, as we have mentioned previously, we can also reformulate the relativistic Lagrangian in Sec. \[sec:relativistic\] in terms of a Newton-Cartan geometry with an additional external field. Therefore, in the bulk of this section, we make that precise and use the energy transport machinery to relate the relativistic stress-energy tensor of Type-I modes to its non-relativistic counterparts.
We begin by noting that the variations in the geometry are not independent as they must satisfy the constraints imposed by Newton-Cartan geometry that $n_\mu v^\mu = 1$ and $n_\mu h^{\mu\nu} = 0$. Parameterizing the variations so as to respect these constraints is done by introducing the perturbations $\delta n_\mu$, $\delta u^\mu$ and $\delta \eta^{\mu\nu}$ such that $$\begin{split}
\delta v^\mu & = - v^\mu v^\lambda \delta n_\lambda + \delta u^\mu, \\
\delta h^{\mu \nu} & = - (v^\mu h^{\nu \lambda} + v^\nu h^{\mu \lambda}) \delta n_\lambda - \delta \eta^{\mu\nu},
\end{split}$$ where $n_\mu \delta u^\mu = 0$, and $n_\mu \delta \eta^{\mu \nu} = 0$ so that $\delta u^\mu$ and $\delta \eta^{\mu\nu}$ are orthogonal to the clock one-form $n_\mu$.
To find the full Lagrangian it is useful to formally define a non-degenerate metric in the full spacetime by $$g^{\mu\nu} \equiv v^\mu v^\nu + h^{\mu\nu}.$$ Note that unlike relativistic metrics, this Newton-Cartan has no invariant distinction between space-like and time-like separations (simultaneity is a global concept imposed by $n_\mu$). As $g^{\mu\nu}$ is non-degenerate, we may proceed to take the inverse which is defined by $$g_{\mu \alpha} g^{\alpha \nu} = \delta_\mu^\nu,$$ where $\delta_\mu^\nu$ is the usual Kronecker delta. This also serves to define the inverse of the degenerate metric $h^{\mu\nu}$ by $$g_{\mu\nu} \equiv n_\mu n_\nu + h_{\mu\nu}.$$ Note that the constraints on the geometry then imply $h_{\mu\nu}$ obeys $$h^{\mu\sigma}h_{\sigma\nu} = \delta^{\mu}_\nu - v^\mu n_\nu.$$ The right hand side essentially acts to project onto the manifold upon which $h^{\mu\nu}$ is not degenerate. These are the “spatial" three-surfaces which are in some sense “iso-temporal."
Introducing $g$ is helpful in particular because we then find that if take the determinant $g= \det(g_{\mu\nu})$, we find that $\sqrt{g} = n_0 \sqrt{h}$ [^1]. This is exactly the volume measure of the Lagrangian Eq. . This assists in taking the variation $$\delta[\sqrt{g}] = \sqrt{g}[v^\mu \delta n_\mu + \tfrac12 h_{\mu\nu}\delta\eta^{\mu\nu}].$$ We can then use the variations to find the stress tensor $T_{\mu\nu}$, energy current $\epsilon^{\mu}$, and momentum density $p_\mu$ via [@geracieSpacetimeSymmetriesQuantum2015] $$\delta S = \int d^{d+1} x \sqrt{g} \left(\tfrac12 T^{\mu\nu} \delta \eta_{\mu \nu} - \epsilon^\mu \delta n_\mu - p_\mu \delta u^\mu \right).$$ Due to the constraints on $\delta u^\mu$ and $\delta \eta_{\mu\nu}$, these values of $p_\mu$ and $T^{\mu\nu}$ are not unique. In fact, we can make any substitution $p_\mu\rightarrow p_\mu + a n_\mu$ or $T^{\mu\nu}\rightarrow T^{\mu\nu} + b^\mu v^\nu + b^\nu v^\mu$. We impose uniqueness by requiring $p_\mu v^\mu = 0$ and $T^{\mu\nu} n_\nu = 0$. Lastly, one can derive continuity equations for these quantities by considering how these objects change under a diffeomorphism (see Ref. [@geracieSpacetimeSymmetriesQuantum2015]).
We now compute these quantities for both the Type-I and Type-II modes. It is worth noting that these models describe the free propagation of Goldstone modes and thus are in a sense “non-interacting." By this, we mean there are no additional terms due to interactions [@liao2019drag]. For Type-II modes, the resulting transport quantities are known [@son2013newtoncartan; @bradlynLowenergyEffectiveTheory2015; @gromovThermalHallEffect2015]. We briefly recapitulate this calculation here.
Energy transport for Type-II modes
----------------------------------
We proceed to vary the Newton-Cartan geometry in action Eq. . This straightforwardly yields the momentum density as $$p_\mu = -\tfrac{i}{2} \left[ \bar{\psi} (\partial_\mu - n_\mu v^\alpha \partial_\alpha) \psi - \psi (\partial_\mu - n_\mu v^\alpha \partial_\alpha) \bar{\psi} \right].$$ The limit works out as expected: if we let $n_\mu = (1,\mathbf{0})$ and $v^\mu = (1,\mathbf{0})^T$, only the spatial components survive and we obtain the momentum current for a non-relativistic theory with conserved density $|\psi|^2$. Next, we compute the stress tensor, which describes the momentum flux. We find $$\begin{gathered}
T^{\mu\nu} = -\tfrac{i}{4} v^\alpha\left[ \bar{\psi} \partial_\alpha \psi - \psi \partial_\alpha \bar{\psi} \right] h^{\mu \nu} \\ + \tfrac1{4m} \partial_\alpha \bar\psi \partial_\beta \psi (h^{\alpha\mu} h^{\beta \nu} + h^{\alpha\nu} h^{\beta \mu} - h^{\mu\nu} h^{\alpha\beta})\end{gathered}$$ and the energy current as $$\epsilon^\mu = -\tfrac{1}{2m}(\partial_\alpha \bar{\psi})(\partial_\beta \psi) \left[ v^\alpha h^{\beta \mu} + v^{\beta}h^{\alpha\mu} -v^\mu h^{\alpha\beta} \right].$$ Both have sensible flat-space limits as well.
Energy transport for Type-I modes
---------------------------------
For Type-I modes, an analogue relativistic theory emerges from a nonrelativistic theory, and in both the cases, we can compute energy densities, momentum densities, and the stress-tensor. The objective of this section is to compute how the quantities in the analogue relativistic system are related to their nonrelativistic counterparts, motivated by the spacetime relations derived in Sec. \[sec:spacetimes\].
We have shown the Type-I modes can be thought of as residing in a relativistic analogue spacetime, equipped with an analogue metric tensor $\mathcal{G}_{\mu\nu}$. If we vary with respect to this tensor, we obtain a Lorentz-invariant stress-energy-momentum tensor, $\mathcal{T}^{\mu\nu}$. Note Lorentz invariance constrains this to be symmetric, relating the energy current and momentum densities to each other.
On the other hand, we have shown that one can obtain the Type-I modes by gapping out one of the generators of a Type-II mode. Thus, we can also consider varying the Newton-Cartan geometry that the Type-II mode resides in before including a mass gap. This yields for us the Newton-Cartan stress tensor, momentum density, and energy current and provide for us a general relationship between the relativistic energy-momentum tensor and the non-relativistic counterparts.
First, we return to Eq. and rewrite the Lagrangian in terms of the Newton-Cartan geometry [*prior*]{} to integrating out the massive mode (recall that unlike a Type-II mode, a Type-I mode is canonically conjugate to a massive mode). We obtain $$\begin{gathered}
\mathcal{L} = \sqrt{g} \big( -2 \beta v^\mu \partial_\mu \phi - \tfrac{h^{\mu\nu}}{2m}[\partial_\mu \phi \partial_\nu \phi +\partial_\mu \beta \partial_\nu \beta ]\\ - 2mC^2(x) \beta^2 \big),\end{gathered}$$ where $c^2 = \rho^{2/d} C^2$ is the speed of sound of the Goldstone mode (the factor of density essentially accounts for the units of $h^{\mu\nu}$). If we integrate out the massive mode $\beta$ in the limit where we can neglect the dispersion (i.e. at long wavelengths), we recover the Type-I relativistic Lagrangian $$\mathcal L_{\mathrm{eff}} = \frac{\sqrt{g}}{2m}\left( \frac{(v^\mu \partial_\mu \phi)^2}{C^2} - h^{\mu \nu} \partial_\mu \phi \partial_\nu \phi\right).$$ From this, we can identify the relativistic metric $\mathcal G_{\mu\nu}$ by observing that this Lagrangian must be of the form in Eq. such that $$\sqrt{-\mathcal G} \mathcal G^{\mu\nu} = \tfrac{\sqrt{g}}{m}\left(\tfrac{v^\mu v^\nu}{C^2} - h^{\mu\nu} \right).$$ This yields an equation relating the relativistic metric to the Newton-Cartan object and the gap of the massive mode. We find $$\begin{split}
\mathcal G_{\mu \nu} & = (m C)^{-\frac2{d-1}} \left( C^2 n_\mu n_\nu - h_{\mu\nu}\right),\\
\mathcal G^{\mu \nu} & = ( m C)^{\frac2{d-1}} \left(\tfrac{v^\mu v^\nu}{C^2} - h^{\mu\nu}\right),
\end{split}$$ where $d$ is the spatial dimension. As we can see, the relativistic metric depends crucially on the potential $C(x)$.
This is helpful since, on the one hand, we can easily obtain the stress-energy tensor in the relativistic theory by varying $\delta \mathcal{G}_{\mu\nu}$. On the other hand, we can use the above formulae to connect this result to the actual stress tensor and energy current/momentum density of the non-relativistic model. In particular, $$\begin{gathered}
\delta \mathcal G^{\mu\nu} = ( m C)^{\frac2{d-1}} [( v^\mu h^{\nu\lambda} + v^\nu h^{\mu\lambda} - 2\tfrac{v^\mu v^\nu}{C^2} v^\lambda) \delta n_\lambda \\
+ \tfrac{1}{C^2}(v^\mu \delta^\nu_\lambda + v^\nu \delta^\mu_\lambda) \delta u^\lambda + \delta \eta^{\mu\nu}].\end{gathered}$$ Thus, we can directly relate the relativistic energy-momentum tensor $\mathcal T_{\mu\nu}$ to its non-relativistic counterparts by expanding $$\delta S = \int d^{d+1} x \tfrac12 \sqrt{-\mathcal G} \mathcal T_{\mu\nu} \delta \mathcal G^{\mu\nu}$$ in terms of the geometric objects in the NC geometry. Doing so, we obtain $$\begin{split}
T^{\mu\nu} & =\tfrac1{m(m C)^{\frac 4{d-1}}} (\delta_\alpha^\mu - n_\alpha v^\mu) \mathcal T^{\alpha\beta} (\delta_\beta^\nu - n_\beta v^\nu), \\
\epsilon^\lambda & = \tfrac{1}{m (m C)^{\frac 2{d-1}} } v^\mu \tensor{\mathcal T}{_\mu^{\lambda}}, \\
p_\lambda & = -\tfrac1{mC^2} (\mathcal T_{\lambda \mu} v^\mu - v^\mu \mathcal T_{\mu\nu} v^\nu n_\lambda).
\end{split}$$ where the indices on $\mathcal T^{\mu\nu}$ and $\tensor{\mathcal T}{_\mu^{\lambda}}$ are raised with $\mathcal G^{\mu\nu}$ while all Newton-Cartan objects use the metric $g_{\mu\nu}$. Ignoring the factors in front of these expressions, one can think of $v^\nu$ as a timelike vector with respect to the metric $\mathcal G_{\mu\nu}$. In this case, $v^{\mu}$ is directly related to the field of the fluid flow and the object $\mathcal E^\lambda \propto v^\nu \tensor{\mathcal T}{_\mu^{\lambda}}$ is the energy current measured by an observer comoving with that flow (*not* the lab observer). By the same token $\mathcal P_\lambda \propto \mathcal T_{\lambda\nu} v^\nu$ is the momentum density measured by the comoving observer as well. Relativistically, these are strictly related $\mathcal E^\lambda = \mathcal G^{\lambda \mu} \mathcal P_\mu$. However, momentum is imposed by the underlying non-relativistic field theory to be orthogonal to flow $v^\lambda p_\lambda = 0$. The form of $p_\lambda$ that accomplishes this includes the comoving energy density $v^\mu \mathcal T_{\mu\nu} v^\nu$ and subtracts it off. Lastly, $T^{\mu\nu}$ is directly related to $\mathcal T^{\alpha \beta}$ projected to live only on spatial slices $n_\mu T^{\mu\nu} = 0$, again as imposed by the underlying non-relativistic theory.
In effective, relativistic, analogue systems, there is a preferred (lab) frame that is captured by the Newton-Cartan geometry (in particular $n_\mu$ specifies the lab frame’s “clock”). This preference is hidden in the high frequency dispersion of the type-I modes and, as we have shown here, results in non-trivial momentum currents and stress-tensors.
As a particular example, a Hawking flux against the flow in an analogue system should result in a real energy and momentum current away from the analogue black hole. Far from the horizon (considering the effective 1+1D problem where the other two spatial dimensions are trivial) we obtain $$\mathcal T_{\mu\nu} =\begin{pmatrix}
\mathcal T_{\mathrm{H}} & -\mathcal T_{\mathrm{H}} \\
-\mathcal T_{\mathrm{H}} & \mathcal T_{\mathrm{H}}
\end{pmatrix},$$ for a constant $\mathcal T_{\mathrm{H}}$ [@daviesEnergymomentumTensorEvaporating1976] (for the radiation flowing to $+\infty$). If we apply this to the above, and assume that at $+\infty$ we have no velocity so that $v^\mu = (v^0, \mathbf{0})$ and a flat $h^{ij} = \delta^{ij}/h_0^{1/3}$, we have $$\begin{split}
T^{xx} & = \tfrac1{m h_0^{2/3}} \mathcal T_{\mathrm H}, \\
\epsilon^\lambda & = \tfrac{v^0}{m}\mathcal T_{\mathrm{H}} \left[ \tfrac{(v^0)^2}{C^2},h_0^{-1/3},0,0\right],\\
p_\lambda & = \tfrac{v^0}{m C^2} \mathcal T_{\mathrm{H}} [0,1,0,0].
\end{split}$$ Importantly, we see that there is a finite energy current $\epsilon^1$ and momentum $p_1$ away from the horizon; there is no $p_0$ component due to the constraint $p_\mu v^\mu = 0$. While related to what is computed relativistically, these quantities are not exactly the same.
\[sec:conclusion\]Discussion and Conclusions
============================================
The primary result of this paper is establishing the connection between the different types of Goldstone modes and different types of analogue spacetimes, as summarized in Table \[tab:Key\_results\]. This is done by revisiting the proof of the non-relativistic Goldstone theorem and allowing for the possibility of an inhomogeneous mean-field solution. We then find that the conventional Type-I Goldstone modes come equipped with an Einstein-Hilbert metric as appears in general relativity while Type-II Goldstone modes couple to a Newton-Cartan geometry. The geometry itself is determined by the spacetime dependence of symmetry-breaking mean-fieldinhomogeneous symmetry breaking ultimately produces the non-trivial spacetime metric. In this work we have restricted ourselves to the case where only the overall $U(1)$ symmetry is inhomogeneously broken. This corresponds to an overall condensate flow.
Another key result of this paper is establishing the connection between quantum phase transitions and changes in the nature of the spacetime. To elucidate this, we present a simple model where the analogue geometry can be tuned by a single parameter. This drives a quantum phase transition which accompanies the transition between the Einstein-Lorentz geometry and Newton-Cartan geometry. As the phase transition is approached, the Hawking radiation produced by an event horizon changes, as encapsulated in Fig. \[fig:totalflux\]. One key result is that the Newton-Cartan geometry exhibits no Hawking radiation, even though all fluid flows are supersonic (the group velocity of Goldstone modes vanishes at long wavelengths).
While Sec. \[sec:model\] is a minimal theoretical model, the experimental system that most readily realizes these geometries are spin-1 condensates. In this case, for the scattering lengths $a_0$ and $a_2$ (for $s$-wave collisions into the spin-0 and spin-2 channels respectively), there are two phases that break the spin SU(2) symmetry: $a_0>a_2$ gives a ferromagnetic phase with one Type-II magnon and $a_0<a_2$ gives a polar phase (antiferromagnetic interactions) with two Type-I magnons. Upon flow, these two phases naturally realize the two different spacetimes described here. In fact, ${}^7$Li, ${}^{41}$K, and ${}^{87}$Rb realize the ferromagnetic phase [@stamper-kurnSpinorBoseGases2013] with ${}^{87}$Rb specifically already being used for Hawking-like experiments with the phonon mode [@steinhauerObservationSelfamplifyingHawking2014; @*steinhauerObservationQuantumHawking2016]. Additionally, ${}^{23}$Na realizes the polar phase and critical spin superflow has been studied [@kimCriticalSpinSuperflow2017] (necessary for Hawking-like experiments). The magnon excitations in these systems can be probed by observing correlations in the spin-density, and the most basic proposal, would be to establish the vanishing Hawking radiation in the ferromagnetic phase. The progress in current spinor condensate experiments highlights that these more exotic analogue spacetimes may already be in reach.
Finally, by considering the response of the Goldstone modes to variations in the analogue geometries, we relate the analogue stress-energy-momentum tensor in relativistic geometries directly to their non-relativistic counterpart. This is summarized by the equations below, which shows how the metric tensor in both analogue spacetimes may be constructed from the underlying geometric objects of the Newton-Cartan geometry along with an additional field $C = C(x)$: $$\begin{split}
g_{\mu\nu} & = n_\mu n_\nu + h_{\mu\nu}, \quad \text{Non-relativistic},\\
\mathcal G_{\mu\nu} & \propto C^2 n_\mu n_\nu - h_{\mu\nu}, \quad \text{Relativistic}.
\end{split}$$ We also provide a direct connection between the energy and momentum currents of an analogue relativistic system and the more fundamental Newton-Cartan geometry which describes the lab-frame.
Within spinor Bose-Einstein condensates, there are other phenomena to include such as inhomogeneous broken non-Abelian symmetry (including textures like spiral magnetization, Bloch domain walls, and skyrmions) and synthetic gauge fields. The construction presented here also considers just the quadratic excitations, but these Goldstone modes realize more complicated nonlinear sigma models for which there is extra *intrinsic* geometry at play and would need to be incorporated into a full theory of these excitations. This new analogue also raises questions of the so-called back-reaction effects of quantum fields on the corresponding analogue spacetime. This has been studied in the relativistic case [@fischerDynamicalAspectsAnalogue2007; @keserAnalogueStochasticGravity2018], and the non-relativistic case leaves us with the tantalizing prospect of a system with a dynamical Newtonian gravity. Finally, while in this work we exclusive focused on the context of flowing spinor Bose-Einstein condensates, the phenomenon should be more general. An interesting future direction to pursue would be to try and extend these results to include more diverse platforms including electrons in solid-state systems, liquid Helium, superconductors, magnetic systems. The wide variety of systems which exhibit symmetry-breaking means there is a wide variety of systems which might exhibit this analogue spacetime and its consequences.
We would like to thank Gil Refael for crucial discussions which lead to this work. We also thank Andrey Gromov and Luca Delacrétaz for indispensable suggestions. This work was supported the U. S. Army Research Laboratory and the U. S. Army Research Office under contract number W911NF1810164, NSF DMR-1613029, and the Simons Foundation (J.B.C. and V.G.). J.H.W. and V.G. performed part of this work at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. J.B.C. and V.G. performed part of this work at the Kavli Institute for Theoretical Physics and thank KITP for hospitality and support. J.B.C. was supported in part by the Heising-Simons Foundation, the Simons Foundation, and National Science Foundation Grant No. NSF PHY-1748958. J.H.W. thanks the Air Force Office for Scientific Research for support.
Calculating the fluctuation Lagrangian {#app:fluctuations}
======================================
In this section, we put all of the algebra and Lagrangian manipulation that we left out of Section \[sec:spacetimes\].
Our starting point is Eq. upon substituting $\Psi = \Psi_0 + \delta\Psi$ where $\Psi_0$ solves the Euler-Lagrange equations Eq. and $\delta \Psi$ can be written in terms of broken generators and massive fields Eq. .
Most of the simplifying algebra comes from $g(\bm \sigma \Psi, \xi) = 0$ and integration-by-parts. To facilitate the integration by parts, all equalities are be understood to be *up to a full derivative*. Furthermore, by construction the linear terms cancel, so we keep second-order terms only, indicated by $\stackrel{\mathrm{fluc}}{=}$.
To deal with the term linear in derivatives, we use the object $$f \overleftrightarrow{\partial_t} g \equiv f (\partial_t g) - (\partial_t f) g,$$ and for simplicity we sometimes replace $\partial_t f$ with $\dot f$ for time derivatives. We further take advantage of the Einstein summation convention (sum over indices is implied) for simplicity. The first term we investigate is $$\begin{gathered}
\tfrac{i}{2}\Psi^\dagger\overleftrightarrow{\partial_t}\Psi \stackrel{\mathrm{fluc}}{=} -\tfrac{i}{2} \Psi_0^\dagger (\bm\sigma \overleftrightarrow{\partial_t} \bm\sigma) \Psi_0 \\+ i(-\Psi_0^\dagger \bm\sigma \dot \xi + \xi^\dagger \dot{\bm\sigma} \Psi_0 + \xi^\dagger \bm\sigma \dot\Psi_0) \\- \tfrac{i}2 \Psi_0^\dagger \bm\sigma^2\dot\Psi_0 + \tfrac{i}2 \dot\Psi_0^\dagger \bm\sigma^2\Psi_0.\end{gathered}$$ Performing integration-by-parts on the $\Psi_0^\dagger \bm\sigma \dot \xi$ term, we get $$\begin{gathered}
\tfrac{i}{2}\Psi^\dagger\overleftrightarrow{\partial_t}\Psi \stackrel{\mathrm{fluc}}{=} -\tfrac{i}{2} \Psi_0^\dagger (\bm\sigma \overleftrightarrow{\partial_t} \bm\sigma) \Psi_0 + i(\Psi_0^\dagger \dot{\bm\sigma} \xi + \xi^\dagger \dot{\bm\sigma} \Psi_0) \\
+ i\dot\Psi_0^\dagger (\tfrac12 \bm\sigma^2\Psi_0 + \bm\sigma \xi )
- i (\tfrac12\Psi_0^\dagger \bm\sigma^2 - \xi^\dagger \bm\sigma)\dot\Psi_0. \label{eq:wzmterm}\end{gathered}$$
The kinetic energy term takes the form $$\begin{gathered}
\partial_j \Psi^\dagger \partial_j \Psi \stackrel{\mathrm{fluc}}{=} -\tfrac12 \partial_j \Psi_0^\dagger \bm\sigma^2 \partial_j \Psi_0 - \partial_j \Psi_0^\dagger \bm \sigma \partial_j \bm \sigma \Psi_0 - \partial_j \Psi_0^\dagger \bm \sigma \partial_j \xi -\tfrac12 \partial_j \Psi_0^\dagger \bm\sigma^2 \partial_j \Psi_0 - \Psi_0^\dagger (\partial_j \bm \sigma) \bm\sigma \partial_j \Psi_0 + \partial_j \xi^\dagger \bm \sigma \partial_j \Psi_0 \\
- \Psi_0^\dagger \partial_j \bm \sigma \partial_j \bm \sigma \Psi_0 + \partial_j \xi^\dagger \partial_j \xi - \Psi_0^\dagger \partial_j \bm\sigma \partial_j \xi + \partial_j \xi^\dagger \partial_j\bm\sigma \Psi_0.\end{gathered}$$ We perform integration by parts on the two instances of $-\tfrac12 \partial_j \Psi_0^\dagger\bm\sigma^2 \partial_j\Psi_0$ above in opposite ways to obtain $$\begin{gathered}
\partial_j \Psi^\dagger \partial_j \Psi \stackrel{\mathrm{fluc}}{=}
\tfrac12 \nabla^2 \Psi_0^\dagger \bm\sigma^2 \Psi_0
- \tfrac12 \partial_j \Psi_0^\dagger \bm \sigma \partial_j \bm \sigma \Psi_0
+ \tfrac12 \partial_j \Psi_0^\dagger (\partial_j\bm\sigma) \bm \sigma \Psi_0
+\tfrac12 \Psi_0^\dagger \bm\sigma^2 \nabla^2 \Psi_0
- \tfrac12\Psi_0^\dagger (\partial_j \bm \sigma) \bm\sigma \partial_j \Psi_0
+ \tfrac12\Psi_0^\dagger \bm\sigma (\partial_j \bm\sigma) \partial_j\Psi_0\\
- \partial_j \Psi_0^\dagger \bm \sigma \partial_j \xi
+ \partial_j \xi^\dagger \bm \sigma \partial_j \Psi_0
- \Psi_0^\dagger \partial_j \bm \sigma \partial_j \bm \sigma \Psi_0
+ \partial_j \xi^\dagger \partial_j \xi
- \Psi_0^\dagger \partial_j \bm\sigma \partial_j \xi
+ \partial_j \xi^\dagger \partial_j\bm\sigma \Psi_0.\end{gathered}$$ If we further use integration by parts on $-\partial_j\Psi_0^\dagger \bm\sigma \partial_j\xi$ and $\partial_j\xi^\dagger \bm \sigma \partial_j \Psi_0$, we obtain (after some reordering) $$\begin{gathered}
\partial_j \Psi^\dagger \partial_j \Psi \stackrel{\mathrm{fluc}}{=}
- \Psi_0^\dagger \partial_j \bm \sigma \partial_j \bm \sigma \Psi_0
- \tfrac12 \partial_j \Psi_0^\dagger \bm \sigma \partial_j \bm \sigma \Psi_0
+ \tfrac12 \partial_j \Psi_0^\dagger (\partial_j\bm\sigma) \bm \sigma \Psi_0
- \tfrac12\Psi_0^\dagger (\partial_j \bm \sigma) \bm\sigma \partial_j \Psi_0
+ \tfrac12\Psi_0^\dagger \bm\sigma (\partial_j \bm\sigma) \partial_j\Psi_0\\
+ \partial_j \xi^\dagger \partial_j \xi
+ \partial_j \Psi_0^\dagger \partial_j\bm \sigma \xi
- \xi^\dagger \partial_j\bm\sigma \partial_j \Psi_0
- \Psi_0^\dagger \partial_j \bm\sigma \partial_j \xi
+ \partial_j \xi^\dagger \partial_j\bm\sigma \Psi_0 \\
+ \nabla^2 \Psi_0^\dagger (\tfrac12 \bm\sigma^2 \Psi_0 +\bm\sigma \xi)
+(\tfrac12 \Psi_0^\dagger \bm\sigma^2 - \bm\sigma\xi) \nabla^2 \Psi_0. \label{eq:kineticterm}\end{gathered}$$ We observe that, along with Eq. , the equation of motion cancels the last lines in Eqs. and with the first line of Eq. .
All together, we can combine these equations to get the full fluctuation Lagrangian $$\begin{gathered}
\mathcal L \stackrel{\mathrm{fluc}}{=}
-\tfrac{i}{2} \Psi_0^\dagger (\bm\sigma \overleftrightarrow{\partial_t} \bm\sigma) \Psi_0 + i(\Psi_0^\dagger \dot{\bm\sigma} \xi + \xi^\dagger \dot{\bm\sigma} \Psi_0)
- \tfrac12 \partial_j \Psi_0^\dagger \bm \sigma \partial_j \bm \sigma \Psi_0
+ \tfrac12 \partial_j \Psi_0^\dagger (\partial_j\bm\sigma) \bm \sigma \Psi_0
- \tfrac12\Psi_0^\dagger (\partial_j \bm \sigma) \bm\sigma \partial_j \Psi_0
+ \tfrac12\Psi_0^\dagger \bm\sigma (\partial_j \bm\sigma) \partial_j\Psi_0\\
+ \partial_j \xi^\dagger \partial_j \xi
+ \partial_j \Psi_0^\dagger \partial_j\bm \sigma \xi
- \xi^\dagger \partial_j\bm\sigma \partial_j \Psi_0
- \Psi_0^\dagger \partial_j \bm\sigma \partial_j \xi
+ \partial_j \xi^\dagger \partial_j\bm\sigma \Psi_0
- \Psi_0^\dagger \partial_j \bm \sigma \partial_j \bm \sigma \Psi_0
+ \partial_j \xi^\dagger \partial_j \xi \\
- \frac12 \xi^*_a \left.\frac{\partial^2 V}{\partial \Psi^\dagger_a \partial \Psi^\dagger_b}\right|_0 \xi^*_b - \xi^*_a \left.\frac{\partial^2 V}{\partial \Psi^\dagger_a \partial \Psi_b}\right|_0 \xi_b - \frac12 \xi_a \left.\frac{\partial^2 V}{\partial \Psi_a \partial \Psi_b}\right|_0 \xi_b .\end{gathered}$$ We can now expand our fluctuations in terms of their fields $\bm\sigma \Psi_0 = \theta_n \sigma_n \Psi_0$ and $\xi = \beta_n \xi_n$, and we obtain $$\begin{gathered}
\mathcal L \stackrel{\mathrm{fluc}}{=}
-\tfrac{i}{2}\Psi_0^\dagger[\sigma_m,\sigma_n]\Psi_0 \theta_m\partial_t \theta_n
+\tfrac1{4m} \theta_m \partial_j \theta_n(\partial_j \Psi_0^\dagger [\sigma_m,\sigma_n]\Psi_0 - \Psi_0^\dagger [\sigma_m,\sigma_n]\partial_j\Psi_0) \\
+i \beta_n \partial_t \theta_n (\Psi_0^\dagger \sigma_m \xi_n + \xi_n^\dagger \sigma_m \Psi_0)
+ \tfrac{1}{2m}\beta_m\partial_j\theta_n(\xi_m^\dagger \sigma_n\partial_j\Psi_0 - \partial_j \Psi^\dagger_0 \sigma_n \xi_m + \Psi_0^\dagger\sigma_n \partial_j\xi_m - \partial_j \xi^\dagger_m \sigma_n \Psi_0) \\
+\tfrac{1}{2m}\Psi_0^\dagger \sigma_n\sigma_m \Psi_0 \partial_j\theta_n\partial_j\theta_m \\
+\tfrac{i}2 \beta_m \partial_t\beta_n(\xi_m^\dagger \xi_m - \xi_n^\dagger\xi_m)
+\tfrac{i}2 \beta_n\beta_m (\xi_m^\dagger \partial_t \xi_n - \partial_t\xi_m^\dagger \xi_n)
+\beta_m\partial_j\beta_n(\xi_n^\dagger \partial_j\xi_m + \partial_j\xi_m^\dagger \xi_n) \\
-\tfrac1{2m} \xi_n^\dagger \xi_m \partial_j\beta_m\partial_j\beta_n
-\tfrac12\beta_n\beta_m\left[\xi_{n}^\dagger\left.\frac{\partial^2 V}{\partial\Psi^\dagger\partial\Psi^\dagger}\right|_0 \xi_{m}^* + \xi_n^T \left.\frac{\partial^2 V}{\partial\Psi\partial\Psi}\right|_0 \xi_m + 2 \xi_m^\dagger \left.\frac{\partial^2 V}{\partial\Psi^\dagger\partial\Psi}\right|_0\xi_n\right]. \label{eq:entire-fluctuation-Lagrangian}\end{gathered}$$ The first three lines of Eq. lead to the Lagrangian presented in the text Eq. while the last two lines represent the massive modes neglected in the main text.
One can then easily check that once the full Lagrangian in Eq. is derived that the massive modes conjugate to Goldstone modes no longer have the term that goes as $\beta_m\partial_\mu\beta_n$, only keeping the kinetic term and mass matrix (which we diagonalize to find the type-I basis states).
Bogoliubov Theory for Hawking Emission {#app:BogoliubovHawking}
======================================
As per Eq. , the magnon field (written in terms of the complexified spinor $\Phi_3(x) = (\zeta, \zeta^*)^T$) obeys the BdG equation $$\left[ i\tau_3 \hat{D}_t + \frac{1}{2m\rho}\nabla\cdot \rho\nabla - g_3\rho\left(\tau_0 + \tau_1\right) \right] \Phi_3(x) =0, \label{eq:appendix-bdg}$$ written in terms of the co-moving frame material derivative $\hat{D}_t = \partial_t + \mathbf{v}_s \cdot \nabla $.
Before proceeding, there are two properties of this equation that prove useful. First is the charge conjugation symmetry: if $\Upsilon$ solves Eq. , then so does $$\overline{\Upsilon} \equiv \tau_1 \Upsilon^*.$$ In particular, this is important since the Nambu spinor should obey the self-conjugate property that $\Phi_3 = (\zeta ,\zeta^*)^T = \overline{\Phi}_3$. Thus, it is important that this is respected by the equations of motion, which we see it is.
Furthermore, provided the density $\rho(x)$ is time independent, we can define the conserved pseudo-scalar product on the solution space $$(\Upsilon_1,\Upsilon_2) \equiv \int d^d r \,\rho(\mathbf{r}) \Upsilon_1^\dagger(\mathbf{r}) \tau_3 \Upsilon_2(\mathbf{r}). \label{eq:inner-product}$$ This scalar product has a number of useful features including that the charge conjugation operation changes the sign, so that $$(\overline{\Upsilon}_1,\overline{\Upsilon}_2) = - (\Upsilon_2,\Upsilon_1).$$ We use this pseudo-inner product to define a notion of norm for solutions. Because of the $\tau_3$, this is not the usual $L_2(\mathbb{R}^d)$ norm, and in fact is not a norm at all since it is not positive semi-definite. There are non-trivial negative norm states which we loosely refer to as “hole-like" states, in contrast to the “particle-like" solutions with positive norm. As remarked earlier, hole-like solutions can be related to particle-like solutions by charge conjugation since if $\Upsilon$ has negative norm we find $$(\Upsilon,\Upsilon ) < 0 \Rightarrow (\overline{\Upsilon},\overline{\Upsilon}) > 0 .$$
To proceed further, we utilize the (assumed) time-independence of the kernel to further separate the solution $\Upsilon(x) = \Upsilon(\mathbf{r},t) $ into energy eigenmodes $$\Upsilon(x) = \int \frac{d\omega}{2\pi} W_\omega(\mathbf{r})e^{-i\omega t},$$ where $W_\omega(\mathbf{r}) = [U_\omega(\mathbf{r}),V_\omega(\mathbf{r})]^T$ is a two-component spinor which obeys the eigenvalue problem $$\label{eqn:BdG-1D}
\left[\omega + i\mathbf{v}_s \cdot \nabla + \frac{1}{2m\rho}\nabla\cdot\rho\nabla \tau_3 - g_3\rho\left(\tau_3 + i\tau_2\right) \right]W_{\omega}(\mathbf{r})=0.$$ We refer to [@curtisEvanescentModesSteplike2019; @macherBlackholeRadiationBoseEinstein2009] for more details of solving this system. What is important for our discussion are the details of the dispersion relation, which are used to analyze the asymptotic scattering states at spatial infinity.
We now focus on the case of a one-dimensional homogeneous flow. In this case both the momentum $k$ and lab-frame frequency $\omega$ are good quantum numbers and obey the standard Bogoliubov dispersion relation (using that $mc^2 = g_3 \rho$) of $$\label{eqn:BdG-dispersion}
\omega = v_sk \pm \sqrt{c^2 k^2 + \left(\frac{k^2}{2m}\right)^2 } \equiv \omega_{\pm}(k),$$ where the last equality is used to define the lab frequency $\omega_{\pm}(k)$. At a particular frequency $\omega>0$, we may determine which scattering states are available by finding the real momenta $k$ which obey $\omega = \omega_{\pm}(k)$.
Considering a step-like variation in the flow, the flow profile is as given in Eq. . For $x<0$ and $x>0$ the solutions to the BdG equations are still plane-waves which obey the Bogoliubov dispersion relation, albeit with different parameters $\rho$ and $v$. These two dispersion relations are shown Figs. \[fig:SuperToSubHawking\] and \[fig:SuperToSuperHawking\] for fixed values of the condensate velocities $|v_l| > |v_r|$ and densities $\rho_l,\rho_r$.
Instead of the lab frame, we may measure frequency with respect to the frame co-moving with the fluid flow. This is implemented by Doppler shifting to the (positive) comoving frequency $$\Omega(k) \equiv \sqrt{c^2 k^2 + \tfrac{k^4}{4m^2}},$$ so that $\omega_\pm(k) = v k \pm \Omega(k)$ ($vk$ amounts to a Galilean boost).
For $|v|<c$ (right dispersion in Fig. \[fig:SuperToSubHawking\]), there are only two real-momenta at any positive frequency, which correspond to a right- and left-moving quasiparticle. For $|v|>c$ (left dispersion in Fig. \[fig:SuperToSubHawking\]) a new scattering channel opens whereby a wavepacket with negative free-fall frequency \[$\omega_-(k)$\] may have positive lab-frame frequency $\omega$.
We find the eigenfunctions for the step potential by employing matching equations at the step. These impose the continuity requirements $$\label{eqn:matching}
\begin{aligned}
&\left[W_\omega(x)\right]_{x=0^-}^{x=0^+} = 0 \\
&\left[\rho\partial_x W_\omega(x)\right]_{x=0^-}^{x=0^+} = 0. \\
\end{aligned}$$ Additionally, we choose them to satisfy $(W_\omega,\overline{W}_\omega) = 0$ and can be normalized such that $(W_\omega,W_\omega) > 0$ if $\omega = \omega_+(k)$ (positive comoving frequency) and $(W_\omega,W_\omega) < 0$ if $\omega = \omega_-(k)$ (negative comoving frequency).
Combining all of this, we can express the full solution in terms of positive-frequency components only via $$\begin{gathered}
\label{eqn:mode-expansion}
\Phi_3(x,t) = \int_0^{\infty} \frac{d\omega}{2\pi} \sum_\alpha \bigg[A(W_{\omega\alpha}) W_{\omega\alpha}(x)e^{-i\omega t} \\
+ A^*(W_{\omega\alpha}) \overline{W}_{\omega\alpha}e^{+i\omega t}\bigg],\end{gathered}$$ where the $A(W_{\omega,\alpha})$ are the Fourier coefficients of the expansion and $\alpha$ is a set of quantum numbers which are used to label the different degenerate modes at each energy $\omega > 0$. At this point, we can second quantize the system and promote $\Upsilon$ to an operator. In such a case, the operator equation looks like $$\begin{gathered}
\label{eqn:mode-expansion2}
\hat \Upsilon(x,t) = \int_0^{\infty} \frac{d\omega}{2\pi} \sum_\alpha \bigg[a(W_{\omega\alpha}) W_{\omega\alpha}(x)e^{-i\omega t} \\
+ a^\dagger(W_{\omega\alpha}) \overline{W}_{\omega\alpha}e^{+i\omega t}\bigg],\end{gathered}$$ where now $a(W_{\omega\alpha})$ are operators satisfying $$[a(W_{\omega\alpha}),a^\dagger(W_{\omega'\alpha'})] = (W_{\omega\alpha},W_{\omega',\alpha'}).$$ All $W_{\omega\alpha}$ are orthogonal with respect to this inner product, and so $a(W_{\omega\alpha})$ is either a creation *or* annihilation operator based on the sign of the norm.
The system may be exactly solved when the flow is homogeneous, in which case the momentum $k$ is also a good quantum number. Assuming a solution of the form $$W_{\omega}(x) = w_{k}e^{ikx}$$ produces the momentum space eigenvalue problem $$\label{eqn:BdG-momentum}
\left[\omega - vk - \frac{1}{2m}k^2 \tau_3 - g_3\rho\left(\tau_3 + i\tau_2\right) \right]w_{k}=0.$$ In principle, the momentum $k$ depends in the energy $\omega$, but we usually suppress this dependence for brevity.
To evaluate $(W_{\omega\alpha},W_{\omega'\alpha'})$, we establish a couple of facts. If we let $w_k = [u_k, v_k]^T$, then we have $$mc^2 v_k = \left( \pm \Omega(k) - \frac{k^2}{2m} - m c^2 \right) u_k,$$ and hence $$m^2 c^4 |v_k|^2 = \left\{m^2 c^4 \mp 2\Omega(k)\left[mc^2 + \frac{k^2}{2m} \mp \Omega(k) \right] \right\}|u_k|^2,$$ this relation between $|u_k|^2$ and $|v_k|^2$ allows us to evaluate
$$\begin{split}
(W_{\omega\alpha},W_{\omega'\alpha'}) & = \pm \Omega(k) \frac{2 \rho}{m^2 c^4} \left[ mc^2 + \frac{k^2}{2m} \mp \Omega(k) \right] |u_k|^2 \delta_{\alpha\alpha'}\delta[k_\alpha(\omega) - k_{\alpha'}(\omega')] \\
& = \pm \Omega(k) \frac{2 \rho |v_g|}{m^2 c^4} \left[ mc^2 + \frac{k^2}{2m} \mp \Omega(k) \right] |u_k|^2 \delta_{\alpha\alpha'}\delta(\omega - \omega').
\end{split}$$
The term in brackets $m c^2 + \frac{k^2}{2m} - \Omega(k) > 0$, so the *sign* of the normalization depends exclusively on whether we have positive ($+\Omega(k)$) or negative ($-\Omega(k)$) comoving frequency. The terms with negative comoving frequency (or negative norm) are represented by the blue curves in Figs. \[fig:SuperToSubHawking\] and \[fig:SuperToSuperHawking\].
We can now perform the Hawking calculation to determine the Bogoliubov transformation giving rise to excitation production. This is presented first in Fig. \[fig:SuperToSubHawking\], where we consider a wavepacket moving away from the horizon to $+\infty$ and frequency $\omega$, this is the Hawking mode. If we trace it back in time, it was related to a scattering process at the horizon itself, so in terms of three other positive frequency modes $$W_{\mathrm{H}} = \alpha_R W_{R,1} + \alpha_L W_{L,2} + \beta_L \overline{W}_{L,1},$$ where $ W_{\mathrm{H}} $ includes the far propagating right-moving mode along with the evanescent near horizon solution, $W_{R,1}$ is the left-moving mode on the right, and $W_{L,(1,2)}$ are the right-moving modes on the left (counted left-to-right in Fig. \[fig:SuperToSubHawking\]). This immediately gives us how to relate the creation operators of the out-vacuum to the in-vacuum $$a(W_\mathrm{H}) = \alpha_R a(W_{R,1}) + \alpha_L a(W_{L,2}) + \beta_L a^\dagger(W_{L,1}).$$ This implies that for $W_H$ at a particular frequency $\omega$, we can find the number of Hawking modes leaving the horizon by considering the expectation value $$\braket{0_{\mathrm{in}}| a(W_\mathrm{H})^\dagger a(W_\mathrm{H}) | 0_\mathrm{in}} = |\beta_L|^2 (W_{L,1}, W_{L,1}).$$ With the proper normalization and putting back in the dependence on frequency, the number of particles leaving the horizon at frequency $\omega$ is $$N(\omega) = |\beta_L(\omega)|^2 \frac{(W_{L,1}(\omega),W_{L,1}(\omega))}{(W_{H}(\omega),W_{H}(\omega))}. \label{eq:Nomega1}$$
This same analysis can be done for the supersonic-to-supersonic case presented in Fig. \[fig:SuperToSuperHawking\]. For lack of a better term, we call the region where there are multiple positive and negative norm channels the “super-Hawking” region. In this case, we have two modes in the Hawking process that need to be backwards scattered: one positive norm and the other negative norm. The result of the scattering process is $$\begin{split}
W_{\mathrm H} & = \beta_R \overline W_{R,1} + \alpha_R W_{R, 2} + \beta_L \overline W_{L,1} + \alpha_L W_{L,2}, \\
\overline{W}_{\mathrm H'} & = \alpha_R' \overline W_{R,1} + \beta_R' W_{R, 2} + \alpha_L' \overline{W}_{L,1} + \beta_L' W_{L,2}.
\end{split}$$ These equations can be similarly related to a Bogoliubov transformation, and we can find the number of Hawking particles leaving the horizon at frequency $\omega$ by considering
$$N(\omega) = |\beta_L(\omega)|^2 \tfrac{(W_{L,1}(\omega),W_{L,1}(\omega))}{(W_{H}(\omega),W_{H}(\omega))} + |\beta_R(\omega)|^2 \tfrac{(W_{R,1}(\omega),W_{R,1}(\omega))}{(W_{H}(\omega),W_{H}(\omega))}
+ |\beta_L'(\omega)|^2 \tfrac{(W_{L,2}(\omega),W_{L,2}(\omega))}{(W_{H'}(\omega),W_{H'}(\omega))} + |\beta_R'(\omega)|^2 \tfrac{(W_{R,2}(\omega),W_{R,2}(\omega))}{(W_{H'}(\omega),W_{H'}(\omega))}. \label{eq:Nomega2}$$
Despite there being more terms, there is generally less of a Hawking flux due to a decoupling of the negative and positive norm channels as we can see in Fig. \[fig:totalflux\].
[^1]: This is derived more directly using $g^{-1}$ defined by $g^{\mu\nu}$. If one locally takes $n_\mu = (n_0, \mathbf 0)$, then $g^{00} = (v^0)^2 \equiv A^{00}$, $g^{0i} = g^{i0} = v^0 v^i \equiv B^{0i}$ and $g^{ij} = v^i v^j + h^{ij} \equiv D^{ij}$. One can take the Schur complement of this inverse metric $g^{-1}/A$ to compute the determinant; then $1/g = \det(g^{-1}) = \det(A) \det(D - B^T A^{-1} B) = 1/(n_0^2 h)$.
|
---
abstract: 'The Projection Congruent Subset (PCS) is new method for finding multivariate outliers. PCS returns an outlyingness index which can be used to construct affine equivariant estimates of multivariate location and scatter. In this note, we derive the finite sample breakdown point of these estimators.'
address:
- '*Corresponding author*. Afdeling Statistiek, Celestijnenlaan 200b - bus 2400, 3001 Leuven. Tel +32 16 37 23 40.'
- 'Faculty of Business and Economics, ORSTAT, KU Leuven, Belgium.'
- 'Afdeling Statistiek, Celestijnenlaan 200b - bus 2400, 3001 Leuven.'
author:
- Eric Schmitt
- Viktoria Öllerer
- Kaveh Vakili
title: The Finite Sample Breakdown Point of PCS
---
breakdown point ,robust estimation ,multivariate statistics.
Introduction
============
Outliers are observations that depart from the pattern of the majority of the data. Identifying outliers is a major concern in data analysis because a few outliers, if left unchecked, can exert a disproportionate pull on the fitted parameters of any statistical model, preventing the analyst from uncovering the main structure in the data.
To measure the robustness of an estimator to the presence of outliers in the data, [@ppcs:D82] introduced the notion of finite sample breakdown point. Given a sample and an estimator, this is the smallest number of observations that need to be replaced by outliers to cause the fit to be arbitrarily far from the values it would have had on the original sample. Remarkably, the finite sample breakdown point of an estimator can be derived without recourse to concepts of chance or randomness using geometrical features of a sample alone [@ppcs:D82]. Recently, [@hcs:VS13] introduced the Projection Congruent Subset (PCS) method. PCS computes an outlyingness index, as well as estimates of location and scatter derived from it. The objective of this paper is to establish the finite sample breakdown of these estimators and show that they are maximal.
Formally, we begin from the situation whereby the data matrix $\pmb X$, is a collection of $n$ so called $genuine$ observations drawn from a $p$-variate model $F$ with $p>1$. However, we do not observe $\pmb X$ but an $n \times p$ (potentially) corrupted data set $\pmb X^{\varepsilon}$ that consists of $g<n$ observations from $\pmb X$ and $c=n-g$ arbitrary values, with $\varepsilon={\left.c\middle/n\right.}$, denoting the (unknown) rate of contamination.
Historically, the goal of many robust estimators has been to achieve high breakdown while obtaining reasonable efficiency. PCS belongs to a small group of robust estimators that have been designed to also have low bias (see [@ppcs:MA92], [@ppcs:AY02] and [@ppcs:AY10]). In the context of robust estimation, a low bias estimator reliably finds a fit close to the one it would have found without the outliers, when $c\leqslant n-h$ with $h=\lceil(n+p+1)/2\rceil$. To the best of our knowledge, PCS is the first member of this group of estimators to be supported by a fast and affine equivariant algorithm (FastPCS) enabling its use by practitioners.
The rest of this paper unfolds as follows. In Section \[s2\], we detail the PCS estimator. In Section \[s3\], we formally detail the concept of finite sample breakdown point of an estimator and establish the notational conventions we will use throughout. Finally, in Section \[s4\], we prove the finite sample breakdown point of PCS.
The PCS criterion {#s2}
=================
Consider a potentially contaminated data set $\pmb X$ of $n$ vectors $\pmb x_i\in\mathbb{R}^p$, with $n>p+1>2$. Given all $M=\binom{n}{h}$ possible $h$-subsets $\{H^m\}_{m=1}^M$, PCS looks for the one that is most [*congruent*]{} along many univariate projections. Formally, given an $h$-subset $H^m$, we denote $B(H^m)$ the set of all vectors normal to hyperplanes spanning a $p$-subset of $H^m$. More precisely, all directions $\pmb a \in B(H^m)$ define hyperplanes $\{\pmb x\in\mathbb{R}^p: \pmb x'\pmb a=1\}$ that contain $p$ observations of $H^m$. For $\pmb a \in B(H^m)$ and $\pmb x_i\in\pmb X$, we can compute the squared orthogonal distance, $d_i^2$, of $\pmb x_i$ to the hyperplane defined by $\pmb a$ as $$\label{pcs:crit0}
d_i^2(\pmb a) =\frac{(\pmb a'\pmb x_i-1)^2}{||\pmb a||^2}\;.$$The set of the $h$ observations with smallest $d_{i}^2(\pmb a)$ is then defined as $$H^{\pmb a}=\{i:d_{i}^2(\pmb a)\leqslant d_{(h)}^2(\pmb a)\},$$ where $d_{(h)}$ denotes the $h$th-order statistic of a vector $\pmb d$.
We begin by considering the case in which $d_{(h)}^2(\pmb a)>0$. For a given subset $H^m$ and direction $\pmb a$ we define the [*incongruence index*]{} of $H^m$ along $\pmb a$ as $$\label{pcs:crit1}
I(H^m,\pmb a):=
\log\left(\frac{\displaystyle\operatorname*{ave}_{i\in H^m}d^2_i(\pmb a)}{\displaystyle\operatorname*{ave}_{i\in H^{\pmb a}}d_i^2(\pmb a)}\right)\;$$with the conventions that $\log(0/0):=0$. This index is always positive and will be smaller the more members of $H^m$ correspond with, or are similar to, the members of $H^{\pmb a}$. To remove the dependency of Equation on $\pmb a$, we measure the incongruence of $H^m$ by considering the average over many directions $\pmb a\in B(H^m)$ as $$\label{mpcs}
I(H^m):=\operatorname*{ave}_{\pmb a\in B(H^m)} I(H^m,\pmb a).\;$$ The optimal $h$-subset, $H^*$, is the one satisfying the PCS criterion: $$H^*=\underset{\{H^m\}_{m=1}^M}{\operatorname*{\arg\!\min}}\;I(H^m).\nonumber$$ Then, the [*PCS estimators of location and scatter*]{} are the sample mean and covariance of the observations with indexes in $H^*$: $$\left(\pmb t^*(\pmb X),\pmb S^*(\pmb X)\right)=\left(\operatorname*{ave}_{i\in H^*}\pmb x_i,\operatorname*{cov}_{i\in H^*}\pmb x_i\right). \nonumber$$
Finally, we have to account for the special case where $d_{(h)}^2(\pmb a)=0$. In this case, we enlarge $H^*$ to be the subset of all observations lying on $\pmb a$. More precisely, if $\exists\;\pmb a^*\in B(H^*):|\{i:d_i^2(\pmb a^*)=0\}|\geqslant h$, then $H^*=\{i:d_i^2(\pmb a^*)=0\}$.\
Illustrative Example
--------------------
To give additional insight into PCS and the characterization of a cloud of point in terms of congruence, we provide the following example. Figure \[fig:dataPlot\] depicts a data set $\pmb X^\varepsilon$ of 100 observations, 30 of which come from a cluster of outliers on the right. For this data set, we draw two $h$-subsets of 52 observations each.
![Bivariate data example. The members of $H^1$ ($H^2$) are depicted as dark blue diamonds (light orange circles).[]{data-label="fig:dataPlot"}](dataPlot2.pdf){width="90.00000%"}
Subset $H^1$ (dark blue diamonds) contains only genuine observations, while subset $H^2$ (light orange circles) contains 27 outliers and 25 genuine observations. Finally, the 17 observations belonging to neither $h$-subset are depicted as black triangles. For illustration’s sake, we selected the members of $H^2$ so that their covariance has smaller determinant than $any$ $h$-subsets formed of genuine observations. Consequently, robust methods based on a characterization of $h$-subsets in terms of density alone will always prefer the contaminated subset $H^2$ over any uncontaminated $h$-subset (and in particular $H^1$).
The outlyigness index computed by PCS differs from that of other robust estimators in two important ways. First, in PCS, the data is projected onto directions given by $p$ points drawn from the members of a given subset, $H^m$, rather than indiscriminately from the entire data set. This choice is motivated by the fact that when $\varepsilon$ and/or $p$ are high, the vast majority of random $p$-subsets of $\{1,\dots,n\}$ will be contaminated. If the outliers are concentrated, this yields directions almost parallel to each other. In contrast, for an uncontaminated $H^m$, our sampling strategy always ensures a wider spread of directions and this yields better results.
The second feature of PCS is that the congruence index used to characterize an $h$-subset depends on all the data points in the sample. We will illustrate this by considering all $\binom{52}{2}=1326$ members of $B(H^1)$. For each, we compute the corresponding value of $I(H^1, \pmb a)$. Then, we sort these and plot them in Figure \[fig:Icompare\]. We do the same for $H^2$. We note in passing that $I(H^1)<I(H^2)$.
![Sorted values of $I(H^m,\pmb a)$ for $H^1$ and $H^2$, shown as dark blue (light orange) lines.[]{data-label="fig:Icompare"}](Icompare2.pdf){width="90.00000%"}
Consider now in particular the values of the $I$-index corresponding to $H^1$ and starting at around 1050 on the horizontal axis of Figure \[fig:Icompare\]. These higher values of $I(H^1,\pmb a)$ correspond to members of $B(H^1)$ that are aligned with the vertical axis (i.e. they correspond to horizontal hyperplanes), and are much larger than the remaining values of $I(H^1,\pmb a)$. This is because, for the data configuration shown in Figure \[fig:dataPlot\], the outliers do not stand out from the good data in terms of their orthogonal distances to hyperplanes defined by the vertical directions. As a result, this causes many outliers to enter the sets $H^{\pmb a}$ and this deflates the values of the $\displaystyle\operatorname*{ave}_{i\in H^{\pmb a}}d_i^2(\pmb a)$ corresponding to these directions. Since the set $H^1$ is fixed, there is no corresponding effect on $\displaystyle\operatorname*{ave}_{i\in H^1}d^2_i(\pmb a)$ so that the outliers will influence the values of $I(H^1,\pmb a)$ for some directions $\pmb a$, even though $H^1$ itself is uncontaminated. This apparent weakness is an inherent feature of PCS. In the remainder of this note, we prove the following counter-intuitive fact: outliers influence the value of $I(H^m)$, even when $H^m$ is free of outliers, yet, so long as there are fewer than $n-h$ of them, their influence on the PCS fit will always remain bounded. In other words, breakdown only occurs if $c>n-h$ (see Section \[s4\]).
Finite sample breakdown point {#s3}
=============================
To lighten notation and without loss of generality, we arrange the observed data matrix $\pmb X^{\varepsilon}=((\pmb X^g)', (\pmb X^{c})')'$ with rows $\{\pmb x_i^{\varepsilon}\}_{i=1}^n$ so that the $\varepsilon\%$ of contaminated observations $\pmb X^c$ are in the last $c$ rows and the uncontaminated observations $\pmb X^g$ in the first $g$ rows. Then, $\mathcal{X}^{\varepsilon}$ will refer to the set of all corrupted data sets $\pmb X^{\varepsilon}$ and $\mathcal{H}$ is the set of all $h$-subsets of $\{1,\ldots,n\}$, $\mathcal{H}^c=\{H\in\mathcal{H}:H\cap \{g+1,\ldots,n\}\neq\varnothing\}$ the set of all $h$-subsets of $\{1,\ldots,n\}$ with at least one contaminated observation, and $\mathcal{H}^g=\{H\in\mathcal{H}:H\cap \{g+1,\ldots,n\}=\varnothing\}$ the set of all uncontaminated $h$-subsets of $\{1,\ldots,n\}$.
The following assumptions (as per, for example [@ppcs:T94]) all pertain to the original, uncontaminated, data set $\pmb X$. In the first part of this note, we will consider the case whereby the point cloud formed by $\pmb X^g=\{\pmb x_i^g\}_{i=1}^g$ lies in *general position* in $\mathbb{R}^p$. The following definition of *general position* is adapted from [@mcs:RL87]:
<span style="font-variant:small-caps;">Definition</span> 1: *General position in $\mathbb{R}^p$*. $\pmb X$ is in general position in $\mathbb{R}^p$ if no more than $p$-points of $\pmb X$ lie in any $(p-1)$-dimensional affine subspace. For $p$-dimensional data, this means that there are no more than $p$ points of $\pmb X$ on any hyperplane, so that any $p+1$ points of $\pmb X$ always determine a $p$-simplex with non-zero determinant.
Throughout, we will also assume that $$\begin{aligned}
\underset{\pmb a\in\mathbb{R}^p_{\neq 0}}{\sup}\max\left(\frac{\pmb X'\pmb a}{||\pmb a||}\right)^2\text{ is bounded}\end{aligned}$$ and that the genuine observations contain no duplicates: $$\begin{aligned}
||\pmb x_i-\pmb x_j||>0\forall\;1\leqslant i<j\leqslant n. \end{aligned}$$ For any $h$-subset $H^m\in\mathcal{H}$ and $\pmb X^{\varepsilon}$, we will denote the sample mean and covariance of the observations with indexes in $H^m$ as $$\left(\pmb t^m(\pmb X^{\varepsilon}),\pmb S^m(\pmb X^{\varepsilon})\right)=\left(\operatorname*{ave}_{i\in H^m}\pmb x_i^{\varepsilon},\operatorname*{cov}_{i\in H^m}\pmb x_i^{\varepsilon}\right). \nonumber$$ Then, given $\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}$, and an affine equivariant estimator of location $\pmb t$, we define the bias of $\pmb t$ at $\pmb X$ as $$\begin{aligned}
\text{bias}(\pmb t,\pmb X,\varepsilon)=\underset{\pmb X^\varepsilon\in\mathcal{X}^{\varepsilon}}{\sup}\;||\pmb t(\pmb X^{\varepsilon}) - \pmb t(\pmb X)||. \nonumber\end{aligned}$$ Furthermore, given $\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}$, genuine data $\pmb X$ and an affine equivariant estimator of scatter $\pmb S$ with $\pmb S(\pmb X)$ positive definite (denoted from now on by $\pmb S(\pmb X)\succ 0$), we define the bias of $\pmb S$ at $\pmb X$ as $$\begin{aligned}
\text{bias}(\pmb S,\pmb X,\varepsilon)=\underset{\pmb X^\varepsilon\in\mathcal{X}^{\varepsilon}}{\sup}\;
\frac{\lambda_1(\pmb Q^{\varepsilon})}{\lambda_p(\pmb Q^{\varepsilon})},\nonumber\end{aligned}$$ where $\pmb Q^{\varepsilon}=(\pmb S(\pmb X)^{-1/2}\pmb S(\pmb X^{\varepsilon})\pmb S(\pmb X)^{-1/2})\succ0$ and $\lambda_1(\pmb Q^{\varepsilon})$ ($\lambda_p(\pmb Q^{\varepsilon})$) denotes the largest (smallest) eigenvalue of a matrix $\pmb Q^{\varepsilon}$. Since PCS is affine equivariant (see Appendix 1), w.l.o.g., we can set $\pmb t(\pmb X) = \pmb 0$ so that the expression of bias reduces to $$\begin{aligned}
\text{bias}(\pmb t,\pmb X,\varepsilon)=\underset{\pmb X^\varepsilon\in\mathcal{X}^{\varepsilon}}{\sup}\;||\pmb t(\pmb X^{\varepsilon})||. \nonumber\end{aligned}$$ Furthermore, if that data is in general position and $\pmb S(\pmb X)$ is affine equivariant then we can w.l.o.g. set $\pmb S(\pmb X)=\pmb I_p$ ($\pmb I_p$ is the rank $p$ identity matrix) so that the expression of the bias reduces to $$\begin{aligned}
\text{bias}(\pmb S,\pmb X,\varepsilon)= \underset{\pmb X^\varepsilon\in\mathcal{X}^{\varepsilon}}{\sup}\;\frac{\lambda_1(\pmb S(\pmb X^{\varepsilon}))}{\lambda_p(\pmb S(\pmb X^{\varepsilon}))}. \nonumber\end{aligned}$$ The finite sample breakdown points $\varepsilon^*_n$ [@pcs:MMY06] of $\pmb t$ and $\pmb S$ are then defined as $$\begin{aligned}
\varepsilon^*_n(\pmb t,\pmb X)&=&\underset{1\leqslant c\leqslant n}{\min}\left\{\varepsilon=\frac{c}{n}:\text{bias}(\pmb t,\pmb X,\varepsilon)=\infty\right\} \\
\varepsilon^*_n(\pmb S,\pmb X)&=&\underset{1\leqslant c\leqslant n}{\min}\left\{\varepsilon=\frac{c}{n}: \text{bias}(\pmb S,\pmb X,\varepsilon)=\infty\right\}.
\label{eq:breakdownexp}\end{aligned}$$ Finally, for point clouds $\pmb X$ lying in general position in $\mathbb{R}^p$, [@ppcs:D87] gives a strict upper bound for the finite sample breakdown point for any affine equivariant location and scatter statistics $(\pmb t,\pmb S)$, namely: $$\begin{aligned}
\label{eq:maxBP}
\varepsilon^*_n(\pmb t,\pmb X)&\leqslant&(n-h+1)/n\notag\\
\varepsilon^*_n(\pmb S,\pmb X)&\leqslant&(n-h+1)/n
\end{aligned}$$
Finite sample breakdown point of PCS {#s4}
====================================
To establish the breakdown point of $\pmb S^*$, we first introduce two lemmas describing properties of the $I$-index. Both deal with the case where $\pmb X$ lies in general position in $\mathbb{R}^p$. Then, we discuss the case where $\pmb X$ does not lie in general position.
In the first lemma, we show that the incongruence index of a clean $h$-subset is bounded.
Let $c\leqslant n-h$ and $\pmb X$ lies in general position in $\mathbb{R}^p$. Then $$\begin{aligned}
\label{pcs:l1_state}
\underset{\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}}{\sup}\;\underset{H^g\in\mathcal{H}^g}{\max}\;\underset{\pmb a\in B(H^g)}{\max}I(H^g,\pmb a)\leqslant k(\pmb X)\end{aligned}$$ for any fixed, positive scalar $k(\pmb{X})$ not depending on the outliers.
Consider first the numerator of $I(H^g,\pmb a)$. For a fixed $H^g\in\mathcal{H}^g$, we can find for each $\pmb a\in B(H^g)$, the $p$ observations of $\pmb X^g$ that lie furthest away from the hyperplane defined by $\pmb a$. The average of their distances (as given by Equation ) to the hyperplane $\pmb a$ is finite and constitutes an upper bound on the average distance of any $p$ observations of $\pmb X^g$ to the hyperplane $\pmb a$. As we have at most $\binom{h}{p}$ different directions $\pmb a\in B(H^g)$ and only $\binom{n-c}{h}$ uncontaminated subsets $H^g\in\mathcal{H}^g$, the upper bound of the average distances stays finite $$\begin{aligned}
\underset{H^g\in\mathcal{H}^g}{\max}\;\underset{\pmb a\in B(H^g)}{\max}\operatorname*{ave}_{i\in H^g}\;d_i^2(\pmb a)\leqslant U(\pmb{X})\nonumber\end{aligned}$$ for any positive, fixed, finite scalar $U(\pmb{X})$ not depending on the outliers. Since the contaminated observations have no influence on the distance $d_i^2(\pmb a)$ with $i \in H^g(\pmb a)$ for $\pmb a\in B(H^g)$, we can say that $$\begin{aligned}
\label{pcs:l1_aux1}
\underset{\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}}{\sup}\;\underset{H^g\in\mathcal{H}^g}{\max}\;\underset{\pmb a\in B(H^g)}{\max}\operatorname*{ave}_{i\in H^g}\;d_i^2(\pmb a)\leqslant U(\pmb{X}).\end{aligned}$$
Consider now the denominator of $I(H^g,\pmb a)$. For any $H^g\in\mathcal{H}^g$ and $\pmb a\in B(H^g)$, let $H^{\pmb a}$ denote the subset that consists of the indexes of the $h$ observations of the observed data matrix $\pmb X^\varepsilon$ that lie closest to the hyperplane spanned by $\pmb a$. As $c\leqslant n-h$ and $h=\lceil(n+p+1)/2\rceil$, $H^{\pmb a}$ contains at least $p+1$ uncontaminated observations. In total, when $H^g\in\mathcal{H}^g$ is not fixed, there are at most $\binom{n-c}{p}$ different directions $\pmb a$ defined by a $H^g\in\mathcal{H}^g$. For any $\pmb a$, the smallest value of $\operatorname*{ave}_{i\in H^{\pmb a}} d_i^2(\pmb a)$ is attained if the contaminated observations of $H^{\pmb a}$ achieve $d_i^2(\pmb a)=0$. As the uncontaminated observations lie in general position, we know that the $p+1$ uncontaminated observations in $H^{\pmb a}$ cannot lie within the same $p$-dimensional subspace, i.e. $$\begin{aligned}
\exists i\in H^{\pmb a}: d_i^2(\pmb a)>0. \nonumber\end{aligned}$$ As the number of uncontaminated observations is fixed, we have that $$\begin{aligned}
\label{pcs:l1_aux2}
\underset{H^g\in\mathcal{H}^g}{\min}\;\underset{\pmb a\in B(H^g)}{\min}\operatorname*{ave}_{i\in H^{\pmb a}}\;d_i^2(\pmb a)\geqslant l(\pmb{X})>0\end{aligned}$$ for any fixed positive scalar $l(\pmb X)$ not depending on the outliers. This inequality holds even if the outliers have the smallest average distance that is possible (i.e. when $\pmb a:d_i^2(\pmb a)=0$ for the contaminated observations). Thus, Inequality (\[pcs:l1\_aux2\]) holds for any $\varepsilon$-contaminated data set $\pmb X^{\varepsilon}$ yielding $$\begin{aligned}
\label{pcs:l1_aux3}\small
\underset{\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}}{\inf}\;\underset{H^g\in\mathcal{H}^g}{\min}\;\underset{\pmb a\in B(H^g)}{\min}\operatorname*{ave}_{i\in H^{\pmb a}}\;d_i^2(\pmb a)\geqslant l(\pmb{X})>0.\end{aligned}$$ Using Equation and the Inequalities and , we get .
The second lemma shows the unboundedness of the incongruence index of contaminated subsets.
Let $c\leqslant n-h$ and assume that $\pmb X$ lies in general position in $\mathbb{R}^p$. Take a fixed h-subset $H^c\in \mathcal{H}^c$. Then $$\begin{aligned}
\forall U_1 >0: \; \exists \pmb X^\varepsilon \in \mathcal{X}^\varepsilon: \;I(H^c,\pmb a)>U_1
\label{eq:l2_aux1}\end{aligned}$$ for at least one $\pmb a\in B(H^c)$. In other words, for a given set of indexes $H^c$, there exists a data set $\pmb X^\varepsilon$ with contaminated observations with indexes in $H^c$ such that $I(H^c,\pmb a)$ is unbounded.
Consider first the numerator of $I(H^c,\pmb a)$. For a fixed $H^c\in\mathcal{H}^c$, denote $G^+=\{G\cap H^c\}$. Since $c\leqslant n-h$, as already mentioned in Lemma 1 above, any $h$-subset contains at least $p+1$ uncontaminated observations, i.e. $|G^+|\geqslant p+1$. Let $B^+(H^c)\subseteq B(H^c)$ be the set of all directions defining a hyperplane spanned by a $p$-subset of $G^+$. $|G^+|\geqslant p+1$ yields $|B^+(H^c)|\geqslant p+1$. As the uncontaminated observations $G\supseteq G^+$ lie in general position, the members of $B^+(H^c)$ are, by definition, linearly independent. As a result, the outliers can belong to (at most) the subspace spanned by $p$ uncontaminated observations. Hence, for every $U_2>0$, there exists at least one member $\pmb a^c_+$ of $B^+(H^c)$, at least one $i\in H^c$ and at least one $\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}$ such that $$\begin{aligned}
\;d_i^2(\pmb a^c_+)>U_2.
\label{eq:l2_aux2}\end{aligned}$$
Consider now the denominator of $I(H^c,\pmb a)$: Since the members of $B^+(H^c)$ all pass through members of $\pmb X^g$ only, we have that $$\begin{aligned}
d_{(h)}^2(\pmb a^c_+)\leqslant U_3\leqslant\underset{i\leqslant h}{\max}\;d_i^2(\pmb a^c_+).
\label{eq:l2_aux3}\end{aligned}$$ Using Equation , and Inequalities and , we get .
With Lemmas 1 and 2, we are now able to derive the finite sample breakdown point of the PCS of $\pmb S^*$ and $\pmb t^*$.
For $n>p+1>2$ and $\pmb X$ in general position, the finite sample breakdown point of $\pmb S^*$ is $$\varepsilon_n^*(\pmb S,\pmb X)=\frac{n-h+1}{n}. \nonumber
\label{eq:PCSbreakdown}$$
Consider first the situation where $c\leqslant n-h$. Then any $h$-subset $H^m$ of $\pmb X^{\varepsilon}$ contains at least $p+1$ members of $G$. In particular, for the chosen $h$-subset $H^*$, denote $G^*=\{H^*\cap \{1,\ldots,g\}\}$ with $|G^*|\geqslant p+1$. The members of $G^*$ are in general position so that $\operatorname*{ave}_{i\in G^*}(\pmb x_i^{\varepsilon}-\pmb t)(\pmb x_i^{\varepsilon}-\pmb t)'\succ0$ for any $\pmb t\in\mathbb{R}^p$. But $\pmb S^*(\pmb X^{\varepsilon})=\operatorname*{ave}_{i\in H^*}(\pmb x_i^{\varepsilon}-\pmb t^*)(\pmb x_i^{\varepsilon}-\pmb t^*)'$ and $\operatorname*{ave}_{i\in H^*\setminus G^*}(\pmb x_i^{\varepsilon}-\pmb t^*)(\pmb x_i^{\varepsilon}-\pmb t^*)'\succeq 0$ so that $\pmb S^*(\pmb X^{\varepsilon})\succ 0$ [@ppcs:Seber 10.58] which implies that $\underset{\pmb X^\varepsilon \in \mathcal{X^\varepsilon}}{\sup} \lambda_p(\pmb S^*(\pmb X^{\varepsilon}))>0$. Thus for breakdown to occur, the numerator of Equation , $\lambda_1(\pmb S^*(\pmb X^\varepsilon))$, must become unbounded. Now, suppose that $\pmb S^*(\pmb X^\varepsilon)$ breaks down. This means that for any $U_4>\max_{i=1}^n||\pmb x_i||^2$, $$\underset{\pmb X^\varepsilon \in \mathcal{X^\varepsilon}}{\sup}\lambda_1(\pmb S^*(\pmb X^\varepsilon)) > U_4.
\label{eq:contrabase}$$ We will show that this leads to a contradiction. In Appendix 2 we show that $$\lambda_1(\pmb S^*(\pmb X^\varepsilon)) \leqslant \underset{i \in H^*}{\max}||\pmb x^\varepsilon_i||^2.
\label{eq:append1}$$ By Equations and , it follows that $\underset{\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}}{\sup} \underset{i \in H^*}{\max}||\pmb x^\varepsilon_i||^2 >U_4$. Then, by Lemma 2 we have that $\underset{\pmb X^{\varepsilon}\in\mathcal{X}^{\varepsilon}}{\sup}I(H^*) > U_1/K$ with $K=\binom{h}{p}$, the number of all directions $\pmb a\in B(H^*)$. In particular, this is also true for $U_1 > k(\pmb X)$, and by Lemma 1, $k(\pmb X) \geqslant I(H^g)$, implying that $I(H^*)>I(H^g)$ $\forall H^g \in \mathcal{H}^g$, which is a contradiction to the definition of $H^*$. Since PCS is affine and shift equivariant, when $c > n-h$, we have by Equation that $\pmb S^{*}(\pmb X^{\varepsilon})$ breaks down.
Equation and Theorem 1 show that the breakdown point of $\pmb S^*$ is maximal. The following theorem shows that the breakdown point of $\pmb t^*$ is also maximal.
For $n>p+1>2$ and $\pmb X$ in general position, the finite sample breakdown point of $\pmb t^*$ is $$\varepsilon_n^*(\pmb t,\pmb X)=\frac{n-h+1}{n}. \nonumber
\label{eq:meanbreakdown}$$
Consider first the situation where $c\leqslant n-h$. In Theorem 1, we showed that under this condition $\underset{i \in H^*}{\max}||\pmb x^\varepsilon_i|| < \infty$. Denote $\pmb x^\varepsilon_u$, where $u=\operatorname*{\arg\!\max}_{i\in H^\ast} \|\pmb x_i^\varepsilon \|$. Then, we have that $\pmb t^*(\pmb X^\varepsilon)$ does not break down since, by homogeneity of the norm and the triangle inequality, $$||\pmb t^*(\pmb X^\varepsilon)|| \leqslant \frac{1}{h }\sum_{i\in H^\ast}\|\pmb x^\varepsilon_i \| \leqslant\frac{1}{h}\sum_{i=1}^h\|\pmb x^\varepsilon_u \| =\|\pmb x^\varepsilon_u\| <\infty. \nonumber$$ For the case of $c > n-h$, Equation and affine equivariance imply that $\pmb t^*(\pmb X^\varepsilon)$ breaks down.
We now relax the assumption that the members of $\pmb X^g$ lie in general position in $\mathbb{R}^{p}$ and substitute it by the weaker condition that they all lie in general position on a common subspace in $\mathbb{R}^q$ for some $q<p$. Then PCS has the so-called exact fit property. Recall that $\pmb a$ are hyperplanes defined by $p$ points drawn from an $h$-subset $H\in\mathcal{H}$. If there are at least $h$ points lying on a subspace, then there exists an $h$-subset of points from this subspace. Let $\tilde{H}$ be this subset. Then, for any $\pmb a^+\in B(\tilde{H})$, both the numerator and denominator of Equation equal zero and so $I(\tilde{H})=0$. Thus, we have without loss of generality that $H^*=\{i:d_{i}^2(\pmb a^+)=0\}$. In summary this means that if $h$ or more observations lie exactly on a subspace, the fit given by the observations in $H^*$ will coincide with this subspace, which is the defintion of the so-called *exact fit* property. Of course, since $|H^*|\geqslant h$, $H^*$ may contain outliers. Given $H^*$, one may proceed with the much simpler task of identifying the at most $|H^*|-h$ outliers in this smaller set of observations on a rank $q$ subspace spanned by the members of $H^*$.
**Appendix 1: Proof of affine equivariance of $(\pmb t^*(\pmb X^{\varepsilon}),\pmb S^*(\pmb X^{\varepsilon}))$**\
Recall that a location vector $\pmb t(\pmb X^{\varepsilon})$ and a scatter matrix $\pmb S(\pmb X^{\varepsilon})$ are affine equivariant if for any non-singular $p\times p$ matrices $\pmb B$ and $p$-vector $\pmb b$ it holds that: $$\begin{aligned}
\pmb t(\pmb B\pmb X^{\varepsilon}+\pmb 1_n\pmb b')&=&\pmb B\pmb t(\pmb X^{\varepsilon})+\pmb b\\
\pmb S( \pmb B \pmb X^{\varepsilon} +\pmb 1_n\pmb b')&=& \pmb B \pmb S(\pmb X^{\varepsilon}) \pmb B'.\end{aligned}$$ Consider now affine transformations of $\pmb X^{\varepsilon}$: $$\begin{aligned}
\label{ae1}
\pmb y^{\varepsilon}_i=\pmb B\pmb x^{\varepsilon}_i+\pmb b',\quad i=1,\ldots,n.\end{aligned}$$ for any non-singular $p\times p$ matrix $\pmb B$ and $p$-vector $\pmb b$. The directions $\pmb a_x$ ($\pmb a_y$) are orthogonal to hyperplanes through $p$-subsets of $\pmb X^{\varepsilon}$ ($\pmb Y^{\varepsilon}$). Since $||\pmb x_i^{g}-\pmb x_j^{g}||>0\;\forall\;1\leqslant i<j\leqslant g$, we can disregard all duplicated rows of $\pmb X^{\varepsilon}$ (and their partner duplicates in $\pmb Y^{\varepsilon}$), so that, w.l.o.g. all $p$-subsets of $\pmb X^{\varepsilon}$ $(\pmb Y^{\varepsilon})$ yield a $p \times p$ matrix with unique rows. Let $p^0$ be any such $p$-subset of $\{1:n\}$, and $\pmb a_x^0$ and $\pmb a_y^0$ the hyperplanes through $\{\pmb x_{i}\}_{i\in p^0}$ and $\{\pmb y_{i}\}_{i\in p^0}$. Since Equation describes an affine transformation, it preserves collinearity: $$\begin{aligned}
\label{ae2}
\{i:\pmb x_i'\pmb a^0_x=1\}=\{i:\pmb y_i'\pmb a^0_y=1\},\end{aligned}$$ and the ratio of lengths of intervals on univariate projections [@hcs:W02 sec. 36]: $$\begin{aligned}
\label{ae3}
\frac{\displaystyle\operatorname*{ave}_{i\in H^m}||\pmb x_i'\pmb a^0_x-\operatorname*{ave}_{i\in p^0}\pmb x_i'\pmb a^0_x||}{\displaystyle\operatorname*{ave}_{i\in H(\pmb a_x^0)}||\pmb x_i'\pmb a^0_x-\operatorname*{ave}_{i\in p^0}\pmb x_i'\pmb a^0_x||}=\frac{\displaystyle\operatorname*{ave}_{i\in H^m}||\pmb y_i'\pmb a^0_y-\operatorname*{ave}_{i\in p^0}\pmb y_i'\pmb a^0_y||}{\displaystyle\operatorname*{ave}_{i\in H(\pmb a_y^0)}||\pmb y_i'\pmb a^0_y-\operatorname*{ave}_{i\in p^0}\pmb y_i'\pmb a^0_y||},\end{aligned}$$ where for readability we denote $H^{\pmb a}$ as $H(\pmb a)$. Equation and imply $$\begin{aligned}
\label{ae4}
I(H^m,\pmb a_x^0)=I(H^m,\pmb a_y^0).\end{aligned}$$ Equation holds for any $p$-subset of $H^m$. Therefore, denoting $B_x(H^m)$ all directions perpendicular to hyperplanes through $p$ elements of $\{\pmb x^{\varepsilon}_i\}_{i\in H^m}$, and $B_y(H^m)$ the same but for $\{\pmb y^{\varepsilon}_i\}_{i\in H^m}$), it holds that $$\begin{aligned}
\operatorname*{ave}_{\pmb a_x\in B_x(H^m)} I(H^m,\pmb a_x)=\operatorname*{ave}_{\pmb a_y\in B_y(H^m)} I(H^m,\pmb a_y),\quad m=1\ldots,M\end{aligned}$$ and in particular for $H^m=H^*$. Since $\#\{H^*\}\geqslant p+1$, we have that if the members of $H^*$ lie in G.P. in $\mathbb{R}^p$, $$\begin{aligned}
\operatorname*{ave}_{i\in H^*}( \pmb B \pmb x_i^{\varepsilon}+\pmb b')&=& \pmb B \operatorname*{ave}_{i\in H^*}( \pmb x_i^{\varepsilon})\pmb b, \nonumber\\
\operatorname*{cov}_{i\in H^*}( \pmb B \pmb x_i^{\varepsilon}+\pmb b')&=& \pmb B \operatorname*{cov}_{i\in H^*}( \pmb x_i^{\varepsilon}) \pmb B'. \nonumber\end{aligned}$$ Hence, $(\pmb t^*(\pmb X^{\varepsilon}),\pmb S^*(\pmb X^{\varepsilon}))$ are affine equivariant.
**Appendix 2: Proof of Equation \[eq:append1\]**\
Here, we show that $\lambda_1(\pmb S^*(\pmb X^\varepsilon)) \leqslant \underset{i \in H^*}{\max}||\pmb x^\varepsilon_i||^2$. The first eigenvalue of $\pmb S^*(\pmb X^{\varepsilon})$ is defined as $$\lambda_1(\pmb S^*(\pmb X^{\varepsilon}))=\operatorname*{cov}_{i\in H^*}((\pmb x_i^{\varepsilon})'d)$$ for $d=\underset{||\tilde{d}||=1}{\operatorname*{\arg\!\max}}\underset{i\in H^*}{\operatorname*{cov}}((\pmb x_i^{\varepsilon})'\tilde{d})$. Furthermore, $$\operatorname*{cov}_{i\in H^*}((\pmb x_i^{\varepsilon})'d)=\operatorname*{ave}_{i\in H^*}(((\pmb x^{\varepsilon}_i)'d)^2)-(\operatorname*{ave}_{i\in H^*}((\pmb x_i^{\varepsilon})'d))^2.$$ Hence, we have that $$\operatorname*{cov}_{i\in H^*}((\pmb x_i^{\varepsilon})'d)\leqslant\operatorname*{ave}_{i\in H^*}(((\pmb x^{\varepsilon}_i)'d)^2).$$ We also have that $$\operatorname*{ave}_{i\in H^*}(((\pmb x^{\varepsilon}_i)'d)^2)\leqslant\max_{i\in H^*}(((\pmb x^{\varepsilon}_i)'d)^2)=\max_{i\in H^*}||(\pmb x^{\varepsilon}_i)'d||^2.$$ Using Cauchy-Schwartz,
$$\max_{i\in H^*}||(\pmb x^{\varepsilon}_i)'d||^2\leqslant(\max_{i\in H^*}||d||||\pmb x^{\varepsilon}_i)||)^2,$$ and $||d||=1$. Thus, $\lambda_1(\pmb S^*(\pmb X^{\varepsilon}))\leqslant\underset{i\in H^*}{\max}||\pmb x^{\varepsilon}_i||^2.$\
Acknowledgements
================
The authors wish to acknowledge the helpful comments from three anonymous referees and the editor for improving this paper.
Viktoria Öllerer would like to acknowledge the support of Research Fund KU Leuven GOA/12/014.
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abstract: |
We present results of the one-body density matrix $\rho_1(r)$ and the condensate fraction $n_0$ of liquid $^4$He calculated at zero temperature by means of the Path Integral Ground State Monte Carlo method. This technique allows to generate a highly accurate approximation for the ground state wave function $\Psi_0$ in a totally model-independent way, that depends only on the Hamiltonian of the system and on the symmetry properties of $\Psi_0$. With this unbiased estimation of $\rho_1(r)$, we obtain precise results for the condensate fraction $n_0$ and the kinetic energy $K$ of the system. The dependence of $n_0$ with the pressure shows an excellent agreement of our results with recent experimental measurements. Above the melting pressure, overpressurized liquid $^4$He shows a small condensate fraction that has dropped to $0.8\%$ at the highest pressure of $p = 87 \, {\rm bar}$.\
[**keywords**]{}: Liquid Helium, Bose-Einstein condensation, Quantum Monte Carlo.\
author:
- 'R. Rota$^1$ and J. Boronat$^1$'
date: '28.09.2011'
title: 'Condensate fraction in liquid $^{\bf 4}$He at zero temperature'
---
Introduction
============
Several microscopic theories point out that the phenomenon of superfluidity in liquid $^4$He has to be seen as a consequence of Bose-Einstein condensation (BEC).[@TilleyTilley] Having total spin $S = 0$, $^4$He atoms behave like bosons and, below the critical temperature $T_{\lambda} =
2.17 \, {\rm K}$, they can occupy macroscopically the same single-particle state. Nevertheless, the strong interaction between $^4$He atoms does not allow all of them to occupy the lowest energy state and, even at zero temperature, only a small fraction $n_0 = N_0/N$ of the $N$ particles is in the condensate.
The macroscopic occupation of the lowest energy state, in a strongly correlated system like $^4$He, appears in the momentum distribution $n(\bf{k})$ as a delta-peak at $\bf{k} = 0$ and a divergent behavior $n(k)
\sim 1/k$ when $k \to 0$. In the coordinate space, the presence of BEC in a homogeneous system can be deduced from the asymptotic behavior of the one-body density matrix $\rho_1(\bf{r})$ ($n_0 =
\lim_{r\to\infty}\rho_1(r)$), which is the inverse Fourier transform of $n(\bf{k})$.
Experimental estimates of $n_0$ can be obtained from the dynamic structure factor, $S(q,\omega)$, measured by neutron inelastic scattering at high energy and momentum transfer. These measurements have a long history:[@GlydeBook; @SilverSokol; @SvenssonSeara] in the 80s, the first experiments gave estimates for $n_0$ slightly above 10%, but they were affected by a poor instrumental resolution and by some difficulties in describing the final states effects of the scattering experiment. Recently, with the advances in the experimental technology and in the method of analysis of the scattering data, Glyde [*et al.*]{}[@Glyde00] have been able to give improved estimations of $n_0$ at very low temperature. At saturated vapor pressure (SVP), they found $n_0 = (7.25 \pm 0.75)
\%$,[@Glyde00] and more recently they have measured the dependence of $n_0$ with pressure $p$.[@Glyde11]
Because of the strong correlations between $^4$He atoms, the calculation of the one-body density matrix in superfluid $^4$He cannot be obtained analytically via a perturbative approach. It is necessary the use of microscopic simulations to provide accurate estimations of the condensate fraction. In particular, the Path Integral Monte Carlo (PIMC) method has been widely used in the study of $^4$He at finite temperature, thanks to its capability of furnishing in principle exact numerical estimates of physical observables relying only on the Hamiltonian of the system.[@CeperleyRevPIMC] The first calculations of $n_0$ with this method date back to 1987,[@Ceperley87] but most recent simulations based on an improved sampling algorithm provide very accurate results for $\rho_1(r)$, showing a condensate fraction $n_0 = 0.081 \pm 0.002$ at temperature $T = 1 \, {\rm K}$. [@BoninsegniWorm] At zero temperature, ground-state projection techniques are widely used in the study of BEC properties of $^4$He. Diffusion Monte Carlo technique, for instance, has provided estimations of $n_0$ in liquid $^4$He on a wide range of pressures.[@Boronat94; @Moroni97; @Vranjes05] This method, however, suffers from the choice of a variational ansatz necessary for the importance sampling whose influence on $\rho_1(r)$ cannot be completely removed. Reptation Quantum Monte Carlo (RQMC) has also been used for this purpose,[@Moroni04] but the calculated value of $n_0$ at SVP lies somewhat below the recent PIMC value [@BoninsegniWorm] at $T = 1 \,
{\rm K}$ noted above.
Motivated by recent accurate experimental data on $n_0(p)$, our aim in the present work is to perform new calculations of $\rho_1(r)$ and of $n_0$ in liquid $^4$He at zero temperature using a completely model-independent technique based on path integral formalism. The Path Integral Ground State (PIGS) Monte Carlo method is able to compute exact quantum averages of physical observables without importance sampling, that is without taking into account any [*a priori*]{} trial wave function.[@Rossi09] Using a good sampling scheme in our Monte Carlo simulations, we are able to provide very precise calculations of the one-body density matrix at different densities. We fit our numerical data for $\rho_1(r)$ with the model used in previous experimental works, [@Glyde00] highlighting the merits and the faults of this model, and finally we give our estimations for the condensate fraction when changing the pressure of the liquid, showing an excellent agreement with experimental data.[@Glyde11]
The PIGS method and the computational details of our simulation are discussed in Sec. \[Sec\_PIGS\]. The results are presented in Sec. \[Sec\_Results\] and Sec. \[Sec\_Conclusions\] comprises the main conclusions.
The PIGS method {#Sec_PIGS}
===============
The PIGS approach to the study of quantum systems consists in a systematic improvement of a trial wave function $\Psi_T$ by repeated application of the evolution operator in imaginary time, which eventually drives the system into the ground state,[@SarsaPIGS] according to the formula $$\label{Eq_PIGSwf}
\Psi_{PIGS}(R_M) = \int \prod_{i=1}^{M} dR_{i-1} G(R_i,R_{i-1};\tau) \Psi_T(R_0) \ ,$$ where the $R_i = \{ {\bf r}_{i;1},{\bf r}_{i;2},...,{\bf r}_{i;N} \}$ represent different sets of coordinates the $N$ particles of the system, and $G(R',R;\tau)=\langle R' \vert e^{-\tau\hat{H}}\vert R \rangle$ is the imaginary time propagator.
Given an approximation of $G(R,R';\tau)$ for small $\tau$, the averages of diagonal observables can be calculated mapping the quantum many-body system onto a classical system made up of $N$ interacting polymers composed by $2 M +1$ beads, each of them representing a different evolution in imaginary time of the initial trial state $\Psi_T$. Increasing the number $M$, one is able to reduce the systematic error and therefore to recover ”exactly” the ground-state properties of the system.
A good approximation for the propagator $G$ is important for improving the numerical efficiency of the method: this greatly reduces the complexity of the calculation and ergodicity issues, allowing to simulate the quantum system with few beads, each one with a large time step. Using a high-order approximation for the propagator, it is possible to obtain an accurate description of the exact ground state wave function with little numeric effort, even when the initial trial wave function contains no more information than the bosonic statistics, that is when one starts the imaginary time evolution from $\Psi_T=1$.[@Rota10]
The one-body density matrix can be written as $$\label{Eq_OBDMPsi}
\rho_1({\bf r}_1,{\bf r'}_1) = \frac{\int d{\bf r}_2 ... d{\bf r}_N
\Psi^*_0(R)\Psi_0(R')}{\int d{\bf r}_1 ... d{\bf r}_N
|\Psi_0(R)|^2} \ ,$$ where the configuration $R = \{ {\bf r}_1,{\bf r}_2,...,{\bf r}_N
\}$ differs from $R' = \{ {\bf r'}_1,{\bf r}_2,...,{\bf r}_N \}$ only by the position of one of the $N$ atoms. In the PIGS approach, the expectation value of non-diagonal observables, like $\rho_1$, is computed mapping the quantum system in the same classical system of polymers as in the diagonal case, but cutting one of these polymers in the mid point. Building the histogram of the frequencies of the distances between the cut extremities of the two half polymers, one can compute the numerator in Eq. (\[Eq\_OBDMPsi\]). The calculation of the normalization factor at the denominator is not strictly necessary since the histogram can be normalized imposing the condition $\rho_1(0)=1$. However, this [*a posteriori*]{} normalization procedure is not easy, because of the small occurrences of the distances close to zero, and may introduce systematic errors in the estimation of the condensate fraction $n_0$. In our work, we have avoided this problem incorporating in the sampling the worm algorithm (WA), a technique previously developed for path integral Monte Carlo simulations at finite temperature.[@BoninsegniWorm] The key aspect of WA is to work in an extended configuration space, containing both diagonal (all polymers with the same length) and off-diagonal (one polymer cut in two separate halves) configurations, and one of its main advantages is its capability of evaluating the normalization factor in off-diagonal estimators. We have extended this technique to zero-temperature calculations and we have been able to get automatically the properly normalized $\rho_1$, and therefore very precise estimations of $n_0$. In our simulations, the sampling also contains movements involving the principle of indistinguishability of quantum particles, like the [*swap*]{} update [@BoninsegniWorm]. Even though swaps are not strictly necessary since boson symmetry is fulfilled with a proper choice of the imaginary time propagator, they improve the sampling allowing a larger displacement of the half polymers and thus a better exploration of the long range limit of $\rho_1$.
Results {#Sec_Results}
=======
To compute $\rho_1(r) = \rho_1(|{\bf r}_1 - {\bf r'}_1|)$ in liquid $^4$He at several densities, we have carried out different simulations with a cubic box with periodic boundary conditions containing $N = 128$ atoms interacting through the Aziz pair potential.[@Aziz] At first we study the system at the equilibrium density $\rho = 0.02186 \,$ Å$^{-3}$: our result for $\rho_1(r)$ is shown in Fig. \[FigOBDM\_eq\]. We have checked that our results starting with $\Psi_T = 1$ or with a Jastrow-McMillan wave function are statistically indistinguishable. In order to check how the finite size of the box affects our results, we have performed a simulation of the same system in a larger box containing $N=256$ $^4$He atoms. In Fig. \[FigOBDM\_eq\], we have compared $\rho_1(r)$ obtained in this last simulation with the one estimated using a smaller number of particles: we can see that, up to the distances reachable with the smaller system, these two results agree within the statistical error. Furthermore, the two functions reach the same plateau at the largest available distances, indicating that the asymptotic regime for $\rho_1(r)$ is already achieved using $N=128$ $^4$He atoms.
![(Color online) One-body density matrix $\rho_1(r)$ at the equilibrium density $\rho = 0.02186$ Å$^{-3}$. The symbols represents the result of the PIGS simulations for the system containing $N = 128$ (green circles) and $N = 256$ (blue squares) $^4$He atoms. The red line is the curve obtained fitting these data with Eq. \[Eq\_OBDM\_fit\] with optimal values: $k_c=1.369 \pm 0.020\ \textrm{\AA}^{-1}$, $\alpha_2=0.794
\pm 0.005\ \textrm{\AA}^{-2}$, $ \alpha_4=0.355 \pm 0.050\ \textrm{\AA}^{-4}$, $\alpha_6=0.680 \pm 0.080\ \textrm{\AA}^{-6}$, and $n_0=0.0801 \pm 0.0022$. The inset shows the same data for $r$ between 3 Åand 11 Åon an expanded scale.[]{data-label="FigOBDM_eq"}](obdm.eps){width="\linewidth"}
To fit our data we use the model proposed by Glyde in Ref. [@GlydeBook] that has been used in the analysis of experimental data,[@Glyde00] $$\label{Eq_OBDM_fit}
\rho_1(r) = n_0[1+f(r)]+ A \rho_1^*(r) \ .$$ The function $f(r)$ represents the coupling between the condensate and the non-zero momentum states. In momentum space, one can express $f(k)$ in terms of the phonon response function [@GlydeBook], $$\label{Eq_fk}
f(k) = \left[\frac{m c}{2 \hbar (2 \pi)^3 \rho} \frac{1}{k} \coth \left(
\frac{c \hbar k}{2 k_B T} \right)\right] e^{-k^2/(2 k_c^2)} \ ,$$ with $c$ the speed of sound. Since we work in the coordinate space, we are interested in its 3D Fourier transform $f(r)$, which at zero temperature can be written as $$f(r) = \frac{m c}{\hbar (2 \pi)^2 \rho}
\frac{\sqrt{2} k_c}{r} D\left( \frac{k_c r}{\sqrt{2}} \right) \ ,$$ where $D(x) = e^{-x^2} \int_{0}^{x}{dt e^{t^2}}$ is the Dawson function. To describe the contribution to $\rho_1$ from the states above the condensate, which we denote by $\rho_1^*$, we use the cumulant expansion of the intermediate scattering function, that is the Fourier Transform of the longitudinal momentum distribution,[@GlydeBook] $$\label{Eq_OBDM_noncond} \rho_1^*(r) = \exp
\left[ -\frac{\alpha_2 r^2}{2!}+ \frac{\alpha_4 r^4}{4!}
-\frac{\alpha_6 r^6}{6!}\right] \ .$$ The constant $A$ appearing in Eq. (\[Eq\_OBDM\_fit\]) is fixed by the normalization condition $\rho_1(0) = 1$. Therefore, the model we used has five parameters: $n_0$, $k_c$, $\alpha_2$, $\alpha_4$ and $\alpha_6$. It has to be noticed that, unlike what is done in the treatment of the experimental data, where $k_c$ is chosen as a cut-off parameter to make the term $f(k)$ vanish out of the phonon region, we have considered $k_c$ as a free parameter of the fit.
The best fit we get using the model of Eq. (\[Eq\_OBDM\_fit\]) is shown in Fig. \[FigOBDM\_eq\]. This model is able to reproduce the behavior of $\rho_1(r)$ for short distances and in the asymptotic regime, but cannot describe well the numerical data in the range of intermediate $r$. Indeed, for distances above 3 Å, $\rho_1(r)$ obtained with PIGS presents oscillations which are damped at larger $r$, as observed also in previous theoretical calculations.[@Moroni04; @BoninsegniWorm] This non monotonic behavior, which can be attributed to coordination shell oscillations,[@BoninsegniWorm] is difficult to describe within the model of Eq. \[Eq\_OBDM\_fit\].
Nevertheless, despite of these difficulties in describing the oscillations of the $\rho_1(r)$ obtained with PIGS, the fit we gave using the model of Eq. (\[Eq\_OBDM\_fit\]) contains important informations about the ground state of liquid $^4$He. First of all, from the long range behavior, we can obtain the value of the condensate fraction $n_0$. From our analysis, we get $n_0 = 0.0801
\pm 0.0022$, in complete agreement with the value $n_0 = 0.081 \pm
0.002$ obtained by Boninsegni [*et al.*]{}[@BoninsegniWorm] in a path integral Monte Carlo simulation at temperature $T = 1 \,
{\rm K}$, and in good agreement with the experimental result $n_0
= 0.0725 \pm 0.0075$.[@Glyde00] Furthermore, from the behavior of $\rho_1(r)$ at short distances, we can obtain an estimation of the kinetic energy per particle $K/N$. In particular, the term $\alpha_2$ appearing in Eq. (\[Eq\_OBDM\_noncond\]) is the second moment of the struck atom wave vector projected along the direction of the incoming neutron [@GlydeBook] and is related to the kinetic energy per particle by the formula $K/N = 3
(\hbar^2/2m) \alpha_2$. Using the value $\alpha_2 = (0.794 \pm
0.005)$ Å$^{-2}$ obtained in our fit, we get $K/N = (14.43 \pm
0.09) \, {\rm K}$ which has to be compared with the value obtained in the PIGS simulation, $K/N = (14.37 \pm 0.03) \, {\rm K}$.
In Fig. \[Fig\_nk\_eq\], we show results of $n(k)$ obtained performing a numerical Fourier transform of $\rho_1(r)$ and we compare it with the Fourier transform of Eq. \[Eq\_OBDM\_fit\], $$n(k) = n_0 \delta(k) + n_0 f(k) + A n^*(k) \ .
\label{fullnk}$$ The PIGS data are plotted from $k_{min} = 2\pi/L \simeq
0.4 \,$ Å$^{-1}$ , $L$ being the length of the simulation box, and are not able to reproduce the $1/k$ behavior of $n(k)$ at low $k$ because of finite size effects; for $k > k_{min}$ the effect of $f(k)$ vanishes and $n(k)=n^*(k)$. We notice that the disagreement between the two curves is larger in the region between $k \simeq 1 \,$ Å$^{-1}$ and $k \simeq 2.5 \,$ Å$^{-1}$. In this range, indeed, $n(k)$ obtained with the PIGS method presents a change of curvature, not seen in $n(k)$ obtained from the fit.
![(Color online) The momentum distribution $n(k)$ at equilibrium density $\rho = 0.02186$ Å$^{-3}$: the black circles represents the numerical result obtained from the PIGS simulation, the red line represents the FT of the fit for $\rho_1(r)$ obtained according to Eq. (\[Eq\_OBDM\_fit\]).[]{data-label="Fig_nk_eq"}](nk_highk.eps){width="\linewidth"}
This discrepancy can be explained considering the coupling between the condensate and the states out of it. The term $f(k)$, defined in Eq. (\[Eq\_fk\]), is obtained considering only pure density excitations in the system and, therefore, is valid only in the limit of small momenta.[@GlydeBook] At higher $k$, one should consider even the contributions due to the coupling of the condensate to the excited states out of the phonon region. However, little is known about these contributions and it is difficult to include them in a more complete form for $f(k)$ in order to give a more reliable model for the momentum distribution.
![(Color online) The momentum distribution, plotted as $k n(k)$, in liquid $^4$He at two different pressure: black circles and red squares are the PIGS results for $n(k)$, respectively, at saturated vapor pressure ($\rho = 0.02186 \,$ Å$^{-3}$) and at a pressure close to the freezing, $p \simeq 24 \, {\rm bar}$ ($\rho = 0.02539 \,$ Å$^{-3}$). The black and the red dashed lines represent the experimental results for the momentum distribution above the condensate $n^*(k)$ at the same pressures.[@Glyde11][]{data-label="Fig_nk_press"}](knk_denshighk.eps){width="\linewidth"}
Our results for $n(k)$ are compared with recent experimental measurements at $T = 0.06 \, {\rm K}$ of the momentum distribution $n^*(k)$ for states above the condensate in Fig. \[Fig\_nk\_press\]. Even in this case, we can notice a good agreement between the two curves, except for the intermediate range of $k$, where our results include contributions arising from the coupling between the condensate and excited states. From the comparison between the two curves, we can deduce that this coupling contributes in depleting the states at higher $k$. In the same figure, in addition to $n(k)$ at the saturated vapor pressure, we also show $n(k)$ for a higher pressure, close to the freezing transition. We can see that the effect of the pressure in the momentum distribution is to decrease the occupancy of the low-momenta states and to make smoother the shoulder at $k \simeq
2 \,$ Å$^{-1}$.
[CCCC]{} $\rho[$ Å$^{-3}]$ & $p$\[bar\] & $n_0$ & $K/N$\[K\]\
0.01964 & -6.23 & 0.1157(19) & 12.01(3)\
0.02186 & -0.04 & 0.0801(22) & 14.37(3)\
0.02264 & 3.29 & 0.0635(16) & 15.35(3)\
0.02341 & 7.36 & 0.0514(16) & 16.28(3)\
0.02401 & 11.07 & 0.0436(11) & 17.02(4)\
0.02479 & 16.71 & 0.0350(7) & 18.08(4)\
0.02539 & 21.76 & 0.0333(8) & 18.82(5)\
0.02623 & 29.98 & 0.0278(8) & 19.99(4)\
0.02701 & 38.95 & 0.0208(6) & 21.08(5)\
0.02785 & 50.23 & 0.0155(6) & 22.24(4)\
0.02869 & 63.37 & 0.0115(4) & 23.65(4)\
0.02940 & 76.28 & 0.0093(4) & 24.83(5)\
0.02994 & 87.06 & 0.0083(4) & 25.61(6)\
Finally, we report our results for the condensate fraction $n_0$ over a wide range of densities, including also densities in the negative pressure region and in the regime of the overpressurized metastable fluid. In this range of high densities, we have been able to frustrate the formation of the crystal by starting the simulation from an equilibrated disordered configuration. The metastability of this phase is checked by monitoring how the total energy per particle $E/N$ changes with the number of Monte Carlo steps. As the simulation goes on, we notice that $E/N$ reaches a plateau for a value above the corresponding value of $E/N$ computed in a perfect crystal at the same density. For instance, at density $\rho = 0.02940 \,$ Å$^{-3}$ we get in our simulation $E/N =
(-5.48 \pm 0.03) \, {\rm K}$. If we perform a PIGS simulation at the same density and with the same choice for the initial trial wave function (in both cases, we choose in Eq. \[Eq\_PIGSwf\] $\Psi_T = 1$) but starting the computation from a hcp crystalline configuration, we get $E/N = (-5.95 \pm 0.02) \, {\rm K}$. The disagreement of the two results for $E/N$ indicates that, in PIGS simulations, initial conditions for the atomic configuration influence the evolution of the system: in particular, a sensible choice of the initial conditions speed up the convergence of the system to the real equilibrium state. In the simulation of $^4$He at high densities, if we use a disordered configuration as the initial one, the system evolves towards the equilibrium crystalline phase, but, since crystallization is a very slow process in PIGS simulations, we see that the overpressurized liquid phase is metastable for a number of Monte Carlo steps sufficiently large to give good statistics for the ground state averages of the physical observables. If the density is increased even more ($p>90$ bars), one starts to observe the formation of crystallites and the stabilization of the liquid becomes more difficult.
Another evidence of the metastability of the liquid configuration in our simulations can be given computing the static structure factor $S(k)$. In all the calculations performed, we notice the absence of Bragg peaks in $S(k)$, which indicates clearly that the system does not present crystalline order.
Our results for $n_0$ at different $p$ are contained in Table \[Tab\_n0\], together with our estimates for the kinetic energy $K/N$. It is interesting to notice that the condensate fraction of the overpressurized liquid is finite also for densities above the melting ($\rho \ge 0.02862 \,$ Å$^{-3}$). This evidence supports our hypothesis that the system has reached a metastable non-crystalline phase, since recent PIGS simulations show that, in commensurate hcp $^4$He crystals, the one-body density matrix decays exponentially to zero at large distances and therefore BEC is not present[@Galli08; @Rota11]. In particular, we obtain that in the overpressurized fluid at the melting density the condensate fraction is $n_0 \simeq 1.2\%$. This result, even though cannot provide any deeper understanding concerning the quest of supersolidity in $^4$He,[@Galli08] can be thought as an upper limit for the condensate fraction in solid $^4$He at melting. It is also interesting to notice that, even at the freezing pressure, the condensate fraction is already quite small, $n_0 = 2.9\%$.
In Fig. \[Fig\_n0\_press\], we plot our results for $n_0$ as a function of $p$ on the range of pressures where the liquid phase is stable. Our results follow well an inverse proportionality law $n_0(p) = A + B/(p-p_0)$, with $p$ and $p_0$ measured in bar: the best fit we got has parameters $A = -0.0068 \pm 0.0012$, $B = 1.56
\pm 0.10$, $p_0 = -19.0 \pm 0.9 \, {\rm bar}$. In Fig. \[Fig\_n0\_press\], we also compare our estimates for $n_0$ with the experimental ones[@Glyde11] and with the ones obtained in previous numerical simulations.[@Boronat94; @Moroni97; @Moroni04; @BoninsegniWorm] It is easy to notice that our results provide an excellent description of the experimental dependence of the condensate fraction as a function of pressure in all the range of stability of the liquid phase of $^4$He, improving previous calculations which focus especially on the equilibrium density and do not explore in detail the physically interesting pressure range where the experimental data can be measured. Notice that the experimental value of $n_0$ at zero pressure reported in the more recent experiment [@Glyde11] is slightly smaller ($7.01 \pm 0.75$%), but still statistically compatible within the error bars, than the previous one by the same team. [@Glyde00]
![(Color online) The condensate fraction $n_0$ in liquid $^4$He at zero temperature as a function of pressure $p$ in the region of stability of liquid phase. Our PIGS results (black squares) are compared with the experimental ones (red diamonds) [@Glyde11] and previous theoretical calculations, obtained with Diffusion Monte Carlo (green up triangles),[@Boronat94] Diffusion Euler Monte Carlo (violet left triangles),[@Moroni97] Reptation Quantum Monte Carlo (blue circle) [@Moroni04] and PIMC at $T = 1 \, {\rm
K}$.[@BoninsegniWorm] The dashed line represents the curve obtained fitting our results with the equation $n_0 = A +
B/(p-p_0)$.[]{data-label="Fig_n0_press"}](n0vsp.eps){width="\linewidth"}
Conclusions {#Sec_Conclusions}
===========
We have computed the one-body density matrix of liquid $^4$He at zero temperature and different densities by means of the PIGS Monte Carlo method. Although it is not easy to give an analytic model to fit the data for $\rho_1(r)$, because of the difficulty of describing the coupling between the condensate and the excited states in strongly correlated quantum systems such as $^4$He, it is possible to extrapolate very precise estimates of the condensate fraction and of the kinetic energy of the system even from a simplified model for $\rho_1$. Our calculations provide an improvement with respect to the other ground state projection techniques used in the past, since the PIGS method allows us to remove completely the influence of any input trial wave function. Indeed, we have performed calculations of $\rho_1(r)$ and $n_0$ in liquid $^4$He at zero temperature using a model for the ground state wave function which depends uniquely on the Hamiltonian and on the symmetry properties of the system. At the equilibrium density of liquid $^4$He, we have recovered the value of $n_0$ obtained with the unbiased PIMC method at temperature $T
= 1 \, {\rm K}$.[@BoninsegniWorm] Simulating the system at several densities, the dependence of $n_0$ with pressure $p$ obtained from the calculation agrees nicely with the recent experimental measurements of Ref.
Authors would like to thank Henry Glyde for helpful discussion and for sending us unpublished data. This work was partially supported by DGI (Spain) under Grant No. FIS2008-04403 and Generalitat de Catalunya under Grant No. 2009-SGR1003.
[99]{}
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R.N. Silver and P.E. Sokol, [*Momentum Distributions*]{} (Plenum, New York, 1989)
E.C. Svensson and V.F. Sears, in [*Frontiers of Neutron Scattering*]{}, edited by R. J. Birgeneau, D.E. Moncton, and D. Zilinger (North-Holland, Amsterdam, 1986)
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M. Boninsegni, N. V. Prokof’ev, B. V. Svistunov, Phys. Rev. E, **74**, 036701, (2006).
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Appendix: full tables for the momentum distribution
===================================================
The aim of this appendix is to provide the full table of the momentum distribution $n(k)$ in liquid $^4$He at zero temperature and different densities, obtained from Path Integral Ground State caluclations.
[CCCCCCC]{} & & & & & &\
$\rho $ (Å$^{-3}$)& 0.01964 & 0.02186 & 0.02264 & 0.02341 & 0.02401 & 0.02479\
& & & & & &\
& & & & & &\
k (Å$^{-1}$) &\
& & & & & &\
0.4 & 0.10415 & 0.08758 & 0.08041 & 0.07708 & 0.07685 & 0.07078\
0.5 & 0.09226 & 0.07838 & 0.07355 & 0.07082 & 0.06972 & 0.06509\
0.6 & 0.08157 & 0.07009 & 0.06697 & 0.06448 & 0.06292 & 0.05937\
0.7 & 0.07238 & 0.06298 & 0.06092 & 0.05833 & 0.05676 & 0.05390\
0.8 & 0.06441 & 0.05687 & 0.05532 & 0.05244 & 0.05127 & 0.04877\
0.9 & 0.05711 & 0.05130 & 0.04996 & 0.04682 & 0.04628 & 0.04397\
1.0 & 0.04996 & 0.04581 & 0.04461 & 0.04141 & 0.04154 & 0.03941\
1.1 & 0.04269 & 0.04011 & 0.03912 & 0.03619 & 0.03684 & 0.03499\
1.2 & 0.03536 & 0.03417 & 0.03351 & 0.03116 & 0.03209 & 0.03066\
1.3 & 0.02823 & 0.02818 & 0.02793 & 0.02637 & 0.02733 & 0.02644\
1.4 & 0.02165 & 0.02244 & 0.02262 & 0.02186 & 0.02272 & 0.02238\
1.5 & 0.01596 & 0.01727 & 0.01781 & 0.01771 & 0.01843 & 0.01858\
& & & & & &\
1.6 & 0.01135 & 0.01291 & 0.01367 & 0.01401 & 0.01465 & 0.01514\
1.7 & 0.00792 & 0.00948 & 0.01032 & 0.01082 & 0.01149 & 0.01215\
1.8 & 0.00561 & 0.00699 & 0.00777 & 0.00824 & 0.00898 & 0.00965\
1.9 & 0.00425 & 0.00533 & 0.00594 & 0.00630 & 0.00709 & 0.00767\
2.0 & 0.00356 & 0.00433 & 0.00472 & 0.00495 & 0.00573 & 0.00617\
2.1 & 0.00325 & 0.00374 & 0.00393 & 0.00408 & 0.00476 & 0.00506\
2.2 & 0.00303 & 0.00334 & 0.00341 & 0.00354 & 0.00406 & 0.00424\
2.3 & 0.00272 & 0.00296 & 0.00300 & 0.00314 & 0.00350 & 0.00361\
2.4 & 0.00225 & 0.00252 & 0.00260 & 0.00276 & 0.00300 & 0.00308\
2.5 & 0.00170 & 0.00201 & 0.00217 & 0.00232 & 0.00251 & 0.00258\
& & & & & &\
2.6 & 0.00118 & 0.00149 & 0.00171 & 0.00184 & 0.00204 & 0.00210\
2.7 & 0.00078 & 0.00106 & 0.00128 & 0.00138 & 0.00161 & 0.00166\
2.8 & 0.00055 & 0.00077 & 0.00092 & 0.00100 & 0.00124 & 0.00127\
2.9 & 0.00045 & 0.00061 & 0.00065 & 0.00074 & 0.00095 & 0.00097\
3.0 & 0.00040 & 0.00053 & 0.00048 & 0.00058 & 0.00075 & 0.00076\
3.1 & 0.00036 & 0.00048 & 0.00038 & 0.00049 & 0.00062 & 0.00062\
3.2 & 0.00029 & 0.00042 & 0.00032 & 0.00041 & 0.00052 & 0.00053\
3.3 & 0.00021 & 0.00027 & 0.00027 & 0.00032 & 0.00044 & 0.00046\
3.4 & 0.00016 & 0.00021 & 0.00022 & 0.00023 & 0.00037 & 0.00039\
3.5 & 0.00015 & 0.00017 & 0.00017 & 0.00015 & 0.00030 & 0.00031\
[CCCCCCCC]{} & & & & & & &\
$\rho $ (Å$^{-3}$)& 0.02539 & 0.02623 & 0.02701 & 0.02785 & 0.02869 & 0.02940 & 0.02994\
& & & & & & &\
& & & & & & &\
k (Å$^{-1}$) &\
& & & & & & &\
0.4 & 0.06464 & 0.06170 & 0.05811 & 0.05400 & 0.04904 & 0.04542 & 0.04310\
0.5 & 0.06013 & 0.05751 & 0.05403 & 0.04993 & 0.04600 & 0.04291 & 0.04076\
0.6 & 0.05554 & 0.05328 & 0.04990 & 0.04589 & 0.04286 & 0.04025 & 0.03828\
0.7 & 0.05106 & 0.04916 & 0.04590 & 0.04210 & 0.03976 & 0.03753 & 0.03578\
0.8 & 0.04673 & 0.04520 & 0.04211 & 0.03862 & 0.03677 & 0.03483 & 0.03330\
0.9 & 0.04252 & 0.04134 & 0.03849 & 0.03541 & 0.03389 & 0.03217 & 0.03086\
1.0 & 0.03834 & 0.03752 & 0.03499 & 0.03239 & 0.03107 & 0.02954 & 0.02843\
1.1 & 0.03416 & 0.03367 & 0.03153 & 0.02942 & 0.02830 & 0.02694 & 0.02602\
1.2 & 0.02995 & 0.02980 & 0.02807 & 0.02644 & 0.02552 & 0.02435 & 0.02359\
1.3 & 0.02579 & 0.02594 & 0.02463 & 0.02343 & 0.02276 & 0.02178 & 0.02118\
1.4 & 0.02177 & 0.02219 & 0.02127 & 0.02043 & 0.02003 & 0.01927 & 0.01879\
1.5 & 0.01801 & 0.01864 & 0.01806 & 0.01751 & 0.01740 & 0.01684 & 0.01649\
& & & & & & &\
1.6 & 0.01462 & 0.01539 & 0.01510 & 0.01479 & 0.01491 & 0.01456 & 0.01430\
1.7 & 0.01171 & 0.01253 & 0.01247 & 0.01234 & 0.01264 & 0.01245 & 0.01229\
1.8 & 0.00931 & 0.01010 & 0.01021 & 0.01023 & 0.01063 & 0.01056 & 0.01048\
1.9 & 0.00743 & 0.00814 & 0.00835 & 0.00848 & 0.00890 & 0.00892 & 0.00891\
2.0 & 0.00602 & 0.00662 & 0.00687 & 0.00707 & 0.00746 & 0.00752 & 0.00756\
2.1 & 0.00500 & 0.00549 & 0.00571 & 0.00595 & 0.00629 & 0.00636 & 0.00644\
2.2 & 0.00425 & 0.00466 & 0.00481 & 0.00506 & 0.00533 & 0.00540 & 0.00550\
2.3 & 0.00365 & 0.00402 & 0.00410 & 0.00433 & 0.00455 & 0.00461 & 0.00473\
2.4 & 0.00312 & 0.00348 & 0.00350 & 0.00370 & 0.00388 & 0.00395 & 0.00407\
2.5 & 0.00261 & 0.00296 & 0.00297 & 0.00314 & 0.00330 & 0.00337 & 0.00349\
& & & & & & &\
2.6 & 0.00211 & 0.00246 & 0.00248 & 0.00262 & 0.00278 & 0.00286 & 0.00297\
2.7 & 0.00165 & 0.00198 & 0.00203 & 0.00215 & 0.00230 & 0.00240 & 0.00250\
2.8 & 0.00126 & 0.00155 & 0.00162 & 0.00173 & 0.00188 & 0.00200 & 0.00208\
2.9 & 0.00096 & 0.00121 & 0.00129 & 0.00139 & 0.00152 & 0.00164 & 0.00171\
3.0 & 0.00076 & 0.00097 & 0.00102 & 0.00111 & 0.00122 & 0.00134 & 0.00140\
3.1 & 0.00064 & 0.00081 & 0.00081 & 0.00090 & 0.00100 & 0.00111 & 0.00114\
3.2 & 0.00055 & 0.00070 & 0.00066 & 0.00074 & 0.00082 & 0.00092 & 0.00094\
3.3 & 0.00047 & 0.00061 & 0.00054 & 0.00062 & 0.00069 & 0.00077 & 0.00078\
3.4 & 0.00039 & 0.00052 & 0.00044 & 0.00053 & 0.00058 & 0.00065 & 0.00066\
3.5 & 0.00031 & 0.00043 & 0.00035 & 0.00044 & 0.00049 & 0.00055 & 0.00055\
|
---
author:
- '[^1]'
- '[^2]'
-
title: The correction of hadronic nucleus polarizability to hyperfine structure of light muonic atoms
---
Introduction {#intro}
============
Precise investigation of the Lamb shift and hyperfine structure of light muonic atoms is a fundamental problem for testing the Standard model and establishing the exact values of its parameters, as well as searching for effects of new physics. At present, the relevance of these studies is primarily related to experiments conducted by the collaboration CREMA (Charge Radius Experiments with Muonic Atoms) [@crema1; @crema2; @crema3; @crema4] with muonic hydrogen and deuterium by methods of laser spectroscopy. So, as a result of measuring the transition frequency $ 2P^{F=2}_{3/2}-2S^{F=1}_{1/2}$ a more accurate value of the proton charge radius was found to be $r_E = 0.84087(39)$ fm, which is different from the value recommended by CODATA for $7\sigma$ [@mohr]. The CODATA value is based on the spectroscopy of the electronic hydrogen atom and on electron-nucleon scattering. The measurement of the transition frequency $2P^{F=1}_{3/2}-2S^{F=0}_{1/2}$ for the singlet $2S$ of the state $ (\mu p)$ allowed to obtain the hyperfine splitting of the $2S$ energy level in muonic hydrogen, and also the values of the Zemach’s radius $r_Z=1.082(37)$ fm and magnetic radius $r_M=0.87(6)$ fm. The first measurement of three transition frequencies between energy levels $2P$ and $2S $ for muonic deuterium $(2S_{1/2}^{F=3/2}-2P_{3/2}^{F=5/2})$, $(2S_{1/2}^{F =1/2}-2P_{3/2}^{F =3/2})$, $ (2S_{1/2}^{F=1/2}-2P_{3/2}^{F=1/2})$ allowed to obtain in 2.7 times the more accurate value of the charge radius of the deuteron, which is also less than the value recommended by CODATA [@mohr], by $7.5 \sigma$ [@crema4]. As a result, a situation emerges when there is an inexplicable discrepancy between the values of such fundamental parameters, like the charge radius of a proton and deuteron, obtained from electronic and muonic atoms. In the process of searching for possible solutions of the proton charge radius “puzzle” various hypotheses were formulated, including the idea of the nonuniversality of the interaction of electrons and muons with nucleons. Preliminary experimental data for muonic helium ions show that there is no large discrepancy in obtained charge radii in comparison with CODATA.
In the experiments of the CREMA collaboration one very important task is solved: to obtain an order of magnitude more accurate values of the charge radii of the simplest nuclei (proton, deuteron, helion, alpha particle ....) that enter into one form or another into theoretical expressions for intervals of fine or hyperfine structure of the spectrum. In this case, high sensitivity of characteristics of the bound muon to distribution of charge density and magnetic moment of the nucleus is used. Successful realization of this program is possible only in combination with precise theoretical calculations of various energy intervals, measured experimentally. In this way, the problem of a more accurate theoretical construction of the particle interaction operator in quantum electrodynamics, the calculation of new corrections in the energy spectrum of muonic atoms acquires a special urgency [@paper1; @paper2; @paper3]. The aim of this work consists in the calculation of the deuteron, helion and triton polarizability correction to the hyperfine splitting (HFS). We perform a calculation of hadronic polarizability contribution using the isobar model describing the processes of photo- and electroproduction of $\pi$, $\eta$ mesons, nucleon resonances on the nucleon in the resonance region, and experimental data on the nucleon and deuteron polarized structure functions obtained in non-resonance region.
General formalism {#sec-1}
=================
The leading order polarizability contribution to HFS is determined by two-photon exchange diagrams, shown in Fig. \[fig-1\]. The corresponding amplitudes of virtual Compton scattering on the nucleus can be represented as a convolution of antisymmetric parts of the lepton and hadron tensors which have the following form [@Z; @cfm]: $$\label{eq:1}
L_{\mu\nu}^{A}=\frac{1}{4}Tr\Bigl\{(1+\gamma^0)\gamma_5\hat s_1\Bigl[\gamma_1^\mu \frac{\hat p_1+\hat k+m_1}
{(p_1+k)^2-m_1^2}\gamma_1^\nu+\gamma_1^\nu \frac{\hat p_1-\hat k+m_1}
{(p_1-k)^2-m_1^2}\gamma_1^\mu\Bigr]\Bigr\},$$ $$\label{eq:2}
W_{\mu\nu}^{A}=i\epsilon_{\mu\nu\alpha\beta}k^\alpha\Bigl\{s_2^\beta \frac{H_1(\nu,Q^2)}{(p_2\cdot k)}+
\frac{[(p_2k)s_2^\beta-(s_2k)p_2^\beta]}{(p_2\cdot k)^2}H_2(\nu,Q^2)\Bigr\},$$ where $m_1$, $m_2$ are the lepton and nucleus masses, the nucleus four-momentum $p_2=(m_2,0)$, $\epsilon_{\mu\nu\alpha\beta}$ is the totally antisymmetric tensor in four dimensions. $s_1$, $s_2$ are spin four vectors of the lepton and nucleus. $H_1$, $H_2$ are the structure functions of polarized scattering. The invariant quantity $p_2\cdot k$ is related to the energy transfer $\nu$ in the nucleus rest frame: $p_2\cdot k=m_2 \nu$. The invariant mass of the electroproduced hadronic system $W$ is then $W^2=m_2^2+2m_2\nu-Q^2=m_2^2+Q^2(1/x-1)$. After computing the convolution of two tensors and , we obtain: $$\label{eq:3}
L^{A}_{\mu\nu}W^{A}_{\mu\nu}=\frac{4}{3}({\bf s}_1{\bf s}_2)\frac{m_2k^2}{k^4-4m_1^2k_0^2}
\left[(k_0^2+2k^2)\frac{H_1}{(p_2\cdot k)}+3k_0^2k^2\frac{H_2}{(p_2\cdot k)^2}\right].$$ According to the optical theorem the imaginary part of the forward Compton amplitude is related to the cross section of inelastic scattering of off-shell photons from protons: $Im H_1(\nu,Q^2)=g_1(\nu,Q^2)/\nu$, $Im H_2(\nu,Q^2)=m_2g_2(\nu,Q^2)/\nu^2$. As a result, neglecting the lepton mass the nucleus polarizability contribution to HFS can be presented in the form [@mf2002; @mf2002a; @carlson1; @carlson]: $$\label{eq:4}
\Delta E^{hfs}_{pol}=\frac{Z\alpha m_1}{2\pi m_2\mu_N}E_F(\Delta_1+\Delta_2)=
(\delta_1^p+\delta_2^p)E_F=\delta_{pol}E_F,$$ $$\label{eq:5}
\Delta_1=\int_0^\infty\frac{dQ^2}{Q^2}\Bigl\{\frac{9}{4}F_2^2(Q^2)-
4m_2\int_{\nu_{th}}^\infty\frac{d\nu}{\nu^2}\beta_1(\theta)g_1(\nu,Q^2)\Bigr\},$$ $$\label{eq:6}
\Delta_2=-12m_2\int_0^\infty\frac{dQ^2}{Q^2}
\int_{\nu_{th}}^\infty \frac{d\nu}{\nu^2}\beta_2(\theta)g_2(\nu,Q^2),$$ where $\nu_{th}$ determines the pion-nucleus threshold: $$\label{eq:7}
\nu_{th}=m_\pi+\frac{m_\pi^2+Q^2}{2m_2},$$ and the functions $\beta_{1,2}$ have the form: $$\label{eq:8}
\beta_1(\theta)=3\theta-2\theta^2-2(2-\theta)\sqrt{\theta(\theta+1)},$$ $$\label{eq:9}
\beta_2(\theta)=1+2\theta-2\sqrt{\theta(\theta+1)},~\theta=\nu^2/Q^2.$$ $F_2(Q^2)$ is the Pauli form factor of the nucleus. The dependence on the mass of the lepton in - can also be taken into account, which leads to a certain modification of the functions $\beta_i(\nu,Q^2)$ [@cfm; @carlson]. This is important for increasing the accuracy of calculations in the case of muonic atoms.
![Two-photon Feynman amplitudes determining the correction of the nucleus polarizability to the hyperfine splitting of muonic atom.[]{data-label="fig-1"}](hat1.eps)
Having information on the polarization structure functions of the nuclei, one can integrate into and obtain this correction. The nucleus spin-dependent structure functions $g_1(\nu,Q^2)$, $g_2(\nu,Q^2)$ can be measured in the inelastic scattering of polarized leptons on polarized nuclei. Accurate measurements of proton and deuteron polarized structure functions were made in SLAC, CERN and DESY [@Abe1; @Abe2; @Anthony; @Mitchell; @Adams; @Adeva]. These experimental data can be used for the calculation the contribution in the nonresonance region where the invariant mass $W$ must be greater than the mass of any resonance $N^\ast$ in the reaction $\gamma^\ast+N\to N^\ast$. The threshold between the resonance region and the deep-inelastic region is not well defined, but it is usually taken to be at about $W^2=4~GeV^2$. On the other hand in the resonance region we need theoretical model describing polarized structure functions $g_{1,2}(\nu,Q^2)$ since experimental data in this region are clearly insufficient. First of all, we can express functions $g_{1,2}(\nu,Q^2)$ in terms of virtual photon absorption cross sections as follows: $$\label{eq:10}
g_1(\nu,Q^2)=\frac{m_2\cdot K}{8\pi^2\alpha(1+Q^2/\nu^2)}\left[\sigma^T_{1/2}
(\nu,Q^2)-\sigma^T_{3/2}(\nu,Q^2)+\frac{2\sqrt{Q^2}}{\nu}\sigma^{TL}_{1/2}(\nu,Q^2)\right],$$ $$\label{eq:11}
g_2(\nu,Q^2)=\frac{m_2\cdot K}{8\pi^2\alpha(1+Q^2/\nu^2)}\left[-\sigma^T_{1/2}
(\nu,Q^2)+\sigma^T_{3/2}(\nu,Q^2)+\frac{2\nu}{\sqrt{Q^2}}\sigma^{TL}_{1/2}(\nu,Q^2)\right],$$ where $K$ is the kinematical flux factor for virtual photons. The virtual photon absorption cross sections have superscripts referring to the initial and final photon polarization being longitudinal L or transverse T. The superscript TL is for the case when the photon polarization direction changes during the interaction. The subscripts refer to the total spin of the photon-nucleus system.
Let us briefly describe the basic formulas that underlie the numerical results. To construct the polarized structure functions - in resonance region we use the Breit-Wigner parameterization for the photoabsorption cross sections [@Walker; @Arndt; @Teis1; @Teis2; @Krusche; @Bianchi; @D; @Dong]. In the considered region of the variables $k^2$, $W$ the most important contribution is given by five resonances: $P_{33} (1232)$, $S_{11} (1535)$, $D_{13} (1520)$, $P_{11} (1440)$, $F_{15} (1680)$. Accounting for the resonance decays to the $N\pi-$ and $N\eta-$ states we can express the absorption cross sections $\sigma^T_{1/2}$ and $\sigma^T_{3/2}$ as follows: $$\label{eq:12}
\sigma^T_{1/2,3/2}=\left(\frac{k_R}{k}\right)^2\frac{W^2\Gamma_\gamma\Gamma_{R
\rightarrow N\pi}}{(W^2-M_R^2)^2+W^2\Gamma_{tot}^2}\frac{4m_N}{M_R\Gamma_R}
|A_{1/2,3/2}|^2,$$ where $A_{1/2,3/2}$ are transverse electromagnetic helicity amplitudes, $$\label{eq:13}
\Gamma_\gamma=\Gamma_R\left(\frac{k}{k_R}\right)^{j_1}\left(\frac{k_R^2+X^2}
{k^2+X^2}\right)^{j_2},~~X=0.3~GeV.$$ The resonance parameters $\Gamma_R$, $M_R$, $j_1$, $j_2$, $\Gamma_{tot}$ are taken from [@PDG; @Teis1; @Teis2]. In accordance with [@Teis1; @Krusche] the parameterization of one-pion decay width is $$\label{eq:14}
\Gamma_{R\rightarrow N\pi}(q)=\Gamma_R\frac{M_R}{M}\left(\frac{q}{q_R}\right)^3
\left(\frac{q_R^2+C^2}{q^2+C^2}\right)^2,~C=0.3~GeV$$ for the $P_{33}(1232)$ and $$\label{eq:15}
\Gamma_{R\rightarrow N\pi}(q)=\Gamma_R\left(\frac{q}{q_R}\right)^{2l+1}
\left(\frac{q_R^2+\delta^2}{q^2+\delta^2}\right)^{l+1},$$ for resonances $D_{13}(1520)$, $P_{11}(1440)$, $F_{15}(1680)$. $l$ is the pion angular momentum and $\delta^2$ = $(M_R-$ $m_N-m_\pi)^2$ + $\Gamma_R^2/4$. Here $q$ $(k)$ and $q_R$ $(k_R)$ denote the c.m.s. pion (photon) momenta of resonances with the mass M and $M_R$ respectively. In the case of $S_{11}(1535)$ we take into account $\pi N$ and $\eta N$ decay modes [@Krusche]: $$\label{eq:16}
\Gamma_{R\rightarrow\pi,\eta}=\frac{q_{\pi,\eta}}{q}b_{\pi,\eta}\Gamma_R
\frac{q_{\pi\eta}^2+C_{\pi,\eta}^2}{q^2+C_{\pi,\eta}^2},$$ where $b_{\pi,\eta}$ is the $\pi$ ($\eta$) branching ratio.
The cross section $\sigma^{TL}_{1/2}$ is determined by an expression similar to , containing the product $(S^\ast_{1/2}\cdot A_{1/2}+A_{1/2}^\ast S_{1/2})$ [@Abe1]. The calculation of helicity amplitudes $A_{1/2}$, $A_{3/2}$ and longitudinal amplitude $S_{1/2}$, as functions of $Q^2$, was done on the basis of constituent quark model (CQM) in [@Dong2; @Isgur; @CL; @Capstick; @LBL; @Warns]. The program of numerical calculation of cross sections was successfully realized by the authors of [@maid1; @maid2] within the unitary isobar model framework known as the MAID package (http://www.kph-uni-mainz.de/MAID). In the unitary isobar model [@maid1; @maid2] accounting for the Born terms, the vector meson, nucleon resonance contributions and interference terms we calculate the cross sections $\sigma^{T}_{1/2,3/2}$, $\sigma^{TL}_{1/2}$ by means of numerical program MAID in the resonance region as the functions of two variables $W$ and $Q^2$. The obtained nucleon polarized structure functions $g_{1,2}(W,Q^2)$ are then used for a construction of nucleus structure functions and calculation the polarizability contribution. In Figs. \[fig-2\],\[fig-3\] we show the obtained structure functions $g_{1,2}(W,Q^2)$.
![The deuteron polarized structure function $g_1^d(W,Q^2)$ as function of $Q^2$ $(0\div 2.0.~GeV^2$ and W $(1.1\div 2.0)~GeV$.[]{data-label="fig-2"}](g1d.eps)
Our calculation of polarizability contribution in deep inelastic region is based on experimental data from [@Abe1; @Abe2; @Anthony; @Mitchell; @Adams; @Adeva]. As was shown in previous paper [@cfm] nucleon polarized structure functions can be expressed through polarized quark and gluon distributions which obey to the evolution equations [@altarelli; @hirai]. Solving $Q^2$ evolution equations we can construct nucleon functions $g_1(\nu,Q^2)$, $g_2(\nu,Q^2)$ which agree well with experimental data and the following parameterization [@Abe1; @Abe2; @Anthony; @Mitchell; @Adams; @Adeva; @erbacher]: $$\label{eq:17}
g_1^{p,d}(x,Q^2)=a_1x^{a_2}(1+a_3x+a_4x^2)[1+a_5f(Q^2)]F_1^{p,d}(x,Q^2),$$ where the superscript index p,d corresponds to the proton or deuteron. Numerical integration is performed with $f(Q^2)=-\ln Q^2$ corresponding to the perturbative QCD behaviour. The calculation of the second part of the correction $\delta_{pol}$ in in nonresonance region is carried out by means of Wandzura-Wilchek relation as in [@cfm].
![The deuteron polarized structure function $g_2^d(W,Q^2)$ as function of $Q^2$ $(0\div 2.0.~GeV^2$ and W $(1.1\div 2.0)~GeV$.[]{data-label="fig-3"}](g2d.eps)
Numerical results {#sec-2}
=================
Most part of numerical calculation is devoted to muonic deuterium. To the hadronic contribution we include contributions that are determined by the nuclear reactions of the production of $\pi$-, $\eta$, and $K$-mesons on nucleons, the production of nucleon resonances. In the approximation, which is then used for the calculation the deuteron appears as a loosely coupled system of the proton and neutron, so the deuteron polarized structure function can be presented as a sum of the proton and neutron structure functions: $$\label{eq:18}
g_i^d(W,Q^2)=g_i^p(W,Q^2)+g_i^n(W,Q^2).$$ The MAID program allows us to calculate separately the proton and neutron structure functions $g_i^{p,n}(W,Q^2)$.
----------------------------------- ---------------------- ----------------------
Contribution to GDH integral $I_{GDH}^p$, $\mu b$ $I_{GDH}^n$, $\mu b$
Contribution of $N\pi$ states 165.9 133.2
Contribution of $N\eta$ states -8.9 -5.7
Contribution of K-mesons -1.8 -3.0
Contribution of $ N\pi\pi$ states 47.8 50.5
Total contribution 203.0 175.0
The GDH value 204.8 232.5
----------------------------------- ---------------------- ----------------------
: The contributions to the GDH sum rule for the proton and neutron.[]{data-label="tb1"}
For the integration in an important role is played by the Gerasimov-Drell-Hern (GDH) sum rule [@sbg; @dh; @krein; @costanza; @aa] which connects an energy-weighted integral of the difference of the helicity dependent real-photon absorption cross sections with the anomalous contribution $\kappa=\frac{\mu_N m_2}{es_2}-Z$ to the magnetic moment $\mu$ of the nucleus: $$\label{eq:19}
I_{GDH}=\int_{\nu_{th}}^\infty\frac{\sigma_p-\sigma_a}{\nu}d\nu=4\pi^2\kappa^2\frac{\alpha s_2}{m_2^2},$$ where $\nu$ is the photon energy, $\sigma_p$ and $\sigma_a$ are the total photoabsorption cross sections for parallel and antiparallel orientation of photon and nucleus spins, respectively. The lower limit of the integral, $\nu_{th}$, corresponds to pion production and photodisintegration threshold for a nucleonic and nuclear target, respectively. Strictly speaking, in order for the integral in to converge over the variable $Q$, the sum rule must exactly be satisfied. The GDH sum rule is satisfied for any nucleus, including a proton and a neutron. Since the deuteron structure function satisfies , it is necessary to achieve the sum rules for the proton and neutron. Since the neutron and the proton have large anomalous magnetic moments $\kappa_p=1.79$, $\kappa_n=-1.91$), we obtain large values for the integral $I^{p,n}_{GDH}$ . In turn, in the case of a deuteron, a small value of AMM ($\kappa_d=-0.143$) leads to a small value $I^{d}_{GDH}=-0.65~\mu b$ , which is two orders of magnitude smaller than the corresponding values for nucleons. When the deuteron is represented in the form of a state of two almost free nucleons, we find that the quantity in right part of is equal to 437 $\mu b$. There are also other channels ($\gamma d\to pn$) which can not be treated in the quasi-free approximation but which contribute to . An analysis carried out in [@ha1; @ha2] showed that this value decreases significantly when taking into account the photodisintegration channel, which is not considered in this paper. Therefore, representing the deuteron in the form of the sum of a proton and a neutron, we add in two terms with the Pauli form factors of the proton and the neutron to ensure the fulfilment of the sum rule and the subsequent integration into . Calculations in the MAID show that the sum rule for a proton is satisfied with a sufficiently high accuracy, whereas for a neutron the difference between the left and right parts in reaches $30\%$ (see Table \[tb1\]).
To avoid this difficulty and obtain however an estimate of hadronic contribution to the neutron and deuteron polarizability we introduce a cutoff of the momentum integral over Q in at some value $\kappa\approx 0.01$ GeV, supposing that the GDH sum rule for the neutron holds exactly and the region of small Q $(0\div 0.01)$ GeV does not give essential contribution to general value of correction . Similar cutoff procedure at small values Q was used in [@khmil] with $\kappa\approx 0.045$ GeV, in which different corrections to deuterium hyperfine structure were considered from the two-photon exchange amplitudes. As a result total value of the polarizability correction for muonic deuterium including the resonance and nonresonance regions is equal to 0.13 meV. In the case of He-3 total polarizability contribution 0.06 meV is determined by unpaired neutron because two protons have opposite spins and do not contribute to hyperfine splitting. The similar situation occurs for the triton in which two neutrons form closed shell. In muonic tritium the polarizability correction is equal to 0.05 meV. Total error of our calculation is estimated in $30~\%$ which is determined mainly by the uncertainty from two-pion contribution and the cutoff procedure used above. The obtained values of polarizability corrections should be used for obtaining total values of hyperfine splitting in light muonic atoms [@apm2004; @apm2008; @paper1; @apm2017].
The work is supported by Russian Foundation for Basic Research (grant No. 16-02-00554).
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[^1]:
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---
abstract: 'We have experimentally studied the pump-probe Kerr rotation dynamics in WSe$_2$ monolayers. This yields a direct measurement of the exciton valley depolarization time $\tau_v$. At $T= 4$ K, we find $\tau_v\approx 6$ ps, a fast relaxation time resulting from the strong electron-hole Coulomb exchange interaction in bright excitons. The exciton valley depolarization time decreases significantly when the lattice temperature increases with $\tau_v$ being as short as 1.5 ps at 125 K. The temperature dependence is well explained by the developed theory taking into account exchange interaction and fast exciton scattering time on short-range potential.'
author:
- 'C.R. Zhu$^1$'
- 'K. Zhang$^1$'
- 'M. Glazov$^2$'
- 'B. Urbaszek$^3$'
- 'T. Amand$^3$'
- 'Z. W. Ji$^4$'
- 'B.L. Liu$^1$'
- 'X. Marie$^3$'
title: ' Exciton Valley Dynamics probed by Kerr Rotation in WSe$_2$ Monolayers'
---
Monolayers of transition metal dichalcogenides (TMDC), such as MoS$_2$ and WSe$_2$, are two-dimensional (2D) semiconductors with strong light absorption and emission associated to direct optical transitions [@Butler:2013a; @Mak:2010a; @Splendiani:2010a]. The optical properties of these 2D crystals are strongly influenced by excitons, Coulomb bound electron-hole pairs, with experimentally determined binding energies of up to 0.6 eV ($\sim 1/3$ of the optical bandgap $\sim 1.7$ eV) [@Cheiwchanchamnangij:2012a; @Ye:2014a; @Zhu:2014a; @Klots:2014a; @Ugeda:2014a; @He:2014a; @Wang:2014a]. Due to the combined effect of inversion symmetry breaking and strong spin-orbit interaction, the interband transitions are governed by chiral selection rules which allow efficient optical initialization of an electron-hole pair in a specific $K$-valley in momentum space [@Xiao:2012a; @Cao:2012a; @Mak:2012a; @Sallen:2012a; @Zeng:2012a]. The circular polarization ($\sigma^+$ or $\sigma^-$) of the absorbed or emitted photon can be directly associated with selective carrier excitation in one of the two non-equivalent valleys, $K_+$ or $K_-$, respectively, formed at the edges of the Brillouin zone. Hence, the spin state of an exciton $S_z=\pm1$ is correlated with its valley state $K_\pm$. Recent time-resolved studies demonstrate photoluminescence (PL) lifetimes in the picosecond range indicating high exciton oscillator strengths [@Korn:2011a; @Lagarde:2014a; @Wang:2014b]. Pump-probe absorption and reflectivity measurements in monolayer (ML) MoS$_2$ have also shown polarization decay times in the picosecond range [@Shi:2013b; @Wang:2013d; @Mai:2014a] corresponding to fast relaxation of the valley polarization, which is surprising as the valley degree of freedom is expected to be protected by the considerable single particle spin splittings in the valence and conduction bands [@Liu:2013a].
In this work we use for the first time a powerful technique, namely time-resolved Kerr Rotation (TRKR) [@Dyakonov:2008a], to investigate exciton dynamics in ML WSe$_2$. TRKR allows, in contrast to time-resolved PL spectroscopy, to address the spin states of both photocreated and resident carriers polarized by a $\sigma^+$ or $\sigma^-$ polarized pump laser. We detect spin-Kerr signal for the neutral exciton, well separated from the charged exciton signal, and uncover a valley relaxation time of $\tau_v=6$ ps at $T=4$ K, which we attribute to the strong Coulomb exchange interaction between the electron and the hole. The strongly bound excitons in ML WSe$_2$ [@Wang:2014b] give access in TRKR to a novel temperature-dependent regime of exciton spin dynamics: in standard quasi-2D semiconductor systems like GaAs quantum wells excitons are ionized as the temperature increases since their binding energy is small ($\sim10$ meV) [@Vinattieri:1994a; @Dareys:1993a; @Maialle:1993a]. The temperature dependence of the exciton spin dynamics is therefore inaccessible. This is in contrast to very robust excitons in ML WSe$_2$, where we can investigate this evolution. We measure a temperature induced decrease of $\tau_v$ that we interpret in terms of the temperature dependence of the exchange interaction induced exciton spin relaxation. Our theory describes very satisfactorily the experimental data. Moreover, we found no evidence of transfer of spin/valley polarization to resident carriers in WSe$_2$ ML contrary to similar time-resolved Kerr experiments performed in III-V or II-VI semiconductors [@Awschalom:1997a; @Glazov:2012a].
![\[fig:fig1\] (a) Optical reflection image of the sample indicating the investigated WSe$_2$ ML flake (the white zone corresponds to multiple layers). (b) Temperature dependence of the WSe$_2$ ML PL spectra. (c) Right ($\sigma^+$) and left ($\sigma^-$) circularly polarized PL components at $T=4$ K under *cw* $\sigma^+$ polarized He-Ne laser excitation ($E_l=1.96$ eV). ](Figure1){width="40.00000%"}
![\[fig:fig2\] Kerr rotation dynamics at $T=4$ K for $\sigma^+$ and $\sigma^-$ pump beam. The laser excitation energy is $E_l=1.735$ eV. Inset: schematics of the optical selection rules of the excitons in $K_\pm$ valleys and their coupling induced by the long-range exchange interaction.](Figure2){width="40.00000%"}
The investigated monolayer WSe$_2$ flakes are obtained by micro-mechanical cleavage of a bulk WSe$_2$ crystal (from SPI Supplies, USA) on 90 nm SiO$_2$ on a Si substrate. The 1ML region is identified by optical contrast, Fig. \[fig:fig1\]a, and very clearly in PL spectroscopy [@Mak:2010a]. The sample is excited near normal incidence with degenerate pump and delayed probe pulses from a mode-locked Ti:Sapphire laser ($\sim$120 fs pulse duration, 76 MHz repetition frequency). The laser beams are focused to a spot size of $\sim$ 5$\mu$m, and the pump and probe beams have average power of 300 $\mu$W and 30 $\mu$W, respectively. This corresponds to a typical pump-generated exciton density of about 10$^{12}$ cm$^{-2}$. The circularly polarized pump pulse incident normal to the sample creates spin-polarized electronic excitations with the spin vector perpendicular to the flake plane. The temporal evolution of the carrier spins is recorded by measuring the Kerr rotation angle $\theta(\Delta t)$ of the reflected linearly polarized probe pulse while sweeping the delay time $\Delta t$, which correspond to the net spin component normal to the sample plane [@Liu:2007a; @Dyakonov:2008a; @Glazov:2012a]. Continuous-wave (*cw*) micro-PL is performed with a laser excitation energy $E_l=1.96$ eV with a standard monochromator coupled to a CCD detection system. Figure \[fig:fig1\]b displays the temperature dependence of the PL spectra from $T=4$ K to 300 K. Similarly to previous studies, three features can be identified at $T=4$ K [@Jones:2013a; @Wang:2014b]: the emission peaks at $E=1.742$ eV and $E=1.714$ eV correspond to the recombination of neutral exciton and charged exciton (trion) respectively, Fig. \[fig:fig1\]c. Considering the commonly observed residual *n*-type doping [@Radisavljevic:2013a; @Wang:2014b], the trion charge is assumed to be negative but this assumption is not critical for the present study and the discussion below. The identification of these transitions is based on the emission polarization analysis [@Jones:2013a; @Wang:2014b]. Under $\sigma^+$ polarized excitation light, the exciton and trion PL peaks are characterized a significant circular polarization degree (typically 33% and 23% respectively at $T=4$ K), note the different intensities of the $\sigma^+$ and $\sigma^-$ PL components in Fig. \[fig:fig1\]c, which demonstrate the optical initialization of valley polarization [@Xiao:2012a; @Cao:2012a; @Mak:2012a; @Sallen:2012a; @Zeng:2012a]. In contrast, only the exciton line exhibits linear polarization degree following a linearly polarized light excitation (not shown) as a consequence of the creation of a coherent superposition of valley states [@Jones:2013a]. The clear separation by 30 meV of the trion and the neutral exciton in WSe$_2$ ML is a major advantage compared to state-of-the-art MoS$_2$ ML samples where the two lines can not be resolved [@Plechinger:2014a]. Below the trion emission several emission peaks are observed at low temperature (labelled L in Fig. \[fig:fig1\]c); these peaks already observed in MoS$_2$ or WSe$_2$ MLs [@Xiao:2012a; @Cao:2012a; @Mak:2012a; @Wang:2014b] have been assigned to localized exciton complexes. In this work we focus on the temperature dependence of the neutral exciton dynamics. As shown in Fig. \[fig:fig1\]b, the exciton emission remains strong as the temperature increases (we observe clearly the red shift of the line) whereas both the trion and localized state emission vanish for $T\gtrsim 100$ K.
Figure \[fig:fig2\] shows the Kerr rotation dynamics $\theta(\Delta t)$ measured at 4 K for both $\sigma^+$ and $\sigma^-$ polarized pump pulses. The pump energy $E_{l}= 1.735$ eV is set to the maximum of the Kerr signal, which is very close to the neutral exciton transition identified in the PL spectra Fig. \[fig:fig1\]c. The observed sign reversal of the Kerr signal in Fig. \[fig:fig2\] at the reversal of pump helicity is a consequence of the optical initialization of the $K_+$ and $K_-$ valley, respectively. We have measured in the same conditions the transient reflectivity using linearly cross-polarized pump and probe pulses. As shown in Fig. \[fig:fig3\]b, the reflectivity decay time is about ten times longer than the one observed in TRKR. Thus the mono-exponential decay time $\tau_v=(6 \pm 0.1)$ ps of the Kerr rotation dynamics at $T=4$ K in Fig. \[fig:fig2\], probes directly the fast exciton valley depolarization. In agreement with recent numerical estimations, it results from the strong long-range exchange interaction [@Glazov:2014a; @Yu:2014a]. Due to limited time resolution, previous investigations by time-resolved PL spectroscopy could not yield the measurement of the exciton valley depolarization time [@Lagarde:2014a; @Wang:2014b]. Kerr rotation dynamics used here is characterized by a higher time resolution ($\sim$100 fs) and allows us a strictly resonant excitation of the exciton.
![\[fig:fig3\] (a) Kerr rotation dynamics after a $\sigma^+$ polarized pump pulse for different lattice temperatures; (b) Transient reflectivity dynamics for different temperatures. The laser excitation energies are identical to the ones used in (a), see text; (c) Temperature dependence of the measured (symbol) and calculated after Eq. (solid and dashed lines) exciton valley polarization relaxation time, see text for details. ](Figure3){width="50.00000%"}
Figure \[fig:fig3\]a displays the variation of the TRKR dynamics as a function of the temperature. For each temperature, the laser excitation energy is set at the maximum Kerr rotation signal which follows well the energy of the neutral exciton PL shown in Fig. \[fig:fig1\]b. For $T\gtrsim 30$ K, we observe a clear decrease of the exciton valley polarization decay time $\tau_v$ down to 1.5 ps at $T=125$ K; for higher temperatures the signal-to-noise ratio is too small to get reliable data. We stress that the transient reflectivity measurements performed in the same temperature range exhibit much longer decay times with very weak temperature dependence, Fig. \[fig:fig3\]b [@Shi:2013b]. We have also investigated the excitation power dependences of the exciton dynamics. In the studied power range corresponding to variation of exciton density from $\sim 1.5 \times 10^{11}$ to $10^{12}$ cm$^{-2}$, both the TRKR and reflectivity dynamics do not depend on the exciton photo-generated density within the experimental accuracy (not shown). This demonstrates that the exciton-exciton interactions play a minor role in exciton valley dynamics presented in Figs. \[fig:fig2\] and \[fig:fig3\].
In order to describe quantitatively the experimental findings we note that the Kerr rotation angle $\theta$ is proportional to the total spin of the exciton ensemble $\bm S=\sum_{\bm K} \bm S_{\bm K}$ (the mechanisms of Kerr rotation by exciton spins in WSe$_2$ MLs are similar to that in quasi-2D semiconductors and can be described as in Ref. [@Glazov:2012a]); $\bm S_{\bm K}$ is the spin distribution function (with $S_z=\pm 1$ corresponding to $K_{\pm}$ valleys) and $\bm K$ is the wavevector of exciton. The spin/valley dynamics of excitons is governed by the long-range exchange interaction between an electron and a hole [@Glazov:2014a; @Yu:2014a] which acts as an effective magnetic field $\bm \Omega_{\bm K} = \alpha K (\cos{2\vartheta}, \sin{2\vartheta})$ on the exciton spin. Here $\vartheta$ is the polar angle of $\bm K$ and the constant $\alpha$ depends on the oscillator strength of exciton transition and system geometry. Following Ref. [@Glazov:2014a] for the system “vacuum – 1ML of WSe$_2$ – substrate” we obtain, assuming the same background refractive index $n$ of the ML and the substrate, $\alpha = c\Gamma_0 (n+1)/[(n^2+1)\omega_0]$, where $\omega_0$ is the exciton resonance frequency, $\Gamma_0$ is its radiative lifetime [^1]. This effective field causes spin precession of excitons which is randomized by the scattering and described by the kinetic equation [@Maialle:1993a; @Glazov:2014a] ${\partial \bm S_{\bm K}}/{\partial t} + S_{\bm K} \times \bm \Omega_{\bm K} = \bm Q\{ \bm S_{\bm K}\}$, where $Q\{ \bm S_{\bm K}\}$ is the collision integral. Assuming that the excitons are thermalized and the scattering is caused by short-range potential we obtain for spin/valley relaxation rate $\tau_{zz}^{-1}$ $$\label{tau:zz}
\frac{1}{\tau_{zz}} = \frac{2\alpha^2 \tau M k_B T}{\hbar^2},$$ where $M=0.67m_0$ is the exciton mass [@Wang:2014b], $\tau$ is the scattering time. Equation is valid provided that the radiative lifetime of excitons exceeds $\tau_{zz}$, $k_B T \tau/\hbar \gg 1$ and $\alpha^2 M k_B T \tau^2/\hbar^2 \ll 1$. The second condition means that the average kinetic energy must be larger than the homogeneous broadening and the latter that the spin relaxation occurs in the spin diffusive regime. Figure \[fig:fig3\]c shows the experimentally measured exciton polarization decay times (points are extracted from the monoexponential decay presented in Fig. \[fig:fig3\]a) and theoretical calculations carried out for $\Gamma_0 = 0.16$ ps$^{-1}$ (corresponding to the radiative lifetime of the states in the light cone $1/(2\Gamma_0) = 3$ ps, a value consistent with measurements [@Wang:2014b]) and $n=\sqrt{10}$. Qualitatively, the drop of the exciton spin/valley relaxation time when $T$ increases can be well explained by the increase with the temperature of the effective magnetic field $\bm \Omega_{\bm K}$, which makes spin precession and decoherence faster, Eq. . The solid and dashed lines in Fig. \[fig:fig3\]c correspond to the calculated exciton spin/valley relaxation time for two different scattering times $\tau=0.066$ ps and $\tau=0.25$ ps, respectively. The latter value of $\tau$ corresponds to exciton energy uncertainty equivalent to $30$ K corresponding to Ioffe-Regel criterion of delocalization where the product of the thermal wavevector $k_T=(2Mk_BT)^{1/2}/\hbar$ and the mean free path $l=\tau \hbar k_T/M$ is on the order of $1$. Remarkably, we observe a nice agreement between the calculated and measured exciton relaxation times in Fig. \[fig:fig3\]c for $T> 30$ K with the scattering time $\tau=0.066$ ps, but formal criterion of kinetic equation is fulfilled only for $T\gtrsim 100$ K [^2]
For $4<T<30$ K, the measured exciton spin relaxation time is temperature independent (the sample temperature was carefully checked). In this temperature range, the PL spectra in Fig. \[fig:fig1\]b are also identical whereas shifts of the peaks energies and relative changes of intensities are clearly observed for larger temperatures. We believe that this behavior could be either due to (i) a regime where K$_BT$ is smaller than the collision broadening leading to a temperature independent spin relaxation time [@Glazov:2014a] or (ii) a localized character of the exciton below 30 K .
Finally we emphasize that the Kerr rotation signal for delay times longer than $\sim$ 25 ps vanishes within our experimental accuracy whatever the temperature is. This means that we get no evidence of the transfer of spin polarization to the resident carriers though they are clearly present as shown by the detection of the trion line in Fig. \[fig:fig1\]. This behavior is probably due to the very robust spin/valley polarization of single particles, electrons and holes, in TMDC 2D layers [@Xiao:2012a], because the polarization transfer from photogenerated carriers to resident ones *n*-doped III-V or II-VI semiconductors usually requires a single particle spin-flip [@Glazov:2012a].
In conclusion, we have demonstrated that the exciton valley dynamics in WSe$_2$ monolayer can be directly probed by time-resolved Kerr rotation dynamics performed in resonant excitation conditions. The temperature dependence of the exciton depolarization time is well described by the Coulomb exchange interaction induced exciton spin/valley dephasing assuming efficient scattering on short-range potential.
This work was supported by the National Science Foundation of China Grant No. 11174338, Programme Investissements d’Avenir ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT and ERC Grant No. 306719. X.M. acknowledges the support by the Chinese Academy of Sciences Visiting Professorship program for Senior International Scientists. Grant No. 2011T1J37. M.G. acknowledges the support by Russian Science Foundation (project 14-12-01067).
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[^1]: In the electrodynamical approach of Ref. [@Glazov:2014a] to the exciton LT-splitting, the reflection at the boundary “vacuum – 1 ML” yields the replacement $\Gamma_{0,\alpha} \to \Gamma_{0,\alpha}(1+r_\alpha)$, where $\alpha=s,p$ and $r_\alpha$ are given by the Fresnel formula with background refractive index $n$.
[^2]: The possible origins of the scattering are: (i) short-range defects, (ii) exchange scattering with resident electrons. In the latter case for resident electron density $n_e \sim 10^{12}$ cm$^{-2}$ the exchange exciton-electron [@tarasenko98] scattering yields $\tau \sim 10^{-13}$ s.
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---
author:
- 'Richard Mould[^1]'
title: '**Auxiliary nRules of Quantum Mechanics**'
---
Standard quantum mechanics makes use of four auxiliary rules that allow the Schrödinger solutions to be related to laboratory experience – such as the Born rule that connects square modulus to probability. These rules (here called the *sRules*) lead to some unacceptable results. They do not allow the primary observer to be part of the system. They do not allow individual observations (as opposed to ensembles) to be part of the system. They make a fundamental distinction between microscopic and macroscopic things, and they are ambiguous in their description of secondary observers such as Schrödinger’s cat.
The *nRules* are an alternative set of auxiliary rules that avoid the above difficulties. In this paper we look at a wide range of representative experiments showing that the nRules adequately relate the Schrödinger solutions to empirical experience. This suggests that the sRules should be abandoned in favor of the more satisfactory nRules, or a third auxiliary rule-set called the *oRules*.
Introduction {#introduction .unnumbered}
============
Quantum mechanics traditionally places the observer ‘outside’ of the system being studied, and refers only to ensembles of data. The theory does not refer to the primary data obtained in individual trials. So standard quantum mechanics is incomplete for two reasons – it excludes the primary observer *and* the primary data. For a theory that claims to be a fundamental law of nature, it excludes too much of nature.
Standard theory also makes a distinction between microscopic and macroscopic systems, and this is ambiguous at best. In addition, if one tries to use a quantum mechanical superposition of states to define an ontology in an individual trial, and if a secondary observer like Schrödinger’s cat is included in the system, then the result will not only be paradoxical, it will be wrong. The cat is not in a dual state of consciousness at any time during that famous experiment.
Standard quantum mechanics posits four auxiliary rules that lead to these unacceptable results. These rules (called the *sRules*) are: (**1**) Quantum measurement occurs only when a quantum mechanical microscopic (or possibly mesoscopic) system engages a macroscopic measuring device. (**2**) Primary data is collected during individual trials of this kind in which single eigenvalues are chosen by a stochastic process. (**3**) This choice is followed by a state reduction (i.e., the collapse) in which a stochastically chosen eigenvalue is the sole survivor. And (**4**), the probability of choosing a given eigenvalue is equal to its square modulus (i.e., the Born rule). These auxiliary rules are not contained in the Schrödinger equation. They are supplementary instructions that tell us how to use the Schrödinger equation[^2].
Other auxiliary rules are possible – rules that avoid the difficulties described above. There are at least two such rule-sets called the *nRules* and the *oRules* that correctly relate Schrödinger’s equation to observation. Both are ontologically based, for they place the primary observer ‘inside’ the system – or at least they allow for that possibility. With either of these rule-sets (as in classical physics) the observer of an external system can extend the system to include himself so he can become a continuous part of the wider system. That is not possible with the auxiliary rules of standard quantum mechanics, where the observer is only allowed to “peek" at the system from time to time.
In addition, the nRules provide an ontological description of individual trials, so they include the primary data obtained in each trial. They give us a running description of each trial. They are sequentially deterministic except for the time interval between stochastic choices, inasmuch as they cannot predict when the next stochastic choice will occur. The nRules are more deterministic than the sRules, but they are less deterministic than classical physics. The oRules are the least deterministic. Both the nRules and the oRules present an ontology for individual trials that removes the ambiguity that is now associated with Schrödinger’s cat.
The proposed rule-sets do not make a fundamental distinction between microscopic and macroscopic things. Each has four rules that apply equally to all parts of nature. As a result, both microscopic and macroscopic systems may be said to experience quantum jumps; and in the nRule case, microscopic states can sometimes undergo state reduction. Examples are given in this and related papers. Neither one of the proposed rule-sets directly includes the Born Rule that connects probability with square modulus; for in both cases, probability is introduced only through *probability current*.
This paper is concerned only with the nRules. Their adequacy is demonstrated in a number of cases, and their properties (described above) are made apparent.
Ontology and Epistemology {#ontology-and-epistemology .unnumbered}
=========================
The method of this paper differs from that of traditional quantum mechanics in that it sees the observer in an ontological rather than an epistemological context. Traditional or standard quantum theory (i.e., Copenhagen) places the observer outside of the system where operators and/or operations are used to obtain information about the system. This is the epistemological model shown in .
The large OP in Fig. 1 might be a mathematical ‘operator’ or a corresponding physical ‘operation’. The observer makes a measurement by choosing a formal operator that is associated with a chosen laboratory operation. As a result, the observer is forever outside of the observed system – making operational choices. The observer is forced to act apart from the system as one who poses theoretical and experimental questions to the system, and he can only get answers through ‘instantaneous’ contacts with the system at a given time. This model is both useful and epistemologically sound.
However, the special rules developed in this paper apply to the system by itself, independent of the possibility that an observer may be inside, and disregarding everything on the outside. This is the ontological model shown in .
A measurement occurring inside this system is not represented by a formal operator. Rather, it is represented by a measuring device that is itself part of the system. If the sub-system being measured is $S$ and a detector is $D$, then a measurement interaction is given by $\Phi = SD$. If an observer joins the system in order to look at the detector, then the system becomes $\Phi = SDB$, where $B$ is the brain state of the observer. Contact between the observer and the observed is continuous in this case.
The ontological model is able to place the observer inside the universe of things and give a full account of his conscious experience there. It is a departure from traditional quantum mechanics and has three defining characteristics: (1) It includes observations given by $\Phi = SDB$ as described above, (2) it allows all conscious observations to be continuous, and (3) it rejects the long-standing Born interpretation of quantum mechanics by introducing probability (only) through the notion of *probability current*.
Quantum mechanical measurement is said to refer to ensembles of observations but not to individual observations. In this paper I propose a set of four *nRules (1-4)* that apply to individual measurements in the ontological model. I claim that they are a consistent and complete set of rules that can give an ontological description of any individual measurement or interaction in quantum mechanics. These rules are not themselves a formal theory of measurement. I make no attempt to understand *why* they work, but strive only to insure that they do work. Presumably, a formal theory can one day be found to explain these rules in the same way that atomic theory now explains the empirically discovered rules of atomic spectra, or in the way that current theories of measurement aspire to merge with standard quantum mechanics, or make the neurological connection with conscious observation.
The oRules {#the-orules .unnumbered}
==========
Other papers [@RM1; @RM2; @RM3] propose another set of rules called the oRules (1-4). These are similar to the nRules except that the basis states of reduction are confined to observer brain states, reflecting the views of Wigner and von Neumann. Like the nRules, they introduce probability through the notion of ‘probability current’ rather than through square modulus, and they address the state reduction of conscious individuals in an ontological context, thereby giving us an alternative quantum friendly ontology. In Ref. 3 they are called simply the *rules (1-4)*. Like the nRules, the oRules are not a formal theory of measurement for they require a wider theoretical framework to be understood. I do not finally choose one of the rule-sets or propose an explanatory theory. I am only concerned with how state reduction might occur in each case. For the above stated reasons, both of these rule-sets are more acceptable than the sRules; and so far as I am aware, there is no observation that can distinguish between the two.
The Interaction: Particle and Detector {#the-interaction-particle-and-detector .unnumbered}
======================================
Before introducing the nRules, we will apply Schrödinger’s equation to a ‘microscopic’ particle interacting with a ‘macroscopic’ detector in order to see what difficulties arise. These two objects are assumed to be initially independent and given by the equation
$$\Phi(t)=exp(-iHt)\psi_i\otimes d_i$$
where $\psi_i$ is the initial particle state and $d_i$ is the initial detector state. The particle is then allowed to pass over the detector, where the two interact with a cross section that may or may not result in a capture. After the interaction begins at a time $t_0$, the state is an entanglement in which the particle variables and the detector variables are not separable.
The first component of the resulting system is the detector $d_0$ in its ground state prior to capture, and the second, third, and fourth components are the detector in various states of capture given by $d_w$, $d_{m}$, and $d_d$. $$\Phi(t \ge t_0) =\psi(t)d_0 + d_w(t) \rightarrow d_{m}(t) \rightarrow d_d(t)$$ where $d_w(t)$ represents the entire detector immediately after a capture when only the window side of the detector is affected, and $d_d(t)$ represents the entire detector when the result of a capture has worked its way through to the display side of the detector. The middle state $d_{m}(t)$ represents the entire detector during stages in between, when the effects of the capture have found their way into the interior of the detector, but not as far as the display.
The state $\psi(t)$ is a free particle as a function of time, including all the incoming and scattered components. It does no harm and it is convenient to let $\psi(t)$ carry the total time dependence of the first component, and to let $d_0$ be normalized throughout The first component in Eq. 2 is a superposition of all possible scattered waves of $\psi(t)$ in product with all possible recoil states of the ground state detector, so $d_0$ is a spread of detector states including all the recoil possibilities together with their correlated environments. Subsequent components are also superpositions of this kind. They include all of the recoil components of the detector that have captured the particle.
There is a clear discontinuity or “quantum jump" between the two components $d_0$ and $d_w$ at the detector’s window interface. This discontinuity is represented by a “plus" sign and can only be bridged by a stochastic jump. The remaining evolution from $d_w(t)$ to $d_d(t)$ is connected by “arrows" and is continuous and classical. These three detector states develop in time and may be represented by the single component $$d_1(t) = d_w(t) \rightarrow d_{m}(t)\rightarrow d_d(t)$$ so $$\Phi(t \ge t_0)=\psi(t)d_0 + d_1(t)$$ The capture state $d_1(t)$ in Eq. 3 is equal to zero at $t_0$ and increases with time[^3]$^,$[^4]$^,$[^5]$^,$[^6].
Add an Observer {#add-an-observer .unnumbered}
===============
Assume that an observer is looking at the detector in Eq. 1 from the beginning. $$\Phi(t)=exp(-iHt)\psi_i\otimes D_iB_i$$ where $B_i$ is the observer’s initial brain state that is entangled with the detector $D_i$. This brain is understood to include *only* higher order brain parts – that is, the physiology of the brain that is directly associated with consciousness after all image processing is complete. All lower order physiology leading to $B_i$ is assumed to be part of the detector. The detector is now represented by a capital $D$, indicating that it includes the bare detector *plus* the low-level physiology of the observer.
Following the interaction between the particle and the detector, we have $$\begin{aligned}
\Phi(t \ge t_0) &=& \psi(t)D_0B_0 + D_w(t)B_0 \rightarrow D_{m}(t)B_0 \rightarrow D_d(t)B_1 \hspace{.5cm} \\
\mbox{or} \hspace{.5cm} \Phi(t \ge t_0) &=& \psi(t)D_0B_0 + D_1(t)B_1 \nonumber\end{aligned}$$ where $B_0$ is the observer’s brain when the detector is observed to be in its ground state $D_0$, and $B_1$ is the brain when the detector is observed (only at the display end) to be in its capture state $D_1$. As before, a discontinuous quantum jump is represented by a plus sign, and the continuous evolution of a single component is represented by an arrow.
If the interaction is long lived compared to the time it takes for the signal to travel through the detector in Eq. 4 (as in the case of a long lived radioactive decay), then the superposition in that equation might exist for a long time before a capture. This means that there can be two active brain states of this observer in superposition, where one sees the detector in its ground state and the other simultaneously sees the detector in its capture state. Equation 4 therefore invites a paradoxical interpretation like that associated with Schrödinger’s cat. This ambiguity cannot be allowed. It is not acceptable on empirical grounds. An observer who watches a detector in these circumstances will *not* experience dual states of consciousness.
The nRules of this paper must not only provide for a stochastic trigger that gives rise to a state reduction, and describe that reduction, they must also insure than an empirical ambiguity of this kind will not occur.
The nRules {#the-nrules .unnumbered}
==========
The first rule establishes the existence of a stochastic trigger. This is a property of the system that has nothing to do with the kind of interaction taking place or its representation. Apart from making a choice, the trigger by itself has no effect on anything. It initiates a state reduction only when it is combined with nRules 2 and 3.
**nRule (1)**: *For any subsystem of n complete components in a system having a total square modulus equal to s, the probability per unit time of a stochastic choice of one of those components at time t is given by $(\Sigma_nJ_n)/s$, where the net probability current $J_n$ going into the $n^{th}$ component at that time is positive.*
\[**note**: A *complete component* is a solution of Schrödinger’s equation that includes all of the (symmetrized) objects in the universe. It is made up of *complete states* of those objects including all their state variables. If $\psi(x_1, x_2)$ is a two particle system with inseparable variables $x_1$ and $x_2$, then $\psi$ is considered to be a single object. All such objects are included in a complete component. A component that is a sum of less than the full range of a variable (such as a partial Fourier expansion) is not complete. This is why representation does not matter to the stochastic trigger.\]
\[**note**: Functions are not normalized in this treatment. Instead, probability currents are normalized at each moment of time by dividing $J$ by the value of $s$ at each moment of time.
The second rule specifies the conditions under which *ready states* appear in solutions of Schrödinger’s equation. These are understood to be the basis states of a state reduction. Ready states are always “underlined" in this treatment.
**nRule (2)**: *If a noncyclic interaction produces complete components that are discontinuous with the initial component, then all of the new states that appear in these components will be ready states.*
\[**note**: A cyclic interaction between two component is one that produces continuous oscillations. A *noncyclic interaction* goes in one direction only.
\[**note**: Continuous means continuous in all variables. Although solutions to Schrödinger’s equation change continuously in time, they can be *discontinuous* in other variables – e.g., the separation between the $n^{th}$ and the $(n +
1)^{th}$ orbit of an atom with no orbits in between. A discontinuity can also exist between macroscopic states that are locally decoherent. For instance, the displaced detector states $d_0$ (ground state) and $d_w$ (window capture state) appearing in are discontinuous with respect to detector variables. There is no state in between. Like atomic orbits, these detector states are a ‘quantum jump’ apart.\]
\[**note**: The *initial component* is the first complete component that appears in a given solution of Schrödinger’s equation. A solution is defined by a specific set of boundary conditions. So Eqs. 1 and 3 are both included in the single solution that contains the discontinuity between $d_0$ and $d_1$, where Eq. 1 (together with its complete environment) is the initial state. However, boundary conditions change with the collapse of the wave function. The component that survives a collapse will be complete, and will be the initial component of the new solution.\]
\[**note**: If a noncyclic interaction does not produce complete components that are discontinuous with the initial component, then the Hamiltonian will develop the state in the usual way, independent of these rules.\]
The collapse of a wave function and the change of a ready state to a *realized* state is provided for by nRule (3). If a complete state is not ‘ready’ it will be called ‘realized’. We therefore introduce dual state categories where ready states are the basis states of a collapse. They are on stand-by, ready to be stochastically chosen and converted by nRule (3) to realized states. In this paper, realized states are *not* underlined.
**nRule (3)**: *If a component is stochastically chosen during an interaction, then all of the ready states that result from that interaction (using nRule 2) will become realized, and all other components in the superposition will be immediately reduced to zero.*
\[**note**: The claim of an immediate (i.e., discontinuous) reduction is the simplest possible way to describe the collapse of the state function. A collapse is brought about by an instantaneous change in the boundary conditions of the Schrödinger equation, rather than by the introduction of a new ‘continuous’ mechanism of some kind.\]
\[**note**: This collapse does not preserve normalization. That does not alter probability of subsequent reductions because of the way probability per unit time is defined in nRule (1); that is, current $J$ is divided by the total square modulus. Again, currents are normalized in this treatment – not functions.\]
\[**note**: If the stochastic trigger selects a component that does not contain ready states, then there will be no nRule (3) state reduction.\]
Only positive current going into a *ready component* (i.e., a component containing ready states) is physically meaningful because it represents positive probability. A negative current (coming out of a ready component) is not physically meaningful and is not allowed by nRule (4). Without this restriction, probability current might flow in-and-out of one ready component and into another. The same probability current would then be ‘used’ and ‘reused’. Given the above rules, this would distort the total probability of a process. If the nRules are to work, the total integrated positive current (divided by $s$) must be no greater than 1.0. To insure this we say
**nRule (4)**: *A ready component cannot transmit probability current.*
Although it can receive current that increases its square modulus, a ready state is dynamically terminal. It cannot develop beyond itself. If a ready state $\underline{S}_1(t)$ evolves from a state $S_0(t)$, then Schrödinger’s equation $H[S_0(t) +
\underline{S}_1(t)] = i\hbar\frac{d}{dt}[S_0(t) + \underline{S}_1(t)]$ will change $\underline{S}_1(t)$ in the usual way. However, nRule (4) will prevent the creation of a second order component. The state $\underline{S}_1(t)$ is time dependent because its square modulus increases *and* because it reflects the dynamical changes coming from $S_0(t)$ at every moment of time, but it does not advance dynamically on its own. It is the launch state of a new solution of Schrödinger’s equation if and when it is stochastically chosen, and it will contain all the initial conditions of that solution. Those initial conditions are not applied until the moment of choice.
While no theoretical reason can be given to explain Rule(4), its important consequences are demonstrated throughout the rest of this paper.
Particle/Detector Revisited {#particledetector-revisited .unnumbered}
===========================
When the nRules are applied to the particle/detector interaction, Eq. 2 becomes $$\Phi(t \ge t_0) = \psi(t)d_0 + \underline{d}_w(t)$$ where the quantum jump (+ sign) is discontinuous and noncyclic, where requires that $\underline{d}_w(t)$ is a ready state, and where the other components of the detector ($d_m$ and $d_d$) are zero because nRule (4) will not allow $\underline{d}_w(t)$ to pass current to them. At $t_0$, the first component is maximum and the second is zero, but $\underline{d}_w(t)$ increases in time because of the probability current flow from $\psi(t)d_0$. These components are complete because the environment of each (not shown) is assumed to be present.
The ready component in this equation is time dependent because of its increase in square modulus *and* because it duplicates the moment-to-moment changes that occur within the detector (such as molecular changes) *although* it will not advance dynamically to $d_m$. So long a $\underline{d}_w(t)$ is a ready state, it will not pass current along to its successors. As indicated above, it is a function that contains the boundary conditions of the next solution of Schrödinger’s equation – that is, the ‘collapsed’ solution that is realized when $\underline{d}_w(t)$ is stochastically chosen. So the effect of nRule (4) is to put the boundary conditions of the ‘next’ solution on hold until there is a stochastic hit. The nRules then launch the new solution with boundary conditions that are inherited at that moment from the old solution.
If there is a stochastic hit at time $t_{sc}$, then a continuous classical evolution will give $$\Phi(t\ge t_{sc} > t_0)= d_w(t) \rightarrow d_m(t) \rightarrow d_d(t)$$ so the capture is eventually registered at the display end of the detector.
If there is *no* stochastic hit on $\underline{d}_w(t)$ it will become a *phantom* component. A component is a phantom when there is no longer probability current flowing into it (in this case because the interaction is complete), and when there can be no current flowing out of it because it is a ready component that complies with nRule (4). A phantom component can be dropped out of the equation without consequence. Doing so only changes the definition of the system – it changes the total square modulus $s$ that normalizes the current in nRule (1). This is the same kind of redefinition that occurs in standard practice when one chooses to renormalize a system at some new starting time. Keeping a phantom is like keeping the initial system. Because of nRule (3), kept phantoms are reduced to zero whenever another component is stochastically chosen.
The nRules place this particle/detector system in an ontological setting, for they give an *insider’s* answer to the probability question. The nRules ask: What is the probability that $\underline{d}_w(t)$ will be stochastically chosen during the next time interval $dt$; and then, what is the probability that it will be stochastically chosen during the time interval $dt$ after that, etc? The nRules are always concerned with what happens next. Probabilities that are associated with the second order states like $d_m(t)$ and $d_d(t)$ are ruled out by nRule (4) until the fate of $\underline{d}_w(t)$ is determined. This is the most important consequence of nRule (4) – it does not allow a stochastic leap over the *next* (ready) state of the system. The insider is concerned with the temporal ordering of states, and the nRules address that concern.
In contrast, the sRules ask: What is the probability of finding $d_0$, $d_w$, $d_m$, or $d_d$ at some finite time $T$ after $t_0$? This is an outsider’s question. It is the question asked by one who can only observer the system at distinct and finitely separated times like $t_0$ and $T$.
Particle/Detector/Observer Revisited {#particledetectorobserver-revisited .unnumbered}
====================================
To see how the nRules carry out a particle capture when an observer is a witness, we apply them to the first row of Eq. 4. As before, this only affects the first two components $$\Phi(t \ge t_0)=\psi(t)D_0B_0 + \underline{D}_w(t)B_0$$ because nRule (4) will not allow $\underline{D}_w(t)$ to pass probability current to the other components. Component $\underline{D}_w(t)$ is ‘ready’ because it is a discontinuous and noncyclic quantum jump away from $\psi(t)D_0B_0$ and it is complete because it includes the (not shown) entangled environment. Again, the time dependence of $\underline{D}_w(t)$ does not represent a dynamical evolution beyond the changes given to it by the first component. It does not evolve dynamically on its own. And again, the sub-0 on $B_0$ indicates an awareness of the ground state $D_0$ of the detector. Since the second brain state in Eq. 5 is the same as the first, there is only one brain state $B_0$ in this superposition. A cat-like ambiguity is thereby avoided. Equation 5 now *replaces* Eq. 4.
Equation 5 is the state of the system before there is a stochastic hit that produces a state reduction. If there is a capture, then there will be a stochastic hit on the second component of Eq. 5 at a time $t_{sc}$. This will reduce the first component to zero according to nRule (3), and convert the ready state in the second component to a realized state. $$\Phi(t = t_{sc} > t_0) = D_w(t)B_0$$ The observer is still conscious of the detector’s ground state in this equation because the capture has only affected the window end of the detector. But after $t_{sc}$, a continuous evolution will produce $$\Phi(t \ge t_{sc} > t_0) = D_w(t)B_0 \rightarrow D_{m}(t)B_0 \rightarrow D_d(t)B_1$$ Since this equation represents a *single component* that evolves in time as shown, there is no time at which both $B_0$ and $B_1$ appear simultaneously. There is therefore no cat-like ambiguity in this equation.
Standard quantum mechanics gives us Eq. 4 by the same logic that it gives us Schrödinger’s cat and Everett’s many worlds. (top or bottom row) is a single equation that simultaneously presents two different conscious brain states, resulting in an unacceptable ambiguity. But with the nRules, each Schrödinger solution is separately grounded by its own stochastically selected boundary, allowing the rules to correctly and unambiguously predict the continuous experience of the observer in two stages. This is accomplished by replacing ‘one’ equation in with ‘two’ equations in Eqs. 5 and 6. Equation 5 describes the state of the system *before* capture, and Eq. 6 describes the state of the system *after* capture. Before and after are two *different* solutions to Schrödinger’s equation, specified by different boundary conditions. Remember we said that the stochastic trigger selects the new boundary that applies to the reduced state. So it is the stochastic event that separates the two solutions before and after.
If there is no stochastic hit on the second component in Eq. 5 it will become a phantom component. The new system is then just the first component of that equation. This corresponds to the observer continuing to see the ground state detector $D_0$, as he should in this case[^7].
A Terminal Observation {#a-terminal-observation .unnumbered}
======================
An observer who is inside a system must be able to confirm the validity of the Born rule that is normally applied from the outside. To show this, suppose our observer is not aware of the detector during the interaction with the particle, but he looks at the detector after a time $t_f$ when the primary interaction is complete. Assume initial normalization equal to 1.0. During that interaction we have $$\Phi(t_f > t \ge t_0) = [\psi(t)d_0 + \underline{d}_w(t)]\otimes X$$ where $X$ is the unknown state of the observer prior to the physiological interaction.
Assume there has *not* been a capture. Then after the interaction is complete and before the observer looks at the detector we have $$\Phi(t \ge t_f > t_0) = [\psi(t)d_0 + \underline{d}_w(t_f)]\otimes X$$ where there is no longer a probability current flow inside the bracket. The second component in the bracket is therefore a phantom. There is no current flowing into it, and none can flow out of it because of nRule (4). So the equation is essentially $$\Phi(t \ge t_f > t_0) = \psi(t)d_0\otimes X$$ When the observer finally observes the detector at $t_{ob}$ he will get $$\Phi(t \ge t_{ob} > t_f > t_0) = \psi(t)d_0\otimes X \rightarrow \psi(t)D_0B_0$$ where the physiological process (represented by the arrow) carries $\otimes X$ into $B_0$ and $d_0$ into $D_0$ by a continuous classical progression leading from independence to entanglement. This corresponds to the observer coming on board to witness the detector in its ground state as he should when is no capture. The probability of this happening (according to the sRules) is equal to the square modulus of $\psi(t)d_0\otimes X$ in Eq. 7.
If the particle *is* captured during the primary interaction, there will be a stochastic hit on the second component inside the bracket of Eq. 7 at some time $t_{sc} < t_f$. This results in a capture given by $$\Phi(t_f > t = t_{sc} > t_0) = d_w(t)\otimes X$$ after which $$\Phi(t_f > t \ge t_{sc} > t_0) = [d_w(t) \rightarrow d_m(t) \rightarrow d_d(t)]\otimes X$$ $$\Phi(t \ge t_f > t_{sc} > t_0) = d_1(t)\otimes X$$ as a result of the classical progression inside the detector. When the observer does become aware of the detector at $t_{ob} > t_f$ we finally get $$\Phi(t \ge t_{ob} > t_f > t_{sc} > t_0) = d_1\otimes X \rightarrow D_1B_1$$ So the observer comes on board to witness the detector in its capture state with a probability (according to the sRules) equal to the square modulus of $\underline{d}_w(t_f)\otimes X$ in Eq. 7. The nRules therefore confirm the Born rule, in this case as a theorem.
An Intermediate Case {#an-intermediate-case .unnumbered}
====================
In Eq. 5 the observer is assumed to interact with the detector from the beginning. Suppose that the incoming particle results from a long half-life decay, and that the observer’s physiological involvement only *begins* in the middle of the primary interaction. Before that time we will have $$\Phi(t \ge t_0) = [\psi(t)d_0 + \underline{d}_w(t)]\otimes X$$ where again $X$ is the unknown brain state of the observer prior to the physiological interaction. Primary probability current here flows between the detector components inside the bracket. Let the physiological interaction begin at a time $t_{look}$ after $\underline{d}_w(t)$ has gained some amplitude, and suppose it lasts for a period of time equal to $\pi$. Then $$\begin{aligned}
&\Phi(t=t_{look} > t_0) = \psi(t)d_0\otimes X + \underline{d}_w(t) \otimes X& \\
&\hspace{3.2cm}\downarrow\hspace{1.6cm}\downarrow&\nonumber\\
&\Phi(t=t_{look} + \pi > t_{look} > t_0) = \psi(t)D_0B_0 + \underline{D}_w(t) B_0& \nonumber\end{aligned}$$ where the arrows mean that the evolution from the first to the second row is classical and continuous. That evolution carries $\otimes X$ into $B_0$ and $d_0$ into $D_0$ by a process that leads from independence to entanglement. So while the second component in each row gains square modulus because of the primary interaction (plus signs), the observer is simultaneously coming on board by a continuous process (arrows).
The second component at each moment of time during the physiological interaction in Eq. 8 is the launch state of the new solution of the Schrödinger equation – if there is a stochastic hit at that moment. This component establishes the boundary conditions of any newly launched solution.
After the time $t_{ob} = t_{look} + \pi$ we can write Eq. 8 as $$\Phi(t \ge t_{ob} > t_0) = \psi(t)D_0B_0 + \underline{D}_w(t)B_0$$ This equation is identical with Eq. 5; so from this moment on, it is as though the observer has been on board from the beginning.
If there is a subsequent capture at a time $t_{sc}$, this will become like Eq. 6. $$\Phi(t \ge t_{sc} > t_{ob} > t_0) = D_w(t)B_0 \rightarrow D_m(t)B_0 \rightarrow D_d(t)B_0$$
If a stochastic hit occurs between $t_{look}$ and $t_{ob}$, then the ready component at that moment (the second component in Eq.8) will be chosen and made a realized state. It will then proceed classically and continuously to $D_dB_1$ as in Eq. 10.
A Second Observer {#a-second-observer .unnumbered}
=================
If a second observer is standing by while the first observer interacts with the detector during the primary interaction, the state of the system will be $$\Phi(t \ge t_0) = [\psi(t)D_0B_0 + \underline{D}_w(t)B_0]\otimes X$$ where $X$ is an unknown state of the second observer prior to his interacting with the system. The detector $D$ here includes the low-level physiology of the first observer. A further expansion of the detector will include the second observer’s low-level physiology when he comes on board. The detector will then split into two parallel paths, one connecting to the first observer and the other connecting to the second observer. When a product of brain states appears in the form $BB$ or $B\otimes X$, the first term will refer to the first observer and the second to the second observer.
The result of the second observer looking at the detector will be the same as that found for the first observer in the previous section, except now the first observer will be present in each case. In particular, the equations similar to are now $$\begin{aligned}
&\Phi(t=t_{look} + t) = \psi(t)d_0\otimes X + \underline{d}_w(t) \otimes X \rightarrow& \nonumber\\
&\Phi(t = t_{look} + \pi > t_{look} > t_0) = [\psi(t)D_0B_0B_0 + \underline{D}_w(t)B_0B_0]& \nonumber\\
&\Phi(t \ge t_{ob} > t_0) = \psi(t)D_0B_0B_0 + \underline{D}_w(t)B_0B_0& \nonumber \\
&\Phi(t \ge t_{sc} > t_{ob} > t_0) = D_w(t)B_0B_0 \rightarrow D_m(t)B_0B_0 \rightarrow D_d(t)B_1B_1& \nonumber \end{aligned}$$ These will all yield the same result for the new observer as they did for the old observer. In no case will the nRules produce a result like $B_1B_2$ or $B_2B_1$.
Up to this point we have seen how the nRules go about including observers inside a system in an ontological model. These rules describe when and how the observer becomes conscious of measuring instruments, and replicate common empirical experience in these situations. The nRules are also successfully applied in another paper [@RM5] where two versions of the Schrödinger cat experiment are examined. In the first version a conscious cat is made unconscious by a stochastically initiated process; and in the second version an unconscious cat is made conscious by a stochastically initiated process.
In the following sections we turn attention to another problem – the requirement that macroscopic states must appear in their normal sequence. This sequencing chore represents a major application of nRule (4) that is best illustrated in the case of a macroscopic counter.
A Counter {#a-counter .unnumbered}
=========
If a beta counter that is exposed to a radioactive source is turned on at time $t_0$, its state function will be given by $$\Phi(t \ge t_0) = C_0(t) + \underline{C}_1(t)$$ where $C_0$ is a counter that reads zero counts, $C_1$ reads one count, and $C_2$ (not shown) reads two counts, etc. The second component $\underline{C}_1(t)$ is zero at $t_0$ and increases in time. The underline indicates that it is a ready state as required by nRule (2). $C_2$ and higher states do not appear in this equation because forbids current to leave $\underline{C}_1(t)$. Ignore the time required for the capture effects to go from the window to the display end of the counter.
The $0^{th}$ and the $1^{st}$ components are the only ones that are initially active, where the current flow is $J_{01}$ from the $0^{th}$ to the $1^{st}$ component. The resulting distribution at some time $t$ before $t_{sc}$ is shown in Fig. 3, where $t_{sc}$ is the time of a stochastic hit on the second component.
This means that the $1^{st}$ component *will* be chosen because all of the current from the (say, normalized) $0^{th}$ component will pore into the $1^{st}$ component making $\int J_{01}dt = 1.0$. Following the stochastic hit on the $1^{st}$ component, there will be a collapse to that component because of . The first two dial readings will therefore be sequential, going from 0 to 1 without skipping a step such as going directly from 0 to 2. It is nRule (4) that enforces the no-skip behavior of a counter; for without it, any component in the superposition might be chosen as a result of probability current flowing into it. It is empirically mandated that these states should always follow in sequence without skipping a step.
With the stochastic choice of the $1^{st}$ component at $t_{sc}$, the process will begin again as shown in Fig. 4. This leads with certainty to a stochastic choice of the $2^{nd}$ component. That certainty is accomplished by the wording of nRule (1) that requires current normalization at each moment of time; that is, the current $J_{12}$ is divided by the total square modulus at each moment. The total integral $\int J_{12}dt$ is less than 1.0 in Fig. 4 because of the reduction that occurred in ; but it is restored to 1.0 when divided by the (new) total square modulus. It is therefore certain that the $2^{nd}$ component will be chosen.
And finally, with the choice of the $2^{nd}$ component, the process will resume again with current $J_{23}/s$ going from the $2^{nd}$ to the $3^{rd}$ component. This leads with certainty to a stochastic choice of the $3^{rd}$ component.
If an observer watches the counter from the beginning he will be able to see it go sequentially from $C_0$ to $C_1$, to $C_2$, etc., for he is himself a continuous part of the system. He does not have to ‘peek’ intermittently like an epistemological observer who is not part of the system as has been said. Although the empirical experience of the ontological (nRule) observer is different from the epistemological (sRule) observer, there is no contradiction between the two. That’s because the nRules and the sRules answer two different questions about probability as previously noted. The nRules ask: What is the probability that the system will jump to the *next* counter state in the next differentially small increment of time; and the sRules ask: What it the probability that an outside observer will find the system in any one of its possible counter states if he looks after some finite time $T$?
The Parallel Case {#the-parallel-case .unnumbered}
=================
Now imagine parallel states in which a quantum process may go either clockwise or counterclockwise as shown in Fig. 5. Each component includes a macroscopic piece of laboratory apparatus $A$, where the Hamiltonian provides for a discontinuous clockwise interaction going from the $0^{th}$ to the $r^{th}$ state, and another one going from there to the final state $f$; as well as a comparable counterclockwise interaction from the $0^{th}$ to the $l^{th}$ state and from there to the final . The Hamiltonian does not provide a direct route from the $0^{th}$ to the final state, so the system will choose stochastically between a clockwise and a counterclockwise route. Ready states $\underline{A}_l$ and $\underline{A}_r$ are the *eigenstates* of that choice, and contain the boundary conditions for each separate path.
With nRule (4) in place, probability current cannot initially flow from either of the intermediate states to the final state, for that would require current flow from a ready state. The dashed lines in Fig. 5 indicate the forbidden transitions. But once the state $\underline{A}_l$ (or $\underline{A}_r$) has been stochastically chosen, it will become a realized state $A_l$ (or $A_r$) and a subsequent transition to $\underline{A}_f$ can occur that realizes $A_f$.
The effect of nRule (4) is therefore to force this macroscopic system into a classical sequence that goes either clockwise or counterclockwise. Without it, the system might make a second order transition (through one of the intermediate states) to the final state, without the intermediate state being realized. The observer would then see the initial state followed by the final state, without knowing (in principle) which pathway was followed. This is a familiar property of continuous microscopic evolution. In Heisenberg’s famous example formalized by Feynman, a microscopic particle observed at point $a$ and later at point $b$ will travel over a quantum mechanical superposition of all possible paths in between. Without nRule (4), macroscopic objects facing discontinuous and noncyclic parallel choices would do the same thing. But that should not occur. The fourth nRule forces this parallel system into one or the other classical path, so it is not a quantum mechanical superposition of both paths.
A Continuous Variable {#a-continuous-variable .unnumbered}
=====================
In the above examples nRule (4) guarantees that sequential discontinuous steps in a superposition are not passed over. If the variable itself is classical and continuous, then continuous observation is possible without the necessity of stochastic jumps. In that case we do not need or any of the , for they do not prevent or in any way qualify the motion.
However, a classical variable may require a quantum mechanical jump-start. For instance, the mechanical device that is used to seal the fate of Schrödinger’s cat (e.g., a falling hammer) begins its motion with a stochastic hit. That is, the decision to begin the motion (or not) is left to a $\beta$-decay. In this case forces the motion to begin at the beginning, insuring that no value of the classical variable is passed over; so the hammer will always fall from its *initial* angle with the horizontal. Without nRule (4), the hammer might begin its fall at some other angle because probability current will flow into angles other than the initial one.
Microscopic Systems {#microscopic-systems .unnumbered}
===================
The discussion so far has been limited to experiments or procedures whose outcome is empirically known. Our claim has been that the nRules are chosen to work without regard to a theory as to ‘why’ they work. Therefore, our attention has always gone to macroscopic situations in which the results are directly available to conscious experience. However, if the nRules are correct we would also want know how they apply to microscopic systems, even though the predicted results in these cases are more speculative. In this section we will consider the implications of the nRules in three microscopic cases. The important question to ask in each case is: Under what circumstances will these rules result in a state reduction of a microscopic system?
$$\mbox{\textbf{Case 1 -- spin states}}$$
No state reduction will result from changing the representation. In particular, replacing the spin state $+z$ with the sum of states $+x$ and $-x$ will not result in either one of the $x$-states becoming a ready state.
This will be true for a spin $+z$ particle even if the common environment of $+x$ and $-x$ includes a magnetic field that is continuously applied in (say) the $x$-direction. So long as the magnetic field is the same for both $+x$ and $-x$, the result will be the same (i.e., neither one will become a ready state).
More generally, let the environments $E^p(t)$ of $+x$ and $E^n(t)$ of $-x$ change continuously over time such that $$\Phi(\tau > t \ge t_0) = \frac{1}{\sqrt{2}}(+z)E^p(t) + \frac{1}{\sqrt{2}}(-z)E^n(t)$$ where $E^p(0) = E^n(0) = E$, and $E^p(\tau) = E_+, E^n(\tau) = E_-$. States $E_+$ and $E_-$ are the final environments of $+x$ and $-x$ at a time $\tau$ that will be either the time at which the process is complete, or complete to some desired extent. It may be that the environments $E_+$ and $E_-$ are *similar* to $E$ and to each other (i.e., same temperature and pressure, same particles, same radiation field, etc.), but time will change $E_+$ and $E_-$ so they can no longer be *identical* with $E$ or with each other. Of course, it might also be that $E_+$ and $E_-$ are not even similar .
Taking the first and last term in Eq. 11 we write $$\begin{aligned}
\Phi(\tau > t \ge t_0) &=& \frac{1}{\sqrt{2}}[(+z) + (-z)]E \rightarrow \frac{1}{\sqrt{2}}[(+z)E_+ + (-z)E_-] \\
&& \hspace{1cm} t =t_ 0 \hspace{2.8cm} t = \tau \nonumber\end{aligned}$$ where the arrow indicates a continuous transition. Therefore, there will be *no state reduction* in this case. It may be noncyclic, but this process does not lead to a collapse of the wave to either $+x$ or $-x$.
For example, if the magnetic field is non-homogeneous there will be a physical separation of the $+x$ and $-x$ states. One will move into a stronger magnetic field and the other will move into a weaker magnetic field. Assuming that this field is continuously (i.e., classically) applied, there will be no ready state and no state reduction. Ready states will appear only when the $+x$ and $-x$ components are picked up by different detectors at different locations, resulting in a *detector related* discontinuity. According to nRule (3), a state reduction can only occur if and when that happens.
There are many other ‘quantum’ processes that are continuous and therefore not subject to stochastic reduction, such as scattering, interference, diffraction, and tunneling. Vacuum fluctuations are cyclic.
$$\mbox{\textbf{Case 2 -- Free neutron decay}}$$
A free neutron decay is written $\Phi(t) = n(t) + \underline{ep\overline{\nu}}(t)$, where the second component is zero at $t = 0$ and increases in time as probability current flows into it. This component contains three entangled particles making a whole object, where all three are ‘ready’ states as indicated by the underline . Each component is multiplied by a term representing the environment (not shown). Each is complete for this reason and also because the variables of each particle take on all of the values that are allowed by the boundary conditions. Following there will be a stochastic hit on $\underline{ep\overline{\nu}(t)}$ at some time $t_{sc}$, reducing the system to the realized correlated states $ep\overline{\nu}(t_{sc})$.
Specific values of, say, the electron’s momentum are not stochastically chosen by this reduction because all possible values of momentum are included in $ep\overline{\nu}(t_{sc})$ and its subsequent evolution. For the electron’s momentum to be determined in a specific direction away from the decay site, a detector in that direction must be activated. That will require another stochastic hit on the component that includes that detector.
This case provides a good example of how $\underline{ep\overline{\nu}}(t)$ is a function time beyond its increase in square modulus. Assume that the neutron moves across the laboratory in a wave packet of finite width. At each moment the ready component $\underline{ep\overline{\nu}}(t)$ will ride with the packet, having the same size, shape, and group velocity. It is the launch component that contains the boundary conditions of the next solution of Schrödinger’s equation – the solution that appears when $\underline{ep\overline{\nu}}(t)$ is stochastically chosen at $t_{sc}$. Before this ‘collapse’, $\underline{ep\overline{\nu}}(t)$ is time dependent because it increases in square modulus *and* because it follows the motion of the neutron; however, nRule (4) insures that it will not evolve dynamically beyond itself until it becomes a realized component at the time of stochastic choice. The neutron $n(t)$ will then disappear and the separate particles $e(t)p(t)\overline{\nu}(t)$ will spread out on their own, still correlated in conserved quantities.
$$\mbox{\textbf{Case 3 -- Atomic absorption and emission}}$$
If an atom is raised to an excited state by a passing photons, the absorption part of this interaction is cyclic because the atom might fall back to its ground state by a stimulated emission. However, the excited state atom might also emit ‘another’ photon by simultaneous emission and that is noncyclic. That could lead to a stochastic hit and collapse of the wave to the newly formed decay state. The pulse of $N$ photons is represented by $\gamma_N(t)$ in the first component of Eq. 13, and the spontaneously emitted photon is $\underline{\gamma}(t)$ in the third component.
$$\Phi(t \ge t_0) = \gamma_N(t)A_0(t) \leftrightarrow \gamma_{N-1}(t)A_1(t)+ \gamma_{N-1}(t)\underline{\gamma}(t)\underline{A}_0(t)$$
The continuous (cyclic) oscillation is represented by the reversible arrow $\leftrightarrow$ between the first and second components.
If the incoming photons have a small cross section with the atom, the second component in Eq. 13 will not fully discharge into the third component. In that case the latter will become a phantom that can be subsequently ignored. There will be no collapse. The second component will then dampen out and the first component, modified by the encounter, will be the only survivor.
Otherwise, there will be a stochastic hit on the third component at some time $t_{sc}$, resulting in $$\Phi(t \ge t_{sc} < t_0) = \gamma_{N-1}(t)\gamma(t)A_0(t)$$ The newly emitted photon $\gamma(t)$ will evolve until it has a final pulse width $\Delta T_f$ that is associated with its final spread of energy $\Delta E$. There is no necessary relationship between $\Delta T_f$ and the half-life $T_{1/2}$ of the decay from the second to the third component in Eq. 13. There might be any number of influences affecting $T_{1/2}$, including the number of photons in the stimulating pulse $\gamma_N(t)$. If there are no incoming photons, and if the excited state $A_1(t)$ is realized by a quantum jump from a higher energy level at some stochastically chosen moment $t_{scc}$, then $$\Phi(t \ge t_{scc}) = A_1(t)+ \underline{\gamma}(t)\underline{A}_0(t)$$ where the second component is zero at $t_{scc}$ and increases in time. The state $A_1(t)$ decays *only* into the second component in this equation; whereas in it shares time with the initial state $\gamma_N(t)A_0(t)$. In Eq. 13 the average current flow into $\underline{\gamma}(t)\underline{A}_0(t)$ is therefore decreased, so the half-life $T_{1/2}$ of the spontaneous decay is increased.
In contrast, $\Delta T_f$ is determined only by the time dependent solutions of the excited state $A_1(t)$. So $\Delta T_f$ is not causally connected to $T_{1/2}$. The value of $\Delta T_f$ is transmitted from $A_1(t)$ to the launch component $\underline{\gamma} (t)\underline{A}_0(t)$ at each moment of time, setting the boundary conditions of that launch site at that moment. Therefore, it will not matter to the properties of the emitted photon (in particular $\Delta T_f$) if a stochastic hit on $\underline{\gamma}(t)\underline{A}_0(t)$ in Eq. 13 or Eq. 14 aborts the decay before it is complete.
Decoherence {#decoherence .unnumbered}
===========
Suppose that two states $A$ and $B$ that are initially in coherent Rabi oscillation and are exposed to a phase disrupting environment. This may be expressed by the equation $$\Phi(\tau > t \ge t_0) = (A \leftrightarrow B)E(t) \rightarrow [AE_A(t) + BE_B(\tau)]$$ where $A$ is at a higher energy level than $B$. The right pointing arrow indicates a continuous process. When the environments $E_A$ and $E_B$ are sufficiently different to be orthogonal, then states $A$ and $B$ in Eq. 15 will become totally decoherent. Statistically, this leaves a local mixture of $A$ and $B$ with no current flow between them. The subsequent decay of $AE_A$ is generally much slower than decoherence, so decoherence will be essentially complete before there is a stochastic interruption. Evidence for this is found in low temperature experiments [@DV; @YY] where Rabi oscillations undergo decoherent decay without any sign of interruption due to state reduction.
The ammonia molecule is generally found in a partially decoherent state. In a rarified atmosphere the molecule will most likely be in its symmetric coherent form $(A\leftrightarrow B)$, where $A$ has the nitrogen atom on one side of the hydrogen plane, and $B$ has the nitrogen atom placed symmetrically on the other side. This is the lowest energy state available to the molecule. In this case the states $A$ and $B$ (Eq. 15) taken by themselves are equally energetic at a higher energy level.
Collisions with other molecules in the environment will tend to destroy the coherence between $A$ and $B$, causing the ammonia molecule to become decoherent to some extent. This decoherence can be reversed by decreasing the pressure. Since an ammonia molecule wants to fall into its lowest energy level, it will tend to return to the symmetric state when outside pressure is reduced. In general, equilibrium can be found between a given environment and some degree of decoherence.
The ammonia molecule cannot assume the symmetric form $(A \leftrightarrow B)$ if the environmental collisions are too frequent – i.e., if the pressure exceeds about 0.5 atm. at room temperature [@JZ]. At low pressures the molecule is a stable coherent system, and at high pressures it is a stable decoherent system. It seems to change from a microscopic object to a macroscopic object as a function of its environment. This further supports the idea that the micro/macroscopic distinction is not fundamental.
To accommodate the main example of this paper, which is a detector that may or may not capture a particle, a more general form is adopted. We now use time dependent coefficients $m(t)$ and $n(t)$, where $m(t_0) = 1, n(t_0) = 0$, and where $m(t)$ decreases in time keeping $m(t)^2 + n(t)^2 = 1$. These coefficients describe the progress of the primary interaction, giving $$\begin{aligned}
\Phi(\tau > t \ge t_0) &=& [m(t)A + n(t)\underline{B}]E \rightarrow [m(t)AE_A(t) + n(t)\underline{B}E_B(t)] \nonumber\\
&& \hspace{.95cm} t = t_0 \hspace{3.3cm} t = \tau \nonumber\end{aligned}$$ where the transitions *inside* the brackets are now discontinuous and noncyclc, making $\underline{B}$ a ready state as required by nRule (2). This can be written $$\begin{aligned}
\Phi(t = t_0) &=& AE \hspace{.5cm}\mbox{(intial component)}\hspace{1.8cm} m(t_0) = 1 \\
\Phi(\tau \ge t \ge t_0) &=& AE \rightarrow [m(t)AE^m(t) + n(t)\underline{B}E^n(t)]\hspace{.4cm} m(t) \rightarrow 0
\nonumber\end{aligned}$$ Since state $\underline{B}$ in this equation is a ‘ready’ state and the primary current flows from $A$ to $\underline{B}$ inside the square bracket, $\underline{B}$ is a candidate for state reduction according to nRule (3).
Equation 16 applied to our example in Eq. 3 with $m(t)A = \psi(t)d_0$ and $n(t)\underline{B} = \underline{d}_w(t)$ says that initially coherent states $\psi(t)d_0$ and $\underline{d}_w$ rapidly become decoherent. Because of the macroscopic nature of the detector, decoherence at time $\tau$ may be assumed to be complete, and the decay time from $t_0$ to $\tau$ extremely short lived. The time for any newly created pair of macroscopic objects to approach full decoherence is so brief that it is not measurable in practice. Still, we see that decoherence does not happen immediately when a new macroscopic state is created.
Grounding the Schrödinger Solutions {#grounding-the-schrödinger-solutions .unnumbered}
===================================
Standard quantum mechanics is not completely grounded because it does not recognize all of the boundary conditions (beyond the initial conditions) that are stochastically chosen during the lifetime of the system. With either one of the proposed rule-sets, every stochastic hit sets a new boundary (i.e., the chosen eigenvalue) for a new solution of Schrödinger’s equation. On the other hand, traditional quantum theory accumulates all the possible solutions as though they were all simultaneously valid; and as a result, this model encourages bizarre speculations such as the many-world interpretation of Everett or the cat paradox of Schrödinger. With the proposed rule-sets, these empirical distortions disappear. It is because standard quantum mechanics fails to respond to the system’s ongoing state reductions that these fanciful excursions seem plausible.
Limitation of the Born Rule {#limitation-of-the-born-rule .unnumbered}
===========================
Using the Born rule in standard theory, the observer can only record an observation at a given instant of time, and he must do so consistently over an ensemble of observations. He cannot himself become part of the system for any finite period of time. When discussing the Zeno effect it is said that continuous observation can be simulated by rapidly increasing the number of instantaneous observations; but of course, that is not really continuous.
On the other hand, the observer in an ontological model can *only* be continuously involved with the observed system. Once he is on board and fully conscious of a system, the observer can certainly try to remove himself “immediately". However, that effort is not likely to result in a truly instantaneous conscious observation. So the epistemological observer claims to make instantaneous observations but cannot make continuous ones; and the ontological observer makes continuous observations but cannot (in practice) make instantaneous ones. Evidently the Born rule would require the ontological observer to do something that cannot be realistically done. Epistemologically we can ignore this difficulty, but a consistent ontology should not match a continuous physical process with continuous observation by using a discontinuous rule of correspondence. Therefore, no ontological model should make fundamental use of the Born interpretation that places unrealistic demands on an observer.
Status of the Rules {#status-of-the-rules .unnumbered}
===================
No attempt has been made to relate conscious brain states to particular neurological configurations. The nRules are an empirically discovered set of macro relationships that exist on another level than microphysiology, and there is no need to connect these two domains. These rules preside over physiological detail in the same way that thermodynamics presides over molecular detail. It is desirable to eventually connect these domains as thermodynamics is now connected to molecular motion; and hopefully, this is what a covering theory will do. But for the present we are left to investigate the rules by themselves without the benefit of a wider theoretical understanding of state reduction or of conscious systems. There are two rule-sets of this kind giving us two different quantum friendly ontologies – the nRules of this paper and the oRules of Refs. 1-3.
The question is, which if either of these two rule-sets is correct (or most correct)? Without the availability of a wider theoretical structure or a discriminating observation, there is no way to tell. Reduction theories that are currently being considered may accommodate a conscious observer, but none are fully accepted. So the search goes on for an extension of quantum mechanics that is sufficiently comprehensive to cover the collapse associated with an individual measurement. I expect that any such theory will support one of the ontological rule-sets, so these rules might server as a guide for the construction of a .
[99]{}
R. A. Mould, “Quantum Brains: The oRules"; *AIP Conf. Proc.*, **750**, ; FPP3, Växjö University, Sweden, June 2004
R. A. Mould, “Quantum Brain oRules", physics/0406016
R. A. Mould, “Quantum Brain States", *Found. Phys.* **33** (4) 591-612 (2003),
quant-ph/0303064
J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, J. E. Lukens, “Quantum superposition of distinct macroscopic states", *Nature* **406**, 43 (2000)
R. A. Mould, “The Cat nRules", quant-ph/0410147
D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, H. H. Devoret, “Manipulating the Quantum State of an Electrical Circuit”, *Science*, **296**, 886 (2002)
Y. Yu, S. Han, X. Chu, S-I. Chu, Z. Wang, “Coherent Temporal Oscillations of Macroscopic Quantum States in a Josephson Junction ”, *Science*, **296**, 889 (2002)
E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I. -O. Stamatescu, *Decoherence and the Appearance of a Classical World in Quantum Mechanics*, p. 106, Springer-Verlag, Berlin (2003)
[^1]: Department of Physics and Astronomy, State University of New York, Stony Brook, 11794-3800; http://ms.cc.sunysb.edu/\~rmould
[^2]: These are Copenhagen sRules. There may be any number of variations. Their essential feature is the Born Rule that establishes the *only* link between observation and the symbols of the theory.
[^3]: Each component in Eq. 3 has an attached environmental term $E_0$ and $E_1$ that is not shown. These are orthogonal to one another, insuring local decoherence. But even though Eq. 3 may be decoherent locally, we assume that the macroscopic states $d_0$ and $d_1$ are fully coherent when $E_0$ and $E_1$ are included. So Eq. 3 and others like it in this paper are understood to be coherent when universally considered. We call them “superpositions", reflecting their global rather than their local properties.
[^4]: Superpositions of macroscopic states have been found at low temperatures [@JRF]. The components of these states are locally coherent for a measurable period of time.
[^5]: Equation 3 can be written with coefficients $c_0(t)$ and $c_1(t)$ giving $\Phi(t \ge t_0) = c_0(t)\psi(t)d_0 + c_1(t)d_1$, where the states $\psi(t)$, $d_0$, and $d_1$ are normalized throughout. We let $c_0(t)\psi(t)$ in this expression be equal to $\psi(t)$ in Eq. 3, and let $c_1(t)d_1$ be equal to $d_1(t)$ in Eq. 3.
[^6]: It is important to realize that the interaction Hamiltonian can only connect $\psi(t)d_0$ with the window state $d_w(t)$, which is the *launch state* of the activated detector. It cannot pour probability current directly into a state that is more dynamically advanced, like $d_m(t)$ or $d_d(t)$. Therefore, at every moment during the current flow, a new state $d_w(t)$ is launched, since the state that was launched immediately before that time has moved on (dynamically) to become $d_m(t)$. This means that the component $d_1(t)$ in Eq. 3 is a superposition of a continuum of all the detector states that have been launched at all previous times during the interaction. For the time being we will ignore this complication and come back to it (in Footnote 6) after we have examined this case from the point of view of the nRules.
[^7]: Had the reasoning of Footnote 5 been applied to Eq. 4, the component $D_1(t)B_1$ would be a superposition of the continuum of launch possibilities. It would include a superposition of all the brain states $B_0$ that existed before the signal could have traveled through the detector to reach the brain, plus all the brain states $B_1$ that were reached after that time. This would have produced a massive cat-like paradox prior to a stochastic hit or state reduction of any kind. However, the nRules also avoid this difficulty because they produce only *one* launch component $D_1(t_{sc})B_1$.
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abstract: 'Complementing our Washington photometric studies on Galactic open clusters (OCs), we now focus on four poorly studied OCs located in the first and fourth Galactic quadrants, namely BH84, NGC5381, BH211 and Czernik37. We have obtained CCD photometry in the Washington system $C$ and $T_1$ passbands down to $T_1$ $\sim$ 18.5 magnitudes for these four clusters. Their positions and sizes were determined using the stellar density radial profiles. We derived reddening, distance, age and metallicity of the clusters from extracted $(C-T_1,T_1)$ color-magnitude diagrams (CMDs), using theoretical isochrones computed for the Washington system. There are no previous photometric data in the optical band for BH84, NGC 5381 and BH211. The CMDs of the observed clusters show relatively well defined main sequences, except for Czernik37, wherein significant differential reddening seems to be present. The red giant clump is clearly seen only in BH211. For this cluster, we estimated the age in (1000$^{+260}_{-200}$) Myr, assuming a metallicity of $Z$ = 0.019. BH84 was found to be much older than it was previously believed, while NGC5381 happened to be much younger than previously reported. The heliocentric distances to these clusters are found to range between 1.4 and 3.4 kpc. BH84 appears to be located at the solar galactocentric distance, while NGC5381, BH211 and Czernik37 are situated inside the solar ring.'
address:
- 'Observatorio Astronómico, Universidad Nacional de Córdoba, Laprida 854, Córdoba, Argentina'
- 'Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET, Argentina'
author:
- 'N. Marcionni'
- 'J. J. Clariá'
- 'M. C. Parisi'
- 'T. Palma'
- 'M. Oddone'
- 'A. V. Ahumada'
title: CCD Washington photometry of four poorly studied open clusters in the two inner quadrants of the galactic plane
---
Galaxy: open clusters and associations: individual: BH84, NGC5381, BH211, Czernik37 - Techniques: photometric
Introduction
============
Galactic open clusters (OCs) have long been considered excellent targets not only to probe the Galactic disc properties [@l82; @f95; @pietal06; @bietal06] but also to trace its chemical evolution (see, e.g., Chen et al., 2003 and references therein). Because it is relatively simple to estimate ages, distances and metallicities of OCs fairly accurately, their basic parameters constitute excellent tracers to the structure and chemical evolution of the Galactic disc. The proximity of most OCs to the Galactic plane, however, usually restricts this analysis to the most populous ones and/or to those located within a few kpc from the Sun (Bonatto et al., 2006). Although there are at present estimates of a total of about $25\times10^3$ OCs in the Milky Way [@poetal10], there is not yet an estimation of fundamental parameters such as reddening, distance and age for nearly 30% of the $\sim$ 2200 catalogued Galactic OCs [@dietal02; @bietal03; @duetal03].
The present work is part of a current project of photometric observation of Galactic OCs in the Washington system that is being developed at the Observatorio Astronómico de la Universidad Nacional de Córdoba (Argentina). This project aims at determining the fundamental parameters or at refining the quality of observationally determined properties for some unstudied or poorly studied OCs, located in different regions of the Milky Way. Washington photometry has proved to be a valuable tool to determine the fundamental parameters of OCs since information on cluster membership, reddening, distance, age and metallicity is obtained through the analysis of the $(C-T_1,T_1)$ color-magnitude diagram (CMD). We have already reported results based on Washington CCD $CT_1$ photometric data on several young (e.g., Piatti et al. 2003a), intermediate-age (e.g., Clariá et al., 2007) and old Galactic OCs (e.g., Piatti et al. 2004). These studies have contributed not only to the individual characterization of these stellar systems but also to the global understanding of some properties of the Galactic disc (e.g., Parisi et al. 2005).
In this study we provide new high-quality photometric CCD data obtained with the Washington system $C$ and $T_1$ passbands down to $T_1$ $\sim$ 18.5 magnitudes in the fields of four faint, poorly studied OCs, namely BH84, NGC5381, BH211 and Czernik37. The equatorial and Galactic coordinates of the cluster centers taken from the WEBDA Open Cluster Database [@me05] are listed in Table 1, together with the angular sizes given by @ah03. The selected clusters are located in the first and fourth Galactic quadrants (280$^\circ$ $<$ [*l*]{} $<$ 3$^\circ$) near the Galactic plane ($\mid$b$\mid$ $\leq$ 3$^\circ$). As far as we know, no previous photometric data in the optical band exist for BH84, NGC5381 and BH211. The four selected clusters have been examined by @buetal11 and @ta08 [@ta11] using Two-Micron All-Sky Survey (2MASS) data. Some preliminary results about BH84, NGC5381 and BH211 are presented in @maetal13. A brief description of these objects as well as a summary of previous results for the fields under investigacion is given below:
[*BH84*]{}. First recognized as an OC by @vh75, this object (IAU designation C0959-579) seems to be a detached, relatively poor and faint OC in the Carina constellation (Fig. 1). It shows the typical morphology of a Trumpler class II-1p cluster, which is characterized by a slight concentration of member stars of similar brightness bf and relatively small population. The only observational data for this object are those given in the 2MASS catalog and discussed by @buetal11. These authors derived a reddening $E(B-V)$ = 0.60 and suggest that BH84 is a young cluster ($\sim$ 18 Myr), located at a heliocentric distance $d$ = (2.92 $\pm$ 0.19) kpc.
[*NGC5381*]{}. This is a cluster in Centaurus, also designated as BH156 [@vh75]. @ah03 refer to this object as belonging to Trumpler class II-2m, i.e., a moderately rich, detached cluster with little central concentration and medium-range bright stars (Fig. 1). According to these authors, NGC5381 has a comparatively large angular diameter of 11$'$. A search for variable stars in the cluster field was carried out by @pietal97. Using 2MASS data, @ta11 suggests that NGC5381, slightly reddened by $E(B-V)$ = 0.06, is an intermediate-age cluster ($\sim$ 1.6 Gyr) located at 1.2 kpc from the Sun.
[*BH211*]{}. This object (C1658-410) appears to be somewhat elongated in the East-West direction (Fig. 1). BH211 is a detached, moderately poor and relatively faint group of stars, first recognized as an OC in Scorpius by @vh75. It is a small-sized OC situated very near the Galactic center direction, practically on the Galactic plane (Table 1). The only observational data-set for this object is the one given in the 2MASS catalog and discussed by @buetal11 who found the following results: $E(B-V)$ = 0.48, $d$ = (1.38 $\pm$ 0.09) kpc and $\sim$ 1.6 Gyr.
[*Czernik37*]{}. Also known as BH253 [@vh75], this is a relatively faint cluster (C1750-273) first recognized in Sagittarius by @cz66. As indicated by its Trumpler class (II-1m), it shows a slight central concentration but can be identified by its relatively dense population compared to that of the field stars (Fig. 1). This cluster is projected on to the central bulge of the Galaxy, only 2$^\circ$ from the Galactic center direction. @caetal05 presented CCD $BVI$ photometry in the field of Czernik37. Although they conclude that this may be a sparse but real cluster superimposed on the Galactic bulge population, they do not provide its physical parameters. Using 2MASS data, @ta08 derived a heliocentric distance of 1.7 kpc, $E(B-V)$ = 1.03 and an age of 0.6 Gyr.
The layout of this paper is as follows. Section 2 provides details on our observations and the data reduction procedure. In Section 3 we determine the cluster centers and the stellar density radial profiles. Section 4 deals with the determination of cluster fundamental parameters through the fitting of theoretical isochrones. A brief description of the results, including a comparison with previous findings, is presented in Section 5, while the final conclusions are summarized in Section 6.
Observations and reductions
===========================
The observations of the selected clusters were carried out with the Cerro Tololo Inter-American Observatory (CTIO, Chile) 0.9 m telescope, during the nights of 2008 May 9-11, with a 2048$\times$2048 pixel Tektronix CCD. The scale on the chip is 0.396“ pixel$^{-1}$ (focal ratio f/13.5) yielding a visual field of 13.6’$\times$13.6’. We controlled the CCD through the CTIO ARCON 3.3 data acquisition system in the standard quad amplifier mode, with a mean gain and readout noise of 1.5 e$^-$/ADU and 4.2 e$^-$, respectively. The filters used were the Washington system $C$ [@ca76] and Kron-Cousins $R_{KC}$. The latter has a significantly higher through-put as compared with the standard Washington $T_1$ filter but $R_{KC}$ magnitudes can be accurately transformed to yield $T_1$ magnitudes [@ge96]. From here onwards, we will use indistinctly the words $R_{KC}$ or $T_1$. Typically 20 standard stars taken from the list of @ge96, covering a wide range in color, were nightly observed. Table 2 shows the log of the observations with dates, filters, exposure times and airmasses. In addition, a series of 10 bias and five dome and sky flat-field exposures per filter were obtained nightly. The weather conditions kept very stable at CTIO, with a typical seeing of 1.0”-1.2", although some images have slightly larger full-widths at half maximum (FWHMs) due to temperature changes of up to 2 $^\circ$C. Fig. 1 shows schematic finding charts of the observed cluster fields built with all the measured stars in the $T_1$ band.
The $CT_1$ images were reduced at the Observatorio Astronómico de la Universidad Nacional de Córdoba (Argentina) with IRAF[^1], using the QUADPROC package. After applying the overscan-bias subtraction for the four amplifiers independently, we carried out flat-field corrections using a combined sky-flat frame, which was previously checked for nonuniform illumination pattern with the averaged dome-flat frame. Then, we did aperture photometry for the standard star fields using the PHOT task within DAOPHOT II [@st91]. The relationships between instrumental and standard magnitudes were obtained by fitting the equations:
$$c = a_1 + T_1 + (C-T_1) + a_2\times X_C + a_3\times (C-T_1),$$
$$r = b_1 + T_1 + b_2\times X_{T_1} + b_3 \times (C-T_1),$$
where $X$ represents the effective airmass and capital and lowercase letters stand for standard and instrumental magnitudes, respectively. The coefficients $a_i$ and $b_i$ ($i$ = 1, 2 and 3) were nightly derived through the IRAF routine FITPARAM. The resulting mean coefficients together with their errors are shown in Table 3; the typical rms errors of equations (1) and (2) are 0.017 and 0.015 mag, respectively, indicating the nights were of good photometric quality.
Point spread function (PSF) photometry for the selected fields was performed using the stand-alone version of the DAOPHOT II package [@st94], which provided us with $x$ and $y$ coordinates and instrumental $c$ and $r$ magnitudes for all the stars identified in each field. The PSFs were generated from two samples of 35-40 and $\sim$ 100 stars interactively selected. For each frame, a quadratically varying PSF was derived by fitting the stars in the larger sample, once their neighbors were eliminted using a preliminary PSF obtained from the smaller star sample, which contained the brightest, least contaminated stars. We then used ALLSTAR program to apply the resulting PSF to the identified stellar objects and create a subtracted image, which was used to find and measure magnitudes of additional fainter stars. The PSF magnitudes were determined using as zero points the aperture magnitudes yielded by PHOT. This procedure was repeated three times on each frame. Next, we computed aperture corrections from the comparison of PSF and aperture magnitudes using the subtracted neighbor PSF star sample. The resulting aperture corrections were on average less than 0.02 mag (absolute value) for $c$ and $r$, respectively. Note that PSF stars are distributed throughout the whole CCD frame, so that variations of aperture corrections should be negligible. Finally, the standard magnitudes and colors for all the measured stars were computed by inverting equations (1) and (2), once positions and instrumental $c$ and $r$ magnitudes of stars in the same field were matched using Stetson’s DAOMATCH and DAOMASTER programs.
Once we obtained the standard magnitudes and colors, we built a master table containing the average of $T_1$ and $C-T_1$, their errors $\sigma$$(T_1)$ and $\sigma$$(C-T_1)$ and the number of observations for each star, respectively. We derived magnitudes and colors for 1884, 2439, 979 and 1129 stars in the fields of BH84, NGC5381, BH211 and Czernik37, respectively. These values are provided in Tables 4-7. Magnitude and color errors are the standard deviation of the mean or the observed photometric errors for stars with only one measurement. Tables 4-7 are only partially presented here as guidance, regarding their form and content. The complete tables can be found on the on-line version of the journal.
Fig. 2 shows the behavior of the $T_1$ and $C-T_1$ photometric errors as a function of $T_1$ for stars measured in the field of NGC5381, the most populated observed field. Note that $\sigma$$T_1$ and $\sigma$$(C-T_1)$ appear to be smaller than 0.04 magnitudes for stars brighter than $T_1$ $\sim$ 18.5 and 17.5 magnitudes, respectively. The bright stars having large associated errors are stars saturated in all frames, bits of stars or even failures in the detector erroneously taken as stars by DAOPHOT. It can be observed in Section 4 that these stars do not affect at all the fundamental parameter determination of the observed clusters. Bearing in mind the behavior of the photometric errors with the $T_1$ magnitude for the observed stars in Fig. 2, we trust the accuracy of the morphology and position of the main clster features in the $(C-T_1,T_1)$ CMDs. Fig. 3 shows these CMDs for all the observed stars in the different fields. They appear to be contaminated by field stars, as should be expected since all the studied OCs are projected on to the Galactic center direction. As can be seen in Fig 3, the faintest $T_1$ magnitudes obtained for each cluster are such that they allow the mapping of most of the cluster main sequences (MSs). If there are still fainter stars in the clusters, then they practically do not add any information to the one already obtained. Cluster MSs appear as broad sequences of stars among crowded field features. On the other hand, a group of comparatively bright late-type stars in BH211 seems to form the cluster red giant clump (RGC) centered at $T_1$ $\sim$ 12.5 and $(C-T_1)$ $\sim$ 3.2 magnitudes, which indicates that we are possibly dealing with an intermediate-age OC.
Determining the center and size of the clusters
===============================================
To determine the central position of the clusters on a firm basis, we applied a statistical method consisting in tracing the stellar density profiles projected on to the directions of the $x$ and $y$-axes. The coordinates of the clusters’ centers and their estimated uncertainties were determined by fitting Gaussian distributions to the star counts in the $x$ and $y$ directions for each cluster. The fits of the Gaussians were performed using the NGAUSSFIT routine in the STSDAS/IRAF package. We adopted a single Gaussian and decided to fix the constant and the linear terms to the corresponding background levels and to zero, respectively. We used as variables the center of the Gaussian, its amplitude and its FWHM. After eliminating a couple of scattered points, the fitting procedure converged after one iteration on average. The stars projected along the $x$ and $y$ directions were counted within intervals of 50 pixels in BH84 and NGC5381 and within intervals of 75 pixels in BH211 and Czernik37. We determined the central position of the clusters with a typical NGAUSSFIT standard deviation of $\pm$ 10 pixels ($\sim$ 4") in all cases. The centers of the Gaussians were finally fixed at ($X_C,Y_C$) = (969,1040), (940,1012), (964,1145) and (894,1105) pixels for BH84, NGC5381, BH211 and Czernik37, respectively. These values were adopted for the analysis that follows. Cluster centers are marked by a cross in Fig. 1.
We then built clusters’ radial profiles, from which we estimated the cluster radii, generally used as an indicator of cluster dimensions. Cluster stellar density radial profiles are usually built by counting the number of stars distributed in concentric rings around the cluster center and normalizing the sum of stars in each ring to the unit area. Although this procedure allows us to stretch the radial profile to its utmost, until complete circles can no longer be traced in the observed field, we preferred to follow another method in order to move even further away from the cluster center. This method is based on counts of stars located in boxes of 50 pixels a side, distributed throughout the whole field of each cluster. Thus, the number of stars per unit area at a given $r$ can be directly calculated through the expression:
$$(n_{r+25} - n_{r-25})/[(m_{r+25}-m_{r-25})\times 50^{2}],$$
where $n_j$ and $m_j$ represent the number of counted stars and boxes included in a circle of radius $j$, respectively. Note that this procedure does not necessarily require a complete circle of radius $r$ within the observed field to estimate the mean stellar density at such distance. It is important to consider this fact since having a stellar density profile which extends far away from the cluster center allows us to estimate the background level with higher precision. This is necessary to derive the cluster radius ($r_{cl}$), defined as the distance from the cluster center where the stellar density profile intersects the background level. It is also helpful to measure the FWHM of the stellar density profile, which plays an important role in the construction of the cluster CMDs. The resulting density profiles expressed as number of stars per unit area in arcmin$^2$ are presented in Fig. 4. The uncertainties estimated at various distances from the cluster centers follow Poisson statistics. The new equatorial coordinates derived for the clusters in this study are listed in Table 1, while columns 2-4 of Table 8 list the radii at the FWHM ($r_{FWHM}$), the estimated radii which yield the best enhanced cluster fiducial features ($r_{clean}$) and the cluster radii ($r_{cl})$. The different linear radii in parsecs were determined using the heliocentric distances derived in Section 4.
Cluster fundamental parameter estimates
=======================================
CMDs covering different circular extractions around each cluster center were constructed, as shown in Figs. 5-8. The panels in the figures are arranged, from left to right and from top to bottom, in such a way that they exhibit the variations in stellar population from the innermost to the outermost regions of the cluster fields. We started with the CMD for stars distributed within $r$ $<$ $r_{FWHM}$, followed by those of the cluster regions delimited by $r$ $<$ $r_{clean}$ and $r$ $<$ $r_{cl}$ and finally by the adopted field CMD. We used the CMDs corresponding to the stars within $r_{FWHM}$ as the cluster fiducial sequence references. Then, we varied the distance from the cluster centers, starting at $r_{FWHM}$. Next, we built different series of extracted CMDs. Finally, we chose those CMDs - one per cluster - which maximize the star cluster population and minimize the field star contamination in the CMDs. The main cluster features can be identified by inspecting the right top panels of Figs. 5-8. Note that the cluster MSs look well populated, particularly in NGC5381, all of them showing clear evidence of evolution. These MSs develop along $\sim$ 5 magnitudes in BH84 and Czernik37 and along $\sim$ 5-6 magnitudes in NGC5381 and BH211. The hook at the MS turnoff of BH211’s CMD and the RGC centered at $T_1$ $\sim$ 12.5 and $C-T_1$ $\sim$ 3.2 magnitudes indicate that we are dealing with an intermediate-age OC. The width of the clusters’ MSs is clearly not the result of photometric errors, since these hardly reach a tenth of magnitude at any $T_1$ level. Thus, such width could be caused by intrinsic effects (binarity, rotation, evolution, etc.), by differential reddening and/or by field star contamination. To derive the cluster fudamental parameters, we will use the CMDs with $r$ $<$ $r_{clean}.$
Although, as we mentioned above, the broadness of the clusters’ MSs is certainly due to several effects, in the particular case of Czernik37, a clear variation of the interstellar reddenning across the cluster field seems to be present. In case this effect indeed existed in the remaining clusters, it is by far less evident. The lower limit estimated by @b75 for clusters with differential reddening is $\Delta$$(B-V)$ = 0.11, which corresponds to $\Delta$$(C-T_1)$ = 0.22, if a value of 1.97 for the $E(C-T_1)$/$E(B-V)$ ratio [@ge96] is adopted. From the right top panel of Fig. 8, we estimated $\Delta$$(C-T_1)$ $\sim$ 1.0 mag for Czernik37, a value which largely exceeds the limit established by @b75. The existence of differential reddening in Czernik37 makes the determination of its fundamental parameters very difficutl, particularly its reddening and its heliocentric distance. It is for this reason that the resulting parameters for Czernik37 present associated errors larger than in the other clusters.
The widely used procedure of fitting theoretical isochrones to observed CMDs was employed to estimate the $E(C-T_1)$ color excess, the $T_1$-M$_{T_1}$ apparent distance modulus and the age and metallicity of the clusters. It is well known that the metallicity of a cluster plays an important role when its age is estimated from the fit of theoretical isochrones. Indeed, isochrones with the same age but with different metallicities can range from slightly to remarkably distinguishable, depending on their sensitivity to metal content. The $(C-T_1,T_1)$ CMD, for example, is nearly three times more metal sensitive than the $(V-I,V)$ CMD [@gs99]. The distinction is particularly evident for the evolved phases of the RGC and the giant branch. As far as zero-age main sequences (ZAMSs) are concerned, they are often less affected by metallicity effects, and can even exhibit imperceptible variations for a specific metallicity range within the photometric errors. This is the case of Galactic OCs, therefore including the four clusters here studied. Since there are no previous available estimates of their metallicities, we followed the general rule of starting by the adoption of both solar ($Z$ = 0.019) and sub-solar ($Z$ = 0.008) values for the clusters’ metal content.
As for the isochrone sets, we used those computed by the Padova group [@gietal02] in steps of $\Delta$log age = 0.05 dex. As shown in previous studies (e.g., Piatti et al., 2003b), these isochrones lead to results similar to those derived from the Geneve group’s isochrones [@ls01]. We preferred to use the Padova group’s isochrones because they reach fainter magnitudes, thus allowing a better fit to the cluster fainter portions of the MSs. Then, we first selected ZAMSs with $Z$ = 0.019 and 0.008 (\[Fe/H\] = 0.0 and -0.4 dex) and fitted these ZAMSs to the cluster CMDs to derive color excesses and apparent distance moduli for each selected metallicity. Since the fits are performed through the lower envelope of the cluster’s MSs, the presence of binaries practically does not affect the choice of the best isochrones. Note that when $Z$ = 0.019 is used in the fits, the resulting cluster reddenings and distances turn out to be slightly larger than the values obtained using Z = 0.008. The increases in distance vary from 250 pc to 500 pc depending on the cluster, while the increase in reddening is within error limits. Then, using each of the derived \[$E(C-T_1)$,$T_1$-M$_{T_1}$\]$_{Z}$ sets, we performed isochrone fits. We repeated the fits for a larger number of isochrones covering appropriate age ranges according to each cluster. The brightest magnitude in the MS, the bluest point of the turnoff and the locus of the RGC, when visible, were used as reference points during the fits. Finally, we chose the best fit for each \[$E(C-T_1)$,$T_1$-M$_{T_1}$\]$_{Z}$ set and compared all the individual best fits to choose the one which best reproduced the cluster features. In all cases, the best fits were done only by eye and they were obtained using solar metallicity isochrones. We would like to point out that the bright stars with very large associated errors in Fig. 2 correspond to objects located beyond NGC5381 $r_{clean}$. For this reason, these stars do not affect the cluster parameter determination in NGC5381. Something similar occurs with the other three studied clusters.
Fig. 9 illustrates the results of our task, while Table 9 lists the estimated $E(B-V)$ color excesses, heliocentric distances, ages and metallicities of the clusters. Their errors were derived taking into account the broadness of the cluster MSs and, in the case of the cluster distances, the expression 0.46$\times$\[$\sigma$($V-M_{V}$) + 3.2$\times$$\sigma$$E(B-V)$\]$\times$d, where $\sigma$$(V-M_{V})$ and $\sigma$$(E(B-V))$ represent the estimated errors in $V-M_{V}$ and $E(B-V)$, respectively (see, e.g., Clariá et al., 2007). The expressions $E(C-T_1)$ = 1.97$\times$$E(B-V)$ and M$_{T_1}$ = $T_1$ + 0.58$\times$$E(B-V)$ - (V-M$_{V}$) taken from @ge96 were used to relate both color excesses and distance moduli. We also list in Table 9 the Galactocentric rectangular coordinates $X$,$Y$,$Z$ and the Galactocentric distances $R_{GC}$ of the clusters, derived assuming the Sun’s distance from the center of the Galaxy to be 8.5 kpc. The adopted reference system is centered on the Sun, with the $X$ and $Y$-axes lying on the Galactic plane and $Z$ perpendicular to the plane. $X$ points towards the Galactic center, being positive in the first and fourth quadrants; $Y$ points in the direction of the Galactic rotation, being positive in the first and second Galactic quadrants. $Z$ is positive towards the north Galactic pole.
Results
=======
The 2MASS catalog has been widely used to determine the fundamental parameters of hundreds of OCs (see, e.g., Kronberger et al., 2006; Bonatto & Bica 2007; Tadross 2011). The results obtained, however, do not always yield consistent results with those derived using optical data. As shown by @dietal12, the accuracy in determining the color excess $E(J-H)$ using only 2MASS is mainly limited by structural uncertainty in the MS and/or narrow range of magnitude sampling of the MS. From the observational point of view, the main reason for this discrepancy is the limiting magnitude, particularly for the oldest clusters, which reflects in the sampling of the MS, together with photometric errors. Note that 2MASS photometric errors typically reach 0.10 magnitudes at $J$ $\leq$ 16.2 and $H$ $\leq$ 15.0 [@sb02], while in the optical bands ($UBV$ or $CT_1$, for example), they are typically lower than 0.05 magnitudes at $V$ $\leq$ 18.0 and $T_1$ $\leq$ 19.0. In the following subsections, we compare the current results with those obtained in previous studies using 2MASS.
BH84
----
The equatorial coordinates we derived for the center of BH84 differ only by 33" in declination from the WEBDA value (Table 1). The radial number density profile of BH84 is shown in Fig.4, from which we derive a radius of 3.6’, slightly lower than the value reported by @ah03. The cluster MS is easily identifiable in Fig. 5, extending along $\sim$ 5 magnitudes and with clear evidence of some evolution. Outer Galactic disc stars remarkably contaminate the cluster MS, particularly its fainter half portion. The solar metallicity ZAMS fitted to the cluster CMD yields a reddening $E(C-T_1)$ = 1.25 $\pm$ 0.10, equivalent to $E(B-V)$ = 0.63 $\pm$ 0.05, and a true distance modulus $(T_1)$$_{o}$-M$_{T_1}$ = 12.64 $\pm$ 0.15, which implies a distance of (3.37 $\pm$ 0.48) kpc from the Sun. For $Z$ = 0.019, the best-fitting isochrone corresponds to an age of (560$^{+150}_{-110}$) Myr (Fig. 9). Therefore, BH84, situated at 142 pc below the Galactic plane and $\sim$ 8.58 kpc from the Galactic center (Table 9), is found to be a cluster only slightly younger than the Hyades. Both the reddening and the distance here derived show a reasonable agreement with the results from 2MASS data [@buetal11]. Surprisingly, however, the age we find is significantly larger than @buetal11 estimate of 18 Myr. Although some of the 2MASS error must be due to the larger pixels, we believe the authors’ gross underestimation of BH84 age is clearly the result of the impossibility to see the turnoff point in the $(J-K,J)$ diagram. Hence, @buetal11 derived the cluster’s age just by fitting the ZAMS.
NGC5381
-------
This appears to be a large cluster covering nearly the entire observed field (Fig.1). We derived for NGC5381 practically the same equatorial coordinates as those listed in WEBDA (Table 1). It is difficult, however, to estimate the cluster radius from the stellar density radial profile (Fig. 4), because it does not present a typical cluster-like structure. If the area for $r$ $>$ 1200 pixels is considered to be the “star field area”, then NGC5381 seems to have a relatively small but conspicuous nucleus and a low-density extended corona (see also Fig. 1). We estimate the angular core and corona radii as $\sim$ 330 pixels ($\sim$ 2.2’) and 870 pixels ($\sim$ 5.7’), respectively. Fig. 3 reveals a crowded broad sequence of stars that traces the cluster MS along $\sim$ 5-6 magnitudes and with clear signs of evolution. This fact suggests that the age of the cluster is some hundred million years. A number of stars visible in Fig. 9 with $T_1$ magnitudes between 18 and 19 and $(C-T_1)$ colors ranging from 0.5 up to 1.5 magnitudes are clearly background stars. No clump of red stars is visible. However, the analysis of 2MASS data by @ta11 reveals that NGC5381 is an intermediate-age (1.6 Gyr) cluster, suffering low interstellar extinction. This is not confirmed by the current optical data. In fact, our best fit of the solar metallicity ZAMS in the $(C-T_1,T_1)$ CMD yields $E(C-T_1)$ = 0.90 $\pm$ 0.08, equivalent to $E(B-V)$ = 0.46 $\pm$ 0.04, and a heliocentric distance of (2.63 $\pm$ 0.40) kpc, while an age of about (250$^{+65}_{-50}$) Myr is derived from the best-fitting isochrone (Fig. 9). Therefore, NGC5381 now appears to be significantly younger than previously believed. The age difference between ours and @ta11 may depend at least partially on the reddening difference. Our reddening estimate, $E(B-V)$ = 0.46, is quite larger than the value of $E(B-V)$ = 0.06 reported by @ta11. Therefore, a smaller age value is obtained when a larger reddening value is adopted. NGC5381 lies at $\sim$ 100 pc above the Galactic plane and $\sim$ 7.04 kpc from the Galactic center (Table 9).
BH211
-----
The coordinates for the cluster center derived in the current study differ by $\sim$ 15“ in right ascension and by only 3” in declination from the WEBDA values (Table 1). From the stellar density radial profile (Fig. 4), we determined the radius of BH211 to be $\sim$ 3.9’, in very good agreement with the value reported by @ah03. The $(C-T_1,T_1)$ CMD obtained using all the measured stars in the cluster field is depicted in Fig. 3, wherein the main cluster features can be identified. What first calls our attention is the cluster MS, which looks well populated, has clear signs of evolution and develops along $\sim$ 5-6 magnitudes. It is relatively broad, especially in its lower envelope, partly due to field star contamination. Another interesting feature of the $(C-T_1,T_1)$ CMD is the presence of a number of good candidates for giant clump stars centered at $(C-T_1,T_1)$ = (3.2,12.5). Since these stars lie within 400 pixels (2.6’) from the cluster center, we may reasonably expect many of them to be cluster giants. The reddening and age we find, $E(B-V)$ = 0.61 $\pm$ 0.05 and $\sim$ (1000$^{+260}_{-200}$) Myr (Table 9), turn out to be somewhat larger and lower, respectively, than @buetal11 estimates. In any case, there seems to be no doubt that BH211 is an intermediate-age cluster located at about (1.44 $\pm$ 0.21) kpc from the Sun, at scarcely $\sim$ 12 pc out of the Galactic plane and $\sim$ 7.12 kpc from the Galactic center (Table 8).
Czernik37
---------
The equatorial coordinates we derived for this cluster are different by 15“ in right ascension and by 36” in declination (Table 1) from the WEBDA values. The radial number density profile of Czernik37 is shown in Fig. 4, from which we derived a radius of 3.6’, in good agreement with the value reported by @ah03. This cluster appears to be embedded in the dense stellar population towards the Galactic bulge so that their properties are difficult to be determined. Although the cluster CMD (Fig. 3) is profusely contaminated by field stars, there is an appearance of a broad MS with evidence of some evolution. As mentioned in Section 4, the broadness of the MS of Czernik37 is caused not only by field star contamination and other effects (binarity, rotation, photometric errors, ets.) but also by differential reddening. Such effect is probably due to the presence of irregularly distributed dark clouds, projected towards the cluster. Different isochrone fittings using $Z$ = 0.019 in Fig. 9 allow us to estimate the variation range of $E(C-T_1)$ as $\sim$ 1.0 mag. The solar metallicity ZAMS which best fits the cluster CMD (Fig. 9) yields a reddening $E(C-T_1)$ = 2.90 $\pm$ 0.50, equivalent to $E(B-V)$ = 1.47 $\pm$ 0.25, and a true distance modulus $(T_1)$$_{o}$- M$_{T_1}$ = 10.80 $\pm$ 0.50. The isochrone corresponding to an age of (250$^{+100}_{-65}$) Myr reasonably reproduces the main cluster features in Fig. 9. These results place Czernik37 at a distance of (1.44 $\pm$ 0.86) kpc from the Sun and $\sim$ 7.1 kpc from the Galactic center (Table 9).The current cluster parameters exhibit a rather poor agreement with those derived by Tadross (2008) from 2MASS data. As in NGC5381, our larger E(B-V) value compared to that of Tadross (2008) may partly explain our lower age estimate.
Conclusions
===========
We have presented new CCD Washington $CT_1$ photometry in the field of four Galactic OCs projected onto the two inner quadrants of the Galactic plane. These data were obtained with the main purpose of estimating the cluster fundamental astrophysical parameters. We performed a star count analysis of the cluster fields to assess the clusters’ reality as over-densities of stars with respect to the field and estimated the cluster radii. We determined the center of the clusters by finding the maximum surface number density of the stars in each cluster. New equatorial coordinates for the 2000.0 epoch are now provided. We outlined possible solutions for cluster fundamental parameters by matching theoretical isochrones, which reasonably reproduce the main cluster features in the $(C-T_1,T_1)$ CMDs. In all cases, the best fits were obtained using solar metallicity isochrones. BH211 was found to be the oldest object of our sample with an age of around 1.0 Gyr. Czernik37, the most heavily reddened cluster of the sample, with a mean colour excess $E(B-V)$ = 1.47, is very likely affected by differential reddening. BH84 is located in the fourth Galactic quadrant just before the tangent to the Carina branch of the Carina-Sagittarius spiral arm. This cluster turned out to be much older than previously believed. Conversely, NGC5381 was found to be much younger than previously reported. It appears to have a relatively small but conspicuous nucleus and a low-density extended corona. We estimated the angular core and corona radii as $\sim$ 2.2’ and $\sim$ 5.7’, respectively. The derived fundamental properties for the studied clusters are listed in Table 9. Previous estimates of cluster parameters are listed in Table 10, for easy comparison with the present results. Since two of the studied clusters, BH211 and Czernik37, are in the VISTA Variables in the Via Lactea (VVV) survey [@mietal10], additional observational information about these two objects can be found in this database.
----------- ----------------------- --------------------------------------------------------------------------- --------- ------- ------- ----------------------- -----------------------
Cluster $\alpha$$_{\rm 2000}$ $\delta$$_{\rm 2000}$ [*l*]{} $b$ Diam. $\alpha$$_{\rm 2000}$ $\delta$$_{\rm 2000}$
(h m s) ($^\circ$ ’ “) & ($^\circ$) & ($^\circ$) & (’) & (h m s) & ($^\circ$ ’ ”)
BH84 10 01 19 -58 13 00 280.06 -2.42 4.5 10 01 19 -58 13 33
NGC5381 14 00 41 -59 35 12 311.60 2.11 11.0 14 00 41 -59 35 20
BH211 17 02 11 -41 06 00 344.97 0.46 4.0 17 02 10 -41 05 57
Czernik37 17 53 16 -27 22 00 2.22 -0.64 3.0 17 53 17 -27 22 36
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: Basic parameters of the four open clusters
----------- -------------- -------- ---------- -----------------------------------------------
Cluster Date Filter Exposure Airmass
(sec) (“)\
\
Cluster & Date & Filter & Exposure & Airmass\
& & & (sec) & (”)
BH84 May 9, 2008 $C$ 30 1.13
$C$ 45 1.13
$C$ 300 1.13
$C$ 450 1.13
$R$ 5 1.13
$R$ 7 1.12
$R$ 30 1.12
$R$ 45 1.12
NGC5381 May 11, 2008 $C$ 90 1.15
$C$ 120 1.15
$C$ 600 1.15
$C$ 30 1.16
$R$ 3 1.16
$R$ 120 1.16
$R$ 120 1.16
BH211 $C$ 30 1.02
$C$ 45 1.02
$C$ 300 1.02
$C$ 450 1.02
$R$ 5 1.02
$R$ 7 1.02
$R$ 30 1.02
$R$ 45 1.02
Czernik37 May 10, 2008 $C$ 30 1.00
$C$ 45 1.00
$C$ 300 1.00
$C$ 450 1.00
$R$ 30 1.00
$R$ 45 1.00
$R$ 5 1.00
$R$ 7 1.00
----------- -------------- -------- ---------- -----------------------------------------------
: Observation log of observed clusters
$C$ $T_1$
-------------------------- --------------------------
$a_1$ = 3.61 $\pm$ 0.03 $b_1$ = 3.04 $\pm$ 0.02
$a_2$ = 0.56 $\pm$ 0.01 $b_2$ = 0.33 $\pm$ 0.02
$a_3$ = -0.19 $\pm$ 0.01 $b_3$ = -0.03 $\pm$ 0.01
: Standard system mean calibration coefficients
------ ---------- --------- --------- --------------- --------- ------------------- ---
Star $X$ $Y$ $T_{1}$ $\sigma$$T_1$ $C-T_1$ $\sigma$$(C-T_1)$ n
(pixel) (pixel) (mag) (mag) (mag) (mag)
495 1870.105 584.378 17.590 0.094 3.237 0.071 1
496 926.421 585.257 15.910 0.018 1.955 0.015 2
497 1568.217 585.395 13.705 0.010 1.653 0.009 1
- - - - - - - -
- - - - - - - -
------ ---------- --------- --------- --------------- --------- ------------------- ---
: CCD $CT_1$ data of stars in the field of BH84
------ ---------- --------- --------- --------------- --------- ------------------- ---
Star $X$ $Y$ $T_{1}$ $\sigma$$T_1$ $C-T_1$ $\sigma$$(C-T_1)$ n
(pixel) (pixel) (mag) (mag) (mag) (mag)
499 918.737 415.590 13.625 0.005 1.874 0.005 2
500 501.666 418.198 14.553 0.005 1.225 0.005 1
501 1000.223 422.144 16.461 0.022 1.801 0.017 2
- - - - - - - -
- - - - - - - -
------ ---------- --------- --------- --------------- --------- ------------------- ---
: CCD $CT_1$ data of stars in the field of NGC5381
------ ---------- ---------- --------- --------------- --------- ------------------- ---
Star $X$ $Y$ $T_{1}$ $\sigma$$T_1$ $C-T_1$ $\sigma$$(C-T_1)$ n
(pixel) (pixel) (mag) (mag) (mag) (mag)
496 1255.054 1053.165 16.858 0.043 2.626 0.034 2
497 1137.926 2039.064 16.899 0.052 2.597 0.040 1
498 983.109 1055.830 15.363 0.016 2.018 0.014 1
- - - - - - - -
- - - - - - - -
------ ---------- ---------- --------- --------------- --------- ------------------- ---
: CCD $CT_1$ data of stars in the field of BH211
------ ---------- --------- --------- --------------- --------- ------------------- ---
Star $X$ $Y$ $T_{1}$ $\sigma$$T_1$ $C-T_1$ $\sigma$$(C-T_1)$ n
(pixel) (pixel) (mag) (mag) (mag) (mag)
500 1480.077 796.860 15.898 0.038 3.126 0.033 1
501 961.983 801.728 15.552 0.021 2.819 0.018 2
502 1114.297 805.020 15.474 0.041 2.794 0.03 1
- - - - - - - -
- - - - - - - -
------ ---------- --------- --------- --------------- --------- ------------------- ---
: CCD $CT_1$ data of stars in the field of Czernik37
------------ ------ ------ ------ ------ ------- ------
Cluster
(px) (pc) (px) (pc) (pix) (pc)
BH84 150 1.0 200 1.3 550 3.6
NGC5381 150 0.8 280 1.4 1200 6.1
BH211 150 0.4 250 0.7 580 1.6
Czernick37 220 0.6 320 0.9 550 1.5
------------ ------ ------ ------ ------ ------- ------
: Cluster sizes
----------- ----------------- ----------------- ---------------------- ---------- ------- ------- ------- ----------
Cluster $E(B-V)$ $d$ Age \[Fe/H\] $X$ $Y$ $Z$ $R_{GC}$
(mag) (kpc) (Myr) (dex) (kpc) (kpc) (kpc) (kpc)
BH84 0.63 $\pm$ 0.05 3.37 $\pm$ 0.48 560$^{+150}_{-110}$ 0.0 7.91 -3.32 -0.14 8.58
NGC5381 0.46 $\pm$ 0.04 2.63 $\pm$ 0.40 250$^{+65}_{-50}$ 0.0 6.76 -1.97 0.10 7.04
BH211 0.61 $\pm$ 0.05 1.44 $\pm$ 0.21 1000$^{+260}_{-200}$ 0.0 7.11 -0.37 0.01 7.12
Czernik37 1.47 $\pm$ 0.25 1.44 $\pm$ 0.86 250$^{+100}_{-65}$ 0.0 7.06 0.06 -0.02 7.06
----------- ----------------- ----------------- ---------------------- ---------- ------- ------- ------- ----------
: Fundamental properties of the observed clusters
----------- ---------- ----------------- ------- -----------
Cluster $E(B-V)$ $d$ Age Reference
(mag) (kpc) (Myr) \#
BH84 0.60 2.92 $\pm$ 0.19 18 1
NGC5381 0.06 1.17 $\pm$ 0.05 1600 2
BH211 0.48 1.38 $\pm$ 0.09 1600 1
Czernik37 1.03 1.73 $\pm$ 0.08 600 3
----------- ---------- ----------------- ------- -----------
: Previous stimates of cluster parameters
References: (1) Bukowiecki et al.(2011); (2) Tadross (2011); (3) Tadross (2008)
![Schematic finding charts of the stars observed in BH84 (top left), NGC5381 (top right), BH211 (bottom left) and Czernik37 (bottom right). North is up and East is to the left. The sizes of the plotting symbols are proportional to the $T_1$ brightness of the stars. Two circles $r_{clean}$ and $r_{cl}$ wide are shown around the cluster centers ([*crosses*]{}).[]{data-label="fig1"}](Fig1.ps){width="12cm"}
![$T_1$ magnitude and C-T$_1$ color photometric errors as a function of $T_1$ for stars measured in the field of NGC5381.[]{data-label="fig2"}](Fig2.ps){width="12cm"}
![($C-T_1$,$T_1)$ CMDs for stars observed in the field of BH84, NGC5381, BH211 and Czernik37.[]{data-label="fig3"}](Fig3.ps){width="12cm"}
![Cluster stellar density radial profiles normalized to a circular area of 50 pixel radius. The radius at the FWHM (r$_{FWHM}$) and the adopted cluster radius (r$_{cl}$) are indicated by green vertical lines. The red horizontal lines represent the measured background levels.[]{data-label="fig4"}](Fig4.ps){width="12cm"}
![CMDs for stars observed in different extracted circular regions around BH84 center as indicated in each panel.[]{data-label="fig5"}](Fig5.ps){width="12cm"}
![CMDs for stars observed in different extracted circular regions around NGC5381 center as indicated in each panel.[]{data-label="fig6"}](Fig6.ps){width="12cm"}
![CMDs for stars observed in different extracted circular regions around BH211 center as indicated in each panel.[]{data-label="fig7"}](Fig7.ps){width="12cm"}
![CMDs for stars observed in different extracted circular regions around Czernik37 center as indicated in each panel.[]{data-label="fig8"}](Fig8.ps){width="12cm"}
{width="12cm"}
Acknowledgements
================
J.J. Clariá, T. Palma and A.V. Ahumada are gratefully indebted to the CTIO staff for their hospitality and support during the observing run. We also thank the anonymous referee for his/her valuable comments and suggestions. This research was partially supported by the Argentinian institutions CONICET, SECYT (Universidad Nacional de Córdoba) and Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT). We have used both the SIMBAD database, operated at CDS, Strasbourg, France, and the NASA’s Astrophysics Data System. This work is based on observations made at Cerro Tololo Inter-American Observatory, which is operated by AURA, Inc., under cooperative agreement with the NSF.
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[^1]: IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation
|
---
abstract: 'New results on hadron spectra have been appearing in abundance in the past few years as a result of improved experimental techniques. These include information on states made of both light quarks (u, d, and s) and with one or more heavy quarks (c, b). The present review, dedicated to the memory of R. H. Dalitz, treats light-quark states, glueballs, hybrids, charmed and beauty particles, charmonium, and $b \bar b$ states. Some future directions are mentioned.'
author:
- 'Jonathan L. Rosner'
title: Hadron Spectroscopy in 2006
---
[ address=[Enrico Fermi Institute and Department of Physics, University of Chicago, 5640 S. Ellis Avenue, Chicago IL 60637 USA]{} ]{}
INTRODUCTION
============
Quantum Chromodynamics (QCD) is our theory of the strong interactions. However, we are far from understanding how it works in many important cases. Many hadrons discovered recently have puzzling properties. Hadron spectra often are crucial in separating electroweak physics from strong-interaction effects. QCD may not be the only instance of important non-perturbative effects; one should be prepared for surprises at the Large Hadron Collider (LHC). Sharpening spectroscopic techniques even may help understand the intricate structure of masses and transitions at the quark and lepton level.
The QCD scale is $\sim 200$ MeV (momentum) or $\sim 1$ fm (distance), where perturbation theory cannot be used. Although lattice gauge theories are the eventual tool of choice for describing effects in this regime, several other methods can provide information, especially for multi-quark and multi-hadron problems not yet feasible with lattice techniques. These include chiral dynamics (treating soft pions, chiral solitons, and possibly parity doubling in spectra [@Jaffe06]), heavy quark symmetry (describing hadrons with one charm or beauty quark as QCD “hydrogen” or “deuterium” atoms), studies of correlations among quarks [@KL; @JW; @SW] and new states they imply (such as a weakly decaying $bq \bar c \bar q'$ state [@KL]), potential descriptions (including relativistic and coupled-channel descriptions), and QCD sum rules. I will describe phenomena to which these methods might be applied.
In the present review I treat light-quark (and no-quark) states, charmed and beauty hadrons, and heavy quarkonium ($c \bar c$ and $b \bar b$), and conclude with some future prospects.
LIGHT-QUARK STATES
==================
Several issues are of interest these days in light-quark spectroscopy. These include (1) the nature of the low-energy S-wave $\pi \pi$ and $K \pi$ interactions; (2) the proliferation of interesting threshold effects in a variety of reactions, and (3) the interaction of quark and gluonic degrees of freedom. -0.3in
Low-energy $\pi \pi$ S-wave
---------------------------
-0.1in
An S-wave $\pi \pi$ low-mass correlation in the $I=0$ channel (“$\sigma$”) has been used for many years to describe nuclear forces. Is it a resonance? What is its quark content? What can we learn about it from charm and beauty decays? This particle, otherwise known as $f_0(600)$ [@PDG], can be described as a dynamical $I=J=0$ resonance in elastic $\pi \pi$ scattering using current algebra, crossing symmetry, and unitarity . It appears as a pole with a large imaginary part with real part at or below $m_\rho$. Its effects differ in $\pi \pi \to \pi \pi$, where an Adler zero suppresses the low-energy amplitude, and inelastic processes such as $\gamma \gamma \to \pi \pi$ [@GR], where the lack of an Adler zero leads to larger contributions at low $m_{\pi \pi}$.
![Fits to $\gamma \gamma \to \pi^0 \pi^0$ in various models. From Ref. [@MP06]. \[fig:ggpzpz\]](ggpzpz.eps){width="75.00000%"}
Modern treatments of the low-energy $\pi \pi$ interaction implement crossing symmetry using an elegant set of exact low-energy relations [@Roy71]. In one approach [@CCL] a $\sigma$ pole is found at $441 - i 272$ MeV, corresponding to a full width at half maximum of 544 MeV; another [@vanBeveren:2006ua] finds the pole at $555 - i 262$ MeV. Such a $\sigma$ provides a good description of $\gamma \gamma \to \pi^0 \pi^0$ [@MP06], as shown in Fig. \[fig:ggpzpz\], with $\Gamma(\sigma \to \gamma \gamma) =
(4.1 \pm 0.3)$ keV. While this large partial width might be viewed as favoring a $q \bar q$ interpretation of $\sigma$ [@MP06], a $\pi \pi$ dynamical resonance seems equally satisfactory [@GR]. Other recent manifestations of a $\sigma$ include the decays $D^+ \to \sigma \pi^+ \to \pi^+ \pi^- \pi^+$ [@E791sigma] and $J/\psi \to \omega \sigma \to \omega \pi^+ \pi^-$ [@BESsigma], where the $\sigma$ pole appears at $(541\pm 39) - i
(252 \pm 42)$ MeV (or $(500 \pm 30) - i(264 \pm 30)$ MeV in an independent analysis [@Bugg:2006gc]). Successful fits without a $\sigma$ have been performed, but have been criticized in Ref. [@Bugg:2005nt]. -0.3in
Low-energy $K \pi$ S-wave
-------------------------
-0.1in
Is there a low-energy $K \pi$ correlation (“$\kappa$”)? Can it be generated dynamically in the same manner as the $\sigma$? Some insights are provided in [@Dobado:1992ha; @Oller].
The low-energy $K \pi$ interaction in the $I=1/2,~J=0$ channel is favorable to dynamical resonance generation: The sign of the scattering length is the same as for the $I=J=0~\pi \pi$ interaction. A broad scalar resonance $\kappa$ is seen in the $I=1/2,~J=0~K^- \pi^+$ subsystem in $D^+ \to K^-\pi^+ \pi^+$, and a model-independent phase shift analysis shows resonant $J=0$ behavior in this subsystem [@Aitala:2002kr]. The $\kappa$ is also seen by the BES II Collaboration in $J/\psi \to \bar K^{*0}(892) K^+ \pi^-$ decays [@Ablikim:2005ni]. An independent analysis of the BES II data [@vanBeveren:2006ua] finds a $\kappa$ pole at $745 - i 316$ MeV, while a combined analysis of $D^+ \to K^-\pi^+ \pi^+$, elastic $K \pi$ scattering, and the BES II data [@Bugg:2005xx] finds a pole at $M(\kappa) =
(750^{+30}_{-55}) - i(342 \pm 60)$ MeV.
![Dalitz plot for $D^0 \to K^+ K^- \pi^0$. From Ref. [@Paras06]. \[fig:KKpi\]](KKpiDal.eps){height="0.47\textheight"}
The $\kappa$, like the $\sigma$, is optional in many descriptions of final-state interactions. An example is a recent fit to the $D^0 \to K^+ K^-
\pi^0$ Dalitz plot based on CLEO data [@Paras06], shown in Fig.\[fig:KKpi\]. The bands correspond to $K^{*-}$ (vertical), $K^{*+}$ (horizontal), and $\phi$ (diagonal). One can see the effect of an S-wave (nonresonant or $\kappa$) background interfering with $K^{*+}$ and $K^{*-}$ with opposite signs on the left and bottom of the plot.
Depopulated regions at $m(K^\pm \pi^0) \simeq$ 1 GeV/$c^2$ may be due to the opening of the $K \pi^0 \to K \eta$ S-wave threshold (a $D^0 \to K^+ K^- \eta$ Dalitz plot would test this) or to a vanishing S-wave $K \pi$ amplitude between the $\kappa$ and a higher $J^P = 0^+$ resonance.
Dips and edges
--------------
-0.1in
With the advent of high-statistics Dalitz plots for heavy meson decays one is seeing a number of dips and edges which often are evidence for thresholds [@JLRthr]. An example is shown in a recent $D^0 \to K_S^0 \pi^+ \pi^-$ plot (Fig. 3) from BaBar [@BaKspipi] (see also results from Belle [@BeKspipi] and CLEO [@CLKspipi]). The vertical band corresponds to $K^{*-}$ and the diagonal to $\rho^0$. The sharp edges along the diagonal in the $\pi^+ \pi^-$ spectrum correspond to $\rho$-$\omega$ interference \[around $M(\pi \pi) = 0.8$ GeV/$c^2$\] and to $\pi^+ \pi^- \leftrightarrow K \bar K$ \[around $M(\pi \pi) =
1$ GeV/$c^2$\]. Rapid variation of an amplitude occurs when a new S-wave channel opens because no centrifugal barrier is present.
![Dalitz plot for $D^0 \to K_S^0 \pi^+ \pi^-$. From Ref.[@BaKspipi]. \[fig:KsPiPi\]](KsPiPi.eps){height="0.4\textheight"}
Further dips are seen in $6 \pi$ photoproduction just at $p \bar p$ threshold; in $R_{e^+ e^-}$ just below the threshold for S-wave production of $D(1865)
+D_1(2420)$; and in the Dalitz plot for $B^\pm \to K^\pm K^\mp K^\pm$ around $M(K^+ K^-) = 1.6$ GeV/$c^2$ [@BaKKK], which could be a threshold for vector meson pair production. -1in
Glueballs and hybrids
---------------------
-0.1in
In QCD, quarkless “glueballs” may be constructed from pure-glue configurations: $F^a_{\mu \nu} F^{a \mu \nu}$ for $J^{PC} = 0^{++}$ states, $F^a_{\mu \nu} \tilde{F}^{a \mu \nu}$ for $J^{PC} = 0^{-+}$ states, etc., where $F^a_{\mu \nu}$ is the gluon field-strength tensor. All such states should be flavor-singlet with isospin $I=0$, though couplings of spinless states to $s
\bar s$ could be favored [@Chanowitz:2005du]. Lattice QCD calculations predict the lowest glueball to be $0^{++}$ with $M \simeq 1.7$ GeV [@Campbell:1997]. The next-lightest states, $2^{++}$ and $0^{-+}$, are expected to be several hundred MeV/$c^2$ heavier. Thus it is reassuring that the lightest mainly flavor-singlet state, the $\eta'$, is only gluonic $(8 \pm 2)\%$ of the time, as indicated by a recent measurement of ${\cal B}(\phi \to \eta' \gamma)$ by the KLOE Collaboration [@giov].
Many other $I=0$ levels, e.g., $q \bar q$, $q \bar q g$ ($g$ = gluon), $q q
\bar q \bar q, \ldots$, can mix with glueballs. One must study $I=0$ levels and their mesonic couplings to separate out glueball, $n \bar n \equiv (u \bar
u + d \bar d)/ \sqrt{2}$, and $s \bar s$ components. Understanding the rest of the [*flavored*]{} $q \bar q$ spectrum for the same $J^P$ thus is crucial. The best $0^{++}$ glueball candidates (mixing with $n \bar n$ and $s \bar s$) are at 1370, 1500, and 1700 MeV. One can explore their flavor structure through production and decay, including looking for their $\gamma(\rho,\omega,
\phi)$ decays [@ClZh]. A CLEO search for such states in $\Upsilon(1S)
\to \gamma X$ finds no evidence for them but does see the familiar resonance $f_2(1270)$ [@CLf].
QCD predicts that in addition to $q \bar q$ states there should be $q \bar q g$ (“hybrid”) states containing a constituent gluon $g$. One signature of them would be states with quantum numbers forbidden for $q \bar q$ but allowed for $q \bar q g$. For $q \bar q$, $P = (-1)^{L+1},~C = (-1)^{L+S}$, so $CP =
(-1)^{S+1}$. The forbidden $q \bar q$ states are then those with $J^{PC}=
0^{--}$ and $0^{+-},~1^{-+},~2^{+-},\ldots$. A consensus in quenched lattice QCD is that the lightest exotic hybrids have $J^{PC} = 1^{-+}$ and $M(n \bar n
g) \simeq 1.9$ GeV, $M(s \bar s g) \simeq 2.1$ GeV, with errors 0.1–0.2 GeV [@McNeile:2002az]. (Unquenched QCD must treat mixing with $qq \bar q \bar
q$ and meson pairs.) Candidates for hybrids include $\pi_1(1400)$ (seen in some $\eta \pi$ final states, e.g., in $p \bar p$ annihilations) and $\pi_1(1600)$ (seen in $3 \pi$, $\rho \pi$, $\eta' \pi$). Brookhaven experiment E-852 published evidence for a $1^{-+}$ state called $\pi_1(1600)$ [@Adams:1998ff]. A recent analysis by a subset of E-852’s participants [@Dzierba:2005sr] does not require this particle if a $\pi_2(1670)$ contribution \[an orbital excitation of the $\pi(140)$\] is assumed. The favored decays of a $1^{-+}$ hybrid are to a $q \bar q
(L=0)$ + $q \bar q (L=1)$ pair, such as $\pi b_1(1235)$. A detailed review of glueballs and hybrids has been presented by C. Meyer at this Conference [@Meyer].
CHARMED STATES
==============
The present status of the lowest S-wave states with a single charmed quark is shown in Fig. \[fig:charm\]. We will discuss progress on orbitally-excited charmed baryons [@Mallik; @Tsuboyama] and charmed-strange mesons, with brief remarks on $D^+$ and $D_s^+$ decay constants which are treated in more detail in Ref. [@Briere]. -0.3in
Charmed $L > 0$ baryons
-----------------------
-0.1in
For many years CLEO was the main source of data on orbitally-excited charmed baryons. Now BaBar and Belle are discovering new states, denoted by the outlined levels in Fig. \[fig:lexb\]. The Belle Collaboration observed an excited $\Sigma_c$ candidate decaying to $\Lambda_c \pi^+$, with mass about 510 MeV above $M(\Lambda_c)$ [@Mizuk:2004yu]. The value of its $J^P$ shown in Fig. \[fig:lexb\] is a guess, using the diquark ideas of [@SW]. The highest $\Xi_c$ levels were reported by Belle in Ref. [@Becb]. The highest $\Lambda_c$ is seen by BaBar in the decay mode $D^0 p$ [@Bacb].
In Fig. \[fig:lexb\] the first excitations of the $\Lambda_c$ and $\Xi_c$ are similar, scaling well from the first $\Lambda$ excitations $\Lambda(1405,1/2^-)$ and $\Lambda(1520,3/2^-)$. They have the same cost in $\Delta L$ (about 300 MeV), and their $L \cdot S$ splittings scale as $1/m_s$ or $1/m_c$. Higher $\Lambda_c$ states may correspond to excitation of a spin-zero $[ud]$ pair to $S=L=1$, leading to many allowed $J^P$ values up to $5/2^-$. In $\Sigma_c$ the light-quark pair has $S=1$; adding $L=1$ allows $J^P \le 5/2^-$. States with higher $L$ may be narrower as a result of inreased barrier factors affecting their decays, but genuine spin-parity analyses would be very valuable.
![Lowest S-wave states with a single charmed quark. Only the $\Omega_c^*$ (dashed line denotes predicted mass) has not yet been reported. \[fig:charm\]](charm.ps){height="0.39\textheight"}
![Singly-charmed baryons and some of their orbital excitations. \[fig:lexb\]](lexb.ps){width="0.46\textheight"}
Lowest charmed-strange $0^+$, $1^+$ states
------------------------------------------
-0.1in
![Charmed-strange mesons with $L=0$ (negative-parity) and $L=1$ (positive-parity). Here $j^P$ denotes the total light-quark spin + orbital angular momentum and the parity $P$. \[fig:ds\]](ds.ps){width="95.00000%"}
In the past couple of years the lowest $J^P = 0^+$ and $1^+$ $c \bar s$ states turned out to have masses well below most expectations. If they had been as heavy as the already-seen $c \bar s$ states with $L=1$, the $D_{s1}(2536)$ \[$J^P = 1^+$\] and $D_{s2}(2573)$ \[$J^P = 2^+$\]), they would have been able to decay to $D \bar K$ (the $0^+$ state) and $D^* \bar K$ (the $1^+$ state). Instead several groups [@Aubert:2003fg] observed a narrow $D_s(2317) \equiv D_{s0}^*$ decaying to $\pi^0 D_s$ and a narrow $D_s(2460)
\equiv D_{s1}^*$ decaying to $\pi^0 D_s^*$, as illustrated in Fig.\[fig:ds\]. Their low masses allow the isospin-violating and electromagnetic decays of $D_{s0}^*$ and $D_{s1}^*$ to be observable. The decays $D_s(2460)
\to D_s \gamma$ and $D_s(2460) \to D_s \pi^+ \pi^-$ also have been seen [@Mallik; @Marsiske], and the absolute branching ratios ${\cal B}(D_{s1}^* \to \pi^0 D_s^*) = (0.56 \pm 0.13 \pm 0.09)\%,$ ${\cal B}(D_{s1}^* \to \gamma D_s) = (0.16 \pm 0.04 \pm 0.03)\%,$ ${\cal B}(D_{s1}^* \to \pi^+ \pi^- D_s^*) = (0.04 \pm 0.01)\%$ measured.
The selection rules in decays of these states show their $J^P$ values are consistent with $0^+$ and $1^+$. Low masses are predicted [@Bardeen:2003kt] if these states are viewed as parity-doublets of the $D_s(0^-)$ and $D^*_s(1^-)$ $c \bar s$ ground states in the framework of chiral symmetry. The splitting from the ground states is 350 MeV in each case. Alternatively, one can view these particles as bound states of $D^{(*)}K$, perhaps bound by the transitions $(c \bar q)(q \bar s) \leftrightarrow (c \bar
s)$ (the binding energy in each case would be 41 MeV), or as $c \bar s$ states with masses lowered by coupling to $D^{(*)}K$ channels [@vanBeveren:2003kd; @Close:2004ip; @NewDs].
$D^+$ and $D_s$ decay constants
-------------------------------
-0.1in
CLEO has reported the first significant measurement of the $D^+$ decay constant: $f_{D^+} = (222.6 \pm 16.7^{+2.8}_{-3.4})$ MeV [@Artuso:2005ym]. This is consistent with lattice predictions, including one [@Aubin:2005ar] of $(201 \pm 3 \pm 17)$ MeV. The accuracy of the previous world average [@PDG] $f_{D_s} = (267 \pm 33)$ MeV has been improved by a BaBar value $f_{D_s} = 283 \pm 17 \pm 7 \pm 14$ MeV [@Aubert:2006sd] and a new CLEO value $f_{D_s} = 280.1 \pm 11.6 \pm 6.0$ MeV [@Stone06]. The latter, when combined with CLEO’s $f_D$, leads to $f_{D_s}/f_D = 1.26 \pm 0.11 \pm
0.03$. A lattice prediction for $f_{D_s}$ [@Aubin:2005ar] is $f_{D_s} =
249 \pm 3 \pm 16$ MeV, leading to $f_{D_s}/f_D = 1.24 \pm 0.01 \pm 0.07$. One expects $f_{B_s}/f_B \simeq f_{D_s}/f_D$ so better measurements of $f_{D_s}$ and $f_D$ by CLEO will help validate lattice calculations and provide input for interpreting $B_s$ mixing. A desirable error on $f_{B_s}/f_B
\simeq f_{D_s}/f_D$ is $\le 5\%$ for useful determination of CKM element ratio $|V_{td}/V_{ts}|$, needing errors $\le 10$ MeV on $f_{D_s}$ and $f_D$. The ratio $|V_{td}/V_{ts}| = 0.208^{+0.008}_{-0.006}$ is implied by a recent CDF result on $B_s$–$\overline{B}_s$ mixing [@CDFmix] combined with $B$–$\overline{B}$ mixing and $\xi \equiv (f_{B_s} \sqrt{B_{B_s}}/f_B
\sqrt{B_B}) = 1.21^{+0.047}_{-0.035}$ from the lattice [@Okamoto]. A simple quark model scaling argument anticipated $f_{D_s}/f_D \simeq
f_{B_s}/f_B \simeq \sqrt{m_s/m_d} \simeq 1.25$, where $m_s \simeq 485$ MeV and $m_d \simeq 310$ MeV are constituent quark masses [@Rosner90].
BEAUTY HADRONS
==============
The spectrum of ground-state hadrons containing a single $b$ quark is shown in Fig. \[fig:beauty\]. The following are a few recent high points of beauty hadron spectroscopy.
![S-wave hadrons containing a single beauty quark. Dashed lines denote predicted levels not yet observed. \[fig:beauty\]](beauty.ps){width="98.00000%"}
The CDF Collaboration has identified events of the form $B_c \to J/\psi
\pi^\pm$, allowing for the first time a precise determination of the mass: $M$=(6276.5$\pm$4.0$\pm$2.7) MeV/$c^2$ [@Aoki:2006]. This is in reasonable accord with the latest lattice prediction of 6304$\pm$12$^{+18}_{-0}$ MeV [@Allison:2004be].
The long-awaited $B_s$–$\overline{B}_s$ mixing has finally been observed [@CDFmix; @D0mix]. The CDF value, $\Delta m_s = 17.31^{+0.33}_{-0.18} \pm
0.07$ ps$^{-1}$, constrains $f_{B_s}$ and $|V_{td}/V_{ts}|$, as mentioned earlier.
The Belle Collaboration has observed the decay $B \to \tau \nu_\tau$ [@Btaunu], leading to $f_B |V_{ub}| = (7.73^{+1.24+0.66}_{-1.02-0.58})
\times 10^{-4}$ GeV. When combined with an estimate [@HL06] $f_{B_d} =
(191 \pm 27)$ MeV, this leads to $|V_{ub}| = (4.05 \pm 0.89) \times 10^{-3}$, which is squarely in the range of recent averages [@HFAG].
A new CDF value for the $\Lambda_b$ lifetime, $\tau(\Lambda_b) = (1.59
\pm 0.08 \pm 0.03)$ ps, was reported at this Conference [@CDFLam]. Whereas the previous world average of $\tau(\Lambda_b)$ was about 0.8 that of $B^0$, below theoretical predictions, the new CDF value substantially increases the world average to a value $\tau(\Lambda_b) = (1.410 \pm 0.054)$ ps which is $0.923 \pm 0.036$ that of $B^0$ and quite comfortable with theory.
CHARMONIUM
==========
-0.1in
Observation of the $h_c$
------------------------
-0.1in
The $h_c(1^1P_1)$ state of charmonium has been observed by CLEO [@Rosner:2005ry; @Rubin:2005px] via $\psi(2S) \to \pi^0 h_c$ with $h_c
\to \gamma \eta_c$ (transitions denoted by red (dark) arrows in Fig. \[fig:ccour\] [@ccour]).
![Transitions among low-lying charmonium states. From Ref.[@ccour]. \[fig:ccour\]](0140406-002.eps){width="90.00000%"}
Hyperfine splittings test the spin-dependence and spatial behavior of the $Q
\bar Q$ force. Whereas these splittings are $M(J/\psi) - M(\eta_c) \simeq 115$ MeV for 1S and $M[\psi'] - M(\eta'_c) \simeq $49 MeV for 2S levels, P-wave splittings should be less than a few MeV since the potential is proportional to $\delta^3(\vec{r})$ for a Coulomb-like $c \bar c$ interaction. Lattice QCD [@latt] and relativistic potential [@Ebert:2002pp] calculations confirm this expectation. One expects $M(h_c) \equiv M(1^1P_1) \simeq
\langle M(^3P_J) \rangle = 3525.36 \pm 0.06$ MeV.
Earlier $h_c$ sightings [@Rosner:2005ry; @Rubin:2005px] based on $\bar p p$ production in the direct channel, include a few events at $3525.4
\pm 0.8$ MeV seen in CERN ISR Experiment R704; a state at $3526.2 \pm
0.15 \pm 0.2$ MeV, decaying to $\pi^0 J/\psi$, reported by Fermilab E760 but not confirmed by Fermilab E835; and a state at $3525.8 \pm 0.2 \pm 0.2$ MeV, decaying to $\gamma \eta_c$ with $\eta_c \to \gamma \gamma$, reported by E835 with about a dozen candidate events [@Andreotti:2005vu].
In the CLEO data, both inclusive and exclusive analyses see a signal near $\langle M(^3P_J) \rangle$. The exclusive analysis reconstructs $\eta_c$ in 7 decay modes, while no $\eta_c$ reconstruction is performed in the inclusive analysis. The exclusive signal is shown on the left in Fig. \[fig:hc\]. A total of 19 candidates were identified, with a signal of $17.5 \pm 4.5$ events above background. The mass and product branching ratio for the two transitions are $M(h_c) = (3523.6 \pm 0.9 \pm 0.5)$ MeV; ${\cal B}_1(\psi' \to \pi^0 h_c)
{\cal B}_2(h_c \to \gamma \eta_c) = (5.3 \pm 1.5 \pm 1.0) \times 10^{-4}$. The result of one of two inclusive analyses is shown on the right in Fig. \[fig:hc\]. These yield $M(h_c) = (3524.9 \pm 0.7 \pm 0.4)$ MeV, ${\cal B}_1 {\cal B}_2 = (3.5 \pm 1.0 \pm 0.7) \times 10^{-4}$. Combining exclusive and inclusive results yields $M(h_c) = (3524.4 \pm 0.6 \pm 0.4)$ MeV, ${\cal B}_1 {\cal B}_2 = (4.0 \pm 0.8 \pm 0.7) \times 10^{-4}$. The $h_c$ mass is $(1.0 \pm 0.6 \pm 0.4)$ MeV below $\langle M(^3P_J) \rangle$, barely consistent with the (nonrelativistic) bound [@Stubbe:1991qw] $M(h_c) \ge
\langle M(^3P_J) \rangle$ and indicating little P-wave hyperfine splitting in charmonium. The value of ${\cal B}_1 {\cal B}_2$ agrees with theoretical estimates of $(10^{-3} \cdot 0.4)$. -0.3in
Decays of the $\psi'' \equiv \psi(3770)$
----------------------------------------
The $\psi''(3770)$ is a potential “charm factory” for present and future $e^+
e^-$ experiments. At one time $\sigma(e^+ e^- \to \psi'')$ seemed larger than $\sigma(e^+ e^- \to \psi'' \to D \bar D)$, raising the question of whether there were significant non-$D \bar D$ decays of the $\psi''$ [@Rosner:2004wy]. A new CLEO measurement [@CLDDbar], $\sigma(\psi'') =
(6.38 \pm 0.08 ^{+0.41}_{-0.30})$ nb, appears very close to the CLEO value $\sigma(D \bar D) = 6.39\pm0.10^{+0.17}_{-0.08})$ nb [@Briere], leaving little room for non-$D \bar D$ decays. Some question has nonetheless been raised by two very new BES analyses [@BESsig] in which a significant non-$D
\bar D$ component could still be present.
One finds that ${\cal B}(\psi''\to \pi \pi J/\psi,~\gamma \chi_{cJ}, \ldots)$ sum to at most 1–2%. Moreover, both CLEO and BES [@LP123], in searching for enhanced light-hadron modes, find only that the $\rho \pi$ mode, suppressed in $\psi(2S)$ decays, also is [*suppressed*]{} in $\psi''$ decays.
Some branching ratios for $\psi'' \to X J/\psi$ [@Adam:2005mr] are ${\cal B}(\psi'' \to \pi^+ \pi^- J/\psi) =(0.189\pm0.020\pm0.020)\%$, ${\cal B}(\psi'' \to \pi^0 \pi^0 J/\psi) =(0.080\pm0.025\pm0.016)\%$, ${\cal B}(\psi'' \to \eta J/\psi) = (0.087\pm0.033\pm0.022)\%$, and ${\cal B}(\psi'' \to \pi^0 J/\psi) < 0.028\%$. The value of ${\cal B}[\psi''(3770) \to \pi^+ \pi^- J/\psi]$ found by CLEO is a bit above 1/2 that reported by BES [@Bai:2003hv]. These account for less than 1/2% of the total $\psi''$ decays.
-------------------- ----- ------ ------------ ------------------
Mode CLEO
(a) (b) (c) [@Briere:2006ff]
$\gamma \chi_{c2}$ 3.2 3.9 24$\pm$4 $<21$
$\gamma \chi_{c1}$ 183 59 $73\pm9$ $75\pm18$
$\gamma \chi_{c0}$ 254 225 523$\pm$12 $172\pm30$
-------------------- ----- ------ ------------ ------------------
: CLEO results on radiative decays $\psi'' \to \gamma \chi_{cJ}$. Theoretical predictions of [@Eichten:2004uh] are (a) without and (b) with coupled-channel effects; (c) shows predictions of [@Rosner:2004wy]. \[tab:psipprad\]
CLEO has recently reported results on $\psi'' \to \gamma \chi_{cJ}$ partial widths, based on the exclusive process $\psi'' \to \gamma \chi_{c1,2} \to
\gamma \gamma J/\psi \to \gamma \gamma \ell^+ \ell^-$ [@Coan:2005] and reconstruction of exclusive $\chi_{cJ}$ decays [@Briere:2006ff]. The results are shown in Table \[tab:psipprad\], implying $\sum_J{\cal B}(\psi'' \to \gamma \chi_{cJ}) = {\cal O}$(1%).
Several searches for $\psi''(3770) \to ({\rm light~ hadrons})$, including VP, $K_L K_S$, and multi-body final states have been performed. Two CLEO analyses [@Adams:2005ks; @Huang:2005] find no evidence for any light-hadron $\psi''$ mode above expectations from continuum production except $\phi \eta$, indicating no obvious signature of non-$D \bar D$ $\psi''$ decays. -0.3in
$X(3872)$: A $1^{++}$ molecule
------------------------------
-0.1in
Many charmonium states above $D \bar D$ threshold have been seen recently. Reviews may be found in Refs. [@GodfreyFPCP; @Swanson]. The $X(3872)$, discovered by Belle in $B$ decays [@Choi:2003ue] and confirmed by BaBar [@Aubert:2004ns] and in hadronic production [@Acosta:2003zx], decays predominantly into $J/\psi \pi^+
\pi^-$. Evidence for it is shown in Fig. \[fig:X3872\] [@Abe:2005iy]. Since it lies well above $D \bar D$ threshold but is narrower than experimental resolution (a few MeV), unnatural $J^P = 0^-,1^+, 2^-$ is favored. It has many features in common with an S-wave bound state of $(D^0 \bar D^{*0} + \bar D^0 D^{*0})/
\sqrt{2} \sim c \bar c u \bar u$ with $J^{PC} = 1^{++}$ [@Close:2003sg]. The simultaneous decay of $X(3872)$ to $\rho J/\psi$ and $\omega J/\psi$ with roughly equal branching ratios is a consequence of this “molecular” assignment.
Analysis of angular distributions [@Rosner:2004ac] in $X \to \rho J/\psi,
\omega J/\psi$ favors the $1^{++}$ assignment [@Abe:2005iy]. (See also [@Marsiske; @Swanson].) The detection of the $\gamma J/\psi$ mode ($\sim
14\%$ of $J/\psi \pi^+ \pi^-$) [@Abe:2005ix] confirms the assignment of positive $C$ and suggests a $c \bar c$ admixture in the wave function. BaBar [@Bapipipsi] finds ${\cal B}[X(3872) \to \pi^+ \pi^- J/\psi] > 0.042$ at 90% c.l.
![Belle distribution in $M(\pi^+ \pi^- J/\psi)$ for the $X(3872)$ region [@Abe:2005iy]. \[fig:X3872\]](mpipijpsi_x_half-box.eps){width="99.00000%"}
Additional states around 3940 MeV
---------------------------------
-0.1in
Belle has reported a candidate for a $2^3P_2(\chi'_{c2})$ state in $\gamma
\gamma$ collisions [@Abe:2005bp], decaying to $D \bar D$ (left panel of Fig. \[fig:3940\]). The angular distribution of $D \bar D$ pairs is consistent with $\sin^4 \theta^*$ as expected for a state with $J=2, \lambda =
\pm2$. It has $M = 3929 \pm 5 \pm 2$ MeV, $\Gamma = 29 \pm 10 \pm 3$ MeV, and $\Gamma_{ee} {\cal B}(D \bar D) = 0.18 \pm 0.06 \pm 0.03$ eV, all reasonable for a $\chi'_{c2}$ state.
A charmonium state $X(3938)$ (the right-most peak in the right panel of Fig.\[fig:3940\]) is produced recoiling against $J/\psi$ in $e^+ e^- \to J/\psi +
X$ [@Pakhlov:2004au] and is seen decaying to $D \bar D^*$ + c.c. Since all lower-mass states observed in this recoil process have $J=0$ (these are the $\eta_c(1S),
\chi_{c0}$ and $\eta'_c(2S)$; see the Figure), it is tempting to identify this state with $\eta_c(3S)$ (not $\chi'_{c0}$, which would decay to $D \bar D$).
The $\omega J/\psi$ final state in $B \to K \omega J/\psi$ shows a peak above threshold at $M(\omega J/\psi) \simeq 3940$ MeV [@Abe:2004zs]. This could be a candidate for one or more excited P-wave charmonium states, likely the $\chi'_{c1,2}(2^3P_{1,2})$. The corresponding $b \bar b$ states $\chi'_{b1,2}$ have been seen to decay to $\omega \Upsilon(1S)$ [@Severini:2003qw]. -0.3in
The $Y(4260)$
-------------
-0.1in
![Evidence for the $Y(4260)$ [@Aubert:2005rm]. \[fig:Ba4260\]](Y4260.eps){width="96.00000%"}
Last year BaBar reported a state $Y(4260)$ produced in the radiative return reaction $e^+ e^- \to \gamma \pi^+ \pi^- J/\psi$ and seen in the $\pi^+ \pi^-
J/\psi$ spectrum [@Aubert:2005rm] (see Fig. \[fig:Ba4260\]). Its mass is consistent with being a $4S$ level [@Llanes-Estrada:2005vf] since it lies about 230 MeV above the $3S$ candidate (to be compared with a similar $4S$-$3S$ spacing in the $\Upsilon$ system). Indeed, a $4S$ charmonium level at 4260 MeV/$c^2$ was anticipated on exactly this basis [@Quigg:1977dd]. With this assignment, the $nS$ levels of charmonium and bottomonium are remarkably congruent to one another, as shown in Fig. \[fig:comp\]. Their spacings would be identical if the interquark potential were $V(r) \sim {\rm
log}(r)$, which may be viewed as an interpolation between the short-distance $\sim -1/r$ and long-distance $\sim r$ behavior expected in QCD. Other interpretations of $Y(4260)$ include a $c s \bar c \bar s$ state [@Maiani:2005pe] and a hybrid $c \bar c g$ state [@Zhu:2005hp], for which it lies in the expected mass range.
![Congruence of charmonium and bottomonium spectra if the $Y(4260)$ is a 4S level. \[fig:comp\]](comp.ps){height="0.44\textheight"}
![Evidence for $Y(4260)$ from a direct scan by CLEO [@Coan:2006rv]. \[fig:cleo4260\]](cleo4260.eps){width="0.4\textheight"}
The CLEO Collaboration has confirmed the $Y(4260)$, both in a direct scan [@Coan:2006rv] and in radiative return [@Blusk]. Results from the scan are shown in Fig. \[fig:cleo4260\], including signals for $Y(4260)
\to \pi^+ \pi^- J/\psi$ 11$\sigma$), $\pi^0 \pi^0 J/\psi$ (5.1$\sigma$), and $K^+ K^- J/\psi$ (3.7$\sigma$). There are also weak signals for $\psi(4160)
\to \pi^+ \pi^- J/\psi$ (3.6$\sigma$) and $\pi^0 \pi^0 J/\psi$ (2.6$\sigma$), consistent with the $Y(4260)$ tail, and for $\psi(4040) \to \pi^+ \pi^- J/\psi$ (3.3$\sigma$).
The hybrid interpretation of $Y(4260)$ deserves further attention. One consequence is a predicted decay to $D \bar D_1 +$ c.c., where $D_1$ is a P-wave $c \bar q$ pair. Now, $D \bar D_1$ threshold is 4287 MeV/$c^2$ if we consider the lightest $D_1$ to be the state noted in Ref. [@PDG] at 2422 MeV/$c^2$. In this case the $Y(4260)$ would be a $D \bar D_1 +$ c.c. [*bound state*]{}. It would decay to $D \pi \bar D^*$, where the $D$ and $\pi$ are not in a $D^*$. The dip in $R_{e^+ e^-}$ lies just below $D \pi \bar
D^*$ threshold, which may be the first S-wave meson pair accessible in $c \bar c$ fragmentation [@Close:2005iz].
Charmonium: updated
-------------------
Remarkable progress has been made in the spectroscopy of charmonium states above charm threshold in the past few years. Fig. \[fig:charmon\] summarizes the levels (some of whose assignments are tentative). Even though such states can decay to charmed pairs (with the possible exception of $X(3872)$, which may be just below $D \bar D^*$ threshold), other decay modes are being seen. I have not had time to discuss much other interesting work by BES and CLEO on exclusive decays of the $\chi_{cJ}$ and $\psi(2S)$ states, including studies of strong-electromagnetic interference in $\psi(2S)$ decays.
![Charmonium states including levels above charm threshold. \[fig:charmon\]](charmon0509.ps){width="98.00000%"}
![$b \bar b$ levels and some decays. Electric dipole (E1) transitions $S \leftrightarrow P \leftrightarrow D$ are not shown. \[fig:ups\]](ups.ps){height="0.4\textheight"}
THE $\Upsilon$ FAMILY (BOTTOMONIUM)
===================================
Some properties and decays of the $\Upsilon$ ($b \bar b$) levels are summarized in Fig. \[fig:ups\]. Masses are in agreement with unquenched lattice QCD calculations, a triumph of theory [@Lepage]. Direct photons have been observed in 1S, 2S, and 3S decays, implying estimates of the strong fine-structure constant consistent with others [@Besson:2005jv]. The transitions $\chi_b(2P) \to \pi \pi \chi_b(1P)$ have been seen [@Cawlfield:2005ra; @Tati]. In addition to the $\Upsilon(4S) \to \pi^+ \pi^-
\Upsilon(1S,2S)$ transitions noted in Fig. \[fig:ups\] [@BaUps], Belle has seen $\Upsilon(4S) \to \pi^+ \pi^- \Upsilon(1S)$, with a branching ratio ${\cal B} = (1.1 \pm 0.2 \pm 0.4) \times 10^{-4}$ [@BeUps].
Remeasurement of $\Upsilon(nS)$ properties
------------------------------------------
-0.1in
New values of ${\cal B}[\Upsilon(1S,2S,3S) \to \mu^+ \mu^-] = (2.39 \pm 0.02
\pm 0.07,~2.03\pm0.03\pm0.08,~2.39\pm0.07\pm0.10)\%$ [@Adams:2004xa], when combined with new measurements $\Gamma_{ee}(1S,2S,3S) = (1.354\pm0.004
\pm0.020,~0.619\pm0.004,\pm0.010,~0.446\pm0.004\pm0.007)$ keV imply total widths $\Gamma_{\rm tot}(1S,2S,3S) = (54.4\pm0.2\pm0.8\pm1.6,~30.5\pm0.2\pm
0.5\pm1.3,~18.6\pm0.2\pm0.3\pm0.9)$ keV. The values of $\Gamma_{\rm tot}
(2S,3S)$ are significantly below world averages [@PDG], which will lead to changes in comparisons of predicted and observed transition rates. As one example, the study of $\Upsilon(2S,3S) \to \gamma X$ decays [@Artuso:2004fp] has provided new branching ratios for E1 transitions to $\chi_{bJ}(1P),~\chi'_{bJ}(2P)$ states. These may be combinedwith the new total widths to obtain updated partial decay widths \[line (a) in Table \[tab:E1\]\], which may be compared with one set of non-relativistic predictions [@KR] \[line (b)\]. The suppression of transitions to $J=0$ states by 10–20% with respect to non-relativistic expectations agrees with relativistic predictions [@rel]. The partial width for $\Upsilon(3S)
\to \gamma 1^3P_0$ is found to be $56 \pm 20$ eV, about eight times the highly-suppressed value predicted in Ref. [@KR]. That prediction is very sensitive to details of wave functions; the discrepancy indicates the importance of relativistic distortions.
----- --------------- --------------- --------------- --------------- --------------- ---------------
$J=0$ $J=1$ $J=2$ $J=0$ $J=1$ $J=2$
(a) 1.14$\pm$0.16 2.11$\pm$0.16 2.21$\pm$0.16 1.26$\pm$0.14 2.71$\pm$0.20 2.95$\pm$0.21
(b) 1.39 2.18 2.14 1.65 2.52 2.78
----- --------------- --------------- --------------- --------------- --------------- ---------------
: Comparison of observed (a) and predicted (b) partial widths for $2S \to 1 P_J$ and $3S \to 2 P_J$ transitions in $b \bar b$ systems. \[tab:E1\]
$b \bar b$ spin singlets
------------------------
-0.1in
Decays of the $\Upsilon(1S,2S,3S)$ states are potential sources of information on $b \bar b$ spin-singlets, but none has been seen yet. One expects 1S, 2S, and 3S hyperfine splittings to be approximately 60, 30, 20 MeV/$c^2$, respectively [@Godfrey:2001eb]. The lowest P-wave singlet state (“$h_b$”) is expected to be near $\langle M(1^3P_J) \rangle \simeq 9900$ MeV/$c^2$ [@Godfrey:2002rp].
Several searches have been performed or are under way in 1S, 2S, and 3S CLEO data. One can search for the allowed M1 transition in $\Upsilon(1S) \to \gamma
\eta_b(1S)$ by reconstructing exclusive final states in $\eta_b(1S)$ decays and dispensing with the soft photon, which is likely to be swallowed up in background. Final states are likely to be of high multiplicity.
One can search for higher-energy but suppressed M1 photons in $\Upsilon(n'S)
\to \gamma \eta_b(nS)~(n \ne n')$ decays. These searches already exclude many models. The strongest upper limit obtained is for $n'=3$, $n=1$: ${\cal B} \le
4.3 \times 10^{-4}$ (90% c.l.). $\eta_b$ searches using sequential processes $\Upsilon(3S) \to \pi^0 h_b(1^1P_1) \to \pi^0 \gamma \eta_b(1S)$ and $\Upsilon(3S) \to \gamma \chi'_{b0} \to \gamma \eta \eta_b(1S)$ (the latter suggested in Ref. [@Voloshin:2004hs]) are being conducted but there are no results yet. Additional searches for $h_b$ involve the transition $\Upsilon(3S) \to \pi^+ \pi^- h_b$ \[for which a typical experimental upper bound based on earlier CLEO data [@Brock:1990pj] is ${\cal O}(10^{-3}$)\], with a possible $h_b \to \gamma \eta_b$ transition expected to have a 40% branching ratio [@Godfrey:2002rp].
FUTURE PROSPECTS
================
Two main sources of information on hadron spectroscopy in the past few years have been BES-II and CLEO. BES-II has ceased operation to make way for BES-III. CLEO’s original goals of 3 fb$^{-1}$ at $\psi(3770)$, 3 fb$^{-1}$ above $D_s$ pair threshold, and $10^9$ $J/\psi$ now appear unrealistic in light of attainable CESR luminosity. Consequently, it was agreed to focus CLEO on 3770 and 4170 MeV, split roughly equally, yielding about 750 pb$^{-1}$ at each energy if current luminosity projections hold. The determination of $f_D$, $f_{D_s}$, and form factors for semileptonic $D$ and $D_s$ decays will provide incisive tests for lattice gauge theories and measure CKM factors $V_{cd}$ and $V_{cs}$ with unprecedented precision. A sample of 30 million $\psi(2S)$ (about 10 times the current number) is planned to be taken, with at least 10 million this summer. Some flexibility to explore new phenomena will be maintained. CLEO-c running will end at the end of March 2008; BES-III and and PANDA will carry the torch thereafter.
Belle has taken 3 fb$^{-1}$ of data at $\Upsilon(3S)$; it is anyone’s guess what they will find with such a fine sample. For comparison, CLEO has (1.1,1.2,1.2) fb$^{-1}$ at (1S,2S,3S). Both BaBar and Belle have shown interest in hadron spectroscopy and are well-positioned to study it. There have been significant contributions from CDF and D0 as well, and we look forward to more.
SUMMARY
=======
Hadron spectroscopy is providing both long-awaited states like $h_c$ (whose mass and production rate confirm theories of quark confinement and isospin-violating $\pi^0$-emission transitions) and surprises like low-lying P-wave $D_s$ mesons, X(3872), X(3940), Y(3940), Z(3940) and Y(4260). Decays of $\psi''(3770)$ shed light on its nature as a $1^3D_1$ $c \bar c$ state with a small S-wave admixture.
Upon reflection, some properties of the new hadron states may be less surprising but we are continuing to learn about properties of QCD in the strong-coupling regime. There is evidence for molecules, 3S, 2P, 4S or hybrid charmonium, and interesting decays of states above flavor threshold.
QCD may not be the last strongly coupled theory with which we have to deal. The mystery of electroweak symmetry breaking or the very structure of quarks and leptons may require related techniques. It is important to realize that insights on hadron spectra are coming to us in general from experiments at the frontier of intensity and detector capabilities rather than energy, and illustrate the importance of a diverse approach to the fundamental structure of matter.
We owe a collective debt to R. H. Dalitz for teaching us that in order to learn about fundamental physics such as parity violation in the weak interactions or the existence of quarks it is often necessary to deal with phenomenological techniques of strong-interaction physics such as “phase space plots” or baryon resonance descriptions. Let us keep Dalitz’s legacy alive in our approach to particle physics.
I am grateful to R. Faccini and other colleagues on BaBar, Belle, and CLEO for sharing data and for helpful discussions, and to E. van Beveren, D. Bugg, F. Close, J. Peláez, and G. Rupp for some helpful references. This work was supported in part by the United States Department of Energy under Grant No.DE FG02 90ER40560.
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|
---
abstract: 'Change detection in heterogeneous multitemporal satellite images is an emerging and challenging topic in remote sensing. In particular, one of the main challenges is to tackle the problem in an unsupervised manner. In this paper we propose an unsupervised framework for bitemporal heterogeneous change detection based on the comparison of affinity matrices and image regression. First, our method quantifies the similarity of affinity matrices computed from co-located image patches in the two images. This is done to automatically identify pixels that are likely to be unchanged. With the identified pixels as pseudo-training data, we learn a transformation to map the first image to the domain of the other image, and vice versa. Four regression methods are selected to carry out the transformation: Gaussian process regression, support vector regression, random forest regression, and a recently proposed kernel regression method called homogeneous pixel transformation. To evaluate the potentials and limitations of our framework, and also the benefits and disadvantages of each regression method, we perform experiments on two real data sets. The results indicate that the comparison of the affinity matrices can already be considered a change detection method by itself. However, image regression is shown to improve the results obtained by the previous step alone and produces accurate change detection maps despite of the heterogeneity of the multitemporal input data. Notably, the random forest regression approach excels by achieving similar accuracy as the other methods, but with a significantly lower computational cost and with fast and robust tuning of hyperparameters.'
author:
- 'Luigi T. Luppino, Filippo M. Bianchi, Gabriele Moser, and Stian N. Anfinsen [^1][^2]'
bibliography:
- 'references.bib'
title: Unsupervised Image Regression for Heterogeneous Change Detection
---
unsupervised change detection, multimodal image analysis, heterogeneous data, image regression, affinity matrix, random forest, Gaussian process, support vector machine, kernel smoothing
[^1]: L.T. Luppino, F.M. Bianchi and S.N. Anfinsen are with the Machine Learning Group, Department of Physics and Technology, UiT The Arctic University of Norway, e-mail: luigi.t.luppino@uit.no.
[^2]: G. Moser is with DITEN Department, University of Genoa, Italy.
|
---
author:
- |
Saskia Metzler\
\
\
- |
Stephan Günnemann\
\
\
- |
Pauli Miettinen\
\
\
bibliography:
- 'bibliography.bib'
title: |
Hyperbolae Are No Hyperbole:\
Modelling Communities That Are Not Cliques
---
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by the German Research Foundation (DFG), Emmy Noether grant GU 1409/2-1, and by the Technical University of Munich - Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement no 291763, co-funded by the European Union.
|
---
abstract: |
In this paper, we first prove generalizations of Fujita vanishing theorems for $q$-ample divisors. We then apply them to study positivity of subvarieties with nef normal bundle in the sense of intersection theory.
After Ottem’s work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties.
We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. This cone contains the movable cone and shares similar intersection-theoretic properties with it, thanks to the aforementioned properties of nef subvarieties.
On the other hand, we prove that if $Y\subset X$ is an ample subscheme of codimension $r$ and $D|_Y$ is $q$-ample, then $D$ is $(q+r)$-ample. This is analogous to a result proved by Demailly-Peternell-Schneider.
address: 'Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA'
author:
- Chung Ching Lau
title: 'Fujita vanishing theorems for q-ample divisors and applications on subvarieties with nef normal bundle'
---
Introduction
============
The concept of ampleness of a divisor is central in the subject of algebraic geometry. It plays an important role in intersection theory (Nakai-Moishezon, Kleiman) and various vanishing theorems on cohomologies (Serre, Kodaira, Fujita etc.).
Weakening the Serre vanishing condition, a line bundle $\mathscr{L}$ is defined to be *$q$-ample* if given any coherent sheaf $\mathscr{F}$, there is an $m_0$ such that $$H^{i}(X,\mathscr{F}\otimes\mathscr{L}^{\otimes m})=0$$ for $i>q$ and $m>m_0$. Here we assume $X$ is projective over a field of characteristic zero. After the works of Andreotti-Grauert [@Andreotti], Sommese [@Sommese] and Demailly-Peternell-Schneider [@DPS] on $q$-ample divisors, Totaro established the basic, yet not elementary properties of $q$-ample divisors [@Totaro]. There is another approach to partial ampleness of a line bundle ([@dFKL], [@Kuronya1]) that we don’t pursue it here.
In this paper, we first prove two generalized versions of Fujita vanishing theorem for $q$-ample divisors (Theorem \[theorem: fujita\] and Proposition \[lemma: uniform2\]), improving one of the main results in Küronya’s paper [@Kuronya Theorem C]. However, our methods of proof are completely different. Here we use the technique of resolution of the diagonal, as developed by Arapura [@Arapura] and Totaro [@Totaro]. We state a simplified version of theorem \[theorem: fujita\] here. The original version is a bit cumbersome to state.
Let $X$ be a projective scheme of dimension $n$. Let $\mathscr{L}$ be a $q$-ample line bundles on $X$ and let $\mathscr{F}$ be a coherent sheaf on $X$. Then there is an $M$, such that $$H^{i}(X,\mathscr{F}\otimes\mathscr{L}^{\otimes m}\otimes\mathscr{P})=0$$ for $i>q$, $m\geq M$ and any nef line bundle $\mathscr{P}$ on $Z$.
Our second version of Fujita vanishing theorem focuses on the vanishing of the top cohomology (Proposition \[lemma: uniform2\]). The remaining of this paper is about applying these results to study subvarieties with nef normal bundle.
After the extensive work of Hartshorne [@Hartshorne], where he studied positivity properties of higher codimension subvarieties, Ottem discovered what is probably the right notion of an ample subscheme [@Ottem]. He defined a subscheme of $Y$ of codimension $r$ of a projective scheme to be *ample* if the exceptional divisor in the blowup of $X$ along $Y$ is $(r-1)$-ample. It is a natural definition that generalizes many properties of ample divisors [@Ottem Corollary 5.6], which were predicted in Hartshorne’s work, while at the same time includes the zero locus of a global section of an ample vector bundle [@Ottem Proposition 4.5].
Our first result sheds more light on the connection between $q$-ample divisors and ample subschemes:
\[thm: 1\] Let $X$ be a projective scheme of dimension $n$. Let $Y$ be an ample subscheme of $X$ of codimension $r$. Suppose $\mathscr{L}$ is a line bundle on $X$, and that its restriction $\mathscr{L}|_{Y}$ to $Y$ is $q$-ample, then $\mathscr{L}$ is $(q+r)$-ample.
This result can be compared to a result by Demailly-Peternell-Schneider [@DPS Theorem 3.4]. Given a chain of codimension $1$ subvarieties $Y_{n-r}\subset Y_{n-r+1}\subset \cdots \subset Y_{n-1}\subset Y_n = X$, such that for $n-r\leq i\leq n-1$, there exists an ample divisor $Z_i$ in the normalization of $Y_{i+1}$, with $Y_{i}$ being the image of $Z_i$ under the normalization map. They showed if $\mathscr{L}|_{Y_{n-r}}$ is ample, then $\mathscr{L}$ is $r$-ample, assuming Totaro’s results on $q$-ample divisors.
We now move on to study a weaker positivity condition of a subscheme. Given an lci subvariety $Y\subset X$ with nef normal bundle, we would like to understand its positivity properties in terms of intersection theory. Fulton and Lazarsfeld [@FulLaz] gave an answer to this: They showed that if $\dim Y +\dim Z\geq \dim X$, then $\deg_H(Y\cdot Z)\geq0$. Here $H$ is an ample divisor.
Now let $Y\subset X$ be an arbitrary subscheme of codimension $r$ and let $E$ be the exceptional divisor in $\operatorname{Bl}_Y X$. We say that $Y$ is *nef* if $(E+\epsilon A)|_E$ is $(r-1)$-ample, where is $A$ is an ample divisor and $0<\epsilon \ll 1$. This definition is inspired by Ottem’s definition of an ample subscheme [@Ottem]. If $Y$ is lci in $X$, $Y$ is nef if and only if $Y$ has nef normal bundle. We show that
\[thm: 2\] Let $\iota: Y\hookrightarrow X$ be a nef subvariety of codimension $r$ of a projective variety $X$. Then the natural map $\iota^{*}: \operatorname{N}^{1}(X)_{\mathbf{R}} \rightarrow \operatorname{N}^{1}(Y)_{\mathbf{R}}$ induces $\iota^{*}: \overline{\operatorname{Eff}}^{1}(X)\rightarrow \overline{\operatorname{Eff}}^{1}(Y)$ and $\iota^{*}: \operatorname{Big}(X)\rightarrow \operatorname{Big}(Y)$.
When $Y$ is a curve with nef normal sheaf, this is a result of Demailly-Peternell-Schneider [@DPS Theorem 4.1]. We also show that nefness and ampleness are transitive properties without any assumptions on smoothness, thus generalizes Ottem’s result [@Ottem Proposition 6.4].
\[thm: 3\] let $X$ be a projective scheme of dimension $n$. If $Y$ is an ample (resp. nef) subscheme of $X$ and $Z$ is an ample (resp. nef) subscheme of $Y$, then $Z$ is ample (resp. nef) in $X$.
We then study the cycle classes of nef subvarieties. We use this new notion of nef subvarieties to introduce the notion of the weakly movable cone, $\overline{\operatorname{WMov}}_d(X)$. We define it as the closure of the convex cone that is generated by pushforward of cycle classes of nef subvarieties of dimension $d$ via proper surjective morphisms. We show that the weakly movable cone shares similar properties to that of the movable cone of $d$-cycles, $\overline{\operatorname{Mov}}_d(X)$.
Let $X$ be a projective variety of dimension $n$. For $1\leq d \leq n-1$,
1. $\overline{\operatorname{Mov}}_d(X)\subseteq\overline{\operatorname{WMov}}_d(X)$ and $\overline{\operatorname{Mov}}_1(X)=\overline{\operatorname{WMov}}_1(X)$.
2. \[item: eff1\] $\overline{\operatorname{Eff}}^{1}(X)\cdot \overline{\operatorname{WMov}}_d(X)\subseteq \overline{\operatorname{Eff}}_{d-1}(X)$.
3. \[item: bignef1\] Let $H$ be a big Cartier divisor, $\alpha \in \overline{\operatorname{WMov}}_d(X)$. If $H\cdot \alpha=0$, then $\alpha =0$.
4. \[item: nef1\] $\operatorname{Nef}^{1}(X)\cdot \overline{\operatorname{WMov}}_d(X)\subseteq\overline{\operatorname{WMov}}_{d-1}(X)$.
Analogous statements of \[item: eff1\], \[item: bignef1\] and \[item: nef1\] hold for the movable cone [@FL Lemma 3.10]. One can ask whether in general the two cones $\overline{\operatorname{Mov}}_d(X)$ and $\overline{\operatorname{WMov}}_d(X)$ are the same. This is true if and only if the cycle class of any nef subvariety lies in the movable cone. This question is closely related to the Hartshorne’s conjecture A. Hartshorne’s conjecture A states that if $Y$ is a smooth subvariety with ample normal bundle of a smooth projective variety $X$, $nY$ moves in a large algebraic family for $n$ large. This was disproved by Fulton and Lazarsfeld [@FulLaz1].
It is unclear what kind of intersection theoretic statements one should expect if we further assume that if $Y$ has ample normal bundle. Voisin gave an example of a subvariety with ample normal bundle such that its cycle class lies on the boundary of the pseudoeffective cone of cycles [@Voisin]. On the other hand, Ottem showed that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves [@Ottem1]. In an upcoming work, we shall study the numerical dimension of a pseudoeffective divisor by restricting it to a subvariety with ample normal bundle.
It is interesting to note that the cone dual to the pseudoeffective cone of $d$-cycles is not in general pseudoeffective, this is a result by Debarre, Ein, Lazarsfeld and Voisin [@DELV].
All schemes in this work are over a field of characteristic $0$.
The author would like to thank his advisor, Tommaso de Fernex, for many hours of discussions on this project, as well as his kindness and support. The author would not have completed this work without his help. He would also like to thank John Christian Ottem and Burt Totaro for their interests in this project. This is part of the author’s PhD’s thesis.
q-ample divisors and ample subschemes
=====================================
In this section, we shall first gather some useful facts about $q$-ample divisors, then we shall recall Ottem’s definition of an ample subscheme and some of its properties.
Let us recall the definition of the definition of a $q$-ample line bundle.
Let $X$ be a projective scheme. A line bundle bundle $\mathscr{L}$ is *$q$-ample* if for any coherent sheaf $\mathscr{F}$ on $X$, there is an $m_{0}$ such that $$H^{i}(X,\mathscr{F}\otimes\mathscr{L}^{\otimes m})=0$$ for $i>q$ and $m>m_0$.
\[lemma: Ottem\] Let $X$ be a projective scheme and fix an ample line bundle $\mathscr{O}(1)$ on $X$. A line bundle $\mathscr{L}$ is $q$-ample if and only if for any $l\geq 0$, $$H^{i}(X,\mathscr{L}^{\otimes m}\otimes \mathscr{O}(-l))=0$$ for $m\gg 0$.
We shall start with the definition of a Koszul-ample line bundle. The details are not very important in this paper, but they are included for the sake of completeness. One useful fact is that any large tensor power of an ample line bundle is $2n$-Koszul-ample, where $n$ is the dimension of the underlying projective scheme [@Backelin].
Let $X$ be a projective scheme of dimension $n$, and that the ring of regular function $\mathscr{O}(X)$ on $X$ is a field (e.g. $X$ is connected and reduced). Given a very ample line bundle $\mathscr{O}_X(1)$, we say that it is *$N$-Koszul ample* if the homogeneous coordinate ring $A=\bigoplus_{j} H^{0}(X,\mathscr{O}_X(j))$ is $N$-Koszul, i.e. there is a resolution $$\cdots\rightarrow M_1 \rightarrow M_0\rightarrow k\rightarrow 0$$ where $M_i$ is a free $A$-module, generated in degree $i$, where $i\leq N$.
Let $X$ be a projective scheme of dimension $n$. Suppose the ring of regular functions of $X$, $\mathscr{O}(X)$ is a field. We fix a $2n$-Koszul-ample line bundle $\mathscr{O}_{X}(1)$ on $X$. We say that a line bundle $\mathscr{L}$ is *$q$-T-ample* if there is a positive integer $N$, such that $$H^{q+i}(X,\mathscr{L}^{\otimes N}(-n-i))=0,$$ for $0\leq i\leq n-q$.
Totaro showed that $q$-T-ampleness is the same as $q$-ampleness [@Totaro Theorem 6.3]. Even though the $q$-T-ampleness notion may appear technical, the equivalence is the key result of his paper. It reduces the problem of showing a line bundle being $q$-ample to checking the vanishing of finitely many cohomology groups. Using the notion of $q$-T-ampleness, Totaro showed that $q$-ampleness is Zariski open [@Totaro Theorem 8.1]. We can extend the definition to $\mathbf{R}$-Cartier divisors.
Let $X$ be a projective scheme. An $\mathbf{R}$-Cartier divisor on $X$ is *$q$-ample* if $D$ is numerically equivalent to $cL+A$ with $L$ a $q$-ample line bundle, $c\in \mathbf{R}_{>0}$, $A$ an ample $\mathbf{R}$-Cartier divisor.
Based on the work of Demailly, Peternell and Schneider, Totaro also proved that
\[thm: open\] An integral divisor is $q$-ample if and only if its associated line bundle is $q$-ample. The $q$-ample $\mathbf{R}$-divisors in $N^{1}(X)_{\mathbf{R}}$ defines an open cone (but not convex in general) and that the sum of a $q$-ample $\mathbf{R}$-divisor and a $r$-ample $\mathbf{R}$-divisor is $(q+r)$-ample.
These facts are non-trivial. We shall use the notion of $q$-T-ampleness to prove proposition \[prop: pullback\].
We note that $(n-1)$-ampleness admits a pleasant geometric interpretation, which we shall use a few times in this paper.
\[theorem: n-1 ample\] Let $X$ be a projective variety of dimension $n$. A line bundle $\mathscr{L}$ on $X$ is $(n-1)$-ample if and only if $[\mathscr{L}^{\vee}]\in N^{1}(X)$ does not lie in the pseudoeffective cone.
We need the following result on the positivity of the pullback of a $q$-ample divisor.
\[prop: pullback\] Let $f:X'\rightarrow X$ be a morphism of projective schemes. Let $D$ be a $q$-ample divisor on $X$, and let $A$ be a relatively (to $f$) ample divisor on $X'$. Then $mf^{*}D+A$ is $q$-ample, for $m\gg0$.
First, let us show that it suffices to prove the proposition in the case when both $X$ and $X'$ are irreducible and reduced. Note that a line bundle is $q$-ample on $X'$ if and only if it is $q$-ample when restricting to each irreducible component of $X'$ [@Ottem Proposition 2.3.i,ii]. We can now assume $X'$ is integral. Let $X_{1}$ be an irreducible component of $X$ that contains the image of $X'$. The map $X'\rightarrow X$ factors through $X_{1}$, and $D|_{X_{1}}$ is again $q$-ample.
Now we can assume both $X$ and $X'$ are integral. In fact, we shall prove that $mf^{*}D+A$ is $q$-T-ample, for $m\gg 0$. In other words, we shall show that for $m\gg 0$, there is a positive integer $r$, such that $$H^{q+a}(X', \mathscr{O}_{X'}(r(mf^{*}D+A))\otimes \mathscr{O}_{X'}(-n-a))=0$$ for $1\leq a\leq n-q$. Here $\mathscr{O}_{X'}(1)$ is a $2n$-Koszul-ample line bundle on $X'$, where $n=\dim X'$.
Using the relative ampleness of $A$, one can find an integer $r$ such that $$R^{j}f_{*}(\mathscr{O}_{X'}(rA)\otimes \mathscr{O}_{X'}(-n-a)) =0,$$ for $j>0$ and $1\leq a\leq n-q$. The Leray spectral sequence then says $$\begin{gathered}
H^{q+a}(X', \mathscr{O}_{X'}(r(mf^{*}D+A))\otimes \mathscr{O}_{X'}(-n-a))
\\
\cong
H^{q+a}(X,\mathscr{O}_{X}(rmD)\otimes
f_{*}(\mathscr{O}_{X'}(rA)\otimes
\mathscr{O}_{X'}(-n-a))).\end{gathered}$$ The right hand side group vanishes for all big $m$, by the $q$-ampleness of $rD$.
We now review the definition of ample subscheme, given by Ottem:
\[definition: ample\] Let $X$ be a projective scheme. Let $Y$ be a closed subscheme of $X$ of codimension $r$ and let $\pi: \operatorname{Bl}_Y X\rightarrow X$ be the blowup of $X$ with center $Y$. We say that $Y$ is an *[ample subscheme]{} of $X$ if the exceptional divisor $E$ of $\pi$ is $(r-1)$-ample in $\operatorname{Bl}_Y X$.*
We shall follow his definition in this paper. An example of an ample subscheme would be the zero locus (of codimension $r$) of a section of an ample vector bundle of rank $r$ [@Ottem Proposition 4.5]. On the other hand, many good properties listed in Hartshorne’s book [@Hartshorne p.XI] are satisfied under this definition. Before stating some of these properties, we need the definition of *cohomological dimension* of a scheme $U$: it refers to the number $$\operatorname{cd}(U):=\max\{i\in \mathbb{Z}_{\geq 0}|\, H^{i}(U,\mathscr{F})\neq 0 \textit{, for some coherent sheaf }\mathscr{F}.\}$$
\[theorem: ample subvariety\] Let $Y$ be a smooth closed subscheme of a smooth projective scheme $X$.
1. $Y$ is ample if and only if its normal bundle is ample and the cohomological dimension of the complement is $r-1$. \[theorem: normal bundle\]
Assume further that $Y$ is an ample subscheme in $X$. Then
2. Generalized Lefschetz hyperplane theorem with rational coefficient holds, i.e. $H^{i}(X,\mathbb{Q})\rightarrow H^{i}(Y,\mathbb{Q})$ is an isomorphism for $i<\dim Y$ and is an injection for $i=\dim Y$. \[theorem: lefschetz\]
3. $Y$ is numerically positive, i.e. $Y\cdot Z>0$ for any effective cycle $Z$ of dimension $r$. \[theorem: intersection\]
4. \[theorem: H\^[i]{}\] $H^{i}(X,\mathscr{F})\rightarrow H^{i}(\hat{X},\hat{\mathscr{F}})$ is an isomorphism for $i<\dim Y$ and is injective for $i=\dim Y$. Here $\hat{X}$ is the formal completion of $X$ along $Y$, $\mathscr{F}$ is a locally free sheaf on $X$ and $\hat{\mathscr{F}}$ is its restriction to $\hat{X}$.
[@Ottem Theorem 5.4], [@Ottem Corollary 5.3] and [@Hartshorne Chapter III, Theorem 3.4] give \[theorem: normal bundle\], \[theorem: lefschetz\] and \[theorem: H\^[i]{}\] respectively. For \[theorem: intersection\], since the cohomological dimension of $(X-Y)=r-1$, $Y$ meets any effective cycle of dimension $r$. We can then apply the result of Fulton and Lazarsfeld [@Laz Corollary 8.4.3], which says if $Y$ has ample normal bundle and $Y$ meets $Z$, where $Z$ is for an effective cycle of complementary dimension to that of $Y$, then $Y\cdot Z>0$.
The above list of properties is incomplete, for a more complete picture, c.f. [@Ottem].
Partial regularity and a Fujita-type vanishing theorem for *q*-ample divisors
=============================================================================
In this section, we shall quickly go through the results in section 2 and 3 in Totaro’s paper [@Totaro]. There Totaro developed on Arapura’s idea [@Arapura] on using resolution of the diagonal to study Castelnuovo-Mumford regularity of a sheaf. Using these ideas, we shall provide a weak extension of a vanishing theorem for $q$-ample line bundles proved by Totaro [@Totaro Theorem 6.4] (theorem \[theorem: unif\]). From this, we prove a generalization of the Fujita vanishing theorem (theorem \[theorem: fujita\]) to the $q$-ample divisors setting. It also generalizes the Fujita-type vanishing theorem that Küronya proved [@Kuronya Theorem C]. We shall later apply this theorem to prove theorem \[theorem: restriction\], as well as theorems \[theorem: trans ample\] and \[theorem: trans nef\].
In this section, we assume $X$ to be a projective scheme of dimension $n$ over a field, with the ring of regular functions on $X$ being a field. Furthermore, we fix a $2n$-Koszul-ample line bundle $\mathscr{O}_{X}(1)$ on $X$.
On $X\times_k X$, we have the following exact sequence of coherent sheaves: $$\label{resolution}
\mathscr{R}_{2n-1}\boxtimes\mathscr{O}_{X}(-2n+1)\rightarrow
\cdots\rightarrow\mathscr{R}_1\boxtimes\mathscr{O}_X(-1)
\rightarrow\mathscr{R}_0\boxtimes\mathscr{O}_X
\rightarrow\mathscr{O}_{\Delta}
\rightarrow 0,$$ where $\Delta\subset X\times_k X$ is the diagonal. Here all the $\mathscr{R}_i$’s are locally free sheaves on $X$ that can be fit into short exact sequences: $$\label{R}
0
\rightarrow \mathscr{R}_{i+1}\otimes \mathscr{O}_X(-1)
\rightarrow B_{i+1}\otimes_{k} \mathscr{O}_X(-1)
\rightarrow \mathscr{R}_{i} \rightarrow 0,$$ where the $B_{i+1}$’s are $k$-vector spaces.
\[lemma: vanishing of tensor product\][@Totaro Lemma 3.1] Let $\mathscr{E}$ and $\mathscr{F}$ be a locally free sheaf and a coherent sheaf on $X$ respectively. Suppose that for each pair of integers $0\leq a\leq 2n-i$ and $b\geq 0$, either $H^{b}(\mathscr{E}\otimes \mathscr{R}_a)=0$ or $H^{i+a-b}(\mathscr{F}(-a))=0$. Then $H^{i}(\mathscr{E}\otimes\mathscr{F})=0$.
After tensoring with $\mathscr{E}\boxtimes\mathscr{F}$, the sequence (\[resolution\]) remains exact, we now apply Künneth’s formula.
\[def: reg\] We fix a $2n$-Koszul ample line bundle $\mathscr{O}_{X}(1)$. Let $\mathscr{G}$ be a coherent sheaf on $X$ and let $q$ be any integer greater than or equal to $0$. We say that $\mathscr{G}$ is *$q$-regular* if the following holds: $$\label{q-regular}
H^{q+i}(X,\mathscr{G}\otimes \mathscr{O}_{X}(-i))= 0$$ for all $\ 1\leq i \leq n-q$.
We set $$\operatorname{reg}^{q}(\mathscr{F})= \inf\{m\in \mathbb{Z}\,| \, \mathscr{F}\otimes\mathscr{O}_{X}(m) \text{ is } q \text{-regular.}\}$$
When $q=0$, this is just the usual Castelnuovo-Mumford regularity of the sheaf $\mathscr{F}$, relative to $\mathscr{O}_{X}(1)$. It is clear that $\operatorname{reg}^{q}(\mathscr{F})\in[-\infty,+\infty)$, by the ampleness of $\mathscr{O}_{X}(1)$.
\[lemma: regularity\] If $\mathscr{F}$ is $q$-regular, then $\mathscr{F}\otimes\mathscr{O}_{X}(1)$ is also $q$-regular.
\[lemma: R\_i\] If $\mathscr{F}$ is a $q$-regular coherent sheaf on $X$, then $$H^{j}(X,\mathscr{F}\otimes \mathscr{R}_{i})=0,$$ for $j>q$ and $i<n+j$. Here, we are referring to the $\mathscr{R}_{i}$’s that appear in lemma \[lemma: vanishing of tensor product\].
We next generalize [@Totaro Theorem 3.4].
\[theorem: subadditivity\] Let $\mathscr{E}$ and $\mathscr{F}$ be a locally free sheaf and a coherent sheaf on $X$ respectively, then $$reg^{q}(\mathscr{E}\otimes \mathscr{F})\leq reg^{l}(\mathscr{E}) + reg^{q-l}(\mathscr{F})$$ for any $0\leq l\leq q$.
Replacing $\mathscr{E}$ and $\mathscr{F}$ by $\mathscr{E}\otimes\mathscr{O}_{X}(k)$ and $\mathscr{F}\otimes\mathscr{O}_{X}(k')$ respectively, where $k$ and $k'$ are sufficiently large, we may assume $\mathscr{E}$ and $\mathscr{F}$ are $l$- and $(q-l)$- regular, respectively. We want to show $$H^{q+i}(X,\mathscr{E}\otimes\mathscr{F}\otimes\mathscr{O}_X(-i))=0,$$ for $1\leq i\leq n-q$. We now apply lemma \[lemma: vanishing of tensor product\].
$b>l$ and $a<n+b$.
By lemma \[lemma: R\_i\], $$H^{b}(X,\mathscr{E}\otimes \mathscr{R}_{a})=0.$$
$b>l$ and $n+b\leq a\leq 2n-(q+i)$.
Since $q+i+a-b\geq q+i+n>n$, $$H^{(q+i)+a-b}(X,\mathscr{F}\otimes\mathscr{O}_{X}(-a-i))=0,$$ for dimensional reason.
$0\leq b\leq l$ and $0\leq a \leq 2n-(q+i)$.
We have $q-b\geq q-l$, and $$H^{(q-b)+a+i}(X,\mathscr{F}\otimes\mathscr{O}_{X}(-a-i))=0,$$ by $(q-l)$-regularity of $\mathscr{F}$ and lemma \[lemma: regularity\].
We next prove an analogue of [@Totaro Theorem 6.4]. This will play a crucial role in proving theorem \[theorem: restriction\].
\[theorem: unif\] Let $\mathscr{L}$ be a $q$-ample line bundle on $X$. Then for any $N$, there is an integer $m_{N}$, such that, for any coherent sheaf $\mathscr{F}$ on $X$ with $reg^{q'}(\mathscr{F})\leq N$, $$H^{i}(X,\mathscr{F}\otimes\mathscr{L}^{\otimes m})=0$$ for $i>q+q'$ and $m>m_{N}$.
Fix an integer $i$ such that $q+q'< i \leq n$, by lemma \[lemma: vanishing of tensor product\], it is enough to show that there is an $M$, depending only on the choice of $N$, but not the coherent sheaf $\mathscr{F}$, such that for $m>M$, $0\leq a \leq 2n-i$ and $b\geq 0$, either $H^{b}(X,\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-N)\otimes \mathscr{R}_{a})=0$ or $H^{i+a-b}(X,\mathscr{F}\otimes\mathscr{O}_{X}(N-a))$. Here $\mathscr{F}$ is any coherent sheaf with $reg^{q'}(\mathscr{F})\leq N$.
$b> q$ and $0\leq a < n+b$.
Using the $q$-ampleness of $\mathscr{L}$, there is an $m_N$, such that we have $$H^{q+j}(X,\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-N-j))=0$$ for all $1\leq j\leq n-q$ and $m>m_N$, i.e. $\mathscr{L}^{\otimes m}\otimes \mathscr{O}_X(-N)$ is $q$-regular for all $m>m_N$. Now lemma \[lemma: R\_i\] says $$H^{b}(X,\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-N)\otimes \mathscr{R}_{a})=0$$ for all $m>m_N$, $b> q$ and $a< n+b$.
$b>q$ and $n+b\leq a \leq 2n-i$.
We have $i+a-b\geq i+n > n$, and $$H^{i+a-b}(X,\mathscr{F}\otimes\mathscr{O}_{X}(N-a))=0$$ for dimensional reason.
$0\leq b\leq q$ and $0\leq a\leq 2n-i$.
We have $i-b>q'$, and $H^{(i-b)+a}(X,\mathscr{F}\otimes\mathscr{O}_{X}(N-a))=0$ by the partial regularity assumption of $\mathscr{F}$ and lemma \[lemma: regularity\]. This proves the theorem.
\[lemma: reg of nef\] There is an $N$ such that $\operatorname{reg}^{0}(\mathscr{P})\leq N$ for any nef line bundle $\mathscr{P}$ on $X$.
By the Fujita vanishing theorem, there is an $N$ such that $$H^{a}(X,\mathscr{O}_X(N-a)\otimes \mathscr{P})=0$$ for $a>0$ and any nef line bundle $\mathscr{P}$.
We prove a Fujita-type vanishing theorem for $q$-ample divisors. It is a generalization of the Fujita-type vanishing theorem that Küronya proved in [@Kuronya Theorem C], thanks to the fact that a divisor $D$ is $q$-ample if and only if its restriction to its augmented base locus $D|_{\mathbf{B}_{+}(D)}$ is $q$-ample [@Brown]. Note that we do not assume $\mathscr{O}(Z)$ is a field in the following.
\[theorem: fujita\] Let $Z$ be a projective scheme of dimension $n$. Let $\mathscr{L}_j$ be $q_j$-ample line bundles on $Z$, $1\leq j\leq k$ and let $\mathscr{F}$ be a coherent sheaf on $Z$. Then for any $(k-1)$-tuple $(M_2,\cdots,M_k)\in \mathbb{Z}^{k-1}$, there is an $M_1$, such that $$H^{i}(Z,\mathscr{F}\otimes\mathscr{L}_1^{\otimes m_1}\otimes \mathscr{L}_2^{\otimes m_2}\otimes\cdots\otimes \mathscr{L}_k^{\otimes m_k}\otimes \mathscr{P})=0$$ for $i>\sum_{j=1}^k q_j$, $m_j\geq M_j$, where $1\leq j\leq k$, and any nef line bundle $\mathscr{P}$ on $Z$.
We can assume that $Z$ is connected. It suffices to prove the lemma assuming that $Z$ is also reduced. Indeed, let $\mathscr{N}$ be the nilradical ideal sheaf of $Z$, and chase through the following exact sequence: $$\begin{gathered}
0\rightarrow
\mathscr{N}^{e+1}\cdot\mathscr{F}\otimes\mathscr{L}_1^{\otimes m_1}\otimes \mathscr{L}_2^{\otimes m_2}\otimes\cdots\otimes \mathscr{L}_k^{\otimes m_k}\otimes \mathscr{P}\\
\rightarrow
\mathscr{N}^{e}\cdot\mathscr{F}\otimes\mathscr{L}_1^{\otimes m_1}\otimes \mathscr{L}_2^{\otimes m_2}\otimes\cdots\otimes \mathscr{L}_k^{\otimes m_k}\otimes \mathscr{P}\\
\rightarrow
(\mathscr{N}^{e}\cdot\mathscr{F}/\mathscr{N}^{e+1}\cdot\mathscr{F})\otimes\mathscr{L}_1^{\otimes m_1}\otimes \mathscr{L}_2^{\otimes m_2}\otimes\cdots\otimes \mathscr{L}_k^{\otimes m_k}\otimes \mathscr{P}
\rightarrow 0.\end{gathered}$$ Note that $(\mathscr{N}^{e}\cdot\mathscr{F}/\mathscr{N}^{e+1}\cdot\mathscr{F})\otimes\mathscr{L}_1^{\otimes m_1}\otimes \mathscr{L}_2^{\otimes m_2}\otimes\cdots\otimes \mathscr{L}_k^{\otimes m_k}\otimes \mathscr{P}$ is a coherent sheaf on $Z_{red}$, and that $\mathscr{N}^{e}=0$ for $e\gg0$.
Since $\mathscr{L}_j$ is $q_j$-ample, $$H^{q_i+a}(Z,\mathscr{L}_j^{\otimes m_j}\otimes\mathscr{O}(-a))=0$$ for $m_j\gg0$ and $1\leq a \leq n-q_i$. This says $\operatorname{reg}^{q_j}(\mathscr{L}_j^{\otimes m_j})\leq 0$ for all $m_j\gg0$. Therefore, there are $N_j$ such that $\operatorname{reg}^{q_j}(\mathscr{L}_j^{\otimes m_j})\leq N_j$ for all $m_j\geq M_j$. We apply theorem \[theorem: subadditivity\] and lemma \[lemma: reg of nef\] to see that $\operatorname{reg}^{\sum_{i=2}^k q_j}(\mathscr{F}\otimes \mathscr{L}_2^{\otimes m_2}\otimes\cdots\otimes \mathscr{L}_k^{\otimes m_k}\otimes \mathscr{P})\leq \operatorname{reg}(\mathscr{F})+\sum_{j=2}^k N_j + N$ for all $m_j\geq M_j$, where $2\leq j\leq k$ and $N$ is the one mentioned in lemma \[lemma: reg of nef\]. Now, we may apply theorem \[theorem: unif\] to get the desired result.
Suppose we are only interested in the vanishing of the top cohomology group only, we may relax the assumption in theorem \[theorem: fujita\] a bit. We shall use this to prove theorem \[theorem: pseudoeffective\].
\[lemma: uniform2\] Let $Z$ be a projective scheme of dimension $n$. Let $\mathscr{L}_1$ and $\mathscr{L}_i$ be line bundles on $Z$ that are $q_1$-ample and $q_i$-almost ample respectively, where $2\leq i \leq k$ and $\sum_{i=1}^{k} q_i\leq n-1$. Then for any coherent sheaf $\mathscr{F}$ on $Z$ and any $(k-1)$-tuple $(M_i)_{2\leq i\leq k}\in \mathbb{Z}^{k-1}$, there is an $M_1$ such that, $$H^{n}(Z,\mathscr{F}\otimes\mathscr{L}_1^{\otimes m_1}\otimes \bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes m_i})=0$$ for $m_i\geq M_i$.
Let us first reduce to the case where $Z$ is integral. Indeed, argue as in the proof of theorem \[theorem: fujita\], we may assume $Z$ is reduced. Suppose $Z=\bigcup_{i=1}^{k}Z_i$, where $Z_i$ are the irreducible components of $Z$. Let $\mathscr{I}$ be the ideal sheaf of $Z_1\subset Z$. Consider the short exact sequence $$0\rightarrow
\mathscr{I}\cdot \mathscr{F}
\rightarrow
\mathscr{F}
\rightarrow
\mathscr{F}/\mathscr{I}\cdot \mathscr{F}
\rightarrow 0.$$ Note that $\mathscr{I}\cdot \mathscr{F}$ and $\mathscr{F}/\mathscr{I}\cdot \mathscr{F}$ are supported on $\bigcup_{i=2}^k Z_i$ and $Z_1$ respectively. We then tensor the above short exact sequence with $\mathscr{L}_1^{\otimes m_1}\otimes \bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes m_i}$ and induct on the number of irreducible components of $Z$. Therefore, we may assume that $Z$ is irreducible as well.
Now we assume $Z$ is a projective variety. We can find a surjection $\oplus\mathscr{O}_Z(a)\twoheadrightarrow\mathscr{F}$, where $\mathscr{O}_Z(1)$ is an ample line bundle on $Z$. Thus it suffices to prove the case when $\mathscr{F}$ is a line bundle $\mathscr{M}$. Let $\omega_Z$ be the dualizing sheaf of $Z$ [@AG III.7]. We have $$H^{n}(Z,\mathscr{M}\otimes\mathscr{L}_1^{\otimes m_1}\otimes \bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes m_i})
\cong
H^{0}(Z,\mathscr{M}^{\vee}\otimes\mathscr{L}_1^{\otimes -m_1}\otimes \bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes -m_i}\otimes\omega_Z)^{\vee}$$ We can embed $\omega_Z\hookrightarrow\mathscr{O}(j)$ [@Totaro Proof of Theorem 9.1]. This reduces to proving the vanishing of $H^{0}(Z,\mathscr{M}^{\vee}\otimes\mathscr{O}(j)\otimes\mathscr{L}_1^{\otimes -m_1}\otimes \bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes -m_i})$. We may find an $M_1$ such that $\mathscr{L}_1^{\otimes m_1}\otimes\bigotimes_{i=2}^k\mathscr{L}_i^{\otimes M_i}\otimes\mathscr{M}\otimes\mathscr{O}(-j)$ is $q_1$-ample for $m_1\geq M_1$, by theorem \[thm: open\]. By theorem \[thm: open\] again, $\bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes m_i}\otimes\mathscr{L}_1^{\otimes m_1}\otimes\mathscr{M}\otimes\mathscr{O}(-j)$ is $(n-1)$-ample for $m_i\geq M_i$ and $m_1\geq M_1$. By theorem \[theorem: n-1 ample\], $\bigotimes_{i=2}^{k}\mathscr{L}_i^{\otimes -m_i}\otimes\mathscr{L}_1^{\otimes -m_1}\otimes\mathscr{M}^{\vee}\otimes\mathscr{O}(j)$ is not pseudoeffective for $m_i\geq M_i$. Therefore, it cannot have any global sections.
Nef subschemes
==============
In this section, we shall define the notion of nef subschemes. We shall show that ampleness and nefness are transitive properties: If $Z$ is an ample (resp. nef) subscheme of $Y$ and $Y$ is an ample (resp. nef) subscheme of $X$, then $Z$ is an ample (resp. nef) subscheme of $X$ (theorems \[theorem: trans ample\] and \[theorem: trans nef\]). We shall study them more closely in later sections. To streamline the arguments, we first make the following definition, which generalizes the notion of a nef divisor.
Let $X$ be a projective scheme, $D$ an $\mathbf{R}$-Cartier divisor on $X$, $A$ an ample Cartier divisor on $X$. We say that $D$ is *$q$-almost ample* if $D+\epsilon A$ is $q$-ample for $0<\epsilon\ll 1$.
The definition is clearly independent of the choice of $A$ and $D$ is $0$-almost ample if and only if $D$ is nef.
Ottem observed that ampleness of a vector bundle $\mathscr{E}$ can be expressed in terms of $q$-ampleness of $\mathbb{P}(\mathscr{E}^{\vee})$ [@Ottem Proposition 4.1]. We give the straightforward generlization to the case when the vector bundle is nef.
\[prop: vector bundle\] Let $\mathscr{E}$ be a vector bundle of rank $r$ on a projective scheme $X$. Then $\mathscr{E}$ is ample (resp. nef) if and only if $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-1)$ is $(r-1)$-ample. (resp. $(r-1)$-almost ample.)
Let $\pi': \mathbb{P}(\mathscr{E}^{\vee})\rightarrow X$ and $\pi: \mathbb{P}(\mathscr{E})\rightarrow X$ be the natural projection maps. Using [@AG Exercise III.8.4], we have for $m>0$, $$R^{j}\pi'_{*}\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-m-r)
\cong
\begin{cases}
\operatorname{Sym}^{m}\mathscr{E}\otimes\det(\mathscr{E}) \text{ for } j=r-1\\
0 \text{ otherwise.}
\end{cases}$$ Here we implicitly used the isomorphism $(\operatorname{Sym}^{m}\mathscr{E}^{\vee})^{\vee}\cong\operatorname{Sym}^{m}\mathscr{E}$ which holds when the ground field is of characteristic $0$.
Therefore we have the isomorphisms $$\begin{gathered}
\label{eq: vb}
H^{r-1+i}(\mathbb{P}(\mathscr{E}^{\vee}),\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-m-r)\otimes \pi{'}^{*}(\mathscr{F}\otimes \det \mathscr{E}^{\vee}))
\cong
H^{i}(X,\operatorname{Sym}^{m}\mathscr{E}\otimes \mathscr{F})\\
\cong
H^{i}(\mathbb{P}(\mathscr{E}),\mathscr{O}_{\mathbb{P}(\mathscr{E})}(m)\otimes\pi^{*}\mathscr{F}),\end{gathered}$$ where $\mathscr{F}$ is locally free on $X$, $i>0$ and $m>0$.
If $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-1)$ is $(r-1)$-ample, then the above observation shows that $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)$ is ample. Indeed, any line bundle on $\mathbb{P}(\mathscr{E})$ can be expressed as $\pi^{*}\mathscr{L}\otimes\mathscr{O}_{\mathbb{P}(\mathscr{E})}(l)$.
Suppose $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-1)$ is $(r-1)$-almost ample. Choose an ample divisor $A$ on $X$, to show that $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)$ is nef, we want to check that $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(k)\otimes\pi^{*}\mathscr{O}(A)$ is ample for all $k>0$. By replacing $A$ with a large multiple, we may assume $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)\otimes\pi^{*}\mathscr{O}(A)$ is ample. We apply lemma \[lemma: Ottem\] and fix an $l\geq0$. Observe that we have the following isomorphisms given by (\[eq: vb\]): $$\begin{gathered}
H^{i}(\mathbb{P}(\mathscr{E}), \mathscr{O}_{\mathbb{P}(\mathscr{E})}(mk-l)\otimes\pi^{*}\mathscr{O}((m-l)A))\\
\cong
H^{r-1+i}(\mathbb{P}(\mathscr{E}^{\vee}),\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-mk-r+l)\otimes \pi{'}^{*}(\mathscr{O}((m-l)A)\otimes \det \mathscr{E}^{\vee})),\end{gathered}$$ where $i,m>0$. The latter term vanishes for $m\gg 0$ since $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-k)\otimes\pi{'}^{*}\mathscr{O}(A)$ is $(r-1)$-ample for any $k>0$. This shows that $\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)$ is nef.
Similarly, we may also assume $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(1)\otimes\pi{'}^{*}\mathscr{O}(A)$ is ample.
If $\mathscr{E}$ is ample, we fix an $l\geq 0$, we have the following isomorphisms of cohomology groups, $$\begin{gathered}
H^{r-1+i}(\mathbb{P}(\mathscr{E}^{\vee}),\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-m-l)\otimes \pi{'}^{*}\mathscr{O}(-lA))\\
\cong
H^{i}(\mathbb{P}(\mathscr{E}), \mathscr{O}_{\mathbb{P}(\mathscr{E})}(m+l-r)\otimes\pi^{*}(\mathscr{O}(-lA)\otimes\det\mathscr{E})),\end{gathered}$$ the latter term vanishes for $i>0$ and $m\gg0$, which says $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-1)$ is $(r-1)$-ample.
If $\mathscr{E}$ is nef, we fix $l\geq0$ again, we observe that for any $k>0$, we have the following isomorphism of cohomology groups, $$\begin{gathered}
H^{r-1+i}(\mathbb{P}(\mathscr{E}^{\vee}),\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-mk-l)\otimes \pi{'}^{*}\mathscr{O}((m-l)A)))\\
\cong
H^{i}(\mathbb{P}(\mathscr{E}), \mathscr{O}_{\mathbb{P}(\mathscr{E})}(mk+l-r)\otimes\pi^{*}(\mathscr{O}((m-l)A)\otimes\det\mathscr{E})),\end{gathered}$$ for $i,m>0$. The latter term vanishes for $m\gg0$. This says that $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-k)\otimes\mathscr{O}(A)$ is $(r-1)$-ample for any $k>0$, which means $\mathscr{O}_{\mathbb{P}(\mathscr{E}^{\vee})}(-1)$ is $(r-1)$-almost ample.
The augmented base locus gives us another measure how far a divisor is being ample.
\[def: aug\] The augmented base locus of an $\mathbf{R}$-divisor D on X is the Zariski-closed subset: $$\mathbf{B}_{+}(D)=\bigcap_{D=A+E} \operatorname{Supp}E$$ where the intersection is taken over all decompositions $D=A+E$ such that $A$ is ample and $E$ is effective.
\[prop: exceptional\] Let $Y\subset X$ be a subscheme of codimension $r$. Then the normal bundle of the exceptional divisor $E$ in $\operatorname{Bl}_Y X$, $\mathscr{O}_E(E)$ is $(r-1)$-almost ample if and only if $E\subset \operatorname{Bl}_Y X$ is $(r-1)$-almost ample.
The “if” part of the statment is clear, since restriction of a $q$-ample divisor to a subscheme is always $q$-ample. For the “only if” part, observe that $\mathbf{B}_{+}(E+\epsilon A)\subseteq \operatorname{Supp}E$ for $0<\epsilon\ll1$, where $A$ is an ample divisor on $X$. Now we can apply Brown’s theorem [@Brown Theorem 1.1] to $E+\epsilon A$, which says that an $\mathbf{R}$-divisor $D$ is $q$-ample if and only if $D|_{\mathbf{B}_{+}(D)}$ is $q$-ample.
Let $Y$ be a closed subscheme of codimension $r$ of $X$, a projective scheme, and let $E$ be the exceptional divisor in $\operatorname{Bl}_Y X$. Then we say that $Y$ is *nef* if $\mathscr{O}_E(E)$ is $(r-1)$-almost ample.
Proposition \[prop: exceptional\] says that $Y$ is a nef subscheme if and only if $E$ lies in the closure of the $(r-1)$-ample cone of $X$. If $Y$ is l.c.i. in $X$, then $Y$ is nef if and only if the normal bundle $\mathscr{N}_{Y/X}$ is nef (proposition \[prop: vector bundle\]). The advantage of making this more general definition, without requiring $Y$ to be lci, is to include more subschemes that are apparently “positive”, for example, a closed point that is not necessarily nonsingular, or if $Y$ is a smooth subvariety with nef normal bundle, the subscheme of $X$ defined by a power of ideal sheaf of $Y$ is also considered as nef in this definition.
The following proposition is the direct generalization of [@Ottem Proposition 3.4].
\[prop: equi\] Suppose $Y$ is a nef subscheme of $X$. Then the restriction of the blowup morphism to $E$, $\pi|_{E}:E\rightarrow Y$, is equidimensional. In particular, $Y$ is pure dimensional.
Suppose $Y\subset X$ has codimension $r$. Let $y\in Y$ be a closed point, we want to show $Z:=\pi^{-1}(y)$ is of dimension $(r-1)$. Note that $E$ has dimension $n-1$, where $n=\dim X$. This implies $\dim Z\geq r-1$.
On the other hand, $-E$ is $\pi$-ample. In particular, $(-E-\epsilon A)|_Z$ is ample for $1\gg \epsilon >0$, where $A$ is an ample divisor on $E$. We also know that $\mathscr{O}_{E}(E+\epsilon A)$ is $(r-1)$-ample, for $1\gg \epsilon >0$. By theorem \[thm: open\], this forces $Z$ to have dimension $(r-1)$.
\[prop: pullback of nef\] Suppose $Y$ is a nef subscheme of $X$ of codimension $r$, $p: X'\rightarrow X$ a morphism from an equidimensional projective scheme $X'$. If $p^{-1}(Y)$ has codimension $r$ in $X'$, then $p^{-1}(Y)$ is nef in $X'$. In particular, if $p$ is equidimensional, $p^{-1}(Y)$ is nef.
We have the following commutative diagram:
$$\xymatrix{
\operatorname{Bl}_{p^{-1}(Y)}(X') \ar[r]^{\tilde{p}} \ar[d] & \operatorname{Bl}_Y(X) \ar[d] \\
X' \ar[r]_{p} & X,
}$$ with $\tilde{p}$ induced by the universal property of blowup and $\tilde{p}^{*}(E)=E'$, where $E$ and $E'$ are exceptional divisors in the respective blowups. We can now apply proposition \[prop: pullback\] to conclude the proof.
\[prop: ample intersect\] Let $Y$ be an ample (resp. nef) subscheme of codimension $r$ of $X$. Let $Z$ be a closed subscheme of $X$. If $Y\cap Z$ has codimension $r$ in $Z$, then $Y\cap Z$ is an ample (resp. nef) subscheme of $Z$.
Indeed, we have the following commutative diagram $$\xymatrix{
\operatorname{Bl}_{Y\cap Z} Z \ar@{^{(}->}[r] \ar[d]_{\pi_Z}
& \operatorname{Bl}_{Y}X \ar[d]^{\pi_X}\\
Z \ar@{^{(}->}[r] & X.
}$$
Note that the exceptional divisor of $\pi_Z$ is the restriction of the exceptional divisor $E$ of $\pi_X$. If $E$ is $(r-1)$-ample (resp. $(r-1)$-almost ample), so is $E|_{\operatorname{Bl}_{Y\cap Z}Z}$.
The following theorem generalizes the transitivity property of ample subschemes [@Ottem Proposition 6.4] in the sense that we do not require $Y$ (resp. $Z$) to be lci in $X$ (resp. $Y$). This gives further evidence that Ottem’s definition of an ample subscheme is a natural one. First, we need a lemma:
\[lemma: pass to blowup\] Let $X$ be a projective scheme and let $Y$ be a closed subscheme of $X$ of codimension $r$. Suppose the blowup of $X$ along $Y$, $\pi:\operatorname{Bl}_Y X\rightarrow X$, has fiber dimension at most $r-1$. If a line bundle $\mathscr{L}$ on $X$ is $q$-ample on $Y$, and for if any $l\geq 0$ $$H^{i}(\operatorname{Bl}_Y X, \pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l)))=0$$ for $i>q+r$ and $m\gg 0$, then $\mathscr{L}$ is $(q+r)$-ample. Here $\mathscr{O}_X(1)$ is an ample line bundle on $X$.
Applying the Leray spectral sequence, we have $$E^{p,s}_{2}=H^{p}(X, R^{s}\pi_{*}\mathscr{O}_{\operatorname{Bl}_Y X}\otimes \mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))
\Rightarrow H^{p+s}(\operatorname{Bl}_Y X, \pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))).$$
Since the fiber dimension of $\pi$ is at most $r-1$ [@Ottem Proposition 3.4], $R^{s}\pi_{*}\mathscr{O}_{\operatorname{Bl}_Y X}=0$ and $E^{p,s}_{2}=0$ for $s>r-1$.
For $s>0$, $R^{s}\pi_{*}\mathscr{O}_{\operatorname{Bl}_Y X}$ is a coherent sheaf on $Y$. Indeed, this follows by considering the long exact sequence $$\cdots\rightarrow R^{s}\pi_{*}\mathscr{O}_{\operatorname{Bl}_Y X}(-jE)
\rightarrow R^{s}\pi_{*}\mathscr{O}_{\operatorname{Bl}_Y X}((-j+1)E)
\rightarrow R^{s}\pi_{*}\mathscr{O}_{E}((-j+1)E)\rightarrow \cdots,$$ where $E$ is the exceptional divisor, and the fact that $R^{s}\pi_{*}\mathscr{O}_{\tilde{X}}(-jE)=0$ for $j\gg 0$, since $-E$ is $\pi$-ample.
By the $q$-ampleness of $\mathscr{L}|_{Y}$, we have $E^{p,s}_2=H^{p}(X,R^{s}\pi_{*}\mathscr{O}_{\operatorname{Bl}_Y X}\otimes \mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))=0$ for $p>q$, $s>0$ and $m\gg0$.
These two vanishing results imply that $E^{p-h,h-1}_{h}=E^{p-h,h-1}_{2}=0$ for $h \geq 2$, $p>q+r$ and $m\gg0$.
By the hypothesis, $$E^{p,0}_{\infty}=H^{p}(\operatorname{Bl}_Y X, \pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l)))=0$$ for $p>q+r$ and $m\gg 0$. Hence we arrive at the desired vanishing $E^{p,0}_{2}=H^{p}(X, \mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))=0$ for $p>q+r$ and $m\gg 0$.
\[theorem: trans ample\] Let $Y\subset X$ be an ample subscheme of codimension $r_1$, $Z\subset Y$ be an ample subscheme of codimension $r_2$. Then $Z\subset X$ is also an ample subscheme of codimension $r_1+r_2$.
\[lemma: trans ample\] Under the same hypothesis as in the theorem, we have the following commutative diagram. $$\xymatrix{
\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X
\ar@/^/[drr]^{\pi_Y}
\ar@/_/[ddr]_{\pi_Z}
\ar@{.>}[dr]|-{\iota} & & \\
& \operatorname{Bl}_{\mathscr{I}_{Y}}X \times_X \operatorname{Bl}_{\mathscr{I}_{Z}}X \ar[r]^{p} \ar[d]_q
& \operatorname{Bl}_{\mathscr{I}_{Z}}X \ar[d]^{\pi'_Z} \\
& \operatorname{Bl}_{\mathscr{I}_{Y}}X \ar[r]^{\pi'_Y}
& X.
}$$
Here $\pi'_Z$ (resp. $\pi'_Y$) is the blowup of $X$ along $\mathscr{I}_Z$ (resp. $\mathscr{I}_Y$), with exceptional divisor $E'_Z$ (resp. $E'_Y$); $\pi_Z$ and $\pi_Y$ are blowups along the ideal sheaves $\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y}X}$ and $\mathscr{I}_Y\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\otimes\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(E'_Z)$, with exceptional divisor $E_Z$ and $E_Y$ respectively. The composition $\pi_Y\circ\pi'_Z=\pi_Z\circ\pi'_Y$ is the blowup map of $X$ along $\mathscr{I}_Y\cdot\mathscr{I}_Z$. The square in the above diagram is a fiber diagram, with $\iota$ induced by the maps $\pi_Z$ and $\pi_Y$. Moreover,
1. \[item: producta\] $\pi_Y^{*}E'_Z=E_Z$ and $\pi_Z^{*}E'_Y=E_Y +E_Z$.
2. \[item: productb\] $\iota$ is a closed immersion.
First, let us check that the blowup of $X$ along $\mathscr{I}_Y\cdot\mathscr{I}_Z$ factors through the maps $\pi'_Y$ and $\pi'_Z$. By the universal property of blowup, it suffices to check that the inverse image ideal sheaves $\mathscr{I}_Y\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ and $\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ are invertible. Let $\mathscr{J}$ be the inverse of $(\mathscr{I}_Y\cdot\mathscr{I}_Z)\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$, i.e. the fractional ideal sheaf such that $(\mathscr{I}_Y\cdot\mathscr{I}_Z)\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}\cdot \mathscr{J}=\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$. We check locally that $\mathscr{I}_Y\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ is invertible. Let $\mathfrak{a}$ and $\mathfrak{b}$ be the stalk of $\mathscr{I}_Y\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ and $(\mathscr{I}_Z\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X})\cdot \mathscr{J}$ at a scheme-theoretic point $x\in \operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X$ respectively, and let $R=\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X,x}$ be the local ring at $x$. Since $\mathfrak{a}\cdot\mathfrak{b}=R$. We may write $\sum_i a_i b_i=1$, where $a_i\in \mathfrak{a}$ and $b_i\in\mathfrak{b}$. Note that each $a_i b_i\in R$, so there must be some $j$ such that $a_jb_j$ is a unit. Let $u=(a_j b_j)^{-1}$. Let $f:R\rightarrow \mathfrak{a}$ be the $R$-module homomorphism that sends $r\mapsto ra_i$. We shall see that $f$ is an isomorphism. For any $a\in\mathfrak{a}$, we can write $a=(ab_j u )a_j$. Note that $(ab_j u )\in R$. Thus, $f$ is onto. Suppose there is an $r\in R$ such that $f(r)=ra_j=0$. Then $r=r(a_j b_j u)=0$. Therefore, $f$ is injective. We conclude that $\mathscr{I}_Y\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ is locally free of rank $1$, hence is invertible. Applying a similar argument, we see that $\mathscr{I}_Z\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ is also invertible. This gives us the maps $\pi_Z$ and $\pi_Y$.
Next, let us check that $\mathscr{I}_Y\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\otimes\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(E'_Z)$ is an ideal sheaf. Indeed, we have the inclusion $\mathscr{I}_Y\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\subset
\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\cong
\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(-E'_Z)$. We then tensor the terms in the inclusion by $\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(E'_Z)$ to see that $\mathscr{I}_Y\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\otimes\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(E'_Z)\subset \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}$. Applying the universal property of blowup again, we see that $\pi_Z:\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X\rightarrow\operatorname{Bl}_{\mathscr{I}_Y}X$ and $\pi_Y:\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X\rightarrow\operatorname{Bl}_{\mathscr{I}_Z}X$ are the same as the blowup of $\operatorname{Bl}_{\mathscr{I}_Y}X$ and $\operatorname{Bl}_{\mathscr{I}_Z}X$ along $\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y}X}$ and $\mathscr{I}_Y\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\otimes\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(E'_Z)$.
For \[item: producta\], note that $\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}\cong \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(-E'_Z)$. Therefore we have the surjection $\pi^{*}_Y\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}(-E'_Z)\twoheadrightarrow
(\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X})\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}\cong
\mathscr{I}_Z\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}\cong \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}(-E_Z)$. This is also an injection, since the pullback of a local generator of $\mathscr{I}_Z\cdot \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}X}$ to $\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X$ is not a zero divisor, thanks to the fact that $\mathscr{I}_Z\cdot\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X}$ is invertible. A similar argument leads to the second statement in \[item: producta\].
For \[item: productb\], let $W$ be the scheme-theoretic image of $\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X$ under $\iota$. It suffices to show that $\mathscr{I}_Y\cdot\mathscr{O}_{W}$ (resp. $\mathscr{I}_Z\cdot \mathscr{O}_W$) is invertible. Note that the natural surjection $q^*\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y}X}(-E'_Y)\rightarrow\mathscr{I}_Y\cdot\mathscr{O}_{W}$ is injective if and only if the pullback of a local generator of $\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y}X}(-E'_Y)$ is not a zero divisor, which follows from the fact that the natural map $\mathscr{O}_W\rightarrow \mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}Z}}X$ is an injection [@Stacks Lemma 28.6.3]. We can use the same argument to show that $\mathscr{I}_Z\cdot \mathscr{O}_W$ is also invertible.
Note that $\pi_Y$ has fiber dimension at most $r_1 -1$. This follows from \[item: productb\] of lemma \[lemma: trans ample\] and the fact that $\pi'_Y$ has fiber dimension at most $r_1-1$ (proposition \[prop: equi\]).
Let $\tilde{Y}$ be the strict transform of $Y$ in $\operatorname{Bl}_{\mathscr{I}_Z}X$. Since $Z$ is an ample subscheme of $Y$, $E'_Z|_{\tilde{Y}}$ is $(r_2 -1)$-ample.
By lemma \[lemma: pass to blowup\], it suffices to prove that given any $l\in \mathbb{Z}_{\geq0}$, $$H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X}(mE_Z)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH)))=0$$ for $i>r_1 +r_2 -1$ and $m\gg 0$. Here $H$ is an ample divisor on $\operatorname{Bl}_{\mathscr{I}_{Z}}X$. We fix an $l\in \mathbb{Z}_{\geq0}$ from now on.
$(E_Z -\delta E_Y)|_{E_Y}$ is $(r_2-1)$-ample for $0<\delta\ll 1$.
Since $-E_Y$ is $\pi_Y$-ample, $(\pi_Y^*E'_Z -\delta E_Y)|_{E_Y}=(E_Z -\delta E_Y)|_{E_Y}$ is $(r_2-1)$-ample, for $0<\delta\ll1$, by proposition \[prop: pullback\].
$E_Z +E_Y -\epsilon E_Z$ is $(r_1-1)$-ample for $0<\epsilon\ll1$.
Indeed, $E_Z +E_Y=\pi_{Y}^{*}E'_Y$ and $E'_Y$ is $(r_1 -1)$-ample by ampleness of $Y\subset X$. Note that $-E_Z$ is $\pi_{Y}$-ample. The claim then follows from proposition \[prop: pullback\].
By the above claims, we may choose a big enough $k\in \mathbb{Z}$ such that $(kE_Z - E_Y)|_{E_Y}$ is $(r_2 -1)$-ample and $kE_Z + (k+1)E_Y$ is $(r_1-1)$-ample.
Write $$m_1 E_Y+m_2 E_Z =\lambda_1(kE_Z - E_Y) +\lambda_2(kE_2 +(k+1)E_Y) + j_1 E_Y +j_2 E_Z,$$ where $\lambda_2=\lfloor\frac{m_1+\lfloor \frac{m_2}{k}\rfloor}{k+2}\rfloor$; $\lambda_1= \lfloor\frac{m_2}{k}\rfloor-\lambda_2$; $j_1=((m_1+\lfloor\frac{m_2}{k}\rfloor)\bmod (k+2))$ and $j_2= (m_2\bmod k)$. Note that $0\leq j_1<k+2$ and $0\leq j_2<k$. The precise formulae for $\lambda_1$ and $\lambda_2$ are not very important. The plan is to choose a big $m_2$, then let $m_1$ increases. As $m_1$ grows, $\lambda_1$ decreases and $\lambda_2$ increases. We then use the positivity of $(kE_Z- E_Y)|_{E_Y}$ and $kE_Z + (k+1)E_Y$ to prove the required vanishing statement.
Since $kE_Z +(k+1)E_Y$ is $(r_1-1)$-ample, we may find $\Lambda_2$ such that $$\label{eq: a}
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(\lambda_2(kE_2 +(k+1)E_Y) + j_1 E_Y +j_2 E_Z)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH)))=0$$ for $i>r_1 -1$, $\lambda_2\geq \Lambda_2$, $0\leq j_1<k+2$ and $0\leq j_2 <k$.
Applying theorem \[theorem: fujita\] to the scheme $E_Y$, there is an $\Lambda'_2$ such that $$H^{i}(E_Y, \mathscr{O}_{E_Y}(\lambda_1(kE_Z - E_Y) +\lambda_2(kE_2 +(k+1)E_Y) + j_1 E_Y +j_2 E_Z)\otimes \pi^{*}_Y\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_Z}}(-lH))=0$$ for $i>(r_2 -1) + (r_1 -1)$, $\lambda_1\geq 0$, $\lambda_2 \geq \Lambda'_2$, $0\leq j_1<k+2$ and $0\leq j_2<k$. This implies $$\begin{gathered}
\label{eq: b}
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(m_2 E_Z + m_1 E_Y)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH)))\\
\cong
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(m_2 E_Z + (m_1 +1)E_Y)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH)))\end{gathered}$$ for $i>r_1 + r_2 -1$, $0<m_1+1< (k+1)\lfloor\frac{m_2}{k}\rfloor + k+2$ and $\lfloor\frac{m_1 +1+\lfloor \frac{m_2}{k}\rfloor}{k+2}\rfloor\geq \Lambda'_2$.
Choose some big $M_2$ such that $\lfloor\frac{\lfloor \frac{M_2}{k}\rfloor}{k+2}\rfloor\geq \max\{\Lambda_2,\Lambda'_2\}$. Applying (\[eq: b\]) repeatedly, we have for $m_2>M_2$, $$\begin{gathered}
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(m_2 E_Z)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH)))\\
\cong
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(m_2 E_Z + (k+1)\lfloor \frac{m_2}{k}\rfloor E_Y)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH)))\end{gathered}$$ for $i>r_1 + r_2 -1$. The above cohomology group can be rewritten as $$H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(\lfloor\frac{m_2}{k}\rfloor(kE_Z +(k+1)E_Y) +(m_2 -k\lfloor\frac{m_2}{k}\rfloor) E_Z)\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-lH))),$$ which is $0$ by (\[eq: a\]).
We then prove the analogue of theorem \[theorem: trans ample\] for nef subschemes. The idea of the proof is essentially the same, although we have to use the full statement of theorem \[theorem: fujita\] by allowing a nef term, as well as take extra care with the variables.
\[theorem: trans nef\] Let $Y\subset X$ be a nef subscheme of codimension $r_1$, $Z\subset Y$ be a nef subscheme of codimension $r_2$. Then $Z\subset X$ is also a nef subscheme of codimension $r_1+r_2$.
Lemma \[lemma: trans ample\] still holds under the hypothesis of the theorem. We shall use the same notation as in lemma \[lemma: trans ample\]. Since $-E'_Z$ and $-E'_Y$ is $\pi'_Z$-ample and $\pi'_Y$-ample respectively, we may choose an ample divisor $A'$ on $X$ such that $\pi_Z^{\prime*} A'-E'_Z$ and $\pi^{\prime *}_Y A'-E'_Y$ are ample. Let $A=\pi_Y^*\pi_Z^{\prime*}A'$ be the pullback of $A$ to $\operatorname{Bl}_{\mathscr{I}_Y\cdot\mathscr{I}_Z}X$, note that $A$ is nef.
Note that we can write $kE_Z+ A$ as $\pi_Y^{*}((k+1)E'_Z+(\pi_Z^{\prime*} A'-E'_Z))$. By lemma \[lemma: pass to blowup\], it suffices to prove that given any $l\geq0$, for $k\gg0$ $$H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(m_2(kE_Z+ A))\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-l)))=0$$ for $i>r_1 +r_2 -1$ and $m_2\gg 0$. We fix $l$ and $k$ from this point on.
Note that $F'_1:=(E_Z + \frac{1}{3k}\pi^{*}_Y(\pi_Z^{\prime *}A'-E'_Z)-\frac{1}{k_1}E_Y)$ is $(r_2-1)$-ample when restricted to $E_Y$ and $F'_2:=E_Z +E_Y + \frac{1}{3k}\pi_Z^{*}(\pi_Y^{\prime *}A' -E'_Y)-\frac{1}{k_1}E_Z$ is $(r_1 -1)$-ample for $k_1\gg0$. We fix such a $k_1$. Let $\alpha = 3kk_1 -k_1$ and $\beta = 3kk_1-k_1-3k$. Let $F_1= 3kk_1\beta F'_1$ and $F_2=3kk_1\alpha F'_2$. They are both integral divisors. In fact, $F_1=\beta(\alpha E_Z-3kE_Y+k_1A)$ and $F_2=\alpha(\beta E_Z+\alpha E_Y +k_1A)$.
Write $$m_1 E_Y+m_2 (kE_Z+A) =\lambda_1 F_1 +\lambda_2 F_2 + \lambda_3 A+ j_1 E_Y +j_2 E_Z,$$ where $\lambda_2=\lfloor\frac{m_1 + 3\beta k\lfloor\frac{m_2 k}{\alpha \beta}\rfloor}{\alpha^2 +3\beta k}\rfloor$; $\lambda_1= \lfloor\frac{m_2 k}{\alpha\beta}\rfloor-\lambda_2$; $\lambda_3=m_2 -\lambda_1\beta k_1-\lambda_2\alpha k_1$; $j_1=((m_1 + 3\beta k \lfloor\frac{m_2 k}{\alpha \beta}\rfloor) \bmod (\alpha^2 + 3\beta k))$ and $j_2=(m_2 k \bmod \alpha\beta)$. Note that if $0\leq m_1\leq \alpha^2\lfloor\frac{m_2 k}{\alpha \beta}\rfloor$, then $\lambda_1\geq 0$ and $\lambda_3\geq 0$.
Since $F_2$ is $(r_1 -1)$-ample and $A$ is nef, there is a $\Lambda_2$ such that $$\label{eq: trans nef a}
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X, \mathscr{O}(\lambda_2 F_2 + \lambda_3 A +j_2 E_Z)\otimes \pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-l)))=0$$ for $i>r_1 -1$, $\lambda_2>\Lambda_2$, $\lambda_3\geq0$ and $0\leq j_2<\alpha\beta$.
Since $F_1|_{E_Y}$ is $(r_2 -1)$-ample, $F_2$ is $(r_1-1)$-ample and $A$ is nef, there is a $\Lambda'_2$ such that $$H^{i}(E_Y, \mathscr{O}_{E_Y}(\lambda_1 F_1 +\lambda_2 F_2 + \lambda_3 A+ j_1 E_Y +j_2 E_Z)\otimes \pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-l)))=0$$ for $i>(r_2 -1) + (r_1 -1)$, $\lambda_2>\Lambda'_2$, $\lambda_1\geq 0$, $\lambda_3\geq 0$, $0\leq j_1<\alpha^2 + 3\beta k$ and $0\leq j_2<\alpha\beta$. This implies if $\lfloor\frac{m_1 + 3\beta k\lfloor\frac{m_2 k}{\alpha \beta}\rfloor}{\alpha^2 +3\beta k}\rfloor>\Lambda'_2$, $$\begin{gathered}
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(m_2(kE_Z+A))\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-l)))\\
\cong
H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X,\mathscr{O}(\alpha^2\lfloor\frac{m_2 k}{\alpha \beta}\rfloor E_Y + m_2(kE_Z+A))\otimes\pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-l)))\end{gathered}$$ for $i>r_1 + r_2 -1$.
The above cohomology groups can be rewritten as $$H^{i}(\operatorname{Bl}_{\mathscr{I}_{Y}\cdot\mathscr{I}_{Z}}X, \mathscr{O}(\lfloor\frac{m_2 k }{\alpha \beta}\rfloor F_2 + \lambda_3 A +j_2 E_Z)\otimes \pi_Y^*(\mathscr{O}_{\operatorname{Bl}_{\mathscr{I}_{Z}}X}(-l)))$$ where $\lambda_3\geq0$ and $0\leq j_2<\alpha\beta$. By (\[eq: trans nef a\]), the above cohomology groups vanish for $m_2\gg0$.
The following corollary says that intersection of $2$ ample (resp. nef) subschemes is ample (resp. nef), assuming the intersection has the desired codimension. It is the generalization of [@Ottem Proposition 6.3], in the sense that we do not assume that $X$ is smooth and the subschemes are lci in $X$.
\[cor: intersection of ample\] If $Y$ and $Z$ are both ample (resp. nef) subschemes of $X$, of codimension $r$ and $s$ respectively and $Y\cap Z$ has codimension $r+s$ in $X$, then $Y\cap Z$ is an ample (resp. nef) subscheme of $X$.
By proposition \[prop: ample intersect\], $Y\cap Z$ is an ample (resp. nef) subscheme of $Z$. We now conclude using the transitivity property of ample (resp. nef) subschemes (theorem \[theorem: trans ample\] or theorem \[theorem: trans nef\] respectively).
Positivity of a line bundle upon restriction to an ample subscheme
==================================================================
If a line bundle is ample after restricting to an ample subscheme, it is reasonable to expect the line bundle to exhibit some positivity features. The following theorem demonstrates a nice interplay between ample subschemes and $q$-ample divisors.
\[theorem: restriction\] Let $X$ be a projective scheme of dimension $n$. Let $Y$ be an ample subscheme of $X$ of codimension $r$. Suppose $\mathscr{L}$ is a line bundle on $X$, and that its restriction $\mathscr{L}|_{Y}$ to $Y$ is $q$-ample. Then $\mathscr{L}$ is $(q+r)$-ample.
We fix an ample line bundle $\mathscr{O}_X(1)$ on $X$. Let $\pi: \tilde{X}\rightarrow X$ be the blowup of $X$ along $Y$.
\[step: blowup\] Pass to the blowup.
By lemma \[lemma: pass to blowup\], it suffices to to prove that $$\label{blowup}
H^{i}(\tilde{X}, \pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l)))=0$$ for $i>q+r$ and $m\gg 0$.
\[step: exceptional\] Pass to the exceptional divisor.
We claim that it is enough to show that there is an $m_0$ such that $$\label{exceptional}
H^{i}(E,\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{E}(kE))=0$$ for $i>r+q-1$, $m>m_0$ and $k\geq 1$. Here $E$ is the exceptional divisor on the blowup $\tilde{X}$.
Indeed, let us consider the short exact sequence: $$\begin{gathered}
0\rightarrow \pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{\tilde{X}}((k-1)E)
\rightarrow
\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{\tilde{X}}(kE)
\\
\rightarrow
\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{E}(kE)
\rightarrow
0.\end{gathered}$$
By looking at the long exact sequence of cohomology groups induced from the above short exact sequence and using the hypothesis (\[exceptional\]), we observe that $$\label{iso}
H^{i}(\tilde{X},\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{\tilde{X}}((k-1)E))
\cong H^{i}(\tilde{X},\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{\tilde{X}}(kE))$$ for $i>r+q$, $m>m_0$ and $k\geq 1$.
Since $E$ is $(r-1)$-ample, for any fixed $m$, $$H^{i}(\tilde{X},\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_{\tilde{X}}(kE))=0$$ for $k\gg0$ and $i>r-1$. Together with the isomorphisms in (\[iso\]), we have the desired vanishing result (\[blowup\]).
Rewrite the line bundles of interest in ($\ref{exceptional}$) in terms of $q$- and $(r-1)$- ample line bundles.
Note that $-E$ is $\pi$-ample, there is an $N>0$ such that $\pi^{*}(\mathscr{L}^{\otimes N})\otimes \mathscr{O}_{\tilde{X}}(-E)$ is $q$-ample, by proposition \[prop: pullback\]. We can replace $\mathscr{L}$ by $\mathscr{L}^{\otimes N}$ and assume that $\pi^{*}(\mathscr{L})\otimes \mathscr{O}_E(-E)$ is $q$-ample. We now rewrite the line bundle on $E$ in (\[exceptional\]): $$\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_X(-l))\otimes\mathscr{O}_E(kE))\cong \pi^{*}(\mathscr{O}_X(-l))
\otimes\mathscr{O}_E((k+m)E)
\otimes(\pi^{*}(\mathscr{L})\otimes \mathscr{O}_E(-E))^{\otimes m}$$ with the second term $\mathscr{O}_E((k+m)E)$ on the right hand side being an $(r-1)$-ample line bundle, and the third term $(\pi^{*}(\mathscr{L})\otimes \mathscr{O}_E(-E))^{\otimes m}$ being an $q$-ample line bundle.
We now apply theorem \[theorem: fujita\] with $\mathscr{L}_1:=\pi^{*}(\mathscr{L})\otimes \mathscr{O}_E(-E)$, $\mathscr{L}_2=\mathscr{O}_E(E)$ and $M_2=1$ to conclude.
One may ask whether we have a converse to theorem \[theorem: restriction\], i.e., given an $r$-ample line bundle $\mathscr{L}$ on a projective scheme $X$, there is a codimension $r$ ample subscheme $Y$, such that $\mathscr{L}|_{Y}$ is ample. Demailly, Peternell and Schneider gave a counter-example to this in [@DPS Example 5.6]:
\[example: DPS\] Let $S$ be a general quartic surface in $\mathbb{P}^3$. Let $X=\mathbb{P}(\Omega_{S}^1)$. They showed that $-K_X$ is $1$-ample, and yet for any ample divisor $Y$ in $X$, $(-K_X)^2\cdot Y<0$, thus $-K_X$ cannot be ample when it is restricted to any ample divisor.
For the reader’s convenience, we shall include the proof of $-K_X$ being $1$-ample in example [@DPS]. In fact, it might be worthwhile to extract from the argument of [@DPS Example 5.6] the following general property.
Let $$\label{ses vector bundle}
0\rightarrow \mathscr{E}' \rightarrow \mathscr{E} \rightarrow \mathscr{L} \rightarrow 0$$ be a short exact sequence of vector bundles on a projective scheme $X$. We assume $\mathscr{E}$ to be a $q$-ample vector bundle of rank $r$, $\mathscr{E}'$ is of rank $(r-1)$ and $\mathscr{L}$ is of rank $1$. Then $\mathscr{E}'$ is $(q+1)$-ample.
We first dualize (\[ses vector bundle\]), then take symmetric product, and dualize again. This will give us the following short exact sequence $$0\rightarrow \operatorname{Sym}^k \mathscr{E}'
\rightarrow \operatorname{Sym}^k \mathscr{E}
\rightarrow \operatorname{Sym}^{k-1} \mathscr{E}\otimes \mathscr{L}
\rightarrow 0.$$ Fix an ample line bundle $\mathscr{O}_X(1)$ on $X$, and tensor the above short exact sequence with $\mathscr{O}_X(-l)$, for $l\geq0$. Note that $H^{i}(X,\operatorname{Sym}^k\mathscr{E}\otimes\mathscr{O}_X(-l))= H^{i}(X,\operatorname{Sym}^{k-1}\mathscr{E}\otimes \mathscr{L}\otimes\mathscr{O}_X(-l))=0$, for $i>q$ and $k\gg 0$. Hence $H^{i}(X,\operatorname{Sym}^k\mathscr{E}'\otimes \mathscr{O}_X(-l))=0$, for $i>q+1$ and $k\gg0$.
Going back to the example \[example: DPS\], note that $\Omega_{S}^1\cong \mathscr{T}_S$, where $\mathscr{T}_S$ is the tangent sheaf of $S$. We have the following short exact sequence of locally free sheaves on $S$. $$0\rightarrow \mathscr{T}_S \rightarrow
\mathscr{T}_{\mathbb{P}^3}|_{S} \rightarrow
\mathscr{O}_S(S) \rightarrow 0.$$ The tangent bundle of a projective space is ample, therefore the tangent bundle of $S$ is $1$-ample by the lemma. Since $\mathscr{O}_X(-K_X)\cong \mathscr{O}_{\mathbb{P}(\Omega_{S}^1)}(2)$, $-K_X$ is $1$-ample. It is not ample since the tangent bundle of $S$ is not ample ($S$ is a K3-surface).
Interestingly, we note that $$H^{2}(X,K_X -K_X)\cong
H^{2}(S,\mathscr{O}_S)\neq 0,$$ Hence, Kodaira-type vanishing theorem fails for $-K_X$, which is $1$-ample. Ottem also gave a counterexample to Koadaira-type vanishing theorem for $q$-ample divisors [@Ottem Chapter 9].
One may also ask if we can relax the positivity assumption on $Y$ in theorem \[theorem: restriction\]. For example, if we only assume that the normal bundle of $Y$ is ample, we shall see the conclusion of the theorem does not hold in general. Let us start with a smooth ample subvariety $Y\subset X$ of a smooth projective variety. We blowup a closed point $p$ in $X\setminus Y$. Observe that the normal bundle of $Y\subset \operatorname{Bl}_{p} X$ is still ample. Let $E\cong\mathbb{P}^{n-1}$ be the exceptional divisor, and let $A$ be an ample divisor on $\operatorname{Bl}_{p}(X)$. Then $E+\epsilon A$ is not $(n-2)$-ample, for $0<\epsilon\ll1$, since it is anti-ample when restricted to the exceptional divisor. But $(E+\epsilon A)|_Y=\epsilon A|_{Y}$ is ample. On the other hand, as we shall see in the following section, a small yet interesting part of the theorem still holds if we assume $Y$ is a nef subvariety.
Restriction of a pseudoeffective divisor to a nef subvariety
============================================================
There are not many results regarding the positivity of subvariety with nef normal bundle, in terms of intersection theory. Here are two of such results the author is aware of.
In Fulton-Lazarsfeld’s work [@FulLaz] (see also [@Laz Theorem 8.4.1]), they proved that if $Y$ is a closed, lci subvariety of a projective variety $X$ and the normal bundle of $Y$ is nef, then for any closed subscheme $Z\subset X$ with $\dim Y + \dim Z \geq \dim X$, $\deg_H(Y\cdot Z)\geq 0$. (Here $H$ is an ample divisor on $X$.) On the other hand, it is not hard to show that if $Y$ has globally generated normal bundle, then restriction of any effective cycle to $Y$ is either effective or $0$ [@Fulton Theorem 12.1.a)].
We show that the restriction of a pseudoeffective divisor to a nef subvariety is still pseudoeffective.
\[theorem: pseudoeffective\] Let $Y$ be a nef subvariety of codimension $r$ of a projective variety $X$. Then $$\iota^{*}\overline{\operatorname{Eff}}^{1}(X)\subseteq\overline{\operatorname{Eff}}^{1}(Y)$$ and $$\iota^{*}\operatorname{Big}(X)\subseteq\operatorname{Big}(Y).$$ Here $\iota:Y\hookrightarrow X$ is the inclusion map, $\iota^{*}:\operatorname{N}^1(X)_{\mathbf{R}}\rightarrow\operatorname{N}^1(Y)_{\mathbf{R}}$ is the induced map on the Néron-Severi group with $\mathbf{R}$-coefficients and $\overline{\operatorname{Eff}}^{1}(X)$ (resp. $\operatorname{Big}(X)$) is the cone of pseudoeffective (resp. big) $\mathbf{R}$-Cartier divisors.
Before proving the theorem, let us point out it is rather straightforward to obtain the conclusion under the stronger assumptions in theorem \[theorem: restriction\] and the added assumption that $X$ and $Y$ are integral. Let $D$ be a pseudoeffective divisor on $X$, i.e. $-D$ is not $(n-1)$-ample (theorem \[theorem: n-1 ample\]). Suppose on the contrary $D|_{Y}$ is not pseudoeffective. Then $-D|_{Y}$ is $(n-r-1)$-ample. This gives a contradiction to theorem \[theorem: restriction\].
A divisor is big if and only if it can be written as the sum of a pseudoeffective divisor and an ample divisor. Therefore, we can focus on the pseudoeffective case. We shall follow the steps in the proof of theorem \[theorem: restriction\] closely. Recall that a Cartier divisor $D$ is $(n-1)$-ample if and only if $-D$ is not pseudoeffective (theorem \[theorem: n-1 ample\]). Given a line bundle $\mathscr{L}$ on $X$ such that $\mathscr{L}|_Y$ is $(n-r-1)$-ample, we need to show $\mathscr{L}$ is $(n-1)$-ample. Fix an ample line bundle $\mathscr{O}_{X}(1)$ on $X$.
It suffices to show for any $l\geq 0$, there is an $m_0$ such that $H^{n}(\tilde{X},\pi^{*}(\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-l)))=0$ for $m\geq m_0$.
This is true by lemma \[lemma: pass to blowup\].
We now fix $l$.
It is enough to show that there is an $m_0$ such that $$H^{n-1}(E, \pi^{*}(\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-l))\otimes\mathscr{O}_E(kE))=0$$ for $m\geq m_0$ and $k\geq 1$.
We just have to repeat the argument in step \[step: exceptional\] in the proof of theorem \[theorem: restriction\], i.e. consider the long exact sequence of cohomologies associated to $$\begin{gathered}
0\rightarrow
\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_{X}(-l))\otimes \mathscr{O}_{\tilde{X}}((k-1)E)\rightarrow
\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_{X}(-l))\otimes \mathscr{O}_{\tilde{X}}(kE)\\
\rightarrow
\pi^{*}(\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-l))\otimes\mathscr{O}_E(kE)
\rightarrow 0.\end{gathered}$$
Also note that for a fixed $m$, $$H^{n}(\tilde{X},\pi^{*}(\mathscr{L}^{\otimes m}\otimes\mathscr{O}_{X}(-l))\otimes \mathscr{O}_{\tilde{X}}(kE))=0$$ for $k\gg 0$. Indeed, $E$ is $(n-1)$-ample ($-E$ is not pseudoeffective!).
Replacing $\mathscr{L}$ with $\mathscr{L}^{\otimes N}$ for $N$ large enough, we may assume $\pi^{*}\mathscr{L}\otimes\mathscr{O}_{E}(-E)$ is $(n-r-1)$-ample, by proposition \[prop: pullback\]. Now we can write $$\pi^{*}(\mathscr{L}^{\otimes m}\otimes \mathscr{O}_{X}(-l))\otimes \mathscr{O}_{E}(kE)
\cong
\pi^*\mathscr{O}_{X}(-l)\otimes
(\pi^{*}\mathscr{L}\otimes\mathscr{O}_{E}(-E))^{\otimes m}
\otimes
\mathscr{O}_E((k+m)E).$$
By proposition \[lemma: uniform2\], there is an $m_0$ such that $$H^{n-1}(E,\pi^*\mathscr{O}_{X}(-l)\otimes
(\pi^{*}\mathscr{L}\otimes\mathscr{O}_{E}(-E))^{\otimes m}\otimes
\mathscr{O}_E((k+m)E))=0$$ for $k\geq 1$ and $m\geq m_0$. This proves the theorem.
Suppose the conclusion of theorem \[theorem: pseudoeffective\] holds, the normal bundle of $Y$ is not necessarily nef. Take a $3$-fold with Picard number $1$ that contains a rational curve $C$ with normal bundle $\mathscr{O}(-1)\oplus \mathscr{O}(-1)$. The condition $D\cdot C>0$ for any pseudoeffective divisor $D$ is obvious due to the Picard number $1$ condition on the $3$-fold. This example is taken from Ottem’s paper [@Ottem1 Example 1.2.vii].
Boucksom, Demailly, P[ă]{}un and Peternell showed that the dual cone of the pseudoeffective cone is the cone of movable curves [@BDPP]. Hence we have the equivalent statement:
\[cor: movable curves\] With the same assumptions as in theorem \[theorem: pseudoeffective\], the map on the numerical equivalence classes of $1$-cycles, $\iota_{*}:\operatorname{N}_1(Y)\rightarrow \operatorname{N}_1(X)$, induces $\iota_*: \overline{\operatorname{Mov}}_{1}(Y)\rightarrow \overline{\operatorname{Mov}}_{1}(X)$, where $\overline{\operatorname{Mov}}_{1}(Y)$ and $\overline{\operatorname{Mov}}_{1}(X)$ are the cones of movable curves in $Y$ and $X$ respectively.
We apply the adjunction formula to get
If both $X$ and $Y$ are non-singular, $Y$ has nef normal bundle and $K_X$ is pseudoeffective, then $K_Y$ is also pseudoeffective. If $K_X$ is big, then $K_Y$ is also big.
The first assertion in the above corollary follows also from [@BDPP] and the theory of deformation of rational curve. More specifically, Boucksom-Demailly-Păun-Peterenell showed that on a smooth projective variety $Z$, $K_Z$ is pseudoeffective if and only if $Z$ is not uniruled. If $Y$ is uniruled, take a smooth rational curve $C$ that covers $Y$. By considering the short exact sequence of normal bundles on $C$, we see that the normal bundle of $C$ in $X$ is nef. Thus, $X$ is uniruled.
Weakly movable cone
===================
We shall define and study the weakly movable cone. In this section, we assume the ground field $k$ is algebraically closed and of characteristic zero. On a smooth projective variety, we know that the movable cone of divisors is the smallest closed convex cone that contain all the pushforwards of nef divisors from $X_{\pi}$, where $\pi:X_{\pi}\rightarrow X$ is projective and birational. With this in mind, we define the weakly movable cone as the closure of the cone that is generated by pushforward of cycles of nef subvariety via generically finite morphism. We find that it contains the movable cone and satisfies some desirable intersection theoretic properties.
First, let us recall the definition of a family of effective cycles. We shall follow Fulger-Lehmann’s definition [@FL].
Let $X$ be a projective variety over $k$. A *family of effective $d$-cycles* on $X$ with $\mathbb{Z}$-coefficient, $(g: U\rightarrow W)$, consists of a closed reduced subscheme $\operatorname{Supp}U$ of $W\times_{k} X$, where $W$ is a variety over $k$; a coefficient $a_i\in \mathbb{Z}_{>0}$ for each irreducible component $U_i$ of $\operatorname{Supp}U$; and the projection morphisms $g_i: U_i\rightarrow W$ is proper and flat of relative dimension $d$.
Over a closed point $w\in W$, $g_i^{-1}(w)$ is a closed subscheme of $X$. Its fundamental cycle $[g_i^{-1}(w)]$ is a $d$-cycle of $X$. We define the *cycle theoretic fiber* over $w$ to be $\sum a_i [g_i^{-1}(w)]$.
We say that the family of effective $d$-cycles is *irreducible* if $\operatorname{Supp}U$ is irreducible.
Kollár’s definition [@Kollar Definition I.3.11] of a *well-defined family of $d$-dimensional proper algebraic cycles* is more general. By [@Kollar Lemma I.3.14], given an effective, well-defined family of proper algebraic cycles of a projective variety $X$ over a variety $W$ (both are over $k$), there is a proper surjective morphism $W'\rightarrow W$ from a variety $W'$ such that there is a family of effective cycles (in the sense of Fulger-Lehmann) over $W'$ that “preserves” the cycle theoretic fibers over the closed points of the original family. Therefore for our purpose, it is enough to use Fulger-Lehmann’s definition.
We say that a family of effective $d$-cycles of $X$ $(g: U\rightarrow W)$ is *strictly movable* if each of the irreducible component $U_i$ of $\operatorname{Supp}U$ dominates $X$ via the second projection.
We say that an effective $d$-cycle of $X$ (with $\mathbb{Z}$-coefficient) is *strictly movable* if it is the cycle theoretic fiber over a closed point of a strictly movable family of $d$-cycles on $X$.
We define the *[movable cone of $d$-cycles]{} $\overline{\operatorname{Mov}}_d(X)\subset \operatorname{N}_{d}(X)$ to be the closure of the convex cone generated by strictly movable $d$-cycles.*
\[prop: irreducible\] The movable cone of $d$-cycles is the closure of the convex cone generated by irreducible, strictly movable $d$-cycles.
Suppose $\sum a_i Z_i$ is the cycle theoretic fiber over a closed point of a family of strictly movable $d$-cycles $(g: U\rightarrow W)$ with irreducible components $U_i$. It suffices to show that $Z_i$ is algebraically equivalent to a sum of irreducible strictly movable $d$-cycles. If the generic fiber of $p_i:U_i\rightarrow W$ is geometrically integral, then the fiber over a general closed point is also (geometrically) integral [@EGAIV Théorème 9.7.7], and we are done.
Suppose the generic fiber of $p_i$ is not geometrically integral. Let $\eta_W$ be the generic point of $W$, let $\overline{k(\eta_W)}$ be the algebraic closure of $k(\eta_W)$ and let $U_{ij}'\subset X_{\overline{k(\eta_W)}}$ be the irreducible components of $\operatorname{Spec}\overline{k(\eta_W)}\times_{\operatorname{Spec}k(\eta_W)}U_i$. We may take a finite field extension $k(\eta_W)\subset K$, such that the generators of the ideal sheaves of $U_{ij}'$ are defined over $K$. Then all the irreducible components of $\operatorname{Spec}K\times_{\operatorname{Spec}k(\eta_W)}U_i$ are geometrically integral. These components dominate the generic fiber of $p_i$. Take a variety $V$ with function field $K$ such that the map $\operatorname{Spec}K\rightarrow \operatorname{Spec}k(\eta_W)$ extends to $V\rightarrow W$. By generic flatness, we may replace $V$ by a smaller open set and assume that each irreducible components $U_{ij}$ of $V\times_W U_i$ is flat over $V$. Note that all $U_{ij}$ dominates $U_i$, hence also $X$. Thus, each $U_{ij}$ is a strictly movable family of $d$-cycles of $X$ over $V$ (with coefficient $1$), and the cycle theoretic fiber over a general closed point of $V$ is (geometrically) integral, by [@EGAIV Théorème 9.7.7] again. Then $Z_i$ is algebraically equivalent to the sum of the cycle theoretic fibers of $U_{ij}$’s, with $\mathbb{Z}$-coefficient, over a general closed point of $V$.
\[prop: movable is nef\] An irreducible, strictly movable cycle can be realized as the pushforward of a multiple of the cycle class of a nef subvariety via a proper, surjective morphism, up to numerical equivalence.
From the proof of proposition \[prop: irreducible\], we may assume the irreducible, strictly movable cycle is the cycle theoretic fiber over a closed point of an irreducible, strictly movable family of $(g: U\rightarrow W)$, with the fiber of $g':\operatorname{Supp}U \rightarrow W$ over a general closed point of $W$ integral. Using the argument in [@FL Remark 2.13] or [@Kollar Proposition I.3.14], we may assume $W$ is projective. We note that a closed point $w\in W$ is nef, hence $g^{'-1}(w)$ is also nef, by proposition \[prop: pullback of nef\], and that $g^{'-1}(w)$ is integral if $w$ is general.
Let $X$ be a projective variety over $k$. We define the *weakly movable cone* $\overline{\operatorname{WMov}}_d(X)\in\operatorname{N}_d(X)$ to be the closure of the convex cone generated by $\pi_{*}[Z]$, where $\pi:Y\rightarrow X$ is proper, surjective morphism from a projective variety and $Z$ is a nef subvariety of dimension $d$ in $Y$.
We shall compare the movable cone and the weakly movable cone.
\[prop: mov vs wmov\] Let $X$ be a projective variety over $k$. We have $$\overline{\operatorname{Mov}}_d(X)\subseteq\overline{\operatorname{WMov}}_d(X).$$ In particular, $\overline{\operatorname{WMov}}_d(X)$ is a full dimensional cone in $\operatorname{N}_d(X)$.
This follows from proposition \[prop: movable is nef\] and [@FL Proposition 3.8].
The following proposition is an analogue of the first statement of [@FL Lemma 3.6].
Let $X'$ and $X$ be projective variety over $k$. Suppose $h:X'\rightarrow X$ is a proper surjective morphism. Then $h_{*}\overline{\operatorname{WMov}}_d(X')\subseteq\overline{\operatorname{WMov}}_d(X)$.
It follows from the definition of the weakly movable cone.
The following theorem is an analogue of [@FL Lemma 3.10].
\[theorem: weakly movable\] Let $X$ be a projective variety over $k$ and let $\alpha \in \overline{\operatorname{WMov}}_d(X)$. Then
1. \[item: eff\] If $\beta\in \overline{\operatorname{Eff}}^1(X)$, then $\beta\cdot \alpha\in \overline{\operatorname{Eff}}_{d-1}(X)$.
2. \[item: bignef\] Let $H$ be a big Cartier divisor. If $H\cdot \alpha=0$, then $\alpha =0$.
3. \[item: nef\]If $\beta\in \operatorname{Nef}^{1}(X)$ then $\beta\cdot \alpha\in \overline{\operatorname{WMov}}_{d-1}(X)$.
For (\[item: eff\]), we may assume $\alpha=\pi_{*}[Z]$, where $\pi:Y\rightarrow X$ is a proper, surjective map and $Z$ a nef subvariety of $Y$. By projection formula, we have $\beta\cdot \pi_{*}[Z]=\pi_{*}(\pi^{*}\beta\cdot [Z])$. We know that $\pi^{*}\beta$ is pseudoeffective. By theorem \[theorem: pseudoeffective\], $\pi^{*}\beta\cdot [Z]\in \overline{\operatorname{Eff}}_{d-1}(Y)$. Since $\pi_{*}\overline{\operatorname{Eff}}_{d-1}(Y)\subseteq \overline{\operatorname{Eff}}_{d-1}(X)$, we have $\beta\cdot \pi_{*}[Z]\in\overline{\operatorname{Eff}}_{d-1}(X)$.
For (\[item: bignef\]), we follow Fulger-Lehmann’s argument [@FL Proof of Lemma 3.10]. We write $H=A+E$, where $A$ is ample and $E$ is effective. By (\[item: eff\]), $A\cdot \alpha ,E\cdot \alpha\in \overline{\operatorname{Eff}}_{d-1}(X)$. In particular, $H\cdot \alpha=0$ implies $A\cdot \alpha=0$ [@FL1 Corollary 3.8], which can only happen when $\alpha=0$ [@FL1 Corollary 3.16].
For (\[item: nef\]), we may again assume $\alpha=\pi_{*}[Z]$, where $\pi:Y\rightarrow X$ is a proper, surjective map and $Z$ a nef subvariety of $Y$. We also assume $d\geq 2$, otherwise the result already follows from (\[item: eff\]). Note that $\pi_{*}\overline{\operatorname{WMov}}_{d-1}(Y)\subseteq\overline{\operatorname{WMov}}_{d-1}(X)$ by the definition of weakly movable cone. It suffices to show that $H\cdot [Z]\in \overline{\operatorname{WMov}}_{d-1}(Y)$, where $H$ is a very ample divisor on $Y$. We may assume that $H\cap Z$ is of dimension $d-1$ and is integral [@Jouanolou Corollaire 6.11]. By corollary \[cor: intersection of ample\], $H\cap Z$ is a nef subvariety in $Y$.
\[prop: weakly movable 1\] Let $X$ be a projective variety of dimension $n$ over $k$. Then $$\overline{\operatorname{WMov}}_{1}(X)=\overline{\operatorname{Mov}}_{1}(X)$$
Let $\pi:Y\rightarrow X$ be a proper, surjective map, $Z\subset Y$ be a nef subvariety of dimension $1$. To show that $\pi_{*}[Z]\in\overline{\operatorname{Mov}}_1(X)$, it suffices to show that $D\cdot \pi_{*}[Z]=\pi^{*}D\cdot [Z]\geq0$ for any pseudoeffective divisor on $X$, since the dual cone of $\overline{\operatorname{Mov}}_1(X)$ is the cone of pseudoeffective divisors [@BDPP]. This follows from theorem \[theorem: pseudoeffective\].
Let us recall Hartshorne’s conjecture A:
Let $X$ be a smooth projective variety, and let $Y$ be a smooth subvariety with ample normal bundle. Then $n[Y]$ moves in a large algebraic family for $n\gg0$.
This was disproved by Fulton and Lazarsfeld. They constructed an ample rank $2$ vector bundle on $\mathbb{P}^2$, such that any multiple of the zero section in the total space of the vector bundle does not move.
In view of proposition \[prop: mov vs wmov\], theorem \[theorem: weakly movable\] and proposition \[prop: weakly movable 1\], it seems reasonable for us to ask the following
Let $X$ be a projective variety of dimension $n$. Do we have $$\overline{\operatorname{WMov}}_d (X)=\overline{\operatorname{Mov}}_d(X),$$ for $1\leq d \leq n-1$?
If the answer is yes, the cycle class of any nef subvariety of $X$ will lie in the movable cone. The key point in the question is that we only consider the cycle classes up to numerical equivalence; the movable cone is also defined to be the closure of the cone generated by movable cycles. This seems to be one of the weakest possible ways of stating the conjecture that relates positivity of the normal bundle of subvarieties and their movability. However, it is possible that the two cones are different in general.
One might want to study the closure of the convex cone generated by the cycle class of nef subvarieties of dimension $d$ (in $\operatorname{N}_d(X)$) instead. We now give an example where it is not of full dimension, when $d=\dim X-1$.
Let $X$ be a normal projective variety over $k$. Let $Y\subset X$ be a nef subscheme of codimension $1$. Then $Y$ is a (nef) Cartier divisor.
Let $\pi:\operatorname{Bl}_Y X\rightarrow X$ be the blowup of $X$ along $Y$, with exceptional divisor $E$. Then $\pi|_E: E\rightarrow Y$ is equidimensional of relative dimension $0$, by proposition \[prop: equi\]. Therefore, $\pi$ is quasi-finite. A proper and quasi-finite morphism is finite, so $\pi$ is finite and birational, with $X$ normal. This implies that $\pi$ is in fact an isomorphism.
Let $X$ be a projective variety of dimension $n$ over $k$. By [@Fulton Example 19.3.3], the natural map $\operatorname{N}^{1}(X)\xrightarrow{\cdot [X]}\operatorname{N}_{n-1}(X)$ is injective. Fulger-Lehmann gave an example [@FL1 Example 2.7] where $\operatorname{N}^{1}(X)\xhookrightarrow{\cdot [X]}\operatorname{N}_{n-1}(X)$ is not surjective. We may assume that $X$ is normal in their example. By the above lemma, the closure of the convex cone generated by the cycle class of nef subschemes of codimension $1$ lies in the subspace $\operatorname{N}^{1}(X)\subsetneq\operatorname{N}_{n-1}(X)$, hence is not full dimensional.
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author:
- 'Hilding R. Neilson'
- 'Fabian R. N. Schneider'
- 'Robert G. Izzard'
- 'Nancy R. Evans'
- Norbert Langer
bibliography:
- 'ceph\_bin.bib'
title: The occurrence of classical Cepheids in binary systems
---
Introduction
============
Classical Cepheids are intermediate-mass, evolved stars which have been observed for centuries ever since Cepheid variability was discovered by John Goodricke in 1784 [@Good1785]. The source of the variability was a long-standing mystery, one suggestion was that Cepheids are eclipsing binary stars. However, the first distance measurements to nearby Cepheids implied that a Cepheid light curve could only be consistent with an eclipsing binary if the orbital separation was less than the stellar radius, negating the binary hypothesis. While disproved, the binary hypothesis persisted until the 1920s [@Jean19xx], even though [@Eddington1918] demonstrated that Cepheid variability is due to radial pulsation.
It is radial pulsation that makes Cepheid variable stars important for stellar physics, extragalactic astronomy and cosmology. [@Leavitt1908] discovered the Cepheid period-luminosity relation (Leavitt Law) from observations of Large Magellanic Cloud Cepheids, which was applied to extragalactic observations to measure the Hubble Constant [@Lemaitre1927; @Hubble1929], thus founding modern cosmology.
Measurements of Cepheid properties such as mass, radius and luminosity constrain the detailed physics of stellar evolution and pulsation. For instance, theories of helium fusion [@Morel2010] and mass loss [@Neilson2012b; @Neilson2012c] have been constrained by measurements of period change, while mass measurements constrain theories of convective core overshooting [@Keller2006; @Neilson2011]. These results have implications for late-stage stellar evolution and supernovae, as well as constraining the main sequence evolution of their progenitors.
While their variability is not a result of binary eclipses, Cepheids with binary companions are also of interest for understanding stellar physics. Observing eclipsing binary Cepheids provides a direct measure of stellar masses that can be compared to predictions from stellar evolution and stellar pulsation models and independently constrains the long-standing mass discrepancy [@Cox1980; @Keller2008]. Similarly, observing the orbital period and mass distribution of binary Cepheids constrains the binary fraction of intermediate-mass main sequence stars and the initial mass function [@Evans2013]. This is of particular importance as [@Mayor1991] and [@Raghavan2010] measured the binary fraction of low-mass stars to be approximately $50\%$ while [@Sana2012] and [@Chini2012] suggested the binary fraction of high-mass stars is $70 - 90\%$. [@Kouwenhoven2005; @Kouwenhoven2007] suggested the binary fraction of intermediate-mass stars in the cluster Scorpius OB2 is about $50\%$, leaving the binary fraction of intermediate-mass stars in the Galaxy an important and open question [@Kouwenhoven2008; @Raghavan2010].
Observations are providing insight into the orbital period and eccentricity distributions of binary Cepheids in the Galaxy. [@Evans2005] presented a list of single-line spectroscopic binaries with measured orbital periods and eccentricities, as well as updated orbital parameters for the Cepheid Polaris [@Evans2008], based on a literature review of all known Galactic Cepheids. That binary sample is complete with the exception of small mass ratios ($< 0.2$) and long-period orbits ($>10~$yr) and is the most complete sample of Galactic Cepheid binarity. [@Szabados2012] argued that the Galactic Cepheids X Pup and XX Sge are also spectroscopic binaries based on 75 years of observations. [@Szabados2013a; @Szabados2013b] analyzed spectroscopic observations to detect binary companions for a sample of Galactic Cepheids in the southern hemisphere. [@LiCausi2012] observed the companion of the Cepheid X Sgr using interferometric observations. Similarly, [@Gallenne2013; @Gallenne2014] discovered, using interferometric observations, that the Cepheids V1334 Cyg and AX Cir also have companions. [@Evans2005] noted that about 35% of Galactic Cepheids have at least one spectroscopic companion, and about 44% of those have more than one companion. There are no known Galactic Cepheids with a companion with orbital period less than one year [@Evans2011].
Observational studies of Cepheid duplicity in the Large Magellanic Cloud are also beginning to yield results. Three eclipsing binaries were discovered in the OGLE-III survey of LMC Cepheids [@Soszynski2008] while [@Szabados2012b] noted there are four known spectroscopic Cepheid binaries. [@Szabados2012b] also attempted to use velocity and $B$- and $V$-band amplitudes to search for new binary Cepheids. From their sample of 43 Cepheids, [@Szabados2012b] detected seven additional binary Cepheids. They suggested that the binary fraction of Cepheids in the LMC may be less than that of the Milky Way, but the result still suffers from significant observational bias because of their method.
The purpose of this work is to explore Cepheid duplicity from the perspective of theoretical binary stellar evolution models and to compare predicted Cepheid binary properties with those observed in the Galaxy and LMC. As such we compute the probable minimum orbital period for Galactic Cepheids from the modeled orbital period distribution. We also predict the number of Cepheids in eclipsing binary systems that would be detectable in the OGLE-III survey to compare with the observed number. This work is a first step in understanding the role of binarity in Cepheid stellar evolution. We explore the distribution of binary systems containing a Classical Cepheid as the primary component from population synthesis calculations. We discuss the synthesis code and describe our assumptions in Section 2. In Section 3, we present our results for Galactic and Large Magellanic Cloud metallicities. We compare these results to the observed binary fraction in Section 4 and discuss the role of tidal evolution and mass transfer on the evolution of stars from the red giant branch to the blue loop. We conclude in Section 5.
Binary population synthesis
===========================
We use the single and binary star population nucleosynthesis code of [@Izzard2004; @Izzard2006; @Izzard2009] which is based on the rapid binary evolution code of [@Hurley2002]. Formulae fitted to detailed single star models with convective core overshooting describe stellar evolution across the whole Hertzsprung-Russell diagram [@Pols1998; @Hurley2000]. Stellar wind mass loss is given by [@Nieuwenhuijzen1990] for stars with luminosities $L \ge 4000~L_{\sun}$ and is also a function of metallicity [@Kudritzki1989]. For giant branch evolution, we apply the [@Kudritzki1978] mass-loss prescription with a wind modification factor $\eta = 0.5$ [@Hurley2000]. Wind mass loss during other evolutionary stages is insignificant for Cepheid stars. Tidal evolution, however, is important because it synchronizes the orbital period and the spins of stars, and circularizes the orbit. For stars with convective envelopes we use the equilibrium tidal model of [@Hut1981], while for stars with radiative envelopes, we employ the dynamical tide models of [@Zahn1975; @Zahn1977] based on the [@Hurley2002] implementation.
We set the mass of the primary star to be $M_1 = 3$ – $15~M_\odot$ and the mass ratio to be $q \equiv M_2/M_1= 0.1$ – $1$, where $M_1$ is the primary mass and $M_2$ is the secondary mass. The initial separation range is $a = 20$ – $10^5~R_\odot$ and the eccentricity varies from zero to unity. We assume a flat distribution for the eccentricity, a uniform distribution for the logarithm of the initial separation and a [@Kroupa1993] initial mass function. Furthermore, we assume a constant star formation rate.
We label a star as a Cepheid when one of the stars evolves onto the Cepheid instability strip as determined by [@Bono2000]. This includes core helium-burning stars, and stars evolving across the Hertzsprung gap. However the latter population is expected to be small because their evolutionary timescale is much shorter than for later crossings of the instability strip. These first crossing Cepheids contribute less than a few percent to the total Cepheid population. We also count binary systems in which the secondary star is a Cepheid. If a Cepheid is the secondary then the primary is initially more massive, hence is more likely to have evolved into a white dwarf star or exploded as a supernova. Such a Cepheid binary system will be rare and difficult to observe because the relic companion will be much dimmer than the Cepheid. However, in rare cases the initially more massive primary star will accrete onto the secondary during main sequence evolution, and the initial primary may not evolve to be a white dwarf or neutron star before the other star becomes a Cepheid. These systems are also counted, but not considered in Sect. 3.
Shortest orbital period Cepheid binary
======================================
The two phenomena that are most important for determining the number of Cepheid binary systems are: 1) the minimum separation at which Roche-lobe overflow starts while evolving along the red giant branch and 2) the initial binary distribution of intermediate-mass stars. The most probable minimum orbital period for a system with a Cepheid primary star will be dictated by the Roche-lobe separation of the binary system, i.e., the radius of the Cepheid when it was a red giant star, while tides play a role also. We use the term ‘most probable minimum orbital period’ because we assume that the Cepheid component of the binary is evolving along its blue loop, which occurs after the red giant stage, i.e., most likely to be observed. However, Cepheid pulsation also occurs when a star crosses the Hertzsprung gap after main sequence evolution, and a binary system at this stage can have a much shorter orbital separation without interacting. We account for this stage later, but as this evolutionary stage has a negligible life time relative to a star evolving on the blue loop, these systems will be rare.
Roche-lobe overflow and tidal interaction both depend on the ratio of the stellar radius, $R$, to the orbital separation, $a$. Tidal circularization and synchronization timescales are proportional to $(R/a)^{-8} $ and $(R/a)^{-6}$, respectively [@Zahn1977; @Hut1981]. The primary undergoes mass transfer when its stellar radius is greater than the Roche-lobe radius [@Paczynski1971; @Eggleton1983]. For $q = 1$, mass transfer starts when the radius is about one-half the orbital separation.
Mass transfer from a red giant star is unstable in our models, hence leads to common envelope evolution even though mass transfer depends on the mass ratio and angular momentum of the binary system. Unstable mass transfer occurs when the mass donor cannot readjust to hydrostatic equilibrium on a dynamical time scale. Once common envelope evolution ends, the primary is left as a helium star which, because it has negligible envelope material, is never a classical Cepheid. But, even if mass transfer is stable, the primary star’s envelope will still be stripped. Therefore, we estimate the minimum orbital period for a Cepheid evolving on the blue loop as the primary star.
Following [@Evans2011], we assume that a typical binary system with a Cepheid primary has a primary mass $M_1 = 5~M_\odot$ and secondary mass $M_2 = 2~M_\odot$. We compute a $5~M_\odot$ stellar evolution model using the [@Yoon2005] stellar evolution code assuming mass loss from [@Kudritzki1989] for hot stars and [@dejager1989] for cool stars, convective core overshooting $\alpha_c = 0.2$ [for details see @Neilson2012b; @Neilson2012c] and the [@Grevesse1998] solar metallicity, $Z = 0.02$. We do not include rotation in the models. [@Anderson2014] and [@Neilson2014b] both suggest that Cepheids cannot evolve from rapidly-rotating main sequence progenitor stars with a equatorial rotational velocity $v_{\rm{rot}} \ge 0.4v_{\rm{crit}}$. The stellar radius at the tip of the red giant branch is $R= 125~R_\odot$.
{width="50.00000%"}{width="50.00000%"}
The Roche-lobe radius of a system with mass ratio $q = 0.4$ is $r_L \approx
0.4a$ [@Eggleton1983] and we assume that $R < r_L$ else the star begins mass transfer. The minimum orbital separation that allows a red giant to evolve to a Cepheid is $a_{\rm{min}} = 312~R_\odot$ or an orbital period $P_{\rm{orb}} > 240~$days or 0.66 years, if we ignore tides.
Tides are an additional constraint on the minimum orbital period for the binary system. We compute the change of orbital separation due to angular momentum transfer from the orbit to the rotation of the $R = 125~R_{\sun}$ red giant star. If tides are important during red giant evolution, then tides will reduce the orbital separation and can lengthen the expected minimum orbit. We assume the red giant is non-rotating and becomes synchronized with the minimum orbital period at which Roche lobe overflow begins, assuming conservation of angular momentum. This is an extreme estimate because if the primary star is already rotating then less angular momentum is transferred by tides. The change of rotational angular momentum of the red giant star is, $$\Delta J \approx I\Delta \omega = I \frac{2\pi}{P_{\rm{RL}} },$$ while the change of orbital angular momentum, for $e = 0$, is $$\Delta J = \left(\frac{GM_1^2M_2^2}{M_1+M_2}\right)^{1/2} (a^{1/2}_{\rm{min}} - a^{1/2}_{\rm{RL}}).$$ In Eq. 1, $I = kMR^2$ is the moment of inertia, $k \approx 0.2$ depends on the density structure of the star and $P_{\rm{RL}}$ is the orbital period at which Roche lobe overflow begins. In Eq. 2, $a_{\rm{RL}}$ is the orbital separation at which the stars begin Roche lobe overflow and $a_{\rm{min}}$ is the minimum initial orbital separation for which tides decrease the orbital separation to $a_{\rm{RL}}$. Combining Eqs. 1 and 2, we find, $$\label{et}
a_{\rm{min}} = \left[a_{\rm{RL}}^{1/2} + I\left(\frac{2\pi}{P_{\rm{RL}}}\right)\left(\frac{M_1+M_2}{GM_1^2M_2^2}\right)^{1/2}\right]^2.$$ Assuming $R = 125~R_{\sun}$, $M_1 = 5~M_{\sun}$, $M_2 = 2~M_{\sun}$, $a_{\rm{RL}}= 312~R_{\sun}$, $P_{\rm{RL}} = 0.66$ years and that $k \sim 0.2$ then the minimum orbital separation, during red giant evolution, is $a_{\rm{min}} = 388~R_{\sun}$ and $P_{\rm{min}} = 333~$days $= 0.91~$years, similar to the minimum observed period for Galactic Cepheids. This minimum period is not an absolute limit, there may exist Cepheid binary systems with orbital periods between 240 and 333 days, but they become increasingly rare for shorter orbital periods. Because of the non-linear nature of Eq. \[et\], the change of orbital separation decreases with increasing initial separation. Therefore, we expect a Gaussian-like period distribution about the characteristic time scale $P = 333~$days based on interactions during the red giant stage of evolution.
Binary systems with initial separations shorter than $388~R_{\sun}$ will interact and undergo mass transfer during the red giant evolution. Systems with longer separations will not interact significantly, making this a reasonable estimate for the most probable minimum separation. In principle, if the star is rotating with non-zero velocity then the minimum separation could be smaller, however, Cepheids are slow rotators, $\le 10~$km s$^{-1}$ [@Bersier1996], so rotation is not likely to play a significant role during Cepheid and red giant evolution. Similarly, mass loss rates are not large enough to significantly widen the binary orbit. It is remarkable that our predicted most probable minimum orbital period agrees so well with the observed distribution of Galactic binary Cepheids and suggests that tidal evolution is important for binary evolution. Of particular interest is the recent observation that X Sgr is a binary system with an orbital separation of $385~R_\odot$ [@LiCausi2012], coincident with the results here. Our analysis is limited to the mass ratio $q = 0.4$ and $e = 0$, different mass ratios and non-zero eccentricities lead to different minimum orbital periods. If the mass ratio is greater than 0.4 then $a_{RL}$ increases, however assuming $M_1$ is still $5~M_\odot$ then decreasing the mass ratio decreases the orbital angular momentum and the ratio $(M_1 + M_2)/(M_1^2M_2^2)$ increases. Because that term is divided by the orbit period at the moment Roche lobe overflow begins, then the changes due to different mass ratios will mostly cancel and we will be left with a similar minimum orbital separation. If, instead, we assume a non-zero eccentricity then $a_{RL}$ is unchanged but the orbital angular momentum changes. Hence, $a_{\rm{min}}$ will increase. Also, as discussed above, there may also exist a small number of close binary Cepheids in which the Cepheid is evolving along the first crossing of the Cepheid instability strip and is not yet a red giant star. We will show that this scenario occurs in about 3 - 5% of binary Cepheids.
Cepheid binary period distribution
==================================
Our estimate for the binary Cepheid with the shortest orbital period depends on the assumed masses of the two stars, yet Classical Cepheids range in mass from about $4~M_\odot$ to $15~M_\odot$. We study the shortest orbital period and number of Cepheid binary systems for a population of stars evolving on the Cepheid instability strip by computing population synthesis models at $Z=0.02$ and $Z=0.008$. This sample includes stars evolving through the blue loop and the first crossing of the Hertzsprung gap.
Galactic Cepheid distribution
-----------------------------
In Fig. \[f1\] we plot the period distribution of binary systems in which the primary, star 1, or the secondary, star 2, resides within the Cepheid instability strip as represented by the dotted histogram. We find that about 8% of these systems have an orbital period shorter than one year and a significant fraction of those systems have an orbital period shorter than $0.1$ years. However, the vast majority of them have undergone mass transfer. These systems are not expected to produce Classical Cepheids because the primary star loses most of its envelope. When we ignore those systems in which the primary loses more than $10\%$ of its initial mass, the orbital period distribution (the solid histogram, Fig. \[f1\]) still contains a few systems ($\approx5\%$) with orbital period shorter than one year and an even smaller number of systems with orbital period shorter than 0.5 year. One constraint also includes mass lost in a wind, but for these stars total mass lost before the Cepheid blue loop is less than $1\%$ of the total mass. Also, about 3% of binary Cepheids have an orbital period shorter than the minimum predicted in Sect. 3, though this number depends weakly on the maximum binary separation assumed in our model. These short orbital period systems correspond to Cepheids evolving along the Hertzsprung gap and those Cepheids that are secondary components with a more evolved companion. The latter scenario occurs when the primary donates material to the secondary, so much so that the mass ratio inverts and the secondary becomes more massive than the primary. This causes the binary orbit to widen and what was the secondary evolves to become a Cepheid.
Comparison to observed binaries
-------------------------------
For further comparison, we plot the probability map of the orbital period and eccentricity for Galactic Cepheids in Fig. \[fig:map\], along with the periods and eccentricities for a sample of Galactic Cepheid binaries from [@Evans2005]. Many of these spectroscopic binaries were detected from ultraviolet observations. All Cepheid binaries in the sample are spectroscopic binaries with the exception of Polaris which is an astrometric binary [@Kamper1996; @Evans2008]. One-half of the Cepheids in the sample have orbital periods between one and two years and most have eccentricities less than $0.5$. There may be an observational bias towards short-orbital period Cepheids because they are more easily identified as spectroscopic binaries.
There is some agreement between our model and the distribution of Galactic Cepheid binaries. The observed number ratio between binary Cepheids with orbital periods $< 2$ yrs and those $2<P_{\rm{orb}}<8$ yrs is 50:50. From the computed models, the ratio is 41:59, consistent with the observed sample of 18 Cepheids. While the orbital period distribution appears to be consistent, there are distinct differences between the observed and predicted eccentricities for binary Cepheids with orbital periods $> 2$ yrs. Observed binaries with $P_{\rm{orb}} >2$ yrs appear to have more circular orbits than our model predicts. Specifically, there appear to be five systems with orbital periods longer than two years and small eccentricity ($<0.25$), otherwise the agreement between our model and observations is remarkable. The difference for those five systems is likely because of the assumed initial eccentricity distribution and resolution of the models but, without a greater number of observed binary Cepheids to which we can compare, it is difficult to quantify the eccentricity discrepancy between the predictions and observations and whether it is significant.
![Comparison of the orbital period and eccentricity distributions from our population synthesis models and properties of observed Galactic Cepheid binary systems [@Evans2005], denoted with dots. The colour scale represents the total probability that a binary of given properties exists assuming constant star formation.[]{data-label="fig:map"}](pvse.eps){width="50.00000%"}
There is a peak in the probability distribution function near an orbital period $P_{\rm{orb}} \approx 1$ yr which is consistent with the observed binary sample [@Evans1995]. This peak occurs for eccentricities ranging from $e = 0$ – $0.6$ for orbital periods ranging from one to three years, also in agreement with the observed sample, confirming our predictions from the previous section. However, it is unclear whether our model agrees with observations of systems with longer orbital periods.
{width="50.00000%"}{width="50.00000%"}
LMC Cepheid period distribution
-------------------------------
We also compare population synthesis models assuming metallicity $Z=0.008$ with the OGLE-III survey of classical Cepheids and plot the predicted period distributions in Fig. \[f2\]. The period distribution of LMC stars evolving along the Cepheid instability strip differs from Galactic stars. The LMC binary systems do not appear to interact as often and fewer stars lose a significant fraction of their initial mass. There are more Cepheid binaries with orbital period shorter than one year relative to all systems including mass transfer. By considering those binary Cepheids with orbital period shorter than $1$ yr, we find a peak in orbital periods at about $0.4$ yr as opposed to between one and two years for Galactic Cepheids. The radius of an LMC star at the tip of the RGB is smaller than for a star with the same mass but with Galactic metallicity suggesting that, for a given separation, $a$, tides are also weaker. Furthermore, stellar evolution models form Cepheid blue loops at lower mass at smaller metallicity [@Pols1998; @Bono2000b], implying that Cepheid binary systems can evolve to have smaller separations. Binary separations must also be shorter to begin Roche-lobe overflow.
[@Soszynski2008] presented OGLE-III survey observations of 1848 fundamental-mode, 1228 first-overtone and 285 other types of Classical Cepheids in $V$ and $I$-band wavelengths for seven years. Three confirmed eclipsing binary systems with a fundamental-mode Cepheid component were observed, with subsequent analysis by [@Piet2011a; @Piet2011b] for two of them. Perhaps the third system, OGLE-LMC-CEP1718, is particularly unusual, being composed of two apparently equal mass Cepheids pulsating in the first overtone, but with different pulsation periods [@Gieren2014]. Four more Cepheids are suspected to be in eclipsing binary systems. We compare to the observed LMC Cepheid eclipsing binary fraction of $3/3361 = 0.089\%$ to $7/3361 = 0.2\%$. We limit our analysis to the orbital-period range $< 3$ years, because only these binary systems would show multiple ($> 1$) eclipses in the seven-year OGLE-III survey.
[@Evans2013] found that 55% of Cepheid binaries with q $>$ 0.4 are spectroscopic binaries. Because the observed percentage of spectroscopic binaries over all mass ratios is about 35%, then we suggest that a total binary fraction of Cepheids is $35\%/0.55 = 64\%$. The spectroscopic binary fraction is compiled from a broad sample of Galactic Cepheids and may suffer some biases, but is the most complete sample available. [@Szabados2012b] found a smaller binary fraction for LMC Cepheids but admit an observational bias. As such, we assume that Cepheids in the LMC have a binary fraction between $50\%$ and $64\%$. Therefore, of the 3361 fundamental-mode Cepheids observed in the OGLE-III survey, between 1680 and 2185 Cepheids have companions. The fraction of LMC Cepheid binaries with orbital periods shorter than $3$ yr is about $ 27\%$ in our model, suggesting 454 - 590 Cepheids observed in the LMC have companions with orbital period less than $3$ yr.
To convert the expected number of binary Cepheids to the number of eclipsing binary Cepheids, we estimate the number of systems with an inclination that gives eclipses. The two LMC binary Cepheids that have been analyzed have orbital inclinations $i = 87^{\circ}$ and $90^{\circ}$. If we consider a typical binary system with a $5~M_\odot$ Cepheid and a $2~M_\odot$ companion and orbital period of $0.4$ yr and if we assume the limiting inclination to produce eclipses is $i=87^{\circ}$ then same binary with a $3$ yr orbit has a limiting inclination of $i = 89.2 ^{\circ}$.
An inclination range between $87^{\circ}$ and $89.2^{\circ}$ suggests an eclipsing binary fraction of $0.9\%$ and $3.3\%$, i.e. between $4$ and $20$ Cepheids observed in the OGLE-III survey should be in eclipsing binary systems. This is consistent with the number presented by [@Soszynski2008] of three to seven Cepheid eclipsing binaries.
Discussion
==========
We compute population synthesis models of Cepheid binary systems for both Galactic and LMC metallicities. Our synthesis models of Galactic Cepheids predict a minimum orbital period of about 300 days (0.82 years), consistent with $P = 381$ days for $Z$ Lac [@Evans1995]. The shortest orbital period for a binary Cepheid evolving along the blue loop depends, primarily, on the Roche lobe separation and the efficiency of tides when the Cepheid was evolving previously as a red giant when its radius was greatest. We also find agreement between our predicted number of LMC eclipsing binary Cepheids and those discovered in the OGLE-III survey.
We predict a period distribution that agrees with observations, but we argue that a significant fraction of binary systems with stars evolving on the instability strip do not form Cepheids because of mass transfer from the star that would be a Cepheid. This suggests that the binary fraction of Cepheids is smaller than the binary fraction of main sequence B-type stars that are Cepheid progenitors. [@Evans2005] suggest the spectroscopic binary fraction of Cepheids is 35% while other works suggest the spectroscopic binary fraction of main sequence B-type stars range from $30\%$ to $40\%$ for Galactic stars [@Chini2012] and $50\%$ for the cluster Sco OB2 [@Kouwenhoven2005; @Kouwenhoven2008]. It should be noted that [@Chini2012] considered only binaries with mass ratios $> 0.2$. Our results suggest a binary fraction for Cepheids should be much smaller than that for main sequence B-type stars. If we assume that 35% of Galactic Cepheids are in spectroscopic binary systems, we infer from our models that the spectroscopic binary fraction of their intermediate-mass main-sequence progenitors is about 40 – 45%. This is based on computing the total probability fractions of Cepheids relative to all systems shown in Fig. \[f1\]. Furthermore, this suggests a total binary fraction of intermediate-mass main sequence stars to be about 73 – 82%, consistent with [@Sana2012] and [@Chini2012]. We note that [@Evans2013] estimate a total Cepheid binary fraction to be about 64%, consistent with our results.
These results, though, depend on our understanding of the structure of the Cepheid blue loop. For instance, the minimum orbital period of about one year depends on the assumed masses and radii of the stars. If smaller-mass stars evolve along the Cepheid blue loop, then the minimum separation of stars in a binary system can be shorter. The observed period distribution of binary Cepheids potentially provides insight into the structure of blue loops even though blue loop structures are sensitive to convective core overshooting [@Bono2000], metallicity [@Keller2008], mass loss [@Neilson2012a; @Neilson2012b] and other model physics. While promising, we require measurements of many more binary Cepheids to draw conclusions regarding the blue loop.
While our results are consistent with the number of LMC eclipsing binary Cepheids, it is difficult to directly predict the evolutionary path of OGLE-LMC-CEP1812 [@Piet2011b]. The system consists of a Cepheid primary and a red giant companion in a 552 day orbit but single star models suggest that the companion must be 100 Myr older than the Cepheid [@Piet2011b]. The authors suggest that the binary system formed by a stellar capture. Another possibility is the system evolved from a hierarchical triple system. This particular binary system is unexplained by our binary evolution models. The binary system OGLE-LMC-CEP1718 with two first-overtone Cepheids of equal mass, but different pulsation periods, also appears strange and may require some small amount of mass transfer to explain the systems properties [@Gieren2014]. However, the system OGLE-LMC-CEP0227 [@Piet2011a] is robustly modelled [@Cassisi2011; @Neilson2012a; @Prada2012], suggesting that not all of the LMC eclipsing binary systems are strange. Therefore, we need to consider alternative stellar evolution scenarios to explain the existence of the system OGLE-LMC-CEP1812 that will be discussed in a forthcoming article.
In summary, we find that the combination tides and Roche-lobe overflow prevent the formation of short-orbital period binary systems containing a Cepheid and a companion that would be detected in various surveys such as [@Evans2005; @Szabados2012]. However, a small fraction of binary systems may interact and the primary accretes mass onto the secondary star. The primary loses its envelope and will evolve to become white dwarf stage without ever being a Cepheid, while the mass gainer will be rejuvenated and eventually evolve as Cepheid with a white dwarf companion. that type of binary system is difficult to detect, even with precision measurements of Cepheid light curves [@Derekas2012; @Neilson2014]. We will explore further in future work.
HRN and RGI thank the Alexander von Humboldt Foundation and HRN thanks the National Science Foundation (AST-0807664) for funding. FRNS acknowledges funding by BCGS (DFG). NRE acknowledges funding from the Chandra X-ray Center NASA Contract NAS8-03060. We also thank the referee for helpful comments that have improved this work.
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SLAC-PUB-8408\
June 2000\
[**FIRST DIRECT MEASUREMENT OF THE PARITY-VIOLATING COUPLING OF THE Z$^{0}$ TO THE $s$-QUARK$^*$**]{}
[**The SLD Collaboration$^{**}$**]{}
Stanford Linear Accelerator Center\
Stanford University, Stanford, CA 94309\
We have made the first direct measurement of the parity-violating coupling of the $Z^0$ boson to the strange quark, $A_s$, using $\sim$550,000 $e^+e^-\!\rightarrow \!Z^0\!\!\rightarrow$hadrons events produced with a polarized electron beam and recorded by the SLD experiment. $Z^0\!\!\rightarrow \!s\bar{s}$ events were tagged by the absence of $B$ or $D$ hadrons and the presence in each hemisphere of a high-momentum $K^\pm$ or $K^0_s$. From the polar angle distributions of the strangeness-signed thrust axis, we obtained $A_s \!=\!0.895\!\pm\!0.066 (stat.)\!\pm\!0.062 (syst.)$. The analyzing power and $u\bar{u}\!+\!d\bar{d}$ background were constrained using the data.
[*Submitted to Physical Review Letters.*]{}
$^*$Work supported in part by Department of Energy contract DE-AC03-76SF00515.
The extent of parity violation in the electroweak coupling of the $Z^0$ boson to an elementary fermion $f$ can be specified by the parameter $A_f\!=\!2v_f a_f/(v_f^2\!+\!a_f^2)$, where $v_f$ ($a_f$) is the vector (axial-vector) $Zf\bar{f}$ coupling. In the Standard Model (SM), universal couplings are expected for the leptons ($A_e\!=\!A_\mu\!=\!A_\tau$), the down-type quarks ($A_d\!=\!A_s\!=\!A_b$) and the up-type quarks ($A_u\!=\!A_c\!=\!A_t$). Precise measurements of the $A_f$ allow stringent tests of the SM and sensitivity through radiative corrections to e.g.: the top quark and Higgs boson masses ($A_{e,\mu,\tau}$); new physics that affects primarily the right-handed couplings ($A_{d,s,b}$); and new physics that couples more strongly to heavier quarks (deviations from universality).
All except $A_t$ can be measured in $e^+e^-$ annihilations at the $Z^0$ resonance via forward-backward production asymmetries in $\theta_f$, the polar angle of the outgoing $f$ with respect to the incoming $e^-$ beam. At the SLC, the $e^-$ beam has longitudinal polarization $P_e$, the $e^+$ beam is unpolarized, and the Born-level differential cross-section for the process $e^+e^-\!\rightarrow\!Z^0\!\!\rightarrow\!f\bar{f}$ is: $$d\sigma_f / dx \propto (1\!-\!A_e P_e)(1\!+\!x^2) + 2A_f(A_e\!-\!P_e)x ,$$ where the last term is antisymmetric in $x\!=\!\cos\theta_f$. Using both left- ($P_e\!<\!0$) and right-polarized ($P_e\!>\!0$) beams of magnitude $|P_e|$, one can measure both the initial- ($A_e$) and final-state ($A_f$) couplings directly [@epol; @sldab]; for $P_e\!=\!0$ one can measure only their product, or $A_{FB}^f \equiv 3 A_e A_f/4$.
The most precisely measured coupling is $A_e$, with a relative error of 1.3% [@epol; @ewlepslc], and lepton universality is verified at the 8% level [@ewlepslc]. In the quark sector, several measurements of $A_b$ and $A_c$ that use properties of the leading $B$ and $D$ hadrons can be combined to yield precisions of 2.0% and 4.4%, respectively [@ewlepslc]. However there are few measurements of $A_u$, $A_d$ or $A_s$ [@opal; @delphi] because the leading particles in $u$, $d$ and $s$ jets are more difficult to identify experimentally; they have relatively low energy, are not unique to events of a particular flavor, and nonleading particles of the same species are produced in hadronic jets of all flavors. Furthermore, these aspects of jet fragmentation are not well measured, and previous indirect measurements either relied on imprecise constraints from their data (OPAL: $A^u_{FB}\!=\!0.040\!\pm\!0.073$; $A^{ds}_{FB}\!=\!0.068\!\pm\!0.037$ [@opal]) or are model-dependent (DELPHI: $A^s_{FB}\!=\!0.101\!\pm\!0.012$ [@delphi]).
In this Letter we present the first direct measurement of $A_s$. We used high-momentum $K^\pm$ and $K^0_s$ to tag $Z^0\!\!\rightarrow\!s\bar{s}$ events, and the $K^\pm$ charge to separate $s$ jets from $\bar{s}$ jets. The heavy flavor ($c\bar{c}\!+\!b\bar{b}$) event background was suppressed by identifying $B$ and $D$ decay vertices. The $u\bar{u}\!+\!d\bar{d}$ background was suppressed and the $s$-$\bar{s}$ separation enhanced by requiring an $s$/$\bar{s}$-tag in each event hemisphere, reducing any model dependence. The remaining $u\bar{u}\!+\!d\bar{d}$ background and the $s$-$\bar{s}$ separation were constrained using related observables in the data.
We used the sample of approximately 550,000 hadronic $Z^0$ decays recorded by the SLD [@sld] experiment at the SLAC Linear Collider, with $\left< |P_e| \right>\!=\!0.735\pm 0.005$ [@epol], from 1993–1998. Charged tracks were measured in the Central Drift Chamber (CDC) [@cdc] and the original (upgraded) Vertex Detector (VXD) [@vxd] in 26.5% (73.5%) of the data; the resolution on the impact parameter $d$ in the plane perpendicular to the beam direction, including the uncertainty on the interaction point, was $\sigma_d =$11$\oplus$70/$(p \sin^{3/2}\theta)$ $\mu$m (8$\oplus$29/$(p \sin^{3/2}\theta)$ $\mu$m), where $p$ is the track momentum in GeV/c and $\theta$ its polar angle with respect to the beamline. Tracks were identified as $\pi^\pm$, $K^\pm$ or p/$\bar{\rm p}$ in the Cherenkov Ring Imaging Detector (CRID) [@crid], which allowed the identification with high efficiency and purity of $\pi^\pm$ with $0.3\!<\!p\!<\!35$ GeV/c, $K^\pm$ with $0.75\!<\!p\!<\!6$ GeV/c or $9\!<\!p\!<\!35$ GeV/c, and p/$\bar{\rm p}$ with $0.75\!<\!p\!<\!6$ GeV/c or $10\!<\!p\!<\!46$ GeV/c [@pprod]. The event thrust axis [@thrust] was calculated using energy clusters measured in the Liquid Argon Calorimeter [@lac].
After selecting hadronic $Z^0$ decays [@homer], we removed $c\bar{c}$ and $b\bar{b}$ events by requiring no more than one well-measured [@homer] track with $d/\sigma_d\!>\!2.5$ in the event. The efficiency for selecting light-flavor events with $|\cos\theta_{thrust}|\!<$0.71 and the VXD, CDC and CRID operational was estimated to be over 95%; the selected sample comprised 205,708 events, with an estimated contribution of 14.2% (3.4%) from $c\bar{c}$ ($b\bar{b}$) events. Such performance parameters were estimated from a detailed Monte Carlo (MC) simulation [@homer; @sldrb] of the SLD based on the JETSET 7.4 [@jetset] event generator, tuned to reproduce many measured properties of hadronic $Z^0$ decays, including the momentum-dependent production of $K^\pm$, $K^0$, $K^*$ and $\phi$ mesons.
Each selected event was divided into two hemispheres by the plane perpendicular to the thrust axis, and in each hemisphere we searched for high-momentum strange particles $K^\pm$, $K^0_s$ and $\Lambda^0/\bar{\Lambda}^0$. Candidate $K^\pm$ tracks were required to have $p\!>\!9$ GeV/c, $d\!<\!1$ mm, to extrapolate through an active region of the CRID gaseous radiator system, and to have a log-likelihood [@pprod] for the $K^\pm$ hypothesis ${\cal L}_K$ that exceeded both ${\cal L}_\pi$ and ${\cal L}_{\rm p}$ by at least 3 units. For $p\!>\!9$ GeV/c, the estimated $K^\pm$ selection efficiency (purity) was 48% (91.5%).
Candidate $K_s^0\!\rightarrow\! \pi^+\pi^-$ and $\Lambda^0$/$\bar\Lambda^0\!\rightarrow$p$\pi^-$/$\bar{\rm p}\pi^+$ decays were reconstructed as described in [@pprod; @hermann] from tracks not identified as $K^\pm$. We required $p\!>\!5$ GeV/c and a reconstructed invariant mass $m_{\pi\pi}$ or $m_{{\rm p}\pi}$ within two standard deviations of the $K_s^0$ or $\Lambda^0$ mass. If CRID information was available for the p/$\bar{\rm p}$ track in a $\Lambda^0$/$\bar\Lambda^0$ candidate, we required ${\cal L}_{\rm p} > {\cal L}_{\pi}$; otherwise we required that the $\Lambda^0$/$\bar\Lambda^0$ not be a $K^0_s$ candidate and that the flight distance exceed 10 times its uncertainty. The estimated $\Lambda^0$/$\bar\Lambda^0$ reconstruction efficiency (purity) was 12% (90.7%). These $\Lambda^0$/$\bar\Lambda^0$ were removed from the $K^0_s$ sample, for an estimated $K^0_s$ efficiency (purity) of 24% (90.7%).
We considered only the selected strange particle with the highest momentum in each hemisphere (5.5% of those tagged contained more than one), and tagged the event as $s\bar{s}$ if one hemisphere contained a $K^\pm$ and the other contained either an oppositely charged $K^\pm$ or a $K^0_s$. The $\Lambda^0$/$\bar{\Lambda}^0$ tags provided a useful veto in multiply tagged hemispheres and important checks of the simulation; however their inclusion did not improve the total error on $A_s$. The thrust axis, signed so as to point into the hemisphere containing (opposite) the $K^-$ ($K^+$), was used as an estimate of the initial $s$-quark direction. Table \[dtags\] shows the number of events tagged in each mode, along with the predictions of the simulation, which are consistent. Also shown are the simulated $s\bar{s}$ event purities and analyzing powers $a_s\!\equiv\! (N_r\!-\!N_w)/(N_r\!+\!N_w)$, where $N_r$ ($N_w$) is the number of events in which the signed thrust axis pointed into the true $s$ ($\bar{s}$) hemisphere.
----------------------------------------- ----------- ------------ ------------ -----------
\# Events MC $s \bar s$ Analyzing
\[-0.1cm\] Mode in Data Prediction Purity Power
$K^+ K^-$ 1290 1312 0.73 0.95
$K^\pm K_s^0$ 1580 1617 0.60 0.70
$K^+ \Lambda^0, K^- \bar \Lambda^0$ 219 213 0.66 0.89
$\Lambda^0 \bar \Lambda^0$ 17 14 0.57 0.70
$\Lambda^0 K_s^0, \bar \Lambda^0 K_s^0$ 193 194 0.50 0.32
----------------------------------------- ----------- ------------ ------------ -----------
: \[dtags\] Summary of the selected event sample for the two tagging modes and the three cross-check modes.
Figure \[asymm\] shows the distributions of the measured $s$-quark polar angle $\theta_s$ for the $K^+K^-$ and $K^\pm K^0_s$ modes. In each case, production asymmetries of opposite sign and different magnitude for left- and right-polarized $e^-$ beams are visible. The content of the largest $|\cos\theta_s|$ bins is reduced by the detector acceptance. The estimated backgrounds (discussed below) are indicated: those from $c\bar{c}\!+\!b\bar{b}$ events exhibit asymmetries of the same sign and similar magnitude to those of the signal, so the measured $A_s$ is largely insensitive to them; those from $u\bar{u}\!+\!d\bar{d}$ events exhibit asymmetries of opposite sign, and $A_s$ is more sensitive to the associated uncertainties.
A simultaneous maximum likelihood fit to these four distributions was performed using the function: $$L = \prod_{k = 1}^{N_{data}} \sum_{q=udscb}^{} N_q
\{ (1\!-\!A_e P_e)(1\!+\!x^2_k) +
2 (A_e\!-\!P_e) (1\!+\!\delta) a_q A_q x_k\}. \label{likfcn}$$ Here, the number of tagged $q\bar{q}$ events $N_q\!=\!N_{events}R_q\epsilon_q$, $R_q\!=\!\Gamma(Z^0\!\!\rightarrow \!q\bar{q})/\Gamma(Z^0\!\!\rightarrow$hadrons$)$, $\epsilon_q$ is the tagging efficiency, $a_q$ is the analyzing power for tagging the $q$ direction, and the correction for hard gluon radiation $\delta\!=\!-0.013$ was derived [@shinya] as in [@sldab]. The values of the $\epsilon_q$ and $a_q$ depend on the tagging mode. World average values [@ewlepslc] of $A_e$, $A_c$, $A_b$, $R_c$ and $R_b$ were used, along with SM values of $A_u$, $A_d$, $R_u$, $R_d$ and $R_s$. Simulated values of $\epsilon_c$, $\epsilon_b$, $a_c$ and $a_b$ were used, as they depend primarily on measured quantities with well defined uncertainties.
For the light flavors, the relevant parameter values were derived where possible from the data. The number of events $N_u\!+\!N_d\!+\!N_s$ was determined by subtracting the simulated $N_c$ and $N_b$ from the total observed. The values of $a_s$ and the ratio $(N_u\!+\!N_d)/N_s$ were constrained (see below) using the data; since the simulation was consistent with the data, the simulated values of $a_s$ were used and the simulated $\epsilon_u$, $\epsilon_d$ and $\epsilon_s$ were scaled by a common factor to give the measured $N_u\!+\!N_d\!+\!N_s$. The average $a_{ud}\equiv (N_u a_u\!+\!N_d a_d)/(N_u\!+\!N_d)$ can also be constrained from the data; however our constraint is less precise than the range $-a_s\!<\!a_u,a_d\!<\!0$, obtained by noting that a $u$ ($d$) jet can produce a leading $K^+$ ($K^{*0}\!\rightarrow\!K^+\pi^-$), giving $a_u$($a_d$)$<$0, but with an associated $K^-$ or $\bar{K}^0$, and a $K^-$ can be selected with reduced probability, giving $|a_u|$($|a_d|$)$<\!|a_s|$. We scaled the simulated $a_u$ and $a_d$ by a common factor such that $a_{ud}\!=\!-a_s/2$ for each mode. The fit yielded $A_s\!=\!0.895\!\pm\!0.066$ (stat.). Histograms corresponding to this value are shown in fig. \[asymm\] and are consistent with the data; the binned $\chi^2$ is 42 for 48 bins.
16.1cm
We considered several sources of systematic uncertainty, summarized in table \[systerr\]. The values of $R_c$, $R_b$, $A_c$ and $A_b$ were varied by the uncertainties on their world averages [@ewlepslc]. A large number of quantities in the simulation of heavy flavor events and detector performance were varied as in [@sldrb] with negligible effect on the measured value of $A_s$. The yield and analyzing power of true $K^\pm$ from $D$ ($B$) decays have been derived from SLD data in the context of a measurement [@twright] of $A_c$ ($A_b$), and our simulation reproduces them within the measurement errors. We applied corresponding relative variations of $\pm$9% to $\epsilon_c$, $\pm$3.3% to $\epsilon_b$, $\pm$5% (15%) to $a_c$ and $\pm$3.6% (4.4%) to $a_b$ for the $K^+K^-$ ($K^\pm K^0_s$) mode. The sum in quadrature of the uncertainties due to heavy flavor background is listed in Table \[systerr\]; the largest contribution is from $\delta a_c$.
The key to this measurement is the understanding of the light-flavor parameters, for which there are few experimental constraints, and these gave rise to the dominant systematic uncertainties [@hermann]. In order to minimize model dependence, we used our data to constrain the largest uncertainties in these parameters within the context of our simulation, which reproduces existing measurements of relevant quantities such as leading particle production and strange-antistrange correlations [@hermann].
To constrain the analyzing power $a_s$, we note that to mistag an $s$ jet as an $\bar{s}$ jet we must either identify a true $K^+$ or misidentify a $\pi^+$ or p as a $K^+$. A true high-momentum $K^+$ in an $s$ jet must be produced in association with an antistrange particle, yielding a jet with three high-momentum particles of nonzero strangeness. In our data we found 61 hemispheres containing three selected $K^{\pm}$ and/or $K_s^0$; the MC prediction of 67 is consistent. We quantified this as a constraint on $a_s$ by subtracting the simulated ($c\!+\!\bar{c}\!+\!b\!+\!\bar{b}$) contribution of 9.3, scaling by the simulated ratio $(s\!+\!\bar{s})/(u\!+\!\bar{u}\!+\!d\!+\!\bar{d}\!+\!s\!+\!\bar{s})=0.74$, and comparing with the MC prediction for ($s\!+\!\bar{s})$. Propagating the data and MC statistical errors yielded an 18.5% relative constraint on the wrong-sign fraction, $w_s=(1\!-\!a_s)/2$, in $s$/$\bar{s}$ hemispheres. This constraint is not entirely model-independent but any further uncertainties are small compared with 18.5%. Assuming equal production of charged and neutral kaons, this procedure delivers a calibration of $w_s$ for both tagging modes, which we varied simultaneously by $\pm$18.5% relative. To account for misidentified particles we varied the production of $>$9 GeV/c p and $\pi^+$ in $s$ jets by $\pm$100%, and varied the misidentification probability by its measured relative uncertainty of $\pm$25% [@pprod]. The sum in quadrature of these three effects is shown in table \[systerr\] and is dominated by the 3-kaon calibration.
The relative $u\bar{u}\!+\!d\bar{d}$ background $B_{ud}=(N_u\!+\!N_d)/N_s$ was constrained in a similar manner, by exploiting the fact that an even number of particles with nonzero strangeness must be produced in a $u$ or $d$ jet. The three quantities, the number $N_1\!=\!1262$ of hemispheres in the data containing an identified $K^+K^-$ pair, $N_2\!=\!983$ hemispheres containing a $K^\pm K^0_s$ pair, and $N_3\!=\!503$ events with an identified $K^\pm$ of the same charge in both hemispheres, constrain $B_{ud}$ in complementary ways: $N_1$ and $N_2$ are primarily sensitive to $K\bar{K}$ production in $u$/$d$ jets; $(N_1\!-\!N_2)$ to $\phi$ production in $s$ jets; and $N_3$ to these and also the production and misidentification of $\pi^\pm$ and p/$\bar{\rm p}$. Furthermore, all are sensitive to deviations from the assumed values of $R_u$, $R_d$ and $R_s$. The MC predictions of $N_1\!=\!1218$, $N_2\!=\!1002$ and $N_3\!=\!559$ are consistent, and relative constraints on $B_{ud}$ of 4.6%, 5.1% and 8.1%, respectively, were derived. Since $N_3$ constrains the sum of all contributions, we varied $B_{ud}$ by $\pm$8.1%.
These quantities are also sensitive to $a_u$ and $a_d$, however our limited event sample did not allow us to obtain a useful constraint. We therefore took $-a_s\!<\!a_u,a_d\!<\!0$ as hard limits and scaled $a_u$ and $a_d$ simultaneously such that $a_{ud}\!=\!-a_s/2\!\pm\!a_s/\sqrt{12}$. This yielded the dominant systematic error on $A_s$ and is a quantity that must be understood experimentally before a more precise measurement can be made. Since the product $A_q a_q$ appears in Eqn. \[likfcn\], this is equivalent to varying $A_u$ and $A_d$ down to half of their SM values and up to well over unity; we considered no additional variation of $A_u$ or $A_d$. The uncertainties listed in table \[systerr\] were added in quadrature to yield a total relative systematic error of $\pm$0.069.
------------------------------------ --------------- ---------- -------------------- -------
\[-0.1cm\] Source $\delta A_s / A_s$
0.014
$-0.013$ $0.006$ 0.006
$\left< |P_e| \right>$ $0.735$ $0.005$ 0.006
MC statistics 0.014
$a_s$ $K^+K^-$ $0.949$ $0.012$
\[-0.1cm\] $K^\pm K^0_s$ $0.701$ $0.060$ 0.032
$(N_u+N_d)/N_s$ $K^+K^-$ $0.190$ $0.015$
\[-0.1cm\] (incl. $(R_u+R_d)/R_s$) $K^\pm K^0_s$ $0.316$ $0.026$ 0.021
$a_u, a_d$ $-a_s/2$ $a_s/\sqrt{12}$ 0.054
\[-0.1cm\] $A_u, A_d$
Total 0.069
------------------------------------ --------------- ---------- -------------------- -------
: \[systerr\] Summary of the systematic uncertainties.
Several systematic checks were also performed. Ad hoc corrections [@hermann] to the simulation of the kaon momentum distributions and identification efficiencies, and the charged track reconstruction efficiency and impact parameter resolution were removed and the analysis repeated; changes in the measured value of $A_s$ were much smaller than the systematic error. We fitted each tagging mode separately, including those involving $\Lambda^0$ tags, with consistent results. We repeated the analysis using all $K^\pm$, and all $\Lambda^0/\bar{\Lambda}^0$, hemispheres with no tag required in the opposite hemisphere; results were consistent. This $K^\pm$ analysis is similar to that in [@delphi]; it has a relative statistical precision of 0.03, but of 0.18 systematic.
In conclusion, we have made the first direct measurement of the parity-violating coupling of the $Z^0$ boson to the strange quark, $$A_s \!=\! 0.895 \!\pm\! 0.066 (stat.) \pm 0.062 (syst.),$$ using high-momentum identified $K^\pm$ and $K^0_s$ to tag $Z^0\!\!\rightarrow\!s\bar{s}$ decays and determine the $s$-quark direction. Our high $K^\pm$ identification efficiency allowed the use of a relatively high-purity, double-tagged event sample, and the extraction from the data of constraints on the analyzing power of the method and the $u\bar{u}\!+\!d\bar{d}$ background, using events with same-charge double tags and jets with two or three identified kaons. This result is consistent with the Standard Model expectation, $A_s\!=\!0.935$, and with less precise, previous measurements of $A_{FB}^s$ [@opal; @delphi]. It is also consistent with a recent world average $b$-quark asymmetry, $A_b\!=\!0.881\!\pm\!0.018$ [@ewlepslc], providing a 10% test of down-type quark universality.
We thank the personnel of the SLAC accelerator department and the technical staffs of our collaborating institutions for their outstanding efforts on our behalf. This work was supported by the U.S. Department of Energy, the UK Particle Physics and Astronomy Research Council (Brunel, Oxford and RAL); the Istituto Nazionale di Fisica Nucleare of Italy (Bologna, Ferrara, Frascati, Pisa, Padova, Perugia); the Japan-US Cooperative Research Project on High Energy Physics (Nagoya, Tohoku); and the Korea Science and Engineering Foundation (Soongsil).
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SLD Collab., K. Abe [*et al.*]{}, Phys. Rev. Lett. [**83**]{} (1999) 3384.
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SLD Collab., K. Abe [*et al.*]{}, SLAC-PUB-8199, SLAC-PUB-8200, unpublished.
$^{**}$List of Authors {#list-of-authors .unnumbered}
======================
=.75
**
(The SLD Collaboration)
=.75 Aomori University, Aomori, 030 Japan, University of Bristol, Bristol, United Kingdom, Brunel University, Uxbridge, Middlesex, UB8 3PH United Kingdom, Boston University, Boston, Massachusetts 02215, University of Colorado, Boulder, Colorado 80309, Colorado State University, Ft. Collins, Colorado 80523, INFN Sezione di Ferrara and Universita di Ferrara, I-44100 Ferrara, Italy, INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy, Johns Hopkins University, Baltimore, Maryland 21218-2686, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, University of Massachusetts, Amherst, Massachusetts 01003, University of Mississippi, University, Mississippi 38677, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russia, Nagoya University, Chikusa-ku, Nagoya, 464 Japan, University of Oregon, Eugene, Oregon 97403, Oxford University, Oxford, OX1 3RH, United Kingdom, INFN Sezione di Perugia and Universita di Perugia, I-06100 Perugia, Italy, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United Kingdom, Rutgers University, Piscataway, New Jersey 08855, Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, Soongsil University, Seoul, Korea 156-743, University of Tennessee, Knoxville, Tennessee 37996, Tohoku University, Sendai, 980 Japan, University of California at Santa Barbara, Santa Barbara, California 93106, University of California at Santa Cruz, Santa Cruz, California 95064, Vanderbilt University, Nashville,Tennessee 37235, University of Washington, Seattle, Washington 98105, University of Wisconsin, Madison,Wisconsin 53706, Yale University, New Haven, Connecticut 06511.
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[***Keywords:*** composite membrane, dynamic load, numerical modeling]{}
Introduction
============
Fabric composites are becoming an essential part of impact shields that protect spacecrafts from micrometeoroids and orbital debris. Present protection systems are based on the original Whipple shield [@whipple] that consists of two rigid (typically metal) layers spaced some distance from each other. This design allows to deal with particles with velocities up to 10-18 kilometres per second. An additional protection is provided by stuffed Whipple shields [@christiansen1995], that include high-strength materials between rigid layers. The International Space Station uses different types of Whipple shields widely [@nasa]. Moreover, the International Space Station starts to use experimental inflatable modules, that use flexible fabric protection system without rigid layers that presented in the original Whipple shield. Computer design and optimization of stuffed Whipple shields and flexible protection system requires modeling of high-strength fabric composites under dynamic loading.
There is a number of works on numerical and experimental studies of fabric materials under a shock load, naming [@kobylkin_selivanov; @walker1999; @walker2001; @porval] as an examples. Most of them consider a single relatively large impactor moving with the velocity 300-700 meters per second. However, for stuffed Whipple shields the load is completely different – the initial particle is destroyed by the outer rigid layer, and the fabric layer is exposed to a cloud of small fragments distributed over time and space and moving with the velocity around 5-9 kilometres per second.
From mathematical model point of view this load profile means that it is not possible to consider the problem to be quasi static, since the speed of the fragments is comparable with the sound speed in the composite membrane. The stress-strain state of the composite layer should be calculated as the dynamic problem, and the elastic waves caused by the impact should be analyzed. Modeling of composite layer under asymmetric load requires also using anisotropic tensor of material elastic parameters and having three degrees of freedom for each point.
This work aims to provide all these features still describing composite layer as 2D object, since this allows to reduce computational time compared with 3D models. The same mathematical model and numerical method can be used later for numerical study of different processes in thin composite layers, such as ultrasound propagation, non-destructive testing procedures, vibrations.
This work is based on the models for thin fibers and membranes from [@rakhmatulin]. This work uses the same approach as [@rakhmatulin] to describe thin flexible structures, after that we do not obtain analytical solutions for particular cases, but solve the equations numerically and study a convergence rate of the numerical scheme for an arbitrary membrane and load.
Mathematical model
==================
Notation
--------
All the computations below are performed in Cartesian coordinates with $OX$, $OY$ axes lying in the plane of the undeformed membrane and $OZ$ axis is orthogonal to that plane. Vector $(x_0, y_0, z_0)^T$ denotes the initial coordinates of the point of the specimen in the undeformed state, $(x, y, z)^T$ coordinates of the same point at the given moment. Let $$\mathbf{u} = \begin{bmatrix}
u \\
v \\
w
\end{bmatrix}
= \begin{bmatrix}
x - x_0 \\
y - y_0 \\
z - z_0
\end{bmatrix}$$ be the displacement vector
Assumptions
-----------
The membrane is considered to be an object with a characteristic size in one dimension ($z$) several orders of magnitude less than in other two. Due to this fact, only the displacement of the midsurface of the membrane in considered [@liu2013]. Based on this, we suggest that difference in displacement in $z$-direction can be neglected: $$\label{no_z}
\mathbf{u}(x, y, z) = \mathbf{u}(x, y)$$ Thus, the membrane effectively is a 2D object in 3D space.
Only small strains are considered. Cauchy’s strain tensor in the following form is used: $$\varepsilon_{ij} = \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$$ We suppose that the materials are subject to Hooke’s law in deformation rates considered: $$\sigma_{ij} = C_{ijkl}\varepsilon_{kl}$$ The assumptions of small strains and linear elasticity up to destruction are generally valid for rigid composites during high speed interactions [@beklemysheva2016]. Other materials may show different behaviour, in this case they will not be covered by the model presented in this work.
Stress-strain and strain-displacement relations
-----------------------------------------------
We use the following vector notation for displacements: $$\label{eq:deformation}
\varepsilon = \begin{bmatrix}
\frac{\partial u}{\partial x} &
\frac{\partial v}{\partial y} &
\frac{\partial w}{\partial z} &
\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} &
\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} &
\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}
\end{bmatrix}^T = \mathbf{Su}$$ with linear differential operator $\mathbf{S}$ defined as $$\label{eq:strain}
\mathbf{S} =\begin{bmatrix}
\frac{\partial}{\partial x} & 0 & 0 \\
0 & \frac{\partial}{\partial y} & 0 \\
0 & 0 & \frac{\partial}{\partial z} \\
\frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \\
0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y} \\
\frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x}
\end{bmatrix}$$ The Hooke’s law then takes the form: $$\label{eq:stress}
\mathbf{\sigma} = \begin{bmatrix}
\sigma_{xx} &
\sigma_{yy} &
\sigma_{zz} &
\tau_{xy} &
\tau_{yz} &
\tau_{xz}
\end{bmatrix}^T = \mathbf{D\varepsilon}$$ with $D$ being the compliance matrix. In the most general case $\mathbf{D}$ is symmetric and depends on $21$ independent elastic moduli.
Equations of motion
-------------------
We derive the equations of motion using virtual work principle. Let us first consider the equilibrium conditions for the unit volume under external loading in the static case. Let $\delta u, \delta \varepsilon$ be variations of corresponding values. $\mathbf{\overline{b}}$ denotes the distributed external force per unit volume. Then the virtual work per unit volume is given by: $$\label{eq:virtual_work}
\delta w = \delta\mathbf{\varepsilon^T \sigma} - \delta\mathbf{u^T \bar{b}}$$ With $u$ sufficiently smooth $\delta\varepsilon = S\delta\mathbf{u}$. For transition to dynamic case, according to d’Alembert’s principle, we shall add distributed inertia force: $$\bar{\mathbf{b}} = \mathbf{b} -\rho\ddot{\mathbf{u}},$$ where $\mathbf{b}$ is distributed load, $-\rho\ddot{\mathbf{u}}$ — inertia force. Taking this into account along with , , we get the equation $$\delta\mathbf{u^T\left(S^TDSu + \rho\ddot{u} - b\right)} = 0,$$ that must be satisfied at any variation $\delta\mathbf{u}$. Thus the term in brackets must be equal to zero. We finally get the equation of motion: $$\label{eq:motion}
\mathbf{S^TDSu + \rho\ddot{u} - b} = 0$$ The further expansion of this equation appears encumbering, in particular when $\mathbf{D}$ is given by more than two independent parameters, as for isotropic material. On the other hand, precisely those cases are of interest for modelling composite materials. Thus, in the following section, we shall derive the governing equations for the computational element.
Numerical method
================
Finite Element Method (FEM) was chosen for the task. One of the advantages of this approach is the possibility for treating complex geometries, laying down the basis for future study of the movement of perforated constructions or screens with holes created by the impacts. Formulation of the computational algorithm follows the general methodology described in [@zie00]. This paper is different from the traditional approach in the fact that material points are parametrized with two coordinates $(x_0, y_0)$, but has three degrees of freedom $(u, v, w)$. Traditionally, for 2D problems only formulations with either one (normal, $w$) or two (in-plane, $(u, v)$) degrees of freedom are studied. This is because for the isotropic material due to particular structure of matrix $\mathbf{D}$, the system of equations of motion is split into two systems: wave equation for $w$ and Lamé equations for in-plane motion, which can be solved independently. This, though, is not true for composite materials, so the need for simultaneous solution of equation for all three degrees of freedom arises. Compared with direct modelling with 3D FEM the proposed methods saves computational time, as we don’t need to mesh throgh thickness direction $z$.
Domain decomposition
--------------------
The computational domain corresponds with the physical membrane. Unstructured grid of triangular elements is used. We suppose thickness $h$, density $\rho$ and elastic moduli $E, \nu$ of the material to be constant around the element. The external force $\mathbf{b}$ is suggested to be distributed uniformly in the element. The following values are associated with the verices: their initial coordinates $(x_0, y_0)$, displacements $\mathbf{a}$ and velocities $\mathbf{v}$.
Displacement approximation
--------------------------
Consider a triangular element with vertices $(x_0^m, y_0^m), (x_0^n, y_0^n), (x_0^p, y_0^p)$. Let us denote the displacement vector at vertex $i$ as $$\mathbf{a}_i = \mathbf{u}(x_0^i, y_0^i)$$ Displacements are approximated in the domain of the element as linear functions of nodal displacements by defining the *shape functions* $\mathbf{N_i}(x, y, z)$: $$\label{shape}
\mathbf{u}(x,y,z) = \sum_{i \in \{m,n,p\}}{\mathbf{N_ia_i}} = \mathbf{Na}$$ with ${\mathbf{N}}= \left[ \begin{smallmatrix} {\mathbf{N}}_m & {\mathbf{N}}_n & {\mathbf{N}}_p
\end{smallmatrix} \right]$, ${\mathbf{a}}= \left[ \begin{smallmatrix} {\mathbf{a}}_m \\ {\mathbf{a}}_n \\ {\mathbf{a}}_p
\end{smallmatrix} \right]$ — displacements of the vertices, as only linear approximation is considered. Taking (\[no\_z\]) into account one gets: $$\label{eq:shape_function}
{\mathbf{N}}_i = (\alpha_i + \beta_i x + \gamma_i y)\mathbf{E}$$ with coefficients defined by: $${\mathbf{N}}_i(x_j, y_j) = \begin{cases}
\mathbf{E},\ i = j \\
\mathbf{0},\ i \neq j
\end{cases}\ i,\ j \in \{m,n,p\}$$ Final coefficients for given shape functions are given by: $$\begin{gathered}
\alpha_i = \frac1{S_e}\begin{vmatrix}
x_j & y_j \\
x_k & y_k
\end{vmatrix} \quad
\beta_i = -\frac1{S_e} (y_k - y_j) \quad
\gamma_i = \frac1{S_e} (x_k - x_j) \quad
S_e = \begin{vmatrix}
1 & x_i & y_i \\
1 & x_j & y_j \\
1 & x_k & y_k
\end{vmatrix}\notag
\end{gathered}$$
Approximate stress and strain
------------------------------
Let us approximate by substituting displacement with approximation from $$\label{eps}
\mathbf{\varepsilon} = \mathbf{Su} = \mathbf{SNa} = \mathbf{Ba}$$ Taking into account the formulas for ${\mathbf{N}}$ the matrix ${\mathbf{B}}$ becomes: $$\mathbf{B} = \begin{bmatrix}
\mathbf{B}_1^1 & \mathbf{B}_2^1 & \mathbf{B}_3^1 \\
\mathbf{B}_1^2 & \mathbf{B}_2^2 & \mathbf{B}_3^2
\end{bmatrix}$$ $$\mathbf{B}_i^1 = \begin{bmatrix}
\beta_i & 0 & 0 \\
0 & \gamma_i & 0 \\
0 & 0 & 0
\end{bmatrix}, \quad
\mathbf{B}_i^2 = \begin{bmatrix}
\gamma_i & \beta_i & 0 \\
0 & 0 & \gamma_i\\
0 & 0 & \beta_i
\end{bmatrix}$$ Stress and strain are connected via Hooke’s law . Substituting with the equatiom for deformation from above, one gets: $$\label{sigma}
\sigma = \mathbf{D\varepsilon} = \mathbf{DBa}$$
In the current implementation it is possible to define either all $21$ independent elastic constants, or, for the case of isotropic material, Young’s modulus $E$ and Poisson’s ratio $\nu$. The compliance matrix in the former case has the following form: $$\begin{gathered}
\mathbf{D} =
\begin{bmatrix}
\mathbf{D}_1 & \mathbf{0} \\
\mathbf{0} & \mathbf{D}_2
\end{bmatrix}.
\mathbf{D}_1 = \frac{E}{(1 + \nu)(1 - 2\nu)}
\begin{bmatrix}
1 - \nu & \nu & \nu \\
\nu & 1 - \nu & \nu \\
\nu & \nu & 1 - \nu
\end{bmatrix} \\ \notag
\mathbf{D}_2 = \frac{E}{(1 + \nu)(1 - 2\nu)}
\begin{bmatrix}
\frac{1-2\nu}2 & 0 & 0 \\
0 & \frac{1-2\nu}2& 0 \\
0 & 0 & \frac{1-2\nu}2
\end{bmatrix}
\end{gathered}$$
Virtual work for the element {#local}
----------------------------
Let us get the approximation for by substituting the variations with finite element approximations (\[shape\]), (\[eps\]): $$\delta\mathbf{u} = {\mathbf{N}}\delta{\mathbf{a}}, \quad \delta \mathbf{\varepsilon} = {\mathbf{B}}\delta{\mathbf{a}}$$
Taking these into account along with (\[sigma\]), the virtual work equation becomes: $$\label{eq:elem_virt_work}
\delta w = \delta\mathbf{\varepsilon^T \sigma} - \delta\mathbf{u^T \bar{b}}
= \mathbf{\delta a^T\left(B^T D Ba - N^T \bar{b}\right)}$$
Let us integrate the equation over the element volume and introduce fictive nodal forces $\mathbf{q_i}$, that balance internal elastic forces and external loads: $$\label{eq:elem_virt_work_integral}
\delta {\mathbf{a}}^T \left(\int_{V_e}\mathbf{B^T D Ba}dV
- \int_{V_e} \mathbf{N^T\bar{b}}dV \right) = \delta{\mathbf{a}}^T\mathbf{q}_e$$ Transition to the dynamic case is done according to the D’Alembert principle: $$\label{eq:inertion}
\bar{\mathbf{b}} = \mathbf{b} - \rho \ddot{\mathbf{u}}$$ where $\mathbf{b}$ is external force, $-\rho\ddot{\mathbf{u}}$ — force of inertia. For acceleration we shall use the same approximation as for displacement: $$\label{eq:acceleration}
\ddot{\mathbf{u}}(x, y) = \mathbf{N}(x, y)\ddot{{\mathbf{a}}}.$$ Taking into account and for computational element from one finally gets: $$\label{eq:single_elem_movement}
{\mathbf{M}}_e\ddot{{\mathbf{a}}} + {\mathbf{K}}_e{\mathbf{a}}+ {\mathbf{f}}_e = \mathbf{q}_e$$ Here the following matrices have been intriduced $$\begin{gathered}
{\mathbf{K}}_e = \int_{V_e}{\mathbf{B^TDB}}dV
\text{~---~stiffnes matrix}\\
{\mathbf{M}}_e = \rho\int_{V_e}{\mathbf{N^TN}}dV
\text{~---~mass matrix}\\
{\mathbf{f}}_e = - \int_{V_e} \mathbf{N^Tb}dV_e
\text{~---~load vector}
\end{gathered}$$
Global equations assembly
-------------------------
We now have to account for the effect from all elements that a certain vertex belongs to. Instead of local displacement vector, defined in \[shape\], consider a $3N_{nodes}$-dimensional vector, consisting of stacked displacement vectors of each vertex.
Thus we introduce global matrices ${\mathbf{K}}, {\mathbf{M}}$ with dimensions $3N_{nodes}\times3N_{nodes}$ and vector ${\mathbf{f}}$ with dimension $3N_{nodes}$. They are assembled by following rules: $$\begin{gathered}
{\mathbf{K}}_{ij} = \sum_{e}{\mathbf{K}}^e_{ij} \\
{\mathbf{M}}_{ij} = \sum_{e}{\mathbf{M}}^e_{ij} \\
{\mathbf{f}}_i = \sum_{e}{\mathbf{f}}^e_{i}
\end{gathered}$$ Here ${\mathbf{K}}_{ij}$ denotes a $3\times 3$ block of ${\mathbf{K}}$, standing at the intesections of row and column corresponding to $i$-th and $j$-th vertex. ${\mathbf{K}}_{ij}^e$ is a local stiffness matrix defined above in section \[local\]. Summation is done for all elements containing both i-th and j-th vertex. Fictive nodal forces are summed to, and, obviously, the sum equals to zero in case of equilibrium. Eventually one arrives at the following equation for the whole domain, analogous to : $$\label{eq:movement}
\boxed{{\mathbf{M}}\ddot{\mathbf{a}} + {\mathbf{K}}{\mathbf{a}}+ {\mathbf{f}}= 0}$$
Applying constraints {#sec:constraints}
--------------------
To study pinpoint strikes we need to constrain the velocity of the strike point. A problem of studying the membrane with fixed border may also arise. We shall demonstrate that such constraints can be taken into account without changing the structure of the governing equation (\[eq:movement\]). Let us fix node $i$. In ${\mathbf{K}}$ we fill the line ${\mathbf{K}}_{ik}, k\in \overline{1, N_{nodes}}$ with zeros. In ${\mathbf{M}}$ let $M_{ii} = \mathbf{E}$, and the rest of the blocks $M_{ik}, k \neq i$ filled with zeros. Finally, in ${\mathbf{f}}$ vector we fill ${\mathbf{f}}_i$ with zeros. As a result, the system now contains an equation $\ddot{\mathbf{a}}_i = \mathbf{0}$, т.е. $\mathbf{v}_i(t) = \mathbf{v}_{fix}$. To constrain displacement we shall determine $\mathbf{v}_{fix} = \mathbf{0}$
Time integration
----------------
The procedure above allowed us to reduce an initial problem for PDE to an ODE problem . We then integrate this equation numerically using Newmark method [@newmark1959method]: $$\begin{gathered}
\dot{\overline{{\mathbf{a}}}}_n = \dot{{\mathbf{a}}}_n + \tau(1-\beta_1)\ddot{{\mathbf{a}}}_n \notag\\
\overline{{\mathbf{a}}}_n = {\mathbf{a}}_n + \tau\dot{{\mathbf{a}}}_n
+ \frac12\tau^2(1 - \beta_2)\ddot{{\mathbf{a}}}_n \notag\\
\ddot{{\mathbf{a}}}_{n+1} = - A^{-1}\left({\mathbf{f}}_{n+1} +{\mathbf{K}}\overline{{\mathbf{a}}}_n
\right),\quad A = {\mathbf{M}}+ \frac{1}{2}\tau^2\beta_2{\mathbf{K}}\\
\dot{{\mathbf{a}}}_{n+1} = \dot{\overline{{\mathbf{a}}}}_n + \beta_1\tau\ddot{{\mathbf{a}}}_{n+1} \notag\\
{\mathbf{a}}_{n+1} = \overline{{\mathbf{a}}} + \frac{1}{2}\tau^2\beta_2\ddot{{\mathbf{a}}}_{n+1} \notag
\end{gathered}$$ If $\beta_2 \geq \beta_1 \geq \frac12$, the method is unconditionally stable. If $\beta_1 = \frac12$, then the method has the second order of approximation [@zie00]. For all the numerical experiments below, parameters $\beta_1 = \beta_2 = \frac12$ were used.
Numerical results
-----------------
The method described works on arbitrary irregular grids. The figure \[fig:non\_uniform\_mesh\_sample\] shows an example of a grid constructed using the gmsh mesh generator [@gmsh].
The figure \[fig:normal\_strike\_waves\] shows an example of calculation using this grid. A single central mesh element is impacted at a constant speed directed normal to the membrane plane. The involvement of membrane material in movement is shown. It can be seen that for a symmetric formulation the solution is symmetric and does not contain numerical artifacts.
[.45]{} ![Exposure to a point load impulse. The dynamics of the involvement of the membrane material in motion. The color shows the velocity module at different points in time.[]{data-label="fig:normal_strike_waves"}](img/step_00_upd.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Exposure to a point load impulse. The dynamics of the involvement of the membrane material in motion. The color shows the velocity module at different points in time.[]{data-label="fig:normal_strike_waves"}](img/step_20_upd.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Exposure to a point load impulse. The dynamics of the involvement of the membrane material in motion. The color shows the velocity module at different points in time.[]{data-label="fig:normal_strike_waves"}](img/step_40_upd.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Exposure to a point load impulse. The dynamics of the involvement of the membrane material in motion. The color shows the velocity module at different points in time.[]{data-label="fig:normal_strike_waves"}](img/step_60_upd.png "fig:"){width="1.0\linewidth"}
The implemented method allows to model asymmetric load profiles. The figure \[fig:skew\_strike\_waves\] shows an example of a calculation in which a blow with a constant speed is applied at an angle to the normal. The involvement of the membrane material in motion and its substantially asymmetric deformation are seen.
[.45]{} ![Exposure to a point load impulse at an angle of 30 degrees to the normal. The color shows the velocity module at different points in time.[]{data-label="fig:skew_strike_waves"}](img/skew_step_10_upd.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Exposure to a point load impulse at an angle of 30 degrees to the normal. The color shows the velocity module at different points in time.[]{data-label="fig:skew_strike_waves"}](img/skew_step_60_upd.png "fig:"){width="1.0\linewidth"}
The figure \[fig:aniso\_material\_waves\] shows an example of a calculation using anisotropic material model. A single central mesh element is impacted at a constant speed directed normal to the membrane plane. Elastic properies of the material are presented in table \[tab:material\]. Elliptic wave pattern formed due to material anisotropy is seen of the figure.
--------------- -----
$c_{11}$, GPa 150
$c_{12}$, GPa 40
$c_{13}$, GPa 10
$c_{22}$, GPa 150
$c_{23}$, GPa 80
$c_{33}$, GPa 150
$c_{44}$, GPa 80
$c_{55}$, GPa 20
$c_{66}$, GPa 30
--------------- -----
: Anisotropic material elasticity matrix non-zero elements[]{data-label="tab:material"}
[.45]{} ![Anisotropic material exposured to a point load impulse. The color shows the velocity module at different points in time. Elliptic wave pattern formed due to material anisotropy.[]{data-label="fig:aniso_material_waves"}](img/aniso_step_0_upd.png "fig:"){width="1.0\linewidth"}
[.45]{} ![Anisotropic material exposured to a point load impulse. The color shows the velocity module at different points in time. Elliptic wave pattern formed due to material anisotropy.[]{data-label="fig:aniso_material_waves"}](img/aniso_step_3_upd.png "fig:"){width="1.0\linewidth"}
The method allows not only to obtain the initial membrane dynamics and elastic waves from the shock load, but also can be used to calculate of deformations that develop over a considerable time. For example, the figure \[fig:large\_deformation\_sample\] shows the result of calculating the deformation of the membrane under the action of a distributed load, which is directed normal to the undeformed membrane and has the modulus $$\label{distrib_force}
b(r) = \begin{cases}
b_0\cos^2{r}, \quad r \leq L \\
0, \quad \text{ otherwise}
\end{cases} \\
r = \frac{\pi}{2L}\sqrt{\left(x - \frac{1}{2}L \right)^2
+ \left(y - \frac{1}{2}L \right)^2},\
L\text{~---~membrane size}$$
Actual convergence rate
=======================
Scheme’s actual convergence rate was examined with a method similar to presented in [@khokhlov2019]. First, a rather coarse regular grid, composed of rectangles, each split in two triangles by a diagonal, is generated. The choice of regular grid is dictated by the need to easily control the coarseness of the grid. The solution is calculated up to some fixed time $T$. Velocities and displacements in all the points of this coarse grid are saved as a baseline solution. Then the calculations are repeated on a grid refined by splitting each rectangle into four smaller ones. Timestep is also scaled down accordingly. The norms of difference of displacements and velocities for consequetive solutions on finer grids are calculated. Displacements and velocities for norm calculation are taken at the same space and time points where they were taken during the baseline solution.
Several test cases were examined, their descriptions summarized in Table \[tab:experiments\]. In cases $1, 2$ the loading force is applied to one central square (a pair of triangular elements) in the middle of the membrane. In cases $3, 4$ the velocity in the point closest to the geometric center of the membrane is constrained, the velocity vector being respectievely normal and inclined at an angle $\frac{\pi}6$ to normal.
Test case Description
----------- -------------------------------------------------------------
$1$ Normal strike with fixed load
$2$ Strike with fixed load, inclined $\frac{\pi}{6}$ to normal
$3$ Normal strike with fixed speed
$4$ Strike with fixed speed, inclined $\frac{\pi}{6}$ to normal
$5$ Normal load defined by formula
: Test case formulations[]{data-label="tab:experiments"}
To demonstrate the convergence rate study, we present the plot used in convergence rate assessment for test case $3$. The number of grid refinements $k$ is plotted over $OX$ axis, over $OY$ is plotted the norm of difference between solutions on consequtive grids in logarithmic scale. Three norms: $L_{1}, L_{2}, L_{inf}$ are considered. The slope of the fitted line shall represent the actual convergence rate.
The results of convergence rate study are presented in \[tab:conv\_results\]
----- ------- ------- ------------
$L_1$ $L_2$ $L_\infty$
$1$ 2.558 2.504 2.511
$2$ 1.534 1.482 1.445
$3$ 2.024 1.990 1.668
$4$ 1.599 1.660 1.964
$5$ 1.748 1.678 1.561
----- ------- ------- ------------
: Convergence rates[]{data-label="tab:conv_results"}
Conclusions
===========
The paper describes the mathematical model and the numerical method for modeling a thin anisotropic composite membrane under a dynamic shock load of an arbitrary time and space profile. The results presented show that the convergence rate of the numerical scheme in the worst case is 1.4 (oblique strike when using the norm $L_{inf}$), and in the best case it is 2.5 (symmetric load using any norm). This order allows using the described numerical method for applied calculations carried out in conjunction with field experiments. Anisotropic materials supported by the numerical method can be used to describe multilayer fabric membranes using their effective parameters.
The model and the method are designed to consider the composite membrane as 2D object in 3D space still having an arbitrary material rheology and load profile, this approach allows to reduce computational time compared with direct modelling using 3D solvers.
An obvious limitation of the described model and method is the lack of the possibility of a detailed calculation of problems associated with a destruction of the membrane. Using the presented approach, it is only possible to determine a start of a failure using strain or stress thresholds. Enhacing the method to cover the destruction is the topic of future work.
Acknowledgements {#acknowledgements .unnumbered}
================
The work was supported by RFBR project 18-29-17027.
The authors are grateful to Beklemysheva K.A., Ph.D. for long thoughtful discussions of the results and Bot.Cafe for a warm atmosphere and great coffee.
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|
---
author:
- Keiichi Ohnaka
- 'David A. Boboltz'
date: 'Received / Accepted '
title: ' Imaging the oxygen-rich disk toward the silicate carbon star EU And [^1] '
---
[We present multi-epoch high-angular resolution observations of 22 GHz masers toward the silicate carbon star EU And to probe the spatio-kinematic distribution of oxygen-rich material. ]{} [EU And was observed at three epochs (maximum time interval of 14 months) with the Very Long Baseline Array (VLBA). ]{} [ Our VLBA observations of the 22 GHz masers have revealed that the maser spots are distributed along a straight line across $\sim$20 mas, with a slight hint of an S-shaped structure. The observed spectra show three prominent velocity components at = $-42$, $-38$, and $-34$ , with the masers in SW redshifted and those in NE blueshifted. The maser spots located in the middle of the overall distribution correspond to the component at = $-38$ , which approximately coincides with the systemic velocity. These observations can be interpreted as either an emerging helical jet or a disk viewed almost edge-on (a circumbinary or circum-companion disk). However, the outward motion measured in the VLBA images taken 14 months apart is much smaller than that expected from the jet scenario. Furthermore, the mid-infrared spectrum obtained with the Spitzer Space Telescope indicates that the 10 silicate emission is optically thin and the silicate grains are of sub-micron size. This lends support to the presence of a circum-companion disk, because an optically thin circumbinary disk consisting of such small grains would be blown away by the intense radiation pressure of the primary (carbon-rich) star. If we assume Keplerian rotation for the circum-companion disk, the mass of the companion is estimated to be 0.5–0.8 . We also identify emission features at 13–16 in the Spitzer spectrum of EU And—the first unambiguous detection of in silicate carbon stars. ]{}
Introduction {#sect_intro}
============
Oxygen-rich circumstellar material (i.e., silicate and/or Al$_2$O$_3$) is usually associated with M-type asymptotic giant branch (AGB) stars, reflecting their photospheric chemical composition. Surprisingly, however, carbon stars showing silicate emission (so-called “silicate carbon stars”) were discovered by IRAS (Little-Marenin [@little-marenin86]; Willems & de Jong [@willems86]). The subsequent detection of H$_2$O and OH masers toward some of the silicate carbon stars, if not all, confirmed the oxygen-rich nature of the circumstellar material (e.g., Nakada et al. [@nakada87], [@nakada88]; Benson & Little-Marenin [@little-marenin87]; Little-Marenin et al. [@little-marenin88]; Barnbaum et al. [@barnbaum91]; Engels [@engels94]). On the other hand, optical spectroscopic studies show that $^{12}$C/$^{13}$C ratios in silicate carbon stars are as low as 4–5 and thus they are classified as “J-type” carbon stars (e.g., Ohnaka & Tsuji [@ohnaka99]), which is difficult to explain by the standard stellar evolution theory.
One currently accepted hypothesis suggests that silicate carbon stars have a low-luminosity companion (possibly a main-sequence star or a white dwarf) and that oxygen-rich material was shed by mass loss when the primary star was an oxygen-rich giant. The oxygen-rich material is stored in a circumbinary disk even after the primary star becomes a carbon star (Morris [@morris87]; Lloyd-Evans [@lloyd-evans90]). Alternatively, Yamamura et al. ([@yamamura00]) propose that the oxygen-rich material is stored in a circumstellar disk around the companion. In this scenario, the observed silicate emission cannot originate directly from the circum-companion disk heated by the companion itself, because its luminosity is too low. Instead, Yamamura et al. ([@yamamura00]) argue that radiation pressure from the primary, carbon-rich, AGB star drives and heats an outflow from the circum-companion disk. The observed silicate emission originates from this outflow.
Our recent $N$-band (8–13 ) spectro-interferometric observations of the silicate carbon star IRAS08002-3803 using the ESO’s Very Large Telescope Interferometer (VLTI) have spatially resolved the dusty environment of a silicate carbon star for the first time, and our radiative transfer modeling shows that the $N$-band visibilities can be fairly explained by an optically thick circumbinary disk in which small ($\sim$0.1 ), amorphous silicate and a second grain species—amorphous carbon, large silicate grains ($\sim$5 ), or metallic iron—coexist (Ohnaka et al. [@ohnaka06]). The $N$-band visibilities observed on another silicate carbon star IRAS18006-3213 show a wavelength dependence very similar to IRAS08002-3803, suggesting that IRAS18006-3213 also has a circumbinary disk (Deroo et al. [@deroo07]).
Some evidence of long-lived disks around silicate carbon stars has been discovered by radio observations as well. Jura & Kahane ([@jura99]) detected narrow CO ($J$=2–1, 1–0) emission lines toward two silicate carbon stars EU And and BM Gem, which indicate the presence of a reservoir of orbiting gas. Szczerba et al. ([@szczerba06]) and Engels (priv. comm.) obtained high-resolution 22 GHz maser maps toward the silicate carbon star V778 Cyg using MERLIN and VLBA, respectively. The maser distributions are linearly aligned with a slightly S-shaped structure, which Szczerba et al. ([@szczerba06]) and Babkovskaia et al. ([@babkovskaia06]) interpret as a warped circum-companion disk viewed almost edge-on. Therefore, as discussed in Yamamura et al. (2000) and Ohnaka et al. (2006), there may be two classes of silicate carbon stars: systems with optically thick circumbinary disks and those with circum-companion disks.
While direct detection of companions of silicate carbon stars is still difficult due to the huge luminosity contrast ($\sim \! \!
10^4$ and $\la$1 for the primary star and the companion, respectively), Izumiura ([@izumiura03]) and Izumiura et al. ([@izumiura07]) detected blue continuum emission at $\la$4000 Å and Balmer lines showing P Cygni profiles with an outflow velocity of $\ga$400 toward BM Gem, which strongly suggests the presence of an accretion disk around an unseen companion.
Despite this observational progress, the formation mechanisms of the circumbinary or circum-companion disks as well as the peculiar photospheric chemical composition of silicate carbon stars are little understood. Radio interferometry observations of the maser emission produced by oxygen-bearing molecules provide us with an excellent opportunity to study the spatio-kinematic distribution of oxygen-rich gas around silicate carbon stars. In this paper, we present the results of multi-epoch, high-resolution observations of the 22 GHz masers toward the silicate carbon star EU And.
Observations {#sect_obs}
============
The maser emission associated with EU And ($\alpha$=23$^h$19$^{m}$58.8814$^{s}$, $\delta$=471434.567, J2000.0, Epoch 2000.0, NOMAD Catalog, Zacharias et al. [@zacharias03])[^2] was observed at a rest frequency of 22.23508 GHz, using the Very Long Baseline Array (VLBA) operated by the National Radio Astronomy Observatory (NRAO)[^3]. The observations were carried out over three epochs spanning approximately 14 months: 2005 November 5, 2006 March 19, and 2007 January 13. VLBA’s 10 antennas were used at the first and second epochs, while only 9 antennas could be used at the third epoch due to a technical problem of the St. Croix station. For each 5-hr epoch, EU And and two extragalactic calibrator sources (3C454.3 and J2322+5057) were observed. We used the technique of rapid switching between the target source and the nearby phase reference source, J2322+5057, in order to remove residual phase offsets in the target source due to the atmosphere. The data were recorded in dual circular polarization with the 8-MHz band centered on a local standard of rest (LSR) velocity of $-36.0$ . The system temperatures and the sensitivities were of the order of 100 K and 10 Jy K$^{-1}$, respectively, for all three epochs.
The data were correlated at the VLBA correlator in Socorro, New Mexico, which produced auto- and cross-correlation spectra with 512 channels, corresponding to a channel spacing of 15.63 kHz (0.22 ). For the calibration of the correlated data, we followed the standard reduction procedures for VLBA spectral line experiments using the Astronomical Image Processing System (AIPS) maintained by NRAO. A bandpass calibration was carried out using intermittent (every 30 minutes) scans on 3C453.3. A fringe fit was performed on the phase-reference source in order to remove effects due to instrumentation and atmosphere on the phase not removed in the correlation. The resulting residual phase delays, phase rates, and phases were applied to the target source. At this point a preliminary image cube was produced for EU And. Because the signal-to-noise in these images was not optimal, we performed a second fringe fit on a strong reference maser feature (at = $-34$ ) in the spectrum of EU And and applied the residual phase rates to the target. Finally, an iterative self-calibration and imaging procedure was performed on the reference channel and the solutions for the residual phase and amplitude corrections were applied to all spectral channels. Images with 1024$\times$1024 pixels (61$\times$61 mas) were produced for all velocity channels between = $-24$ and $-52$ using beam sizes of 0.71$\times$0.29 mas, 0.73$\times$0.36 mas, and 0.82$\times$0.31 mas for the first, second, and third epochs, respectively. Typical rms off-source noise in the final images is 7–10 mJy beam$^{-1}$.
Maser components were identified, and their positions were measured by fitting the emission in the final images of each spectral channel with two-dimensional (2-D) Gaussians using the AIPS task SAD. The errors of the positions (within an epoch) are 10–20 $\mu$as for the strong maser features and approach $\sim$50 $\mu$as for the weaker features. The absolute astrometric information obtained by phase-referencing is lost in the above self-calibration procedure. To restore this information to the final component positions, we fit 2-D Gaussians to the strongest maser feature (assumed to be the same for all three epochs at = $-34$ ) prior to any self-calibration. The resulting shifts were applied to all maser components identified in the final images. The absolute positions of the strongest maser obtained for the first, second, and third epochs are (23$^{h}$19$^{m}$58.8822$^{s}$, 471434.550), (23$^{h}$19$^{m}$58.8823$^{s}$, 471434.548), and (23$^{h}$19$^{m}$58.8824$^{s}$, 471434.546), respectively (all in J2000.0).
The accuracy of the absolute position of the strongest maser feature is primarily limited by errors resulting from the transfer of the residual phase corrections from the extragalactic reference source to the target source and by the error in the absolute position of the reference source itself. In order to estimate the error involved in the transfer of the residual phase corrections, we essentially reversed the procedure used to determine the absolute position of the target maser source. We applied the residual phase solutions resulting from the self-calibration on the target source (strongest maser at = $-34$ ) to the extragalactic reference source. We then imaged the reference source and performed a 2-D Gaussian fit to determine its position. The separation between the positions of the target/calibrator pair (i.e., the arc length) was computed for 1) both sources calibrated with the extragalactic reference source solutions, and 2) both sources calibrated with the target source solutions. The difference between the arc lengths yields an estimate of the error involved in the phase transfer. The computation of the arc lengths is unnecessary in experiments specifically designed for astrometry where a second extragalactic check source is typically observed. This procedure was repeated for all three epochs and the standard deviation of the difference in arc lengths was found to be 0.12 mas. This error in the phase transfer compares favorably with the error in the absolute position of the reference source itself, which is a defining source of the International Celestial Reference Frame (ICRF) with an error of 0.26 mas (Fey et al. [@fey04]). The total error in the absolute position of the strongest maser feature is estimated from the root-sum-square of the two errors described above to be $\sim$0.3 mas.
Results {#sect_results}
=======
Figure \[obs\_map\] shows the spatio-kinematic distributions (left panel) and spectra (right panels) of the masers toward EU And as observed over the three epochs. At all epochs, the maser spectra are characterized by two strong peaks at = $-34$ and $-42$ . In the first and second epochs, weak features were also detected at $-38$ , approximately in the middle of the two peaks. The velocity of these weak maser features is in agreement with that of the narrow CO lines ($\sim \! -37$ ) measured by Jura & Kahane ([@jura99]). The velocity of the weak maser features also coincides with the radial velocities derived from the optical spectra, which represent the velocity of the primary carbon-rich AGB star (Barnbaum [@barnbaum91]). However, as discussed below, the putative companion, instead of the primary star, is more likely to be located at the position of these weak maser components. We note that while the spectra taken over 14 months appear rather stable, the masers toward EU And exhibit remarkable temporal variations in strength and velocity. For example, Little-Marenin ([@little-marenin88]) detected maser components near = $-30$ as strong as 8 Jy, which are non-existent in our VLBA data.
The masers at all three epochs are linearly aligned across $\sim$20 mas, with a slight hint of an S-shaped structure. As explained in Sect. \[sect\_obs\], the offsets among the epochs shown in Fig. \[obs\_map\] are real and represent the proper motion and parallax. In fact, the slight curvature seen in the direction of the positional displacement over the three epochs, which is discernible particularly for the strong features (the one at $(0, 0)$ at the first epoch), shows the annual parallax component. The observed S-shaped structure can be interpreted as an edge-on disk or an emerging helical jet. Such a nearly linear distribution of the masers is similar to those observed toward the so-called water fountain sources (e.g., Imai et al. [@imai02]; Boboltz & Marvel [@boboltz07] and references therein), which show well-collimated, fast jets of masers. However, the outflow velocities of the masers toward the water fountain sources are much higher ($\sim$60–150 ) than that observed toward EU And ($\sim$5 ), although the projection effect is uncertain.
In order to examine the jet scenario, we measured the increase of the separation (arc length) between the masers associated with the two highest peaks of emission in the spectrum. Since a single maser spans multiple channels in the spectrum, we performed a flux density squared weighted average over velocity and position prior to measuring the separations between the two peak features. The difference in the arc lengths between the first and third epochs is $\sim$0.3 mas. Using this measurement, we estimate the velocity $V$ and the inclination angle $i$ (angle toward us out of the plane of the sky) of a water-fountain-like outflow for EU And as follows. The observed velocity separation of the two strongest masers is 8 , which means that $2 V \sin i = 8$ . The spatial separation of the two masers increases by $2 V \cos i \times \Delta t /d = 0.3$ mas, where $\Delta t$ is the time interval between the first and third epochs (14 months), and $d$ is the distance of EU And. With distances of 1.5–2.6 kpc adopted (Jura & Kahane [@jura99]; Engels [@engels94]), we obtain $V$ = 4.1–4.3 and $i$ = 68–77. These outflow velocities are much lower than those observed toward the water-fountain sources, making the helical jet interpretation unfavorable, although the distance of EU And may be remarkably larger than the above estimates and/or the outflow velocity of EU And may indeed be so low due to some driving mechanism different from that in the water-fountain sources. A further long-term monitoring as well as a measurement of proper motion with VLBA would be necessary to settle this issue.
The maser distributions observed toward EU And are similar to that observed toward V778 Cyg by Szczerba et al. ([@szczerba06]) and Engels (priv. comm.). Our observations of another silicate carbon star IRAS07204-1032 using VLBA and the Very Large Array (VLA) have also revealed a similar spectrum and spatial distribution of the masers (Boboltz et al. in prep.). The expectation of observing linear distributions of masers toward the three silicate carbon stars imaged to date may seem statistically unlikely, given a random distribution of disk inclination angles. However, it is quite possible that this is due to a selection effect in which masers are preferentially detected in objects with the most favorable geometry for maser amplification (i.e., nearly edge-on disks). A survey of more objects with VLA and VLBA would be useful for obtaining a definitive answer to the interpretation of the masers toward silicate carbon stars.
Discussion {#sect_discuss}
==========
Although our maser maps of EU And lend support to the presence of an edge-on disk, these observations alone cannot clarify whether the disk is located around the carbon-rich primary star (circum-primary disk) or the whole binary system (circumbinary disk) or an unseen companion (circum-companion disk). The optical position of EU And, which corresponds to that of the primary star, is ($\alpha$, $\delta$) = (23$^{h}$19$^{m}$58.8814$^{s}$, +471434.567, J2000.0) with proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = ($5.5 \pm 1.7$, $-1.5 \pm 2.0$) mas yr$^{-1}$ (NOMAD catalog, Zacharias et al. [@zacharias03]). The position of the primary star expected for the first epoch, which is shown in Fig. \[euand\_optpos\], appears to be offset from the masers. However, the uncertainty of the absolute position ($\pm 15$ mas, Zacharias et al. [@zacharias04]) as well as that of the proper motion is too large to examine whether the primary star is located at the center of the masers or indeed offset from the maser distribution.
Constraints from the mid-infrared spectrum
------------------------------------------
On the other hand, as Yamamura et al. ([@yamamura00]) argue, a circum-primary or circumbinary disk would be unstable against the intense radiation pressure from the primary carbon-rich star ($\sim$10$^3$–10$^4$ ), if the disk is entirely optically thin. In order to examine whether or not the silicate dust around EU And is optically thin, we obtained its mid-infrared spectrum from the data archive of the Spitzer Space Telescope (Werner et al. [@werner04]). EU And was observed on 2004 December 9 (Program ID: P03235, P.I.: C. Waelkens) with the InfraRed Spectrograph[^4] (IRS, Houck et al. [@houck04]) in the Short-High (SH) and Long-High (LH) modes with a spectral resolution of $\sim$600 . We downloaded the Basic Calibrated Data (BCD) processed with the S15-3 pipeline and extracted the spectrum using SMART v.6.2.5 (Higdon et al. [@higdon04])[^5]. Figure \[euand\_irs\] shows the Spitzer/IRS spectrum of EU And (no sky subtraction was performed) together with the spectra of the silicate carbon star V778 Cyg and the oxygen-rich AGB star $o$ Cet obtained with the Short Wavelength Spectrometer (SWS) onboard the Infrared Space Observatory (ISO). As Yamamura et al. ([@yamamura00]) show, the 10 silicate features of the latter two stars indicate that the silicate emission is optically thin and silicate grains are of sub-micron size. Obviously, the spectrum of EU And closely resembles those of V778 Cyg and $o$ Cet. A comparison with the IRAS 12 flux of EU And and its Spitzer/IRS spectrum reveals that the flux level has little changed for the last 21 years, and EU And is similar to V778 Cyg in this aspect as well. This means that the silicate emission of EU And is optically thin with sub-micron grain sizes, rendering the possibility of a circum-primary or circumbinary disk unlikely. Therefore, the disk-like structure discovered by the masers is likely to represent a circum-companion disk.
The predominance of sub-micron-sized grains appears to contradict the conclusion of Jura & Kahane ([@jura99]) that the grain size around EU And should be as large as $\ge$0.2 cm to be gravitationally bound against radiation pressure. However, they assumed a circum-primary or circumbinary disk around the central star with a luminosity of $10^4$ . In the case of a disk around a low-luminosity companion ($\sim$1 ), grains as small as 0.2 are stable against the radiation pressure of the companion, although this does not entirely exclude the presence of larger grains due to grain growth in dense regions (e.g., mid-plane) of the circum-companion disk.
In the Spitzer/IRS spectrum of EU And shown in Fig. \[euand\_irs\], we can identify the emission features. To illustrate this identification, we also show the (scaled) absorption cross sections of $^{12}$CO$_2$ and $^{13}$CO$_2$, which were calculated using the line list of the HITRAN database (Rothman et al. [@rothman05]) for 1000 K (the column densities of the both isotopic species are set to be equal). The features are clearly seen in the ISO/SWS spectrum of the oxygen-rich star $o$ Cet, and also marginally in V778 Cyg. Yamamura et al. ([@yamamura00]) could only tentatively identify these features in V778 Cyg because of the low S/N ratio, and the case of EU And is the first unambiguous detection of toward silicate carbon stars. This is another piece of evidence for oxygen-rich gas around silicate carbon stars, and the mid-infrared features will enable us to estimate the density and temperature if the disk structure is more tightly constrained in the future. It would also be possible to derive the $^{12}$C/$^{13}$C ratio of the circumstellar gas from an analysis of the mid-infrared $^{12}$CO$_2$ and $^{13}$CO$_2$ features, which would shed new light on the origin of the abnormally low $^{12}$C/$^{13}$C ratios in silicate carbon stars.
It is worth noting that $o$ Cet (Mira AB), whose mid-infrared spectrum closely resembles that of EU And, is a well-known binary system (separation $\sim$05) with a main-sequence or white dwarf companion. The high-resolution mid-infrared images of Mira AB recently obtained by Ireland et al. ([@ireland07]) suggest that the edge of an accretion disk around Mira B (companion) is heated by Mira A (the primary star at the AGB). As Izumiura et al. ([@izumiura07]) suggest, this may resemble the circumstellar environment of some — if not all — silicate carbon stars (including EU And). Unlike silicate carbon stars, the circum-companion disk around Mira B consisting of silicate is embedded in the oxygen-rich outflow from Mira A, which masks the mid-infrared spectroscopic signature of the circum-companion disk.
Nature of the companion {#subsect_pv}
-----------------------
In order to draw a $p$-$V$ diagram for the observed masers toward EU And, we determined the center of the overall spatial distribution by taking the mid-point of a line connecting the mean positions of the redshifted and blueshifted masers at each epoch. The center velocity was also derived by halving the difference between the mean velocities of the redshifted and blueshifted masers. The resulting $p$-$V$ diagram for EU And is shown in Fig. \[pv\_diagram\]. It is very similar to that derived for V778 Cyg by Szczerba et al. ([@szczerba06]), although the number of maser spots is much smaller than for V778 Cyg. Both objects show a linear part with very large gradients at both ends of the velocity range. This is consistent with the presence of a Keplerian disk, although it is not definitive evidence. If we assume that the (circum-companion) disk is in Keplerian rotation, and the center of the disk is located in the middle of the maser distribution at = $-37.9$ , we can estimate the mass of the companion at the disk center. As shown in Pestalozzi et al. ([@pestalozzi04]), maser emission observed toward an edge-on disk has three maxima—on the line of sight toward the disk center and near the lines of sight tangential to the outer edge of the disk. If the strong maser components to NE and SW correspond to the maxima near the outer edge of the disk, it follows that the Keplerian velocity is $\sim$5 at a radius of 10 mas = 10 $\times$ ($d$/kpc) AU, where $d$ is the distance of EU And. This translates into 0.3 ($d$/kpc) for the central mass (0.5–0.8 for $d$ = 1.5–2.6 kpc). This mass would be reasonable for an unevolved main-sequence star or a white dwarf.
It should be noted, however, that the strong maser components at $-42$ and $-34$ may not represent the outer edge of the disk. As mentioned above, the strong maser components at $\sim \! -30$ were detected by Little-Marenin et al. ([@little-marenin88]), but the lack of spatial information does not allow us to take these observations into account in estimating the mass of the secondary star. Moreover, there may be masers fainter than the VLBA’s detection limit and/or more diffuse emission which would be resolved out by VLBA. Therefore, the above estimate of the companion mass based on Keplerian rotation is a lower limit. The estimate of the companion mass also depends on the disk geometry assumed in the analysis. For example, for the MERLIN data on V778 Cyg presented by Szczerba et al. ([@szczerba06]), Babkovskaia et al. ([@babkovskaia06]) applied a doubly warped disk model and derived a companion mass of 1.7 , which is significantly larger than the 0.06 derived by Szczerba et al. ([@szczerba06]). This highlights, once again, the importance of long-term monitoring of the masers in order to derive the properties of the disk and the secondary star.
Conclusion {#sect_concl}
==========
Our VLBA observations of the 22 GHz masers toward the silicate carbon star EU And have revealed that the masers are linearly aligned with a slight hint of an S-shaped structure, with the masers in SW redshifted and those in NE blueshifted. Such a spatio-kinematic structure can be interpreted either as an edge-on disk or an emerging jet. The maser maps obtained at three epochs over 14 months show a little outward motion of 0.3 mas, but the outflow velocity of $\sim$4 estimated from this outward motion is too low compared to those observed toward the water-fountain sources. This lends support to the disk interpretation, although some kind of outflow (possibly with a velocity much lower than in the water-fountain sources) cannot be entirely ruled out due to the uncertainty in the distance. The mid-infrared spectrum of EU And obtained with the Spitzer/IRS shows that the silicate emission is optically thin and emanates from sub-micron-sized grains, which suggests that the masers originate in a circum-companion disk seen nearly edge-on. Furthermore, we unambiguously identified the features at 13–16 in the Spitzer spectrum for the first time.
If we assume that the disk is in Keplerian motion, the mass of the putative secondary star is estimated to be 0.5–0.8 . However, given the remarkable variability of the water masers and the possible presence of fainter or more diffuse emission not detected by VLBA, the estimated mass should be regarded as a lower limit. Observations with a more compact, high-sensitivity, array (e.g., the VLA or MERLIN) might be useful in characterizing any such extended emission. Future VLBA observations will be necessary to put stronger constraints on the geometry of the disk and the properties of the secondary star. Astrometric monitoring with the VLBA should also provide more accurate estimates of the parallax and proper motion of EU And, thus improving our understanding of the three-dimensional structure of the oxygen-rich gas surrounding the system.
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[^1]: This work is based \[in part\] on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.
[^2]: This is offset by 32 from the position used for the VLA observation on 1990 June 2 by Colomer et al. ([@colomer00]), who detected no maser toward EU And. The negative detection may have been due to this positional offset, although it is still possible that the masers were indeed absent in June 1990.
[^3]: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[^4]: The IRS was a collaborative venture between Cornell University and Ball Aerospace Corporation funded by NASA through the Jet Propulsion Laboratory and Ames Research Center.
[^5]: SMART was developed by the IRS Team at Cornell University and is available through the Spitzer Science Center at Caltech.
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